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Cooperative multi-agent planning (MAP) is a relatively recent research field that combines technologies, algorithms and techniques developed by the Artificial Intelligence Planning and Multi-Agent Systems communities. While planning has been generally treated as a single-agent task, MAP generalizes this concept by considering multiple intelligent *agents* that work cooperatively to develop a course of action that satisfies the goals of the group.
This paper reviews the most relevant approaches to MAP, putting the focus on the solvers that took part in the 2015 Competition of Distributed and Multi-Agent Planning, and classifies them according to their key features and relative performance.
author:
- ALEJANDRO TORREÑO EVA ONAINDIA ANTONÍN KOMENDA MICHAL ŠTOLBA
bibliography:
- 'references.bib'
title: 'Cooperative Multi-Agent Planning: A Survey'
---
<ccs2012> <concept> <concept\_id>10010147.10010178.10010199.10010202</concept\_id> <concept\_desc>Computing methodologies Multi-agent planning</concept\_desc> <concept\_significance>500</concept\_significance> </concept> <concept> <concept\_id>10010147.10010178.10010219.10010223</concept\_id> <concept\_desc>Computing methodologies Cooperation and coordination</concept\_desc> <concept\_significance>500</concept\_significance> </concept> <concept> <concept\_id>10010147.10010178.10010219.10010220</concept\_id> <concept\_desc>Computing methodologies Multi-agent systems</concept\_desc> <concept\_significance>300</concept\_significance> </concept> <concept> <concept\_id>10010147.10010178.10010199.10010200</concept\_id> <concept\_desc>Computing methodologies Planning for deterministic actions</concept\_desc> <concept\_significance>300</concept\_significance> </concept> <concept> <concept\_id>10010147.10010178.10010205.10010206</concept\_id> <concept\_desc>Computing methodologies Heuristic function construction</concept\_desc> <concept\_significance>300</concept\_significance> </concept> <concept> <concept\_id>10002978.10002991.10002995</concept\_id> <concept\_desc>Security and privacy Privacy-preserving protocols</concept\_desc> <concept\_significance>300</concept\_significance> </concept> </ccs2012>
This work is supported by the Spanish MINECO under project TIN2014-55637-C2-2-R, the Prometeo project II/2013/019 funded by the Valencian Government, and the 4-year FPI-UPV research scholarship granted to the first author by the Universitat Politècnica de València. Additionally, this research is partially supported by the Czech Science Foundation under grant 15-20433Y.
Author’s addresses: A. Torreño ([email protected]) [and]{} E. Onaindia ([email protected]), Universitat Politècnica de València, Camino de Vera, s/n, Valencia, 46022, Spain; A. Komenda ([email protected]) [and]{} M. Štolba ([email protected]), Czech Technical University in Prague, Zikova 1903/4, 166 36, Prague, Czech Republic.
Introduction
============
Automated Planning is the field devoted to studying the reasoning side of acting. From the restricted conceptual model assumed in classical planning to the extended models that address temporal planning, on-line planning or planning in partially-observable and non-deterministic domains, the field of Automated Planning has experienced huge advances [@Ghallab04].
Multi-Agent Planning (MAP) introduces a new perspective in the resolution of a planning task with the adoption of a distributed problem-solving scheme instead of the classical single-agent planning paradigm. Distributed planning is required “when planning knowledge or responsibility is distributed among agents or when the execution capabilities that must be employed to successfully achieve objectives are inherently distributed” [@Desjardins99a].
The authors of [@Desjardins99a] analyze distributed planning from a twofold perspective; one approach, named *Cooperative Distributed Planning*, regards a MAP task as the process of formulating or executing a plan among a number of participants; the second approach, named *Negotiated Distributed Planning*, puts the focus on coordinating and scheduling the actions of multiple agents in a shared environment. The first approach has evolved to what is nowadays commonly known as *cooperative and distributed MAP*, with a focus on extending planning into a distributed environment and allocating the planning task among multiple agents. The second approach is primarily concerned with controlling and coordinating the actions of multiple agents in a shared environment so as to ensure that their local objectives are met. We will refer to this second approach, which stresses the coordination and execution of large-scale multi-agent planning problems, as *decentralized planning for multiple agents*. Moreover, while the first planning-oriented view of MAP relies on deterministic approaches, the study of decentralized MAP has yielded an intensive research work on coordination of activities in contexts under uncertainty and/or partial observability with the development of formal methods inspired by the use of Markov Decision Processes [@Seuken08].
This paper surveys deterministic cooperative and distributed MAP methods. Our intention is to provide the reader with a broad picture of the current state of the art in this field, which has recently gained much attention within the planning community thanks to venues such as the Distributed and Multi-Agent Planning workshop[^1] and the 2015 Competition of Distributed and Multi-Agent Planning[^2] ([[CoDMAP]{}]{}). Interestingly, although there was a significant amount of work on planning in multi-agent systems in the 90’s, most of this research was basically aimed at developing coordination methods for agents that adopt planning representations and algorithms to carry out their tasks. Back then, little attention was given to the problem of formulating collective plans to solve a planning task. However, the recent [[CoDMAP]{}]{}initiative of fostering [[MA-STRIPS]{}]{}a classical planning model for multi-agent systems [@Brafman08]$\;$ has brought back a renewed interest.
Generally speaking, cooperative MAP is about the collective effort of multiple planning agents to develop solutions to problems that each could not have solved as well (if at all) alone [@Durfee99]. A cooperative MAP task is thus defined as the collective effort of multiple agents towards achieving a *common goal*, irrespective of how the goals, the knowledge and the agents’ abilities are distributed in the application domain. In [@deWeerdt09], authors identify several phases to address a MAP task that can be interleaved depending on the characteristics of the problem, the agents and the planning model. Hence, MAP solving may require allocation of goals, formulating plans for solving goals, communicating planning choices and coordinating plans, and execution of plans. The work in [@deWeerdt09] is an overview of MAP devoted to agents that plan and interact, presenting a rough outline of techniques for cooperative MAP and decentralized planning. A more recent study examines how to integrate planning algorithms and Belief-Desire-Intention (BDI) agent reasoning [@MeneguzziS15]. This survey puts the focus on the integration of agent behaviour aimed at carrying out predefined plans that accomplish a goal and agent behaviour aimed at formulating a plan that achieves a goal.
This paper presents a thorough analysis of the advances in cooperative and distributed MAP that have lately emerged in the field of Automated Planning. Our aim is to cover the wide and fragmented space of MAP approaches, identifying the main characteristics that define tasks and solvers and establishing a taxonomy of the main approaches of the literature. We explore the great variety of MAP techniques on the basis of different criteria, like agent distribution, communication or privacy models, among others. The survey thus offers a deep analysis of techniques and domain-independent MAP solvers from a broad perspective, without adopting any particular planning paradigm or programming language. Additionally, the contents of this paper are geared towards reviewing the broad range of MAP solvers that participated in the 2015 [[CoDMAP]{}]{}competition.
This survey is structured in five sections. Section \[related\_work\] offers a historical background on distributed planning with a special emphasis on work that has appeared over the last two decades. Section \[MAP\_task\] discusses the main modelling features of a MAP task. Section \[aspects\] analyzes the main aspects of MAP solvers, including distribution, coordination, heuristic search and privacy. Section \[taxonomy\] discusses and classifies the most relevant MAP solvers in the literature. Finally, section \[ongoing\] concludes and summarizes the ongoing and future trends and research directions in MAP.
Related Work: historical background on MAP {#related_work}
==========================================
The large body of work on distributed MAP started jointly with an intensive research activity on multi-agent systems (MAS) at the beginning of the 90’s. Motivated by the distributed nature of the problems and reasoning of MAS, decentralized MAP focused on aspects related to distributed control including activities like the decomposition and allocation of tasks to agents and utilization of resources [@Durfee91; @Wilkins98]; reducing communication costs and constraints among agents [@Decker92; @Wolvertond98]; or incorporating group decision making for distributed plan management in collaborative settings (group decisions for selecting a high-level task decomposition or an agent assignation to a task, group processes for plan evaluation and monitoring, etc.) [@GroszHK99]. From this Distributed Artificial Intelligence (DAI) standpoint, MAP is fundamentally regarded as multi-agent *coordination of actions in decentralized system*s.
The inherently distributed nature of tasks and systems also fostered the appearance of techniques for *cooperative formation of global plans*. In DAI, this form of MAP puts greater emphasis on reasoning, stressing the deliberative planning activities of the agents as well as how and when to coordinate such planning activities to come up with a global plan. Given the cooperative nature of the planning task, where all agents are aimed at solving a common goal, this MAP approach features a more centralized view of the planning process. Investigations in this line have yield a great variety of planning and coordination methods such as techniques to merge the local plans of the agents [@EphratiR94; @Desjardins99b; @Cox04], heuristic techniques for agents to solve their individual sub-plans [@Ephrati97], mechanisms to coordinate concurrent interacting actions [@BoutilierB01] or distributed constraint optimization techniques to coordinate conflicts among agents [@Cox09]. In this latter work, the authors propose a general framework to coordinate the activities of several agents in a common environment such as partners in a military coordination, subcontractors working on a building, or airlines operating in an alliance.
Many of the aforementioned techniques and approaches were actually used by some of the early MAP tools. [[Distributed NOAH]{}]{} [@Corkill79] is one of the first Partial-Order Planning (POP) systems that generates gradual refinements in the space of (abstract) plans using a representation similar to the Hierarchical Task Networks (HTNs). The scheme proposed in [@Corkill79] relies on a distributed conflict-solving process across various agents that are able to plan without complete or consistent planning data; the limitation of [[Distributed NOAH]{}]{}is the amount of information that must be exchanged between planners and the lack of robustness to communication loss or error. In the domain-specific Partial Global Planning ([[PGP]{}]{}) method [@Durfee91], agents build their partial global view of the planning problem, and the search algorithm finds local plans that can be then coordinated to meet the goals of all the agents. Generalized PGP ([[GPGP]{}]{}) is a domain-independent extension of [[PGP]{}]{}[@Decker92; @Lesser04] that separates the process of coordination from the local scheduling of activities and task selection, which enables agents to communicate more abstract and hierarchically organized information and has smaller coordination overhead. [[DSIPE]{}]{} [@Desjardins99b] is a distributed version of [[SIPE-2]{}]{} [@Wilkins88] closely related to the [[Distributed NOAH]{}]{}planner. [[DSIPE]{}]{}proposes an efficient communication scheme among agents by creating partial views of sub-plans. The plan merging process is centralized in one agent and uses the conflict-resolution principle originally proposed in [[NOAH]{}]{}. The authors of [@WeerdtBTW03] propose a plan merging technique that results in distributed plans in which agents become dependent on each other, but are able to attain their goals more efficiently.
HTN planning has also been exploited for coordinating the plans of multiple agents [@Clement99]. The attractiveness of approaches that integrate hierarchical planning in agent teams such as [[STEAM]{}]{} [@Tambe97] is that they leverage the abstraction levels of the plan hierarchies for coordinating agents, thus enhancing the efficiency and quality of coordinating the agents’ plans. [[A-SHOP]{}]{}[@DixMNZ03] is a multi-agent version of the [[SHOP]{}]{}HTN planner [@NauAIKMWY03] that implements capabilities for interacting with external agents, performs mixed symbolic/numeric computations, and makes queries to distributed, heterogeneous information sources without requiring knowledge about how and where these resources are located. Moreover, authors in [@KabanzaSG04] propose a distributed version of [[SHOP]{}]{}that runs on a network of clusters through the implementation of a simple distributed backtrack search scheme.
As a whole, cooperative MAP approaches devoted to the construction of a plan that solves a common goal are determined by two factors, the underlying planning paradigm and the mechanism to coordinate the formation of the plan. The vast literature on multi-agent coordination methods is mostly concerned with the task of combining and adapting local planning representations into a global consistent solution. The adaptability of these methods to cooperative MAP is highly dependent on the particular agent distribution and the plan synthesis strategy of the MAP solver. Analyzing these aspects is precisely the aim of the following sections.
Cooperative Multi-Agent Planning Tasks {#MAP_task}
======================================
We define a (cooperative) MAP task as a process in which several agents that are not self-interested work together to synthesize a joint plan that solves a common goal. All agents wish thereby the goal to be reached at the end of the task execution.
First, this section presents the formalization of the components of a cooperative MAP task. Next, we discuss the main aspects that characterize a MAP task by means of two illustrative examples. Finally, we present how to model a MAP task with [*MA-PDDL*]{}, a multi-agent version of the well-known Planning Domain Description Language ([*PDDL*]{}) [@Ghallab98].
Formalization of a MAP Task
---------------------------
Most of the cooperative MAP solvers that will be presented in this survey use a formalism that stems from [[MA-STRIPS]{}]{}[@Brafman08] to a lesser or greater extent. For this reason, we will use [[MA-STRIPS]{}]{}as the baseline model for the formalization of a MAP task. [[MA-STRIPS]{}]{}is a minimalistic multi-agent extension of the well-known [[STRIPS]{}]{}planning model [@Fikes71], which has become the most widely-adopted formalism for describing cooperative MAP tasks.
In [[MA-STRIPS]{}]{}, a MAP task is represented through a finite number of situations or *states*. States are described by a set of *atoms* or *propositions*. States change via the execution of planning *actions*. An action in [[MA-STRIPS]{}]{}is defined as follows:
\[action\] A **planning action** is a tuple $\alpha = \langle {pre(\alpha)}, {add(\alpha)}, {del(\alpha)}\rangle$, where ${pre(\alpha)}$, ${add(\alpha)}$ and ${del(\alpha)}$ are sets of atoms that denote the preconditions, *add effects*, and *delete effects* of the action, respectively.
An action $\alpha$ is executable in a state $S$ if and only if all its preconditions hold in $S$; that is, $\forall p \in {pre(\alpha)}$, $p \in S$. The execution of $\alpha$ in $S$ generates a state $S'$ such that $S' = S \setminus {del(\alpha)}\cup {add(\alpha)}$.
\[MAP\_task\_def\] A **MAP task** is defined as a 5-tuple ${{\mathcal T}}=\langle {{\mathcal{AG}}}, {{\mathcal P}}, \{ {{\mathcal A}}^i \}^n_{i=1},{{\mathcal I}},{{\mathcal G}}\rangle$ with the following components:
- ${{\mathcal{AG}}}$ is a finite set of $n$ planning entities or agents.
- ${{\mathcal P}}$ is a finite set of atoms or propositions.
- ${{\mathcal A}}^i$ is the finite set of planning actions of the agent $i \in AG$. We will denote the set of actions of ${{\mathcal T}}$ as ${{\mathcal A}}= \bigcup_{\forall i \in {{\mathcal{AG}}}} {{\mathcal A}}^i$.
- ${{\mathcal I}}\subseteq {{\mathcal P}}$ defines the *initial state* of ${{\mathcal T}}$.
- ${{\mathcal G}}\subseteq {{\mathcal P}}$ denotes the *common goal* of ${{\mathcal T}}$.
A *solution plan* is an ordered set of actions whose application over the initial state ${{\mathcal I}}$ leads to a state $S_g$ that satisfies the task goals; i.e., ${{\mathcal G}}\subseteq S_g$. In [[MA-STRIPS]{}]{}a solution plan is defined as a sequence of actions $\Pi_g=\{\Delta, \prec\}$ that attains the task goals, where $\Delta \subseteq {{\mathcal A}}$ is a non-empty set of actions and $\prec$ is a total-order relationship among the actions of $\Delta$. However, other MAP models assume a more general definition of a plan; for example, as a set of of sequences of actions (one sequence per agent), as in [@Kvarnstrom11]; or as a partial-order plan [@Torreno12ECAI]. In the following, we will consider a solution plan as a set of partially-ordered actions.
The action distribution model of [[MA-STRIPS]{}]{}, introduced in Definition \[MAP\_task\_def\], classifies each atom $p \in {{\mathcal P}}$ as either *internal* (private) to an agent $i \in {{\mathcal{AG}}}$, if it is only used and affected by the actions in ${{\mathcal A}}^i$, or *public* to all the agents in ${{\mathcal{AG}}}$. ${{\mathcal P}}^i_{int}$ denotes the atoms that are internal to agent $i$, while ${{\mathcal P}}_{pub}$ refers to the public atoms of the task. The distribution of the information of a MAP task ${{\mathcal T}}$ configures the *local view* that an agent $i$ has over ${{\mathcal T}}$, ${{\mathcal T}}^i$, which is formally defined as follows:
\[Local\_view\] The **local view** of a task ${{\mathcal T}}=\langle {{\mathcal{AG}}}, {{\mathcal P}}, {{\mathcal A}},{{\mathcal I}},{{\mathcal G}}\rangle$ by an agent $i \in {{\mathcal{AG}}}$ is defined as ${{\mathcal T}}^i = \langle{{\mathcal P}}^i,{{\mathcal A}}^i,{{\mathcal I}}^i,{{\mathcal G}}\rangle$, which includes the following elements:
- ${{\mathcal P}}^i={{\mathcal P}}^i_{int}\cup{{\mathcal P}}_{pub}$ denotes the atoms accessible by agent $i$.
- ${{\mathcal A}}^i \subseteq {{\mathcal A}}$ is the set of planning actions of $i$.
- ${{\mathcal I}}^i \subseteq {{\mathcal P}}^i$ is the set of atoms of the initial state accessible by agent $i$.
- ${{\mathcal G}}$ denotes the common goal of the task ${{\mathcal T}}$. An agent $i$ knows all the atoms of ${{\mathcal G}}$ and it will contribute to their achievement either directly (achieving a goal $g \in {{\mathcal G}}$) or indirectly (reaching effects that help other agents achieve $g$).
Note that Definition \[Local\_view\] does not specify ${{\mathcal G}}^i$, a set of individual goals of an agent $i$, because in a cooperative MAP context the common goal ${{\mathcal G}}$ is shared among all agents and it is never assigned to some particular agent (see next section for more details).
Characterization of a MAP Task {#characterization}
------------------------------
This section introduces a brief example based on a logistics domain [@Torreno14] in order to illustrate the characteristics of a MAP task.
![MAP task examples: task ${{\mathcal T}}_1$ (left) and task ${{\mathcal T}}_2$ (right)[]{data-label="ExampleFig"}](Examples4.pdf){width="10.5cm"}
Consider the transportation task ${{\mathcal T}}_1$ in Figure \[ExampleFig\] (left), which includes three different agents. There are two transport agencies, $ta1$ and $ta2$, each of them having a truck, $t1$ and $t2$, respectively. The two transport agencies work in two different geographical areas, $ga1$ and $ga2$, respectively. The third agent is a factory, $ft$, located in the area $ga2$. In order to manufacture products, factory $ft$ requires a package of raw materials, $p$, which must be collected from area $ga1$. In this task, agents $ta1$ and $ta2$ have the same planning capabilities, but they operate in different geographical areas; i.e., they are *spatially* distributed agents. Moreover, the factory agent $ft$ is *functionally* different from $ta1$ and $ta2$.
The goal of task ${{\mathcal T}}_1$ is for $ft$ to manufacture a final product $fp$. For solving this task, $ta1$ will use its truck $t1$ to load the package of raw materials $p$, initially located in $l2$, and then it will transport $p$ to a storage facility, $sf$, that is located at the intersection of both geographical areas. Then, $ta2$ will complete the delivery by using its truck $t2$ to transport $p$ from $sf$ to the factory $ft$, which will in turn manufacture the final product $fp$. Therefore, this task involves three specialized agents, which are spatially or functionally distributed, and must cooperate to accomplish the common goal of having a final manufactured product $fp$.
Task ${{\mathcal T}}_1$ defines a [*group goal*]{}; i.e., a goal that requires the participation of all the agents in order to solve it. Given a task ${{\mathcal T}}_g=\langle {{\mathcal{AG}}}, {{\mathcal P}}, {{\mathcal A}}, {{\mathcal I}}, \{g\}\rangle$ which includes a single goal $g$, we say that $g$ is a [group goal]{}if for every solution plan $\Pi_g=\{\Delta, \prec\}$, $\exists \alpha, \beta \in \Delta: \alpha \in {{\mathcal A}}^i, \beta \in {{\mathcal A}}^j$ and $i \neq j$. We can thus distinguish between [group goals]{}, which require the participation of more than one agent, and non-[group goals]{}, which can be independently achieved by a single agent.
The presence or absence of [group goals]{}often determines the complexity of a cooperative MAP task. Figure \[ExampleFig\] (right) depicts task ${{\mathcal T}}_2$, where the goal is to deliver the package $p$ into the factory $ft$. This is a non-group goal because agent $ta1$ is capable of attaining it by itself, gathering $p$ in $l1$ and transporting it to $ft$ through the locations of its area $ga1$. However, the optimal solution for this task is that agent $ta1$ takes $p$ to $sf$ so that agent $ta2$ loads then $p$ in $sf$ and completes the delivery to the factory $ft$ through location $l7$. These two examples show that cooperative MAP involves multiple agents working together to solve tasks that they are unable to attain by themselves (task ${{\mathcal T}}_1$), or tasks that are accomplished better by cooperating (task ${{\mathcal T}}_2$) [@deWeerdt09].
\[views\]
Tasks ${{\mathcal T}}_1$ and ${{\mathcal T}}_2$ emphasize most of the key elements of a MAP context. The spatial and/or functional distribution of the participants gives rise to *specialized agents* that have different capabilities and knowledge of the task. Information of the MAP tasks is distributed among the specialized agents as summarized in Table \[views\]. Atoms of the form $\mathit{(pos\;t1\;*)}$ (note that $*$ acts as a wildcard) are accessible to agent $ta1$, since they model the position of truck $t1$. Atoms of the form $\mathit{(pos\;t2\;*)}$, which describe the location of truck $t2$, are accessible to agent $ta2$. Finally, $\mathit{(pending\;fp)}$ belongs to agent $ft$ and denotes that the manufacturing of $fp$ (the goal of task ${{\mathcal T}}_1$) is still pending.
The atoms related to the location of the product $p$, $\mathit{(at\;p\;*)}$, as well as $\mathit{(manufactured\;fp)}$, which indicates that the final product $fp$ is already manufactured, are accessible to the three agents, $ta1$, $ta2$ and $ft$. Since agents ignore the configuration of the working area of the other agents, the knowledge of agent $ta1$ regarding the location of $p$ is restricted to the atoms $\mathit{(at\;p\;l1)}$, $\mathit{(at\;p\;l2)}$, $\mathit{(at\;p\;sf)}$ and $\mathit{(at\;p\;t1)}$, while agent $ta2$ knows $\mathit{(at\;p\;sf)}$, $\mathit{(at\;p\;l3)}$, $\mathit{(at\;p\;l4)}$, $\mathit{(at\;p\;t2)}$ and $\mathit{(at\;p\;ft)}$. The awareness of agent $ft$ with respect to the location of $p$ is limited to $\mathit{(at\;p\;ft)}$.
The information distribution of a MAP task stresses the issue of *privacy*, which is one of the basic aspects that must be considered in multi-agent applications [@Such13]. Agents manage information that is not relevant for their counterparts or sensitive data of their internal operational mechanisms that they are not willing to disclose. For instance, $ta1$ and $ta2$ cooperate in solving tasks ${{\mathcal T}}_1$ and ${{\mathcal T}}_2$ but they could also be potential competitors since they work in the same business sector. For these reasons, providing privacy mechanisms to guarantee that agents do not reveal the internal configuration of their working areas to each other is a key issue.
In general, agents in MAP seek to minimize the information they share with each other, thus exchanging only the information that is relevant for other participating agents to solve the MAP task.
Modelling of a MAP Task with [*MA-PDDL*]{} {#specification}
------------------------------------------
The adoption of a common language for modelling planning domains allows for a direct comparison of different approaches and increases the availability of shared planning resources, thus facilitating the scientific development of the field [@Fox03]. Modelling a cooperative MAP task involves defining several elements that are not present in single-agent planning tasks. Widely-adopted single-agent planning task specification languages, such as [*PDDL*]{}[@Ghallab98], lack the required machinery to specify a MAP task. Recently, [*MA-PDDL*]{}[^3], the multi-agent version of [*PDDL*]{}, was developed in the context of the 2015 [[CoDMAP]{}]{}competition [@Komenda16codmap] as the first attempt to create a *de facto* standard specification language for MAP tasks. We will use [*MA-PDDL*]{}as the language for modelling MAP tasks.
MAP solvers that accept an *unfactored* specification of a MAP task use a single input that describes the complete task ${{\mathcal T}}$. In contrast, other MAP approaches require a *factored* specification; i.e., the local view of each agent, ${{\mathcal T}}^i$. Additionally, modelling a MAP task may require the specification of the private information that an agent cannot share with other agents.
[*MA-PDDL*]{}allows for the definition of both factored ([:factored-privacy]{} requirement) and unfactored ([:unfactored-privacy]{} requirement) task representations. In order to model the transportation task ${{\mathcal T}}_1$ (see Figure \[ExampleFig\] (left) in Section \[characterization\]), we will use the factored specification. Task ${{\mathcal T}}^i_1$ of agent $i$ is encoded by means of two independent files: the *domain* file describes general aspects of the task (${{\mathcal P}}^i$ and ${{\mathcal A}}^i$, which can be reused for solving other tasks of the same typology); the *problem* file contains a description of the particular aspects of the task to solve (${{\mathcal I}}^i$ and ${{\mathcal G}}$). For the sake of simplicity, we only display fragments of the task ${{\mathcal T}}_1^{ta1}$.
The domain description of agents of type [[transport-agency]{}]{}, like $ta1$ and $ta2$, is defined in Listing \[domain\_agency\].
(define (domain transport-agency)
(:requirements :factored-privacy :typing :equality :fluents)
(:types transport-agency area location package product - object
truck place - location
factory - place)
(:predicates
(manufactured ?p - product) (at ?p - package ?l - location)
(:private
(area ?ag - transport-agency ?a - area) (in-area ?p - place ?a - area)
(owner ?a - transport-agency ?t - truck) (pos ?t - truck ?l - location)
(link ?p1 - place ?p2 - place)
)
)
(:action drive
:parameters (?ag - transport-agency ?a - area ?t - truck ?p1 - place ?p2 - place)
:precondition (and (area ?ag ?a) (in-area ?p1 ?a) (in-area ?p2 ?a)
(owner ?a ?t) (pos ?t ?p1) (link ?p1 ?p2))
:effect (and (not (pos ?t ?p1)) (pos ?t ?p2))
)
[...]
)
The domain of [[transport-agencies]{}]{}starts with the type hierarchy, which includes the types [[transport-agency]{}]{}and [factory]{} to define the agents of the task. Note that the type [factory]{} is defined as a subtype of [place]{} because a [factory]{} is also interpreted as a [place]{} reachable by a truck. The remaining elements of the task are identified by means of the types [location]{}, [package]{}, [truck]{}, etc.
The [:predicates]{} section includes the set of first-order *predicates*, which are patterns to generate the agent’s propositions, ${{\mathcal P}}^i$, through the instantiation of their parameters. The domain for [[transport-agencies]{}]{}includes the public predicates [at]{}, which models the position of the [packages]{}, and [manufactured]{}, which indicates that the task goal of manufacturing a [product]{} is fulfilled. Despite the fact that only the [factory]{} agent $ft$ has the ability to manufacture [products]{}, all the agents have access to the predicate [manufactured]{} so that the [[transport-agencies]{}]{}will be informed when the task goal is achieved.
The remaining predicates in Listing \[domain\_agency\] are in the section [:private]{} of each [[transport-agency]{}]{}, meaning that agents will not disclose information concerning the topology of their working [areas]{} or the status of their [trucks]{}.
Finally, the [:action]{} block of Listing \[domain\_agency\] shows the *action schema* [drive]{}. An action schema represents a number of different actions that can be derived by instantiating its variables. Agents $ta1$ and $ta2$ have three action schemas: [load]{}, [unload]{} and [drive]{}.
Regarding the *problem* description, the factored specification includes three problem files, one per agent, that contain ${{\mathcal I}}^i$, the initial state of each agent, and the task goals ${{\mathcal G}}$.
(define (problem ta1)
(:domain transport-agency)
(:objects
ta1 - transport-agency
ga1 - area l1 l2 sf - place
p - package fp - product
(:private t1 - truck)
)
(:init
(area ta1 ga1) (pos t1 l1) (owner t1 ta1) (at p l1)
(link l1 l2) (link l2 l1) (link l1 sf)
(link sf l1) (link l2 sf) (link sf l2)
(in-area l1 ga1) (in-area l2 ga1) (in-area sf ga1)
)
(:goal (manufactured fp))
)
Listing \[problem\_ta1\] depicts the problem description of task ${{\mathcal T}}_1^{ta1}$. The information managed by $ta1$ is related to its [truck t1]{} (which is defined as [:private]{} to prevent $ta1$ from disclosing the location or cargo of [t1]{}), along with the [places]{} within its working [area]{}, [ga1]{}, and the [package p]{}. The [:init]{} section of Listing \[problem\_ta1\] specifies ${{\mathcal I}}^{ta1}$; i.e., the location of [truck t1]{} and the position of [package p]{}, which is initially located in [ga1]{}. Additionally, $ta1$ is aware of the [links]{} and [places]{} within its [area]{} [ga1]{}. The [:goal]{} section is common to the three agents in ${{\mathcal T}}_1$ and includes a single goal indicating that the [product fp]{} must be [manufactured]{}.
This modelling example shows the flexibility of [*MA-PDDL*]{}for encoding the specific requirements of a MAP task, such as the agents’ distributed information via factored input and the private aspects of the task. These functionalities make [*MA-PDDL*]{}a fairly expressive language to specify MAP tasks.
Main Aspects of a MAP Solver {#aspects}
============================
Solving a cooperative MAP task requires various features such as information distribution, specialized agents, coordination or privacy. The different MAP solving techniques in the literature can be classified according to the mechanisms they use to address these functionalities. We identify six main features to categorize cooperative MAP solvers:
- **Agent distribution**: From a conceptual point of view, MAP is regarded as a task in which multiple agents are involved, either as entities participating in the plan synthesis (planning agents) or as the target entities of the planning process (actuators or execution agents).
- **Computational process**: From a computational perspective, MAP solvers use a *centralized* or *monolithic* design that solves the MAP task through a central process, or a *distributed* approach that splits the planning activity among several processing units.
- **Plan synthesis schemes**: There exist a great variety of strategies to tackle the process of synthesizing a plan for the MAP task, mostly characterized by how and when the *coordination* activity is applied. Coordination comprises the distributed information exchange processes by which the participating agents organize and harmonize their activities in order to work together properly.
- **Communication mechanisms**: Communication among agents is an essential aspect that distinguishes MAP from single-agent planning. The type of communication enabled in MAP solvers is highly dependent on the type of computational process (centralized or distributed) of the solver. Thus, we will classify solvers according to the use of internal or external communication infrastructures.
- **Heuristic search**: As in single agent planning, MAP solvers commonly apply heuristic search to guide the planning process. In MAP, we can distinguish between *local* heuristics (each agent $i$ calculates an estimate to reach the task goals ${{\mathcal G}}$ using only its accessible information in ${{\mathcal T}}^i$) or *global* heuristics (the estimate to reach ${{\mathcal G}}$ is calculated among all the agents in ${{\mathcal{AG}}}$).
- **Privacy preservation**: Privacy is one of the main motivations to adopt a MAP approach. Privacy means coordinating agents without making sensitive information publicly available. Whereas this aspect was initially neglected in former MAP solvers [@Krogt09], the most recent approaches tackle this issue through the development of robust privacy-preserving algorithms.
The following subsections provide an in-depth analysis of these aspects, which characterize and determine the performance of the existing MAP solvers.
Agent Distribution
------------------
Agents in MAP can adopt different roles: planning agents are *reasoning* entities that synthesize the course of action or plan that will be later executed by a set of *actuators* or *execution agents*. An execution agent can be, among others, a robot in a multi-robot system, or a software entity in an execution simulator.
From a conceptual point of view, MAP solvers are characterized by the agent distribution they apply; i.e., the number of planning and execution agents involved in the task. Typically, it is assumed that one planning agent from the set ${{\mathcal{AG}}}$ of a MAP task ${{\mathcal T}}$ is associated with one actuator in charge of executing the actions of this planning agent in the solution plan. However, some MAP solvers alter this balance between planning and acting agents.
Table \[conceptual\_schemes\] summarizes the different schemes according to the relation between the number of planning and execution agents. Single-agent planning is the simplest mapping: the task is solved by a single planning agent, i.e., $|{{\mathcal{AG}}}|=1$, and executed by a single actuator. We can mention Fast Downward ([[FD]{}]{}) [@Helmert06] as one of the most utilized single-agent planners within the planning community.
\[conceptual\_schemes\]
MAP solvers like [[Distoplan]{}]{}[@Fabre10], [[A\#]{}]{}[@Jezequel12] and [[ADP]{}]{}[@Crosby13] follow a *factored* agent distribution inspired by the factored planning scheme [@Amir03]. Under this paradigm, a single-agent planning task is decomposed into a set of independent factors (agents), thus giving rise to a MAP task with $|{{\mathcal{AG}}}|>1$. Then, factored methods to solve the agents’ local tasks ${{\mathcal T}}^i$ are applied, and finally, the computed local plans are pieced together into a valid solution plan [@Brafman06]. Factored planning exploits locality of the solutions and a limited information propagation between components.
The second row of Table \[conceptual\_schemes\] outlines the classification of MAP approaches that build a plan that is conceived to be then executed by several actuators. Some solvers in the literature regard MAP as a single planning agent working *for* a set of actuators ($|{{\mathcal{AG}}}|=1$), while other approaches regard MAP as planning *by* multiple planners ($|{{\mathcal{AG}}}|>1$).
### Planning *for* multiple agents
Under this scheme, the actions of the solution plan are distributed among actuators typically via the introduction of constraints. [[TFPOP]{}]{}[@Kvarnstrom11] applies single-agent planning for multi-agent domains where each execution agent is associated with a sequential thread of actions within a partial-order plan. The combination of forward-chaining and least-commitment of [[TFPOP]{}]{}provides flexible schedules for the acting agents, which execute their actions in parallel. The work in [@Crosby14] transforms a MAP task that involves multiple agents acting concurrently and cooperatively in a single-agent planning task. The transformation compels agents to select joint actions associated with a single subset of objects at a time, and ensures that the concurrency constraints on this subset are satisfied.The result is a single-agent planning problem in which agents perform actions individually, one at a time.
The main limitation of this planning-for-multiple-agents scheme is its lack of privacy, since the planning entity has complete access to the MAP task ${{\mathcal T}}$. This is rather unrealistic if the agents involved in the task have sensitive private information they are not willing to disclose [@Sapena08]. Therefore, this scheme is not a suitable solution for privacy-preserving MAP tasks like task ${{\mathcal T}}_1$ described in section \[characterization\].
### Planning *by* multiple agents
This scheme distributes the MAP task among several planning agents where each is associated with a local task ${{\mathcal T}}^i$. Thus, planning-by-multiple-agents puts the focus on the coordination of the planning activities of the agents. Unlike single-planner approaches, the planning decentralization inherent to this scheme makes it possible to effectively preserve the agents’ privacy.
In general, solvers that follow this scheme, such as [[FMAP]{}]{}[@Torreno12ECAI], maintain a one-to-one correspondence between planning and execution agents; that is, planning agents are assumed to solve their tasks, which will be later executed by their corresponding actuators. There exist, however, some exceptions in the literature that break this one-to-one correspondence such as [[MARC]{}]{}[@marc-codmap15], which rearranges the $n$ planning agents in ${{\mathcal{AG}}}$ into $m$ *transformer agents* ($m < n$), where a transformer agent comprises the planning tasks of several agents in ${{\mathcal{AG}}}$. All in all, [[MARC]{}]{}considers $m$ reasoning entities that plan for $n$ actuators, where $m < n$.
![Centralized (monolithic) vs. distributed (agent-based) implementation[]{data-label="FigDistribution"}](Distribution2.pdf){width="10.4cm"}
Computational process
---------------------
From a computational standpoint, MAP solvers are classified as *centralized* or *distributed*. Centralized solvers draw upon a monolithic implementation in which a central process synthesizes a global solution plan for the MAP task. In contrast, distributed MAP methods are implemented as multi-agent systems in which the problem-solving activity is fully decentralized.
#### Centralized MAP
In this approach, the MAP task ${{\mathcal T}}$ is solved on a single machine regardless the number of planning agents conceptually considered by the solver. The main characteristic of a centralized MAP approach is that tasks are solved in a monolithic fashion, so that all the processes of the MAP solver, $\{P_1,\ldots,P_n\}$, are run in one same machine (see Figure \[FigDistribution\] (left)).
The motivation for choosing a centralized MAP scheme is twofold: 1) external communication mechanisms to coordinate the planning agents are not needed; and 2) robust and efficient single-agent planning technology can be easily reused.
Regarding agent distribution, MAP solvers that use a single planning agent generally apply a centralized computational scheme, as for example [[TFPOP]{}]{}[@Kvarnstrom11] (see Table \[conceptual\_schemes\]). On the other hand, some algorithms that conceptually rely on the distribution of the MAP task among several planning agents do not actually implement them as software agents, but as a centralized procedure. For example, [[MAPR]{}]{}[@Borrajo13] establishes a sequential order among the planning agents and applies a centralized planning process that incrementally synthesizes a solution plan by solving the agents’ local tasks in the predefined order.
#### Distributed MAP
Many approaches that conceive MAP as planning by multiple agents (see Table \[conceptual\_schemes\]), are developed as multi-agent systems (MAS) defined by several independent *software agents*. By software agent, we refer to a computer system that 1) makes decisions without any external intervention (autonomy), 2) responds to changes in the environment (reactivity), 3) exhibits goal-directed behaviour by taking the initiative (pro-activeness), and 4) interacts with other agents via some communication language in order to achieve its objectives (social ability) [@Wooldridge97].
In this context, a software agent of a MAS plays the role of a planning agent in ${{\mathcal{AG}}}$. This way, in approaches that follow the *planning by multiple agents* scheme introduced in the previous section, a software agent encapsulates the local task ${{\mathcal T}}^i$ of a planning agent $i \in {{\mathcal{AG}}}$. Given a task, where $|{{\mathcal{AG}}}|=n$, distributed MAP solvers can be run on up to $n$ different hosts or machines (see Figure \[FigDistribution\] (right)).
The emphasis of the distributed or agent-based computation lies in the coordination of the concurrent activities of the software planning agents. Since agents may be run on different hosts (see Figure \[FigDistribution\] (right)), having a proper communication infrastructure and message-passing protocols is vital for the synchronization of the agents.
Distributed solvers like [[FMAP]{}]{}[@Torreno14] launch $|{{\mathcal{AG}}}|$ software agents that seamlessly operate on different machines. [[FMAP]{}]{}builds upon the MAS platform [[Magentix2]{}]{}[@Such12], which provides the messaging infrastructure for agents to communicate over the network.
Plan synthesis schemes {#planning-coordination}
----------------------
In most MAP tasks, there are dependencies between agents’ actions and none of the participants has sufficient competence, resources or information to solve the entire problem. For this reason, agents must coordinate with each other in order to cooperate and solve the MAP task properly.
Coordination is a multi-agent process that harmonizes the agents’ activities, allowing them to work together in an organized way. In general, coordination involves a large variety of activities, such as distributing the task goals among the agents, making joint decisions concerning the search for a solution plan, or combining the agents’ individual plans into a solution for a MAP task. Since coordination is an inherently distributed mechanism, it is only required in MAP solvers that conceptually draw upon multiple planning agents (see right column of Table \[conceptual\_schemes\]).
The characteristics of a MAP task often determine the coordination requirements for solving the task. For instance, tasks that feature [group goals]{}, like the task ${{\mathcal T}}_1$ depicted in section \[characterization\], usually demand a stronger coordination effort. Therefore, the capability and efficiency of a MAP solver is determined by the coordination strategy that governs its behaviour.
The following subsections analyse the two principal coordination strategies in MAP; namely, *unthreaded* and *interleaved* planning and coordination.
![Plan synthesis schemes in unthreaded planning and coordination[]{data-label="FigUnthreaded"}](Unthreaded_schemes2.pdf){width="13.8cm"}
### Unthreaded Planning and Coordination
This strategy defines planning and coordination as *sequential* activities, such that they are viewed as two separate black boxes. Under this strategy, an agent $i \in {{\mathcal{AG}}}$ synthesizes a plan to its local view of the task, ${{\mathcal T}}^i$, and coordination takes place *before* or/and *after* planning.
#### Pre-planning coordination
Under this plan synthesis scheme, the MAP solver defines the necessary constrains to guarantee that the plans that solve the local tasks of the agents are properly combined into a consistent solution plan for the whole task ${{\mathcal T}}$ (see Figure \[FigUnthreaded\] (a)).
[[ADP]{}]{}[@Crosby13] follows this scheme by applying an *agentification* procedure that distributes a [[STRIPS]{}]{}planning task among several planning agents (see Table \[conceptual\_schemes\]). More precisely, [[ADP]{}]{}is a fully automated process that inspects the multi-agent nature of the planning task and calculates an agent decomposition that results in a set of $n$ decoupled local tasks. By leveraging this agent decomposition, [[ADP]{}]{}applies a centralized, sequential and total-order planning algorithm that yields a solution for the original [[STRIPS]{}]{}task. Since the task ${{\mathcal T}}$ is broken down into several local tasks independent from each other, the local solution plans are consistently combined into a solution for ${{\mathcal T}}$. Ultimately, the purpose of pre-planning coordination is to guarantee that the agents’ local plans are seamlessly combined into a solution plan that attains the goals of the MAP task, thus avoiding the use of plan merging techniques at post-planning time.
#### Post-planning coordination
Other unthreaded MAP solvers put the coordination emphasis *after* planning. In this case, the objective is to *merge* the plans that solve the agents’ local tasks, $\{{{\mathcal T}}^1, \ldots, {{\mathcal T}}^n\}$, into a solution plan that attains the goals ${{\mathcal G}}$ of the task ${{\mathcal T}}$ by removing inconsistencies among the local solutions (see Figure \[FigUnthreaded\] (b)).
In [[PMR]{}]{}(Plan Merger by Reuse) [@Luis14], the local plans of the agents are concatenated into a solution plan for the MAP task. Other post-planning coordination approaches apply an information exchange between agents to come up with the global solution. For instance, [[PSM]{}]{}[@Tozicka15] draws upon a set of finite automata, called Planning State Machines (PSM), where each automaton represents the set of local plans of a given agent. In one iteration of the [[PSM]{}]{}procedure, each agent $i$ generates a plan that solves its local task ${{\mathcal T}}^i$, incorporating this plan into its associated PSM. Then, agents exchange the public projection of their PSMs, until a solution plan for the task ${{\mathcal T}}$ is found.
#### Iterative response planning
This plan synthesis scheme, firstly introduced by [[DPGM]{}]{}[@Pellier10], successively applies a planning-coordination sequence, each corresponding to a planning agent. An agent $i$ receives the local plan of the preceding agent along with a set of constraints for coordination purposes, and *responds* by building up a solution for its local task ${{\mathcal T}}^i$ on top of the received plan. Hence, the solution plan is incrementally synthesized (see Figure \[FigUnthreaded\] (c)).
Multi-Agent Planning by Reuse ([[MAPR]{}]{}) [@Borrajo13] is an iterative response solver based on *goal allocation*. The task goals ${{\mathcal G}}$ are distributed among the agents before planning, such that an agent $i$ is assigned a subset ${{\mathcal G}}^i \subset {{\mathcal G}}$, where $\bigcap_{i \in {{\mathcal{AG}}}} {{\mathcal G}}^i = \emptyset$. Additionally, agents are automatically arranged in a sequence that defines the order in which the iterative response scheme must be carried out.
In unthreaded planning and coordination schemes, agents do not need communication skills because they do not interact with each other during planning. This is the reason why the unthreaded strategy is particularly efficient for solving tasks that do not require a high coordination effort. In contrast, it presents several limitations when solving tasks with [group goals]{}, due to the fact that agents are unable to discover and address the cooperation demands of other agents. The needs of cooperation that arise when solving [group goals]{}are hard to discover at pre-planning time, and plan merging techniques are designed only to fix inconsistencies among local plans, rather than repairing the plans to satisfy the inter-agent coordination needs. Consequently, unthreaded approaches are more suitable for solving MAP task that do not contain [group goals]{}; that is, every task goal can be solved by at least one single agent.
![Multi-agent search in interleaved planning and coordination[]{data-label="FigInterleaved"}](Interleaved_scheme.pdf){width="6.2cm"}
### Interleaved Planning and Coordination {#interleaved}
A broad range of MAP techniques *interleave* the planning and coordination activities. This coordination strategy is particularly appropriate for tasks that feature [group goals]{}since agents explore the search space jointly to find a solution plan, rather than obtaining local solutions individually. In this context, agents continuously coordinate with each other to communicate their findings, thus effectively intertwining planning and coordination.
Most interleaved solvers, such as [[MAFS]{}]{}[@Nissim12] and [[FMAP]{}]{}[@Torreno14], commonly rely on a *coordinated multi-agent search* scheme, wherein nodes of the search space are contributed by several agents (see Figure \[FigInterleaved\]). This scheme involves selecting a node for expansion (planning) and exchanging the successor nodes among the agents (coordination). Agents thus jointly explore the search space until a solution is found, alternating between phases of planning and coordination.
Different forms of coordination are applicable in the interleaved resolution strategy. In [[FMAP]{}]{}[@Torreno14], agents share an open list of plans and jointly select the most promising plan according to global heuristic estimates. Each agent $i$ expands the selected node using its actions ${{\mathcal A}}^i$, and then, agents evaluate and exchange all the successor plans. In [[MAFS]{}]{}[@Nissim12], each agent $i$ keeps an independent open list of states. Agents carry out the search simultaneously: an agent $i$ selects a state $S$ to expand from its open list according to a local heuristic estimate, and synthesizes all the nodes that can be generated through the application of the actions ${{\mathcal A}}^i$ over $S$. Out of all the successor nodes, agents only share the states that are relevant to other agents.
Interleaving planning and coordination is very suitable for solving complex tasks that involve [group goals]{}and a high coordination effort. By using this strategy, agents learn the cooperation requirements of other participants during the construction of the plan and can immediately address them. Hence, the interleaved scheme allows agents to efficiently address [group goals]{}.
The main drawback of this coordination strategy is the high communication cost in a distributed MAP setting because alternating planning and coordination usually entails exchanging a high number of messages in order to continuously coordinate agents.
Communication mechanisms {#comm}
------------------------
Communication among agents plays a central role in MAP solvers that conceptually define multiple planners (see Table \[conceptual\_schemes\]), since planning agents must coordinate their activities in order to accomplish the task goals. As shown in Figure \[FigDistribution\], different agent communication mechanisms can be applied, depending on the computational process followed by the MAP solver.
#### Internal communication
Solvers that draw upon a centralized implementation resort to *internal* or simulated multi-agent communication. For example, the centralized solver [[MAPR]{}]{}[@Borrajo13] distributes the task goals, ${{\mathcal G}}$, among the planning agents, and agents solve their local tasks sequentially. Once a local plan $\Pi^i$ for an agent $i \in AG$ is computed, the information of $\Pi^i$ is used as an input for the next agent in the sequence, $j$, thus simulating a message passing between agents $i$ and $j$. This type of simple and simulated communication system is all that is required in centralized solvers that run all planning agents in a single machine.
#### External communication
As displayed in Figure \[FigDistribution\] (right), distributed MAP solvers draw upon *external* communication mechanisms by which different processes (agents), potentially allocated on different machines, exchange messages in order to interact with each other. External communication can be easily enabled by linking the different agents via network sockets or a messaging broker. For example, agents in [[MAPlan]{}]{}[@maplan-codmap15] exchange data over the TCP/IP protocol when the solver is executed in a distributed manner. A common alternative to implement external communication in MAS implies using a message passing protocol compliant with the IEEE FIPA standards [@FIPA02protocol], which are intended to promote the interoperation of heterogeneous agents. The [[Magentix2]{}]{}MAS platform [@Such12] used by [[MH-FMAP]{}]{}[@Torreno15] facilitates the implementation of agents with FIPA-compliant messaging capabilities.
The use of external communication mechanisms allows distributed solvers to run the planning agents in decentralized machines and to coordinate their activities by exchanging messages through the network. The flexibility provided by external communication mechanisms comes at the cost of performance degradation. The results of the 2015 [[CoDMAP]{}]{}competition [@Komenda16codmap] show that centralized solvers like [[ADP]{}]{}[@Crosby13] outperform solvers executed in a distributed setting, such as [[PSM]{}]{}[@psm-codmap15]. Likewise, an analysis performed in [@Torreno14] reveals that communication among agents is the most time-consuming activity of the distributed approach [[FMAP]{}]{}, thus compromising the overall scalability of this solver. Nevertheless, the participants in the distributed track of the [[CoDMAP]{}]{}exhibited a competitive performance, which proves that the development of fully-distributed MAP solvers is worth the overhead caused by external communication infrastructures.
Heuristic Search {#heuristic}
----------------
Ever since the introduction of [[HSP]{}]{} [@Long00], the use of heuristic functions that guide the search by estimating the quality of the nodes of the search tree has proven to be one of the most robust and reliable problem-solving strategies in single-agent planning. Over the years, many solvers based on heuristic search, such as [[FF]{}]{} [@Hoffmann01] or [[LAMA]{}]{} [@Richter10], have consistently dominated the International Planning Competitions[^4].
Since most MAP solvers stem from single-agent planning techniques, heuristic search is one of the most common approaches in the MAP literature. In general, in solvers that synthesize a plan for multiple executors, the single planning agent (see Table \[conceptual\_schemes\]) has complete access to the MAP task ${{\mathcal T}}$, and so it can compute heuristics that leverage the *global* information of ${{\mathcal T}}$, in a way that is very similar to that of single-agent planners. However, in solvers that feature multiple planning agents, i.e., $|{{\mathcal{AG}}}| > 1$, each agent $i$ is only aware of the information defined in ${{\mathcal T}}^i$ and no agent has access to the complete task ${{\mathcal T}}$. Under a scheme of planning by multiple agents, one can distinguish between *local* and *global* heuristics.
In local heuristics, an agent $i$ estimates the cost of the task goals, ${{\mathcal G}}$, using only the information in ${{\mathcal T}}^i$. The simplicity of local heuristics, which do not require any interactions among agents, contrasts with the low accuracy of the estimates they yield due to the limited task view of the agents. Consider, for instance, the example task ${{\mathcal T}}_1$ presented in section \[characterization\]: agent $ta1$ does not have sufficient information to compute an accurate estimate of the cost to reach the goals of ${{\mathcal T}}_1$ since ${{\mathcal T}}^{ta1}_1$ does not include the configuration of the area $ga2$. In general, if ${{\mathcal T}}^i$ is a limited view of ${{\mathcal T}}$, local heuristics will not yield informative estimates of the cost of reaching ${{\mathcal G}}$.
In contrast, a global heuristic in MAP is the application of a heuristic function “carried out by several agents which have a different knowledge of the task and, possibly, privacy requirements” [@Torreno15]. The development of global heuristics for multi-agent scenarios must account for additional features that make heuristic evaluation an arduous task [@Nissim12]:
- Solvers based on distributed computation require robust communication protocols for agents to calculate estimates for the overall task.
- For MAP approaches that preserve agents’ privacy, the communication protocol must ensure that estimates are computed without disclosing sensitive private data.
The application of local or global heuristics is also determined by the characteristics of the plan synthesis scheme of the MAP solver. Particularly, in an *unthreaded* planning and coordination scheme, agents synthesize their local plans through the application of local heuristic functions. For instance, in the sequential plan synthesis scheme of [[MAPR]{}]{}[@Borrajo13], agents apply locally $h_{FF}$ [@Hoffmann01] and $h_{Land}$ [@Richter10] when solving their allocated goals.
Local heuristic search has also been applied by some *interleaved* MAP solvers. Agents in [[MAFS]{}]{}and [[MAD-A\*]{}]{}[@Nissim14] generate and evaluate search states locally. An agent $i$ shares a state $S$ and local estimate $h^i(S)$ only if $S$ is relevant to other planning agents. Upon reception of $S$, agent $j$ performs its local evaluation of $S$, $h^j(S)$. Then, depending on the characteristics of the heuristic, the final estimate of $S$ by agent $j$ will be either $h^i(S)$, $h^j(S)$, or a combination of both. In [@Nissim12], authors test [[MAD-A\*]{}]{}with two different optimal heuristics, [[*LM-Cut*]{}]{}[@Helmert09] and [[*Merge&Shrink*]{}]{}[@Helmert07], both locally applied by each agent. Despite heuristics being applied only locally, [[MAD-A\*]{}]{}is proven to be cost-optimal.
Unlike unthreaded solvers, the interleaved planning and coordination strategy makes it possible to accommodate global heuristic functions. In this case, agents apply some heuristic function on their tasks ${{\mathcal T}}^i$ and then they exchange their local estimates to come up with an estimate for the global task ${{\mathcal T}}$.
[[GPPP]{}]{}[@Maliah14] introduces a distributed version of a privacy-preserving *landmark* extraction algorithm for MAP. A landmark is a proposition that must be satisfied in every solution plan of a MAP task [@Hoffman04]. The quality of a plan in [[GPPP]{}]{}is computed as the sum of the local estimates of the agents in ${{\mathcal{AG}}}$. [[GPPP]{}]{}outperforms [[MAFS]{}]{}thanks to the accurate estimates provided by this landmark-based heuristic. [[MH-FMAP]{}]{}[@Torreno15], the latest version of [[FMAP]{}]{}, introduces a multi-heuristic alternation mechanism based on Fast Downward ([[FD]{}]{}) [@Helmert06]. Agents alternate two global heuristics when expanding a node in their tasks: $h_{DTG}$, which draws upon the information of the *Domain Transition Graphs* [@Helmert04] associated with the state variables of the task, and the landmark-based heuristic $h_{Land}$, which only evaluates the *preferred successors* [@Torreno15]. Agents jointly build the DTGs and the landmarks graph of the task and each of them stores its own version of the graphs according to its knowledge of the MAP task.
Some recent work in the literature focuses on the adaptation of well-known single-agent heuristic functions to compute global MAP estimators. The authors of [@Stolba14] adapt the single-agent heuristic $h_{FF}$ [@Hoffmann01] by means of a compact structure, the *exploration queue*, that optimizes the number of messages exchanged among agents. This multi-agent version of $h_{FF}$, however, is not as accurate as the single-agent one. The work in [@Stolba15] introduces a global MAP version of the admissible heuristic function [[*LM-Cut*]{}]{}that is proven to obtain estimates of the same quality as the single-agent [[*LM-Cut*]{}]{}. This multi-agent [[*LM-Cut*]{}]{}yields better estimates at the cost of a larger computational cost.
In conclusion, heuristic search in MAP, and most notably, the development of global heuristic functions in a distributed context, constitutes one of the main challenges of the MAP research community. The aforementioned approaches prove the potential of the development and combination of global heuristics towards scaling up the performance of MAP solvers.
Privacy
-------
The preservation of agents’ sensitive information, or *privacy*, is one of the basic aspects that must be enforced in MAP. The importance of privacy is illustrated in the task ${{\mathcal T}}_1$ of section \[characterization\], which includes two different agents, $ta1$ and $ta2$, both representing a transport agency. Although both agents are meant to cooperate for solving this task, it is unlikely that they are willing to reveal sensitive information to a potential competitor.
Privacy in MAP has been mostly neglected and under-represented in the literature. Some paradigms like Hierarchical Task Network (HTN) planning apply an form of implicit privacy when an agent delegates subgoals to another agent, which solves them by concealing the resolution details from the requester agent. This makes HTN a very well-suited approach for practical applications like composition of web services [@Sirin04]. However, formal treatment of privacy is even more scarce. One of the first attempts to come up with a formal privacy model in MAP is found in [@vanderKrogt07b], where authors quantify privacy in terms of the Shannon’s information theory [@Shannon48]. More precisely, authors establish a notion of uncertainty with respect to plans and provide a measure of privacy loss in terms of the data uncovered by the agents along the planning process. Unfortunately, this measure is not general enough to capture details such as heuristic computation. Nevertheless, quantification of privacy is an important issue in MAP, as it is in distributed constraint satisfaction problems [@Faltings08]. A more recent work, also based on Shannon’s information theory [@Stolba16b], quantifies privacy leakage for [[MA-STRIPS]{}]{}according to the reduction of the number of possible transition systems caused by the revealed information. In this work, the main sources of privacy leakage are identified, but not experimentally evaluated.
\[tab:privacy\]
The next subsections analyze the privacy models adopted by the MAP solvers in the literature according to three different criteria (see summary in Table \[tab:privacy\]): the *modelling* of private information, the *information sharing* schemes, and the privacy *practical privacy guarantees* offered by the MAP solver.
### Modelling of Private Information
This feature is closely related to whether the language used to specify the MAP task enables explicit modelling of privacy or not.
Early approaches to MAP, such as [[MA-STRIPS]{}]{}[@Brafman08], manage a notion of *induced privacy*. Since the [*MA-STRIPS*]{}language does not explicitly model private information, the agents’ private data are inferred from the task structure. Given an agent $i \in {{\mathcal{AG}}}$ and a piece of information $p^i \in {{\mathcal T}}^i$, $p^i$ is defined as private if $\forall_{j \in {{\mathcal{AG}}}| j \neq i} p^i \not \in {{\mathcal T}}^j$; that is, if $p^i$ is known to $i$ and ignored by the rest of agents in ${{\mathcal T}}$.
[[FMAP]{}]{}[@Torreno14] introduces a more general *imposed privacy* scheme, explicitly describing the private and shareable information in the task description. [*MA-PDDL*]{} [@Kovacs12], the language used in the [[CoDMAP]{}]{}competition [@Komenda16codmap], follows this imposed privacy scheme and allows the designer to model the private elements of the agents’ tasks.
In general, both induced and imposed privacy schemes are commonly applied by current MAP solvers. The induced privacy scheme enables the solver to automatically identify the naturally private elements of a MAP task. The imposed privacy scheme, by contrast, offers a higher control and flexibility to model privacy, which is a helpful tool in contexts where agents wish to occlude sensitive data that would be shared otherwise.
### Information Sharing
Privacy-preserving algorithms vary accordingly to the number of agents that share a particular piece of information. In general, we can identify two information sharing models, namely [[MA-STRIPS]{}]{}and *subset privacy*.
#### [[MA-STRIPS]{}]{}
The [[MA-STRIPS]{}]{}information sharing model [@Brafman08] defines as public the data that are shared among all the agents in ${{\mathcal{AG}}}$, so that a piece of information is either known to all the participants, or only to a single agent. More precisely, a proposition $p \in {{\mathcal P}}$ is defined as *internal* or *private* to an agent $i \in {{\mathcal{AG}}}$ if $p$ is only used and affected by the actions of ${{\mathcal A}}^i$. However, if the proposition $p$ is also in the preconditions and/or effects of some action $\alpha \in {{\mathcal A}}^j$, where $j \in {{\mathcal{AG}}}$ and $j\neq i$, then $p$ is publicly accessible to all the agents in ${{\mathcal{AG}}}$.
An action $\alpha \in {{\mathcal A}}^i$ that contains only public preconditions and effects is said to be public and it is known to all the participants in the task. In case $\alpha$ includes both public and private preconditions and effects, agents share instead $\alpha_p$, the *public projection* of $\alpha$, an abstraction that contains only the public elements of $\alpha$.
This simple dichotomic privacy model of [[MA-STRIPS]{}]{}does not allow for specifying MAP tasks that require some information to be shared only by a subset of the planning agents in ${{\mathcal{AG}}}$.
#### Subset privacy
Subset privacy is introduced in [@Bonisoli14] and generalizes the [[MA-STRIPS]{}]{}scheme by establishing pairwise privacy. This model defines a piece of information as private to a single agent, publicly accessible to all the agents in ${{\mathcal{AG}}}$ or known to a *subset of agents*. This approach is useful in applications where agents wish to conceal some information from certain agents.
For instance, agent $ta2$ in task ${{\mathcal T}}_1$ of section \[characterization\] notifies the factory agent $ft$ whenever the proposition $(pos\;t2\;ft)$ is reached. This proposition indicates that the truck $t2$ is placed at the factory $ft$, a location that is known to both $ta2$ and $ft$. Under the [[MA-STRIPS]{}]{}model, agent $ta1$ would be notified that truck $t2$ is at the factory $ft$, an information that $ta2$ may want to conceal. However, the subset privacy model allows $ta2$ to hide $(pos\;t2\;ft)$ from $ta1$ by defining it as private between $ta2$ and $ft$.
Hence, subset privacy is a more flexible information sharing model compared to the more conservative and limited approach of [[MA-STRIPS]{}]{}, which enables representing more complex and realistic situations concerning information sharing.
### Privacy Practical Guarantees {#practical_privacy}
Recent studies devoted to a formal treatment of practical privacy guidelines in MAP [@Nissim14; @Shani16] conclude that some privacy schemes allow agents to infer private information from other agents through the transmitted data. According to these studies, it is possible to establish a four-level taxonomy to classify the practical privacy level of MAP solvers. The four levels of the taxonomy, from the least to the most secure one, are: *no privacy*, *weak privacy*, *object cardinality privacy* and *strong privacy*.
#### No privacy
Privacy has been mostly neglected in MAP but has been extensively treated within the MAS community [@Such12]. The 2015 [[CoDMAP]{}]{}competition introduced a more expressive definition of privacy than [[MA-STRIPS]{}]{}and this was a boost for many planners to model private data in the task descriptions. Nevertheless, we can cite a large number of planners that completely disregard the issue of privacy among agents such as early approaches like [[GPGP]{}]{}[@Decker92] or more recent approaches like [[$\mu$-SATPLAN]{}]{}[@Dimopoulos12], [[A\#]{}]{} [@Jezequel12] or [[DPGM]{}]{}[@Pellier10].
#### Weak privacy
A MAP system is weakly privacy-preserving if agents do not explicitly communicate their private information to other agents at execution time [@Brafman15]. This is accomplished by either *obfuscating* (encrypting) or *occluding* the private information they communicate to other agents in order to only reveal the public projection of their actions. In a weak privacy setting, agents may infer private data of other agents through the information exchanged during the plan synthesis.
*Obfuscating* the private elements of a MAP task is an appropriate mechanism when agents wish to conceal the meaning of propositions and actions. In obfuscation, the proposition names are encrypted but the number and unique identity of preconditions and effects of actions are retained, so agents are able to reconstruct the complete isomorphic image of their tasks. In [[MAPR]{}]{}[@Borrajo13], [[PMR]{}]{}[@Luis14] and [[CMAP]{}]{}[@mapr-cmap-codmap15], when an agent communicates a plan, it encrypts the private information in order to preserve its sensitive information. Agents in [[MAFS]{}]{}[@Nissim14], [[MADLA]{}]{}[@madla-codmap15], [[MAPlan]{}]{}[@maplan-codmap15] and [[GPPP]{}]{}[@Maliah14] encrypt the private data of the relevant states that they exchange during the plan synthesis.
Other weak privacy-preserving solvers in the literature *occlude* the agents’ private information rather than sharing obfuscated data. Agents in [[MH-FMAP]{}]{}[@Torreno15] only exchange the public projection of the actions of their partial-order plans, thus occluding private information like preconditions, effects, links or orderings.
#### Object cardinality privacy
Recently, the [[DPP]{}]{}planner [@Shani16] introduced a new level of privacy named object cardinality privacy. A MAP algorithm preserves object cardinality privacy if, given an agent $i$ and a type $t$, the cardinality of $i$’s private objects of type $t$ cannot be inferred by other agents from the information they receive [@Shani16]. In other words, this level of privacy strongly preserves the number of objects of a given type $t$ of an agent $i$, thus representing a middle ground between the weak and strong privacy settings.
Hiding the cardinality of private objects is motivated by real-world scenarios. Consider, for example, the logistics task ${{\mathcal T}}_1$ of section \[characterization\]. One can assume that the transport agencies that take part in the MAP task, $ta1$ and $ta2$, know that packages are delivered using trucks. However, it is likely that each agent would like to hide the number of trucks it possesses or the number of transport routes it uses.
#### Strong privacy
A MAP algorithm is said to strongly preserve privacy if none of the agents in ${{\mathcal{AG}}}$ is able to infer a private element of an agent’s task from the public information it obtains during planning. In order to guarantee strong privacy, it is necessary to consider several factors, such as the nature of the communication channel (synchronous, asynchronous, lossy) or the computational power of the agents.
[[Secure-MAFS]{}]{}[@Brafman15] is a theoretical proposal to strong privacy that builds upon the [[MAFS]{}]{}[@Nissim14] model. In [[Secure-MAFS]{}]{}, two states that only differ in their private elements are not communicated to other agents in order to prevent them from deducing information through the non-private or public part of the states. [[Secure-MAFS]{}]{}is proved to guarantee strong privacy for a sub-class of tasks based on the well-known *logistics* domain.
In summary, weak privacy is easily achievable through obfuscation of private data, but provides little security. On the other hand, the proposal of [[Secure-MAFS]{}]{}lays the theoretical foundations to strong privacy in MAP but the complexity analysis and the practical implementation issues of this approach have not been studied yet. Additionally, object cardinality privacy accounts for a middle ground between weak and strong privacy. In general, the vast majority of MAP methods are classified under the no privacy or weak privacy levels: the former approaches to MAP do not consider privacy at all, while most of the recent proposals, which claim to be privacy preserving, resort in most cases to *obfuscation* to conceal private information.
\[tab:features\]
Distributed and Multi-Agent Planning Systems Taxonomy {#taxonomy}
=====================================================
As discussed in section \[introduction\], MAP is a long-running research field that has been covered in several articles [@Desjardins99a; @deWeerdt09; @MeneguzziS15]. This section reviews the large number of domain-independent cooperative MAP solvers that have been proposed since the introduction of the [[MA-STRIPS]{}]{}model [@Brafman08]. This large body of research was recently crystallized in the 2015 [[CoDMAP]{}]{}competition [@Komenda16codmap], the first attempt to directly compare MAP solvers through a benchmark encoded using the standardized [*MA-PDDL*]{}language.
The cooperative solvers analyzed in this section cover a wide range of different plan synthesis schemes. As discussed in section \[aspects\], one can identify several aspects that determine the features of MAP solvers; namely, *agent distribution*, *computational process*, *plan synthesis scheme*, *communication mechanism*, *heuristic search* and *privacy preservation*. This section presents an in-depth taxonomy that classifies solvers according to their main features and analyzes their similarities and differences (see Table \[tab:features\]).
This section also aims to critically analyze and compare the strengths and weaknesses of the planners regarding their applicability and experimental performance. Given that a comprehensive comparison of MAP solvers was issued as a result of the 2015 [[CoDMAP]{}]{}competition, Table \[tab:features\] arranges solvers according to their positions in the coverage ranking (number of problems solved) of this competition. The approaches included in this taxonomy are organized according to their plan synthesis scheme, an aspect that ultimately determines the types of MAP tasks they can solve. Section \[taxonomy\_unthreaded\] discusses the planners that follow an unthreaded planning and coordination scheme, while section \[taxonomy\_interleaved\] reviews interleaved approaches to MAP.
Unthreaded Planning and Coordination MAP Solvers {#taxonomy_unthreaded}
------------------------------------------------
The main characteristic of unthreaded planners is that planning and coordination are not intertwined but handled as two separate and independent activities. Unthreaded solvers are labelled as *UT* in the column *Coordination strategy* of Table \[tab:features\]. They typically apply local single-agent planning and a combinatorial optimization or satisfiability algorithm to coordinate the local plans.
### [[Planning First]{}]{}, 2008 (implemented in 2010) {#planning-first-2008-implemented-in-2010 .unnumbered}
[[Planning First]{}]{} [@Nissim10] is the first MAP solver that builds upon the [[MA-STRIPS]{}]{}model. It is an early representative of the unthreaded strategy that inspired the development of many subsequent MAP solvers, which are presented in the next paragraphs. [[Planning First]{}]{}generates a local plan for each agent in a centralized fashion by means of the [[FF]{}]{}planner [@Hoffmann01], and coordinates the local plans through a distributed Constraint Satisfaction Problem (DisCSP) solver to come up with a global solution plan. More precisely, [[Planning First]{}]{}distributes the MAP task among the agents and identifies the coordination points of the task as the actions whose application affects other agents. The DisCSP is then used to find consistent coordination points between the local plans. If the DisCSP solver finds a solution, the plan for the MAP task is directly built from the local plans since the DisCSP solution guarantees compatibility among the underlying local plans.
The authors of [@Nissim10] empirically evaluate [[Planning First]{}]{}over a set of tasks based on the well-known *rovers*, *satellite* and *logistics* domains. The results show that a large number of coordination points among agents derived from the number of public actions limits the scalability and effectiveness of [[Planning First]{}]{}. Later, the [[MAP-POP]{}]{}solver outperformed [[Planning First]{}]{}in both execution time and coverage [@Torreno12ECAI].
### [[DPGM]{}]{}, 2010 (implemented in 2013) {#dpgm-2010-implemented-in-2013 .unnumbered}
[[DPGM]{}]{}[@Pellier10] makes also use of CSP techniques to coordinate the agents’ local plans. Unlike [[Planning First]{}]{}, the CSP solver in [[DPGM]{}]{}is explicitly distributed across agents and it is used to extract the local plans from a set of distributed planning graphs. Under the *iterative response planning* strategy introduced by [[DPGM]{}]{}, the solving process is started by one agent, which proposes a local plan along with a set of coordination constraints. The subsequent agent uses its CSP to extract a local plan compatible with the prior agent’s plan and constraints. If an agent is not able to generate a compliant plan, [[DPGM]{}]{}backtracks to the previous agent, which puts forward an alternative plan with different coordination constraints.
### [[$\mu$-SATPLAN]{}]{}, 2010 {#mu-satplan-2010 .unnumbered}
[[$\mu$-SATPLAN]{}]{}[@Dimopoulos12] is a MAP solver that extends the satisfiability-based planner [[SATPLAN]{}]{}[@Kautz06] to a multi-agent context. [[$\mu$-SATPLAN]{}]{}performs an *a priori* distribution of the MAP task goals, ${{\mathcal G}}$, among the agents in ${{\mathcal{AG}}}$. Similarly to [[DPGM]{}]{}, agents follow an iterative response planning strategy, where each participant takes the previous agent’s solution as an input and extends it to solve its assigned goals via [[SATPLAN]{}]{}. This way, agents progressively generate a solution.
[[$\mu$-SATPLAN]{}]{}is unable to attain tasks that include [group goals]{}because it assumes that each agent can solve its assigned goals by itself. [[$\mu$-SATPLAN]{}]{}is experimentally validated on several multi-agent tasks of the *logistics*, *storage* and *TPP* domains. Although these tasks feature only two planning agents, the authors claim that [[$\mu$-SATPLAN]{}]{}is capable of solving tasks with a higher number agents.
### [[MAPR]{}]{}, [[PMR]{}]{}, [[CMAP]{}]{}, 2013-2015 {#mapr-pmr-cmap-2013-2015 .unnumbered}
Multi-Agent Planning by Reuse ([[MAPR]{}]{}) [@Borrajo13] allocates the goals ${{\mathcal G}}$ of the task among the agents before planning through a relaxed reachability analysis. The private information of the local plans is encrypted, thus preserving weak privacy by obfuscating the agents’ local tasks. [[MAPR]{}]{}also follows an iterative response plan synthesis scheme, wherein an agent takes as input the result of the prior agent’s solution plan and runs the [[LAMA]{}]{}planner [@Richter10] to obtain an extended solution plan that attains its allocated agent’s goals. The plan of the last agent is a solution plan for the MAP task, which is parallelized to ensure that execution agents perform as many actions in parallel as possible. [[MAPR]{}]{}is limited to tasks that do not feature specialized agents or [group goals]{}. This limitation is a consequence of the assumption that each agent is able to solve its allocated goals by itself, which renders [[MAPR]{}]{}incomplete.
Plan Merging by Reuse ([[PMR]{}]{}) [@Luis14; @pmr-codmap15] draws upon the goal allocation and obfuscation privacy mechanisms of [[MAPR]{}]{}. Unlike [[MAPR]{}]{}, agents carry out the planning stage simultaneously instead of sequentially and each agent generates local plans for its assigned goals. In the post-planning plan merging strategy of [[PMR]{}]{}, the resulting local plans are concatenated, yielding a sequential global solution. If the result of the merging process is not a valid solution plan, local plans are merged through a repair procedure. If a merged solution is not found, the task is solved via a single-agent planner.
Although [[CMAP]{}]{}[@mapr-cmap-codmap15] follows the same goal allocation and obfuscation strategy of [[MAPR]{}]{}and [[PMR]{}]{}, the plan synthesis scheme of [[CMAP]{}]{}transforms the encrypted local tasks into a single-agent task ($|{{\mathcal{AG}}}| = 1$), which is then solved through the planner [[LAMA]{}]{}. [[CMAP]{}]{}was the best-performing approach of this family of MAP planners in the 2015 [[CoDMAP]{}]{}competition as shown in Table \[tab:features\]. [[CMAP]{}]{}ranked 7th in the centralized track and exhibited a solid performance over the 12 domains of the [[CoDMAP]{}]{}benchmark (approximately 90% coverage). [[PMR]{}]{}and [[MAPR]{}]{}ranked 14th and 15th, with roughly 60% coverage. The plan synthesis scheme of [[MAPR]{}]{}affects its performance in domains that feature [group goals]{}, such as *depots* or *woodworking*, while [[PMR]{}]{}offers a more stable performance over the benchmark.
### [[MAP-LAPKT]{}]{}, 2015 {#map-lapkt-2015 .unnumbered}
[[MAP-LAPKT]{}]{}[@map-lapkt-codmap15] conceives a MAP task as a problem that can be transformed and solved by a single-agent planner using the appropriate encoding. More precisely, [[MAP-LAPKT]{}]{}compiles the MAP task into a task that features one planning agent ($|{{\mathcal{AG}}}|=1$) and solves with the tools provided in the repository [[LAPKT]{}]{}[@lapkt]. The authors of [@map-lapkt-codmap15] try three different variations of best-first and depth-first search that result in algorithms with different theoretical properties and performance. The task translation performed by [[MAP-LAPKT]{}]{}offers weak privacy preservation guarantees.
As shown in Table \[tab:features\], two of the three versions of [[MAP-LAPKT]{}]{}that participated in the [[CoDMAP]{}]{}ranked 2nd and 3rd in the centralized track. The coverage of [[MAP-LAPKT]{}]{}and of [[CMAP]{}]{}is roughly to 90% of the benchmark problems, an indication that shows the efficiency of the scheme that compiles a MAP task into a single-agent task.
### [[MARC]{}]{}, 2015 {#marc-2015 .unnumbered}
The Multi-Agent Planner for Required Cooperation ([[MARC]{}]{}) [@marc-codmap15] is a centralized MAP solver based on the theory of required cooperation [@Zhang14]. [[MARC]{}]{}analyzes the agent distribution of the MAP task and comes up with a different arrangement of planning agents. Particularly, [[MARC]{}]{}compiles the original task into a task with a set of *transformer agents*, each one being an ensemble of various agents; i.e., $|{{\mathcal{AG}}}_{MARC}| < |{{\mathcal{AG}}}|$. A transformer agent comprises the representation of various agents of the original MAP task including all their actions. The current implementation of [[MARC]{}]{}compiles all the agents in ${{\mathcal{AG}}}$ into a single transformer agent ($|{{\mathcal{AG}}}_{MARC}| = 1$). Then, a solution plan is computed via [[FD]{}]{}[@Helmert06] or the portfolio planner [[IBACOP]{}]{}[@Cenamor14], and the resulting plan is subsequently translated into a solution for the original MAP task. [[MARC]{}]{}preserves weak privacy since private elements of the MAP task are occluded in the transformer agent task.
Regarding experimental performance, [[MARC]{}]{}ranks at the 4th position of the centralized [[CoDMAP]{}]{}with 90% coverage, thus being one of the best-performing MAP approaches. The experimental results also reveal the efficiency of this multi-to-one agent transformation.
### [[ADP]{}]{}, 2013 {#adp-2013 .unnumbered}
The Agent Decomposition-based Planner ([[ADP]{}]{}) [@Crosby13] is a factored planning solver that exploits the inherently multi-agent structure (agentization) of some [[STRIPS]{}]{}-style planning tasks and comes up with a MAP task where $|{{\mathcal{AG}}}| > 1$. [[ADP]{}]{}applies a state-based centralized planning procedure to solve the MAP task. In each iteration, [[ADP]{}]{}determines a set of subgoals that are achievable from the current state by one of the agents. A search process, guided through the well-known $h_{FF}$ heuristic, is then applied to find a plan that achieves these subgoals, thus resulting in a new state. This mechanism iterates successively until a solution is found.
Experimentally, [[ADP]{}]{}outperforms several state-of-the-art classical planners (e.g., [[LAMA]{}]{}) and is the top-ranked solver at the centralized track of the [[CoDMAP]{}]{}, outperforming other approaches that compile the MAP task into a single-agent planning task, such as [[MAP-LAPKT]{}]{}or [[CMAP]{}]{}.
### [[Distoplan]{}]{}, 2010 {#distoplan-2010 .unnumbered}
[[Distoplan]{}]{}[@Fabre10] is a factored planning approach that exploits independence within a planning task. Unlike other factored methods [@Kelareva07], [[Distoplan]{}]{}does not set any bound on the number of actions or coordination points of local plans. In [[Distoplan]{}]{}, a component or abstraction of the global task is represented as a finite automaton, which recognizes the regular language formed by the local valid plan of the component. This way, all local plans are manipulated at once and a generic distributed optimization technique enables to limit the number of compatible local plans. With this unbounded representation, all valid plans can be computed in one run but stronger conditions are required to guarantee polynomial runtime. [[Distoplan]{}]{}is the first optimal MAP solver in the literature (note that [[Planning First]{}]{}is optimal with respect to the number of coordination points, but local planning is carried out through a suboptimal planner).
[[Distoplan]{}]{}was experimentally tested in a factored version of the *pipesworld* domain. However, the solver is unable to solve even the smallest instances of this domain in a reasonable time. The authors claim that the reason is that [[Distoplan]{}]{}scales roughly as $n^3$, where $n$ is the number of components of the global task. For obvious reasons, [[Distoplan]{}]{}has not been empirically compared against other MAP solvers in the literature.
### [[A\#]{}]{}, 2012 (not implemented) {#a-2012-not-implemented .unnumbered}
In the line of factored planning, [[A\#]{}]{}[@Jezequel12] is a multi-agent A\* search that finds a path for the goal in each local component of a task and ensures that the component actions that must be jointly performed are compatible. [[A\#]{}]{}runs in parallel a modified version of the A\* algorithm in each component, and the local search processes are guided towards finding local plans that are compatible with each other. Each local A\* finds a plan as a path search in a graph and informs its neighbors of the common actions that may lead to a solution. Particularly, each agent searches its local graph or component while considering the constraints and costs of the rest of agents, received through an asynchronous communication mechanism. The authors of [@Jezequel12] do not validate [[A\#]{}]{}experimentally; however, the soundness, completeness and optimality properties of [[A\#]{}]{}are formally proven.
### [[PSM]{}]{}, 2014 {#psm-2014 .unnumbered}
[[PSM]{}]{}[@Tozicka15; @psm-codmap15] is a recent distributed MAP solver that follows [[Distoplan]{}]{}’s compact representation of local agents plans into Finite Automata, called Planning State Machines (PSMs). This planner defines two basic operations: obtaining a public projection of a PSM and merging two different PSMs. These operations are applied to build a public PSM consisting of merged public parts of individual PSMs. The plan synthesis scheme gradually expands the agents’ local PSMs by means of new local plans. A solution for the MAP task is found once the public PSM is not empty. [[PSM]{}]{}weakly preserves privacy as it obfuscates states of the PSMs in some situations. [[PSM]{}]{}applies an efficient handling of communication among agents, which grants this solver a remarkable experimental performance in both the centralized and distributed setting. In the centralized [[CoDMAP]{}]{}track, [[PSM]{}]{}ranks 12th (solving 70% of the tasks), and it is the top performer at the distributed track of the competition.
### [[DPP]{}]{}, 2016 {#dpp-2016 .unnumbered}
The DP-Projection Planner ([[DPP]{}]{}) [@Shani16], is a centralized [[MA-STRIPS]{}]{}solver that uses the Dependency-Preserving (DP) projection, a novel and accurate public projection of the MAP task information with object cardinality privacy guarantees. The single planning agent of [[DPP]{}]{}uses the [[FD]{}]{}planner to synthesize a high-level plan which is then extended with the agents’ private actions via the [[FF]{}]{}planner, thus resulting in a multi-agent solution plan.
The authors of [@Shani16] provide a comprehensive experimental evaluation of [[DPP]{}]{}through the complete benchmark of the 2015 [[CoDMAP]{}]{}competition. The results show that [[DPP]{}]{}outperforms most of the top contenders of the competition (namely, [[GPPP]{}]{}, [[MAPR]{}]{}, [[PMR]{}]{}, [[MAPlan]{}]{}and [[PSM]{}]{}). All in all, [[DPP]{}]{}can be considered the current top [[MA-STRIPS]{}]{}-based solver, as well as one of the best-performing MAP approaches to date.
### [[TFPOP]{}]{}, 2011 {#tfpop-2011 .unnumbered}
[[TFPOP]{}]{}[@Kvarnstrom11] is a hybrid approach that combines the flexibility of partial-order planning and the performance of forward-chaining search. Unlike most [[MA-STRIPS]{}]{}-based solvers, [[TFPOP]{}]{}supports temporal reasoning with durative actions. [[TFPOP]{}]{}is a centralized approach that synthesizes a solution for multiple executors. It computes *threaded partial-order plans*; i.e., non-linear plans that keep a thread of sequentially-ordered actions per agent, since authors assume that an execution agent performs its actions sequentially.
[[TFPOP]{}]{}is tested in a reduced set of domains, which include the well-known *satellite* and *zenotravel* domains, as well as a UAV delivery domain. The objective of this experimentation is to compare [[TFPOP]{}]{}against several partial-order planners. [[TFPOP]{}]{}is not compared to any MAP solver in this taxonomy.
Interleaved Planning and Coordination MAP Solvers {#taxonomy_interleaved}
-------------------------------------------------
Under the interleaved scheme, labelled as *IL* in the column *Coordination strategy* of Table \[tab:features\], agents jointly explore the search space intertwining their planning and coordination activities. The development of interleaved MAP solvers heavily relies on the design of robust communication protocols to coordinate agents during planning.
### [[MAP-POP]{}]{}, [[FMAP]{}]{}, [[MH-FMAP]{}]{}, 2010-2015 {#map-pop-fmap-mh-fmap-2010-2015 .unnumbered}
In this family of MAP solvers, agents apply a distributed exploration of the plan space. Agents locally compute plans through an embedded partial-order planning (POP) component and they build a joint search tree by following an A\* search scheme guided by global heuristic functions.
[[MAP-POP]{}]{}[@Torreno12ECAI; @Torreno12KAIS] performs an incomplete search based on a classical backward POP algorithm and POP heuristics. [[FMAP]{}]{}[@Torreno14] introduces a sound and complete plan synthesis scheme that uses a forward-chaining POP [@Benton12] guided through the $h_{DTG}$ heuristic. [[MH-FMAP]{}]{}[@fmap-codmap15] applies a multi-heuristic search approach that alternates $h_{DTG}$ and $h_{Land}$, building a Landmark Graph (LG) to estimate the number of pending landmarks of the partial-order plans. The three planners guarantee weak privacy since private information is occluded throughout the planning process and heuristic evaluation. The $h_{Land}$ estimator uses some form of obfuscation during the construction of the LG.
Regarding experimental results, [[FMAP]{}]{}is proven to outperform [[MAP-POP]{}]{}and [[MAPR]{}]{}in terms of coverage over 10 MAP domains, most of which are included in the [[CoDMAP]{}]{}benchmark. Results in [@Torreno15] indicate that [[MH-FMAP]{}]{}obtains better coverage than both [[FMAP]{}]{}and [[GPPP]{}]{}. Interestingly, this planner exhibits a much worse performance in the [[CoDMAP]{}]{}(see Table \[tab:features\]), ranking 17th with only 42% coverage. This is due to the lose of accuracy of the $h_{DTG}$ heuristic when the internal state-variable representation of the tasks in [[MH-FMAP]{}]{}is transformed to a propositional representation to be tested in the [[CoDMAP]{}]{}benchmark, thus compromising the performance of the solver [@fmap-codmap15].
### [[MADLA]{}]{}, 2013 {#madla-2013 .unnumbered}
The Multiagent Distributed and Local Asynchronous Planner ([[MADLA]{}]{}) [@Stolba15] is a centralized solver that runs one thread per agent on a single machine and combines two versions of the $h_{FF}$ heuristic, a projected (local) variant ($h_L$) and a distributed (global) variant ($h_D$) in a multi-heuristic state-space search. The main novelty of [[MADLA]{}]{}is that the agent which is computing $h_D$, which requires contributions of the other agents for calculating the global heuristic estimator, is run asynchronously and so it can continue the search using $h_L$ while waiting for responses from other agents that are computing parts of $h_D$. [[MADLA]{}]{}evaluates as many states as possible using the global heuristic $h_D$, which is more informative than $h_L$. This way, [[MADLA]{}]{}can use a computationally hard global heuristic without blocking the local planning process of the agents, thus improving the performance of the system.
Experimentally, [[MADLA]{}]{}ranks 13th in the centralized [[CoDMAP]{}]{}, reporting 66% coverage. It outperforms most of the distributed MAP solvers of the competition, but it is not able to solve the most complex tasks of the [[CoDMAP]{}]{}domains, thus not reaching the figures of the top performers such as [[ADP]{}]{}, [[MAP-LAPKT]{}]{}or [[MARC]{}]{}.
### [[MAFS]{}]{}, [[MAD-A\*]{}]{}, 2012-2014 {#mafs-mad-a-2012-2014 .unnumbered}
[[MAFS]{}]{}[@Nissim14] is an updated version of [[Planning First]{}]{}that implements a distributed algorithm wherein agents apply a heuristic state-based search (see section \[heuristic\]). In [@Nissim12], authors present [[MAD-A\*]{}]{}, a cost-optimal variation of [[MAFS]{}]{}. In this case, each agent expands the state that minimizes $f=g+h$, where $h$ is estimated through an admissible heuristic. Particularly, authors tested the landmark heuristic *LM-Cut* [@Helmert09] and the abstraction heuristic *Merge&Shrink* [@Helmert07]. [[MAD-A\*]{}]{}is the first distributed and interleaved solver based on [[MA-STRIPS]{}]{}.
[[MAFS]{}]{}is compared against [[MAP-POP]{}]{}and [[Planning First]{}]{}over the *logistics*, *rovers* and *satellite* domains, notably outperforming both solvers in terms of coverage and execution time [@Nissim14]. On the other hand, the authors of [@Nissim14] only compare [[MAD-A\*]{}]{}against single-agent optimal solvers.
### [[Secure-MAFS]{}]{}, 2015 (not implemented) {#secure-mafs-2015-not-implemented .unnumbered}
[[Secure-MAFS]{}]{}[@Brafman15] is an extension of [[MAFS]{}]{}towards secure MAP, and it is currently the only solver that offers *strong privacy* guarantees (see section \[practical\_privacy\]). Currently, [[Secure-MAFS]{}]{}is a theoretical work that has not been yet implemented nor experimentally evaluated.
### [[GPPP]{}]{}, 2014 {#gppp-2014 .unnumbered}
The Greedy Privacy-Preserving Planner ([[GPPP]{}]{}) [@Maliah14; @Maliah16] builds upon [[MAFS]{}]{}and improves its performance via a global landmark-based heuristic function. [[GPPP]{}]{}applies a *global planning* stage and then a *local planning* stage. In the former, agents agree on a joint coordination scheme by solving a relaxed MAP task that only contains public actions (thereby preserving privacy) and obtaining a skeleton plan. In the *local planning* stage, agents compute private plans to achieve the preconditions of the actions in the skeleton plan. Since coordination is done over a relaxed MAP task, the individual plans of the agents may not succeed at solving the actions’ preconditions. In this case, the global planning stage is executed again to generate a different coordination scheme, until a solution is found. In [[GPPP]{}]{}, agents weakly preserve privacy by obfuscating the private information of the shared states through private state identifiers.
[[GPPP]{}]{}provides a notable experimental performance, ranking 10th in the centralized [[CoDMAP]{}]{}track. [[GPPP]{}]{}reaches 83% coverage and is only surpassed by the different versions of [[ADP]{}]{}, [[MARC]{}]{}, [[MAP-LAPKT]{}]{}, [[CMAP]{}]{}and [[MAPlan]{}]{}, which proves the accuracy of its landmark based heuristic and the overall efficiency of its plan synthesis scheme.
### [[MAPlan]{}]{}, 2015 {#maplan-2015 .unnumbered}
[[MAPlan]{}]{}[@maplan-codmap15] is a heuristic [[MAFS]{}]{}-based solver that adapts several concepts from [[MAD-A\*]{}]{}and [[MADLA]{}]{}. [[MAPlan]{}]{}is a distributed and flexible approach that implements a collection of state-space search methods, such as best first or A[\*]{}, as well as several local and global heuristic functions ($h_{FF}$, *LM-Cut*, potential heuristics and others), which allows the solver to be run under different configurations. [[MAPlan]{}]{}can be executed in a single-machine, using local communication, or in a distributed fashion, where each agent is in a different machine and communication among agents is implemented through network message passing. Regarding privacy, [[MAPlan]{}]{}applies a form of obfuscation, replacing private facts in search states by unique local identifiers, which grants weak privacy.
[[MAPlan]{}]{}exhibits a very solid performance in the centralized and distributed tracks of the 2015 [[CoDMAP]{}]{}competition, ranking 9th and 2nd, respectively. In the centralized track, [[MAPlan]{}]{}obtains 83% coverage, outperforming [[GPPP]{}]{}and reaching similar figures than the top-performing centralized solvers.
Conclusions {#ongoing}
===========
The purpose of this article is to comprehensively survey the state of the art in cooperative MAP, offering an in-detail overview of this rapidly evolving research field, which has experienced multiple key advances over the last decade. These contributions crystallized in the 2015 [[CoDMAP]{}]{}competition, where MAP solvers were compared through an exhaustive benchmark testing encoded with [*MA-PDDL*]{}, the first standard modelling language for MAP tasks.
In this paper, the topic of MAP was studied from a twofold perspective: from the representational structure of a MAP task and from the problem-solving standpoint. We formally defined a MAP task following the well-known MA-STRIPS model and provided several examples which illustrate the features that distinguish MAP tasks from the more compact single-agent planning tasks. We also presented the modelling of these illustrative tasks with [*MA-PDDL*]{}.
MAP is a broad field that allows for a wide variety of problem-solving approaches. For this reason, we identified and thoroughly analyzed the main aspects that characterize a solver, from the architectural design to the practical features of a MAP tool. Among others, these aspects include the computational process of the solvers and the plan synthesis schemes that stem from the particular combination of planning and coordination applied by MAP tools, as well as other key features, such as the communication mechanisms used by the agents to interact with each other and the privacy guarantees offered by the existing solvers.
Finally, we compiled and classified the existing MAP techniques according to the aforementioned criteria. The taxonomy of MAP techniques presented in this survey prioritizes recent domain-independent techniques in the literature. Particularly, we focused on the approaches that took part in the 2015 [[CoDMAP]{}]{}competition, comparing their performance, strengths and weaknesses. The classification aims to provide the reader with a clear and comprehensive overview of the existing cooperative MAP solvers.
The body of work presented in this survey constitutes a solid foundation for the ongoing and future scientific development of the MAP field. Following, we summarize several research trends that have recently captured the attention of the community.
#### Theoretical properties
The aim of the earlier cooperative MAP solvers was to contribute with a satisficing approach capable of solving a relatively small number of problems in a reasonable time but without providing any formal properties [@Nissim10; @Borrajo13]. The current maturity of the cooperative MAP field has witnessed the introduction of some models that focus on granting specific theoretical properties, such as completeness [@Torreno14], optimality [@Nissim14] or stronger privacy preservation guarantees [@Brafman15; @Shani16].
#### Privacy {#privacy}
The state of the art in MAP shows a growing effort in analyzing and formalizing privacy in MAP solvers. Nowadays, various approaches to model private information and to define information sharing can be found in the literature, which reveals that privacy is progressively becoming a key topic in MAP. However, the particular implementation of a MAP solver may jeopardize privacy, if it is possible for an agent to infer private information from the received public data. Aside from the four-level classification exposed in section \[practical\_privacy\], other recent approaches attempt to theoretically quantify the privacy guarantees of a MAP solver [@Stolba16b]. In the same line, the authors of [@Tozicka17] analyze the implications and limits of strong privacy and present a novel [[PSM]{}]{}-based planner that offers strong privacy guarantees.
On the other hand, one can also find work that proposes a smart use of privacy to increase the performance of MAP solvers, like [[DPP]{}]{}[@Shani16], which calculates an accurate public projection of the MAP task in order to obtain a robust high-level plan that is then completed with private actions. This scheme minimizes the communication requirements, resulting in a more efficient search. In [@Maliah17], authors introduce a novel weak privacy-preserving variant of [[MAFS]{}]{}which ensures that two agents that do not share any private variable never communicate with each other, significantly reducing the number of exchanged messages. In general, the study of privacy in MAP is gaining much attention and more and more sophisticated approaches have been recently proposed.
#### MAP with self-interesed agents
The mainstream in MAP with self-interested planning agents is handling situations which involve interactive decision making with possibly conflicting interests. Game theory, the study of mathematical models of conflict and cooperation between rational self-interested agents, arises naturally as a paradigm to address human conflict and cooperation within a competitive situation. Game-theoretic MAP is an active and interesting research field that reflects many real-world situations, and thus, it has a broad variety of applications, among which we can highlight congestion games [@JonssonR11], cost-optimal planning [@NissimB13], conflict resolution in the search of a joint plan [@JordanO15] or auction systems [@RobuNPS11].
From a practical perspective, game-theoretic MAP has been successfully applied to ridesharing problems on timetabled transport services [@HrncirRJ15]. In general, strategic approaches to MAP are very appropriate to model smart city applications like traffic congestion prevention: vehicles can be accurately modelled as rational self-interested agents that want to reach their destinations as soon as possible, but they are also willing to deviate from their optimal routes in order to avoid traffic congestion issues that would affect all the involved agents.
#### Practical applications
MAP is being used in a great variety of applications, like in product assembly problems in industry (e.g., car assembly). Agents plan the manufacturing path of the product through the assembly line, which is composed of a number of interconnected resources that can perform different operations. [[ExPlanTech]{}]{}, for instance, is a consolidated framework for agent-based production planning, manufacturing, simulation and supply chain management [@Pechoucek07].
MAP has also been used to control the flow of electricity in the Smart Grid [@Reddy11]. The agents’ actions are individually rational and contribute to desirable global goals such as promoting the use of renewable energy, encouraging energy efficiency and enabling distributed fault tolerance. Another interesting application of MAP is the automated creation of workflows in biological pathways like the Multi-Agent System for Genomic Annotation ([[BioMAS]{}]{}) [@Decker02]. This system uses [[DECAF]{}]{}, a toolkit that provides standard services to integrate agent capabilities, and incorporates the [[GPGP]{}]{}framework [@Lesser04] to coordinate multi-agent tasks.
In decentralized control problems, MAP is applied in coordination of space rovers and helicopter flights, multi-access broadcast channels, and sensor network management, among others [@Seuken08]. MAP combined with argumentation techniques to handle belief changes about the context has been used in applications of ambient intelligence in the field of healthcare [@Pajares13].
Aside from the aforementioned trends, there is still a broad variety of unexplored research topics in MAP. The solvers presented in this survey do not support tasks with advanced requirements. Particularly, handling temporal MAP tasks is an unresolved matter that should be addressed in the years to come. This problem will involve both the design of MAP solvers that explicitly support temporal reasoning and the extension of [*MA-PDDL*]{}to incorporate the appropriate syntax to model tasks with temporal constraints.
Cooperative MAP, as exposed in this paper, puts the focus on offline tasks, without paying much attention to the problematic of plan execution. Online planning carried out by several agents poses a series of challenges derived from the integration of planning and execution and the need to respond in complex, real-time environments. Real-time cooperative MAP is about planning and simultaneous execution by several cooperative agents in a changing environment. This interesting and exciting research line is very relevant in applications that involve, for example, soccer robots.
Additionally, the body of work presented in this survey does not consider agents with individual preferences. Preference-based MAP is an unstudied field that can be interpreted as a middle ground between cooperative and self-interested MAP, since it involves a set of rational agents that work together towards a common goal while having their own preferences concerning the properties of the solution plan.
All in all, we believe that the steps taken over the last years towards the standardization of MAP tasks and tools, such as the 2015 [[CoDMAP]{}]{}competition or the introduction of [*MA-PDDL*]{}, will decisively contribute to foster a rapid expansion of this field in a wide variety of research directions.
[^1]: http://icaps16.icaps-conference.org/dmap.html
[^2]: http://agents.fel.cvut.cz/codmap
[^3]: Please refer to <http://agents.fel.cvut.cz/codmap/MA-PDDL-BNF-20150221.pdf> for a complete BNF definition of the syntax of [*MA-PDDL*]{}.
[^4]: http://www.icaps-conference.org/index.php/Main/Competitions
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'The Faddeev-Yakubovsky equations for the $\alpha$ particle are solved. Accurate results are obtained for several modern NN interaction models, which include charge-symmetry breaking effects in the NN force, nucleon mass dependences as well as the Coulomb interaction. These models are augmented by three-nucleon forces of different types and adjusted to the 3N binding energy. Our results are close to the experimental binding energy with a slight overbinding. Thus there is only little room left for the contribution of possible 4N interactions to the $\alpha$ particle binding energy. We also discuss model dependences of the binding energies and the wave functions.'
author:
- 'A. Nogga$^{1}$, H. Kamada$^{2}$, W. Glöckle$^{3}$, B.R. Barrett$^{1}$'
bibliography:
- 'literatur.bib'
title: 'The $\alpha$ particle based on modern nuclear forces'
---
Introduction
============
In spite of the tremendously increased computational power of today’s super computers, numerical investigations of nuclear bound states are still a challenging problem, even for systems of few nucleons. Investigations promise insights into the rich structure of nuclear interactions. To this aim one requires reliable solutions of the dynamical equations. In this article we would like to present results for the $\alpha$ particle, which are based on realistic microscopic nuclear forces including three-body interactions.
In recent years forces could be adjusted accurately to the huge amount of available NN low energy scattering data [@cdbonn; @av18; @nijm93]. The overall agreement of the predictions of these model forces with the data is essentially perfect. As a result it has been shown that most of the observables in the low energy regime of the 3N continuum could be predicted model independently [@report; @witala01a], though the interactions themselves are quite different. On the other side, it is known since quite a long time that the 3N binding energies (BE) are quite model dependent and, moreover, are generally smaller than the experimental value [@friar88b; @wu93; @stadler91; @nogga97]. It is assumed that most of this underbinding is due to three-nucleon forces (3NF) and modifications of the NN interaction in the presence of a third nucleon. The latter one is of course also part of a three-nucleon force mechanism.
The nature of these 3NF’s is still not completely understood. It is clear that such forces should already arise because of the composite structure of the nucleons, what is partially taken into account by allowing for intermediate $\Delta$-excitation. Other mechanisms of various meson-exchange types will also contribute (for a review see [@robilotta87]). In recent years there has been new progress in understanding the form of nuclear forces, because of the application of chiral perturbation theory ($\chi$PT) [@weinberg90; @weinberg91; @kolck94; @ordonez96; @kaiser97; @epelbaum00; @epelbaumphd]. From this developments one can expect a more systematic understanding of the form of NN and 3N forces in the near future. However, $\chi$PT implies [*a priori*]{} unknown constants, the low-energy constants, which have to be determined from experimental data. The bound states of few-nucleons seem to be an ideal laboratory to determine 3NF parameters, as the BE’s are sensitive to the 3N interaction and they are expected to be governed by the low-energy regime of nuclear physics [@bedaque00]. Therefore, the understanding of nuclear bound states is an important contribution to the understanding of the 3NF.
At present these chiral interactions are not as accurate as the traditional, phenomenological NN forces [@cdbonn; @av18; @nijm93]. At the order of the chiral expansion parameter considered up to now [@epelbaum01c], they do not yet describe the NN phase shifts with the same accuracy. Allowing, however, for additional fine tuning a high accuracy description can be achieved [@entem01]. The aim of this article is to pin down model dependences of predictions for the $\alpha$ particle BE and wave function (WF) properties. To insure that differences in the predictions are not due to an inaccurate description of the NN system, but are due to the more fundamental differences of the interaction models, we restrict ourselves to the traditional models in this paper. The techniques developed, however, will help to apply also the upcoming chiral interactions. First investigations, using chiral interactions, have already been undertaken [@epelbaum01a; @epelbaum01c].
Our approach leads immediately to a basic problem. It has been shown that the 3N interaction cannot be determined uniquely and, moreover, that each NN interaction has to be accompanied by a different 3NF [@polyzou90]. For the traditional NN interactions, there are no 3NF’s available, which have been derived consistently to them. Therefore, we have to rely on 3NF’s, which just take parts of the mechanisms into account, which are expected to contribute to the 3NF. These models are, for example, the Tucson-Melbourne (TM) [@coon79] and the Urbana IX (Urb-IX) [@pudliner97] 3NF’s. For the different NN interactions these models have been adjusted separately to the experimental $^3$H BE, as described in Section \[sec:resa\]. This scheme is justified for two reasons:
- It has been shown that many 3N scattering observables in the low energy regime (below $\approx 10$ MeV nucleon lab energy) scale with the $^3$H BE. This means that predictions for different model Hamiltonians are equal, when the models predict the same $^3$H BE [@huber98b; @witala98c; @friar86b]. An adjustment of the 3NF’s exclude model dependences related to this phenomenon. We will see that these effects are also visible for the $\alpha$ particle.
- In the high energy regime (above $\approx 100$ MeV nucleon lab energy) the predictions for 3N scattering observables are sensitive to the 3NF showing that the available 3NF models are quite different [@witala01a]. This insures that the models applied in this paper cover a wide range of possible 3NF’s. Therefore, our results show the model dependences of our current understanding of the $\alpha$ particle, which are related to the structure of the 3NF.
Bound states of light nuclei have been investigated by several groups using different techniques [@wiringa00; @pieper01; @navratil00; @suzukilec; @viviani92; @kameyama89; @barnea00; @ciesielski98; @kamada92a; @nogga00; @kamada01c]. But much of the work is still restricted to somewhat simplified interactions. Perhaps the most advanced calculations covering several nuclei have been performed by the Argonne-Los Alamos collaboration [@wiringa00; @pieper01]. Using the Greens functions Monte Carlo (GFMC) technique, they were able to predict BE’s for the light nuclei up to $A=8$. However, their work is restricted to the AV18 NN interaction model and the class of Urb-IX 3NF’s (new terms not considered here have been added in [@pieper01]). This leads to the question, whether the other available interactions give similar or different results for these nuclei. In this respect the “no-core” shell model approach (NCSM) [@navratil00] might be more flexible. But the work on 3NF’s has not been finished yet. Therefore, we think that a study of the 4N system can provide important new information on the nuclear interactions, if one can investigate a wide range of NN and 3N models in this system.
In this paper we use the Faddeev-Yakubovsky scheme to solve the non-relativistic Schrödinger equation (SE) for four nucleons. This has been started already in Refs. [@kamada92a; @glockle93a; @glockle93b; @kamada92b; @kamada94b; @kamada94c]. With this method we are able to get reliable results for the BE and the WF of the $\alpha$ particle for several NN and 3N interactions. The calculations are restricted to $A=3$ and $A=4$, but we were able to pin down the dependence on today’s interaction models. The highly accurate WF, which result from the calculations, are necessary for the analysis of several on-going or planned experiments on the $\alpha$ particle, which might reveal the short-range correlations in nuclei [@e97-111] or give insights into the charge independence breaking of the nuclear interaction [@kolckpriv]. Exact WF’s are also necessary to understand the results of parity violating e$^-$ scattering experiments [@ramavataram94]. Therefore, we will also give first results of calculations including the isospin $T=1$ and $T=2$ component of the $\alpha$ particle ground state WF.
Another important issue is a first estimate of the size of a possible 4N interaction. We expect that it should show up especially prominent in the $\alpha$ particle, because of its high density. Our calculations give some hints, as to whether there is room for an important contribution of the 4N interaction in nuclei given today’s NN and 3N interaction models.
In Section \[sec:form\] we briefly review the 4N Faddeev-Yakubovsky formalism. The calculations are based on adjusted 3NF’s. The adjustment procedure is described in Section \[sec:resa\]. Our results for the BE’s of the $\alpha$ particle based on various nuclear force combinations are given in Section \[sec:resb\] and the properties of the obtained WF’s are presented in Section \[sec:resc\]. Finally we summarize in Section \[sec:concl\].
The 4N Yakubovsky formalism {#sec:form}
===========================
![Definition of the 1A and 2A type of Jacobi coordinates.[]{data-label="fig:jacobi"}](jakobi.eps){width="7cm"}
![The three parts of a meson exchange 3NF, which differ only by an exchange of the particles.[]{data-label="fig:3nf"}](3nf.eps){width="7cm"}
The technical challenge in all investigations of nuclear bound states is the accurate inclusion of all short-range correlations in the nuclear WF. Due to these short-range correlations, the partial wave decomposition of nuclear WF’s is very slowly converging. This hold especially for the very tightly bound $\alpha$ particle WF. Therefore, a rewriting of the Schrödinger equation for the 4N system $$H \ \Psi = \left( \ T + \sum_{i<j} V_{ij} + \sum_{i<j<k} V_{ijk} \
\right)
\ \Psi = E \ \Psi$$ according to the formalism of Yakubovsky [@yakubovsky67] is useful. We take NN pair potentials $V_{ij}$ and 3N potentials $V_{ijk}$ into account. $T$ denotes the kinetic energy operator, $H$ the full 4N Hamiltonian and $\Psi$ the 4N WF. We will use Jacobi coordinates (see Fig. \[fig:jacobi\]) to represent our WF and dynamical equations. These separate the center of mass motion and, at the same time, guarantee a kinetic energy operator independent from angular variables. But these coordinates do not include [ *all*]{} kinds of pair coordinates at the same time and it is hard to describe the short range correlations of pairs in other coordinates than their own relative coordinate. Other coordinates unavoidably lead to strong angular dependences or, in other words, to a very slowly converging series of partial waves. On the other hand, the Jacobi coordinates include the relative coordinates of [*some*]{} pairs. Correlations of those pairs are easily described. The WF contains the correlations of all pairs and is hard to expand in Jacobi coordinates. This makes the decomposition of the WF in Yakubovsky components (YC) highly advisable. The YC’s single out clusters of the four particles. The way they are defined guarantees that they are driven by correlations within these clusters only. Therefore, they are efficiently expanded in Jacobi coordinates, which single out the same clusters.
In the isospin formalism nucleons are identical particles. This implies several symmetry properties, which connect the different YC’s and reduce the number of independent coupled equations and YC’s to two. The following set of Yakubovsky equations (YE’s) are obtained for the two YC’s $\psi_{1A}$ and $\psi_{2A}$ [@kamada92a; @glockle93a; @nogga01b] $$\begin{aligned}
\label{eq:alphaeq1}
\psi_{1A} & \equiv & \psi_{(12)3,4} = G_0 t_{12} P
\ [ (1-P_{34}) \ \psi_{1A} + \psi_{2A} ] + (1+G_0 t_{12}) \ G_0 \ V_{123}^{(3)} \ \Psi \\
\label{eq:alphaeq4}
\psi_{2A} & \equiv & \psi_{(12)34} = G_0 t_{12}
\tilde P [ ( 1- P_{34}) \psi_{1A} + \psi_{2A} ] \end{aligned}$$ The other YC’s are replaced by transposition operators $P_{ij}$ and combinations $P=P_{13}P_{23} + P_{12}P_{23}$ and $\tilde P = P_{13} P_{24}$, acting on the two remaining YC’s. The kinetic energy enters through the free propagator $G_0 = { 1 \over E - T }$ and the pair interaction by means of the pair $t$-matrix $t_{12}$. The 3NF’s show up in the interaction term $V_{123}^{(3)}$. This defines a part of the 3NF in the cluster (123), which is symmetric in the pair (12) and which can be related by an interchange of the three particles to two other parts $V_{123}^{(1)}$ and $V_{123}^{(2)}$ that sum up to the total 3NF of particles 1,2 and 3: $ V_{123} = V_{123}^{(1)} + V_{123}^{(2)} + V_{123}^{(3)}$. For 3NF’s based on a meson-exchange picture, $ V_{123}^{(3)}$ describes the interaction induced by a meson interchanged between particles 1 and 2 and, on the way, re-scattered by the third particle, as indicated in Fig. \[fig:3nf\].
Applying a combination of transpositions to the set of YC’s, one obtains the WF as $$\label{eq:alphawavef}
\Psi = [1-(1+P)P_{34}] (1+P) \psi_{1A} + (1+P)(1+\tilde P) \psi_{2A}$$ The YC’s $\psi_{1A}$ and $\psi_{2A}$ are anti-symmetric in the pairs (12) or (12) and (34), respectively [@nogga01b]. This guarantees the total anti-symmetry of the WF $\Psi$.
The YC’s are expanded in their “natural” Jacobi coordinates. This means that $\psi_{1A}$ is represented in the coordinates shown in the top of Fig. \[fig:jacobi\], because both single out the pair (12) and the cluster (123). $\psi_{2A}$ singles out both pairs, (12) and (34), and is simplest, when expanded in the coordinates, shown at the bottom of Fig. \[fig:jacobi\]. Each of the coordinates involves three relative momenta $p_{12}$, $p_{3}$ and $q_{4}$ or $p_{12}$, $p_{34}$ and $q$, respectively. The angular dependence is expanded in partial waves, leading to three orbital angular momentum quantum numbers for each kind of coordinate: $l_{12}$, $l_{3}$ and $l_{4}$ or $l_{12}$, $l_{34}$ and $\lambda$. We use $jj$ coupling. Therefore, we couple, as indicated in the figure, the orbital angular momenta and corresponding spin quantum numbers to the intermediate quantum numbers $j_{12}$, $I_3$ and $I_4$ or $j_{12}$ and $j_{34}$, and these are coupled to the total angular momentum $J$ and its third component $M$, using two additional intermediate angular momenta $j_3$ and $I$: $ (( j_{12} I_3 ) j_3 I_4 ) J M $ or $ (( j_{12} \lambda ) I j_{34}) JM $. For the isospin quantum numbers (see Fig. \[fig:jacobi\]) similar coupling schemes to total isospin $TM_T$ involve only one intermediate quantum number $\tau$: $((t_{12} \ {1 \over 2 } ) \tau \ {1 \over 2 } ) T M_T $ or $(t_{12} t_{34}) T M_T $.
As we already pointed out, the partial wave decomposition requires a huge number of partial waves, whenever one needs to represent correlations of pairs and clusters in coordinates, which don’t single them out. Unfortunately, this is still necessary in the intermediate states in Eqs. (\[eq:alphaeq1\]) and (\[eq:alphaeq4\]) (for $P \ \psi_{1A}$, etc.). Therefore, we still need a tremendous number of partial waves to find converged results. However, we are going to show numerically in Section \[sec:resb\] that we can speed up the convergence greatly by using the Yakubovsky decomposition. For our calculations, we decided to truncate the orbital angular momenta, requiring that $j_{ij} \le 6$ and $l_i, \lambda \le 8$. Additionally, we constrain the expansion for both kinds of coordinates by another parameter $l_{sum}^{max}$, requiring that $l_{12}+l_{3}+l_{4} \le l_{sum}^{max}$ and $l_{12}+l_{34}+ \lambda \le l_{sum}^{max}$.
We use $l_{sum}^{max}=14$. Our most sophisticated calculations, including the $T=1$ and $T=2$ isospin channels, need a total number of 4200 partial waves for the first kind of coordinates and 2000 for the second kind. We require 36-40 mesh points to discretize the magnitudes of each of the momenta $p_{12}$, $p_{3}$ and $q_{4}$ or $p_{12}$, $p_{34}$ and $q$. This insures that we obtain results for the binding energy of the $\alpha$ particle, which are converged within 50 keV.
Using this partial wave truncation, we find that the discretized integral kernel for the set of Eqs. (\[eq:alphaeq1\]) and (\[eq:alphaeq4\]) is of the dimension $3 \cdot 10^8 \times 3 \cdot 10^8$. Clearly this can no longer be treated by standard techniques of numerical linear algebra, like the QR-algorithm, and one is forced to use an iterative scheme. A Lanczos type method [@saake; @stadler91] has turned out to be very powerful in the past and also here. Succinctly, for an arbitrary $N$-component starting vector for the unknown amplitude, one applies the kernel leading to a new vector. This is repeated several times by applying the kernel always to the new vectors. That set of vectors is then orthonormalized and the unknown amplitude expanded into those elements. Inserting this expansion again into the eigenvalue equation Eqs. (\[eq:alphaeq1\]) and (\[eq:alphaeq4\]), one ends up with a small set of linear algebraic eigenvalue equations of dimension $n$, where $n$ counts the number of applications of the kernel. $n$ is typically 10-20. The energy eigenvalue $E$, which is buried as a parameter in the kernel is determined in such a manner that the eigenvalue of the kernel is 1.
Another challenge is the application of the three-nucleon force. In momentum space and partial-wave decomposed, this is a huge matrix of typical dimension $5 \times 10^4 \times 5 \times 10^4$ for each total 3N angular momentum and parity. In case of the $\alpha$ particle, the 3N subsystem total angular momenta have to be taken into account up to $17 \over 2$. Instead of preparing these matrices, we handle the 3N forces differently. They can be naturally broken up into a sequence of pseudo two-body forces with a change of Jacobi momenta in between (transpositions). This has been described, for the first time in [@huber97b]. The generalization to the 4N system is described in Appendix \[app:3nf\]. This technique is much more efficient and even allows one to evaluate the 3NF’s in each new iteration of the kernel – no storage of huge intermediate matrices related to 3NF’s is required.
A typical run on a massively parallel T3E with 128 processors takes 2 hrs to get one eigenvalue and the corresponding eigenvector. For our method of parallelization we refer to [@noggaphd].
Results
=======
Adjustment of 3NF’s {#sec:resa}
-------------------
In this paper we restrict ourselves to the modern realistic NN interactions, which are all fitted to the NN data with the same high accuracy and also provide a nn force, which predicts a reasonable nn scattering length. These interactions are the AV18 [@av18] and the CD-Bonn [@cdbonn]. Additionally we show results for the Nijm I, Nijm II and Nijm 93 interactions [@nijm93], which are not adjusted to the nn scattering length and in case of Nijm 93 give a slightly less accurate fit to the NN data. The results for the $^3$He and $^3$H BE’s are shown in Table \[tab:3nbound\]. They are based on calculations, which take two-body angular momenta up to $j_{12}=6$ into account and are converged up to 2 keV. The full charge dependence of the interaction as well as the n-p mass difference are considered. Also the Coulomb force is included exactly as described in [@noggaphd; @nogga00].
[l|rr|rr|r]{} & & & interaction & $E_B$ & $T$ & $E_B$ & $T$ & $\Delta E_B$ CD-Bonn & -8.013 & 37.43 & -7.288 & 36.62 & 0.725 AV18 & -7.628 & 46.76 & -6.917 & 45.69 & 0.711 Nijm I & -7.741 & 40.74 & -7.083 & 40.01 & 0.658 Nijm II & -7.659 & 47.55 & -7.008 & 46.67 & 0.651 Nijm 93 & -7.668 & 45.65 & -7.014 & 44.79 & 0.654 Exp. & -8.482 & — & -7.718 & — & 0.764
As well known [@stadler91; @sauer86; @friar93; @wu93; @kievsky95] all NN model interactions lead to an underpredicted 3N BE. The underprediction is strongly model dependent and ranges from 0.8 MeV to 0.5 MeV for the most modern interactions (see Table \[tab:3nbound\]) though their description of the NN data is comparable. For benchmark purposes, we also show results for the expectation value of the kinetic energy. These tend to be smaller for the non-local interaction Nijm I and CD-Bonn. This behavior can be traced back to the softer repulsive core of non-local NN forces. We also show the binding energy difference of the two mirror nuclei $\Delta E_B$. One sees that all models underpredict the experimental value. The deviation is somehow larger for the Nijmegen interactions, which do not describe the nn scattering length correctly. The additional differences for the Nijmegen interactions are, therefore, likely a result of an inadequate description of the NN scattering data. We will address the issue of the $^3$H-$^3$He binding energy difference in Ref. [@nogga01c]; therefore, we do not want to go in details here.
Two possible dynamical ingredients are still missing in our calculations: relativistic effects and 3NF’s. We will not address the interesting question of including relativity in few-nucleon dynamics here. Attempts to understand this issue can be found in Refs. [@rupp92; @sammarruca92; @stadler97a; @stadler97b; @glockle86; @forest99]. The results of those calculations are varying. Whereas approaches based on field equations, like Bethe-Salpeter or Gross equations generally predict an increased binding energy compared to the non-relativistic solution, the calculations based on a relativistic Schrödinger equation predict a decreased binding energy. In the latter case the relativistic effects are driven by boost properties, whereas in field theoretical approaches additional dynamical effects also occur. The magnitude of the predicted effects is of the order of 200 keV. The problem is not yet solved. It has also been observed that relativistic effects and 3NF effects are related and cannot be separated in field equation approaches [@stadler97b]. In this paper we neglect all relativistic effects, hoping that part of them are included in effective 3NF terms.
The knowledge regarding 3NF’s is similarly scarce, as for the relativistic effects. It has been shown in Ref. [@polyzou90] that 3NF’s are not defined independently from the accompanying NN interactions. Two 3N Hamiltonians based on two different, but phase equivalent NN interactions, can be augmented by a properly chosen 3N interaction to be equivalent in the 3N system. In [@nogga00] we formulated the more inclusive statement that one could, in principle, always find NN interactions, which replace a 3N interaction completely in a 3N Hamiltonian. Ref. [@polyzou90] does not conclude that this is [*always*]{} possible. It is clear anyhow that the transformation are complicated and therefore it is not practicable to use them to get rid of the 3NF’s. As soon as one includes relativistic features the Poincar[é]{} algebra inevitably enforces 3NF’s [@foldy74], which cannot be transformed away.
In view of this connection of 3NF models and NN force models, a phenomenological approach to the 3NF is justified: given a 3NF model, one adjusts its parameters in conjunction with one NN interaction model to 3N or other nuclear data leading to different parameter sets of the 3NF for each NN interaction.
For the Urb-IX 3NF the parameters have been fixed in conjunction with the AV18 interaction using the $^3$H BE and the nuclear matter density predicted by this combination [@pudliner97]. The TM force originally has not been adjusted in this way. Its parameters have been deduced from model assumptions and using $\pi$N scattering data [@coon79; @coon93; @coon01]. It is clear that a complete 3NF based on meson-exchange should include not only $\pi$-$\pi$, but also $\rho$-$\pi$, $\rho$-$\rho$ and so on exchanges. Attempts to include these processes have been done, but conclusive results, fixing the parameter sets, could not be obtained [@stadler95]. Therefore, we assume in our study that we can effectively include the effects of heavier mesons in the $\pi$-$\pi$ exchange TM model by a variation of the $\pi$NN form factor parameter $\Lambda$. It has been observed [@mckellar84; @stadler91] that the $^3$H BE is sensitive to this cut-off. The original value $\Lambda=5.8 m_\pi$ has been fixed by matching the Goldberger Treiman discrepancy [@coon01]. However, as has been argued in [@robilotta86], the form factors are ill-defined, because they strongly influence the long-range part of the 3NF. Therefore an adjustment is justified. We emphasize that the aim of this paper is the investigation of model dependences due to the different 3NF’s. To this aim we only require 3NF models, which are different and which have a sufficiently rich spin-isospin dependence. An adjustment of the 3NF does not spoil these requirements.
We combined in Refs. [@nogga97; @nogga00] the available NN interactions with the TM 3NF and tuned $\Lambda$ to reproduce the $^3$H or $^3$He BE’s. The resulting $\Lambda$ values are shown in Table \[tab:3nbind3nf\]. The table also includes results for a modified TM interaction. It has been argued in [@friar98] that the long-range/short-range part of the $c$-term is not consistent with chiral symmetry. Dropping it leads to a changed set of parameters, which we refer to as TM’. The parameters of TM and TM’ are summarized in Table \[tab:3nfpara\] of Appendix \[app:3nf\].
[l|r|rr|r]{} interaction & $\Lambda$ & $E$($^3$H) & $E$($^3$He) & $\Delta E_B$ CD-Bonn+TM & 4.784 & -8.478 & -7.735 & 0.743 AV18+TM & 5.156 & -8.478 & -7.733 & 0.744 AV18+TM’ & 4.756 & -8.448 & -7.706 & 0.742 AV18+Urb-IX & — & -8.484 & -7.739 & 0.745 AV18+Urb-IX (Pisa) [@kievskypriv] & — & -8.485 & -7.742 & 0.743 AV18+Urb-IX (Argonne) [@wiringa00] & — & -8.47(1) & — & — Exp. & — & -8.482 & -7.718 & 0.764
The fits have been done using less accurate BE calculations not including the isospin $T={3 \over 2}$ component and not including the effect of the n-p mass difference. Therefore, the new results for the BE’s, shown in the table, do not exactly match the experimental values. The deviations are non-significant for the following study, so we refrain from refitting the $\Lambda$’s. We adjusted TM to the $^3$H BE and TM’ to the $^3$He BE. The table also shows our results using the Urb-IX interaction, as defined in [@pudliner97].
Table \[tab:3nbind3nf\] confirms at the same time a well-known scaling behavior of the Coulomb interaction with the BE of $^3$He [@friar87]. The adjusted 3N Hamiltonians predict very similar 3N binding energies and $\Delta E_B$’s. This removes the model dependence of $\Delta E_B$ found in Table \[tab:3nbound\]. We observe that the model independent prediction for these energy difference deviates from the experimental value by about 20 keV. Again, we refer to Ref. [@nogga01c] for a more detailed discussion of this issue. In the same reference, a detailed comparison with hyper-spherical variational calculations is given. In Table \[tab:3nbound\], for comparison, we only show the BE’s obtained by the Pisa and Argonne group. We note that the calculation by the Pisa group is in full agreement with our results. The small deviation from the Argonne result is not significant in view of the comparably large statistical error bar of the GFMC calculation.
We are now ready to apply the 3N model Hamiltonians, given by the $\Lambda$ values in Table \[tab:3nbind3nf\], to the 4N system. By using the models from Table \[tab:3nbind3nf\], we insure that dependences due to scaling effects, as visible for example in $\Delta E_B$, are excluded. Given the very different functional forms of the Urb-IX, TM and TM’ interactions, we can expect to see any remaining model dependences in our calculations.
$\alpha$ particle binding energies {#sec:resb}
----------------------------------
Based on these model Hamiltonians, we solved the YE’s (\[eq:alphaeq1\]) and (\[eq:alphaeq4\]) with no uncontrolled approximation. The following results are based on a partial wave decomposition truncated using $l_{sum}^{max}=14$. It has been verified that this is sufficient to obtain converged BE’s with an accuracy of 50 keV. The binding energies given were found varying the energy parameter in Eqs. (\[eq:alphaeq1\]) and (\[eq:alphaeq4\]) until the eigenvalue 1 appears in the spectrum of the set of YE’s.
Independently, one can check the results with a calculation of the expectation value of the Hamiltonian. We emphasize that this is an important feature of our method, which minimizes the possibility of errors in the codes or unexpected numerical difficulties.
For these checks one faces the problem to represent the WF with high accuracy. We already pointed out that the WF of the $\alpha$ particle is extremely slowly converging, because there is no set of Jacobi momenta suitable to describe the short range correlation in [*all*]{} NN pairs. In Table \[tab:av18conv\] we exemplify the convergence behavior of the WF for the AV18 interaction. The normalization and the expectation values of the kinetic energy, potential energy and Hamiltonian are shown. The WF’s have been derived from the same set of YC’s, using Eq. (\[eq:alphawavef\]). The calculation of the WF is based on a partial wave decomposition truncated with $l_{sum}^{max}=14$. In this way we obtained the WF in the two different representations, depicted in Fig. \[fig:jacobi\]. For the expectation values shown in the table, we truncated the WF in a second step to the partial waves given by the $l_{sum}^{max}$ parameter in the first column. It turned out that the evaluation of the kinetic energy is difficult, because $T$ amplifies the slowly converging high momentum components of the WF. The kinetic-energy expectation values, shown in the fourth and fifth columns of the table, do not converge within the chosen partial-wave truncation. However, one can rewrite the kinetic energy using Eq. (\[eq:alphawavef\]) and the fact that the transposition operators commute with the kinetic energy and simply result in a sign change, if applied to a fully anti-symmetrized WF: $$\label{eq:kintrick}
<\Psi | T | \Psi > = 12 <\Psi | T | \psi_{1A} > + 6 <\Psi | T | \psi_{2A} >$$ The right-hand side involves mixed matrix elements with the YC’s. The first term has to be evaluated in the 1A representation, because $\psi_{1A}$ is given in these coordinates, and the second term in the 2A ones because of the coordinates of $\psi_{2A}$. The results for $T$ based on this equation are shown in the column labeled $T(mix)$ and show a promising convergence behavior. We observe a much faster convergence for the YC’s, which was expected and which justifies the YE’s approach to the 4N Schrödinger equation. Based on this experience, we normalize our WF and the YC’s using a similar formula for the norm. Consequently, the deviation of directly calculated norms of the WF, shown in columns 2 and 3, from one is a measure of the numerical error of our anti-symmetization of the full WF.
[c|cc|ccc|cc|cc]{} $l_{sum}^{max}$ & $<\Psi | \Psi > ^{1A}$ & $<\Psi | \Psi > ^{2A}$ & $<\Psi | T | \Psi > ^{1A}$ & $<\Psi | T | \Psi > ^{2A}$ & $T(mix)$ & $<\Psi | V | \Psi > ^{1A}$ & $<\Psi | V | \Psi > ^{2A}$ & $<H>^{1A}$ & $<H>^{2A}$ 2 & 0.9117 & 0.9084 & 61.27 & 62.14 & 91.80 & -110.20 & -110.44 &-18.40 &-18.65 4 & 0.9662 & 0.9582 & 79.10 & 76.11 & 96.85 & -117.55 & -118.12 &-20.70 &-21.27 6 & 0.9820 & 0.9766 & 86.36 & 83.49 & 97.56 & -120.66 & -120.71 &-23.09 &-23.15 8 & 0.9927 & 0.9890 & 92.41 & 90.09 & 97.75 & -121.43 & -121.39 &-23.67 &-23.63 10 & 0.9961 & 0.9939 & 94.59 & 92.93 & 97.79 & -121.84 & -121.84 &-24.05 &-24.05 12 & 0.9982 & 0.9969 & 96.10 & 95.04 & 97.80 & -121.97 & -121.96 &-24.16 &-24.16 14 & 0.9990 & 0.9986 & 96.51 & 95.70 & 97.80 & -122.03 & -122.01 &-24.23 &-24.21
Unfortunately, a similar approach is not possible for the expectation values of the potential. However, the interaction does not overemphasize the high-momentum tail and its expectation value is much faster converging. We find a reasonable agreement of 0.02 % of the 1A and 2A results and convergence of both values to uncertainty of 60 keV. For completeness we show the expectation value of the Hamiltonian based on $T(mix)$ and the 1A or 2A expectation value of $V$. These values agree within 0.1 %. The expectation values differ from the binding energy result of $-24.25$ MeV by only 20 to 40 keV. This is well within the error of 60 keV, which has to be expected from the convergence behavior of $V$ and verifies the accuracy of our results. In the following, we will only present the binding energies, the $T(mix)$ values and the expectation values of $H$ and $V$ based on the 1A representation. We consider it more accurate than the $2A$ representation, because the norm is closer to one.
In Table \[tab:alphabindnn\] our $\alpha$ particle binding energies are summarized for Hamiltonians based on NN forces only. The results are identical to the ones published in Ref. [@nogga00] except for AV18, where we present a new calculation, based on a more accurate grid and taking $T=1$ and $T=2$ components into account. Due to the more accurate momentum grid, our binding energy changed by 30 keV, well within our estimated numerical error of 50 keV. Therefore, we did not redo the calculation for the other interactions. The table also shows a result obtained using the NCSM approach [@navratil00]. Our result agrees with their number within the numerical errors estimated.
[l|rrrr]{} interaction & $E_\alpha$ & $H$ & $T$ & $V_{NN}$ Nijm 93 & -24.53 & -24.55 & 95.34 & -119.89 Nijm I & -24.98 & -24.99 & 84.19 & -109.19 Nijm II & -24.56 & -24.55 & 100.31 & -124.86 AV18 & -24.25 & -24.23 & 97.80 & -122.03 CD-Bonn & -26.26 & -26.23 & 77.15 & -103.38 CD-Bonn [@navratil00] & -26.4(2) & — & — & — Exp. & -28.30 & — & — & —
As in the case of the 3N BE’s, the 4N BE’s are also underpredicted by all modern NN force models.The underbinding ranges from 2 to 4 MeV, showing that the results are also strongly model dependent. Once again the non-local forces predict more binding and, similarily, a reduced kinetic energy. The expectation values of $H$ agree within the numerical accuracy of 50 keV with the BE’s $E_\alpha$, which have been directly obtained from the YE’s.
In Ref. [@tjon75] an fascinating linear correlation of the $\alpha$ particle and $^3$H BE’s has been observed, known as the Tjon-line. Our new results confirm this correlation for the newest NN forces. This is displayed in Fig. \[fig:tjonline\]. One sees that all predictions based on only NN forces are situated on a straight line. However, the experimental point slightly deviates from this line hinting to dynamical ingredients beyond the NN interaction and the non-relativistic Schrödinger equation. We also observe a strong dependence of this result on the accuracy of the NN force. Omitting the electromagnetic part of the AV18 NN interaction leads to 16 keV overbinding for the deuteron. A calculation based on this potential resulted in a visible deviation from the Tjon-line.
![Tjon-line: $\alpha$ particle binding-energy predictions $E$($^4$He) dependent on the predictions for the $^3$H binding energies for several realistic interaction models. Predictions of interaction models without (crosses) and with (diamonds) a 3NF are shown. The experimental point is marked by a star. The line represents a least square fit to the predictions of models without a 3NF. []{data-label="fig:tjonline"}](tjon-line.b.eps){width="7cm"}
In the next step we also include 3NF’s into our Hamiltonian. As discussed above, we adjusted these force in conjunction with the different NN interactions. We expect a much smaller dependence of the BE’s on the 3N Hamiltonians in this case, because we remove in this way model dependences, which are correlated to the 3N BE. As one learns from the Tjon-line these are the dominant ones. Our results are given in Table \[tab:alphabind3nf\]. Again we obtained an accuracy of the BE’s $E_\alpha$ of 50 keV. The convergence is slower for these calculations. Therefore, we do not find the same accuracy for the expectation values as for the BE’s. For these we estimate an error of 100 keV, which is still within 0.2 % of the kinetic energy.
[l|rrrrr]{} interaction & $E_\alpha$ & $H$ & $T$ & $V_{NN}$ & $V_{3NF}$ CD-Bonn+TM & -29.15 & -29.09 & 83.92 & -106.16 & -6.854 AV18+TM & -28.84 & -28.81 & 111.84 & -132.62 & -8.033 AV18+TM’ & -28.36 & -28.40 & 110.14 & -133.36 & -5.178 AV18+Urb-IX & -28.50 & -28.53 & 113.21 & -135.81 & -5.929 AV18+Urb-IX (Argonne) [@wiringa00] & -28.34(4) & — & 110.7(7) & -135.3(7) & -6.3(1) Exp. & -28.30 & — & — & — & —
For the NN and 3N forces used, we observe a small overbinding of 60 to 800 keV. These results are also included in Fig. \[fig:tjonline\]. For the TM’ and Urb-IX results we find only small deviations of our results from the Tjon-line. For TM we see more deviations. The TM force seems to destroy the correlation between the $^3$H and $\alpha$ particle BE’s. Though the TM’ force and the Urb-IX interaction are quite different, their BE predictions seem to be comparable. Unfortunately the adjustment of the 3N force has not been done with the same accuracy for TM’. In view of the very expensive calculations necessary to improve the TM’ results and in view of the expected agreement of the TM’ and Urb-IX BE’s, we did not re-calculate for TM’, but omit its results in the following argumentation. The average BE for the $\alpha$ particle using only a NN interaction (based on the restricted choice shown in Table \[tab:alphabindnn\]) is -24.9 MeV or 88 % of the $\alpha$ particle BE. Based on the TM and Urb-IX results in Table \[tab:alphabind3nf\], we estimate an average 3NF contribution to the $\alpha$ particle binding of 3.9 MeV or 14 % of the experimental BE. From the same results we find an average overbinding of 500 keV or 2 % of the BE. The contribution of the 3NF is strongly dependent on the NN interaction due to the adjustment of these forces to the 3N BE. The model dependence of the overbinding is much smaller, but depends on the NN and the 3N force. One can consider this overbinding as the effect of a missing repulsive 4N force. The average size of this force can be expected to be 2 % of the BE in the $\alpha$ particle. Certainly the size of this force will be related to the NN and 3N forces used. The approach employed in [@polyzou90] shows that these 4N forces are related to the 3N Hamiltonians in the same way as the 3N forces to the 2N Hamiltonians. We conclude from our results that we have found numerical evidence that 4N forces are, indeed, much smaller than 3N forces, at least in conjunction with today’s NN and 3N interactions. We do not exclude that new additional 3NF terms could be found, which reduce the necessary contribution of 4N forces. The results support the generally accepted assumption that meaningful nuclear-structure calculations can be performed utilizing bare NN and 3N interactions in a microscopically self-consistent manner. We expect that 4N forces probably show up in heavier nuclei in the same order of magnitude (2 % of the BE). We, therefore, suggest to take an error of this size into account, when one discusses BE’s for systems with $A>4$, based on present NN and 3N forces.
Properties of the $\alpha$ particle WF {#sec:resc}
--------------------------------------
Besides the BE’s, we are also interested in the WF of the 4N system, because it serves as input to several analyses of experiments involving the $\alpha$ particle. Most of these calculations are based on plane wave impulse approximation (PWIA). These calculations are directly sensitive to the WF. Model dependences of the WF are hints to model dependences of these observables. However, because WF’s are not observable themselves, we emphasize that these dependences might disappear once the full dynamics are taken into account.
We start with a contribution of the different isospin states to the WF. Because we made a full, isospin breaking calculation for AV18 only, there is only one result shown in Table \[tab:isoprops\]. The results for the 3N system do not depend on the interaction used [@nogga01c]. Therefore, we do not expect model dependences here.
[l|rrr]{} interaction & $T=0$ & $T=1$ & $T=2$ AV18 & 99.992 & 0.003 & 0.005
One sees an extremely small contribution of the $T=1$ and $T=2$ component to the WF. However, it is of interest that our $T=1$ probability, based on realistic nuclear forces, is larger than the one estimated in [@ramavataram94]. There the $T=1$ admixture has been found to be about $7\cdot 10 ^ {-4}$ % and the $T=2$ state has not been considered. We found the $T=2$ component nearly twice as large as the $T=1$ admixture. Moreover the form of our $T=1$ state will also be different from the one in [@ramavataram94]. As a consequence the isospin admixture correction to the asymmetry as given in [@ramavataram94] will change. A renewed evaluation of that correction, also including the larger $T=2$ state has not been carried through, but appears interesting in view of ongoing experiments.
WF properties are also important for comparisons to other calculational schemes for treating the 4N system. Among the most simple of these properties are the $S$, $P$ and $D$-wave probabilities of the WF. These are given in Table \[tab:spdalpha\] for the models based on the CD-Bonn and AV18 interactions. The values given in Table \[tab:spdalpha\] are based on overlaps between the YC’s and the WF’s, similar to those for the kinetic energies. These numbers are more accurate than the results given in [@nogga00]. However, the differences are not significant, as they affect only the last digit of the results.
As expected the orbital $S$-state is dominant. The $D$-state probability is sizable, very similar to results for $^3$H [@nogga01c], and the $P$-state gives only a small contribution. The $D$-state probability for 3N Hamiltonians, based on CD-Bonn, is smaller than those for models based on the AV18. This is related to the smaller tensor force of non-local interactions. It sticks out that all 3NF’s lead to an increase of the $P$-wave probability by a factor of 2.
[l|rrr]{} interaction & $S$ & $P$ & $D$ CD-Bonn & 89.06 & 0.22 & 10.72 CD-Bonn+TM & 89.65 & 0.45 & 9.90 AV18 & 85.87 & 0.35 & 13.78 AV18+TM & 85.36 & 0.77 & 13.88 AV18+TM’ & 83.58 & 0.75 & 15.67 AV18+Urb-IX & 83.23 & 0.75 & 16.03
This raises the question, whether the considered 3NF really act differently in the 4N system. Because of the scarce knowledge on 3NF’s, this issue is very important. It insures that we get insight into possible impacts of 3NF’s in general, only if our models cover a wide range of interactions. To verify this issue, we decompose the WF’s into parts with different total orbital angular momentum, namely $S$, $P$ and $D$-states. Based on these components, we calculate the expectation values of the Urb-IX and TM 3NF’s for three different WF’s. One is based on the AV18 interaction only, one on the AV18+Urb-IX and the last on the AV18+TM. Four kinds of matrix elements dominate the total expectation value of the 3NF: the diagonal $S$-$S$-state and $D$-$D$-state matrix elements and the overlaps of $S$-state with $P$- and $D$-state. Table \[tab:orbang3nfexp\] shows our results. In the second and fourth columns expectation values for Urb-IX and TM are shown for the same WF, based on AV18. One observe a strong disagreement of these matrix elements. The diagonal elements for Urb-IX are strongly repulsive. They seem to be driven by the isospin and spin independent, phenomenological short-range core of the Urb-IX model. In strong contrast, the $S$-$S$ matrix element contributes most of the attraction in the case of the TM. The attraction of Urb-IX is contributed by the $S$-$D$ overlap. This is a major difference in the action of both models in the 4N system. It insures that we used, indeed, very different 3NF models though both are based on the 2$\pi$ exchange mechanism. Additionally, we see in columns three and five of the table the expectation values based on WF’s for the full Hamiltonian. These expectation values differ sizably from the ones based on the AV18 WF. We confirm for both 3NF’s that a perturbative treatment of them is impossible. For the 3N system this was already emphasized in Refs. [@friar88a; @sasakawa86; @bomelburg86]. Especially interesting is the $S$-$D$ overlap of the TM force. The AV18 WF result is strongly repulsive, whereas the full calculation leads to a slightly attractive contribution. This suggests interesting changes of the 3N configurations in the $\alpha$ particle due to this force.
[l|r|r|r|r]{} 3NF & & WF & AV18 & AV18+Urb-IX & AV18 & AV18+TM S-S & 3.16 & 2.74 & -2.34 & -4.09 S-P/P-S & -0.96 & -2.10 & -1.22 & -3.56 S-D/D-S & -5.44 & -7.46 & 2.08 & -0.14 D-D & 0.59 & 0.85 & 0.01 & 0.06
Are these changes in the 3N configuration visible in momentum distributions? We start in Fig. \[fig:momnn\] with a comparison of the nucleon momentum distribution $$\label{eq:distrfunc}
D(p) = {1 \over 4 \pi} \
< \Psi \ J=0 \ M=0 | \ \delta(p-q_{4}) \ | \Psi \ J=0 \ M=0 >$$ for WF’s based on different NN interactions. The momentum distributions are angular independent. We only consider the $T=0$ components here. Therefore, the proton and neutron distributions are equal. Because we include in both calculations a 3NF, the WF’s are the result of calculations, which roughly give the same BE’s. This insures that we do not find differences, which can be traced back to a higher density of the nucleus. The distributions are equal for momenta below $p=1$ fm$^{-1}$ for both WF’s. For momenta between $p=1$ fm$^{-1}$ and $p=2$ fm$^{-1}$ the deviations are moderate. Above this momentum the AV18 WF is much bigger. We find a clear difference between CD-Bonn and AV18 in this momentum region.
![Nucleon momentum distributions in $^4$He on a logarithmic scale. The distribution functions are based on calculations using the AV18+TM (solid line) and CD-Bonn+TM (dashed line) potentials. The functions are normalized to $\int D(p) dp = { 1 \over 4 \pi } $.[]{data-label="fig:momnn"}](mom.dist.nndep.eps){width="7cm"}
We do not see similar deviations comparing the momentum distributions for different 3NF’s. This is shown in Fig. \[fig:mom3nf\]. The WF’s shown there are based on the same NN interaction, AV18, but differ in the 3NF used. Again the BE’s are comparable and there can be no deviations expected because of density differences. In fact one observes that the distributions are nearly equal for all models in the whole momentum range. This indicates a remarkable stability of momentum distributions with respect to the 3NF. This is in accord with the same independence of the 3NF choice for $T$, the second moment of the momentum distribution, as shown in Table \[tab:alphabind3nf\].
![Nucleon momentum distributions in $^4$He on a logarithmic scale. The distribution functions are based on calculations using the AV18+TM (solid line), AV18+Urb-IX (dotted line) and AV18+TM’ (dashed line) potentials. The functions are normalized to $\int D(p) dp = { 1 \over 4 \pi } $.[]{data-label="fig:mom3nf"}](mom.dist.3ndep.eps){width="7cm"}
The correlations of two nucleons in nuclei are of great theoretical interest. Defined as the probability that two nucleons have a certain distance inside the nucleus, one finds very similar correlations for nuclei with different $A$ [@forest96; @noggaphd]. The correlation is characterized by the strong short-range repulsion of nuclear forces, leading to a small probability that two nucleons are close to each other. However, the quantitative results depend on the force model used. For the 3N system this has been shown in [@nogga97], and we find similar results for the 4N system [@noggaphd]. These correlations are not observable. Therefore, difference in this WF property might not show up in observables. Electron induced scattering experiments, which intent to see these correlations also see effects of meson exchange currents (MEC’s) and final state interactions (FSI’s). Therefore a complete dynamical description of these processes is necessary [@glockle01a].
Nevertheless, we want to show those correlations here. Two-nucleon knock-out experiments are expected to provide information on relative momentum distributions (see for instance [@glockle01a; @ryckebusch97; @rosner00]). In the PWIA these are sensitive to the distribution of relative momenta in the nucleus. Consequently, we show in the following momentum correlations, defined as $$\label{eq:momcorr}
C^{SM_S}(\vec p) = { p^2 \over 4 \pi } \
< \Psi \ J=0 \ M=0 | \ \delta(\vec p_{12}-\vec p )
\ |S \ M_S > <S \ M_S | \
| \Psi \ J=0 \ M=0 >$$ They are the probabilities to find a pair of nucleons in a spin state $|S \ M_S >$ and with a relative momentum $\vec p$. A similar definition in configuration space is given in Ref. [@forest96], where it has been observed numerically that this function has a simple angular dependence, which can be expanded in two Legendre polynomials $P_f ( \hat p \cdot \hat e_z )$ for $f=0$ and $f=2$: $$\label{eq:corrangle}
C^{SM_S}(\vec p) = C^S_{f=0} (p) + C^{SM_S}_{f=2} (p)
\ P_2 ( \hat p \cdot \hat e_z )$$ It only depends on the angle between the momentum and the quantization axis $\hat e_z$. In Appendix \[app:c2npw\] we give an analytical proof of this relation.
The probabilty for two nucleons to be in a fixed spin state $S$ is given by $$\label{eq:spinprob}
N_{S}= \sum_{M_S} < \Psi \ J=0 \ M=0 | \ S \ M_S > <S \ M_S \
| \Psi \ J=0 \ M=0 >$$ For completeness these values are given in Table \[tab:nnpairprobalpha\]. In the following we will always normalize the correlations to $4\pi \ \int dp \ C(p)=1$. The probabilties show the importance of individual channels to the total correlation.
[r|ll]{} interaction & $S=0$ & $S=1$ CD-Bonn & 44.60 & 55.40 CD-Bonn+TM & 44.98 & 55.02 AV18 & 43.07 & 56.93 AV18+TM & 42.95 & 57.05 AV18+TM’ & 42.05 & 57.95 AV18+Urb-IX & 41.87 & 58.13
In Figs. \[fig:c2pnn\] and \[fig:c2n3nf\] spin independent momentum correlations are shown, which have been obtained by summing over all $SM_S$ states. Obviously, because of no fixed quantization axis, they are angular independent. The first figure shows the momentum correlation for the CD-Bonn+TM and AV18+TM interactions. Similar to the distribution functions, they show discrepancies above $p=1$ fm$^{-1}$. In contrast, we did not find a similar model dependence for different 3NF forces. This is shown in Fig. \[fig:c2n3nf\] and suggests that 3NF models to not affect observables, which are considered to be sensitive to NN correlations. A search for kinematical regions, where FSI’s and MEC’s are suppressed, might reveal these correlations. In this case they should show up for momenta greater than $p=1$ fm$^{-1}$.
![Spin-averaged NN momentum correlations in $^4$He for the AV18+TM (solid line) and the CD-Bonn+TM (dashed line) interactions. The functions are normalized to $\int C(p) dp = { 1 \over 4 \pi } $.[]{data-label="fig:c2pnn"}](mom.c2p.nndep.eps){width="7cm"}
![Spin-averaged NN momentum correlations in $^4$He for the AV18+TM (solid line), the AV18+Urb-IX (dotted line) and the AV18+TM’ (dashed line) interactions. The functions are normalized to $\int C(p) dp = { 1 \over 4 \pi } $.[]{data-label="fig:c2n3nf"}](mom.c2p.3ndep.eps){width="7cm"}
We also show the angular dependence of these momentum correlations. In Figs. \[fig:c2pangnn\] and \[fig:c2pang3nf\] both parts of the correlation, as defined in Eq. (\[eq:corrangle\]), are displayed for $S=1$ and $M_S=0$. The angular dependent part does not depend on the 3NF, but for higher momenta, on the NN interaction. Around $p=1$ fm$^{-1}$ the $f=2$ part is comparable in size to the $f=0$ part. In this region one can expect a visible angular dependence of the correlation. This is related to the toroidal structures found in configuration space correlations in [@forest96].
![Angular independent ($f=0$) and dependent ($f=2$) parts of the NN correlations $C^{S\ M_S}$ in $^4$He for spin $S=1$ and its third component $M_S=0$, as defined in the text, compared for different interactions on a logarithmic scale. The correlation functions are based on calculations using the AV18+TM (solid lines) and CD-Bonn+TM (dashed lines) potentials. The functions are normalized, such that the angular independent part fulfills $\int dp \ C(p)={1 \over 4\pi}$. The magnitude $|C|$ is shown. $+$($-$) indicates positive (negative) $C_{f=2}$.[]{data-label="fig:c2pangnn"}](mom.c2p.s=1.andep.eps){width="7cm"}
![Same as Fig. \[fig:c2pangnn\], except that the correlation functions are based on calculations using the AV18+TM (solid lines), AV18+Urb-IX (dotted lines) and AV18+TM’ (dashed lines) potentials.[]{data-label="fig:c2pang3nf"}](mom.c2p.s=1.an3ndep.eps){width="7cm"}
In recent years a knock-out reaction on $^4$He with $^3$H in the final state has received a great deal of attention [@leeuwe98; @e97-111; @sofianos99; @benhar00]. It has been shown that this reaction might be sensitive to the short-range correlations in nuclei [@morita91]. A first experiment has not shown the expected dip in the cross section [@leeuwe98], which has been tracked back to effects of MEC’s and FSI’s. Ongoing experiments probe this reaction in different kinematical configurations, which are expected to be more sensitive to the correlations. The cross sections in the PWIA or in the more reliable Generalized Eikonal Approximation approach [@frankfurt97] are connected to the $^4$He/$^3$H overlap functions $$\label{eq:toverdef}
T ( p ) = \sum_{m_t}
< \Psi \ J=0 \ M=0 | \ \delta ( q_4 - p ) \
| \phi_t \ j_t m_t > < \phi_t \ j_t m_t | \Psi \ J=0 \ M=0 >$$ The momentum of the fourth particle is fixed to $p$ and the state of the other three is projected on the triton state $\phi_t$ with spin $j_t={1 \over 2}$ and third component $m_t$. Because of the sum over different orientations of the $^3$H state, $T$ is angular independent. One can show that this is still true, if one fixes $m_t$ [@noggaphd]. The probability to find a $^3$H inside the $\alpha$ particle is given by $N_t = \int dp \ T(p)$. For completeness we give our results for $N_t$ in Table \[tab:taovernorm\]. The results depend slightly on the interaction model, but are of the order of 80 %. Thus, one observes a definite change in the 3N configuration in the presence of the fourth nucleon.
[l|r]{} interaction & $N_t$($^4$He) CD-Bonn & 84.46 CD-Bonn +TM & 83.49 AV18 & 82.40 AV18+TM & 80.84 AV18+Urb-IX & 80.33 AV18+TM’ & 80.54
Fig. \[fig:overnn\] shows the dependence of $T$ on the NN interaction. The function exhibits a dip structure around $p=2$ fm$^{-1}$. The structure is the result of a node in the momentum space $s$-wave function of the fourth particle relative to the other three. This node is a necessary consequence of the short-range repulsion. Parity and angular momenta for $^3$H and $^4$He guarantee that only the $s$-wave contributes to $T$. The figure shows that $T$, indeed, depends on the NN interaction.
![$^3$H-p momentum distribution $ T $ in $^4$He on a logarithmic scale. The distribution functions are based on calculations using the AV18+TM (solid line) and CD-Bonn+TM (dashed line) potentials. The functions are normalized to $\int T (p) dp = { 1 \over 4 \pi } $. []{data-label="fig:overnn"}](mom.over.nndep.eps){width="7cm"}
The comparision in Fig. \[fig:over3nf\] of the results for different 3NF’s show that $T$ does not depend on the 3NF’s. Therefore, our results confirm that the measurement of $T$ might be valuable to pin down the correlations of two nucleons due to different NN forces (if FSI and MEC effects would be negligible).
![Same as Fig. \[fig:overnn\], except that the distribution functions are based on calculations using the AV18+TM (solid line), AV18+Urb-IX (dotted line) and AV18+TM’ (dashed line) potentials.[]{data-label="fig:over3nf"}](mom.over.3ndep.eps){width="7cm"}
Conclusions and outlook {#sec:concl}
=======================
We solved the Faddeev-Yakubovsky equations for the bound 4N system in momentum space and obtained converged results. Two-nucleon interactions, by themselves, underbind the $\alpha$ particle and leave room for considerable model dependences. Taking properly adjusted 3NF’s into account, one can considerably reduce the model dependences of the BE’s. The combinations of NN and 3N forces lead, in general, to a small overbinding, suggesting that 4N forces are repulsive and much smaller than 3N forces.
We also investigated model dependences of the WF. For momenta below $p=1$ fm$^{-1}$ we do not observe any model dependences in the momentum distributions and correlations. For higher momenta, only effects of the NN interaction show up, because the 3NF’s do not affect these single nucleon and NN properties.
In contrast, we found a huge effect of 3N forces on 3N correlations visible in the matrix elements of the 3N force. These effects require further visualization in future studies. We also found that that the $\alpha$ particle ground state is an extremely pure $T=0$ isospin state. The admixtures of $T=1$ and $T=2$ states are of the order of 0.003 % and 0.005 %, respectively. This sharpens and questions the result found before in [@ramavataram94].
These calculations provide a baseline for the analysis of experiments involving the $\alpha$ particle, which require highly accurate WF’s and insight into NN-force model dependences. The technical developments presented are also important for further studies of nuclear interactions based on $\chi$PT. First studies have already been started [@epelbaum01a; @epelbaum01c; @epelbaum01d]. $\chi$PT allows a systematic derivation of 3NF’s, which are consistent with the NN forces. An investigation of these 3N forces requires accurate techniques for solving the 3N and 4N Schrödinger equation, in order to fix the parameters of the force and to see their effects. The bound states are an intersting object for these studies, because they are the physical quantities very sensitive to 3NF effects and are dominated by the low-energy properties of the nuclear interaction.
Acknowledgments {#acknowledgments .unnumbered}
===============
A.N. and B.R.B. acknowledge partial support from NSF grant\# PHY0070858. The numerical calculations have been performed on the Cray T3E of the NIC in Jülich, Germany.
Partial wave decomposition of correlation functions {#app:c2npw}
===================================================
The spin-dependent correlation functions are angular dependent. In momentum space and for a general nuclear $A$-body bound-state $\Psi$ with angular momentum $J\ M$, it is defined as $$\label{eq:momcorrb}
C^{SM_S}(\vec p) = { p^2 \over 4 \pi } \sum_M \
< \Psi \ J\ M| \ \delta(\vec p_{12}-\vec p )
\ |S \ M_S > <S \ M_S | \
| \Psi \ J\ M >$$ The operator $\delta(\vec p_{12}-\vec p )
\ |S \ M_S > <S \ M_S |$ acts only on the subsystem of particles 1 and 2, i.e., (12). Therefore we choose coordinates, which single out this subsystem and denote the coordinates of the remaining particles by $\alpha_{A-2}\ J_{A-2}\ M_{A-2}$, where we have separated the angular momentum quantum numbers. $\alpha_{A-2}$ also includes the motion of the (12) subsystem relative to the $A-2$ spectators. The two-body subsystem is described by the usual momentum $p_{12}$ and quantum numbers $l_{12}$, $s_{12}$ and $j_{12}$ and the third component $m_j$. Resolving the coupling of the angular momemtum of the (12) subsystem and the spectators to the total angular momentum, one obtains for the correlation $$\begin{aligned}
\label{eq:defcorra2}
C^{S \ M_S}( \vec p ) & = &
{ 1 \over 2 J +1 } \ \sum_{M}
\sum_{\begin{array}{c} \alpha_{A-2} \\ J_{A-2} M_{A-2} \end{array}}
\sum_{\begin{array}{c} l_{12}l_{12}'s_{12}s_{12}' \\ j_{12}j'_{12} m_j m_j ' \end{array}}
\int dp_{12} \ p_{12}^2 \ \int dp'_{12} \ {p'_{12}} ^2 \nonumber \\[5pt]
& & \quad ( j_{12} J_{A-2} J, m_j M_{A-2} M ) \ ( j'_{12} J_{A-2} J, m'_j M_{A-2} M )
\nonumber \\[5pt]
& & \quad < \Psi \ JM | p_{12} \ \alpha_{A-2} \ ((l_{12}s_{12})j_{12} J_{A-2}) JM >
< p'_{12} \ \alpha_{A-2} \ ((l'_{12}s'_{12})j'_{12} J_{A-2}) JM
| \Psi \ JM > \nonumber \\[5pt]
& & \quad < p_{12} (l_{12} s_{12}) j_{12} m_j |
\ \delta^{3} (\vec p - \vec p_{12}) \ |S\ M_S >< S\ M_S|
\ | p'_{12} (l'_{12} s'_{12}) j'_{12} m'_j >\end{aligned}$$ The nuclear bound state WF $< p'_{12} \ \alpha_{A-2} \ ((l'_{12}s'_{12})j'_{12} J_{A-2}) JM
| \Psi \ JM >$ is independent of $M$. We choose $M=J$ in these matrix elements and perform the $M$ and $M_{A-2}$ summation, using the orthogonality relations for the Clebsch-Gordan coefficients. This leads to $$\begin{aligned}
\label{eq:defcorra3}
C^{S \ M_S}( \vec p ) & = &
\sum_{ \alpha_{A-2} J_{A-2}}
\sum_{\begin{array}{c} l_{12}s_{12} j_{12} \\ l_{12}' s_{12}' \end{array} }
\int dp_{12} \ p_{12}^2 \ \int dp'_{12} \ {p'_{12}} ^2 \nonumber \\[5pt]
& & \quad \ { 1 \over 2 j_{12} +1 } \sum_{m_j}
< p_{12} (l_{12} s_{12}) j_{12} m_j |
\ \delta^{3} (\vec p - \vec p_{12}) \ |S\ M_S >< S\ M_S|
\ | p'_{12} (l'_{12} s'_{12}) j_{12} m_j >
\nonumber \\[5pt]
& & \quad < \Psi \ JJ | p_{12} \ \alpha_{A-2} \ ((l_{12}s_{12})j_{12} J_{A-2}) JJ >
< p'_{12} \ \alpha_{A-2} \ ((l'_{12}s'_{12})j_{12} J_{A-2}) JJ
| \Psi \ JJ > \end{aligned}$$ In this form the problem is reduced for arbitrary nuclei to the matrix element $$M_{12} \equiv
{ 1 \over 2 j_{12} +1 } \sum_{m_j}
< p_{12} (l_{12} s_{12}) j_{12} m_j |
\ \delta^{3} (\vec p - \vec p_{12}) \ |S\ M_S >< S\ M_S|
\ | p'_{12} (l'_{12} s'_{12}) j_{12} m_j > ,$$ which is diagonal in $j_{12}$ and $m_{j}$.
By inserting the unity operator in states of 3D momentum and resolving the coupling of spins and orbital angular momenta, we are able to simplify the expression to $$\begin{aligned}
\label{eq:part2ncorrb}
M_{12} & = & \delta_{s_{12} s'_{12}} \delta_{s_{12} S}
{ \delta (p_{12}-p) \over p_{12} p} \ { \delta (p'_{12}-p) \over p'_{12} p} \
{ 1 \over 2 j_{12} +1 } \sum_{m_j} \nonumber \\[5pt]
& & \quad (l_{12} S j_{12} , m_j-M_S \ M_S )
\ (l'_{12} S j_{12} , m_j-M_S \ M_S ) \
Y^*_{l_{12}m_j-M_S}(\hat p) \ Y_{l'_{12}m_j-M_S}(\hat p) \end{aligned}$$ Using standard techniques, one can recouple the angular momenta to obtain a coupled spherical harmonic ${\cal Y}_{l_{12} l'_{12}}^{f\mu}(\hat p \hat p ) $. It turns out that only $\mu=0$ contributes, which is expected, because fixing the spin only fixes the $z$-axis. The matrix elements depend only on $x=\hat p \cdot \hat e_z$. This dependence can be expanded in Legendre polynomials and one ends up with $$\begin{aligned}
\label{eq:part2ncorre}
M_{12} & = & \delta_{s_{12} s'_{12}} \delta_{s_{12} S}
{ \delta (p_{12}-p) \over p_{12} p} \ { \delta (p'_{12}-p) \over p'_{12} p} \
\sum_f (-)^{S-j_{12}} (-)^{l_{12}+l'_{12}-f} \
\sqrt{ (2 l_{12} +1 ) (2 l'_{12} +1 ) (2 f + 1 )\over 2 S + 1 } \
\nonumber \\[5pt]
& & \quad \left\{
\begin{array}{ccc}
S & l_{12} & j_{12} \\
l'_{12} & S & f
\end{array} \right\}
(S \ f \ S , M_S \ 0 ) \ (l_{12} l'_{12} f,00) \ { 1 \over 4 \pi } \ P_f (x)\end{aligned}$$ Because $S$ is restricted to 0 and 1, the order of the Legendre polynomial $f$ can only take the values 0, 1 and 2. Parity conservation fixes the phase $(-)^{l_{12}+l'_{12}}=1$. Therefore, the Clebsch-Gordan coefficient $(l_{12}l'_{12}f,00)$ demands even $f$’s. Because of this, the expansion of the angular dependence contains only two Legendre polynomials: $P_0(x)$ and $P_2(x)$. This proves the form of Eq. (\[eq:corrangle\]). From the explicit form of $M_{12}$ one also reads off that the $M_S$ dependence is given by an overall Clebsch-Gordan coefficient. This justifies the fact that we only present results for $M_S=0$ in Section \[sec:resc\]. We also see that the $f=0$ part of $C$ is independent of $M_S$. Finally, we would like to note that the expressions are also valid in configuration space, replacing the momenta by the corresponding distances.
Treatment of the 3NF embedded in the 4N Hilbert space {#app:3nf}
=====================================================
TM-like forces
--------------
We consider the 3NF as the successive applications of NN-like interactions, which, however, do not respect parity and rotational invariance. Only the full 3NF repects these symmetries [@huber97b]. The YE’s \[eq:alphaeq1\] and \[eq:alphaeq4\] require the matrix elements $$\label{eq:3nfmat}
< (12)3,4 | V_{123}^{(3)} | \Psi >$$ where we can assume that the state $\Psi$ is antisymmetric in the nucleons 123.
One distinguishes four terms in the TM force, the so called $a$-, $b$-, $c$- and $d$-term, which are given by their individual strength constants. These constants are listed together with $V_0$ in Table \[tab:3nfpara\]. $$\begin{aligned}
\label{eq:tmform}
& & V_{123}^{(3)} = V_0 \ \big( \
a \ \vec {\tau_1} \cdot \vec {\tau_2} \ W^a_{23} \ \ W^a_{31}
+b \ \vec {\tau_1}
\cdot \vec {\tau_2} \ \vec W^b_{23} \
\cdot \vec W^b_{31}
+c \ \vec {\tau_1} \cdot \vec {\tau_2}
\ (W^a_{23} \ \ W^c_{31} + W^c_{23} \ W^a_{31})
+d \ \vec {\tau_3} \cdot \vec {\tau_1}
\times \vec {\tau_2} \ \vec W^d_{23} \
\cdot \vec W^b_{31} \big) \nonumber \\[5pt]
& & \end{aligned}$$ where we have separated the isospin operators (Pauli isospin matrices $\tau_i$) and the spin-orbital operators $W$.
[l|r|rrrr]{} & $4 (2\pi)^6 \ V_0$ \[$m_N^{-2}$\] & a \[$m_\pi^{-1}$\] & b \[$m_\pi^{-3}$\] & c \[$m_\pi^{-3}$\] & d \[$m_\pi^{-3}$\] TM & 179.7 & 1.13 & -2.58 & 1.00 & -0.753 TM’ & 179.7 & -0.87 & -2.58 & 0.00 & -0.753
The $W$’s can be read off from the definition of the TM force in momentum space, as given in Ref. [@huber97b]. $$\label{eq:tmws}
\begin{array}{ll}
\displaystyle
W_{23}^a = F({\vec {Q'}}^2) { \vec \sigma _2 \cdot \vec {Q'} \over {\vec {Q'}}^2 + m_\pi^2 } &
\displaystyle
W_{31}^a = F({\vec Q}^2) { \vec \sigma _1 \cdot \vec Q \over {\vec Q}^2 + m_\pi^2 } \cr
\displaystyle
\vec W_{23}^b = F({\vec {Q'}}^2) { \vec \sigma _2 \cdot \vec {Q'}
\over {\vec {Q'}}^2 + m_\pi^2 } \vec {Q'} &
\displaystyle
\vec W_{31}^b = F({\vec Q}^2) { \vec \sigma _1 \cdot \vec Q
\over {\vec Q}^2 + m_\pi^2 } \vec Q \cr
\displaystyle
W_{23}^c = F({\vec {Q'}}^2) { \vec \sigma _2 \cdot \vec {Q'}
\over {\vec {Q'}}^2 + m_\pi^2 } {\vec {Q'}}^2 &
\displaystyle
W_{31}^c = F({\vec Q}^2) { \vec \sigma _1 \cdot \vec Q \over {\vec Q}^2 + m_\pi^2 } {\vec Q}^2 \cr
\displaystyle
\vec W_{23}^d = F({\vec {Q'}}^2) { \vec \sigma _2 \cdot \vec {Q'}
\over {\vec {Q'}}^2 + m_\pi^2 }
\vec \sigma_3 \times \vec {Q'} & \cr
\end{array}$$ with the momentum transfers $ \vec Q = \vec k_1 - \vec k_1 ^{\ \prime} $ and $ \vec Q^{\ \prime} = \vec k_2^{\ \prime} - \vec k_2 $, as indicated in Fig. \[fig:3nfqmom\]. The $\sigma_i$’s are Pauli spin matrices and the form factors are chosen to be $ F({\vec Q}^2) = { \Lambda^2 - m_\pi^2 \over \Lambda^2 + {\vec Q}^2} $.
![\[fig:3nfqmom\] Symbolic representation of a 3NF, like the TM force, and the definition of the momentum transfers $\vec Q$ and $\vec Q^{\ \prime}$ within the two subsystems ](3nf.qdef.eps){width="7cm"}
Applied to a state vector $\psi$, all four terms have the form $$\label{eq:tmformb}
\psi ' \sim W_{23} \ I \ W_{31} \ \Psi ,$$ where we have abbreviated the isospin operators by $I$.
By introducing the unit operator in the coordinates, which are natural for the $W$ potentials, we are able to turn Eq. (\[eq:tmformb\]) into $$\begin{aligned}
\label{eq:tmmatform}
< (12)3,4 | \psi ' > &\sim & < (12)3,4 | (23)1,4 ' > < (23)1,4 ' | W_{23} | (23)1,4 ''> \cr
& & <(23)1,4 '' | I | (31)2,4 ''' >
< (31)2,4 ''' | W_{31} | (31)2,4^* > < (31)2,4^* | \Psi > .\end{aligned}$$ We omit the integrals and sums over momenta and quantum numbers of the intermediate states, in order to simplify the expressions and denote by $(ij)k,l$ Jacobi coordinates, which single out the pair $ij$, the 3-body cluster $ijk$ and the spectator $l$. $\Psi$ originally enters in $(12)3,4$ coordinates. But because of the antisymmetry of $\Psi$ in the $(123)$ subsystem, the $(31)2,4$ coordinates are equivalent in this case.
Because the $W_{23}$’s do not respect the symmetries of nuclear interactions, the sum over $''$- and $'''$-states have to include other parities or other total angular momenta, depending on whether $a/c$ of $b/d$-terms are considered. The $'$-, $''$- and $'''$-sums have also to include unphysical symmetric states of the (31) or (23) subsystems. The matrix elements of the coordinate transformations $ <(23)1,4 '' | I | (31)2,4 ''' > $ are given in Ref. [@nogga01b; @noggaphd]. The isospin operator leads to a change of the isospin part of the transformation. The new isospin matrix elements have been derived in Ref. [@huber97b] for the 3N system and are given below for the 4N system for completeness.
The matrix elements for the different $W$’s are summarized below. For the $a$-term one finds $$\begin{aligned}
\label{eq:waterm}
& & <(31)2,4 | W_{31}^a | (31)2,4 ' > \nonumber \\[5pt]
& & \qquad ={ \delta(q_4 - q_4 ' ) \over q_4 q_4' } \ { \delta(p_2 - p_2 ' ) \over p_2 p_2' } \
\delta_{I_4 I_4'} \delta_{I_2 I_2'} \delta_{l_4 l_4'} \delta_{l_2 l_2'}
\delta_{J J'} \delta_{M M'}\delta_{j_3 j_3'}
\delta_{j_{31}j_{31}'}\delta_{|l_{31}-l_{31}'|,1} \cr
&& \quad \qquad \ 2 \pi \sqrt{6} \ (-)^{j_{31}+1+{\rm max}(l_{31},l_{31}')}
\sqrt{\hat s_{31} \hat s_{31}'}
\left\{
\begin{array}{ccc}
{ 1 \over 2 } & { 1 \over 2 } & s_{31}' \cr
1 & s_{31} & { 1 \over 2 } \cr
\end{array}
\right\}
\left\{
\begin{array}{ccc}
l_{31}' & s_{31}' & j_{31}' \cr
s_{31} & l_{31} & 1 \cr
\end{array}
\right\} \cr
& & \quad \qquad \sqrt{{\rm max}(l_{31},l_{31}')}
\ \left( p_{31} \ H_{l_{31}'}(p_{31},p_{31}')
-p_{31}' \ H_{l_{31}}(p_{31},p_{31}') \right) \end{aligned}$$ The operator has no isospin dependence; therefore, it is diagonal in isospin space. The momentum dependence is given in terms of the function $H$, which is a combination of Legendre polynomials of the second kind $Q_l$ and their derivatives $Q'_l$. $$\label{eq:hfunc}
H_l (p,p') = { 1 \over p p' } \ ( Q_l(B_{m_\pi}) - Q_l(B_{\Lambda}) )
+ { \Lambda^2 - m_{\pi}^2 \over 2 ( p p' )^2 } \ Q_l' (B_\Lambda)$$ with $B_{m_\pi} = { p^2 + {p'}\; ^2 + m_\pi^2 \over 2 p p ' } $ and $ B_{\Lambda} = { p^2 + {p'}\; ^2 + \Lambda ^2 \over 2 p p ' } $.
The $c$-term looks very similar as the $a$-term and follows, if one replaces $H$ in Eq. (\[eq:waterm\]) by $$\label{eq:htildefunc}
\tilde H_l (p,p') = - { m_\pi^2 \over p p' } \ ( Q_l(B_{m_\pi}) - Q_l(B_{\Lambda}) )
- { \Lambda^2 - m_{\pi}^2 \over 2 ( p p' )^2 } \ \Lambda^2
\ Q_l' (B_\Lambda) .$$ For our convenience, we used the abbreviation $\hat k = 2k +1$ in these expressions. The notation of the different quantum numbers is an obvious generalization of the notation in Fig. \[fig:jacobi\].
Because the $W_{23}$- and $W_{31}$-operators are equivalent up to a renumbering of the particles, the matrix elements are equal up to a phase factor $$\begin{aligned}
\label{eq:wacphase}
<(23)1,4 | W^{a,c}_{23} | (23)1,4 '>
& = & (-)^{l_{31}+s_{31}+t_{31}+l_{31}'+s_{31}'+t_{31}'+1}
\ <(31)2,4 | W^{a,c}_{31} | (31)2,4 '> .\end{aligned}$$
As we already mentioned, we replace the simple isospin transformation matrix element by a combination of the transformation and the isospin operator $$\label{eq:isotau12}
< (23)1,4 | \vec \tau_1 \cdot \vec \tau_2 | (31)2,4 '>
= \delta_{TT'} \delta_{M_T M_T'} \delta_{\tau \tau'}
(-6) \ (-)^{t_{23}} \sqrt{ \hat t_{23} \hat t_{31}'}
\ \left\{
\begin{array}{ccc}
{1 \over 2 } & {1 \over 2 } & t_{31}' \cr
{1 \over 2 } & 1 & {1 \over 2 } \cr
t_{23} & {1 \over 2 } & \tau \cr
\end{array}
\right\} .$$
The $b$- and $d$-terms are a slightly more complicated, because the NN-like potentials are now vector operators. The matrix elements of the spherical components $W^{1} = - { 1 \over \sqrt{2} } \ ( W^x+iW^y )$, $W^{0} = W^z$ and $W^{-1} = { 1 \over \sqrt{2} } \ ( W^x-iW^y )$ decompose into a Clebsch-Gordan coefficient and a reduced matrix element $$\label{eq:reduced}
<(31)2,4 | W^\mu_{31}|(31)2,4 '> = (J'1J,M' \mu M ) \ <(31)2,4 || W_{31}||(31)2,4 '> .$$ The scalar product in spherical coordinates reads $$\label{eq:skalarsphaerical}
\vec W_{23} \cdot \vec W_{31} = \sum_{\mu} (-)^\mu W_{23}^\mu W_{31}^{-\mu}.$$
Because there is no dependence on the third component of the total angular momentum, neither in the transformation matrix elements nor in the incoming state, we can analytically perform the $M''$ and $\mu$-summations $$\label{eq:mmusum}
\sum_{M'' \mu} (-)^\mu (J'' 1 J',M'' \mu M' ) \ (J^* 1 J '' , M^* -\mu M'')
= \delta_{J'J^*} \delta_{M'M^*} \sqrt { \hat J '' \over \hat J' } (-)^{J''-J'}$$ and recover the conservation of the total angular momentum. The NN-like potentials effectively requires only the application of the reduced matrix elements and the additional factor $\sqrt { \hat J '' \over \hat J' } (-)^{J''-J'}$. The intermediate states are also $M$ independent.
The generalization of the formulas given in Ref. [@huber97b] to the four-nucleon system yields $$\begin{aligned}
\label{eq:btermredmat}
& & < (31)2,4 || W_{31}^b || (31)2,4 >' \nonumber \\[5pt]
& & \quad = { \delta(q_4 - q_4 ' ) \over q_4 q_4' }
\ { \delta(p_2 - p_2 ' ) \over p_2 p_2' } \
\delta_{I_4 I_4'} \delta_{I_2 I_2'} \delta_{l_4 l_4'} \delta_{l_2 l_2'}
(-)^{J'+j_3'+j_3+I_4+I_2+s_{31}+s_{31}'}
\sqrt{\hat J' \hat j_3' \hat j_3 \hat s_{31}' \hat s_{31} \hat j_{31}' \hat j_{31}} \cr
& & \qquad \qquad
\left\{
\begin{array}{ccc}
j_3' & 1 & j_3 \cr
J & I_4 & J' \cr
\end{array}
\right\} \
\left\{
\begin{array}{ccc}
1 & j_{31} & j_{31}' \cr
I_2 & j_3' & j_3 \cr
\end{array}
\right\}\
\left\{
\begin{array}{ccc}
{1 \over 2} & {1 \over 2} & s_{31} \cr
1 & s_{31}' & {1 \over 2} \cr
\end{array}
\right\} \cr
& & \qquad \times \left[ \delta_{l_{31}l_{31}'} \ { 2 \pi \over 3 } \ \sqrt{6} \ (-)^{l_{31}+1} \
\left\{
\begin{array}{ccc}
j_{31}' & j_{31} & 1 \cr
s_{31} & s_{31}' & l_{31}\cr
\end{array}
\right\} \ \tilde H_{l_{31}} (p_{31} p_{31}') \right. \nonumber \\[5pt]
& & \qquad \quad
- 40 \pi \sqrt{6} \ (-)^{s_{31}'+j_{31}} \
\left\{
\begin{array}{ccc}
2 & 1 & 1 \cr
l_{31} & s_{31} & j_{31} \cr
l_{31}' & s_{31}' & j_{31}' \cr
\end{array}
\right\} \ \sum_{\bar l } \hat {\bar l} H_{\bar l} (p_{31} p_{31}') \cr
& & \left. \qquad \qquad
\times \sum_{a+b=2} { {p_{31}'} \! \! ^a {p_{31}} \! \! ^b \over \sqrt{ (2a)! (2b)!}} \
\left\{
\begin{array}{ccc}
b & a & 2 \cr
l_{31}' & l_{31} & \bar l\cr
\end{array} \right\}
(a \ \bar l \ l_{31}',00) \ (b \ \bar l \ l_{31},00) \right]\end{aligned}$$ and $$\begin{aligned}
\label{eq:dtermredmat}
& & < (23)1,4 || W_{23}^d || (23)1,4 >' \nonumber \\[5pt]
& & \quad = { \delta(q_4 - q_4 ' ) \over q_4 q_4' }
\ { \delta(p_1 - p_1 ' ) \over p_1 p_1' } \
\delta_{I_4 I_4'} \delta_{I_1 I_1'} \delta_{l_4 l_4'} \delta_{l_1 l_1'}
(-)^{J'+j_3'+j_3+I_4+I_1+s_{23}+s_{23}'+1}
\sqrt{\hat J' \hat j_3' \hat j_3 \hat s_{23}' \hat s_{23} \hat j_{23}' \hat j_{23}} \cr
& & \qquad \qquad
\left\{
\begin{array}{ccc}
j_3' & 1 & j_3 \cr
J & I_4 & J' \cr
\end{array}
\right\} \
\left\{
\begin{array}{ccc}
1 & j_{23} & j_{23}' \cr
I_1 & j_3' & j_3 \cr
\end{array}
\right\} \cr
& & \qquad \times \left[ \delta_{l_{23}l_{23}'} \ i \ 4\pi \ \sqrt{6} \ (-)^{l_{23}+s_{23}} \
\left\{
\begin{array}{ccc}
l_{23} & s_{23} & j_{23} \cr
1 & j_{23}' & s_{23}'\cr
\end{array}
\right\} \
\left\{
\begin{array}{ccc}
1 & 1 & 1 \cr
{1\over 2} & {1\over 2} & s_{23}' \cr
{1\over 2} & {1\over 2} & s_{23} \cr
\end{array}
\right\} \ \tilde H_{l_{23}} (p_{23} p_{23}') \right. \nonumber \\[5pt]
& & \qquad \quad
+ \ i \ 240 \pi \sqrt{6} \ (-)^{j_{23}'} \
\sum_{\chi} (-)^{\chi} \ \hat \chi \
\left\{
\begin{array}{ccc}
2 & \chi & 1 \cr
1 & 1 & 1 \cr
\end{array}
\right\} \
\left\{
\begin{array}{ccc}
2 & \chi & 1 \cr
l_{23}' & s_{23}' & j_{23}'\cr
l_{23} & s_{23} & j_{23} \cr
\end{array}
\right\} \
\left\{
\begin{array}{ccc}
1 & 1 & \chi \cr
{1 \over 2}&{1 \over 2}& s_{23}'\cr
{1 \over 2}&{1 \over 2}& s_{23} \cr
\end{array}
\right\} \ \cr
& & \left. \qquad \qquad
\times \sum_{\bar l } \hat {\bar l} H_{\bar l} (p_{23} p_{23}') \
\sum_{a+b=2} { {p_{23}'} \! \! ^a {p_{23}} \! \! ^b \over \sqrt{ (2a)! (2b)!}} \
\left\{
\begin{array}{ccc}
b & a & 2 \cr
l_{23}' & l_{23} & \bar l\cr
\end{array} \right\}
(a \ \bar l \ l_{23}',00) \ (b \ \bar l \ l_{23},00) \right].\end{aligned}$$ The momentum dependent functions $H$ and $\tilde H$ are given in Eqs. (\[eq:hfunc\]) and (\[eq:htildefunc\]).
Again, there is a simple phase relation between $W^b_{23}$ and $W^b_{31}$ $$\begin{aligned}
\label{eq:wbdphase}
<(23)1,4 | W^{b,d}_{23} | (23)1,4 '>
& = & (-)^{l_{31}+s_{31}+t_{31}+l_{31}'+s_{31}'+t_{31}'}
\ <(31)2,4 | W^{b,d}_{31} | (31)2,4 '> \end{aligned}$$
The isospin matrix element of the $d$-term differs from the one for the $a$-, $b$- and $c$-term, given in Eq. (\[eq:isotau12\]). It reads $$\begin{aligned}
\label{eq:isotaucross}
< (23)1,4 | \vec \tau_3 \cdot ( \vec \tau_1 \times \tau_3) | (31)2,4 '> & = & \delta_{TT'} \delta_{M_T M_T'} \delta_{\tau \tau'}
24i \ (-)^{2 \tau } \sqrt{ \hat t_{23} \hat t_{31}'} \cr
& & \sum_{\lambda} (-)^{3 \lambda + { 1 \over 2 }} \
\ \left\{
\begin{array}{ccc}
\lambda & {1 \over 2 } & 1 \cr
{1 \over 2 } & {1 \over 2 } & t_{23} \cr
\end{array}
\right\}
\ \left\{
\begin{array}{ccc}
\tau & {1 \over 2 } & t_{23} \cr
{1 \over 2 } & 1 & \lambda \cr
t_{31}' & {1 \over 2 } & {1 \over 2 } \cr
\end{array}
\right\}. \end{aligned}$$
Urbana type forces
------------------
The functional form of the Urbana 3NF is much simpler. One usually expresses the Urbana interaction in terms of commutator and anticommutator parts. This reads $$\begin{aligned}
\label{eq:vurbcomm}
V_{123}^{(3)} & = & \phantom{ + } A_{2\pi} \ [ \ \{ X_{23},X_{31} \} \
\{ \vec \tau _2 \cdot \vec \tau _3 ,
\vec \tau _3 \cdot \vec \tau _1 \} \cr
& & \phantom{ + A_{2\pi} } + { 1 \over 4 } \ [ X_{23},X_{31} ] \
[ \vec \tau _2 \cdot \vec \tau _3 ,
\vec \tau _3 \cdot \vec \tau _1 ] \ ] \cr
& & + U_0 \ T_\pi ^2 (r_{23}) \ T_\pi ^2 (r_{31}) \end{aligned}$$
The force is explicitly defined in terms of NN interactions $$\label{eq:xterms}
X_{ij} = Y_\pi ( r_{ij} ) \ \vec \sigma_i \cdot \vec \sigma_j
+ T_\pi (r_{ij}) \ S_{ij}$$ $X_{ij}$ is derived from the $\pi$ exchange NN force. Therefore, it has a spin-spin part $\vec \sigma_i \cdot \vec \sigma_j$ and a tensor part $$\label{eq:tensorop}
S_{ij} = 3 \ \vec \sigma_i \cdot \hat r_i \ \vec \sigma_j \cdot \hat r_j
- \vec \sigma_i \cdot \vec \sigma_j .$$ The radial dependence is given as $$\begin{aligned}
\label{eq:radialfunc}
Y_\pi (r) & = & { e ^ { - m_\pi r } \over m_\pi r } \ \left( 1 - e ^ { - c r^2 } \right) \cr
T_\pi (r) & = & \left[ 1 + { 3 \over m_\pi r } + { 3 \over (m_\pi r)^2 } \right]
{ e ^ { - m_\pi r } \over m_\pi r } \ \left( 1 - e ^ { - c r^2 } \right)^2 \end{aligned}$$ The parameters $A_{2\pi}$, $U_0$ and $c$ for the Urb-IX are given in Ref. [@pudliner97].
In [@witala01a] it is shown that the application of the Urbana force can be rewritten as $$\begin{aligned}
\label{eq:urballterms}
< (12)3,4 | \psi ' > &= & 2 A_{2\pi} \
< (12)3,4 | (23)1,4 ' > < (23)1,4 ' | X_{23} | (23)1,4 ''> \cr
& & \quad <(23)1,4 '' | I^- | (31)2,4 ''' >
< (31)2,4 ''' | X_{31} | (31)2,4^* > < (31)2,4^* | \psi > \cr
& & + U_0 \ < (12)3,4 | (23)1,4 ' > < (23)1,4 ' | T^2_\pi(r_{23}) | (23)1,4 ''> \cr
& & <(23)1,4 '' | (31)2,4 ''' >
< (31)2,4 ''' | T^2_\pi(r_{31}) | (31)2,4^* > < (31)2,4^* | \psi > .\end{aligned}$$ The isospin operators are very similar to the ones encountered in the TM force $$\begin{aligned}
\label{eq:isoplusminus}
I^- & \equiv & 2 ( \vec \tau _1 \cdot \vec \tau _2
- { i \over 4 } \ \vec \tau_3 \cdot \vec \tau _1 \times \vec \tau _2 )\cr
I^+ & \equiv & 2 ( \vec \tau _1 \cdot \vec \tau _2
+ { i \over 4 } \ \vec \tau_3 \cdot \vec \tau _1 \times \vec \tau _2 ) .\end{aligned}$$ It is easy to combine Eqs. (\[eq:isotau12\]) and (\[eq:isotaucross\]) to find their matrix elements.
The matrix elements of the NN-like interactions $X_{31}$ and $T^2_\pi (r_{31})$ read in momentum space $$\begin{aligned}
\label{eq:fourx31}
& & <(31)2,4 | X_{31} | (31)2,4' > \nonumber \\[5pt]
& & \quad = { \delta(q_4 - q_4 ' ) \over q_4 q_4' }
\ { \delta(p_2 - p_2 ' ) \over p_2 p_2' } \
\delta_{I_4 I_4'} \delta_{I_2 I_2'} \delta_{l_4 l_4'} \delta_{l_2 l_2'}
\delta_{J J'} \delta_{M M'} \delta_{j_{31} j_{31}'} \cr
& & \qquad \left[ \tilde Y _{l_{31}} (p_{31},p_{31}')
\ \delta_{l_{31}l_{31}'} \ \delta_{s_{31}s_{31}'} \ (-3+4s_{31})
+ \tilde T _{l_{31}l_{31}'} (p_{31},p_{31}')
\ \delta_{s_{31}s_{31}'} \ \delta_{s_{31}1} \ S_{l_{31}l_{31}'j_{31}}
\right] \end{aligned}$$ and $$\begin{aligned}
\label{eq:shorttq}
& & <(31)2,4 | T_\pi^2(r_{31}) | (31)2,4' > \nonumber \\[5pt]
& & = { \delta(q_4 - q_4 ' ) \over q_4 q_4' }
\ { \delta(p_2 - p_2 ' ) \over p_2 p_2' } \
\delta_{I_4 I_4'} \delta_{I_2 I_2'} \delta_{l_4 l_4'} \delta_{l_2 l_2'}
\delta_{J J'} \delta_{M M'} \delta_{j_{31} j_{31}'} \delta_{l_{31} l_{31}'}
\delta_{s_{31} s_{31}'} \ \bar T_{l_{31}} (p_{31},p_{31}') .\end{aligned}$$ Here the tensor operator can be expressed in simple rational functions of the quantum numbers $$\label{eq:tensoroppart}
\raisebox{3ex}{$S_{l_{31}l_{31}'j_{31}}$ = }
\begin{array}{lc}
\begin{array}{l}
l_{31}=j_{31}-1 \cr
l_{31}=j_{31} \cr
l_{31}=j_{31}+1 \cr
\end{array}
& \left[
\begin{array}{ccc}
- 2 \ { j_{31}-1 \over 2 j_{31}+1 }
& 0 & 6 \ { \sqrt{ j_{31}(j_{31}+1)} \over 2j_{31}+1 } \cr
0 & 2 & 0 \cr
6 \ { \sqrt{ j_{31}(j_{31}+1)} \over 2j_{31}+1 } & 0
& - 2 \ { j_{31}+2 \over 2 j_{31}+1 } \cr
\end{array} \right] \cr
& \cr
&
\begin{array}{ccc}
l_{31}'=j_{31}-1 &
l_{31}'=j_{31} &
l_{31}'=j_{31}+1 \cr
\end{array}
\end{array}$$ We numerically perform the Fourier transformations $$\begin{aligned}
\label{eq:fouryt}
\tilde Y _{l_{31}} (p_{31},p_{31}') & = & { 2 \over \pi }
\ \int_{0}^{\infty} dr \ r^2 \ {\rm j}_{l_{31}} (p_{31}r) \ Y_\pi (r)
\ {\rm j}_{l_{31}} (p_{31}'r) \cr
\tilde T_{l_{31}l_{31}'} (p_{31},p_{31}') & = & (-) ^{l_{31}-l_{31}'}
\ { 2 \over \pi }
\ \int_{0}^{\infty} dr \ r^2 \ {\rm j}_{l_{31}} (p_{31}r) \ T_\pi (r)
\ {\rm j}_{l_{31}'} (p_{31}'r) \cr
\bar T_{l_{31}} (p_{31},p_{31}') & = &
{ 2 \over \pi }
\ \int_{0}^{\infty} dr \ r^2 \ {\rm j}_{l_{31}} (p_{31}r) \ T^2_\pi (r)
\ {\rm j}_{l_{31}} (p_{31}'r) \end{aligned}$$ with the usual spherical Bessel functions ${\rm j}_{l}(x)$.
Because these NN-like interactions are all symmetric with respect to an interchange of the subsystem particles, the matrix elements for the $X_{23}$ and $T^2(r_{23})$ equal those for $X_{31}$ and $T^2(r_{31})$, respectively
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We realize the two dimensional anti-de Sitter ($AdS_2$) space as a Kaluza-Klein reduction of the $AdS_3$ space in the framework of the discrete light cone quantization (DLCQ). Introducing DLCQ coordinates which interpolate the original (unboosted) coordinates and the light cone coordinates, we discuss that $AdS_2/CFT$ correspondence can be deduced from the $AdS_3/CFT$. In particular, we elaborate on the deformation of WZW model to obtain the boundary theory for the $AdS_2$ black hole. This enables us to derive the entropy of the $AdS_2$ black hole from that of the $AdS_3$ black hole.'
address: |
[*${}^{1}$Department of Physics, Kyung Hee University, Seoul 130-701, Korea\
${}^2$Asia Pacific Institute for Theoretical Physics, Seoul 130-012, Korea\
${}^{3}$Department of Physics, Kangwon National University, Chuncheon 200-701, Korea\
${}^{4}$School of Physics, Korea Institute for Advanced Study, Seoul 130-012, Korea\
${}^{5}$Department of Physics, University of British Columbia, Canada* ]{}
author:
- 'Jin-Ho Cho$^{1,2}$ [^1], Taejin Lee$^{3,4}$ [^2] and Gordon Semenoff$^{5}$ [^3]'
title: |
Two Dimensional Anti-de Sitter Space and\
Discrete Light Cone Quantization
---
One of the main progresses achieved recently in the string theory is the $AdS/CFT$ duality [@mal97; @adscft], which connects the gravity in the $D$-dimensional anti-de Sitter ($AdS_D$) space and the $(D-1)$ dimensional conformal field theory ($CFT$) on its boundary. Among the $AdS/CFT$ dualities in the various dimensions are the $AdS_3/CFT$ and $AdS_2/CFT$ dualities relevant for the black hole physics, since most of the black holes in the string theories are known to contain either $AdS_3$ space or $AdS_2$ space in their near horizon geometries [@mal97; @hyun]. Thus, the $AdS/CFT$ dualities in low dimensions would play a key role in understanding the quantum aspects of the black holes. However, compared with the case of the $AdS_3/CFT$ duality [@Mal98; @tlee98; @Giveon] the $AdS_2/CFT$ duality is less well discussed in the literature. Observing that the near horizon geometry of the three dimensional BTZ (Bañados-Teitelboim-Zanelli) black hole [@btz] becomes effectively $AdS_2$ in the low energy regime, one may attempt to derive the $AdS_2/CFT$ duality from the known $AdS_3/CFT$. This approach was taken by Strominger in his recent work on $AdS_2/CFT$ duality [@Stro98].
Here in this paper we will employ a different strategy to derive the $AdS_2/CFT$ duality, namely the DLCQ (discrete light cone quantization) [@dlcq], which reveals the relationship between two dualities more transparently. If the BTZ black hole is viewed in the light cone frame along the circle direction, the metric components in the light like directions are constant and can be scaled by boosting the frame. Thus, if the light cone coordinate, $x^-$ is taken to be periodic, the Kaluza-Klein compactification can be easily performed. In order to have a periodic light cone coordinate, we employ the DLCQ procedure, which has been discussed recently [@Seib] in the context of the Matrix M-theory [@Suss]. It is found useful to introduce DLCQ coordinates, which interpolate the original (unboosted) coordinates and the light cone ones when we apply the DLCQ procedure to the $AdS_3$ black hole. One advantage of this approach is that we do not need to confine ourselves to the near horizon region.
Let us begin with the well-known $D1$-$D5$ black hole in ten dimensions, which has its near horizon geometry as $M_{BTZ} \times S^3 \times T^4$ [@horo] $$\begin{aligned}
\label{d1d5}
{ds^2\over\alpha'}&=&{U^2 \over l^2}(-dt^2+dx_5{}^2)+
{U_0{}^2 \over l^2}(\cosh{\sigma}\; dt+\sinh{\sigma}\; dx_5)^2\nonumber\\
&+&{l^2 \over U^2-U_0{}^2}dU^2
+l^2d\Omega_3+{r_1 \over r_5}\sum_{i=6}^{9}{(dx^i)^2\over\alpha'},\end{aligned}$$ where $x_5\sim x_5+2\pi R_s$, $x_{6,7,8,9}\sim x_{6,7,8,9}+2\pi
V^{{1 \over 4}}\alpha'^{{1 \over 2}}$. Hereafter we will take $\alpha' = 1$ for the sake of convenience. $M_{BTZ}$ corresponds to the well known three dimensional BTZ black hole, which has mass and angular momentum as follows $$\begin{aligned}
\label{mj}
Ml &=& \frac{1}{8Gl}(\rho^2_++\rho^2_-)=
{R_s^2 \over 8Gl^3}U_0{}^2\cosh{2\sigma}, \\
|J| &=& 2\frac{1}{8Gl} \rho_+\rho_- =
{R_s^2 \over 8Gl^3}U_0{}^2|\sinh{2\sigma}|.\end{aligned}$$ Here we note that the BTZ black hole becomes BPS object as we take the limit $\sigma \rightarrow \pm \infty$ while keeping $M$, $J$ finite by making $U_0 \rightarrow 0$; in this limit $M \mp J/l \rightarrow 0$.
It is tempting to reduce the three dimensional black hole to a two dimensional $AdS$ black hole by the Kaluza-Klein (KK) reduction. The early attempt was made in ref.[@Lowe], where the four dimensional extremal Reissner-Nordström black hole is described as $AdS_2 \times S^2$: The BTZ black hole can be viewed as a two dimensional dilatonic $AdS$ black hole with a $U(1)$ charge, when we apply the KK reduction along the spatial $S^1$ direction to the BTZ black hole. A more elaborated description of $AdS_2$ is given by Strominger in his recent work [@Stro98], by taking the very near horizon limit of the extremal black hole. In this paper we take the KK reduction along nearly lightlike circle rather than along the spatial circle. We first give naive idea on this. The metric of the BTZ black hole in the light cone frame becomes $$\begin{aligned}
\label{light}
ds^2&=&{U_0{}^2e^{2\sigma} \over 2l^2}dx^+{}^2+
{U_0{}^2e^{-2\sigma} \over 2l^2}dx^-{}^2\nonumber\\
&+&{2U^2-U_0{}^2 \over l^2}dx^+dx^-+{l^2 \over U^2-U_0{}^2}dU^2,\end{aligned}$$ where $x^\pm = (x^5 \pm t)/\sqrt{2}$. It suggests that the light cone coordinate, $x^-$ may be chosen to be compactified, since the dilaton is constant. Here we need to employ the DLCQ procedure in order to have a periodic light like coordinate. The DLCQ procedure has been recently discussed [@Seib] in the context of the Matrix M-theory [@Suss]. So some part of the analysis to be presented also will be useful to study the Matrix M-theory. Let us suppose that $x^5$ is periodic, $x^5 \sim x^5 + 2\pi R_s$, i.e., $(x^+, x^-)
\sim(x^++ \sqrt{2}\pi R_s,x^-+\sqrt{2}\pi R_s)$. Then by a Lorentz boost, we have $({x'}^+,{x'}^-)\sim({x'}^++\sqrt{2}\pi R_se^\alpha,
{x'}^-+\sqrt{2}\pi R_se^{-\alpha}), \quad
R_s / R=(\cosh{2\alpha})^{-\frac{1}{2}}$, where ${x'}^\pm=e^{\pm\alpha}x^\pm$ are the boosted light cone coordinates. In the limit of the large boosting, i.e. when the boosting parameter $\alpha \rightarrow - \infty$, (equivalently $R_s \rightarrow 0$ with $R$ kept finite), ${x'}^-$ becomes periodic; $({x'}^+,{x'}^-)\sim({x'}^+,{x'}^-+ 2\pi R)$.
The metric reads in terms of the boosted coordinates as, $$\begin{aligned}
ds^2 &=& -{2U^2(U^2-U_0{}^2) \over U_0{}^2 l^2}
e^{2(\sigma-\alpha)}
%\left(\frac{R_s}{R}\right)^2
({dx'}^+)^2+{l^2 \over U^2-U_0{}^2}dU^2 \nonumber\\
&+&{U_0{}^2e^{-2(\sigma-\alpha)} \over 2l^2}
%\left(\frac{R_s}{R}\right)^2
\left({dx'}^-+ \left(2\frac{U^2}{U_0^2} -1\right)
e^{2(\sigma-\alpha)}{dx'}^+\right)^2.
\label{boost}\end{aligned}$$ Here we observe that the dilaton factor, i.e., the compactification radius is constant and ${x'}^+$ plays the role of time coordinate. In the DLCQ limit, the geometry becomes $AdS_2\times S^1$. However in order to be more transparent on the deformation of a spatial circle to make a nearly light like circle, we introduce new coordinates $(\hatt,U,\hatx)$, called DLCQ coordinates, which interpolate the original coordinates $(t,U,x^5)$ (when $\alpha=0$) and the infinitely boosted light cone coordinates $({x'}^+,U,{x'}^-)$ (when $\alpha\rightarrow-\infty$): $(\hatt, \hatx) \sim (\hatt, \hatx + 2\pi R_s \sqrt{\cosh{2\alpha}})$, $$\begin{aligned}
\hatx =
{e^{\alpha}{x'}^++e^{-\alpha}{x'}^-\over\sqrt{e^{2\alpha}+e^{-2\alpha}}}, \qquad
\hatt =
{e^{-\alpha}{x'}^+-e^{\alpha}{x'}^-\over \sqrt{e^{2\alpha}+e^{-2\alpha}}}.\end{aligned}$$ In the boosted frame $\hatx$ is identified as a periodic coordinate and the time coordinate $\hatt$ is chosen such that $\partial_{\hatt}$ is orthogonal to $\partial_{\hatx}$. In the infinite boosting limit ($\alpha\rightarrow-\infty$), $\hatt\rightarrow {x'}^{+}=e^\alpha x^+$ and $\hatx\rightarrow {x'}^{-}=e^{-\alpha}x^-$.
In this DLCQ coordinates, the metric reads as $$\begin{aligned}
ds^2&=&{U^2+U_0^2\sinh^2(\sigma'+\alpha)\over l^2\cosh{2\alpha}}
\left(d\hatx+{U_0^2\sinh{2\sigma'}
-(2U^2-U_0^2)\sinh{2\alpha}
\over 2(U^2+U_0^2\sinh^2(\sigma'+\alpha))}d\hatt\right)^2\nonumber\\
&&-{U^2(U^2-U_0^2)\cosh{2\alpha}
\over l^2(U^2+U_0^2\sinh^2(\sigma'+\alpha))}d\hatt^2
+{l^2\over U^2-U_0^2}dU^2.\end{aligned}$$ Now it becomes clear that we should keep $\sigma' = \sigma-\alpha$ finite in order to obtain the metric for the space $AdS_2 \times S^1$ in the limit, $\alpha \rightarrow -\infty$, $$\begin{aligned}
\label{boost2}
ds^2 &=& -{2U^2(U^2-U_0{}^2) \over U_0{}^2 l^2}
e^{2\sigma'}d\hatt^2+{l^2 \over U^2-U_0{}^2}dU^2\nonumber\\
&+&{U_0{}^2e^{-2\sigma'} \over 2l^2} \left(d\hatx+ a_\hatt d\hatt\right)^2,\end{aligned}$$ where $a_\hatt = \left(2\frac{U^2}{U_0^2} -1\right)e^{2\sigma'}$. It depicts an $AdS_2$ black hole with $U(1)$ charge and coincides with the metric Eq.(\[boost\]) obtained by the naive DLCQ procedure. The mass and angular momentum of the black hole are given as $$\begin{aligned}
\label{mj2}
Ml = -J = \frac{R^2 U_0^2 e^{-2\sigma'}}{8Gl^3}.\end{aligned}$$ Thus, the $AdS_2$ black hole can be described in terms of the extremal $AdS_3$ black hole in the framework of DLCQ. In passing we note that the way the DLCQ procedure results in the extremal limit is different from the usual one; $\sigma \rightarrow -\infty$, and $R_s \rightarrow 0$ so that $R\,e^{-\sigma^\prime} =
R_s\, \sqrt{\cosh 2\alpha}\, e^{-\sigma+\alpha}$ is kept finite.
The DLCQ coordinates are more useful when we discuss the boundary conformal field theory. In ref.[@tlee98], one of the authors explicitly showed that the boundary conformal field theory is given by a $SL(2,R)\otimes SL(2,R)$ WZW model, which is equivalent to the three dimensional gravity on $AdS_3$, resorting to the Faddeev-Shatashvili procedure [@Faddeev]. Since $AdS_2$ space is obtained by the Kaluza-Klein reduction from $AdS_3$, the same procedure would lead us to the boundary conformal theory corresponding to the $AdS_2$ space. We first give the DLCQ reduction of the bulk Chern-Simons action [@Achu86], which may be rewritten as $$\begin{aligned}
\label{cs}
I_{CS}&=&{k \over 4\pi}\int_{ M}{{\rm tr}\epsilon^{\mu\nu}
(A_{\hatx}F_{\mu\nu}-A_\mu\partial_{\hatx}A_\nu)}
+{k \over 4\pi}\int_{\partial M}{{\rm tr} A_{\hatt}A_{\hatx}}\nonumber\\
&-&{k \over 4\pi}\int_{ M}{{\rm tr}\epsilon^{\mu\nu}(\bar{A}_{\hatx}
\bar{F}_{\mu\nu}-\bar{A}_\mu\partial_{\hatx}\bar{A}_\nu)}
-{k \over 4\pi}\int_{\partial M}{{\rm tr} \bar{A}_{\hatt}\bar{A}_{\hatx}},\nonumber\end{aligned}$$ where $A =(\omega+{e \over l})_I{}^Adx^IJ_A$, $\bar{A}=(\omega-{e \over l})_I{}^Adx^I\bar{J}_A$, $\left[J_A, J_B\right] = \epsilon_{AB}{}^{C} J_C$, $\left[\bar{J}_A, \bar{J}_B\right] = \epsilon_{AB}{}^{C} \bar{J}_C$, $\mu,\nu,\rho\in\{\hatt,U\}$, $A,B,C\in\{0,1,2\}$, and $I,J,K\in\{\hatt,U, \hatx^5\}$. We may cast the dreibein and the spin connection into the form of Kaluza-Klein ansatz as $$\begin{aligned}
e_I{}^A=\pmatrix{
e_\mu{}^a & -le^{\psi}a_\mu \cr
0 & -{l\over R}e^\psi \cr
},\qquad\omega_I{}^A=\pmatrix{
\omega_\mu{}^a & \omega_\mu{}^2 \cr
\omega_{\hatx}{}^a & \omega_{\hatx}{}^2 \cr
},
\nonumber\end{aligned}$$ where $a\in \{0,1\}$. Then, imposing the torsion free conditions, which are obtained as part of the equations of motion we find that the Chern-Simons action reduces to the action for the two dimensional gravity as expected $$\begin{aligned}
\label{2dg}
I_{2D}={k \over 2}\int\sqrt{-g}\left(e^\psi({ R}+{2\over l^2})-
{e^{3\psi}l^2 \over 4}f_{\mu\nu}f^{\mu\nu}
\right),\end{aligned}$$ where $f_{\mu\nu}=\partial_\mu a_\nu-\partial_\nu a_\mu$.
From the metric for the BTZ black hole in the light cone frame Eq.(\[light\]), we obtain the black hole solution in terms of the gauge fields as follows. $$\begin{aligned}
A^0 &=& 0, \quad A^1 = 0, \nonumber\\
A^2 &=& {U_0e^{-\sigma'}\over l^2}
{-e^{\alpha}d\hatt+e^{-\alpha}d\hatx
\over\sqrt{\cosh{2\alpha}}},\\
\bar{A}^0 &=& -2{Ue^{\sigma'}\over U_0l^2}
\left(U^2-U_0^2\right)^{1\over 2}
{e^{-\alpha}d\hatt+e^{\alpha}d\hatx
\over\sqrt{\cosh{2\alpha}}},\nonumber\\
\bar{A}^1 &=& - 2 \left(U^2 - U_0{}^2\right)^{-{1\over 2}}dU \nonumber\\
\bar{A}^2 &=& -2{e^{\sigma'}\over U_0l^2}\left(U^2-{U_0^2\over 2}\right)
{e^{-\alpha}d\hatt+e^{\alpha}d\hatx
\over\sqrt{\cosh{2\alpha}}}.\nonumber\end{aligned}$$ Among those, only the components $A_\hatx$ and $\bar{A}_\hatx$ are physically important because the nontrivial geometrical structure of the three dimensional space is completely encoded by holonomies or Wilson loops of the Chern-Simons gauge fields [@holo], $W[C] = {\cal P} \exp \left( \oint_C A_\hatx d\hatx \right), \quad
{\bar W}[C] = {\cal P} \exp \left( \oint_C {\bar A}_\hatx d\hatx \right)$, where $C$ is a closed curve and ${\cal P}$ denotes a path ordered product. Since in the limit where $\alpha \rightarrow -\infty$, $ A_\hatx \rightarrow \sqrt{2}{U_0e^{-\sigma'}\over l^2},\quad
{\bar A}_\hatx \rightarrow 0$, we see that the the right $SL(2,R)$ sector becomes trivial while the left $SL(2,R)$ sector remains relevant. It is consistent with the observation that the DLCQ and scaling procedure results in the BPS limit. This also explains how the isometry group of $AdS_3$, $SL(2,R)
\otimes SL(2,R)$ reduces to the isometry group of $AdS_2$, $SL(2,R)$ in the framework of DLCQ.
Rewriting the boundary action obtained in ref.[@tlee98] in terms of the DLCQ coordinates, we find that the boundary theory for $AdS_2$ may be given by the WZW model, deformed by the DLCQ procedure. The total action is composed of the followings; $$\begin{aligned}
\label{baction}
I &=& {k\over 4\pi}\int_{M}{\rm tr}\epsilon^{ij}\partial_{\hatt}A_iA_j
+{k\over 4\pi}\int_{M}{\rm tr}\epsilon^{ij}A_{\hatt}F_{ij}
\nonumber\\
&-&{k\over 4\pi}\int_{\partial M}
{\rm tr}\left[e^{2\alpha}(A_{\hatx}+\partial_{\hatx}gg^{-1})^2
+2A_{\hatt}(A_{\hatx}+\partial_{\hatx}gg^{-1})\right] \nonumber\\
&-&{k\over 4\pi}\int_{\partial M}
{\rm tr}\partial_{\hatt}gg^{-1}\partial_{\hatx}gg^{-1}
-{k \over 12\pi} \int_{M}{\rm tr}(g^{-1}dg)^3,\nonumber\\
{\bar I} &=& -{k\over 4\pi}\int_{M}
{\rm tr}\epsilon^{ij}\partial_{\hatt}\bar{A}_i\bar{A}_j
-{k\over 4\pi}\int_{M}{\rm tr}\epsilon^{ij}\bar{A}_{\hatt}\bar{F}_{ij}\nonumber\\
&-&{k\over 4\pi}\int_{\partial M}
{\rm tr}\left[e^{-2\alpha}(\bar{A}_{\hatx}+\partial_{\hatx}\bar{g}\bar{g}^{-1})^2
-2\bar{A}_{\hatt}(\bar{A}_{\hatx}
+\partial_{\hatx}\bar{g}\bar{g}^{-1})\right]
\nonumber\\
&+&{k\over 4\pi}\int_{\partial M}
{\rm tr}\partial_{\hatt}\bar{g}\bar{g}^{-1}
\partial_{\hatx}\bar{g}\bar{g}^{-1}
+{k \over 12\pi}\int_{M}{\rm tr}(\bar{g}^{-1}d\bar{g})^3,\end{aligned}$$ where $i,j\in (U,\hatx)$. With $(\tau, \theta)=\frac{1}{R}(e^{2\alpha}\hatt, \hatx)$ we may identify $I$ as the left sector of the action for the BTZ black hole given in ref.[@tlee98]. We also find that ${\bar I}$ can be understood as the right sector of the action for the BTZ black hole in terms of $(\tau', \theta)=\frac{1}{R}(e^{-2\alpha}\hatt, \hatx)$. The right sector is confined in the extreme low energy regime, thus, suppressed. The left $SL(2,R)$ sector only becomes relevant to the $AdS_2$.
The DLCQ procedure presented here illustrates explicitly how $AdS_2/CFT$ correspondence can be deduced from the $AdS_3/CFT$. It would be interesting to explore its consequences in various contexts. One may find an application of the DLCQ procedure readily in evaluation of the entropy for the $AdS_2$ black hole. To evaluate the entropy, we adopt the result of ref.[@tlee98], where the entropy of the three dimensional BTZ black hole is evaluated in accord with the proposal of Strominger [@stro]. Comparing the boundary action for the $AdS_2$ (\[baction\]) with that for the BTZ black hole, given in ref.[@tlee98], we find that the the expectation values of the Virasoro generators $L_0$ and ${\bar L}_0$ for the $AdS_2$ black hole are given by $$\begin{aligned}
&&n_L=<L_0> = \frac{k{U_0}^2 R^2 e^{-2\sigma '}}{2l^4} ,\nonumber\\
&&n_R=<{\bar L}_0> =
\frac{k{U_0}^2 R^2 e^{2\sigma '}}{2l^4}e^{4\alpha}.\end{aligned}$$ Here we assume that $n_L = n_R= 0$ for an appropriately chosen vacuum state of the black hole. Then it follows from the Cardy formula that the entropy for the $AdS_2$ black hole is evaluated as $$\begin{aligned}
\label{ent}
S
%= 2\pi \sqrt{\frac{cn_L}{6}} + 2\pi \sqrt{\frac{cn_R}{6}}
= \sqrt{2}\pi k \frac{RU_0 e^{-\sigma'}}{l^2} +
\sqrt{2}\pi k \frac{RU_0 e^{\sigma'}}{l^2} e^{2\alpha}.\end{aligned}$$ As $\alpha \rightarrow -\infty$, the right sector does not contributes to the entropy, $S \rightarrow 4\pi k \sqrt{GM}$. (Some difficulty in evaluating the entropy by applying the Cardy formula to the boundary conformal theory was pointed out by Carlip [@carlip]. This difficulty may be resolved if we consider the space-time Virasoro algebra, regarding the boundary conformal theory as the worldsheet action for the string on $AdS_3$ [@Giveon]. It would be worth while to extend the work of ref.[@tlee98] along this direction.)
We conclude this paper with a few brief remarks. We show that the $AdS_2$ black hole can be described as a DLCQ limit of the $AdS_3$ black hole. The boundary theory for $AdS_2$ is found to be a $SL(2,R)$ WZW model defined on the light like coordinate. If we are concerned with the gravity on $AdS_2$, the zero mode sector of the $AdS_3$ suffices in the framework of DLCQ. In accordance with it, the zero mode sector of the WZW model given by the action Eq.(\[baction\]) defines the boundary theory, hence a $SL(2,R)$ quantum mechanics. The correspondence between the gravity on $AdS_2$ and the $SL(2,R)$ boundary conformal quantum mechanics will be discussed in detail elsewhere. We note that the entropy of the $AdS_2$ black hole evaluated as Eq.(\[ent\]) is the same as that of the $AdS_3$ black hole in the original frame. Thus, the entropy is preserved by the DLCQ procedure.
The present paper may be extended along various directions. One of the direction would be the DLCQ reduction of the string on $AdS_3$ [@Giveon]. It would be interesting to see if the DLCQ reduction leads us to a point source on $AdS_2$ and the entropy of $AdS_2$ black hole can be given by the quantum mechanics. This may clarify some subtle issues associated with the entropy of black holes in $AdS_3$ and $AdS_2$. The two dimensional black holes have been one of the important subjects in string theory and gravity. The present work may enable us to discuss the various aspects of the two dimensional black holes [@2dbh1; @2dbh2] from the viewpoint of the three dimensional black hole. According to the conjecture of $AdS/CFT$ correspondence [@mal97] the (2+1) dimensional gravity on $AdS_3$ is supposed to be equivalent to the boundary (1+1) dimensional conformal Yang-Mills theory on its boundary. The present work also suggests that the boundary conformal theory corresponding to the gravity on $AdS_2$ may be obtained by the DLCQ reduction of the (1+1) dimensional Yang-Mills theory. Work along this direction is now in progress [@chl99]. The most important application of the present work may be found in the Matrix M-theory. DLCQ reduction of the M-brane configurations may shed some light on the Matrix M-theory. After completing the work, we found that reduction of $AdS_3$ to $AdS_2$ along the light like coordinate also has been suggested in the study of M-brane configurations [@sken].
Acknowledgement {#acknowledgement .unnumbered}
===============
The work of TL was supported in part by the Basic Science Research Institute Program, Ministry of Education of Korea (BSRI-98-2401). Part of the work of TL was done during his visit to PIMS, KEK and YITP. We would like to thank K. Skenderis for informing us of ref.[@sken] and useful comments.
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[^1]: [email protected]
[^2]: [email protected]
[^3]: [email protected]
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'In this paper, we introduce two alternative extensions of the classical univariate *Value-at-Risk* (VaR) in a multivariate setting. The two proposed multivariate VaR are vector-valued measures with the same dimension as the underlying risk portfolio. The *lower-orthant VaR* is constructed from level sets of multivariate distribution functions whereas the *upper-orthant VaR* is constructed from level sets of multivariate survival functions. Several properties have been derived. In particular, we show that these risk measures both satisfy the positive homogeneity and the translation invariance property. Comparison between univariate risk measures and components of multivariate VaR are provided. We also analyze how these measures are impacted by a change in marginal distributions, by a change in dependence structure and by a change in risk level. Illustrations are given in the class of Archimedean copulas.'
author:
- 'Areski Cousin[^1], Elena Di Bernardino[^2]'
bibliography:
- 'biblio.bib'
nocite:
- '[@Tasche]'
- '[@Gauthier]'
- '[@Zhou]'
- '[@Jouini]'
- '[@Galichon]'
- '[@Masse]'
- '[@Koltchinskii]'
- '[@Chaouch]'
- '[@Vero]'
- '[@Serfling]'
- '[@Embrechts]'
- '[@Nappo]'
- '[@Prekopa]'
- '[@Embrechts]'
- '[@Artzner]'
- '[@Mullerbook]'
- '[@Muller]'
- '[@kendalorder]'
- '[@Rivest]'
- '[@Nelsen]'
- '[@Genest1]'
- '[@Genest2]'
- '[@Rivest]'
- '[@Sklar]'
- '[@Rivest]'
- '[@McNeil_Neslehova]'
- '[@Barbe]'
- '[@Barbe]'
- '[@Rivest]'
- '[@Rivest]'
- '[@Denuit]'
- '[@Mullerbook]'
- '[@kendalorder]'
- '[@McNeil_Neslehova]'
- '[@NelsenLibro]'
- '[@NelsenLibro]'
- '[@Wei]'
- '[@Embrechts]'
- '[@Embrechts]'
- '[@Nappo]'
- '[@Tibiletti3]'
- '[@Rossi]'
- '[@Tibiletti4]'
- '[@Tibiletti1990]'
- '[@Feller]'
- '[@Imlahi]'
- '[@Chakak]'
- '[@Tibiletti1990]'
- '[@Artzner]'
- '[@NelsenLibro]'
- '[@Sklar]'
- '[@Tibiletti2]'
- '[@NelsenLibro]'
- '[@Tibiletti2]'
- '[@Tibiletti2]'
- '[@NelsenLibro]'
- '[@charpentierbook]'
- '[@Denuit]'
- '[@Joe]'
- '[@Lehmann]'
- '[@NelsenLibro]'
- '[@Joe]'
- '[@Laurent]'
title: 'On Multivariate Extensions of Value-at-Risk'
---
Multivariate risk measures, Level sets of distribution functions, Multivariate probability integral transformation, Stochastic orders, Copulas and dependence.
Introduction {#introduction .unnumbered}
============
During the last decades, researchers joined efforts to properly compare, quantify and manage risk. Regulators edict rules for bankers and insurers to improve their risk management and to avoid crises, not always successfully as illustrated by recent events.\
Traditionally, risk measures are thought of as mappings from a set of real-valued random variables to the real numbers. However, it is often insufficient to consider a single real measure to quantify risks created by business activities, especially if the latter are affected by other external risk factors. Let us consider for instance the problem of solvency capital allocation for financial institutions with multi-branch businesses confronted to risks with specific characteristics. Under Basel II and Solvency II, a bottom-up approach is used to estimate a “top-level” solvency capital. This is done by using risk aggregation techniques who may capture risk mitigation or risk diversification effects. Then this global capital amount is re-allocated to each subsidiaries or activities for internal risk management purpose (“top-down approach”). Note that the solvability of each individual branch may strongly be affected by the degree of dependence amongst all branches. As a result, the capital allocated to each branch has to be computed in a multivariate setting where both marginal effects and dependence between risks should be captured. In this respect, the “Euler approach” (e.g., see Tasche, 2008) involving vector-valued risk measures has already been tested by risk management teams of some financial institutions.\
Whereas the previous risk allocation problem only involves internal risks associated with businesses in different subsidiaries, the solvability of financial institutions could also be affected by external risks whose sources cannot be controlled. These risks may also be strongly heterogeneous in nature and difficult to diversify away. One can think for instance of systemic risk or contagion effects in a strongly interconnected system of financial companies. As we experienced during the 2007-2009 crisis, the risks undertaken by some particular institutions may have significant impact on the solvability of the others. In this regard, micro-prudential regulation has been criticized because of its failure to limit the systemic risk within the system. This question has been dealt with recently by among others, Gauthier *et al.* (2010) and Zhou (2010) who highlights the benefit of a “macro-prudential” approach as an alternative solution to the existing “micro-prudential” one (Basel II) which does not take into account interactions between financial institutions.\
In the last decade, much research has been devoted to risk measures and many extensions to multidimensional settings have been investigated. On theoretical grounds, Jouini *et al.* (2004) proposes a class of set-valued coherent risk measures. Ekeland *et al.*, (2012) derive a multivariate generalization of Kusuoka’s representation for coherent risk measures. Unsurprisingly, the main difficulty regarding multivariate generalizations of risk measures is the fact that vector preorders are, in general, partial preorders. Then, what can be considered in a context of multidimensional portfolios as the analogous of a “worst case” scenario and a related “tail distribution”? This is why several definitions of quantile-based risk measures are possible in a higher dimension. For example, Massé and Theodorescu (1994) defined multivariate quantiles as half-planes and Koltchinskii (1997) provided a general treatment of multivariate quantiles as inversions of mappings. Another approach is to use geometric quantiles (see, for example, Chaouch *et al.*, 2009). Along with the geometric quantile, the notion of depth function has been developed in recent years to characterize the quantile of multidimensional distribution functions (for further details see, for instance, Chauvigny *et al.*, 2011). We refer to Serfling (2002) for a large review on multivariate quantiles. When it turns to generalize the *Value-at-Risk* measure, Embrechts and Puccetti (2006), Nappo and Spizzichino (2009), Prékopa (2012) use the notion of quantile curve which is defined as the boundary of the upper-level set of a distribution function or the lower-level set of a survival function.\
In this paper, we introduce two alternative extensions of the classical univariate *Value-at-Risk* (VaR) in a multivariate setting. The proposed measures are based on the Embrechts and Puccetti (2006)’s definitions of multivariate quantiles. We define the *lower-orthant Value-at-Risk* at risk level $\alpha$ as the conditional expectation of the underlying vector of risks $\mathbf{X}$ given that the latter stands in the $\alpha$-level set of its distribution function. Alternatively, we define the *upper-orthant Value-at-Risk* of $\mathbf{X}$ at level $\alpha$ as the conditional expectation of $\mathbf{X}$ given that $\mathbf{X}$ stands in the $(1-\alpha$)-level set of its survival function. Contrarily to Embrechts and Puccetti (2006)’s approach, the extensions of *Value-at-Risk* proposed in this paper are real-valued vectors with the same dimension as the considered portfolio of risks. This feature can be relevant from an operational point of view.\
Several properties have been derived. In particular, we show that the *lower-orthant Value-at-Risk* and the *upper-orthant Value-at-Risk* both satisfy the positive homogeneity and the translation invariance property. We compare the components of these vector-valued measures with the univariate VaR of marginals. We prove that the *lower-orthant Value-at-Risk* (resp. *upper-orthant Value-at-Risk*) turns to be more conservative (resp. less conservative) than the vector composed of univariate VaR. We also analyze how these measures are impacted by a change in marginal distributions, by a change in dependence structure and by a change in risk level. In particular, we show that, for Archimedean families of copulas, the *lower-orthant Value-at-Risk* and the *upper-orthant Value-at-Risk* are both increasing with respect to the risk level whereas their behavior is different with respect to the degree of dependence. In particular, an increase of the dependence amongts risks tends to lower the *lower-orthant Value-at-Risk* whereas it tends to widen the *upper-orthant Value-at-Risk*. In addition, these two measures may be useful for some applications where risks are heterogeneous in nature. Indeed, contrary to many existing approaches, no arbitrary real-valued aggregate transformation is involved (sum, min, max,$\ldots$).\
The paper is organized as follows. In Section \[Notation\], we introduce some notations, tools and technical assumptions. In Section \[Multivariate generalization VAR\], we propose two multivariate extensions of the *Value-at-Risk* measure. We study the properties of our multivariate VaR in terms of Artzner *et al.* (1999)’s invariance properties of risk measures (see Section \[Invariance properties VAR\]). Illustrations in some Archimedean copula cases are presented in Section \[Archimedean copula section\]. We also compare the components of these multivariate risk measures with the associated univariate Value-at-Risk (see Section \[proprieta VAR bidimensionale\]). The behavior of our ${\rm VaR}$ with respect to a change in marginal distributions, a change in dependence structure and a change in risk level $\alpha$ is discussed respectively in Sections \[stochastic order copulas var\], \[dependence structure copulas\] and \[proprieta VAR PRD\]. In the conclusion, we discuss open problems and possible directions for future work.
Basic notions and preliminaries {#Notation}
===============================
In this section, we first introduce some notation and tools which will be used later on.
Stochastic orders {#stochastic-orders .unnumbered}
-----------------
From now on, let $Q_X(\alpha)$ be the univariate quantile function of a risk $X$ at level $\alpha \in (0,1)$. More precisely, given an univariate continuous and strictly monotonic loss distribution function $F_X$, $Q_X(\alpha)= F_X^{-1}(\alpha)$, $\forall\, \alpha \, \in \, (0,1)$. We recall here the definition and some properties of useful univariate and multivariate stochastic orders.
\[def st order\] Let $X$ and $Y$ be two random variables. Then $X$ is said to be smaller than $Y$ in stochastic dominance, denoted as $\, X \preceq_{st} Y$, if the inequality $Q_X(\alpha) \leq Q_Y(\alpha)$ is satisfied for all $\alpha \in (0, 1).$
\[def sl order\]
Let $X$ and $Y$ be two random variables. Then $X$ is said to be smaller than $Y$ in the stop-loss order, denoted as $\, X \preceq_{sl} Y$, if for all $t \in {\mathbb{R}},$ ${ {\operatorname{\mathbb{E}}}[(X- t)_+] \leq {\operatorname{\mathbb{E}}}[(Y- t)_+]},$ with $x_+ :=\max\{x,0\}$.
\[def icx order\] Let $X$ and $Y$ be two random variables. Then $X$ is said to be smaller than $Y$ in the increasing convex order, denoted as $\, X \preceq_{icx} Y$, if $\,{\operatorname{\mathbb{E}}}[f(X)] \leq {\operatorname{\mathbb{E}}}[f(Y)],$ for all non-decreasing convex function $f$ such that the expectations exist.
The stop-loss order and the increasing convex order are equivalent (see Theorem 1.5.7 in M[ü]{}ller and Stoyan, 2001). Note that stochastic dominance order implies stop-loss order. For more details about stop-loss order we refer the interested reader to M[ü]{}ller (1997).\
Finally, we introduce the definition of supermodular function and supermodular order for multivariate random vectors.
\[def sm fucntion\] A function $f: {\mathbb{R}}^d \rightarrow {\mathbb{R}}$ is said to be supermodular if for any $\textbf{x}, \textbf{y} \in {\mathbb{R}}^d $ it satisfies $$f(\textbf{x})+ f(\textbf{y}) \leq f(\textbf{x} \wedge \textbf{y}) + f(\textbf{x} \vee \textbf{y}),$$ where the operators $\wedge$ and $\vee$ denote coordinatewise minimum and maximum respectively.
\[def sm order\] Let $\textbf{X}$ and $\textbf{Y}$ be two $d-$dimensional random vectors such that $\, {\operatorname{\mathbb{E}}}[f(\textbf{X})] \leq {\operatorname{\mathbb{E}}}[f(\textbf{Y})],\,$ for all supermodular functions $f: {\mathbb{R}}^d \rightarrow {\mathbb{R}}$, provided the expectation exist. Then $\textbf{X}$ is said to be smaller than $\textbf{Y}$ with respect to the supermodular order (denoted by $\, \textbf{X} \preceq_{sm} \textbf{Y}$).
This will be a key tool to analyze the impact of dependence on our multivariate risk measures.
Kendall distribution function {#kendall-distribution-function .unnumbered}
-----------------------------
Let $\textbf{X}= (X_1, \ldots, X_d)$ be a $d-$dimensional random vector, $d\geq 2$. As we will see later on, our study of multivariate risk measures strongly relies on the key concept of Kendall distribution function (or multivariate probability integral transformation), that is, the distribution function of the random variable $F(\textbf{X})$, where $F$ is the multivariate distribution of random vector $\textbf{X}$. From now on, the Kendall distribution will be denoted by $K$, so that $K(\alpha) = {\mathbb{P}}[F(\textbf{X})\leq \alpha]$, for $\alpha \in [0,1]$. We also denote by $\overline{K}(\alpha)$ the survival distribution function of $F(\textbf{X})$, i.e., $\overline{K}(\alpha)= {\mathbb{P}}[F(\textbf{X}) > \alpha]$. For more details on the multivariate probability integral transformation, the interested reader is referred to Cap[é]{}ra[à]{} *et al.*, (1997), Genest and Rivest (2001), Nelsen *et al.* (2003), Genest and Boies (2003), Genest *et al.* (2006) and Belzunce *et al.* (2007).\
In contrast to the univariate case, it is not generally true that the distribution function $K$ of $F(\textbf{X})$ is uniform on $[0, 1]$, even when $F$ is continuous. Note also that it is not possible to characterize the joint distribution $F$ or reconstruct it from the knowledge of $K$ alone, since the latter does not contain any information about the marginal distributions $F_{X_{1}}, \ldots, F_{X_{d}}$ (see Genest and Rivest, 2001). Indeed, as a consequence of Sklar’s Theorem, the Kendall distribution only depends on the dependence structure or the copula function $C$ associated with $\textbf{X}$ (see Sklar, 1959). Thus, we also have $K(\alpha) = {\mathbb{P}}[C(\textbf{U})\leq \alpha]$ where $\textbf{U}= (U_1, \ldots, U_d)$ and $U_1= F_{X_{1}}(X_1), \ldots, U_d= F_{X_{d}}(X_d)$.\
Furthermore:
For a $d-$dimensional random vector $\textbf{X}= (X_1, \ldots, X_d)$ with copula $C$, the Kendall distribution function $K(\alpha)$ is linked to the Kendall’s tau correlation coefficient via: $\tau_C = \frac{2^d\, {\operatorname{\mathbb{E}}}[C(\textbf{U})]-1}{2^{d-1}-1}$, for $d \geq 2$ (see Section 5 in Genest and Rivest, 2001).
The Kendall distribution can be obtain explicitly in the case of multivariate Archimedean copulas with generator[^3] $\phi$, i.e., $C(u_1, \ldots, u_d) = \phi^{-1}\left(\phi(u_1)+\cdots+\phi(u_d)\right)$ for all $(u_1, \ldots, u_d)\in [0,1]^d.$ Table \[kendall classiche\] provides the expression of Kendall distributions associated with Archimedean, independent and comonotonic $d-$dimensional random vectors (see Barbe *et al.*, 1996). Note that the Kendall distribution is uniform for comonotonic random vectors.
[| c | c| c | c | c |c |]{} Copula & Kendall distribution $K(\alpha)$\
Archimedean case & $\alpha + \, \sum_{i=1}^{d-1} \, \frac{1}{i!}\left(-\phi(\alpha)\right)^{i} \, \left(\phi^{-1}\right)^{(i)}\left(\phi(\alpha)\right)$\
Independent case & $\alpha +\alpha\, \sum_{i=1}^{d-1} \left(\frac{\ln(1/\alpha)^i}{i!}\right)$\
Comonotonic case & $\alpha$\
For further details the interested reader is referred to Section 2 in Barbe *et al.* (1996) and Section 5 in Genest and Rivest (2001). For instance, in the bivariate case, the Kendall distribution function is equal to $\alpha-\frac{\phi(\alpha) }{\phi'(\alpha)},$ $\alpha\in(0,1),$ for Archimedean copulas with differentiable generator $\phi.$ It is equal to $ \alpha\left(1-\ln(\alpha)\right),$ $\alpha\in(0,1)$ for the bivariate independence copula and to $1$ for the counter-monotonic bivariate copula.
It holds that $\alpha \leq K(\alpha) \leq 1$, for all $\alpha \, \in (0,1),$ i.e., the graph of the Kendall distribution function is above the first diagonal (see Section 5 in Genest and Rivest, 2001). This is equivalent to state that, for any random vector $\textbf{U}$ with copula function $C$ and uniform marginals, $C(\textbf{U}) \preceq_{st}C^{\text{c}}(\textbf{U}^{\text{c}}) $ where $\textbf{U}^{\text{c}}=(U_{1}^{\text{c}},\ldots, U_{d}^{\text{c}})$ is a comonotonic random vector with copula function $C^{\text{c}}$ and uniform marginals.
This last property suggests that when the level of dependence between $X_1, \ldots, X_d$ increases, the Kendall distribution also increases in some sense. The following result, using definitions of stochastic orders described above, investigates rigorously this intuition.
\[super modal order\] Let $\textbf{U}= (U_1, \ldots, U_d)$ [(]{}resp. $\textbf{U}^*= (U_{1}^*, \ldots, U_{d}^*)$[)]{} be a random vector with copula $C$ (resp. $C^*$) and uniform marginals.
If $ \, \, \textbf{U} \preceq_{sm} \textbf{U}^*, \, \,$ then $ \, \,C(\textbf{U}) \preceq_{sl} C^*(\textbf{U}^*).$
*Proof:* Trivially, $\textbf{U} \preceq_{sm} \textbf{U}^* \Rightarrow C(\textbf{u}) \leq C^*(\textbf{u})$, for all $\textbf{u} \in [0,1]^d$ (see Section 6.3.3 in Denuit *et al.*, 2005). Let $f: [0,1] \rightarrow {\mathbb{R}}$ be a non-decreasing and convex function. It holds that $f(C(\textbf{u})) \leq f(C^*(\textbf{u}))$, for all $\textbf{u} \in [0,1]^d$, and ${\operatorname{\mathbb{E}}}[f(C(\textbf{U}))]\leq {\operatorname{\mathbb{E}}}[f(C^*(\textbf{U}))]$. Remark that since $C^*$ is non-decreasing and supermodular and $f$ is non-decreasing and convex then $f \circ C^*$ is a non-decreasing and supermodular function (see Theorem 3.9.3 in M[ü]{}ller and Stoyan, 2001). Then, by assumptions, ${\operatorname{\mathbb{E}}}[f(C(\textbf{U}))] \leq {\operatorname{\mathbb{E}}}[f(C^*(\textbf{U}))] \leq {\operatorname{\mathbb{E}}}[f(C^*(\textbf{U}^*))]$. This implies $C(\textbf{U}) \preceq_{sl} C^*(\textbf{U}^*).$ Hence the result. $\Box$\
From Proposition \[super modal order\], we remark that $\textbf{U}\preceq_{sm} \textbf{U}^*$ implies an ordering relation between corresponding Kendall’s tau : $\tau_{C} \leq \tau_{C^*}$. Note that the supermodular order between $\textbf{U}$ and $\textbf{U}^*$ does not necessarily yield the stochastic dominance order between $C(\textbf{U})$ and $C^*(\textbf{U}^*)$ (i.e., ${C(\textbf{U}) \preceq_{st} C^*(\textbf{U}^*)}$ does not hold in general). For a bivariate counter-example, the interested reader is referred to, for instance, Cap[é]{}ra[à]{} *et al.* (1997) or Example 3.1 in Nelsen *et al.* (2003).\
Let us now focus on some classical families of bivariate Archimedean copulas. In Table \[kendall achimedeans\], we obtain analytical expressions of the Kendall distribution function for Gumbel, Frank, Clayton and Ali-Mikhail-Haq families.
[| c | c| c | c | c |c |]{} Copula & $\theta \in $ & Kendall distribution $K(\alpha, \theta)$\
Gumbel & $[1, \infty )$ & $\alpha\left(1-{\frac {1 }{\theta}\ln\alpha}\right)$\
Frank & $(-\infty,\infty) \setminus \{0\}$ & ${ \alpha+ \frac{1}{\theta}\left(1-{\rm e}^{\theta\alpha}\right)\ln \left( {\frac {{1-{\rm e}^{-\theta\,\alpha}}}{{1-{\rm e}^{-\theta}}}} \right)}$\
Clayton & $[-1,\infty) \setminus \{0\}$ & $ \alpha\left(1+ \frac{1}{\theta}{\left(1-\alpha^{\theta}\right)}\right)$\
Ali-Mikhail-Haq & $[-1,1)$ & $\frac {\alpha\, -1+\theta+ (1- \theta + \theta \alpha) (\ln \left( 1-\theta+\theta\,\alpha \right) + \ln\alpha) }{\theta-1}$\
\[theta e dimensione\] Bivariate Archimedean copula can be extended to $d$-dimensional copulas with $d> 2$ as far as the generator $\phi$ is a $d$-monotone function on $[0, \infty)$ (see McNeil and Nešlehová, 2009 for more details). For the $d$-dimensional Clayton copulas, the underlying dependence parameter must be such that $ \theta > - \frac{1}{d-1}$ (see Example 4.27 in Nelsen, 1999). Frank copulas can be extended to $d$-dimensional copulas for $\theta>0$ (see Example 4.24 in Nelsen, 1999).
Note that parameter $\theta$ governs the level of dependence amongst components of the underlying random vector. Indeed, it can be shown that, for all Archimedean copulas in Table \[kendall achimedeans\], an increase of $\theta$ yields an increase of dependence in the sense of the supermodular order, i.e., $\theta \leq \theta^* \Rightarrow \textbf{U} \preceq_{sm}\textbf{U}^*$ (see further examples in Joe, 1997 and Wei and Hu, 2002). Then, as a consequence of Proposition \[super modal order\], the following comparison result holds $$\theta \leq \theta^* \Rightarrow C(\textbf{U}) \preceq_{sl} C^*(\textbf{U}^*).
$$ In fact, a stronger comparison result can be derived for Archimedean copulas of Table \[kendall achimedeans\], as shown in the following remark.
\[St\_Kendall\_Archimedean\]
For copulas in Table \[kendall achimedeans\], one can check that $\frac{\partial K(\alpha, \theta)}{\partial \theta} \leq 0$, for all $\alpha \in (0,1)$. This means that, for these classical examples, the associated Kendall distributions actually increase with respect to the stochastic dominance order when the dependence parameter $\theta$ increases, i.e., $$\label{eq_St_Dominance}
{\theta \leq \theta^* \Rightarrow C(\textbf{U}) \preceq_{st} C^*(\textbf{U}^*)}.$$ In order to illustrate this property we plot in Figure \[Kendall alpha theta gumbel\] the Kendall distribution function $K(\cdot, \theta)$ for different choices of parameter $\theta$ in the bivariate Clayton copula case and in the bivariate Gumbel copula case.
![[Kendall distribution $K(\cdot, \theta)$ for different values of $ \theta$ in the Clayton copula case (left) and the Gumbel copula case (right). The dark full line represents the first diagonal and it corresponds to the comonotonic case.]{}[]{data-label="Kendall alpha theta gumbel"}](kendallCLAYTON){width="8.2cm"}
![[Kendall distribution $K(\cdot, \theta)$ for different values of $ \theta$ in the Clayton copula case (left) and the Gumbel copula case (right). The dark full line represents the first diagonal and it corresponds to the comonotonic case.]{}[]{data-label="Kendall alpha theta gumbel"}](kendallGUMBEL){width="8.4cm"}
Multivariate generalization of the Value-at-Risk measure {#Multivariate generalization VAR}
========================================================
From the usual definition in the univariate setting, the *Value-at-Risk* is the minimal amount of the loss which accumulates a probability $\alpha$ to the left tail and $1 - \alpha$ to the right tail. Then, if $F_X$ denotes the cumulative distribution function associated with risk $X$ and $\overline{F}_X$ its associated survival function, then $$\VAR_\alpha(X) := \inf \left\{x\in {\mathbb{R}}: F_X(x) \geq \alpha \right\}$$ and equivalently, $$\VAR_\alpha(X) := \inf \left\{x\in {\mathbb{R}}: \overline{F}_X(x) \leq 1-\alpha \right\}.$$
Consequently, the classical univariate VaR can be viewed as the boundary of the set $\left\{x\in {\mathbb{R}}:\right.$ $\left. F_X(x) \geq \alpha \right\}$ or, similarly, the boundary of the set $\left\{x\in {\mathbb{R}}: \overline{F}_X(x) \leq 1-\alpha \right\}$.\
This idea can be easily extended in higher dimension, keeping in mind that the two previous sets are different in general as soon as $d\geq 2$. We propose a multivariate generalization of *Value-at-Risk* for a portfolio of $d$ dependent risks. As a starting point, we consider Definition 17 in Embrechts and Puccetti (2006). They suggest to define the multivariate lower-orthant Value-at-Risk at probability level $\alpha$, for a increasing function $\underline{G}$ : ${\mathbb{R}}^d \rightarrow [0,1]$, as the boundary of its $\alpha$–upper-level set, i.e., $\partial \{\textbf{x} \in \mathbb{R}^{d} : \underline{G}(\textbf{x}) \geq \alpha\}$ and analogously, the multivariate upper-orthant Value-at-Risk, for a decreasing function $\overline{G}$ : ${\mathbb{R}}^d \rightarrow [0,1]$, as the boundary of its $(1-\alpha)$–lower-level set, i.e., $\partial \{\textbf{x} \in \mathbb{R}^{d} : \overline{G}(\textbf{x}) \leq 1- \alpha\}$.\
Note that the generalizations of *Value-at-Risk* by Embrechts and Puccetti (2006) (see also Nappo and Spizzichino, 2009; Tibiletti, 1993) are represented by an infinite number of points (an hyperspace of dimension $d-1$, under some regularly conditions on the functions $\underline{G}$ and $\overline{G}$). This choice can be unsuitable when we face real risk management problems. Then, we propose more parsimonious and synthetic versions of the Embrechts and Puccetti (2006)’s measures. In particular in our propositions, instead of considering the whole hyperspace $\partial \{\textbf{x} : \underline{G}(\textbf{x}) \geq \alpha\}$ (or $\partial \{\textbf{x} : \overline{G}(\textbf{x}) \leq 1- \alpha\}$) we only focus on the particular point in $\mathbb{R}_+^d$ that matches the conditional expectation of $\textbf{X}$ given that $\textbf{X}$ stands in this set. This means that our measures are real-valued vectors with the same dimension as the considered portfolio of risks.\
In addition, to be consistent with the univariate definition of $\rm{VaR}$, we choose $\underline{G}$ (resp. $\overline{G}$) as the $d-$dimensional loss distribution function $F$ (resp. the survival distribution function $\overline{F}$) of the risk portfolio. This allows to capture information coming both from the marginal distributions and from the multivariate dependence structure, without using an arbitrary real-valued aggregate transformation (for more details see Introduction).\
In analogy with the Embrechts and Puccetti’s notation we will denote $\underline{{\rm VaR}}$ our multivariate lower-orthant Value-at-Risk and $\overline{{\rm VaR}}$ the upper-orthant one.\
In the following, we will consider non-negative absolutely-continuous random vector[^4] ${\textbf{X}= (X_1, \ldots, X_d)}$ (with respect to Lebesgue measure $\lambda$ on $\mathbb{R}^d$) with partially increasing multivariate distribution function[^5] $F$ and such that ${{\operatorname{\mathbb{E}}}(X_i)< \infty},$ for $i=1, \ldots, d$. These conditions will be called *regularity conditions*.
However, extensions of our results in the case of multivariate distribution function on the entire space $\mathbb{R}^{d}$ or in the presence of plateau in the graph of $F$ are possible. Starting from these considerations, we introduce here a multivariate generalization of the [VaR]{} measure.
\[VAR multivariate\] Consider a random vector $\textbf{X}=(X_1,\ldots, X_d)$ with distribution function $F$ satisfying the regularity conditions. For $\alpha \in (0, 1)$, we define the multidimensional lower-orthant Value-at-Risk at probability level $\alpha$ by $$\underline{{\rm VaR}}_\alpha(\textbf{X})={\operatorname{\mathbb{E}}}[\textbf{X}|\, \textbf{X} \in \partial \underline{L}(\alpha) ]=\left(
\begin{array}{ll}
{\operatorname{\mathbb{E}}}[\,X_1\,|\,\textbf{X} \in \partial \underline{L}(\alpha)\,] \vspace{0.04cm} \\
\quad \quad \quad \quad \vdots \vspace{0.04cm} \\
{\operatorname{\mathbb{E}}}[\,X_d\,|\,\textbf{X} \in \partial \underline{L}(\alpha)\,]
\end{array}
\right).$$ where $\partial \underline{L}(\alpha)$ is the boundary of the set $\underline{L}(\alpha):= \{\textbf{x} \in \mathbb{R}^{d}_+ : F(\textbf{x}) \geq \alpha\}$. Under the regularity conditions, $\partial \underline{L}(\alpha)$ is the $\alpha$-level set of $F$, i.e., $\partial \underline{L}(\alpha)=\{\textbf{x} \in \mathbb{R}^{d}_+ : F(\textbf{x}) = \alpha\}$ and the previous definition can be restated as $$\underline{{\rm VaR}}_\alpha(\textbf{X}) ={\operatorname{\mathbb{E}}}[\textbf{X}|\,F(\textbf{X}) = \alpha]=\left(
\begin{array}{ll}
{\operatorname{\mathbb{E}}}[\,X_1\,|\,F(\textbf{X})= \alpha\,]\vspace{0.04cm} \\
\quad \quad \quad \quad \vdots \vspace{0.04cm} \\
{\operatorname{\mathbb{E}}}[\,X_d\,|\,F(\textbf{X})= \alpha\,]
\end{array}
\right).$$
Note that, under the *regularity conditions*, $\partial \underline{L}(\alpha)= \{\textbf{x} \in \mathbb{R}^{d}_{+}: F(\textbf{x}) = \alpha\}$ has Lebesgue-measure zero in $\mathbb{R}^{d}_{+}$ (e.g., see Property 3 in Tibiletti, 1990). Then we make sense of Definition \[VAR multivariate\] using the limit procedure in Feller (1966), Section 3.2: $$\begin{gathered}
\label{limit procedure}
{\operatorname{\mathbb{E}}}[\,X_i \,|\, F(\textbf{X}) =\alpha\,]= \lim_{h \rightarrow 0}\, {\operatorname{\mathbb{E}}}[\,X_i \,|\,\alpha <F(\textbf{X}) \leq \alpha +h \,] \\ =
\lim_{h \rightarrow 0}\, \frac{ \int_{Q_{X_i}(\alpha)}^\infty x \left( \int_{\alpha}^{\alpha +h } f_{(X_i,F(\textbf{X}))}(x,y) \, {{\mathrm{d}}}y\right) {{\mathrm{d}}}x}{ \int_{\alpha}^{\alpha +h } f_{F(\textbf{X})}(y) \, {{\mathrm{d}}}y},\end{gathered}$$ for $i= 1,\ldots, d$.
Dividing numerator and denominator in by $h$, we obtain, as $h \rightarrow 0$ $$\label{VAR multivariate formula}
{\operatorname{\mathbb{E}}}[X_i \, | F(\textbf{X}) =\alpha]= \frac{ \int_{Q_{X_i}(\alpha)}^\infty x\, f_{(X_i,F(\textbf{X}))}(x, \alpha) \, {{\mathrm{d}}}x} {K'(\alpha)},$$ for $i= 1,\ldots, d$, where $K'(\alpha) = \frac{{{\mathrm{d}}}K(\alpha)}{{{\mathrm{d}}}\alpha}$ is the Kendall distribution density function. This procedure gives a rigorous sense to our $\underline{{\rm VaR}}_\alpha(\textbf{X})$ in Definition \[VAR multivariate\]. Remark that the existence of $f_{(X_i,F(\textbf{X}))}$ and $K'$ in is guaranteed by the *regularity conditions* (for further details, see Proposition 1 in Imlahi *et al.*, 1999 or Proposition 4 in Chakak and Ezzerg, 2000).\
In analogy with Definition \[VAR multivariate\], we now introduce another possible generalization of the [VaR]{} measure based on the survival distribution function.
\[VAR multivariate upper\] Consider a random vector $\textbf{X}=(X_1,\ldots, X_d)$ with survival distribution $\overline{F}$ satisfying the regularity conditions. For $\alpha \in (0, 1)$, we define the multidimensional upper-orthant Value-at-Risk at probability level $\alpha$ by $$\overline{{\rm VaR}}_\alpha(\textbf{X})={\operatorname{\mathbb{E}}}[\textbf{X}|\, \textbf{X} \in \partial \overline{L}(\alpha) ]=\left(
\begin{array}{ll}
{\operatorname{\mathbb{E}}}[\,X_1\,|\,\textbf{X} \in \partial \overline{L}(\alpha)\,] \vspace{0.04cm} \\
\quad \quad \quad \quad \vdots \vspace{0.04cm} \\
{\operatorname{\mathbb{E}}}[\,X_d\,|\,\textbf{X} \in \partial \overline{L}(\alpha)\,]
\end{array}
\right).$$ where $\partial \overline{L}(\alpha)$ is the boundary of the set $\overline{L}(\alpha):= \{\textbf{x} \in \mathbb{R}^{d}_+ : \overline{F}(\textbf{x}) \leq 1 - \alpha\}$. Under the regularity conditions, $\partial \overline{L}(\alpha) $ is the $(1-\alpha)$-level set of $\overline{F}$, i.e., $\partial \overline{L}(\alpha) = \{\textbf{x} \in \mathbb{R}^{d}_+ : \overline{F}(\textbf{x}) = 1 - \alpha\}$ and the previous definition can be restated as $$\overline{{\rm VaR}}_\alpha(\textbf{X}) ={\operatorname{\mathbb{E}}}[\textbf{X}|\, \overline{F}_{\textbf{X}}(\textbf{X})=1-\alpha ]=\left(
\begin{array}{ll}
{\operatorname{\mathbb{E}}}[\,X_1\,|\, \overline{F}(\textbf{X})=1-\alpha\,]\vspace{0.04cm} \\
\quad \quad \quad \quad \vdots \vspace{0.04cm} \\
{\operatorname{\mathbb{E}}}[\,X_d\,|\, \overline{F}(\textbf{X})=1-\alpha\,]
\end{array}
\right).$$
As for $\partial \underline{L}(\alpha)$, under the *regularity conditions*, $\partial \overline{L}(\alpha)= \{\textbf{x} \in \mathbb{R}^{d}_{+}: \overline{F}(\textbf{x}) = 1- \alpha\}$ has Lebesgue-measure zero in $\mathbb{R}^{d}_{+}$ (e.g., see Property 3 in Tibiletti, 1990) and we make sense of Definition \[VAR multivariate upper\] using the limit Feller’s procedure (see Equations -).\
From now on, we denote by $\underline{{\rm VaR}}^1_\alpha(\textbf{X})$, $\ldots$, $\underline{{\rm VaR}}^d_\alpha(\textbf{X})$ the components of the vector $\underline{{\rm VaR}}_\alpha(\textbf{X})$ and by $\overline{{\rm VaR}}^1_\alpha(\textbf{X})$, $\ldots$, $\overline{{\rm VaR}}^d_\alpha(\textbf{X})$ the components of the vector $\overline{{\rm VaR}}_\alpha(\textbf{X})$.\
Note that if $\textbf{X}$ is an exchangeable random vector, ${\underline{\mbox{VaR}}_{\alpha}^i(\textbf{X})= \underline{\mbox{VaR}}_{\alpha}^j(\textbf{X})}$ and ${\overline{\mbox{VaR}}_{\alpha}^i(\textbf{X})= \overline{\mbox{VaR}}_{\alpha}^j(\textbf{X})}$ for any $i,j =1, \ldots, d$. Furthermore, given a univariate random variable $X$, ${\operatorname{\mathbb{E}}}[X \, | F_X(X) = \alpha] = {\operatorname{\mathbb{E}}}[X \, | \overline{F}_X(X) = 1-\alpha] = \rm{VaR}_\alpha(X),$ for all $\alpha$ in $(0,1)$. Hence, lower-orthant VaR and upper-orthant VaR are the same for (univariate) random variables and Definitions \[VAR multivariate\] and \[VAR multivariate upper\] can be viewed as natural multivariate versions of the univariate case. As remarked above, in Definitions \[VAR multivariate\]-\[VAR multivariate upper\] instead of considering the whole hyperspace $\partial \underline{L}(\alpha)$ (or $\partial \overline{L}(\alpha)$), we only focus on the particular point in $\mathbb{R}_+^d$ that matches the conditional expectation of $\textbf{X}$ given that $\textbf{X}$ falls in $\partial \underline{L}(\alpha)$ (or in $\partial \overline{L}(\alpha)$).\
Invariance properties {#Invariance properties VAR}
---------------------
In the present section, the aim is to analyze the lower-orthant VaR and upper-orthant VaR introduced in Definitions \[VAR multivariate\]-\[VAR multivariate upper\] in terms of classical invariance properties of risk measures (we refer the interested reader to Artzner *et al.*, 1999). As these measures are not the same in general for dimension greater or equal to $2$, we also provide some connections between these two measures.\
We now introduce the following results (Proposition \[passaggio tra le VAR upper et lower con h\] and Corollary \[passaggio tra le VAR\]) that will be useful in order prove invariance properties of our risk measures.
\[passaggio tra le VAR upper et lower con h\] Let the function $h$ be such that $h(x_1, \ldots, x_d) = (h_1(x_1), \ldots, h_d(x_d))$.
- If $h_1, \ldots, h_d$ are non-decreasing functions, then the following relations hold $$\underline{{\rm VaR}}^{i}_\alpha(h(\textbf{X})) = {\operatorname{\mathbb{E}}}[\,h_i(X_i)\,|\,F_{\textbf{X}}(\textbf{X})= \alpha\,],\;\; i =1, \ldots, d.$$
- If $h_1, \ldots, h_d$ are non-increasing functions, then the following relations hold $$\underline{{\rm VaR}}^{i}_\alpha(h(\textbf{X})) = {\operatorname{\mathbb{E}}}[\,h_i(X_i)\,|\,\overline{F}_{\textbf{X}}(\textbf{X})= \alpha\,],\;\; i =1, \ldots, d.$$
*Proof:* From Definition \[VAR multivariate\], $\underline{{\rm VaR}}^{i}_\alpha(h(\textbf{X})) = {\operatorname{\mathbb{E}}}[\,h_i(X_i)\,|\, {F}_{h(\textbf{X})}(h(\textbf{X}))= \alpha\,]$, for $i =1, \ldots, d$. Since $$F_{h(\textbf{X})}(y_1, \ldots, y_d)= \left\{
\begin{array}{ll}
F_{\textbf{X}}(h^{-1}(y_1), \ldots, h^{-1}(y_d)), & \mbox{ if } h_1, \ldots, h_d \mbox{ are non-decreasing functions,} \\
\overline{F}_{\textbf{X}}(h^{-1}(y_1), \ldots, h^{-1}(y_d)), & \mbox{ if } h_1, \ldots, h_d \mbox{ are non-increasing functions,}
\end{array}
\right.$$ then we obtain the result. $\Box$
From Proposition \[passaggio tra le VAR upper et lower con h\] one can trivially obtain the following property which links the multivariate upper-orthant Value-at-Risk and lower-orthant one.
\[passaggio tra le VAR\] Let $h$ be a linear function such that $h(x_1, \ldots, x_d) = (h_1(x_1), \ldots, h_d(x_d))$.
- If $h_1, \ldots, h_d$ are non-decreasing functions then then it holds that
$\underline{{\rm VaR}}_\alpha(h(\textbf{X})) = h(\underline{{\rm VaR}}_\alpha(\textbf{X})) \quad $ and $\quad \overline{{\rm VaR}}_\alpha(h(\textbf{X})) = h(\overline{{\rm VaR}}_\alpha(\textbf{X}))$.
- If $h_1, \ldots, h_d$ are non-increasing functions then it holds that
$\underline{{\rm VaR}}_\alpha(h(\textbf{X})) = h(\overline{{\rm VaR}}_{1-\alpha}(\textbf{X})) \quad $ and $\quad \overline{{\rm VaR}}_\alpha(h(\textbf{X})) = h(\underline{{\rm VaR}}_{1-\alpha}(\textbf{X}))$.
\[Link\_LO\_UO\_VaR\] If $X = (X_1, \ldots, X_d)$ is a random vector with uniform margins and if, for all $i=1,\ldots, d$, we consider the functions $h_i$ such that $h_i(x) = 1-x$, $x\in [0,1]$, then from Corollary \[passaggio tra le VAR\], $$\label{Eq:Link_VaR_UO_LU}
\overline{{\rm VaR}}^i_\alpha(\textbf{X}) = 1 -\underline{{\rm VaR}}^i_{1-\alpha}(\textbf{1}-\textbf{X})$$ for all $i=1, \ldots, d$, where $\textbf{1}-\textbf{X} = (1-X_1, \ldots, 1-X_d)$. In this case, $\overline{{\rm VaR}}_\alpha(\textbf{X})$ is the point reflection of $\, \underline{{\rm VaR}}_{1-\alpha}({\rm\textbf{1}}-\textbf{X})$ with respect to point $\mathcal{I}$ with coordinates $(\frac{1}{2}, \ldots, \frac{1}{2})$. If $\textbf{X}$ and $\textbf{1}-\textbf{X}$ have the same distribution function, then $X$ is invariant in law by central symmetry and additionally the copula of $X$ and its associated survival copula are the same. In that case $\overline{{\rm VaR}}_\alpha(\textbf{X})$ is the point reflection of $\underline{{\rm VaR}}_{1-\alpha}(\textbf{X})$ with respect to $\mathcal{I}$. This property holds for instance for elliptical copulas or for the Frank copula.
Finally, we can state the following result that proves positive homogeneity and translation invariance for our measures.
\[invarianza var biv\] Consider a random vector $\textbf{X}$ satisfying the regularity conditions. For $\alpha \in (0,1)$, the multivariate upper-orthant and lower-orthant Value-at-Risk satisfiy the following properties:
Positive Homogeneity: $\quad \forall \,\,\, \textbf{c} \in {\mathbb{R}}^d_+,$ $$\underline{{\rm VaR}}_\alpha(\textbf{c} \textbf{X}) = \textbf{c} \underline{{\rm VaR}}_\alpha(\textbf{X}), \quad
\overline{{\rm VaR}}_\alpha(\textbf{c} \textbf{X}) = \textbf{c} \overline{{\rm VaR}}_\alpha(\textbf{X})
$$ Translation Invariance: $\quad \forall \, \,\, \textbf{c} \in {\mathbb{R}}^d_+,$ $$\underline{{\rm VaR}}_\alpha(\textbf{c} + \textbf{X}) = \textbf{c} + \underline{{\rm VaR}}_\alpha(\textbf{X}), \quad
\overline{{\rm VaR}}_\alpha(\textbf{c} + \textbf{X}) = \textbf{c} + \overline{{\rm VaR}}_\alpha(\textbf{X})$$
The proof comes down from Corollary \[passaggio tra le VAR\].
Archimedean copula case {#Archimedean copula section}
-----------------------
Surprisingly enough, the and $\overline{\VAR}$ introduced in Definitions \[VAR multivariate\]-\[VAR multivariate upper\] can be computed analytically for any $d-$dimensional random vector with an Archimedean copula dependence structure. This is due to McNeil and Nešlehová’s stochastic representation of Archimedean copulas.
(McNeil and Nešlehová, 2009) \[McNeil\_Neslehova\] Let $\textbf{U}= (U_1, \ldots, U_d)$ be distributed according to a $d$-dimensional Archimedian copula with generator $\phi$, then $$\label{Eq_Representation_Archi_1}
\left(\phi(U_1), \ldots, \phi(U_d)\right) \eqd R\textbf{S}\,,$$ where $\textbf{S}=(S_1, \ldots, S_d)$ is uniformly distributed on the unit simplex $\left\{\textbf{x}\geq 0 \mid \sum_{k=1}^d x_k = 1\right\}$ and $R$ is an independent non-negative scalar random variable which can be interpreted as the radial part of $\left(\phi(U_1), \ldots, \phi(U_d)\right)$ since $\sum_{k=1}^d S_k = 1$. The random vector $\textbf{S}$ follows a symmetric Dirichlet distribution whereas the distribution of $R \eqd \sum_{k=1}^d \phi(U_k)$ is directly related to the generator $\phi$ through the inverse Williamson transform of $\phi^{-1}.$
Recall that a $d$-dimensional Archimedean copula with generator $\phi$ is defined by $C(u_1, \ldots, u_d) = \phi^{-1}(\phi(u_1)+ \cdots+ \phi(u_d))$, for all $(u_1, \ldots, u_d) \in [0,1]^d.$ Then, the radial part $R$ of representation (\[Eq\_Representation\_Archi\_1\]) is directly related to the generator $\phi$ and the probability integral transformation of $\textbf{U}$, that is, $$R \eqd \phi(C(\textbf{U})).$$
As a result, any random vector $\textbf{U}=\left(U_1, \ldots, U_d\right)$ which follows an Archimedean copula with generator $\phi$ can be represented as a deterministic function of $C(\textbf{U})$ and an independent random vector $\textbf{S}=(S_1, \ldots, S_d)$ uniformly distributed on the unit simplex, i.e.,
$$\label{Eq_Representation_Archi_2}
\left(U_1, \ldots, U_d\right) \eqd \left(\phi^{-1}\left(S_1\phi\left(C(\textbf{U})\right)\right), \ldots, \phi^{-1}\left(S_d\phi\left(C(\textbf{U})\right)\right) \right).$$
The previous relation allows us to obtain an easily tractable expression of $\underline{{\rm VaR}}_\alpha(\textbf{X})$ for any random vector $\textbf{X}$ with an Archimedean copula dependence structure.
\[VaR\_Archimedean\] Let $\textbf{X}$ be a $d$-dimensional random vector with marginal distributions $F_1, \ldots, F_d$. Assume that the dependence structure of $\textbf{X}$ is given by an Archimedian copula with generator $\phi$. Then, for any $i=1,\ldots, d,$ $$\label{Expression_VaR_Arch}
\underline{{\rm VaR}}_\alpha^i(\textbf{X}) = {\operatorname{\mathbb{E}}}\left[F_i^{-1}\left(\phi^{-1}(S_i \phi(\alpha))\right)\right]$$ where $S_i$ is a random variable with ${\rm Beta}(1, d-1)$ distribution.
*Proof:* Note that $\textbf{X}$ is distributed as $(F_1^{-1}(U_1), \ldots, F_d^{-1}(U_d))$ where $\textbf{U} = (U_1, \ldots, U_d)$ follows an Archimedean copula $C$ with generator $\phi$. Then, each component $i=1,\ldots, d\,$ of the multivariate risk measure introduced in Definition \[VAR multivariate\] can be expressed as $\underline{{\rm VaR}}_\alpha^i(\textbf{X}) = {\operatorname{\mathbb{E}}}\left[F_i^{-1}(U_i) \mid C(\textbf{U})=\alpha \right]$. Moreover, from representation (\[Eq\_Representation\_Archi\_2\]) the following relation holds $$\label{Archi_Cond_Distr}
\left[\textbf{U}\mid C(\textbf{U}) = \alpha\right] \eqd \left(\phi^{-1}\left(S_1\phi\left(\alpha\right)\right), \ldots, \phi^{-1}\left(S_d\phi\left(\alpha\right)\right) \right)$$ since $\textbf{S}$ and $C(\textbf{U})$ are stochastically independent. The result comes down from the fact that the random vector $(S_1, \ldots, S_d)$ follows a symmetric Dirichlet distribution. $\Box$\
Note that, using (\[Archi\_Cond\_Distr\]), the marginal distributions of $\textbf{U}$ given $C(\textbf{U})=\alpha$ can be expressed in a very simple way, that is, for any $k=1,\ldots, d,$ $$\label{Marginal_Cond_Distr_Archi}
{\mathbb{P}}(U_k\leq u \mid C(\textbf{U})=\alpha) = \left(1 - \frac{\phi(u)}{\phi(\alpha)}\right)^{d-1} \,\, \quad \mbox{ for } \,\, 0<\alpha<u<1.$$ The latter relation derives from the fact that $S_k,$ which is ${\rm Beta}(1, d-1)$- distributed, is such that $S_k \eqd 1-V^{\frac{1}{d-1}}$ where $V$ is uniformly-distributed on $(0,1)$.\
We now adapt Corollary \[VaR\_Archimedean\] for the multivariate upper-orthant Value-at-Risk, i.e., $\overline{{\rm VaR}}_\alpha$.
\[VaR\_Archimedean upper\] Let $\textbf{X}$ be a $d$-dimensional random vector with marginal survival distributions $\overline{F}_1, \ldots, \overline{F}_d$. Assume that the survival copula of $\textbf{X}$ is an Archimedean copula with generator $\phi$. Then, for any $i=1,\ldots, d,$ $$\label{Expression_VaR_Arch upper}
\overline{{\rm VaR}}_\alpha^i(\textbf{X}) = {\operatorname{\mathbb{E}}}\left[\overline{F}_{i}^{-1}\left(\phi^{-1}(S_i \phi(1-\alpha))\right)\right]$$ where $S_i$ is a random variable with ${\rm Beta}(1, d-1)$ distribution.
*Proof:* Note that $\textbf{X}$ is distributed as $(\overline{F}_1^{-1}(U_1), \ldots, \overline{F}_d^{-1}(U_d))$ where $\textbf{U} = (U_1, \ldots, U_d)$ follows an Archimedean copula $\overline{C}$ with generator $\phi$. Then, each component $i=1,\ldots, d\,$ of the multivariate risk measure introduced in Definition \[VAR multivariate upper\] can be expressed as $\overline{{\rm VaR}}_\alpha^i(\textbf{X}) = {\operatorname{\mathbb{E}}}\left[\overline{F}_i^{-1}(U_i) \mid \overline{C}(\textbf{U})=1-\alpha \right]$. Then, relation \[Archi\_Cond\_Distr\] also holds for $\mathbf{{U}}$ and $\overline{C}$, i.e., $\left[\textbf{U}\mid \overline{C}(\textbf{U}) = 1-\alpha\right] \eqd \left(\phi^{-1}\left(S_1\phi\left(1-\alpha\right)\right), \ldots, \phi^{-1}\left(S_d\phi\left(1-\alpha\right)\right) \right)$. Hence the result. $\Box$\
In the following, from and , we derive analytical expressions of the lower-orthant and the upper-orthant Value-at-Risk for a random vector $\mathbf{X}=(X_1, \ldots, X_d)$ distributed as a particular Archimedean copula. Let us first remark that , as Archimedean copulas are exchangeable, the components of $\underline{\rm{VaR}}$ (resp. $\overline{\rm{VaR}}$) are the same. Moreover, as far as closed-form expressions are available for the lower-orthant $\underline{\rm{VaR}}$ of $\mathbf{X}$, it is also possible to derive an analogue expression for the upper-orthant $\overline{\rm{VaR}}$ of $\mathbf{\tilde{X}} = (1-X_1, \ldots, 1-X_d)$ since from Example \[Link\_LO\_UO\_VaR\] $$\label{Link_Arch_Copula}
\overline{{\rm VaR}}^i_{\alpha}(\mathbf{\tilde{X}}) = 1 - \underline{{\rm VaR}}^i_{1-\alpha}(\mathbf{X}).$$\
As a matter of example, let us now consider the Clayton family of bivariate copulas. This family is interesting since it contains the counter-monotonic, the independence and the comonotonic copulas as particular cases. Let $(X,Y)$ be a random vector distributed as a Clayton copula with parameter $\theta \geq -1$. Then, $X$ and $Y$ are uniformly-distributed on $(0,1)$ and the joint distribution function $C_\theta$ of $(X,Y)$ is such that
$$\label{clayton}
C_\theta(x,y)= (\max\{x^{-\theta}+y^{-\theta}-1, 0\})^{-\frac{1}{\theta}}, \quad \mbox{ for } \theta \geq -1, \,\,\, \,\, (x,y) \in [0,1]^2.$$
Table \[VAR different copula\] gives analytical expressions for the first (equal to the second) component of $\underline{\rm{VaR}}$ as a function of the risk level $\alpha$ and the dependence parameter $\theta$. For $\theta = -1$ and $\theta = \infty $ we obtain the Fréchet-Hoeffding lower and upper bounds: $W(x,y)= \max\{x +y -1, 0\}$ (counter-monotonic copula) and $M(x,y)= \min\{x, y\}$ (comonotonic random copula) respectively. The settings $\theta=0$ and $\theta=1$ correspond to degenerate cases. For $\theta = 0$ we have the independence copula $\Pi(x,y)=x\,y$. For $\theta=1$, we obtain the copula denoted by $\frac{ \Pi}{\Sigma - \Pi}$ in Nelsen (1999), where $\frac{ \Pi}{\Sigma - \Pi}(x,y)= \frac{x \,y}{x + y - x \,y}$.
[| c | c| c | c | c |c |]{} Copula & $\theta$ & $\underline{{\rm VaR}}^1_{\alpha, \theta}(X,Y)$\
Clayton $C_\theta$ & $(-1, \infty)$ & $\frac{\theta}{\theta-1}{\frac {{\alpha}^{\theta} -\alpha}{ {\alpha}^{\theta} -1 }}$\
Counter-monotonic $W$ & $-1$ & $\frac{1+ \alpha}{2}$\
Independent $\Pi$ & $0$ & ${\frac {\alpha-1}{\ln \alpha }}$\
$\frac{ \Pi}{\Sigma - \Pi} $ & $1$ & ${\frac {\alpha\,\ln \alpha }{\alpha-1}}$\
Comonotonic $M$ & $\infty$ & $ \alpha$\
Interestingly, one can readily show that $\frac{\partial \underline{{\rm VaR}}^1_{\alpha, \theta}}{\partial \alpha} \geq 0$ and $\frac{\partial \underline{{\rm VaR}}^1_{\alpha, \theta}}{\partial \theta} \leq 0$, for $\theta \geq -1$ and $\alpha \in (0,1)$. This proves that, for Clayton-distributed random couples, the components of our multivariate are increasing functions of the risk level $\alpha$ and decreasing functions of the dependence parameter $\theta$. Note also that the multivariate in the comonotonic case corresponds to the vector composed of the univariate VaR associated with each component. These properties are illustrated in Figure \[VAR bivariate Clayton grafico\] (left) where $\underline{{\rm VaR}}^1_{\alpha, \theta}(X,Y)$ is plotted as a function of the risk level $\alpha$ for different values of the parameter $\theta$. Observe that an increase of the dependence parameter $\theta$ tends to lower the up to the perfect dependence case where $\underline{{\rm VaR}}_{\alpha, \theta}^1(X,Y)= \underline{{\rm VaR}}_{\alpha}(X)=\alpha$. The latter empirical behaviors will be formally confirmed in next sections.\
![[ Behavior of $\underline{{\rm VaR}}^1_{\alpha, \theta}(X,Y)= \underline{{\rm VaR}}^2_{\alpha, \theta}(X,Y)$ (left) and $\overline{{\rm VaR}}^1_{\alpha, \theta}(1-X,1-Y)= \overline{{\rm VaR}}^2_{\alpha, \theta}(1-X,1-Y)$ (right) with respect to risk level $\alpha$ for different values of dependence parameter $\theta$. The random vector $(X,Y)$ follows a Clayton copula distribution with parameter $\theta$. Note that, due to Equation \[Link\_Arch\_Copula\], the two graphs are symmetric with respect to the point $(\frac{1}{2},\frac{1}{2})$]{}[]{data-label="VAR bivariate Clayton grafico"}](varClayton2){width="8.1cm"}
![[ Behavior of $\underline{{\rm VaR}}^1_{\alpha, \theta}(X,Y)= \underline{{\rm VaR}}^2_{\alpha, \theta}(X,Y)$ (left) and $\overline{{\rm VaR}}^1_{\alpha, \theta}(1-X,1-Y)= \overline{{\rm VaR}}^2_{\alpha, \theta}(1-X,1-Y)$ (right) with respect to risk level $\alpha$ for different values of dependence parameter $\theta$. The random vector $(X,Y)$ follows a Clayton copula distribution with parameter $\theta$. Note that, due to Equation \[Link\_Arch\_Copula\], the two graphs are symmetric with respect to the point $(\frac{1}{2},\frac{1}{2})$]{}[]{data-label="VAR bivariate Clayton grafico"}](varClayton2upper){width="8cm"}
In the same framework, using Equation \[Link\_Arch\_Copula\], one can readily show that $\frac{\partial \overline{{\rm VaR}}^1_{\alpha, \theta}}{\partial \alpha} \geq 0$ and $\frac{\partial \overline{{\rm VaR}}^1_{\alpha, \theta}}{\partial \theta} \geq 0$, for $\theta \geq -1$ and $\alpha \in (0,1)$. This proves that, for random couples with uniform margins and Clayton survival copula, the components of our multivariate $\overline{\VAR}$ are increasing functions both of the risk level $\alpha$ and of the dependence parameter $\theta$. Note also that the multivariate $\overline{\VAR}$ in the comonotonic case corresponds to the vector composed of the univariate VaR associated with each component. These properties are illustrated in Figure \[VAR bivariate Clayton grafico\] (right) where $\overline{{\rm VaR}}^1_{\alpha, \theta}(X,Y)$ is plotted as a function of the risk level $\alpha$ for different values of the parameter $\theta$. Observe that, contrary to the lower-orthant $\underline{\rm{VaR}}$, an increase of the dependence parameter $\theta$ tends to increase the $\overline{\VAR}$. The upper bound is represented by the perfect dependence case where $\overline{{\rm VaR}}_{\alpha, \theta}^1(X,Y)= {\rm VaR}_{\alpha}(X)=\alpha$. The latter empirical behaviors will be formally confirmed in next sections.\
\
Let $(X,Y)$ be a random vector distributed as a Ali-Mikhail-Haq copula with parameter $\theta \in [-1, 1)$. In particular, the marginal distribution of $X$ and $Y$ are uniform. Then, the distribution function $C_\theta$ of $(X,Y)$ is such that $$\label{Ali}
C_\theta(x,y)= \frac{x\,y}{1- \theta\,(1-x)(1-y)}, \quad \mbox{ for } \theta \in [-1, 1), \,\,\, \,\, (x,y) \in [0,1]^2.$$ Using Corollary \[VaR\_Archimedean\], we give in Table \[VAR different copula Ali\] analytical expressions for the first (equal to the second) component of the $\underline{\VAR}$, i.e., $\underline{{\rm VaR}}^1_{\alpha, \theta}(X,Y)$. When $\theta = 0$ we obtain the independence copula $\Pi(x,y)=x\,y$.
[| c | c| c | c | c |c |]{} Copula & $\theta$ & $\underline{{\rm VaR}}^1_{\alpha, \theta}(X,Y)$\
Ali-Mikhail-Haq copula $C_\theta$ & $[-1, 1)$ & ${\frac {\left( \theta -1
\right) \ln \left( 1-\theta(1-\alpha) \right) }{ \theta \left( \ln \left( 1-\theta(1-
\alpha) \right) -\ln \left( \alpha \right) \right)}}
$\
Independent $\Pi$ & $0$ & ${\frac {\alpha-1}{\ln \left( \alpha \right) }}$\
\
We now consider a $3$-dimensional vector $\textbf{X}= (X_1,X_2, X_3)$ with Clayton copula and parameter $ \theta > - \frac{1}{2}$ (see Remark \[theta e dimensione\]) and uniform marginals. In this case we give an analytical expression of $\underline{{\rm VaR}}^i_{\alpha, \theta}(X_1,X_2, X_3)$ for $i=1,2,3$. Results are given in Table \[VAR different copula dim 3\].\
[| c | c| c | c | c |c |]{} Copula & $\theta$ & $\underline{{\rm VaR}}^i_{\alpha, \theta}(X_1,X_2, X_3)$\
Clayton $C_\theta$ & $(-1/2, \infty)$ & $2\,{\frac {\theta\, \left( (\theta-1) {\alpha}^{2\,
\theta} + (1-2\theta) {\alpha}^{\theta} + \theta \alpha \right) }{ \left( 2\,\theta-1 \right) \left( \theta -1 \right)
\left( {\alpha}^{2\,\theta} -2\,{\alpha}^{\theta} + 1\right) }}$\
Independent $\Pi$ & $0$ & $-2\,{\frac {1-\alpha+\ln \left( \alpha \right) }{ \left( \ln \left(
\alpha \right) \right) ^{2}}}$\
As in the bivariate case above, one can readily show that $\frac{\partial \underline{{\rm VaR}}^1_{\alpha, \theta}}{\partial \alpha} \geq 0, $ $\frac{\partial \underline{{\rm VaR}}^1_{\alpha, \theta}}{\partial \theta} \leq 0$ when $\mathbf{X}$ is distributed as a $3-$dimensional Clayton copula. In addition, using Equation \[Link\_Arch\_Copula\], $\frac{\partial \overline{{\rm VaR}}^1_{\alpha, \theta}}{\partial \alpha} \geq 0$ and $\frac{\partial \overline{{\rm VaR}}^1_{\alpha, \theta}}{\partial \theta} > 0$ when $\mathbf{X}$ admits a trivariate Clayton survival copula. Then, the results obtained above in the bivariate case are confirmed also in higher dimension. These empirical behaviors will be formally confirmed in next sections.\
Comparison of univariate and multivariate VaR {#proprieta VAR bidimensionale}
---------------------------------------------
Note that, using a change of variable, each component of the multivariate VaR can be represented as an integral transformation of the associated univariate VaR. Let us denote by $F_{X_i}$ the marginal distribution functions of $X_i$ for $i=1, \ldots, d$ and by $C$ (resp. $\overline{C}$) the copula (resp. the survival copula) associated with $\textbf{X}$. Using the Sklar’s theorem we have $F(x_1,\ldots, x_d)=C(F_{X_1}(x_1), \ldots, F_{X_d}(x_d))$ (see Sklar, 1959). Then the random variables $U_i$ defined by $U_i=F_{X_i}(X_i)$, for $i=1, \ldots, d$, are uniformly distributed and their joint distribution is equal to $C$. Using these notations and since $F^{-1}_{X_i}(\gamma)= \VAR_\gamma(X_i)$, we get
$$\label{VAR XY integrale 1}
\underline{\VAR}_{\alpha}^i(\textbf{X}) = \frac{1}{K'(\alpha)}\int_{\alpha}^1 \VAR_\gamma(X_i) f_{(U_i,C(\textbf{U}))}(\gamma, \alpha) \, d\gamma,$$
for $i=1, \ldots, d,$ where $K'$ is the density of the Kendall distribution associated with copula $C$ and $f_{(U_i,C(\textbf{U}))}$ is the density function of the bivariate vector $(U_i, C(\textbf{U}))$.
As for the upper-orthant VaR, let $V_i= \overline{F}_{X_{i}}(X_i)$, for $i =1, \ldots, d$. Using these notations and since $\overline{F}^{-1}_{X_{i}}(\gamma) = \VAR_{1-\gamma}(X_i)$, we get $$\label{VAR XY integrale 1 upper}
\overline{\VAR}_{\alpha}^i(\textbf{X}) =
\frac{1}{K'_{\overline{C}}(1-\alpha)}\int_{0}^{1-\alpha} \VAR_{1-\gamma}(X_i) f_{(V_i,\overline{C}(\textbf{V}))}(\gamma, 1-\alpha) \, d\gamma,$$
for $i=1, \ldots, d$, where $K'_{\overline{C}}$ the density of the Kendall distribution associated with the survival copula $\overline{C}$ and $f_{(V_i,\overline{C}(\textbf{V}))}$ is the density function of the bivariate vector $(V_i, \overline{C}(\textbf{V}))$.
Remark that the bounds of integration in and derive from the geometrical properties of the considered level curve, i.e., $\partial \underline{L}(\alpha)$ (resp. $\partial \overline{L}(\alpha)$) is inferiorly (resp. superiorly) bounded by the marginal univariate quantile functions at level $\alpha$.
The following proposition allows us to compare the multivariate lower-orthant and upper-orthant *Value-at-Risk* with the corresponding univariate VaR.
\[VAR XY respect to the univariate VaR\] Consider a random vector $\textbf{X}$ satisfying the regularity conditions. Assume that its multivariate distribution function $F$ is quasi concave[^6]. Then, for all $\, \alpha \in (0,1),$ the following inequalities hold
$$\label{confronto var univ1}
\overline{{\rm VaR}}^i_{\alpha}(\textbf{X}) \leq {\rm VaR}_\alpha(X_i) \leq \underline{{\rm VaR}}^i_{\alpha}(\textbf{X}),$$
for $i=1, \ldots, d$.
*Proof:* Let $\alpha \, \in \, (0,1)$. From the definition of the accumulated probability, it is easy to show that $\partial \underline{L}(\alpha)$ is inferiorly bounded by the marginal univariate quantile functions. Moreover, recall that $\underline{L}(\alpha)$ is a convex set in $\mathbb{R}^d_+$ from the quasi concavity of $F$ (see Section 2 in Tibiletti, 1995). Then, for all $\textbf{x}=(x_1,\ldots, x_d) \in \partial \underline{L}(\alpha)$, $x_1 \geq \underline{\VAR}_{\alpha}(X_1), \cdots, x_d \geq \underline{\VAR}_{\alpha}(X_d)$ and trivially, $\underline{{\rm VaR}}^i_{\alpha}(\textbf{X})$ is greater than ${\rm VaR}_\alpha(X_i)$, for $i=1, \ldots, d$. Then $\underline{{\rm VaR}}^i_{\alpha}(\textbf{X}) \geq{\rm VaR}_\alpha(X_i)$, for all $\, \alpha \in (0,1)$ and $i=1, \ldots, d$. Analogously, from the definition of the survival accumulated probability, it is easy to show that $\partial \overline{L}(\alpha)$ is superiorly bounded by the marginal univariate quantile functions at level $\alpha$. Moreover, recall that $\overline{L}(\alpha)$ is a convex set in $\mathbb{R}^d_+$. Then, for all $\textbf{x}=(x_1,\ldots, x_d) \in \partial \overline{L}(\alpha)$, $x_1 \leq {\VAR}_{\alpha}(X_1), \cdots, x_d \leq {\VAR}_{\alpha}(X_d)$ and trivially, $ \overline{{\rm VaR}}^i_{\alpha}(\textbf{X})$ is smaller than ${\rm VaR}_{\alpha}(X_i)$, for all $\, \alpha \in (0,1)$ and $i=1, \ldots, d$. Hence the result $\Box$
Proposition \[VAR XY respect to the univariate VaR\] states that the multivariate lower-orthant $\underline{{\rm VaR}}_{\alpha}(\textbf{X})$ (resp. the multivariate upper-orthant $\overline{{\rm VaR}}_{\alpha}(\textbf{X})$) is more conservative (resp. less conservative) than the vector composed of the univariate $\alpha$-*Value-at-Risk* of marginals. Furthermore, we can prove that the previous bounds in are reached for comonotonic random vectors.
\[comonotonic caseVAR XY respect to the univariate VaR\] Consider a comonotonic non-negative random vector $\textbf{X}$. Then, for all $\, \alpha \in (0,1),$ it holds that $$\overline{{\rm VaR}}^i_{\alpha}(\textbf{X}) = {\rm VaR}_\alpha(X_i) = \underline{{\rm VaR}}^i_{\alpha}(\textbf{X}),$$ for $i=1, \ldots, d$.
*Proof:* Let $\alpha \, \in \, (0,1)$. If $\textbf{X}=(X_1, \ldots, X_d)$ is a comonotonic non-negative random vector then there exists a random variable $Z$ and $d$ increasing functions $g_1, \ldots, g_d$ such that $\textbf{X}$ is equal to $(g_1(Z), \ldots, g_d(Z))$ in distribution. So the set $\{(x_1,\ldots, x_d): F(x_1,\ldots, x_d)= \alpha\}$ becomes $\{(x_1,\ldots, x_d): $ $\min\{g^{-1}_1(x_1), \ldots, g^{-1}_d(x_d)\}$ $= Q_Z(\alpha) \},$ where $Q_Z$ is the quantile function of $Z$. So, trivially, $\underline{{\rm VaR}}^i_{\alpha}(\textbf{X})= {\operatorname{\mathbb{E}}}[\,X_i\,|\,F(\textbf{X})= \alpha\,] = Q_{X_i}(\alpha)$, for $i=1, \ldots, d$ and $\overline{{\rm VaR}}^i_{\alpha}(\textbf{X}) = {\operatorname{\mathbb{E}}}[g_i(Z) | \overline{F}_\textbf{X}(\textbf{X})= 1-\alpha]= {\operatorname{\mathbb{E}}}[g_i(Z) | \overline{F}_{(Z,\ldots,Z)}(Z,\ldots,Z)= 1-\alpha]$. Since $\overline{F}_{(Z,\ldots,Z)}(u_1, \ldots, u_d)= \overline{F}_{Z}(\max_{i=1,\ldots, d} u_i),$ then $\overline{{\rm VaR}}^i_{\alpha}(\textbf{X}) = {\operatorname{\mathbb{E}}}[g_i(Z) | \overline{F}_{Z}(Z)= \alpha] = {\rm VaR}_\alpha(X_i),$ for $i=1, \ldots, d$. Hence the result. $\Box$
\[independente VAR\] For bivariate independent random couple $(X,Y)$, Equations and become respectivley $$\begin{aligned}
\underline{\VAR}_{\alpha}^1(X,Y) &=& \frac{1}{- \ln(\alpha)}\int_{\alpha}^1 \frac{\VAR_{\gamma}(X)}{\gamma} \, d\gamma, \\
\overline{\VAR}_{\alpha}^1(X,Y) &= & \frac{1}{- \ln(1-\alpha)}\int_{0}^{1-\alpha} \frac{\VAR_{1-\gamma}(X)}{\gamma} \, d\gamma ,\end{aligned}$$ then, obviously, in this case the $X$-related component only depends on the marginal behavior of $X$. For further details the reader is referred to Corollary 4.3.5 in Nelsen (1999).
Behavior of the multivariate VaR with respect to marginal distributions {#stochastic order copulas var}
-----------------------------------------------------------------------
In this section we study the behavior of our VaR measures with respect to a change in marginal distributions. Results presented below provide natural multivariate extensions of some classical results in the univariate setting (see, e.g., Denuit and Charpentier, 2004).
\[VAR invarinati con copula\] Let $\textbf{X}$ and $\textbf{Y}$ be two $d$–dimensional continuous random vectors satisfying the regularity conditions and with the same copula $C$. If $X_i \buildrel d \over = Y_i $ then it holds that $$\underline{{\rm VaR}}^i_{\alpha}(\textbf{X})= \underline{{\rm VaR}}^i_{\alpha}(\textbf{Y}), \quad \mbox{ for all } \alpha \in (0,1),$$ and $$\overline{{\rm VaR}}^i_{\alpha}(\textbf{X})= \overline{{\rm VaR}}^i_{\alpha}(\textbf{Y}), \quad \mbox{ for all } \alpha \in (0,1).$$
The proof of the previous result directly comes down from Equation and . From Proposition \[VAR invarinati con copula\], we remark that, for a fixed copula $C$, the $i$-th component $\underline{\VAR}_{\alpha}^i(\textbf{X})$ and $\overline{{\rm VaR}}^i_{\alpha}(\textbf{X})$ do not depend on marginal distributions of the other components $j$ with $j\neq i$.
In order to derive the next result, we use the definitions of stochastic orders presented in Section \[Notation\].
\[st order for VaR\] Let $\textbf{X}$ and $\textbf{Y}$ be two $d$–dimensional continuous random vectors satisfying the regularity conditions and with the same copula $C$. If $X_i \preceq_{st} Y_i$ then it holds that $$\underline{{\rm VaR}}^i_{\alpha}(\textbf{X})\leq \underline{{\rm VaR}}^i_{\alpha}(\textbf{Y}), \quad \mbox{ for all } \alpha \in (0,1),$$ and $$\overline{{\rm VaR}}^i_{\alpha}(\textbf{X})\leq \overline{{\rm VaR}}^i_{\alpha}(\textbf{Y}), \quad \mbox{ for all } \alpha \in (0,1).$$
*Proof:* The proof comes down from formulas - and Definition \[def st order\]. Furtheremore, we remark that if $X_i \preceq_{st} Y_i$ then ${F}^{-1}_{X_i}(x) \leq{F}^{-1}_{Y_i}(x)$ for all $x$, and $\overline{F}^{-1}_{X_i}(y) \leq \overline{F}^{-1}_{Y_i}(y)$ for all $y$. Hence the result. $\Box$
Note that, the result in Proposition \[st order for VaR\] is consistent with the one-dimensional setting (see Section 3.3.1 in Denuit *et al.*, 2005). Indeed, as in dimension one, an increase of marginals with respect to the first order stochastic dominance yields an increase in the corresponding components of ${\rm VaR}_{\alpha}(\textbf{X})$.
As a result, in an economy with several interconnected financial institutions, capital required for one particular institution is affected by its own marginal risk. But, for a fixed dependence structure, the solvency capital required for this specific institution does not depend on marginal risks bearing by the others. Then, our multivariate VaR implies a “fair” allocation of solvency capital with respect to individual risk-taking behavior. In other words, individual financial institutions may not have to pay more for risky business activities undertook by the others.
Behavior of multivariate VaR with respect to the dependence structure {#dependence structure copulas}
---------------------------------------------------------------------
In this section we study the behavior of our VaR measures with respect to a variation of the dependence structure, with unchanged marginal distributions.
\[st order for VaR avec U conditionel\] Let $\textbf{X}$ and $\textbf{X}^*$ be two $d$–dimensional continuous random vectors satisfying the regularity conditions and with the same margins $F_{X_i}$ and $F_{X^*_i}$, for $i=1, \ldots, d$, and let $C$ [(]{}resp. $C^*$[)]{} denote the copula function associated with $\textbf{X}$ [(]{}resp. $\textbf{X}^*$[)]{} and $\overline{C}$ [(]{}resp. $\overline{C}^*$[)]{} the survival copula function associated with $\textbf{X}$ [(]{}resp. $\textbf{X}^*$[)]{}.
Let $U_i=F_{X_i}(X_i)$, $U_{i}^*=F_{{X_i}^*}(X_{i}^*)$, $\textbf{U}= (U_1, \ldots, U_d)$ and $\textbf{U}^*= (U_1^*, \ldots, U_d^*)$.
If $\,\,[U_i | C(\textbf{U}) = \alpha] \preceq_{st} [U_{i}^* | C^*(\textbf{U}^*) = \alpha]\,\, \mbox{ then }\,\, \underline{{\rm VaR}}^i_{\alpha}(\textbf{X})\leq \underline{{\rm VaR}}^i_{\alpha}(\textbf{X}^*).$
Let $V_i=\overline{F}_{X_i}(X_i)$, $V_{i}^*=\overline{F}_{{X_i}^*}(X_{i}^*)$, $\textbf{V}= (V_1, \ldots, V_d)$ and $\textbf{V}^*= (V_1^*, \ldots, V_d^*)$.
If $\,\,[V_i | \overline{C}(\textbf{V}) = 1-\alpha] \preceq_{st} [V_{i}^* | \overline{C}^*(\textbf{V}^*) = 1-\alpha]\,\, \mbox{ then }\,\, \overline{{\rm VaR}}^i_{\alpha}(\textbf{X}) \geq \overline{{\rm VaR}}^i_{\alpha}(\textbf{X}^*).$
*Proof:* Let $U_1 \buildrel d \over = [U_i | C(\textbf{U}) = \alpha]$ and $U_2 \buildrel d \over = [U_{i}^{*} | C^*(\textbf{U}^*) = \alpha]$. We recall that $U_1 \preceq_{st} U_2$ if and only if ${\operatorname{\mathbb{E}}}[f(U_1)]\leq {\operatorname{\mathbb{E}}}[f(U_2)]$, for all non-decreasing function $f$ such that the expectations exist (see Denuit *et al.*, 2005; Proposition 3.3.14). We now choose $f(u)= F^{-1}_{X_i}(u)$, for $u \in (0,1)$. Then, we obtain
${\operatorname{\mathbb{E}}}[\, F^{-1}_{X_i}(U_i) | C(\textbf{U}) = \alpha \,] \leq {\operatorname{\mathbb{E}}}[\, F^{-1}_{X_i}(U_{i}^{*}) | C^*(\textbf{U}^*) = \alpha \,]$,
But the right-hand side of the previous inequality is equal to ${\operatorname{\mathbb{E}}}[\, F^{-1}_{X^{*}_{i}}(U_{i}^{*}) | C^*(\textbf{U}^*) = \alpha \,]$ since $X_i$ and $X_i^*$ have the same distribution. Finally, from formula we obtain $\underline{{\rm VaR}}^i_{\alpha}(\textbf{X})\leq \underline{{\rm VaR}}^i_{\alpha}(\textbf{X}^*)$.\
Let now $V_1 \buildrel d \over = [V_i | \overline{C}(\textbf{V}) = 1-\alpha]$ and $V_2 \buildrel d \over = [V_{i}^{*} | \overline{C}^*(\textbf{V}^*) = 1-\alpha]$. We now choose the non-decreasing function $f(u)= - \overline{F}^{-1}_{X_i}(u)$, for $u \in (0,1)$. Since $X_i$ and $X_i^*$ have the same distribution, we obtain
${\operatorname{\mathbb{E}}}[\, \overline{F}^{-1}_{X_i}(V_i) | \overline{C}(\textbf{V}) = 1-\alpha \,] \geq {\operatorname{\mathbb{E}}}[\, \overline{F}^{-1}_{X_i}(V_i^{*}) | \overline{C}^*(\textbf{V}^*) = 1-\alpha \,]$,
Hence the result. $\Box$\
We now provide an illustration of Proposition \[st order for VaR avec U conditionel\] in the case of $d-$dimensional Archimedean copulas.
\[dependence\_impact\_Archi\] Consider a $d$–dimensional random vector $\textbf{X}$, satisfying the regularity conditions, with marginal distributions $F_{X_i}$, for $i=1, \ldots, d$, copula $C$ and survival copula $\overline{C}$.
: If $C$ belongs to one of the $d$-dimensional family of Archimedean copulas introduced in Table \[kendall achimedeans\], an increase of the dependence parameter $\theta$ yields a decrease in each component of $\underline{{\rm VaR}}_{\alpha}(\textbf{X})$.
: If $\overline{C}$ belongs to one of the $d$-dimensional family of Archimedean copulas introduced in Table \[kendall achimedeans\], an increase of the dependence parameter $\theta$ yields an increase in each component of $\overline{{\rm VaR}}_{\alpha}(\textbf{X})$.
*Proof:* Let $C_{\theta}$ and $C_{\theta^*}$ be two Archimedean copulas of the same family with generator $\phi_{\theta}$ and $\phi_{\theta^*}$ such that $\theta\leq \theta^*$. Given Proposition \[st order for VaR avec U conditionel\], we have to check that the relation $[U_i^* | C_{\theta^*}(\textbf{U}^*) = \alpha] \preceq_{st} [U_i | C_{\theta}(\textbf{U}) = \alpha]$ holds for all $i=1,\ldots, d$ where $(U_1, \ldots, U_d)$ and $(U_1^*, \ldots, U_d^*)$ are distributed (resp.) as $C_{\theta}$ and $C_{\theta^*}$. However, using formula , we can readily prove that the previous relation can be restated as a decreasing condition on the ratio of generators $\phi_{\theta^*}$ and $\phi_{\theta}$, i.e., $$[U_i^* | C_{\theta^*}(\textbf{U}^*) = \alpha] \preceq_{st} [U_i | C_{\theta}(\textbf{U}) = \alpha] \text{ for any }\alpha\in(0,1) \iff \; \frac{\phi_{\theta^*}}{\phi_{\theta}} \text{ is a decreasing function.}$$ Eventually, we have check that, for all Archimedean family introduced in Table \[kendall achimedeans\], the function defined by $\frac{\phi_{\theta^*}}{\phi_{\theta}}$ is indeed decreasing when $\theta \leq \theta^*$. We immediately obtain from Proposition \[st order for VaR avec U conditionel\] that each component of $\underline{{\rm VaR}}_{\alpha}(\textbf{X})$ is a decreasing function of $\theta$. The proof of the second statement of Corollary \[dependence\_impact\_Archi\] follows trivially usign the same arguments. $\Box$\
From Corollary \[dependence\_impact\_Archi\] the multivariate $\overline{\VAR}$ (resp. $\underline{\VAR}$) for copulas in Table \[kendall achimedeans\] is increasing (resp. decreasing) with respect to the dependence parameter $\theta$ (coordinate by coordinate). In particular, this means that, in the case of Archimedean copula, limit behaviors of dependence parameters are associated with bounds for our multivariate risk measure. For instance, let $(X,Y)$ be a bivariate random vector with a Clayton dependence structure and fixed margins and $(\tilde{X},\tilde{Y})$ be a bivariate random vector with a Clayton survival copula and the same margins as $(X,Y)$. If we denote by $\underline{{\rm VaR}}^1_{(\alpha, \theta)}(X, Y)$ (resp. $\overline{{\rm VaR}}^1_{(\alpha, \theta)}(\tilde{X},\tilde{Y})$) the first component of the lower-orthant VaR (resp. upper-orthant VaR) when the dependence parameter is equal to $\theta$, then the following comparison result holds for all ${\alpha \, \in \,(0,1)}$ and all $\theta \in (-1, \infty)$: $$\begin{aligned}
&& \overline{{\rm VaR}}^1_{(\alpha, -1)}(\tilde{X},\tilde{Y}) \leq \overline{{\rm VaR}}^1_{(\alpha, \theta)}(\tilde{X},\tilde{Y}) \leq \overline{{\rm VaR}}^1_{(\alpha, +\infty)}(\tilde{X},\tilde{Y})\\
&=& \underline{{\rm VaR}}^1_{(\alpha, +\infty)}(X, Y) \leq \underline{{\rm VaR}}^1_{(\alpha, \theta)}(X, Y) \leq \underline{{\rm VaR}}^1_{(\alpha, -1)}(X, Y). \end{aligned}$$ Note that the upper bound corresponds to comonotonic random variables, so that $\overline{{\rm VaR}}^1_{(\alpha, + \infty)}(X, Y)$ $=$ $\underline{{\rm VaR}}^1_{(\alpha, + \infty)}(X, Y)$ $=$ ${\rm VaR}_{\alpha}(X)= \alpha$, for a random vector ($X, Y$) with uniform marginal distributions.
Behavior of multivariate VaR with respect to risk level {#proprieta VAR PRD}
-------------------------------------------------------
In order to study the behavior of the multivariate lower-orthant *Value-at-Risk* with respect to risk level $\alpha$, we need to introduce the *positive regression dependence* concept. For a bivariate random vector $(X,Y)$ we mean by positive dependence that $X$ and $Y$ are likely to be large or to be small together. An excellent presentation of positive dependence concepts can be found in Chapter 2 of the book by Joe (1997). The positive dependence concept that will be used in the sequel has been called *positive regression dependence* (PRD) by Lehmann (1966) but most of the authors use the term *stochastically increasing* (SI) (see Nelsen, 1999; Section 5.2.3).
\[PRD\]
A bivariate random vector ${(X,Y)}$ is said to admit positive regression dependence with respect to $X$, [PRD(]{}$Y | X$[)]{}, if $$\label{PRD formula}
[Y | X=x_1] \preceq_{st} [Y | X=x_2],\quad \forall \,x_1 \leq x_2.$$
Clearly condition in is a positive dependence notion (see Section 2.1.2 in Joe, 1997). From Definition \[PRD\], it is straightforward to derive the following result.
\[PRD Prop\] Consider a $d$–dimensional random vector $\textbf{X}$, satisfying the regularity conditions, with marginal distributions $F_{X_i}$, for $i=1, \ldots, d$, copula $C$ and survival copula $\overline{C}$. Let $U_i=F_{X_i}(X_i)$, $\textbf{U}= (U_1, \ldots, U_d)$, $V_i= \overline F_{X_i}(X_i)$ and $\textbf{V}= (V_1, \ldots, V_d)$. Then it holds that :
- If $(U_i,C(\textbf{U}))$ is $\mbox{PRD}(U_i| C(\textbf{U}))$ then $\underline{{\rm VaR}}_{\alpha}^i(\textbf{X})$ is a non-decreasing function of $\alpha$.\
- If $(V_i, \overline{C}(\textbf{V}))$ is $\mbox{PRD}(V_i | \overline{C}(\textbf{V}))$ then $\overline{{\rm VaR}}_{\alpha}^i(\textbf{X})$ is a non-decreasing function of $\alpha$.
*Proof:* If $\alpha_1\leq \alpha_2$, we have $[U_i | C(\textbf{U})=\alpha_1] \preceq_{st} [U_i | C(\textbf{U})=\alpha_2]$ and $[V_i | \overline{C}(\textbf{V})=1-\alpha_2] \preceq_{st} [V_i | \overline{C}(\textbf{V})=1-\alpha_1]$. As in the proof of Proposition \[st order for VaR avec U conditionel\],
${\operatorname{\mathbb{E}}}[\, F^{-1}_{X_i}(U_i) | C(\textbf{U}) = \alpha_1\,] \leq {\operatorname{\mathbb{E}}}[\, F^{-1}_{X_i}(U_i) | C(\textbf{U}) = \alpha_2 \,]$.
and
${\operatorname{\mathbb{E}}}[\, \overline{F}^{-1}_{X_i}(V_i) | \overline{C}(\textbf{V}) = 1-\alpha_1\,] \leq {\operatorname{\mathbb{E}}}[\, \overline{F}^{-1}_{X_i}(V_i) | \overline{C}(\textbf{V}) = 1-\alpha_2 \,]$.
Then $\underline{\VAR}_{\alpha_1}^i(\textbf{X})\leq \underline{\VAR}_{\alpha_2}^i(\textbf{X})$ and $\overline{\VAR}_{\alpha_1}^i(\textbf{X})\leq \overline{\VAR}_{\alpha_2}^i(\textbf{X})$, for any $\alpha_1 \leq \alpha_2$ which proves that $\underline{\VAR}_{\alpha}^i(\textbf{X})$ and $\overline{\VAR}_{\alpha}^i(\textbf{X})$ are non-decreasing functions of $\alpha$. $\Box$
Note that behavior of the multivariate with respect to a change in the risk level does not depend on marginal distributions of $\textbf{X}$.
The following result proves that assumptions of Proposition \[PRD Prop\] are satisfied in the large class of $d$-dimensional Archimedean copulas.
\[PRD Prop Archimediennes\] Consider a $d$–dimensional random vector $\textbf{X}$, satisfying the regularity conditions, with marginal distributions $F_{X_i}$, for $i=1, \ldots, d$, copula $C$ and survival copula $\overline{C}$.
If $C$ is a $d$-dimensional Archimedean copula, then $\underline{{\rm VaR}}_{\alpha}^i(\textbf{X})$ is a non-decreasing function of $\alpha$.
If $\overline{C}$ is a $d$-dimensional Archimedean copula, then $\overline{{\rm VaR}}_{\alpha}^i(\textbf{X})$ is a non-decreasing function of $\alpha$.
*Proof:* Let $U_i=F_{X_i}(X_i)$, $\textbf{U}= (U_1, \ldots, U_d)$, $V_i= \overline F_{X_i}(X_i)$ and $\textbf{V}= (V_1, \ldots, V_d)$. If $C$ is the copula of $X$, then $\textbf{U}$ is distributed as $C$ and if $C$ is Archimedean, ${\mathbb{P}}[U_i > u\, | \, C(\textbf{U}) = \alpha]$ is a non-decreasing function of $\alpha$ from formula . In addition, if $\overline{C}$ is the survival copula of $X$, then $\textbf{V}$ is distributed as $\overline{C}$ and if $\overline{C}$ is Archimedean, ${\mathbb{P}}[V_i > u\, | \, \overline{C}(\textbf{V}) = \alpha]$ is a non-decreasing function of $\alpha$ from the same argument. The result then derives from Proposition \[PRD Prop\]. $\Box$
Conclusion and perspectives {#Conclusions .unnumbered}
===========================
In this paper, we proposed two multivariate extensions of the classical Value-at-Risk for continuous random vectors. As in the Embrechts and Puccetti (2006)’s approach, the introduced risk measures are based on multivariate generalization of quantiles but they are able to quantify risks in a much more parsimonious and synthetic way: the risk of a $d$-dimensional portfolio is evaluated by a point in ${\mathbb{R}}^d_+$. The proposed multivariate risk measures may be useful for some applications where risks are heterogeneous in nature or because they cannot be diversify away by an aggregation procedure.\
We analyzed our multivariate risk measures in several directions. Interestingly, we showed that many properties satisfied by the univariate VaR expand to the two proposed multivariate versions under some conditions. In particular, the *lower-orthant* VaR and the *upper-orthant* VaR both satisfy the positive homogeneity and the translation invariance property which are parts of the classical axiomatic properties of Artzner *et al.* (1999). Using the theory of stochastic ordering, we also analyzed the effect of some risk perturbations on these measures. In the same vein as for the univariate VaR, we proved that an increase of marginal risks yield an increase of the multivariate VaR. We also gave the condition under which an increase of the risk level tends to increase components of the proposed multivariate extensions and we show that these conditions are satisfied for $d$-dimensional Archimedean copulas. We also study the effect of dependence between risks on individual contribution of the multivariate VaR and we prove that for different families of Archimedean copulas, an increase of the dependence parameter tends to lower the components of the *lower-orthant* VaR whereas it widens the components of the *upper-orthant* VaR. At the extreme case where risks are perfectly dependent or comonotonic, our multivariate risk measures are equal to the vector composed of univariate risk measures associated with each component.\
Due to the fact that the Kendall distribution is not known analytically for elliptical random vectors, it is still an open question whether components of our proposed measures are increasing with respect to the risk level for such dependence structures. However, numerical experiments in the case of Gaussian copulas support this hypothesis. More generally, the extension of the McNeil and Nešlehová’s representation (see Proposition \[McNeil\_Neslehova\]) for a generic copula $C$ and the study of the behavior of distribution $[U | C(\textbf{U})=\alpha]$, with respect to $\alpha$, are potential improvements to this paper that will be investigated in a future work.\
In a future perspective, it should also be interesting to discuss the extensions of our risk measures to the case of discrete distribution functions, using “discrete level sets” as multivariate definitions of quantiles. For further details the reader is referred, for instance, to Laurent (2003). Another subject of future research should be to introduce a similar multivariate extension but for *Conditional-Tail-Expectation* and compare the proposed VaR and CTE measures with existing multivariate generalizations of risk measures, both theoretically and experimentally. An article is in preparation in this sense.
**Acknowledgements:** The authors thank an anonymous referee for constructive remarks and valuable suggestions to improve the paper. The authors thank Véronique Maume-Deschamps and Clémentine Prieur for their comments and help and Didier Rulli[è]{}re for fruitful discussions. This work has been partially supported by the French research national agency (ANR) under the reference ANR-08BLAN-0314-01. Part of this work also benefit from the support of the MIRACCLE-GICC project.
**References:**
[^1]: Université de Lyon, Université Lyon $1$, ISFA, Laboratoire SAF, $50$ avenue Tony Garnier, $69366$ Lyon, France, Tel.: $+33 4 37 28 74 39$, [email protected], http://www.acousin.net/.
[^2]: CNAM, Paris, Département IMATH, 292 rue Saint-Martin, Paris Cedex 03, France. elena.di\[email protected]. http://isfaserveur.univ-lyon1.fr/elena.dibernardino/.
[^3]: Note that $\phi$ generates a $d-$dimensional Archimedean copula if and only if its inverse $\phi^{-1}$ is a $d-$ monotone on $[0, \infty)$ (see Theorem 2.2 in McNeil and Nešlehová, 2009).
[^4]: We restrict ourselves to $\mathbb{R}^d_+$ because, in our applications, components of $d-$dimensional vectors correspond to random losses and are then valued in $\mathbb{R}_+$.
[^5]: A function $F(x_1,\ldots, x_d)$ is partially increasing on $\mathbb{R}^{d}_{+} \setminus (0,\ldots, 0)$ if the functions of one variable $g(\cdot)= F(x_1, \ldots, x_{j-1}, \cdot, x_{j+1}, \ldots, x_d)$ are increasing. About properties of partially increasing multivariate distribution functions we refer the interested reader to Rossi (1973), Tibiletti (1991).
[^6]: A function $F$ is quasi concave if the upper level sets of $F$ are convex sets. Tibiletti (1995) points out families of distribution functions which satisfy the property of quasi concavity. For instance, multivariate elliptical distributions and Archimedean copulas are quasi concave functions (see Theorem 4.3.2 in Nelsen, 1999 for proof in dimension $2$; Proposition 3 in Tibiletti, 1995, for proof in dimension $d$).
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'This article is devoted to completing some aspects of the classical Cauchy-Lipschitz (or Picard-Lindelöf) theory for general nonlinear systems posed on time scales, that are closed subsets of the set of real numbers. Partial results do exist but do not cover the framework of general dynamics on time-scales encountered e.g. in applications to control theory. In the present work, we first introduce the notion of absolutely continuous solution for shifted and non shifted $\DD$-Cauchy problems, and then the notion of a maximal solution. We state and prove a Cauchy-Lipschitz theorem, providing existence and uniqueness of the maximal solution of a given $\DD$-Cauchy problem under suitable assumptions like regressivity and local Lipschitz continuity, and discuss some related issues like the behavior of maximal solutions at terminal points.'
author:
- 'Loïc Bourdin[^1] , Emmanuel Trélat[^2]'
title: 'General Cauchy-Lipschitz theory for shifted and non shifted $\DD$-Cauchy problems on time scales'
---
**Keywords:** Time scale; Cauchy-Lipschitz (Picard-Lindelöf) theory; existence; uniqueness; shifted problems.
**AMS Classification:** 34N99; 34G20; 39A13; 39A12.
Introduction {#section0}
============
The *time scale* theory was introduced by S. Hilger in his PhD thesis [@hilg] in 1988 in order to unify discrete and continuous analysis, with the general idea of extending classical theories on an arbitrary non empty *closed* subset $\T$ of $\R$. Such a closed subset $\T$ is called a *time scale*. The objective is to establish the validity of some results both in the continuous case $\T =\R$ and in the purely discrete case $\T =\N$, but also to treat more general models of processes involving both continuous and discrete time elements. We refer the reader e.g. to [@gama; @may] where the authors study a seasonally breeding population whose generations do not overlap or to [@ati] for applications to economy. By considering $\T = \{ 0 \} \cup \lambda^\N$ with $0 < \lambda < 1$, time scale concept also allows to cover quantum calculus [@kac]. Since S. Hilger defined the $\DD$-derivative and the $\DD$-integral on a time scale, many authors have extended to time scales various results from the continuous or discrete standard calculus theory. We refer the reader to the surveys [@agar2; @agar3; @bohn; @bohn3]. However, due to the recency of the field, the basic nonlinear theory is yet to be developed and refined.
Some Cauchy-Lipschitz (Picard-Lindelöf) type results on time scales are provided in [@bohn; @cich2; @hilg2; @kaym; @kubi; @laks] where the authors prove the existence and uniqueness of solutions for $\DD$-Cauchy problems of the form: $$\label{introcp}
q^\DD = f(q,t), \quad q(t_0) = q_0,$$ where $t_0 \in \T$. Note that papers are devoted to $\DD$-Cauchy problems with parameter in [@hils4] and with time delays in [@karp]. Many authors are also interested in shifted $\DD$-Cauchy problems $$\label{introcps}
q^\DD = f(q^\sigma,t),\quad q(t_0) = q_0,$$ where $q^\sigma = q \circ \sigma$ (see further for the precise definitions of these notions). Such shifted problems are often used as models in the existing literature (see e.g. [@torr8; @hils2; @torr7], [@hils3 Remark 3.9] and [@hils4 Remark 3.6]), because they emerge in adjoint equations accordingly to the *shifted* Leibniz formula [@bohn] $$(q_1 q_2)^\DD = q_1^\DD q_2^\sigma + q_1 q_2^\DD = q_1^\DD q_2 + q_1^\sigma q_2^\DD.$$ Nevertheless, to the best of our knowledge, there does not exist a general Cauchy-Lipschitz theory on time scales that is fully complete in order to be applied to problems arising for example in control theory[^3].
Let us recall briefly the bibliographical context on the Cauchy-Lipschitz theory on time scales. The first result on $\DD$-Cauchy problems is due to S. Hilger in [@hilg2 Paragraph 5], who derived the existence and uniqueness of $\CC^1_\mathrm{rd}$-solutions for continuous dynamics. This framework is not suitable for general control problems where controls are measurable functions that have discontinuities in general. Note that similar frameworks and results are provided in [@bohn Paragraph 8.2], in [@kaym; @laks; @tisd] and references therein. In [@cich2; @kubi], the authors respectively treat weak continuous and Carathéodory dynamics living in a general Banach space. Note that they only treat the non shifted case where $q_0$ is an initial condition, that is, solutions are only defined for $t \geq t_0$. In view of considering adjoint equations, it is of interest to study backward $\DD$-Cauchy problems where $q_0$ is a final condition, for which solutions are def ined for $t \leq t_0$. As is very well known in time scale calculus, the solvability of such backward non shifted $\DD$-Cauchy problems requires a *regressivity* assumption on the dynamics (see e.g. [@bohn; @hilg2] and [@hils3 Remark 3.8]). This important issue is not addressed in these two articles. Another issue which is not addressed is the fact that the usual Cauchy-Lipschitz theory treats Cauchy problems constraining the solutions to take values in an open subset $\Omega$ of $\R^n$ (see e.g. [@codd; @smal]). Finally, up to our knowledge, the notion of extension of a solution on time scales, and the behavior of the maximal solution at terminal points, have not been studied. Similarly, we are not aware of articles treating both shifted and non shifted general nonlinear $\DD$-Cauchy problems.
This article is thus devoted to fill an existing gap of the literature, and to provide a general Cauchy-Lipschitz theory on time scales generalizing the basic notions and results of the classical continuous theory surveyed e.g. in [@codd; @smal]. Precisely, we first introduce the notion of an *absolutely continuous solution*. Then we define the concept of *extension* of a solution, and of *maximal* and *global* solutions in the time scale context. We establish a general version of the Cauchy-Lipschitz theorem (existence and uniqueness of the maximal solution, also referred to as Picard-Lindelöf theorem) for dynamics posed on a time scale, under regressivity and local Lipschitz continuity assumptions, for shifted and non shifted general nonlinear $\DD$-Cauchy problems in the following framework:
- $f$ is a general $\DD$-Carathéodory function, where $\DD$-measure $\mu_\DD$ on a time scale $\T$ is defined in terms of Carathéodory extension in [@bohn3 Chapter 5];
- $q_0$ is not necessarily an initial or a final condition;
- the solutions take their values in an open subset $\Omega$ of $\R^n$.
We also investigate the globality feature of the maximal solution. Our results are established first for general non shifted $\DD$-Cauchy problems and then for shifted ones .
Our study uses the work of A. Cabada and D. Vivero in [@caba2], who proved a criterion for absolutely continuous functions written as the $\DD$-integral of their $\DD$-derivatives. Their result allows us to give a $\DD$-integral characterization of the solutions of $\DD$-Cauchy problems which is instrumental in our proofs.
Notice that analogous results on $\nabla$-Cauchy problems ($\rho$-shifted or not) can be derived in a similar way.
The article is structured as follows. Section \[section1\] is devoted to recall basic notions of time scale calculus. In Section \[section2\], we define the notions of a solution, of an extension of a solution, of a maximal and a global solution for general non shifted $\DD$-Cauchy problems. Under suitable assumptions on the dynamics, we establish a Cauchy-Lipschitz theorem and then investigate the behavior of the maximal solution at its terminal points. Section \[section3\] is devoted to establish similar results for *shifted* $\DD$-Cauchy problems.
Preliminaries on time scale calculus {#section1}
====================================
In this section, we recall basic results in time scale calculus. The first part concerns the structure of time scales and the notion of $\DD$-differentiability (see [@bohn]). The second part concerns the $\DD$-Lebesgue measure defined in terms of Carathéodory extension (see [@bohn3; @guse]) and surveys results on $\DD$-integrability proved in [@caba]. The last part gathers properties of absolutely continuous functions borrowed from [@caba2].
Let $n\in\N^*$. Throughout, the notation $\Vert \cdot \Vert$ stands for the Euclidean norm of $\R^n$. For every $x \in \R^n$ and every $R \geq 0$, the notation $\overline{B}(x,R)$ stands for the closed ball of $\R^n$ centered at $x$ and with radius $R$.
Time scale and $\DD$-differentiability {#section11}
--------------------------------------
Let $\T$ be a time scale, that is, a closed subset of $\R$. We assume that $\mathrm{card}(\T) \geq 2$. For every $A \subset \R$, we denote $A_\T = A \cap \T$. An interval of $\T$ is defined by $I_\T$ where $I$ is an interval of $\R$.
The backward and forward jump operators $\rho,\sigma:\T\rightarrow\T$ are respectively defined by $$\begin{split}
\rho (t) &= \sup \{ s \in \T\ \vert\ s < t \},\\
\sigma (t) &= \inf \{ s \in \T\ \vert\ s > t \},
\end{split}$$ for every $t \in \T$, where $\rho (\min \T) = \min \T$ (resp. $\sigma(\max \T) = \max \T$) whenever $\T$ admits a minimum (resp. a maximum).
A point $t \in \T$ is said to be a left-dense (respectively, left-scattered, right-dense or right-scattered) point of $\T$ if $\rho (t) = t$ (respectively, $\rho (t) < t$, $\sigma (t) = t$ or $\sigma (t) > t$). The graininess function $\mu:\T\rightarrow\R^+$ is defined by $\mu(t) = \sigma (t) -t$ for every $t \in \T$.
We set $\T^\kappa = \T \backslash \{ \max \T \}$ whenever $\T$ admits a left-scattered maximum, and $\T^\kappa = \T$ otherwise. A function $q:\T\rightarrow\R^n$ is said to be $\DD$-differentiable at $t \in \T^\kappa$ if the limit $$q^\DD (t) = \lim\limits_{\substack{s \to t \\ s \in \T}} \dfrac{q^\sigma (t) -q(s)}{\sigma (t) -s}$$ exists in $\R^n$, where $q^\sigma = q \circ \sigma$. We recall the following well known results (see [@bohn]):
- if $t \in \T^\kappa$ is a right-dense point of $\T$, then $q$ is $\DD$-differentiable at $t$ if and only if the limit $$q^\DD (t) = \lim\limits_{\substack{s \to t \\ s \in \T}} \dfrac{q(t)-q(s)}{t-s}$$ exists in $\R^n$;
- if $t \in \T^\kappa$ is a right-scattered point of $\T$ and if $q$ is continuous at $t$, then $q$ is $\DD$-differentiable at $t$, and $$q^\DD (t) = \dfrac{q^\sigma(t) - q(t)}{\mu(t)}.$$
Lebesgue $\DD$-measure and Lebesgue $\DD$-integrability {#section12}
-------------------------------------------------------
Recall that the set of right-scattered points $\RR \subset \T$ is at most countable (see [@caba Lemma 3.1]).
Let $\mu_\DD$ be the Lebesgue $\DD$-measure on $\T$ defined in terms of Carathéodory extension in [@bohn3 Chapter 5]. We also refer the reader to [@agar; @caba; @guse] for more details on the $\mu_\DD$-measure theory. In particular, for all elements $a,b$ of $\T$ such that $a \leq b $, one has $\mu_\DD ([a,b[_\T) = b-a$. Recall that $A \subset \T$ is a $\mu_\DD$-measurable set of $\T$ if and only if $A$ is an usual $\mu_L$-measurable set of $\R$, where $\mu_L$ denotes the usual Lebesgue measure (see [@caba Proposition 3.1]). Moreover, if $A \subset \T \backslash \{ \sup \T \}$, then $$\mu_\DD ( A ) = \mu_L (A) + \di \sum_{r \in A \cap \RR} \mu (r).$$ Let $A \subset \T$. A property is said to hold $\DD$-almost everywhere (shortly $\DD$-a.e.) on $A$ if it holds for every $t \in A \backslash A_0$, where $A_0 \subset A$ is some $\mu_\DD$-measurable subset of $\T$ satisfying $\mu_\DD (A_0) = 0$. In particular, since $\mu_\DD (\{ r \}) = \mu (r) > 0$ for every $r \in \RR$, we conclude that if a property holds $\DD$-a.e. on $A$, then it holds for every $r \in A \cap \RR$.
Let $A \subset \T \backslash \{ \sup \T \}$ be a $\mu_\DD$-measurable set of $\T$. Consider a function $q$ defined $\DD$-a.e. on $A$ with values in $\R^n$. Let $\tilde{A}=A \cup ]r,\sigma(r)[_{r \in A \cap \RR}$, and let $\tilde{q}$ be the extension of $q$ defined $\mu_L$-a.e. on $\tilde{A}$ by $$\tilde{q} (t) = \left\lbrace \begin{array}{rcl}
q(t) & \textrm{if} & t \in A \\
q(r) & \textrm{if} & t \in ]r,\sigma(r)[ \ \textrm{for every} \; r \in A \cap \RR.
\end{array} \right.$$ We recall that $q$ is $\mu_\DD$-measurable on $A$ if and only if $\tilde{q}$ is $\mu_L$-measurable on $\tilde{A}$ (see [@caba Proposition 4.1]).
The functional space $\L^\infty_\T (A,\R^n)$ is the set of all functions $q$ defined $\DD$-a.e. on $A$, with values in $\R^n$, that are $\mu_\DD$-measurable on $A$ and such that $$\operatorname*{sup\,ess}\limits_{\tau \in A} \Vert q(\tau) \Vert < + \infty.$$ Endowed with the norm $\Vert q \Vert_{\L^\infty_\T (A)} = \operatorname*{sup\,ess}\limits_{\tau \in A} \Vert q(\tau) \Vert$, it is a Banach space (see [@agar Theorem 2.5]). The functional space $\L^1_\T (A,\R^n)$ is the set of all functions $q$ defined $\DD$-a.e. on $A$, with values in $\R^n$, that are $\mu_\DD$-measurable on $A$ and such that $$\int_{A} \Vert q(\tau) \Vert \; \DD \tau < + \infty.$$ Endowed with the norm $\Vert q \Vert_{\L^1_\T (A,\R^n)} = \int_{A} \Vert q(\tau) \Vert \; \DD \tau$, it is a Banach space (see [@agar Theorem 2.5]). Recall that if $q \in \L^1_\T (A,\R^n)$ then $$\int_{A} q(\tau) \; \DD \tau = \int_{\tilde{A}} \tilde{q}(\tau) \; d\tau = \di \int_{A} q(\tau) \; d\tau + \sum_{r \in A \cap \RR} \mu (r) q(r)$$ (see [@caba Theorems 5.1 and 5.2]). Note that if $A$ is bounded then $\L^\infty_\T (A,\R^n) \subset \L^1_\T (A,\R^n)$.
Properties of absolutely continuous functions {#section13}
---------------------------------------------
Let $a$ and $b$ be two elements of $\T$ such that $a < b$. Let $\CC ([a,b]_\T,\R^n)$ denote the space of continuous functions defined on $[a,b]_\T$ with values in $\R^n$. Endowed with its usual norm $\Vert \cdot \Vert_\infty$, it is a Banach space. Let $\AC ([a,b]_\T,\R^n)$ denote the subspace of absolutely continuous functions. We recall the two following results.
\[prop13-1\] Let $t_0 \in [a,b]_\T$ and $q:[a,b]_\T\rightarrow\R^n$. Then $q \in \AC([a,b]_\T,\R^n)$ if and only if the two following conditions are satisfied:
1. $q$ is $\DD$-differentiable $\DD$-a.e. on $[a,b[_\T$ and $q^\DD \in \L^1_\T ([a,b[_\T,\R^n)$;
2. For every $t \in [a,b]_\T$, there holds $$q(t) = q(t_0) + \int_{[t_0,t[_\T} q^\DD (\tau) \; \DD \tau$$ whenever $t \geq t_0$, and $$q(t) = q(t_0) - \int_{[t,t_0[_\T} q^\DD (\tau) \; \DD \tau$$ whenever $ t \leq t_0$.
This result can be easily derived from [@caba2 Theorem 4.1]. By combining Proposition \[prop13-1\] and the usual Lebesgue’s point theory in $\R$, we infer the following result (see also [@zhan3] for a similar result).
\[prop13-2\] Let $t_0 \in [a,b]_\T$ and $q \in \L^1_\T ([a,b[_\T,\R^n)$. Let $Q$ be the function defined on $[a,b]_\T$ by $$Q(t) = \int_{[t_0,t[_\T} q (\tau) \; \DD \tau$$ if $ t \geq t_0$, and by $$Q(t) = - \int_{[t,t_0[_\T} q (\tau) \; \DD \tau$$ if $t \leq t_0$. Then $Q \in \AC([a,b]_\T)$ and $Q^\DD = q$ $\DD$-a.e. on $[a,b[_\T$.
General non shifted $\DD$-Cauchy problem {#section2}
========================================
Throughout this section we consider the general non shifted $\DD$-Cauchy problem $$\mathrm{(\DD\text{-}CP)} \begin{split}
q^\DD(t) &= f (q(t),t), \\
q(t_0)&=q_0,
\end{split}$$ where $t_0 \in \T$, $q_0 \in \Omega$, where $\Omega$ is a non empty open subset of $\R^n$, and $f:\Omega \times \T \setminus \{ \sup \T \}\rightarrow\R^n$ is a $\DD$-Carathéodory function. The notation $\KK$ stands for the set of compact subsets of $\Omega$.
Preliminaries {#section20}
-------------
In what follows it will be important to distinguish between three cases:
1. $t_0 = \min \T$;
2. $t_0 = \max \T$;
3. $t_0 \neq \inf \T$ and $t_0 \neq \sup \T$.
Indeed, the interval of definition of a solution of $\mathrm{(\DD\text{-}CP)}$ will depend on the specific case under consideration. If $t_0 = \min \T$, then a solution can only *go forward* since $]-\infty,t_0[_\T = \emptyset$. If $t_0 = \max \T$, then a solution can only *go backward*. If $t_0 \neq \inf \T$ and $t_0 \neq \sup \T$, then a solution can go *backward* and *forward*.
For all $(a,b) \in \T^2$, we say that $\ab$ if
- $a = t_0 < b$ in the case $t_0 = \min \T$;
- $a < t_0 = b$ in the case $t_0 = \max \T$;
- $a < t_0 < b$ in the case $t_0 \neq \inf \T$ and $t_0 \neq \sup \T$.
If $\ab$ then $[a,b]_\T$ is a potential interval of definition for a solution of $\mathrm{(\DD\text{-}CP)}$. Due to this difference of intervals, it is required to make different assumptions on $f$ accordingly, whence the following series of definitions.
The function $f$ is said to be *locally bounded on $\Omega \times \T \backslash \{ \sup \T \}$* if, for every $K \in \KK$, for all $(a, b) \in \T^2$ such that $ a < b$, there exists $M \geq 0$ such that $$\label{eqcondinfini}\tag{H${}_\infty$}
\Vert f(x,t) \Vert \leq M,$$ for every $x\in K$ and for $\DD$-a.e. $t \in [a,b[_\T$.\
In what follows this property will be referred to as .
The function $f$ is said to be *locally Lipschitz continuous with respect to the first variable at right-dense points* if, for every $\overline{x} \in \Omega$ and every right-dense point $\overline{t}\in\T \backslash \{ \sup \T \}$, there exist $R > 0$, $\delta > 0$ and $L \geq 0$ such that $\overline{B}(\overline{x},R) \subset \Omega$ and $\overline{t}+\delta \in \T$, and such that $$\label{eqcondrdloc-Lip}\tag{H${}^{\mathrm{rd}}_{\mathrm{loc-Lip}}$}
\Vert f(x_1,t) - f(x_2,t) \Vert \leq L \Vert x_1 - x_2 \Vert,$$ for all $ x_1, x_2 \in \overline{B}(\overline{x},R)$ and for $\DD$-a.e. $t \in [\overline{t},\overline{t}+\delta[_\T$.\
In what follows this property will be referred to as .
The function $f$ is said to be *forward $\Omega$-stable at right-scattered points* if the mapping $$\label{eqcondfsta}\tag{H${}^{\mathrm{forw}}_{\mathrm{stab}}$}
{\begin{array}[t]{lrcl}G^+(t) :&\Omega &\rightarrow &\R^n\\&x& \mapsto &x+\mu (t) f(x,t) \end{array}}$$ takes its values in $\Omega$, for every $t\in\RR$.\
In what follows this property will be referred to as .
The function $f$ is said to be *locally Lipschitz continuous with respect to the first variable at left-dense points* if, for every $\overline{x} \in \Omega$ and every left-dense point $\overline{t}\in\T \backslash \{ \inf \T \}$, there exist $R > 0$, $\delta > 0$ and $L \geq 0$ such that $\overline{B}(\overline{x},R) \subset \Omega$ and $\overline{t}-\delta \in \T$ and such that $$\label{eqcondldloc-Lip}\tag{H${}^{\mathrm{ld}}_{\mathrm{loc-Lip}}$}
\Vert f(x_1,t) - f(x_2,t) \Vert \leq L \Vert x_1 - x_2 \Vert,$$ for all $x_1, x_2 \in \overline{B}(\overline{x},R)$ and for $\DD$-a.e. $t \in [\overline{t}-\delta,\overline{t}[_\T$.\
In what follows this property will be referred to as .
The function $f$ is said to be *backward regressive at right-scattered points* if $$\label{eqcondbreg}\tag{H${}^{\mathrm{back}}_{\mathrm{regr}}$}
G^+(t) \; \text{is invertible},$$ for every $t\in\RR$.\
In what follows this property will be referred to as .
Assumption will be instrumental to provide a $\DD$-integral characterization of the solutions of $\mathrm{(\DD\text{-}CP)}$ (see Lemma \[prop21-1\] in Section \[app11\]). The other assumptions play a role in order to *go forward* or *backward* for a solution of a non shifted $\DD$-Cauchy problem. More precisely, and allow to go forward, and and allow to go backward (see the proofs of Propositions \[prop21-3\] and \[prop21-4\] in Section \[app11\] for more details).
In view of investigating global solutions, the following definition will be also useful.
The function $f$ is said to be *globally Lipschitz continuous* if there exists $L\geq 0$ such that $$\label{eqcondgloblip}\tag{H$^{\mathrm{glob}}_{\mathrm{Lip}}$}
\Vert f(x_1,t) - f(x_2,t) \Vert \leq L \Vert x_1 - x_2 \Vert.$$ for all $x_1,x_2\in\Omega$ and for $\DD$-a.e. $t \in \T \backslash \{ \sup \T \}$.\
In what follows this property will be referred to as .
Definition of a maximal solution {#section21}
--------------------------------
We first define the notion of a solution of $\mathrm{(\DD\text{-}CP)}$ on an interval $[a,b]_\T$ with $\ab$.
Let $(a, b) \in \T^2$ be such that $\ab$ and let $q:[a,b]_\T\rightarrow\Omega$. The couple $(q,[a,b]_\T)$ is said to be a solution of $\mathrm{(\DD\text{-}CP)}$ if $q \in \AC([a,b]_\T)$, if $q(t_0) = q_0$, and if $q^\DD (t) = f(q(t),t)$ for $\DD$-a.e. $t \in [a,b[_\T$.
Note that, if $(q,[a,b]_\T)$ is a solution of $\mathrm{(\DD\text{-}CP)}$, then $(q,[a',b']_\T)$ is as well a solution of $\mathrm{(\DD\text{-}CP)}$ for all $a',b' \in [a,b]_\T$ satisfying $a' \trianglelefteq t_0 \trianglelefteq b'$.
In view of defining the notion of a solution of $\mathrm{(\DD\text{-}CP)}$ on more general intervals, we set $$\I = \{ I_\T\ \vert\ \exists a, b \in I_\T, \; \ab \}.$$ The set $\I$ is the set of potential intervals of $\T$ for a solution of $\mathrm{(\DD\text{-}CP)}$.
Let $I_\T \in \I$ and let $q:I_\T\rightarrow\Omega$. The couple $(q,I_\T)$ is said to be a solution of $\mathrm{(\DD\text{-}CP)}$ if $(q,[a,b]_\T)$ is a solution of $\mathrm{(\DD\text{-}CP)}$ for all $a,b \in I_\T$ satisfying $\ab$.
Finally, we define the concept of a maximal solution.
Let $(q,I_\T)$ and $(q_1,I^1_\T)$ be two solutions of $\mathrm{(\DD\text{-}CP)}$. The solution $(q_1,I^1_\T)$ is said to be an extension of the solution $(q,I_\T)$ if $I_\T \subset I^1_\T$ and $q_1 = q$ on $I_\T$. A solution $(q,I_\T)$ of $\mathrm{(\DD\text{-}CP)}$ is said to be maximal if, for every extension $(q_1,I^1_\T)$ of $(q,I_\T)$, there holds $I^1_\T = I_\T$. A solution $(q,I_\T)$ of $\mathrm{(\DD\text{-}CP)}$ is said to be global if $I_\T= \T$.
Note that, if $(q,I_\T)$ is a global solution of $\mathrm{(\DD\text{-}CP)}$, then $(q,I_\T)$ is a maximal solution of $\mathrm{(\DD\text{-}CP)}$.
Main results {#section22}
------------
Recall that we consider the general non shifted $\DD$-Cauchy problem $$\mathrm{(\DD\text{-}CP)} \begin{split}
q^\DD(t) &= f (q(t),t), \\
q(t_0)&=q_0,
\end{split}$$ where $t_0 \in \T$, $q_0 \in \Omega$, where $\Omega$ is a non empty open subset of $\R^n$, and $f:\Omega \times \T \setminus \{ \sup \T \}\rightarrow\R^n$ is a $\DD$-Carathéodory function. We have the following general Cauchy-Lipschitz result.
\[thm21-1\] We make the following assumptions on the dynamics $f$, depending on $t_0$.
1. If $t_0 = \min \T$, then we assume that
- $f$ satisfies , that is, $f$ is locally bounded on $\Omega \times \T \backslash \{ \sup \T \}$;
- $f$ satisfies , that is, $f$ is locally Lipschitz continuous with respect to the first variable at right-dense points;
- $f$ satisfies , that is, $f$ is forward $\Omega$-stable at right-scattered points.
2. If $t_0 = \max \T$, then we assume that
- $f$ satisfies , that is, $f$ is locally bounded on $\Omega \times \T \backslash \{ \sup \T \}$;
- $f$ satisfies , that is, $f$ is locally Lipschitz continuous with respect to the first variable at left-dense points;
- $f$ satisfies , that is, $f$ is backward regressive in right-scattered points.
3. If $t_0 \neq \inf \T$ and $t_0 \neq \sup \T$, then we assume that
- $f$ satisfies , that is, $f$ is locally bounded on $\Omega \times \T \backslash \{ \sup \T \}$;
- $f$ satisfies , that is, $f$ is locally Lipschitz continuous with respect to the first variable at right-dense points;
- $f$ satisfies , that is, $f$ is forward $\Omega$-stable at right-scattered points;
- $f$ satisfies , that is, $f$ is locally Lipschitz continuous with respect to the first variable at left-dense points;
- $f$ satisfies , that is, $f$ is and backward regressive in right-scattered points.
Then, the non shifted $\DD$-Cauchy problem $\mathrm{(\DD\text{-}CP)}$ has a unique maximal solution $(q,I_\T)$. Moreover, $(q,I_\T)$ is the maximal extension of any other solution of $\mathrm{(\DD\text{-}CP)}$.
This theorem is proved in Section \[app11\]. The following result gives information on the behavior of a maximal solution at its terminal points.
\[thm21-2\] Under the assumptions of Theorem \[thm21-1\], let $(q,I_\T)$ be the maximal solution of the non shifted $\DD$-Cauchy problem $\mathrm{(\DD\text{-}CP)}$. Then either $I_\T = \T$, that is, the maximal solution $(q,I_\T)$ is global, or the maximal solution is not global and then
1. if $t_0 = \min \T$ then $I_\T = [t_0,b[_\T$ where $b \in ]t_0,+\infty[_\T$ is a left-dense point of $\T$;
2. if $t_0 = \max \T$ then $I_\T = ]a,t_0]_\T$ where $a \in ]-\infty,t_0[_\T$ is a right-dense point of $\T$;
3. if $t_0 \neq \inf \T$ and $t_0 \neq \sup \T$ then $I_\T = ]a,+\infty[_\T$ or $I_\T = ]-\infty,b[_\T$ or $I_\T = ]a,b[_\T$, where $a \in ]-\infty,t_0[_\T$ is a right-dense point of $\T$ and $b \in ]t_0,+\infty[_\T$ is a left-dense point of $\T$;
and moreover, for every $K \in \KK$ there exists $t\in I_\T$ (close to $a$ or $b$ depending on the cases listed above) such that $q(t)\in\Omega\setminus K$.
This theorem is proved in Section \[app12\]. It states that the maximal solution must go out of any compact of $\Omega$ near its terminal points whenever it is not global.
The following last result states that, under global Lipschitz assumption, the maximal solution is global.
\[thm22-1\] If $t_0 = \min \T$, $\Omega= \R^n$, if $f$ satisfies , that is, $f$ is locally bounded on $\R^n \times \T \backslash \{ \sup \T \}$, and if $f$ satisfies , that is, $f$ is globally Lipschitz continuous, then the non shifted $\DD$-Cauchy problem $\mathrm{(\DD\text{-}CP)}$ has a unique maximal solution $(q,I_\T)$, which is moreover global.
The proof is done in Section \[app13\].
\[rem1\] As an application of Theorem \[thm22-1\], we recover the well known fact that, in the linear case $$q^\DD(t) = h(t) \times q(t),$$ where $h \in \L^\infty_\T (\T \backslash \{ \sup \T \},\R^{n\times n})$, solutions are global.
Further comments {#section24}
----------------
In this section, we provide simple examples (in the one-dimensional case) showing the sharpness of the assumptions made in Theorem \[thm21-1\]. Indeed, if one of these assumptions is not satisfied, then the existence or the uniqueness of the maximal solution is no more guaranteed.
Let $\T = [0,+\infty[$, $\Omega = \R$, $t_0 = 0$, $q_0 = 0$ and $f:\R\times\T\rightarrow\R$ be defined by $f(x,t)=2 \sqrt{\vert x \vert}$. The function $f$ obviously satisfies since $\RR = \emptyset$, however it does not satisfy . The corresponding $\DD$-Cauchy problem $\mathrm{(\DD\text{-}CP)}$ has two global solutions $q_1$ and $q_2$ given by $q_1(t) = 0$ and $q_2(t) = t^2$, for every $t\in\T$.
This example shows that, in the absence of Assumption , the uniqueness of the maximal solution is not guaranteed.
Let $\T = \{ 0,1 \}$, $\Omega = ]-1,1[$, $t_0 = 0$, $q_0 = 0$ and $f:\Omega\times\{0\}\rightarrow\R$ be defined by $f(x,t)=1$. The function $f$ obviously satisfies since $\T \backslash \{ \sup \T \} = \{ 0 \}$ does not admit any right-dense point of $\T$, however it does not satisfy since $x+1 \notin \Omega$ for $x \in [0,1[$. Since $q(0) = 0$ and $q(1)=q(0)+\mu(0) f(q(0),0)$ imply $q(1) = 1 \notin \Omega$, we conclude that $\mathrm{(\DD\text{-}CP)}$ does not admit any solution.
Therefore, in the absence of Assumption , $\mathrm{(\DD\text{-}CP)}$ may fail to have a solution.
Let $\T = ]-\infty,0]$, $\Omega = \R$, $t_0 = 0$, $q_0 = 0$ and $f:\R\times\T\rightarrow\R$ be defined by $f(x,t)=-2 \sqrt{\vert x \vert}$. The function $f$ obviously satisfies since $\RR = \emptyset$, however it does not satisfy . The corresponding $\DD$-Cauchy problem $\mathrm{(\DD\text{-}CP)}$ ha two global solutions $q_1$ and $q_2$ given by $q_1(t) = 0$ and $q_2(t) = t^2$ for every $t\in\T$.
This example shows that, in the absence of Assumption , the uniqueness of the maximal solution is not guaranteed.
Let $\T = \{ 0,1 \}$, $\Omega = \R$, $t_0 = 1$, $q_0 \in \R$ and $f:\R \times \{ 0 \}\rightarrow\R$ be defined by $f(x,t)=-x$. The function $f$ obviously satisfies since $\T \backslash \{ \inf \T \} = \{ 1 \}$ does not admit any left-dense point of $\T$, however it does not satisfy since $G^+(0) = 0$. As a consequence, if $q_0 \neq 0$, $\mathrm{(\DD\text{-}CP)}$ does not admit any solution. Indeed, $q(1)=q_0$ and $q(1)=q(0)+\mu (0) f(q(0),0)$ imply $q(1)=0$, which is a contradiction. If $q_0 = 0$, we obtain an infinite number of global solutions. Indeed, any function $q$ defined on $\T$ with $q(1) = 0$ is then a global solution of $\mathrm{(\DD\text{-}CP)}$.
General shifted $\DD$-Cauchy problem {#section3}
====================================
Throughout this section we consider the general *shifted* $\DD$-Cauchy problem $$\mathrm{(\DD\text{-}CP^\sigma)} \begin{split}
q^\DD(t) & = f (q^\sigma(t),t), \\
q(t_0)&=q_0,
\end{split}$$ where $t_0 \in \T$, $q_0 \in \Omega$, where $\Omega$ is a non empty open subset of $\R^n$ and $f:\Omega \times \T \backslash \{ \sup \T \}\rightarrow \R^n$ is a $\DD$-Carathéodory function.
The results of the section follow the same lines as in the previous section. Therefore we do not give any proof nor counterexamples as above. Some comments are however done in Section \[app2\].
Preliminaries {#section30}
-------------
As in Section \[section20\], it will be important to distinguish between three cases:
1. $t_0 = \min \T$;
2. $t_0 = \max \T$;
3. $t_0 \neq \inf \T$ and $t_0 \neq \sup \T$.
With respect to Section \[section20\], we introduce two additional concepts.
The function $f$ is said to be *backward $\Omega$-stable at right-scattered points* if the mapping $$\label{eqcondbsta}\tag{H${}^{\mathrm{back}}_{\mathrm{stab}}$}
{\begin{array}[t]{lrcl}G^-(t) :&\Omega &\rightarrow &\R^n\\&x& \mapsto &x-\mu (t) f(x,t) \end{array}}$$ takes its values in $\Omega$, for every $t \in \RR$.\
In what follows this property will be referred to as .
The function $f$ is said to be *forward regressive at right-scattered points* if $$\label{eqcondfreg}\tag{H${}^{\mathrm{forw}}_{\mathrm{regr}}$}
{G^-(t) : \Omega \rightarrow \R^n} \; \text{is invertible},$$ for every $t \in \RR$.\
In what follows this property will be referred to as .
These above assumptions play a role in order to *go forward* or *backward* for a solution of a shifted $\DD$-Cauchy problem. Precisely, and allow to *go forward*. Similarly, and allow to *go backward*.\
Definition of a maximal solution {#section31}
--------------------------------
Let $(a,b) \in \T^2$ satisfying $\ab$ and let ${q : [a,b]_\T \rightarrow \Omega}$. The couple $(q,[a,b]_\T)$ is said to be a solution of $\mathrm{(\DD\text{-}CP^\sigma)}$ if $q \in \AC([a,b]_\T)$, $q(t_0) = q_0$, and $q^\DD (t) = f(q^\sigma(t),t)$ for $\DD$-a.e. $t \in [a,b[_\T$.
Let $I_\T \in \I$ and let ${q : I_\T \rightarrow \Omega}$. The couple $(q,I_\T)$ is said to be a solution of $\mathrm{(\DD\text{-}CP^\sigma)}$ if $(q,[a,b]_\T)$ is a solution of $\mathrm{(\DD\text{-}CP^\sigma)}$ for all $a,b \in I_\T$ satisfying $\ab$.
Let $(q,I_\T)$ and $(q_1,I^1_\T)$ be two solutions of $\mathrm{(\DD\text{-}CP^\sigma)}$. The solution $(q_1,I^1_\T)$ is said to be an extension of the solution $(q,I_\T)$ if $I_\T \subset I^1_\T $ and $q_1 = q$ on $I_\T$. A solution $(q,I_\T)$ of $\mathrm{(\DD\text{-}CP^\sigma)}$ is said to be *maximal* if, for every extension $(q_1,I^1_\T)$ of $(q,I_\T)$, there holds $I^1_\T = I_\T$. A solution $(q,I_\T)$ of $\mathrm{(\DD\text{-}CP^\sigma)}$ is said to be *global* if $I_\T= \T$.
Main results {#section32}
------------
Recall that we consider the general shifted $\DD$-Cauchy problem $$\mathrm{(\DD\text{-}CP^\sigma)} \begin{split}
q^\DD(t) & = f (q^\sigma(t),t), \\
q(t_0)&=q_0,
\end{split}$$ where $t_0 \in \T$, $q_0 \in \Omega$ where $\Omega$ is a non empty open subset of $\R^n$ and $f:\Omega \times \T \backslash \{ \sup \T \}\rightarrow \R^n$ is a $\DD$-Carathéodory function.
\[thm31-1\] We make the following assumptions on the dynamics $f$, depending on $t_0$.
1. If $t_0 = \min \T$, then we assume that
- $f$ satisfies , that is, $f$ is locally bounded on $\Omega \times \T \backslash \{ \sup \T \}$;
- $f$ satisfies , that is, $f$ is locally Lipschitz continuous with respect to the first variable at right-dense points;
- $f$ satisfies , that is, $f$ is forward regressive in right-scattered points.
2. If $t_0 = \max \T$, then we assume that
- $f$ satisfies , that is, $f$ is locally bounded on $\Omega \times \T \backslash \{ \sup \T \}$;
- $f$ satisfies , that is, $f$ is locally Lipschitz continuous with respect to the first variable at left-dense points;
- $f$ satisfies , that is, $f$ is backward $\Omega$-stable in right-scattered points.
3. If $t_0 \neq \inf \T$ and $t_0 \neq \sup \T$, then we assume that
- $f$ satisfies , that is, $f$ is locally bounded on $\Omega \times \T \backslash \{ \sup \T \}$;
- $f$ satisfies , that is, $f$ is locally Lipschitz continuous with respect to the first variable at right-dense points;
- $f$ satisfies , that is, $f$ is forward regressive at right-scattered points;
- $f$ satisfies , that is, $f$ is locally Lipschitz continuous with respect to the first variable at left-dense points;
- $f$ satisfies , that is, $f$ is backward $\Omega$-stable at right-scattered points.
Then the shifted $\DD$-Cauchy problem $\mathrm{(\DD\text{-}CP^\sigma)}$ has a unique maximal solution $(q,I_\T)$. Moreover, $(q,I_\T)$ is the maximal extension of any other solution of $\mathrm{(\DD\text{-}CP^\sigma)}$
\[thm31-2\] Under the assumptions of Theorem \[thm31-1\], let $(q,I_\T)$ be the maximal solution of the shifted $\DD$-Cauchy problem $\mathrm{(\DD\text{-}CP^\sigma)}$. Then either $I_\T = \T$, that is, the maximal solution $(q,I_\T)$ is global, or the maximal solution is not global and then
1. if $t_0 = \min \T$ then $I_\T = [t_0,b[_\T$ where $b \in ]t_0,+\infty[_\T$ is a left-dense point of $\T$;
2. if $t_0 = \max \T$ then $I_\T = ]a,t_0]_\T$ where $a \in ]-\infty,t_0[_\T$ is a right-dense point of $\T$;
3. if $t_0 \neq \inf \T$ and $t_0 \neq \sup \T$ then $I_\T = ]a,+\infty[_\T$ or $I_\T = ]-\infty,b[_\T$ or $I_\T = ]a,b[_\T$ where $a \in ]-\infty,t_0[_\T$ is a right-dense point of $\T$ and $b \in ]t_0,+\infty[_\T$ is a left-dense point of $\T$;
and moreover, for every $K \in \KK$ there exists $t\in I_\T$ (close to $a$ or $b$ depending on the cases listed above) such that $q(t)\in\Omega\setminus K$.
\[thm32-1\] If $t_0 = \max \T$, $\Omega= \R^n$, if $f$ satisfies , that is, $f$ is locally bounded on $\R^n \times \T \backslash \{ \sup \T \}$, and if $f$ satisfies , that is, $f$ is globally Lipschitz continuous, then, the shifted $\DD$-Cauchy problem $\mathrm{(\DD\text{-}CP^\sigma)}$ has a unique maximal solution $(q,I_\T)$, which is moreover global.
As in Remark \[rem1\], in the linear case the maximal solution of any shifted $\DD$-Cauchy problem is automatically global.
Proofs of the results {#app1}
=====================
Proof of Theorem \[thm21-1\] {#app11}
----------------------------
If $f$ satisfies , then for all $(a,b)\in\T^2$ such that $a<b$, there holds $$\label{eq666}
f(q,t) \in \L^\infty_\T ([a,b[_\T,\R^n) \subset \L^1_\T ([a,b[_\T,\R^n),$$ for every $q \in \CC ([a,b]_\T,\R^n)$. Then, from Section \[section13\], we have the following $\DD$-integral characterization of the solutions of $\mathrm{(\DD\text{-}CP)}$.
\[prop21-1\] Let $I_\T \in \I$ and let $q:I_\T\rightarrow\Omega$. If $f$ satisfies , then the couple $(q,I_\T)$ is a solution of $\mathrm{(\DD\text{-}CP)}$ if and only if for all $a,b \in I_\T$ satisfying $\ab$, one has $q \in \CC([a,b]_\T)$ and $$q(t) = \left\lbrace \begin{array}{lcc}
q_0 + \int_{[t_0,t[_\T} f(q (\tau),\tau) \; \DD \tau & \text{if} & t \geq t_0, \\
q_0 - \int_{[t,t_0[_\T} f(q (\tau),\tau) \; \DD \tau & \text{if} & t \leq t_0. \\
\end{array} \right.$$ for every $ t \in [a,b]_\T$.
This characterization allows one to prove the following result.
\[prop21-2\] If $f$ satisfies , then every solution of $\mathrm{(\DD\text{-}CP)}$ can be extended to a maximal solution.
Let $(q,I_\T)$ be a solution of $\mathrm{(\DD\text{-}CP)}$. Let us define the non empty set $\mathscr{F}$ of extensions of $(q,I_\T)$. The set $\mathscr{F}$ is ordered by $$(q_1,I^1_\T) \leq (q_2,I^2_\T) \; \text{if and only if} \; (q_2,I^2_\T) \; \text{is an extension of} \; (q_1,I^1_\T).$$ Let us prove that $\mathscr{F}$ is inductive. Let $\GG = \cup_{p \in \mathscr{P}} \{ (q_p,I^p_\T) \}$ be a non empty totally ordered subset of $\mathscr{F}$. Let us prove that $\GG$ admits an upper bound.
Let us define $\overline{I} = \cup_{p \in \mathscr{P}} I^p $. This is an interval of $\R$, since $t_0 \in \cap_{p \in \mathscr{P}} I^p$. Then $\overline{I}_\T = \cup_{p \in \mathscr{P}} I^p_\T \in \I$. For every $t \in \overline{I}_\T$, there exists $p \in \mathscr{P}$ such that $t \in I^p_\T$ and, since $\GG$ is totally ordered, if $t \in I^{p_1}_\T \cap I^{p_2}_\T$ then $q_{p_1} (t) = q_{p_2} (t)$. Consequently, we can define $\overline{q}$ by $$\forall t \in \overline{I}_\T, \; \overline{q}(t) = q_p (t) \in \Omega \; \text{where} \; t \in I^p_\T.$$ Our aim is to prove that $(\overline{q},\overline{I}_\T)$ is a solution of $\mathrm{(\DD\text{-}CP)}$. Let $a,b \in \overline{I}_\T$ satisfying $\ab$. Since $\GG$ is totally ordered, there exists $p \in \mathscr{P}$ such that $[a,b]_\T \subset I^p_\T$ and $\overline{q} = q_p$ on $[a,b]_\T$. Since $(q_p,I^p_\T) $ is a solution of $\mathrm{(\DD\text{-}CP)}$, we obtain that $q_p$ satisfies the necessary and sufficient condition of Lemma \[prop21-1\] on $[a,b]_\T$. Consequently, this holds true as well for $\overline{q}$ on $[a,b]_\T$. Finally, since this last sentence is true for all $a, b \in \overline{I}_\T$ satisfying $\ab$, we infer from Lemma \[prop21-1\] that $(\overline{q},\overline{I}_\T)$ is a solution of $\mathrm{(\DD\text{-}CP)}$. Since $(\overline{q},\overline{I}_\T)$ is obviously an extension of any element of $\GG$, we obtain that $\GG$ admits an upper bound and then, $\mathscr{F}$ is inductive.
Finally, $\mathscr{F}$ is a non empty ordered inductive set and consequently, from Zorn’s lemma, admits a maximal element. The proof is complete.
\[prop21-3\] There exist $a, b \in \T$ satisfying $\ab$ and $q : [a,b]_\T\rightarrow\Omega$ such that $(q,[a,b]_\T) $ is a solution of $\mathrm{(\DD\text{-}CP)}$.
We only prove this proposition in the third case of Theorem \[thm21-1\] (the two first cases are derived similarly) for which $t_0 \neq \inf \T$ and $t_0 \neq \sup \T$. We distinguish between four situations.
#### First case:
$t_0$ is a left- and a right-scattered point of $\T$. In this case, it is sufficient to consider $a= \rho (t_0) \in ]-\infty,t_0[_\T$, $b=\sigma(t_0) \in ]t_0,+\infty[_\T$ and the function $q$ defined on $[a,b]_\T = \{a, t_0,b \}$ with values in $\Omega$ by $q(a) = G^+(a)^{-1}(q_0)$, $q(t_0) = q_0$ and $q(b) = G^+(t_0)(q_0)$. We note that $q(a)$ is well-defined in $\Omega$ from and $q(b) \in \Omega$ from .
#### Second case:
$t_0$ is a left- and a right-dense point of $\T$. Let $R'$, $\delta'$ and $L'$ associated with $q_0$ and $t_0$ in and let $R''$, $\delta''$ and $L''$ associated with $q_0$ and $t_0$ in . We define $R = \min (R',R'') > 0$ and $L = \max (L',L'') \geq 0$. Let $M$ associated with $\overline{B}(q_0,R) \in \KK$ and $[t_0-\delta',t_0+\delta''[_\T$ in . Consider $0 <\delta_1 \leq \delta'$ and $0 <\delta_2 \leq \delta''$ such that $a=t_0-\delta_1 \in ]-\infty,t_0[_\T$, $b=t_0+\delta_2 \in ]t_0,+\infty[_\T$ and $\delta_1$ and $\delta_2$ are sufficiently small in order to have $\max(\delta_1,\delta_2) M \leq R$ and $\max(\delta_1,\delta_2) L < 1$. Then, we can construct the $\max(\delta_1,\delta_2)L$-contraction map with respect to the norm $\Vert \cdot \Vert_{\infty}$ $${\begin{array}[t]{lrcl}F :&\CC ([a,b]_\T,\overline{B}(q_0,R)) &\rightarrow &\CC ([a,b]_\T,\overline{B}(q_0,R))\\&q& \mapsto &F(q), \end{array}}$$ with $${\begin{array}[t]{lrcl}F(q) :&[a,b]_\T &\rightarrow &\overline{B}(q_0,R)\\&t& \mapsto &\left\lbrace \begin{array}{lcc}
q_0 + \int_{[t_0,t[_\T} f(q (\tau),\tau) \; \DD \tau & \text{if} & t \geq t_0 \\
q_0 -\int_{[t,t_0[_\T} f(q (\tau),\tau) \; \DD \tau & \text{if} & t \leq t_0. \\
\end{array} \right. \end{array}}$$ It follows from the Banach fixed point theorem that $F$ has a unique fixed point denoted by $q$, and then $(q,[a,b]_\T)$ is a solution of $\mathrm{(\DD\text{-}CP)}$.
#### Third case:
$t_0$ is a left-scattered and a right-dense point of $\T$. Let $R$, $\delta$ and $L$ associated with $q_0$ and $t_0$ in . Let $M$ associated with $\overline{B}(q_0,R) \in \KK$ and $[t_0,t_0+\delta[_\T$ in . Consider $0 <\delta_1 \leq \delta$ such that $b=t_0+\delta_1 \in ]t_0,+\infty[_\T$ and $\delta_1$ is sufficiently small in order to have $\delta_1 M \leq R$ and $\delta_1 L < 1$. Then, we can construct the $\delta_1 L$-contraction map with respect to the norm $\Vert \cdot \Vert_{\infty}$ $${\begin{array}[t]{lrcl}F :&\CC ([t_0,b]_\T,\overline{B}(q_0,R)) &\rightarrow &\CC ([t_0,b]_\T,\overline{B}(q_0,R))\\&q& \mapsto &F(q) \end{array}}$$ with $${{\begin{array}[t]{lrcl}F(q) :&[t_0,b]_\T &\rightarrow &\overline{B}(q_0,R)\\&t& \mapsto &q_0 + \di \int_{[t_0,t[_\T} f(q(\tau),\tau) \; \DD \tau. \end{array}}}$$ It follows from the Banach fixed point theorem that $F$ has a unique fixed point denoted by $q$ defined on $[t_0,b]_\T$. Finally, since $t_0$ is a left-scattered point of $\T$ and from , we define $a= \rho (t_0) \in ]-\infty,t_0[_\T$ and $q(a) = G^+(a)^{-1}(q_0) \in \Omega$. We have thus obtained a solution $(q,[a,b]_\T)$ of $\mathrm{(\DD\text{-}CP)}$.
#### Fourth case:
$t_0$ is a left-dense and a right-scattered point of $\T$. Let $R$, $\delta$ and $L$ associated with $q_0$ and $t_0$ in . Let $M$ associated with $\overline{B}(q_0,R) \in \KK$ and $[t_0-\delta,t_0[_\T$ in . Consider $0 <\delta_1 \leq \delta$ such that $a=t_0-\delta_1 \in ]-\infty,t_0[_\T$ and $\delta_1$ is sufficiently small in order to have $\delta_1 M \leq R$ and $\delta_1 L < 1$. Then, we can construct the $\delta_1 L$-contraction map with respect to the norm $\Vert \cdot \Vert_{\infty}$ $${\begin{array}[t]{lrcl}F :&\CC ([a,t_0]_\T,\overline{B}(q_0,R)) &\rightarrow &\CC ([a,t_0]_\T,\overline{B}(q_0,R))\\&q& \mapsto &F(q) \end{array}}$$ with $${{\begin{array}[t]{lrcl}F(q) :&[a,t_0]_\T &\rightarrow &\overline{B}(q_0,R)\\&t& \mapsto &q_0 - \di \int_{[t,t_0[_\T} f(q(\tau),\tau) \; \DD \tau. \end{array}}}$$ It follows from the Banach fixed point theorem that $F$ admits a unique fixed point denoted by $q$ defined on $[a,t_0]_\T$. Since $t_0$ is a right-scattered point of $\T$, and from , we define $b= \sigma (t_0) \in ]t_0,+\infty[_\T$ and $q(b) = G^+(t_0)(q_0) \in \Omega$. We have thus obtained a solution $(q,[a,b]_\T)$ of $\mathrm{(\DD\text{-}CP)}$.
From Lemma \[prop21-2\], we can extend the solution given in Proposition \[prop21-3\] and we obtain the existence of a maximal solution. The following result proves that it is unique.
\[prop21-4\] Let $(q_1,I^1_\T)$ and $(q_2,I^2_\T)$ be two solutions of $\mathrm{(\DD\text{-}CP)}$. Then, $q_1 = q_2$ on $I^1_\T \cap I^2_\T$.
As before, we only prove this proposition in the third case of Theorem \[thm21-1\]. We denote by $I = I^1 \cap I^2$ (interval of $\R$). One can easily prove that $I_\T = I^1_\T \cap I^2_\T \in \I$. It is sufficient to prove $q_1 = q_2$ on $[a,b]_\T$ for all $a,b \in I_\T$ satisfying $\ab$. Let $a$, $b \in I_\T$ satisfying $\ab$. Set $$A = \{ t \in [a,t_0]_\T, \; q_1 (t) \neq q_2 (t) \},$$ and $$B = \{ t \in [t_0,b]_\T, \; q_1 (t) \neq q_2 (t) \}.$$ Let us prove by contradiction that $A \cup B = \emptyset$. Assume that $A \neq \emptyset$ and let $\overline{t} = \sup A$. Note that $\overline{t} \in [a,t_0]_\T$ (since $\T$ is closed) and that $q_1 = q_2$ on $]\overline{t},t_0]_\T$. In order to raise a contradiction, we first derive the four following facts.
1. *Fact 1: $\overline{t} < t_0$.* If $t_0$ is a left-scattered point of $\T$, this claim is obvious since $q_1(t_0) = q_2(t_0) = q_0$ and $q_1 (\rho(t_0)) = q_2 (\rho(t_0)) = G^+(\rho(t_0))^{-1}(q_0)$ from . If $t_0$ is a left-dense point of $\T$, let $R$, $\delta$ and $L$ associated with $q_0$ and $t_0$ in . Let $M$ associated with $\overline{B}(q_0,R) \in \KK$ and $[t_0-\delta,t_0[_\T$ in . Consider $0 <\delta_1 \leq \delta$ such that $c=t_0-\delta_1 \in [a,t_0[_\T$ and $\delta_1$ is sufficiently small in order to have $\delta_1 M \leq R$, $\delta_1 L < 1$ and $q_1$, $q_2 \in \CC ([c,t_0]_\T,\overline{B}(q_0,R))$. Since $q_1$ and $q_2$ are solutions of $\mathrm{(\DD\text{-}CP)}$ on $[a,b]_\T$, they are in particular fixed points of the $\delta_1 L$-contraction map $${\begin{array}[t]{lrcl}F :&\CC ([c,t_0]_\T,\overline{B}(q_0,R)) &\rightarrow &\CC ([c,t_0]_\T,\overline{B}(q_0,R))\\&q& \mapsto &F(q) \end{array}}$$ with $${{\begin{array}[t]{lrcl}F(q) :&[c,t_0]_\T &\rightarrow &\overline{B}(q_0,R)\\&t& \mapsto &q_0 - \di \int_{[t,t_0[_\T} f(q(\tau),\tau) \; \DD \tau. \end{array}}}$$ Since $F$ has a unique fixed point from the Banach fixed point theorem, we conclude that $q_1 = q_2$ on $[c,t_0]_\T$. Hence $\overline{t} < t_0$.
2. *Fact 2: $q_1(\overline{t})=q_2(\overline{t})$.* If $\overline{t}$ is a right-scattered point of $\T$, then $\sigma(\overline{t})$ is a left-scattered point of $\T$ and $q_1(\sigma(\overline{t})) = q_2(\sigma(\overline{t}))$. As a consequence, $q_1(\overline{t}) = q_2(\overline{t}) = G^+(\overline{t})^{-1} (q_1(\sigma(\overline{t})))$. If $\overline{t}$ is a right-dense point of $\T$, then $q_1(\overline{t}) = q_2(\overline{t})$ from the continuity of $q_1$ and $q_2$ and since $q_1=q_2$ on $]\overline{t},t_0]_\T$.
3. *Fact 3: $\overline{t} > a$.* Indeed, if $\overline{t} = a$ then $A = \emptyset$ since $q_1(\overline{t})=q_2(\overline{t})$;
4. *Fact 4: $\overline{t}$ is a left-dense point of $\T$.* Indeed, if $\overline{t}$ were to be a left-scattered point of $\T$, since $q_1(\overline{t})=q_2(\overline{t})$, then $q_1(\rho(\overline{t}))=q_2(\rho(\overline{t})) = G^+(\rho(\overline{t}))^{-1}(q_1(\overline{t}))$ and then it would raise a contradiction with the definition of $\overline{t}$.
Let us denote by $ \overline{x} = q_1(\overline{t})=q_2(\overline{t})$. Let $R$, $\delta$ and $L$ associated with $\overline{t}$ and $\overline{x}$ in . Let $M$ associated with $B(\overline{x},R) \in \KK$ and $[\overline{t}-\delta,\overline{t}[_\T$ in . Consider $0 <\delta_1 \leq \delta$ such that $c_0=\overline{t}-\delta_1 \in [a,\overline{t}[_\T$ and $\delta_1$ is sufficiently small in order to have $\delta_1 M \leq R$, $\delta_1 L < 1$ and $q_1$, $q_2 \in \CC ([c_0,\overline{t}]_\T,B(\overline{x},R))$. Since $q_1$ and $q_2$ are solutions of $\mathrm{(\DD\text{-}CP)}$ on $[a,b]_\T$, they are in particular fixed points of the $\delta_1 L$-contraction map $${\begin{array}[t]{lrcl}F_0 :&\CC ([c_0,\overline{t}]_\T,B(\overline{x},R)) &\rightarrow &\CC ([c_0,\overline{t}]_\T,B(\overline{x},R))\\&q& \mapsto &F_0(q) \end{array}}$$ with $${{\begin{array}[t]{lrcl}F_0(q) :&[c_0,\overline{t}]_\T &\rightarrow &B(\overline{x},R)\\&t& \mapsto &\overline{x} - \di \int_{[t,\overline{t}[_\T} f(q(\tau),\tau) \; \DD \tau. \end{array}}}$$ Since $F_0$ has a unique fixed point from the Banach fixed point theorem, we conclude that $q_1 = q_2$ on $[c_0,\overline{t}]_\T$, and this is a contradiction. Consequently $A = \emptyset$.
In the same way, we prove that $B = \emptyset$ and the proof is complete.
Theorem \[thm21-1\] follows from Lemma \[prop21-2\], Propositions \[prop21-3\] and \[prop21-4\].
Proof of Theorem \[thm21-2\] {#app12}
----------------------------
\[prop21-5\] Under the assumptions of Theorem \[thm21-1\], let $(q,I_\T)$ be the maximal solution of $\mathrm{(\DD\text{-}CP)}$. Then either $I_\T = \T$, that is, the solution $(q,I_\T)$ is global, or
1. if $t_0 = \min \T$ then $I_\T = [t_0,b[_\T$ where $b \in ]t_0,+\infty[_\T$ is a left-dense point of $\T$;
2. if $t_0 = \max \T$ then $I_\T = ]a,t_0]_\T$ where $a \in ]-\infty,t_0[_\T$ is a right-dense point of $\T$;
3. if $t_0 \neq \inf \T$ and $t_0 \neq \sup \T$ then $I_\T = ]a,+\infty[_\T$ or $I_\T = ]-\infty,b[_\T$ or $I_\T = ]a,b[_\T$, where $a \in ]-\infty,t_0[_\T$ is a right-dense point of $\T$ and $b \in ]t_0,+\infty[_\T$ is a left-dense point of $\T$.
We only prove this proposition in the first case of Theorem \[thm21-1\] (the other ones are derived similarly).
Let us first prove that if $I_\T = [t_0,b]_\T$ then $b=\max \T$ (and thus $I_\T = \T$). By contradiction, assume that $I_\T = [t_0,b]_\T$ with $b < \sup \T$. Consider the $\DD$-Cauchy problem $$z^\DD(t) = f (z(t),t), \quad z(b)=q(b).$$ As in Proposition \[prop21-3\], we can prove that it has a solution $(z,[b,b_1]_\T)$ with $b_1 \in ]b,+\infty[_\T$. Then, we define $q_1$ by $$q_1 (t) = \left\lbrace \begin{array}{rcl}
q(t) & \text{if} & t \in [t_0,b]_\T, \\
z(t) & \text{if} & t \in [b,b_1]_\T,
\end{array} \right.$$ for every $ t \in [t_0,b_1]_\T$. Then $q_1 \in \CC([t_0,b_1]_\T)$ and one can easily prove that $$q_1 (t) = q_0 + \di \int_{[t_0,t[_\T} f(q_1(\tau),\tau) \; \DD \tau.$$ for every $ t \in [t_0,b_1]_\T$. It follows from Lemma \[prop21-1\] that $(q_1,[t_0,b_1]_\T)$ is a solution of $\mathrm{(\DD\text{-}CP)}$ and is a strict extension of $(q,[t_0,b]_\T)$. It is a contradiction with the maximality of $(q,[t_0,b]_\T)$.
If $I_\T = [t_0,b[_\T$ with $b$ a left-scattered point of $\T$, then $I_\T = [a,\rho(b)]_\T$ with $\rho(b) < \sup \T$ and we recover to the previous contradiction.
\[prop21-6\] Under the assumptions of Theorem \[thm21-1\], let $(q,I_\T)$ be the maximal solution of $\mathrm{(\DD\text{-}CP)}$. If $(q,I_\T)$ is not global, then $q$ cannot be continuously extended with a value in $\Omega$ at $t=a$ or at $t=b$ (see Proposition \[prop21-5\] for $a$ and $b$).
We only prove this lemma in the first case of Theorem \[thm21-1\]. By contradiction, let us assume that $q$ can be continuously extended with a value in $\Omega$ at $t=b$, that is, $\lim_{t \to b, \; t \in [t_0,b[_\T } q(t) = q_b \in \Omega$. Then, we define $q_1$ by $$q_1 (t) = \left\lbrace \begin{array}{rcl}
q(t) & \text{if} & t \in [t_0,b[_\T \\
q_b & \text{if}& t=b,
\end{array} \right.$$ for every $ t \in [t_0,b]_\T$. In particular $q_1:[t_0,b]_\T\rightarrow\Omega$ and $q_1 \in \CC ([t_0,b]_\T)$. Our aim is to prove that $(q_1,[t_0,b]_\T)$ is a solution of $\mathrm{(\DD\text{-}CP)}$.
Since $(q,[t_0,b[_\T)$ is a solution of $\mathrm{(\DD\text{-}CP)}$, it follows from Lemma \[prop21-1\] that $$\label{eq671}
q_1(t) = q(t) = q_0 + \di \int_{[t_0,t[_\T} f(q(\tau),\tau) \; \DD \tau = q_0 + \di \int_{[t_0,t[_\T} f(q_1(\tau),\tau) \; \DD \tau ,$$ for every $b' \in ]t_0,b[_\T$ and every $t \in [t_0,b']_\T$. Since $f(q_1,t) \in \L^1_\T ([t_0,b[_\T,\R^n)$ (see ), we infer from Lebesgue’s dominated convergence theorem that $$q_1 (b) = q_b = q_0 + \di \int_{[t_0,b[_\T} f(q_1(\tau),\tau) \; \DD \tau.$$ Therefore also holds for $b'=b$. It follows from Lemma \[prop21-1\] that $(q_1,[t_0,b]_\T)$ is a solution of $\mathrm{(\DD\text{-}CP)}$ and is a strict extension of $(q,[t_0,b[_\T)$. It is a contradiction with the maximality of $(q,[t_0,b[_\T)$.
\[prop21-7\] Under the assumptions of Theorem \[thm21-1\], let $(q,I_\T)$ be the maximal solution of $\mathrm{(\DD\text{-}CP)}$. If $(q,I_\T)$ is not global, then for every $K \in \KK$ there exists $t\in I_\T$ (close to $a$ or $b$ depending on the cases listed in the theorem) such that $q(t)\in\Omega\setminus K$.
We only prove this lemma in the first case of Theorem \[thm21-1\]. By contradiction, assume that there exists $K \in \KK $ such that $q$ takes its values in $K$ on $I_\T = [t_0,b[_\T$ with $b$ a left dense point of $\T$. Consider $M \geq 0$ associated with $K \in \KK$ and $[t_0,b[_\T$ in . For all $t_1 \leq t_2$ elements of $[t_0,b[_\T$, one has $$\Vert q(t_2) - q(t_1) \Vert \leq \di \int_{[t_1,t_2[_\T} \Vert f(q(\tau),\tau) \Vert \; \DD \tau \leq M (t_2 - t_1).$$ Therefore $q$ is Lipschitz continuous and thus uniformly continuous on $[t_0,b[_\T$ with $b$ a left-dense point of $\T$. Hence $q$ can be continuously extended at $t=b$ with a value $q_b \in \R^n$. Moreover, since $q$ takes its values in the compact $K \subset \Omega$, it follows that $q_b \in \Omega$. Using Lemma \[prop21-6\], this raises a contradiction.
The proof of Theorem \[thm21-2\] follows from Proposition \[prop21-5\] and Lemma \[prop21-7\].
Proof of Theorem \[thm22-1\] {#app13}
----------------------------
Note that since $\Omega = \R^n$ and since $f$ satisfies , $f$ automatically satisfies and . Since $t_0 = \min \T$, $\mathrm{(\DD\text{-}CP)}$ admits a unique maximal solution $(q,I_\T)$ from Theorem \[thm21-1\]. Proving that $I_\T = \T$ requires the following result.
\[lem22-1\] If $t_0 = \min \T$ then $$\int_{[t_0,t[_\T} (\tau - t_0)^k \; \DD \tau \leq \dfrac{(t-t_0)^{k+1}}{k+1},$$ for every $k \in \N$ and every $t \in \T$.
One has $$\int_{[t_0,t[_\T} (\tau - t_0)^k \; \DD \tau = \int_{[t_0,t[_\T} (\tau - t_0)^k \; d\tau + \di \sum_{r \in [t_0,t[_\T \cap \RR} \mu (r) (r -t_0)^k ,$$ for every $k \in \N$ and every $t \in \T$. Since $$\begin{split}
\sum_{r \in [t_0,t[_\T \cap \RR} \mu (r) (r -t_0)^k &= \di \sum_{r \in [t_0,t[_\T \cap \RR} \di \int_{]r,\sigma(r)[} (r -t_0)^k \; d\tau \\
& \leq \di \sum_{r \in [t_0,t[_\T \cap \RR} \di \int_{]r,\sigma(r)[} (\tau -t_0)^k \; d\tau,
\end{split}$$ it follows that $$\di \int_{[t_0,t[_\T} (\tau - t_0)^k \; \DD \tau \leq \di \int_{[t_0,t[} (\tau - t_0)^k \; d\tau = \dfrac{(t-t_0)^{k+1}}{k+1},$$ and the proof is complete.
We define the mapping $${\begin{array}[t]{lrcl}F :&\CC (\T,\R^n) &\rightarrow &\CC (\T,\R^n)\\&q& \mapsto &F(q) \end{array}}$$ with $${{\begin{array}[t]{lrcl}F(q) :&\T &\rightarrow &\R^n\\&t& \mapsto &q_0 + \di \int_{[t_0,t[_\T} f ( q(\tau),\tau) \; \DD \tau. \end{array}}}$$ From Lemma \[lem22-1\], one can easily prove by induction that $$\Vert F^k (q_1)(t) - F^k (q_2)(t) \Vert \leq \dfrac{L^k}{k!} \Vert q_1 - q_2 \Vert_\infty (t-t_0)^k,$$ for every $k\in \N^*$, all $q_1, q_2 \in \CC(\T,\R^n)$, and every $t \in \T$. Then, $$\Vert F^k (q_1) - F^k (q_2) \Vert_\infty \leq \dfrac{(L(b-a))^k}{k!} \Vert q_1 - q_2 \Vert_\infty ,$$ for every $k\in \N^*$, all $q_1, q_2 \in \CC(\T,\R^n)$. Therefore $F$ admits a contraction iterate and thus has a unique fixed point that is a global solution of $\mathrm{(\DD\text{-}CP)}$. This concludes the proof of Theorem \[thm22-1\].
Further comments for the shifted case {#app2}
-------------------------------------
An important remark in the *shifted* case is the following. Let $(a,b) \in \T^2$ satisfying $\ab$ and let $q:[a,b]_\T\rightarrow\Omega$. Since $\sigma(t) \in [a,b]_\T$ for every $t \in [a,b[_\T$, $q^\sigma$ is well defined on $[a,b[_\T$. This remark permits to derive all results of Section \[section2\] in a similar way since $\DD$-integrals are considered on intervals of the form $[a,b[_\T$.
For example, if $f$ satisfies , then for all $(a,b)\in\T^2$ such that $a<b$, $$f(q^\sigma,t) \in \L^\infty_\T ([a,b[_\T,\R^n) \subset \L^1_\T ([a,b[_\T,\R^n),$$ for every $q \in \CC ([a,b]_\T,\R^n)$. This remark permits to prove (from Section \[section13\]) the following $\DD$-integral characterization of the solutions of $\mathrm{(\DD\text{-}CP^\sigma)}$.
\[prop31-1\] Let $I_\T \in \I$ and $q:I_\T\rightarrow\Omega$. If $f$ satisfies , then the couple $(q,I_\T)$ is a solution of $\mathrm{(\DD\text{-}CP^\sigma)}$ if and only if for all $a,b \in I_\T$ satisfying $\ab$, one has $q \in \CC([a,b]_\T,\R^n)$ and $$q(t) = \left\lbrace \begin{array}{lcc}
q_0 + \int_{[t_0,t[_\T} f(q^\sigma (\tau),\tau) \; \DD \tau & \text{if} & t \geq t_0, \\
q_0 - \int_{[t,t_0[_\T} f(q^\sigma (\tau),\tau) \; \DD \tau & \text{if} & t \leq t_0. \\
\end{array} \right.$$ for every $t \in [a,b]_\T$.
All results permitting to prove Theorems \[thm31-1\] and \[thm31-2\] can be derived as in Section \[app1\]. Nevertheless, in order to derive Theorem \[thm32-1\], the following result is required.
\[lem32-1\] If $t_0 = \max \T$ then $$\int_{[t,t_0[_\T} (t_0-\sigma(\tau))^k \; \DD \tau \leq \dfrac{(t_0-t)^{k+1}}{k+1},$$ for every $k \in \N$ and every $t \in \T$.
One has $$\int_{[t,t_0[_\T} (t_0-\sigma(\tau))^k \; \DD \tau = \di \int_{[t,t_0[_\T} (t_0-\tau)^k \; d\tau + \di \sum_{r \in [t,t_0[_\T \cap \RR} \mu (r) (t_0-\sigma(r))^k ,$$ for every $k \in \N$ and every $t \in \T$. Since $$\begin{split}
\sum_{r \in [t,t_0[_\T \cap \RR} \mu (r) (t_0-\sigma(r))^k &= \di \sum_{r \in [t,t_0[_\T \cap \RR} \di \int_{]r,\sigma(r)[} (t_0-\sigma(r))^k \; d\tau \\
&\leq \di \sum_{r \in [t,t_0[_\T \cap \RR} \di \int_{]r,\sigma(r)[} (t_0-\tau)^k \; d\tau,
\end{split}$$ we infer that $$\int_{[t,t_0[_\T} (t_0-\sigma(\tau))^k \; \DD \tau \leq \di \int_{[t,t_0[} (t_0-\tau)^k \; d\tau = \dfrac{(t_0-t)^{k+1}}{k+1},$$ and the statement follows.
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[^1]: Laboratoire de Mathématiques et de leurs Applications - Pau (LMAP). UMR CNRS 5142. Université de Pau et des Pays de l’Adour. `[email protected]`
[^2]: Université Pierre et Marie Curie (Univ. Paris 6) and Institut Universitaire de France, CNRS UMR 7598, Laboratoire Jacques-Louis Lions, F-75005, Paris, France. `[email protected]`
[^3]: Actually, the present article was motivated by the needs of completing the existing results on Cauchy-Lipschitz theory on time scales, in order to investigate general non linear control systems on time scales, and more precisely to derive a general version of the Pontryagin Maximum Principle in optimal control.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We study topological entropy of exactly Devaney chaotic maps on totally regular continua, i.e. on (topologically) rectifiable curves. After introducing the so-called $P$-Lipschitz maps (where $P$ is a finite invariant set) we give an upper bound for their topological entropy. We prove that if a non-degenerate totally regular continuum $X$ contains a free arc which does not disconnect $X$ or if $X$ contains arbitrarily large generalized stars then $X$ admits an exactly Devaney chaotic map with arbitrarily small entropy. A possible application for further study of the best lower bounds of topological entropies of transitive/Devaney chaotic maps is indicated.'
address: 'Department of Mathematics, Faculty of Natural Sciences, Matej Bel University, Tajovského 40, 974 01 Banská Bystrica, Slovakia'
author:
- Vladimír Špitalský
title: |
Entropy and exact Devaney chaos\
on totally regular continua
---
[^1]
Introduction {#S:intro}
============
A *(discrete) dynamical system* is a pair $(X,f)$ where $X$ is a compact metric space and $f:X\to X$ is a continuous map. For $n\in{\mathbb N}$ we denote the composition $f\circ f\circ \dots\circ f$ ($n$-times) by $f^n$. A point $x\in X$ is a *periodic point* of $f$ if $f^n(x)=x$ for some $n\in{\mathbb N}$. We say that a dynamical system $(X,f)$ is *(topologically) transitive* if for every nonempty open sets $U,V\subseteq X$ there is $n\in{\mathbb N}$ such that $f^n(U)\cap V\ne\emptyset$. If every iterate $f^n$ ($n\ge 1$) is transitive we say that $f$ is *totally transitive*. If $(X\times X, f\times f)$ is transitive then $(X,f)$ is called *(topologically) weakly mixing*. A system $(X,f)$ is *(topologically) mixing* provided for every nonempty open sets $U,V\subseteq X$ there is $n_0\in{\mathbb N}$ such that $f^n(U)\cap
V\ne\emptyset$ for every $n\ge n_0$. Further, $(X,f)$ is *(topologically) exact* or *locally eventually onto* if for every nonempty open subset $U$ of $X$ there is $n\in{\mathbb N}$ such that $f^n(U)=X$. Finally, a system $(X,f)$ is *Devaney chaotic* (*totally Devaney chaotic*, *exactly Devaney chaotic*) provided $X$ is infinite, $f$ is transitive (totally transitive, exact, respectively) and has dense set of periodic points.
One of the main quantitative characteristics of a dynamical system $(X,f)$ is the *topological entropy*, denoted by $h(f)$. A challenging topic in discrete dynamics is the study of relationships between entropy and qualitative properties of dynamical systems, such as transitivity, exactness and (exact) Devaney chaos. This goes back to 1982 when Blokh in [@Blo82] showed that every transitive map on the unit interval $[0,1]$ has entropy at least $(1/2) \log
2$ and that this is the best possible bound, i.e. ${I^{\operatorname{T}}}([0,1])=(1/2) \log 2$, where $${I^{\operatorname{T}}}(X)=\inf \{h(f):\quad f:X\to X \text{ is transitive}\}.$$ Moreover, this infimum is in fact the minimum, i.e. the infimum is attainable.
Instead of ${I^{\operatorname{T}}}(X)$ one can study infima/minima of entropies of more restrictive classes of dynamical systems on $X$. Denote by ${I^{\operatorname{D}}}(X)$ and ${I^{\operatorname{ED}}}(X)$ the infima of entropies of Devaney chaotic and exactly Devaney chaotic systems on $X$, respectively. We see at once that ${I^{\operatorname{T}}}(X)\le {I^{\operatorname{D}}}(X)\le {I^{\operatorname{ED}}}(X)$. For (connected) *graphs*, these infima and the existence of corresponding minima were studied e.g. in [@Blo87; @AKLS; @ABLM; @ARR; @ALM; @Ye; @Bal01; @Ruette; @KM; @HKO11]; see Table \[tab:ResultsOnGraphs\] for some of the results. (For the definition of classes $\mathcal{P}(i)$ of trees see [@Ye].) Every transitive graph map which is not conjugate to an irrational rotation of the circle has dense periodic points [@Blo84], so ${I^{\operatorname{D}}}(X)={I^{\operatorname{T}}}(X)$ whenever $X$ is a graph. Notice also that any exact map on a non-degenerate space $X$ has positive entropy, see e.g. [@KM]; hence the infimum is never attained if ${I^{\operatorname{ED}}}(X)=0$.
[![width 1pt]{}l![width 1pt]{}c|c![width 1pt]{}c|c![width 1pt]{}]{} The space $X$ & ${I^{\operatorname{T}}}(X)$ / ${I^{\operatorname{D}}}(X)$ & Attainable? & ${I^{\operatorname{ED}}}(X)$ & Att.?\
interval & $\frac 12\log 2$ & yes & $\frac 12\log 2$ & no\
circle & $0$ & yes/no & $0$ & no\
$n$-star & $\frac 1n \log 2$ & yes & $\frac 1n \log 2$ & ?\
tree with $n$ ends & $\ge \frac 1n \log 2$ & depends on $X$ & $\ge \frac 1n \log 2$ & ?\
tree $\in\mathcal{P}(i)$, $n$ ends & $\le \frac 1{n-i} \log 2$ & depends on $X$ & $\le \frac 1{n-i} \log 2$ & ?\
graph $\ne$ tree, circle & $0$ & no & $0$ & no\
In the present paper we study ${I^{\operatorname{ED}}}(X)$ for [totally regular continua]{} $X$. Recall that a *continuum* is a connected compact metric space. A continuum $X$ is *totally regular* [@Nik] if for every point $x\in X$ and every countable set $P\subseteq X$ there is a basis of neighborhoods of $x$ with finite boundary not intersecting $P$. Equivalently one can define totally regular continua as those continua $X$ which admit a compatible metric $d$ such that $(X,d)$ is a *rectifiable curve* (i.e. $(X,d)$ has finite one-dimensional Hausdorff measure ${{\mathcal{H}}^1_{d}(X)}$). Every *graph* and even every *local dendrite* (a locally connected continuum containing only finitely many simple closed curves) is totally regular. As another example one can mention the Hawaiian earrings (an infinite wedge of circles), which is a totally regular continuum but not a local dendrite.
Recall that an *arc* is a homeomorphic image of $[0,1]$. A *free arc* in $X$ is an arc $A\subseteq X$ such that every non-end point of $A$ is an interior point of $A$ (in the topology of $X$). We say that a free arc $A\subseteq X$ *does not disconnect* $X$ if no non-end point of $A$ disconnects $X$; for locally connected continua (hence also for totally regular ones) this is equivalent to the fact that $A$ is a subset of a simple closed curve in $X$, see e.g. [@Why42 IV.9.3]. We say that $X$ *contains arbitrarily large generalized stars* if for any $k\in{\mathbb N}$ there are a point $a\in X$ and $k$ components of $X\setminus\{a\}$ such that the closures of the components are homeomorphic relative to $a$ (i.e. the corresponding homeomorphisms fix the point $a$).
In [@SpA] we introduced the so-called length-expanding Lipschitz maps (briefly LEL-maps) and we showed that for every two non-degenerate totally regular continua $X,Y$ there exists a LEL-map $f:X\to Y$; see Section \[SS:tentLike\] for details. LEL-maps form “building blocks” in our constructions of exactly Devaney chaotic systems with small entropy. Our main results are summarized in the following theorem.
For every non-degenerate totally regular continuum $X$ it holds that ${I^{\operatorname{ED}}}(X) < \infty$. Moreover, if $X$ contains arbitrarily large generalized stars or $X$ contains a free arc which does not disconnect $X$, then $X$ admits
- a Devaney chaotic map which is not totally Devaney chaotic, and
- an exactly Devaney chaotic map,
both with arbitrarily small positive entropy; hence ${I^{\operatorname{ED}}}(X)=0$.
Thus, in particular, ${I^{\operatorname{ED}}}(X)$ and hence also ${I^{\operatorname{D}}}(X)$ and ${I^{\operatorname{T}}}(X)$ are zero for many dendrites, e.g. for the $\omega$-star or the Wazewski’s universal dendrite. Still, the following general question is open.
Is it true that ${I^{\operatorname{T}}}(X)=0$ for every dendrite $X$ which is not a tree?
Our result gives ${I^{\operatorname{ED}}}(X)=0$ also for many spaces $X$ which are not dendrites, say for the Hawaiian earrings. Since every graph which is not a tree contains a free non-disconnecting arc, Main Theorem is a generalization of [@ARR Theorems 3.7 and 4.1], cf. the last row of Table \[tab:ResultsOnGraphs\]. Let us note that ${I^{\operatorname{ED}}}(X)=0$ is true also for other subclasses of the class of totally regular continua, see e.g. Lemma \[T:I(free-arc)\] and Remark \[R:generalizations\]. We even believe that the following conjecture is true.
${I^{\operatorname{ED}}}(X)=0$ for every totally regular continuum $X$ which is not a tree.
A potential application of our results lies in the fact that the estimates of ${I^{\operatorname{T}}}(X)$, ${I^{\operatorname{D}}}(X)$ and ${I^{\operatorname{ED}}}(X)$ for one-dimensional continua can be used to estimate the corresponding quantities for some higher-dimensional continua. To do that, one needs two tools. The first one is an appropriate extension theorem. For instance, in [@AKLS] and [@KM11] it was proved that a transitive map $f:X\to X$ on a compact metric space $X$ without isolated points can be extended to a transitive map $F$ on $X\times [0,1]$ without increasing the entropy. An analogous result is true for Devaney chaotic maps [@BS]. The second tool are procedures enabling to go from $X\times[0,1]$ to the cone or the suspension over $X$, or to the space $X\times{\mathbb S}^1$, similarly as in [@BS]. However, it is not the topic of the present paper to continue in this direction.
Finally let us notice that the study of infima of entropies of transitive/Devaney chaotic maps on a given space is naturally connected with the question whether the space admits a *zero entropy* transitive/Devaney chaotic map. Using techniques developed in [@BS] one can easily construct a Devaney chaotic map on the Cantor fan (the cone over the Cantor set) which has zero entropy. On the other hand, if $X$ is a compact metric space containing a free arc and $X$ is not a union of finitely many disjoint circles, then every transitive map on $X$ has positive entropy [@DSS]. However, the following question is still open.
Is there a dendrite admitting a transitive map with zero entropy?
The paper is organized as follows. In Section \[S:preliminaries\] we recall all the needed definitions and facts. In Section \[S:entropyBound\] we introduce the so-called $P$-Lipschitz maps and we give an upper bound for their topological entropy. Finally, in Section \[S:applications\] we prove [Main Theorem]{}.
Preliminaries {#S:preliminaries}
=============
Here we briefly recall all the notions and results which will be needed in the rest of the paper. The terminology is taken mainly from [@Kur2; @Nad; @Macias; @Fal].
If $M$ is a set, its cardinality is denoted by ${\#M}$. The cardinality of infinite countable sets is denoted by $\aleph_0$. If $M$ is a singleton set we often identify it with its only point. We write ${\mathbb N}$ for the set of positive integers $\{1,2,3,\dots\}$, ${\mathbb R}$ for the set of reals and ${I}$ for the unit interval $[0,1]$. By an interval we mean any nonempty connected subset of ${\mathbb R}$ (possibly degenerate to a point). For intervals $J,J'$ we write $J\le J'$ if $t\le s$ for every $t\in J$, $s\in J'$.
By a *space* we mean any nonempty metric space. A space is called *degenerate* provided it has only one point; otherwise it is called *non-degenerate*. If $E$ is a subset of a space $X=(X,d)$ we denote the closure, the interior and the boundary of $E$ by ${\overline{E}}$, ${\operatorname{int}(E)}$ and ${\partial {E}}$, respectively, and we write $d(E)$ for the diameter of $E$. We say that two sets $E,F\subseteq X$ are *non-overlapping* if they have disjoint interiors. For $x\in X$ and $r>0$ we denote the closed ball with the center $x$ and radius $r$ by $B(x,r)$.
Let $(X,f)$ be a dynamical system. A set $A\subseteq X$ is called *$f$-invariant* if $f(A)\subseteq A$. For the definitions of (total) transitivity, (weak, strong) mixing, exactness and (exact) Devaney chaos see Section \[S:intro\].
Minkowski dimension and topological entropy {#SS:minkDim}
-------------------------------------------
A subset $S$ of a metric space $(X,d)$ is called *${\varepsilon}$-separated* if $d(x,y)>{\varepsilon}$ for every points $x\ne y$ from $S$. A subset $R$ of $X$ is said to *${\varepsilon}$-span* $X$ if for every $x\in X$ there is $y\in R$ with $d(x,y)\le {\varepsilon}$. We will denote by $s(X,{\varepsilon})$ and $r(X,{\varepsilon})$ the maximal cardinality of ${\varepsilon}$-separated set and the minimal cardinality of ${\varepsilon}$-spanning set, respectively. We see at once that $r(X,{\varepsilon}) \le s(X,{\varepsilon})\le r(X,{\varepsilon}/2)$. The *upper* and *lower Minkowski (ball) dimensions* of $X$ are defined by (see e.g [@Mattila 5.3] or [@Katok Definition 3.2.7]) $${\overline{\dim}}(X)
= \limsup\limits_{{\varepsilon}\to 0} \frac{\log r(X,{\varepsilon})}{-\log {\varepsilon}}
\qquad\text{and }\qquad
{\underline{\dim}}(X)
= \liminf\limits_{{\varepsilon}\to 0} \frac{\log r(X,{\varepsilon})}{-\log {\varepsilon}}
\,.$$
Let $(X,f)$ be a dynamical system. For $n\in{\mathbb N}$ define a metric $d_n$ on $X$ by $d_n(x,y) = \max\limits_{0\le i<n} d(f^i(x),f^i(y))$. We say that a set $S\subseteq X$ is *$(n,{\varepsilon})$-separated* if it is an ${\varepsilon}$-separated subset of $(X,d_n)$, i.e. if for every distinct $x,y\in S$ there is $i<n$ with $d(f^i(x),f^i(y))>{\varepsilon}$. A set $R\subseteq X$ is said to *$(n,{\varepsilon})$-span* $X$ if it ${\varepsilon}$-spans $(X,d_n)$. Let $s_n(X,{\varepsilon},f)$ and $r_n(X,{\varepsilon},f)$ denote the maximal cardinality of $(n,{\varepsilon})$-separated subsets and the minimal cardinality of $(n,{\varepsilon})$-spanning subsets of $X$, respectively. Then, by [@Bowen71] and [@Dinaburg], the *topological entropy $h(f)$* of the system $(X,f)$ is given by $$\label{EQ:BowenDefEntropy}
h(f)
= \lim\limits_{{\varepsilon}\to 0} \limsup\limits_{n\to\infty} \frac 1n \log r_n(X,{\varepsilon},f)
= \lim\limits_{{\varepsilon}\to 0} \limsup\limits_{n\to\infty} \frac 1n \log s_n(X,{\varepsilon},f) \,.$$
Continua {#SS:continua}
--------
A *continuum* is a connected compact metric space. A *cut point* (or a *separating point*) of a continuum $X$ is any point $x\in X$ such that $X\setminus\{x\}$ is disconnected. A point $x$ of a continuum $X$ is called a *local separating point* of $X$ if there is a connected neighborhood $U$ of $x$ such that $U\setminus\{x\}$ is not connected. If $a, b$ are points of $X$ then any cut point of $X$ such that $a,b$ belong to different components of $X\setminus\{x\}$ is said to *separate $a,b$*. The set of all such points is denoted by ${\operatorname{Cut}}(a,b)$ or ${\operatorname{Cut}}_X(a,b)$. If $a=b$ then obviously ${\operatorname{Cut}}(a,b)=\emptyset$.
Let $X$ be a continuum. A metric $d$ on $X$ is said to be *convex* provided for every distinct $x,y\in X$ there is $z\in X$ such that $d(x,z)=d(z,y)=d(x,y)/2$. By [@BingPartSet Theorem 8] every locally connected continuum admits a compatible convex metric.
If $X$ is a continuum, we introduce the following notion. A *splitting* of $X$ is any system ${\mathcal{A}}=\{X_1,\dots,X_n\}$ of non-degenerate subcontinua covering $X$ such that $P_{\mathcal{A}}=\bigcup_{i\ne j} X_i\cap X_j$ is finite. In a splitting ${\mathcal{A}}$, every $X_i\in{\mathcal{A}}$ is *regular closed* (i.e. $X_i$ is closed and ${\operatorname{int}(X_i)}$ is dense in it), ${\operatorname{int}(X_i)}=X_i\setminus\left( \bigcup_{j\ne i} X_j \right)\supseteq
X_i\setminus P_{\mathcal{A}}$ and $\partial{X_i}\subseteq P_{\mathcal{A}}$.
\[L:convexMetricOnUnion\] Let $X$ be a continuum and ${\mathcal{A}}=\{X_1,\dots,X_n\}$ be a splitting of $X$. For $i=1,\dots,n$ let $d_i$ be a convex metric on $X_i$ satisfying $d_i(x,y)=d_j(x,y)$ whenever $i\ne j$ and $x,y\in X_i\cap X_j$. For $x,y\in X$ put $$\label{EQ:convexMetricOnUnion}
\begin{split}
d(x,y) = \inf\Bigg\{
&\sum_{k=1}^m d_{i_k}(x_{k-1},x_k):\
m\ge 1,\
x_0=x,\ x_m=y
\\
&\text{ and }
x_{k-1},x_k\in X_{i_k} \text{ for } k=1,\dots,m
\Bigg\}.
\end{split}$$ Then $d$ is a convex metric on $X$ equivalent with the original one and the infimum in (\[EQ:convexMetricOnUnion\]) is in fact the minimum. Moreover, if $d(x,y)=d_i(x,y)$ whenever $x,y\in P_{\mathcal{A}}\cap X_i$, then $$d|_{X_i\times X_i} = d_i.$$
Using convexity of every $d_i$ we may assume that, in the infimum from (\[EQ:convexMetricOnUnion\]), $i_{k}\ne i_{k+1}$, $x_k\in P_{\mathcal{A}}$ for $1\le k<m$ and $x_k\ne x_l$ for every $k\ne l$. Since $P_{\mathcal{A}}$ is finite we have that, for any fixed $x,y$, the infimum is in fact the minimum and there are $m\ge 1$, $i_1,\dots,i_m$, $x_0=x,x_1,\dots,x_m=y$ such that $x_0,\dots,x_{m}$ are pairwise distinct, $x_0\in X_{i_1}$, $x_m\in X_{i_m}$, $$\label{EQ:convexMetricOnUnion2}
d(x,y) =
\sum_{k=1}^m d_{i_k}(x_{k-1},x_k)
\quad
\text{and}
\quad
x_k\in X_{i_k}\cap X_{i_{k+1}}\cap P_{\mathcal{A}}\quad \text{for } 1\le k< m.$$
The fact that $d$ is a convex pseudometric is straightforward and, by (\[EQ:convexMetricOnUnion2\]), $d(x,y)=0$ implies $x=y$. To show that $d$ is equivalent with the original metric, fix a sequence $(x_k)_k$ and a point $x$; since $X$ is compact we only need to prove that $x_k\to x$ implies $d(x_k,x)\to 0$. It is no loss of generality in assuming that, for some $i$, $x_k\in X_i$ for every $k$. Since $x_k\to x$, also $x\in X_i$ and $d(x_k,x)\le d_i(x_k,x)\to 0$.
Now fix $i$ and assume that $d(x,y)=d_i(x,y)$ for every $x,y\in P_{\mathcal{A}}\cap X_i$. Take arbitrary $x,y\in X_i$. Then $d(x,y)\le d_i(x,y)$ by (\[EQ:convexMetricOnUnion\]). To show the opposite inequality take $i_1,\dots,i_m$ and $x_0=x,x_1,\dots,x_m=y$ satisfying (\[EQ:convexMetricOnUnion2\]); we can assume that $i_1=i_m=i$ (so possibly $x_0=x_1$ or $x_{m-1}=x_m$). Since $x_1,x_{m-1}\in P_{\mathcal{A}}\cap X_i$, we get $d(x_1,x_{m-1})=d_i(x_1,x_{m-1})$ and $$d_i(x,y)
\le d_i(x,x_1)+d_i(x_1,x_{m-1}) + d_i(x_{m-1},y)
\le \sum_{k=1}^m d_{i_k}(x_{k-1},x_k)=d(x,y) \,.$$ Thus $d(x,y)=d_i(x,y)$ for every $x,y\in X_i$.
Hausdorff one-dimensional measure {#SS:hausdorffMeasure}
---------------------------------
For a Borel subset $B$ of a metric space $(X,d)$ the *one-dimensional Hausdorff measure* of $B$ is defined by $${{\mathcal{H}}^1_{d}(B)}=\lim_{\delta\to 0} {{\mathcal{H}}^1_{d,\delta}(B)},
\quad
{{\mathcal{H}}^1_{d,\delta}(B)} = \inf\left\{
\sum_{i=1}^\infty d(E_i):\ B\subseteq \bigcup_{i=1}^\infty E_i,\ d(E_i)<\delta
\right\}.$$ We say that $(X,d)$ has *finite length* if ${{\mathcal{H}}^1_{d}(X)}<\infty$. By e.g. [@Fre92 Proposition 4A], $$\label{EQ:length-diam}
{{\mathcal{H}}^1_{d}(C)}\ge d(C)
\qquad
\text{whenever }
C
\text{ is a connected Borel subset of }
X.$$ If $A\subseteq X$ is an arc then ${{\mathcal{H}}^1_{d}(A)}$ is equal to the length of $A$ [@Fal Lemma 3.2]. In the case when $(X,d)$ is the Euclidean real line ${\mathbb R}$ and $J\subset {\mathbb R}$ is an interval, ${{\mathcal{H}}^1_{d}(J)}$ is equal to the length of $J$ and we denote it simply by ${\lvertJ\rvert}$.
If $(X,d)$ is a continuum of finite length endowed with a convex metric $d$, then it has the so-called *geodesic property* (see e.g. [@Fre92 Corollary 4E]): for every distinct $x,y\in X$ there is an arc $A$ with end points $x,y$ such that $d(x,y)={{\mathcal{H}}^1_{d}(A)}$; any such arc $A$ is called a *geodesic arc* or shortly a *geodesic*. Every subarc of a geodesic is again a geodesic. If $x,y$ are the end points of a geodesic $A$ and $z\in A$ then $d(x,y)=d(x,z)+d(z,y)$.
Lipschitz maps {#SS:lipschitzMaps}
--------------
A map $f:(X,d)\to (Y,\varrho)$ between metric spaces is called *Lipschitz* with a Lipschitz constant $L\ge 0$, shortly *Lipschitz-$L$*, provided $\varrho(f(x),f(x')) \le L\cdot
d(x,x')$ for every $x,x'\in X$; the smallest such $L$ is denoted by ${\operatorname{Lip}}(f)$ and is called the *Lipschitz constant* of $f$. If $f:X\to Y$ is Lipschitz-$L$ then ${{\mathcal{H}}^1_{\varrho}(f(B))} \le L\cdot {{\mathcal{H}}^1_{d}(B)}$ for every Borel set $B\subset X$ such that $f(B)$ is Borel-measurable [@Fal p. 10]. We will use the following simple fact.
\[L:pwLip\] Let $X$ be a continuum with a convex metric $d$ and let ${\mathcal{A}}=\{X_1,\dots,X_n\}$ be a splitting of $X$. Let $(Y,\varrho)$ be a metric space. Then any map $f:(X,d)\to(Y,\varrho)$ with Lipschitz-$L$ restrictions $f|_{X_i}$ ($i=1,\dots,n$) is Lipschitz-$L$.
By Lemma \[L:convexMetricOnUnion\], for any $x,y\in X$ there are $x_0=x,x_1,\dots,x_{m-1},x_m=y$ such that $x_{k-1},x_k\in X_{i_k}$ ($k=1,\dots,m$) and $d(x,y)=\sum_{k=1}^m d(x_{k-1},x_k)$. So $$\varrho(f(x),f(y))
\le \sum_{k=1}^m \varrho(f(x_{k-1}),f(x_k))
\le L\cdot \sum_{k=1}^m d(x_{k-1},x_k)
= L\cdot d(x,y).$$
Totally regular continua {#SS:totallyRegular}
------------------------
By e.g. [@Kur2; @Nik], a continuum $X$ is called
- a *dendrite* if it is locally connected and contains no simple closed curve;
- a *local dendrite* if it is locally connected and contains at most finitely many simple closed curves;
- *completely regular* if it contains no non-degenerate nowhere dense subcontinuum;
- *totally regular* if for every $x\in X$ and every countable set $P\subseteq X$ there is a basis of neighborhoods of $x$ with finite boundary not intersecting $P$;
- *regular* if every $x\in X$ has a basis of neighborhoods with finite boundary;
- *hereditarily locally connected* if every subcontinuum of $X$ is locally connected;
- a *curve* if it is one-dimensional.
Notice that (local) dendrites as well as completely regular continua are totally regular and (totally) regular continua are hereditarily locally connected, hence they are locally connected curves.
We will need the following simple estimate of the maximal cardinality of ${\varepsilon}$-separated sets in totally regular continua $X$. Notice that an analogous estimate follows from the fact that the $1$-dimensional Minkowski content is equal to ${{\mathcal{H}}^1(X)}$ for continua $X\subseteq {\mathbb R}^n$ of finite length ([@Federer Theorem 3.2.39], see also [@Mattila p. 75]).
\[L:separatedSetsInContinuaOfFiniteLength\] Let $X=(X,d)$ be a non-degenerate totally regular continuum. Then $${\underline{\dim}}(X)={\overline{\dim}}(X)=1
\quad\text{and}\quad
s(X,{\varepsilon}) \le \frac{2\cdot{{\mathcal{H}}^1(X)}}{{\varepsilon}}
\quad\text{for every }0<{\varepsilon}<d(X)\,.$$
Let us first prove the inequality for $s(X,{\varepsilon})$. There is no loss of generality in assuming ${{\mathcal{H}}^1(X)}<\infty$. By (\[EQ:length-diam\]), $$\label{EQ:lengthOfXcapBall}
{{\mathcal{H}}^1(B(x,r))} \ge r
\qquad
\text{for every } x\in X
\text{ and } 0<r<d(X)/2 \,.$$ Take $0<{\varepsilon}<d(X)$ and an ${\varepsilon}$-separated set $S=\{x_1,\dots,x_k\}$ of maximal cardinality $k=s(X,{\varepsilon})$. Since $d(x_i,x_j)>{\varepsilon}$ for every $i\ne j$, the closed balls $B(x_i,{\varepsilon}/2)$ are disjoint. Applying (\[EQ:lengthOfXcapBall\]) we conclude that $
{{\mathcal{H}}^1(X)}
\ge \sum_i {{\mathcal{H}}^1(B(x_i,{\varepsilon}/2))}
\ge k {\varepsilon}/2
$, so $s(X,{\varepsilon})\le 2{{\mathcal{H}}^1(X)}/{\varepsilon}$.
The equality ${\underline{\dim}}(X)={\overline{\dim}}(X)=1$ is a special case of [@Federer Theorem 3.2.39]. (Notice that ${\overline{\dim}}(X)\le 1$ immediately follows from the upper estimate for $s(X,{\varepsilon})$.)
The following lemma will be used in the proof of Lemma \[P:exactLipschitz\].
\[L:totRegCircle\] If $X$ is a totally regular continuum which is not a dendrite, then there are points $a,b\in X$ and subcontinua $X_0,X_1$ of $X$ such that $X_0\cup X_1=X$, $X_0\cap X_1=\{a,b\}$ and ${\operatorname{Cut}}_{X_i}(a,b)$ is uncountable for $i=0,1$.
By [@BNT] we can write $X$ as the inverse limit ${\varprojlim}(X_n,f_n)$, where $X_n$ are graphs and $f_n:X_{n+1}\to X_n$ are monotone; let $\pi_n:X\to X_n$ ($n\in{\mathbb N}$) be the natural projections. Since $X$ is not a dendrite, by [@Nad Theorem 10.36] there is $m$ such that $X_m$ contains a circle $S$. Let $C$ be the set of all points $x_m\in S$ which are not vertices of $X_m$ and are such that $\pi_m^{-1}(x_m)$ are singletons. Then $S\setminus C$ is countable since every system of disjoint non-degenerate subcontinua of $X$ is countable. Take $a_m\ne b_m$ from $C$ and put $a=\pi_m^{-1}(a_m)$, $b=\pi_m^{-1}(b_m)$. Write $X_m$ as the union $X_{m,0}\cup X_{m,1}$ of two non-degenerate subgraphs with $X_{m,0}\cap X_{m,1}=\{a_m,b_m\}$. Then $X_i=\pi_m^{-1}(X_{m,i})$ ($i=0,1$), being inverse limits of continua [@Macias Proposition 2.1.17], are subcontinua of $X$ covering $X$ and $X_0\cap X_1=\pi_m^{-1}(X_{m,0}\cap X_{m,1}) = \{a,b\}$.
Length-expanding Lipschitz maps on totally regular continua {#SS:tentLike}
-----------------------------------------------------------
Here we recall the main results of [@SpA]. We say that a family ${\mathcal{C}}$ of non-degenerate subcontinua of $X$ is *dense* if every nonempty open set in $X$ contains a member of ${\mathcal{C}}$. By ${\mathcal{C}}_I$ we denote the system of all non-degenerate closed subintervals of $I$; the Euclidean metric on $I$ is denoted by $d_I$.
Let $X=(X,d)$, $X'=(X',d')$ be non-degenerate (totally regular) continua of finite length and let ${\mathcal{C}},{\mathcal{C}}'$ be dense systems of subcontinua of $X,X'$, respectively. We say that a continuous map $f:X\to X'$ is *length-expanding* with respect to ${\mathcal{C}},{\mathcal{C}}'$ if there exists $\varrho>1$ (called *length-expansivity constant* of $f$) such that, for every $C\in {\mathcal{C}}$, $f(C)\in{\mathcal{C}}'$ and $$\label{EQ:defLengthExpanding}
\text{if} \quad
f(C)\ne X'
\qquad\text{then}\quad
{{\mathcal{H}}^1_{d'}(f(C))} \ge \varrho\cdot {{\mathcal{H}}^1_{d}(C)}.$$ Moreover, if $f$ is surjective and Lipschitz-$L$ we say that $f:(X,d,{\mathcal{C}})\to (X',d',{\mathcal{C}}')$ is *$(\varrho,L)$-length-expanding Lipschitz*. Sometimes we briefly say that $f$ is *$(\varrho,L)$-LEL* or only *LEL*.
\[P:tentLikeIsExact\] [ Let $f:(X,d,{\mathcal{C}})\to (X,d,{\mathcal{C}})$ be a LEL map. Then $f$ is exact and has finite positive entropy. Moreover, if $f$ is the composition $\varphi\circ\psi$ of some maps $\psi:X\to I$ and $\varphi:I\to X$, then $f$ has the specification property and so it is exactly Devaney chaotic. ]{}
For $k\ge 1$ let $f_k:I\to I$ be the piecewise linear continuous map which fixes $0$ and maps every $[(i-1)/k, i/k]$ onto $I$.
\[T:MAINa\]
For every non-degenerate totally regular continuum $X$ and every $a,b\in X$ we can find a convex metric $d=d_{X,a,b}$ on $X$ and Lipschitz surjections $\varphi_{X,a,b}:I\to X$, $\psi_{X,a,b}:X\to I$ with the following properties:
1. ${{\mathcal{H}}^1_{d}(X)}= 1$;
2. the system ${\mathcal{C}}={\mathcal{C}}_{X,a,b}=\{\varphi_{X,a,b}(J):\ J\text{ is a closed subinterval of } I\}$ is a dense system of subcontinua of $X$;
3. for every $\varrho>1$ there are a constant $L_\varrho$ (depending only on $\varrho$) and $(\varrho,L_\varrho)$-LEL maps $$\varphi:(I,d_I,{\mathcal{C}}_I)\to (X,d,{\mathcal{C}})
\quad\text{and}\quad
\psi:(X,d,{\mathcal{C}})\to (I,d_I,{\mathcal{C}}_I)$$ with $\varphi(0)=a$, $\varphi(1)=b$ , $\psi(a)=0$ and such that $\varphi=\varphi_{X,a,b}\circ f_k$, $\psi=f_l\circ \psi_{X,a,b}$ for some $k,l\ge 3$.
Moreover, if ${\operatorname{Cut}}_X(a,b)$ is uncountable, $d,\varphi,\psi$ can be assumed to satisfy
1. $d(a,b)>1/2$ and $\psi(b)=1$.
\[T:MAIN\] [ Keeping the notation from Proposition \[T:MAINa\], for every $\varrho>1$, every non-degenerate totally regular continua $X,X'$ and every points $a,b\in X$, $a',b'\in X'$ there are a constant $L_\varrho$ (depending only on $\varrho$) and $(\varrho,L_\varrho)$-LEL map $$f:(X,d_{X,a,b},{\mathcal{C}}_{X,a,b})\to (X',d_{X',a',b'},{\mathcal{C}}_{X',a',b'})$$ with $f(a)=a'$ and, provided ${\operatorname{Cut}}_X(a,b)$ is uncountable, $f(b)=b'$. Moreover, $f$ is equal to the composition $\varphi\circ\psi$ of two LEL-maps $\psi:X\to I$ and $\varphi:I\to X'$. ]{}
We will need Proposition \[T:lipschitzXtoX\], which is a simple generalization of Proposition \[T:MAIN\]. Before giving the formulation of it we need to introduce the following notation. If $p\ge 1$, $X_i$ ($i=1,\dots,p$) are non-degenerate subcontinua of $X$ and $a_i$ ($i=0,\dots,p$) are points of $X$ such that $a_0\in X_1, a_p\in X_p$, $a_i\in X_i\cap X_{i+1}$ for $1\le i<p$ and ${\mathcal{C}}_i={\mathcal{C}}_{X_i,a_{i-1},a_i}$ for $1\le i \le p$, we write $$(X,d,{\mathcal{C}}) = \coprod_{i=1}^p (X_i,d_i,{\mathcal{C}}_i)$$ if $X=\bigcup_i X_i$, $d$ is a convex metric on $X$ such that $d|_{X_i\times X_i}=d_{X_i,a_{i-1},a_i}$ for $i=1,\dots,p$ and ${\mathcal{C}}$ is the system of all unions $C_i\cup X_{i+1}\cup \dots\cup X_{j-1}\cup C_j$ where $i\le j$, $C_i\in{\mathcal{C}}_{i}$, $C_j\in{\mathcal{C}}_{j}$, and, if $i<j$, $C_i=\varphi_{X_i,a_{i-1},a_i}([s,1])$ and $C_j=\varphi_{X_j,a_{j-1},a_j}([0,t])$ for some $s,t\in{I}$. Particularly, if $p=1$ then $d=d_1$ and ${\mathcal{C}}={\mathcal{C}}_{X,a_0,a_1}$.
\[T:lipschitzXtoX\] Let $X,X'$ be non-degenerate totally regular continua. Let $a,b\in X$ and $d=d_{X,a,b}$, ${\mathcal{C}}={\mathcal{C}}_{X,a,b}$. Let $a'_0, a'_1, \dots, a'_p$ be points of $X'$ such that $$(X',d',{\mathcal{C}}') = \coprod_{i=1}^p (X_i',d_{X_i',a_{i-1}',a_i'},{\mathcal{C}}_{X_i',a_{i-1}',a_i'}).$$ Then for every $\varrho>1$ there is a map $f:X\to X'$ such that:
1. $f:(X,d)\to (X', d')$ is a Lipschitz-$\tilde{L}$ surjection, where $\tilde{L}=\tilde{L}_{p,\varrho}$ depends only on $p$ and $\varrho$;
2. for every $C\in{\mathcal{C}}$ we have $f(C)\in {\mathcal{C}}'$ and $$\text{if} \quad
f(C)\not\supseteq X'_i \quad\text{for every }i
\qquad \text{then} \quad
{{\mathcal{H}}^1_{d'}(f(C))}\ge\rho\cdot{{\mathcal{H}}^1_{d}(C)};$$
3. $f$ is the composition $\varphi\circ \psi$ of two Lipschitz-$\tilde{L}$ surjections $\psi:(X,d)\to {I}$ and $\varphi:{I}\to (X', d')$ such that $\psi(a)=0$, $\varphi(0)=a_0'$, $\varphi(1)=a_p'$ and, provided ${\operatorname{Cut}}_X(a,b)$ is uncountable, $\psi(b)=1$.
Notice that, by (b), $f(a)=a'_0$ and, provided ${\operatorname{Cut}}_X(a,b)$ is uncountable, $f(b)=a'_p$;
We may assume that $\varrho\ge 2$. For $i=1,\dots,p$ put $d_i'=d'_{X_i',a_{i-1}',a_i'}$ and ${\mathcal{C}}_i'={\mathcal{C}}'_{X_i',a_{i-1}',a_i'}$. Let $L_\varrho$ be as in Proposition \[T:MAIN\] and let $\psi:(X,d,{\mathcal{C}})\to (I,d_I,{\mathcal{C}}_I)$, $\varphi_i'=\varphi_{X_i',a_{i-1}',a_i'}\circ f_{k_i}:(I,d_I,{\mathcal{C}}_I)\to (X_i',d_i',{\mathcal{C}}_i')$ ($i=1,\dots,p$) be $(\varrho,L_\varrho)$-LEL maps with $\varphi_i'(0)=a_{i-1}'$, $\varphi_i'(1)=a_{i}'$, $\psi(a)=0$ and, provided ${\operatorname{Cut}}_X(a,b)$ is uncountable, also $\psi(b)=1$.
For $i=1,\dots,p$ let $I_i = [ (i-1)/p,\, i/p \,]$ and $g_i:I_i\to I$ be the increasing linear surjection. Define $f:X\to X'$ by $f = \varphi' \circ \psi$, where $\varphi':I\to X'$ is given by $\varphi'|_{I_i}=\varphi'_i \circ g_i$ for $i=1,\dots,p$. Then $\varphi'(i/p)=a_i'$ for $0\le i\le p$, $\varphi'$ is continuous and Lipschitz with ${\operatorname{Lip}}(\varphi')\le pL_\varrho$ by Lemma \[L:pwLip\]. Since ${\operatorname{Lip}}(f) \le p L_\varrho^2$, $f$ satisfies (a1) and (b) with $\tilde{L}=p L_\varrho^2$.
To show that $f$ satisfies also (a2) fix any $C\in{\mathcal{C}}$ and put $L=\psi(C)$ and $L_i=L\cap I_i$ for $i=1,\dots,p$; notice that $L$ and nonempty $L_i$’s are closed intervals. The fact that $f(C)=\varphi'(L)\in {\mathcal{C}}'$ immediately follows from the definitions of $f$ and ${\mathcal{C}}'$. If $L$ contains some $I_i$ then $f(C) \supseteq X_i'$. Otherwise ${\lvertL\rvert}\ge\varrho\cdot {{\mathcal{H}}^1_{d}(C)}$ since $L\ne I$ and there is $i$ such that $L=L_i\cup L_{i+1}$. We may assume that ${\lvertL_i\rvert}\ge {\lvertL\rvert}/2$. Then $${{\mathcal{H}}^1_{d'}(f(C))}
\ge {{\mathcal{H}}^1_{d'}(\varphi'_i \circ g_i (L_i))}
\ge \varrho p \cdot {\lvertL_i\rvert}
\ge \frac{\varrho^2 p}{2}\cdot {{\mathcal{H}}^1_{d}(C)}
\ge \varrho\cdot{{\mathcal{H}}^1_{d}(C)}$$ since $\varrho\ge 2$. So $f$ satisfies also (a2) and the proof is finished.
$P$-Lipschitz maps and an entropy bound {#S:entropyBound}
=======================================
In this section we introduce a special class of maps called $P$-Lipschitz maps (do not confuse with Lipschitz-$L$ maps) and we show a relatively easy way of obtaining an upper bound for their entropy (see Proposition \[P:entropyOfPLipschitz\]), which will be used in Section \[S:applications\]. For the definition of a splitting see Section \[SS:continua\].
\[D:generalizedMarkovMap\] Let $X$ be a non-degenerate continuum and $f:X\to X$ be a continuous map. Let $P$ be a finite $f$-invariant subset of $X$, ${\mathcal{A}}$ be a splitting of $X$ with $P_{\mathcal{A}}\subseteq P$ and $(L_A)_{A\in{\mathcal{A}}}$ be positive constants. Then we say that $f$ is a *$P$-Lipschitz map (w.r.t. the splitting ${\mathcal{A}}$ and the constants $(L_A)_{A\in{\mathcal{A}}}$)* if, for every $A\in{\mathcal{A}}$, $f(A)$ is non-degenerate and ${\operatorname{Lip}}(f|_A)\le L_A$.
Roughly speaking, $P$-Lipschitz maps are “piecewise-Lipschitz” maps with the “pieces” being subcontinua intersecting only in a finite invariant subset $P$. Trivially, every non-constant Lipschitz map is $P$-Lipschitz with $P=\emptyset$ and ${\mathcal{A}}=\{X\}$. On the other hand, if the metric $d$ of $X$ is convex then every $P$-Lipschitz map is Lipschitz with ${\operatorname{Lip}}(f)\le\max_{A\in{\mathcal{A}}} L_A$ (see Lemma \[L:pwLip\]).
Let $f$ be a $P$-Lipschitz map w.r.t. ${\mathcal{A}}$. For $A,B\in{\mathcal{A}}$ we write $A\to B$ or, more precisely, $A{\stackrel{f}{\rightarrow}} B$ provided $f(A)$ intersects the interior of $B$. So if $A\to B$ then $f(A)\cap B$ is a finite union of subcontinua of $X$ at least one of which is non-degenerate; the number of components of $f(A)\cap B$ is smaller than or equal to the cardinality of $\partial B$. Since every $f(A)$ is non-degenerate, for every $A\in{\mathcal{A}}$ there is $B\in{\mathcal{A}}$ with $A\to
B$ and, provided $f$ is surjective, also $C\in{\mathcal{A}}$ with $C\to A$.
The *$P$-transition graph $G_f$ of $f$* is the directed graph the vertices of which are the sets $A\in{\mathcal{A}}$ and the edges of which correspond to $A\to B$. The corresponding $01$-transition matrix will be called the *$P$-transition matrix* and will be denoted by $M_f$. For an integer $n\ge 1$ put $${\mathcal{A}}^n = \{{\mathbb{A}}=(A_0,A_1,\dots, A_{n-1}):\
A_i\in {\mathcal{A}},\ A_0\to A_1\to \dots\to A_{n-1}
\}.$$ So ${\mathcal{A}}^n$ is the set of all paths of length $n-1$ in $G_f$.
\[L:MarkovMapTrajectories\] Let $f$ be a $P$-Lipschitz map w.r.t. ${\mathcal{A}}$ and let $x\in X$. Then for every integer $n\ge 1$ there is ${\mathbb{A}}=(A_0,A_1,\dots, A_{n-1})\in{\mathcal{A}}^n$ such that $$f^i(x)\in A_i
\qquad
\text{for every } i=0,1,\dots, n-1.$$
We prove the lemma by induction. If $n=1$ then the assertion follows from the fact that ${\mathcal{A}}={\mathcal{A}}^1$ covers $X$. Assume now that the assertion of the lemma is true for some $n\ge 1$; we are going to show that it is true also for $n+1$. To this end take any point $x\in X$; we want to find ${\mathbb{A}}\in{\mathcal{A}}^{n+1}$ with $f^i(x)\in A_i$ for $i\le n$. By the induction hypothesis there is $(A_0,A_1,\dots, A_{n-1})\in{\mathcal{A}}^n$ such that $f^i(x)\in A_i$ for every $i\le n-1$. Put $y=f^{n-1}(x)\in A_{n-1}$. If $f(y)\not\in P$ then $f(y)\in{\operatorname{int}(A_n)}$ for some $A_n\in{\mathcal{A}}$; hence $A_{n-1}\to A_n$ and we can put ${\mathbb{A}}=(A_0,A_1,\dots, A_{n-1},A_n)\in{\mathcal{A}}^{n+1}$. Now assume that $f(y)\in P$. Let $B_1,\dots, B_{k}$ ($k\ge 1$) be the collection of all sets from ${\mathcal{A}}$ containing $f(y)$. Since $f(A_{n-1})$ has no isolated point (it is a non-degenerate continuum) and contains $f(y)$, it must intersect the interior of some $B_i$. If we put ${\mathbb{A}}=(A_0,A_1,\dots, A_{n-1},B_i)$, we are done.
Let $X$ be a continuum with a convex metric $d$ and let $f$ be a $P$-Lipschitz map on $X$ w.r.t. ${\mathcal{A}}$ and $(L_A)_{A\in{\mathcal{A}}}$. For every nonempty subset ${\mathcal{B}}$ of ${\mathcal{A}}$ put $$L_{{\mathcal{B}}} = \max_{A\in{\mathcal{B}}} L_A.$$ Since any $P$-Lipschitz map $f$ has ${\operatorname{Lip}}(f)\le L_{{\mathcal{A}}}$, we immediately have (see e.g. [@Katok Theorem 3.2.9]) $$h(f) \le {\overline{\dim}}(X)\cdot \log^+ L_{\mathcal{A}}$$ where $\log^+ L=\max\{\log L, 0\}$.
Assume now that $X$ is a non-degenerate totally regular continuum and the metric $d$ of $X$ is convex with ${{\mathcal{H}}^1_{d}(X)}<\infty$. Then ${\overline{\dim}}(X)=1$ (see Lemma \[L:separatedSetsInContinuaOfFiniteLength\]), hence $h(f)\le \log^+ L_{\mathcal{A}}$. But this upper bound is often too pessimistic. Consider e.g. the following example. Let $f$ and ${\mathcal{A}}=\{X_0,\dots,X_{k-1}\}$ be such that $f(X_i)\subseteq X_{(i+1) \operatorname{mod} k}$ and that $L_{X_0}=L>1$, $L_{X_i}\le 1$ for $i\ge 1$. Then $L_{\mathcal{A}}=L$ and we have $h(f)\le \log L$. But since $X_i$’s are $f^k$-invariant and ${\operatorname{Lip}}(f^k|_{X_i})\le L$, we have a much better estimate $h(f)\le (1/k) \log L$. The key point here is the fact that the sets $A\in{\mathcal{A}}$ with “large” $L_A$ occurs “rarely” in paths ${\mathbb{A}}\in{\mathcal{A}}^n$ of $f$.
To formalize this idea, for nonempty ${\mathcal{B}}\subseteq {\mathcal{A}}$ put $$\label{EQ:thetaB}
\theta_{\mathcal{B}}=\limsup\limits_{n\to\infty} \frac{k_n^{{\mathcal{B}}}}{n}
\qquad
\text{where}
\quad
k^{{\mathcal{B}}}_n
=\max\limits_{{\mathbb{A}}\in {\mathcal{A}}^{n}} {\# \{k=0,\dots, n-1: A_k\not\in{\mathcal{B}}}\} \,.$$ This quantity measures the maximal “asymptotic frequency” of occurrences of $A\in {\mathcal{A}}\setminus{\mathcal{B}}$ in paths of $f$. The following proposition asserts that if $L_{\mathcal{B}}$ is close to $1$ and $\theta_{\mathcal{B}}$ is sufficiently small, then the entropy of $f$ can be small even for large $L_{\mathcal{A}}$.
\[P:entropyOfPLipschitz\] Let $X$ be a non-degenerate totally regular continuum endowed with a convex metric $d$ such that ${{\mathcal{H}}^1_{d}(X)}<\infty$. Let $f:X\to X$ be $P$-Lipschitz w.r.t. ${\mathcal{A}}$ and $(L_A)_{A\in{\mathcal{A}}}$. Then, for every nonempty subsystem ${\mathcal{B}}$ of ${\mathcal{A}}$, $$h(f) \le \log^+ L_{\mathcal{B}}+ 2\theta_{\mathcal{B}}\log^+ L_{\mathcal{A}}.$$
Since the assertion of the proposition does not change if we replace every $L_A$ by $\max\{L_A,1\}$, we may assume that $L_A\ge 1$ for every $A\in{\mathcal{A}}$. Fix any $0<{\varepsilon}< d(X)$ such that ${\varepsilon}<{\operatorname{dist}}(A,B)$ for every disjoint $A,B\in{\mathcal{A}}$ and ${\varepsilon}< d(x,y)$ for every distinct $x,y\in P$.
Let $n\in{\mathbb N}$ and let $S$ be an $(n,{\varepsilon})$-separated subset of $X$ of maximal cardinality ${\#S}=s_n(X,{\varepsilon},f)$. Take any $x\ne y$ from $S$. By Lemma \[L:MarkovMapTrajectories\] there are ${\mathbb{A}}^x=(A_i^x)_{i<n}, {\mathbb{A}}^y=(A_i^y)_{i<n}\in{\mathcal{A}}^n$ such that $x_i = f^i(x)\in A_i^x$ and $y_i = f^i(y)\in A_i^y$ for $i<n$. Put $d_i=d(x_i,y_i)$ for $i<n$ and $j = \min\{i: d_i>{\varepsilon}\}$; then $0\le j\le n-1$.
Fix any $0\le i< j$. Let $B$ be a geodesic arc from $x_i$ to $y_i$. Then, by (\[EQ:length-diam\]), $d(B)\le {{\mathcal{H}}^1(B)}=d(x_i,y_i)\le {\varepsilon}$. By the choice of ${\varepsilon}$ the set $B$ contains at most one point from $P$. If $B\cap P=\emptyset$ then $B$ does not intersect the boundaries of $A_i^x$, $A_i^y$ and so $B\subseteq {\operatorname{int}(A_i^x)} \cap {\operatorname{int}(A_i^y)}$. Hence $A_i^x = A_i^y$ and $$\label{EQ:PLip1}
d_{i+1}\le L_{A_i^x} \cdot d_i.$$ Otherwise $B\cap P=\{z\}$ is a singleton. We claim that $z\in A_i^x$. Indeed, if $z\ne x_i$ then $x_i$ belongs to the interior of $A_i^x$. Since $B$ is connected, intersects the interior of ${A_i^x}$ and $\partial A_i^x\subseteq P$, the subarc $x_i z$ of $B$ is contained in $A_i^x$ and so $z\in A_i^x$. If $z=x_i$ then again $z\in A_i^x$. Analogously $z\in A_i^y$ and thus $z\in A_i^x\cap A_i^y$. Using the facts that $f|_A$ is Lipschitz on every $A\in{\mathcal{A}}$ and that $B$ is geodesic (hence $d(x_i,z)+d(z,y_i)=d_i$) we obtain $$\label{EQ:PLip2}
\begin{split}
d_{i+1}
&\le d(x_{i+1},f(z)) + d(f(z), y_{i+1})
\le L_{A_i^x} \cdot d(x_i,z) + L_{A_i^y} \cdot d(z,y_i)
\\
&\le \max\{L_{A_i^x}, L_{A_i^y}\} \cdot d_i \,.
\end{split}$$ Combination of (\[EQ:PLip1\]) and (\[EQ:PLip2\]) for $i<j$ gives $$d_{j}
\le d_0 \cdot \prod\limits_{i<j} \max\{L_{A_i^x}, L_{A_i^y}\}
\le d_0 \cdot \prod\limits_{i<n} \max\{L_{A_i^x}, L_{A_i^y}\} \,.$$ Thus, by the definition (\[EQ:thetaB\]) of $k_n=k_n^{\mathcal{B}}$, $${\varepsilon}< d_{j}
\le d_0 \cdot L_{{\mathcal{A}}}^{2k_n} \cdot L_{{\mathcal{B}}}^{n-2k_n}
\le d_0 \cdot L_{{\mathcal{A}}}^{2k_n} \cdot L_{{\mathcal{B}}}^{n} \,.$$ So the set $S$ is ${\varepsilon}'$-separated with ${\varepsilon}'={\varepsilon}/ (L_{{\mathcal{A}}}^{2k_n} \cdot L_{{\mathcal{B}}}^{n})$ and Lemma \[L:separatedSetsInContinuaOfFiniteLength\] gives $$s_n(X,{\varepsilon},f)
\le
s(X,{\varepsilon}')
\le \frac{2 {{\mathcal{H}}^1(X)}}{{\varepsilon}} \cdot L_{{\mathcal{A}}}^{2k_n} \cdot L_{{\mathcal{B}}}^{n} \,.$$ From this and (\[EQ:BowenDefEntropy\]), $h(f) \le 2\theta_{\mathcal{B}}\log L_{\mathcal{A}}+ \log L_{\mathcal{B}}$.
In the following example we show how Proposition \[P:entropyOfPLipschitz\] can be used to obtain a “good” upper estimate of the topological entropy.
Consider a graph $X$ which is not a tree. We can write $X=I\cup G$, where $I$ is a free arc in $X$ identified with $[0,1]$ and $G$ is a subgraph of $X$ such that $I\cap G=\{0,1\}$. Let $n\ge 3$ and $g=g_n:X\to X$ be the map constructed in the proof of [@ARR Theorem 4.1]. Recall that if we put $X_i=\left[i/n, (i+1)/n \right]$ for $0\le i<n$, $i\ne 1$ and $X_1^-=\left[ \frac{1}{n}, \frac{3}{2n} \right]$, $X_1^+=\left[ \frac{3}{2n}, \frac{2}{n} \right]$, then $g$ maps linearly $X_0$ onto $X_1^-\cup X_1^+$, $X_1^-$ onto $X_1^+$, $X_1^+$ onto $X_1^+\cup X_2$ and $X_i$ onto $X_{i+1}$ for $i=2,\dots,n-2$. Moreover, $g|_{X_{n-1}}:X_{n-1}\to G$ and $g|_G:G\to X_0$ are piecewise linear maps with the number of pieces not depending on $n$.
Put $\lambda=\sqrt[n-2]{2}$ and define the convex metric $d=d_n$ on $X$ in such a way that $d(0,1)=d\left( 0, \frac{1}{n} \right)=1$, $d\left( \frac{1}{n}, \frac{3}{2n} \right)=d\left( \frac{3}{2n}, \frac{2}{n} \right)
=\frac 12$, $d\left( \frac{i}{n}, \frac{i+1}{n} \right)=\frac 12 \lambda^{i-1}$ for $i=2,\dots,n-1$, and the Lipschitz constants of $g|_{X_{n-1}}$, $g|_G$ are bounded from above by a constant $L$ not depending on $n$. Then $g$ is $P$-Lipschitz with $P=\left\{ \frac{i}{n},\ i=0,\dots,n\right\} \cup
\left\{ \frac{3}{2n}\right\}$, ${\mathcal{A}}=\{X_0,X_1^-,X_1^+,X_2,\dots,X_{n-1},G\}$ and $L_{X_0}=L_{X_1^-}=1$, $L_{X_1^+}=L_{X_2}=\dots=L_{X_{n-2}}=\lambda$, $L_{X_{n-1}}=L_G=L$. Put ${\mathcal{B}}={\mathcal{A}}\setminus \{G,X_{n-1}\}$. Then $L_{\mathcal{A}}=L$, $L_{\mathcal{B}}=\lambda$ and an easy computation gives that $\theta_{\mathcal{B}}\le\frac{2}{n}$. According to Proposition \[P:entropyOfPLipschitz\] we have $$h(g_n)
\le \log \lambda + \frac {2}{n} \log L
\le \frac{\log(2L^2)}{n-2} \,,$$ and thus $\limsup_n h(g_n)=0$.
We finish this section with a simple lemma giving a sufficient condition for transitivity and exactness of $P$-Lipschitz maps. Recall that a $01$-square matrix $M$ is called *irreducible* if for every indices $i,j$ there is $n>0$ with $M^n_{ij}>0$; i.e. the corresponding directed graph $G_M$ is *strongly connected* (there is a path from each vertex to every other vertex). The *period $p_M$* of an irreducible matrix $M$ is the greatest common divisor of those $n>0$ for which $M^n_{ii}>0$ for some $i$. Equivalently, $p_M$ is the greatest common divisor of the lengths of cycles (closed paths) in $G_M$. An irreducible matrix $M$ and the corresponding graph $G_M$ are called *primitive* provided $p_M=1$.
\[L:exactnessOfPLipschitz\] Let $X$ be a non-degenerate totally regular continuum endowed with a convex metric $d$ such that ${{\mathcal{H}}^1_{d}(X)}<\infty$. Let a map $f:X\to X$ be $P$-Lipschitz w.r.t. ${\mathcal{A}}$ such that $A\to B$ implies $f(A)\supseteq B$ for $A,B\in{\mathcal{A}}$. Assume that there is a dense system ${\mathcal{D}}$ of subcontinua of $X$ such that for every $D\in{\mathcal{D}}$ there are $n\in{\mathbb N}$, $A\in{\mathcal{A}}$ with $f^n(D)\supseteq A$. Then the following hold:
1. $f$ is transitive but not totally transitive if and only if $M_f$ is irreducible but not primitive;
2. $f$ is exact if and only if $M_f$ is primitive.
If $M_f$ is not irreducible there are $A,B\in{\mathcal{A}}$ such that there is no path from $A$ to $B$. Hence $f^n({\operatorname{int}(A)})\cap {\operatorname{int}(B)} = \emptyset$ for every $n>0$, which contradicts transitivity of $f$. If $M_f$ is irreducible with the period $p>1$, then there are $A,B\in{\mathcal{A}}$ such that there is no path from $A$ to $B$ of length $n p$. I.e. $f^p$ is not transitive and thus $f$ is not totally transitive.
Assume now that $M_f$ is irreducible and take any nonempty open sets $U,V$. Since $\bigcup_{A\in {\mathcal{A}}} {\operatorname{int}(A)}\supseteq X\setminus P$ is dense in $X$, there is $B\in{\mathcal{A}}$ intersecting $V$. By the assumption on ${\mathcal{D}}$ there are $D\in {\mathcal{D}}$, $n\in {\mathbb N}$ and $A\in{\mathcal{A}}$ such that $D\subseteq U$ and $f^{n}(D)\supseteq A$. Since $M_f$ is irreducible there is $m$ with $f^{m}(A)\supseteq B$. Hence $f^{n+m}(D)\supseteq B$ and so $f^{n+m}(U)$ intersects $V$. Thus $f$ is transitive. Moreover, if $M_f$ is primitive there is $p$ such that, for every $C\in{\mathcal{A}}$, there is a path of length $p$ from $A$ to $C$. So $f^{n+p}(U)\supseteq
f^{n+p}(D)\supseteq f^{p}(A)=X$ and $f$ is exact.
Proof of [Main Theorem]{} {#S:applications}
=========================
In this section we prove [Main Theorem]{}. We use the notation from Section \[SS:tentLike\].
Lipschitz, exactly Devaney chaotic maps
---------------------------------------
\[P:exactLipschitz\] There is $L>0$ such that any non-degenerate totally regular continuum $X$ admits a convex metric $d$ with ${{\mathcal{H}}^1_{d}(X)}<\infty$ and Lipschitz-$L$, Devaney chaotic maps $f,g:(X,d)\to (X,d)$ with positive entropies such that $f$ is not totally transitive and $g$ is exact.
The existence of a map $g:(X,d)\to (X,d)$ was shown in [@SpA], so we only need to construct $f$. If $X$ contains a circle, there are points $a\ne b$ such that $X$ is the union of two subcontinua $X_0,X_1$ with $X_0\cap X_1=\{a,b\}$ and ${\operatorname{Cut}}_{X_i}(a,b)$ is uncountable for $i=0,1$. (Such points $a,b$ can be chosen as follows. By [@BNT] we can write $X={\varprojlim}(X_n,f_n)$, where $X_n$ are graphs and $f_n:X_{n+1}\to X_n$ are monotone; let $\pi_n:X\to X_n$ ($n\in{\mathbb N}$) be the natural projections. Since $X$ is not a dendrite, by [@Nad Theorem 10.36] there is $m$ such that $X_m$ contains a circle $S$. Let $C$ be the set of all points $x_m\in S$ which are not vertices of $X_m$ and are such that $\pi_m^{-1}(x_m)$ are singletons. Then $S\setminus C$ is countable since every system of disjoint non-degenerate subcontinua of $X$ is countable. Take $a_m\ne b_m$ from $C$ and put $a=\pi_m^{-1}(a_m)$, $b=\pi_m^{-1}(b_m)$. Write $X_m$ as the union $X_{m,0}\cup X_{m,1}$ of two non-degenerate subgraphs with $X_{m,0}\cap X_{m,1}=\{a_m,b_m\}$. Then $X_i=\pi_m^{-1}(X_{m,i})$ ($i=0,1$), being inverse limits of continua [@Macias Proposition 2.1.17], are non-degenerate subcontinua of $X$ covering $X$ and $X_0\cap X_1=\pi_m^{-1}(X_{m,0}\cap X_{m,1}) = \{a,b\}$.)
Without loss of generality we may assume that $d_{X_0,a,b}(a,b)=d_{X_1,a,b}(a,b)$ (otherwise we replace one of the metrics by a constant multiple of it). Let $d$ be the convex metric on $X$ defined by (\[EQ:convexMetricOnUnion\]) for the splitting $\{X_0,X_1\}$ and metrics $d_0=d_{X_0,a,b}$, $d_1=d_{X_1,a,b}$; by Lemma \[L:convexMetricOnUnion\], $d$ coincides with $d_{X_i,a,b}$ on $X_i$ for $i=0,1$. Let $f_0:X_0\to X_1$, $f_1:X_1\to X_0$ be maps from Proposition \[T:lipschitzXtoX\] (with $\varrho>1$, $p=1$) fixing points $a$ and $b$. Define $f:X\to X$ by $f(x)=f_i(x)$ if $x\in X_i$. Then, for $i=0,1$, $f^2|_{X_i}:X_i\to X_i$ is LEL and so, by Proposition \[P:tentLikeIsExact\], $f^2|_{X_i}$ is exactly Devaney chaotic. Thus $f$ is Devaney chaotic and has positive entropy. Since $f^2$ is not transitive, $f$ is not totally transitive.
If $X$ does not contain a circle, then $X$ is a dendrite. Take any non-end point $a\in X$ and write $X=X_0\cup X_1$, where $X_i$’s are non-degenerate subdendrites with $X_0\cap X_1=\{a\}$. Fix $b_i\in X_i\setminus\{a\}$ and notice that, since $X_i$’s are dendrites, the sets ${\operatorname{Cut}}_{X_i}(a,b_i)$ are uncountable. Now we can proceed analogously as in the non-dendrite case; the only change is that instead of $f_i(b)=b$ we require $f_0(b_0)=b_1$ and $f_1(b_1)=b_0$.
\[C:IT(X)\_is\_finite\] Every non-degenerate totally regular continuum admits a (not totally) Devaney chaotic, as well as an exactly Devaney chaotic map with finite positive entropy. Every compact metric space which is the disjoint union of finitely many non-degenerate totally regular continua admits a Devaney chaotic map with finite positive entropy.
Exactly Devaney chaotic maps with arbitrarily small entropy {#SS:exactSmallEntropy}
-----------------------------------------------------------
In what follows we show that under some conditions a totally regular continuum $X$ admits (exactly) Devaney chaotic maps with arbitrarily small entropy. We consider two subclasses of totally regular continua: those containing arbitrarily large generalized stars and those containing a non-disconnecting free arc. The constructions are modifications of those for stars [@AKLS Theorem 1.2] and for graphs which are not trees [@ARR Theorems 3.7 and 4.1]. The key difference is that instead of constructing piecewise linear Markov maps we construct $P$-Lipschitz maps using Proposition \[T:lipschitzXtoX\].
Let $X$ be a continuum and $k\ge 2$ be an integer. We say that $X$ *contains a generalized $k$-star* if there are a point $a\in X$ and $k$ components $C_1,\dots,C_k$ of $X\setminus\{a\}$ such that the closures $X_i={\overline{C_i}}=C_i\cup\{a\}$ ($i=1,\dots,k$) are homeomorphic relative to $a$; i.e. for every $i,j$ there is a homeomorphism $h_{ij}:X_i\to X_j$ fixing the point $a$ (see [Figure \[Fig:kstar\]]{}). Recall that $X$ *contains arbitrarily large generalized stars* if it contains a generalized $k$-star for every $k\ge 2$.
\[T:I(omega-star)\] There is a constant $c>0$ such that any totally regular continuum $X$ containing a generalized $k$-star ($k\ge 2$) admits a (not totally) Devaney chaotic map $f$ as well as an exactly Devaney chaotic map $g$ with positive topological entropies bounded from above by $c/k$. Consequently, if $X$ contains arbitrarily large generalized stars then ${I^{\operatorname{ED}}}(X)=0$.
![A continuum containing a generalized $k$-star[]{data-label="Fig:kstar"}](figA.eps)
Since $X$ contains a generalized $k$-star, there is $a\in X$ such that we can write $
X = X_0\cup X_1\cup\dots \cup X_{k},
$ where $X_i$’s are subcontinua of $X$, $X_i\cap X_j=\{a\}$ for every $i\ne j$ and $X_i,X_j$ are homeomorphic relative to $a$ for $i,j\ge 1$. Further, $X_1,\dots,X_k$ are non-degenerate and, by replacing $k$ with $k-1$ if necessary, we may assume that also $X_0$ is such. For $i=0,\dots,k$ put $d_i=d_{X_i,a,a}$ and ${\mathcal{C}}_i={\mathcal{C}}_{X_i,a,a}$.
We may assume that the metrics $d_i$ are such that for every $1\le i<k$ there is an isometry $f_i:(X_i,d_i)\to (X_{i+1},d_{i+1})$ fixing $a$. Since $\bigcup_{i\ne j} X_i\cap X_j=\{a\}$ is a singleton, by Lemma \[L:convexMetricOnUnion\] there is a convex metric $d$ on $X$ such that $d|_{X_i\times X_i}=d_i$ for every $i$. Let $f_{k}:X_{k}\to X_0$, $f_0:X_0\to X_1$ be maps from Proposition \[T:lipschitzXtoX\] (with $p=1$, $\varrho>1$) such that $f_{k}(a)=f_0(a)=a$. Without loss of generality we may assume that if $C\in {\mathcal{C}}_{i}$ ($i=1,\dots,k-1$) then $f_i(C)\in {\mathcal{C}}_{i+1}$. We define the map $f:X\to X$ by $f(x)=f_i(x)$ for $x\in X_i$, $i=0,\dots,k$. Proposition \[P:tentLikeIsExact\] implies that $f^{k+1}|_{X_0}: X_0\to X_0$, being LEL, is exactly Devaney chaotic. Hence $h(f)>0$, $f$ is a (not totally) Devaney chaotic map, and, since $d$ is convex, $f$ is Lipschitz with ${\operatorname{Lip}}(f)\le\tilde{L}$ by Lemma \[L:pwLip\]. Moreover, since ${\operatorname{Lip}}(f^{k+1}) \le \tilde{L}^2$ we have $h(f)\le \tilde{L}^2/(k+1)\le c/k$ for some constant $c$ depending only on $\tilde{L}$.
To construct an exactly Devaney chaotic map $g:X\to X$ we take the metric $d$ on $X$ as in the previous case and we define $g|_{X_i}=f_i$ for $i=1,\dots,k$. The only difference is the definition of $g$ on $X_0$: we put $g|_{X_0}=g_0$, where $g_0:X_0\to (X_1\cup X_2)$ is a map obtained from Proposition \[T:lipschitzXtoX\] with $p=2$, $\varrho'>2$, $X'=X_1\cup X_2$ and $a'_i=a$ for $i=0,1,2$. Then $g_0(a)=a$. By Proposition \[T:lipschitzXtoX\] we have that, for $C\in{\mathcal{C}}_{0}$, if $g_0(C)$ contains neither $X_1$ nor $X_2$, then ${g_0(C)}=C_1\cup C_2$, where $C_1\in{\mathcal{C}}_{1}$ and $C_2\in{\mathcal{C}}_{2}$ are continua and ${{\mathcal{H}}^1(g_0(C))}={{\mathcal{H}}^1(C_1)}+{{\mathcal{H}}^1(C_2)}\ge \varrho' \cdot{{\mathcal{H}}^1(C)}$. Hence $$\label{EQ:kstar1}
{{\mathcal{H}}^1(C')}
\ge
\frac{\varrho'}{2}\cdot {{\mathcal{H}}^1(C)}
\qquad
\text{for some}\quad
C'\in {\mathcal{C}}_{1}\cup {\mathcal{C}}_{2},\
C'\subseteq g(C).$$ Notice that ${\operatorname{Lip}}(g)$ is bounded from above by an absolute constant $L$, see Lemma \[L:pwLip\].
We are going to show the entropy bound for $g$ using Proposition \[P:entropyOfPLipschitz\]. Put $P=\{a\}$, ${\mathcal{A}}=\{X_0,\dots,X_k\}$, ${\mathcal{B}}=\{X_1,\dots,X_{k-1}\}$, $L_{X_i}=1$ for $1\le i<k$ and $L_{X_0}=L_{X_k}=L$. Then $g$ is $P$-Lipschitz w.r.t. ${\mathcal{A}}$ and $(L_{X_i})_i$, $L_{\mathcal{A}}=L$ and $L_{\mathcal{B}}=1$. Moreover, $X_i\to X_j$ if and only if $j=(i+1) \operatorname{mod} (k+1)$ or $i=0$, $j=2$. Fix any $n\in{\mathbb N}$ and ${\mathbb{A}}=(A_0,\dots,A_{n-1})\in{\mathcal{A}}^n$ with exactly $k_n=k_n^{\mathcal{B}}$ members from ${\mathcal{A}}\setminus{\mathcal{B}}$, see (\[EQ:thetaB\]). Let $l_1<l_2<\dots<l_{k_n}$ be the indices $l$ with $A_l\not\in{\mathcal{B}}$, i.e. with $A_l\in\{X_0,X_k\}$. Then, for any $l_j$, $1\le i\le k-1$ and $l_j+i<n$, it holds that $A_{l_j+i}\in\{X_{i-1},X_i,X_{i+1}\}$ and thus $A_{l_j+2},\dots,A_{l_j+k-2}\in {\mathcal{B}}$. So $l_{j+2}-l_j\ge k-1$ for every $j\le k_n-2$. We see at once that $n-1\ge l_{k_n} \ge (k_n-1)(k-1)/2$, i.e. $\theta_{\mathcal{B}}= \limsup_{n} k_n/n
\le
2/(k-1)$. By Proposition \[P:entropyOfPLipschitz\], $h(g)\le \log^+ L_{\mathcal{B}}+ 2\theta_{\mathcal{B}}\log^+ L_{\mathcal{A}}\le (4/(k-1)) \log L
\le c/k$ for some absolute constant $c$.
To prove that $g$ is exact we use Lemma \[L:exactnessOfPLipschitz\] with ${\mathcal{D}}= \cup_i {\mathcal{C}}_{i}$. To this end take any $i$ and $C\in {\mathcal{C}}_{i}$ and suppose that for every $n$ the set $g^n(C)$ does not contain any $X_i$. Then Proposition \[T:lipschitzXtoX\] and (\[EQ:kstar1\]) give that, for every $n\ge 1$, $${{\mathcal{H}}^1(g^n(C))} \ge q^n \cdot {{\mathcal{H}}^1(C)}
\qquad\text{with}\quad
q=\min\{\varrho,\varrho'/2\}>1.$$ (The inequality can be shown by induction as follows. Trivially it holds for $n=0$. Put $C_0=C$ and assume that, for some $n\ge 0$, there is $i$ such that the set $g^n(C)$ contains some $C_n\in {\mathcal{C}}_{i}$ with ${{\mathcal{H}}^1(C_n)} \ge q^n \cdot{{\mathcal{H}}^1(C)}$. If $i\ne 0$ then put $C_{n+1}=g(C_n)\in {\mathcal{C}}_{(i+1) \operatorname{mod} \, (k+1)}$ and use Proposition \[T:lipschitzXtoX\]. If $i=0$, put $C_{n+1}=C'$, where $C'$ is the continuum from (\[EQ:kstar1\]).) But this contradicts the fact that ${{\mathcal{H}}^1(X)}<\infty$. Hence ${\mathcal{D}}$ satisfies the assumption from Lemma \[L:exactnessOfPLipschitz\]. Notice also that $A\to B$ implies $g(A)\supseteq B$. And since the $P$-transition graph of $g$ contains cycles $X_0\to X_1\to\dots\to X_k\to X_0$ and $X_0\to X_2\to X_3\to\dots\to X_k\to X_0$ of lengths $k+1$ and $k$, the $P$-transition matrix of $g$ is primitive. Hence, by Lemma \[L:exactnessOfPLipschitz\], $g$ is exact.
To finish it suffices to show that $g$ has dense periodic points. Let $\tilde{X}=X\cup A$ be a totally regular continuum obtained from $X$ by adding to it an arc $A$ such that $A\cap X=\{a\}$. Take a convex metric $\tilde{d}$ on $\tilde{X}$ such that it coincides with $d$ on $X$ and the length of $A$ is $1$. Then there is an isometry from ${I}$ onto $A$ mapping $0$ to $a$. So, by the choice of $g_0$, there are Lipschitz maps $\psi:X_0\to A$ and $\varphi:A\to X_1\cup X_2$ with $\varphi\circ\psi=g_0$ and $\psi(a)=\varphi(a)=a$. Define the map $\tilde{g}:\tilde{X}\to \tilde{X}$ by $$\tilde{g}({x}) = \begin{cases}
g({x}) & \text{if } x\in X_1\cup \dots\cup X_k;
\\
\psi(x) & \text{if } x\in X_0;
\\
\varphi(x) & \text{if } x\in A.
\end{cases}$$ Analogously as before, $\tilde{g}$ is exact. Since $\tilde{X}$ contains a disconnecting interval (an open set homeomorphic to $(0,1)$ such that any point of it disconnects $X$ into exactly two components), $\tilde{g}$ has dense periodic points by [@AKLS Theorem 1.1]. Trivially, any periodic point of $\tilde{g}$ in $X$ is also a periodic point of $g$. So also the periodic points of $g$ are dense.
\[T:I(free-arc)\] There is $c>0$ with the following property. Let $X$ be a totally regular continuum which can be written as the union of non-degenerate subcontinua $$X = X_0\cup X_1\cup \dots\cup X_k,$$ where $k\ge 2$ and the following hold:
1. there are distinct points $a_0,\dots,a_k$ with $X_i\cap X_{(i+1) \operatorname{mod} (k+1)} = \{a_i\}$ for $i=0,\dots,k$;
2. $X_i\cap X_j=\emptyset$ whenever $2\le {\lverti-j\rvert}<k$;
3. for $i=1,\dots,k-1$ there is a homeomorphism $h_i:X_i\to X_{i+1}$ such that $h_i(a_{i-1})=a_{i}$ and $h_i(a_i)=a_{i+1}$;
4. ${\operatorname{Cut}}_{X_1}(a_0,a_1)$ is uncountable.
Then $X$ admits a (not totally) Devaney chaotic map $f$ and an exactly Devaney chaotic map $g$ with positive topological entropies bounded from above by $c/k$.
See [Figure \[Fig:freeArc\]]{} for an illustration of a continuum satisfying the assumptions of the lemma.
![A continuum from Lemma \[T:I(free-arc)\][]{data-label="Fig:freeArc"}](figB.eps)
We may assume that ${\operatorname{Cut}}_{X_0}(a_k,a_0)$ is uncountable, for if not, we replace $k$ by $k'=k-1$ and $X_{k-1}$ by $X'_{k-1}=X_{k-1}\cup X_k$ (if $k'=1$ we simply apply Lemma \[P:exactLipschitz\]).
Consider the metrics $d_i=d_{X_i,a_{i-1},a_i}$ for $i=0,\dots,k$ (with $a_{-1}:=a_k$); in view of (c) we may assume that $f_i:(X_i,d_i)\to (X_{i+1},d_{i+1})$ are isometries for $1\le i<k$. Let $d$ be the convex metric on $X$ defined by (\[EQ:convexMetricOnUnion\]). Since $\bigcup_{i\ne j} X_i\cap X_j = \{a_i:\ i=0,\dots,k\}$, $1>d_i(a_{i-1},a_i)>1/2$ and $X_i\cap X_j$ are empty or singletons for $i\ne j$, Lemma \[L:convexMetricOnUnion\] gives that $d|_{X_i\times X_i}=d_i$ for every $i$.
Let $f_k:X_k\to X_0$, $f_0:X_0\to X_1$ and $g_0:X_0\to (X_1\cup X_2)$ be maps from Proposition \[T:lipschitzXtoX\] with ($\varrho>2$) such that $f_0(a_k)=g_0(a_k)=a_0$, $f_0(a_0)=g_0(a_0)=a_1$, $f_k(a_{k-1})=a_k$ and $f_k(a_k)=a_0$. Define $f,g:X\to X$ by $$f(x)=\begin{cases}
f_0(x) &\text{if } x\in X_0;
\\
h_i(x) &\text{if } x\in X_i,\ 1\le i<k;
\\
f_k(x) &\text{if } x\in X_k;
\end{cases}
\qquad
g(x)=\begin{cases}
g_0(x) &\text{if } x\in X_0;
\\
h_i(x) &\text{if } x\in X_i,\ 1\le i<k;
\\
f_k(x) &\text{if } x\in X_k.
\end{cases}$$ Analogously as in the proof of Lemma \[T:I(omega-star)\] we can show that $f$ is (not totally) Devaney chaotic, $g$ is exact and the entropies of $f,g$ are positive and bounded from above by $c/k$, where $c$ is an absolute constant. What is left is to prove that $g$ has dense periodic points.
To this end put $P=\{a_i:\ i=0,\dots,k\}$ and realize that $g(P)=P$. Let ${\mathcal{D}}$ be the decomposition of $X$ into $P$ and singletons $\{x\}$, $x\not\in P$. Let $X'=X/{\mathcal{D}}$ (i.e. we collapse $P$ into a point), $\pi:X\to X'$ be the natural projection and $\{a'\}=\pi(P)$, $X_i'=\pi(X_i)$. Then ${X'}$ is a compact metric space since the decomposition ${\mathcal{D}}$ is upper semicontinuous. The map $g:X\to X$ induces ${g'}:{X'}\to {X'}$ which fixes ${a'}$ and, being a factor of $g$, is exact. Notice that $g'|_{X_0'}:X_0'\to X_1'\cup X_2'$ can be written as the composition $\varphi\circ \psi$ with continuous $\psi:X_0'\to{I}$ and $\varphi:{I}\to X_1'\cup X_2'$. So, analogously as in the proof of Lemma \[T:I(omega-star)\], $g'$ has dense periodic points. Hence also $g$ has dense periodic points, since whenever $x'\ne a'$ is a periodic point of $g'$, then its only $\pi$-preimage $x$ is a periodic point of $g$.
Finally, we are ready to prove our main result.
For every non-degenerate totally regular continuum $X$ it holds that ${I^{\operatorname{ED}}}(X) < \infty$. Moreover, if $X$ contains arbitrarily large generalized stars or $X$ contains a free arc which does not disconnect $X$, then $X$ admits
- a Devaney chaotic map which is not totally Devaney chaotic, and
- an exactly Devaney chaotic map,
both with arbitrarily small positive entropy; hence ${I^{\operatorname{ED}}}(X)=0$.
Note that if $X$ contains a non-disconnecting free arc $A$ then, for every $k\ge 2$, $X$ can be written as in Lemma \[T:I(free-arc)\] with $X_1,\dots,X_k$ being subarcs of $A$. So the theorem immediately follows from Corollary \[C:IT(X)\_is\_finite\] and Lemmas \[T:I(omega-star)\], \[T:I(free-arc)\].
\[R:generalizations\] Lemma \[T:I(free-arc)\] gives ${I^{\operatorname{ED}}}(X)=0$ also for some continua containing no non-disconnecting free arc. For example if, for some $m\in\{3,4,\dots,\aleph_0\}$, $X$ contains the universal dendrite $D_m$ of order $m$ with two point boundary and with connected $X\setminus {\operatorname{int}(D_m)}$, we again have ${I^{\operatorname{ED}}}(X)=0$. (Recall that $D_m$ is the topologically unique dendrite such that the branch points of it are dense and all have the order $m$.)
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[^1]: The author wishes to express his thanks to [L-0.08cm39]{}ubomír Snoha for his help with the preparation of the paper. The author was supported by the Slovak Research and Development Agency under the contract No. APVV-0134-10 and by the Slovak Grant Agency under the grant number VEGA 1/0978/11.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'In this article we show that a Dirac Hamiltonian in a curved background spacetime can be interpreted, when discretized, as a tight binding Fermi-Hubbard model with non unitary tunnelings. We find the form of the nonunitary tunneling matrices in terms of the metric tensor. The main motivation behind this exercise is the feasibility of such Hamiltonians by means of laser assisted tunnelings in cold atomic experiments. The mapping thus provide a physical interpretation of such Hamiltonians. We demonstrate the use of the mapping on the example of time dependent metric in 2+1 dimensions. Studying the spin dynamics, we find qualitative agreement with known theoretical predictions, namely the particle pair creation in expanding universe.'
author:
- Jiří Minář
- Benoît Grémaud
bibliography:
- 'CurvedDirac\_paper\_arXiv\_v2.bib'
title: 'Dirac fields in curved spacetime as Fermi-Hubbard model with non unitary tunnelings'
---
Introduction {#sec Introduction}
============
Recently, much interest was devoted to a study of many body physics of quantum gases [@Ketterle_2008; @Bloch_2008]. High degree of control of the experimental parameters has allowed for designing specific Hamiltonians [@Lewenstein_2007]. A special category is then a fabrication of synthetic gauge fields [@Dalibard_2011], where a remarkable experimental progress has been achieved in last couple of years, including the realization of synthetic electric [@Lin_2011] and magnetic [@Lin_2009] fields in the bulk as well as on the lattice [@Aidelsburger_2011]. A non abelian synthetic gauge field of the spin orbit Rashba type has been demonstrated in the bulk [@Lin_2011b; @ZhangJY_2012]. In the case of a lattice, which will be of main interest, laser assisted hoppings [@Jaksch_2003; @Gerbier_2010] allow for a simulation of a (non abelian) lattice gauge theory [@Kogut_1979] with cold atoms [@Osterloh_2005]. Different works adressed the question of non abelian background fields with cold atoms. Such situation occurs e.g. in the case of electrons with spin orbit coupling and can be studied with cold atomic systems, in both non interacting [@Goldman_2009] and interacting [@Cocks_2012; @Cole_2012] cases. In those scenarios, however, the tunnelings of different spin components between two adjacent lattice sites are described by unitary matrices, due to the hermiticity of the gauge field [@Kogut_1979; @Makeenko_2002]. Moreover, an explicit form of these matrices is determined by the theory one wants to simulate, e.g. the mentioned spin orbit coupling. It is thus an interesting question, what happens, if these tunnelings become non unitary, which can be, in principle, done in the cold atomic experiments using known techniques, as explained below.
The article is structured as follows. In Sec. \[sec Theoretical model\] we derive the mapping between continuous Dirac fields in curved background spacetime and the Fermi-Hubbard model with general tunnelings (we also derive the non-relativistic limit of the mapping in the Appendix). In Sec. \[sec Expanding universe\] we demonstrate the use of the mapping on a specific example of the expanding universe in 2+1 dimensions. We discuss in detail the dispersion relations in continuum and on the lattice and how they are related using adiabaticity criteria. In Sec. \[sec Implementations\] we discuss how such systems can be implemented with cold atoms. We show how the toy example of expanding universe can be observed through the spin dynamics. Finally we conclude in Sec. \[sec Conclusion\].
In a simple case of a static diagonal metric, the tunnelings become unitary. Alternative interpretation of the Fermi-Hubbard Hamiltonian is that of a Pauli Hamiltonian, i.e. a non relativistic limit of the Dirac Hamiltonian. In this case the tunnelings remain, in general, non unitary even for the static diagonal metric.
Theoretical model {#sec Theoretical model}
=================
Continuum - lattice mapping
---------------------------
Driven by this motivation, let us start with a kinetic fermionic Hamiltonian in $d-1$ spatial dimensions of a form H(t) = \_[[**x**]{}, k, s, s’]{} \^\_s([**x**]{})T\_[s,s’]{}(x,x+a\_k) \_[s’]{}([**x**]{}+a\_k) + [h.c.]{}, \[eq H kin latt\] where the sum runs over all lattice sites ${\bf x}$ and directions $k=1..d-1$ and the fermionic operators satisfy the usual anticommutation relations {\_s([**x**]{}), \^\_[s’]{}([**x’**]{}) } = \_[s,s’]{} \_[**x, x’**]{}. \[eq acomm rel psi\] Throughout the paper, we denote the spacetime coordinates as $x$ $(=(t,{\bf x}))$, while the space coordinates as $\bf x$. The matrices $T(x,x+a_k)$ represent a parallel transporter of a quantum field $\psi$ between sites $x$ and $x + a_k$. In the case of lattice gauge theories, the matrix $T$ belongs to a representation of a gauge group, which is typically compact, such as $U(n)$ with $n$ being the number of “flavor” components of the field $\psi$. In such case, the matrices $T$ become unitary. The relevant question is thus, what if the $T$ belong to a non compact gauge group? This question has actually been already adressed in the past [@Cahill_1979; @Cahill_1983; @Wu_1975] and more recently [@Lehmann_2005], but such interpretation seems to be problematic and was not actively pursued. Lets return to the Hamiltonian [Eq.(\[eq H kin latt\])]{}. In what follows, we will be interested in physics of fermions, so that $\psi$ is a spinor. As discussed later, a general matrix $T \in Gl(n,\mathbb{C})$ can be engineered in cold atomic systems ($n$ is the number of spin components). We would like to emphasize, that for such a novel situation, the Hamiltonian [Eq.(\[eq H kin latt\])]{} is interesting on its own right. However, it is also interesting to look, whether some physical significance can be given to it.
A starting point of our discussion will be a classical (not quantized) fermionic field in a curved spacetime, which can be described by a Lagrangian density [@Birell_1982; @Kibble_1961] (x) = { i |(x) \^D\_(x) + [h.c.]{} - m|(x) (x) }. \[eq L\] Let us recall [@Weinberg_1972], that working in a coordinate basis $e_\mu$, in which the spacetime vector $x$ is defined in terms of its components $x^\mu$, $x = x^\mu e_\mu$, one may construct a local orthonormal basis $e_\alpha$. The two bases are related through vielbeins $e_\alpha = e^\mu_\alpha e_\mu$. The metric tensor $g$ is defined as $\eta_{\alpha \beta} = e^\mu_\alpha e^\nu_\beta g_{\mu \nu}$, where $\eta$ is the Minkowski, i.e. flat metric. Then, the curved spacetime $\gamma$ matrices are defined as [@Parker_2009] $\underline{\gamma}^\mu = \gamma^\alpha e^\mu_\alpha$, where $\gamma^\alpha$ are the usual (flat spacetime) Dirac matrices, for which $\{\gamma^\alpha, \gamma^\beta\} = 2 \eta^{\alpha \beta}$ and we adopt the sign convention $\eta = (+,-,-,...)$. The covariant derivative acting on the spinor is $D_\mu = \partial_\mu - \Gamma_\mu$, where $\Gamma_\mu(x) = \frac{1}{8}\left[\gamma^\alpha, \gamma^\beta \right] e^\nu_\alpha (\nabla_\mu e_{\beta \nu})$ and $\nabla_\mu e_{\beta \nu} = \partial_\mu e_{\beta \nu} - \Gamma^\sigma_{\mu \nu} e_{\beta \sigma}$ (for a brief overview with essential technical details, see [@Yepez_2011]). The canonically conjugate momentum to $\psi$ can be found in a usual way as (x) = = i | \^0 \[eq pi\] and similarly for $\bar{\pi}$ which is conjugate to $\bar{\psi}$. One then obtains the Hamiltonian density $\mathcal{H} = \pi (\partial_0 \psi) + (\partial_0 \bar{\psi}) \bar{\pi} - \mathcal{L}$, (x) = - { i | + [h.c.]{} - m| }. \[eq H dens\] Lets now cosider an isotropic square lattice in coordinate basis with lattice spacing $a$. We introduce a covariant derivative on the lattice as D\_(x) = , where $P(x,x + a_k)$ is the parallel transporter from $x + a_k$ to $x$ and reads (here $\mu$ is fixed) P(x,x + a\_) = \[eq par prop\] and $\mathcal{P}$ stands for the path ordering. One can then formally discretize the Hamiltonian $H(t) = \int {\rm d}^{d-1} x \mathcal{H}(x)$, H(t) &=& \_[[**x**]{}, k]{} \^(x) M\^k P(x,x+a\_k) (x+a\_k)\
&& \^(x) (x) + [h.c.]{}, \[eq H Dirac latt\] where $M^\mu(x) \equiv -i \sqrt{g} \gamma^0 \underline{\gamma}^\mu$. At first sight, the structure of the Hamiltonian [Eq.(\[eq H Dirac latt\])]{} is similar to [Eq.(\[eq H kin latt\])]{}, but there are two major differences. First, in the former case, the fields are not quantized and second, they are time dependent, so it is not obvious, whether one can obtain the desired anticommutation relation [Eq.(\[eq acomm rel psi\])]{} for the space dependent operators. In order to proceed, we shall rely on the arguments exposed in [@Huang_2009]. We summarize the main steps crucial for our purpose. The classical field $\psi(x)$ is a projection of ket $\ket{\psi}$ to a spatiotemporal basis $\ket{x}$ (x) = . \[eq psi xt\] Similarly, one obtains a different field which is only space dependent on some constant time hypersurface as ([**x**]{}) = . \[eq psi x\] The relationship between the two fields $\psi(x)$ and $\Psi({\bf x})$ can be found from the equivalence $\braket{\phi | \psi} = (\phi, \psi)$, where the scalar product in curved spacetime is defined as [@Huang_2009] (, ) = \^[d-1]{}x \^\^0 \^0 . \[eq scal prod\] From the resolution of identity $\int {\rm d}^{d-1} x \ket{x} \sqrt{g} \gamma^0 \underline{\gamma}^0 \bra{x} = \mathds{1}$ one obtains the evolution of the ket $\ket{x}$ \_0 = - (\_0 \^0 \^0 ) ( \^0 \^0 )\^[-1]{}, which can be formally integrated to give = = ( \^0 \^0 )\^[-]{}. \[eq ket xt\] The factor $\sqrt{2}$ is an integration constant coming from the hermitian definition of the Lagrangian [Eq.(\[eq L\])]{}.
The quantization of the space and time dependent field [Eq.(\[eq psi xt\])]{}, $\psi(x)$, then proceeds by imposing equal time anticommutation relations for the canonically conjugate operators [@Parker_2009], namely {\_s(x), \_[s’]{}(x’) } = i ([**x - x**]{}’) \_[s, s’]{}, \[eq anticomm psi pi\] where $\pi(x)$ is given by [Eq.(\[eq pi\])]{}. We can use the relations [Eq.(\[eq psi xt\], \[eq psi x\])]{} and [Eq.(\[eq ket xt\])]{} to find the relationship between the two fields $\psi(x)$ and $\Psi({\bf x})$ to be (x) = ( \^0 \^0 )\^[-]{} ([**x**]{}). \[eq psi xt psi x\] Plugging [Eq.(\[eq pi\])]{} into [Eq.(\[eq anticomm psi pi\])]{} and using [Eq.(\[eq psi xt psi x\])]{} we obtain for the anticommutator of the constant time hypersurface fields $$\{\Psi_s({\bf x}), \Psi^\dag_{s'}({\bf x}') \} = \delta({\bf x - x}') \delta_{s, s'},$$ which is precisely the relation [Eq.(\[eq acomm rel psi\])]{}.
Lets now take the lattice Dirac Hamiltonian [Eq.(\[eq H Dirac latt\])]{} and write it as H(t) = \_[[**x**]{}, k]{} \^(x) (x, x + a\_k) (x + a\_k) + [h.c.]{} + \^(x) (x) (x), \[eq H t temp\] We now use the relation [Eq.(\[eq psi xt psi x\])]{} to substitute for $\psi(x)$ and write the Hamiltonian as && H(t) =\
&& \_[[**x**]{}, k]{} \^([**x**]{}) T(x, x + a\_k) ([**x**]{} + a\_k) + [h.c.]{} + \^([**x**]{}) V(x) ([**x**]{}), \[eq H t\] where && T(x,x+a\_i) =\
&& (iM\^0)\^[-1/2]{}(x) M\^i P(x,x+a\_i) ((iM\^0)\^)\^[-1/2]{}(x+a\_i)\
&& V(x) =\
&& (iM\^0)\^[-1/2]{} ((iM\^0)\^)\^[-1/2]{},\
&& \[eq T V\] where we have put the lattice spacing $a=1$ for simplicity. Now, the Hamiltonian [Eq.(\[eq H t\])]{} has the same structure with the correct anticommutation relations for the operators as [Eq.(\[eq H kin latt\])]{} (plus the local term). The price to pay in order to achieve this goal was to absorb the spatiotemporal dependence of the fields $\psi(x)$ to the elements of the Hamiltonian and thus spoiling its covariance.
*Fermion doubling.* Due to the naive discretization of the Hamiltonian [Eq.(\[eq H dens\])]{} one obtains the lattice formulation with doublers. Since in the present work we are interested only in noninteracting theory, and taking into account the fact, that one can address experimentally individual $\bf k$ vectors (cf. below), we don’t elaborate on this issue further. Proposals how to deal with the fermion doubling in cold atomic experiments exist in the literature, see e.g. [@Banerjee_2013] for staggered fermions formulation.
Physical interpretation
-----------------------
We just provided a possible interpretation of the kinetic Hamiltonian of the form [Eq.(\[eq H t\])]{} with general non unitary tunnelings $T$. In Appendix, we provide an alternative mapping, corresponding to the nonrelativistic limit of the Dirac Hamiltonian leading to the Pauli Hamiltonian, yielding formally the same expression for the lattice Hamiltonian [Eq.(\[eq H t\])]{}.
The question is then what field theory is actually simulated. Lets take an example of simulation, where the Hamiltonian [Eq.(\[eq H kin latt\])]{} describes a motion in a (two dimensional) plane for a two component field $\psi_s$, $s=1,2$. If we want to interpret it as a Dirac Hamiltonian, our simulator would correspond to a Dirac Hamiltonian in 2+1 dimensions. If we are to interpret it, however, as a Pauli Hamiltonian, the simulator corresponds to a spin half fermion living in 3+1 dimensions, but whose motion is confined to a plane.
We would like to mention, that a simulation of a Dirac field in curved spacetime with cold atoms was already adressed [@Boada_2011], but the discretization was carried out in the limit of small lattice spacing, such that the approximation $P \approx 1 + a \Gamma$ is valid. For stationary metrics, considered in [@Boada_2011], it results in unitary tunnelings.
Case study: Expanding FLRW universe {#sec Expanding universe}
===================================
In order to show the above derived mapping on some physically relevant scenario, we choose a textbook example of an expanding universe described by a Friedmann-Lemaître-Robertson-Walker (FLRW) metric in 2+1 dimensions with a line element s\^2 = [d]{}t\^2 - b\_x(t)\^2 [d]{}x\^2 - b\_y(t)\^2 [d]{}y\^2. \[eq line el\]
Using the relations [Eq.(\[eq T V\])]{} leads to the result T\_x && T(x,x+a\_x) = -\
T\_y && T(x,x+a\_y) = -\
V(x) &=& m \_z. \[eq T V in FW\] It then clear, that in the lattice formulation the FLRW metric maps to a generalized time dependent spin-orbit coupling.
Continuum dispersion
--------------------
Lets consider the continuum Hamiltonian density [Eq.(\[eq HD dens\])]{} \_D = (\^0)\^[-1]{} (m - i\^k D\_k) + i\_0. One can derive from the line element [Eq.(\[eq line el\])]{} the following relations = b\_x b\_y &\
\^0 = \^0 & \^0 = 0\
\^1 = \^1 & \^1 = w\_[011]{} = 2 \_x \_0 b\_x\
\^2 = \^2 & \^2 = w\_[022]{} = -2 \_y \_0 b\_y Using the Dirac representation of $\gamma$ matrices $\gamma^0 = \sigma_z$, $\gamma^1 = i \sigma_y$, $\gamma^2 = i \sigma_x$ and the above relations, one gets \_D = \^0 m - (\_x \_x - 2 (\_0 b\_x) \_2 ) - (-\_y \_y + 2 (\_0 b\_y) \_2 ). \[eq HD dens FW\] Next, going to the Fourier space (t,[**x**]{}) = \^2 k [e]{}\^[-i[**k**]{} ]{} (t,[**k**]{}) \[eq FT\] the final Hamiltonian takes the form $H(t) = \int {\rm d}^2 k \mathcal{H}$, where &=& (t,[**k**]{})\^ ( \_D + [h.c.]{} ) (t,[**k**]{})\
&=& (t,[**k**]{})\^ (
[cc]{} m & - -\
- + & -m
) (t,[**k**]{})\
&=& ([**k**]{})\^(
[cc]{} m & - -\
- + & -m
) ([**k**]{}), where in the last equality we have used the relation [Eq.(\[eq psi xt psi x\])]{} between the two fields $\psi(t,{\bf k}) = \sqrt{2} (\sqrt{g} \gamma^0 \underline{\gamma}^0)^{-\frac{1}{2}} \Psi({\bf k}) = \sqrt{2}/\sqrt{b_x b_y} \Psi({\bf k})$. We thus obtain the instantaneous eigenvalues of the total Hamiltonian in continuum (this result is compatible with the one of Ref. [@Huang_2009]) \_= \[eq disp cont\]
Lattice dispersion
------------------
Combining [Eq.(\[eq H t\])]{}, [Eq.(\[eq T V in FW\])]{} and the lattice version of [Eq.(\[eq FT\])]{}, one gets for the lattice Hamiltonian H(t) = \_[k\_x,k\_y]{} \^\_[**k**]{} (
[cc]{} m - \_[**k**]{} & -\_[**k**]{}\
-\^\*\_[**k**]{} & -m - \_[**k**]{}
) \_[**k**]{} , \[eq Hk latt\] where the coefficients read \_[**k**]{} &=& +\
\_[**k**]{} &=& + i. \[eq mu gamma\] The instantaneous eigenenergies on the lattice are \_= - \_[**k**]{} . \[eq disp latt\] One can also verify, that the lattice dispersion relation [Eq.(\[eq disp latt\])]{} yields the dispersion relation [Eq.(\[eq disp cont\])]{} in the continuum limit. So far we have been working with $\hbar = c = a = 1$. In order to find the continuum limit, we need to restore the lattice spacing $a$ in the equations. Noting, that the lattice spacing enter the lattice version of the Hamiltonian through the definition of the covariant derivative only ($\psi_x^\dag D_j \psi_x = \psi_x^\dag 1/a \left[ P_j \psi_{x+j} - \psi_x \right]$ with $P_j = {\rm exp}(-\Gamma_j a)$). In our case it translates to the multiplication of tunneling matrices $T$, [Eq.(\[eq T V in FW\])]{}, by a factor $1/a$ ($T \rightarrow \frac{1}{a} T$) and since $\Gamma_j \propto \partial_0 b_j$, by multiplication of the occurences of $\partial_0 b_j$ by $a$ ($\partial_0 b_j \rightarrow a \partial_0 b_j$). Written explicitly, the coefficients $\mu_{\bf k}, \gamma_{\bf k}$ ([Eq.(\[eq mu gamma\])]{}) become \_[**k**]{} &=& +\
\_[**k**]{} &=& + i, which in the continuum limit gives \_[a 0\^+]{} \_[**k**]{} &=& 0\
\_[a 0\^+]{} \_[**k**]{} &=& + i , which in turn yields the continuum dispersion relation as expected.
Note on adiabaticity {#subsec Note on adiabaticity}
--------------------
We have shown, that in the limit $a \rightarrow 0^+$, one recovers the correct continuum dispersion relation. In practice, however, the lattice spacing $a$ is finite and thus care must be taken when interpreting the results of the simulation in terms of continuum theory. Intuitively, one expects to recover the continuum theory in the limit of small wavevectors $k$, since the details of the underlying lattice should not be important. In our specific example we thus consider the limit ($a$ nonzero) $ak = \frac{2\pi}{k_{\rm max}}k \ll 1$. In this limit, the dispersion relation [Eq.(\[eq disp latt\])]{} becomes \_ + , \[eq disp latt approx\] where we assume sufficiently slow changes in $b$, such that only leading term in the expansions of functions $\sinh (a\partial_0 b)$, $\cosh (a \partial_0 b)$ is dominant. In order to recover the continuum dispersion, the $\sqrt{\phantom{cau}}$ term has to be dominant. Explicitly, we can consider two limiting cases (i) $m \approx 0 \ll k_{j}/b_{j}$ and (ii) $m \gg k_{j}/b_{j}$. In these two cases we thus have m &&\
m && m , \[eq ad criteria\] where $j=x,y$. These conditions can be once again understood intuitively such that the characteristic rate $\omega$ of change of $b$ has to be much smaller than the maximum frequency supported by the lattice $\omega_{\rm max} = k_{\rm max} = \frac{2\pi}{a}$ (with $c=1$).
An important remark to make here, which is relevant for the implementations with cold atoms, is that a particularly important case is half filling. In the massless case, the dispersion relation [Eq.(\[eq disp latt\])]{} yields Dirac cones at $k_{x,y} = 0, \pm \pi$ with Fermi energy $\epsilon_{\rm F}=0$. The development around the Dirac points ${\bf k} \rightarrow {\bf k}_{\rm Dirac} + {\bf k}$ yields the dispersion relation for small $k$ [Eq.(\[eq disp latt approx\])]{}. In other words, it is natural to work at half filling, where the relevant wavevectors lie in the vicinity of the Dirac points, which approximates well the continuum theory.
Implementation with cold atoms {#sec Implementations}
==============================
In the experiments with cold atoms, the internal degrees of freedom are usually played by the hyperfine states of the atoms [@Bloch_2008]. These allow for a laser assisted tunnelings between adjacent sites, say $i,i'$ of the optical lattice. Lets denote the internal degrees of freedom $s$. In order to engineer an arbitrary $T(x) \in Gl(n,\mathbb{C})$, it is necessary to control each of the tunneling rates $(i,s) \leftrightarrow (i',s')$ independently in all spatial directions and moreover, the rates in general vary in spacetime. Different techniques and their combination can be used in order to achieve this goal. For example, bichromatic lattices can be combined with an independent Raman laser for each transition $s \leftrightarrow s'$ [@Mazza_2012]. The spatial dependence is then given by a transverse profile of each Raman laser. It can be given e.g. by a (typically) gaussian laser profile which varies slowly on the lattice spacing or it can be designed using a specific phase masks [@Bakr_2009] or array of microlenses [@Itah_2010], which allow for the modulation on the scale of lattice spacing and were already used in the cold atomic experiments. Another comment is, that the potential $V(x)$ in [Eq.(\[eq H t\])]{} is non diagonal and might be difficult to engineer. The way around is that since $V$ is hermitian, it can be diagonalized by unitary transformation. It amounts to redefine the tunneling matrix $T$ (analogous to a local gauge transformation in the case of gauge fields) and the spinors $\Psi$. Since the transformation is unitary, the anticommutation relations [Eq.(\[eq acomm rel psi\])]{} are preserved.
Expanding FLRW universe: Time evolution of the spins
----------------------------------------------------
In this section we will investigate the dynamics of a typical observable accessible with cold atomic systems. We would like to emphasize, that we are interested only in qualitative features of our mapping with respect to the continuum theory. More formal and quantitative comparison between the continuum theory and its lattice counterpart is in principle possible (using the analysis in [@Parker_1971; @Parker_1989; @Birell_1982]), however it requires a significant amount of additional work, which is not crucial for the conclusions presented in the following.
We consider the spin $S$ defined either in the physical spin space (spanned by operators $\Psi_{{\bf k},s}, \Psi_{{\bf k},s}^\dag$) or the spin $\mathcal{S}$ defined in the local diagonal basis spanned by operators $d_{{\bf k},s}, d_{{\bf k},s}^\dag$. They are defined as S\^a\_[**k**]{}(t) &=&\
\^a\_[**k**]{}(t) &=& , \[eq spins\] where $\sigma$ are the usual Pauli matrices. Working in the Heisenberg picture, an evolution of an arbitrary operator $O$ is governed by the Heisenberg equations of motion $\dot{O} = -i \left[ O, H \right]$ with the Hamiltonian [Eq.(\[eq Hk latt\])]{}. Written in components the equation of motion reads \_[[**k**]{},s]{} = -i H\_[[**k**]{},ss’]{} O\_[[**k**]{},s’]{}. \[eq dot O\] Defining spinors as $\Psi_{\bf k} = (\Psi_{{\bf k},1},\Psi_{{\bf k},2})^{\rm T}$ and $d_{\bf k} = (d_{{\bf k},+},d_{{\bf k},-})^{\rm T}$ one can introduce the relation between the two bases as d\_[**k**]{}(t) = U\_[**k**]{}\^(t) \_[**k**]{}(t) and the time evolution operator as \_[**k**]{}(t) = \_[**k**]{}(t,t\_0) U\_[**k**]{}(t\_0) d\_[**k**]{}(t\_0) K\_[**k**]{}(t,t\_0) d\_[**k**]{}(t\_0). Plugging these relations to the definitions of the spin observables [Eq.(\[eq spins\])]{}, one gets S\^a\_[**k**]{}(t) &=&\
&=& K\^\_[[**k**]{},sr]{}(t,t\_0) \^a\_[rr’]{} K\_[[**k**]{},r’s’]{}(t,t\_0) n\_[[**k**]{},s]{}(t\_0) \_[ss’]{}\
&=& ( K\^\_[[**k**]{}]{}(t,t\_0) \^a K\_[[**k**]{}]{}(t,t\_0) n\_[[**k**]{}]{}(t\_0) ), where we have used the relation $\braket{d_{{\bf k},s}^\dag d_{{\bf k},s'}} = n_{{\bf k},s} \delta_{ss'}$ and $n_{{\bf k}} \equiv {\rm diag}(n_{{\bf k},+},n_{{\bf k},-})$. Similarly, one gets for the spins in the local diagonal basis \^a\_[**k**]{}(t) = ( K\^\_[[**k**]{}]{}(t,t\_0) U\_[**k**]{}(t) \^a U\^\_[**k**]{}(t) K\_[[**k**]{}]{}(t,t\_0) n\_[[**k**]{}]{}(t\_0) )
In order to investigate the spin dynamics, we solve the equation [Eq.(\[eq dot O\])]{} numerically using the Runge-Kutta integrator. For initial conditions, we assume thermal distribution $n_{{\bf k},\pm} = ({\rm exp}(\beta \epsilon_{{\bf k},\pm})+1)^{-1}$ where $\beta$ is the inverse temperature.
For $b_{x,y}(t)$ one could choose any smooth functions with some asymptotic values for $t \rightarrow \pm \infty$. For numerics related reasons, we choose a function such that $\partial_0 b_{x,y} = 0$ at the beginning and the end of the expansion, namely b\_j(t) = {
[l l]{} 1 & t<0\
1 + B\_j \^2 ( ) & 0t \_j\
1+B\_j & t>\_j
. where $j=x,y$, $B_j$ is the amplitude and $\tau_j$ the duration of the expansion. This gives \_0 b\_j(t) = B\_j ( ) B\_j , \[eq ad criteria num\] which can be directly used to evaluate the adiabaticity criteria [Eq.(\[eq ad criteria\])]{}.
We consider three characteristic cases
(i) [massless isotropic ($m=0, b_x = b_y$)]{}
(ii) [massless anisotropic ($m=0, b_x \neq b_y$)]{}
(iii) [massive isotropic ($m \neq 0, b_x = b_y$)]{}
The massless isotropic case is trivial, because the field is conformally invariant and there is no associated dynamics of the spins, which we have verified in our simulation. This is in qualitative agreement with the fact, that there are no particle creations in the massless isotropic case [@Parker_2009]. The same argument holds for case (ii) for field evaluated at the Dirac points. In order to observe the spin dynamics, one thus has to look in the vicinity, but not directly at the Dirac point.
![(Color online) Massless anisotropic case. z component of the spin in (a) the fermionic basis, $S^z_{\bf k}$ and (b) the diagonal basis, $\mathcal{S}^z_{\bf k}$ vs. time and $k_y$, i.e. the wavevector in the direction which does not undergo the expansion. Used parameters: $m=B_y=0, B_x=1, \tau=5, \beta=5, k_x/\pi=0.02$. See text for details. []{data-label="fig spin vs time vs k"}](sz_vs_tk_beta5.png){width="9cm"}
Motivated by the realization in cold atomic experiments, we choose in the following relatively large value of the inverse temperature $\beta=5$.
*Massless anisotropic case.* In [Fig.(\[fig spin vs time vs k\])]{} we show the spin dynamics for the massless anisotropic case for different wavevectors $k_y$, i.e. the direction which does not undergo the expansion. The spins in the diagonal basis do not evolve after the expansion is finished ($t/\tau=1$), which is the case for all $\bf k$ (the effect is not clearly visible in [Fig.(\[fig spin vs time vs k\])]{} due to the strong thermal background, see also [Fig.(\[fig spin vs time ii\])]{}). This is in contrast to the spin evolution in the fermionic basis where the spins continue to evolve under the action of the free propagator. This is an important fact with respect to the experimental signatures of the expansion (cf. below).
An example of spin dynamics for different amplitudes of the expansion is shown in [Fig.(\[fig spin vs time ii\])]{}, where we evaluate $S^z_{\bf k}$ and $\mathcal{S}^z_{\bf k}$ in the vicinity of the Dirac point (0,0), namely $(k_x/\pi,k_y/\pi)=(0.02,0.02)$. One can see, as discussed above, that in the diagonal basis, the spin evolution stops when the expansion is finished, as opposed to the fermionic basis. For $B_x = (0.2,1,10)$ using [Eq.(\[eq ad criteria num\])]{} gives $\partial_0 b_x \leq (0.063, 0.31, 3.1)$. According to [Eq.(\[eq ad criteria\])]{}, the cases $B_x = (0.2,1)$ are thus well adiabatic, while $B_x = 10$ is not.
To complete the discussion of the massless anisotropic case, we show in [Fig.(\[fig spin vs time Beta\])]{} the effect of the thermal background for different temperatures ($\beta = \infty, 40, 5$). The qualitative features of the expansion are not affected by the thermal background, however they may be strongly suppressed (e.g. for $\beta=5$).
*Massive isotropic case.* Another situation yielding non trivial spin dynamics is an isotropic expansion, but where the field is massive, since the mass term explicitly breaks the conformal invariance. In this case, for small masses the mass term further enforces the restriction to the small $k$ values (cf. the discussion in Sec. \[subsec Note on adiabaticity\]), $m \gg \frac{k_j}{b_j} \geq k_j$. An example of the spin dynamics for $B_x = B_y = 1$ and $m = (0.1, 0.25, 0.5)$ is shown in [Fig.(\[fig spin vs time iii\])]{}. One can see, that for increasing mass, the expansion has smaller effect on the spin dynamics (decrease in amplitude), i.e. the particle pair creation is suppressed. Another comment is that in the lattice formulation the mass term plays the role of effective magnetic field along $z$ direction, which induces spin precession. This is clearly visible in the fermionic basis, where the precession rate is proportional to the mass (i.e. effective magnetic field), however the amplitude of the precession decreases with increasing mass.
![(Color online) Massless anisotropic case. z component of the spin in the diagonal basis (red lines), $\mathcal{S}^z_{\bf k}$, and in the fermionic basis (blue lines), $S^z_{\bf k}$, vs. time, evaluated at $(k_x/\pi,k_y/\pi)=(0.02, 0.02)$. Three cases ($B_x = 0.2, 1, 10$) are shown (solid, dashed and dash-dotted lines respectively). Used parameters: $m=B_y=0, \tau=5, \beta=5$. See text for details. []{data-label="fig spin vs time ii"}](sz_vs_t_m0_by0_q0p02_beta5.png){width="9cm"}
![(Color online) Massless anisotropic case, temperature dependence. z component of the spin in the diagonal basis (red lines), $\mathcal{S}^z_{\bf k}$, and in the fermionic basis (blue lines), $S^z_{\bf k}$, vs. time, evaluated at $(k_x/\pi,k_y/\pi)=(0.02, 0.02)$. The effect of thermal background for three different temperatures ($\beta = \infty, 40, 5$) is shown (solid, dashed and dash-dotted lines respectively). Used parameters: $m=B_y=0, B_x=1, \tau=5$. See text for details. []{data-label="fig spin vs time Beta"}](sz_vs_t_Beta_m0_by0bx10_q0p02.png){width="9cm"}
![(Color online) Massive isotropic case. z component of the spin in the diagonal basis (red lines), $\mathcal{S}^z_{\bf k}$, and in the fermionic basis (blue lines), $S^z_{\bf k}$, vs. time, evaluated at $(k_x/\pi,k_y/\pi)=(0.02, 0.02)$. Three cases ($m = 0.1, 0.25, 0.5$) are shown (solid, dashed and dash-dotted lines respectively). Used parameters: $B_x=B_y=1, \tau=5, \beta=5$. See text for details. []{data-label="fig spin vs time iii"}](sz_vs_t_m_bxby1_q0p02_beta5_mod.png){width="9cm"}
The spin provides an ideal experimental signature of the expansion since it is routinely measured in nowadays cold atomic experiments by probing the populations of atomic levels [@Bloch_2008]. Moreover, the time of flight measurements allow to address a specific wavevector $\bf k$ of the Brillouin zone, namely the neighbourhood of the Dirac points.
Interactions
------------
So far, we were considering only non interacting theory. Although non abelian lattice gauge theories are non trivial already at this level, the most interesting physics can be obtained in the presence of interactions. A natural interaction term for spin half fermions in optical lattice is $H_{\rm int} \propto U \sum_{s \neq s'} n_s n_{s'}$, where $n_s = \psi^\dag_s \psi_s$ is the density operator. Once again, one entirely legitimate approach is to consider a Hamiltonian $H = H_{\rm kin} + H_{\rm int}$, with $H_{\rm kin}$ [Eq.(\[eq H kin latt\])]{} and $T \in Gl(2,\mathbb{C})$ as such and study its properties ($H$ could also describe interacting bosons instead of fermions or both bosons and fermions. The interspecies interaction might lead to interesting physical phenomena, such as particle number fractionalization [@Ruostekoski_2002; @Javanainen_2003; @Ruostekoski_2008]). The other approach is to design directly a given field theory. For example, a proposal of simulation of a Thirring model (i.e. 1+1 dimensional field theory) with cold atoms was made [@Cirac_2010], where the interaction term reads $J^\mu J_\mu$ with $J^\mu = \bar{\psi} \gamma^\mu \psi$. In curved spacetime, the replacement $\gamma^\mu \rightarrow \underline{\gamma}^\mu$ makes the interaction term spacetime dependent. One can thus try to modify the proposal [@Cirac_2010] in a way that creates the correct interaction term, which might be an interesting test bed situation, since as one dimensional theory, the massless Thirring model is soluble also in curved spacetime [@Birell_1978].
Conclusion {#sec Conclusion}
==========
In this article we have shown how to map the continuous Dirac fields in curved background spacetime to the Fermi-Hubbard model with general nonunitary tunnelings both in the relativistic and non-relativistic cases. Next, we have demonstrated the mapping on the example of an expanding FLRW universe in 2+1 dimensions. Motivated by the experimental feasibility of such Hamiltonian in cold atomic experiments with laser assisted tunnelings, we could explicitly demonstrate the effect of time dependent non unitary tunnelings on the spin dynamics. We found, that the dynamics of the spin (representing the particle mode occupation) shares the same qualitative features as those predicted by the quantum field theory in FLRW spacetime, namely the dependence on mass and no particle creation in the massless isotropic (conformally invariant) case.
Acknowledgements {#acknowledgements .unnumbered}
----------------
We would like to thank J. Baez, T. Brauner, A. Kempf, T. Lappi, V. Scarani and M.C. Tan for useful discussions. J.M. acknowledges the support by the National Research Foundation and the Ministry of Education, Singapore. The Centre for Quantum Technologies is a Research Centre of Excellence funded by the Ministry of Education and the National Research Foundation of Singapore.
Appendix {#appendix .unnumbered}
========
Nonrelativistic limit
---------------------
Next, we discuss a non relativistic limit of the Dirac equation. After all, the kinetic part of the usual Hubbard model for electrons in tight binding approximation ([Eq.(\[eq H kin latt\])]{} with $T \propto \mathds{1}$) is obtained from the non relativistic quantum mechanical Hamiltonian $H \propto {\bf p}^2/(2m)$. It is thus interesting to see, how similar derivation works for fermions in curved spacetime background. We should use a systematic method, known as Foldy-Wouthouysen transformation [@Bjorken_1964] (also used in the context of quantum fields in curved spacetimes [@Bakke_2008]), which perturbatively decouples the electron and positron modes. One can derive the Dirac equation from [Eq.(\[eq L\])]{} (i\^D\_- m) (x) = 0, which can be rewritten as Schrödinger equation i\_0 (x) = \_D (x), where \_D = (\^0)\^[-1]{} (m - i\^k D\_k) + i\_0 \[eq HD dens\] is the Dirac Hamiltonian. The non relativistic limit can be obtained from the Dirac Hamiltonian of the form \_D = \^0 m + + O, \[eq HD OE\] where $\mathcal{E}$ and $O$ are even and odd operator, defined by the property $\left[ \gamma^0, \mathcal{E} \right] = 0$ and $\{\gamma^0, O\} = 0$ and $\gamma^0$ is in the Dirac representation. The lowest order expression for the nonrelativistic Hamiltonian is \_P = \^0 m + + \^0 O\^2, where the subscript $P$ stands for the Pauli Hamiltonian. One can identify $\gamma^0, \mathcal{E}$ and $O$ by comparing [Eq.(\[eq HD OE\])]{} with [Eq.(\[eq HD dens\])]{}. In the most general case it yields rather lengthy expressions. In order to proceed with the calculation, we will thus consider a simple, yet non-trivial scenario with a static diagonal metric of the form g = (
[cc]{} 1 & 0\
0 & h
), \[eq g stat\] and $h = {\rm diag}(h_{ii}(x^k))$, where $i = 1..d-1$ and the diagonal terms depend only on spatial coordinates $x^k$. First thing we note, is that in this case, the vielbein fields are also diagonal, $e^\mu_\alpha = 0$ for $\mu \neq \alpha$. In particular $e^0_0 = 1$ implying $(\underline{\gamma}^0)^{-1} = \underline{\gamma}^0 = \gamma^0$. Also, $\Gamma_0 = 0$. We then obtain for the Dirac Hamiltonian \_D = \^0 m - i \^0 \^k D\_k = \^0 m + O, since the term $\gamma^0 \underline{\gamma}^k D_k$ is odd for the metric considered. We then obtain for the Pauli Hamiltonian density \_P = \^0 m - \^0 (\^0 \^k D\_k) (\^0 \^j D\_j). The total Hamiltonian, expressed in terms of field variables, then reads H\_P = . \[eq H Pauli def\] The scalar product can be evaluated by integrating per parts in curved spacetime. The reason why one wants to do that is to obtain terms of type $(D \psi^\dag) (D \psi)$ rather than $\psi^\dag D^2 \psi$, since the former can be mapped to a Hubbard model with only nearest neighbor hopping. Evaluating [Eq.(\[eq H Pauli def\])]{}, we get && H\_P = \^[d-1]{} x | \^k \^j D\_k D\_j + [h.c.]{}\
&& + \^[d-1]{} x m | At this point $\psi$ is still $2^{\left[d/2\right]}$ component spinor, where $\left[n\right]$ is the integer part of $n$. By construction, the Hamiltonian $H_P$ contains only even operators and we can thus split the spinor into two parts, say $\psi = (\chi, \varphi)$. Each of the spinors $\chi, \varphi$ has $2^{\left[d/2\right]-1}$ components, which will have independent dynamics. In case of the diagonal static metric and $d=4$, we find \^k \^j D\_k D\_j = - (
[cc]{} e\^k\_k e\^j\_j \^k \^j \_k \_j & 0\
0 & e\^k\_k e\^j\_j \^k \^j \_k \_j
), where $\nabla_k = \partial_k - \tilde{\Gamma}_k$, $\tilde{\Gamma}_k = \left. -1/4 \sigma^j \sigma^l e^\nu_j({\bf x}) (\nabla_k e_{l \nu}({\bf x})) \right|_{j<l}$. Lets write the Pauli Hamiltonian for one of the spinor components, say $\chi$, which we write as && \^[3]{} x \^\^k \^j \_k \_j =\
&& \^[3]{} \^f\_[ii]{} \_i \_i + \^[d-1]{} . \^f\_[kj]{} \^k \^j \[\_k, \_j\] |\_[k<j]{} , where $f_{kj} = \sqrt{g} e^k_k e^j_j$. The commutator in the second term is familiar from non-abelian gauge theories and we have $[\nabla_k, \nabla_j] = \partial_{\left[ j \right.} \tilde{\Gamma}_{\left. k \right]} - \left[ \tilde{\Gamma}_k, \tilde{\Gamma}_j \right]$, which acts locally on the spinor $\chi$. The first term can be integrated per parts to yield (using $\tilde{\Gamma}_i^\dag = -\tilde{\Gamma}_i$) && \^[3]{} x \^f\_[ii]{} \_i \_i =\
&& - \^[3]{} x { f\_[ii]{} (\_i )\^(\_i ) - (\_i f\_[ii]{}) \^(\_i ) }. We thus write the Pauli Hamiltonian as
H\_P = \^[3]{} x + [h.c.]{} + m \^. \[eq H Pauli\]
We are now in the position to discretize the Pauli Hamiltonian, which is to follow exactly the same steps as in the case of Dirac Hamiltonian. Using again the prescription [Eq.(\[eq psi xt psi x\])]{}, which now takes a simple form $\chi(x) = \sqrt{2}\sqrt{g}^{-\frac{1}{2}} X({\bf x})$, we arrive at a Hamiltonian, which can be formally written as [Eq.(\[eq H t\])]{}, where we have to replace $\Psi \rightarrow X$ and the matrices $T, V$ now depends only on spatial coordinates $\bf x$ and read && T([**x**]{},[**x**]{} + a\_i) = - (f\_[ii]{} + \_i f\_[ii]{}) P([**x**]{},[**x**]{} + a\_i)\
&& V([**x**]{}) = 2m + , where $f^{-}_{ii} = f_{ii}({\bf x} - a_i )$ and we have to replace $\Gamma \rightarrow \tilde{\Gamma}$ in the definition of the parallel propagator [Eq.(\[eq par prop\])]{}.
It is interesting to notice, that in the case of relativistic Dirac Hamiltonian taking static diagonal metric [Eq.(\[eq g stat\])]{}, $\underline{\gamma}$ are unitary, $\Gamma_k$ are antihermitian, $\Gamma_k^\dag = -\Gamma_k$ and the parallel propagators [Eq.(\[eq par prop\])]{} become unitary. We thus have $T$ which is also unitary, contrary to the Pauli Hamiltonian case.
|
{
"pile_set_name": "ArXiv"
}
|
---
author:
- 'Yuxiao Wang$^1$, Chee-Ming Ting$^2$ and Hernando Ombao$^{1,3}$'
bibliography:
- 'ywang.bib'
title: Exploratory Analysis of High Dimensional Time Series with Applications to Multichannel Electroencephalograms
---
Introduction
============
Neuronal populations behave in a coordinated manner in order to execute learning, memory retention and even during resting state. In fact, disruptions in connectivity between brain regions is associated with a number of neurological diseases such as schizophrenia, obsessive compulsive disorder and Alzheimer’s disease. In this project, we shall investigate connectivity between brain regions using electroencephalograms (EEGs) which indirectly measure cortical neuronal activity. The key challenge in estimating connectivity brain networks is the high dimensionality of these signals. Another is the high multicollinearity between these channels that is primarily due to the spatial filtering and volume conduction. Finally, since these EEG signals are recorded over several epochs in an experiment, it is vital to understand how connectivity patterns may change across the experiment. Empirical inspection of the EEGs in each region show a high degree of multicollinearity (see Figure \[fig:eeg\_two\_regions\]). Therefore it is sensible to perform dimensionality reduction at region level. One approach to modeling the connectivity is to first derive low dimensional representation or simpler structure of the original signal and secondly, analyze the connectivity structure or other properties in the lower dimensional space. The approach is illustrated in Figure \[fig:demo\_factor\_model\]. In this paper, we focus on the first step of the approach and study two models that can be used to reduce the dimensionality of the high dimensional EEG time series. This is a first step towards dimension reduction which could lead to better statistical modeling and inference. It could also help to identify any potential irregularities in the signals (e.g. outliers, non-stationarities across epochs).
The remainder of the paper is organized as follows. Section \[sec:dimension\_reduction\] describes the methods for dimensionality reduction for time series data. Section \[sec:simulation\] presents the evaluation results of the models based on simulated data. Section \[sec:data\_analysis\] presents the results of an exploratory analysis on a real resting-state EEG dataset followed by a conclusion in Section \[sec:conclusion\].
Dimensionality reduction of time series {#sec:dimension_reduction}
=======================================
In this section, we describe the approach to performing exploratory analysis of high-dimensional EEG time series. Let $R$ be the number of regions on the scalp area. The EEG signals at region $r \in \{1, \dots, R\}$ can be represented using ${\bf z}_r(t)$, where the dimension of ${\bf z}_r(t)$ is $n_r$, which is equal to the number of channels within region $r$. EEG signals within a region appear to be highly correlated, which indicates that the variability can be well captured by a more compressed time series (the factors) of lower dimension. Besides that, modeling through low-dimensional representation or simplified structure has advantages in the sense that (1.) it takes advantage of the structure of data, e.g. high correlation for EEG signals within the same region, and hence the low dimensional embedding approach serves similar purpose as imposing regularization; (2.) it enables the modeling training using a relatively small training sample size. Problem with reduced dimension enables us to train a less constrained parametric model, for example, allowing larger lags in VAR model allows us to capture more complex dependence structure; and (3.) models with reduced size can be trained much faster, especially when the training time is higher than quadratic order of the dimension. For example, in computing partial coherence, inversion operation is far less painful for matrix of lower dimension.
A substantial amount of research has been done on dimensionality reduction of high dimensional time series. Such methods include frequency domain approaches [@brillinger1964frequency; @brillinger2001time]; dynamic factor models and state space approach with low dimensional state [@durbin2001time; @harvey1990forecasting; @lam2012factor]; canonical analysis [@box1977canonical] and principal component analysis [@wang2016modeling]. Here, we build on these foundations in two directions: we develop exploratory methods that handle multiple-epochs (rather than just a single epoch) and then package these into a toolbox that we hope would help neuroscientists to use a more data-adaptive approach to investigating connectivity in high dimensional brain signals.
In this paper, we will be mainly focusing on two classes of approaches based on principal component analysis. In this section, we derive factor activities for each region $r$, denoted by ${\bf f}_r(t)$, which has a lower dimension $m_r$ comparing to the original space, i.e. $m_r$ $\ll$ $n_r$. For consistency, we use lower case letter for scalar number, bold lowercase letter for column vector, and uppercase letter for matrix. For convenience purpose, we dropped the subscript $r$ when dealing with the time series signals.
Auto encoder for time series data
---------------------------------
The auto encoder algorithm is a general approach to learning representations of the input data (in this case the high dimensional time series). The algorithm was first introduced by [@rumelhart1985learning]. It can be used to reduce the dimension of the time series via learning a low dimensional representation. The algorithm consists of two parts, the encoder and the decoder. The encoder function, defined as $F_{en}: Z \rightarrow {\bf f}$, is a transformation from the original high dimensional time series $Z$ into some low dimensional space. The decoder function, defined as $F_{de}: {\bf f} \rightarrow Z$ is a transformation from the encoded (low-dimensional space) to the original high dimensional space. In this paper, we will consider a special case of encoders and decoders which as linear transformations (instantaneous mixing or filtered versions) of the original time series. Denote the parameters of the encoder and the decoder to be $\Theta$, then the best encoder and the decoder are the ones that minimize the reconstruction error, i.e., $$\widehat{\Theta} = \argmin_{\Theta} ||Z - F_{de}(F_{en}(Z))||^2_F$$ where the $|| E ||_F$ is the Frobenius norm which is defined as $|| E ||_F = \sqrt{\text{Trace}(EE^T)}$. For the time series data ${\bf z}(t)$ generated from distribution $P(Z)$ where $Z = [{\bf z}(1)', \dots, {\bf z}(T)']'$, we consider the expected loss, that is, the best parameters are computed as $$\widehat{\Theta} = \argmin_{\Theta} {{\mathbb E}}_{P(Z)}||Z - F_{de}(F_{en}(Z))||^2_F$$ For linear encoder and decoders, i.e., both of $F_{en}(Z)$ and $F_{de}({\bf f})$ are linear functions, the solution is strongly related to principal component analysis. In this paper, we will focus on two types of linear encoders and decoders: the first is the instantaneous mixing encoder and the second is a linear filter of the time series and thus captures the entire temporal dynamics of the time series.
Method 1: The factor is an instantaneous linear mixture of the time series ${\bf z} (t)$
----------------------------------------------------------------------------------------
For a given time series process ${\bf z}(t) \in \mathbb{R}^{n}$, we will consider the problem of learning a representation of lower dimension. The optimal representation is considered as the one that gives the best reconstruction accuracy. In this approach, we derive the factor ${\bf f}(t) \in \mathbb{R}^{m}$ using the instantaneous linear mixture of the original time series ${\bf z} (t)$, as described in Equation \[eq:approach\_1\]. The dimensionality reduction can be achieved when $m$ is smaller than $n$. $$\label{eq:approach_1}
{\bf f}(t) = {\bf A}^T {\bf z}(t)$$ For the purposes of keeping the parameters identifiable, we shall assume that (a.) ${A}^T {A} = I_{m}$ and (b.) $\Cov[{\bf f}(t)]$ is a diagonal matrix, i.e., the factors are uncorrelated. We reconstruct ${\bf z}(t)$ using the instantaneous linear mixture of ${\bf f}(t)$ in the form of $\widehat{\bf z}(t) = {B} {\bf f}(t)$. The goal is to find ${A}$ and ${B}$ that minimize the reconstruction error defined in Equation . $$\label{eq:approach_1_error_function}
{L}( A, B ) = \text{Trace} ( \ex \left [ {\bf z}(t) \ - \ {\widehat{\bf z} (t)} \right ] \ \left [ {\bf z}(t) \ - \ {\widehat{\bf z}(t)} \right ]^T )$$ The solution can be derived using the following two steps.
- Step 1. Compute the eigenvalues-eigenvectors of $\Sigma^{\bf z}(0)$ as $\{ (\lambda_{s}, {\bf e}_{s} )\}_{s=1}^{n}$ where $\lambda_{1} > \ldots, > \lambda_{n}$ and $\| {\bf e}_{s} \| = 1$. When $\Sigma^{\bf z}(0)$ is not known, we use an estimator instead, which can be computed as $\widehat{\Sigma}^{\bf z}(0) = \frac{1}{T}\sum_{t=1}^T {\bf z}(t) {\bf z}(t)^T$ assuming ${\bf z}(t)$ has zero mean.
- Step 2. The solution can be represented by $$\widehat{{A}} = \widehat{{B}} = [{\bf e}_{1}, \ldots, {\bf e}_{m}]
\ \ \ {\mbox{and}} \ \ \
\widehat{\bf f}(t) = \widehat{{A}}^T {\bf z}(t) \ .$$
The solution is closely related to principal components analysis (PCA) of the covariance matrix of the input signals at the zero lag, i.e., $\Sigma^{\bf z}(0) = \cov ({\bf z}(t), {\bf z}(t))$. It is the one that accounts for the most of the variation of the time series, among all the instantaneous linear projections with the same dimension.
Method 2: The factor is a linear filter of the time series ${\bf z}(t)$
-----------------------------------------------------------------------
Alternative to approach 1, if we restrict the form of the representation to linear functions of all ${\bf z}(t) \in \mathbb{R}^n$ rather than merely an instantaneous linear mixture, the lower dimensional representation denoted by ${\bf f}(t) \in \mathbb{R}^m$ can be written as $$\label{eq:compression}
{\bf f}(t) = \sum_{h=\infty}^{\infty} A(h)^T {\bf z}(t-h)$$ where $A(h) \in \mathbb{C}^{n \times m}$ with $m < n$, and ${f}_i(t)$ and ${f}_j(t)$ has zero coherency for $i \neq j$. We consider the reconstruction of ${\bf z}(t)$ using linear function of ${\bf f}(t)$ in the following form $$\label{eq:reconstruction}
\widehat{\bf z}(t) = \sum_{j=-\infty}^{\infty} B(j) {\bf f}(t-j)$$ where $B(j) \in \mathbb{C}^{n \times m}$ is the transformation coefficient matrix. The reconstruction error (loss function) is defined by the expected squared loss. That is $$\begin{aligned}
L(\{A(h)\}, \{B(j)\}) &=& \text{Trace} ( \ex \left [ {\bf z}(t) \ - \ {\widehat{\bf z} (t)} \right ] \ \left [ {\bf z}(t) \ - \ {\widehat{\bf z}(t)} \right ]^T )\\
&=& \text{Trace}(\Cov[\widehat{\bf z}(t) - {\bf z}(t)])\end{aligned}$$ The best transformation is defined as the $A(h)$ and $B(j)$ values that minimize $L(\{A(h)\}, \{B(j)\})$ $$\label{eq:loss_function_2}
\{\widehat{A}(h)\}, \{\widehat{B}(j)\} = \argmin_{\{A(h)\}, \{B(j)\}} L(\{A(h)\}, \{B(j)\})$$ The solution to the criterion in Equation \[eq:loss\_function\_2\] is obtained via principal components analysis of the spectral matrix which is described in detail in Algorithms Algorithms (\[algo:spectrum\]) and (\[algo:pca\_freq\]). Note that this dimension reduction procedure was originally described in [@brillinger1964frequency].
Comparison of the two linear encoders
-------------------------------------
Both encoder methods are based on projecting the original high dimensional signal onto a space of lower dimension. Both methods are similar in the sense that the factors are constrained to be linear functions of the original signal. However they differ in this respect: method 1 produces factor ${\bf f}(t)$ which only explicitly depends on the signal at time $t$. Under this approach, temporal dynamics of ${\bf z}(t)$ is ignored. The second method gives factors which are low dimensional filtered versions of the original signal. The factors at time point $t$ is obtained by using [*all*]{} data points ${\bf z}(t \pm \ell)$. Thus it captures temporal dynamics and lead-lag relationships in the original time series. We note here that the first method is a special case of the second. In fact, by constraining $A(h) = 0$ and $B(j) = 0$ for all $h \ne 0$ and $j \ne 0$ then the linear filtered series is reduced to the instantaneously-mixed signal.
The key advantage of the second method is that it is likely to give lower reconstruction error because it uses all the information about the signal. The first method is particularly problematic when there is some lead-lag relationships between the original signals which could be completely washed out with the simplistic instantaneous mixing. It is also supported by the simulation results, where the second model has better performance when the time series has time shift in some channels (Figure \[fig:simu\_4\] and \[fig:simu\_5\]).
In terms of computational complexity, model 2 needs to compute the eigenvalue decomposition of all the frequency matrices while model 1 only needs to decompose the zero-lag covariance matrix. That means the second model requires more computational resources (in both space and time), comparing to model 1. It would be helpful to identify the model with suitable complexity for the problem. In the simulation study, we applied the two models on time series data generated from different distributions to gain better understanding of their performance.
Simulation {#sec:simulation}
==========
In this section, we apply the algorithms on simulated time series of various properties and evaluate the performance. The goal of the simulation is to provide comprehensive evaluation of the performance of the models including application scope, capability and computational complexity. In particular, we perform simulations to analyze the performance of the approaches in terms of reconstructing the original time series. The step of the simulations are described as follows.
- Step 1: model training. Generate training time series data $Z_{tr}$ from distribution $F(Z)$ and fit the model using $Z_{tr}$.
- Step 2. model evaluation. Generate $K$ iid test datesets $Z_{te_1}, \dots, Z_{te_K}$. For each test dataset $Z_{te}$, we evaluate the model using the normalized reconstruction error in the form of $\frac{||Z_{te} - \widehat{Z}_{te}||_F^2}{||Z_{te}||_F^2}$, where the reconstructed time series $\widehat{Z}_{te}$ is computed by applying the trained model obtained in Step 1. The mean and the standard deviation of the test error are computed and compared across models.
We performed multiple simulations using data generated from different distributions $F(Z)$ and models with different complexities.
Spatial independent, temporal independent {#sec:independent_time_location}
-----------------------------------------
In this section, we consider the distribution of the data $F(Z)$ to be spatial independent and temporal independent. Specifically, the generated time series ${\bf z}(t)$ has dimension $20$, for $t = 1, \dots, 1000$, and that ${\bf z}(t)$’s are iid random variables from multivariate Gaussian distribution with mean zero and variance $I_{20}$. It can be observed form Figure \[fig:simu\_1\] that comparing between two methods, the reconstruction errors evaluated on the test datasets are similar. For both models, the reconstruct error appears to decrease linearly as the number of factors increases, which indicates that in the iid Gaussian case, all factors account for the same amount of the total variation. The result is as expected because (1.) for iid Gaussian random variable with identity covariance matrix, the projection on any direction will account for the same amount of variation and (2.) there is no lead-lag relationship between observations at different time points therefore in terms of predicting current observation, there is no gain of using observations from past or in the future. It can also be shown that in theory, two models will have the same solution when the signals are iid Gaussian with zero mean and identity covariance matrix.
Temporal independent, spatially highly correlated
-------------------------------------------------
Similar to Section \[sec:independent\_time\_location\], we consider white noise time series where ${\bf z}(t)$ and ${\bf z}(t+h)$ are independent Gaussian variables for $h \neq 0$. In this simulation, the covariance $\Cov {\bf z}(t)$ has low rank (rank is 2), which means that at time $t$, the channels are highly correlated. Figure \[fig:simu\_2\] displays reconstruction error as a function of number of factors. It shows that two factors are capable of capturing all the dynamics of the input time series which is reasonable since since the generated time series has no temporal dynamics (they are temporally independent even though they have high spatial correlation).
Temporal correlated, spatially highly correlated {#sec:simulation_both_correlated}
------------------------------------------------
In this simulation, we consider the time series ${\bf z}(t)$ that has both high spatial correlation and high temporal correlation. That is to say, ${\bf z}(t_1)$ and ${\bf z}(t_2)$ are correlated and ${z}_i(t)$ and ${z}_j(t)$ are also correlated. The time series is generated by first simulating data from autoregressive model $f_t = 0.9 f_{t-1} + \epsilon_t$ and then linearly projecting $f(t)$ to the observation space, which has dimension $20$. Standard normal white noises are then added to the observations. The training time series plot and the reconstruction errors are displayed in Figure \[fig:simu\_3\]. In terms of the reconstruction error on the test data, two models have similar performance. The variation of the reconstruction error shows a decreasing trend as the number of factors increases. When the number of factors reaches $20$, which is the dimension of the observations, the encoding-to-decoding procedure is equivalent to an identity transformation.
Phase-shifted time series
-------------------------
In this section we evaluate the model using the same data that is used in Section \[sec:simulation\_both\_correlated\], except that the time series from some of the channels are shifted. It is important to investigate shifts in time series because it is possible to have lead-lag relationships between EEGs in a region. The time series is generated by first simulating time series ${\bf z}(t)$ following the same distribution as in Section \[sec:simulation\_both\_correlated\], and then shifting the channels. We perform two simulation studies, where in the first simulation, the time series is shifted using $z_i(t) \leftarrow z_i(t+40)$ for $i = 1, \dots, 10$, and in the second simulation, the time series is shifted using $z_i(t) \leftarrow z_i(t+40)$ for $i = 1, \dots, 6$ and $z_j(t) \leftarrow z_j(t+80)$ for $j = 7, \dots, 12$.
It is observable from both Figure \[fig:simu\_4\] and \[fig:simu\_5\] that (1.) the time series plots show clearly clustered pattern, where within each cluster the signals are more synchronized, (2.) the second model, where the factor is a filtered version of the signal at all time points, gives a lower reconstruction error when the number of factors is smaller than the number of shifted clusters, and (3.) after the number of factors reaches the number of shifted time series clusters, the decreasing rate of the reconstruction error drops dramatically and the two models have similar performance.
The result is expected because model 2 is using information of all time lags to make prediction while model 1 only uses the instantaneous information, and hence in the shifted case, model 2 is more capable in capturing the temporal dynamics in the time series. The decreasing rate of the reconstruction error can be useful in estimating the number of synchronized clusters appeared in the data.
Exploratory analysis of the EEG data {#sec:data_analysis}
====================================
In this section, we perform exploratory analysis on real EEG data. The key challenge in analyzing EEG is the high dimensionality of the data. Computing dependence between regions or channels can be difficult due to the high dimensionality. Our goal here is address the dimensionality problem by deriving signal summaries (factors) of the EEGs in each region (e.g. SMA, left Pf) and then characterizing the dynamics and connectivity using the factor signals and the encoding/decoding functions.
The data were recorded during a motor learning study performed in the Stroke Rehab laboratory of our collaborator. The dataset contains EEG recordings for multiple subjects, where for each subject, 180 trails of 1 second EEG signals were recorded. The sampling rate of the data is 1000 Hz and number of channels is 256. The raw EEG data have been pre-processed by (1.) applying low pass filter at 50 Hz and (2.) using visual inspection and independent component analysis (ICA) to remove artifacts due to muscle activity, eye blinks and heart rhythms. Various analysis have been performed on the dataset, including using brain connectivity as predictor for ability of motor skill acquisitions [@wu2014resting] and the analysis of curves of log periodograms using functional boxplots [@ngo2015exploratory]. In this paper, the goal of the exploratory analysis is to gain better understanding of the EEG data as well as the models that we used.
Computing regional summaries (factors)
--------------------------------------
Figure \[fig:eeg\_two\_regions\] displays the EEG signals recorded for one subject at one trial. It can be observed that the EEG signals are very highly synchronized, which means EEG at channel $i$ is highly correlated with EEG at channel $j$ at the same time $t$. It also appears that signals within the same region (e.g. SMA and left Pr) have higher correlation, comparing to that of the signals in different regions. Due to these high spatial correlations, it is sensible to represent these EEGs in terms of low dimensional summaries that capture the most variation in these EEG signals. Figure \[fig:Demo\_signal\_compression\_specPCA\] shows the reconstruction of EEGs at SMA region using the linear convolution encoder. It can been seen that as the number of factors increases, the magnitude of the residuals decrease. The top two summary signals (factors) computed using EEGs from SMA region and left Pre-frontal region are plotted in Figure \[fig:factor\_time\_series\] and the proportion of total variation accounted by theses factors are shown in Figure \[fig:var\_accounted\_by\_factors\]. The results for both regions show that factors with very low dimension (less than 3) can represent most of the variation of the original signals. This is consistent with the fact that the EEGs are highly correlated spatially due to volume conduction.
Properties of the summary factors
---------------------------------
Figure \[fig:power\_spectrum\_density\_of\_factors\] shows the estimated power spectrum density of the top factors computed for SMA region and left Pf region. It shows that factor 1 in both the SMA and Left Pre-frontal regions capture the alpha oscillations (8-12 Hertz) and low beta (16-30 Hertz). Factor 2 has more power in the delta and theta band oscillations (1-8 Hertz). The power spectrum across 100 EEG epochs are estimated and visualized in Figure \[fig:power\_spectrum\_density\_of\_factors\_multiple\_trials\]. The results show that the spectrum pattern for the top factors are consistent across trails, where factor 1 concentrates more on alpha oscillations (8-12 Hertz) and factor 2 concentrates more on the delta and theta band oscillations (1-8 Hertz). In order to study the temporal dependence between the factors, we plot the cross-correlation between the top two factors, evaluated for multiple epochs (Figure \[fig:factors\_cross\_correlation\_multiple\_trial\]). The cross-correlation between factor 1 in SMA region and factor 1 in left Pre-frontal region appears to be very consistent across epochs. The cross-correlation that involves factor 2 also shows some consistent patterns across epochs, although the consistency is weaker comparing to the cross-correlation between factor 1’s in two regions.
Interactive Matlab toolbox for exploratory analysis
---------------------------------------------------
We implemented and actively maintain a Matlab toolbox (Exploratory High-Dimensional Time Series (XHiDiTS) toolbox https://goo.gl/uXc8ei) with a graphical interface that allows users to performance exploratory analysis easily. Figure \[fig:matlab\_toolbox\] shows a screen shot of the toolbox interface.
The option panels provide a rich set of options that allow users to select from by just one-clicking. The options include (1.) subject-specific data; (2.) experimental conditions (resting state vs task); (3.) specific regions (users can load their own channel location/ grouping files) (4.) methods for learning lower dimensional representations and the complexity of the model (e.g. number of factors); and (5.) methods for computing connectivity (e.g. partial directed coherence, correlation matrix, coherence matrix and block coherence).
The visualization panels show (1.) the 2-d scalp, with selected regions highlighted and colored, where the coloring is consistent with the title of the signal plot, allowing users to match the plot and region easily; (2.) the signals and factors (low dimensional representations) for selected regions; and (3.) the spectrum of the signals and the connectivity maps.
The toolbox has low latency in updating the results for datasets with reasonable sizes. For example, for a 256-channel EEG data that contains 1000 time points for each channel and 200 epochs, the latency for updating the results for a new setup is within seconds (<1s for most of the methods). Users can also load their own datasets or add their own definition of functions for connectivity and other quantities.
Conclusion and future work {#sec:conclusion}
==========================
In this paper, we developed exploratory procedures for high dimensional EEGs under the presence of high multi-collinearity by using low-dimensional representations. We evaluated (benchmark) the performance of the dimension-reduction methods via numerical experiments by applying the models on time series generated form different distributions, thus provided guidelines for the application scope of the methods. We performed exploratory analysis on a real EEG dataset to gain deeper understanding of the methods. The results for both of the simulation and exploratory analysis show that learning low-dimensional representations (factors) has potential benefits for subsequent modeling of the connectivity in high dimensional time series because the factors are capable of preserving the dynamics of the data (i.e., temporal dynamics, variation) while reducing dimension (complexity) of the original problem. We also implemented the methods in a Matlab toolbox with graphical interface that allows users to interactively explore, process and analyze the data in a convenient way.
Our future work in this area includes a comprehensive evaluation of the methods. For example, we would like evaluate the ability of the models in capturing the temporal dynamics and at the same time, quantify the artifact that might be induced by the mixing. Moreover, to make the package more comprehensive, we shall include other emerging measures of dependence such as isolated coherence [@pascual2014isolated; @ombao2008evolutionary; @fiecas2011generalized; @yuxiao2016bookchapter] and other more general (possibly non-linear) methods for obtaining summary signals [@pe2015generalized].
Algorithms
==========
set nfft = $T$, $m = [\sqrt T]$, $h_\ell = \frac{1}{2m+1}$ //Remark 1: $h_{\ell} \geq 0$, $h_{-\ell} = h_{\ell}$ and $\sum_\ell h_\ell = 1$ //Remark 2: In order to make $\widehat{S}_{zz}$ positive definite we need $2m+ 1 > p$ compute ${\bf z}_{\omega}(k) \gets \sum_{t=0}^{T-1}{\bf z}(t)\exp(-i2\pi t \frac{k}{T})$ transfer function: fft compute $I_{\omega}(k) \gets {\bf z}_{\omega}(k){\bf z}^*_{\omega}(k)$ raw periodogram padding $I_{\omega}(-k) \gets I^*_{\omega}(k)$ $I_{\omega}(T+k) \gets I^*_{\omega}(T-k)$
smoothing compute $\widehat{S}_{zz}(k) \gets \sum_{\ell=-m}^{m} h_\ell I_{\omega}(k+\ell)$
{${\bf z}_\omega(k)$}, {$\widehat{S}_{zz}(k)$}
compute eigen values $\lambda_1(k) > \lambda_2(k), \dots, \lambda_p(k)$ compute corresponding eigen vectors ${\bf e}_1(k), {\bf e}_2(k), \dots, {\bf e}_p(k)$
{$\lambda_j(k)$}, {${\bf e}_j(k)$}
${\bf C}(k) \gets [{\bf e}_1(k), \dots, {\bf e}_m(k)]$
${\bf C}(k) \gets {\bf C}^*(T-k)$
${\bf f}_\omega(k) = {\bf C}^* (T-k){\bf z}_\omega(k)$ transfer function for ${\bf f}(t)$
${\bf f}(t) = \frac{1}{T} \sum \limits_{k=0}^{T-1} {\bf f}_\omega(k) \exp(i2\pi t \frac{k}{T})$ ifft
{${\bf f}(t)$}, {${\bf C}(k)$}
Figures
=======
![Illustration of the modeling procedure. The goal is to characterize the dependence between three different regions: SMA, left Pre-frontal cortex and left parietal. As the first step, summary factors are obtained. Then the dependence between the summary factors are computed. []{data-label="fig:demo_factor_model"}](Demo_Factor_Model.pdf){width="100.00000%"}
\
![Signal compression using dynamic PCA (model 2). Column one shows the original EEG time series at SMA region, column two shows the factors computed via dynamic PCA, column three shows the reconstructed time series using different number of factors and column four shows the difference between the original signals and the reconstructed signals. Note that as the number of factors increases, the magnitude of the residuals decrease (i.e., the squared error of reconstruction decreases).[]{data-label="fig:Demo_signal_compression_specPCA"}](Demo_signal_compression_specPCA.pdf){width="100.00000%" height="10cm"}
![Top: SMA region. Estimated power spectrum across 100 epochs of the first factor (left) and the second factor (right). Bottom: Left Pre-frontal region. Estimated power spectrum across 100 epochs of the first factor (left) and the second factor (right).[]{data-label="fig:power_spectrum_density_of_factors_multiple_trials"}](spectrum_factor_multiple_trial_log_scale_specPCA.pdf){width="100.00000%" height="10cm"}
![Cross correlation between factors in SMA region and Left Pre-frontal region across 180 epochs. Top left: correlation between factor 1 in SMA region and factor 1 in Left Pre-frontal region; top right: correlation between factor 1 in SMA region and factor 2 in Left Pre-frontal region; bottom left: correlation between factor 2 in SMA region and factor 1 in left pre-frontal region; bottom right: correlation between factor 2 in SMA region and factor 2 in left pre-frontal region. The correlations show consistent patterns across epochs. []{data-label="fig:factors_cross_correlation_multiple_trial"}](factors_cross_correlation_multiple_trial_specPCA.pdf){width="100.00000%" height="10cm"}
![The interface of the XHiDiTS Toolbox: exploratory high dimensional time series toolbox in Matlab. This is an interactive toolbox where the user selects the dataset to be analyzed. From the dataset, the user can select specific regions of interest (ROIs) to be analyzed. This toolbox supports a rich set of options and methods for visualizing and analyzing high dimensional time series, including the methods presented in this paper. This toolbox is actively developed and maintained. It can be downloaded from [https://goo.gl/uXc8ei]{}.[]{data-label="fig:matlab_toolbox"}](matlab_toolbox.png){width="100.00000%" height="10cm"}
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Differentiable render is widely used in optimization-based 3D reconstruction which requires gradients from differentiable operations for gradient-based optimization. The existing differentiable renderers obtain the gradients of rendering via numerical technique which is of low accuracy and efficiency. Motivated by this fact, a differentiable mesh renderer with analytical gradients is proposed. The main obstacle of rasterization based rendering being differentiable is the discrete sampling operation. To make the rasterization differentiable, the pixel intensity is defined as a double integral over the pixel area and the integral is approximated by anti-aliasing with an average filter. Then the analytical gradients with respect to the vertices coordinates can be derived from the continuous definition of pixel intensity. To demonstrate the effectiveness and efficiency of the proposed differentiable renderer, experiments of 3D pose estimation by only multi-viewpoint silhouettes were conducted. The experimental results show that 3D pose estimation without 3D and 2D joints supervision is capable of producing competitive results both qualitatively and quantitatively. The experimental results also show that the proposed differentiable renderer is of higher accuracy and efficiency compared with previous method of differentiable renderer.'
author:
- 'Zaiqiang Wu, Wei Jiang\*'
bibliography:
- 'egbib.bib'
title: 'Analytical Derivatives for Differentiable Renderer: 3D Pose Estimation by Silhouette Consistency '
---
16SubNumber[\*\*\*]{}
Introduction
============
In recent years, convolutional neural networks (CNNs) have achieved appealing results in image understanding, such as single image based 3D reconstruction. It is generally known that differentiable operations are essential for back-propagation algorithm to train the neural networks. For instance, 3D reconstruction in generative manner requires differentiable renderer to construct the loss for supervision. However, due to the discrete sampling operation, the traditional rendering algorithms, (e.g., rasterization and ray tracing [@whitted2005improved]) are not differentiable and can not be directly applied in the framework of 3D reconstruction.
Many researchers paid a lot of attention to differentiate the process of rendering to make it feasible to incorporate the rendering operation into gradient-based optimization framework. Loper *et al.* [@loper2014opendr] proposed a general-purpose differentiable renderer named OpenDR which is capable of rendering triangular meshes into images and automatically acquiring derivatives with respect to the model parameters. However the derivatives of OpenDR are computed by numerical method which is lack of accuracy. Further more, OpenDR is not compatible with existing deep learning framework. Kato *et al.* [@kato2018neural] proposed a differentiable renderer designed for neural networks, but this method still relies on numerical methods to compute derivatives. Liu *et al.* [@liu2019soft] proposed a differentiable renderer called SoftRas which only focuses on the rendering of silhouette, however this method requires to generate probability maps for each triangle in the mesh, which results in high memory consumption and blurry rendering results.
To address these issues mentioned above, the first differentiable silhouette renderer with analytical derivatives which is of higher efficiency and accuracy compared with previous methods. It is worth mentioning that it is not necessary to utilize a general-purpose renderer in 3D reconstruction tasks since the illumination and material parameters are usually unknown, thus a differentiable renderer focusing on synthesizing silhouettes is enough for supervision. The forward pass of our renderer is similar to rasterization with anti-aliasing. However the backward pass is different from previous methods which depend on accessing to rendered frame buffers and obtaining derivatives by numerical methods. The high light of our work is that the derivatives of pixel intensities with respect to the coordinates of vertices are obtained by our proposed analytical method without the need of accessing to the frame buffers and applying any numerical method.
To obtain the derivatives of rasterization, the pixel intensities are defined as the average value of the certain area within the pixel region. The average value can be obtained by a double integral over pixel region of the pixel intensity function. Since only silhouette is considered in this paper, there is no need to deal with self-occlusion. Based on the integral expression of pixel intensity, the expression of derivatives could be obtained and simplified to an analytical expression without integral forms. With the analytical expression of derivatives, it is convenient and efficient to implement the backward pass of rendering. Our main contributions are summarized below.
- The analytical expressions of derivatives of rasterization are derived and a novel non-numerical approach is proposed to implement the backward pass of differentiable renderer efficiently.
- Experiments were conducted to demonstrated that our proposed method is of higher accuracy and efficiency compared with previous state-of-the-art method.
- The potential of 3D pose estimation by silhouette consistency without 2D and 3D joints is shown in the experiments we conducted.
Related Work
============
Differentiable Renderer
-----------------------
Computer vision problems have been viewed as inverse graphics in a long literature. Computer graphics aims to render an image from the object shape, texture and illumination. In contrary to computer graphics, inverse graphics aims to estimate the object shape, texture and illumination from an input image. Differentiable rendering offers a straightforward and practical technique to infer the parameters of 3D models by gradient-based methods.
Gkioulekas *et al.* [@gkioulekas2016evaluation] developed an algorithmic framework to infer internal scattering parameters for heterogeneous materials. Gradients are leveraged for optimization to solve this inverse problem, however this approach is limited to specific illumination problems. Mansinghka *et al.* [@mansinghka2013approximate] proposed a probabilistic graphics model to estimate scene parameters from observations. Loper and Black [@loper2014opendr] introduced an approximate differentiable renderer called OpenDR that makes it easy to render 3D model and automatically obtain derivatives w.r.t. the model parameters. However OpenDR has no interfaces to popular deep learning library which makes it difficult to be incorporated into deep learning framework. Kato *et al.* [@kato2018neural] introduced a differentiable rendering pipeline which approximate the rasterization gradient with a hand-designed function. More recently, Li *et al.* [@li2018differentiable] presented a differentiable ray tracer which is able to compute derivatives of scalar function over the rendered image w.r.t. arbitrary scene parameters. However the forward pass and backward pass of this method are performed by Monte Carlo ray tracing which makes it time consuming and impractical to be incorporated into learning-based framework.
With the development of deep learning and CNNs, there is a growing trend for researchers to achieve the froward pass and backward pass of differentiable rendering in a deep learning framework [@zienkiewicz2016real; @liu2017material; @richardson2017learning; @tewari2017mofa; @tewari2018self; @deschaintre2018single; @kundu20183d; @genova2018unsupervised]. Nguyen-Phuoc *et al.* [@nguyen2018rendernet] presented RenderNet, a convolutional network which learns the direct map from scene parameters to corresponding rendered images. However the shortcoming of RenderNet is that it is computational expensive since it is composed of convolutional networks.
In this paper, we focus on exploring a rasterization-based differentiable renderer with analytical derivatives. The main difference between our work and Neural 3D Mesh Render [@kato2018neural] is that instead of approximating the derivatives with hand-designed functions we derived a analytical expression to obtain derivatives with significantly higher efficiency and accuracy.
Single-image 3D reconstruction
------------------------------
Inferring 3D shape from images is a traditional and challenging problem in computer vision. With the surge of deep learning, 3D reconstruction from a single image has become an active research topic in recent years.
Most of learning-based approaches learn the mapping from 2D image to 3D shape with 3D supervision. Some of these methods predict a depth map to reconstruct 3D shape [@eigen2014depth; @saxena20083], while others predict 3D shapes directly [@kato2018neural; @wang2018pixel2mesh; @choy20163d; @fan2017point; @tatarchenko2017octree; @tulsiani2017multi; @wu2016learning].
When it comes to 3D pose estimation, statistical body shape models such as SMPL [@loper2015smpl] and SCAPE [@anguelov2005scape] are frequently employed due to their low dimensional representation. Bogo *et al.* [@bogo2016keep] proposed a iteratively optimization-based approach to reconstruct 3D human pose and shape from single image by minimizing the reprojection error between the 2D image and the statistical body shape model. Pavlakos *et al.* [@pavlakos2018learning] presented an end-to-end framework to predict the parameters of the statistical body shape model by training CNNs with single image and 3D ground truth.
Since 3D ground truth models are hard to obtain, 3D reconstruction without 3D supervision also attracts increasing attention. Yan *et al.* [@yan2016perspective] proposed perspective transformer nets (PTN) to infer 3D voxels from silhouette images from multiple viewpoints. Recent works predict 3D polygon meshes using differentiable renderer with 2D silhouettes supervision only. We follow these works in supervision, but we use a statistical body shape model named SMPL to represent 3D shape of human body and optimize the 3D pose with the gradients obtained by our proposed differentiable renderer.
Analytical derivatives for rasterization
========================================
Rasterization is a process of computing the mapping from scene geometry described in vector graphics format to raster images. The main obstacle that impedes rasterization from being differentiable is the discrete sampling operation that pixel intensities are sampled only at the central points of each pixel. Due to the discrete sampling operation and limited resolution, aliasing effect often appears in the rendered images. Anti-aliasing techniques are proposed to remove the aliasing effect and smooth the rendered images. In traditional anti-aliasing techniques, an image with higher resolution is rendered and down-sampled to the expected resolution with a average filter. Inspired by this approach, it is natural for us to assume that if a 3D model is rendered into an image with infinite resolution and down-sampled to the expected resolution using average filter, the sampling operation will be continuous and derivable. Since infinite resolution can not be achieved, the resolution of rendering is set to a higher and finite value to approximate the ideal situation. the forward pass of our renderer works the same as standard graphics pipeline with anti-aliasing but the backward derivatives are derived under the hypothesis that the image is rendered in infinite resolution and down-sampled into expected resolution by average filter.
Forward rendering
-----------------
The forward pass of our proposed differentiable renderer follows the standard graphics method [@marschner2015fundamentals]. To ensure the consistency between the forward and backward propagation, anti-aliasing is applied to smooth the rendered images.
Rendering a model with infinite resolution and down-sampling to the expected resolution, i.e., the pixel intensities equal to the double integral of a scalar function $p(x,y)$ of two variables $x$ and $y$ over the region within the pixel. The scalar function $p(x,y)$ represents the continuous distribution of intensity in the screen space.
Since we only focus on synthesizing silhouettes, i.e., there are only two possible values for $p(x,y)$: the foreground intensity $p_1$ and the background intensity $p_0$. Consider a image with $H$ rows and $W$ columns, the pixel intensity $I(i,j)$ of pixel in $i$-th row and $j$-th column can be represented as: $$\begin{aligned}
\label{eq:continuous}
I(i,j)=\frac{1}{S}\iint_{\Omega_{i,j}}p(x,y)\,dx\,dy\end{aligned}$$ where $\Omega_{i,j}$ represents the region of the pixel in row $i$ and column $j$, $S$ denotes the area of region within the pixel.
However the value of the integral expression in Equation \[eq:continuous\] is hard to compute in computer, so we use anti-aliasing to approximate this integral value as shown in Figure \[fig:forward\_render\]. The anti-aliasing we adopt is fairly rudimentary compared to more modern techniques. With this approach, individual pixels are divided into multiple coverage samples. By analyzing the intensity of the pixels surrounding each of these samples, an average intensity is produced, which determines the intensity of the original pixel. $F$ times anti-aliasing is applied in rendering, then the pixel intensity can be obtained as: $$\begin{aligned}
I(i,j)=\frac{1}{F^2}\sum_{k=1}^{F^2}p(x_k,y_k)\end{aligned}$$ where $x_k$ and $y_k$ represent the coordinate of the $k$-th sampling point in screen space.
It is obvious that: $$\begin{aligned}
\lim_{F\to \infty}\frac{1}{F^2}\sum_{k=1}^{F^2}p(x_k,y_k)=\frac{1}{S}\iint_{\Omega_{i,j}}p(x,y)\,dx\,dy\end{aligned}$$
When implementing the code, we set $F$ to $4$ for the tradeoff between accuracy and speed.
Derivatives computation
-----------------------
With the continuous definition of pixel intensity in Equation \[eq:continuous\], the derivatives with respect to the vertices can be derived. Considering a edge consisted of vertices $v_a$ and $v_b$ located at the boundary of the silhouette, the coordinates of $v_a$ and $v_b$ are denoted as $(x_0,y_0)$ and $(x_1,y_1)$. Assuming that this edge is intersected with the region of pixel in $i$-th row and $j$-th column. The partial derivative $\frac{\partial I(i,j)}{\partial x_0}$ can be written as: $$\begin{aligned}
\frac{\partial I(i,j)}{\partial x_0}&=\frac{\partial \frac{1}{S}\iint_{\Omega_{i,j}}p(x,y)\,dx\,dy}{\partial x_0}\\
&=\frac{1}{S}\iint_{\Omega_{i,j}}\frac {\partial p(x,y)}{\partial x_0}\,dx\,dy\end{aligned}$$
For notational convenience we denote that $A=y_1-y_0$, $B=x_0-x_1$, $C=x_1y_0-x_0y_1$. The equation of the edge can be represented as: $$\begin{aligned}
\alpha(x,y)=Ax+By+C \label{eq:line}\end{aligned}$$
Assuming that if $\alpha(x,y)<0$, then the point $(x,y)$ is in the region of foreground, and vice versa. Let $\Omega_0$ be a appropriate sub region of $\Omega_{i,j}$ s.t. $\Omega_0$ only covers the edge connecting $v_a$ and $v_b$, thus the intensity distribution function $p(x,y)$ can be written as: $$\begin{aligned}
p(x,y)=
\begin{cases}
p_1, & \mbox{if }\alpha(x,y)<0 \mbox{ and } (x,y)\in\Omega_0 \\
p_0, & \mbox{if }\alpha(x,y)>0 \mbox{ and } (x,y)\in\Omega_0
\end{cases}\end{aligned}$$
The equation above can be simplified with the Heaviside step function $h$: $$\begin{aligned}
p(x,y)=p_0h(\alpha(x,y))+p_1h(-\alpha(x,y)), (x,y)\in\Omega_0 \label{eq:step}\end{aligned}$$
The partial derivative $\frac{\partial I(i,j)}{\partial x_0}$ can be rewritten as: $$\begin{aligned}
\frac{\partial I(i,j)}{\partial x_0}&=\frac{1}{S}\iint_{\Omega_0}\frac {\partial p(x,y)}{\partial x_0}\,dx\,dy + \frac{1}{S}\iint_{\Omega_{i,j}-\Omega_0}\frac {\partial p(x,y)}{\partial x_0}\,dx\,dy \\
&=\frac{1}{S}\iint_{\Omega_0}\frac {\partial p(x,y)}{\partial x_0}\,dx\,dy \label{eq:partial}\end{aligned}$$
From Equation \[eq:step\] and Equation \[eq:partial\] we can obtain the partial derivative $\frac{\partial I(i,j)}{\partial x_0}$ as: $$\begin{aligned}
\frac{\partial I(i,j)}{\partial x_0}&=\frac{1}{S}\iint_{\Omega_0}p_0\delta(\alpha(x,y))\frac{\partial \alpha(x,y)}{\partial x_0} -p_1\delta(\alpha(x,y))\frac{\partial \alpha(x,y)}{\partial x_0}\,dx\,dy \\
&=\frac{p_1-p_0}{S}\iint_{\Omega_0}\delta(\alpha(x,y))(-\frac{\partial \alpha(x,y)}{\partial x_0})\,dx\,dy \label{eq:derivative}\end{aligned}$$ where $\delta$ denotes the Dirac delta function.
Substituting Equation \[eq:line\] into Equation \[eq:derivative\], the partial derivative $\frac{\partial I(i,j)}{\partial x_0}$ can be represented as: $$\begin{aligned}
\frac{\partial I(i,j)}{\partial x_0}=\frac{p_1-p_0}{S}\iint_{\Omega_0}\delta(Ax+By+C)(y_1-y)\,dx\,dy \label{eq:derivative1}\end{aligned}$$
To eliminate the Dirac delta function, we perform the following variable substitution: $$\begin{aligned}
\begin{cases}
t=Ax+By \\
k=-Bx+Ay
\end{cases}\end{aligned}$$
After variable substitution, Equation \[eq:derivative1\] can be rewritten as: $$\begin{aligned}
\frac{\partial I(i,j)}{\partial x_0}&=\frac{p_1-p_0}{S(A^2+B^2)}\iint\delta(t+C)(y_1-\frac{Bt+Ak}{A^2+B^2})\,dt\,dk \\
&=\frac{p_1-p_0}{S(A^2+B^2)}\int_{k_0}^{k_1}(y_1-\frac{Ak-BC}{A^2+B^2})\,dk \\
&=\frac{p_1-p_0}{S(A^2+B^2)}((y_1+\frac{BC}{A^2+B^2})(k_1-k_0)-\frac{A(k_1^2-k_0^2)}{2(A^2+B^2)})\end{aligned}$$ where $A^2+B^2$ is the $L^2$ length of the edge, which takes the Jacobian of the variable substitution into account. $k_0$ and $k_1$ are the lower and upper limits of integral obtained by Liang-Barsky algorithm [@liang1984new].
To illustrate the procedure of determining the lower and upper limits, the two new endpoints after clipping are denoted as $v_a^\prime$ and $v_b^\prime$ as shown in Figure \[fig:Liang-Barsky\], the coordinates are denoted as $(x_0^\prime,y_0^\prime)$ and $(x_1^\prime,y_1^\prime)$ respectively. Then the lower and upper limits can be obtained as: $$\begin{aligned}
\begin{cases}
k_0&=-Bx_0^\prime+Ay_0^\prime \\
k_1&=-Bx_1^\prime+Ay_1^\prime
\end{cases}\end{aligned}$$
![Illustration of how to determine the lower and upper limits of the integral by Liang-Barsky algorithm. After clipping, two new endpoints $v_a^\prime$ and $v_b^\prime$ are obtained. Following the variable substitution, the coordinates of $v_a^\prime$ and $v_b^\prime$ can be transformed to the lower limit $k_0$ and upper limit $k_1$.[]{data-label="fig:Liang-Barsky"}](Figures/Liang-Barsky){width="8cm"}
The same procedure can be easily adapted to obtain the partial derivatives $\frac{\partial I(i,j)}{\partial y_0}$, $\frac{\partial I(i,j)}{\partial x_1}$ and $\frac{\partial I(i,j)}{\partial y_1}$ as follows. $$\begin{aligned}
\frac{\partial I(i,j)}{\partial y_0}&=-\frac{p_1-p_0}{S(A^2+B^2)}((x_1+\frac{AC}{A^2+B^2})(k_1-k_0)+\frac{B(k_1^2-k_0^2)}{2(A^2+B^2)})\\
\frac{\partial I(i,j)}{\partial x_1}&=\frac{p_1-p_0}{S(A^2+B^2)}(-(y_0+\frac{BC}{A^2+B^2})(k_1-k_0)+\frac{A(k_1^2-k_0^2)}{2(A^2+B^2)})\\
\frac{\partial I(i,j)}{\partial y_1}&=\frac{p_1-p_0}{S(A^2+B^2)}((x_0+\frac{AC}{A^2+B^2})(k_1-k_0)+\frac{B(k_1^2-k_0^2)}{2(A^2+B^2)})\end{aligned}$$
It is feasible to obtain the derivatives without any numerical method with the analytical expressions of derivatives above, which brings space for improvement in accuracy and efficiency.
Backward gradients flow
-----------------------
Considering a 3D mesh consisting of a set of vertices $\{v_1^o,v_2^o,\dots,v_{N_v}^0\}$ and faces $\{f_1,f_2,\dots,f_{N_f}\}$. $v_k^o\in\mathbb{R}^3$ represents the position of the $k$-th vertex in the 3D object space and $f_k\in\mathbb{N}^3$ represents the the indices of the three vertices corresponding to the $k$-th triangle face. For rendering this 3D mesh, vertices $\{v_k^o\}$ in the object space are projected into screen space as vertices $\{v_k\}, v_k\in\mathbb{R}^2$.
The scalar loss function over the rendered image for optimization is denoted as $L$. The partial derivatives $\{\frac{\partial L}{\partial I(i,j)}|i=1,\dots,H,j=1,\dots,W\}$ can be computed through automatic differentiable library. Our task is that: given the partial derivatives of loss function $L$ with respect to pixel intensities $\{\frac{\partial L}{\partial I(i,j)}\}$, our goal is to compute derivatives of pixel intensities with respect to vertices $\{\frac{\partial I(i,j)}{\partial v_k}\}$. Thus the derivatives $\{\frac{\partial L}{\partial v_k}\}$ can be obtained by chain rule, after which the gradient backward flow will be completed.
It should be noted that the gradients flow is sparse since $\frac{\partial I(i,j)}{\partial v_k}\ne0$ only if there is at least one edge consisted of $v_k$ intersected with the pixel region of $I(i,j)$. We only have to focus on specific $i$, $j$ and $k$ such that $\frac{\partial I(i,j)}{\partial v_k}\ne0$, this allows skipping pixels that have no contribution of gradient to current triangle when traversing the arrays of triangles and improves the efficiency.
In order to achieve efficient retrieval of pixels that have contribution of gradient to current triangle, pixels out of the bounding box of current triangle are excluded first. The Liang-Barsky clipping algorithm [@liang1984new] is adopted to determine wether a pixel is intersected with current triangle. As shown in Figure \[fig:Quadrant\], a pixel is intersected with the triangle only if there is at least one edge of the triangle intersected with the pixel.
a quadtree is recursively constructed to collections of triangle faces. For each node in the quadtree, the collection of triangles only contains triangles which have at least one edge intersected with the quadrant of the node, as shown in Figure \[fig:Quadrant\]. We use Morton codes [@morton1966computer] to encode each quadrant such that we can obtain the memory location within constant time given a coordinate represented as float point numbers.
![Several intuitive examples of wether a triangle is intersected with the pixel.[]{data-label="fig:Quadrant"}](Figures/QuadTree_quadrant){width="12cm"}
It is obvious that gradients only flow at the boundary pixel of the silhouette image, so edge detection is performed on the rendered image to determine pixels that gradients can flow into, computation is required only at the boundary of silhouette.
Considering a pixel at the boundary and it is in the $i$-th row and $j$-th column, we need to determine the partial derivatives of pixel intensity with respect to the location of $k$-th vertices $v_k$, denoted as $\frac{\partial I(i,j)}{\partial v_k}$. It is assumed that there are $N_e$ edges consisted of $v_k$ intersected with the pixel in row $i$, column $j$. The derivatives of the pixel intensity $I(i,j)$ with respect to the position of $v_k$ can be represented as: $$\begin{aligned}
\frac{\partial I(i,j)}{\partial v_k}=
\begin{cases}
\sum_{n=1}^{N_e}\frac{\partial I(i,j)}{\partial v_k^n}, & \mbox{if }N_e>0 \\
0, & \mbox{if }N_e=0
\end{cases}\end{aligned}$$ where $\frac{\partial I(i,j)}{\partial v_k^n}$ represents the derivatives computed by the $n$-th edge.
To efficiently retrieve the edges which are consisted of $v_k$ and intersected with the region of the pixel in $i$-th row and $j$-th column, the row number $i$ and column number $j$ are transformed to the Morton code to obtain the memory location of the leaf node which contains the collection of triangles that may potentially be intersected with the current pixel. Intersection test is required to perform only in the small subset rather than the whole triangles in the polygon mesh.
To verify our method, experiments of our differentiable renderer on generating per-pixel gradient with respect to translation, rotation and scaling were conducted. The visualized results are presented in Figure \[fig:gradient\_map\]. From the visualized per-pixel gradient images, conclusion can be draw that our proposed differentiable renderer is able to generate correct gradients with respect to vertices location, which enables the gradient-based optimization for 3D pose estimation.
3D pose estimation
==================
To show the effectiveness of our method, experiments of 3D pose estimation based on statistical body shape model by our proposed differentiable silhouette renderer were performed. Following the work of [@bogo2016keep], an iteratively optimization-based method is presented to estimate the pose parameters of statistical body shape model by minimizing the error between reprojected silhouettes and ground truth silhouettes. The images and 3D ground truth leveraged in the experiments are from a 3D pose dataset named UP-3D [@lassner2017unite]. Unlike previous works, ground truth 2D and 3D joints truth are not necessary for experiments of 3D pose estimation in this paper.
Statistical body shape model
----------------------------
A statistical body shape model named SMPL [@loper2015smpl] is employed as our representation of 3D body model. Essential notations of SMPL model are provided here. The SMPL model can be view as s function $\mathcal{M}(\beta,\theta;\Phi)$, where $\beta$ is the shape parameters, $\theta$ is the pose parameters and $\Phi$ are fixed parameters learned from a dataset with body scans [@robinette2002civilian]. The output of the SMPL function are vertices $P\in \mathbb{R}^{N\times3}$ with $N=6890$ of a body mesh. The shape parameters $\beta\in \mathbb{R}^{10}$ are the linear coefficients of a low number of principal body shapes. The pose parameters $\theta\in\mathbb{R}^{24\times3}$ are expressed in axis and angle representation and define the relative rotation between parts of the skeleton. Additionally, the 3D joints $J\in\mathbb{R}^{24\times3}$ obtained conveniently by a sparse linear combination of mesh vertices.
In our experiments, the shape parameters $\beta$ are fixed and our goal is optimizing the pose parameters $\theta$ to minimize the errors between the ground truth silhouettes and reprojected silhouettes.
Data preparation
----------------
It is assumed that only images and multi-viewpoints silhouettes are available in the 3D pose estimation task. The ground truth silhouettes are generated by rendering the 3D ground truth meshes of UP-3D [@lassner2017unite] from $4$ azimuth angles (with step of $90^\circ$) with fixed elevation angles ($0^\circ$) under the same camera setup as illustrated in Figure \[fig:data\_generation\]. The resolution of silhouettes is set to $64\times64$.
![The ground truth silhouettes for supervision are generated by projecting the ground truth 3D model to the image plane by cameras in different viewpoints.[]{data-label="fig:data_generation"}](Figures/data_generation){width="12cm"}
Method
------
Given a single image $I$ and its multi-viewpoints 2D silhouettes $\{S_i\}$, the 3D body model is fitted by minimizing a weighted sum of error terms.
The differentiable silhouette rendering process is denoted as $\mathcal{R}$, then the silhouette error term $E_{sl}$ can be represented as: $$\begin{aligned}
E_{sl}&=\sum_{i=1}^{N_s}\lVert \mathcal{R}_i(\hat{P})-\mathcal{R}_i(P) \rVert_2^2\\
&=\sum_{i=1}^{N_s}\lVert \mathcal{R}_i(\hat{P})-S_i \rVert_2^2 \\
&=\sum_{i=1}^{N_s}\lVert \mathcal{R}_i(\mathcal{M}(\beta,\theta;\Phi))-S_i \rVert_2^2\end{aligned}$$ where $P$ and $\hat{P}$ denote the ground truth vertices and estimated vertices, $N_s$ denotes the total number of silhouettes, $\mathcal{R}_i$ denotes the camera in the $i$-th position, $S_i$ denotes the $i$-th ground truth silhouette.
To discourage the body model from self-intersection, a self-intersection penalty term $E_{spt}$ from [@wu2019novel] is adopted. This self-intersection penalty term can be represented as: $$\begin{aligned}
E_{spt}=\frac{N_{sec}}{N_v}\end{aligned}$$ where $N_{sec}$ denotes the number of vertices in self-intersection region, $N_v$ denotes the total number of vertices.
The backward gradients of $E_{spt}$ is obtained by a hand-designed algorithm which can produce gradients to pull vertices out of region of self-intersection. The details of this algorithm are beyond the scope of this paper, we refer the interested readers to [@wu2019novel] for more details.
The objective function can be written as the weighted sum of the two error terms above: $$\begin{aligned}
E=E_{sl}+\lambda E_{spt}
\label{eq:objective}\end{aligned}$$ where $\lambda$ is a scalar weight.
Experiments
===========
In this section, experiments of 3D pose estimation are performed to evaluate the effectiveness of our method. The details of our experiments setup are provided. The results of qualitative comparison and quantitative comparison are presented to demonstrate the effectiveness of our method.
Experimental setup
------------------
### Dataset
Our proposed method is tested on UP-3D [@lassner2017unite] for evaluation. This dataset contains color images and corresponding ground truth 3D pose represented as pose parameters of SMPL model. Noting that our iterative optimization-based method is sensitive to the initial pose, results on the subset of UP-3D selected by Tan *et al.* [@tan2018indirect] aiming to limit the range of global rotation of SMPL models are reported.
### Evaluation metric
![Detail of SMPL mesh model. The SMPL mesh model is a vertex-based model that accurately represents body shapes by vertices and triangles.[]{data-label="fig:mesh"}](Figures/mesh){width="8cm"}
For quantitative evaluation, per-vertex error from [@pavlakos2018learning] is used as metric for evaluating the accuracy of 3D pose when comparing with other methods. As shown in Figure \[fig:mesh\], the surface of body mesh is represented as vertices and triangles. The accuracy of pose estimation can be effectively evaluated by measuring error of each vertex, the per-vertex error $E_p$ can be represented as: $$\begin{aligned}
E_p=\frac{1}{N_v}\sum_{i=1}^{N_v}\lVert \hat{P}_i-P_i\rVert_2\end{aligned}$$ where $N_v$ denotes the total number of vertices, $\hat{P}_i$ denotes the estimated location of vertices, $P_i$ denotes the ground truth location of vertices.
### Implementation details
The resolution of output images of differentiable renderer is set to $64\times64$, and the multiple of anti-aliasing $F$ is set to $4$. The number of silhouettes $N_s$ is set to $4$. The code is implemented in C++ with interface to the automatic differentiation library PyTorch [@paszke2017automatic], which allows us to employ their built-in optimizers and optimize the pose parameters of SMPL model easily. The objective function is minimized with Adam optimizer [@kingma2014adam] with $\alpha=1.5\times10^{-4}$, $\beta_1=0.9$ and $\beta_2=0.999$. $\lambda$ in Equation \[eq:objective\] is set to $0.001$ across all experiments.
Qualitative comparison
----------------------
Comparison between the proposed differentiable renderer with Neural 3D Mesh Render (N3MR) [@kato2018neural] is performed by conducting 3D pose estimation in same experimental setup. To demonstrate the effectiveness of our approach, we also compare our results with that of direct prediction method named Learning to Estimate 3D Human Pose and Shape from a Single Color Image (L2EPS) by Pavlakos *et al.* [@pavlakos2018learning].
![Visualized results of 3D pose estimation by different methods. From left to right, we show the input images, ground truth, the results obtained by our method, the results obtained by N3MR [@kato2018neural] and results of L2EPS [@pavlakos2018learning].[]{data-label="fig:qualitative"}](Figures/qualitative){width="12cm"}
From the results shown in Figure \[fig:qualitative\], it is apparent that the Neural 3D Mesh Render suffers from local minimums which often result in failed prediction. Due to the discontinuous forward rendering pass without any smooth filter and the inconsistency between forward and backward propagations, the process of optimization is unstable and tends to fall in local minimums. In contrast, we apply anti-aliasing in the forward rendering to make the intensity of each pixel as much as possible close to the continuous definition in Equation \[eq:continuous\], which achieves the consistency between forward and backward propagations and stability of optimization.
Though our method performs 3D pose estimation without any 2D joint error term, the results are comparable with the learning-based method [@pavlakos2018learning] whose model is trained with 3D ground truth. Since 3D ground truth and 2D location are apparently more difficult to obtain than silhouette, our method offers possibility for 3D pose estimation without any 2D joint location and 3D ground truth.
Quantitative comparison
-----------------------
We show the quantitative evaluation on per vertex error with different approaches. Results are given in Table \[table:quantitative\]. As seen in Table \[table:quantitative\], our differentiable renderer outperforms N3MR [@kato2018neural] in 3D pose estimation. The result of our method is worse than that of L2EPS [@pavlakos2018learning] since the method in [@pavlakos2018learning] leverages 3D ground truth but our method only leverages 2D silhouettes and predict 3D pose in an unsupervised manner.
-------------------------------------------- ----------------------- --
Method Per-vertex error (mm)
L2EPS [@pavlakos2018learning] (supervised) 117.7
N3MR [@kato2018neural] (unsupervised) 172.2
Ours (unsupervised) 142.8
-------------------------------------------- ----------------------- --
: Quantitative results compared with other state-of-the-art methods[]{data-label="table:quantitative"}
Ablation analysis
-----------------
In this section, we conduct controlled experiments to validate the necessity of different components.
### Self-intersection penalty term.
We investigate the influence of Self-intersection penalty term in 3D pose estimation by conducting experiment without the self-intersection penalty term [@wu2019novel] (SPT). In Figure \[fig:self-intersection\] we visually compare the results of 3D pose estimation with and without SPT. As shown in Figure \[fig:self-intersection\], the result without SPT suffers from self-intersection. However the experiment with SPT obtains more reasonable result.
![Results of 3D pose estimation with and without SPT term. From left to right: input image, ground truth, prediction with SPT term and prediction without SPT term.[]{data-label="fig:self-intersection"}](Figures/self-intersection){width="11cm"}
### Anti-aliasing.
To demonstrate the importance of anti-aliasing in the forward pass of our differentiable renderer, we conduct quantitative comparison of 3D pose estimation by differentiable renderer with and without anti-aliasing. The result is given in Table \[table:anti-aliasing\], As seen in Table \[table:anti-aliasing\], anti-aliasing improves the accuracy of 3D pose estimation, especially when the resolution is quite low.
-------------- --------------- -----------------------
Resolution Anti-aliasing Per-vertex error (mm)
$32\times32$ No 181.9
$32\times32$ Yes 173.7
$64\times64$ No 153.4
$64\times64$ Yes 142.8
-------------- --------------- -----------------------
: Quantitative results of different rendering resolution and wether anti-aliasing is applied caption should end without a full stop[]{data-label="table:anti-aliasing"}
Running time analysis
---------------------
To demonstrate the efficiency of our differentiable renderer, we carried out experiments of our method with different resolution and different number of SMPL models compared with N3MR [@kato2018neural]. For a fair comparison, we implemented the CPU version of N3MR from their released GPU version. All experiments in this section were performed on a laptop with Intel(R) Core(TM) i7-8750H processer. We recorded the elapsed time of a single forward and backward pass of the two different renderer in Table \[table:running-time0\] and Table \[table:running-time1\]. As seen in Table \[table:running-time0\] and Table \[table:running-time1\], with the increasing number of triangles and resolution, it is more and more obvious that our method runs faster than N3MR.
---------------- ------- -------
Resolution Ours N3MR
$16\times16$ 17.11 16.08
$32\times32$ 17.21 16.11
$64\times64$ 17.63 18.00
$128\times128$ 18.50 23.64
---------------- ------- -------
: Elapsed time in ms of one iteration of our method and N3MR in different resolution setup. The number of SMPL models is set to 1[]{data-label="table:running-time0"}
---------------- ------- -------
Number of SMPL Ours N3MR
1 17.63 18.00
2 28.35 29.20
3 36.72 38.96
4 46.65 49.14
---------------- ------- -------
: Elapsed time in ms of one iteration of our method and N3MR with different number of SMPL model. The rendering resolution is set to $64\times64$[]{data-label="table:running-time1"}
Conclusion
==========
In this paper, we proposed a novel method to obtain analytical derivatives for differentiable silhouette renderer. We demonstrate experiments of 3D pose estimation by silhouette consistency to show the effectiveness efficiency of our proposed method. Unlike pervious works like N3MR [@kato2018neural] using numerical approach to obtain derivatives, we proposed a continuous definition of pixel intensity and derived the analytical derivatives based on the continuous definition. We adopt anti-aliasing to make sure the intensity of each pixel is close to the continuous definition. Experiments have shown that accuracy and stability of optimization benefit from the consistency between forward and backward propagations of our differentiable renderer. Since we only focus on synthesizing silhouettes, only a few pixels and edges need to be considered. We employ quadtree to accelerate the process of retrieving edges which the gradient of current pixel may back-propagate into. As shown in the experiment, the efficiency of our implementation is higher than that of N3MR [@kato2018neural].
There are two main limitations of our method. One is that our differentiable renderer is not general-purpose which means that our method can not obtain derivatives with respect to texture and lighting parameters and limits the application in inverse graphic. The other is that our implementation requires constructing a quadtree recursively which leads to lower efficiency compared with previous method when the mesh is too simple or the resolution of output image is quite low.
Future direction of this work may include deriving analytical derivatives for general-purpose renderer to enable the gradients back-propagate into arbitrary scene parameters. It may also include developing a parallelizable algorithm to enable efficient implementation on GPU.
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We derive the detailed balance condition as a solution to the Hamilton-Jacobi equation in the Ho[v r]{}ava-Lifshitz gravity. This result leads us to propose the existence of the $d$-dimensional quantum field theory on the future boundary of the $(d+1)$-dimensional Ho[ř]{}ava-Lifshitz gravity from the viewpoint of the holographic renormalization group. We also obtain a Ricci flow equation of the boundary theory as the holographic RG flow, which is the Hamilton equation in the bulk gravity, by tuning parameters in the theory.'
---
KUNS-2206\
=cmr10 scaled4 =cmmi10 scaled4 =cmmi7 scaled4
Ho[ř]{}ava-Lifshitz Holography
[[ Tatsuma Nishioka ]{}]{}\
[*Department of Physics, Kyoto University, Kyoto 606-8502, Japan*]{}\
\
5em
A renormalizable theory of gravity, so called the Ho[ř]{}ava-Lifshitz gravity, has been attracting great interest since the advent of the seminal works of Ho[ř]{}ava [@Horava1; @Horava2]. This theory does not have the full diffeomorphism invariance, but the following anisotropic scaling with dynamical critical exponent $z$ larger than the spatial dimensions $d$ exists $$\begin{aligned}
x^i &\to bx^i \quad(i=1,\dots, d)\ ,\qquad t \to b^z t \ .\end{aligned}$$ This theory has a remarkable property such that it describes the non-relativistic theory of gravity in the UV regime, but becomes the Einstein gravity in the IR region. It was extensively applied to a resolution of the cosmological problem including inflation and non-Gaussianity in [@TaSo; @Cal; @KiKo; @Mu; @Br; @Pi; @Gao; @MNTY], and new solutions were constructed in [@LMP; @Nas; @CoYa; @CCO; @CLS]. For the other recent progress, refer the reader to [@Vi; @MTML; @CaLe; @Horava3; @VoWe; @Je; @Kl; @Pal; @ChHu; @MyKi; @CHZ; @OrSu].
It becomes convenient in the following discussion to take the Wick rotation to the Euclidean space. The ADM decomposition of the $(d+1)$-dimensional Euclidean space is $$\begin{aligned}
\label{ADM}
ds^2 &= N^2dt^2 + g_{ij}(dx^i + N^idt)(dx^j + N^jdt) \ ,\end{aligned}$$ and the extrinsic curvature for the spacelike slice is given by $$\begin{aligned}
K_{ij}&= \f{1}{N}(\dot g_{ij} - \nabla_i N_j - \nabla_j N_i ) \ .\end{aligned}$$ The action of the Ho[ř]{}ava-Lifshitz gravity is defined as [@Horava1; @Horava2] $$\begin{aligned}
\label{HLaction}
S&= \f{2}{\kappa^2}\int^{t_0}_{-\infty}dt \int_{\Sigma_{t}}d^d
x\s{g}N\left( K_{ij}G^{ijkl}K_{kl} +
\f{\kappa^4}{16}E^{ij}\CG_{ijkl}E^{kl} \right) \ , \end{aligned}$$ where we set a future boundary at $t=t_0$ and $\CG_{ijkl}$ is the inverse of the DeWitt metric $G^{ijkl}$ of the space of metrics $$\begin{aligned}
G^{ijkl}&=\f{1}{2}(g^{ik}g^{jl}+g^{il}g^{jk}) - \lambda g^{ij}g^{kl} \
,\\
\CG_{ijkl}&=\f{1}{2}(g_{ik}g_{jl}+g_{il}g_{jk}) - \xi g_{ij}g_{kl} \ ,\quad
\xi= \f{\lambda}{d\lambda - 1} \ .\end{aligned}$$ The first and second terms in (\[HLaction\]) are the kinetic and potential terms, respectively, and $E^{ij}$ does not contain the time derivative of $g_{ij}$. When we require $E^{ij}$ to be a gradient of some function $W[g]$ with respect to the metric $g_{ij}$ $$\begin{aligned}
\label{DBC}
E^{ij}=\f{1}{\s{g}}\f{\d W[g]}{\d g_{ij}} \ ,\end{aligned}$$ it is called a “detailed balance condition” [@Horava1; @Horava2]. In general, in the context of condensed matter physics, it is known that theories which satisfy the detailed balance condition have simpler quantum properties than a generic theory, and that the renormalization properties in $(d+1)$ dimensions are often inherited from the simpler renormalization of the theory in $d$ dimensions with the action $W$. It still remains to be understood what the detailed balance condition means in the context of the Ho[ř]{}ava-Lifshitz gravity, however. We would like to get a deep insight into a role of it.
In this paper we derive the detailed balance condition as a solution to the Hamilton-Jacobi equation in the Ho[ř]{}ava-Lifshitz gravity. This result leads us to propose [*the existence of the $d$-dimensional quantum field theory with the effective action $W$ on the future boundary of the $(d+1)$-dimensional Ho[ř]{}ava-Lifshitz gravity*]{} from the viewpoint of the holographic renormalization group [@DVV]. This proposal reminds us of the dS/CFT correspondence [@St], while the detailed balance condition forces the cosmological constant to be always negative[^1]. In addition, we obtain a Ricci flow equation of the boundary theory as the holographic RG flow, which is the Hamilton equation in the bulk gravity.
Although this proposal is the most salient feature of our work, we should emphasize that we can derive the detailed balance condition and the Ricci flow equation based on the Hamiltonian formulation of the Ho[ř]{}ava-Lifshitz gravity without the holography. The Hamiltonian formulation in the bulk gravity affords us the renormalization group flow of the field theory on the future boundary $t=t_0$ along the time direction as the Hamilton equation. Such a flow is termed a holographic renormalization group flow initiated by [@DVV] (for a comprehensive review see [@FMS] and references therein).
The ADM decomposition (\[ADM\]) is suitable for the Hamiltonian formulation. The conjugate momentum associated with $g_{ij}$ in the Ho[ř]{}ava-Lifshitz action (\[HLaction\]) is ( $\dot{}\equiv \p/\p t$) $$\begin{aligned}
\pi^{ij}&= \f{1}{\s{g}}\f{\d S}{\d \dot g_{ij}} =
G^{ijkl} K_{kl} \ ,\end{aligned}$$ and the momenta conjugate to $N$ and $N_i$ are identically zero. The Hamiltonian is $$\begin{aligned}
\label{Ham}
H & = \int_{\Sigma_t}d^d x\s{g} \left[ N \CH + N_a \CP^a + 2\nabla_j(\pi^{ij}N_i)
\right] \ , \end{aligned}$$ with $$\begin{aligned}
\CH &\equiv \pi^{ij}\CG_{ijkl}\pi^{kl} -
\f{\kappa^2}{16}E^{ij}\CG_{ijkl}E^{kl} \ , \\
\CP^i &\equiv -2\nabla_j\pi^{ij} \ .\end{aligned}$$ The last term in the second line of (\[Ham\]) vanishes when $\Sigma_t$ is compact space; this is the case we focus on here.
In the Hamiltonian formulation, the conjugate momentum $\pi^{ij}$ becomes the independent variable instead of $\dot g_{ij}$. Using the momentum, we can take the action to the one in the first-order form $$\begin{aligned}
\label{actHam}
S[g_{ij},\pi^{ij},N,N^i]&= \int dt \int_{\Sigma_t}
d^dx\s{g} \left[ \pi^{ij}\dot g_{ij} - N\CH -
N_i\CP^i \right] \ .\end{aligned}$$ Varying this action, we obtain the Hamilton equation $$\begin{aligned}
\label{RGflow}
\dot g_{ij} &= 2N \CG_{ijkl}\pi^{kl} +
\nabla_i N_j + \nabla_j N_i \ ,\end{aligned}$$ with the Hamiltonian and momentum constraints $$\begin{aligned}
\label{constraint}
\CH&= \CP = 0 \ .\end{aligned}$$
Substituting the classical solution $g_c$ into the action (\[actHam\]) and integrating it along the time direction, one can express $S[g]$ as a surface integral with respect to $g_c(x,t_0)$[^2] $$\begin{aligned}
S[g=g_c]&= S_{bdy}[g_c(x,t_0)]\ . $$ It follows from this relation that $S_{bdy}$ is the effective action of the $d$-dimensional quantum field theory on the future boundary $\Sigma_{t_0}$ using the bulk/boundary relation $Z_{gravity}=Z_{QFT}$ and $Z_{gravity}=\exp (-S)$ in a manner similar to the AdS/CFT correspondence [@Ma; @GKP; @Wi].
In this case, the momentum is expressed in terms of the boundary action (see [@FMS] for a careful derivation) $$\begin{aligned}
\label{MomBound}
\pi^{ij}(x,t_0)&= \f{1}{\s{g}}\f{\d
S_{bdy}[g_{ij}]}{\d g_{ij}}\ .\end{aligned}$$ We use the same notation $g_{ij}$ to denote $g_c(x,t_0)$ for simplicity hereafter. Inserting these relations into the constraints (\[constraint\]), we obtain the momentum constraint $$\begin{aligned}
\nabla_j \left[ \f{1}{\s{g}}\f{\d S_{bdy}}{\d g_{ij}}\right] = 0 \ ,\end{aligned}$$ which indicates the conservation law of the energy momentum tensor in the $d$-dimensional QFT, and the Hamilton-Jacobi equation from the Hamiltonian constraint $$\begin{aligned}
\left( \f{1}{\s{g}} \f{\d S_{bdy}}{\d g_{ij}}\right) \CG_{ijkl}
\left( \f{1}{\s{g}}\f{\d S_{bdy}}{\d g_{kl}}\right)
=
\f{\kappa^4}{16}E^{ij}\CG_{ijkl}E^{kl} \ .\end{aligned}$$ This equation is easily solved $$\begin{aligned}
\label{E}
E^{ij}&=
\f{1}{\s{g}}\f{\d W[g]}{\d g_{ij}} \ ,
$$ where $W$ stands for the rescaled effective action of the $d$-dimensional QFT $$\begin{aligned}
W[g]=\f{4}{\kappa^2}S_{bdy}[g] \ .\end{aligned}$$ The solution (\[E\]) to the Hamiltonian constraint results in the detailed balance condition (\[DBC\]) and we find that $W$ is the (rescaled) effective action of QFT on the future boundary of the Ho[ř]{}ava-Lifshitz gravity.
It is worth mentioning that the Hamilton equation (\[RGflow\]) gives us the holographic renormalization group flow after substituting (\[MomBound\]) into it $$\begin{aligned}
\label{HolRG}
\dot g_{ij}|_{t=t_0}&= \f{\kappa^2}{2}N \CG_{ijkl}\f{1}{\s{g}}\f{\d W}{\d g_{kl}} +
\nabla_i N_j + \nabla_j N_i \ .\end{aligned}$$ This is a simpler equation in first-order as opposed to the original equation of motion which is second-order in time derivatives. The Hamilton equation for $\pi^{ij}$ is automatically satisfied and it is enough to solve (\[RGflow\])[^3]. As was mentioned in [@Horava1; @Horava2], we can easily find classical solutions in the Ho[ř]{}ava-Lifshitz gravity by just solving this equation.
Moreover, this equation has a remarkable property as follows. When we take the $d$-dimensional effective action $W$ for the Einstein-Hilbert action $$\begin{aligned}
W&= \f{1}{\kappa_W^2}\int d^dx\s{g} (-R+\Lambda_W) \ ,\end{aligned}$$ the $(d+1)$-dimensional theory becomes the Ho[ř]{}ava-Lifshitz gravity with dynamical critical exponent $z=2$. The holographic RG flow (\[HolRG\]) is given by [@Horava1] $$\begin{aligned}
\label{Ricciflow}
\dot g_{ij}|_{t=t_0}&= -\f{\kappa^2}{2\kappa_W^2}N \left[ R_{ij} +
\f{1-2\lambda}{2(d\lambda -1)}(R-2\Lambda_W)g_{ij}\right] +
\nabla_i N_j + \nabla_j N_i \ .\end{aligned}$$ If we take $\lambda =1/2$, $\kappa_W=\kappa/2$, $N=1$ and $N_i=0$, this becomes the Ricci flow equation[^4]. From this viewpoint one may say that [*the Ricci flow in $d$ dimensions is the holographic RG flow to the $(d+1)$-dimensional Ho[v r]{}ava-Lifshitz gravity with $z=2$ and $\lambda=1/2$*]{}.
One simple but interesting application of our results is that static solutions in the Ho[ř]{}ava-Lifshitz gravity are obtainable by solving[^5] $$\begin{aligned}
\label{dW}
\f{\d W[g]}{\d g_{ij}}&= 0 \ .\end{aligned}$$ In the case of the four-dimensional Ho[ř]{}ava-Lifshitz gravity with $z=3$, $W$ is the Einstein-Hilbert action with the gravitational Chern-Simons term in three dimensions ([*i.e.*]{}the topologically massive gravity). (\[dW\]) is just the Einstein equation in TMG and the interesting solutions were constructed in [@Nu; @BoCl; @ALPSS]. For example, we can construct the four-dimensional solitonic solution by the use of the Euclidean warped AdS$_3$ black hole. It would be of wide interest to investigate such a solution in higher dimensions with different dynamical exponent $z$.
**Acknowledgements**
We are grateful to K. Izumi, S. Minakami, S. Mukohyama, K. Murata, T. Kobayashi for valuable discussions, and S. Horiuchi for careful reading of this manuscript. This work is supported by JSPS Grant-in-Aid for Scientific Research No.19$\cdot$3589.
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J. de Boer, E. P. Verlinde and H. L. Verlinde, “On the holographic renormalization group,” JHEP [**0008**]{}, 003 (2000) \[arXiv:hep-th/9912012\]. A. Strominger, “The dS/CFT correspondence,” JHEP [**0110**]{}, 034 (2001) \[arXiv:hep-th/0106113\].
M. Fukuma, S. Matsuura and T. Sakai, “Holographic renormalization group,” Prog. Theor. Phys. [**109**]{}, 489 (2003) \[arXiv:hep-th/0212314\]. J. M. Maldacena, “The large N limit of superconformal field theories and supergravity,” Adv. Theor. Math. Phys. [**2**]{}, 231 (1998) \[Int. J. Theor. Phys. [**38**]{}, 1113 (1999)\] \[arXiv:hep-th/9711200\]. S. S. Gubser, I. R. Klebanov and A. M. Polyakov, “Gauge theory correlators from non-critical string theory,” Phys. Lett. B [**428**]{}, 105 (1998) \[arXiv:hep-th/9802109\]. E. Witten, “Anti-de Sitter space and holography,” Adv. Theor. Math. Phys. [**2**]{}, 253 (1998) \[arXiv:hep-th/9802150\]. Y. Nutku, “Exact solutions of topologically massive gravity with a cosmological constant,” Class. Quant. Grav. [**10**]{}, 2657 (1993). A. Bouchareb and G. Clement, “Black hole mass and angular momentum in topologically massive gravity,” Class. Quant. Grav. [**24**]{}, 5581 (2007) \[arXiv:0706.0263 \[gr-qc\]\]. D. Anninos, W. Li, M. Padi, W. Song and A. Strominger, “Warped AdS$_3$ Black Holes,” JHEP [**0903**]{}, 130 (2009) \[arXiv:0807.3040 \[hep-th\]\].
[^1]: We are grateful to S. Mukohyama for pointing out this feature.
[^2]: We are grateful to K. Murata for discussion of this point.
[^3]: We are grateful to K. Izumi for informing us of this point.
[^4]: We can set $N_i=0$ using the $d$-dimensional diffeomorphism, but we are not sure if we can take $N=1$ by the reparametrization of the time $t\to f(t)$.
[^5]: One can find the same discussion in [@Horava1] under the assumption of (\[HolRG\]).
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'The recent reports of temporal and spectral peculiarities in the early stages of some afterglows suggest that we may be wrong in postulating a central engine which becomes dormant after the burst itself. A continually decreasing postburst relativistic outflow, such as put out by a decaying magnetar, may continue to be emitted for periods of days or longer, and we argue that it can be efficiently reprocessed by the ambient soft photon field radiation. Photons produced either by the postexplosion expansion of the progenitor stellar envelope or by a binary companion provide ample targets for the relativistic outflow to interact and produce high energy $\gamma$-rays. The resultant signal may yield luminosities high enough to be detected with the recently launched [*Integral*]{} and the [*Glast*]{} experiment now under construction. Its detection will surely offer important clues for identifying the nature of the progenitor and possibly constraining whether some route other than single star evolution is involved in producing a rapidly rotating helium core which in turn, at collapse, triggers a burst.'
author:
- |
Enrico Ramirez-Ruiz$^{1,2}$\
$^{1}$School of Natural Sciences, Institute for Advanced Study, Einstein Drive, Princeton, NJ 08540, USA.\
$^{2}$Chandra Fellow.
title: 'Identifying young gamma-ray burst fossils'
---
\[firstpage\]
radiation mechanisms:non-thermal; hydrodynamics; gamma-rays: bursts; stars: neutron
Introduction
============
The typical GRB model assumes that the energy input episode is brief, typically $t_{\rm grb} \le 1- 10^2$s (see Mészáros 2002 for a recent review). However, peculiarities in the early stages of some afterglows, e.g. GRB 021004, have served as motivation for considering a more extended input period in which the energy injection continues well beyond $t_{\rm grb}$ (Fox et al. 2003). The enduring activity could in principle emanate from the sluggish drain of orbiting matter into a newly formed black hole (e.g. MacFadyen & Woosley 1999) or from a spin down millisecond super-pulsar (e.g. Usov 1994), which could produce a luminosity that was, still one day after the burst, as high as $L \sim 10^{47}$ erg s$^{-1}$. Collapsar (Woosley 1993, Paczyński 1998; MacFadyen & Woosley 1999; Aloy et al. 2000) or magnetar-like GRB models (Usov 1994; Thompson 1994, Wheeler et al. 2000) provide a natural scenario for a sudden burst succeeded by a more slowly decaying energy release (Rees & Mészáros 2000; Ramirez-Ruiz, Celotti & Rees 2002).
The power output would be primarily in a magnetically driven relativistic wind, which would be hugely super-Eddington during the time scales discussed here. Its luminosity may not dominate the afterglow continuum (Dai & Lu 1998; Zhang & Mészáros 2001), but we argue that it could be efficiently reprocessed into $\gamma$-rays. This could be due to the interaction of the postburst relativistic outflow with the dense soft photon bath arising either from a stellar companion or from the postexplosion expansion of the remnant shell or supernova (Hjorth et al. 2003; Stanek et al. 2003). The detection of such scattered hard $\gamma$-ray radiation would offer the possibility of diagnosing the nature of the precursor star and the compact object that triggers the burst.
binary system with a decaying magnetar
======================================
In the generic pulsar model the field is assumed to maintain a steady value, and the luminosity declines as the spin rate slows down. However, during the early stages, the magnetic field strength might decline more rapidly than the slowing down timescale. The power output in this case declines in proportion to $B^2$. The spindown law is given by $-I\Omega\dot\Omega= {B^2R^6\Omega^4}/{(6c^3)} +
{32GI^2\varepsilon^2\Omega^6}/{(5c^5)}$ (Shapiro & Teukolsky 1983), where $\Omega$ is the angular frequency, $B$ is the dipolar field strength at the poles, $R$ is the radius of the light cylinder, $I$ is moment of inertia, and $\varepsilon$ is ellipticity of the neutron star. The above decay solution includes both electromagnetic (EM) and gravitational wave (GW) losses. At various times the spin-down will be dominated by one of the loss terms, and one can get approximate solutions. When EM dipolar radiation losses dominate the spin-down, we have $\Omega=\Omega_0(1+t/t_{\rm m})^{-1/2}$. Here $$t_{\rm m}={3c^3I \over B^2R^6\Omega_0^2}\simeq 10^3~{\rm s}~
I_{45}B_{15}^{-2}P_{0,-3}^2R_6^{-6}
\label{tm}$$ is the characteristic time scale for dipolar spindown (e.g Dai & Lu 1998), $B_{15}=B/(10^{15}{\rm G})$, $P_0$ is the initial rotation period in milliseconds. When GW radiation losses dominate the spin-down, the evolution is $\Omega \approx \Omega_0(1+t/t_{\rm
gw})^{-1/4}$, where $t_{\rm gw} \simeq 1 {\rm s}
I_{45}^{-1}P_{0,-3}^{4}(\varepsilon/0.1)^{-2}$. GW spindown is important only when the neutron star is born with an initial $\Omega_0
\ge \Omega_{\ast} \sim 10^4 {\rm s}^{-1}$ (e.g. Blackman & Yi 1998). When $\Omega_0 < \Omega_{\ast}$, the continuous injection luminosity is given by $$L_{\rm m}(t) =L_{{\rm m},0}(1+ t/t_{\rm m})^{-2},
\label{spin}$$ where $$L_{{\rm m},0}={I\Omega_0^2 \over 2t_{\rm m}} \simeq 10^{49} {\rm
erg~s^{-1}}B_{15}^2P_{0,-3}^{-4}R_6^6.
\label{lm}$$ If the neutron star is born with $\Omega_0 > \Omega_{\ast}$, the timescale for GW-dominated regime is short so that $\Omega$ will be damped to below $\Omega_{\ast}$ promptly in a time $t_\ast
=[(\Omega_0/\Omega_{\ast})^{4}-1]t_{\rm gw}$. After $\Omega <
\Omega_{\ast}$, GW losses decrease sharply, and the spin-down becomes dominated by the EM losses. The injection luminosity can therefore be divided into two phases, i.e. $L=L_{{\rm m},0}/(1+t/t_{\rm gw})$ for $t < t_\ast$ or $L=L_{{\rm m},\ast}/[1+(t-t_\ast)/t_{m,\ast}]^2$ otherwise, where $L_{{\rm m},\ast} = I\Omega_\ast^2/(2t_{\rm m})$ and $t_{m,\ast}=3c^3I/(B^2 R^6 \Omega_\ast)$.
The bulk of the magnetar energy $L_{\rm m}$ would be primarily in the form of a magnetically driven, highly-relativistic wind consisting of $e^{-}$, $e^+$ and probably heavy ions with $L_\Omega \simeq \zeta
L_{\rm m}$ and $\zeta \le 1$. We envisage that the burst is triggered by the collapse of a massive star whose helium core is presumed to be kept rapidly rotating by spin-orbit tidal interactions with its binary companion. The relativistic (postburst) wind would then escape the compact remnant while interacting with the soft photon field of the companion with typical energy $\theta=kT/(m_ec^2) \sim
10^{-6}-10^{-5}$ (i.e. optical and UV frequencies). The scattered photons whose energy is boosted by the square of the bulk Lorentz factor of the magnetized wind (i.e. $\Theta=2\gamma^2 \theta$) propagate in a narrow $\gamma^{-1}$ beam owing to relativistic aberration. In the case of a mono-energetic isotropic magnetar wind (which is likely to be undisturbed even in the presence of a strong mass outflow from the binary companion; i.e. $L_\Omega/c \gg
v\dot{M}$), and assuming that the relativistic particles and soft photons are emitted radially from their source, the energy extracted from a relativistic particle by the inverse Compton process in the Thompson regime is given by $d\gamma (r,\varphi) \simeq - L_\ast
\sigma_T \gamma^2 (1-\cos \omega)^2 dr /(4\pi m_e \delta^2 c^3)$, where $L_\ast$ is the luminosity of the stellar companion, $\omega$ is the angle between the photon and particle directions before scattering, and $r$ denotes the distance to the magnetar from the volume region where the $\gamma$-ray radiation is generated. Here $\delta=\sqrt{\Lambda^2 \sin^2 \varphi + (r-\Lambda \cos\varphi)^2}$ defines the distance from this volume to the companion, $\Lambda$ the binary separation, $\varphi$ the angle between the lines connecting the binary members and the observer, and $\delta \cos \omega=r-\Lambda
\cos\varphi$.
The decrease of the particle’s Lorentz factor from the magnetar to infinity is $${\Delta \gamma \over \gamma} \simeq 1-(1+ \varsigma \Psi)^{-1},
\label{deltag}$$ where $$\varsigma={\gamma \sigma_T L_\ast \over 4\pi \Lambda m_ec^3} \sim 30
\left({\gamma \over 10^6}\right) \left({L_\ast \over 10^3
L_\odot}\right) \left({\Lambda \over R_\odot}\right)^{-1}
\label{eff}$$ denotes the efficiency in extracting energy from the relativistic outflow (mainly composed of $e^{\pm}$), and $\Psi(\varphi)=(3[\pi
-\varphi]- {1 \over 2}\sin 2\varphi - 4\sin \varphi )/(2\sin\varphi)$ is the beaming function of the generated $\gamma$-ray radiation[^1]. The total luminosity emitted by the relativistic outflowing wind through the Compton-drag process in the direction of the observer is highly anisotropic: $L_\Gamma (\varphi) =
[\Delta\gamma (\varphi)/ \gamma] L_\Omega$. $L_\Gamma$ strongly varies with $\varphi$ which in turn changes periodically during orbital motion (i.e. $\cos \varphi=\sin i \cos \nu$ for a circular orbit, where $i$ is the inclination angle and $\nu$ is the true anomaly). Fig. \[fig1\] shows the luminosity carried by the scattered photons for various positions of the companion and different assumptions regarding the magnetar luminosity.
The Compton drag process can be very efficient in extracting energy from the outflowing relativistic wind provided that $\varsigma \ge 1$ (see equation \[eff\]). $\varsigma$ is of course uncertain, but this number does not seem unreasonably high for a progenitor star with a massive binary companion, and suggests that our fiducial value of $L_\Omega$ for the overall luminosity need not be an overestimate (Fig. \[fig2\]). PSR B1259 – 63 is an example of a system where the mildly relativistic outflow from the aged pulsar is thought to interact with the soft photon field radiation produced by its high mass binary companion (Chernyakova & Illarionov 1999). Since $\varsigma$ could be less than $10^{-3}$ for a stellar companion with modest luminosity (or a binary with $\Lambda \gg R_\ast$), this suggests that we cannot rule out the possibility that part of the $\gamma$-ray continuum could still come from the intrinsic pulsed emission from the magnetar itself. For a typical aged pulsar this mechanism yields $L_\Gamma / L_\Omega \sim 10^{-2}-10^{-3}$ (Arons 1996). The time dependence and spectral properties, in this latter case, will be very distinctive and such effects should certainly be looked for.
Spectral attributes
-------------------
If the scenario proposed here is indeed relevant to an understanding of the nature of GRB progenitors, then its existence becomes inextricably linked to both the evolutionary history of the companion and the characteristics of the hydromagnetic wind. The postburst outflow from the compact remnant (which we assume to be highly relativistic) propagates in the soft photon field of the stellar companion where $U(r,\varpi)=m_ec^2 \varpi n_{\varpi}(r,\varpi)$ is the photon energy density at the location $r$. As the stellar companion emits a black body spectrum, of effective temperature $\theta_\ast$, the local photon energy density is given by $$n(r,\varpi)={2\pi \over h c^3}\left({m_e c^2 \over h}\right)^2
\left({R_* \over \delta[r]} \right)^2 {\varpi^2 \over
\exp[\varpi/\theta_\ast]-1},
\label{den}$$ where $\varpi$ is the soft photon energy in units of $m_ec^2$. The scattered photons are boosted by the square of the Lorentz factor so that the local spectrum has a black body shape enhanced by $\gamma^2$.
The luminosity of the scattered emission moving along a radial trajectory at an angle $\varphi$ is $L_\Gamma
(\gamma,\varphi,\epsilon)=\int_0^{\infty} \epsilon n_\gamma n_\varpi
(1- \beta \cos \omega) \sigma_{\rm KN} dV$, where $\epsilon$ is the energy of the scattered photons in units of $m_ec^2$, and $\sigma_{\rm
KN}$ is the Klein-Nishina cross-section (Jauch & Rohrlich 1976). As can be seen in Fig. \[fig3\], the resulting spectrum is the convolution of all the locally emitted spectra (i.e. $\int_0^{\infty}\;dn_\varpi[r,\varpi]$) and it is not one of a blackbody. Note that Klein-Nishina effects are important for incoming photon energies such that $\varpi \gamma (1 + \cos \varphi)>1$. The maximum energy of the scattered photons in this regime is $\gamma
m_ec^2$.
A further effect which may strongly affect the observed spectrum is the production of $e^{\pm}$ pairs through photon-photon collisions. $e^{\pm}$ pairs can be produced by scattered photons interacting with the isotropic companion emission or with each other. Photon collisions within the beam itself occur between photons of equal age and can only affect the high energy tail of the spectrum provided that $\gamma
\theta_* >1/3$ (Svensson 1987). Let us thus consider in turn the role of scattered and companion radiation as seed photons for this process. The interaction between the $\gamma$-rays produced by the Compton-drag process and photons emitted by the companion star would occur at large angles, resulting in an average energy threshold of $\epsilon_{\gamma\gamma}>1/\gamma$. The radiation flux produced at the location $r_\tau$ will then decreased by $\exp[-\tau_{\gamma\gamma}(r_\tau,\epsilon)]$, where $\tau_{\gamma\gamma} (r_\tau, \epsilon) =
\int_{\epsilon_{\gamma\gamma}}^{\infty}
d\varpi\int_{r_{\tau}}^{\infty}\sigma_{\gamma\gamma} (\varpi,\epsilon)
n_{\varpi} (r,\varpi) dr$ is the photon-photon optical depth and $\sigma_{\gamma\gamma}(\varpi,\epsilon)$ is the corresponding cross-section. As $\sigma_{\gamma\gamma}(\varpi,\epsilon)$ is peaked at the threshold energy (Svensson 1987), the above expression can be simplified to $\tau_{\gamma\gamma} (r_\tau, \epsilon) = (\sigma_T/
5)\int_{r_{\tau}}^{\infty} \epsilon_{\gamma\gamma}n_{\varpi}
(r,\epsilon_{\gamma\gamma})dr$, where $n_\varpi(r,\epsilon_{\gamma\gamma})$ is the photon density at threshold at the location $r$ (see equation \[den\]).
As $\varpi \ll \gamma$, this absorption mechanism would be important as long as the companion star produces a sufficient number of photons with energies $\epsilon_{\gamma\gamma}>1/\gamma$. This limit is illustrated in Fig. \[fig2\] for various evolutionary histories of the stellar companion, which has been assumed to be in the main sequence (Izzard et al. 2003). It can be seen that the number of soft photons able to interact with the high-energy $\gamma$-rays to produce $e^{\pm}$ pairs strongly increases as the bulk Lorentz factor of the relativistic wind exceeds $10^5$. The absorbed radiation will subsequently be reprocessed by the pairs and redistributed in energy. Each electron and positron will have an energy $\gamma_\pm
\sim \epsilon/2$ at birth, and will cool as a consequence of the Compton-drag process. The positrons will in turn annihilate in collisions with electrons in the wind, producing a blueshifted annihilation line at $\epsilon \sim \gamma$.
Direct measurements of the characteristics of this hard-energy radiation are frustrated by the fact that in traversing intergalactic distances, $\gamma$-rays may be absorbed by photon-photon pair production on the background field radiation (Gould & Schréder 1967). Photons of energy near 1 TeV interacting with background photons of $\sim 0.5$ eV have the highest cross section, although a broad range of optical-infrared wavelengths can be important absorbers because the cross section for pair production is rather broad in energy and, in addition, spectral features in the extragalactic background density can make certain wave bands more important than the cross section alone would indicate (Biller et al. 1998). The current generation of ground-based, $\gamma$-ray telescopes have a typical lower energy threshold of $\sim 0.5$ TeV and are thus expected to be able to see sources possessing redshifts up to $z=0.1$. The next generation of instruments (e.g. [*Glast*]{}[^2]) is expected to have an energy threshold in the region 0.05-0.1 TeV, and will therefore be able to see out to a redshift of at least $z=0.5$. Fig. \[fig2\] and its caption summarise the above limits along with the spectral attributes of the observed $\gamma$-ray signal.
A decaying magnetar in an asymmetric SNR
========================================
The success of the previous model is inseparably linked with the assumed residence of a companion star. Here we consider a related and less restrictive scenario in which the ambient photons are produced by the postexplosion expansion of the disrupted envelope. Photons emitted from the supernova remnant (SNR) shell provide copious targets for the relativistic outflow to interact and produce high energy $\gamma$-rays.
GRBs are thought to be produced when the evolved core of a massive star collapses either to a fast-spinning neutron star or a newly formed black hole. In the latter case, a GRB is likely to be triggered if the remaining star has sufficient angular momentum to form a centrifugally supported disk (e.g. MacFadyen & Woosley 1999). A funnel along the rotation axis would have been blasted open during the 1-100 s duration of the original burst; it would subsequently enlarge owing to the postexplosion expansion of the envelope of the progenitor star (e.g. Woosley 1993). The ram pressure of the continuing MHD outflow would further enlarge the funnel. Besides asymmetrically ejecting the SNR envelope (e.g., no ejecta in the polar direction), the supernova explosion may leave behind parts of the He core which take longer in falling back. Additional target photons may arise from parts of the disrupted He core, no longer in hydrostatic equilibrium and moving outwards inside the SNR shell. For a nominal subrelativistic shell speed $v=10^9 v_9$ cm s$^{-1}$, the typical distance reached is $r_{\rm snr} \approx 10^{14}v_9 t_{\rm d}$ cm in $t_{\rm d}$ days. The outflowing wind from the compact remnant (which we assume to be relativistic) would propagate inside the funnel cavity (of conical shape with semi-aperture angle $\psi$) while interacting with the SNR target photons with typical energy $\theta_{\rm snr}
\approx 10^{-6}-10^{-5}$ (i.e. optical frequencies).
Under the foregoing conditions the particles in the postburst relativistic outflow see blue-shifted photons pouring in from the forward direction. The rate of energy loss of a relativistic particle moving in a radiation with an energy density $w_{\rm snr}\approx
L_{\rm snr}/ 2\pi\psi^2r_{\rm snr}^2c$ is about $m_e c^2 d \gamma/dt
\approx - w_{\rm snr}\sigma_T c \gamma^2$ in the Thompson limit. The total luminosity of scattered hard photons $L_\Gamma$ is equal to the total particle energy losses in the course of motion from the magnetar to infinity: $L_\Gamma (\theta_{\rm snr})=[\Delta \gamma(\theta_{\rm
snr})/\gamma]L_\Omega$. $L_\Gamma$ varies with $\theta_{\rm snr}$ which in turn evolves with the expansion history of the stellar envelope. Fig. \[fig4\] shows the total luminosity of scattered hard photons $L_\Gamma$ as a function of the age of the remnant. The bolometric luminosity of SN 1998bw, as derived by Woosley et al. (1999), is used here to calculated the photon field energy by assuming a constant expansion velocity of $v=10^9$ cm s$^{-1}$. The resulting radiation pressure on electrons in the ejecta will brake any outflow whose initial Lorentz factor exceeds some critical value $\gamma_{\rm cd} \leq (L_\Omega/L_{\rm snr})^{1/2}$, converting the excess kinetic energy into a directed beamed of scattered photons. In reality, the external parts of the postburst relativistic outflow, which are in closer contact with the funnel walls, are dragged more efficiently since the soft photons arising from the walls can penetrate only a small fraction of the funnel before being upscattered by relativistic electrons. The outflow itself, is then likely to develop a velocity profile with higher Lorentz factor along the symmetry axis, gradually decreasing as the polar angle increases. Moreover, the outflow power may fluctuate; so also may its baryon content, due to entrainment, or to unsteadiness in the acceleration process at the base of the outflow. In this case, additional $\gamma$-rays can be produced in relativistic shocks that developed when fast material overtakes slower material (i.e. internal shocks).
Discussion
==========
The initial, energetic portion of the relativistic jet, with a typical burst duration of 10 s, will rapidly expand beyond the envelope of the progenitor star, leading in the usual fashion to shocks and a decelerating blast wave. A continually decreasing fraction of energy may continue being emitted for periods of days or longer, and its reprocessing by the soft photon field radiation can yield a continuum luminosity extending into the $\gamma$-ray band. Photons from a binary companion and/or photons entrapped from a supernova explosion provide ample targets for the wind to interact and produce high energy $\gamma$-rays. With a sensitivity of $5 \times 10^{-5}$ ph cm$^{-2}$ s$^{-1}$ ($6 \times 10^{-9}$ ph cm$^{-2}$ s$^{-1}$) in a $\sim 10^5$ s exposure, the $\gamma$-ray instrument on board of [*Integral[^3]*]{} ([*Glast*]{}) will detect a $10^{47}$ erg s$^{-1}$ signal out to a distance of $z \sim 0.5$ ($\sim 1$). A magnetar with $P_0=0.8$ ms and $B\sim
10^{14}$ G would lead to $L_\Omega \sim 10^{47}$ erg s$^{-1}$ after 1 day. This could also be a consequence of a drop in $B$ from $10^{15}$ to $3 \times 10^{12}$ G in a compact structure with stored energy of at least $10^{52}$ erg whose characteristic spin period remained constant (at a fraction of a millisecond). A similar argument can be developed based on the concept of $\alpha$ viscosity. For a hot dense torus around a black hole resulting from collapse of the core of the progenitor star, the viscous accretion time for a torus of radius $10^{9}r_9$ cm is $t_{\rm vis} \sim 1.5r_9^{3/2}B_{12.5}^{-2}$ for $nT
\sim$ constant, and the accretion of $\le 10^{-2}M_\odot$ in $t \sim
1$ day is sufficient to provide a characteristic $L \sim 10^{47}$ erg s$^{-1}$. This luminosity may not dominate the continuum GeV afterglow if it is only modestly reprocessed (i.e $L_\Gamma \le 10^{44}$ erg s$^{-1}$). In this case, we cannot rule out the possibility that much of the GeV emission is due to inverse Compton (Mészáros & Rees 1994) or synchrotron self-Compton (Derishev et al. 2001) losses in the standard decelerating blast wave, which could produce an afterglow luminosity that was still, 1 day after the original explosion, as high as $10^{45}$ erg s$^{-1}$ (see e.g. Dermer et al. 2000 for a blast wave expanding into a medium with $n_0=100$ cm$^{-3}$).
In closing, the scattered $\gamma$-rays from a peculiar Ib/c SNR of greater than usual brightness or a high luminosity stellar companion (i.e. $> 10 M_\odot$) should be easily detectable (at $t_{\rm obs} \le
1.5$ days) by the recently launched [*Integral*]{}. For a less luminous accompanying star (i.e. $< 5 M_\odot$), the expected $\gamma$-ray flux can be detected up to $z=0.1$, although may be difficult to disentangle from the scattered SNR emission whose intensity is likely to dominate at $t \le 100$ days. The $\gamma$-ray signals for “mean” events (i.e. $z \sim 1$) would stand out with high statistical significance above the background provided that $t_{\rm obs} < 0.1 - 0.5$ days. The planned [*Glast*]{} experiment may also soon provide relevant limits (i.e. $z\le 0.5$) for individual events at higher threshold energies (Figs. 2 and 3). Its detection would unveiled the presence of a postburst relativistic wind, and the spectral signatures of the $\gamma$-ray emission would help constrain the nature of the precursor star.
Acknowledgements {#acknowledgements .unnumbered}
================
I gratefully acknowledge very helpful discussions with Martin J. Rees and Aldo Serenelli. I also thank the referee for valuable comments. This research has been supported by NASA through a Chandra Postdoctoral Fellowship award PF3-40028.
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[^1]: Owing to the fact that the optical star is not a point source, $\Psi(\varphi)$ is clearly only accurately for $R_\ast/\delta < \varphi < \pi -R_\ast/\delta$, where $R_\ast$ is the radius of the stellar companion.
[^2]: http://www-glast.slac.stanford.edu/
[^3]: http://sci.esa.int/home/integral/
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{
"pile_set_name": "ArXiv"
}
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---
abstract: 'We design games for truly concurrent bisimilarities, including strongly truly concurrent bisimilarities and branching truly concurrent bisimilarities, such as pomset bisimilarities, step bisimilarities, history-preserving bisimilarities and hereditary history-preserving bisimilarities.'
title: Truly Concurrent Bisimilarities are Game Equivalent
---
\[firstpage\]
Games; Two-person Games; Bisimilarity; Formal Theory.
\[lastpage\]
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{
"pile_set_name": "ArXiv"
}
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---
abstract: 'Dialogue quality assessment is crucial for evaluating dialogue agents. An essential factor of high-quality dialogues is coherence – what makes dialogue utterances a whole. This paper proposes a novel dialogue coherence model trained in a hierarchical multi-task learning scenario where coherence assessment is the primary and the high-level task, and dialogue act prediction is the auxiliary and the low-level task. The results of our experiments for two benchmark dialogue corpora (i.e. SwitchBoard and DailyDialog) show that our model significantly outperforms its competitors for ranking dialogues with respect to their coherence. Although the performance of other examined models considerably varies across examined corpora, our model robustly achieves high performance. We release the source code and datasets defined for the experiments in this paper.'
author:
- 'Mohsen Mesgar, Sebastian B[ü]{}cker, Iryna Gurevych'
- |
**Mohsen Mesgar, Sebastian B[ü]{}cker, Iryna Gurevych**\
Ubiquitous Knowledge Processing Lab (UKP)\
Department of Computer Science, Technische Universit[ä]{}t Darmstadt\
www.ukp.tu-darmstadt.de\
bibliography:
- 'lit.bib'
- 'my\_lit.bib'
title: A Neural Model for Dialogue Coherence Assessment
---
Conclusions
===========
We propose a coherence model which is trained in a hierarchical multi-task learning scenario. We use coherence assessment as the primary task and dialogue act prediction as the auxiliary task. Our coherence method outperforms its counterparts in ranking dialogues concerning their coherence on several perturbations for dialogues from the [DailyDialog]{} and [SwitchBoard]{} corpora. We also observe that our MTL approach for coherence modeling yields a more robust model on the examined perturbations compared with a recent state-of-the-art coherence model [@cervone18]. For future work, we improve the quality of the dialogue act prediction part of our model, utilize this model for training a dialogue agent to produce coherent dialogues, and use recent contextualize word embeddings, e.g. BERT, to obtain utterance representations.
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{
"pile_set_name": "ArXiv"
}
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---
address: |
Escuela de Física, Universidad Autónoma de Zacatecas\
Antonio Dovalí Jaime s/n, Zacatecas 98068, ZAC., México\
Internet address: [email protected]
author:
- 'VALERI V. DVOEGLAZOV$^{\,\dagger}$'
title: 'FERMION-FERMION AND BOSON-BOSON AMPLITUDES: SURPRISING SIMILARITIES$^{\,\ast}$'
---
The scattering amplitude for the two-fermion interaction had been obtained in the Lobachevsky space in the second order of perturbation theory long ago, ref. \[1a,Eq.(31)\]: $$\begin{aligned}
\label{eq:TF}
\lefteqn{T^{(2)}_V ({\vec q} (-) {\vec p}, {\vec p}) =
-g_v^2 \frac{4m^2}{\mu^2 +4 {\vec \kappa}^{\,2}} -
4g_v^2\frac{({\vec \sigma}_1 {\vec \kappa})({\vec \sigma}_2
{\vec \kappa}) - ({\vec \sigma}_1 {\vec \sigma}_2)
{\vec \kappa}^2}{\mu^2 +4{\vec \kappa}^{\,2}} -\nonumber}\\
&-& {8g_v^2 p_0 \kappa_0 \over m^2}\,
\frac{i{\vec \sigma}_1 [{\vec p} \times {\vec \kappa} ] +i{\vec \sigma}_2
[{\vec p} \times {\vec \kappa} ]}{\mu^2 +4 {\vec \kappa}^{\,2}} -
{8g_v^2 \over m^2}\,\frac{p_0^2 \kappa_0^2 +2p_0 \kappa_0 ({\vec p}
\cdot {\vec \kappa}) - m^4}{\mu^2 +4{\vec \kappa}^{\,2}} -\nonumber\\
&-& \frac{8g_v^2}{m^2}\,\frac{({\vec \sigma}_1 {\vec p})
({\vec \sigma}_1 {\vec \kappa}) ({\vec \sigma}_2 {\vec p})
({\vec \sigma}_2 {\vec \kappa})}{\mu^2 +4{\vec \kappa}^{\,2}}\quad,\end{aligned}$$ $g_v$ is the coupling constant. The treatment is based on the use of the formalism of separation of Wigner rotations and parametrization of currents by means of the Pauli-Lyuban’sky vector, developed in the sixties. [@Shirokov] The quantities $$\kappa_0 = \sqrt{\frac{m(\Delta_0 +m)}{2}}\quad,\quad
{\vec \kappa} = {\vec n}_\Delta \sqrt{\frac{m(\Delta_0 -m)}{2}}$$ are the components of the 4-vector of momentum “half-transfer". This concept is closely connected with a notion of the half-velocity of a particle. The 4-vector $\Delta_{\mu}$: $$\begin{aligned}
{\vec \Delta} &=& \Lambda^{-1}_{{\vec p}} {\vec q}
= {\vec q} (-) {\vec p} = {\vec q}
-\frac{{\vec p}}{m} (q_0 - \frac{{\vec q}\cdot
{\vec p}}{p_0 +m})\quad,\\
\Delta_0 &=& (\Lambda^{-1}_{p} q)_0 = (q_0 p_0
-{\vec q}\cdot{\vec p})/m = \sqrt{m^2\,+ \,{\vec \Delta}^2}\end{aligned}$$ could be regarded as the momentum transfer vector in the Lobachevsky space. The amplitude (\[eq:TF\]) has been successfully applied for describing bound states of two fermions in the framework of the Kadyshevsky version of the quasipotential approach. Moreover, the use of the Shapiro [@Shapiro] technique of expansion in the plane-waves on hyperboloid and of the supplementary series of unitary representations of the Lorentz group led Prof. Skachkov to the very interest model of the quark confinement. [@Skachkov1]
In order to obtain the 4-vector current for the interaction of a Joos-Weinberg $j=1$ boson with the external 4-potential field one can use the known formulas of refs. [@Skachkov; @Shirokov], which are valid for any spin: $${\cal U}^\sigma({\vec p}) =
{S}_{{\vec p}} \,{\cal U}^\sigma({\vec 0})\quad, \quad {S}_{{\vec
p}}^{-1} {S}_{{\vec q}} = {S}_{{\vec q}(-){\vec p}}\cdot I\otimes
D^{J}\left \{ V^{-1}(\Lambda_p, q)\right \}\quad,$$ $$W_\mu({\vec p})\cdot D^J\left \{ V^{-1}(\Lambda_{p}, q)\right \}
= D^J\left \{ V^{-1}(\Lambda_{p}, q)\right \}
\cdot\left [ W_\mu({\vec q})
-\frac{p_\mu+q_\mu}{M(\Delta_0+M)}p^\nu W_\nu ({\vec q})\right ],$$ $$q^\mu W_\mu ({\vec p})\cdot D^J\left \{ V^{-1}(\Lambda_{p}, q)\right \} =
-D^J\left \{ V^{-1}(\Lambda_{p}, q)\right \}\cdot p^\mu W_\mu ({\vec q})
\quad.$$ $W_\mu$ is the Pauli-Lyuban’sky 4-vector of relativistic spin; the matrix $D^J \left \{ V^{-1} (\Lambda_{p}, q)\right \}$ is the matrix of the Wigner rotation for spin $j$.
Thus, we come to the 4-current vector for a $j=1$ Joos-Weinberg boson: $$\begin{aligned}
j_{\mu}^{\sigma_{p}\nu_{p}}({\vec p}, {\vec q}) &=&
j_{\mu \,(S)}^{\sigma_{p}\nu_{p}}({\vec p}, {\vec q}) +
j_{\mu \,(V)}^{\sigma_{p}\nu_{p}}({\vec p}, {\vec q}) +
j_{\mu \,(T)}^{\sigma_{p}\nu_{p}}({\vec p}, {\vec q})\quad,\\
j_{\mu \,(S)}^{\sigma_{p}\nu_{p}}({\vec p}, {\vec q}) \,&=&\,
-\,g_S \xi^\dagger_{\sigma_p} \left \{ (p+q)_\mu \left (
1+ \frac{({\vec J}\cdot {\vec \Delta})^2}{M (\Delta_0 + M)} \right )\right
\} \xi_{\nu_p}\quad,\label{curs}\\
j_{\mu \,(V)}^{\sigma_{p}\nu_{p}}({\vec
p}, {\vec q}) \,&=&\, -\,g_V \xi^\dagger_{\sigma_p} \left \{ (p+q)_{\mu}+
{1\over M}W_{\mu}({\vec p})({\vec J}\cdot{\vec \Delta})-
{1\over M}({\vec J}\cdot{\vec \Delta}) W_{\mu}({\vec p})\right \}
\xi_{\nu_p}\nonumber\\
&&\label{cur}\\
j_{\mu \,(T)}^{\sigma_{p}\nu_{p}}({\vec
p}, {\vec q}) \,&=&\, -\, g_T \xi_{\sigma_p}^\dagger \left \{ - (p+q)_\mu
\frac{({\vec J}\cdot {\vec \Delta})^2}{M (\Delta_0 + M)}+
\right.\label{curt}\\
&& \left. \qquad\qquad\qquad + {1\over M} W_{\mu}({\vec p})({\vec
J}\cdot{\vec \Delta})- {1\over M}({\vec J}\cdot{\vec \Delta})
W_{\mu}({\vec p})\right \} \xi_{\nu_p}\quad.\nonumber\end{aligned}$$ Let us note an interesting feature. The 6-spinors ${\cal U} ({\vec p})$ and ${\cal V} ({\vec p})$ in the old Weinberg formulation do not form a complete set: $${1\over M} \left \{{\cal U} ({\vec p}) \overline {\cal U} ({\vec p})
+{\cal V} ({\vec p}) \overline {\cal V} ({\vec p}) \right \} \,=\,
\pmatrix{I & {S}_{{\vec p}}\otimes {S}_{{\vec p}} \cr
{S}^{-1}_{{\vec p}}\otimes {S}^{-1}_{{\vec p}} &
I\cr}\quad.$$ But, if regard $\tilde{{\cal V}} ({\vec p})=
\gamma_5 {\cal V} ({\vec p})$ one can obtain the complete set. Fortunately, $\overline{\tilde {\cal V}} ({\vec 0}) {\cal U} ({\vec 0}) = 0$ , what permits us to keep the parametrization $$\label{chalf}
j^\mu_{\sigma\sigma^\prime} ({\vec p}, {\vec q}) =
\sum_{\sigma_p=-1/2}^{1/2} j^\mu_{\sigma\sigma_p} ({\vec q} (-) {\vec p},
{\vec p}) \,\,D^{1/2}_{\sigma_p \sigma^\prime}
\left \{ V^{-1} (\Lambda_p, q)\right \}\quad.$$ As a matter of fact, this definition of negative-energy spinors follows from the explicit construct of the theory of the Bargmann-Wightman-Wigner type, which has been proposed by Ahluwalia, [@DVA] and it presents itself a generalization (and a correction) of the Joos-Weinberg model.
As a result we obtain the amplitude for interaction of two $j=1$ Joos-Weinberg particles, mediated by the vector potential: $$\begin{aligned}
\label{212}
\lefteqn{\hat T^{(2)} ({\vec q}(-){\vec p}, {\vec p})
\,=\, g^2 \left\{ \frac{\left [p_0
(\Delta_0 +M) + ({\vec p}\cdot {\vec \Delta})\right ]^2
-M^3 (\Delta_0+M)}{M^3 (\Delta_0 -M)}+\right.}\nonumber\\
&+&\left.\frac{i ({\vec J}_1+{\vec J}_2)\cdot\left [{\vec p}
\times{\vec \Delta}\right ]}
{\Delta_0-M}\left [ \frac{p_0 (\Delta_0 +M)+{\vec p}\cdot
{\vec \Delta}}{M^3} \right ] + \right.\nonumber\\
&+& \left.\frac{({\vec J}_1\cdot {\vec \Delta})({\vec
J}_2\cdot {\vec \Delta})-({\vec J}_1\cdot{\vec J}_2) {\vec \Delta}^2}{2M
(\Delta_0-M)}-\frac{1}{M^3}\frac{{\vec J}_1\cdot\left [{\vec p}
\times{\vec \Delta}\right ] \,\,{\vec J}_2\cdot \left [{\vec
p} \times{\vec \Delta}\right ]}{\Delta_0-M}\right\}.\end{aligned}$$ We have assumed $g_S = g_V = g_T$ above, what is motivated by group-theoretical reasons. The expression (\[212\]) reveals the advantages of the $2(2j+1)$- formalism, since it looks like the amplitude for interaction of two spinor particles within the substitutions $1/\left [ 2M(\Delta_0 - M)\right ] \Rightarrow 1/{\vec \Delta}^2$ and ${\vec J}\Rightarrow {\vec \sigma}$. Calculations hint that many analytical results produced for a Dirac fermion could be applicable for describing a $2(2j+1)$ particle. Nevertheless, it is required adequate explanation of the obtained difference. The reader could note: its origin lies at the kinematical level. Free-space (without interaction) Joos-Weinberg equations admit acausal tachyonic solutions, [*e.g.*]{}, ref. [@DVA0]. “Interaction introduced in the massive Weinberg equations will couple to both the causal and acausal solutions and thus cannot give physically reasonable results". However, let us not forget that we have used the Tucker-Hammer approach, indeed, that does not possess tachyonic solutions.
In the straightforward manner we also obtain amplitudes for interactions: 1) spin-0 boson and spin-1/2 fermion; 2) spin-0 boson and spin-1 Joos-Weinberg boson; 3) spin-1/2 fermion and spin-1 Joos-Weinberg boson. Like the previous ones the Wigner rotations are separated out and all spin indices are “re-setted" on the momentum ${\vec p}$. The detailed discussion of the presented topics could be found in refs. [@DVO1; @DVO2; @DVO3].
Acknowledgments {#acknowledgments .unnumbered}
===============
I appreciate very much discussions with Prof. D. V. Ahluwalia and Prof. A. F. Pashkov. I should thank Prof. N. B. Skachkov for his help in analyzing several topics. I am grateful to Zacatecas University for professorship.
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{
"pile_set_name": "ArXiv"
}
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---
abstract: 'The periodic Anderson lattice model for the crystalline electric field (CEF) split 4f quartet states is used to describe the Yb-based Kondo insulators/semiconductors. In the slave-boson mean-field approximation, we derive the hybridized quasiparticle bands, and find that decreasing the hybridization difference of the two CEF quartets may induce an insulator-to-metal phase transition. The resulting metallic phase has a hole and an electron Fermi pockets. Such a phase transition may be realized experimentally by applying pressure, reducing the difference in hybridization of the two CEF quartets.'
author:
- 'Guang-Bin Li and Guang-Ming Zhang'
- Lu Yu
title: 'Insulator-to-metal phase transition in Yb-based Kondo insulators'
---
Kondo insulators or semiconductors, such as YbB$_{12}$, belong to strongly correlated electron systems[@Aeppli1992; @Riseborough2000], in which the conduction electrons hybridize with the localized 4f-electrons and the strong Coulomb repulsion results in highly renormalized quasiparticle bands with a small indirect energy gap[Susaki1996,Okamura1998,Mignot2005,Nemkovski2007]{}. To study the characteristic properties of these materials at low temperatures, some experiments have been attempted to make the insulating gap vanish by applying an external magnetic field[@Sugiyama1988; @Jaime2000] or pressure[@Gabani2003], leading to an insulator-metal phase transition. Such a transition under the external magnetic field has been considered in the previous studies[@Beach2003; @Milat2004; @Ohashi2004; @Izumi2007], however, the microscopic mechanism for the pressure induced insulator-to-metal transition remains far from being fully understood.
For the Yb-based Kondo insulators, an external pressure can affect the hybridization between 5d band electrons and the more atomic-like 4f electrons, giving rise to the intermediate valence behavior. The Yb valence is directly related to the number of 4f-holes $n_{h}$ by $v=2+n_{h}$. At the ambient pressure, $n_{h}$ spans a broad range between $0$ and $1$ in Yb-based compounds, and the intermediate valence reflects the hybridization of the energetically close Yb$^{2+}$ (4f$^{14}$) and Yb$^{3+}$ (4f$^{13}$) configurations. The electronic configuration of Yb$^{3+}$ (4f$^{13}$) can be regarded as a single hole in the 4f-shell, while the configuration of Yb$%
^{2+}$ (4f$^{14}$) corresponds to the closed 4f-shell. Taking into account the much larger strength (1.3eV) of the spin-orbit coupling[Martensson1982]{}, a $j=7/2$ f-hole state is split into a quartet and two doublet states by the crystalline electric field (CEF) under the cubic symmetry, which is the usual lattice structure of YbB$_{12}$. These two doublets are almost degenerate and may be treated as a quasi-quartet. Thus, a periodic Anderson lattice model with $U\rightarrow \infty $ for the CEF split 4f states can be used to describe these Yb-based Kondo insulators or semiconductors[@Akbari2009].
It has been further pointed out that the anisotropic hybridizations of the two CEF quartets play an important role in the formation of the two dispersive spin resonances at the continuum threshold[@Akbari2009], the most salient features observed by inelastic neutron scattering experiments in YbB$_{\text{12}}$ (Ref.). Motivated by this analysis, we further notice that, above a threshold of the CEF splitting, decreasing the difference in the hybridization of the two CEF quartets may cause an overlap between the middle lower and upper hybridized quasiparticle bands, leading to an insulator-to-metal phase transition. Experimentally, this phase transition can be realized by applying pressure, reducing the difference in the hybridization of the two CEF quasi-quartets. Such a pressure induced insulator-to-metal transition has been observed in the Kondo insulator SmB$_{6}$, where the electrical resistivity has been measured below $80$ K and under pressure between $1$ bar and $70$ kbar (Ref.[@Gabani2003]). Above the critical pressure $40$ kbar, a transition occurs from a Kondo insulator to a metallic heavy fermion liquid and a non-Fermi liquid behavior has been found[@Gabani2003].
In this paper, we will carefully study the periodic Anderson model with $%
U\rightarrow \infty $ for the CEF split 4f quartet states. Using the slave-boson mean-field approximation, we will derive four quasiparticle bands resulting from the hybridization between the conduction electrons and localized 4f-hole states, and find that decreasing the hybridization difference of the two CEF quartets indeed induce an insulator-to-metal phase transition. The resulting metallic phase has a hole and an electron Fermi pockets. By including the Coulomb interaction between the localized and conduction electrons, we discuss the possible instability of the resulting metallic phase.
To describe the Yb-based Kondo insulators or semiconductors, the periodic Anderson lattice model with $U\rightarrow \infty $ for the CEF split 4f states has been introduced[@Akbari2009]$$\begin{aligned}
\mathcal{H}& =\sum_{\mathbf{k},\gamma }\epsilon _{\mathbf{k}}d_{\mathbf{k}%
\gamma }^{\dag }d_{\mathbf{k}\gamma }+\sum_{\mathbf{k},\gamma }(\varepsilon
_{f}+\Delta _{\gamma })f_{i\gamma }^{\dag }f_{i\gamma } \notag \\
& +\frac{1}{\sqrt{\mathcal{N}}}\sum_{i,\mathbf{k},\gamma }(V_{\mathbf{k}%
\gamma }e^{i\mathbf{k\cdot R}_{i}}f_{i,\gamma }^{\dag }d_{\mathbf{k}\gamma
}b_{i}+h.c.), \label{model}\end{aligned}$$where the first term denotes the conduction electron band, the second term stands for the binding energy of the 4f-hole, and $\Delta _{\gamma }$ ($%
\Delta _{1}=0,\Delta _{2}=\Delta $) is the CEF splitting energy for the two quasi-quartets with $\gamma =(\Gamma ,m)$, where $\Gamma =1,2$ denotes the quartets and $m=1-4$ represents the four-fold orbital degeneracy. Due to the exclusion of the double occupancy, a projection has been implemented by using the slave-boson representation[@Coleman1984]. Then the Yb$^{2+}$ (4f$^{14}$) configuration without a 4f-hole state can be accounted for by an auxiliary boson state $b_{i}^{\dag }\left| 0\right\rangle $, while the Yb$%
^{3+}$ (4f$^{13}$) configuration with a 4f-hole state is represented by a fermion state $f_{i\gamma }^{\dag }\left| 0\right\rangle $. The conduction electrons hybridize with the $f$-hole at each lattice site in both quartets with different strengths. At each lattice site the constraint $%
Q_{i}=b_{i}^{\dag }b_{i}+\sum_{\gamma }f_{i\gamma }^{\dag }f_{i\gamma }=1$ has to be enforced, and the total Hamiltonian is $\mathcal{H}%
+\sum_{i}\lambda _{i}(Q_{i}-1)$, where $\lambda _{i}$ is the Lagrange multiplier.
Now the slave-boson mean-field approximation is performed by neglecting the fluctuation of the Bose field $\langle b_{i}^{\dagger }\rangle =\left\langle
b_{i}\right\rangle =b$ and the site dependence of the local field $\lambda
_{i}=\lambda $. Within these approximations, the mean-field model Hamiltonian can be written as $$\mathcal{H}_{mf}=\sum_{\mathbf{k},\gamma }[\epsilon _{\mathbf{k}}d_{\mathbf{k%
}\gamma }^{\dag }d_{\mathbf{k}\gamma }+\tilde{\varepsilon}_{\gamma }f_{%
\mathbf{k}\gamma }^{\dag }f_{\mathbf{k}\gamma }+\tilde{V}_{\gamma }(d_{%
\mathbf{k}\gamma }^{\dag }f_{\mathbf{k}\gamma }+h.c.)]+\mathcal{N}\epsilon
_{0},$$where $\tilde{\varepsilon}_{\gamma }=\varepsilon _{f}+\Delta _{\gamma
}+\lambda $ is the renormalized energy level of the localized states, $%
\tilde{V}_{\gamma }=bV_{\gamma }$, and $\epsilon _{0}=\lambda (b^{2}-1)$. It should be noticed that the dependence of the hybridization strength on $%
\mathbf{k}$ has been neglected, *i.e.*, $V_{\mathbf{k}\gamma
}=V_{\gamma }$. Furthermore, we will replace $V_{\gamma }$ by $V_{\Gamma }$ for simplicity. By performing the Bogoliubov transformation $$\alpha _{\mathbf{k}\gamma }=\mu _{\mathbf{k\gamma }}d_{\mathbf{k}\gamma
}+\nu _{\mathbf{k}}f_{\mathbf{k}\gamma },\text{ }\beta _{\mathbf{k}\gamma
}=-\nu _{\mathbf{k\gamma }}d_{\mathbf{k}\gamma }+\mu _{\mathbf{k\gamma }}f_{%
\mathbf{k}\gamma },\text{\ }$$we can diagonalize the quadratic Hamiltonian and obtain $$\mathcal{H}_{MF}=\sum_{\mathbf{k},\gamma }\left( E_{\mathbf{k}\gamma
}^{+}\alpha _{\mathbf{k}\gamma }^{\dag }\alpha _{\mathbf{k}\gamma }+E_{%
\mathbf{k}\gamma }^{-}\beta _{\mathbf{k}\gamma }^{\dag }\beta _{\mathbf{k}%
\gamma }\right) ,$$with four hybridized quasiparticle bands are
$$E_{\mathbf{k}\gamma }^{\pm }=\frac{1}{2}\left[ \epsilon _{\mathbf{k}}+\tilde{%
\varepsilon}_{\gamma }\pm \sqrt{(\epsilon _{\mathbf{k}}-\tilde{\varepsilon}%
_{\gamma })^{2}+4\tilde{V}_{\gamma }^{2}}\right] ,$$
while the Bogoliubov parameters $\mu _{\mathbf{k}\gamma }$ and $\nu _{%
\mathbf{k}\gamma }$ are given by$$\left(
\begin{array}{c}
\mu _{\mathbf{k}\gamma } \\
\nu _{\mathbf{k}\gamma }%
\end{array}%
\right) =\frac{1}{\sqrt{2}}\left[ 1\pm \frac{\epsilon _{\mathbf{k}}-\tilde{%
\varepsilon}_{\gamma }}{\sqrt{(\epsilon _{\mathbf{k}}-\tilde{\varepsilon}%
_{\gamma })^{2}+4\tilde{V}_{\gamma }^{2}}}\right] ^{1/2}.$$These two parameters describe the contributions of the conduction electron band and localized $f$-hole band to the hybridized quasiparticles, respectively.
Moreover, the ground-state energy per site is given by $$E_{g}=\frac{1}{\mathcal{N}}\sum_{\mathbf{k},\gamma }\left[ E_{\mathbf{k}%
\gamma }^{+}\theta (E_{\mathbf{k}\gamma }^{+})+E_{\mathbf{k}\gamma
}^{-}\theta (E_{\mathbf{k}\gamma }^{-})\right] +\epsilon _{0},$$where $\theta (E_{\mathbf{k}\gamma }^{\pm })$ is the step function. The chemical potential $\mu $ and the Lagrange multiplier $\lambda $ have to be determined self-consistently according to the conservation of the total number of particle per lattice site $n_{c}+n_{f}=2$. Depending on the parameter values $\varepsilon _{f}$, $\Delta $, and$V_{\Gamma }$, the variational parameters $b$ and $\lambda $ are also determined self-consistently. From the hybridized quasiparticle band structure, the ground state of the system can be an insulating state, where the two lower bands are filled completely, leaving an indirect energy gap. As the $\mathbf{%
k}$ dependence in $E_{g}$ appears through the conduction electron energy $%
\epsilon _{\mathbf{k}}$, summations over $\mathbf{k}$ can be transformed into an integral over energy $\epsilon $ in the interval $[-D,D]$. By assuming a constant density of states, the ground-state energy is thus evaluated as$$\begin{aligned}
E_{g}& =\frac{1}{8D}\sum_{\Gamma }\left\{ 4D\tilde{\varepsilon}_{\Gamma }-4%
\tilde{V}_{\Gamma }^{2}\ln \frac{\Lambda _{\Gamma }^{-}(D)+D-\tilde{%
\varepsilon}_{\Gamma }}{\Lambda _{\Gamma }^{+}(D)-D-\tilde{\varepsilon}%
_{\Gamma }}\right. \notag \\
& -\left. [(D-\tilde{\varepsilon}_{\Gamma })\Lambda _{\Gamma }^{-}(D)+(D+%
\tilde{\varepsilon}_{\Gamma })\Lambda _{\Gamma }^{+}(D)]\right\} +\epsilon
_{0},\end{aligned}$$where $\Lambda _{\Gamma }^{\pm }(x)=\sqrt{(x\pm \tilde{\varepsilon}_{\Gamma
})^{2}+4\tilde{V}_{\Gamma }^{2}}$, $\tilde{\varepsilon}_{2}=\tilde{%
\varepsilon}_{1}+\Delta ,$ and $\tilde{V}_{2}=bV_{2}=b(V_{1}+\delta V)$. Minimizing the ground-state energy density with respect to $b$ and $\lambda $, respectively, we obtain the following self-consistent equations$$\begin{gathered}
b^{2}=\frac{1}{4D}\sum_{\Gamma }\left[ \Lambda _{\Gamma }^{+}(D)-\Lambda
_{\Gamma }^{-}(D)\right] , \notag \\
\lambda =\frac{1}{2D}\sum_{\Gamma }V_{\Gamma }^{2}\ln \frac{\Lambda _{\Gamma
}^{-}(D)+D-\tilde{\varepsilon}_{\Gamma }}{\Lambda _{\Gamma }^{+}(D)-D-\tilde{%
\varepsilon}_{\Gamma }}. \label{insulator}\end{gathered}$$
However, we notice that there exists another possible structure of the quasiparticle bands, where the chemical potential $\mu $ cuts through the two middle hybridized quasiparticle bands $E_{\mathbf{k}1}^{+}$ and $E_{%
\mathbf{k}2}^{-}$ at $\xi _{1}$ and $\xi _{2}$, respectively. Both these energy parameters are determined by the equation $E_{\mathbf{k}1}^{+}=E_{%
\mathbf{k}2}^{-}=\mu $. From the condition of the total number of particles per lattice site $n_{c}+n_{f}=2$, we can derive the result $\xi _{1}=-\xi
_{2}\equiv -\xi $ and $$2\xi +\Delta =\Lambda _{1}^{+}(\xi )+\Lambda _{2}^{-}(\xi ).
\label{order-para}$$Here $\xi $ can be used to characterize the insulator-to-metal transition. When $0<\xi <D$, the ground state should be metallic, while for $\xi =D$ the ground state corresponds to a critical point. The corresponding ground-state energy density in the metallic phase is thus expressed as $$\begin{aligned}
E_{g}& =\frac{1}{4D}[(3D-\xi )\tilde{\varepsilon}_{1}+(D+\xi )\tilde{%
\varepsilon}_{2}+\xi ^{2}-D^{2}] \notag \\
& +\frac{\tilde{V}_{1}^{2}}{2D}\ln \frac{\Lambda _{1}^{+}(\xi )-\xi -\tilde{%
\varepsilon}_{1}}{\Lambda _{1}^{-}(D)+D-\tilde{\varepsilon}_{1}}-\frac{%
\tilde{V}_{2}^{2}}{2D}\ln \frac{\Lambda _{2}^{-}(\xi )+\xi -\tilde{%
\varepsilon}_{2}}{\Lambda _{2}^{+}(D)-D-\tilde{\varepsilon}_{2}} \notag \\
& -\frac{1}{8D}[(\xi +\tilde{\varepsilon}_{1})\Lambda _{1}^{+}(\xi )+(D-%
\tilde{\varepsilon}_{1})\Lambda _{1}^{-}(D)] \notag \\
& -\frac{1}{8D}[(\xi -\tilde{\varepsilon}_{2})\Lambda _{2}^{-}(\xi )+(D+%
\tilde{\varepsilon}_{2})\Lambda _{2}^{+}(D)]+\epsilon _{0}.\end{aligned}$$By minimizing $E_{g}$ with respect to $b$ and $\lambda $, the corresponding self-consistent equations can be deduced to $$\begin{gathered}
b^{2}=\frac{1}{4D}[\Lambda _{1}^{+}(\xi )-\Lambda _{1}^{-}(D)-\Lambda
_{2}^{-}(\xi )+\Lambda _{2}^{+}(D)], \notag \\
\lambda =\frac{V_{1}^{2}}{2D}\ln \frac{\Lambda _{1}^{-}(D)+D-\tilde{%
\varepsilon}_{1}}{\Lambda _{1}^{+}(\xi )-\xi -\tilde{\varepsilon}_{1}}+\frac{%
V_{2}^{2}}{2D}\ln \frac{\Lambda _{2}^{-}(\xi )+\xi -\tilde{\varepsilon}_{2}}{%
\Lambda _{2}^{+}(D)-D-\tilde{\varepsilon}_{2}}. \label{metal}\end{gathered}$$
In order to deduce the ground state phase diagram, we should first numerically solve Eq.(\[insulator\]) for the insulating phase and Eqs.([order-para]{}) and (\[metal\]) for the metallic phase, respectively. The hybridized quasiparticle band energy versus the momentum along the diagonal direction $\Gamma $ $(0,0,0)->M$ $(\pi ,\pi ,\pi )$ are plotted in Fig.[spectra]{} with $V_{1}=0.4D$, $\epsilon _{f}=-0.5D$, and $\Delta =0.1D$ for three different values of $\delta V$. As shown in Fig.\[spectra\](a) for $%
\delta V=0.18D$, there opens an indirect gap between the middle upper and lower bands, corresponding to an insulating phase. In Fig.\[spectra\](b) for $\delta V=0.126D$, the middle upper and lower bands just meet at the chemical potential, corresponding the critical point of the transition. Since we have $\xi =D$ at the critical point, the ground-state energies of the metallic and insulating phases are equal. So the insulator-metal transition is a continuous second-order phase transition. Finally, in Fig.\[spectra\](c) for $\delta V=0.01D$, the middle lower and upper bands overlap, and the chemical potential cuts through these two bands, which corresponds to the metallic phase.
The critical condition under which the insulator-metal transition occurs can be determined from Eq.(\[order-para\]) and Eq.(\[metal\]) by setting $%
\xi =D$. Then the ground-state phase diagram can constructed for $V_{1}=0.4D$ and $\epsilon _{f}=-0.5D$ and is shown in Fig.\[phasediagram\](a). Clearly there exists a threshold of the CEF splitting energy $\Delta _{c}$, and only when $\Delta >\Delta _{c}$ the insulator-to-metal phase transition occurs by turning the difference in hybridization of the two CEF quasi-quartets. The change of the indirect gap is another evidence to characterize the insulator-to-metal phase transition, and can be also calculated and displayed in Fig.\[phasediagram\](b) for $\Delta =0.1D$. It shows that the indirect quasiparticle gap decreases almost linearly with decreasing the hybridization difference of the two CEF quartets, and this energy gap finally vanishes at $\delta V_{c}$. There is another critical value $\delta
V^{\ast }$, where the top energy levels of the two lower quasiparticle bands interchange with each other around the Brillouin zone boundary. Then the indirect energy gap has a cusp.
Actually, such an insulator-to-metal phase transition can be realized experimentally. There exists a strong CEF splitting estimated in YbB$_{12}$, and we believe that increasing pressure can continuously reduce the difference in hybridization of the two CEF quasi-quartets. So below the critical value $\delta V_{c}$, YbB$_{12}$ is an insulator with an indirect gap as observed in experiments [Susaki1996,Okamura1998,Mignot2005,Nemkovski2007]{}, while above this critical value $\delta V_{c}$ this material can transform into a heavy electron metal with an enhanced effective mass due to the presence of heavy charge carriers. Thus, our theory may provide a general microscopic mechanism of the pressure induced insulator-to-metal transition in Yb-based Kondo insulators/semiconductors.
Since a constant density of states for the conduction electron band was assumed in the above slave-boson mean-field calculation, the obtained results are independent of the dimensionality of the model. In order to see the special Fermi surface structure of the metallic phase, the model Hamiltonian Eq.(\[model\]) is redefined on a two-dimensional square lattice system with the conduction electron band$$\epsilon _{\mathbf{k}}=-2t(\cos k_{x}+\cos k_{y})+4t^{\prime }\cos k_{x}\cos
k_{y},$$where $t$ denotes the nearest neighbor hopping and $t^{\prime }$ denotes the next-nearest neighbor hopping. Then the same slave-boson mean field calculation can be performed, and the insulator-to-metal phase transition also takes place for a set of parameters $V_{1}=0.4D$, $\epsilon _{f}=-0.5D$, $t=0.25D$, and $t^{\prime }=0.3t$ when decreasing the parameter $\delta V$. In the metallic phase, we have calculated the corresponding Fermi surface structure shown in Fig.\[fermisurface\]. There exist two Fermi pockets: one electron-like in the center of the Brillouin zone and one hole-like in the corners of the Brillouin zone. These two Fermi pockets have exactly the same area in the Brillouin zone. Such a heavy electron metal corresponds to a semi-metal. Decreasing the hybridization difference $\delta
V$ below the critical value, the sizes of the electron and hole Fermi pockets become larger and larger, as displayed in Fig.\[fermisurface\].
For the three dimensional Yb-based Kondo insulators/semiconductors, the Fermi surface of the resulting metallic phase should still be given by a hole and an electron pockets. As the temperature is lowered enough, some instabilities may further appear. Due to the presence of the strong mixed valence effect in such systems, the additional on-site Coulomb interaction between the conduction electrons and localized $f$-hole should be taken into account. In the slave-boson representation, it is given by $$\mathcal{H}_{I}=U_{fc}\sum_{i}\sum_{\gamma \gamma ^{\prime }}f_{i\gamma
}^{\dag }f_{i\gamma }d_{i\gamma ^{\prime }}^{\dag }d_{i\gamma ^{\prime }}.$$When the coupling strength $U_{fc}$ is assumed to be small, we can rewrite this additional interaction in terms of the hybridized quasiparticles as$$\begin{gathered}
\mathcal{H}_{I}=\frac{U_{fc}}{\mathcal{N}}\sum_{\mathbf{\mathbf{k}}_{1}%
\mathbf{k}_{2}\mathbf{\mathbf{k}}_{3}\mathbf{\mathbf{k}}_{4}}\left( \nu _{%
\mathbf{k}_{1}1}\mu _{\mathbf{k}_{2}1}\nu _{\mathbf{\mathbf{k}}_{3}1}\mu _{%
\mathbf{\mathbf{k}}_{4}1}\alpha _{\mathbf{\mathbf{k}}_{3}1}^{\dag }\alpha _{%
\mathbf{k}_{1}1}\alpha _{\mathbf{\mathbf{k}}_{4}1}^{\dag }\alpha _{\mathbf{k}%
_{2}1}\right. \text{ } \notag \\
+\mu _{\mathbf{k}_{1}2}\nu _{\mathbf{k}_{2}2}\mu _{\mathbf{\mathbf{k}}%
_{3}2}\nu _{\mathbf{\mathbf{k}}_{4}2}\beta _{\mathbf{\mathbf{k}}_{3}2}^{\dag
}\beta _{\mathbf{k}_{1}2}\beta _{\mathbf{\mathbf{k}}_{4}2}^{\dag }\beta _{%
\mathbf{k},2} \notag \\
+\nu _{\mathbf{k}_{1}1}\nu _{\mathbf{k}_{2}2}\nu _{\mathbf{\mathbf{k}}%
_{3}1}\nu _{\mathbf{\mathbf{k}}_{4}2}\alpha _{\mathbf{\mathbf{k}}%
_{3}1}^{\dag }\alpha _{\mathbf{k}_{1}1}\beta _{\mathbf{\mathbf{k}}%
_{4}2}^{\dag }\beta _{\mathbf{k}_{2}2} \notag \\
\text{ \ \ \ \ \ }+\left. \mu _{\mathbf{k}_{1}2}\mu _{\mathbf{k}_{2}1}\mu _{%
\mathbf{\mathbf{k}}_{3}2}\mu _{\mathbf{\mathbf{k}}_{4}1}\beta _{\mathbf{%
\mathbf{k}}_{3}2}^{\dag }\beta _{\mathbf{k}_{1}2}\alpha _{\mathbf{\mathbf{k}}%
_{4}1}^{\dag }\alpha _{\mathbf{k}_{2}1}\right) ,\end{gathered}$$where $\mathbf{\mathbf{k}}_{1}+\mathbf{k}_{2}\mathbf{=\mathbf{k}}_{3}+%
\mathbf{\mathbf{k}}_{4}$ should be satisfied and *only* the two quasiparticle bands, crossing the Fermi energy, have been taken into account. $\alpha _{\mathbf{k},1}$ and $\alpha _{\mathbf{k},1}^{\dagger }$ are defined on the electron Fermi pocket, while $\beta _{\mathbf{k},2}$ and $%
\alpha _{\mathbf{k},2}^{\dagger }$ are defined on the hole Fermi pocket. Among these residual quasiparticle interactions, the first two terms represent the intra-pocket scatterings with a small momentum transfer, while the last two terms correspond to the inter-pocket scatterings with a large momentum transfer.
According to the recent renormalization group analysis for a two-band interacting model with electron and hole Fermi pockets[@Chubukov2008], the inter-pocket quasiparticle interactions will determine the possible instabilities at low temperatures. When we set $\mathbf{q}$ as a small momentum and $\mathbf{Q}$ as a large momentum which is the distance between the centers of two Fermi pockets, then inter-pocket quasiparticle interactions can be approximated as$$\begin{aligned}
&&-\frac{U_{fc}}{\mathcal{N}}\sum_{\mathbf{qq}^{\prime }}\text{ }(\mu _{%
\mathbf{q}1}\mu _{\mathbf{q}^{\prime }1}\mu _{\mathbf{q}2}\mu _{\mathbf{q}%
^{\prime }2}+\nu _{\mathbf{q}1}\nu _{\mathbf{q}^{\prime }1}\nu _{\mathbf{q}%
2}\nu _{\mathbf{q}^{\prime }2}) \notag \\
&&\text{ \ \ \ \ \ \ \ \ }\times \alpha _{\mathbf{q\mathbf{,}}1}^{\dag
}\beta _{\mathbf{Q+q},2}\beta _{\mathbf{Q+q}^{\prime }2}^{\dag }\alpha _{%
\mathbf{q}^{\prime },1}.\end{aligned}$$If there is a strong nesting between the hole and electron Fermi pockets, this inter-pocket repulsive interaction will further induce a particle-hole pairing instability, corresponding to an orbital-density wave ordering. The corresponding order parameter is given by $\langle \alpha _{\mathbf{q}%
1}^{\dag }\beta _{\mathbf{Q+q}2}\rangle $ or $\langle \beta _{\mathbf{Q+q}%
^{\prime }2}^{\dag }\alpha _{\mathbf{q}^{\prime }1}\rangle $. Such a new type of ordering in heavy fermion materials will be discussed in our further investigations.
In conclusion, we have studied the Yb-based Kondo insulators with a strong CEF splitting in the framework of the periodic Anderson lattice model by using the slave-boson mean-field approximation. The obtained ground-state phase diagram and the indirect gap have demonstrated that a second-order insulator-to-metal transition occurs via reducing the hybridization difference of the two CEF quasi-quartets. Our theory provides a general microscopic mechanism of the pressure induced insulator-to-metal transition, because increasing the external pressure can effectively reduce the anisotropy of the hybridization strengths of the two CEF quartets experimentally. The resulting metallic phase has a hole and an electron Fermi pockets, which may exhibit an instability of an orbital-density wave ordering at low temperatures when the inter-pocket quasiparticle residual interactions are taken into account. These theoretical results are certainly needed to be confirmed experimentally in the future.
The authors would like to thank Dung-Hai Lee for his stimulating discussions and Yu Liu for his helps in the numerical calculations. This work is partially supported by NSF-China and the National Program for Basic Research of MOST, China.
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: |
We report on the second multiwavelength campaign of the blazar 3C 454.3 during the first half of December 2007. This campaign involved AGILE, [*Spitzer*]{}, [*Swift*]{}, [ *Suzaku*]{}, the WEBT consortium, the REM and MITSuME telescopes, offering a broad band coverage that allowed for a simultaneous sampling of the synchrotron and inverse Compton (IC) emissions. The 2-week monitoring was accompanied by radio to optical monitoring by WEBT and REM and by sparse observations in mid-Infrared and soft/hard X-ray energy bands performed by means of Target of Opportunity observations by [*Spitzer*]{}, [*Swift*]{} and [*Suzaku*]{}, respectively. The source was detected with an average flux of $\sim 250
\times 10^{-8}$ above 100 MeV, typical of its flaring states. The simultaneous optical and $\gamma$-ray monitoring allowed us to study the time-lag associated with the variability in the two energy bands, resulting in a possible $\lesssim$ 1-day delay of the gamma-ray emission with respect to the optical one. From the simultaneous optical and gamma-ray fast flare detected on December 12, we can constrain the delay between the gamma-ray and optical emissions within 12 hours. Moreover, we obtain three Spectral Energy Distributions (SEDs) with simultaneous data for 2007 December 5, 13, 15, characterized by the widest multifrequency coverage. We found that a model with an external Compton on seed photons by a standard disk and reprocessed by the Broad Line Regions does not describe in a satisfactory way the SEDs of 2007 December 5, 13 and 15. An additional contribution, possibly from the hot corona with $T
= 10^{6}$ K surrounding the jet, is required to account simultaneously for the softness of the synchrotron and the hardness of the inverse Compton emissions during those epochs.
author:
- 'I. Donnarumma, G. Pucella, V. Vittorini, F. D’Ammando, S. Vercellone, C. M. Raiteri, M. Villata, M. Perri, W.P. Chen, R.L. Smart, J. Kataoka, N. Kawai, Y. Mori, G. Tosti, D. Impiombato, T. Takahashi, R. Sato, M. Tavani, A. Bulgarelli, A. W. Chen, A. Giuliani, F. Longo, L. Pacciani, A. Argan, G. Barbiellini, F. Boffelli, P. Caraveo, P. W. Cattaneo, V. Cocco, T. Contessi, E. Costa, E. Del Monte, G. De Paris, G. Di Cocco, Y. Evangelista, M. Feroci, A. Ferrari, M. Fiorini, T. Froysland, M. Frutti, F. Fuschino, M. Galli, F. Gianotti, C. Labanti, I. Lapshov, F. Lazzarotto, P. Lipari, M. Marisaldi, M. Mastropietro, S. Mereghetti, E. Morelli, E. Moretti, A. Morselli, A. Pellizzoni, F. Perotti, P. Picozza, M. Pilia, G. Porrovecchio, M. Prest, M. Rapisarda, A. Rappoldi, A. Rubini, S. Sabatini, E. Scalise, P. Soffitta, E. Striani, M. Trifoglio, A. Trois, E. Vallazza, A. Zambra, D. Zanello, C. Pittori, P. Santolamazza, F. Verrecchia, P. Giommi, L. A. Antonelli, S. Colafrancesco, L. Salotti'
nocite: '[@*]'
title: 'Multiwavelength observations of 3C 454.3 II. The 2007 December campaign'
---
Introduction
============
3C 454.3 is a flat spectrum radio quasar at redshift $z=0.859$. It is one of the brightest extragalactic radio sources with superluminal motion hosting a radio and X-ray jet. It has been observed in almost all the electromagnetic spectrum from radio up to energies; the SED has the typical double-humped shape of the blazars, the first peak occurring at mid-far infrared frequencies and the second one at MeV–GeV energies (see Ghisellini et al. 1998).
The first peak is commonly interpreted as synchrotron radiation from high energy electrons in a relativistic jet, while the second component is due to electrons scattering on soft seed photons. In the context of a simple, homogeneous scenario, the emission at the synchrotron and IC peaks is produced by the same electrons population that can self scatter the same synchrotron photons (Synchrotron Self Compton, SSC). Alternatively, the jet-electrons producing the synchrotron flux can Compton-scatter seed photons produced outside of the jet (External Compton, EC).
3C 454.3 is a highly variable blazar source. In spring 2005, 3C 454.3 experienced a strong outburst in the optical band reaching its historical maximum with $R=12.0 $ mag (Villata et al. 2006). The exceptional event triggered observations at higher energies from Chandra (Villata et al. 2006), [*Swift*]{} (Giommi et al. 2006) and INTEGRAL (Pian et al. 2006). The available data allowed to build the spectrum of the source up to 200 keV. In particular, INTEGRAL detected (15-18 May 2005) the source from 3 up to 200 keV in a very bright state ($\sim 5\times
10^{-10}$ erg cm$^{-2}$ s$^{-1}$), being almost a factor 2-3 higher than the previously observed fluxes (see Tavecchio et al. 2002). Pian et al. (2006) compared the SED in 2000 with that obtained during the 2005 outburst. They were able to describe both observed SEDs with minimal changes in the jet power, assuming that the dissipation region (where most of the radiation is produced) was inside the Broad Line Region (BLR) in 2000 and outside of it in 2005. On the other hand, Sikora et al. (2008) argued that X-rays and could be produced via inverse Compton scattering of near- and mid-IR photons emitted by the hot dust: a very moderate energy density of the dust radiation is sufficient to provide the dominance of the EC luminosities over the SSC ones.
In July 2007 the source woke up again in the optical band, reaching a maximum at $R=12.6$ mag (Raiteri et al. 2008b). Such an increase in the optical activity triggered observations with and . Although still in its Science Verification Phase, repointed at 3C 454.3 and detected it in high $\gamma$-ray activity. The average flux detected by was the highest $\gamma$-ray flux ever detected from this blazar, being $(280 \pm 40) \times 10^{-8}$ (see Fig. 3 lower panel in Vercellone et al. 2008).
Ghisellini et al. (2007) reproduced the three source states in 2000, 2005, 2007 with the model proposed in Katarzynski $\&$ Ghisellini (2007). The model assumes that the relative importance of synchrotron and SSC luminosity with respect to the EC one is controlled by the value of the bulk Lorentz factor $\Gamma$, which is associated to the compactness of the source.
Villata et al. (2007) and Raiteri et al. (2008b) suggested an alternative interpretation involving changes of the viewing angle of the different emitting regions of the jet. In both cases, a strong degeneracy of parameters exists in both SSC and EC models especially when the data are missing. This is the case of both SEDs in 2000, 2005 in which historical EGRET data were used to constrain the models.
3C 454.3 exhibited outbursts several times between 2007 and 2008 (see Vercellone et al. 2008, Tosti et al. 2008, Raiteri et al. 2008a, Raiteri et al. 2008b, Vercellone et al. 2009) posing stringent constraints on its gamma-ray duty cycle.
This strengthened the need for simultaneous observations in different energy bands. In the case of 3C 454.3 (and other MeV blazars) it is clear that the dominant contribution in the SED comes from IR-optical bands, where the synchrotron peak lies and from both X-rays and energy range where the inverse Compton emission lies.
In this paper we present and discuss the result of a multiwavelength campaign on 3C 454.3 during a period of intense $\gamma$-ray activity occurred between 2007 December 1 and 16. In Section 2 we present the multiwavelength campaign, in Sect. 3 - 7 we present the , , , $Spitzer$, REM, WEBT and observations and data analysis; in Sect. 8 we analyse the $\gamma$-optical correlation and present broad-band SEDs built with simultaneous data, discussing in details how they are modelled in the framework of SSC and EC scenarios. Throughout this paper the photon indexes are parametrized as $N(E)\propto E^{-\Gamma}$ (keV$^{-1}$ or MeV$^{-1}$). The uncertainies are given at 1-$\sigma$ level, unless otherwise stated.
The multiwavelength campaign
============================
During the period of intense $\gamma$-ray activity showed in November 2007 (Vercellone et al. 2009), continued the pointing towards 3C 454.3 for the first half of December 2007. The persistent high $\gamma$-ray activity of the source stimulated us to activate a new multiwavelength campaign.
data were collected between 2007 December 1 and 2007 December 16. data were collected during a dedicated Target of Opportunity (ToO) performed on December 5, whereas the [*Spitzer*]{} data were collected on December 13 and 15 thanks to a granted Director’s Discretionary Time (DDT) observation.
During these two days a ToO with [*Swift*]{} data was activated for a total exposure of 9 ks.
During the whole observations the source was monitored in radio-to-optical bands by WEBT (see Raiteri et al. 2008a). In addition, observations in the NIR and optical energy bands by REM occurred between December 1 and 8. Moreover, optical data from telescope are available on this source until December 6. In the following sections we report on the details of the observations and the data analysis for each instrument.
Observation
============
(Astrorivelatore Gamma a Immagini Leggero, Tavani et al. 2008, 2009) is a mission of the Italian Space Agency (ASI) for the exploration of $\gamma$-ray sky, operating in a low Earth orbit since 2007 April 23. The scientific Instrument (Prest et al. 2003, Perotti et al. 2006, Labanti et al. 2009) is very compact and combines four active detectors yielding simultaneous coverage in gamma-rays, 30 MeV-30 GeV and in hard X-ray energy band 18-60 keV (Feroci et al. 2007).
The observations of 3C 454.3 were performed between 2007 December 1 and 16, for a 2-week total pointing duration. In the first period, between December 1 and 5, the source was located $\sim 45^\circ$ off the pointing direction. In the second period, between December 5 and 16, after a satellite re-pointing, the source was located at $\sim 30^{\circ}$ off-axis (variable by $\sim 1$ degree per day due to the pointing drift) thus to increase the significance of the detection.
-GRID (AGILE-Gamma Rays Imaging Detector) data were analyzed using the Standard Analysis Pipeline. Counts, exposure, and Galactic background maps were created with a bin-size of $0.3^{\circ} \times 0.3^{\circ}$ for photons with energy greater than 100 MeV. To reduce the particle background contamination we selected only events flagged as confirmed $\gamma$-ray events, and all events collected during the South Atlantic Anomaly were rejected. We also reduced the $\gamma$-ray Earth albedo contamination by excluding regions within $\sim 10^{\circ}$ from the Earth limb.
The 2-week data have been divided in 2 sets taking into account the two different pointings during which the source shifted from $\sim 45$ to $\sim
30$ degrees off-axis: the first set between UTC 2007-12-01 13:21 and UTC 2007-12-05 12:34; the second set between UTC 2007-12-05 12:35 and 2007-12-16 10:27 (in Fig. 1, top panel, we also report the flux of 1 day before). The first set required a more detailed analysis due to uncertainty on calibration for large off-axis angles in the Field of View (FoV).
We ran the Maximum Likelihood procedure (ALIKE) on each data set, in order to obtain the average flux as well as the daily values in the $\gamma$-ray band, according to Mattox et al. (1996). The average fluxes obtained integrating separately the two data sets are $(280\pm 50) \times 10^{-8}$ ($\sqrt(TS) \sim 8$) and $(210\pm16) \times 10^{-8}$ ($\sqrt(TS)\sim 20$) for the first and second periods, respectively. The source was always detected on the 2-week period with a daily integration time. The 1-day binned light curve shows three enhancements of the emission around December 4, December 7 and December 13 (see Fig. 1 top panel). In particular, the last two enhancements are characterized by a sharp increase of the emission followed by a slow recovery. We accumulated the spectrum over the second set of data in which the source was positioned within 30 degrees in the -GRID Field of View, where the most significant energy spectrum can be extracted, due to the higher statistical quality. The spectral fit was performed by using only data between 100 MeV and 1 GeV (which are better calibrated) although in Fig. 2 we report also the energy bin below 100 MeV. It resulted in a power law with a photon index $\Gamma = 1.78\pm 0.14$. We note that the current AGILE response is calibrated up to 1 GeV, and that the energy flux above 1 GeV is underestimated by a factor of 2-3. This prevented us to discuss any possible spectral break above 1 GeV as found by [*Fermi*]{} (Abdo et al. 2009). However, we note that the AGILE spectrum seems to be harder than the one inferred by [ *Fermi*]{} below $\sim 3$ GeV. The different energy range and the non-simultaneity of the data could explain the difference between the photon indexes.
Super did not detect the source during the 2-week pointing. A deep 3-$\sigma$ upper limit of $\sim 10$ mCrab was derived integrating all the data in which the source was within 30 degrees in the FoV (net source exposure of 360 ks).
Observation
============
Following the detection of the flaring state, 3C 454.3 was observed with (Mitsuda et al. 2007) on December 2007 as a ToO, with a total duration of 40 ks. carries four sets of X-ray telescopes (Serlemitsos et al. 2007) each one equipped with a focal-plane X-ray CCD camera (XIS, X-ray Imaging Spectrometer; Koyama et al. 2007) that is sensitive in the energy range of 0.3$-$12keV, together with a non-imaging Hard X-ray Detector (HXD; Takahashi et al. 2007; Kokubun et al. 2007), which covers the 10$-$600keV energy band with Si PIN photo-diodes and GSO scintillation detectors. 3C 454.3 was focused on the nominal center position of the XIS detectors.
For the XIS, we analyzed the screened data, reduced via software version 2.1. The reduction followed the prescriptions described in ‘The Data Reduction Guide’ provided by the guest observer facility at the NASA/GSFC[^1]. The screening was based on the following criteria: (1) only ASCA-grade 0, 2, 3, 4, 6 events are accumulated, while hot and flickering pixels were removed from the XIS image using the <span style="font-variant:small-caps;">cleansis</span> script, (2) the time interval after the passage through the South Atlantic Anomaly (T\_SAA\_HXD) is greater than 500s, (3) the object is at least 5$^\circ$ and 20$^\circ$ above the rim of the Earth (ELV) during night and day, respectively. In addition, we also selected the data with a cut-off rigidity (COR) larger than 6GV. After this screening, the net exposure for good time intervals is 35.1ks. The XIS events were extracted from a circular region with a radius of $4.3'$ centred on the source peak, whereas the background was accumulated in an annulus with inner and outer radii of $5.0'$ and $7.0'$ pixels, respectively. The response (RMF) and auxiliary (ARF) files are produced using the analysis tools <span style="font-variant:small-caps;">xisrmfgen</span> and <span style="font-variant:small-caps;">xissimarfgen</span>, which are included in the software package HEAsoft version 6.4.1.
The HXD/PIN event data (version 2.1) are processed with basically the same screening criteria as those for the XIS, except that ELV$\ge$5$^\circ$ through night and day, and COR$\ge$8GV. The HXD/PIN instrumental background spectra were generated from a time dependent model provided by the HXD instrument team for each observation (see Kokubun et al. 2007). Both the source and background spectra were made with identical good time intervals (GTIs) and the exposure was corrected for a detector deadtime of 6.9$\%$. We used the response files version <span style="font-variant:small-caps;">ae\_hxd\_pinxinome\_20070914.rsp</span>, provided by the HXD instrumental team. Similarly, the HXD/GSO event data (version 2.1) were processed with a standard analysis technique described in the cited ‘The Data Reduction Guide’. Despite the relatively high instrumental background of the HXD/GSO, the source was marginally detected at 5.5 $\sigma$ level between 80 and 120 keV. We used the response files version <span style="font-variant:small-caps;">ae\_hxd\_gsoxinom\_20080129.rsp</span>. Spectral analysis was performed using the Xspec fitting package 12.3.1. and we fitted both the soft and hard X-ray spectra with a power law with Galactic absorption free to vary. The XIS spectra are well fitted with a power law with $\Gamma = 1.63$ absorbed with N$_{H}= 1.1 \times 10^{21}$ cm$^{-2}$, which infers the absorbed fluxes of $4.51^{+0.07}_{-0.03}\times 10^{-11}$ erg cm$^{-2}$ s$^{-1}$ and $3.20^{+0.04}_{-0.01}\times 10^{-11}$ erg cm$^{-2}$ s$^{-1}$ in the energy bands 0.3-10 keV and 2-10 keV, respectively. The hard X-ray spectrum determined by HXD/PIN and GSO seems to be a bit flatter than those determined by the XIS only below 10 keV, as it is shown in the residuals reported in Fig. 3 (where a model with a single power law is assumed). We found that it is better fitted by a power-law photon index $\Gamma$ = 1.35$\pm$0.14, which gives F$_{10-100
\rm keV}= 1.37^{+0.1}_{-0.08}\times 10^{-10}$ erg cm$^{-2}$ s$^{-1}$. The uncertainties reported above are at 90$\%$ confidence level.
observations
=============
During our campaign (Gehrels et al. 2004) performed two ToO observations of 3C 454.3: the first on 2007 December 13, the second on 2007 December 15. Both observations were performed using all on-board experiments: the X-ray Telescope (XRT; Burrows et al. 2005, 0.2-10 keV), the UV and Optical Telescope (UVOT; Roming et al. 2005, 170-600 nm) and the Burst Alert Telescope (BAT; Barthelmy et al. 2005, 15-150 keV). The hard X-ray flux of this source is below the sensitivity of the BAT instrument for short exposure and therefore the data from this instruments will not be used. We refer to Raiteri et al. (2008a) for a detailed description of data reduction and analysis of the UVOT data.
XRT observations were carried out using the instrument in Photon Counting (PC) readout mode (see Burrows et al. 2005 and Hill 2004, for details of the XRT observing modes). The XRT data were processed with the XRTDAS software package (v.2.2.2) developed at the ASI Science Data Center (ASDC) and distributed by HEASARC within the HEASoft package (v. 6.4). Event files were calibrated and cleaned with standard filtering criteria with the [*xrtpipeline*]{} task using the latest calibration files available in the Swift CALDB distributed by HEASARC. Both observations showed an average count rate $>$ 0.5 counts s$^{-1}$ and therefore pile-up correction was required. We extracted the source events from an annulous extraction region with inner, outer radii of 3, 30 pixels. To account for the background, we also extracted events within a circular region centred on a region free from background sources and with radius of 80 pixels. The ancillary response files were generated with the task xrtmkarf. We used the latest spectral redistribution matrices (RMF, v011) in the Calibration Database maintained by HEASARC. The adopted energy range for spectral fitting is 0.3-10 keV, and all data are rebinned with a minimum of 20 counts per energy bin to use the $\chi^2$ statistics. /XRT uncertainties are given at 90$\%$ confidence level for one interesting parameter, unless otherwise stated.
Spectral analysis was performed using the Xspec fitting package 12.3.1 and we fitted the spectra with a power law model with galactic absorption left free to vary. In Table 1 we summarize the best fit parameters and the derived absorbed fluxes in the energy ranges 0.3-10 keV, 2-10 keV. We note that the best-fit N$_{H}$ values in Table 1 are in agreement with the value $1.34 \times 10^{21}$ cm$^{-2}$ derived by Villata et al. (2006) when analysing Chandra observations in May 2005, and adopted by Raiteri et al. (2007) and Raiteri et al. (2008b) when fitting the X-ray spectra acquired by XMM-[*Newton*]{} in 2006–2007. In Fig. 4 we show the data and the folded models for these observations.
Optical monitoring
==================
WEBT Observation
----------------
The Whole Earth Blazar Telescope (WEBT; http://www.oato.inaf.it/blazars/webt/) is an international collaboration including tens of optical, near-IR, and radio astronomers devoted to blazar studies. An extensive monitoring effort on 3C 454.3 was carried out by the WEBT from 2005 to 2008, to follow the large 2005 outburst and post-outburst phases (Villata et al. 2006, 2007; Raiteri et al. 2007), and the new flaring phase started in mid 2007 (Raiteri et al. 2008a, 2008b). A detailed presentation and discussion of the radio, mm, optical and [*SWIFT*]{}-UVOT data collected in December 2007 can be found in Raiteri et al. (2008a). Here we adopt their data analysis in the context of our multifrequency study.
REM Observation
---------------
The photometric optical observations were carried out with the Rapid Eye Mount (REM, Zerbi et al. 2004), a robotic telescope located at the ESO Cerro La Silla observatory (Chile). The REM telescope has a Ritchey-Chretien configuration with a 60 cm f/2.2 primary and an overall f/8 focal ratio in a fast moving alt-azimuth mount providing two stable Nasmyth focal stations. Two cameras are simultaneously used at the focus of the telescope, by means of a dichroic filter, REMIR for the NIR (Conconi et al. 2004) and ROSS for the optical (Tosti et al. 2004), in order to obtain nearly simultaneous data.
The telescope REM has continuously observed 3C 454.3 between 2007 December 1 and 2007 December 8, overlapping with the observation period. The light curve produced by REM in the $R$-band is shown in Fig. 1 (bottom panel, red points).
MitSuME Observation
-------------------
A contribution of the optical follow-up observations was given also by MITSuME (Multicolor Imaging Telescopes for Survey and Monstrous Explosions), composed of 3 robotic telescopes (of 50 cm diameter each) located at the ICCR (Institute of Cosmic-Ray Research) Akeno Observatory, Yamanashi, Japan and the OAO (Okayama Astrophysical Observatory). Each MITSuME telescope has a Tricolor Camera, which allows to take simultaneous images in $g'$, $R_c$ and $I_c$ bands. The camera employs three Alta U-6 cameras (Apogee Instruments Inc.) and KAF-1001E CCD (Kodak) with 1024$\times$1024 pixels. The pixel size is 24$\mu$m$\times$24$\mu$m, or 1.6”$\times$1.6” at the focal plane. It is designed to have a wide field view of 28’$\times$28’. The primary motivation of MITSuME project is a multi-band photometry of gamma-ray bursts and their afterglows at very early phases, but the telescopes are also actively used for multi-color optical monitoring of more than 30 blazars and other interesting Galactic or extragalactic sources. These telescopes are automatically operated and respond to GRB alerts and transient events like AGN flares.
MITSuME observed 3C 454.3 almost every day from Nov 22 to Dec 6, 2007, so as to provide simultanous data with and $Suzaku$. All raw $g'$,$R_c$ and $I_c$ frames were corrected for dark, bias and flat field by using IRAF ver 2.12 software. Instrumental magnitudes were obtained via aperture photometery using <span style="font-variant:small-caps;">DAOPHOT</span> (Stetson 1987) and <span style="font-variant:small-caps;">SExtractor</span> (Bertin & Arnouts 1996). Calibration of the optical source magnitude was conducted by differential photometry with respect to the comparison stars sequence reported by Raiteri et al. (1998) and Gonzalez-Perez et al. (2001). The fluxes are corrected for the Galactic extinction corresponding to a reddening of $E(B-V)$ = 0.108 mag (Schlegel et al. 1998). The $R_{c}$-optical light curve between November 30 and December 6 is showed in Fig. 1 (bottom panel, green circles).
Mid-Infrared observations
=========================
Given the high $\gamma$-ray activity detected by from 3C 454.3, we also requested and obtained a Director’s Discretionary Time for a mid-Infrared follow-up by Spitzer (Werner et al. 2004). The DDT was approved for 2 epochs for a total duration of 0.8 hours of the Infrared Spectrograph (IRS, Houck et al. 2004) providing short-low and long-low observations of 3C 454.3 scheduled for December 13 (starting at MJD 54447.410) and 15 (starting at MJD 54449.403). Both observations provided us with a low resolution spectrum ($\Delta\lambda/\lambda \sim 80$) in the energy range $\sim 5-38$ $ \mu m$. Data were acquired in the IRS standard staring mode: observations were obtained at two positions along the slit to enable sky subtraction. Each ramp duration was set to 14.68 s with a number of cycles equal to 5. Each set of data was processed with the IRS Standard Pipeline [*SMART*]{} developed at the Spitzer Science Center to produce calibrated data frames (Basic Calibrated Data, BCD files). Moreover, the BCD files covering the same spectral range were coadded and then sky-subtracted spectra were obtained. The absolute flux calibration was estimated by using the electron-to-Jy conversion polynomial given in the appropriate [*Spitzer*]{} calibration file. In Fig. 5 we present the two spectra obtained on December 13 and 15. We performed a linear fit of the two, obtaining a flux equal to $(1.59\pm 0.02)\times 10^{-10}$ $(1.38\pm 0.02)\times 10^{-10}$ for December 13 and 15, respectively.
Discussion
==========
Timing analysis
---------------
We investigated the emission of the blazar 3C 454.3 during a multifrequency campaign performed in the first half of December 2007. The source was found to be in flaring state with an average $\gamma$-ray flux above 100 MeV of $\sim 250\times 10^{-8}$ , which is typical of its high gamma-ray state (Vercellone et al. 2008, Anderhub et al. 2008, Vercellone et al. 2009). As in the case of the previous multifrequency campaign (November 2007, Vercellone et al. 2009), the source was continuously monitored in $\gamma$-rays as well as in the optical energy bands. In both energy bands the source exhibited comparable flux variations of the order of $\sim 4$: this argues for an EC model. Moreover, we deeply studied the optical-$\gamma-ray$ correlation by means of a Discrete Correlation Function (DCF; Edelson & Krolik 1988; Hufnagel & Bregman 1992) applied to the optical and gamma-ray light curves reported in Fig. 1. This analysis revealed $\lesssim$ 1 day delay of the gamma-ray emission with respect to the optical one (see Fig. 6). Indeed, the DCF maximum at a time lag $\tau=-1$ day corresponds to a centroid $\tau=-0.56$ day, whose uncertainty can be estimated by means of the Monte Carlo method known as “flux redistribution/random subset selection” (Peterson et al. 1998, Raiteri et al. 2003). By running 1000 simulations we found $\tau= -0.6^{+0.7}_{-0.5}$ day at 1 sigma confidence level. We also performed the DCF reducing the data binning down to 12 hr between December 5 and 16, keeping the 1-day binned light curve for the data before December 5 (MJD=54439.524). This shows a peak at -1 day with centroid at -0.54 which is in agreement with the result obtained with the 1-day binned light curve. In this case, the Monte Carlo method is not able to provide a reliable estimate of the error on the time lag due to the larger uncertainties on the fluxes.
The evidence of this time lag again suggests the dominance of the EC model: such a delay is compatible with the typical blob dimensions and the corresponding crossing time of the external seed photons (Sokolov et al. 2004). We note that this evidence agrees with what was found by Bonnig et al. (2009).
Particularly interesting is the source optical flare recorded by WEBT on December 12 (Raiteri et al. 2008a). The source experienced an exceptional variability in less than 3 hr. Raiteri et al. (2008a) interpreted this event as a variation in the properties of the jet emission. This unusual event clearly required an intra-day analysis of the $\gamma$-ray data. This analysis depends on the source brightness and the instrumental sensitivity. Given the $\gamma$-ray flux level of 3C 454.3 reached between 2007-12-05 and 2007-12-16, we obtained a data binning not smaller than 12 hr (Fig. 7). This analysis showed an enhancement of more than a factor of 2 of the $\gamma$-ray flux during the second half of 2007 December 12, that remarkably includes the time of the optical event (see vertical lines in Fig. 7). The enhancement by a factor of $\sim 2$ of the $\gamma$-ray flux was comparable with the 1.1 mag optical brightening. This could support the evidence of a change in the jet emission in the EC scenario. The 12-hour light curve could constrain a possible delay between the emission and the optical one within 12 hours, shorter than ever observed before for this source.
Spectral Modelling
------------------
As described in the previous section, the December 2007 multifrequency campaign was characterized by ToO carried out in mid-Infrared ([*Spitzer*]{}), soft X-ray (*Suzaku*, *Swift*), hard X-ray (*Suzaku*) and radio-to-optical and monitoring. These observations allowed us to obtain the SED of this blazar with a wide multi-frequency coverage for three different epochs: December 5, 13, 15. At these dates the SED in X-rays shows a *softening* towards lower frequencies that can be due to two causes: 1) a contribution from bulk comptonization by cold electrons in the jet (Celotti, Ghisellini & Fabian 2007), 2) the emergence of the SSC contribution in soft X-rays from the more energetic EC component due to the disk and the BLR. The mid-Infrared *Spitzer* data and optical data available in December 13 and 15 (which well define the synchrotron peak), combined with the resolved X-ray spectrum and the gamma-ray data constrain the model parameters, arguing for the latter cause; the SSC emergence is a natural and inevitable consequence of the simultaneous modelling of the broad-band SED. Nevertheless, some contribution from bulk comptonization cannot be ruled out.
We first considered the state of December 13 and 15 in which we have radio, mid-Infrared, optical, X and $\gamma$-ray simultaneous data. In these epochs was in a different state with respect to the one analyzed in November: optical and UV fluxes appeared lower by a factor 2-3, suggesting the synchrotron bump peaking at a frequency 5-10 times lower than the one observed in November, as confirmed also from the mid-Infrared data. On the other hand, the soft X-ray data were only a little bit lower than in November. Despite the softer synchrotron bump, data showed in the SED the *persistence* of a hard peak at $\simeq 1$ GeV, similar to the higher states observed by in July 2007 and November 2007 (Anderhub et al. 2009, Vercellone et al. 2009). In fact, the December $\gamma$-ray spectrum (characterized by a photon index of $\sim 1.78$) is consistent with those obtained during the 2 previous observations.
We attempted to fit the SEDs with a one-zone SSC model, adding the contribution of external seed photons coming from an accretion disk and a BLR (Raiteri et al. 2007). With this model, we succeeded to fit the synchrotron peak as well as the X-ray data assuming parameters similar to the November ones, but a *lower* $\gamma_b\simeq 350$ was required to account for the softness of the synchrotron bump: with this $\gamma_b$ the EC from a standard BLR peaks at $h\nu\simeq
h\nu_{soft}\Gamma\gamma_b^2\delta/(1+z)\sim10^{8}\,$eV. This is in contrast with the observed hardness of the spectrum up to 1 GeV ($h\nu_{soft}\simeq 10$eV is the typical energy of the external source as seen by the observer). We note that the EC by the disk can account for the rising hard X-ray portion of the SED, which did not show clear variability. Nevertheless, we note that both the disk and BLR components cannot account for the hardness of the spectrum. Thus, we consider a further external source of seed photons.
A possibile candidate for this source is the hot extended corona that must be consistently produced in steady accretion/ejection flows as shown by MHD numerical simulations (Tzeferacos et al. 2009). Hence, we considered a one-zone SSC model plus the contribution by external seed photons coming from the accretion disk, the BLR and the hot corona. We adopted a spherical blob with radius $R=2.2\times
10^{16}$cm and a broken power law for the electron energy density in the blob, $$n_{e}(\gamma)=\frac{K\gamma_{b}^{-1}}{(\gamma
/\gamma_{b})^{a_{l}}+(\gamma /\gamma_{b})^{a_{h}}}\,$$ where $\gamma$ is the electron Lorentz factor assumed to vary between $10<\gamma<10^{4}$, while $a_{l}=2.3$ and $a_{h}=4.2$ are the pre– and post–break electron distribution spectral indices, respectively. We assumed that the blob contained a random magnetic field $B=2$ Gauss and that it moved with bulk Lorentz Factor $\Gamma=18$ at an angle $\Theta_{0}=1^{\circ}$ ($\delta\simeq33$) with respect to the line of sight. The density parameter into the blob is $K=52$ cm$^{-3}$.
The bolometric luminosity of the accretion disk is $L_{\rm
d}=3\,10^{46}$ erg $\, $s$^{-1}$, and it is assumed to lie at 0.01pc from the blob; we assumed a BLR distant 1.5 pc, reprocessing a 10$\%$ of the irradiating continuum. We assumed for the disk a black-body spectrum peaking in UV (see Tavecchio and Ghisellini 2008, Raiteri et al. 2008b). Finally, we added the hot corona photons surrounding the jet as a black body spectrum of $T=10^6\, K$ and $L_h=10^{45}$ erg $\,
$s$^{-1}$, and distant 0.5 pc from the blob. The SEDs of both December 13 and 15 could be fitted with almost the same parameters (see red and blue solid lines in Fig. 8). The high energy portion of the electron density becomes softer in December 15 as the same electrons should be accelerated with less efficiency than in December 13.
Remarkably, the lower $\gamma_b$ required in the epochs considered here, makes the BLR a too soft contributor at GeV energies, while the contribution of the hot corona succeeded to account for the persistence of the hard spectra measured by .
On December 5, the low energy peak of the SED is less constrained with respect to the December 13 and 15 ones due to the lack of the mid-Infrared data. On the other hand the [*Suzaku*]{} X-ray data (green points in Fig. 8) better constrain the rise of the IC emission. We fitted this SED with almost the same model assumed for the other two epochs, but the higher optical flux and the lower flux detected with respect to December 13 required a higher magnetic field and a lower $\gamma_{b}$ (see Table 2).\
Given the different $\gamma$-ray state of the source analyzed in the November and December campaigns, we compared the particle injection luminosity, $L_{\rm inj}$ measured during the two multiwavelength campaigns. This is expressed by means of the following formula: $$L_{\rm inj} = \pi\,R^{2}\,\Gamma^{2}\,c\,\int[d\gamma\,\,m_{\rm e}\,c^{2}\gamma\,n(\gamma)].$$ We found the particle injection luminosity of December to be $6 \times 10^{43} \rm erg \rm
s^{-1}$, a factor of 5 lower than the November one. This difference is due to both the lower $\gamma_{b}$ and $\gamma_{min}$ values needed to reproduce the SED in the states of December.
Conclusions
===========
We reported in this paper the main results of a multifrequency campaign on the blazar 3C 454.3 performed in December 2007. The source was simultaneously observed in mid-Infrared, optical, X-ray and energy bands, which provided us with a wide dataset aimed to study the correlation between the emission properties at lower and higher frequencies. We summarize below the major results.
- The emission from 3C 454.3 shows variations on a daily time scale.
- The simultaneous monitoring of the source in the optical and energy bands allowed us to determine a possible $\lesssim$ 1 day delay of the emission with respect to the optical one.
- The extraordinary optical activity (lasting less than 3 hours), observed on December 12 has a counterpart in the data. A possible delay between the emission and the optical one is constrained within 12 hours.
- We found that a leptonic model with an External Compton on seed photons from disk and BLR does not succeed to account for both the “hardeness” of the spectrum and the “softness” of the Synchrotron emission, requiring an additional component. We argued that a possible candidate for it is the hot Corona ($T \sim 10^{6}$ K) surrounding the disk.
[ is a mission of the Italian Space Agency, with co-participation of INAF (Istituto Nazionale di Astrofisica) and INFN (Istituto Nazionale di Fisica Nucleare). This work was partially supported by ASI grants I/R/045/04, I/089/06/0, I/011/07/0 and by the Italian Ministry of University and Research (PRIN 2005025417). INAF personnel at ASDC are under ASI contract I/024/05/1. This work is partly based on data taken and assembled by the WEBT collaboration and stored in the WEBT archive at the Osservatorio Astronomico di Torino-INAF ([http://www.oato.inaf.it/blazars/webt/]{}). This work is based in part on observations made with the Spitzer Space Telescope, which is operated by the Jet Propulsion Laboratory, California Institute of Technology under a contract with NASA. The IRS was a collaborative venture between Cornell University and Ball Aerospace Corporation funded by NASA through the Jet Propulsion Laboratory and Ames Research Center. SMART was developed by the IRS Team at Cornell University and is available through the Spitzer Science Center at Caltech. ]{}
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![[*Spitzer*]{} spectra of 3C 454.3 for the observation carried out in 2007 December 13 (red points) and 15 (blue points).](f5.eps)
{width="10cm" height="7cm"}
------------- ----------------- --------------------------------- ---------------------------------- ----------------- --------- --
13-Dec-2007 $0.13 \pm 0.03$ $(4.38\pm 0.25)\times 10^{-11}$ $(3.04 \pm 0.24)\times 10^{-11}$ $1.74 \pm 0.10$ 1.28/54
15-Dec-2007 $0.14 \pm 0.03$ $(3.60\pm 0.22)\times 10^{-11}$ $(2.49 \pm 0.22)\times 10^{-11}$ $1.76 \pm 0.12$ 1.14/44
------------- ----------------- --------------------------------- ---------------------------------- ----------------- --------- --
\[tab.spectral\_results\]
------------- ---- ----- -------------------- ---- ----------------- ---- ----- -----
5-Dec-2007 18 2.5 $2.2\times10^{16}$ 50 $3\times10^2$ 30 2.3 4.2
13-Dec-2007 18 2 $2.2\times10^{16}$ 52 $3.5\times10^2$ 38 2.3 4.2
15-Dec-2007 18 2 $2.2\times10^{16}$ 52 $3.2\times10^2$ 35 2.3 4.2
------------- ---- ----- -------------------- ---- ----------------- ---- ----- -----
[^1]: http://suzaku.gsfc.nasa.gov/docs/suzaku/analysis/abc. See also seven steps to the data analysis at http://www.astro.isas.jaxa.jp/suzaku/analysis
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'The Kuramoto model is a system of nonlinear differential equations that models networks of coupled oscillators and is often used to study synchronization among the oscillators. In this paper we study steady state solutions of the Kuramoto model by assigning to each steady state a tuple of integers which records how the state twists around the cycles in the network. We then use this new classification of steady states to obtain a “Weyl” type of asymptotic estimate for the number of steady states as the number of oscillators becomes arbitrarily large while preserving the cycle structure. We further show how this asymptotic estimate can be maximized, and as a special case we obtain an asymptotic lower bound for the number of stable steady states of the model.'
author:
- Timothy Ferguson
bibliography:
- 'TopologicalStatesInTheKuramotoModel.bib'
title: Topological states in the Kuramoto Model
---
**Keywords.** Kuramoto model, fixed points, graph topology, Weyl asymptotic\
\
**AMS subject classifications.** 34C15, 34D06
Introduction
============
The Kuramoto model is used to describe the behavior of a network of coupled oscillators. If we let $\theta_i$ denote the position of the $i$th oscillator, $\omega_i$ its natural frequency, and $\gamma_{ij}$ the strengh of the coupling between the $i$th and $j$th oscillators, then the Kuramoto model on a network of $N$ coupled oscillators is given by the system of $N$ nonlinear differential equations $$\begin{aligned}
\label{model}
\frac{d\theta_i}{dt} = \omega_i - \sum_{j=1}^N \gamma_{ij} \sin(\theta_i - \theta_j) \quad \text{for} \quad i \in \{1,\dots,N\}.\end{aligned}$$ This model was first proposed by Kuramoto in [@MR762432] to study the phenomenon of synchronization. There are many interesting systems exhibiting synchronization including, for example, the flashing of interacting fireflies [@Buck1988SynchronousRhythmicFlashing], the phase synchronization of arrays of lasers [@PhysRevA.52.4089], and the synchronization of pacemaker cells in the heart [@peskin75]. For a review of the history of the Kuramoto model and a more complete list of synchronization phenomena see [@MR1783382].
Unless stated otherwise we will work with the simplified model $$\begin{aligned}
\label{simplified model}
\frac{d\theta_i}{dt} = - \sum_{j=1}^N \gamma_{ij} \sin(\theta_i - \theta_j) \quad \text{for} \quad i \in \{1,\dots,N\}\end{aligned}$$ obtained from $\eqref{model}$ by setting $\omega_i = 0$. Since $\eqref{simplified model}$ is a gradient system on the compact $n$-torus, we see that every solution converges to a steady state solution and for this reason steady state solutions are of key interest. (Note that although $\eqref{model}$ is also a gradient system the phase space is not compact.) A particular class of steady state solutions which has received a lot attention [@MR2220552; @MR3388491; @MR3129708; @MR3415044; @MR3600363] are the so called twisted states. These are steady states for which the positions of successive oscillators wrap around the unit circle. For example, if we consider a cyclic network with unit edge weights, then one can check that $\theta_i = \frac{2\pi qi}{N} \pmod{1}$ is a steady state solution of $\eqref{simplified model}$ for any integer $q$. This solution wraps around the unit circle $q$ times and is referred to as a $q$-twisted state. These steady states were first introduced by Wiley, Strogatz, and Girvan in [@MR2220552]. Since then their existence and stability on various networks and under various perturbations has received a lot of attention. After introducing $q$-twisted states in [@MR2220552] they studied their stability properties in the continuum Kuramoto model for $k$-nearest neighbor graphs. They found a sufficient condition for linear stability. In [@MR3388491], Girnyk, Hasler, and Maistrenko again studied $q$-twisted states for $k$-nearest neighbor graphs. They showed that if a $q$-twisted state is stable for the continuum model then it is for the finite model for sufficiently large $N$. Furthermore they numerically investigated a new class of steady state solutions which they called multi-twisted states. These are states for which angle differences between neighbors are not restricted to only $2\pi q/N$. On two different sectors of the network, the angle differences between neighbors are allowed to be close to $2\pi q/N$ and $-2\pi q/N$ respectively with intermediate values in between. They further numerically showed that the number of such states appears to grow exponentially in $N$. In [@MR3129708], Medvedev showed the existence of $q$-twisted states in small world graphs. He further showed that random long-range connections tend to increase the chance of synchronization while simultaneously decreasing the chance of stability. He did this by showing that as he increased a parameter, which statistically would result in more long range connections, that the chance of synchronization went up while the chance of any given $q$-twisted state being stable went down. In [@MR3415044], Medvedev and Tang investigated the existence and stability properties of twisted states for Cayley graphs. In particular they showed that there exist Cayley graphs with similar properties, such as the distribution of edges, which exhibit different stability for $q$-twisted states. Finally, in [@MR3600363], Medvedev and Wright showed that linear stability in the continuum model implies stability under perturbations by weakly differentiable and bounded variation periodic functions.
In this paper we investigate topological states for an arbitrary network. As a special case this will allow us to study the multiplicity of stable topological states. In section $\ref{main}$ we make definitions and state our main results without proof. We leave the proofs for an appendix at the end. For our first result we demonstrate a one-to-one correspondence between steady states of $\eqref{model}$ and the lattice points in a certain set, $W(A)$, depending on the underlying network. Here both $W(A)$ and the lattice have dimension $c = |E| - |V| + 1$ which arises from the cycle structure of the graph. Furthermore each component of a lattice point in the set records how the corresponding steady state twists around a particular cycle in the graph. This is the motivation for referring to the steady states of $\eqref{model}$ as topological states. It is not surprising that the number of steady states should depend on the cycle structure. For example, consider the continuum Kuramoto model on a figure eight quantum graph. In this case a steady state corresponds a local minimum of a certain functional. Since a solution is a continuous function on each loop, each loop must have at least one local minimum. However these loops are not homotopic so generically we expect that their local minimums are distinct. Thus we expect to have a local minimum for each loop which in our example is two.
In our second result we look at the how the set $A$ depends on the network. In our third result we show that if we allow the number of vertices in our network to increase while preserving the network structure, that the number of steady states has a “Weyl” type asymptotic estimate $|W(A)| N^c$. This result is surprising since this would suggest that the number of steady states increases as we add more cycles, equivalently edges, to the network. However Taylor showed in [@MR2878025] that the complete graph has only the trivial stable steady state $\theta_i = 0$. Therefore we see that either the asymptotics of the number of stable steady states with respect to the number of vertices and edges does not commute or that $|W(A)|$ must decrease rapidly as $c$ increases.
In our fourth and last main result we show how to maximize $|W(A)|$. We finish by applying our results to networks with one and two cycles.
Definitions and main results {#main}
============================
In this section we make definitions that will be used throughout the rest of the paper. Let $G = (V,E,\Gamma)$ be a simple weighted graph. We will assume for simplicity that for every edge $e$ there exists a spanning tree $T$ which does not contain $e$. In other words we don’t have a graph with an attached tree. We will let $\operatorname{spt}(G)$ denote the set of spanning trees of $G$. Given such a graph we can order and assign an orientation to the edges and therefore define the corresponding incidence matrix $B : \R^E \rightarrow \R^V$ by $$\begin{aligned}
B_{ie} =
\begin{cases}
1 & \mbox{if the vertex $i$ is the head of the edge $e$,}
\\
-1 & \mbox{if the vertex $i$ is the tail of the edge $e$,}
\\
0 & \mbox{otherwise.}
\end{cases}\end{aligned}$$ The integral vectors in the kernel of $B$ form an additive group called the cycle space since each such vector corresponds to a cycle in the graph $G$. Let $v_1,\dots,v_c$ denote a basis for the cycle space. It is well known that $c = |E| - |V| + 1$. It is not hard to see that any two such basis can be obtained from the other by an element of $SL_c(\Z)$, the special linear group of degree $c$ over $\Z$.
If $e$ is an edge with head $i$ and tail $j$ we define the angle difference $\theta_e = \theta_i - \theta_j$. When we consider fixed points of $\eqref{model}$ and $\eqref{simplified model}$ we will restrict these angle differences so that $\theta_e \in I_e + 2\pi \Z$ for some choice of sets $I_e$. Throughout we will assume that each set is such that the restriction of the sine function to $I_e$, $\sin_{I_e}$, is injective. Given a collection of such sets $\mathcal{I} = \{ I_e \}_{e \in E}$ we define the function $\sin_\I$ to be the $|E|$ vector valued sine function restricted to the set $I_{e_1} \times \dots \times I_{e_{|E|}}$. Finally if $\gamma_e$ is the weight of the edge $e$ we let $D_\gamma$ be the diagonal matrix with diagonal entries $\gamma_{e_1}, \dots, \gamma_{e_{|E|}}$.
We can now define two functions on $\R^c$ by $$\begin{aligned}
L(\alpha) = D_\gamma^{-1} B^{-1} {\bf \omega} + \sum_{i=1}^c \alpha_i D_\gamma^{-1} v_i \quad \text{and} \quad W(\alpha) = \frac{1}{2\pi} (\langle v_1, \sin_\mathcal{I}^{-1} L(\alpha) \rangle, \dots, \langle v_c, \sin_\mathcal{I}^{-1} L(\alpha) \rangle)\end{aligned}$$ where $B^{-1}$ is the pseudoinverse of $B$. We further let $A$ denote the subset of $\R^c$ on which $\sin_\mathcal{I}^{-1} L(\alpha)$ is well-defined. In fact $A$ is a polytope, and we will study it further in Theorem $\ref{polytope}$. One will notice that both $L$ and hence $W$ are injective on $A$. We can now state our first main result.
\[equivalence\] There is a one-to-one correspondence between the fixed points of $\eqref{model}$ with $\theta_e \in I_e + 2\pi \Z$ and the lattice points in $W(A)$.
Let $G$ be the expanded diamond graph with unit edge weights shown in Figure 1 below. Also let $I_e = [-\pi/2,\pi/2]$ and $$\begin{gathered}
v_1 = (0, -1, 1, 0, 0, 0, 0, 0, 0, 0, 1, -1, 1, 1, 1, 1, 1, 1, 1, 1), \\
v_2 = (-1, 0, 1, -1, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 0, 0, 0, 0, 1, 1), \\
K = (1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0).\end{gathered}$$ We will see the meaning of $K$ momentarily. From Figure 1 we see that $W(A)$ contains the lattice point $(2,-1)$ and therefore by Theorem $\ref{equivalence}$ there is a corresponding fixed point of $\eqref{model}$. For simplicity we will work with $\eqref{simplified model}$. We will compute this fixed point which will also allow us to sketch the proof of Theorem $\ref{equivalence}$ which will be given in complete detail in the appendix.
To begin we rewrite the fixed point equation of $\eqref{simplified model}$ as $B \sin B^\top \theta = 0$. Since $v_1$ and $v_2$ span the kernel of $B$ we see that $\sin B^\top \theta = L(\alpha,\beta)$ for some $(\alpha,\beta) \in A$ and therefore that $B^\top \theta = \sin_\I^{-1} L(\alpha,\beta) - 2\pi K$ for some $K \in \Z^{20}$. If we let $B^{-\top}$ denote the pseudo inverse of $B^\top$ then we obtain the formula $\theta = B^{-\top}(\sin_\I^{-1} L(\alpha,\beta) - 2\pi K)$. Thus it remains to compute $(\alpha,\beta)$ and $K$. Since $v_1$ and $v_2$ are orthogonal to the image of $B^\top$ we see that $W(\alpha,\beta) = (\langle v_1,K \rangle,\langle v_2,K \rangle) = (1,0)$. This works with our choice for $K$, and we numerically solve to find that $(\alpha,\beta) = (0.994148 \dots, -0.779356 \dots)$. This finally results in $$\begin{gathered}
\theta = (0.724472, 1.61811, 2.51175, 3.40538, 4.29902, 5.19266, 6.0863, 0.696749, 1.59039, 3.05294, \\ 4.5155, 5.97806, 1.15743, 2.61999, 4.08254, 5.5451, 0.94095, 1.15743, 1.37391)\end{gathered}$$ which is displayed in Figure 1 below.
[.5]{} {width=".65\linewidth"}
[.5]{} {width="1\linewidth"}
Furthermore since we chose $I_e = [-\pi/2,\pi/2]$ and positive edge weights we know that our fixed point must be stable which we illustrate in Figure 2 below. Of course this holds in general.
{width=".5\linewidth"}
It is worth noting that since $W$ is always injective on $A$ we can in fact associate a unique $\alpha \in A$ to any steady state $\theta$ of $\eqref{model}$ via the equation $L(\alpha) = \sin(B^\top \theta)$. We will use this association in Theorem 5 in Section 3.
One will notice that the fixed points of $\eqref{model}$ with $\theta_e \in I_e +2\pi \Z$ are completely independent of our choice of cycle basis whereas $W(A)$ explicitly depends on it. We will briefly explain why the number of lattice points in $W(A)$ is independent of the choice of cycle basis directly from the definition of $W$ and $A$. To do this let $G$ be a graph with two cycle basis $v_1,\dots,v_c$ and $v_1',\dots,v_c'$. Let $A$ and $A'$, $L$ and $L'$, and $W$ and $W'$ be the corresponding polytopes and functions respectively. As mentioned earlier these two cycle basis are related by an element of $SL_c(\Z)$. In particular there exists an $M \in SL_c(\Z)$ such that $(v_1',\dots,v_c') = (v_1,\dots,v_c)M$. From this we see that $L'(\alpha) = (v_1',\dots,v_c') \alpha = (v_1,\dots,v_c)M\alpha = L(M\alpha)$ and therefore that $L'(M^{-1} A) = L(A) \subseteq \operatorname{Im}(\sin_\I)$ which implies that $M^{-1}A \subseteq A'$. Reasoning again in this way we are ultimately lead to the identities $A = MA'$ and $A' = M^{-1}A$. Just like $L$ and $L'$ are related by $M$ we see that $W'(\alpha) = L'(\alpha)(v_1',\dots,v_c') = L(M\alpha)(v_1,\dots,v_c)M = W(M\alpha)M$. From this we are finally lead to the identity $W'(A') = W(A)M$. Since $M$ is an element of $SL_c(\Z)$ we see that it is a bijection between the lattice points in $W(A)$ and $W'(A')$. Thus again we see that our result is independent of the choice of cycle basis.
Along the same lines we see that the polytope $A$ is not well-defined until we specify a cycle basis. However, as we have already seen, any two polytopes are related by an element of the special linear group and therefore there ought to be properties of $A$ which are independent of the particular choice of cycle basis. We give an example of such a property in Theorem $\ref{polytope}$, but first we need to discuss the smoothing of a graph.
If a graph has a 2-valent vertex it is possible to obtain a new graph by removing this vertex and replacing the two edges with a single edge. When this is done we say that the vertex has been smoothed. We can repeat this process and in so doing we obtain a new graph which potentially contains self loops and edges with multiplicity greater than two. Our only restriction is that if our graph is only a single self loop we do not remove that 2-valent vertex.
\[polytope\] Suppose that $\sin(I_e) = [-1,1]$. Then the number of faces of the polytope $A$ is equal to twice the number of edges of the graph obtained from $G$ by smoothing all 2-valent vertices.
Notice that if we smooth out all 2-valent vertices that we don’t change the number of distinct components of $L(\alpha)$. Thus the theorem is simply saying that each of the hyperplanes $L(\alpha)_e = \pm 1$ results in a face of $A$.
Consider the left most graph shown in Figure 1 below. After smoothing the 2-valent vertices 3, 4, 5, and 6 we obtain the right most graph. Thus according Theorem $\ref{polytope}$, $A$ has eight faces which we can easily confirm see from Figure 3.
Furthermore, with a suitable choice of orientation and numbering of the edges we can choose the cycle basis $v_1 = (1,1,1,0,0,0,0,0)$, $v_2 = (0,0,1,1,1,0,0,0)$, and $v_3 = (0,0,0,0,0,1,1,1)$ for the left most graph. Thus $L(\alpha,\beta,\gamma) = (\alpha,\alpha,\alpha+\beta,\beta,\beta,\gamma,\gamma,\gamma)$ and we see that we get the eight hyperplanes $\alpha = \pm1$, $\beta = \pm 1$, $\gamma = \pm1$, and $\alpha + \beta = \pm 1$.
Now we consider an application of Theorem $\ref{equivalence}$. Let $G$ be a graph and let $G_M$ be a sequence of graphs such that each edge $e$ in $G$ is replaced by a path of edges of length $r_e M + o(M)$ for some $r_e > 0$. For each new edge $e'$ set $I_{e'} = I_e$ where $e$ is the edge that was replaced by the path of edges containing $e'$. Essentially we are doing the exact opposite of smoothing 2-valent vertices which is referred to as subdivision. An example of a few graphs in such a sequence is shown in Figure 4 below.
\[asymptotics\] Let $G_M$ be a sequence of graphs as described above and let $D_r$ be the diagonal matrix with entries $r_e$. Then $$\begin{aligned}
\lim_{M \rightarrow \infty} \frac{\#(W_M(A_M) \cap \Z^c)}{M^c} = |W_r(A_0)|\end{aligned}$$ where $$\begin{aligned}
W_r(\alpha) = \frac{1}{2\pi} (\langle v_1, D_r \sin_\mathcal{I}^{-1} L_0(\alpha) \rangle, \dots, \langle v_c, D_r \sin_\mathcal{I}^{-1} L_0(\alpha) \rangle).\end{aligned}$$ In particular, if $r_e = 1$, then the limit equals $|W(A)|$.
Essentially since $W_M(A_M)$ expands proportionally to $M$ we rescale and instead count the number of elements of $(\frac{1}{M} \Z)^c$ in $W_M(A_M)/M$. Since the sequence of sets $W_M(A_M)/M$ approaches a “limiting set” we can estimate this number by $M^c$ times a Riemann sum of the indicator function for this “limiting set”. In the case special case that $r_e = 1$ the “limiting set” is in fact $W(A)$.
Theorem $\ref{asymptotics}$ indicates that the main contribution to the number of fixed points of $\eqref{model}$ when $r_e = 1$ comes from $|W(A)|$. Therefore if we seek a large number of fixed points we ought to try to maximize $|W(A)|$. The only way we can do this without changing the graph is by changing the intervals $I_e$. This motivates the following theorem.
\[angle differences\] The maximum value of $|W(A)|$ is achieved if we choose $$\begin{aligned}
I_e =
\begin{cases}
[-\pi/2,\pi/2] & \mbox{if $\gamma_e > 0$,}
\\
[\pi/2,3\pi/2] & \mbox{if $\gamma_e < 0$,}
\end{cases}
\quad \text{or} \quad
I_e =
\begin{cases}
[\pi/2,3\pi/2] & \mbox{if $\gamma_e > 0$,}
\\
[-\pi/2,\pi/2] & \mbox{if $\gamma_e < 0$.}
\end{cases}\end{aligned}$$ In particular, if all of the edge weights are positive, then the maximum is achieved if all of the angle differences are less than or equal to $\pi/2$ or all greater than or equal to $\pi/2$.
This theorem follows from the identity $|W(A)| = \int_A |\det(W')|$ and the fact that the given interval choices prevent cancellation among the terms in the determinant. In applications such as power systems the special case $I_e = [-\pi/2,\pi/2]$ is particularly useful. Furthermore if all of the edge weights are positive and all of the angle differences are less than $\pi/2$, the steady state is automatically stable. Thus our estimates can be used to obtain a lower bound for the number of stable steady states. More generally, if the intervals are chosen as in the left most case of Theorem $\ref{angle differences}$, then the steady state is automatically stable.
Graphs with one cycle
=====================
In this section we study single cycle graphs with the additional assumption that $\sin(I_e) = (-1,1)$. If $G$ has one cycle, then it is a ring. (Recall that we are assuming that $G$ doesn’t have any edges which are contained in every spanning tree of $G$.) Thus we can orient all edges in one direction so that a cycle basis is given by the single vector $v = (1,\dots,1)$. Furthermore we note that $\alpha$ is a scalar and no longer a vector. Thus $L(\alpha) = (\alpha/\gamma_{e_1},\dots,\alpha/\gamma_{e_{|E|}})$ and $W(\alpha) = \sum_{e \in E} \sin_{I_e}^{-1} (\alpha/\gamma_e)$. In addition $A = (-\min_{e \in E} |\gamma_e|, \min_{e \in E} |\gamma_e|)$.
\[extreme winding\] Define $I_+ = (0,\pi/2) \cup (\pi,3\pi/2)$ and $I_- = (\pi/2,\pi) \cup (3\pi/2,2\pi)$. Given a fixed point $\theta$ of $\eqref{simplified model}$ let $\alpha$ be such that $L(\alpha) = \sin(B^\top \theta)$. If one of the inequalities $$\begin{aligned}
W(\alpha) > \max_{e \in E} \biggr\{ \sin_{I_-}^{-1} \left( \frac{\alpha}{\gamma_e} \right) + \sum_{e' \ne e} \sin_{I_+}^{-1} \left( \frac{\alpha}{\gamma_{e'}} \right) \biggr\}, \quad \alpha \ge 0,
\\
W(\alpha) < \min_{e \in E} \biggr\{ \sin_{I_+}^{-1} \left( \frac{\alpha}{\gamma_e} \right) + \sum_{e' \ne e} \sin_{I_-}^{-1} \left( \frac{\alpha}{\gamma_{e'}} \right) \biggr\}, \quad \alpha \le 0,\end{aligned}$$ holds, then $\theta$ is unstable.
By Theorem 2.9 of [@MR3513871] $\theta$ is unstable if there are at least two edges with $\gamma_e \cos \theta_e < 0$. As we shall see in the proof of Theorem $\ref{equivalence}$ we have that $\theta_e$ is in fact equal to $\sin_{I_e}^{-1} (\alpha/\gamma_e)$ modulo an integer multiple of $2\pi$. If $\alpha \ge 0$ one can easily check that $\sin_{I_+}^{-1}(\alpha/\gamma_e) < \sin_{I_-}^{-1}(\alpha/\gamma_e)$, $\gamma_e \cos(\sin_{I_+}^{-1}(\alpha/\gamma_e)) > 0$, and $\gamma_e \cos(\sin_{I_-}^{-1}(\alpha/\gamma_e)) < 0$. Thus the first inequality can only hold if $\gamma_e \cos \theta_e < 0$ for at least two edges which results in instability. The second inequality follows by the same reasoning since if $\alpha \le 0$ we have that $\sin_{I_+}^{-1}(\alpha/\gamma_e) < \sin_{I_-}^{-1}(\alpha/\gamma_e)$, $\gamma_e \cos(\sin_{I_+}^{-1}(\alpha/\gamma_e)) < 0$, and $\gamma_e \cos(\sin_{I_-}^{-1}(\alpha/\gamma_e)) > 0$.
Consider the cycle graph with unit edge weights and the $q$-twisted state $\theta_i = 2\pi iq/N$. If $q \in (N/4,N/2)$, then $\theta_e = 2\pi q/N \in (\pi/2,\pi) \subseteq I_-$. As we will see in the proof of Theorem $\ref{equivalence}$ we have that $B^\top \theta = \sin_\I^{-1} L(\alpha) - 2\pi K$ for some $K \in \Z^E$ hence $\alpha = \sin(2\pi q/N) \ge 0$. Therefore since $W(\alpha) = N \sin_{I_-}^{-1} (\alpha) > \sin_{I_-}^{-1} (\alpha) + (N -1) \sin_{I_+}^{-1} (\alpha)$ we see that this twisted state is unstable by Theorem $\ref{extreme winding}$. Similarly, if $q \in (N/2,3N/4)$, then $\theta_e \in (\pi,3\pi/2) \subseteq I_+$ and $\alpha = \sin(2\pi q/N) \le 0$. We again conclude that the twisted state is unstable since $W(\alpha) = N \sin_{I_+}^{-1} (\alpha) < \sin_{I_+}^{-1} (\alpha) + (N-1) \sin_{I_-}^{-1} (\alpha)$. It is worth noting that $\cos (2\pi q/N) < 0$ in both cases. One can in fact show directly that such a twisted state is unstable if and only if $\cos (2\pi q/N) < 0$.
Graphs with two cycles
======================
In this section we will study two cycle graphs with the additional assumption that $\sin(I_e) = [-1,1]$. Due to the translation invariance of fixed points of $\eqref{simplified model}$ we observe that if we glue two graphs together at a single vertex that the number of fixed points of this graph is simply the product of the number of fixed points of the two subgraphs. Therefore we assume our graph is composed of two cycles which intersect nontrivially. Let $v_1$ and $v_2$ be a cycle basis. We can assume that all of the components of $v_1$ are zero or one and that the components of $v_2$ are zero or one outside of the intersection of the two cycles and negative one on this intersection. With this choice of cycle basis our assumption about our graph implies that $L(\alpha,\beta)_e \in \{\alpha,\beta,\alpha-\beta \}$ with $\alpha -\beta$ being realized for at least one edge. Thus we see that $A = \{ (\alpha,\beta) : |\alpha| \le 1, |\beta | \le 1, |\alpha - \beta | \le 1 \}$ which is shown in Figure 5 above.
{width="150px" height="150px"}
It turns out that the set $\{\alpha,\beta,\alpha-\beta \}$ has a lot of symmetry which allows us to prove the following theorem.
\[cycle intersection matrix\] Suppose that all of the edge weights are equal to a common value $\gamma$ and that $I_e = [-\pi/2,\pi/2]$ for all edges or $I_e = [\pi/2,3\pi/2]$ for all edges. Then $$\begin{aligned}
|W(A)| = \frac{N |\operatorname{spt}(G)|}{|\gamma|^2} \left(\frac{\pi^2}{2} + 2 \int_0^1 \frac{\sin^{-1} (1-\beta)}{\sqrt{1-\beta^2}} d\beta \right).\end{aligned}$$
Since $W$ is injective on $A$ we have that $|W(A)| = \int_A |\det(W'(\alpha))| d\alpha$. By Theorem 2.8 of [@MR3513871] we have the identity $$\begin{aligned}
|\det(W'(\alpha))| = \frac{N}{|\gamma|^c} \sum_{T \in \operatorname{spt}(G)} \prod_{e \notin E_T} \frac{1}{\sqrt{1-L(\alpha)_e^2}}\end{aligned}$$ for any number of cycles. In the notation of [@MR3513871], $W'(\alpha)$ is a cycle intersection matrix for the graph with edge weights $\gamma_e \sqrt{1-L(\alpha)_e^2}$. We will now argue that in the case of two cycles the integrals $$\begin{aligned}
I_T = \int_A \prod_{e \notin E_T} \frac{1}{\sqrt{1-L(\alpha)_e^2}} d\alpha\end{aligned}$$ are independent of the spanning tree. To see this notice that on the complement of any spanning tree $L(\alpha,\beta)$ takes exactly two distinct values from the set $\{\alpha,\beta,\alpha-\beta \}$. However, there exists a linear transformation $P(\alpha,\beta) = (\beta,-\alpha+\beta)$ which along with its square maps any two such values to any other two with at most a sign change. Moreover the determinant of $P$ has magnitude one. Thus we can use $P$ and its square to switch between spanning trees without changing the value of the integral. Thus to compute this integral we can choose a spanning tree whose complement contains an edge from $v_1$ not contained in $v_2$ and an edge from $v_2$ not contained in $v_1$. This results in the integral $$\begin{aligned}
I = \int_A \frac{d\alpha d\beta}{\sqrt{(1-\alpha^2)(1-\beta^2)}} = \frac{\pi^2}{2} + 2 \int_0^1 \frac{\sin^{-1} (1-\beta)}{\sqrt{1-\beta^2}} d\beta = 0.64368 \dots\end{aligned}$$
Unfortunately the independence of the integral on the spanning tree does not seem to hold in general for graphs with more than two cycles. For example consider the graph in Figure 6 below along with two of its spanning trees.
{width="400px" height="150px"}
Then $I_{T_1} \approx 36 \ne 44 \approx I_{T_2}$. These numerical values were obtained with Mathematica by means of a Monte Carlo method with precision goal two. Of course we do have the inequalities $$\begin{aligned}
\frac{N |\operatorname{spt}(G)|}{|\gamma|^c} \min_{T \in \operatorname{spt}(G)} I_T \le |W(A)| \le \frac{N |\operatorname{spt}(G)|}{|\gamma|^c} \max_{T \in \operatorname{spt}(G)} I_T\end{aligned}$$ which hold in general.
Acknowledgements
================
The author gratefully acknowledges support under NSF grant DMS1615418.
Appendix
========
In this section we provide proofs of our theorems. We start with a proof of Theorem $\ref{equivalence}$ which requires the following lemma.
\[lattice\] Let $v_1,\dots,v_c$ be a cycle basis. Then $$\begin{aligned}
\operatorname{span}_\mathbb{Z}
\begin{pmatrix}
v_1
\\
\vdots
\\
v_c
\end{pmatrix}
= \mathbb{Z}^c\end{aligned}$$ where the span consists of all integral linear combinations of the columns of the matrix.
We start by making two simple observations. First, notice that the lemma is independent of the particular choice of cycle basis since any two cycle basis are related by an element of $SL_c(\mathbb{Z})$. Second, the lemma is trivial for graphs with a single cycle. From these two observations we see that the lemma will follow by induction once we show that a graph with a cycle basis $v_1, \dots, v_c$ satisfying the lemma implies that a graph with one more edge and one new cycle basis vector transversing this new edge exactly once satisfies the lemma.
In other words the new cycle basis $v_1',\dots, v_{c+1}'$ satisfies $v_i' = (v_i,0)$ for $i \in \{1,\dots,c\}$ and $v_{c+1}' = (v_{c+1}, \pm1)$ for some integral $v_{c+1}$. Thus $$\begin{aligned}
\operatorname{span}_\mathbb{Z}
\begin{pmatrix}
v_1'
\\
\vdots
\\
v_{c+1}'
\end{pmatrix}
=
\operatorname{span}_\mathbb{Z}
\begin{pmatrix}
v_1 & 0
\\
\vdots
\\
v_c & 0
\\
v_{c+1} & \pm 1
\end{pmatrix}
= \mathbb{Z}^{c+1}\end{aligned}$$ by the inductive hypothesis which completes the proof.
We start by noting that $\eqref{model}$ is equivalent to the equation $\omega = BD \sin B^\top \theta$. Recalling that a cycle basis is also a basis for $\ker(B)$ we easily find that $\theta$ is a solution of $\eqref{model}$ if and only if $\sin B^\top \theta = L(\alpha)$ for some $\alpha \in \R^c$. Since $\theta_e \in I_e + 2\pi \mathbb{Z}$ we see that in fact $\alpha \in A$ and also that $B^\top \theta = \sin_\mathcal{I}^{-1} L(\alpha) - 2\pi K$ for some $K \in \mathbb{Z}^E$. Note that $\theta$ is uniquely determined up to translation by $\alpha$. Of course this equation holds if and only if $\frac{1}{2\pi} \langle v_i, \sin_\mathcal{I}^{-1} L(\alpha) \rangle = \langle v_i, K \rangle$. In other words $\theta$ is a solution of $\eqref{model}$ if and only if there exists an $\alpha \in A$ such that $$\begin{aligned}
W(\alpha) \in \operatorname{span}_\mathbb{Z}
\begin{pmatrix}
v_1
\\
\vdots
\\
v_c
\end{pmatrix}
= \mathbb{Z}^c\end{aligned}$$ where the equality holds by Lemma $\ref{lattice}$.
Now we prove Theorem $\ref{polytope}$.
We do this by induction. If $G$ has only one cycle the result is obvious. Thus let $G$ be a graph with at least two cycles. If we choose the cycle basis for $G$ to be a fundamental cycle basis, then there exists two edges $e'$ and $e''$ which are only transversed by exactly one member of the cycle basis. This allows us to define two subgraphs $G'$ and $G''$ which are obtained from $G$ by deleting the edges $e'$ and $e''$ respectively. We will show by applying the inductive hypothesis to $G'$ that the intersection of $A$ with the hyperplane $L(\alpha)_e = h_e \in \{-1,1\}$ has non-empty interior for all edges $e \ne e'$. By applying the same reasoning to the subgraph $G''$ we will complete the inductive step. To do this let $A$ and $A'$ be the corresponding polytopes and $L'(\alpha)$ and $L(\alpha,\beta)$ be the corresponding linear functions defined on these polytopes. Furthermore let $v$ denote the cycle basis vector which transverses $e'$ only once and completes a cycle basis for $G$ when added to the cycle basis for $G'$. Note that we can assume that $v_e \in \{-1,0,1\}$ and therefore it suffices to consider the two cases $v_e = 0$ and $v_e \in \{-1,1\}$.
We first consider the case when $v_e = 0$ which implies that $L(\alpha,\beta)_e = L'(\alpha)_e$. Let $\epsilon > 0$ be a small parameter and define the set $C_\epsilon= \{ \alpha \in A' : L'(\alpha)_e = h_e, \min_{e^* \ne e} |L'(\alpha)_{e^*} \pm 1| \ge \epsilon \}$. By the inductive hypothesis we know that this set has non-empty interior for all sufficiently small $\epsilon$. Now the set $C_\epsilon \times [-\epsilon,\epsilon]$ lies in the intersection of $A$ and the hyperplane $L(\alpha,\beta)_e = h_e$ and has non-empty interior.
Next we consider the case when $v_e \in \{-1,1\}$ which implies that $L(\alpha,\beta)_e = L'(\alpha)_e + v_e \beta$. Define $\beta(\alpha) = v_e(h_e - L'(\alpha)_e)$, then the graph $(\alpha,\beta(\alpha))$ lies on the hyperplane $L(\alpha,\beta)_e = h_e$. Define the set $C = \{ \alpha \in A' : |L'(\alpha)_e - h_e| \le \min_{e^* \neq e} |L'(\alpha)_{e^*} \pm 1| \}$. Note that this set has non-empty interior by the inductive hypothesis. If $v_{e^*} = 0$, then $L(\alpha,\beta(\alpha))_{e^*} \in [-1,1]$ trivially. Thus suppose that $v_{e^*} \in \{-1,1\}$. Then on $C$ we have that $$\begin{aligned}
1-h_eL'(\alpha)_e \le 1-v_e h_e v_{e^*} L'(\alpha)_{e^*}\end{aligned}$$ which implies that $$\begin{aligned}
L(\alpha,\beta(\alpha))_{e^*} = v_{e^*}(v_e h_e -v_e L'(\alpha)_e + v_{e^*} L'(\alpha)_{e^*}) \in [-1,1]\end{aligned}$$ by considering the two cases $v_e h_e \in \{-1,1\}$. Therefore the graph $(\alpha,\beta(\alpha))$ on $C$ lies in the intersection of $A$ and the hyperplane $L(\alpha,\beta)_e = h_e$ and has non-empty interior.
We now prove Theorem $\ref{asymptotics}$.
First notice that $A_M = A$. By construction we have that $W_M(\alpha) = MW_r(\alpha) + E_M(\alpha)$ where $\sup_{\alpha \in A} |E_M(\alpha)| = o(M)$. Therefore $$\begin{aligned}
\#(W_M(A) \cap \Z^c) = \#(\{ W_r(\alpha) + E_M(\alpha)/M : \alpha \in A\} \cap (\frac{1}{M}\Z)^c )\end{aligned}$$ If we let $R_M$ denote a Riemann sum for the indicator function of the set $\{ W_r(\alpha) + E_M(\alpha)/M : \alpha \in A\}$ with mesh $(\frac{1}{M}\Z)^c$, then the above quantity equals $R_M M^c$. Thus it suffices to show that $R_M$ approaches $|W_r(A)|$ as $M$ tends to infinity. Notice that the boundary $\partial W_r(A)$ is compact and that $$\begin{aligned}
|R_M - |W_r(A)|| \le |\{ \beta : \operatorname{dist}(\beta,\partial W_r(A)) \le \sup_{\alpha \in A} |E_M(\alpha)|/M + \sqrt{c}/M \}|.\end{aligned}$$ where $\operatorname{dist}(\beta,W_r(A)) = \inf_{\alpha \in A} |\beta - W_r(\alpha)|$. Thus we obtain our desired result since the quantity bounding the distance between $\beta$ and $\partial W_r(A)$ tends to zero.
Finally we prove our last theorem which is Theorem $\ref{angle differences}$.
As noted before we have that $|W(A)| = \int_A |\det(W'(\alpha)| d\alpha$ since $W$ is injective on $A$. If we define $\epsilon_e(\alpha) \in \{-1,1\}$ such that $$\begin{aligned}
\frac{d}{d\alpha} \sin_{I_e}^{-1} L(\alpha)_e = \frac{\epsilon_e(\alpha)}{\sqrt{1-L(\alpha)_e^2}} \quad \text{then} \quad (W')_{ij} = \sum_{e \in E} \frac{(v_i)_e (v_j)_e}{\epsilon_e(\alpha) \gamma_e \sqrt{1-L(\alpha)_e^2}}.\end{aligned}$$ Again $W'(\alpha)$ is a cycle intersection matrix in the notation of [@MR3513871] with edge weights $\epsilon_e(\alpha) \gamma_e \sqrt{1-L(\alpha)_e^2}$. Therefore by Theorem 2.8 of [@MR3513871] we find that $$\begin{aligned}
\det(W'(\alpha)) = N \sum_{T \in \operatorname{spt}(G)} \prod_{e \notin E_T} \frac{1}{\epsilon_e(\alpha) \gamma_e \sqrt{1-L(\alpha)_e^2}}.\end{aligned}$$ Thus the absolute value of the determinant is maximized when $\prod_{e \notin E_T} \epsilon_e(\alpha) \gamma_e$ has the same sign for all spanning trees $T$. It is not difficult to convince oneself that this implies that $\epsilon_e(\alpha) \gamma_e$ must have the same sign for any edge $e$ which is in the complement of some spanning tree $T$. By assumption this is all of $E$ and therefore for each $\alpha$ we see that $\epsilon_e(\alpha) \gamma_e$ must have a fixed sign for all edges $e$. Finally we easily see that this occurs when the intervals $I_e$ are chosen as given in the theorem.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We reexamine the dynamical generation of mass for fermions charged under various Lie groups with equal charge and mass at a high Grand Unification scale, extending the Renormalization Group Equations in the perturbative regime to two-loops and matching to the Dyson-Schwinger Equations in the strong coupling regime.'
address: 'Depto. Física Teórica, Universidad Complutense de Madrid, Plaza de las Ciencias 1, 28040 Madrid, Spain.'
author:
- 'Felipe J. Llanes-Estrada and Alexandre Salas-Bernárdez'
title: Chiral symmetry breaking for fermions charged under large Lie groups
---
Introduction
============
The Standard Model is a gauge field theory based on the gauged symmetry $$\label{eq:SM}
SU(3)_C\otimes SU(2)_L\otimes U(1)_Y\;.$$ Here $SU(3)_C$ denotes the color interaction responsible for the strong force, $SU(2)_L$ the isospin coupling of left-handed fermions and $U(1)_Y$ the hypercharge group. The spontaneous breaking of the electroweak symmetry by the Higgs mechanism suggested the possibility of higher symmetries at yet higher scales that would also be spontaneously broken, providing strong and electroweak force unification at higher scales; these symmetries would also have to be spontaneously broken [^1].
In the SM, the Higgs vacuum expectation value breaks the global symmetry $SU(2)_L\times U(1)_Y$ of the Higgs sector in the SM down to $U(1)_{em}$ [@SymSM] (or, considering the $U(1)$ as a perturbation, and the approximate global custodial $SU(2)$, it breaks $SU(2)\times SU(2)\to SU(2)_c$). This generates masses for the $W^\pm$ and $Z$ bosons, and for fermions, leaving us the symmetry $$SU(3)_C\otimes U(1)_{em}\;$$ (and the approximate custodial $SU(2)$). A feature of the symmetry group of the Standard Model that stands out is the small size of the numbers 1-2-3. Why are we confronted by such symmetry groups? Why not larger groups like $SU(6)$ or $Sp(10)$?
To address these question we study in Section \[FromGUT\] how hypothetical quarks colored under different groups acquire masses from a Grand Unified Theory (GUT) scale where all groups under consideration are chosen to have the same couplings and quark masses, down to lower energies where the interaction becomes strong. For this task we will use the Renormalization Group Equations (RGE).
Then, section \[sec:DSE\] treats the Dyson-Schwinger Equations (DSE) for the lowest scales when the interactions become strong. Any workable truncation of the DSE typically fails to satisfy local gauge invariance, while respecting global symmetry. This is however enough to discuss its breaking in view of Elitzur’s theorem. While realistic models [@GG] that embed the SM such as $SU(5)$ or $SO(10)$ are often discussed [^2], we are here less ambitious and keep the discussion at a general level, considering multiple groups.
In addition to a brief discussion in section \[sec:outlook\], the article has an appendix addressing the computation of color factors for almost all of the continuous Lie groups (results for E8 are not at hand). We have kept the article as short as is compatible with its being self-contained, since the theory behind our approach has already been laid out in a previous publication [@Llanes]. We have striven to extend that calculation as explained next.
From Grand Unification to strong interaction scale with the Renormalization Group Equations {#FromGUT}
===========================================================================================
Our motivation in this work is to extend the one-loop RGE computation of [@Llanes] to two loops. This was the aspect that introduced the most uncertainty to predict the mass of the fermions charged under large groups. In doing so we have unveiled partial errors in the original publication that we here correct. An erratum has also been issued to warn the reader of the earlier article.
We evolve the masses of one single color-charged fermion for the different color groups from the Grand Unification scale of $\mu_{GUT}=10^{15}\;GeV$ to the point where interactions become strong (at a scale $\sigma$) for each group, that is, when $C_F\,\alpha_s(\sigma)=0.4$. Once this happens we use Dyson-Schwinger equations for the non-perturbative regime in order to obtain the constituent masses for these fermions: this step is explained in the next section \[sec:DSE\].
An efficient way of keeping track of the parameter evolution needed for the physical predictions of a theory to be invariant under $\mu$ scale choice is the use of RGEs. We generalize those of Quantum Chromodynamics to an arbitrary gauge group $G$. The running of the coupling constant $g_s$ with $\mu$ is [@TMuta] determined by the $\beta(g_s)$ function, $$\beta(g_s)\equiv -\mu\frac{dg_s}{d\mu}=\beta_1g_s^3+\beta_2 g_s^5+...\;.$$ The one-loop correction $\beta_1$ is $$\beta_1=\frac{1}{(4\pi)^2}\left(\frac{11C_G-2T_RN_f}{3}\right)\;,$$ where $C_G$ is the adjoint Casimir [^3], $T_R$ the normalization of the generators $T^a$ of the group $G$ defined as $Tr(T^aT^b)\equiv T_R\delta^{ab}$ and $N_f$ the number of colored fermions [^4].
The two loop contribution to the $\beta(g_s)$ function, $\beta_2$, entails a larger effort in perturbation theory, but can also be easily found in the literature [@TMuta], $$\beta_2=\frac{1}{(4\pi)^4}\left(\frac{34}{3}C_G^2-4\left(\frac{5}{3}C_G+C_F\right)T_RN_f\right)\;,$$ where $C_F$ is the Casimir of the fundamental representation(see appendix). Using the color coefficients listed there, we obtain the running couplings of $SU(N)$, $SO(N)$, $Sp(N)$ and the exceptional groups $G2$, $F4$, $E6$ and $E7$, shown in Figure \[fig:runningcoupling\].
![Running couplings for the families $SU(N)$, $SO(N)$, $Sp(N)$ and most of the Exceptional Lie groups. \[fig:runningcoupling\]](FIGS.DIR/SU_N_CA.png "fig:"){width="1\columnwidth"} ![Running couplings for the families $SU(N)$, $SO(N)$, $Sp(N)$ and most of the Exceptional Lie groups. \[fig:runningcoupling\]](FIGS.DIR/SO_N_CA.png "fig:"){width="1\columnwidth"} ![Running couplings for the families $SU(N)$, $SO(N)$, $Sp(N)$ and most of the Exceptional Lie groups. \[fig:runningcoupling\]](FIGS.DIR/Sp_N_CA.png "fig:"){width="1\columnwidth"} ![Running couplings for the families $SU(N)$, $SO(N)$, $Sp(N)$ and most of the Exceptional Lie groups. \[fig:runningcoupling\]](FIGS.DIR/ConstanteAgruposespeciales.png "fig:"){width="1\columnwidth"}
The result of [@Llanes], that for small groups and one flavor $\sigma \propto e^{N}$ stands out. The very large groups have strongly interacting scales $\sigma$ clustering around the GUT scale, since they run very fast. We are then ready to start employing the DSEs down from the scale $\sigma$.
Simultaneously, running of the current mass $m_c$ is set by the self energy correction to the quark propagator that implies an anomalous mass dimension $\gamma_m$ $$\gamma_m(g_s)\equiv -\frac{\mu}{m}\frac{dm}{d\mu}=\gamma_1g_s^2+\gamma_2 g_s^4+...\;.$$ The one loop contribution to the $\gamma_2(g_s)$ function for the quarks, $\gamma_1$, amounts to $$\gamma_1=\frac{6C_F}{(4\pi)^2}\;.$$
The two loop contribution to $\gamma_m(g_s)$ (see [@Tarrach]), $\gamma_2$, is $$\gamma_2=\frac{C_F}{(4\pi)^4}\Big(3C_F+\frac{97}{3}C_G-\frac{20}{3}T_R N_f\Big)\label{eq:gamma2}\;.$$
At the GUT starting scale of the RGEs we choose a fermion mass $m_{c}(\mu_{GUT})=1\;MeV$ and fix the coupling $\alpha_{s}(\mu_{GUT})\equiv g_s({\mu_{GUT}})^2/4\pi=0.0165$ to broadly reproduce the isospin average mass for the $SU(3)_C$ quarks of the first generation at the scale $\mu=2\;GeV$, $$\overline{m}(2\,GeV)=\frac{m_u(2\;GeV)+m_d(2\;GeV)}{2}\simeq3.5\;\;MeV\;.$$ These initial conditions are taken to be the same for all Lie groups, as suggested by the concept of GUT. Then, the mass running for the various Lie groups, with color factors taken from \[sec:CF\] is plotted in Figure \[fig:runningmass\].\
![Running masses for the families $SU(N)$, $SO(N)$, $Sp(N)$ and four out of five Exceptional Lie groups.\[fig:runningmass\]](FIGS.DIR/SU_N_Masas.png "fig:"){width="1\columnwidth"} ![Running masses for the families $SU(N)$, $SO(N)$, $Sp(N)$ and four out of five Exceptional Lie groups.\[fig:runningmass\]](FIGS.DIR/SO_N_Masas.png "fig:"){width="1\columnwidth"} ![Running masses for the families $SU(N)$, $SO(N)$, $Sp(N)$ and four out of five Exceptional Lie groups.\[fig:runningmass\]](FIGS.DIR/Sp_N_Masas.png "fig:"){width="1\columnwidth"} ![Running masses for the families $SU(N)$, $SO(N)$, $Sp(N)$ and four out of five Exceptional Lie groups.\[fig:runningmass\]](FIGS.DIR/GruposEspecialesMasas.png "fig:"){width="1\columnwidth"}
$ $
Running at the strong interaction scale with the Dyson-Schwinger Equations {#sec:DSE}
==========================================================================
Once the interactions become strong, perturbation theory breaks down and resummation becomes necessary: we thus adopt the simplest possible DSE for the quark self energy. The free propagator of a fermion with current mass $m_c$ [@TMuta], $ S_0(p^2)=\frac{1}{m_c-\slashed p}\;,$ becomes a fully dressed one $ \tilde{S}(p^2)=\frac{1}{B(p^2)-A(p^2)\slashed p}\;$. Being only interested in qualitative features of spontaneous mass generation, we can approximate $A(p^2)=1$ which leaves the physical mass as $M(p^2)\equiv B(p^2)$. Denoting $\Sigma(p)$ as the sum of all one-particle irreducible diagrams, the DSE takes the form (omitting the $p$ dependence) $$\tilde{S}(p^2)=S_0(p^2)\,(1-\Sigma(p) S_0(p^2))^{-1}\;.$$ Inverting, we see that $ \tilde{S}^{-1}(p^2)=S_0(p^2)^{-1}-\Sigma(p) \Rightarrow M(p^2)=m_c-\Sigma(p)\;. $
To illustrate the possibilities, we will employ the *rainbow truncation* that sums only “rainbow shaped” diagrams, with great simplification (Fig. \[fig:rainbow\]).
![Rainbow DSE for the full quark propagator (filled circle).\[fig:rainbow\]](FIGS.DIR/DysonSchwinger1loop.png "fig:"){width="0.9\columnwidth"}\
![Rainbow DSE for the full quark propagator (filled circle).\[fig:rainbow\]](FIGS.DIR/Rainbowselfenergy.png "fig:"){width="0.7\columnwidth"}
\
The one-loop self energy is then, passing to Euclidean space with $k^0\to ik^0$, $p^0\to ip^0$, given by $$\begin{aligned}
\Sigma_{\rm rainbow}(p)=&g^2_s\int\frac{d^4k}{i(2\pi)^4}\gamma^\mu(T^a)\frac{1}{M(k^2)-\slashed k}\gamma^\nu(T^a) \frac{\eta_{\mu\nu}}{(k-p)^2}\nonumber\\
= C_F&g^2_s\int_0^\infty\frac{dk_E \,k_E^3}{\pi^3}\frac{-M(k^2)}{M^2(k^2)+k_E^2}\nonumber\times\\
\times&\int_{-1}^{+1}\sqrt{1-x^2}\frac{dx}{(k_E^2-2|k_E||p_E|\,x+p^2_E)}\;.\end{aligned}$$ We define the last integral in $x$ as the averaged gluon propagator $D^0_{k-p}$ (in the Feynman Gauge) over the four dimensional polar angle. Hence, we conclude that the Dyson-Schwinger equation in the rainbow approximation for the quark propagator is $$\label{eq:DSE}
M(p^2)=m_c+C_Fg^2_s\int_0^\infty\frac{dq \,q^3}{\pi^3}\frac{M(q^2)}{M^2(q^2)+q^2}D^0_{q-p}\;.$$
Note that the integral in (\[eq:DSE\]) is divergent and must be regularized. We could employ a simple cutoff regularization cutting this integral at a scale $\Lambda$; instead we would like to preserve Lorentz invariance and exhibit renormalizability. Following again [@Llanes], we introduce renormalization constants $Z(\Lambda^2,\mu^2)$ to absorb infinities and any dependence on the cutoff $\Lambda$, $$\tilde{S}^{-1}(p^2,\mu^2)\equiv Z_2 S_0^{-1}(p^2)-\Sigma(p^2,\mu^2)\;,$$ where the dependence of $\Sigma$ on $\mu$ is given by the fermion and gluon propagators. Apart from the wavefunction renormalization $Z_2$ we introduce $Z_m$ for the bare quark mass. The relation between the (cutoff dependent) unrenormalized mass $m_c(\Lambda^2)$ and the renormalized mass at the renormalization scale $\mu$, $m_R(\mu^2)$, is $$m_c(\Lambda^2)=Z_m(\Lambda^2,\mu^2)m_R(\mu^2).$$ Since we will maintain the restriction $A(p^2)=1$, renormalization of the quark wavefunction is not necessary, therefore $Z_2=1$. The only renormalization condition is to fix the mass function at $p^2=\mu^2$. The DSE is then $$M(p^2)=Z_m m_R(\mu^2)-\Sigma(p^2,\mu^2)\;.\label{EQ:RMM}$$ Evaluating (\[EQ:RMM\]) at $p^2=\mu^2$ and subtracting it again to (\[EQ:RMM\]) we obtain, $$\begin{aligned}
\label{eq:DSEMOM}
M(p^2)&=M(\mu^2)\nonumber\\
&+C_Fg^2_s\int_0^\infty\frac{dq \,q^3}{\pi^3}\frac{M(q^2)}{M^2(q^2)+q^2}\Big(D^0_{q-p}-D^0_{q-\mu}\Big)\;.\end{aligned}$$ It is easy to show, taking $\mu$ parallel to $p$, that asymptotically [@Llanes], $$\frac{\partial M(p^2)}{\partial \Lambda}\propto\frac{M(\Lambda^2)(p-\mu)}{\Lambda^2}\;.$$ Therefore, for large $\Lambda$, $M(p^2)$ stops depending on the cutoff, which can be taken [*e.g.*]{} to $\Lambda=10^{10}\; GeV$ and renormalization is achieved.
Now we are ready to obtain the quark constituent masses for all the groups studied. We match the RGE solution (high scales) to the DSE solution (low scales) at the matching energy $\sigma$ where interactions become strong, $C_F\alpha_s(\sigma)=0.4$ for each group, as advertised. For SU(3) ($C_F=\frac{4}{3}$), the scale where $\alpha_s(\sigma)=0.3$ is $\sigma=2.09\; GeV$. From this point down in scale we freeze $\alpha_s$. A constant vertex factor of order 7 is applied to the DSE to guarantee sufficient chiral symmetry breaking at low scales, requiring the constituent quark mass $M(0)$ to be close to $300\;MeV$ using the substracted DSE (\[eq:DSEMOM\]). This is supposed to mock up the effect of vertex-corrections not included, and is known to scale with $N$ [@Alkofer:2008tt] for large $N$, the group’s fundamental dimension. Finally, the $M(p)$ obtained is plotted in Figure \[fg:DSERGE\].
![Matching of RGE and DSE solutions of the Mass Running for $SU(3)$.[]{data-label="fg:DSERGE"}](FIGS.DIR/RGEDSEsolutionscopia.png){width="1\columnwidth"}
To obtain the constituent fermion masses for the different Lie Groups we use a trick presented in [@Llanes]: to perform a scale transformation $$p^2\to \lambda^2p^2 \ ; \ \sigma^2\to \lambda^2\sigma^2$$ on the DSE (\[eq:DSEMOM\]). Changing the integration variable $q^2\to \lambda^2q^2$, giving $d^4q\to \lambda^4d^4q$, the modified DSE equation is satisfied by a modified $\tilde{M}$ and the relation between the constituent masses is simply $ M(0)=\frac{\tilde{M}(0)}{\lambda}\;.$ Now, taking $\lambda$ as the ratio of the scales where interactions become strong for $SU(3)$ and another group, $$\frac{\sigma_{group}}{\sigma_{SU(3)}}=\lambda\;,$$ the mass function scales in the same way, $$\frac{M_{group}(0)}{M_{SU(3)}(0)}=\lambda\;.$$ Hence, eliminating the auxiliary $\lambda$, we find $$\frac{M_{group}(0)}{M_{SU(3)}(0)}=\frac{\sigma_{group}}{\sigma_{SU(3)}}\,.$$ Using these results we compute the constituent masses for the quarks charged under the different groups (Fig. \[constituent\]).
![Constituent Masses for the groups which break chiral symmetry in RGE before $10^{-5}\;GeV$.[]{data-label="constituent"}](FIGS.DIR/MasasConstituyentes1.png){width="1.1\columnwidth"}
The outcome is that the special Lie groups examined do not spontaneously generate fermion mass at a high scale: their interactions, running at two loops from the GUT, are too weak to do so. This is because the $C_G$ Casimir of the adjoint representation, though proportional to the group dimension, carries a small numerical factor that reduces the intensity of coupling running.
Large $SU(N)$ and $Sp(N)$ groups, on the other hand, behave as advanced in [@Llanes], and generate a mass for the fermions that puts them beyond reach of past accelerators. The exceptions are $Sp(4)$, for which the mass generation is similar to QCD; and $Sp(2)$ , which is too weak. As for the special orthogonal groups, for $SO(N>10)$, once more the fermion mass generated is too large to be accessible at accelerators.
Conclusions and Outlook {#sec:outlook}
=======================
We have examined mass generation for different Lie groups with an arbitrary number of colours. As a definite starting point, we have adopted the philosophy of Grand Unification in which fermion masses as well as coupling constants, for all groups, coincide at a high scale, namely $10^{15}$ GeV.
We have run the couplings and masses for each group to lower scales employing two-loop Renormalization Group Equations, using as an input the Cuadratic Casimirs obtained in \[sec:CF\]. We chose the initial conditions at $\mu=\mu_{\rm GUT}$ to be the same for all groups and selected so that $SU(3)_C$ at the scale of 2 GeV yields a rough approximation of the strong force coupling and first-generation isospin-averaged quark mass.
Typically, for all but the smallest groups, a scale arises where interactions become strong (discernable as a Landau pole in perturbation theory). We stop running at the scale $\mu$ such that $\alpha_s(\mu)C_F=0.4$; below that, we employ a non-perturbative treatment, namely Dyson-Schwinger Equations in the rainbow approximation to assess the masses down to yet lower scales.
Combining the methods of RGE and DSE and requiring that the constituent masses of $SU(3)_C$ colored quarks to be 300 MeV has allowed us to obtain the constituent masses of hypothetical fermions charged under different groups from a Grand Unification Scale of $10^{15}\;\text{GeV}$. From this treatment we can conclude that groups belonging to the $SU(N)$ and $Sp(N)$ families, with $N>4$, generate masses of order or above the few TeV. Notwithstanding the crude approximations we have employed, our computation gives about 5 TeV to $SU(4)$-charged fermions, which would not be far out of reach of mid-future experiments provided the GUT conditions apply. It appears from our simple work that larger groups (except the Exceptional Groups and $SO(N)$ with $N<10$) might endow fermions with a mass too high to make them detectable in the foreseeable future. In case these superheavy fermions would have been coupled to the Standard Model, they would have long decayed in the early universe due to the enormous phase space available. If they existed and be decoupled from the SM, they would appear to be some form of dark matter. We have also provided a partial answer to the question “*Why the symmetry group of the Standard Model, $S U (3)_C \otimes SU(2)_L \otimes U(1)_Y$ , contains only small-dimensional subgroups?*” It happens that, upon equal conditions at a large Grand Unification scale, large-dimensioned groups in the classical $SO(N)$, $SU(N)$ and $Sp(N)$ families force dynamical mass generation at higher scales because their coupling runs faster. Should fermions charged under these groups exist, they would appear in the spectrum at much higher energies than hitherto explored [@Odense].\
Interestingly for collider phenomenology, we find the masses of fermions charged under the following groups are within reach of the energy frontier: $M_{\rm SU(4)}\simeq 5$ TeV; $M_{Sp(6)}\simeq4.4$ TeV; $M_{\rm SO(10)}\simeq 7$ TeV. The LHC might be able to exclude those [^5].
However, the following groups $SO(N<10)$, $E_6$, $E_7$, $G2$ and $F4$ yield masses that are below the TeV scale and should already have been seen if they coupled to the rest of the Standard Model (one could argue that those isomorphic to groups present in the SM have already been sighted). Their absence from phenomenology thus suggests that fermions charged under any of those groups , if at all existing, belong to a decoupled dark sector.
Color Factors {#sec:CF}
=============
We here present some of the calculations carried out to obtain the quadratic Casimirs needed in both RGE and DSE. Such quadratic Casimirs are elements in the Lie Algebra which commute with all the other elements (See [@Cvit; @GTN; @FFS] for the necessary group theory).\
We will focus on the Casimir invariant in the fundamental representation of the group $G$, $C_F\delta^{ij}=(T^a T^a)^{ij}$, and the Casimir invariant in the adjoint representation, $C_G\delta^{ab}=f^{acd}f^{bcd}$. Normalization of the algebra generators is chosen as $Tr(T^aT^b)=\frac{1}{2}\delta^{ab}$.
Special Unitary Groups $SU(N)$
------------------------------
We start with the special unitary family $SU(N)$. Its generators $T^{a}$ are traceless hermitian. Therefore every Hermitean $N\times N$ matrix $A$ can be written as, $$A=A^\dagger=c_0\mathbb{I}+c_a T^a\,.$$ From this we find $$c_0=\frac{Tr(A)}{N}\;\;\;\;\;\;c_a=2\, Tr(AT^a)\,.$$ Having then $$\begin{aligned}
&A_{ij}=A_{lk}\delta^{li}\delta^{kj}=A_{lk}\Big(2(T^a)_{ij}(T^a)^{kl}+\frac{1}{N}\delta^{kl}\delta_{ij}\Big)\\
&\Rightarrow A_{lk}\Big(2(T^a)_{ij}(T^a)^{kl}+\frac{1}{N}\delta^{kl}\delta_{ij}-\delta^{li}\delta^{kj}\Big)=0\;.\end{aligned}$$ Since $A$ is arbitrary, we find for the generators the useful relation $$(T^a)_{ij}(T^a)_{kl}=\frac{1}{2}\Big(\delta_{li}\delta_{kj}-\frac{1}{N}\delta_{kl}\delta_{ij}\Big)\;.$$ Contracting $j$ and $k$ we obtain the fundamental representation Casimir or Color Factor $$(T^aT^a)_{ij}=\frac{1}{2}\Big(\frac{N^2-1}{N}\Big)\delta_{ij}=C_F\delta_{ij}\;.$$ Now we compute the following combination, $$\begin{aligned}
(T^a)_{i}^j(T^b)_{jk}(T^a)^k_{l}=\frac{1}{2}\Big((T^b)_{jk}\delta_{li}\delta^{kj}-\frac{1}{N}(T^b)_{jk}\delta^k_{l}\delta_{i}^j\Big)\nonumber\\
=-\frac{1}{2N}(T^b)_{il}\;.\end{aligned}$$ Noting the following identity and using the results already computed, we obtain the adjoint Casimir for $SU(N)$, $$\begin{aligned}
\label{eq:CG}
f^{acd}f^{bcd}&=-2\,Tr\Big([T^a,T^c][T^b,T^c]\Big)\nonumber\\
&=-2 Tr\Big(2T^aT^cT^bT^c-(T^aT^b+T^bT^a)T^cT^c\Big)\nonumber\\
&=N\,\delta^{ab}=C_G\delta^{ab}\,.\end{aligned}$$
Special Orthogonal Groups $SO(N)$
---------------------------------
We will follow now the same steps for the Special Orthogonal family $SO(N)$. Its generators are antisymmetric and traceless and they form a basis for the antisymmetric $N\times N$ matrices. Thus, taking an antisymmetric matrix $A$, we have $$A=-A^T=c_a T^a\;\;\;\; \Rightarrow\;\;\;\; c_a=2 Tr\Big(A T^a\Big)\;.$$ Then we have $$A_{ij}=A_{kl} \delta^k_i\delta^l_j=\frac{1}{2}A_{kl}\Big( \delta^k_i\delta^l_j-\delta^k_j\delta^l_i\Big)=A_{lk}\Big(2 (T^a)_{ij}(T^a)^{kl}\Big)\;.$$ Finding $$A_{kl}\Big[ \frac{1}{2}\Big( \delta^k_i\delta^l_j-\delta^k_j\delta^l_i\Big)+\Big(2 (T^a)_{ij}(T^a)^{kl}\Big)\Big]=0\;,$$ and since A is an arbitrary antisymmetric matrix we get $$(T^a)_{ij}(T^a)^{kl}=\frac{1}{4}\Big(\delta^k_j\delta^l_i- \delta^k_i\delta^l_j\Big)\;.$$ Here, since the group is real there is no need for distinction between upper and lower indices. Contracting in the previous expression $j$ with $k$ we obtain the Color Factor $$(T^a)_{ij}(T^a)^{jl}=\frac{1}{4}\Big(\delta^j_j\delta^l_i- \delta^j_i\delta^l_j\Big)=\frac{N-1}{4}\delta_i^l=C_F\delta_i^l\;.$$ As before, we compute $$\begin{aligned}
(T^a)_{ij}(T^b)^{jk}(T^a)_{kl}&=\frac{1}{4}\Big((T^b)^{jk}\delta_{il}\delta_{kj}-(T^b)^{jk}\delta_{ik}\delta_{lj}\Big)\nonumber\\
&=-\frac{1}{4}(T^b)_{li}=\frac{1}{4}(T^b)_{il}\;.\end{aligned}$$ We are able now to obtain the adjoint Casimir for $SO(N)$. Similar to (\[eq:CG\]) $$\begin{aligned}
f^{acd}f^{bcd}&=2\,Tr\Big(\frac{1}{2}T^aT^b-(T^aT^b+T^bT^a)\frac{N-1}{4}\Big)\nonumber\\
&=\frac{1}{2}(N-2)\delta^{ab}=C_G\delta^{ab}\,.\end{aligned}$$
Simplectic Groups $Sp(N)$
-------------------------
The elements $M\in Sp(N)$ (with $N$ even) are $N\times N$ matrices which preserve the antisymmetric tensor $$\Omega=
\begin{pmatrix}
0 & \mathbb{I}_{\frac{N}{2}\times\frac{N}{2}}\\
-\mathbb{I}_{\small{\frac{N}{2}\times\frac{N}{2}}} & 0
\end{pmatrix},$$ in the sense $$\Omega=M^T\Omega \,M\Rightarrow\;\;M^{-1}=\Omega^TM^T\Omega\,.$$ Using this relation it is possible to prove that the generators of the group take the form $$\label{eq:Spgen}
-T^a=\Omega^T(T^a)^T\Omega\;\;\Rightarrow\;\;T^a=
\begin{pmatrix}
A & B\\
C & -A^T
\end{pmatrix},$$ where $B$ and $C$ are symmetric matrices. It is now possible to show that the generators satisfy $$(T^a)_{ij}(T^a)_{kl}=\frac{1}{4}\Big(\delta_{il}\delta_{jk}+\Omega_{ik}(\Omega^{-1})_{jl}\Big)\;.$$ Therefore $$(T^a)_{ij}(T^a)^j_{l}=\frac{1}{4}(N+1)\delta_{ij}=C_F\delta_{ij}\;.$$ Noticing $\Omega^T=\Omega^{-1}=-\Omega$, we compute the usual combination $$\begin{aligned}
(T^a)_{ij}(T^b)^{jk}(T^a)_{kl}&=\frac{1}{4}\Big(\delta_{il}(T^b)^{jk}\delta_{jk}+\Omega_{ik}(T^b)^{jk}(\Omega^{T})_{jl}\Big)\nonumber\\
&=\frac{1}{4}(\Omega^{T})_{ik}((T^b)^T)^{kj}\Omega_{jl}=-\frac{1}{4}(T^b)_{il}\;,\end{aligned}$$ where in the last equality we have used (\[eq:Spgen\]). The adjoint Casimir now falls down easily $$\begin{aligned}
f^{acd}f^{bcd}&=-2\,Tr\Big(-\frac{1}{2}T^aT^b-(T^aT^b+T^bT^a)\frac{N+1}{4}\Big)\nonumber\\
&=\frac{1}{2}(N+2)\delta^{ab}=C_G\delta^{ab}\end{aligned}$$ To obtain the Color Factors and adjoint Casimirs for $G2$, $F4$, $E6$ and $E7$ we refer to the article of P. Cvitanović [@Cvit]. The results obtained are presented in Table I.\
Group Color Factor $(C_F)$ Adjoint Casimir $(C_G)$ $N_c$
--------- ---------------------------------------- ------------------------------ ---------------------------------
$SU(N)$ $\frac{1}{2}\Big(N-\frac{1}{N}\Big)$ $N$ $\forall N\in\mathbb{N}$
$SO(N)$ $\frac{1}{4}\Big(N-1\Big)$ $\frac{1}{2}\Big(N-2\Big)$ $\forall N\in\mathbb{N}$
$Sp(N)$ $\frac{1}{4}\Big(N+1\Big)$ $\frac{1}{2}\Big(N+2\Big)$ $N=2n$ $\forall n\in\mathbb{N}$
$G2$ $\frac{1}{4}\Big(N-3\Big)$ $\frac{1}{2}\Big(N-3\Big)$ $N=7$
$F4$ $\frac{1}{18}\Big(N-8\Big)$ $\frac{1}{18}\Big(N+1\Big)$ $N=26$
$E6$ $\frac{1}{12}\Big(N-\frac{29}{3}\Big)$ $\frac{1}{12}\Big(N-3\Big)$ $N=27$
$E7$ $\frac{1}{48}\Big(N+1\Big)$ $\frac{1}{48}\Big(N+16\Big)$ $N=56$
[21]{} S. Elitzur, Phys. Rev. D. [**12**]{} (1975) 3978-3982.
S. Willenbrock, Physics in D >= 4 proceedings, Theoretical Advanced Study Institute in elementary particle physics 2004, Boulder, USA, June 6-July 2, 2004, pages 3-38. Published by World Scientific, Hackensack, USA. hep-ph/0410370.
H. Georgi and S. L. Glashow, Phys. Rev. Lett. [**32**]{} (1974) 438. doi:10.1103/PhysRevLett.32.438 B. Fornal and B. Grinstein, arXiv:1808.00953 \[hep-ph\].
G. García Fernández, J. Guerrero Rojas and F. J. Llanes-Estrada, Nucl. Phys. B [**915**]{} (2017) 262 doi:10.1016/j.nuclphysb.2016.12.010 \[arXiv:1507.08143 \[hep-ph\]\].
T. Muta, *Foundations of Quantum Chromodynamics*, World Scientific Lecture notes in Physics-vol. 57, (1984).
R. Alkofer [*et al.*]{}, Annals Phys. [**324**]{} (2009) 106 doi:10.1016/j.aop.2008.07.001; A. Kizilersu [*et al.*]{}, Eur. Phys. J. C [**50**]{} (2007) 871 doi:10.1140/epjc/s10052-007-0250-6; A. C. Aguilar [*et al.*]{}, Phys. Rev. D [**90**]{} (2014) no.6, 065027 doi:10.1103/PhysRevD.90.065027.;
P. Cvitanovic, Phys. Rev. D [**14**]{} (1976) 1536. doi:10.1103/PhysRevD.14.1536 A. Zee, *Group Theory in a Nutshell for Physicists*, Princeton University Press (2016). F. Wilzeck, A. Zee, *Families from Spinors*, Physical Review D 25 2 (1982).
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This work was presented at the Odense $CP^3$ Origin of Mass at the High intensity Frontier conference in May 2018.\
http://cp3-origins.dk/content/uploads/2018/05/parallel program22052018.pdf
[^1]: It is usually and superficially stated that the gauge symmetry $SU(2)_L\times U(1)_Y$ is spontaneously broken. However, Elitzur’s theorem [@Elitz] states that gauge symmetries cannot be spontaneously broken. First they must be broken explicitly by a gauge fixing term leaving only the global symmetry and then this remaining symmetry can be spontaneously broken. The modern viewpoint is that gauge symmetries are just a redundancy in the description of the theory on which expectation values of observables must not depend. The actual symmetry from which consecuences such as degeneracies in the spectrum, couplings or conserved currents appear is the true global symmetry. We will continue using “spontaneous symmetry breaking” without specifying, though in the understanding that it is the global group which is affected.
[^2]: While the absence of proton decay rules out some classic implementations of the GUT idea, models keep being constructed that evade the constraints [@Fornal:2018aqc]
[^3]: For $SU(N)$, $C_G=N$, the group dimension. But in general, $C_G=aN+b$ with $a,b$ depending on the particular group, as listed in the appendix. This detail was in error in [@Llanes] and is being corrected.
[^4]: In this article we take $N_f=1$, but a brief discussion in [@Llanes] reminds us that there is a critical number of colors $N_f^c$ that shuts off the vacuum antiscreening and thwarts spontaneous symmetry breaking.
[^5]: For comparison, the one-loop results are $M_{\rm SU(4)}\simeq 2$ TeV; $M_{Sp(6)}\simeq1.5$ TeV; $M_{\rm SO(10)}\simeq 3$ TeV., which indicates fair convergence.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We propose a novel actor-critic, model-free reinforcement learning algorithm which employs a Bayesian method of parameter space exploration to solve environments. A Gaussian process is used to learn the expected return of a policy given the policy’s parameters. The system is trained by updating the parameters using gradient descent on a new surrogate loss function consisting of the Proximal Policy Optimization ‘Clipped’ loss function and a bonus term representing the expected improvement acquisition function given by the Gaussian process. This new method is shown to be comparable to and at times empirically outperform current algorithms on environments that simulate robotic locomotion using the MuJoCo physics engine.'
author:
- |
Ashish Rao [email protected]\
Cupertino High School\
Cupertino, CA 95014, USA Bidipta Sarkar [email protected]\
Cupertino High School\
Cupertino, CA 95014, USA Tejas Narayanan [email protected]\
Cupertino High School\
Cupertino, CA 95014, USA
bibliography:
- 'citations.bib'
title: Gaussian Process Policy Optimization
---
Deep Reinforcement Learning, Gaussian Processes, Parameter-space Exploration
Introduction
============
Reinforcement learning involves the study of how intelligent systems (or agents) can be made to learn behaviors (or policies) that yield high rewards in a given environment. Popular algorithms to tackle reinforcement learning problems include Policy Gradient methods like Proximal Policy Optimization (PPO) [@DBLP:journals/corr/SchulmanWDRK17] and Trust Region Policy Optimization (TRPO) [@DBLP:journals/corr/SchulmanLMJA15], as well as techniques such as Deep Q Learning [@mnih2013playing]. Often times, deep neural networks are employed to represent certain quantities specific to each type of algorithm: the policy for Policy Gradient Methods, the policy and value function for Actor-Critic Methods (specific type of Policy Gradient methods), and the Q values for Q learning.
Despite various successes of these methods on a wide variety of environments, they possess certain shortcomings as described in [@DBLP:journals/corr/SchulmanWDRK17]. Deep Q Learning is constrained to problems involving discrete and low dimensional action spaces, can fail on simple problems, and is poorly understood. TRPO and some other policy gradient methods are complicated to implement and incompatible with certain model architectures. Simpler policy gradient methods, such as the vanilla policy gradient, are not data efficient.
Besides the learning algorithm itself, it is vital for an agent to strike an effective balance between exploring its environment and gathering further information as to which actions in which states yield high rewards, and exploiting that knowledge to achieve those high rewards. Deep Reinforcement Learning methods often use entropy regularization to achieve this balance, but this technique can be ineffective in higher dimensional action spaces involving sparse rewards [@nachum2016improving]. In addition, recent work has shown that rather than noise injection in the action space, adding noise noise in the parameter space can yield richer behaviors [@plappert2017parameter].
We propose a new algorithm which we call GPPO, or Gaussian Process Policy Optimization. It utilizes a Gaussian Process, a Bayesian Model, to encourage exploration in the parameter space of parameters that maximize a given acquisition function, a weighted sum of the predicted return of a policy with those parameters and the variance of the Gaussian Process. Adding this acquisition function as a bonus to the PPO Clipped loss provides at times improved performance over current state-of-the-art methods.
Preliminaries
=============
Notation
--------
The traditional Reinforcement Learning problem is considered, where agents interact with their environment at each timestep with an action to receive a reward and transition to a new state. The environment is stochastic and modeled by a Markov Decision Process (MDP). A trajectory consists of a state, an action taken by the policy $\pi_\theta$ parameterized by $\theta$, the reward due to the action, and the next state.
$$\tau = (s_0, a_0, r_0, s_1, a_1, r_1, \ldots, s_{T-1}, a_{T-1}, r_{T-1}, s_T)$$
The policy $\pi_\theta$ maps a given state to a probability distribution over possible actions. $R(\tau)$ gives the total reward over a trajectory.
A common quantity considered in Reinforcement Learning algorithms is the Q value, or the expected return of taking an action $a_t$ at state $s_t$ while using policy $\pi_\theta$:
$$Q^{\pi_\theta}(s_t, a_t) = \mathbb{E}[R(t)|s_t, a_t]$$
The estimate for this quantity is given as $\hat{Q}^{\pi_\theta}(s_t, a_t)$. The value function gives the expected reward from a given state.
$$V^{\pi_\theta}(s_t) = \mathbb{E}[R(t)|s_t]$$
The advantage function is given by $$A^{\pi_\theta}(s_t, a_t) = Q^{\pi_\theta}(s_t, a_t) - V^{\pi_\theta}(s_t)$$
The estimated advantage function is given as $\hat{A}^{\pi_\theta}(s_t, a_t)$. The reinforcement learning problem seeks to maximize:
$${\eta}(\pi) = \mathbb{E}_{s_0, a_0, ...}\left[\sum_{t=0}^{\infty}{\gamma}^t r(s_t)\right]$$
Policy Gradient Methods
-----------------------
As proven in [@Sutton:1999:PGM:3009657.3009806], the Policy Gradient theorem allows for a convenient method through which we can compute gradients and optimize our policy.
$$\nabla_\theta \mathbb{E}_{\tau \sim \pi_\theta}[R(\tau)] = \mathbb{E}_{\tau \sim
\pi_\theta} \left[R(\tau) \cdot \nabla_\theta \left(\sum_{t=0}^{T-1}\log
\pi_\theta(a_t|s_t)\right)\right]$$
However, this gradient needs to be computed by sampling trajectories from the environment. These trajectories have extremely high variance as each transition between states is probabilistic (the environment is formulated as a Markov Decision Process). A *baseline function* is introduced into the estimate to reduce variance while still providing an unbiased signal.
$$\nabla_\theta \mathbb{E}_{\tau \sim \pi_\theta}[R(\tau)]
{\approx}\; \mathbb{E}_{\tau \sim \pi_\theta} \sum_{t=0}^{T-1} \nabla_\theta \log \pi_\theta(a_t|s_t) \cdot \hat{A}(s_t,a_t)$$
For use with automatic differentiation libraries, a surrogate loss is constructed whose gradient equals the above expression.
$$L^{PG}=\mathbb{E}_{\tau \sim \pi_\theta}[\log \pi_\theta(a_t|s_t) \cdot \hat{A}(s_t,a_t)]$$
Though the introduction of the baseline aids in lowering variance, a key assumption in current policy gradient methods is that the distribution of states visited does not change upon changing the policy. This causes the surrogate loss functions to only be local approximations to the expected return of a policy. Due to the surrogate loss being a local approximation, state-of-the-art methods constrain the magnitude by which the parameters of the policy network are perturbed while training.
### Trust Region Policy Optimization
TRPO [@DBLP:journals/corr/SchulmanLMJA15] involves a surrogate loss function with a constraint:
$$\max_\theta{\mathbb{E}_t[{\frac{\pi_\theta(a_t|s_t)}{\pi_{\theta_old}(a_t|s_t)}}]\hat{A}_t}$$ subject to $$\mathbb{E}_t[KL[\pi_{\theta_old}(\cdot|s_t), \pi_{\theta}(\cdot|s_t)]] \leq \delta$$
This surrogate loss avoids overly large parameter updates that can irreversibly damage the policy’s performance. However, it requires complex second order computations and is difficult to implement.
### Proximal Policy Optimization
There exist two versions of PPO [@DBLP:journals/corr/SchulmanWDRK17]: one involving a penalty based on the KL divergence between the old and new policy, and another that relies on clipping the surrogate loss function when the ratio between the old and current policy grows past a set hyper-parameter.
$$L^{CLIP} = \hat{\mathbb{E}}[min(r_t(\theta)\hat{A}_t, clip(r_t(\theta), 1-\epsilon, 1+\epsilon)\hat{A}_t]$$
where
$$r_t(\theta) = \frac{\pi_\theta(a_t|s_t)}{\pi_{\theta_{old}}(a_t|s_t)}$$
This surrogate loss avoids overly large parameter updates that can irreversibly damage the policy’s performance.
Gaussian Processes
------------------
As explained in [@rasmussen2004gaussian], a Gaussian process is an infinite-dimensional generalization of a Gaussian distribution. They can be used to model distributions over possible functions in a Bayesian fashion. Gaussian Processes are parameterized by a mean function $m(x)$ and a covariance (kernel) function $k(x,x')$. We can express a sample function from the GP as:
$$f(\cdot) \sim GP(m(\cdot), k(\cdot, \cdot)),$$ which means: the function f is distributed as a GP with mean function m and covariance function k.
A set of $n$ observations, $\mathbf{y}=\{y_1,...,y_n\}$, can be interpreted as a single sample from a multivariate Gaussian distribution that can be paired with a GP. This set of observations are derived from a function $f(\mathbf{x})$ with some noise variance for a set of inputs, $X$, such that: $$y_i = f(\mathbf{x_i}) + \epsilon,\quad \epsilon \sim \mathcal{N}(0,\sigma^2).$$
The kernel defines the relation between the inputs; if $x$ is distant from $x'$, $k(x,x') \approx 0$. The radial basis function, as expressed in [@ebden2015gaussian]: $$k(\mathbf{x}, \mathbf{x}') = \sigma_f^2 e^{-\frac{1}{2} (\mathbf{x}-\mathbf{x}')^T M (\mathbf{x}-\mathbf{x}')} + \sigma_n^2 \delta_{\mathbf{x},\mathbf{x}'},$$ is a popular choice for the kernel. Here, $\sigma_f$ is the maximum allowable covariance, $\sigma_n$ is the noise in the underlying function, and $\delta_{\mathbf{x},\mathbf{x}'}$ is the Kronecker delta function. If the structure of the underlying function is unknown, the mean of the partner GP is often assumed to be zero everywhere.
The GP can have predictive power as a prior for Bayesian inference. Let $\mathbf{y}$ be the function values of the training cases $X$, as shown above, and let $y_*$ be the function value of the test set input $\mathbf{x_*}$. The kernel function yields the following three matrices:
$$K =
\begin{bmatrix}
k(\mathbf{x_1},\mathbf{x_1}) & k(\mathbf{x_1},\mathbf{x_2}) & \cdots & k(\mathbf{x_1},\mathbf{x_n}) \\
k(\mathbf{x_2},\mathbf{x_1})& k(\mathbf{x_2},\mathbf{x_2}) & \cdots & k(\mathbf{x_2},\mathbf{x_n}) \\
\vdots & \vdots & \ddots & \vdots \\
k(\mathbf{x_n},\mathbf{x_1}) & k(\mathbf{x_n},\mathbf{x_2}) & \cdots & k(\mathbf{x_n},\mathbf{x_n})
\end{bmatrix}$$ $$K_*=
\begin{bmatrix}
k(\mathbf{x_*},\mathbf{x_1}) & k(\mathbf{x_*},\mathbf{x_2}) & \cdots & k(\mathbf{x_*},\mathbf{x_n})
\end{bmatrix}$$ $$K_{**}=k(\mathbf{x_*},\mathbf{x_*}).$$
Since the GP is a generalization of the multivariate Gaussian distribution,
$$\begin{bmatrix}
\mathbf{y} \\
y_*
\end{bmatrix}
\sim
\mathcal{N}
\begin{pmatrix}
\mathbf{0},
\begin{bmatrix}
K & K_*^T\\
K_* & K_{**}
\end{bmatrix}
\end{pmatrix}.$$
One of the properties of the multivariate Gaussian distribution is that conditional probabilities also follow a Gaussian distribution, so
$$y_* | y \sim \mathcal{N}(K_*K^{-1}\mathbf{y}, K_{**}-K_*K^{-1}K_*^T).$$
The Gaussian Process gives the following posterior prediction distribution:
$$\mu_{y_*} = K_*K^{-1}\mathbf{y}$$
$$\Sigma_{y_*} = K_{**}-K_*K^{-1}K_*^T.$$
Algorithm
=========
Details
-------
As described above, we use a Gaussian Process to directly model the expected return of a given policy. We use a Gaussian Process with a 0 mean function and the Radial Basis Function kernel. A surrogate loss function is constructed by adding a bonus $B$ to the PPO Clipped loss function, where $B$ is the expected improvement acquisition function:
$$L^{GPPO} = L^{CLIP} + B$$
The expected improvement acquisition function was used because of recent results showing the effectiveness of a measure of ’curiosity’ in improving performance such as in [@pathakICMl17curiosity] and [@DBLP:journals/corr/PlappertHDSCCAA17]. Experimentation showed that a lengthscale parameter of $5*10^{-4}$ and a noise parameter of $10^{-2}$ led to good policies being learned.
The PPO objective provides a lower bound, local approximation to the true return for a given policy. It also disincentivizes larger updates, as it is only a local approximation and optimizing the objective beyond a range where it is accurate hurts performance. By using the Gaussian Process to explicitly learn the expected advantage of any given set of parameters, we are essentially allowing for less conservative updates while optimizing a the policy. This process also reflects recent work done in learning loss functions, in that we use a Gaussian Process to explicitly learn the function describing the returns of parameters [@DBLP:journals/corr/abs-1802-04821].
For training the system, following [@DBLP:journals/corr/SchulmanWDRK17], stochastic gradient descent is conducted on a modified objective that incorporates the loss from the critic as well:
$$L_t^{CLIP+VF+GP}(\theta) = L_t^{CLIP}(\theta) + c_1 L_t^{VF}(\theta) + B_t(\theta)$$
It is important to note that no entropy is used to facilitate exploration; exploration is conducted solely through the expected acquisition bonus that has been added.
Gaussian Process Update Rule
----------------------------
As shown in [@Kakade02approximatelyoptimal] and described in @DBLP:journals/corr/SchulmanLMJA15, the expected discounted reward from policy $\pi_\theta$ can be written in terms of a different policy $\pi_{\theta_k}$ as follows:
$${\eta}(\pi_\theta) = {\eta}(\pi_{\theta_k}) + \sum_{s}{\rho_{\pi_\theta} \sum_{a}\pi_{\theta}(a|s) A_{\pi_{\theta_k}}(s, a)}$$
Where
$$\rho_{\pi_\theta} = \rho_{\pi_\theta}(s) = P(s_0 == s) + P(s_1 == s) + P(s_2 == s) + ...$$
Thus, it follows that
$${\eta}(\pi_\theta) = {\eta}(\pi_{\theta_k}) + \sum_{t=0}^{T}{\gamma^t(\sum_{s}{P_{\pi_\theta}(s_t == s) \sum_{a}\pi_{\theta}(a|s) A_{\pi_{\theta_k}}(s, a)})}$$
$$= {\eta}(\pi_{\theta_k}) + \sum_{t=0}^{T}{\gamma^t}\mathbb{E}_{s \sim P_{\pi_{\theta_k}}, a \sim {\pi_{\theta_k}}}\left[ \frac{P_{\pi_\theta}(s_t == s)}{P_{\pi_{\theta_k}}(s_t == s)} \frac{\pi_{\theta}(a|s)}{\pi_{\theta_k}(a|s)} A_{\pi_{\theta_k}}(s, a) \right]$$
Assuming that the probabilities of the actions we take are independent of the probability we are in that state:
$$= {\eta}(\pi_{\theta_k}) + \sum_{t=0}^{T}{\gamma^t}\mathbb{E}_{s \sim P_{\pi_{\theta_k}}, a \sim {\pi_{\theta_k}}}\left[ \frac{P_{\pi_\theta}(s_t == s)}{P_{\pi_{\theta_k}} (s_t == s)} \right] \mathbb{E}_{s \sim P_{\pi_{\theta_k}}, a \sim {\pi_{\theta_k}}}\left[ \frac{\pi_{\theta}(a|s)}{\pi_{\theta_k}(a|s)} A_{\pi_{\theta_k}}(s, a) \right]$$
Note that:
$$\mathbb{E}_{s \sim P_{\pi_{\theta_k}}, a \sim {\pi_{\theta_k}}}\left[ \frac{P_{\pi_\theta}(s_t == s)}{P_{\pi_{\theta_k}} (s_t == s)} \right] = \sum_{s}{P_{\pi_{\theta_k}} (s_t == s) \frac{P_{\pi_\theta}(s_t == s)}{P_{\pi_{\theta_k}} (s_t == s)}}$$
$$= \sum_{s}{P_{\pi_{\theta_k}} (s_t == s)} = 1$$
Thus,
$${\eta}(\pi_\theta) = \sum_{t=0}^{T}{\gamma^t} \mathbb{E}_{s \sim P_{\pi_{\theta_k}}, a \sim {\pi_{\theta_k}}}\left[\frac{\pi_{\theta}(a|s)}{\pi_{\theta_k}(a|s)} A_{\pi_{\theta_k}}(s, a) \right] + {\eta}(\pi_{\theta_k})$$
$${\eta}(\pi_\theta) = \frac{1-\gamma^{T+1}}{1-\gamma} \mathbb{E}_{s \sim P_{\pi_{\theta_k}}, a \sim {\pi_{\theta_k}}}\left[\frac{\pi_{\theta}(a|s)}{\pi_{\theta_k}(a|s)} A_{\pi_{\theta_k}}(s, a) \right] + {\eta}(\pi_{\theta_k})$$
This result inspires an update rule on how a Gaussian Process can be trained to predict the expected discounted reward of any given policy. The Gaussian Process shall be trained to map
$$\theta \rightarrow \frac{1-\gamma^{T+1}}{1-\gamma} \mathbb{E}_{s \sim P_{\pi_{\theta_k}}, a \sim {\pi_{\theta_k}}}\left[\frac{\pi_{\theta}(a|s)}{\pi_{\theta_k}(a|s)} A_{\pi_{\theta_k}}(s, a) \right] + {\mu}(\pi_{\theta_k})$$
Where ${\mu}(\pi_{\theta_k})$ represents the mean prediction of ${\eta}(\pi_{\theta_k})$ by the Gaussian Process. In the algorithm, $\pi_{\theta_k}$ represents the old policy.
Other Implementation Details
----------------------------
We built our implementation of the proposed GPPO algorithm on top of the OpenAI Baselines repository [@baselines]. We used the implementations of other algorithms in the same repository to compare GPPO’s performance against current methods. The policy is represented by a Multilayer Perceptron with two hidden layers of 64 units and $tanh$ nonlinearities.
Pseudocode
----------
An algorithm involving $N$ actors collecting data of $T$ timesteps, and then optimizing the above objective using the Adam optimization process is described below. The Gaussian Process needs to store each set of parameters and the mean advantage they earned, but since this can become too much and slow down the system, we limit the size of the Gaussian Process’ memory to $S$ points. The most recent $S$ points are used as these are most likely the points in the neighborhood of our current parameters. Our implementation used $S = 20$ and $T = 10^6$.
Experiments
===========
We compare our algorithm’s performance to current state-of-the-art methods in the literature on OpenAI environments [@1606.01540]. Specifically, we use 6 included MuJoCo environments, which feature continuous action spaces and simulate robotic locomotion: Ant, HalfCheetah, Hopper, InvertedDoublePendulum, InvertedPendulum, and Swimmer, all v2. For each environment, we run the system with 3 random seeds and present results averaged across those same seeds.
![Some of the environments used for evaluating algorithm’s performance[]{data-label="fig:roboticsEnvs"}](RoboticsEnvs.png){width="\linewidth"}
Results
-------
The above graphs present the reward of each episode over time averaged across three trials. The table below gives the mean episode reward across the algorithm’s training. A noise parameter of 1e-2 and a lengthscale of 5e-4 were used. As shown, GPPO gives comparable, and at times, superior results to current methods.
--------------------------- ---------- ---------- ----------
Environment GPPO PPO TRPO
Ant-v2 261.813 231.259 26.366
HalfCheetah-v2 1119.417 1076.696 958.878
Hopper-v2 1237.472 1308.940 1296.983
InvertedDoublePendulum-v2 1706.657 1695.973 2306.271
InvertedPendulum-v2 500.694 525.656 621.293
Swimmer-v2 49.340 62.348 86.588
--------------------------- ---------- ---------- ----------
: Mean return of all episodes averaged across three trials
Future Work
===========
Several improvements could be made over the GPPO algorithm presented. For one, as described in [@hensman2013gaussian], instead of our method of limiting the Gaussian Process ’memory’ to just the recent samples, the exact inducing points that are most relevant to the predictions can be used. Additionally, performance is dependent on good parameters being set for the lengthscale and noise parameter. Some method to dynamically learn these quantities while training in the environment would likely significantly boost performance. Further experimentation with different kernels may also yield a more precise kernel that reflects the structure of a deep neural network model.
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{
"pile_set_name": "ArXiv"
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---
abstract: 'We introduce a data-based approach to estimating key quantities which arise in the study of nonlinear control systems and random nonlinear dynamical systems. Our approach hinges on the observation that much of the existing linear theory may be readily extended to nonlinear systems – with a reasonable expectation of success – once the nonlinear system has been mapped into a high or infinite dimensional feature space. In particular, we develop computable, non-parametric estimators approximating controllability and observability energy functions for nonlinear systems, and study the ellipsoids they induce. In all cases the relevant quantities are estimated from simulated or observed data. It is then shown that the controllability energy estimator provides a key means for approximating the invariant measure of an ergodic, stochastically forced nonlinear system.'
author:
- 'Jake Bouvrie [^1]'
- 'Boumediene Hamzi [^2]'
title: '**<span style="font-variant:small-caps;">Empirical Estimators for Stochastically Forced Nonlinear Systems: Observability, Controllability and the Invariant Measure[^3]</span>**'
---
Introduction
============
Personal computing has developed to the point where in many cases it ought to be easier to simulate a dynamical system and analyze the empirical data, rather than attempt to study the system analytically. Indeed, for large classes of nonlinear systems, numerical analysis may be the only viable option. Yet the mathematical theory necessary to analyze dynamical systems on the basis of observed data is still largely underdeveloped. In previous work the authors proposed a linear, data-based approach for model reduction of nonlinear control systems [@allerton]. The approach is based on lifting simulated trajectories of the system into a high or infinite dimensional feature (Hilbert) space where the evolution of the original system may be reasonably modeled as linear. One may then implicitly carry out linear balancing, truncation, and model reduction in the feature space while retaining nonlinearity in the original statespace. In this paper, we continue under this setting and explore data-based definitions of key concepts for nonlinear control and random dynamical systems. We propose an empirical approach for estimation of the controllability and observability energies for stable nonlinear control systems, as well as invariant measures and their supports for ergodic nonlinear stochastic differential equations. Our methodology applies the relevant linear theory in a feature space where it is assumed that the original nonlinear system behaves approximately linearly. In this case we leverage the well-known connection between the controllability gramian of a linear control system and the invariant measure associated to the corresponding linear stochastic differential equation. This relationship was previously identified as useful for finding the controllability energy for certain nonlinear control systems given the invariant measure of the corresponding randomly forced dynamical system [@newman]. The approach in [@newman], however, requires solving a Fokker-Planck equation and so applies to only a narrow class of systems. Our approach takes the reverse direction in a data-driven setting: given an empirical estimate of the controllability energy function, one can obtain an estimate of the invariant measure. In particular, we will propose a consistent, data-based estimator for the controllability energy function of a nonlinear control system, and show how it can be used to estimate the invariant measure and its support for the corresponding stochastic differential equation (SDE). The essential point of this paper is to illustrate that it is possible to find data-based estimates of linear objects in order to understand nonlinear control and random dynamical systems, without having to solve a Hamilton-Jacobi-Bellman or Lyapunov equation in the case of nonlinear control systems, or a Fokker-Planck equation in the case of nonlinear SDEs. The approach proposed here also highlights the close interaction between control and random dynamical systems and demonstrates how control theoretic objects can be useful for studying random dynamical systems. Our contribution should be seen as a step towards developing a mathematical, data-driven theory for dynamical systems which can be used to analyze and predict random dynamical systems, as well as offer data-driven control strategies for nonlinear systems on the basis of observed data rather than a pre-specified model.
Linear Systems as a Paradigm for Working in RKHS: Background {#sec:bkgnd}
============================================================
In this section we give a brief overview of some important background concepts in linear control, random dynamical systems and reproducing kernel Hilbert spaces (RKHS). We will make use of the linear theory that follows after mapping the state variable of a nonlinear system into a suitable RKHS, thereby harnessing RKHS theory as a framework for extending linear tools to nonlinear systems. The following background material closely follows [@dullerud; @risken; @brockett].
Linear Control Systems {#sec:linear-control}
----------------------
Consider a linear control system $$\label{linsys}\begin{array}{rcl}\dot{x}&=&Ax+Bu\\y&=&Cx \end{array},$$ where $x\in{\mathbb{R}}^n$, $u \in {\mathbb{R}}^q$, $y \in {\mathbb{R}}^p$, $(A,B)$ is controllable, $(A,C)$ is observable and $A$ is Hurwitz. We define the controllability and the observability Gramians as, respectively, $W_c=\int_0^{\infty}e^{At}BB^{{\!\top\!}}e^{A^{{\!\top\!}}t}\, dt,$ $W_o=\int_0^{\infty}e^{A^{{\!\top\!}}t}C^{{\!\top\!}}Ce^{At}\, dt$. These two matrices can be viewed as a measure of the controllability and the observability of the system [@moore]. For instance, consider the past energy [@scherpen_thesis], $L_c(x_0)$, defined as the minimal energy required to reach $x_0$ from $0$ in infinite time $$\label{L_c}
L_c(x_0)=\inf_{\substack{
u \in { L}_2(-\infty,0),\\ x(-\infty)=0, x(0)=x_0}}
\frac{1}{2}\int_{-\infty}^0\|u(t)\|^2\,dt,$$ and the future energy [@scherpen_thesis], $L_o(x_0)$, defined as the output energy generated by releasing the system from its initial state $x(t_0)=x_0$, and zero input $u(t)=0$ for $t\ge0$, i.e. $$\label{L_o}
L_o(x_0)=\frac{1}{2}\int_{0}^{\infty}\|y(t)\|^2\,dt,$$ for $x(t_0)=x_0$ and $u(t)=0, t\ge0$.
In the linear case, it can be shown that $$\begin{aligned}
L_c(x_0)&=\tfrac{1}{2}x_0^{{\!\top\!}}W_c^{-1}x_0,\label{eqn:lin_Lc}\\
L_o(x_0)&=\tfrac{1}{2}x_0^{{\!\top\!}}W_o x_0 \label{eqn:lin_Lo} .\end{aligned}$$ Moreover, $W_c$ and $W_o$ satisfy the following Lyapunov equations [@dullerud]: $$\label{lyap_lin}\begin{array}{rcl}
AW_c+W_cA^{{\!\top\!}}&=&-BB^{{\!\top\!}},\\
A^{{\!\top\!}}W_o+W_oA&=&-C^{{\!\top\!}}C .
\end{array}$$ These energies are directly related to the controllability and observability operators.
[@dullerud] Given a matrix pair $(A,B)$, the controllability operator $\Psi_c$ is defined as $$\begin{array}{rcl} \Psi_c: L_2(-\infty,0)
&\rightarrow& {\mathbb{C}}^n \\
u &\mapsto& \int_{-\infty}^0 e^{-A\tau}Bu(\tau) d\tau \nonumber
\end{array}$$
The significance of this operator is made evident via the following optimal control problem: Given the linear system $\dot{x}(t)=Ax(t)+Bu(t)$ defined for $t \in (-\infty,0)$ with $x(-\infty)=0$, and for $x(0) \in {\mathbb{C}}^n$ with unit norm, what is the minimum energy input $u$ which drives the state $x(t)$ to $x(0)=x_0$ at time zero? That is, what is the $u \in L_2(-\infty,0]$ solving $\Psi_cu=x_0$ with smallest norm $\|u\|_2$? If $(A,B)$ is controllable, then $\Psi_c\Psi_c^{\ast}=:W_c$ is nonsingular, and the answer to the preceding question is $$\label{uopt} u_{opt}:=\Psi_c^{\ast}W_c^{-1}x_0.$$ The input energy is given by $$\begin{aligned}
\|u_{opt}\|_2^2 &=& \langle \Psi_c^{\ast}W_c^{-1}x_0, \Psi_c^{\ast}W_c^{-1}x_0 \rangle \nonum \\
&=& \langle W_c^{-1}x_0, \Psi_c\Psi_c^{\ast}W_c^{-1}x_0 \rangle \nonum \\
&=& x_0^{\ast}W_c^{-1}x_0 \;.\nonum\end{aligned}$$
Moreover, the reachable set through $u_{opt}$, i.e. the final states $x_0=\Psi_c u$ that can be reached given an input $u \in L_2(-\infty,0]$ of unit norm, $\{\Psi_cu: u \in L_2(-\infty,0] \;\mbox{and} \; \|u\|_2 \le 1
\}$ may be defined as $${\mathcal{R}}:=\{W_c^{\frac{1}{2}}x_c: x_c \in {\mathbb{C}}^n \; \mbox{and} \; \|x_c\|\le 1 \}.$$
Similarly, for the autonomous system $$\begin{array}{rcl}
\dot{x}&=& Ax, \quad x(0)=x_0\in {\mathbb{C}}^n,\\
y&=&Cx \nonum
\end{array}$$ where $A$ is Hurwitz, the observability operator is defined as follows.
[@dullerud] Given a matrix pair $(A,C)$, where $A$ is Hurwitz, the observability operator $\Psi_o$ is defined as $$\begin{array}{rcl} \Psi_o: {\mathbb{C}}^n
&\rightarrow& L_2(0,\infty) \\
x_0 &\mapsto& \left\{
\begin{array}{rcl}
Ce^{At}x_0, & \text{for } t \ge 0\\ 0, & \text{otherwise}
\end{array}\right.
\end{array}\nonum$$ The corresponding observability ellipsoid is given by $${\cal E}:=\{W_o^{\frac{1}{2}}x_0: x_0 \in {\mathbb{C}}^n \; \mbox{and} \; \|x_0\|=1 \}.$$
The energy of the output signal $y=\Psi_o x_0 $, for $x_0 \in {\mathbb{C}}^n$ can then be computed as $$\|y\|_2^2=\langle \Psi_ox_0, \Psi_ox_0
\rangle=\langle x_0, \Psi_o^{\ast}\Psi_ox_0
\rangle=\langle x_0, W_0x_0
\rangle$$ where $\Psi_o^{\ast}: L_2[0,\infty) \rightarrow {\mathbb{C}}^n $ is the adjoint of $\Psi_o$.
Linear Stochastic Differential Equations {#sec:linear-sdes}
----------------------------------------
In this section, we review the relevant background for stochastically forced differential equations (see e.g. [@risken; @brockett] for more detail). Here we will consider stochastically excited dynamical control systems affine in the input $u\in{\mathbb{R}}^q$ $$\label{control_nonlin}
\dot{x}=f(x)+G(x)u \;,$$ where $G:{\mathbb{R}}^n\to{\mathbb{R}}^{n\times q}$ is a smooth matrix-valued function and $x\in{\mathbb{R}}^n$. We replace the control inputs by sample paths of white Gaussian noise processes, giving the corresponding stochastic differential equation $$\label{sde_nonlin}
d{X_t}=f(X_t)dt + G(X_t)dW_t^{(q)}$$ with $W_t^{(q)}$ a $q-$dimensional Brownian motion. The solution $X_t$ to this SDE is a Markov stochastic process with transition probability density $\rho(t,x)$. The time evolution of the probability density $\rho(t,x)$ is described by the *Fokker-Planck (or Forward Kolmogorov) equation* $$\label{ito}\frac{\partial \rho}{\partial t}=-\langle \frac{\partial}{\partial x}, f\rho \rangle+
\frac{1}{2}\sum_{j,k=1}^{n}\frac{\partial^2}{\partial x_j \partial x_k}[(GG^{{\!\top\!}})_{jk}\rho]=:{\mathcal{L}}\rho \;.$$ The differential operator ${\mathcal{L}}$ on the right-hand side is referred to as the *Fokker-Planck operator* associated to (\[sde\_nonlin\]). The *steady-state probability density* for (\[sde\_nonlin\]) is a solution of the equation $$\label{steady}{\mathcal{L}}{\rho_{\infty}}=0.$$
In the context of linear Gaussian theory where we are given an $n-$dimensional system of the form $$\label{lin_sde}dX_t=AX_tdt+BdW_t^{(q)},$$ with $A \in {\mathbb{R}}^{n \times n}$, $B \in {\mathbb{R}}^{n \times q}$, the transition density is Gaussian. It is therefore sufficient to find the mean and covariance of the solution $X(t)$ in order to uniquely determine the transition probability density. The mean satisfies $\frac{d}{dt}{{\mathbb{E}}}[X]=A {{\mathbb{E}}}[X]$ and thus ${{\mathbb{E}}}[X(t)]=e^{At}{{\mathbb{E}}}[X(0)]$. If $A$ is Hurwitz, $\lim_{t \rightarrow \infty}{{\mathbb{E}}}[X(t)]=0$. The covariance satisfies $\frac{d}{dt}{{\mathbb{E}}}[XX^{{\!\top\!}}]=A{{\mathbb{E}}}[XX^{{\!\top\!}}]+{{\mathbb{E}}}[XX^{{\!\top\!}}]A
+BB^{{\!\top\!}}$. This formulation gives the steady-state distribution’s covariance matrix as ${\cal Q}=\lim_{t \rightarrow \infty}{{\mathbb{E}}}[X_tX_t^{{\!\top\!}}]$ so that we may find ${\cal Q}$ by solving the Lyapunov system $A{\cal Q}+{\cal Q}A^{{\!\top\!}}=-BB^{{\!\top\!}}$. Thus the solution ${\cal Q}$ is exactly the controllability gramian $${\cal Q}=W_c=\int_0^{\infty}e^{At}BB^{{\!\top\!}}e^{A^{{\!\top\!}}t}\, dt$$ which is positive iff. the pair $(A,B)$ is controllable [@brockett]. Combining the above facts, the steady-state probability density is given by $$\label{eqn:rho_linear_theory}
\rho_{\infty}(x)=Z^{-1}e^{-\frac{1}{2}x^TW_c^{-1}x} = Z^{-1} e^{-L_c(x)}$$ using (\[L\_c\]) and letting $Z=\sqrt{(2\pi)^n \mbox{det}(W_c)}$.
Equation suggests the following key observations in the linear setting:
- [*Given an approximation $\hat{L}_c$ of $L_c$ we obtain an approximation for $\rho_{\infty}$ of the form*]{} $$\label{rho_approx}
\hat{\rho}_{\infty} \propto e^{-\hat{L}_c(x)}$$
- [*Given an approximation $\hat{\rho}_{\infty}$ of $\rho_{\infty}$ we obtain an approximation for $L_c(x)$ by solving*]{} $$\label{Lc_approx}
\hat{L}_c(x)= -\ln[\hat{\rho}_{\infty}(x)]+C .$$
We note that these approximations have been used in different contexts to study nonlinear control and random dynamical systems. For instance, in [@butchart], Equation was used to find explicit solutions of the Fokker-Planck equation for systems where a Lyapunov equation for the unforced system can be found and solved. In [@newman], Equation was used, given an explicit solution to the Fokker-Planck equation, to approximate the controllability energy and subsequently applied to the problem of model reduction for nonlinear control systems.
Although the above relationship between $\rho_{\infty}$ and $L_c$ holds for only a small class of systems (e.g. linear and some Hamiltonian systems), by mapping a nonlinear system into a suitable reproducing kernel Hilbert space we may reasonably extend this connection to a broad class of nonlinear systems. We will return to this topic in Section \[sec:nonlinear-sdes\] after defining kernel Hilbert spaces and introducing gramians in RKHS.
Reproducing Kernel Hilbert Spaces
---------------------------------
We give a brief overview of reproducing kernel Hilbert spaces as used in statistical learning theory. The discussion here borrows heavily from [@cucker; @smola; @Wahba]. Early work developing the theory of RKHS was undertaken by N. Aronszajn [@AronRKHS].
Let ${\cal H}$ be a Hilbert space of functions on a set ${\cal X}$. Denote by $\langle f, g \rangle$ the inner product on ${\cal H}$ and let $\|f\|= \langle f, f \rangle^{1/2}$ be the norm in ${\cal H}$, for $f$ and $g \in {\cal H}$. We say that ${\cal H}$ is a reproducing kernel Hilbert space (RKHS) if there exists a function $K:{\cal X} \times {\cal X} \rightarrow {\mathbb{R}}$ such that
- $K_x:=K(x,\cdot)\in{\mathcal{H}}$ for all $x\in{\mathcal{X}}$.
- $K$ spans ${\cal H}$: ${\cal H}=\overline{\mbox{span}\{K_x~|~x \in {\cal X}\}}$.
- $K$ has the [*reproducing property*]{}: $\forall f \in {\cal H}$, $f(x)=\langle f,K_x \rangle$.
$K$ will be called a reproducing kernel of ${\cal H}$. ${\cal H}_K$ will denote the RKHS ${\cal H}$ with reproducing kernel $K$ where it is convenient to explicitly note this dependence.
A function $K:{\cal X} \times {\cal X} \rightarrow {\mathbb{R}}$ is called a Mercer kernel if it is continuous, symmetric and positive definite.
The important properties of reproducing kernels are summarized in the following proposition.
\[prop1\] If $K$ is a reproducing kernel of a Hilbert space ${\cal H}$, then
- $K(x,y)$ is unique.
- $\forall x,y \in {\cal X}$, $K(x,y)=K(y,x)$ (symmetry).
- $\sum_{i,j=1}^q\alpha_i\alpha_jK(x_i,x_j) \ge 0$ for $\alpha_i \in {\mathbb{R}}$, $x_i \in {\cal X}$ and $q\in\mathbb{N}_+$ (positive definiteness).
- $\langle K(x,\cdot),K(y,\cdot) \rangle=K(x,y)$.
Common examples of Mercer kernels defined on a compact domain ${\mathcal{X}}\subset{\mathbb{R}}^n$ are $K(x,y)=x\cdot y$ (Linear), $K(x,y)=(1+x\cdot y)^d$ for $d \in \NN_+$ (Polynomial), and $K(x,y)=e^{-\|x-y\|^2_2/\sigma^2}, \sigma >0$ (Gaussian).
\[thm1\] Let $K:{\cal X} \times {\cal X} \rightarrow {\mathbb{R}}$ be a symmetric and positive definite function. Then there exists a Hilbert space of functions ${\cal H}$ defined on ${\cal X}$ admitting $K$ as a reproducing Kernel. Conversely, let ${\cal H}$ be a Hilbert space of functions $f: {\cal X} \rightarrow {\mathbb{R}}$ satisfying $\forall x \in {\cal X}, \exists \kappa_x>0,$ such that $|f(x)| \le \kappa_x \|f\|_{\cal H},
\quad \forall f \in {\cal H}. $ Then ${\cal H}$ has a reproducing kernel $K$.
\[thm4\] Let $K(x,y)$ be a positive definite kernel on a compact domain or a manifold $X$. Then there exists a Hilbert space $\mathcal{F}$ and a function $\Phi: X \rightarrow \mathcal{F}$ such that $$K(x,y)= \langle \Phi(x), \Phi(y) \rangle_{{\mathcal{F}}} \quad \mbox{for} \quad x,y \in X.$$ $\Phi$ is called a feature map, and $\mathcal{F}$ a feature space[^4].
Given Theorem \[thm4\], and property \[iv.\] in Proposition \[prop1\], note that we can take $\Phi(x):=K_x:=K(x,\cdot)$ in which case $\mathcal{F}={\mathcal{H}}$ – the “feature space” is the RKHS itself, as opposed to an isomorphic space. We will make extensive use of this feature map. The fact that Mercer kernels are positive definite and symmetric is also key; these properties ensure that kernels induce positive, symmetric matrices and integral operators, reminiscent of similar properties enjoyed by gramians and covariance matrices. Finally, in practice one typically first chooses a Mercer kernel in order to choose an RKHS: Theorem \[thm1\] guarantees the existence of a Hilbert space admitting such a function as its reproducing kernel.
A key observation however, is that working in RKHS allows one to immediately find nonlinear versions of algorithms which can be expressed in terms of inner products. Consider an algorithm expressed in terms of the inner product $\langle x, x^{\prime} \rangle_{{\mathcal{X}}}$ with $x, x^{\prime} \in {\mathcal{X}}$. Now assume that instead of looking at a state $x$, we look at its $\Phi$ image in ${\mathcal{H}}$, $$\label{eqn:phi-mapped-data}
\begin{array}{rccl}
\Phi&:& X & \rightarrow {\cal H} \\
& & x &\mapsto \Phi(x) \;.
\end{array}$$ In the RKHS, the inner product $\langle \Phi(x), \Phi(x^{\prime}) \rangle$ is $$\langle \Phi(x), \Phi(x^{\prime}) \rangle= K(x,x^{\prime})$$ by the reproducing property. Hence, a nonlinear variant of the original algorithm may be implemented using kernels in place of inner products on ${\mathcal{X}}$.
Empirical Gramians in RKHS
==========================
In this Section we recall empirical gramians for linear systems [@moore], as well as a notion of empirical gramians for nonlinear systems in RKHS introduced in [@allerton]. The goal of the construction we describe here is to provide meaningful, data-based empirical controllability and observability gramians for nonlinear systems. In [@allerton], observability and controllability gramians were used for balanced model reduction, however here we will use these quantities to analyze nonlinear control properties and random dynamical systems. We note that a related notion of gramians for nonlinear systems is briefly discussed in [@gray], however no method for computing or estimating them was given.
Empirical Gramians for Linear Systems {#sec:linear_gramians}
-------------------------------------
To compute the Gramians for the linear system (\[linsys\]), one can attempt to solve the Lyapunov equations (\[lyap\_lin\]) directly although this can be computationally prohibitive. For linear systems, the gramians may be approximated by way of matrix multiplications implementing primal and adjoint systems (see the method of snapshots, e.g. [@Rowley05]). Alternatively, for any system, linear or nonlinear, one may take the simulation based approach introduced by B.C. Moore [@moore] for reduction of linear systems, and subsequently extended to nonlinear systems in [@lall]. The method proceeds by exciting each coordinate of the input with impulses from the zero initial state $(x_0=0)$. The system’s responses are sampled, and the sample covariance is taken as an approximation to the controllability gramian. Denote the set of canonical orthonormal basis vectors in ${\mathbb{R}}^n$ by $\{e_i\}_{i}$. Let $u^i(t) = \delta(t)e_i$ be the input signal for the $i$-th simulation, and let $x^i(t)$ be the corresponding response of the system. Form the matrix $X(t) = \bigl[x^1(t)
~\cdots~ x^q(t)\bigr] \in {\mathbb{R}}^{n\times q}$, so that $X(t)$ is seen as a data matrix with column observations given by the respective responses $x^i(t)$. Then the $(n\times n)$ controllability gramian is given by $$W_{c,{\text{lin}}} = \frac{1}{q}\int_0^{\infty}X(t)X(t)^{{\!\top\!}} dt.$$ We can approximate this integral by sampling the matrix function $X(t)$ within a finite time interval $[0,T]$ assuming for instance the regular partition $\{t_i\}_{i=1}^N, t_i = (T/N)i$. This leads to the [*empirical controllability gramian*]{} $$\label{eqn:Wchat_lin}
\widehat{W}_{c,{\text{lin}}} = \frac{T}{Nq}\sum_{i=1}^N X(t_i)X(t_i)^{{\!\top\!}} .$$
The observability gramian is estimated by fixing $u(t) = 0$, setting $x_0 = e_i$ for $i=1,\ldots,n$, and measuring the corresponding system output responses $y^i(t)$. Now assemble the output responses into a matrix $Y(t) = [y^1(t) ~\cdots~ y^n(t)]\in
{\mathbb{R}}^{p\times n}$. The $(n\times n)$ observability gramian $W_{o,{\text{lin}}}$ and its empirical counterpart $\widehat{W}_{o,{\text{lin}}}$ are respectively given by $$W_{o,{\text{lin}}} = \frac{1}{p}\int_0^{\infty}Y(t)^{{\!\top\!}}Y(t) dt$$ and $$\label{eqn:Wohat_lin}
\widehat{W}_{o,{\text{lin}}} = \frac{T}{Np}\sum_{i=1}^N \widetilde{Y}(t_i)\widetilde{Y}(t_i)^{{\!\top\!}}$$ where $\widetilde{Y}(t) = Y(t)^{{\!\top\!}}$. The matrix $\widetilde{Y}(t_i)\in{\mathbb{R}}^{n\times p}$ can be thought of as a data matrix with column observations $$\label{eqn:obs_data}
d_j(t_i) = \bigl(y_j^1(t_i), \ldots, y_j^n(t_i)\bigr)^{\!{\!\top\!}} \in{\mathbb{R}}^n,$$ for $j=1,\ldots,p, \,\,i=1,\ldots, N$ so that $d_j(t_i)$ corresponds to the response at time $t_i$ of the single output coordinate $j$ to each of the (separate) initial conditions $x_0=e_k, k=1,\ldots,n$.
Empirical Gramians in RKHS Characterizing Nonlinear Systems {#sec:rkhs-gramians}
-----------------------------------------------------------
Consider the generic nonlinear system $$\label{sigma}
\left\{\begin{array}{rcl}\dot{x}&=&F(x,u)\\ y &=& h(x), \end{array}\right.$$ with $x \in {\mathbb{R}}^n$, $u \in {\mathbb{R}}^q$, $y\in {\mathbb{R}}^p$, $F(0)=0$ and $h(0)=0$. Assume that the linearization of around the origin is controllable, observable and $A=\frac{\partial F}{\partial x}|_{x=0}$ is asymptotically stable.
RKHS counterparts to the empirical quantities , defined above for the system can be defined by considering feature-mapped lifts of the simulated samples in ${\mathcal{H}}_K$. In the following, and without loss of generality, [*we assume the data are centered in feature space*]{}, and that the observability samples and controllability samples are centered separately. See ([@smola], Ch. 14) for a discussion on implicit data centering in RKHS with kernels.
First, observe that the gramians $\widehat{W}_c, \widehat{W}_o$ can be viewed as the sample covariance of a collection of $N\cdot q, N\cdot p$ vectors in ${\mathbb{R}}^n$ scaled by $T$, respectively. Then applying $\Phi$ to the samples as in , we obtain the corresponding gramians in the RKHS associated to $K$ as bounded linear operators on ${\mathcal{H}}_K$: $$\begin{aligned}
\widehat{W}_c &= \frac{T}{Nq}\sum_{i=1}^N\sum_{j=1}^q \Phi(x^j(t_i))\otimes \Phi(x^j(t_i)) \label{eqn:emp_Wc_rkhs}\\
\widehat{W}_o &= \frac{T}{Np}\sum_{i=1}^N\sum_{j=1}^p \Phi(d_j(t_i))\otimes\Phi(d_j(t_i))\nonumber\end{aligned}$$ where the samples $x_j,d_j$ are as defined in Section \[sec:linear\_gramians\], and $a\otimes b=a{\left\langle{b},{\cdot}\right\rangle}$ denotes the tensor product in ${\mathcal{H}}$. From here on we will use the notation $W_c, W_o$ to refer to RKHS versions of the true (integrated) gramians, and $\widehat{W}_c, \widehat{W}_o$ to refer to RKHS versions of the empirical gramians.
Let ${\boldsymbol{\Psi}}$ denote the matrix whose columns are the (scaled) observability samples mapped into feature space by $\Phi$, and let ${\boldsymbol{\Phi}}$ be the matrix similarly built from the feature space representation of the controllability samples. Then we may alternatively express the gramians above as $\widehat{W}_c={\boldsymbol{\Phi}}{\boldsymbol{\Phi}}^{{\!\top\!}}$ and $\widehat{W}_o={\boldsymbol{\Psi}}{\boldsymbol{\Psi}}^{{\!\top\!}}$, and define two other important quantities:
- The *controllability kernel matrix* $K_c\in{\mathbb{R}}^{Nq\times Nq}$ of kernel products $$\begin{aligned}
K_c &= {\boldsymbol{\Phi}}^{{\!\top\!}}{\boldsymbol{\Phi}}\\
(K_c)_{\mu\nu} &= K(x_\mu, x_\nu) = {\left\langle{\Phi(x_\mu)},{\Phi(x_\nu)}\right\rangle}_{{\mathcal{F}}}\end{aligned}$$ for $\mu,\nu=1,\ldots,Nq$ where we have re-indexed the set of vectors $\{x^{j}(t_i)\}_{i,j} =
\{x_{\mu}\}_{\mu}$ to use a single linear index.
- The *observability kernel matrix* $K_o\in{\mathbb{R}}^{Np\times Np}$, $$\begin{aligned}
K_o &= {\boldsymbol{\Psi}}^{{\!\top\!}}{\boldsymbol{\Psi}}\\
(K_o)_{\mu\nu} &= K(d_\mu, d_\nu) = {\left\langle{\Phi(d_\mu)},{\Phi(d_\nu)}\right\rangle}_{{\mathcal{F}}}\end{aligned}$$ for $\mu,\nu=1,\ldots,Np$, where we have again re-indexed the set $\{d_j(t_i)\}_{i,j}=\{d_\mu\}_{\mu}$ for simplicity.
Note that $K_c,K_o$ may be highly ill-conditioned. The SVD may be used to show that $\widehat{W}_c$ and $K_c$ ($\widehat{W}_o$ and $K_o$) have the same singular values (up to zeros).
Nonlinear Control Systems in RKHS
=================================
In this section, we introduce empirical versions of the controllability and observability energies - for stable nonlinear control systems of the form , that can be estimated from observed data. Our underlying assumption is that a given nonlinear system may be treated as if it were linear in a suitable feature space. That reproducing kernel Hilbert spaces provide rich representations capable of capturing strong nonlinearities in the original input (data) space lends validity to this assumption.
In general little is known about the energy functions in the nonlinear setting. However, Scherpen [@scherpen_thesis] has shown that the energy functions $L_c(x)$ and $L_o(x)$ defined in and satisfy a Hamilton-Jacobi and a Lyapunov equation, respectively.
\[thm:scherp1\][@scherpen_thesis] Consider the nonlinear control system (\[sigma\]) with $F(x,u)=f(x)+G(x)u$. If the origin is an asymptotically stable equilibrium of $f(x)$ on a neighborhood $W$ of the origin, then for all $x \in W$, $L_o(x)$ is the unique smooth solution of $$\label{Lo_hjb} \frac{\partial L_o}{\partial x}(x)f(x)+\frac{1}{2}h^{{\!\top\!}}(x)h(x)=0,\quad L_o(0)=0$$ under the assumption that (\[Lo\_hjb\]) has a smooth solution on $W$. Furthermore for all $x \in W$, $L_c(x)$ is the unique smooth solution of $$\label{Lc_hjb} \frac{\partial L_c}{\partial x}(x)f(x)+\frac{1}{2} \frac{\partial L_c}{\partial
x}(x)g(x)g^{{\!\top\!}}(x) \frac{\partial^{{\!\top\!}}L_c}{\partial x}(x)=0,\; L_c(0)=0$$ under the assumption that (\[Lc\_hjb\]) has a smooth solution $\bar{L}_c$ on $W$ and that the origin is an asymptotically stable equilibrium of $-(f(x)+g(x)g^{{\!\top\!}}(x) \frac{\partial \bar{L}_c}{\partial x}(x))$ on $W$.
We would like to avoid solving explicitly the PDEs (\[Lo\_hjb\])- (\[Lc\_hjb\]) and instead find good estimates of their solutions directly from simulated or observed data.
Energy Functions {#sec:energy_fns}
----------------
Following the linear theory developed in Section \[sec:linear-control\], we would like to define analogous controllability and observability energy functions paralleling -, but adapted to the nonlinear setting. We first treat the controllability function. Let $\mu_{\infty}$ on the statespace ${\mathcal{X}}$ denote the unknown invariant measure of the nonlinear system when driven by white Gaussian noise. We will consider here the case where the controllability samples $\{x_i\}_{i=1}^m$ are i.i.d. random draws from $\mu_{\infty}$, and ${\mathcal{X}}$ is a compact subset of ${\mathbb{R}}^n$. The former assumption is implicitly made in much of the empirical balancing literature, and if a system is simulated for long time intervals, it should hold approximately in practice. If we take $\Phi(x)=K_x$, the infinite-data limit of is given by $$\label{eqn:covop_gramian}
W_c = {\mathbb{E}}_{\mu_{\infty}} [\widehat{W}_{c}] = \int_{{\mathcal{X}}}{\left\langle{\cdot},{K_x}\right\rangle}K_xd\mu_{\infty}(x).$$
In general neither $W_c$ nor its empirical approximation $\widehat{W}_c$ are invertible, so to define a controllability energy similar to one is tempted to define $L_c$ on ${\mathcal{H}}$ as , where $A^{\dag}$ denotes the pseudoinverse of the operator $A$. However, the domain of $W_c^{\dag}$ is equal to the range of $W_c$, and so in general $K_x$ may not be in the domain of $W_c^{\dag}$. We will therefore introduce the orthogonal projection $W_c^{\dag}W_c$ mapping ${\mathcal{H}}\mapsto\text{range}(W_c)$ and define the nonlinear control energy on ${\mathcal{H}}$ as $$\label{eqn:best_lc}
L_c(h) = {\left\langle{W_c^{\dag}(W_c^{\dag}W_c)h},{h}\right\rangle}.$$ We will consider finite sample approximations to , however a further complication is that $\widehat{W}_c^{\dag}\widehat{W}_c$ may not converge to $W_c^{\dag}W_c$ in the limit of infinite data (taking the pseudoinverse is not a continuous operation), and $\widehat{W}_c^{\dag}$ can easily be ill-conditioned in any event. Thus one needs to impose regularization, and we replace the pseudoinverse $A^{\dag}$ with a regularized inverse $(A + \lambda I)^{-1}, \lambda > 0$ throughout. We note that the preceding observations were also made in [@RosascoDensity]. Intuitively, regularization prevents the estimator from overfitting to a bad or unrepresentative sample of data. We thus define the estimator (that is, on the domain $\{K_x~|~ x\in{\mathcal{X}}\}\subseteq{\mathcal{H}}$) to be $$\label{eqn:rkhs_lc_def}
\hat{L}_c(x)=\tfrac{1}{2}\bigl\langle(\widehat{W}_c + \lambda I)^{-2}\widehat{W}_c K_x,K_x\bigr\rangle, \quad x\in{\mathcal{X}}$$ with infinite-data limit $$L_c^{\lambda}(x) = \tfrac{1}{2}{\left\langle{(W_c + \lambda I)^{-2}W_c K_x},{K_x}\right\rangle},$$ where $\lambda > 0$ is the regularization parameter.
Towards deriving an equivalent but computable expression for $\hat{L}_c$ defined in terms of kernels, we recall the sampling operator ${S_{\mathbf{x}}}$ of [@SmaleIntegral] and its adjoint. Let ${\mathbf{x}}= \{x_i\}_{i=1}^{m}$ denote a generic sample of $m$ data points. To ${\mathbf{x}}$ we can associate the operators $$\begin{aligned}
{4}
{S_{\mathbf{x}}}&: {\mathcal{H}}&\to &\, {\mathbb{R}}^{m}, &\quad h &\in{\mathcal{H}}&\mapsto&\, \bigl(h(x_1),\ldots,h(x_{m})\bigr)\\
{S_{\mathbf{x}}^{\ast}}&:{\mathbb{R}}^{m} &\to &\, {\mathcal{H}}, &\quad c &\in{\mathbb{R}}^{m} &\mapsto&\, \textstyle\sum_{i=1}^{m}c_iK_{x_i}\,.\end{aligned}$$ If ${\mathbf{x}}$ is the collection of $m=Nq$ controllability samples, one can check that $\widehat{W}_c = \tfrac{1}{m}{S_{\mathbf{x}}^{\ast}}{S_{\mathbf{x}}}$ and $K_c={S_{\mathbf{x}}}{S_{\mathbf{x}}^{\ast}}$. Consequently, $$\begin{aligned}
\hat{L}_c(x) &=\tfrac{1}{2}{\left\langle{(\tfrac{1}{m}{S_{\mathbf{x}}^{\ast}}{S_{\mathbf{x}}}+ \lambda I)^{-2}\tfrac{1}{m}{S_{\mathbf{x}}^{\ast}}{S_{\mathbf{x}}}K_{x}},{K_{x}}\right\rangle}\\
&=\tfrac{1}{2m}{\left\langle{{S_{\mathbf{x}}^{\ast}}(\tfrac{1}{m}{S_{\mathbf{x}}}{S_{\mathbf{x}}^{\ast}}+ \lambda I)^{-2}{S_{\mathbf{x}}}K_{x}},{K_{x}}\right\rangle}\\
&= \tfrac{1}{2m}{\bf k_c}(x)^{{\!\top\!}}(\tfrac{1}{m}K_c + \lambda I)^{-2}{\bf k_c}(x),\end{aligned}$$ where ${\bf k_c}(x):={S_{\mathbf{x}}}K_x = \bigl(K(x,x_{\mu})\bigr)_{\mu=1}^{Nq}$ is the $Nq$-dimensional column vector containing the kernel products between $x$ and the controllability samples.
Similarly, letting ${\mathbf{x}}$ now denote the collection of $m=Np$ observability samples, we can approximate the future output energy by $$\begin{aligned}
\hat{L}_o(x) &= \tfrac{1}{2}\bigl\langle\widehat{W}_oK_x,K_x\bigr\rangle \\\label{eqn:Lo_rkhs}
&= \tfrac{1}{2m}\bigl\langle{S_{\mathbf{x}}^{\ast}}{S_{\mathbf{x}}}K_x, K_x\bigr\rangle \nonumber \\
&= \tfrac{1}{2m}{\bf k_o}(x)^{{\!\top\!}}{\bf k_o}(x)
= \tfrac{1}{2m}{\left\|{{\bf k_o}(x)}\right\|}_2^2\nonumber\end{aligned}$$ where ${\bf k_o}(x):=\bigl(K(x,d_{\mu})\bigr)_{\mu=1}^{Np}$ is the $Np$-dimensional column vector containing the kernel products between $x$ and the observability samples. We collect the above results into the following definition:
\[def:rkhs\_energies\] Given a nonlinear control system of the form , we define the kernel controllability energy function and the kernel observability energy function as, respectively, $$\begin{aligned}
\hat{L}_c(x) &= \tfrac{1}{2Nq}{\bf k_c}(x)^{{\!\top\!}}(\tfrac{1}{Nq}K_c + \lambda I)^{-2}{\bf k_c}(x) \\ \label{eqn:lc_hat}
\hat{L}_o(x) &= \tfrac{1}{2Np}{\left\|{{\bf k_o}(x)}\right\|}_2^2 \;.\end{aligned}$$
Note that the kernels used to define $\hat{L}_c$ and $\hat{L}_o$ need not be the same.
Consistency
-----------
We’ll now turn to showing that the estimator $\hat{L}_c$ is consistent, but note that [*we do not address the approximation error*]{} between the energy function estimates and the true but unknown underlying functions. Controlling the approximation error requires making specific assumptions about the nonlinear system, and we leave this question open. In the following we will make an important set of assumptions regarding the kernel $K$ and the RKHS ${\mathcal{H}}$ it induces.
\[ass:rkhs\] The reproducing kernel $K$ defined on the compact statespace ${\mathcal{X}}\subset{\mathbb{R}}^n$ is locally Lipschitz, measurable and defines a completely regular RKHS. Furthermore the diagonal of $K$ is uniformly bounded, $$\label{eqn:kappa}
\kappa^2 = \sup_{x\in{\mathcal{X}}}K(x,x) <\infty.$$
Separable RKHSes are induced by continuous kernels on separable spaces ${\mathcal{X}}$. Since ${\mathcal{X}}\subset{\mathbb{R}}^n$ is separable and locally Lipschitz functions are also continuous, ${\mathcal{H}}$ will always be separable. [*Completely regular*]{} RKHSes are introduced in [@RosascoDensity] and the reader is referred to this reference for details. Briefly, complete regularity ensures recovery of level sets of [*any*]{} distribution, in the limit of infinite data. The Gaussian kernel does not define a completely regular RKHS, but the $L_1$ exponential and Laplacian kernels do [@RosascoDensity].
We introduce some additional notation. Let $W_{c,m}$ denote the empirical RKHS gramian formed from a sample of size $m$ observations, and let the corresponding control energy estimate in Definition \[def:rkhs\_energies\] involving $W_{c,m}$ and regularization parameter $\lambda$ be denoted by $L_{c,m}^{\lambda}$.
The following preliminary lemma provides finite sample error bounds for Hilbert-Schmidt covariance matrices on real, separable reproducing kernel Hilbert spaces.
\[lem:cov\_conc\]
- The operators $W_c, W_{c,m}$ are Hilbert-Schmidt.
- Let $\delta\in(0,1]$. With probability at least $1-\delta$, $${\left\|{W_c-W_{c,m}}\right\|}_{HS} \leq \frac{2\sqrt{2}\kappa^2}{\sqrt{m}}\log^{1/2}\frac{2}{\delta}.$$
The following theorem establishes consistency of the estimator $L_{c,m}^{\lambda}$, the proof of which follows the method of integral operators developed by [@SmaleIntegral; @AndreaFastRates] and subsequently adopted in the context of density estimation by ([@RosascoDensity], Theorem 1).
- Fix $\lambda > 0$. For each $x\in{\mathcal{X}}$, with probability at least $1-\delta$, $$\bigl|L_{c,m}^{\lambda}(x) - L_c^{\lambda}(x)\bigr| \leq \frac{2\sqrt{2}\kappa^4(\lambda^2 + \kappa^4)}{\lambda^4\sqrt{m}}\log^{1/2}\frac{2}{\delta} .$$
- If $(K,{\mathcal{X}},\mu_{\infty})$ is such that $$\label{eqn:bounded_pinv}
\sup_{x\in{\mathcal{X}}}\|W_c^{\dag}(W_c^{\dag}W_c)K_x\|_{{\mathcal{H}}}<\infty,$$ then for all $x\in{\mathcal{X}}$, $$\displaystyle\lim_{\lambda\to 0}|L_c^{\lambda}(x) - L_c(x)| = 0.$$
- If the condition holds and the sequence $\{\lambda_m\}_m$ satisfies $\displaystyle\lim_{m\to\infty}\lambda_m=0$ with $\displaystyle\lim_{m\to\infty}\tfrac{\log^{1/2}m}{\lambda_m\sqrt{m}} = 0$, then $$\lim_{m\to\infty}\bigl|L_{c,m}^{\lambda}(x) - L_c(x)\bigr| = 0,\quad\text{almost surely.}$$
For (i), the sample error, we have $$\begin{aligned}
2\bigl|L_{c,m}^{\lambda}(x) - L_c^{\lambda}(x)\bigr|
& \leq {\left\|{(W_{c,m} + \lambda I)^{-2}W_{c,m} - (W_c+ \lambda I)^{-2}W_c }\right\|}{\left\|{K_{x}}\right\|}^2_{{\mathcal{H}}} \\
& \leq \bigl\|(W_{c} + \lambda I)^{-2}[\lambda^2(W_{c,m} - W_c) + W_c(W_c - W_{c,m})W_{c,m}](W_{c,m} + \lambda I)^{-2} \bigr\|\kappa^2\\
& \leq \frac{\kappa^2(\lambda^2 + \kappa^4)}{\lambda^4}{\left\|{W_{c,m} - W_c}\right\|}_{HS}\end{aligned}$$ where ${\left\|{\cdot}\right\|}$ refers to the operator norm. The second inequality follows from spectral calculus and . The third line follows making use of the estimates ${\left\|{(W_{c,m} + \lambda I)^{-2}}\right\|}\leq \lambda^{-2}, {\left\|{(W_c + \lambda I)^{-2}}\right\|}\leq \lambda^{-2}, \|W_c\|_{HS}\leq \kappa^2, \|W_{c,m}\|_{HS}\leq\kappa^2$ (and the fact that $\lambda > 0$ so that the relevant quantities are invertible). Part (i) then follows applying Lemma \[lem:cov\_conc\] to the quantity ${\left\|{W_{c,m} - W_c}\right\|}_{HS}$. For (ii), the approximation error, note that the compact self-adjoint operator $W_c$ can be expanded onto an orthonormal basis $\{\sigma_i,\phi_i\}$. We then have $$\begin{aligned}
2\bigl|L_{c}^{\lambda}(x) - L_c(x)\bigr|
& = \bigl|\bigl\langle[(W_{c} + \lambda I)^{-2}W_{c} - W_c^{\dag}(W_c^{\dag}W_c)]K_x,K_x\bigr\rangle\bigr| \\
& = \left| \sum_i\frac{\sigma_i}{(\sigma_i + \lambda)^2}|\langle \phi_i,K_x\rangle|^2 -
\sum_{i:\sigma_i>0}\frac{1}{\sigma_i}|\langle\phi_i,K_x\rangle|^2\right| \\
& \leq \lambda\sum_{i:\sigma_i>0}\frac{2\sigma_i + \lambda}{(\sigma_i + \lambda)^2\sigma_i}|\langle \phi_i,K_x\rangle|^2 .\end{aligned}$$ The last quantity above can be seen to converge to 0 as $\lambda\to 0$ since the sum converges for all $x$ under the condition . Lastly for part (iii), we see that if $m\to\infty$ and $\lambda^2\to 0$ slower than $\sqrt{m}$ then the sample error (i) goes to 0 while (ii) also holds. For almost sure convergence in part (i), we additionally require that for any $\varepsilon\in(0,\infty)$, $$\sum_m{\mathbb{P}}\bigl(|L_{c,m}^{\lambda}(x) - L_c^{\lambda}(x)| > \varepsilon\bigr)\leq
\sum_m e^{-{\mathcal{O}}(m\lambda^4_m\varepsilon^2)} < \infty.$$ The choice $\lambda_m = \log^{-1/2}m$ satisfies this requirement, as can be seen from the fact that for large enough $M<\infty$, $\sum_{m>M} e^{-m/\log^2 m} \leq \sum_{m>M} e^{-\sqrt{m}} < \infty$.
We note that the condition required in part (ii) of the theorem has also been discussed in the context of support estimation in forthcoming work from the authors of [@RosascoDensity].
Observability and Controllability Ellipsoids
--------------------------------------------
Given the preceding, we can estimate the reachable and observable sets of a nonlinear control system as level sets of the RKHS energy functions $\hat{L}_c, \hat{L}_o$ from Definition \[def:rkhs\_energies\]:
Given a nonlinear control system , its reachable set can be estimated as $$\label{reachable_estimate}
\widehat{\cal R}_{\tau}=\{x\in{\mathcal{X}}~|~\hat{L}_c(x) \leq \tau \}$$ and its observable set can be estimated as $$\label{observable_estimate}
\widehat{\cal E}_{\tau'}=\{x\in{\mathcal{X}}~|~\hat{L}_o(x) \geq \tau' \}$$ for suitable choices of the threshold parameters $\tau, \tau'$.
If the energy function estimates above are replaced with the true energy functions, and $\tau=\tau'=1/2$, one obtains a finite sample approximation to the controllability and observability ellipsoids defined in Section \[sec:linear-control\] if the system is linear. In general, $\tau$ may be chosen empirically based on the data, using for instance a cross-validation procedure. Note that in the linear setting, the ellipsoid of strongly observable states is more commonly characterized as $\{x~|~x^{{\!\top\!}}W_o^{-1}x\leq 1\} = \{W_o^{\frac{1}{2}}x ~|~ {\left\|{x}\right\|}\leq 1\}$; hence the definition .
Estimation of Invariant Measures for Ergodic Nonlinear SDEs {#sec:nonlinear-sdes}
===========================================================
In this Section we consider [*ergodic*]{} nonlinear SDEs of the form , where the invariant (or “stationary”) measure is a key quantity providing a great deal of insight. In the context of control, the support of the stationary distribution corresponds to the reachable set of the nonlinear control system and may be estimated by . Solving a Fokker-Planck equation of the form is one way to determine the probability distribution describing the solution to an SDE. However, for nonlinear systems finding an explicit solution to the Fokker-Planck equation –or even its steady-state solution– is a challenging problem. The study of existence of steady-state solutions can be traced back to the 1960s [@fuller; @zakai], however explicit formulas for steady-state solutions of the Fokker-Planck equation exist in only a few special cases (see [@butchart; @da_prato1992; @fuller; @guinez; @liberzon; @risken] for example). Such systems are often conservative or second order vector-fields. Hartmann [@hartmann:08] among others has studied balanced truncation in the context of linear SDEs, where empirical estimation of gramians plays a key role.
We propose here a data-based non-parametric estimate of the solution to the steady-state Fokker-Planck equation for a nonlinear SDE, by combining the relation with the control energy estimate . Following the general theme of this paper, we make use of the theory from the linear Gaussian setting described in Section \[sec:linear-sdes\], but in a suitable reproducing kernel Hilbert space. Other estimators have of course been proposed in the literature for approximating invariant measures and for density estimation from data more generally (see e.g. [@biau; @froyland; @froyland1; @kilminster; @RosascoDensity]), however to our knowledge we are not aware of any estimation techniques which combine RKHS theory and nonlinear dynamical control systems. An advantage of our approach over other non-parametric methods is that an invariant density is approximated by way of a regularized fitting process, giving the user an additional degree of freedom in the regularization parameter.
Our setting adopts the perspective that the nonlinear stochastic system behaves approximately linearly when mapped via $\Phi$ into the RKHS ${\mathcal{H}}$, and as such may be modeled by an infinite dimensional linear system in ${\mathcal{H}}$. Although this system is [*unknown*]{}, we know that it is linear and that we can estimate its gramians and control energies from observed data. Furthermore, we know that the invariant measure of the system in ${\mathcal{H}}$ is zero-mean Gaussian with covariance given by the controllability gramian. Thus the original nonlinear system’s invariant measure on ${\mathcal{X}}$ should be reasonably approximated by the pullback along $\Phi$ of the Gaussian invariant measure associated with the linear infinite dimensional SDE in ${\mathcal{H}}$.
We summarize the setting in the following [*modeling Assumption*]{}:
\[ass:ou\_proc\] Let ${\mathcal{H}}$ be a real, possibly infinite dimensional RKHS satisfying Assumption \[ass:rkhs\].
- Given a suitable choice of kernel $K$, if the ${\mathbb{R}}^d$-valued stochastic process $x(t)$ is a solution to the (ergodic) stochastically excited nonlinear system , the ${\mathcal{H}}$-valued stochastic process can be reasonably modeled as an Ornstein-Uhlenbeck process $$\label{eqn:infdim-sde}
dX(t) = AX(t)dt + \sqrt{C}dW(t), \quad X(0)=0\in{\mathcal{H}}$$ where $A$ is linear, negative and is the infinitesimal generator of a strongly continuous semigroup $e^{tA}$, $C$ is linear, continuous, positive and self-adjoint, and $W(t)$ is the cylindrical Wiener process.
- The measure $P_{\infty}$ is the invariant measure of the OU process and $P_{\infty}$ is the pushforward along $\Phi$ of the unknown invariant measure $\mu_{\infty}$ on the statespace ${\mathcal{X}}$ we would like to approximate.
- The measure $\mu_{\infty}$ is absolutely continuous with respect to Lebesgue measure, and so admits a density.
We will proceed in deriving an estimate of the invariant density under these assumptions, but note that there are interesting systems for which the assumptions may not always hold in practice. For example, uncontrollable systems may not have a unique invariant measure. In these cases one must interpret the results discussed here as heuristic in nature.
It is known that a mild solution $X(t)$ to the SDE exists and is unique ([@da_prato1992], Thm. 5.4. pg. 121). Furthermore, the controllability gramian associated to $$\label{eqn:infdim-gramian}
W_ch = \int_0^{\infty}e^{tA}Ce^{tA^*}hdt,\quad h\in{\mathcal{H}}$$ is trace class ([@da_prato2006], Lemma 8.19), and the unique measure $P_{\infty}$ invariant with respect to the Markov semigroup associated to the OU process has characteristic function ([@da_prato2006], Theorem 8.20) $$\label{eqn:inv-meas}
\widetilde{P}_{\infty}(h) = \exp\Bigl(-\tfrac{1}{2}{\left\langle{W_ch},{h}\right\rangle}\Bigr),\quad h\in{\mathcal{H}}\;.$$ We will use the notation $\widetilde{P}$ to refer to the Fourier transform of the measure $P$. The law of the solution $X(t)$ to problem given initial condition $X(0)=0$ is Gaussian with zero mean and covariance operator $Q_t=\int_{0}^t e^{sA}Ce^{sA^*}ds$. Thus $$\begin{aligned}
W_c &= \lim_{t\to\infty}{\mathbb{E}}[X(t)\otimes X(t)]\\
& = \int_{{\mathcal{H}}}{\left\langle{\cdot},{h}\right\rangle}{h}dP_{\infty}(h)\\
&= \int_{{\mathcal{X}}}{\left\langle{\cdot},{K_x}\right\rangle}K_xd\mu_{\infty}(x)\end{aligned}$$ where the last integral follows pulling $P_{\infty}$ back to ${\mathcal{X}}$ via $\Phi$, establishing the equivalence between and .
Given that the measure $P_{\infty}$ has Fourier transform and by Assumption \[ass:ou\_proc\] is interpreted as the pushforward of $\mu_{\infty}$ (that is, for Borel sets $B\in\mathcal{B}({\mathcal{H}})$, $P_{\infty}(B)=(\Phi_*\mu_{\infty})(B)=\mu_{\infty}(\Phi^{-1}(B))$ formally), we have that $\widetilde{\mu}_{\infty}(x) = \exp\bigl(-\tfrac{1}{2}{\left\langle{W_cK_x},{K_x}\right\rangle}\bigr)$.
The invariant measure $\mu_{\infty}$ is defined on a finite dimensional space, so together with part (iii) of Assumption \[ass:ou\_proc\], we may consider the corresponding (Radon-Nikodym) density $$\rho_{\infty}(x) \propto \exp\bigl(-\tfrac{1}{2}{\left\langle{W_c^{\dag}(W_c^{\dag}W_c)K_x},{K_x}\right\rangle}\bigr)$$ whenever the condition holds. If does not hold or if we are considering a finite data sample, then we regularize to arrive at $$\rho_{\infty}(x) \propto \exp\bigl(-\tfrac{1}{2}{\left\langle{(W_c + \lambda I)^{-1}K_x},{K_x}\right\rangle}\bigr)$$ as discussed in Section \[sec:linear-sdes\] (see Eq. \[rho\_approx\]) and Section \[sec:energy\_fns\]. This density may be estimated from data $\{x_i\}_{i=1}^N$ since the controllability energy may be estimated from data: at a new point $x$, we have $$\hat{\rho}_{\infty}(x) = Z^{-1}\exp\bigl(-\hat{L}_c(x)\bigr)$$ where $\hat{L}_c$ is the empirical approximation computed according to Definition \[def:rkhs\_energies\], and the constant $Z$ may be either computed analytically in some cases or simply estimated from the data sample to enforce summation to unity. We may also estimate, for example, level sets of $\rho_{\infty}$ (such as the support) by considering level sets of the regularized control energy function estimator, $\{x\in{\mathcal{X}}~|~ L_{c,m}(x) \leq \tau\}$.
Conclusion
==========
To summarize our contributions, we have introduced estimators for the controllability/observability energies and the reachable/observable sets of nonlinear control systems. We showed that the controllability energy estimator may be used to approximate the stationary solution of the Fokker-Planck equation governing nonlinear SDEs (and its support).
The estimators we derived were based on applying linear methods for control and random dynamical systems to nonlinear control systems and SDEs, once mapped into an infinite-dimensional RKHS acting as a “linearizing space”. These results collectively argue that there is a reasonable passage from linear dynamical systems theory to a data-based nonlinear dynamical systems theory through reproducing kernel Hilbert spaces.
We leave for future work the formulation of data-based estimators for Lyapunov exponents and the controllability/observability operators $\Psi_c,\Psi_o$ associated to nonlinear systems.
Acknowledgements {#acknowledgements .unnumbered}
================
We thank Lorenzo Rosasco and Jonathan Mattingly for helpful discussions. BH thanks the European Union for financial support received through an International Incoming Marie Curie Fellowship, and JB gratefully acknowledges support under NSF contracts NSF-IIS-08-03293 and NSF-CCF-08-08847 to M. Maggioni.
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[^1]: J. Bouvrie is with the Department of Mathematics, Duke University, Durham, NC 27708, USA [[email protected]]{}
[^2]: B. Hamzi is with the Department of Mathematics, Imperial College London, London, SW7 2AZ, UK [[email protected]]{}
[^3]: An abbreviated version of this report will appear in Proc. American Control Conference (ACC), 2012.
[^4]: The dimension of the feature space can be infinite, for example in the case of the Gaussian kernel.
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{
"pile_set_name": "ArXiv"
}
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---
abstract: 'In this paper, we study the control of the traffic-driven epidemic spreading by immunization strategy. We consider the random, degree-based and betweeness-based immunization strategies, respectively. It is found that the betweeness-based immunization strategy can most effectively prevent the outbreak of traffic-driven epidemic. Besides, we find that the critical number of immune nodes above which epidemic dies out is increased with the enhancement of the spreading rate and the packet-generation rate.'
author:
- 'Han-Xin Yang$^{1}$'
- 'Bing-Hong Wang$^{2}$'
title: 'Immunization of traffic-driven epidemic spreading'
---
Keywords: traffic-driven epidemic spreading; immunization strategy; betweeness
Introduction
============
Both epidemic spreading [@1; @2; @3; @4; @5; @6; @7; @8; @9; @10; @11] and traffic transportation [@12; @13; @14; @15; @16; @17; @18] on complex networks have attracted much attention in the past decade. For a long time, the two types of dynamical processes have been studied independently. The first attempt to incorporate traffic into epidemic spreading is based on metapopulation model [@m1; @m2; @m3; @m4; @m5; @m6]. This framework describes a set of spatially structured interacting subpopulations as a network, whose links denote the traveling path of individuals across different subpopulations. Each subpopulation consists of a large number of individuals. An infected individual can infect other individuals in the same subpopulation. The metapopulation model is often used to simulate the spread of human and animal diseases (such as SARS and H1N1) among different cities. In a recent work, Meloni $et$ $al.$ proposed another traffic-driven epidemic spreading model [@Meloni], in which each node of a network represents a router in the Internet and the epidemic can spread between nodes by the transmission of packets. A susceptible node will be infected with some probability every time it receives a packet from an infected neighboring node. Meloni model can be applied to study the propagation of computer virus in the Internet.
Meloni model has received increasing attention in recent years. It has been found that the routing strategy plays an important role in Meloni model. Epidemic spreading can be effectively controlled by a local routing strategy [@yang1] or an efficient routing protocol [@yang3]. For a given routing strategy, epidemic spreading is affected by network structures. The increase of the average network connectivity can slow down the epidemic outbreak [@yang4]. Besides, the epidemic threshold can be enhanced by the targeted cutting of links among large-degree nodes or edges with the largest algorithmic betweenness [@yang5].
Previous studies have shown that immunization is an effective way to inhibit traditional epidemic spreading in which infections are transmitted as a reaction process from nodes to all neighbors [@immune1; @immune2]. In this paper, we will study how different immunization strategies affect the traffic-driven epidemic spreading in Meloni model. Three immunization strategies: the random immunization, the targeted immunization of nodes with the largest degree and the targeted immunization of nodes with the largest algorithmic betweenness are considered, respectively. It is found that, the targeted immunization of nodes with the largest algorithmic betweenness can most effectively inhibit the traffic-driven epidemic spreading.
Traffic-driven epidemic spreading model and immunization strategies {#sec:model}
===================================================================
Following the work of Meloni $et$ $al.$ [@Meloni], we incorporate the traffic dynamics into the classical susceptible-infected-susceptible model [@SIS] of epidemic spreading as follows. In a network of size $N$, at each time step, $\lambda N$ new packets are generated with randomly chosen sources and destinations (we call $\lambda$ as the packet-generation rate), and each node can deliver at most $C$ packets towards their destinations. Packets are forwarded according to a given routing algorithm. The queue length of each agent is assumed to be unlimited. The first-in-first-out principle applies to the queue. Each newly generated packet is placed at the end of the queue of its source node. Once a packet reaches its destination, it is removed from the system. After a transient time, the total number of delivered packets at each time will reach a steady value, then an initial fraction of nodes $\rho_{0}$ is set to be infected (we set $\rho_{0}=0.1$ in numerical experiments). The infection spreads in the network through packet exchanges. Each susceptible node has the probability $\beta$ of being infected every time it receives a packet from an infected neighbor. The infected nodes recover at rate $\mu$ (we set $\mu=1$ in this paper).
Once a node becomes immune, it cannot be infected and thus does not transmit the infection to their neighbors. We consider three immunization strategies respectively. (I) The random strategy (RS): we randomly set $n$ nodes to be immunized from the network. (II) The degree-based strategy (DS): we select $n$ nodes with the largest degree to be immunized. (III) The betweenness-based strategy (BS): we choose $n$ nodes with the largest algorithmic betweenness to be immunized.
Results and discussions {#sec:results}
=======================
\[0.45\][![The dependence of the algorithmic betweenness $b_k$ on degree $k$. []{data-label="fig1"}](Graph1.eps "fig:")]{}
In the following, we carry out simulations systematically by employing traffic-driven epidemic spreading on top of the Internet maps at the autonomous system level [@real], where the network size $N = 6474$, the average degree $\langle k \rangle = 3.88$, and the degree distribution follows a power law form $P(k) \sim k
^{-\gamma}$ with $\gamma \approx 2.2 $. Without special mention, we use the the shortest-path routing algorithm to deliver packets. Moreover, we assume that the node-delivering capacity $C$ is infinite, so that traffic congestion will not occur in the network.
The algorithmic betweenness of a node $k$ is defined as [@bc]: $$b_{k}=\frac{1}{N(N-1)}\sum_{i\neq
j}\frac{\sigma_{ij}(k)}{\sigma_{ij}},$$ where $\sigma_{ij}$ is the total number of possible paths going from $i$ to $j$ according to a specific routing algorithm, $\sigma_{ij}(k)$ is the number of such paths running through node $k$, and the sum runs over all pairs of nodes. The algorithmic betweenness of a node represents the average number of packets passing through that node at each time step when the packet-generation rate $\lambda=1/N$.
Figure \[fig1\] shows the dependence of the algorithmic betweenness $b_k$ on degree $k$. A general trend is that, $b_k$ increases with $k$, and the relationship between $b_k$ and $k$ approximatively follows a power-law form as $b_k \sim k^{\nu}$. We need to point out that, a larger-degree node may have lower algorithmic betweenness. For example, $b_k$ is $8.7\times10^{-3}$ for $k=73$, which is smaller than that for $k=58$, whose algorithmic betweenness is $2.0\times10^{-2}$ .
\[0.45\][![ (Color online) Density of infected nodes $\rho$ as a function of the spreading rate $\beta$ in RS, DS, and BS cases. The packet-generation rate $\lambda=0.3$ and the number of immune nodes $n=50$. Each curve is an average of 100 different realizations.[]{data-label="fig2"}](Graph2.eps "fig:")]{}
Figure \[fig2\] shows the density of infected nodes $\rho$ as a function of the spreading rate $\beta$ in RS, DS, and BS cases. One can observe that for each case, there exists an epidemic threshold $\beta_{c}$, beyond which the density of infected nodes is nonzero and increases as $\beta$ is increased. For $\beta<\beta_{c}$, the epidemic goes extinct and $\rho=0$.
Figure \[fig3\] shows the epidemic threshold $\beta_{c}$ as a function of the number of immune nodes $n$ in RS, DS, and BS cases. For RS case, the epidemic threshold $\beta_{c}$ is almost unchanged when the number of immune nodes is small, indicating that the random immunization is useless. For DS and BS cases, $\beta_{c}$ increases as $n$ increases, indicating that more immune nodes can better suppress the outbreak of epidemic. Moreover, from Fig. \[fig3\], one can find that for the same value of $n$, $\beta_{c}$ is the highest in BS case. For BS case, only ninety immune nodes (about 1.4% nodes) can completely prevent the outbreak of epidemic even the spreading rate is 1. The above phenomena manifest that, BS is the most effective immunization strategy in the traffic-driven epidemic spreading.
\[0.45\][![(Color online) The epidemic threshold $\beta_{c}$ as a function of the number of immune nodes $n$ in DS and BS cases. The packet-generation rate $\lambda=0.3$. Each data point results from an average over 100 different realizations.[]{data-label="fig3"}](Graph3.eps "fig:")]{}
\[0.45\][![(Color online) The density of infected nodes $\rho$ as a function of the number of immune nodes $n$ in DS and BS cases. The packet-generation rate $\lambda=0.3$ and the spreading rate $\beta=0.5$. Each curve results from an average over 100 different realizations.[]{data-label="fig4"}](Graph4.eps "fig:")]{}
\[0.6\][![(Color online) (a) The critical number of immune nodes $n_{c}$ as a function of the spreading rate $\beta$ in DS and BS cases. The packet-generation rate $\lambda=0.3$. (b) The critical number of immune nodes $n_{c}$ as a function of the packet-generation rate $\lambda$ in DS and BS cases. The spreading rate $\beta=0.3$. Each data point results from an average over 100 different realizations. []{data-label="fig5"}](Graph5.eps "fig:")]{}
Figure \[fig4\] shows the density of infected nodes $\rho$ as a function of the number of immune nodes $n$ in DS and BS cases. One can observe that there exists a critical number of immune nodes $n_{c}$, above which the epidemic goes extinct and $\rho=0$. Undoubtedly, the smaller $n_{c}$ reduces the cost of immunization. Figure \[fig5\](a) shows that $n_{c}$ as a function of the spreading rate $\beta$ in DS and BS cases. One can see that $n_{c}$ increases with $\beta$. Besides, for the same value of $\beta$, $n_{c}$ is smaller in the case of BS than that of DS. Figure \[fig5\](b) shows that $n_{c}$ as a function of the packet-generation rate $\lambda$ in DS and BS cases. One can see that $n_{c}$ increases with $\lambda$, manifesting that the increase of traffic flow is unfavorable for the control of epidemic spreading. For the same value of $n$, $n_{c}$ is smaller in the case of BS than that of DS. From Fig. \[fig5\], one can find that, compared to the degree-based immunization strategy, the betweenness-based immunization strategy is more effective in suppressing the traffic-driven epidemic spreading.
\[0.45\][![(Color online) The epidemic threshold $\beta_{c}$ as a function of the number of immune nodes $n$ in DS and BS cases when the efficient routing algorithm is applied. The packet-generation rate $\lambda=0.3$. Each data point results from an average over 100 different realizations.[]{data-label="fig6"}](Graph6.eps "fig:")]{}
In all the above studies, packets are forwarded following the shortest-path routing algorithm. In fact, our finding that betweenness-based immunization strategy can most effectively prevent the outbreak of traffic-driven epidemic is robust with respect to different kinds of routing algorithms. Figure \[fig6\] shows the epidemic threshold $\beta_{c}$ as a function of the number of immune nodes $n$ in RS, DS, and BS cases when the efficient routing algorithm is applied. The efficient routing algorithm is described as follows [@ef]. For any path between nodes $i$ and $j$, $P(i
\rightarrow j) : = i \equiv x_{1}, x_{2}, \cdots , x_{n}\equiv j$, we define $$\label{Eq2}
L\left(P(i \rightarrow
j):\alpha\right)=\sum_{l=1}^{n}k(x_{l})^{\alpha},$$ where $k(x_{l})$ is the degree of node $x_{l}$ and $\alpha$ is a tunable parameter. For any given $\alpha$, the efficient path between $i$ and $j$ is corresponding to the route that makes the sum $L\left(P(i \rightarrow j):\alpha\right)$ minimum. In this paper, we set the routing parameter $\alpha=1$. From Fig. \[fig6\], one can find that for the same number of immune nodes $n$, $\beta_{c}$ is the largest in BS case while $\beta_{c}$ keeps almost unchanged in RS case.
Conclusion {#sec:conclusion}
==========
In conclusion, we have studied the effects of immunization strategies on traffic-driven epidemic spreading. Our results show that, the outbreak of traffic-driven epidemic can be effectively suppressed when a small fraction of nodes with the largest algorithmic betweenness are immunized. This finding is robust with respect to different kinds of routing algorithms including the shortest-path routing algorithm and the efficient routing algorithm. For traditional epidemic spreading where infections are transmitted as a reaction process from nodes to all neighbors, an effective immunization strategy is to vaccinate the largest-degree nodes [@immune1]. However, compared to the betweenness-based immunization strategy, the degree-based immunization strategy is less efficient in the suppression of traffic-driven epidemic spreading. This is because the larger-degree nodes may not have higher algorithmic betweenness. Moreover, we find that more immune nodes are needed to prevent the outbreak of epidemic as the spreading rate and the packet-generation rate are increased. We hope our results can be useful to control traffic-driven epidemic spreading.
This work was supported by the National Science Foundation of China (Grant Nos. 61403083, 11275186, 91024026 and 71301028), and the Natural Science Foundation of Fujian Province, China (Grant No. 2013J05007).
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{
"pile_set_name": "ArXiv"
}
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---
author:
- Chunlan Jiang
title: '**Reduction to dimension two of local spectrum for $AH$ algebra with ideal property**'
---
\
Abstract: A $C^{*}$-algebra $A$ has ideal property if any ideal $I$ of $A$ is generated as a closed two sided ideal by the projections inside the ideal. Suppose that the limit $C^{*}$-algebra $A$ of inductive limit of direct sums of matrix algebras over spaces with uniformly bounded dimension has ideal property. In this paper we will prove that $A$ can be written as an inductive limit of certain very special subhomogeneous algebras, namely, direct sum of dimension drop interval algebras and matrix algebras over 2-dimensional spaces with torsion $H^{2}$ groups.
**§1. Introduction**
An $AH$ algebra is a nuclear $C^{*}$-algebra of the form $A=\lim\limits_{\longrightarrow}(A_{n}, \phi_{n,m})$ with $A_{n}=\bigoplus_{i=1}^{t_{n}}P_{n,i}M_{[n,i]}(C(X_{n,i}))P_{n,i}$, where $X_{n,i}$ are compact metric spaces, $t_{n}$, $[n,i]$ are positive integers, $M_{[n,i]}(C(X_{n,i}))$ are algebras of $[n,i]\times[n,i]$ matrices with entries in $C(X_{n,i})$—the algebra of complex - valued functions on $X_{n,i}$, and finally $P_{n,i}\in M_{[n,i]}(C(X_{n,i}))$ are projections (see \[Bla\]). If we further assume that $\sup\limits_{n,i}\dim(X_{n,i})<+\infty$ and $A$ has ideal property—each ideal $I$ of $A$ is generated by the projections inside the ideal, then it is proved in \[GJLP1-2\] that $A$ can be written as inductive limit of $B_{n}=\bigoplus_{i=1}^{s_{n}}P^{'}_{n,i}M_{[n,i]^{'}}(C(Y_{n,i}))P^{'}_{n,i}$. In this paper, we will further reduce the dimension of local spectra (that is the spectra of $A_{n}$ or $B_{n}$ above) to 2 (instead of 3). Namely, the above $A$ can be written as inductive limit of direct sum of matrix algebras over the $\{pt\}, [0,1], S^{1}, T_{\uppercase\expandafter{\romannumeral2}, k}$ (no $T_{\uppercase\expandafter{\romannumeral3}, k}$ and $S^{2}$) and $M_{l}(I_{k})$, where $I_{k}$ is dimension drop interval algebra, $$I_{k}=\{f\in C([0,1],M_{k}(\mathbb{C})), f(0)=\lambda\textbf{1}_{k}, f(1)=\mu\textbf{1}_{k}, \lambda,\mu\in\mathbb{C}\}.$$ In this paper, we will also call $\bigoplus^{s}_{i=1}M_{l_{i}}(I_{k_{i}})$ a dimension drop-algebra.\
This result unifies the theorems of \[DG\] and \[EGS\] (for the rank zero case) and \[Li4\] (for the simple case). Note that Li$^{'}$s reduction theorem was not used in the classification of simple $AH$ algebra, and Li$^{'}$s proof depends on the classification of simple $AH$ algebra (see \[Li4\] and \[EGL\]). For our case, the reduction theorem is an important step toward the classification (see \[GJL\]). The proof is more difficult than Li$^{'}$s case. For example, in the case of $AH$ algebra with ideal property, one can not remove the space $S^{2}$ without introduce $M_{l}(I_{k})$ (for the simple case, the space $S^{2}$ is removed from the list of spaces in \[EGL\] without introduced dimension drop algebras). Another point is that, in the simple $AH$ algebras, one can assume each partial map $\phi^{i,j}_{n,m}$ is injective, but in $AH$ algebras with ideal property, we can not make such assumption. For the classification of real rank zero $AH$ algebras, we refer to the readers \[Ell1\], \[EG1-2\], \[D1-2\], \[G1-4\] and \[DG\]. For the classification simple $AH$ algebra, we refer to the readers \[Ell2-3\], \[Li1-3\], \[G5\] and \[EGL1-2\].
The paper is organized as follows.\
In section 2, we will do some necessary preparation. In section 3, we will prove our main theorem.
**§2. Preparation**
We will adopt all the notation from section 2 of \[GJLP2\]. For example we refer the reader to \[GJLP2\] for the concepts of $G-\delta$ multiplicative maps (see definition 2.2 there), spectral variation $SPV(\phi)$ of a homomorphism $\phi$ (see 2.12 there) weak variation $\omega(F)$ of a finite set $F\subset QM_{N}(C(X))Q$ (see 2.16 there).\
As in 2.17 of \[GJLP2\], we shall use $\bullet$ to denote any possible integer.
**2.1.** In this article, without lose of generality we will assume the $AH$ algebras $A$ are inductive limit of $$A=\lim(A_{n}=\bigoplus_{i=1}^{t_{n}}M_{[n,i]}(C(X_{n,i})), \phi_{n,m}),$$ where $X_{n,i}$ are the spaces of $\{pt\}, [0,1], S^{1}, T_{II, k}, T_{III, k}$, and $S^{2}$. (Note by the main theorem of \[GJLP2\], all $AH$ algebras with ideal property and with no dimension growth are corner subalgebras of the above form (also see 2.7 of \[GJLP2\]).)
**2.2.** Recall a projection $P\in M_{k}(C(X))$ is called a trivial projection if it is unitarily\
equivalent to $\begin{pmatrix}\textbf{1}_{k_{1}}& 0\\0 & 0\end{pmatrix}$ for $k_{1}=rank(P)$. If $P$ is a trival projection and $rank(P)=k_{1}$, then $$PM_{k}(C(X))P\cong M_{k_{1}}(C(X)).$$
**2.3.** Let $X$ be a connected finite simplicial complex, $A=M_{k}(C(X))$, a unital $\ast$ homomorphism $\phi: A\longrightarrow M_{l}(A)$ is called a (unital) simple embedding if it is homotopic to the homomorphism $id\oplus\lambda$, when $\lambda: A\longrightarrow M_{l-1}(A)$ is defined by $$\lambda(f)=diag(\underbrace{f(x_{0}),f(x_{0}),\cdots,f(x_{0})}\limits_{l-1}),$$ for a fixed base point $x_{0}\in X$.
The following two lemmas are special cases of Lemma 2.15 of \[EGS\] (also see 2.12 of \[EGS\]).
**Lemma 2.4.** (c.f 2.12 or case 2 of 2.15 in \[EGS\]). For any finite set $F\subset A=M_{n}(C(T_{\uppercase\expandafter{\romannumeral3}, k}))$ and $\varepsilon>0$, there is a unital simple embedding $\phi: A\longrightarrow M_{l}(A)$ (for $l$ large enough) and a $C^{*}$-algebra $B\subset A$, which is a direct sum of dimension drop algebras and a finite dimensional $C^{*}$-algebra such that $$dist(\phi(f),B)<\varepsilon,~~~~~~\forall\;f\in F.$$
**Lemma 2.5.**(see case 1 of 2.15 in \[EGS\]) For any finite set $F\subset M_{n}(C(S^{2}))$ and $\varepsilon>0$, there is a unital simple embedding $\phi:A\longrightarrow M_{l}(A)$ (for $l$ large enough) and a $C^{*}$-algebra $B\subset A$, which is a finite dimensional $C^{*}$-algebra such that $$dist(\phi(f),B)<\varepsilon,~~~~~~\forall\;f\in F.$$
The following lemma is well known:
**Lemma 2.6.**(see \[G5,4.40\]) For any $C^{*}$-algebra $A$ and finite set $F\subset A$, $\varepsilon>0$, there is a finite set $G\subset A$ and $\eta>0$ such that if $\phi: A\longrightarrow B$ is a homomorphism and $\psi: A\longrightarrow B$ is a completely positive linear map, satisfing the following $$\|\phi(g)-\psi(g)\|<\eta,~~~~~\forall\;g\in G,$$ then $\psi$ is the $F-\varepsilon$ multiplicative.\
**Lemma 2.7.** Let $A=M_{n}(C(T_{\uppercase\expandafter{\romannumeral3}, k}))$ or $M_{n}(C(S^{2}))$. And let a finite set $F\subset A$ and $\varepsilon>0$, there is a diagram $$\xymatrix{
A\ar[rr]^{\phi}\ar[rdrd]^{\beta} & & M_{l}(A) & & \\
& & & & & & \\
& & B\ar[uu]_{\iota} & & \\
}$$ with the following conditions.\
(1) $\phi$ is a simple embedding,\
(2) if $A=M_{n}(C(S^{2}))$, then $B$ is a finite dimensional $C^{*}$-algebra, and if $A=M_{n}(C(T_{\uppercase\expandafter{\romannumeral3}, k}))$, then $B$ is a direct sum of dimension drop $C^{*}$-algebras and a finite dimensional $C^{*}$-algebra, and $\iota$ is an inclusion,\
(3) $\|\iota\circ\beta(f)-\phi(f)\|<\varepsilon$, $\forall f\in F$, and $\beta$ is $F-\varepsilon$ multiplicative.
Let $G$ and $\eta$ be as Lemma 2.6 for $F$ and $\varepsilon$. Apply Lemma 2.4 or Lemma 2.5 to $A$, $F\cup G\subset A$ and $\frac{1}{3}\min(\varepsilon,\eta)$. One can find a unital simple embedding $\phi: A\longrightarrow M_{l}(A)$, and an sub-$C^{*}$-algebra $B\subset M_{l}(A)$ as required in condition (2) such that $$dist(\phi(f),B)<\frac{1}{3}\min(\varepsilon,\eta),~~~~~~\forall\;f\in F.$$ Choose a finite $\widetilde{F}\subset B$ such that $$dist(\phi(f),\widetilde{F})<\frac{1}{3}\min(\varepsilon,\eta),~~~~~~\forall\;f\in F.$$ Since $B$ is a nuclear $C^{*}$-algebra, there are two completely positive linear maps $$\lambda_{1}: B\longrightarrow M_{N}(\mathbb{C})~~~\mbox{and}~~~\lambda_{2}: M_{N}(\mathbb{C})\longrightarrow B,$$ such that $$\|\lambda_{2}\circ\lambda_{1}(g)-g\|<\frac{1}{3}\min(\varepsilon,\eta),~~\forall g\in\widetilde{F}$$ Using Arveson extension theorem, one can extend $\lambda_{1}: B\longrightarrow M_{N}(\mathbb{C})$ to a map\
$\beta_{1}: M_{l}(A)\longrightarrow M_{N}(\mathbb{C})$. Then it is a straight forward to prove that $$\beta=\lambda_{2}\circ\beta_{1}\circ\phi: A\longrightarrow B,$$ is as desired.
The following is modification of Theorem 3.8 of \[GJLP2\].
**Proposition 2.8.** Let $\lim\limits_{n\rightarrow\infty}(A_{n}=\bigoplus_{i=1}^{t_{n}}M_{[n,i]}(C(X_{n,i})), \phi_{n,m})$ be $AH$ inductive limit with ideal property, with $X_{n,i}$ being one of $\{pt\}, [0,1], S^{1}, T_{II, k}, T_{III, k}$, or $S^{2}$. Let $B=\bigoplus_{i=1}^{s}B^{i}$, where $B^{i}=M_{l_{i}}(C(Y_{i}))$, with $Y_{i}$ being $\{pt\}, [0,1], S^{1}$, or $T_{II, k}$, (no $T_{\uppercase\expandafter{\romannumeral3}, k}$ or $S^{2}$) or $B^{i}=M_{l_{i}}(I_{k_{i}})$—dimension drop $C^{*}$-algebras. Suppose that $$\widetilde{G}(=\oplus\widetilde{G}^{i})\subset G(=\oplus G^{i})\subset B(=\oplus B^{i}),$$ is a finite set, $\varepsilon_{1}$ is a positive number with $\omega(\widetilde{G}^{i})<\varepsilon_{1}$, if $Y_{i}=T_{\uppercase\expandafter{\romannumeral2}, k}$, and $L$ is any positive integer. Let $\alpha: B\longrightarrow A_{n}$ be any homomorphism. Denote $$\alpha(\textbf{1}_{B}):= R(=\oplus R^{i})\in A_{n}(=\oplus A^{i}_{n}).$$ Let $F\subset RA_{n}R$ be any finite set and $\varepsilon<\varepsilon_{1}$ be any positive number. It follows that there are $A_{m}$, and mutually orthogonal projections $Q_{0},Q_{1},Q_{2}\in A_{m}$ with $$\phi_{n,m}(R)=Q_{0}+Q_{1}+Q_{2},$$ a unital map $\theta_{0}\in Map(RA_{n}R,Q_{0}A_{m}Q_{0})_{1}$, two unital homomorphisms $\theta_{1}\in Hom(RA_{n}R,Q_{1}A_{m}Q_{1})_{1}$, $\xi\in Hom(RA_{n}R,Q_{2}A_{m}Q_{2})_{1}$ such that\
(1) $\parallel \phi_{n,m}(f)-(\theta_{0}(f)\oplus\theta_{1}(f)\oplus\xi(f))\parallel<\varepsilon$, for all $f\in F$,\
(2) there is a unital homomorphism $$\alpha_{1}: B\longrightarrow(Q_{0}+Q_{1})A_{m}(Q_{0}+Q_{1}),$$ such that $$\parallel \alpha_{1}(g)-(\theta_{0}+ \theta_{1})\circ\alpha(g)\parallel<3\varepsilon_{1}~~~~\forall g\in \widetilde{G}_{i},~~~~~~~\mbox{if}~~ B^{i} ~~\mbox{is of form }~~M_{\bullet}(T_{\uppercase\expandafter{\romannumeral2}, k})$$ and $$\parallel \alpha_{1}(g)-(\theta_{0}+\theta_{1})\circ\alpha(g)\parallel<\varepsilon,~~~~\forall g\in G^{i},~~~~~~~\mbox{if}~~ B^{i} ~~\mbox{is not of form }~~M_{\bullet}(T_{\uppercase\expandafter{\romannumeral2}, k}).$$ (3) $\theta_{0}$ is $F-\varepsilon$ multiplicative and $\theta_{1}$ satisfies that $$\theta^{i,j}_{1}([e])\geqslant L\cdot[\theta^{i,j}_{0}(R^{i})].$$ (4) $\xi$ factors through a $C^{\ast}$-algebra C—a direct sum of matrix algebras over C\[0,1\] as $$\xi:RA_{n}R\xrightarrow{\xi_{1}}C\xrightarrow{\xi_{2}}Q_{2}A_{m}Q_{2}.$$ **Proposition 2.9.** Let $\lim\limits_{n\rightarrow\infty}(A_{n}=\bigoplus_{i=1}^{t_{n}}M_{[n,i]}(C(X_{n,i})), \phi_{n,m})$ be $AH$ inductive limit with ideal property, with $X_{n,i}$ being one of $\{pt\}, [0,1], S^{1}, T_{\uppercase\expandafter{\romannumeral2}, k}, T_{\uppercase\expandafter{\romannumeral3}, k}$ or $S^{2}$. Let $B=\bigoplus_{i=1}^{s}B^{i}$, where $B^{i}=M_{l_{i}}(C(Y_{i}))$, with $Y_{i}$ being $\{pt\}, [0,1], S^{1}$ or $T_{\uppercase\expandafter{\romannumeral2}, k}$, (no $T_{\uppercase\expandafter{\romannumeral3}, k}$ or $S^{2}$) or $B^{i}=M_{l_{i}}(I_{k_{i}})$—dimension drop $C^{*}$-algebras. Suppose that $$\widetilde{G}(=\oplus\widetilde{G}^{i})\subset G(=\oplus G^{i})\subset B(=\oplus B^{i}),$$ is a finite set, $\varepsilon_{1}$ is a positive number with $\omega(\widetilde{G}^{i})<\varepsilon_{1}$, if $Y_{i}=T_{\uppercase\expandafter{\romannumeral2}, k}$, and $L>0$ is any positive integer. Let $\alpha: B\longrightarrow A_{n}$ be any homomorphism. Let $F\subset A_{n}$ be any finite set and $\varepsilon<\varepsilon_{1}$ be any positive number. It follows that there are $A_{m}$ and mutually orthogonal projections $P,Q\in A_{m}$ with $\phi_{n,m}(\textbf{1}_{A_{n}})=P+Q$, a unital map $\theta\in Map(A_{n},PA_{m}P)_{1}$, and a unital homomorphism $\xi\in Hom(A_{n},QA_{m}Q)_{1}$ such that\
(1) $\parallel \phi_{n,m}(f)-(\theta(f)\oplus\xi(f))\parallel<\varepsilon$, for all $f\in F$,\
(2) there is a homomorphism $\alpha_{1}: B\longrightarrow PA_{m}P$ such that
$\parallel \alpha^{i,j}_{1}(g)-(\theta\circ\alpha)^{i,j}(g)\parallel<3\varepsilon_{1}$ $\forall g\in \widetilde{G}^{i}$, if $B^{i}$ is of form $M_{\bullet}(C(T_{\uppercase\expandafter{\romannumeral2}, k}))$,
$\parallel \alpha^{i,j}_{1}(g)-(\theta\circ\alpha)^{i,j}(g)\parallel<\varepsilon$ $\forall g\in G^{i}$, if $B^{i}$ is not of form $M_{\bullet}(C(T_{\uppercase\expandafter{\romannumeral2}, k})).$
\(3) $\omega(\theta(F))<\varepsilon$ and $\theta$ is $F-\varepsilon$ multiplicative.\
(4) $\xi$ factors through a $C^{\ast}$-algebra $C$—a direct sum of matrix algebras over $C[0,1]$ or $\mathbb{C}$ as $$\xi: A_{n}\xrightarrow{\xi_{1}}C\xrightarrow{\xi_{2}}QA_{m}Q.$$\
The proof is similar to Proposition 2.8 and is omitted.
**2.10.** Let $\alpha:\mathbb{Z}\longrightarrow\mathbb{Z}/k_{1}\mathbb{Z}$ be the group homomorphism defined by $\alpha(\textbf{1})=[\textbf{1}]$, where the right hand side is the equivalent class $[\textbf{1}]$ of $\textbf{1}$ in $\mathbb{Z}/k_{1}\mathbb{Z}$. Then it is well known from homological algebra that for the group $\mathbb{Z}/k\mathbb{Z}$, $\alpha$ induces a surjective map $$\alpha_{\ast}: Ext(\mathbb{Z}/k\mathbb{Z},\mathbb{Z})(=\mathbb{Z}/k\mathbb{Z})\longrightarrow Ext(\mathbb{Z}/k\mathbb{Z},\mathbb{Z}/k_{1}\mathbb{Z})(=\mathbb{Z}/(k,k_{1})\mathbb{Z}),$$ where $(k,k_{1})$ is the greatest common factor of $k$ and $k_{1}$.\
Recall, as in \[DN\], for two connected finite simplicial complexes $X$ and $Y$, we use $kk(Y,X)$ to denote the group of equivalent classes of homomorphisms from $C_{0}(X\backslash\{pt\})$ to $C_{0}(Y\backslash\{pt\})\otimes {\mathcal K}(H)$. Please see \[DN\] for details.
**Lemma 2.11.** (a) Any unital homomorphism $$\phi: C(T_{\uppercase\expandafter{\romannumeral2}, k})\longrightarrow M_{\bullet}(C(T_{III,k_{1}})),$$ is homotopy equivalent to unital homomorphism $\psi$ factor as $$C(T_{\uppercase\expandafter{\romannumeral2}, k})\xrightarrow{\psi_{1}}C(S^{1})\xrightarrow{\psi_{2}}M_{\bullet}(C(T_{III,k_{1}})).$$ (b) Any unital homomorphism $\phi: C(T_{\uppercase\expandafter{\romannumeral2}, k})\longrightarrow PM_{\bullet}(C(S^{2}))P$ is homotopy equivalent to unital homomorphism $\psi$ factor as $$C(T_{\uppercase\expandafter{\romannumeral2}, k})\xrightarrow{\psi_{1}}\mathbb{C}\xrightarrow{\psi_{2}}PM_{\bullet}(C(S^{2}))P.$$
Part (b) is well known (see chapter 3 of \[EG2\]). To prove part (a), we note that $$KK(C_{0}(S^{1}\backslash \{1\}),C_{0}(T_{III,k_{1}}\backslash \{x_{1}\})=kk(T_{III,k_{1}},S^{1})=\mathbb{Z}/k_{1}\mathbb{Z}=Hom(\mathbb{Z},\mathbb{Z}/k_{1}\mathbb{Z})),$$ where $x_{1}\in T_{III,k_{1}}$ is a base point. The map $\alpha: \mathbb{Z}\longrightarrow\mathbb{Z}/k_{1}\mathbb{Z}$ in 2.10 can be induced by a homomorphism: $\psi_{2}: C(S^{1})\longrightarrow M_{\bullet}(C(T_{\uppercase\expandafter{\romannumeral3}, k}))$.\
Let $$[\phi]\in kk(T_{III,k_{1}}\; T_{\uppercase\expandafter{\romannumeral2}, k})=Ext(K_{0}(C_{0}(T_{\uppercase\expandafter{\romannumeral2}, k}\backslash\{x_{0}\})), K_{1}(C_{0}(T_{\uppercase\expandafter{\romannumeral3}, k}))),$$ be the element induced by homomorphism $\phi$, where $\{x_{0}\}$ is the base point. By 2.10 $$[\phi]=\beta\times[\psi_{2}],\;for\;\beta\in kk(S^{1},T_{\uppercase\expandafter{\romannumeral2}, k})=Ext(K_{0}(C(T_{\uppercase\expandafter{\romannumeral2}, k}\backslash\{x_{0}\})),K_{1}(C(S^{1}))),$$ on the other hand $\beta$ can be realized by unital homomorphism $$\psi_{1}: C(T_{\uppercase\expandafter{\romannumeral2}, k})\longrightarrow C(S^{1}).$$ (see section 3 of \[EG2\]).\
The following result is modification of Theorem 3.12 of \[GJLP2\].
**Theorem 2.12.** Let $B_{1}=\bigoplus_{i=1}^{s}B_{1}^{i}$, each $B^{i}$ is either matrix algebras over $\{pt\}, [0,1], S^{1}$ or $\{T_{\uppercase\expandafter{\romannumeral2}, k}\}^{\infty}_{k=2}$ or a dimension drop algebra. Let $\varepsilon_{1}>0$ and let $$\widetilde{G}_{1}(=\oplus\widetilde{G}^{i}_{1})\subset G_{1}(=\oplus G^{i}_{1})\subset B_{1}(=\oplus B^{i}_{1}),$$ be a finite set with $\omega(G^{i}_{1})<\varepsilon_{1}$ for $B^{i}_{1}=M_{\bullet}(C(T_{\uppercase\expandafter{\romannumeral2}, k}))$.
Let $A=M_{N}(C(X))$, where $X$ is one of form $\{pt\}, [0,1], S^{1}, \{T_{\uppercase\expandafter{\romannumeral2}, k}\}^{\infty}_{k=2}, \{T_{\uppercase\expandafter{\romannumeral3}, k}\}^{\infty}_{k=2}$ or $S^{2}$. Let $\alpha_{1}: B_{1}\longrightarrow A$ be a homomorphism. Let $F_{1}\subset A$ be a finite set, and let $\varepsilon(<\varepsilon_{1})$ and $\delta$ be any positive number. Then there exists a diagram. $$\xymatrix{
A\ar[rdrd]^{\beta}\ar[rr]^{\phi} & & A^{'} \\
& & & & & & \\
B_{1}\ar[uu]_{\alpha_{1}}\ar[rr]^{\psi} & & B_{2}\ar[uu]_{\alpha_{2}} \\
}$$ where $A^{'}=M_{K}(A)$, $B_{2}$ is described as below. $B_{2}$ is a direct sum of finite dimensional $C^{*}$-algebra and dimension drop algebra if $X=T_{\uppercase\expandafter{\romannumeral3}, k}$. $B_{2}$ is a finite dimensional algebra if $X=S^{2}$. And $B_{2}=M_{\bullet}(A)$ if $X=\{pt\}, [0,1], S^{1}, T_{\uppercase\expandafter{\romannumeral2}, k}$. Furthermore the diagram satisfies the following conditions\
(1) $\psi$ is a homomorphism, $\alpha_{2}$ is a unital injective homomorphism and $\phi$ is a unital simple embedding;\
(2) $\beta\in Map(A,B_{2})_{1}$ is $F_{1}-\delta$ multiplicative;\
(3) if $B^{i}_{1}$ is of form $M_{\bullet}(C(T_{\uppercase\expandafter{\romannumeral2}, k}))$, then $$\parallel\psi(g)-\beta\circ\alpha_{1}(g)\parallel<10\varepsilon_{1},\; ~~\forall g\in\widetilde{G}^{i}_{1};$$ and if $B^{i}_{1}$ is not of form $M_{\bullet}(C(T_{\uppercase\expandafter{\romannumeral2}, k}))$, then $$\parallel \psi(g)-\beta\circ\alpha_1(g)\parallel<\varepsilon,\; ~~\forall g\in G^{i}_{1};$$ (4) If $X=T_{\uppercase\expandafter{\romannumeral2}, k}$, then $\omega(\beta(F_{1})\cup\psi(G_{1}))<\varepsilon.$\
(Note that, we only required that the weak variation of finite set in $M_{\bullet}(C(T_{\uppercase\expandafter{\romannumeral2}, k})$ to be small. In particular, we do not need to introduce the concept of weak variation for a finite subset of a dimension drop algebra.)
. For $X=T_{\uppercase\expandafter{\romannumeral2}, k},\{pt\}, [0,1]$ or $S^{1}$, one can choose $B_{2}=M_{K}(A)=A^{'}$ and homomorphism $\phi=\beta: A\longrightarrow B_{2}$ being simple embedding such that $$\omega(\beta(F_{1})\cup\alpha_{1}(G_{1})))<\varepsilon.$$ This can be done by choosing $K$ large enough. Choose $\psi=\beta\circ\alpha_{1}$, and $\alpha_{2}=id: B_{2}\longrightarrow A^{'}$.\
For the case $X=T_{\uppercase\expandafter{\romannumeral3}, k}$, or $S^{2}$, the requirement (4) is an empty requirement.
We will deal with each block of $B_{1}$ separately. For the block $B^{i}_{1}$ other than $M_{\bullet}(C(T_{\uppercase\expandafter{\romannumeral2}, k})$, the construction can be done easily by using Lemma 2.7, since $B^{i}_{1}$ is stably generated, which implies that any sufficiently multiplicative map from $B^{i}_{1}$ is close to a homomorphism. So we assume that $B^{i}_{1}=M_{\bullet}(C(T_{\uppercase\expandafter{\romannumeral2}, k})$. Recall we already assume $A$ is of form $M_{\bullet}(C(T_{\uppercase\expandafter{\romannumeral3}, k}))$ or $M_{\bullet}(C(S^{2})$. By Lemma 2.11, the homomorphism $\alpha_{1}: B^{i}_{1}\longrightarrow A$ is a homotopy to $\alpha^{'}: B^{i}_{1}\longrightarrow A$ with $\alpha^{'}(\textbf{1}_{B^{i}_{1}})=\alpha_{1}(\textbf{1}_{B^{i}_{1}})$ and $\alpha^{'}$ factor as $$B_{1}^i\xrightarrow{\xi_{1}}C\xrightarrow{\xi_{2}}A,$$ where $C$ is a finite dimensional $C^{*}$-algebra for the case $X=S^{2}$ or $C=M_{\bullet}(C(S^1)$ for the case $X=T_{\uppercase\expandafter{\romannumeral3}, k}$ (note that $B^{i}_{1}=M_{\bullet}(C(T_{\uppercase\expandafter{\romannumeral2}, k})$). Since $C$ is stably generated, there is a finite set $E_{1}\subset A$ and $\delta_{1}>0$ such that if a complete positive map $\beta: A\longrightarrow D$ (for any $C^{*}$-algebra $D$) is $E_{1}-\delta_{1}$ multiplicative, then the map $\beta\circ\xi_{2}: C\longrightarrow D$ can be perturbed to a homomorphism $$\widetilde{\xi}: C\longrightarrow D$$ such that $$\|\widetilde{\xi}(g)-\beta\circ\xi_{2}(g)\|<\varepsilon_{1},\;\forall\;g\in\xi_{1}(\widetilde{G}^{i}_{1}).$$ Apply Theorem 1.6.9 of \[G5\] to two homotopic homomorphism $$\alpha_{1},\alpha^{'}: B^{i}_{1}\longrightarrow A,$$ and $G^{i}_{1}\subset B^{i}_{1}$ which is approximately constant to within $\varepsilon_{1}$, to obtain a finite set $E_{2}\subset A$, $\delta_{2}>0$ and positive integer $L^{'}>0$ (in places of $G,\delta$ and $L$ in Theorem 1.6.9 of \[G5\]). Apply Lemma 2.7 to the set $\widetilde{E}=E_{1}\cup E_{2}\cup F_{1}$ and $\widetilde{\delta}=\frac{1}{3}\min(\varepsilon,\delta,\delta_{1},\delta_{2})$ to obtain the diagram $$\xymatrix{
A\ar[rr]^{\phi^{'}}\ar[rdrd]^{\beta^{'}} & & M_{L_{1}}(A) & & \\
& & & & & & \\
& & B^{'}\ar[uu]_{\iota} & & \\
}$$ with $\beta^{'}$ being $\widetilde{E}-\widetilde{\delta}$ multiplicative and $$\parallel\iota\circ\beta^{'}(f)-\phi^{'}(f)\parallel<\widetilde{\delta},\;\forall\;f\in\widetilde{E}.$$ Let $L=L^{'}\cdot rank(\textbf{1}_{A})$, and let $\beta_{1}: A\longrightarrow M_{L}(B^{'})$ be any unital homomorphism defined by point evaluation. Then by Theorem 1.6.9 of \[G5\], there is a unitary $u\in M_{L+1}(B)$ such that $$\|u((\beta^{'}\oplus\beta_{1})\circ\alpha^{'}(f))u^{\ast}-(\beta^{'}\oplus\beta_{1})\circ\alpha_{1}(f)\|<8\varepsilon_{1},\;\forall\;f\in\widetilde{G}^{i}_{1}.$$ By the choice of $E_{1}$, there is a homomorphism $$\widetilde{\xi}: C\longrightarrow M_{L+1}(B^{'}),$$ such that $$\|\widetilde{\xi}(f)-u((\beta^{'}\oplus\beta)\circ\xi_{2}(f))u^{\ast}\|<\varepsilon_{1},\;\forall\;f\in\xi_{1}(\widetilde{G}^{i}_{1}).$$ Define $B_{2}=M_{L+1}(B^{'}), K=L_{1}(L+1), A^{'}=M_{K}(A)=M_{L+1}(M_{L_{1}}(A))$,
$\psi: B^{i}_{1}\longrightarrow B_{2}$ by $\psi=\widetilde{\xi}\circ\xi_{1}: B^{i}_{1}\xrightarrow{\xi_{1}}C\xrightarrow{\widetilde{\xi}}B_{2}$,
$\beta: A\longrightarrow M_{L+1}(B^{'})$ by $\beta=\beta^{'}\oplus\beta_{1}$,
and
$\phi: A\longrightarrow M_{L+1}(M_{L_{1}}(A))$ by $\phi=\phi^{'}\oplus((\iota\otimes id_{L})\circ\beta_{1})$,
(note that $\beta_{1}$ is a homomorphism) to finish the proof.\
**2.13.** Recall for $A=\bigoplus^{t}_{i=1}M_{k_{i}}(C(X_{i}))$, where $X_{i}$ are path connected simplicial complexs, we use the notation $r(A)$ to denote $\bigoplus^{t}_{i=1}M_{k_{i}}(\mathbb{C})$ which could be considered to be the subalgebra consisting of all t-tuples of constant function from $X_{i}$ to $M_{k_{i}}(\mathbb{C})$ ($i=1,2,\cdots,t$). Fixed a base point $x^{0}_{i}\in X_{i}$ for each $X_{i}$, one defines a map $r: A\longrightarrow r(A)$ by $$r(f_{1},f_{2},\cdots,f_{t})=(f_{1}(x^{0}_{1}),f_{2}(x^{0}_{2}),\cdots,f_{t}(x^{0}_{t}))\in r(A).$$ We have the following Corollary
**Corollary 2.14.** Let $B_{1}=\oplus B^{j}_{1}$, where $B^{j}_{1}$ are either of form $M_{k(j)}(C(X_{j}))$, with $X_{j}$ being one of $\{pt\}$, \[0,1\], $S^{1}$, $\{T_{\uppercase\expandafter{\romannumeral2}, k}\}^{\infty}_{k=2}$ or one of form $M_{k(j)}(I_{l(j)})$. Let $\alpha_{1}; B_{1}\longrightarrow A$ be a homomorphism, where $A$ is a direct sum of matrix algebras over $\{pt\}, [0,1], S^{1}, \{T_{\uppercase\expandafter{\romannumeral2}, k}\}^{\infty}_{k=2}, \{T_{\uppercase\expandafter{\romannumeral3}, k}\}^{\infty}_{k=2}$ and $S^{2}$. Let $\varepsilon_{1}>0$ and let $\widetilde{E}(=\oplus\widetilde{E}^{i})\subset E(=\oplus E^{i})\subset B_{1}(=\oplus B^{i}_{1})$ be two finite subset with the condition
$\omega(\widetilde{E}^{i})<\varepsilon_{1}$, if $B^{i}_{1}=M_{\bullet}(C(Y_{i}))$ with $Y_{i}\in\{T_{\uppercase\expandafter{\romannumeral2}, k}\}^{\infty}_{k=2}$.
Let $F\subset A$ be any finite set, $\varepsilon_{2}>0$, $\delta>0$. Then there exists a diagram $$\xymatrix{
A\ar[rdrd]^{\beta\oplus r}\ar[rr]^{\phi\oplus r} & & A^{'}\oplus r(A) \\
& & & & & & \\
B_{1}\ar[uu]_{\alpha_{1}}\ar[rr]^{\psi\oplus(r\circ\alpha_{1})} & & B_{2}\oplus r(A)\ar[uu]_{\alpha_{2}\oplus id} \\
}$$ where $A^{'}=M_{L}(A)$, and $B_{2}$ is a direct sum of matrix algebras over space: $\{pt\}$, \[0,1\], $S^{1}$, $\{T_{\uppercase\expandafter{\romannumeral2}, k}\}^{\infty}_{k=2}$ and dimension drop algebras, with the following properties.\
(1) $\psi$ is a homomorphism, $\alpha_{2}$ is a injective homomorphism and $\phi$ is a unital simple embedding.\
(2) $\beta\in Map(A,B_{2})_{1}$ is $F_{1}-\delta$ multiplicative.\
(3) for $g\in\widetilde{E}^{i}$ with $B^{i}_{1}=M_{\bullet}(C(X_{i}))$, $X_{i}\in\{T_{\uppercase\expandafter{\romannumeral2}, k}\}^{\infty}_{k=2}$ we have $$\|(\beta\oplus r)(g)-(\psi\oplus(r\circ\alpha_{1}))(g)\|\leqslant10\varepsilon_{1};$$ and for $g\in E^{i}(\supset\widetilde{E}^{i})$ where $B^{i}_{1}$ is not of form $M_{\bullet}(C(T_{\uppercase\expandafter{\romannumeral2}, k}))$, we have $$\|(\beta\oplus r)(g)-(\psi\oplus(r\circ\alpha_{1}))(g)\|<\varepsilon_{1};$$ and for $f\in F$ $$\|(\alpha_{2}\oplus id)\circ(\beta\oplus r)(f)-(\phi\oplus r)(f)\|<\varepsilon_{1}.$$ (4) For $B^{i}_{2}$ of form $M_{\bullet}(C(T_{\uppercase\expandafter{\romannumeral2}, k}))$, $$\omega(\pi_{i}(\beta(F)\cup\psi(E)))<\varepsilon_{2},$$ where $\pi_{i}$ is the canonical projection from $B_{2}$ to $B^{i}_{2}$.\
[**Remark**]{}: In the application of the above Corollary, we will denote the map $\beta\oplus r$ by $\beta$ and $\psi\oplus(r\circ\alpha_{1})$ by $\psi$.
**§3. The proof of main theorem**
In this section, we prove the following main theorem.
**Theorem 3.1.** Suppose $\lim(A_{n}=\bigoplus_{i=1}^{t_{n}}M_{[n,i]}(C(X_{n,i})), \phi_{n,m})$ is an $AH$ inductive limit with $X_{n,i}$ being among the spaces $\{pt\}, [0,1], S^{1}, \{T_{\uppercase\expandafter{\romannumeral2}, k}\}^{\infty}_{k=2}$, $\{T_{\uppercase\expandafter{\romannumeral3}, k}\}^{\infty}_{k=2}$, such that the limit algebra $A$ has ideal property. Then there is another inductive system ($B_{n}=\oplus B^{i}_{n}, \psi_{n,m}$) with same limit algebra, where $B^{i}_{n}$ are either $M_{[n,i]^{'}}(C(Y_{n,i}))$ with $Y_{n,i}$ being one of $\{pt\}, [0,1], S^{1}, \{T_{\uppercase\expandafter{\romannumeral2}, k}\}^{\infty}_{k=2}$ (but without $T_{\uppercase\expandafter{\romannumeral3}, k}$ and $S^{2}$), or dimension drop algebra $M_{[n,i]^{'}}(I_{k(n,i)})$.
Let $\varepsilon_{1}>\varepsilon_{2}>\varepsilon_{3}>\cdots$ be a sequence of positive number with $\sum\varepsilon_{n}<+\infty$. We need to construct the intertwining diagram
$$\xymatrix@R=0.5ex{
F_{1} & & F_{2} & & & & F_{n} & & F_{n+1}\\
\bigcap & & \bigcap & & & & \bigcap & & \bigcap\\
A_{s(1)} \ar[rdrd]^{\beta_{1}} \ar[rr]^{\phi_{s(1),s(2)}} & & A_{s(2)} \ar[rdrd]^{\beta_{2}} \ar[rr]^{\phi_{s(2),s(3)}} & &\cdots \ar[rr] & & A_{s(n)} \ar[rdrd]^{\beta_{n}} \ar[rr]^{\phi_{s(n),s(n+1)}} & & A_{s(n+1)} \ar[rr] \ar[rdrd] & & \cdots\\
& & & & & &\\
B_{1} \ar[rr]^{\psi_{1,2}} \ar[uu]^{\alpha_{1}} & & B_{2} \ar[uu]^{\alpha_{2}} \ar[rr]^{\psi_{2,3}} & &\cdots \ar[rr] & & B_{n}\ar[rr]^{\psi_{n,n+1}} \ar[uu]^{\alpha_{n}} & & B_{n+1} \ar[uu]^{\alpha_{n+1}} \ar[rr] & & \cdots\\
\bigcup & & \bigcup & & & & \bigcup & & \bigcup\\
E_{1} & & E_{2} & & & & E_{n} & & E_{n+1}\\
\bigcup & & \bigcup & & & & \bigcup & & \bigcup\\
\widetilde{E_{1}} & & \widetilde{E_{2}} & & & & \widetilde{E_{n}} & & \widetilde{E_{n+1}}
}$$ satisfying the following conditions\
(0.1) $(A_{s(n)},\phi_{s(n),s(m)})$ is a sub-inductive system of $(A_{n}, \phi_{n,m}), (B_{n}, \psi_{n,m})$ is an inductive system of direct sum of matrix algebras over the spaces $\{pt\}, [0,1], S^{1}, {T_{\uppercase\expandafter{\romannumeral2},k}}$ and dimension drop algebra $M_{\bullet}(I_{k(n,i)})$.\
(0.2) Choose $\{a_{i,j}\}^{\infty}_{j=1}\subset A_{s(i)}$ and $\{b_{i,j}\}^{\infty}_{j=1}\subset B_{i}$ to be countable dense subsets of unit balls of $A_{s(i)}$ and $B_{i}$, respectively. $F_{n}$ are subsets of unit balls of $A_{s(n)}$, and $\widetilde{E_{n}}\subset E_{n}$ are both subsets of unit balls of $B_{n}$ satisfying $$\phi_{s(n),s(n+1)}(F_{n})\cup\alpha_{n+1}(E_{n+1})\cup\bigcup^{n+1}_{i=1}\phi_{s(i),s(n+1)}(\{a_{i1},a_{i2},\cdot\cdot\cdot, a_{in+1}\})\subset F_{n+1},$$ $$\psi_{n,n+1}(E_{n})\cup\beta_{n}(F_{n})\subset \widetilde{E}_{n+1}\subset E_{n+1},$$ and $$\bigcup^{n+1}_{i=1}\psi_{i,n+1}(\{b_{i1},b_{i2},\cdot\cdot\cdot, b_{in+1}\})\subset E_{n+1}.$$ (Here $\phi_{n,n}:A_{n}\longrightarrow A_{n}$, and $\psi_{n,n}:B_{n}\longrightarrow B_{n}$ are understand as identity maps.)\
(0.3) $\beta_{n}$ are $F_{n}-2\varepsilon_{n}$ multiplicative and $\alpha_{n}$ are homomorphism.\
(0.4) for all $g\in\widetilde{E}_{n}$, $$\|\psi_{n,n+1}(g)-\beta_{n}\circ\alpha_{n}(g)\| < 14\varepsilon_{n},$$ and for all $f\in F_{n}$, $$\|\phi_{s(n),s(n+1)}(f)-\alpha_{n+1}\circ\beta_{n}(f)\| < 14\varepsilon_{n}.$$ (0.5) For any block $B^{i}_{n}$ with spectrum $T_{\uppercase\expandafter{\romannumeral2},k},$ we have $\omega(\widetilde{E}_{n}^{i}) < \varepsilon_{n}$, where $\widetilde{E}_{n}^{i}=\pi_{i}(\widetilde{E}_{n})$ for $\pi_{i}:B_{n}\longrightarrow B^{i}_{n}$ the canonical projections.
The diagram will be constructed inductively. First, let $B_{1}=\{0\}, A_{s(1)}=A_{1}, \alpha_{1}=0$. Let $b_{1j}=0\in B_{1}$ for $j=1,2,...$ and let $\{a_{1j}\}^{\infty}_{j=1}$ be a countable dense subset of the unit ball of $A_{s(1)}$. And let $\widetilde{E}_{1}=E_{1}=\{b_{11}\}=B_{1}$ and $F_{1}=\bigoplus^{t_{1}}_{i=1}F^{i}_{1}$, where $F^{i}_{1}=\pi_{i}(\{a_{11}\})\subset A^{i}_{1}$.
As inductive assumption, assume that we already have the diagram
$$\xymatrix@R=0.4ex{
F_{1} & & F_{2} & & & & F_{n}\\
\bigcap & & \bigcap & & & & \bigcap\\
A_{s(1)} \ar[rdrd]^{\beta_{1}} \ar[rr]^{\phi_{s(1),s(2)}} & & A_{s(2)} \ar[rdrd]^{\beta_{2}} \ar[rr]^{\phi_{s(2),s(3)}} & &\cdots \ar[rr] \ar[rdrd]^{\beta_{n-1}} & & A_{s(n)}\\
& & & & & &\\
B_{1} \ar[rr]^{\psi_{1,2}} \ar[uu]^{\alpha_{1}} & & B_{2} \ar[uu]^{\alpha_{2}} \ar[rr]^{\psi_{2,3}} & &\cdots \ar[rr] & & B_{n}\ar[uu]^{\alpha_{n}}\\
\bigcup & & \bigcup & & & & \bigcup\\
E_{1} & & E_{2} & & & & E_{n}\\
\bigcup & & \bigcup & & & & \bigcup\\
\widetilde{E_{1}} & & \widetilde{E_{2}} & & & & \widetilde{E_{n}}
}$$
and for each $i=1,2,\cdot\cdot\cdot,n$, we have dense subsets $\{a_{ij}\}^{\infty}_{j=1}\subset$ the unit ball of $A_{s(i)}$ and $\{b_{ij}\}^{\infty}_{j=1}\subset$ the unit ball of $B_{i}$ satisfying the conditions (0.1)-(0.5) above. We have to construct the next piece of the diagram $$\xymatrix@!C{
F_{n}\subset A_{s(n)}\ar[rr]^{\phi_{s(n),s(n+1)}}\ar[rdrd]^{\beta_{n}} & & A_{s(n+1)}\supset F_{n+1} & & \\
& & & & & & \\
\widetilde{E}_{n}\subset E_{n}
\subset B_{n}\ar@<-8mm>[uu]^{\alpha_{n}}\ar[rr]_{\psi_{n,n+1}} & &
\;\;B_{n+1}\ar@<10mm>[uu]_{\alpha_{n+1}}\supset
E_{n+1}\supset\widetilde{E}_{n+1} & &
}$$ to satisfy the condition (0.1)-(0.5).
Among the conditions for induction assumption, we will only use the conditions that $\alpha_{n}$ is a homomorphism and (0.5) above.\
**Step 1**. We enlarge $\widetilde{E}_{n}$ to $\bigoplus_{i}\pi_{i}(\widetilde{E}_{n}^{i})$ and $E_{n}$ to $\bigoplus_{i}\pi_{i}(E_{n})$. Then $\widetilde{E}_{n}(=\oplus\widetilde{E}_{n}^{i})\subset E_{n}(=\oplus E_{n})$ and for each $B^{i}_{n}$ with spectrum $T_{\uppercase\expandafter{\romannumeral2},k}$, we have $\omega(E_{n}^{i})<\varepsilon_{n}$ from induction assumption (0.5). By proposition 2.9 applied to $\alpha_{n}:B_{n}\rightarrow A_{s(n)},\widetilde{E}_{n}\subset E_{n}\subset B_{n},F_{n}\subset A_{s(n)}$ and $\varepsilon_{n}>0$, there are $A_{m_{1}}(m_{1}>s(n))$, two othogonal projections $P_{0},P_{1}\in A_{m_{1}}$ with $\phi_{s(n),m_{1}}(\textbf{1}_{A_{s(n)}})=P_{0}+P_{1}$ and $P_{0}$ trivial, a $C^{*}$-algebra $C$—a direct sum of matrix algebras over $C[0,1]$ or $\mathbb{C}$, a unital map $\theta\in Map(A_{s(n)},P_{0}A_{m_{1}}P_{0})_{1}$, a unital homomorphism $\xi_{1}\in Hom(A_{s(n)},C)_{1}$, a unital homomorphism $\xi_{2}\in Hom(C,P_{1}A_{m_{1}}P_{1})_{1}$ such that\
(1.1) $\|\phi_{s(n),m_{1}}(f)-\theta(f)\oplus(\xi_{2}\circ\xi_{1})(f)\|<\varepsilon_{n}$ for all $f\in F_{n}$.\
(1.2) $\theta$ is $F_{n}-\varepsilon$ multiplicative and $F:=\theta(F_{n})$ satisfy $\omega(F)<\varepsilon_{n}$.\
(1.3) $\|\alpha(g)-\theta\circ\alpha_{n}(g)\|<3\varepsilon_{n}$ for all $g\in\widetilde{E}_{n}$.
Let all the blocks of C be parts of $C^{\ast}$-algebra $B_{n+1}$. That is
$B_{n+1}=C\oplus$ (some other blocks).
The map $\beta_{n}:A_{s(n)}\rightarrow B_{n+1}$, and the homomorphism $\psi_{n,n+1}:B_{n}\rightarrow B_{n+1}$ are defined by $\beta_{n}=\xi_{1}:A_{s(n)}\rightarrow C(\subset B_{n+1})$ and $\psi_{n,n+1}=\xi_{1}\circ\alpha_{n}:B_{n}\rightarrow C(\subset B_{n+1})$ for the blocks of $C(\subset B_{n+1})$. For this part, $\beta_{n}$ is also a homomorphism.\
**Step 2**. Let $A=P_{0}A_{m_{1}}P_{0},F=\theta(F_{n})$. Since $P_{0}$ is a trivial projection, $$A\cong\oplus M_{l_{i}}(C(X_{m_{1},i})).$$ Let $r(A):=\oplus M_{l_{i}}(\mathbb{C})$ and $r:A\rightarrow r(A)$ be as in 2.13. Applying corollary 2.14 and its remark to $\alpha:B_{n}\rightarrow A,\widetilde{E}_{n}\subset E_{n}\subset B_{n}$ and $F\subset A$, we obtain the following diagram $$\xymatrix{
A\ar[rr]^{\phi\oplus r}\ar[rdrd]^{\beta} & &M_{L}(A)\oplus r(A) & & \\
& & & & & & \\
B_{n}\ar[rr]^{\psi}\ar[uu]^{\alpha} & & B\ar[uu]_{\alpha^{'}} & & \\
}$$ such that\
(2.1) $B$ is a direct sum of matrix algebras over $\{pt\},[0,1],S^{1},T_{\uppercase\expandafter{\romannumeral2},k}$ and dimension drop algebra.\
(2.2) $\alpha^{'}$ is an injective homomorphism and $\beta$ is $F-\varepsilon_{n}$ multiplicative.\
(2.3) $\phi:A\rightarrow M_{L}(A)$ is a unital simple embedding and $r:A\rightarrow r(A)$ is as in 2.13.\
(2.4) $\|\beta\circ\alpha(g)-\psi(g)\|<10\varepsilon_{n}$ for all $g\in \widetilde{E}_{n}$ and $\|(\phi\oplus r)(f)-\alpha^{'}\circ\beta(f)\|<\varepsilon_{n}$ for all $f\in F(:=\theta(F_{n}))$.\
(2.5) $\omega(\pi_{i}(\psi(E_{n}))\cup\beta(F))<\varepsilon_{n+1}$ (note that $\beta(F)=\beta\circ\theta(F_{n})$), for $B_n^{i}$ being of form $M_{\bullet}(C(X))$ with $X\in\{T_{\uppercase\expandafter{\romannumeral2}, k}\}^{\infty}_{k=2}$.
Let all the blocks $B$ be also part of $B_{n+1}$, that is
$B_{n+1}=C\oplus B\oplus$ (some other blocks).
The maps $\beta_{n}:A_{s(n)}\longrightarrow B_{n+1},\psi_{n,n+1}:B_{n}\longrightarrow B_{n+1}$ are defined by $$\beta_{n}:=\beta\circ\theta:A_{s(n)}\xrightarrow{\theta}A\xrightarrow{\beta}B(\subset B_{n+1}),$$ and $$\psi_{n,n+1}:=\psi:B_{n}\rightarrow B(\subset B_{n+1}),$$ for the blocks of $B(\subset B_{n+1})$. This part of $\beta_{n}$ is $F_{n}-2\varepsilon_{n}$ multiplicative, since $\theta$ is $F_{n}-\varepsilon_{n}$ multiplicative, $\beta$ is $F-\varepsilon_{n}$ multiplicative and $F=\theta(F_{n})$.\
**Step 3**. By Lemma 3.15 of \[GJLP2\] applied to $\phi\oplus r:A\rightarrow M_{L}(A)\oplus r(A)$, there is an $A_{m_{2}}$ and there is a unital homomorphism $$\lambda:M_{L}(A)\oplus r(A)\rightarrow RA_{m_{2}}R,$$ where $R=\phi_{m_{1},m_{2}}(P_{0})$ (write $R$ as $\bigoplus_{j}R^{j}\in\bigoplus_{j}A^{j}_{m}$) such that the diagram $$\xymatrix{
A(=P_{0}A_{m}P_{0})\ar[rr]^{\phi_{m_{1},m_{2}}}\ar[rdrd]^{\phi\oplus r} & & RA_{m_{2}}R & & \\
& & & & & & \\
& & M_{L}(A)\oplus r(A)\ar[uu]_{\lambda} & &
}$$ satisfies the following conditions:\
(3.1) $\lambda\circ(\phi\oplus r)$ is homotopy equivalent to $$\phi^{'}:=\phi_{m_{1},m_{2}}|_{A}.$$ **Step 4**. Applying Theorem 1.6.9 of \[G5\] to finite set $F\subset A$ (with $\omega(F)<\varepsilon_{n}$) and to two homotopic homomorphisms $\phi^{'}$ and $\lambda\circ(\phi\oplus r):A\rightarrow RA_{m_{2}}R$ (with $RA_{m_{2}}R$ in place of $C$ in Theorem 1.6.9 of \[G5\]), we obtain a finite set $F^{'}\subset RA_{m_{2}}R,\; \delta>0$ and $L>0$ as in the Theorem.
Let $G=\oplus\pi_{i}(\psi(E_{n})\cup\beta(F))=\oplus G^{i}$. Then by (2.5), we have $\omega(G^{i})<\varepsilon_{n+1}$, if $B^{i}$ is of form $M_{\bullet}(C(T_{\uppercase\expandafter{\romannumeral2}, k}))$. By Proposition 2.8 applied to $RA_{m_{2}}R$ and $$\lambda\circ\alpha^{'}:B\rightarrow RA_{m_{2}}R,$$ finite set $G\subset B$, $F^{'}\cup(\phi_{m_{1}m_{2}}\mid_{A}(F))\in RA_{m_{2}}R$, $min(\varepsilon_{n},\delta)>0$ (in place of $\varepsilon$) and $L>0$, there are $A_{s(n+1)}$, mutually orthogonal projections $Q_{0},Q_{1},Q_{2}\in A_{s(n+1)}$ with $\phi_{m_{2},s(n+1)}(R)=Q_{0}\oplus Q_{1}\oplus Q_{2}$, a $C^{*}$-algebra $D$—a direct sum of matrix algebras over C\[0,1\] or $\mathbb{C}$—, a unital map $\theta_{0}\in$ Map($RA_{m_{2}}R,Q_{0}A_{s(n+1)}Q_{0}$), and four unital homomorphisms $$\theta_{1}\in Hom(RA_{m_{2}}R,Q_{1}A_{s(n+1)}Q_{1})_{1},\xi_{3}\in Hom(RA_{m_{2}}R,D)_{1},\xi_{4}\in Hom(D,Q_{2}A_{s(n+1)}Q_{2})_{1}$$ and $\alpha^{''}\in Hom(B,(Q_{0}+Q_{1})A_{s(n+1)}(Q_{0}+Q_{1}))_{1}$ such that the following is true.\
(4.1) $\|\phi_{m_{2},s(n+1)}(f)-((\theta_{0}+\theta_{1})\oplus\xi_{4}\circ\xi_{3})(f)\|<\varepsilon_{n}$, for all $f\in\phi_{m_{1},m_{2}}|_{A}(F)\subset RA_{m_{2}}R$.\
(4.2) $\|\alpha^{''}(g)-(\theta_{0}+\theta_{1})\circ\lambda\circ\alpha^{'}(g)\|<3\varepsilon_{n+1}<3\varepsilon_{n},\;\forall\;g\in G$.\
(4.3) $\theta_{0}$ is $F^{'}-min(\varepsilon_{n},\delta)$ multiplicative and $\theta_{1}$ satisfies that $$\theta^{i,j}_{1}([q])>L\cdot[\theta^{i,j}_{0}(R^{i})],$$ for any non zero projection $q\in R^{i}A_{m_{1}}R^{i}$.\
By Theorem 1.6.9 of \[G5\], there is a unitary $u\in(Q_{0}\oplus Q_{1})A_{s(n+1)}(Q_{0}+Q_{1})$ such that $$\|(\theta_{0}+\theta_{1})\circ\phi^{'}(f)-Adu\circ(\theta_{0}+\theta_{1})\circ\lambda\circ(\phi\oplus r)(f)\|<8\varepsilon_{n},$$ for all $f\in F$.\
Combining with second inequality of (2.4), we have\
(4.4)$\|(\theta_{0}+\theta_{1})\circ\phi^{'}(f)-Adu\circ(\theta_{0}+\theta_{1})\circ\lambda\circ\alpha^{'}\circ\beta(f)\|<9\varepsilon_{n}$ for all $f\in F$.\
**Step 5**. Finally let all blocks of $D$ be the rest of $B_{n+1}$. Namely, let $$B_{n+1}=C\oplus B\oplus D,$$ where $C$ is from Step 1, $B$ is from Step 2 and $D$ is from Step 4.
We already have the definition of $\beta_{n}:A_{s(n)}\rightarrow B_{n+1}$ and $\psi_{n,n+1}: B_{n}\rightarrow B_{n+1}$ for those blocks of $C\oplus B\subset B_{n+1}$ (from Step 1 and Step 2). The definition of $\beta_{n}$ and $\psi_{n,n+1}$ for blocks of $D$ and the homomorphism $\alpha_{n+1}:C\oplus B\oplus D\rightarrow A_{s(n+1)}$ will be given below.
The part of $\beta_{n}:A_{s(n)}\rightarrow D(\subset B_{n+1})$ is defined by $$\beta_{n}=\xi_{3}\circ\phi^{'}\circ\theta:A_{s(n)}\xlongrightarrow{\theta}A\xlongrightarrow{\phi}RA_{m_{2}}R\xlongrightarrow{\xi_{3}}D.$$ (Recall that $A=P_{0}A_{m_{2}}P_{0}$ and $\phi^{'}=\phi_{m_{1},m_{2}}|_{A}.)$ Since $\theta$ is $F_{n}-\varepsilon_{n}$ multiplicative, and $\phi^{'}$ and $\xi_{3}$ are homomorphism, we know this part of $\beta_{n}$ is $F_{n}-\varepsilon_{n}$ multiplicative.
The part of $\psi_{n,n+1}:B_{n}\rightarrow D(\subset B_{n+1})$ is defined by $$\psi_{n,n+1}=\xi_{3}\circ\phi^{'}\circ\alpha:B_{n}\xlongrightarrow{\alpha}A\xlongrightarrow{\phi^{'}}RA_{m}R\xlongrightarrow{\xi_{3}}D,$$ which is a homomorphism.
The homomorphism $\alpha_{n+1}:C\oplus B\oplus D\rightarrow A_{s(n+1)}$ is defined as following.
Let $\phi^{''}=\phi_{m_{1},s(n+1)}|_{P_{1}A_{m_{1}}P_{1}}:P_{1}A_{m_{1}}P_{1}\longrightarrow
\phi_{m_{1},s(n+1)}(P_{1})A_{s(n+1)}\phi_{m_{1},s(n+1)}(P_{1})$, where $P_{1}$ is from Step 1. Define $$\alpha_{n+1}|_{C}=\phi^{''}\circ\xi_{2}:C\xlongrightarrow{\xi_{2}}P_{1}A_{m_{1}}P_{1}\xlongrightarrow{\phi^{''}}\phi_{m_{1},s(n+1)}(P_{1})A_{s(n+1)}\phi_{m_{1},s(n+1)}(P_{1}),$$ where $\xi_{2}$ is from Step 1.
$\alpha_{n+1}|_{B}=Adu\circ\alpha^{''}:B\xlongrightarrow{\alpha^{''}}(Q_{0}\oplus Q_{1})A_{s(n+1)}(Q_{0}+Q_{1})\xlongrightarrow{Ad u}(Q_{0}\oplus Q_{1})A_{s(n+1)}(Q_{0}+Q_{1})$
where $\alpha^{''}$ is from Step 4, and define $$\alpha_{n+1}|_{D}=\xi_{4}:D\rightarrow Q_{2}A_{s(n+1)}Q_{2}.$$ Finally choose $\{a_{n+1,j}\}^{\infty}_{j=1}\subset A_{s(n+1)}$ and $\{b_{n+1,j}\}^{\infty}_{j=1}\subset B_{n+1}$ to be countable dense subsets of the unit balls of $A_{s(n+1)}$ and $B_{n+1}$, respectively. And choose\
$$F^{'}_{n+1}=\phi_{s(n),s(n+1)}(F_{n})\cup\alpha_{n+1}(E_{n+1})\cup\bigcup^{n+1}_{i=1}\phi_{s(i),s(n+1)}(\{a_{i1},a_{i2},\cdot\cdot\cdot, a_{in+1}\}),$$ $$E^{'}_{n+1}=\psi_{n,n+1}(E_{n})\cup\beta_{n}(F_{n})\cup\bigcup^{n+1}_{i=1}\psi_{i,n+1}(\{b_{i1},b_{i2},\cdot\cdot\cdot, b_{in+1}\}),$$ $$\widetilde{E}_{n+1}^{'}=\psi_{n,n+1}(E_{n})\cup\beta_{n}(F_{n})\subset E_{n+1}^{'}.$$ Define $F^{i}_{n+1}=\pi_{i}(F_{n+1}^{'})$ and $F_{n+1}=\bigoplus_{i}F^{i}_{n+1}$, $E^{i}_{n+1}=\pi_{i}(E^{'}_{n+1})$ and $E_{n+1}=\oplus_{i}E^{i}_{n+1}$. For those block $B^{i}_{n+1}$ inside the algebra $B$ define $\widetilde{E}_{n+1}^{i}=\pi_{i}(\widetilde{E}_{n+1})$. For those blocks inside $C$ and $D$, define $\widetilde{E}_{n+1}^{i}=E^{i}_{n+1}$. And finally let $E_{n+1}=\bigoplus_{i}\widetilde{E}_{n+1}^{i}$. Note all the blocks with spectrum $T_{\uppercase\expandafter{\romannumeral2},k}$ are in $B$. And (2.5) tells us that for those blocks $\omega(\widetilde{E}_{n+1}^{i})<\varepsilon_{n+1}$. Thus we obtain the following diagram $$\xymatrix@!C{
F_{n}\subset A_{s(n)}\ar[rr]^{\phi_{s(n),s(n+1)}}\ar[drdr]^{\beta_{n}} & & A_{s(n+1)}\supset F_{n+1} & & \\
& & & & & & \\
\widetilde{E}_{n}\subset E_{n}
\subset B_{n}\ar@<-8mm>[uu]^{\alpha_{n}}\ar[rr]_{\psi_{n,n+1}} & &
\;\;B_{n}\ar@<12mm>[uu]_{\alpha_{n+1}}\supset
E_{n+1}\supset\widetilde{E}_{n+1} & & \\
}$$ **Step 6**.Now we need to verify all the condition (0.1)-(0.5) for the above diagram.
From the end of Step 5, we know (0.5) holds, (0.1)-(0.2) hold from the construction (see the construction of $B, C, D$ in Step 1, 2 and 4, and $\widetilde{E}_{n+1}\subset E_{n+1}, F_{n+1}$ is the end of Step 5). (0.3) follows from the end of Step 1, the end of Step 2 and the part of definition of $\beta_{n}$ for $D$ from Step 5.
So we only need to verify (0.4).
Combining (1.1) with (4.1), we have $$\|\phi_{s(n),s(n+1)}(f)-[(\phi^{''}\circ\xi_{2}\circ\xi_{1})\oplus(\theta_{0}+\theta_{1})\circ\phi^{'}\circ\theta\oplus(\xi_{4}\circ\xi_{3}\circ\phi^{'}\circ\theta)(f)](f)\|
<\varepsilon_{n}+\varepsilon_{n}=2\varepsilon_{n}$$ for all $f\in F_{n}$ (recall that $\phi^{''}=\phi_{m_{1},s(n+1)}|_{P_{1}A_{m_{1}}P_{1}},\phi^{'}:=\phi_{m_{1},m_{2}}|_{P_{0}A_{m_{1}}P_{o}}$).
Combining with (4.2) and (4.4), and definition of $\beta_{n}$ and $\alpha_{n+1}$, the above inequality yields $$\|\phi_{s(n),s(n+1)}(f)-(\alpha_{n+1}\circ\beta_{n+1})(f)\| < 9\varepsilon_{n}+3\varepsilon_{n}+2\varepsilon_{n}=14\varepsilon_{n},\;\forall\; f\in F_{n}.$$ Combining (1.3), the first inequality of (2.4) and the definition of $\beta_{n}$ and $\psi_{n,n+1}$, we have $$\|\psi_{n,n+1}(g)-(\beta_{n}\circ\alpha_{n})(g)\|<10\varepsilon_{n}+3\varepsilon_{n}<14\varepsilon_{n},\;\forall\; g\in \widetilde{E}_{n}.$$ So we obtain(0.4). The theorem follows from Proposition 4.1 of \[GJLP2\].\
Note that if $q\in M_{l}(I_{k})$, then $qM_{k}(I_{k})q$ isomorphic to $M_{l_{1}}(I_{k})$. Combining with the main theorem of \[GJLP2\] (see Theorem 4.2, and 2.7 of \[GJLP2\]) we have
**Theorem 3.2.** Suppose that $A=\lim(A_{n}=\oplus P_{n, i}M_{[n,i]}(C(X_{n,i}))P_{n,i})$ is an $AH$ inductive limit with $dim(X_{n,i})\leqslant M$ for a fixed positive integer $M$ such that limit algebra $A$ has ideal property. Then $A$ can be rewrite as inductive limit $\lim(B_{n}=\oplus B^{i}_{n}, \psi_{n,m})$, where either $B^{i}_{n}=Q_{n,i}M_{[n,i]^{'}}(C(Y_{n,i}))Q_{n,i}$ with $Y_{n,i}$ being one of the spaces $\{pt\},[0,1],S^{1},\{T_{\uppercase\expandafter{\romannumeral2},k}\}^{\infty}_{k=2}$ or $B^{i}_{n}=M_{[n,i]^{'}}(I_{l_{(n,i)}})$ a dimension drop algebra.
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{
"pile_set_name": "ArXiv"
}
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author:
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$^a$, N. Shakura$^a$, A. Kochetkova$^a$, L. Hjalmarsdotter$^a$\
Moscow M.V. Lomonosov State University, Sternberg Astronomical Institute, 13, Universitetskij pr., 119992 Moscow, Russia\
E-mail: , , ,
title: 'Quasi-spherical accretion in low-luminosity X-ray pulsars: Theory vs. observations'
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Two regimes of quasi-spherical accretion in X-ray pulsars
=========================================================
There can be two different regimes of quasi-spherical accretion (see e.g. [@Bozzo_ea08] for a recent review of previous studies of wind accretion). The captured stellar wind heated up in the bow shock at Bondi radius $\sim R_B=2GM/v^2$ (where $v$ is the relative stellar wind velocity) to high temperatures $k_BT\sim m_p v^2$. If the characteristic cooling time of plasma $t_{cool}$ is smaller than the free-fall time $t_{ff}= R_B/\sqrt{2GM/R_B}$, the gas falls supersonically toward the magnetosphere with the formation of a shock. This regime is usually considered in connection with bright XPSRs [@AronsLea76], [@Burnard_ea83]. The role of X-ray photons generated near the NS surface is two-fold: first, they heat up plasma in the bow-shock zone via photoionization, and second, they cool down the hot plasma near the magnetosphere (with $k_BT\sim GM/R_A$) by Compton processes thus allowing matter to enter the magnetosphere via the Rayleigh-Taylor instability [@ElsnerLamb77].
In the free-fall accretion regime, the X-ray luminosity (the mass accretion rate $\dot M$) is determined by the rate of gravitational capture of stellar wind at $R_B$ (Bondi-Hoyle-Littleton formula, $\dot M\sim \rho v R_B^2$). The accretion torque exerted on the NS due to plasma-magnetopshere interaction is always of the same sign as the specific angular momentum of the gravitationally captured stellar wind $j_m$, and the NS can spin-up or spin down.
If the relative wind velocity $v$ at $R_B$ is slow ($\lesssim 80$ km/s), the photoionization heating of plasma is important, but the radiation cooling time of plasma is shorter than the free-fall time, so the free-fall accretion regime is realized. If the wind velocity is larger than $\sim 80-100$ km/s, the post-shock temperature is higher than $5 \times 10^5$ K (the maximum photoionization temperature for a photon temperature of several keV); for $L_x\lesssim 4\times 10^{36}$ erg/s, the plasma radiative cooling time is longer than the free-fall time, so a hot quasi-spherical shell is formed above the magnetosphere with temperature determined by hydrostatic equilibrium [@DaviesPringle81], [@Shakura_ea12]. Accretion of matter through such a shell is subsonic, so no shock is formed above the magnetosphere. The accretion rate $\dot M$ is now determined by the ability of hot plasma to enter magnetosphere. This is the settling accretion regime.
Settling accretion regime: theory
=================================
Theory of settling accretion regime was elaborated in [@Shakura_ea12]. In this regime, the accreting matter subsonically settles down onto the rotating magnetosphere forming an extended quasi-static shell. This shell mediates the angular momentum transfer to/from the rotating NS magnetosphere by viscous stresses due to large-scale convective motions and turbulence. The settling regime of accretion can be realized for moderate accretion rates $\dot M< \dot M_*\simeq 4\times 10^{16}$ g/s. At higher accretion rates, a free-fall gap above the neutron star magnetosphere appears due to rapid Compton cooling, and accretion becomes highly non-stationary.
**Mass accretion rate** through the hot shell is determined by mean velocity of matter entering the magnetosphere, $u(R_A)=f(u)\sqrt{2GM/R_A}$. The dimensionless factor $f(u)$ is determined by the Compton cooling of plasma above magnetosphere and the critical temperature for Rayleigh-Taylor instability to develop [@ElsnerLamb77], and is found to be $f(u)\sim (t_{ff}/t_{cool})^{1/3}$. In the case of Compton cooling $$f(u)\approx 0.4 \dot M_{16}^{4/11}\mu_{30}^{-1/11}\,,$$ where $\dot M_{16}=\dot M/[10^{16} \hbox{g/s}]$ and $\mu_{30}=\mu/[10^{30} \hbox{G}\,\hbox{cm}^3]$ is the NS magnetic moment. The definition of the Alfven radius in this case is different from the value used for disk accretion: $$R_A\approx 1.6\times 10^9[\hbox{cm}]
\left(\frac{\mu_{30}^3}{\dot M_{16}}\right)^{2/11}\,.$$ **Accretion torques** applied to NS in this regime are determined not only by the specific angular momentum of captured matter $j_m\sim \Omega_b R_B^2$ ($\Omega_b$ is the orbital angular velocity of the NS), as is the case of the free-fall accretion, but also by the possibility to transfer angular momentum to/from the rotating magnetosphere through the shell by large-scale convective motions. The plasma-magnetosphere interaction results in emerging of the toroidal magnetic field $B_t/B_p=(K_1/\zeta)(\omega_m-\omega^*)/\omega_K(R_A)$, where $\omega_m$ is the angular frequency of matter at the Alfven radus, $K_1\sim 1$ the dimensionless coupling coefficient which is different in different sources, $\omega_K(R_A)$ is the Keplerian angular frequecy, and $\zeta$ is the size of the region of plasma-magnetopshere angular moentum coupling in units of the Alfven radius $R_A$. The NS spin evolution equation reads: $$\label{sd1}
I\dot\omega^*=\frac{K_1}{\zeta}K_2\frac{\mu^2}{R_A^3}\frac{\omega_m-\omega^*}{\omega_K(R_A)}+
z\dot M R_A^2\,,$$ where the second term takes into acount the angular momentum braught to the NS with the infalling matter ($z<1$). This formula can be rearraged to the form $$\label{sd_eq}
I\dot \omega^*=Z \dot M R_A^2(\omega_m-\omega^*)+z \dot M R_A^2\omega^*\,,$$ where the coupling coefficient is $Z\approx 0.36 (K_1/\zeta)\dot M_{16}^{-4/11}\mu_{30}^{1/11}$.
The gas-dynamical treatment of the problem of angular momentum transfer through the shell by viscous turbulence stresses [@Shakura_ea12] showed that $\omega_m\approx \Omega_b(R_A/R_B)^n$, where the index $n$ depends on the character of turbulence in the shell. For example, in the case of isotropic near-sonic turbulence we obtain $n\simeq 3/2$, i.e. quasi-Keplerian rotation distribution. In the more likely case of strongly anisotropic turbulence (because of strong convection) we find $n\approx 2$, i.e. an iso-angular-momentum distribution[^1].
Settling accretion regime: observations
=======================================
**Equilibrium X-ray pulsars**. In equilibrium XPSRs $<\dot\omega^*>=0$ (e.g. Vela X-1 and GX 301-2). In this case, measurements of spin-up/spin-down near the equilibrium pulsar period $P_{eq}^*$ (or frequency $\omega^*_{eq}$) provides additional quantity $\partial \omega^*/\partial\dot M$ (or $\partial \omega^*/\partial y$, where $y\equiv \dot M\dot M_{eq}$ is mass accretion rate or X-ray luminosity noprmalized to the equilibrium value). In this case (see [@Shakura_ea12], [@Shakura_ea13] for more details) for the convective shell ($n=2$) we find: a) equilibrium NS period via binary orbital period $P_b$, mass accretion rate $\dot M$ (or X-ray luminosity $L_x=0.1\dot M c^2$), NS magnetic field $\mu_{30}$ and relative stellar wind velocity $v_8\equiv v/(1000 \hbox{km/s})$ $$\label{Peq}
P_{eq}^*\approx 1300 [\hbox{s}]\mu_{30}^{12/11}(P_b/10\hbox{d})
\dot M_{16}^{-4/11}v_8^4\,;$$ b) estimate of the coupling parameters $Z_{eq}$ or $(K_1/\zeta)$ via $P^*$ and $\partial \omega^*/\partial y$: $$\label{Zeqrho}
Z_{eq}\approx\frac{I\frac{\partial \dot\omega^*}{\partial \dot M}|_{eq}}{\frac{4}{11}\omega^*R_A^2}\approx 1.8\left(\frac{\partial \dot\omega^*/\partial y|_{y=1}}{10^{-12}\hbox{rad/s}}\right)(P^*/100s)\dot M_{16}^{-7/11}\mu_{30}^{-12/11}\,;$$ c) estimate of the NS magnetic field via $P^*$ and $\partial \omega^*/\partial y$ $$\label{mueqnew}
\mu_{30,eq}\approx 5 \left(\frac{\partial \dot\omega^*/\partial y|_{y=1}}{10^{-12}\hbox{rad/s}}\right)(P^*/100s)\left(\frac{K_1}{\zeta}\right)^{-1}\dot M_{16}^{-3/11}\,.$$ d) estimate of the stellar wind velocity $$\label{e:v8min}
v_8\approx 0.53 (1-z/Z_{eq})^{-1/4}
\dot M_{16}^{1/11}\mu_{30,eq}^{-3/11}
\left(\frac{P_*/100 \hbox{s}}{P_b/10 \hbox{d}}\right)^{1/4}$$ (note here the weak dependence on $\dot M$ and $\mu$). The observed and calculated parameters of the equilibrium wind-fed pulsars Vela X-1 and GX 301-2 are listed in Table 1. Note close values of the coupling parameter $Z_{eq}\sim 3$ (or $\zeta\sim 1/10$) for both pulsars, and the independent measurement of the stellar wind velocity similar to the observed values.
**Non-equilibrium X-ray pulsars**. From Eq. (\[sd1\]) it can be shown that $\dot \omega^*$ as a function of $\dot M$ reaches a mimimum at some accretion rate $y_{cr}=\dot M_{cr}/\dot M_{eq}=(3/2n+3)^{11/2n}<1$. For $n=2$ we find: $$\label{e:omegadotsdmax}
\dot\omega^*_{sd,max}\approx -1.13\times 10^{-12}[\hbox{rad/s}] (1-z/Z)^{7/4} \left(\frac{K_1}{\zeta}\right)
\mu_{30}^{2}v_8^3\left(\frac{P^*}{100\hbox{s}}\right)^{-7/4}
\left(\frac{P_b}{10\hbox{d}}\right)^{3/4}\,.$$ At $y<y_{cr}$ the spin-down torque should anti-correlate with the the X-ray flux variations, $\partial \dot\omega^*/\partial y<0$, with $\dot\omega_{sd}\sim -R_A^{-3}\sim -\dot M^{6/11}$. This is the case observed in long-term spinning-down XPSR GX 1+4 [@Chakrabarty_ea97], [@GonzalezGalan_ea12]. From the condition $|\dot \omega^*_{sd}|\le |\dot\omega^*_{sd,max}|$ we obtain the lower limit of the NS magnetic field: $$\label{e:mulim1}
\mu_{30}>\mu_{30, min}'\approx 0.94
\left|\frac{\dot\omega^*_{sd}}{10^{-12}\hbox{rad/s}}\right|
\left(\frac{K_1}{\zeta}\right)^{-1/2}
v_8^{-3/2}
\left(\frac{P^*}{100\hbox{s}}\right)^{7/8}
\left(\frac{P_b}{10\hbox{d}}\right)^{-3/8}.$$ If spin-up torque can be neglected, we find another estimate of the lower limit to the NS magnetic field $$\label{e:mulim2}
\mu_{30}>\mu_{30, min}''\approx 1.66
\left|\frac{\dot\omega^*_{sd}}{10^{-12}\hbox{rad/s}}\right|^{11/13}
\left(\frac{K_1}{\zeta}\right)^{-11/13}
\dot M_{16}^{-3/13}
\left(\frac{P^*}{100\hbox{s}}\right)^{11/13}.$$ Note that in contrast to Eq. (\[e:mulim1\]), this estimate is independent of the poorly known stellar wind velocity $v_8$ and binary orbital period $P_b$.
With decreasing $\dot M$ in non-equilibrium pulsars, the ratio of the toroidal to poloidal magnetic field components increases, reaching $B_t\sim B_p$ at $\dot M^*_{16}\approx 0.27 \left|\frac{\dot\omega^*_{sd}}{10^{-12}\hbox{rad/s}}\right|^{11/6}\mu_{30}^{-2/3}$. Below this luminosity acrretion becomes more non-stationary (likely the case of GX 1+4), but it is not still centrifugally prohibited. The propeller stage begins once $R_A>R_c$ at much smaller luminosities: $\dot M^{**}_{16}\approx 0.008 \mu_{30}^{3}
(P^*/100\hbox{s})^{-11/3}$.
\[T2\]
$$\begin{array}{lcc|ccc}
\hline
\hbox{Pulsars}&\multicolumn{2}{c}{\hbox{Equilibrium}}&
\multicolumn{3}{c}{\hbox{Non-equilibrium}}\\
\hline
& {\rm GX 301-2} & {\rm Vela X-1}
& {\rm GX 1+4} &{\rm SXP1062}&{\rm 4U 2206+54}\\
\hbox{Ref.}& \cite{Doroshenko_ea10} & \cite{Doroshenko11}& \cite{GonzalezGalan_ea12}
& \cite{Haberl_ea12} &\cite{Reig_ea12}\\
\hline
\multicolumn{5}{c}{\hbox{Measured parameters}}\\
\hline
P^*{\hbox{(s)}} & 680 & 283 & 140 & 1062 &5560\\
P_B {\hbox{(d)}} & 41.5 & 8.96 & 1161 & \sim 300^\dag& 19\\
v_{w} {\hbox{(km/s)}} & 300 & 700 & 200 & \sim 300^\ddag& 350\\
\mu_{30}& 2.7 & 1.2 & ? & ? & 1.7\\
\dot M_{16} & 3 & 3 & 1 & 0.6 & 0.2\\
%%%%%%<\dot \omega_{sd}> {\rm(rad/s)} & & & -1.5\times 10^{-12} \\
\frac{\partial \dot \omega}{\partial y} \arrowvert_{y=1}{\hbox{(rad/s)}}
& 1.5\cdot10^{-12} & 1.2\cdot10^{-12} & n/a & n/a & n/a\\
\dot\omega^*_{sd} & 0 & 0 & - 2.34 \cdot 10^{-11} & - 1.63 \cdot 10^{-11} &
-9.4 \cdot 10^{-14}\\
\hline
\multicolumn{5}{c}{\hbox{Obtained parameters}}\\
\hline
f(u) & 0.53 & 0.57 \\
\frac{K_1}{\zeta}& 14 &10& & & \gtrsim 8\\
Z& 3.7 & 2.6\\
B_t/B_p & 0.17 & 0.22\\
R_A{\hbox{(cm)}}& 2\cdot 10^9 & 1.4\cdot 10^9\\
\omega^*/\omega_K(R_A)& 0.07 & 0.08\\
v_{w,min} (km/s)& 500 & 740\\
\mu_{30,min}& & &\mu_{min}'\approx4&\mu_{min}''\approx20&\mu_{min}'\approx
3.6 \\
\hline
\end{array}$$ $^\dag$ Estimated from the Corbet diagram\
$^\ddag$ Typical velocity assumed in Be X-ray binaries
**Very slowly rotating XPSRs** There are known several very slowly rotating XPSRs, including some in HMXB (SXP 1062 with $P^*=1062$ s [@Haberl_ea12], 4U 2206+54 with $P^*=5550$ s [@Reig_ea12]) and some in LMXB (e.g. 3A 1954+319, $P^*=19400$ s [@Marcu_ea11]). Assuming disk accretion in such pulsars would require incredibly high NS magnetic fields (see e.g. discussion in [@Wang_12]). However, application of our model to these and other non-equilibrium XPSRs (see Table 1) gives the low limits of the NS surface magnetic field in the usual range $10^{12}-10^{13}$ G, and it is too preliminary to classify these objects as accreting magnetars. Note also that if at small X-ray luminosities convection is not developed in the shell, a quasi-Keplerian rotation law with $n=3/2$ can be established. The equilibrium NS spin period in this case is $P_{eq}^{(n=3/2)} =
P_{eq}^{(n=2)}\sqrt{R_B/R_A}\sim 10 P_{eq}^{(n=2)}$. That means that NS periods in such XPSRs can easily reach a few 10000 s for the standard NS magnetic field.
**Other applications** A possible implication of the settling accretion theory can be for non-stationary phenomena in XPSRs. The theory explains the observed temporary ’off’-states in Vela X-1, GX 301-2, 4U 1907+09, when the plasma cooling near the magnetospheric equator occurs due to radiative processes [@Shakura_ea12b]. The Compton cooling turns out to be ineffective due to X-ray pattern changing from fan-beam to the pencil-beam (as suggested by the observed X-ray pulse shape changes during the off-state in Vela X-1 [@Doroshenko_ea11]).
A dynamical instability of the shell on the time scale of the order of the free-fall time from the magnetosphere can appear due to increased Compton cooling and hence increased mass accretion rate in the shell, leading to an X-ray outburst with duration lasting about the free-fall time scale of the entire shell ($\sim 1000$ s). Such a transient behaviour is observed in supergiant fast X-ray transients (SFXTs) (see e.g. [@Ducci_ea10]), in which slow X-ray pulsations are found (e.g. IGRJ16418-4532, $P^*\approx 1212$ s [@Sidoli_ea12]). The observed flaring behavior can be the manifestation of a Rayleigh-Taylor instability from the magnetospheric radius occurring on the dynamical time scale $\sim R_A^{3/2}/\sqrt{GM}$.
Conclusion
==========
At X-ray luminosities $<4\times 10^{36}$ erg/s wind-fed X-ray pulsars can be at the stage of subsonic settling accretion. In this regime, accretion rate onto NS is determined by the ability of plasma to enter magnetosphere via Rayliegh-Taylor instability. The angular momentum can be transferred through the quasi-static shell via large-scale convective motions initiating turbulence cascade. The theory explains long-term spin-down in wind- fed accreting pulsars and properties of short-term torque-luminosity correlations. Long-period low-luminosity X-ray pulsars are most likely experiencing settling accretion too. Spectral and timing measurements of slowly rotating X-ray pulsars can be used to further test this accretion regime.
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[^1]: If there is no convection in the shell, the magnetosphere interacts with the shell in a turbulent boundary layer. In that case the spin-down torque is $\sim \mu^2/R_c^3$ ($R_c=(GM/\omega*^2)^{1/3}$ is the corotation radius) and is independent on $\dot M$. This case of weak coupling can be realized for very faint slowly rotating XPSRs.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We investigate the scale dependence of fluctuations inside a realistic model of an evolving turbulent HII region and to what extent these may be studied observationally. We find that the multiple scales of energy injection from champagne flows and the photoionization of clumps and filaments leads to a flatter spectrum of fluctuations than would be expected from top-down turbulence driven at the largest scales. The traditional structure function approach to the observational study of velocity fluctuations is shown to be incapable of reliably determining the velocity power spectrum of our simulation. We find that a more promising approach is the Velocity Channel Analysis technique of Lazarian & Pogosyan (2000), which, despite being intrinsically limited by thermal broadening, can successfully recover the logarithmic slope of the velocity power spectrum to a precision of $\pm 0.1$ from high resolution optical emission line spectroscopy.'
author:
- |
S.-N. X. Medina$^{1,2}$, S. J. Arthur$^{2,4}$[^1], W. J. Henney$^{2,4}$, G. Mellema$^{3,4}$, A. Gazol$^{2}$\
$^{1}$Departamento de Astronomía, Universidad de Guanajuato, Guanajuato, México.\
$^{2}$Centro de Radioastronomía y Astrofísica, Universidad Nacional Autónoma de México, Campus Morelia, 58090 Morelia, Michoacán, México.\
$^3$Dept. of Astronomy and Oskar Klein Centre, AlbaNova, Stockholm University, SE-106 91 Stockholm, Sweden.\
$^4$Nordita, KTH Royal Institute of Technology and Stockholm University, Roslagstullsbacken 23, SE-106 91 Stockholm, Sweden.
title: Turbulence in simulated HII regions
---
\[firstpage\]
hydrodynamics — HII regions — ISM: kinematics and dynamics — turbulence
Introduction
============
The line broadening in excess of thermal broadening seen in optical spectroscopic studies of [[ $\mathrm{H\,\scriptstyle\Roman{ionstage}}$]{}]{} regions has been attributed to turbulence in the photoionized gas. There are many examples in the literature of attempts to identify the presence and characterize this turbulence, for example @1958RvMP...30.1035M, @1985ApJ...288..142R, @1987ApJ...317..686O, @1995ApJ...454..316M, @1997ApJ...487..163M, @2011MNRAS.413..721L and references cited by these papers. In these studies, the variation of the point-to-point radial velocities with scale was investigated using structure functions following .
A general finding of these studies is that the structure function derived from these observations does not follow that predicted by Kolmogorov’s law [@1941DoSSR..30..301K] . The interpretation of this result is that Kolmogorov’s law for incompressible and subsonic turbulent flows cannot be strictly applied to the photoionized gas in [[ $\mathrm{H\,\scriptstyle\Roman{ionstage}}$]{}]{} regions, which will be compressible and possibly mildly supersonic [@1985ApJ...288..142R], or that energy does not cascade from larger to smaller scales but is input at many scales . Moreover, the results differ according to which emission lines are used in the study, since the topology of the emitting gas will be different. For instance, the [\[[ $\mathrm{O\,\scriptstyle\Roman{ionstage}}$]{}\]]{}$\lambda$5007 line comes from gas in the interior of an [[ $\mathrm{H\,\scriptstyle\Roman{ionstage}}$]{}]{} region, whereas the [\[[ $\mathrm{S\,\scriptstyle\Roman{ionstage}}$]{}\]]{}$\lambda$6731 line comes from the vicinity of the ionization front, essentially a two-dimensional surface. The suggested mechanisms responsible for the generation and maintenance of the turbulent velocities in [[ $\mathrm{H\,\scriptstyle\Roman{ionstage}}$]{}]{} regions include photoevaporated flows from globules and stellar winds .
In order to construct the structure functions for the velocity fields, observations at many points in an [[ $\mathrm{H\,\scriptstyle\Roman{ionstage}}$]{}]{} region are needed. This can be achieved either by multiple longslit spectroscopic observations at many positions across a nebula , or by Fabry-Perot interferometry . Longslit observations with high velocity resolution have enabled several velocity components to be identified for emission lines of metal ions for which the thermal widths are small, e.g., [\[[ $\mathrm{O\,\scriptstyle\Roman{ionstage}}$]{}\]]{}$\lambda$5007. These observations have been used to determine the radial velocities of the principal components of the emitting gas at hundreds of positions within the nebula. Fabry-Perot observations produce datasets of thousands of radial velocities, but without the velocity resolution to distinguish between different velocity components. Obviously, in order to obtain a structure function over a wide range of scales, very high quality data with an ample spatial coverage are required. In the case of longslit spectra, this is a non-trivial task, not least the calibrating of the positions of the slits .
@1949ApJ...110..329C suggested that turbulence is ubiquitous in astrophysics and that it could be analysed statistically via the spectrum of the velocity field, which expresses the correlations between instantaneous velocity components at all possible pairs of points in the medium. Turbulence develops in a fluid when velocity advection is dominant over dissipation (high Reynolds number), then continuous injection of energy is required in order to maintain the turbulent state. One of the main assumptions of the Kolmogorov theory of turbulence is that energy is only injected at large scales and is dissipated only at small scales. This implies that the energy cascades from large to small scales without dissipation. Another assumption of this theory is that in the range between injection and dissipation scales the energy is transferred at a constant rate, and this range is called the inertial range. Turbulence is often described in terms of the energy spectrum $E(v)$, and the inertial range is represented by a power-law relationship $E(v) \propto k^\beta$, where $k$ is the wave number. For incompressible, homogeneous, isotropic, 3D turbulence [@1941DoSSR..30..301K], we have $\beta = -5/3$, while in the limit of high Mach number, shock-dominated turbulence in one dimension [@1974Burgers], the power law is $\beta = -2$. It has proved more difficult to obtain exact results for the general case of 3D compressible, hydrodynamic turbulence. Scaling laws suggest that the original Kolmogorov energy spectrum scaling should be preserved even for highly compressible turbulence if the density weighted velocity $\rho^{1/3}v$ is taken instead of the pure velocity $v$ . Recent work by @2011PhRvL.107m4501G predicts the relation $E(\rho^{1/3}v) \propto k^{-19/9}$ for compressible, isothermal turbulence with compressive driving, with a turnover around the sonic scale to $E(\rho^{1/3}v)\propto
k^{-5/3}$. This has been confirmed by extremely high resolution numerical experiments [@2013MNRAS.436.1245F].
Another important prediction of the Kolmogorov theory is the scaling of the structure function of any order. The universality of such scaling laws has been tested by detailed numerical experiments . Even in numerical experiments of incompressible, homogeneous and isotropic turbulence with energy injection at a fixed large scale, it is found that the exponents of the higher order structure functions have an anomalous behaviour, which has been interpreted as the result of spatial and/or temporal intermittency. Intermittency is a sparseness in space and time of strong structures associated with the dissipation or injection of energy [@1994PhRvL..72..336S]. Since astrophysical turbulence driven by real physical phenomena is almost certainly intermittent, it is important to understand the possible effects of intermittency on the derived statistical properties of real [[ $\mathrm{H\,\scriptstyle\Roman{ionstage}}$]{}]{} regions.
The three-dimensional properties of the density and velocity fields that describe the turbulence in the interstellar medium, in this case an [[ $\mathrm{H\,\scriptstyle\Roman{ionstage}}$]{}]{} region, must be deduced from the statistical properties of the observed quantities. For photoionized gas, the observations generally consist of spectra at different positions, from which centroid velocities and linewidths can be obtained. For optical line emission, the emissivity depends on density, temperature and ionization state and therefore an analysis of the emission lines will not just be providing information on the turbulent velocity fluctuations in the gas. The observations are a two-dimensional projection of the three-dimensional properties and much effort has been dedicated to the problem of recovering 3D information from 2D data . Generally, some simplifying assumptions, such as isothermal gas and statistical isotropy of the turbulence, must be made.
Recent modeling of [[ $\mathrm{H\,\scriptstyle\Roman{ionstage}}$]{}]{} regions has addressed the origin of the irregular structures, filaments, and globules seen within and around the borders in optical images . Morphologically, the simulated emission-line images are very reminiscent of observed [[ $\mathrm{H\,\scriptstyle\Roman{ionstage}}$]{}]{} regions and the global dynamics, as measured by the r.m.s. velocity, is also similar to observationally derived values . The internal dynamics of the simulated [[ $\mathrm{H\,\scriptstyle\Roman{ionstage}}$]{}]{} regions is due mainly to the interaction of photoevaporated flows from the heads of the filaments and clumps, which flow into the interior of the [[ $\mathrm{H\,\scriptstyle\Roman{ionstage}}$]{}]{} region, superimposed on the general expansion of the [[ $\mathrm{H\,\scriptstyle\Roman{ionstage}}$]{}]{} region . However, the dynamics of real [[ $\mathrm{H\,\scriptstyle\Roman{ionstage}}$]{}]{} regions could also be affected at large scales by the action of stellar winds from the ionizing star or stars, and at small scales by outflows from young, low-mass stars.
In this paper we investigate to what extent our numerical simulations of [[ $\mathrm{H\,\scriptstyle\Roman{ionstage}}$]{}]{} regions model observed statistical properties of real [[ $\mathrm{H\,\scriptstyle\Roman{ionstage}}$]{}]{} regions. In § \[sec:nummod\] we briefly describe the numerical methods used in the radiation-hydrodynamics simulations and in the calculation of the simulated emission-line radiation, and how the statistical information is obtained from these calculations. In § \[sec:results\] we describe our results for the expansion of an [[ $\mathrm{H\,\scriptstyle\Roman{ionstage}}$]{}]{} region in a turbulent, clumpy molecular cloud. We discuss these results in § \[sec:discuss\], and comment on the extent to which the statistical properties of the numerical simulations agree with those of observed [[ $\mathrm{H\,\scriptstyle\Roman{ionstage}}$]{}]{} regions. In § \[sec:summary\] we summarize and present our conclusions.
Numerical Model {#sec:nummod}
===============
We perform radiation-hydrodynamic simulations of the expansion of photoionized regions in a non-uniform initially neutral medium. The central ionizing source has an effective temperature of 37,500 K and an ionizing photon rate of $10^{48.5}$ s$^{-1}$, which corresponds approximately to an O7 star.
The simulation takes as an initial condition a clumpy medium with mean density $\langle n_0 \rangle= 1000$ cm$^{-3}$, which results from a self-gravitating forced turbulence simulation by @2005ApJ...630L..49V. The computational cube represents a spatial volume of 4 pc$^3$ and the initial temperature is 5 K.
The radiation-hydrodynamics code used in the present paper does not include self gravity. However, although the initial conditions would be gravitationally unstable in the absence of photoionization heating, on the relatively short timescales of the simulations ($\sim 3\times10^5$ yrs) this will not be important since the global free-fall time of the computational cube is $\sim
6\times10^5$ yrs.
Radiation-hydrodynamics Code {#subsec:rhdcode}
----------------------------
The radiation-hydrodynamics code used in this paper is the same as that used by @2006ApJ...647..397M. The hydrodynamics is calculated using the nonrelativistic Roe solver PPM scheme described in with the addition of a local oscillation filter to suppress numerical odd-even decoupling behind radiatively cooling shock waves. Heating and cooling in the ionized and neutral gas is dealt with in the same manner as described in detail by @2009MNRAS.398..157H. The radiation transport and photoionization makes use of the C$^2$-RAY (Conservative-Causal ray) code developed by @2006NewA...11..374M. The calculations are all performed on a fixed, uniform Cartesian grid in three dimensions with a resolution of $512^3$ cells.
Simulated Emission Lines {#subsec:simemiss}
------------------------
The emissivity cubes for the H$\alpha$ recombination line and the [\[[ $\mathrm{O\,\scriptstyle\Roman{ionstage}}$]{}\]]{}$\lambda$5007, [\[[ $\mathrm{N\,\scriptstyle\Roman{ionstage}}$]{}\]]{}$\lambda$6584 and [\[[ $\mathrm{S\,\scriptstyle\Roman{ionstage}}$]{}\]]{}$\lambda$6731 collisionally excited lines are calculated as described in , with the assumption that the heavy element ionization fractions are fixed functions of the hydrogen ionization fractions, calibrated with the Cloudy photoionization code [@2013RMxAA..49..137F].
Turbulence Statistics {#subsec:turbstats}
---------------------
The emissivity statistics have contributions from velocity, density, temperature and ionization state fluctuations. Obviously, the velocity fluctuations are the most dynamically important statistic and spectral line data, both real and simulated, can be used to extract statistical information for the different contributions. In particular, the second-order structure function of the velocity centroids and the technique of velocity-channel analysis (VCA; [@2000ApJ...537..720L]) have been widely used.
In this paper, we calculate the second-order structure function of the velocity centroids of the simulated emission lines and also apply VCA to the corresponding position-position-velocity (PPV) datacubes. In addition, we calculate the power spectrum of the velocity field, both weighted and unweighted, of the 3D hydrodynamic simulation datacube in an attempt to relate the observed statistics to the underlying hydrodynamics.
### Power Spectrum
The power spectrum is a statistical tool that is useful for describing the intrinsic properties of, for example, velocity and density fields or any other physical property. It is the Fourier transform of the auto-correlation function of the physical quantity. For example, the N-dimensional auto-correlation function of the physical quantity $a$ can be written $$\xi_N(\boldsymbol{l}) = \langle
a(\mathbf{r})a(\mathbf{r}+\boldsymbol{l})\rangle \ ,
\label{eq:corrfn}$$ where $\mathbf{r}$ is the spatial position and $\boldsymbol{l}$ is the spatial separation. The N-dimensional power spectrum is then $$p_N(k) = \int e^{i\mathbf{k}\cdot\boldsymbol{l}} \xi_N({\boldsymbol{l}}) \,
d\boldsymbol{l} \ ,$$ where $\mathbf{k} = (k_x,k_y,k_z)$ is the wavenumber, which is related to the scale by $k = 2\pi/l$, and the integration is performed over all N-dimensional space. The energy spectrum is the angle integral of the power spectrum over shells of radius $k = |\mathbf{k} |$, such that $E_N(k) \propto k^{N-1} p_N(k)$.
Often the power spectrum can be represented by a power law $p_3(k)
\propto k^{n}$. For instance, the velocity field for incompressible, homogeneous [@1941DoSSR..30..301K] turbulence has $n = -11/3$, while for shock-dominated turbulence [@1974Burgers], the 3D power-law index is $n = -4$.
### Velocity centroid statistics {#subsubsec:centroid}
The use of positional fluctuations in the velocity centroids of spectral lines as probes of turbulent gas motions was developed in the 1950s by, for example, @1958RvMP...30.1035M, and has been applied to observations of molecular emission lines in molecular cloud complexes and to optical emission lines in the Orion Nebula, galactic and extragalactic [[ $\mathrm{H\,\scriptstyle\Roman{ionstage}}$]{}]{} regions .
The plane-of-the-sky velocity centroid map is calculated from the first two velocity moments $M_j$ of the simulated line intensity maps $I(v_r)$, where $$M_j = \int_{v_{s1}}^{v_{s2}} v_s^j I(v_s) dv_s \ ,$$ and $v_s$ is the radial (line-of-sight) velocity, for convenience taken to be along one of the principal axes of the 3D data cube. The centroids are then $V_c(x,y) = M_1/M_0$, where $(x,y)$ is the projection plane (plane of the sky) when the line of sight is along the $z$-axis. The limits of the integration over velocity are $v_{s1}$ and $v_{s2}$, which represent the full range of velocities produced by the simulations plus thermal broadening, where the thermal width is $v_T = (kT/m_p)^{1/2}$ and $m_p$ is the atomic mass.
The observed second-order structure function is $$\label{eq:strucfunc}
S_2(\boldsymbol{l}) = \frac{\Sigma[V_c(\mathbf{r}) - V_c(\mathbf{r} +
\boldsymbol{l})]^2}{\sigma_c^2 N(\boldsymbol{l})} \ ,$$ where the variance of centroid velocity fluctuations is $$\label{eq:variance}
\sigma^2_c \equiv \frac{\Sigma [V_c(\mathbf{r}) - \langle V_c \rangle
]^2}{N} \ ,$$ and $\langle V_c \rangle$ is the mean centroid velocity $$\label{eq:mean}
\langle V_c \rangle \equiv \frac{\Sigma V_c(\mathbf{r})}{N} \ .$$ In this definition, $\mathbf{r}$ is the two-dimensional position vector in the plane of the sky and $\boldsymbol{l}$ is the lag, or separation vector. The summation in Equation \[eq:strucfunc\] is over all data pairs for each separation, $N(\boldsymbol{l})$, while the summations in the centroid variation and mean are over the total number of array elements, i.e., pixels in the $(x,y)$-plane.
For homogeneous, incompressible (Kolmogorov) turbulence, the velocity fluctuations scale as $l^{1/3}$ and hence the second-order structure function scales like $l^{2/3}$. For isotropic velocity fields, the structure function and the auto-correlation function (see Eq. \[eq:corrfn\]) have the same scaling and differ only by a constant. The structure function is therefore related to the power spectrum. If the power-law index of the 3D structure function is $m$, then the power-law index of the 3D power spectrum is $n = -3 -m$.
### Velocity channel analysis {#sec:stats-vca}
The relationship between the velocity centroids and the statistics of the velocity field is only reliable if the density fluctuations are negligible. Velocity channel analysis was developed to extract the separate contributions of density and velocity from spectral line data cubes.
This is a technique for analyzing position-position-velocity (PPV) cubes developed by @2000ApJ...537..720L. With this method, spectroscopic observations are not reduced to velocity centroids as a function of position on the plane of the sky. Instead, the PPV cubes are analyzed in terms of velocity channels, or slices, as a function of the velocity resolution used. As the width of the velocity slices increases, the relative contribution of a velocity fluctuation to the total intensity fluctuations decreases, because the contributions from many velocity fluctuations will be averaged out in thicker velocity slices. A slice is described as *thick* when the dispersion of turbulent velocities is less than the velocity slice thickness on the turbulence scale studied, otherwise the slice is *thin* . In the thickest velocity channels, we obtain only information about the density fluctuations, since the velocity information is averaged out. Conversely, the velocity fluctuations dominate in thin channels.
Velocity channel analysis (VCA) consists of obtaining the 2D power spectrum for each velocity channel and then averaging over all velocity channels for each PPV cube. @2003MNRAS.342..325E stress that whether a slice in velocity space is considered *thin* or *thick* depends not only on the slice width $\delta v = (v_\mathrm{max} - v_\mathrm{min})/N$, where $N$ is the number of channels, but also on the scale of the turbulence. For power-law statistics, the velocity dispersion scales as $\sigma_r
\propto r^{m/2}$, where $m$ is the velocity structure function index. The criterion for a channel to be considered thin is $\delta v
< \sigma_r$, hence we do not expect a pure power-law result from the averaged 2D power-spectra analysis. This is because the largest scales, of the order of the size of the cube, are almost always in the *thin* regime. At the smallest scales, numerical dissipation plays a rôle, and there is an additional deviation from the power law.
Taking into account the thermal width of the optically thin emission line introduces an additional limitation on the resolution that can be used to discriminate between the *thick* and *thin* regimes. For a fixed velocity resolution, a velocity channel will remain *thin* up to wavenumber $$k \leq 2\pi \left[ \frac{1}{\sigma_L^2} \left( \delta v^2 + 2v_T^2
\right) \right]^{-1/m} \ ,$$ where $\sigma_L$ is the velocity dispersion over the scale $L$, equivalent to the size of the computational domain, and $v_T$ is the thermal width. The thermal width smears velocity fluctuations on smaller scales.
### Projection Smoothing {#subsec:projsmooth}
The effect of projecting a three-dimensional correlation function onto a two-dimensional space has been studied by , @1958RvMP...30.1035M, @1987ApJ...317..686O and @2004ApJ...604..196B, among others. For the velocity field, the two-dimensional case corresponds to the emission-line velocity centroid map.
Previous authors have established that for an isotropic, power-law, three-dimensional power spectrum, the spectral index does not change on going from three dimensions to two (projected) dimensions, i.e. $\kappa_{2D} = \kappa_{3D}$, where $\kappa_{ND}$ is the power spectrum spectral index in $N$ dimensions (defined as $p_N(k) \propto k^{-\kappa_{ND}}$, ). This is because the line-of-sight contribution to the line-of-sight velocity has no amplitude on the $k_z = 0$ plane, where $z$ is the line-of-sight direction.
The relationship between the power spectrum spectral index and the exponent of the second-order structure function for homogeneous turbulence is $\kappa_{ND} = m_\mathrm{ND} + N$, where $m_\mathrm{ND}$ is the power-law index of the $N$-dimensional second-order structure function. For projection from 3 to 2 dimensions, we therefore have $m_\mathrm{2D} = m_\mathrm{3D} + 1$, and this is known as *projection smoothing*.
In the case of incompressible, homogeneous (Kolmogorov) turbulence, we have $\kappa_{3D} = 11/3$ and $m_\mathrm{3D} = 2/3$, hence the relationship $\kappa_{2D} = \kappa_{3D}$ leads to $m_\mathrm{2D} =
5/3$. For the compressible turbulence [@1974Burgers] case, we have $\kappa_{3D} = \kappa_{2D} = 4$ and $m_\mathrm{3D} = 1$, leading to $m_\mathrm{2D} = 2$. When the line-of sight depth is small compared to the plane-of-the sky size of the emitting region, the distribution of emittors is essentially two-dimensional (sheet-like) and we would expect $m_\mathrm{2D} \simeq m_\mathrm{3D}$ .
In order to compensate for the effects of density inhomogeneities, @2004ApJ...604..196B introduced a correction term $\delta\kappa$ such that $\kappa_{2D} = \kappa_{3D} + \delta\kappa$, where $\delta\kappa$ tends to $-1$ when density fluctuations are important. This idea was applied by @2011MNRAS.413..721L in an effort to use 2D observational statistics to infer the 3D velocity field of photoionized gas in giant extragalactic [[ $\mathrm{H\,\scriptstyle\Roman{ionstage}}$]{}]{} regions.
Results {#sec:results}
=======
General Properties {#subsec:genprop}
------------------
{width="45.00000%"} {width="45.00000%"}\
{width="45.00000%"} {width="45.00000%"}
We begin by considering the morphological appearance of the simulated [[ $\mathrm{H\,\scriptstyle\Roman{ionstage}}$]{}]{} regions. Figure \[fig:HIIimages\] shows the evolutionary sequence from 150,000 to 300,000 years. Each image is a composition of the three optical emission lines [\[[ $\mathrm{N\,\scriptstyle\Roman{ionstage}}$]{}\]]{}$\lambda 6584$ (red), H$\alpha$ (green) and [\[[ $\mathrm{O\,\scriptstyle\Roman{ionstage}}$]{}\]]{}$\lambda 5007$ (blue), where we have employed the classical *Hubble Space Telescope* red-green-blue colour scheme. Emission from the dense, neutral zones is negligible and for the production of these images we also include dust extinction in the radiative transfer for the projection to the plane of the sky, with the assumption that the dust-to-gas mass ratio is $1 \%$.
The images show that the ionized gas distribution is not spherical and around the corrugated edge of the [[ $\mathrm{H\,\scriptstyle\Roman{ionstage}}$]{}]{} region there are many neutral clumps and fingers of gas. This is a consequence of the [[ $\mathrm{H\,\scriptstyle\Roman{ionstage}}$]{}]{} region evolving in a clumpy neutral medium. The fingers and clumps at the edge of the [[ $\mathrm{H\,\scriptstyle\Roman{ionstage}}$]{}]{} region are the remnants of denser regions in the initial turbulent cloud. We can also see that the different ions are important in different regions of the photoionized gas. The [\[[ $\mathrm{O\,\scriptstyle\Roman{ionstage}}$]{}\]]{}$\lambda 5007$ (blue) emission is strongest closer to the photoionizing source, while the [\[[ $\mathrm{N\,\scriptstyle\Roman{ionstage}}$]{}\]]{}$\lambda 6584$ (red) emission is most prominent around the edge of the [[ $\mathrm{H\,\scriptstyle\Roman{ionstage}}$]{}]{} region and the H$\alpha$ is distributed throughout the nebula. Although not shown in Figure \[fig:HIIimages\], we also calculate the [\[[ $\mathrm{S\,\scriptstyle\Roman{ionstage}}$]{}\]]{}$\lambda 6716$ emission, which is important only around the edge of the [[ $\mathrm{H\,\scriptstyle\Roman{ionstage}}$]{}]{} region in the vicinity of the ionization front. This is consistent with the phenomenon of ionization stratification .
The neutral clumps and fingers around the edge of the [[ $\mathrm{H\,\scriptstyle\Roman{ionstage}}$]{}]{} region are the source of photoevaporated flows, which flow away from the ionization front into the [[ $\mathrm{H\,\scriptstyle\Roman{ionstage}}$]{}]{} region . These flows are mildly supersonic and can reach velocities of up to two or three times the sound speed in the ionized gas. They shock against each other in the interior of the [[ $\mathrm{H\,\scriptstyle\Roman{ionstage}}$]{}]{} region. Neutral clumps inside the [[ $\mathrm{H\,\scriptstyle\Roman{ionstage}}$]{}]{} region are accelerated outwards by the rocket effect in the opposite direction to the photoevaporated material flowing off their ionized skins. In Figure \[fig:HIIimages\] these photoevaporated flows can be discerned as shadowy regions ahead of the convex bright cusps that mark the position of the ionization front. As the photoevaporated gas flows away from the ionization front, the density drops in the diverging flow and this is why the flows appear darker than the surrounding photoionized gas.
These colliding photoevaporated flows are responsible for the velocity dispersion of the ionized gas, which we show in Figure \[fig:veldisp\]. This figure shows both the mass-weighted velocity dispersion of the ionized gas and the volume-weighted velocity dispersion. At early times ($t < 10^5$ yrs) the volume-weighted velocity dispersion is higher than that of the mass-weighted velocity dispersion. This is because the [[ $\mathrm{H\,\scriptstyle\Roman{ionstage}}$]{}]{} region breaks out of the dense clump where it is formed after about 50,000 yrs and a low-density, relatively high velocity champagne flow results. After about 100,000 yrs, the main ionization front has just about caught up and thereafter the two different velocity dispersions show the same behaviour, remaining roughly constant with $V_\mathrm{rms} \sim 8$ km s$^{-1}$. In Figure \[fig:veldisp\] we also show the r.m.s. velocity for the analytical expansion [@1978Spitzer] in a uniform medium, where the gas velocity behind the ionization front is one half that of the ionization front and the internal velocities are linear with radius. In this case, the velocity dispersion falls steadily as a function of time, and after 100,000 yrs is less than 2 km s$^{-1}$. The differences between the velocity dispersions in the numerical and analytic cases are mainly due to the interaction of the photoevaporated flows produced in the simulations described above. In the analytic expansion, the velocities are radial away from the central source. In the clumpy medium, the photoevaporated flows lead to large non-radial velocities, which increase the velocity dispersion in the photionized region. Finally, Figure \[fig:veldisp\] also shows that the mass-weighted mean radial velocity peaks just before 200,000 yrs at about 7 km s$^{-1}$ and thereafter falls off. This indicates that the global expansion will not be an important influence on the dynamics, compared to the velocity dispersion, after 200,000 yrs.
![Velocity dispersion as a function of time. The thick, black solid line shows the mass-weighted velocity dispersion of the ionized gas, the dotted line is the volume-weighted velocity dispersion of the ionized gas, while the thick grey line is the r.m.s. velocity of the analytical expansion [@1978Spitzer] in a uniform medium of density $n_0$. The thin, black continuous line is the mass-weighted mean radial expansion velocity of the ionized gas.[]{data-label="fig:veldisp"}](fig2){width="\linewidth"}
![Sound-crossing time as a function of evolution time. The sound-crossing time is $\langle R \rangle / c_\mathrm{i}$, where $\langle R \rangle$ is the radius of an equivalent sphere with volume equal to that of the [[ $\mathrm{H\,\scriptstyle\Roman{ionstage}}$]{}]{} region and $c_\mathrm{i} =
11$ km s$^{-1}$ is the sound speed in the photoionized gas. The diagonal grey line has slope of unity. The horizontal dashed line is the sound-crossing time for a distance of 2 pc, which is the half-side length of the computational box.[]{data-label="fig:scross"}](fig3){width="\linewidth"}
In Figure \[fig:scross\] we compare the sound-crossing time of the ionized volume with the evolution time of the simulation. During the breakout of the [[ $\mathrm{H\,\scriptstyle\Roman{ionstage}}$]{}]{} region from its initial dense clump (between about 80,000 and 100,000 yrs), the expansion is supersonic and the sound-crossing time exceeds the evolution time. This coincides with the peak in the velocity dispersion (see Fig. \[fig:veldisp\]). After this time, the expansion is subsonic and the sound-crossing time becomes shorter than the evolution time. A statistically steady state then becomes possible for the turbulence within the ionized volume.
Statistical Properties {#subsec:statprop}
----------------------
### Power Spectra {#sssec:pspec}
{width="\textwidth"}
![Time variation of 3D power spectra power-law indices. Top panel: 3D physical quantities; solid line—ionized gas velocity $v_i$, dashed line—ionized density $d_i$, dotted line—square of ionized density $d_i^2$, dot-dashed line—temperature $T$. Bottom panel: emission-line volume emissivities; filled circles—$j(\mathrm{H}\alpha)$, open squares—$j(\mathrm{[OIII]})$, filled triangles—$j(\mathrm{[NII]})$, open diamonds—$j(\mathrm{[SII]})$.[]{data-label="fig:psevol"}](fig5a "fig:"){width="\linewidth"}\
![Time variation of 3D power spectra power-law indices. Top panel: 3D physical quantities; solid line—ionized gas velocity $v_i$, dashed line—ionized density $d_i$, dotted line—square of ionized density $d_i^2$, dot-dashed line—temperature $T$. Bottom panel: emission-line volume emissivities; filled circles—$j(\mathrm{H}\alpha)$, open squares—$j(\mathrm{[OIII]})$, filled triangles—$j(\mathrm{[NII]})$, open diamonds—$j(\mathrm{[SII]})$.[]{data-label="fig:psevol"}](fig5b "fig:"){width="\linewidth"}
In Figure \[fig:ps\] we show the 3D power spectra of the ionized gas velocity, the ionized density, and the ionized velocity weighted by the cube root of the density . In the following sections we use a dimensionless $k$ that is normalised to the size of our computational grid. Thus $k = 1$ corresponds to a physical scale of $4$ pc. The power spectra all exhibit a break in the power law at about wavenumber $k \sim 32$, equivalent to a scale of 16 computational cells and consistent with the numerical dissipation scale. We fit power laws between $k = 4$ (corresponding to a length scale of one parsec) and $k =32$. For times later than 200,000 yrs, the power-law fits for a given quantity are essentially constant (see Fig. \[fig:psevol\]). Note that here we use $n$ to represent the 3D power spectrum spectral index,[^2] which corresponds to the slope or gradient in log-log space, where $n = -\kappa_{3D}$ and $\kappa_{3D}$ is defined in § \[subsec:projsmooth\]. At earlier times, the power laws are steeper and evolving. The settling down of the power spectra power-law indices correlates with the time at which the global expansion ceases to be important for the internal dynamics (see Fig. \[fig:veldisp\]). This occurs after about 200,000 yrs (as evidenced in Fig. \[fig:psevol\]), which corresponds to the evolution time being equal to approximately 1.5 times the sound-crossing time (see Fig. \[fig:scross\]).
### Second-order Structure Functions {#sssec:s2func}
{width="80.00000%"}
{width="0.6\linewidth"}
We use the procedure described in Section \[subsubsec:centroid\] to calculate velocity centroid maps for the H$\alpha$, [\[[ $\mathrm{O\,\scriptstyle\Roman{ionstage}}$]{}\]]{}$\lambda
5007$, [\[[ $\mathrm{N\,\scriptstyle\Roman{ionstage}}$]{}\]]{}$\lambda 6584$ and also [\[[ $\mathrm{S\,\scriptstyle\Roman{ionstage}}$]{}\]]{}$\lambda 6716$ emission lines and then calculate the corresponding second-order structure functions according to Equation \[eq:strucfunc\]. Results for representative evolutionary times are shown in Figures \[fig:sfunc\] to \[fig:sfuncyz\] of Appendix \[app:sf\], where fits to the power-law index ([$m_{\mathrm{2D}}$]{}) of the structure function, which corresponds to the slope or gradient in log-log space, are carried out for the inertial range of scales. A description of the procedure for identifying the inertial range is given in Appendix \[app:sf\] and illustrated by the accompanying Figure \[fig:sfauto\].
In Figure \[fig:sftrends\] we show the evolution of [$m_{\mathrm{2D}}$]{} with time for the different lines and for the three principal viewing directions of the simulation cube. For the line of sight along the $z$-axis (first column of Fig. \[fig:sftrends\]), one sees for all lines a consistent steepening of the structure function graph with time (increase in [$m_{\mathrm{2D}}$]{}). But for other viewing directions no such trend is apparent: both rising and falling behavior of [$m_{\mathrm{2D}}$]{} is seen, with little consistency between different lines.
In order to understand why one particular viewing direction is different, we produced histograms of the emission-line velocity centroid values binned into narrow $<2$ km s$^{-1}$ bins for the three different lines of sight at the four different times. The histograms are presented in Figure \[fig:histogram\], from which we see that for the $z$-axis line of sight, the values of $V_c$ are not distributed symmetrically about the mean value and, in fact, for the H$\alpha$ and [\[[ $\mathrm{O\,\scriptstyle\Roman{ionstage}}$]{}\]]{}$\lambda$5007 emission lines, a “wing” develops for negative values of $V_c$ that extends to more negative values as time progresses. This tendency is not seen for the $y$- and $x$- axis lines of sight. We attribute this wing to a “champagne” flow towards the observer along the $z$-axis. This flow would be perpendicular to the line of sight for observations along the other axes.
### Velocity Channel Analysis {#sssec:vca}
{width="\linewidth"}
{width="80.00000%"}
The velocity channel analysis consists of calculating the two-dimensional power spectrum of the brightness distribution in isovelocity channels of varying thickness. We consider two cases: thick slices, which are wide enough ($\sim 100~\mathrm{km\ s^{-1}}$) to include all the emission in the line, and thin slices, with width $5~\mathrm{km\ s^{-1}}$. Because the velocity spectrum in our simulations is rather shallow (see above), the line-of-sight turbulent velocity dispersion $\delta v$ exceeds the width of these thin slices over the full range of length scales that we can usefully study, from $0.1$ pc ($\delta v \approx 8~\mathrm{km\ s^{-1}}$) to $1$ pc ($\delta v \approx 10~\mathrm{km\
s^{-1}}$). Figure \[fig:o3-thick-thin\] shows typical examples of the [\[[ $\mathrm{O\,\scriptstyle\Roman{ionstage}}$]{}\]]{} brightness in thick and thin slices.
To use thinner slices would not be useful for a variety of reasons. First, $5~\mathrm{km\ s^{-1}}$ corresponds to the highest resolution that can be achieved with optical spectrographs that are optimised for studying extended sources, such as Keck HIRES or VLT UVES. Second, thinner slices are increasingly subject to “shot noise” due to the finite resolution of the numerical simulations, which produces spurious small-scale power, as discussed by @2003MNRAS.342..325E and @2003ApJ...593..831M. Third, thermal broadening would smoothe out any structure on scales $< 5~\mathrm{km\ s^{-1}}$ for all but the heaviest ions.
Figure \[fig:vcatrends\] shows the evolution with time of the VCA power-law indices from thin ([$\gamma_{\mathrm{t}}$]{}) and thick ([$\gamma_{\mathrm{T}}$]{}) channels (shown by filled circle and cross symbols, respectively) for different ions and for different viewing directions. The individual VCA power spectra from which these power-law indices, which correspond to the slope or gradient in log-log space, were extracted are presented in Appendix \[app:vca\]. It can be seen that both [$\gamma_{\mathrm{t}}$]{} and [$\gamma_{\mathrm{T}}$]{} are remarkably stable with time during the latter part of the evolution ($t >
200,000$ years). Although thermal broadening means that there is no clear distinction between [$\gamma_{\mathrm{t}}$]{} and [$\gamma_{\mathrm{T}}$]{} for the H$\alpha$ line, the two values are clearly distinguished for the heavier ions, with the thin slices showing a significantly shallower slope, especially for [\[[ $\mathrm{O\,\scriptstyle\Roman{ionstage}}$]{}\]]{}. The implications for diagnosing turbulence statistics are discussed in § \[sssec:vca2\].
Discussion {#sec:discuss}
==========
Characterization of the turbulence from optical emission lines {#subsec:charac}
--------------------------------------------------------------
At times later than 150,000 years, our second-order structure function results for the H$\alpha$ and [\[[ $\mathrm{O\,\scriptstyle\Roman{ionstage}}$]{}\]]{}$\lambda$5007 emission-line velocity centroids strongly suggest the presence of turbulence with an inertial range between 1 pc and the numerical dissipation scale of about 8 cells (equivalent to 0.0625 pc). The [\[[ $\mathrm{N\,\scriptstyle\Roman{ionstage}}$]{}\]]{}$\lambda$6584 and [\[[ $\mathrm{S\,\scriptstyle\Roman{ionstage}}$]{}\]]{}$\lambda$6716 structure function results also suggest turbulence, with a smaller upper limit to the inertial range, which is consistent with these ions being confined to relatively thin layers near the ionization front. However, it is difficult to characterize this turbulence since the slope of the structure function for a given emission line varies with time in an unpredictable manner and also depends on the line of sight. Although the results for the $z$-axis line of sight suggest an increase in slope with time, an examination of the distributions of the velocity centroids (see Fig. \[fig:histogram\]) shows that this is due to a champagne-type flow in that direction, and other lines of sight do not show a definite trend with time.
Different emission lines originate in different volumes of ionized gas, and this is reflected in the different slopes for the structure functions from different emitters. The H$\alpha$ line is produced throughout the volume and is brightest close to the ionization front around the bright edges of the photoionized gas. There is therefore a wide range of densities associated with the H$\alpha$-emitting gas. The emissivity of the H$\alpha$ recombination line depends on the square of the density and only weakly on the temperature in the photoionized gas [see e.g., @2006agna.book.....O]. Indeed, the gradients of the 3D power spectra of the H$\alpha$ emissivity and the square of the density are essentially the same. On the other hand, the emissivity of the [\[[ $\mathrm{O\,\scriptstyle\Roman{ionstage}}$]{}\]]{}$\lambda$5007 collisional line depends more strongly on temperature. This line originates in the interior of the [[ $\mathrm{H\,\scriptstyle\Roman{ionstage}}$]{}]{} region, where the density is more uniform but weak shocks due to the collision of photoevaporated flows cause fluctuations in the temperature. A more uniform density distribution corresponds to a steeper density power spectrum and, indeed, the structure functions, 2D velocity channel power spectra, and 3D power spectrum of the [\[[ $\mathrm{O\,\scriptstyle\Roman{ionstage}}$]{}\]]{}$\lambda$5007 line are all steeper than those for the H$\alpha$ recombination line.
The stellar parameters for the simulations presented in this work correspond to a relatively hot (37,500 K) O7 star. For these parameters, the [\[[ $\mathrm{N\,\scriptstyle\Roman{ionstage}}$]{}\]]{}$\lambda$6584 and [\[[ $\mathrm{S\,\scriptstyle\Roman{ionstage}}$]{}\]]{}$\lambda$6716 collisionally ionized lines come from regions close to the ionization front, where the density variations are strong, and this is reflected in the less steep structure function and 2D velocity channel power spectra gradients. In particular, the [\[[ $\mathrm{S\,\scriptstyle\Roman{ionstage}}$]{}\]]{} emission will come from very close to the ionization front where the acceleration of the ionized gas is strongest, and as a result the structure function and 2D velocity channel power-spectra gradients are shallowest for this line. Other stellar parameters, e.g., a cooler B0 star or a much hotter white dwarf, would produce photoionized regions with different ionization stratifications.
### Intrinsic Power Spectra Of Physical Quantities {#sssec:ips}
Figure \[fig:ps\] shows that the power spectra of physical quantities are very well approximated by power laws over the range from $k = 4$ to 32 (scales of 1 pc to 0.125 pc). In particular, the ionized gas velocity shows a power-law slope of $n = -3.2 \pm 0.1$ once the turbulence is fully developed. This is significantly shallower than the Kolmogorov ($n = -3.667$) or Burgers ($n = -4$) value, indicating more velocity structure at small scales than would be seen in a simple turbulent cascade of energy injected at the largest scale. As a consequence, the turbulent velocity dispersion is relatively insensitive to scale, varying as $\sigma \sim l^{0.5 (-3 - n)} \sim l^{0.1}$.
One possible reason for the shallow velocity power spectrum may be that energy is injected over a variety of scales, corresponding to the different sizes of clumps and filaments responsible for the photoevaporated flows in the simulated [[ $\mathrm{H\,\scriptstyle\Roman{ionstage}}$]{}]{} regions. Moreover, the energy injection will vary with time due to the global expansion of the [[ $\mathrm{H\,\scriptstyle\Roman{ionstage}}$]{}]{} region, which moves the sources of the photoevaporated flows generally outwards, and the destruction of the clumps and filaments as they are eroded by the ionizing radiation.
The density power spectrum has a very similar slope to that of the velocity: $n = -3.2 \pm 0.1$, but of greater relevance are the slopes of the emissivities of the different emission lines, which are $n = -3.4 \pm 0.1$ for [\[[ $\mathrm{O\,\scriptstyle\Roman{ionstage}}$]{}\]]{}, $n = -2.9 \pm 0.1$ for [$\mathrm{H\alpha}$]{}, $n = -2.7 \pm 0.1$ for [\[[ $\mathrm{N\,\scriptstyle\Roman{ionstage}}$]{}\]]{}, and $n = -2.6 \pm 0.1$ for [\[[ $\mathrm{S\,\scriptstyle\Roman{ionstage}}$]{}\]]{}. These span the critical value of $n = -3$ that divides “steep” from “shallow” power spectra. [\[[ $\mathrm{O\,\scriptstyle\Roman{ionstage}}$]{}\]]{} has a steep slope, indicating that large-scale fluctuations dominate, while [\[[ $\mathrm{N\,\scriptstyle\Roman{ionstage}}$]{}\]]{} and [\[[ $\mathrm{S\,\scriptstyle\Roman{ionstage}}$]{}\]]{} have shallow slopes, indicating that small-scale fluctuations dominate. The [$\mathrm{H\alpha}$]{} slope is very close to the critical value, indicating roughly equal contributions from fluctuations on all size-scales.
It is interesting to study the question of whether the known power-law indices of the velocity and emissivity power spectra in our simulations can in practice be recovered from observational diagnostics. If this is not the case for a given diagnostic, then it would call into question its utility for studying real [[ $\mathrm{H\,\scriptstyle\Roman{ionstage}}$]{}]{} regions. In particular, we will concentrate on two commonly used diagnostics: the second-order structure function of the line velocity centroids, and the power spectra of the surface brightness in isovelocity channel maps (Velocity Channel Analysis).
### Structure Function {#sssec:strfunc}
The structure function of the velocity centroids is an observationally attractive diagnostic because it is relatively immune to the effects of thermal broadening and poor spectral resolution, so long as sufficiently high signal-to-noise spectra are used. However, it has the disadvantage that relating the observed slope to the 3-dimensional velocity statistics depends on the geometry of the emitting region, see § \[subsec:projsmooth\]. For transverse separations larger than the characteristic line-of-sight depth of the emitting gas, the two-dimensional gradient should be equal to the three-dimensional one: $$m_{\mathrm{2D}} = m_{\mathrm{3D}} = -3 - n,$$ whereas at smaller separations than this, projection smoothing, as described above, means that the two-dimensional gradient is steeper: $$m_{\mathrm{2D}} = 1 + m_{\mathrm{3D}} = -2 - n.$$ Based on our simulation’s velocity power spectrum index at late times of $n \approx -3.2$ (see Figs. \[fig:ps\] and \[fig:psevol\]), the structure function slope should be $m_{\mathrm{2D}} = 0.2$ in the large-scale limit and $m_{\mathrm{2D}} = 1.2$ in the small-scale limit.
In fact, all of the measured slopes lie between these two limits, with a systematically increasing value from low to high-ionization lines: $m_{\mathrm{2D}}({[{\setcounter{ionstage}{2} \ensuremath{\mathrm{S\,\scriptstyle\Roman{ionstage}}}}]}) = 0.33 \pm 0.02$, $m_{\mathrm{2D}}({[{\setcounter{ionstage}{2} \ensuremath{\mathrm{N\,\scriptstyle\Roman{ionstage}}}}]}) = 0.49 \pm 0.03$, $m_{\mathrm{2D}}({\ensuremath{\mathrm{H\alpha}}}) = 0.59 \pm 0.04$, $m_{\mathrm{2D}}({[{\setcounter{ionstage}{3} \ensuremath{\mathrm{O\,\scriptstyle\Roman{ionstage}}}}]}) = 0.74 \pm 0.04$, where the averages were performed for $t > 200,000$ years. This is qualitatively consistent with expectations because the emission from lower-ionization lines is confined to thin layers near the ionization front, whereas higher ionization emission is more distributed over the volume and therefore subject to greater projection smoothing.
If the line-of-sight depth were constant over the face of the [[ $\mathrm{H\,\scriptstyle\Roman{ionstage}}$]{}]{} region, then the structure function would show a break at a scale equal to that depth, but in reality the depth varies from point to point, so there will not be a sharp break. Instead, the structure function is expected to show negative curvature, with the gradient gradually decreasing as one passes from smaller to larger scales. A small such effect is seen in the structure functions derived from our simulations (Fig. \[fig:sfunc\] to \[fig:sfuncyz\]): the fit to a power law is generally not so good as in the case of the power spectra, with negative residuals at both ends of the fitted range, indicative of a negative curvature. That the observed effect is so small is probably due to the fact that the distribution of line-of-sight depths strongly overlaps with the limited dynamic range in separations available from our simulations, bounded at small scales by numerical dissipation, and at large scales by the size of the ionized region.
It is disappointing that none of the measured slopes reach either of the limiting cases discussed above. All that can be deduced from the structure function is that $1 + m_{\mathrm{3D}} >
m_{\mathrm{2D}}({[{\setcounter{ionstage}{3} \ensuremath{\mathrm{O\,\scriptstyle\Roman{ionstage}}}}]})$ and $m_{\mathrm{3D}} <
m_{\mathrm{2D}}({[{\setcounter{ionstage}{2} \ensuremath{\mathrm{S\,\scriptstyle\Roman{ionstage}}}}]})$, which implies $n = -2.74$ to $-3.33$. Although this is a rather wide range of allowed velocity power spectrum slopes, it does serve to clearly rule out the Kolmogorov value of $n
= -3.667$. Furthermore, the “true” value of $n = -3.12 \pm 0.03
$ lies close to the middle of the allowed range.
A further proviso to the use of the structure function is that systematic anisotropic flows can affect the measured slopes when the viewing angle is along the direction of the flow. Such an effect is seen at later times for our simulation when viewed along the $z$-axis (Fig. \[fig:sfunc\]). In this case, the structure function tends to steepen at the large-scale end of our fitting range, producing a positive curvature, which is opposite to the more typical case of negative curvature discussed above. Such cases may also be identified by the presence of a significant skew in the PDF of the line-of-sight velocity (see Fig. \[fig:histogram\]).
Figure \[fig:sf-vs-n\] illustrates these points by graphing the correlation between the structure function slope [$m_{\mathrm{2D}}$]{} and the slope $n$ of the underlying 3D velocity fluctuations. The theoretical relation is shown by black diagonal lines, both with (continuous line) and without (dashed line) projection smoothing. It is apparent that a large part of the variation in [$m_{\mathrm{2D}}$]{} is not driven by changes in $n$. Indeed, [$m_{\mathrm{2D}}$]{} shows a larger or equal variation in the latter stages of evolution, when $n$ is approximately constant, than it does in the earlier stages, when $n$ is varying.
{width="\linewidth"}
Note that the additional complication identified by @2004ApJ...604..196B, whereby correlations between density and velocity fluctuations affect the translation between $m_{\mathrm{2D}}$ and $n$, is likely of minor importance in our case. @2007MNRAS.381.1733E show that this is most important for high Mach number turbulence, where $\delta\rho/\langle \rho
\rangle > 1$, whereas the transonic turbulence inside our simulated [[ $\mathrm{H\,\scriptstyle\Roman{ionstage}}$]{}]{} regions produces more modest density contrasts.
### Velocity Channel Analysis {#sssec:vca2}
{width="\linewidth"}
Figure \[fig:vca-thin-vs-n\] shows the correlations between the slope of the velocity fluctuation power spectrum and the VCA slopes found in § \[sssec:vca\] above (see Fig. \[fig:vcatrends\]). The theoretical procedure [@2000ApJ...537..720L] for deriving one from the other is slightly different, depending on whether the power spectrum of the emissivity fluctuations is “steep” or “shallow” (see § \[sec:stats-vca\] above). In the steep case, which applies to [\[[ $\mathrm{O\,\scriptstyle\Roman{ionstage}}$]{}\]]{} in our simulation, the slope of the average power spectrum of the brightness maps in the thin isovelocity channels is given by ${\ensuremath{\gamma_{\mathrm{t}}}}{} = -3 + \frac12 m_{\mathrm{3D}}$, where $m_{\mathrm{3D}} = -3 - n = 0.2 \pm 0.1$ for our simulation. The derived value from the [\[[ $\mathrm{O\,\scriptstyle\Roman{ionstage}}$]{}\]]{} thin channel maps for $t >
200,000$ is ${\ensuremath{\gamma_{\mathrm{t}}}}{} = -2.80 \pm 0.07 $, which compares well with the value $-2.9 \pm 0.05$ that is implied by the simulation’s value of $n$.
In the shallow case, it is the difference in slope between the thin and thick slices that is predicted to depend on the velocity fluctuations: ${\ensuremath{\gamma_{\mathrm{t}}}}{} - {\ensuremath{\gamma_{\mathrm{T}}}}{} = \frac12 m_{\mathrm{3D}}$. The derived values are ${\ensuremath{\gamma_{\mathrm{t}}}}{} - {\ensuremath{\gamma_{\mathrm{T}}}}{} = 0.07 \pm 0.05$, $0.19 \pm 0.02$, and $0.17 \pm 0.02$ for [$\mathrm{H\alpha}$]{}, [\[[ $\mathrm{N\,\scriptstyle\Roman{ionstage}}$]{}\]]{}, and [\[[ $\mathrm{S\,\scriptstyle\Roman{ionstage}}$]{}\]]{}, respectively. These also compare tolerably well with the value of $0.1 \pm 0.05$ that is implied by the simulation’s value of $n$.
Note, however, that the large Doppler width of the [$\mathrm{H\alpha}$]{} line means that the thin velocity slices are not useful in this case, since the thick and thin slices have identical slopes. The fact that this agrees with the theoretical prediction is merely a coincidence, due to our velocity spectrum having a slope that is close to $-3$. For the lines from heavier ions, [\[[ $\mathrm{O\,\scriptstyle\Roman{ionstage}}$]{}\]]{}, [\[[ $\mathrm{N\,\scriptstyle\Roman{ionstage}}$]{}\]]{} and [\[[ $\mathrm{S\,\scriptstyle\Roman{ionstage}}$]{}\]]{}, the difference between the thin and thick velocity slices is not erased by thermal broadening, but in these three cases there is a consistent difference of $\approx 0.1$ between the measured VCA slope and the theoretically expected one. The origin of this difference is unclear, but it is small enough that it is not a significant impediment to the application of the VCA method.
The slopes of the power spectra of the thick slices themselves, which are simply the velocity-integrated surface brightness images[^3] are predicted [@2000ApJ...537..720L] to be equal to the slopes of the 3D power spectra of their respective emissivities. The comparison between these two quantities is shown in Figure \[fig:vca-thick-vs-n\], from which it is clear that only in the case of [\[[ $\mathrm{O\,\scriptstyle\Roman{ionstage}}$]{}\]]{} are the two slopes equal. In the case of the other lines, [$\gamma_{\mathrm{T}}$]{} is shallower than the emissivity’s spectral index $n$ by 0.36, 0.19, 0.61 for [$\mathrm{H\alpha}$]{}, [\[[ $\mathrm{N\,\scriptstyle\Roman{ionstage}}$]{}\]]{}, and [\[[ $\mathrm{S\,\scriptstyle\Roman{ionstage}}$]{}\]]{}, respectively. The reason for this discrepancy is the increasingly “sheet-like” morphology of the emission in the lower ionization lines. As shown in § 4.1 of @2003ApJ...593..831M, one should see a transition from ${\ensuremath{\gamma_{\mathrm{T}}}}{} = n$ to the shallower slope ${\ensuremath{\gamma_{\mathrm{T}}}}{} = n + 1$ at transverse scales larger than the line-of-sight depth of the emitting region.
{width="\linewidth"}
Comparison with observational results
-------------------------------------
Non-thermal linewidths in [[ $\mathrm{H\,\scriptstyle\Roman{ionstage}}$]{}]{} regions have been interpreted as evidence for turbulence in the photoionized gas. Galactic [[ $\mathrm{H\,\scriptstyle\Roman{ionstage}}$]{}]{} regions generally exhibit subsonic turbulent widths , while giant extragalactic [[ $\mathrm{H\,\scriptstyle\Roman{ionstage}}$]{}]{} regions can have supersonic turbulent motions . The majority of observational studies consider only the H$\alpha$ line, but @1988AA...198..283O and examine the components of the [\[[ $\mathrm{O\,\scriptstyle\Roman{ionstage}}$]{}\]]{}$\lambda$5007 line in the Orion Nebula, M42, while @1992ApJ...387..229O and @1993ApJ...409..262W investigate the kinematics of the \[S[iii]{}\] and \[O[i]{}\] lines of this same object. The giant extragalactic [[ $\mathrm{H\,\scriptstyle\Roman{ionstage}}$]{}]{} region NGC-595 was studied by who analyzed the H$\alpha$, [\[[ $\mathrm{O\,\scriptstyle\Roman{ionstage}}$]{}\]]{} and [\[[ $\mathrm{S\,\scriptstyle\Roman{ionstage}}$]{}\]]{} kinematics. The heavier ions have the advantage that their emission lines have low thermal broadening compared to the H$\alpha$ line.
Observational studies of the spatial scales of velocity fluctuations have mostly focused on the structure function of velocity centroids. The results are rather disparate, partly because the methodology varies considerably between different studies. For instance, some authors attempt to filter out “ordered” large-scale motions before analysing the fluctuations [@1995ApJ...454..316M; @2011MNRAS.413..721L], whereas others analyse the unfiltered observations [@1992ApJ...387..229O; @1997ApJ...487..163M]. Also, in some cases multiple Gaussian velocity components are fitted to the line profiles [@1988ApJS...67...93C; @1993ApJ...409..262W], which are then assigned to a small number of velocity “systems” that are each analysed separately, whereas in most studies the mean velocity of the entire line profile is used.
Despite these differences, there are interesting commonalities in the results: a rising structure function with $m_{\mathrm{2D}} =
0.5$–$1.0$ is nearly always found at the smallest scales, which transitions to a flat structure function with $m_{\mathrm{2D}} \sim
0$ at larger scales. However, the scale at which the transition occurs varies enormously from object to object, from $0.02$–$0.2$ pc in compact ($R = 1$ to 5 pc) Galactic [[ $\mathrm{H\,\scriptstyle\Roman{ionstage}}$]{}]{} regions [@1987ApJ...317..676O; @1988ApJS...67...93C; @1993ApJ...409..262W; @1995ApJ...454..316M], up to 50 pc in giant ($R
\sim 400$ pc) extragalactic regions [@2011MNRAS.413..721L]. We comment that the sound-crossing time for a region of size 50 pc is about 5 Myr, roughly the same as the estimated age of the NGC 595 nebula [@1990ApJ...364..496D]. For the full extent of NGC 595, the sound crossing time is about 40 Myr. It is therefore unlikely that the turbulence in such large regions is in a statistically steady state unless it is highly supersonic. Indeed, @1988AA...198..283O suggest that in the case of large extragalactic [[ $\mathrm{H\,\scriptstyle\Roman{ionstage}}$]{}]{} regions, the large linewidths could be due to multiple velocity components, that is, parts of the [[ $\mathrm{H\,\scriptstyle\Roman{ionstage}}$]{}]{} region with separation greater than the distance a sound wave could travel within the current lifetime of the object are kinematically distinct. Such giant [[ $\mathrm{H\,\scriptstyle\Roman{ionstage}}$]{}]{} regions show velocity centroid dispersions of $\sigma_{\mathrm{c}} > 10~\mathrm{km\ s^{-1}}$ on the largest scales, which is several times larger than is seen in compact single-star regions or in our simulations. We will therefore not consider them further since they are governed by additional physical processes, such as powerful stellar winds and the cluster gravitational potential, which are beyond the scope of the current paper.
The explanations that have been offered for the break in the structure function slope are also varied. In the case of compact [[ $\mathrm{H\,\scriptstyle\Roman{ionstage}}$]{}]{} regions, it is often taken to indicate the characteristic line-of-sight depth of the emission zone [@1951ZA.....30...17V; @1987ApJ...317..686O], with projection smoothing steepening the slope at the smaller separations (see § \[subsec:projsmooth\] above). If that were the case, then the correct three-dimensional structure function slope is the flat one: $m_{\mathrm{3D}} \sim 0$, corresponding to a velocity power spectrum slope of $n = -3$. This interpretation would be broadly consistent with our simulation results, which show a very similar velocity power spectrum (Fig. \[fig:ps\]). However, our simulated structure functions rarely show a clear break in the same way as the observations do, although they do show a slight negative curvature in many cases. This is probably because of the very limited useful dynamic range, roughly a factor of 10, that the simulations allow between the small scales that are affected by numerical diffusion and the large scales, that are affected by systematic flows, anisotropies, and edge-effects.
An alternative explanation for the observed break in the structure function is that it represents the scale of the largest turbulent eddies [@1988ApJS...67...93C; @1995ApJ...454..316M] and that the fluctuations at larger scales are simply uncorrelated. In such a picture it would still be necessary to postulate a velocity spectrum considerably shallower than Kolmogorov in order to explain the small-scale slope.
Based on the discussion of our simulation results above (§ \[sssec:vca2\]), it seems that Velocity Channel Analysis would be a very useful complement to the structure function, since it is less affected by uncertainties in projection smoothing and gives a more consistent result between different emission lines (at least, for our simulations).
In a forthcoming paper, we will present such an analysis of recent high-resolution echelle spectroscopy of multiple emission lines in the Orion Nebula [@2008RMxAA..44..181G; @2008AJ....136.1566O].
Summary {#sec:summary}
=======
1. We have investigated the statistics of fluctuations in physical conditions within a radiation-hydrodynamic simulation of the evolution of an [[ $\mathrm{H\,\scriptstyle\Roman{ionstage}}$]{}]{} region inside a highly inhomogeneous molecular cloud. We find that steady-state turbulence, corresponding to time-independent profiles of the 3D power spectra, is only established after about 1.5 sound-crossing times of the [[ $\mathrm{H\,\scriptstyle\Roman{ionstage}}$]{}]{} region. In these simulations, this corresponds to about 200,000 years (§ \[sssec:pspec\]).
2. We find a power-law behaviour for the 3D power spectra in the range from about 1 pc down to 0.125 pc, equivalent to 16 computational cells. The larger scale can be interpreted as the size of the largest photoevaporated flows, while the smaller scale is about twice the numerical dissipation scale. The power-spectrum slopes of the velocity and density fluctuations are very similar and always lie in the range $-3.1 \pm 0.1$. This is significantly shallower than the slope predicted for the inertial range of either incompressible or compressible turbulence ($-3.667$ to $-4.1$). This suggests that turbulent driving is occuring over all scales in our simulation, unlike the case of classical turbulence where energy is injected only at the largest scales. The power-spectrum slopes of the emissivities of optical lines are even shallower, increasingly so for lower ionization lines, indicating that the smallest scale fluctuations are dominant (§ \[sssec:ips\]).
3. We investigate in detail the utility of observational diagnostics for inferring the power spectra of emissivity and velocity fluctuations in our simulation. We find that the traditional velocity centroid structure function technique gives ambivalent results because of the effects of projection smoothing, combined with the fact that the effective line-of-sight depth of the emitting gas does not have a single well-defined value. In addition, the presence of anisotropic motions such as champagne flows can yield misleading structure function slopes when the simulation is viewed from certain directions (§ \[sssec:strfunc\]).
4. The more recently developed technique of Velocity Channel Analysis is found to offer a more robust diagnostic of the three-dimensional velocity statistics of our simulation. The slope of the velocity power spectrum can be correctly recovered to a precision of $\pm 0.1$ from either high or low ionization lines, and with no significant dependence on viewing direction (§ \[sssec:vca2\]).
Acknowledgments {#acknowledgments .unnumbered}
===============
We would like to thank the referee for constructive comments, which improved the presentation of this paper. SNXM acknowledges a CONACyT, Mexico student fellowship. SJA would like to thank DGAPA-UNAM for financial support through project IN101713. SJA, WJH and GM thank Nordita for support during the program Photo-Evaporation in Astrophysical Systems. This work has made use of NASA’s Astrophysics Data System.
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Example second-order structure functions of the line-of-sight velocity centroids {#app:sf}
================================================================================
Figures \[fig:sfunc\] to \[fig:sfuncyz\] show the second-order structure functions of the line-of-sight velocity centroid maps (see §§ \[sssec:strfunc\] and \[subsubsec:centroid\]) for the four emission lines at the four evolutionary times depicted in Figure \[fig:HIIimages\]. If turbulence is present, the second-order structure function should exhibit an inertial range over which it is a power law with length scale. Accordingly, we perform a least-squares fit to the data points. However, it is not immediately clear what the limits for the fit should be.
At small scales, the lower limit for the inertial range should be defined by the scale at which numerical dissipation effects cease to be important [@2004ApJ...604..196B]. For the present simulations, we tested several values and the size scale equivalent to 10 computational cells proved to be adequate for all emission lines and evolution times studied. For the upper limit, we examined the projected emission maps and calculated the area occupied by the pixels having greater than the mean intensity. This method is independent of the resolution of the image and could equally be applied to images obtained from observations. We then took the radius of the circle having the same area to be the upper limit for the least-squares fit. This procedure appears to work very well, as can be seen in Figures \[fig:sfunc\] and \[fig:sfuncyz\]. If a different line of sight is chosen, the radius of this circle will be different and needs to be calculated self-consistently for every projection. Note that the inertial range for each combination of line and view tends to become broader with time due to the expansion of the [[ $\mathrm{H\,\scriptstyle\Roman{ionstage}}$]{}]{} region. At the latest time, 300,000 yrs, both the H$\alpha$ and [\[[ $\mathrm{O\,\scriptstyle\Roman{ionstage}}$]{}\]]{}$\lambda
5007$ structure functions appear to develop a break, which would be better fit by two power laws, one below a scale of about 0.3 pc and a steeper one for larger scales. However, we have fit just a single power law to both of these cases. We speculate that this apparent break in the power law at late times could be due to energy injection at size scales associated with the photoevaporated flows emanating from the ubiquitous clumps and filaments seen in the emission-line images (see Fig. \[fig:HIIimages\]).
An alternative criterion for the upper limit was used by @2011MNRAS.413..721L who used the theoretical result for isotropic, homogeneous turbulence that decorrelation of the second-order structure function occurs when the auto-correlation function changes sign from positive to negative. In the theory, this corresponds to the scale for which the second-order structure function is equal to 2. Figure \[fig:sfauto\] shows the structure function and auto-correlation function obtained from our simulations at times 150,000 yrs and 250,000 yrs for the [\[[ $\mathrm{O\,\scriptstyle\Roman{ionstage}}$]{}\]]{}$\lambda 5007$ emission line velocity centroids projected onto the $yz$-plane. The other emission lines and viewing directions give very similar results. From this figure, we see that at early times, when the structure function does indeed rise above 2, this corresponds approximately to the length scale at which the auto-correlation function changes sign. However, at later times the auto-correlation function changes sign at a length scale much smaller than the size of the computational box but the structure function remains less than 2. This suggests that the assumptions of the simple theory (isotropic, homogeneous turbulence) do not apply to the emission-line velocity centroids obtained from our simulations. Figure \[fig:sfauto\] also indicates the fitting range suggested by the procedure described earlier. At early times, our procedure results in a slightly larger upper length scale for the power-law fit than using the zero point of the auto-correlation function. At later times, the two more-or-less coincide and this is true for all the emission lines, viewing angles and times we have examined. Since there is no clear reason to prefer the auto-correlation approach, we will use our computationally simpler procedure to determine the upper length scale for the power-law fit.
{width="\textwidth"}
{width="\textwidth"}
{width="\textwidth"}
![Second-order structure function (thick solid line) and auto-correlation function (thick dashed line) against logarithmic length scale. The horizontal lines at 2 and 0 are included as reference values.[]{data-label="fig:sfauto"}](figAppA4){width="0.8\linewidth"}
Example power spectra from Velocity Channel Analysis {#app:vca}
====================================================
Figures \[fig:vca\] to \[fig:vcayz\] show the power spectra resulting from the velocity channel analysis (see § \[sec:stats-vca\]). Each of the three figures is for a different viewing direction and shows the four emission lines at four different times. For each combination of line and time, there are two panels: an upper panel without including thermal Doppler broadening and a lower panel with the broadening effects included. In each graph, two power spectra are plotted: one representing a very thick velocity slice (i.e., encompassing all the emission) and the other averaged over thin velocity slices of width $\delta v \sim 5$ km s$^{-1}$. Also shown on each panel are the power-law indices obtained from least-squares power-law fits to the thin ([$\gamma_{\mathrm{t}}$]{}) and very thick slice ([$\gamma_{\mathrm{T}}$]{}) spectra and the range in wavenumber over which the fit is calculated. This wavenumber range corresponds to the length-scale range used for the structure function fits (see § \[sssec:s2func\]). The very thick velocity slice is equivalent to the total intensity along the line of sight and its power spectrum does not vary with the addition of thermal broadening.
It is clear that the thermal broadening has a large effect on the VCA of the H$\alpha$ line, effectively erasing the difference in slope between the thin and thick slices. For photoionized gas at $T_e=
10^4$ K, the FWHM of the H$\alpha$ line is $\sim 22$ km s$^{-1}$, while that of an oxygen line is a quarter of this, $\sim
5.5$ km s$^{-1}$. Indeed, the heavier ions are less affected by thermal broadening, but a slight steepening of the thin-slice power spectra can still be seen, amounting to a reduction in [$\gamma_{\mathrm{t}}$]{} of $\sim 0.1$.
For the thermally broadened case, the variation with time of the slopes of these fits, [$\gamma_{\mathrm{T}}$]{} for the thick slices and [$\gamma_{\mathrm{t}}$]{} for the thin slices, is shown in Fig \[fig:vcatrends\] and discussed in § \[sssec:vca\].
{width="\textwidth"}
{width="\textwidth"}
{width="\textwidth"}
\[lastpage\]
[^1]: E-mail: [email protected]
[^2]: This is to facilitate comparison with the theoretical literature, such as @2000ApJ...537..720L.
[^3]: Although for simplicity, extinction is not included.
|
{
"pile_set_name": "ArXiv"
}
|
---
author:
- Felix Kempf
- Romain Mueller
- Erwin Frey
- 'Julia M. Yeomans'
- Amin Doostmohammadi
bibliography:
- './all\_reports.bib'
title: Active Matter Invasion
---
Introduction\[section:intro\]
=============================
Understanding the mechanisms by which living (active) systems such as cells and bacteria invade and navigate through their surroundings is of pivotal importance in many physiological and pathological processes. Depending on the physical and chemical properties of their microenvironment active matter can show distinct modes of invasion, from single cell migration to groups of cells moving as a collective. The latter has been identified as the primary mode of cancer cell invasion [@Clark15; @Clark19]. Similarly collective migration is the prime mode of growth and invasion by bacterial biofilms [@Wu15; @Even17; @Hartmann18].
Among physical environmental factors, geometrical constraints have a strong influence on the dynamics of collective migration in biological matter. Various physiological processes such as bacterial filtering rely on active matter living within pores, cavities, and constrictions [@Yang2010]. [*In vivo*]{}, cancerous cells are known to preferentially move along pre-existing tracks of least resistance, such as myelinated axons or blood vessels, when invading into healthy tissue [@Gritsenko2011; @Weigelin2012]. Similarly, during biofilm formation interaction between the bacterial colony and the surface it grows on is of major importance [@Hall-Stoodley2004; @Conrad18]. The relevance of the interaction between biological matter and confining geometries in many physiological processes has prompted extensive experiments. In particular, [*in-vitro*]{} experiments on epithelial cells [@Vedula2012; @Marel2014; @Marel2014_2; @Yang2016; @Gauquelin2019; @Duclos2017; @Duclos2018], bacteria [@Wioland2016; @Conrad2018], and mixtures of biofilaments and molecular motors [@Suzuki2017; @Wu2017] have shown that confinement to channel-like geometries generically alters the flow of biological matter in a significant way. Striking examples are the crossover from chaotic flows of bacteria [@Wioland2013; @Wioland2016] and microtubule/motor protein mixtures [@Hardouin2019; @Opathalage2019] to polarised movement along the long channel axis and the emergence of shear flow in fibroblast cells [@Duclos2018] confined within rectangular geometries. Moreover, theoretical and computational studies of active matter interacting with a periodic array of obstacles have shown a reduction in the effective diffusion coefficient of active particles with increasing density of obstacles [@Brun18] and modification of the active dispersion within periodic arrays in the presence of applied flows [@Alonso19].
While recent research has been primarily focused on the crosstalk between the confinement and the activity of the particles in order to determine patterns of motion, a complete understanding of the patterns of collective invasion and spreading also requires a detailed investigation of growth dynamics. From the physical perspective, such interplay between activity, confinement, and growth can be accompanied by additional complexities arising from hydrodynamic interactions between growing active matter and the confinement, orientation dynamics of elongated active particles, and emergent collective phenomena such as active turbulence [@Wensink2012; @Thampi2014; @Giomi2015; @James2018; @Bratanov2015] and active topological defects [@Elgeti2011; @Giomi2013; @Giomi2014; @Saw2017]. Due to this interconnection of complex physical processes, the mechanistic understanding of active matter invasion within geometrical constraints remains largely unexplored. Recent numerical analyses of a self-propelled particle model [@Tarle2017], neglecting orientational dynamics and hydrodynamic effects, have shown that the physics of growing active matter can explain some of the observed experimental phenomena [@Vedula2012], such as caterpillar motion of the advancing front and the enhancement of collective migration speed in thin capillaries, by assuming a coupling between the curvature of the front and the motility of the particles at the leading edge.
Here we use a generic model of active matter that accounts for hydrodynamics, orientational effects, and growth to investigate collective patterns of active matter invasion of capillaries. We identify three different regimes of invasion, each with distinct interface shapes, flow patterns, orientational ordering, and topological defect dynamics. In particular, we show that above a certain threshold in the strength of the active forces, highly dynamic deformations are formed at the interface between the invading active matter and the surrounding medium. At higher activities, we also find a second threshold beyond which blobs of active matter begin to detach from the growing active column to enhance the invasion of free space. We explain the first crossover in terms of intrinsic hydrodynamic instabilities of the bulk active matter accompanied by additional instabilities arising from the presence of the growing interface. The latter can be understood by the ability of active stresses to overcome the stabilising effect of the surface tension and pinching off the growing interface. Together, our study reveals several possible invasion patterns and collective dynamics when active matter invades a capillary constriction thus providing a framework for further experimental investigations.
![*Simulation setup* (A) Schematic drawing of simulation setup. The active phase (yellow) invades from the broader reservoir (region below black dashed line) where it can grow into the narrower capillary (width $d$) that is initially filled with isotropic liquid (green). The broad gray arrow illustrates the invasion direction. Different heights are marked for later discussion of observables: $h_{\text{max}}$, the highest point of the active phase which is connected to the reservoir, $h_{\text{inv}}$, the highest point that any patch of active phase reaches, and $h_{\text{min}}$, the highest point where the capillary is filled with active fluid to the full width. (B) A representative simulation snapshot to illustrate the dynamics. Black bars mark the director field; the two green speckles inside the nematic phase are cores of +1/2 topological defects.[]{data-label="fig:setup"}](./newfigures/figure1_rendered.pdf)
Model\[section:model\]
======================
We employ a two-phase model of active nematohydrodynamics [@Blow2014] to explore the growth of an active layer into otherwise isotropic surroundings within a confined channel and use a hybrid lattice Boltzmann (LB) method to numerically solve the equations [@Marenduzzo2007]. This choice was motivated by the generality of this class of continuum models which is due to the fact that only local conservation laws, a nematic interaction between the systems’ constituents, and perpetual injection of energy at the smallest length-scale are assumed. Models of this kind reproduce a variety of non-equilibrium flows such as stable arrays of vortices or chaotic flows, together with diverse stationary and non-stationary patterns of nematic ordering and topological defects [@Simha2002; @Thampi2013; @Giomi2013; @Giomi2015; @Thampi2014; @DeCamp2015; @Cortese2018; @Doostmohammadi2018]. On a phenomenological level, these are successful in modelling the collective dynamics in biological systems in cases such as microtubule/motor-protein mixtures [@Sanchez2012; @Giomi2013; @Thampi2013], cellular monolayers [@Bittig2008; @Duclos2014; @Doostmohammadi2015; @Saw2017], or bacteria [@Volfson2008]. In addition to the bulk dynamics, the interaction with walls or obstacles of different shape can add further complexity. Theoretically, using the active nematohydrodynamics framework, a transition to spontaneous flows in channels has been predicted [@Voituriez2005; @Edwards2009], as well as more complex states with intricate interplay of defects and vortices, and a transition to active turbulence [@Shendruk2017; @Doostmohammadi2017; @Norton2018].
Governing equations\[section:equations\]
----------------------------------------
We consider a two-dimensional model of a two-phase system consisting of an isotropic fluid phase and an active nematic phase [@Blow2014]. The orientational order in the nematic phase is characterised by the symmetric and traceless nematic tensor $Q_{\alpha\beta}=S\left(2n_{\alpha}n_\beta-\delta_{\alpha\beta}\right)$ [@deGennes] with $S$ (magnitude of the nematic order) and $n_\alpha$ (director) indicating the magnitude and the direction of the nematic order, respectively. The relative density of the nematic phase is measured by a scalar phase field $\phi$ which is 0 in the purely isotropic and 1 in the purely nematic phase.
The free energy density of the system is given by (using the Einstein summation convention): $$\begin{gathered}
\label{eq:freeenergydensity}
f=\frac12{D}\phi^2\left(1-\phi\right)^2+\frac12{C_{LQ}}\left(\phi-\frac12Q_{\alpha\beta}Q_{\alpha\beta}\right)^2\\+\frac12{K_{\phi}}\partial_\gamma\phi\partial_\gamma\phi+\frac12{K_Q}\partial_\gamma Q_{\alpha\beta}\partial_\gamma Q_{\alpha\beta},\end{gathered}$$ where ${D}$, ${C_{LQ}}$, ${K_{\phi}}$, and ${K_Q}$ are positive material constants. Since $Q_{\alpha\beta}Q_{\alpha\beta}=2S^2$, the second term ensures a tight coupling between the magnitude of the nematic order $S$, and the phase field $\phi$.. Together with the first term, which corresponds to a Cahn-Hilliard free energy [@ChaikinLubensky; @Orlandini1995], this leads to well-defined interfaces betweens a nematic ($S=1,\phi=1$) and an isotropic ($S=0,\phi=0$) phase (Fig. \[fig:setup\]). The third and fourth terms are elastic energies. The first, third, and fourth terms contribute to the surface energy and the fourth term also penalises bulk deformations in $Q_{\alpha\beta}$. The free energy then reads $$\begin{aligned}
\mathcal{F}=\int\text{d}^2{\mathbf}{r}\;f.\end{aligned}$$
The order parameters evolve according to the following equations: $$\begin{aligned}
\partial_t\phi+\partial_\beta\left(\phi u_\beta\right)=&\Gamma_\phi\Delta\mu,\label{eq:phi}\\
\left(\partial_t+u_\kappa\partial_\kappa\right)Q_{\alpha\beta}=&-{\xi}\Sigma_{\alpha\beta\kappa\lambda}E_{\kappa\lambda}-T_{\alpha\beta\kappa\lambda}\Omega_{\kappa\lambda}+\Gamma_Q H_{\alpha\beta},\label{eq:Q}\end{aligned}$$ with $$\begin{aligned}
\mu=&\frac{\partial f}{\partial\phi}-\partial_\gamma\frac{\partial f}{\partial\left(\partial_\gamma\phi\right)}\\
H_{\alpha\beta}=&\left(\delta_{\alpha\beta}\delta_{\kappa\lambda}-\delta_{\alpha\kappa}\delta_{\beta\lambda}-\delta_{\alpha\lambda}\delta_{\beta\kappa}\right)\nonumber\\&
\left\lbrace
\frac{\partial f}{\partial Q_{\kappa\lambda}}-
\partial_\gamma\left(\frac{\partial f}{\partial\left(\partial_\gamma Q_{\kappa\lambda}\right)}\right)
\right\rbrace,\\
E_{\alpha\beta}=&\frac12\left(\partial_\beta u_\alpha + \partial_\alpha u_\beta \right),\:\:\:\:\:
\Omega_{\alpha\beta}=\frac12\left(\partial_\beta u_\alpha-\partial_\alpha u_\beta\right),\\
\Sigma_{\alpha\beta\kappa\lambda}=&Q_{\alpha\beta}Q_{\kappa\lambda}-\delta_{\alpha\kappa}\left(Q_{\lambda\beta}+\delta_{\lambda\beta}\right)\nonumber\\
&-\left(Q_{\alpha\lambda}+\delta_{\alpha\lambda}\right)\delta_{\kappa\beta}+\delta_{\alpha\beta}\left(Q_{\kappa\lambda}+\delta_{\kappa\lambda}\right),\\
T_{\alpha\beta\kappa\lambda}=&Q_{\alpha\kappa}\delta_{\beta\lambda}-\delta_{\alpha\kappa}Q_{\beta\lambda}.\end{aligned}$$ The l.h.s. of Eqs. and are convective derivatives with the underlying velocity field $u_\alpha$. Advection and diffusion drive the dynamics in $\phi$; $\Gamma_\phi$ is the corresponding diffusion constant. $\Gamma_Q$ is a rotational diffusion constant which, together with the molecular field $H_{\alpha\beta}$, controls diffusive relaxation in $Q_{\alpha\beta}$. In addition, the interplay of flow $u_\alpha$ and order $Q_{\alpha\beta}$ is less trivial. The first and second terms on the r.h.s. of the Eq. form the co-rotational derivative which accounts for the response of the orientation field to the extensional ($E_{\alpha\beta}$) and rotational ($\Omega_{\alpha\beta}$) components of the velocity gradients respectively. ${\xi}$ is the tumbling parameter which determines the relative influence of the rate of strain on the director orientation. It depends on the geometry of the active particles, for prolate ellipsoids ${\xi}>0$, while for oblate ellipsoids ${\xi}<0$, and for spherical particles ${\xi}=0$ [@Larson99].
The velocity field $u_\alpha$ obeys the Navier-Stokes equations: $$\begin{aligned}
\partial_t\rho+\partial_{\beta}\left(\rho u_{\beta}\right)=&0,\\
\rho\left(\partial_t+u_{\beta}\partial_{\beta}\right)u_{\alpha}=&\partial_{\beta}\Pi_{\alpha\beta},\end{aligned}$$ where $\rho$ is the density of the fluid and $\Pi_{\alpha\beta}$ is the stress tensor comprising viscous stress, pressure contribution, elastic stresses, and the active stress: $$\begin{aligned}
\Pi^{visc}_{\alpha\beta}=&2\rho\eta E_{\alpha\beta},\\
\Pi^{p}_{\alpha\beta}=&-\frac\rho3\delta_{\alpha\beta},\\
\Pi^{el,1}_{\alpha\beta}=&\left(f-\mu\phi\right)\delta_{\alpha\beta}\nonumber\\&-\frac{\partial f}{\partial\left(\partial_\beta\phi\right)}\partial_\alpha\phi-\frac{\partial f}{\partial\left(\partial_\beta Q_{\kappa\lambda}\right)}\partial_\alpha Q_{\kappa\lambda},\\
\Pi^{el,2}_{\alpha\beta}=&\left({\xi}\Sigma_{\alpha\beta\kappa\lambda}+T_{\alpha\beta\kappa\lambda}\right)H_{\kappa\lambda},\\
\Pi^{act}_{\alpha\beta}=&-\zeta \phi Q_{\alpha\beta}.\end{aligned}$$ Here, $\eta$ is the viscosity, and the elastic stresses $\Pi^{el,i}_{\alpha\beta}$ describe feedback from variations in the order parameters on the fluid flow [@Blow2014]. The definition of the active stress $\Pi^{act}_{\alpha\beta}$ is such that any gradient in $Q_{\alpha\beta}$ generates a flow field and drives the system at small length-scales, with strength determined by the magnitude of the activity $\zeta$. A positive (negative) $\zeta$ corresponds to an extensile (contractile) material. The dipole flow-fields generated with this ansatz correspond to those of microswimmers - “pushers” generate extensile stresses, “pullers” contractile stresses [@Ramaswamy10]. The active stress continuously drives the system out of thermodynamic equilibrium. It will be of major importance in the following discussions.
Simulation setup\[section:setup\]
---------------------------------
In Fig. \[fig:setup\], we show the geometry for which we will solve the equations. This simulation setup corresponds to the experimental configurations for cell monolayers in references [@Vedula2012; @Marel2014; @Marel2014_2; @Yang2016; @Gauquelin2019]. It consists of a wider reservoir that feeds a narrower capillary. At $t=0$, the active phase is restricted to the large reservoir.
In order to generate new active material in the reservoir, we locally increase $\phi$ to values higher than $1.0$ at random positions. For every point in the active phase in the reservoir a *growth event* takes place with probability $r\phi(1 - \phi/\phi_{c})$. This means that for a given time $\tau_{g}$, a source term $\alpha\phi$ is added to the r.h.s. of Eq. in a circle of radius $r_{g}$ around this site. Diffusion and convection then lead to a spreading of the active phase, causing it to rise into the thinner capillary. This local implementation leads to additional flow in the reservoir as growth events generate dipole-like flow fields and a growth pressure through the isotropic part of $\Pi^{el,1}$. The logistic-growth-like saturation in the probability $r\phi(1 - \phi/\phi_{c})$ prevents unbounded growth of $\phi$. We find that the details of this implementation or even the geometry of a reservoir are not important for the qualitative dynamics in the capillary (see ESI section [A]{}).
In this study, we use the following parameters: ${D}=0.08$, ${C_{LQ}}=0.15$, ${K_Q}=0.02$, ${K_{\phi}}=0.08$, $\Gamma_\phi=0.2$, $\Gamma_Q=0.4$, ${\xi}=0.7$, $\eta=1/6$, $r=0.001$, $\tau_{g}=10000$, $r_{g}=5$, $\alpha=0.01$, $\phi_{c}=1.2$, unless otherwise noted. We vary $\zeta$ from $0.0020$ to $0.0100$ to investigate how invasion depends on the strength of the active driving. The free energy coefficients are chosen such that there is a well-defined interface between the active and the passive phase. ${\xi}=0.7$ leads to alignment of the director to an external flow in a passive nematic. Parameter fitting of the continuum equations to physical active systems remains a topic of research; therefore, we consider a generic parameter set that has been shown to reproduce the flow vortex-lattice generated by a dense assembly of endothelial cells [@Rossen2014; @Doostmohammadi2016NC], and the flow fields of dividing Madin-Darby Canine Kidney cells[@Doostmohammadi2015]. Independent samples for parameter sets are obtained by varying the initial condition randomly. Boundaries are no-slip with respect to the fluid and von-Neumann for $\phi$ and $Q_{\alpha\beta}$, meaning that there is no preferred anchoring of the director in the absence of activity. Variation of the boundary conditions can potentially lead to the emergence of additional exotic phases of invasion, but their systematic study lies beyond the scope of this work. For the reservoir, we have periodic boundary conditions in the direction perpendicular to the capillary axis.
In the following, we characterise the strength of the activity by the dimensionless activity number $A = d/(\sqrt{K_{\text{Q}}/\zeta}) = d/\Lambda_{\zeta}$. The activity-induced length-scale ${\Lambda_{\zeta}}=\sqrt{{K_Q}/\zeta}$ emerges from the competition between the activity driving the dynamics and the elastic resistance against deformations in the director field [@Thampi2014; @Hemingway2016; @Guillamat17], while the capillary width $d$ imposes an upper limit for hydrodynamic interactions across the capillary. The activity number $A$ relates these two length-scales [@Shendruk2017; @Doostmohammadi2017].
Results\[section:results\]
==========================
Distinct invasion regimes
-------------------------
For a first characterisation of the invasion behaviour of active matter into a capillary, we will focus on how the phenomenological changes in the structure and the dynamics of the interface between the active phase and the isotropic fluid depend on the dimensionless activity $A$. Surprisingly, we observe that this can be categorised into three different regimes separated by two well-defined crossovers under variations of this single quantity alone.
Figure \[fig:regimes\]A shows characteristic snapshots for the three different regimes. For small activities, below a certain threshold ($A\sim16$), the system is characterised by a flat interface between the isotropic and active phase that advances steadily from the reservoir. We denote this as regime I. Due to the extensile activity of the particles (i.e. $\zeta>0$), the active stresses generate a preferential orientation of the director parallel to the interface, an effect called ‘active anchoring’ [@Blow2014]. For low $A$, the director field remains homogeneous throughout the nematic phase and the hydrodynamic instabilities are suppressed.
When approaching the crossover from regime I to regime II by increasing the activity, bend deformations of the director field arise in the bulk without influencing the shape of the interface (see ESI Movies [flows0020.mp4]{} and [flows0030.mp4]{}). The defining property of regime II is the crossover to a state where the interface is deformed to an S-shape while keeping a $90^{\circ}$ contact angles at both walls. As discussed in more detail in the next section, within this regime the deformed interface initially advances with the leading point on one side of the capillary wall (see ESI Movie [flows0035.mp4]{}). With a further increase in activity, we observe a periodic switching of the S-shape of the interface from one side of the capillary to the other side as the system approaches the third invasion regime.
In both regimes, I and II, the active phase remains as one single coherent phase which is connected to the reservoir. This changes with the crossover to regime III at $A\sim20$, where a higher activity leads to the dispatching of small clusters of active material from the main body of the active phase, that protrude deeper into the capillary. In addition, the active interface and the director field are strongly deformed and highly dynamic. Defects are regularly created at the boundary and move throughout the bulk of the active phase (see ESI Movie [flows0060.mp4]{}).
![*Comparison of flows across regimes.* (A) Root mean-squared velocity in units of $d$ over the active time scale $\tau_\zeta=\eta/\zeta$ [@Orlandini1995] plotted against $A$ . Crossover from regime I to II coincides with the appearance of finite flows. Strength of flows increases inside regimes II and III with a small plateau at the crossover from regime II to regime III. Lower case letters indicate the values at which examples in (B-D) were taken. (B-D) Characteristic velocity and orientation fields near the interface. (B) For $A=15.5$, the system is in regime I, the interface is flat with no flows near it. (C) At $A=16.7$, in regime II, the interface is bent and the characteristic flow that turns at the interface is visible. (D) In regime III ($A=20.4$), the interface can take various forms, $+1/2$-defects (green dot) appear in the bulk and the flow becomes highly dynamical and complex. ESI Movies corresponding to panels B-D are [flows0030.mp4]{}, [flows0035.mp4]{}, and [flows0052.mp4]{}, respectively, see ESI section [C]{}.[]{data-label="fig:flows"}](./newfigures/figure3_rendered.pdf)
Detailed characterisation of invasion regimes and crossovers\[section:results\_quantitative\]
---------------------------------------------------------------------------------------------
To understand the connection between the different interface dynamics and the overall invasion process, we further characterise the defining features of the three different regimes. Quantitatively, the crossovers between the regimes can be read off from two observables: (i) ${\ensuremath{\left(h_{\text{max}}-h_{\text{min}}\right)/d}}$, which characterises deformations of the isotropic-active interface in the capillary (see Fig. \[fig:setup\]A for definition of $h_{\text{max}}$ and $h_{\text{min}}$), and (ii) $N_c$ which is the number of clusters of the active phase that are in the system in addition to the main body of the active phase connected to the reservoir. As the activity number is increased, the crossover from regime I to regime II is reflected by a jump in the mean value of $h_{\text{max}}-h_{\text{min}}$ from 0 to the width $d$ of the capillary, or equivalently a jump of ${\ensuremath{\left(h_{\text{max}}-h_{\text{min}}\right)/d}}$ from 0 to 1 (Fig. \[fig:regimes\]B), which mirrors the S-shape of the interface in regime II. Within the parameter range studied here, we did not observe any variation in the positioning of the crossover from regime I to regime II under variation of ${D}$. The second crossover, from regime II to regime III, is also marked by a further jump in ${\ensuremath{\left(h_{\text{max}}-h_{\text{min}}\right)/d}}$, but is even more evident in $N_c$ which becomes finite (Fig. \[fig:regimes\]C), reflecting the emergence of additional smaller clusters that detach from the main active phase. It is noteworthy that a similar quantitative behaviour is observed when varying the width of the channel (see ESI Fig. [2]{} in ESI section [B]{}), indicating that the activity number is the relevant dimensionless parameter in this setup.
### Properties of regime II.
The crossover from regime I to II is also accompanied by sharp changes in the flow and director fields in the capillary. This not only sheds light on the underlying mechanism, but also has consequences for the overall speed of invasion as we will see later (section \[section:invasion\]). Figures \[fig:flows\]A,B show that below the first crossover, there are no flows in the capillary ($v_{\text{rms}}=0$). For higher activities, with a non-zero ${\ensuremath{\left(h_{\text{max}}-h_{\text{min}}\right)/d}}$, finite flows are generated (Figs. \[fig:flows\]A,C). This is reminiscent of the well-established spontaneous flow transitions in confined active matter [@Voituriez2005; @Marenduzzo2007], which arise due to the formation of hydrodynamic instabilities in active nematics above a certain threshold of activity. While in those cases the active force alone is responsible for the transition, in the present situation the growth dynamics in the reservoir induces long-range effects which influence the position of the crossover (for a system without reservoir, see ESI section [A]{}). The advection of active material along with these flows causes the characteristic S-shape of the interface in this regime. Interestingly, $v_{\text{rms}}$ is increasing with higher activity numbers within regime II while ${\ensuremath{\left(h_{\text{max}}-h_{\text{min}}\right)/d}}\sim1$ throughout.
Moreover, because of the impact of the extensile activity that keeps the director aligned with the interface, the increasing bend deformation looks similar to a backwards pointing $+1/2$-topological defect at the less advanced side of the interface (Fig. \[fig:flows\]C; right panel). Corresponding to the characteristic self-motility of $+1/2$-topological defects [@Putzig2014; @Shendruk2017; @Saw2017], the interface on this side continues to move backwards relative to the mean interface-position stretching out the interface until the defect detaches from the interface and moves into the active phase. Concomitantly with this defect absorption into the nematic, the interface re-attaches to the wall at a higher position. Repeatedly, this leads to a periodic shape-change in the interface (see Fig. \[fig:dynamics\]A and the ESI Movie [flows0035.mp4]{}). As the activity is increased further, the initial spontaneous transition to flows is followed by the generation of counter-rotating vortices along the capillary. As the active phase grows and advances through the capillary, a lattice of vortices is formed behind the interface, reminiscent of vortex-lattices in a non-growing confined active matter [@Doostmohammadi2017; @Theillard17], which appear when the activity-induced vorticity length scale becomes comparable to the capillary width. Interestingly, however, in this growing active matter, the orientation of the vortex closest to the interface also determines the parity of the interface. As a consequence, a change in the orientation of the most forward vortex leads to a switching at the interface, with the S-shape of the progressive front flipping from one side of the channel to the other with respect to the capillary axis (see Fig. \[fig:dynamics\]B and the ESI Movie [flows0046.mp4]{}).
Within regime II, the rms-velocity $v_{\text{rms}}$ grows approximately linearly with the activity number $A$ and the flow field remains structured until transitioning to active turbulence in regime III (Fig. \[fig:flows\]D).
![*Interface dynamics in regimes II and III.* (A) Characteristic periodic motion in regime II ($A=17.9$). The $+1/2$-defect like bend-deformation on the right wall moves backwards relative to the interface until it is pinched off and dissolved in the nematic bulk. Simultaneously the interface relaxes back to a less stretched state and the cycle begins anew. (B) Switching in the S-shape for higher activities in regime II ($A=19.2$). When approaching the second crossover, vortices emerge in the capillary (panels on the right). When the most forward vortex changes its rotation direction, the director follows and bend deformations as well as interface shape adapt accordingly. (C) Time series of a cluster-formation event in regime III ($A=20.4$). The nematic layer at the wall becomes unstable until finally a cluster detaches from the main active phase. In each of (A-C) the simulation time between all sequential snapshots is equal and is measured in units of the active time scale $\tau_\zeta=\eta/\zeta$ [@Thampi2015], A: $\delta t=480\tau_{\zeta}$, B: $\delta t=2760\tau_{\zeta}$, C: $\delta t=1248\tau_{\zeta}$. ESI Movies corresponding to panels B and C are [flows0046.mp4]{} and [flows0052.mp4]{}, respectively, see ESI section [C]{}.[]{data-label="fig:dynamics"}](./newfigures/figure3_2_rendered.pdf)
### Properties of regime III.
While the crossover to regime II was solely controlled by the hydrodynamic instability to spontaneous flow formation within the bulk of the active matter in the capillary, the crossover from regime II to III, where active clusters appear, is strongly influenced by the competition between activity and surface tension at the interface. This interfacial effect can be clearly seen in a close-up view of the interface at the onset of cluster detachment (Fig. \[fig:dynamics\]C): At a sufficiently high activity number relatively long and thin layers of active matter with an approximately uniform director field are formed at the sides of the capillary wall. As the simulation time goes by, a small deformation at the tip of the layer develops and grows until it breaks away from the active layer, forming a detached cluster (see ESI Movie [flows0052.mp4]{}). The formation of clusters means an increase in total interface length which is increasingly unfavourable for higher surface tension and thus requires higher active stresses to be generated. Therefore, the breakup of the thin layer can be understood as the destabilising effect of the activity that leads to the interface deformation dominating over the stabilising effect of the surface tension, working to keep the interface straight. However, especially in this complex setup, it is not possible to decide to what degree this effect is purely interfacial and whether dynamics in the bulk are important. The simultaneous appearance of $+1/2$ topological defects in the nematic phase (see ESI Fig. [3]{}) hints that changes in the bulk dynamics are also involved in the crossover as flow dynamics become more reminiscent of turbulent dynamics and higher active stresses are generated. Interestingly, directly after the crossover the number of additional clusters peaks. With a further increase in $A$ the clusters are still present, however the number of clusters goes down. A close look at the interface reveals that at very high activities the active material forms extremely long and thin layers on the capillary walls and the detached clusters quickly reattach to the main body of the active nematic, leading to a smaller number of clusters on average.
Invasion capability\[section:invasion\]
---------------------------------------
In the context of our original research question we now investigate the influence of the different regimes on the capability of the active system to claim new territories. To this end, we show in Fig. \[fig:invasion\]A the total amount of active material in the capillary at the end of the simulation ($9.5\:10^6$ simulation steps) ${\Phi_T}:=\int \phi\left(\mathbf{r},T\right)\:\text{d}^2\mathbf{r}/d^2$, which we denote as the invasion index. In the absence of flows (regime I, when the process is purely diffusive), the invasion index $\Phi_T$ is almost independent of $A$ and it is significantly lower than in regimes II and III, where activity-induced flows enhance the advective transport of active material from the reservoir to the capillary. After the crossover from regime I to II, ${\Phi_T}$ follows a linear dependence on $A$ similar to the linear increase of $v_{rms}$ in this region. The crossover from regime II to III does not alter this quantity in a qualitative way, i.e. the additional clusters have no influence on the total amount of active material that invades the capillary. However, they can be shown to invade about $0.5$ to $1.0$ capillary widths deeper into the capillary than the tip of the coherent active phase connected to the reservoir. This is best illustrated by plotting the relative difference of the maximum invasion height $h_{\text{inv}}$ (see Fig. \[fig:setup\]A) and the maximum height of the interface $h_{\text{max}}$: ${\ensuremath{\left(h_{\text{inv}}-h_{\text{max}}\right)/d}}$ (Fig. \[fig:invasion\]B). Together, these results indicate that the flows in the capillary and the interfacial dynamics control the rate at which active material enters the capillary and thus determine the invasion speed.
![*Propensity of the system to invade the capillary.* (A) Invasion index ${\Phi_T}:=\int \phi\left(\mathbf{r},T\right)\:\text{d}^2\mathbf{r}/d^2$ defined as the amount of active material inside the capillary after $9.5\:10^6$ simulation steps plotted versus activity number $A$. In regime I, the invasion shows little dependence on $A$ until the appearance of flows at the crossover from regime I to II leads to a sharp increase. (B) Relative difference of the maximum invasion height and the maximum height of the interface ${\ensuremath{\left(h_{\text{inv}}-h_{\text{max}}\right)/d}}$ against $A$. In regimes I and II, this difference is trivially zero. In regime III it shows an increase of up to one capillary-width meaning the clusters protrude significantly deeper into the capillary than the main body of the growing active matter.[]{data-label="fig:invasion"}](./newfigures/figure4_rendered.pdf)
Conclusions\[section:conclusion\]
=================================
In this work, we have presented a two-phase computational framework for studying the growth of active matter within an isotropic fluid medium. Our approach accounts for hydrodynamic effects and the orientational dynamics of active particles, and is able to reproduce emergent active phenomena including spontaneous flow generation and active turbulence. Motivated by the prevalence of active matter growing within confined spaces and constrictions, we investigated the dynamics in capillaries of different sizes, focusing on the combined effects of growth and the active stresses that are continuously exerted by active particles on their surroundings.
We find three distinct invasion patterns as the activity is varied: (i) at small activities invasion is completely controlled by the growth dynamics and the interface between the active phase and the surroundings remains stable with no, or slight, deformations. (ii) Increasing activity beyond a given threshold results in spontaneous flow generation which significantly enhances the invasion within the capillary and is accompanied by deformations of the interface. (iii) At yet higher strengths of activity, a second crossover appears, where active clusters begin to detach from the main body of the invading active phase, further enhancing the invasion of the surrounding space.
Our analysis shows that the crossovers between the various invasion regimes are controlled by different mechanisms. The crossover between regimes I and II is governed by the well-established spontaneous flow formation in confined active matter [@Voituriez2005; @Edwards2009], which is due to a hydrodynamic instability of the active nematic. We show that the onset of flow within the bulk is accompanied by the deformation of the interface and can even result in a periodic switching of the active protrusions from one side of the capillary to the other, as flow vortices start to form behind the advancing front. When activity becomes stronger, turbulent flows and defects are observed in the bulk of the active material and, dependent on its surface tension, the system also becomes able to rip apart the active-to-passive interface marking the third invasion regime.
Our results thus highlight the significant differences in invasion patterns when the activity of growing matter is accounted for. From a biological perspective, this could be linked to the invasive behaviour of biological matter such as growing colonies of cells or bacteria. It suggests that qualitative changes in the spread of these organisms into confined spaces can be caused by changes in the availability and conversion rate of chemical energy into mechanical stresses. One could conjecture that more invasive cell lines or bacterial strands could regulate their invasion of pores or cavities by tuning their strength of motion. For example, inducing the formation of additional clusters might be beneficial for cell-lines that aim at aggressive invasion of surrounding tissue while, for bacteria, it might be beneficial to remain coherent in order to profit from the benefits of cooperative behaviour.
Considering possible experimental realisations, our model predictions apply to active growing systems, where hydrodynamic interactions and nematic orientational order play an important role. Such nematic ordering has been reported in various biological systems including bacterial colonies [@Volfson2008; @Li19], cultures of amoeboid cells [@Gruler99], fibroblast cells [@Duclos2014; @Duclos2017], human bronchial cells [@Blanch18], neural progenitor stem cells [@Kawaguchi2017], and Madin-Darby Kanine Kidney (MDCK) epithelial monolayers [@Saw2017; @Saw18]. Investigating the degree to which experiment and theory can be matched by the adaptation of parameters and boundary conditions in the model or the reasons for deviations is a powerful way of unraveling the role of mechanics in determining the behaviour of such active biological matter. It would also be interesting to compare to models that resolve individual particles such as models of self-propelled particles [@Tarle2017], phase field approaches [@Shao2010; @AransonBook; @Mueller2019], or cellular Potts models [@Thueroff2019]. In contrast to these, our model by construction cannot resolve phenomena or implement mechanisms that act on the single particle-level, for example dynamics of cell size or shape, but due to its generality could contribute to a deeper understanding of the generic fundamental mechanisms that underlie the invasion of microscopic biological systems.
Finally, several improvements can be envisaged for the current model. In the context of growing tissues, studies have highlighted the importance of leader cells at the progression front [@Poujade07; @Trepat09]. Within particle-based models such an effect is shown to be captured by introducing a curvature-dependent motility for the cells at the interface [@Mark10; @Tarle2017]. Within our proposed framework, a similar effect could be modelled by introducing a curvature dependence to the activity coefficient. In addition, the natural environment of active material is often more complicated than the isotropic fluid considered here and active invasion happens through viscoelastic media. For example, cells invade the interconnected networks of collagen matrices and bacteria secrete their own extracellular matrices. The continuum framework presented here could be extended to include viscoelasticity by introducing additional order parameters representing polymer conformation.
Conflicts of interest {#conflicts-of-interest .unnumbered}
=====================
There are no conflicts to declare.
Acknowledgements {#acknowledgements .unnumbered}
================
E.F. and F.K. would like to thank Sophia Schaffer, Matthias Zorn, and Joachim Rädler for stimulating discussions. A.D, F.K., and J.M.Y. would like to thank Kristian Thijssen for helpful discussions. This research was supported by the German Excellence Initiative via the program “Nanosystems Initiative Munich” (NIM) and the Deutsche Forschungsgemeinschaft (DFG) via project B03 within the Collaborative Research Center (SFB 1032) “nanoagents”. R.M. was supported by grant P2EZP2\_165261 of the Swiss National Science Foundation. A.D. was supported by the Royal Commission for Exhibition of 1851 Research Fellowship.
The preprint template has been taken from <https://github.com/brenhinkeller/preprint-template.tex/>
Supplementary Materials {#supplementary-materials .unnumbered}
=======================
Role of reservoir and growth\[appendix:reservoir\]
==================================================
In this section we examine the importance of the cell division and the presence of a reservoir on the dynamics reported in the main text. ESI Figure \[fig:reservoir\] shows simulation results for two systems that differ from the one considered in the main text, by the absence of a wider reservoir, and by the absence of cell division, respectively. We see that growth is an essential factor to create the phenomena reported in the main text as in the absence of growth the behaviour is qualitatively different. If the reservoir is absent, meaning that there is only a capillary of uniform width where growth takes place in the lower region, we in contrast observe the first crossover at the same values for the activity $A$ (ESI Fig. \[fig:reservoir\]A), together with the same change in the invasion speed (ESI Fig. \[fig:reservoir\]B). The second crossover is however significantly shifted to higher values of $A$ (see ESI Fig. \[fig:reservoir\]C,D), indicating that the reservoir has long-range effects on the dynamics in the capillary. These, however, do not change the qualitative picture.
Universality for different channel widths\[appendix:width\]
===========================================================
As described in the main text, the activity number $A$ characterises the ratio of the channel width $d$ to the intrinsic length scale of the active phase. In ESI Figure \[fig:universality\] the observables ${\ensuremath{\left(h_{\text{max}}-h_{\text{min}}\right)/d}}$ and $N_c$ are plotted against $A$ for varying channel width $d$. Both crossovers happen at the same values for $A$, independent of $d$. In addition, both observables are of the same order of magnitude and within regime II ${\ensuremath{\left(h_{\text{max}}-h_{\text{min}}\right)/d}}\sim1$. These results indicate that the activity number is the relevant dimensionless parameter in this setup and that ${\ensuremath{\left(h_{\text{max}}-h_{\text{min}}\right)/d}}$ is a meaningful observable.
Description of Movies
=====================
All movies show the time-evolution of the velocity field on the left hand side and the orientation field on the right hand side, in the same way as panels B-D in Figure [3]{}. We choose different values of the activity $\zeta$ to display the qualitatively different phenomena. There are the following files:
Higher quality movies can be found on [](https://doi.org/10.1039/c9sm01210a).
flows0020.mp4\[movie:0020\]
: $\zeta=0.0020$. System is in regime I, director homogeneous and parallel to the interface. Corresponds to Figure [2]{}A, leftmost panel.
flows0030.mp4\[movie:0030\]
: $\zeta=0.0030$. System is in regime I, but the director starts to show deviations from the perfectly homogeneous configuration, especially close to the reservoir. Corresponds to Figure [3]{}B.
flows0035.mp4\[movie:0035\]
: $\zeta=0.0035$. System is in regime 2, showing characteristic flow- and director-fields. At later times the periodic dynamics described in section [3.2.1 ]{}can be observed. Corresponds to Figures [2]{}A (middle panel) and [3]{}C.
flows0046.mp4\[movie:0046\]
: $\zeta=0.0046$. System is the higher activity region of regime II. The video shows two examples of the switching behaviour described in section [3.2.2 ]{}and Figure [4]{}B.
flows0052.mp4\[movie:0052\]
: $\zeta=0.0052$. System is in regime III. This example shows the typical birth of small active clusters, corresponding to the situations shown in Figures [3]{}D and [4]{}C.
flows0060.mp4\[movie:0060\]
: $\zeta=0.0060$. System is deep in regime III. The dynamics is more turbulent than in the video for $\zeta=0.0052$; there are more defects and clusters. Corresponds to the example shown in Figure [2]{}A on the right panel.
Supplementary figures
=====================
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'In terms of the fully relativistic screened Korringa-Kohn-Rostoker method we investigate the effect of stacking faults on the magnetic properties of hexagonal close-packed cobalt. In particular, we consider the formation energy and the effect on the magnetocrystalline anisotropy energy (MAE) of four different stacking faults in hcp cobalt – an intrinsic growth fault, an intrinsic deformation fault, an extrinsic fault and a twin-like fault. We find that the intrinsic growth fault has the lowest formation energy, in good agreement with previous first-principles calculations. With the exception of the intrinsic deformation fault which has a positive impact on the MAE, we find that the presence of a stacking fault generally reduces the MAE of bulk Co. Finally, we consider a pair of intrinsic growth faults and find that their effect on the MAE is not additive, but synergic.'
address:
- '$^1$ Department of Physics, University of York, York YO10 5DD, United Kingdom'
- '$^2$ Department of Theoretical Physics and Condensed Matter Research Group of Hungarian Academy of Sciences, Budapest University of Technology and Economics, Budafoki út 8., H1111 Budapest, Hungary'
author:
- 'C.J. Aas$^1$, L. Szunyogh$^2$, R.F.L. Evans$^1$, R.W. Chantrell$^1$'
title: 'Effect of stacking faults on the magnetocrystalline anisotropy of hcp Co: a first-principles study'
---
Introduction
============
Within the magnetic recording industry, cobalt alloys such as CoPt and CoPd are of great interest due to their large magnetocrystalline anisotropy energies (MAE) [@luklemmer]. For the purpose of magnetic recording, a large MAE of the recording medium is crucial in order to maintain stability of the written information as larger areal information storage densities require smaller grain sizes [@weller]. In close-packed metals and alloys, stacking faults are known to form relatively easily [@chetty]. This is one of the contributing factors to the relatively large ductility and malleability that are observed in many such materials [@chetty]. For a magnetic recording medium, the presence of stacking faults is generally considered detrimental, as disturbances in the microstructure will generally worsen the signal-to-noise ratio of the medium [@sokalski]. Stacking faults may also break the local lattice symmetry and, therefore, drastically influence the MAE.\
Experimentally, the effects of stacking faults are generally measured in terms of the stacking fault density, which can be determined from X-ray diffraction spectra (see e.g. [@luklemmer; @sokalski; @mitra]). There are a large number of experimental studies into stacking fault formation energies and the effect of the stacking fault density on the magnetocrystalline anisotropy for various magnetic recording alloys [@luklemmer; @saito]. However, in experiment, the real effect of a stacking fault might be obscured by other phenomena, such as migration of impurities along the stacking fault, synergies of closely spaced stacking faults, etc. Consequently, a number of theoretical methods have been developed for determining the properties and effects of stacking faults, see e.g. [@berliner]. In particular, there is a large number of first-principles studies of the formation energies of given types of stacking faults in metals [@chetty; @crampin]. It has been suggested that stacking fault formation energies determined from first-principles may be more accurate than experimental measurements [@chetty] as theoretical calculations separate the formation energy from any other correlated effects on the total energy. The effect on the MAE of a particular stacking fault is, however, less commonly explored. In this work, we aim to determine from first principles the effect on the MAE of four different types of stacking faults in hcp cobalt.
The stacking faults {#sec:stackingfaults}
===================
Hexagonal planes can be packed either in an ...ABAB... sequence, yielding a hexagonal close-packed lattice structure, or in an ...ABCABC... sequence, yielding a face-centred cubic lattice structure [@physmetbook]. In the hexagonal close-packed lattice structure, the stacking direction corresponds to the $(0 0 0 1)$ axis of the lattice, whereas for the face-centred cubic lattice structure, the stacking direction is parallel to the $(1 1 1)$ axis of the lattice. In a hcp lattice, a stacking fault is defined as an interruption in the ...ABAB... stacking of the hexagonal planes. While there are of course any number of conceivable stacking faults, their varying degrees of formation energies and formation mechanisms mean they have different probabilities of occurrence [@crampin]. In line with previous work [@chetty; @zope], we consider the following four different stacking faults, denoted in standard notation as I$_1$, I$_2$, E and T$_2$ [@sfbook1; @frank]:
- I$_1$ (intrinsic): $\cdots$ B A B A ${\mathrm{\bf B}}$ C B C B $\cdots$
- I$_2$ (intrinsic): $\cdots$ A B A B ${\bf \mid}$ C A C A $\cdots$
- E (extrinsic): $\cdots$ A B A B ${\mathrm{\bf C}}$ A B A B $\cdots$
- T$_2$ (twin-like): $\cdots$ A B A B ${\mathrm{\bf C}}$ B A B A $\cdots$
Here the bold face letters or vertical line denote the plane of reflection symmetry of the stacking fault. In an *intrinsic* stacking fault (I$_1$ and I$_2$), the stacking fault is simply a shift of one lattice parameter and the stacking on either side is correct all the way up to the very fault [@physmetbook]. The stacking fault I$_1$ is a growth fault while the stacking fault I$_2$ is a deformation fault [@chetty]. In the *extrinsic* stacking fault (E), a plane has been inserted so that it is incorrectly stacked with respect to the planes on either side of it [@physmetbook; @Hammer]. In a *twin*-like fault (T$_2$), the stacking sequence is reflected in the fault layer [@chetty]. In the following, we refer to the centre of reflection symmetry as the zeroth layer. The two layers adjacent to the centre of reflection symmetry are then indexed $\pm 1$, and so on. Note that in the case of a stacking fault of type I$_2$, the plane of reflection symmetry lies in between two atomic layers. Therefore, in the following, for type I$_2$ the atomic layers will be labelled by $\pm \frac{1}{2}, \pm \frac{3}{2}, \dots$, rather than by $0, \pm 1, \pm 2, \dots$ as for the other types of stacking faults.
Computational details
=====================
For our theoretical study we employed the fully relativistic Screened Korringa-Kohn-Rostoker (SKKR) method, in which the Kohn-Sham scheme is performed in terms of the Green’s function of the system (rather than the wavefunctions) and the treatment of extended layered systems is particularly efficient [@laszlo1; @screening1; @screening2]. We used the local spin density approximation (LSDA) of density functional theory (DFT) as parametrised by Vosko and co-workers [@voskoCJP80]. The effective potentials and fields were treated within the atomic sphere approximation (ASA) and an angular momentum cut-off of $\ell_{max}=2$ was used. The magnetocrystalline anisotropy energy (MAE) is calculated using the magnetic force theorem [@jansen], within which the MAE is defined as the difference in the band energy of the system when magnetised along the easy axis $( 0 0 0 1)$ and perpendicular to the easy axis. Alternatively, the uniaxial MAE can be calculated from the derivative of the band energy, for more details see Ref. [@aas-PtCo]. Only when calculating the MAE, we used an angular momentum cut-off of $\ell_{max}=3$.\
The LSDA+ASA fails in describing the orbital moment and the MAE of Co accurately. Similar to our previous work [@aas-PtCo] we, therefore, employed the orbital polarisation (OP) correction [@orbcorr; @eschrig; @eschrig2], as implemented within the KKR method by Ebert and Battocletti [@orbcorrKKR]. Note that the OP correction was applied only for the $\ell=2$ orbitals. Excluding the OP correction we obtained an easy-plane magnetisation and a MAE of 6.7 $\mu$eV per cobalt atom, while including the OP correction we obtained an easy axis perpendicular to the hexagonal cobalt planes and a MAE of 84.4 $\mu$eV per cobalt atom. This is in good agreement with the experimental value of 65.5 $\mu$eV [@stearns] and with the experimental easy axis being parallel to the $(0 0 0 1)$ direction. Our result also compares well with that of Trygg [*et al.*]{} [@trygg95], who calculated $K=110$ $\mu$eV for hcp cobalt using a full-potential LMTO method including the OP correction.\
In this study we consider an infinite cobalt system, consisting of two semi-infinite bulk cobalt systems and an internal region (region $I$). Region $I$ contains the stacking fault and is positioned in between the semi-infinite regions. The combined system is periodic and infinite in the plane normal to the $(0 0 0 1)$ direction. Due to the long-ranged nature of the stacking fault effects on the MAE (see section \[sec:aggMAECo\]), the region $I$ in this study had to be kept at a size of around 80 atomic layers. More specifically, for stacking faults I$_1$ and I$_2$, systems of 80 atomic layers were required, while for stacking faults E and T$_2$, 74 atomic layers were required. In order to keep the calculations tractable we limited the self-consistent calculations only for a number of atomic layers near the stacking fault, and then appended the bulk potentials for the atomic layers further away from the stacking fault. We found that it was sufficient to treat only the 20 centremost layers self-consistently. In line with previous first-principles studies of stacking faults in close-packed metals, we ignored any atomic and volume relaxations (see e.g. [@crampin]). The effects of such relaxations are normally negligible because atoms in the faulted part of the system tend to retain their close-packed coordination numbers despite the presence of the fault [@chetty; @Hammer; @denteneer; @twins; @schweizer; @wrightSF]. Throughout, therefore, we have used the experimental lattice parameter for cobalt, $a=2.507$ Å.
Results
=======
Stacking Fault Formation Energies
---------------------------------
Before exploring how the stacking faults influence the MAE of bulk Co, we would like to gain an idea of their formation energy. Within the SKKR-ASA scheme, the LSDA total energy can be cast into contributions related to individual atomic cells, $E_i$, comprising the kinetic energy, the intracell Hartree energy and the exchange-correlation energy, and into the two-cell Madelung (or intercell Hartree) energy, $E_{\rm Mad}$ [@laszlo1]. For a simple bulk metal, like hcp Co, $E_{\rm Mad}$ is, in practice, negligible, while in the presence of stacking faults it gives a non-negligible contribution due to charge redistributions. However, from our self-consistent calculations we found that $E_{\rm Mad}$ is in the order of $0.1-0.2$ meV per stacking fault. Since the typical formation energy of stacking faults are by about two orders higher in magnitude than this value, in the following we consider only the layer-resolved (cell-like) contributions to the the total energy. In order to check these contributions for artefacts of the appending of the bulk potential, we compare $E_i$ for $i=-10$ (layer with effective potential from a self-consistent stacking fault calculation) to $E_i$ for $i=10$ (layer with appended bulk potential), since due to the mirror symmetry, these two contributions should be identical. Reassuringly enough, they agreed to within a relative error of $10^{-9}$, which is well within intrinsic and numerical error of our computational method.\
The layer-resolved contributions to the total energy across the systems containing the stacking faults I$_1$, I$_2$, E and T$_2$ is shown in Fig. \[fig:layer-totEres\]. Herein we observe the expected mirror symmetry and that the layer-resolved total energy contributions approach the bulk total energy, $E_{Co}=-37839.459$ eV, towards the edges of each system. From this figure it is obvious that the deviation of $E_i$ from $E_{Co}$ is significant up to about 15 layers away from the centre of stacking fault.\
![The layer-resolved contributions to the total energy in four hcp cobalt systems, each exhibiting one of the four different types of stacking fault. The label $0$ refers to the plane of mirror symmetry. Solid lines serve as a guide for the eyes. \[fig:layer-totEres\]](fig1.pdf)
The stacking fault formation energy is defined as the difference in the total energy caused by the presence of the stacking fault. In order to ensure we include enough atomic layers on either side of the stacking fault, we consider the cumulative sums: $$\Delta E_{(\mathrm{I}_1, \mathrm{E},\mathrm{T}_2)}(N) =
\sum_{-N}^{N}E_{i} - (2N+1) E_{Co} \: , \label{eq:Esum1}$$ and $$\Delta E_{\mathrm{I}_2}(N) = \sum_{-N+\frac{1}{2}}^{N-\frac{1}{2}}E_{i} - 2N E_{Co} \: .
\label{eq:Esum2}$$ The formation energy of a given stacking fault $X=\mathrm{I}_1,\mathrm{I}_2, \mathrm{E},\mathrm{T}_2$, $E_{form}^{(X)}$, is then defined as $$E_{form}^{(X)} = \lim_{N \rightarrow \infty} \left( \Delta E_X (N) \right) \: .$$
![The cell-like part of the formation energy, $\Delta E_X (N)$, see Eqs. (\[eq:Esum1\]) and (\[eq:Esum2\]), of stacking faults I$_1$, I$_2$, E and T$_2$ in hcp cobalt, displayed as a function of the number of layers, $N$, considered in the system on either side of the stacking fault. Solid lines serve as guides for the eyes. \[fig:lmax2totE\]](fig2.pdf)
The calculated values of $\Delta E_{X}(N)$ are shown in Fig. \[fig:lmax2totE\]. Quite obviously, for all types of stacking faults, nearly 30 atomic layers (i.e., 15 layers on either side of the stacking fault) are required in order to obtain well-converged stacking fault formation energies. The fact that the effect of the stacking fault is relatively long-ranged could have significant impact on nano-sized systems as the formation energy and, consequently, the likelihood of occurrence of a stacking fault could be different depending on its location in relation to, e.g., other imperfections as well as surfaces or interfaces in the sample. We obtain the following formation energies, with a possible error of $\sim 0.1-0.2$ meV due to the Madelung energy not being included: $$\begin{aligned}
E_{form}^{(I_1)} & \approx & 16 \mathrm{\ meV \ } \approx 40 \mathrm{\ mJ}\cdot\mathrm{m}^{-2} \nonumber \\
E_{form}^{(I_2)} & \approx & 48 \mathrm{\ meV \ } \approx 122 \mathrm{\ mJ}\cdot\mathrm{m}^{-2} \nonumber \\
E_{form}^{(E)} & \approx & 62 \mathrm{\ meV \ } \approx 160 \mathrm{\ mJ}\cdot\mathrm{m}^{-2} \nonumber \\
E_{form}^{(T_2)} & \approx & 39 \mathrm{\ meV \ } \approx 100 \mathrm{\ mJ}\cdot\mathrm{m}^{-2} \: . \nonumber\end{aligned}$$ As expected, all stacking faults incur a positive change in the total energy. Of the four types of stacking faults considered here, the intrinsic stacking fault I$_1$ has the lowest formation energy and the stacking fault E exhibits the highest one. While there is no available experiment in literature, the overall results agree well with e.g. Ref. [@crampin]: the extrinsic stacking fault formation energy for the close-packed fcc metals in this study is generally significantly larger than that of the intrinsic and twin faults. Moreover, our calculated values for the hcp Co growth stacking fault I$_1$ and the hcp Co extrinsic fault E are close to those obtained by Crampin and co-workers for Ni (which is next to Co in the periodic table) [@crampin]: 28 $\mathrm{ mJ}\cdot\mathrm{m}^{-2}$ for the intrinsic stacking fault and 180 $\mathrm{ mJ}\cdot\mathrm{m}^{-2}$ for the extrinsic fault.
Layer-Resolved Contributions to the Magnetocrystalline Anisotropy Energy
------------------------------------------------------------------------
Because it is calculated directly from the band energy, the MAE can naturally be resolved into layer-dependent contributions, $D_i$, see Ref. [@aas-PtCo]. These layer-resolved contributions are depicted in Fig. \[fig:layer-res\] for the different types of stacking faults. Note that the mirror symmetry is well reproduced in the layer-resolved MAE contributions for all stacking faults. Moreover, the MAE approaches the bulk MAE, $K_{Co}=84.4$ $\mu$eV, towards the edges of all four systems. For stacking faults of type I$_1$, I$_2$ and T$_2$, the MAE contributions become negative at the centre of the fault, favoring thus an in-plane easy axis in these layers. This could indicate that these types of stacking faults may act as pinning sites. For the type E stacking fault, the layer-resolved MAE contributions near the centre are also reduced, retaining, however, very small positive values.\
![Calculated layer-resolved contributions to the MAE for stacking faults I$_1$ and $I_2$ (upper panel) and E and T$_2$ (lower panel). The horizontal line refers to the bulk MAE, 84.4 $\mu$ eV/atom. Solid lines serve as guide for the eyes. \[fig:layer-res\]](fig3.pdf)
Furthermore, we note that all stacking faults induce long-ranged oscillations in the MAE. For layers of about $\left| i \right| > 15$, the four stacking faults exhibit very similar trends in the layer-resolved MAE contributions. In other words, at about 15 layers away from the stacking fault, the presence of a stacking fault still influences the MAE, while the particular type of the stacking fault is less significant. This will, however, obviously depend on the size of the sample.
Finite Size Effect on the Magnetocrystalline Anisotropy Energy {#sec:aggMAECo}
--------------------------------------------------------------
The long-ranged oscillations in the MAE could cause significant finite-size effects in the experimental determination of the MAE of nano-sized samples. We therefore consider the following cumulative sums, $$K_{(\mathrm{I}_1, \mathrm{E},\mathrm{T}_2)}(N) =
\sum_{-N}^{N}D_{i} - (2N+1) K_{Co} \: , \label{eq:Ksum1}$$ and $$K_{\mathrm{I}_2}(N) = \sum_{-N+\frac{1}{2}}^{N-\frac{1}{2}}D_{i} - 2N K_{Co} \: ,
\label{eq:Ksum2}$$ where the MAE of the stacking fault systems of finite width is related to the MAE of hcp Co.\
![Cumulative sums of layer-resolved contributions to the MAE, $K_{X}(N)$ ($X=$ I$_1$, I$_2$, E and T$_2$), see Eqs. (\[eq:Ksum1\]) and (\[eq:Ksum2\]). Solid lines serve as guide for the eyes. \[fig:friedel-tot\]](fig4.pdf)
Fig. \[fig:friedel-tot\] shows $K_{X}(N)$ for the four different stacking faults as a function of $N$. Surprisingly, for $N \ge 5$ the I$_2$ type stacking fault appears to increase the MAE, i.e., to strengthen the easy axis $(0 0 0 1)$ ([*positive effect*]{}). As seen from Fig. \[fig:layer-res\], this is due to the positive MAE contributions induced by the stacking fault on neighbouring layers. These apparently outweigh the strongly negative MAE contributions induced in the centre of stacking fault type I$_2$. This is an unexpected result as stacking faults are typically reported to lower the MAE (see e.g. [@sokalski]). It should be noted, however, that, of the stacking faults studied here, type I$_2$ has the next highest formation energy and it is therefore less likely to occur in an equilibrated sample. For stacking faults of types I$_1$, E and T$_2$, the overall change in the MAE with respect to hcp Co is negative ([*negative effect*]{}). As noted earlier, in the vicinity of these stacking faults, the easy dirction is rotated normal to the $(0001)$ axis. This is consistent with the reduction in the total MAE observed experimentally by Sokalski et al. in[@sokalski].\
It is quite a remarkable feature that, as seen from Fig. \[fig:friedel-tot\], the layer-resolved MAE contributions do not settle until at about approximately 35 layers on either side of the stacking fault. This long-ranged behaviour could give rise to significant finite-size effects in nano-sized samples. Moreover, this might have consequences for theoretical investigations into the formation and effects of stacking faults on magnetic properties. Typically, in Monte Carlo simulations of stacking faults, interactions between stacking faults is kept to around three neighbouring planes [@sokalski]. In light of our results, this appears to be an uncertain assumption.
Magnetocrystalline Anisosotropy of a Composite Stacking Fault
-------------------------------------------------------------
Experimentally, the presence of stacking faults is normally quantified in terms of the stacking fault density, which is partly a measure of how close the stacking faults are located. As the simplest assumption, the change in the MAE due to the presence of a number stacking faults in a sample is approximated by the sum of the changes in the MAE due to each individual stacking fault. If this were the case, the effect of an isolated stacking fault on the MAE could quite straightforwardly be transformed into the change in MAE as a function of the stacking fault density. However, the long-ranged oscillations in the MAE caused by the presence of each stacking fault indicates that the situation is far more complex.\
In particular, we considered two stacking faults of type I$_1$, separated by three atomic layers. In other words, the system exhibits the composite stacking fault:
$\cdots$ A B A ${\mathrm{\bf B}}$ C B C ${\mathrm{\bf B}}$ A B A $\cdots$
Note that by removing one of the two C-B pairs of atomic layers, a twin-like stacking fault T$_2$ is obtained. We have chosen three atomic layers between the centres of the two stacking faults, since in dynamical models it is often used as the distance beyond which the interaction between stacking faults is neglected (see e.g. Ref. [@sokalski]). Moreover, we deal with a pair of I$_1$ type stacking faults because this type of stacking fault has the lowest formation energy and is, therefore, expected to occur more commonly than the other three types of stacking faults.\
The difference between the layer-resolved MAE contributions and the MAE of bulk hcp Co, $$\Delta D_i^{(\mathrm{I}_1\mathrm{I}_1)} = D_{i}^{(\mathrm{I}_1\mathrm{I}_1)} - K_{Co} \: ,
\label{eq:DeltaDII}$$ is shown in Fig. \[fig:averagecomp\] for the composite stacking fault. As a comparison, we also show the average deviations in the layer-resolved MAE contributions from the bulk MAE of two independent type I$_1$ stacking faults, $$\Delta D_i^{(\mathrm{I}_1+\mathrm{I}_1)} = \frac{1}{2} \left(D_{i+2}^{(\mathrm{I}_1)} + D_{i-2}^{(\mathrm{I}_1)}\right) - K_{Co} \: .
\label{eq:DeltaDI-I}$$ If $\Delta D_i^{(\mathrm{I}_1\mathrm{I}_1)}$ and $\Delta D_i^{(\mathrm{I}_1+\mathrm{I}_1)}$ were equal for each atomic layer $i$, the change of the MAE due to the presence of the composite stacking fault would be exactly twice that of a single I$_1$ stacking fault. However, as shown in Fig. \[fig:averagecomp\], $\Delta D_i^{(\mathrm{I}_1\mathrm{I}_1)}$ and $\Delta D_i^{(\mathrm{I}_1+\mathrm{I}_1)}$ deviate significantly, particularly in the layers $\left|i\right| \le 2$, i.e., in the layers between the two stacking faults. Beyond $\left|i\right| >3$, the magnitudes of the MAE contributions are similar for $\Delta D_i^{(\mathrm{I}_1\mathrm{I}_1)}$ and $\Delta D_i^{(\mathrm{I}_1+\mathrm{I}_1)}$, but with a phase shift of approximately one layer.\
![$\bullet:$ Calculated deviations in the layer-resolved MAE contributions, $\Delta D_i^{(\mathrm{I}_1 \mathrm{I}_1)}$, of the composite stacking fault from the bulk Co MAE, see Eq. (\[eq:DeltaDII\]), and $+ :$ the corresponding average deviations, $\Delta D_i^{(\mathrm{I}_1+\mathrm{I}_1)}$, of two superposed I$_1$ type stacking faults centred on atomic layers $i \pm 2$, see Eq. (\[eq:DeltaDI-I\]). Solid lines serve as guide for the eyes. \[fig:averagecomp\]](fig5.pdf)
Similar to the case of single stacking faults, we calculate the cumulative sum of the MAE contributions of the composite stacking fault, $$K_{\mathrm{I}_1 \mathrm{I}_1}(N) = \sum_{i=-N}^{N} D_i^{(\mathrm{I}_1 \mathrm{I}_1)} - (2N+1) K_{Co}
\: ,
\label{eq:compaggMAE}$$ and plot it in Fig. \[fig:KiiM2Ki\]. Apparently, $K_{\mathrm{I}_1 \mathrm{I}_1}(N)$ converges to approximately $-1.18$ meV for large $N$, which is almost three times the change of the MAE of the single type I$_1$ stacking fault ($\sim -0.40$ meV, see Fig. \[fig:friedel-tot\]). Also shown in Fig. \[fig:KiiM2Ki\] is the difference $K_{\mathrm{I}_1 \mathrm{I}_1}(N) -2 K_{\mathrm{I}_1}(N)$, which appears to settle at approximately $-0.38$ meV. In other words, the two stacking faults interact to yield a stronger negative effect on the total MAE as compared to two isolated type I$_1$ stacking faults. This appears to be mainly due to MAE contributions from the atomic layers located in between the two type I$_1$ stacking faults. This could have significant consequences for predicting the resulting MAE in dynamical models used to explain experimental data. To draw any definite conclusions, a systematic study of the stacking fault types and separations would be required. We expect that such a study would be computationally extremely intensive as interlayers (or supercells) of up to approximately 160 atomic layers would be required in order to reach the limit in which the two stacking faults are far enough apart not to interact.
![$\star:$ Change in the MAE of hcp Co due the composite stacking fault, $K_{\mathrm{I}_1 \mathrm{I}_1}(N)$, as defined in Eq. \[eq:compaggMAE\]). $\bullet :$ Interaction term of the two stacking faults in the MAE, $K_{I_1 I_1}(N) - 2K_{I_1}(N)$, see. Eqs. (\[eq:Ksum1\]) and (\[eq:compaggMAE\]) Solid lines serve as guide for the eyes. . \[fig:KiiM2Ki\]](fig6.pdf)
Summary and Conclusions
=======================
Using the fully relativistic screened Korringa-Kohn-Rostoker method, we have studied the MAE of bulk hcp cobalt in the vicinity of four different types of stacking faults. We find that, in accordance with experiment, most stacking faults have a detrimental overall effect on the MAE. The one exception to this overall conclusion is the type $I_2$ intrinsic stacking fault, which, however, exhibits a relatively high formation energy and which may, consequently, occur relatively infrequently under standard experimental conditions. The effect of a stacking fault on the layer-resolved contributions to the MAE is long-ranged, in the order of 15 atomic layers on either side of each stacking fault. Motivated by this observation, we investigated a particular composite stacking fault and concluded that the MAE of the composite stacking fault is not identical to the sum of the MAE of the two isolated stacking faults. A further challenging study is proposed regarding the dependence of the ’interaction’ of two stacking faults on the separation between them.\
CJA is grateful to EPSRC and to Seagate Technology for the provision of a research studentship. Support of the EU under FP7 contract NMP3-SL-2012-281043 FEMTOSPIN is gratefully acknowledged. Financial support was in part provided by the New Széchenyi Plan of Hungary (TÁMOP-4.2.2.B-10/1–2010-0009) and the Hungarian Scientific Research Fund (OTKA K77771).\
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: |
Let $X_0$ be a compact connected Riemann surface of genus $g$ with $D_0\, \subset\, X_0$ an ordered subset of cardinality $n$, and let $E_G$ be a holomorphic principal $G$–bundle on $X_0$, where $G$ is a reductive affine algebraic group defined over $\mathbb C$, that is equipped with a logarithmic connection $\nabla_0$ with polar divisor $D_0$. Let $(\mathcal{E}_G\, , \nabla)$ be the universal isomonodromic deformation of $(E_G\, ,\nabla_0)$ over the universal Teichmüller curve $(\mathcal{X}, \mathcal{D})\,\longrightarrow\, \mathrm{Teich}_{g,n}$, where $\mathrm{Teich}_{g,n}$ is the Teichmüller space for genus $g$ Riemann surfaces with $n$–marked points. We prove the following (see Section \[SecMain\]):
1. Assume that $g\, \geq\, 2$ and $n\,=\, 0$. Then there is a closed complex analytic subset $\mathcal{Y}\, \subset\,
\mathrm{Teich}_{g,n}$, of codimension at least $g$, such that for any $t\,\in\,
\mathrm{Teich}_{g,n} \setminus \mathcal{Y}$, the principal $G$–bundle $\mathcal{E}_G\vert_{{\mathcal X}_t}$ is semistable, where ${\mathcal X}_t$ is the compact Riemann surface over $t$.
2. Assume that $g\,\geq\, 1$, and if $g\,=\, 1$, then $n\, >\, 0$. Also, assume that the monodromy representation for $\nabla_0$ does not factor through some proper parabolic subgroup of $G$. Then there is a closed complex analytic subset $\mathcal{Y}'\, \subset\,
\mathrm{Teich}_{g,n}$, of codimension at least $g$, such that for any $t\,\in\,
\mathrm{Teich}_{g,n} \setminus \mathcal{Y}'$, the principal $G$–bundle $\mathcal{E}_G\vert_{{\mathcal X}_t}$ is semistable.
3. Assume that $g\,\geq\, 2$. Assume that the monodromy representation for $\nabla_0$ does not factor through some proper parabolic subgroup of $G$. Then there is a closed complex analytic subset $\mathcal{Y}''\, \subset\,
\mathrm{Teich}_{g,n}$, of codimension at least $g-1$, such that for any $t\,\in\, \mathrm{Teich}_{g,n} \setminus \mathcal{Y}'$, the principal $G$–bundle $\mathcal{E}_G\vert_{{\mathcal X}_t}$ is stable.
In [@Viktoria1], the second–named author proved the above results for $G\,=\, \text{GL}(2,{\mathbb C})$.
address:
- 'School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400005, India'
- 'Institut de Recherche Mathématique Avancée, 7 rue René-Descartes, 67084 Strasbourg Cedex, France'
- 'Department of Mathematics, McGill University, Burnside Hall, 805 Sherbrooke St. W., Montreal, Que. H3A 0B9, Canada'
author:
- Indranil Biswas
- Viktoria Heu
- Jacques Hurtubise
title: Isomonodromic deformations of logarithmic connections and stability
---
Introduction
============
Take any quadruple of the form $(E \longrightarrow X\, , D\, ,\nabla)$, where $E$ is a holomorphic vector bundle over a smooth connected complex variety $X$, and $\nabla$ is an integrable logarithmic connection on $E$ singular over a simple normal crossing divisor $D\, \subset\, X$. The monodromy functor associates to it a representation $\rho_\nabla \,:\, \pi_1(X\setminus D,\, x_0)\,\longrightarrow\,
\mathrm{GL}(E_{x_0})$, where $x_0\, \in\, X\setminus D$. Altering the connection by a holomorphic automorphism $A$ of $E$ leads to a representation conjugated by $A(x_0)$. The monodromy functor produces an equivalence between the category of logarithmic connections $(E\, ,\nabla)$ on $(X,D)$ such that the real parts of the residues lie in $[0\, ,1)$ and the category of equivalence classes of linear representations of $\pi_1(X\setminus D,\, x_0)$. Given a monodromy representation $\rho$, one can consider the set of all logarithmic connections $(E\longrightarrow X\, , D\, ,\nabla)$ (with no condition on the residues) up to holomorphic isomorphisms that produce the same monodromy representation $\rho\,=\,\rho_\nabla$ up to conjugation. All these connections are conjugated to each other by meromorphic gauge transformations with possible poles over $D$ (see for example [@Sabbah]).
The classical Riemann–Hilbert problem can be formulated as follows:
*Given a representation $\rho \,:\, \pi_1(\mathbb{P}^1_{\mathbb C}\setminus D_0,\, x)
\,\longrightarrow\,
\mathrm{GL}(r,\mathbb{C})$, is there a logarithmic connection $(E\longrightarrow
\mathbb{P}^1_{\mathbb C}\,
, D_0\, ,\nabla)$ such that $\rho\,=\,\rho_\nabla$ and $E$ is holomorphically trivial?*
The answer to this problem is
1. positive if rank $r=2$ [@Plemelj], [@Dekkers],
2. negative in general ($r\geq 3$) [@Bolibruch1],
3. positive if $\rho$ is irreducible [@Bolibruch2], [@Kostov].
On the other hand, the fundamental group $ \pi_1(\mathbb{P}^1_{\mathbb C}\setminus D_0,
\, x)$ depends only on the topological and not the complex structure of $\mathbb{P}^1_{
\mathbb C}\setminus D_0$. So given an integrable connection on $\mathbb{P}^1_{\mathbb C}
\setminus D_0$, one can consider variations of the complex structure without changing the monodromy representation. More precisely, consider the Teichmüller space $\mathrm{Teich}_{0,n}$ of the $n$–pointed Riemann sphere together with the corresponding universal Teichmüller curve $$\tau\,:\,(\mathcal{X}\,=\,\mathbb{P}^1_{\mathbb C}\times \mathrm{Teich}_{0,n}\, ,
\mathcal{D})\,\longrightarrow\, \mathrm{Teich}_{0,n}\, ,$$ where $n\,=\,{\rm degree}(D_0)$. Since $\mathrm{Teich}_{0,n}$ is contractible, the inclusion $$(\mathbb{P}^1_{\mathbb C}\, , D_t)\,:=\,\tau^{-1}(t)\,\hookrightarrow\, (\mathcal{X}
\, ,\mathcal{D})\, ,\ \ t\,\in\, \mathrm{Teich}_{0,n}\, ,$$ induces an isomorphism $\pi_1(\mathbb{P}^1_{\mathbb C}\setminus D_t, \, x_t)
\,\simeq\, \pi_1(\mathcal{X}\setminus \mathcal{D}, \, x_t)$. Hence by the Riemann–Hilbert correspondence, we can associate to any logarithmic connection $(E_0\, ,\nabla_0)$ on $\mathbb{P}^1_{\mathbb C}$, with polar divisor $D_0$, its *universal isomonodromic deformation*: a flat logarithmic connection $(\mathcal{E}\, , \nabla)$ over $\mathcal{X}$ with polar divisor $\mathcal{D}$ that extends the connection $(E_0,\nabla_0)$. The conjugacy class of the monodromy representation for $\nabla\vert_{\tau^{-1}(t)}$ does not change as $t$ moves over $\mathrm{Teich}_{0,n}$ (see for example [@Viktoria2]).
We are led to another Riemann–Hilbert problem:
*Given a logarithmic connection $(E_0\, ,\nabla_0)$ on $\mathbb{P}^1_{\mathbb C}$ with polar divisor $D_0$ of degree $n$, is there a point $t\,\in\, \mathrm{Teich}_{0,n}$ such that the holomorphic vector bundle $E_t\,=\,\mathcal{E}\vert_{\mathbb{P}^1_{\mathbb C}
\times\{t\}}$ underlying the universal isomonodromic deformation $(\mathcal{E}\, , \nabla)$ is trivial?*
A partial answer to this question is given by the following theorem of Bolibruch:
\[thmBolibruch\] Let $(E_0\, ,\nabla_0)$ be an irreducible trace–free logarithmic rank two connection with $n\,\geq\, 4$ poles on $\mathbb{P}^1_{\mathbb C}$ such that each singularity is resonant. There is a proper closed complex analytic subset $\mathcal{Y}\,\subset\,
\mathrm{Teich}_{0,n}$ such that for all $t\,\in\, \mathrm{Teich}_{0,n}\setminus
\mathcal{Y}$, the holomorphic vector bundle $E_t\,=\,\mathcal{E}\vert_{\mathbb{P}^1_{
\mathbb C}\times\{t\}}$ underlying the universal isomonodromic deformation $(\mathcal{E}\, , \nabla)$ of $(E_0,\nabla_0)$ is trivial.
It should be mentioned that the condition in Theorem \[thmBolibruch\], that each singularity is resonant, can actually be removed [@Viktoria1].
From the Birkhoff–Grothendieck classification of holomorphic vector bundles on $\mathbb{P}^1_{\mathbb C}$ it follows immediately that the only semistable holomorphic vector bundle of degree zero and rank $r$ on $\mathbb{P}^1_{\mathbb C}$ is the trivial bundle $\mathcal{O}_{\mathbb{P}^1_{\mathbb C}}^{\oplus r}$. This leads to the following more general question:
*Given a representation $\rho \,:\, \pi_1(X\setminus D,\, x)
\,\longrightarrow\, \mathrm{GL}_r(\mathbb{C})$, where $X$ is a compact connected Riemann surface, is there a logarithmic connection $(E\longrightarrow X\,, D\, ,\nabla)$ such that $\rho\,=\, \rho_\nabla$ and $E$ is semistable of degree zero?*
The answer to this problem is
1. negative in general [@Helene1],
2. positive if $\rho$ is irreducible [@Helene2].
Let $\tau\,:\,(\mathcal{X}\, , \mathcal{D})\,\longrightarrow\, \mathrm{Teich}_{g,n}$ be the universal Teichmüller curve. In view of the above, it is natural to ask the following:
*Given a logarithmic connection $(E_0\, ,
\nabla_0)$, with polar divisor $D_0$ of degree $n$ on a compact connected Riemann surface $X_0$ of genus $g$, is there an element $t\,\in\, \mathrm{Teich}_{g,n}$ such that the holomorphic vector bundle $E_t\,=\,\mathcal{E}\vert_{{\mathcal X}_t}\,
\longrightarrow\, {\mathcal X}_t\,=\, \tau^{-1}(t)$ underlying the universal isomonodromic deformation $(\mathcal{E}\, , \nabla)$ of $(E_0\, , \nabla_0)$ is semistable?*
Note that we necessarily have ${\rm degree}(E_t)\,=\,{\rm degree}(E_0)$. Again, Theorem \[thmBolibruch\] can be generalized as follows.
\[thmViktoria\] Let $(E_0\, ,\nabla_0)$ be an irreducible logarithmic rank two connection with polar divisor $D_0$ of degree $n$ on a compact connected Riemann surface $X_0$ of genus $g$ such that $3g-3+n\, >\, 0$. Consider its universal isomonodromic deformation $(\mathcal{E}\, , \nabla)$ over $\tau\,:\,
(\mathcal{X}\, , \mathcal{D})\,\longrightarrow\, \mathrm{Teich}_{g,n}$. There is a closed complex analytic subset $\mathcal{Y}\, \subset\, \mathrm{Teich}_{g,n}$ (respectively, $\mathcal{Y}_s\, \subset\, \mathrm{Teich}_{g,n}$) of codimension at least $g$ (respectively, $g-1$) such that for any $t\,\in\,
\mathrm{Teich}_{g,n}\setminus\mathcal{Y}$, the vector bundle $E_t\,=\,
\mathcal{E}\vert_{{\mathcal X}_t}$, where $({\mathcal X}_t\, ,D_t)\,=\,
\tau^{-1}(t)$, is semistable (respectively, stable).
Our aim here is to prove an analog of Theorem \[thmViktoria\] in the more general context of logarithmic connections on principal $G$–bundles over a compact connected Riemann surface (see [@Philip] for logarithmic connections on principal $G$–bundles).
Let $X_0$ be a compact connected Riemann surface of genus $g$, and let $D_0\,
\subset\, X_0$ be an ordered subset of it of cardinality $n$. Let $G$ be a reductive affine algebraic group defined over $\mathbb C$. Let $E_G$ be a holomorphic principal $G$–bundle on $X_0$ and $\nabla_0$ a logarithmic connection on $E_G$ with polar divisor $D_0$. Let $(\mathcal{E}_G\, , \nabla)$ be the universal isomonodromic deformation of $(E_G\, ,\nabla_0)$ over the universal Teichmüller curve $\tau\,:\, (\mathcal{X}\, , \mathcal{D})\,\longrightarrow\, \mathrm{Teich}_{g,n}$. For any point $t\, \in\, \mathrm{Teich}_{g,n}$, the restriction $\mathcal{E}_G\vert_{\tau^{-1}(t)}\,\longrightarrow\, {\mathcal X}_t\,:=\,
\tau^{-1}(t)$ will be denoted by $\mathcal{E}^t_G$.
We prove the following (see Section \[SecMain\]):
\[Result\]
1. Assume that $g\, \geq\, 2$ and $n\,=\, 0$. Then there is a closed complex analytic subset $\mathcal{Y}\, \subset\,
\mathrm{Teich}_{g,n}$ of codimension at least $g$ such that for any $t\,\in\,
\mathrm{Teich}_{g,n} \setminus \mathcal{Y}$, the holomorphic principal $G$–bundle $\mathcal{E}^t_G\,\longrightarrow\, {\mathcal X}_t$ is semistable.
2. Assume that $g\,\geq\, 1$, and if $g\,=\, 1$, then $n\, >\, 0$. Also, assume that the monodromy representation for $\nabla_0$ does not factor through some proper parabolic subgroup of $G$. Then there is a closed complex analytic subset $\mathcal{Y}'\, \subset\,
\mathrm{Teich}_{g,n}$ of codimension at least $g$ such that for any $t\,\in\,
\mathrm{Teich}_{g,n} \setminus \mathcal{Y}'$, the holomorphic principal $G$–bundle $\mathcal{E}^t_G$ is semistable.
3. Assume that $g\,\geq\, 2$. Assume that the monodromy representation for $\nabla_0$ does not factor through some proper parabolic subgroup of $G$. Then there is a closed complex analytic subset $\mathcal{Y}''\, \subset\,\mathrm{Teich}_{g,n}$ of codimension at least $g-1$ such that for any $t\,\in\, \mathrm{Teich}_{g,n}\setminus
\mathcal{Y}'$, the holomorphic principal $G$–bundle $\mathcal{E}^t_G$ is stable.
It is known that if a holomorphic principal $G$–bundle $E_G$ over a complex elliptic curve admits a holomorphic connection, then $E_G$ is semistable. Therefore, a stronger version of Theorem \[Result\](1) is valid when $g\,=\,1$.
Infinitesimal deformations
==========================
We first recall some classical results in deformation theory, and in the process setting up our notation.
Deformations of a $n$–pointed curve {#se2.1}
-----------------------------------
Let $X_0$ be an irreducible smooth complex projective curve of genus $g$, with $g\, >\, 0$, and let $$D_0\,:=\, \{x_1\, , \cdots\, , x_n\}\, \subset\, X_0$$ be an ordered subset of cardinality $n$ (it may be zero). We assume that $n\, >\, 0$ if $g\,=\,1$. This condition implies that the pair $(X_0\, ,D_0)$ does not have any infinitesimal automorphism, equivalently, the automorphism group of $(X_0\, ,D_0)$ is finite.
Let $B\,:=\, \text{Spec}({\mathbb C}[\epsilon]/\epsilon^2)$ be the spectrum of the local ring. An *infinitesimal deformation of $(X_0\, ,D_0)$* is given by a quadruple $$\label{e1}
({\mathcal X}\, ,q \, , {\mathcal D}\, ,f)\, ,$$ where
- $q\, :\, {\mathcal X}\, \longrightarrow\, B$ is a smooth proper holomorphic morphism of relative dimension one,
- $\mathcal{D}\,=\, (\mathcal{D}_1\, ,\cdots\, , \mathcal{D}_n)$ is a collection of $n$ ordered disjoint sections of $q$, and
- $f\, :\, X_0\,\longrightarrow\, {\mathcal X}$ is a holomorphic morphism such that $$f(X_0)\, \subset\, {\mathcal X}_0 \,:=\, q^{-1}(0) \ \ \text{ with }\ \
f(x_i)\,=\, \mathcal{D}_i(0) ~ \ \ \forall \ \ 1\,\leq\,i\, \leq\, n\, ,$$ and the morphism $X_0\,\stackrel{f}{\longrightarrow}\, {\mathcal X}_0$ is an isomorphism.
The divisor $\sum_{i=1}^n \mathcal{D}_i(B)$ on $\mathcal X$ will also be denoted by $\mathcal{D}$. For a vector bundle $\mathcal{V}$ on $\mathcal X$, the vector bundle $\mathcal{V}\otimes_{{\mathcal O}_{\mathcal X}} {\mathcal O}_{\mathcal X}(-
{\mathcal D})$ will be denoted by $\mathcal{V}(-\mathcal{D})$.
The differential of $f$ $$\mathrm{d}f\, :\, \mathrm{T}X_0\, \longrightarrow\,
f^*\mathrm{T}{\mathcal X}$$ produces a homomorphism $$\mathrm{T}X_0(-D_0)\,:=\,
\mathrm{T}X_0\otimes_{{\mathcal O}_{X_0}}{\mathcal O}_{X_0}(-D_0)
\,\longrightarrow\, f^*\mathrm{T}{\mathcal X}(-\mathcal{D})$$ which will also be denoted by $\mathrm{d}f$. Consider the following short exact sequence of coherent sheaves on $X_0$: $$\label{ti}
0\, \longrightarrow\, \mathrm{T}X_0(-D_0)\, \stackrel{\mathrm{d}f}{\longrightarrow}\,
f^*\mathrm{T}{\mathcal X}(-\mathcal{D})
\, \stackrel{h}{\longrightarrow}\, {\mathcal O}_{X_0} \, \longrightarrow\, 0\, ;$$ note that the normal bundle to ${\mathcal X}_{0}\, \subset\, \mathcal X$ is the pullback of $\mathrm{T}_{0}B$ by $q\vert_{{\mathcal X}_{0}}$, and hence this normal bundle is identified with ${\mathcal O}_{X_0}$. Consider the connecting homomorphism $$\label{ch}
{\mathbb C} \,=\, \mathrm{H}^0(X_0,\, {\mathcal O}_{X_0})\,
\stackrel{\phi}{\longrightarrow}\,\mathrm{H}^1(X_0,\, \mathrm{T}X_0(-D_0))$$ in the long exact sequence of cohomologies associated to the short exact sequence in . Let $1_{X_0}$ denote the constant function $1$ on $X_0$. The cohomology element $$\label{e2}
\phi(1_{X_0})\, \in\, \mathrm{H}^1(X_0,\, \mathrm{T}X_0(-D_0))\, ,$$ where $\phi$ is the homomorphism in , is the *Kodaira–Spencer infinitesimal deformation class* for the family in . If this infinitesimal deformation class is zero, then the family $(\mathcal{X}\, , \mathcal{D})\,\longrightarrow\, B$ is isomorphic to the trivial family $(X_0\times B \, , D_0\times B)\,\longrightarrow\, B$.
Deformations of a curve together with a principal bundle {#SecBundleDef}
--------------------------------------------------------
Take $(X_0\, ,D_0)$ as before. Let $G$ be a connected reductive affine algebraic group defined over $\mathbb C$. The Lie algebra of $G$ will be denoted by $\mathfrak g$. Let $$\label{g1}
p\, :\, E_G\,\longrightarrow\, X_0$$ be a holomorphic principal $G$–bundle on $X_0$. The infinitesimal deformations of the triple $$\label{tr}
(X_0\, ,D_0\, ,E_G)$$ are guided by the Atiyah bundle $\text{At}(E_G)\,\longrightarrow\, X_0$, the construction of which we shall briefly recall (see [@At] for a more detailed exposition). Consider the direct image $p_*\mathrm{T}E_G\, \longrightarrow\, X_0$, where $\mathrm{T}E_G$ is the holomorphic tangent bundle of $E_G$, and $p$ is the projection in . It is a quasicoherent sheaf equipped with an action of $G$ given by the action of $G$ on $E_G$. The invariant part $$\text{At}(E_G)\,:=\, (p_*\mathrm{T}E_G)^G \, \subset\, (p_*\mathrm{T}E_G)$$ is a vector bundle on $X_0$ of rank $1+\dim G$ which is known as the Atiyah bundle of $E_G$. Consequently, we have $\text{At}(E_G)\,=\, (TE_G)/G$. Let $$\text{ad}(E_G)\, :=\, E_G\times^G {\mathfrak g}\,\longrightarrow\, X_0$$ be the adjoint vector bundle associated to $E_G$ for the adjoint action of $G$ on its Lie algebra $\mathfrak g$. So the fibers of $\text{ad}(E_G)$ are Lie algebras isomorphic to $\mathfrak g$. Let $$\mathrm{d}p\, :\, \mathrm{T}E_G\, \longrightarrow\, p^*\mathrm{T}X_0$$ be the differential of the map $p$ in . Being $G$–equivariant it produces a homomorphism $\text{At}(E_G)\,
\longrightarrow\,\mathrm{T}X_0$ which will also be denoted by $\mathrm{d}p$. Now, the action of $G$ on $E_G$ produces an isomorphism $E_G\times{\mathfrak g}
\,\longrightarrow\, \text{kernel}(\mathrm{d}p)$. Therefore, we have $\text{kernel}(\mathrm{d}p)/G\,=\, \text{ad}(E_G)$. In other words, the above isomorphism $E_G\times{\mathfrak g}
\,\longrightarrow\, \text{kernel}(\mathrm{d}p)$ descends to an isomorphism $$\text{ad}(E_G)\,\stackrel{\sim}{\longrightarrow}\, (p_*(\text{kernel}(\mathrm{d}p)))^G$$ that preserves the Lie algebra structure on the fibers of the two vector bundles (the Lie algebra structure on the fibers of $(p_*(\text{kernel}(\mathrm{d}p)))^G$ is given by the Lie bracket of $G$–invariant vertical vector fields). Therefore, $\text{At}(E_G)$ fits in the following short exact sequence of vector bundles on $X_0$ $$\label{at1}
0\, \longrightarrow\, \text{ad}(E_G)\, \longrightarrow\,\text{At}(E_G)\,
\stackrel{\mathrm{d}p}{\longrightarrow}\, \mathrm{T}X_0\, \longrightarrow\, 0\, ,$$ which is known as the Atiyah exact sequence for $E_G$. The logarithmic Atiyah bundle $\text{At}(E_G\,,D_0)$ is defined by $$\text{At}(E_G\,,D_0)\,:=\, (\mathrm{d}p)^{-1}(\mathrm{T}X_0(-D_0))
\,\subset\, \text{At}(E_G) \,.$$ From we have the short exact sequence of vector bundles on $X_0$ $$\label{e3}
0\, \longrightarrow\, \text{ad}(E_G)\, \longrightarrow\,
\text{At}(E_G\,,D_0)\,
\stackrel{\sigma}{\longrightarrow}\, \mathrm{T}X_0(-D_0)\,
\longrightarrow\, 0\, ,$$ which is called the [*logarithmic Atiyah exact sequence*]{}. The above homomorphism $\sigma$ is the restriction of $\mathrm{d}p$ to $\text{At}(E_G\,,D_0)\,\subset\, \text{At}(E_G)$.
An *infinitesimal deformation* of the triple $(X_0\, , D_0\, , E_G)$ in is a $6$–tuple $$\label{RelativeBundle}
({\mathcal X}\, ,q \, , {\mathcal D}\, ,f\, , {\mathcal E}_G\, ,\psi)\, ,$$ where
- $({\mathcal X}\, ,q\, , {\mathcal D}\, ,f)$ in an infinitesimal deformation of the $n$–pointed curve $(X_0\, , D_0)$ as in (\[e1\]),
- ${\mathcal E}_G\, \longrightarrow\,{\mathcal X}$ is a holomorphic principal $G$–bundle, and
- $\psi$ is a holomorphic isomorphism $$\label{psi}
\psi\, :\, E_G\, \longrightarrow\, f^*{\mathcal E}_G$$ of principal $G$–bundles.
The logarithmic Atiyah bundle $$\text{At}({\mathcal E}_G\, ,{\mathcal D})\,\longrightarrow\, \mathcal X$$ for $({\mathcal E}_G\, ,{\mathcal D})$ is the inverse image, in $\text{At}
({\mathcal E}_G)$, of the subsheaf $\mathrm{T}{\mathcal X}(-{\mathcal D}))\,\subset\,
\mathrm{T}{\mathcal X}$. We have the following short exact sequence of sheaves on $X_0$: $$\label{f2}
0\, \longrightarrow\, \text{At}(E_G\, ,D_0)\, \longrightarrow\, f^*\text{At}(
{\mathcal E}_G\, ,{\mathcal D})\, \longrightarrow\, {\mathcal O}_{X_0} \,
\longrightarrow\, 0$$ given by $\psi$ in . Let $${\mathbb C} \,=\, \mathrm{H}^0(X_0,\, {\mathcal O}_{X_0})\,
\stackrel{\widetilde\phi}{\longrightarrow}\,\mathrm{H}^1(X_0,\, \text{At}(E_G\,,D_0))$$ be the connecting homomorphism in the long exact sequence of cohomologies associated to . The cohomology element $$\label{e4}
{\widetilde\phi}(1_{X_0})\, \in\, \mathrm{H}^1(X_0,\, \text{At}(E_G\, , D_0))$$ is the *cohomology class* of the infinitesimal deformation of the triple $(X_0\, , D_0\, , E_G)$ given by . Let $$\sigma_*\, :\, \mathrm{H}^1(X_0,\, \text{At}(E_G\,, D_0))\, \longrightarrow\,
\mathrm{H}^1(X_0,\, \mathrm{T}X_0(-D_0))$$ be the homomorphism given by the projection $\sigma$ in . From the commutativity of the diagram $$\begin{matrix}
0 & \longrightarrow & \text{At}(E_G\, ,D_0) & \longrightarrow & f^*\text{At}(
{\mathcal E}_G\, ,{\mathcal D}) &\longrightarrow & {\mathcal O}_{X_0}
&\longrightarrow & 0\\
&&\Big\downarrow &&\Big\downarrow && \Vert\\
0 & \longrightarrow & \mathrm{T}X_0(-D_0) & \longrightarrow &
f^*\mathrm{T}{\mathcal X}(-\mathcal{D})& \longrightarrow & {\mathcal O}_{X_0}
&\longrightarrow & 0
\end{matrix}$$ where the top and bottom rows are as in and respectively, it follows that $$\sigma_*({\widetilde\phi}(1_{X_0}))\,=\, {\phi}(1_{X_0})\, ,$$ where ${\widetilde\phi}(1_{X_0})$ and ${\phi}(1_{X_0})$ are constructed in and respectively. We note that $\sigma_*$ is the forgetful map that sends an infinitesimal deformation of $(X_0\, , D_0\, ,
E_G)$ to the underlying infinitesimal deformation of $(X_0\, , D_0)$ forgetting the principal $G$–bundle.
Obstruction to extension of a reduction of structure group to $P$ {#se2.2}
=================================================================
Given a holomorphic reduction of structure group of $E_G$ to a parabolic subgroup of $G$, our aim in this section is to compute the obstruction for this reduction to extend to a reduction of an infinitesimal deformation ${\mathcal E}_G\,\longrightarrow\, \mathcal X$ as in (see the paragraph after the proof of Lemma \[lem1\]).
A parabolic subgroup of $G$ is a connected Zariski closed subgroup $P$ such that $G/P$ is a complete variety. Fix a parabolic subgroup $
P\, \subset\, G\, .
$ The Lie algebra of $P$ will be denoted by $\mathfrak p$. As before, $E_G$ is a holomorphic principal $G$–bundle on $X_0$. Let $$\label{e5}
E_P\, \subset\, E_G$$ be a holomorphic reduction of structure group of $E_G$ to the subgroup $P\,\subset\, G$. Let $$\text{ad}(E_P)\, :=\, E_P\times^P{\mathfrak p}\, \longrightarrow\, X_0$$ be the adjoint vector bundle associated to $E_P$ for the adjoint action of $P$ on its Lie algebra $\mathfrak p$. The vector bundle over $X_0$ associated to the principal $P$–bundle $E_P$ for the adjoint action of $P$ on the quotient ${\mathfrak g}/
{\mathfrak p}$ will be denoted by $E_P({\mathfrak g}/{\mathfrak p})$. So, $E_P({\mathfrak g}/{\mathfrak p})\,=\, \text{ad}(E_G)/\text{ad}(E_P)$. The logarithmic Atiyah bundle for $(E_P\, , D_0)$ will be denoted by $\text{At}(E_P, D_0)$. We have $$\text{ad}(E_P)\, \subset\, \text{ad}(E_G)\ \ \text{ and }\ \
\text{At}(E_P, D_0)\, \subset\,\text{At}(E_G\, , D_0)\, ;$$ both the quotient bundles above are identified with $E_P({\mathfrak g}/{\mathfrak p})$. In other words, we have the following commutative diagram of vector bundles on $X_0$ $$\label{e6}
\begin{matrix}
&& 0 && 0 \\
&& \Big\downarrow && \Big\downarrow\\
0& \longrightarrow & \text{ad}(E_P) & \longrightarrow & \text{At}(E_P\, , D_0) &
\stackrel{\beta}{\longrightarrow} & \mathrm{T}X_0(-D_0) & \longrightarrow & 0\\
&&\Big\downarrow && ~ \Big\downarrow \xi && \Vert\\
0& \longrightarrow & \text{ad}(E_G) & \longrightarrow & \text{At}(E_G\, , D_0) &
\stackrel{\sigma}{\longrightarrow} & \mathrm{T}X_0(-D_0) & \longrightarrow & 0\\
&&\,\,\,\,\,\, \Big\downarrow \mu_1 && ~ \Big\downarrow \mu \\
&& E_P({\mathfrak g}/{\mathfrak p}) & = & E_P({\mathfrak g}/{\mathfrak p})\\
&& \Big\downarrow && \Big\downarrow\\
&& 0 && 0
\end{matrix}$$ where $\sigma$ is the homomorphism in . Let $$\label{e7}
\widetilde{\xi}\, :\, \mathrm{H}^1(X_0,\, \text{At}(E_P\, , D_0))\, \longrightarrow\,
\mathrm{H}^1(X_0,\,\text{At}(E_G\, , D_0))$$ be the homomorphism induced by the canonical injection $\xi$ in .
Take any $({\mathcal X}\, ,q \, , {\mathcal D}\, ,f\, , {\mathcal E}_G\, ,\psi)$ as in . Assume that the reduction $E_P\, \subset\,
E_G$ in extends to a holomorphic reduction of structure group $${\mathcal E}_P\,\, \subset\, \mathcal E_G$$ to $P\,\subset\, G$ on $\mathcal X$. Consider the short exact sequence on $X_0$ $$\label{f1}
0\, \longrightarrow\, \text{At}(E_P\, , D_0)\, \longrightarrow\,
f^*\text{At}({\mathcal E}_P\, , \mathcal{D})\,
\longrightarrow\, {\mathcal O}_{X_0}\, \longrightarrow\, 0\, ,$$ where $\text{At}({\mathcal E}_P\, , \mathcal{D})\,\longrightarrow
\,{\mathcal X}$ is the logarithmic Atiyah bundle associated to the principal $P$–bundle ${\mathcal E}_P$, and $f$ is the map in . Let $$\label{theta}
\theta\, \in\, \mathrm{H}^1(X_0,\, \text{At}(E_P\, , D_0))$$ be the image of the constant function $1_{X_0}$ by the homomorphism $$\mathrm{H}^0(X_0,\, {\mathcal O}_{X_0})\,\longrightarrow\,
\mathrm{H}^1(X_0,\, \text{At}(E_P\, , D_0))$$ in the long exact sequence of cohomologies associated to .
\[lem1\] The cohomology class $\theta$ in satisfies the equation $$\widetilde{\xi}(\theta)\,=\, \widetilde{\phi}(1_{X_0})\, ,$$ where $\widetilde{\xi}$ and $\widetilde{\phi}(1_{X_0})$ are constructed in and respectively.
Consider the commutative diagram of vector bundles on $X_0$ $$\label{f5}
\begin{matrix}
0& \longrightarrow & \text{At}(E_P\, , D_0) & \longrightarrow &
f^*\text{At}({\mathcal E}_P\, , \mathcal{D}) &
\longrightarrow & {\mathcal O}_{X_0} & \longrightarrow & 0\\
&&~ \Big\downarrow\xi && \Big\downarrow && \Vert\\
0& \longrightarrow & \text{At}(E_G\, , D_0) &
\longrightarrow & f^*\text{At}({\mathcal E}_G\, , \mathcal{D}) & {\longrightarrow} &
{\mathcal O}_{X_0} & \longrightarrow & 0
\end{matrix}$$ where the top and bottom rows are as in and respectively, and $\xi$ is the homomorphism in ; the above homomorphism $f^*\text{At}({\mathcal E}_P\, , \mathcal{D})\,\longrightarrow\,
f^*\text{At}({\mathcal E}_G\, , \mathcal{D})$ is the pullback of the natural homomorphism $\text{At}({\mathcal E}_P\, , \mathcal{D})\,\longrightarrow\,
\text{At}({\mathcal E}_G\, , \mathcal{D})$. In view of , the lemma follows by comparing the constructions of $\theta$ and $\widetilde{\phi}(1_{X_0})$.
Consider the homomorphism $\mu_*\, :\,
\mathrm{H}^1(X_0,\,\text{At}(E_G\, , D_0))\,\longrightarrow\,
\mathrm{H}^1(X_0,\, E_P({\mathfrak g}/{\mathfrak p}))$ induced by the homomorphism $\mu$ in . From Lemma \[lem1\] we conclude that $\mu_*(\widetilde{\phi}(1_{X_0}))\,=\, 0$; to see this consider the long exact sequence of cohomologies associated to the right vertical exact sequence in . Therefore, $\mu_*(\widetilde{\phi}(1_{X_0}))$ is an obstruction for the reduction $E_P\, \subset\, E_G$ to extend to a reduction of ${\mathcal E}_G$ to $P$.
Logarithmic connections and the second fundamental form
=======================================================
In this section we characterize those infinitesimal deformations of the principal bundle $E_G$ on the $n$–pointed curve that arise from the isomonodromic deformations.
Canonical extension of a logarithmic connection {#SecConnections}
-----------------------------------------------
As before, let $p\,:\, E_G\, \longrightarrow\, X_0$ be a holomorphic principal $G$–bundle. A *logarithmic connection* on $E_G$ with polar divisor $D_0$ is a holomorphic splitting of the logarithmic Atiyah exact sequence in . In other words, a logarithmic connection is a homomorphism $$\label{de}
\delta\, :\, \mathrm{T}X_0(-D_0)\, \longrightarrow\, \text{At}(E_G\, , D_0)$$ such that $\sigma\circ\delta\,=\, \text{Id}_{\mathrm{T}X_0(-D_0)}$, where $\sigma$ is the homomorphism in . Note that given such a $\delta$, there is a unique homomorphism $$\label{depp}
\delta''\, :\, \text{At}(E_G\, , D_0)\, \longrightarrow\,
\text{ad}(E_G)$$ such that $\delta''\circ\delta\,=\, 0$, and the composition $$\text{ad}(E_G)\, \hookrightarrow\,\text{At}(E_G\, , D_0)\,
\stackrel{\delta''}{\longrightarrow}\, \text{ad}(E_G)$$ (see ) is the identity map of $\text{ad}(E_G)$. As there are no nonzero $(2\, ,0)$–forms on $X_0$, all logarithmic connections on $E_G$ are automatically integrable.
At the level of first order infinitesimal deformations, given a principal $G$–bundle $${\mathcal E}_G\, \longrightarrow \,{\mathcal X}\, \stackrel{q}{\longrightarrow} \,B\, ,$$ a logarithmic connection on ${\mathcal E}_G$ with polar divisor ${\mathcal D}$ is a homomorphism $\text{At}({\mathcal E}_G\, , {\mathcal D})\,\longrightarrow
\, \text{ad}({\mathcal E}_G)$ that splits the logarithmic Atiyah exact sequence for ${\mathcal E}_G$. We note that a connection on ${\mathcal E}_G$ need not be integrable, as we have added an (infinitesimal) extra dimension. However, the Riemann–Hilbert correspondence for principal $G$–bundles yields the following:
\[LemRH\] Let $({\mathcal X}\, ,q \, , {\mathcal D}\, ,f)$ be an infinitesimal deformation of $(X_0\, ,D_0)$ as in . Let $\delta$ be a logarithmic connection on a holomorphic principal $G$–bundle $E_G$ on $X_0$ with polar divisor $D_0$. Then there exists a unique pair $(\mathcal{E}_G\, , \nabla)$, where
- $\mathcal{E}_G$ is a holomorphic principal $G$–bundle on ${\mathcal X}$, and
- $\nabla$ is an integrable logarithmic connection on $\mathcal{E}_G$ with polar divisor ${\mathcal D}$,
such that $(f^*\mathcal{E}_G\, ,f^*\nabla)\,=\, (E_G\, , \delta)$.
Let us recall a few elements of the proof of this (classical) result. Choose a covering $\mathfrak{U}$ of $X_0\setminus D_0$ by complex discs and a small neighborhood $U_i$ for each $x_i \,\in\, D_0$. Since $\delta$ is integrable, we can choose local charts for $E_G$ over $\mathfrak{U}$ such that all transition functions are constants. Now if the curve fits into an analytic family ${\mathcal X}\,\longrightarrow
\,{\mathcal B}$, one can, restricting ${\mathcal B}$ if necessary, cover ${\mathcal X}$ by open subsets of the form $V_j\times B$, where $V_j$ are the open subsets of $X_0$ in the collection $\mathfrak{U}\cup \{U_i\}_{i=1}^n$. The isomonodromic deformation is then given by simply extending the transition maps by keeping them to be constant in deformation parameters.
The logarithmic connection $\delta$ gives a splitting of the logarithmic Atiyah bundle $$\text{At}(E_G\, , D_0)\, =\, \text{ad}(E_G) \oplus \mathrm{T}X_0(-D_0)\, .$$ The corresponding cohomological decomposition $$\mathrm{H}^1(X_0, \text{At}(E_G\, , D_0) )\,=\,
\mathrm{H}^1(X_0,\text{ad}(E_G)) \oplus \mathrm{H}^1(X_0,\mathrm{T}X_0(-D_0))$$ gives a splitting of the infinitesimal deformations of $(X_0\, ,D_0\, ,E_G)$ into the infinitesimal deformations of $(X_0\, ,D_0)$ and the infinitesimal deformations of $E_G$ (keeping $(X_0\, ,D_0)$ fixed). In other words, let $$\delta'\,:\, \mathrm{H}^1(X_0,\, \mathrm{T}X_0(-D_0))\, \longrightarrow\,
\mathrm{H}^1(X_0,\, \text{At}(E_G\, , D_0))$$ be the homomorphism induced by the homomorphism $$\label{de12}
\delta\, :\, \mathrm{T}X_0(-D_0)
\, \longrightarrow\, \text{At}(E_G\, , D_0)$$ in Lemma \[LemRH\] defining the logarithmic connection on $E_G$. Given an infinitesimal deformation $({\mathcal X}\, ,q
\, , {\mathcal D}\, ,f)$ of $(X_0\, ,D_0)$, the above homomorphism $\delta'$ produces an infinitesimal deformation $({\mathcal X}\, ,q \, , {\mathcal D}\, ,f\, ,
{\mathcal E}_G\, ,\psi)$ of $(X_0\, ,D_0\, ,E_G)$. As explained above, this holomorphic principal $G$–bundle ${\mathcal E}_G$ on $\mathcal X$ coincides with the holomorphic principal $G$–bundle on $\mathcal X$ produced by the isomonodromic deformation in Lemma \[LemRH\].
We will now construct the exact sequence in corresponding to the above infinitesimal deformation $({\mathcal X}\, ,q \, , {\mathcal D}\, ,f\, , {\mathcal E}_G
\, ,\psi)$. Consider the injective homomorphism $$\mathrm{T}X_0(-D_0)\, \longrightarrow\, \text{At}(E_G\, , D_0)\oplus
f^*\mathrm{T}\mathcal{X}(-\mathcal{D})\, , ~\ ~ v\, \longmapsto\,
(\delta(v)\, , -(\mathrm{d}f)(v))\, ,$$ where $\mathrm{d}f$ is the differential in and $\delta$ is the homomorphism in . The corresponding cokernel $$\text{At}^\delta (E_G\, , D_0)\, :=\, (\text{At}(E_G\, , D_0)\oplus
f^*\mathrm{T}\mathcal{X}(-\mathcal{D}))/(\mathrm{T}X_0(-D_0))$$ possesses a canonical projection $$\label{what}
\widehat{\delta}\, :\, \text{At}^\delta (E_G\, , D_0)\, \longrightarrow\,{\mathcal O}_{X_0}\, , ~\ ~
(v\, ,w)\, \longmapsto\, h(w)\, ,$$ where $h$ is the homomorphism in ; note that the above homomorphism $\widehat{\delta}$ is well–defined because $h$ vanishes on the image of $\mathrm{T}X_0(-D_0)$ in $\text{At}(E_G\, , D_0)\oplus
f^*\mathrm{T}\mathcal{X}(-\mathcal{D})$. The kernel of $\widehat{\delta}$ is identified with $\text{At}(E_G\, , D_0)$ by sending any $z\, \in\,
\text{At}(E_G\, , D_0)$ to the image in $\text{At}^\delta (E_G\, , D_0)$ of $(z\, ,0)\,\in\, \text{At}(E_G\, , D_0)\oplus f^*\mathrm{T}\mathcal{X}(-\mathcal{D})$. Therefore, we obtain the following exact sequence of vector bundles over $X_0$: $$0\, \longrightarrow\,{\rm At}(E_G\, , D_0)\, \longrightarrow\,{\rm At}^\delta (E_G\, , D_0) \,
\stackrel{\widehat{\delta}}{\longrightarrow}\, {\mathcal O}_{X_0} \,
\longrightarrow\, 0\, ,$$ This exact sequence coincides with the one in . In particular, we have ${\rm At}^\delta (E_G\, , D_0)\,=\,f^*\text{At}({\mathcal E}_G)$.
Consider the projection $$\text{At}(E_G\, , D_0)\oplus
f^*\mathrm{T}\mathcal{X}(-\mathcal{D})\, \longrightarrow\, \text{ad}(E_G)\, ,
\ \ (z_1\, ,z_2)\, \longmapsto\, \delta''(z_1)\, ,$$ where $\delta''$ is constructed in from $\delta$. It vanishes on the image of $\mathrm{T}X_0(-D_0)$, yielding a projection $$\label{ar}
\lambda\, :\, \text{At}^\delta (E_G\, , D_0)\,\longrightarrow\, \text{ad}(E_G)\, .$$ Let $\nabla''\, :\, \text{At}({\mathcal E}_G\, , {\mathcal D})\,
\longrightarrow\, \text{ad}({\mathcal E}_G)$ be the homomorphism given by the logarithmic connection $\nabla$ in Lemma \[LemRH\]. The homomorphism in fits in the commutative diagram $$\label{comm}
\begin{matrix}
\text{At}^\delta (E_G\, , D_0)& \stackrel{\lambda}{\longrightarrow} & \text{ad}(E_G)\\
\Vert && \Vert\\
f^*\text{At} ({\mathcal E}_G\, , {\mathcal D})
&\stackrel{f^*\nabla''}{\longrightarrow} & f^*\text{ad}({\mathcal E}_G)
\end{matrix}$$ (the vertical identifications are evident).
We summarize the above constructions in the following lemma:
\[lem2\] Given $({\mathcal X}\, ,q\, , {\mathcal D}\, ,f)$ as in , and also given a logarithmic connection $\delta$ on a holomorphic principal $G$–bundle $E_G\, \longrightarrow\, X_0$, the exact sequence in corresponding to the infinitesimal deformation of $(X_0\,,D_0\,, E_G)$ in Lemma \[LemRH\] is $$0\, \longrightarrow\,{\rm At}(E_G\, , D_0)\, \longrightarrow\,{\rm At}^\delta
(E_G\, , D_0) \,\stackrel{\widehat{\delta}}{\longrightarrow}\, {\mathcal O}_{X_0}
\, \longrightarrow\, 0\, ,$$ where $\widehat{\delta}$ is constructed in .
The second fundamental form {#SecFundForm}
---------------------------
Fix a logarithmic connection $\delta$ on $(E_G\,,D_0)$ as in . Take a holomorphic reduction of structure group $E_P\,\subset\, E_G$ to $P$ as in . The composition $$\label{sff}
S(\delta)\,:=\,
\mu\circ\delta\, :\, \mathrm{T}X_0(-D_0)\,\longrightarrow\, E_P({\mathfrak g}/{\mathfrak p})\, ,$$ where $\mu$ is constructed in , is called the *second fundamental form* of $E_P$ for the connection $\delta$. We note that $\delta$ is induced by a logarithmic connection on the holomorphic principal $P$–bundle $E_P$ if and only if we have $S(\delta)\,=\, 0$.
Assume that $E_P$ satisfies the condition that $
S(\delta)\,\not=\, 0\, .
$ Let $$\label{l}
{\mathcal L}\, \subset\, E_P({\mathfrak g}/{\mathfrak p})$$ be the holomorphic line subbundle generated by the image of the homomorphism $S(\delta)$ in . More precisely, ${\mathcal L}$ is the inverse image, in $E_P({\mathfrak g}/{\mathfrak p})$, of the torsion part of the quotient $E_P({\mathfrak g}/{\mathfrak p})/(S(\delta)(\mathrm{T}X_0(-D_0)))$. Now consider the homomorphism $$\label{sdp}
S(\delta)' \,:\, \mathrm{T}X_0(-D_0)\,\longrightarrow\,
{\mathcal L}$$ given by the second fundamental form $S(\delta)$. Let $$\label{sp}
S'\, :\, \mathrm{H}^1(X_0,\, \mathrm{T}X_0(-D_0))\,\longrightarrow\,
\mathrm{H}^1(X_0,\, {\mathcal L})$$ be the homomorphism of cohomologies induced by $S(\delta)'$ in .
\[prop1\] As before, let $\delta$ be a logarithmic connection on $E_G\,\longrightarrow\, X_0$ with polar divisor $D_0$, and let $({\mathcal X}\, ,q \, , {\mathcal D}\, ,f)$ be an infinitesimal deformation of $(X_0,D_0)$. Let ${\mathcal E}_G\,\longrightarrow\,{\mathcal
X}$ be the isomonodromic deformation of $\delta$ along $(\mathcal{X}\, ,
\mathcal{D})$ obtained in Lemma \[LemRH\]. Let $E_P\, \subset\, E_G$ be a holomorphic reduction of structure group to $P$ over $X_0$ that extends to a holomorphic reduction of structure group ${\mathcal E}_P\, \subset\, {\mathcal E}_G$ to $P$ over ${\mathcal X}$. Then $$S'(\phi(1_{X_0}))\,=\, 0\, ,$$ where $\phi(1_{X_0})$ is the cohomology class constructed in corresponding to $({\mathcal X}\, ,q \, , {\mathcal D}\, ,f)$, and $S'$ is constructed in .
Consider the inverse images $$\text{At}_P(E_G\, , D_0)\,:=\, \mu^{-1}({\mathcal L})\,\subset\, \text{At}(E_G\, , D_0)
\ \ \text{ and} \ \ \
\text{ad}_P(E_G )\,:=\, \mu^{-1}_1({\mathcal L})\,\subset\, \text{ad}(E_G)\, ,$$ where $\mu$ and $\mu_1$ are the quotient maps in , and $\mathcal L$ is constructed in . These two vector bundles fit in the following commutative diagram produced from : $$\label{f6}
\begin{matrix}
&& 0 && 0 \\
&& \Big\downarrow && \Big\downarrow\\
0& \longrightarrow & \text{ad}(E_P) & \longrightarrow & \text{At}(E_P\, , D_0) &
\stackrel{\beta}{\longrightarrow} & \mathrm{T}X_0(-D_0) & \longrightarrow & 0\\
&&\Big\downarrow && ~ \Big\downarrow \xi && \Vert\\
0& \longrightarrow & \text{ad}_P(E_G) & \longrightarrow & \text{At}_P(E_G\, , D_0) &
\stackrel{\gamma}{\longrightarrow} & \mathrm{T}X_0(-D_0) & \longrightarrow & 0\\
&& \,\,\,\,\, \Big\downarrow \mu_1 && ~ \Big\downarrow \mu \\
&& {\mathcal L} & = & {\mathcal L}\\
&& \Big\downarrow && \Big\downarrow\\
&& 0 && 0
\end{matrix}$$ By the construction of $\text{At}_P(E_G\, , D_0)$, the connection homomorphism $\delta$ in factors through a homomorphism $$\delta^1\, :\,\mathrm{T}X_0(-D_0)\, \longrightarrow\, \text{At}_P(E_G\, , D_0)\, .$$ Consider the homomorphism $$\label{de1s}
\delta^1_*\, :\, \mathrm{H}^1(X_0,\, \mathrm{T}X_0(-D_0))\, \longrightarrow\,
\mathrm{H}^1(X_0,\,\text{At}_P(E_G\, , D_0))$$ induced by the above homomorphism $\delta^1$, and let $$\label{tau}
\Phi\,:=\, \delta^1_*(\phi(1_{X_0}))\, \in\, \mathrm{H}^1(X_0,\,\text{At}_P(E_G\, , D_0))$$ be the image of the cohomology class $\phi(1_{X_0})$ that characterizes the deformation $(\mathcal{X}\,,\mathcal{D})$ as in .
As in the statement of the proposition, let ${\mathcal E}_P\, \longrightarrow\,
{\mathcal X}$ be a holomorphic extension of the reduction $E_P$. Note that from , and Lemma \[lem2\] we have $$\text{At}(E_G\, , D_0)/\text{At}(E_P\, , D_0)\,=\,
E_P({\mathfrak g}/{\mathfrak p})\,=\, (f^*\text{At}({\mathcal E}_G\, ,
{\mathcal D}))/ (f^*\text{At}({\mathcal E}_P\, ,{\mathcal D}))$$ $$\,=\, {\rm At}^\delta(E_G\, , D_0)/
f^*\text{At}({\mathcal E}_P\, , \mathcal{D})\, .$$ Let $\mu_2\, :\, {\rm At}^\delta (E_G\, , D_0)\,\longrightarrow\,
E_P({\mathfrak g}/{\mathfrak p})$ be the above quotient map. Define $${\rm At}^\delta_P (E_G\, , D_0)\, :=\, \mu^{-1}_2({\mathcal L})
\, \subset\, {\rm At}^\delta (E_G\, , D_0)\, ,$$ where $\mathcal L$ is constructed in .
Now we have the following commutative diagram: $$\label{dia}
\begin{matrix}
0 & \longrightarrow & \mathrm{T}X_0(-D_0) & \longrightarrow &
f^*\mathrm{T}{\mathcal X}(-\mathcal{D})& \longrightarrow & {\mathcal O}_{X_0}
&\longrightarrow & 0\\
&&~\Big\downarrow\delta &&\,\,\,\,\,\,\,\,\, \Big\downarrow f^*\nabla && \Vert\\
0& \longrightarrow & \text{At}_P(E_G\, , D_0) &
\longrightarrow & {\rm At}^\delta_P (E_G\, , D_0) & {\longrightarrow} &
{\mathcal O}_{X_0} & \longrightarrow & 0
\end{matrix}$$ where the bottom exact sequence is obtained from , and the top exact sequence is as in (see also ). Let $$\label{nu}
\nu\, :\, \mathrm{H}^0(X_0, \, {\mathcal O}_{X_0})\, \longrightarrow\,
\mathrm{H}^1(X_0,\, \text{At}_P(E_G\, , D_0))$$ be the connecting homomorphism in the long exact sequence of cohomologies associated to the bottom exact sequence in . From the commutativity of and the construction of $\phi(1_{X_0})$ (see ) it follows that $$\label{phex}
\nu(1_{X_0})\,=\, \delta^1_*(\phi(1_{X_0}))\,=\, \Phi\, ,$$ where $\nu$ and $\delta^1_*$ are the homomorphisms constructed in and respectively, and $\Phi$ is the cohomology class in .
The diagram in produces the diagram $$\label{g5}
\begin{matrix}
&& 0 && 0 \\
&& \Big\downarrow && \Big\downarrow\\
0& \longrightarrow & \text{At}(E_P\, , D_0) & \longrightarrow &
f^*\text{At}({\mathcal E}_P\, , D_0) &
\longrightarrow & {\mathcal O}_{X_0} & \longrightarrow & 0\\
&&~ \Big\downarrow\xi && \Big\downarrow && \Vert\\
0& \longrightarrow & \text{At}_P(E_G\, , D_0) &
\longrightarrow & {\rm At}^\delta_P (E_G\, , D_0) & {\longrightarrow} &
{\mathcal O}_{X_0} & \longrightarrow & 0\\
&&~ \Big\downarrow\mu && ~ \Big\downarrow\\
&& {\mathcal L} & = & {\mathcal L}\\
&& \Big\downarrow && \Big\downarrow\\
&& 0 && 0
\end{matrix}$$ where $\xi$ and $\mu$ are the homomorphisms in . Using this diagram we can check that $$\label{id}
\Phi\,=\, \xi_*(\theta)\, ,$$ where $\theta$ is the cohomology classes in , and $$\xi_*\, :\, \mathrm{H}^1(X_0,\, \text{At}(E_P\, , D_0))\, \longrightarrow\,
\mathrm{H}^1(X_0,\,\text{At}_P(E_G\, , D_0))$$ is the homomorphism induced by $\xi$ in . Indeed, follows from , the commutativity of and the construction of $\theta$.
Let $\mu_*\, :\, \mathrm{H}^1(X_0,\, \text{At}_P(E_G\, , D_0))\,\longrightarrow\,
\mathrm{H}^1(X_0,\, {\mathcal L})$ be the homomorphism induced by the homomorphism $\mu$ in . It is straight–forward to check that $$\mu_*(\Phi)\,=\, S'(\phi(1_{X_0}))$$ (see , and for $S'$, $\phi(1_{X_0})$ and $\Phi$ respectively). Indeed, this follows from and the definition of $S(\delta)$ in . Combining this with , we have $$S'(\phi(1_{X_0}))\,=\, \mu_*(\Phi)\,=\, \mu_*(\xi_*(\theta))\, .$$ Since $\mu\circ\xi\,=\, 0$ (see ), it now follows that $S'(\phi(1_{X_0}))\,=\,0$.
Logarithmic connections and semistability of the underlying principal bundle {#SecMain}
============================================================================
Let $\mathcal{T}_{g,n}$ be the Teichmüller space of genus $g$ compact Riemann surfaces with $n$ ordered marked points. As before, we assume that $g\, >\, 0$, and if $g\,=\,1$, then $n\, >\, 0$. Let $$\tau\, :\, \mathcal{C} \,\longrightarrow\, \mathcal{T}_{g,n}$$ be the universal Teichmüller curve with $n$ ordered sections $\Sigma$. The fiber of $\mathcal{C}$ over any point $t\,\in\, \mathcal{T}_{g,n}$ will be denoted by ${\mathcal C}_t$. The ordered subset ${\mathcal C}_t\cap \Sigma\,\subset\,
{\mathcal C}_t$ will be denoted by $\Sigma_t$.
Take a $n$–pointed Riemann surface $(C_0\, ,\Sigma_0)$ of genus $g$, which is represented by a point of $\mathcal{T}_{g,n}$. Let $$\label{na0}
\nabla_0$$ be a logarithmic connection on a holomorphic principal $G$–bundle $F_G\,\longrightarrow
\, C_0$ with polar divisor $\Sigma_0$. By the Riemann–Hilbert correspondence, the connection $\nabla_0$ produces a flat (isomonodromic) logarithmic connection $\nabla$ on a holomorphic principal $G$–bundle $\mathcal{F}_G\,\longrightarrow\, \mathcal{C}$ with polar divisor $\Sigma$.
The following lemma is a special case of the main theorem in [@Nitsure] (see also [@Sh]). Although the families of $G$–bundles considered in [@Nitsure] are algebraic, all arguments there go through in the analytic case of our interest with obvious modifications.
\[lemNitsure\] Let $\mathcal{F}_G\,\longrightarrow\, \mathcal{C}\,\longrightarrow\,\mathcal{T}_{g,n}$ be as above. For each Harder–Narasimhan type $\kappa$, the set $$\mathcal{Y}_\kappa \,:=\, \{t\,\in\, \mathcal{T}_{g,n} ~\mid ~
\mathcal{F}_G\vert_{{\mathcal C}_t}\ \text{ is\ of\ type }\ \kappa\}$$ is a (possibly empty) locally closed complex analytic subspace of $\mathcal{T}_{g,n}$. More precisely, for each Harder–Narasimhan type $\kappa$, the union $\mathcal{Y}_{\leq \kappa}\,:=\,\bigcup_{\kappa'\leq \kappa}\mathcal{Y}_{\kappa'}$ is a closed complex analytic subset of $\mathcal{T}_{g,n}$. Moreover, the principal $G$–bundle $$\mathcal{F}_G\vert_{\tau^{-1}(\mathcal{Y}_\kappa)}\,\longrightarrow\,
\tau^{-1}(\mathcal{Y}_\kappa)$$ possesses a canonical holomorphic reduction of structure group inducing the Harder–Narasimhan reduction of $\mathcal{F}_G\vert_{{\mathcal C}_t}$ for every $t\,\in \,\mathcal{Y}_\kappa$.
In the following two Sections \[se5.1\] and \[se5.2\], we will see that under certain assumptions, the only Harder-Narasimhan stratum $\mathcal{Y}_\kappa$ of maximal dimension $\dim(\mathcal{T}_{g,n})\,=\,3g-3+n$ is the trivial one, in the sense that the Harder–Narasimhan parabolic subgroup is $G$ itself. In other words, if the principal $G$–bundle $F_G$ is not semistable, and therefore has a non-trivial Harder–Narasimhan reduction $E_P\, \subset\, E_G$ to a certain parabolic subgroup $P\, \subsetneq\, G$, then there is always an isomonodromic deformation in which direction the reduction $E_P$ is obstructed meaning it does not extend.
The case of $n = 0$ {#se5.1}
-------------------
In this subsection we assume that $n\,=\,0$. So, we have $g\, >\, 1$.
\[prop2\] There is a closed complex analytic subset ${\mathcal Y}\, \subset\,
{\mathcal T}_{g,0}$ of codimension at least $g$ such that for every $t\, \in\, {\mathcal T}_{g,0}\setminus {\mathcal Y}$, the holomorphic principal $G$–bundle $\mathcal{F}_G\vert_{{\mathcal C}_t}\,
\longrightarrow\, {\mathcal C}_t$ is semistable.
Let ${\mathcal Y}\, \subset\, {\mathcal T}_{g,0}$ denote the (finite) union of all Harder-Narasimhan strata ${\mathcal Y}_\kappa$ as in Lemma \[lemNitsure\] with nontrivial Harder-Narasimhan type $\kappa$. From Lemma \[lemNitsure\] we know that ${\mathcal Y}$ is a closed complex analytic subset of ${\mathcal T}_{g,0}$.
Take any $t\, \in\, \mathcal{Y}_\kappa\, \subset\, {\mathcal Y}$. Let $E_G\,=\,
\mathcal{F}_G\vert_{{\mathcal C}_t}$ be the holomorphic principal $G$–bundle on $X_0\,:=\, {\mathcal C}_t$. The holomorphic connection on $E_G$ obtained by restricting $\nabla$ will be denoted by $\delta$. Since $E_G$ is not semistable, there is a proper parabolic subgroup $P\, \subsetneq\, G$ and a holomorphic reduction of structure group $E_P\, \subset\, E_G$ to $P$, such that $E_P$ is the Harder–Narasimhan reduction [@Be], [@AAB]; the type of this Harder–Narasimhan reduction is $\kappa$. From Lemma \[lemNitsure\] we know that $E_P$ extends to a holomorphic reduction of structure group of the principal $G$–bundle $\mathcal{F}_G\vert_{\tau^{-1}(\mathcal{Y}_\kappa)}$ to the subgroup $P$.
Let $\text{ad}(E_P)$ and $\text{ad}(E_G)$ be the adjoint vector bundles of $E_P$ and $E_G$ respectively. Consider the vector bundle $$\text{ad}(E_G)/\text{ad}(E_P)\,=\, E_P({\mathfrak g}/{\mathfrak p})$$ (see ). We know that $$\label{deg2}
\mu_{\rm max}(E_P({\mathfrak g}/{\mathfrak p}))\, <\, 0$$ [@AAB p. 705] (see sixth line from bottom). In particular $$\label{deg3}
\text{degree}(E_P({\mathfrak g}/{\mathfrak p})) \, <\, 0\, .$$ A holomorphic connection on $E_G$ induces a holomorphic connection on $\text{ad}(E_G)$, hence $\text{degree}(\text{ad}(E_G))\,=\, 0$. Combining this with it follows that $\text{degree}(\text{ad}(E_P))\,>\, 0$, because $\text{ad}(E_G)/\text{ad}(E_P)\,=\,E_P({\mathfrak g}/{\mathfrak p})$. Since $\text{degree}(\text{ad}(E_P))\,\not=\, 0$, the holomorphic vector bundle $\text{ad}(E_P)$ does not admit any holomorphic connection, hence the principal $P$–bundle $E_P$ does not admit a holomorphic connection. Consequently, the second fundamental form $S(\delta)$ of $E_P$ for $\delta$ (see ) is nonzero.
Using the second fundamental form $S(\delta)$, construct the holomorphic line subbundle $${\mathcal L}\, \subset\, E_P({\mathfrak g}/{\mathfrak p})$$ as done in . From we have $$\label{deg}
\text{degree}({\mathcal L}) \, <\, 0\, .$$ Consider the short exact sequence of sheaves on $X_0$ $$\label{f7}
0\,\longrightarrow\, \mathrm{T}X_0 \,
\stackrel{S(\delta)'}{\longrightarrow}\, {\mathcal L}
\,\longrightarrow\, T^\delta \,:=\, {\mathcal L}/(S(\delta)(\mathrm{T}X_0))
\,\longrightarrow\, 0\, ,$$ where $S(\delta)'$ is constructed in ; note that $T^\delta$ is a torsion sheaf because $S(\delta)'\,\not=\, 0$ (recall that $S(\delta)\,\not=\, 0$). From it follows that $$\text{degree}(T^\delta)\,=\, \text{degree}({\mathcal L}) -
\text{degree}(\mathrm{T}X_0) \, <\, - \text{degree}(\mathrm{T}X_0)\, =\, 2g-2\, .$$ So, we have $$\dim \mathrm{H}^0(X_0,\, T^\delta)\,=\, \text{degree}(T^\delta)\, <\, 2g-2
\, \,=\, \dim \mathrm{H}^1(X_0,\, \mathrm{T}X_0)+1-g\, .$$ This implies that $$\label{li}
\dim \mathrm{H}^1(X_0,\, \mathrm{T}X_0)- \dim \mathrm{H}^0(X_0,\, T^\delta)
\,\geq\, g\, .$$ Consider the long exact sequence of cohomologies $$\mathrm{H}^0(X_0,\, T^\delta)\,\longrightarrow\,\mathrm{H}^1(X_0,\,
\mathrm{T}X_0)\,\stackrel{\zeta}{\longrightarrow}\,\mathrm{H}^1(X_0,\, {\mathcal L})$$ associated to the short exact sequence of sheaves in . From it follows that $$\label{codi}
\dim \zeta(\mathrm{H}^1(X_0,\,
\mathrm{T}X_0))\, \geq\, g\, .$$
Since the reduction $E_P$ extends to a holomorphic reduction of structure group of the principal $G$–bundle $\mathcal{F}_G\vert_{\tau^{-1}(\mathcal{Y}_\kappa)}$ to the subgroup $P$, combining and Proposition \[prop1\] we conclude that the codimension of the complex analytic subset $\mathcal{Y}_\kappa\, \subset\, {\mathcal T}_{g,0}$ is at least $g$. This completes the proof of the theorem.
When $n$ is arbitrary {#se5.2}
---------------------
Now we assume that $n\,>\,0$.
A logarithmic connection $\eta$ on a holomorphic principal $G$–bundle $F_G\,\longrightarrow\, X_0$ is called *reducible* if there is pair $(P\, ,F_P)$, where $P\, \subsetneq\, G$ is a parabolic subgroup and $F_P\, \subset\, F_G$ is a holomorphic reduction of structure group of $F_G$ to $P$, such that $\eta$ is induced by a connection on $F_P$. Note that $\eta$ is induced by a connection on $F_P$ if and only if the second fundamental form of $F_P$ for $\eta$ vanishes identically. A connection is called *irreducible* if it is not reducible or, equivalently, if the monodromy representation of the corresponding flat principal $G$–bundle does not factor through any proper parabolic subgroup of $G$.
\[propo3\] Assume that the logarithmic connection $\nabla_0$ in is irreducible. Then there is a closed complex analytic subset ${\mathcal Y}\, \subset\,
{\mathcal T}_{g,n}$ of codimension at least $g$ such that for every $t\, \in\, {\mathcal T}_{g,n}\setminus {\mathcal Y}$, the holomorphic principal $G$–bundle $\mathcal{F}_G\vert_{{\mathcal C}_t}$ is semistable.
The proof of Theorem \[prop2\] goes through after some obvious modifications. As before, let ${\mathcal Y}\, \subset\, {\mathcal T}_{g,n}$ be the locus of all points $t$ such that the principal $G$–bundle $\mathcal{F}_G\vert_{{\mathcal C}_t}$ is not semistable. Take any point $t\, \in\, {\mathcal Y}_\kappa\, \subset\,{\mathcal Y}$. Let $E_G\,=\,
\mathcal{F}_G\vert_{{\mathcal C}_t}$ be the holomorphic principal $G$–bundle on $X_0\,:=\, {\mathcal C}_t$. The logarithmic connection on $E_G$ with polar divisor $D_0\,:=\,\Sigma_t$ obtained by restricting $\nabla$ will be denoted by $\delta$.
Let $E_P\, \subset\, E_G$ be the Harder–Narasimhan reduction; its type is $\kappa$. Since $\nabla_0$ is irreducible, the second fundamental form $S(\delta)$ of $E_P$ for $\delta$ (see ) is nonzero. We note that for the monodromy of a logarithmic connection, the property of being irreducible is preserved under isomonodromic deformations. Consider the short exact sequence of sheaves on $X_0$ $$\label{l7}
0\,\longrightarrow\, \mathrm{T}X_0(-D_0) \,
\stackrel{S(\delta)'}{\longrightarrow}\, {\mathcal L}
\,\longrightarrow\, T^\delta \,:=\, {\mathcal L}/(S(\delta)(\mathrm{T}X_0(-D_0)))
\,\longrightarrow\, 0\, ,$$ where $S(\delta)'$ is constructed in . As before, $\text{degree}(
{\mathcal L})\, <\, 0$, because $\mu_{\rm max}(E_P({\mathfrak g}/{\mathfrak p}))
\, <\, 0$. So, $$\text{degree}(T^\delta)\,=\, \text{degree}({\mathcal L})- \text{degree}
(\mathrm{T}X_0(-D_0))\, <\, -\text{degree}(\mathrm{T}X_0(-D_0))\,=\,2g-2+n\, .$$ We now have $$\dim \mathrm{H}^0(X_0,\, T^\delta)\,=\, \text{degree}(T^\delta)\, <\, 2g-2+n
\, \,=\, \dim \mathrm{H}^1(X_0,\, \mathrm{T}X_0(-D_0))+1-g\, .$$ Hence the dimension of the image of the homomorphism $$\label{codi2}
\mathrm{H}^1(X_0,\, \mathrm{T}X_0(-D_0))\, \longrightarrow\,
\mathrm{H}^1(X_0,\, {\mathcal L})$$ in the long exact sequence of cohomologies associated to is at least $g$. Since the reduction $E_P$ extends to a holomorphic reduction of $\mathcal{F}_G\vert_{\tau^{-1}(\mathcal{Y}_\kappa)}$ to $P$, and the dimension of the image of the homomorphism in is at least $g$, from Proposition \[prop1\] we conclude that the codimension of $\mathcal{Y}_\kappa \, \subset\, {\mathcal T}_{g,0}$ is at least $g$.
Stability of underlying principal bundle {#se5.3}
----------------------------------------
We now assume that $g\, \geq\, 2$.
\[thm1\] Assume that the logarithmic connection $\nabla_0$ in is irreducible. There is a closed analytic subset ${\mathcal Y}'\, \subset\, {\mathcal T}$ of codimension at least $g-1$ such that for every $t\, \in\, {\mathcal T}_g\setminus {\mathcal Y}$, the holomorphic principal $G$–bundle $\mathcal{F}_G\vert_{{\mathcal C}_t}$ is stable.
The proof is identical to the proof of Proposition \[propo3\]. If $E_G\,=\, \mathcal{F}_G\vert_{{\mathcal C}_t}$ is not stable, there is a maximal parabolic subgroup $P\,\subsetneq\, G$ and a holomorphic reduction of structure group $E_P\, \subset\, E_G$ to $P$, such that the quotient bundle $$\text{ad}(E_G)/\text{ad}(E_P)\,=\, E_P({\mathfrak g}/{\mathfrak p})$$ is semistable of degree zero. Therefore, we have $\text{degree}({\mathcal L})
\,\leq\, 0$. This implies that $$\text{degree}(T^\delta)
\,\leq\, 2g-2-n\, .$$ Hence the dimension of the image of the homomorphism $\mathrm{H}^1(X_0,\, \mathrm{T}X_0(-D_0))\, \longrightarrow\,
\mathrm{H}^1(X_0,\, {\mathcal L})$ in the long exact sequence of cohomologies associated to is at least $g-1$.
Acknowledgements {#acknowledgements .unnumbered}
================
We thank the referee for helpful comments. We thank Université de Brest for hospitality where the work was initiated. The first author is supported by a J. C. Bose Fellowship. The second author is supported by ANR-13-BS01-0001-01 and ANR-13-JS01-0002-01.
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Audio-based cover song detection has received much attention in the MIR community in the recent years. To date, the most popular formulation of the problem has been to compare the audio signals of two tracks and to make a binary decision based on this information only. However, leveraging additional signals might be key if one wants to solve the problem at an industrial scale. In this paper, we introduce an ensemble-based method that approaches the problem from a many-to-many perspective. Instead of considering pairs of tracks in isolation, we consider larger sets of potential versions for a given composition, and create and exploit the graph of relationships between these tracks. We show that this can result in a significant improvement in performance, in particular when the number of existing versions of a given composition is large.'
bibliography:
- 'track-to-work-ismir.bib'
title: 'Ensemble-based cover song detection'
---
Introduction
============
With the rise of online streaming services, it is becoming easier for artists to share their music with the rest of the world. With catalogs that can reach up to tens of millions of tracks, one of the rising challenges faced by music streaming companies is to assimilate ever-better knowledge of their content – a key requirement for enhancing user and artist experience. From a musical perspective, one highly interesting aspect is the detection of composition similarities between tracks, often known as the *cover song* detection problem. This is, however, a very challenging problem from a content analysis point of view, as artists can make their own version of a composition by modifying any number of elements – instruments, harmonies, melody, rhythm, structure, timbre, vocals, lyrics, among others.
Over the years, it has become customary in the Music Information Retrieval (MIR) literature to address the cover song detection problem in what is arguably the most challenging setting. Indeed, most papers attempt to detect composition relationships between pairs of tracks based on their two audio signals only – in other words, completely out of context and without using any metadata information. While this well-defined task makes sense from an academic perspective, it might not be the optimal approach for solving the problem at an industrial scale [@correya2018large].
The second starting point of our work is the fact, often mentioned in cognitive science, that commonly observed patterns are represented and stored in a redundant fashion in the human brain, which makes them more likely to be retrieved, recognised and identified than patterns that are observed less frequently [@kurzweil2013create]. If true, this would apply to our assessment of composition similarities as well. The main idea behind our work is that the corpus of existing versions of a composition can be precisely a substitute for these multiple representations.
Following these guiding intuitions, we turn to a new use case, where we do not just have pairs but a pool of *candidates* that are likely to be instances of some given musical work (according *e.g.* to some first metadata analysis). We then compare these candidates not only to *one* reference version (*e.g.* the original track, if it exists) but also to other candidate versions. We then build a graph of all these versions to identify composition clusters. Sometimes, when hundreds or thousands of versions of a given work exist (which is quite common in the catalogue of a streaming company), this ensemble-based approach can result in substantial improvements on the cover detection task.
In Section \[sec:literature\] we present a review of the literature on cover identification. In Section \[sec:1v1\], we present the 1-vs-1 cover identification algorithm that we use throughout the paper, which is heavily based on [@tralie2017early]. The main contribution of this paper lies in Section \[sec:clustering\], in which we present the new use case for cover identification described in the previous paragraph.We then showcase our method with examples in Section \[sec:results\] and discuss some challenges in Section \[sec:discussion\].
Related work {#sec:literature}
============
A number of possible approaches for cover song identification have been developed in the last decade, with varying levels of performance. Reference [@ellis2007identifyingcover] introduced a first solution to this problem and has been used as a starting point for many subsequent studies. The main idea is to extract a list of beat-synchronous [@durand2017robust] chroma features from two input tracks and quantify their similarity by applying dynamic programming algorithms to a cross-similarity matrix derived from these features. This algorithm has been refined in [@ellis20072007] by the same authors by adding a few modifications such as tempo biasing to improve the results. Harmonic Pitch Class Profile (HPCP) features (chroma features) have proven very useful in cover identification [@ellis2007identifyingcover; @serra2008chroma; @ravuri2010cover; @serra2008transposing] as they capture meaningful musical information for composition. Other features have subsequently been introduced, such as self-similarity matrices (SSM) of Mel-Frequency Cepstral Coefficients (MFCC) [@tralie2015cover; @tralie2017early]. To take advantage of the complementary properties of different types of features, [@tralie2017early] further introduced a method to combine several audio features by fusing the associated cross-similarity matrices, which resulted in a significant increase in performance compared to single-feature approaches.
Having extracted audio features from two tracks to be compared, most methods use dynamic programming (either Dynamic Time Warping or the Smith-Waterman algorithm [@smith81]) to assign a score to the pair [@tralie2017early; @tralie2015cover; @ellis2007identifyingcover]. One drawback of these methods is that they are computationally expensive and cannot be run at scale. Hence other authors have developed solutions that enable cover identification at scale by mapping audio features to smaller latent spaces. For instance, [@bertin2012large; @humphrey2013data] use Principal Component Analysis (PCA) to compute a condensed representation of audio features which they use to perform a large-scale similarity search (*e.g.* a nearest neighbor search). In the same vein, references [@fang2017deep; @qi2017audio] use deep neural networks to learn low-dimensional representations of chroma features.
Pairwise matching {#sec:1v1}
=================
As mentioned above, our ensemble-based cover identification method consists of two steps. For a given work, we proceed to: ***(i)*** a pairwise (1-vs-1) comparison of all the tracks in a pool of potential candidates, ***(ii)*** a clustering of these candidates based on the results of step (i).
In this section we present the 1-vs-1 cover song identification algorithm (i) which will be used as a starting point for our ensemble-based approach, and evaluate its performance on two distinct cover datasets.
The algorithm {#subsec:1v1}
-------------
For the purposes of this work, any 1-vs-1 similarity measure could be used for step (i), as we are mainly interested in quantifying the impact of step (ii) on the overall performance. We have chosen to rely on an implementation of the algorithm introduced in [@tralie2017early], as the algorithm achieves the best results to date on the *Covers80* [@ellis2007covers80] and *MSD* (*Covers1000*) datasets [@bertin2011million]. A high-level overview of the pipeline is shown on Figure \[fig:pipeline\]. As with most algorithms presented in Section \[sec:literature\], it can be decomposed into two stages: first, it extracts a list of meaningful audio features from the two tracks to be compared, then it computes a similarity score based on these. The details of this method are not directly relevant to our work, so we will focus here on a quantitative assessment of its performance, to give the reader a quantitative idea of our starting point. For those interested in the details of how the algorithm works, please refer to [@tralie2017early].
![\[fig:pipeline\] High level overview of the 1-vs-1 matching pipeline. Image reproduced from [@tralie2017early].](CoverSongsFusionPipeline.pdf){width="50.00000%"}
Quantitative evaluation of the 1-vs-1 method {#subsec:results}
--------------------------------------------
We evaluate our implementation of [@tralie2017early] on two different datasets, and compare it with the numbers reported in the original paper as well as with a publicly available implementation of [@tralie2017early] by its authors.[^1] To make the comparison more interpretable, we evaluate two versions of our implementation with two sets of parameters: *Params1* mimics the parameters used in [@tralie2017early], and should therefore produce numbers that very similar to those described in the original paper, while *Params2* uses shorter 8-beats-long blocks.
We first compare the algorithms on the widely used *Covers80* dataset [@ellis2007covers80] to enable comparison with other published methods. The dataset is composed of 160 tracks that are divided into two sets (A and B) of 80 tracks each, with every track in set A matching one (and only one) track in set B. For each of the 160 tracks, we compute its score with all the other 159 tracks and report the rank of its true match. Table \[table:benchmark\] reports the Mean Rank (MR) of the true match (1 is best), the Mean Reciprocal Rank (MRR) [@craswell2009mean], as well as the Recall@1 (R@1) and Recall@10 (R@10). We also compute the so-called *Covers80 scores* by querying each track in set A against all the tracks in set B and reporting the number of matches found with rank 1.[^2] Overall, our results are close to the ones reported in [@tralie2017early] – even though we could not quite reach the numbers given in their paper.
[| c | c | c | c | c | c || c | c |]{} & &\
& MR & MRR &
--------
R@1
(/159)
--------
: \[table:benchmark\] Comparison of our implementations (*Params1* and *Params2*) against the implementations of [@tralie2017early], on the *Covers80* dataset and on our internal dataset. The recall rates for the internal dataset correspond to a false positive rate of 0.5%. For each column, the best performance is printed in bold.
&
--------
R@10
(/159)
--------
: \[table:benchmark\] Comparison of our implementations (*Params1* and *Params2*) against the implementations of [@tralie2017early], on the *Covers80* dataset and on our internal dataset. The recall rates for the internal dataset correspond to a false positive rate of 0.5%. For each column, the best performance is printed in bold.
&
------------
*Covers80*
score
------------
: \[table:benchmark\] Comparison of our implementations (*Params1* and *Params2*) against the implementations of [@tralie2017early], on the *Covers80* dataset and on our internal dataset. The recall rates for the internal dataset correspond to a false positive rate of 0.5%. For each column, the best performance is printed in bold.
& Recall &
-----------
Recall
(no Jazz)
-----------
: \[table:benchmark\] Comparison of our implementations (*Params1* and *Params2*) against the implementations of [@tralie2017early], on the *Covers80* dataset and on our internal dataset. The recall rates for the internal dataset correspond to a false positive rate of 0.5%. For each column, the best performance is printed in bold.
\
[@tralie2017early] paper & **7.8** & **0.85** & **131** & 143 & **68/80** & - & -\
[@tralie2017early] code & 8.6 & 0.77 & 114 & **145** & 62/80 & 73.2% & 81.8%\
Params1 & 10.5 & 0.81 & 125 & 136 & 64/80 & 79.2% & 87.8%\
Params2 & 13.2 & 0.75 & 116 & 128 & 60/80 & **85.8%** & **95.1%**\
To complement this baseline, we have created an internal dataset of 452 pairs of covers grouped into several categories, obtained by metadata filtering based on the keywords *Acoustic Cover, Instrumental Cover, Karaoke, Live, Remix, Tribute* as well as some *Classical* and *Jazz* covers. Such granularity allows us to compare the performance of our algorithm across genres and cover types, providing a new perspective on the problem, as shown in Table \[table:recall\]. We have tested the two versions of our algorithm on the 452 positive pairs and 10,000 negative pairs selected uniformly at random. We selected the classification threshold to ensure a very low false positive rate below $0.5\%$. Results are presented in Table \[table:benchmark\]. Our algorithm reaches $86.3\%$ recall, versus $73.2\%$ for the publicly available implementation of [@tralie2017early][^3]. Note that jazz is the most challenging genre to detect, as jazz covers include a lot of improvisation that can be structurally different from their parent track (see Table \[table:recall\]). If we remove jazz covers from the dataset, the recall increases to $95.1\%$ with the *Params2* implementation.
[-0.5cm]{}
**Type** Acoustic Instr. Karaoke Live Remix Tribute Classical Jazz
----------------- ---------- -------- --------- ------ ------- --------- ----------- ------
**\# of pairs** 57 63 46 57 31 53 77 68
**Recall** 94% 84 % 97% 100% 93 % 96 % 100 % 35 %
: \[table:recall\] Recall rates for each genre in our internal dataset, with a $<0.5\%$ false positive rate.
In view of these results, we will use our own implementation with *Params2* throughout the rest of this paper, as it is faster and performs best on our internal dataset, which is larger and more diverse than *Cover80*.
Distributions of scores {#subsec:threshold}
-----------------------
Figure \[fig:histogramscores\] presents the histogram of pairwise scores for all the positive and negative pairs in our internal dataset. The distribution of scores for the negative pairs is short-tailed and tightly concentrated around $s=2$. This means that above $s\simeq 5$, all the pairs can be matched with high confidence. The distribution of scores for the positive pairs is much wider. As we can see from the histogram, a non-negligible fraction of these pairs lies below the classification threshold (dashed vertical line) and thus cannot be detected with this 1-vs-1 method. The purpose of the next section will be to apply an ensemble method to a pool of candidate versions of a given work, to bring these undetected candidates above the threshold by exploiting the many-to-many relationships between the candidates.
![\[fig:histogramscores\] Histogram of the scores for positive *(blue)* and negative *(green)* pairs on our internal dataset. The threshold corresponds to the threshold used for Table \[table:recall\].](histogram.pdf){width=".45\textwidth"}
Ensemble analysis {#sec:clustering}
=================
While the 1-vs-1 algorithm we presented in Section \[sec:1v1\] gives satisfying results overall, it still struggles on covers that are significantly different from their original track. Here we show how analyzing a large pool of candidate covers for one given reference track can improve the quality of the matching. The intuition behind this idea is that a cover version can match the reference track poorly, but match another intermediate version which is closer to the reference. For instance, an acoustic cover can be difficult to detect on a 1-vs-1 basis, but might match a karaoke version which itself strongly matches the reference track. We therefore turn to a new use case, where we not only compare single pairs (*e.g.* one reference track against one possible cover), but instead start from a pool of candidates that are all likely to be instances of some given composition (or *work*). Usually, this pool corresponds to candidates that have been pre-filtered according to some non-audio related signal, *e.g.* their title, and might comprise up to a few thousands candidates, depending on the popularity of the work and the specificity of the pre-filtering step.
[0.15]{} {width="\textwidth"}
[ ]{}
[0.15]{} {width="\textwidth"}
[ ]{}
[0.15]{} {width="\textwidth"}
Computing all pairwise scores {#sec:NvsN}
-----------------------------
Given a set of $N$ candidate versions of a work, we first compare all possible pairs of candidates within the set, resulting in $\frac{N(N-1)}{2}$ distinct scores $\left\{s_{ij}\right\}_{1\leq i < j \leq N}$. As mentioned above, if the candidates have been pre-filtered using some metadata-matching algorithm, $N$ typically varies from a few dozen to a few thousand candidates.
Scores to distances {#subsubsec:logistic}
-------------------
Figure \[fig:histogramscores\] shows that almost all negative pairs have scores between 0 and 4 while scores above 8 always correspond to positives. Scores above 8 should thus indicate a high probability of a true match regardless of the score, while a variation in score around 4 should have a significant impact on that probability. To account for this fact, we convert our scores into more meaningful distances using a logistic function: $d_{ij} = \left(1 + e^{- \frac{s_{ij}-m}{\sigma}}\right)^{-1}$, where $s_{ij}$ is the score associated to pair $(i, j)$ and $d_{ij}$ is the resulting distance. We have found that the values $\sigma=0.5$ and $m=4.3$ work well with the distance-collapsing algorithm introduced in the next section.
Collapsing the distances {#subsubsec:collapsing}
------------------------
Let $D = \left\{ d_{ij}\right\}$ denote the pairwise distance matrix between all pairs of candidates (see Figure \[fig:floydwarshall\], top left). The idea behind the ensemble-based approach is to exploit the geometry of the data to enhance the accuracy of the classification – for example, the fact that a track can match the reference track better through intermediate tracks than directly. We use a loose version of the Floyd-Warshall algorithm [@Floyd:1962:A9S:367766.368168] to update the distances in $D$, such that the new distances satisfy the triangular inequality most of the time[^4]. The method is presented in Algorithm \[alg:loosefw\].
$\widetilde{D}(i, j) \gets \textbf{min}_{k\neq i, j}^{(2)} D(i, k) + D(k, j) + \eta$ $D(i, j) \gets \textbf{min} \Big(D(i, j),\ \widetilde{D}(i, j) \Big)$
Here $\mathrm{min}^{(k)}(x)$ denotes the $k^{th}$ smallest value of a vector $x$. We have found that the algorithm is slightly more robust when imposing a penalty $\eta>0$ for using an intermediate node, which we have set to $\eta=0.01$ after performing a grid-search optimization.
Figure \[fig:floydwarshall\] shows the distance matrix before (top left) and after (top right) updating the distances using Algorithm \[alg:loosefw\], for a set of candidates versions of *Get Lucky* by Daft Punk. We can see that the updated distance matrix has a more neatly defined division between clusters of tracks. The figure shows one large cluster in which all tracks are extremely close to each other (the white area), a few smaller clusters (white blocks on the first diagonal) and a number of isolated tracks that match only themselves.
![\[fig:floydwarshall\] Top: The Floyd-Warshall algorithm applied to the distance matrix of *Get Lucky*, with (left) original distance matrix and (right) the distance matrix after applying the Floyd-Warshall algorithm. Darker shades correspond to larger distances. Bottom: the corresponding dendrogram obtained using hierarchical clustering on the Floyd-Warshall distance matrix.](distancematrix-2.pdf){width="\linewidth"}
![\[fig:floydwarshall\] Top: The Floyd-Warshall algorithm applied to the distance matrix of *Get Lucky*, with (left) original distance matrix and (right) the distance matrix after applying the Floyd-Warshall algorithm. Darker shades correspond to larger distances. Bottom: the corresponding dendrogram obtained using hierarchical clustering on the Floyd-Warshall distance matrix.](distancematrix-1.pdf){width="\linewidth"}
![\[fig:floydwarshall\] Top: The Floyd-Warshall algorithm applied to the distance matrix of *Get Lucky*, with (left) original distance matrix and (right) the distance matrix after applying the Floyd-Warshall algorithm. Darker shades correspond to larger distances. Bottom: the corresponding dendrogram obtained using hierarchical clustering on the Floyd-Warshall distance matrix.](dendrogram-1.pdf){width=".5\textwidth"}
Hierarchical clustering {#sec:hierarchical}
-----------------------
We then proceed to a clustering of the tracks using the updated distance matrix defined in \[subsubsec:collapsing\], denoted $D'$. We use hierarchical clustering as we have no prior knowledge on the number of clusters in the graph. Figure \[fig:floydwarshall\] (bottom) shows the dendrogram associated with the hierarchical clustering applied to $D'$. In this example, if we apply a relatively selective threshold, we find one major cluster (colored in blue in Figure \[fig:floydwarshall\]) that contains 97% of the true positives and no false positives. Most other clusters contain a single element, which are all the negative tracks and the remaining 3% of the positives. If we set the clustering threshold lower, then we can get more granular clusters within a same work.
Final score {#sec:cophenetic}
-----------
In order to assign each track a final score that measures its similarity to the reference track, we use the *cophenetic distance* to the reference track, *i.e.* the distance along the dendrogram that is produced by the hierarchical clustering. Each track is thus assigned a final score in $0-100$, simply taken equal to 100 $\times$ (1 - cophenetic distance), such that exact matches have a score of 100.
Analysis of real world examples {#sec:results}
===============================
Data
----
We now apply the above to real world data. Our dataset consists of 10 sets of candidates that correspond to 10 works that we want to find the versions of. These 10 works span multiple genres and musical styles, including Hip Hop, R&B, Rap, Pop and Jazz. For a given work, we create the set of candidates by performing a metadata search of the given work’s title on the whole Spotify catalogue. Across the given works that we study, this produces sets of candidates whose sizes vary from a few hundred to a few thousand candidate tracks. Each set includes a reference track, which will be the anchor point for that composition. More details on the dataset can be found in Table \[dataset\].
[| c | c | c | c | c |]{} **Work** & **\# tracks** & **% positives** &
---------------
**Reference**
**artist**
---------------
: \[dataset\] The “10 works” dataset. For each work, we have selected a reference track that will be our anchor point for that composition. *Click on a work to play the reference track in the browser*.
\
[*Airplane*](https://open.spotify.com/track/6lV2MSQmRIkycDScNtrBXO?si=xdSaKceZQS-MqGR8O1J4Og) & 811 & 19% & B.o.B\
[*Believer*](https://open.spotify.com/track/05KfyCEE6otdlT1pp2VIjP?si=nZRyhDndSH6rBmnaRMiMnA) & 2552 & 5% & Imagine Dragons\
[*Blurred Lines*](https://open.spotify.com/track/5PUvinSo4MNqW7vmomGRS7?si=HoWWVLjFSd6V2MpySx-JdA) & 386 & 71% & Robin Thicke\
[*Bodak Yellow*](https://open.spotify.com/track/2771LMNxwf62FTAdpJMQfM?si=QjthOI5dSTy5FZ_WMUXQSg) & 110 & 78% & Cardi B\
[*Brown Sugar*](https://open.spotify.com/track/7rt0kEDWRg3pgTZJKuszoE?si=zGZGkIM0RSmp5FYtHIbBaQ) & 721 & 5.8% & D’Angelo\
[*Embraceable You*](https://open.spotify.com/track/00gLKa8SbzW3SlRwlfh6U6?si=L6AH33NDROeOnp8wNDN3rQ) & 1319 & 94% & Sarah Vaughan\
[*Get Lucky*](https://open.spotify.com/track/2Foc5Q5nqNiosCNqttzHof?si=qCa_9l5mTJGxmF-lR_No3A) & 657 & 83% & Daft Punk\
[*Halo*](https://open.spotify.com/track/0DCdMiGUOlZM8YRRF2kzR7?si=mxnfznyrQEuukrOVQmCA9w) & 2995 & 7.9% & Beyoncé\
[*Heartless*](https://open.spotify.com/track/28siypca4TEqLnQ6Cgbdbe?si=YJO0kJF2QQyA4wallfNwug) & 1747 & 5.3% & Kayne West\
[*Imagine*](https://open.spotify.com/track/7pKfPomDEeI4TPT6EOYjn9?si=OsgDBo7yRwu3NtL_6FE47Q) & 2044 & 50% & John Lennon\
Outline of the analysis
-----------------------
For each of these works, we analyze the set of candidates following the steps outlined in the previous two sections, providing us with two sets of outputs for each work: ***(a)*** the *direct score*, defined as the output of the 1-vs-1 algorithm between each candidate and the reference track, as described in Section \[sec:1v1\] (rescaled between 0 and 100); ***(b)*** the *ensemble-based score*, produced by the method described in Section \[sec:clustering\] (also between 0 and 100). In the next section we start by quantitatively evaluating our ensemble-based approach (b) against the direct approach (a), before turning to some qualitative examples.
Quantitative results {#sec:metrics}
--------------------
We define two different metrics to evaluate the direct and the ensemble-based methods: **Ranking metric:** For each work, we pick the value of the threshold that minimizes the number of classification errors, and report the number of errors. We call this a *ranking metric* as the number of errors is minimized when positives and negatives are perfectly ranked, regardless of their scores. We also report the corresponding recall and false positive rates for this threshold.
**Classification metric:** We fix a universal classification threshold and compute the corresponding number of classification errors.
------------------- --------------- ------ ------- ------ -------- ------ -------
**Work** **Best thr.**
Abs. Rel. Abs. Rel. Abs. Rel.
*Airplane* 12.1 33 21.9% 1 0.2% 34 4.2%
*Believer* 18.2 6 5.2% 0 0.0% 6 0.2%
*Blurred Lines* 10.1 19 7.0% 9 8.3% 28 7.3%
*Bodak Yellow* 6.1 6 7.0 % 6 33.3% 12 10.9%
*Brown Sugar* 12.1 2 4.8% 1 0.1% 3 0.4%
*Embraceable You* 4 0 0% 74 98.7 % 74 5.6 %
*Get Lucky* 10.1 17 3.1% 3 2.6% 20 3.0%
*Halo* 11.1 8 3.4% 8 0.3% 16 0.5%
*Heartless* 12.2 15 16.3% 2 0.1% 17 1.0%
*Imagine* 15.2 72 7.1% 17 1.7% 89 4.4%
------------------- --------------- ------ ------- ------ -------- ------ -------
: \[results:A\] Optimal thresholds and corresponding results for the ranking metric.
------------------- --------------- ------ ------- ------ -------- ------ -------
**Work** **Best thr.**
Abs. Rel. Abs. Rel. Abs. Rel.
*Airplane* 70.7 4 2.6% 3 0.5% 7 0.9%
*Believer* 85.9 0 0.0% 0 0.0% 0 0.0%
*Blurred Lines* 52.5 0 0.0% 0 0.0% 0 0.0%
*Bodak Yellow* 29.3 9 10.5% 0 0.0% 9 8.2%
*Brown Sugar* 70.7 0 0.0% 1 0.1% 1 0.1%
*Embraceable You* 40.4 22 1.8 % 19 25.3 % 41 3.1 %
*Get Lucky* 78.8 5 0.9% 2 1.7% 7 1.1%
*Halo* 98.0 4 1.7 % 20 0.7% 24 0.8%
*Heartless* 83.8 8 8.7% 1 0.1% 9 0.5%
*Imagine* 96.0 1 0.1% 5 0.5% 6 0.3%
------------------- --------------- ------ ------- ------ -------- ------ -------
: \[results:A\] Optimal thresholds and corresponding results for the ranking metric.
Table \[results:A\] shows the results for the ranking metric for each work in our dataset. For the optimal thresholds, we report the number of false negatives, false positives and the sum of both (*i.e.* the total number of classification errors). We also compute the corresponding false negative rate, false positive rate and total error rate.
Table \[table:results-ranking-direct\] shows the ranking results for the direct approach. Interestingly, the number of false negatives tends to be higher than the number of false positives.[^5] This is in line with the histogram in Figure \[fig:histogramscores\], which shows a short-tailed distribution for the negatives and a wider distribution for the positives. Overall, the error rate lies between $0-10\%$, corresponding to a recall rate between $80\%$ and $97\%$ and a false positive rate below $10\%$ (except for *Bodak Yellow* and *Embraceable You* which have a very small number of negatives to begin with – the latter case is in fact degenerate as nearly all tracks are classified as matching).
Table \[table:results-ranking-graphical\] shows the ranking results for our ensemble-based approach. The number of ranking errors is substantially lower than for the direct approach, including both the number of false positives and false negatives, as the total error rate goes down below $1\%$ in most cases. Again, the main exception is *Bodak Yellow*, which has the smallest number of candidates.[^6] *Embraceable You* is the second most challenging work, but remarkably its threshold is no longer degenerate, meaning that the method has now found a way to separate the candidates. Notably, the number of false negatives no longer outnumbers the number of false positives: the ensemble-based approach has successfully caught most of the difficult tracks that poorly matched the reference track. Among the few tracks that are still missed, several are actually very close to the threshold, and only a handful are still completely undetected (cf Table \[misses\]).
------------------- ---------- ------ -------- ------ ------ ------ --------
**Work** **Thr.**
Abs. Rel. Abs. Rel. Abs. Rel.
*Airplane* 12.1 33 21.9% 1 0.2% 34 4.2%
*Believer* 12.1 4 3.5% 91 3.7% 95 3.7%
*Blurred Lines* 12.1 28 10.3% 0 0% 28 7.3%
*Bodak Yellow* 12.1 49 57 % 0 0% 49 44.5%
*Brown Sugar* 12.1 2 4.8% 1 0.1% 3 0.4%
*Embraceable You* 12.1 753 60.5 % 3 4 % 756 57.3 %
*Get Lucky* 12.1 23 4.2% 1 0.9% 24 3.7%
*Halo* 12.1 12 5.1% 4 0.1% 16 0.5%
*Heartless* 12.1 15 16.3% 2 0.1% 17 1.0%
*Imagine* 12.1 49 4.8% 94 9.3% 143 7%
------------------- ---------- ------ -------- ------ ------ ------ --------
: \[results:B\] Universal threshold and corresponding results for the classification metric.
------------------- ---------- ------ ------- ------ ------- ------ -------
**Work** **Thr.**
Abs. Rel. Abs. Rel. Abs. Rel.
*Airplane* 78.8 9 6.0% 1 0.2% 10 1.2%
*Believer* 78.8 0 0.0% 27 1.1% 27 1.1%
*Blurred Lines* 78.8 2 0.7% 0 0.0% 2 0.5%
*Bodak Yellow* 78.8 19 22.1% 0 0.0% 19 17.3%
*Brown Sugar* 78.8 2 4.8% 1 0.1% 3 0.4%
*Embraceable You* 78.8 70 5.6 % 1 1.3 % 71 5.4 %
*Get Lucky* 78.8 5 0.9% 2 1.7% 7 1.1%
*Halo* 78.8 0 0 % 163 5.9% 163 5.4%
*Heartless* 78.8 7 7.6% 4 0.2% 11 0.6%
*Imagine* 78.8 0 0% 158 15.6% 158 7.7%
------------------- ---------- ------ ------- ------ ------- ------ -------
: \[results:B\] Universal threshold and corresponding results for the classification metric.
Table \[results:B\] shows the results for the classification metric. The universal threshold for each approach is defined as the median of the optimal thresholds obtained in the ranking experiment above. Again, we report the number of false negatives, the number of false positives and the sum of both. We also compute the corresponding false negative rate, false positive rate and total error rate. Here again, the results of the ensemble-based approach are overall superior to the direct approach, mostly due to an increase in recall. Although Table \[table:results-classification-direct\] is quite similar to Table \[table:results-ranking-direct\], which is a sign that the threshold on direct scores can be chosen in a nearly universal way, Table \[table:results-classification-graphical\] differs considerably from Table \[table:results-ranking-graphical\] for some specific works (namely *Halo*, *Imagine* and *Believer*). This happens as the optimal threshold is significantly higher on these works (often $>95\%$), letting a large number of false positives above the 78.8% threshold.
Examples {#sec:examples}
--------
For each work, we can identify the cases where the ensemble-based approach has allowed us to detect previously undetected tracks, and trace back the optimal path that joined the reference track and the newly found track. Table \[paths\] shows a few examples of such paths for various works. For each example, the reference track is shown at the top of the cell (depth 0), and the newly found track at the bottom of the cell (depth $>1$), with the intermediate tracks that allowed to bridge the gap in between. All the examples are true positives, except for the last example (*Halo Halo* by Fajters), which has been erroneously matched to a karaoke version of the reference track.
[| c | c | c | c | c | c | c |]{} **Work** & **Depth** & **Main artist (& link)** &
------------
**Direct**
**scores**
------------
: \[paths\] Some examples of tracks that are undetected by the direct approach and captured by the ensemble-based approach, in the ranking experiment. The scores that are above the detection thresholds for each method are displayed in bold (the corresponding detection thresholds can be found in Table \[results:A\]). *Click on an artist to play in the browser*.
&
--------------------
**Ensemble-based**
**scores**
--------------------
: \[paths\] Some examples of tracks that are undetected by the direct approach and captured by the ensemble-based approach, in the ranking experiment. The scores that are above the detection thresholds for each method are displayed in bold (the corresponding detection thresholds can be found in Table \[results:A\]). *Click on an artist to play in the browser*.
\
& 0 & [John Lennon](https://open.spotify.com/track/7pKfPomDEeI4TPT6EOYjn9?si=OsgDBo7yRwu3NtL_6FE47Q) & **100** & **100**\
*Imagine* & 1 & [Classic Gold Hits](https://open.spotify.com/track/4szK84IjHD9sYi2acYQuy2?si=HVjmfKf3RqW6AfE1BHvsgg) & **60.0** & **99.99**\
& 2 & [A Perfect Circle](https://open.spotify.com/track/066N0phJeNFLEiREiP65VG?si=EaCo5Pa1Ss6bjAyz4M2-NQ) & **21.6** & **97.9**\
& 3 & [Yoga Pop Ups](https://open.spotify.com/track/4CZdBLanxIhSElTxSpIH5b?si=JUXL1Rk0R2iKhcT4r-_VyA) & 8.6 & **97.9**\
& 0 & [Kanye West](https://open.spotify.com/track/28siypca4TEqLnQ6Cgbdbe?si=YJO0kJF2QQyA4wallfNwug) & **100** & **100**\
*Heartless* & 1 & [The Fray](https://open.spotify.com/track/0DU2iZx61j4QCsjEJQHeS8?si=zR_4bZH0RHSSm_dzsIFkuA) & **34.1** & **99.85**\
& 2 & [William Fitzsimmons](https://open.spotify.com/track/1OV00OgzTFS8JkAmd7ABR6?si=BKJHJ9XjSiORb8Htto-PyA) & 9.5 & **90.7**\
& 0 & [Daft Punk](https://open.spotify.com/track/2Foc5Q5nqNiosCNqttzHof?si=qCa_9l5mTJGxmF-lR_No3A) & **100** & **100**\
*Get Lucky* & 1 & [Samantha Sax](https://open.spotify.com/track/6fclktfmcORXSsD4Acl5HF?si=p8BhHTc8RLSiG8qwhKgfvQ) & **40.4** & **99.95**\
& 2 & [Dallas String Quartet](https://open.spotify.com/track/0JvSslOE8D16WpkCkABzNE?si=c7W3UB9tTPe8yXvZj2kAwA) & 6.9 & **86.6**\
& 0 & [Beyonce](https://open.spotify.com/track/0DCdMiGUOlZM8YRRF2kzR7?si=mxnfznyrQEuukrOVQmCA9w) & **100** & **100**\
*Halo* & 1 & [LP](https://open.spotify.com/track/35LlWyY6u5ajhNbHkalXkB?si=PBR2fzq6TfabzKtZGzBIgw) & **27.6** & **99.96**\
& 2 & [Dion Lee](https://open.spotify.com/track/3e9WMvJgfFzHy7KDpqJQqh?si=wOM5OYiTTiW0BSHvFigy8A) & 7.96 & **99.16**\
*Halo* & 0 & [Beyonce](https://open.spotify.com/track/0DCdMiGUOlZM8YRRF2kzR7?si=mxnfznyrQEuukrOVQmCA9w) & **100** & **100**\
& 1 & [Karaoke Universe](https://open.spotify.com/track/0ditbVz3cyzR6BQrxcKvOA?si=nQC6vL9aSU2wZTcW5NcCKw) & **20.5** & **99.96**\
*Halo Halo* & 2 & [Fajters](https://open.spotify.com/track/2EO4dhWzJG8HCL4hNgTQW4?si=JgN16LRnSlyIyseCxASX2Q) & 7.45 & **99.35**\
What about the tracks that are still undetected? Table \[misses\] shows examples of tracks that are still undetected by our ensemble-based approach for a couple of works. No clear pattern emerges – apart from the fact that they are often in a very different musical style from the original.
[| c | c | c | c | c | c |]{} **Work**& **Main artist - Title** &
------------
**Direct**
**score**
------------
: \[misses\] Some example of tracks that are undetected by the ensemble-based approach in the ranking experiment, with their scores for both methods. *Click on a title to play in the browser*.
&
--------------
**Ensemble**
-**based**
**score**
--------------
: \[misses\] Some example of tracks that are undetected by the ensemble-based approach in the ranking experiment, with their scores for both methods. *Click on a title to play in the browser*.
\
*Get Lucky* & The Getup - [*Get Lucky*](https://open.spotify.com/track/3W944t5PM0e7cuSj9k01XE?si=Rh_KlK1KT76qQGymsVuvaA) & 6.4 & 24.9\
*Halo* & Polina Kermesh - [*Halo*](https://open.spotify.com/track/562PPTMnlQkDt3O4rxJFmA?si=nxaC9mg7RCW1s-GrEX7txA) & 6.3 & 98.1\
& Amanda Sense - [*Halo*](https://open.spotify.com/track/356d9i2vsYS7zDCg2HATtg?si=uhstl0X9R6uVKO1s4DDX1w) & 12.4 & 94.3\
*Imagine* & Dena De Rose - [*Imagine*](https://open.spotify.com/track/51x7Ia264SWiSmLq1YB1sN?si=RoiFmfG3SHKVVWJBKvoXVg) & 10.4 & 86.2\
*Embraceable* & Earl Hines - [*Embraceable You*](https://open.spotify.com/track/1Kos7DKwmn2frHqZtS72b6) & 9.6 & 36.3\
*You* & Samina - [*Embraceable You*](https://open.spotify.com/track/29INEYnk37kJ9PhY4IZYLr) & 5.8 & 17.0\
*Heartless* & Bright Light - [*Heartless*](https://open.spotify.com/track/5KFmbjXswQefMej9noa35t?si=AyBChL4HTICaosAd4R39Ow) & 11.9 & 50.3\
& Rains - [*Heartless*](https://open.spotify.com/track/01WuRGzdHatPEn2oxiYaQG?si=DmQxSGafQtyyMxsTx5shZA) & 6.4 & 47.7\
*Bodak Yellow* & Josh Vietti - [*Bodak Yellow*](https://open.spotify.com/track/2MLP51NFC6RKSvDRWj8QEX?si=TF1ifhTZRxSYYjUNKSsVkw) & 5.8 & 14.2\
& J-Que Beenz - [*Bodak Yellow*](https://open.spotify.com/track/4NP1uXIqS5dEbxpEIELNP8?si=ZKbJx3DkS-ePUyYLiwPf_A) & 5.6 & 13.0\
*Airplanes* & Em Fresh - [*Airplanes*](https://open.spotify.com/track/3zdHo8xxdM7ovkr2gBQAE6?si=a7z0epr-QFWgeAv54x_bSg) & 5.5 & 66.1\
& Lisa Scinta - [*Airplanes*](https://open.spotify.com/track/50tsm5izJLx7Of5z8l96yG?si=ZFaLKmteQuivw2fUFFhKWA) & 9.6 & 55.0\
Discussion {#sec:discussion}
==========
One main challenge associated with our ensemble-based approach is how to correctly handle transitivity. This issue emerges from the fact that compositions are not mutually exclusive. For example, a medley might constitute a bridge between two distinct composition groups, which our algorithm would then merge together (which is undesirable). There are probably at least two ways around this issue: one is metadata-based (*i.e.* identify these potential outliers from the metadata and exclude them from the graph computation), while another is to detect them directly from the graph structure (identify bridges between otherwise unrelated clusters).
Conclusion {#sec:conclusion}
==========
In this paper, we have introduced a new formulation of the cover song identification problem: among a pool of candidates that are likely to match one given reference track, find the actual positives. We have introduced a two-step approach, with a first step that computes pairwise similarities between every pair of tracks in the pool of candidates (for which any known 1-vs-1 approach can be used), and a second ensemble-based step that exploits the relationships between all the candidates to output final results. We have shown that this second step can significantly improve the performance compared to a pure 1-vs-1 approach, in particular on the ranking task, where the error rate is down from a few percents to less than 1% in general. The classification task is naturally more challenging as the optimal threshold might vary from work to work, suggesting that the method would be best exploited as a complement to human annotations – where the human’s task would mainly be to find the optimal threshold for the classification. Automating this last step turned out to be non-trivial and is left for future work.
[^1]: https://github.com/ctralie/GeometricCoverSongs
[^2]: Each track from set A is now queried against the 80 tracks from set B, instead of all other 159 tracks.
[^3]: As the computational time is much higher for this algorithm, we only computed the false positive rate using 500 negative pairs.
[^4]: The distances that would be obtained by applying the original Floyd-Washall algorithm to $D$ would always satisfy the triangular inequality, but the resulting configuration would be very sensitive to outliers. Our method is more robust to outliers, as it requires to find more than one better path to update the distance between two points.
[^5]: *Embraceable You* is an exception, as its threshold is degenerate and all tracks are classified as matching.
[^6]: It was also a genuinely difficult example and we struggled to annotate it.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We review the $X=K$ conjecture and important ingredients for the proof. We also attach notes on the rank estimate for the $X=K$ theorem to hold and on the strange relation that was found to be valid without the assumption that the rank is sufficiently large. Using the latter one obtains an algorithm to calculate the image of the combinatorial $R$-matrix and the value of the coenergy function.'
author:
- 'Cédric <span style="font-variant:small-caps;">Lecouvey</span>[^1], Masato <span style="font-variant:small-caps;">Okado</span>[^2] and Mark <span style="font-variant:small-caps;">Shimozono</span>[^3]'
title: $X=K$ under Review
---
Review on $X=K$
===============
Let ${\mathfrak{g}}$ be an affine algebra of nonexceptional type and $I=\{0,1,\ldots,n\}$ the index set of its Dynkin nodes. Let $0\in I$ as specified in [@Kac] and set $I_0=I\setminus\{0\}$. For a pair $(r,s)$ ($r\in I_0,
s\in{\mathbb{Z}}_{>0}$) there exists a crystal $B^{r,s}$ called the Kirillov-Reshetikhin (KR) crystal [@FOS]. It is a crystal base in the sense of Kashiwara [@Ka] of the Kirillov-Reshetikhin module $W^{r,s}(a)$ for a suitable parameter $a$ [@O; @OSch] over the quantum affine algebra $U'_q({\mathfrak{g}})$ without the degree operator $q^d$. Let $B$ be a tensor product of KR crystals $B=B^{r_1,s_1}{\otimes}B^{r_2,s_2}{\otimes}\cdots{\otimes}B^{r_L,s_L}$, and for a subset $J$ of $I$ set $\mathrm{hw}_{J}(B)=\{b\in B\mid e_ib=0\text{ for any }
i\in J\}$ where $e_i$ is the Kashiwara operator acting on $B$. We call an element of $\mathrm{hw}_J(B)$ $J$-highest. For an $I_0$-weight ${\lambda}$ we define the 1-dimensional sum ${\overline}{X}_{{\lambda},B}(q)$ by $${\overline}{X}_{{\lambda},B}(q)=\sum_{b\in\mathrm{hw}_{I_0}(B),{\mathrm{wt}\,}b={\lambda}}q^{{\overline}{D}(b)}.$$ Here ${\overline}{D}:B\rightarrow{\mathbb{Z}}$ is the intrinsic coenergy function (see [@LOS §3.5]). Assume now that $n=|I_0|$ is sufficiently large. (We make an attempt to estimate $n$ such that our main theorem holds.) Then it can be shown that ${\overline}{X}_{{\lambda},B}(q)$ depends only on the attachment of the node $0$ to the rest of the Dynkin diagram of ${\mathfrak{g}}$. In the table below we list all possibilities of the attachment of $0$ and enumerate the corresponding nonexceptional affine algebras. $$\begin{array}{|c|c|c|c|} \hline
\text{Dynkin} & {\mathfrak{g}}& {\diamondsuit}\\ \hline \hline
{\xymatrix@R=1ex{
{} & *{\circ}<3pt> \ar@{-}[r] & *{\circ}<3pt> \ar@{..}[r] & \\
*{\circ}<3pt> \ar@{-}[ur]^<{0} \ar@{-}[dr] & & & \\
{} & *{\circ}<3pt> \ar@{-}[r] & *{\circ}<3pt> \ar@{..}[r] &
}}
&
\lower4mm\hbox{$A_n^{(1)}$}
&
\lower4mm\hbox{$\varnothing$}
\\
\hline
{\xymatrix@R=1ex{
*{\circ}<3pt> \ar@{-}[dr]^<{0} & {} & {} & \\
{} & *{\circ}<3pt> \ar@{-}[r]& *{\circ}<3pt> \ar@{..}[r] & \\
*{\circ}<3pt> \ar@{-}[ur] & {} & {} &
}}
&
\lower4mm\hbox{$B_n^{(1)},D_n^{(1)},A_{2n-1}^{(2)}$}
&
\lower4mm\hbox{${{\Yvcentermath1\Yboxdim4pt \,\yng(1,1)\,}}$}
\\
\hline
{\xymatrix@R=1ex{
*{\circ}<3pt> \ar@{=>}[r]^<{0} & *{\circ}<3pt> \ar@{-}[r] & *{\circ}<3pt> \ar@{..}[r] & \\
}}
& C_n^{(1)} & {{\Yvcentermath1\Yboxdim4pt \,\yng(2)\,}}\\ \hline
{\xymatrix@R=1ex{
*{\circ}<3pt> \ar@{<=}[r]^<{0} & *{\circ}<3pt> \ar@{-}[r] & *{\circ}<3pt> \ar@{..}[r] & \\
}}
& A_{2n}^{(2)},D_{n+1}^{(2)} & {{\Yvcentermath1\Yboxdim4pt \,\yng(1)\,}}\\ \hline
\end{array}$$ Hence, we have four kinds of “stable" 1-dimensional sums denoted by ${\overline}{X}^{\diamondsuit}_{{\lambda},B}(q)$ (${\diamondsuit}=
\varnothing,{{\Yvcentermath1\Yboxdim4pt \,\yng(1,1)\,}},{{\Yvcentermath1\Yboxdim4pt \,\yng(2)\,}},{{\Yvcentermath1\Yboxdim4pt \,\yng(1)\,}}$). Then the so-called $X=K$ conjecture proposed by Shimozono and Zabrocki [@Sh; @SZ] is stated as follows.
\[th:main\] For ${\diamondsuit}\ne\varnothing$, $${\overline}{X}^{\diamondsuit}_{{\lambda},B}(q)=q^{\frac{|B|-|{\lambda}|}{|{\diamondsuit}|}}
\sum_{\mu\in\mathcal{P}^{\diamondsuit}_{|B|-|{\lambda}|},\nu\in\mathcal{P}^{{\Yvcentermath1\Yboxdim3pt \,\yng(1)\,}}_{|B|}}c_{{\lambda}\mu}^\nu
{\overline}{X}^\varnothing_{\nu,B}(q^\frac2{|{\diamondsuit}|}).$$ Here $|B|=\sum_{j=1}^Lr_js_j,|{\lambda}|=\sum_i{\lambda}_i$ for ${\lambda}=({\lambda}_1,{\lambda}_2,\ldots)$ where a non-spin weight ${\lambda}$ is identified with a partition by the standard way, $\mathcal{P}_N^{\diamondsuit}=$set of partitions of $N$ tiled by ${\diamondsuit}$, and $c_{{\lambda}\mu}^\nu$ stands for the Littlewood-Richardson coefficient.
We sketch the proof of this theorem from [@LOS]. Since ${\overline}{X}^{\diamondsuit}_{{\lambda},B}(q)$ depends only on the symbol ${\diamondsuit}$, we choose an affine algebra ${\mathfrak{g}}^{\diamondsuit}$ from each kind such that $i\mapsto n-i$ ($i\in I$) gives a Dynkin diagram automorphism. Namely, we set ${\mathfrak{g}}^{\diamondsuit}=A_n^{(1)},D_n^{(1)},C_n^{(1)},
D_{n+1}^{(2)}$ for ${\diamondsuit}=\varnothing,{{\Yvcentermath1\Yboxdim4pt \,\yng(1,1)\,}},{{\Yvcentermath1\Yboxdim4pt \,\yng(2)\,}},{{\Yvcentermath1\Yboxdim4pt \,\yng(1)\,}}$. Let ${\diamondsuit}={{\Yvcentermath1\Yboxdim4pt \,\yng(1,1)\,}},{{\Yvcentermath1\Yboxdim4pt \,\yng(2)\,}}$ or ${{\Yvcentermath1\Yboxdim4pt \,\yng(1)\,}}$ from now on. Then there exists an automorphism $\sigma$ on the KR crystal $B^{r,s}$ for ${\mathfrak{g}}^{\diamondsuit}$ satisfying $$\sigma(e_ib)=e_{n-i}\sigma(b)$$ for any $i\in I,b\in B^{r,s}$. This automorphism $\sigma$ is extended to $B$ by $\sigma(b)=\sigma(b_1){\otimes}\sigma(b_2){\otimes}\cdots{\otimes}\sigma(b_L)$. Then the important facts for the proof are summarized as follows.
- $\sigma$ restricts to the following bijection. $$\left\{
\begin{array}{c}
\hbox{$I_0$-highest elements}\\
\hbox{in $B$ of ${\mathrm{wt}\,}{\lambda}$}
\end{array}
\right\}\overset{\sigma}{\longrightarrow}
\left\{
\begin{array}{c}
\hbox{$I\setminus\{0,n\}$-highest elements}\\
\hbox{in $\max(B)$ of ${\mathrm{wt}\,}{\overline}{{\lambda}}$}
\end{array}
\right\}$$ Here $\max(B)=\bigoplus_\gamma B(\gamma)$, where $B(\gamma)$ is the highest weight $U_q({\mathfrak{g}}^{\diamondsuit}_{I_0})$-crystal of highest weight $\gamma$ and $\gamma$ runs over all weights with $|\gamma|=|B|$ such that $B(\gamma)$ appears in the restriction of $B$. Namely, $\max(B)$ is the disjoint union of classical highest weight crystals of maximal highest weights. We remark that ${\mathfrak{g}}_{I\setminus\{0,n\}}$ is isomorphic to $A_{n-1}$ and set ${\overline}{{\lambda}}=(-{\lambda}_n,\ldots,-{\lambda}_1)$ if ${\lambda}=({\lambda}_1,\ldots,{\lambda}_n)$.
- ${\overline}{D}(b)={\overline}{D}(\sigma(b))+(|B|-|\hbox{wt}\,b\,|)/|{\diamondsuit}|$ for $b\in\hbox{hw}_{I_0}(B)$.
- We have $[\left.V^G(\nu)\right\downarrow^G_{GL_n}:V^{GL_n}(\bar{{\lambda}})]
=\sum_{\mu\in\mathcal{P}^{\diamondsuit}}c_{{\lambda}\mu}^\nu$, where $G=SO_{2n},Sp_{2n},SO_{2n+1}$ for ${\diamondsuit}={{\Yvcentermath1\Yboxdim4pt \,\yng(1,1)\,}},{{\Yvcentermath1\Yboxdim4pt \,\yng(2)\,}},{{\Yvcentermath1\Yboxdim4pt \,\yng(1)\,}}$ and $V^G(\nu)$ stands for the irreducible $G$-module of non-spin highest weight $\nu$.
- If we represent elements of a KR crystal by Kashiwara-Nakashima tableaux [@KN], $I_0$-highest elements in $\max(B)$ contain no barred letters and can therefore be viewed as elements of type $A$. Under this correspondence we have ${\overline}{D}^{\diamondsuit}(b)=\frac{2}{|{\diamondsuit}|}{\overline}{D}^\varnothing(b)$.
Once these properties are established, our theorem can easily be proved as $$\begin{aligned}
{\overline}{X}^{\diamondsuit}_{{\lambda},B}(q)&=\sum_{b\in\hbox{\scriptsize hw}_{I_0}(B),{\mathrm{wt}\,}b={\lambda}}q^{{\overline}{D}(b)}\\
&\overset{(ii)}{=}q^d\sum_bq^{{\overline}{D}(\sigma(b))}\\
&\overset{(i)(iii)}{=}q^d\sum_{\mu\in\mathcal{P}^{\diamondsuit},\nu\in\mathcal{P}}c_{{\lambda}\mu}^\nu
\sum_{\hat{b}\in\hbox{\scriptsize hw}_{I_0}(\max(B)),{\mathrm{wt}\,}\hat{b}=\nu}q^{{\overline}{D}(\hat{b})}\\
&\overset{(iv)}{=}q^d\sum_{\mu,\nu}c_{{\lambda}\mu}^\nu{\overline}{X}^\varnothing_{\nu,B}(q^\frac{2}{|{\diamondsuit}|})\end{aligned}$$ where we have set $d=(|B|-|{\lambda}|)/|{\diamondsuit}|$.
Consider the affine algebra ${\mathfrak{g}}=D_6^{(1)}$ of kind ${{\Yvcentermath1\Yboxdim4pt \,\yng(1,1)\,}}$ and the following three elements of $B=B^{2,2}{\otimes}B^{3,1}{\otimes}B^{1,3}$. They all have weight ${\lambda}=(211)$. Their images by the automorphism $\sigma$ are also given. $$\begin{array}{|c|c|} \hline
b & \sigma(b) \\ \hline\vbox{\vspace{9mm}}
\quad\vcenter{
\tableau[sby]{2\\1}
}
{\otimes}\vcenter{
\tableau[sby]{{\overline}{3}\\3\\1}
}
{\otimes}\vcenter{
\tableau[sby]{1&3&{\overline}{1}}
}\quad&\quad
\vcenter{
\tableau[sby]{6&{\overline}{5}\\ {\overline}{6}&{\overline}{6}}
}
{\otimes}\vcenter{
\tableau[sby]{{\overline}{5}\\{\overline}{6}\\5}
}
{\otimes}\vcenter{
\tableau[sby]{5&{\overline}{5}&{\overline}{4}}
}\quad\\[6mm]
\quad\vcenter{
\tableau[sby]{2\\1}
}
{\otimes}\vcenter{
\tableau[sby]{{\overline}{4}\\4\\3}
}
{\otimes}\vcenter{
\tableau[sby]{1&3&{\overline}{3}}
}\quad&\quad
\vcenter{
\tableau[sby]{6&{\overline}{5}\\ {\overline}{6}&{\overline}{6}}
}
{\otimes}\vcenter{
\tableau[sby]{{\overline}{4}\\ {\overline}{5}\\5}
}
{\otimes}\vcenter{
\tableau[sby]{3&{\overline}{6}&{\overline}{3}}
}\quad\\[6mm]
\quad\vcenter{
\tableau[sby]{2&2\\1&1}
}
{\otimes}\vcenter{
\tableau[sby]{{\overline}{2}}
}
{\otimes}\vcenter{
\tableau[sby]{1&3&{\overline}{1}}
}\quad&\quad
\vcenter{
\tableau[sby]{{\overline}{5}&{\overline}{5}\\ {\overline}{6}&{\overline}{6}}
}
{\otimes}\vcenter{
\tableau[sby]{6\\ {\overline}{6}\\5}
}
{\otimes}\vcenter{
\tableau[sby]{5&{\overline}{5}&{\overline}{4}}
}\quad\\[6mm]\hline
\end{array}$$ By the property (i) each $\sigma(b)$ should belong to $\max(B)$. Actually, by applying the raising operators $e_i$ ($i\in I_0$) one finds that these three elements have the common $I_0$-highest element $$\hat{b}=\vcenter{
\tableau[sby]{2&2\\1&1}
}
{\otimes}\vcenter{
\tableau[sby]{4\\3\\1}
}
{\otimes}\vcenter{
\tableau[sby]{2&3&5}
}$$ of weight $\nu=(33211)$. It is also true that they are the all $I_0$-highest elements in $B$ whose images under $\sigma$ belong to the same $I_0$-component as the above one. We can check the property (iii), since we get $c_{{\lambda}\mu}^\nu
=1$ if $\mu=(33)$, $=2$ if $\mu=(2211)$, $=0$ if $\mu$ are other elements in $\mathcal{P}^{{\Yvcentermath1\Yboxdim4pt \,\yng(1,1)\,}}$ and therefore $$\sum_{\mu\in\mathcal{P}^{{\Yvcentermath1\Yboxdim3pt \,\yng(1,1)\,}}}c_{{\lambda}\mu}^\nu=3.$$ The intrinsic coenergies ${\overline}{D}(b)$ are all equal and can be calculated using the property (ii) as $${\overline}{D}(b)={\overline}{D}(\sigma(b))+\frac{10-4}2.$$ Since ${\overline}{D}$ is constant on each $I_0$-component, we have ${\overline}{D}(\sigma(b))={\overline}{D}(\hat{b})$. The r.h.s. is calculated to be $4$ using the knowledge of the type $A$ crystal [@Sh]. Therefore we obtain ${\overline}{D}(b)=7$.
The so-called $X=M$ conjecture [@HKOTY; @HKOTT] claims that the 1-dimensional sum ${\overline}{X}_{{\lambda},B}(q)$ is equal to the fermionic formula ${\overline}{M}({\lambda},\mathbf{L};q)$. Hence, when $n$ is sufficiently large, one can expect that ${\overline}{M}({\lambda},\mathbf{L};q)$ has a similar formula to Theorem \[th:main\]. This is confirmed in [@OSa]. Combining these results with [@KSS], the $X=M$ conjecture is settled when the affine algebra is of nonexceptional type and its rank is sufficiently large.
Rank Estimate
=============
In this section we make an attempt to estimate $n$ such that Theorem \[th:main\] holds.
Let $\ell$ be the length of ${\lambda}$. Then Theorem \[th:main\] holds if $$n>(2\ell+1)+|B|-|{\lambda}|.$$
The obstacle for the theorem to hold lies in the fact that the property (i) is no longer valid when $n$ is not large enough. In [@LOS] this property is stated as Theorem 7.1. In view of the proof there one recognizes that if $n$ is so large that $\sigma(b)$ for $b\in\mathrm{hw}_{I_0}(B)$ is contained in $\max(B)$, then everything is ok. Using row and box splittings in [@LOS §6] one can also reduce the proof when $B$ is a tensor product of the simplest KR crystal $B^{1,1}$, that is, $B=(B^{1,1})^{{\otimes}L}$. Hence, our task is to estimate $n$ such that $\sigma(b)$ belongs to $\max((B^{1,1})^{{\otimes}L})$ for any $b\in\mathrm{hw}_{I_0}((B^{1,1})^{{\otimes}L})$ of weight ${\lambda}$.
Recall that an element of $(B^{1,1})^{{\otimes}L}$ can be regarded as a word of length $L$ from the alphabet $$\{(\phi,)1,2,\ldots,n,(0,){\overline}{n},{\overline}{n-1},\ldots,{\overline}{1}\}.$$ Here letters in parentheses are only for ${\diamondsuit}={{\Yvcentermath1\Yboxdim4pt \,\yng(1)\,}}$. Let $b$ be a word of length $L$ that is $I_0$-highest. Then the letters $b$ lie in the set $\{(\phi,)1,2,\ldots,m,{\overline}{m},{\overline}{m-1},\ldots,{\overline}{1}\}$ for some $m(\ge\ell)$. Let $c_z$ be the number of letters $z$ in $b$. Then we have $c_j-c_{{\overline}{j}}={\lambda}_j>0$ for $1\le j\le \ell$ and $c_{j}=c_{{\overline}{j}}>0$ for $\ell<j\le m$. Since $\sum_{j=1}^m(c_j+c_{{\overline}{j}})\le L$, we have $\sum_{j=1}^mc_{{\overline}{j}}\le\frac{L-|{\lambda}|}2$. Setting $M=\max_{\ell<j\le m}c_{{\overline}{j}}$, we get $$\label{eq1}
M+(m-\ell-1)\le\frac{L-|{\lambda}|}2.$$
Next recall the insertion algorithms from [@L]. For a word or element of a tensor product of $B^{1,1}$ the insertion algorithm tells us the highest weight of the $I_0$-component the word belongs to. In our case we wish to apply this algorithm to $\sigma(b)$ to see if the shape of the resulting tableau has $L$ nodes. This is equivalent to say that at each step of insertion of a letter to a column the resulting column remains to be admissible. This in particular means that if letter $x$ and ${\overline}{x}$ coexist at position $p$ and $q$ in some column of height $N$, then we have $$\label{eq2}
x\ge p+(N+1-q).$$ Let us obtain the minimal possible unbarred letter $X$ that could appear in the course of insertion algorithms. Note that letters of $\sigma(b)$ lie in $\{n-m+1,n-m+2,\ldots,n,(0,){\overline}{n},{\overline}{n-1},$ $\ldots,{\overline}{n-m+1}\}$. Since plactic relations of [@L] contain $x{\overline}{x}y\equiv({\overline}{x-1})(x-1)y$, a pair $(n-m+1,{\overline}{n-m+1})$ could create $(n-m+1-M,{\overline}{n-m+1-M})$. Hence we can set $X=n-m+1-M$. The worst situation that could break is that there exist pairs $(X+j-1,{\overline}{X+j-1})$ for any $1\le j\le M+m$ in the first column during the insertion procedure. The condition for such a column to be admissible is given by $$\label{eq3}
n\ge2(M+m).$$ In view of we obtain the desired result.
Strange Relation
================
In this section we show the following proposition and apply it to give an algorithm to obtain the image of the combinatorial $R$-matrices and the value of the coenergy function ${\overline}{H}$. As we see in the proof, we do not assume the rank is sufficiently large. So the algorithm can be used for any $n$. However, we need to restrict our affine algebras to ${\mathfrak{g}}^{\diamondsuit}$ (${\diamondsuit}={{\Yvcentermath1\Yboxdim4pt \,\yng(1,1)\,}},
{{\Yvcentermath1\Yboxdim4pt \,\yng(2)\,}},{{\Yvcentermath1\Yboxdim4pt \,\yng(1)\,}}$), since we use the automorphism $\sigma$. From the same reason we exclude the KR crystals $B^{n-1,s}$ and $B^{n,s}$ for ${\mathfrak{g}}^{{\Yvcentermath1\Yboxdim4pt \,\yng(1,1)\,}}=D_n^{(1)}$ and $n$ is odd.
\[prop:strange rel\] Let $B$ be a tensor product of KR crystals. Suppose $b\in\mathrm{hw}_{I_0}(B)$. Then we have $${\overline}{D}(b)-{\overline}{D}(\sigma(b))=\frac{|B|-|{\lambda}(b)|}{|{\diamondsuit}|}.$$ Here ${\lambda}(b)$ stands for the partition corresponding to the weight of $b$.
This proposition is essentially the same as Theorem 8.1 of [@LOS] except that we do not assume $n$ is sufficiently large. We prepare a lemma. Let us extend the definition of ${\lambda}(b)$ to an arbitrary element $b$ by ${\lambda}(b)=({\lambda}_1,{\lambda}_2,\ldots,{\lambda}_n)$ where ${\lambda}_i=({\mathrm{wt}\,}b,\epsilon_i)$ and $\{\epsilon_i\}_{1\le i\le n}$ stands for the standard basis vectors of the weight lattice. We note that ${\lambda}(b)$ is not necessarily a partition. Some ${\lambda}_i$’s may be negative. Hence $|{\lambda}|=\sum_i{\lambda}_i$ may also become negative.
Let $B_1,B_2$ be single KR crystals. Let $b_1{\otimes}b_2$ be an element of $B_1{\otimes}B_2$ and suppose it is mapped to $b'_2{\otimes}b'_1$ by the affine crystal isomorphism. Then we have $$\label{strange H}
{\overline}{H}(b_1{\otimes}b_2)-{\overline}{H}(\sigma(b_1){\otimes}\sigma(b_2))=\frac{|{\lambda}(b'_2)|-|{\lambda}(b_2)|}{|{\diamondsuit}|}.$$
Since $B_1{\otimes}B_2$ is connected, it is sufficient to show
- if $b_1=u(B_1),b_2=u(B_2)$ (see [@LOS §3.4] for the definition of $u(B_i)$), holds, and
- with $b_1{\otimes}b_2$ replaced by $e_i(b_1{\otimes}b_2)$ holds, provided that holds and $e_i(b_1{\otimes}b_2)\ne0$.
For (i) recall $b'_1=b_1,b'_2=b_2$ if $b_1=u(B_1),b_2=u(B_2)$. Since $u(B_1){\otimes}u(B_2)$ can be reached from $\sigma(u(B_1)){\otimes}\sigma(u(B_2))$ by applying $e_i$ ($i\ne0$), we have ${\overline}{H}(u(B_1){\otimes}u(B_2))={\overline}{H}(\sigma(u(B_1)){\otimes}\sigma(u(B_2)))=0$. Hence (i) is verified.
For (ii) recall $|{\lambda}(e_ib)|-|{\lambda}(b)|=-|{\diamondsuit}|\,(i=0),=|{\diamondsuit}|\,(i=n),=0\,(\text{otherwise})$. If $i\ne0,n$, both sides do not change when we replace $b_1{\otimes}b_2$ with $e_i(b_1{\otimes}b_2)$. If $i=0$, the first term of the l.h.s decreases by one in case LL, increases by one in case RR, and does not change in case LR or RL. (For the meaning of LL, etc, see [@LOS Prop. 3.7].) The second term does not change, while the r.h.s varies in the same way as the first term of the l.h.s. The $i=n$ case is similar.
Let $B=B^{r_1,s_1}{\otimes}\cdots{\otimes}B^{r_p,s_p}$. We prove by induction on $p$. When $p=1$, the proof is the same as in [@LOS Th. 8.1].
Let $B=B'{\otimes}B^{r_p,s_p}$ and $b_1{\otimes}b_2\in B'{\otimes}B^{r_p,s_p}$ is mapped to $b'_2{\otimes}b'_1
\in B^{r_p,s_p}{\otimes}B'$ by the affine crystal isomorphism. Then $\sigma(b_1){\otimes}\sigma(b_2)$ should be mapped to $\sigma(b'_2){\otimes}\sigma(b'_1)$. Using (3.52) of [@LOS] we have $$\begin{aligned}
{\overline}{D}(b)&={\overline}{D}(b_1)+{\overline}{D}(b'_2)+{\overline}{H}(b_1{\otimes}b_2),\\
{\overline}{D}(\sigma(b))&={\overline}{D}(\sigma(b_1))+{\overline}{D}(\sigma(b'_2))+{\overline}{H}(\sigma(b_1){\otimes}\sigma(b_2)).\end{aligned}$$ On the other hand, by the previous lemma and [@Memoir Lemma 5.2] we have $${\overline}{H}(b_1{\otimes}b_2)-{\overline}{H}(\sigma(b_1){\otimes}\sigma(b_2))=\frac{|{\lambda}(b'_2)|-|{\lambda}(b_2)|}{|{\diamondsuit}|}.$$ Using the induction hypothesis we obtain $$\begin{aligned}
{\overline}{D}(b)-{\overline}{D}(\sigma(b))&=\frac{|B'|-|{\lambda}(b_1)|}{|{\diamondsuit}|}+\frac{|B^{r_p,s_p}|-|{\lambda}(b'_2)|}{|{\diamondsuit}|}
+\frac{|{\lambda}(b'_2)|-|{\lambda}(b_2)|}{|{\diamondsuit}|}\\
&=\frac{|B|-|{\lambda}(b)|}{|{\diamondsuit}|}\end{aligned}$$ as desired.
Using Proposition \[prop:strange rel\] we can give an algorithm to obtain the image of the combinatorial $R$-matrix and the value of the coenergy function ${\overline}{H}$. This algorithm turns out effective when it is calculated using computer. For the calculation of $\sigma$ see [@LOS Appendix B.2]. Let $B_i=B^{r_i,s_i}$ ($i=1,2$) be KR crystals. The affine crystal isomorphism $$R: B_1{\otimes}B_2\longrightarrow B_2{\otimes}B_1,$$ which is known to exist uniquely, is called the combinatorial $R$-matrix. For an element $b_1{\otimes}b_2
\in B_1{\otimes}B_2$ we wish to calculate the image $R(b_1{\otimes}b_2)$. Since the application of Kashiwara operators $e_i,f_i$ for $i\ne0$ is not difficult, one can reduce its calculation to $I_0$-highest elements of $B_1{\otimes}B_2$. For an element $b$ in an $I_0$-component let $High(b)$ stand for the $I_0$-highest element and set $\Phi=High\circ\sigma$. From Proposition \[prop:strange rel\] and the invariance of ${\overline}{D}$ by classical Kashiwara operators, one has $$\label{eq4}
{\overline}{D}(\Phi(b_1{\otimes}b_2))={\overline}{D}(b_1{\otimes}b_2)-\frac{|B_1{\otimes}B_2|-|{\lambda}(b_1{\otimes}b_2)|}{|{\diamondsuit}|}.$$ Note that the second term of the above relation vanishes, if and only if $b_1{\otimes}b_2\in\max(B_1{\otimes}B_2)$. Since the application of $\Phi$ decreases ${\overline}{D}$ and ${\overline}{D}$ takes a finite number of values, there exists a positive integer $m$ such that $\Phi^m(b_1{\otimes}b_2)\in\max(B_1{\otimes}B_2)$. Namely, there exist sequences $\mathbf{a}_1,\ldots,\mathbf{a}_m$ from $I_0$ and an element $\hat{b}_1{\otimes}\hat{b}_2\in\max(B_1{\otimes}B_2)$ such that $$\hat{b}_1{\otimes}\hat{b}_2=
(e_{\mathbf{a}_m}\circ\sigma\circ\cdots\circ e_{\mathbf{a}_1}\circ\sigma)(b_1{\otimes}b_2),$$ or equivalently, $$b_1{\otimes}b_2=
(\sigma\circ f_{\mathrm{Rev}(\mathbf{a}_1)}\circ\cdots\circ\sigma\circ f_{\mathrm{Rev}(\mathbf{a}_m)})
(\hat{b}_1{\otimes}\hat{b}_2).$$ Here for $\mathbf{a}=(i_1,\ldots,i_l)$ $e_\mathbf{a}$ stands for $e_{i_1}\cdots e_{i_l}$ ($f_\mathbf{a}$ is similar) and $\mathrm{Rev}(\mathbf{a})=(i_l,\ldots,i_1)$. Since $R$ commutes with $e_\mathbf{a},f_\mathbf{a}$ and $\sigma$, we have $$R(b_1{\otimes}b_2)=
(\sigma\circ f_{\mathrm{Rev}(\mathbf{a}_1)}\circ\cdots\circ\sigma\circ f_{\mathrm{Rev}(\mathbf{a}_m)})
R(\hat{b}_1{\otimes}\hat{b}_2).$$ On the other hand, for an $I_0$-highest element in $\max(B_1{\otimes}B_2)$ the image of $R$ is easily calculated (see [@LOS §9.1]). Hence, one can calculate $R(b_1{\otimes}b_2)$.
We proceed to the calculation of ${\overline}{H}(b_1{\otimes}b_2)$. Firstly, one has the relation $$\label{eq5}
{\overline}{D}(b_1{\otimes}b_2)={\overline}{D}(b_1)+{\overline}{D}(b'_2)+{\overline}{H}(b_1{\otimes}b_2),$$ where $R(b_1{\otimes}b_2)=b'_2{\otimes}b'_1$. The l.h.s is has been obtained in the course of the previous process and the known result of the value of ${\overline}{D}$ for an element in $\max(B_1{\otimes}B_2)$. For $I_0$-highest elements $b_1,b'_2$ of a single KR crystal the value of ${\overline}{D}$ is calculated as $${\overline}{D}(b)=\frac{rs-|{\lambda}(b)|}{|{\diamondsuit}|}\quad\text{for }b\in B^{r,s}.$$ Therefore, one obtains ${\overline}{H}(b_1{\otimes}b_2)$.
Consider the affine algebra ${\mathfrak{g}}=D_6^{(1)}$ of kind ${{\Yvcentermath1\Yboxdim4pt \,\yng(1,1)\,}}$ and the following element of $B^{4,3}{\otimes}B^{3,3}$. $$b_1{\otimes}b_2=
\vcenter{
\tableau[sby]{4\\ 3\\ 2&2\\ 1&1}
}
{\otimes}\vcenter{
\tableau[sby]{{\overline}{2}\\ {\overline}{3}\\ 3&4&{\overline}{4}}
}$$ Then $\Phi(b_1{\otimes}b_2)$ and $\Phi^2(b_1{\otimes}b_2)$ are given as follows. $$\Phi(b_1{\otimes}b_2)=
\vcenter{
\tableau[sby]{4&4&4\\ 3&3&3\\ 2&2&2\\ 1&1&1}
}
{\otimes}\vcenter{
\tableau[sby]{3&5&{\overline}{6}\\ 2&2&6\\ 1&1&5}
}\;,\qquad
\Phi^2(b_1{\otimes}b_2)=
\vcenter{
\tableau[sby]{4&4&4\\ 3&3&3\\ 2&2&2\\ 1&1&1}
}
{\otimes}\vcenter{
\tableau[sby]{3&5&6\\ 2&2&5\\ 1&1&1}
}$$ with $$\begin{aligned}
\mathbf{a}_1=
(&64354643215432643215432643564321543264354643215432643546643215432643546),\\
\mathbf{a}_2=
(&66456435464325436643215432646432154326435643215432643546432154326435466\\
&43215432643546643215432643546).\end{aligned}$$ Since one knows the image of $R$ of $\Phi^2(b_1{\otimes}b_2)$ is given by $$\vcenter{
\tableau[sby]{3&3&3\\ 2&2&2\\ 1&1&1}
}
{\otimes}\vcenter{
\tableau[sby]{4&5&6\\ 3&4&5\\ 2&2&4\\ 1&1&1}
}\;,$$ one obtains $$R(b_1{\otimes}b_2)=b'_2{\otimes}b'_1=
\vcenter{
\tableau[sby]{3&3\\ 2&2\\ 1&1&1}
}
{\otimes}\vcenter{
\tableau[sby]{{\overline}{2}\\ {\overline}{3}\\ {\overline}{4}&{\overline}{1}\\ 4&4}
}\;.$$
We proceed to the calculation of ${\overline}{H}(b_1{\otimes}b_2)$. By using twice and ${\overline}{D}(\Phi^2(b_1{\otimes}b_2))=3$ one gets ${\overline}{D}(b_1{\otimes}b_2)=12$. Since ${\overline}{D}(b_1)=3,{\overline}{D}(b'_2)=1$, we obtain ${\overline}{H}(b_1{\otimes}b_2)=8$ from .
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A. N. Kirillov, A. Schilling and M. Shimozono, *A bijection between Littlewood-Richardson tableaux and rigged configurations*, Selecta Math. (N.S.) [**8**]{} (2002) 67–135.
C. Lecouvey, *Schensted-type correspondence, plactic monoid and jeu de taquin for type $C_n$*, J. Algebra **247** (2002) 295–331; *Schensted-type correspondences and plactic monoids for types $B_n$ and $D_n$*, J. Algebraic Combin. **18** (2003) 99–133.
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[^1]: LMPT, Université François Rabelais Tours.e-mail: `[email protected]`
[^2]: Department of Mathematical Science, Graduate School of Engineering Science, Osaka University, Toyonaka, Osaka 560-8531, Japan.e-mail: `[email protected]`
[^3]: Department of Mathematics, Virginia Tech, Blacksburg, VA 24061-0123 USA. e-mail: `[email protected]`
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Recent times have witnessed sharp improvements in reinforcement learning tasks using deep reinforcement learning techniques like Deep Q Networks, Policy Gradients, Actor Critic methods which are based on deep learning based models and back-propagation of gradients to train such models. An active area of research in reinforcement learning is about training agents to play complex video games, which so far has been something accomplished only by human intelligence. Some state of the art performances in video game playing using deep reinforcement learning are obtained by processing the sequence of frames from video games, passing them through a convolutional network to obtain features and then using recurrent neural networks to figure out the action leading to optimal rewards. The recurrent neural network will learn to extract the meaningful signal out of the sequence of such features. In this work, we propose a method utilizing transformer networks which have recently replaced RNNs in Natural Language Processing (NLP), and perform experiments to compare with existing methods.'
address: |
$^{\dagger}$Department of Computer Science and Engineering, Indian Institute of Technology-Bombay\
$^{\star}$Department of Mechanical Engineering, Indian Institute of Technology-Bombay\
$^{\alpha}$Department of Electrical Engineering, Indian Institute of Technology-Bombay
bibliography:
- 'refs.bib'
nocite: '[@a1; @a2; @a3; @a5; @a6; @a7; @a8; @a9; @a10; @a11]'
title: Transformer based Reinforcement Learning for Games
---
Deep Learning, Transformers, Q-Learning, Long Short Term Memory (LSTM), Natural Language Processing (NLP)
Introduction and Related Work {#sec:intro}
=============================
Recent advancements in reinforcement learning have witnessed the heavy use of Deep Neural Networks (DNN) to perform many of the reinforcement learning tasks such as prediction and control. These classes of approaches at the intersection of deep learning and reinforcement learning are known as Deep reinforcement learning (DRL). DRL uses deep learning and reinforcement learning principles to create efficient algorithms that can be applied on areas like robotics, video games, finance, and healthcare [@franccois2018introduction]. Implementing neural networks like deep convolutional networks with reinforcement learning algorithms such as Q-learning, Actor Critic or Policy Search results in a powerful model (DRL) that is capable to scale to previously unsolvable problems [@mnih2013playing]. That is because DRL usually work with raw sensors or image signals as input (for instance in Deep Q Networks - DQN for ATARI games [@Arulkumaran_2017]) and can receive the benefit of end-to-end reinforcement learning as well as that of convolutional neural networks. In other words, neural networks act as function approximators, of action-value functions used in predicting the next best action, which is particularly useful in reinforcement learning when the state or action space is too large to be completely known or to be stored in memory.
Recently, DQN with recurrent layers also called as Deep Recurrent Q Networks (DRQN), have started showing promising results in reinforcement learning problem. They have the capability to outperform the DQN’s outcomes and generate better trained agents in certain scenarios. The primary reason for this being, DQN has limited or no amount of distant history. In practice, DQNs are often trained with just a single state representation corresponding to current time-steps (i.e often times they neglect the temporal aspects of the input), even when fed with the sequence of state representation, standard DQN architecture without RNNs are unable to capture and extract from the temporal aspect of the sequences. Thus DQN will be unable to master games that require the player to remember events more distant in the past. Put differently, any game that requires a memory of past states in the trajectory will appear non-Markovian because the future game states (and rewards) depend on more than just DQN’s current input. Instead of a Markov Decision Process (MDP), the game becomes a Partially-Observable Markov Decision Process (POMDP) [@hausknecht2015deep]. Hence, LSTM along with DQN is used in such cases.
Since LSTM or recurrent neural networks, in general, have had a strong presence in the Natural Language Processing (NLP), we tried to implement a technique inspired from NLP called Transformer to perform the reinforcement learning tasks. Transformer was first introduced in [@vaswani2017attention] to perform sequence-to-sequence translation, however, it has been adapted successfully in various applications spanning language, speech, etc.
NLP uses a sequence-to-sequence architecture at the core in various tasks. In NLP we generally need to analyze a sequence of words and generate another sequence of words based on them, therefore language translation task is one such example where sequence to sequence architecture is applicable. LSTM was a popular choice to be used as an encoder and decoder for the above-specified task and related architecture. LSTM based approach implicitly accounts for ’attention’ which is a mechanism that looks at an input sequence and decides at each step which other parts of the sequence are important. The transformer was a novel technique introduced to replace this attention-mechanism performed by LSTM by more effective and explicit attention-mechanism. It is also an encoder-decoder model but differs from LSTM by avoiding any usage of recurrent neural networks which is common in LSTM, GRU, etc. This improves training time as well as accuracy in NLP tasks.
Methods and Experiments {#sec:method}
=======================
In the following we describe the techniques we used to extend the current framework to accommodate our transformer based proposal and training procedure, we also describe the various experiments performed to compare methods.
Natural Language Processing often deals with the problem of predicting one set of sequences from another set of sequences (for instance, a sequence of words from the sequence of acoustic features for automatic speech recognition tasks, or sequence of words in German from a sequence of words in English for language translation task, etc). As described in the above section, deep reinforcement learning also makes use of the sequence and we take inspiration from recent updates in NLP to propose a new method for DRL.
![Frame from the Cartpole environment of OpenAI Gym. The task is to balance the pole on the cart, by moving the cart left or right[]{data-label="fig:m1"}](pics/cartpole.png){width="\linewidth"}
[0.33]{} {width="0.5\linewidth"}
[0.33]{} {width="\linewidth"}
[0.33]{} {width="0.8\linewidth"}
Environment
-----------
In our experiments, we set up an environment for the “Cartpole” game (using OpenAI gym [@openai]) where the goal is to balance the pole on the cart (i.e., prevent the pole from falling) by moving the cart left or right. Figure \[fig:m1\] shows a frame from the game to visualize the environment. The set of actions $A$ consists of {*left*, *right*} and the environment provides a reward from the set of rewards $R$ consisting of {$+1$, $-1$}. For ever time-step where the pole does not fall the environment provides a reward of $+1$ and when the pole falls (i.e., the angle it makes from the cart crosses a certain threshold) the episode is completed and the agent receives a reward of $-1$. The typical deep reinforcement learning pipeline passes the frames (or sequence of frames) through a deep convolutional neural network to extract the useful features, which are further processed to estimate the *value function ($V$)* or the *state-action value function ($Q$)*. However, this results in model with relatively larger number of parameters, which also requires access to large GPUs to train the models using learning algorithms, not to mention that such algorithms take significantly longer to train. In order to overcome this problem we decided to use the RAM provided by the OpenAI environment describing the state of the game. In this case the state of the game can be described using the:
- position of the cart (on the X-axis, from -4.8 to 4.8)
- velocity of the cart (from $-\infty$ to $\infty$)
- angle the pole makes with the cart (from -24 to 24)
- pole velocity at tip (from $-\infty$ to $\infty$)
However, to make the problem more interesting and difficult we set up a partially observable Markov decision process (POMDP), where the system dynamics are known to follow an MDP but the agent can not directly observe the states. In our experiments, the agent only observes the partial state consisting position of the cart and the angle pole makes with the cart. Different algorithms evaluated in this paper takes in the sequence of partially observed game states (a window of previous states preceding the current state) and predicts the action to be taken at the current state. The intuition is that the sequence of partially observed states should be sufficient for the algorithms to learn about the missing state features and act optimally.
In this work, we evaluate three different classes of algorithms namely, 1) Deep Q-Networks (DQN), 2) Deep Recurrent Q-Networks (DRQN) and 3) Deep Transformer Q-Network (DTQN).
DQN, DRQN, and DTQN
-------------------
Deep Q-Learning (DQN) uses a neural network to approximate the Q-value function. The state is given as the input and the Q-value of all possible actions is generated as the output. In a typical Deep Q learning setup, all the past experiences are first stored in the memory, the next action is then determined by using epsilon greedy policy with respect to current $Q$ values and the final loss is computed using equation \[e1\], where $S_t, a_t, r_t$ is the state, action taken and reward received at time-step $t$ and $S_{t+1}$ is the state at the next time-step. $$\begin{aligned}
L = ||r_t + \gamma \max_{a \in A}Q(S_{t+1},a) - Q(S_t, a_t)||_2^2
\label{e1}\end{aligned}$$ The term $r_t + \gamma \max_{a \in A}Q(S_{t+1},a)$ in equation \[e1\] is known as the *target* and will change erratically at every time-step as the Q values will change erratically at every time-step, in order to make the learning more stable we use a second copy of the deep neural network called *target network*. The target network has the same architecture as the function approximator but with frozen parameters. For every $C$ iterations (a hyperparameter), the parameters from the prediction network are copied to the target network. This leads to more stable training. Figure \[fig:r1\] shows a representative architecture for the DQN. In our experiments, the input to DQN is the feature vector produced by concatenating the partially observed states from current and time-steps with the partially observed states of previous time-steps (4 time-steps in total, including current time-step).
![Transformer: encoder taking input sequence and decoder taking output sequences[]{data-label="m2"}](pics/transformer.png){width="0.6\linewidth"}
Deep Recurrent Q-Learning (DRQN) uses a recurrent neural network to approximate the Q-value function. Sequence of states is given as the input and the network consists of RNN unit (LSTM/GRU), the output of the RNN unit at the final time-step is used to predict the Q-value of all possible actions is generated. Such networks are also often trained using the loss function defined equation \[e1\]. Figure \[fig:r2\] shows a representative architecture for DRQN. In our experiments, the RNN unit (GRU) is fed the sequence of partially observed states, the output from the final time-step is used to predict the Q-value function.
The idea behind Deep Transformer Q-learning (DTQN) is to use a *transformer* instead of RNN to extract the meaningful feature out of the input sequence. However, transformers were designed to work for sequence to sequence (seq2seq) tasks such as automatic speech recognition or language translation. But in this work we do not have s sequence to sequence task, rather we need to predict the $Q$ value function given the input sequence.
![Transformer: encoder taking input sequence and decoder taking output sequences[]{data-label="m3"}](pics/multiheadattn.png){width="0.4\linewidth"}
{width="0.5\linewidth"} {width="0.5\linewidth"}
![Scores vs Episodes for multiple runs of different algorithms. (a) DQN, (b) DRQN, (c) DTQN.[]{data-label="fig:r4"}](pics/DQN_traces.png){width="\linewidth"}
![Scores vs Episodes for multiple runs of different algorithms. (a) DQN, (b) DRQN, (c) DTQN.[]{data-label="fig:r4"}](pics/DRQN_traces.png){width="\linewidth"}
![Scores vs Episodes for multiple runs of different algorithms. (a) DQN, (b) DRQN, (c) DTQN.[]{data-label="fig:r4"}](pics/DTQN_traces.png){width="\linewidth"}
The transformer model consists of encoder and decoder modules, as shown in figure \[m2\]. The encoder module processes the input sequence of embeddings (tokens from input sequence are first embedded and then passed through the network) through a multi-head attention module followed by a feed-forward neural network. In our experiments we only use the encoder module (since our task is not Seq2Seq) to extract the features from input sequence which are further used to predict the Q-values. The multi-head attention module in the encoder refers to the self-attention layer which is a mechanism relating different positions of a single sequence in order to compute a representation of the sequence. Figure \[m3\] shows the multi-head attention module used in the encoder, $V$, $K$, $Q$ denotes the value vector, key vector, query vector. Figure \[fig:r3\] shows a representative architecture for DTQN.
Results and Discussions {#sec:results}
=======================
We did multiple experiments for each of the following DQN, DRQN and DTQN based reinforcement learning algorithms. Each algorithm was trained for 5000 episodes and we ran 10 different instances for each of the algorithms with random initialization. Figure \[fig:r4\] shows the value of the scores over episodes for different runs. We see that DQN and DTQN performed worse compared to our DRQN model. While on average the scores for the DQN and DTQN are decreasing, we see that there are a few experiments where the scores improve during training as shown in the graph, but for a majority of the traces, the scores did not improve when using DQN or DTQN. However, the results show significant improvement in scores when using the DRQN model.
As we used location and angle as the only inputs to our neural networks determine the next action, the scores and loss function did not show very promising trends in all three cases. Still, it was possible to make a distinction as to which one of the three: DQN, DRQN, or DTQN performed better by analyzing their relative scores and loss values. DRQN gave the best score results with the maximum reaching to 135 in one of the test cases. DTQN performed the worst on average.
We believe that a transformer-based approach is indeed not suitable for solving the reinforcement learning problem at least in cases where input is taken in the form of a pair of location value and angle value. Reason for the failure can be pointed out as follows: LSTM based approach (DRQN) captures the temporal attributes of the sequence of inputs whereas transformer-based approach tries to put attention on different time-steps of the sequence to obtain the representation for input sequence and does not explicitly tries to capture the temporal aspect of the sequence. As our problem is more of temporal related as the next state is the state at the next time step, LSTM achieves better results. We would also like to add that during our course of work we found that training RL algorithm, for video games, based on DQN or neural networks, in general, is difficult to train and the performance greatly depends upon the random initialization. As can be seen from the traces, some of the random initialization perform too bad while a few perform well.
Conclusions and Future Work
===========================
In this work, we propose a new transformer based model for reinforcement learning, the inspiration for the same is derived from the fact that the recent advancements in NLP have been achieved by moving away from RNNs and introducing the new transformer based model. Even though the standard transformer consists of both encoder and decoder modules as it’s designed to perform Seq2Seq task, we decided to use only the encoder module with the multi-head attention mechanism from the transformer to extract important features from the states to learn the Q-values. We perform experiments on a POMDP and compared multiple algorithms. We conclude that using the state representation provided by the RAM of the game (X-coordinate of the cart and the angle made by pole with the cart), the conventional DRQN (based on RNNs such as LSTM and GRU) perform better than DQN or the proposed DTQN. As part of future work, we intend to compare different algorithms using different state representation like images instead of RAM values.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'The fundamental relationship $\zeta(s) = \chi(s)\zeta(1-s)$ reveals that no special function is more suitable than the chi function $\chi(s)$ to study the Riemann zeta function $\zeta(s)$. When $t$ is sufficiently large, the modulus and argument of $\chi(\sigma+it)$ are monotone about $\sigma$ and $t$ respectively, which accomodates the construction of a Riemann surface for the multivalued $z = \chi(s)$. The inverse of $\chi(s)$, which is branched and much similiar to the logarithm function, is introduced as tau function $s=\tau(z)$. Then $\zeta(s)=\zeta\circ\tau(z)$ can be studied by its different branches, with a much simpler relationship in single-valued domains, which finally leads to the conclusion about the nontrivial zeros of $\zeta(s)$.'
author:
- Jianyun Zhang
bibliography:
- 'refer.bib'
title: 'The Chi Function, Tau Function, and Riemann Zeta Function'
---
Introduction
============
The Riemann zeta function $\zeta(s)$ is one of the most challenging functions, whose zeros have interested many. The analytic function $\zeta(s)$ satisfies the functional equation $$\zeta(s) = 2^s\pi^{s-1}\sin(\frac{\pi{s}}{2})\Gamma(1-s)\zeta(1-s),\quad s\in \mathbb{C}\backslash\{1\}.$$
The chi function $\chi(s)$ is defined as $$\chi(s) = 2^s\pi^{s-1}\sin(\frac{\pi{s}}{2})\Gamma(1-s),$$ and $$\label{chi0}
\chi(s)\chi(1-s)=1.$$
Considering the zeros of $\sin(\frac{\pi{s}}{2})$ and the poles of $\Gamma(1-s)$, the chi function $\chi(s)$ is meromophic on the entire complex plane, with poles at $s=1,3,5,\dots$ and zeros at $s=0,-2,-4,\dots$ It has been shown that the only function which satisfies $$\label{basicfun}
\zeta(s) = \chi(s)\zeta(1-s)$$ or $$\label{basicfun2}
\zeta(1-s) = \chi(1-s)\zeta(s),$$ and has the same general characteristics as $\zeta(s)$, is $\zeta(s)$ itself[@Titchmarsh86]. The fundamental relationship reveals that no special function is more suitable than $\chi(s)$ to study $\zeta(s)$.
The modulus and argument of the chi function {#china}
============================================
Some properties of the chi function $\chi(s)$ are proposed. One is about its modulus, the other is about its argument, both for the $s$ far away from the real axis in the complex plane. We only discuss the $s$ in the upper half-plane for the symmetry of $t$.
\[chi1\] Let $s=\sigma+it$. Then $|\chi(s)|=1$ for $\sigma=\frac{1}{2}$. There also exists a real number $M_{1}>0$, such that the modulus of $\chi(s)$ is a continuous function of $\sigma$ and $t$ when $t\ge M_{1}$, satisfying the following properties:
1. \[rad:mono\] $|\chi(s)|$ decreases strictly monotonously with increasing $\sigma$, and tends to $0$ as $\sigma\to+\infty$.
2. \[rad:1\] $0<|\chi(s)|<1$ for $\frac{1}{2}<\sigma<+\infty$.
3. \[rad:2\] $1<|\chi(s)|<+\infty$ for $-\infty<\sigma<\frac{1}{2}$.
Taking $\sigma=\frac{1}{2}$ in , then $|\chi(s)|=1$.
All the poles and zeros of $\chi(s)$ are on the real axis, then $0<|\chi(s)|<+\infty$ when $t\ne0$.
The following asymptotic expansion[@Beals10] is adopted for real $x$ and $y$: $$\label{absgamma}
|\Gamma(x+iy)|=\sqrt{2\pi}|y|^{x-\frac{1}{2}}e^{-\frac{1}{2}\pi |y|}\big\{1+O(\frac{1}{|y|})\big\}, \quad |y|\rightarrow\infty.$$ Taking $x=1-\sigma$ and $y=-t$ in , then $$\label{abschi}
\begin{split}
|\chi(s)| &= |2^{\sigma+it}|\cdot|\pi^{\sigma-1+it}|\cdot|\sin\big\{\frac{\pi}{2}(\sigma+it)\big\}|\cdot|\Gamma(1-\sigma-it)|\\
&= 2^{\sigma}\pi^{\sigma-1}\sqrt{\sin^2(\frac{\pi}{2}\sigma)+\sinh^2(\frac{\pi}{2}t)}\sqrt{2\pi}t^{\frac{1}{2}-\sigma}e^{-\frac{1}{2}\pi t}\big\{1+O(\frac{1}{t})\big\}\\
&= \sqrt{2\pi}2^{\sigma}\pi^{\sigma-1}\sqrt{\frac{\sin^2(\frac{\pi}{2}\sigma)+\sinh^2(\frac{\pi}{2}t)}{e^{\pi t}}}t^{\frac{1}{2}-\sigma}\big\{1+O(\frac{1}{t})\big\}.
\end{split}$$ The modulus $|\chi(s)|$ is a continuous function of $\sigma$ and $t$ when $t$ is sufficiently large. It’s obvious that $0<|\chi(s)|<1$ for all $\sigma>\frac{1}{2}$, and $|\chi(s)|>1$ for all $\sigma<\frac{1}{2}$, both as $t\to+\infty$.
For any sufficiently large $t$, we also have $|\chi(s)|\to0$ as $\sigma\to+\infty$, and $|\chi(s)|\to+\infty$ as $\sigma\to-\infty$.
The infinitesimal $\Delta|\chi(s)|$ with respect to $\Delta\sigma$ is $$\begin{split}
\frac{\Delta|\chi(s)|}{|\chi(s)|} &=
\big\{\log (2\pi)-\log t+\frac{\frac{\pi}{4}\sin\pi\sigma}{\sin^2(\frac{\pi}{2}\sigma)+\sinh^2(\frac{\pi}{2}t)}\big\}\big\{1+O(\frac{1}{t})\big\}\Delta\sigma
\end{split}.$$ It’s obvious that $|\chi(s)|$ decreases strictly monotonously with increasing $\sigma$ when $t$ is sufficiently large.
[Lemma \[chi1\]]{} demonstrates that the critical line $\{s|\textup{Re}(s)=\frac{1}{2}\}$ in the $s$-plane is mapped to a unit circle $S^1$ by $\chi(s)$.
\[chi2\] Let $s=\sigma+it$. There exists a real number $M_{2}>0$, such that the argument of $\chi(s)$ is a continuous function of $\sigma$ and $t$ when $t\ge M_{2}$, satisfying the following properties:
1. \[ang:1\] $\arg(\chi(s))$ decreases strictly monotonously with increasing $t$, and tends to $-\infty$ as $t\to+\infty$.
2. \[ang:2\] $\arg(\chi(s))$ remains almost constant with varying $\sigma$.
3. \[ang:3\] $\arg(\chi(s))=\arg(\chi(1-\overline{s}))$.
When $t\ne0$, the logarithm of $\chi(s)$ is obtained as $$\label{logchi}
\begin{split}
\log\chi(\sigma+it) &= (\sigma+it)\log 2+(\sigma-1+it)\log\pi\\
& \quad +\log\sin\big\{\frac{\pi}{2}(\sigma+it)\big\}+\log\Gamma(1-\sigma-it).
\end{split}$$ The third item in is $$\begin{split}
\log\sin\big\{\frac{\pi}{2}(\sigma+it)\big\}
&= \log\frac{e^{-\frac{\pi}{2}t}e^{i\frac{\pi}{2}\sigma}-e^{\frac{\pi}{2}t}e^{-i\frac{\pi}{2}\sigma}}{2i} \\
&= \log\frac{-e^{\frac{\pi}{2}t}e^{-i\frac{\pi}{2}\sigma}}{2i}+\log(1-\frac{e^{i\pi\sigma}}{e^{\pi t}}) \\
&= \frac{\pi}{2}t - \log2 + i\frac{\pi}{2}(1-\sigma) + O(-\frac{e^{i\pi\sigma}}{e^{\pi t}}).
\end{split}$$ The last item in can be expressed by the asymptotic expansion[@Edelyi81] $$\label{loggamma}
\log\Gamma(z+a)=(z+a-\frac{1}{2})\log z-z+\frac{1}{2}\log(2\pi)+O(\frac{1}{z})$$ where $|\arg z|<\pi$ and $a\in\mathbb{C}$. Taking $z=-it$ and $a=\sigma$ in , then $$\log\Gamma(\sigma-it)= (-it+\frac{1}{2}-\sigma)\log (-it) + it +\frac{1}{2}\log(2\pi)+O(i\frac{1}{t}).$$ Finally $$\label{argchi}
\begin{split}
\arg(\chi(\sigma+it)) &= \textup{Im}(\log\chi(\sigma+it)) \\
&= t+t\log\frac{2\pi}{t} + \frac{\pi}{4} + O(\frac{1}{t}) + O(-\frac{\sin\pi\sigma}{e^{\pi t}}).
\end{split}$$ The argument $\arg(\chi(\sigma+it))$, if not restricted in the range of $2\pi$, is a continuous function of $\sigma$ and $t$ when $t$ is sufficiently large. It’s obvious that $$\label{limargchi}
\arg(\chi(\sigma+it))\to t+t\log\frac{2\pi}{t}+\frac{\pi}{4}, \quad t\to+\infty.$$ The $\arg(\chi(\sigma+it))$ decreases strictly monotonously with increasing $t$ when $t$ is sufficiently large, and $\arg(\chi(\sigma+it))$ tends to $-\infty$ as $t\to+\infty$.
It’s also observed that $\arg(\chi(\sigma+it))$ remains almost constant with varying $\sigma$ when $t$ is sufficiently large. Since $\chi(s)\ne 0$ when $t\ne0$, the argument of is $$\begin{split}
2\pi m &=\arg(\chi(\sigma+it))+\arg(\chi(1-\sigma-it))\\
&=\arg(\chi(\sigma+it))-\arg(\chi(1-\sigma+it)), \quad m=0,\pm 1,\pm 2,\dots.
\end{split}$$ And excludes the possibility of any non-zero $m$ for sufficiently large $t$, although $\arg(\chi(s))$ is multivalued.
[Lemma \[chi2\]]{} demonstrates that for any real constant $\sigma$, while $s=\sigma+it$ moves upwards on the vertical line $\{s|\textup{Re}(s)=\sigma\}$, the image $\chi(s)$ wraps around the origin clockwise for $t\ge M_{2}$. Combined with [Lemma \[chi1\]]{}, while $s$ moves upwards on the critical line $\{s|\textup{Re}(s)=\frac{1}{2}\}$, the image $\chi(s)$ loops on the unit circle $S^1$ clockwise, passing through one point such as $\{1\}$ on $S^1$ infinitely many times.
The branch and the inverse of the chi function {#argpresrv}
==============================================
[Lemma \[chi1\]]{} and [Lemma \[chi2\]]{} apply when $t\ge M_1$ and $t\ge M_2$ respectively, and the following domain is focused on:
\[FAR\] The domain $D\subset\mathbb{C}$ is said to be far away from the real axis(FAR for short), if a proper real number $M\ge \max(M_{1},M_{2})$ can be chosen, such that $\arg(\chi(s))\le \arg(\chi(\frac{1}{2}+iM))$ for all $s\in D$.
As a special case, the veritical line $\{s|\sigma=\frac{1}{2},t\ge M\}$ is FAR for any $M\ge \max(M_{1},M_{2})$, since $\arg(\chi(\frac{1}{2}+it))\le \arg(\chi(\frac{1}{2}+iM))$ for all the points on the line by [Lemma \[chi2\]]{}. Definition \[FAR\] can be extended to the lower half-plane as $\arg(\chi(\sigma+it))\ge \arg(\chi(\frac{1}{2}+iM))$ for $t<0$.
For all $s\in D$, we observe that $0<|\chi(s)|<+\infty$ by [Lemma \[chi1\]]{}, and that $\arg(\chi(s))$ is monotonous decreasing about $t$ and tends to $-\infty$ by [Lemma \[chi2\]]{}. Hence a Riemann surface can be constructed as follows to make the range of $z=\chi(s)$ single-valued.
The slit $z$-plane $\mathbb{C}\backslash[0,+\infty)$ is designated as the $m$th sheet $S_{m}$, where $m$ is any integer sufficiently large. And when every two sheets $S_{m}$ and $S_{m+1}$ are attached along the branch cut $(0,+\infty)$, a Riemann surface $R$ spread over the $z$-plane is constructed.
Then the chi function $$\label{Rchi}
z=\chi(s),\quad s\in D $$ where $z\in R$, is the composition of infinitely many branches $$\label{multichi}
z=\chi_{m}(s)=|\chi(s)|e^{\textup{Arg}(\chi(s))-2\pi im} $$ where $z\in S_{m}\subset R$.
Let $\phi\in \mathbb{R}$ be a constant. The arc $\gamma_{\phi}$ in the $s$-plane is said to be argument-preserving for the map $\chi:s\to z$, if $\arg(z)=\phi$ for all $s\in \gamma_{\phi}$.
\[argpreserving\] Let $D=\{s|\sigma_1\le\textup{Re}(s)\le\sigma_2\}$ be a FAR domain where $\sigma_1<\frac{1}{2}<\sigma_2$. For any $s_{0}\in D$, there exists a unique argument-preserving arc $\gamma_{\phi}$ for the map $\chi$, such that $s_{0}\in\gamma_{\phi}$ and the arc $\gamma_{\phi}$ splits the domain $D$ horizontally into two parts.
Let $\chi(\sigma+it)=r(\sigma,t)e^{i\phi(\sigma,t)}$ where $\phi(\sigma,t)=\arg(\chi(\sigma+it))$ and $r(\sigma,t)=|\chi(\sigma+it))|$. Step 1. Choose $M$ for the domain $D$.
From we know that $\phi(\sigma,t)$ is continuous and bounded for any finite $\sigma$ and $t$. Let $$\epsilon=\sup\limits_{\sigma\in[\sigma_1,\sigma_2]}|\phi(\sigma,t)-\phi(\frac{1}{2},t)|.$$ There exists a real number $M_{3}>0$ and an arbitrarily small $\epsilon_{0}<\pi$, such that $\epsilon<\epsilon_{0}$ for all $t\ge M_{3}$, because $\epsilon\to 0$ as $t\to +\infty$ by [Lemma \[chi2\]]{}.
Let $M_{0}=\max(M_{1},M_{2},M_{3})$. Then $|\phi(\sigma,M_{0})-\phi(\frac{1}{2},M_{0})|<\epsilon_{0}$ for all $\sigma\in[\sigma_1,\sigma_2]$. Take a real number $M>M_{0}$ satisfying $\phi(\frac{1}{2},M)=\phi(\frac{1}{2},M_{0})-2\epsilon_{0}$. Then for all $\sigma\in[\sigma_1,\sigma_2]$ and $t\ge M$, $$\label{inequality}
\phi(\sigma,M)<\phi(\frac{1}{2},M)+\epsilon_{0}=\phi(\frac{1}{2},M_{0})-\epsilon_{0}<\phi(\sigma,M_{0}).$$
Choose the domain $D=\{s|\sigma_1\le\sigma\le\sigma_2,t\ge M\}$ as the FAR critical strip.
Step 2. The existance of the unique set $\gamma_{\phi}$.
If $s_{0}=\sigma_{0}+it_{0}\in D$, then $t_{0}\ge M>M_{0}$. For any constant $\sigma\in[\sigma_1,\sigma_2]$, a continuous real function of $t$ is constructed as $$F(t)=\phi(\sigma,t)-\phi(\sigma_{0},t_{0}).$$ When $t=M_{0}$, we obtain that $F(M_{0})=\phi(\sigma,M_{0})-\phi(\sigma_{0},t_{0})>0$ by . When $t\to\infty$, we obtain that $F(t)=\phi(\sigma,t)-\phi(\sigma_{0},t_{0})$ decreases strictly monotonously and tends to $-\infty$ by [Lemma \[chi2\]]{}.
Therefore there exists a unique and bounded $\hat{t}$ satisfying $F(t)=0$ for each $\hat{\sigma}\in[\sigma_1,\sigma_2]$, all of which form the unique set $\gamma_{\phi}=\{\sigma+it|\phi(\sigma,t)=\phi(\sigma_{0},t_{0}),\sigma_1\le\sigma\le\sigma_2\}$.
Step 3. The set $\gamma_{\phi}$ is a continuous and simple arc.
The two-variable function $H(\sigma,t)=\phi(\sigma,t)-\phi(\sigma_{0},t_{0})$ is continuous about $\sigma$ and $t$ for $\sigma\in(\sigma_1-\delta,\sigma_2+\delta)$ and $t\ge M$, where $\delta>0$ is small. Because $H(\sigma,t)$ is monotone decreasing about $t$, a unique implicit function $t=h(\sigma)$ can be established from $H(\sigma,t)=0$ near any of its solution $(\hat{\sigma},\hat{t})$ which is bounded. And $t=h(\sigma)$ is continuous and bounded near any $\hat{\sigma}\in[\sigma_1,\sigma_2]$.
Therefore the single-valued function $t=h(\sigma)$ is continuous and bounded for all $\sigma\in[\sigma_1,\sigma_2]$, which shows that $\gamma_{\phi}$ is a continuous and simple arc, splitting the critical strip $\{\sigma+it|\sigma_1\le\sigma\le\sigma_2\}$ horizontally into two parts.
The argument-preserving arc $\gamma_{\phi}\subset D$ is mapped to $\beta_{\phi}\subset R$ by the map $$\chi:\gamma_{\phi}\to\beta_{\phi},$$ and $\beta_{\phi}$ is a line segment on a ray issuing from the origin.
\[chiarc\] The map $\chi:\gamma_{\phi}\to \beta_{\phi}$ is continuous and one-to-one, satisfying the following properties:
1. \[map:1\] If $s_{0}\in \gamma_{\phi}$, then $(1-\overline{s_{0}})\in \gamma_{\phi}$.
2. \[map:2\] Two arcs $\gamma_{\phi_{1}}$ and $\gamma_{\phi_{2}}$ do not intersect in the $s$-plane, if $\phi_{1}\ne\phi_{2}$.
For any $s\in\gamma_{\phi}$, let $\chi(s)=r(\sigma,t)e^{i\phi}$ where $\phi$ is a constant real. By [Lemma \[chi1\]]{} the radius $r$ is continuous and monotone decreasing with respect to $\sigma$. Then the map $\chi:\gamma_{\phi}\to \beta_{\phi}$ is continuous and one-to-one.
[Lemma \[chi2\]]{} claims that $\arg(\chi(s_{0}))=\arg(\chi(1-\overline{s_{0}}))$, which makes property (i) true.
Suppose $s_{0}$ is the point where $\gamma_{\phi_{1}}$ and $\gamma_{\phi_{2}}$ intersect. The assumption of $\gamma_{\phi_{1}}\ne\gamma_{\phi_{2}}$ with $\phi_{1}=\phi_{2}$ contradicts [Lemma \[argpreserving\]]{}, which makes property (ii) true.
The FAR domain $D$ is mapped to $U$ by the map $$\chi:D\to U$$ where $U$ is the domain in the Riemann surface $R$.
When the branch cut $[0,+\infty)$ is chosen, the domain $U$ can be separated into infinitely many sheets as $$U_m=U(\phi_2,\phi_1)=\bigcup_{\phi_2<\phi< \phi_1}\beta_{\phi}$$ where $m$ is any sufficiently large integer and $$\phi_1=-2\pi m,\quad \phi_2=\phi_1-2\pi.$$ Each $U_m$ is a domain in the slit complex plane $S_m$. Correspondingly the domain $D$ can be separated into infinitely many horizontal strips as $$D_m=D(\phi_2,\phi_1)=\bigcup_{\phi_2<\phi< \phi_1}\gamma_{\phi}.$$ Each $D_m$ is the pre-image of $U_m$.
\[chiPatch\] The map $\chi:D_m\to U_m$ is one-to-one, and the inverse map $\chi^{-1}:U_m\to D_m$ can be defined on the whole slit plane $S_m$.
Suppose $s=\sigma+it\in D_m$ and $z=re^{i\phi}\in U_m$.
If $\chi(s_1)=\chi(s_2)=r_0e^{i\phi_0}$ where $s_1,s_2\in D_m$, then $s_1,s_2\in\gamma_{\phi_0}$. The Corollary \[chiarc\] requires that $s_1=s_2$ for they share a single $r_0$. Then the map $\chi$ is one-to-one from $D_m$ to $U_m$.
Let $\chi:\gamma_{\phi}\to\beta_{\phi}$ where $\gamma_{\phi}$ is the arc preserving the argument $\phi$ for the map $\chi$, and let $$D_0=D_m\cap\{s|\sigma_1\le\textup{Re}(s)\le\sigma_2\}$$ where $\sigma_1<\frac{1}{2}<\sigma_2$.
[Lemma \[argpreserving\]]{} claims that $\gamma_{\phi}$ always intersects the vertical lines $\{s|\sigma=\sigma_1\}$ and $\{s|\sigma=\sigma_2\}$ in the FAR domain $D_0$, even as $\sigma_2\to+\infty$ and $\sigma_1\to-\infty$. [Lemma \[chi1\]]{} continues to claim that $\beta_{\phi}$ is a line segment with one end tending to the origin, and with the other end tending to $\infty$. As $\phi$ decreases by $2\pi$, the domain $U_m$ covers the whole complex plane $\mathbb{C}$ except the branch cut.
The tau function $\tau(z)$, inverse of the $m$-th branch function , can be well defined based on Corollary \[chiPatch\] as $$\label{tau}
s=\tau(z)=\chi^{-1}(z), \quad z\in \mathbb{C}\backslash [0,+\infty)$$ where $-2\pi (m+1)<\arg(z)<-2\pi m$.
\[conformalMap\] The tau function $s=\tau(z)$ is a conformal mapping of the slit plane $S_m$ onto the horizontal strip $D_m$.
Suppose $\chi:s\to z$ where $s=\sigma+it\in D_m$ and $z=re^{i\phi}\in S_m$.
The FAR domain $D_m$ contains neither zeros nor poles of $\chi(s)$. Since $\chi(s)$ is analytic for all $s\in D_m$, the function $\chi(s)$ is differentiable for all $s\in D_m$, and the derivative of the nonzero $\chi(s)=re^{i\phi}$ is $$\begin{split}
\chi'(s)&=\frac{\partial(r\cos\phi)}{\partial\sigma} + i\frac{\partial(r\sin\phi)}{\partial\sigma}\\
&=\frac{\partial r}{\partial\sigma}e^{i\phi} + i\frac{\partial\phi}{\partial\sigma}\chi(s).
\end{split}$$
Both [Lemma \[chi1\]]{} and [Lemma \[chi2\]]{} applies for all $s\in D_m$, which require $\frac{\partial r}{\partial\sigma}<0$ and $\frac{\partial\phi}{\partial\sigma}\to0$. Therefore $\chi'(s)\ne0$ for all $s\in D_m$. And the inverse map $\tau:z\to s$ is analytic on $S_m$ by implicit function theorem.
The tau function $\tau(z)$ can also be defined on the negative $m$-th sheet.
\[conjBranch\] The branch $$\label{tau-}
s=\tau_{-}(z), \quad z\in S_m^*$$ is said to be the conjugated branch of $$\label{tau+}
s=\tau(z), \quad z\in S_m$$ if $S_m^{*}=\{\overline{z}|z\in S_m\}$ is the reflection of $S_m$.
The conjugated branch of is $$\label{tau1-}
s=\tau_-(z)=\chi^{-1}(z), \quad z\in \mathbb{C}\backslash [0,+\infty)$$ where $2\pi m<\arg(z)< 2\pi (m+1)$.
Another pair of conjugated branches are available if $(-\infty,0]$ is chosen to be the branch cut. The corresponding tau functions are $$\label{tau2+}
s=\tau(z)=\chi^{-1}(z), \quad z\in \mathbb{C}\backslash (-\infty,0]$$ where $-\pi-2\pi m<\arg(z)<\pi-2\pi m$, and $$\label{tau2-}
s=\tau_-(z)=\chi^{-1}(z), \quad z\in \mathbb{C}\backslash (-\infty,0]$$ where $-\pi + 2\pi m<\arg(z)< \pi + 2\pi m$.
Generally speaking, the topology of $\chi(s)$ is similiar to the topology of $e^s$, when $s$ is far away from the real axis in the complex plane. And the topology of $\tau(z)$ is similiar to the topology of $\log(z)$.
The logarithmic integral of the branched zeta function {#tauContour}
======================================================
The composite function is introduced based on or as $$\label{multifun}
w=\zeta(s)=\zeta\circ\chi^{-1}(z)=\zeta\circ\tau(z)=G(z),\quad z\in S_m.$$ where $-2\pi (m+1)<\arg(z)<-2\pi m$ or $-\pi-2\pi m<\arg(z)<\pi-2\pi m$, and $m$ is sufficiently large.
The function $G(z)$ is also branched in accordance with the branch of $\tau(z)$. And $G(z)$ is analytic on the slit plane $S_m$ by [Lemma \[conformalMap\]]{}.
By Definition \[conjBranch\] the conjugated branch of is $$\label{G-}
w=\zeta(s)=\zeta\circ\tau_-(z)=G_-(z),\quad z\in S_m^*$$ where $S_m^*$ is the reflection of $S_m$. Let the point $z\in S_m$ and $$\label{Point1}
G:z\stackrel{\tau}{\to}s \stackrel{\zeta}{\to}\zeta(s).$$ Supposing $\eta=\chi(1-s)$, by we have $$\eta=\chi(1-s)=\frac{1}{\chi(s)}=\frac{1}{z}.$$ [Lemma \[chi2\]]{} claims that $\arg(\overline{\eta})=\arg(z)$ and $\arg(\eta)=-\arg(z)$, requiring the points $\overline{\eta}\in S_m$ and $\eta\in S_m^{*}$ respectively and $$\label{Point23}
\begin{split}
G&:\overline{\eta}=\overline{(\frac{1}{z})}\stackrel{\tau}{\to}1-\overline{s} \stackrel{\zeta}{\to}\overline{\zeta(1-s)},\\
G_-&:\eta=\frac{1}{z}\stackrel{\tau_-}{\to}1-s \stackrel{\zeta}{\to}\zeta(1-s).
\end{split}$$
The fundamental functional equation can be rewritten as $$\label{funEquation}
G(z) = z\, G_-(\frac{1}{z}) $$
Let $\tau:S_m\to D_m$ and the domain $D_m$ is a horizontal strip $$\label{HStrip}
D_m=D(\phi_2,\phi_1)=\bigcup_{\phi_2<\phi< \phi_1}\gamma_{\phi}$$ where $\phi_1=-2\pi m$ or $\phi_1=\pi-2\pi m$, and $\phi_2=\phi_1-2\pi$. We come to study the logarithmic integral of nonvanishing zeta function $$\label{logintzeta}
\int_{\partial D_s}\frac{\zeta'(s)}{\zeta(s)}ds$$ where the domain $D_s$ is defined as follows.
\[simpleD\] A domain $D_s$ is said to be simple, if
1. the domain $D_s\subset D_m$ is bounded, simple connected, and symmetric with respect to the vertical line $\{s|\textup{Re}(s)=\frac{1}{2}\}$;
2. the boundary $\partial D_s$ is piecewise smooth, and $\zeta(s)\ne0$ for all $s\in \partial D_s$;
3. the boundary $\partial D_s$ meets the vertical line $\{s|\textup{Re}(s)=\frac{1}{2}\}$ only twice.
Two types of simple domains $D_s$ are studied.
\[simpleD1\] A domain $D_s^1$ is said to be type-one, if
1. the domain $D_s^1\subset D_m$ is simple and $$D_m=D(\phi_2,\phi_1)=\bigcup_{\phi_2<\phi< \phi_1}\gamma_{\phi}$$ where $\phi_1=-2\pi m$ and $\phi_2=-2\pi (m+1)$;
2. the boundary $\partial D_s^1$ meets the vertical line $\{s|\textup{Re}(s)=\frac{1}{2}\}$ at two points $$s_{m+1}^-=\frac{1}{2}+i(t_{m+1}-\epsilon) \quad\text{and}\quad s_m^+=\frac{1}{2}+i(t_m+\epsilon)$$ where $\arg(\chi(\frac{1}{2}+it_m))=-2\pi m$, $\arg(\chi(\frac{1}{2}+it_{m+1}))=-2\pi (m+1)$, and $\epsilon>0$ is arbitrarily small.
\[simpleD2\] A domain $D_s^2$ is said to be type-two, if
1. the domain $D_s^2\subset D_m$ is simple and $$D_m=D(\phi_2,\phi_1)=\bigcup_{\phi_2<\phi< \phi_1}\gamma_{\phi}$$ where $\phi_1=\pi-2\pi m$ and $\phi_2=-\pi-2\pi m$;
2. the boundary $\partial D_s^2$ meets the vertical line $\{s|\textup{Re}(s)=\frac{1}{2}\}$ at two points $$s_m^+=\frac{1}{2}+i(t_m+\epsilon) \quad\text{and}\quad s_m^-=\frac{1}{2}+i(t_m-\epsilon)$$ where $\arg(\chi(\frac{1}{2}+it_m))=-2\pi m$, and $\epsilon>0$ is arbitrarily small.
\[Res1\] The logarithmic integral of nonvanishing zeta function $$\lim\limits_{\epsilon\to0}\int_{\partial D_s^1}\frac{\zeta'(s)}{\zeta(s)}ds = 2\pi i$$ where $D_s^1$ is type-one simple domain defined in Definition \[simpleD1\].
Step 1. Choose $\epsilon$ for the domain $D_s^1$.
When $s=\frac{1}{2}+it$, we have $1-s=\frac{1}{2}-it=\overline{s}$. By we obtain $$\zeta(s)-\overline{\zeta(s)} = (\chi(s)-1)\overline{\zeta(s)},$$ which requires $\zeta(s)$ to be real when $\chi(s)=1$.
Since the zeros of analytic $\zeta(s)$ are isolated, there exists a small $\epsilon_0>0$ such that $\zeta(\frac{1}{2}+i(t_m+\epsilon))\ne0$ and $\zeta(\frac{1}{2}+i(t_{m+1}+\epsilon))\ne0$ for all $0<|\epsilon|<\epsilon_0$.
Choose $0<\epsilon<\epsilon_0$ for the domain $D_s^1$, where $$\label{realValue}
\begin{split}
\lim\limits_{\epsilon\to0}\zeta(s_m^+)=\lim\limits_{\epsilon\to0}\overline{\zeta(s_m^+)}=\lim\limits_{\epsilon\to0}\zeta(\overline{s_m^+}), \\ \lim\limits_{\epsilon\to0}\zeta(s_{m+1}^-)=\lim\limits_{\epsilon\to0}\overline{\zeta(s_{m+1}^-)}=\lim\limits_{\epsilon\to0}\zeta(\overline{s_{m+1}^-}),
\end{split}$$ since $\chi(s)\to1$ and $\zeta(s)\to\overline{\zeta(s)}$ as $\epsilon\to0$.
Let $\gamma$ and $\overline{\gamma}$ be any arc and its conjugate respectively, where $\gamma$ starts at $s_{m+1}^-$ and ends at $s_m^+$. When $\zeta(s)\ne0$ on $\gamma$ or $\overline{\gamma}$, the lorgarithmic integral $$\begin{split}\label{conjInt}
\lim\limits_{\epsilon\to0}\Big\{\int_{\gamma}\frac{\zeta'(s)}{\zeta(s)}ds - \int_{\overline{\gamma}}\frac{\zeta'(s)}{\zeta(s)}ds\Big\} &= \lim\limits_{\epsilon\to0}\Big\{\log\zeta(s)\Big|_{s_{m+1}^-}^{s_m^+} - \log\zeta(s)\Big|_{\overline{s_{m+1}^-}}^{\overline{s_m^+}}\Big\} \\
&= 0
\end{split}$$ for $\epsilon\in(0,\epsilon_0)$.
Step 2. Integrate along the boundary $\partial D_s^1$.
Suppose the boundary $\partial D_s^1\subset D_m$ is separated into two arcs $\gamma^0$ and $\gamma=\{1-\overline{s}|s\in \gamma^0\}$ by the vertical line $\{s|\textup{Re}(s)=\frac{1}{2}\}$. And the arc $\overline{\gamma}=\{\overline{s}|s\in \gamma\}$ is the conjugate of the arc $\gamma$.
Let $$\chi:\gamma^0\to\beta^0,\quad\gamma\to\beta,\quad\overline{\gamma}\to\overline{\beta}.$$ Then the arc $\overline{\beta}=\{z|\frac{1}{z}\in\beta^0\}\subset S_m^*$ and the arc $\beta=\{z|\overline{z}\in\overline{\beta}\}\subset S_m$. And the map $\chi$ sends both $\gamma_{\phi_1}$ and $\gamma_{\phi_2}$ to the branch cut $(0,+\infty)$.
When the point $s\in\gamma^0$ starts at $s_{m+1}^-$ and ends at $s_m^+$ in the counterclockwise direction of $\partial D_s^1$, the point $1-\overline{s}\in\gamma$ starts at $s_{m+1}^-$ and ends at $s_m^+$ in the clockwise direction of $\partial D_s^1$, and the point $1-s\in\overline{\gamma}$ starts at $\overline{s_{m+1}^-}$ and ends at $\overline{s_m^+}$.
As $s$ describes the arc $\gamma^0$, the value $z=\chi(s)\in\beta^0$ moves continuously, starting at $1+i0^+$ on the top edge of the branch cut $(0,+\infty)$ and ending at $1+i0^-$ on the bottom edge of $(0,+\infty)$, with $\arg(z)$ increasing from $\phi_2=-2\pi (m+1)$ to $\phi_1=-2\pi m$.
For nonvanishing $\zeta(s)$, taking the logarithm of , we obtain $$\label{funEquLog}
\log G(z) = \log z + \log G_-(\frac{1}{z})$$ The derivative of with respect to $z$ is $$\label{funEquDlog}
\frac{G'(z)}{G(z)} = \frac{1}{z} - \frac{1}{z^2}\frac{G'_-(\frac{1}{z})}{G_-(\frac{1}{z})},$$ which can be integrated along the arc $\beta^0\subset S_m$ as $$\begin{split}\label{GInt}
\int_{\beta^0}\frac{G'(z)}{G(z)}dz &= \int_{\beta^0}\frac{dz}{z} - \int_{\beta^0}\frac{1}{z^2}\frac{G'_-(\frac{1}{z})}{G_-(\frac{1}{z})}dz \\
&= \int_{1+i0^+}^{1+i0^-}\frac{dz}{z} + \int_{\overline{\beta}}\frac{G'_-(\eta)}{G_-(\eta)}d\eta \\
&\to 2\pi i + \int_{\overline{\beta}}\frac{G'_-(\eta)}{G_-(\eta)}d\eta, \quad\epsilon\to0.
\end{split}$$
For nonvanishing $\zeta(s)$, by and we obtain $$\begin{split}
\lim\limits_{\epsilon\to0}\int_{\partial D_s^1}\frac{\zeta'(s)}{\zeta(s)}ds &= \lim\limits_{\epsilon\to0}\Big\{\int_{\gamma^0}\frac{\zeta'(s)}{\zeta(s)}ds - \int_{\gamma}\frac{\zeta'(s)}{\zeta(s)}ds\Big\} \\
&= \lim\limits_{\epsilon\to0}\Big\{\int_{\gamma^0}\frac{\zeta'(s)}{\zeta(s)}ds - \int_{\overline{\gamma}}\frac{\zeta'(s)}{\zeta(s)}ds\Big\} \\
&= \lim\limits_{\epsilon\to0}\Big\{\int_{\beta^0}\frac{G'(z)}{G(z)}dz - \int_{\overline{\beta}}\frac{G'_-(\eta)}{G_-(\eta)}d\eta\Big\} \\
&= 2\pi i.
\end{split}$$
\[Res2\] The logarithmic integral of nonvanishing zeta function $$\lim\limits_{\epsilon\to0}\int_{\partial D_s^2}\frac{\zeta'(s)}{\zeta(s)}ds = 0$$ where $D_s^2$ is type-two simple domain defined in Definition \[simpleD2\].
We only sketch the proof here.
The branch cut is chosen to be $(-\infty,0]$, such that all the arcs broken from $\partial D_s^2$ or its conjugate $\partial \overline{D_s^2}$ stay inside one sheet $S_m$ or its conjugate $S_m^*$.
When the point $s\in\gamma^0$ starts at $s_m^+$ and ends at $s_m^-$ in the counterclockwise direction of $\partial D_s^2$, the point $1-\overline{s}\in\gamma$ starts at $s_m^+$ and ends at $s_m^-$ in the clockwise direction of $\partial D_s^2$, and the point $1-s\in\overline{\gamma}$ starts at $\overline{s_m^+}$ and ends at $\overline{s_m^-}$.
As $s$ describes the arc $\gamma^0$, the value $z=\chi(s)\in\beta^0$ moves continuously, starting at $1+i0^+$ on the top edge of $(0,+\infty)$ and ending at $1+i0^-$ on the bottom edge of $(0,+\infty)$, without cutting through the branch cut $(-\infty,0]$, and the total increase in $\arg(z)$ is zero.
That makes the difference in , which leads to $$\begin{split}\label{GInt0}
\int_{\beta^0}\frac{G'(z)}{G(z)}dz &= \int_{\beta^0}\frac{dz}{z} - \int_{\beta^0}\frac{1}{z^2}\frac{G'_-(\frac{1}{z})}{G_-(\frac{1}{z})}dz \\
&= \int_{1+i0^+}^{1+i0^-}\frac{dz}{z} + \int_{\overline{\beta}}\frac{G'_-(\eta)}{G_-(\eta)}d\eta \\
&\to \int_{\overline{\beta}}\frac{G'_-(\eta)}{G_-(\eta)}d\eta, \quad\epsilon\to0
\end{split}$$ and $$\begin{split}
\lim\limits_{\epsilon\to0}\int_{\partial D_s^2}\frac{\zeta'(s)}{\zeta(s)}ds &= 0.
\end{split}$$
Finally we give some conclusion about the nontrivial zeros of $\zeta(s)$ when $s$ is far away from the real axis.
\[RH\] Suppose $t$ is sufficiently large. All the zeros of $\zeta(\sigma+it)$ are on the critical line $\{\sigma+it|\sigma=\frac{1}{2}\}$. And each time $\chi(\frac{1}{2}+it)$ loops once around the origin from $\{1\}$ to $\{1\}$ on the unit circle $S^1$ with increasing $t$, there exists one zero of $\zeta(\frac{1}{2}+it)$.
Let $s=\sigma+it$.
The FAR domain $D$ in the $s$-plane can be separated into infinitely many horizontal strips $$D_m=D(\phi_2,\phi_1)=\bigcup_{\phi_2<\phi< \phi_1}\gamma_{\phi}$$ where $m$ is sufficiently large integer and $\phi_2 = \phi_1 - 2\pi$, with their boundaries $\gamma_{\phi_1}$ and $\gamma_{\phi_2}$.
Step 1. Estimate the number of zeros and poles of $\zeta(s)$ in each domain $D_m$ with its boundary.
Let $$\overline{D}_m=\bigcup_{\phi_2\le\phi\le\phi_1}\gamma_{\phi}$$ and $D_0=\overline{D}_m\cap\{s|0<\sigma<1\}$ where $\phi_1=-2\pi m$. The domain $D_0$ is bounded and suppose $T=\sup\limits_{\sigma+it\in D_0}t$.
By von Mangoldt’s result[@Mangoldt05] the number of zeros in the domain $\{\sigma+it|0<\sigma<1,0<t\le T\}$ is $\frac{T}{2\pi}\log\frac{T}{2\pi}-\frac{T}{2\pi}+O(\log T)$, requiring a less number of zeros in each domain $\overline{D}_m$. The only pole of $\zeta(s)$ is at $s=1$.
Therefore each domain $\overline{D}_m$ contains a finite number of zeros and no pole.
Step 2. Count the number of zeros of $\zeta(s)$ in each domain $D_m$.
Suppose $\rho_1,\rho_2,\dots,\rho_n$ are the finite number of zeros of $\zeta(s)$ in the domain $D_m$. And the punctured domain $D_m\backslash\{\rho_1,\rho_2,\dots,\rho_n\}$ is multiply-connected.
Supposing any $\rho_n\notin \{\sigma+it|\sigma=\frac{1}{2}\}$, we could find a domain symmetric with respect to the critical line in the connected $D_m$, with no zeros on its boundary except $\rho_n$ and $\overline{\rho}_n$. We could change the boundaries near $\rho_n$ and $\overline{\rho}_n$, constructing a pair of type-one simple domains, with one containing $\rho_n$ and the other not. The argument principle required that the logarithmic integrals along the boundaries of this pair of domains differ by $4\pi i$, which made a contradiction with [Lemma \[Res1\]]{}.
Therefore all the zeros of $\zeta(\sigma+it)$ in the domain $D_m$ are on the critical line, and [Lemma \[Res1\]]{} rules the number to be one. In other words, each time $\chi(\frac{1}{2}+it)$ loops once around the origin from $\{1\}$ to $\{1\}$ on the unit circle $S^1$ with increasing $t$, there exists one zero of $\zeta(\frac{1}{2}+it)$.
Step 3. Count the number of zeros of $\zeta(s)$ on the boundary of domain $D_m$.
Let $$D_m^2=D(\phi_2,\phi_1)=\bigcup_{\phi_2<\phi< \phi_1}\gamma_{\phi}$$ where $\phi_1=\pi-2\pi m$ and $\phi_2=-\pi-2\pi m$.
Supposing any $\rho\in\gamma_{\phi}$ where $\phi=-2\pi m$ such that $\zeta(\rho)=0$, then type-two simple domain in $D_m^2$ could be constructed to contain $\rho$. The argument principle required that the logarithmic integral along the boundary of the constructed simple domain to be nonzero, which made a contradiction with [Lemma \[Res2\]]{}.
Therefore no zero of $\zeta(s)$ is on the boundary of domain $D_m$. In other words, $\zeta(s)\ne0$ when $\arg\chi(s)=-2\pi m$.
[Theorem \[RH\]]{} claims that the argument of $\chi(\frac{1}{2}+it)$ determines the distribution of zeros of $\zeta(s)$ on the critical line, which can be estimated by . Roughly speaking, there exists one nontrivial zero when $t+t\log\frac{2\pi}{t}$ decreases by $2\pi$ for sufficiently large $t$.
Conclusion
==========
The chi function and tau function, as a pair of special functions similiar to the exponential function and logarithm function, can be expected to play more roles in the complex analysis than to study Riemann zeta function. And the cross-branch technique is also presented for the analysis of multivalued complex function.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We construct the detection rate for particle detectors moving along non-inertial trajectories and interacting with quantum fields. The detectors described here are characterized by the presence of records of observation throughout their history, so that the detection rate corresponds to directly measurable quantities. This is in contrast to past treatments of detectors, which actually refer to probes, i.e., microscopic systems from which we extract information [*only after*]{} their interaction has been completed. Our treatment incorporates the irreversibility due to the creation of macroscopic records of observation. The key result is a [*real-time*]{} description of particle detection and a rigorously defined time-local probability density function (PDF). The PDF depends on the scale $\sigma$ of the temporal coarse-graining that is necessary for the formation of a macroscopic record. The evaluation of the PDF for Unruh-DeWitt detectors along different types of trajectory shows that only paths with at least one characteristic time-scale much smaller than $\sigma$ lead to appreciable particle detection. Our approach allows for averaging over fast motions and thus predicts a constant detection rate for all fast periodic motions.'
author:
- |
Charis Anastopoulos[^1] and Ntina Savvidou[^2]\
[Department of Physics, University of Patras, 26500 Greece]{}
title: 'Real-time particle-detection probabilities in accelerated macroscopic detectors '
---
Introduction
============
A particle detector moving along a non-inertial trajectory in Minkowski spacetime and interacting with a quantum field records particles, even if the quantum field lies in the vacuum state [@Unruh; @Boyer; @Dewitt]. For trajectories with constant proper acceleration $a$, the detected particles are distributed according to a Planckian spectrum, at the Unruh temperature $T = \frac{a}{2\pi}$. However, the thermal response of the accelerated detectors is only local. As shown in Ref. [@AnSav11], a spatially separated pair of particle detectors with the same proper acceleration $a$ has non-thermal temporal correlations of the detection events.
In Ref. [@AnSav11] we established the importance of a distinction between probes and particle detectors, in any discussion of the response of the quantum vacuum to accelerated motion. In particular, probes are microscopic systems that interact with the quantum field through their accelerating motion. After the interaction has been completed, information can be extracted from a probe through a single measurement. The mathematical description of probes is very similar to that of particle scattering. In contrast, particle detectors are characterized by the macroscopic records of detection events. These macroscopic records are present [*throughout the detectors’ history*]{}, so that a detection rate that corresponds to directly measurable quantities can be defined. Such a description of a particle detector is compatible with the standard use of the term in physics. Our approach is distinguished from most existing studies of the issue which employ the word “detector” for systems that are best described as probes. The difference between probes and detectors is further analyzed in Sec. 2: it leads to a different mathematical description and to different expressions for the particle detection rate.
In this article, we undertake a description of particle detectors moving along general spacetime trajectories, by incorporating explicitly the creation of macroscopic records into physical description. The key result is a [*real-time*]{} description of particle detection and the rigorous construction of the associated probabilities.
To this end, we employ a general method for the construction of Quantum Temporal Probabilities (QTP) for any experimental configuration, that was developed in Ref. [@AnSav12]. The key idea is to distinguish between the roles of time as a parameter to Schrödinger’s equation and as a label of the causal ordering of events [@Sav99; @Sav10]. This important distinction leads to the definition of quantum temporal observables. In particular, we identify the time of a detection event as a coarse-grained quasi-classical variable associated with macroscopic records of observation. Besides the construction of particle detectors, the QTP method has been applied to the time-of-arrival problem [@AnSav12; @AnSav06], to the temporal characterization of tunneling processes [@AnSav08; @AnSav13] and to non-exponential decays [@An08].
In Sec. 3, we present a detailed construction of the detection probability for moving particle detectors using the QTP method. This generalizes the results of Ref. [@AnSav11] derived for the case of constant proper acceleration. The detection probability is sensitive to a time-scale $\sigma$ that characterizes the detector’s degree of coarse-graining.That is, $\sigma$ corresponds to the minimal temporal localization in time of a detection event. The time-scale $\sigma$ is a physical parameter that is determined by a detailed knowledge of the detector’s physics.
The most important feature of the derived probability density is that it is local in time for scales of observation much larger than $\sigma$. This property significantly simplifies the calculation of the detection probability for a large class of spacetime paths. Such calculations are presented in Sec. 4. The most important results are (i) the derivation of the adiabatic approximation and of its corrections and (ii) the demonstration that the detection probability of fast periodic motions is constant.
Finally, in Sec. 5 we summarize our results, discussing in particular their relevance to proposed experiments.
Distinction between probes and detectors
=========================================
In order to explain the difference between a probe and a detector, we consider the most common model employed in this context, the Unruh-DeWitt detector [@Unruh; @Dewitt]. This is a quantum system that moves along a trajectory $X^{\mu}(\tau)$ in Minkowski spacetime and that is characterized by a Hamiltonian $\hat{H}_0$. The Unruh-DeWitt detector is coupled to a massless, free quantum scalar field $\hat{\phi}(x)$ through an interaction Hamiltonian $\hat{H}_{int} = \hat{m} \otimes \hat{\phi}[X(\tau)]$, where $\hat{m}$ is an operator analogue of the dipole moment.
Assuming that the detector is initially at the lowest energy state $| 0 \rangle$ and the field is at the vacuum state $|\Omega \rangle$, the probability $\mbox{Prob}(E, \tau)$ that a measurement of the detector will find energy $E > 0$ [*at proper time $\tau$*]{} is $$\begin{aligned}
\mbox{Prob}(E, \tau) = Tr \left[(\hat{P}_E \otimes 1) (T e^{ -i \int_{\tau_0}^{\tau} ds\hat{H}_{int}(s) }) \left(|0 \rangle \langle 0| \otimes |\Omega \rangle \langle \Omega|\right) (T e^{ -i \int_{\tau_0}^{\tau} ds\hat{H}_{int}(s) })^{\dagger}\right], \label{refbad}\end{aligned}$$ where $\hat{P}_E$ is the projector onto the eigenstates of the detector with energy equal to $E$ and $T e^{ -i \int_{\tau_0}^{\tau} ds\hat{H}_{int}(s) }$ is the time-ordered product.
To first order in perturbation theory, $Te^{- i \int_{\tau_0}^{\tau} ds\hat{H}_{int}(s) } = 1 - i \int_{\tau_0}^{\tau} ds\hat{H}_{int}(s) $ and Eq. (\[refbad\]) becomes $$\begin{aligned}
\mbox{Prob}(E, \tau) &=& \alpha(E) \int_{\tau_0}^{\tau} ds \int_{\tau_0}^{\tau} ds' e^{-iE(s'-s)} W[X(s'), X(s)] \nonumber
\\
&=& 2 \alpha(E) \mbox{Re} \int_{\tau_0}^{\tau} ds' \int_0^{s' - \tau_0} ds e^{-iEs} W[X(s'), X(s' - s)] ,\label{refbad2}\end{aligned}$$ where $\alpha(E) = \langle 0|\hat{m} \hat{P}_E \hat{m}|0 \rangle $ and $W(X',X) = \langle \Omega|\hat{\phi}(X')\hat{\phi}(X)|\Omega\rangle$ is the field’s Wightman’s function. In the derivation of Eq. (\[refbad2\]) one assumes that the detector-field interaction is switched on only for a finite time interval of duration $T$ [@SvSv; @Hig93; @SPad96].
The time derivative of $\mbox{Prob}(E, \tau)$ is often identified with the transition probability $P(E, \tau)$, i.e. with a Probability Density Function (PDF) such that $P(E, \tau) \delta t$ equals with the probability that a transition to energy $E$ occurred at some time within the interval $[\tau, \tau + \delta \tau]$. Hence, one often defines $$\begin{aligned}
P(E, \tau) := \frac{d}{d \tau} Prob(E, \tau) = 2 \alpha(E) \mbox{Re} \int_0^{\tau - \tau_0} ds e^{-iEs} W(s', s' - s) \label{petbad} .\end{aligned}$$ With a suitable regularization of the Wightman’s function, Eq. (\[petbad\]) can be made fully causal [@Schlicht; @Langlois; @LoukoSatz; @Milgrom], i.e., variables at time $\tau$ depend only on the trajectory at times prior to $\tau$. However, Eq. (\[petbad\]) is highly non-local in time; one needs to know the full past history of the detector in order to identify the transition probability at the present moment of time.
There are two problems with the line of reasoning that leads to the consideration of Eq. (\[petbad\]) as a PDF for the detection rate. First, the above definition of a PDF is not probabilistically sound, because the derivative of $\mbox{Prob}(E, \tau)$ with respect to $\tau$ does not, in general, define a probability measure. The time $\tau$ in $Prob(E, \tau)$ is not a random variable (unlike $E$), but a [*parameter*]{} of the probability distribution. Thus, the time derivative of $Prob(E, \tau)$ does not define a PDF; in general, it takes negative values. The fact that $P(E, \tau)$ in Eq. (\[petbad\]) is positive-valued is an artifact of first-order perturbation theory. Second order effects include the relaxation of the detector through particle emission, which render $\mbox{Prob}(E, \tau)$ a decreasing function of $\tau$ at later times [@HuLin]. Thus, negative values of $P(E, \tau)$ appear.
Second, Eq. (\[refbad\]) applies to experimental configurations at which information is extracted from the Unruh-DeWitt detector only once, at time $\tau$. Hence, Eq. (\[petbad\]) does not describe a system that actually [*records*]{} particles at times prior to $\tau$; it purports to define a detection rate for fictitious detection events. In fact, the physical system described by Eq. (\[refbad\]) cannot be described as a detector, in any reasonable use of the word in physics. We usually think of a particle detector (e.g. a photodetector) as a physical system that outputs a time-series of detection events, each event being well localized in time. The quantum mechanical description of the detector ought to take into account the [*irreversible*]{} process of outputting information, namely, the creation of macroscopic records of observation.
However, the system described by Eq. (\[refbad\]) outputs information only once, at time $\tau$, and its time evolution prior to $\tau$ is fully unitary. Such a system is best described as a field [*probe*]{}, in the sense of Bohr and Rosenfeld [@BoRo]: a probe is a microscopic system that interacts with the quantum field; information about the field in incorporated in the probe’s final state and we extract this information through a single measurement. For such a system, Eq. (\[refbad\]) provides a good approximation for times much earlier than the system’s relaxation time $\Gamma^{-1}$.
Another key difference between detectors and probes is that in the former it is possible to determine [*temporal*]{} correlation functions for particle detection, by considering several detectors along different spacetime trajectories [@AnSav11]. Measurements of such correlations (for static detectors) are well established in quantum optics [@WM].
Next, we proceed to the construction of models for particle detectors characterized by macroscopic records of observation.
Macroscopic particle detectors
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In this section, we derive the detection rate of an Unruh-Dewitt detector moving along a general trajectory in Minkowski spacetime. First, we briefly review the QTP method for defining PDFs with respect to detection time. Then, we construct explicitly a model for a macroscopic Unruh-Dewitt detector.
A review of the QTP method
--------------------------
We follow the general methodology for constructing detection probabilities with respect to time, that was developed in Ref. [@AnSav12] (the Quantum Temporal Probabilities method). The reader is referred to this article for a detailed presentation. The key result of Ref. [@AnSav12] is the derivation of a general formula for probabilities associated with the time of an event in a general quantum system. Here, the word “event” refers to a definite and persistent macroscopic record of observation. The event time $t$ is a coarse-grained, quasi-classical parameter associated with such records: it corresponds to the reading of an external classical clock that is simultaneous with the emergence of the record.
Let ${\cal H}$ be the Hilbert space associated with the physical system under consideration; ${\cal H}$ describes the degrees of freedom of microscopic particles and of a macroscopic measurement apparatus. We assume that ${\cal H}$ splits into two subspaces: ${\cal H} = {\cal
H}_+ \oplus {\cal H}_-$. The subspace ${\cal H}_+$ describes the accessible states of the system given that a specific event is realized; the subspace ${\cal H}_-$ is the complement of ${\cal H}_+$. For example, if the quantum event under consideration is a detection of a particle by a macroscopic apparatus, the subspace ${\cal H}_+$ corresponds to all accessible states of the apparatus given that a detection event has been recorded. We denote the projection operator onto ${\cal H}_+$ as $\hat{P}$ and the projector onto ${\cal H}_-$ as $\hat{Q} := 1 - \hat{P}$.
We note that the transitions under consideration are always correlated with the emergence of a macroscopic observable that is recorded as a measurement outcome. In this sense, the transitions considered here are [*irreversible*]{}. Once they occur, and a measurement outcome has been recorded, the further time evolution of the degrees of freedom in the measurement device is irrelevant to the probability of transition.
Once a transition has taken place, the values of a microscopic variable are determined through correlations with a pointer variable of the measurement apparatus. We denote by $\hat{P}_\lambda$ projection operators (or, more generally, positive operators) that correspond to different values $\lambda$ of some physical magnitude. This physical magnitude can be measured only if the quantum event under consideration has occurred. For example, when considering transitions associated with particle detection, the projectors $\hat{P}_\lambda$ may be correlated to properties of the microscopic particle, such as position, momentum and energy. The set of projectors $\hat{P}_\lambda$ is exclusive ($\hat{P}_{\lambda} \hat{P}_{\lambda'} = 0, $ if $\lambda \neq \lambda'$). It is also exhaustive given that the event under consideration has occurred; i.e., $\sum_\lambda \hat{P}_\lambda = \hat{P}$.
We also assume that the system is initially ($t = 0$) prepared at a state $|\psi_0 \rangle \in {\cal H}_-$, and that time evolution is governed by the self-adjoint Hamiltonian operator $\hat{H}$.
In Ref. [@AnSav12], we derived the probability amplitude $| \psi; \lambda, [t_1, t_2] \rangle$ that corresponds to (i) an initial state $|\psi_0\rangle$, (ii) a transition occurring at some instant in the time interval $[t_1, t_2]$ and (iii) a recorded value $\lambda$ for the measured observable:
$$\begin{aligned}
| \psi_0; \lambda, [t_1, t_2] \rangle = - i e^{- i \hat{H}T}
\int_{t_1}^{t_2} d t \hat{C}(\lambda, t) |\psi_0 \rangle.
\label{ampl5}\end{aligned}$$
where the [*class operator*]{} $\hat{C}(\lambda, t)$ is defined as $$\begin{aligned}
\hat{C}(\lambda, t) = e^{i \hat{H}t} \hat{P}_{\lambda} \hat{H}
\hat{S}_t, \label{class}\end{aligned}$$ and $\hat{S}_t = \lim_{N \rightarrow \infty}
(\hat{Q}e^{-i\hat{H} t/N} \hat{Q})^N$ is the restriction of the propagator in ${\cal H}_-$. The parameter $T$ in Eq. (\[ampl5\]) is a reference time-scale at which the amplitude is evaluated. It defines an upper limit to $t$ and it corresponds to the duration of an experiment. It cancels out when evaluating probabilities, so it does not appear in the physical predictions.
If $[\hat{P}, \hat{H}] = 0$, i.e., if the Hamiltonian evolution preserves the subspaces ${\cal H}_{\pm}$, then $|\psi_0;
\lambda, t \rangle = 0$. For a Hamiltonian of the form $\hat{H} = \hat{H_0} + \hat{H_I}$, where $[\hat{H}_0, \hat{P}] = 0$, and $H_I$ a perturbing interaction, we obtain $$\begin{aligned}
\hat{C}(\lambda, t) = e^{i \hat{H}_0t} \hat{P}_{\lambda} \hat{H}_I
e^{-i \hat{H}_0t}, \label{perturbed}\end{aligned}$$ to leading order in the perturbation.
The benefit of Eq. (\[perturbed\]) is that it does not involve the restricted propagator $\hat{S}_t$, which is difficult to compute.
The amplitude Eq. (\[ampl5\]) squared defines the probability $\mbox{Prob} (\lambda, [t_1, t_2])\/$ that at some time in the interval $[t_1, t_2]$ a detection with outcome $\lambda$ occurred $$\begin{aligned}
\mbox{Prob}(\lambda, [t_1, t_2]) := \langle \psi_0; \lambda, [t_1, t_2] | \psi_0;
\lambda, [t_1, t_2] \rangle = \int_{t_1}^{t_2} \, dt
\, \int_{t_1}^{t_2} dt' \; Tr [\hat{C}(\lambda, t) \hat{\rho}_0
\hat{C}^{\dagger}(\lambda, t) ], \label{prob1}\end{aligned}$$ where $\hat{\rho}_0 = |\psi_0\rangle \langle \psi_0|$.
However, $\mbox{Prob}(\lambda, [t_1, t_2])$ does not correspond to a well-defined probability measure because it fails to satisfy the Kolmogorov additivity condition for probability measures $$\begin{aligned}
\mbox{Prob}(\lambda, [t_1, t_3]) = \mbox{Prob}(\lambda, [t_1, t_2]) + \mbox{Prob}(\lambda, [t_2,
t_3]). \label{kolmogorov}\end{aligned}$$
Eq. (\[kolmogorov\]) does not hold for generic choices of $t_1, t_2$ and $t_3$. The key point is that in a macroscopic system (or in a system with a macroscopic component) one expects that Eq. (\[kolmogorov\]) holds with a good degree of approximation, given a sufficient degree of coarse-graining [@Omn; @Omn2; @Gri; @GeHa93; @hartlelo]. Thus, if the time of transition is associated with a macroscopic measurement record, there exists a coarse-graining time-scale $\sigma$, such that Eq. (\[kolmogorov\]) holds, for $ |t_2 - t_1| >> \sigma$ and $|t_3 - t_2| >> \sigma$. Then, Eq. (\[prob1\]) does define a probability measure when restricted to intervals of size larger than $\sigma$.
It is convenient to proceed by smearing the amplitudes Eq. (\[ampl5\]) at a time-scale of order $\sigma$. To this end, we introduce a family of probability density functions $f_{\sigma}(s)$, localized around $s = 0$ with width $\sigma$, and normalized so that $\lim_{\sigma \rightarrow 0} f_{\sigma}(s) = \delta(s)$. The only requirement is that $f_{\sigma}$ satisfies approximately the equality
$$\begin{aligned}
\sqrt{f_{\sigma}(t-s) f_{\sigma}(t-s')} = f_{\sigma}(t - \frac{s+s'}{2}) g_{\sigma}(s-s'), \label{eq2}\end{aligned}$$
where $w_{\sigma}(s)$ is a function strongly localized around $s = 0$. From Eq. (\[eq2\]), it trivially follows that $w_{\sigma}(0) = 1$. As an example, we note that the Gaussians $f_{\sigma}(s) = (2 \pi \sigma^2)^{-1/2}
\exp\left[-\frac{s^2}{2\sigma^2}\right]$ satisfy Eq. (\[eq2\]) [*exactly*]{}, with $w_{\sigma}(s) = \exp[-s^2/(8\sigma^2)]$.
We define the smeared amplitude $|\psi_0; \lambda, t\rangle_{\sigma}$ as $$\begin{aligned}
|\psi_0; \lambda, t\rangle_{\sigma} := \int ds \sqrt{f_{\sigma}(s -t)}
|\psi_0; \lambda, s \rangle = \hat{C}_{\sigma}(\lambda, t) |\psi_0
\rangle, \label{smearing}\end{aligned}$$ where $$\begin{aligned}
\hat{C}_{\sigma}(\lambda, t) := \int ds \sqrt{f_{\sigma}(t - s)}
\hat{C}(\lambda, s).\end{aligned}$$ The square amplitudes $$\begin{aligned}
\bar{P}(\lambda, t) = {}_{\sigma}\langle \psi_0; \lambda, t|\psi_0;
\lambda, t\rangle_{\sigma} = Tr \left[\hat{C}^{\dagger}_{\sigma}(\lambda, t)
\hat{\rho}_0 \hat{C}_{\sigma}(\lambda, t)\right] \label{ampl6}\end{aligned}$$ define a PDF with respect to the time $t$ that corresponds to a Positive-Operator-Valued-Measure.
Using Eq. (\[eq2\]), the PDF Eq. (\[ampl6\]) becomes $$\begin{aligned}
\bar{P}(\lambda,t) = \int ds f_{\sigma}(t - s) P(\lambda, s), \label{conv}\end{aligned}$$ where $$\begin{aligned}
P(\lambda, t) = \int ds w_{\sigma}(s) Tr \left[\hat{C}^{\dagger}_{\sigma}(\lambda, t+ \frac{s}{2})
\hat{\rho}_0 \hat{C}_{\sigma}(\lambda, t - \frac{s}{2})\right]. \label{probmain}\end{aligned}$$ This means that the PDF $\bar{P}(\lambda, s)$ is the convolution of $P(\lambda, s)$ with weight corresponding to $f_{\sigma}$. For scales of observation much larger than $\sigma$ the difference between $P(\lambda, s)$ and $\bar{P}(\lambda, s)$ is insignificant. Hence, it is usually sufficient to work with the deconvoluted form of the probability density, Eq. (\[probmain\]), which is simpler. An important exception is Sec. 3.5. Note also that other sources of measurement error (for example, environmental noise) lead to further smearing of the PDF Eq. (\[conv\]). In this sense, the PDF $P(\lambda, t)$ corresponds to the ideal case that (i) any external classical noise vanishes and (ii) the scale of observation is much larger than the temporal coarse-graining scale $\sigma$.
Modeling an Unruh-DeWitt detector
---------------------------------
Next, we employ Eq. (\[probmain\]) for constructing a PDF with respect to time for particle detection along non-inertial trajectories. The system under consideration consists of a quantum field $\hat{\phi}$ that describes microscopic particles and a macroscopic particle detector in motion. The Hilbert space for the total system is the tensor product ${\cal F} \otimes {\cal H}_d$. The Hilbert space ${\cal H}_d$ describes the detector’s degrees of freedom and the Fock space ${\cal F}$ is associated with the field $\hat{\phi}$. Here, we take $\hat{\phi}$ to be a massless scalar field in Minkowski spacetime but the methodology can be easily adapted to more elaborate set-ups.
We assume that the detector is sufficiently small so that its motion is described by a single spacetime path $X^{\mu}(\tau)$, where $\tau$ is the proper time along the path. An important assumption is that the physics at the detector’s rest frame is independent of the path followed by the detector: the evolution operator for an Unruh-DeWitt detector is $e^{-i\hat{H}_d\tau}$, where $\tau$ is the proper time along the detector’s path and $\hat{H}_d$ is the Hamiltonian for a stationary detector. Thus, path-dependence enters into the propagator only through the proper time. For a detector that follows a timelike path $X^{\mu}(\tau)$ in Minkowski spacetime, the equation $X^0(\tau) = t$ can be solved for $\tau$, in order to determine the proper time $\tau(t)$ as a function of the inertial time coordinate $t$ of Minkowski spacetime. The evolution operator for the detector with respect to the coordinate time $t$ is then $e^{-i\hat{H}_d\tau(t)}$.
We model a detector that records the energy of field excitations. The relevant transitions correspond to changes in the detector’s energy, as it absorbs a field excitation (particle). We assume that the initially prepared state of the detector $|\Psi_0 \rangle$ is a state of minimum energy $E_0 = 0$, and that the excited states have energies $E > E_0$. We denote the projectors onto the constant energy subspaces as $\hat{P}_E$. Thus, the projectors $\hat{P}_{\pm}$ on ${\cal H}$ corresponding to the transitions are $\hat{P}_+ = \left(\sum_{E>E_0} \hat{P}_E \right) \otimes \hat{1}$ and $\hat{P}_- = \hat{P}_0 \otimes \hat{1}$, where $\hat{P}_0 = \hat{P}_{E_0}$.
The energy of an absorbed particle is determined within an uncertainty $\Delta E << E_0$ intrinsic to the detector. This measurement is modeled by a family of positive operators on ${\cal H}_d$ $$\begin{aligned}
\hat{\Pi}_E = \int d\bar{E} F_{\Delta E} (E- \bar{E}) \hat{P}_{\bar{E}},
\end{aligned}$$ where $F_{\Delta E}$ is a smearing function of width $\Delta E$.
The scalar field Hamiltonian is $\hat{H}_{\phi} = \int d^3 x \left( \frac{1}{2} \hat{\pi}^2 + \frac{1}{2} (\nabla \phi)^2\right)$. The evolution operator for the field with respect to a Minkowskian inertial time coordinate $t$ is $e^{- i \hat{H}_{\phi}t}$, where $t$ is a Minkowski inertial time parameter.
We also consider a local interaction Hamiltonian $$\begin{aligned}
\hat{H}_{I} = \int dx \hat{\phi}(x) \otimes \hat{J}(x), \label{hint}\end{aligned}$$ where $\hat{J}({\bf x})$ is a current operator on the Hilbert ${\cal H}_a$ of the detector.
Then Eq. (\[probmain\]) yields $$\begin{aligned}
P(E, \tau) = \int dy w_{\sigma}(y) \int ds ds' \int d^3x d^3x' Tr\left[ \hat{\phi}(x', s') \hat{\phi}(x, s) \hat{\rho}_0 \right]
\delta [s - X^0(\tau + \frac{y}{2})] \delta [s' - X^0(\tau - \frac{y}{2})] \nonumber \\
\times \langle \Psi_0|\hat{J}(x',\tau -\frac{y}{2}) \sqrt{\hat{\Pi}_E} e^{-i \hat{H} y} \sqrt{\hat{\Pi}_E} \hat{J}(x,\tau +\frac{y}{2})|\Psi_0 \rangle, \label{pet3}\end{aligned}$$ where $\hat{\rho}_0$ is the initial state of the field, $\hat{\phi}(x, s) = e^{i \hat{H}_0s} \hat{\phi}(x) e^{-i \hat{H}_0s}$ and $\hat{J}(x, \tau) = e^{i \hat{H}_d \tau} \hat{J}(x) e^{-i \hat{H}_d \tau}$ are the Heisenberg-picture operators for the scalar field and the current respectively.
A defining feature of an Unruh-Dewitt detector is that it is point-like, i.e., effects that are related to its finite size do not enter the detection probability. In the present context, this condition is implemented through the approximation $$\begin{aligned}
\hat{J}(x, \tau) \simeq \hat{m} \delta^3 [x - X(\tau)], \label{UdW}\end{aligned}$$ where $\hat{m}$ is an averaged current operator, corresponding to the analogue of the dipole moment for a scalar field.
Using Eq. (\[UdW\]), the detection probability Eq. (\[pet3\]) simplifies $$\begin{aligned}
P(E, \tau) = \alpha(E) \int dy g_{\sigma}(y) e^{-iEy} W[X(\tau+\frac{y}{2}), X(\tau - \frac{y}{2})], \label{petb}\end{aligned}$$
where $$\begin{aligned}
\alpha(E) = Tr_{{\cal H}_d}(\hat{P}_E\hat{m} \hat{P}_0\hat{m})/Tr_{{\cal H}_d} \hat{P}_0, \label{ae}\end{aligned}$$ and $$\begin{aligned}
W(X, X') = Tr\left[ \hat{\phi}(X') \hat{\phi}(X) \hat{\rho}_0 \right]\end{aligned}$$ is the positive-frequency Wightman function. When the field is in the vacuum state, $$\begin{aligned}
W(X, X') = \frac{-1}{4\pi^2 [(X^0-X^{0'}-i \epsilon)^2 - ({\bf X} - {\bf X'})^2]},\end{aligned}$$ where $\epsilon >0$ is the usual regularization parameter. In our approach the issue of carefully regularizing the Wightman function in order to preserve causality [@Schlicht; @Tagaki] does not arise. The detection probability is by construction local in time and causal within an accuracy defined by the temporal coarse-graining. In particular, the value of the regularization parameter $\epsilon$ is irrelevant to the physical predictions. The only use of the term $- i \epsilon$ is to guarantee that poles of $y$ along the real axis do not contribute to the integral Eq. (\[petb\]).
Thus, the PDF for particle detection in the vacuum becomes $$\begin{aligned}
P(E, \tau) = - \frac{\alpha(E)}{4 \pi^2} \int_{-i\epsilon - \infty}^{-i \epsilon + \infty} dy \frac{ g_{\sigma}(y) e^{-i Ey}}{ \Sigma(\tau, y)}, \label{pet5}
\end{aligned}$$ where $$\begin{aligned}
\Sigma(\tau, y) = \eta_{\mu \nu} [X^{\mu}(\tau + \frac{y}{2}) - X^{\mu}(\tau - \frac{y}{2})] [X^{\nu}(\tau + \frac{y}{2}) - X^{\nu}(\tau - \frac{y}{2})] \label{sigma0}
\end{aligned}$$ is the proper distance between two points on the detector’s path characterized by values of the proper time $\tau \pm y/2$.
The function $g_{\sigma}(y)$ in Eq. (\[petb\]) is defined as $$\begin{aligned}
g_{\sigma}(y) = w_{\sigma}(y) \tilde{F}_{\Delta E}(y) , \label{gs}\end{aligned}$$ where $\tilde{F}_{\Delta E}(y) = \int dE F_{\Delta E}(E) e^{-i Ey}$. The term $\tilde{F}_{\Delta E}$ suppresses temporal interferences between amplitudes Eq. (\[ampl5\]) at timescales larger than $(\Delta E)^{-1}$. Hence, the temporal coarse-graining parameter $\sigma$ must be of order $(\Delta E)^{-1}$ or higher for Eq. (\[kolmogorov\]) to be satisfied. Since $\Delta E << E_0 <E$, this implies that $$\begin{aligned}
E \sigma >> 1. \label{fund}\end{aligned}$$ Eq. (\[fund\]) is a defining condition for a physically meaningful particle detector and it specifies the values of energy $E$ to which Eq. (\[petb\]) applies.
The term $g_{\sigma}(y)$ in the detection probability Eq. (\[petb\]) is of extreme importance. This term guarantees that the detector’s response to changes in its state of motion is causal and approximately local in time at macroscopic scales of observation. To see this, we note that $g_{\sigma}(y)$ truncates contributions to the detection probability from all instants $s$ and $s'$ such that $|s-s'|$ is substantially larger than $\sigma$. Thus, at each moment of time $\tau$, the detector’s response is determined solely from properties of the path at times around $\tau$ with a width of order $\sigma$. In particular, properties of the path at the asymptotic past (or future) do not affect the detector’s response at time $\tau$. The response is determined solely by the properties of the path at time $\tau$, within the accuracy allowed by the detector’s temporal resolution.
[*Finite-size detectors.*]{} The approximation Eq. (\[UdW\]) simplifies the PDF for particle detection significantly. Eq. (\[UdW\]) implies that the detection events are sharply localized in space, while their time localization is of order $\sigma$. A more realistic model requires the evaluation of Eq. (\[pet3\]) without the simplifying assumption Eq. (\[UdW\]). Assuming that the dimensions of the detector are small in relation to the wave-length of the absorbed quanta, a meaningful approximation for the current operator is
$$\begin{aligned}
\hat{J}(x, \tau) \simeq \hat{m} u_{\delta}[x - X^i(\tau)], \label{UdW2}\end{aligned}$$
where $u_{\delta}(x)$ is a function localized around $x = 0 $ with a spread equal to $\delta$. In this approximation, a detector is characterized by two scales: a spatial coarse-graining scale $\delta$ and a temporal coarse-graining scale $\sigma$. If $\sigma >> \delta$, the contribution of the temporal coarse-graining dominates and Eq. (\[pet5\]) follows. This is the physically most relevant regime, because as we show in Sec. 3, the unless the coarse-graining parameter $\sigma$ is sufficiently large, the detection rate is strongly suppressed.
Detection probabilities at different regimes
============================================
In this section, we evaluate the detection probability Eq. (\[petb\]) for different trajectories. The cases presented here are chosen in order to clarify the physical interpretation of the coarse-graining scale, and to illustrate various techniques. An important result is the derivation of the adiabatic approximations and of its corrections (Sec. 4.4). Furthermore, we demonstrate that the effective detection probability of fast periodic motions is constant (Sec. 4.5).
General expressions
-------------------
In order to evaluate the PDF Eq. (\[pet5\]) we must first specify the function $g_{\sigma}(x)$. As a matter of fact, the explicit form of $g_{\sigma}$ is of little physical significance because $\sigma$ is a time-scale rather than a sharply defined time-parameter. The most interesting physical predictions correspond to the asymptotic regimes with respect to $\sigma$.
We consider smearing functions $g_{\sigma}$ that satisfy the following conditions
1. $g_{\sigma}(0) = 1$ and this is the single local maximum of the function.
2. $\lim_{\sigma \rightarrow 0} g_{\sigma} = 0$; $\lim_{\sigma \rightarrow \infty} g_{\sigma} = 1$.
3. $g_{\sigma}(y)$ drops to zero for $ y >> \sigma$.
4. In the lower half of the imaginary plane ${\bf C}$, $g_{\sigma}(y)$ is meromorphic and vanishes as $y \rightarrow \infty$.
Conditions 1-3 follow from the definition of $g_{\sigma}$. Condition 4 is a technical assumption that is convenient for calculational purposes. In what follows, we will employ the Lorentzian $$\begin{aligned}
g_{\sigma}(y) = \frac{1}{\left(\frac{y}{\sigma}\right)^2 +1}, \label{lor}
\end{aligned}$$ whenever an explicit form of $g_{\sigma}$ is needed.
Let $y =-i w_n(\tau)$ be the solutions of equation $\Sigma(\tau, y) = 0$ for fixed $\tau$, where $\Sigma(\tau, y)$ is given by Eq. (\[sigma0\]). We restrict to solutions that lie in the lower half of the imaginary plane (for $Re w_n(\tau) >0 $), because in this half-plane $e^{-iEy}$ vanishes as $y$ goes to complex infinity. This is the reason for the assumption 4 above.
We assume that the solutions are simple, and we choose the index $n$ so that the solutions are indexed by their modulus: if $|w_n(t)| > |w_m(\tau)|$ then $n > m$. Then, using Cauchy’s theorem, Eq. (\[pet5\]) becomes $$\begin{aligned}
P(E, \tau) = \frac{\alpha(E)}{2 \pi} \left(i \sum_n \left[\left(\frac{w_n(\tau)}{\sigma}\right)^2 + 1\right]^{-1} \frac{e^{-E w_n(\tau)}}{\partial_y \Sigma[\tau, -iw_n(\tau)]} - \frac{\sigma e^{-E \sigma}}{2 \Sigma(\tau, - i \sigma) } \right), \label{sumsol}
\end{aligned}$$ where Eq. (\[lor\]) was employed.
The modulus of each term in the sum over $n$ in Eq. (\[sumsol\]) is suppressed by a factor of $(\sigma/|w_n|)^2$. Hence, poles with modulus $|w_n| >> \sigma$ contribute little to the detection probability.
Motion along inertial trajectories
----------------------------------
For any straight-line timelike path on Minkowski spacetime we have $\Sigma(\tau, y) = y^2$. Hence, there is no pole in the lower half of the complex plane, and the detection probability Eq. (\[petb\]) becomes $$\begin{aligned}
P(E, \tau) = \frac{\alpha(E) e^{-E \sigma}}{4 \pi \sigma}.\end{aligned}$$
Since $E \sigma >> 1$, the detection probability is strongly suppressed, as expected. It is non-zero, because any localized measurement in relativistic systems is subject to false alarms [@PerTer], i.e., spurious detection events.
Uniform proper acceleration
---------------------------
Next, we consider a detector under uniform proper acceleration $a$, with a trajectory $$\begin{aligned}
X^{\mu}(\tau) = \frac{1}{a}(\sinh(a\tau), \cos(a\tau) - 1, 0, 0).\end{aligned}$$ For this path, the function $$\begin{aligned}
\Sigma(\tau, y) = \frac{4 }{a^2} \sinh^2\left(\frac{ay}{2}\right) \label{sigma1}
\end{aligned}$$ has double zeros at $y = - i \frac{2 \pi}{a} n$, for $n = 1, 2, \ldots$.
The detection probability Eq. (\[pet5\]) becomes $$\begin{aligned}
P(E, \tau) = \frac{\alpha(E)}{2 \pi \sigma} \left\{ \frac{E}{T_a} x \left[\frac{x}{2}\left(\Phi_1(e^{-\frac{E}{T_a}}, x)- \Phi_1(e^{-\frac{E}{T_a}}, -x)\right) -1 \right] \right. \nonumber \\
\left.
+ \frac{x^2}{2} \left(\Phi_2(e^{-\frac{E}{T_a}}, x)- \Phi_2(e^{-\frac{E}{T_a}}, -x)\right) + \frac{\pi^2 x^2 e^{-x\frac{E}{T_a}}}{2 \sin^2(\pi x)} \right\}, \label{pet6}\end{aligned}$$ where $T_a := \frac{a}{2 \pi}$ is the [*acceleration temperature*]{}, $x := T_a \sigma$ and $\Phi_s(z, x)$ is the Lerch-Hurwitz function [@GR] defined by $$\begin{aligned}
\Phi_s(z, x) = \sum_{n = 0}^{\infty} \frac{z^n}{(n + x)^s}.
\end{aligned}$$
For $x >> 1$, the asymptotic expansion of Lerch-Hurwitz functions is relevant [@Ferreira] $$\begin{aligned}
\Phi_1(z, x) = \frac{1}{1 -z} \frac{1}{x} + \sum_{n = 1}^{\infty} \frac{(-1)^n L_n(z)}{x^{n+1}} \\
\Phi_2(z, x) = \frac{1}{1 -z} \frac{1}{x^2} + \sum_{n = 1}^{\infty} \frac{(-1)^n (n+1) L_n(z)}{x^{n+2}},
\end{aligned}$$ where $L_n(z)$ is defined by the recursion relation $L_0(z) = \frac{1}{1 - z}$, $L_n(z) = z \frac{d}{dz} L_{n-1}(z)$. We obtain
$$\begin{aligned}
P(E, \tau) = \frac{\alpha(E) T_a}{2 \pi} \left[ \frac{E/T_a }{e^{\frac{E}{T_a}}-1} + \sum_{k = 1}^{\infty} \frac{L_{2k}(e^{-\frac{E}{T_a}}) (E/T_a) - k L_{2k-1}(e^{-\frac{E}{T_a}}) }{x^{2k}} \right], \label{exp}\end{aligned}$$
where we dropped the last term in Eq. (\[pet6\]). Eq. (\[exp\]) is valid for energies such that $E/T_a >> x^{-1}$, which is the physically relevant regime corresponding to $E \sigma >> 1$.
For $x \rightarrow \infty$, we recover the standard Planck-spectrum response. Including the leading correction we obtain $$\begin{aligned}
P(E, \tau) = \frac{\alpha(E) E}{2 \pi\left(e^{\frac{E}{T_a}}-1\right)} \left[1 + \frac{1}{x^2} \frac{2 - T_a/E + e^{-\frac{E}{T_a}}(1+ T_a/E)}{\left(1 - e^{-\frac{E}{T_a}}\right)^2} + \ldots\right]. \label{corrections}
\end{aligned}$$
In Fig. 1, we plot the relative size $C$ of the correction terms in Eq. (\[exp\]), as a function of $E/T_a$ and for different values of $x$. $C$ is defined as the absolute value of the second term in the parenthesis of Eq. (\[exp\]), modulo the first term in the parenthesis (that corresponds to the Planckian spectrum). We see that the correction to the Planckian spectrum is stronger at low energies but in any case small for $\sigma a $ of order $10^1$.
![ The relative size $C$ of the correction terms to the Planckian spectrum in Eq. (\[exp\]) as a function of $E/T_a$, for different values of $x$: (a) $x = 25$, (b) $x = 50$, (c) $x = 100$. The correction term is stronger at low energies; however one has to keep in mind the condition $E/T_a >> x^{-1}$ for energy measurements. $C$ is defined as the absolute value of the second term, modulo the first term in the parenthesis of Eq. (\[exp\]).](unruh1.eps){height="7cm"}
If $x$ is of the order of unity or smaller then only energies $E$ such that $E >> T_a$ contribute to the detection probability. In this regime, the detection probability is exponentially suppressed for all measurable values of energy.
We conclude that only in the regime $\sigma a >> 1$ is there a significant particle detection rate, and in this regime the rate is well approximated by the Planckian spectrum expression.
Motion along a single axis
--------------------------
Next, we consider a general path along a single Cartesian axis in Minkowski spacetime. The four-velocity is $$\begin{aligned}
\dot{X}^{\mu}(\tau) = (\cosh b(\tau), \sinh b(\tau), 0 ,0), \label{1dim}
\end{aligned}$$ for some function $b(\tau)$; $\alpha(\tau) = \dot{b}(\tau) $ is the proper acceleration of the path. For this path, we find $$\begin{aligned}
\Sigma(\tau, y) = \left( \int_{-y/2}^{y/2} ds e^{b(\tau + s)}\right)\left( \int_{-y/2}^{y/2} ds' e^{-b(\tau + s')}\right). \label{sigma2a}
\end{aligned}$$
### Adiabatic approximation.
The presence of the factor $g_{\sigma}$ in Eq. (\[petb\]) implies that values of $y >> \sigma$ are suppressed. Hence, unless $b(\tau)$ varies too strongly in the scale of $\sigma$, the Taylor expansion of $b(\tau + s)$ around $b(\tau)$ in Eq. (\[sigma2a\]) is a meaningful approximation.
The first order in the Taylor expansion $b(\tau + s) = b(\tau) + s a_{\tau}$ corresponds to the adiabatic approximation. Writing $a_t = \dot{b}(t)$, we obtain $\Sigma(\tau, y) = 4 \sin^2[a_{\tau}y/2]/a^2_{\tau}$. Hence, $$\begin{aligned}
P(E, \tau) = \frac{\alpha(E) E}{2 \pi\left(e^{\frac{2 \pi E}{a_{\tau}}}-1\right)} + O \left( \frac{1}{[\sigma a_{\tau}]^2}\right), \label{timed}\end{aligned}$$ i.e., we obtain a Planckian spectrum with a time-dependent temperature.
### Corrections to the adiabatic approximation.
Eq. (\[timed\]) is a good approximation for paths such that $ |\dot{a}_{\tau}| \sigma << |a_{\tau}|$. In order to calculate the first corrections to Eq. (\[timed\]), we expand $b(\tau +s)$ to second order in $s$, writing $b(\tau +s) = b(\tau) + s a_{\tau} + \frac{1}{2} s^2 \dot{a}_{\tau}$. We obtain $$\begin{aligned}
\Sigma(\tau, y) = \frac{\pi}{\dot{a}_{\tau}} \left\{ \mbox{erfi}\left[\sqrt{\frac{\dot{a}_{\tau}}{2}}\left(\frac{a_{\tau}}{\dot{a}_{\tau}} + \frac{y}{2}\right) \right] - \mbox{erfi}\left[\sqrt{\frac{\dot{a}_{\tau}}{2}}\left(\frac{a_{\tau}}{\dot{a}_{\tau}} - \frac{y}{2}\right) \right] \right\} \nonumber \\
\times \left\{ \mbox{erf}\left[\sqrt{\frac{\dot{a}_{\tau}}{2}}\left(\frac{a_{\tau}}{\dot{a}_{\tau}} + \frac{y}{2}\right) \right] - \mbox{erf}\left[\sqrt{\frac{\dot{a}_{\tau}}{2}}\left(\frac{a_{\tau}}{\dot{a}_{\tau}} - \frac{y}{2}\right) \right] \right\}, \label{sigma2}\end{aligned}$$ where $\mbox{erf}(x) = \frac{2}{\sqrt{\pi}} \int_0^x e^{-t^2} dt$ is the error function and $\mbox{erfi}(x) = -i \mbox{erf}(ix)$.
For $|\dot{a}_{\tau}y/ a_{\tau}| << 1$, the asymptotic regime for the error function $$\begin{aligned}
\mbox{erf} \simeq 1 - \frac{e^{-x^2}}{\sqrt{\pi} x} \label{erfa}
\end{aligned}$$ is relevant. Substituting Eq. (\[erfa\]) in Eq. (\[sigma2\]) we find $$\begin{aligned}
\Sigma(\tau, y) = \frac{4}{a_{\tau}^2} \left[ \sinh\left(\frac{a_{\tau}y}{2}\right) - \frac{\dot{a}_{\tau} y}{2 a_{\tau}} \cosh\left(\frac{a_{\tau}y}{2}\right)\right]\left[ \sinh\left(\frac{a_{\tau}y}{2}\right) + \frac{\dot{a}_{\tau} y}{2 a_{\tau}} \cosh\left(\frac{a_{\tau}y}{2}\right)\right], \label{sigma3}
\end{aligned}$$ to leading order in $|\dot{a}_ty/a_t|$.
We expect that the roots $w_n$ of Eq. (\[sigma3\]) are small corrections to the roots obtained for $\dot{a}_t = 0$. Hence, we write $w_n = -i \frac{2 \pi}{a} n + \epsilon_n$, where $|\epsilon_n|<< \frac{2 \pi}{a} n$. The roots approximately correspond to solutions of the equation $$\begin{aligned}
\tanh(a\epsilon_n/2) = \pm i n \frac{\pi \dot{a}_{\tau}}{a_{\tau}^2}. \label{tanhe}
\end{aligned}$$
According to Eq. (\[sumsol\]), the only roots that contribute significantly to the detection probability are characterized by $n < N_{max}$ where $N_{max} \sim \sigma a_{\tau}/(2 \pi)$. For these values of $n$, the modulus of the right-hand-side of Eq. (\[tanhe\]) is smaller than $| \dot{a}_{\tau} \sigma/a_{\tau}| <<1$. Hence, we can approximate the left-hand-side of Eq. (\[tanhe\]) by setting $\tanh(x) \simeq x$. We obtain, $\epsilon_n = \pm i n 2\pi \dot{a}_{\tau}/a_{\tau}^3$. It follows that
$$\begin{aligned}
w_n(\tau) = - i \frac{2 \pi}{a_{\tau}} \left( 1 \pm \frac{\dot{a}_{\tau}}{a_{\tau}^2} \right) n, \hspace{1cm} n = 1, 2, \ldots, N_{max}.
\end{aligned}$$
Variations in the acceleration results to a split of each double root of Eq. (\[sigma1\]) into two single roots for Eq. (\[sigma3\]).
With the roots above we evaluate the PDF for particle detection $$\begin{aligned}
p(E, \tau) = \frac{\alpha(E) a_{\tau}}{8 \pi^2 \delta_{\tau}} \sum_{n = 1}^{N_{max}} \frac{e^{-\frac{2 \pi E}{a_{\tau}}n} }{n} \left( e^{\frac{2 \pi \delta_{\tau}E}{a_{\tau}}n} - e^{-\frac{2 \pi \delta_{\tau}E}{a_{\tau}}n}\right), \label{pet8}
\end{aligned}$$ where we wrote $\delta_{\tau} = \dot{a}_{\tau}/a_{\tau}^2$.
We extend the summation in Eq. (\[pet8\]) to infinity, as terms of $n > N_{max}$ affect only low values of energy that are physically irrelevant (they fail to satisfy the condition $E \sigma >> 1$). We obtain $$\begin{aligned}
p(E, \tau) = \frac{\alpha(E) a_{\tau}}{8 \pi^2 \delta_{\tau}} \left[ g_1\left( \frac{2 \pi}{a_{\tau}} ( 1 - \delta_{\tau})\right)- g_1\left( \frac{2 \pi}{a_{\tau}} ( 1 + \delta_{\tau})\right) \right], \label{petg}\end{aligned}$$ where $g_1(x) = \sum_{n=1}^{\infty}\frac{e^{-nx}}{n}$.
Eq. (\[petg\]) applies to all energies $E$, such that $E\sigma >> 1$. If we exclude very high energies of order $a_{\tau}/\delta_{\tau}$ we can expand the function $g_1$ with respect to $\delta_{\tau}$. Thus, we obtain the leading correction to the Planckian spectrum due to small variations in the acceleration $$\begin{aligned}
P(E, \tau) = \frac{\alpha(E) E}{2 \pi\left(e^{\frac{2 \pi E}{a_{\tau}}}-1\right)} \left[1 + \frac{2 \pi^2 \dot{a}_{\tau}^2 E^2}{3 a_{\tau}^6} \frac{ 1 + e^{-\frac{2 \pi E}{a_{\tau}}}}{\left(1 - e^{-\frac{2 \pi E}{a_{\tau}}}\right)^2} \right]+ O \left( \frac{1}{[\sigma a_{\tau}]^2} , \frac{\dot{a}_{\tau}\sigma}{a_{\tau}}\right). \label{pet11}\end{aligned}$$
Note that Eq. (\[pet11\]) applies in the regime where both $\sigma a_{\tau} >> 1$ and $|\dot{a}_{\tau} \sigma/a_{\tau}| << 1$. The introduction of $\sigma$ is essential for the separation of timescales leading to the probability density Eq. (\[pet11\]), even though $\sigma$ does not explicitly appear in the equation.
### Non-relativistic limit
In the non-relativistic regime, $b(\tau)$ coincides with the non-relativistic velocity $v_{\tau}$ and satisfies $|v(\tau)|<<1$. We write the exponentials of Eq. (\[sigma2a\]) as $e^{\pm b} \simeq 1 \pm v$ and we Taylor-expand $b(\tau +s)$ around $\tau$. If the coarse-graining time-scale $\sigma$ is sufficiently large then a finite number of terms in the Taylor expansion suffices. Thus, $\Sigma(\tau, y)$ becomes a product of two polynomials, and the full set of its roots can be easily determined.
To the lowest-non trivial order, $$\begin{aligned}
\Sigma(\tau, y) = y^2 \left( 1 + v_{\tau} + \frac{\dot{a}_{\tau}}{24} y^2 \right) \left( 1 - v_{\tau} - \frac{\dot{a}_{\tau}}{24} y^2 \right). \label{sigma6}\end{aligned}$$ The only root of $\Sigma(\tau, y)$ in the lower half of the complex plane is $w_1 = -2 i \sqrt{6(1 + v_{\tau})/|\dot{a}_{\tau}|} \simeq -2 i \sqrt{6/|\dot{a}_{\tau}|}$. However, the Taylor-expansion of $v(\tau + s)$ preserve the condition $|v_{\tau}| << 1$ only if $|\dot{a}_{\tau}|\sigma^2 << 1$. This implies that $|w_1|>> \sigma$, and hence, that the contribution of $w_1$ to the detection probability is suppressed. Thus, the detection probability effectively vanishes.
The conclusion above is not affected when keeping higher order terms in the Taylor expansion of $v(\tau+s)$; the relevant roots of Eq. (\[sigma6\]) are much larger in norm than $\sigma$ because the velocity is restricted in the non-relativistic regime. Hence, particle detection vanishes in the non-relativistic regime. The only possible exceptions are paths that are characterized by rapid variations of their velocity at scales much smaller than $\sigma$. For such paths, any finite number of terms in the Taylor expansion of $v(\tau + s)$ provides an inadequate approximation. Such is the case of rapid oscillations, which we examine next.
Periodic motions
----------------
In what follows, we show that the effective detection rate for periodic motions is constant, provided that the period $T$ is much shorter than the coarse-graining time-scale $\sigma$.
### Time-averaging
Consider a spacetime path of the Unruh-DeWitt detector with strong variations at time-scales that are much shorter than $\sigma$. The details of such variations are unobservable; detection events are localized at a coarser time-scale. The physical probability density involves a smearing of $\tau$ in $P(E, \tau)$ at a scale $\sigma$, as given in Eq. (\[conv\]). We must recall that the smeared form Eq. (\[conv\]) is derived from first principles and that the de-convoluted PDF is a convenient approximation.
The use of the smeared probability density $\bar{P}(E, \tau)$ is equivalent to the substitution of the term $1/ \Sigma(\tau, y)$ in Eq. (\[pet5\]) with a term $1 / \bar{\Sigma}(\tau, y) := \langle 1/ \Sigma(\tau, y)\rangle_{\sigma}$, where
$$\begin{aligned}
\langle A(\tau) \rangle_{\sigma} = \int d \tau' f_{\sigma}(\tau - \tau') A(\tau'),\end{aligned}$$
defines that average of any function $A(\tau)$ with the probability density $f_{\sigma}$ of Eq. (\[conv\]).
To a first approximation, $$\begin{aligned}
\bar{\Sigma}(\tau, y) \simeq \langle \Sigma(\tau, y)\rangle_{\sigma} = \int_{-y/2}^{y/2} ds \int_{-y/2}^{y/2} ds' \langle \dot{X}^{\mu}(\tau + s) \dot{X}_{\mu}(\tau + s')\rangle_{\sigma}\end{aligned}$$
If the detector’s trajectory is exactly periodic with period $T$, and $T << \sigma$, then the $\tau$ dependence drops from $1/\Sigma$ after averaging. This implies that we can substitute averaging with respect to $f_{\sigma}$ with averaging over the period, i.e.,
$$\begin{aligned}
\langle A(\tau) \rangle_{\sigma} = \frac{1}{T} \int_0^T d\tau A(\tau) = \langle A\rangle_T\end{aligned}$$
for any variable $A(\tau)$. Hence, the smeared PDF $\bar{P}(E, \tau)$ becomes time-independent $$\begin{aligned}
\bar{P}(E) = - \frac{\alpha(E)}{4 \pi^2} \int_{-i\epsilon - \infty}^{-i \epsilon + \infty} dy \frac{ g_{\sigma}(y) e^{-i Ey}}{\bar{\Sigma}(y)}, \label{pet10}\end{aligned}$$ where $$\begin{aligned}
\bar{\Sigma}(y) = \int_{-y/2}^{y/2} ds \int_{-y/2}^{y/2} ds' \langle L_s[\dot{X}^{\mu}] L_{s'}[\dot{X}_{\mu}]\rangle_{T} \label{sls}\end{aligned}$$ and $L_s$ stands for the translation operator $L_s[F](\tau) = F(\tau +s)$.
### Non-relativistic systems
Eq. (\[pet10\]) applies to any periodic motion in Minkowski spacetime. It simplifies significantly for non-relativistic systems, where $\dot{X}^0(\tau) \simeq 1$ and $|\dot{X}^i(\tau)| << 1$. To see this, we decompose $X^i(\tau)$ in its Fourier modes
$$\begin{aligned}
X^i(\tau) = \frac{X_0^i}{2} + \sum_{n = 1}^{\infty} c^i_n \cos ( n \omega_0 \tau + \phi^i_n), \label{xin}\end{aligned}$$
where $\omega_0 = 2 \pi/T$. Then, Eq. (\[sls\]) becomes
$$\begin{aligned}
\bar{\Sigma}(y) = y^2 - {\cal F} (y),\end{aligned}$$
where $$\begin{aligned}
{\cal F} (y) = \sum_{n=1}^{\infty} \overrightarrow{c}_n^2 \sin^2\left(\frac{n \omega_0 y}{2} \right).\end{aligned}$$ is a periodic function of period $T$.
Thus, the calculation of the detection probability for an oscillator in periodic non-relativistic motion reduces to finding the roots of the equation $z^2 - {\cal F}(z) = 0$, in the lower half of the imaginary plane with modulus of order $\sigma$ or smaller.
For harmonic motion of frequency $\omega$, $X^i(\tau) = x^i_0 \cos(\omega \tau + \phi^i)$. Hence, $$\begin{aligned}
{\cal F}(y) = \overrightarrow{x}^2_0 \sin^2 \left(\frac{\omega y}{2}\right)\end{aligned}$$ and the detection PDF coincides with that for uniform circular motion at angular frequency $\omega$, as it has been studied in Refs. [@LePf; @Letaw].
### Relativistic harmonic oscillation
We calculate the time-averaged detection probability for relativistic harmonic motion in one dimension, a case that is of interest for experimental reasons. We consider a path characterized by the spatial oscillatory motion $X^1(\tau) = x_0 \sin(\omega \tau)$. This path is of the form Eq. (\[1dim\]), where $b(\tau) = \sinh^{-1}[v_0 \sin(\omega \tau)]$ and $v_0 = \omega x_0 < 1$.
We expand $e^{\pm b(\tau)}$ as a Fourier series $$\begin{aligned}
e^{\pm b(\tau)} = \pm v_0 \cos \left( \omega \tau \right) + \sqrt{1 + v_0^2 \cos^2\left(\omega \tau \right)} = \pm v_0 \cos \left( \omega \tau \right) + \frac{c_0(v_0)}{2} + \sum_{k = 1}^{\infty} c_k(v_0) \cos \left(2k \omega \tau \right),\end{aligned}$$ where $$\begin{aligned}
c_k(v_0) = 2 \sum_{n = k}^{\infty} (-1)^{n-1} \frac{1 \cdot 3 \cdot 5 \cdot 7\cdot \ldots \cdot (2n - 1) }{(n-k)!(n+k)!n!} \left( \frac{v_0^2}{8}\right)^{n}, \hspace{0.5cm} k = 0, 1, 2, \ldots.\end{aligned}$$ Then, we find $$\begin{aligned}
\langle L_s[\dot{X}^{\mu}] L_{s'}[\dot{X}_{\mu}]\rangle_{T} = \langle L_s[e^b] L_{s'}[e^{-b}]\rangle_{T} = \frac{c_0^2}{4} - \frac{v_0^2}{2} \cos\left[\omega(s-s')\right] + \frac{1}{2} \sum_{k = 1}^{\infty} c_k^2 \cos \left[ 2 k \omega (s -s') \right],\end{aligned}$$ and we compute the time-averaged proper distance $$\begin{aligned}
\bar{\Sigma}(y) = \frac{c_0^2}{4} y^2 - \frac{2 v_0^2}{\omega^2} \sin^2\left(\frac{\omega y}{2} \right) + \sum_{k = 1}^{\infty} \frac{2 c_k^2}{\omega^2 k^2} \sin^2\left(k \omega y \right). \label{sigma13}\end{aligned}$$ The time-averaged detection probability $\bar{P}(E)$ associated to Eq. (\[sigma13\]) is computed numerically. Fig.2 is a logarithmic plot of $\bar{P}(E)/\alpha(E)$, for different values of $v_0$ in the relativistic regime.
![ The PDF $\bar{P}(E)$ divided by $\alpha(E)$, Eq. (\[ae\]), as a function of $E/\omega$ for a detector undergoing an harmonic oscillation at frequency $\omega$. The scale on the vertical axis is logarithmic, as the detection probability varies strongly with the maximum velocity $v_0$ of the oscillation; (a) $v_0 = 0.01$, (b) $v_0 = 0.1$, (c) $v_0 = 0.5$, and (d) $v_0= 0.99$. ](unruh2.eps){height="7cm"}
Averaging over fast motions
----------------------------
We saw that the averaged detection PDF is constant for periodic motions with period $T$ much smaller than the coarse-graining scale $\sigma$. We expect that the detection probability will vary slowly in time in quasi-periodic motions, i.e., in motions that are almost periodic at a scale $T$ but non-periodic when examined at a larger scale.
An example of such a quasiperiodic motion is an harmonic motion $X^i(\tau) = x^i_0 \cos(\omega_{\tau} \tau)$, where $\omega(\tau)$ is a function of time that varies at time-scales much large than $\omega^{-1}$. In particular, if $\dot{\omega} \sigma/\omega << 1$, by continuity we expect that to leading order Eq. (\[pet10\]) applies for the detection probability, with a time-dependence due to the variation of $\omega_{\tau}$.
In general, time-averaging allows for the elimination of motions at scales much smaller than $\sigma$, usually resulting in a simpler description at the time-scale of observation. To see this, we consider a path of the form Eq. (\[1dim\]) with $$\begin{aligned}
b(\tau) = a_0 \tau + \frac{a_1}{\omega} \sin (\omega \tau)\end{aligned}$$ where $a_1 << a_0$ and $a_1 << \omega$. This path is characterized by constant acceleration $a_0$ modulated by small but rapid oscillations at frequency $\omega$. It is not well described by Eq. (\[petg\]), because $\dot{a}_{\tau}/a_{\tau}^2 \sim a_1 \omega/a_0^2$ is not necessarily much smaller than unity. For this path, $$\begin{aligned}
\bar{\Sigma}(y) = \int_{-y/2}^{y/2} ds \int_{-y/2}^{y/2} ds' e^{a_0(s -s') }\langle e^{\frac{a_1}{\omega} [\cos(\omega \tau + s) - \cos[\omega \tau + s')]} \rangle_T. \label{sig6}\end{aligned}$$ Since $a_1/\omega << 1$, we expand the exponential in Eq. (\[sig6\]) to obtain $$\begin{aligned}
\bar{\Sigma}(y) = \frac{4}{a_0^2} \sinh^2\left(\frac{a_0y}{2} \right) - \frac{a_1^2}{\omega^2} \frac{(a_0^2 - \omega^2) [\cosh(a_0y) \cos(\omega y) - 1]+ 2 a_0 \omega \sinh(a_0y) \sin(\omega y)}{(a^2_0 + \omega^2)^2}\end{aligned}$$ In the regime $\omega >> a_0$, the roots $w_n(\tau)$ of $\bar{\Sigma}(y)$ are obtained perturbatively as in Sec. 3.4. We find $$\begin{aligned}
w_n(\tau) = -i\frac{2 \pi }{a_0} \left(1 \pm \frac{\sqrt{2}a_1}{\omega} \right)n, \hspace{1cm} n < N_{max} << \frac{\omega}{a_0} \log\left(\frac{\omega}{a_1}\right).\end{aligned}$$ Following the same methodology as the one leading to Eq. (\[pet11\]), we compute the averaged detection probability $$\begin{aligned}
\bar{P}(E) = \frac{\alpha(E) E}{2 \pi\left(e^{\frac{2 \pi E}{a_0}}-1\right)} \left[1 + \frac{4 \pi^2 a_1^2 E^2}{3 a_0^2 \omega^2} \frac{ 1 + e^{-\frac{2 \pi E}{a_0}}}{\left(1 - e^{-\frac{2 \pi E}{a_0}}\right)^2} \right], \label{pet13}\end{aligned}$$ for the corrections to the Planckian spectrum due to rapid oscillations.
General spacetime trajectories
------------------------------
The common feature of all calculations so far has been that the detection probability is non-zero only for trajectories characterized by at least one time-scale $T$ that is much smaller than $\sigma$. This result is physically reasonable. The emergence of macroscopic records at a scale of $\sigma$ implies that quantum processes at time-scales larger than $\sigma$ are decohered. In contrast, quantum processes at time-scales much smaller than $\sigma$ remain unaffected. Therefore, it is essential that at least one of the characteristic time-scales of the detector’s motion be much smaller than $\sigma$.
A generic path $X^{\mu}(\tau)$ in Minkowski spacetime is characterized by three Lorentz-invariant functions of proper time, with values having dimension of inverse time: the acceleration $a(\tau)$, the torsion ${\cal T}(\tau)$ and the hypertorsion $ \upsilon(\tau)$. These are defined as follows $$\begin{aligned}
a(\tau) :&=& \sqrt{-a_{\mu}a^{\mu}}, \\
{\cal T}(\tau) :&=& \frac{\sqrt{a^4 - \dot{a}^2-\dot{a}_{\mu} \dot{a}^{\mu}}}{a},\\
\upsilon(\tau) :&=& \frac{\epsilon_{\mu \nu \rho \sigma}\dot{X}^{\mu} a^{\nu} \dot{a}^{\rho} \ddot{a}^\sigma }{a^3 {\cal T}^2},\end{aligned}$$ where $a^{\mu} = \ddot{X}^{\mu}$.
Our previous analysis suggests that the detection probability will be significant whenever any of the following conditions hold: $ a \sigma >> 1$, ${\cal T} \sigma >> 1$, or $\upsilon \sigma >> 1$. The converse does not hold necessarily; a generic path is characterized by other parameters arising from higher order derivatives of the functions $a, {\cal T}$ and $\upsilon$.
Of particular importance are the so-called [*stationary paths*]{}, i.e., paths characterized by a proper distance $\Sigma(\tau, y)$ that is independent of $\tau$. These paths correspond to constant values of $a, {\cal T}$ and $\upsilon$. They separate naturally into six classes. We have already encountered two classes corresponding to straight line motion ($ a = {\cal T} = \upsilon = 0$) and to constant linear acceleration ($a \neq 0$, $ {\cal T} = \upsilon = 0$). A third class that corresponds to circular motion ($\upsilon = 0$, $|a| < |{\cal T}|$) falls also under the category of periodic motions, described in Sec. 3.5. A representative path of the latter class is $$\begin{aligned}
X^{\mu}(\tau) = \frac{1}{\omega^2} ({\cal T} \omega \tau, a \cos( \omega \tau), a \sin (\omega \tau)),\end{aligned}$$ where $\omega = \sqrt{{\cal T}^2 - a^2}$ is the angular frequency of the circular motion. For this trajectory, $$\begin{aligned}
\Sigma(\tau, y) = \frac{{\cal T}^2}{\omega^2} [y^2 - \frac{a^2}{{\cal T}^2 \omega^2} \sin^2( \omega y)]\end{aligned}$$ is formally similar to the expression for non-relativistic harmonic motion in Sec. 3.5.
Two other classes that correspond to spatially unbounded trajectories also cannot be analytically evaluated. For a numerical evaluation that corresponds to the regime $\sigma \rightarrow \infty$ of our formalism see Ref. [@Letaw].
The last case corresponds to $a = {\cal T}, \upsilon = 0$ and it corresponds to a peculiar cusped motion, as described by the representative path $$\begin{aligned}
X^{\mu}(\tau) = (\tau + \frac{1}{6}a^2 \tau^3, \frac{1}{2} a \tau^2, \frac{1}{6} a^2 \tau^3, 0),\end{aligned}$$ for which $\Sigma(\tau, y) = y^2 \left( 1 + \frac{a^2}{12} y^2 \right)$. Then Eq. (\[pet5\]) becomes $$\begin{aligned}
P(E, \tau) = \frac{\alpha(E) a}{8 \sqrt{3} \pi \left(1 - \frac{12}{(\sigma a)^2}\right)} \left[ e^{-\frac{2\sqrt{3}E}{a}} - \frac{24\sqrt{3}}{(\sigma a)^3} e^{-\sigma E}\right]. \label{cusp}\end{aligned}$$ From Eq. (\[cusp\]) we readily verify that the detection probability is suppressed unless $\sigma a >> 1$ and that the corrections to the asymptotic value at $\sigma \rightarrow \infty$ are of order $(\sigma a)^{-2}$.
For paths $X^{\mu}(\tau)$ characterized by slow changes to the invariants $a, {\cal T}$ and $\upsilon$ (for example $|\dot{a} \sigma/a| <<1$), we expect that the adiabatic approximation will be applicable, and that the corrections to the adiabatic approximation will be of order $\dot{a}/a^2, \dot{{\cal T}}/{\cal T}^2$ and $\dot{\upsilon}/\upsilon^2$—see Sec. 3.4.2.
Conclusions
===========
In this article, we constructed the particle-detection probability for macroscopic detectors moving along general trajectories in Minkowski spacetime. The detectors are macroscopic in the sense that a particle detection is expressed in terms of definite records of observation. We describe the detection process in terms of PDF for the detection time. The derivation of this PDF is probabilistically sound and takes into account the irreversibility due to the emergence of measurement records in a detector.
The resulting PDF is causal and local in time at macroscopic time-scales. We found that a key role is played by the time-scale $\sigma$ of the temporal coarse-graining necessary for the creation of a macroscopic record. Detectors moving along paths with characteristic time-scales of order $\sigma$ or larger do not click. This behavior is physically sensible: particle creation depends strongly on the coherence properties of the quantum vacuum and the emergence of records at a time-scale of $\sigma$ destroys the coherence of all processes at larger timescales.
For paths characterized by multiple time-scales, $\sigma$ provides a scale by which to distinguish between slow and fast variables. Slow variables can be treated with an adiabatic approximation, and corrections to the adiabatic approximation can be systematically calculated. Moreover, the PDF Eq. (\[petb\]) can be averaged over all fast processes. We showed that for [*any*]{} periodic motion of period $T << \sigma$, the effective detection probability is time-independent.
We believe that our results are important for the conceptual clarification of the role of the detector in the Unruh effect. In particular, the consideration of macroscopic detectors is one step towards a thermodynamic description of the Unruh effect, because the recorded energy can be interpreted as heat. Indeed, the detectors presented here can be viewed as a quantum models for a calorimeter.
In order to explain this point, we note that an interpretation of the Unruh temperature as a thermodynamic temperature faces the problem of explaining what is the physical system to which this temperature is attributed. The analogy with the Hawking temperature does not hold here because the Hawking temperature is attributed to a black hole. There are several proposals that the Unruh temperature can be interpreted as a temperature of the spacetime [@thermgrav]. In this viewpoint, general relativity is viewed as an emergent thermodynamic description of an underlying theory.
While unrelated to such proposals, our results strengthen the idea that the Unruh temperature can be interpreted thermodynamically. First, we establish the robustness of the Unruh effect, in the sense that slow changes in a detector’s acceleration correspond to slow changes in the temperature of the Planckian spectrum. Second, our formulation in terms of the macroscopic response of detectors and the emphasis on the role of coarse-graining is structurally compatible with the foundations of statistical mechanics.
[*Experimental tests.*]{} Starting with Bell’s and Leinaas’ proposal of an Unruh effect interpretation of spin depolarization of electrons in circular motion [@BeLe; @Uncir], several proposals have been formulated for measuring the Unruh effect, or more generally particle detection due to non-inertial motion [@rogers; @MaVa; @ChTa; @skbfc; @SSH; @MFM]. Such experiments are mostly concerned with microscopic systems rather than macroscopic detectors, as described here.
In order to identify what kind of physical system could play the role of a macroscopic detector that effectively records particle creation, we recall that the coarse-graining time-scale $\sigma$ must be significantly larger than $(\Delta E)^{-1}$, where $\Delta E$ is the uncertainty in the energy of the pointer variable. Supposing that the pointer variable corresponds to a collective excitation of $N$-particles, we estimate $\Delta E$ as the energy fluctuations in the canonical ensemble for an $N$-particle system at temperature $T$. Then, $\Delta E = k(T) \sqrt{N}$, where $k(T)$ is a function that vanishes as $T \rightarrow 0$. Hence, $\sigma $ decreases with increasing $N$ as $N^{-1/2}$, but increases as $T$ approaches zero. The ideal detector should have the minimum number of particles consistent with the formation of stable records of detection, and it should be prepared at temperatures close to zero.
However, our method also applies when the detector consists of a microscopic probe (such as an atom in a trap) [*and*]{} a macroscopic apparatus (such as a photodetector) that detects the photons emitted by the atom, provided that atom and detector follow the same trajectory. Proposed experiments involve an accelerated microscopic probe together with detectors that are at rest in the laboratory frame, so Eq. (\[petb\]) does not apply. However, a theoretical model that involves only the coupling of the probe to the field misses the fact that the records of observation corresponds to photon emitted by the probe after it has been excited.
The non-equilibrium dynamics of an accelerating probe coupled to the field is, in general, non-Markovian [@Hual], so there is no obvious relation between the state of the probe and the recorded detection rate. Furthermore, as shown here, the temporal coarse-graining inherent in the detection process may destroy the particle detection signal.
For the reasons above, we believe that a complete theoretical analysis of such experiments requires not only the study of the non-equilibrium dynamics of the probe coupled to the field [@Hual; @Hual2], but also a precise modeling of the detection process along the lines presented here.
Finally, we note that our treatment of macroscopic detectors has the additional benefit that it allows for the explicit construction of multi-time correlation functions [@AnSav11] known as [*coherence*]{} in quantum optics. The consideration of the detection correlations for multiple detectors, along different spacetime trajectories, may lead to a new method for verifying phenomena of particle creation due to non-inertial motion.
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[^1]: [email protected]
[^2]: [email protected]
|
{
"pile_set_name": "ArXiv"
}
|
---
author:
- 'Sabrina Nusrat, Md. Jawaherul Alam, Stephen Kobourov'
title: Evaluating Cartogram Effectiveness
---
[[Nusrat, Alam and Kobourov]{}]{}
Introduction
============
Cartograms are maps in which areas of geographic regions, such as countries and states, appear in proportion to some variable of interest, such as population or income. They are popular visualizations for geo-referenced data that have been used for over a century [@Tobler04]. As such visualizations make it possible to gain insight into patterns and trends in the world around us, they have gained a great deal of attention from researchers in computational cartography, geography, computational geometry, and GIS. Many different types of cartograms have been proposed and implemented, optimizing different aspects: statistical accuracy (cartographic error), geographic accuracy (preserving the outlines of geographic shapes), and topological accuracy (maintaining correct adjacencies between countries).
Cartograms provide a compact and visually appealing way to represent the world’s political, social and economic state in pictures. Red-and-blue population cartograms of the United States have become an accepted standard for showing political election predictions and results. Likely due to aesthetic appeal and the possibility to put political and socioeconomic data into perspective, cartograms are widely used in newspapers, magazines, textbooks, and blogs. For example, while geographically accurate maps seemed to show an overwhelming victory for George W. Bush in the 2004 election; the population cartograms used by the New York Times [@NYT_04] effectively communicated the near even split; see Fig. \[fig:red-blue\]. The Los Angeles Times [@LAT12] shows the 2012 election results using cartograms and cartograms are used to show the European Union election results of 2009 in the Dutch daily newspaper NRC [@NRC]. In addition to visualizing elections, cartograms are frequently used to represent other kinds of geo-referenced data. Dorling cartograms are used in the UK Guardian newspaper [@Guar] to visualize social structure and in the New York Times to show the distribution of medals in Olympic Games since 2008 [@NYT16]. Popular TED talks use cartograms to illustrate how the news media can present a distorted view of the world [@Alisa], and to illustrate the progress of developing countries [@Hans2]. Cartograms continue to be used in textbooks, for example, to teach middle-school and high-school students about global demographics and human development [@Class2].
![Geographic map and a cartogram for the 2004 US election [@NYT_04].[]{data-label="fig:red-blue"}](NYT_2004_map.png "fig:"){width="23.00000%"} ![Geographic map and a cartogram for the 2004 US election [@NYT_04].[]{data-label="fig:red-blue"}](NYT_2004_cartogram.png "fig:"){width="23.00000%"}
Despite the popularity of cartograms and the large number of cartogram variants, there are very few studies evaluating cartograms. In order to design effective visualizations we need to compare cartograms generated by different methods on a variety of suitable tasks. In this paper we describe an in-depth evaluation of four major types of cartograms: contiguous, non-contiguous, rectangular, and Dorling cartograms. We first evaluate the effectiveness of these cartogram types by quantitative performance analysis (time and error) with a controlled experiment that covers seven different tasks from a recently developed task taxonomy for cartograms [@Task_C]. Second, we collect qualitative data with an attitude study and by analyzing subjective preferences. Third, we compare the quantitative and qualitative results with the results of a metrics-based cartogram evaluation. Fourth, we analyze the results of our study in the context of cartography, geography, visual perception, and demography. Finally, we consider implications for design and possible improvements.
Related Work
============
Cartograms have a long history; several major types of cartograms are briefly reviewed in Sec. \[sec:types\]. While there is some work on quantitative comparisons between the different types, there is no systematic qualitative evaluation.
In 1975 Dent [@dent1975] considered the effectiveness of cartograms and wrote that “attitudes point out that these (value-by-area) cartograms are thought to be confusing and difficult to read; at the same time they appear interesting, generalized, innovative, unusual, and having – as opposed to lacking – style”. Dent also suggested effective communication strategies when the audience is not familiar with the underlying geography and statistics, e.g., providing an inset map and labeling the statistical units on the cartogram. Griffin [@Gri83] studied the task of identifying locations in cartograms and found that cartograms are effective. Olson [@Olson] designed methods for the construction of non-contiguous cartograms and studied their characteristics. Krauss [@Krauss_ms] also studied non-contiguous cartograms using three evaluation tasks (from very general to specific) in order to find out how well the geographic information is communicated and concluded that non-contiguous cartograms work well for showing general distribution, but not for showing specific information (e.g., ratios between two regions). Sun and Li [@Hui] analyzed the effectiveness of different types of maps by collecting subjective preferences. Two types of experimental tests were conducted: (1) comparison of cartograms with thematic maps (choropleth maps, proportional symbol maps and dot maps), and (2) comparison between cartograms (non-contiguous cartogram, diffusion cartogram, rubber sheet cartogram, Dorling cartogram, and pseudo-cartogram). The participants in this study were asked to select one map that is more effective for the representation of the given dataset and to provide reasons for this choice. The results indicate that cartograms are more effective in the representation of nominal data (e.g., who who won–republicans or democrats?), but thematic maps are more effective in the representation ordinal data (e.g., population growth rates). Note that in both experiments the subjects gave their preferences, but were not asked to perform any specific tasks.
In a more recent study, Kaspar et al. [@kaspar2013empirical] investigated how people make sense of population data depicted in contiguous (diffusion) cartograms, compared to choropleth maps, augmented with graduated circle maps. The subjects were asked to perform tasks, based on Bertin’s map reading levels (*elementary*, *intermediate* and *overall*) [@BERTIN83]. The overall results showed that the augmented choropleth maps are more effective (as measured by accurate responses) and more efficient (as measured by faster responses) than the cartograms. The results seemed to depend on the complexity of the task (simple tasks are easier to perform in both maps compared to complex tasks), and the shape of the polygons. Note that only one type of cartogram (Gastner-Newman diffusion [@GN04]) was used in this study.
In order to improve cartogram design, Tao [@Manting] conducted an online survey to collect suggestions from map users. The majority of the participants found cartograms difficult to understand but at least agreed that cartograms are commonly regarded as members of the map “family”. Jennifer Ware [@Jen] evaluated the effectiveness of animation in cartograms with a user-study in which *locate* and *compare* tasks were considered. The results indicate that although the participants preferred animated cartograms, the response time for the tasks was best in static cartograms.
The studies above indicate an interest in cartograms and their effectiveness. While some specific types of cartograms have been evaluated on some specific tasks, a more comprehensive evaluation of different types of cartograms with a varied set of questions is lacking. In this paper we consider both qualitative and quantitative measurements, covering the spectrum of cartogram tasks, using four of the main types of cartograms.
{width="48.00000%"} {width="48.00000%"}\
(a) Contiguous cartogram, *Compare* task (b) Rectangular cartogram, *Find adjacency* task\
{width="48.00000%"} {width="48.00000%"}\
(c) Non-contiguous cartogram, *Find top-$k$* task (d) Dorling cartogram, *Summarize* task\
Graphical perception of area is relevant to cartograms as different methods generate different shapes (circles, rectangles, irregular polygons). There is a great deal of research in visualization and cartography about the impact of length, area, color, hue, and texture on map visualization and understanding. Bertin [@BERTIN83] was one of the first to provide systematic guidelines to test visual encodings. Cleveland and McGill [@cleveland1984graphical] extended Bertin’s work with human-subjects experiments that established a significant accuracy advantage for position judgments over both length and angle judgments, which in turn proved to be better than area judgments. Stevens [@Steven_law] modeled the mapping between the physical intensity of a stimulus and its perceived intensity as a power law. His experiments showed that subjects perceive length with minimal bias, but underestimate differences in area. This finding is further supported by Cleveland et al. [@cleveland1982judgments]. In a more recent study, Heer and Bostock [@JM10] investigated the accuracy of area judgment between rectangles and circles, both of which provide similar judgment accuracy, but are worse than length judgments. These results were consistent with the findings about “judgment of size” by Teghtsoonian [@teghtsoonian1965judgment]. Dent [@dent1975] surveyed related work in magnitude estimation and suggested that the shapes of the enumeration units in cartograms should be irregular polygons or squares. However, it is difficult to use these experiments directly to determine what would work best in the cartogram setting, as the datasets used, the tasks given, and the experimental conditions vary widely from experiment to experiment.
Cartogram Types {#sec:types}
===============
There is a wide variety of algorithms that generate cartograms and three major design dimensions along which cartograms vary:
- **Statistical accuracy:** how well do the modified areas represent the corresponding statistic shown (e.g., population or GDP). This is measured in terms of “cartographic error.”
- **Geographical accuracy:** how much do the modified shapes resemble the original geographic shapes and how well preserved are their relative positions.
- **Topological accuracy:** how well does the topology (as measured by adjacent regions) of the cartogram match that of the original map.
There is no “perfect” cartogram that is geographically accurate, preserves the topology, and also has zero cartographic error [@AKV15]. Some cartograms preserve shape at the expense of cartographic error, others preserve topology, still others preserve shapes and relative positions. Cartograms can be broadly categorized in four types [@ks07]: contiguous, non-contiguous, Dorling, and rectangular; see Fig. \[fig:four-types\].
**Contiguous Cartograms:** These cartograms deform the regions of a map, so that the desired areas are obtained, while adjacencies are maintained; see Fig. \[fig:four-types\](a). They are also called *deformation cartograms* [@AKV15], since the original geographic map is modified (by pulling, pushing, and stretching the boundaries) to change the areas of the regions on the map. Worldmapper [@WorldMapper] has a rich collection of diffusion-based cartograms. Among deformation cartograms the most popular variant is the ones generated by the diffusion-based algorithms of Gastner and Newman [@GN04], which we use in our evaluation. Others of this type include the rubber-map cartograms by Tobler [@Tobler73], contiguous area cartograms by Dougenik et al. [@DCN85], CartoDraw by Keim et al. [@KNPS03], constraint-based continuous area cartograms by House and Kocmoud [@HK98], and medial-axis-based cartograms by Keim et al. [@KPN05].
In deformation cartograms, since the input map is deformed to realize some given weights, the original map is often recognizable, but the shapes of some countries might be distorted. Recent variants for contiguous cartograms allow for some cartographic error in order to better preserve shape and topology [@zackary_blog].
**Rectangular Cartograms:** Rectangular cartograms schematize the regions in the map with rectangles; see Fig. \[fig:four-types\](b). These are “topological cartograms” where the topology of the map (which country is a neighbor of which other country) is represented by the dual graph of the map, and that graph is used to obtain a schematized representation with rectangles. In rectangular cartograms there is often a trade-off between achieving zero (or small) cartographic error and preserving the map properties (relative position of the regions, adjacencies between them).
Rectangular cartograms have been used for more than 80 years [@Raisz34]. Several more recent methods for computing rectangular cartogram have also been proposed [@BSV12; @hkps04; @ks07]. In our study, we use a state-of-the-art rectangular cartograms algorithm [@BSV12]. There are several options for this type of algorithm and we choose the variant where the generated cartogram preserves topology (adjacencies), at the possible expense of some cartographic error. Note that in addition to possible cartographic errors in this particular variant, rectangular cartograms in general have one major problem. To make a map realizable with a rectangular cartogram, it might be necessary to merge two countries into one (which is highly undesirable in practice), or to split one country into two parts [@ks07]. When recombining them this leads to regions that are no longer rectangular. In our study, we used the variant where the regions remain rectangular, at the expense of some countries getting merged with other countries. In particular 5 states in the map of USA, 3 states in Germany and 2 regions in Italy get merged in this algorithm. While some countries have states and others have provinces and regions, for simplicity we refer to all of them as “regions” in the rest of the paper.
**Non-Contiguous Cartograms:** These cartograms are created by starting with the regions of a map, and scaling down each region independently, so that the desired size/area is obtained; see Fig. \[fig:four-types\](c). They satisfy area and shape constraints, but do not preserve the topology of the original map [@Krauss_ms]. The non-contiguous cartograms method of Olson [@Olson] scales down each region in place (centered around the original geographical centroid), while preserving the original shapes. For each region, the density (statistical data value divided by geographic area) is computed and the region with the highest density is chosen as the anchor, i.e., its area remains unchanged while all other regions become smaller in proportion to the given statistical values. If the highest density region is geographically small, there will be a lot white space in the cartogram. If this is the case, Olson’s method searches for a high density region of reasonable size as an anchor; in this case smaller regions with higher densities are enlarged rather than reduced. In our study, we optimize the choice for an anchor to ensure that no pairs of regions overlap. Despite these efforts to reduce white space, since the size of the final regions depends on their original size and statistic to be shown, some regions may become too small. By definition, non-contiguous cartograms do not preserve the original region adjacencies, however, there is some evidence that the loss of adjacencies might not cause serious perceptual difficulties [@KPN05].
**Dorling Cartograms:** Dorling cartograms represent areas by circles [@dorling96]. Data values are realized by size of the circle: the bigger the circle, the larger the data value; see Fig. \[fig:four-types\](d). However, in order to avoid overlaps, circles might need to be moved (typically as little as possible) away from their original geographic locations. Unlike contiguous and non-contiguous cartograms, Dorling cartograms preserve neither shape nor topology. Dorling cartograms became very popular in the UK where the computer programs for generating Dorling cartograms were first published by its creator Danny Dorling. Dorling-style cartograms have become very popular on the web with JavaScript D3 implementations.
Metric-Based Analysis
=====================
We performed a comparative study on the four major types of cartograms, based on a set of quantitative performance metrics. Various quantitative cartogram measures have been proposed in the literature, and several studies used ad-hoc definitions of performance metrics to compare new algorithms to existing ones [@BSV12; @KNPS03; @ks07; @BMS10]. A recent standard set of such parameters with which to compare and evaluate cartograms [@AKV15], can be categorized based on the three cartogram dimensions:
\
(a) Statistical accuracy metrics\
\
(b) Geographical accuracy metrics\
**Statistical Accuracy:** This measures how well the obtained region areas in the cartogram match the desired statistical values. The *cartographic error* for a region $v$ in the cartogram is defined as $\frac{|o(v)-w(v)|}{max\{o(v),w(v)\}}$, where $o(v)$ and $w(v)$ are the obtained and desired area for the region. After evaluating different options for measuring the cartographic error of a given cartogram [@KNPS03; @KNP04; @BSV12], Alam et al. [@AKV15] advocate for both the average error and the maximum error, as measures of statistical distortion in the cartograms.
\[stat\] **Geographical Accuracy:** Two measures are also proposed in this context: one for region shape preservation and another for the preservation of the relative positions of the regions. Shape preservation is measured using the Hamming distance [@SKI98], also known as the symmetric difference [@MRS10] between two polygons. The polygons for each cartogram region and the corresponding map region are normalized to unit area and superimposed on top of each other; the fraction of the area in exactly one of the polygons is the Hamming distance $\tsdist$. Relative position preservation is measured by the *angular orientation error*, $\angular$, defined by Heilmann *et al.* [@hkps04] and obtained by computing the average change in the slope of the line between the centroids of pairs of regions.
**Topological Accuracy:** Topological accuracy is measured with the *adjacency error* $\toperror$: the fraction of the regional adjacencies that the cartogram fails to preserve, i.e., $\toperror = 1 - \frac{|E_c\cap E_m|}{|E_c\cup E_m|}$, where $E_c$ and $E_m$ are respectively the adjacencies between regions in the cartogram and the original map.
Alam et al. [@AKV15] used these measures to compare five cartogram algorithms. Among these five were contiguous and rectangular cartograms, but not Dorling and non-contiguous cartograms. We add these two cartogram types and evaluate their performance with three different countries (Germany, Italy, USA) and with two different statistics (population and GDP) for each map. Fig. \[fig:metric\] shows the results for statistical and geographical accuracy, for each of the three countries.
Statistical Accuracy: Dorling and non-contiguous cartograms are perfect in that regard, while rectangular cartogram have 3–10 times greater cartographic error than diffusion cartograms; see Fig. \[fig:metric\](a).
Geographical Accuracy: Non-contiguous cartograms are perfect in that regard (zero angular orientation error and Hamming distance), while contiguous cartograms show low errors in both shapes and angles. Rectangular cartograms are a clear outlier with errors in both shapes and angles that are at least 2 times greater than any other cartogram type; see Fig. \[fig:metric\](b).
Topological Accuracy: Contiguous cartograms are perfect, and so are the topology-preserving variant of rectangular cartograms. Non-contiguous cartograms do not maintain any adjacencies. Dorling cartograms have high adjacency error, especially the variant with attraction forces keeping the regions near the correct geographic locations. We note that adjacency error might not be a “fair" metric for non-contiguous and Dorling cartograms, since for both of these two cartogram types, the region becomes non-contiguous and geographical proximity rather than exact adjacency becomes a guide for topological relation.
We discuss these results, together with the results of the task-based study, in Section \[sec:results\].
Visualization Tasks in Cartograms
=================================
Cartograms are employed to simultaneously convey two types of information: geographical and statistical. Our goal is to evaluate different types of cartograms in these two aspects, by conducting experiments that cover the spectrum of possible tasks. In this context, a recent task taxonomy for cartograms is particularly useful, as it categorizes tasks in different dimensions (e.g., goals, means, characteristics) and groups similar tasks together [@Task_C].
In order to cover the spectrum of tasks, and yet to keep the number of tasks low for practical reasons, we selected seven of the ten tasks in the taxonomy. We included basic map tasks, such as *find adjacency* and *recognize*. We also included basic statistical tasks, such as *compare*, as well as composite tasks, such as *summarize*.
The tasks *filter*, *cluster*, and *find top-$k$* are tasks that have similar goals (exploring data), similar means (finding data relation), similar high-level data characteristics, and all three tasks consider “all instances” of the data. Another group of tasks with similar goals and means contains *summarize* and *identify* tasks. We used *find top-$k$* and *summarize* as representatives from these two groups of tasks.
Here we describe all the visualization tasks used in our study; these are also included in Table \[tab:tasks\], where the exact input setting, along with the exact questions given to the participants, are summarized.
**Compare:** The *compare* task has been frequently used in taxonomies and evaluations [@Jen; @RR13; @Weh93]. The task typically asks for similarities or differences between attributes; see Fig. \[fig:four-types\](a) for an example of a *compare* task in our experiment.
**Detect change:** In cartograms the size of a region is changed in order to realize the input weights. Since change in size (i.e., whether a region has grown or shrunk) is a central feature, it is crucial that the viewer be able to detect such change.
**Locate:** The task in this context corresponds to searching and finding the position of a region in a cartogram. In some taxonomies this task is denoted as *locate* and in others as *lookup*, but these are not necessarily the same [@BM13]. Since cartograms often drastically deform an existing map, even if the viewer is familiar with the underlying maps, finding something in the cartogram might not be a simple lookup.
**Recognize:** One of the goals in generating cartograms is to keep the original map recognizable, while distorting it to realize the given statistic. Therefore, this is an important task in our taxonomy. The aim of this task is to find out if the viewer can recognize the shape of a region from the original map when looking at the cartogram.
**Find top-$k$:** This is another commonly used task in visualization. Here the goal is to find $k$ entries with the maximum (or minimum) values of a given attribute. This task generalizes tasks, such as *Find extremum* and *Sort*. In our evaluation, we ask the subjects to find out the region with the highest or second highest value of an attribute; see Fig. \[fig:four-types\](c).
**Find adjacency:** Some cartograms preserve topology, others do not. In order to understand the map characteristics properly, it is important to identify the neighboring regions of a given region.
**Summarize (Analyze / Compare Distributions and Patterns):** Cartograms are most often used to convey the “big picture”. *Summarize* tasks ask the viewer to find patterns and trends in the cartogram.
**Input** **Question** [**Time (s)**]{} **Error %**
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\[tab:tasks\]
Experiment {#sec:experiment}
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We conduct a series of controlled experiments aimed at producing a set of design guidelines for creating effective cartograms. We assess the effectiveness of our visualizations by performance (in terms of accuracy and completion time for visualization tasks) and subject reactions (attitude).
Hypotheses Formulation {#sec:hypotheses}
----------------------
Our hypotheses are informed by prior cartogram evaluations, perception studies, and popular critiques of cartograms. One of the most common criticisms is about shape distortion in cartograms, which makes it hard to recognize familiar geographic regions [@Tobler04]. Dorling [@dorling96] says “A frequent criticism of cartograms is that even cartograms based upon the same variable for the same areas of a country can look very different.” Tobler [@Tobler04] reports “It has been suggested that cartogram are difficult to use, although Griffin does not find this to be the case.” Dent [@dent1975] suggests effective communication strategies such as providing an inset map and labeling. With these comments in mind, in our experiment we added an undistorted map for the relevant tasks (*locate*, *detect change* *find adjacency* and *summarize*). We also labeled the regions for all tasks except *locate* and *recognize*, since labeling the regions for these two tasks would defeat the purpose of the tasks. Before stating our hypotheses we note that we say that one cartogram type is “better" than another for some task, when we expect quantitative differences (e.g., participants make fewer errors, or take less time) or qualitative differences (e.g., the participants prefer one over the other).
[**H1**]{}: For location tasks, contiguous and non-contiguous cartograms will be better than the other cartograms, as these two types preserve the relative position of regions [@AKV15; @Olson; @cartogram-star]. Dorling cartograms will likely be better than rectangular cartograms.
[**H2:**]{} For recognition tasks, non-contiguous cartograms are likely better than the rest since they preserve the original shapes [@Olson]. (For recognizing the shape of a region we only test contiguous and non-contiguous cartograms, because rectangular and Dorling cartograms replace the original shapes with rectangles and circles; testing shape recognizability would lead to predictably high errors and time).
[**H3:**]{} For detecting change (whether a region has grown or shrunk in cartogram), and comparison of areas (size comparison, find top-$k$), contiguous cartograms are likely better than Dorling and rectangular cartograms, since the judgment of size of circles is difficult [@teghtsoonian1965judgment], and potentially large aspect ratios for rectangular cartograms can make the changes/comparisons difficult to perceive.
[**H4:**]{} For finding adjacencies, contiguous and rectangular cartograms are likely better than the rest, because they preserve topology [@AKV15; @cartogram-star], whereas non-contiguous and Dorling cartograms seem to be ill-suited for such tasks.
[**H5:**]{} For summarizing the results and understanding data patterns, Dorling, non-contiguous and contiguous cartograms will work better than rectangular cartograms, as the first three types better preserve the map characteristics (location, shape and topology) [@AKV15]. With respect to subjective preferences, we expect that the participants in our study are likely to prefer contiguous and Dorling cartograms, as they are more frequently used.
Participants
------------
We recruited participants by sending email to students in selected classes at the University of Arizona:
[*“We would like to invite you to take part in a research study to evaluate the usability of cartograms. A cartogram is a map in which some thematic mapping variable (e.g., population, income) is represented by the land area. The study takes 35–40 minutes: you will be asked to perform several tasks using cartograms and to compare different types of cartograms on a computer. All data will be collected anonymously and will be used for research purposes only. Modest compensation (\$10) will be provided for all participants. If you are interested, please find a convenient 1 hour time slot and provide your name and email address below.”*]{}
The participants took part in the experiment one at a time, so that the experimenter could ensure that each participant understood the tasks at hand and had all their questions answered prior to starting the timed portion of the experiment. The participants were encouraged to ask as many questions as needed during the training session as well. All participants completed the experiment successfully, and no data was discarded.
Out of the 33 participants that took part in the study, 24 were male and 9 female; 23 between 18–25 years of age and 10 between 25–40; 9 listed high school, 12 listed undergrad, 8 listed Masters and 4 listed PhD as their highest completed education level. Familiarity with cartograms also differed: 14 participants were familiar with Dorling, 11 were familiar with contiguous, 8 with rectangular, and 3 with non-contiguous cartograms.
Since some of our tasks require the subjects to identify regions highlighted with different colors, all participants were tested for color blindness using an *Ishihara test* [@Ishihara17], and every participant passed the test. We used red and blue colors for highlighting.
Test Environment
----------------
We designed and implemented a simple application software that guided the participants through the experiment, provided task instructions and collected data about time and accuracy. The study was conducted using a computer (with i7 CPU 860 @ 2.80 GHz processor and 24 inch screen with 1600x900 pixel resolution), where the participants interacted with a standard mouse and keyboard to answer the questions. The experiment consisted of preliminary questions, a familiarity and initial ratings survey, task-based questions, and preference and attitude questions.
**Preliminary questions:** At the beginning, the participants filled out a standard human-subjects form confirming their participation in the experiment. They were also briefed about the purpose of the study: what cartograms are and what kind of tasks they will be asked to perform. The participants then completed some training tasks, familiarizing themselves with they software. Before proceeding to the next stage, each participant was given one more chance to ask questions and were told they would be able to take a break or leave the experiment whenever they wanted. All participants completed the experiment.
**Main experiment:** The main experiment had several stages:
*Familiarity and initial rating:* The participants reported their familiarity with each of the four cartogram types. For each type we showed one example, along with a short description, and asked whether they were familiar with this particular type of cartogram. We also asked the participants for an initial rating of the four cartogram types, using a Likert scale (excellent = 5, good = 4, average = 3, poor = 2, very poor =1).
*Task-based questions:* For the task-based part of the study, the participants answered multiple choice questions about different visualizations, using all four types of cartograms under consideration, and showing different statistics for different countries (described in mode detail below). We recorded the number of correct and incorrect answers, as well as the time taken to provide the answers.
*Preference and attitude study:* After all the tasks were completed, we asked the participants to choose one of the four cartogram types for another five questions. The goal of this set of questions was to help us detect whether the initial preferences might change after performing 67 timed tasks. For these five questions we were not interested in the time and error, but just in the choice that was made.
For the attitude study we adapted Dent’s semantic differential technique [@dent1975]. We used a rating scale between pairs of words or phrases that are polar opposites. There were five marks between these phrases and the participants selected the mark that best represented their attitudes for a given map and a given aspect. We used three aspects: general attitude about helpfulness and usability of the visualization, appearance, and readability.
Datasets and Questions
----------------------
We evaluated four different types of cartograms, using seven types of tasks. In order to guard against potential bias introduced by only one or two datasets, we used three different maps (USA, Germany, Italy) and eleven different geo-statistical datasets. Specifically, for the first six tasks we used population and GDP of the USA, Germany and Italy from 2010. For *summarize* tasks we used population of the USA in 1960 and 2010; GDP of Germany in 2010, crime rate in Italy, and three election results (2000, 2004, and 2008) in the USA.
We used a within-subject experimental design. For each subject, questions were selected from all the cartogram types and all the tasks. To guard against adversary effects from the order of the questions, we took a random permutation of the questions for each subject. For each of the tasks, the participants worked with all three country maps (USA, Germany, Italy).
In order to make a fair comparison we also wanted the participants to work with all four cartogram types for each task. Indeed, the participants worked with all four cartogram types for all the tasks, with two unavoidable exceptions. First, for *recognize* tasks we used only contiguous and non-contiguous cartograms, since all the region shapes in Dorling and rectangular cartograms are circles and rectangles. Asking the participants to recognize the shape of a given region, when every region is a circle or a rectangle would be an unreasonably difficult challenge and might affect performance on other (and more meaningful) tasks. Second, for *detect change* we omit non-contiguous cartograms, since they use a different normalization of the areas than the other cartograms. In particular, as described in Section \[sec:types\], the size of a region in a non-contiguous cartogram is not directly related to the statistical data for that region, but it also depends on the distribution of the statistical data across all the regions. Thus, determining whether one region has grown or shrunk in a non-contiguous cartogram would be an unreasonably difficult challenge.
For each task, the questions were drawn from a pool of questions involving all possibly cartograms. Therefore, each participant answered 4 cartograms $\times$ 3 maps = 12 questions for four of the tasks (*locate*, *compare*), *find top-$k$*, and *find adjacency*). Since we evaluated only contiguous and non-contiguous cartograms for *recognize*, this task involved 2 cartograms $\times$ 3 maps = 6 questions. Similarly for *detect change* there were 3 cartograms $\times$ 3 maps = 9 questions. Finally for *summarize*, where the participants compared and analyzed the overall data trends in the map, we used 4 different data sets: crime rate (arson) in Italy, GDP of Germany, population change (from1960 to 2010) in the USA, and Presidential election results in the USA. These four datasets were used on four different cartograms for each subject. In total, there were 4 tasks $\times$ 12 questions + 6 questions + 9 questions + 4 questions = 67 cartogram task-based questions. The order of the tasks, and the cartograms varied for each user.
Results and Data Analysis {#sec:results}
=========================
In this section, we report and analyze the results of our task-based quantitative experiment and qualitative experiment (subjective preferences and attitude study). Finally, we compare and contrast the metric-based data with the quantitative and quantitative data.
Results of the Task-Based Study
-------------------------------
We use ANOVA $F$-tests with significance level $\alpha = 0.05$ to carry out the statistical analysis. The within-subject independent variables are the four cartogram types. The two dependent measures are the average completion times and error percentages by the participants, shown in the last two columns of Table \[tab:tasks\]. The null hypothesis is that the cartogram type does not affect completion times and error rates. When the probability of the null hypothesis ($p$-value) is less than $0.05$ (or, equivalently the $F$-value is greater than the critical $F$-value, $F_{cr}$), the null hypothesis is rejected. For significance level $\alpha = 0.05$, the critical value of $F$ is $F_{cr}=F_{0.05}(3,128)=2.68$ for all tasks except for recognize and detect change. For these two tasks the critical values are $F_{0.05}(1,64)=3.99$ and $F_{0.05}(2,96)=3.09$, respectively.
There is strong evidence for rejecting the null hypotheses in several cases; see Table \[tab:tasks\]. When the null hypothesis is rejected, paired $t$-tests are utilized for the post-hoc analysis, with Bonferroni correction on the significance level $\alpha = 0.05$. For each pair of cartogram types, we conclude that there is a significant difference in the mean completion time (respectively, mean error rate), if the computed $t$-value is greater than the critical $t$-value, $t_{cr}$. In pairwise comparison between 4 algorithms (i.e., 6 different pairs), the critical value of $t$ is $t_{cr}=t_{0.05/6}(32)=2.81$ (for all tasks except detect change and recognize). In pairwise comparison between 3 algorithms (i.e., 3 different pairs), the critical value of $t$ is $t_{cr}=t_{0.05/3}(32)=2.52$ (for detect change task). For the recognition task, only two algorithms are involved and hence a post-hoc analysis is not required.
**Hypothesis 1:** H1 is based on the expectation that cartograms that preserve the relative position of the regions in the map facilitate *locate* tasks. In particular, contiguous and non-contiguous cartograms should outperform the other two, with Dorling cartograms expected to be better than rectangular cartograms. Indeed, there is strong evidence in support of this hypothesis, based on the results of the *locate* task. In particular, there are statistically significant differences (both completion times and error rates) in performance between contiguous and rectangular cartograms, and between non-contiguous and rectangular cartograms.
Dorling cartograms require significantly more time than non-contiguous cartograms, and are associated with significantly more errors than contiguous cartograms. There is also a statistically significant difference in the error rate for Dorling cartograms compared with rectangular cartograms, although the difference in completion times is not significant. In essence, the performance of the four types of cartograms varied as we expected, although in few cases, the differences were not statistically significant.
**Hypothesis 2:** H2 is based on the expectation that non-contiguous cartograms should facilitate *recognize* tasks, since they perfectly preserve the shapes of the regions from the geographic map. Again, there is evidence in support of this hypothesis, based on the results of the *recognize* task. In particular, there is a statistically significant difference in the error rates of contiguous and non-contiguous cartograms. Moreover, the difference in errors is very large, at nearly a factor of four. Although there is no statistically significant difference for completion times, there are notable differences. For example, the range of time required for contiguous cartograms is much larger (5 - 45 seconds). Also note the bimodal distribution in the error plots for contiguous cartograms, with a peak at around 5% error and another peak around 30% error – a different pattern from the unimodal distribution for non-contiguous cartograms, which peaks around 1% error; see Table \[tab:tasks\].
One plausible explanation for the larger time range and the bimodal error distribution for contiguous cartograms, is that some participants took longer time than usual and sometimes found the correct answer, whereas others took little time and had little success finding the correct answer. While the average time is roughly the same time as for non-contiguous cartograms, the pattern is very different. Note that we intentionally did not evaluate Dorling and rectangular cartograms for *recognize* tasks, since recognizing the shape of a given region in a sea of circles or rectangles is impossible. Nevertheless, we can confidently say that non-contiguous cartograms are most suited for *recognize* tasks among the four types under consideration.
**Hypothesis 3:** H3 is based on the expectation that contiguous cartograms should facilitate *detect change* and *compare* tasks, since these kinds of tasks are more difficult with circles and rectangles with possibly poor aspect ratios. There is partial evidence in support of this hypothesis, based on the three tasks used to test it: *compare*, *find top-$k$*, *detect change*. Indeed for all three tasks the errors were the lowest in the contiguous cartogram setting. However, there were statistically significant results only in a subset of the possible pairs. In particular, there is a statistically significant difference in the error rates between contiguous and rectangular cartograms for all three tasks. Even though the time spent was the lowest in the contiguous cartogram setting for two of the three tasks, there was statistical significance between contiguous and rectangular cartograms for only one task.
We used a relative difference in areas for the *compare* task in the range (1.5, 4). We considered factors smaller than 1.5 too difficult and larger than 4 too easy. Although previous cognitive studies show that judgment of circle sizes is not very effective, in our study Dorling cartograms performed well for simple comparison between regions. This could be due to the fact that our *compare* task was too easy (minimum ratio was 1.5), or because we did not ask the participants to estimate the size (area) of circles exactly, but rather to compare two circles and to find the circle with the larger area. For the more complex tasks of *find top-$k$*, and *detect change*, the error rates in Dorling cartograms are indeed significantly higher than contiguous cartograms, although there was no statistically significant difference in the time required.
**Hypothesis 4:** H4 is based on the expectation that cartograms that preserve topology (i.e., contiguous and rectangular cartograms) would facilitate *find adjacency* tasks. There is partial evidence in support of this hypothesis, based on the results for the *find adjacency* task. Specifically, there is a statistically significant difference between the performance on contiguous and rectangular cartograms compared against Dorling and non-contiguous cartograms, in terms of error rates, although the same is not true for completion time.
Note that for this task we provide an undistorted geographical map along with the cartogram, as suggested by Dent [@dent1975] and Griffin [@Gri83]. Despite this, the average error rates for non-contiguous (48.5%) and Dorling cartograms (24.2%) are much larger than the average error rates of rectangular (5%) and contiguous cartograms (11.1%). This implies that even in the presence of the original undistorted map, the cartograms which preserves topology significantly help the viewer finding the correct adjacency.
**Hypothesis 5:** H5 is based on the expectation that Dorling, non-contiguous and contiguous cartograms should be better at showing geographic trends and patterns than rectangular cartogram, since they better preserve the map characteristics. There is partial evidence in support of this Hypothesis, based on the results of the *summarize* task. In particular, the error rate for rectangular cartogram is the highest among all four cartograms. For both non-contiguous and Dorling cartograms, this difference in error rate is statistically significant. While the difference in error rate between contiguous and rectangular cartograms is not statistically significant, the error rate in rectangular cartograms is nearly twice that in contiguous cartograms. The completion time does not vary significantly among these cartograms, perhaps because this is a complex task where the participants spent significant time for each type. It is worth noting the wide distribution of errors and time for all four types. Participants took over 100 seconds to answer one *summarize* question with rectangular and contiguous cartograms, while non-contiguous and Dorling required less than 75 seconds. All four cartograms yielded bimodal distributions of errors.
In general, the results of this part of the study show significant differences in performance (in terms of time and accuracy) between the four types of cartograms. As indicated by our hypotheses, different tasks seem better suited to different types of cartograms. Achieving perfection (with respect to minimum cartographic error, shape recognizability and topology preservation) in cartograms is difficult and no cartogram is equally effective in all three dimensions. Rectangular cartograms preserve adjacency relations, and that is reflected in the results. Non-contiguous cartograms maintain perfect shape, making the *recognize* task easy, but the “sparseness” of the map makes it difficult to understand adjacencies. Dorling cartograms disrupt the adjacency relations but somewhat preserve the relative positions of regions, and are good at getting the “big picture.” Contiguous cartograms more or less preserve localities, region shapes, and adjacencies, and give the best performance for almost all the tasks. The familiarity with contiguous cartograms might play a role in this regard.
Subjective Preferences
----------------------
As described in Section \[sec:experiment\], we asked the participants several preference questions in addition to the visualization tasks. At the beginning of the experiment, after introducing the different types of cartograms, the participants were asked to rate all four cartograms using a Likert scale (excellent = 5, good = 4, average = 3, poor = 2, very poor =1); see Fig. \[fig:first-choice\](a). The results confirm our expectation that Dorling (average 3.84) and contiguous (3.66) are rated higher than non-contiguous (2.75) and rectangular (2.54).
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After performing the visualization tasks, the participants were asked to select one of the four cartograms which would be used for an additional group of five questions. We asked this in order to test which cartograms were selected [*after*]{} performing many tasks and experiencing the different types of cartograms. The actual five questions were selected at random from the previous pool of questions, and the time and error rates for those five questions were not relevant. We were interested in the choices and in any changes from the preliminary ranking. Contiguous and Dorling cartograms remained the most preferred cartograms, although the order of the top two choices changes: out of 33 participants, 17 chose contiguous, 15 chose Dorling, 1 chose non-contiguous, and 0 chose rectangular; see Fig. \[fig:first-choice\](b). In addition to the ease and efficiency in performing tasks with these two cartograms, the preference for contiguous and Dorling cartograms might partially be due to familiarity with these two cartograms in the news and on social media (10 participants reported that they are familiar with contiguous cartograms and 15 were familiar with Dorling cartograms, contrasted with 7 for rectangular cartograms and 2 for non-contiguous cartograms).
Attitude Study
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As described in Section \[sec:experiment\], we collected information about the attitude of the participants, which can be valuable as argued by Stasko [@Stasko06]. In particular, at the end of the experiment, the participants were asked to rate the different cartogram types according to categories such as the helpfulness of the visualization, readability, and appearance, with a rating scale between pairs of polar opposite words and phrases. We considered the mode (most frequent response) and the mean (average response); see Fig. \[fig:attitude\]. This data also indicates a clear preference for contiguous and Dorling cartograms over the rest. The participants found contiguous cartograms to be helpful, well-organized and showing relative magnitude clearly, and Dorling cartograms to be entertaining, elegant, innovative, showing magnitude clearly, and easy to understand. The “Interested to use later?” choices also favor contiguous and Dorling cartograms.
Summary of All Results
----------------------
Table \[tab:summary\] summarizes the results of the metric-based and task-based analyses of all four cartogram types. The results are aggregated in four dimensions. The first three dimensions aggregate results on the measures and tasks related to statistical accuracy, geographical accuracy and topological accuracy; while the last one illustrates each cartogram’s effectiveness in showing the *big picture*, i.e., trends, patterns, and outliers. Considering the results in Table \[tab:summary\], together with they subjective preferences and attitudes of participants, allows us to make several general observations.
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[Statistical Accuracy]{} [**M**]{} [**L**]{} [**H**]{} [**H**]{} [**H**]{} [**L**]{} [**H**]{} [**H**]{}
[Geographical Accuracy]{} [**M**]{} [**L**]{} [**H**]{} [**M**]{} [**H**]{} [**L**]{} [**H**]{} [**M**]{}
[Topological Accuracy]{} [**H**]{} [**H**]{} [**L**]{} [**M**]{} [**H**]{} [**H**]{} [**L**]{} [**M**]{}
[Big Picture]{} [**M**]{} [**L**]{} [**H**]{} [**H**]{}
--------------------------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
: The result for metric-based and task-based analysis for all cartogram types. For metric-based analysis, [[**H**]{}]{}, [[**M**]{}]{}, [[**L**]{}]{} represent high, medium and low accuracy, respectively; for task-based analysis, they represent high, medium and low performance, respectively[]{data-label="tab:summary"}
Comparing the results of the metric-based and task-based analyses shows remarkable consistency in each of the dimensions: in each row of Table \[tab:summary\] a high ([[**H**]{}]{}) or medium ([[**M**]{}]{}) accuracy in the metric-based evaluation corresponds to a high ([[**H**]{}]{}) or medium ([[**M**]{}]{}) accuracy in the task-based evaluation. This indicates a consistency in how the different metrics and different tasks capture the three dimensions of cartogram design: topological accuracy, geographical accuracy and statistical accuracy.
Rectangular cartograms are a clear outlier and they should be used carefully. They performed sub-optimally in both the analysis of quantitative efficiency and in the qualitative subjective preference. This suggests that cartograms that severely distort region shapes and relative positions from the original map should be used very carefully. A promising compromise might be offered by rectilinear cartograms, such as that in Fig. \[fig:red-blue\](b), where instead of a rectangle, a more complex rectilinear polygon represents each region, so that the region shapes and locations are preserved better. Mosaic cartograms are a recent practical method for generating such rectilinear cartograms [@cano2015mosaic].
Non-contiguous cartograms are good performers (many Hs) in both the metric-based and task-based evaluation, but they are not particularly appreciated by the participants (based on subjective preferences and attitude). Although these cartograms preserve perfect shape and relative positions for the regions, this lack of appreciation might be due to the loss of a feel of a map from the lack of contiguity. Further, some regions become too small to recognize and overall there is more white space. One possible way to mitigate this is to compromise the perfect relative position by moving the regions to allow for them to scale up without overlapping and reduce the unused space.
Contiguous cartograms and Dorling are good performers (Ms and Hs) in both the metric-based and task-based evaluations; they are also well liked (subjective preferences and attitude).
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(d) Summarize
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Demographic Analysis
====================
In this section, we consider how participants of different age groups, gender, and education levels perform in the study. In particular, our goal here is to find out how people with different background make sense of geographic data using different cartogram types. For this demographic analysis we selected a subset of the seven tasks from the study. These tasks measure all three dimensions of cartogram design: topological accuracy (*find adjacency*), geographical accuracy (*locate*), statistical accuracy (*compare*), as well as composite tasks (*summarize*).
We analyzed our task-based results as well as the subjective ratings in the context of different demographic groups: participants who are familiar and who are not familiar with a particular cartogram type, male and female participants, participants under and over the age of 25, and undergraduate and graduate students. We discuss several interesting findings; see Figs. \[fig:demographics-time\], \[fig:demographics-error\], and \[fig:rating-demo\].
Task-Based Performance of Demographic Groups
--------------------------------------------
**Familiarity affects performance.** At the beginning of the study, we collected data about the familiarity of the participants with the four cartogram types. We analyzed the impact of familiarity with cartograms on the completion time and error rate; see Figs. \[fig:demographics-time\] and \[fig:demographics-error\]. Subjects who were familiar with cartograms took significantly longer time to perform the tasks (the significance was tested with Welch’s $t$-test), while the error rates seem not to be affected. This seems counter-intuitive, as we expected participants familiar with a particular cartogram type should make fewer errors and be faster in their response. One possible explanation is that familiarity is associated with deeper engagement: participants familiar with a cartogram type might have been more interested and engaged in the visualization.
**Female participants were more accurate.** We did not expect gender of the participants to be a factor in the accuracy of performing the tasks. However, in our study, female participants seem to be more accurate in most tasks involving contiguous, Dorling and non-contiguous cartograms (for contiguous cartograms, the difference in accuracy between the two groups is statistically significant (using Welch’s $t$-test), but there is no such pattern for rectangular cartograms. Completion times are not significantly different for male and female participants.
**Age and education did not affect performance.** We considered the possibility that older participants and participants with higher education level might perform better, since they are likely to be more familiar with more cartograms and maps [@monmonier2014lie]. However, we did not find significant differences for different age groups and education levels. One possible explanation is that by using three different maps (USA, Germany, Italy) and a within-subject experiment design, the participants were not aided by knowledge of a particular map or cartogram type.
Subjective Preferences of Demographic Groups
--------------------------------------------
**Female participants gave higher ratings.** Female participants rated all cartogram types, except rectangular cartograms, higher than their male counterparts. In particular, there is a strong indication (using Welch’s $t$-test) that female participants prefer Dorling cartograms more than the male participants; see Fig. \[fig:rating-demo\](left). Once possible explanation could come from earlier findings that round, circular shapes are preferred over sharp, angular shapes [@silvia2009people; @bertamini2016observers].
**Familiarity affected preferences.** We anticipated that familiarity with a particular cartogram type might make this type more liked. The participants gave similar ratings to unfamiliar cartogram types (around 3.5 on average), but different ratings to cartogram types they were familiar with. In the subjective ratings, contiguous and Dorling cartograms clearly outperform non-contiguous and rectangular cartograms. However, a closer look shows something interesting about rectangular and non-contiguous cartograms. Participants who were familiar with these two types of cartograms rated them lower than those who were unfamiliar; see Fig. \[fig:rating-demo\]. This is consistent with the choices made at the end of the study. After performing 67 tasks, all participants were familiar with all cartogram types, but hardly any participant chose rectangular or non-contiguous cartograms for the final 5 tasks.
Discussion and Design Implication
=================================
Cartograms are good at summarizing data and showing broader trends and patterns, as shown in early research [@Krauss_ms; @Hui]. While partially confirming some of these results, our study also identified significant differences in performance between different cartogram types and different tasks. This is relevant as new cartogram types continue to be created [@cano2015mosaic] and identifying difficult tasks for specific cartogram types can lead to improvements in design.
So, Which Cartogram is Best?
----------------------------
The choice of cartogram type should take into account the expected tasks. All cartogram types, except rectangular, performed well in tasks involving analyzing and comparing trends, with Dorling cartograms giving the best results. The reason might be that the simple circular shapes convey the data pattern easily, whereas the distortion in shape and size for other cartograms distract the viewers. When the geographic locations and adjacencies are important aspects, and the required map-reading is more detailed, contiguous cartograms might be more suitable. This seems to be the case for tasks, such as *locate, find top-$k$,* and *detect change*. On the other hand, rectangular cartograms work well if adjacency relations are important, and having a simple schematic representation is useful. For comparison of polygons, contiguous, non-contiguous, and Dorling all work equally well. We summarize these observations in a flowchart that could be used to guide the choice of a cartogram for a particular application; see Fig. \[fig:picker\].
The choice of cartogram type should also take into account the type of map being shown. Countries with few regions, such as Italy and Germany, are easier to schematize, while still preserving the general outlines. Similarly, most of their regions are on the periphery, making it easier to shrink or grow individual regions. Countries with more regions (and more landlocked regions) are more difficult to deal with.
![Cartogram type flow diagram.[]{data-label="fig:picker"}](cartogram-picker2.pdf){width="30.00000%"}
Design Improvements by Interaction
----------------------------------
One of the design implications from this study is that simple interaction techniques might mitigate some cartogram shortcomings. For example, to reduce the effect of misinterpretation associated with area perception, exact values can be shown using mouse-over/tool-tip labels. Some interactive web visualizations already provide such features [@NYT_O]. Non-contiguous cartograms perfectly preserve region shapes and geographic locations, and their performance is good for almost all the tasks, with the clear exception of finding adjacencies. Note, however, that the high errors for non-contiguous and Dorling cartograms on tasks such as *find adjacency* can be remedied by another simple interaction, such as mouse-over highlighting of all neighbors. For cartograms where identifying a particular region by its shape is difficult (e.g., Dorling, rectangular), link-and-brush type highlighting of the corresponding region in a linked geographic map might alleviate the problem. Such interactions, together with interactions that show exact values with mouse-over/tool-tip labels, will likely lead to improved performance for most cartogram types.
Limitations
-----------
We limited ourselves to one representative from each of four major types of cartograms. There are other types of cartograms and even more variants thereof (e.g., over a dozen contiguous cartograms) that we did not consider. Similarly, while attempting to cover the spectrum of possible cartogram tasks, we limited ourselves to a particular subset of tasks and particular choices for the task settings. There are numerous limitations when considering the types of possible geographic maps (e.g., more countries, continents, or even synthetic maps) and the relationship between the original geographic area and the statistical data shown (e.g., extreme area changes, moderate area changes, insignificant area changes). Since each participants met with the experimenter in person, we had a small number of participants and not as wide a spread over age and background knowledge. Despite such limitations, we believe the results of our study will be of use.
Conclusion
==========
We described a thorough evaluation of four major types of cartograms, going beyond time and error by synthesizing metrics-based, task-based, and subjective evaluations. The results show significant differences between cartogram types and provide insights about the effectiveness of the different types for different tasks. Given the popularity of cartograms in representing geo-spatial data, we believe that cartograms should be studied more carefully. While it is unlikely that a single evaluation study will be complete and will cover all possible issues, we feel that our work can be a useful starting point, while providing directions for future cartogram studies. We provide all details about this study (e.g., datasets, exact questions, answers, statistical analysis) available online at <http://cartogram.cs.arizona.edu/evaluations.html>.
A great deal of interesting future work remains. Cartograms are convenient tools for learning; and they are used in textbooks, for example, to teach middle-school and high-school students about global demographics and human development [@Class2]. It would be worthwhile to study the effect of different cartogram types on engagement in the context of learning. Enjoyment is a concept related to engagement and while enjoyment is extensively studied in psychology and recently of interest in visualization there is little work in the context of cartograms. Intuitively, it seems clear that being engaged with a visualization, enjoying it, and having fun can be beneficial, especially in the context of learning. Similarly, memorability (both in the context of recognition, e.g., “have you seen this visualization before?” and recall of data, e.g., “can you retrieve data from memory about a visualization you have seen before?”) is relevant for cartograms and not well studied yet.
[Sabrina Nusrat]{} is a PhD student at the Department of Computer Science at the University of Arizona. In 2012 she obtained a Masters degree in Computer Science from the University of Saskatchewan. Her current research interest is in visualization and visual analytics, with a focus on geo-referenced visualization.
[Md. Jawaherul Alam]{} is a research scientist at the University of California, Irvine. He received a PhD in Computer Science from the University of Arizona in 2015. His research interests include algorithms for graphs and maps, graph drawing, and information visualization.
[Stephen Kobourov]{} is a Professor at the Department of Computer Science at the University of Arizona. He received a BS degree in Mathematics and Computer Science from Dartmouth College and MS and PhD degrees from Johns Hopkins University. His research interests include information visualization, graph theory, and geometric algorithms.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'While time-dependent perturbation theory shows inefficient carrier-phonon scattering in semiconductor quantum dots, we demonstrate that a quantum kinetic description of carrier-phonon interaction predicts fast carrier capture and relaxation. The considered processes do not fulfill energy conservation in terms of free-carrier energies because polar coupling of localized quantum-dot states strongly modifies this picture.'
author:
- 'J. Seebeck'
- 'T.R. Nielsen'
- 'P. Gartner'
- 'F. Jahnke'
title: 'Polarons in semiconductor quantum-dots and their role in the quantum kinetics of carrier relaxation'
---
INTRODUCTION
============
Applications of semiconductor quantum dots (QDs) in optoelectronic devices rely on fast carrier scattering processes towards and between the discrete confined levels. These carrier transitions determine the dynamics of QD luminescence [@Morris:99] or the operation of QD lasers [@Bhattacharya:99; @Deppe:00]. For low carrier densities, where Coulomb scattering can be neglected, carrier-phonon interaction provides the dominant scattering channel. In QDs only phonons with small momenta can efficiently couple to the confined carriers [@Inoshita:97]. Then interaction with LA phonons does not contribute for large transition energies and only quasi-monochromatic LO phonons need to be considered.
The simplest theoretical approach to electronic scattering processes is based on time-dependent perturbation theory. Fermi’s golden rule for carrier transitions due to phonon emission or absorption contains a delta-function for strict energy conservation in terms of free-carrier energies of initial and final states and the phonon energy. When transition energies of localized QD states do not match the LO-phonon energy, efficient scattering is inhibited (leading to the prediction of a phonon bottleneck) and only higher-order processes, like a combination of LO and LA phonons [@Inoshita:92; @Singh:98], weakly contribute. Attempts to broaden the delta-function “by hand” immediately change the results [@Singh:98] which underlines that this point should be addressed microscopically. The phonon bottleneck effect is still a debated topic, with experimental evidence both for [@Urayama:01; @Minnaert:01; @Xu:02] and against it [@Tsitsishvili:02; @Peronne:03; @Quochi:03].
As in any coupled system, carrier-phonon interaction renormalizes both electronic and vibrational states. However, in bulk semiconductors or quantum wells with weak polar coupling, the net effect can be described by renormalized effective carrier masses, a small polaron shift of the band-edge, and lattice distortions only modify the background dielectric constant for the Coulomb interaction of carriers. The broadening of the transition energies due to carrier-phonon interaction remains weak.
For carriers in QDs, the discrete nature of localized electronic states changes the role of polaronic effects [@Inoshita:97; @Kral:98; @Verzelen:02]. Restricting the analysis to a single QD state coupled to phonons, polaron effects can be obtained from an exact diagonalization of the Hamiltonian [@Mahan:90]. While an extension to several discrete levels has been presented [@Stauber:00], the influence of the energetically nearby continuum of wetting-layer (WL) states, typical for self-assembled QDs, has not been included. Furthermore, only quasiparticle properties have been discussed which provide no direct information about the scattering efficiency for various processes. Calculations of carrier transition rates based on the polaron picture are missing.
We use a quantum kinetic treatment for carrier-phonon interaction in the polaron picture. As a first step, quasi-particle renormalizations due to the polar interaction for both QD and WL carriers are determined. For the QD states, the hybridization of one state with strong satellites of another state leads to a rich multi-peak structure. The WL states exhibit weak LO-phonon satellites. Coupling to the WL states provides a broadening mechanism for the QD states.
Based on the spectral properties of QD and WL polarons, quantum kinetic equations for the capture process (carrier transitions from the WL into the QD) and relaxation processes (transitions between QD states) are solved. For situations where, in terms of free-carrier energies, energy conserving scattering processes are not possible, the quantum-kinetic treatment provides efficient scattering rates. Even for the InGaAs material system with weak polar coupling, sub-picosecond scattering times are obtained.
Quantum dot polarons
====================
The single-particle properties of carriers under the influence of lattice distortions are determined by the retarded Green’s function (GF), $G^r_{\alpha}$, which obeys the Dyson equation $$\begin{gathered}
\Big[ i\hbar{\frac{\partial^{} }{\partial t_1^{}}} - \varepsilon_{\alpha} \Big]
~G^r_{\alpha}(t_1,t_2)
= \delta(t_1-t_2) \\
+ \int\!dt_3 ~~ \Sigma^r_{\alpha}(t_1,t_3) ~ G^r_{\alpha}(t_3,t_2).
\label{eq:Dyson_G_ret}\end{gathered}$$ Here $\alpha$ is an arbitrary (QD or WL) electronic state with free-carrier energy $\varepsilon_{\alpha}$. In the polaron theory one usually considers all possible virtual transitions from this state due to emission or absorption of phonons. This corresponds to a self-energy $\Sigma^r_{\alpha}$ for the carrier-phonon interaction where the population of the involved carrier states is neglected (electron vacuum). The corresponding retarded self-energy in random-phase approximation (RPA) is given by [@note:diag_GF] $$\Sigma^r_{\alpha}(t_1,t_2) = i \hbar \sum_{\beta} ~
G^r_{\beta}(t_1,t_2) ~ D^<_{\beta \alpha}(t_2-t_1).
\label{eq:Sigma_ret}$$ Assuming that the phonon system is in thermal equilibrium, the phonon propagator (combined with the interaction matrix elements) is given by $$\begin{gathered}
i\hbar ~ D^<_{\beta \alpha}(\tau)=\sum_{{\bm q}} ~
|M_{\beta \alpha}({\bm q})|^2 \\ \times
\Big[ n_{LO} ~ e^{-i\omega_{LO}\tau}
~+~(1+n_{LO}) ~ e^{ i\omega_{LO}\tau} \Big]
\label{eq:pn_propagator}\end{gathered}$$ where monochromatic LO-phonons with the frequency $\omega_{LO}$ are considered. The corresponding phonon population is given by $n_{LO}=1/(e^{\hbar\omega_{LO}/kT}-1)$ and the Fröhlich interaction matrix element $$M_{\beta \alpha}({\bm q})=\frac{ M_{LO} }{ q }~
\langle \beta | e^{i{\bm q}{\bm r}} | \alpha\rangle
\label{eq:coupling}$$ contains the overlap between the electronic states and the phonon mode. For localized electronic states this acts as a form factor. The prefactor $M^2_{LO}=4\pi\alpha \frac{\hbar}{\sqrt{2m}}
(\hbar\omega_{LO})^{3/2}$ includes the polar coupling strength $\alpha$ and the reduced mass $m$. As a result of the above assumptions, the retarded GF itself depends only on the difference of time arguments and its Fourier transform can be directly related to the quasi-particle properties.
Due to energy separation between the discrete QD states and the WL continuum, polaronic effects in QDs are often computed by neglecting the presence of the WL [@Inoshita:97; @Kral:98; @Verzelen:02; @Stauber:00]. For a single discrete level this amounts to the exactly solvable independent Boson model [@Mahan:90] and for several discrete levels it was shown to be nearly exactly solvable [@Stauber:00]. In both cases, even for non-zero temperatures, the spectral function contains a series of sharp delta-like peaks. In real QDs, however, the interaction with the WL continuum (which might require multi-phonon processes) leads to a broadening of these peaks. The RPA accounts for this broadening effect while it retains a hybridization effect (see below) characteristic for the full solution. Therefore the RPA is expected to provide an adequate description [*in the presence of the continuum*]{} [@note:RPA]. An additional source of broadening is the finite LO-phonon lifetime due to anharmonic interaction between phonons.
For the numerical results presented in this paper we consider an InGaAs QD-WL system with weak polar coupling $\alpha=0.06$. The effective-mass approximation is assumed to be valid with $m_e=0.067
m_0$ for the conduction band. For flat lens-shaped QDs the in-plane wave-functions of an isotropic two-dimensional harmonic potential are used while for the (strong) confinement in the direction perpendicular to the WL a finite-height potential barrier is considered (see [@Nielsen:04] for parameters and further details). To account for a finite height of the QD confinement potential, the calculations only include the (double degenerate) ground state and the (four-fold degenerate) first excited state, in the following called s and p-shell, respectively with s-p spacing and p-WL separation of 40 meV. For the description of the WL states we use the following steps: (i) the WL states in the absence of QDs are described by plane waves for the in-plane part, multiplied by the state corresponding to the finite-height barrier confinement for the perpendicular direction. (ii) to describe the WL states in the presence of the QDs, the orthogonalized plane wave scheme, described in Appendix A of Ref. [@Nielsen:04], is used to construct WL states orthogonal to the QD states. Calculations are done for a density of QDs on the WL $n_{dot}= 10^{10}$ cm$^{-2}$. Details on the calculation of the interaction matrix elements in Eq. (\[eq:coupling\]) with these wave functions for various combinations of QD and WL states can be found in Appendix B of Ref. [@Nielsen:04]. Convergent results are obtained with 128 points for the in-plane momentum radial integrals and 50 points for the remaining angular integrations entering the interaction matrix elements.
![(a) Spectral function of electrons for the lowest WL state at $k=0$ under the influence of polar coupling in the combined QD-WL system (solid line) and without coupling to the QD states (dotted line). For electrons in the QD p-shell (b) and s-shell (c) full coupling between all states (solid line) is compared to the case without coupling to other QD states (dotted line). Vertical lines show the corresponding free-carrier energies. Energies are given relative to the continuum edge $E_G$ in units of the phonon energy $\hbar\omega_{LO}$. The temperature is 300 K. []{data-label="fig:retardedGF"}](fig1.eps){width=".45\textwidth"}
The Fourier transform of the spectral function, -2 Im $G^r_{\alpha}(\omega)$, is shown in Fig. \[fig:retardedGF\] for the $k=0$ WL state and for the QD p- and s-shell (from top to bottom). In the absence of polar coupling to lattice distortions, the spectral functions are delta functions at the free-particle energies indicated by the vertical lines. The dotted line in Fig. \[fig:retardedGF\](a) is the result for the $k=0$ WL state without coupling to the QD states, indicating that their influence on the WL polarons is weak. The WL spectral function is broadened, the central peak exhibits a small polaron shift, and multiple sidebands due to LO-phonon emission (absorption) appear to the right (left). The polaron broadening is a result of the irreversible decay in the continuous WL density of states.
The LO-phonon sidebands of the localized QD states in Fig. \[fig:retardedGF\](b) and (c) are more pronounced and the hybridization of peaks from one shell with the energetically close sidebands of the other shell can be observed. This effect stems from the discrete nature of the localized states and requires that the coupling strength, which is modified by the form factors in Eq. (\[eq:coupling\]), exceeds the polaron damping. If only coupling matrix elements diagonal in the state index would be considered, a series of sidebands of a state with discrete energy, spaced by the LO-phonon energy, would be obtained. Off-diagonal coupling elements alone would lead to a hybridization of discrete levels as, e.g., in quantum optics where instead of the phonon-field a monochromatic light field coupled to a two-level system is considered [@Inoshita:97]. When the level splitting equals the LO-phonon energy, in the limit of weak damping the splitting of each line is determined by the carrier-phonon coupling strength. Due to the interplay of diagonal and off-diagonal interaction matrix elements, the QD spectral functions in Fig. \[fig:retardedGF\](b) and (c) show a series of satellites, each of them reflecting the hybridization. The asymmetry stems from the difference between level spacing (40 meV) and LO-phonon energy (36 meV). The broadening of peaks stems mainly from the coupling to the WL states. A finite LO-phonon lifetime of 5 ps due to anharmonic interaction between phonons has been included in the calculations.
Carrier kinetics of relaxation and capture processes
====================================================
In this section we study consequences of the renormalized quasi-particle properties on the scattering processes. Fermi’s golden rule, which has been frequently used in the past, describes only transition rates from fully populated initial into empty final states. Proper balancing between in- and out-scattering events, weighted with the population $f$ of the initial states and the blocking $1-f$ of the final states, leads to the kinetic equation $$\begin{aligned}
& \frac{\partial}{\partial t} \: f_\alpha
= \frac{2 \pi}{\hbar} \:
\sum_{\beta,{\bm q}} \:|M_{\beta \alpha}({\bm q})|^2
\label{eq:Boltzmann} \\
& \hspace{0.0cm}
\times \big\{ \: (1 - f_\alpha) f_{\beta} \big[
(1+n_{LO}) \: \delta( \varepsilon_\alpha - \varepsilon_{\beta} + \hbar\omega_{LO} )
\nonumber \\
& \hspace{2.3cm}
+ n_{LO} \: \delta( \varepsilon_\alpha - \varepsilon_{\beta} - \hbar\omega_{LO} )
\big]
\nonumber \\
& \hspace{0.3cm}
- f_\alpha (1-f_{\beta}) \big[
n_{LO} \: \delta( \varepsilon_\alpha - \varepsilon_{\beta} + \hbar\omega_{LO} )
\nonumber \\
& \hspace{2.3cm}
+(1+n_{LO}) \: \delta( \varepsilon_\alpha - \varepsilon_{\beta} - \hbar\omega_{LO} )
\big]
\big\} \nonumber .\end{aligned}$$ Quasiparticle and Markov approximation can also be applied to renormalized polaronic states which results in a rate-equation description for the population of these states [@Verzelen:00].
A quantum-kinetic approach extends Eq. (\[eq:Boltzmann\]) in the sense that the delta-functions with free-carrier energies are replaced by time-integrals over polaronic retarded GFs. Furthermore, the population factors are no longer instantaneous but explicitly depend on the time evolution. This is the time-domain picture for the inclusion of renormalized quasi-particle properties (beyond a quasi-particle approximation and beyond Markov approximation). Using the generalized Kadanoff-Baym ansatz (GKBA) [@Haug:], the quantum-kinetic equation has the form $$\begin{gathered}
{\frac{\partial^{} }{\partial t_1^{}}} f_{\alpha}(t_1) = 2 {\rm Re} \sum_{\beta}
\int_{-\infty}^{t_1} \! dt_2
~~G^r_{\beta}(t_1-t_2) ~\big[G^r_{\alpha}(t_1-t_2)\big]^* \\
\times \Big\{ \big[ 1-f_{\alpha}(t_2) \big] f_{\beta}(t_2)
~i\hbar D^>_{\alpha \beta}(t_2-t_1) \\
- f_{\alpha}(t_2) \big[ 1-f_{\beta}(t_2) \big]
~i\hbar D^<_{\alpha \beta}(t_2-t_1) \Big\} .
\label{eq:qkin}\end{gathered}$$ The phonon propagator $D^>$ follows from Eq. (\[eq:pn\_propagator\]) by replacing $\tau \rightarrow -\tau$. A Markov approximation [*in the renormalized quasi-particle picture*]{} corresponds to the assumption of a slow time-dependence of the population $f_{\alpha}(t_2)$ in comparison to the retarded GFs such that the population can be taken at the external time $t_1$. The Boltzmann scattering integral of Eq. (\[eq:Boltzmann\]) follows if one additionally neglects quasi-particle renormalizations and uses free-carrier retarded GFs [@note:FT].
![Temporal evolution of the QD population due to carrier-phonon scattering between p-shell and s-shell for an energy spacing larger than the LO-phonon energy. The solid lines correspond to a quantum-kinetic calculation whereas for the dotted line the Markov approximation is used together with polaronic spectral functions. []{data-label="fig:relaxation"}](fig2.eps){width=".45\textwidth"}
To demonstrate the influence of quantum-kinetic effects due to QD-polarons, we first study the relaxation of carriers from p-shell to s-shell for the above discussed situation where the level spacing does not match the LO-phonon energy such that both, Fermi’s golden rule and the kinetic equation (\[eq:Boltzmann\]) predict the absence of transitions. A direct time-domain calculation of the polaron GFs from Eq. (\[eq:Dyson\_G\_ret\])-(\[eq:coupling\]) together with Eq. (\[eq:qkin\]) is used. We assume an initial population $f_s(t_0)=0$, $f_p(t_0)=0.3$ and start the calculation at time $t_0$. While this example addresses the relaxation process itself, more advanced calculations would also include the carrier generation via optical excitation or carrier capture discussed below. Then ambiguities due to initial conditions can be avoided since the population vanishes prior to the pump process which naturally provides the lower limit of the time integral in Eq. (\[eq:qkin\]). In practice, we find that within the GKBA results weakly depend on the details of the initial conditions.
The evaluation of the quantum-kinetic theory (solid lines in Fig. \[fig:relaxation\]) yields a fast population increase of the initially empty QD s-shell accompanied by oscillations which reflect in the time-domain the hybridization of coupled carrier and phonon states. The analogy to Rabi oscillations has been pointed out in Ref. [@Inoshita:97]. If one uses the Markov approximation together with polaronic retarded GFs in Eq. (\[eq:qkin\]), such that quasi-particle renormalizations are still included, these transient oscillations disappear. In both cases the same steady-state solution is obtained which corresponds to a thermal population at the renormalized energies. The equilibrium solution can be obtained from the polaron spectral function using the Kubo-Martin-Schwinger (KMS) relation, $
f_{\alpha}=-\int \frac{ d \hbar \omega }{\pi } ~~
f(\omega) ~ {\rm Im} G_{\alpha}^r(\omega)
$ where $f(\omega)$ is a Fermi function with the lattice temperature. Note that particle number conservation is obeyed in Fig. \[fig:relaxation\] since the degeneracy of the p-shell is twice that of the s-shell.
![Time-evolution of the QD p-shell (solid line) and s-shell (dashed line) electron population due to carrier capture from the WL including the effect of carrier relaxation between QD shells. If only direct capture processes are considered, the dashed-dotted and dotted lines are obtained for p-shell and s-shell, respectively. []{data-label="fig:capture"}](fig3.eps){width=".45\textwidth"}
Another important process is the capture of carriers from the delocalized WL states into the localized QD states. For the used QD parameters, where the spacing between the p-shell and the lowest WL state (40 meV) exceeds the LO-Phonon energy, again Fermi’s golden rule and Eq. (\[eq:Boltzmann\]) predict the absence of electronic transitions. For the numerical solution of Eq. (\[eq:qkin\]) we use now as initial condition empty QD states and a thermal population of carriers in the polaronic WL states (obtained from the KMS relation) corresponding to a carrier density 10$^{11}$ cm$^{-2}$ and temperature 300 K [@note:spec_func]. The dashed-dotted and dotted lines in Fig. \[fig:capture\] show the increase of the p- and s-shell population, respectively, when only capture processes are considered (scattering from a WL-polaron to a QD-polaron state due to emission of LO-phonons). Also in this situation the quantum-kinetic theory predicts a fast population of the initially empty p-shell. Albeit the large detuning (exceeding two LO-phonon energies) the direct capture to the s-shell is still possible but considerably slower. When both, direct capture of carriers as well as relaxation of carriers between the QD states are included in the calculation, the solid (dashed) line is obtained for the p-shell (s-shell) population. While faster capture to the p-shell states leads at early times to a p-shell population exceeding the s-shell populations (see inset of Fig. \[fig:capture\]), the subsequent relaxation efficiently populates the s-states. Since the WL states form a quasi-continuum, beating at early times is strongly suppressed.
![(a) Spectral function of electrons in the coupled QD-WL system for the p-shell (solid line) and s-shell (dashed line) using various energy spacings $\Delta E$ between s-shell, p-shell and WL ($k=0$) in units of the LO-phonon energy $\hbar\omega_{LO}$. Vertical lines indicate the positions of the unrenormalized QD states, $E_G$ is the continuum edge of the WL states. The temperature is 300 K. []{data-label="fig:retGFdetun"}](fig4.eps){width=".45\textwidth"}
![Temporal evolution of the QD population due to carrier-phonon scattering between p-shell (initially populated) and s-shell (initially empty) for different energy spacings $\Delta E$ corresponding to Fig. \[fig:retGFdetun\]. []{data-label="fig:relax_detun"}](fig5.eps){width=".38\textwidth"}
![Time-evolution of the QD p-shell (top) and s-shell (bottom) electron population due to carrier capture from the WL including the effect of carrier relaxation between QD shells for different energy spacings $\Delta E$ corresponding to Fig. \[fig:retGFdetun\]. []{data-label="fig:capt_detun"}](fig6.eps){width=".38\textwidth"}
With the results in Figs. \[fig:retardedGF\]-\[fig:capture\] we have demonstrated the ultrafast (subpicosecond) carrier relaxation and fast (picosecond) carrier capture for a material with weak polar coupling and 10% detuning between the transition energies and the LO-phonon energy. This detuning is on the one hand sufficiently large for the alternative LO+LA mechanism proposed by Inoshita and Sakaki [@Inoshita:92] to fail and on the other hand small enough to illustrate the hybridization of one state with sidebands of the other states. We find that the fast scattering is not related to the near resonance condition and in fact relatively insensitive to the detuning between transition energies and LO-phonon energy. The spectral functions of the coupled QD-WL system for various detunings, ranging from resonance to a large mismatch of 40%, are shown in Fig. \[fig:retGFdetun\]. For better visibility only the curves for the s- and p-shells are displayed, while the WL-states are also included in the calculations. As seen in Fig. \[fig:retardedGF\], the spectral function for the WL states is only weakly influenced by the coupling to the QDs. In all three cases of Fig. \[fig:retGFdetun\] there is substantial overlap between the s-shell and p-shell density of states which points to efficient transition processes. This overlap is due to the multi-peak-structure which contains the series of phonon sidebands spaced by the LO-phonon energy and their hybridization. From top to bottom in Fig. \[fig:retGFdetun\], the peak splitting increases with detuning.
The corresponding results for the carrier relaxation, as in Fig. \[fig:relaxation\] but for different detunings $\Delta E$, are shown in Fig. \[fig:relax\_detun\]. The fast carrier relaxation towards an equilibrium situation is retained in all three cases. The main difference is in the oscillation period, which is reduced for larger detuning due to the increased splitting in Fig. \[fig:retGFdetun\].
A stronger influence of the detuning between transition energies and the LO-phonon energy is found for the capture of carriers from the WL into the QD states. As can be seen in Fig. \[fig:capt\_detun\], from the resonance situation to a detuning of 40% the capture efficiency is reduced by about one order of magnitude. Nevertheless, a significant occupancy can be reached within several ten picoseconds. The reduced capture efficiency is related to a reduced overlap between the WL and QD spectral functions for increasing detuning (which is mainly because the WL states are weakly influenced by the QD states). In comparison to this, the strong interaction between s- and p-states maintains a strong overlap between their spectral functions. As a consequence the relaxation time is less sensitive to the detuning.
In summary, the quantum-kinetic treatment of carrier-phonon interaction explains the absence of a phonon bottleneck in terms of scattering between renormalized quasi-particle states. A quasi-equilibrium situation is reached on a ps-timescale at elevated temperatures even in materials with weak polar coupling.
This work was supported by the Deutsche Forschungsgemeinschaft and with a grant for CPU time at the NIC, Forschungszentrum Jülich.
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for a review see: H. Haug and P. Jauho, [*Quantum Kinetics in Transport & Optics of Semiconductors*]{}, Springer-Verlag, Berlin, 1996.
The quantum-kinetic equation provides energy-conserving scattering processes only in the long-time limit. The corresponding spectral function in Fig. \[fig:retardedGF\] is obtained from a complete Fourier transform whereas on a short timescale the peaks shown there are only partially emerged (since only a finite-time contribution of the retarded GF enters in the quantum-kinetic equation at a given time.)
For small carrier densities, we use the polaron GF for the electron vacuum since it is weakly influenced by population effects, see P. Gartner, L. Bányai, and H. Haug, Phys. Rev. B [**60**]{}, 14234 (1999).
|
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---
abstract: 'We present MatrixNets ($x$Nets), a new deep architecture for object detection. $x$Nets map objects with similar sizes and aspect ratios into many specialized layers, allowing $x$Nets to provide a scale and aspect ratio aware architecture. We leverage $x$Nets to enhance single-stage object detection frameworks. First, we apply $xNets$ on anchor-based object detection, for which we predict object centers and regress the top-left and bottom-right corners. Second, we use MatrixNets for corner-based object detection by predicting top-left and bottom-right corners. Each corner predicts the center location of the object. We also enhance corner-based detection by replacing the embedding layer with center regression. Our final architecture achieves mAP of 47.8 on MS COCO, which is higher than its CornerNet [@law2019cornernet] counterpart by +5.6 mAP while also closing the gap between single-stage and two-stage detectors. The code is available at <https://github.com/arashwan/matrixnet>.'
author:
- Abdullah Rashwan
- Rishav Agarwal
- Agastya Kalra
- Pascal Poupart
bibliography:
- 'egpaper.bib'
title: |
MatrixNets: A New Scale and Aspect Ratio Aware Architecture\
for Object Detection
---
Introduction
============
![(a) Shows the FPN architecture [@lin2017feature], where there are different output layers assigned at each scale. Note we do not show the skip connections for the sake of simplicity. (b) Shows the MatrixNet architecture, where the 5 FPN layers are viewed as the diagonal layers in the matrix. We fill in the rest of the matrix by downsampling these layers.[]{data-label="fig:xnets"}](figure5.png){width="0.8\linewidth"}
Object detection is one of the most widely studied tasks in computer vision with many applications to tasks such as object tracking, instance segmentation, and image captioning. Object detection architectures can be grouped into two categories: two-stage detectors [@li2019scale], and one-stage detectors [@lin2017focal; @law2018cornernet]. Two-stage detectors leverage a region proposal network to find a fixed number of object candidates. Then a second network is used to predict a score for each candidate and to refine its bounding box. Furthermore, one-stage detectors can also be split into two categories: anchor-based detectors [@lin2017focal; @zhu2019feature] and corner (or key-points) based detectors [@law2018cornernet; @duan2019centernet]. Anchor-based detectors contain many anchor boxes, and they predict offsets and classes for each anchor. On the other hand, corner based detectors [@law2018cornernet; @duan2019centernet] predict top-left and bottom-right corner heat-maps and match them together using feature embeddings.
Detecting objects at different scales is a major challenge for object detection. One of the biggest advancements in scale aware architectures was Feature Pyramid Networks (FPNs) [@lin2017feature]. FPNs were designed to be scale-invariant by having multiple layers with different receptive fields so that objects are mapped to layers with relevant receptive fields. Small objects are mapped to earlier layers in the pyramid, and larger objects are mapped to later layers. Since the size of the objects relative to the downsampling of the layer is kept nearly uniform across pyramid layers, a single output sub-network can be shared across all layers. Although FPNs provided an elegant way for handling objects of different sizes, they did not provide any solution for objects of different aspect ratios. Objects such as a high tower, a giraffe, or a knife introduce a design difficulty for FPNs: does one map these objects to layers according to their width or height? Assigning the object to a layer according to its larger dimension would result in loss of information along the smaller dimension due to aggressive downsampling, and vice versa. This problem is prevalent in datasets like MS-COCO [@lin2014microsoftcoco]. Fig. \[fig:histogram\] shows the histogram of the number of objects versus the values of the maximum side of an object divided by the minimum side. We found that 50% of the objects have max/min values higher than 1.75, and 14% have max/min values greater than 3. Hence, modelling these rectangular objects efficiently is essential for good detection performance. In this work, we introduce MatrixNets ($x$Nets), a new scale and aspect ratio aware CNN architecture. $x$Nets, as shown in Fig. \[fig:xnets\], have several matrix layers, each layer handles an object of specific size and aspect ratio. $x$Nets assign objects of different sizes and aspect ratios to layers such that object sizes within their assigned layers are close to uniform. This assignment allows a square output convolution kernel to equally gather information about objects of all aspect ratios and scales. $x$Nets can be applied to any backbone, similar to FPNs. We denote this by appending a “-X” to the backbone, i.e. ResNet50-X [@he2016deep]. As an application for $x$Nets, we first use $x$Nets for anchor-based one-stage object detection. Instead of using multiple anchor boxes per feature map, we decided to consider the case where there is only one box per feature map, making it similar to an anchor free architecture. In a second application, we use $x$Net for corner-based object detection. We show how to leverage $x$Net to improve the CornerNet architecture. On MS-COCO, we set the new state-of-the-art performance (47.8 mAP) for human-designed single-stage detectors.
![Histogram of number of boxes vs the ratio of maximum dimension to minimum dimension of the object.[]{data-label="fig:long"}](figures/histogram_13nov.png){width="0.85\linewidth"}
\[fig:histogram\]
The rest of the paper is structured as follows. Section \[sec:related\_work\] reviews related work. Section \[sec:MatrixNets\] formalizes the idea of MatrixNets. Section \[sec:MatrixNetsApplications\] discusses the application of MatrixNets to different mainstream single-stage object detection frameworks. Section \[sec:Experiments\] covers the experiments, including a thorough ablation analysis. We conclude the paper in Section \[sec:Conclusions\].
Related Work {#sec:related_work}
============
**Two-Stage Detectors:** Two-stage detectors generate the final detection by first extracting RoIs, then in a second stage, hence the name, classifying and regressing on top of each RoIs. The two-stage object detection paradigm was first introduced by R-CNN [@girshick2014richrcnn]. R-CNN used the selective search method [@uijlings2013selective] to propose RoIs, then a CNN network is used to score and refine the RoIs. Fast-RCNN [@girshick2015fast] and SPP [@he2015spatial] improved R-CNN by extracting RoIs from feature maps rather than the input image. Faster-RCNN [@ren2015faster] introduced the Region Proposal Network (RPN), which is a trainable CNN that generates the RoIs allowing the two-stage detectors to be trained end-to-end. Several improvements to the Faster-RCNN framework have been proposed since [@cai2018cascade; @lin2017feature; @li2019scale].
**One-Stage Detectors:** Anchor-based detection is the most common framework for single-stage object detectors. Anchor-based detectors generate the detections by directly classifying and regressing the pre-defined anchors. One of the first single-stage detectors, YOLO [@redmon2016you; @redmon2017yolo9000], is still widely used since it can be run in real time. One-stage detectors tend to be superior in speed, but lagging in performance when compared to two-stage detectors. RetinaNet [@lin2017focal] was the first attempt to close the gap between the two paradigms. RetinaNet proposed the focal loss to help correct for the class imbalance of positive to negative anchor boxes. RetinaNet uses a hand-crafted heuristic to assign anchors to ground-truth objects using Intersection-Over-Union (IOU). Recently, it has been found that improving the anchors to ground-truth object assignments can have a significant impact on the performance [@zhu2019feature; @zhang2019freeanchor]. Further, Feature Selective Anchor-Free (FSAF) [@zhu2019feature] ensembles the anchor-based output with an anchor free output head to improve performance. AnchorFree [@zhang2019freeanchor] improved the anchor to the ground-truth matching process by formulating the problem as a maximum likelihood estimation (MLE). Another framework for one-stage detection is corner-based (or keypoint-based) detectors which was first introduced by CornerNet [@law2018cornernet]. CornerNet predicts top-left and bottom-right corner heat-maps and match them together using feature embeddings. CenterNet [@duan2019centernet] substantially improved CornerNet architecture by predicting object centers along with corners.
MatrixNets {#sec:MatrixNets}
==========
MatrixNets ($x$Nets), as shown in Fig. \[fig:xnets\], model objects of different sizes and aspect ratios using a matrix of layers where each entry $i,j$ in the matrix represents a layer, $l_{i,j}$. Each layer $l_{i,j}$ has a width down-sampling of $2^{i-1}$ and height down-sampling of $2^{j-1}$. The top left layer (base layer) is $l_{1,1}$ in the matrix. The diagonal layers are square layers of different sizes, equivalent to an FPN, while the off-diagonal layers are rectangle layers, unique to $x$Nets. Layer $l_{1,1}$ is the largest layer in size, every step to the right cuts the width of the layer by half, while every step-down cuts the height by half. For example, $Width(l_{3,4}) = 0.5 Width(l_{3,3})$. Diagonal layers model objects with square-like aspect ratios, while off-diagonal layers model objects with more extreme aspect ratios. Layers close to the top right or bottom left corners of the matrix model objects with very high or very low aspect ratios. Such objects are scarce, so these layers can be pruned for efficiency.
Layer Generation
----------------
Generating matrix layers is a crucial step since it impacts the number of model parameters. The more parameters, the more expressive the model, but the harder the optimization problem. In our method, we chose to introduce as few new parameters as possible. The diagonal layers can be obtained from different stages of the backbone or using a feature pyramid backbone [@lin2017feature]. The upper triangular layers are obtained by applying a series of shared 3x3 convolutions with stride 1x2 on the diagonal layers. Similarly, the left bottom layers are obtained using shared 3x3 convolutions with stride 2x1. This sharing helps reduce the number of additional parameters introduced by the matrix layers.
Layer Ranges
------------
We define the range of widths and heights of objects assigned to each layer in the matrix to allow each layer to specialize. The ranges need to reflect the receptive field of the feature vectors of the matrix layers. Each step to the right in the matrix effectively doubles the receptive field in the horizontal dimension, and each step down doubles the receptive field in the vertical dimension. Hence, the range of the widths or heights needs to be doubled as we advance to the right or down in the matrix. Once the range for the first layer $l_{1,1}$ is defined, we can generate the ranges for the rest of the matrix layers using the above rule. For example, if the range for layer $l_{1,1}$ (base layer) is $H\in [24px,48px]$, $W\in [24px,48px]$, the range for layer $l_{1,2}$ will be $H\in [24,48]$, $W\in [48,96]$. We show multiple layer ranges in our ablation studies.
Objects on the boundaries of these ranges could destabilize training since layer assignment would change if there is a slight change in object size. To avoid this problem, we relax the layer boundaries by extending them in both directions. This relaxation is accomplished by multiplying the lower end of the range by a number less than one, and the higher end by a number greater than one. In all our experiments, we use 0.8, and 1.3 respectively.
Advantages of MatrixNets
------------------------
The key advantage of MatrixNets is that they allow a square convolutional kernel to accurately gather information about different aspect ratios. In traditional object detection models, such as RetinaNet, a square convolutional kernel is required to output boxes of different aspect ratios and scales. Using a square convolutional kernel is counter-intuitive since boxes of different aspect ratios and scales require different contexts. In MatrixNets, the same square convolutional kernel can be used for detecting boxes of different scales and aspect ratios since the context changes in each matrix layer. Since object sizes are nearly uniform within their assigned layers, the dynamic range of the widths and heights is smaller compared to other architecture such as FPNs. Hence, regressing the heights and widths of objects becomes an easier optimization problem. Finally, MatrixNets can be used as a backbone to any object detection architecture, anchor-based or keypoint-based, one-stage or two-stage detectors.
MatrixNets Applications {#sec:MatrixNetsApplications}
=======================
In this section, we show that MatrixNets can be used as a backbone for two single-shot object detection frameworks; center-based and corner-based object detection. In center-based object detection, we predict the object centers while regressing the top-left and bottom-right corners. In corner-based object detection, we predict the object corners and regress the center of the object. Corners that predict the same center are matched together to form a detection.
![The Centers-$x$Net architecture.[]{data-label="fig:long"}](figure_centers.png){width="1.0\linewidth"}
\[fig:AnxNet\]
Center-based Object Detection
-----------------------------
Anchor-based object detection is a common framework for single-stage object detection. Using MatrixNet as a backbone naturally handles objects of different scales and aspect ratios. Although using multiple anchors of different scales can potentially improve the performance, we decided to simplify the architecture by using a single anchor per location making it anchor free. Hence, ground truth objects can be assigned to the nearest center location during the training.
### Center-based Object Detection Using MatrixNets
As shown in Fig. \[fig:AnxNet\], our Centers-$x$Net architecture consists of 4 stages. (a-b) We use a $x$Net backbone as defined in Section \[sec:MatrixNets\]. (c) Using a shared output sub-network, for each matrix layer, we predict the center heatmaps, top-left corner regressions, and bottom-right corner regressions for objects within their layers. (d) We combine the outputs of all layers with soft non-maximum suppression [@bodla2017softnms] to achieve the final output.
**Center Heatmaps** During the training, ground truth objects are first assigned to layers in the matrix according to their widths and heights. Within the layer, objects are assigned to the nearest center location. To deal with unbalanced classes, we use focal loss [@lin2017focal].
**Corner Regression** Object sizes are bounded with the matrix layers, which makes it feasible to regress object top-left and bottom-right corners. As shown in Fig. \[fig:AnxNet\], for each center, Centers-$x$Net predicts the corresponding top-left and bottom-right corners. During the training, we use the smooth L1 loss for parameter optimization.
**Training** We use a batch size of 23 for all experiments. During the training, we use crops of sizes 640x640, and we use a standard scale jitter of 0.6-1.5. For optimization, we use the Adam optimizer and set an initial learning rate to 5e-5, and cut it by 1/10 after 250k iterations, training for a total of 350k iterations. For our matrix layer ranges, we set $l_{1,1}$ to be \[24px-48px\]x\[24px-48px\] and then scale the rest as described in Section \[sec:MatrixNets\].
**Inference** For single-scale inference, we resize the max side of the image to 900px. We use the original and the horizontally flipped images as an input to the network. For each layer in the network, we choose the top 100 center detections. Corners are computed using the top-left and bottom-right corner regression outputs. The bounding boxes of the original image and the flipped ones are mixed. Soft-NMS [@bodla2017softnms] layer is used to reduce redundant detections. Finally, we choose the top 100 detections according to their scores as the final output of the detector.
![The Corners-$x$Net architecture.[]{data-label="fig:long"}](figure6.png){width="1.0\linewidth"}
\[fig:KPxNet\]
Corner-based Objection Detection
--------------------------------
CornerNet [@law2018cornernet] was proposed as an alternative to anchor-based detectors, CornerNet predicts a bounding box as a pair of corners: top-left, and bottom-right. For each corner, CornerNet predicts heatmaps, offsets, and embeddings. Top-left and bottom-right corner candidates are extracted from the heatmaps. Embeddings are used to group the top-left, and bottom-right corners that belong to the same object. Finally, offsets are used to refine the bounding boxes producing tighter bounding boxes. This approach has three main limitations.
1. CornerNet handles objects from different sizes and aspect ratios using a single output layer. As a result, predicting corners for large objects presents a challenge since the available information about the object at the corner location is not always available with regular convolutions. To solve this challenge, CornerNet introduced the corner pooling layer that uses a max operation on the horizontal and vertical dimensions. The top left corner pooling layer scans the entire right bottom image to detect any presence of a corner. Eventhough experimentally, it is shown that corner pooling stabilizes the model, we know that max operations lose information. For example, if two objects share the same location for the top edge, only the object with the max features will contribute to the gradient. So, we can expect to see false positive predictions due to corner pooling layers.
2. Matching the top left and bottom right corners is done with feature embeddings. Two problems arise from using embeddings in this setting. First, the pairwise distances need to be optimized during the training, so as the number of objects in an image increases, the number of pairs increases quadratically, which affects the scalability of the training when dealing with dense object detection. The second problem is learning embeddings themselves. CornerNet tries to learn the embedding for each object corner conditioned on the appearance of the other corner of the object. Now, if the object is too big, the appearance of both corners can be very different due to the distance between them. As a result, the embeddings at each corner can be different, as well. Also, if there are multiple objects in the image with a similar appearance, the embeddings for their corners will likely be similar. This is why we saw examples where CornerNet merged persons or traffic lights.
3. As a result of the previous two problems, CornerNet is forced to use the Hourglass-104 backbone to achieve state-of-the-art performance. Hourglass-104 has over 200M parameters, very slow and unstable training, requiring 10 GPUs with 12GB memory to ensure a large enough batch size for stable convergence.
### Corner-based Object Detection Using MatrixNets
Fig. \[fig:KPxNet\] shows our proposed architecture for corner-based object detection, Corners-$x$Net. Corners-$x$Net consists of 4 stages. (a-b) We use a $x$Net backbone as defined in Section 2. (c) Using a shared output sub-network, for each matrix layer, we predict the top-left and bottom-right corner heatmaps, corner offsets, and center predictions for objects within their layers. (d) We match corners within the same layer using the center predictions and then combine the outputs of all layers with soft non-maximum suppression to achieve the final output.
**Corner Heatmaps** Using $x$Nets ensures that the context required for objects within a layer is bounded by the receptive field of a single feature map in that layer. As a result, corner pooling is no longer needed; regular convolutional layers can be used to predict the heatmaps for the top left and bottom right corners. Similar to CornerNet, we use the focal loss to deal with unbalanced classes.
**Corner Regression** Due to image downsampling, refining the corners is important to have tighter bounding boxes. When scaling down a corner to $x$, $y$ location in a layer, we predict the offsets so that we can scale up the corner to the original image size without losing precision. We keep the offset values between $-0.5$, and $0.5$, and we use the smooth L1 loss to optimize the parameters.
**Backbone** **Test Image Size** (px) **Inference Times** (ms) **$AP$** **$AP_{50}$** **$AP_{75}$** **$AP_{S}$** **$AP_{M}$** **$AP_{L}$**
---------------------- -------------------------- -------------------------- ---------- --------------- --------------- -------------- -------------- --------------
800 247 42.7 61.9 46.6 22.6 48.0 59.2
Resnet-152-X-Centers 900 265 43.6 62.3 47.5 24.0 48.4 59.1
1000 275 44.0 63.0 48.2 25.0 49.0 57.9
800 340 44.6 63.4 48.8 24.6 49.5 61.0
Resnet-152-X-Corners 900 355 44.7 63.6 48.7 26.0 49.2 59.8
1000 385 44.5 63.5 48.7 27.7 48.9 58.0
\[table:cornersvscenters\]
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**Center Regression** Since the matching is done within each matrix layer, the width and height of the object are guaranteed to be within a certain range. The center of the object can be regressed easily because the range for the centers is small. In CornerNet, the dynamic range for the centers is large, trying to regress centers in a single output layer would probably fail. Once the centers are obtained, the corners can be matched together by comparing the regressed centers to the actual center between the two corners. During the training, center regression scales linearly with the number of objects in the image compared to quadratic growth in the case of learning embeddings. To optimize the parameters, we use smooth L1 loss.
**Corners Matching** For any pair of corners, the correct center is the average of their x and y locations. The relative distance between the correct center and each corner is the correct values for center regression for both corners if they belong to the same object. Hence, if both corners predict the center with an error rate of $30\%$ or lower, we match these corners together.
**Training** We use a batch size of 23 for all experiments. During the training, we use crops of sizes 512x512, and we use a standard scale jitter of 0.6-1.5. For optimization, we use the Adam optimizer and set an initial learning rate of 5e-5, and cut it by 1/10 after 250k iterations, training for a total of 350k iterations. For our matrix layer ranges, we set $l_{1,1}$ to be \[24px-48px\]x\[24px-48px\] and then scale the rest as described in Section \[sec:MatrixNets\].
**Inference** For single-scale inference, we resize the max side of the image to 900px. We use the original and the horizontally flipped images as an input to the network. For each layer in the network, we choose the top 50 top-left and bottom-right corners. The corner location is refined using corner regression outputs. Then, each pair of corners are matched together, as we discussed above. The bounding boxes of the original image and the flipped ones are mixed. Soft-nms [@bodla2017softnms] layer is used to reduce redundant detections. Finally, we choose the top 100 detections according to their scores as the final output of the detector.
Corners-$x$Net solves the problem (1) of CornerNets because all the matrix layers represent different scales and aspect ratios rather than having them all in a single layer. This also allows us to get rid of the corner pooling operation. (2) is solved since we no longer predict embeddings. Instead, we regress centers directly. By solving the first two problems of CornerNets, we will show in the experiments that we can achieve significantly higher results than CornerNet.
Experiments {#sec:Experiments}
===========
We train all of our networks on a server with 8 Titan XP GPUs. Our implementation is done in PyTorch [@paszke2017automatic], and the code will be made publicly available. For evaluation, we used the MS COCO detection dataset [@lin2014microsoftcoco]. We trained our models on MS COCO ’train-2017’ set, validated on ’val-2017’ and tested on the ’test-dev2017’ set. For comparisons between our models, and ablation study, we reported the numbers on the ’val-2017’ set. For our comparison to other detectors, we reported the numbers on ’test-dev2017’.
In the following subsections, we make a comparison between the performance of the Centers-$x$Net and Corners-$x$Net detectors. Then, we compare our detectors to other detectors. Finally, we show an ablation study through a set of experiments for evaluating different parts of the models.
Centers-$x$Net vs Corners-$x$Net
--------------------------------
In this experiment, we wanted to compare the performance of Centers-$x$Net and Corners-$x$Net. As far as we know, this is the first fair comparison between both frameworks since both are sharing the same backbone (Resnet-152-X), training and inference settings.
Table \[table:cornersvscenters\] shows the performance of both architectures at different test image sizes. mAP numbers are reported on MS COCO ’val-2017’ set. Centers-$x$Net performs the best at a test image size of $1000$px, while Corners-$x$Net performs the best at a test image size of $900$px. Overall, Corners-$x$Net performs better than Centers-$x$Net in terms of mAP numbers. Corners-$x$Net seems more robust to varying image sizes, and the mAP drops by $<$0.2 mAP when varying the test image size by $\pm 100$px. On the other hand, Centers-$x$Net is very sensitive to test image size, and there is a performance drop of $<$1.3 mAP when varying the test image size by $\pm 100$px. The reason for such a drop in performance in the case of Centers-$x$Net is that an object can be entirely missed if two centers of the same objects collide at the same location. Since Centers-$x$Net is equivalent to using one anchor per location, the probability of collisions increases as the test image size decreases. Hence, we can see a drop in performance as we decrease the test image size. For Corners-$x$Net, an object can be missed if collisions happen at both corners, which has a much lower probability compared to the Centers-$x$Net case.
As per inference time, Centers-$x$Net architecture is $100$ms faster than Corners-$x$Net at all test image sizes. Corners-$x$Net uses more prediction outputs than Centers-$x$Net. Also, corner matching uses GPU and CPU time. Hence, there is an overhead of $100$ms using Corners-$x$Net.
One more observation from Table \[table:cornersvscenters\], test image sizes directly impact AP for small, medium, and large objects. This observation can be used to tune the mAP on the set of objects we are chiefly interested in. For consistency, and from this point until the end of this paper, we fix the test image sizes at $900$px.
\[table:comparison\_to\_others\]
Apart from the mAP numbers and inference times, we studied the difference in performance between the center-based and corner-based object detection based on visual inspection. As shown in Fig. \[fig:centervscorner\], there are three main differences that we observed by examining the detection results of the two detectors. First, the corner-based detector generally produces better detections, while center-based sometimes misses visible objects within the image. Fig. \[fig:centervscorner\]a shows some examples that demonstrate such difference. Second, the corner-based detector, as shown in Fig. \[fig:centervscorner\]b, produces refined detections with tighter bounding boxes around the objects compared to the center-based detector. Finally, the center-based detector performs better when detecting objects that are occluded, while the corner-based detector tends to split the detection into smaller bounding boxes. For example, in the first image of Fig. \[fig:centervscorner\]c, the bus is occluded by trees, yet the center-based detector was able to detect the bus correctly. On the other hand, the corner-based detector splits the detection into two smaller bounding boxes.
Comparison To Other Detectors
-----------------------------
We compared our best detectors for both Centers-$x$Net and Corners-$x$Net to other single stage detectors. We report mAP numbers on MS COCO ’test-dev2017’ set. Table \[table:comparison\_to\_others\] shows a comprehensive comparison of our top-performing human-crafted architectures when using single and multi-scale input images to the rest of the one-stage detectors. Corners-$x$Net comes on top for both single and multi-scale input images. It also closes the gap between single-stage and two-stage detectors. Centers-$x$Net performs on-par with other anchor-based architectures while only using a single scale per anchor and without using any object-to-anchor assignment optimization. These results demonstrate the effectiveness of using a MatrixNet as a backbone for object detection architectures.
Ablation Study
--------------
### MatrixNet Design
A 5 layer MatrixNet is equivalent to an FPN, so we use that as a baseline for evaluating adding more matrix layers to the backbone. Table \[table:ablation\_study\]a shows the mAP numbers for different choices of the numbers of the matrix layers. Using 19 layers MatrixNet improves the performance by 5.1 points compared to FPN (5 layers MatrixNet). The extra layers in the 19 layers MatrixNet are much smaller than the FPN layers since each step right or down in the matrix cuts the width or height by half. As a result, the total number of anchors in the 19 layers MatrixNet is 2.5 times those for FPNs.
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We also did a visual inspection for the detection results for both detectors. Fig. \[fig:fpnvsmatrixnet\] shows qualitative examples for the center-based detector when using FPN (5 layers MatrixNet) compared to using 19 layers MatrixNet as backbones. Generally, we observed that using MatrixNet results in better handling of rectangular objects.
### Layer Ranges
In this experiment, we wanted to examine the effect of the base layer ($l_{1,1}$) range choice on the performance of the detector. We used Centers-$x$Net architecture to evaluate the effect of this hyper-parameter. Table. \[table:ablation\_study\]b shows that using the range of $24$px-$48$px is more effective. The goal for selecting this range is to have a balanced object assignment to all of the matrix layers. Selecting a larger range than $24$px-$48$px (e.g., $32$px-$64$px) would require using a larger training image crops to have enough examples to train the bottom right layers in the matrix. This will require more GPUs and longer training times. We also found that the choice of layer ranges is as essential for Corners-$x$Net architectures as it is for Centers-$x$Net.
### Training Image Crop Sizes
During the training, we use scale jitter to scale the image randomly, then we use crops of fixed sizes to train the model. The choice of crop sizes mainly affects the bottom right layers of the MatrixNet. Smaller crop sizes would prevent these layers from having enough objects that span their entire ranges. For Centers-$x$Net, the training crop sizes would impact the performance of the corner regression outputs, and hence the overall performance of the detector. Table \[table:ablation\_study\]c shows the effect of the crop sizes on the overall performance of the Centers-$x$Net architecture. For Corners-$x$Net, the training crop sizes would impact the performance of the center regression outputs. Since center regression output only impacts corner matching, and since we allow for an error of $30\%$, we found that the choice of image crops has little impact on the performance of Corners-$x$Net.
### Backbones
Backbones act as feature extractors. Hence a better and a larger backbone usually results in a better overall performance for an architecture. Table \[table:ablation\_study\]d shows the effect of using Resnet50, Resnet101, and Resnet152 on the overall performance of both Centers-$x$Net and Corners-$x$Net architectures.
Conclusion {#sec:Conclusions}
==========
In this work, we introduced MatrixNet, a scale and aspect ratio aware architecture for object detection. We used MatrixNets to solve fundamental limitations of corner based object detection. We also used MatrixNet as a backbone for anchor-based object detection. In both applications, we showed significant improvements in mAP over the baseline.
We view MatrixNet as a backbone that is an improvement over FPN. We demonstrated the impact of using MatrixNet for one-stage object detection, which can be extended in the future to two-stage object detection. MatrixNets can also replace FPNs in other computer vision tasks such as instance segmentation, key-point detection, and panoptic segmentation tasks. Code for the paper is available at <https://github.com/arashwan/matrixnet>.
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---
abstract: |
[We derive stringent upper bounds on all the $(\lambda''_{ijk}
\mu_l)$-type combinations from the consideration of proton stability, where $\lambda''_{ijk}$ are baryon-number-violating trilinear couplings and $\mu_l$ are lepton-number-violating bilinear mass parameters in a $R$-parity-violating supersymmetric theory.]{}
author:
- |
[**Gautam Bhattacharyya**]{}[^1] and [**Palash B. Pal**]{}[^2]\
[*Saha Institute of Nuclear Physics, 1/AF Bidhan Nagar, Calcutta 700064, India*]{}
title: |
May 1998 SINP/TNP/98-13\
[hep-ph/9806214]{}
[**New constraints on $R$-parity violation from proton stability**]{}
---
-10pt
In the standard model (SM) of particle interactions, baryon and lepton numbers ($B$ and $L$ respectively) appear as accidental global symmetries which are not violated in any order of perturbation theory. On the other hand, in the minimally supersymmetrized version of this model (MSSM), unless one [*assumes*]{} that those are indeed conserved quantities, $R$-parity-violating (${{\not\!{R}}}$) couplings are naturally allowed [@rpar]. Defined as $R = (-1)^{(3B+L+2S)}$ (where $S$ is the spin of the particle), $R$-parity is a discrete symmetry under which SM particles are even while their superpartners are odd. In the MSSM, the most general superpotential is expressed as $$W = W_0 + W',$$ where $W_0$ and $W'$ represent the $R$-parity-conserving and $R$-parity-violating interactions, respectively. In terms of the MSSM superfields, $W_0$ and $W'$ can be expressed as $$\begin{aligned}
\label{W0}
W_0 & = & f_e^{ij} L_i H_d E_j^c + f_d^{ij} Q_i H_d D_j^c
+ f_u^{ij} Q_i H_u U_j^c + \mu H_d H_u, \\
\label{W'}
W' & = & {1\over{2}}\lambda_{ijk} L_i L_j E^c_k +
\lambda'_{ijk} L_i Q_j D^c_k +
{1\over{2}}\lambda''_{ijk} U^c_i D^c_j D^c_k +
\mu_i L_i H_u,
\end{aligned}$$ where $i$, $j$, $k$ are generation indices, which are assumed to be summed over. In eqs. (\[W0\]) and (\[W’\]), $L_i$ and $Q_i$ are SU(2)-doublet lepton and quark superfields, $E^c_i,
U^c_i, D^c_i$ are SU(2)-singlet charged lepton, up- and down-quark superfields, and $H_d$ and $H_u$ are Higgs superfields that are responsible for the down- and up-type masses respectively. In eq. (\[W’\]), $\lambda$, $\lambda'$ and $\mu_i$ are $L$-violating while $\lambda''$ are $B$-violating parameters. $\lambda_{ijk}$ is antisymmetric under the interchange of the first two family indices, while $\lambda''_{ijk}$ is antisymmetric under the interchange of the last two. Thus there could be 27 $\lambda'$-type, 9 each of $\lambda$- and $\lambda''$-type couplings and 3 $\mu_i$ parameters when $R$-parity is explicitly broken.
Since no evidence of $L$ or $B$ violation has been found in experiments to date, one can take either of the two attitudes. The first is to impose $R$-parity that forbids all the terms in $W'$. The second is to take the view that since there is no good theoretical motivation for applying such a symmetry [*a priori*]{}, perhaps it is more general to admit all interactions invariant under supersymmetry and gauge symmetry, and to examine what are the bounds on the couplings in $W'$ from various phenomenological considerations [@review]. We take the latter approach in this paper.
The prime phenomenological concern attached to any ${{\not\!{R}}}$ theory is the question of proton stability. Proton decay requires [*both*]{} $L$ and $B$ violations. Therefore the nonobservation of proton decay could be translated to bounds on the simultaneous presence of $L$- and $B$-violating terms. Thus what could be obtained in this way are correlated bounds involving the $\lambda''$ with one of the $\lambda'$, $\lambda$ or $\mu_i$. Investigations along this direction have already revealed the following constraints.
($i$) The simultaneous presence of $\lambda'$ and $\lambda''$ couplings involving the lighter generations drives proton decay ($p
\rightarrow \pi^0 e^+$) at tree level yielding extremely tight constraints. From the bound on the proton lifetime $\tau_p \gtap
10^{32}$y in the given channel, one obtains [@HK], for an exchanged squark mass of 1 TeV, $$\lambda'_{11k} \lambda''_{11k} \ltap 10^{-24},
\label{lpldp}$$ where $k = 2, 3$. These constraints weaken linearly with the squark mass as the latter increases.
($ii$) One can always find at least one diagram at one loop level in which any $\lambda'_{ijk}$ in conjunction with any $\lambda''_{lmn}$ contributes to proton decay [@SmVi98]. It follows that for an exchanged scalar mass of 1 TeV, $$\lambda'_{ijk} \lambda''_{lmn} \ltap 10^{-9}.$$ If one admits tree level flavour-changing squark mixing, the bounds are strengthened by two orders of magnitude.
($iii$) The combination of $\lambda''$ and $\lambda$, in association with the charged lepton Yukawa coupling $f_e$, drives $p \rightarrow K^+ \ell^{\pm} \ell'^{\mp}\bar{\nu}$ at tree level [@LoPa97]. It should be noted though that the analysis in [@LoPa97] is based on an extended ${\rm SU(3)}_{\rm c} \times {\rm
SU(3)}_{\rm L} \times {\rm U(1)}$ gauge model. However the argument applies also to the usual ${\rm SU(3)}_{\rm c}
\times {\rm SU(2)}_{\rm L} \times {\rm U(1)}$ gauge theory, and the same bound holds. In absence of any direct experimental limits on the above decay modes, this bound is $$\lambda''_{112}\lambda_{ijk} \ltap 10^{-16} \qquad (k\neq 3) \,,$$ assuming a benchmark value $\tau_p \gtap 10^{31}~y$ and an exchanged scalar mass of 1 TeV.
(190,100)(0,0) (10,10)(50,50) (25,30)\[r\][$d^c$]{} (10,90)(50,50) (25,70)\[r\][$s^c$]{} (100,50)(50,50)[3]{} (75,55)\[b\][$\widetilde u^c$]{} (51,40)\[bl\][$\lambda''_{112}$]{} (100,50)(100,90) (97,70)\[r\][$q_1$]{} (100,50)(140,50) (120,55)\[b\][$H_u$]{} (180,50)(140,50) (165,55)\[b\][$\psi_l$]{} (100,40)\[b\][$f_u$]{} (140,40)\[b\][$\mu_l$]{} (135,45)(145,55) (135,55)(145,45)
What therefore remains to be done is to examine the bounds on the third and final type of $L$- and $B$-violating product couplings $\lambda''_{ijk}\mu_l$. This is precisely our goal in the present paper.
Consider the mechanism of nucleon decay shown in Fig. \[f:pdk\]. Here one utilizes the $B$-violating $\lambda''_{112}$, the $L$-violating $\mu_l$ and the canonical $R$-parity-conserving Yukawa coupling $f_u$. For the sake of convenience, we assume that all flavour indices correspond to particles in their mass basis. Recalling that $\lambda''_{ijk}$ is antisymmetric under the interchange of the last two indices, at the quark level we obtain the processes $$d^c s^c \rightarrow u \bar{\nu}_l, \qquad
d^c s^c \rightarrow d \bar{l} \,.$$ They imply the following decay modes of the proton: $$p \rightarrow K^+ \nu_l, ~~~~ p \rightarrow K^+ \pi^+ l^- \,,
\label{dmode}$$ where the charged lepton in the final state can only be $e$ or $\mu$. The relevant dimension-6 operators have an effective coupling $$G_{{{\not\!{R}}}} \simeq \frac{\lambda''_{112} f_u}{m^2_{\tilde{u}_R}}
\left(\frac{\mu_l}{m_{\tilde{H}_u}}\right),$$ leading to an approximate proton lifetime $$\tau_p^{{{\not\!{R}}}} \simeq \left(m_p^5 G^2_{{{\not\!{R}}}}\right)^{-1}.$$ Among the decay modes in eq. (\[dmode\]), the channel $p\to K^+\nu$ ($\tau_p\gtap 10^{32}\,y$ [@pdg]) offers the most stringent constraints. For a squark mass $m_{\tilde{u}_R} = 1$TeV, we obtain $$\lambda''_{112} \epsilon_l \ltap 10^{-21},$$ where $\epsilon_l$ is defined through $\mu_l \equiv \epsilon_l
\mu$. For our order of magnitude estimate, we have made a simple approximation $\mu \simeq m_{\tilde{H}_u} \simeq 1~{\rm TeV}$.
(200,100)(-10,0) (20,20)(50,50) (35,30)\[tl\][$u^c_i$]{} (20,20)(-10,-10) (5,5)\[tl\][$d$]{} (20,80)(50,50) (35,70)\[bl\][$d^c_j$]{} (20,80)(-10,110) (5,100)\[bl\][$u$]{} (20,20)(20,80)[3]{} (17,50)\[r\][$h^+$]{} (100,50)(50,50)[3]{} (75,58)\[b\][$\widetilde d^c_k$]{} (51,40)\[bl\][$\lambda''_{ijk}$]{} (100,50)(100,90) (97,70)\[r\][$q_k$]{} (100,40)\[b\][$f_{d_k}$]{} (100,50)(130,50) (115,55)\[b\][$H_d$]{} (160,50)(130,50) (146,55)\[b\][$H_u$]{} (160,50)(180,50) (175,55)\[b\][$\psi_l$]{} (130,40)\[b\][$\mu$]{} (160,40)\[b\][$\mu_l$]{} (125,45)(135,55) (125,55)(135,45) (155,45)(165,55) (155,55)(165,45)
In the mass basis that we have employed, $\lambda''_{112}$ is the only independent baryon number violating coupling that can be bounded in conjunction with $\mu_l$ at the tree level. However, the other $\lambda''$ couplings also drive proton decay at loop levels, and therefore are also constrained. This can be seen from the 1-loop diagram of Fig. \[f:pdkloop\], where $h^+$ denotes the physical charged Higgs, given by $$\begin{aligned}
h^+ = \sin\beta \; H_d^+ - \cos\beta \; H_u^+ \,,
\end{aligned}$$ where $\tan\beta\equiv\langle H_u^0 \rangle/\langle H_d^0 \rangle$. Let us consider the case of a neutrino in the final state. In this case, the quark on the line $q_k$ must be a down-type quark. The index $k$ cannot be 3 since there cannot be a $b$-quark in the final state. For $k=1$ and 2, the final states will be $\pi^+\bar\nu$ and $K^+\bar\nu$. Indeed, with $k=1$ and 2, we can cover all the independent $\lambda''_{ijk}$-couplings because of the antisymmetry of these couplings in the last two indices. The effective coupling for the dimension-6 operator is given by $$G'_{{{\not\!{R}}}} \simeq \left( {f_u^i f_d^j \over
16\pi^2} \right) \left( V_{i1}^* V_{1j} \right)
\frac{\lambda''_{ijk} f_d^k}
{m^2_{\tilde{d}_R}}
\left(\frac{\mu^2}{m_{\tilde{H}}^2}\right) \epsilon_l \,,$$ where $\tilde H$ is a linear combination of $\tilde H_u$ and $\tilde
H_d$. Although this diagram exists for $\lambda''_{112}$ as well, more stringent bounds on $\lambda''_{112}\epsilon_l$ originate from the tree level diagram discussed earlier. For all other combinations, the bounds are displayed in Table \[t:bounds\].
21 31 32
--- ------------ ------------ ------------
1 $10^{-21}$ $10^{-10}$ $10^{-11}$
2 $10^{-13}$ $10^{-12}$ $10^{-13}$
3 $10^{-14}$ $10^{-13}$ $10^{-14}$
: Summary of upper bounds on the products $\lambda''_{ijk}\epsilon_l$ derived in this paper. The index $i$ appears as row-headings. The indices $j,k$ appear as column headings. The table entry gives the upper bound, which are independent of the index $l$, for superparticle masses of order 1TeV.\[t:bounds\]
Before we conclude, a few comments are in order. Although it has been argued [@HaSu] that the $L_iH_u$ terms can be rotated away by suitable redefinitions of the $L_i$ and $H_d$ fields in the superpotential, they reappear through quantum corrections after supersymmetry is broken. Moreover, $\lambda$- and $\lambda'$-couplings generate bilinear terms via one-loop graphs [@RoMu]. Phenomenological constraints on $\mu_l$ originate from the fact that the bilinear terms trigger sneutrino-Higgs and consequently neutrino-neutralino mixing, leading to non-zero neutrino masses [@deC]. Similarly, individual constraints on various $\lambda''$ exist. They arise, e.g., from $n$-$\bar n$ oscillation [@nnbar] and precision electroweak observables [@bcs].
To conclude, we have used the constraints on proton stability to obtain stringent upper limits on all products of the form $\lambda''_{ijk}\mu_l$. These are complementary to the bounds on the products of the $L$- and $B$-violating couplings obtained before. The combinations $\lambda''_{112}\mu_l$ contribute to proton decay at the tree level and are therefore very suppressed. The other combinations drive proton decay through loop diagrams and the resulting constraints are also displayed. Most importantly, the parameter space we have examined in this paper has never been explored before.
[\[W\]]{}
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H. N. Long, P.B. Pal, `hep-ph/9711455`.
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[^1]: E-mail address: [email protected]
[^2]: E-mail address: [email protected]
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We report the first [*ab initio*]{} density-functional study of $\langle 111\rangle$ screw dislocations cores in the bcc transition metals Mo and Ta. Our results suggest a new picture of bcc plasticity with symmetric and compact dislocation cores, contrary to the presently accepted picture based on continuum and interatomic potentials. Core energy scales in this new picture are in much better agreement with the Peierls energy barriers to dislocation motion suggested by experiments.'
author:
- |
Sohrab Ismail-Beigi$^\dag$ and T.A. Arias$^\ddag$\
$\dag$ Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139\
$\ddag$ Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, NY 14853
date:
title: '[*Ab Initio*]{} Study of Screw Dislocations in Mo and Ta: A new picture of plasticity in bcc transition metals'
---
=10000
The microscopic origins of plasticity are far more complex and less well understood in bcc metals than in their fcc and hcp counterparts. For example, slip planes in fcc and hcp metals are almost invariably close-packed, whereas in bcc materials many slip systems can be active. Moreover, bcc metals violate the Schmid law that the resistance to plastic flow is constant and independent of slip system and applied stress[@Schmid].
Detailed, microscopic observations have established that in bcc metals at low temperatures, long, low-mobility $\langle 111\rangle$ screw dislocations control the plasticity[@Vitek; @Duesbery]. Over the last four decades, the dominant microscopic picture of bcc plasticity involves a complex core structure for these dislocations. The key ingredient of this intricate picture is an extended, non-planar sessile core which must contract before it moves. The first such proposed structure respected the symmetry of the underlying lattice and extended over many lattice constants[@Hirsch]. More recent and currently accepted theories, based on interatomic potentials, predict extension over several lattice constants and spontaneously broken lattice symmetry[@Vitek; @Duesbery; @MGPT; @MGPTTa]. While these models can explain the overall non-Schmid behavior, their predicted magnitude for the critical stress required to move dislocations ([*Peierls stress*]{}) is uniformly too large by a factor of about three when compared to experimental yield stresses extrapolated to zero-temperature[@Duesbery; @Duesbery2].
We take the first [*ab initio*]{} look at dislocation core structure in bcc transition metals. Although we study two metals with quite different mechanical behavior, molybdenum and tantalum, a consistent pattern emerges from our results which, should it withstand the test of time, will require rethinking the presently accepted picture. Specifically, we find screw dislocation cores with compact structures, without broken symmetry, and with energy scales which appear to be in much better accord with experimental Peierls barriers.
[*Ab initio methodology –* ]{} Our [*ab initio*]{} calculations for Mo and Ta are carried out within the total-energy plane-wave density functional pseudopotential approach[@RMP], using the Perdew-Zunger[@PerdewZunger] parameterization of the Ceperly-Alder[@CeperlyAlder] exchange-correlation energy. Non-local pseudopotentials of the Kleinman-Bylander form[@KB] are used with $s$, $p$, and $d$ channels. The Mo potential is optimized according to[@Rappe] and the Ta potential is from[@Tapot]. We use plane wave basis sets with energy cutoffs of 45 Ryd for Mo and 40 Ryd for Ta to expand the wave functions of the valence (outermost $s$ and $d$) electrons. Calculations in bulk show these cutoffs to give total system energies to within 0.01 eV/atom. We carry out electronic minimizations using the analytically continued approach[@ACPRL] within the DFT++ formalism[@DFT++].
To gauge the reliability of the pseudopotentials, Table \[table:bulkprops\] displays our [*ab initio*]{} results for the materials’ lattice constants and those elastic moduli most relevant for the study of $\langle 111\rangle$ screw dislocations. The tabulated moduli describe the long-range elastic fields of the dislocations ($K$), the coupling of displacement gradients along the dislocation axis $z$ to core-size changes in the orthogonal $x,y$ plane ($c_{xx,zz}=(c_{11}+5c_{12}-2c_{44})/6$), and the coupling of core-size changes to themselves in the plane ($c_{xx,xx}=(c_{11}+c_{12}+2c_{44})/2$ and $c_{xx,yy}=(c_{11}+2c_{12}-2c_{44})/3$). These results indicate that our predicted core energy differences should be reliable to within better than , which suffices for the purposes of our study.
[*Preparation of dislocation cells –*]{} The cell we use for dislocation studies has lattice vectors $\vec{a}_1 = 5a[1,-1,0]$, $\vec{a}_2=3a[1,1,-2]$, and $\vec{a}_3=a[1,1,1]/2$, where $a$ is the lattice constant. We call this ninety-atom cell the “5$\times$3” cell in reference to the lengths of $\vec{a}_1$ and $\vec{a}_2$, and the Burgers vectors of all of the dislocations in our work are along $\vec{a}_3$. Eight $k$-points $k_1=k_2={1 \over 4},k_3\in\pm\{{1\over
16},{3\over 16},{5\over 16},{7\over 16}\}$ sample the Brillouin zone in conjunction with a non-zero electronic temperature of $k_BT=0.1$ eV, which facilitates the sampling of the Fermi surface. These choices give total energies to within 0.01 eV/atom.
Given the relatively small cell size, we wish to minimize the overall strain and the effects of periodic images. We therefore follow [@PayneSi] and employ a quadrupolar arrangement of dislocations (a rectangular checkerboard pattern in the $\vec{a}_1,\vec{a}_2$ plane). This ensures that dislocation interactions enter only at the quadrupolar level and that the net force on each core is zero by symmetry, thereby minimizing perturbations of core structure due to the images. As was found in[@PayneSi] and as we explore in detail below, we find very limited impact of finite-size effects on the cores when following this approach.
In bcc structures, screw dislocations are known to have two inequivalent core configurations, termed “easy” and “hard”[@Vitek; @MGPT; @MGPTTa]. These cores can be obtained from one another by reversing the Burgers vector of a dislocation line while holding the line at a fixed position. We produce cells with either only easy or only hard cores in this way. To create atomic structures for the cores, we proceed in three stages. First, we begin with atomic positions determined from isotropic elasticity theory for our periodic array of dislocations. Next, we relax this structure to the closest local energy minimum within the interatomic MGPT model for Mo[@MGPT]. Since we do not have an interatomic potential for Ta and expect similar structures in Ta and Mo[@MGPTTa], we create suitable Ta cells by scaling the optimized MGPT Mo structures by the ratio of the materials’ lattice constants. Finally, we perform standard [*ab initio*]{} atomic relaxations on the resulting MGPT structures until all ionic forces in all axial directions are less than 0.06 eV/Å.
[*Extraction of core energies –* ]{} The energy of a long, straight dislocation line with Burgers vector $\vec b$ is $E = E_c(r_c) +
Kb^3\ln(L/r_c)$ per $b$ along the line[@HirthLothe], where $L$ is a large-length cutoff, and $K$ is an elastic modulus (see Table \[table:bulkprops\]) computable within anisotropic elasticity theory[@Head]. The core radius $r_c$ is a short-length cutoff inside of which the continuum description fails and the discrete lattice and electronic structure of the core become important. $E_c(r_c)$ measures the associated “core energy”, which, due to severe distortions in the core, is most reliably calculated by [*ab initio*]{} methods.
The energy of our periodic cell contains both the energy of four dislocation cores and the energy stored outside the core radii in the long-range elastic fields. To separate these contributions, we start with the fact that two straight dislocations at a distance $d$ with equal and opposite Burgers vectors have an anisotropic elastic energy per $b$ given by $E = 2E_c(r_c) + 2Kb^3\ln(d/r_c)$. Next, by regularizing the infinite sum of this logarithmically divergent pair interaction, we find that the energy per dislocation per $b$ in our cell is given by $$E = E_c(r_c) + Kb^3\left[\ln\left({|\vec{a}_1|/2\over r_c}\right) +
A\left({|\vec{a}_1|\over |\vec{a}_2|}\right)\right].
\label{eq:ewaldsum}$$ The function $A(x)$ contains all the effects of the infinite Ewald-like sums of dislocation interactions and has the value $A =
-0.598\,846\,386$ for our cell. Subtracting the long-range elastic contribution (the second term of (\[eq:ewaldsum\])) from the total energy, we arrive at the core energy $E_c$.
To test the feasibility of this approach, we compare $E_c(r_c)$ for the MGPT potential as extracted with the above procedure from cells of two different sizes: the 5$\times$3 cell and the corresponding 9$\times$5 cell. (The MGPT is [*fit*]{} to reproduce experimental elastic moduli, so $K$ is given in Table \[table:bulkprops\].) With the choice $r_c=2b$, Table \[table:MGPT5x39x5\] shows that our results, even for the 5$\times$3 cell, compare quite favorably with those of[@MGPT; @MGPT2], especially given that our 5$\times$3 and 9$\times$5 cells contain only ninety and 270 atoms respectively, whereas the cited works used cylindrical cells with a single dislocation and [*two thousand*]{} atoms or more. Given the suitability of the 5$\times$3 cell, all [*ab initio*]{} results reported below are carried out in this cell.
[*Ab initio core energies –* ]{} Except for the Mo hard core, all the core structures relax quite readily from their MGPT configurations to their equilibrium [*ab initio*]{} structures. The Mo hard-core configuration, however, spontaneously relaxes into easy cores, strongly indicating that the hard core, while meta-stable within MGPT by only 0.02 eV/$b$[@MGPT2], is not stable in density functional theory. We do not believe that this instability is due to finite-size effects, which appear to be quite small for the reasons outlined previously.
Table \[table:compareabiMGPT\] compares our [*ab initio*]{} results to available MGPT results for core energies in Mo and Ta. To make comparison with the MGPT, for the unstable Mo hard core we evaluate the [*ab initio*]{} core energy at the optimal MGPT atomic configuration (column AI$^*$ in Table \[table:compareabiMGPT\]). Note that, in computing hard–easy core energy differences, the long-range elastic contributions cancel so that these differences are much better converged than the absolute core energies.
=3.0in
Table \[table:compareabiMGPT\] shows that the MGPT hard–easy core energy differences are much larger than the corresponding [*ab initio*]{} values by approximately a factor of three. The accuracy of the elastic moduli of Table \[table:bulkprops\], combined with the high transferability of the local-density pseudopotential approach, indicates that this factor of three is not an artifact of our approximations. We believe that the reason for this discrepancy is that the MGPT is less transferable. Having been forced to fit [*bulk*]{} elastic moduli and thus long-range distortions, the MGPT may not describe the short wavelength distortions in the cores with high accuracy. An examination of Mo phonons along $[100]$ provides poignant evidence: the MGPT frequencies away from the zone center are too large when compared to experimental and band-theoretic values[@MGPT] and translate into spring constants that are up to approximately three times too large.
The magnitude of the core energy difference has important implications for the magnitude of the Peierls energy barrier and Peierls stress for the motion of screw dislocations in Mo and Ta. In a recent Mo MGPT study[@MGPT2], the most likely path for dislocation motion was identified to be the $\langle 112\rangle$ direction: the moving dislocation core changes from easy to hard and back to easy as it shifts along $\langle 112\rangle$. The energy barrier was found to be $0.26$ eV/$b$, very close to the MGPT hard–easy energy difference itself. The fact that the [*ab initio*]{} hard–easy energy differences in Mo and Ta are smaller by about a factor of three than the respective interatomic values suggests that the [*ab initio*]{} energy landscape for the process has a correspondingly smaller scale. If so, the Peierls stress in Mo and Ta should also be correspondingly smaller and in much better agreement with the values suggested by experiments.
[*Dislocation core structures –* ]{} Figure \[fig:MGPTDDmaps\] shows differential displacement (DD) maps[@Vitek] of the core structures we find in our ninety-atom supercell when working with the interatomic MGPT potential for Mo. Our DD maps show the atomic structure projected onto the (111) plane. The vector between a pair of atomic columns is proportional to the change in the $[111]$ separation of the columns due to the presence of the dislocations. The maps show that both easy and hard cores have approximate 3-fold rotational ($C_3$) point-group symmetry about the out-of-page $[111]$ axis through the center of each map. The small deviations from this symmetry reflect the weakness of finite-size effects in our quadrupolar cell. The hard core has three additional 2-fold rotational ($C_2$) symmetries about the three $\langle 110\rangle$ axes marked in the maps, increasing its point-group symmetry to the dihedral group $D_3$ which is shared by the underlying crystal. The easy core, however, shows a strong spontaneous breaking of this symmetry: its core spreads along only three out of the six possible $\langle 112\rangle$ directions. Our results reproduce those of[@MGPT; @MGPT2] who employed much larger cylindrical cells with open boundaries, underscoring the suitability of our cell for determining core structure. This symmetry-breaking core extension is that which has been theorized to explain the relative immobility of screw dislocations and violation of the Schmid law in bcc metals.
Figure \[fig:abiDDmaps\] displays DD maps of our [*ab initio*]{} core structures. Contrary to the atomistic results, we find that the low-energy easy cores in Mo and Ta have full $D_3$ symmetry and do not spread along the $\langle 112\rangle$ directions. Combining this with the above results concerning core energetics, we have two examples for which our pseudopotentials are sufficiently accurate to disprove the conventional wisdom that generic bcc metallic systems [*require*]{} broken symmetry in the core to explain the observed immobility of screw dislocations.
Turning to the hard core structures, the [*ab initio*]{} resuts for Ta show a significant distortion when compared to the atomistic core (contrast Figure \[fig:MGPTDDmaps\]b and Figure \[fig:abiDDmaps\]b). As the [*ab initio*]{} Mo hard core was unstable, we believe that this distortion of the Ta hard core suggests that this core is much less stable within density functional theory than in the atomistic potentials.
To complete the specification of the three-dimensional [*ab initio*]{} structure of easy cores in Mo and Ta, Figure \[fig:horizmaps\] presents maps of the atomic displacement in the (111) plane. The small atomic shifts, which are due entirely to anisotropic effects, are shown as in-plane vectors centered on the bulk atomic positions and magnified by a factor of [*fifty*]{}. To reduce noise in the figure, before plotting we perform $C_3$ symmetrization of the atomic positions about the $[111]$ axis passing through the center of the figure. As all the dislocation cores in our study have a minimum of $C_3$ symmetry, this procedure does not hinder the identification of possible spontaneous breaking of the larger $D_3$ symmetry group. Our maps indicate that the easy cores in both Mo and Ta have full $D_3$ symmetry.
=3.0in
Recent high-resolution electron microscopy explorations of the symmetry of dislocations in Mo have focused on the small shifts in the (111) plane of columns of atoms along $[111]$[@Sigle]. This pioneering work reports in-plane displacements extending over a range much greater than the corresponding MGPT results and also much greater than what we find [*ab initio*]{}. In[@Sigle] this is attributed to possible stresses from thickness variations and foil bending. We believe this makes study of the internal structure of the core difficult, and that cleaner experimental results are required to resolve the nature of the symmetry of the core and its extension.
In conclusion, our first principles results show no preferential spreading or symmetry breaking of the dislocation cores and exhibit an energy landscape with the proper scales to explain the observed immobility of dislocations. Atomistic models which demonstrate core spreading and symmetry breaking, both of which tend to reduce the mobility of the dislocations, are well-known to over-predict the Peierls stress. The combination of these two sets of observations argues strongly in favor of much more compact and symmetric bcc screw dislocation cores than presently believed.
This work was supported by an ASCI ASAP Level 2 grant (contract \#B338297 and \#B347887). Calculations were run primarily at the Pittsburgh Supercomputing Center with support of the ASCI program and also on the MIT Xolas prototype SMP cluster. We thank members of the H-division at Lawrence Livermore National Laboratories for providing the Ta pseudopotential, the Mo MGPT code, and many useful discussions.
=3.0in
[0]{}
---------------- ------- -------- ---------- ------- -------- ----------
AI Expt Error AI Expt Error
$a$ 3.10 3.15 -1.6% 3.25 3.30 -1.5%
$K$ 1.60 1.36 18% 0.65 0.62 5%
$c_{xx,zz}$ 2.17 1.91 14% 1.72 1.39 24%
$c_{xx,xx}$ 5.48 4.25 29% 3.02 2.98 1.3%
$c_{xx,yy}$ 2.21 1.77 25% 1.72 1.49 15%
---------------- ------- -------- ---------- ------- -------- ----------
$E_c$ (eV/$b$) 5$\times$3 9$\times$5 Cylindrical[@MGPT; @MGPT2]
--------------------- -------------------- -------------------- ----------------------------
hard 2.57 2.57 2.66
easy 2.35 2.31 2.42
$\Delta$ 0.22 0.26 0.24
---------------- ------ -------- ------ --------------- ------
$E_c$ (eV/$b$) MGPT AI$^*$ AI MGPT[@MGPTTa] AI
hard 2.57 2.94 – – 0.91
easy 2.35 2.86 2.64 – 0.86
$\Delta$ 0.22 0.08 – 0.14 0.05
---------------- ------ -------- ------ --------------- ------
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Exit times for stochastic Ginzburg-Landau classical field theories with two or more coupled classical fields depend on the interval length on which the fields are defined, the potential in which the fields deterministically evolve, and the relative stiffness of the fields themselves. The latter is of particular importance in that physical applications will generally require different relative stiffnesses, but the effect of varying field stiffnesses has not heretofore been studied. In this paper, we explore the complete phase diagram of escape times as they depend on the various problem parameters. In addition to finding a transition in escape rates as the relative stiffness varies, we also observe a critical slowing down of the string method algorithm as criticality is approached.'
bibliography:
- 'Reference.bib'
---
[ p ]{}
[Noisy Classical Field Theories with Two Coupled Fields: Dependence of Escape Rates on Relative Field Stiffnesses ]{}\
[**Lan Gong$^{1}$ and D. L. Stein$^{1,2}$**]{}\
[<[email protected]>]{} [<[email protected]>]{}\
$^{1}$[*Department of Physics, New York University, New York, NY 10003*]{}\
$^{2}$[*Courant Institute of Mathematical Sciences, New York University, New York, NY 10003*]{}
Introduction {#sec:introduction}
============
In a previous paper [@GS10] (hereafter GS), the authors introduced and solved a system of two coupled nonlinear stochastic partial differential equations. Such equations are useful for modelling noise-induced activation processes of spatially varying systems with multiple basins of attraction. Examples of such processes include micromagnetic domain reversal [@MSK05; @MSK06], pattern nucleation [@Cross93; @Tu97; @Bisang98], transitions in hydrogen-bonded ferroelectrics [@Dikande97], dislocation motion across Peierls barriers [@GB97], and structural transitions in monovalent metallic nanowires [@BSS05; @BSS06]. It is the last problem in particular that the model introduced in GS was constructed to analyze.
The GS model provided a mathematical realization of a stochastic Ginzburg-Landau field theory consisting of two coupled classical fields, denoted $\phi_1(z)$ and $\phi_2(z)$, defined on a linear domain of finite extent $L$. Stochastic partial differential equations of this type are constructed to model noise-driven transitions between locally stable states. In the especially important case of weak noise, where the transition rate is of the Arrhenius form $\Gamma_0e^{-\Delta E/\epsilon}$, with prefactor $\Gamma_0$ and activation barrier $\Delta E$ independent of the noise $\epsilon$, the transition path occurs near (i.e., within a lengthscale of order $O(\epsilon^{1/2})$) the saddle (or col) of least action connecting the two stable states.
The two-field model displayed several interesting features, including a type of “phase transition” in activation behavior as $L$ varied. The transition was driven by a change in the saddle state, from a uniform configuration at small $L$ to a spatially varying one (“instanton”) at larger $L$. This transition has been noticed and analyzed for the case of a single field [@MS01b; @MS03], but had not been seen in the rarely studied case of a system with [*two*]{} coupled fields. Perhaps more remarkably, the system admitted an exact solution for the instanton state; such exact solutions are rare in the case of nonlinear field theories with a single field, much less a nontrivial system of coupled fields.
The introduction of two fields was required to study transitions among different quantized conductance states in non-axisymmetric nanowires. The axisymmetric case had previously been treated theoretically in [@BSS04; @BSS05]. However, detailed studies using linear stability analysis by Urban [*et al.*]{} [@UBZSG04] indicated that roughly 1/4 of all such transitions involved either non-axisymmetric initial or final states, or else a least-action transition passing through a non-axisymmetric saddle. To describe such transitions, one field ($\phi_1(z)$) describes radial variations along the wire length and the other ($\phi_2(z)$) describes deviations from axisymmetry.
One restriction of the analysis in GS was that the respective bending coefficients $\kappa_1$ and $\kappa_2$ of the two fields were taken to be equal. However, this is generally not the case in real nanowires [@UBZSG04]. Therefore, in order to apply the model to actual transitions, as well as to provide a complete picture of the activation behavior in such systems, we need to consider the case where $\kappa_1\ne\kappa_2$. In such cases analytical solutions cannot be found and we need to rely on numerical methods. The study of this more general problem is the subject of this paper.
The Model {#sec:model}
=========
Consider two coupled classical fields $\phi_1(z)$, $\phi_2(z)$ on the interval $[-L/2,L/2]$, subject to the energy functional $$\begin{aligned}
{\cal H}=\int_{-L/2}^{L/2} \!
(\frac{1}{2}\kappa_{1}(\phi_{1}'(z))^{2}+\frac{1}{2}\kappa_{2}(\phi_{2}'(z))^{2}+U(\phi_{1},\phi_{2}))
\, dz\, .
\label{eq:H}
\intertext{where}
U(\phi_1,\phi_2)=-\frac{\mu_1}{2}\phi_1^{2}+\frac{1}{4}\phi_1^{4}-\frac{\mu_2}{2}\phi_{2}^{2}+\frac{1}{4}\phi_2^{4}+\frac{1}{2}\phi_{1}^{2}\phi_{2}^{2}
\label{eq:U}\end{aligned}$$
The bending coefficients $\kappa_1$, $\kappa_2$ can be related to the wire surface tension [@BSS04; @BSS05]. The arbitrary positive constants $\mu_1$, $\mu_2$ are chosen such that $\mu_1 \neq \mu_2$, breaking rotational symmetry. (The case $\mu_1=\mu_2$ has been investigated analytically by Tarlie [*et al.*]{} [@Goldbart94] in the context of phase slippage in conventional superconductors.)
If the system is subject to additive spatiotemporal white noise, then their time evolution is governed by the pair of stochastic partial differential equations: $$\begin{aligned}
\label{eq:timeevolution}
\dot{\phi_{1}}=\kappa_1\phi_{1}''+\mu_{1}\phi_{1}-\phi_{1}^{3}-\phi_{1}\phi_{2}^{2}+\sqrt{2\epsilon}\,\xi_{1}(z,t)\nonumber\\
\dot{\phi_{2}}=\kappa_2\phi_{2}''+\mu_{2}\phi_{2}-\phi_{2}^{3}-\phi_{1}^{2}\phi_{2}+\sqrt{2\epsilon}\,\xi_{2}(z,t)\, ,\end{aligned}$$ where $\xi_{1,2}$ are the spatiotemporal noise terms satisfying $<\xi_{i}(z_{1},t_{1})\xi_{j}(z_{2},t_{2})>=\delta(z_{1}-z_{2})\delta(t_{1}-t_{2})\delta_{ij},~i,j=1,2$. If the noise is due to thermal fluctuations, then by the fluctuation-dissipation theorem $\epsilon = k_BT$.
The activation energy $\Delta E$ and prefactor $\Gamma_0$ in the Arrhenius rate formula depend not only on the details of the potential (\[eq:U\]), but also on the interval length $L$ on which the fields are defined, and on the choice of boundary conditions at the endpoints $z=-L/2$ and $z=L/2$. It was shown in [@Burki03] that Neumann boundary conditions are appropriate for the nanowire problem, and we will employ them here.
In the usual case of a single field, the bending coefficient $\kappa$ plays a role in determining the intrinsic lengthscale (and therefore the transition length at which the saddle state changes) on which field variations occur; but once it is absorbed into a dimensionless lengthscale by rescaling along with the parameters determining the potentials, it plays no further role. Now, however, there are [*two*]{} bending coefficients, and varying their relative magnitudes can in principle lead to new phenomena. The aim of this paper is to study the effects of these variations.
The metastable and saddle states are time-independent solutions of the zero-noise equations: $$\begin{aligned}
\kappa_1\phi_1''=-\mu_1\phi_1+\phi_1^{3}+\phi_1\phi_2^{2}\nonumber\\
\kappa_2\phi_2''=-\mu_2\phi_2+\phi_2^{3}+\phi_2\phi_1^{2}
\label{eq:zero-noise}\end{aligned}$$ Without loss of generality, we choose $\mu_{1}>\mu_{2}$.
Then there are two metastable states: $\phi_{1,s}=\pm\sqrt{\mu_1}$, $\phi_{2,s}=0$; two spatially uniform saddle states: $\phi_{1,u}=0$, $\phi_{2,u}=\pm\sqrt{\mu_2}$; and spatially nonuniform saddle states, or instantons. When $\kappa_1=\kappa_2(=1)$, analytical solutions for the instanton saddle states can be found: $$\begin{gathered}
\phi^{\rm
inst}_{1,m}(z)=\pm\sqrt{m}\sqrt{(2\mu_{1}-\mu_{2})-m(\mu_{1}-\mu_{2})}{\rm sn}(\sqrt{\mu_{1}-\mu_{2}}\,z|m)
\label{eq:ins1}\\ \phi^{\rm
inst}_{2,m}(z)=\pm\sqrt{\mu_{2}-m(\mu_{1}-\mu_{2})}{\rm dn}(\sqrt{\mu_{1}-\mu_{2}}\,z|m)
\label{eq:ins2}\end{gathered}$$ where ${\rm sn}(.|m)$and ${\rm dn}(.|m)$ are the Jacobi elliptic functions with parameter $m$, whose periods are $4K(m)$ and $2K(m)$ respectively, with $K(m)$ the complete elliptic integral of the first kind [@Abramowitz65]. The parameter $m \in [0, 1]$ is related to interval length $L$ through the Neumann boundary conditions, with $m\to 0^+$ corresponding to $L\to L_c^+$, where $L_c$ is the critical length, and $m\to 1$ corresponding to $L\to\infty$ [@MS01b; @MS03; @GS10]. When $\kappa_1=\kappa_2=1$, $$\label{eq:lc}
L=\frac{2K(m)}{\sqrt{\mu_{1}-\mu_{2}}}$$
We found in GS that varying $L$ triggers a transition between the uniform and instanton saddle states; the critical length $L_c$ is determined by: $$\label{eq:Lc}
L_c = \frac{2K(0)}{\sqrt{\mu_{1}-\mu_{2}}}\\
=\frac{\pi}{\sqrt{\mu_{1}-\mu_2}}$$ This results in a transition in the activation behavior, including anomalous behavior at the critical length. Such a transition may have already been seen experimentally [@BSS06], in a crossover from ohmic to nonohmic conductance as the voltage across gold quantum point contacts increases [@YOT05]. We will show below that the same effect occurs when the ratio $\kappa_1/\kappa_2$ is varied.
As noted above, the transition rate in the low-noise ($\epsilon\to 0$) limit is given by the Kramers formula: $$\label{eq:rate}
\Gamma\sim\Gamma_0 \exp(-\Delta E /\epsilon)\, .$$ Here $\Delta E$ is the activation barrier, that is, the difference in energy between the saddle and the starting metastable states, while $\Gamma_0$ is the rate prefactor: $$\label{eq:prefactor}
\Gamma_{0}=\frac{1}{\pi}\sqrt{\left|\frac{\det\,{\bf\Lambda}_{s}}{\det\,{\bf\Lambda}_{u}}\right|}|\lambda_{u,1}|\, .$$ In the above equation ${\bf\Lambda}_s$ is the linearized dynamical operator describing perturbations about the metastable state; similarly ${\bf\Lambda}_u$ describes perturbations about the saddle. $\lambda_{u,1}$ is the single negative eigenvalue of ${\bf\Lambda}_{u}$, corresponding to the direction along which the most probable transition path approaches the saddle state. The behavior of $\Gamma_0$ becomes anomalous at the critical point $L_c$, where fluctuations around the most probable path become large.
Calculation of the Minimum Energy Path {#sec:mep}
======================================
Computation of exit behavior requires knowledge of the transition path(s), in particular behavior near the local minimum and the saddle. In our model, both are found as solutions of two coupled nonlinear differential equations [@GS10]. A powerful numerical technique constructed explicitly for this type of problem is the “string method” of E, Ren, and Vanden-Eijnden[@ERenEric02; @ERenEric07]. The algorithm proceeds by evolving smooth curves, or strings, under the zero-noise dynamics. These strings connect the beginning and final locally stable states, and in between the two ends each string contains a series of intermediate states called “images”. The method is constructed so that the string evolves towards the most probable transition path. The evolution proceeds until the condition for equilibrium is reached: $$[\delta{\cal H}]^{\perp}=0$$ \[eq:string equation\] where ${\cal H}$ is given by (\[eq:H\]) and $[\delta{\cal H}]^{\perp}$ is its component perpendicular to the string.
Once equilibrium is reached, the string images correspond to the configurations sampled by the system at different stages of the activation process. The image with highest energy is the one nearest the saddle state. In order to get an accurate result, the distribution of images needs to be sufficiently fine-grained. In our computation, we used 61 images (including the two ends of the string); because of the symmetry of our energy functional, the image in the middle corresponds to the saddle.
When such symmetry is absent and the location of the saddle needs to be determined with high precision, one can use an alternative method to the brute force one of simply increasing the number of images along the string. This alternative requires first finding a rough approximation of the most probable transition path, again using the string method but with a small number of images, and then switching to a “climbing image” algorithm in which one picks up an image that is believed close to the saddle and then drive it towards the saddle. The climbing force is obtained from inverting the energy gradient along the direction of the unstable eigenvector of the saddle state. Details can be found in [@ERenEric02; @ERenEric07].
We have found an analogue to critical slowing down in the current context: near criticality convergence of the string method becomes increasingly slow. Expanding the energy functional around the saddle reveals that the lowest stable eigenvalue vanishes to second order, leading to a rapid increase in relaxation time. This phenomenon will be further investigated in the following sections.
Results {#sec:results}
=======
We now turn to the case $\kappa_{1} \neq \kappa_{2}$. To begin, we fix $\kappa_{2}=1$ and vary $\kappa_{1}$. We consider the cases where $\kappa_1$ is both less than and greater than 1. Because the critical length now depends on $\kappa_1$, we denote it $L_c(\kappa_1)$.
As noted earlier (cf. (\[eq:Lc\]), $L_{c}(1)=\frac{\pi}{\sqrt{\mu_{1}-\mu_{2}}}$: below $L_c(1)$ the saddle is spatially uniform, and above $L_c(1)$ it is spatially varying [@GS10]. The situation becomes more complicated when $\kappa_1 \neq 1$. Fig. \[fig:transition state evolution\] summarizes our results when $\mu_1=3$, $\mu_2=2$ and $L>L_c(1)=\pi$. In this and Fig. \[fig:transition state evolution 250\], the saddle state (whether uniform or instanton) is denoted $\phi_i^{\rm
saddle}(z), i=1,2$. We find that as $\kappa_{1}$ increases, the spatial variation of the instanton becomes increasingly suppressed until the instanton finally collapses to the uniform state. Conversely, when $L<L_c(1)$, the instanton state is retrieved for $\kappa_1<1$ (cf. Fig. \[fig:transition state evolution 250\]).
This effect can be understood as follows. In the limit of low noise, the transition occurs over the saddle state of least energy. An increase in $\kappa_1$ raises the bending energy of any nonuniform configuration, and therefore that of the instanton, while leaving the energy of the uniform saddle unchanged. When the energies of these two states cross, the activation process switches from one saddle state to the other. This is seen explicitly in Fig. \[fig:energy\], where we plot the energy of the saddle state against $\kappa_1$ for both $L=0.25$ and $L=4.51$. In these figures the curve to the left of the dashed line is the instanton branch, which increases monotonically until it reaches a constant value: the energy of the uniform saddle state.
We next investigated the question of whether the transition from uniform saddle to instanton (or vice-versa) as $\kappa_1$ varies occurs as a continuous crossover or as an abrupt phase transition. If the latter, then we also need to determine the order of the transition.
In [@GS10], the uniform $\to$ instanton saddle transition was induced by changing $L$ at fixed $\kappa_1$. There we concluded that the transition was reminiscent of a second-order phase transition, in the asymptotic $\epsilon\to 0$ limit. This follows from the continuity of the activation energy at all values of $L$, including $L_c(1)$. (For examples of potentials where the activation energy jumps at a precise value of $L$, corresponding to a first-order transition, see [@BSS08].) In fact, it can be shown analytically that the first derivative of the activation energy curve with respect to $L$ is also continuous everywhere, but the second derivative is discontinuous at $L_c(1)$.
Similarly, Fig. \[fig:energy\] suggests that there is indeed a continuous phase transition, in that the activation energy changes continuously as one passes through the transition, as $\kappa_1$ varies for fixed $L$. This continuity leads to a divergence in the transition rate prefactor, shown in Fig. \[fig:prefactor\_L451\] (similar to that induced by changing $L$ at fixed $\kappa_1$ in [@GS10]). The value of $\kappa_{1c}(L=4.51)$ where the prefactor diverges and that where the energies of the respective saddles cross agree to within a numerical error of $10^{-2}$.
What causes this divergence? Away from criticality, the spectrum of the linearized dynamical operator $\Lambda_u$ about the saddle consists of a single negative eigenvalue, whose corresponding eigenvector determines the unstable direction, with all other eigenvalues positive. As criticality is approached, the smallest positive eigenvalue, denoted $\lambda_{u,2}$, approaches zero. This signals the mathematical divergence on the “normal” lengthscale of $O(\epsilon^{1/2})$ of fluctuations about the saddle, and by (\[eq:prefactor\]) is seen to lead to divergence of the prefactor. (For a discussion of how to interpret this “divergence”, see [@Stein04].)
The eigenvalue spectrum about the uniform saddle can be analytically calculated [@GS10]. The eigenvalue $\lambda_{u,2}$ is found to be $$\label{eq:lambda2}
\lambda_{u,2}=\frac{\kappa_1\pi^{2}}{L^{2}}-(\mu_1-\mu_2)\, .$$ At fixed $L$, this switches from negative (unstable) to positive (stable) as $\kappa_1$ increases, as shown in Fig. \[fig:lambda2\]. This change of sign corresponds to a transition from an instanton saddle to a uniform one as $\kappa_1$ varies. Using this approach, the curve $L_c$ vs. $\kappa_{1c}$ can be derived analytically as the locus of points where $\lambda_{u,2}=0$ and thus the full phase diagram determined as represented by the solid curve in Fig. \[fig:phasediagram\].
We have also studied the behavior of the transition rate prefactor in a wide range of values of $L$ numerically, all of which lead to the same conclusion as described above. Fig. \[fig:prefactor\_all\] shows the divergence of $\Gamma_0$ at different $L_c$’s and their corresponding $\kappa_{1c}$’s.
Discussion {#sec:discussion}
==========
We have solved the general two-field model of (\[eq:H\]) and (\[eq:U\]) for its full parameter space. We have uncovered a new mechanism for the transition in the switching rate, and shown that it has features of a second-order phase transition.
In the one-field case, the mechanism behind the transition is not difficult to understand. At smaller $L$ (recall that this is a [*dimensionless*]{} lengthscale, in units of a reduced length that includes the single bending coefficient $\kappa$), bending costs (in conformity with the boundary condition) are prohibitive, and the uniform saddle therefore has lower energy than the instanton. At large lengthscales, the uniform saddle has a prohibitive bulk energy (i.e., potential difference with the stable state), whereas the instanton differs from one or the other stable state only within the domain wall region, whose lengthscale remains of $O(1)$. What is perhaps less intuitive is that the transition in saddle states should be asymptotically (as $\epsilon\to 0$) sharp.
Here we have uncovered a second mechanism for the transition to occur: as noted in Sec. \[sec:results\], increase of $\kappa_1$ when $L>L_c(1)$ suppresses spatial variation, causing the instanton (again in a sharp transition) to collapse to the uniform saddle. Conversely, the instanton state can be retrieved for $L<L_c(1)$ when $\kappa_1$ [*decreases*]{}; of course, bending becomes increasingly favorable energetically. The result is a phase diagram in $(L,\kappa_1/\kappa_2)$ space, as in Fig. \[fig:phasediagram\], where a phase separation line denotes the boundary between the uniform and instanton “phases”.
We close with some remarks about the string method as applied to this problem.
A randomly placed string will relax towards the most probable transition path along the stable direction of the saddle. In Sec. \[sec:results\] we defined the smallest positive eigenvalue (corresponding to the stable direction) of the linearized operator $\Lambda_u$. As a second-order phase transition is approached, $\lambda_{u,2}$ drops to $0^{+}$, so that the energy landscape curvature in the stable direction becomes very small. When the string arrives in its neighborhood, the restoring force exerted along its normal direction correspondingly becomes small leading to slow convergence. If one sits right at the critical point, the string will not arrive at the saddle.
The string method assumes that most of the probability flux from the reactant to the product state is carried by one (or more generally a few) paths through the saddle state, in each of which the probability flux is tightly confined to a narrow quasi - 1D region in state space. However, near criticality the path splays out in the direction normal to the longitudinal transition path. In this case one needs to consider transition “tubes”, inside which most of probability flux is concentrated. The equilibrium condition (\[eq:string equation\]) corresponds to conditions away from criticality, where the transition tube is thin.
The equation for the path of maximum flux is derived in [@EEric10], where it is noted that the reaction flux intensity must be maximized along the thin transition tube (or the string, when using the string method). An alternative derivation can be found in [@BerkowitzMorgan83].
[*Acknowledgments.*]{} The authors are grateful to Weiqing Ren and Ning Xuan for helpful discussions. We are especially grateful to Gabriel Chaves for his help in programming the string method. This work was supported in part by NSF Grant PHY-0965015.
![Behavior of the prefactor $\Gamma_0$ as calculated numerically using (\[eq:prefactor\]) when $\mu_{1}=3$, $\mu_{2}=2$, and $L=4.51$.[]{data-label="fig:prefactor_L451"}](Fig4){width="\linewidth"}
![Smallest nonnegative eigenvalue $\lambda_{u,2}$ (solid line) of the saddle state for $\mu_{1}=3$, $\mu_{2}=2$ and $L=4.51$. For $\kappa_1<2.06$ this eigenvalue corresponds to the instanton saddle, and for $\kappa_1>2.06$ to the uniform saddle. The extended dashed line shows the continuation of this eigenvalue for the uniform saddle in its unstable regime below $\kappa_{1c}$.[]{data-label="fig:lambda2"}](Fig5){width="\linewidth"}
![The phase diagram at $\mu_{1}=3$, $\mu_{2}=2$. The dots represent numerically determined values where the energies of the uniform and instanton saddles cross. The solid line was computed analytically using (\[eq:lambda2\]), that is, by determining the relation between $\kappa_1$ and $L$ along which the smallest nonnegative eigenvalue of the uniform saddle is zero.[]{data-label="fig:phasediagram"}](Fig6){width="\linewidth"}
![The divergence of $\Gamma_0$ at $L$ ranging from $2.6$ to $5.0$. Here $\mu_{1}=3$, $\mu_{2}=2$, and the values of $\kappa_{1c}$ (or the corresponding $L_c$) correspond to the dots in Fig. \[fig:phasediagram\]. The peaks are of different heights because the speeds of divergence are not necessarily the same for every $L$. []{data-label="fig:prefactor_all"}](Fig7){width="\linewidth"}
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Colliders have been at the forefront of scientific discoveries in high-energy particle physics since the inception of the colliding beams method in the middle of the 20th century. The field of accelerators is very dynamic and many innovative concepts are currently being considered such future facilities as Higgs factories and energy frontier colliders beyond the LHC. Here we briefly overview leading proposals and studies towards the next generation colliders and discuss their major challenges as well as directions of corresponding accelerator R&D programs needed to address their cost and performance risks.'
author:
- Vladimir Shiltsev
title: Future High Energy Frontier Colliders
---
INTRODUCTION
============
The needs of modern high energy physics (HEP) call for two types of future accelerator facilities –- Higgs Factories (HF) and Energy Frontier (EF) colliders. There are four feasible concepts for these machines: linear $e^+e^-$ colliders, circular $e^+e^-$ colliders, $pp/ep$ colliders, and muon colliders. They all have limitations in energy, luminosities, efficiencies, and costs which in turn depend on five basic underlying accelerator technologies: normal-conducting (NC) magnets, superconducting (SC) magnets, NC RF, SC RF and plasma. The technologies are at different level of performance and readiness, cost efficiency and required R&D. Comprehensive reviews of colliders can be found in [@Handbook], [@VS2012], [@BM2016], [@RASTv7], [@SZ]. Below we overview the Higgs factory implementation options, their accelerator physics and technology challenges, readiness, cost and power; possible paths towards the highest energies, how to achieve the ultimate energy and performance, and required R&D; as well as promises, challenges and expectations of new acceleration techniques. There is an extended literature for each of the considered future colliders but in this paper, for reasons of brevity, references will be given the corresponding brief Inputs to the 2019 European Particle Physics Strategy Update symposium (EPPSU, May 2019, Granada, Spain) where all the proposals were presented [@EPPSU]. More details and references on each of the proposals can be found therein. Table \[future\] summarizes main parameters of the future facilities.
![Luminosity of the proposed Higgs factories.[]{data-label="HFlumis"}](HFlumis.png){width="80mm"}
---------------- ---------- -------------- ---------------- --------------- ----------- --------------- ---------------- -----------------
**Project** **Type** **Energy** **Int. Lumi.** **Oper.Time** **Power** **Cost** **Cost/Lumi.** **Lumi./Power**
TeV, c.m.e. ab$^{-1}$ years MW BCHF/ab$^{-1}$ ab$^{-1}$/TWh
**ILC** $e^+e^-$ 0.25 2 11 129 4.8-5.3GILCU 2.65 0.24
$L$-upgr. 150-200 + upgrade
0.5 4 10 163(204) 7.8GILCU 1.3 0.4
1 300 ?
**CLIC** $e^+e^-$ 0.38 1 8 168 5.9BCHF 5.9 0.12
1.5 2.5 7 370 + 5.1BCHF 3.1 0.16
3 5 8 590 +7.3BCHF 2.0 0.18
**CEPC** $e^+e^-$ 0.091+0.16 16+26 4 149 5 G\$ 0.27 5.1
0.24 5.6 7 266 +? 0.21 0.5
**FCC-ee** $e^+e^-$ 0.091+0.16 150+10 4+1 259 10.5BCHF 0.065 20.5
0.24 5 3 282 0.064 0.9
0.365(+0.35) 1.5(+0.2) 4(+1) 340 +1.1 BCHF 0.07 0.15
**LHeC** $ep$ 0.06/7 1 12 (+100) 1.75 GCHF 1.75 0.14
**HE-LHC** $pp$ 27 20 20 220 7.2 GCHF 0.36 0.75
**FCC-hh** $pp$ 100 30 25 580 17(+7) GCHF 0.8 0.35
**Muon.Coll.** $\mu\mu$ 14 50$^*$ 15$^*$ 230 10.7$^*$ GCHF 0.21$^*$ 2.4$^*$
---------------- ---------- -------------- ---------------- --------------- ----------- --------------- ---------------- -----------------
\[future\]
HIGGS FACTORIES
===============
There are several well studied approaches to produce copious number of Higgs particles: $e^+e^-$ linear colliders such as ILC (the EPPSU Input 77) and CLIC (Input 146); $e^+e^-$ circular colliders like FCC-ee (Input 132) and CepC (Input 51) and a $\mu^+\mu^-$ circular HF. Less traditional and yet not that well studied options such as $\gamma\gamma$-colliders are described elsewhere [@HF2012] and are not discussed here. The most critical high level requirement for HFs is high luminosity $O(10^{34})$ cm$^{-2}$s$^{-1}$ at the Higgs energy scale. Usually, these machines are compared to the LHC which, as an accelerator, is 27 km long, is based on 8 T SC magnets, requires some 150 MW total AC power (out of 200MW for the entire site of CERN which consumes some 1-1.2 TWh of electric energy annually), took about 10 years to build after 10-15 years of the design and development studies, and did cost about 5 BCHF (excluding the cost of the existing LEP tunnel and of the proton injector complex).
Linear Colliders
----------------
The International Linear Collider (ILC, EPPSU Inputs 66 and 77) is about 20 km long, including 5 km of the Final Focus system, it employs 1.3 GHz SRF cavities operating at 2 K and providing 31.5 MV/m of the accelerating gradient. The ILC requires some 130 MW site power when operates at 250 GeV c.m.e. It is estimated to cost 700 BJPY (with $\pm$25% error, including cost of labor).
The 380 GeV c.m.e. Compact Linear Collider (CLIC) is 11 km long, its two-beam NC RF accelerator cavities operate at 72 MV/m. Out of the total 168 MW site power some 9MW goes into electron and positron beams. The CLIC cost estimate is 5.9 BCHF with some $\pm$25% accuracy.
Luminosity of the linear colliders scales as: $$L = H_d \Big( \frac{N_e}{\sigma_x} \Big) \Big( N_e N_b f_r \Big) \Big( \frac{1}{\sigma_y} \Big).
\label{LumiHiggs}$$ Correspondingly, the luminosity challenges are associated with each of three limiting factors in the parentheses: the first one $(N_e / \sigma_x)$ defines the luminosity spectrum which reaches $\delta E/E$ $\sim$1.5% in ILC and grows with energy, so some 40% of the CLIC luminosity is out of 1% c.m.e. (due to so called $beamstrahlung$); the second factor is nothing but the total beam current which is limited by the total available RF power, by the beam coherent instability concerns, by challenging positron production via two proposed schemes, etc; and the third one calls for ultra small vertical beam size that in turns requires record small beam emittances from damping rings, stabilization of focusing magnets and accelerating cavities [@Shi1], [@Shi2], 0.1 $\mu$m resolution BPMs in order to obtains the rms beam sizes of 8nm (vertical) and 500nm (horizontal) at the ILC interaction point(IP) and 3nm/150nm in the CLIC IP. One should note remarkable progress of the linear collider projects: beam accelerating gradients met the ILC specs of 31.5 MeV/m (at the Fermilab FAST facility in 2017 [@BROEMM] and KEK in 2019) and the CLIC specs of 100 MeV/m at the CLEX facility at CERN; final focusing into 40 nm verstical rms beam size has been demonstrated at the ATF2 in KEK in 2016, etc.
In general, one should consider numerous advantages of the linear $e^+e^-$ colliders as HFs: they are based on mature technologies of NC RF and SRF; their designs are quite mature (ILC has a TDR, CLIC has CDR, there are several beam test facilities); beam polarization of 80%-30% in the ILC and 80% - 0% in CLIC helps the HEP research; they are expandable to higher energies (ILC to 0.5 and 1 TeV, CLIC to 3 TeV); both have well-organized established international collaborations (LCC) which indicate readiness to start costruction soon; the AC wall plug power of 130-170 MW is less than that of the LHC. At the same time one has to pay attention to following factors: the cost of these facilities is 1-1.5 times the LHC cost; the ILC and CLIC luminosity projections are in general lower than that for rings (see Fig.\[HFlumis\] and discussion below), and luminosity upgrades (such as via two-fold increase of the number bunches $N_b$ and doubling the repetition rate from 5 Hz to 10 Hz in the ILC) will probably come at an additional cost; operation experience with LCs is limited to one past machine (SLAC’s SLC), CLIC’s two-beam scheme is quite novel (so, klystrons are assumed as a backup RF source option); and the wall plug power may grow beyond that of the LHC for the proposed LC luminosity and energy upgrades.
Circular colliders
------------------
The most advanced circular electron-positron Higgs factories designs are CERN’s FCC-ee (Input 132) and Chinese CepC (Input 51). Both proposals call for 100 km tunnels to host three rings ($e^-, e^+$ and a fast cycling booster ring), very high SRF power transfer to beams (100 MW in FCC-ee and 60 MW in CepC), that leads to the total site power of about 300MW. Cost estimate of the FCC-ee is 10.5 BCHF (plus additional 1.1BCHF for option to operate at the higher $tt$ energy) and 5 to 6 B\$ for the CepC (“less than 6BCHF” cited in the CepC CDR).
The key accelerator physics challenge of the circular HFs is that the synchrotron radiation power from both beams has to be limited to about $P$=100 MW and the maximum allowed beam current $I$ scales as inverse fourth power of the beam energy: $$I = P \Big( \frac{e\rho}{2 C_\gamma E^4} \Big).
\label{SRcurrent}$$ Correspondingly, the luminosity scales as the product of the ring radius $\rho$, beam-beam parameter $\xi_y$, beta-function at the IP $\beta_y^*$ and the RF power $P$ and inverse of $E^3$. The beam-beam parameter is limited to about $\xi_y$=0.13 by a new type of beam-beam instability. Beam energy loss per turn due to the synchrotron radiation is about 0.1-5% (from $Z$ energy to 360 GeV) is significantly more than the energy spread due to beamstrahlung 0.1-0.2%, but the latter occurs instanteneously at the IPs and the tails of the resulting energy distribution reach $\pm$2.5% or upto 10 times the rms value. Correspondingly, these tails determine 18 min beam lifetime even in a sophisticated large energy acceptance $crab-waist$ optics with $\beta^*_y$ = 0.8-1.6 mm – that in turn, calls for a full energy fast ramping booster ring.
The advantages of the circular HFs include quite mature technology of the SRF acceleration, vast experience of other rings suggests lower performance risk; they have higher luminosity and luminosity/cost ratio and can host upto 4 detectors at IPs that could make them sort of [*EW*]{} (electroweak) factories. 100 km long tunnels can be reused by follow-up future $pp$ colliders. Transverse polarization occurs naturally after about 18 min at the $tt$ energies and can be employed for precise energy calibration $O$(100keV). Very strong and broad [*Global FCC Collaboration*]{} came up with comprehensive CDR that addresses key design points and indicates possible ca.2039 start (the CepC schedule is more aggressive with the machine start some 7-9 years sooner). Before that, the R&D program is expected to address several important items, such high efficiency RF sources (e.g. over over 85% for 400/800 MHz klystrons, up from thecurrent 65%); high efficiency SRF cavities (to achieve 10-20 MV/m CW gradient and high $Q_0$; or use of new technologies like Nb-on-Cu, Nb$_3$Sn); exploration of the crab-waist collision scheme (the Super KEK-B nanobeams experience will be very helpful in that regard); energy storage and release (so energy in magnets can be re-used for more that 20,000 cycles) and on the efficient use of excavated materials (some 10 million cu.m. will need to be taken out of a 100 km tunnel).
Finally, muon collider HF (Inputs 41, 120, 141) might offer a very economical approach as the luminosity required in $\mu^+\mu^-$ collisions can be about 1/100th of that in $e^+e^-$ due to large cross-section in $s$-channel; beam energy is also only one half of the $e^+e^-$ case (i.e., 2$\times$63 GeV for $\mu^+\mu^- \rightarrow H_0$) and, therefore, a small footprint of less than 10 km and low cost of the collider; very small energy spread in non-radiating muon beams $O$(3 MeV) and low total site power $\sim$200MW. The biggest challenge of all $\mu^+\mu^-$ collider proposals is that the method, though based mostly on conventional technologies, is quite novel conceptually, and some key techniques, such as efficient muon cooling, are still being explored. So, the muon colliders offer great cost savings for future HF and EF machines and substantial R&D needs to be carried out to prove their practicality to be considered on equal footing with the above mentioned proposals (see also discussion in the next section).
![Energy reach of muon-muon collisions: the energy at which the proton collider cross-section equals that of a muon collider (from the EPPSU Input 120).[]{data-label="MuonProton"}](MuonProton.png){width="80mm"}
FUTURE ENERGY FRONTIER FACILITIES
=================================
Key challenges of all EF collider proposals are mostly about having affordable cost and, simultaneously, high luminosity. Usually, the scale of civil construction grows with beam energy, the cost of accelerator components grows with energy, required site power grows with energy. So, the total project cost grows with energy, but, thankfully, not linearly [@VScost]. Take the ILC as an example: the costs of 0.25 TeV vs 0.5 TeV vs 1 TeV facilities relate as 0.69 : 1 : 1.67. Still, there are huge financial challenges for collision energies an order of magnitude beyond the LHC’s 14 TeV.
Linear $e^+e^-$ colliders face the most severe challenges: both ILC and CLIC offer staged approach to ultimate energies (1 TeV and 3 TeV c.m.e., respectively), but their lengths grow to 40-50 km, the AC power requirements become 300-600 MW, the beamstrahlung leaves only 30-40% of the luminsity within 1% of the maximum energy and the project cost grows to 17 B\$ (ILC 1 TeV, TDR) and 18.3 BCHF (CLIC 3 TeV, CDR).
EF $pp$ colliders such as HE-LHC (Input 133), FCC-hh (Input 136) and SppC (Input 51) require long tunnels (27km, 100km and 100 km, respectively), high field SC magnets (16T, 16T and 12 T, respectively), the total AC site power of 200 MW (HE-LHC) to $\sim$500 MW and cost 7.2 BCHF (HE-LHC) to 17.1 BCHF (FCC-hh, assuming that 7 BCHF tunnel is available). In all these options, the detectors will need to operate at luminosities of $O(10^{35}$ cm$^{-2}$s$^{-1}$) and corresponding pileup will be $O$(500).
Serious R&D program that might take 12-18 years is needed to address most critical technical issues, such as development of accelerator quality 16 T dipole magnets based on Nb$_3$Sn (or 12 T iron-based HTS magnets for the SppC); effective intercept of the synchrotron radiation (5 MW in FCC-hh and 1 MW in SppC); beam halo collimation with circulating beam power 7 times that of the LHC; choice of optimal injector (eg., 1.3TeV scSPS, or 3.3 TeV ring either in the LHC tunnel or the FCC tunnel); overall machine design issues (IRs, pileup, vacuum, etc); power and cost reduction, etc. To be noted that such machines can also be used for ion-ion/ion-proton collisions; also high energy proton beams can be collided with high intensity $O$(60) GeV electrons out of an ERL (Input 159).
![Total length (upper scale) and total AC power requirements of multi-TeV $e^+e^-$ collider schemes based on plasma wakefield acceleration (data from the EPPSU Input 7). []{data-label="PWApower"}](PWApower.png){width="80mm"}
Muon colliders offer in some aspects moderately conservative, in others - moderately innovative path to cost affordable energy frontier colliders. Again, their major advantages are a) muons do not radiate as readily as electrons and not affected by the beamstrahlung that allows efficient acceleration in rings at low cost and with great power efficiency; and b) the energy reach in muon collisions is about factor of 7 compared to the same energy $pp$ collisions - see Fig.\[MuonProton\].
The muon collider R&D had a number of remarkable advances in the past decade: the ionization cooling of muons was demonstrated in the MICE experiment at RAL (4D emittance reduction by $O$(10%)); NC RF gradients 50 MV/m obtained in 3 T fields at Fermilab; also at Fermilab - rapid cycling HTS magnets achieved record ramping rate of 12 T/s; the first RF acceleration of muons demonstrated at JPARC MUSE RFQ (90 KeV); no cooling low emittance LEMMA concept is proposed (to use 45 GeV positrons to generate muon pairs at threshold). The [*US MAP Collaboration*]{} and its international partners have successfully carried out feasibility studies and showed that muon colliders can be built with the present day SC magnet and RF technologies; there is a well-defined path forward and initial designs exist for 1.5 TeV, 3 TeV, 6 TeV and 14 TeV colliders (the later, in the LHC tunnel and with the US MAP type proton driver[@Neuffer] is marked by asterisk in the Table \[future\]). Still, there are many remaining issues to resolve which call for test facilities to demonstrate effective muon production and 6D cooling, to study the LEMMA scheme [@LEMMA] as well as to further develop concepts of fast acceleration, to deal with high detector background rates and the neutrino radiation issue.
Last but not least, one has to mention impressive progress of new methods of acceleration in plasmas (Inputs 7, 109, 58. 95). There are three ways to excite plasma wake-fields: by lasers (demonstrated electron energy gain of about 8 GeV over 20 cm of plasma with density 3$\cdot$10$^{17}$ cm$^{-3}$ at the BELLA facility in LBNL); by short electron bunches (9 GeV gain over 1.3m $\sim$10$^{17}$ cm$^{-3}$) plasma at FACET facility in SLAC) and by proton bunch (some 2 GeV gain over 10 m of 10$^{15}$ cm$^{-3}$ plasma at AWAKE experiment at CERN). In principle, the method of plasma wake field acceleration (PWFA) can make possible multi-TeV $e^+e^-$ colliders. There is a number of critical issues to resolve along that paths: power efficiency of the laser/beam PWFA schemes; acceleration of positrons (which are defocused when accelerated in plasma); efficiency of staging (beam transfer and matching from one short plasma accelerator cell to another); beam emittance control in scattering media; beamstrahlung (that leads to the rms energy spread at IP of about 30% for 10 TeV machines and 80% for 30 TeV collider - see Fig.\[PWApower\]). The last four of these issues can possibly be addressed by accelerating muons (instead of electrons) in crystals or carbon nanotubes (density $\sim$10$^{22}$ cm$^{-3}$) –- the maximum theoretical accelerating gradient scales as square root of the plasma and can reach 1-10 TeV/m allowing to envision a compact 1 PeV linear crystal muon collider [@VS2012], [@2019WRKSH].
The PWFA technology is being actively explored becuase of the opportunities it offers for a number of non-HEP applications. There are several proposals of plasma-based electron injectors for operational circular machines (a 100 MeV injector to the IOTA ring at FNAL [@IOTA]; a 700 MeV injector to the PETRA-IV booster in DESY). Several collaborations are formed worldwide (EuPRAXIA, ALEGRO study, ATHENA) and facilities built or being built (in Europe - PWASC, ELBE/HZDR, AWAKE, CILEX, CLARA and SCAPA, EuPRAXIA at SPARCLAB at INFN-LNF, Lund, JuSPARC at FZJ and FLASHForward and SINBAD at DESY; also ImPACT in Japan , SECUF in China and FACET-II and BELLA in the US). Roadmap of the advanced acceleration concepts R&D in the US aims at the PWFA collider CDR by 2035.
Summary
=======
At present, aspirations of the high energy particle physics community include future Higgs factories and the Energy Frontier colliders. There are four feasible concepts: linear $e^+e^-$ colliders, circular $e^+e^-$ colliders, $pp/ep$ colliders and muon colliders. They all have limitations in energy, luminosity, efficiency and cost. The most critical (“Tier 0”) requirement for a Higgs factory is high luminosity and there are four proposals which generally satisfy it: ILC, CLIC-0.38, CEPC and FCC-ee. Te next level “Tier I” criteria include (in order): facility cost, the required AC power and readiness. The construction cost, if calibrated to performance (i.e., in GCHF/ab$^{-1}$) is the lowest for the FCC-ee, followed by CepC ($\times$4), then ILC (another $\times$10), then CLIC (another $\times$2) - see Table \[future\]. The required AC site power consumption, if calibrated to performance (i.e., in TWh/ab$^{-1}$) is the lowest for the FCC-ee, followed by CepC ($\times$2), then ILC (another $\times$2), then CLIC (another $\times$2). As for the readiness, the ILC as a project is somewhat ahead of other proposals (it has TDR vs CDRs for CLIC, CepC, and FCC-ee) and its technical readiness is quite matured and includes industrial participation. The FCC-ee and CEPC proposals are based on the concepts and beam dynamics parameters that have already been proven at many past and presently operating circular colliders.
The most critical “Tier 0” requirement for the energy frontier colliders is the center-of-mass energy reach and there are four proposals which generally satisfy it (in order of the higher energy reach): CLIC-3 TeV, HE-LHC, 6/14 TeV Muon Collider and FCC-hh/SppC. The next level “Tier I” criteria for the EF machines are (in order); cost, AC power and R&D effort. The construction cost is the lowest for the HE-LHC and the Muon Collider, followed by CLIC-3 ($\times$2) and FCC-hh (another $\times$1.5). The required AC site power consumption, is the lowest for the HE-LHC and the Muon Collider, followed by CLIC ($\times$2), then by FCC-hh (another $\times$1.5). As for the required duration/scale of the R&D effort to reach the TDR level of readiness, the CLIC-3 project is ahead of other proposals as it requires $\sim$10 years of R&D vs about twice that for the HE-LHC and FCC-hh/SppC, and for the Muon Collider (the latter being at present the only concept without a comprehensive CDR).
Arguably the hardest challenge for the EF hadron and muon colliders is development of representing magnets with maximum 16T field. There are fundamental challenges in getting the required current density in superconductors and in dealing with the ultimate magnetic pressures and mechanical stresses in the superconductor and associated components. Experts estimate that 20 to 30 years might be needed to innovate new approaches and technology to overcome the above-mentioned limits through continuous R&D efforts. Lowering the maximum field requirement to 12-14 T or even to 6-9 T can greatly reduce the time needed for short-model R&D, prototyping and pre-series work with industry. To realize even higher fields - beyond 16 T, if needed - High Temperature Superconductor (HTS) technology will be inevitably required. At present, the most critical constraint for the HTS is its much higher cost, even compared with the Nb$_3$Sn superconductor.
Impressive advances of the exploratory PWFA R&D over the past decade make it important to find out whether a feasible “far future” lepton collider option for particle physics can be based on that technology. One should note that PWFA has potential for non-HEP applications and has drawn significant interest and support from broader community, most notably, because of its possible use for X-ray production. Several research and test facilities are already built and operated, and many more are being planned. It will be important for the HEP accelerator designers to learn from the corresponding experience, understand applicability of the PWFA advances for particle colliders and encourage further technological development of the method.
This presentation with slight modifications was also given at the European Particle Physics Strategy Update Symposium (May 13-16, 2019, Granada, Spain). I would like to acknowledge input from and fruitful discussions on the subject of this presentation and thank M.Benedikt (CERN), P.Bhat (FNAL), M.Benedikt (CERN), C.Biscari (ALBA), A.Blondel (CERN), J.Brau (UO), O.Bruning(CERN), A.Canepa (FNAL), W.Chou (IHEP, China), M.Klein (CERN), J.P.Delahaye (CERN), D.Denisov (BNL), V.Dolgashev (SLAC), E.Gschwendtner (CERN), A.Grasselino (FNAL), W.Leemans (DESY), E.Levichev (BINP), B.List (DESY), H.Montgomery (JLab), P.Muggli (MPG), D.Neuffer (FNAL), H.Padamsee (Cornell), M.Palmer (BNL), N.Pastrone (INFN), Q.Qin (IHEP), T.Raubenheimer (SLAC), L.Rivkin (EPFL/PSI), A.Romanenko (FNAL), M.Ross (SLAC), D.Schulte (CERN), A.Seryi (Jlab), T.Sen (FNAL), F.Willeke (BNL), V.Yakovlev, A.Yamomoto (KEK), F.Zimmermann (CERN), A.Zlobin (FNAL).
Fermi National Accelerator Laboratory is operated by Fermi Research Alliance, LLC under Contract No. DE-AC02-07CH11359 with the United States Department of Energy.
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We present a multidimensional flow exhibiting a Rovella-like attractor: a transitive invariant set with a non-Lorenz-like singularity accumulated by regular orbits and a multidimensional non-uniformly expanding invariant direction. Moreover, this attractor has a physical measure with full support and persists along certain submanifolds of the space of vector fields. As in the $3$-dimensional Rovella-like attractor, this example is not robust. The construction introduces a class of multidimensional dynamics, whose suspension provides a Rovella-like attractor, which is partially hyperbolic, and whose quotient over stable leaves is a multidimensional endomorphism to which Benedicks-Carleson type arguments are applied to prove non-uniform expansion.'
address:
- 'Vítor Araújo, Instituto de Matemática, Universidade Federal do Rio de Janeiro C. P. 68.530, 21.945-970, Rio de Janeiro, RJ-Brazil'
- |
Armando de Castro, Departamento de Matemática, Universidade Federal da Bahia\
Av. Ademar de Barros s/n, 40170-110 Salvador, Brazil.
- 'Maria José Pacifico, Instituto de Matemática, Universidade Federal do Rio de Janeiro, C. P. 68.530, 21.945-970 Rio de Janeiro, Brazil'
- |
Vilton Pinheiro, Departamento de Matemática, Universidade Federal da Bahia\
Av. Ademar de Barros s/n, 40170-110 Salvador, Brazil.
author:
- 'V. Araujo, A. Castro, M. J. Pacifico and V. Pinheiro'
title: 'Multidimensional Rovella-like attractors'
---
Introduction
============
Flows in compact (two-dimensional) surfaces are very well understood since the groundbreaking work of Peixoto [@peixoto59; @peixoto62]. A theory of three-dimensional flows has steadily been developing since the characterization of robust invariant sets in [@MPP04]. Extensions of these techniques for higher-dimensional flows are well under way, see e.g. [@MeMor06]. For non-robust but persistent attractors, like the Rovella attractor presented in [@Ro93], there is still no higher-dimensional analogue.
In a rather natural way, since low dimensional dynamics is much better understood than the dynamics of general systems, the techniques used in the mathematical analysis of three-dimensional attractors for flows frequently depend on a dimensional reduction through projection along a stable manifold inside some Poincaré cross-section. This method yields a one-dimensional system whose dynamics can be nearly completely understood as well as the dynamics of its small perturbations.
In this work we start a rigorous study of a proposed higher dimensional analogue of the three-dimensional Rovella (or *contracting Lorenz*) attractor. Other examples of higher dimensional chaotic attractors have been recently presented, e.g. by Bonatti, Pumariño and Viana in [@BPV97] and by [@ST98], but these are robust sets, while our construction leads to a persistent attractor.
In [@BPV97], Bonatti, Pumarino and Viana define a uniformly expanding map on a higher-dimensional torus, suspend it as a time-one map of a flow, and then singularize the flow adding a singularity in a convenient flow-box. This procedure creates a new dynamics on the torus presenting a multidimensional version of the one-dimensional expanding Lorenz-like map, and a flow with *robust multidimensional Lorenz-like attractors*: the singularity contained in the attractor may have any number of expanding eigenvalues, and *the attractor remains transitive in a whole neighbourhood of the initial flow.* The construction in [@ST98] is also robust but yields a quasi-attractor: the attracting invariant set is not transitive but it is the maximal chain recurrence class in its neighborhood. For the case of the Lorenz attractor and singular-hyperbolic attractors in general see e.g. [@viana2000i] and [@AraPac07].
![On the left, the geometric Lorenz attractor with the contracting directions on the cross-section $\Sigma$; on the right, the Lorenz one-dimensional transformation.[]{data-label="fig:geom-lorenz"}](lorenzgeom4.eps "fig:"){height="5cm" width="8cm"} $\quad$ ![On the left, the geometric Lorenz attractor with the contracting directions on the cross-section $\Sigma$; on the right, the Lorenz one-dimensional transformation.[]{data-label="fig:geom-lorenz"}](L1D.eps "fig:"){width="5cm"}
The quotient of the return map to the global cross-section over the stable directions (see Figure \[fig:geom-lorenz\]) is the one-dimensional Lorenz transformation. Our goal is to construct a flow such that “after the identification by the stable directions”, the first return map in a certain cross section $M$ is a multidimensional version of the one-dimensional Rovella-map [@Ro93].
A Rovella-like attractor is the maximal invariant set of a geometric flow whose construction is very similar to the one that gives the geometric Lorenz attractor, [@GW79; @ABS77; @AraPac07], except for the fact that the eigenvalues relation $\lambda_u + \lambda_s > 0$ there is replaced by $\lambda_u + \lambda_s < 0$, where $\lambda_u>
0$ and $\lambda_s$ is the weakest contractive eigenvalue at the origin. We remark that, unlike the one-dimensional Lorenz map obtained from the usual construction of the geometric Lorenz attractor, a one-dimensional Rovella map has a criticality at the origin, caused by the eigenvalue relation $\lambda_u + \lambda_s<0$ at the singularity. In Figure \[fig:contractingLorenz\] we present some possible “Rovella one-dimensional maps” obtained through quotienting out the stable direction of the return map to the global cross-section of the attractor, as in Figure \[fig:geom-lorenz\].
![Several possible cases for the one-dimensional map for the contracting Lorenz model[]{data-label="fig:contractingLorenz"}](mps23.eps){width="14cm"}
The interplay between expansion away from the critical point with visits near the criticality prevents this system to have uniform expansion and also prevents robustness, that is, the attractor is not transitive on a whole neighborhood of the original flow. However the Rovella attractors are *persistent*. That is, *for generic parameterized families of vector fields passing through the original vector field $X_0$, the parameters corresponding to transitive attractors, with the same features of the Rovella attractor, form a positive Lebesgue measure subset and have positive density at $X_0$.* See [@Ro93] for the full statement.
Rovella-like and Lorenz-like attractors appear naturally in the generic unfolding of resonant double homoclinic loops, see e.g. [@Ro89; @Ro92; @Ro2000; @MPS05; @MPsM].
In our construction we follow the same strategy of [@BPV97] with two main differences.
On the one hand, since we aim at a multidimensional Rovella-like map, we have to deal with critical regions, that is, regions where the derivative of the return map to a global cross-section vanishes. Because of this, proving the existence of non-trivial attractors for the flow arising from such construction requires a more careful analysis. Indeed, as in the one-dimensional case, depending on the dynamics of the critical region, every attractor for the return map may be periodic (trivial).
On the other hand, our construction leads to the presence of a pair of hyperbolic saddle equilibria accumulated by regular orbits inside the attractor but *with different indexes* (the dimension of their stable manifolds). This feature creates extra difficulties for the analysis of the possible dynamics arising under small perturbations of the flow.
Typically, when the critical region is non-recurrent (Misiurewicz maps in one-dimensional dynamics), most of the difficulties introduced by the critical region can be bypassed. That is one of the main reasons for us to construct a kind of multidimensional Misiurewicz dynamics. In general, such critical regions in dimension greater than one are sub-manifolds, and one cannot rule out that they intersect each other under the action of the dynamics. Albeit this, we shall exhibit a class of multidimensional Misiurewicz endomorphisms that appears naturally in a flow dynamics.
We start with a basic dynamics presenting an expanding invariant torus ${\mathbb{T}}_1^k$ that will absorve the image of the critical region after the singularization of the associated flow. By topological reasons, this map can not be seen as a time-one map of a suspension flow: locally its degree is not constant. To bypass this new difficulty, we realize this map as a first return map of a flow with singularities (after identification by stable directions). Afterwards, we singularize a periodic orbit of this flow, i.e. we introduce a new singularity $s$ of saddle-type, with $(k+1)$-dimensional unstable manifold and $4$-dimensional stable manifold. Moreover, all the eigenvalues of $s$ are real and if $\sigma_{s,i}$, and $\sigma_{u,j}$ denote the stable and the unstable eigenvalues at $s$ respectively, then $\max\{\sigma_{s,i}\} + \max\{\sigma_{u,j}\} < 0$ for $1\le
i \le 4$ and $1 \le j \le k+1$. We say that this kind of singularity is a [*Rovella-like singularity*]{}. The resulting flow will present a multidimensional transitive *Rovella-like attractor*, supporting a physical measure.
The existence of the physical/SRB measure for the original flow is obtained taking advantage of the fact that, through identification of stable leaves, we can project the dynamics of the first return map of the flow to a global cross-section into a one-dimensional transformation with a Misiurewicz critical point.
Moreover we point out that *the analysis of the dynamics of most perturbations of our flow cannot be easily reduced (perhaps not at all) to a one-dimensional model*. This indicates that intrinsic multidimensional tools should be developed to fully understand this class of flows.
In addition, considering the perturbation of this flow along parametrized families we can show the existence of many parameters for which nearby flows exhibit an attractor with a unique physical measure by using the construction of a Poincaré return map to a cross-section of the flow exhibiting partial hyperbolicity and taking advantage of this weak hyperbolicity through a multidimensional extension of arguments of Benedicks-Carleson type. That is, we analyze the return of the orbits of the multidimensional critical set to itself under certain natural assumptions of slow recurrence and asymptotic expansion.
Preliminary definitions {#sec:preliminaryresults}
-----------------------
In what follows $M$ is a compact boundaryless finite dimensional manifold and ${\fX}^1(M)$ is the set of $C^1$ vector fields on $M$, endowed with the $C^1$ topology. From now on we fix some smooth Riemannian structure on $M$ and an induced normalized volume form $m$ that we call Lebesgue measure. We write also ${\operatorname{dist}}$ for the induced distance on $M$ and $\|\cdot\|$ for the induced Riemannian norm on $TM$. Given $X \in {\fX}^1(M)$, we denote by $X^t$, $t \in {{\mathbb R}}$ the flow induced by $X$, and if $x \in M$ and $[a,b]\subset
{{\mathbb R}}$ then $X^{[a,b]}(x)= \{X^t(x), a\leq t \leq b\}$.
We say that a differentiable map $f:M\circlearrowleft$ is $C^{1+}$ if the derivative $Df$ of $f$ is *Hölder*: there are $\alpha,C>0$ such that for every $x\in M$ we can find parametrized neighborhoods $U=\phi(U_0)$ of $x$ and $V=\psi(V_0)$ of $f(x)$ in $M$, where $U_0,V_0$ are neighborhoods of $0$ in ${{\mathbb R}}^{\dim(M)}$ and $\phi,\psi$ are parametrizations of $M$, such that for all $y_1,y_2\in U$ $$\begin{aligned}
\|D(\psi^{-1}\circ f\circ \phi)(y_1)-D(\psi^{-1}\circ
f\circ \phi)(y_2)\|
\le
C {\operatorname{dist}}(y_1,y_2)^\alpha.\end{aligned}$$
A point $p\in M$ is a *periodic point* for $X^t$ if $X(p)\neq0$ and there exists $\tau>0$ such that $X^\tau(p)=p$. The minimal value of $\tau$ such that $X^\tau(p)=p$ is the *period* of $p$. If $p$ is a periodic point, we also say that the orbit ${\EuScript{O}}(p)=\{
X^t(p): t\in{{\mathbb R}}\}$ of $p$ is a *periodic orbit*. A *singularity* $\sigma$ is an equilibrium point of $X^t$, that is, $X(\sigma)=0$. If $X(p)\neq0$ then we say that $p$ is a *regular point* and its orbit ${\EuScript{O}}(p)$ is a regular orbit.
Let $\Lambda$ be a compact invariant set of $X\in
{\fX}^1(M)$. We say that $\Lambda$ is an *attracting set* if there exists an *trapping region*, i.e. an open set $U\supset \Lambda$ such that $\overline{X^t(U)}\subset U$ for $t>0$ and $ \Lambda
=\bigcap_{t\in {{\mathbb R}}}X^t(U)$. Here $\overline{A}$ means the topological closure of the set $A$ in the manifold we are considering.
We say that an attracting set $\Lambda$ is *transitive* if it is equal to the $\omega$-limit set of a regular $X$-orbit. We recall that the $\omega$-limit set of a given point $x$ with respect to the flow $X^t$ of $X$ is the set $\omega(x)$ of accumulation points of $(X^t(x))_{t>0}$ when $t\to+\infty$. An *attractor* is a transitive attracting set and a *singular-attractor* is an attractor which contains some equilibrium point of the flow. An attractor is *proper* if it is not the whole manifold. An invariant set of $X$ is *non-trivial* if it is neither a periodic orbit nor a singularity.
\[d.dominado\] Let $\Lambda$ be a compact invariant set of a $C^{1+}$ map $f:M\circlearrowleft$ , $c>0$, and $0 < \lambda < 1$. We say that $\Lambda$ has a $(c,\lambda)$-dominated splitting if the bundle over $\Lambda$ splits as a $Df$-invariant sum of sub-bundles $ T_\Lambda M=E^s\oplus E^{cu}, $ such that for all $n\in{{\mathbb Z}}^+$ and each $x \in \Lambda$ $$\label{eq.domination}
\|Df^n \mid E^s_x\| \cdot \|(Df^{n} \mid E^{cu}_{x})^{-1}\| < c \,
\lambda^n.$$
We say that a $f$-invariant subset $\Lambda$ of $M$ is *partially hyperbolic* if it has a $(c,\lambda)$-dominated splitting, for some $c>0$ and $\lambda\in(0,1)$, such that the sub-bundle $E^s$ is uniformly contracting: for all $n\in{{\mathbb Z}}^+$ and each $x \in
\Lambda$ we have $$\begin{aligned}
\label{eq:unif-contr}
\|Df^n \mid E^s_x\| < c \, \lambda^n.\end{aligned}$$
We denote by $\overline{A}$ the topological closure of the set $A\subset M$ in what follows.
\[def:NUE\] A $C^{1+}$ map $g:M\to M$ is non-uniformly expanding if there exists a constant $c>0$ such that $$\begin{aligned}
\liminf_{n\to+\infty}
\frac1n\sum_{j=0}^{n-1}\log\|(Dg(g^j(x))^{-1}\| \le -c<0 \quad\text{for
Lebesgue almost every}\quad x\in M.
\end{aligned}$$
Non-uniform expansion ensures the existence of absolutely continuous invariant probability measures under some mild extra assumptions on $g$, see below. One property of such measures is that they have a “large ergodic basin”.
[(physical measure and Ergodic basin for flows.)]{} We say that an invariant probability measure $\mu$ is a [*physical measure*]{} for:
- the flow given by a field $X:M \to TM$ if there exists a positive Lebesgue measure $B(\mu)\subset M$ such that for all $x \in B(\mu)$ we have $
\frac{1}{T} \int_{0}^T {\varphi}\circ X^t(x) dt
\xrightarrow[T\to+\infty]{} \int_M {\varphi}d\mu
$ for all continuous functions ${\varphi}:M\to{{\mathbb R}}$;
- the map $g:M\to M$ if there exists a positive Lebesgue measure $B(\mu)\subset M$ such that for each $x
\in B(\mu)$ we have $
\frac{1}{n} \sum_{j=0}^{n-1} {\varphi}(g^j(x))
\xrightarrow[n\to+\infty]{} \int_M {\varphi}\, d\mu
$ for all continuous functions ${\varphi}:M\to{{\mathbb R}}$.
The set $B(\mu)$ is called the [*ergodic basin*]{} of $\mu$.
Statement of results {#sec:statem-results}
--------------------
To construct the attractor for the flow we first obtain
\[thm:seg\] For any $k\in{{\mathbb Z}}^+$, there exist a non-uniformly expanding map $g$ of class $C^{1+}$ on a $(k+1)$-manifold $N$, without any uniformly expanding directions, admitting an absolutely continuous invariant probability measure with full ergodic basin in $N$, whose Lyapunov exponents are all positive.
The map in the theorem above is obtained by quotienting out along stable directions the Poincaré map from the vector field $X$ in the statement of the next theorem, as a higher dimensional version of Rovella’s construction we describe in the rest of the paper.
\[thm:pri\] For any dimension $m=k+5$, $k \in{{\mathbb Z}}^+$, there exist a $C^{\infty}$ vector field $X \in {\fX}^\infty(M^m)$ on a $m$-dimensional manifold exhibiting a singular-attractor $\Lambda$, containing a pair of hyperbolic singularities $s_0,s_1$ *with different indexes* in a trapping region $U$. Moreover
1. there exists a map $R:\Sigma\circlearrowleft$ on a $(k+4)$-dimensional cross-section $\Sigma$ of the flow of $X$ such that
1. the set $\Lambda$ is the suspension of an attractor $\Lambda_\Sigma\subset\Sigma$ with respect to $R$;
2. $R$ admits a $3$-dimensional stable direction $E^s$ and $(k+1)$-dimensional center-unstable direction $E^c$ such that $E^s\oplus E^c$ is a partially hyperbolic splitting of $T\Sigma\mid\Lambda_\Sigma$;
3. $\Lambda_\Sigma$ supports a physical measure $\nu$ for $R$.
2. $\Lambda$ is the support of a physical hyperbolic measure $\mu$ for the flow $X^t$: the ergodic basin of $\mu$ is a positive Lebesgue measure subset of $U$ and every Lyapunov exponent of $\mu$ along the suspension of the bundle $E^c$ is positive, except along the flow direction.
This singular-attractor $\Lambda$ is not robustly transitive not even robust: there exist arbitrary small perturbations $Y$ of the vector field $X$ such that the orbits of the flow of $Y$, in a full Lebesgue measure set of points in $U$, converge to a periodic attractor (a periodic sink for the flow). The singular-attractor $\Lambda$ is not partially hyperbolic in the usual sense and adapted to the flow setting, since we can only define the splitting on the points of $\Lambda$ which do not converge to the singular points $s_0,s_1$, that is, we exclude the stable set of the singularities within $\Lambda$. The remaining set, however, has full measure with respect to $\mu$. Since the equilibria $s_0,s_1$ contained in $\Lambda$ are hyperbolic with different indexes (i.e. the dimension of their stable manifolds) we believe the following can be proved.
\[conj:noextension\] It is not possible to extend the dominated splitting on $\Lambda$ away from equilibria to the equilibria $s_0,s_1$ which belong to $\Lambda$.
In addition, the attractor $\Lambda$ for $X$ in $U$ is such that the Jacobian along any $2$-plane $P_x$, inside the central subbundle $E^c_x$ for Lebesgue almost all $x\in U$, is *asymptotically expanded but not uniformly expanded*, that is we can find a constant $c>0$ such that $
\lim_{t\to+\infty}\frac1t\log|\det DX^t\mid P_x|\ge c,
$ but $\frac1t\log|\det DX^t\mid P_x|$ can take an arbitrary long time (depending on $x$) to become positive.
In a similar way to [@Ro93], we have the following result concerning to the persistence of our Rovella-like attractor along certain submanifolds of the space $\fX^2(M)$ of $C^2$ vector fields endowed with the $C^2$ topology.
\[mthm:codimension\] For any $k\in{{\mathbb Z}}^+$, there exists a family ${\EuScript{P}}$ of $C^2$ vector fields on a $(k+2)$-codimension submanifold in $\fX^2(M)$ such that
1. the vector field $X$ from Theorem \[thm:pri\] belongs to ${\EuScript{P}}$ and, for all $Y\in{\EuScript{P}}$ in a neighborhood ${\EuScript{U}}$ of $X$ in $\fX^2(M)$, the Poincaré return map $R_Y$ to the cross-section $\Sigma$ admits a $3$-dimensional strongly contracting $C^\gamma$ foliation ${\EuScript{F}}$, for some $\gamma>1$. The induced map $g_Y$ on the quotient of $\Sigma$ over ${\EuScript{F}}$ is a $C^{\gamma}$ endomorphism on a cylinder $[-1,1]\times{{\mathbb T}}^k$.
2. for vector fields $Y\in{\EuScript{U}}\setminus{\EuScript{P}}$, the Poincaré return map $R_Y$ to the cross-section $\Sigma$ admits a one-dimensional strongly contracting $C^\gamma$ foliation ${\EuScript{F}}$, for some $\gamma>1$. The induced map $g_Y$ on the quotient of $\Sigma$ over ${\EuScript{F}}$ is a $C^{\gamma}$ endomorphism on a manifold diffeomorphic to the unit ball in ${{\mathbb R}}^{k+3}$.
For all vector fields $Y\in{\EuScript{U}}$ considered in Theorem \[mthm:codimension\], the open set $U$ of $M$ is still a trapping region of the flow $Y^t$ and contains the orbit of a periodic sink $p(Y)$ (a hyperbolic periodic attracting orbit).
Regarding the ergodic properties of the set $\Lambda_\Sigma(Y)=\cap_{n\in{{\mathbb Z}}^+} R_Y^n(\Sigma)$, which induce the ergodic properties of $\Lambda(Y):=\cap_{t>0}
Y^t(U)$, we prove the following.
\[mthm:fullbasinsink\] For a vector field $Y\in{\EuScript{U}}\cap{\EuScript{P}}$, where ${\EuScript{U}}$ is the neighborhood of $X$ and ${\EuScript{P}}$ the submanifold in $\fX^2(M)$ introduced at Theorem \[mthm:codimension\], *if* the quotient map $g_Y$ sends the critical set inside the stable manifold of the sink $p(Y)$, *then* this stable manifold contains the trapping region $U$ except for a zero Lebesgue measure subset.
For the remaining cases we conjecture that the quotient map behaves in a similar way to the typical perturbations of a smooth one-dimensional multimodal map.
\[conj:physicalmeas\] For a vector field $Y\in{\EuScript{U}}$, where ${\EuScript{U}}$ is the neighborhood of $X$ in $\fX^2(M)$ introduced in Theorem \[mthm:codimension\], if the quotient map $g_Y$ *does not send* the critical set inside the stable manifold of the sink $p(Y)$, then the complement of this stable manifold contains the basin of a physical measure for $g_Y$ whose Lyapunov exponents are positive.
To make any progress in the understanding of this conjecture one needs to study the interplay between the critical set, the expanding behavior in some regions of the space, and the stable manifold of the sink, in a higher dimensional setting. We believe this will demand the development of new ideas in dynamics and ergodic theory.
Organization of the text {#sec:open-questi}
------------------------
We present the construction of the vector field $X$ in stages in Section \[sec:theconstruction\]. We start with the non-uniformly expanding higher dimensional endomorphism in Section \[sec:an-example-nue\], which proves Theorem \[thm:seg\]. Then we adapt this first construction to become the quotient of the return map of the flow we will construct, in Section \[sec:unpert-basic-dynamic\]. In Section \[sec:unpert-singul-flow\] we start the construction of the singular flow we are interested in. This is done again in stages, and here we obtain a first candidate. Next we obtain the vector field $X$ after perturbing the candidate in Section \[sec:pert\_flow\].
We study the properties of $X$ and its unfolding in Section \[sec:conseq-unfold-x\]. We prove the existence of the dominated splitting for the return map to the cross-section and describe the construction of the physical measure with positive multidimensional Lyapunov exponents in Section \[sec:x-chaotic-with\], completing the proof of Theorem \[thm:pri\]. The details on the existence of the physical measure for the return map are left for Section \[sec:higher-dimens-misiur\], where “Benedicks-Carleson type” arguments are adapted to our higher-dimensional setting.
The unfolding of the vector field $X$ is studied in Section \[sec:unfold-x\], where Theorem \[mthm:codimension\] is proved and the argument for the proof of Theorem \[mthm:fullbasinsink\] is described. The proof of Theorem \[mthm:fullbasinsink\] is given in Section \[sec:full-basin-attract\].
We end with two appendixes. Appendix \[sec:hyperb-neighb-constr\] provides an adaptation of a major technical tool to our setting to prove the existence of an absolutely continuous measure for higher dimensional non-uniformly expanding maps. Appendix \[sec:isotopy\] provides a topological argument for the existence of a certain isotopy we need during the construction of the vector field $X$.
Acknowledgments {#acknowledgments .unnumbered}
---------------
This paper started among discussions with A.C.J. and V.P. during a (southern hemisphere) summer visit V.A. and M.J.P. payed to Universidade Federal da Bahia, at Salvador. We thank the hospitality and the relaxed and inspiring atmosphere of Bahia.
The construction {#sec:theconstruction}
================
Now we give an example of non-uniformly expading dynamics in higher dimensions which is interesting by itself since it *does not exhibit any uniformly expanding direction*. At the end of this subsection, we show how this yields a construction of an attractor with $k+1$ non-uniformly expanding directions on a compact boundaryless $(k+5)$-dimensional manifold $M$.
We start by defining a non-uniformly expanding endomorphism of a $(k+1)$- dimensional manifold $N$.
Let $\Upsilon:{{\mathbb R}}\times {{\mathbb C}}^{k} \to {{\mathbb R}}\times {{\mathbb C}}^{k}$ be given by $(t, z)\mapsto (\cos(\pi t), z \sin(\pi t))$. We consider ${{\mathbb T}}^k={{\mathbb S}}^1 \times \overset{k}{\dots} \times
{{\mathbb S}}^1$ where ${{\mathbb S}}^1=\{z\in{{\mathbb C}}: |z|=1\}$ and let $N=
\Upsilon([-1,1]\times {{\mathbb T}}^{k})$. Clearly $N$ has a natural differential structure: $N=G^{-1}(\{1,\dots,1\})$ for $$G:{{\mathbb R}}\times{{\mathbb C}}^{k}\to{{\mathbb R}}^k:
(t,z_1,\dots,z_k)=(t^2+|z_1|^2,\dots,t^2+|z_k|^2)$$ which we name “torusphere” and is a manifold of dimension $k+1$, see Figure \[fig:torusph-each-meridi\].
![The “torusphere”: each parallel is a $k$-torus.[]{data-label="fig:torusph-each-meridi"}](torusphere.eps){width="6cm"}
We remark that this manifold is the boundary of $M:=G^{-1}([0,1]^k)$, which is a “solid torusphere”, that is $M\cap(\{t\}\times{{\mathbb R}}^k) \simeq {{\mathbb T}}^k\times{{\mathbb D}}$ for all $-1<t<1$, where ${{\mathbb D}}$ is the unit disk in ${{\mathbb C}}$. In what follows we write $I= [-1, 1]$.
An example of NUE dynamics in high dimension {#sec:an-example-nue}
--------------------------------------------
Let $g_0: I \circlearrowleft$ be a $C^{1+}$ non-flat unimodal map with the critical points $c_0=0$ and $c_1=g_0(c_0)=1$ as follows $$g_0(x)=
\begin{cases}
\varsigma^+\left(
\left|\frac{x}2\left(1-\frac{x}2\right)\right|^\alpha
\right)
& \text{if } x\in[0,1]\\
\varsigma^-(|x|^\alpha) & \text{if } x\in[-1,0)
\end{cases};$$ for $C^\infty$ diffeomorphisms $\varsigma^\pm:[0,1]\to I$ such that both $\varsigma^+$ and $\varsigma^-$ are monotonous decreasing. Moreover we assume that *the critical order $\alpha$ is strictly between $1$ and $2$*, $1<\alpha<2$, and that $g_0$ satisfies (see Figure \[fig:1dnue\]):
![The one-dimensional map $g_0$.[]{data-label="fig:1dnue"}](1dnue.eps)
1. $g_0(\pm1)=-1$ and $g_0$ has exactly two fixed points, namely, $p_0=-1$ and $0<p_1<1$;
2. $g_0'( p_0) > 1$ and $g_0'(p_1)<-1$.
It is well known that these maps are non-uniformly expanding and conjugated to the tent map (see the proof of Lemma \[le:alamane\] in what follows). Thus, in particular, they are topologically transitive and admit a unique absolutely continuous probability measure supported on the entire interval.
Now we define $f_1: {{\mathbb C}}^k\to {{\mathbb C}}^k$ by $(w_1,\dots,w_k)\mapsto (w_1^2,\dots,w_k^2).$ Then the map $g_0\times f_1: [-1, 1]\times {{\mathbb T}}^k\circlearrowleft $ induces a $C^{1+\alpha}$ map $g : N \circlearrowleft$ by $
g:=\Upsilon \circ (\hat g_0\times f_1 ) \circ \Upsilon^{-1}.
$ It is easy to see that $g$ takes meridians onto meridians and parallels onto parallels of the torusphere. This means that the derivative of $g$ preserves the directions associated to such foliations.
\[claim:NUE\] The map $g$ is a non-uniformly expanding map.
Using polar coordinates, we can take a parametrization $h:
(-1,1) \times (0, 2\pi)^k \to N$ given by $ h(t,\Theta)=
h(t, \theta_1, \dots, \theta_k):= (\sin(\frac\pi2 t),
z(\Theta)\cos(\frac\pi2 t)), $ where $\Theta:= (\theta_1,
\dots, \theta_k)$ and $$z(\Theta):=
(\cos(\theta_1), \sin(\theta_1), \dots, \cos(\theta_k),
\sin(\theta_k)) \in {{\mathbb R}}^{2k}.$$ The image of $h$ covers $N$ except for a null Lebesgue measure set. Therefore, the expression of $g$ in these coordinates is $$\underbrace{\Big(\sin(\frac\pi2 t), \cos(\frac\pi2 t) z(\Theta)\Big)}_{:= x}
\longmapsto \Big(\sin( \frac\pi2 g_0(t)), \cos(\frac\pi2
g_0(t)) z(2\Theta) \Big).$$ In the meridian directions, this implies that the derivative $Dg(x): T_x N \to T_{g(x)} N$ takes the vector $v:=
\big(\cos(\frac\pi2 t), \sin(\frac\pi2 t) z(\Theta)\big)$ to $Dg(x) \cdot v=
g'_0(t) \cdot \big(\cos(\frac\pi2 g_0(t)), \sin(\frac\pi2 g_0(t))
z(2\Theta)\big)$.
We adopt in ${{\mathbb R}}\times {{\mathbb R}}^{2k}$ (where $N$ is embedded) the norm ${\interleave(t, z)\interleave}:= \sqrt{{\lvertt\rvert}^2+ \|z\|^2/k}$, where $\|z\|$ is the standard Euclidean norm in ${{\mathbb R}}^{2k}$. Therefore $$\begin{aligned}
\frac{{\interleaveDg(x) \cdot v\interleave}}{{\interleavev\interleave}}
&= | g'_0(t)| \cdot
\sqrt{\frac{\cos^2(\frac\pi2 g_0(t))+ \sin^2(\frac\pi2 g_0(t)) \cdot
\|z(2\Theta)\|^2/k}{\cos^2(\frac\pi2 t)+ \sin^2(\frac\pi2 t) \cdot
\|z(\Theta)\|^2/k} }
\\
&=
| g'_0(t)| \cdot \sqrt{\frac{\cos^2(\frac\pi2 g_0(t))+
\sin^2(\frac\pi2 g_0(t))}{\cos^2(\frac\pi2 t)+
\sin^2(\frac\pi2 t) } }= |g'_0(t)|.\end{aligned}$$ Along the directions of the parallels, given any $j\in \{1,
\dots, k\}$ the derivative $Dg(x)$ takes the vector $v_j:= (0,
\dots, -\cos(\frac\pi2 t)\sin(\theta_j), \cos(\frac\pi2 t)\cos(\theta_j), 0,
\dots 0)$ to the vector $$Dg(x) \cdot v_j= (0, \dots, -2 \cos(\frac\pi2 g_0(t))\sin(2 \theta_j),
2\cos(\frac\pi2 g_0(t))\cos(2 \theta_j), 0, \dots 0).$$ Therefore $$\frac{\||Dg(x) \cdot v_j|\|}{\||v_j|\|}
=
2 \frac{|\cos(\frac\pi2 g_0(t))|
\cdot \sqrt{\sin^2(2\theta_j)+ \cos^2(2\theta_j)}}
{|\cos(\frac\pi2 t)|\cdot \sqrt{\sin^2(\theta_j)+
\cos^2(\theta_j)}} =
2 \Big|\frac{\cos(\frac\pi2 g_0(t))}{\cos(\frac\pi2 t)}\Big|.$$ We note that for $w= \sum_{j= 1}^k \alpha_j v_j$, the relation $ \frac{{\interleaveDg(x) \cdot w\interleave}}{{\interleavew\interleave}}= 2
\Big|\frac{\cos(\frac\pi2 g_0(t))}{\cos(\frac\pi2 t)}\Big| $ also holds. Let us call $E_x$ the space generated by the directions of the parallels through $x$ in $T_x N$, and $m(x):= \inf_{w \in E_x} \frac{{\interleaveDg(x) \cdot
w|\interleave}}{{\interleavew\interleave}}$ the minimum norm (or conorm) of $Dg(x)$ restricted to $E_x$. We have $$\begin{aligned}
\frac 1 n \sum_{j= 0}^{n-1} \log m(g^j(x))
&=
\log(2) +
\frac 1 n \sum_{j= 0}^{n-1} \log\Big|\frac{\cos(\frac\pi2
g_0^{j+1}(t))}{\cos(\frac\pi2 g_0^{j}(t))}\Big|
= \log(2) +
\frac1n
\log\left|
\frac{\cos(\frac\pi2 g_0^n(t))}{\cos(\frac\pi2 t)}
\right|.\end{aligned}$$ Thus we will have $n^{-1} \sum_{j= 0}^{n-1} \log
m(g^j(x)) \ge \log 2$ whenever $|\cos(\frac\pi2 g_0^n(t))|\ge
|\cos(\frac\pi2 t)|$, which is true if, and only if, $|g_0^n(t)|\le |t|$.
To conclude the proof of Claim \[claim:NUE\] we use Lemma \[le:alamane\] below, whose proof we postpone to the end of this subsection.
\[le:alamane\] Given a neighborhood $U$ of $c_0$, Lebesgue almost every orbit visits $U$ infinitely often.
Since $c_0=0$, Lemma \[le:alamane\] implies that for every given $t\in I\setminus\{0\}$ the inequality $|g_0^n(t)|\le|t|$ is true for infinitely many values of $n\ge1$. This implies that $$\begin{aligned}
\label{eq:NUEinf0}
\liminf_{n\to+\infty}
\frac1n\sum_{j=0}^{n-1}\log{\interleave(Dg\mid
E_{g^j(x)})^{-1}\interleave} \le -\log 2<0 \quad\text{for Lebesgue
almost every}\quad x.\end{aligned}$$ Denoting $F_x$ the direction of the meridian at $T_x N$, for $x=(t,\Theta)$, we also showed that $$\begin{aligned}
\label{eq:NUEinf1}
\liminf_{n\to+\infty}
\frac1n\sum_{j=0}^{n-1}\log{\interleave(Dg\mid F_{g^j(x)})^{-1}\interleave}
=
-\limsup\frac1n\sum_{j=0}^{n-1}\log|g_0^\prime(g_0^j(t))|<0\end{aligned}$$ for Lebesgue almost every $t\in I$, which is strictly negative by known results on unimodal maps (see e.g. [@MS93]). From and we obtain $$\begin{aligned}
\label{eq:NUEsup}
\limsup_{n\to+\infty} \frac 1 n \sum_{j= 0}^{n-1} \log
{\interleaveDg(g^j(x))^{-1}\interleave}^{-1} > 0 \quad\text{for Lebesgue
almost every}\quad x.\end{aligned}$$ Hence, according to [@Pinheiro05], $g$ is a non-uniformly expanding map since, besides , the orbit of the critical set clearly *does not accumulate* the critical set. This ensures that $g$ admits an absolutely continuous invariant probability measure, which is unique because $g$ is a transitive map (see the proof of Lemma \[le:alamane\] below).
It is not difficult to see that $g_0$ is topologically conjugated to the Tent Map $T(x):=1-2|x|$ for $x\in I$ under some homeomorphism $h$ of the interval $I$. Indeed, searching for $h$ of the form $h(x)=x+u(x)$ on each interval $[-1,0]$ and $[0,1]$ for some continuous $u:\pm[0,1]\to I$ with small $C^0$-norm, we get the relation $$\begin{aligned}
h(g_0(x))=1-2|h(x)|
\quad\text{or}\quad
g_0(x) + u(g_0(x))
= 1-2|x+u(x)|.
\end{aligned}$$ For $x\in [0,1]$ we have $h(x)\ge0$ and so we obtain $$\begin{aligned}
\underbrace{2^{-1}
u(g_0(x))+u(x)}_{{\EuScript{L}}(u)x}=
\frac12\big(1-2x-g_0(x)\big).
\end{aligned}$$ We remark that from this relation it follows that $u(0)=u(1)=0$ and, moreover, the right hand side is strictly smaller than $1/2$ uniformly on $[0,1]$, that is, ${\EuScript{L}}(u)\le1/2-\xi$ for some $0<\xi<1/2$. Clearly ${\EuScript{L}}(u)={\EuScript{I}}+L$ where ${\EuScript{I}}$ is the identity on $C^0([0,1],I)$ and $(Lu)x=2^{-1}u( g_0(x) )$ has $C^0$ norm $\le1/2$. Thus the linear operator ${\EuScript{L}}:
C^0([0,1],I)\to C^0([0,1],I)$ admits an inverse. Analogously for the conjugation equation on $[-1,0]$. So we can find $h={\EuScript{I}}+u$ with $u$ having $C^0$ norm $<1$, as needed to ensure that $h$ is invertible, thus a homeomorphism of $I$.
*This guarantees that $g_0$ is transitive and, in particular, has no attracting periodic orbits.*
The subset $K:=\cap_{n\ge0} T^{-n}([-1,1-{\varepsilon}])$ is a $T$-invariant Cantor set with zero Lebesgue measure, for each $0<{\varepsilon}<1$. Thus $K_0:=h(K)$ is also a $g_0$-invariant Cantor set such that, for some $\delta>0$ and every $z\in K_0$ satisfies $g_0^n(z)\in
I\setminus\big( (-\delta,\delta)\cup (1-\delta,1] \big)$ for all $n\ge0$.
That is, $K_0$ is the set of points whose future orbits under $g_0$ do not visit a neighborhood $V_0$ of the critical set, so that $g_0\mid (I\setminus
\overline{V_0})$ is a $C^\infty$ map acting on $K_0$, by the definition of $g_0$. Hence in $K_0$ we have no critical points and no attracting periodic orbits, thus the restriction $g_0\mid K_0: K_0\to K_0$ is a uniformly expanding local diffeomorphism by Mañé’s results in [@Man85].
Therefore the Lebesgue measure of $K_0$ is zero, for otherwise this set would have a Lebesgue density point $p$. Since $g_0$ is $C^\infty$ is a neighborhood of $K_0$, this would imply that $K_0$ would contain some interval $J$ (see e.g. [@Vi97b] or [@alves-luzzatto-pinheiro2005]). But $g_0$ is uniformly expanding on $K_0$, thus the length of the successive images $g_0^k(J)$ of $J$ would grow to at least the length of one of the domains of monotonicity of $g_0$, in a finite number of iterates. The next iterate would contain the critical point, contradicting the definition of $K_0$.
It follows that the set $E(\delta)$ of points of $I$ which do not visit a $\delta$-neighborhood of $c_0$ under the action of $g_0$ has zero Lebesgue measure, for all small $\delta>0$. This ensures that $\cup_{n>N}\cap_{k>n}
g_0^{-k}\big( E(k^{-1})\big)$ has zero volume for every big $N>1$. Consequently the set $\cap_{n>N}\cup_{k>n}
g_0^{-k}\big(M\setminus E(k^{-1})\big)$ has full measure, and for points in this set there are infinitely many iterates visiting any given neighborhood of $c_0$.
The unperturbed basic dynamics {#sec:unpert-basic-dynamic}
------------------------------
We now adapt the example in Section \[sec:an-example-nue\] in order to obtain a map $f$ which will be a kind of Poincaré return map for a singular flow, that we will perturb later to obtain a Rovella-like flow. We again construct a map $f$ in the torusphere by defining its action in the meridians and parallels.
Let $f_0: I \to I$ be a $C^{1+}$ non-flat unimodal map with the critical point $c=0$, that is, $$f_0(x)=\begin{cases}
\psi^+(x^\alpha) & \text{if } x\in(0,1]\\
\psi^-(|x|^\alpha) & \text{if } x\in[-1,0)
\end{cases};$$ for smooth monotonous increasing diffeomorphisms $\psi^\pm:[0,1]\to I$. Moreover we assume that the critical order $\alpha$ is at least $2$ and that $f$ satisfies (see Figure \[fig:1dcritical\]):
![The one-dimensional map $f_0$.[]{data-label="fig:1dcritical"}](1dcritical.eps)
1. $f_0(\pm1)=-1$, $f_0$ has exactly three fixed points: $p_0=-1< p_1 <c=0 < p_2<1$;
2. $f_0'(p_2)<-1<0 \leq f_0'(-1) < 1< f_0'(p_1)$.
The map $f_0\times
f_1$ induces a $C^{1+\alpha}$ map $f : N\to N$ by $f=\Upsilon \circ (f_0\times f_1 ) \circ \Upsilon^{-1}.$ Let ${{\mathbb T}}_1^k=\Upsilon(\{p_1\}\times {{\mathbb T}}^k)$. Note that $f({{\mathbb T}}_1^k)=
{{\mathbb T}}^k_1$, in other words, ${{\mathbb T}}^k_1$ is positively invariant by $f$ and a uniform repeller, see Figure \[fig:repel\].
![\[fig:repel\]The parallel corresponding to the expanding fixed point is a uniform repeller for $f$, a vector $\vec u$ tangent to a meridian, and a vector $\vec v$ tangent to a parallel.](torusrepeller.eps)
The unperturbed singular flow {#sec:unpert-singul-flow}
-----------------------------
Here we build a geometric model for a $(k+6)$-dimensional flow $X^t_0$, $t \ge 0$. We write $B^n$ for the $n$-dimensional unit ball in ${{\mathbb R}}^n$, that is $B^n:=\{ x=(x_1,\dots,x_n)\in{{\mathbb R}}^n:
\sum_{i=1}^n x_i^2<1\}$.
Recall that $f_1: {{\mathbb T}}^k \to {{\mathbb T}}^k$ is the expanding map defined in section \[sec:an-example-nue\]. Such map has a lift to an inverse limit which is a higher dimensional version of a Smale solenoid map, see [@Sm67].
More precisely, as shown in Appendix \[sec:isotopy\], given a solid $k$-torus ${\EuScript{T}}:={{\mathbb T}}^k\times{{\mathbb D}}$, where ${{\mathbb D}}=\{z\in{{\mathbb C}}:|z|\le1\}$, there exists a map $S: e({\EuScript{T}})\to
e({\EuScript{T}})$ defined on the image of a smooth embedding $e:{\EuScript{T}}\to
B^{k+2}$ such that $\pi_{{{\mathbb D}}}\circ S = f_1\circ\pi_{{\mathbb D}},$ where $\pi_{{\mathbb D}}:e({\EuScript{T}})\to e({{\mathbb T}}^k\times 0) \simeq{{\mathbb T}}^k$ is the projection along the leaves of the foliation ${\EuScript{F}}^s:=\{
e(\Theta\times{{\mathbb D}}) \}_{\Theta\in{{\mathbb T}}^k}$ of ${\EuScript{T}}$. Moreover, $F$ contracts the disks in ${\EuScript{F}}^s$ by a uniform contraction rate $\lambda\in(0,1)$. We remark that ${{\mathbb T}}^k$ is the quotient of $e({\EuScript{T}})$ over ${\EuScript{F}}^s$. From now on we identify $e({\EuScript{T}})$ with ${\EuScript{T}}$ and ${{\mathbb T}}^k$ with $e({{\mathbb T}}^k\times0)$. The map $S$ is the higher dimensional version of a Smale solenoid map we mentioned above.
We also have (see Appendix \[sec:isotopy\]) that there exists an smooth isotopy $\phi_t:B^{k+2}\to B^{k+2}$ between $S=\phi_1$ and the identity map $\phi_0$ on $B^{k+2}$.
We define a flow between two cross sections $\Sigma_j=\pi_1(I^2\times\{3-j\}\times B^{k+2}), j=1,2$, by $$Y^t\big( \pi_1 (x_1,x_2,2,W)\big)=
\pi_1\big(x_1,x_2,2-t,\phi_t(W)\big),\quad 0\le t\le1$$ where $(x_1,x_2,2-t,W)\in
I^2\times[1,2]\times B^{k+2}$ and $$\pi_1: {{\mathbb R}}\times{{\mathbb R}}\times{{\mathbb R}}\times B^{k+2}\circlearrowleft,
\quad (x_1,x_2,x_3,W)\mapsto(x_1,x_2,x_3,(1-x_1^2)^{1/2}W).$$ We note that $\Sigma_j\simeq I\times\overline{B^{k+3}}$ naturally for any $j=1,2$. We extend this flow so that there exists a Poincaré return map from $\Sigma_2$ to $\Sigma_1$ with the properties we need. For this we take a linear flow on ${{\mathbb R}}^3\times {{\mathbb R}}^{k+2}$ with a singularity $s_0$ at the origin having real eigenvalues $\lambda_1 >0$, and $\lambda_1+\lambda_j< 0$ for $2\le j \le k+5$. For simplicity, we also assume that $\lambda_j=\lambda_3$ for all $j\ge4$ and that $\alpha:=-\lambda_3/\lambda_1$ and $\beta:=-\lambda_2/\lambda_1$ satisfy $\beta>\alpha+2$. This last *strong dissipative condition* on the saddle $s_0$ ensures that the foliation corresponding to the $x_2$ direction is dominated, and so persists for all $C^2$ nearby flows.
We note that the subspace $\{0\}\times{{\mathbb R}}^2\times B^{k+2}$, excluded from further considerations, is contained in the stable manifold of $s_0$ and so its points never return no $\Sigma_j$.
We write $(x_1,x_2,1,W)$ for a point on $\Sigma_2$, with $x_i\in{{\mathbb R}}$ and $W\in B^{k+2}$, and consider the cross-sections $\Sigma^\pm=\{(\pm1,x_2,x_3,W): (x_2,x_3)\in
I^2, W\in{{\mathbb R}}^{k+2}\}$ to the flow and the Poincaré first entry transformations given by (see Figure \[fig:startflow\]): $$\begin{aligned}
\label{eq:thrusing}
L^\pm: \Sigma_1\cap\{\mp x_1>0\}\to \Sigma^\pm: \quad
(x_1,x_2,1,W)\mapsto (\pm 1, x_2 |x_1|^\beta,
|x_1|^{\alpha}, |x_1|^{\alpha}W).\end{aligned}$$
\[rmk:Holder-thru\] The maps $L^\pm$ given in are clearly Hölder maps in their domain of definition. Moreover the time the flow needs to take the point $(x_1,x_2,1,W)$ to $\Sigma^\pm$ is given by $-\log |x_1|$, where $|x_1|$ is the distance to the local stable manifold of $s_0$ on $\Sigma_2$.
![The starting singular flow with the transformations $Y^1$, $L^\pm$ and the projections $\pi_1$ and $\pi_2$.[]{data-label="fig:startflow"}](singflow.eps)
Now we define diffeomorphisms from the image of $L^\pm$ to $\Sigma_1$ which can be realized as the first entry maps from $\Sigma^\pm$ to $\Sigma_1$ under a flow defined away from the origin in ${{\mathbb R}}^{k+5}$. Since we want to define a flow with an attractor containing the origin in ${{\mathbb R}}^{k+5}$, we need to ensure that the return map defined on $\Sigma_1$ through the composition of all the above transformations does preserve the family of tori together with their stable foliations. Moreover the quotient of this return transformation over the stable directions should be the map $f$. We write these transformations as $T^\pm:\Sigma^\pm\to\Sigma_2$ given by $$(\pm1,z_2,z_3,V)
\mapsto\left(\psi^\pm(z_3),
\pm\frac12+\frac{z_2}C,
2,
\Psi^\pm(z_3)V\right)$$ where $C>0$ is big enough so that the map restricted to the first $3$ coordinates is injective and thus a diffeomorphism with its image. We recall that $\psi^\pm$ is part of the definition of the one-dimensional map $f_0$. The diffeomorphisms $\Psi^\pm:[0,1]\to I$ are chosen to ensure that the quotient map is well defined on $N$, as follows: $$z_3\mapsto
\Psi^\pm(z_3):=
\frac1{z_3}\sqrt{\frac{1-\psi^\pm(z_3)^2}{1-z_3^{2/\alpha}}},
\quad\text{for } z_3\in(0,1)$$ and we set $\Psi^\pm(0):=\mp1$ and $\Psi^\pm(1)=\pm1$. We remark that $\Psi^\pm$ are diffeomorphisms, see Figure \[fig:startflow\]. Indeed both the numerator and denominator inside the square root can be written as a Taylor series around $z_3=1$ with an expression like $
const\cdot(1-z_3) + o(1-z_3)
$ for some non-zero constant, thus $\Psi^\pm$ is differentiable at $1$. Now $\psi^\pm$ can be expanded around $z_3=0$ as $
1+const\cdot z_3 + o(|z_3|)
$ for some positive constant. Hence $\sqrt{1-(\psi^\pm)^2}$ can be expanded as $
\big(1-(1-const\cdot z_3 + o(|z_3|))^2\big)^{1/2}=
{\operatorname{const}}\cdot z_3 + o(|z_3|)
$ thus $\Psi^\pm$ is also differentiable at $0$.
Now we check that the return map $R_0$ given by $
R_0:= T^\pm \circ L^\pm \circ Y^1:\Sigma_2\circlearrowleft
$ can be seen as a map on $N$. Indeed notice that $Y^1$ commutes with $\pi_1$ by construction and that for $x_1>0$ $$\begin{aligned}
(T^+\circ L^+\circ \pi_1)(x_1,x_2,1,W) &= (T^+\circ
L^+)\big(x_1,x_2,1,(1-x_1^2)^{1/2} W\big)
\\
&= T^+\Big( \psi^+\big(|x_1|^\alpha\big), x_2
|x_1|^\beta, |x_1|^\alpha, |x_1|^\alpha
(1-x_1^2)^{1/2} W\Big)
\\
&= \left( f_0(x_1), \frac12+\frac{x_2
|x_1|^\beta}C , 2, W
\sqrt{1-\psi^+(|x_1|^\alpha)^2} \right)
\\
&= \pi_1\Big( f_0(x_1), \frac12+\frac{x_2
|x_1|^\beta}C , 2, W\Big).\end{aligned}$$ Hence we get $$\begin{aligned}
R_0(x_1,x_2,2,W)&= (Y^1\circ T^+\circ L^+ \circ
\pi_1)(x_1,x_2,2,W)
\nonumber
\\
&= \pi_1
\Big( f_0(x_1), \frac12+\frac{x_2
|x_1|^\beta}C ,
2, \phi_1(W)\Big), \label{eq:returnmap}\end{aligned}$$ where $x_1>0$ and $\phi_1=F$ by the definition of $Y^1$. Analogous calculations are valid for $x_1<0$ taking the maps $T^-$ and $L^-$ into account. By the definition of $\phi_t$ we get that $R_0$ maps $\hat \Sigma =
\Sigma_2\cap\pi_1(I^2\times\{1\}\times{\EuScript{T}})$ inside itself.
\[rmk:domination\] The contracting direction along the eigendirection of the eigenvalue $\lambda_2$ can be made dominated by the other directions in this construction by increasing the contraction rates given by $C$ and $\lambda_2$. This is very important to ensure the persistence of the stable lamination, see e.g. [@HPS77] and Section \[sec:x-chaotic-with\]. This is why we assumed the strong dissipative condition $\beta>\alpha+2$ on the saddle equilibrium $s_0$. We note that this domination is for the action of the flow we are constructing.
We now remark that quotienting out the contracting directions of ${\EuScript{F}}^s$ we obtain the map $f$ on $N$. Indeed, each element of the quotient can be seen as the image of the following projection $
\pi_2:\hat\Sigma_2\to N$ given by $(x_1,x_2,1,W)\mapsto(x_1,(1-x_1^2)^{1/2}\pi_{{\mathbb D}}(W))$ where, we recall, $\pi_{{\mathbb D}}: {\EuScript{T}}\to{{\mathbb T}}^k={{\mathbb S}}^1 \times
\overset{k}{\dots} \times {{\mathbb S}}^1 \subset {{\mathbb C}}^k \simeq {{\mathbb R}}^{2k}$. It is easy to see that *the return map $R_0$ is semiconjugated to $f$ through $\pi_2$*, that is $
\pi_2\circ R_0 = f\circ \pi_2. $
\[rmk:cylinder\] We take advantage of the fact that the hyperplanes $\{x_1=\pm1\}$ can be identified to a single point due to the dynamics of $\hat g_0$. If we do not perform this type of identification, i.e. if we consider instead the projection $\pi_3:\hat\Sigma_2\to I\times{{\mathbb T}}^k$ given by $(x_1,x_2,1,W)\mapsto(x_1,\pi_{{\mathbb D}}(W))$, then we get a cylinder $I\times{{\mathbb T}}^k$ as the domain of the quotient map, instead of the torusphere $N$, and likewise $\pi_3\circ R_0 = (f_0\times f_1)\circ \pi_3$, where $f_1:{{\mathbb T}}^k\circlearrowleft$ is the expanding map on the $k$-torus defined in Section \[sec:an-example-nue\].
### Localizing some periodic orbits of the flow {#sec:some-period-orbits}
The vector field $X_0$ just constructed has a flow with an attracting periodic orbit, the orbit of $P_0=\pi_1(-1,y_*,2,0)$ for some $y_*\in I$. Indeed, we note that $-1$ is an attracting fixed point for the one-dimensional map $f_0$ and that the $f_0$ orbit of almost every $x\in I$ tends to $-1$. Then, since on the second coordinate in $\Sigma_2$ we have a strong contraction under the action of $R_0$, this ensures that there exists $y_*$ as above satisfying $R_0(\pi_1(-1,y_*,2,0))=\pi_1(-1,y_*,2,0)$. We note that along the $x_1$ and $x_2$ directions the flow clearly is a contraction. Moreover, for $x$ in the interval $(-1,p_1)$, the “toruspherical coordinates” of $\pi_1(x,y,2,W)$ (that is, the last coordinate of dimension $k+2$) tend to $0$ since they are multiplied by $\sqrt{1-f_0^k(x)^2}$ and $f_0^k(x)\to-1$ as $k\to+\infty$.
In addition the fixed point $p_1$ of $f_0$ corresponds to a hyperbolic invariant subset for the flow inside the solid torus $P_1=\pi_1((p_1,y^*,2)\times{\EuScript{T}})$ for some $y^*\in I$. We observe that quotienting out the stable directions we get the invariant torus ${{\mathbb T}}^k_1$ as already mentioned before, see Figure \[fig:startflow\].
Finally the orbit of $P_2=\pi_1(1,\bar y,2,0)$ is in the stable manifold of $P_0$ for any $\bar y\in I$. Indeed it returns to the stable leaf $\{\pi_1(-1,y,2,0):y\in I\}$ and never leaves this leaf in all future returns. In the quotient $\pi_2(\hat\Sigma_2)$ the point $P_0$ is fixed and $P_2$ is one of its preimages under $f$.
Perturbing the original singular flow {#sec:pert_flow}
-------------------------------------
Now we make a perturbation $X$ of the vector field $X_0$ constructed in the previous section to obtain a Rovella-like attractor.
From now on we assume that $P_2$ is the point where the component of the unstable manifold of $s_0$ through $\Sigma^+$ first arrives at $I^2\times\{2\}\times B^{k+2}$. We consider then the positive orbit of $P_2$ up until it returns to $\hat\Sigma_2$. By construction the return point $R_0(P_2)$ has the expression $\hat P=\pi_1(-1,\hat y,2,0)$, so it returns in $\pi_2(-1,0)$, after the quotient through $\pi_2$.
![\[fig:beforeperturb\] This represents the initial flow before the perturbation and the one-dimensional quotient.](tubularperturb0.eps)
This is a regular orbit, so it admits a tubular neighborhood. We can assume that the orbit chosen above returns to $\hat\Sigma_2$ close enough to $P_0$, that is $\hat
y$ is close to $y_*$. We also assume that $P_1$ is also very close to $P_0$ so that the tubular neighborhood contains both $P_0$ and $P_1$, see Figure \[fig:beforeperturb\].
We note that since $P_0$ is a hyperbolic attracting orbit it is easy to extend the tubular neighborhood to its local basin of attraction whose topological closure contains $P_1$.
![The perturbed singular flow seen through the projections $\pi_1$ and $\pi_2$.[]{data-label="fig:tubularperturb"}](tubularperturb.eps)
In this setting we can now perturb the flow inside the tubular neighborhood in the same way as to produce a kind of Cherry flow, introducing two hyperbolic saddle singularities $\hat s$ and $s_1$. Here the extra dimensions are very useful to enable us to introduce such saddle fixed point with the adequate dimensions of stable and unstable manifolds. The saddle $s_1$ has $k+1$ expanding eigenvalues $\bar\lambda_0,\dots,\bar\lambda_{k}$, the remaining $4$ contracting eigenvalues $\bar\lambda_j, j=k+1,\dots,k+4$, and $s_1$ is sectionally dissipative: $\bar\lambda_i+\bar\lambda_j<0$ for all $i\le k$ and $j>k$. In fact the extra dimensions are essential to allow the construction of such a saddle.
The other saddle $\hat s$ has $k+2$ expanding eigenvalues and $3$ contracting ones. We assume that this perturbation is done in such a way that the $(k+1)$-dimensional unstable manifold of $s_1$ *contains* ${{\mathbb T}}^k_1$, see Figure \[fig:tubularperturb\], and the stable manifold of $s_1$ is everywhere tangent to the subspace given by the direction of the stable manifolds of the solid tori together with the $x_2$ direction. In this way we can still quotient out the stable leaves, which are preserved by the perturbed flow.
\[rmk:keepsaddleconnection\] Since $P_2$ is part of the unstable manifold of $s_0$, then we have constructed a saddle connection between $s_0$ and $s_1$. Because the stable manifold of $s_1$ is $3$-dimensional, we can keep the connection for nearby vector fields restricted to a $(k+2)$-codimension submanifold of the space of all smooth vector fields: all we have to do is to keep one branch of the one-dimensional unstable manifold of $s_0$ contained in the $3$-dimensional local stable manifold of $s_1$, and this submanifold has codimension $k+2$. See Figure \[fig:saddle-connection\].
![The connection between $s_0$ and $s_1$.[]{data-label="fig:saddle-connection"}](connection.eps)
The action of the first return map $R$ of the new flow $X^t$ to $\Sigma_2$, on the family of stable leaves which project to the interval $[p_1,1]$, equals (except for a linear change of coordinates) the map $g$ presented in Section \[sec:an-example-nue\], see Figure \[fig:tubularperturb\]. That is, we have a multidimensional non-uniformly expanding transformation as the quotient of the first return map. This transformation sends each $k$-torus in $N$ to another $k$-torus and so it can be further reduced to the interval map $g_0$, by considering its action on the tori.
\[rmk:poincare-time\] The introduction of the saddle-connection changes the return time of the points in $\Sigma_2$ under the new flow, but we can assume that the distance from the stable manifold of $s_1$ of the orbit of $(x_1,x_2,2,W)\in\Sigma_2$ near $s_1$ is given by the $x_3$ coordinate of $L^\pm(x_1,x_2,2,W)$, that is, by $|x_1|^\alpha$. Therefore the Poincaré return time is now bounded, after Remark \[rmk:Holder-thru\], by some uniform constant plus $const\cdot\log|x_1|$. The value $|x_1|$ equals the distance of $(x_1,x_2,2,W)$ to the critical set of $g$.
This reduction to a one-dimensional model will be essential to our analysis of the existence of a absolutely continuous invariant probability measures for $g$ and the existence of a physical probability measure for the attracting set of the flow near the origin.
### The attracting set and an invariant stable foliation {#sec:attract-set-its}
We note that the set $\widetilde\Lambda_\Sigma:=\cap_{n>0}
R^n(\Sigma_2)$ contains a subset $\Lambda_\Sigma:=\cap_{n>0}
R^n(\widetilde U)$, where $\widetilde U$ is a small neighborhood in $\Sigma_2$ of $\pi_2^{-1}[p_1,1]$, which is an attracting set for $R$, that is, $\overline{R(\widetilde
U)}\subset\widetilde U$. Hence the saturation $\Lambda=\cup_{t\in{{\mathbb R}}}X^t(\Lambda_\Sigma)$ is also an attracting set for $X^t$ such that $\Lambda_\Sigma= \Lambda\cap\Sigma_2$.
It is easy to see that every point $z=\pi_1(x_1,x_2,1,W)$ of $\Lambda_\Sigma$ belongs to the solenoid attractor in $\{(x_1,x_2,1)\}\times B^{k+2}$. Therefore it is straightforward to define a local stable foliation ${\EuScript{F}}^{ss}$ through the points of $\Lambda_\Sigma$: we define ${\EuScript{F}}^{ss}_z$ to be the stable disk through $z$ of the solid torus $\{(x_1,x_2,1)\}\times \phi_1({\EuScript{T}})$. Hence we get a $DR$-invariant continuous and uniformly contracting subbundle $E^{ss}$ of $T\Lambda_\Sigma$ given by $E^{ss}_z=T_z{\EuScript{F}}^{ss}_z$.
From the existence of the invariant contracting subbundle $E^{ss}$ over $T\Lambda_\Sigma$, we can define the normal subbundle $G=(E^{ss})^\perp\cap {{\mathbb R}}\times\{(0,0)\}\times{{\mathbb R}}^{k+2}$ to $E^{ss}$ in the tangent space to $I\times\{(x_2,1)\}\times B^{k+2}$, and use this pair of continuous bundles to define a stable cone field and the complementary cone field $$\begin{aligned}
C^s(z)=\{(u,v)\in E^{ss}_z\oplus G_z : \|u\|\ge\|v\|\}
,\quad
C^u(z)=\{(u,v)\in E^{ss}_z\oplus G_z : \|u\|\le\|v\|\}\end{aligned}$$ for all points $z\in\Lambda_\Sigma$.
Is is clear that the bundle of tangent spaces to $\{I\times\{(x_2,1)\}\times
B^{k+2}\}_{x_2\in I}$ is invariant under $DR$, under both forward and backward iteration. Moreover, we have that $C^s$ is strictly invariant under backward iteration by $DR$ and, consequently, the complementary cone is invariant under forward iteration by $DR$. Therefore vectors in $C^u(z)$ make an angle with vectors in $E^{ss}_z$ uniformly bounded away from zero. We use this in the next section to prove the existence of a dominated splitting over $T\Lambda_\Sigma$.
Consequences for the vector field and its unfolding {#sec:conseq-unfold-x}
===================================================
We now show that the flow of $X$ is chaotic with multidimensional expansion, and consider its perturbations.
$X$ is chaotic with multidimensional nonuniform expansion {#sec:x-chaotic-with}
---------------------------------------------------------
We observe that, since we have a one-dimensional quotient map where the critical point is mapped to a repelling fixed point, we are in the setting of “Misiurewicz maps”, see the right hand side of Figure \[fig:tubularperturb\].
We note that the map on the torusphere for the perturbed flow is *non-uniformly expanding in all directions*, because the non-uniformity is seen on every direction away from the repelling torus ${{\mathbb T}}_1^k$, and because the singularity contracts also in every direction, see Figure \[fig:symperturb0\]. This was already proved in Section \[sec:an-example-nue\].
![\[fig:symperturb0\] The last perturbation defining the flow $X$. We can project the dynamics into a one-dimensional map.](tubularskewperturb0.eps)
### Invariant probability measure {#sec:invari-probab-measur}
The results in Section \[sec:higher-dimens-misiur\] ensure that there exists an ergodic absolutely continuous invariant probability measure $\upsilon$ for the transformation $g$, which is the action of the first return map $R$ of the flow of $X$ to $\Sigma_2$ on the stable leaves. Indeed $g$ satisfies all the conditions of Theorem \[thm:Misiurew\] since its quotient $g_0$ is a Misiurewicz map of the interval. Moreover $\upsilon$ is an *expanding measure*: all Lyapunov exponents for $\upsilon$-a.e. point are strictly positive, as shown in Section \[sec:an-example-nue\].
From standard arguments using the uniformly contracting foliation through $\Lambda_\Sigma$ (recall Section \[sec:attract-set-its\]), see e.g. [@APPV Section 6], we can construct an $R$-invariant ergodic probability measure $\nu$, whose basin has positive Lebesgue measure on $\Sigma_2$, and which projects to $\upsilon$ along stable leaves. Finally, using a suspension flow construction over the transformation $R$ we can easily obtain (see [@APPV Section 6] for details) a corresponding ergodic physical probability measure $\mu$ for the flow of $X$, which induces $\nu$ as the associated $R$-invariant probability on $\Sigma_2$.
We claim that $\mu$ is a hyperbolic measure for $X^t$ with $k+1$ positive Lyapunov exponents. To prove this, we first obtain a dominated splitting for the tangent $DR$ of the return map $R$ which identifies the bundle of directions with nonuniform expansion.
### Strong domination and smooth stable foliation {#sec:dominat-splitt}
We now consider the return map $R$ to $\Sigma_2$ and show that the subbundle $E^{s}$ corresponding to the tangent planes to the local stable leaves of the solenoid maps is “strongly $\gamma$-dominated” by the non-uniformly expanding direction “parallel” to the toruspheres, for some $\gamma>1$.
Let ${\EuScript{F}}^{s}_{a,b}$ be the bidimensional stable foliation of the solenoid map acting on the section $\{x_1=a,x_2=b\}\cap (I^2\{1\}\times B^{k+2})$ for each $(a,b)\in I^2$. Then, from the expression of $R_0$, which is only modified to $R$ by putting $g_0$ in the place of $f_0$, we see that the derivative of $R_0$ along some leaf $\gamma\in{\EuScript{F}}^{s}_{a,b}$ equals the derivative of $\phi_1$ along the same direction multiplied by $(1-g_0(a)^2)^{1/2}$. Since $\|D\phi_1\mid\gamma\|\le\lambda$ for some constant (not dependent on $a,b$ nor on the particular leaf of ${\EuScript{F}}^{s}_{a,b}$) $\lambda\in(0,1)$, we get that $\|DR\mid\gamma\|\le\lambda(1-g_0(a)^2)^{1/2}.$
We can estimate expansion/contraction rates of the derivative of the quotient map $g$ as in Section \[sec:an-example-nue\]. Since this map is obtained through a projection of $R$ along ${\EuScript{F}}^{s}$, the real expansion and contraction rates of the derivative of $R$ along any direction in the complementary cone $C^u$ are bounded by the corresponding rates of $Dg$ up to constants. These constants depend on the angle between the stable leaves and the direction on the complementary cone field, which is uniformly bounded away from zero.
Hence, recalling that $m(x)$ is the minimum norm of $Dg(x)$ for $x=\pi_1(t,b,1,z,\Theta)$ with $t\in I$, $z\in{{\mathbb D}}$ and $\Theta\in[0,2\pi)^k$, to obtain the smoothness of the foliation, it is enough to get that $$\begin{aligned}
\label{eq:domination}
d(x):=\frac{(1-g_0(t)^2)^{1/2}}{\|Dg(x)\|^\gamma m(x)^\omega}\end{aligned}$$ is bounded by some constant uniformly on every point $x$, for some $\gamma>1$ and $\omega>0$. This implies that for a small enough $\lambda>0$ we have $ \lambda(1-g_0(t)^2)^{1/2}\|Dg(x)\|^{-\gamma}<
\lambda m(x)^\omega < m(x)$, since $\omega>0$ and $m(x)$ is bounded from above. *This ensures (see [@HP70 Theorem 6.2]) that ${\EuScript{F}}^{s}$ is a $C^\gamma$ foliation, so that holonomies along the leaves of ${\EuScript{F}}^{s}$ are of class $C^\gamma$.*
From Section \[sec:an-example-nue\] we know that $$\begin{aligned}
\label{eq:minimum}
m(x)&=m(t)=\min\left\{|g_0^\prime(t)|,
2 \Big|\frac{\cos(\frac\pi2 g_0(t))}{\cos(\frac\pi2 t)}\Big|
\right\},\quad\text{and}
\\
\|Dg(x)\|&=\max\left\{|g_0^\prime(t)|,
2 \Big|\frac{\cos(\frac\pi2 g_0(t))}{\cos(\frac\pi2 t)}\Big|
\right\}.\end{aligned}$$ Hence $d(x)=d(t)$ only depends on $t\in I$.
Now we note that for $t\in I\setminus\{-1,0,1\}$ the quotient $d(t)$ is continuous. Therefore, if we show that $d$ can be continuously extended to $\{-1,0,1\}$, then $d$ is bounded on $I$ and $\lambda d(t)$ can be made arbitrarily small letting the contraction rate $\lambda$ be small enough, which can be done without affecting the rest of the construction. Having this concludes the proof of the smoothness of ${\EuScript{F}}^{s}$.
Finally, we compute, on the one hand $\lim_{t\to\pm1}|{\cos(\frac\pi2
g_0(t))/\cos(\frac\pi2 t)}|
=|g_0^\prime(\pm1)| \neq0
$ which shows that $d$ can be continuously extended to $\pm1$. On the other hand, by the choice of the map $g_0$ in Section \[sec:an-example-nue\], we have that both $m(x)$ and $\|Dg(x)\|$ are of the order of $|t|^{\alpha-1}$ for $t$ near $0$ (ignoring multiplicative constants). Thus $$\begin{aligned}
\label{eq:crucial}
d(x)
=O\left(\frac{(|t|^\alpha)^{1/2}}{(|t|^{\alpha-1})^\gamma
(|t|^{\alpha-1})^\omega}\right)
=O(|t|^{\alpha/2-(\gamma+\omega)(\alpha-1)}).\end{aligned}$$ For $1<\alpha<2$ we have $\alpha/(2(\alpha-1)) > 1$ so that, in this setting, we can take $\gamma>1$ and $\omega>0$ in order that $\alpha/2-(\gamma+\omega)(\alpha-1)\ge0$. Hence $d$ can also be extended continuously to $0$. This concludes the proof that ${\EuScript{F}}^{s}$ is strongly dominated by the action of $DR$ along the directions on the complement $C^u$ of the stable cone field, so that it becomes a $C^\gamma$ foliation.
Now we define the subspace $E$ to be the sum of the tangent space $E^{s}:=T{\EuScript{F}}^{s}$ to ${\EuScript{F}}^{s}$ with the $x_2$ direction $E^{ss}$ on $\Sigma_2$, i.e. $E:= E^{ss}\oplus E^s$. This bundle $E$ is a $DR$-invariant contracting subbundle which is also strongly dominated by the directions on the center-unstable cone $C^u$ (see Section \[sec:attract-set-its\]), since the direction $E^{ss}$ can be made even more strongly contracted than the bundle $E^{s}$, see Remark \[rmk:domination\]. In fact, the same argument as above, especially the relation , is analogous. The foliation ${\EuScript{F}}$ tangent to $E$ is then uniformly contracting, with $3$-dimensional $C^1$-leaves and $C^\gamma$ holonomies, for some $\gamma>1$.
\[rmk:subfoliation\] The foliation ${\EuScript{F}}^{ss}$ is a subfoliation of ${\EuScript{F}}$ in the sense that every leaf $\gamma\in{\EuScript{F}}$ admits a foliation $\gamma\cap{\EuScript{F}}^{ss}$ by leaves of ${\EuScript{F}}^{ss}$ tangent to $E^{ss}$ at every point.
\[rmk:1dfoliation\] *Arguing with the flow of $X$*, there exists a one-dimensional foliation ${\EuScript{F}}^{ss}$ tangent to the one-dimensional field of directions $\cup_{t\ge0}DX^t(E^{ss}_z)$, since this line bundle is uniformly contracted by $X^t$. This one-dimensional bundle is also strongly dominated by the “saturated” bundle $\cup_{t\ge0}DX^t(E^{s}\oplus E^X)$, by the choice of the constants $C$ and $\beta$ in the construction of the vector field, see Section \[sec:unpert-singul-flow\].
We can now complete the construction of the center subbundle $E^c$. The domination just obtained shows that the complementary cone field $C^u$ through the points of $\Lambda_\Sigma$ is strictly invariant by forward iteration under $DR$, so there exists a unique $DR$-invariant subbundle $E^c$ contained in $C^u$ and defined on all points of $\Lambda_\Sigma$. We thus obtain a dominated splitting $E^s\oplus E^c$ of the tangent bundle of $\Sigma_2$ over $\Lambda_\Sigma$.
### Hyperbolicity of the physical measure {#sec:hyperb-physic-measur}
The suspension $\mu$ of the ergodic and physical invariant probability measure $\nu$ for $R$ is also an ergodic and physical measure for $X^t$ on $U$. In addition, denoting $\tau(z)$ the Poincaré return time for $z\in\Sigma_2$ (which a well defined smooth function except on $\{0\}\times
I\times\{1\}\times B^{k+2}$) and $\tau^n(z)=\tau(R^{n-1}(z))+\dots +\tau(z)$ for all $n\in{{\mathbb Z}}^+$ such that $\tau^n(z)<\infty$, we have $DR^n(z)=P_{R^n(z)}\circ DX^{\tau^{n}(z)}\mid T_z\Sigma_2$, where $P_z:T_zM\to T_z\Sigma_2$ is the projection parallel to the direction $X(z)$ of the flow at $z\in\Sigma_2$. Therefore we can write, for $z\in\Sigma_2$ such that $R^n(z)$ is never in the local stable manifold of $s_0$ for $n\in{{\mathbb Z}}^+$ and $v\in E^c(z)\subset T_z\Sigma_2$ $$\begin{aligned}
\label{eq:exponent}
0<\limsup_{n\to+\infty}\frac1n\log\|DR^n(z)v\| \le
\limsup_{n\to+\infty}
\frac{\tau^n(z)}n\cdot\frac1{\tau^n(z)}\log\|DX^{\tau^n(z)}(z)v\|.\end{aligned}$$ But $\tau^n(z)/n\xrightarrow[n\to\infty]{}\nu(\tau)$ is finite for $z$ in the basin of $\nu$. Indeed, by Remark \[rmk:poincare-time\], $\tau(z)$ is essentially the logarithm of the distance to the critical set of $R$, i.e. the intersection of the local stable manifold of $s_0$ with $\Sigma_2$. Indeed, the main contribution to the return time comes from the time it takes $z\in\Sigma_2$ to pass near the singularities, which is given by the logarithm of the distance to $\Sigma_2\cap W^s_{loc}(s_0)$ and this same value controls the time it takes the point to pass near $s_1$ also, except for a multiplication by a positive constant, because of the local expression of the flow near a hyperbolic equilibrium and by the form of the connection between $s_0$ and $s_1$, see Figure \[fig:saddle-connection\] and Remark \[rmk:poincare-time\].
In the quotient dynamics of $R$, i.e. for the map $f$ on $N$, the function $\tau$ is comparable to the logarithm of the distance to the critical set. This function is $\nu$-integrable as a consequence of the non-uniformly expanding properties of the map $f$, as stated in Theorem \[thm:Misiurew\] of Section \[sec:higher-dimens-misiur\].
Hence implies that the Lyapunov exponents along $\mu$-almost every orbit of $X^t$ are positive along the directions of the bundle $E^{cu}(X^t(z)):=DX^t(E^c(z)\oplus {{\mathbb R}}X(z))$ for $z\in\Lambda$ with the exception of the flow direction (along which the Lyapunov exponent is zero).
In this way we show that the attractor for the flow of $X$ is chaotic, in the sense that it admits a physical probability measure with $k+1$ positive Lyapunov exponents.
Unfolding $X$. {#sec:unfold-x}
--------------
The $E^{ss}$ direction on $\Sigma_2$ can be made uniformly contracting with arbitrarily strong contraction rate (see Section \[sec:unpert-singul-flow\] and Remark \[rmk:domination\]). Moreover, the $x_2$ direction is dominated by all the other directions under the flow $X^t$. Thus $E^{ss}$ is a stable direction for the flow over $\Lambda$ which is dominated by any complementary direction.
Hence this uniformly contracting foliation ${\EuScript{F}}^{ss}_X$ admits a continuation ${\EuScript{F}}^{ss}_Y$ for all flows $Y^t$ where $Y$ is close to $X$ in $\fX^2(M)$, that is, in the $C^2$ topology. Since it is a one-dimensional foliation whose contraction rate can be made arbitrarily small, the holonomies along its leaves are of class $C^{\gamma}$ for some $\gamma>1$, see [@HP70 Theorem 6.2].
This ensures that the return map $R_Y$ to the cross-section $I_{\varepsilon}\times I_{\varepsilon}\times\{1\}\times B^{k+2}$, where $I_{\varepsilon}:=(-1-{\varepsilon},1+{\varepsilon})$, admits a one-dimensional invariant foliation such that the quotient map $g_Y$ of $R_Y$ on the leaves of this foliation is a $(k+3)$-dimensional $C^{\gamma}$ map, for *any vector field $Y$ close to $X$ in the $C^2$ topology*. In addition, the leaves of ${\EuScript{F}}^{ss}_Y$ are $C^1$-close to the leaves of the original ${\EuScript{F}}^{ss}_X$ foliation.
Considering the set $I_{\varepsilon}\times\{(0,1)\}\times
B^{k+2}$ diffeomorphic to $I_{\varepsilon}\times B^{k+2}$, we see that ${\EuScript{F}}^{ss}_X$ is transverse to this set and thus the continuation remains transverse. Hence we can see the map $g_Y$ as a map between subsets of $I_{\varepsilon}\times
B^{k+2}$. Let $\pi^{ss}_Y:\Sigma_2\to I_{\varepsilon}\times
B^{k+2}$ be the projection along the leaves of ${\EuScript{F}}^{ss}_Y$ in what follows and let $\ell:=\Sigma_2\cap
W^s_{loc}(s_0(Y))$ be the connected component of the local stable manifold of the continuation of $s_0$ for $Y$ on the cross-section $\Sigma_2$.
We have scarce information about the dynamics of this map: it has a sink $p(Y)$ (the continuation of the sink of $R$) and is $C^\gamma$ close to the quotient of the map $R$ over the foliation ${\EuScript{F}}^{ss}_X$. Thus, for points outside a neighborhood of $\pi_Y^{ss}(\ell)$ and away from the stable manifold of $p(Y)$, we should have “hyperbolicity” for $g_Y$ due to proximity of $R_Y$ to $R$, that is, there is a pair of complementary directions on the tangent space such that one is expanded and the other contracted by the derivative of $g_Y$. The interplay of this hyperbolic-like behavior with the behavior near $\ell$ is unknown to us.
\[conj:physicalViana\] Similarly to the one-dimensional setting, *$g_Y$ admits a physical hyperbolic measure $\mu_Y$ for all those vector fields $Y$ which are $C^2$ close to $X$ and the stable manifold of the sink does not contain the critical region.* Moreover the basin of $\mu_Y$ should be the complement of the stable manifold of the sink.
However, we can be more specific along certain submanifolds of the space of vector fields, as follows.
### Keeping the domination on $\Sigma_2$ under perturbation {#sec:keeping-dominat}
The argument presented in the Section \[sec:dominat-splitt\], proving smoothness of a $3$-dimensional stable foliation ${\EuScript{F}}$ after quotienting by $\pi_1$, strongly depends on the fact that $\pi_1$ identifies every point whose first coordinate is $1$, which is represented by an infinite contraction there. We just have to consider , which holds because the point $0$, corresponding to the intersection of $\Sigma_2$ with the local stable manifold of $s_0$, is sent by $g_0$ to $1$ on each side, that is $g_0(0^\pm)=\lim_{t\to0^\pm}g_0(t)=1$.
In order to keep the strong domination for a perturbation $Y$ of $X$ in the $C^2$ topology, *we restrict the perturbation in such a way that the corresponding points $P_2(Y)$ and $P_3(Y)$ are in the same stable leaf $\xi\in{\EuScript{F}}_Y$*. This is well defined according to Remark \[rmk:1dfoliation\], see Figure \[fig:startflow\] for the positions of $P_2$ and $P_3$. Here $P_2(Y)$ and $P_3(Y)$ are the points of first intersection of each branch $W^u(s_0)\setminus\{s_0\}$ of the one-dimensional unstable manifold of the equilibrium $s_0$. We note that we can write each branch as an orbit of the flow of $Y$ and so the notion of first intersection with $\Sigma_2$ is well defined. This restriction on the vector field corresponds to restricting to a $(k+4)$-codimension submanifold ${\EuScript{P}}$ of the space of vector fields $\fX^2(M)$.
In this way, on the one hand, the same arguments of Section \[sec:dominat-splitt\] can be carried through and the strong domination persists for vector fields $Y\in{\EuScript{P}}$. On the other hand, this implies that there exists a $R_Y$-invariant $3$-dimensional contracting $C^\gamma$ foliation ${\EuScript{F}}_Y$ of $\Sigma_2$, for some $\gamma>1$, with $C^1$ leaves, for all vector fields $Y$ close enough to $X$ within ${\EuScript{P}}$.
We can then quotient $R_Y$ over the leaves of ${\EuScript{F}}_Y$ to obtain a $(k+1)$-dimensional map $g_Y$. We note that, defining the cylinder ${\EuScript{C}}:=I_{\varepsilon}\times\{0\}\times\{1\}\times
e({{\mathbb T}}^k\times0)$ (diffeomorphic to $I_{\varepsilon}\times{{\mathbb T}}^k$) inside $I_{\varepsilon}\times I\times\{1\}\times B^{k+2}$ (recall that $I_{\varepsilon}=(-1-{\varepsilon},1+{\varepsilon})$) we have that the initial foliation ${\EuScript{F}}$ is everywhere transverse to ${\EuScript{C}}$. Therefore, since ${\EuScript{C}}$ is a proper submanifold, the continuation ${\EuScript{F}}_Y$ is still transverse to ${\EuScript{C}}$ for $Y\in{\EuScript{N}}\cap{\EuScript{P}}$. Hence we can define a corresponding quotient map $g_Y:I_{\varepsilon}\times{{\mathbb T}}^k\circlearrowleft$ which will *not be*, in general, either a direct or skew-product along the $I_{\varepsilon}$ and ${{\mathbb T}}^k$ directions.
We observe that $g_Y$ is close to $f_0\times f_1$ on $I\times{{\mathbb T}}^k$, recall Remark \[rmk:cylinder\] during the construction of the original flow. Hence for pieces of orbits which remain away from a neighborhood of the critical set and away from the basin of the sink, we have uniform expansion in all directions (akin to condition C on the statement of Theorem \[thm:Misiurew\]).
### Keeping the saddle-connection {#sec:nearby-vector-fields}
In addition to keeping the foliation ${\EuScript{F}}$, we may impose the restriction already mentioned in Remark \[rmk:keepsaddleconnection\] to *keep also the connection between $s_0$ and $s_1$*: the component of the unstable manifold of $s_0$ through $\Sigma^+$ (i.e. the orbit of $P_2(Y)$) is contained in the stable manifold of $s_1$, a $(k+1)$-codimension condition on the family of all vector fields $Y$ $C^2$ close to $X$. Let ${\EuScript{N}}$ be the submanifold of such vector fields in a neighborhood of $X$.
Therefore we can ensure that there exists a stable foliation ${\EuScript{F}}_Y$ nearby ${\EuScript{F}}_X$ for every vector field $Y\in{\EuScript{N}}\cap{\EuScript{P}}$, invariant under the corresponding return map $R_Y$. We can again quotient $R_Y$ over the leaves of ${\EuScript{F}}_Y$ to obtain a $(k+1)$-dimensional map $g_Y$. We observe that ${\EuScript{N}}\cap{\EuScript{P}}$ will have codimension $2k+5$ since the conditions defining ${\EuScript{N}}$ and ${\EuScript{P}}$ are independent.
### Keeping the one-dimensional quotient map {#sec:symmetr-unfold-x}
We can also perturb the vector field $X$ within the manifold ${\EuScript{N}}\cap{\EuScript{P}}$ keeping the saddle-connection in such a way that we obtain a one-dimensional $C^{1+}$ quotient map. In this setting we can apply Benedicks-Carleson exclusion of parameters techniques along these families of flows, exactly in the same way Rovella proved his main theorem in [@Ro93]. *So we have an analogous result to Rovella’s if we perturb the flow keeping the symmetry which allows us to project to a one-dimensional map, that is, if $g_Y$ is a skew-product over $I_{\varepsilon}$.*
![\[fig:symperturb\]A symmetrical unfolding. We can still project the dynamics into a one-dimensional map.](tubularskewperturb1.eps)
Since we have a well defined strong stable foliation for the Poincaré map, we can quotient out along such stable foliation obtaining a map in the torusphere. By [*keeping the symmetry*]{} we mean that we unfold our Misiurewicz type flow preserving the invariance of each parallel torus in the torusphere. We can consider families which unfold the criticality introduced by the singularity $s_1$ (as in the right hand side of Figure \[fig:contractingLorenz\]) and/or unfold the intersection of the unstable manifold of $s_1$ with ${{\mathbb T}}_1^k$, see Figure \[fig:symperturb\]. Once more, this permits us to reduce the study of the attractor to a one-dimensional problem.
In the very small manifold within ${\EuScript{P}}\cap{\EuScript{N}}$ where the one-dimensional quotient map is kept, we can argue just like Rovella in [@Ro93] obtaining the same results.
### Loosing the one-dimensional quotient map {#sec:non-symmetr-unfold}
We note also that, even if we keep the saddle-connection, it is very easy to *perturb this flow* to another arbitrarily close flow such that *the quotient to a one-dimensional map is not defined*, as the left hand side of Figure \[fig:skewperturb\] suggests.
![\[fig:skewperturb\] On the left: a non-symmetrical perturbation. We cannot reduce the analysis to a one-dimensional map. On the right: An open region nearby the singularity is sent into the basin of a sink. ](tubularskewperturb2.eps "fig:") ![\[fig:skewperturb\] On the left: a non-symmetrical perturbation. We cannot reduce the analysis to a one-dimensional map. On the right: An open region nearby the singularity is sent into the basin of a sink. ](tubularskewperturb3.eps "fig:")
### Breaking the saddle-connection {#sec:breaking-saddle-conn}
Another way to break the one-dimensional quotient is to break the saddle-connection. Then the orbit of $P_2(Y)$ is no longer contained in the stable manifold of $s_1$. So the positive orbit of $P_2(Y)$ under $Y^t$ follows the unstable manifold of $s_1$ and crosses $\Sigma_2$ at some point, but points in the image $T^+_Y(\Sigma^+)$ near $P_2(Y)$ may no longer be mapped preserving the stable foliation of the solenoids (see Figure \[fig:startflow\]). In this case we still have a quotient map $g_Y:I_{\varepsilon}\times{{\mathbb T}}^k\circlearrowleft$ but it is no longer a skew-product over $I_{\varepsilon}$.
### Returning away from the sink {#sec:return-away-from}
We conjecture that, in the situation depicted in the left side of Figure \[fig:skewperturb\], that is, we have a non-skew-product map $g_Y$ but the image of $P_2(Y)$ under $R_Y$ does not intersect the basin of the sink $P_0$, *either with or without a saddle-connection* between $s_0$ and $s_1$, then it is *always true that there exists an expanding (all Lyapunov exponents are positive) physical measure with full basin outside the basin of the sink*.
The motivation behind this conjecture is that $g_Y$ is close to $g_0$ and away from the singular set and away from the basin of the sink we have uniform expansion, since this behavior was present in the original map $g_0$ and is persistent. Therefore, we have the interplay between expansion and a critical region which will be approximately a circle on the cylinder $I_{\varepsilon}\times {{\mathbb T}}^k$. This setting is similar to the one introduced by Viana in [@Vi97 Theorem A] and carefully studied in e.g. [@alves-viana2002; @AA03; @alves-luzzatto-pinheiro2005].
### Returning inside the basin of the sink {#sec:return-inside-basin}
On the one hand, the perturbation can send points nearby $P_2$ into the basin of the sink $P_0$, *either with or without a saddle-connection* between $s_0$ and $s_1$, as shown in the right hand side of Figure \[fig:skewperturb\]. We prove, in Section \[sec:full-basin-attract\], that this implies that the *basin of the sink grows to fill the whole manifold* Lebesgue modulo zero, for such vector fields in the submanifold ${\EuScript{P}}\cap{\EuScript{N}}$.
Higher dimensional Benedicks-Carleson conditions {#sec:higher-dimens-misiur}
================================================
To present the statement and proof of the existence of an absolutely continuous invariant probability measure on the setting of higher dimensional maps, we need to recall some notions from non-uniformly expanding dynamics.
Non-flat critical or singular sets {#sec:non-flat-critic}
----------------------------------
Let $f$ be a $C^{1+}$ local diffeomorphism outside a compact proper submanifold ${\EuScript{S}}$ of $M$ with positive codimension. The set ${\EuScript{S}}$ may be taken as some set of critical points of $f$ or a set where $f$ fails to be differentiable. The submanifold ${\EuScript{S}}$ has an at most countable collection of connected components $\{{\EuScript{S}}_i\}_{i\in{{\mathbb N}}}$, which may have different codimensions. This is enough to ensure that the volume or Lebesgue measure of ${\EuScript{S}}$ is zero and, in particular, that $f$ is a *regular map*, that is, if $Z\subset M$ has zero volume, then $f^{-1}(Z)$ has zero volume also.
*In what follows we assume that the number of connected components of ${\EuScript{S}}$ is finite.* It should be possible to drop this condition if we impose some global restrictions on the behavior of the map $f$, see [@PRV98] for examples with one-dimensional ambient manifold $M$ of non-uniformly expanding maps with infinitely many critical points. We do not pursue this issue in this work.
We say that $ {\EuScript{S}}\subset M$ is a [*non-flat critical or singular set*]{} for $f$ if the following conditions hold for each connected component ${\EuScript{S}}_i$. The first one essentially says that $f$ [*behaves like a power of the distance*]{} to $ {\EuScript{S}}_i $: there are constants $B>1$ and real numbers $\alpha_i>\beta_i$ such that $\alpha_i-\beta_i<1, 1+\beta_i>0$ and *on a neighborhood $U_i$ of ${\EuScript{S}}_i$* (where $U_i\cap{\EuScript{S}}_j=\emptyset$ if $j\neq i$ and we also take $U_i\subset B({\EuScript{S}}_i,1/2)$) for every $x\in U_i$
1. $\displaystyle{\frac{1}{B}{\operatorname{dist}}(x,{\EuScript{S}})^{\alpha_i}\leq
\frac {\|Df(x)v\|}{\|v\|}\leq B{\operatorname{dist}}(x,{\EuScript{S}})^{\beta_i}}$ for all $v\in T_x M$.
Moreover, we assume that the functions $ \log|\det Df(x)| $ and $ \log \|Df(x)^{-1}\| $ are *locally Lipschitz* at points $ x\in U_i $ with Lipschitz constant depending on ${\operatorname{dist}}(x, {\EuScript{S}})$: for every $x,y\in
U_i$ with ${\operatorname{dist}}(x,y)<{\operatorname{dist}}(x,{\EuScript{S}}_i)/2$ we have
1. $\displaystyle{\left|\log\|Df(x)^{-1}\|-
\log\|Df(y)^{-1}\|\:\right|\leq
\frac{B}{{\operatorname{dist}}(x,{\EuScript{S}}_i)^{\alpha_i}}{\operatorname{dist}}(x,y)}$;
2. $\displaystyle{\left|\log|\det Df(x)|- \log|\det
Df(y)|\:\right|\leq
\frac{B}{{\operatorname{dist}}(x,{\EuScript{S}}_i)^{\alpha_i}}{\operatorname{dist}}(x,y)}$.
The assumption that the number of connected components is finite implies that there exists $\beta>0$ such that $\max_i\{|\alpha_i|,|\beta_i|\}\le\beta$ and we also assume that for all $x\in M\setminus{\EuScript{S}}$
- $\displaystyle{\frac{1}{B}{\operatorname{dist}}(x,{\EuScript{S}})^{\beta}\leq
\frac {\|Df(x)v\|}{\|v\|}\leq B{\operatorname{dist}}(x,{\EuScript{S}})^{-\beta}}$ for all $v\in T_x M$.
The case where ${\EuScript{S}}$ is equal to the empty set may also be considered. The assumption $1+\beta_i>0$ prevents that the image of arbitrary small neighborhoods around the singular set accumulates every point of $M$ (consider e.g. the Gauss map $[0,1]\ni x\mapsto x^{-1}\bmod1$) which would prevent a meaningful definition of “singular value”, see the statement of Theorem \[thm:Misiurew\] in what follows.
Hyperbolic times {#sec:hyperb-times}
----------------
We write $x_i=f^i(x)$ for $x\in M$, $i\ge0$, and also $S_n^f\varphi(x)=S_n\varphi(x)=\sum_{i=0}^{n-1}
\varphi(x_i), n\ge1$ for the ergodic sums of a function $\varphi:M\to{{\mathbb R}}$ with respect to the action of $f$, in what follows. For the next definition it will be useful to introduce $ {\operatorname{dist}}_{\delta}(x,{\EuScript{S}}) $, the $ \delta
$-*truncated* distance from $ x $ to ${\EuScript{S}}$, defined as $ {\operatorname{dist}}_{\delta}(x,{\EuScript{S}}) = {\operatorname{dist}}(x,{\EuScript{S}}) $ if $
{\operatorname{dist}}(x,{\EuScript{S}}) \leq \delta$, and $ {\operatorname{dist}}_{\delta}(x,{\EuScript{S}}) =1
$ otherwise. From now on we write $\psi(x)=\log\|Df(x)^{-1}\|$ and $\fD_r(x)=-\log
{\operatorname{dist}}_r(x,{\EuScript{S}})$ for $x\in M\setminus{\EuScript{S}}$ and $r>0$.
Let $\beta>0$ be given by the non-flat conditions on ${\EuScript{S}}$, and fix $b>0$ such that $b < \min\{1/2,1/(4\beta)\}$. Given $c>0$ and $\delta>0$, we say that $h$ is a [ *$(e^c,\delta)$-hyperbolic time*]{} for a point $x\in M$ if, for all $0\le k \le h$, $$\label{d.ht}
S_k\psi(x_{n-k})\le -ck
\quad\text{and}\quad
S_k\fD_\delta(x_{n-k})\le b c k
$$ We convention that an empty sum evaluates to $0$ so that the above inequalities make sense for all indexes in the given range.
We say that the [*frequency of $(e^c,\delta)$-hyperbolic times*]{} for $x\in M$ is greater than $\theta>0$ if, for infinitely many times $n$, there are $h_1<h_2\dots <h_\ell\le n$ which are $(e^c,\delta)$–hyperbolic times for $x$ and $\ell
\ge\theta n$.
The following statement summarizes the main properties of hyperbolic times. For a proof the reader can consult [@ABV00 Lemma 5.2] and [@ABV00 Corollary 5.3].
\[p.contr\] There are $0<\delta_1<\delta/4$ and $C_1>0$ (depending only on $\delta$ and $\sigma$) such that if $h$ is a $(e^c,\delta)$-hyperbolic time for $x$, then there is a *hyperbolic neighborhood* $V_x$ of $x=x_0$ in $M$ for which
1. $f^h$ maps $V_x$ diffeomorphically onto the ball of radius $\delta_1$ around $x_n$;
2. for $1\le k \le h$ and $y, z\in V_x$, $
{\operatorname{dist}}(y_{n-k},z_{n-k}) \le
C_1 e^{ck/2}{\operatorname{dist}}(y_h,z_h)$;
3. for all $y\in V_x$, $S_h\fD_\delta(y)\le
S_h\fD_\delta(x) + o(\delta)$ where $o(\delta)/\delta\to0$ when $\delta\to0$;
4. $f^h\vert V_x$ has distortion bounded by $C_1$: if $y, z\in V_x$, then $ \frac{|\det Df^h (y)|}{|\det Df^h (z)|}\le C_1. $
\[rmk:preball-away\] The image of $V_x$ by $f^h$ is away from $B({\EuScript{S}},3\delta/4)$: for $y\in V_x$ $$\begin{aligned}
{\operatorname{dist}}(y_h,{\EuScript{S}})
\ge
{\operatorname{dist}}(x_h,{\EuScript{S}}) - {\operatorname{dist}}(y_h,x_h)
\ge \delta-\delta_1>\frac34 \delta
\end{aligned}$$ since ${\operatorname{dist}}_\delta(x_h,{\EuScript{S}})\ge1$.
Item (3) above, which is not found in [@ABV00], is an easy consequence of item (2): for every $1\le k\le h$ and $y\in V_x$ $$\begin{aligned}
-\log\frac{{\operatorname{dist}}_\delta(y_{h-k},{\EuScript{S}})}
{{\operatorname{dist}}_\delta(x_{h-k},{\EuScript{S}})} &\le
-\log\frac{{\operatorname{dist}}_\delta(x_{h-k},{\EuScript{S}})-
{\operatorname{dist}}(y_{h-k},x_{h-k})}
{{\operatorname{dist}}_\delta(x_{h-k},{\EuScript{S}})}
\\
&= -\log\left( 1-\frac{{\operatorname{dist}}(y_{h-k},x_{h-k})}
{{\operatorname{dist}}_\delta(x_{h-k},{\EuScript{S}})} \right)
\\
&\le -\log\left(1-\frac{\delta_1\sigma^{k/2}}
{e^{bck}} \right) \le
-\log\left(1-\frac34\delta e^{(1/2-b)ck}\right)
\\
&\le \frac{(3/4)\delta}{1-(3/4)\delta} \cdot \frac34\delta
e^{(1/2-b)ck}.
\end{aligned}$$ So each point $y$ in a hyperbolic neighborhood as above satisfies $$\begin{aligned}
S_h\fD_\delta(y)
&\le
S_h\fD_\delta(x)
+
\frac{(3/4)^2\delta^2}{1-(3/4)\delta}
\sum_{j=0}^h e^{(1/2-b)cj}
\le
S_h\fD_\delta(x)
+
\frac{(3/4)^2\delta^2}{1-(3/4)\delta}\cdot
\frac1{1-e^{bc}},
\end{aligned}$$ and the last term is $o(\delta)$.
In what follows we say that an open set $V$ is a *hyperbolic neighborhood* of any one of its points $x\in V$, *with $(e^{c/2},\delta)$-distortion time $h$* (or, for short, a $(e^{c/2},\delta)$-*hyperbolic neighborhood*), if the properties stated in items (1) through (4) of Proposition \[p.contr\] are true for $x$ and $k=h$.
We also say that a given point $x$ has *positive frequency of hyperbolic neighborhoods* bounded by $\theta>0$ if there exist $c,\delta>0$ and neighborhoods $V_{h_k}$ of $x$ with $(e^{c},\delta)$-distortion time $h_k$, for all $k\ge1$, such that for every big enough $k$ we have $k\ge\theta h_k$.
Existence of many hyperbolic neighborhoods versus absolutely continuous invariant probability measures {#sec:existence-hyperb-tim}
------------------------------------------------------------------------------------------------------
Hyperbolic times appear naturally when $f$ is assumed to be [*non-uniformly expanding*]{} in some set $H\subset M$: there is some $c>0$ such that for every $x\in H$ one has $$\label{liminf1}
\liminf_{n\to +\infty}\frac{1}{n}
S_n\psi(x)<-c,$$ and points in $H$ satisfy some [*slow recurrence to the critical or singular set*]{}: given any ${\varepsilon}>0$ there exists $\delta>0$ such that for every $x\in H$ $$\label{eq:slow-recurrence}
\limsup_{n\to+\infty} \frac1{n} S_n\fD_{\delta}(x)
\le{\varepsilon}.$$
The next result has been proved in [@ABV00 Theorem C & Lemma 5.4]. It provides sufficient conditions for the existence of many hyperbolic times along the orbit of points satisfying the non-uniformly expanding and slow recurrence conditions.
\[thm:abv0\] Let $f: M\to M$ be a $C^{1+}$ local diffeomorphism outside a non-flat critical or singular set ${\EuScript{S}}\subset M$. If there is some set $H\subset M\setminus{\EuScript{S}}$ such that and hold for all $x\in H$, then for any given $0<\xi<1$ and $0<\zeta<bc$ there exist $\delta>0$ and $\theta>0$ such that the frequency of $(e^{-c\xi},\delta)$-hyperbolic times for each point $x\in H$ is bigger than $\theta$. Moreover for such hyperbolic times $h$ we have $S_h\fD_\delta(x) \le \zeta h.$
Together with Proposition \[p.contr\] the results from Theorem \[thm:abv0\] ensure the existence of positive frequency of hyperbolic neighborhoods around Lebesgue almost every point.
This will imply the existence of absolutely continuous invariant probability measures for the map $f$ through the following result from [@Pinheiro05].
\[thm:abv1\] Let $f: M\to M$ be a $C^{1+}$ local diffeomorphism outside a non-degenerate exceptional set ${\EuScript{S}}\subset M$. If there are $c,\delta>0$ such that the frequency of $(e^c,\delta)$-hyperbolic neighborhoods is bigger than $\theta>0$ for Lebesgue almost every $x\in M$, then $f$ has some absolutely continuous invariant probability measure.
Since we are using a modified definition of hyperbolic time, we present a proof of Theorems \[thm:abv0\] and \[thm:abv1\] in Section \[sec:hyperb-neighb-constr\] for completeness.
Existence of absolutely continuous probability measures {#sec:existence-absolut-co}
-------------------------------------------------------
The following theorem provides higher dimensional existence result for physical measures which applies to our setting.
\[thm:Misiurew\] Let $f:M\setminus{\EuScript{S}}\to M$ be a $C^{1+}$ local diffeomorphism, where ${\EuScript{S}}$ is a compact sub-manifold of $M$ which is a non-flat critical/singular set for $f$. We define $$f({\EuScript{S}}):=\cap_{n\ge1}\overline{f\big(B_{1/n}({\EuScript{S}})\big)},$$ the set of all accumulation points of sequences $f(x_n)$ for $x_n$ converging to ${\EuScript{S}}$ as $n\to+\infty$, and assume that $f$ also satisfies:
- *$f$ is non-uniformly expanding along the orbits of critical values*: there exist $c_0>0$ and $N\ge1$ such that for all $x\in f({\EuScript{S}})$ and $n\ge N$ we have $S_n\psi(x) \le -c_0 n$;
- *the critical set has slow recurrence to itself*: given ${\varepsilon}>0$ we can find $\delta>0$ such that for all $x\in f({\EuScript{S}})$ there exists $N=N(x)$ satisfying $S_n\fD_\delta(x)\le{\varepsilon}n$ for every $n\ge N$;
- *$f$ is uniformly expanding away from the critical/singular set*: for every neighborhood $U$ of ${\EuScript{S}}$ there exist $c=c(U)>0$ and $K=K(U)>0$ such that for any $x\in M$ and $n\ge1$ satisfying $x=x_0, x_1,\dots, x_{n-1}\in M\setminus U$, then $S_n \psi(x) \le K- c n$.
- *$f$ does not contract too much when returning near the critical/singular set*: that is, there exists $\kappa>0$ and a neighborhood $\hat U$ of ${\EuScript{S}}$ such that for every open neighborhood $U\subset
\hat U$ of ${\EuScript{S}}$ and for $x=x_0$ satisfying $x_1,
\dots, x_{n-1}\in M\setminus U$ and either $x_0\in U$ or $x_n \in U$, then $S_n\psi(x_0)\le\kappa$.
Then $f$ has an absolutely continuous invariant probability measure $\nu$ such that $\fD_d$ is $\nu$-integrable for some (and thus all) $d>0$.
We observe that condition B above ensures, in particular, that $f({\EuScript{S}})\cap{\EuScript{S}}=\emptyset$, for otherwise $\fD_\delta(x)$ is not defined for $x\in{\EuScript{S}}\cap
f({\EuScript{S}})$. Moreover, condition C above is just a convenient translation to this higher dimensional setting of the conclusion of the one-dimensional theorem of Mañé [@Man85], ensuring uniform expansion away from the critical set and basins of periodic attractors. It can be read alternatively as: given $\delta>0$ there are $C,\lambda>0$ such that if $x_i\in M\setminus B({\EuScript{S}},\delta)$ for $i=0,\dots,n-1$, then $\|Df^n(x)^{-1}\|\le C e^{-\lambda
n}$. In addition, condition D is a translation to our setting of a similar property that holds for unidimensional multimodal “Misiurewicz maps”, that is, for maps whose critical orbits are non-recurrent, ensuring a minimal lower bound for the derivative of the map along orbits which return near ${\EuScript{S}}$.
Now we show that under the conditions in the statement of Theorem \[thm:Misiurew\], we can find a full measure subset of points of $M$ having positive density of hyperbolic times.
\[thm:dense-hyp-times\] Let $f:M\setminus{\EuScript{S}}\to M$ be a $C^{1+}$ local diffeomorphism away from a non-flat critical/singular set ${\EuScript{S}}$, satisfying all conditions in the statement of Theorem \[thm:Misiurew\]. Then for every small enough $0<\xi<1$ there exists $\delta=\delta(\xi)>0$ and $\theta=\theta(\xi,\delta)>0$ such that Lebesgue almost every $x\in M$ admits positive frequency bounded by $\theta$ of $(e^{-\xi c_0},\delta)$-hyperbolic neighborhoods.
From this result we deduce Theorem \[thm:Misiurew\] appling Theorem \[thm:abv1\]. So all we need to do is prove Theorem \[thm:dense-hyp-times\]. For the integrability of $\fD$ see Remark \[rmk:log-dist-integrable\] in what follows.
### Existence of hyperbolic neighborhoods {#sec:existence-hyperb-nei}
Fix $\xi_0,{\varepsilon},\tilde\delta>0$ and small enough so that condition B is satisfied in what follows and $\xi_0c_0<b$. Let $\zeta>0$ be small enough in order that $$\begin{aligned}
\label{eq:zeta}
\frac{\xi_0 c_0}{1+\alpha_i} + 2\zeta
< \frac{\xi_0 c_0}{1+\zeta} \quad\text{for all } i.
\end{aligned}$$ Depending on $f$ and $\xi_0$, we can choose the pair $({\varepsilon},\tilde\delta)$ so that, from conditions A and B above together with Theorem \[thm:abv0\], every point $z\in f({\EuScript{S}})$ has infinitely many $(e^{-2\xi_0
c_0},\tilde\delta)$-hyperbolic times $h_1<h_2<h_3<\dots$ satisfying $$\begin{aligned}
\label{eq:Daverage}
S_{h_i}\fD_{\tilde\delta}(z)\le\zeta h_i
\quad\text{for}\quad i\ge1.
\end{aligned}$$ Then there are corresponding hyperbolic neighborhoods $V_i$ of $z$ satisfying the conclusions of Proposition \[p.contr\] for each hyperbolic time $h_i$ of $z$, where $\tilde\delta>0$ does not depend on $z\in{\EuScript{S}}$, that is $f^{h_i}\mid V_i: V_i\to
B(z_{h_i},\delta_1)$ is a diffeomorphism with a ball of radius $\delta_1 \in (0,\tilde\delta/4)$ whose inverse is a contraction with rate bounded by $e^{-\xi_0 c_0 h_i}$.
We will consider, instead of $V_i$, the subset $B_i\subset
V_i$ given by $$\begin{aligned}
\label{eq:halfball}
B_i = \big(f^{n_i}\mid
V_i\big)^{-1}(B(z_n,\delta_2))
\end{aligned}$$ where we set $2\delta_2=\delta_1$. Since $f({\EuScript{S}})$ is compact, we can cover this set by finitely many hyperbolic neighborhoods of the type $B_i$.
Now we fix a connected component ${\EuScript{S}}_i$ of ${\EuScript{S}}$.
### Hyperbolic neighborhoods near the critical/singular set {#sec:hyperb-neighb-near}
Let $T_i$ be the smallest distortion (or hyperbolic) time associated to the balls covering $f({\EuScript{S}}_i)$. We remark that $T_i$ can be taken arbitrarily big, independently of $\xi_0,\zeta,\tilde\delta$, because every $z\in f({\EuScript{S}}_i)$ has positive frequency of hyperbolic neighborhoods and, consequently, the open neighborhoods in the above covering can be made arbitrarily small.
We observe that, by definition of hyperbolic neighborhoods, if ${\operatorname{dist}}(z,f({\EuScript{S}}_i))<e^{-b j \xi_0 c_0}$ for some $j\ge1$ and there exists $x\in f({\EuScript{S}}_i)$ such that $z\in V_i(x)$, then the corresponding distortion time $h_i$ satisfies $h_i\ge j$.
We note also that, by the non-flat condition (S1) on $f$ near ${\EuScript{S}}_i$, a ${\varrho}$-neighborhood of ${\EuScript{S}}$ is sent into a ${\varrho}^{1+\beta_i}$-neighborhood of $f({\EuScript{S}}_i)$, for each ${\varrho}>0$. Indeed, since ${\EuScript{S}}_i$ is assumed to be a submanifold, we can find for $x$, on a tubular neighborhood of ${\EuScript{S}}_i$ with radius ${\varrho}$, a curve $\gamma:[0,{\varrho}]\to
M$ from $\gamma(0)\in{\EuScript{S}}_i$ to $\gamma(1)=x$ such that ${\operatorname{dist}}(\gamma(t),{\EuScript{S}}_i)=t$ and $\|\dot\gamma(t)\|=1$ for $t\in[0,{\varrho}]$. Hence $$\begin{aligned}
{\operatorname{dist}}(f(x),f({\EuScript{S}}_i))
&\le\nonumber
\int_0^{\varrho}\|Df(\gamma(t))\dot\gamma(t)\|\,dt
\le
\int_0^{\varrho}B {\operatorname{dist}}(\gamma(t),{\EuScript{S}})^{\beta_i}\,dt
\\
&=
\frac{B{\varrho}^{1+\beta_i}}{1+\beta_i}
=\frac{B}{1+\beta_i}{\operatorname{dist}}(x,{\EuScript{S}})^{1+\beta_i}.
\label{eq:distS1}
\end{aligned}$$ Therefore we can find $C_2=C_2(\delta_2)>0$ such that $C_2^{1+\beta_i} B/(1+\beta_i)=\delta_2$ and if, for some $j\ge0$ $$\begin{aligned}
\label{eq:dist-time}
{\operatorname{dist}}(x,{\EuScript{S}}_i)\le C_2 e^{-\xi_0 c_0
(T_i+j)/(1+\alpha_i)}:=d_j,
\end{aligned}$$ then the smallest possible distortion time $h$ of $f(x)$ is at least $\frac{1+\beta_i}{1+\alpha_i}(T_i+j)$, since $$\begin{aligned}
\delta_2 e^{-\xi_0 c_0 h}
\le
{\operatorname{dist}}(f(x),f({\EuScript{S}}_i))
\le
\delta_2 e^{- \xi_0 c_0
\frac{1+\beta_i}{1+\alpha_i}(T_i+j)}
\le
\delta_2 e^{-b \xi_0 c_0
\frac{1+\beta_i}{1+\alpha_i}(T_i+j)}.
\end{aligned}$$ Note that by the previous observations and Remark \[rmk:preball-away\] we have $$\begin{aligned}
f(B({\EuScript{S}}_i,\tilde\delta)\subset
B\left(f({\EuScript{S}}_i),\frac{B}{1+\beta_i}
\tilde\delta^{1+\beta_i}
\right)
\end{aligned}$$ and we can assume without loss of generality that $$\begin{aligned}
\label{eq:d_0}
d_0<\tilde\delta-\delta_1,
\end{aligned}$$ letting $T_i$ grow if necessary.
In the opposite direction, we can find an upper bound for the distortion time associated with a given distance to ${\EuScript{S}}_i$ reversing the inequality in as follows. Letting $\gamma$ denote a smooth curve $\gamma:[0,1]\to M$ such that $\gamma(0)\in{\EuScript{S}}_i$ and $\gamma(1)=x$ for any given fixed $x$ near ${\EuScript{S}}_i$, we get $$\begin{aligned}
{\operatorname{dist}}(f(x),f({\EuScript{S}}_i))
&=
\inf_\gamma \int_0^1
\|Df(\gamma(t))\dot\gamma(t)\|\,dt
\ge
\frac1B\inf_\gamma
\int_0^1 {\operatorname{dist}}(\gamma(t),{\EuScript{S}}_i)^{\alpha_i}
\|\dot\gamma(t)\|\,dt \nonumber
\\
&\ge
\frac1B \inf_\gamma
\int_0^1 {\operatorname{dist}}(\gamma(t),{\EuScript{S}}_i)^{\alpha_i}
\left|\frac{d}{dt}{\operatorname{dist}}(\gamma(t),{\EuScript{S}})\right| \, dt
\nonumber
\\
&=
\frac1B \inf_\gamma
\int_0^1\left|
\frac{d}{dt}
\frac{{\operatorname{dist}}(\gamma(t),{\EuScript{S}}_i)^{1+\alpha_i}}{1+\alpha_i}
\right| \, dt
\nonumber
\\
&=
\frac1{B(1+\alpha_i)} \inf_\gamma
{\operatorname{var}}_{t\in[0,1]}{\operatorname{dist}}(\gamma(t),{\EuScript{S}})^{1+\alpha_i}
\ge\label{eq:distS2}
\frac{{\operatorname{dist}}(x,{\EuScript{S}})^{1+\alpha_i}}{B(1+\alpha_i)},
\end{aligned}$$ where we use, beside the non-flat condition (S1), the relation $$\begin{aligned}
\left|\frac{d}{dt}{\operatorname{dist}}(\gamma(t),{\EuScript{S}}_i)\right|=
\|\pi_t(\dot\gamma(t))\|=\|\dot\gamma(t)\|\cdot |\cos
\measuredangle(\dot\gamma(t),N_t)|\le \|\dot\gamma(t)\|
\end{aligned}$$ and write $N_t$ for the normal direction to the level submanifold $$\begin{aligned}
S_t=\{z\in M:{\operatorname{dist}}(z,{\EuScript{S}}_i)={\operatorname{dist}}(\gamma(t),{\EuScript{S}}_i)\}
\quad\text{at}\quad \gamma(t);
\end{aligned}$$ and $\pi_t$ for the orthogonal projection from $T_{\gamma(t)}M$ to $N_t$. We also use the well known relation $ {\operatorname{var}}_{[0,1]} {\varphi}= \int_0^1 |D{\varphi}(t)|
\,dt$ for the total variation of a differentiable function ${\varphi}:[0,1]\to{{\mathbb R}}$.
From the inequality we see that if $d_j\ge{\operatorname{dist}}(x,{\EuScript{S}})\ge d_{j+1}$ and $f(x)\in B_i$ for some hyperbolic neighborhood with distortion time $h$, then $$\begin{aligned}
\delta_2 e^{-\xi_0 c_0 h}
&\ge
{\operatorname{dist}}(f(x),f({\EuScript{S}}_i))
\ge
\frac{C_2^{1+\alpha_i}}{B(1+\alpha_i)}
e^{ - \xi_0 c_0 (T_i+j+1)}
=
\frac{C_2^{1+\beta_i+\alpha_i-\beta_i}}{B(1+\alpha_i)}
e^{ - \xi_0 c_0 (T_i+j+1)}
\\
&=
\delta_2^{\alpha_i-\beta_i}
\frac{(1+\beta_i)^{\alpha_i-\beta_i}}
{B^{1+\alpha_i-\beta_i}(1+\alpha_i)}
e^{ - \xi_0 c_0 (T_i+j+1)}
\ge
C_3 \delta_2^{\alpha_i-\beta_i} e^{- \xi_0 c_0 (T_i+j+1)}
\quad \text{or}
\\
h
&\le
(T_i+j+1)
- \frac{\log C_3}{\xi_0 c_0}
+\frac{1-(\alpha_i-\beta_i)}{\xi_0 c_0}\log\delta_2
\le
(1+\frac\zeta2)(T_i+j+1)
\end{aligned}$$ as long as $T_i$ is big enough, depending on $f$ and $\xi_0,c_0,\delta_2$. We remark that we have used the condition $0<\alpha_i-\beta_i<1$ in the inequalities above. For future reference we write this inequalities (for a big enough $T_i$) in the convenient format $$\begin{aligned}
\label{eq:distS3}
d_j\ge{\operatorname{dist}}(x,{\EuScript{S}}_i)\ge d_{j+1} , j\ge0
\implies
\frac{1+\beta_i}{\xi_0 c_0}\fD_{d_0}(x)\le h\le
\frac{(1+\zeta/2)(1+\alpha_i)}{\xi_0 c_0}\fD_{d_0}(x).
\end{aligned}$$ We are now ready for the main arguments.
\[claim:a\] There are $(e^{\xi c_0},\delta)$-hyperbolic neighborhoods for each point in $B({\EuScript{S}}_i,\delta)\setminus{\EuScript{S}}_i$, for suitable constants $\xi,\delta\in(0,1)$.
Indeed, for each $y_0\in B({\EuScript{S}}_i,d_0)\setminus{\EuScript{S}}_i$ there exists a unique integer $k$ such that $y_0\in
B({\EuScript{S}}_i,d_k)\setminus B({\EuScript{S}}_i,d_{k+1})$. By the choice of $k$ we know that $y_1=f(y_0)$ has some distortion time $\frac{1+\beta_i}{1+\alpha_i}(T_i+k) \le h \le
(1+\zeta/2)(T_i+k+1)$. Using the non-flat conditions on ${\EuScript{S}}_i$ we can estimate the norm of the derivative and the volume distortion in a suitable neighborhood of $y_0$, and show that $y_0$ has a hyperbolic neighborhood with $h+1$ as distortion time, with slightly weaker constants of expansion and distortion, as follows.
Let $B_i$ be the hyperbolic neighborhood containing $y_1$ with distortion time $h$ (as defined in ). The image of $B_i$ under $f^h$ is the $\delta_2$-ball around $z_h$ for some point $z_0\in
f({\EuScript{S}}_i)$ and $y_{h+1}$ is inside this ball. Note that since ${\operatorname{dist}}(y_1,f({\EuScript{S}}_i))\ge
d_{k+1}^{1+\alpha_i}/(B(1+\alpha_i))$ from , then $$\begin{aligned}
\label{eq:distB}
{\operatorname{dist}}(y_{h+1},z_h)\ge
\frac{d_{k+1}^{1+\alpha_i}}{B(1+\alpha_i)}
e^{\xi_0 c_0 h}
=
\frac{C_2^{1+\alpha_i}}
{B(1+\alpha_i)}e^{\xi_0 c_0 h
(1- \frac{T+k+1}{h})}
\ge C_3 \delta_2^{\alpha_i-\beta_i}.
\end{aligned}$$ Thus if we set $2\delta_3=C_3
\delta_2^{\alpha_i-\beta_i}$, then we can take a hyperbolic neighborhood of $y_1$ defined by $W:=(f^h\mid
V_i)^{-1}(B(y_{n+1},\delta_3))$, where $V_i$ is the original neighborhood associated to $B_i$, see . Observe that because ${\operatorname{diam}}W\le
\delta_3 e^{-\xi_0 c_0 h}$ $$\begin{aligned}
{\operatorname{dist}}(W,f({\EuScript{S}}_i))
&\ge
\frac{C_2^{1+\alpha_i}}{B(1+\alpha_i)}
d_{k+1}^{1+\alpha_i}
- \delta_3 e^{-\xi_0 c_0 h}
\\
&=
\left(C_3\delta_2^{\alpha_i-\beta_i}d_{k+1}^{1+\alpha_i}
e^{\xi_0 c_0 h} -\delta_3 \right)e^{-\xi_0 c_0 h}
\ge
\delta_3 e^{-\xi_0 c_0 h}
\end{aligned}$$ by the definition of $\delta_3$.
Now we find a radius ${\varrho}>0$ such that the image $f(B(y_0,{\varrho}))$ covers the hyperbolic neighborhood $W$ of $y_1$. By the definition of $k$ and the non-flatness condition (S1) we have that $f(B(y_0,{\varrho}))\supset
B(y_1,{\varrho}_1)$ for all small enough ${\varrho}>0$, where ${\varrho}_1\ge {\varrho}\|Df(y_0)^{-1}\|^{-1} \ge B^{-1}{\varrho}d_{k+1}^{-\alpha_i}$. Since we want ${\varrho}_1\ge \delta_3
e^{-\xi_0 c_0 h}$ it is enough that $$\begin{aligned}
\frac{\varrho}{B} e^{-\frac{\alpha_i}{1+\alpha_i}\xi_0 c_0
(T+k+1)}
\ge
\delta_3 e^{-\xi_0 c_0 h}
\iff
{\varrho}\ge B\delta_3 e^{-\xi_0 c_0\frac{T+k+1}{1+\alpha_i}
\big( (1+\alpha_i)\frac{h}{T+k+1} - \alpha_i\big)}.\end{aligned}$$ But using the relations obtained between $h$ and $T+k+1$ we get $$\begin{aligned}
{\varrho}\ge B\delta_3 e^{-\xi_0 c_0\frac{T+k+1}{1+\alpha_i}
\big( (1+\alpha_i)(1+\zeta/2) - \alpha_i\big)}
=
B\delta_3 d_{k+1}^{1+(1+\alpha_i)\zeta/2}.\end{aligned}$$ We note that $\gamma:=(1+\alpha_i)\zeta/2$ is positive due to the non-flatness conditions. If we take $w,\tilde w\in
B(y_0, d_{k+1}^{1+\gamma})$, then $$\begin{aligned}
\label{eq:V0-dentro}
{\operatorname{dist}}(w,{\EuScript{S}}_i)\ge{\operatorname{dist}}(y_0,{\EuScript{S}}_i) - {\operatorname{dist}}(w,y_0)
\ge d_{k+1}-d_{k+1}^{1+\gamma}
= d_{k+1}\big(1-d_{k+1}^{\gamma}\big)
>d_{k+2}\end{aligned}$$ whenever $T_i$ is big enough in order that $1-d_{k+1}^\gamma> d_{k+2}/d_{k+1}=e^{-\xi_0 c_0
/(1+\alpha_i)}$. Thus if $w_1=f(w) $ and $ \tilde w_1=f(\tilde w)$ are both in $W$, then using the bound for the inverse of the derivative provided also by the non-flatness condition (S1) $$\begin{aligned}
{\operatorname{dist}}(w,\tilde w)
&\le
B d_{k+2}^{-\alpha_i}{\operatorname{dist}}(w_1,\tilde w_1)
\le
B d_{k+2}^{-\alpha_i} e^{-\xi_0 c_0 h}{\operatorname{dist}}(w_{h+1},\tilde w_{h+1})\nonumber
\\
&=
B\exp\left(-\xi_0 c_0 (h+1) \big(\frac{h}{h+1} -
\frac{\alpha_i}{1+\alpha_i}\frac{T_i+k+2}{h+1}\big)\right)
{\operatorname{dist}}(w_{h+1},\tilde w_{h+1})\nonumber
\\
&\le \label{eq:xi-1}
e^{-\xi_0 c_0 (1-\frac{\alpha_i}{1+\beta_i}) (h+1)}
{\operatorname{dist}}(w_{h+1},\tilde w_{h+1})\end{aligned}$$ for all big enough $h$ since $h+1>\frac{1+\beta_i}{1+\alpha_i}(T_i+k)+1>
\frac{1+\beta_i}{1+\alpha_i}(1+o(T_i))(T_i+k+2)$ where $o(T_i)$ is a small quantity which tends to zero when $T_i$ is taken arbitrarily large. We recall that condition (S1) on $\alpha_i,\beta_i$ ensures that $1+\beta_i>\alpha_i$, so that $0<\xi_1:=\xi_0\big(1-\frac{\alpha_i}{1+\beta_i}\big)$ and the contraction rate is bounded by $e^{-\xi_1 c_0}$. In particular this shows that we can indeed take a neighborhood with the radius ${\varrho}$ we estimated above. This means that every point $w_1$ in $W$ is the image of some point $w\in B(y_0,d_{k+1}^{1+\gamma})$, and the connected component $W_0$ containing $y_0$ of the pre-image of $W$ under $f$ is fully contained in $M\setminus
B({\EuScript{S}}_i,d_{k+2})$, by the relation . Thus the neighborhood $W_0$ of $y_0$ satisfies items (1) and (2) of Proposition \[p.contr\] for $n=h+1$ iterates and for a $e^{-\xi_1 c_0}$ contraction rate. Therefore item (3) of the same proposition also holds as a consequence with $\delta=d_0$, see Section \[sec:hyperb-times\].
In addition we can estimate $$\begin{aligned}
S_{h+1}\fD_{d_0}(y_0)
&=
\fD_{d_0}(y_0) + S_h\fD_{d_0}(y_1)
\le
-\log d_{k+1} + \zeta h + o(\tilde\delta)\nonumber
\\
&\le
\left( \xi_0 c_0 \frac{T_i+k+1}{(h+1)(1+\alpha_i)}
+\zeta\frac{h}{h+1} + \frac{o(\tilde\delta)}{h+1}
\right) (h+1)\nonumber
\\
&\le
\left(\frac{\xi_0 c_0}{1+\alpha_i} + 2\zeta\right)(h+1)
\le
\frac{\xi_0 c_0}{1+\zeta} (h+1) , \label{eq:SR}
\end{aligned}$$ where we have used item (3) of Proposition \[p.contr\] applied to the hyperbolic neighborhood $W$ of $y_1$ and the choice of $\zeta$ in and . This bound is essential to obtain positive frequency of hyperbolic neighborhoods in the final stage of the proof. We use here the assumption B of the statement of Theorem \[thm:Misiurew\] with appropriately chosen constants.
Moreover we also have the following estimate for the bounded distortion of volume, again using the non-flat conditions together with the above inequalities for $w,\tilde w\in W_0$ $$\begin{aligned}
\log\frac{|\det Df(w)|}{|\det Df(\tilde w)|}
&\le
B {\operatorname{dist}}(W_0,{\EuScript{S}}_i)^{-\alpha_i} {\operatorname{dist}}(w,\tilde w)
\le
e^{-\xi_1 c_0 (h+1)}{\operatorname{dist}}(w_{h+1},\tilde w_{h+1}).
\end{aligned}$$ Therefore we can bound (using that $f(w),f(\tilde w)\in
B_i$ and Proposition \[p.contr\]) $$\begin{aligned}
\log\frac{|\det Df^{h+1}(w)|}{|\det Df^{h+1}(\tilde w)|}
&\le
\sum_{j=0}^{h} \log\frac{|\det Df(w_j)|}{|\det
Df(\tilde w_j)|}
\\
&\le
\sum_{j=0}^{h} e^{-\xi_1 c_0 j}
{\operatorname{dist}}(w_{h+1},\tilde w_{h+1} )
\le \frac{\delta_3}{1-e^{-\xi_1 c_0}} = C_1.
\end{aligned}$$ Hence $W_0$ satisfies all the conditions of Proposition \[p.contr\] with appropriate constants.
We stress that the value of $T_i$ (thus the value of $d_0$) depends only on $\alpha_i,\beta_i$ and $\xi_0,c_0,\delta_1$.
This completes the proof of Claim \[claim:a\].
\[rmk:highexpnearsing\] The minimal backward contraction in $B({\EuScript{S}}_i,d_0)$ near a singularity can be made arbitrarily big in a single iterate taking $T_i$ large enough or, which is the same, taking the neighborhood of the singularities small enough.
\[rmk:exp-const\] The constant of average expansion $\xi c_0$ is the same in all cases for each connected component of ${\EuScript{S}}$, but in principle the radius $\delta_3=\delta_3(i)$ and the distance $d_0=d_0(i)$ to ${\EuScript{S}}_i$ ensuring existence of distortion times depend on the connected component. However we assume that the values of $T_i$ are big enough so that all the inequalities above are satisfied for all connected components ${\EuScript{S}}_i$ and such that the corresponding neighborhoods of ${\EuScript{S}}_i$ be contained in the $\tilde \delta$-neighborhood of ${\EuScript{S}}$.
In what follows we write $\delta$ (which is smaller than $\tilde\delta$) for the smallest value $d_0(i)$ over all connected components.
### Hyperbolic neighborhoods for almost every point {#sec:hyperb-neighb-almost}
We use the statement of Claim \[claim:a\] from now on and show that all points whose orbit does not fall into the singular/critical set admit some distortion time with well defined contraction rate and slow approximation rate.
\[claim:b\] Lebesgue almost every point admits a $(e^{-\xi_2
c_0},\delta)$-hyperbolic neighborhood for some iterate $\ell$ and the frequency of visits of the $\ell$-iterates to a $\delta$-neighborhood of ${\EuScript{S}}$ is bounded by $\frac{\xi_0c_0}{1+\zeta}$.
In a similar way to the one-dimensional multimodal case, we use condition D on $\hat U$ to show that we can obtain a minimal derivative when an orbit returns to any fixed arbitrarily small neighborhood $U\subset\hat U$ of the critical/singular set. From this intermediate result we deduce that most points on $M\setminus U$ have some distortion time. We prove first an auxiliary result.
\[claim:b1\] There exists a minimal average expansion rate $e^{- c/2}$ either for the first return of $x_0$ to a small enough neighborhood $U$ of ${\EuScript{S}}$, or for the first $N$ iterates, where $N$ does not depend on the starting point $x_0\in U$.
Let $U=B({\EuScript{S}},\delta)$ be an open neighborhood of ${\EuScript{S}}$ compatible with the choices of the $T_i$ as in the proofs of the previous claims. So $U$ is the union of a number of connected open sets, one for each connected component of ${\EuScript{S}}$.
The first $h$ iterates of $x_0$ correspond to a $(e^{-\xi
c_0},\delta)$-distortion time. From condition (C) there are $K,c>0$ such that for $n\ge h$ satisfying $x_h,\dots,
x_{n-1}\in M\setminus U$ we have that $S_n\psi(x_0)\le -\xi
c_0 h + K - c (n-h)$.
We shrink the neighborhood $U$ so that the smaller distortion time $\underbar h$ for points $x_0\in U\setminus{\EuScript{S}}$ satisfies $$\begin{aligned}
\label{eq:cond-h0}
\frac{\kappa}{\underbar h}
<
\min\left\{\frac{\xi c_0}4,\frac{c}8\right\}
\quad\text{and}\quad
\frac{\underbar h}{\underbar h+2 K/ c}\ge \frac12.\end{aligned}$$ We write $[t]:=\max\{i\in{{\mathbb Z}}:i\le t\}$ for all $t\in{{\mathbb R}}$ in what follows and set $N=[2 K/ c]$.
Case (i) – the orbit returns to $U$ in more than $N$ iterates
: In this case we have for $n > N$ that $
S_n\psi(x_0) \le -\frac{ c}2 n$, since we can assume without loss that $c < \xi c_0$. Notice that we have the same conclusion if the orbit of $x_0$ *never returns* to $U$.
Case (ii) – the return to $U$ occurs in less than $N$ iterates
: We now use condition D to get, since the first return iterate $n$ satisfies $n\ge h\ge \underbar h$ $$\begin{aligned}
S_n\psi(x_0)\le -\xi c_0 h +\kappa
=
n (-\xi c_0\frac{h}n + \frac\kappa{n})
\le
n (-\frac{\xi c_0}2 + \frac{\xi c_0}4)
= -\frac{\xi c_0}4 n,
\end{aligned}$$ since $\underbar h$ was chosen as in .
This completes the proof of Claim \[claim:b1\], setting $\min\{ c/2, \xi c_0/4\}$ as the average expansion rate.
Now we prove Claim \[claim:b\].
Again fix an arbitrary $x_0\in M\setminus U$ such that the orbit of $x_0$ never falls into ${\EuScript{S}}$. We divide the argument in two cases.
Case (iii) – the orbit takes more than $N$ iterates to enter $U$
: Arguing as in the proof of Case (i) above we get for all $n > N$ satisfying $x_0,\dots, x_{n-1}\in
M\setminus U$ that $ S_n(x_0)\le K - cn \le -\frac{c}2 n.$ This implies that there exists some distortion time $h\le
n$ for $x_0$ by the proof of Theorem \[thm:abv0\]. We note that in this case $S_n\fD_{\delta}(x_0)=0$.
Case (iv) – the orbit enters $U$ in at most $N$ iterates
: Let $j\le N$ be the first entrance time of the orbit of $x_0$ in $U$, $h$ the distortion time associated to $x_j$ and use Claim \[claim:b1\] to get $$\begin{aligned}
S_{j+h}\psi(x_0)
&\le
\kappa + S_{h}\psi(x_{j})
\le
\kappa- \frac{c}2 h
=
(j+h)\Big( \frac{\kappa}{j+h}
- \frac{c}2\frac{h}{j+h} \Big)
\\
&\le
(j+h)\Big( \frac{c}8 - \frac{c}4\Big)=-\frac{c}8 (j+h)\end{aligned}$$ since $h\ge\underbar h$ by the choice of $U$ in .
We also obtain that $S_{j+h}\fD_{\delta}(x_0)=S_h\fD_\delta(x_j)\le
\frac{\xi_0c_0}{1+\zeta} h\le \frac{\xi_0c_0}{1+\zeta} (j+h)$.
Moreover the distortion of $f^{j+h}$ on the connected component of $(f^j)^{-1}(V_{x_j})$ containing $x_0$ is bounded from above by a constant dependent on $N$ only (since we are dealing with a local diffeomorphism away from a given fixed neighborhood of ${\EuScript{S}}$), where $V_{x_j}$ is the hyperbolic neighborhood of $x_j$ given by Claim \[claim:a\].
This completes the proof of Claim \[claim:b\] if we set $\xi_2$ such that $\xi_2 c_0 = c/8$.
### Positive frequency of hyperbolic neighborhoods almost everywhere {#sec:positive-frequency-h}
Here we finish the proof of Theorem \[thm:dense-hyp-times\].
\[claim:c\] The frequency of $(e^{-\xi_2 c_0},\delta)$-hyperbolic neighborhoods is positive and bounded away from zero Lebesgue almost everywhere.
We now define an auxiliary induced map $F:\tilde M\to
M$, so that the $F$-iterates correspond to iterates of $f$ at distortion times, as follows. For $x_0\in\tilde M$ we have two cases
Expansion without shadowing
: the $\ell\ge N$ iterates of $x$ belong to $M\setminus U$. In this case there exists some $(e^{-\xi_2 c_0},\delta)$-distortion time $\ell\le N$ and we define $F(x)=f^\ell(x)$ and $\tau(x)=\ell=q(x)$.
Expansion with shadowing
: let $0\le q<N$ be the least non-negative integer such that $x_q\in U$ and $p$ be the $(e^{-\xi_0 c_0},\delta)$-distortion time associated to $x_q$ from Claim \[claim:a\]. We define $F(x)=f^{p+q}(x)=x_{p+q}$ and $\tau(x)=p+q$ and $q(x)=q$ in this case.
The images of $F$ always belong to $M\setminus
B({\EuScript{S}},\delta)$ by the choice of $\delta$ from following Remark \[rmk:preball-away\]. In addition, for any given $x_0\in\tilde M$ the map $F(x)$ is defined and the iterate $\tau(x)$ has all the properties of a $(e^{-\xi_2
c_0},\delta)$-distortion time for $x_0$ according to Claim \[claim:b\].
Now we observe the statement of Claim is a consequence of the following property: there exists $\theta>0$ such that for every $x\in\tilde M$ $$\begin{aligned}
\label{eq:positive-freq}
\limsup_{n\to+\infty}\frac1n\#\{ 0\le j < n :
f^j(x)\in{\EuScript{O}}^+_F(x)\} \ge\theta,
\end{aligned}$$ where ${\EuScript{O}}^+_F(x)=\{F^i(x), i\ge0\}$ is the positive orbit of the induced map $F$. Moreover it is easy to see that for each $x\in M$ and each $n\in{{\mathbb N}}$ $$\begin{aligned}
\label{eq:updown}
\#\{ 0\le j < n : f^j(x)\in{\EuScript{O}}^+_F(x)\}
=
\sup\big\{
k\ge0: S_{k+1}^F\tau(x)=\sum_{i=0}^k
\tau\big(F^i(x)\big) < n
\big\}.
\end{aligned}$$ We remark that from to obtain it is enough to show that $S_{k+1}^F \tau(x)< \frac1{\theta}k$, at least for every big enough $k$. Indeed $\big\{
k\ge0: S_{k+1}^F\tau(x) < n
\big\}
\supset
\big\{
k\ge0: \frac1\theta k < n
\big\}$ so $ \sup\big\{
k\ge0: S_{k+1}^F\tau(x) < n
\big\} \ge \theta n.$
The bounds and together with the definition of $F$ ensure that we are in the conditions of the following result.
\[le:positive-freq\] Assume that we have an induced map $F=f^\tau$ for some $\tau:K\to{{\mathbb N}}$ defined on a positive invariant subset $K$ such that for every $x\in K$:
1. $\tau(x)=q(x)+p(f^{q(x)}(x))$ for well defined integer functions $q$ on $K$ and $p$ on $f^{q(x)}(x)$ for all $x\in K$;
2. there exists $N\in{{\mathbb N}}$ such that $q\le N$;
3. there exists $0<d<1$ and $C,{\varrho}>0$ such that $0<{\varrho}C<1$ and for all $x\in K$
1. the iterates $x,f(x),\dots,f^{q(x)-1}(x)$ are outside $B_{d}({\EuScript{S}})$;
2. $p(f^{q(x)}(x)) \le C \fD_d(f^{q(x)}(x))$;
3. $S_p^f \fD_d(f^{q(x)}(x))\le{\varrho}p$.
Then $S_k^F\tau(x)\le\frac{N}{1-{\varrho}C} k$ for each $x\in K$ and every $k\ge1$.
In addition, we observe that since $f$ is a regular map (that is $f_*{\operatorname{Leb}}\ll{\operatorname{Leb}}$) then we can further assume that the full Lebesgue measure set $\tilde M$ is $f$-invariant, since $\cap_{i\ge0}f^{-i}(\tilde M)$ also has full Lebesgue measure. Hence we can apply Lemma \[le:positive-freq\] with $K=\tilde M$, $C=\frac{1+\zeta/2}{\xi_0 c_0}$ and ${\varrho}=\frac{\xi_0 c_0}{1+\zeta}$ to obtain $\theta\ge\frac{\zeta}{2(1+\zeta)\ell}$.
\[rmk:log-dist-integrable\] According to item (3b) of the statement of Lemma \[le:positive-freq\] above, if we have an absolutely continuous invariant probability measure $\nu$ for $f$, then $\nu(\fD_d)\le{\varrho}<\infty$ and so $\fD_d$ is $\nu$-integrable, as claimed in Theorem \[thm:Misiurew\].
This completes the proof of Theorem \[thm:dense-hyp-times\] except for the proof of Lemma \[le:positive-freq\].
Using the definition of $F$ and the assumptions on $\tau$, for every given $k\ge0$ and $x\in K$ we can associate a sequence $q_0,p_0,q_1,p_1,\dots,q_k,p_k$ such that for each $i=0,\dots,k$ we have $ q_i=q(F^i(x))
\quad\text{and}\quad q_i+p_i=\tau(F^i(x)).$ This together with the assumptions of item (3) in the statement of the lemma allows us to estimate $$\begin{aligned}
S_{k}^F\tau(x) &= \sum_{i=0}^k ( q_i + p_i) \le
\sum_{i=0}^{k-1}\Big( N +
C\fD_{d_0}\big(f^{q_i}(F^i(x))\big) \Big) \nonumber
\\
&\le k N + C\sum_{i=0}^{k-1}
S^f_{q_i+p_i} \fD_{d_0} (F^i(x)) \label{eq:noreturn}
\\
&\le \label{eq:contraction1} k N
+ C{\varrho}\sum_{i=0}^{k-1}(q_i+p_i) = k N +
C{\varrho}S_{k}^F\tau(x),
\end{aligned}$$ where in we have used that $\fD_{d_0}\ge0$ and that this function equals zero at each of the $q_i$ iterates before each visit to $B({\EuScript{S}},d)$. The contraction in implies $
S_k^F\tau(x) \le \frac{N}{1-{\varrho}C}k $ as stated.
Periodic attractor with full basin of attraction {#sec:full-basin-attract}
================================================
Here we show that a perturbation of $X$, like the one depicted in the right side of Figure \[fig:skewperturb\], for a flow $Y\in{\EuScript{P}}\cap{\EuScript{N}}$ is such that $U$ is as a trapping region which coincides Lebesgue modulo zero with the basin of a periodic attracting orbit (a sink) of $Y$. This is a consequence of the smoothness of the first return map to $\Sigma_1$ after quotienting out the stable leaves, which is the reason why we assume the flow is of class at least $C^2$ and restrict the vector field to a submanifold ${\EuScript{P}}\cap{\EuScript{N}}$ of all possible vector fields nearby $X$, together with the robustness of property C in the statement of Theorem \[thm:Misiurew\].
Indeed, let $g=g_Y$ be the action on the stable leaves of the first return map $R_Y$ of the flow of $Y\in{\EuScript{P}}\cap{\EuScript{N}}$ to $\Sigma_2$. Recall that we constructed $X$ having on $\Lambda_\Sigma=\cap_{n\in{{\mathbb Z}}^+}g^n(\Sigma_2)$ a partially hyperbolic splitting so that the stable foliation does persist under perturbations. The projection along the leaves of this foliation in $\Sigma_2$ is absolutely continuous with Hölder Jacobian, as a consequence of the strong domination obtained in Section \[sec:dominat-splitt\].
The map $g$ sends a $\delta$-neighborhood of ${\EuScript{S}}$ into the local basin of attraction of a periodic sink $p$. We denote by $B$ the stable set of the orbit of $p$. On $N\setminus B$ the map $g$ is uniformly expanding, since $B\supset
B({\EuScript{S}},\delta)$ and condition C on the statement of Theorem \[thm:Misiurew\] is persistent under small perturbations.
\[le:fullbasinsink\] Lebesgue almost every point of $N$ belongs to the basin of the periodic sink.
Arguing by contradiction, assume that the basin $B$ of the sink is such that $E:=N\setminus B$ has positive volume. Since $B$ contains a neighborhood of $\sigma_1$ and a neighborhood of the sink, then $E$ is an invariant subset satisfying condition C of the statement of Theorem \[thm:Misiurew\]. Hence $f_1\mid E$ is uniformly expanding: there exists $N\ge1$ and $\lambda\in(0,1)$ such that $\|(Dg_Y^N)^{-1}\|\le\lambda$. Therefore, since $g_Y$ is $\log$-Hölder and expanding, and $E$ is closed, invariant and has positive volume, we can apply the arguments in [@AAPP] to show that there exists a ball $U$ of radius $r>0$ fully contained in $E$.
We claim that for $g=g_Y^N$ there exists ${\varrho}>1$ such that $g^k(U)$ contains a ball of radius ${\varrho}^k r$ for all $k\ge1$, which yields a contradiction, since the ambient manifold is compact and $E$ is by assumption a proper subset.
To prove the claim, recall that $g$ is a local diffeomorphism on a neighborhood of $E$, since $E$ is far from the singularities of the stable foliation. We assume that $B(x_0,s_0)$ is a ball centered at $x_0$ with radius $s_0$ contained in $E$ and consider $g(B(x_0,s_0))$.
Let us take $y_1$ in the boundary of $g(B(x_0,s_0))$ and a smooth curve $\gamma_1:[0,1]\to N$ such that $\gamma_1(0)=x_1:=g(x_0)$ and $\gamma_1(1)=y_1$. Let $\gamma_0$ be a lift of $\gamma_1$ under $g$, that is, $\gamma_1=g\circ\gamma_0$ such that $\gamma_0(0)=x_0$. We then define $s:=\sup\{ t\in[0,1] : \gamma_1([0,t])\subset
g(B(x_0,s_0)) \}.$ Clearly $s>0$ and by its definition and the expansion properties of $g$ in $E$ we get $$\begin{aligned}
\lambda \times (\text{length of } \gamma_1([0,s]))
\ge \text{length of } \gamma_0([0,s]) \ge
{\operatorname{dist}}(\gamma_0(0),\gamma_0 (s)).
\end{aligned}$$ However, $\gamma_1(s)$ is at the boundary of $g(B(x_0,s_0))$ so that $\gamma_0(s)$ is also at the boundary of $B(x_0,s_0)$, because $g$ is a local diffeomorphism. Thus we get $$\begin{aligned}
{\operatorname{dist}}(y_1,x_1)\ge \text{length of } \gamma_1([0,s]) \ge
\frac1\lambda\times {\operatorname{dist}}(\gamma_0(0),\gamma_0 (s)) \ge
\frac1\lambda s_0
\end{aligned}$$ and the claim is proved with ${\varrho}=\lambda^{-1}$.
Non-uniform expansion and existence of hyperbolic times {#sec:hyperb-neighb-constr}
=======================================================
Here we prove Theorem \[thm:abv0\]. The proof is very similar to [@ABV00 Lemma 5.4] but our definition of hyperbolic times/hyperbolic neighborhoods is slightly different, in a crucial way, from the definition on [@ABV00], and we include a proof for completeness.
We start with the following extremely useful technical result will be the key for several arguments.
\[le:Pliss\] Let $H\ge c_2 > c_1 >0$ and $\zeta={(c_2-c_1)}/{(H-c_1)}$. Given real numbers $a_1,\ldots,a_N$ satisfying $$\sum_{j=1}^N a_j \ge c_2 N \quad\text{and}\quad a_j\le H
\;\;\mbox{for all}\;\; 1\le j\le N,$$ there are $l>\zeta N$ and $1<n_1<\ldots<n_l\le N$ such that $$\sum_{j=n+1}^{n_i} a_j \ge c_1\cdot(n_i-n)
\;\;\mbox{for each}\;\; 0\le n < n_i, \; i=1,\ldots,l.$$
See [@Man87 Lemma 11.3].
The proof uses Lemma \[le:Pliss\] twice, first for the sequence $a_j=-\psi(x_{j-1})$ (properly cut off so that it becomes bounded from above), and then for $a_j=\fD_\delta(x_j)$ for an adequate $\delta>0$.
Let $H\subset M\setminus{\EuScript{S}}$ be such that conditions and hold for all $x\in H$ and let $0<\xi<1$ and $\zeta>0$ be given, and take $x=x_0\in H$ and $\gamma_0:=(2+\xi)/3\in(\xi,1)$, $\gamma_2:=(1-\xi)/3$ and $\gamma_3=\gamma_1-\gamma_2=(1+2\xi)/3$. Then for every large $N$ we have $S_N\psi(x) \le - \gamma_0 c N.$ Moreover since $f({\EuScript{S}})\cap{\EuScript{S}}=\emptyset$ we can assume that $\zeta<\inf\{{\operatorname{dist}}(x,y): x\in{\EuScript{S}}, y\in f({\EuScript{S}})\}$.
For any fixed ${\varrho}>\beta$, by non-degeneracy condition (S1), we can find a neighbourhood $V$ of ${\EuScript{S}}$ such that $$\begin{aligned}
\label{eq:bound2}
|\psi(z)|
\le {\varrho}\fD(z)\quad\text{for every}\quad
x\in V.\end{aligned}$$ Setting $\varepsilon_1>0$ such that ${\varrho}\varepsilon_1
\le \gamma_1$, we can use the slow approximation condition to find $r_1>0$ so small that $$\begin{aligned}
\label{eq:bound1}
S_N\fD_{r_1}(x) \le \varepsilon_1 N.\end{aligned}$$ We may assume without loss that $V=B({\EuScript{S}},r_1)$ in what follows. Now we fix $H_1 \ge \max\{ c,{\varrho}|\log r_1|,
\sup_{M\setminus V} |\psi| \}$ and define the set $E=\{z\in
M: \psi(z)<-H_1\}$ and the sequence $a_j=-(\psi\chi_{M\setminus E})(x_{j-1})$.
*We remark that there is a shift between the index of $a_j$ and that of $x_{j-1}$ in the above definition.*
We note that by construction $a_j\le H_1$ and that $x_j\in
E$ implies $x_j\in V$, $\fD(x_j)<-\log r_1$, because ${\varrho}|\log r_1| \le H_1 < -\psi(x_{j}) < {\varrho}\fD(x_j).$
This means that $\fD_{r_1}(x_j)=\fD(x_j)<|\log r_1|$ whenever $x_j\in E$. From and we get that $
-S_N(\psi\chi_E)(x) \le {\varrho}S_N(\fD\chi_E)(x)\le
{\varrho}{\varepsilon}_1 N \le \gamma_1 c N.$
Therefore $\sum_{j=1}^{N} a_j = -S_N\psi(x) -
S_N(\psi\chi_E)(x) \ge (\gamma_0-\gamma_1)c N = \gamma_3 c
N$. Hence we can apply Lemma \[le:Pliss\] with $c_2=\gamma_3 c$, $c_1=\xi c$, $H=H_1$, obtaining $\theta_1=\gamma_2 c/(H-\xi c)\in(0,1)$ and $l_1\ge \theta_1
N$ times $1 \le p_1< \cdots <p_{l_1}\le N$ such that $$\begin{aligned}
\label{eq:conclusion1}
\sum_{j=n+1}^{p_i} \psi(x_{j-1}) \le -\sum_{j=n+1}^{p_i}
a_j \le -\xi c (p_i-n)
\end{aligned}$$ for every $0 \le n < p_i$ and $1\le i\le l_1$.
Let now $\varepsilon_2>0$ be small enough so that $\varepsilon_2 < \min\{\zeta, b c \theta_1\}$, and let $r_2>0$ be such that $-S_N\fD_{r_2}(x_1) \ge -{\varepsilon}_2 N$ from the slow recurrence condition (we note that ${\operatorname{dist}}(x_0,{\EuScript{S}})>\zeta>r_2$). Taking $c_1= b c$, $c_2=-\varepsilon_2$, $A=0$, and $
\theta_2=\frac{c_2-c_1}{A-c_1}=1-\frac{\varepsilon_2}{bc}
$ we can apply again Lemma \[le:Pliss\] to $a_j=-\fD_{r_2}(x_j)$.
*We remark that now there is no shift between the index of $a_j$ and $x_j$ in the previous definition.*
In this way we obtain $l_2\ge\theta_2 N$ times $1\le q_1 <
\cdots < q_{l_2}\le N$ such that $$\begin{aligned}
\label{eq:conclusion2}
\sum_{j=n+1}^{q_i} \fD_{r_2}(x_j)\le {\varepsilon}_2
(q_i - n) \le \zeta (q_i-n)\end{aligned}$$ for every $0 \le n < q_i$ and $1\le i \le l_2$.
Finally since our choice of $\theta_2$ ensures that $\theta=\theta_1+\theta_2-1>0$, then there must be $l=(l_1+l_2-N)\ge \theta N$ and $1\le n_1 <\ldots <n_l \le
N$ such that and simultaneously hold.
This exactly means that we have condition with $\delta=r_2$, because $\fD_{r_2}(x_0)=0$ by the choice of $r_2$ above, and $\xi c$ as the logarithm of the contraction rate, as in the statement of Theorem \[thm:abv0\].
Solenoid by isotopy {#sec:isotopy}
===================
We recall that ${{\mathbb S}}^1=\{z\in{{\mathbb C}}: |z|=1\}$, ${{\mathbb T}}=({{\mathbb S}}^1)^k$, $B^k:=\{ x=(x_1,\dots,x_k)\in{{\mathbb R}}^n: \sum_{i=1}^k x_i^2=1\}$ for all $k\ge1$, and ${\EuScript{T}}^k={{\mathbb T}}^k\times{{\mathbb D}}$, where ${{\mathbb D}}=\{z\in{{\mathbb C}}: |z|<1\}$.
Here we prove the results needed in Section \[sec:unpert-singul-flow\] ensuring the existence of a smooth family of embeddings of ${\EuScript{T}}^k$ into $B^{k+2}$, for all $k\ge1$, which deforms a tubular neighborhood in $B^{k+2}$ of the usual embedding of ${{\mathbb T}}^k$ into $B^{k+1}\simeq B^{k+1}\times\{0\}\subset B^{k+2}$, into the embedding of ${\EuScript{T}}$ into the image of the Smale solenoid map.
More precisely, consider the identity map $i$ on ${\EuScript{T}}$ and the solenoid map $$\begin{aligned}
s:{\EuScript{T}}\to{\EuScript{T}}, \quad
(\Theta=(\theta_1,\dots,\theta_k),z)\mapsto
(\Theta^2=(\theta_1^2,\dots,\theta_k^2), A_\Theta(z))
$$ where $A_\Theta$ is a contraction with contraction rate bounded by $0<\lambda<1$, and the map in the coordinate $\Theta$ is the expanding torus endomorphism $f_1$ defined in Section \[sec:an-example-nue\], but restricted to ${{\mathbb T}}^k\subset{{\mathbb C}}^k$.
\[pr:embedding\] There exists an embedding $e:{\EuScript{T}}\to B^{k+2}$ such that the projections $\pi_{{\mathbb D}}:{\EuScript{T}}\to{{\mathbb T}}^k$ on the first coordinate and $\tilde\pi_{{\mathbb D}}:e({\EuScript{T}})\to e({{\mathbb T}}^k\times\{0\})$ along the leaves of the foliation ${\EuScript{F}}^s:=\{ e(\Theta\times{{\mathbb D}})
\}_{\Theta\in{{\mathbb T}}^k}$ of $e({\EuScript{T}})$ define the solenoid map $S$ by the commutative diagram:
$\begin{CD} {\EuScript{T}}@>>{s}> {\EuScript{T}}\\
@VV{e}V @V{e}VV
\\
e({\EuScript{T}}) @>{S}>> e({\EuScript{T}})
\end{CD}.$
Moreover there exists a smooth family $e_t:{\EuScript{T}}^k\to B^{k+2}$ of embeddings for all $t\in[0,1]$ such that $e_0=e\circ i$ and $e_1=e\circ s$.
We prove this statement in the following steps.
### An embedding of ${{\mathbb T}}^k$ in $B^{k+2}$ {#sec:embedd-ttk-bk+1}
We argue by induction on $k\ge1$. We know how to embed ${{\mathbb T}}^1$ in $B^3$. Let us denote by $e^1$ this embedding and fix a small number $d>0$ and $\lambda\in(0,1/2)$.
We assume that we have an embedding $e^{l}$ of ${{\mathbb T}}^{l}$ on $B^{l+2}$ for all $l=1,\dots,k-1$, in such a way that the image of $e^{l+1}$ is in a tubular neighborhood of the image of $e^l$ inside $B^{l+2}$ with size $\le d\lambda^{l+1}$, for $1\le l < k-1$.
For each $w\in e^{k-1}({{\mathbb T}}^{k-1})\subset {{\mathbb R}}^{k+1}\simeq
{{\mathbb R}}^{k+1}\times 0 \subset {{\mathbb R}}^{k+2}$ let $N_w=(T_w{{\mathbb S}}_{k})^\perp$ be the normal space to $e^{k-1}({{\mathbb T}}^{k-1})$ at $w$ in ${{\mathbb R}}^{k+2}$ (where we take in ${{\mathbb R}}^{k+2}$ the usual Euclidean inner product). This is a $3$-dimensional space.
We know we can embed ${{\mathbb T}}^1$ into $B^3$ through $e^1$. So by a simple rescaling we can assume that $e^1_w$ embeds ${{\mathbb T}}^{1}$ into a small neighborhood of $w$ in $w+N_w$. To keep the inclusion in the tubular neighborhood of the image of $e^{k-1}$, we take this neighborhood around $w$ to have radius $\lambda^{k+1}d$.
Hence letting $\hat w\in{{\mathbb T}}^{k-1}$ be the unique element such that $e^{k-1}(\hat w)=w$ and considering the map $e^k:{{\mathbb T}}^{k-1}\times{{\mathbb T}}^1\to B^{k+2}$ given by $(\hat
w,\theta)\mapsto e^1_w(\theta)$ we easily see that
- $D_1 e^k (\hat w,\theta) ({{\mathbb R}}^{k-1}) =
De^{k-1}(\hat w)({{\mathbb R}}^{k-1})$ is the tangent space of $e^{k-1}({{\mathbb T}}^{k-1})$ at $w$;
- $D_2 e^k (\hat w,\theta) ({{\mathbb R}})$ is a subspace of $N_w$.
Therefore the tangent map $De^k$ to $e^k$ always has maximal rank and clearly is injective with a compact domain, thus $e^k$ is an embedding. This completes the induction argument and proves the existence of an embedding $e^k:{{\mathbb T}}^k\to B^{k+2}$.
\[rmk:embedd-B2\] This argument is also true if we start with an embedding of ${{\mathbb T}}^1$ into $B^2$ so that we obtain an embedding of ${{\mathbb T}}^k$ into $B^{k+1}$ for all $k\ge1$. But we need one extra dimension to deal with the solid torus in what follows.
*It is crucial to observe that the entire inductive construction just presented is built over nested tubular neighborhoods.* Indeed, the image of $e^{k+1}$ is contained in a tubular neighborhood of the image of $e^k$ for each $k\ge1$. In the above construction we consider the tubular neighborhood of $e^k$ in ${{\mathbb R}}^{k+3}$. However since we proceed inductively using orthogonal bundles of the successive images of $e^{k+1}, e^{k+2},\dots$, and we contract the diameter of the tubular neighborhood at each step by a constant factor $0<\lambda<1$, we have in fact that *the image of $e^{k+l}$ is in a tubular neighborhood of $e^{k}$ for all $k,l\ge1$.*
Therefore, the distance between the image $e^{k+l}$ and the image of $e^k$ is bounded by $d\sum_{i=k}^{k+l}\lambda^i<d$, always *inside* a tubular neighborhood $U^{k+l}_k$ of $e^k({{\mathbb T}}^k)$ in ${{\mathbb R}}^{k+l+2}$. Hence there exists a projection $\pi_k^{k+l}:U^{k+l}_k\to e^k({{\mathbb T}}^k)$ associated to this tubular neighborhood, for each $k\ge1$ and $l\ge0$.
### The embedding of ${\EuScript{T}}^k$ into $B^{k+2}$ {#sec:embedd-ct-into}
The previous discussion provides an embedding $e^k$ of ${{\mathbb T}}^k$ into $B^{k+2}$ for each $k\ge1$. Therefore considering a tubular neighborhood $U^k$ of the compact submanifold $e^k({{\mathbb T}}^k)$ in $B^{k+2}$ we obtain a projection $\pi:U\to e({{\mathbb T}}^k)$ such that $\pi^{-1}(w)$ is a $2$-disk. Thus we obtain an embedding $\hat e^k$ of ${{\mathbb T}}^k\times{{\mathbb D}}$ into $B^{k+2}$.
We can assume that the tubular neighborhood has radius smaller than $\lambda^k d$ and that $U^k\subset U^k_l$ for all $l<k$. Then we can consider the projections $\pi^k:U^k\to e^k({{\mathbb T}}^k)$ and $\pi^k_l:U^k_l\to e^l({{\mathbb T}}^l)$. In what follows we assume without loss of generality that $U^k$ is the image of $\hat e^k$.
### The solenoid map through an isotopy of the identity {#sec:soleno-map-through}
The previous construction of the embeddings $e^k$ of ${{\mathbb T}}^k$ depends on the initial embedding $e^1$ of ${{\mathbb T}}^1$ on $B^3$. Moreover it is clear that each of the embeddings $e^k$ for $k>1$ depend smoothly on $e^1$. Hence a smooth family $e^1_t$ of embeddings, for $t\in[0,1]$, defines a smooth family $e^k_t$ of corresponding higher dimensional embeddings.
We argue again by induction on $k\ge1$. For $k=1$ we consider the family of embeddings $e^1_t$ described in Figure \[fig:isotopy\].
![\[fig:isotopy\] The isotopy $e^1_t$ and the tubular neighborhood $U^1_1$ in the end.](isotopy.eps)
We can construct this family as depicted so that the extreme elements of the family satisfy $\pi^1_1\circ
e_1^1(\theta)=e_0^1(\theta^2)$ for $\theta\in{{\mathbb T}}^1$. If we choose a small tubular neighborhood of the image of $e^1_t$, then we obtain a family of embeddings $\hat e^1_t$ of ${\EuScript{T}}^1$ such that $(\pi^1_1\circ\hat e^1_1)(\theta,z)=
e^1_0(\theta^2)$.
Now we can use the family $e^1_t$ to construct families $e^k_t$ of embeddings following the inductive procedure explained before, for each $t\in[0,1]$ and each fixed $k\ge1$.
Again taking a small tubular neighborhood of the image of $e^k_t$ we obtain a family of embeddings $\hat e^k_t$ of ${\EuScript{T}}^k$ such that the image of $\hat e^k_1$ of ${\EuScript{T}}^k$ is compactly contained inside the image of $\hat e^k_0$.
Due to the nested construction, for each $k\ge1$ we have $
(\pi^k_1\circ\hat e^k)(\theta_1,\dots,\theta_k,z)=
e^1_0(\theta_1^2) $ after projecting into the lower dimensional image. Moreover projecting on the previous stage of the construction we get $ (\pi^k_{k-1}\circ\hat
e^k)(\theta_1,\dots,\theta_k,z)=
e^{k-1}_0(\theta_2^2,\dots,\theta_{k}^2)
\cap(\pi^k_1)^{-1}\{e^1_0(\theta_1^2)\}$, which can easily be proved by induction on $k\ge1$ following the nested construction presented above.
This is enough to prove that (independently of the definition of $A_\Theta$ in $s$) $$\begin{aligned}
\label{eq:round}
\pi^k_k\circ \hat e^k_1 = \pi^k_k \circ \hat e^k_0\circ s
\quad\text{for all}\quad k\ge1.\end{aligned}$$ Finally, since by definition $\hat e^k_1$ is a small tubular neighborhood of the image of $e^k_1$, and this set is contained inside a tubular neighborhood of the image of $e^k_0$, then from we see that in fact there exists a family of contractions $(A_\Theta)_{\Theta\in{{\mathbb T}}^k}$ such that $\hat e^k_1=\hat
e^k_0\circ s$. This completes the proof of Proposition \[pr:embedding\].
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{
"pile_set_name": "ArXiv"
}
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---
bibliography:
- '../inputs/992.bib'
---
**Logarithmic CFT at generic central charge:\
from Liouville theory to the $Q$-state Potts model\
**
------------------------------------------------------------------------
height .6mm
[ **Rongvoram Nivesvivat and Sylvain Ribault** ]{}
[* Université Paris-Saclay, CNRS, CEA, Institut de physique théorique* ]{}
[* E-mail:*` [email protected], [email protected] `]{}
<span style="font-variant:small-caps;">Abstract:</span> Using derivatives of primary fields (null or not) with respect to the conformal dimension, we build infinite families of non-trivial logarithmic representations of the conformal algebra at generic central charge, with Jordan blocks of dimension $2$ or $3$. Each representation comes with one free parameter, which takes fixed values under assumptions on the existence of degenerate fields. This parameter can be viewed as a simpler, normalization-independent redefinition of the logarithmic coupling. We compute the corresponding non-chiral conformal blocks, and show that they appear in limits of Liouville theory four-point functions.
As an application, we describe the logarithmic structures of the critical two-dimensional $O(n)$ and $Q$-state Potts models at generic central charge. The validity of our description is demonstrated by semi-analytically bootstrapping four-point connectivities in the $Q$-state Potts model to arbitrary precision. Moreover, we provide numerical evidence for the Delfino–Viti conjecture for the three-point connectivity. Our results hold for generic values of $Q$ in the complex plane and beyond.
------------------------------------------------------------------------
------------------------------------------------------------------------
Introduction
============
Following the lead of its non-logarithmic counterpart, the study of two-dimensional chiral logarithmic CFT has been focusing on fairly complicated cases: higher symmetry algebras, rational values of the central charge, representations with intricate structures [@CR13]. However, powerful results on chiral logarithmic CFT are not always easy to translate into a good understanding of bulk logarithmic CFT. Bulk CFT involves coupling left- and right-moving chiral structures, and this coupling can be nontrivial [@rgw12]. Then is it possible to study the bulk theory without starting with the chiral theory? We propose a positive answer to this question, in the case of CFTs based on the Virasoro algebra at generic central charge. This case is relatively simple algebraically, and it is motivated by the $O(n)$ model and the $Q$-state Potts model. These models are known to be logarithmic at certain rational central charges, and have long been suspected of being logarithmic at generic central charge.
Of course, at any central charge and even in any dimension, we can obtain simple logarithmic fields by taking derivatives of primary fields with respect to the conformal dimension [@hpv16]. We will focus on the next simplest case, and build logarithmic fields from primary fields that have null vectors. Our main technical tool will still be derivatives with respect to the conformal dimension. Actually, we can not only differentiate, but also apply any linear operation: in particular, we will linearly combine a primary field with its null descendant.
This approach is particularly effective for computing correlation functions and conformal blocks. Four-point conformal blocks need not be computed by summing over states in logarithmic representations: they can be obtained by combining standard conformal blocks for Verma modules. This will allow us to determine four-point conformal blocks for representations whose logarithmic features appear at arbitarily high level, and ultimately to compute connectivities in the Potts model to arbitrary precision.
Another important technical idea is that the existence of degenerate fields can determine the structure of logarithmic representations [@mr07]. In the cases of the $O(n)$ model and of the $Q$-state Potts model, this idea was recently used in [@GZ20], although it was only worked out for a subset of logarithmic representations. We will write a conjecture for the structures of all logarithmic representations in these models at generic central charge. (See Section \[sec:onqp\].)
The determination of the structures of logarithmic representations includes the determination of their parameters, sometimes called logarithmic couplings. We will find explicit expressions for these parameters, which not only hold at generic central charge, but also seem to make sense at rational central charge, as we will show in examples. (See Section \[sec:2pt\].)
Our logarithmic structures may seem somewhat speculative, as they rely on using derivative fields in theories with discrete spectrums, and assuming the existence of degenerate fields. In order to validate our techniques and assumptions, we will first show that our constructions make sense in the context of Liouville theory, whose continuous spectrum allows us to differentiate physical fields, and where degenerate fields are known to exist. (See Section \[sec:liou\].) Then we will bootstrap four-point connectivities in the $Q$-state Potts model. A recent attempt at bootstrapping these connectivities was broadly successful [@hjs20], although its accuracy was limited by the presence of unknown logarithmic contributions. We will overcome this limitation, and solve crossing symmetry equations to a high accuracy. (See Section \[sec:cipm\].) Our Python code for computing four-point connectivities is available at GitLab [@bV2].
According to forthcoming work by Gräns-Samuelsson et al, predictions from the lattice discretization of the $Q$-state Potts model also seem to converge towards the same logarithmic structures [@glhj+20].
Logarithmic fields as derivatives of primary fields
===================================================
Primary fields play a central role in conformal field theory, because they generate the simplest and most common representations of the conformal algebra: Verma modules and quotients thereof. In such representations, the dilation generator is diagonalizable. We will now investigate logarithmic representations, where by definition the dilation generator is not diagonalizable.
We will build logarithmic representations from derivatives of primary fields with respect to the conformal dimension. This approach is technically convenient, because it allows us to easily compute correlation functions, conformal blocks and operator product expansions: since these objects are solutions of linear Ward identities, their derivatives are solutions as well. In two dimensions, this approach has the added advantage of guaranteeing the single-valuedness of correlation functions: if we separately studied left- and right-moving fields and representations, we would face the additional problem of combining them in a single-valued way, in other words of building bulk CFT from chiral CFT.
From the derivative of a primary field, we build a logarithmic representation whose structure depends on the primary field’s properties. We will first focus on the simplest case of a non-degenerate primary field, i.e. a primary field that generates an irreducible Verma module. This case is well-known, and has even been investigated in higher-dimensional CFT [@hpv16]. We will then move on to the case where a null vector (= a singular vector) is present. This case has already been investigated in two dimensions [@kr09] and higher dimensions [@bh16]. Things cannot get more complicated at generic central charge, as representations with several null vectors only appear at rational central charge.
Derivatives of primary fields {#sec:dpf}
-----------------------------
### Jordan blocks from derivatives
Let us start with a primary field $V_\Delta$ with the conformal dimension $\Delta$. By definition, this is a field on which the dilation generator $L_0$ and annihilation modes $L_{n>0}$ act as $$\begin{aligned}
L_0 V_\Delta &= \Delta V_\Delta\ ,
\label{lzv}
\\
L_{n>0} V_\Delta &= 0 \ .\end{aligned}$$ Our notations for the symmetry generators $L_n$ come from the Virasoro algebra which is relevant to the two-dimensional case. Let us take $\Delta$-derivatives of these equations, while considering the operators $L_0,L_{n>0}$ as $\Delta$-independent. Let $$\begin{aligned}
\hat V_\Delta = \left[\frac{1}{n!} V_\Delta^{(n)}, \cdots , \frac12 V_\Delta'',V_\Delta',V_\Delta\right]^T
\label{hvd}\end{aligned}$$ be the vector of derivatives of $V_\Delta$ up to $V_\Delta^{(n)}$. We then have $$\begin{aligned}
\hat L_0 \hat V_\Delta = \hat\Delta \hat V_\Delta \ ,
\\
\hat L_{n>0} \hat V_\Delta = 0\ , \end{aligned}$$ where $\hat L_n$ is the diagonal matrix with $L_n$ on the diagonal, and we introduce $$\begin{aligned}
\hat\Delta = \left[\begin{array}{ccccc}\Delta & 1 & 0 & \cdots & 0
\\ 0 & \Delta & 1 & \ddots & 0
\\ 0 & 0 & \Delta & \ddots & 0
\\ \vdots & \ddots & \ddots & \ddots & 1
\\ 0 & 0 & 0 & 0 & \Delta
\end{array} \right]\ .
\label{hdelta}\end{aligned}$$ This describes a Jordan block of dimension $n+1$.
The primary field $V_\Delta$ is defined up to a $\Delta$-dependent normalization factor, and this leads to ambiguities in the definition of $V_\Delta'$ too: $$\begin{aligned}
V_\Delta \to \lambda(\Delta) V_\Delta \quad \implies \quad V'_\Delta \to \lambda(\Delta)V'_\Delta + \lambda'(\Delta) V_\Delta\ .
\label{vlv}\end{aligned}$$ More generally, in the module generated by $V_\Delta^{(n)}$, this change of normalization leads to a change of bases that preserves the action of $L_0$. A change of bases is not a change of structure, and the module has no free parameters.
### Two-dimensional case: diagonal primary fields
In two dimensions, the conformal algebra factorizes into a product of two Virasoro algebras, called left-moving and right-moving. A field that is primary for both Virasoro algebras is characterized by a left-moving and a right-moving conformal dimensions called $\Delta$ and $\bar\Delta$. Single-valuedness of correlation functions on the sphere requires $\Delta-\bar\Delta \in \frac12 \mathbb{Z}$. (See [@rib14] for a review.) The simplest way to fulfil this constraint is to consider diagonal fields, i.e. fields with $\Delta =\bar \Delta$.
There are known conformal field theories, such as Liouville theory, that involve diagonal primary fields whose dimensions can vary continuously. This encourages us to consider derivatives of diagonal primary fields. On the other hand, we do not know of any theory that would involve non-diagonal primary fields with continuously varying dimensions. We will therefore refrain from building logarithmic representations from derivatives of non-diagonal primary fields.
From now on, $V_\Delta$ will denote a diagonal primary field whose left and right dimensions are both $\Delta$. Then the derivative field $V'_\Delta$ obeys $$\begin{aligned}
L_0 V'_\Delta = \bar L_0 V'_\Delta = \Delta V'_\Delta + V_\Delta\ .
\label{lvp}\end{aligned}$$ It follows that the representation generated by $V'_\Delta$ cannot be factorizable. To prove this, let us concentrate on the two-dimensional subspace $\text{Span}(V_\Delta,V'_\Delta)$, which we view as a representation of the subalgebra $\text{Span}(L_0,\bar L_0)$. If this subspace could be factorized as a tensor product of representations of $L_0$ and $\bar L_0$, one of the two factors would have dimension one, and $V'_\Delta$ would be an eigenvector of either $L_0$ or $\bar L_0$.
Derivatives of null fields {#sec:dnf}
--------------------------
### Null fields
Let us rewrite the central charge $c$ in terms of a coupling constant $\beta$, and the conformal dimension $\Delta$ in terms of a momentum $P$: $$\begin{aligned}
c = 1 - 6\left(\beta -\frac{1}{\beta}\right)^2 \quad , \quad \Delta = \frac{c-1}{24} + P^2\ .
\label{dp}\end{aligned}$$ The condition that the Verma module $\mathcal{V}_\Delta$ with the conformal dimension $\Delta$ has a null vector is $$\begin{aligned}
\mathcal{V}_\Delta \text{ has a null vector } \iff \ \Delta\in \left\{\Delta_{(r, s)}\right\}_{r,s\in\mathbb{N}^*} \ ,
\label{hanv}\end{aligned}$$ where the degenerate dimensions $\Delta_{(r,s)}$ correspond to the momentums $$\begin{aligned}
P_{(r, s)} = \frac12 \left(\beta r - \beta^{-1} s\right)\ .
\label{prs}\end{aligned}$$ We will use the notation $V_{(r,s)} = V_{\Delta_{(r,s)}}$ for a diagonal primary field of dimension $\Delta_{(r,s)}$. Let us write the null fields in the corresponding Verma module as $\mathcal{L}V_{(r,s)}=\mathcal{L}_{(r,s)}V_{(r,s)}$, where $\mathcal{L}_{(r,s)}$ is a creation operator. For example, $\mathcal{L}_{(1,1)} = L_{-1}$ and $\mathcal{L}_{(2,1)} = L_{-1}^2 - \beta^2L_{-2}$. The same primary field also has a right-moving null descendant $\bar{\mathcal{L}}V_{(r,s)}$. Then $\mathcal{L}\bar{\mathcal{L}}V_{(r,s)}$ is a diagonal primary field of dimension $\Delta_{(r, -s)}$, and we make the identification $$\begin{aligned}
V_{(r, -s)} = \mathcal{L}\bar{\mathcal{L}}V_{(r,s)} \ .
\label{vllv}\end{aligned}$$ Null fields can consistently be set to zero, and they do vanish in CFTs such as minimal models. However they do not have to vanish, and we assume that our null fields do not vanish, in particular $\mathcal{L}\bar{\mathcal{L}}V_{(r,s)}\neq 0$. We plot the four primary fields $V_{(r, s)},\mathcal{L}V_{(r,s)}, \bar{\mathcal{L}}V_{(r,s)}, V_{(r,-s)} $ according to their left and right conformal dimensions: $$\begin{aligned}
\begin{tikzpicture}[baseline=(current bounding box.center), scale = .45]
\filldraw[red!10] (0, -4) -- (9, -13) -- (-9, -13) -- cycle;
\draw[latex-latex] (-13, -13) node[above left]{$\Delta$} -- (0, 0) -- (13, -13) node[above right]{$\bar\Delta$};
\filldraw (0, -4) circle (.2);
\filldraw (-4, -8) circle (.2);
\filldraw (4, -8) circle (.2);
\filldraw (0, -12) circle (.2);
\draw[-latex, red, ultra thick] (-.5, -4.5) -- (-3.5, -7.5);
\node at (-2.8, -5.2) {$\mathcal{L}_{(r,s)}$};
\draw[-latex, red, ultra thick] (3.5, -8.5) -- (.5, -11.5);
\node at (3, -10.6) {$\mathcal{L}_{(r,s)}$};
\draw[-latex, red, ultra thick] (.5, -4.5) -- (3.5, -7.5);
\node at (2.8, -5.2) {$\bar{\mathcal{L}}_{(r,s)}$};
\draw[-latex, red, ultra thick] (-3.5, -8.5) -- (-.5, -11.5);
\node at (-3, -10.6) {$\bar{\mathcal{L}}_{(r,s)}$};
\node at (0, -3) {$V_{(r,s)}$};
\node at (0, -13) {$V_{(r,-s)}$};
\node[left] at (-4.1, -8) {$\mathcal{L}V_{(r,s)}$};
\node[right] at (4.1, -8) {$\bar{\mathcal{L}}V_{(r,s)}$};
\draw (1.8, -2.2) node[above right]{$\Delta_{(r,s)}$} -- (2.2, -1.8);
\draw (5.8, -6.2) node[above right]{$\Delta_{(r,-s)}$} -- (6.2, -5.8);
\draw (-1.8, -2.2) node[above left]{$\Delta_{(r,s)}$} -- (-2.2, -1.8);
\draw (-5.8, -6.2) node[above left]{$\Delta_{(r,-s)}$} -- (-6.2, -5.8);
\end{tikzpicture}\end{aligned}$$
### Combinations of derivatives
We have considered a representation that contains two diagonal primary fields, namely $V_{(r, s)}$ and $V_{(r, -s)}$. From their derivatives $V'_{(r, s)}$ or $V'_{(r, -s)}$, we could generate logarithmic representations that would not differ much from the representations from Section \[sec:dpf\]. To build something new, we introduce the linear combination $$\begin{aligned}
\boxed{W^\kappa_{(r, s)} = (1-\kappa) V'_{(r, -s)} + \kappa \mathcal{L}\bar{\mathcal{L}}V'_{(r,s)}} \ .
\label{wk}\end{aligned}$$ We call $\mathcal{W}^\kappa_{(r,s)}$ the representation of the product of two Virasoro algebras that is generated by the field $W^\kappa_{(r, s)}$. We have fixed the sum of the coefficients to one by a choice of normalization, but $\kappa$ is a normalization-independent parameter of the representation.
Let us investigate the properties of the representation $\mathcal{W}^\kappa_{(r,s)}$. We first compute $$\begin{aligned}
\left(L_0-\Delta_{(r,-s)}\right) W^\kappa_{(r, s)} = \left(\bar L_0-\Delta_{(r,-s)}\right) W^\kappa_{(r, s)} = V_{(r, -s)}\ , \end{aligned}$$ where we used the equations and . This shows that the representation $\mathcal{W}^\kappa_{(r,s)}$ is logarithmic for any finite value of $\kappa$. On the other hand, $W^\infty_{(r,s)}$ is an eigenvector of $L_0$, and generates a non-logarithmic representation.
From the field $W^\kappa_{(r, s)}$, we can climb back to the primary fields $\mathcal{L}V_{(r,s)}$ and $\bar{\mathcal{L}}V_{(r,s)}$ using annihilation operators. In order to demonstrate this explicitly, let us consider an annihilation operator $\mathcal{D}$ of degree $rs$ (for example $\mathcal{D}=L_1^{rs}$), and define a polynomial $P_{\mathcal{D}}(\Delta)$ such that $$\begin{aligned}
\mathcal{D}\mathcal{L}_{(r,s)}V_\Delta = P_{\mathcal{D}}(\Delta) V_\Delta\ .
\label{dlv}\end{aligned}$$ Since $\mathcal{D}\mathcal{L}V_{(r,s)}=0$, we have $$\begin{aligned}
P_{\mathcal{D}}(\Delta_{(r,s)}) = 0\ .\end{aligned}$$ This zero of the polynomial $P_{\mathcal{D}}(\Delta)$ is simple if the central charge is generic. Differentiating with respect to $\Delta$, we then find $$\begin{aligned}
\frac{1}{P'_{\mathcal{D}}(\Delta_{(r,s)})} \mathcal{D}\mathcal{L}V_{(r,s)}' = V_{(r, s)}\ .\end{aligned}$$ For simplicity, we now assume that $\mathcal{D}$ is normalized such that $$\begin{aligned}
P'_{\mathcal{D}}(\Delta_{(r,s)})=1\ .
\label{ppd}\end{aligned}$$ We then have $$\begin{aligned}
\mathcal{D} W^\kappa_{(r, s)} = \kappa\bar{\mathcal{L}}V_{(r,s)} \quad , \quad \bar{\mathcal{D}} W^\kappa_{(r, s)} = \kappa\mathcal{L}V_{(r,s)}\ .
\label{dwk}\end{aligned}$$ We deduce closed equations for $W^\kappa_{(r, s)}$, $$\begin{aligned}
\boxed{\mathcal{L}_{(r,s)}\mathcal{D} W^\kappa_{(r, s)} = \bar{\mathcal{L}}_{(r,s)}\bar{\mathcal{D}} W^\kappa_{(r, s)}
= \kappa\left(L_0-\Delta_{(r,-s)}\right) W^\kappa_{(r, s)} = \kappa\left(\bar L_0-\Delta_{(r,-s)}\right) W^\kappa_{(r, s)}} \ .
\label{kalg}\end{aligned}$$ These equations provide an algebraic definition of the parameter $\kappa$, which depends neither on the normalization of $\mathcal{L}_{(r,s)}$ nor on the choice of the annihilation operator $\mathcal{D}$, provided the condition is obeyed. It is also possible to write an algebraic definition of $W^\kappa_{(r, s)}$ itself: in addition to Eq. , we would have to prescribe how general annihilation operators act on $W^\kappa_{(r, s)}$.
Let us plot the resulting representation $\mathcal{W}^\kappa_{(r,s)}$, with black dots for primary fields, and a blue circle for $W^\kappa_{(r,s)}$: $$\begin{aligned}
\begin{tikzpicture}[baseline=(current bounding box.center), scale = .55]
\filldraw[red!10] (4, -8) -- (9, -13) -- (-9, -13) -- (-4, -8) -- (0, -12) -- cycle;
\filldraw (-4, -8) circle (.2);
\filldraw (4, -8) circle (.2);
\filldraw (0, -12) circle (.2);
\draw[blue, thick] (0, -12) circle (.35);
\draw[-latex, red, ultra thick] (3.5, -8.5) -- (.5, -11.5);
\node at (3, -10.6) {$\mathcal{L}_{(r,s)}$};
\draw[-latex, red, ultra thick] (-3.5, -8.5) -- (-.5, -11.5);
\node at (-3, -10.6) {$\bar{\mathcal{L}}_{(r,s)}$};
\draw[-latex, blue, ultra thick] (.3, -11.3) -- (3.3, -8.3);
\node[blue] at (1.2, -9.6) {$\mathcal{D}$};
\draw[-latex, blue, ultra thick] (-.3, -11.3) -- (-3.3, -8.3);
\node[blue] at (-1.2, -9.6) {$\bar{\mathcal{D}}$};
\node at (0, -13.1) {$V_{(r,-s)},W^\kappa_{(r, s)}$};
\node[left] at (-4.1, -8) {$\mathcal{L}V_{(r,s)}$};
\node[right] at (4.1, -8) {$\bar{\mathcal{L}}V_{(r,s)}$};
\end{tikzpicture}
\label{picw}\end{aligned}$$ Although the diagonal primary field $V_{(r,s)}$ does not belong to $\mathcal{W}^\kappa_{(r,s)}$, we use the notations $\mathcal{L}V_{(r,s)},\bar{\mathcal{L}}V_{(r,s)}$ for two non-diagonal primary fields that do. The subrepresentation that they generate can be built by joining the Verma modules generated by each field along their common submodule, which is generated by the diagonal primary field $V_{(r,-s)}$: $$\begin{aligned}
\frac{\mathcal{V}_{(r, s)}\otimes \bar{\mathcal{V}}_{(r, -s)}
\oplus \mathcal{V}_{(r, -s )}\otimes \bar{\mathcal{V}}_{(r, s)}}{
\mathcal{V}_{(r, -s)}\otimes \bar{\mathcal{V}}_{(r, -s)}} \subset \mathcal{W}^\kappa_{(r,s)} \ .\end{aligned}$$ Finally, let us consider changes of normalization . In order to preserve the structure of the representation, a change of normalization must not change $\kappa$, equivalently it must be compatible with Eq. , i.e. $$\begin{aligned}
\lambda(\Delta_{(r,s)})=\lambda(\Delta_{(r,-s)}) \ .
\label{ldld}\end{aligned}$$ If this constraint is respected, our first derivative field changes as $$\begin{aligned}
W^\kappa_{(r, s)}\to \lambda(\Delta_{(r,s)}) W^\kappa_{(r,s)} + \left[(1-\kappa)\lambda'(\Delta_{(r,-s)})+\kappa \lambda'(\Delta_{(r,s)})\right] V_{(r,-s)}\ ,
\label{normW}\end{aligned}$$ which amounts to a change of bases of the representation $\mathcal{W}^\kappa_{(r,s)}$.
### Degenerate fields and special values of $\kappa$
We will now argue that two special values of $\kappa$ are singled out by degenerate fields. By a degenerate field we mean a diagonal primary field with not only a degenerate dimension of the type , but also such that the left and right null vectors vanish. We will use the notation $V_{\langle r_0,s_0\rangle}$ for the degenerate field of conformal dimension $\Delta_{(r_0,s_0)}$.
Operator product expansions involving degenerate fields are constrained by fusion rules, which are simpler when written in terms of the momentum rather than the conformal dimension: $$\begin{aligned}
V_{\langle r_0,s_0\rangle} V_P \sim \sum_{i=-\frac{r_0-1}{2}}^{\frac{r_0-1}{2}} \sum_{j=-\frac{s_0-1}{2}}^\frac{s_0-1}{2} V_{P+i\beta +j\beta^{-1}}\ ,
\label{vrsvp}\end{aligned}$$ where the sums run by increments of $1$. Given $r,s\in\mathbb{N}^*$, we now consider a degenerate field of the type $V_{\langle 1, s_0\rangle}$ and a diagonal primary field $V_{P_{(r,0)}}$ such that their OPE includes the two primary field $V_{(r,\pm s)}$. (This happens if $s_0\in s+1+2\mathbb{N}$.) We consider the OPE $$\begin{gathered}
V_{\langle 1, s_0\rangle} V_{P_{(r,0)}+\epsilon}
= C_1(\epsilon)\Big[ 1+ f(\epsilon)\mathcal{L}_{(r,s)} + f(\epsilon) \bar{\mathcal{L}}_{(r,s)} + f(\epsilon)^2 \mathcal{L}_{(r,s)}\bar{\mathcal{L}}_{(r,s)}\Big] V_{P_{(r,s)}+\epsilon}
\\
+ C_2(\epsilon) V_{P_{(r,-s)}+\epsilon} + \cdots \ .
\label{vvp}\end{gathered}$$ Here $C_i(\epsilon)$ and $f(\epsilon)$ are OPE coefficients, which also depend on the fields’ positions. We omit contributions of other primary fields, and of all descendant fields except for the three $\mathcal{L}_{(r,s)},\bar{\mathcal{L}}_{(r,s)}$ and $\mathcal{L}_{(r,s)}\bar{\mathcal{L}}_{(r,s)}$-descendants, which become null vectors as $\epsilon\to 0$. Therefore, the coefficient $f(\epsilon)$ has a simple pole at $\epsilon=0$.
The behaviour of our OPE will now follow from two facts:
- *The OPE is finite.* This is a consequence of the associativity of the double OPE $V_{\langle 1, s_0\rangle} V_{P_{(r,0)}+\epsilon}V_{P'}$ for a generic $P'$.
- *$C_1(\epsilon)$ has a simple zero at $\epsilon=0$.* The coefficients $C_1(\epsilon), C_2(\epsilon)$ are determined by crossing symmetry of $\left< V_{P_{(r,0)}+\epsilon} V_{\langle 1, s_0\rangle} V_{P_{(r,0)}+\epsilon} V_{\langle 1, s_0\rangle} \right>$ via standard analytic bootstrap methods [@rib14]. They do not depend from the particular CFT we are considering, provided we impose the normalization condition . One way to determine their behaviour is to read it from their known expressions in Liouville theory, see Section \[sec:liou\].
We deduce that the first term of the OPE has a simple pole, which must cancel with a simple pole of the second term. This implies that $C_2(\epsilon)$ has a simple pole, such that $$\begin{aligned}
\lim_{\epsilon\to 0} \frac{C_1(\epsilon)f(\epsilon)^2}{C_2(\epsilon)} = -1 \ .
\label{cfc}\end{aligned}$$ It follows that the leading behaviour of the terms includes a contribution from the derivative field $$\begin{aligned}
W_{(r, s)}^- = \partial_P V_{P_{(r, -s)}} - \mathcal{L}_{(r,s)}\bar{\mathcal{L}}_{(r,s)} \partial_P V_{P_{(r, s)}} \ .\end{aligned}$$ This is a special case of the derivative field $W_{(r, s)}^\kappa$ , whose value of $\kappa$ is found by translating $P$-derivatives into $\Delta$-derivatives, $$\begin{aligned}
\boxed{\kappa^-_{(r, s)} = \frac{P_{(r,s)}}{P_{(r, s)}-P_{(r,-s)}} = \frac{s-r\beta^2}{2s}}\ .
\label{km}\end{aligned}$$ Had we started with degenerate fields of the type $V_{\langle r_0, 1\rangle}$ instead of $V_{\langle 1, s_0\rangle}$, we would have found derivative fields of the type $$\begin{aligned}
W_{(r, s)}^+ = \partial_P V_{P_{(-r, s)}} - \mathcal{L}_{(r,s)}\bar{\mathcal{L}}_{(r,s)} \partial_P V_{P_{(r, s)}} \ ,\end{aligned}$$ whose parameter $\kappa$ is $$\begin{aligned}
\boxed{\kappa^+_{(r, s)} = \frac{P_{(r, s)}}{P_{(r, s)}-P_{(-r,s)}} = \frac{r-s\beta^{-2}}{2r}} \ .
\label{kp}\end{aligned}$$ In theories with both types of degenerate fields, both types of derivative fields can exist.
Second derivatives of null fields
---------------------------------
We will now consider second derivative fields. Like our first derivative fields $W^\kappa_{(r,s)}$, second derivative fields appear in OPEs of degenerate fields. Higher derivative fields only appear in subleading terms of these OPEs, and we will refrain from considering them.
### Combinations of second derivatives
We introduce the combination $$\begin{aligned}
\boxed{\widetilde{W}^\kappa_{(r, s)} = \frac{1-\kappa}{2} V''_{(r, -s)} + \frac{\kappa}{2} \mathcal{L}\bar{\mathcal{L}}V''_{(r,s)} }\ ,
\label{wkt}\end{aligned}$$ and we call $\widetilde{\mathcal{W}}^\kappa_{(r,s)}$ the corresponding representation of the product of two Virasoro algebras. The field $\widetilde{W}^\kappa_{(r, s)}$ obeys $$\begin{aligned}
&\left(L_0-\Delta_{(r,-s)}\right)\widetilde{W}^\kappa_{(r, s)}
= \left(\bar L_0-\Delta_{(r,-s)}\right) \widetilde{W}^\kappa_{(r, s)}
= W^\kappa_{(r,s)} \ ,
\\
&\left(L_0-\Delta_{(r,-s)}\right)^2\widetilde{W}^\kappa_{(r, s)}
= V_{(r, -s)}\ , \end{aligned}$$ Using an annihilation operator $\mathcal{D}$ normalized as in Eq. , and taking the second derivative of Eq., we obtain $$\begin{aligned}
\mathcal{D}\widetilde{W}^\kappa_{(r,s)}=\kappa \bar{\mathcal{L}}V'_{(r,s)} \quad , \quad
\bar{\mathcal{D}}\widetilde{W}^\kappa_{(r,s)}=\kappa \mathcal{L}V'_{(r,s)} \quad ,
\quad
\mathcal{D}\bar{\mathcal{D}}\widetilde{W}^\kappa_{(r,s)}= \kappa V_{(r,s)} \ .\end{aligned}$$ This leads to a closed equation for the field $\widetilde{W}^\kappa_{(r, s)} $, $$\begin{aligned}
\boxed{\mathcal{L}_{(r,s)}\bar{\mathcal{L}}_{(r,s)}
\mathcal{D}\bar{\mathcal{D}}\widetilde{W}^\kappa_{(r,s)} =
\kappa \left(L_0-\Delta_{(r,-s)}\right)^2 \widetilde{W}^\kappa_{(r, s)}} \ .\end{aligned}$$ This also leads to $$\begin{aligned}
\mathcal{L}_{(r,s)}\mathcal{D}\widetilde{W}^\kappa_{(r,s)}
=
\bar{\mathcal{L}}_{(r,s)}\bar{\mathcal{D}}\widetilde{W}^\kappa_{(r,s)}
=
\kappa W^1_{(r,s)}\ .\end{aligned}$$ Therefore, the representation $\widetilde{\mathcal{W}}^\kappa_{(r,s)}$ contains both fields $W^1_{(r,s)}$ and $W^\kappa_{(r,s)}$. By linearly combining these fields, we could obtain $W^{\kappa'}_{(r,s)}$ for any value of $\kappa'$. We collectively denote these fields as $W^*_{(r,s)}$ in the following plot, which uses the same conventions as the plot : $$\begin{aligned}
\begin{tikzpicture}[baseline=(current bounding box.center), scale = .55]
\filldraw[red!10] (0, -4) -- (9, -13) -- (-9, -13) -- cycle;
\filldraw (0, -4) circle (.2);
\filldraw (-4, -8) circle (.2);
\filldraw (4, -8) circle (.2);
\filldraw (0, -12) circle (.2);
\draw[-latex, red, ultra thick] (-.5, -4.5) -- (-3.5, -7.5);
\node at (-2.8, -5.2) {$\mathcal{L}_{(r,s)}$};
\draw[-latex, red, ultra thick] (3.5, -8.5) -- (.5, -11.5);
\node at (3, -10.6) {$\mathcal{L}_{(r,s)}$};
\draw[-latex, red, ultra thick] (.5, -4.5) -- (3.5, -7.5);
\node at (2.8, -5.2) {$\bar{\mathcal{L}}_{(r,s)}$};
\draw[-latex, red, ultra thick] (-3.5, -8.5) -- (-.5, -11.5);
\node at (-3, -10.6) {$\bar{\mathcal{L}}_{(r,s)}$};
\node at (0, -3) {$V_{(r,s)}$};
\draw[blue, thick] (0, -12) circle (.35);
\draw[blue,thick] (0, -12) circle (.5);
\draw[blue, thick] (-4, -8) circle (.35);
\draw[blue, thick] (4, -8) circle (.35);
\draw[-latex, blue, ultra thick] (.3, -11.3) -- (3.3, -8.3);
\draw[latex-, blue, ultra thick] (.3, -4.7) -- (3.3, -7.7);
\node[blue] at (1.2, -9.6) {$\mathcal{D}$};
\node[blue] at (-1.2, -6.4) {$\mathcal{D}$};
\draw[-latex, blue, ultra thick] (-.3, -11.3) -- (-3.3, -8.3);
\draw[latex-, blue, ultra thick] (-.3, -4.7) -- (-3.3, -7.7);
\node[blue] at (-1.2, -9.6) {$\bar{\mathcal{D}}$};
\node[blue] at (1.2, -6.4) {$\bar{\mathcal{D}}$};
\node at (0, -13.1) {$V_{(r,-s)}, W^*_{(r, s)}, \widetilde{W}^\kappa_{(r, s)}$};
\node[left] at (-4.2, -8) {$\mathcal{L}V_{(r,s)},\mathcal{L}V'_{(r,s)}$};
\node[right] at (4.2, -8) {$\bar{\mathcal{L}}V_{(r,s)},\bar{\mathcal{L}}V'_{(r,s)}$};
\end{tikzpicture}
\label{twpic}\end{aligned}$$ In terms of subrepresentations, we have $$\begin{aligned}
\mathcal{V}_{(r, s)}\otimes \bar{\mathcal{V}}_{(r, s)}
\subset \widetilde{\mathcal{W}}^\kappa_{(r,s)} \quad , \quad \mathcal{W}_{(r,s)}^* \subset \widetilde{\mathcal{W}}^\kappa_{(r,s)}\ .\end{aligned}$$ Under a change of normalization that respects Eq. , the second derivative field changes as $$\begin{gathered}
\widetilde{W}^\kappa_{(r, s)} \to \lambda(\Delta_{(r,s)}) \widetilde{W}^\kappa_{(r, s)} + (1-\kappa)\lambda'(\Delta_{(r,-s)}) V'_{(r,-s)} +\kappa\lambda'(\Delta_{(r,s)}) \mathcal{L}\bar{\mathcal{L}}V'_{(r,s)}
\\
+\frac12\left[(1-\kappa)\lambda''(\Delta_{(r,-s)})+\kappa \lambda''(\Delta_{(r,s)})\right] V_{(r,-s)}\ .
\label{wtt}\end{gathered}$$ In particular, there appear fields $W^{\kappa'}_{(r,s)}$ with parameters $\kappa'$ that depend on the function $\lambda(\Delta)$. Since all these fields belong to the representation $\widetilde{\mathcal{W}}^\kappa_{(r,s)}$, changes of normalization amount to changes of bases.
### Degenerate fields and special values of $\kappa$
Given $r,s\in\mathbb{N}^*$, let us consider a momentum $P_0$ such that the degenerate fusion rule for the OPE $V_{\langle r_0,s_0\rangle}V_{P_0}$ allows the four field $V_{P_{(\pm r,\pm s)}}$. For simplicity, we choose $P_0=P_{(0,0)}=0$ and assume $r_0\in r+1+2\mathbb{N},s_0\in s+1+2\mathbb{N}$, but the results do not depend on this choice. We focus on the behaviour of a few terms in the OPE $V_{\langle r_0,s_0\rangle}V_{\epsilon}$ as $\epsilon\to 0$, $$\begin{gathered}
V_{\langle r_0,s_0\rangle} V_\epsilon = \sum_\pm \Bigg\{C_1(\epsilon)\Big[1+ f(\epsilon)\mathcal{L}_{(r,s)} + f(\epsilon) \bar{\mathcal{L}}_{(r,s)} + f(\epsilon)^2 \mathcal{L}_{(r,s)}\bar{\mathcal{L}}_{(r,s)}\Big] V_{P_{(r,s)}\pm \epsilon}
\\
+ C_2(\epsilon) V_{P_{(r,-s)}\pm \epsilon} \Bigg\} + \cdots\ ,\end{gathered}$$ where we assume that the fields are normalized such that $V_P=V_{-P}$, which implies that the coefficients $C_i(P),f(P)$ are even functions of $P$. As in the case with first derivatives, the leading terms of the OPE cancel. The proof of that cancellation still works, and Eq. is still valid for our coefficients $C_i(P),f(P)$, with the only difference that $C_1(\epsilon)$ now has a finite limit instead of a simple zero. The cancelling leading terms are now double poles; the simple poles also cancel by parity in $\epsilon$. The limit of our OPE therefore involves the double derivative field $$\begin{aligned}
\widetilde{W}_{(r, s)}^0 = \partial^2_P V_{P_{(r, -s)}} - \mathcal{L}_{(r,s)}\bar{\mathcal{L}}_{(r,s)} \partial^2_P V_{P_{(r, s)}} \ .\end{aligned}$$ Transforming $P$-derivatives into $\Delta$-derivatives, and using the ambiguity for getting rid of first derivative terms, we find that $\widetilde{W}_{(r, s)}^0$ is a field of the type $\widetilde{W}_{(r, s)}^\kappa$ , for the special value of the parameter $$\begin{aligned}
\boxed{\kappa^0_{(r,s)} = \frac{P_{(r,s)}^2}{P_{(r,s)}^2-P_{(r,-s)}^2} = \frac12 - \frac{r}{4s}\beta^2 -\frac{s}{4r}\beta^{-2}}\ .
\label{kz}\end{aligned}$$
Correlation functions and conformal blocks
==========================================
Let us compute correlation functions of our logarithmic fields. We will pay particular attention to the cases of two-point functions and four-point conformal blocks on the sphere. Two point functions are simple observables that capture the structure of representations, including their parameters. Four-point functions are necessary and sufficient for establishing consistency of a CFT on the sphere.
Two-point functions {#sec:2pt}
-------------------
In contrast to higher correlation functions, two-point functions of derivative fields cannot be computed by just differentiating two-point functions of primary fields. In a CFT with continuously varying conformal dimensions, the two-point function involves a Dirac delta function $\left<V_{\Delta_1}(z_1)V_{\Delta_2}(z_2)\right> \propto \frac{\delta(\Delta_1-\Delta_2)}{|z_{12}|^{4\Delta_1}}$. While we can easily deduce the simple identities $$\begin{aligned}
\Delta_1\neq \Delta_2 \quad \implies \quad \left< V_{\Delta_1}^{(i)} V_{\Delta_2}^{(j)}\right> = 0 \ ,
\label{dnd}\end{aligned}$$ there is no well-defined procedure to recover two-point functions of $V_\Delta$ and its derivatives, which would require setting $\Delta_1=\Delta_2$.
Instead of just differentiating two-point functions, we therefore have to differentiate the Ward identities for the two-point functions, and solve the differentiated Ward identities [@F00].
### Derivatives of primary fields {#derivatives-of-primary-fields}
Let us determine the matrix $\hat B_\Delta$ of two-point functions of the derivatives of $V_\Delta$ up to order $n$: $$\begin{aligned}
\hat B_\Delta^{ij} = \left< \frac{1}{(n-i)!}V_\Delta^{(n-i)} \frac{1}{(n-j)!} V_\Delta^{(n-j)}\right> \ .\end{aligned}$$ Here and in the rest of Section \[sec:2pt\], the first field is at $z_1$ and the second field at $z_2$, i.e. we use the notation $$\begin{aligned}
\left<XY\right> = \left<X(z_1)Y(z_2)\right>\ .\end{aligned}$$ Two-point functions of arbitrary fields obey the global Ward identities [@rib14] $$\begin{aligned}
\left(\partial_{z_1}+\partial_{z_2}\right) \left<XY\right> & = 0 \ ,
\label{ward1}
\\
\left(z_1\partial_{z_1}+z_2\partial_{z_2}+L_0^{(X)} + L_0^{(Y)}\right) \left<XY\right> & = 0 \ ,
\label{ward2}
\\
\left(z_1^2\partial_{z_1}+z_2^2\partial_{z_2}+2z_1L_0^{(X)}+2z_2L_0^{(Y)}+L_1^{(X)}+L_1^{(Y)}\right) \left<XY\right> & = 0 \ ,
\label{ward3}\end{aligned}$$ where the $L_1$-terms vanish when the fields are primaries or derivatives thereof. In this case, the global Ward identities imply $(L_0^{(X)}-L_0^{(Y)})\left<XY\right>=0$. In matrix form, this equation reads $$\begin{aligned}
\hat \Delta \hat B_\Delta = \hat B_\Delta \hat \Delta^T \ , \end{aligned}$$ where $\hat \Delta$ is the matrix form of the action of $L_0$ on the vector of derivative field $\hat V_\Delta$ . The general solution of this equation is $$\begin{aligned}
\hat B_\Delta = f(\hat \Delta) \hat B_0 = \hat B_0 f(\hat \Delta^T)\ ,\end{aligned}$$ where we introduced $$\begin{aligned}
\hat B_0 = \begin{bmatrix}
b_{n} & b_{n-1}& \ldots && b_{0}
\\
b_{n-1}&\iddots && b_{0} &0
\\
\vdots & & \iddots && \vdots
\\
&b_{0} &&&
\\
b_{0}& 0 &\cdots & & 0
\end{bmatrix} \ ,\end{aligned}$$ for some coefficients $b_0,\ldots, b_{n}$, and the function $f$ is undetermined. To determine it, we use the rest of the Ward identities, and find $f(\Delta)=|z_{12}|^{-4\Delta}$, plus the requirement that the coefficients $b_i$ are $z_k$-independent. For example, in the case $n=1$, we recover the well-known result [@CR13] $$\begin{aligned}
\renewcommand{\arraystretch}{1.5}
\begin{bmatrix}
\left< V_\Delta'V_\Delta'\right> & \left<V_\Delta'V_\Delta\right>
\\
\left<V_\Delta V_\Delta' \right> & \left< V_\Delta V_\Delta \right>
\end{bmatrix}
=
\frac{1}{|z_{12}|^{4\Delta}}
\begin{bmatrix}
b_1-b_0\log|z_{12}|^4 & b_0 \\
b_0 & 0
\end{bmatrix}
\ .
\label{kele}\end{aligned}$$ In the case $n=2$, the result is $$\begin{gathered}
\renewcommand{\arraystretch}{1.5}
\begin{bmatrix}
\left<\frac12 V_\Delta'' \frac12 V_\Delta''\right> & \left<\frac12 V_\Delta'' V_\Delta'\right> & \left<\frac12 V_\Delta'' V_\Delta\right>
\\
\left< V_\Delta' \frac12 V_\Delta''\right> & \left< V_\Delta' V_\Delta' \right> & \left<V_\Delta' V_\Delta\right>
\\
\left< V_\Delta \frac12 V_\Delta''\right> & \left< V_\Delta V_\Delta' \right> & \left<V_\Delta V_\Delta\right>
\end{bmatrix}
\\
=
\frac{1}{|z_{12}|^{4\Delta} }
\begin{bmatrix}
b_2 -b_1\log|z_{12}|^4 +\tfrac12 b_0 \left(\log|z_{12}|^4\right)^2 & b_1-b_0\log|z_{12}|^4 & b_0
\\
b_1-b_0\log|z_{12}|^4 & b_0 & 0
\\
b_0 & 0 & 0
\end{bmatrix}\ .\end{gathered}$$ Notice that the existence of derivative fields constrains the two-point functions of the original primary field to vanish. More generally, $i+j<n \implies \left< V_\Delta^{(i)} V_\Delta^{(j)}\right> =0$.
Recall that two-point functions of descendants of a primary field $V_\Delta$ obey the identity $$\begin{aligned}
\Big< \mathcal{L}_1 V_\Delta \mathcal{L}_2V_\Delta\Big>
= (-1)^{|\mathcal{L}_1|+|\mathcal{L}_2|} \Big< \mathcal{L}_2 V_\Delta \mathcal{L}_1V_\Delta\Big>\ ,
\label{lvlv}\end{aligned}$$ where $\mathcal{L}_1$ and $\mathcal{L}_2$ are creation operators, and $|\mathcal{L}|$ is the level of $\mathcal{L}$, in particular $|\mathcal{L}_{(r,s)}|=rs$. For derivative fields, we observe $\left< V_\Delta^{(i)} V_\Delta^{(j)}\right> = \left< V_\Delta^{(j)} V_\Delta^{(i)}\right>$. Therefore, the identity still holds if $\mathcal{L}_1,\mathcal{L}_2$ are combinations of creation operators and derivatives with respect to $\Delta$.
### Null fields
Since the field $V_{(r,s)}$ and its null descendant $\mathcal{L}V_{(r,s)}$ are two primary fields of different dimensions, their two-point function vanishes, and so do two-point functions of their descendants. In particular, $$\begin{aligned}
\left<V_{(r,s)} \mathcal{L}V_{(r,s)} \right> = \left<\mathcal{L}V_{(r,s)} \mathcal{L}V_{(r,s)} \right> = 0 \ . \end{aligned}$$ The non-trivial and non-vanishing quantities that will appear in two-point functions of derivative fields are $$\begin{aligned}
\rho_{(r,s)} &= \left.\frac{\partial}{\partial\Delta}
\frac{\left<\mathcal{L}_{(r,s)}V_\Delta \mathcal{L}_{(r,s)}V_\Delta \right>}{z_{12}^{-rs}z_{21}^{-rs}\left<V_\Delta V_\Delta\right>}\right|_{\Delta=\Delta_{(r,s)}}
=
2\frac{\left<\mathcal{L}V_{(r,s)} \mathcal{L}V'_{(r,s)} \right>}{z_{12}^{-rs}z_{21}^{-rs}\left<V_{(r,s)} V_{(r,s)}\right>}
\ ,
\label{rhodef}
\\
\omega_{(r,s)} &= \left.\frac{\partial}{\partial\Delta}
\frac{\left<V_\Delta \mathcal{L}_{(r,s)}V_\Delta \right>}{z_{12}^{-rs}\left<V_\Delta V_\Delta\right>}\right|_{\Delta=\Delta_{(r,s)}}
=
\frac{\left<V_{(r,s)} \mathcal{L}V'_{(r,s)} \right>}{z_{12}^{-rs}\left<V_{(r,s)} V_{(r,s)}\right>}
\ .
\label{omegadef}\end{aligned}$$ The second expression for $\rho_{(r,s)}$ directly follows from the first expression, and from the identity $\left<\mathcal{L}_{(r,s)}V'_\Delta \mathcal{L}_{(r,s)}V_\Delta\right> =\left<\mathcal{L}_{(r,s)}V_\Delta \mathcal{L}_{(r,s)}V'_\Delta\right>$ which follows from Eq. . Similarly, the second expression for $\omega_{(r,s)}$ follows from $$\begin{aligned}
\left.\frac{\partial}{\partial\Delta}
\left<V_\Delta \mathcal{L}_{(r,s)}V_\Delta \right>\right|_{\Delta=\Delta_{(r,s)}}
= \left<V'_{(r,s)} \mathcal{L}V_{(r,s)} \right>
+ \left<V_{(r,s)} \mathcal{L}V'_{(r,s)} \right>
\ ,\end{aligned}$$ where the first term vanishes due to Eq. . One may worry that we are using the field $V'_{(r,s)}$, whose existence in principle implies $\left<V_{(r,s)}V_{(r,s)}\right>=0$. However, the particular correlation function where we insert $V'_{(r,s)}$ is not related to $\left<V_{(r,s)}V_{(r,s)}\right>$ by Ward identities, so there is no inconsistency.
The quantity $\rho_{(r,s)}$ is actually well-known, and it coincides with the denominator of the conformal block’s residue at $\Delta =\Delta_{(r,s)}$. Assuming the normalization $$\begin{aligned}
\mathcal{L}_{(r,s)} = L_{-1}^{rs} + \cdots \ ,
\label{llrs}\end{aligned}$$ we have [@zam03; @zam84] $$\begin{aligned}
\rho_{(r,s)} = -\frac{\prod^r_{i=1-r}\prod^s_{j=1-s}2P_{(i,j)}}{2P_{(0,0)}P_{(r,s)}}\ .
\label{rrs}\end{aligned}$$ The quantity $\omega_{(r,s)}$ does not seem so well-known. Computer-assisted calculations of examples with $rs\leq 20$ suggest $$\begin{aligned}
\omega_{(r,s)} = \frac{2P_{(r+1,s+1)}}{2P_{(1,1)}}
\frac{\prod_{i=0}^r\prod_{j=0}^s 2P_{(i,j)}}{2P_{(0,0)}2P_{(r,0)}2P_{(0,s)}2P_{(r,s)}}
\prod_{i=1}^{r-1}\prod_{j=1}^{s-1} 2P_{(i,j)} \ .\end{aligned}$$ In practice, it is convenient to repeatedly use the global Ward identity , which reduces $\omega_{(r,s)}$ to the purely algebraic quantity $$\begin{aligned}
\omega_{(r,s)} = \frac{P'_{L_1^{rs},(r,s)}(\Delta_{(r,s)})}{(rs)!}\ ,\end{aligned}$$ where the polynomial $P_{L_1^{rs},(r,s)}(\Delta)$ was defined in Eq. . In the formulas for $\rho_{(r,s)}$ and $\omega_{(r,s)}$, the total powers of $\Delta \sim P^2$ coincides with what we would expect from the heuristic rule $\mathcal{L}_{(r,s)}\propto \Delta^{rs}$.
### Derivatives of null fields {#derivatives-of-null-fields}
Let us compute two-point functions in the representation $\mathcal{W}^\kappa_{(r,s)}$ . We will focus on the representation-generating logarithmic field $W^\kappa_{(r,s)}$, and on the primary fields $V_{(r,-s)}$ and $\bar{\mathcal{L}}V_{(r,s)}$. (The primary field $\mathcal{L}V_{(r,s)}$ can be dealt with similarly.) We normalize the creation operator $\mathcal{L}_{(r,s)}$ via Eq. . Let us show that the two-point functions are of the type $$\begin{gathered}
\begin{bmatrix}
\left<W^\kappa_{(r,s)} W^\kappa_{(r,s)}\right> &
\left<W^\kappa_{(r,s)} V_{(r,-s)}\right> &
\left<W^\kappa_{(r,s)} \bar{\mathcal{L}}V_{(r,s)}\right>
\\
\left<V_{(r,-s)} W^\kappa_{(r,s)}\right> &
\left<V_{(r,-s)} V_{(r,-s)}\right> &
\left<V_{(r,-s)} \bar{\mathcal{L}}V_{(r,s)}\right>
\\
\left<\bar{\mathcal{L}}V_{(r,s)} W^\kappa_{(r,s)}\right> &
\left<\bar{\mathcal{L}}V_{(r,s)} V_{(r,-s)}\right> &
\left<\bar{\mathcal{L}}V_{(r,s)} \bar{\mathcal{L}}V_{(r,s)}\right>
\end{bmatrix}
\\
= \frac{1}{|z_{12}|^{4\Delta_{(r,-s)}}}
\begin{bmatrix}
b_1 -\log |z_{12}|^4 & 1 & \frac{2\omega_{(r,s)}}{\rho_{(r,s)}} z_{12}^{rs}
\\
1 & 0 & 0
\\
\frac{2\omega_{(r,s)}}{\rho_{(r,s)}} z_{21}^{rs} & 0 & \frac{2}{\kappa\rho_{(r,s)}} z_{12}^{rs}z_{21}^{rs}
\end{bmatrix}
\ .
\label{2j2}\end{gathered}$$ We start with the Jordan block of dimension $2$ whose basis is $(V_{(r,-s)}, W^\kappa_{(r,s)})$. Since $L_1W^\kappa_{(r,s)}\neq 0$, we may fear that we cannot directly apply Eq. . However, thanks to Eq. , the $L_1$ terms actually cancel in the Ward identiy , so that is applicable after all. This yields the top left submatrix of size two, where we have normalized $W^\kappa_{(r,s)}$ so that $b_0=1$, and $b_1$ cannot be determined as it is not invariant under changes of bases .
Starting from the definition of $\rho_{(r,s)}$ , let us insert the action of $\bar{\mathcal{L}}_{(r,s)}$. Noticing that the relations and imply $\left<V'_{(r,-s)}V_{(r,-s)}\right>=0$, we can rewrite the numerator as a two-point function of $W^\kappa_{(r,s)}$ , $$\begin{aligned}
\rho_{(r,s)} = \frac{2\left<W^\kappa_{(r,s)} V_{(r,-s)} \right>}{\kappa z_{12}^{-rs}z_{21}^{-rs}\left<\bar{\mathcal{L}}V_{(r,s)} \rule{0pt}{1em}\bar{\mathcal{L}}V_{(r,s)}\right>}\ .\end{aligned}$$ Next, we consider the definition of $\omega_{(r,s)}$ . Inserting again the action of $\bar{\mathcal{L}}_{(r,s)}$ on all involved fields, we obtain $$\begin{aligned}
\omega_{(r,s)} = \frac{\left<\bar{\mathcal{L}}V_{(r,s)} W^\kappa_{(r,s)} \right>}{\kappa z_{12}^{-rs}\left<\bar{\mathcal{L}}V_{(r,s)} \rule{0pt}{1em}\bar{\mathcal{L}}V_{(r,s)}\right>}\ , \end{aligned}$$ which completes the proof of Eq. .
Two-point functions of fields in the representation $\widetilde{W}^\kappa_{(r,s)}$ could be computed along the same lines. They would involve more complicated coefficients of the type of $\omega_{(r,s)}$ and $\rho_{(r,s)}$.
### Logarithmic couplings
Logarithmic representations are usually parametrized by a number called the logarithmic coupling [@CR13], which has to be related to our parameter $\kappa$. The main difference is that $\kappa$ is fully normalization-independent, while the logarithmic coupling is not. As a result, $\kappa$ is much simpler. On the other hand, the logarithmic coupling has the advantage that it can be directly read off from the two-point functions. Let us use this for relating it to $\kappa$.
In the particular normalization where $\mathcal{L}_{(r, s)} = L_{-1}^{rs} + \cdots $ and $\mathcal{D}=L_1^{rs}+\cdots$, the logarithmic coupling should coincide with the ratio of the $\log |z_{12}|^2$ term with the bottom right coefficient of the matrix two-point function . Calling $\gamma$ the logarithmic coupling (since the usual notation $\beta$ for that coupling already has another meaning for us), we therefore find $$\begin{aligned}
\boxed{\gamma = \rho_{(r,s)} \kappa}\ ,\end{aligned}$$ where $\rho_{(r,s)}$ should be considered a normalization prefactor.
Let us compare this with some known logarithmic couplings from logarithmic minimal models [@mr07]. We first focus on the logarithmic minimal model $LM(2, 3)$, which corresponds to $\beta^2 = \frac32$. Due to the existence of degenerate fields, the logarithmic coupling should be $\gamma^-_{(r,s)} = \rho_{(r,s)} \kappa_{(r,s)}^-$, where $\kappa^-_{(r,s)}$ is given in Eq. . We should beware of two subtleties
- The normalization in [@mr07] is not $\mathcal{L}_{(r, s)} = L_{-1}^{rs} + \cdots $ but $\mathcal{L}_{(r, s)} = L_{-rs} + \cdots $, so their couplings have to be multiplied with a normalization factor.
- Our labels $(r, s)$ are the indices of a non-vanishing null vector, whereas the labels in [@mr07] are the indices of a vanishing null vector.
Modulo these subtleties, we find that the logarithmic couplings agree: $$\begin{aligned}
\renewcommand{\arraystretch}{1.4}
\begin{array}{|c|c|c|}
\hline
\text{Coupling from \cite{mr07}(3.9)} & \text{Normalization factor} & \gamma^-_{(r,s)}
\\
\hline \hline
\beta_{1,4} = -\frac12 & 1 & \gamma^-_{(1,1)} = -\frac12
\\
\hline
\beta_{1,5} = -\frac58 & \frac49 & \gamma^-_{(1,2)} = -\frac{5}{18}
\\
\hline
\beta_{1,7} = -\frac{35}{3} & 36 & \gamma^-_{(3,1)} = -420
\\
\hline
\end{array}
\label{betagamma}\end{aligned}$$ Let us also discuss the vacuum module, which includes an identity field $V_{\langle 1,1\rangle}=V_{(1,2)}$ of dimension zero whose level one null vector vanishes, but whose level two null vector $\mathcal{L}_{(1,2)}V_{\langle 1,2\rangle}$ does not. The representation $\mathcal{W}^-_{(1,2)}$, which is characterized by the coupling $-\frac58$, obviously differs from the vacuum module, as it does not contain the identity field. Actually, the vacuum module should contain not only the identity field, but also a Jordan block of dimension $3$ [@rid12], so our module $\widetilde{\mathcal{W}}_{(1,2)}^0$ is a better candidate. The usual definition of the vacuum module’s logarithmic coupling is however not based on the Jordan block of dimension $3$, but on the Jordan block of dimension $2$ whose generators appear as $(\mathcal{L}V_{(r,s)},\mathcal{L}V_{(r,s)}'=W^1_{(r,s)})$ in the diagram . The $\kappa$-parameter for this Jordan block is not the parameter $\kappa^0_{(r,s)}$ of the full module, rather it is simply $\kappa=1$. (Notice that the identity [@rid12](4.6) just reads in our notations $1=\frac12(\frac{1}{\kappa_{(r,s)}^+}+\frac{1}{\kappa_{(r,s)}^-})$.) Therefore, the usually defined logarithmic coupling is just the normalization prefactor $\rho_{(1,2)}= -\frac{20}{9}$, divided by the normalization factor $\frac49$ from Table . The resulting coupling $-5$ is not a particularly exciting quantity, as it is made of nothing but normalizations, and ignores the structural parameter $\kappa^0_{(1,2)}=-\frac{1}{288}$.
There is also a conjecture for some of the logarithmic couplings of $LM(2,p)$ [@mr07](5.11), which corresponds to $\beta^2=\frac{p}{2}$. These couplings are $$\begin{aligned}
\hat\beta_{1,p+n}=(-1)^n\frac{p}{8}\prod_{i=-n}^{n-1}(p+2i) = \frac{p^{2n}}{4(n-1)!^2}\gamma^-_{(1,n)}\ ,\end{aligned}$$ where the prefactor of the last expression comes from a particular choice of normalization. This equality is valid for any $1\leq n< p$, i.e. whenever the formula for $\hat\beta_{1,p+n}$ holds. Our expression for $\gamma^-_{(1,n)}$ should also be valid for more general values of $n$.
Four-point conformal blocks {#sec:fpcb}
---------------------------
If we wanted to compute four-point functions of logarithmic fields, we would of course need the corresponding logarithmic conformal blocks. These would just be derivatives of non-logarithmic conformal blocks with respect to conformal dimensions. Computing such derivatives is straightforward, because conformal blocks are analytic functions of the fields’ dimensions.
However, in CFTs such as the Potts model, we are not that interested in correlation functions of logarithmic fields – fields whose existence and properties we are establishing only now. More interesting observables, which have been studied for a long time, are correlation functions of non-logarithmic fields fields, for example correlation functions of spin fields, or the cluster connectivities of Section \[sec:cipm\]. Logarithmic fields can appear as *channel fields* when we decompose such correlation functions into conformal blocks.
### Derivatives of primary fields {#derivatives-of-primary-fields-1}
Starting from a three-point function of diagonal primary fields, $$\begin{aligned}
\left<\prod_{i=1}^3 V_{\Delta_i}(z_i)\right>
= C_{\Delta_1,\Delta_2,\Delta_3}
|z_{12}|^{2(\Delta_3-\Delta_1-\Delta_2)} |z_{13}|^{2(\Delta_2-\Delta_1-\Delta_3)}
|z_{23}|^{2(\Delta_1-\Delta_2-\Delta_3)}\ ,\end{aligned}$$ we deduce a three-point function involving the derivative field $V'_{\Delta_1}(z_1)$, $$\begin{gathered}
\Big<V'_{\Delta_1}(z_1) V_{\Delta_2}(z_2)V_{\Delta_3}(z_3)\Big>
=
\Big(C'_{\Delta_1,\Delta_2,\Delta_3} - C_{\Delta_1,\Delta_2,\Delta_3} \log|z_{12}z_{13}|^2 \Big)
\\
\times
|z_{12}|^{2(\Delta_3-\Delta_1-\Delta_2)} |z_{13}|^{2(\Delta_2-\Delta_1-\Delta_3)}
|z_{23}|^{2(\Delta_1-\Delta_2-\Delta_3)}
\ .\end{gathered}$$ This shows that the coupling of the logarithmic field $V'_{\Delta_1}$ to two diagonal primary fields $V_{\Delta_2},V_{\Delta_3}$ involves two structure constants $C'_{\Delta_1,\Delta_2,\Delta_3}$ and $C_{\Delta_1,\Delta_2,\Delta_3}$. Let us now recall the decomposition of a four-point function into conformal blocks, $$\begin{aligned}
\left<\prod_{i=1}^4V_{\Delta_i}\right> = \int d\Delta\ C_{\Delta,\Delta_1,\Delta_2}C_{\Delta,\Delta_3,\Delta_4} \mathcal{F}_\Delta\bar{\mathcal{F}}_\Delta\ .
\label{4pt}\end{aligned}$$ Here we omit the dependence on positions $z_i$, which are spectators in our reasoning. We introduce the $s$-channel conformal block $\mathcal{F}_\Delta$ for a Verma module of dimension $\Delta$, and its value $\bar{\mathcal{F}}_\Delta$ at complex conjugate positions. For positions $(z_1,z_2,z_3,z_4)=(z,0,\infty,1)$, the conformal block is $$\begin{aligned}
\mathcal{F}_\Delta(z) = z^{\Delta-\Delta_1-\Delta_2}\left( 1 + \frac{(\Delta+\Delta_1-\Delta_2)(\Delta+\Delta_4-\Delta_3)}{2\Delta} z+ O(z^2) \right) \ .
\label{fdz}\end{aligned}$$ Let us differentiate the four-point function’s integrand with respect to the channel dimension: $$\begin{gathered}
\frac{\partial}{\partial\Delta} \Big(C_{\Delta,\Delta_1,\Delta_2}C_{\Delta,\Delta_3,\Delta_4} \mathcal{F}_\Delta\bar{\mathcal{F}}_\Delta\Big)
=
C_{\Delta,\Delta_1,\Delta_2}C_{\Delta,\Delta_3,\Delta_4} \Big(\mathcal{F}_\Delta\bar{\mathcal{F}}'_\Delta + \mathcal{F}'_\Delta\bar{\mathcal{F}}_\Delta\Big)
\\
+
\Big(C_{\Delta,\Delta_1,\Delta_2}'C_{\Delta,\Delta_3,\Delta_4}
+ C_{\Delta,\Delta_1,\Delta_2}C_{\Delta,\Delta_3,\Delta_4}'\Big)
\mathcal{F}_\Delta\bar{\mathcal{F}}_\Delta \ .
\label{ppdb}\end{gathered}$$ We interpret this expression as the contribution of the channel representation generated by the field $V'_\Delta$. This contribution involves two distinct non-chiral conformal blocks $\mathcal{F}_\Delta\bar{\mathcal{F}}'_\Delta + \mathcal{F}'_\Delta\bar{\mathcal{F}}_\Delta$ and $\mathcal{F}_\Delta\bar{\mathcal{F}}_\Delta$, whose coefficients are combinations of three-point structure constants. For the channel representation generated by $V^{(n)}_\Delta$, we would similarly obtain a linear combination of the conformal blocks $\mathcal{F}_\Delta\bar{\mathcal{F}}_\Delta, (\mathcal{F}_\Delta\bar{\mathcal{F}}_\Delta)', \dots , (\mathcal{F}_\Delta\bar{\mathcal{F}}_\Delta)^{(n)}$.
We insist that taking derivatives is only a technical trick, and that the resulting expressions are also valid if the spectrum is discrete, i.e. if the decomposition into conformal blocks is a discrete sum rather than an integral. In this context, the structure constants $C'_{\Delta_1,\Delta_2,\Delta_3}$ and $C_{\Delta_1,\Delta_2,\Delta_3}$ should be understood as independent quantities.
### Derivatives of null fields {#derivatives-of-null-fields-1}
The conformal block $\mathcal{F}_\Delta$ has simple poles at degenerate values of the channel dimension $\Delta\in\{\Delta_{(r,s)}\}_{r,s\in\mathbb{N}^*}$. The first pole $\Delta=\Delta_{(1,1)}=0$ can be seen in Eq. . The behaviour near a simple pole is $$\begin{aligned}
\mathcal{F}_{\Delta_{(r,s)}+\epsilon} = \frac{R_{r,s}}{\epsilon}\mathcal{F}_{\Delta_{(r,-s)}} + \mathcal{F}^\text{reg}_{\Delta_{(r,s)}} +\epsilon\mathcal{F}^{(1)}_{\Delta_{(r,s)}}+ O\left(\epsilon^2\right)\ ,
\label{freg}\end{aligned}$$ where $R_{r,s}$ is a known, $z$-independent constant. This is the basis for Al. Zamolodchikov’s recursive representation of conformal blocks. This will now allow us to determine the conformal blocks that correspond to the logarithmic representation $\mathcal{W}^\kappa_{(r,s)}$.
Let us consider the formal combination of values of the four-point function’s integrand at dimensions $\Delta_{(r,s)}$ and $\Delta_{(r,-s)}$: $$\begin{gathered}
\mathcal{Z}^\kappa_\epsilon = \kappa C_{\Delta_{(r,s)}+\epsilon,\Delta_1,\Delta_2}C_{\Delta_{(r,s)}+\epsilon,\Delta_3,\Delta_4}
\mathcal{F}_{\Delta_{(r,s)}+\epsilon} \bar{\mathcal{F}}_{\Delta_{(r,s)}+\epsilon}
\\
+ (1-\kappa) C_{\Delta_{(r,-s)}+\epsilon,\Delta_1,\Delta_2}C_{\Delta_{(r,-s)}+\epsilon,\Delta_3,\Delta_4} \mathcal{F}_{\Delta_{(r,-s)}+\epsilon} \bar{\mathcal{F}}_{\Delta_{(r,-s)}+\epsilon}\ .
\label{zk}\end{gathered}$$ Due to the relation between $V_{(r,s)}$ and $V_{(r,-s)}$, the two terms of the combination (without their $\kappa$-dependent prefactors) must coincide in the limit $\epsilon\to 0$. Given the behaviour of conformal blocks, this means that the structure constants must be such that $$\begin{aligned}
\frac{C_{\Delta_{(r,-s)}+\epsilon,\Delta_1,\Delta_2}C_{\Delta_{(r,-s)}+\epsilon,\Delta_3,\Delta_4}}{C_{\Delta_{(r,s)}+\epsilon,\Delta_1,\Delta_2}C_{\Delta_{(r,s)}+\epsilon,\Delta_3,\Delta_4} } \ \underset{\epsilon\to 0}{\sim}\ \frac{R_{r,s}^2}{\epsilon^2}\ .
\label{cccc}\end{aligned}$$ We then have the Taylor expansion $$\begin{gathered}
\mathcal{Z}^\kappa_\epsilon = C_{(r,s),\Delta_1,\Delta_2}C_{(r,s),\Delta_3,\Delta_4} \mathcal{F}_{\Delta_{(r,-s)}} \bar{\mathcal{F}}_{\Delta_{(r,-s)}}
+ \Bigg\{ C_{(r,s),\Delta_1,\Delta_2}C_{(r,s),\Delta_3,\Delta_4} \mathcal{G}^\kappa_{(r,s)}
\\
+
\Big(C_{(r,s),\Delta_1,\Delta_2}C'_{(r,s),\Delta_3,\Delta_4} + C'_{(r,s),\Delta_1,\Delta_2}C_{(r,s),\Delta_3,\Delta_4} \Big)\mathcal{F}_{\Delta_{(r,-s)}} \bar{\mathcal{F}}_{\Delta_{(r,-s)}}
\Bigg\}\epsilon + O\left(\epsilon^2 \right) \ ,
\label{zke}
\end{gathered}$$ where $C_{(r,s),\Delta_1,\Delta_2}$ and $C_{(r,s),\Delta_1,\Delta_2}'$ are combinations of three-point structure constants and their derivatives, and we introduced the non-chiral logarithmic conformal block $$\begin{aligned}
\mathcal{G}^\kappa_{(r,s)}=
\frac{\kappa}{R_{r,s}}
\left[ \mathcal{F}_{\Delta_{(r,-s)}}\bar{\mathcal{F}}^\text{reg}_{\Delta_{(r,s)}} +
\mathcal{F}^\text{reg}_{\Delta_{(r,s)}}\bar{\mathcal{F}}_{\Delta_{(r,-s)}}\right]
+(1-\kappa)\left(\mathcal{F}_{\Delta_{(r,-s)}}\bar{\mathcal{F}}_{\Delta_{(r,-s)}}\right)'\ .
\label{gk}\end{aligned}$$ We interpret the $O(\epsilon)$ term of $\mathcal{Z}_\epsilon^\kappa$ as the contribution of the representation $\mathcal{W}^\kappa_{(r,s)}$ to a four-point function. This contribution has the same overall structure as Eq. , with two independent structure contants $C, C'$ for each three-point coupling, and two conformal blocks $\mathcal{F}_{\Delta_{(r,-s)}} \bar{\mathcal{F}}_{\Delta_{(r,-s)}}$ and $\mathcal{G}^\kappa_{(r,s)}$. The latter conformal block is however more complicated than in Eq. , and it depends on the parameter $\kappa$.
Since the logarithms come from differentiating the $z^\Delta$ prefactor of the conformal block $\mathcal{F}_\Delta(z)$ , they cancel in the combination $\frac{\bar{\mathcal{F}}^\text{reg}_{\Delta_{(r,s)}}}{R_{r,s}} - \bar{\mathcal{F}}'_{\Delta_{(r,-s)}}$. Therefore $\mathcal{G}^\infty_{(r,s)}$ is not logarithmic, whereas $\mathcal{G}^\kappa_{(r,s)}$ is logarithmic for any finite $\kappa$. Since the field $W^\kappa_{(r,s)}$ is logarithmic if and only if $\kappa\neq \infty$, this is a basic check of our formula for $\mathcal{G}^\kappa_{(r,s)}$.
Computing the $O(\epsilon^2)$ term in the Taylor expansion of $\mathcal{Z}^\kappa_\epsilon$, we would obtain the conformal block for the representation $\widetilde{\mathcal{W}}^\kappa_{(r,s)}$, namely $$\begin{gathered}
\widetilde{ \mathcal{G}}^{\kappa}_{(r,s)}
=
\kappa
\Bigg\{
\frac{\mathcal{F}^{\text{reg}}_{\Delta_{(r,s)}}\bar{\mathcal{F}}^{\text{reg}}_{\Delta_{(r,s)}}}{R_{r,s}^2} + \frac{1}{R_{r,s}}\Big(
\mathcal{F}^{(1)}_{\Delta_{(r,s)}}\bar{ \mathcal{F}}_{\Delta_{(r,-s)}}+\bar{\mathcal{F}}^{(1)}_{\Delta_{(r,s)}} \mathcal{F}_{\Delta_{(r,-s)}}
\Big)
\Bigg\}
\\
+\frac{(1-\kappa)}{2}\left(\mathcal{F}_{\Delta_{(r,s)}}\bar{\mathcal{F}}_{\Delta_{(r,-s)}}\right)''\ ,
\label{gkrs}
\end{gathered}$$ with the coefficient $C_{(r,s),\Delta_1,\Delta_2}C_{(r,s),\Delta_3,\Delta_4}$. The $O(\epsilon^2)$ term also includes contributions of $\mathcal{G}^{\kappa'}_{(r,s)}$ and $\mathcal{F}_{\Delta_{(r,-s)}}\bar{\mathcal{F}}_{\Delta_{(r,-s)}}$, which we interpret as coming from the representation $\widetilde{\mathcal{W}}^\kappa_{(r,s)}$ again.
### Case with degenerate fields
Let us now introduce the combination $$\begin{aligned}
\mathcal{Z}^-_{\epsilon} = D(P_{(r,s)}+\epsilon) \mathcal{F}_{P_{(r,s)}+\epsilon}\bar{\mathcal{F}}_{P_{(r,s)}+\epsilon}
+
D(P_{(r,-s)}+\epsilon) \mathcal{F}_{P_{(r,-s)}+\epsilon}\bar{\mathcal{F}}_{P_{(r,-s)}+\epsilon}\ .
\label{zme}\end{aligned}$$ In contrast to the formal combination $\mathcal{Z}^\kappa_{\epsilon}$ , we do not introduce the parameter $\kappa$. Instead, we make the assumption that degenerate fields of the type $V_{\langle 1,s_0\rangle}$ exist. This assumption leads to relations between structure constants whose arguments differ by integer multiples of $\beta^{-1}$, such as $
D(P_{(r,s)}+\epsilon)$ and $D(P_{(r,-s)}+\epsilon)$. Even though our conformal blocks come from a four-point function $\left<\prod_{i=1}^4V_{\Delta_i}(z_i)\right>$ that needs not involve any degenerate field, the combination $\mathcal{Z}^-_{\epsilon}$ is analogous to the OPE . And like in that OPE, the leading terms will cancel, thanks to the relation $$\begin{aligned}
\frac{D(P_{(r,-s)}+\epsilon)}{D(P_{(r,s)}+\epsilon)}\underset{\epsilon\to 0}{\sim} -\frac{R_{r,s}\bar R_{r,s}}{4P_{(r,s)}^2\epsilon^2}\ .\end{aligned}$$ For greater generality, we allowed our four fields to be non-diagonal, in which case the residues $R_{r,s}$ and $\bar R_{r,s}$ of the left- and right-moving conformal blocks may differ. This relation is ultimately a consequence of the analytic bootstrap equations of [@mr17], and it can be deduced from Eqs. (2.17) and (3.8) in [@rib18]. The factor $4P_{(r,s)}^2$ comes from translating the $\Delta$-residues $R_{r,s}$ into $P$-residues.
The analytic bootstrap equations determine not just the leading behaviour of the ratio of structure constants as $\epsilon\to 0$, but also its value for any finite $\epsilon$. This allows us to compute the leading non-vanishing term $$\begin{gathered}
\mathcal{Z}^-_{\epsilon} \underset{\epsilon\to 0}{\propto} \mathcal{G}^-_{(r,s)} =
2P_{(r,s)}
\left[ \mathcal{F}_{\Delta_{(r,-s)}}\frac{\bar{\mathcal{F}}^\text{reg}_{\Delta_{(r,s)}}}{\rule{0pt}{1em}\bar R_{(r,s)}} +
\frac{ \mathcal{F}^\text{reg}_{\Delta_{(r,s)}}}{R_{r,s}}\bar{\mathcal{F}}_{\Delta_{(r,-s)}}\right]
\\
-2P_{(r,-s)}\left(\mathcal{F}_{\Delta_{(r,-s)}}\bar{\mathcal{F}}_{\Delta_{(r,-s)}}\right)'
- \ell^{(1)-}_{(r,s)}\mathcal{F}_{\Delta_{(r,-s)}} \bar{\mathcal{F}}_{\Delta_{(r,-s)}} \ ,
\label{zem}\end{gathered}$$ where the coefficient $\ell^{(1)-}_{(r,s)}$ comes from the Taylor expansion $$\begin{aligned}
\log \left(\epsilon^2\frac{D(P_{(r,-s)}+\epsilon)}{D(P_{(r,s)}+\epsilon)}\right) = \sum_{n=0}^\infty \ell^{(n)-}_{(r,s)}\epsilon^n\ .
\label{delln}\end{aligned}$$ Explicitly, $$\begin{gathered}
\beta\ell^{(1)-}_{(r,s)} = -4\sum_{j=1-s}^s \Big\{ \psi(-2\beta^{-1}P_{(r,j)}) +\psi(2\beta^{-1}P_{( r,-j)}) \Big\}
-4\pi \cot(\pi s \beta^{-2})
\\
+\sum_{j\overset{2}{=}1-s}^{s-1}\sum_{\pm,\pm}\Big\{
\psi\left(\tfrac12-\beta^{-1}(P_{( r,j)}\pm P_1\pm P_2)\right)
+ \psi\left(\tfrac12+\beta^{-1}(P_{( r,j)}\pm \bar P_1\pm \bar P_2)\right)
\Big\}
\\
+\sum_{j\overset{2}{=}1-s}^{s-1}\sum_{\pm,\pm}\Big\{
\psi\left(\tfrac12-\beta^{-1}(P_{( r,j)}\pm P_3\pm P_4)\right)
+ \psi\left(\tfrac12+\beta^{-1}(P_{( r,j)}\pm \bar P_3\pm \bar P_4)\right)
\Big\} \end{gathered}$$ where $\psi(x) = \frac{\Gamma'(x)}{\Gamma(x)}$ is the digamma function, regularized such that $\psi(-r)=\psi(r+1)$ for $r\in\mathbb{N}$.
Comparing the non-chiral conformal block $\mathcal{G}^-_{(r,s)}$ with $\mathcal{G}^\kappa_{(r,s)}$ , we first see that the logarithmic terms do correspond to the expected value $\kappa^-_{(r,s)}$ of the parameter $\kappa$. In addition to fixing the value of $\kappa$, the existence of degenerate fields determines the coefficient of $\mathcal{F}_{\Delta_{(r,-s)}} \bar{\mathcal{F}}_{\Delta_{(r,-s)}} $, which used to be an independent structure constant in Eq. .
Instead of degenerate fields of the type $V_{\langle 1,s_0\rangle}$, we could assume the existence of degenerate fields of the type $V_{\langle r_0, 1\rangle}$. In this case, we would find conformal blocks $\mathcal{G}^+_{(r,s)}=\theta\cdot \mathcal{G}^-_{(r,s)}$ for the representation $\mathcal{W}^+_{(r,s)}$, where we introduce the operation $$\begin{aligned}
\theta \ : \ \left\{\begin{array}{l} \beta \to -\beta^{-1}\ , \\ r\leftrightarrow s \ , \end{array}\right.\end{aligned}$$ in particular $\theta\cdot P_{(r,s)} = P_{(r,s)}$ and $\theta\cdot P_{(r,-s)}=-P_{(r,-s)}$. Assuming the existence of both types of degenerate fields, or equivalently of general degenerate fields $V_{\langle r_0,s_0\rangle}$, we can obtain both types of conformal blocks, and also conformal blocks $\widetilde{\mathcal{G}}^0_{(r,s)}$ for the representations $\widetilde{\mathcal{W}}^0_{(r,s)}$ with Jordan blocks of dimension $3$. The conformal block $\widetilde{\mathcal{G}}^0_{(r,s)}$ is the leading term of $\mathcal{Z}^0_\epsilon = \sum_\pm \mathcal{Z}^-_{\pm\epsilon}$ as $\epsilon\to 0$, and we find $$\begin{gathered}
\widetilde{\mathcal{G}}^0_{(r,s)} = \left(\mathcal{F}\bar{\mathcal{F}}\right)''_{P_{(r,-s)}} - \frac{4P_{(r,s)}^2}{R_{r,s}\rule{0pt}{1em}\bar R_{r,s}} \left((P-P_{(r,s)})^2\mathcal{F}\bar{\mathcal{F}}\right)''_{P_{(r,s)}}
\\
+\left(\ell^{(1)-}_{(r,s)}-\ell^{(1)+}_{(r,s)}\right) \left(\mathcal{F}\bar{\mathcal{F}}\right)'_{P_{(r,-s)}}
+ \frac{4P_{(r,s)}^2}{R_{r,s}\rule{0pt}{1em}\bar R_{r,s}} \left(\ell^{(1)-}_{(r,s)}+\ell^{(1)+}_{(r,s)}\right)
\left((P-P_{(r,s)})^2\mathcal{F}\bar{\mathcal{F}}\right)'_{P_{(r,s)}}
\\
+\left(2\ell^{(2)}_{(r,s)} - \ell^{(1)+}_{(r,s)}\ell^{(1)-}_{(r,s)}\right) \left(\mathcal{F}\bar{\mathcal{F}}\right)_{P_{(r,-s)}}\ .
\label{tgz}\end{gathered}$$ In this formula only, the primes are derivatives with respect to $P$, not $\Delta$. We have introduced $\ell^{(1)+}_{(r,s)}=\theta\cdot \ell^{(1)-}_{(r,s)}$ and $$\begin{gathered}
\ell^{(2)}_{( r,s)}=\ell^{(2)+}_{(r,s)}=\ell^{(2)-}_{(r,s)}=
-8\Bigg\{\sum_{\substack{j=1-s}}^{s}\sum_{\substack{i=1-r}}^{r}
\frac{1}{(2P_{(i,j)})^2}-\frac{1}{(2P_{(0,0)})^2}\Bigg\}
\\
-
\frac{1}{2}\sum_{j\overset{2}{=}1-s}^{s-1}\sum_{j\overset{2}{=}1-r}^{r-1}\sum_{\pm,\pm}\Bigg\{
\frac{1}{(P_1\pm P_2\pm P_{(i,j)})^2}+\frac{1}{\rule{0pt}{1em}(\bar P_1\pm \bar P_2\pm P_{(i,j)})^2}
\\
+\frac{1}{( P_3\pm P_4\pm P_{(i,j)})^2}+\frac{1}{\rule{0pt}{1em}( \bar P_3\pm \bar P_4\pm P_{(i,j)} )^2}
\Bigg\}
\ .
\label{l2sym}\end{gathered}$$ The statement here is that the coefficient $\ell^{(n)-}_{(r,s)}$ in the expansion is a linear combination of values of $\psi^{(n-1)}$, which however simplifies to a rational function of the momentums if $n$ is even. Moreover, we have $\ell^{(n)-}_{(r,s)}\underset{n\in 2\mathbb{N}^*}{=}\ell^{(n)+}_{(r,s)}$, where $\ell^{(n)+}_{(r,s)}=\theta\cdot\ell^{(n)-}_{(r,s)}$.
The conformal block $\widetilde{\mathcal{G}}^0_{(r,s)}$ coincides with the conformal block $\widetilde{\mathcal{G}}^\kappa_{(r,s)}$ at $\kappa=\kappa^0_{(r,s)}$ , plus terms of the type $\mathcal{G}^{\kappa'}_{(r,s)}$ and $\mathcal{F}_{\Delta_{(r,-s)}}\bar{\mathcal{F}}_{\Delta_{(r,-s)}}$, which are now completely fixed by constraints from degenerate fields. By definition of $\widetilde{\mathcal{G}}^0_{(r,s)}$ we must have the identities $\theta\cdot\kappa^0_{(r,s)}=\kappa^0_{(r,s)}$ and $\theta\cdot\widetilde{\mathcal{G}}^0_{(r,s)}=\widetilde{\mathcal{G}}^0_{(r,s)}$, which are indeed satisfied.
Limits of Liouville theory four-point functions {#sec:liou}
-----------------------------------------------
We have been building logarithmic fields as formal derivatives of diagonal fields. In CFTs where conformal dimensions take continuous values, these derivatives need not be formal, and can be performed in actual correlation functions, leading to specific values for the parameter $\kappa$ for logarithmic fields. We will now consider the case of Liouville theory, a nontrivial CFT with a continuous spectrum, whose structure constants are known analytically. (See [@rib14] for a review.) In this case, the presence of degenerate fields will determine the values of the parameter $\kappa$.
### Two flavours of Liouville theory
The properties of Liouville theory, and the analytic expression for the structure constants, depend on whether $c\leq 1$ or $c\in \mathbb{C}-(-\infty, 1)$.
In the case $c\leq 1$, the null descendants of $V_{(r,s)}=V_{\Delta_{(r,s)}}$ do not vanish [@rs15], and the relation holds, provided the primary fields are properly normalized. It is still possible to introduce degenerate fields $V_{\langle r_0,s_0\rangle}$ whose null descendants vanish, but they are not obtained as limits of the primary fields $V_\Delta$. Our statements on behaviour of OPE coefficients and structure constants in Sections \[sec:dnf\] and \[sec:fpcb\] then hold.
In the case $c\in \mathbb{C}-(-\infty, 1)$, which we will call DOZZ–Liouville theory, the null descendants of $V_{(r,s)}$ vanish, i.e. $\mathcal{L}V_{(r,s)} = \bar{\mathcal{L}}V_{(r,s)}=0$, so that $V_{(r,s)} = V_{\langle r,s\rangle}$ is a degenerate field. Nevertheless, a diagonal primary field of dimension $\Delta_{(r,-s)}$ is obtained as [@zam03] $$\begin{aligned}
\text{(DOZZ--Liouville theory)} \qquad V_{(r, -s)} = \mathcal{L}\bar{\mathcal{L}}V_{(r,s)}'\ ,\end{aligned}$$ instead of our relation . Equivalently, $$\begin{aligned}
\text{(DOZZ--Liouville theory)} \qquad \mathcal{L}_{(r,s)}\bar{\mathcal{L}}_{(r,s)}V_\Delta \ \underset{\Delta\to \Delta_{(r,s)}}{\sim}\ (\Delta-\Delta_{(r,s)}) V_{(r,-s)}\ .\end{aligned}$$ A zero at $\Delta=\Delta_{(r,s)}$ is present in the DOZZ three-point structure constant. Nevertheless, the OPE $\lim_{\Delta\to \Delta_{(r,s)}} V_\Delta V_{\Delta_0}$ does not vanish, thanks to poles of the structure constant. In order to build the logarithmic field $W^\kappa_{(r,s)}$ in DOZZ-Liouville theory, we should therefore do the replacement $V'_{(r,s)}\to V''_{(r,s)}$ in Eq. . Our analyses of representations, OPEs and conformal blocks then hold.
### From Liouville theory to logarithmic conformal blocks
This article starts with the structure of logarithmic representations, and deduces conformal blocks and correlation functions. It is however possible to reverse the logic, and start with logarithmic correlation functions before looking for the corresponding representations. The reverse logic has the advantage of starting with well-known quantities, namely correlation functions in Liouville theory. And this is how we actually found the representations’ structure in the first place.
Let us describe this reverse logic in more detail. There is no need to redo any calculations: whatever the logic, the formulas are the same. We start with Liouville four-point functions that involve one degenerate field, $$\begin{aligned}
\mathcal{Z}^\text{Liouville}_\epsilon = \left< V_{\langle r_0,s_0\rangle} V_{P_{(r_2,s_2)}+\epsilon} V_{P_3} V_{P_4}\right> \ ,
\label{zle}\end{aligned}$$ where $V_{\langle r_0,s_0\rangle}$ with $r_0,s_0\in\mathbb{N}^*$ is a degenerate field, and $r_2,s_2\in\mathbb{Z}$. We consider the $s$-channel decomposition of this four-point function into conformal blocks. The $s$-channel momentums are the type $P_{(r,s)}+\epsilon$, where the possible values of the integers $r,s\in\mathbb{Z}$ are dictated by the degenerate fusion rules . Given a value of $(r,s)$, the situation depends on how many terms of the type $(\pm r,\pm s)$ are present:
- One-term case: the conformal block $\mathcal{F}_{P_{(r,s)}+\epsilon}\bar{\mathcal{F}}_{P_{(r,s)}+\epsilon}$ remains non-logarithmic as $\epsilon \to 0$.
- Two-term case $(r,0)\& (-r,0)$ or $(0,s)\& (0,-s)$: the leading behaviour of the sum of the two terms as $\epsilon \to 0$ gives rise to the logarithmic non-chiral conformal block $\left(\mathcal{F}_{P_{(r,0)}}\bar{\mathcal{F}}_{P_{(r,0)}}\right)'$ or $\left(\mathcal{F}_{P_{(0,s)}}\bar{\mathcal{F}}_{P_{(0,s)}}\right)'$. This corresponds to simple logarithmic representations, with no null fields involved.
- Two-term case $(r,s)\& (r,-s)$ with $r,s\neq 0$: the leading behaviour of the sum of the two terms as $\epsilon \to 0$ gives rise to the logarithmic non-chiral conformal block $\mathcal{G}^-_{(r,s)}$.
- Two-term case $(r,s)\& (-r,s)$ with $r,s\neq 0$: the leading behaviour of the sum of the two terms as $\epsilon \to 0$ gives rise to the logarithmic non-chiral conformal block $\mathcal{G}^+_{(r,s)}$.
- Four-term case $(r,s)\&(r,-s)\&(-r,s)\&(-r,-s)$: the leading behaviour of the sum of the four terms as $\epsilon \to 0$ gives rise to the logarithmic non-chiral conformal block $\widetilde{\mathcal{G}}^0_{(r,s)}$.
All these conformal blocks involve at most second derivatives of the chiral Virasoro conformal blocks $\mathcal{F}_P$. This is why we limited our investigations to second derivative fields.
### From conformal blocks to OPEs and representations
The reader may worry that we have only obtained very special examples of our logarithmic conformal blocks, since we started with four-point functions $\mathcal{Z}^\text{Liouville}_\epsilon$ where two fields have discrete parameters. However, the other two fields are generic, and we obtain logarithmic contributions to their OPE $V_{P_3}V_{P_4}$. This is enough for characterizing the structure of the corresponding representations. This is also enough for reconstructing logarithmic contributions to correlation functions $\left<\prod_{i=1}^4 V_{P_i}\right>$ of generic diagonal fields, using the OPEs $V_{P_1}V_{P_2}$ and $V_{P_3}V_{P_4}$.
In Liouville theory, how do we understand the logarithmic contributions to $\left<\prod_{i=1}^4 V_{P_i}\right>$? This four-point function has an $s$-channel decomposition as a integral over momentums $P\in\mathbb{R}$, with no logarithmic terms involved. In order to get logarithmic terms, we could deform the contour until it goes through values of the type $P_{(r,s)}$ with $r,s\in\mathbb{Z}$. Such contorsions would not be natural, and the logarithmic values of $P$ would anyway have measure zero.
Therefore, Liouville theory should certainly not be considered a logarithmic CFT. Rather, we found one more class of interesting limits of Liouville theory correlation functions. Other interesting limits include minimal model correlation functions [@rib14].
The $O(n)$ model and the $Q$-state Potts model {#sec:onqp}
==============================================
### Torus partition functions
The main known source of information on the spaces of states of the $O(n)$ model and of the $Q$-state Potts model is their torus partition functions. This information has limitations:
- There can be interesting fields that do not contribute to the torus partition function: for example, in the Ising model, disorder fields [@fms97] or the fields that describe cluster connectivities [@dv11].
- In the case of fields that do contribute, the torus partition function determines the structures and multiplicities of representations in simple cases such as minimal models [@fms97], but not in more complicated cases.
In the $O(n)$ model and the $Q$-state Potts model, the torus partition function is a combination of characters with multiplicities that are generally not positive integers [@fsz87]. And since the torus partition function is defined as a trace of a function of the dilation generator, it does not know about off-diagonal components of that generator. Therefore, it does not know about logarithmic structures.
However, as we saw in Section \[sec:dnf\], we can obtain logarithmic fields by fusing two non-logarithmic fields, including one degenerate field. This will allow us to predict the existence of logarithmic representations, even if their structure is not directly captured by the partition function.
### Primary fields
Let us review what we know about the primary fields that appear in the $O(n)$ model and of the $Q$-state Potts model, based on their respective partition functions. The question has been analyzed in the original article [@fsz87], and more recently in [@grz18] for the $Q$-state Potts model and in [@GZ20] for the $O(n)$ model. We do not worry about the multiplicities of the fields: rather, we focus on whether they are degenerate.
The relations between the models’ parameters and the central charge are $$\begin{aligned}
n = 2\cos (\pi \beta^{-2}) \quad , \quad Q = 4\cos^2(\pi \beta^2)\ ,
\label{nqb}\end{aligned}$$ where $\beta$ was defined in Eq. . We still denote as $V_{\langle r_0,s_0\rangle}$ with $r_0,s_0\in\mathbb{N}^*$ a diagonal degenerate primary field whose left and right momentums are $P_{(r_0,s_0)}$ . Moreover, we denote as $V^N_{(r,s)}$ with $r,s\in\mathbb{Q}$ a primary field whose left and right dimensions are $(\Delta,\bar\Delta)=(\Delta_{(r,s)},\Delta_{(r,-s)})$. The superscript $N$ stands for non-diagonal, although the spin $rs$ of $V^N_{(r,s)}$ vanishes if $r=0$ or $s=0$. With these notations, the primary fields are $$\begin{aligned}
O(n)\text{ model: }& \left\{V_{\langle r_0,1\rangle}\right\}_{r_0\in 2\mathbb{N}+1}
\cup \left\{ V^N_{(r,s)}\right\}_{\substack{s\in\frac12\mathbb{N}^* \\ r \in\frac{1}{s}\mathbb{Z}}} \ ,
\label{opf}
\\
Q\text{-state Potts model: }& \left\{V_{\langle 1,s_0\rangle}\right\}_{s_0\in \mathbb{N}^*}
\cup \left\{ V^N_{(r,s)}\right\}_{\substack{r\in\mathbb{N}^*\\ s\in\frac{1}{r}\mathbb{Z}}}
\cup \left\{ V^N_{(0,s)}\right\}_{s\in\mathbb{N}+\frac12} \ .
\label{ppf}\end{aligned}$$ In both models, the set of degenerate fields is closed under fusion, and generated by one basic degenerate field: $V_{\langle 3,1\rangle}$ in the $O(n)$ model, and $V_{\langle 1,2\rangle}$ in the $Q$-state Potts model.
Unless $r,s\in\mathbb{Z}^*$, there are no null vectors among the descendants of the primary field $V^N_{(r,s)}$, and the corresponding representation must be the product of left and right Verma modules $\mathcal{V}_{\Delta_{(r,s)}}\otimes \bar{\mathcal{V}}_{\Delta_{(r,-s)}}$. If however $r,s\in\mathbb{Z}^*$, then $V^N_{(r,s)}$ has a null descendant on the left if $rs>0$ or on the right if $rs<0$. In this case, there can be several distinct indecomposable representations that contain $V^N_{(r,s)}$. In the case $r,s>0$, the possibilities include
- $\mathcal{V}_{\Delta_{(r,s)}}\otimes \bar{\mathcal{V}}_{\Delta_{(r,-s)}}$,
- $\frac{\mathcal{V}_{\Delta_{(r,s)}}}{\mathcal{V}_{\Delta_{(r,-s)}}}\otimes \bar{\mathcal{V}}_{\Delta_{(r,-s)}}$, i.e. the product of a degenerate representation with a Verma module,
- $\mathcal{W}^\kappa_{(r,s)}$, our logarithmic representation with Jordan blocks of dimension $2$.
We will now determine which representation is the right one.
### Logarithmic structures
Let us start with the $Q$-state Potts model. We have primary fields of the type $V_{\langle 1,s_0\rangle}$ and $V^N_{(r, 0)}$. These are the fields that appear in the $\epsilon\to 0$ limit of the OPE , which leads to logarithmic fields of the type $W^-_{(r,s)}$. This suggests that for any $(r,s)\in \mathbb{N}^*\times \mathbb{Z}^*$, the fields $V^N_{(r,s)}$ and $V^N_{(r,-s)}$ of the Potts model are part of the same logarithmic representation $\mathcal{W}^-_{(r,s)}$ , where they are called $\mathcal{L}V_{(r,s)}$ and $\bar{\mathcal{L}}V_{(r,s)}$.
In the $O(n)$ model, we have primary fields of the type $V_{\langle r_0,1\rangle}$ and $V_{(0,s)}^N$. We therefore expect logarithmic representations of the type $\mathcal{W}^+_{(r,s)}$. In the case $(r,s)=(1,1)$, let us compare this with the results of [@GZ20]. Our field $W^+_{(1,1)}$ is called $A$ in [@GZ20] (Section 5.2), and it obeys $\frac12 L_{-1}L_1A = \kappa (L_0-1)A$ in agreement with Eq. . The parameter $\kappa$ is called $-s^2$ in [@GZ20], and it takes the value $\kappa^+_{(1,1)} =\frac{1-\beta^{-2}}{2}$ in agreement with Eq. .
To summarize, we propose the following logarithmic subspaces of the spectrums of the $Q$-state Potts model and $O(n)$ model, $$\begin{aligned}
\boxed{\bigoplus_{r,s=1}^\infty \mathcal{W}^-_{(r,s)}\subset \mathcal{S}^\text{$Q$-state Potts model}}
\qquad ,\qquad
\boxed{\bigoplus_{r,s=1}^\infty \mathcal{W}^+_{(r,s)}\subset \mathcal{S}^\text{$O(n)$ model}} \ ,\end{aligned}$$ where we however do not know the multiplicities of the representations.
### Values of the parameters $n$ and $Q$
The allowed values of the parameters $n$ and $Q$, and the corresponding values of $\beta^2$ , are traditionally given as [@fsz87] $$\begin{aligned}
-2 \leq c \leq 1 \quad , \quad \frac12 \leq \beta^2 \leq 1 \quad , \quad
\left\{\begin{array}{r}
-2\leq n \leq 2 \ ,
\\
0\leq Q\leq 4\ ,
\end{array}\right.\end{aligned}$$ although it is known that analytic continuations are possible [@grz18]. From the point of view of conformal field theory, the only hard limit is the convergence of the operator product expansions, which requires that conformal dimensions be bounded from below. Given the models’ primary fields, this condition amounts to [@prs19] $$\begin{aligned}
\Re c < 13 \iff \Re \beta^2 > 0 \ .\end{aligned}$$ The allowed region is therefore vastly larger than the complex $n$-plane or the complex $Q$-plane. In order to cover these complex planes, it is enough to consider the following fundamental domains: $$\begin{aligned}
n\in\mathbb{C} \quad \iff \quad 1\leq \Re \beta^{-2} < 2 \quad \iff \quad \Re\Delta_{(1,2)} < 1 \leq\Re\Delta_{(1,3)} \ ,
\\
Q \in \mathbb{C} \quad \iff \quad \frac12 < \Re \beta^2 \leq 1 \quad \iff \quad \Re\Delta_{(3,1)} \leq 1 < \Re \Delta_{(5,1)}\ .\end{aligned}$$ (See [@prs19] for a picture of the $Q$-state Potts model’s fundamental domain in the complex $c$-plane.) We have rewritten the boundaries of the fundamental domains in terms of conditions for certain fields to be relevant. Curiously, the $Q$-state Potts model’s fundamental domain is related to the relevance of degenerate fields that appear in the $O(n)$ model, and vice-versa.
It is therefore possible to analytically continue the models beyond the complex $n,Q$-planes. Nothing dramatic happens to the conformal field theories, but their statistical interpretations may change.
Four-point connectivities in the $Q$-state Potts model {#sec:cipm}
======================================================
Our determination of logarithmic structures in the $Q$-state Potts model relies on plausible but unproven assumptions on the existence and properties of degenerate fields. In order to test these assumptions and the logarithmic structures themselves, we will look for solutions of the crossing symmetry equations based on our logarithmic conformal blocks.
Four-point connectivities in the $Q$-state Potts model were recently computed using a semi-analytic conformal bootstrap approach [@hjs20]. Crossing symmetry could be checked to a good precision, which was only limited by the lack of knowledge of logarithmic conformal blocks. If our logarithmic structures are correct, they should allow us to bootstrap connectivities to a precision that is only limited by numerical artefacts.
Logarithmic structures in four-point connectivities
---------------------------------------------------
According to a longstanding conjecture [@jac12], connectivities in the critical $Q$-state Potts model coincide with correlation functions of primary fields with the left and right dimension $\Delta_{(0,\frac12)}$. This conjecture was of little practical help for determining four-point connectivities, until it was complemented with another conjecture on the decompositions of the four-point functions into conformal blocks [@js18].
### Decomposing four-point connectivities into conformal blocks
Let us schematically write a decomposition of a four-point connectivity $P(z_1, z_2,z_3, z_4)$: $$\begin{aligned}
P(z_k) = \sum_{i \in \mathcal{S}^{(c)}} D^{(c)}_i \mathcal{G}_i^{(c)}(z_k) \ ,
\label{pdec}\end{aligned}$$ where $D^{(c)}_i$ is a four-point structure constant, $\mathcal{G}_i^{(c)}(z_k)$ is a four-point conformal block in the channel $c\in \{s,t,u\}$ for four fields of dimension $\Delta_{(0,\frac12)}$, and $\mathcal{S}^{(c)}$ is the spectrum of the connectivity in that channel, i.e. a set of representations of the product of the left and right Virasoro algebras.
There are four independent connectivities $P_{aaaa}(z_k),P_{aabb}(z_k),P_{abab}(z_k),P_{abba}(z_k)$. Permutations of the positions $z_k$ leave $P_{aaaa}(z_k)$ invariant, and exchange the other connectivities. Let us call $\mathcal{S}^\sigma$ the $s$-channel spectrum of the connectivity $P_\sigma$. By permutation symmetry, we have $\mathcal{S}^{abab}=\mathcal{S}^{abba}$, and the connectivity $P_{aabb}$ has the spectrums $\mathcal{S}^{aabb},\mathcal{S}^{abab}$ and $\mathcal{S}^{abab}$ in the $s$, $t$ and $u$ channels respectively.
The conjecture for the spectrums is based on a lattice discretization of the model [@js18], and suffers from the same shortcoming as the determination of the $Q$-state Potts model’s spectrum based on the torus partition function: it does not predict the full structure of the representations, but only their primary fields. We therefore have the following three subsets of the model’s set of primary fields : $$\begin{aligned}
\mathcal{S}^{aaaa} &=
\left\{ V^N_{(r,s)}\right\}_{\substack{r\in 2\mathbb{N}^*\\ s\in\frac{2}{r}\mathbb{Z}}}
\cup \left\{ V^N_{(0,s)}\right\}_{s\in\mathbb{N}+\frac12} \ ,
\label{aaaa}
\\
\mathcal{S}^{aabb} &=
\left\{ V^N_{(r,s)}\right\}_{\substack{r\in 2\mathbb{N}^*\\ s\in\frac{2}{r}\mathbb{Z}}}
\cup \left\{ V^N_{(0,s)}\right\}_{s\in\mathbb{N}+\frac12}
\cup
\left\{V_{\langle 1,s_0\rangle}\right\}_{s_0\in \mathbb{N}^*} \ ,
\label{aabb}
\\
\mathcal{S}^{abab},\mathcal{S}^{abba} &=
\left\{ V^N_{(r,s)}\right\}_{\substack{r\in 2\mathbb{N}^*\\ s\in\frac{1}{r}\mathbb{Z}}} \ .
\label{abab}\end{aligned}$$
### Singularities of conformal blocks
Recall that the $s$-channel conformal block $\mathcal{F}_\Delta$ for a Verma module of dimension $\Delta$ has simple poles at degenerate values $\Delta\in\{\Delta_{(r,s)}\}_{r,s\in\mathbb{N}^*}$. In a four-point function of fields with dimensions $\Delta_{(0,\frac12)}$, some of these poles have vanishing residues, and are therefore actually absent. The pole at $\Delta=\Delta_{(r,s)}$ has a vanishing residue whenever the fusion rule of the degenerate field $V_{\langle r,s\rangle}$ allows the fusion $V_{\langle r,s\rangle}V_{P_{(0,\frac12)}} \to V_{P_{(0,\frac12)}}$ or $V_{-P_{(0,\frac12)}}$, i.e. whenever $r$ is odd.
Any conformal block that appears in the decomposition of a four-point connectivity must of course be finite. We will now see that this basic criterion gives us hints on the structure of the spectrum.
The degenerate fields $V_{\langle 1,s_0\rangle}$ that are conjectured to appear in the spectrum $\mathcal{S}^{aabb}$ have an odd first index, so that $\mathcal{F}_{\Delta_{(1,s_0)}}$ is finite, and the corresponding conformal block in the decomposition of the connectivity $P^{aabb}(z_k)$ may well be of the type $$\begin{aligned}
\mathcal{G}^{aabb}_{\langle 1,s_0\rangle} = \mathcal{F}_{\Delta_{(1,s_0)}} \bar{\mathcal{F}}_{\Delta_{(1,s_0)}}\ .\end{aligned}$$ This is consistent with the claim that $V_{\langle 1,s_0\rangle}$ is a diagonal degenerate field, i.e. a primary field that generates a degenerate representation $\frac{\mathcal{V}_{\Delta_{(1,s_0)}}}{\mathcal{V}_{\Delta_{(1,-s_0)}} }\otimes \frac{\bar{\mathcal{V}}_{\Delta_{(1,s_0)}}}{\rule{0pt}{.8em}\bar{\mathcal{V}}_{\Delta_{(1,-s_0)}} }$. (In contrast to characters, four-point conformal blocks do not see the difference between Verma modules and their degenerate quotients.)
The conjectured spectrums $\mathcal{S}^{aaaa}$, $\mathcal{S}^{aabb}$ and $\mathcal{S}^{abab}$ also contain primary fields of the type $V^N_{(r,s)}$ with $(r,s)\in 2\mathbb{N}^*\times \mathbb{Z}^*$. Since the first index is even, the conformal block $\mathcal{F}_{\Delta_{(r,s)}}$ is infinite. This rules out the possibility that the corresponding representation could be a Verma module or a degenerate representation, and suggests that more complicated structures are required. Our claim is that the correct conformal blocks are of the type $\mathcal{G}^-_{(r,s)}$ .
Linear relations between four-point structure constants {#sec:linrel}
-------------------------------------------------------
The basic idea of the conformal bootstrap method is that the decomposition of a given four-point function should not depend on the channel. The equality between the $s$, $t$ and $u$-channels is called crossing symmetry. Assuming that we know the spectrum, crossing symmetry amounts to linear equations for the four-point structure constants, which we will solve numerically. This flavour of the conformal bootstrap was introduced in [@prs16], and may be called semi-analytic in contrast to situations where the spectrum is itself an unknown (numerical bootstrap) or where the structure constants are known analytically too (analytic bootstrap).
However, in our case, the structure constants are not quite independent unknowns. The degenerate fields that allowed us to predict the nontrivial conformal blocks, also lead to linear relations between certain structure constants [@mr17]. These relations may be viewed as emanating from an “interchiral” symmetry algebra that is larger than the product of the left and right Virasoro algebra [@hjs20].
### The relations
Let us call $D^\sigma_{(r,s)}$ the four-point structure constants for the primary fields $V^N_{(r,s)}$ in the spectrums $\mathcal{S}^\sigma$ -, and $D^{aabb}_{\langle 1,s_0\rangle}$ the structure constants for the diagonal degenerate fields in $\mathcal{S}^{aabb}$.
Due to the known permutation properties of structure constants and conformal blocks [@rib14], we have the relations $$\begin{aligned}
D^{abab}_{(r,s)} = (-1)^{rs} D^{abba}_{(r,s)} \ ,
\label{drsd}\end{aligned}$$ since $rs = \Delta_{(r,-s)}-\Delta_{(r,s)}$ is the conformal spin of $V^N_{(r,s)}$. Moreover, we are considering four-point functions of four spinless primary fields. This implies that the crossing symmetry equations are invariant under the exchange of the left and right quantities, and therefore the relations $$\begin{aligned}
D^\sigma_{(r,s)} = D^\sigma_{(r,-s)}\ .
\label{sms}\end{aligned}$$ From the definition of four-point connectivities, it is also possible to predict [@prs19] $$\begin{aligned}
D^{aabb}_{(0,\frac12)} = -D^{aaaa}_{(0,\frac12)}\ .
\label{dmd}\end{aligned}$$ Let us now move to the relations that follow from the existence of the degenerate field $V_{\langle 1,2\rangle}$. In general four-point functions, these relations would determine the ratios $\frac{D^\sigma_{(r,s+2)}}{D^\sigma_{(r,s)}}$ [@mr17]. However, in a four-point function of fields with the particular dimension $\Delta_{(0,\frac12)}$, we have slightly more powerful relations, which determine how structure constants behave under shifts of the second index by one unit, rather than the usual two units [@hjs20]: $$\begin{aligned}
\frac{D^\sigma_{(r,s+1)}}{D^\sigma_{(r,s)}}
= 2^{(2+2\omega)r-\frac{4s+2}{\beta^2}}
\frac{\Gamma(\frac{1-r}{2}+\frac{s}{2\beta^2})}{\Gamma(\frac{2-r}{2}+\frac{s}{2\beta^2})}
\frac{\Gamma(\frac{\omega r}{2}-\frac{s}{2\beta^2})}{\Gamma(\frac{1+\omega r}{2}-\frac{s}{2\beta^2})}
\frac{\Gamma(\frac{1-r}{2}+\frac{s+1}{2\beta^2})}{\Gamma(\frac{-r}{2}+\frac{s+1}{2\beta^2})}
\frac{\Gamma(\frac{2+\omega r}{2}-\frac{s+1}{2\beta^2})}{\Gamma(\frac{1+\omega r}{2}-\frac{s+1}{2\beta^2})}\ ,
\label{dd}\end{aligned}$$ where $\omega = -1$ for our non-diagonal fields $V^N_{(r,s)}$, and $\omega=1$ for diagonal fields with dimensions $\Delta=\bar\Delta=\Delta_{(r,s)}$, such as $V_{\langle 1,s_0\rangle}$. For $s\in\{0,-1\}$, this shift equation involves the value of the Gamma function at its poles. In the non-diagonal case however, these singularities cancel among the factors, and the ratio can be rewritten in a manifestly finite manner, $$\begin{aligned}
\frac{D^\sigma_{(r,s+1)}}{D^\sigma_{(r,s)}}\ \underset{s\in \{0, -1\}}{=} \ \left(\frac{2^{1-\frac{2}{\beta^2}}}{|r|}
\frac{\Gamma(\frac{1-r}{2}+\frac{1}{2\beta^2})}{\Gamma(\frac{-r}{2}+\frac{1}{2\beta^2})}
\frac{\Gamma(\frac{2-r}{2}-\frac{1}{2\beta^2})}{\Gamma(\frac{1-r}{2}-\frac{1}{2\beta^2})}
\right)^{2s+1}\ .
\label{ddzo}\end{aligned}$$ We insist that the shift equations also hold in the case $r,s\in\mathbb{Z}^*$ i.e. if $V^N_{(r,s)}$ belongs to a logarithmic representation. The validity of these equations indeed only depends on $V^N_{(r,s)}$ being a primary field.
Our formulas for the ratios are equivalent to the formulas in [@hjs20] (Section 4.1). Cosmetic differences come down to notations and to our use of the Gamma function duplication formula. We will now rederive these ratios by following the logic of [@hjs20], while trying to streamline the derivation.
### The derivation
Let us denote two- and three-point structure constants as $\left<V_iV_i\right>\sim B_i$ and $\left<V_iV_jV_k\right> \sim C_{ijk}$. Then an OPE reads $V_iV_j\sim \sum_k \frac{C_{ijk}}{B_k} V_k$, and four-point structure constants that appear in $s$ and $t$-channel decompositions of four-point functions read $$\begin{aligned}
\text{$s$-channel: }&
\begin{tikzpicture}[baseline=(current bounding box.center), very thick, scale = .25]
\draw (-1,2) node [left] {$2$} -- (0,0) -- node [above] {$\epsilon$} (4,0) -- (5,2) node [right] {$3$};
\draw (-1,-2) node [left] {$1$} -- (0,0);
\draw (4,0) -- (5,-2) node [right] {$4$};
\end{tikzpicture}
\ \rightarrow\ d^{(s)}_\epsilon = \frac{C_{12\epsilon}C_{\epsilon 34}}{B_\epsilon}
\ ,
\\
\text{$t$-channel: }&
\begin{tikzpicture}[baseline=(current bounding box.center), very thick, scale = .2]
\draw (-2,1) node [left] {$2$} -- (0,0) -- node [right] {$\eta$} (0,-4) -- (2,-5) node [right] {$4$};
\draw (-2,-5) node [left] {$1$} -- (0,-4);
\draw (0,0) -- (2,1) node [right] {$3$};
\end{tikzpicture}
\ \rightarrow\ d^{(t)}_\eta = \frac{C_{23\eta }C_{41\eta}}{B_\eta}\ .\end{aligned}$$ We normalize the degenerate field $V_{\langle 1,1\rangle}$ as an identity field, i.e. $C_{\langle 1,1\rangle ii} =B_i$ and $B_{\langle 1,1\rangle}=1$.
We consider four-point functions that involve at least one degenerate field, such as $\left<V_{\langle 1,2\rangle}\prod_{i=2}^4 V^N_{(r_i,s_i)}\right>$ or $\left<V_{\langle 1,2\rangle} V^N_{(r,s)}V^N_{(r,s)}V_{\langle 1,2\rangle}\right>$. In these cases, only two fields can appear in each channel, as dictated by the OPEs $V_{\langle 1,2\rangle} V^N_{(r,s)} \sim \sum_{\epsilon=\pm } V^N_{(r,s+\epsilon)}$ and $V_{\langle 1,2\rangle}V_{\langle 1,2\rangle}\sim V_{\langle 1,1\rangle}+V_{\langle 1,3\rangle}$. Crossing symmetry and single-valuedness determine the ratios of all involved four-point structure constants, and in particular [@mr17] $$\begin{aligned}
\frac{d^{(t)}_{\eta}}{d^{(s)}_{\epsilon}}
= -\epsilon\omega\eta\frac{F_{\epsilon,\eta}}{\rule{0pt}{1em}\bar F_{\epsilon,-\omega\eta}}
\det \bar F
\qquad \text{with}\qquad
\det \bar F = -\frac{\bar P_2}{\rule{0pt}{1em}\bar P_4}\ .
\label{dede}\end{aligned}$$ Here the $s$- and $t$-channel fields are labelled by discrete indices $\epsilon,\eta = \pm $. We define $\omega = +$ if the fourth field is $V_{\langle 1,2\rangle}$, and $\omega=-$ if the fourth field is $V^N_{(r_4,s_4)}$. The fusing matrix elements are $$\begin{aligned}
F_{\epsilon,\eta} &= \frac{\Gamma(1-2\beta^{-1}\epsilon P_2)\Gamma(2\beta^{-1}\eta P_4)}{\prod_{\pm}\Gamma(\tfrac{1}{2}-\beta^{-1}\epsilon P_2 \pm \beta^{-1} P_3+\beta^{-1}\eta P_4)}\ . \label{fusmat}\end{aligned}$$ We first consider the four-point function $\left< V_{\langle 1,2\rangle} V^N_{(r,s)} V^N_{(0,\frac12)} V^N_{(0,\frac12)}\right>$, and focus on the following $s$- and $t$-channel terms: $$\begin{aligned}
\begin{tikzpicture}[baseline=(current bounding box.center), very thick, scale = .4]
\draw (-1,2) node [left] {$(r,s)$} -- (0,0) -- node [above] {$(r,s+ 1)$} (4,0) -- (5,2) node [right] {$(0,\frac12)$};
\draw[dashed] (-1,-2) node [left] {$\langle 1,2\rangle$} -- (0,0);
\draw (4,0) -- (5,-2) node [right] {$(0,\frac12)$};
\end{tikzpicture}
\qquad , \qquad
\begin{tikzpicture}[baseline=(current bounding box.center), very thick, scale = .3]
\draw (-2,1) node [left] {$(r,s)$} -- (0,0) -- node [right] {$(0,\frac12)$} (0,-4) -- (2,-5) node [right] {$(0,\frac12)$};
\draw[dashed] (-2,-5) node [left] {$\langle 1,2\rangle$} -- (0,-4);
\draw (0,0) -- (2,1) node [right] {$(0,\frac12)$};
\end{tikzpicture} \end{aligned}$$ In this case, Eq. gives us $\frac{d^{(t)}_{-}}{d^{(s)}_+}
=
-\frac{\bar P}{P_{(0,\frac12)}} \frac{F_{+-}}{\rule{0pt}{.8em}\bar F_{+-}}$, explicitly $$\begin{gathered}
\frac{C_{\langle 1,2\rangle (0,\frac12)(0,\frac12)} C_{(r,s)(0,\frac12)(0,\frac12)}}
{C_{\langle 1,2\rangle (r,s)(r,s+1)} C_{(r,s+1)(0,\frac12)(0,\frac12)}} \frac{B_{(r,s+1)}}{B_{(0,\frac12)}}
\\
=
2^{1-2\beta^{-1}(P-\bar P)}
\gamma(\tfrac{1}{2\beta^2})
\frac{\Gamma(1-\beta^{-1}P)}{\Gamma(-\beta^{-1}\bar P)}
\frac{\Gamma(\frac12 -\beta^{-1}\bar P -\frac{1}{2\beta^2})}{\Gamma(\frac12-\beta^{-1}P+\frac{1}{2\beta^2})}\ ,
\label{dc1}\end{gathered}$$ where we denote the left and right momentums of $V^N_{(r,s)}$ as $(P,\bar P)=(P_{(r,s)},P_{(r,-s)})$, and introduce $\gamma(x)=\frac{\Gamma(x)}{\Gamma(1-x)}$. We next consider the four-point function $ \left< V_{\langle 1,2\rangle} V^N_{(r,s)} V^N_{(r,s)} V_{\langle 1,2\rangle}\right>$, and focus on the following $s$- and $t$-channel terms: $$\begin{aligned}
\begin{tikzpicture}[baseline=(current bounding box.center), very thick, scale = .4]
\draw (-1,2) node [left] {$(r,s)$} -- (0,0) -- node [above] {$(r,s+ 1)$} (4,0) -- (5,2) node [right] {$(r,s)$};
\draw[dashed] (-1,-2) node [left] {$\langle 1,2\rangle$} -- (0,0);
\draw[dashed] (4,0) -- (5,-2) node [right] {$\langle 1,2\rangle$};
\end{tikzpicture}
\qquad , \qquad
\begin{tikzpicture}[baseline=(current bounding box.center), very thick, scale = .3]
\draw (-2,1) node [left] {$(r,s)$} -- (0,0);
\draw[dashed] (0,0) -- node [right] {$\langle 1,1\rangle$} (0,-4) -- (2,-5) node [right] {$\langle 1,2\rangle$};
\draw[dashed] (-2,-5) node [left] {$\langle 1,2\rangle$} -- (0,-4);
\draw (0,0) -- (2,1) node [right] {$(r,s)$};
\end{tikzpicture} \end{aligned}$$ In this case, Eq. gives us $\frac{d^{(t)}_{-}}{d^{(s)}_+}
=
-\frac{\bar P}{P_{\langle 1,2\rangle}} \frac{F_{+-}}{\rule{0pt}{.8em}\bar F_{++}}$, explicitly $$\begin{aligned}
\frac{B_{\langle 1,2\rangle} B_{(r,s)}B_{(r,s+1)}}{C_{\langle 1,2\rangle (r,s)(r,s+1)}^2 }
= 2^{4\beta^{-2}-3}\gamma(\beta^{-2}-\tfrac12)
\frac{\Gamma(1-2\beta^{-1}P)}{\Gamma(-2\beta^{-1}P+\beta^{-2})}
\frac{\Gamma(1-2\beta^{-1}\bar P -\beta^{-2})}{\Gamma(-2\beta^{-1}\bar P)} \ .
\label{dc2}\end{aligned}$$ We finally consider the four-point function $\left< V_{\langle 1,2\rangle} V^N_{(0,\frac12)} V^N_{(0,\frac12)} V_{\langle 1,2\rangle}\right> $, and focus on the following $s$- and $t$-channel terms: $$\begin{aligned}
\begin{tikzpicture}[baseline=(current bounding box.center), very thick, scale = .4]
\draw (-1,2) node [left] {$(0,\frac12)$} -- (0,0) -- node [above] {$(0,\frac12)$} (4,0) -- (5,2) node [right] {$(0,\frac12)$};
\draw[dashed] (-1,-2) node [left] {$\langle 1,2\rangle$} -- (0,0);
\draw[dashed] (4,0) -- (5,-2) node [right] {$\langle 1,2\rangle$};
\end{tikzpicture}
\qquad , \qquad
\begin{tikzpicture}[baseline=(current bounding box.center), very thick, scale = .3]
\draw (-2,1) node [left] {$(0,\frac12)$} -- (0,0);
\draw[dashed] (0,0) -- node [right] {$\langle 1,1\rangle$} (0,-4) -- (2,-5) node [right] {$\langle 1,2\rangle$};
\draw[dashed] (-2,-5) node [left] {$\langle 1,2\rangle$} -- (0,-4);
\draw (0,0) -- (2,1) node [right] {$(0,\frac12)$};
\end{tikzpicture} \end{aligned}$$ In this case, Eq. gives us $\frac{d^{(t)}_{-}}{d^{(s)}_-}
=
-\frac{P_{(0,\frac12)}}{P_{\langle 1,2\rangle}} \frac{F_{--}}{\rule{0pt}{.8em}\bar F_{-+}}$, explicitly $$\begin{aligned}
\frac{B_{(0,\frac12)}^2B_{\langle 1,2\rangle}}{C_{\langle 1,2\rangle (0,\frac12)(0,\frac12)}^2}
=
2^{4\beta^{-2}-3}\gamma(\beta^{-2}-\tfrac12) \gamma(1-\tfrac{1}{2\beta^{2}})^{2} \ .
\label{dc3}\end{aligned}$$ We are interested in the four-point structure constant $D_{(r,s)} = \frac{C_{(r,s)(0,\frac12)(0,\frac12)}^2}{B_{(r,s)}}$. Combining the square of Eq. with Eq. and Eq. , we obtain the expression for $\frac{D_{(r,s+1)}}{D_{(r,s)}}$ that was written in Eq. with $\omega = -1$. For the case $\omega=1$, it suffices to set $(P,\bar P)=(P_{(r,s)},-P_{(r,s)})$ instead of $(P_{(r,s)},P_{(r,-s)})$.
### Interchiral conformal blocks and reduced spectrums
Using the linear relations and between four-point structure constants, we can rewrite $s$-channel decompositions of four-point connectivities such that the only unknown coefficients are $D^\sigma_{(r,s)}$ with $0\leq s\leq \frac12$. In this rewriting, $D^\sigma_{(r,s)}$ is the coefficient of an infinite linear combinations of conformal blocks, which was called an interchiral conformal block in [@hjs20]: $$\begin{aligned}
\mathcal{H}_{(r,s)} \ &\underset{0<s\leq\frac12}{=} \
\sum_{s'\in (s+\mathbb{Z})\cup (-s+\mathbb{Z})}
\frac{D^\sigma_{(r,s')}}{D^\sigma_{(r,s)}} \mathcal{F}_{\Delta_{(r,s')}}\bar{\mathcal{F}}_{\Delta_{(r,-s')}}\ ,
\\
\mathcal{H}_{(r,0)} &= \mathcal{F}_{\Delta_{(r,0)}}\bar{\mathcal{F}}_{\Delta_{(r,0)}} +\sum_{s'=1}^\infty \frac{D^\sigma_{(r,s')}}{D^\sigma_{(r,0)}} \mathcal{G}^-_{(r,s')}\ ,
\\
\mathcal{H}_{\langle 1,1\rangle} &= \sum_{s'=1}^\infty \frac{D^\sigma_{\langle 1,s'\rangle}}{D^\sigma_{\langle 1,1\rangle}} \mathcal{F}_{\Delta_{(1,s')}}\bar{\mathcal{F}}_{\Delta_{(1,s')}}\ ,\end{aligned}$$ where $\mathcal{F}_\Delta$ is a standard Virasoro conformal block, and $\mathcal{G}^-_{(r,s)}$ is a logarithmic conformal block. Neither the ratios of structure constants, nor therefore the interchiral conformal blocks, depend on $\sigma$. The spectrums - can then be reduced to the fields whose second indices obey $0\leq s\leq \frac12$, $$\begin{aligned}
\mathcal{S}^{aaaa}_\text{reduced} &=
\left\{ V^N_{(r,s)}\right\}_{\substack{r\in 2\mathbb{N}^*\\ s\in \frac{2}{r}\mathbb{N}\cap [0,\frac12]}}
\cup \left\{ V^N_{(0,\frac12)}\right\}\ ,
\label{raaaa}
\\
\mathcal{S}^{aabb}_\text{reduced} &=
\left\{ V^N_{(r,s)}\right\}_{\substack{r\in 2\mathbb{N}^*\\ s\in \frac{2}{r}\mathbb{N}\cap [0,\frac12]}}
\cup \left\{ V^N_{(0,\frac12)}\right\}
\cup
\left\{V_{\langle 1,1\rangle}\right\}\ ,
\label{raabb}
\\
\mathcal{S}^{abab}_\text{reduced},\mathcal{S}^{abba}_\text{reduced} &=
\left\{ V^N_{(r,s)}\right\}_{\substack{r\in 2\mathbb{N}^*\\ s\in \frac{1}{r}\mathbb{N}\cap [0,\frac12]}} \ ,
\label{rabab}\end{aligned}$$ and we write the four-point connectivities as $$\begin{aligned}
P^\sigma = \sum_{V\in \mathcal{S}^\sigma_{\text{reduced}}} D^\sigma_V \mathcal{H}_V\ ,\end{aligned}$$ where we abuse notations by identifying a primary field $V$ with its indices.
Semi-analytic bootstrap
-----------------------
Let us determine the four-point structure constants $D_i^\sigma$ by solving crossing symmetry equations for the connectivities $P_\sigma$ with $\sigma\in\{aaaa,aabb,abab,abba\}$. Given a connectivity and two different channels $c,c'\in \{s,t,u\}$, we have the crossing symmetry equation $$\begin{aligned}
\sum_{i\in \mathcal{S}^{(c)}_\text{reduced}} D_i^{(c)} \mathcal{H}_i^{(c)}(z_k) =
\sum_{i'\in \mathcal{S}^{(c')}_\text{reduced}} D_{i'}^{(c')} \mathcal{H}_{i'}^{(c')}(z_k) \ ,
\label{dhdh}\end{aligned}$$ for any values of the positions $z_k$. We are not so much interested in the solutions themselves as in their existence, which tests several things at once:
- the identification of connectivities with correlation functions,
- the existence of degenerate fields,
- Jacobsen and Saleur’s spectrums $\mathcal{S}^{aaaa},\mathcal{S}^{aabb}$ and $\mathcal{S}^{abab}$,
- our conformal blocks $\mathcal{G}^-_{(r,s)}$ and the structure of logarithmic representations.
### Numerical implementation
Using Zamolodchikov’s recursive representation, the interchiral conformal blocks have a series expansion of the type $$\begin{aligned}
\mathcal{H}_i^{(c)}(z_k) = \mathcal{H}_0^{(c)}(z_k)
\left|q^{(c)}\right|^{\Delta_i+\bar \Delta_i}
\sum_{N=0}^\infty h_{i,N}\left(q^{(c)}\right)\left|q^{(c)}\right|^N\ .\end{aligned}$$ Here $\mathcal{H}_0^{(c)}(z_k)$ is an $i$-independent prefactor, the nome $q^{(c)}$ is a function of $(z_1,z_2,z_3,z_4)$ that depends on the channel $c$, and the coefficients $h_{i,N}(q)$ is a polynomially bounded function of $\frac{q}{|q|}$ and $\log q$. In order to write the connectivities as finite sums, we introduce a cutoff $N_\text{max}$ and truncate the conformal blocks’ expansions to $N\leq N_\text{max}$, while also truncating the sums over reduced spectrums to $\Re(\Delta_i+\bar\Delta_i)\leq N_\text{max}$. After truncation, the crossing symmetry equation involves a finite number of unknown four-point structure constants, $$\begin{aligned}
X(N_\text{max}) = \#\left(\left\{D_i^{(c)}\right\}_{\Re(\Delta_i+\bar \Delta_i)\leq N_\text{max}} \cup \left\{D_{i'}^{(c')}\right\}_{\Re(\Delta_{i'}+\bar \Delta_{i'})\leq N_\text{max}}\right) .\end{aligned}$$ We then normalize one of these structure constants to be $1$, and determine the rest by solving crossing symmetry for a number $E\geq X(N_\text{max})-1$ of randomly chosen positions $Z^e=(z_1^e,z_2^e,z_3^e,z_4^e)$. We compute the averages and the relative deviations of the resulting structure constants over a number $A$ of random draws of the positions, $$\begin{aligned}
\bar D_i^{(c)} = \frac{1}{A}\sum_{a=1}^A D_i^{(c)}(Z_a^e) \quad , \quad \text{Deviation}\left(D_i^{(c)}\right) = \max_a\left(\frac{\big|D_i^{(c)}(Z_a^e) - \bar D_i^{(c)}\big|}{\max\left(\big|D_i^{(c)}(Z_a^e)\big|,\big|\bar D_i^{(c)}\big|\right)}\right) .\end{aligned}$$ If the crossing symmetry equation has a unique solution, the deviations should tend to zero as $N_\text{max}$ increases, except for structure constants that are in fact zero.
### Results
We focus on two specific examples of the crossing symmetry equations : $$\begin{aligned}
& \sum_{i\in \mathcal{S}^{abab}_\text{reduced}} D_i^{abab} \mathcal{H}_i^{(s)}(z_k) =
\sum_{i'\in \mathcal{S}^{aabb}_\text{reduced}} D_{i'}^{aabb} \mathcal{H}_{i'}^{(u)}(z_k)\ ,
\label{suc}
\\
& \sum_{i\in \mathcal{S}^{aaaa}_\text{reduced}} D_i^{aaaa} \left(\mathcal{H}_i^{(s)}(z_k) - \mathcal{H}_{i}^{(t)}(z_k)\right) = 0\ .
\label{stc}\end{aligned}$$ In the first equation, we can use the normalization condition $D_{\langle 1,1\rangle}^{aabb}=1$, and determine the rest of the structure constants. In the second equation, we can then normalize the structure constants such that the relation is obeyed. We find that both equations have unique solutions. For example, let us display how the deviations for $\mathcal{S}^{aabb}_\text{reduced}$ behave as the cutoff $N_\text{max}$ increases [@bV2]: $$\begin{aligned}
\begin{array}{ccc}
N_\text{max} = 16 & N_\text{max}=24 & N_\text{max}=32 \\
\begin{array}[t]{|l|l|} \hline
(r, s) & \text{Deviation} \\
\hline\hline (1, 1)& 7.61 \times 10^{-12}\\
\hline (0, 1/2)& 8.19 \times 10^{-12}\\
\hline (2, 0)& 4.37 \times 10^{-11}\\
\hline (4, 0)& 6.19 \times 10^{-8}\\
\hline (4, 1/2)& 6.12 \times 10^{-8}\\
\hline (6, 0)& 0.269\\
\hline (6, 1/3)& 0.802\\
\hline
\end{array}
&
\begin{array}[t]{|l|l|} \hline
(r, s) & \text{Deviation} \\
\hline\hline (1, 1)& 3.47 \times 10^{-18}\\
\hline (0, 1/2)& 3.78 \times 10^{-18}\\
\hline (2, 0)& 9.85 \times 10^{-18}\\
\hline (4, 0)& 1.04 \times 10^{-14}\\
\hline (4, 1/2)& 9.88 \times 10^{-15}\\
\hline (6, 0)& 4.44 \times 10^{-8}\\
\hline (6, 1/3)& 8.25 \times 10^{-8}\\
\hline (8, 0)& 0.211\\
\hline (8, 1/4)& 0.118\\
\hline (8, 1/2)& 0.574\\
\hline
\end{array}
&
\begin{array}[t]{|l|l|} \hline
(r, s) & \text{Deviation} \\
\hline\hline (1, 1)& 2.3 \times 10^{-24}\\
\hline (0, 1/2)& 2.38 \times 10^{-24}\\
\hline (2, 0)& 2.28 \times 10^{-24}\\
\hline (4, 0)& 3.17 \times 10^{-21}\\
\hline (4, 1/2)& 2.62 \times 10^{-21}\\
\hline (6, 0)& 1.2 \times 10^{-14}\\
\hline (6, 1/3)& 2.14 \times 10^{-14}\\
\hline (8, 0)& 4.86 \times 10^{-7}\\
\hline (8, 1/4)& 3.41 \times 10^{-7}\\
\hline (8, 1/2)& 2.37 \times 10^{-7}\\
\hline (10, 0)& 0.489\\
\hline (10, 1/5)& 1.19\\
\hline (10, 2/5)& 1.76\\
\hline
\end{array}
\end{array}
\label{ddd}\end{aligned}$$ Here and in our other numerical examples, we chose a generic value of the parameter $\beta=0.8+0.1i$ i.e. $Q\simeq -0.121+1.725i$. The code runs in $O(10^3)$ seconds on a standard laptop. The deviation of $D^{aabb}_{\langle 1,1\rangle}$ is nonzero because the code uses the normalization condition $D^{abab}_{(0,\frac12)}=1$ rather than $D^{aabb}_{\langle 1,1\rangle}=1$.
In order to determine the structure constants $D_i^{abab}$, it may seem easier to focus on a crossing symmetry equation with fewer unknowns, $$\begin{aligned}
\sum_{i\in \mathcal{S}^{abab}_\text{reduced}} D_i^{abab} \left(\mathcal{H}_i^{(s)}(z_k) - (-1)^{\text{Spin}(i)}\mathcal{H}_{i}^{(t)}(z_k)\right) = 0 \ ,
\label{stm}\end{aligned}$$ where the spin-dependent sign is due to the relation . However, while the deviation of $D^{abab}_{(2,\frac12)}$ does tend to zero as $N_\text{max}$ increases, the deviations of $D^{abab}_{(r\geq 4,s)}$ remain large. The intepretation is that our equation has a solution, which is however not unique. For the deviation of a structure constant to tend to zero, that structure constant must be nonvanishing in one solution only. We find an infinite series of subsets of the spectrum $$\begin{aligned}
(r_0\in 2\mathbb{N}^*) \qquad \mathcal{S}^{r_0}_\text{reduced} = \left\{V^N_{(r_0,0)}\right\}\cup \left\{V^N_{(r,s)}\right\}_{\substack{r\geq r_0\in 2\mathbb{N}^*\\ s\in\frac{1}{r}\mathbb{N}^* \cap [0,\frac12]}}\subset \mathcal{S}^{abab}_\text{reduced} \ ,
\label{sr0}\end{aligned}$$ such that the crossing symmetry equation for $\mathcal{S}^{r_0}_\text{reduced}$ has a unique solution. These solutions provide a basis of solutions for the original crossing symmetry equation . Let us display the deviations in the cases $r_0=2,4,6$ with $N_\text{max}=32$ [@bV2]: $$\begin{aligned}
\begin{array}{ccc}
r_0 = 2 & r_0 = 4 & r_0 = 6
\\
\begin{array}[t]{|l|l|} \hline
(r, s) & \text{Deviation} \\
\hline\hline (2, 0)& 0\\
\hline (2, 1/2)& 2.98 \times 10^{-28}\\
\hline (4, 1/4)& 2.01 \times 10^{-23}\\
\hline (4, 1/2)& 3.1 \times 10^{-24}\\
\hline (6, 1/6)& 3.56 \times 10^{-17}\\
\hline (6, 1/3)& 1.21 \times 10^{-16}\\
\hline (6, 1/2)& 4.71 \times 10^{-16}\\
\hline (8, 1/8)& 7.42 \times 10^{-5}\\
\hline (8, 1/4)& 0.000244\\
\hline (8, 3/8)& 0.000406\\
\hline (8, 1/2)& 0.000243\\
\hline
\end{array}
&
\begin{array}[t]{|l|l|} \hline
(r, s) & \text{Deviation} \\
\hline\hline (4, 0)& 0\\
\hline (4, 1/4)& 3.57 \times 10^{-25}\\
\hline (4, 1/2)& 3.25 \times 10^{-25}\\
\hline (6, 1/6)& 2.8 \times 10^{-15}\\
\hline (6, 1/3)& 2.52 \times 10^{-15}\\
\hline (6, 1/2)& 2.51 \times 10^{-15}\\
\hline (8, 1/8)& 0.00143\\
\hline (8, 1/4)& 0.0014\\
\hline (8, 3/8)& 0.00137\\
\hline (8, 1/2)& 0.00136\\
\hline
\end{array}
&
\begin{array}[t]{|l|l|} \hline
(r, s) & \text{Deviation} \\
\hline\hline (6, 0)& 0\\
\hline (6, 1/6)& 1.76 \times 10^{-17}\\
\hline (6, 1/3)& 4.62 \times 10^{-17}\\
\hline (6, 1/2)& 6.39 \times 10^{-17}\\
\hline (8, 1/8)& 0.105\\
\hline (8, 1/4)& 0.105\\
\hline (8, 3/8)& 0.105\\
\hline (8, 1/2)& 0.105\\
\hline
\end{array}
\end{array}\end{aligned}$$ (For conciseness we do not display the deviations for the structure constants $D^{r_0}_{(10,s)}$, which are of order $1$.)
### Comparison with analytic formulas
Using a lattice regularization of the $Q$-state Potts model, analytic formulas for a few ratios of four-point structure constants have been conjectured [@hgjs20](Section 5.2): $$\begin{aligned}
\frac{D^{aabb}_{(2, 0)}}{D^{aaaa}_{(2, 0)}} &= \frac{1}{1-Q}\ ,
\\
\frac{D^{aabb}_{(4, 0)}}{D^{aaaa}_{(4, 0)}} &= -\frac{Q^5 -7Q^4+15Q^3-10Q^2+4Q-2}{2(Q^2-3Q+1)}\ ,
\\
\frac{D^{aabb}_{(4,\frac12)}}{D^{aaaa}_{(4,\frac12)}} &= \frac{2-Q}{2}\ ,
\\
\frac{D^{abab}_{(2, 0)}}{D^{aaaa}_{(2,0)}} &= \frac{2-Q}{2}\ ,
\\
\frac{D^{abab}_{(4, 0)}}{D^{aaaa}_{(4, 0)}} &= -\frac14(Q^2-4Q+2)(Q^2-3Q-2)\ ,
\\
\frac{D^{abab}_{(4,\frac12)}}{D^{aaaa}_{(4,\frac12)}} &= \frac14(Q-1)(Q-4)\ .\end{aligned}$$ We have checked these formulas to a high precision [@bV2]. With our code, it is possible to look for analytic formulas for other ratios of structure constants. For instance, we have found the formulas $$\begin{aligned}
\frac{D^{aabb}_{(6,\frac13)}}{D^{aaaa}_{(6,\frac13)}}
&=\frac{2-Q}{2}\ ,
\\
\frac{D^{abab}_{(6,\frac13)}}{D^{aaaa}_{(6,\frac13)}}
&= \frac{1}{4} \left(Q^5-9 Q^4+27 Q^3-28 Q^2+Q+4\right)\ ,
\\
\frac{D^{aabb}_{(6,0)}}{D^{aaaa}_{(6,0)}}
&=
\frac{2 Q^8-26 Q^7+134 Q^6-348 Q^5+479 Q^4-337 Q^3+112 Q^2-23 Q+3}{\left(1-6Q+5Q^2-Q^3\right) \left(3 Q^6-24 Q^5+64 Q^4-66 Q^3+24 Q^2-8 Q+3\right)}\ ,
\\
\frac{D^{abab}_{(6,0)}}{D^{aaaa}_{(6,0)}}
&=
\frac{(2-Q) \left(Q^2-4 Q+1\right) \left(Q^6-9 Q^5+30 Q^4-40 Q^3+13 Q^2+4 Q+3\right)}{2 \left(3 Q^6-24 Q^5+64 Q^4-66 Q^3+24 Q^2-8
Q+3\right)}\ ,\end{aligned}$$ which hold at high precision.
The Delfino–Viti conjecture
---------------------------
Our semi-analytic bootstrap calculations give us access to the three-point structure constant $C_{(0,\frac12)(0,\frac12)(0,\frac12)} = \sqrt{D_{(0,\frac12)}^{aaaa}}$, which can be interpreted as the three-point connectivity [@prs19]. Let us compare this with the conjectured exact expression of this connectivity.
Delfino and Viti’s conjectured expression [@dv10] is the three-point structure constant of Liouville theory with $c\leq 1$, times a combinatorial prefactor $\sqrt{2}$ whose lattice origin was elucidated in [@ijs15]: $$\begin{aligned}
C_{(0,\frac12)(0,\frac12)(0,\frac12)} = \sqrt{2} C^{c\leq 1\text{ Liouville}}_{\Delta_{(0,\frac12)},\Delta_{(0,\frac12)},\Delta_{(0,\frac12)}}\ .
\label{dvconj}\end{aligned}$$ As far as we understand, Liouville theory appeared here not because it has any particular relation with the $Q$-state Potts model, but because it exists at generic central charges, has a diagonal field of dimension $\Delta_{(0,\frac12)}$, and is analytically solvable. Based on this information alone, we would a priori not expect the conjecture to be exactly true. And consistency with Monte-Carlo simulations of the $Q$-state Potts model [@prs19; @jps19] only tests the conjecture to a relatively low precision.
However, it was recently observed that connectivities of the $Q$-state Potts model are related to correlation functions of the RSOS model [@hgjs20]. In the critical limit, the latter model is described by analytically solvable CFTs of the type of Liouville theory and minimal models. This suggests that the Delfino–Viti conjecture may be exactly true. And this is what our numerical results confirm, with a precision of about $26$ significant digits [@bV2].
Let us emphasize that the $Q$-state Potts model is nevertheless not related to Liouville theory proper. The former has a discrete spectrum, the latter a continuous spectrum. The analytic expression for $C^{c\leq 1\text{ Liouville}}_{\Delta_{(0,\frac12)},\Delta_{(0,\frac12)},\Delta_{(0,\frac12)}}$ is valid in Liouville theory for $c\leq 1$ only [@rs15], while the three-point connectivity is valid under the much weaker condition $\Re c< 13$. The analytic expression for $C^{c\leq 1\text{ Liouville}}_{\Delta_{(0,\frac12)},\Delta_{(0,\frac12)},\Delta_{(0,\frac12)}}$ is the unique solution of certain crossing symmetry equations, and should be considered a universal quantity, although it happened to be first discovered in the context of Liouville theory.
Outlook
=======
Let us point out a few questions that arise from our results.
### Logarithmizing CFT
Using derivative fields, we found it relatively simple to derive logarithmic from non-logarithmic CFT objects: we have *logarithmized* Verma modules with zero or one null vector, as well as the associated correlation functions and conformal blocks. It would be interesting to understand this operation at a more formal level, in order to apply it to more complicated situations, including higher-dimensional CFT.
In two dimensions, *logarithmizing* Verma modules with null vectors leads to so-called staggered Virasoro modules [@kr09]. For $\beta^2\in \mathbb{Q}_{<0}$, there are infinitely many null vectors, and the resulting representations are relevant in CFTs such as critical percolation. Some of our results can be directly applied to these staggered Verma modules, as we saw in Section \[sec:2pt\] when studying logarithmic couplings in a few examples. It would be interesting to study the applicability of our results for rational $\beta^2$ more systematically.
An alternative to *logarithmizing* non-logarithmic CFT at rational central charge would be to *rationalize* logarithmic CFT at generic central charge, by taking limits $\beta^2\to \frac{p}{q}$. This would be straightforward if we wanted to numerically compute connectivities in critical percolation: we would just need to do it in the $Q$-state Potts model with a parameter $Q\approx 1$ i.e. $c\approx 0$. However, it would be more difficult to derive the space of states, structure constants and conformal blocks at $c=0$: *rationalizing* is already known to be non-trivial in simpler, non-logarithmic CFTs [@rib18].
If *logarithmizing* and *rationalizing* could be defined precisely, a natural question would be whether they commute.
### Towards a solution of the $Q$-state Potts model
With our high-precision checks of analytic conjectures for structure constants or ratios thereof, we provided additional evidence that the $Q$-state Potts model may be analytically solvable. Of course, solving the model involves computing not just four-point connectivities, but also more general correlation functions. Connectivities have a few nice peculiarities, for instance their structure constants obey slightly stronger linear relations as we saw in Section \[sec:linrel\], but this should not make an essential difference.
Meanwhile, the problem of numerically computing four-point connectivities to arbitrary precision is now effectively solved. After numerically determining four-point structure constants by solving crossing symmetry equations, we can use the structure constants for computing connectivities. An estimate of the precision of such computations is given by the lowest nonzero deviation of a structure constant, for example $O(10^{-24})$ for the last table of Eq. .
### New crossing-symmetric four-point functions
When solving the crossing symmetry equation , we found that the connectivity $P_{abab}$ is just one element of a space of solutions, which we believe to be infinite-dimensional. This echos a speculation of [@hjs20] (Section 3.3), which predicted the existence of multiple solutions of crossing symmetry equations, based on the freedom to change the weights of non-contractible loops in the discretized model. Nevertheless, it is not yet clear what our solutions describe, or to which CFT they belong. There are not too many known solutions of crossing symmetry at generic central charge [@wvcs], so these solutions may be worth investigating.
Acknowledgements {#acknowledgements .unnumbered}
================
We are grateful to Linnea Grans-Samuelsson, Yifei He, Jesper Jacobsen, and Hubert Saleur, for discussing their work on the $Q$-state Potts model. We are also indebted to them, as well as to Miguel Paulos, for comments on the draft of this article. We would like to thank Riccardo Guida for help with speeding up Python code. We are grateful to Raoul Santachiara and Nina Javerzat for stimulating discussions. Many thanks to David Ridout for very stimulating exchanges on the draft of this article, which in particular led us to pay more attention to the logarithmic coupling.
We acknowledge support from the ERC grant ReNewQuantum, ERC-2018-SyG 810573.
|
{
"pile_set_name": "ArXiv"
}
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---
abstract: 'It is well-known that extreme ultraviolet (EUV) emission emitted at the solar surface is absorbed by overlying cool plasma. Especially in active regions dark lanes in EUV images suggest that much of the surface activity is obscured. Simultaneous observations from the Interface Region Imaging Spectrograph (IRIS), consisting of UV spectra and slit-jaw images (SJI) give vital information with sub-arcsecond spatial resolution on the dynamics of jets not seen in EUV images. We studied a series of small jets from recently formed bipole pairs beside the trailing spot of active region 11991, which occurred on 2014 March 5 from 15:02:21 UT to 17:04:07 UT. There were collimated outflows with bright roots in the SJI 1400 [Å]{} (transition region) and 2796 [Å]{} (upper chromosphere) that were mostly not seen in AIA 304 [Å]{} (transition region) and AIA 171 Å (lower corona) images. The Si IV spectra show strong blue-wing but no red-wing enhancements in the line profiles of the ejecta for all recurrent jets indicating outward flows without twists. We see two types of Mg II line profiles produced by the jets spires: reversed and non-reversed. Mg II lines remain optically thick but turn into optically thin in the highly Dopper shifted wings.The energy flux contained in each recurrent jet is estimated using a velocity differential emission measure technique which measures the emitting power of the plasma as a function of line-of-sight velocity. We found that all the recurrent jets release similar energy (10$^8$ erg cm$^{-2}$ s$^{-1}$ ) toward the corona and the downward component is less than 3%.'
author:
- 'N.-H. Chen$^{1\ast,2}$ and D. E. Innes$^{2}$'
bibliography:
- 'jet\_draft.bib'
title: Undercover EUV solar jets observed by the Interface Region Imaging Spectrograph
---
Introduction
============
Jet-like eruptions, occur in coronal holes, active regions, and the quiet Sun over a broad range of spatial and temporal scales [@1992shibata; @1996Shimojo; @2007cirtain; @2010Moore; @2016innes]. They have been observed at wavelengths ranging from hard X-ray (HXR) to white light, revealing their multi-thermal plasma constitution [@2013chen]. An inversed-Y or Eiffel-tower structure characterizes their standard morphology and they are typically seen near cancelling and newly emerging magnetic fields [@1996Shimojo; @1996canfield; @2015cheung]. Therefore they are thought to be initiated by magnetic reconnection [@1992shibata; @2007shibata].
Jets are often associated with cool plasma surges [@1996canfield; @1999chae; @2007jiang; @2008chifor; @2015sterling; @2016mulay]. @1996canfield observed blue-shifted H$\alpha$ surges adjacent to X-ray jets, associated with moving magnetic features around a sunspot. @1999chae found recurring extreme ultraviolet (EUV) jets with correlated H$\alpha$ surges near cancelling magnetic flux in a decaying active region. Recently, @2016zeng reported co-spatial H$\alpha$, EUV and HXR footpoint emission and well-correlated H$\alpha$ and UV spires in small-scale chromospheric jets observed by the New Solar Telescope (NST). In these coordinated observations, X-ray/EUV jets and H$\alpha$ surges are seen from regions with evolving magnetic fields and are characterized by common features, such as a point-like brightening at the footpoints and a nearly collimated spire. These studies have shown the close association between X-ray/EUV jets and H$\alpha$ surges, although a few observational discrepancies were also noted, such as a time delay between the H$\alpha$ surges and EUV jets [@2007jiang]. Other cool eruptions, such as Ca II jets, are notable particularly because they share a similar inversed Y-shaped scheme as the hot X-ray jets in the corona but are much smaller and occur in the chromosphere [@2008nishizuka]. Furthermore, most of the UV/EUV jets have been seen at the edge of H$\alpha$ surges [@1996canfield; @1999chae]. This could be due to inhomogenous density or temperature distributions along the outflows or that the cool and hot plasma were ejected along adjacent field lines [@1995yokoyama; @2005anzer]. The acceleration process may be similar both in cool and hot eruptions, but the detailed processes have not been clearly determined.
To investigate the nature of the jet-like eruptions, we use data from the two solar missions: SDO [@pesnell2015solar] and IRIS [@2014IRIS] which provide images, spectra and magnetic field strengths from features in the photosphere, chromosphere, transition region and corona. The Atmospheric Imaging Assenbly [AIA; @2012lemen] on SDO consists of seven EUV and two UV channels with a 12 s and 24 s cadence and $1\arcsec.5$ spatial resolution of the entire solar disk. The lines are centered at 94, 131, 171, 193, 211, 304, 335, 1600, and 1700 Å. We use images from the Heliospheric and and Magnetic Imager [HMI; @2012schou] on SDO to investigate the underlying photospheric line-of-sight (LOS) magnetic field and intensity. The IRIS spectrograph observes in two bands: FUV (1332-1406 [Å]{}) and NUV (2783-2834 [Å]{}) with a spatial and spectral resolution of 0.33 and 26 m[Å]{}. The slit-jaw images (SJI) can be obtained simultaneously with spectra in the wavelength regions 1330, 1400, 2796, and 2830 [Å]{} with a spatial resolution of 0.16 pixel$^{-1}$. and a cadence of a few seconds.
The aim of this paper is to describe the initiation and evolution of recurrent jets from a region with moving magnetic features just outside the sunspot penumbra. In the following, we present a multi-wavelength analysis of the recurrent jets by using images from IRIS and AIA and spectra from IRIS. In section 2, the detailed spatial evolution is investigated from the simultaneous imaging of IRIS and AIA. The corresponding line profiles and the energy of the recurrent jets are described in section 3. Finally, the summary and discussion are given in section 4.
Observations
============
Overview of observations
------------------------
The events under study were a series of jets that recurred near the periphery of the trailing sunspot of AR11991 when it was near the disk center on 2014 March 5. The IRIS slit crossed the jets from east to west in a large, sparse, 4-step raster covering a field-of-view (FOV) $3\arcsec\times119\arcsec$ in 64 s with $1\arcsec$ step size and exposure time 15 s (Table 1). We concentrate on the IRIS Si IV spectra, IRIS SJI 1400 and 2796 images, and AIA 171, and 304 [Å]{} (hereafter, A171 and A304) observations with the objective of investigating the behaviour of each recurrent jet. The SJI 1400 [Å]{} images are dominated by Si IV line emission from the transition region and continuum emission from the low chromosphere. The 2796 [Å]{} images, dominated by Mg II k lines, are used to study the corresponding upper chromospheric behaviour of the jets. As for the two channels of AIA: A171 is emitted primarily from plasma with a formation temperature at $6\times10^{5}$ K so should be good for tracking the coronal evolution of the jets and A304, formed around the same temperature as the Si IV lines, is dominated by He II emission. Absorption by overlying neutral plasma may cause a darkening of the EUV emission from the jet and footpoint. To coalign these images, first, we scaled the AIA and HMI images to match the FOV and pixel size of the IRIS SJI images. Second, we coaligned the 2832 [Å]{} and HMI continuum images on prominent features, such as the sunspot. We further matched the 1600 [Å]{} with the HMI continuum images, and also the 2832 [Å]{} with the other SJI bandpass images separately. Then the SJI images can be coaligned to the rest of co-temporal AIA images. We also examined the LOS HMI magnetograms for clues on the initiation of the jets.
Here we use the strong Si IV lines and also Mg II lines for the spectral study and present the line profiles in the jet and loop. The shape of the Si IV line profile in the jet’s spire often consists of two components: a highly blue-shifted component, namely the moving outflow, and a slight red-shifted component, representing the background bulk plasma. To begin with the characteristics of the line profiles, such as the line width are obtained from double-Gaussian fits. More results will be discussed in section 3.
Recurrent jets
--------------
The development of the jets is best studied in the 1400 [Å]{} images. Figure 1 shows all the recurrent jets (labeled as Jet 1, 2 etc) from 15:02:21 UT to 17:04:07 UT in the 1400, 2796, 304, 171, and 171 [Å]{} running-difference images. The LOS magnetogram contours are overlaid on the leftmost bottom frame to show the associated magnetic bipoles. Eight jets were observed coming from the same location during the three hours observation. Jet 1 was the faintest. All of them show a clear collimated long spire with bright footpoints, namely a typical inversed-Y shaped geometry, which is in agreement with the standard solar jet model [@1996Shimojo]. Although the contrast of the 2796 [Å]{} images is lower, the identical bright footpoints are visible in these images as well. The spire itself in the 2796 [Å]{} images is actually dimmer than the surroundings. They have the same length as the bright ones but are morphologically wider. The dissimilar appearance is due to the line formation process. While the strong emission of Si IV line is proportional to the column density of transition region plasma along the LOS, the Mg II k line, emitted by cooler plasma, is optically thick and its intensity is not a direct reflection of the column of Mg II along the line-of-sight. The intensity is given by the source function at optical depth unity. Since the jets are darker than the surroundings, the source function and possibly the temperature of the Mg II plasma is cooler in the jet than in the surrounding. Investigation of the line profiles is required to understand the Mg II emission. The reason that most the jets are not visible in the EUV images is probably because the jet EUV emission was absorbed by overlying cool plasma.
These observations demonstrate the importance of the UV observations for revealing heated jet plasma below the overlying cold material and for identifying the jet footpoints. Before the IRIS-era, EUV together with H$\alpha$ images were often used to diagnose the evolution of solar jets and the accompanying surges [@2015chenJ; @2014zhang; @2014adams] so it is interesting to see how much additional information on the jet morphology is revealed by including the UV images. In the jets described here, typical features such as the bright footpoints and spire, are only partially visible in the EUV channels, which is also evidenced in the sequences of running-difference (RD) A171 images (column 5). The RD images are created from two consecutive images taken 12 s apart and show the intensity enhancement/decrease of the observed A171 emission in the eruptions. Only two of the jets (Jet 1 & 2) show simultaneous enhancements of the EUV with the 1400 [Å]{} jets. The other jets show no obvious response (mostly grey) in the EUV, even at their footpoints. The most likely interpretation is surge plasma absorbing the EUV as indicated by the darkening in the 2796 Å images. It is noted that there is also an abnormal intensity enhancement (white arrow) in Jet 2 resulting from the superposition of an additional jet launched from a different starting point along the path of the other jets (see movie aia\_2.mp4).
Figure 2 shows the temporal evolution of the photospheric LOS magnetic fields near the base of the recurrent jets. There were two obvious bipole regions labelled as A and B (white circles) on the south west of the trailing spot of AR 11991 (see bottom left image in Figure 1). The positive (negative) flux concentrations are outlined in the red (green) contours with the 10[%]{} (solid), 30[%]{} (dot) of the maximum magnetic field strength. In the region A, the positive flux separated and migrated gradually in a west-south direction as they emerged. Their migration was terminated when they cancelled with the negative flux. The recurrent jets were launched from region A which was also the site of the flux cancellation. Region B was more compact and slightly decomposed at the edge of the negative pole where the footpoint of the loops were situated.
To estimate the speed of each jet, we make a distance-time map, shown in Figure 3(a). The jets are traced via a line-cut along the axis of the spire in the SJI 1400 Å and appear as dark elongations (indicated by blue dashed lines) in the intensity-reversed map. The speed of each jet ranged from 50 to 200 which is identical to the speed of the jets observed by TRACE in the EUV [@1999chae]. We also measure their LOS velocity in the averaged 1402 [Å]{} line intensity map shown in figure 3(b). The Si IV line intensity is averaged over the fourth exposure over the selected pixels (green box in the lower right hand panel) for the jet’s spire. As each raster consists four 15 s exposures (plus CCD read-out time $\sim$1.5 s), the plot has a 64 s cadence. The Si IV 1402.773 [Å]{} was used as the reference wavelength for the measurement of Doppler shifts. The negative Doppler velocity gives the blue shift, namely the direction away from the Sun while the positive shows the red shift. Each jet instance was highly blueshifted ($>$100 ) except for the first jet which was missed by the slit (in the 4th exposure). The LOS velocity from spectra and the speed derived from the temporal evolution of SJI images are roughly equal, so the combination gives a jet speed between 70 and 300 . The bottom frame of figure 3 shows an example of a SJI image and co-temporal 1400 [Å]{} spectral image.
Time-dependent Spectroscopy
===========================
Spectra of jets
---------------
Figure 4 gives an overview of the line profiles seen in Si IV and Mg II lines at selected times. To study the spectra during the jets, we selected the pixels where the slit crossed the spire and the loops connecting the footpoint of the jet to the nearby region B (see previous section). The cyan and purple lines in figure 4 show the selected pixels and the line profile taken simultaneously with SJI is shown on the right together with the spectral image. We use 2796.34 [Å]{} and 2803.52 [Å]{} as the rest wavelengths of the Mg II k and h lines respectively for measuring the Doppler shifts. The results of fitting the Si IV spectra are shown with a double-Gaussian at jet’s spire after background emission from the continuum has been subtracted. In the Si IV spectra, every jet (blue solid line) had a highly blue-shifted asymmetric profile or component and almost no enhancement in the red wing, indicating that this is a collimated outflow without twists. The dominant blue component of each spire’s profile extends to 200 and the width ranged from 23-48 , suggesting a strong non-thermal upward outflow. The width of their red wings is bounded between 3-40 . During the jets, a single-Gaussian method is used to fit the narrow line profile at the corresponding loop pixels (red solid line). The loops’ profiles are slightly red-shifted and some have intense peak emission which is about a factor two stronger than the spires’ emission.
To better understand the behavior of the Mg II lines during the jets, we consider the k to h intensity ratio in the line core and wings. The Mg II line formation was discussed extensively in @2013a_leenaarts [@leenaarts_2013b] with emphasis on the structure of the lines near the core but the profiles from the jet spire are far from typical due to flows along the jet (Fig. 4), so it is difficult to select line core and peaks as done by Leenaarts. Instead we have measured ratios of the k to h at specific Doppler shift in order to determine whether the emission is optically thin in the jet. As noted, the intensity in the jet core is significantly less than in the surroundings. This is because the jet is above the typical chromosphere and in the core, the line is optically thick so one only sees emission from the optical depth equals one layer (e.g. @2013a_leenaarts). In the wings, the intensities in the jet are higher than in the surroundings and it is hard to tell if the Mg II lines from the jet is optically thin or thick. If it is optically thin then the whole column in the jet would contribute emission on top of the background which is coming from the temperature minimum. If it is optically thick then the emission is coming from the optical depth equal one layer in the jet and the emission from the temperature minimum is not seen.
We therefore plot the ratio of the two lines at 0, $\pm50$ and $\pm80$ in the jet and at 0 in the neighboring loops. Away from the line core we plot both with and without background subtraction. The background values are taken as the intensity 0, $\pm50$ and $\pm80$ before jetting. We take the average value of intensity at the different velocities (both k and h lines) in three selected rasters ($\sim$3 mins) and the standard deviation is used to determine the minimum/maximum background values used in the line ratio computations. The ratio is very sensitive to the height of the lines above the background when the difference is small. Therefore the accuracy of the ratio for weak emission is much less than for stronger emission.The error bars on the ratios reflect the difference between ratios computed with a maximum and minimum background level. The error bar is only applicable and shown when the line intensity is higher than the background level. In the core (0 ) of the jet’s spire, the k to h ratio is between 1.2 to 1.5 and reduces slightly (1.1$\sim$1.28) at loop pixels (see figure 5a). In the wings of the jets, the ratio with background ranged from 1.19 to 1.8 at $\pm50$ and from 0.7 to 1.5 at $\pm80$ . After the background subtraction, it is bounded by same range at -50 but drops down to less than 1 at +50 . In the high Doppler velocity jet ($\pm80$ ), several rise up to 2, indicating that the spires are possibly optically thin. The relation of the k and h lines intensity (figure 5d) reveals that Mg II lines at the core and wings of the jet are mostly optically thick even though the central reversal feature sometimes vanishes (e.g. Jet 6). But it could turn into optically thin in the high Doppler shifted wings.
The Mg II spectra at the loops show the standard line profiles with peaks on either side of the line core. Most cases have a stronger blueward peak (two nearly equivalent peaks at jet 6 and 8) for both the h and k lines, signifying downflowing material above the optical depth equals one height [@leenaarts_2013b]. The peak separation is between 25 and 50 which is larger than the separation width of Mg II peaks in a plage region reported by @2015carlsson.
Energetics of the recurrent jets
--------------------------------
In this section, we address the energy possibly contained in these recurrent jets. The analysis method adopted here is the Velocity Differential Emission Measure (VDEM) introduced by @1995newton for flare events and @1999winebarger [@2002winebarger] for explosive events in SOHO/SUMER observations. The VDEM gives a measure of the energy of the emitting plasma moving at the LOS velocity in the observed profiles (for more information, see @1995newton & @1999winebarger). It is defined by $$\hbox{VDEM}=n^{2}_{e} G(T_e) \frac{ds}{dv}$$ where *$n_{e}$* is the electron density, $G(T_e)$ is the temperature, $T_e$, dependent emissivity function of the line, *s* is the distance along the LOS and *v* is the velocity. When deriving the VDEM from the observed spectrum, we assume that the plasma is moving along the LOS direction with uniform density and pressure during the exposure. Also the observed emission is assumed to have been emitted from the plasma at the temperature of the peak in the emissivity function. Since the propagation direction of the jets here is oblique and not only radially outward, the true energy flux would be larger. Before applying the method to IRIS observations, we increased the photon counts (the minimum required photon counts suggested in @2002winebarger is $>$ 1000 counts) by first summing the line profiles of three consecutive spatial pixels centered at the position with the maximum width along the slit (mostly around the jet’s spire). We also took the thermal broadening of Si IV lines (6.88 at $T_e = 8\times10^4$ K), non-thermal broadening (decided in each profile) and the instrumental broadening of IRIS (3.9 , suggested in @2014IRIS) into account. The corresponding VDEM function is then produced after deconvolution of the observed spectrum from which the moments of the velocity can be calculated. Next, we are able to estimate the kinetic energy flux in the plasma outflow/infow by using equation (1) in @2002winebarger.
Table 2 summarizes the results of the estimated energy flux of each jet. Three directional components have been determined, including the energy flux, kinetic energy flux and skewed energy flux. The directional energy flux is the total energy carried by the (in-/out-) flows which consists of the total thermal enthalpy and the total non-thermal energy flux in the certain direction. The kinetic energy component represents the energy contained in the flows moving with an average velocity while the skewed component indicates the high energy contribution, e.g. the high velocity term of the spectral line profiles (two ends of red and blue wings). Our calculations reveal that the average energy flux contributed toward the corona (upward) was $2\times $10$^8$ ergs cm$^{-2}$ s$^{-1}$ and toward the chromosphere (downward) was $2.2\times $10$^6$ ergs cm$^{-2}$ s$^{-1}$. Among the average upward energy flux, 39.6% was from the kinetic energy flux. The downward kinetic energy was 1% of the average downward energy flux. It is noted that the kinetic is $<$ 50% of the total upward energy in each jet, suggesting that more than half of energy goes to thermal energy that heated the surrounding transition region plasma. The energy in the downward direction is very small.
1: $E_{up}$, $E_{down}$, $Ek_{up}$ and $Ek_{down}$ are the upward/downward total energy and upward/downward kinetic enegy separately.
2: The value of each energy flux is the multiples of 10$^6$, except for the $Ek_{down}$ that is for 10$^4$.
3: The precentage of $Ek_{up}$ to $E_{up}$ is gaven underneath and so does the downward direction.
Discussion and summary
======================
We studied the spatial and temporal evolution of recurrent jets both with AIA and IRIS observations in detail. These jets recurred above the active flux cancellation regions in the periphery of AR 11991. The spectroscopic studies of these jets suggest collimated outflows without twists. Although the formation of Si IV and Mg II lines are not the same, the regular features of a standard jet were commonly visible in the slit-jaw images. We also found discrepancies in the morphologies of the jets in the UV transition region and EUV transition region and corona emission (as shown in figure 1). One explanation is that pre-existing cool material lay between the jets and observer which caused the absorption of the EUV at short wavelengths [@2003innes; @2005anzer; @2009pontieu]. Another possibility is that the cool jets/ material were accelerated and ejected because of the slingshot effect caused by the magnetic reconnection if the cool plasma was situated around the reconnection sites [@1995yokoyama; @2008nishizuka]. The coexistence of cool and hot plasma ejections could also be due to the cooling of previous hot ejections/ material suggested in previous works [@1994schmieder; @1999alex; @2007jiang]. For our case here, the dim coronal emission appeared cospatially and cotemporally in the observed wavelengths which makes the cooling mechanism unlikely. And most of the cool/hot plasma around the reconnection sites would be exhausted after first ejection (drop by factor of 2 in the following ejection [@2010archontis]) so it could be difficult to supply enough cool plasma in the following ejections within the repetition period (5-15 mins) of jets 2-8 here. It is worth noting that dim coronal ejections recurred at the same region of AR 11991 throughout the whole day. The driving mechanism of such long duration dark/dim ejections still needs further investigations.
The line formation of Mg II lines involves complicated physics which is greatly influenced by the inhomogeneous solar stratification. The appearance of the central reversal behaviour in Mg II lines might be related to the ratio between collision to radiative de-excitation, the density and the temperature if a simplified radiative transfer model is assumed [@2013a_leenaarts]. We found some profiles in spire position were non-reversed with more symmetric profiles, e.g. the core in emission, but some were reversed. It might relate to the enhancement of the density or temperature in each jet-associated reconnection process. Some flare observations [@2015liu] suggest that the non-reversed profiles can be due to high coronal pressures that could be attributed to the evaporation flows of jets and the reversed ones is often associated with strong heating. We found the correlation of k to h line intensity ratio at the line core and wings remains optically thick, however, it might become optically thin under the highly Dopper shifted condition.
As noted in Si IV lines spectra (figure 4), the broader blueward component is, the stronger energy flux would be. Energetics of jets derived in Si IV lines demonstrated a similar magnitude in the upward energy flux with a downward energy flux of less than 3%. Our case show the average energy flux of these jets is two-order-magnitude larger than the explosive events reported in SUMER observations. The energy of those transition region jets or small explosive events, however, is not the major source to heat the corona or chromosphere. It is still significantly evidenced that these undercover solar jets, contained almost similar energy (or mass) as other observed jets, might be occulted frequently in current EUV observations.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We consider the problem of <span style="font-variant:small-caps;">locc</span> discrimination between two bipartite pure states of Fermionic systems. We show that, contrarily to the case of quantum systems, for Fermionic systems it is generally not possible to achieve the ideal state discrimination performances through <span style="font-variant:small-caps;">locc</span> measurements. On the other hand, we show that an ancillary system made of two Fermionic modes in a maximally entangled state is a sufficient additional resource to attain the ideal performances via <span style="font-variant:small-caps;">locc</span> measurements. The stability of the ideal results is studied when the probability of preparation of the two states is perturbed, and a tight bound on the discrimination error is derived.'
author:
- Matteo
- Paolo
- Alessandro
bibliography:
- 'addendum.bib'
title: Fermionic state discrimination by LOCC
---
The very concept of quantum information theory requires encoding distinguishable pieces of information on quantum states. In the simplest instance of encoding of classical information, the decoding procedure corresponds to the widely studied task of [*quantum state discrimination*]{} [@Helstrom1969; @Helstrom1976; @Ivanovic1987; @Dieks1988; @Peres1988; @Chefles2000; @Bergou2007; @Barnett2009]. In turn, the state discrimination task has been extensively studied in the scenario where states are shared by distant agents that are only allowed to use [*Local Operations and Classical Communication*]{} (<span style="font-variant:small-caps;">locc</span>) [@Walgate2000; @Virmani2001; @Jonathan1999]. These tasks are now exhaustively understood in the quantum realm.
On the other hand, real physical systems are Bosons or Fermions, and the latter are ruled by a theory that is a slight variation of the standard quantum one. The study of information processing in Fermionic theory has then various reasons, that are both practical and fundamental [@Bravyi2002]. Of particular importance is establishing analogies and differences between quantum and Fermionic implementation of specific information processing tasks. For example, it is known that quantum and Fermionic computation are equivalent, meaning that every quantum algorithm can be efficiently mapped to a Fermionic one, and viceversa [@Bravyi2002]. This implies e.g. that Fermionic processes are efficiently simulated by quantum computers [@Jordan2014]. In many other respects, however, the two theories present significant differences [@Wolf2006; @Banuls2007; @DAriano2014a].
In the present Letter, we study the task of <span style="font-variant:small-caps;">locc</span> state discrimination in the Fermionic theory. We show that, unlike the quantum case, in the typical situation <span style="font-variant:small-caps;">locc</span> discrimination is strictly suboptimal. We also derive conditions where ideal discrimination performances can be achieved via a <span style="font-variant:small-caps;">locc</span> protocol. These conditions are very sensitive to prior information about the probability of occurrence of the two states. Therefore, we study the behavior of <span style="font-variant:small-caps;">locc</span> protocols in the presence of a small perturbation of the ideal conditions. Moreover, we show that a pair of Fermionic systems in a maximally entangled state is a sufficient resource in addition to <span style="font-variant:small-caps;">locc</span> to achieve discrimination performances equivalent to the optimal one.
We briefly introduce the Fermionic quantum theory as the theory dealing with systems made of local Fermionic modes [@Bravyi2002; @Wolf2006; @Banuls2007; @Friis2013; @DAriano2014]. A Fermionic mode represents the counterpart of a qubit in the quantum theory and can be either empty or occupied by a single “excitation.” The states of Fermionic systems satisfy the *parity superselection rule* [@Schuch2004; @Kitaev2004; @Schuch2004a], i.e. superpositions of vectors having even or odd excitation numbers are forbidden. The latter can be derived as a consequence of the assumption that the elements of the Fermionic algebra are Kraus operators of local Fermionic transformations [@DAriano2014]. The generators of the Fermionic algebra $\varphi_i$, $i$ running over arbitrary sets of $N$ modes, fulfil the canonical anticommutation relations $\{\varphi_i, \varphi_j^\dagger\} = \delta_{ij}$ and $\{\varphi_i, \varphi_j\} = \{\varphi_i^\dagger, \varphi_j^\dagger\} = 0 \ \forall i, j$. Once we define the vacuum state $\ket{\Omega}$ as the common eigenvector of operators $\varphi^\dag_i\varphi_i$ with null eigenvalues, the Fermionic operators enable us to define the Fock states as $\ket{n_1\ldots n_N} \coloneqq (\varphi_1^\dagger)^{n_1} \cdots (\varphi_N^\dagger)^{n_N} \ket{\Omega}$ and the antisymmetrized Fock space $\FockSpace$ through the linear combination of all Fock states. We may label with the lowercase letters $e$, $o$ those sectors of the Fock space featuring even and odd parity, respectively. The Jordan-Wigner isomorphism [@Jordan1928; @Verstraete2005; @Pineda2010] is a crucial tool to handle the transformations and informational protocols in Fermionic theory. Indeed, it maps non-locally the Fermionic operator algebra to an algebra of transformations on qubits, thus allowing us to proceed with the usual quantum notation.
*The orthogonal case.*—In quantum theory, we may perfectly discriminate between any two orthogonal states $\ket\psi$, $\ket\phi$ of a bipartite system $\rA\rB$ through <span style="font-variant:small-caps;">locc</span> measurements [@Walgate2000]. We remind that the most general case of a quantum measurement is represented by a positive-operator valued measure (<span style="font-variant:small-caps;">povm</span>), i.e. a collection of effects (positive operators $0 \le S \le I$) that sum to the identity operator $I$. A necessary condition for a <span style="font-variant:small-caps;">povm</span> to represent a <span style="font-variant:small-caps;">locc</span> measurement is to be separable (<span style="font-variant:small-caps;">sep</span>). The effect $S$ is separable if there exists some operators $0 \le A_i, B_i \le I$ such that $S = \sum_i A_i \otimes B_i$, and a <span style="font-variant:small-caps;">povm</span> represents a separable measurement if it is exclusively made of separable effects. Moreover, we recall that <span style="font-variant:small-caps;">locc</span> <span style="font-variant:small-caps;">povm</span>s are a proper subset of <span style="font-variant:small-caps;">sep</span> <span style="font-variant:small-caps;">povm</span>s [@Bennett1999]. In the following we will use the acronyms <span style="font-variant:small-caps;">locc</span> and <span style="font-variant:small-caps;">sep</span> to denote the corresponding subsets of <span style="font-variant:small-caps;">povm</span>s.
We now give a sketchy summary of the result of Ref. [@Walgate2000]. Let us introduce the orthonormal basis $\{\ket{i}_\rA\}$ for Alice party. Due to the Schmidt decomposition, every bipartite pure state can be written as $$\label{eq:walg}
\ket{\psi} =\sum_{i = 1}^n \ket{i}_\rA \ket{\eta_i}_\rB,\quad \ket{\phi} =\sum_{i = 1}^n \ket{i}_\rA \ket{\nu_i}_\rB ,$$ where $\{\ket{\eta_i}_\rB\}$ and $\{\ket{\nu_i}_\rB\}$ are suitable sets of orthogonal states in Bob’s Hilbert space. Alice has to measure her system in a properly selected basis and send the outcome to Bob, who in turn menages to locally discriminate between two orthogonal states and infer the correct result. Such a basis always exists, as shown in [@Walgate2000].
We follow here a strategy similar to the quantum one in order to distinguish between two pure orthogonal states $\ket\psi,\ket\phi$ of a bipartite Fermionic system. First of all, we notice that whenever the two preparations have different parity, e.g. $\ket{\psi} \in \FockSpace_e (\rA\rB)$ and $\ket{\phi} \in \FockSpace_o (\rA\rB)$, it is always possible to perfectly discriminate between the two just through local measurements. Indeed, Alice and Bob have to locally measure the parity of their subsystems and if their outcomes match, then the provided state was even, otherwise it was the odd one. The nontrivial case then is that of two pure states with the same parity. Since the even and odd sector are equivalent under <span style="font-variant:small-caps;">locc</span>, it is not restrictive to focus on even vectors only. We introduce the following convenient notation for the even vectors $\ket{\psi}, \ket{\phi} \in \FockSpace_e(\rA\rB)$ $$\begin{aligned}
\begin{split}
&\ket{\psi} = \ket{\psi_E}+\ket{\psi_O},\\
& \ket{\phi} = \ket{\phi_E} + \ket{\phi_O},
\end{split}\label{eq:fermi-walg}\end{aligned}$$ and recalling the decomposition in Ref. [@Walgate2000], we decompose $\ket{\psi_E}=\sum_{i = 1}^n \ket{e_i}_\rA \ket{\eta^e_i}_\rB $, $\ket{\psi_O}=\sum_{i = 1}^n \ket{o_i}_\rA \ket{\eta^o_i}_\rB $, $\ket{\phi_E}=\sum_{i = 1}^n \ket{e_i}_\rA \ket{\nu^e_i}_\rB $, $\ket{\phi_O}=\sum_{i = 1}^n \ket{o_i}_\rA \ket{\nu^o_i}_\rB $, where $\{\ket{e_i}\}$, $\{\ket{o_i}\}$ are Alice orthonormal bases of even and odd vectors, respectively, while $\{\ket{\eta^x_i}_\rB\}$, and $\{\ket{\nu^x_i}_\rB\}$, for $x=e,o$ are Bob vectors resulting from the Schmidt decomposition. In general, the latter are not normalized and $\braket{\eta_i^x|\nu_i^x}_\rB\neq0$. We may indicate with the capitalized letters $E$ or $O$ those entities pertaining to the $E$- and $O$-spaces of $\FockSpace_e (\rA\rB)$, i.e. those subspaces where the parities of Alice’s and Bob’s subsystems are both even or odd, respectively. E.g. the $E$-part of vector $\ket{\psi}$ is defined as $\ket{\psi_E} = \sum_{i = 1}^n \ket{e_i}_\rA \ket{\eta^e_i}_\rB$, whereas the $O$-part is $\ket{\psi_O} = \sum_{i = 1}^n \ket{o_i}_\rA \ket{\nu^o_i}_\rB$. The orthogonality condition $\braket{\psi|\phi}=0$ generally reads $$\begin{aligned}
\label{eq:orthogen}
\braket{\psi_E|\phi_E}+\braket{\psi_O|\phi_O}=0.\end{aligned}$$
Let us consider as the first case the scenario where the two preparations have only one component. Then they have components either in complementary subspaces, e.g. $\ket{\psi} = \ket{\psi_E}$ and $\ket{\phi} = \ket{\phi_O}$, and it is trivially possible to discriminate via <span style="font-variant:small-caps;">locc</span> by measuring the local parities, or in the same subspace, e.g. $\ket{\psi} = \ket{\psi_E}$ and $\ket{\phi} = \ket{\phi_E}$. In the latter case the protocol reduces to the quantum one. Indeed, Alice selects the right basis $\{\ket{e_i}\}$ and lets Bob perfectly discriminate between $\ket{\eta^e_i}$ and $\ket{\nu^e_i}$, which are now orthogonal thanks to the result of Ref. [@Walgate2000]. Moreover, as proved in Ref. [@DAriano2014], product <span style="font-variant:small-caps;">povm</span>s in the Jordan-Wigner representation correspond to <span style="font-variant:small-caps;">locc</span> Fermionic <span style="font-variant:small-caps;">povm</span>s.
As the second case, we consider the situation where only one component out of the four $\ket{\psi_E},\ket{\psi_O},\ket{\phi_E},\ket{\phi_O}$ is null. Perfect discrimination is implementable through <span style="font-variant:small-caps;">locc</span> in this case as well. Let us take for instance the vectors $\ket{\psi}=\ket{\psi_E}+\ket{\psi_O}$, and $\ket{\phi} = \ket{\phi_O}$; Alice and Bob firstly measure the parity of their subsystem and if the outcome is even, they know for sure that the system has been prepared in the state $\ket{\psi}$. Otherwise, the state after the measurement is either $\ket{\psi_O}/\Norm{\psi_O}$ or $\ket{\phi_O}$, and the above strategy for the first case applies.
In the most general case all four components are non-null. If the two $E$- and $O$-parts are orthogonal—that is when $\braket{\psi_E|\phi_E} = \braket{\psi_O | \phi_O} = 0$—Alice and Bob can measure locally the parity of their systems, thus obtaining the post-measurement states $\ket{\psi'}=\ket{\psi_E}/\Norm{\psi_E}$ and $\ket{\phi'}=\ket{\phi_E}/\Norm{\phi_E}$ if the outcomes are both even, $\ket{\psi'}=\ket{\psi_O}/\Norm{\psi_O}$ and $\ket{\phi'}=\ket{\phi_O}/\Norm{\phi_O}$ if the outcomes are both odd. Consequently they reduced to the first case.
There is one situation left fulfilling condition , i.e. when $\braket{\psi_E|\phi_E}\neq0$ and $\braket{\psi_O|\phi_O}\neq0$. This case exhibits the main difference with respect to quantum theory. Consider for instance the states $1/\sqrt2(\ket{00}_\rA \ket{00}_\rB \pm \ket{01}_\rA \ket{01}_\rB)$. In this case, the decompositions in Eq. involves bases $\{\ket{\eta_i}_\rB\}$ and $\{\ket{\nu_i}_\rB\}$ where superpositions forbidden by the Fermionic superselection rule appear. Indeed, one has $i=\pm$ and $$\begin{aligned}
&\ket{\pm}_\rA\coloneqq\frac1{\sqrt2}(\ket{00}\pm\ket{01}),\\
&\ket{\eta_\pm}_\rB=\ket{\nu_\mp}_\rB\coloneqq\frac1{\sqrt2}(\ket{00}\pm\ket{01}).\end{aligned}$$
The last case can thus not be treated by straightforwardly applying the quantum strategy of Ref. [@Walgate2000]. The following theorem summarizes what we discussed so far, and shows that it is not possible to perfectly discriminate two states with $\braket{\psi_E|\phi_E}\neq0$ and $\braket{\psi_O|\phi_O}\neq0$ through <span style="font-variant:small-caps;">povm</span>s in <span style="font-variant:small-caps;">sep</span>, thus neither by means of <span style="font-variant:small-caps;">locc</span>.
\[th:orthogonal\_discriminability\] Let $\ket{\psi}$ and $\ket{\phi}$ be two pure, normalized and orthogonal states. Then the following statements are equivalent:
The even and odd parts are separately orthogonal, i. e. \[itm:orthogonal\_Fermion\]
$$\label{eq:even_discriminable_state}
\braket{\psi_E | \phi_E} = \braket{\psi_O | \phi_O} = 0 .$$
The two states are perfectly discriminable through <span style="font-variant:small-caps;">locc</span>. \[itm:orthogonal\_LOCC\]
The two states are perfectly discriminable through <span style="font-variant:small-caps;">sep</span>. \[itm:orthogonal\_SEP\]
It is trivial to see that *\[itm:orthogonal\_LOCC\]* $\Rightarrow$ *\[itm:orthogonal\_SEP\]*, whereas we have already shown above that *\[itm:orthogonal\_Fermion\]* $\Rightarrow$ *\[itm:orthogonal\_LOCC\]* thanks to [@Walgate2000]. We now focus on the implication *\[itm:orthogonal\_SEP\]* $\Rightarrow$ *\[itm:orthogonal\_Fermion\]* and wonder under what conditions one has $$\label{eq:operational_norm}
\max_{S \in \SEPSet} \Tr[(\ket{\psi}\bra{\psi} - \ket{\phi}\bra{\phi}) S]=1,$$ namely the condition for perfect discriminability via <span style="font-variant:small-caps;">sep</span>. The expression in Eq. clearly involves only the component of $S$ supported on the even subspace $\FockSpace_e(\rA\rB)$. Now, a necessary condition for a Fermionic effect $S$ supported on $\FockSpace_e(\rA\rB)$ to be <span style="font-variant:small-caps;">sep</span> it that $S = S_E + S_O$, where $S_E$ and $S_O$ have their support on the $E$-space and $O$-space, respectively (see the Supplemental Material). Consequently, the condition in Eq. is equivalent to $$\begin{aligned}
\begin{aligned}
\Tr\left[\left(\ket{\psi_E} \bra{\psi_E} - \ket{\phi_E} \bra{\phi_E}\right) S_E\right] &= 1,\\
\Tr\left[\left(\ket{\psi_O} \bra{\psi_O} - \ket{\phi_O} \bra{\phi_O}\right) S_O\right] &= 1 ,
\end{aligned}
\label{eq:perfect_condition}
\end{aligned}$$ for $S=S_E+S_O$ representing an effect in <span style="font-variant:small-caps;">sep</span>. Thus, it is possible to perfectly discriminate the two states through separable effects only if the $E$- and $O$-parts are perfectly discriminable separately, as required in Eq. .
*Ancilla assisted discrimination.*—We now show that one can overcome the limits of Theorem \[th:orthogonal\_discriminability\] by providing the two parties with an ancillary system prepared in a suitable pure entangled state $\ket\omega$. Let us take $$\label{eq:ancilla}
\ket{\omega}_{\rm AB} \coloneqq a \ket{00} + b \ket{11} \quad\text{for}\quad a, b \neq 0 ,$$ and consider the task of discriminating the new vectors $\ket{\psi'} \coloneqq \ket{\psi} \otimes \ket{\omega}$ and $\ket{\phi'} \coloneqq \ket{\phi} \otimes \ket{\omega}$. In particular, we will see that only a *maximally entangled* ancillary state—i.e. with $\abs a^2=\abs b^2=1/2$—enables perfect discrimination between every two pure Fermionic states, regardless of condition .
\[th:ancilla\_assisted\] It is always possible to perfectly discriminate between every two pure, normalized and orthogonal preparations $\ket{\psi}$ and $\ket{\phi}$ with <span style="font-variant:small-caps;">locc</span> and an ancillary system in a pure maximally entangled state $$\ket{\omega}_{\rm AB} = \frac{1}{\sqrt{2}} \left(\ket{00} + e^{i\varphi} \ket{11}\right) , \quad \varphi \in [0, 2\pi) .
\label{eq:ancimax}$$ Moreover, the same does not hold if the ancillary state is not maximally entangled.
We show here a sketch of the proof, the full rigorous derivation being given in the Supplemental Material. Let us consider the states $$\begin{aligned}
&\ket{\psi'} = \ket{\psi} \otimes \ket{\omega} =\ket{\psi'_O}+\ket{\psi'_E},\\
&\ket{\phi'} = \ket{\psi} \otimes \ket{\omega} =\ket{\phi'_O}+\ket{\phi'_E},
\end{aligned}$$ with $\ket{\psi'_E}=
a \ket{\psi_E00} + b \ket{\psi_O 11}$, $\ket{\psi'_O}
=b \ket{\psi_E 11} + a \ket{\psi_O 00}$, $\ket{\phi'_E}=a \ket{\phi_E 00} + b \ket{\phi_O 11}$, and $\ket{\phi'_O}=b \ket{\phi_E 11} + a \ket{\phi_O 00}$, and evaluate for $\abs{a}^2 = \abs{b}^2 = \frac{1}{2}$ the scalar products $$\braket{\psi_E' | \phi_E'} = \braket{\psi_O' | \phi_O'} = \frac{1}{2} \braket{\psi | \phi} = 0 .$$ The vectors $\ket{\psi'}$ and $\ket{\phi'}$ do satisfy Eq. , even if $\ket{\psi}$ and $\ket{\phi}$ may not. Thus, we are now able to apply the protocol of Theorem \[th:orthogonal\_discriminability\] to the new states as shown above. Condition is also necessary for perfect discrimination, as shown in the Supplemental Material.
*Optimal discrimination.*—If the orthogonality condition $\braket{\psi|\phi}=0$ is relaxed, the two states are clearly not perfectly discriminable. Hence, one looks for the protocol which minimizes the error probability—i.e. the probability of wrong detection. For this purpose, it is necessary to introduce our prior probabilities for the two states, given by the distribution $\{p,q\}$. In this case, the error probability reads $$\Perr \coloneqq \Tr[p\ket{\psi} \bra{\psi} \Pi_\phi + q\ket{\phi} \bra{\phi} \Pi_\psi] ,$$ where $\{\Pi_\psi,\Pi_\phi\}$ is the binary <span style="font-variant:small-caps;">povm</span> describing the discrimination protocol. We remind that by definition the <span style="font-variant:small-caps;">povm</span> satisfies $\Pi_\psi, \Pi_\phi \ge 0$ and $\Pi_\psi + \Pi_\phi = I$. In the quantum case, the optimal discrimination strategy corresponds to the <span style="font-variant:small-caps;">povm</span> $\{\ket{+}\bra+,\ket{-}\bra-\}$ diagonalizing the operator $$\label{eq:operator_delta}
\Delta \coloneqq p \ket{\psi} \bra{\psi} - q \ket{\phi} \bra{\phi} = \lambda_+ \ket{+} \bra{+} + \lambda_- \ket{-} \bra{-} ,$$ where $\lambda_+ > 0$, $\lambda_- < 0$ are the eigenvalues of $\Delta$, and $\braket{+ | -} = 0$ (see [@Helstrom1969; @Helstrom1976]). The corresponding error probability is [@Helstrom1976] $$\begin{aligned}
\Perr=\frac12\left(1-{\Norm\Delta_1}\right).
\label{eq:hell}\end{aligned}$$ In [@Virmani2001], the authors observe that optimal discrimination through <span style="font-variant:small-caps;">locc</span> of $\ket\psi$ and $\ket\phi$ with prior probabilities $p$ and $q$, respectively, is equivalent to perfect <span style="font-variant:small-caps;">locc</span> discrimination between $\ket{+}$ and $\ket{-}$ (see also Ref. [@Helstrom1976]), thus reducing the optimal case to an instance of perfect discrimination. While the latter is always possible in quantum theory, we know from Theorem \[th:orthogonal\_discriminability\] that in Fermionic theory this is true only if the eigenvectors satisfy $$\label{eq:optimal_LOCC_condition}
\braket{+_E | -_E} = \braket{+_O | -_O} = 0.$$ Otherwise, by Theorem \[th:ancilla\_assisted\] perfect <span style="font-variant:small-caps;">locc</span> discrimination requires a maximally entangled ancilla. As for the perfect discrimination case, also the conditions for optimal <span style="font-variant:small-caps;">locc</span> discrimination in Fermionic theory differ from the quantum ones only when the $E$- and $O$-components of $\ket+$ and $\ket-$ are all non-zero, and $\braket{+_E|-_E},\braket{+_O|-_O}\neq0$. For the latter case, we now prove a necessary and sufficient condition for achievability of optimal discrimination with <span style="font-variant:small-caps;">locc</span> that does not require diagonalization of $\Delta$.
\[th:Optimal\_LOCC\] Let $\rho = p \ket{\psi} \bra{\psi}$ and $\sigma = q \ket{\phi} \bra{\phi}$ be two pure and sub-normalized states for $p, q > 0$ and $p + q = 1$. They are optimally discriminable through <span style="font-variant:small-caps;">locc</span> if and only if they satisfy $$\label{eq:optimal_condition}
[\Delta, P_E] = 0 ,$$ where $\Delta$ is defined in Eq. and $P_E$ is the projector onto the $E$-subspace.
Since optimal discrimination between $\ket\psi$ and $\ket\phi$ is equivalent to perfect discrimination between $\ket+$ and $\ket-$, by Eq. optimal discriminability of the states $\ket\psi$ and $\ket\phi$ by <span style="font-variant:small-caps;">locc</span> is equivalent to the condition $$\label{eq:Fermionic_optimal_condition_v2}
\braket{+ | P_E | -} = \braket{+ | P_O | -} = 0 ,$$ where $P_O$ is the projector onto the $O$-subspace. Now, taking the difference of the first two members of Eq. , we can then express the <span style="font-variant:small-caps;">locc</span>-discriminability condition through the single expression $$\label{eq:Fermionic_optimal_condition_v3}
\braket{+ |( P_E - P_O )| -} = 0 .$$ Indeed, since $P_O = P_e - P_E$, where $P_e$ is the projection on the even subspace $\FockSpace_e(\rA\rB)$ of system $\rA\rB$, Eq. is equivalent to the requirement that the restriction of the projector $P_E$ onto the space $\Span\{\ket{\psi}, \ket{\phi}\}$ is diagonal in the basis $\{\ket{+}, \ket{-}\}$. The operators $\Delta$ and $P_E$ are simultaneously diagonalizable if and only $[\Delta, P_E] = 0$. Eq. is then equivalent to attainability of optimal discrimination between the two states $\rho$ and $\sigma$ via <span style="font-variant:small-caps;">locc</span>.
We may wonder what happens when condition is not satisfied. As we show in the next theorem, the best discrimination strategy through <span style="font-variant:small-caps;">sep</span> corresponds to measuring in the basis of eigenvectors of $\Delta_E$ and $\Delta_O$, defined as the restriction of the operator $\Delta$ onto the $E$- and $O$-subspaces, respectively. Such a strategy is <span style="font-variant:small-caps;">locc</span>.
\[th:optimal\_LOCC\_error\] Let $\rho = p\ket\psi \bra\psi$ and $\sigma = q \ket\phi \bra\phi$ be two pure sub-normalized states for $p, q > 0$ and $p + q = 1$. The optimal <span style="font-variant:small-caps;">sep</span> discrimination protocol is locally implementable through <span style="font-variant:small-caps;">locc</span> and its error probability reads $$\label{eq:optimal_LOCC_error}
\Perr^\SEPSet = \Perr^\LOCCSet = \frac{1}{2} (1 - \Norm{\Delta_E + \Delta_O}_1) ,$$ where $\Delta_E = P_E \Delta P_E$ and $\Delta_O = P_O \Delta P_O$.
This result can be obtained considering that $$ \Perr^\SEPSet = p - \max_{\Pi_\psi\in\SEPSet(\rA\rB)} \Tr[\Pi_\psi \Delta],$$ where $\Pi_\psi$ must be of the form $\Pi_\psi = \Pi_\psi^E + \Pi_\psi^O$ in order to comply with the separability condition, as observed in the proof of Theorem \[th:orthogonal\_discriminability\]. The result then follows.
The above result allows us to treat the case where we are restricted only to local measurements and Eq. does not hold for the preparations $\rho$, $\sigma$. Once we are given the pure states $\ket\psi$ and $\ket\phi$, the condition for optimal <span style="font-variant:small-caps;">locc</span>-discrimination of Eq. is fulfilled either for the vectors laying in the $E$- or $O$-space, i.e. $[\ket\psi\bra\psi, P_E] = [\ket\phi\bra\phi, P_E] = 0$, or if the probability $p$ satisfies $$\label{eq:unstable}
[\ket\psi\bra\psi, P_E] = \frac{1 - p}{p} [\ket\phi\bra\phi, P_E] .$$ Condition can be satisfied by a unique value of the prior probability $p$, unless $[\ket\psi\bra\psi,P_E]=[\ket\phi\bra\phi,P_E]=0$. However, we now show that optimal <span style="font-variant:small-caps;">locc</span> discrimination can achieve the performances of unconstrained protocols, provided that two ancillary Fermionic systems are used in a maximally entangled state. As discussed above, indeed, the problem of optimal discrimination between two pure states reduces to that of the orthogonal vectors $\ket+,\ket-$ in Eq. . Considering Theorem \[th:ancilla\_assisted\], we know that orthogonal states can be perfectly discriminated via <span style="font-variant:small-caps;">locc</span> provided a maximally entangled ancillary system is available. These two observations immediately lead to our last result.
Let $\rho = p\ket\psi \bra\psi$ and $\sigma = q \ket\phi \bra\phi$ be two pure sub-normalized states for $p, q > 0$ and $p + q = 1$. It is always possible to optimally discriminate between the two preparations via <span style="font-variant:small-caps;">locc</span> if we use an ancillary system in a pure maximally entangled state.
Equation introduces a strict condition on the prior probability of the preparations, which are always subject to noise. We show hereafter that if we introduce a small perturbation $\epsilon$ on the preparation probabilities of pair of states satisfying Eq. , the discrimination error probability increases at most linearly in $\epsilon$ with respect to the appropriate optimal <span style="font-variant:small-caps;">locc</span> protocol. Thus, we map $p \mapsto p + \epsilon$ and attain $$\begin{split}
\Delta^\epsilon \coloneqq& (p + \epsilon) \ket{\psi} \bra{\psi} - (q - \epsilon) \ket{\phi} \bra{\phi} \\
=& \Delta^{0} + \epsilon (\ket{\psi} \bra{\psi} + \ket{\phi} \bra{\phi}) ,
\end{split}$$ where $[\Delta^0, P_E] = 0$. At this stage, we estimate the error difference between the optimal <span style="font-variant:small-caps;">povm</span> $\mathbb{P}^0 \coloneqq \{\Pi_\psi, \Pi_\phi\}$ for $\epsilon = 0$, which is <span style="font-variant:small-caps;">locc</span> thanks to Theorem \[th:Optimal\_LOCC\], and the <span style="font-variant:small-caps;">locc</span>-optimal <span style="font-variant:small-caps;">povm</span> for the perturbed case $\Delta^\epsilon$. The error increases as $\delta\Perr \coloneqq \Perr(\mathbb{P}^0 | \Delta^\epsilon) - \Perr^\LOCCSet(\Delta^\epsilon) \ge 0$ where $\Perr(\mathbb{P}^0 | \Delta^\epsilon) = \Tr[(p + \epsilon) \ket\psi\bra\psi \Pi_\phi + (q - \epsilon) \ket\phi\bra\phi \Pi_\psi]$ and $\Perr^\LOCCSet(\Delta^\epsilon) = \frac{1}{2} (1 - \Norm{\Delta_E^\epsilon + \Delta_O^\epsilon}_1)$ as in Eq. . Suitably manipulating the expression for $\delta\mathcal{P}_{\text{err}}$ one obtains $$\begin{aligned}
\delta\Perr \leq k\abs\epsilon+g\epsilon,
\label{eq:bound}\end{aligned}$$ where $k,g$ are suitable constants depending only on $\ket\psi, \ket\phi$. The former inequality is as tight as possible: let us take indeed the states $\ket\psi = 1/\sqrt{2} \ket{00} + 1/\sqrt{2} \ket{11}$ and $\ket\phi = \alpha \ket{00} + \sqrt{1-\alpha^2}\ket{11}$, where $\alpha \coloneqq (1/\sqrt{2} + \xi)$, and $\xi$ belongs to a neighborhood of zero. In such a case, we have numerically assessed that the error difference $\delta\Perr$ exhibits a corner in $\epsilon = 0$ as $\xi \to 0$ (more details can be found in the Supplemental Material).
We also investigate the performance of the optimal <span style="font-variant:small-caps;">locc</span> protocol for $\epsilon\neq0$ in the neighborhood of a prior probability $p$ satisfying condition , by comparing its efficiency to that of the optimal unconstrained (i.e. entanglement-assisted <span style="font-variant:small-caps;">locc</span>) <span style="font-variant:small-caps;">povm</span>. Thus we estimate $\delta\Perr' \coloneqq \Perr^\LOCCSet(\Delta^\epsilon) - \Perr(\Delta^\epsilon) $ by means of Eqs. and , obtaining $$\begin{split}
\delta\Perr' &\leq \kappa\abs\epsilon,
\end{split}\label{eq:boundlocc}$$ for a suitable $\kappa$, thanks to the triangle inequality.
We remark that, in the case of a mismatch in the assessment of the prior probability $p$, also for unconstrained optimal discrimination—coinciding with ancilla-assisted <span style="font-variant:small-caps;">locc</span>—one has the same bound as in Eq. , with possibly different constants $k$ and $g$. This feature, however, must not be considered as an artefact of Fermionic theory. Indeed, the technique used to derive the bound in Eq. is very general and leads to the same behavior in the quantum case as well.
*Discussion.*—As in the quantum case, discrimination with separable and <span style="font-variant:small-caps;">locc</span> <span style="font-variant:small-caps;">povm</span>s in the Fermionic case achieve the same performances. Unlike in quantum theory, on the other hand, in Fermionic theory ideal state discrimination through <span style="font-variant:small-caps;">locc</span> is subject to non-trivial conditions. In this Letter, we derived the conditions under which <span style="font-variant:small-caps;">locc</span> discrimination achieves the ideal performances of unconstrained discrimination protocols. However, in the Fermionic case, ancilla-assisted <span style="font-variant:small-caps;">locc</span> protocols achieve ideal discrimination. One has to remark, though, that this is the case only for maximally entangled ancillary states. We finally studied the behavior of optimal protocols—which depend on prior probabilities of the states to be discriminated—if the prior conditions are subject to perturbation. A remarkable instability is observed, corresponding to a corner point in the curve representing the error probability excess due to non-optimized <span style="font-variant:small-caps;">povm</span>s. We stress that the latter phenomenon is not exclusive of Fermionic theory, as it occurs also in the quantum case.
We thank Massimiliano F. Sacchi for useful discussions and comments.
Separable effects
=================
In order to implement the parity superselection rule, the operator $0\leq S\leq I$ representing a separable effect supported on $\FockSpace_e(\rA\rB)$ must be of the form $$S = S_E + S_O,$$ where $S_E= \sum_i e_i \otimes e_i'$, $S_O=\sum_j o_j \otimes o_j'$, and $e_i, e_i', o_j, o_j' \ge 0$, with $$\begin{aligned}
\operatorname{Supp}(e_i) &\subseteq \FockSpace_e (\rA) & \operatorname{Supp}(e_i') &\subseteq \FockSpace_e (\rB) \\
\operatorname{Supp}(o_j) &\subseteq \FockSpace_o (\rA) & \operatorname{Supp}(o_j') &\subseteq \FockSpace_o (\rB) .\end{aligned}$$ Once the effect is applied to a Fermionic state $\tau \in \Stset(\rA\rB)$, the Born rule returns $$\Tr[\tau S] = \Tr[P_E \tau P_E S_E + P_O \tau P_O S_O] .$$ The above expression shows that any separable <span style="font-variant:small-caps;">povm</span> operates on the $E$- and $O$-parts of $\tau$ independently. In particular, in the proof of Theorem \[th:orthogonal\_discriminability\] we seek the maximum of $r \coloneqq \Tr[(\rho - \sigma) S]$, with $\rho = \ket\psi\bra\psi$ and $\sigma = \ket\phi
\bra\phi$, i. e. $$\begin{aligned}
r =& \Tr\Big[\left( \ket{\psi_E} \bra{\psi_E} - \ket{\phi_E} \bra{\phi_E}\right) S_E \\
&+ \left( \ket{\psi_O} \bra{\psi_O} - \ket{\phi_O} \bra{\phi_O}\right) S_O\Big].\end{aligned}$$ The latter achieves unit value iff one can find $S_E$ and $S_O$ such that $$\begin{aligned}
& \Tr[\rho S] = \Norm{\psi_E}^2 \braket{\tilde\psi_E | S_E | \tilde\psi_E} + \Norm{\psi_O}^2 \braket{\tilde\psi_O | S_O |\tilde\psi_O} = 1,
\\
& \Tr[\sigma S] = \Norm{\phi_E}^2 \braket{\tilde\phi_E | S_E | \tilde\phi_E} + \Norm{\phi_O}^2 \braket{\tilde\phi_O | S_O | \tilde\phi_O} = 0 ,
$$ where $\ket{\tilde\eta}\coloneqq\ket\eta/\Norm\eta$. However, due to the hypotheses assumed so far, we achieve the above conditions if and only if Eq. is satisfied.
Proof of Theorem \[th:ancilla\_assisted\]
=========================================
Alice and Bob are provided with an entangled ancilla in the state $\ket{\omega}$, as in Eq. . They now share two bipartite systems in the possible states $\ket{\psi'} = \ket{\psi} \otimes \ket{\omega}$ or $\ket{\phi'} = \ket{\phi} \otimes \ket{\omega}$, whose full expression can be obtained from $$\begin{aligned}
\begin{aligned}
\ket{\psi'_E} &=a \sum_{i = 0}^n \ket{e_i 0}_\rA \ket{\eta_i^e 0}_\rB + b \sum_{j = 0}^n \ket{o_j 1}_\rA \ket{\eta_j^o 1}_\rB\\
\ket{\phi'_E} &=a \sum_{i = 0}^n \ket{e_i 0}_\rA \ket{\nu_i^e 0}_\rB + b \sum_{j = 0}^n \ket{o_j 1}_\rA \ket{\nu_j^o 1}_\rB \\
\ket{\psi'_O}&=b \sum_{i = 0}^n \ket{e_i 1}_\rA \ket{\eta_i^e 1}_\rB + a \sum_{j = 0}^n \ket{o_j 0}_\rA \ket{\eta_j^o 0}_\rB\\
\ket{\phi_O'}&= b \sum_{i = 0}^n \ket{e_i 1}_\rA \ket{\nu_i^e 1}_\rB + a\sum_{j = 0}^n \ket{o_j 0}_\rA \ket{\nu_j^o 0}_\rB.
\end{aligned}
\label{eq:ancilla_pxi_prime}
\end{aligned}$$ Let $\Sigma_E\coloneqq \braket{\psi_E|\phi_E}$ and $\Sigma_O\coloneqq \braket{\psi_O|\phi_O}=-\Sigma_E$, where the last equality follows from the fact that $\Sigma_E+\Sigma_O=\braket{\psi|\phi}=0$. As shown in the body, there are cases where the ancilla is not needed, and clearly its presence cannot reduce the performances of <span style="font-variant:small-caps;">locc</span> discrimination. The remaining case is that where $\Sigma_E\neq0$. The necessary and sufficient condition for perfect <span style="font-variant:small-caps;">locc</span> discrimination between $\ket{\psi'}$ and $\ket{\phi'}$ of Eq. can then be written using Eq. as $$\begin{gathered}
\braket{\psi_E'|\phi_E'} = (\abs{a}^2 - \abs{b}^2)\Sigma_E=0.
$$ For $\abs{a}^2 = \abs{b}^2 = \frac{1}{2}$ the above condition is clearly satisfied. On the other hand, if $\Sigma_E\neq0$, discrimination by <span style="font-variant:small-caps;">locc</span> is not possible for $\abs{a}\neq\abs{b}$.
Extremal case for $\delta\Perr$ bound
=====================================
In the Letter we investigated the behavior of the discrimination error in the case where the prior probabilities slightly differ from the ideal ones. We are given two pure states $\ket\psi$, $\ket\phi$ and if there exists a probability distribution $\{p, q\}$ such that condition is satisfied, we proved that such a solution is unique and the optimal discrimination strategy is <span style="font-variant:small-caps;">locc</span>-implementable, unless $[P_E,\ket\psi\bra\psi]=[P_E,\ket\phi\bra\phi]=0$. Therefore, a small perturbation $\epsilon$ in the prior probability $p$ produces an increase of error probability of the protocol—which is optimized for the unperturbed case—with respect to the optimal <span style="font-variant:small-caps;">locc</span> one. For this purpose, we introduce the quantity $$\label{eq:Perr}
\delta\Perr \coloneqq \Perr(\mathbb{P}^0 | \Delta^\epsilon) - \Perr^\LOCCSet(\Delta^\epsilon).$$ Thanks to the triangle inequality we have that $$\begin{aligned}
\delta\Perr =&\frac12\left(\abs{\Norm{\Delta_E^\epsilon + \Delta_O^\epsilon}_1 - \Norm{\Delta^0}_1}\right.\\
&\left.+\epsilon\Tr[(\ket\psi\bra\psi+\ket\phi\bra\phi)(\Pi_\phi-\Pi_\psi)]\right) \\
&\le k\abs\epsilon+g\epsilon, \\
\intertext{for}
k\coloneqq&\frac12\norm{\delta^\epsilon_E+\Delta^\epsilon_O-\Delta^0},\\
g\coloneqq&\frac12\Tr[(\ket\psi\bra\psi+\ket\phi\bra\phi)(\Pi_\phi-\Pi_\psi)].\end{aligned}$$ Hence, the error difference $\delta\mathcal{P}_{\text{err}}$ is *sublinear*.
We numerically assessed that the bound above is indeed achieved by the states $$\begin{aligned}
\ket\psi &= \frac{1}{\sqrt{2}} \ket{00} + \frac{1}{\sqrt{2}} \ket{11} \\
\ket\phi &= \left(\frac{1}{\sqrt{2}} + \xi\right) \ket{00} + \frac{\gamma}{\sqrt2} \ket{11},\end{aligned}$$ where $\gamma \coloneqq \sqrt{1 - 2\sqrt{2} \xi - 2 \xi^2}$ and $\xi$ belongs to a neighborhood of zero. The condition for optimality of Eq. is fulfilled by $$p(\xi) = \frac{\gamma + \sqrt2 \gamma \xi}{1 + \gamma + \sqrt2 \gamma \xi} \quad \text{for} \quad \xi \in [0, 1 - \sqrt2/2)$$ and the terms of Eq. read $$\begin{aligned}
\Perr(\mathbb{P}^0 | \Delta^\epsilon) &= \Tr[(p + \epsilon) \ket\psi\bra\psi \Pi_\phi + (q - \epsilon) \ket\phi\bra\phi \Pi_\psi] \\
\Perr^\LOCCSet(\Delta^\epsilon) &= \frac{1}{2} (1 - \Norm{\Delta_E^\epsilon + \Delta_O^\epsilon}_1) .\end{aligned}$$ In Fig. \[fig:delta\_error\] we show a plot of the quantity $\delta\Perr$ versus $\epsilon$ and $\xi$. We observe that, letting $\xi$ vary in a neighborhood of 0 one gets arbitrarily close to the bound in Eq. . On the other hand, the same analysis shows that one cannot find any lower bound for $\delta\Perr$ better than $\delta\Perr\geq0$.
![The plot shows the difference $\delta\Perr$ between the error probability $\Perr(\mathbb{P}^0 | \Delta^\epsilon)$ in discrimination between $\rho = (p+\epsilon)\ket\psi\bra\psi$ and $\sigma = (1-p-\epsilon)\ket\phi\bra\phi$ with the <span style="font-variant:small-caps;">povm</span> that is optimal for discrimination between $p\ket\psi\bra\psi$ and $(1-p)\ket\phi\bra\phi$ and the error probability $\Perr^\LOCCSet(\Delta^\epsilon)$ in discrimination between the same states $\rho$ and $\sigma$ with the correct <span style="font-variant:small-caps;">locc</span>-optimal <span style="font-variant:small-caps;">povm</span>, as a function of $\epsilon$ and $\xi$, where $\ket\psi=1/{\sqrt{2}} (\ket{00} +\ket{11})$ and $\ket\phi = \alpha \ket{00} + \sqrt{1-\alpha^2}\ket{11}$, and $\alpha=1/{\sqrt{2}} + \xi$. The special value $p$ of the prior probability, corresponding to $\epsilon=0$, is such that $p\ket\psi\bra\psi$ and $(1-p)\ket\phi\bra\phi$ are ideally discriminable via <span style="font-variant:small-caps;">locc</span>. All the other values of $\epsilon$, on the other hand, lead to pairs of states $\rho$ and $\sigma$ that cannot be ideally discriminated via <span style="font-variant:small-caps;">locc</span>. For values of $\xi$ in a neighborhood of 0 the function $\delta\Perr$ gets arbitrarily close to the bound $\delta\Perr\leq k\abs\epsilon+g\epsilon$ for suitable constants $k$ and $g$.[]{data-label="fig:delta_error"}](DeltaError){width="8.6cm"}
Following exactly the same line as in the above derivation of the bound in Eq. , one can derive the bound in Eq. .
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We report first-principles density functional theory studies of native defects in lithium borohydride (LiBH$_{4}$), a potential material for hydrogen storage. Based on our detailed analysis of the structure, energetics, and migration of lithium-, boron-, and hydrogen-related defects, we propose a specific mechanism for the decomposition and dehydrogenation of LiBH$_{4}$ that involves mass transport mediated by native defects. In this mechanism, LiBH$_{4}$ releases borane (BH$_{3}$) at the surface or interface, leaving the negatively charged hydrogen interstitial (H$_{i}^{-}$) in the material, which then acts as the nucleation site for LiH formation. The diffusion of H$_{i}^{-}$ in the bulk LiBH$_{4}$ is the rate-limiting step in the decomposition kinetics. Lithium vacancies and interstitials have low formation energies and are highly mobile. These defects are responsible for maintaining local charge neutrality as other charged defects migrating along the material, and assisting in the formation of LiH. In light of this mechanism, we discuss the effects of metal additives on hydrogen desorption kinetics.'
author:
- Khang Hoang
- 'Chris G. Van de Walle'
title: Mechanism for the decomposition of lithium borohydride
---
\[sec:intro\]Introduction
=========================
Lithium borohydride (LiBH$_{4}$) is a potential candidate for hydrogen storage because of its high hydrogen density (18.4 wt%) [@Zuttel20031]. The major drawbacks of LiBH$_{4}$ that prevent it from practical use are its high decomposition temperature (370$^\circ$C) [@mauron2008] and slow hydrogen desorption kinetics. Although it has been reported that incorporation of some metal additives into the material can lower the decomposition temperature and enhance the kinetics [@Zuttel20031; @en4010185; @li2011], the mechanism behind the decomposition and dehydrogenation processes and the role of the metal additives are not really understood. As demonstrated in our previous work for other complex hydrides, first-principles calculations based on density functional theory can produce such an understanding by providing insights on the atomistic mechanism involved in mass transport and hydrogen release in the material [@peles_prb_2007; @hoang_prb_2009; @wilson-short_prb_2009; @hoang_angew].
LiBH$_{4}$ undergoes a polymorphic transformation from the ordered low-temperature orthorhombic phase to the disordered high-temperature hexagonal phase around 104$^\circ$C [@Soulie2002], and exhibits a slight hydrogen desorption of 0.3 wt% between 100 and 200$^\circ$C [@Zuttel20031]. The melting occurs around 270$^\circ$C. The first and second significant hydrogen desorption peaks start at 320 and 400$^\circ$C, respectively, and the desorption reaches its maximum around 500$^\circ$C [@Zuttel20031]. The overall reaction for LiBH$_{4}$ decomposition can be expressed as the following equation: $$\label{eq;decomp}
\mathrm{Li}\mathrm{B}\mathrm{H}_{4} \rightarrow \mathrm{LiH} + \mathrm{B} + \frac{3}{2}\mathrm{H}_{2},$$ which releases 13.8 wt% hydrogen when LiBH$_{4}$ is heated up to 900$^\circ$C [@Zuttel20031]. The decomposition of LiBH$_{4}$ may, however, involve several intermediate steps. It has been reported that hydrogen desorption in LiBH$_{4}$ is accompanied by the release of gaseous diborane (B$_{2}$H$_{6}$) [@jp073783k], which subsequently decomposes into B and H$_{2}$ at high temperatures [@cm100536a]. Some diboranes or higher boranes may, however, react with the not-yet-decomposed LiBH$_{4}$ to form Li$_{2}$B$_{12}$H$_{12}$ and possibly Li$_{2}$B$_{10}$H$_{10}$, which have been detected in experiments [@cm100536a; @orimo:021920; @jp710894t].
Experimental data suggested that the decomposition and dehydrogenation of LiBH$_{4}$ involve hydrogen and/or boron mass transport mediated by native defects. However, there is no consensus among these experimental reports about the diffusing species involved in the decomposition process, which could be single H atoms, (BH$_{4}$)$^{-}$ units, or BH$_{3}$ units [@Borgschulte2008; @shane2009; @gremaud2009; @borg_2010]. Regarding the activation energy for decomposition, Pendolino [*et al.*]{} [@pendolino2009] reported a value of 1.22 eV for pure LiBH$_{4}$. For comparison, Züttel [*et al.*]{} [@Zuttel20031] reported an activation energy of 1.62 eV for the decomposition of LiBH$_{4}$ mixed with SiO$_{2}$; other authors reported values of 1.36 and 1.06 eV for the ball-milled (3LiBH$_{4}$+MnCl$_{2}$) mixture [@Choudhury2009; @varinLiBH4].
In this paper we report first-principles studies of native point defects and defect complexes in LiBH$_{4}$. Based on our detailed analysis of the structure, energetics, and migration of the defects, we propose a specific atomistic mechanism that explains the decomposition and dehydrogenation of LiBH$_{4}$, and the effects of metal additives on hydrogen desorption kinetics. Comparison with the experimental work will be made throughout. Some results for hydrogen-related defects were reported previously [@hoang_prb_2009], but are included here for completeness.
\[sec:metho\]Methodology
========================
Our calculations are based on density functional theory, using the generalized-gradient approximation (GGA) [@GGA] and the projector-augmented wave method [@PAW1; @PAW2] as implemented in the VASP code [@VASP1; @VASP2; @VASP3]. For defect calculations in LiBH$_{4}$ (orthorhombic $Pnma$, 24 atoms/unit cell) [@Soulie2002], we used a (2$\times$2$\times$2) supercell which contains 192 atoms, and a 2$\times$2$\times$2 Monkhorst-Pack $\mathbf{k}$-point mesh [@monkhorst-pack]. The plane-wave basis-set cutoff was set to 400 eV and convergence with respect to self-consistent iterations was assumed when the total energy difference between cycles was less than 10$^{-4}$ eV and the residual forces were less than 0.01 eV/[Å]{}. The migration of selected point defects in LiBH$_{4}$ was studied using the climbing-image nudged elastic band method (NEB) [@ci-neb].
Throughout the paper we use defect formation energies to characterize different native defects in LiBH$_{4}$. The formation energy ($E^{f}$) of a defect is a crucial factor in determining its concentration. In thermal equilibrium, the concentration of the defect X at temperature $T$ can be obtained via the relation [@walle:3851; @janotti2009] $$\label{eq;concen}
c(\mathrm{X})=N_{\mathrm{sites}}N_{\mathrm{config}}\mathrm{exp}[-E^{f}(\mathrm{X})/k_BT],$$ where $N_{\mathrm{sites}}$ is the number of high-symmetry sites in the lattice per unit volume on which the defect can be incorporated, and $N_{\mathrm{config}}$ is the number of equivalent configurations per site. Note that the energy in Eq. (\[eq;concen\]) is, in principle, a free energy; however, the entropy and volume terms are often neglected because they are negligible at relevant experimental conditions [@janotti2009]. It is evident from Eq. (\[eq;concen\]) that defects with low formation energies will easily form and occur in high concentrations.
The formation energy of a defect X in charge state $q$ is defined as [@walle:3851] $$\begin{aligned}
\label{eq;eform}
\nonumber
E^f({\mathrm{X}}^q)=E_{\mathrm{tot}}({\mathrm{X}}^q)&-&E_{\mathrm{tot}}({\mathrm{bulk}})-\sum_{i}{n_i\mu_i} \\
&+&q(E_{\mathrm{V}}+\mu_{e}),\end{aligned}$$ where $E_{\mathrm{tot}}(\mathrm{X}^{q})$ and $E_{\mathrm{tot}}(\mathrm{bulk})$ are, respectively, the total energies of a supercell containing the defect X, and of a supercell of the perfect bulk material; $\mu_{i}$ is the chemical potential of species $i$ (and is referenced to the standard state), and $n_{i}$ denotes the number of atoms of species $i$ that have been added ($n_{i}$$>$0) or removed ($n_{i}$$<$0) to form the defect. $\mu_{e}$ is the electron chemical potential, i.e., the Fermi level, referenced to the valence-band maximum in the bulk ($E_{\mathrm{V}}$).
The atomic chemical potentials $\mu_{i}$ are variables and can be chosen to represent experimental conditions. For defect calculations in LiBH$_{4}$, based on Eq. (\[eq;decomp\]) one can assume equilibrium of LiBH$_{4}$ with LiH and H$_{2}$, and the chemical potentials of Li, B, and H can be obtained from the equations that express the stability of LiH, H$_{2}$, and LiBH$_{4}$, which gives rise to $\mu_{\mathrm{Li}}$=$-$0.825, $\mu_{\mathrm{B}}$=$-$1.183, and $\mu_{\mathrm{H}}$=0 eV. This condition, however, corresponds to assuming equilibrium with LiH and H$_{2}$ at 0 K and 0 bar, and thus does not reflect the actual experimental conditions [@Zuttel20031]. One can also assume equilibrium with B and H$_{2}$, but this scenario is unlikely to occur during the decomposition of LiBH$_{4}$, given that both B and H$_{2}$ may not be formed directly from LiBH$_{4}$ but through the decomposition of the intermediates such as B$_{2}$H$_{6}$ [@cm100536a]. In the following presentation of defect formation energies, we assume equilibrium with LiH and B, which gives rise to $\mu_{\mathrm{Li}}$=$-$0.431, $\mu_{\mathrm{B}}$=0, and $\mu_{\mathrm{H}}$=$-$0.394 eV. This condition corresponds to assuming equilibrium with LiH and H$_{2}$ gas at 610 K and 1 bar [@H2gas], which is close to the decomposition temperature (643 K) of LiBH$_{4}$ [@mauron2008].
\[sec:defects\]Point Defects and Complexes
==========================================
The compound can be regarded as an ordered arrangement of (Li)$^{+}$ and (BH$_{4}$)$^{-}$ units. In its electronic structure, the valence-band maximum (VBM) consists of the bonding states of B $p$ and H $s$, whereas the conduction-band minimum (CBM) consists predominantly of the antibonding states of B $p$ and H $s$. The calculated band gap is 7.01 eV [@hoang_prb_2009]. In such an insulating, large band gap material, native point defects are expected to exist in charged states other than neutral, and charge neutrality requires that defects with opposite charge states coexist in equal concentrations [@peles_prb_2007; @hoang_prb_2009; @wilson-short_prb_2009; @hoang_angew]. We therefore investigated native defects in LiBH$_{4}$ in all possible charge states. Defect complexes are also considered, with special attention devoted to Frenkel pairs, i.e., interstitial-vacancy pairs of the same species. In the following, we analyze the structure, energetics, and migration of the defects in detail. The role of these defects in ionic conduction and the decomposition of LiBH$_{4}$ will be discussed in Sec. \[sec:disc\].
Hydrogen-related defects
------------------------
![(Color online) Calculated formation energies of hydrogen-related defects in LiBH$_{4}$, plotted as a function of Fermi energy with respect to the valence-band maximum.[]{data-label="FE;H"}](LiBH4_FE_H){width="3.2in"}
![(Color online) Structure of (a) $V_{\mathrm{H}}^{-}$, (b) $V_{\mathrm{H}}^{+}$, (c) H$_{i}^{+}$, and (d) H$_{i}^{-}$ in LiBH$_{4}$. Large (gray) spheres are Li, medium (blue) spheres B, and small (red) spheres H.[]{data-label="struct"}](LiBH4_H){width="3.2in"}
Figure \[FE;H\] shows the calculated formation energies of hydrogen vacancies ($V_{\mathrm{H}}$), interstitials (H$_{i}$), and interstitial molecule (H$_{2}$)$_{i}$ in LiBH$_{4}$. We find that the positively charged hydrogen vacancy ($V_{\mathrm{H}}^{+}$) and negatively charged hydrogen interstitial (H$_{i}^{-}$) have the lowest formation energies over a wide range of Fermi-level values. The neutral hydrogen vacancy ($V_{\mathrm{H}}^{0}$) and interstitial (H$_{i}^{0}$) are energetically less favorable than their respective charged defects, which is a characteristic of negative-$U$ centers [@negativeU]. Near $\mu_{e}$=4.06 eV, where the formation energies of $V_{\mathrm{H}}^{+}$ and H$_{i}^{-}$ are equal, (H$_{2}$)$_{i}$ actually has the lowest formation energy. The creation of $V_{\mathrm{H}}^{-}$ in LiBH$_{4}$ involves removing a proton (H$^{+}$) from the LiBH$_{4}$ supercell, and this results in a BH$_{3}$ unit; see Fig. \[struct\](a). $V_{\mathrm{H}}^{+}$ is created by removing one H atom and an extra electron from the system, and this leads to formation of a BH$_{3}$-H-BH$_{3}$ complex (two BH$_{4}$ units sharing a common H atom); see Fig. \[struct\](b). The addition of a proton to create H$_{i}^{+}$ results in a BH$_{5}$ complex; see Fig. \[struct\](c). The creation of H$_{i}^{-}$, on the other hand, involves adding a H atom and an extra electron to the system, and the resulting H$_{i}^{-}$ stands next to three Li atoms with the average Li-H distance being 1.79 [Å]{}; see Fig. \[struct\](d). Finally, (H$_{2}$)$_{i}$ involves adding an H$_{2}$ molecule to the system. This interstitial molecule prefers to stay in the interstitial void with the calculated H$-$H bond length of 0.75 [Å]{}, being equal to that calculated for an isolated H$_{2}$ molecule.
For the migration of H$_{i}^{+}$, H$_{i}^{-}$, $V_{\rm{H}}^{+}$, and $V_{\rm{H}}^{-}$, we find energy barriers of 0.65, 0.41, 0.91, and 1.32 eV, respectively. The energy barriers for H$_{i}^{+}$, $V_{\rm{H}}^{+}$, and $V_{\rm{H}}^{-}$ are relatively high because the diffusion of these defects involves breaking B$-$H bonds. For example, the diffusion of $V_{\rm{H}}^{-}$ involves moving an H atom from a BH$_{4}$ unit to the vacancy. The saddle-point configuration in this case consists of a H atom located midway between two BH$_{4}$ units (i.e., BH$_{4}$-H-BH$_{4}$), an energetically favorable situation for $V_{\rm{H}}^{+}$ but unfavorable for $V_{\rm{H}}^{-}$. H$_{i}^{-}$, on the other hand, loosely bonds to three Li atoms and therefore can diffuse more easily. The barrier for H$_{i}^{-}$ given here is slightly lower than the preliminary value reported previously [@hoang_prb_2009].
Since hydrogen vacancies and interstitials are stable as oppositely charged defects, charge and mass conservation conditions suggest that these native defects may be created in the interior of the material in form of Frenkel pairs. Therefore, we have investigated the formation of hydrogen Frenkel pairs (H$_{i}^{+}$,$V_{\mathrm{H}}^{-}$) and (H$_{i}^{-}$,$V_{\mathrm{H}}^{+}$). We find that in these complexes the configurations of the individual defects are preserved. (H$_{i}^{+}$,$V_{\mathrm{H}}^{-}$) has a formation energy of 3.89 eV, and a binding energy of 0.28 eV with respect to the isolated constituents. The distance between the two defects in the pair (as measured by the B$-$B distance) is 3.55 [Å]{}, compared to the B$-$B distance of 3.64 [Å]{} in the bulk. (H$_{i}^{-}$,$V_{\mathrm{H}}^{+}$), on the other hand, has a formation energy of 2.28 eV and a binding energy of 0.73 eV. The distance from H$_{i}^{-}$ to the H atom near the center of $V_{\mathrm{H}}^{+}$ \[[*cf.*]{} Fig. \[struct\](b)\] is 4.02 [Å]{}. Thus, (H$_{i}^{-}$,$V_{\mathrm{H}}^{+}$) has a much lower formation energy than (H$_{i}^{+}$,$V_{\mathrm{H}}^{-}$), which is consistent with the results presented in Fig. \[FE;H\] where H$_{i}^{-}$ and $V_{\mathrm{H}}^{+}$ both have lower formation energies.
Hao and Sholl [@hao2009] recently reported first-principles calculations for hydrogen-related defects in LiBH$_{4}$, using a methodology similar to ours. The formation energies (evaluated with $\mu_{\mathrm{H}}$=0 eV) of the defects, except H$_{i}^{-}$, are in close agreement with our results obtained under condition (1) as reported in Table \[tab:libh\] (to within 0.1 eV). For H$_{i}^{-}$, our calculated formation energy is lower by 0.3$-$0.4 eV, suggesting that the H$_{i}^{-}$ configuration we identified is more stable. Hao and Sholl also considered neutral hydrogen divacancies in LiBH$_{4}$ (denoted as $V_{\rm 2H}$) by removing two H atoms from the system, resulting in a B$_{2}$H$_{6}$ unit with an ethane-like geometry. The formation energy of $V_{\rm 2H}$ was reported to be 1.14 eV [@hao2009], comparable to that (1.11 eV) obtained under the condition with $\mu_{\mathrm{H}}$=0 eV in our calculations. This divacancy can be regarded as a complex of $V_{\rm H}^{+}$ and $V_{\rm H}^{-}$ with a binding energy of 2.84 eV with respect to its individual constituents, although the structures of the individual defects are not preserved in the divacancy. Hao and Sholl [@hao2009] also noted that the hydrogen divacancy and interstitial hydrogen molecule have formation energies much lower than those of $V_{\rm H}^{+}$ and H$_{i}^{-}$. The sum of their calculated formation energies for $V_{\rm 2H}$ and (H$_{2}$)$_{i}$ is 1.56 eV, close to the value of 1.51 eV in our calculations. We have also explicitly calculated a $V_{\rm 2H}$-(H$_{2}$)$_{i}$ complex, finding a formation energy of 1.49 eV. Note that the formation energy of this complex is independent of the chemical potentials. The $V_{\rm 2H}$-(H$_{2}$)$_{i}$ complex therefore has almost zero binding energy (0.02 eV) with respect to its constituents, suggesting that, once created, it would readily dissociate into $V_{\rm 2H}$ and (H$_{2}$)$_{i}$.
Lithium-related defects
-----------------------
![(Color online) Calculated formation energies of lithium-related defects in LiBH$_{4}$, plotted as a function of Fermi energy with respect to the valence-band maximum.[]{data-label="LiBH4;FE;Li"}](LiBH4_FE_Li){width="3.2in"}
Figure \[LiBH4;FE;Li\] shows the calculated formation energies of lithium vacancies ($V_{\mathrm{Li}}$), lithium interstitials (Li$_{i}$), and $V_{\mathrm{LiH}}$ (removing Li and H) in LiBH$_{4}$. Among these defects, we find that Li$_{i}^{+}$ and $V_{\mathrm{Li}}^{-}$ have the lowest formation energies for the entire range of Fermi-level values. These two defects have equal formation energies at $\mu_{e}$=4.32 eV. The creation of $V_{\mathrm{Li}}^{-}$ involves removing a Li$^{+}$ ion from the system. This causes very small changes to the lattice geometry near the void created by the removed ion. On the contrary, $V_{\mathrm{Li}}^{+}$, created by removing a Li atom and an extra electron, strongly disturbs the system. Besides the void, there are two BH$_{4}$ units that come close and form a B$_{2}$H$_{8}$ complex which can be identified as H$_{i}^{+}$ plus $V_{\mathrm{H}}^{+}$. $V_{\mathrm{Li}}^{+}$ thus can be regarded as a complex of $V_{\mathrm{Li}}^{-}$, H$_{i}^{+}$, and $V_{\mathrm{H}}^{+}$, with a binding energy of 0.56 eV with respect to its isolated constituents. Regarding the interstitials, Li$_{i}^{+}$ is created by adding a Li$^{+}$ ion to the system. Like $V_{\mathrm{Li}}^{-}$, Li$_{i}^{+}$ does not cause much disturbance to the lattice geometry of LiBH$_{4}$. Li$_{i}^{-}$, however, strongly disturbs the system by breaking B$-$H bonds and forming H and BH$_{3}$ units which can be identified as H$_{i}^{-}$ and $V_{\mathrm{H}}^{-}$, respectively. This defect can, therefore, be considered as a complex of Li$_{i}^{+}$, $V_{\mathrm{H}}^{-}$, and H$_{i}^{-}$, with a binding energy of 0.52 eV. Finally, $V_{\mathrm{LiH}}^{0}$ can be regarded as a complex of $V_{\mathrm{Li}}^{-}$ and $V_{\mathrm{H}}^{+}$.
The migration of Li$_{i}^{+}$ involves an energy barrier as low as 0.30 eV, and the migration of $V_{\mathrm{Li}}^{-}$ involves a barrier of 0.29 eV. These values are small, suggesting that these two defects are highly mobile. For Li$_{i}^{-}$, which can be considered as a complex of Li$_{i}^{+}$, $V_{\mathrm{H}}^{-}$, and H$_{i}^{-}$, the migration barrier is given by the least mobile constituent [@wilson-short_prb_2009], i.e., 1.32 eV, the value for $V_{\mathrm{H}}^{-}$. Similarly, the estimated migration barrier of $V_{\mathrm{Li}}^{+}$ and $V_{\mathrm{LiH}}^{0}$ is 0.91 eV, the value for $V_{\mathrm{H}}^{+}$.
We also investigated possible formation of lithium Frenkel pairs. Since Li$_{i}^{-}$ and $V_{\mathrm{Li}}^{+}$ are complex defects, only (Li$_{i}^{+}$,$V_{\mathrm{Li}}^{-}$) was considered. This pair is, however, unstable at short pair distances because there is no energy barrier between Li$_{i}^{+}$ and the neighboring $V_{\mathrm{Li}}^{+}$. In order to avoid recombination, Li$_{i}^{+}$ and $V_{\mathrm{Li}}^{-}$ should be about 4.20 [Å]{} away from each other. At this distance, the binding energy of (Li$_{i}^{+}$,$V_{\mathrm{Li}}^{-}$) with respect to its isolated constituents is 0.04 eV (almost zero), and the formation energy is 0.95 eV. This indicates that once created (Li$_{i}^{+}$,$V_{\mathrm{Li}}^{-}$) will readily recombine or dissociate into Li$_{i}^{+}$ and $V_{\mathrm{Li}}^{-}$.
Boron-related defects
---------------------
![(Color online) Calculated formation energies of boron-related defects in LiBH$_{4}$, plotted as a function of Fermi energy with respect to the valence-band maximum.[]{data-label="LiBH4;FE;B"}](LiBH4_FE_B){width="3.2in"}
![(Color online) Structure of $V_{\mathrm{BH}_{3}}^{0}$ in LiBH$_{4}$. The defect can be regarded as a complex of $V_{\mathrm{BH_{4}}}^{+}$ (presented by empty spheres) and H$_{i}^{-}$ (staying near three Li$^{+}$ units).[]{data-label="struct;VBH3"}](LiBH4_VBH3n){width="3.0in"}
Figure \[LiBH4;FE;B\] shows the calculated formation energies of boron vacancies ($V_{\mathrm{B}}$), BH vacancies ($V_{\mathrm{BH}}$), BH$_{2}$ vacancies ($V_{\mathrm{BH_{2}}}$), BH$_{3}$ vacancies ($V_{\mathrm{BH_{3}}}$), and BH$_{4}$ vacancies ($V_{\mathrm{BH_{4}}}$) in LiBH$_{4}$. Only stable and low-energy defects are presented. We find that $V_{\mathrm{BH_{3}}}^{2+}$, $V_{\mathrm{BH_{4}}}^{+}$, $V_{\mathrm{BH_{3}}}^{0}$, and $V_{\mathrm{B}}^{3-}$ have the lowest formation energies for certain ranges of Fermi-level values. $V_{\mathrm{BH_{4}}}^{+}$ corresponds to the removal of an entire (BH$_{4}$)$^{-}$ unit from the system. There are very small changes in the local lattice structure surrounding this defect. $V_{\mathrm{BH_{3}}}^{0}$, on the other hand, involves removing one B and three H from a (BH$_{4}$)$^{-}$ unit in LiBH$_{4}$. This results in a void formed the the removed atoms and a H$^{-}$ staying near three Li$^{+}$ units with the average Li-H distance being 1.92 [Å]{}; see Fig. \[struct;VBH3\]. $V_{\mathrm{BH_{3}}}^{0}$, therefore, can be regarded as a complex of $V_{\mathrm{BH_{4}}}^{+}$ and H$_{i}^{-}$, with a binding energy of 1.31 eV. With such a high binding energy, even higher than the formation energy of H$_{i}^{-}$ (1.24 eV at $\mu_{e}$=4.32 eV), $V_{\mathrm{BH_{3}}}^{0}$ can occur with a concentration larger than either of its constituents under thermal equilibrium [@walle:3851]. The defect is, however, expected to dissociate into $V_{\mathrm{BH_{4}}}^{+}$ and H$_{i}^{-}$ at high temperatures. Like $V_{\mathrm{BH_{3}}}^{0}$, other boron-related defects can be regarded as complexes of $V_{\mathrm{BH_{4}}}^{+}$ and hydrogen-related defects. For example, $V_{\mathrm{BH}}^{0}$ is a complex of $V_{\mathrm{BH_{4}}}^{+}$, H$_{i}^{-}$, and (H$_{2}$)$_{i}$; $V_{\mathrm{BH_{2}}}^{+}$ a complex of $V_{\mathrm{BH_{4}}}^{+}$ and (H$_{2}$)$_{i}$; $V_{\mathrm{BH_{2}}}^{-}$ a complex of $V_{\mathrm{BH_{4}}}^{+}$ and 3H$_{i}^{-}$; $V_{\mathrm{BH_{3}}}^{2+}$ a complex of $V_{\mathrm{BH_{4}}}^{+}$ and H$_{i}^{+}$; and $V_{\mathrm{B}}^{3-}$ a complex of $V_{\mathrm{BH_{4}}}^{+}$ and 4H$_{i}^{-}$. The migration of $V_{\mathrm{BH_{4}}}^{+}$ involves an energy barrier of 0.27 eV, whereas for $V_{\mathrm{BH_{3}}}^{0}$, the barrier is at least 0.41 eV, given by the migration of H$_{i}^{-}$.
\[sec:disc\]Discussion
======================
----------------------- ------ ------------ -------------- -----------------------------------
Defect $E_m$ (eV) Constituents
(1) (2)
H$_{i}^{+}$ 1.99 2.78 0.65
H$_{i}^{-}$ 1.24 1.24 0.41
$V_{\rm{H}}^{+}$ 1.77 1.77 0.91
$V_{\rm{H}}^{-}$ 2.18 1.40 1.32
(H$_{2}$)$_{i}$ 0.40 1.19 -
$V_{\rm 2H}$ 1.11 0.33 - $V_{\rm H}^{+}$+$V_{\rm H}^{-}$
Li$_{i}^{+}$ 0.50 0.50 0.30
$V_{\rm{Li}}^{-}$ 0.50 0.50 0.29
$V_{\rm{BH_{4}}}^{+}$ 0.61 0.61 0.27
$V_{\rm{BH_{3}}}^{0}$ 0.54 0.54 0.41^*a*^ $V_{\rm{BH}_{4}}^{+}$+H$_{i}^{-}$
----------------------- ------ ------------ -------------- -----------------------------------
: Calculated formation energies ($E^{f}$) and migration energies ($E_{m}$) for selected defects in LiBH$_{4}$. Atomic chemical potentials are chosen to reflect equilibrium with LiH and H$_{2}$ gas at (1) 0 K and 0 bar and (2) 610 K and 1 bar. The formation energies for charged defects are taken at the Fermi-level position where Li$_{i}^{+}$ and $V_{\rm{Li}}^{-}$ have equal formation energies.[]{data-label="tab:libh"}
^*a*^Lower bound, estimated by considering the defect as a complex and taking the highest of the barriers of the constituents.
It emerges from our analyses in Sec. \[sec:defects\] that the structure and energetics of all possible native defects in LiBH$_{4}$ can be interpreted in terms of H$_{i}^{+}$, H$_{i}^{-}$, $V_{\rm{H}}^{+}$, $V_{\rm{H}}^{-}$, (H$_{2}$)$_{i}$, Li$_{i}^{+}$, $V_{\rm{Li}}^{-}$, and $V_{\rm{BH_{4}}}^{+}$, which can be regarded as elementary native defects. Table \[tab:libh\] summarizes key information for the most relevant native defects in LiBH$_{4}$. Defect formation energies are obtained with respect to two sets of atomic chemical potentials, assuming equilibrium with LiH and H$_{2}$ gas at (1) 0 K and 0 bar and (2) 610 K and 1 bar. Condition (1) is given here only for comparison since it does not reflect the actual experimental condition, as discussed in Sec. \[sec:metho\]. We find that Li$_{i}^{+}$ and $V_{\rm{Li}}^{-}$ are the charged native point defects with the lowest formation energies in both conditions. Therefore, in the absence of electrically active impurities or when such impurities occur in much lower concentrations than charged native defects, the Fermi-level position of LiBH$_{4}$ is determined by these two defects, which is at $\mu_{e}$=3.93 eV under condition (1) or 4.32 eV under condition (2) (hereafter this level will be referred to as $\mu_{e}^{\rm int}$, the Fermi-level position determined by intrinsic/native defects). We also find that the calculated formation energies of the defects, except H$_{i}^{+}$, $V_{\rm{H}}^{-}$, and (H$_{2}$)$_{i}$, are all independent of the hydrogen partial pressure and temperature, [*cf.*]{} Table \[tab:libh\]. Since Li$_{i}^{+}$ and $V_{\rm{Li}}^{-}$ have comparable migration barriers (0.30 and 0.29 eV), we expect that they contribute almost equally to lithium-ion conductivity. The activation energy for ionic conductivity is estimated to be 0.79 eV, the summation of the formation energy and migration barrier of $V_{\rm{Li}}^{-}$, which is in agreement with the reported experimental value (0.69 eV) [@matsuo:224103]. Note that our calculated migration barriers of Li$_{i}^{+}$ and $V_{\rm{Li}}^{-}$ are almost equal to that (0.31 eV) for Li migration obtained in first-principles molecular dynamics simulations by Ikeshoji [*et al.*]{} [@ikeshoji2011].
Decomposition mechanism
-----------------------
Let us now discuss the role of native defects in the decomposition of LiBH$_{4}$ into LiH, B, and H$_{2}$ \[i.e., Eq. (\[eq;decomp\])\]. It is important to note that the decomposition necessarily involves hydrogen and/or boron mass transport in the bulk of LiBH$_{4}$. In addition, local and global charge neutrality must be maintained while charged defects are migrating. Keeping these constraints in mind, we identify the following native defects as essential to the decomposition process:
First, H$_{i}^{-}$, which is expected to act as the nucleation site for the formation of LiH as discussed in Sec. \[sec:defects\]. The activation energy for self-diffusion of H$_{i}^{-}$ is 1.65 eV, the sum of its formation energy and migration barrier. Note that, given the relatively high formation energy of the (H$_{i}^{-}$,$V_{\rm H}^{+}$) pair, H$_{i}^{-}$ is not likely to be created inside the material via the Frenkel pair mechanism. Other hydrogen-related defects such as H$_{i}^{+}$, $V_{\rm{H}}^{+}$, and $V_{\rm{H}}^{-}$ have very high formation energies ([*cf.*]{} Table \[tab:libh\]) and hence low concentrations, and are expected not to play an important role. (H$_{2}$)$_{i}$ can occur with a high concentration in the bulk but it does not form by itself; below we comment on its formation in conjunction with $V_{\rm 2H}$, as suggested by Hao and Sholl [@hao2009].
Second, $V_{\mathrm{BH_{4}}}^{+}$ and $V_{\mathrm{BH_{3}}}^{0}$, which are needed for boron mass transport. These two defects have the lowest formation energies in the range of Fermi-level values near $\mu_{e}^{\rm int}$, [*cf.*]{} Fig. \[LiBH4;FE;B\]. The activation energies for the formation and migration of $V_{\mathrm{BH_{4}}}^{+}$ and $V_{\mathrm{BH_{3}}}^{0}$ are, respectively, 0.88 eV and 0.95 eV. Note that boron-related defects such as $V_{\mathrm{BH_{4}}}^{+}$ and $V_{\mathrm{BH_{3}}}^{0}$ can only be created at the surface or interface since the creation of such defects inside the material requires creation of the corresponding boron-related interstitials which are too high in energy.
Third, Li$_{i}^{+}$ and $V_{\mathrm{Li}}^{-}$, which can be created in the interior of the material in the form of a (Li$_{i}^{+}$,$V_{\mathrm{Li}}^{-}$) Frenkel pair. These low-energy and mobile point defects can act as accompanying defects in hydrogen/boron mass transport, providing local charge neutrality as H$_{i}^{-}$ and $V_{\mathrm{BH_{4}}}^{+}$ migrating in the bulk. Li$_{i}^{+}$ can also participate in mass transport that assists the formation of LiH. The activation energy for the formation and diffusion of (Li$_{i}^{+}$,$V_{\mathrm{Li}}^{-}$) in the bulk is 1.24 eV, which is the formation energy of the Frenkel pair plus the migration barrier of $V_{\mathrm{Li}}^{-}$.
Given these native defects and their properties, the decomposition of LiBH$_{4}$ can be described in terms of the following mechanism: $V_{\mathrm{BH_{4}}}^{+}$ and/or $V_{\mathrm{BH_{3}}}^{0}$ are created at the surface or interface. The creation of $V_{\mathrm{BH_{4}}}^{+}$ corresponds to removing one (BH$_{4}$)$^{-}$ unit from the bulk. Since (BH$_{4}$)$^{-}$ is not stable outside the material, it dissociates into BH$_{3}$ and H$^{-}$ where the latter stays near the surface or interface. The formation of $V_{\mathrm{BH_{3}}}^{0}$ at the surface or interface also leaves H$^{-}$ in the material and releases BH$_{3}$. It is well known that BH$_{3}$ can decompose into B and H$_{2}$, or dimerize to form diborane (B$_{2}$H$_{6}$). At room temperature, diboranes decompose to produce H$_{2}$ and higher boranes; whereas at higher temperatures (about 300$^\circ$C), diboranes decompose into B and H$_{2}$. Some diboranes or higher boranes may react with LiBH$_{4}$ to form Li$_{2}$B$_{12}$H$_{12}$ and Li$_{2}$B$_{10}$H$_{10}$ as detected in experiments [@cm100536a; @orimo:021920; @jp710894t].
From the surface or interface, H$^{-}$ diffuses into the bulk in form of H$_{i}^{-}$ which acts as the nucleation site for the formation of LiH from LiBH$_{4}$ according to Eq. (\[eq;decomp\]). Here, the highly mobile Li$_{i}^{+}$ will help maintain local charge neutrality in the region near H$_{i}^{-}$. In order to maintain the reaction, (BH$_{4}$)$^{-}$ and/or BH$_{3}$ has to be transported to the surface/interface, which is equivalent to $V_{\mathrm{BH_{4}}}^{+}$ and/or $V_{\mathrm{BH_{3}}}^{0}$ diffusing into the bulk. As $V_{\mathrm{BH_{4}}}^{+}$ is migrating, local charge neutrality condition is maintained by having the highly mobile $V_{\mathrm{Li}}^{-}$ in the vacancy’s vicinity. Note that, although the formation energy of $V_{\mathrm{BH_{3}}}^{0}$ is slightly lower than that of $V_{\mathrm{BH_{4}}}^{+}$ ([*cf.*]{} Table \[tab:libh\]) and thus can occur with a higher concentration, the defect is likely to dissociate into $V_{\mathrm{BH_{4}}}^{+}$ and $V_{\rm{H}}^{-}$ at high temperatures as discussed in Sec. \[sec:defects\], suggesting that these constituents may diffuse independently in the bulk of LiBH$_{4}$ during the decomposition process.
In this mechanism, the hydrogen-related diffusing species involved in the decomposition process are H$^{-}$ and (BH$_{4}$)$^{-}$ and/or BH$_{3}$ units, and the possible rate-limiting step is the formation and migration of H$_{i}^{-}$, $V_{\mathrm{BH_{4}}}^{+}$, $V_{\mathrm{BH_{3}}}^{0}$, or the (Li$_{i}^{+}$,$V_{\mathrm{Li}}^{-}$) pair in the bulk. Since H$_{i}^{-}$ gives the highest activation energy, we believe that the decomposition of LiBH$_{4}$ is rate-limited by the formation and migration of this defect. The activation energy for the decomposition of LiBH$_{4}$ is therefore 1.65 eV, the activation for self-diffusion of H$_{i}^{-}$. Our calculated value is thus higher than the reported experimental values (ranging from 1.06 to 1.36 eV) for pure LiBH$_{4}$ and the ball-milled (3LiBH$_{4}$+MnCl$_{2}$) mixture [@pendolino2009; @Choudhury2009; @varinLiBH4], but comparable to that (1.62 eV) for LiBH$_{4}$ mixed with SiO$_{2}$ reported by Züttel [*et al.*]{} [@Zuttel20031]. Note that, because the decomposition process occur at the surface or interface, ball milling that enhances the specific surface area and/or shortens the diffusion paths is expected to slightly enhance the hydrogen desorption kinetics via the kinetic prefactor.
Hao and Sholl [@hao2009] pointed out that the concentrations of $V_{\rm 2H}$ and (H$_{2}$)$_{i}$ can be much larger than those of $V_{\rm H}^{+}$ and H$_{i}^{-}$, and concluded that the former are the dominant hydrogen-defects in LiBH$_{4}$. Indeed, these neutral defects can have very low formation energies, as seen in Table \[tab:libh\]. However, $V_{\rm 2H}$ and (H$_{2}$)$_{i}$ alone cannot explain the formation of LiH in Eq. (\[eq;decomp\]) and as observed in experiment. In the bulk of LiBH$_{4}$, mass (and charge) conservation is required, and therefore $V_{\rm 2H}$ and (H$_{2}$)$_{i}$ can only be created in the form of a $V_{\rm 2H}$-(H$_{2}$)$_{i}$ complex, similar to a Frenkel-pair mechanism for the formation of charged hydrogen and lithium vacancies and interstitials in the bulk as presented in Sec. \[sec:defects\]. The activation energy associated with this complex is about 1.70 eV, obtained by adding the formation energy of $V_{\rm 2H}$-(H$_{2}$)$_{i}$ and the migration barrier of (H$_{2}$)$_{i}$, with the latter taken from Ref. [@hao2009]. This value is comparable to that associated with self-diffusion of H$_{i}^{-}$. The formation of $V_{\rm 2H}$-(H$_{2}$)$_{i}$ complexes in the bulk is thus in principle an alternative pathway for hydrogen transport, provided there are no additional barriers involved in the formation of the $V_{\rm 2H}$-(H$_{2}$)$_{i}$ pair, an issue not addressed by Hao and Sholl [@hao2009]. We note, however, that the decomposition of LiBH$_4$ via the reaction of Eq. (\[eq;decomp\]) necessarily involves the breaking of B-H bonds, which requires the involvement of a single-atom hydrogen-related point defect (for which we propose H$_i^-$).
Effects of metal additives
--------------------------
It should be noted that the Fermi level $\mu_{e}$ can be shifted away from $\mu_{e}^{\rm int}$, e.g., as electrically active impurities are incorporated into the material [@peles_prb_2007; @hoang_prb_2009; @wilson-short_prb_2009]. If this occurs, the formation energy, hence concentration, of charged native defects will be enhanced or reduced, depending on how the Fermi level is shifted. For $\mu_{e}$ $<$ 4.26 eV ([*cf.*]{} Fig. \[LiBH4;FE;B\]), $V_{\mathrm{BH_{4}}}^{+}$ has a lower formation energy than $V_{\mathrm{BH_{3}}}^{0}$, and is thus expected to be the dominant defect involved in boron mass transport. Lowering $\mu_{e}$ in this range will lead to an increase (decrease) in the activation energy associated with H$_{i}^{-}$ ($V_{\mathrm{BH_{4}}}^{+}$). Since the diffusion of H$_{i}^{-}$ is the rate-limiting step, this will increase the activation energy for decomposition. Increasing $\mu_{e}$, on the other hand, will decrease the activation energy. For $\mu_{e}$ $>$ 4.26 eV, $V_{\mathrm{BH_{3}}}^{0}$ is the dominant boron-related defect. Since the activation energy associated with $V_{\mathrm{BH_{3}}}^{0}$ is independent of $\mu_{e}$, lowering (increasing) $\mu_{e}$ in this range will only increase (decrease) the activation energy associated with H$_{i}^{-}$.
As reported previously [@hoang_prb_2009], transition impurities such as Ti, Zr, Fe, Ni, Pd, Pt, and Zn can be electrically active in LiBH$_{4}$ and effective in shifting the Fermi level. For example, we have found that, once Ti is incorporated into LiBH$_{4}$ at a certain lattice site with a concentration higher than that of the charged native defects, the Fermi level of the system will be determined by this impurity (hereafter referred to as $\mu_{e}^{\rm ext}$, the Fermi-level position determined by extrinsic impurities), which is at 4.16 eV (if Ti is incorporated on the B site), 4.37 eV (on the Li site), or 4.81 eV (at interstitial sites) [@hoang_prb_2009]. In light of the mechanism proposed above, Ti is effective in lowering the decomposition activation energy, hence enhancing the kinetics, if incorporated on the Li site or at interstitial sites (where $\mu_{e}^{\rm ext}$$>$$\mu_{e}^{\rm int}$$\equiv$4.32 eV), and ineffective if incorporated on the B site (where $\mu_{e}^{\rm ext}$$<$$\mu_{e}^{\rm int}$). Zr is expected to also enhance the kinetics of LiBH$_{4}$ because we find that the incorporation of Zr on the Li and B sites and at the interstitial sites give rise to $\mu_{e}^{\rm ext}$=4.60, 4.44, and 4.45 eV [@hoang_prb_2009], which are all higher than $\mu_{e}^{\rm int}$. On the other hand, we find that Fe, Ni, Pd, Pt, and Zn give rise to $\mu_{e}^{\rm ext}$$<$$\mu_{e}^{\rm int}$ [@hoang_prb_2009], and are thus expected not to be effective additives. These impurities may, however, lower the formation energy of $V_{\mathrm{BH_{4}}}^{+}$ and thus enhance the release of BH$_{3}$ and probably reduce the onset decomposition temperature.
Experimentally, it has been found that Ti- and Zn-containing additives such as TiO$_{2}$ [@Yu_LiBH4_2008], TiCl$_{3}$ or TiF$_{3}$ [@Au2008; @fang_2011], and ZnF$_{2}$ are effective in enhancing the kinetics [@Au2008], whereas FeCl$_{3}$ is ineffective [@Au2008]. Our results are therefore in agreement with the available experimental data in the case of Ti and Fe, whereas there is a disagreement in the case of Zn. One possible explanation for this discrepancy is that we have not yet investigated all possible interactions between Zn and the constituents of the host material. The halogen anion in ZnF$_{2}$ may also play a role, as it was reported to be the case for titanium halides [@fang_2011]. Finally, the addition of Ni to nanoconfined LiBH$_{4}$ has been found to slightly lower the onset temperature of hydrogen release but does not have a significant effect on the hydrogen desorption [@ngene_2011], which appears to be consistent with our conclusions for Ni discussed above. Note that it was reported more recently that mixing or ball milling of LiBH$_{4}$ with FeCl$_{2}$ resulted in a significant decrease in the decomposition temperature [@zhang2010], which seems to contradict to the previous report of LiBH$_{4}$ mixing with FeCl$_{3}$ [@Au2008]. Further theoretical and experimental studies are therefore needed to clarify this situation.
It is important to note that the incorporation of metal impurities and the formation of native point defects are processes that are very distinct and occur in different stages of preparation or use of the material. The point defects are formed during decomposition, in a process that is close to equilibrium, such that their concentration will be determined by their formation energy $-$ which is quite low, as seen in Table \[tab:libh\]. The metal impurities, on the other hand, are incorporated during initial processing of the material, often in a process such as ball milling, which can be highly energetic and potentially introduce impurities in non-equilibrium concentrations not directly related to their formation energy. This allows for incorporation of impurities with formation energies higher than those of the point defects.
\[sec:sum\]Summary
==================
We have carried out a comprehensive first-principles study of native point defects and defect complexes in LiBH$_{4}$. We find that lithium vacancies and interstitials have low formation energies and are highly mobile. These defects can participate in lithium-ion conduction, and act as accompanying defects in hydrogen and boron mass transport. We have proposed a specific mechanism for the decomposition of LiBH$_{4}$ that involves the formation and migration of H$_{i}^{-}$, $V_{\mathrm{BH_{4}}}^{+}$, $V_{\mathrm{BH_{3}}}^{0}$, and (Li$_{i}^{+}$,$V_{\mathrm{Li}}^{-}$) in the bulk LiBH$_{4}$. Based on this atomistic mechanism, we explain the decomposition and dehydrogenation, the rate-limiting step in hydrogen desorption kinetics, and the effects of metal additives on the kinetics of LiBH$_{4}$.
This work was supported by General Motors Corporation and by the U.S. Department of Energy (Grant No. DE-FG02-07ER46434). It made use of NERSC resources supported by the DOE Office of Science under Contract No. DE-AC02-05CH11231 and of the CNSI Computing Facility under NSF Grant No. CHE-0321368. The writing of this paper was partly supported by the Naval Research Laboratory through Grant No. NRL-N00173-08-G001.
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{
"pile_set_name": "ArXiv"
}
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---
author:
- 'T. Hendrix'
- 'R. Keppens'
- 'P. Camps'
bibliography:
- 'aa25498-14.bib'
title: Modelling ripples in Orion with coupled dust dynamics and radiative transfer
---
=1
[In light of the recent detection of direct evidence for the formation of Kelvin-Helmholtz instabilities in the Orion nebula, we expand upon previous modelling efforts by numerically simulating the shear-flow driven gas and dust dynamics in locations where the H$_{II}$ region and the molecular cloud interact. We aim to directly confront the simulation results with the infrared observations.]{} [To numerically model the onset and full nonlinear development of the Kelvin-Helmholtz instability we take the setup proposed to interpret the observations, and adjust it to a full 3D hydrodynamical simulation that includes the dynamics of gas as well as dust. A dust grain distribution with sizes between 5-250 nm is used, exploiting the gas+dust module of the MPI-AMRVAC code, in which the dust species are represented by several pressureless dust fluids. The evolution of the model is followed well into the nonlinear phase. The output of these simulations is then used as input for the SKIRT dust radiative transfer code to obtain infrared images at several stages of the evolution, which can be compared to the observations.]{} [We confirm that a 3D Kelvin-Helmholtz instability is able to develop in the proposed setup, and that the formation of the instability is not inhibited by the addition of dust. Kelvin-Helmholtz billows form at the end of the linear phase, and synthetic observations of the billows show striking similarities to the infrared observations. It is pointed out that the high density dust regions preferentially collect on the flanks of the billows. To get agreement with the observed Kelvin-Helmholtz ripples, the assumed geometry between the background radiation, the billows and the observer is seen to be of critical importance.]{}
Introduction
============
Sometimes a little push is all that is needed to make a seemingly stable fluid evolve into a turbulent state. Typically this transition is caused by a fluid instability, and many of these mechanisms have been studied extensively in the past decades (see e.g. @1961hhs..book.....C). The Kelvin-Helmholtz instability (KHI) is a notable example of this as it plays an important role in a wide range of different fluid applications such as for example oceanic circulation [@Haren], winds on planet surfaces [@1997QJRMS.123.1433C], the flanks of expanding coronal mass ejections [@2011ApJ...729L...8F], magnetic reconnection in the solar corona [@2003SoPh..214..107L], interaction between comet tails and the solar wind [@1980SSRv...25....3E], mixing of solar wind material into Earth’s magnetosphere [@2004Natur.430..755H], astrophysical jets and many others. While the KHI is a hydrodynamical instability, magnetic fields can alter its dynamics and cause stabilisation or further destabilise the setup. As the previous range of examples demonstrates, many of the relevant astrophysical fluids in the KHI is of importance display magnetic effects. In molecular clouds, the KHI has been linked to the formation of filamentary structures , as well as to turbulence formation. While the source of turbulence, observed in molecular clouds through the detection of non-thermal line-widths around $1\times10^5$ - $2\times10^5$ cm s$^{-1}$, is still debated, it has been linked at least partially to the KHI allowing to transfer energy to smaller scale structures . While the occurrence of the KHI in space is clearly established, direct evidence of ongoing instabilities are harder to obtain. At a distance of 412 pc [@2009ApJ...700..137R], the Orion nebula is the closest H$_{II}$ region. Its association with young massive stars and its apparent brightness make it an intensively investigated region over a large range of frequencies . As such, it is an ideal laboratory for investigation of smaller scale structure development. Recently @2010Natur.466..947B discussed mid-infrared observations of ripple-like structures on the edge of the Orion nebula’s H$_{II}$ region and the surrounding giant molecular clouds. The wave-like nature of this observation (see figure \[fig:berne\]), points to a mechanism with fixed periodicity in time or space. This periodic structure, in combination with the detection of a strong velocity gradient resulting in velocity differences up to $7\times10^5$ - $9\times10^5$ cm s$^{-1}$ leads @2010Natur.466..947B to propose that these ripples are manifestations of the KHI.\
Because of the high research interest in the Orion nebula and the surroundings regions, the physical conditions in the neighbourhood of the observed ripples are fairly well documented, providing an ideal case to numerically model the observed system. In @2012ApJ...761L...4B an effort was undertaken to numerically study the linear growth phase of a KHI with physical values deduced from observations. It was found that the used setup was indeed Kelvin-Helmholtz unstable for setups with magnetic field orientations close to perpendicular to the flow, and parallel to the separation layer between the H$_{II}$ and cloud region.\
In this work, our goal is to expand the numerical modelling of the ripples in Orion in a way in which the observations can be directly compared to the modelling itself. To do so, several ingredients are needed. First, the proposed setup (see sections \[physSetup\] and \[magp\]) is simulated using a 3D numerical hydrodynamical simulation from the start of the instability, through the linear phase and into the nonlinear phase. To perform these simulations we use the [MPI-AMRVAC]{} code [@2012JCoPh.231..718K; @2014ApJS..214....4P], with numerical properties as described in section \[NumMeth\]. In the mid-infrared observation a significant part of the radiation is due to dust emission. Therefore we use the gas+dust module of the [MPI-AMRVAC]{} code to model the dynamics of dust particles, which are drag-coupled to the gas. We use a range of dust sizes and model it self-consistently with the gas dynamics. Finally, to connect the dynamical simulations to the observations we use the [SKIRT]{} dust radiative transfer code [@2011ApJS..196...22B; @Camps201520] to emulate the radiation by the dust particles and the effect of the actual geometry of the observed system, as explained in section \[RadTrans\]. The properties of the outcome of these simulations are described in section \[results\] and the conclusions are discussed in section \[conclusions\].
![Observation of the ripples in Orion at 8 $\mu$m, taken with the Spitzer Infrared Array Camera. The spatial wavelength $\lambda$, the orientation of the phase velocity $V_{\phi}$, and the linear regime length $L_{lin}$ are identified in the image. Credit: figure (1) from @2012ApJ...761L...4B, reproduced by permission of the AAS.[]{data-label="fig:berne"}](./berne.jpg){width="\columnwidth"}
Model
=====
Physical setup {#physSetup}
--------------
The setup used here is similar to that of the 2D setup of @2012ApJ...761L...4B, but here adjusted to a full 3D configuration. The domain of the simulation is a cube with $L=0.33$ pc sides, and is initially divided in three regions along the $y$-axis: the upper part corresponds to the hot, low density H$_{II}$ region (n$_{II} = 3.34 \times 10^{-23}$ g cm$^{-3}$, T$_{II}$ = 10$^4$ K), the lower part represents the cold, high density molecular cloud (n$_c = 1.67 \times 10^{-20}$ g cm$^{-3}$, T$_{c} = 20$ K) and both are separated by a thin middle layer with thickness $D=0.01$ pc. This boundary layer is thus oriented perpendicular to the $y$-axis. Note that the choice of density and temperature result in thermal pressure equilibrium between the upper and lower region as $$\qquad p = \rho \frac{k_b T}{m_H \mu},$$ with $p$ the pressure, $k_b$ the Boltzmann constant, $m_H$ the mass of hydrogen and $\mu$ the average molecular weight, set to $\mu = 1$ here. The energy density of the gas , $e$, can be calculated using the equation of state, and gives $$\qquad e = \frac{p}{\gamma - 1} + \frac{\rho v^2}{2},$$ with $\gamma = 5/3$ the adiabatic constant and $v$ the velocity of the flow.\
To initialise the dust content in the simulation domain, we assume that the dust-to-gas mass density ratio has the canonical value of 0.01 [@1954ApJ...120....1S] in the molecular cloud region, and no dust is present in the hot H$_{II}$ region. We assume that the size distribution of dust particles, $n$, can be approximated as $n(a) \propto a^{-3.5}$ with the size of the particles, $a$, between 5 nm and 250 nm as was determined from excitation in the interstellar medium (ISM) by @1994ApJ...422..164K. We use four dust fluids to represent this power law size distribution with each fluid representing a part of the size distribution, chosen in a way in which the total dust mass in each dust fluid is the same (see ). In this way, the resulting representative size of dust grain in the four dust fluids are 7.9 nm, 44.2 nm, 105 nm, and 189 nm, respectively. The grain density of all dust fluids is set to that of silicate grains, i.e. 3.3 g cm$^{-3}$ [@1984ApJ...285...89D].\
The H$_{II}$ region has an initially uniform velocity of magnitude $v_0 = 10^6$ cm s$^{-1}$ in the direction parallel to our $x$-axis. @2012ApJ...761L...4B propose that this high velocity is due to *champagne flow*, the resulting high velocity flow when the expanding H$_{II}$ breaks trough the molecular cloud. This velocity is similar to the shear velocity derived from observation in @2010Natur.466..947B. In the molecular cloud region the velocity is initially set to zero. In contrast to @2012ApJ...761L...4B, where a hyperbolic tangent profile is used for both velocity and density, we use a linear profile in the middle layer that continuously links up with the constant velocities and densities on both sides of the layer. This is done in analogy with our previous work , as it allows to better quantify the linear stability properties.\
A perturbation is added by introducing an initial velocity component perpendicular to the boundary layer: $$\begin{aligned}
\qquad v_{y,0}(x,y,z) =& 10^{-3} v_0 \exp \left( -\frac{(y-M_y )^2}{2 \sigma_y^2} -\frac{(z-M_z)^2}{2 \sigma_z^2} \right) \sin{(k_x x)} \nonumber \\
\qquad + &10^{-4} v_0 \, \textrm{rect} (\frac{y}{5D}) (1 - 2\textrm{rand} ()) \label{perturb},\end{aligned}$$ with $\sigma_y = 5D$, $\sigma_z = L / 5$ and $M_y$ and $M_z$ being the $y$- and $z$- coordinates of the middle point of the separation layer. The first part on the right side of equation (\[perturb\]) adds a sine perturbation with wavelength $\lambda = k_x / 2\pi$. We adopt $\lambda = 0.11$ pc in accord with the observations in @2010Natur.466..947B. The second part on the right side of equation (\[perturb\]) adds random velocities[^1] between $-10^{-4} v_0$ and $10^{-4} v_0$ in a layer of thickness $5D$ around the middle of the separation layer. The velocity in the $z$-direction is seeded with a similar random term: $$\qquad v_{z,0}(x,y,z) = 10^{-4} v_0 \, \textrm{rect} (\frac{y}{5D}) (1 - 2\textrm{rand} ()).$$ The purpose of the exponential part in equation (\[perturb\]) in the $y$-direction is to preferentially locate the perturbation around the middle layer. The exponential part in the $z$-direction centres the perturbation around the middle of the $z$-axis to confine the instability development region. These random perturbations in the velocity break the symmetry of the setup, and allow in essence all unstable modes to develop spontaneously, although the fixed $\lambda$ wavelength in the $x$-direction gets preference.
Magnetic pressure {#magp}
-----------------
@2012ApJ...761L...4B take into account a magnetic contribution in their 2D setup as well, assuming a uniform magnetic field with a strength of $B = 200$ $\mu$G in the entire domain based on observations of surrounding regions [@2004ApJ...609..247A; @2005ASPC..343..183B]. Using the values of the physical setup (section \[physSetup\]) this results in a ratio between thermal and magnetic pressure $\beta_{pl} = p_t / p_M = 0.0173$, with $\beta_{pl} $ the plasma beta value, meaning that the magnetic pressure is dominant over the thermal pressure contribution. The dominance of magnetic over thermal pressure is confirmed by observations in the orion molecular cloud [@2014ApJ...795...13B], both for large and small scale structures. @2012ApJ...761L...4B note that the setup is most unstable when the magnetic field is perpendicular to the flow and parallel to the contact layer. In this configuration, a uniform magnetic field only contributes as an additional magnetic pressure $$\qquad p_M = \frac{B^2}{8\pi}.$$ This means that one can actually substitute the full MHD treatment by a HD treatment with an additional pressure term, in which the total pressure is raised while keeping the density fixed (thus artificially increasing the temperature). When calculating the thermal energy of the gas to quantify the coupling to the dust (see [@2014ApJS..214....4P]), this artificial term is subtracted to obtain the relevant temperature. To demonstrate that this approximation is valid, we compare evolution of an MHD setup with that of a HD + $p_M$ simulation in section \[2Dcomp\].
Numerical method {#NumMeth}
----------------
We use the [MPI-AMRVAC]{} code [@2012JCoPh.231..718K; @2014ApJS..214....4P] for all the hydrodynamical (HD) and magnetohydrodynamical (MHD) simulations. The dust module of [MPI-AMRVAC]{}, discussed in detail in , allows to add dust to a HD simulation by adding multiple dust fluids. These fluids follow the Euler equations with vanishing pressure [@rjl:dust] and couple to the gas fluid through a drag force term. Each dust fluid has its own physical properties such as grain size and grain material density. Typically we use multiple dust fluids with the same grain material density and different grain sizes to model the size distribution in the ISM.\
For the 3D simulations we use four levels of adaptive mesh refinement (AMR), resulting in an effective resolution of $448\times 1792\times448$ cells. The triggering of extra refinement levels is based on a combination of the gradients in the gas fluid and those in the dust fluid representing the largest grains. Because the actual physical domain is cube shaped, this resolution results in a four time higher resolution perpendicular to the flow (see section \[physSetup\]). This is necessary to resolve all small-scale variations that develop during the linear (and also the nonlinear) phase of the instability. The solution of the coupled gas+dust fluid equations is advanced using a total variation diminishing Lax-Friedrich (TVDLF) scheme with a two-step predictor-corrector time discretisation and a monotonised central (MC) type limiter [@1977JCoPh..23..263V]. To ensure stable time-stepping the timestep is limited by using a CFL number of 0.6 for gas and dust, as well a separate dust acceleration criterion based on the stopping time of dust grains [@2012MNRAS.420.2345L].
Radiative transfer {#RadTrans}
------------------
To be able to directly compare the output from the 3D hydrodynamical simulations with observations, post-processing of the data is performed with the Monte Carlo radiative transfer code SKIRT [@2011ApJS..196...22B; @Camps201520]. SKIRT simulates continuum radiation transfer in dusty astrophysical systems by launching a set of photon packages in a given wavelength range through the dust distribution obtained from our dynamical simulations. These packages are followed for several cycles of multiple anisotropic scattering, absorption and (re-)emission by interstellar dust, including non-local thermal equilibrium dust emission by transiently heated small grains. Emission from stochastically heated grains is used in all the results in this work and typically around 4 dust emission cycles are needed to come to equilibrium.\
To launch the packages into the domain, we use a (stellar) point-source at a given distance outside of the simulated domain as our source of initial photons. Photon packages in a wavelength range between 0.01 $\mu$m and 1000 $\mu$m are incorporated. In SKIRT we use exactly the same distribution of dust species as the one obtained from MPI-AMRVAC, meaning that the mass density distribution of the four dust fluids is used for each representative part of the grain size distribution and that, just like in the HD simulations, we adopt silicate properties for the grains in the radiative transfer.
Results
=======
2D analysis {#2Dcomp}
-----------
![Growth of the kinetic energy perpendicular to the bulk flow. The MHD and HD simulation that take into account the magnetic pressure are similar, while the HD simulation without magnetic pressure behaves differently. The 3D setup is also shown up to $t=0.01$ and has a growth rate similar to that of the 2D setup.[]{data-label="fig:linGrowth"}](./linGrowth.jpg){width="\columnwidth"}
![Gas density plots of the KHI in 2D and 2.5D after the end of the linear phase. The density units are in g cm$^{-3}$. In all figures the entire domain (0.33 pc $\times$ 0.33 pc) is shown. **Left:** A 2D simulation of the KHI in HD with dust and an artificial magnetic pressure term $p_M$ added to the total pressure at $t=0.007$ (6.84$\times10^4$ years). **Centre:** The same setup, but in 2.5D MHD with a magnetic field perpendicular to the plane, also at $t=0.007$. **Right:** A 2D HD simulation without the effect of a magnetic field added into the total gas pressure, at $t=0.02$ (1.95$\times10^5$ years). Note this figure is taken at a different time as the linear phase end later in this case.[]{data-label="fig:magNomag"}](./endLinear2.jpg){width="\columnwidth"}
To prove that an MHD setup with the magnetic field component perpendicular to the flow direction and parallel to the boundary layer can be reasonably approximated by a similar setup in HD but with added pressure, we simulate the setup discussed in sections \[physSetup\] and \[magp\] first in 2D, but in three variations: a HD simulation without a magnetic contribution, an MHD setup with magnetic field, and an HD simulation with the magnetic field contribution added to the pressure. The MHD setup is actually simulated in 2.5D, as it includes the information of the velocity and magnetic field perpendicular to the simulated plane. The simulated plane in 2D corresponds to a slice in the 3D simulation perpendicular to the $x-y$ plane and through the centre of the simulated domain. In figure \[fig:linGrowth\] the buildup of kinetic energy perpendicular to the flow direction is shown for all three 2D setups, and for the 3D run discussed further on. Clearly, for the MHD setup and the HD plus magnetic pressure setup the growth rate in the linear regime (up to $t=0.006$ in code units, or $\sim$ 5.87$\times10^4$ years) is the same. The growth rate is significantly slower when the magnetic pressure is ignored. Also, figure \[fig:magNomag\] shows that the formed structures are of similar size and shape in the two simulations where the magnetic pressure is taken into account. Small differences include the formation of small-scale structures on top of the larger structure. These small-scale perturbations are also present in the HD setup, but develop faster in the MHD simulation. The reason that they are less apparent in the HD simulation is because in the MHD case they seemingly grow faster due to small inhomogeneities (a decrease by $\approx 2\%$) in the magnetic field, leading to numerical differences that accumulate over time. When the magnetic pressure is not taken into account, it can be seen in figure \[fig:magNomag\] that the morphology is very different. Because the total pressure is lower, the Mach number for the flow at the boundary is higher, causing shocks to propagate. These shocks also cause the striped structure in the high density region. We will now further discuss a full 3D gas plus dust setup that has the pressure adjusted to account for the magnetic pressure effects.
3D model
--------
In figure \[fig:linGrowth\] it can be seen that the growth rate of the 3D simulation is comparable to that of the 2D simulations in which the effect of the magnetic field is taken into account. Due to the added computational cost in 3D, this simulation is only followed until $t=0.01$ in code units, or up to about 9.78$\times10^4$ year.
### Dust distribution {#dustDistri}
In previous work we found that in a 3D setup with the same density on both sides of the separation layer, the KHI can cause the dust density to increase by almost two orders of magnitude. These strong increases in dust density occur in filament-like locations between the vortices when dust is swirled out of the vortices and compressed into these regions. This process if strengthened further by additional 3D instabilities. Also, it was found that the process of dust density enhancement is stronger for larger dust particle sizes. Figure \[fig:maxDens\] shows that in the setup used here the growth in local dust density is less strong. During the end of the linear phase, i.e. up to time $t=0.006$ in figure \[fig:maxDens\], the maximal density increases gradually, and the rate of increase is proportional to the grain size. In the further nonlinear stage the densities still increase, however the relation between instantaneous local maximal density and grain size gets modified. Similarly to what was seen in , the density enhancements are significantly stronger in 3D than in 2D, where the maximum increase is less than $15\%$ for all dust species in the 2D case with magnetic pressure added. Clearly, 3D effects are paramount when studying dust growth.\
![Time evolution of the maximal density enhancements in the 3D simulation for all four dust fluids, with *dust 1* representing the smallest grains (7.9 nm) and *dust 4* the largest grains (189 nm).[]{data-label="fig:maxDens"}](./maxDens.jpg){width="0.9\columnwidth"}
The dust density enhancements are strongest in three distinct regions, which are indicated in figure \[fig:regions\]. Chronologically dust first accumulates in the convex outer region of the KH wave (the region labeled with 1 in figure \[fig:regions\]). This is due to the acceleration of dust by gas in the concave region when the gas swirls around the low pressure region created by the KHI. Next, the arc-like structure below the surface of the wave, i.e. region number 2 in figure \[fig:regions\], is formed. This region forms when the KHI accelerates the bulk of the gas upward into the low density region, and the dust is dragged with it. The location of the region is caused by a gradient in the drag strength, as the velocity difference between gas and dust is stronger under the region than above, causing the underlying dust to overtake the dust above it. The third dust gathering region is along the boundary between high and low density regions in between two successive waves or KHI rolls. A dust pile-up is seen here in the nonlinear stage when the velocity of the gas around the low pressure vortex is highest. In animated views one can see how the end point of the flow that passes over the crest of the waves moves from location 1 to a spread out region all along the density boundary, i.e. up to location 3 as indicated.\
While dust density increases up to a factor 10 are observed in these three regions for the four dust species, the actual location of these dust-gathering regions does not necessarily fully coincide for all dust species, similar to the findings in where a clear size-separation was evident. Also, the actual importance of the three regions is distinct for different grain sizes. Therefore, the increase of the total dust density will be less strong and distributed over a larger region. Furthermore, the strongest increases can be found in small local clumps, as can be seen in figure \[fig:rhodTot\], visualising the total dust density concentrations. Quantitatively speaking, while 14.76$\%$ of the total volume experiences a total dust density enhancement of more than 5$\%$, in only 0.03$\%$ of the total volume the total dust density more than doubles (regions indicated in orange and red in figure \[fig:rhodTot\]). This is in contrast with the 3D simulations in , where the high density dust is found in long filamentary structures and more than 4.5 $\%$ of the volume exhibits a doubling of the total dust density. The main differences reside in the adopted initial density contrast, as well as the fact that here only the molecular cloud region initially had dust.
![Density of the largest dust species ($a=189$ nm) in a slice from the 3D simulation ($z=0.165$pc) at $t=0.0065$ (6.36$\times10^4$ years). Only a part of the simulated region with an extend of 0.138 pc in the $x$-direction is shown. Three distinct regions of dust density enhancement are indicated with labels 1, 2 and 3 discussed in the text. The velocity field of the largest dust species in the $x-y$ plane is indicted with the use of vectors, the largest velocity are around $6 \times 10^5$ cm s$^{-1}$.[]{data-label="fig:regions"}](./flowkh2.jpg){width="0.8\columnwidth"}
![Volume plot of the total dust density at $t=0.01$ (9.78$\times10^4$ years). Only densities higher than the initial maximum density ($\rho_d = 1.67\times 10^{-22}$ g cm$^{-3}$) are visualised.[]{data-label="fig:rhodTot"}](./rhodTot_frame50_quad.jpg){width="\columnwidth"}
Modelling observations
----------------------
In the previous section we have outlined how the model setup from section \[physSetup\] evolves into a nonlinear 3D KHI. Next, we investigate how the simulated structures would look in synthetic observations. As described in section \[RadTrans\], the dust distribution of our 3D simulations is used as input for the SKIRT radiative transfer code. To see to which degree our simulations correspond to the actual observed structures (figure \[fig:berne\]), in addition to the hydrodynamical setup one has to take into account the orientation in relation to the observer, as well as the location of the light source(s). @2010Natur.466..947B indicated that the star $\theta^1$ Orionis C, a massive type O7V star located in the H$_{II}$ Trapezium region at a distance of $\sim$ 3.4 pc from the cloud, illuminates the ripples from behind with respect to the observer. In SKIRT the radiation of this star is simulated by adding a point source of photons at $d=3.4$ pc and inclination $\alpha$ with respect to the initial separation layer in the HD simulation, as illustrated in figure \[fig:geo\]. For the radiation of the star we use a model spectrum from with corresponds to a star with physical properties comparable to those of $\theta^1$ Orionis C[^2]. The location of the observer with respect to the simulated domain must also be specified in SKIRT. As shown in figure \[fig:geo\], the observer is placed at an angle $\beta$ with respect to the initial separation layer in the HD simulation.\
![Geometry of the stellar object (photon source) and observer location with respect to the structures in Orion, designated by independent angles $\alpha$ and $\beta$, respectively. In this image, the location of the source and observer are shown with respect to the KH features at t=0.084 (8.21$\times10^4$ years). The black-white image is actually a SKIRT image at 54 $\mu$m, where we see the radiation which is coming from dense and heated dust in the billow structures formed by the KHI. In this image, the observer is located perpendicular to the $x-y$ plane.[]{data-label="fig:geo"}](./geo.jpg){width="\columnwidth"}
Because the actual inclination between the observer, the billows and the background radiation source are hard to gauge from the observation, several different values of $\alpha$ and $\beta$ were tried to investigate their role. Table \[table:1\] gives an overview of several SKIRT geometries we will discuss here. An interesting setup to look at first is case D (figure \[fig:BDAC\], top right). With this arbitrary choice for the geometry ($\alpha=60^{\degree}$ and $\beta=90^{\degree}$) the result is rather different from the observations. While some periodicity is observable, no sharp elongated structures are seen. The diffuseness of the radiation in case D can be seen to be inherent to an observer angle of $90^{\degree}$. Figure \[fig:perp\] demonstrates that when going from $t=0.0082$ in E to $t=0.01$ in G, while the onset of the nonlinear phase increases the development of small-scale features (as discussed in section \[dustDistri\]), the emission in the nonlinear phase remains diffuse in both cases.\
In figure \[fig:geo\] we see that the emission at 54 $\mu$m is strongest where the dust is directly radiated by the source, but the colder dust inside the KH billows also radiates at this wavelength. At shorter wavelengths such as 8.25 $\mu$m, the direct light is the more important and only dust close to the edges of the billows radiates. To get features more reminiscent of the observations we can use this knowledge to consider two changes to the geometry of the source and the observer. On the one hand, the angle $\alpha$ can be chosen to maximise the photons from the source reaching the protruding billows and not the rest of the cloud, which increases the amount of observed photons in a more compact location. Nevertheless, the effect of changing $\alpha$ is small at 8.25 $\mu$m, as demonstrated by comparing cases A to C and B to D in figure \[fig:BDAC\]. On the other hand the observers angle $\beta$ can be chosen to be along the billows, maximising the perceived compactness. The change in observer angle has a much stronger impact. Changing $\beta$ from $90^{\degree}$ in case B to $\beta=128^{\degree}$ in case A clearly decreases the thickness of the features, increases the flux in the elongated regions, and enhances the contrast between the bright en dark regions. The choice for “optimal angles" is illustrated in figure \[fig:geo\]. The values we find are $\alpha=51^{\degree}$ and $\beta=128^{\degree}$. These values are used in cases F and H (figure \[fig:opti\]). Using this geometry, a fair approximation of the real observations can be made, at a comparable wavelength. The evolution from case F into H again displays the formation of the small scale structures in the nonlinear phase, on a scale which is comparable to the local bends in the infrared observations.
Case $\alpha$ $\beta$ time
------ ---------- --------- --------
A 40 128 0.0082
B 40 90 0.0082
C 60 128 0.0082
D 60 90 0.0082
E 51 90 0.0082
F 51 128 0.0082
G 51 90 0.01
H 51 128 0.01
: Summary of the SKIRT radiative transfer models, with $\alpha$ the angle between the star and the cloud, and $\beta$ the angle between the cloud and the observer (see figure \[fig:geo\]), and the time in code units.[]{data-label="table:1"}
![SKIRT simulations of the same dataset with different geometries. From left to right and top to bottom: B, D, A, C. Horizontally the observers angle $\beta$ is the same ($\beta = 90^{\degree}$ on top, $\beta = 128^{\degree}$ below) and the same scaling is used. Note that the flux quantification is arbitrary here and no effort has been taken to compare these to real values. Vertically the irradiation angle is constant ($\alpha = 40^{\degree}$ left, $\alpha = 60^{\degree}$ right). All images are observed at 8.25 $\mu$m. []{data-label="fig:BDAC"}](./BDAC_sameScale_tag.jpg){width="1.03\columnwidth"}
![Synthetic observation of the KHI at 8.25 $\mu$m, with fixed observational angle $\beta=90^{\degree}$ and $\alpha=128^{\degree}$ (cases E and G). Two different times are shown, left: $t=0.0084$, right: $t=0.01$ or 8.21$\times10^4$ and 9.78$\times10^4$ year, respectively). During this interval the development of small-scale perturbations in the nonlinear phase can be seen. A linear scale is used for the intensity of the images.[]{data-label="fig:perp"}](./perp_hori.jpg){width="1.05\columnwidth"}
![Synthetic observation of the KHI at 8.25 $\mu$m, with observational angle $\beta=128^{\degree}$ and $\alpha=51^{\degree}$ (cases F and H). Two different times are shown, left: $t=0.0084$, right: $t=0.01$ or 8.21$\times10^4$ and 9.78$\times10^4$ year, respectively). In comparison to the images at $\beta=90^{\degree}$, the features of the KHI are more pronounced and clearly distinguishable from the background. A linear scale is used for the intensity of the images. []{data-label="fig:opti"}](./opti_hori.jpg){width="1.05\columnwidth"}
Conclusions
===========
In the previous sections, we have modelled a region of the Orion molecular cloud in which elongated ripple features are observed. To do so, we have built upon previous numerical models, and expanded these to full 3D dusty hydrodynamics coupled to a radiation transfer code designed for simulating dusty astrophysical systems. The synthetic images allow a direct comparison with the observations. In the infrared observations, the ripples are thin, elongated features that have a clear periodicity and are sharp and bright compared to the background radiation. All these features can also be reproduced by our model. The hydrodynamical simulations confirm that the previously proposed setup is indeed KH unstable for the observed spatial wavelength. We find that the dynamical contribution of dust with a size distribution typical for the ISM does not inhibit the formation of the KHI, and the growth rate in 3D is similar to that of the 2D simulation. We see that the presence of a background star is able to light up the features of the KH billows. Also, the synthetic images demonstrate clearly that the geometry is of great importance in distinguishing the KH features from the background. Observers located in a direction perpendicular to the shearing layer would observe some periodicity, however with shallow features over a continuous background, while observers which look along the formed billows observe them very sharp and bright compared to the background. Nevertheless, even when considering the most optimal geometry, the ripples are still somewhat wider than the sharp ripples of the observations. Additional to geometrical effects, the sharp features may point to strong local density increases in the dust, however in contrast to our previous investigation of dusty KHI only small increases in dust density are seen here, and the highest increases are found in small and compact clumps and not elongated regions. The treatment of additional physics such as self gravity and magnetic fields may lead to these additional density increases as was shown for larger scale structures in @2014ApJ...789...37V. It is unclear if a significant effect would also be expected here, as in section \[2Dcomp\] the magnetic field only causes minor deviations in the 2D setup. For simulations in 3D, the strong magnetic field (plasma $\beta_{pl}=0.0173$) may somewhat alter the outcome of the simulations in the nonlinear phase, when secondary 3D instabilities break the earlier quasi-2D behaviour. @2000ApJ...545..475R demonstrated that even weak magnetic fields can be of importance in the nonlinear regime. While a strong magnetic fields may suppress the growth of hydrodynamical perturbations perpendicular to the fields, @2007JGRA..112.6223M find that in cases with plasma beta as low as $\beta_{pl}=0.1$ secondary 3D instabilities also occur and cause small scale fragmentation along the initial magnetic field, however at a stage far in the nonlinear regime. The resulting influence of the 3D magnetic field on the dynamics of the dust grains, and thus also the observed structures, is further complicated by the unknown charge of the dust grains. While for example @2012ApJ...747...54H have calculated mean grain charging as function of grain sizes for different ISM phases, the charging of grains can be location dependant due to for example interaction with a radiation field, as is the case here. Fully taking into account the magnetic field would thus also require further assumptions to be made with regard to dust distribution as a function of the both the size and the charge. Furthermore, the strength of the magnetic field is one of the less constrained parameters in the model; while the value in the model ($B = 200$ $\mu$G) is representative for surrounding regions, no local measurements of orientation and strength exist to our knowledge. As the magnetic pressure is shown to be of importance in finding the correct value for the growth rate (section \[2Dcomp\]), the outcome would be different if a different magnetic field was assumed. This would especially be the case for different relative orientations of this field and the flow shear.\
Another important factor which may change the outcome of the simulations is the actual width of the shearing layer between the hot medium and the molecular cloud. The width is an important parameter in the evaluation of the stability and growth of the KHI instability. The value used here ($D = 0.01$ pc) is in analogy with the value of @2012ApJ...761L...4B where it is argued that this value represents the width of the photodissociation region (PDR), where molecular gas is dissociated by the far ultraviolet photons of the background star $\theta^1$ Orionis C. Nevertheless, as discussed in the supplement of @2010Natur.466..947B, actually a broader ($\sim 0.1$ pc) photo-ablation region forms between the PDR and the hot medium. Due to its thickness this region may inhibit the formation of the KHI with wavelengths in the range of the observed periodicity in the ripples or shorter, as a boundary layer of thickness $D$ inhibits the growth of perturbations with $\lambda < 4.91 D$ . Additionally it should be noted that the effect of heat conduction, which has not been included in this work, can be of importance in the formation of the shearing layer between the hot medium and the molecular cloud. Indeed, demonstrate that heat conduction can reduce the steepness of the velocity gradient between the cloud and a streaming flow, stabilising the surface of the cloud against the development of the KHI.\
While these remarks demonstrate that additional physics may be needed to understand the full range of interactions occurring in the Orion nebula, in this work we tried to model the observations of its KH ripples in full detail. We demonstrated that a full treatment of gas and dust dynamics, including a range of dust sizes, coupled with radiative transfer provides a promising approach to explaining the observations. Even though the physical values in the models are prone to intrinsic observational uncertainties or assumptions, we see that these values are reasonable in reproducing most of the features when the most optimal geometrical model is used.
We acknowledge financial support from project GOA/2015-014 (KU Leuven) and by the Interuniversity Attraction Poles Programme initiated by the Belgian Science Policy Office (IAP P7/08 CHARM). Part of the simulations used the infrastructure of the VSC - Flemish Supercomputer Center, funded by the Hercules Foundation and the Flemish Government - Department EWI.
[^1]: The random function $\textrm{rand}$ generates a random floating point value between 0 and 1, while the $\textrm{rect}$ function (also called “rectangular function”) is one between $-0.5$ and $0.5$ and zero elsewhere.
[^2]: Model T46p1\_logg4p05.sed from <http://www.mpe.mpg.de/~martins/SED.html>
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{
"pile_set_name": "ArXiv"
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---
abstract: 'Parent compounds of Fe-based superconductors undergo a structural phase transition from a tetragonal to an orthorhombic structure. We investigated the temperature dependence of the frequencies of transverse acoustic (TA) phonons that extrapolate to the shear vibrational mode at the zone center, which corresponds to the orthorhombic deformation of the crystal structure at low temperatures in [BaFe$_{2}$As$_{2}$]{} and [SrFe$_{2}$As$_{2}$]{}. We found that acoustic phonons at small wavevectors soften gradually towards the transition from high temperatures, tracking the increase of the size of slowly fluctuating magnetic domains. On cooling below the transition to base temperature the phonons harden, following the square of the magnetic moment (which we find is proportional to the anisotropy gap). Our results provide evidence for close correlation between magnetic and phonon properties in Fe-based superconductors.'
author:
- 'D. Parshall'
- 'L. Pintschovius'
- 'J. L. Niedziela'
- 'J.-P. Castellan'
- 'D. Lamago'
- 'R. Mittal'
- 'Th. Wolf'
- 'D. Reznik'
title: 'Close correlation between magnetic properties and the soft phonon mode of the structural transition in [BaFe$_{2}$As$_{2}$]{} and [SrFe$_{2}$As$_{2}$]{}'
---
Introduction
============
The parent compounds of the ferropnictide superconductors are metallic but not superconducting, and order antiferromagnetically. Just like the cuprates, they become superconducting upon doping, which gradually suppresses antiferromagnetic order.[@Dai2012] There is strong coupling between the magnetic degrees of freedom and the lattice. The most important experimental evidence for this coupling is a structural transition from tetragonal to orthorhombic, which is thought to be driven by magnetism [@Yildirim09-1]. Further, it was reported that the arsenic A$_{1g}$ mode can generate antiferromagnetic (AFM) order dynamically [@KimKW2012], and Raman scattering shows strong splitting of the in-plane E$_g$ mode in the ordered state [@Chauviere2009]. These observations have led several groups to suggest enhanced electron-phonon coupling via the spin channel [@Egami10-1; @Yndurain2011].
On the other hand, density-functional theory (DFT) predicts only very weak electron-phonon coupling [@Boeri09], which is not directly contradicted by any experimental result so far. Previous work using inelastic x-ray scattering (IXS) [@Reznik09; @Hahn09] found no measurable changes in most optic phonons across the magnetic transition, in contrast to what has been predicted by DFT calculations making allowance for spin-phonon coupling. Therefore, the importance of spin-phonon coupling for the occurrence of high-[T$_c$]{} superconductivity in the pnictides remains an open question.
The structural transition is accompanied by a transition into an antiferromagnetically ordered state. DFT calculations predict that the AFM state is energetically more stable even in the absence of a structural distortion, and that the structural distortion is the consequence of AFM order via magnetoelastic coupling. The structural transition temperature (T$_s$) always occurs coincident with or at a higher temperature that the magnetic transition (T$_N$) in the electron-doped pnictides, which has led to the proposition of a nematic phase which does not show static magnetic order, in which magnetic fluctuations have a preferred direction [@Fernandes2010]. Alternatively, the nematic phase might be characterized by orbital ordering with unequal occupations of the d$_{xz}$ and d$_{yz}$ orbitals [@CCLee2009; @Yi2011]. The magnetic, orbital, and structural order parameters break the tetragonal symmetry in the same way, but the parameter primarily responsible for the symmetry breaking is still a matter of debate (e.g., recent work [@Liang2013] has claimed that the nematic state is driven by spin, but strongly enhanced by orbital fluctuations). However as pointed out in Ref. , if the parameters are strongly coupled the “primary” parameter loses some distinction.
In this paper, we investigate the coupling of the crystal lattice to the magnetic degrees of freedom by a study of transverse acoustic phonons and the magnetic properties of undoped [SrFe$_{2}$As$_{2}$]{} and [BaFe$_{2}$As$_{2}$]{}. More precisely, phonons were studied whose eigenvectors correspond to the structural distortion in the long-wavelength limit. These phonons propagate along the tetragonal direction and are polarized along . An early IXS study [@Niedziela2011] showed a pronounced softening of such phonons at low [$\mathbf{q}$]{} (reduced momentum transfer) on approaching on approaching [T$_s$]{} from above. On further cooling, these modes hardened gradually to some extent. The softening on cooling is corroborated by measurements of the elastic modulus by ultrasound techniques [@Fernandes2010]. The softening was found to be closely correlated with a previous report [@Fernandes2013] of magnetic susceptibility as deduced from NMR above [T$_N$]{}.
The motivation for our present study was twofold. In the first place, whereas the behavior of the shear modulus[@Fernandes2010] and phonon frequencies above [T$_N$]{} reported so far looks quite plausible, this is not really the case for the behavior below [T$_N$]{}: since the phase transition in undoped pnictides is largely first-order, one would expect strong and sudden changes on cooling below [T$_N$]{}/[T$_s$]{}. In the second place, to gain a better understanding of the phonon behavior below the Néel temperature (the uncertainties associated with the lowest-temperature points in the previous report [@Niedziela2011] were large). We used inelastic neutron scattering for our study which allowed more precise measurement of phonon energies over a wide range of temperatures. Neutron scattering was further used to explore the magnetic properties of our samples above and below [T$_N$]{} with the aim to correlate the magnetic properties with the phonon properties. We show that the magnetic correlation length above [T$_N$]{} and the magnetic order parameter squared below [T$_N$]{} track the phonon softening at all temperatures.
Experimental details
====================
We investigated two single-crystal samples [@Hardy2010] of [BaFe$_{2}$As$_{2}$]{} (Ba122) and of [SrFe$_{2}$As$_{2}$]{} (Sr122), each of mass [$\approx$]{} 1 g, and mosaic spread of < [$0.5^\circ$]{}. The samples were mounted in closed-cycle refrigerators with a temperature range of 10 K to 300 K ([BaFe$_{2}$As$_{2}$]{}) or 10 K to 350 K ([SrFe$_{2}$As$_{2}$]{}). The experiments were carried out on the 4F cold triple-axis spectrometer and on the 1T thermal triple-axis spectrometer at the ORPHÈE reactor at the Laboratoire Léon Brillouin at Saclay, France. On 1T, final energies were chosen between 8 meV and 14.7 meV, depending on the energy resolution needed for the particular part of the experiments. The open collimations of the standard configuration were used for some scans (effectively 35’-35’-35’-35’), but tighter collimations were also often used to improve the resolution in energy and momentum transfer. On 4F, measurements were made with initial energy of 8 meV, and used 50’ in-pile collimation, 50’ collimation between the pair of monochromators, and 40’ collimation elsewhere. Resolutions of 0.4 meV and 0.05 Å$^{-1}$ were easily achieved. Measurements of the phonon branch were made in the tetragonal HK0 plane. Most measurements of the magnetic fluctuations were made in the tetragonal HHL plane, but for some measurements we mounted the sample in the HK0 plane and tilted the sample to reach finite values of L. Peaks were fit to Gaussian functions; error bars are derived from the statistical uncertainty. Follow-up measurements were conducted using the BT-7 instrument at the NIST Center for Neutron Research.
![(color online) Lattice deformation and phonon renormalization across the magnetic ordering transition. a) Schematic of spin-phonon coupling in iron pnictides. Dots represent Fe positions. Straight lines/arrows on top are boundaries of the unit cell/unit vectors used for the notation in this paper respectively. Parallel/antiparallel alignment of the spins on near-neighbor Fe sites favor longer/shorter bonds. b) Schematic of scans shown in (c) and (d). c) Constant-energy scans of [SrFe$_{2}$As$_{2}$]{} at 2 meV showing that the acoustic phonon peaks move closer to the zone center upon cooling below [T$_s$]{} = 200 K, which indicates that the low-energy dispersion becomes steeper (schematic in (b), lines are a guide to the eye). d) constant-Q scans reflecting phonon hardening on cooling through [T$_N$]{} = 135 K in [BaFe$_{2}$As$_{2}$]{}(schematic in (b), lines are a guide to the eye). All error bars represent one standard deviation, and are derived from the statistical uncertainty. []{data-label="fig:Fig1"}](Fig1.png)
Results
=======
At room temperature these materials are tetragonal, but as has been amply documented in the literature [@Tegel08-1; @Rotter08], they undergo a structural distortion from tetragonal-to-orthorhombic structure at [T$_s$]{}. This transition is closely accompanied by a transition to an antiferromagnetically ordered state at [T$_N$]{}. [T$_N$]{} is very close to [T$_s$]{}, if different at all. We did not observe any separation of the structural and AFM transitions in either of our samples. The transition was observed at T = 200.0(5) K ([SrFe$_{2}$As$_{2}$]{}) and 135(1) K ([BaFe$_{2}$As$_{2}$]{}). The transition can be understood within the framework of DFT calculations, which show that magnetic order as observed in experiment with antiparallel spin alignment on neighboring Fe sites in one direction and parallel alignment in the other direction leads to a lowering of the total energy [@Yildirim09-1]. A further energy gain is obtained by shortening the bonds with parallel spin alignment and stretching the bonds with antiparallel spin alignment, as illustrated in Fig. \[fig:Fig1\]a. The transition into the orthorhombic state leads to twinning of the samples, which results in a noticeable broadening of the mosaic distribution. In the following, we will adopt the tetragonal notation, and will make use of the orthorhombic notation only in a few places.
As discussed in the introduction, the long-wavelength transverse acoustic phonons are expected to soften on approaching [T$_s$]{} from above, followed by a gradual and moderate hardening on cooling below [T$_s$]{}. To our surprise, we observed a strong and sudden hardening of this branch in [SrFe$_{2}$As$_{2}$]{} (typically phonon energies might change by 1% over a temperature change of 100 K; the energy of the low-[$\mathbf{q}$]{} portion of this branch changes by over 20% within just a few degrees). The previous investigation made by IXS was, however, performed on [BaFe$_{2}$As$_{2}$]{}. In order to see whether the two compounds behave in a different way, or whether the effect observed in [SrFe$_{2}$As$_{2}$]{} is a generic feature of the ferropnictides, complementary measurements were performed on [BaFe$_{2}$As$_{2}$]{}. It turned out that the two compounds show very similar behavior. The phonon effect is clearly evident from both constant-[$\mathbf{Q}$]{} (constant momentum transfer) and constant-energy scans through the phonon dispersions (see Fig. \[fig:Fig1\]c,d). The constant-[$\mathbf{Q}$]{} scan in Fig. \[fig:Fig1\]d illustrates that in [BaFe$_{2}$As$_{2}$]{} the phonon hardens strongly on cooling across [T$_N$]{} = 135 K by just 10 K from 140 K to 130 K. The same effect is reflected in the constant-energy scan on Sr122 in Fig. \[fig:Fig1\]c. Here [$\mathbf{q}$]{} = 0 is the zone center and the peaks on both sides at [$\mathbf{q}$]{} = 0.07 r.l.u. (reciprocal lattice units) originate from acoustic phonons dispersing on both sides of the zone center. The steeper (shallower) the dispersions, the closer (further) the peaks in the constant-energy scan are to the zone center (see schematic in Fig. \[fig:Fig1\]b). On cooling through [T$_N$]{} = 200 K, the peaks move closer together, which reflects the same hardening as observed in Fig. \[fig:Fig1\]c. There is some broadening in both [$\mathbf{Q}$]{} and energy (some part of which is certainly due to the broader mosaic arising from twinning in the orthorhombic phase), but the change in energy is nonetheless clear.
Fig. \[fig:Fig2\]a shows varying behavior at different [$\mathbf{q}$]{}. For small [$\mathbf{q}$]{}, the phonon softens substantially upon cooling towards [T$_N$]{}, which is followed by abrupt hardening at lower temperature. At intermediate [$\mathbf{q}$]{} (e.g. [$\mathbf{q}$]{} = 0.25) there is no softening above [T$_N$]{}, but there is still the hardening below [T$_N$]{}. And at high [$\mathbf{q}$]{} (e.g. [$\mathbf{q}$]{} = 0.45), no temperature dependence is observed across [T$_N$]{}. The ratio of the frequency just below [T$_N$]{} divided by the frequency above [T$_N$]{} increases strongly towards the zone center (Fig. \[fig:Fig2\]a,b). As is evident from Fig. \[fig:Fig2\]c, the hardening below [T$_N$]{} not only undoes the softening above [T$_N$]{}, but overshoots it by a substantial amount (see Fig. \[fig:Fig2\]d). Further, the hardening below [T$_N$]{} extends much farther in [$\mathbf{q}$]{}, pointing to a different origin of the two phenomena.
Since we expected the phonon effect to be related to the formation of magnetic order, we carefully investigated the temperature dependence of magnetic Bragg peak intensity below [T$_N$]{} and of magnetic fluctuations above [T$_N$]{} in the same samples. In order to correlate the phonon softening above [T$_N$]{} with magnetic properties, we measured the width in momentum transfer of the magnetic fluctuations at several low energies (up to 6 meV). The [$\mathbf{q}$]{}-resolution at each energy was determined empirically from the spin wave spectrum below [T$_N$]{} (which is nearly resolution-limited at the energies considered). We found that the temperature dependence of the linewidths was essentially the same at all energies investigated. The data shown are those for which we obtained the best quality. Fig. \[fig:Fig3\]a shows that well-defined peaks from magnetic scattering at 4 meV are already present at 300 K. They become much sharper on approaching [T$_N$]{} from above, although they never become resolution-limited, even close to [T$_N$]{}. As generally expected, the in-plane widths are considerably smaller than those measured along c, but even in the c-direction, linewidth narrowing begins already at 300 K (Fig. \[fig:Fig3\]b). This highlights the three-dimensional character of low energy magnetic fluctuations, and is in qualitative agreement with previous reports on the spin dynamics in the [BaFe$_{2}$As$_{2}$]{} and [SrFe$_{2}$As$_{2}$]{} systems [@Ewings2011; @Matan10]. None of the widths taken alone correlates well with the phonon softening above [T$_N$]{}, however, the product of all three does (Fig. \[fig:Fig4\]b). Since each linewidth is proportional to the inverse of a correlation length in a particular direction, the product of the linewidths is proportional to the inverse of the volume of the correlated domains.
Below [T$_N$]{}, we determined both the temperature dependence of the magnetic order parameter and that of the spin gap. For the magnetic order parameter, we measured the intensity of the magnetic Bragg peak at [$\mathbf{Q}$]{} = (2.5, 2.5, 1) (or (5 0 1) in orthorhombic notation). To determine the spin gap, we made a series of constant-energy scans through the magnetic Bragg peak at [$\mathbf{Q}$]{} = (0.5, 0.5, 5) (or (105) in orthorhombic notation, see Fig. \[fig:Fig3\]a). On approaching the spin gap energy from above, the peak intensities decrease sharply for reasons of phase space. We made careful measurements of the spin gap energy, and found that the temperature dependence of the spin gap tracks the square of the magnetic moment quite closely, for both Sr122 and Ba122 (see Fig. \[fig:Fig3\]d).
Discussion
==========
![(color online) Correspondence between the renormalized phonon energies and magnetic properties in [BaFe$_{2}$As$_{2}$]{} and [SrFe$_{2}$As$_{2}$]{}. Open markers represent phonon frequencies, and red dashed lines represent peak intensities of the magnetic Bragg peaks. a) For [BaFe$_{2}$As$_{2}$]{}, the solid blue line is a guide to the eye. b) for [SrFe$_{2}$As$_{2}$]{}, the filled blue circles represent 1/$\xi$ (scaled to the phonon data), where $\xi$ is the size of slowly-fluctuating magnetic domains, determined by multiplying inverses of all three linewidths shown in Fig. \[fig:Fig3\]b. All error bars represent one standard deviation, and are derived from the statistical uncertainty. []{data-label="fig:Fig4"}](Figure4_20150112_dot.pdf)
For temperatures below [T$_N$]{}, it is clear that our data show a very large hardening that exceeds the softening above [T$_N$]{}. In the [$\mathbf{q}$]{} = 0 limit, these results should extrapolate to the shear modulus that was previously reported for [BaFe$_{2}$As$_{2}$]{} based on resonant ultrasound experiments (RUS)[@Fernandes2010]. However, RUS shows only a slight hardening on cooling (small in comparison to the softening above [T$_N$]{}). This disagreement is likely due to the twin domain boundaries inside the crystal in the orthorhombic phase, which are known to interfere with RUS measurement, but not with phonon measurements (beyond a slight broadening). In addition, there is a significant quantitative disagreement between our data and the previous IXS report [@Niedziela2011] for T $<$ [T$_N$]{}, in that the low-[$\mathbf{q}$]{} IXS results for [$\mathbf{q}$]{} $\leq$ 0.1 do not capture the sudden and very strong increase of phonon frequencies just below [T$_N$]{}. We do not have any plausible explanation for the disagreement, but emphasize that we observe the strong and sudden increase of phonon frequencies below [T$_N$]{} under a wide range of experimental conditions (multiple samples, growers, instruments, sample environments, etc.).
We found that below [T$_N$]{}, the phonon hardening tracks the square of the magnetic order parameter very closely in both [SrFe$_{2}$As$_{2}$]{} and in [BaFe$_{2}$As$_{2}$]{} (Fig. \[fig:Fig4\]) (although the effect is somewhat smaller in [BaFe$_{2}$As$_{2}$]{}, possibly due to the greater mass). Since we have found that the low-[$\mathbf{q}$]{} phonon frequency is biquadratically coupled to the magnetic order parameter, it is essentially linearly coupled to the structural order parameter (in the undoped 122-compounds the magnetic and the structural order parameters are biquadratically coupled [@Avci2011; @Cano2010]). But since we also found that the temperature evolution of the spin gap is quite similar to that of the square of the magnetic order parameter, the correlation seen in Fig. \[fig:Fig3\]d might instead be related to the opening of the spin gap. There thus seem to be three possibilities for the correlation observed in Fig. \[fig:Fig4\]: that the phonon hardening could track either the square of the structural distortion, the square of the magnetic order parameter, or opening of the spin gap. To determine which is this case, further studies are necessary on samples where the structural and the magnetic order parameters do not follow the same temperature dependence, as in, e.g., Co-doped Ba122.
At temperatures above [T$_N$]{} the softening of the investigated phonon modes is proportional to the volume of fluctuating magnetic domains. Our data are qualitatively consistent with the RUS results above [T$_N$]{}: the slope at the lowest [$\mathbf{q}$]{} becomes rapidly softer upon approaching [T$_N$]{}from above. A direct quantitative comparison is not possible, because the softening is strongly dependent on [$\mathbf{q}$]{}, and imperfect [$\mathbf{q}$]{}-resolution does not allow us to probe [$\mathbf{q}$]{}-values which are very close to [$\mathbf{q}$]{} = 0.
When considering finite values of [$\mathbf{q}$]{}, we find that for [$\mathbf{q}$]{} $\geq$ 0.2 r.l.u., the energy of the phonon branch is nearly unaffected when approaching [T$_N$]{} from above. For [$\mathbf{q}$]{} $\leq$ 0.2 r.l.u., the branch softens rapidly upon approaching [T$_N$]{}. The absolute value of the softening is greatest close to [$\mathbf{q}$]{} = 0.1 r.l.u., and the relative value of the softening is greatest at the lowest measured [$\mathbf{q}$]{} ([$\approx$]{} 0.05 r.l.u.). These results are also fully consistent with the IXS results for [BaFe$_{2}$As$_{2}$]{} above [T$_N$]{} reported in Ref. , after taking into account the somewhat coarser [$\mathbf{q}$]{}-resolution used here.
We also find that above [T$_N$]{} there is a close correlation between the volume of the short-lived magnetic domains and the phonon frequencies (Fig. \[fig:Fig4\]). As previously shown [@ChuJH2010], only a slight uniaxial strain is required to produce a resistivity anisotropy in the paramagnetic state, which corresponds to the soft phonon seen here. Since a local lattice distortion will entail an elastic strain field, which is inherently long-range in 3D, it is plausible that the volume of the dynamic domains is closely correlated to the phonon frequencies. It follows that in order to understand coupling of magnetic (and for that matter nematic) fluctuations to the atomic lattice, it is necessary to consider not only the in-plane magnetic correlation length but also the correlation length along the c-axis.
Although it is clear that there exists a strong coupling between the lattice and the electronic degrees of freedom, it can be difficult to distinguish the primary order parameter (given that the magnetic, orbital, and structural order parameters break the tetragonal symmetry in the same way). In this context it may be illuminating to consider the [$\mathbf{q}$]{}-dependence of phonon softening, as its correlation with the [$\mathbf{q}$]{}-dependence of magnetic or orbital fluctuations make it possible to differentiate between the two mechanisms. This question requires a theoretical investigation that is outside the scope of the current work. In any event, our finite-[$\mathbf{q}$]{} data provide detailed information relating a microscopic quantity (the size of the magnetic domains) to a macroscopic quantity (the elastic constant), and may prove to be a useful basis for comparison with theory.
It seems likely that there are two competing effects; while the softening above [T$_N$]{} is primarily confined to [$\mathbf{q}$]{} $\leq$ 0.1, the hardening extends to much higher [$\mathbf{q}$]{}. Both effects are associated with the (110) transverse shear mode. We took measurements of the longitudinal acoustic phonon along the (110) using the BT-7 instrument (data not shown) and found no evidence of either effect there, nor did we see any strong changes of the high-energy phonons in our previous inelastic neutron scattering study[@Parshall2014] of [SrFe$_{2}$As$_{2}$]{}.
The previous IXS study suggested the softening might occur as a result of a Kohn anomaly due to two-dimensional Fermi surface nesting, because the range of the softening corresponds roughly to the size of inner hole pocket in [BaFe$_{2}$As$_{2}$]{} ([$\approx$]{} 0.1 r.l.u.). In that case, there should be an enhancement of the electron-phonon coupling in the AFM state (previous theoretical work found that the electron-phonon coupling was significantly stronger in the stripe AFM state than in the paramagnetic state [@Boeri2010]). It is difficult for us to determine if the electron-phonon coupling for this mode (as measured by the energy linewidth) has become significantly enhanced in the AFM phase. This is because the orthorhombic distortion causes a broadening of the mosaic spread, which in turn causes a broadening of the TA phonon. Separating these effects is challenging.
It may be possible to understand the detailed [$\mathbf{q}$]{}-dependence of the hardening using DFT calculations. In order to capture both the low-[$\mathbf{q}$]{} deviation from linear dispersion, and then the gradual return to the paramagnetic behavior at high-[$\mathbf{q}$]{}, one would need to calculate a very dense mesh in [$\mathbf{q}$]{} for both the AFM and paramagnetic phases. Such a calculation would be time-consuming, and given the challenges associated with DFT in this system [@Mazin08-1], we would have only limited faith in the results.
Conclusions
===========
We found that in undoped 122 compounds, the TA phonons propagating along with polarization along soften gradually on approaching [T$_N$]{} from above, followed by an abrupt hardening just below [T$_N$]{} and further gradual hardening on cooling to very low temperatures. Interestingly, the hardening below [T$_N$]{} exceeds significantly that of the softening above [T$_N$]{} in both magnitude and [$\mathbf{q}$]{}-range, which remains to be understood. While the softening above [T$_N$]{} was known already from previous publications at least qualitatively, the pronounced hardening below [T$_N$]{} was not. The temperature evolution of the phonon frequencies both below and above [T$_N$]{} correlates closely with that of magnetic properties observed on the same samples. However, this correlation does not necessarily imply a direct spin-phonon coupling. The coupling between the phonons and the static/dynamic magnetic order might only be indirect via magnetoelastic coupling.
The most promising candidates for further insight into this issue are samples where magnetic order and structural distortions do not appear at the same temperature, as in Ba122 samples lightly doped with Co. In such samples, [T$_s$]{} $>$ [T$_N$]{}. Just like in the undoped sample, we can expect gradual softening of the phonons on cooling towards the structural transition. It is, however, difficult to predict how the phonon frequencies will evolve on further cooling through the magnetic transition. Answering this question will be a subject of a different paper, which will provide additional insight into understanding the coupling of phonons to magnetic properties in Fe-based superconductors.
The research at ORNL’s Spallation Neutron Source was sponsored by the Scientific User Facilities Division, Office of Basic Energy Sciences, U.S. Department of Energy. D.P. and D.R. were supported by the U.S. Department of Energy, Office of Basic Energy Sciences, Office of Science, under Contract No. DE-SC0006939. The authors thank A. Alatas for valuable discussions.
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|
---
abstract: 'Can the topology of a network that consists of many particles interacting with each other change in complexity when a phase transition occurs? The answer to this question is particularly interesting to understand the nature of phase transitions if the distinct phases do not break any symmetry, such as topological phase transitions. Here we present a novel theoretical framework established by complex network analysis for demonstrating that across a transition point of the topological superconductors, the network space experiences a homogeneous-heterogeneous transition invisible in real space. This transition is nothing but related to the robustness of a network to random failures. We suggest that the idea of the network robustness can be applied to characterizing various phase transitions whether or not the symmetry is broken.'
author:
- 'Chung-Pin Chou'
title: 'Network Robustness: Detecting Topological Quantum Phases'
---
Phases of matter can be distinguished by using Landau’s approach, which characterizes phases in terms of underlying symmetries that are spontaneously broken. The information we need to understand phase transitions is usually encoded in appropriate correlation functions, e.g. the correlation length would diverge close to a quantum critical point. Particularly, the low-lying excitations and the long-distance behavior of the correlations near the critical phase are believed to be well described by quantum field theory. A major problem is, however, that in some cases it is unclear how to extract important information from the correlation functions if these phases do not break any symmetries, such as topological phase transitions [@WenBook06; @NayakRMP08; @AliceaRPP12; @BernevigBook13]. Complex network theory has become one of the most powerful frameworks for understanding network structures of many real-world systems [@AlbertRMP02; @DorogovtsevAIP02; @NewmanSIAM03; @BoccalettiPR06; @DorogovtsevRMP08]. According to graph theory, the elements of a system often are called nodes and the relationships between them, which a weight is associated with, are called links. Decades ago, this unnoticed idea constructing a weighted network from condensed matters had been proposed in quantum Hall systems [@SenthilPRL99]. In the language of network analysis, therefore, each network of $N$ nodes can be described by the $N\times N$ adjacency matrix $\hat{A}$. In what follows, we consider lattice sites as the nodes of the weighted network of which each weighted link between nodes $i$ and $j$ is expressed by the element of the adjacency matrix $\hat{A}_{ij}$. In Fig.\[fig1\](a), we start from a square-lattice example of size $N=9$ in which the nodes form a simple regular network. Following the procedure \[Fig.\[fig1\](b) and (c)\], we then reconnect them by using some correlation functions as weights of network links to generate a complete network with the link-weight distribution (see Fig.\[fig1\](d)). The links now carry the weights containing information about important relationship between particles in many-body systems. In this letter, focusing on the topological superconductors in one (1D) and two (2D) dimensions [@KitaevPU01; @ReadPRB00], we explore the possibilities of detecting the topological phase transitions by using the novel network analysis. Our analysis reveals that (i) a homogeneous-heterogeneous transition occurs in network space from a topologically trivial phase to a topologically non-trivial phase, which is accompanied by a hidden symmetry breaking (namely, a reduction of the network robustness), and (ii) the complex many-body network analysis can be applied to other phase transitions without a prior knowledge of the system’s symmetry.
![An example of the construction of complex many-body networks: (a) The system is a $3\times3$ square lattice with periodic boundary conditions; (b) The system is replotted based on the same topology as (a); (c) All links are removed from (b); (d) All nodes are reconnected by using some correlations between particles. The thickness of links denotes the magnitude of the correlation.[]{data-label="fig1"}](fig1){height="2.6in" width="2.8in"}
*1D $p$-wave superconductor.*$-$The first model we consider was introduced by Kitaev [@KitaevPU01]. The Hamiltonian for $L$ spinless fermions in a chain with periodic boundary conditions is $$\begin{aligned}
H_{1D}=-\sum_{i}\left(\hat{c}_{i}^{\dag}\hat{c}_{i+1}+\hat{c}_{i}^{\dag}\hat{c}_{i+1}^{\dag}+H.c.\right)+\mu\hat{c}_{i}^{\dag}\hat{c}_{i},\label{KitaevR}\end{aligned}$$ where $\mu$ is chemical potential. The simplest superconducting (SC) model system shows the two-fold ground-state degeneracy stemming from an unpaired Majorana fermion at the end of the chain with open boundary conditions. This model has two phases sharing the same physical symmetries: a topologically trivial (strong pairing) phase for $\mu>\mu_{c}$($=2$) and a topologically non-trivial (weak pairing) phase for $\mu<\mu_{c}$. The transition between them is the topological phase transition identified by the presence or absence of unpaired Majorana fermions localized at each end. The Hamiltonian in momentum space is quadratic of fermionic operators $\hat{c}_{\mathbf{k}}$, given by $$\begin{aligned}
\sum_{\mathbf{k}}\left(
\begin{array}{cc}
\hat{c}_{\mathbf{k}}^{\dag} & \hat{c}_{-\mathbf{k}} \\
\end{array}
\right)\left(
\begin{array}{cc}
\epsilon_{\mathbf{k}} & -i\sin{\mathbf{k}} \\
i\sin{\mathbf{k}} & -\epsilon_{\mathbf{k}} \\
\end{array}
\right)\left(
\begin{array}{c}
\hat{c}_{\mathbf{k}} \\
\hat{c}_{-\mathbf{k}}^{\dag} \\
\end{array}\right),\label{KitaevK}\end{aligned}$$ where $\epsilon_{\mathbf{k}}=-\frac{\mu}{2}-\cos{\mathbf{k}}$. By using the standard Bogoliubov transformation, $\gamma_{\mathbf{k}}=\cos{(\theta_{\mathbf{k}}/2)}\hat{c}_{\mathbf{k}}-i\sin{(\theta_{\mathbf{k}}/2)}\hat{c}_{-\mathbf{k}}^{\dag}$ where $\tan{\theta_{\mathbf{k}}}=\sin{\mathbf{k}}/\epsilon_{\mathbf{k}}$, the Hamiltonian can be diagonalized. The excitation spectrum of the form, $E_{\mathbf{k}}=\sqrt{\left(2\epsilon_{\mathbf{k}}\right)^{2}+\sin^{2}{\mathbf{k}}}$, remains fully gapped except at the critical point $\mu_{c}$. The SC ground state is the state annihilated by all $\gamma_{\mathbf{k}}$: $$\begin{aligned}
|\Psi_{GS}\rangle=e^{\frac{1}{2}\sum_{i,j}G_{ij}\hat{c}_{i}^{\dag}\hat{c}_{j}^{\dag}}|0\rangle,\label{KitaevGS}\end{aligned}$$ where $G_{ij}$ represents the pairing amplitude given by [@ChungPRB00] $$\begin{aligned}
G_{ij}=\frac{1}{L}\sum_{\mathbf{k}}\tan{(\theta_{\mathbf{k}}/2)}e^{i\mathbf{k}\cdot(\mathbf{r}_{i}-\mathbf{r}_{j})}.\label{KitaevGij}\end{aligned}$$ A possible choice of the adjacency matrix of the 1D superconductor is the normalized pairing amplitude in which the non-local property between spinless fermions is concealed. The adjacency matrix can serve as an intuitive definition for the network of spinless fermions with $p$-wave Cooper pairing: $$\begin{aligned}
\hat{A}_{ij}=\frac{|G_{ij}|}{\max{G_{ij}}}.\label{AijGij}\end{aligned}$$ The node $i$ or $j$ stands for a given lattice site. The weights of links contain information about the pairing strength between spinless fermions.
![(a) The evolution of network topologies of the 1D $p$-wave superconductor from chemical potential $\mu=0.2$ to $4.2$. The chain length $L=10$. The thickness and color of links represent their weights. Color scale: Blue (Red) indicates the largest (smallest) weights. (b) Pairing amplitude $A_{r}$ as a function of distance $r$ for different $\mu$. (c) The probability distribution $p(w)$ of the weights $w$ of network links for different $\mu$. The bin size is chosen for clear demonstrations. The chain length $L=1000$.[]{data-label="fig2"}](fig2){height="2.5in" width="3.2in"}
In Fig.\[fig2\](a), one can see the change of network topologies for different $\mu$ in a short chain of $L=10$. Below $\mu_{c}$, the topologically non-trivial phase displays irregular patterns of the complete network, where each node is connected to all other nodes with different link weights. As increasing $\mu$ above $\mu_{c}$, the topologically trivial phase demonstrates a ring structure comprised of the nodes with the largest link weight in the network pond. There are only few links with the strongest weight that is called “highways” of the network. The obvious change of topologies of the network across the critical point is intimately related to the critical behavior observed in real space. We now recall the pairing amplitude in real space shown in Eq.(\[KitaevGij\]). Consider translational invariance, Figure \[fig2\](b) shows how the normalized pairing amplitude $A_{r\equiv|\mathbf{r}_{i}-\mathbf{r}_{j}|}$($=\hat{A}_{ij}$) changes as the topological phase transition occurs. For $\mu<\mu_{c}$, the weak pairing phase indicates that the size of the Cooper pair is infinite, leading to $A_{r}\sim const$. At the critical point, the critical phase has power-law correlations at large distances. Above the critical point, [*i.e.*]{} $\mu>\mu_{c}$, the strong pairing phase instead shows that the pairing amplitude is exponentially decaying with distances: $A_{r}\sim e^{-r/\xi}$. The Cooper pairs form molecules from two fermions bound in real space over a length scale $\xi$. The exponentially decaying pairing amplitude in real space results in a ring structure in network space. Similar physics would also appear in the well-known phenomena of BEC-BCS crossover in $s$-wave superconductors [@ChenPR05]. To further analyze the network structure, we examine how the probability distribution $p(w)$ of the weights $w$ of network links evolves over contiguous topological phases. In Fig.\[fig2\](c), the weights distribute like a delta function for $\mu<\mu_{c}$. Namely, the weights homogeneously distribute in network space. As further increasing $\mu$, the distributions begin to lose weight but still remain nearly homogeneous. It is noteworthy that the distribution at the critical point possesses a decaying function with a heavy tail. Hence the weight distribution of network links becomes more heterogeneous. In the strong pairing regime ($\mu>\mu_{c}$), the distribution moves to almost zero weight and recovers a sharp peak at $w\sim0$. The tail of the distribution in the strong pairing regime thus looks much more heterogeneous than the weak pairing regime. This observation reminds us of a well-known fact in real-world networks that a network with the heterogeneous weight distribution of links is robust to random failures [@AlbertNat00; @WangCM05]. The robustness of the network originating from its heterogeneity seems to indicate a hidden symmetry in network space. More precisely, the hidden symmetry describes a phenomenon that the network function and structure remain unchanged or invariant under random removal of its links. Thus, there exists a homogeneous-heterogeneous network transition hidden in the topological phase transition. It may allow us to define a topological order parameter in network space for identifying the phase transition without any local order parameter.
*2D $p$+i$p$ superconductor.*$-$Consider now a 2D time-reversal symmetry breaking superconductor, the $p$+i$p$ superconductor for $N$ spinless fermions: $$\begin{aligned}
H_{2D}=\sum_{\mathbf{k}}\varepsilon_{\mathbf{k}}\hat{c}_{\mathbf{k}}^{\dag}\hat{c}_{\mathbf{k}}+\left(\Delta_{\mathbf{k}}\hat{c}_{\mathbf{k}}^{\dag}\hat{c}_{-\mathbf{k}}^{\dag}+H.c.\right),\label{pipH}\end{aligned}$$ where the single-particle dispersion $\varepsilon_{\mathbf{k}}=-2(\cos k_{x}+\cos k_{y})-\mu$ and the gap function $\Delta_{\mathbf{k}}=\sin k_{x}+i\sin k_{y}$. For the spinless fermions, the gap function has odd parity symmetry, $\Delta_{-\mathbf{k}}=-\Delta_{\mathbf{k}}$. One can see that the excitation spectrum has gapless nodes at time-reversal invariant momenta: $(0,0)$, $(0,\pi)$, $(\pi,0)$, $(\pi,\pi)$.
![(a) The evolution of network topologies of the 2D $p$+i$p$ superconductor from chemical potential $\mu=-2.5$ to $-6$. The lattice size $N=16$. (b) Pairing amplitude $A_{r}$ as a function of distance $r$ for different $\mu$. (c) The probability distribution $p(w)$ of the weights $w$ of network links for different $\mu$. The lattice size $N=1600$.[]{data-label="fig3"}](fig3){height="2.5in" width="3.2in"}
Following the same analysis as the 1D example, the SC ground-state wave function has the same form as Eq.(\[KitaevGS\]). However, the pairing amplitude $G_{ij}$ is now given by $$\begin{aligned}
G_{ij}=\frac{1}{N}\sum_{\mathbf{k}}\frac{v_{\mathbf{k}}}{u_{\mathbf{k}}}e^{i\mathbf{k}\cdot(\mathbf{r}_{i}-\mathbf{r}_{j})},\label{pipGS}\end{aligned}$$ where $v_{\mathbf{k}}$ and $u_{\mathbf{k}}$ are BCS coherence factors (More details can be found in Ref. ). Note that the system preserves the particle-hole symmetry so that only $\mu\leq0$ will be considered later. This model also has two topological distinct phases: a topologically trivial phase for $\mu<\mu_{c}$($=-4$) and a topologically non-trivial phase for $\mu>\mu_{c}$. Other than the 1D case, however, there is the other transition point at $\mu'_{c}=0$ due to the bulk gap closure at $(\pi,0)$ and $(0,\pi)$. We now investigate how the network analysis performs in the face of the 2D topological phase transition. The definition for the adjacency matrix $\hat{A}_{ij}$ in the 2D superconductor is still the same as Eq.(\[AijGij\]). The complex topologies of the weighted network for the topological trivial and non-trivial phases are shown in Fig.\[fig3\](a) in a square lattice of $N=16$. The topologically non-trivial phase ($\mu>\mu_{c}$) gives rise to a weighted complete network with the link-weight distribution. It would be just a trivial complete network if the network were unweighted. For $\mu<\mu_{c}$, each node only has four highways to its neighbors, hence the network topology is equivalent to a torus which corresponds to a square lattice with periodic boundary conditions. As in the 1D example the difference between the strong pairing phase and the weak pairing phase can be distinguished by examining the pairing amplitude. In Figure \[fig3\](b), with $\mu>\mu_{c}$ a weakly paired condensate forms from Cooper pairs loosely bound in real space, which gives $A_{r}\sim r^{-1}$ at large distance. As for $\mu<\mu_{c}$, the pairing amplitude falls exponentially for large $r$, $A_{r}\sim e^{-r/\xi}$, because the pairs of the strong pairing phase are tightly bound in real space. Thus, the exponentially decaying pairing amplitude gives rise to the torus structure in network space. Let us turn to discussing the probability distribution of the weights of network links. For $\mu>\mu_{c}$, Fig.\[fig3\](c) shows that the distribution still shows a bell-like function due to power-law decaying pairing amplitude. Most links centering around a moderate weight behave homogeneous in network space. Similar to the 1D case, the heterogeneity of the distribution appears as further approaching $\mu_{c}$. For $\mu<\mu_{c}$, a majority of links lose their weights, hence, the distribution becomes much more heterogeneous and exhibits a long tail. The 2D superconductor shows much broader weight distribution of network links than the 1D case. Even so, a homogeneous-heterogeneous transition is still observed in the network space.
*Network measures.*$-$So far we have not introduced the topological order parameter for the topological superconductors in 1D and 2D yet. We turn our attention to two network measures that could be used in these topological quantum systems. One is the so-called small-world phenomena [@WattsBook99]. Many of real-world networks have the property of relatively short average path length defined by a shortest route running along the links of a network. A small-world network including not only strong clustering but also short path length has also been introduced to describe real-world networks [@WattsNat98]. Instead of the weighted clustering coefficient $C$ [@OnnelaPRE05] and the average path length $D$ [@NewmanBook10] commonly used in network analysis (see more details in the Supplemental Material [@suppl]), a measure of the small-world property called “small-worldness” has been recently proposed [@HumphriesPLos08; @CPCArXiv13]. It is defined as $$\begin{aligned}
S\equiv\frac{C}{D},\label{smallworld}\end{aligned}$$ which is based on the maximal tradeoff between high clustering and short path length. A network with larger $S$ has a higher small-world level. The small-worldness seems to be appropriate to describing the universal critical properties because it can extract information about both locality (weighted clustering coefficient) and non-locality (average path length) from network space. In Fig.\[fig4\](a), we illustrate the critical behavior of the small-worldness in the 1D $p$-wave superconductor. One can see that the small-worldness drops to zero when the 1D superconductor comes from the topologically non-trivial phase to the topologically trivial phase. This coincidence convinces us that the network topology enables the small-worldness, akin to an order parameter in the theory of conventional phase transitions, to expose the change of nontrivial topology inherent in the weak pairing regime [@CPCArXiv13]. For the 2D $p$+i$p$ superconductor, however, the small-worldness displays the notorious finite size effect for the topologically non-trivial phase as a result of its much broader weight distribution. In order to overcome this hassle, we define the normalized small-worldness as $S^{*}=S/\max S(\mu)$. In Fig.\[fig4\](b), near the critical point $\mu_{c}$ the normalized small-worldness vanishes in the strong pairing regime as well. Surprisingly, the other critical point $\mu'_{c}$ seems to be also characterized by a decline in the normalized small-worldness. There is an alternative way to understand the disappearance of the small-worldness in the strong pairing regime. As approaching the strong pairing phase, most links start to lose weights and the weight distribution shows more heterogeneous, thus leading to the reduction of the small-worldness. The phenomenon that most links are like slow traffic lanes results in vanishing small-worldness. In other words, the regular network of the strong pairing phase has a very small weighted clustering coefficient and much longer average path length as the system size goes to infinity. The same reasoning from the weight distribution of network links can be also applied to other many-body systems with/without local order parameters. Hence, this result strongly suggests that the small-worldness can be considered as a topological order parameter in the network space. We have to mention a point now in passing. In the network representation, there exists a hidden symmetry corresponding to a heterogeneous network with a long-tail weight distribution. The homogeneous-heterogeneous transition of the network topology observed is intimately related to the hidden symmetry breaking. The hidden symmetry is nothing but the robustness of a complex network. The heterogeneity of network links implies that a weighted network is more robust against random failures [@WangCM05], accompanied by higher hidden symmetry in network space. Conversely, the homogeneity of network links means that a weighted network becomes fragile to random failures, and thus breaks the hidden symmetry. The other network measure we have to take is just to quantify the hidden symmetry.
![(a) Small-worldness $S$ and (c) network robustness $R$ of the 1D $p$-wave superconductor vs chemical potential $\mu$ for different chain length $L$. (b) Normalized small-worldness $S^{*}$ and (d) network robustness $R$ of the 2D $p$+i$p$ superconductor for different lattice size $N$ as a function of $\mu$. The blue lines indicate the critical points.[]{data-label="fig4"}](fig4){height="2.3in" width="3.5in"}
The network structure and function strongly rely on its structural robustness, i.e. the ability of a network to maintain the connectivity when a fraction of nodes or links are randomly removed. A variety of network measures have been proposed to detect the structural robustness [@HararyNet83; @KrisCMA87; @EsfIPL88; @BauerCC90]. Recently the concept of natural connectivity derived from the graph spectrum has been introduced to measure the structural robustness [@BarahonaCPL10; @BarahonaChao12; @EstradaPR12] (also see the details in the Supplemental Material [@suppl]). We can further extend the concept of natural connectivity to the weighted network. The natural connectivity in a weighted network represent the “strength” of loops of all lengths instead of the number of loops. Thus we call the natural connectivity as the network robustness $R$, that can be given by $$\begin{aligned}
R=\ln\left(\frac{1}{n}\sum_{i=1}^{n}e^{\bar{\lambda}_{i}}\right),\label{robustR}\end{aligned}$$ where $\bar{\lambda}_{i}$($\equiv\lambda_{i}/\max\lambda_{i}$) stands for the normalized eigenvalue of the adjacency matrix $\hat{A}$ to avoid the enhancement of the network robustness as increasing the number of nodes $n$($=L$ in 1D and $N$ in 2D). In Fig.\[fig4\](c) and (d), we analyze the homogeneous-heterogeneous transition of network topology by plotting the network robustness $R$ for the topological superconductors. As we expect, in both 1D and 2D cases the strong pairing phase always exhibits more robust network structure than the weak pairing phase, owing to its more heterogeneous weight distribution. For the homogeneous-heterogeneous transition in network space, the hidden symmetry breaking at the critical point indicates that the network loses its robustness ($R=0$, namely, the hidden symmetry is broken), further leading to an appearing order parameter: the small-worldness ($S\neq0$). This is a clear picture that Landau’s symmetry breaking theory works well even in network space. More interestingly, the symmetry-breaking idea is successfully applied to identifying the topological phase transitions in the topological superconductors. It is worthy to be mentioned that the concept behind the hidden symmetry breaking can also provides significant information to comprehend traditional phase transitions with Landau’s symmetry breaking in condensed matter systems.
*Conclusions.*$-$By using complex network analysis we have addressed how to read useful information from the pairing amplitude to characterize the topological phases in 1D and 2D topological superconductors. We have illustrated that a network measure, small-worldness, plays a significant role as a topological order parameter in network space, relied on Landau symmetry-breaking picture. The evolution of the weight distribution of network links across the critical point is responsible for the change of the small-worldness, which is analogous to the change of the speed limit on a road network from the highway to the slow traffic lane. The phenomenon that the structure of the weighted network varies from heterogeneity to homogeneity implies a hidden symmetry broken$-$or, to put it another way, the disappearance of the network robustness to random failures. The hidden symmetry breaking has been successfully described by another network measure, network robustness. The robustness of a complex network is able to uncover a wealth of topological information underneath the pairing amplitude, and further comprehend the mechanism of the phase transitions without local order parameters. We thus suggest that complex network analysis can be a valuable tool to investigate quantum or classical phase transitions in condensed matters.
We would like to thank Ting-Kuo Lee and Ming-Chung Chang for helpful discussion and comments. Our special thanks go to Xiao-Sen Yang for fruitful collaborations. This work is supported by CAEP and MST.
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|
{
"pile_set_name": "ArXiv"
}
|
---
author:
- 'Christoph Rothe[^1]'
bibliography:
- 'bibl.bib'
title: '**Combining Population and Study Data for Inference on Event Rates**'
---
Introduction
============
In a recent study, @Streeck2020 estimate the infection fatality rate (IFR) of SARS-CoV-2 infection in a German town that experienced a super-spreading event in mid-February 2020. The study features prominently in Germany’s current political discussion, and has been covered extensively by major German and international news outlets. Several newspaper articles raised the question, however, whether the study reports an accurate confidence interval (CI) for its IFR estimate.
To explain the issue, consider a stylized version of the setup in @Streeck2020. There is a population of total size $N_T$, in which $N_I$ individuals are infected, and $N_D$ units have died from the infection. The values $N_T$ and $N_D$ are known from administrative records, but $N_I$ is not directly observed. Instead, the researcher collects a random sample of $N_S$ individuals, and observes that $N_P$ of them test positive for the disease. If the test is always accurate, the IFR can then be estimated by $$\widehat{\theta} = \frac{N_D}{\widehat N_I}, \quad\textnormal{ where }\quad \widehat N_I = \frac{N_P}{N_S}\cdot N_T$$ is an estimate of the number of infected units in the population. Now, the CI for the IFR reported in @Streeck2020 only takes the sampling uncertainty about $\widehat N_I$ into account, but treats the number of deaths $N_D$ as fixed. The question is whether doing so is appropriate, or if $N_D$ should be treated as random. We argue that the answer depends on whether $\widehat{\theta}$ is interpreted as an estimate of the IFR among the $N_I$ infected individuals, or an estimate of the IFR among all $N_T$ members of the population.
To clarify this point, we postulate the existence of vectors $\mathbf{D} =(D_1,\ldots, D_{N_T})$, $\mathbf{I} =(I_1,\ldots, I_{N_T})$ and $\mathbf{S} =(S_1,\ldots, S_{N_T})$, with $D_j\in\{0,1\}$ an indicator for the (possibly counterfactual) event that the $j$th individual in the population would have died in the study period if s/he had been infected with SARS-CoV-2, $I_j\in\{0,1\}$ an indicator for the $j$th individual actually being infected, and $S_j\in\{0,1\}$ an indicator for the $j$th individual being included in the sample. These indicators are in principle unobserved, and such that $$N_S = \sum_{j=1}^{N_T}S_j, \quad N_P = \sum_{j=1}^{N_T}S_jI_j, \quad N_I = \sum_{j=1}^{N_T}I_j, \quad N_D = \sum_{j=1}^{N_T}I_jD_i, \quad N_{D,C} = \sum_{j=1}^{N_T}D_j,$$ with the last term being a new notation for the counterfactual number of deaths one would have observed if the entire population had been infected at the time of the study.
We consider $\mathbf{D}$ to be a fixed feature of the population, and both $\mathbf{S}$ and $\mathbf{I}$ to be random vectors whose distribution is determined by the sampling design used in the study and the process that governs the spread of the infection, respectively. This means that $N_I$ and $N_D$ are also random, through their dependence on $\mathbf{I}$. There are then two plausible candidates for the parameter of interest : the IFR among the individuals that were infected at the time of the study, given by $$\theta_1 = \frac{N_D}{N_I},$$ and the IFR for the entire population, given by $$\theta_2 = \frac{N_{D,C}}{N_T}.$$
Now consider a CI that only accounts for the uncertainty in $\widehat{\theta}$ through its dependence on $\widehat{N}_I$, which can be obtained by scaling a conventional $(1-\alpha)$ CI for the proportion of infected individuals. For example, if $(L_\alpha,U_\alpha)$ is a conventional $(1-\alpha)$ Clopper-Pearson CI for the proportion $N_I/N_T$, such a $(1-\alpha)$ CI is given by $$\mathcal{C}^\alpha_1 = \left(\frac{N_D}{N_T\cdot L_\alpha}, \frac{N_D}{N_T\cdot U_\alpha}\right).$$ This type of CI is reported in @Streeck2020, and it is easily seen to have correct coverage for $\theta_1$ conditional on $\mathbf{I}$, and therefore it must also have correct coverage unconditionally: $$P( \theta_1 \in \mathcal{C}^\alpha_1|\mathbf{I}) = 1-\alpha \Rightarrow
P( \theta_1 \in \mathcal{C}^\alpha_1) = 1-\alpha.$$ In that sense, the CI in @Streeck2020 is not wrong, but it is a CI for a very particular target parameter.
In general, inference on $\theta_2$ is going to be more practically relevant since IFR estimates are typically used to design policy measures that affect the entire population. The CI $\mathcal{C}^\alpha_1$ clearly does not have correct coverage for $\theta_2$ though, with or without conditioning on $\mathbf{I}$. Intuitively, an appropriate CI for $\theta_2$ should be wider than $\mathcal{C}^\alpha_1$, but it is not immediately obvious how such a CI should be constructed. In the remainder of this note, we propose two approaches that both result in good coverage properties. To avoid modeling the number of infections, we seek CIs $\mathcal{C}_2^\alpha$ that are valid conditional on $N_I$, $$P( \theta_2 \in \mathcal{C}^\alpha_2|N_I) \approx 1-\alpha,$$ and any CI that has such approximately correct conditional coverage must again also have approximately correct unconditional coverage. Note that the distinction between $\theta_1$ and $\theta_2$ is similar in spirit to that of sampling-based and design-based uncertainty in @abadie2020sampling, but the details of their framework are very different from ours.
Assumptions
===========
We impose the following assumptions for our analysis.
The sampling and infection indicators are independent conditional on $N_I$: $$\mathbf{S}\bot \mathbf{I}|N_I$$
The infection status of each individual is as good as randomly assigned conditional on $N_I$, in the sense that for all $N_T$-vectors $\mathbf{i}=(i_1,\ldots, i_{N_T})$ of dummy variables with $\sum_{j=1}^{N_T} i_j = N_I$ we have that: $$P(\mathbf{I}=\mathbf{i}|N_I) = {N_T \choose N_I}^{-1}.$$
The individuals included in the study sample are determined by simple random sampling independently of $N_I$, in the sense that for all $N_T$-vectors $\mathbf{s}=(s_1,\ldots, s_{N_T})$ of dummy variables with $\sum_{j=1}^{N_T} s_j=N_S$ we have that $$P(\mathbf{S}=\mathbf{s}|N_I) = {N_T \choose N_S}^{-1}.$$
Assumption 1 is natural, and likely to hold even unconditionally. It would be violated, for example, if individuals with knowledge of their infection status are more or less like to participate in the study. Assumption 2 implies that the individuals infected at the time of the study are representative for the entire population. This rules out, for example, different age groups being affected more or less severely over the course of the pandemic. Note that the “success” probability $N_I/N_T$ can be changed to accommodate infection testing with less than 100% sensitivity and specificity. Assumption 3 can easily be adapted if the sample of $N_S$ individuals is obtained though a different sampling scheme, such as cluster sampling. Note that an equivalent definition of $\theta_2$ under the above assumptions is given by $$\theta_2 = {\mathbb{E}}\left(\frac{N_D}{N_I}\right),$$ so that this parameter can be interpreted as the “average” IFR, where the averaging is done with respect to the distribution of $\mathbf{I}$. This representation also makes it more apparent that $\widehat{\theta}$ is actually a suitable estimate of $\theta_2$.
Since $\widehat{\theta}$ depends on $\mathbf{S}$ and $\mathbf{I}$ through $N_P$ and $N_D$ only, it is also useful to state the implications of the above assumptions for the joint distribution of the latter two quantities conditional on $N_I$. Simple calculations show that this joint conditional distribution corresponds to two independent binomials: $$N_P \bot N_D|N_I, \qquad N_P|N_I\sim \textnormal{Binomial}\left(N_S, \frac{N_I}{N_T}\right), \qquad
N_D|N_I \sim \textnormal{Binomial}\left(N_I , \theta_2 \right).$$ These distributions should be kept in mind for the following arguments.
Confidence Sets
===============
Consider a test of the null hypothesis $H_0: \theta_2 = \theta^o$ that uses the estimated IFR $\widehat{\theta}$ as the test statistic. We propose to construct $(1-\alpha)$ CIs for $\theta_2$ by collecting all values of $\theta^o$ for which the $p$-value of such a test is less than $\alpha$. With conditioning on $N_I$, the number of infections effectively becomes a nuisance parameter in this testing problem; and since $N_I$ is unknown no exact $p$-value is feasible in this setup. However, we can still use existing statistical approaches to obtain CIs with good coverage properties. We specifically consider one based on the parametric bootstrap, and one based on varying $N_I$ over a “large” preliminary CI.
To describe these two approaches in our context, we introduce some notation. For constants $n_I$ and $\theta^o$, let $N_P^*$ and $N_D^*$ be independent random variables that each follow particular binomial distributions that only depend on the constants and other observable quantities: $$N_P^* \bot N_D^*, \quad N_P^* \sim \textnormal{Binomial}\left(N_S, \frac{n_I}{N_T}\right),
\quad N_D^* \sim \textnormal{Binomial}\left(n_I, \theta^o \right).$$ We also put $\widehat{N}^*_I =N_T N_P^*/N_S$, and denote the CDF of the ratio $N_P^*/\widehat{N}^*_I$ by $$G(c|n_I,\theta^o) = P\left( \frac{N_D^*}{\widehat{N}_I^*} \leq c\right).$$ There is no simple closed form expression for this distribution function, but it can easily be computed through standard numerical methods for any value of the constants $n_I$ and $\theta^o$. For example, one can compute $G(c|n_I,\theta^o)$ to desired accuracy by simulating a sufficiently large number of draws from the distribution of $(N_P^*,N_D^*)$, and then taking the empirical CDF of the resulting realizations of $N_P^*/\widehat{N}^*_I$.
The function $G(c|N_I,\theta_2)$ is the CDF of $\widehat{\theta}$ conditional on $N_I$ under the statistical model described above, and $G(c|N_I,\theta^o)$ is the CDF under $H_0: \theta_2 = \theta^o$. If $N_I$ was observed, an equal-tailed $p$-value for a test of $H_0$ based on $\widehat{\theta}$ would be given by $$p(\theta^o,N_I) = 2\min\left\{\widehat G(\widehat{\theta}|N_I,\theta^o), 1-\widehat G(\widehat{\theta}|N_I,\theta^o)\right\}.$$
Using a “plug-in” or parametric bootstrap approach [e.g. @horowitz2001bootstrap; @hall2013bootstrap], we can substitute the estimator $\widehat{N}_I$ into the $p$-value formula to construct a feasible CI for $\theta_2$: $$\mathcal{C}_{2,PB}^{\alpha} = \{\theta^o: p(\theta^o,\widehat{N}_I) \geq \alpha \}.$$ This CI is easily seen to have correct asymptotic coverage of $\theta_2$ conditional on $N_I$ under any sequence for which $\widehat N_I / N_I = 1 + o_P(1)$. That is, it holds that $$P( \theta_2 \in \mathcal{C}^\alpha_{2,PB}|N_I) = 1-\alpha + o_P(1) \quad\textnormal{ if }\quad\widehat N_I / N_I = 1 + o_P(1).$$ If the sample size $N_S$ is rather large, it can be reasonable to treat $\widehat N_I$ as a consistent estimate of $N_I$, in which case the above result implies that $\mathcal{C}_{2,PB}^{\alpha}$ has approximately correct finite sample coverage of $\theta_2$.
If the goal is to have a CI with guaranteed finite sample coverage, a different method can be used to compute a $p$-value. Let $[ L_\beta; U_\beta ]$ be a standard $(1-\beta)$ Clopper-Pearson CI for the share $N_I/N_T$ of infected individuals in the population, so that $\mathcal{C}^\beta = [N_T L_\beta; N_T U_\beta ]$ is a $(1-\beta)$ CI for the number of infections $N_I$, for some $\beta$ substantially smaller than $\alpha$. We can then obtain a new $p$-value by maximizing $p(\theta^o,n_I)$ over $n_I\in\mathcal{C}^\beta$, and correcting the result for the fact that $\beta$ is not zero [@berger1994; @Silvapulle1996]. This yields the following CI for $\theta_2$: $$\mathcal{C}_{2,CS}^{\alpha} = \left\{\theta^o: \sup_{n_I\in \mathcal{C}^\beta} p(\theta^o, n_I) + \beta \geq \alpha \right\}.$$ This CI has conditional coverage of at least $1-\alpha$ in finite samples: $$P( \theta_2 \in \mathcal{C}_{2,CS}^\alpha|N_I) \geq 1-\alpha.$$ The CI is conservative, however, in that the last inequality is generally strict. Exact coverage only occurs in the unlikely scenario that the supremum in the definition of the $p$-value is attained at $N_I$, which happens only if $N_I$ coincides with one of the boundaries of $\mathcal{C}^\beta$.
Numerical Illustration
======================
We illustrate methods described above with numerical values taken from @Streeck2020. The town investigated in that study has $N_T = 12,597$ inhabitants, of which $N_D = 7$ died in the study period with a SARS-CoV-2 infection. Out of a sample of $N_S=919$ individuals, $N_P = 138$ tested positive for SARS-CoV-2. This corresponds to an infection rate of $N_P/N_S = 15.0\%$ in the sample, an estimated $\widehat{N}_I = 1892$ infected individuals in the population, and an estimated IFR of $\widehat{\theta}=0.37\%$. Setting $\alpha =.05$ and $\beta=.01$, we obtain the CIs $$\mathcal{C}_1^{\alpha} = [0.32\%; 0.43\%], \qquad \mathcal{C}_{2,PB}^{\alpha} = [0.16\%; 0.74\%], \qquad \mathcal{C}_{2,CS}^{\alpha} = [ 0.14\%; 0.81\%].$$ Recall that the first of these CIs has $\theta_1$ as the target parameter, while the latter two aim for coverage of $\theta_2$. As expected, the latter two CIs are substantially wider than the first. We would argue that they are also more appropriate measures of uncertainty about the IFR estimate, since this quantity is used to design policy measures that affect the entire population.
We note that @Streeck2020 report an estimated 1,956 infected individuals, an IFR of .36%, and a CI for the IFR of $[0.29\%; 0.45\%]$. These results differ from the $\widehat{N}_I$, $\widehat{\theta}$ and $\mathcal{C}_1^{\alpha}$ given above for two reasons: first, @Streeck2020 apply an adjustment factor to the raw infection rate in their sample to account for the sensitivity and specificity of their test for SARS-CoV-2 infection; and second, their sample is generated through a form of cluster sampling, which leads to a slightly wider CI relative to simple random sampling. Such adjustments should also slightly widen our CIs for $\theta_2$.
Discussion
==========
While this note is motivated by research on the current SARS-CoV-2 pandemic, the CIs proposed here could also be used in other contexts in which researchers want to combine sample and population data in a similar fashion. To give an economic example, suppose that there is a group of individuals that qualify for benefits from some public program, and that the researcher is interested in the share of these individuals that actually receive benefits (this share could be small if the program is not well-known, difficult to apply for, or comes with social stigma). This then fits into the framework of this note if the number of benefit recipients is known to administrators, but the number of qualifying individuals needs to be estimated from survey data.
[^1]: Department of Economics, University of Mannheim, 68131 Mannheim, Germany. E-Mail: [email protected]. Website: http://www.christophrothe.net.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'A dedicated front-end electronics has been developed for the trigger chambers of the ALICE muon spectrometer under construction at the future LHC at CERN. These trigger chambers are based on RPCs (Resistive Plate Chambers) working in streamer mode. The number of electronics channels (about 21000) and the fact that RPC pulsed signals have specific characteristics have led to the design of a 8 channels front-end ASIC using a new discrimination technique. The principle of the ASIC is described and the radiation hardness is discussed. Special emphasis is put on production characteristics of about 4000 chips.'
author:
- 'Philippe Rosnet and Laurent Royer, for the ALICE Collaboration [^1]'
title: 'Front-end electronics for the ALICE dimuon trigger RPCs'
---
LHC, ALICE, QGP, quarkonia, RPC, streamer, front-end, ASIC.
Introduction to physics challenge
=================================
ALICE [@ALICE_TP] (A Large Ion Collider Experiment) is a detector designed for the study of nucleus-nucleus collisions at the future LHC (Large Hadron Collider). Its physics program will address question concerning QCD (Quantum ChromoDynamics) of hot and dense nuclear matter produced in central heavy ion collisions at a center of mass energy of 5.5 TeV. The main goal of ALICE is to characterize a deconfined state of matter called the Quark-Gluon Plasma (QGP) [@ALICE_PPR1]. One of its most promising probes is the production of quarkonia (J/$\Psi$ or $\Upsilon$), which is expected to be decreased by color screening in the QGP [@Vogt].
The role of the ALICE forward spectrometer [@ALICE_TDRmuon] is to reconstruct quarkonia in their dimuon channel in an angular acceptance of $[2^0,9^0]$. To reach this physics goal, the spectrometer consists of a front and small angle (beam shielding) absorber, a set of high resolution tracking chambers, a dipole magnet, an iron wall (muon filter) and a trigger system.
Muon trigger system
===================
The trigger system is composed by two stations of two detector planes of about $6 \times 6$ m$^2$ separated by one meter. The detectors are RPCs (Resistive Plate Chambers) [@Santonico] working in streamer mode at a high voltage value of about 8 kV [@RPC_streamer]. The necessary granularity implies to use strips of about 1, 2 and 4 cm width projective to the interaction point in each of the four planes, for a total of twelve different strip widths. Two perpendicular strip planes are used on each RPC (one on each side of the gas gap) to allow a three dimensional hits reconstruction. The total number of channels is then close to 21000.
The muon trigger is involved in the level 0 (L0) of the general ALICE trigger system [@ALICE_TDRtrigger]. The timing constraint required by the L0 is a muon trigger signal delivered each 25 ns (40 MHz) to the central trigger processor less than 800 ns after the collision. The architecture of the muon trigger electronics consists of the front-end electronics which picks-up and processes the RPC pulses and sends logical signals to the local trigger electronics (234 boards). The role of the local trigger is to store all signals (in a sequence of bit-patterns) in a pipeline memory and to identify single tracks with a transverse momentum above pre-defined cuts (by using a look-up-table) by means of a dedicated algorithm located in FPGAs (Field Programmable Gate Arrays). The regional (16 boards) and next the global (1 board) triggers collect the information from the local boards in order to select single or dimuon events from the full system.
Front-end electronics principle
===============================
As shown in Fig. \[fig-RPC\_pulses\], the RPC pulses are composed of two peaks, called precursor and streamer. The precursor has the particularity to be synchronous with the particle crossing the RPC. The second peak is characterized by a variable time jitter with respect to the precursor, but with a large amplitude as expected for a streamer pulse.
![Typical pulses picked-up on a single gas gap RPC operating in streamer mode, with a digital scope (1 GHz bandwidth) via a short BNC cable (50 $\Omega$ impedance) synchronized with particles crossing scintillator plates readout by photo-multipliers.[]{data-label="fig-RPC_pulses"}](fig-RPC_pulses.eps){width="2.5in"}
ASIC main characteristics
-------------------------
To take advantage of these properties a new discrimination technique, called ADULT [@FEE_ADULT] (A DUaL Threshold), has been developed and implemented in a 8 channels ASIC using the AustriaMicroSystems BiCMOS 0.8 $\mu$m technology [@FEE_ASIC]. This bipolar technology, with very low offset, is well adapted to trigger on small amplitude signals. The necessary number of ASIC for equipping all the muon trigger chambers is 2624 (not including spares).
The schematic of a single electronics channel is shown in Fig. \[fig-ASIC\_principle\]. The precursor detection with a low threshold (typically 10 mV) provides a good time reference, and the streamer validation with a high threshold (about 100 mV) takes advantage of this RPC working mode: a large signal/noise ratio and a small cluster size (defined as the mean number of adjacent strips fired). A coincidence of these two discriminator output signals defines a hit of a strip. The time resolution obtained with this technique is comparable to the avalanche, which is about 1 ns as compared to 3 ns typically obtained with a single discriminator in streamer mode. The delayed (15 ns) low threshold comparator defines the reference time by using the precursor as long as the streamer jitter is less than about 15 ns. The oneshot function is used to latch the two comparators during 100 ns, via a monostable, when a streamer signal has been validated to avoid re-triggering. A remote control delay, up to 50 ns, common to the 8 channels of the ASIC is tuned by an external DC voltage and allows to adjust the timing of the ASIC. The signal is then converted into a (22-23) ns logical LVDS signal in order to drive the signal through a twisted pair cable from the RPCs to the local electronics trigger boards.
{width="7.0in"}
The main characteristics of an ADULT ASIC, as shown in Fig. \[fig-ASIC\_pictures\], are the following:
- AustriaMicroSystems BiCMOS 0.8 $\mu$m technology,
- 8 electronics channels,
- die surface equal to about 8 mm$^2$,
- plastic packaging type PLCC 52 pins,
- power consumption: 10 mW/channel for $-2$ V and 80 mW/channel for $+3.5$ V.
![ASIC after packaging (upper picture) and view of the silicon part with a microscope (lower picture).[]{data-label="fig-ASIC_pictures"}](fig-ASIC_pictures.eps){width="2.5in"}
Front-end boards supporting the ASIC
------------------------------------
The ASIC is implemented on a dedicated board developed to pick-up the signals provided by the RPC strips, as shown in Fig. \[fig-FEB\_picture\]. Twelve different strip widths are necessary to equip the full detector, but only six boards with different mechanical characteristics are needed. Two ASICs are used on the boards corresponding to strips of 1 cm width (16 channels) while the boards associated with strips of width 2 cm or 4 cm (8 channels) contain only one ASIC. Each single gas gap RPC provides positive pulses on one side and negative pulses on the other side of the gas gap. Then, the two polarities are implemented at the cabling level on the front-end boards. Furthermore, the different distances between the boards and the trigger electronics located in racks above the chambers (6 meters high) lead to different output cable length. These differences were compensated for on the boards by implementing delays in step of 7.5 ns corresponding to 1.5 m of cable. To summarize, the total number of different boards after cabling is equal to ten and each board can be configured with five possible delays, but the same ASIC is used on all boards.
In addition, a test system has been implemented on each front-end board: a LVDS signal is received on the board, translated in TTL which, and inverted, if appropriate, following the polarity associated to the board. Then, a buffer allows to send an analogue pulse to each of the 8 ASIC channels, simulating a RPC pulse with a width of 20 ns.
An adaptation board is associated to each front-end board at the other extremity of the strips. It is a simple 50 $\Omega$ resistor which avoids signal reflection along the strips.
{width="6.0in"}
Radiation tests
===============
Tests have been performed to check the radiation hardness in conditions close (or worse) than the expected ALICE working environment:
- Cumulative effects due to low energy neutrons and to ionizing particles (such as photons) which damage progressively the electronics components. The simulations indicates 2.6 Gy over 10 LHC years for the boards closest to the beam.
- Single event effects due to energy hadrons ($> 30$ MeV) which induce malfunctioning or may even destroy components. The simulations indicates $2 \times 10^{11}$ hadrons/cm$^2$ over 10 LHC years (and integrated over the energy spectrum) for the boards closest to the beam.
The first test was done at the Gamma Irradiation Facility (GIF) at CERN providing $\gamma$ of 662 keV from a Cs source. The total dose received by the electronics was about 0.25 Gy. A second test was performed at the neutron generator of the LPC Clermont-Ferrand which delivers neutrons of 14.1 MeV. The total neutron fluence received by the electronics components was $3.8 \times 10^{10}$ n/cm$^2$ corresponding to a dose of 2.5 Gy which was measured with the help of PIN diodes. The third test was done with protons of 60 MeV at the Paul Scherrer Institute (PSI) at Zürich. The fluence measured on the board was $2.3 \times 10^{11}$ p/cm$^2$.
During all these tests, the front-end boards were active and pulsed signals from a generator (or the internal test system of the board) were sent (or activated) to check the ASIC (or the whole chain: test system plus ASIC) for each electronics channel. For the three campaigns, no problem has been observed, as illustrated for example by Fig. \[fig-PSI\_radiation\] showing the response time obtained with two ASICs of the same board before and after irradiation to protons at PSI.
![Response time for 16 channels (2 ASICs) before and after irradiation with the proton beam at PSI-Zürich.[]{data-label="fig-PSI_radiation"}](fig-PSI_radiation.eps){width="3.5in"}
Front-end electronics production
================================
The main constraint on this front-end electronics is to deliver the signal of any of the 21000 channels to the muon trigger electronics in a time window less than 25 ns (corresponding to the LHC clock). By taking into account all the time dispersion sources coming mainly from the RPC itself, the strip length (up to 72 cm), the output signal cables (up to 20 meters), the requirement for the front-end electronics is a time dispersion less than 4 ns.
To check this time requirements and others parameters of each electronics channels, the test of the production is divided in two steps:
- a working test and characterization of each ASIC (whole production done end of the year 2003 corresponding to 3880 chips),
- a tuning of the response time of the board (mean value) with the help of a potentiometer and a measurement of each parameters (about 12) of each electronics channel of the front-end boards (production done during summer 2004).
Automatic test bench design
---------------------------
The test bench is based on a special card equipped with relays which allow to switch to any electronics input and output channel of a board (the six different mechanical front-end boards can be handled). The input signal is generated by a pulse generator simulating RPC signals, including precursor and streamer peaks, with the possibility to vary each parameters of the signal (amplitude of each peak, time jitter between precursor and streamer, ...). A copy of this signal is sent to a scope (500 MHz bandwidth) to measure exactly its characteristics. The scope is also used to measure the output LVDS signal generated by the front-end board. The apparatus of the test bench are controlled by Labview via GPIB and the relays via a DIO card.
Test of the ASIC production
---------------------------
In practice, for the test of the ASICs, the same front-end board equipped with a socket allowing to plug and unplug easily each chip was used. For each ASIC, the 8 output signals were characterized by measuring the amplitude of the LVDS signal $A_{ch}$, its width $w_{ch}$, and the time difference between the slowest and the fastest channels $\Delta t_{ASIC}$ (which can be associated to the response time dispersion of the ASIC). Fig. \[fig-ASIC\_production\] shows the channel output signal width distribution and the ASIC time dispersion. The requirements for each parameter are: $A_{ch} = (800 \pm 100)$ mV, $w_{ch} = (23.0 \pm 1.5)$ ns and $\Delta t_{ASIC} < 3$ ns.
![ASIC production characteristics: channel output signal width $w_{ch}$ (upper plot) and ASIC dispersion time $\Delta t_{ASIC}$ (lower plot). The dotted lines on each plot represent the requirements.[]{data-label="fig-ASIC_production"}](fig-ASIC_1.eps "fig:"){width="3.5in"} ![ASIC production characteristics: channel output signal width $w_{ch}$ (upper plot) and ASIC dispersion time $\Delta t_{ASIC}$ (lower plot). The dotted lines on each plot represent the requirements.[]{data-label="fig-ASIC_production"}](fig-ASIC_2.eps "fig:"){width="3.5in"}
Over 3880 ASICs, the results are the following:
- 5.7% were not working (due to short-circuits, ...),
- 11.8% were working but with at least one parameter outside limits (about half due to output signal width and half due to response time dispersion),
- 82.5% were within specifications.
This means that 3280 ASICs are available for ALICE front-end board production (while 2624 are needed).
Test of the pre-production boards
---------------------------------
The whole production of the boards (including spares) was done during summer 2004. At the time of the writing, the tests have just started and are scheduled to finish in summer 2005.
However, the test of a pre-production representing about 12% the whole production (286 boards) was performed at the beginning of 2004. These boards are devoted to the test bench of the RPCs located at Torino. Each board has been tested and all the parameters measured. Fig. \[fig-FEB\_production\] displays the results obtained for three parameters: the response time to the internal test system, the low and high threshold discriminator.
![Front-end boards pre-production characteristics: response time with the internal test system (upper plot), low threshold discriminator (middle plot) and high threshold discriminator (lower plot). The dotted lines on each plot represent the requirements.[]{data-label="fig-FEB_production"}](fig-FEB_3.eps "fig:"){width="3.5in"} ![Front-end boards pre-production characteristics: response time with the internal test system (upper plot), low threshold discriminator (middle plot) and high threshold discriminator (lower plot). The dotted lines on each plot represent the requirements.[]{data-label="fig-FEB_production"}](fig-FEB_2.eps "fig:"){width="3.5in"} ![Front-end boards pre-production characteristics: response time with the internal test system (upper plot), low threshold discriminator (middle plot) and high threshold discriminator (lower plot). The dotted lines on each plot represent the requirements.[]{data-label="fig-FEB_production"}](fig-FEB_4.eps "fig:"){width="3.5in"}
The most crucial parameter, [*e.g.*]{} the response time, shows good behavior (the RMS of the distribution is less than 1 ns) with respect to the ALICE requirement (all 21000 channels in a time window of 4 ns due to the front-end dispersion alone). The value of the discriminator thresholds are relatively well peaked around the expectations.
During the test, it appeared that 5 ASICs were not working over 316 (among the 286 boards, 30 have two ASICs), namely about 2%. The other problems met are attributed to the cabling and concerned about 4% of the boards (only one problem on printed circuit was reported). After intervention, only 2% of the board have at least one channel with one or more parameters outside specifications. In this case, the only possibility to solve the problem was to change the ASIC.
Conclusion
==========
The dedicated front-end electronics of the ALICE muon trigger, for RPC in streamer mode, is based on a 8 channels ASIC which has been produced and tested. Its performance fulfils the ALICE requirements, both in terms of timing and concerning the low threshold discriminator sensitivity needed for the ADULT technique. The yield of bad chips (less than 6%) is acceptable. The pre-production of front-end boards supporting the ASIC shows that it is possible to tune each board such that all channels are within a time window of 4 ns. The fraction of chips to be changed after the cabling phase is about 4%, leaving a sufficient number of spares for the lifetime of the ALICE experiment.
Acknowledgment {#acknowledgment .unnumbered}
==============
The authors would like to thank the Région Auvergne (France) for funding support.
[1]{}
ALICE Collaboration, ALICE Technical Proposal, CERN/LHCC 95-71 (1995).\
ALICE Collaboration, ALICE Technical Proposal (Addendum), CERN/LHCC 96-32 (1996).
ALICE Collaboration, ALICE Physics Performance Report (Volume 1), CERN/LHCC 2003-049 (2003).
R. Vogt, Phys. Rept. 310 (1999) 197-260.
ALICE Collaboration, ALICE Technical Design Report, CERN/LHCC 99-22 (1999).
R. Santonico and R. Cardarelli, Nucl. Instr. and Meth. A 187 (1981) 377-380.
R. Arnaldi et al., Nucl. Instr. and Meth. A 456 (2000) 462-473.
ALICE Collaboration, ALICE Technical Design Report, CERN/LHCC 2003-062 (2003).
R. Arnaldi et al., Nucl. Instr. and Meth. A 457 (2001) 117-125.\
P. Dupieux, Nucl. Instr. and Meth. A 508 (2003) 185-188.
L. Royer et al., Proceeding of the 6th Workshop on electronics for the LHC experiments, CERN/LHCC 2000-041 (2000) 323-327.
[^1]: Talk given by Ph. Rosnet, LPC Clermont-Ferrand, CNRS/IN2P3 and Université Blaise Pascal, 63177 Aubière Cedex, France (Email: [email protected]).
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Two phenomena can affect the transmission of a weak signal field through an absorbing medium in the presence of a strong additional field: electromagnetically induced transparency (EIT) and Autler-Townes splitting (ATS). Being able to discriminate between the two is important for various applications. Here we present an experimental investigation into a method that allows for such a disambiguation as proposed in \[Phys. Rev. Lett. **107**, 163604 (2011)\]. We apply the proposed test based on Akaike’s information criterion to a coherently driven ensemble of cold cesium atoms and find a good agreement with theoretical predictions, therefore demonstrating the suitability of the method. Additionally, our results demonstrate that the value of the Rabi frequency for the ATS/EIT model transition in such a system depends on the level structure and on the residual inhomogeneous broadening.'
author:
- 'L. Giner'
- 'L. Veissier'
- 'B. Sparkes'
- 'A. S. Sheremet'
- 'A. Nicolas'
- 'O. S. Mishina'
- 'M. Scherman'
- 'S. Burks'
- 'I. Shomroni'
- 'D. V. Kupriyanov'
- 'P. K. Lam'
- 'E. Giacobino'
- 'J. Laurat'
title: |
Experimental Investigation of the Transition between Autler-Townes Splitting\
and Electromagnetically-Induced Transparency
---
Fine engineering of interactions between light and matter is critical for various purposes, including information processing and high-precision metrology. For more than two decades, coherent effects leading to quantum interference in the amplitudes of optical transitions have been widely studied in atomic media, opening the way to controlled modifications of their optical properties [@Fleischhauer]. More specifically, such processes as coherent population trapping [@Alzetta1976; @Arimondo] or electromagnetically induced transparency (EIT) [@Harris1990; @Harris1991; @Marangos] allow one to take advantage of the modification of an atomic system by a so-called control field to change the transmission characteristics of a probe field. These features are especially important for the implementation of optical quantum memories [@Tittel] relying on dynamic EIT [@Hau], or for coherent driving of a great variety of systems, ranging from superconducting circuits [@Kelly] to nanoscale optomechanics [@Safavi].
However, if in general the transparency of an initially absorbing medium for a probe field is increased by the presence of a control field, two very different processes can be invoked to explain it in a $\Lambda$-type configuation. One of them is a quantum Fano interference between two paths in a three-level system [@Fano1961], which occurs even at very low control intensity and gives rise to EIT [@Harris97]. The other one is the appearance of two dressed states in the excited level at large control intensity, corresponding to the Autler-Townes splitting (ATS) [@Autler1955; @Cohen77; @Cohen]. Discerning whether a transparency feature observed in an absorption profile is the signature of EIT or ATS is therefore crucial [@Anisimov; @Salloum; @Zhang]. A recent paper by P.M. Anisimov, J.P. Dowling and B.C. Sanders [@Sanders2011] introduced a versatile and quantitative test to discriminate between these two phenomena.
In this paper, we report an experimental study of the proposed witness, relying on a detailed analysis of the absorption profile of a probe field in an atomic ensemble in the presence of a control field. In order to analyze the quantum interferences in detail and avoid any inhomogeneous broadening, our study is performed with an ensemble of cold cesium atoms in a well-controlled magnetic environment. We show that the general behavior is in agreement with Ref. [@Sanders2011], but we identify some quantitative differences. We finally interpret the characteristics of the EIT to ATS model transition by taking into account the multilevel structure of the atomic system and some residual inhomogeneous broadening.
{width="1.85\columnwidth"}
The experimental setup is illustrated in Fig. \[exp\](a). The optically thick atomic ensemble is obtained from cold cesium atoms in a magneto-optical trap (MOT). The three-level $\Lambda$ system involves the two ground states, $|g\rangle=|6S_{1/2},F=3\rangle$ and $|s\rangle=|6S_{1/2},F=4\rangle$, and one excited state $|e\rangle=|6P_{3/2},F=4\rangle$. The control field is resonant with the $|s\rangle$ to $|e\rangle$ transition, while the probe field is scanned around the $|g\rangle$ to $|e\rangle$ transition, with a detuning $\delta$ from resonance.
Each run of the experiment involves a period for the cold atomic cloud to build up and a period for measurement. This sequence is repeated every 25 ms and controlled with a FPGA board. After the build-up of the cloud in the MOT, the current in the coils generating the trapping magnetic field and then the MOT trapping beams are switched off. In order to transfer the atoms from the $|s\rangle$ to the $|g\rangle$ ground state, the MOT is illuminated with a $\sigma^{-}$-polarized 1 ms-long depump pulse with a power of 900 $\mu$W and resonant with the $|s\rangle$ to $|e\rangle$ transition. After this preparation stage, the optical depth at resonance for atoms in $|s\rangle$ is zero within our experimental precision. The remaining spurious magnetic fields have been canceled down to 5mG using a RF spectroscopy technique.
The measurement period starts 3 ms after the extinction of the MOT magnetic field. The atomic ensemble is illuminated with a 30 $\mu$s-long control pulse and a probe pulse lasting 15 $\mu$s is sent during this time. The probe field is emitted by an extended cavity grating stabilized laser diode, whereas the control field is generated by a Ti:Sapphire laser locked on resonance using saturated absorption spectroscopy. The two lasers are phase-locked. The control field is $\sigma^{-}$-polarized, with a 200 $\mu$m waist in the MOT and a 2$^\circ$ angle relative to the direction of the probe beam. The probe field is $\sigma^{+}$-polarized, with a waist of 50 $\mu$m and a power of 30 nW. To measure absorption profiles, the probe beam frequency is swept over a few natural linewidths by changing the locking frequency point. Its absorption is measured with a high-gain photodiode. The optical depth in the $|g\rangle$ state is chosen to be around 3 to avoid any profile shape distortion due to the limited dynamic range of the photodiode.
Figure \[exp\](b) gives the absorption of the probe field, $A=\textrm{ln}(I_{\textrm{ref}}/I)$, as a function of its detuning $\delta$ from resonance for different values of the control power (0.1 to 200 $\mu$W), i.e. for different values of the control Rabi frequency $\Omega$. The quantity $I_{\textrm{ref}}$, which gives the transmission in the absence of atoms, is measured by sending an additional probe pulse when all the atoms are still in the $|s\rangle$ ground state. The Rabi frequency $\Omega$ of the control field is changed from very weak values (at the back) to four times the natural linewidth $\Gamma$ (at the front). Each profile results from an averaging over twenty repetitions of the experiment. The narrow transparency dip appearing for low Rabi frequencies gets wider when the Rabi frequency increases, to finally give two well-separated resonances corresponding to the two excited dressed states.
Let us note that the Rabi frequency $\Omega$ is a linear function of the electric field, and can be expressed as $\Omega=\alpha\,\sqrt{P}$, with $P$ the power of the control field. An effective value of $\Omega$ can be inferred from the experimental splittings (i.e. the distance between the two maxima) observed in the absorption profiles for low-power control field (Fig. \[exp\](c)). For a three-level system this splitting is indeed equal to the Rabi frequency within a very good approximation for low decoherence in the ground state [@Arimondo]. We find $\alpha=1670\pm 100
\,\textrm{MHz}/\sqrt{\textrm{W}}$.
We now turn to the detailed analysis of the absorption profiles. For a three-level $\Lambda$ system, to first order in the probe electric field, the atomic susceptibility on the probe transition for a control field on resonance is given by [@Anisimov]: $$\begin{aligned}
\label{khi}
\chi(\delta)=-\frac{n_{g}
|d_{eg}|^{2}}{\mathbf{\hbar}\epsilon_{0}}\,
\frac{\delta+i\gamma_{gs}}{\delta^2-|\Omega_{0}|^2/4-\gamma_{eg}\gamma_{gs}+i\delta(\gamma_{eg}+\gamma_{gs})}\end{aligned}$$ ${n_{g}}$ stands for the atomic density in state $|g \rangle$ and $d_{eg}$ denotes the electric dipole moment between $|e\rangle$ and $|g\rangle$. Here, the Rabi frequency of the control field is $\Omega_{0}=2\,|d_{es}|\varepsilon_c/\hbar$, with $\varepsilon_c$ the amplitude of the positive frequency part of the control field. The optical coherence relaxation rate is $\gamma_{eg}=\Gamma/2$ where $\Gamma/2\pi=5.2 $ MHz. $\gamma_{gs}$ is the dephasing rate of the ground state coherence, $\gamma_{gs}= 10^{-2}\Gamma$ in our experimental case.
Depending on the value of the control Rabi frequency $\Omega_{0}$, Eq. \[khi\] can be rewritten in different ways [@Anisimov; @Salloum; @Zhang]. For Rabi frequencies $\Omega_{0}<\Omega_t=\gamma_{eg}-\gamma_{gs}$, the spectral poles of the susceptibility are imaginary. Then, the linear absorption $A\propto\textrm{Im}[\chi]$ can be expressed as the difference between two Lorentzian profiles centered at zero frequency, a broad one and a narrow one. For $\Omega_{0}>\Omega_t$, this decomposition is not possible anymore. For large Rabi frequencies, $\Omega_{0}\gg\Gamma$, Eq. \[khi\] can be written as the sum of two well separated Lorentzian profiles with similar widths. Absorption profiles for these two model can thus be written as: $$\begin{aligned}
A\sb{\textrm{EIT}}&=\frac {C_{+}} {1 + (\delta -
\epsilon)^{2}/(\gamma_{+}^{2} /4)} - \frac {C_{-} } {1 +
\delta^{2}/(\gamma_{-}^{2} /4)} \label{EIT}\\
A\sb{\textrm{ATS}}&=\frac {C_{1}} {1 + (\delta +
\delta_{1})^{2}/(\gamma_{1}^{2}/4)} + \frac {C_{2}} {1 + (\delta -
\delta_{2})^{2}/(\gamma_{2}^{2}/4)} \label{ATS}\end{aligned}$$ where $C_{+}$, $C_{-}$, $C_{1}$, $C_{2}$ are the amplitudes of the Lorentzian curves, $\gamma_{+}$ and $\gamma_{-}$, $\gamma_{1}$ and $\gamma_{2}$ are their widths, $\epsilon$, $\delta_{1}$ and $\delta_{2}$ are their shifts from zero frequency. Equation \[EIT\] describes a Fano interference and corresponds to the EIT model, while Eq. \[ATS\] corresponds a strongly-driven regime with a splitting of the excited state, i.e. ATS.
For a three-level system, the various parameters introduced in the two above expressions can be calculated from Eq. \[khi\]. Conversely, in our experimental system, we use functions $A\sb{\textrm{EIT}}$ and $A\sb{\textrm{ATS}}$ to fit the experimental absorption curves, adjusting all the aforementioned parameters. The test proposed in [@Sanders2011] aims at determining which of these generic models is the most likely for given experimental data.
Figure \[curves\] shows the measured probe absorption as a function of the detuning $\delta$ (blue dots) together with the fits to $A\sb{\textrm{EIT}}$ (red curves) and $A\sb{\textrm{ATS}}$ (green curves). A low value of the control Rabi frequency, $\Omega=0.2\Gamma$ is shown in panel (a), and a larger one, $\Omega=2.3\Gamma$ in panel (b). Let us note that for the EIT model a detuning parameter $\epsilon$ was introduced between the atomic line center and the EIT dip to account for a possible experimental inaccuracy in the frequency locking reference of the lasers. For the ATS model the parameters describing each Lorentzian curve are independent of each other (contrary to what would be deduced from Eq. \[khi\]) in order to account for their experimentally different widths and heights. These asymmetries are discussed below. As expected, the EIT model fits better the low-power control field region (panel (a)) while the ATS model fits better the strong-power control field region (panel (b)).
![(color online). Absorption profiles and model fits for two values of the control Rabi frequency $\Omega$. Experimental data (blue dots) are presented together with the best fits of functions $A\sb{\textrm{EIT}}(C_{+},C_{-},\epsilon,\gamma_{+},\gamma_{-})$ (red solid lines) and $A\sb{\textrm{ATS}}(C_{1},C_{2},\delta_{1},\delta_{2},\gamma_{1},\gamma_{2})$ (green solid lines). Parameters $C_{+},C_{-},C_{1},C_{2}$ representing the amplitudes of the absorption curves are in dimensionless units, while the parameters $\epsilon,\gamma_{+},\gamma_{-}$ and $\delta_{1},\delta_{2},\gamma_{1},\gamma_{2}$ representing detunings and widths are in MHz. (a) $\Omega = 0.2
\Gamma$. In this case $A\sb{\textrm{EIT}}(3.52, 3.14, 1.45 \times
10^{-2},
5.71, 0.239)$ fits the experimental data much better than $A\sb{\textrm{ATS}}( 2.01, 2.04 ,1.84, 1.84,
4.14, 4.08)$. (b) $\Omega= 2.3 \Gamma$. Here $A\sb{\textrm{ATS}}(
2.05, 1.64, 5.86, 5.67, 3.94, 4.68)$ fits the data better than $A\sb{\textrm{EIT}}(1.59 \times 10^{5}, 1.59 \times 10^{5}, 1.45
\times 10^{-7}, 8.17, 8.17)$.[]{data-label="curves"}](figure2.pdf){width="0.95\columnwidth"}
As proposed in [@Sanders2011], in order to quantitatively test the quality of these model fits, we then calculate the *Akaike information criterion* (AIC) [@Akaike]. This criterion, directly provided by the function <span style="font-variant:small-caps;">NonLinearModelFit</span> in <span style="font-variant:small-caps;">Mathematica</span>, is equal to $I_{j}=2 k - \textrm{ln}(L_{j})$ where $k$ is the number of parameters used and $L_{j}$ the maximum of the likelihood function obtained from the considered model, labeled with $j$ ($j=\textrm{EIT}$ or $\textrm{ATS}$). The relative weights $w_{\textrm{EIT}}$ and $w_{\textrm{ATS}}$ that give the relative probabilities of finding one of the two models can be calculated from these quantities and are given by: $$w_{\textrm{EIT}}=\dfrac{e^{-I_{\textrm{EIT}}/2}}{ e^{-I_{\textrm{EIT}}/2}+ e^{-I_{\textrm{ATS}}/2}},\,\,w_{\textrm{ATS}}=1-w_{\textrm{EIT}}.\nonumber$$ These weights are plotted in Fig. \[criterion\] (curves (1) and (2)), as a function of the experimentally determined Rabi frequency. They exhibit a binary behavior. They are close to 0 or 1 and there is an abrupt transition from EIT model to ATS model.
![(color online). Experimental Akaike weights $w_{j}$ as a function of the Rabi frequency $\Omega$ for ATS model (green triangles, curve (1)) and for EIT model (red triangles, curve (2)). Experimental per-point weights $\overline{w}_{j}$ for ATS model (green dots, curve (3)) and for EIT model (red dots, curve (4)). The grey area indicates the EIT/ATS model transition. Error bars include the uncertainty on the coefficient $\alpha$ and on the measured power. The solid lines give the theoretical per-point weights for a pure 3-level system (curves (5) and (6)).[]{data-label="criterion"}](figure3.pdf){width=".95\columnwidth"}
We then investigate the second criterion proposed in Ref. [@Sanders2011], also based on Akaike’s information criterion but with a mean per-point weight $\overline{w}$. It can be obtained by dividing $I_{j}$ by the number of experimental points . The weights for the EIT and the ATS models are now given respectively by $$\overline{w}_{\textrm{EIT}}=\dfrac{e^{-I_{\textrm{EIT}}/2N}}{ e^{-I_{\textrm{EIT}}/2N}+
e^{-I_{\textrm{ATS}}/2N}}, \,\, \overline{w}_{\textrm{ATS}}=1-\overline{w}_{\textrm{EIT}}.\nonumber$$ The resulting curves are presented in Fig. \[criterion\] (curves (3) and (4)). Starting from a per-point weight equal to 0.5 for both models in the absence of control field (the two models are equally likely), the EIT model first dominates in the low Rabi frequency region. Then the likelihood of the EIT model decreases and a crossing is observed for the same value as for the previous criterion. The ATS model then dominates for larger Rabi frequency, as expected.
For the Akaike weights, as well as for the per-point weights, the behavior is in good qualitative agreement with the predictions given in Ref. [@Sanders2011] and with our simulations for a three-level system. However, the transition between the two models is obtained experimentally for $\Omega/\Gamma=1.23\pm 0.10$, while a value of $\Omega/\Gamma=0.91$ is obtained for the per-point weights of a pure three-level system, calculated for the same Rabi frequencies, as shown in Fig. \[criterion\] (solid grey lines). Moreover, for large Rabi frequencies the per-point weights corresponding to ATS and EIT saturate at 0.7 and 0.3 respectively instead of going to 1 and 0 as in the theoretical three-level model. For low Rabi frequencies, the shape of the curves also differs significantly. These various features suggest that the system cannot be described by a simple three-level model. Below, we proceed to theoretical simulations including additional parameters that influence the ATS/EIT model transition and the general shape of the per-point weight curves.
First, we take into account the other hyperfine sublevels of the $6P_{3/2}$ manifold, based on a previous theoretical model [@Oxy2011; @Michael2012]. We find that these contributions explain the asymmetry between the two dressed-state resonances observed in Fig. \[curves\](b) at large Rabi frequencies but they do not significantly influence the per-point weight curves. The latter are shown in Fig. \[fig4\](b) (solid lines), with a crossing point for $\Omega/\Gamma=0.91$.
We then consider the effect of the Zeeman structure. Several Zeeman sublevels are involved in each atomic level as shown in Fig. \[fig4\](a). We have determined the atomic distribution in the Zeeman sublevels from the optical pumping due to the depump field (Fig. \[fig4\](a), inset). Since the control and probe fields have opposite circular polarizations, we can consider that the atomic scheme is a superposition of six independent $\Lambda$ subsystems with different Rabi frequencies. The susceptibility is calculated as the sum of the corresponding susceptibilities. The per-point weights for theoretical absorption curves calculated from this model (including the hyperfine structure) are shown in Fig. \[fig4\](b) (dotted lines). For the horizontal axis, as the system does not have a single Rabi frequency, we have used an effective Rabi frequency obtained from the splitting between the maxima of the theoretical absorption curves. The transition point is found for $\Omega/\Gamma=0.98$, close to the value obtained for a three-level system. These simulations show that taking into account the Zeeman sublevels does not lead to a large enough alteration of the crossing point as compared to the three-level model. However a significant change in the values of the per-point weights for large Rabi frequencies is obtained for the model including the Zeeman sublevels, and it is comparable to the experimental one.
We finally include a residual inhomogeneous Doppler broadening $\Gamma_D$. By fitting the experimental absorption profile in the absence of control field, we obtain $\Gamma_D/2\pi=0.6$ MHz. The per-point weights for theoretical absorption curves including this residual broadening are given in Fig. \[fig4\](b) (dashed lines). The crossing point is found for a value $\Omega/\Gamma=1.05$, which is in better agreement with the experimental value. The slightly larger value of the experimental transition point is very likely to be due to heating and additional broadening caused by the control laser. If we assume an inhomogeneous broadening $\Gamma_D/2\pi=1.3$ MHz (dash-dotted lines), the per-point weights for the theoretical absorption curves cross each other for $\Omega/\Gamma=1.23$. Moreover the shape of the curves including even a small Doppler broadening agrees much better with the experimental results for the low control Rabi frequency region. Thus, including in the model both the Zeeman structure of the atomic system and a residual Doppler broadening due to the finite temperature of the atoms allows us to explain the observed experimental behaviour when the EIT-ATS discrimination criterion is applied.
![(color online). Theoretical simulations with Zeeman sublevels and Doppler broadening. (a) Level scheme for the Cs $D_{2}$ line and the six three-level transitions involving Zeeman sublevels; inset: population distribution. (b) Per-point weights for a three-level system (grey solid lines), for a system involving Zeeman sublevels (dotted lines), for a system involving Zeeman sublevels and residual Doppler broadenings $\Gamma_D/2\pi=0.6$ MHz (dashed lines) and $\Gamma_D=1.3/2\pi$ MHz (dash-dotted lines).[]{data-label="fig4"}](figure4.pdf){width=".9\columnwidth"}
In summary, we have tested and analyzed in detail the transition from the ATS model to the EIT model proposed in Ref. [@Sanders2011] in a well controlled experimental situation. The criteria have been calculated and give a consistent conclusion for discerning between the two regions. The observed differences from the three-level model have been interpreted by a refined model taking into account the specific level structure and some residual inhomogeneous broadening. This study confirms the sensitivity of the proposed test to the specific properties of the medium and opens the way to a new tool for characterizing complex systems involving coherent processes.\
This work was supported by the CHIST-ERA ERA-NET (QScale project), by the Australian Research Council Centre of Excellence for Quantum Computation and Communication Technology (CE110001027), and by the CNRS-RFBR collaboration (CNRS 6054 and RFBR P2-02-91056). A.S. acknowledges the support from the Foundation “Dynasty” and O.S.M. from the Ile-de-France program IFRAF. J.L. is a member of the Institut Universitaire de France.
[99]{}
|
{
"pile_set_name": "ArXiv"
}
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---
abstract: 'A transmitter without channel state information (CSI) wishes to send a delay-limited Gaussian source over a slowly fading channel. The source is coded in superimposed layers, with each layer successively refining the description in the previous one. The receiver decodes the layers that are supported by the channel realization and reconstructs the source up to a distortion. In the limit of a continuum of infinite layers, the optimal power distribution that minimizes the expected distortion is given by the solution to a set of linear differential equations in terms of the density of the fading distribution. In the optimal power distribution, as SNR increases, the allocation over the higher layers remains unchanged; rather the extra power is allocated towards the lower layers. On the other hand, as the bandwidth ratio $b$ (channel uses per source symbol) tends to zero, the power distribution that minimizes expected distortion converges to the power distribution that maximizes expected capacity. While expected distortion can be improved by acquiring CSI at the transmitter (CSIT) or by increasing diversity from the realization of independent fading paths, at high SNR the performance benefit from diversity exceeds that from CSIT, especially when $b$ is large.'
author:
- |
\
Email: {ngctk,andrea}@wsl.stanford.edu, [email protected], [email protected] [^1]
bibliography:
- 'ieeebib/IEEEabrv.bib'
- 'bib/wrlscomm.bib'
title: Minimum Expected Distortion in Gaussian Layered Broadcast Coding with Successive Refinement
---
Introduction
============
We consider the transmission of a delay-limited Gaussian source over a slowly fading channel in the absence of channel state information (CSI) at the transmitter. As the channel is non-ergodic, source-channel separation is not necessarily optimal. We consider the layered broadcast coding scheme in which each superimposed source layer successively refines the description in the previous one. The receiver decodes the layers that are supported by the channel realization and reconstructs the source up to a distortion. We are interested in minimizing the expected distortion of the reconstructed source by optimally allocating the transmit power among the layers of codewords.
The broadcast strategy is proposed in [@cover72:broadcast_ch] to characterize the set of achievable rates when the channel state is unknown at the transmitter. In the case of a Gaussian channel under Rayleigh fading, [@shamai97:bc_strat_slow_fade; @shamai03:bc_app_slow_fade_mimo] describe the layered broadcast coding approach and derive the optimal power allocation that maximizes the expected capacity. In the transmission of a Gaussian source over a Gaussian channel, uncoded transmission is optimal [@goblick65:lim_tx_analog_src] in the special case when the source bandwidth equals the channel bandwidth [@gastpar03:to_code_or_not]. For other bandwidth ratios, hybrid digital-analog joint source-channel transmission schemes are studied in [@shamai98:sys_lossy_src_ch; @mittal02:hda_src_ch_bc_rc; @reznic06:dstrn_bnd_bc_bw_exp], where the codes are designed to be optimal at a target SNR but degrade gracefully should the realized SNR deviate from the target. The distortion exponent, defined as the exponential decay rate of the expected distortion in the high SNR regime, is investigated in [@laneman05:src_ch_parl_ch] in the transmission of a source over two independently fading channels. For quasi-static multiple-antenna Rayleigh fading channels, distortion exponent upper bounds and achievable joint source-channel schemes are studied in [@gunduz05:src_ch_code_quasi_fading; @gunduz06:jt_src_ch_code_mimo; @caire05:snr_expn_hybrid_st]. The expected distortion of the layered source coding with progressive transmission (LS) scheme proposed in [@gunduz06:jt_src_ch_code_mimo] is analyzed in [@etemadi06:opt_layered_tx] for a finite number of layers at finite SNR. Concatenation of broadcast channel coding with successive refinement [@equitz91:sus_refn_info; @rimoldi94:sus_refn_info_ach] source coding is shown in [@gunduz05:src_ch_code_quasi_fading; @gunduz06:jt_src_ch_code_mimo] to be optimal in terms of the distortion exponent for multiple input single output (MISO) and single input multiple output (SIMO) channels. Numerical optimization of the power allocation with constant rate among the layers is examined in [@sesia05:pro_sup_hyb], while [@zachariadis05:src_fid_fading] considers the optimization of power and rate allocation and presents approximate solutions in the high SNR regime. The optimal power allocation that minimizes the expected distortion at finite SNR in layered broadcast coding is derived in [@ng07:recur_pow_lbc] when the channel has a finite number of discrete fading states. This work extends [@ng07:recur_pow_lbc] and considers the minimum expected distortion for channels with continuous fading distributions. In a related work in [@tian07:exp_dist_gaus_src_bc], the optimal power distribution that minimizes the expected distortion is derived using the calculus of variations method.
The remainder of the paper is organized as follows. Section \[sec:sys\_mod\] presents the system model, and Section \[sec:layered\_bc\_code\] describes the layered broadcast coding scheme with successive refinement. The optimal power distribution that minimizes the expected distortion is derived in Section \[sec:opt\_pow\_dist\]. Section \[sec:ray\_div\] considers Rayleigh fading channels with diversity, followed by conclusions in Section \[sec:conclu\].
System Model {#sec:sys_mod}
============
Consider the system model illustrated in Fig. \[fig:src\_ch\_coding\_pdf\]: A transmitter wishes to send a Gaussian source over a wireless channel to a receiver, at which the source is to be reconstructed with a distortion. Let the source be denoted by $s$, which is a sequence of independent identically distributed (iid) zero-mean circularly symmetric complex Gaussian (ZMCSCG) random variables with unit variance: $s\in\mathbb{C}\sim\mathcal{CN}(0,1)$. The transmitter and the receiver each have a single antenna and the channel is described by: $y = Hx + n$, where $x\in\mathbb{C}$ is the transmit signal, $y\in\mathbb{C}$ is the received signal, and $n\in\mathbb{C}\sim\mathcal{CN}(0,1)$ is iid unit-variance ZMCSCG noise.
\[\]\[\][$\gamma \sim f(\gamma)$]{} \[\]\[\][$x^N$]{} \[\]\[\][$y^N$]{} \[\]\[\][$s^K$]{} \[\]\[\][$\hat{s}^K$]{} \[\]\[\][$\mathcal{CN}(0,1)$]{} ![Source-channel coding without CSI at the transmitter.[]{data-label="fig:src_ch_coding_pdf"}](src_ch_coding_pdf.eps "fig:")
Suppose the distribution of the channel power gain is described by the probability density function (pdf) $f(\gamma)$, where $\gamma \triangleq {{\lvert{h}\rvert}^2}$ and $h\in\mathbb{C}$ is a realization of $H$. The receiver has perfect CSI but the transmitter has only channel distribution information (CDI), i.e., the transmitter knows the pdf $f(\gamma)$ but not its instantaneous realization. The channel is modeled by a quasi-static block fading process: $H$ is realized iid at the onset of each fading block and remains unchanged over the block duration. We assume decoding at the receiver is *delay-limited*; namely, delay constraints preclude coding across fading blocks but dictate that the receiver decodes at the end of each block. Hence the channel is non-ergodic.
Suppose each fading block spans $N$ channel uses, over which the transmitter describes $K$ of the source symbols. We define the *bandwidth ratio* as $b\triangleq N/K$, which relates the number of channel uses per source symbol. At the transmitter there is a power constraint on the transmit signal ${\mathrm{E}}\bigl[{{\lvert{x}\rvert}^2}\bigr] \leq P$, where the expectation is taken over repeated channel uses over the duration of each fading block. We assume a short-term power constraint and do not consider power allocation across fading blocks. We assume $K$ is large enough to consider the source as ergodic, and $N$ is large enough to design codes that achieve the instantaneous channel capacity of a given fading state with negligible probability of error.
At the receiver, the channel output $y$ is used to reconstruct an estimate $\hat{s}$ of the source. The distortion $D$ is measured by the mean squared error ${\mathrm{E}}[(s-\hat{s})^2]$ of the estimator, where the expectation is taken over the $K$-sequence of source symbols and the noise distribution. The instantaneous distortion of the reconstruction depends on the fading realization of the channel; we are interested in minimizing the expected distortion ${\mathrm{E}_H}[D]$, where the expectation is over the fading distribution.
Layered Broadcast Coding with\
Successive Refinement {#sec:layered_bc_code}
==============================
We build upon the power allocation framework derived in [@ng07:recur_pow_lbc], and first assume the fading distribution has $M$ discrete states: the channel power gain realization is $\gamma_i$ with probability $p_i$, for $i=1,\dotsc,M$, as depicted in Fig. \[fig:src\_ch\_layers\]. Accordingly there are $M$ virtual receivers and the transmitter sends the sum of $M$ layers of codewords. Let layer $i$ denote the layer of codeword intended for virtual receiver $i$, and we order the layers as $\gamma_M>\dotsb>\gamma_1\geq0$. We refer to layer $M$ as the highest layer and layer 1 as the lowest layer. Each layer successively refines the description of the source $s$ from the layer below it, and the codewords in different layers are independent. Let $P_i$ be the transmit power allocated to layer $i$, then the transmit symbol $x$ can be written as $$\begin{aligned}
x &= \sqrt{P_1}\,x_1 + \sqrt{P_2}\,x_2 + \dotsb +\sqrt{P_M}\,x_M,\end{aligned}$$ where $x_1,\dotsc,x_M$ are iid ZMCSCG random variables with unit variance. Suppose the layers are evenly spaced, with $\gamma_{i+1}-\gamma_i = \Delta\gamma$. In Section \[sec:opt\_pow\_dist\] we consider the limiting process as $\Delta\gamma\rightarrow0$ to obtain the power distribution: $$\begin{aligned}
\rho(\gamma) \triangleq \lim_{\Delta\gamma \rightarrow 0}
\dfrac{1}{\Delta\gamma} P_{\lceil \gamma/\Delta\gamma \rceil},\end{aligned}$$ where for discrete layers the power allocation $P_i$ is referenced by the integer layer index $i$, while the continuous power distribution $\rho(\gamma)$ is indexed by the channel power gain $\gamma$.
\[\]\[\][$p_1:\gamma_1$]{} \[\]\[\][$p_2:\gamma_2$]{} \[\]\[\][$p_M:\gamma_M$]{} \[\]\[\][$(P_1,R_1)$]{} \[\]\[\][$(P_2,R_2)$]{} \[\]\[\][$(P_M,R_M)$]{} \[\]\[\][$s^K$]{} \[\]\[\][$\hat{s}^K$]{} ![Layered broadcast coding with successive refinement.[]{data-label="fig:src_ch_layers"}](src_ch_layers.eps "fig:")
With successive decoding [@cover91:eoit], each virtual receiver first decodes and cancels the lower layers before decoding its own layer; the undecodable higher layers are treated as noise. Thus the rate $R_i$ intended for virtual receiver $i$ is $$\begin{aligned}
R_i &= \log\biggl(1+\frac{\gamma_i P_i}{1+\gamma_i \sum_{j=i+1}^{M}P_j}\biggr),\end{aligned}$$ where the term $\gamma_i \sum_{j=i+1}^{M}P_j$ represents the interference power from the higher layers. Suppose $\gamma_k$ is the realized channel power gain, then the original receiver can decode layer $k$ and all the layers below it. Hence the realized rate $R_{\operatorname{rlz}}(k)$ at the original receiver is $R_1+\dotsb+R_k$.
From the rate distortion function of a complex Gaussian source [@cover91:eoit], the mean squared distortion is $2^{-bR}$ when the source is described at a rate of $bR$ per symbol. Thus the realized distortion $D_{\operatorname{rlz}}(k)$ of the reconstructed source $\hat{s}$ is $$\begin{aligned}
D_{\operatorname{rlz}}(k) &= 2^{-bR_{\operatorname{rlz}}(k)}
= 2^{-b(R_1+\dotsb+R_k)},\end{aligned}$$ where the last equality follows from successive refinability [@equitz91:sus_refn_info; @rimoldi94:sus_refn_info_ach]. The expected distortion ${\mathrm{E}_H}[D]$ is obtained by averaging over the fading distribution: $$\begin{aligned}
{\mathrm{E}_H}[D] &= \sum_{i=1}^{M} p_i D_{\operatorname{rlz}}(i)
\label{eq:ED_sum_prods}
= \sum_{i=1}^M p_i \Bigl(\prod_{j=1}^i \frac{1+\gamma_j T_j}{1+\gamma_j T_{j+1}}\Bigr)^{-b},\end{aligned}$$ where $T_i$ represents the cumulative power in layers $i$ and above: $T_i \triangleq \sum_{j=i}^{M}P_j$, for $i=1,\dotsc,M$; $T_{M+1}\triangleq 0$. In the next section we derive the optimal cumulative power allocation $T_2^*,\dotsc,T_M^*$ to find the minimum expected distortion ${\mathrm{E}_H}[D]^*$.
Optimal Power Distribution {#sec:opt_pow_dist}
==========================
To derive the minimum expected distortion, we factor the sum of cumulative products in (\[eq:ED\_sum\_prods\]) and rewrite the expression as a set of recurrence relations: $$\begin{aligned}
D_M^* &\triangleq \bigl(1+\gamma_M T_M\bigr)^{-b}p_M\\
\label{eq:recur_min_dist}
D_i^*
&=\min_{0\leq T_{i+1} \leq T_i}
\Bigl(\frac{1+\gamma_i T_i}{1+\gamma_i T_{i+1}}\Bigr)^{-b}\bigl(p_i+D_{i+1}^*\bigr),\end{aligned}$$ where $i$ runs from $M-1$ down to 1. The term $D_i^*$ can be interpreted as the cumulative distortion from layers $i$ and above, with $D_1^*$ equal to the minimum expected distortion ${\mathrm{E}_H}[D]^*$. Note that $D_i$ depends on only two adjacent power allocation variables $T_i$ and $T_{i+1}$; therefore, in each recurrence step $i$ in (\[eq:recur\_min\_dist\]), we solve for the optimal $T_{i+1}^*$ in terms of $T_i$.
Specifically, consider the optimal power allocation between layer $\gamma$ and its lower layer $\gamma-\Delta\gamma$ as shown in Fig. \[fig:two\_delta\_layers\]. Let $T(\gamma-\Delta\gamma)$ denote the available transmit power for layers $\gamma-\Delta\gamma$ and above, of which $T(\gamma)$ is allocated to layers $\gamma$ and above; the remaining power $T(\gamma)-T(\gamma-\Delta\gamma)$ is allocated to layer $\gamma-\Delta\gamma$. Under optimal power allocation, it is shown in [@ng07:recur_pow_lbc] that the cumulative distortion from layers $\gamma$ and above can be written in the form: $$\begin{aligned}
\label{eq:Dr_W_form}
D^*(\gamma) = \bigl(1+\gamma T(\gamma)\bigr)^{-b} W(\gamma),\end{aligned}$$ where $W(\gamma)$ is interpreted as an equivalent probability weight summarizing the aggregate effect of the layers $\gamma$ and above. For the lower layer in Fig. \[fig:two\_delta\_layers\], $f(\gamma)\Delta\gamma$ represents the probability that layer $\gamma-\Delta\gamma$ is realized.
\[\]\[\][$W(\gamma):\gamma$]{} \[\]\[\][$f(\gamma)\Delta\gamma:\gamma-\Delta\gamma$]{} \[\]\[\][$-\,T(\gamma)$]{} \[\]\[\][$T(\gamma)$]{} \[\]\[\][$T(\gamma-\Delta\gamma)$]{} ![Power allocation between two adjacent layers.[]{data-label="fig:two_delta_layers"}](two_delta_layers.eps "fig:")
In the next recurrence step as prescribed by (\[eq:recur\_min\_dist\]), the cumulative distortion for the lower layer is $$\begin{aligned}
&D^*(\gamma-\Delta\gamma) = \min_{0\leq T(\gamma)\leq T(\gamma-\Delta\gamma)} D(\gamma-\Delta\gamma)\\
\begin{split}
\label{eq:Drd_min_Tr}
& \quad= \min_{0\leq T(\gamma)\leq T(\gamma-\Delta\gamma)}\Bigl(\frac{1+(\gamma-\Delta\gamma) T(\gamma-\Delta\gamma)}{1+(\gamma-\Delta\gamma) T(\gamma)}\Bigr)^{-b}\\
&\hspace{7.25em}\cdot\Bigl[f(\gamma)\Delta\gamma+\bigl(1+\gamma T(\gamma)\bigr)^{-b} W(\gamma)\Bigr].
\end{split}\end{aligned}$$ We solve the minimization by forming the Lagrangian: $$\begin{aligned}
\begin{split}
&L(T(\gamma),\lambda_1,\lambda_2) = \\
&\quad D(\gamma-\Delta\gamma) + \lambda_1\bigl(T(\gamma)-T(\gamma-\Delta\gamma)\bigr) - \lambda_2 T(\gamma).
\end{split}\end{aligned}$$ The Karush-Kuhn-Tucker (KKT) conditions stipulate that the gradient of the Lagrangian vanishes at the optimal power allocation $T^*(\gamma)$, which leads to the solution:
[\[eq:Tr\_opt\] T\^\*() = ]{} \[eq:Tr\_opt\_uncon\] U() & if $U(\gamma) \leq T(\gamma-\Delta\gamma)$\
\[eq:Tr\_opt\_con\] T(-) & else,
where
[U() ]{} \[eq:Ur\_o\] 0 &\
\[eq:Ur\_nz\] (\^-1) &
We assume there is a region of $\gamma$ where the cumulative power allocation is not constrained by the power available from the lower layers, i.e., $U(\gamma)\leq U(\gamma-\Delta\gamma)$ and $U(\gamma)\leq P$. In this region the optimal power allocation $T^*(\gamma)$ is given by the unconstrained minimizer $U(\gamma)$ in (\[eq:Tr\_opt\_uncon\]). In the solution to $U(\gamma)$ we need to verify that $U(\gamma)$ is non-increasing in this region, which corresponds to the power distribution $\rho^*(\gamma)$ being non-negative. With the substitution of the unconstrained cumulative power allocation $U(\gamma)$ in (\[eq:Drd\_min\_Tr\]), the cumulative distortion at layer $\gamma-\Delta\gamma$ becomes: $$\begin{aligned}
\begin{split}
\label{eq:Drd_U_W}
D^*(\gamma-\Delta\gamma) &=
\Bigl(\frac{1+(\gamma-\Delta\gamma) T(\gamma-\Delta\gamma)}{1+(\gamma-\Delta\gamma) U(\gamma)}\Bigr)^{-b}\\
&\hspace{1.75em}\cdot\Bigl[f(\gamma)\Delta\gamma+\bigl(1+\gamma U(\gamma)\bigr)^{-b} W(\gamma)\Bigr],
\end{split}\end{aligned}$$ which is of the form in (\[eq:Dr\_W\_form\]) if we define $W(\gamma-\Delta\gamma)$ by the recurrence equation: $$\begin{aligned}
\begin{split}
\label{eq:Wrd_Wr}
W(\gamma-\Delta\gamma) &=
\bigl(1+(\gamma-\Delta\gamma) U(\gamma)\bigr)^b\\
&\hspace{2em}\cdot\bigl[f(\gamma)\Delta\gamma+\bigl(1+\gamma U(\gamma)\bigr)^{-b} W(\gamma)\bigr].
\end{split}\end{aligned}$$
Next we consider the limiting process as the spacing between the layers condenses. In the limit of $\Delta\gamma$ approaching zero, the recurrence equations (\[eq:Drd\_U\_W\]), (\[eq:Wrd\_Wr\]) become differential equations. The optimal power distribution $\rho^*(\gamma)$ is given by the derivative of the cumulative power allocation: $$\begin{aligned}
\rho^*(\gamma) &= -{T^*}'(\gamma),\end{aligned}$$ where $T^*(\gamma)$ is described by solutions in three regions:
[\[eq:Tr\_all\_opt\]T\^\*() =]{} \[eq:Tr\_ro\] 0 & $\gamma > \gamma_o$\
\[eq:Tr\_rP\_ro\] U() & $\gamma_P \leq \gamma \leq \gamma_o$\
\[eq:Tr\_rP\] P & $\gamma < \gamma_P$.
In region (\[eq:Tr\_ro\]) when $\gamma > \gamma_o$, corresponding to cases (\[eq:Tr\_opt\_uncon\]) and (\[eq:Ur\_o\]), no power is allocated to the layers and (\[eq:Wrd\_Wr\]) simplifies to $W(\gamma) = 1-F(\gamma)$, where $F(\gamma) \triangleq \int_0^\gamma f(s)\,ds$ is the cumulative distribution function (cdf) of the channel power gain. The boundary $\gamma_0$ is defined by the condition in (\[eq:Ur\_o\]) which satisfies: $$\begin{aligned}
\label{eq:ro_rf_F}
\gamma_o f(\gamma_o) + F(\gamma_o) - 1 = 0.\end{aligned}$$ Under Rayleigh fading when $f(\gamma) = \bar{\gamma}^{-1}e^{-\gamma/\bar{\gamma}}$, where $\bar{\gamma}$ is the expected channel power gain, (\[eq:ro\_rf\_F\]) evaluates to $\gamma_o = \bar{\gamma}$. For other fading distributions, $\gamma_o$ may be computed numerically.
In region (\[eq:Tr\_rP\_ro\]) when $\gamma_P \leq \gamma \leq \gamma_o$, corresponding to cases (\[eq:Tr\_opt\_uncon\]) and (\[eq:Ur\_nz\]), the optimal power distribution is described by a set of differential equations. We apply the first order binomial expansion $(1+\Delta\gamma)^b\cong1+b\Delta\gamma$, and (\[eq:Wrd\_Wr\]) becomes: $$\begin{aligned}
W'(\gamma) &= \lim_{\Delta\gamma \rightarrow 0} \frac{W(\gamma) - W(\gamma-\Delta\gamma)}{\Delta\gamma}\\
\label{eq:dW_W}
&= b\frac{W(\gamma)}{\gamma} - (1+b)\Bigl[f(\gamma)\Big(\frac{W(\gamma)}{\gamma}\Bigr)^b\Bigr]^{\frac{1}{1+b}},\end{aligned}$$ which we substitute in (\[eq:Ur\_nz\]) to obtain: $$\begin{aligned}
\label{eq:dU_U}
U'(\gamma) &= -\Big(\frac{2/\gamma + f'(\gamma)/f(\gamma)}{1+b}\Bigr) \Big[U(\gamma)+1/\gamma\Bigr].\end{aligned}$$ Hence $U(\gamma)$ is described by a first order linear differential equation. With the initial condition $U(\gamma_o) = 0$, its solution is given by $$\begin{aligned}
\label{eq:Ur_int_fr}
U(\gamma) &= \frac{\displaystyle -\int_{\gamma_o}^{\gamma}
\dfrac{1}{s}\Bigl(\dfrac{2}{s}+\dfrac{f'(s)}{f(s)}\Bigr) \bigl[s^2 f(s)\bigr]^{\frac{1}{1+b}} \,ds}
{(1+b)\bigl[\gamma^2 f(\gamma)\bigr]^{\frac{1}{1+b}}},\end{aligned}$$ and condition (\[eq:Tr\_opt\_con\]) in the lowest active layer becomes the boundary condition $U(\gamma_P) = P$. In [@tian07:exp_dist_gaus_src_bc], the power distribution in (\[eq:Ur\_int\_fr\]) is derived using the calculus of variations method.
Similarly, as $\Delta\gamma\rightarrow0$, the evolution of the expected distortion in (\[eq:Drd\_U\_W\]) becomes: $$\begin{aligned}
D'(\gamma) &= -\dfrac{b\gamma U'(\gamma)}{1+\gamma U(\gamma)}D(\gamma) - f(\gamma)\\
&= \Bigl[\dfrac{b}{1+b}\Bigl(\dfrac{2}{\gamma}+\dfrac{f'(\gamma)}{f(\gamma)}\Bigr)\Bigr]D(\gamma)- f(\gamma),\end{aligned}$$ which is again a first order linear differential equation. With the initial condition $D(\gamma_o) = W(\gamma_o) = \gamma_o f(\gamma_o)$, its solution is given by $$\begin{aligned}
D(\gamma) &= \frac{\displaystyle -\int_{\gamma_o}^{\gamma} f(s)
\Bigl[\Bigl(\dfrac{s}{\gamma_o}\Bigr)^2\dfrac{f(s)}{f(\gamma_o)}\Bigr]^{\frac{-b}{1+b}} \,ds
+ \gamma_o f(\gamma_o)}
{\Bigl[\Bigl(\dfrac{\gamma}{\gamma_o}\Bigr)^2\dfrac{f(\gamma)}{f(\gamma_o)}\Bigr]^{\frac{-b}{1+b}}}.\end{aligned}$$
Finally, in region (\[eq:Tr\_rP\]) when $\gamma < \gamma_P$, corresponding to case (\[eq:Tr\_opt\_con\]), the transmit power $P$ has been exhausted, and no power is allocated to the remaining layers. Hence the minimum expected distortion is $$\begin{aligned}
{\mathrm{E}_H}[D]^* = D(0) = F(\gamma_P) + D(\gamma_P),\end{aligned}$$ where the last equality follows from when $\gamma < \gamma_P$ in region (\[eq:Tr\_rP\]), $\rho^*(\gamma)=0$ and $D(\gamma) = \int_\gamma^{\gamma_P} f(s) \,ds + D(\gamma_P)$.
Rayleigh Fading with Diversity {#sec:ray_div}
==============================
In this section we consider the optimal power distribution and the minimum expected distortion when the wireless channel undergoes Rayleigh fading with a diversity order of $L$ from the realization of independent fading paths. Specifically, we assume the fading channel is characterized by the Erlang distribution: $$\begin{aligned}
f_L(\gamma) = \frac{(L/\bar{\gamma})^L \gamma^{L-1} e^{-L\gamma/\bar{\gamma}} }
{(L-1)!}, \qquad\gamma > 0,\end{aligned}$$ which corresponds to the average of $L$ iid channel power gains, each under Rayleigh fading with an expected value of $\bar{\gamma}$. The $L$-diversity system may be realized by having $L$ transmit antennas using isotropic inputs, by relaxing the decode delay constraint over $L$ fading blocks, or by having $L$ receive antennas under maximal-ratio combining when the power gain of each antenna is normalized by $1/L$.
Fig. \[fig:erlang\_pow\_dist\] shows the optimal power distribution $\rho^*(\gamma)$, which is concentrated over a range of active layers. A higher SNR $P$ or a larger bandwidth ratio $b$ extends the span of the active layers further into the lower layers but the upper boundary $\gamma_o$ remains unperturbed. It can be observed that a smaller bandwidth ratio $b$ reduces the spread of the power distribution. In fact, as $b$ approaches zero, the optimal power distribution that minimizes expected distortion converges to the power distribution that maximizes expected capacity. To show the connection, we take the limit in the distortion-minimizing cumulative power distribution in (\[eq:Ur\_int\_fr\]): $$\begin{aligned}
\lim_{b\rightarrow0}U(\gamma) &= \frac{1-F(\gamma)-\gamma f(\gamma)}{\gamma^2 f(\gamma)},\end{aligned}$$ which is equal to the capacity-maximizing cumulative power distribution as derived in [@shamai03:bc_app_slow_fade_mimo]. Essentially, from the first order expansion $e^b\cong1+b$ for small $b$, ${\mathrm{E}_H}[D]\cong 1- b {\mathrm{E}_H}[C]$ when the bandwidth ratio is small, where ${\mathrm{E}_H}[C]$ is the expected capacity in nats/s, and hence minimizing expected distortion becomes equivalent to maximizing expected capacity. For comparison, the capacity-maximizing power distribution is also plotted in Fig. \[fig:erlang\_pow\_dist\]. Note that the distortion-minimizing power distribution is more conservative, and it is more so as $b$ increases, as the allocation favors lower layers in contrast to the capacity-maximizing power distribution.
![Optimal power distribution ($P=0~\operatorname{dB}$).[]{data-label="fig:erlang_pow_dist"}](fig_erlang_pow_dist_bw.eps){width="8cm"}
Fig. \[fig:erlang\_ED\_P\] shows the minimum expected distortion ${\mathrm{E}_H}[D]^*$ versus SNR for different diversity orders. With infinite diversity, the channel power gain becomes constant at $\bar{\gamma}$, and the distortion is given by $$\begin{aligned}
D\vert_{L=\infty} = (1+\bar{\gamma}P)^{-b}.\end{aligned}$$ In the case when there is no diversity ($L=1$), a lower bound to the expected distortion is also plotted. The lower bound assumes the system has CSI at the transmitter (CSIT), which allows the transmitter to concentrate all power at the realized layer to achieve the expected distortion: $$\begin{aligned}
{\mathrm{E}_H}[D_{\operatorname{CSIT}}] &= \int_0^\infty e^{-\gamma}(1+\gamma P)^{-b}\,d\gamma.\end{aligned}$$ Note that at high SNR, the performance benefit from diversity exceeds that from CSIT, especially when the bandwidth ratio $b$ is large. In particular, in terms of the distortion exponent $\Delta$ [@laneman05:src_ch_parl_ch], it is shown in [@gunduz06:jt_src_ch_code_mimo] that in a MISO or SIMO channel, layered broadcast coding achieves: $$\begin{aligned}
\Delta \triangleq - \lim_{P\rightarrow\infty} \frac{\log {\mathrm{E}_H}[D]}{\log P}
= \min(b,L),\end{aligned}$$ where $L$ is the total diversity order from independent fading blocks and antennas. Moreover, the layered broadcast coding distortion exponent is shown to be optimal and CSIT does not improve $\Delta$, whereas diversity increases $\Delta$ up to a maximum as limited by the bandwidth ratio $b$.
![Minimum expected distortion ($b=2$).[]{data-label="fig:erlang_ED_P"}](fig_erlang_ED_P_bw.eps){width="8cm"}
Conclusion {#sec:conclu}
==========
We considered the problem of source-channel coding over a delay-limited fading channel without CSI at the transmitter, and derived the optimal power distribution that minimizes the end-to-end expected distortion in the layered broadcast coding transmission scheme with successive refinement. In the case when the channel undergoes Rayleigh fading with diversity order $L$, the optimal power distribution is congregated around the middle layers, and within this range the lower layers are assigned more power than the higher ones. As SNR increases, the power distribution of the higher layers remains unchanged, and the extra power is allocated to the idle lower layers. Furthermore, increasing the diversity $L$ concentrates the power distribution towards the expected channel power gain $\bar{\gamma}$, while a larger bandwidth ratio $b$ spreads the power distribution further into the lower layers. On the other hand, in the limit as $b$ tends to zero, the optimal power distribution that minimizes expected distortion converges to the power distribution that maximizes expected capacity. While the expected distortion can be improved by acquiring CSIT or increasing the diversity order, it is shown that at high SNR the performance benefit from diversity exceeds that from CSIT, especially when the bandwidth ratio $b$ is large.
[^1]: This work was supported by the US Army under MURI award W911NF-05-1-0246, the ONR under award N00014-05-1-0168, DARPA under grant 1105741-1-TFIND, a grant from Intel, and the NSF under grant 0430885.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We have carried out *ab-initio* calculations of local structure of Mn and Ni in Mn$_{2}$Ni$_{1.5}$In$_{0.5}$ alloy with different site occupancies in order to understand the similarities in martensitic and magnetic properties of Mn$_{2}$Ni$_{1+x}$In$_{1-x}$ and Ni$_2$Mn${1+x}$In$_{1-x}$ alloys. Our results show that in Mn$_{2}$Ni$_{1+x}$In$_{1-x}$ alloys there is a strong possibility of Mn atoms occupying all the three, X, Y and Z sites of X$_2$YZ Heusler structure while Ni atoms preferentially occupy the X sites. Such a site occupancy disorder of Mn atoms is in addition to a local structural disorder due to size differences between Mn and In atoms which is also present in Ni$_2$Mn$_{1+x}$In$_{1-x}$ alloys. Further, a comparison of the calculations with experimental XAFS at the Mn and Ni K edges in Mn$_{2-y}$Ni$_{1.6+y}$In$_{0.4}$ ($-0.08 \le y \le 0.08$) indicate a strong connection between martensitic transformation and occupancy of Z sites by Mn atoms.'
address:
- 'Department of Physics, Goa University, Taleigao Plateau, Goa 403 206 India'
- 'Department of Physics, Goa University, Taleigao Plateau, Goa 403 206 India'
- 'Department of Materials Molecular Science, Institute of Molecular Science, Myodaiji-cho, Okazaki, Aichi 444-8585, Japan'
- 'Institute of Scientific and Industrial Research, Osaka University, 8-1 Mihogaoka, Ibaraki,Osaka 567-0047, Japan'
author:
- 'D. N. Lobo'
- 'K. R. Priolkar'
- 'A. Koide'
- 'S. Emura'
bibliography:
- 'Mn2NiIn-abinitio.bib'
title: 'Effect of site occupancy disorder on Martensitic properties of Mn$_{2}$NiIn type alloys: x-ray absorption fine structure study'
---
Introduction
============
Mn rich Ni-Mn-Z (Z = Ga, In, Sn or Sb) type shape memory alloys have been studied for their novel properties like giant reverse magneto-caloric effect [@krenke-natmat; @preeti; @ali; @du; @liu; @khan; @dub; @nayak; @han; @sharma; @sham; @buchel1; @li; @xuan; @bruno; @chen; @chen2], large magnetic field induced strain [@kainuma; @kainuma-nat; @kainuma2; @haldar; @karaca; @karaca2], magnetic superelasticity [@krenke] and complex magnetic order [@khov; @buchel2; @kano; @khov1; @ye; @kata; @monroe]. The origin of all these effects lie in a strong coupling between structural and magnetic degrees of freedom. Therefore understanding the magnetic interactions between the constituent atoms as the alloys transform structurally gains importance. Despite several attempts, the understanding of the magnetism of martensitic state is still elusive. Though the magnetic moment in Ni-Mn based Heusler alloys is almost entirely due to Mn atoms[@webster; @hurd; @brown], factors like antisite disorder[@felser], changes in bond distances due to structural transformation[@krp-epl] as well as local structural disorder[@krp-prb] bring in newer magnetic interactions and add to the complexities of the problem.
Increasing the Mn content in Ni$_{2-x}$Mn$_{1+x}$Z (Z= Ga, In, Sn, Sb) alloys and at the same time preserving the Heusler structure results in alloys of type Mn$_{2}$NiZ. Band structure calculations have indicated such alloys to be ferrimagnetic due to unequal magnetic moments of antiferromagnetically coupled Mn atoms occupying the X and Y sites of X$_2$YZ Heusler structure [@GDL; @gdliu; @Barman; @aparna; @Souvik]. Though martensitic transformation has been observed in Mn$_2$NiGa (T$_M$ $\sim$ 270K, T$_C$ = 588K) [@gdliu] the same is not observed in other Mn$_2$NiZ alloys where Z = In or Sn. However, increasing of Ni content at the expense of Z atoms to realize alloys of the type Mn$_2$Ni$_{1+x}$Z$_{1-x}$ leads to martensitic instability in them [@LMa; @Sanchez; @nelson1].
In L2$_1$ Heusler composition, Ni atoms are believed to prefer X sites [@ayuela]. According to this premise then in Mn$_2$Ni$_{1+x}$Z$_{1-x}$ alloys, if Ni atoms preferentially occupy X sites, a proportionate amount of Mn atoms would be forced to occupy Z sites leading to what is known as site occupancy disorder. Such an disorder is known to introduce competing ferromagnetic and antiferromagnetic interactions between Mn atoms occupying Y sublattice (Mn(Y)) and Mn atoms occupying Z sublattice (Mn(Z)) leading to observation of exotic properties like exchange bias effect in zero field cooled state, spin valve effect, etc. in these Mn rich martensitic alloys [@12singh; @12nayak; @tan].
In our recent work [@nelson1], the magnetic properties of Mn$_2$Ni$_{1+x}$Z$_{1-x}$ alloys in the martensitic state were found to be similar to those of Ni$_2$Mn$_{1+x}$In$_{1-x}$ and this was conjectured to be due to site occupancy disorder arising out of preferential occupation of X sites by Ni atoms. But this general picture could not explain the complete suppression of martensitic transformation in Mn$_{2-y}$Ni$_{1.6+y}$In$_{0.4}$ due to small increase in Mn concentration at the expense of Ni ($-0.1 < y < 0$) [@nelson1]. This is especially important because, the alloy with $y$ = 0 is martensitic with a transformation temperature of $\sim$ 230K which increases with increase in Ni concentration ($ y > 0$). Hence it becomes necessary to understand the correlation between the perceived site occupancy disorder in Mn$_{2} $Ni$_{1+x}$In$_{1-x}$ alloys and the observed similarity in martensitic and magnetic properties of these alloys with those of Ni$_{2}$Mn$_{1+x}$In$_{1-x}$ alloys at a microscopic level. To achieve this objective, here we report *ab-initio* calculations of Ni and Mn K edge x-ray absorption fine structure (XAFS) in prototypical Mn$_{2}$Ni$_{1.5}$In$_{0.5}$ using FEFF 8.4 program and its comparison with experimental results obtained in Mn$_2$Ni$_{1.5}$In$_{0.5}$ and Mn$_{2-y}$Ni$_{1.6+y}$In$_{0.4}$ ($-0.08 \le y \le 0.08$).
Methods
=======
The samples of above composition were prepared by arc melting the weighed constituents in argon atmosphere followed by encapsulating in a evacuated quartz tube and annealing at 750 $^\circ$C for 48 hours and subsequent quenching in ice cold water. The prepared alloys were cut in suitable sizes using a low speed diamond saw and part of the sample was powdered and re-annealed in the same procedure above. X-ray diffraction (XRD) patterns were recorded at room temperature in the angular range of $20^\circ \leq 2\theta \leq 100^\circ$ and were found to be single phase [@nelson1]. Magnetization measurements were performed in the temperature interval 5 K - 400 K using a vibrating sample magnetometer in 100 Oe applied field during the zero field cooled (ZFC) and subsequent field cooled cooling (FCC) and field cooled warming (FCW) cycles.
XAFS at Ni K and Mn K edges were recorded at Photon Factory using beamline 12C at room temperature. For XAFS measurements the samples to be used as absorbers, were ground to a fine powder and uniformly distributed on a scotch tape. These sample coated strips were adjusted in number such that the absorption edge jump gave $\Delta\mu t \le 1$ where $\Delta\mu$ is the change in absorption coefficient at the absorption edge and $t$ is the thickness of the absorber. The incident and transmitted photon energies were simultaneously recorded using gas-ionization chambers as detectors. Measurements were carried out from 300 eV below the edge energy to 1000 eV above it with a 5 eV step in the pre-edge region and 2.5 eV step in the XAFS region. At each edge, at least three scans were collected to average statistical noise.
FEFF 8.4 software based on the self-consistent real-space multiple-scattering formalism [@Rehr] was employed for calculation of XAFS oscillations at the Mn K and Ni K edge in a prototypical alloy, Mn$_2$Ni${1.5}$In$_{0.5}$. This alloy composition is not only close to the experimentally studied compositions but also the constituent atoms have non fractional number of near neighbors. For the FEFF calculations spherical muffin tin potentials were self consistently calculated over a radius of 5Å. A default overlapping muffin tin potentials and Hedin-Lunqvist exchange correlations were used to calculate x-ray absorption transitions to a fully relaxed final state in presence of a core hole. Calculations were carried out for Mn K and Ni K edges assuming L2$_1$ type Heusler structure. Two possible structural models and their variations which are explained in detail in next section were considered. XAFS was calculated for absorbing atoms at occupying X, Y and Z sites of the Heusler structure and combined together by multiplying each site XAFS with appropriate weighting fraction. During calculations the amplitude reduction factor, $S_0^2$ was fixed to 0.8 and the $\sigma^2$ for respective paths were calculated considering a Debye temperature of 320K [@chun] and the spectrum temperature of 300K
Results and Discussion
======================
Figure \[expt\] presents the magnetization measurements carried out in the temperature range 5 K $\le$ T $\le$ 400K in Mn$_{2-y}$Ni$_{1.6+y}$In$_{0.4}$ (y = 0.08, 0 and -0.08). It can be clearly seen that the alloys with y = 0 and 0.08 undergo martensitic transformation at about 230 K and 270 K respectively. While Mn$_{2.08}$Ni$_{1.52}$In$_{0.4}$ does not show any martensitic instability down to 5K thus highlighting a drastic change in martensitic transformation temperature with small changes in alloy composition. A detailed study of magnetic properties of these alloys along with Mn$_2$Ni$_{1+x}$In$_{1-x}$ (x = 0.5, 0.6 and 0.7) has been already presented in Ref. . To understand these changes in martensitic transformation temperature, experimental XAFS data recorded at the Mn K and Ni K edges at room temperature in each of these alloys have been compared with calculated Mn and Ni XAFS data using FEFF.
![\[expt\] Magnetization as a function of temperature for Mn$_{2-y}$Ni$_{1.6+y}$In$_{0.4}$ (y = 0.08, 0, -0.08) measured in applied field of 100 Oe during ZFC, FCC and FCW cycles.](magnetization.eps){width="\columnwidth"}
For the *ab-initio* calculations of Mn K and Ni K edge XAFS in Mn$_2$Ni$_{1.5}$In$_{0.5}$ alloy, two structural models, designated as MODEL A and MODEL B were considered. In MODEL A, the X sites of X$_{2}$YZ are occupied equally by Ni and Mn, while all the Y sites are occupied by Mn and In and the remaining Ni atoms occupy the Z sites. In MODEL B, entire fraction of Ni atoms occupy the X sites, forcing the proportionate amount of Mn atoms to occupy the Z sites along with Y sites. Thus resulting in Mn occupying all the three X, Y and Z sites in different fractions. The site occupancies in both these models is tabulated in Table \[feff\].
Sites MODEL A MODEL B MODEL B1 MODEL B2
------- --------- --------- ---------- ----------
X Ni Ni1.5 Ni1.5 Ni
Mn Mn0.5 Mn0.5 Mn
Y Mn Mn Mn0.75 Mn0.5
In0.25 Ni0.5
Z Ni0.5 Mn0.5 Mn0.75 Mn0.5
In0.5 In0.5 In0.25 In0.5
: \[feff\] Assumed site occupancies of X, Y and Z sites of X$_2$YZ Heusler structure in MODEL A and MODEL B used for XAFS calculations of Mn$_2$Ni$_{1.5}$In$_{0.5}$.
![\[ModelA-B-Ni-Mn.eps\] Calculated Ni and Mn K edge XAFS for MODEL A and MODEL B along with experimental data.](ModelA-B-Ni-Mn.eps){width="\columnwidth"}
In figure \[ModelA-B-Ni-Mn.eps\], calculated spectra at the Ni and Mn K edges according to MODEL A and MODEL B are compared with the experimental data recorded at room temperature. It is observed that the oscillatory parts of the experimental Mn and Ni K edge XAFS spectra are reproduced by the two theoretical models. The calculated Ni K XAFS spectra of MODEL B gives a much better description with the experimental data for the entire $k$ range under consideration than MODEL A. A similar conclusion could also be drawn for the calculated Mn K spectra although one can observed a mismatch between experimental and calculated Mn K XAFS spectra especially in the region between 5 to 8 Å$^{-1}$. The in general better agreement of calculated spectra from MODEL B with the experimental XAFS spectra augers well with the literature reports that support the case of Ni atoms preferentially occupying X sites [@gdliu; @12nayak; @burch]. These reports also indicate a disorder in occupancy of Mn atoms at different sites. Such a site occupancy disorder could be the reason for observed mismatch between the experimental and calculated spectra as per MODEL B. The site disorder in occupancy of Mn atoms was introduced in the MODEL B in two different ways and are referred to as MODEL B1 and MODEL B2. Their site occupancies of the X, Y and Z sites are detailed in Table \[feff\].
Another reason for observed mismatch between experimental and calculated XAFS spectra could be incorrect estimation of $\sigma^2$. This is because the Debye temperature of Mn$_2$NiIn alloys have not been reported in literature and the value of 320K chosen for the present calculations is the one reported for Ni-Mn-Ga alloys. It must be mentioned here that the measured values of Debye temperature for isostructural Heusler alloys containing Mn is reported to lie between 220 K to 320 K [@mism] and hence the present choice of Debye temperature may not be far from the true value. Also the other extreme choice of 220K does not significantly affect the calculated spectra.
The calculated Mn K edge EXAFS of MODELS B, B1 and B2 along with the experimental data have been plotted in Fig.\[ModelB-B1-B2-Mn.eps\](a), Fig.\[ModelB-B1-B2-Mn.eps\](b) and Fig.\[ModelB-B1-B2-Mn.eps\](c) respectively. A comparison of the calculated Mn K edge XAFS spectra with the experimental data does not conclusively suggest any one of these models to be a better descriptor of experimental data.
![\[ModelB-B1-B2-Mn.eps\] Comparison of experimental XAFS data at the Mn K edge with the calculated Mn K edge XAFS for MODEL’s B, B1 and B2 for undistorted lattice ((a) - (c)), for a lattice with local structural distortion wherein Mn atoms at the Z site are displaced closer to X sites by 0.1 Å ((d)-(f)) and 0.2 Å ((g)-(i)).](ModelB-B1-B2-Mn.eps){width="\columnwidth"}
In Ni$_{2}$Mn$_{1+x}$In$_{1-x} $ alloys, a local structural distortion especially in the position of Mn atoms at Z site (Mn(Z)) was shown to be responsible for the martensitic transformation for $x > 0.3$ [@nelson-APL]. Since in MODEL B both, Mn and In atoms occupy the Z sites, a similar local structural distortion can exist in Mn$_2$Ni$_{1+x}$In$_{1-x}$ alloys resulting in a shorter Mn(Z)-X bond as compared to In-X bond. Such a distortion was introduced by tweaking the coordinates of Mn(Z) atoms in the FEFF input file. The coordinates of Mn(Z) atoms in MODELS B, B1 and B2 were changed in such a way that they were closer to X site atoms by 0.1 Å and 0.2 Å as compared to the In atoms occupying the Z sites. Fig. \[ModelB-B1-B2-Mn.eps\](d)-(f) and Fig. \[ModelB-B1-B2-Mn.eps\](g)-(i) shows the comparison for MODEL B, B1 and B2 at the Mn K edge EXAFS with experimental data for local structure distortion of 0.1Å and 0.2Å respectively in the k range from 2 to 12 Å$ ^{-1} $.
![\[ModelB-Ni-k.eps\] Ni K edge XAFS data calculated as for MODEL B for (a) undistorted lattice and (b) lattice with local structural distortion wherein Mn atoms at In site are displaced closer to X site atoms by 0.1 Å.](ModelB-Ni-k.eps){width="\columnwidth"}
From Fig. \[ModelB-B1-B2-Mn.eps\], it is observed that with increasing disorder from 0.1 Å to 0.2 Å for all three models, the amplitude of calculated EXAFS oscillations for all models reduce and tend towards the experimental data which is an indication of presence of local structure disorder in these alloys. Since all the alloys have long range structural order as evidenced from Bragg reflections in x-ray diffraction data, a displacement of a particular atom by 0.2Å from its crystallographic site position may be a bit unrealistic. Hence models with 0.1Å displacement of Mn(Z) atoms were taken to provide the most realistic description of site occupancies in such Mn$_2$Ni$_{1+x}$In$_{1-x}$ alloys. Of the three MODEL B was preferred over MODELS B1 and B2 due to its relative simplicity. Irrespective of the choice of models, the present analysis clearly suggests antisite disorder along with local structural disorder to be primarily responsible for physical properties of Mn$_{2}$NiIn type alloys. Antisite disorder has also been reported to be responsible for exotic properties like spin valve effect and zero field exchange bias in related Mn$_{2} $NiGa and Mn$_{2} $PtGa [@12singh; @12nayak].
Calculated Ni K edge XAFS plot for MODEL B with a local structure distortion of 0.1 Å also presents a better agreement with the experimental data as compared to the undistorted MODEL B (See Fig. \[ModelB-Ni-k.eps\]). This confirms the presence of antisite disorder along with a local structural disorder in these Mn$_{2}$NiIn type alloys. Presence of Mn at the Z sites could be the reason for similarity of magnetic properties in martensitic state of Mn$_{2}$Ni$_{1+x}$In$_{1-x}$ and the magnetic properties of Ni$_{2}$Mn$_{1+x}$In$_{1-x}$ alloys in their martensitic state. In other words, an increase in Ni content at the expense of In in Mn-Ni-In alloys causes Mn to occupy the Z sites due to preference of Ni atoms for X sites. Such a site occupancy coupled with local structural disorder favors formation of Ni-Mn(Z) hybridization which is responsible for martensitic transformation and antiferromagnetism in the martensitic state.
Although above calculations give a fair understanding of magnetic properties of the martensitic state in Mn$_2$NiIn type alloys, it does not give explain the complete suppression of martensitic transformation with small changes in Mn:Ni ratio. In Mn$_{2}$Ni$_{1+x}$In$_{1-x}$, though alloy with $x = 0.6$ undergoes martensitic transformation at 232K, the alloy Mn$_{2.08}$Ni$_{1.52}$In$_{0.4}$ alloy does not exhibit any martensitic transformation down to 4K. At the same time Mn$_{1.92}$Ni$_{1.68}$In$_{0.4}$ exhibits martensitic transformation at a higher temperature of 250K. To understand the possible cause of such drastic variation of martensitic transformation temperatures in these alloys, a linear component fitting (LCF) analysis was performed on the experimental data recorded at the Mn and Ni K edge XAFS in the above three compositions using the FEFF calculated XAFS of Ni and Mn occupying different site position as per MODEL A and MODEL B with distortion of 0.1Å. As per MODEL A, Ni would be found at X and Z sites while Mn would be present at X and Y sites. In case of MODEL B, Ni would be present only at X sites and Mn would occupy all the three sites. Calculated XAFS of Ni and Mn for each site were used as standards and Athena was employed to give a best possible combination that describes the experimental data in Mn$_{2-y}$Ni$_{1.6+y}$In$_{0.4}$. Based on this LCF analysis the obtained site occupancies of Ni and Mn are presented in Table \[LCF-Ni\] and Table \[LCF-Mn\] respectively.
---------------------------------- ------------ --------------- ---------------
Sample
Model Ni-Mn(Z) species
(position) bond distance concentration
disorder
Mn$_{2.08}$Mn$_{1.52}$In$_{0.4}$ A(X) - 79 $\pm$ 20
A(Z) - 14 $\pm$ 07
B(X) 0.1 12 $\pm$ 14
Mn$_{2}$Ni$_{1.6}$In$_{0.4}$ A(X) - 83 $\pm$ 13
B(X) 0.1 17 $\pm$ 11
Mn$_{1.92}$Ni$_{1.68}$In$_{0.4}$ A(X) - 86 $\pm$ 14
B(X) 0.1 14 $\pm$ 13
---------------------------------- ------------ --------------- ---------------
: \[LCF-Ni\]LCF analysis for Ni K edge XAFS The bracketed letters indicates the crystallographic site positions.
---------------------------------- ------------ --------------- ---------------
Sample
Model Ni-Mn(Z) species
(position) bond distance concentration
disorder
Mn$_{2.08}$Ni$_{1.52}$In$_{0.4}$ A(X) - 62 $\pm$ 05
B(Y) 0.1 38 $\pm$ 05
Mn$_2$Mn$_{1.6}$In$_{0.4}$ A(X) - 49 $\pm$ 6
B(Y) 0.1 31 $\pm$ 10
B(Z) 0.1 20 $\pm$ 07
Mn$_{1.92}$Ni$_{1.68}$In$_{0.4}$ A(X) - 50 $\pm$ 25
B(Y) 0.1 23 $\pm$ 5.4
B(Y) 0.1 27 $\pm$ 7.7
---------------------------------- ------------ --------------- ---------------
: \[LCF-Mn\]LCF analysis for Mn K edge XAFS The bracketed letters indicates the crystallographic site positions.
It is interesting to note that in case of Mn$_{2.08}$Ni$_{1.92}$In$_{0.4}$ alloy which does not undergo martensitic transformation, Mn is found to be only at X and Y sites and while Ni is present at the Z site. While in case of the other two alloys, Ni primarily occupies X sites while Mn is found to occupy all the three sites. Presence of Mn at the Z site along with a local structural distortion in its position gives rise to a shorter Ni-Mn bond as compared to Ni-In and Ni(3d) - Mn(3d) hybridization which plays an important role in martensitic transformation in Ni$_2$Mn$_{1+x}$In$_{1-x}$ alloys. A clear differentiation between site occupancies of Mn$_2$NiIn type alloys undergoing martensitic transformation and non-martensitic alloys highlights the importance of antisite disorder along with local structural distortion in inducing martensitic transformation in these alloys.
Conclusion
==========
We have carried out ab-initio calculations at the Ni and Mn K edge to understand the driving force for martensitic transformation in Mn$_{2}$Ni$_{1+x}$In$_{1-x}$ alloys. Presence of Mn at Z sites appears to be the main requirement for the alloy composition to undergo martensitic transformation. The ab-initio XAFS calculations indicate preferential occupation of X sites by Ni atoms while Mn occupy all X, Y and Z sites of the X$_{2} $YZ Heusler structure. Such a site occupancy disorder of Mn atoms is in addition to a local structural disorder due to size differences between Mn and In atoms which is also present in Ni$_2$Mn$_{1+x}$In$_{1-x}$ alloys. This augers well with the observed similarities in magnetic properties of martensitic state of Mn$_{2}$Ni$_{1+x}$In$_{1-x}$ and Ni$_2$Mn$_{1+x}$In$_{1-x}$ alloys. Further the drastic suppression of martensitic transformation with small changes in composition in Mn$_{2-y}$Ni$_{1.6+y}$In$_{0.4}$ can also be understood based on occupancy of Mn at Z sites. Mn$_{2.08}$Ni$_{1.52}$In$_{0.4}$ which has no Mn atoms at the Z site, does not undergo martensitic transformation while Mn$_2$Ni$_{1.6}$In$_{0.4}$ which has about 20% Z site occupancy of Mn, undergoes martensitic transformation at about 230K.
Acknowledgements {#acknowledgements .unnumbered}
================
Authors would like to acknowledge the financial assistance from Science and Engineering Research Board (SERB), DST, Govt. of India under the project SB/S2/CMP-096/2013. The work at Photon Factory was performed under the Proposal No. 2011G0077
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Traditional compression methods including network pruning, quantization, low rank factorization and knowledge distillation all assume that network architectures and parameters are one-to-one mapped. In this work, we propose a new perspective on network compression, i.e., network parameters can be disentangled from the architectures. From this viewpoint, we present the Neural Epitome Search (NES), a new neural network compression approach that learns to find compact yet expressive epitomes for weight parameters of a specified network architecture end-to-end. The complete network to compress can be generated from the learned epitome via a novel transformation method that adaptively transforms the epitomes to match weight shapes of the [given architecture]{}. Compared with existing compression methods, NES allows the weight tensors to be independent of the architecture design and hence can achieve a good trade-off between model compression rate and performance given [a specific model size constraint]{}. Experiments demonstrate that, on ImageNet, when taking MobileNetV2 as backbone, our approach improves the full-model baseline by 1.47% in top-1 accuracy with 25% MAdd reduction, and with the same compression ratio, improves AutoML for Model Compression (AMC) by 2.5% in top-1 accuracy. Moreover, taking EfficientNet-B0 as baseline, our NES yields an improvement of 1.2% but has 10% less MAdd. In particular, our method achieves a new state-of-the-art results of 77.5% under mobile settings ($<$350M MAdd). Code will be made publicly available.'
author:
- |
Daquan Zhou$^1$, Xiaojie Jin$^2$, Qibin Hou$^1$, Kaixin Wang$^1$, Jianchao Yang$^2$, Jiashi Feng$^1$\
\
$^1$Department of Electrical and Computer Engineering, National University of Singapore\
$^2$Bytedance Inc., Menlo Park, USA\
`{e0357894,kaixin.wang}@u.nus.edu`\
`[email protected]`\
`{jinxiaojie,yangjianchao}@bytedance.com`\
`[email protected]`\
title: 'Neural Epitome Search for Architecture-Agnostic Network Compression'
---
Introduction {#sec:introduction}
============
Despite the remarkable performance achieved in many applications, powerful deep convolutional neural networks (CNNs) typically suffer from high complexity [@han2015deep]. The large model size and computation cost hinders their deployment on resource limited devices, such as mobile phones. Very recently, huge efforts have been made to compress powerful CNNs. Existing compression techniques can be generally categorized into four categories: network pruning [@han2015deep; @collins2014memory; @han2015learning], low rank factorization [@jaderberg2014speeding], quantization [@jacob2018quantization; @hubara2017quantized; @rastegari2016xnor], and knowledge distillation [@hinton2015distilling; @papernot2016semi]. Network pruning targets on removing unimportant connections or weights to reduce the number of parameters and multiply-adds (MAdd). Low rank factorization decomposes an existing layer into lower-rank and smaller layers to reduce the computation cost. Weights quantization aims to use less number of bits to store the weights and activation maps. Knowledge distillation uses a well trained teacher network to train a lightweight student network. All of those compression methods assume that the model parameters (weight tensors) must have one-to-one correspondence to the architectures. As a result, they suffer from performance drop since changing architectures will inevitably lead to loss of informative parameters. In this paper, we consider the network compression problem from a new perspective where the shape of the weight tensors and the architecture are designed independently. The key insight is that the network parameters can be disentangled from the architecture and can be compactly represented by a small-sized parameter set (called epitome), inspired by success of epitome methods in image/video modeling and data sparse coding [@jojic2003epitomic; @cheung2008video; @aharon2008sparse]. As shown in Figure \[convolution comparison\], unlike conventional convolutional layers that use the architecture tied weight tensors to convolve with the input feature map, our proposed neural epitome search (NES) approach first learns a compact yet expressive epitome along with an adaptive transformation function to expand the epitomes. The transformation function is able to generate a variety of parameters from epitomes via a novel learnable transformation function, which also guarantees the representation capacity of the resulting weight tensors to be large. Our transformation function is differentiable and hence enables the NES approach to search for optimal epitome end-to-end, achieving a good trade-off between required model size and performance. [ In addition, [we propose a novel routing map to record the index mapping used for the transformation between the epitome and the convolution kernel.]{} During inference, this routing map enables the model to reuse computations when the expanded weight tensors are formed based on the same set of elements in the epitomes and therefore effectively reduces the computation cost. ]{}
Benefiting from the learned epitomic network parameters and transformation method, compared to existing compression approaches, NES has less performance drop. To the best of our knowledge, this is the first work that automatically learns compact epitomes of network parameters and the corresponding transformation function for network compression. To sum up, our work offers the following attractive properties:
- Our method is flexible. It allows the weight tensors to be independent of the architecture design. We can easily control the model size by defining the size of the epitomes given a specified network architecture. This is especially beneficial in the context of edge devices.
- Our method is effective. The learning-based transformation method empowers the epitomes with highly expressive capability and hence incurs less performance drop even with large compression ratios.
- Our method is easy to use. It can be encapsulated as a drop in replacement to the current convolutional operator. There is no dependence on specialized platforms/frameworks for NES to conduct compression.
To demonstrate the efficacy of the proposed approach, We conduct extensive experiments on CIFAR-10 [@krizhevsky2009learning] and ImageNet [@deng2009imagenet]. On CIFAR-10 dataset, our method outperforms the baseline model by 1.3%. On ImageNet, our method outperforms MobileNetV2 full model by $1.47\%$ in top-1 accuracy with $25\%$ MAdd reduction, and MobileNetV2-0.35 baseline by $3.78\%$. Regarding MobileNetV2-0.7 backbone, our method improves AMC [@he2018amc] by 2.47%. Additionally, when taking EfficientNet-b0 [@tan2019efficientnet] as baseline, we have an improvement of 1.2% top-1 accuracy with 10% MAdd reduction.
[figures/tisser.pdf]{} (20, 0)[(a)]{} (70, 0)[(b)]{}
Related work {#gen_inst}
============
Traditional model compression methods include network pruning [@collins2014memory; @han2015learning], low rank factorization [@jaderberg2014speeding], quantization [@jacob2018quantization; @hubara2017quantized; @rastegari2016xnor] and knowledge distillation [@hinton2015distilling; @papernot2016semi]. For all of those methods, as mentioned in Section \[sec:introduction\]， extensive expert knowledge and manual efforts are needed and the process might need to be done iteratively and hence is time consuming.
[Recently, AutoML based methods have been proposed to reduce the experts efforts for model compression ]{} [@he2018amc; @zoph2018learning; @noy2019asap; @li2019partial] and efficient convolution architecture design [@liu2018darts; @FBNet; @tan2018mnasnet]. As proposed in AutoML for model compression (AMC [@he2018amc]), reinforcement learning can be used as an agent to remove redundant layers by adding resource constraints into the rewards function which however is highly time consuming. Later, gradient based search method such as DARTS [@liu2018darts] is developed for higher search efficiency over basic building blocks. There are also methods that use AutoML based method to search for efficient architectures directly [@FBNet; @he2018amc]. All of those methods are searching for optimized network architecture with an implicit assumption that the weights and the model architecture have one-to-one correspondence. [Different from all of the above mentioned methods, our method provides a new search space by separating the model weights from the architecture. The model size can thus be controlled precisely by nature. ]{}
Our method is also related to the group-theory based network transformation. [Based on the group theory proposed in [@cohen2016group], recent methods try to design a compact filter to reduce the convolution layer computation cost such as WSNet [@jin2017wsnet] and CReLU [@shang2016understanding]. WSNet tries to reduces the model parameters and computations by allowing overlapping between adjacent filters of 1D convolution kernel. This can be seen as a simplified version of our method as the overlapping can be regarded as a fixed rule transformation. CReLu tries to learn diversified features by concatenating ReLU output of original and negated inputs. However, as the rule is fixed, the design of those schemes are application specific and time consuming. Besides, the performance typically suffers since the scheme is not optimized during the training. In contrast, our method requires negligible human efforts and the transformation rule is learned end-to-end. ]{}
Method
======
Overview
--------
A convolutional layer is composed of a set of learnable weights (or kernels) that are used for feature transformation. The observation of this paper is that the learnable weights in CNNs can be disentangled from the architecture. Inspired by this fact, we provide a new perspective on the network compression problem, i.e., finding a compact yet expressive parameter set, called *epitome*, along with a proper transformation function to fully represent the whole network, as illustrated in Figure \[convolution comparison\].
Formally, consider a convolutional network with a fixed architecture consisting of a stack of convolutional layers, each of which is associated with a weight tensor $\theta_i$ ($i$ is layer index). Further, let $\mathcal{L}(X, Y; \theta)$ be the loss function used to train the network, where $X$ and $Y$ are the input data and label respectively and $\theta=\{\theta_i\}$ denotes the parameter set in the network. Our goal is to learn epitomes $E=\{E_i\}$ which have smaller sizes than $\theta$ and the transformation function ${\tau}=\{{\tau}_i\}$ to represent the network parameter $\theta$ with the compact epitome as $\tau(E)$. In this way, network compression is achieved. The objective function of neural epitome search (NES) for a given architecture is to learn optimal epitomes $E^*$ and transformation functions $\tau^*$: $$\label{eqn:transformer}
\{E^*, \tau^* \} = \argmin_{E, \tau} \mathcal{L}(X,Y;\tau(E)), ~~~~~~~s.t.~~~ |E| < |\theta|,$$ where $|\cdot|$ calculates the number of all elements.
The above NES approach provides a flexible way to achieve network compression since the epitomes can be defined to be of any size. By learning a proper transformation function, the epitomes of predefined sizes can be adaptively transformed to match and instantiate the specified network architecture. In the following sections, we will elaborate on how to learn the transformation functions $\tau(\cdot)$, the epitome ($E \in \mathbb{R}^{W^E\times H^E\times C^E_{in} \times C^E_{out}}$) and how to compress models via NES in an end-to-end manner. In this paper, we use “a sub-tensor in the epitome” to describe a patch of the epitome that will be selected to construct the convolution weight tensor. The sub-tensor, $E_s$, is represented with the starting index and the length along each dimension in the epitome as shown below: $$\label{eqn:sub_tensor_epitome}
E_s = E[p:p+w, q:q+h, c_{in}:c_{in}+\beta_1, c_{out}:c_{out}+\beta_2],$$ where $(p,q,c_{in},c_{out})$ denote the starting index of the sub-tensor and $(w,h,\beta_1, \beta_2)$ denotes the length of the sub-tensor along each dimension.
Differentiable Search for Epitome Transformation {#sec:nes_search}
------------------------------------------------
As aforementioned, NES generates convolution weight tensors from the epitome $E$ via a transformation function $\tau$. In this section, we explain how the transformation $\tau$ is designed and optimized. We start with the formulation of a conventional convolution operation. Then we introduce how the transformed epitome is deployed to conduct the convolution operations, with a reduced number of parameters and calculations. A conventional 2D convolutional layer transforms an input feature tensor $F \in \mathbb{R}^{W \times H \times C_{in}}$ to an output feature tensor $G \in \mathbb{R}^{W\times H \times C_{out}}$ through convolutional kernels with weight tensor $\theta \in \mathbb{R}^{w\times h \times C_{in} \times C_{out}}$. Here $(W,H,C_{in}, C_{out})$ denote the width, height, input and output channel numbers of the feature tensor; $w$ and $h$ denote width and height of the convolution kernel. The convolution operation can be formulated as: $$\label{eqn:2d_convolution}
G_{t_w,t_h,c} = \sum_{i=0}^{w-1}{\sum_{j=0}^{h-1}\sum_{m=0}^{C_{in}-1}{ {F_{t_w + i, t_h + j, m} } \theta_{i,j,m,c}}}, \forall\ \ t_w \in [0, W), t_h \in [0, H), c \in [0, C_{out}).$$ Instead of maintaining the full weight tensor $\theta$, NES maintains a much smaller epitome $E$ that can generate the weight tensor and thus achieves model compression. To make sure the generated weights $\tau(E)$ can conduct the above convolution operation without incurring performance drop, we carefully design the transformation function $\tau$ with following three novel components: (1) a learnable indexing function $\eta$ to determine starting indices of the sub-tensor within the epitome $E$ to sample the weight tensor; (2) a routing map $\mathcal{M}$ that records the location mapping from the sampled sub-tensors epitome to the generated weight tensor; (3) an interpolation-based sampler to perform the sampling from $E$ even when the indices are fractional. We now explain their details.
![Our proposed compression process along the spatial dimension. We only show the transformation along spatial dimensions for easy understanding and the transformation along the channel dimension can be found in Figure \[Sampling\_process\_channel\_wise\]. The indexing learner learns the position mapping function, $\mathcal{M}: (i,j,m) \xrightarrow[]{} (p,q)$, between the convolution kernel elements and the sub-tensor in the epitome $E_{s}$. The learned starting indices and the epitome are fed into Eqn. (\[eqn:interpolation\_implement\_spatial\]) to sample the weight tensor. The outputs of Eqn. (\[eqn:interpolation\_implement\_spatial\]) are concatenated together to form the weight tensor. Note that we use a moving average way to update $\mathcal{M}$ so that the indexing learner can be removed during inference. This is shown in details in section \[sec:nes\_search\] in the paragraph ‘Routing map’. The whole training process is end-to-end trainable and hence can be used for any specified network architecture. Here, we abuse the notion for the starting index pair $(p_n,q_n)$ by using subscript $n$ to denote the $n^{th}$ pair of the starting index during sampling(Eqn. (\[eqn:interpolation\_implement\_spatial\])) while in the main text, we use subscript $t$ to denote the training epoch. The learned indices and the epitome are fed into the interpolation-based sampler (Eqn. (\[eqn:interpolation\_implement\_spatial\])) and the outputs of Eqn. (\[eqn:interpolation\_implement\_spatial\]) are concatenated together to form the convolution weights tensor. []{data-label="Sampling_process"}](figures/arch.pdf){width="\textwidth"}
#### Indexing function
The indexing function $\eta$ is used to localize the sub-tensor within the epitome that is used to generate the weight tensor, as illustrated in Figure \[Sampling\_process\]. Concretely, given the input feature tensor $F \in \mathbb{R}^{W \times H \times C_{in}}$, the function generates indices as follows, $$\label{eqn:idex_learner}
(\mathbf{p},\mathbf{q},\mathbf{c_{in}},\mathbf{c_{out}}) = S(\eta(F)),$$ where $\mathbf{p},\mathbf{q},\mathbf{c_{in}}, \mathbf{c_{out}}$ are vectors of learned starting indices along the spatial, input channel and filter dimensions respectively to sample the sub-tensor within epitome to generate the model weight tensors. Each vector contains a set of starting indices and the element number inside each vector is equal to the number of transformations that will be applied along each dimension. Note they are all non-negative real numbers. $\eta$ is the index learner and it outputs the normalized indices $(\mathbf{p',q',c'_{in},c'_{out}})$ through a sigmoid function, each ranging from 0 to 1. These outputs are further up-scaled by a scaling function $S(\cdot)$ to the corresponding dimension of the epitome by $S(\cdot)$ $$\label{eqn:upsclaing}
S(\mathbf{p',q',c'_{in},c'_{out}}) = [W^E, H^E, C^E_{in}, C^E_{out}] \otimes [\mathbf{p',q',c'_{in},c'_{out}}],
$$ where $W^E, H^E, C^E_{in}, C^E_{out}$ are dimensions of the epitome and $\otimes$ denotes element-wise multiplication. The learned indices are then fed into the following interpolation based sampler to generate the weight tensor. We implement the indexing learner by a two-layer convolution module that can be jointly optimized with the backbone network end-to-end. In particular, we use separate indexing learners and epitomes for each layer of the network. More implementation details are given in Appendix \[appendix:search\_space\_index\].
#### Routing map
The routing map is constructed to record the position correspondence between the convolution weight tensor and the sub-tensor in the epitome. It takes a position within the weight tensor as input and returns the corresponding starting index of the sub-tensor in the epitome. The mapped starting index of the sub-tensor in the epitome can thus be retrieved from the routing map fast. More importantly, the indexing learner can be removed during inference with the help of the routing map. The routing map is built as a look-up table during the training phase by recording the moving average of the output index from the index learner $\eta$. For example, the starting index pair as shown in Figure \[Sampling\_process\] can be fetched via $(p_t,q_t) = \mathcal{M}(i,j,m)$ where $(i,j, m)$ is the spatial location in the weight tensor and $(p_{t},q_{t})$ is the starting index of the selected sub-tensor in the epitome at training epoch $t$. The routing map $\mathcal{M}$ is constructed via Eqn. (\[eqn:index\_update\]) as shown below with momentum $\mu$ during the training phase. $\mu$ is treated as a hyper-parameter and is decided empirically[^1]: $$\label{eqn:index_update}
\mathcal{M}(i,j,m) = (p_t,q_t) = (p_{t-1},q_{t-1}) + \mu \cdot \eta(x).$$
![Transformation along the input channel dimension of NES. In the figure, we only show three dimensions of the epitome with $E \in \mathbb{R}^{W^E \times H^E \times C^E_{in} \times 1}$. To simplify the illustration, we set $W^E = w$ and $H^E = h$ where $w$ and $h$ are the size of the convolution kernel. Thus, the starting indices along the spatial dimension,(p,q), are not shown in the figure. The generated weight tensor has input channel number equal to 8. $R_{cin}=\lceil{C_{in}}/{\beta_1}\rceil = 2$ denotes the number of samplings applied along the input channel dimension. During each transformation, sub-tensor with shape ${w \times h \times \beta_1}$ is selected each time based on Eqn. (\[eqn:interpolation\_implement\_channel\]) by replacing the starting index (p,q) in Eqn. (\[eqn:interpolation\_implement\_spatial\]) with ($c_{in}$) and enumerating over the input channel dimension. In this example, $\beta_1$ is set to 4. []{data-label="Sampling_process_channel_wise"}](figures/channel_illu.pdf){width="\textwidth"}
#### Interpolation based sampler
The learned starting index and the pre-defined dimension $(w, h, \beta_1, \beta_2)$ of the sub-tensor in the epitome is then fed into the sampler function. The sampler function samples the sub-tensor within the epitome to generate the weight tensor. To simplify the illustration on the sampler function, we use the transformation along the spatial dimension as an example as shown in Figure \[Sampling\_process\]. The weight tensor is generated via the equation as shown below: $$\label{eqn:interpolation_implement_spatial}
\theta_{(:,:,m)} = {\tau}(E|(p, q)) = \sum_{n_w=0}^{W^E -1}\sum_{n_h=0}^{H^E - 1}{{\mathcal{G}(n_w, p)\mathcal{G}(n_h, q)E_{(n_w : n_w + w,n_h :n_h + h)}}} ,$$ where $\mathcal{G}(a,b) = \max(0,1-|a-b|)$; $n_w$ and $n_h$ enumerate over all integral spatial locations within $E$. Following Eqn. (\[eqn:interpolation\_implement\_spatial\]), we first find all sub-tensors in the epitome whose starting indices $(n_w, n_h)$ along spatial dimensions satisfy: $\mathcal{G}(n_w, p_t)\mathcal{G}(n_h, q_t) > 0$. Then a weighted summation (or interpolation) over the involved sub-tensors is computed according to Eqn. (\[eqn:interpolation\_implement\_spatial\]). In the case of applying the sampling along input channel dimension, $(p,q)$ in the equation is replaced with the learned starting index $c_{in}$ and the weight tensor is generated by iterating along the input channel dimension as shown below: $$\label{eqn:interpolation_implement_channel}
\theta_{(:,:,m:m+\beta_1)} = {\tau}(E|c_{in}) = \sum_{n_c=0}^{R_{cin} - 1}{{\mathcal{G}(n_w, c_{in})E_{( : , :, c_{in} : c_{in} + \beta_1)}}},$$ Where $R_{cin}=\lceil{C_{in}}/{\beta_1}\rceil$ is the number of samplings applied along the input channel dimension.An example of the generation process with Eqn. (\[eqn:interpolation\_implement\_channel\]) along the channel dimension can be found in Figure \[Sampling\_process\_channel\_wise\].
####
The transformation function can be applied in any dimension of the weight tensor. Figure \[Sampling\_process\] illustrates the transformation along the spatial dimension and Figure \[Sampling\_process\_channel\_wise\] shows the transformation along the input channel dimension. The transformation along the filter dimension is the same as the transformation along the input channel dimension. However, transformation along the filter dimension is easier for the computation reuse. We will show this in details in section \[sec:storage\_and\_computation\_saving\].
Learning to Search Epitomes End-to-end
--------------------------------------
Benefiting from the differentiable Eqn. (\[eqn:interpolation\_implement\_spatial\]), the elements in $E$ and the transformation learner $\eta$ can be updated together with the convolutional layers through back propagation in an end-to-end manner. For each element in the epitome $E$, as its transformed weight parameter can be used in multiple positions in weight tensor, the gradients of the epitome are thus the summation of all the positions where the weight parameters are used. Here, for clarity, we use {${\tau}^{-1}(p,q)$} to denote the set of the indices in the convolution kernel that are mapped from the same position $(p,q)$ in $E$. Note that here we abuse the notion of $(p,q)$ to denote the integer spatial position in the epitome. The gradients of $E_{(p,q)}$ can, thus, be calculated as: $$\label{eqn:gradient_updates}
{\nabla_{E_{(p,q)}} \mathcal{L}} =\sum_{z \in \{{\tau}^{-1}(p,q)\}} {\alpha_z {\nabla_{\theta_{z} }\mathcal{L}} },$$ where $\theta_z$ is the kernel parameters that are transformed from $E_{(p,q)}$, and $\alpha_z$ are the fractions that are assigned to $E_{(p,q)}$ during the transformation. The epitome can thus be updated via Eqn. (\[eqn:epitome\_update\]): $$\label{eqn:epitome_update}
E_{(p_t,q_t)}^t = E_{(p_t,q_t)}^{t-1} - \epsilon \nabla_{E_{(p_{t},q_{t})}^{t-1}}\mathcal{L},$$ where $\epsilon$ denotes the learning rate and subscript $t$ denotes the training epoch. Eqn. (\[eqn:gradient\_updates\]) and (\[eqn:epitome\_update\]) use the parameter updating rule along the spatial dimension as an example. The above equations can be applied on any dimension by replacing the index mapping. The indexing learner $\eta$ can be simply updated according to the chain rule.
Compression Efficiency {#sec:storage_and_computation_saving}
----------------------
[**Parameter reduction**]{} By using the routing map which records location mappings from the sub-tensor in the epitome to the convolution weight tensor, the indexing learner can be removed during the inference phase. Thus, the total number of parameters during inference is decided by the size of the epitome and the routing map. Recall that the epitome $E$ is a four dimensional tensor with shape $(W^E, H^E, C^E_{in}, C^E_{out})$. The size of an sub-tensor in the epitome is denoted as $(w, h, \beta_1, \beta_2)$[^2] where $\beta_1 \leq C^E_{in}$ and $\beta_2 \leq C^E_{out}$. The size of the epitome can be calculated as $W^E \times H^E \times C^E_{in} \times C^E_{out}$. The size of the routing map $\mathcal{M}$ is calculated as $3 \times R_{cin} + R_{cout}$ where $R_{cout}={\lceil C_{out}}/{\beta_2}\rceil$ is the number of starting indices learned along the output channel dimension, and $3 \times R_{cin}= 3 \times \lceil{C_{in}} / \beta_1\rceil$ is the number of starting index learned along the spatial and input channel dimension. Note that we can enlarge the size of the sub-tensor in the epitome to reduce the size of the routing map. Here, the size is referring to the number of parameters. Detailed explanations of how $R_{cin}$ is calculated can be found in the Figure \[Sampling\_process\_channel\_wise\]. The parameter compression ratio $r$ can thus be calculated via Eqn. (\[eqn:parameter\_reduction\]): $$\label{eqn:parameter_reduction}
r=\frac{ { w \times h \times C_{in} \times C_{out}}}{{ W^E\times H^E \times C^E_{in} \times C^E_{out}} + 3\times R_{cin} + R_{cout}} \approx \frac{C_{out} \times C_{in} \times w \times h}{C^E_{out} \times C^E_{in} \times W^E \times H^E},$$
From Eqn. (\[eqn:parameter\_reduction\]), it can be seen that the compression ratio is nearly proportional to the ratio between the size of the epitome and the generated weight tensor. Detailed proof can be found in Appendix \[appendix:computation\_reduction\]. The above analysis demonstrates that NES provides a *precise control of the model size* via the proposed transformation function.
[**Computation reduction**]{} As the weight tensor $\theta$ is generated from the epitome $E$, the computation in convolution can be reused when different elements in $\theta$ are from the same portion of elements in $E$.
![NES transformation along the input channel dimension with channel wrapping. To simplify the illustration, we choose an epitome with shape $\mathbb{R}^{w \times h \times 3 \times 1}$ and $C^E_{in} = 3$. In this example, the transformation is applied twice and the two learned starting indices are 0 . The input feature map $\mathcal{F}$ is first added based on the learned interpolation position of the kernels. Input feature map $\mathcal{F}_1$ and $\mathcal{F}_4$ are both multiplied with the first channel in the epitome since $W_1$ and $W_4$ are both generated with $E_1$. To reuse the multiplication, feature map $\mathcal{F}_1$ and $\mathcal{F}_4$ are first added together before multiplying with the weights kernel $E_1$. The figure uses integer index to simplify the illustration. When the learned indices are fractions, the feature maps are the weighted summation of the two nearest integer indexed sub-tensors in the epitome as shown in Eqn. (\[eqn:interpolation\_implement\_channel\]). For example, if the two starting indices in this figure are 0.6 and 0.3, the calculation becomes (0.6$F_1$ + 0.3$F_4$ + 0.4$F_3$ + 0.7$F_6$)$\otimes E1$ + (0.4$F_1$ + 0.7$F_4$ + 0.6$F_2$ + 0.3$F_5$) $\otimes E_2$ + (0.4$F_2$ + 0.7$F_5$ + 0.6$F_3$ + 0.3$F_6$) $\otimes E_3$. Since we group the feature map first before the convolution, the computation cost is reduced. []{data-label="channel wrap"}](figures/channel.pdf){width="\textwidth"}
Concretely, we propose two novel schemes to reuse the computation along the input channel dimension and the filter dimension respectively. *Channel wrapping*. During the inference, the computation along the input channel dimension is reduced with channel wrapping as illustrated in Figure \[channel wrap\]. For the elements in the input feature map that are multiplied with the same element in the epitome, we group the feature map elements first and then multiplied with the weight tensor in the epitome as follows: $$\label{eqn:channel_wrapping}
\Tilde{F}(i,j,m) = \sum_{c'=0}^{R_{cin}-1}{F(p,q,m + c' \times C^E_{in} + c_{in})},$$ where $R_{cin}=\lceil{C_{in}}/{C^E_{in}}\rceil$ is the number of samplings (Eqn. (\[eqn:interpolation\_implement\_channel\])) applied along the input channel dimension and $m + c_{in} + c'\times C^E_{in}$ is the learned position with $(p,q,c_{in}) = \mathcal{M}(i,j,m)$ and $c_{in} \in [0,C^E_{in})$. This process is also illustrated in Figure \[channel wrap\]. *Product map and integral map*. For the reuse along the filter dimension, given the routing map $\mathcal{M}$ for the transformation, we first calculate the convolution results between the epitome and the input feature map once and then save the results as a product map $P$.
During inference, given $P$, the multiplication in convolution can be reused in a lookup table manner with $O(1)$ complexity: $$\label{eqn:modified_2d_convolution}
G_{t_w,t_h,c} = \sum_{i=0}^{W-1}{\sum_{j=0}^{H-1}\sum_{m=0}^{C_{in}-1}{ {F_{t_w + i, t_h + j, m} } \theta_{\mathcal{M}({i,j,m,c})}}} = \sum_{i=0}^{W-1}{\sum_{j=0}^{H-1}\sum_{m=0}^{C_{in}-1}{ P_{\mathcal{M}({i,j,m,c})}}}.$$ The additions in Eqn. (\[eqn:modified\_2d\_convolution\]) can also be reused via an integral map $I$ [@crow1984summed] as done in [@viola2001rapid]. With the product map and the integral map, the MAdd can be calculated as: $$\label{eqn:reduced_MAdd}
\text{Reduced MAdd} = (2C_{in}W^EH^E-1)WHC^E_{out} + WHW^EH^EC^E_{out} + 2R_{cin}WH\beta_1 + 2R_{cout}\beta_2.$$ Hence, the computation cost reduction ratio can be written as:$$\label{eqn:MAdd_reduction}
\begin{split}
\text{MAdd Reduction Ratio} = \frac{C_{out}HW(2C_{in}wh-1)}{C^E_{out}HW(W^EH^E+2C^E_{in}W^EH^E-1)+2R_{cin}WH\beta_1+2 \beta_2 R_{cout}} \\
\approx \frac{C_{out}C_{in}wh}{C^E_{out}C^E_{in}W^EH^E}
\end{split}$$ See more details and analysis in Appendix \[appendix:computation\_reduction\]. [**Discussion.**]{} We make a few remarks on the advantages of our proposed method as follows. The proposed NES method disentangles the weight tensors from the architecture by using a learnable transformation function. This provides a new research direction for model compression by bringing in better design flexibility against the traditional compression methods on both sides of software and hardware. On the software side, NES does not require re-implementation of acceleration algorithms. All the operations employed by NES are compatible with popular neural network libraries and can be encapsulated as a drop in operator. On the hardware side, the memory allocation of NES is more flexible by allowing easily adjust the epitome size. This is especially helpful for hardware platform where the off chip memory access is the main power consumption as demonstrated in [@han2016eie]. NES provides a way to balance the computation/memory-access ratio in hardware: a smaller epitome with a complex transformation function results in a computation intense model while a large epitome with simple transformation function results in a memory intensive model. Such ratio is an important hardware optimization criteria which however is not covered by most previous compression methods.
Method Conv1 Conv2 Conv3 Conv4 Conv5 Conv{6-8} Acc. (%) Params
---------- ------- ------- ------- ------- ------- ----------- ---------- ----------- --
Config. SC SC SC SC SC SC
baseline 11 11 11 11 S1 11 66.0 1$\times$
WSNet 81 41 22 12 S4 18 66.5 $4\times$
Ours 81 41 22 12 S4 18 **73.0** $4\times$
: Comparison with WSNet on ESC-50 dataset. We use the same configuration but change the sampling stride to be learnable. ‘S’ denotes the stride and ‘C’ denotes the repetition times along the input channel dimension. We use the ‘S’ in WSNet as initial values and learns the offsets.
\[ESC-50\]
Methods MAdd(M) Parameters Param Compression Rate Top-1 Accuracy(%)
---------------------- --------- ------------ ------------------------ -------------------
MobilenetV2-1.0 301 3.4M $1.00\times$ 71.8
MobilenetV2-$0.75^*$ 217 2.94M $1.17\times$ 69.14
MobilenetV2-$0.5^*$ 153 2.52M $1.36\times$ 67.22
MobilenetV2-$0.35^*$ 115 2.26M $1.54\times$ 65.18
MobilenetV2-$0.18^*$ 71 1.98M $1.80\times$ 60.70
Our method-0.75 220 2.94M $1.17\times$ 71.54
Our method-0.5 157 2.52M $1.36\times$ 69.42
Our method-0.35 120 2.26M $1.54\times$ 67.01
Our method-0.18 79 1.95M $1.80\times$ 64.48
: Results of ImageNet classification. Our method uses vanilla MobileVetV2 as backbone. For a fair comparison, we evaluate multiple width multiplier values of $0.75, 0.5, 0.35$ and $0.18$ and only apply it on the filter dimension of the first $1 \times 1$ convolution. We apply the proposed method on all the invert residual blocks equally to disentangle the architecture affects on the performance. MAdd are calculated based on all convolution blocks with an assumption that the batch normalization layers are merged. ‘$*$’ denotes our own implementation.
\[Ablation\]
Experiments {#others}
===========
We first evaluate the efficacy of our method in 1D convolutional model compression on the sound dataset ESC-50 [@piczak2015esc] for the comparison with WSNet. We then test our method with MobileNetV2 and EfficientNet as the backbone on 2D convolutions on ImageNet dataset [@deng2009imagenet] and CIFAR-10 dataset [@krizhevsky2009learning]. Detailed experiments settings can be found in Appendix \[appendix:experiments\]. For all experiments, we do not use additional training tricks including the squeeze-and-excitation module [@hu2018squeeze] and the Swish activation function [@ramachandran2017swish] which can further improve the results unless those are used in the bachbone model originally. The calculation of MAdd is performed for all convolution blocks. We evaluate our methods in terms of three criteria: model size, multiply-adds(MAdd) and the classification performance. For 1D convolution compression, we compare with WSNet. Similar to WSNet [@jin2017wsnet], we use the same 8-layer CNN model for a fair comparison. The compression ratio in WSNet is decided by the stride ($S$) and the repetition times along the channel dimension ($C$), as shown in Table \[ESC-50\]. From Table \[ESC-50\], one can see that with the same compression ratio, our method outperforms WSNet by 6.5% in classification accuracy. This is because our method is able to learn proper weights and learn a transformation rules that are adaptive to the dataset of interest and thus overcome the limitation of WSNet where the sampling stride is fixed. More results can be found in Appendix \[appendix:esc50\].
GROUP Methods MAdd (M) Params Top-1 Acc. (%)
------- --------------------------------------------- -------------- ---------------- -----------------
MobilenetV2-0.35 [@sandler2018mobilenetv2] 59 1.7M 60.3
S-MobilenetV2-0.35 [@yu2018slimmable] 59 3.6M 59.7
US-MobilenetV2-0.35 [@yu2019universally] 59 3.6M 62.3
MnasNet-A1 (0.35x) [@tan2018mnasnet] 63 1.7M 62.4
Our method-0.18 79 2.0M $\bold{64.48}$
MobilenetV2-0.5 [@sandler2018mobilenetv2] 97 2.0M 65.4
S-MobilenetV2-0.5 [@yu2018slimmable] 97 3.6M 64.4
US-MobilenetV2-0.5 [@yu2019universally] 97 3.6M 65.1
Our method-0.35 120 2.2M $\bold{67.01}$
MobilenetV2-0.75 [@sandler2018mobilenetv2] 209 2.6M 69.8
S-MobilenetV2-0.75 [@yu2019network] 209 3.6M 68.9
US-MobilenetV2-0.75 [@yu2019universally] 209 3.6M 69.6
FBNet-A [@FBNet] 246 4.3M 73
AUTO-S-MobilenetV2-0.75 [@yu2019network] 207 4.1M 73
Our method-0.5 $\bold{157}$ $\bold{2.5M}$ $ \bold{69.42}$
Our method-0.75 $\bold{220}$ $\bold{2.9M}$ $\bold{71.54}$
Our method-0.75-A $\bold{225}$ $\bold{3.7M}$ $\bold{73.27}$
Our method-0.5 $\bold{240}$ $\bold{3.92M}$ $\bold{75.55}$
Our method-0.5 $\bold{350}$ $\bold{5.46M}$ $\bold{77.5}$
: Comparison of our method with other state-of-the-art models on ImageNet where our method shows superior performance over all other methods. MAdd are calculated based on all convolution blocks with an assumption that the batch normalization layers are merged. Suffix ‘-A’ means we use larger compression ratio for front layers. Our method does not modify the backbone model architecture and applies a uniform compression ratio, unless specified with suffix ‘-A’. All experiments are using MobileNetV2 as backbone unless labeled with EfficientNet as suffix.
\[ImageNet-cross comparison\]
[**Implementation details.**]{} We use both MobilenetV2 [@sandler2018mobilenetv2] and EfficientNet [@tan2019efficientnet] as our backbones to evaluate our approach on 2D convolutions. Both models are the most representative mobile networks very recently and have achieved great performance on ImageNet and CIFAR-10 datasets with much fewer parameters and MAdd than ResNet [@he2016deep] and VGGNet [@simonyan2014very]. For a fair comparison, we follow the experiment settings as in the original papers.
[**Results on ImageNet.**]{} We first conduct experiments on the ImageNet dataset to investigate the effectiveness of our method. We use the same width multiplier as in [@sandler2018mobilenetv2] as our baseline. We choose four common width multiplier values, *i.e.*, $0.75, 0.5, 0.35 \text{ and } 0. 18$ [@sandler2018mobilenetv2; @zhang2018shufflenet; @yu2018slimmable] for a fair comparison with other compression approaches. The performance of our method and the baseline is summarized in Table \[Ablation\]. For all the width multiplier values, our method outperforms the baseline by a significant margin. It can be observed that under higher compression ratio the performance gain is also larger. Moreover, the performance of MobileNetV2 drops significantly when the compression ratio is larger than $3\times$. However, our NES method increases the performance by 3.78% at a large compression ratio. This is because when the compression ratio is high, each layer in the baseline model does not have enough capacity to learn good representations. Our method is able to generate more expressive weights from the epitome with a learned transformation function.
We also compare our method with the state-of-the-art compression methods in Table \[ImageNet-cross comparison\]. Since an optimized architecture tends to allocate more channels to upper layers [@he2018amc; @yu2019network], we also run experiments with larger size of the epitome for upper layers. The results is denoted with suffix ‘-A’ in Table \[ImageNet-cross comparison\]. Comparison with more models are shown in Figure \[acc plot\]. As shown, NES outperforms the current SOTA mobile model (less than 400M MAdd model) EfficientNet-b0 by 1.2% with 40M less MAdd. Obviously, our method performs even better than some NAS-based methods.Although our method does not modify the model architecture, the transformation from the epitome to the convolution kernel optimizes the parameter allocation and enriches the model capacity through the learned weight combination and sharing.
![ImageNet classification accuracy of our method, EfficientNet, MobileNetV2 baselines and other NAS based methods including AMC [@he2018amc], IGCV3 [@sun2018igcv3], MNasNet [@tan2018mnasnet], ChamNet [@dai2018chamnet] and ChannelNet [@gao2018channelnets]. Our method outperforms all the methods within the same level of MAdd. Here, $\text{MobileNetV2}^*$ is our implementation of baseline models and MobileNetV2 is the original model with width multiplier of 0.35,0,5,0,75 and 1. The backbone model for our results are EfficientNet\_b1 and b0 [@tan2019efficientnet] with multiplier 0.5 and MobileNetV2 with multiplier of 0.75, 0.5, 0.35 and 0.2, respectively (from top to bottom). Note that we do not use additional training tricks including the squeeze-and-excitation module [@hu2018squeeze] and the Swish activation function [@ramachandran2017swish].[]{data-label="acc plot"}](plot_acc_v7.png){width="70.00000%"}
Methods Parameters MAdd (M) Top-1 Accuracy(%)
----------------------------- ------------ ---------- ----------------------------
MobilenetV2-1 2.2M 93.52 $\text{94.06}^*$
$\text{MobilenetV2-0.18}^*$ 0.4 M 21.59 91.70
Auto-Slim 0.7M 59 93.00
Auto-Slim 0.3M 28 92.00
Our method 0.39M 26.8 $\mathbf{93.22 \pm 0.013}$
: Comparison with other state-of-the-art models in classification accuracy on CIFAR-10. Our method outperforms the recently proposed AUTO-SLIM [@yu2019network] which is a compression method using AutoML method.
\[CIFAR10\]
[**Results on CIFAR-10.**]{} We also conduct experiments on CIFAR-10 dataset to verify the efficiency of our method as shown in Table \[CIFAR10\]. Our method achieves $3.5\times$ MAdd reduction and 5.64$\times$ model size reduction with only 1% accuracy drop, outperforming NAS-based AUTO-SLIM [@yu2019network]. More experiments and implementation details are shown in the supplementary material.
[**Discussion.**]{} From the above results, one can observe significant improvements of our method over competitive baselines, even the latest architecture search based methods. The improvement of our method mainly comes from alleviating the performance degradation due to insufficient model size by learning richer and more reasonable combination of weights parameters and allocating suitable weight parameters sharing among different filters. The learned transformation from the epitome to the convolution kernel increases the weight representation capability with less increase on the memory footprint and the computation cost. This distinguishes our method from previous compression methods and improves the model performance significantly under the same compression ratio.
Conclusion
==========
[We present a novel neural epitome search method which can reuse the parameters efficiently to reduce the model size and MAdd with minimum classification accuracy drop or even increased accuracy in certain cases. Motivated by the observation that the parameters can be disentangled form the architecture, we propose a novel method to learn the transformation rule between the filters to make the transformation adaptive to the dataset of interest. We demonstrate the effectiveness of the method on CIFAR-10 and ImageNet dataset with extensive experimental results.]{}
Implementation details {#appendix:experiments}
======================
MobileNetV2 Settings
--------------------
[**Epitome dimensions for MobileNetV2 bottleneck.**]{} With our NES method, the shape of the feature map produced by each layer can be kept the same as the ones from the original model before compression. However, the number of channels in the feature map is reduced using the width multiplier method for MobileNetV2. Hence, for a fair comparison, we only apply the width multiplier on the output dimension of the first $1\times 1$ convolutional layer and the input channel dimension of the second $1 \times 1$ convolutional layer within the bottleneck blocks of MobileNetV2 for obtaining the same feature map shape between blocks as our method. Based on this principle, we generate weight tensor based on the epitome along the filter dimension for the first $1\times 1$ convolutional layers within the bottleneck and along the input channel dimension for the second $1\times 1$ convolutional layers.
Specifically, we set the epitome shape per layer as $(\# \mathrm{in\_channels},\frac{\# \mathrm{out\_channels}\times \mathrm{expansion}} {\mathrm{multiplier}},1,k)$ for the first $1\times 1$ convolution layer and $(\frac{\# \mathrm{in\_channels} \times expansion } {\mathrm{multiplier}},\# \mathrm{out\_channels},1,k)$ for the second $1\times 1$ layer as shown in Table \[bottleneck structure\]. Here, expansion is referring to the ratio between the input size of the bottleneck and the inner size as detailed in Figure 2 of [@sandler2018mobilenetv2]. The shape represents the number of input channels, the number of output channels and the kernel size, respectively. The compression ratio for each layer, $c$, can thus be calculated as $c=\frac{1}{multiplier}$.
Input Operators Output Epitome Comp. ratio
--------------------------------------------- ------------------------------------------ -------------------------------------------- --------------------------------------- ----------------------------
$h \times w \times k$ $1\times 1$, conv2d, ReLU6 $ h \times w \times tk $ $w_c \times h_c \times k \times ctk $ $\frac{w_c\times h_c}{c}$
$\frac{h}{s} \times \frac{w}{s} \times tk$ $3\times 3$, depth-wise separable, ReLU6 $\frac{h}{s} \times \frac{w}{s} \times tk$ – $1$
$\frac{h}{s} \times \frac{w}{s} \times tk$ $1\times 1$, conv2d, linear $\frac{h}{s} \times \frac{w}{s} \times k'$ $w_c \times h_c\times ctk \times k'$ $\frac{w_c \times h_c}{c}$
: Epitome dimensions for the inverted residual blocks of the MobileNetV2 backbone. Here $w,h,k,k'$ denotes the spatial size, input channels and output channels of the input feature map respectively. Variables $w_c$ and $h_c$ denote the spatial size of the epitome and are set to 1 for $1\times1$ convolutional layer. $c$ is used to set the compression ratio for each layer and is similar to the concept of width multiplier as defined in MobileNet [@howard2017mobilenets]. [$t$ is the expansion ratio as defined in MobileNetV2.]{}
\[bottleneck structure\]
Epitome Dimension Design
------------------------
The size of the epitome can be calculated precisely by the original model and the desired compression ratio $r$. For a CNN with $C$ $n$-dimensional convolutional layers and $K$ fully-connected layers, its number of parameters can be calculated as $\sum_{i=1}^C { \prod_d{L^i_d} +\sum_{k=1}^K N_{in}^k {N^k_{out}}}$, where $L^i_d$ denotes the length of the convolution weight tensor along the $d^{th}$ dimension of the $i^{th}$ convolutional layer. $N^k_{in}$ and $N^k_{out}$ denote the input and output dimension of the $k^{th}$ fully-connected layer.
We assign an epitome $E^j \in \mathbb{R}^{W^E_j \times H^E_j}$ for each layer $j$, and a routing map $\mathcal{M}: (x_1,x_2,\ldots,x_n) \rightarrow (p,q)$, *i.e.*, the weight value of an $n$-d filter at location $(x_1,x_2,\ldots,x_n)$ being equal to $E(p,q)$. We define the dimension of the epitome to be 2D here to illustrate the general case and later, we will show that in practice, the dimension of the epitome can be increased to save the computation memory. The size of the total epitome is, thus, $\sum_{j=1}^{C+K} { W^E_j \times H^E_j}$. After learning the routing map for layer $j$, we store the location mapping as a lookup table of size $M_j$. The size of all the mapping tables is $\sum_{j=1}^{C+K}M_j$. Hence, the compression ratio can be calculated as $$\label{eqn:compression ratio}
r=\frac{\sum_{i}^C { \prod_d{L^d_i} +\sum_{k}^K N_{in}^k {N^k_{out}}}}{\sum_{j}^{C+K} { (W^E_j \times H^E_j} + {M}_j ) }.$$ In our analysis, we use a uniform compression ratio for all the layers. Therefore, given a compression ratio $r$, the size of the epitome for each layer can be calculated accordingly. This deterministic design of the epitome size is hardware friendly and can be used to control the memory allocation.
#### Epitome patch design
We set the patch size along each dimension to be $w, h, \beta_1$ and $\beta_2$. Note that the transformation along each dimension are independent and hence can be conducted separately. The starting index of the transformation along the spatial dimension and the input channel dimension are learned in pairs. This is because the transformation along the spatial dimension will also increase the input channel dimension.
Index search space in Epitome {#appendix:search_space_index}
-----------------------------
The selection space of the starting index for the transformation is not all the indices available along the channel dimension. We partition the channels into groups to build a super-index with a group length $l_g$. For example, for an epitome that has C channels along the input channel dimension, the potential channel index ranges from 0 to C - 1. With our super-index scheme, adjacent $l_g$ channels are grouped as a single index $C_g$ and thus, the selection space of the index ranges from 0 to $C/l_g -1$. The range of the super-index is used to scale the output from the transformation learner.
More Results on 1D Convolution Compression {#appendix:esc50}
==========================================
We also conduct experiments to examine the highest compression ratio that our method can achieve without performance drop compared to WSNet [@jin2017wsnet]. For a fair comparison, we choose the same 8-layer CNN model backbone as used in WSNet. Configuration details have been demonstrated in Table \[ESC-50\] in the formal paper. The results are shown in Table \[ESC-50 more results\].
\[ESC50-ext-experiments\]
Methods Compression Rate Accuracy (%) (top1)
-------------- ------------------ ---------------------
WSNet 1.00$\times$ 66.5
Our method-1 1$\times$ 73.0
Our method-2 2.35$\times$ 69.25
Our method-3 3.16$\times$ 65.5
: Comparison with WSNet on ESC-50 dataset. We choose the compressed model by WSNet method as our baseline. By decreasing the size of the epitome, we can achieve higher compression ratio. We apply uniform compression ratio for all layers. It is observed that the highest compression ratio we can achieve before our method’s performance become smaller than WSNet is $3.16\times$.
\[ESC-50 more results\]
NES for Fully-Connected Layer
=============================
Let $D\in \mathbb{R}^{N_{in}\times N_{out}}$ denotes the parameter matrix for a fully-connected (FC) layer. We could use Eqn. (\[eqn:feature wrapping\]) to sampling along the input dimension and Eqn. (\[eqn:filter reuse\]) to sample along the output dimension. We also conduct experiments on CIFAR-10 dataset to verify the efficacy of our method on FC layers as shown in Table \[CIFAR10 FC Compress\]. We take MobileNetV2-1.0 as our baseline.
Methods Parameters Model Comp. Rate FC Param. Comp. Rate Top-1 Acc. (%)
------------------------------- ------------ ------------------ ---------------------- ----------------
MobileNetV2-1 2.2M 1.00$\times$ 1.00$\times$ 94.06
$\text{MobileNetV2-0.5 FC}^*$ 0.4 M 5.55$\times$ 2.00$\times$ 91.8
Our method-0.5 FC 0.39M 5.64$\times$ 2.00$\times$ $\bold{92.96}$
: Experiment results of our method applied on fully connected layer of MobileNetV2.
\[CIFAR10 FC Compress\]
Proof on Parameter and Computation Reduction {#appendix:computation_reduction}
============================================
With the epitome $E$ as defined in the main text, our method introduces a novel transformation function where the convolution filter weights are transformed from $E$ with $\tau(\cdot)$. In this section, we start with the most general situation where the epitome is two-dimensional,$E \in \mathbb{R}^{W^E \times H^E}$, and we will show that increasing the dimension of epitome can reduce the computation memory cost aggressively. Our method can be extended to $n$-dimension convolution and fully-connected layers straightforwardly. The transformation process along the input channel dimension is illustrated in Figure \[Sampling\_process\_channel\_wise\] In our method, all the weight parameters $\theta_{i,j,m,c}$ are transformed from the compact epitome $E$. The convolution with NES is shown as below: $$\label{eqn:mapped 2d convolution}
G_{t_w,t_h,c}=f_{t_w,t_h} \ast k_c = \sum_i^W{\sum_{j}^H\sum_m^{C_{in}}{ {F_{t_w + i, t_h + j, m} } E_{\mathcal{M}({i,j,m,c})}}}.$$
#### Proof on computational cost reduction.
Since the weight elements per convolutional layer are formed based on the same $E$, there is computational redundancy when two convolution kernels are selected from the same portion of $E$ as shown in Figure \[Sampling\_process\]. To reuse the multiplication in convolution, we first compute the multiplication between each element in the epitome $E$ and the input feature map. The results are saved as a *product map* $P$ such that the computations are done only once. However, given the epitome $E \in \mathbb{R}^{W^E \times H^E}$, the product map size is $W\times H \times C_{in} \times W^E \times H^E \times C_{out}$ which consumes large computational memory. To reduce its size, we propose to increase the dimensions of the epitome to $E \in \mathbb{R}^{W^E \times H^E \times C^E_{in} \times C^E_{out}}$ in order to group the computation results in the product map. Each entry in the product map $P$ is calculated as the dot product between the channel dimension along the input feature map and the third dimension along the compact weight matrix. We set $C^E_{in}$ to be smaller than $C_{in}$ to further boost the compression and $\beta_1$ equal to $C^E_{in}$. As illustrated in Figure \[channel wrap\], the input channels of the feature map is first grouped by $$\label{eqn:feature wrapping}
\Tilde{F}(i,j,m) = \sum_{c'=0}^{R_{cin} - 1}{F(p,q,m + c'\times C^E_{in} + c_{in})},$$ where $R_{cin}=\lceil{C_{in}}/{C^E_{in}}\rceil$ is the compression ratio along the input channel dimension and $m + c_{in} + c\times C^E_{in}$ is the learned position with $(p,q,c_{in},n) = \mathcal{M}(i,j,c,m)$, where $c_{in} \in [1,C^E_{in}]$. The transformed input feature map $\Tilde{F}$ has the same number of channels as $C^E_{in}$. Let $(i,j,m)$ index the transformed feature map location. The product map is then calculated as $$\label{eqn:product map}
P_{i,j,p,q,n} = \mathcal{\Tilde{F}}_{i,j,:} \cdot E_{p,q,:,n}',
$$ where $\cdot$ denotes the dot product operation.
![NES transformation along the input channel dimension with channel wrapping. To simplify the illustration, we choose an epitome with shape $\mathbb{R}^{w \times h \times 3 \times 1}$ and $\beta_1 = 3$. The transformation is applied twice and the two starting indices are 0. The input feature map $\mathcal{F}$ is first grouped based on the learned interpolation position of the kernels. Input feature map $\mathcal{F}_1$ and $\mathcal{F}_4$ are both multiplied with the first channel in the epitome since $W_1$ and $W_4$ are both generated with $E_1$. To reuse the multiplication, feature map $\mathcal{F}_1$ and $\mathcal{F}_4$ are first added together before multiplying with the weights kernel $E_1$. The figure uses integer index to simplify the illustration. When the learned indices are fractions, the feature maps are the weighted summation of the two nearest integer indexed sub-tensors in the epitome as shown in Eqn. (\[eqn:interpolation\_implement\_spatial\]). For example, if the two starting indices in this figure are 0.6 and 0.3, the calculation becomes (0.6$F_1$ + 0.3$F_4$ + 0.4$F_3$ + 0.7$F_6$)$\otimes E1$ + (0.4$F_1$ + 0.7$F_4$ + 0.6$F_2$ + 0.3$F_5$) $\otimes E_2$ + (0.4$F_2$ + 0.7$F_5$ + 0.6$F_3$ + 0.3$F_6$) $\otimes E_3$. Since we group the feature map first before the convolution, the cost is reduced. []{data-label="channel wrap appendix"}](figures/channel.pdf){width="\textwidth"}
Now, the multiplications can be reused by replacing the convolution kernels with the product map $P$. Based on Eqns. (\[eqn:mapped 2d convolution\]), (\[eqn:feature wrapping\]), (\[eqn:product map\]), the convolution can be reduced by $$\label{eqn:product map interpolation implement}
G_{t_w,t_h,n} = \sum^{W-1}_{i=0}\sum^{H-1}_{j=0}{P_{(t_w + i,t_h + j,p,q,n)}}.$$ To reuse the additions, we adopt an integral image $I$ which is proposed in [@crow1984summed] but differently we extend the integral image dimension to make it suitable for 2D convolution based on $P$. Our integral image can be constructed by $$\label{eqn:integral image}
I(t_w,t_h,p,q,n) =
\begin{cases}
P_{0,t_h,p,q,n}, & t_w = 0 \\
P_{t_w,0,p,q,n} , & t_h = 0\\
P_{t_w,t_h,0,q,n}, & p = 0\\
P_{t_w,t_h,p,0,n}, & q = 0\\
P_{t_w,t_h,p,q,0}, & n = 0\\
I(t_w-1, t_h-1, p-1, q -1,n-1) + P(t_w,t_h,p,q,n), & \text{else}.
\end{cases}$$ From Eqn. (\[eqn:integral image\]), the 2D convolution results can be retrieved in a similar way to [@jin2017wsnet] as follows: $$\label{eqn:cal convolution}
G_{t_w,t_h,p,q,n} = I(t_w + w -1, t_h + h -1, p + w -1, q + h -1, n) - I(t_w -1, t_h -1, p -1, q -1, n -1).$$
As we set $C^E_{out}$ to be smaller than the output channel dimension of the convolution kernel, we reuse the computation results from the epitome via $$\label{eqn:filter reuse}
\Tilde{G}_{t_w,t_h,r_{out} \times \beta_2:(r_{out}+1) \times \beta_2} = G_{t_w,t_h,n:n + \beta_2},$$ where $r_{out} \in \{0,1,...,R_{cout}-1\}$, $R_{cout}={\lceil C_{out}}/{\beta_2}\rceil$ is the number of samplings conducted along the output channel dimension, and $n = \mathcal{M}(r_{out} \times \beta_2)$ is the learned mapping along the filter dimension. The filter length $\beta_2\in \{1,2,...,C^E_{out}\}$ is a hyper-parameter and is decided empirically. In our experiments, we choose $\beta_2 = C_{out}$. Thus, the MAdds can be calculated as $$\label{eqn:reduced MAdd}
\text{Reduced MAdd} = \underbrace{(2C_{in}W^EH^E-1)WHC^E_{out}}_{\text{From}\,\text{Eqn}.~ (\ref{eqn:product map})} + \underbrace{WHW^EH^EC^E_{out}}_{\text{From\, Eqn}.~(\ref{eqn:integral image})} + \underbrace{2R_{cin}WH\beta_1}_{\text{From Eqn}.~(\ref{eqn:feature wrapping})} + \underbrace{2R_{cout}\beta_2}_{\text{From~Eqn}.(\ref{eqn:filter reuse})}.$$ Suppose we use sliding window with a stride of 1 and no bias term, the MAdds of the conventional convolution can be calculated based on Eqn. (\[eqn:2d\_convolution\]) as shown below: $$\label{eqn:MAdd num}
\text{MAdd} = (2\times C_{in} \times w \times h - 1) \times H \times W \times C_{out}. $$ Therefore, the total MAdd reduction ratio is $$\label{eqn:compression ratio improved}
\text{MAdd Reduction Ratio} = \frac{C_{out}HW(2C_{in}wh-1)}{C^E_{out}HW(W^EH^E+2C^E_{in}W^EH^E-1)+2R_{cin}WH\beta_1 + 2 R_{cout}\beta_2}.$$
#### Proof on parameter reduction.
The parameter compression ratio for a 2D convolution layer can be calculated as follows: $$\label{eqn:2d compress}
r=\frac{ { whC_{in}C_{out}}}{{ W^E\times H^E \times C^E_{in} \times C^E_{out}} + 3\times R_{cin} + R_{cout}},$$ where $w$ and $h$ denote the width and height of the kernel of the layer. $W_c$ and $H_c$ denote the spatial size of the corresponding epitome. From Eqn. (\[eqn:compression ratio improved\]) and Eqn. (\[eqn:2d compress\]), it can be observed that the compression ratio is mainly decided by $\frac{C_{out}C_{in}wh}{C^E_{out}C^E_{in}W^EH^E}$.
[^1]: We set $\mu$ to be 0.97 in our experiments.
[^2]: We set $\beta_1$ to $C^E_{in}$ and $\beta_2$ to $C^E_{out}$ in this paper.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Transport phenomena, including diffusion, mixing, spreading, and mobility, are crucial to understand and model dynamical features of complex systems. In particular, the study of geophysical flows attracted a lot of interest in the last decades as fluid transport has proven to play a fundamental role in climatic and environmental research across a wide range of scales. Two theoretical frameworks have been effectively used to investigate transport phenomena in complex systems: Dynamical Systems Theory (DST) and Network Theory (NT). However, few explicit connections between these two different views have been established. Here, we focus on the betweenness centrality, a widely used local measure which characterizes transport and connectivity in NT. By linking analytically DST and NT we provide a novel, continuous-in-time formulation of betweenness, called Lagrangian Betweenness, as a function of Lyapunov exponents. This permits to quantitatively relate hyperbolic points and heteroclinic connections in a given dynamical system to the main transport bottlenecks of its associated network. Moreover, using modeled and observational velocity fields, we show that such bottlenecks are present and surprisingly persistent in the oceanic circulation illustrating their importance in organizing fluid motion. The link between DST and NT rooted in the definition of the Lagrangian Betweenness has the potential to promote further theoretical developments and applications at the interface between these two fields. Finally, the identification of such circulation hotspots provides new crucial information about transport processes in geophysical flows and how they control the redistribution of various tracers of climatic (e.g. heat, carbon, moisture), biological (e.g. larvae, pathogens) and human (e.g. pollutants, plastics) interests.'
author:
- 'Enrico Ser-Giacomi$^{*}$'
- Alberto Baudena
- Vincent Rossi
- Ruggero Vasile
- Cristóbal López
- 'Emilio Hernández-García'
bibliography:
- 'references.bib'
title: 'From network theory to dynamical systems and back: Lagrangian Betweenness reveals bottlenecks in geophysical flows'
---
Keywords {#keywords .unnumbered}
========
Network Theory, Betweenness Centrality, Dynamical Systems Theory, Geophysical Flows, Lagrangian Transport, Hyperbolic Points, Heteroclinic Connections, Lyapunov Exponents, Fluid Dispersion and Mixing
Introduction
============
During the last decades the study of Complex Systems was significantly boosted by the development of Dynamical Systems Theory [@ott2002chaos] (DST) and Network Theory [@newman2009networks] (NT). In fact, while DST fostered the characterization of chaotic, strongly non-linear dynamics arising in highly interacting systems, NT permitted to associate these dynamical behaviors to the geometry of connections existing among their elementary components.
However, looking for explicit connections between these two theories is a not trivial task, even if they are often applied to similar scientific objectives. Nevertheless, transport phenomena are good candidates for being studied from both NT and DST perspectives and relationships between the two descriptions should bring mutual benefits. Indeed, Lagrangian approaches in DST explicitly resolve transport dynamics by following trajectories associated to different initial conditions [@haller2000lagrangian; @ott2002chaos; @wiggins2005dynamical]. At the same time, processes like diffusion, spreading and mobility, which are strongly related to transport dynamics, are efficiently characterized on networks [@moreno2002epidemic], especially when spatially-embedded [@barthelemy2011spatial].
Importantly, transport processes play a fundamental role in several flow systems (from the smallest scales of turbulence to planetary scales) influencing many research topics such as biodiversity [@kool2013population; @bauer2014migratory], climate [@dijkstra2019networks], human physiology [@hendabadi2013topology], or engineering [@bogus2005misfire] and some of such systems can be studied from both DST and NT perspectives. Driven by DST, the last two decades have seen indeed important advances in the Lagrangian characterization of fluid flows based on geometric principles [@wiggins2005dynamical; @mancho2006lagrangian], Lyapunov exponents [@shadden2005definition] or set-oriented descriptions [@froyland2003detecting; @miron2017lagrangian; @mcadam2018surface]. Only very recently, NT approaches were applied to fluid transport [@ser2015flow], turbulence [@iacobello2018visibility], and pollutants or invasive species spreading [@seebens2016predicting; @fellini2019propagation] bringing innovative concepts and generating new results into each field. Yet, only few connections between the DST and NT paradigms has been explicitly proved.
Moreover, especially from the DST side, most of the attention has been given to the detection of transport barriers and coherent regions while a few studies focused on regions that enhance fluid exchanges across the system [@ser2015most; @koltai2018large]. Whatever the complex system studied, fulfilling such objective allows indeed the identification of hotspots which are crucial for the maintenance of the system dynamics and for its resilience.
Conversely, in the NT field in general, the identification of groups of nodes responsible for most of the network connectivity received a broad attention [@wang2003complex]. In particular, a fundamental measure of how much a single node is able to control and promote connections across network parts, hence ensuring the stability of the whole network, is the betweenness centrality [@freeman1977set; @newman2009networks]. Its use permitted to highlight bottlenecks in a variety of different systems, form air transportation networks [@guimera2004modeling] to the human brain [@hagmann2008mapping].
By bridging a gap between DST and NT, we here propose a novel quantifier of the concept of betweenness, called *Lagrangian Betweenness*, that is expressed in terms of the widely recognized Lyapunov exponents [@ott2002chaos; @shadden2005definition]. Such formulation is derived for any network in which a transport processes can be defined permitting to export the concept of betweenness, that has facilitated important discoveries in network sciences, directly into DST. After investigating the theoretical implications of the Lagrangian Betweenness introduction, we show that we can correlate it with an explicit, network-derived definition of betweenness based on most probable paths (MMPs) [@ser2015most; @aiello2018complex] proving appropriately our analogy. Finally, we use the Lagrangian Betweenness to highlight and characterize hidden circulation bottlenecks in realistic geophysical flows, focusing on oceanic transport.
![Betweenness centrality in a simple network: node $A$, despite its low degree, would have the maximum value of betweenness due its role of linking the red and blue sub-networks. Indeed, all the paths connecting both sub-networks must go trough it.[]{data-label="fig:beetwgraph"}](betw_sketch.pdf){width="\columnwidth"}
Results
=======
Background: Network Theory and Dynamical Systems
------------------------------------------------
The theoretical part of this work aims at advancing in the building of a dictionary between Dynamical Systems Theory and Network Theory as started by Ser-Giacomi et al. [@ser2015flow] within the Lagrangian Flow Networks (LFNs) framework. In particular, we focus on transport dynamics (e.g. fluid advection, information spreading, human mobility) across systems that can be modeled as a network embedded in a metric space. For such systems it is possible indeed to establish parallels and complementarities between the network geometry and the flow that steers transport processes on top of it. In some cases, such as the ocean or the atmosphere, the associated flows are directly driving mass transport, while in others they are instead defined in abstract phase spaces representing different states of the system.
Therefore, on the one hand, from the perspective of DST, we characterize transport by tracking particle trajectories in space or phase space [@ott2002chaos; @shadden2005definition; @haller2000lagrangian; @wiggins2005dynamical] and, following a fluid-dynamics perspective, such approach is often referred to as Lagrangian, in contrast to the Eulerian view where the system is characterized by quantities given at fixed locations in space. Hence, for a specific interval of time, we can associate to any initial condition $(\mathbf{x}_0,t)$ a particle following a Lagrangian trajectory across the system. Such Lagrangian perspective is the most natural when we intend to study transport processes across a dynamical system. In particular, measuring the local rate of separation of infinitesimally close initial conditions, the Finite-Time Lyapunov Exponent (FTLE) [@ott2002chaos; @shadden2005definition] is used to quantify the strength of dispersion and mixing across a given time interval (see Methods \[sec:ftle\] for details).
On the other hand, we consider networks in which nodes represent discrete sub-regions of a spatial domain of interest and the geometry of the links describes a transport process taking place on it during a precise interval of time. Such networks, in the most general case, are thus directed, weighted and temporal, and each link weight quantifies the importance of a transport event occurred between a pair of nodes starting from time $t_0$ and for a duration of $\tau$. Local measures computed on single nodes of the network (e.g. degrees, strengths, etc.) are thus expected to highlight transport patterns at different spatio-temporal scales [@ser2015flow; @newman2009networks] characterized by $\tau$ and by the spatial sub-region’s size.
In this paper we look for the corresponding formulation and signification of a fundamental NT measure, the betweenness centrality [@freeman1977set; @newman2009networks], in DST. Betweenness centrality is a widely used local metric defined as the proportion of paths passing through a node of the network. A path is characterized by a set of contiguous links, called steps, that are necessary to connect an initial node to a destination one. Depending on the kind of network studied, betweenness centrality can be calculated from the whole set of paths, the shortest, the fastest or the most probable ones [@newman2005measure; @estrada2008communicability; @holme2012temporal; @ser2015most; @aiello2018complex]. Hence, betweenness measures the extent to which a node lies on the existing paths linking other nodes. In this view, high betweenness can be associated to those nodes that behave like “bottlenecks", bridging parts of the network that otherwise would be significantly less connected (see Fig. \[fig:beetwgraph\]).
Though many kinds of systems can be studied with the paradigm proposed in the previous paragraphs, here we concentrate on transport dynamics in fluid flows and in particular, in the ocean. However, our theoretical considerations and most of our conclusions would apply to any system which can be approached through NT or DST.
Introducing Lagrangian Betweenness {#sec:lagbetwdef}
----------------------------------
In order to give a network representation of fluid flows we adopt the Lagrangian Flow Network (LFN) approach [@ser2015flow; @lindner2017spatio; @ser2017lagrangian] building weighted and directed networks that describe fluid transport (see Methods \[sec:lfn\] for details on LFNs definition and construction).
To simplify notation, without loss of generality, we now take $t_0=0$ (this corresponds to considering $[0,\tau]$ as time interval). In Methods \[sec:lfn\] the out-degree of node $i$, $ K_{i}^{O}(t_0,\tau)$, and the corresponding in-degree $K_{i}^{I}(t_0,\tau)$ are defined. Starting from Eq. (\[eq:inoutdegree\]) it is easy to show that the number of two-steps paths crossing the network node $i$ at time $t$ (with $0 \leq t \leq \tau$) is the product of the temporal in- and out-degree: $$\label{eq:betw2step}
K_{i}^{I}(0, t) \, K_{i}^{O}(t,\tau - t) .$$ Each step is associated to a precise time interval, $[0,t]$ and $[t,\tau]$ respectively, that matches the interval in which the degrees $K_{i}^{I}(0, t)$ and $K_{i}^{O}(t,\tau - t)$ are calculated. Our goal is to use Eq. (\[eq:betw2step\]) to define a quantity with the meaning of a *betweenness*. Note that the product of degrees in Eq. (\[eq:betw2step\]) differs from the classical betweenness centrality formulations from Network Theory in two aspects:
1. It counts all the paths crossing the node $i$, not only the shortest, fastest, or most probable ones.
2. It considers only the paths composed of two temporal steps that pass through node $i$ exactly at time $t$.
We argue that point 1 is not an issue for our definition, on the contrary, there is an increasing interest in considering centrality measures that take into account the information from all the paths across the network [@newman2005measure; @estrada2008communicability]. Regarding point 2, we need to overcome the limitation of forcing the path-crossing to occur exactly and only at time $t$ and the solution relies in the intrinsic temporal features of the LFNs. We can indeed change the paradigm in the way betweenness is calculated: instead of building paths of arbitrary number of steps and fixed step duration, we look at paths of just two steps but we can vary the duration in time of such steps. This allows the variable $t$, that is the time at which the two steps connect in $i$, to take all the possible values in the interval $[0;\tau]$. Hence, Eq. (\[eq:betw2step\]) can be generalized to consider all the two-step paths crossing the node $i$ at any $t \in [0;\tau]$: $$\label{eq:betw2stepintegral}
\frac{1}{\tau} \int_{0}^{\tau} K_{i}^{I}(0, t) \, K_{i}^{O}(t,\tau - t) dt.$$ Eq. (\[eq:betw2stepintegral\]) represents thus a candidate for a novel continuous-in-time definition of betweenness centrality for any network where the time duration associated to each link can be tuned. However, a similar definition of betweenness can also be derived for time-independent networks and/or networks with fixed link-duration by using $k$-neighbor degrees (see Methods \[sec:timeindepcase\] for the details).
To realize the connection with DST, we use the relation between in/out-degrees and backward/forward-in-time stretching factors (see Eq. (\[eq:stretching\])), which are exponential functions of the Finite-Time Lyapunov Exponents (FTLEs). If we take the limit of sufficiently small nodes, we can omit from Eq. (\[eq:degreestretching\]) the average over points inside the node and write the following relations: $$\begin{aligned}
\label{eq:degftlerel2}
&K_{i}^{O}(t_0,\tau) \approx \, \, \, e^{\tau\lambda(\mathbf{x}_{i},t_0,\tau)} \nonumber , \\
&K_{i}^{I}(t_0,\tau) \approx \, \, \, e^{\tau\lambda(\mathbf{x}_{i},t_0 + \tau,-\tau)} ,\end{aligned}$$ where $\lambda$ is the the standard FTLE (see Methods \[sec:ftle\] for its definition) and $\mathbf{x}_i$ denotes the center of node $i$.
The last step is to use Eq. (\[eq:degftlerel2\]) into Eq. (\[eq:betw2stepintegral\]) to link the betweenness centrality of a network with the Finite Time Lyapunov Exponents of the associated flow. In such way we finally define the *Lagrangian Betweenness* of node $i$ as: $$\label{eq:betwftle}
B^{L}_{i}(0,\tau) = \frac{1}{\tau} \int_{0}^{\tau} e^{t\lambda(\mathbf{x}_i,t ,-t)} \, e^{(\tau - t) \lambda (\mathbf{x}_i,t,\tau-t)} \,dt .$$ The integrand in Eqs. (\[eq:betwftle\]) corresponds to a product of forward and backward stretching factors associated to $\mathbf{x}_i$ at time $t$ and, consistently, $B^{L}$ is dimensionless (see Methods \[sec:ftle\]). Note that, even without taking the limit of small nodes, we can still provide an expression for $B^{L}$ but it would involve an additional spatial integration to average the stretching factors inside the node $i$. Moreover, since relative stretching measures such as $\lambda$ remain invariant under coordinate transformation, $B^{L}$ is frame-invariant too.
In order to numerically compare $B^L$ with an explicit, network-derived definition of betweenness (Eq. (\[eq:symmppbetw\]) in the next section), a discretized version of Eq. (\[eq:betwftle\]) is necessary. It can be written as: $$\begin{aligned}
\label{eq:discrint}
&B^{L}_{i}(0,\tau) = \frac{1}{\tau} \sum_{\alpha=0}^{N} e^{t_{\alpha}\lambda (\mathbf{x}_i,t_{\alpha} ,-t_{\alpha})} \, e^{(\tau - t_{\alpha}) \lambda (\mathbf{x}_i,t_{\alpha},\tau-t_{\alpha})} \,\Delta t_{\alpha} \,\, ,\end{aligned}$$ where $\alpha$ is the discrete index labeling contiguous intermediates times $t_{\alpha}$ and $N$ is the total number of time steps of durations $\Delta t_{\alpha}$ used in the discretization of the integrals.
Even though in the rest of the paper we will always use the formulation of Eq. (\[eq:betwftle\]) or (\[eq:discrint\]), making some approximations we can evaluate the integral of Eq. (\[eq:betwftle\]) to obtain approximated expressions for $B^L$. Indeed, if we assume the stretching dynamics to be purely exponential with almost constant rates $c_f$, $c_b$ we can write (see Methods \[sec:ftle\]): $$\frac{\;\,||\delta\mathbf{x}_i(t,\tau)||_{\max}}{||\delta\bar{\mathbf{x}_i}(t)||} \simeq \begin{cases} e^{c_f \tau} & \text{for $T>0$} \nonumber \\
e^{c_b \tau} & \text{for $T<0$} \end{cases} ,$$ and, consequently, we would have $\lambda(\mathbf{x}_i,t ,T) \simeq c_f$ for $T>0$ and $\lambda(\mathbf{x}_i,t ,T) \simeq c_b$ for $T<0$. Under this assumption we can evaluate Eq. (\[eq:betwftle\]) as: $$\label{eq:betwftlesolution}
B^{L}_{i}(0,\tau) \simeq \frac{e^{c_b \tau} - e^{c_f \tau}}{\tau (c_b - c_f)} .$$ If $c_f = c_b$, as appropriate, for instance, for incompressible two-dimensional flows, using the l’Hôpital rule in Eq. (\[eq:betwftlesolution\]) we find: $$B^{L}_{i}(0,\tau) \simeq e^{c\tau}\,,
\label{eq:expoB}$$ where we defined $c = c_f = c_b$. This shows that, under the stated conditions, the Lagrangian Betweenness $B^{L}_{i}(0,\tau)$ increases with the transport time $\tau$ and with the intensity of stretching occurring in node $i$.
![Schematic representations of a hyperbolic point (a) and a heteroclinic connection (b). Lines with converging/diverging arrows denote stable/unstable manifolds of the hyperbolic points (represented as black dots), respectively. Note that in (b) the manifold that realizes the connection is unstable for the left-hand side hyperbolic point and stable for the right-hand side one. Due to time-dependence, which is weak but present in geophysical flows, patterns like (a) or (b) are weakly perturbed and transformed in so-called moving hyperbolic points and connections. In particular, panels (c) and (d) sketch two examples of circulation patterns that can be often found in the ocean, respectively associated to a hyperbolic point (selfconnecting homoclinics loops) and to a heteroclinic connection. They exemplify two gyres sharing a common point or segment of their boundaries that can be associated to the elementary sketches (a) and (b) and are therefore expected to display high Lagrangian Betweenness values.[]{data-label="fig:sketchhyperhetero"}](sketch_hyper_hetero.pdf){width="\columnwidth"}
After defining $B_i^L$ in terms of FTLEs, we investigate the meaning of such relation from a Dynamical Systems Theory perspective. From Eq. (\[eq:betwftle\]) we see that nodes presenting in average high values of both backward and forward FTLEs during the interval $[0,\tau]$ are characterized by high $B^L$. Interestingly, in dynamical systems, large values of forward or backward FTLEs highlight the locations of strongly repelling or attracting material surfaces, related to stable or unstable manifolds of relevant dynamical objects, respectively [@haller2000lagrangian; @shadden2005definition]. Considering for definiteness the case of two-dimensional motion, their intersections define, at each instant of time, hyperbolic points with eventual heteroclinic and homoclinic connections among them [@guckenheimer2013nonlinear; @ott2002chaos; @wiggins2005dynamical] (see Fig. \[fig:sketchhyperhetero\]). In time-dependent flows, such objects move in space spanning hyperbolic trajectories (points) or areas (connections) making their detection more difficult. Moreover, heteroclinic and homoclinic connections become intrinsically unstable and are transformed in the so-called heteroclinic/homoclinic tangles that are a perturbed version of the original objects in which some trajectories are transported within lobes and filaments across sides of the structure [@mancho2006lagrangian; @wiggins2005dynamical]. When time-dependence becomes important such perturbation can finally disrupt the tangles completely. Nevertheless, in situations in which the typical time scale of variation of the velocity field is much slower than the time scale of particle advection, as is the case in geophysical flows, tangles remain relatively well-confined and in the rest of the paper we will simply denote these weakly time dependent structures as (moving) connections. We will also refer to the hyperbolic trajectories as (moving) hyperbolic points.
Hence, thanks to Eq. (\[eq:betwftle\]), an explicit correspondence emerges between hyperbolic points, heteroclinic/homoclinic connections and the main bottlenecks of the system. In this sense, $B^L$ provides a clear intuition of the role of these features in organizing, limiting and eventually controlling any transport process across a dynamical system. We also stress that there is a crucial distinction between hyperbolic points and connections in terms of transport: while relative velocities of trajectories in the neighborhood of hyperbolic points are close to zero, velocities along connections can be significantly large [@shadden2005definition]. In fact, bottlenecks of transport are not determined locally by the magnitude of fluxes but rather by the entire topology of the system that amalgamate around them trajectories coming from diverse origins and going to several other destinations [@newman2009networks; @ser2015most].
Lagrangian Betweeenness in a theoretical model {#sec:numericalval}
----------------------------------------------
In this section we calculate the Lagrangian Betweenness in a two-dimensional theoretical flow, called the double-gyre system (see Methods \[subsec:dgyre\] for the description of the velocity field). Such flow is defined analytically, it is time-dependent and represents a well-known benchmark to study mixing and transport in fluid dynamics [@shadden2005definition; @farazmand2012computing].
First, we compare the Lagrangian Betweenness calculated from Eq. (\[eq:discrint\]) and the betweenness explicitly calculated from most probable paths (MPPs) in LFNs (see Methods \[sec:lfn\] for details). The latter MPP-betweenness is defined as: $$\label{eq:symmppbetw}
\bar{B}^{MPP}_{i} = \sum_{l,m} g_i(l;m) + \sum_{l,m} h_i(l;m) \ ,$$ where $g_i(l;m) = 1$ or $h_i(l;m) = 1$ when the node $i$ is crossed by the forward or backward MPP between nodes $l$ and $m$ respectively, and zero otherwise.
Note that any network-based formulations of betweenness, as the one of Eq. (\[eq:symmppbetw\]), implies inherently a discrete description of the dynamics since network paths are discontinuously composed by different steps. As such, to perform properly the aforementioned comparison, it is necessary to match the temporal discretization scales of $B^{L}$ and $\bar{B}^{MPP}$ by setting the number of time steps $N$ of Eq. (\[eq:discrint\]) equal to the number of steps $M$ used for the calculation of $\bar{B}^{MPP}$. Supplementary Fig. 1 illustrates $B^{L}$ and $\bar{B}^{MPP}$ fields for $N=M=2,3,5$ in the $[0,15]$ time interval: high betweenness regions are organized in narrow lines that, for a given $N=M$, create identical spatial patterns both for $B^{L}$ and $\bar{B}^{MPP}$. While the characteristic values of $B^{L}$ do not depend on the number of steps, $\bar{B}^{MPP}$ values increase for larger $M$, becoming noisier. Possibly, such discrepancy is related to two main factors: (i) $\bar{B}^{MPP}$ uses only the MPPs while $B^{L}$ accounts for all paths, (ii) the added numerical diffusion due to the discretization of space [@froyland2013analytic], which is less important in the two-step paths used for $B^{L}$ because of the smaller number of steps. To quantitatively investigate the similarity among $B^{L}$ and $\bar{B}^{MPP}$ patterns, since the spatial resolution of $B^{L}$ is higher than the one of $\bar{B}^{MPP}$, we averaged the values of $B^{L}$ inside each network node and we compared such averages with the corresponding values of $\bar{B}^{MPP}$. The resulting Spearman correlation coefficients are: 0.90, 0.90, 0.86 for $N=M=2,3,5$ respectively, confirming the good agreement between both quantities.
![Plot of $B^{L}$ with linear (top) and normalized logarithmic (bottom) color map. Here we performed a fine numerical integration (with $N=300$) to converge to the analytical expression for $B^{L}$ of Eq. (\[eq:betwftle\]) . Higher values of Lagrangian Betweenness are found at the boundaries and across the narrow semi-circular line splitting the domain vertically. The region spanned by the moving line separating both gyres is clearly highlighted by intermediate values of betweenness.[]{data-label="fig:lagbetwlinlog"}](fig4merged.png){width="\columnwidth"}
Once we tested explicitly the relation between $B^{L}$ and $\bar{B}^{MPP}$, we focus on Lagrangian Betweenness. By shortening the time step of Eq. (\[eq:discrint\]), we retrieve the continuous definition of Eq. (\[eq:betwftle\]). In Fig. \[fig:lagbetwlinlog\] we show the $B^L$ field computed by using $N=300$ time steps plotted with a linear color map (top) and a logarithmic one (bottom). High values of Lagrangian Betweenness are found close to the boundaries of the system and in a narrow semi-circular pattern splitting the domain vertically. Intermediate values are mainly found into a rectangular band centered on the mid-line of the domain. Its width matches the region spanned by the line separating the two gyres in the flow that oscillates with an amplitude that is proportional to the parameter $\gamma$ of the flow (see Methods \[subsec:dgyre\] and [@shadden2005definition]).
![Superimposition (by difference) of forward (red) and backward (blue) FTLE fields for different intermediate times and matching the time interval $[0,15]$. The dashed black line is the $B^L$ ridge of Fig. \[fig:lagbetwlinlog\]. We clearly see the moving location of a hyperbolic trajectory, detected by the intersection of both main ridges in backward and forward FTLE, changing its position for different intermediate times. Such hyperbolic trajectory matches perfectly the $B^L$ ridge.[]{data-label="fig:ftlediff"}](fig5merged.png){width="\columnwidth"}
While high values of $B^L$ at the borders are clearly due to boundary effects, the semi-circular pattern in the middle of the domain needs further analysis to be properly understood. To this aim, in Fig. \[fig:ftlediff\] we plot snapshots of the difference between the exponents of the two factors inside the integral of Eq. (\[eq:betwftle\]), i.e. $\lambda(\mathbf{x}_i,t,\tau-t) -\lambda(\mathbf{x}_i,t ,-t)$, for $t=5,6,7,8,9,10$ and keeping $\tau=15$. Hence, red and blue regions present high values of forward and backward FTLE respectively and can be ultimately related to stable and unstable manifolds of the system. As illustrated in Fig. \[fig:ftlediff\], the crossing point of backward- and forward-in-time FTLE ridges identifies the position for different values of $t$ of a hyperbolic point (like the one of Fig. \[fig:sketchhyperhetero\] (a)), drawing thus a hyperbolic trajectory.
The narrow semi-circular line of very high Lagrangian Betweenness is highlighting thus a thin region of the system that, averaging over the time interval \[0,15\], results to be strongly hyperbolic. Interpretation of the features in Supplementary Fig. 1 in terms of the high-stretching lines of Fig. \[fig:ftlediff\] is more difficult because the former is an average over products of stretching factors of several two-step paths with different intermediate times. But still we can recognize many features in Supplementary Fig. 1 from the location of the high-stretching lines in Fig. \[fig:ftlediff\] at appropriate times. For example one can identify some of the lines in Supplementary Fig. 1b, computed as an average of quantities at times 0, 5, 10 and 15 ($N=3$), with lines in Figs. \[fig:ftlediff\]a and \[fig:ftlediff\]f, computed at times 5 and 10. We can also relate Supplementary Fig. 1a with an intermediate situation (at time 7.5) between Figs. \[fig:ftlediff\]c and \[fig:ftlediff\]d (at times 7 and 8). All these comparisons finally confirm our previous statement that strongly-stretching hyperbolic points or regions can be associated to high values of betweenness in weakly time-dependent flows.
Lagrangian Betweenness in the real ocean {#sec:oceanapp}
----------------------------------------
To test our framework in a realistic geophysical setting, we exploit state-of-the-art gridded velocity fields of the ocean as modeled by a high-resolution hydrodynamic model and as measured by satellite altimetry (see Methods \[sec:data\]). By doing so, we illustrate numerically our approach for velocity fields widely exploited in oceanography while considering different effective resolutions: altimetry originates from remote-sensing observations but resolves only the upper mesoscale [@amores2018up]; the high-resolution model is eddy-resolving, spanning the full mesoscale and possibly the upper submesoscale [@poulain2013transit]. Note that while we focus on a few examples of specific regions in the following, our analyses and conclusions would also hold to other similar structures that are found elsewhere in the surface global ocean.
The paradigmatic areas on which we focus our attention are the Adriatic Sea and the Kerguelen region. These two regions are currently the subject of intense research efforts to characterize ocean transport and dispersion, a fundamental driver of ecosystems functioning and a key pre-requisite for sound marine spatial planning.
### The Adriatic Sea
The Adriatic Sea is a relatively shallow sub-basin of the Mediterranean Sea whose surface circulation is mainly constrained by topographic features and seasonally-varying atmospheric forcing processes, dominated by two large wind-driven gyres [@poulain2001adriatic; @carlson2016observed]. On top of this simplified view, significant levels of sub/mesoscale variability super-impose their dynamical signatures, hampering our ability to predict how oceanic tracers would mix around. This complex surface circulation as well as the diversity of marine activities developing there have stimulated strong research interests in the last years [@bray2017assessing; @legrand2019multidisciplinary]. Understanding how any particulate or dissolved tracer such as pollutants, fish larvae or debris may be mixed and redistributed across this small -yet dynamical- sea can have critical consequences for conservation stakeholders, environmental managers and politicians.
![On the top: $B^L$ field calculated for the 1st of December 2013 and with an integration time $\tau$ of 15 days from model data. The black arrows sketch the circulation pattern: two regional cyclonic gyres sharing a contact point exactly where the “Pelagosa peak" of $B^L$ is located. On the bottom, four panels: Evolution of particle patches seeded in the interior of the gyres for different intermediate times (see also the Supplementary Video 1 for the full animation) from model data. The white patch occupies initially the Middle Adriatic Gyre and the red one the South Adriatic Gyre while the contour of the $B^L$ peak is denoted by a solid grey line. Note that exchanged water between the two gyres always flows in a small region around the $B^L$ peak.[]{data-label="fig:adriatic"}](fig6+8.png){width="\columnwidth"}
We plot in Fig. \[fig:adriatic\] (top) the $B^L$ field computed via Eq. (\[eq:discrint\]) from the high-resolution simulated velocity fields in the Adriatic Sea, calculated for the 1st of December 2013 and with an integration time $\tau$ of 15 days (see Methods \[sec:data\]). It shows a small circular area with extremely large values of $B^L$ (almost one order of magnitude greater than the surrounding) located south-east of the Pelagosa Islands.
To better understand the origins of such strong pattern, we focus on the main circulation by computing (see Supplementary Fig. 2) the average sea surface height (SSH) on the same period from the same model data. Under the geostrophic approximation, isolines of the average SSH give the paths of the main currents. Interestingly, we note that the region is characterized by two cyclonic gyres that present a contact point in the same approximate location of the “Pelagosa peak" of $B^L$, reminding the hyperbolic geometry [@rypina2009transport] of Fig.\[fig:sketchhyperhetero\] (c).
In order to quantify explicitly the influence of the Pelagosa $B^L$ peak on the surrounding circulation we fill the interior of the two gyres with tagged Lagrangian particles and we simulate their trajectories in between the 1st and the 15th of December. To draw the boundary between the gyres interior and the exterior, we set a SSH threshold of -20 cm and we seed particles only for SSH values smaller than the threshold, associated thus to the core of both gyres (see Supplementary Fig. 2). In Fig. \[fig:adriatic\] (bottom, four panels) we also show the evolution of the aforementioned particle patches at different intermediate times (see also the Supplementary Video 1 for the full animation). The white patch is associated to the Middle Adriatic Gyre and the red one to the South Adriatic Gyre while the contour of the peak is denoted by a solid grey line. The patches evolution confirms that the Pelagosa peak is associated to the presence of a surprisingly stable and steady hyperbolic point, crossed by a stable manifold in the east/west direction and by a unstable one in the north-west/south-east direction.
Moreover, looking at the water origins in the interior of the Pelagosa peak, we find constantly both red and white particles (representing water parcels) during the entire period. Such high Lagrangian Betweenness area corresponds indeed to the only place in the basin where it is possible to encounter “white water” advected toward the northern gyre and, at the same time, “red water” transported to the southern gyre (Supplementary Video 1). This means that, similarly to the yellow node of Fig. \[fig:beetwgraph\], the Pelagosa peak correctly exemplifies what an oceanic bottleneck represents: a tiny portion of the ocean surface that permits the exchange and subsequent mixing of two water masses which occupied two, otherwise disconnected, large oceanic regions.
To prove the robustness of this structure to various velocity field of different origins and resolutions, we repeat the $B^L$ calculation varying the integration time $\tau$ and using another dataset. In Supplementary Fig. 3 we show four snapshots of $B^L$ fields for the 1st of December 2013, for $\tau$ = 15, 30 days. These were computed from the high resolution model velocity field already used (panel (a) and (b)) and a regional altimetry-derived velocity field (panel (c) and (d)), see Methods \[sec:data\] for details on these two products used. We find a remarkable regularity in the position of the peak and a consistent increase of its absolute value with $\tau$, both for model and satellite observations, as the approximation of Eq. (\[eq:expoB\]) would suggest.
Finally, motivated by these results, in Supplementary Fig. 4 we show a temporal average of the $B^L$ field from the regional altimetry-derived velocity field across the years 2002-2013, starting each calculation the 1st of December and using an integration time $\tau$ of 30 days. Again, the Pelagosa peak is clearly distinguishable confirming a striking regularity of this pattern also from simulations based on satellite observations and across several years.
### The Kerguelen region
To validate the existence of such high betweenness patterns in other dynamical regimes we now focus on the Kerguelen region. Located in the Indian sector of the Southern Ocean, Kerguelen is characterized by intricate circulation patterns due to the interaction of the energetic Antarctic Circumpolar Current (driven by large-scale forcing) with a complex topography [@park2014polar]. This region constitutes also one of the ten largest marine protected areas in the world and understanding its circulation properties is a pivotal step to characterize all the marine biological processes inside it [@koubbi2016ecoregionalisation1].
Using observed altimetry fields to compute trajectories we show in Fig. \[fig:kerg\] (top) the $B^L$ field in the region north-east of the Kerguelen Islands for the 1st of December 2007 with integration time $\tau=20$. We can clearly identify a marked high-betweenness region which is shaped as an elongated strip centered around 47.7S 75.0E with an approximate length of 200 km and a width of 25 km (we define it as the locations having $B^L > 100$). Its location likely coincides with the area where the Polar Front meets the fast eastward flowing Antarctic Circumpolar Current delimited by the Subantarctic Front [@park2014polar].
In Fig. \[fig:kerg\] (bottom, four panels) we also plot the evolution of two colored patches of water that flow across the high betweenness strip at different intermediate times between the 1st and the 20th of December 2007 (see also the Supplementary Video 2 for the full animation). The two patches in the figure denote indeed all the surface water parcels that, in the time window considered here, touches a point with a betweenness value equal or greater than the threshold used to delimit the strip.
To distinguish distinct water origins and to delineate both patches, we apply a threshold on the backward-in-time drifts computed for the 7 previous days for all Lagrangian particles (see the bimodal drift distribution in Supplementary Fig. 5). Specifically, the patch presenting a drift larger than 200 km is associated directly to the Circumpolar Current (red color). Conversely, the patch of particles characterized by drifts smaller than 200 km is associated to water coming from south-east of the Polar Front (white color). This choice is supported by the fact that the Polar Front is much more stagnating and meandering that the Circumpolar Current in the Kerguelen region and this is also reflected in the strong bimodality of the drift distribution (see Supplementary Fig. 5).
![ On the top: $B^L$ field from altimetry data in the region north-east of the Kerguelen Islands for the 1st of December 2007 with integration time $\tau=20$. A marked high betweenness strip is located in the area where the Polar Front meets the Antarctic Circumpolar Current. On the bottom, four panels: Evolution of the Circumpolar Current patch (red) and the Polar Front patch (white) flowing across the high betweenness strip (delimited by the solid grey contour) at different intermediate times from altimetry data (see also the Supplementary Video 2 for the full animation). Waters from different current systems are funneled through the 20 km wide strip being partially mixed; they separate and disperse shortly after leaving this high-betweeness strip.[]{data-label="fig:kerg"}](fig_kerg_new.png){width="\columnwidth"}
The “hourglass” shape formed by these two Lagrangian patches (Fig. \[fig:kerg\], panels (b) and (c)) indicates an underlying mean circulation resembling the one sketched in panels (b) and (d) of Fig. \[fig:sketchhyperhetero\], suggesting thus the presence of a heteroclinic-like structure. Consistently with this interpretation, three fundamental features characterize the particle’s evolution close to the high $B^L$ area: (i) a strong convergence towards the strip of the particles initialized in the west, equivalent to a backward-in-time dispersion for particles entering in the strip from the northwestern edge, (ii) a similarly strong forward-in-time dispersion for particle exiting from the southeastern edge and (iii) a rapid and coherent southeasterly flow along the entire strip.
Consequently, the tracer separation distance in the transversal direction of the main eastward flow is of the order of 250 km prior and after being funneled into a mere 25 km wide strip (Supplementary Video 2). Similarly to the Adriatic Sea example, the high-betweenness strip represents thus a tiny oceanic region that sees different water masses converging together to rapidly spread away after being partially mixed. Such dense congestion of trajectories illustrates perfectly the concept of bottleneck sketched in Fig. \[fig:beetwgraph\]: a narrow passage in the ocean surface that sees water parcels coming from disparate origins and going to many different destinations.
Finally, to investigate the robustness of the pattern to changes of the integration time, in Supplementary Fig. 6 we show the Lagrangian Betweenness field in the same region but for different values of $\tau$. We find, both in the $\tau=15$ and $\tau=25$ field, the high $B^L$ area in the same location and with small variations in intensity (consistently with the use of larger $\tau$).
Discussion
==========
From Network Theory to Dynamical Systems and back
-------------------------------------------------
Even if NT and DST are used to approach a broad variety of common subjects in the complex systems field, there exists only few theoretical connections between them.
In the present work we contribute to bridge this gap by introducing the concept of betweenness [@freeman1977set] in the context of DST, allowing the quantitative identification of bottlenecks of transport and unveiling their role in the connectivity of dynamical systems. On the other hand, from the NT side, Eq. (\[eq:betwftle\]) provides a novel interpretation of betweenness in complex networks connecting it to hyperbolic, homoclinic and heteroclinic dynamics [@ott2002chaos]. Hence, sensitivity to initial conditions and chaotic behavior can be related to high-betweenness nodes of the network representing a given dynamical system.
Moreover, the continuous-in-time definition that we propose permits to obtain a betweenness measure directly from degrees without passing through the definition of paths. This poses the question if betweenness centrality is a property that can be simply ascribed to degree’s topology. Indeed, also in networks where the link duration is fixed, there is still the possibility of using a formulation similar to Eq. (\[eq:betw2stepintegral\]) in which the degrees are replaced by $k$-neighbor degrees (see Methods \[sec:timeindepcase\]).
Our definition of betweenness accounts for all the paths crossing a given node, not just a subset of them (e.g. most probable, fastest, shortest ones) [@newman2005measure; @estrada2008communicability; @holme2012temporal; @ser2015most]. Indeed, such approach seems to be the most natural when there is no possibility for the transported quantity to actively chose the most convenient pathways [@newman2005measure; @estrada2008communicability].
All in all, if betweenness has proved to be a fundamental measure to assess locally the vulnerability and controllability of complex networks, these properties can be also linked now with DST concepts of predictability and chaos associated to the underlying system dynamics.
Bottlenecks in fluid flows and beyond
-------------------------------------
From a geophysical perspective, this paper gives a new perspective on the role, in terms of fluid transport, of hyperbolic points and heteroclinic connections.
Even though a wide variety of structures, including some resembling those revealed by Lagrangian Betweenness, have been already detected in laboratory and in geophysical flows [@mancho2006lagrangian; @rypina2009transport], they have not been explicitly associated to transport “bottlenecks”. Here we show that high $B^L$ regions see fluid masses coming from several origins and going to many other destinations, with the difference that, for hyperbolic points (Adriatic Sea), the fluid exchange is typically modest, while for heteroclinic connections (Kerguelen), it is larger.
Surprisingly, in our oceanic examples, we also find that such points or connections are sometimes much more stable and persistent than expected, despite the time variability of realistic flows that, evidently, is not sufficiently strong to completely disrupt them. The robustness of such structures is confirmed by a series of considerations: (i) they are detectable both from high-resolution models and SSH measurements, (ii) they result to be robust for different $\tau$’s and different resolutions of the initialization grid, (iii) they can be found across several years close to the same position, possibly constrained by the bathymetry.
In the ocean, high Lagrangian Betweenness regions represent the optimal compromise between “source” and “sinks” in terms on number of water origins and destinations, and could be associated thus to ecological hot-spots promoting marine ecosystems biodiversity [@huston1979general; @dubois2016linking]. Such marked heterogeneity of water histories could indeed favor immigration, species turnover and aggregation across the entire trophic chain, from plankton to top predators [@angel1993biodiversity].
Another potential application would be to use these bottlenecks to deploy primarily pollution monitoring program or cleaning solutions. Moreover, the effectiveness of the betweenness measures in assessing systems sensitivity and vulnerability could also contribute to the design of optimized observing systems [@baehr2008optimization]. High betweenness hot-spots could indeed provide early-warning signals of how geophysical flows will be affected by multiple stressors such as heat-waves and pollutants spreading. This suggests that these regions could represent good candidates for the application of protection strategies and help substantially marine spatial planning.
Beyond the applications to fluid flows at different scales, the concept of bottleneck has clear applications to any type of transportation network (from urban mobility networks or internet routing, to transport in cells or communication in neurons). The connection of this network concept with well-studied behaviors in dynamical systems such as divergence of trajectories or controllability [@boccaletti2000control] opens novel and promising avenues of research.
Material and Methods
====================
Finite-Time Lyapunov Exponents {#sec:ftle}
------------------------------
In Dynamical System Theory, a quantity to characterize locally dispersion and mixing is the Finite-Time Lyapunov Exponent (FTLE) [@ott2002chaos]. It is defined as: $$\label{eq:ftledef}
\lambda(\mathbf{x}_0,t;T)=\frac{1}{2|T|}\log |\Lambda_{max}| ,$$ with $\Lambda_{max}$ the largest eigenvalue of the right Cauchy-Green strain tensor [@shadden2005definition]. Eq. (\[eq:ftledef\]) can be expressed also as: $$\label{eq:ftledef2}
\lambda(\mathbf{x}_0,t;T)=\frac{1}{|T|}\log \bigg( \frac{\;||\delta\mathbf{x}_0(t,T)||_{\max}}{||\delta\bar{\mathbf{x}_0}(t)||} \bigg) ,$$ where $||\delta\bar{\mathbf{x}_0}(t)||$ is the initial separation between infinitesimally-close initial conditions located around $\mathbf{x}_0$ at time $t$ and aligned with the eigenvector of $\Lambda_{max}$; while $||\delta\mathbf{x}_0(t,T)||$ is the final separation of those particles at time $t+T$, being the maximum possible separation resulting from all the directions of particle separation $\delta\mathbf{x}_0(t)$. FTLEs characterize thus the maximum logarithmic separation rate, over an interval of time $T$, around $\mathbf{x}_0$; for $T>0$ and $T<0$ we obtain the forward and backward in time FTLE respectively. Hence, an initial sphere or circle of diameter $d$ located in $\mathbf{x}_0$ at time $t$ would be elongated at time $t+T$ by a stretching factor $s(\mathbf{x}_0,t;T)$ defined as: $$\label{eq:stretching}
s(\mathbf{x}_0,t;T)=e^{\tau \lambda(\mathbf{x}_0,t;T)} .$$ Practically, for a given system, FTLEs are obtained from Lagrangian trajectories of a set of initial conditions during a fixed time interval. Such trajectories are usually reconstructed using modeled or observed gridded velocity fields, or real trajectories from Lagrangian drifters. Then, from the initial and final distances between different initial conditions, the local rate of separation is calculated.
Lagrangian Flow Networks {#sec:lfn}
------------------------
Lagrangian Flow Networks (LFNs) construction is based on the discretization of a metric domain $D$ in a fine partition in boxes, $\{\mathcal{B}_i, i=1,2,...,L\}$, characterized by a linear size $\chi$. This set of boxes are identified uniquely with the nodes of the network. Then, to each pair of nodes $i$ and $j$ a directed link with a weight $\mathbf{A}(t_0,\tau)_{ij}$ is assigned and it corresponds to the amount of volume $m$ present in $\mathcal{B}_i$ at time $t_0$ that is found in $\mathcal{B}_j$ after a time $\tau$: $$\mathbf{A}(t_0,\tau)_{ij} = m\left(\mathcal{B}_i \cap
\Phi_{t_0+\tau}^{-\tau}(\mathcal{B}_j)\right) \ , \label{eq:PF}$$ where $\Phi_{t_0}^{\tau}$ is the time evolution operator from time $t_0$ to $t_0+\tau$. Numerical estimations of $\mathbf{A}(t_0,\tau)$ can be done by seeding in $\mathcal{B}_i$ a large number of initial conditions, i.e. Lagrangian particles, following their trajectories for a time $\tau$, and counting how many ended into each $\mathcal{B}_j$. We define the network out-degree and in-degree of node $i$ respectively, as: $$\begin{aligned}
K^{O}_{i}(t_0,\tau) &= \sum_j \begin{cases} 1 & \text{if $\mathbf{A}(t_0,\tau)_{ij} > 0$} \nonumber \\
0 & \text{otherwise} \end{cases}, \\
K^{I}_{i}(t_0,\tau) &= \sum_j \begin{cases} 1 & \text{if $\mathbf{A}(t_0,\tau)_{ji} > 0$} \\
0 & \text{otherwise} \end{cases} . \label{eq:inoutdegree}\end{aligned}$$ Similarly we also define the out-strength and in-strength of node $i$ as: $$\begin{aligned}
S^{O}_{i}(t_0,\tau) &= \sum_j \mathbf{A}(t_0,\tau)_{ij}, \nonumber \\
S^{I}_{i}(t_0,\tau) &= \sum_j \mathbf{A}(t_0,\tau)_{ji}. \label{eq:inoutstreng}\end{aligned}$$ Using Eq. (\[eq:inoutstreng\]), two normalizations for the matrix $\mathbf{A}(t_0,\tau)_{ij}$ can be defined: $$\begin{aligned}
\mathbf{P}^{f}(t_0,\tau)_{ij} &= \frac{\mathbf{A}(t_0,\tau)_{ij}}{S^{O}_{i}(t_0,\tau)}, \nonumber \\
\mathbf{P}^{b}(t_0,\tau)_{ij} &= \frac{\mathbf{A}(t_0,\tau)_{ij}}{S^{I}_{j}(t_0,\tau)}. \label{eq:inoutnormmat}\end{aligned}$$ Since $\mathbf{A}(t_0,\tau)_{ij}\geq0$, $\mathbf{P}^{f}(t_0,\tau)_{ij}$ is a row-stochastic matrix while $\mathbf{P}^{b}(t_0,\tau)_{ij}$ is column-stochastic. Hence, $\mathbf{P}^{f}(t_0,\tau)_{ij}$ can be interpreted as the probability for a Lagrangian particle to reach the box $\mathcal{B}_j$ at time $t_0+\tau$, under the condition that it started from a uniformly random position within box $\mathcal{B}_i$ at time $t_0$. Analogously, $\mathbf{P}^{b}(t_0,\tau)_{ij}$ corresponds to the probability for a particle of having started from $\mathcal{B}_i$ at time $t_0$, under the condition of being found in a random position within $\mathcal{B}_j$ at time $t_0+\tau$. Thus, $\mathbf{P}^{f}(t_0,\tau)_{ij}$ is also the forward-in-time probability for a random walker to jump from node $i$ at $t_0$ to $j$ in a time $\tau$ while $\mathbf{P}^{b}(t_0,\tau)_{ij}$ is the backward-in-time probability to go from $j$ to $i$.
An interesting relationship between the degrees defined in this Methods and the FTLEs defined in Methods \[sec:ftle\] was found in [@ser2015flow]. The degree of a node turns out to be given, to a good approximation, to an average of the stretching factor (\[eq:stretching\]) over the initial conditions contained in the node: $$\begin{aligned}
K^{O}_{i}(t_0,\tau) &\approx \frac{1}{m(\mathcal{B}_i)} \int_{\mathcal{B}_i} d\mathbf{x}_0 e^{\tau \lambda(\mathbf{x}_0,t_0,\tau)} , \nonumber \\
K^{I}_{i}(t_0,\tau) &\approx \frac{1}{m(\mathcal{B}_i)} \int_{\mathcal{B}_i} d\mathbf{x}_0 e^{\tau \lambda(\mathbf{x}_0,t_0+\tau,-\tau)} .
\label{eq:degreestretching}\end{aligned}$$ These relationships are used, in the limit of sufficiently small nodes, to derive Eq. (\[eq:degftlerel2\]).
Moving towards a multi-step description of the dynamics, we denote a generic path $\mu$ of $M$-steps between nodes $i$ and $j$ as a $(M+1)$-uplet $\mu\equiv\bigl\{i, k_{1},\, ...\, , k_{M-1}, j \bigl\}$ providing a sequence of nodes crossed to reach $j$ at time $t_M$ from $i$ at time $t_0$. Assuming a Markovian dynamics, the forward-in-time probability for a random walker to take the path $\mu$ under the condition of starting at $i$ is [@ser2015most; @ser2015dominant; @aiello2018complex]: $$\label{eq:mostprobpathprobfor}
(p^{f}_{ij})_{\mu} =
\mathbf{P}^{f(1)}_{ik_{1}} \Bigg[
\prod_{l=2}^{M-1} \mathbf{P}^{f(l)}_{k_{l-1}k_{l}}
\Bigg] \mathbf{P}^{f(M)}_{k_{M-1}j} \ ,$$ where $\mathbf{P}^{f(l)}$ corresponds to $\mathbf{P}^{f}(t_{l-1}, \Delta t)$ with $t_l = (t_0 + l\Delta t)$ and $l=\{1, ... ,M\}$. Hence, $\Delta t$ is the duration of a single step and, without loss of generality, is assumed to be constant in the whole path. Consequently, the backward-in-time probability to take the path $\mu$ under the condition of starting at $j$ is: $$\label{eq:mostprobpathprobback}
(p^{b}_{ij})_{\mu} =
\mathbf{P}^{b(1)}_{ik_{1}} \Bigg[
\prod_{l=2}^{M-1} \mathbf{P}^{b(l)}_{k_{l-1}k_{l}}
\Bigg] \mathbf{P}^{b(M)}_{k_{M-1}j} \ ,$$ where $\mathbf{P}^{b(l)}$ corresponds to $\mathbf{P}^{b}(t_{l-1}, \Delta t)$ with $t_l = (t_0 + l\Delta t)$ and $l=\{1, ... ,M\}$. Maximizing Eq. (\[eq:mostprobpathprobfor\]) and Eq. (\[eq:mostprobpathprobback\]) over the intermediate nodes, we are able to find the *most probable path* (MPP) connecting each pair of nodes $i,j$ forward and backward in time respectively. With the whole set of MPPs at hand, we can now provide a probability-based definition of betweenness centrality. We define the forward- and backward-in-time *MPP-betweenness* at $M$-steps as: $$\begin{aligned}
B^{fMPP}_{i} &= \sum_{l,m} g_i(l;m), \\
B^{bMPP}_{i} &= \sum_{l,m} h_i(l;m), \label{eq:mppbetwdef}\end{aligned}$$ where $g_i(l;m) = 1$ or $h_i(l;m) = 1$ when $i$ is crossed by the forward or backward MPP between $l$ and $m$ respectively, and zero otherwise. Then, we can finally introduce the symmetrized-in-time MPP-betweenness of Eq. (\[eq:symmppbetw\]) as an average of $B^{fMPP}_{i}$ and $B^{bMPP}_{i}$: $$\bar{B}^{MPP}_{i} = \sum_{l,m} g_i(l;m) + \sum_{l,m} h_i(l;m) \ .$$
The time-independent case {#sec:timeindepcase}
-------------------------
For the case of time-independent networks we lose the temporal dimension and the degrees will not depend anymore on time. However, we still have the information of the number of steps needed to build a given path across the network. In this sense, long-range connections will be realized across a larger number of steps than the shorter ones. We denote the weighted, time-independent adjacency matrix of a given time-independent network as $\mathbf{A}$. We also define the correspondent unweighted, time-independent adjacency matrix as: $$\mathbf{U}_{ij} = \begin{cases} 1 & \text{if $\mathbf{A}_{ji} > 0$} \\
0 & \text{otherwise} \end{cases} . \label{eq:unweigadjm}$$ We introduce the time-independent $k$-neighbor in- and out-degrees as: $$\begin{aligned}
\label{eq:kneighdegrees}
&\mathcal{K}_{i}^{O^{(k)}} = \sum_{j_{1}, j_{2},...,j_{k}} \mathbf{U}_{ij_{1}} \mathbf{U}_{j_{1}j_{2}} \, ... \, \mathbf{U}_{j_{k-1}j_{k}} \, , \nonumber \\
&\mathcal{K}_{i}^{I^{(k)}} = \sum_{j_{1}, j_{2},...,j_{k}} \mathbf{U}_{j_{1}i} \mathbf{U}_{j_{2}j_{1}} \, ... \, \mathbf{U}_{j_{k}j_{k-1}} \, ,\end{aligned}$$ where we set $\mathcal{K}_{i}^{O^{(0)}} = \mathcal{K}_{i}^{I^{(0)}} = 1 $.
Following the approach presented in Section \[sec:lagbetwdef\] and using Eq. (\[eq:kneighdegrees\]) we finally find an analogous expression to Eq. (\[eq:betw2stepintegral\]) for the time-independent case: $$\label{eq:timeindepbetween}
\frac{1}{k} \sum_{l=0}^{k} \mathcal{K}_{i}^{I^{(l)}} \mathcal{K}_{i}^{O^{(k-l)}} \,\, .$$
Theoretical and realistic flow fields for numerical evaluations {#sec:data}
---------------------------------------------------------------
In this Section we describe the theoretical flow model used to test Eq. (\[eq:betwftle\]) and the realistic oceanic velocity fields used to compute $B^L$ in two different geophysical contexts.
### The double-gyre flow system {#subsec:dgyre}
The double gyre [@shadden2005definition; @farazmand2012computing] is a two-dimensional time-periodic flow defined in the rectangular region of the plane $\mathbf{x}=(x,y) \in
[0;2]\times[0;1]$. It is characterized by the stream function: $$\psi(x,y,t)= A \sin(\pi f(x,t))\sin(\pi y) \ ,
\label{dg-stream1}$$ with: $$\begin{aligned}
&f(x,t)= a(t)x^2+b(t)x , \\
&a(t) = \gamma \sin(\omega t) \ , \\
&b(t) = 1-2\gamma \sin(\omega t) \ . \label{dg-stream2}\end{aligned}$$ From these expressions, the two components of the velocity are: $$\begin{aligned}
&\dot x = -\frac{\partial\psi}{\partial y}=-\pi A \sin(\pi f(x,t))\cos(\pi y) ,\\
&\dot y = \frac{\partial\psi}{\partial x}=\pi A \cos(\pi f(x,t)) \sin(\pi y) \frac{\partial f(x,t)}{\partial x} \ .\end{aligned}$$ Depending on the value taken by the parameter $\gamma$, this theoretical flow field displays different dynamical behaviors, yet sufficiently simple to carefully analyze the underlying structures. For $\gamma=0$, the flow is steady and fluid particles follow very simple trajectories, rotating along closed streamlines, clockwise in the left-hand side of the rectangular domain, and counterclockwise in its right-hand side. The central streamline $x=1$, a heteroclinic connection between the hyperbolic point at $(1,1)$ and the one at $(1,0)$, acts as a separatrix between the two regions. However, when $\gamma > 0$, more complex behavior, including chaotic trajectories, arises. The periodic perturbation breaks the separatrix, so that some exchange of fluid is possible between the left and the right sub-domains. As parameters, following [@shadden2005definition], we chose $A=0.1$, $\omega=2\pi/10$, $\gamma=0.25$ and we focus our analysis on the time interval $[0,15]$ setting $t_0=0$ and $\tau = 15$. For the calculation of $B^{L}$ we fill the whole domain with $78804$ Lagrangian particles regularly spaced and we reconstruct each trajectory using a Runge-Kutta 4th-order integration algorithm with temporal step of $0.05$. For the calculation of $\bar{B}^{MPP}$ we use instead $2001000$ particles uniformly seeded in $20000$ square boxes representing network nodes and the same Runge-Kutta 4th-order integration scheme.
### Ocean current velocities: Adriatic Sea
First, we use the horizontal near-surface currents simulated by a data-assimilative operational ocean model at (1/16)$\degree$ of resolution over the Mediterranean basin, provided by E.U. Copernicus Marine Environment Service Information website (http://marine.copernicus.eu). Further information on this model can be found in [@simoncelli2014mediterranean]. Among the 72 horizontal layers resolved by the model, we focus on surface ocean dynamics by seeding Lagrangian particles on a regular grid of (1/20)$\degree$ of resolution over the 15 m depth layer and considering only the horizontal velocity. Particles were advected with a 4th-order Runge-Kutta scheme with a time step of 3 hours to generate trajectories that were then used to compute the Finite-Time Lyapunov Exponents and $B^L$, which was calculated according to Eq. (\[eq:discrint\]), by considering $\Delta t_{\alpha} = 1$ day.
Secondly, we exploit a gridded velocity field at (1/8)$\degree$ spatial resolution representing surface geostrophic currents computed from remote-sensed Sea Surface Height (SSH). Altimetric products (SSH, SLA Sea Level Anomaly, and the twenty-year mean geoid) come from the regional SSALTO/DUACS gridded multi-mission altimeter product, processed by SSALTO/DUACS and distributed by Aviso+ (https://www.aviso.altimetry.fr). This horizontal velocity field was used to compute $B^L$ as explained before, while seeding particles over a regular grid of resolution of (1/40)$\degree$.
### Ocean current velocities: Kerguelen region
The horizontal velocity fields used for the Kerguelen region come from the Kerguelen altimetry regional product. This product, specifically calibrated for the region, was also processed by SSALTO/DUACS and distributed by Aviso+ (https://www.aviso.altimetry.fr). The velocity field possesses a (1/8)$\degree$ spatial resolution and were used to compute $B^L$ with the same scheme illustrated for the Mediterranean Sea, with a resolution of (1/40)$\degree$.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: |
We have observed the decay $B^+\rightarrow \bar D^0 K^+$, using 3.3 million $B\bar B$ pairs collected with the CLEO II detector at the Cornell Electron Storage Ring. We find the ratio of branching fractions $R\equiv{\cal B}(B^+\rightarrow \bar D^0 K^+) / {\cal
B}(B^+\rightarrow \bar D^0 \pi^+) = 0.055 \pm 0.014 \pm 0.005$.
author:
- CLEO Collaboration
title: 'First Observation of the Cabibbo Suppressed Decay $B^+\rightarrow \bar D^0 K^+$'
---
6.5 in 9.0 in -0.50in 0.00in 0.00in
M. Athanas,$^{1}$ P. Avery,$^{1}$ C. D. Jones,$^{1}$ M. Lohner,$^{1}$ S. Patton,$^{1}$ C. Prescott,$^{1}$ J. Yelton,$^{1}$ J. Zheng,$^{1}$ G. Brandenburg,$^{2}$ R. A. Briere,$^{2}$ A. Ershov,$^{2}$ Y. S. Gao,$^{2}$ D. Y.-J. Kim,$^{2}$ R. Wilson,$^{2}$ H. Yamamoto,$^{2}$ T. E. Browder,$^{3}$ Y. Li,$^{3}$ J. L. Rodriguez,$^{3}$ T. Bergfeld,$^{4}$ B. I. Eisenstein,$^{4}$ J. Ernst,$^{4}$ G. E. Gladding,$^{4}$ G. D. Gollin,$^{4}$ R. M. Hans,$^{4}$ E. Johnson,$^{4}$ I. Karliner,$^{4}$ M. A. Marsh,$^{4}$ M. Palmer,$^{4}$ M. Selen,$^{4}$ J. J. Thaler,$^{4}$ K. W. Edwards,$^{5}$ A. Bellerive,$^{6}$ R. Janicek,$^{6}$ D. B. MacFarlane,$^{6}$ P. M. Patel,$^{6}$ A. J. Sadoff,$^{7}$ R. Ammar,$^{8}$ P. Baringer,$^{8}$ A. Bean,$^{8}$ D. Besson,$^{8}$ D. Coppage,$^{8}$ C. Darling,$^{8}$ R. Davis,$^{8}$ S. Kotov,$^{8}$ I. Kravchenko,$^{8}$ N. Kwak,$^{8}$ L. Zhou,$^{8}$ S. Anderson,$^{9}$ Y. Kubota,$^{9}$ S. J. Lee,$^{9}$ J. J. O’Neill,$^{9}$ R. Poling,$^{9}$ T. Riehle,$^{9}$ A. Smith,$^{9}$ M. S. Alam,$^{10}$ S. B. Athar,$^{10}$ Z. Ling,$^{10}$ A. H. Mahmood,$^{10}$ S. Timm,$^{10}$ F. Wappler,$^{10}$ A. Anastassov,$^{11}$ J. E. Duboscq,$^{11}$ D. Fujino,$^{11,}$[^1] K. K. Gan,$^{11}$ T. Hart,$^{11}$ K. Honscheid,$^{11}$ H. Kagan,$^{11}$ R. Kass,$^{11}$ J. Lee,$^{11}$ M. B. Spencer,$^{11}$ M. Sung,$^{11}$ A. Undrus,$^{11,}$[^2] A. Wolf,$^{11}$ M. M. Zoeller,$^{11}$ B. Nemati,$^{12}$ S. J. Richichi,$^{12}$ W. R. Ross,$^{12}$ H. Severini,$^{12}$ P. Skubic,$^{12}$ M. Bishai,$^{13}$ J. Fast,$^{13}$ J. W. Hinson,$^{13}$ N. Menon,$^{13}$ D. H. Miller,$^{13}$ E. I. Shibata,$^{13}$ I. P. J. Shipsey,$^{13}$ M. Yurko,$^{13}$ S. Glenn,$^{14}$ Y. Kwon,$^{14,}$[^3] A.L. Lyon,$^{14}$ S. Roberts,$^{14}$ E. H. Thorndike,$^{14}$ C. P. Jessop,$^{15}$ K. Lingel,$^{15}$ H. Marsiske,$^{15}$ M. L. Perl,$^{15}$ V. Savinov,$^{15}$ D. Ugolini,$^{15}$ X. Zhou,$^{15}$ T. E. Coan,$^{16}$ V. Fadeyev,$^{16}$ I. Korolkov,$^{16}$ Y. Maravin,$^{16}$ I. Narsky,$^{16}$ V. Shelkov,$^{16}$ J. Staeck,$^{16}$ R. Stroynowski,$^{16}$ I. Volobouev,$^{16}$ J. Ye,$^{16}$ M. Artuso,$^{17}$ F. Azfar,$^{17}$ A. Efimov,$^{17}$ M. Goldberg,$^{17}$ D. He,$^{17}$ S. Kopp,$^{17}$ G. C. Moneti,$^{17}$ R. Mountain,$^{17}$ S. Schuh,$^{17}$ T. Skwarnicki,$^{17}$ S. Stone,$^{17}$ G. Viehhauser,$^{17}$ J.C. Wang,$^{17}$ X. Xing,$^{17}$ J. Bartelt,$^{18}$ S. E. Csorna,$^{18}$ V. Jain,$^{18,}$[^4] K. W. McLean,$^{18}$ S. Marka,$^{18}$ R. Godang,$^{19}$ K. Kinoshita,$^{19}$ I. C. Lai,$^{19}$ P. Pomianowski,$^{19}$ S. Schrenk,$^{19}$ G. Bonvicini,$^{20}$ D. Cinabro,$^{20}$ R. Greene,$^{20}$ L. P. Perera,$^{20}$ G. J. Zhou,$^{20}$ M. Chadha,$^{21}$ S. Chan,$^{21}$ G. Eigen,$^{21}$ J. S. Miller,$^{21}$ M. Schmidtler,$^{21}$ J. Urheim,$^{21}$ A. J. Weinstein,$^{21}$ F. Würthwein,$^{21}$ D. W. Bliss,$^{22}$ G. Masek,$^{22}$ H. P. Paar,$^{22}$ S. Prell,$^{22}$ V. Sharma,$^{22}$ D. M. Asner,$^{23}$ J. Gronberg,$^{23}$ T. S. Hill,$^{23}$ D. J. Lange,$^{23}$ R. J. Morrison,$^{23}$ H. N. Nelson,$^{23}$ T. K. Nelson,$^{23}$ D. Roberts,$^{23}$ B. H. Behrens,$^{24}$ W. T. Ford,$^{24}$ A. Gritsan,$^{24}$ J. Roy,$^{24}$ J. G. Smith,$^{24}$ J. P. Alexander,$^{25}$ R. Baker,$^{25}$ C. Bebek,$^{25}$ B. E. Berger,$^{25}$ K. Berkelman,$^{25}$ K. Bloom,$^{25}$ V. Boisvert,$^{25}$ D. G. Cassel,$^{25}$ D. S. Crowcroft,$^{25}$ M. Dickson,$^{25}$ S. von Dombrowski,$^{25}$ P. S. Drell,$^{25}$ K. M. Ecklund,$^{25}$ R. Ehrlich,$^{25}$ A. D. Foland,$^{25}$ P. Gaidarev,$^{25}$ L. Gibbons,$^{25}$ B. Gittelman,$^{25}$ S. W. Gray,$^{25}$ D. L. Hartill,$^{25}$ B. K. Heltsley,$^{25}$ P. I. Hopman,$^{25}$ J. Kandaswamy,$^{25}$ P. C. Kim,$^{25}$ D. L. Kreinick,$^{25}$ T. Lee,$^{25}$ Y. Liu,$^{25}$ N. B. Mistry,$^{25}$ C. R. Ng,$^{25}$ E. Nordberg,$^{25}$ M. Ogg,$^{25,}$[^5] J. R. Patterson,$^{25}$ D. Peterson,$^{25}$ D. Riley,$^{25}$ A. Soffer,$^{25}$ B. Valant-Spaight,$^{25}$ and C. Ward$^{25}$
$^{1}$[University of Florida, Gainesville, Florida 32611]{}\
$^{2}$[Harvard University, Cambridge, Massachusetts 02138]{}\
$^{3}$[University of Hawaii at Manoa, Honolulu, Hawaii 96822]{}\
$^{4}$[University of Illinois, Urbana-Champaign, Illinois 61801]{}\
$^{5}$[Carleton University, Ottawa, Ontario, Canada K1S 5B6\
and the Institute of Particle Physics, Canada]{}\
$^{6}$[McGill University, Montréal, Québec, Canada H3A 2T8\
and the Institute of Particle Physics, Canada]{}\
$^{7}$[Ithaca College, Ithaca, New York 14850]{}\
$^{8}$[University of Kansas, Lawrence, Kansas 66045]{}\
$^{9}$[University of Minnesota, Minneapolis, Minnesota 55455]{}\
$^{10}$[State University of New York at Albany, Albany, New York 12222]{}\
$^{11}$[Ohio State University, Columbus, Ohio 43210]{}\
$^{12}$[University of Oklahoma, Norman, Oklahoma 73019]{}\
$^{13}$[Purdue University, West Lafayette, Indiana 47907]{}\
$^{14}$[University of Rochester, Rochester, New York 14627]{}\
$^{15}$[Stanford Linear Accelerator Center, Stanford University, Stanford, California 94309]{}\
$^{16}$[Southern Methodist University, Dallas, Texas 75275]{}\
$^{17}$[Syracuse University, Syracuse, New York 13244]{}\
$^{18}$[Vanderbilt University, Nashville, Tennessee 37235]{}\
$^{19}$[Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061]{}\
$^{20}$[Wayne State University, Detroit, Michigan 48202]{}\
$^{21}$[California Institute of Technology, Pasadena, California 91125]{}\
$^{22}$[University of California, San Diego, La Jolla, California 92093]{}\
$^{23}$[University of California, Santa Barbara, California 93106]{}\
$^{24}$[University of Colorado, Boulder, Colorado 80309-0390]{}\
$^{25}$[Cornell University, Ithaca, New York 14853]{}
Several authors [@ref:b2dk] have devised methods for measuring the phase $\gamma \approx \arg(V^*_{ub})$ of the Cabibbo-Kobayashi-Maskawa (CKM) [@ref:ckm] unitarity triangle, using decays of the type $B\rightarrow DK$. Comparison between these measurements and results from other $B$ and $K$ decays may be used to test the CKM model of $CP$ violation. $CP$ violation could be manifested in $B\rightarrow DK$ in the interference between a $\bar b\rightarrow \bar c$ and a $\bar b\rightarrow \bar u$ amplitude (Figure \[fig:b2dk\]), detected when the $D$ meson is observed in a final state accessible to both $D^0$ and $\bar D^0$.
The data used in this analysis were produced in $e^+e^-$ annihilations at the Cornell Electron Storage Ring (CESR), and collected with the CLEO II detector [@ref:detector]. The data consist of $3.1~{\rm
fb}^{-1}$ taken at the $\Upsilon$(4S) resonance, containing approximately 3.3 million $B\bar B$ pairs. To study the continuum $e^+e^-
\rightarrow q\bar q$ background, we use $1.6~{\rm fb}^{-1}$ of off-resonance data, taken 60 MeV below the $\Upsilon$(4S) peak.
CLEO II is a general-purpose solenoidal magnet detector. The momenta of charged particles are measured in a tracking system, consisting of a 6-layer straw tube chamber, a 10-layer precision drift chamber, and a 51-layer main drift chamber, all operating inside a 1.5 T superconducting solenoid. The main drift chamber also provides measurements of the specific ionization, $dE/dx$, which we use for particle identification. Photons are detected in a 7800-CsI crystal electromagnetic calorimeter inside the magnet coil. Muons are identified using proportional counters placed at various depths in the magnet return iron.
We reconstruct $\bar D^0$ candidates in the decay modes $K^+\pi^-$, $K^+\pi^-\pi^0$, or $K^+\pi^-\pi^+\pi^-$ (reference to the charge-conjugate state is implied). Pion and kaon candidate tracks are required to originate from the interaction point and satisfy criteria designed to reject spurious tracks. Muons are rejected by requiring that the tracks stop in the first five interaction lengths of the muon chambers. Electrons are rejected using $dE/dx$ and the ratio of the track momentum to the associated calorimeter shower energy. The $\bar
D^0$ daughter tracks are required to have $dE/dx$ consistent with their particle hypothesis to within three standard deviations ($\sigma$). Neutral pion candidates are reconstructed from pairs of isolated calorimeter showers with invariant mass within 15 MeV (approximately $2.5\sigma$) of the nominal $\pi^0$ mass. The lateral shapes of the showers are required to be consistent with those of photons. We require a minimum energy of 30 MeV for showers in the barrel part of the calorimeter, and 50 MeV for endcap showers. At least one of the two $\pi^0$ showers is required to be in the barrel. The $\pi^0$ candidates are kinematically fitted with the invariant mass constrained to be the $\pi^0$ mass.
The invariant mass of the $\bar D^0$ candidate, $M(D)$, is required to be within 60 MeV of the nominal $\bar D^0$ mass. The $M(D)$ resolution, $\sigma_{M(D)}$, is 9 MeV in the $K^+\pi^-$ mode, 13 MeV in the $K^+\pi^-\pi^0$ mode, and 7 MeV in the $K^+\pi^-\pi^+\pi^-$ mode. The loose $M(D)$ requirement leaves a broad sideband to assess the background.
$B^+$ candidates are formed by combining a $\bar D^0$ candidate with a “hard” kaon candidate track. For each $B^+$ candidate, we calculate the beam-constrained mass, $M_{bc} \equiv \sqrt{E_{\rm b}^2 - p_B^2}$, where $p_B$ is the $B^+$ candidate momentum and $E_{\rm b}$ is the beam energy. $M_{bc}$ peaks at the nominal $B^+$ mass for signal, with a resolution of $\sigma_{M_{bc}} = 2.6$ MeV, determined mostly by the beam energy spread. We accept candidates with $M_{bc} > 5.230$ GeV. We define the energy difference, $\Delta E \equiv E_D + \sqrt{p_K^2 + M_K^2}- E_{\rm b}$, where $E_D$ is the measured energy of the $\bar D^0$ candidate, $p_K$ is the momentum of the hard kaon candidate, and $M_K$ is the nominal kaon mass. Signal events peak around $\Delta E = 0$, with a resolution of 24 MeV in the $K^+\pi^-$ mode, 27 MeV in the $K^+\pi^-\pi^0$ mode, and 20 MeV in the $K^+\pi^-\pi^+\pi^-$ mode. We require $-100 < \Delta E
< 200$ MeV.
The largest source of background is the Cabibbo allowed decay $B^+\rightarrow \bar D^0 \pi^+$, distributed around $\Delta E =
48$ MeV. Taking into account correlations between $\Delta E$ and $M(D)$, the $\Delta E$ separation between signal and $B^+\rightarrow
\bar D^0 \pi^+$ is about $2.3\sigma$ in all three modes. The only additional variable which provides significant $K-\pi$ separation is $dE/dx$ of the hard kaon candidate. The $dE/dx$ separation between kaons and pions in the relevant momentum range of $2.1-2.5$ GeV is approximately $1.5\sigma$. Our $dE/dx$ variable is chosen such that pions are distributed approximately as a zero-centered, unit-r.m.s. Gaussian, and kaons are centered around $-1.4$, with a width of about 0.9.
Other sources of $B\bar B$ background are $B\rightarrow \bar D^*
\pi^+$, $B^+\rightarrow \bar D^0 \rho^+$, and events with a misreconstructed $\bar D^0$ which pass the selection criteria. Such $B\bar B$ events tend to have low $\Delta E$ and broad $M_{bc}$ distributions. Continuum $e^+e^- \rightarrow q\bar q$ events also contribute to the background. We reject 69% of the continuum and retain 87% of the signal by requiring $|\cos\theta_s| < 0.9$, where $\theta_s$ is the angle between the sphericity axis of the $B^+$ candidate and that of the rest of the event. The sphericity axis, ${\boldmath \bf s}$, of a set of momentum vectors, $\{ {\boldmath \bf
p}_i \}$, is the axis for which $\sum_i | {\boldmath \bf p}_i \times
{\boldmath \bf s}|^2$ is minimized.
In addition to the above variables, discrimination between signal and continuum background is obtained from $\cos\theta_B$, where $\theta_B$ is the angle between the $B^+$ candidate momentum and the beam axis, and by using a Fisher discriminant [@ref:bigrare]. The Fisher discriminant is a linear combination, ${\cal F}\equiv
\sum_{i=1}^{11}\alpha_i y_i$, where the coefficients $\alpha_i$ are chosen so as to maximize the separation between $B\bar B$ and continuum Monte Carlo samples. The eleven variables, $y_i$, are $|\cos\theta_{thr}|$ (the cosine of the angle between the $B^+$ candidate thrust axis and the beam axis), the ratio of the Fox-Wolfram moments $H_2/H_0$ [@ref:fox], and nine variables measuring the scalar sum of the momenta of tracks and showers from the rest of the event in nine, $10^\circ$ angular bins centered about the candidate’s thrust axis. Signal events peak around ${\cal F}= 0.4$, while continuum events peak at ${\cal F}= 2$, both with approximately unit r.m.s.
18.8% of the events have more than one $B^+$ candidate, reconstructed in any of the three modes, which satisfies the selection criteria. In such events we select the best candidate, defined to have the smallest $\chi^2 \equiv [(M_{bc} - M_B)/\sigma_{M_{bc}}]^2 + [(M(D) -
M_D)/\sigma_{M(D)}]^2$, where $M_B$ and $M_D$ are the nominal $B$ and $D$ masses, respectively. We verify that the distribution of the number of candidates per event in the Monte Carlo agrees well with the data.
The efficiency of signal events to pass all the requirements is $0.4412 \pm
0.0029$ for the $K^+\pi^-$ mode, $0.1688\pm 0.0016$ for the $K^+\pi^-\pi^0$ mode, and $0.2186\pm 0.0024$ for the $K^+\pi^-\pi^+\pi^-$ mode. The efficiencies are determined using a detailed GEANT-based Monte Carlo simulation [@ref:geant], and the errors quoted are due to Monte Carlo statistics.
The number of data events that satisfy the selection criteria, $N_e$, is 1221 in the $K^+\pi^-$ mode, 5249 in the $K^+\pi^-\pi^0$ mode , and 7353 in the $K^+\pi^-\pi^+\pi^-$ mode. The fraction of signal events in the data samples is found mode-by-mode using an unbinned maximum likelihood fit. We define the likelihood function $${\cal L} = \prod_{e=1}^{N_e} \left[\sum_{t=1}^7
{\cal P}_t(e) f_t\right], \label{eq:likelihood}$$ where ${\cal P}_t(e)$ is the normalized probability density function (PDF) for events of type $t$, evaluated on event $e$, and $f_t$ is the fraction of such events in the data sample. The seven event types in the sum are 1) signal, 2) $B^+\rightarrow \bar D^0 \pi^+$, 3) $B
\rightarrow \bar D^*\pi^+ + \bar D^0\rho^+$, 4) a hard kaon or 5) pion in combinatoric $B\bar B$ events with a misreconstructed $\bar D^0$, and 6) a hard kaon or 7) pion in continuum events. The fit maximizes ${\cal L}$ by varying the seven fractions, $f_t$, subject to the constraint $\sum_t f_t = 1$.
The PDF’s are analytic, six-dimensional functions of the variables $\Delta E$, $dE/dx$ of the hard kaon candidate, $M(D)$, $M_{bc}$, ${\cal F}$, and $\cos\theta_B$. The PDF’s are mostly products of six one-dimensional functions, except for correlations between $\Delta E$, $M(D)$, and $M_{bc}$ in the $B^+\rightarrow \bar D^0 K^+$ and $B^+\rightarrow \bar D^0 \pi^+$ PDF’s.
The $dE/dx$ distributions of $K^\pm,\pi^\pm$ are parameterized using a Gaussian distribution, whose parameters depend linearly on the track momentum. The parameterization is determined by studying pure samples of kaons and pions in data, tagged in the decay chain $D^{*+}\rightarrow D^0 \pi^+$, $D^0 \rightarrow K^-\pi^+$. The parameterization in the other variables is obtained from the off-resonance data for the continuum PDF’s and from Monte Carlo for the $B\bar B$ PDF’s.
The distribution of $B^+\rightarrow \bar D^0K^+$ and $B^+\rightarrow
\bar D^0\pi^+$ events in $\Delta E-M(D)-M_{bc}$ space is parameterized using the sum of two three-dimensional Gaussians, which are rotated to account for correlations. For $B\rightarrow \bar D^*\pi^+ + \bar
D^0\rho^+$ events we use the sum of two Gaussians to parameterize the $M_{bc}$ and $\Delta E$ distributions, and a Gaussian plus a bifurcated Gaussian for the $M(D)$ distribution. These distributions are essentially uncorrelated due to the requirement $\Delta E >
-100$ MeV. For $B\bar B$ events with a misreconstructed $\bar D^0$ we use a third-order polynomial to parameterize the $\Delta E$ distribution, and a first-order polynomial plus a Gaussian for the $M(D)$ distribution. The Gaussian is about three times broader than the $M(D)$ resolution, and models the peaking which arises due to the selection of the best candidate in the event. The $M_{bc}$ distribution is parameterized using the Argus function [@ref:argus-function] $f(M_{bc})\propto
M_{bc}\sqrt{1-(M_{bc}/E_b)^2}\exp[-a(1-(M_{bc}/E_b)^2)]$, plus a Gaussian, which reflects mostly $B\rightarrow \bar D^{(*)}\pi^+$ or $B^+\rightarrow\bar D^0\rho^+$ events in which we misreconstruct a $\bar D^0$.
We use a first-order polynomial to parameterize the $\Delta E$ distribution of continuum events, and a first-order polynomial plus a Gaussian for their $M(D)$ distribution. The Gaussian peaking is due both to real $\bar D^0$’s and to the selection of the best candidate in the event. The $M_{bc}$ distribution is parameterized using an Argus function whose sharp edge is smeared by adding a bifurcated Gaussian to account for the beam energy spread. We use the function $1-\xi\cos^2\theta_B$ to parameterize the $\cos\theta_B$ distributions, and bifurcated Gaussians for the $\cal F$ distributions.
The results of the maximum likelihood fits are summarized in Table \[tab:results\]. Averaging over the three modes, we find $R\equiv{\cal B}(B^+\rightarrow \bar D^0 K^+) / {\cal
B}(B^+\rightarrow \bar D^0 \pi^+) = 0.055 \pm
0.014$ (statistical). This is consistent with the value $(f_K /
f_\pi)^2 \tan^2 \theta_c \approx 0.07$, expected from factorization, with $a_2\ll a_1$ [@ref:factorization]. The $\chi^2$ of the average is $1.2$ for two degrees of freedom, indicating the consistency among the results obtained with the three decay modes. To illustrate the significance of the signal yield, contour plots of $-2\ln {\cal L}$ vs. the number of $B^+\rightarrow \bar D^0 K^+$ and $B^+\rightarrow \bar D^0 \pi^+$ events are shown in Figure \[fig:contour\]. The curves represent $n\sigma$ contours, corresponding to the increase in $-2\ln {\cal L}$ by $n^2$ over the minimum value.
Mode: $K^+\pi^-$ $K^+\pi^-\pi^0$ $K^+\pi^-\pi^+\pi^-$
----------------------- ------------------- ------------------- ----------------------
$N_{DK}$ $16.5 \pm 5.9$ $13.5 \pm 8.7$ $21.5 \pm 7.8$
$N_{D\pi}$ $240 \pm 15$ $379 \pm 22$ $326 \pm 20$
$N_{DK}$ significance $4.2\sigma$ $1.8\sigma$ $3.8\sigma$
$N_{DK}/N_{D\pi}$ $0.069 \pm 0.026$ $0.035 \pm 0.023$ $0.066 \pm 0.025$
: Results of the maximum likelihood fits. $N_{DK}$ and $N_{D\pi}$ are the numbers of $B^+\rightarrow \bar D^0 K^+$ and $B^+\rightarrow \bar D^0 \pi^+$ events found in the fit, respectively. Errors are statistical only. The statistical significance of the signal yield is determined from $-2\ln
{\cal L}$ by fixing the number of signal events at zero and refitting the data. []{data-label="tab:results"}
The quality of the fit is illustrated in Figure \[fig:projections\]a, showing projections of the data onto $dE/dx$ and $\Delta E$ for events in the $B^+\rightarrow \bar D^0
K^+$ region, defined by ${\cal F} < 1.6$, $|M_{bc} - 5280 \ {\rm MeV}|
< 5\ {\rm MeV}$, $|M(D) - 1864.5 \ {\rm MeV}| < 20 \ {\rm MeV}$, $-50
< \Delta E < 10 \ {\rm MeV}$, $dE/dx < -0.75$. Requiring that events fall within this $B^+\rightarrow \bar D^0 K^+$ region reduces the signal efficiency by about 50%, but strongly suppresses the background. Overlaid on the data are projections of the fit function. The fit function is the sum of the PDF’s, each weighted by the number of corresponding events found in the fit and multiplied by the efficiency of the corresponding event type to be in the $B^+\rightarrow \bar D^0 K^+$ region. In Figure \[fig:projections\]b we show projection plots for events in the $B^+\rightarrow \bar D^0 \pi^+$ region, defined by $0 < \Delta E <
100 \ {\rm MeV}$, $|dE/dx| < 2.5$, and with the same requirements on $\cal F$, $M_{bc}$ and $M(D)$ as in the $B^+\rightarrow \bar D^0
K^+$ region. These projections demonstrate that the fit function agrees well with the data in the regions most highly populated by signal and the most pernicious background, and provides confidence in our modelling of the tails of the $B^+\rightarrow \bar D^0 \pi^+$ distributions.
Projections onto $M_{bc}$ for events in the signal region (Figure \[fig:projections-mb\]) illustrate the relative contributions and distributions of signal and background events. Only $B^+\rightarrow \bar D^0K^+$ and $B^+\rightarrow \bar D^0\pi^+$ events peak significantly around $M_{bc} = M_B$, despite the selection of the best candidate in the event.
We conduct several tests to verify the consistency of our result. The fit is run on off-resonance data and on Monte Carlo samples containing the expected distribution of background events with no signal. In both cases the signal yield is consistent with zero. We also fit the data without making use of $\cal F$ or $dE/dx$, and obtain results consistent with those of Table \[tab:results\], with increased errors. We find the branching fraction ${\cal B}(B^+\rightarrow
\bar D^0 \pi^+) = (4.82 \pm 0.19 \pm 0.31)\times 10^{-3}$, in agreement with previous CLEO measurements [@ref:b2dpi]. The ratio between the $B \rightarrow \bar D^*\pi^+ + \bar D^0\rho^+$ and $B^+\rightarrow \bar D^0 \pi^+$ yields obtained from the fit is consistent with the measured branching fractions of these decays [@ref:pdg]. In addition, our $B^+\rightarrow \bar
D^0K^+$ result is consistent with that of a simpler, though less sensitive method, used to analyze the same data [@ref:warsaw].
Many systematic errors cancel in the ratio $R$. We assess systematic errors due to our limited knowledge of the PDF’s by varying all the PDF parameters by $\pm1$ standard deviation in the basis in which they are uncorrelated, where the magnitude of a standard deviation is determined by the statistics in the data or Monte Carlo sample used to evaluate the PDF parameters. The systematic error in $R$ due to Monte Carlo statistics is 0.0033. The error due to statistics in the data sample used to parameterize the $dE/dx$ distributions is 0.0028, and the error due to statistics in the off-resonance data sample is 0.0017. We assign a systematic error of 0.0005 due to the uncertainty in the average beam energy, which we estimate to be $\pm0.16$ MeV by using the peak of the $M_{bc}$ distribution of $B^+\rightarrow \bar
D^0 \pi^+$ events. The total systematic error is 0.0047.
In summary, we have observed the decay $B^+\rightarrow \bar D^0
K^+$ and determined the ratio of branching fractions $$R = {{\cal B}(B^+\rightarrow \bar D^0 K^+) \over
{\cal B}(B^+\rightarrow\bar D^0 \pi^+)}
= 0.055 \pm 0.014 \pm 0.005.$$ Combining this result with the CLEO II measurement [@ref:b2dpi] ${\cal
B}(B^+\rightarrow \bar D^0 \pi^+) = (4.67 \pm 0.22 \pm
0.40)\times 10^{-3}$, we obtain ${\cal B}(B^+\rightarrow \bar D^0
K^+) = (0.257 \pm 0.065 \pm 0.032)\times 10 ^{-3}$.
We gratefully acknowledge the effort of the CESR staff in providing us with excellent luminosity and running conditions. This work was supported by the National Science Foundation, the U.S. Department of Energy, Research Corporation, the Natural Sciences and Engineering Research Council of Canada, the A.P. Sloan Foundation, and the Swiss National Science Foundation.
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R. Brun [*et al.*]{}, GEANT 3.15, CERN DD/EE/84-1.
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CLEO Collaboration, J. P. Alexander [*et al.*]{}, ICHEP-96 PA05-68, CLEO CONF 96-27.
=5.00in
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[^1]: Permanent address: Lawrence Livermore National Laboratory, Livermore, CA 94551.
[^2]: Permanent address: BINP, RU-630090 Novosibirsk, Russia.
[^3]: Permanent address: Yonsei University, Seoul 120-749, Korea.
[^4]: Permanent address: Brookhaven National Laboratory, Upton, NY 11973.
[^5]: Permanent address: University of Texas, Austin TX 78712.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'It is known that the critical exponent (CE) for conventional, continuous powers of $n$-by-$n$ doubly nonnegative (DN) matrices is $n-2$. Here, we consider the larger class of diagonalizable, entrywise nonnegative $n$-by-$n$ matrices with nonnegative eigenvalues (GDN). We show that, again, a CE exists and are able to bound it with a low-coefficient quadratic. However, the CE is larger than in the DN case; in particular, 2 for $n=3$. There seems to be a connection with the index of primitivity, and a number of other observations are made and questions raised. It is shown that there is no CE for continuous Hadamard powers of GDN matrices, despite it also being $n-2$ for DN matrices.'
author:
-
bibliography:
- 'mybibfile.bib'
title: The critical exponent for generalized doubly nonnegative matrices
---
Critical exponent, nonnegative matrix, index of primitivity, continuous conventional power, generalized doubly nonnegative matrix
Primary 15B48
Introduction
============
An $n$-by-$n$ real symmetric matrix is called *doubly nonnegative* (DN) if it is both positive semidefinite and entrywise nonnegative. Given a doubly nonnegative matrix $A$, the continuous conventional powers of $A$ are defined using the spectral decomposition: if $\alpha>0$, $A = UDU^{T}$, and $D = \text{diag}(d_{11}, . . . , d_{nn})$, then $A^\alpha := UD^\alpha U^T$, with $D^\alpha := \text{diag}(d_{11}^\alpha, . . . , d_{nn}^\alpha)$. The conventional *critical exponent* (CE) for DN matrices is the least real number $m$ such that for any DN matrix $A$, $A^\alpha$ is also DN for all $\alpha > m$. It was shown in [@Johnson] that the critical exponent for DN matrices exists and is no smaller than $n-2$, and is $n-2$ for $n<6$. A low coefficient quadratic upper bound was also given in [@Johnson]. The authors conjectured that the critical exponent is $n-2$, and this conjecture was proven in [@Stanford] by applying a result from [@M]. There is also the concept of critical exponents of DN matrices under Hadamard powering, and interestingly enough, the critical exponent is also shown to be $n-2$ in [@Hadamard].
Here we relax the assumption that the matrix be symmetric while still insisting that the matrix is entrywise nonnegative, diagonalizable, and has nonnegative eigenvalues. We call such matrices *generalized doubly nonnegative* (GDN). Because GDN matrices are diagonalizable, we have $A = SDS^{-1}$, where $D$ is a diagonal matrix, and we can define continuous powers similarly via $A^\alpha := SD^\alpha S^{-1}$.
We show here the critical exponent for GDN matrices also exists, and we give low-coefficient quadratic upper bounds for it. We show that the critical exponent is strictly larger than $n-2$ if $n$ is an odd integer greater than 2. We make observations about the relation between the index of primitivity of a primitive matrix and the critical exponent for GDN matrices of that size. In addition, we make the observation that the GDN critical exponent under Hadamard powering does not exist.
Background
==========
Any diagonalizable matrix $A \in M_n({\mathbf{R}})$ can be decomposed as $$A = SDS^{-1}$$ in which $D$ is a diagonal matrix consisting of the eigenvalues of $A$. If $x_i$ denotes the $i\textsuperscript{th}$-column of $S$ and $y_i$ denotes the $i\textsuperscript{th}$-row of $S^{-1}$, then $A$ can be written as $$A = \lambda_1x_1y_1 + ... + \lambda_nx_ny_n.$$
If all eigenvalues of $A$ are nonnegative, then for $\alpha>0 $, $A^\alpha$ is defined by $$A^\alpha = \lambda_1^\alpha x_1y_1+ ... + \lambda_n^\alpha x_ny_n.$$ Each entry of $A^\alpha$ has the form $$(A^\alpha)_{ij} = \lambda_1^\alpha (x_1y_1)_{ij}+...+ \lambda_n^\alpha (x_ny_n)_{ij}.$$ Any function of the form $$\phi(t) = a_1 e^{b_1 t} + ... +a_n e^{b_n t}$$ where $a_i, b_i \in {\mathbf{R}}$, is called an *exponential polynomial*. In particular, if all eigenvalues of $A$ are nonnegative, then each entry of $A^\alpha$ is an exponential polynomial in $\alpha$. The eigenvalues of the $A^\alpha$ are obvious but the non-negativity of the entries is not obvious. The following version of Descartes’ rule for exponential polynomials is well known and appears as an exercise in [@DR].
\[DR\] Let $ \phi(t) = \sum_{i=1}^{n} a_i e^{b_i t}$ be a real exponential polynomial such that each $a_i \not = 0$ and $b_1 > b_2 > . . . > b_n$. The number of real roots of $\phi(t)$, counting multiplicity, cannot exceed the number of sign changes in the sequence of coefficients $(a_1, a_2, . . . , a_n)$.
The existence and an upper bound for the GDN critical exponent
==============================================================
We follow the strategy of [@Johnson] to show the existence and an upper bound for CE. Lemma \[DR\] leads immediately to the the existence of a GDN CE.
\[existence\] There is a function $m(n)$ such that for any $n$-by-$n$ GDN matrix $A$, $A^\alpha$ is generalized doubly nonnegative for $\alpha \ge m(n)$.
The proof of Theorem 2.1 in [@Johnson] does not rely on the symmetry assumption so that essentially the same proof establishes Theorem \[existence\].
Let $A$ be an $n$-by-$n$ GDN matrix. Since $A$ is entrywise nonnegative, so is $A^k$ for all positive integers $k$. If $A^\alpha$ is entrywise nonnegative for all $\alpha \in [m,m + 1]$, where $m \in {\mathbf{Z}}$, then it follows from repeated multiplication by $A$ that $A^\alpha$ is also entrywise nonnegative for all $\alpha \ge m$. Suppose that $A^\alpha$ has a negative entry for some $\alpha \in [m,m + 1]$, then the exponential polynomial corresponding to that entry must have at least two roots in the interval $[m,m+1]$ by continuity and the fact that $A^m$ and $A^{m+1}$ are both entrywise nonnegative. By Lemma \[DR\], the maximum number of roots each entry may possess depends on $n$. It follows that there is a constant $m(n)$ such that $A^\alpha $ is entrywise nonnegative, and thus GDN, for all $\alpha > m(n)$.
Moreover, we may strengthen the argument in the proof of Theorem \[existence\] to give an upper bound for the CE after developing some tools.
Let $A$ be any $n$-by-$n$ GDN matrix. Following [@Johnson], corresponding to the matrix $A$, we define a matrix $W = [w_{ij}]$ where $w_{ij}$ equals the number of sign changes in the sequence of coefficients of the exponential polynomial $(A^\alpha)_{ij}$ arranged in decreasing order of the corresponding eigenvalues. We refer to $W$ as the *sign change matrix* for $A$. By Lemma \[DR\], each entry $w_{ij}$ of a sign change matrix gives an upper bound on the number of real zeros of the corresponding exponential polynomial $(A^\alpha)_{ij}$, counting multiplicity. Note that $w_{ij} \le n-1$ because there are at most $n$ terms in the exponential polynomial $(A^\alpha)_{ij}$.
\[2.3\]
Let $A$ be an invertible GDN matrix with sign change matrix $W = [w_{ij}]$. Let $\bar{T}_{ij}= \{ \alpha > 1 : A^\alpha_{ij} < 0 \}$. Then the maximum number of connected components of $\bar{T}_{ij}$ is $$\begin{cases}
\left \lfloor{\frac{w_{ij}-1}{2}}\right \rfloor &\text{ if $w_{ij} > 0$ and $i \not = j$} \\
\left \lfloor{\frac{w_{ij}}{2}}\right \rfloor & \text{ if $w_{ij} > 0$ and $i = j$} \\
0 & \text{ if $w_{ij} = 0$}
\end{cases}.$$
By Lemma \[DR\] the maximum number of real roots of the exponential polynomial $A^\alpha_{ij}$ is given by $w_{ij}$. Since $A$ is invertible, the exponential polynomials defining the entries of $A^\alpha$ when $\alpha > 0$ still agree with $A^\alpha$ at $\alpha = 0$. Since $A^0$ is the identity matrix, the exponential polynomial $(A^\alpha)_{ij}$ has at most $w_{ij} - 1$ roots in the interval $[1,\infty)$ when $i \not = j$. Each of the connected components of $\bar{T}_{ij}$ is bounded because $A^k$ is nonnegative for all positive integers $k$. The endpoints of these components are roots of the exponential polynomial $(A^\alpha)_{ij}$. If two adjacent connected components of $\bar{T}_{ij}$ share an endpoint, that endpoint must be a root of degree at least two. Counting multiplicity, the number of real roots of $(A^\alpha)_{ij}$ with $\alpha \ge 1$ must therefore be at least double the number of connected components of $\bar{T}_{ij}$. If $w_{ij}$ is zero, then the exponential polynomial $(A^\alpha)_{ij}$ has all positive coefficients, so $\bar{T}_{ij}$ is empty. And if $w_{ii}$ is not zero, the corresponding exponential polynomial has at most $w_{ij}$ roots counting multiplicity and $0$ is not one of them. So there are at most $\left \lfloor{\frac{w_{ij}}{2}}\right \rfloor$ number of connected components.
From now on, we will denote the GDN critical exponent of $n$-by-$n$ matrices by $CE_n$. We now provide an upper bound for $CE_n$.
\[upper\] We have $$CE_n \le \begin{cases}
\frac{n^2 -3n+4}{2} & \text{n is odd} \\
\frac{n^2 - 2n}{2} & \text{n is even} \\
\end{cases}.$$
\[ob\] For $j\in \{1,...,n\}$, let $\bar{T}_{j} = \{ \alpha > 1 : {A^\alpha}_{ij} < 0, i = 1, ..., n \}$. Note that if $\bar{T}_{j} \cap (m,m + 1) = \emptyset$, for some integer $m$, then every entry in column $j$ of $A^\alpha$ is nonnegative for all powers $\alpha \in [m,m + 1]$. Using repeated left multiplication by $A$, we see that column $j$ of $A^\alpha$ must be nonnegative for all $\alpha \ge m$.
Let $k(n)$ be the proposed upper bound. Assume $A$ is an invertible GDN matrix.
If $n$ is odd, then $\bar{T}_{j}$ can have at most $$\begin{aligned}
(n-1)\left \lfloor{\frac{n-2}{2}}\right \rfloor + \left \lfloor{\frac{n-1}{2}}\right \rfloor = \frac{n^2 -3n+2}{2} \end{aligned}$$ connected components in $[1, \infty)$. By Observation \[ob\], there has to be a connected component in $(0,1)$ as well. But this can be achieved by letting the exponential polynomial for one of the off-diagonal entries to have exactly one simple root in $(0,1)$.
If $n$ is even, then $\bar{T}_{j}$ can have at most $$\begin{aligned}
(n-1)\left \lfloor{\frac{n-2}{2}}\right \rfloor + \left \lfloor{\frac{n-1}{2}}\right \rfloor = \frac{n^2 -2n}{2} \end{aligned}$$ connected components in $[1, \infty)$. Again by Observation \[ob\], there has to be a connected component in $(0,1)$ as well. But in this case, the number of connected components in $[1, \infty)$ has to decrease by at least 1 if we insist that there is a connected component in $(0,1)$. Therefore, there are at most $\frac{n^2 -2n-2}{2}$ connected components in $[1, \infty)$. Finally, by Observation \[ob\], the connected components in $[1, \infty)$ have to lie in intervals with consecutive integers as end points, starting from $[1,2]$. Therefore, there are no such connected components in $(k(n), \infty)$
Now suppose that $A$ is singular. By continuity, $A^\alpha$ cannot have a negative entry for any $\alpha > k(n)$. Therefore the critical exponent $CE_n \le k(n)$.
The GDN critical exponent and the index of primitivity
======================================================
We first note that since a DN matrix is also GDN, the critical exponent for GDN matrices is no smaller than the critical exponent for DN matrices.
We now focus on irreducible matrices and explore the relation between the GDN critical exponent and the index of primitivity. We will address reducible GDN matrices in the next section.
If an $n$-by-$n$ GDN matrix is irreducible, then it is primitive by the Perron-Frobenius theorem. The *index of primitivity* of a primitive matrix $A$ is the least positive integer $k$ such that $A^k$ is entrywise positive. It is known that the index of primitivity of a $n$-by-$n$ matrix is at most $(n-1)n^n$ (NOTE:CITE HORN,JOHNSON MATRIX ANALYSIS HERE). We denote the maximum index of primitivity for primitive $n$-by-$n$ GDN matrices by $MIP_n$. By the definition of index of primitivity, there exists a GDN matrix that has at least one zero entry, say the $(i,j)$-entry, when raised to the power $MIP_n - 1$. If the exponential polynomial $$p(t) = a_1 \lambda_1^t + ... + a_n \lambda_n^t$$ corresponding to the $ij$-th entry has non-vanishing derivative at $t = MIP_n - 1$, that is, $p'(MIP_n - 1) \not = 0$ then, either $p(k)< 0$ for some $k>MIP_n - 1$, or $p(k)<0$ for all $ k \in (MIP_n - 1 -\epsilon, MIP_n - 1)$ with some $\epsilon$ small enough. In either case, the GDN critical exponent is at least $MIP_n-1$. The only case in which the GDN critical exponent is less than $MIP_n-1$ is when the exponential polynomial corresponding to a certain entry has a multiple root at an integer larger than the critical exponent. Since the index of primitivity depends only on the number and positions of zeros in the matrix but not on the numerical values of nonzero entries, if it so happens that $MIP_n-1 < CE_n$ with a certain matrix $A$, then the exponential polynomial corresponding to the entry where $A^{MIP_n-1}$ is zero for all GDN matrices with the same zero-nonzero pattern as $A$ has a multiple root at $MIP_n - 1$, which appears highly unlikely. Therefore, we make the following conjecture.
\[MIP<CE\] $MIP_n - 1 < CE_n$.
If Conjecture \[MIP<CE\] is true, then $MIP_n - 1$ gives a lower bound for the critical exponent, and the following question arises naturally.
\[MIP\_pattern\] What is $MIP_n$ and what are the zero-nonzero patterns that attain $MIP_n$?
Question \[MIP\_pattern\] not only may help improve the lower bound for GDN critical exponent but is also interesting in its own right. Note that as shown in Theorem 3.2 of [@Johnson], the $MIP_n \ge n -1$ because of the tridiagonal DN matrices. By Lemma 2.4 in [@Johnson], the maximum index of primitivity for DN matrices is precisely $n-1$. The next two lemmas gives an upper bound for $MIP_n$ and shows that $MIP_n > n-1$ if $n$ is odd.
\[2n-d-1\] $MIP_n \le 2n - 3$.
Let $A$ be an $n$-by-$n$ matrix and first assume it has strict positive eigenvalues. Let $t_k = \text{Tr}(A^k)$. Then the characteristic polynomial of $A$ is given by: $$p(\lambda) = (-1)^n\big(\lambda^n + c_1\lambda^{n-1}+c_2\lambda^{n-2}+...+c_{n-1}\lambda + c_n\big)$$ where $c_1 = -t_1$ and $c_2 = \frac{1}{2}(t_1^2- t_2)$.
By Descartes’ rule of signs, the number of positive roots of $p(\lambda)$ is at most the number of the sign changes in the sequence $(1, c_1, ..., c_n)$. Hence, if $A$ is GDN, then $c_1<0$ and $c_2 > 0$. Therefore, $A$ has at least two positive diagonal entries. If $A$ has only one positive diagonal entry, then $$2c_2 = \Big (\sum_{i=1}^n a_{ii} \Big ) - \sum_{i,j=1}^n a_{ij}a_{ji} = - \sum_{i\not = j}^n a_{ij}a_{ji} \le 0$$ which is impossble as $c_2>0$. For an irreducible matrix $A$ with at least one positive diagonal entry, it is a routine exercise to verify that the index of primitivity of $A$ is at most $2n - d - 1$, where $d$ is the number of positive diagonal entries (see e.g., Theorem 8.5.9 in [@MA]). Therefore, $MIP_n \le 2n - 3$. Finally, we relax the assumption that $A$ has positive eigenvalues as the general case follows from continuity.
\[n-1\] If $n$ is odd, then $MIP_n > n-1$ and $CE_n \ge n-1$.
Consider the matrix $$A = \begin{bmatrix}
d_1 & \epsilon & 0 & \cdots & \cdots & \cdots & \cdots & 0 \\
0 & d_2 & \epsilon & \ddots & && & \vdots \\
0 & 0 & d_3 & \ddots & \ddots & & & \vdots \\
\vdots & \ddots & \ddots & \ddots & \ddots & \ddots & & \vdots \\
\vdots & & \ddots & \ddots & \ddots & \ddots & \ddots& \vdots\\
\vdots & & & \ddots & \ddots & d_{n-2} & \epsilon& 0\\
0 & & & & \ddots & 0 & d_{n-1} & \epsilon\\
\epsilon & 0 & \cdots & \cdots & \cdots & 0 & 0& d_n \\
\end{bmatrix}$$ where $d_1> d_2 > ... > d_{n-1} > d_n = 0$ and $0< \epsilon< \max\{\frac{d_i - d_{i+1}}{2}\}$. By the Gershgorin circle theorem, all eigenvalues are real and the first $n-1$ eigenvalues are positive. Moreover, since $$\det(A) = \epsilon^n > 0$$ all eigenvalues are positive. Therefore, $A$ is GDN.
Note that $A^k_{nn} = 0$ for $ k = 1,2, ..., n-1$. Hence the index of primitivity of $A$ is at least $n$. Since there are at most $n-1$ roots for the exponential polynomial $p(\alpha) = A^\alpha_{nn}$ and $A^{n}_{nn}>0$, if follows that $A^\alpha_{nn}<0$ if $\alpha \in(n-2,n-1)$. Therefore, the critical exponent is at least $n-1$.
$CE_3 = 2$.
The upper bound for the critical exponent is 2 by Theorem \[upper\], and the lower bound is also 2 by Proposition \[n-1\].
In Lemma 2.2 of [@Johnson], it was shown that the diagonal entries of DN matrices remain positive under continuous powering, while in the proof of Proposition \[n-1\], a negative entry appears on the diagonal under continuous powering.
It should be noted that when $n>3$, better lower bounds for $CE_n$ than $n-1$ exist, as demonstrated in the following examples. Examples of GDN matrices with highest index of primitivity discovered are displayed at the end of the section.
$$A_4 = \begin{bmatrix}
1 & 7 & 0 & 0\\
0 & 17000 & 8500 & 0\\
0 & 0 & 24000 & 1600\\
20 & 0 & 0 & 5\\
\end{bmatrix}$$
The critical exponent for $A_4$ is between 3.99 and 4. $$A_5 = \begin{bmatrix}
10 & 70 & 0 & 0 & 0 \\
0 & 5 & 90 & 0 & 0\\
0 & 0 & 80000 & 15000 & 0\\
0 & 0 & 0 & 120000 & 30\\
150 & 0 & 0 & 0 & 0\\
\end{bmatrix}$$ The critical exponent for $A_5$ is between 5.99 and 6.
$$A_6 = \begin{bmatrix}
156 & 1605 & 0 & 0 & 0 & 0\\
0 & 375 & 7932 & 0 & 0 & 0\\
0 & 0 & 805 & 7840 & 0 & 0\\
0 & 0 & 0 & 13803330 & 224210 & 0\\
0 & 0 & 0 & 0 & 9373900 & 18590\\
105720 & 0 & 0 & 0 & 0 & 25200\\
\end{bmatrix}$$
The critical exponent for $A_6$ is between 6.99 and 7.
Note that in the 4-by-4 case, the upper bound for critical exponent is 4 by Theorem \[upper\] and the matrix $A_4$ in the previous example has critical exponent greater than 3.99. We notice that as row 1 and 4 of $A_4$ decrease in proportion (or equivalently as row 2 and 3 increase in proportion), the critical exponent increases. Therefore, we make the following conjecture.
\[4\] $CE_4 = 4$.
In Lemma \[2n-d-1\], we have shown that there have to be at least 2 positive entries on the diagonal. If there are exactly 2 positive entries on the diagonal, then the maximum index of primitivity is $2n-3$, giving $CE_n \ge 2n-4$ if Conjecture \[MIP<CE\] holds. However, generally the zero-nonzero pattern with exactly two positive diagonal entries do not permit GDN matrices. Hence we perturb the diagonal zero entries and aim to the achieve lower bounds for $CE_n$ that are close to $2n-4$. And we observe that when $n=4$ and $n=5$, we can perform such perturbation and produce CE greater than 3.99 and 5.99 respectively. Therefore, we ask the following question:
Is $CE_n$ = $2(n-2)$?
\[t\]
n $MIP_n$ $CE_n$ Upper bound for $CE_n$ (by Theorem 3.3)
--- --------- --------- -----------------------------------------
2 1 0 0
3 3 2 2
4 4 $>$3.99 4
5 6 $>$5.99 7
6 6 $>$6.99 12
7 7 $>$8.99 16
: Largest $CE_n$ and $MIP_n$ discovered in numerical experiments for small $n$’s[]{data-label="table:t"}
Table \[table:t\] shows the highest $MIP_n$ and GDN CE discovered in numerical experiments.
Notice that for all these low dimension cases with $n>2$, the lower bounds for the critical exponent are strictly larger than $n-2$, the critical exponent for DN matrices.
Now we give examples of 4-by-4 GDN matrices with index of primitivity 4, 5-by-5 GDN matrices with index of primitivity 6, and 6-by-6 GDN matrices with index of primitivity 6.
$$A_4 = \begin{bmatrix}
0 & 0 & 2 & 0\\
0 & 68 & 56 & 21\\
0 & 0 & 0 & 16\\
14 & 72 & 0 & 168\\
\end{bmatrix}$$
The index of primitivity of $A_4$ is 4 and the GDN CE of $A_4$ at least 2.99.
$$A_5 = \begin{bmatrix}
1800 & 405 & 0 & 0 & 0 \\
0 & 916 & 794 & 0 & 0\\
447 & 0 & 0 & 7 & 0\\
0 & 300 & 0 & 0 & 15\\
0 & 0 & 72 & 0 & 0\\
\end{bmatrix}$$
The index of primitivity of $A_5$ is 6 and the GDN CE of $A_5$ is at least 4.99.
$$A_6 = \begin{bmatrix}
2439 & 1020 & 0 & 0 & 0 & 0\\
0 & 1917 & 668 & 0 & 0 & 0\\
509 & 0 & 890 & 213 & 0 & 0\\
0 & 2746 & 0 & 0 & 158 & 0\\
0 & 0 & 270 & 0 & 0 & 2\\
0 & 0 & 0 & 206 & 0 & 0\\
\end{bmatrix}$$
The index of primitivity of $A_6$ is 6 and the GDN CE of $A_6$ is at least 4.99.
Additional observations
=======================
We make a few observations about the reducible GDN matrices and about the Hadamard powering critical exponent of GDN matrices in this section.
Let $A$ be a reducible $n$-by-$n$ GDN matrix. If $$P^TAP = \begin{bmatrix}
B & C\\
0 & D \\
\end{bmatrix}$$ for some permutation matrix $P$, then $$P^TA^\alpha P = \begin{bmatrix}
B' & C'\\
0 & D' \\
\end{bmatrix}$$ for all $\alpha > 0$. The matrix $B$ is $k$-by-$k$, $C$ is $k$-by-$n-k$, $0$ is $n-k$-by-$k$, and $D$ is $n-k$-by-$n-k$ for some integer $1<k<n$.
Let $p(\alpha)$ be the exponential polynomial corresponding to the $ij$-th entry, where $k+1 \le i \le n$ and $1 \le j \le k$, then $p(\alpha)$ has a root at every positive integer because the $ij$-th entry is zero for all integer powers of $A$. But $p(\alpha)$ has at most $n-1$ roots counting multiplicity if it is not identically zero, so $ p(\alpha) \equiv 0$ Therefore, the $ij$-th entry stays zero under all continuous powers of $A$ and $$P^TA^\alpha P = \begin{bmatrix}
B' & C'\\
0 & D' \\
\end{bmatrix}$$ where $B'$ is $k$-by-$k$, $C'$ is $k$-by-$n-k$, $0$ is $n-k$-by-$k$, and $D'$ is $n-k$-by-$n-k$, for all $t > 0$.
Continuous powers of a GDN matrix $A = (a_{ij})$ are also well defined under Hadamard multiplication. Namely, for $\alpha>0$ $$A^{(\alpha)} = (a_{ij}^\alpha).$$ Contrary to the conventional multiplication, in the Hadamard case, entrywise nonnegativity is clear, but the nonnegativity of the eigenvalues is in question. It was shown in [@Hadamard] that the critical exponent for continuous Hadamard powering of doubly nonnegative matrices is also $n-2$. So it is natural to ask whether there exists a critical exponent without the symmetry condition and consider generalized doubly nonnegative matrices; however, the $n-2$ critical exponent does not generalize. In fact the critical exponent does not exist as demonstrated in the case below:
If $$A =
\begin{bmatrix}
2 & 1 & 1 \\
1 & 1 & 1 \\
1 & 5 & 2 \\
\end{bmatrix},$$ then the eigenvalues of $A^{(\alpha)}$ are $\lambda_1(\alpha) = 2^\alpha - 1$, $\lambda_2(\alpha) = 2^\alpha + \frac{1}{2} + \sqrt{5^\alpha + \frac{5}{4}}$ and $\lambda_3(\alpha) = 2^\alpha + \frac{1}{2} - \sqrt{5^\alpha + \frac{5}{4}}$. Because $\lambda_3(\alpha) < 0$ for all $\alpha > 1$, the critical exponent does not exist.
Questions
=========
In this section, we collect some questions that naturally arise when studying the GDN critical exponent. They are not only important and helpful in finding the GDN critical exponent, but are also interesting in their own right.
Are GDN critical exponents for all $n$-by-$n$ matrices integers?
The critical exponent for both conventional and Hadamard powering of DN matrices turn out to be the integer $n-2$. It is natural to ask whether the same holds true in the conventional powering of GDN matrices. If that is indeed the case, then we can conclude $CE_4 = 4$ by the argument from section 4. Moreover, in the case of conventional powering of DN, the maximum critical exponent is achieved by tridiagonal matrices. If $CE_n$ is also an integer and is achieved by a certain class of matrices, then we would have $2n-4$ as an upper bound for the critical exponent. To see that, suppose $A$ is a GDN matrix with the integer critical exponent $CE_n$. Then $A^{CE_n}$ has a zero entry. Because the index of primitivity of GDN matrices is at most $2n-3$ as shown in Lemma \[2n-d-1\], $CE_n \le 2n-4$.
For which zero-nonzero patterns of primitive matrices do there exist GDN matrices?
What is the relation between $MIP_n$ and $CE_n$?
We have seen in section 4 that $MIP_n$ is closely related to $CE_n$, and the knowledge of the relation between $MIP_n$ and $CE_n$ would help us gain information on one given knowledge about the other.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We study the power spectrum of the mass density perturbations in an inflation scenario that includes thermal dissipation. We show that the condition on which the thermal fluctuations dominate the primordial density perturbations can easily be realized even for weak dissipation, [*i.e.*]{}, the rate of dissipation is less than the Hubble expansion. We find that our spectrum of primordial density perturbations follows a power law behavior, and exhibits a “thermodynamical” feature – the amplitude and power index of the spectrum depend mainly on the thermodynamical variable $M$, the inflation energy scale. Comparing this result with the observed temperature fluctuations of the cosmic microwave background, we find that both amplitude and index of the power spectrum can be fairly well fitted if $M \sim 10^{15}-10^{16}$ GeV.'
address: 'Department of Physics, University of Arizona, Tucson, AZ 85721'
author:
- 'Wolung Lee and Li-Zhi Fang'
title: Mass Density Perturbations from Inflation with Thermal Dissipation
---
\#1[[$\backslash$\#1]{}]{}
Introduction {#sec:level1}
============
In the past decade, there has been a number of studies on dissipative processes associated with the inflaton decay during its evolution. These studies have shed light into the possible effects of the dissipative processes. For instance, it was realized that dissipation effectively slows down the rolling of the inflaton scalar field $\phi$ toward the true vacuum. These processes are capable of supporting the scenario of inflation [@AST; @KT]. Recently, inspired by several new developments, the problem of inflation with thermal dissipation has attracted many re-investigations.
The first progress is from the study of the non-equilibrium statistics of quantum fields, which has found that, under certain conditions, it seems to be reasonable to introduce a dissipative term (such as a friction-like term) into the equation of motion of the scalar field $\phi$ to describe the effect of heat contact between the $\phi$ field and a thermal bath. These studies shown that the thermal dissipation and fluctuation will most likely appear [*during*]{} the inflation if the inflaton is coupled to light fields[@GM]. However, to realize sufficient e-folds of inflation with thermal dissipation, this theory needs to introduce tens of thousands of scalar and fermion fields interacting with the inflaton in an [*ad hoc*]{} manner[@BGR]. Namely, it is still far from a realistic model. Nevertheless, this study indicates that the condition necessary for the “standard" reheating evolution – a coupling of inflaton with light fields – is actually also the condition under which the effects of thermal dissipation during inflation should be considered.
Secondly, in the case of a thermal bath with a temperature higher than the Hawking temperature, the thermal fluctuations of the scalar field plays an important and even dominant role in producing the primordial perturbations of the universe. Based on these results, the warm inflation scenario has been proposed. In this model, the inflation epoch can smoothly evolve to a radiation-dominated epoch, without the need of a reheating stage[@BF; @LF]. Dynamical analysis of systems of inflaton with thermal dissipation[@OR] gives further support to this model. It is found that the warm inflation solution is very common. A rate of dissipation as small as $10^{-7}\ H$, $H$ being the Hubble parameter during inflation, can lead to a smooth exit from inflation to radiation.
Warm inflation also provides explanation to the super-Hubble suppression. The standard inflationary cosmology, which is characterized by an isentropic de Sitter expansion, predicts that the particle horizon should be much larger than the present-day Hubble radius $c/H_0$. However, a spectral analysis of the COBE-DMR 4-year sky maps seems to show a lack of power in the spectrum of the primordial density perturbations on scales equal to or larger than the Hubble radius $c/H_0$[@JF; @BFH]. A possible explanation of this super-Hubble suppression is given by hybrid models, where the primordial density perturbations are not purely adiabatic, but mixed with an isocurvature component. The warm inflation is one of the mechanisms which can naturally produce both adiabatic and isocurvature initial perturbations[@LF].
In this paper, we study the power spectrum of mass density perturbations caused by inflation with thermal dissipation. One purpose of developing the model of warm inflation is to explain the amplitudes of the initial perturbations. Usually, the amplitude of initial perturbations from quantum fluctuation of the inflaton depends on some unknown parameters of the inflation potential. However, for the warm inflation model, the amplitude of the initial perturbations is found to be mainly determined by the energy scale of inflation, $M$. If $M$ is taken to be about $10^{15}$ GeV, the possible amplitudes of the initial perturbations are found to be in a range consistent with the observations of the temperature fluctuations of the cosmic microwave background (CMB) [@cobe]. That is, the thermally originated initial perturbations apparently do not directly depend on the details of the inflation potential, but only on some thermodynamical variables, such as the energy scale $M$. This result is not unexpected, because like many thermodynamical systems, the thermal properties including density fluctuations should be determined by the thermodynamical conditions, regardless of other details.
Obviously, it would be interesting to find more “thermodynamical" features which contain only observable quantities and thermodynamical parameters, as these predictions would be more useful for confronting models with observations. Guided by these considerations, we will extend the above-mentioned qualitative estimation of the order of the density perturbations to a quantitative calculation of the power spectrum of the density perturbations. We show that the power spectrum of the warm inflation does not depend on unknown parameters of the inflaton potential and the dissipation, but only on the energy scale $M$. The spectrum is found to be of power law, and the index of the power law can be larger or less than 1. More interestingly, we find that for a given $M$, the amplitude and the index of the power law are not independent from each other. In other words, the amplitude of the power spectrum is completely determined by the power index and the number $M$. Comparing this result with the observed temperature fluctuations of the CMB, we find that both amplitude and index of the power spectrum can be fairly well fitted if $M \sim 10^{15}-10^{16}$ GeV.
This paper is organized as follow: In Sec. II we discuss the evolution of the radiation component for inflationary models with dissipation prescribed by a field-dependent friction term. In particular, we scrutinize the physical conditions on which the thermal fluctuations dominate the primordial density perturbations. Section III carries out the calculations of the power spectrum of the density perturbations of the warm inflations. And finally, in Sec. IV we give the conclusions and discuss further observational tests.
Inflation with Thermal Dissipation
==================================
Basic equations
---------------
Let us consider a flat universe consisting of a scalar inflaton field $\phi$ and a thermal bath. Its dynamics is described by the following equations[@BF]. The equations of the expanding universe are $$2 \dot{H} +3 H^{2} = -\frac{8\pi}{m_{\rm Pl}^2}
\left[\frac{1}{2}\dot{\phi}^2 + \frac{1}{3}\rho_{r} - V(\phi)\right],$$ $$H^{2} = \frac{8\pi}{3}\frac{1}{m_{\rm Pl}^2}\left[\rho_{r} +
\frac{1}{2}\dot\phi^{2} + V(\phi)\right],$$ where $H=\dot{R}/R$ is the Hubble parameter, and $m_{\rm Pl} = \sqrt {1/G}$ the Planck mass. $V(\phi)$ is the effective potential for field $\phi$, and $\rho_{r}$ is the energy density of the thermal bath. Actually the scalar field $\phi$ is not uniform due to fluctuations. Therefore, the field $\phi$ in Eqs.(2.1) and (2.2) should be considered as an average over the fluctuations.
The equation of motion for scalar field $\phi$ in a de-Sitter universe is $$\ddot{\phi} + 3H \dot{\phi} + \Gamma\dot{\phi}
- e^{-2Ht}\nabla^2\phi+ V'(\phi)=0,$$ where the friction term $\Gamma \dot{\phi}$ describes the interaction between the $\phi$ field and a heat bath. Obviously, for a uniformed field, or averaged $\phi$, the term $\nabla^2\phi$ of Eq. (2.3) can be ignored. Statistical mechanics of quantum open systems has shown that the interaction of quantum fields with thermal or quantum bath can be described by a general fluctuation-dissipation relation[@weiss]. It is probably reasonable to describe the interaction between the inflaton and the heat bath as a “decay" of the inflaton [@BHP]. These results support the idea of introducing a damping or friction term into the field equation of motion. In particular, the friction term with the form in Eq. (2.3), $\Gamma\dot{\phi}$, is a possible approximation for the dissipation of $\phi$ field in a heat bath environment in the near-equilibrium circumstances. In principle, $\Gamma$ can be a function of $\phi$. In the cases of polynomial interactions between $\phi$ field and bath environment, one may take the polynomial of $\phi$ for $\Gamma$, [*i.e.*]{}, $\Gamma= \Gamma_m \phi^m$. The friction coefficient must be positive definite, hence $\Gamma_m > 0$, and the dissipative index of friction $m$ should be zero or even integer if $V(\phi)$ is invariant under the transformation $\phi \rightarrow -\phi$.
The equation of the radiation component (thermal bath) is given by the first law of thermodynamics as $$\dot \rho_{r} + 4H\rho_{r} = \Gamma\dot{\phi}^{2}.$$ The temperature of the thermal bath can be calculated by $\rho_{r}=
(\pi^{2}/30) g_{\rm eff}T^{4}$, $g_{\rm eff}$ being the effective number of degrees of freedom at temperature $T$.
The warm inflation scenario is generally defined by a characteristic that the thermal fluctuations of the scalar field dominate over the quantum origin of the initial density perturbations. Because the thermal and quantum fluctuations of the scalar field are proportional to $T$ and $H$ respectively, a necessary condition for warm inflation models is the existence of a radiation component with temperature $$T > H$$ during the inflationary expansion. Eq. (2.5) is also necessary for maintaining the thermal equilibrium of the radiation component. In general, the time scale for the relaxation of a radiation bath is shorter for higher temperature. Accordingly, to have a relaxing time of the bath shorter than the expansion of the universe, a temperature higher than $H$ is generally needed.
As a consequence of Eq. (2.5), warm inflation scenario requires that the solutions of Eqs. (2.1) - (2.4) should contain an inflation era, followed by smooth transition to a radiation-dominated era. Dynamical system analysis also confirmed that for a massive scalar field $V(\phi) = \frac{1}{2}M^2\phi^2$, the warm inflation solution of Eqs. (2.1) - (2.4) is very common. A smooth exit from inflation to radiation era can be established even for a dissipation with $\Gamma$ as small as $10^{-7}H$[@OR]. A typical solution of warm inflation will be given in next section.
Evolution of Radiation component during inflation
-------------------------------------------------
Since warm inflation solution does not rely on a specific potential, we will employ the popular $\phi^4$ potential commonly used for the “new" inflation models. It is $$V(\phi)=\lambda(\phi^{2}-\sigma^{2})^{2}.$$ To have slow-roll solutions, the potential should be flat enough, [*i.e.*]{}, $\lambda \leq (M/m_{\rm Pl})^4$, where $V(0)\equiv M^{4}=\lambda \sigma^{4}$.
For models based on the potential of Eq. (2.6), the existence of a thermal component during inflation seems to be inevitable. In order to maintain the $\phi$ field close to its minimum at the onset of the inflation phase transition, a thermal force is generically necessary. In other words, there is, at least, a weak coupling between $\phi$ field and other fields contributing to the thermal bath. During the slow roll period of inflation, the potential energy of the $\phi$ field is fairly constant, and their kinetic energy is small, so that the interaction between the $\phi$ field with the fields of the thermal bath remains about the same as at the beginning. As such, there is no compelling reason to ignore these interactions.
Strictly speaking, we should use a finite temperature effective potential $V(\phi,T)$. However, the correction due to finite temperature is negligible. The leading temperature correction of the potential (2.6) is $\lambda T^2
\phi^2$. On the other hand, as mentioned above, we have $\lambda \leq
(M/m_{\rm Pl})^4$ for the flatness of the potential. Therefore, $\lambda T^2
\leq M^6/m_{\rm Pl}^4 \sim (M/m_{\rm Pl})^2H^2 \ll H^2$, [*i.e.*]{}, the influence of the finite temperature effective potential can be ignored when $\phi < m_{\rm Pl}$.
Now, we try to find warm inflation solutions of Eqs. (2.1) - (2.4) for weak friction $\Gamma < H$. In this case, Eqs. (2.1) - (2.3) are actually the same as the “standard" new inflation model when $$\rho_r \ll V(0).$$ Namely, we have the slow-roll solution as $$\dot{\phi} \simeq - \frac{V'(\phi)}{3H + \Gamma(\phi)} \simeq
\frac{V'(\phi)}{3H},$$ $$\frac{1}{2} \dot{\phi}^2 \ll V(0),$$ and $$H^2 \simeq H_i^2 \equiv \frac{8\pi}{3}\frac{V(0)}{m_{\rm Pl}^2}
\simeq \left(\frac{M}{m_{\rm Pl}}\right)^2 M^2,$$ where the subscript $i$ denotes the starting time of the inflation epoch.
During the stage of $\phi \ll \sigma$, it is reasonable to neglect the $\phi^{3}$ term in Eq. (2.3). We have then $$\ddot{\phi}+(3H + \Gamma)\dot{\phi}-4\lambda\sigma^2 \phi=0.$$ Considering $\Gamma < H$, an approximate solution of $\phi$ can immediately be found as $$\phi =\phi_i e^{\alpha Ht},$$ where $\alpha \simeq \lambda^{1/2}(m_{\rm Pl}/M)^2/2\pi$ and $\phi_i$ is the initial value of the scalar field.
Substituting solution (2.12) into Eq. (2.4), we have the general solution of (2.4) as $$\rho_r(t) = A e^{(m+2)\alpha Ht} + B e^{-4Ht}$$ where $A=\alpha^{2}H\Gamma_m\phi_i^{m+2}/[(m+2)\alpha+4]$, $B = \rho_r(0) -
A$, and $\rho_r(0)$ is the initial radiation density. Obviously, the term $B$ in Eq. (2.12) describes the blowing away of the initial radiation by the inflationary exponential expansion, and the term $A$ is due to the generation of radiation by the $\phi$ field decay.
According to Eq. (2.13), the evolution of the radiation has two phases. Phase 1 covers the period during which the $B$ term is dominant, and radiation density drops drastically due to the inflationary expansion. The component of radiation evolves into phase 2 when the $A$ term becomes dominant, where the radiation density increases due to the friction of the $\phi$ field. Namely, both heating and inflation are simultaneously underway in phase 2. Therefore, this phase is actually the era of inflation plus reheating.
The transition from phase 1 to phase 2 occurs at time $t_b$ determined by $(d\rho/dt)_{t_b}=0$. We have $$Ht_b \simeq \frac{1}{(m+2)\alpha + 4}
\ln \left\{\frac{4[(m+2)\alpha+4]}{(m+2)\alpha^3H}\cdot
\frac {aM^4}{\Gamma_m\phi_i^{m+2}} \right\},$$ where $a \equiv (\pi^2/30)g_{\rm eff}$. Then the radiation density at the rebound time becomes $$\rho_r(t_b)= \frac{1}{4}
\left[(m+2)\alpha+4\right]A\exp [(m+2)\alpha Ht_b].$$
From Eqs. (2.12) and (2.13), the radiation density in phase 2 is given by $$\rho_r(t) = \frac{1}{4}\alpha^2 \Gamma H \phi^2(t)
\simeq \frac{1}{16\pi^2} \lambda\left(\frac{m_{\rm Pl}}{M} \right )^4
\Gamma H \phi^2(t).$$ Since $H \simeq (M/m_{\rm Pl})M$, Eq. (2.16) can be rewritten as $$\rho_r(t) \sim
\lambda^{1/2} \left(\frac{m_{\rm Pl}}{M} \right )^2 \frac{\Gamma}{H}
\left (\frac{\phi(t)}{\sigma}\right)^2 V(0).$$ On the other hand, from (2.12), we have $$\frac{1}{2}\dot{\phi}(t)^2 \simeq \lambda^{1/2}
\left(\frac{m_{\rm Pl}}{M} \right )^2
\left (\frac{\phi(t)}{\sigma}\right)^2 V(0).$$ Therefore, in the case of weak dissipation $\Gamma <H$, we have $$\rho_r (t) < \dot{\phi}(t)^2 /2.$$ This is consistent with the condition of inflation Eq. (2.7) when Eq. (2.9) holds.
Eqs. (2.7) and (2.9) indicate that the inflation will come to an end at time $t_f$ when the energy density of the radiation components, or the kinetic energy of $\phi$ field, $\dot{\phi}^2/2$, become large enough, and comparable to $V(0)$. From Eqs. (2.17) and (2.18), $t_f$ is given by $$\lambda^{1/2} \left(\frac{m_{\rm Pl}}{M} \right )^2
\left (\frac{\phi(t_f)}{\sigma}\right)^2 \simeq 1.$$
In general, at the time when the phase 2 ends, or a radiation-dominated era starts, the potential energy may not be fully exhausted yet. In this case, a non-zero potential $V$ will remain in the radiation-dominated era, and the process of $\phi$ decaying into light particles is still continuing.
However, considering $\lambda^{1/2} (m_{\rm Pl}/{M})^2(\Gamma/H) <1$, the right hand side of Eq. (2.17) will always be less than 1 when $\phi(t)$ is less than $\sigma$. This means that, for weak dissipation, phase 2 cannot terminate at $\phi(t) < \sigma$, or $V(\phi(t_f)) \neq 0$. Therefore, under weak dissipation, phase 2 will end at the time $t_f$ when the potential energy $V(\phi)$ is completely exhausted, [*i.e.*]{}, $$\phi(t_f) \sim \sigma.$$ This means that no non-zero $V$ remains once the inflation exits to a radiation-dominated era, and the heating of $\phi$ decay also ends at $t_f$.
Temperature of radiation
------------------------
From Eq. (2.13), one can find the temperature $T$ of the radiation in phases 1 ($t<t_b$) and 2 ($t>t_b$) as $$T(t) = \left\{ \begin{array}{ll}
T_{b}e^{-H(t-t_b)}, & \mbox{if \ $t < t_b$}, \\
T_{b} e^{(m+2)\alpha H (t-t_b)/4}, & \mbox{if \ $t_f > t > t_b$},
\end{array}
\right.$$ where $$T_b = (4a)^{-1/4}[(m+2)\alpha+4]^{1/4}A^{1/4}
\exp \left[\left(\frac{m+2}{4}\right)\alpha Ht_b \right].$$ The temperature $T_f$ at the end of phase 2 is $$T_f=T(t_f) = T_b e^{(m+2)\alpha H (t_f-t_b)/4},$$ where $t_f$ is given by Eq. (2.21).
Since $T(t)$ is increasing with $t$ in phase 2, the condition (2.5) for warm inflation can be satisfied if $T(t_f) > H$, or $$\rho_r(t_f) > a H_i^4.$$ Using Eqs. (2.17) and (2.21), condition (2.25) is realized if $$\frac{\Gamma}{H} > \left(\frac{\sigma}{m_{\rm Pl}}\right)^2
\left(\frac{M}{m_{\rm Pl}} \right )^4.$$ Namely, $\Gamma$ can be as small as $10^{-12} H$ for $M \sim 10^{16}$ GeV, and $\sigma \sim 10^{19}$ GeV. Therefore, the radiation solution (2.13), or warm inflation, should be taken into account in a very wide range of dissipation $$10^{-12} H < \Gamma < H.$$ This result is about the same as that given by dynamical system analysis[@OR]: a tiny friction $\Gamma$ may lead the inflaton to a smooth exit directly at the end of the inflation era.
A typical solution of the evolution of radiation temperature $T(t)$ is demonstrated in Fig. 1, for which parameters are taken to be $M = 10^{15}$ GeV, $\sigma = 2.24 \cdot 10^{19}$ GeV, $\Gamma_2 = 10^{-5}H_i$ and $g_{\rm
eff}$ = 100. Actually, $g_{\rm eff}$-factor is a function of $T$ in general. However, as can be seen below, the unknown function $g_{\rm eff}(T)$ has only a slight effect on the problems under investigation. Figure 1 shows that the rebound temperature $T_b$ can be less than $H$. In this case, the evolution of $T(t)$ in phase 2 can be divided into two sectors: $T < H$ for $t < t_e$, and $T >H $ for $t > t_e$, where $t_e$ is defined by $T(t_e) =H$. We should not consider the solution of radiation to be physical if $T < H$ since it is impossible to maintain a thermalized heat bath with the radiation temperature less than the Hawking temperature $H$ of an expanding universe. Nevertheless, the solution (2.13) should be available if $t > t_e$. Therefore, one can only consider the period of $t_e < t < t_f$ as the epoch of the warm inflation.
Figure 1 also plots the Hubble parameter $H(t)$. The evolution of $H(t)$ is about the same as in the standard new inflation model, [*i.e.*]{}, $H(t) \sim H_i$ in both phases 1 and 2. In Fig. 1, it is evident that the inflation smoothly exits to a radiation era at $t_f$. The Hubble parameter $H(t)$ also evolves from the inflation $H(t) \sim $ constant to a radiation regime $H(t) \propto t^{-1}$.
The duration of the warm inflation is represented by ($t_f - t_e$) then. The number of $e$-folding growth of the comoving scale factor $R$ during the warm inflation is given by $$N \equiv \int_{t_e}^{t_f} Hdt \simeq \frac {4}{(m+2)\alpha} \ln \frac
{T_f}{H}.$$ One can also formally calculate the number of $e$-folds of the growth in phase 2 as $$N_2 \equiv \int_{t_b}^{t_f} H dt \simeq
\frac {4}{(m+2)\alpha} \ln \frac {T_f}{T_b},$$ and the number of $e$-folds of the total growth as $$N_t \equiv \int_{0}^{t_f} Hdt \simeq \frac{4}{(m+2)\alpha}
\ln \left(\frac {T_f}{T_b}\right) + Ht_b.$$
It can be found from Eqs. (2.28) - (2.30) that both $N_2$ and $N_t$ depend on the initial value of the field $\phi_i$ via $T_b$, but $N$ does not. The behavior of $T$ at the period $t > t_e$ is completely determined by the competition between the diluting and producing radiation at $t>t_b$. Initial information about the radiation has been washed out by the inflationary expansion. Hence, the initial $\phi_i$ will not lead to uncertainty in our analysis if we are only concerned the problems of warm evolution at the period $t_e < t < t_f$.
The primordial density perturbations
====================================
Density fluctuations of the $\phi$ field
----------------------------------------
The fluctuations of $\phi$ field can be calculated by the similar way as stochastic inflations[@star]. Recall that the coarse-grained scalar field $\phi$ is actually determined from the decomposition between background and high frequency modes, i.e. $$\Phi({\bf x}, t)=\phi({\bf x}, t) + q({\bf x}, t),$$ where $\Phi({\bf x},t)$ is the scalar field satisfying $$\ddot{\Phi} + 3H \dot{\Phi} - e^{-2Ht}\nabla^2\Phi+ V'(\Phi)=0.$$ $q({\bf x}, t)$ in Eq.(3.1) contains all high frequency modes and gives rise to the thermal fluctuations. Since the mass of the field can be ignored for the high frequency modes, we have $$q({\bf x}, t) = \int d^3k W(|{\bf k}|)
\left [ a_{\bf k}\sigma_{\bf k}(t)e^{-i{\bf k}\cdot {\bf x}}
+ a^{\dagger}_{\bf k}\sigma^{*}_{\bf k}(t) e^{i{\bf k}\cdot {\bf x}} \right ]$$ where $k$ is comoving wave vector, and modes $\sigma_{\bf k}(t)$ is given by $$\sigma_{\bf k}(t) = \frac{1}{(2\pi)^{3/2}}\frac{1}{\sqrt{2k}}
\left [ H\tau - i\frac{H}{k}\right ] e^{-ik\tau},$$ and $\tau=-H^{-1}\exp(-Ht)$ is the conformal time. Eq.(3.3) is appropriate in the sense that the self-coupling of the $\phi$ field is negligible. Considering the high frequency modes are mainly determined by the heat bath, this approximation is reasonable. The window function $W(|{\bf k}|)$ is properly chosen to filter out the modes at scales larger than the horizon size $H^{-1}$, [*i.e.*]{}, $W(k)=\theta(k-k_{h}(t))$, where $k_{h}(t)\simeq (1/ \pi)H \exp(Ht)$ [^1] is the lower limit to the wavenumber of thermal fluctuations.
From Eqs.(3.1) and (3.3), with the slow-roll condition, Eq.(3.2) renders $$3H\dot{\phi} - e^{-2Ht}\nabla^2\phi+ V'(\Phi)|_{\Phi=\phi}=
3H\eta({\bf x},t),$$ and $$\eta({\bf x}, t)=\left (-\frac{\partial}{\partial t}
+\frac{1}{3H}e^{-2Ht}\nabla^2\right) q({\bf x}, t).$$ Eq.(3.5) can be rewritten as $$\frac {d\phi({\bf x}, t)}{dt} = - \frac{1}{3H}
\frac{\delta F[\phi({\bf x}, t)]}{\delta \phi} +
\eta({\bf x}, t)$$ where $$F[\bar{\phi}]=\int d^3{\bf x} \left[ \frac{1}{2}
(e^{-Ht}\nabla\bar{\phi})^2 +
V(\bar{\phi}) \right]$$ Eq. (3.7) is, in fact, the rate equation of the order parameter $\phi$ of a system with free energy $F[\phi]$. It describes the approach to equilibrium for the system during phase transition.
Using the expression of free energy (3.8), the slow-roll solution (2.8) can be rewritten as $$\frac {d\phi}{dt} = - \frac{1}{3H + \Gamma}
\frac{dF[\phi]}{d \phi}.$$ Hence, in the case of weak dissipation ($\Gamma < H$), Eq. (3.7) is essentially the same as the slow-roll solution (2.8) or Eq. (3.9) but with fluctuations $\eta$. The existence of the noise field ensures that the dynamical system properly approaches the global minimum of the inflaton potential $V(\phi)$. Strictly speaking, both the dissipation $\Gamma$ and fluctuations $\eta$ are consequences derived from $q({\bf x},t)$. They should be considered together. However, it seems to be reasonable to calculate the fluctuations alone if the dissipation is weak.
Unlike (3.2), the Langevin equation (3.7) is of first order ($\dot{\phi}$) due to the slow-roll condition. Generally, thermal fluctuations will cause both growing and decaying modes[@GM] [^2]. Therefore, the slow-roll condition simplifies the problem from two types of fluctuation modes to one, [*i.e.*]{}, we can directly calculate the total fluctuation as the superposition of various fluctuations. It has been shown [@berera] that during the eras of dissipations, the growth of the structures in the universe is substantially the same as surface roughening due to stochastic noise. The evolution of the noise-induced surface roughening is described by the so-called KPZ-equation [@kardar]. Eqs.(3.5) or (3.7), which includes terms of non-linear drift plus stochastic fluctuations, is a typical KPZ-like equation.
From Eq.(3.6), the two-point correlation function of $\eta({\bf x}, t)$ can be found as $$\langle \eta({\bf x}, t) \eta({\bf x'}, t') \rangle
=\frac{H^3}{4\pi^2}\left [ 1+ \frac{2}{\exp(H/\pi T)-1}\right]
\frac{\sin(k_{h}|{\bf x}-{\bf x'}|)}{k_{h}|{\bf x}-{\bf x'}|}
\delta(t-t'),$$ where $1/[\exp(H/\pi T)-1]$ is the Bose factor at temperature $T$. Therefore, when $T > H$, we have $$\langle \eta({\bf x}, t) \eta({\bf x}, t') \rangle
=\frac{H^2T}{2\pi}\delta(t-t').$$ This result can also be directly obtained via the fluctuation-dissipation theorem[@hohe; @hu]. In order to accord with the dissipation terms of Eq. (3.7), the fluctuation-dissipation theorem requires the ensemble average of $\eta$ to be given by $$\langle \eta \rangle = 0$$ and $$\langle \eta({\bf x},t) \eta({\bf x},t') \rangle = D\delta(t-t') \ .$$ The variance $D$ is determined by $$D=2\frac{1}{U} \frac{T}{3H+\Gamma}~,$$ where $U = (4\pi/3) H^{-3}$ is the volume with Hubble radius $H^{-1}$. In the case of weak dissipation, we then recover the same result as in Eq.(3.11), $$D=H^2T/2\pi.$$ When $T=H$, we obtain $$D=\frac{H^3}{2\pi},$$ which agrees exactly the result derived from quantum fluctuations of $\phi$-field[@star]. Therefore, the quantum fluctuations of inflationary $\phi$ field are equivalent to the thermal noises stimulated by a thermal bath with the Hawking temperature $H$. Eqs. (3.15) and (3.16) show that the condition (2.5) is necessary and sufficient for a warm inflation.
For long-wavelength modes, the $V'(\phi)$ term is not negligible. It may lead to a suppression of correlations on scales larger than $|V''(\phi)|^{-1/2}$. However, before the inflaton actually rolls down to the global minimum, we have $|V''(\phi\ll \sigma)|^{-1/2} \geq H^{-1}$. The so-called abnormal dissipation of density perturbations [@kirk] may produce more longer correlation time than $H$. Therefore in phase 2, [*i.e.*]{}, the warm inflation phase $H< T < M$, the long-wavelength suppression will not substantially change the scenario presented above.
The fluctuations $\delta \phi$ of the $\phi$ field can be found from linearizing Eq. (3.7). If we only consider the fluctuations $\delta \phi$ crossing outside the horizon, [*i.e.*]{}, with wavelength $\sim H^{-1}$, the equation of $\delta \phi$ is $$\frac{d \delta \phi}{dt}= -
\frac{H^2 + V^{''}(\phi)}{3H+\Gamma}\delta \phi + \eta.$$ For the slow-roll evolution, we have $|V^{''}(\phi)| \ll 9H^2$ [@KT]. One can ignore the $V^{''}(\phi)$ term on the right hand side of Eq. (3.17). Accordingly, the correlation function of the fluctuations is $$\langle \delta \phi(t) \delta \phi(t') \rangle
\simeq D \frac{3H+\Gamma}{2H^2} e^{-(t-t')H^2/(3H +\Gamma)},
\ \ \ t > t',$$ hence $$\langle (\delta \phi)^2 \rangle
\sim \frac{3}{4\pi}HT$$ Thus, in the period $t_e< t < t_f$ the density perturbations on large scales are produced by the thermal fluctuations that leave the horizon with a Gaussian-distributed amplitude having a root-mean-square dispersion given by Eq. (3.19).
Principally, the problem of horizon crossing of thermal fluctuations given by Eq. (3.7) is different from the case of quantum fluctuations, because the equations of $H$ and $\dot H$, (2.1) and (2.2) contain terms in $\rho_r$. However, these terms are insignificant for weak dissipation \[Eq. (2.19)\] in phase 2. Thus Eqs.(2.1) and (2.2) depend only nominally on the evolution of $\rho_r$. Accordingly, for weak dissipation, the behavior of thermal fluctuations at horizon crossing can be treated by the same way as the evolutions of quantum fluctuations in stochastic inflation. In that theory, quantum fluctuations of inflaton are assumed to become classical upon horizon crossing and act as stochastic forces. Obviously, this assumption is not necessary for thermal fluctuations. Moreover, we will show that in phase 2 the thermal stochastic force $HT$ is contingent upon the comoving scale of perturbations by a power law \[Eqs. (2.21) and (3.21)\], and therefore the power spectrum of the thermal fluctuations obeys the power law. This make it more easier to estimate the constraint quantity in the super-horizon regime.
Accordingly, the density perturbations at the horizon re-entry epoch are characterized by[@KT] $$\left(\frac{\delta\rho}{\rho} \right)_h =
\frac{-\delta\phi V^{\prime}(\phi)}{\dot\phi^{2} + (4/3)\rho_{r}} \ .$$ All quantities in the right-hand side of Eq. (3.20) are calculated at the time when the relevant perturbations cut across the horizon at the inflationary epoch.
Using the solutions of $\phi$ and $\rho_r$ of warm inflation (2.12) and (2.13), Eq. (3.20) gives $$\left(\frac{\delta\rho}{\rho} \right )_h \simeq
\left(\frac{5\cdot3^{3m/2+4}}{2^{m+3}\cdot\pi^{m/2+3}}\right)^{\frac{1}{m+2}}
\cdot \left(\frac{\gamma_m}{g_{\rm eff}\alpha^m}\right)^{\frac{1}{m+2}}
\left(\frac{T}{H}\right)^{\frac{1}{2}\left(\frac{m-6}{m+2}\right)},$$ where the dimensionless parameter $\gamma_m \equiv \Gamma_m H^{m-1}$, and $T$ is the temperature at the time when the considered perturbations $\delta
\rho_r$ crossing out of the horizon $H^{-1} \sim H_i^{-1}$. Eq. (3.21) shows that the density perturbations are insensitive to the $g_{\rm eff}$-factor.
Power law index
---------------
Since inflation is immediately followed by the radiation dominated epoch, the comoving scale of a perturbation with crossing over (the Hubble radius) at time $t$ is given by $$\frac {k}{H_0} = 2\pi \frac{H}{H_0} \frac{T_0}{T_f}e^{H(t-t_f)},$$ where $T_0$ and $H_{0}$ are the present CMB temperature and Hubble constant respectively. Eq. (3.22) shows that the smaller $t$ is, the smaller $k$ will be. This is the so-called “first out - last in" of the evolution of density perturbations produced by the inflation.
Using Eqs. (2.22) and (3.22), the perturbations (3.21) can be rewritten as $$\left\langle \left ( \frac{\delta\rho}{\rho} \right )^2 \right \rangle_h
\propto k^{(m-6)\alpha/4}, \ \ \ \ {\rm if} \ \ \ k > k_e,$$ where $k_e$ is the wavenumber of perturbations crossing out of horizon at $t_e$. It is $$k_e = 2\pi H\frac{T_0}{T_f} e^{H(t_e-t_f)}
\simeq 2\pi H\frac{T_0}{T_f} e^{-N}.$$ Therefore, the primordial density perturbations produced during warm inflation are of power law with an index $(m-6)\alpha/4$. We may also express the power spectrum of the density perturbations at a given time $t$. It is $$\left\langle \left ( \frac{\delta\rho}{\rho} \right )^2 \right \rangle_t
\propto k^{3+n}, \ \ \ \ {\rm if} \ \ \ k > k_e,$$ where the spectral index $n$ is $$n= 1 + \left(\frac{m-6}{4}\right)\alpha.$$
Clearly, for $m = 6$, the warm inflation model generates a flat power spectrum $n=1$, yet the power spectrums will be tilted for $m \neq 6$. The dissipation models $\Gamma = \Gamma_{m}\phi^{m}$ may not be realistic for higher $m$, but we will treat $m$ like a free parameter in order to show that the results we concerned actually are not very sensitive to these parameters.
The warm inflation scenario requires that all perturbations on comoving scales equal to or less than the present Hubble radius originate in the period of warm inflation. Hence, the longest wavelength of the perturbations (3.24), [*i.e.*]{}, $2\pi/k_{e}$, should be larger than the present Hubble radius $H_0^{-1}$. We have then $$N > \ln \left(\frac{HT_0}{H_0T_f}\right) =
\ln \left(\frac{T_0}{H_0}\right) - \ln \left(\frac{T_f}{H}\right) \sim 55,$$ where we have used $(T_0/H_0)\gg (T_f/H)$, as $T_f \leq M$. Using Eq. (2.28), the condition (3.27) gives an upper bound to $\alpha$ for a given $m$ as $$\alpha_{\rm max} = \left(\frac{4}{m+2}\right) \frac
{\ln(T_f/H)}{\ln(T_0/H_0)}.$$ Thus, the possible area of the index $n$ can be found from Eq. (3.27) as $$n = \left \{ \begin{array}{ll}
1 - (6-m)\alpha_{\rm max}/4 \ {\rm to}\ 1, & \mbox{if \ $m<6$}, \\
1\ {\rm to}\ 1 + (m - 6)\alpha_{\rm max}/4, & \mbox{if \ $m>6$}.
\end{array}
\right.$$ Therefore, the power spectrum is positive-titled ([*i.e.*]{}, $n>1$) if $m>6$, and negative-titled ($n<1$) if $m < 6$. Figure 2 plots the allowed area of $n$ as a function of the inflation mass scale $M$. Apparently, for $M \geq 10^{16}$ GeV, the tilt $|n-1|$ should not be larger than about 0.15 regardless of the values of $m$ from 2 to 12.
Amplitudes of perturbations
---------------------------
To calculate the amplitude of the perturbations we rewrite spectrum (3.25) into $$\left\langle \left ( \frac{\delta\rho}{\rho} \right )^2 \right \rangle_h
= A \left( \frac{k}{k_0}\right )^{n-1}, \ \ \ \ {\rm if} \ \ \ k > k_e,$$ where $k_0=2\pi H_0$. $A$ is the spectrum amplitude normalized on scale $k=k_0$, corresponding to the scale on which the perturbations re-enter the Hubble radius $1/H_0$ at present time. From Eqs. (3.21), and (3.23), we have $$A =
\left(\frac{5\cdot3^{3m/2+4}}{2^{m+3}\cdot\pi^{m/2+3}}\right)^{\frac{2}{m+2}}
\left(\frac{\gamma_m}{g_{\rm eff}\alpha^m}\right)^{\frac{2}
{m+2}}\left(\frac{H_0T_f}{HT_0}\right)^{n-1}
\left(\frac{T}{H}\right)^{\frac{m-6}{m+2}} e^{(n-1)H(t_f - t)}.$$ Applying Eq. (2.21), the radiation temperature at the moment of horizon-crossing, $t$, can be expressed as $T(t) = T_f\exp[(m+2)\alpha
H(t-t_f)/4]$. With the help of Eq. (2.28), we obtain $$\left(\frac{T}{H}\right)^{\frac{m-6}{m+2}} \left(\frac{T_f}{H}\right)^{n-1}
e^{(n-1)H(t_f-t)} = \exp\left\{(n-1)\left[1+\left(\frac{m+2}
{4}\alpha \right)\right]N\right\}.$$ On the other hand, using Eqs. (2.20), (2.23) and (2.28), one has $$\gamma_m = \left(\frac{3}{4}\right)^{1-\frac{m}{2}}\cdot \frac{g_{\rm eff}}
{30}\left(\frac{M}{m_{\rm Pl}}\right)^{2m}\alpha^{-3-\frac{m}{2}}.$$ Substituting Eqs. (3.32) and (3.33) into Eq. (3.31), we have finally $$A = \left(\frac{3^{4-m}}{64\pi^{3+\frac{m}{2}}}\right)^{\frac{2}{m+2}}
\cdot \left(\frac{M}{m_{\rm Pl}}\right)^{\frac{4m}{m+2}}
\left(\frac{H_0}{T_0}\right)^{n-1} \alpha^{-3} \exp \left\{(n-1)
\left[1+\left(\frac{m+2}{4}\right)\alpha\right]N\right\}.$$
Eq. (3.34) shows that the amplitude $A$ does not contain the unknown $g_{\rm
eff}$-factor. Moreover, $\alpha$ can be expressed by $n$ and $m$ through Eq. (3.26), and $N$ can be expressed by $\alpha$ and $M$ via Eq. (2.28). Therefore, the amplitude of the initial density perturbations, $A$, is only a function of $M$, $n$, and $m$.
Figures 3 and 4 plot the relations between the amplitude $A$ and index $n$ for various parameters $M$ and $m$. In the case of $m=6$, $n=1$, the relation of $A$ and $\alpha$ is plotted in Fig. 5. It can be seen from Figs. 3, 4 and 5 that for either $m \geq 6$ or $m < 6$, the amplitude $A$ is significantly dependent on $M$, but not so sensitive to $m$. Namely, the testable $A$-$n$ relationship is mainly determined by a thermodynamical variable, the energy scale $M$. This is a “thermodynamical" feature. The relationship between $A$ and $N$ plotted in Figs. 6 and 7 also show this kind of “thermodynamical" feature: the $A$-$N$ relation depends mainly on $M$.
For comparison, the observed results of $A$ and $n$ derived from the 4-year COBE-DMR data (quadrupole moment $Q_{rms-PS} \sim 15.3_{-2.8}^{+3.7}\mu K$ and $n \sim 1.2 \pm 0.3$[@cobe]) are plotted in Figs. 3, 4 and 5. The observationally allowed $A$-$n$ range is generally in a good agreement with the predicted $A$-$n$ curve if $M \sim 10^{15} -10^{16}$ GeV, regardless the parameter $m$. Figures 3 and 4 also indicate that if the tilt of spectrum $|n-1|$ is larger than 0.1, the parameter area of $M\leq 10^{14}$ GeV will be ruled out. Therefore, the warm inflation seems to fairly well reconcile the initial perturbations with the energy scale of the inflation.
Conclusions and Discussion
==========================
Assuming that the inflaton $\phi$-field undergoes a dissipative process with $\Gamma\dot{\phi}^2$, we have studied the power spectrum of the mass density perturbations. In this analysis, we have employed the popular $\phi^4$ potential. However, only one parameter, the mass scale of the inflation $M$, is found to be important in predicting the observable features of power spectrum, [*i.e.*]{}, the amplitude $A$ and index $n$. Actually, the warm inflation scenario is based on two thermodynamical requirements: (a) the existence of a thermalized heat bath during inflation, and (b) that the initial fluctuations are given by the fluctuation-dissipation theorem. Therefore, we believe that the “thermodynamical" features – $A$ and $n$ depend only on $M$ – would be generic for the warm inflation. This feature is useful for model testing. Hence, the warm inflation can be employed as an effective working model when more precise data about the observable quantities $A$, $n$ etc. become available. The current observed data of $A$ and $n$ from CMB are consistent with the warm inflation scenario if the mass scale $M$ of the inflation is in the range of $10^{15} - 10^{16}$ GeV.
We would like to thank an anonymous referee for a detailed report that improved the presentation of the paper. Wolung Lee would like to thank Hung Jung Lu for helpful discussions.
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[^1]: The coefficient $1/\pi$ actually depends on the details of the cut-off function, which may not be step-function-like. For instance, considering causality, the cut-off function can be soft, and the longest wavelength of fluctuations can be a few times of the size of horizon[@BFH]
[^2]: We thank the referee for pointing this problem out.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract:
- 'We generalize the results in [@MB1] giving a reduction algorithm allowing to compute the index of seaweed subalgebras of classical simple Lie algebras. We thus are able to obtain the index of some interesting families of seaweed subalgebras and to give new examples of large classes of Frobenius Lie algebras among them.'
- 'On généralise les résultats de [@MB1] en donnant un algorithme de réduction pour le calcul de l’indice des sous-algèbres biparaboliques d’une algèbre de Lie simple classique qui permet d’obtenir l’indice de certaines classes intéressantes de ces sous-algèbres et d’en déduire en particulier celles parmi elles qui sont des sous-algèbres de Frobenius.'
address: |
Université Tunis El-Manar\
Faculté des Sciences de Tunis\
Département de Mathématiques\
Campus Universitaire\
2092 El-Manar\
Tunis, Tunisie
author:
- Meher Bouhani
bibliography:
- 'biblio.bib'
title: 'indice des sous-algèbres biparaboliques d’une algèbre de Lie simple classique'
---
Introduction
============
Soit $\mathfrak{g}$ l’algèbre de Lie d’un groupe de Lie algébrique complexe $\mathbold{G}$ et $\mathfrak{g}^{\ast}$ son dual. Pour $f\in \mathfrak{g}^{*}$, on note $\mathfrak{g}_{f}$ le stabilisateur de $f$ pour l’action coadjointe. On appelle indice de $\mathfrak{g}$ et on note $\chi[\mathfrak{g}]$ la dimension minimale de $\mathfrak{g}_{f}$ lorsque $f$ parcourt $\mathfrak{g}^{\ast}$. Si $\chi[\mathfrak{g}]=0$, $\mathfrak{g}$ est dite une algèbre de Frobenius.\
Dans toute la suite, les groupes et les algèbres de Lie considérés sont algébriques définis sur le corps des complexes. Pour toute paire $(r,s)$ d’entiers naturels, on note $r[s]$ le reste de la division euclidienne de $r$ par $s$ et $r \wedge s$ le plus grand commun diviseur de $r$ et $s$. Pour $\underline{a}=(a_{1},\ldots,a_{k})\in\mathbb{N}^{k}$, on pose $|\underline{a}|:=a_{1}+\cdots+a_{k}$.\
Soient $n\in\mathbb{N}^{\times},\;\underline{a}=(a_{1},\ldots,a_{k})$ et $\underline{b}=(b_{1},\ldots,b_{t})$ deux compositions vérifiant $|\underline{a}|\leq n$ et $|\underline{b}|\leq n$. On associe à la paire $(\underline{a},\underline{b})$ une unique sous-algèbre biparabolique de $\mathfrak{sp}(2n)$ (resp. $\mathfrak{so}(2n+1)$, $\mathfrak{so}(2n)$) que l’on note $\mathfrak{q}^{C}_{n}(\underline{a}\mid\underline{b})$ (resp. $\mathfrak{q}^{B}_{n}(\underline{a}\mid\underline{b})$, $\mathfrak{q}^{D}_{n}(\underline{a}\mid\underline{b})$), et toutes les sous-algèbres biparaboliques de $\mathfrak{sp}(2n)$ (resp. $\mathfrak{so}(2n+1)$, $\mathfrak{so}(2n)$) sont ainsi obtenues à conjugaison près par le groupe adjoint connexe de $\mathfrak{sp}(2n)$ (resp. $\mathfrak{so}(2n+1)$, $\mathfrak{so}(2n)$). Cependant, dans le cas de type $D$, comme expliqué dans la section 4, on peut supposer que, si $|\underline{a}|=n$ (resp. $|\underline{b}|=n$), alors $a_{k}>1$ (resp. $b_{t}>1$). Si $|\underline{a}|=|\underline{b}|= n$, on associe à la paire $(\underline{a},\underline{b})$ une unique sous-algèbre biparabolique de $\mathfrak{gl}(n)$ (à conjugaison près par le groupe adjoint connexe de $\mathfrak{gl}(n)$) que l’on note $\mathfrak{q}^{A}(\underline{a}\mid\underline{b})$ (voir [@MB], [@meander.C] et [@meander.D]).\
Pour $\underline{a}=(a_{1},\ldots,a_{k})\in\mathbb{N}^{k}$ et $\underline{b}=(b_{1},\ldots,b_{t})\in\mathbb{N}^{t}$ tels que $|\underline{a}|\leq n$ et $|\underline{b}|\leq n$, on désigne par $\tilde{\underline{a}}$ (resp. $\tilde{\underline{b}}$) la suite obtenue de $\underline{a}$ (resp. $\underline{b}$) en supprimant les termes nuls et on convient que $\mathfrak{q}^{A}(\underline{a}\mid\underline{b})=\mathfrak{q}^{A}(\tilde{\underline{a}}\mid\tilde{\underline{b}})$ si $|\underline{a}|=|\underline{b}|=n$ et $\mathfrak{q}^{I}_{n}(\underline{a}\mid\underline{b})=\mathfrak{q}^{I}_{n}(\tilde{\underline{a}}\mid\tilde{\underline{b}})$, $I=B,\;C\;ou\;D$.\
Pour $n >1$, soit $\Xi_{n}$ l’ensemble des paires de compositions $(\underline{a}=(a_{1},\ldots,a_{k}),\underline{b}=(b_{1},\ldots,b_{t}))$ qui vérifient : $|\underline{a}|=n$, $|\underline{b}|=n-1$ et $a_{k}>1$ ou $|\underline{b}|=n$, $|\underline{a}|=n-1$ et $b_{t}>1$.\
Dans [@D.K], Dergachev et Kirillov associent à chaque sous-algèbre biparabolique $\mathfrak{q}^{A}(\underline{a}\mid\underline{b})$ de $\mathfrak{gl}(n)$ un graphe appelé méandre de $\mathfrak{q}^{A}(\underline{a}\mid\underline{b})$ et noté $\Gamma^{A}(\underline{a}\mid\underline{b})$. Il est construit de la manière suivante : on place $n$ points consécutifs, appelés sommets de $\Gamma^{A}(\underline{a}\mid\underline{b})$ et numérotés de $1$ à $n$, sur une droite horizontale, puis on relie au dessous (resp. au dessus) de cette droite par un arc toute paire de sommets distincts de la forme $(a_{1}+\cdots+a_{i-1}+j_{i},a_{1}+\cdots+a_{i}-j_{i}+1),\;1\leq j_{i}\leq a_{i},\;1\leq i\leq k$ (resp. $(b_{1}+\cdots+b_{i-1}+j_{i},b_{1}+\cdots+b_{i}-j_{i}+1),\;1\leq j_{i}\leq b_{i},\;1\leq i\leq t$). Au moyen des composantes connexes de ce graphe, qui sont des cycles et des segments, les auteurs décrivent l’indice de $\mathfrak{q}^{A}(\underline{a}\mid\underline{b})$ (voir théorème \[thm1\]). Ce résultat a été généralisé au cas des sous-algèbres biparaboliques de $\mathfrak{sp}(2n)$ de deux manières indépendantes dans [@C.meanders] et [@meander.C], et au cas des sous-algèbres biparaboliques de $\mathfrak{so}(n)$ dans [@meander.C] et [@meander.D]. Dans [@meander.C] et [@meander.D], les auteurs associent à la sous-algèbre biparabolique $\mathfrak{q}^{I}_{n}(\underline{a}\mid\underline{b})$ un méandre noté $\Gamma^{I}_{n}(\underline{a}\mid\underline{b})$, où $\;I=B,\;C\;ou\;D$. Lorsque $I=B,\;C\;ou\;I=D$ et $(\underline{a}\mid\underline{b})\notin\Xi_{n}$, le méandre $\Gamma^{I}_{n}(\underline{a}\mid\underline{b})$ vérifie $\Gamma^{I}_{n}(\underline{a}\mid\underline{b})=\Gamma^{A}(a_{1},\ldots,a_{k},2(n-|\underline{a}|),a_{k},\ldots,a_{1}\mid b_{1},\ldots,b_{t},2(n-|\underline{b}|),b_{t},\ldots,b_{1} )$. Lorsque $I=D$ et $(\underline{a}\mid\underline{b})\in\Xi_{n}$, le méandre $\Gamma^{D}_{n}(\underline{a}\mid\underline{b})$ possède deux arcs qui se croisent (voir section 4).\
Dans [@C.meanders], les auteurs donnent une formule de l’indice de la sous-algèbre biparabolique $\mathfrak{q}_{n}^{C}(a,b\mid c)$ lorsque $|a+b-c|=1$ ou $2$ leur permettant de déterminer la famille des sous–algèbres biparaboliques de Frobenius qui sont de la forme $\mathfrak{q}_{n}^{C}(a,b\mid c),\;(a,b,c)\in(\mathbb{N}^{\times})^{3}$. Dans ce travail, nous donnons une formule de l’indice des sous-algèbres biparaboliques de $\mathfrak{sp}(2n)$ (resp. $\mathfrak{so}(2n+1)$, $\mathfrak{so}(2n))$ qui sont de la forme $\mathfrak{q}_{n}^{C}(a,b\mid c)$ (resp. $\mathfrak{q}_{n}^{B}(a,b\mid c)$, $\mathfrak{q}_{n}^{D}(a,b\mid c)$), pour tout $(a,b,c)\in(\mathbb{N}^{\times})^{3}$ (resp. $(a,b,c)\in(\mathbb{N}^{\times})^{3}$, $(a,b,c)\in(\mathbb{N}^{\times})^{3}$ et $b>1$ si $a+b=n$) (voir théorèmes \[I3\] et \[I’3\]). Plus précisément, nous montrons le théorème suivant :
Soient $a,b,c,n\in\mathbb{N}^{\times}$ tels que $s:=\max(a+b,c)\leq n$. On pose $p=(a+b)\wedge(b+c)$ et $r=|a+b-c|$. Alors
- - Si $p>r$, on a $\chi(\mathfrak{q}^{B}_{n}(a,b\mid c))=\chi[\mathfrak{q}^{C}_{n}(a,b\mid c)]=p-r+[\frac{r}{2}]+n-s$
- Si $p\leq r$, on a\
$\chi[\mathfrak{q}^{B}_{n}(a,b\mid c)]=\chi[\mathfrak{q}^{C}_{n}(a,b\mid c)]=\begin{cases}[\frac{r}{2}]+n-s\;\text{si $p$ et $r$ sont de m\^eme pari\'et\'e}\\
[\frac{r}{2}]-1+n-s\;\text{sinon}\\
\end{cases}$\
- Soit $\Gamma^{D}_{n}(a,b\mid c)$ le méandre de $\mathfrak{q}^{D}_{n}(a,b\mid c)$, Alors
- Si $((a,b),c)\notin\Xi_{n}$, on a $\chi[\mathfrak{q}^{D}_{n}(a,b\mid c)]=\chi[\mathfrak{q}^{C}_{n}(a,b\mid c)]+\epsilon$, où $\epsilon$ est donné par:$\\$ $ \epsilon=\begin{cases} 0\;\;\;\;si\;r\;\text{est un entier pair} \\
1\;\;\;\;si\;r\;\text{est un entier impair},\;s=n\;\text{et de plus l'arc de }\Gamma^{D}_{n}(a,b\mid c)\;\text{joignant les }\\
\;\;\;\;\;\;sommets \;n\;et\;n+1\;est\; \text{un arc d'un segment de}\;\Gamma^{D}_{n}(a,b\mid c)\\
-1\;\text{dans les cas restants}\\
\end{cases}$\
- Si $((a,b),c)\in\Xi_{n}$, on a $\chi[\mathfrak{q}_{n}^{D}(a,b\mid c)]=|(a\wedge n)-2|$
En particulier, nous caractérisons les algèbres de Frobenius de cette famille (voir corollaires \[J3\] et \[J’3\]).\
Pour une sous-algèbre biparabolique $\mathfrak{q}^{A}(\underline{a}\mid\underline{b})$ de $\mathfrak{gl}(n)$, on pose\
$\Psi[\mathfrak{q}^{A}(\underline{a}\mid\underline{b})]=\begin{cases}\chi[\mathfrak{q}^{A}(\underline{a}\mid\underline{b})]\;\text{si le sommet $n$ appartient \`a un segment de }\Gamma^{A}(\underline{a}\mid\underline{b})\\
\chi[\mathfrak{q}^{A}(\underline{a}\mid\underline{b})]-2\;\text{sinon}\\
\end{cases}$\
Dans [@MB1], nous avons donné un algorithme de réduction permettant le calcul de l’indice des sous-algèbres biparaboliques $\mathfrak{q}^{A}(\underline{a},\underline{b})$ de $\mathfrak{gl}(n)$. Dans cet article, nous généralisons ce résultat au cas des sous-algèbres biparaboliques $\mathfrak{q}^{C}_{n}(\underline{a},\underline{b})$ (resp. $\mathfrak{q}^{B}_{n}(\underline{a},\underline{b})$, $\mathfrak{q}^{D}_{n}(\underline{a},\underline{b})$) de $\mathfrak{sp}(2n)$ (resp. $\mathfrak{so}(2n+1)$, $\mathfrak{so}(2n)$) (voir théorèmes \[thm0’\], \[thm r1\] et \[thm r2\]). Nous montrons d’abord qu’on peut se ramener au cas $\underline{a}=(t)$ et $|\underline{b}|\leq t\leq n$, $t\in\mathbb{N}^{\times}$. Ensuite, nous donnons les deux théorèmes suivants
Soient $t\in\mathbb{N}^{\times}$ et $\underline{a}=(a_{1},\ldots,a_{k})$ une composition vérifiant $|\underline{a}|\leq t\leq n$. On pose $a_{k+1}=t-|\underline{a}|$ et $d_{i}=(a_{1}+\dots a_{i-1})-(a_{i+1}+\dots+a_{k+1}),\;1\leq i\leq k$. Soit $\mathfrak{q}_{n}(t\mid\underline{a}):=\mathfrak{q}^{B}_{n}(t\mid\underline{a}),\;\mathfrak{q}^{C}_{n}(t\mid\underline{a})$ ou $\mathfrak{q}_{n}(t\mid\underline{a}):=\mathfrak{q}^{D}_{n}(t\mid\underline{a})$ si $(t\mid\underline{a})\notin\Xi_{n}$.
- Pour tout $\;1\leq i\leq k$ tel que $d_{i}\neq 0$ et tout $\alpha\in\mathbb{Z}$ tel que $a_{i}+\alpha|d_{i}|\geq 0$, on a $$\chi[\mathfrak{q}_{n}(t\mid\underline{a})]=\chi[\mathfrak{q}_{n+\alpha |d_{i}|}(t+\alpha |d_{i}|\mid a_{1},\ldots,a_{i-1},a_{i}+ \alpha |d_{i}|,a_{i+1},\ldots,a_{k})]$$ En particulier, on a $$\chi[\mathfrak{q}_{n}(t\mid\underline{a})]=\chi[\mathfrak{q}_{n-a_{i}+a_{i}[|d_{i}|]}(t-a_{i}+a_{i}[|d_{i}|]\mid a_{1},\ldots,a_{i-1},a_{i}[|d_{i}|],a_{i+1},\ldots,a_{k})]$$
- Pour tout $\;1\leq i\leq k$ tel que $d_{i}=0$, on a $$\chi[\mathfrak{q}_{n}(t\mid\underline{a})]=a_{i}+\chi[\mathfrak{q}_{n-a_{i}}(t-a_{i}\mid a_{1},\ldots,a_{i-1},a_{i+1}\ldots,a_{k})]$$
\[thm r2\] Soient $\underline{a}=(a_{1},\ldots,a_{k})$ une composition vérifiant $1\leq |\underline{a}|=n-1$ $(i.e \;(n\mid\underline{a})\in\Xi_{n})$. On pose $\underline{a}^{'}=(a^{'}_{1},\ldots,a^{'}_{k})=(a_{1},\ldots,a_{k-1},a_{k}+1)$, $d_{k}=-(a^{'}_{1}+\cdots+a^{'}_{k-1})$ et $d_{i}=(a^{'}_{1}+\cdots+a^{'}_{i-1})-(a^{'}_{i+1}+\cdots+a^{'}_{k})$, $ 1\leq i\leq k-1$.
- Pour tout $1\leq i\leq k$ tel que $d_{i}\neq 0$ et tout $\alpha\in\mathbb{Z}$ tel que $a^{'}_{i}+\alpha|d_{i}|\geq 0$, on a $$\chi[\mathfrak{q}^{D}_{n}(n\mid a_{1},\ldots,a_{k})]=\Psi[\mathfrak{q}^{A}(n+\alpha|d_{i}|\mid a^{'}_{1},\ldots,a^{'}_{i-1},a^{'}_{i}+\alpha|d_{i}|,a^{'}_{i+1},\ldots,a^{'}_{k})]$$ En particulier, si on pose $t_{i}=a^{'}_{i}-a^{'}_{i}[|d_{i}|]$, on a $$\chi[\mathfrak{q}^{D}_{n}(n\mid a_{1},\ldots,a_{k})]=\Psi[\mathfrak{q}^{A}(n-t_{i}\mid a^{'}_{1},\ldots,a^{'}_{i-1},a^{'}_{i}[|d_{i}|],a^{'}_{i+1}\ldots,a^{'}_{k})]$$
- Pour tout $1\leq i\leq k$ tel que $d_{i}=0$, on a $$\chi[\mathfrak{q}^{D}_{n}(n\mid a_{1},\ldots,a_{k})]=a_{i}+\Psi[\mathfrak{q}^{A}(n-a_{i}\mid a_{1},\ldots,a_{i-1},a_{i+1}\ldots,a_{k})]$$
Comme conséquence de ces deux théorèmes, nous donnons de nouvelles familles de sous-algèbres biparaboliques de $\mathfrak{sp}(2n)$ (resp. $\mathfrak{so}(2n+1)$, $\mathfrak{so}(2n)$) qui sont de Frobenius (voir lemme \[lem F\] et théorème \[thm a\]). Enfin, nous montrons que si $\underline{a}=(a_{1},\ldots,a_{m})$ et $\underline{b}=(b_{1},\ldots,b_{t})$ sont deux compositions de $n$ telles que $\mathfrak{q}_{s}^{A}(\underline{a}\mid\underline{b}):=\mathfrak{q}^{A}(\underline{a}\mid\underline{b})\cap\mathfrak{sl}(n)$ est une sous-algèbre de Frobenius de $\mathfrak{sl}(n)$ (i.e, $\chi[\mathfrak{q}^{A}(\underline{a}\mid\underline{b})]=1$), alors les algèbres $\mathfrak{q}^{D}_{2n}(2a_{1},\ldots,2a_{m}\mid 2b_{1},\ldots,2b_{t-1},2b_{t}-1)$ et $\mathfrak{q}^{D}_{2n}(2a_{1},\ldots,2a_{m-1},2a_{m}-1\mid 2b_{1},\ldots,2b_{t})$ sont deux sous-algèbres de Frobenius de $\mathfrak{so}(4n)$, et toutes les sous-algèbres de Frobenius de $\mathfrak{so}(2n)$ qui sont de la forme $\mathfrak{q}^{D}_{n}(\underline{a}\mid\underline{b})$ où $(\underline{a}\mid\underline{b})\in\Xi_{n}$ sont ainsi obtenues (voir théorème \[th B\]). En particulier, pour tout $n\geq 1$ et pour toute paire de compositions $(\underline{a}\mid\underline{b})\in\Xi_{2n+1}$, $\mathfrak{q}^{D}_{2n+1}(\underline{a}\mid\underline{b})$ n’est pas une sous-algèbre de Frobenius.\
Je remercie les professeurs Pierre Torasso et Mohamed Salah khalgui pour d’utiles conversations qui m’ont aidé dans ce travail.
Rappels
=======
Soit $\mathbold{G}$ un groupe de Lie algébrique complexe, $\mathfrak{g}$ son algèbre de Lie et $\mathfrak{g}^{\ast}$ le dual de $\mathfrak{g}$. Au moyen de la représentation coadjointe, $\mathfrak{g}$ et $\mathbold{G}$ opèrent dans $\mathfrak{g}^{*}$ par : $$(x.f)(y)=f([y,x]), \;\; \forall x,y\in \mathfrak{g}\;\text{et}\;f\in
\mathfrak{g}^{*}$$ $$(x.f)(y)=f(\operatorname{Ad }x^{-1}y),\;\; \forall x\in\mathbold{G},y\in \mathfrak{g}\;\text{et}\;f\in
\mathfrak{g}^{*}$$ Pour $f\in \mathfrak{g}^{*}$, soit $\mathbold{G}_{f}$ le stabilisateur de $f$ pour cette action et $\mathfrak{g}_{f}$ son algèbre de Lie: $$\mathbold{G}_{f}=\{x\in \mathbold{G};f(\operatorname{Ad }x^{-1}y)=f(y),\;\; \forall y\in
\mathfrak{g}\}$$
$$\mathfrak{g}_{f}=\{x\in \mathfrak{g};f([x,y])=0,\;\; \forall y\in
\mathfrak{g}\}$$ On appelle indice de $\mathfrak{g}$ l’entier $\chi[\mathfrak{g}]$ défini par: $$\chi[\mathfrak{g}]=\min\{\dim \mathfrak{g}_{f}\;,\;f\in
\mathfrak{g}^{*}\}$$ Si $\chi[\mathfrak{g}]=0$, $\mathfrak{g}$ est dite une algèbre de Frobenius.$\\$
Supposons $\mathfrak{g}$ une algèbre de Lie semi-simple. Soit $\mathfrak{h}$ une sous-algèbre de Cartan de $\mathfrak{g}$, $\Delta\subset\mathfrak{h}^{*}$ le système de racines de $\mathfrak{g}$ relativement à $\mathfrak{h}$, $\pi:=\lbrace \alpha_{1},\ldots,\alpha_{n}\rbrace$ une base de racines simples numérotée, dans le cas où $\mathfrak{g}$ est simple, conformément à Bourbaki [@Bourbaki]. Alors $\Delta=\Delta^{+}\cup\Delta^{-}$ où $\Delta^{+}$ est l’ensemble des racines positives relativement à $\pi$ et $\Delta^{-}=-\Delta^{+}$. Pour $\alpha\in\Delta$, soit $\mathfrak{g}_{\alpha}:=\lbrace x\in\mathfrak{g};\;[h,x]=\alpha(h)x,\;h\in\mathfrak{h}\rbrace$, $\mathfrak{g}_{\alpha}$ est de dimension un.\
Pour toute partie $\pi^{'}\subset\pi$, soient $\Delta^{+}_{\pi^{'}}=\Delta^{+}\cap\mathbb{N}\pi^{'}$ où $\mathbb{N}\pi^{'}$ désigne l’ensemble des combinaisons linéaires des éléments de $\pi^{'}$ à coefficients dans $\mathbb{N}$, $\Delta^{-}_{\pi^{'}}=-\Delta^{+}_{\pi^{'}}$ et $\mathfrak{n}^{\pm}_{\pi^{'}}=\oplus_{\alpha\in\Delta^{\pm}_{\pi^{'}}}\mathfrak{g}_{\alpha}$.\
Soit $(\pi^{'},\pi^{''})$ une paire de parties de $\pi$, la sous-algèbre $\mathfrak{q}_{\pi^{'},\pi^{''}}:=\mathfrak{n}^{+}_{{\pi}^{'}}\oplus \mathfrak{h} \oplus\mathfrak{n}^{-}_{\pi^{''}}$ est appelée sous-algèbre biparabolique standard de $\mathfrak{g}$.\
On appelle sous-algèbre biparabolique de $\mathfrak{g}$ toute sous-algèbre de $\mathfrak{g}$ conjuguée par $\mathbold{G}$ à une sous-algèbre biparabolique standard de $\mathfrak{g}$.\
Si $\pi^{'}=\pi$ ou $\pi^{''}=\pi$, toute sous-algèbre de $\mathfrak{g}$ conjuguée par $\mathbold{G}$ à $\mathfrak{q}_{\pi^{'},\pi^{''}}$ est dite sous-algèbre parabolique de $\mathfrak{g}$.
Soit $\pi^{'}=\{\alpha_{i_{1}},\ldots,\alpha_{i_{k}}\}$ une partie de $\pi$, on pose $\mathcal{S}_{\pi^{'}}:=(i_{1},i_{2}-i_{1},\ldots,i_{k}-i_{k-1},n+1-i_{k})$, $\mathcal{T}_{\pi^{'}}:=(i_{1},i_{2}-i_{1},\ldots,i_{k}-i_{k-1})$ et on convient que $\mathcal{S}_{\varnothing}=(n+1)$ et $\mathcal{T}_{\varnothing}=\varnothing$. Alors $\mathcal{S}_{\pi^{'}}$ est une composition de $n+1$, $\mathcal{T}_{\pi^{'}}$ est une composition d’un entier $t\leq n$.
Supposons $\mathfrak{g}=\mathfrak{sl}(n+1)$, on associe à toute paire $(\underline{a},\underline{b})$ de compositions de $n+1$ la sous-algèbre biparabolique $\mathfrak{q}^{A}_{s}(\underline{a}\mid\underline{b}):=\mathfrak{q}_{\pi^{'},\pi^{''}}$ telle que $\underline{a}=\mathcal{S}_{\pi\backslash\pi^{'}}$ et $\underline{b}=\mathcal{S}_{\pi\backslash\pi^{''}}$. La sous-algèbre $\mathfrak{q}^{A}(\underline{a}\mid\underline{b}):=\mathfrak{q}^{A}_{s}(\underline{a}\mid\underline{b})\oplus\mathbb{C}I_{n+1}$, $I_{n+1}$ étant la matrice identité d’ordre $n+1$, est une sous-algèbre biparabolique de $\mathfrak{gl}(n+1)$ qui vérifie $\chi[\mathfrak{q}^{A}(\underline{a}\mid\underline{b})]=\chi[\mathfrak{q}^{A}_{s}(\underline{a}\mid\underline{b})]+1$. Toutes les sous-algèbres biparaboliques de $\mathfrak{gl}(n+1)$ sont ainsi obtenues (à conjugaison près par le groupe adjoint connexe de $\mathfrak{gl}(n+1)$). La sous-algèbre $\mathfrak{q}^{A}(\underline{a}\mid\underline{b})$ est une sous-algèbre parabolique de $\mathfrak{gl}(n+1)$ si et seulement si $\underline{a}=(n+1)$ ou $\underline{b}=(n+1)$. Soit $(e_{1},\ldots,e_{n})$ la base canonique de $\mathbb{C}^{n}$, ${\mathscr}{V}=\{V_{0}=\{0\}\subsetneq
V_{1}\subsetneq\cdots\subsetneq V_{m}=\mathbb{C}^{n} \}$ et ${\mathscr}{W}=\{\mathbb{C}^{n}=W_{0}\supsetneq W_{1}\supsetneq\cdots\supsetneq
W_{t}=\{0\}\}$ les drapeaux de sous-espaces tels que $V_{i}=\langle
e_{1},\ldots,e_{a_{1}+\cdots+a_{i}}\rangle$, $1\leq i\leq m$, et $W_{i}=\langle e_{b_{1}+\cdots+b_{i}+1},\ldots,e_{n}\rangle$, $1\leq
i\leq t-1$. Alors $\mathfrak{q}^{A}(\underline{a}\vert\underline{b})$ est le stabilisateur dans $\mathfrak{gl}(n)$ de la paire de drapeaux $({\mathscr}{V},{\mathscr}{W})$.
Supposons $\mathfrak{g}=\mathfrak{sp}(2n)$ (resp. $\mathfrak{so}(2n+1)$, $\mathfrak{so}(2n)$), on associe à toute paire de compositions $(\underline{a},\underline{b})$ vérifiant $|\underline{a}|\leq n$ et $|\underline{b}|\leq n$, la sous-algèbre biparabolique $\mathfrak{q}^{I}_{n}(\underline{a}\mid\underline{b}):=\mathfrak{q}_{\pi^{'},\pi^{''}},\;I=C$ (resp. $B,\;D$) de $\mathfrak{sp}(2n)$ (resp. $\mathfrak{so}(2n+1)$, $\mathfrak{so}(2n)$) telle que $\underline{a}=\mathcal{T}_{\pi\backslash\pi^{'}}$ et $\underline{b}=\mathcal{T}_{\pi\backslash\pi^{''}}$. À conjugaison près par le groupe adjoint connexe de $\mathfrak{sp}(2n)$ (resp. $\mathfrak{so}(2n+1)$, $\mathfrak{so}(2n)$), toutes les sous-algèbres biparaboliques de $\mathfrak{sp}(2n)$ (resp. $\mathfrak{so}(2n+1)$, $\mathfrak{so}(2n)$) sont ainsi obtenues. La sous-algèbre $\mathfrak{q}^{I}_{n}(\underline{a}\mid\underline{b}),\;I=C$ (resp. $B,\;D$) est une sous-algèbre parabolique de $\mathfrak{sp}(2n)$ (resp. $\mathfrak{so}(2n+1)$, $\mathfrak{so}(2n)$) si et seulement si $\underline{a}=\varnothing$ ou $\underline{b}=\varnothing$. Soit encore $(e_{1},\ldots,e_{n})$ la base canonique de $\mathbb{C}^{n}$. On munit $\mathbb{C}^{n}$ de la forme bilinéaire $\langle\,,\rangle$ antisymétrique (cas $\mathfrak{sp}(n)$ avec $n$ pair) (resp. symétrique (cas $\mathfrak{so}(n)$)) telle que $\langle
e_{i},e_{n+1-j}\rangle=\delta_{i,j}$ $1\leq i,j\leq n$, $i+j\leq n+1$. Soit $(\underline{a},\underline{b})$ une paire de compositions vérifiant $\vert\underline{a}\vert\leq[\frac{n}{2}]$ et $\vert\underline{b}\vert\leq[\frac{n}{2}]$, ${\mathscr}{V}=\{V_{0}=\{0\}\subsetneq V_{1}\subsetneq\cdots\subsetneq V_{m} \}$ et ${\mathscr}{W}=\{W_{0}\supsetneq W_{1}\supsetneq\cdots\supsetneq
W_{t}=\{0\}\}$ les drapeaux de sous-espaces isotropes tels que $V_{i}=\langle
e_{1},\ldots,e_{a_{1}+\cdots+a_{i}}\rangle$, $1\leq i\leq m$, et $W_{i}=\langle e_{n-(b_{1}+\cdots+b_{t-i})+1},\ldots,e_{n}\rangle$, $0\leq
i\leq t-1$. Alors $\mathfrak{q}_{[\frac{n}{2}]}^{C}(\underline{a}\vert\underline{b})$ (resp. $\mathfrak{q}_{[\frac{n}{2}]}^{I}(\underline{a}\vert\underline{b})$, $I=B$, si $n$ est impair et $I=D$, si $n$ est pair) est le stabilisateur dans $\mathfrak{sp}(n)$ (resp. $\mathfrak{so}(n)$) de la paire de drapeaux $({\mathscr}{V},{\mathscr}{W})$.\
Soit $\underline{a}=(a_{1},\dots, a_{k})$ une composition d’un entier $n\in\mathbb{N}^{\times}$, on pose $I_{i}=[a_{1}+\dots+a_{i-1}+1,\ldots,a_{1}+\dots+a_{i-1}+a_{i}]\cap\mathbb{N},\;1\leq i\leq k$ et on associe à la composition $\underline{a}$, l’involution $\theta_{\underline{a}}$ de $\{1,\ldots,n\}$, définie par $\theta_{\underline{a}}(x)=2(a_{1}+\dots+a_{i-1})+a_{i}-x+1$, $x\in I_{i},1\leq i\leq k\;$.\
Soit $(\underline{a},\underline{b})$ une paire de compositions d’un entier $n\in\mathbb{N}^{\times}$, On associe à la sous-algèbre biparabolique $\mathfrak{q}^{A}(\underline{a}\mid \underline{b})$ de $\mathfrak{gl}(n)$ un graphe noté $\Gamma^{A}(\underline{a}\mid\underline{b})$ et appelé méandre de $\mathfrak{q}^{A}(\underline{a}\mid \underline{b})$, dont les sommets sont $n$ points consécutifs situés sur une droite horizontale D et numérotés $1,2,\ldots,n$. Il est construit de la manière suivante: on relie par un arc au dessous (resp. au dessus) de la droite D toute paire de sommets distincts de $\Gamma(\underline{a}\mid\underline{b})$ de la forme $(x,\theta_{\underline{a}}(x))$ (resp.$(x,\theta_{\underline{b}}(x))$), $x\in\{1,\ldots,n\}$. Une composante connexe de $\Gamma^{A}(\underline{a}\mid\underline{b})$ est soit un cycle, soit un segment (voir[@MB]). $\\$
\[Ex6.1\] $\Gamma^{A}(2,4,3\mid5,2,2)$= [ ]{}
\[thm1\][@D.K] Soient $\mathfrak{q}^{A}(\underline{a}\mid\underline{b})$ une sous-algèbre biparabolique de $\mathfrak{gl}(n)$ et $\Gamma^{A}(\underline{a}\mid\underline{b})$ son méandre, on a : $$\chi[\mathfrak{q}^{A}(\underline{a}\mid\underline{b})]=2\times (nombre\;de\;cycles)+nombre\;de\;segments$$
[@MB1]\[lem0\] Soient $\underline{a}=(a_{1},\dots, a_{k})$ et $\underline{b}=(b_{1},\ldots,b_{t})$ deux compositions telles que $|\underline{b}|=|\underline{a}|=n$. On pose $\underline{a}^{-1}=(a_{k},\dots, a_{1})$. On a : $$\chi[\mathfrak{q}^{A}(\underline{a}\mid\underline{b})]=\chi[\mathfrak{q}^{A}(2n\mid\underline{a}^{-1},\underline{b})]$$
[@MB1]\[ind\] Soit $a,b,c,d\;et\;n\in\mathbb{N}^{\times}$ tels que $a+b=c+d=n$, on a
- $\chi[\mathfrak{q}^{A}(a,b\mid n)]=a\wedge b$
- $\chi[\mathfrak{q}^{A}(a,b\mid c,d)]=\chi[\mathfrak{q}^{A}(a,b,c\mid n+c)]=(a+b)\wedge (b+c)$
[@MB1]\[thm0\] Soit $\mathfrak{p}^{A}(a_{1},\ldots,a_{k}):=\mathfrak{q}^{A}(a_{1},\ldots,a_{k}\mid n)$ une sous-algèbre parabolique de $\mathfrak{gl}(n)$. On pose $d_{k}=-(a_{1}+\cdots+a_{k-1})$ et $d_{i}=(a_{1}+\cdots+a_{i-1})-(a_{i+1}+\cdots+a_{k})$, $ 1\leq i\leq
k-1$.
- Pour tout $ 1\leq i\leq k$ tel que $d_{i}\neq 0$ et tout $\alpha\in\mathbb{Z}$ tel que $a_{i}+\alpha|d_{i}|\geq 0$, on a $$\chi[\mathfrak{p}^{A}(a_{1},\ldots,a_{k})]=\chi[\mathfrak{p}^{A}(a_{1},\ldots,a_{i}+\alpha|d_{i}|,\ldots,a_{k})]$$ En particulier, on a $$\chi[\mathfrak{p}^{A}(a_{1},\ldots,a_{k})]=\chi[\mathfrak{p}^{A}(a_{1},\ldots,a_{i-1},a_{i}[|d_{i}|],a_{i+1}\ldots,a_{k})]$$
- Pour tout $ 1\leq i\leq k$ tel que $d_{i}=0$, on a $$\chi[\mathfrak{p}^{A}(a_{1},\ldots,a_{k})]=a_{i}+\chi[\mathfrak{p}^{A}(a_{1},\ldots,a_{i-1},a_{i+1}\ldots,a_{k})]$$
sous-algèbres biparaboliques de $sp(2n)$
========================================
Soient $n\in\mathbb{N}^{\times}$ et $(\underline{a},\underline{b})$ une paire de compositions vérifiant $|\underline{a}|\leq n$ et $|\underline{b}|\leq n$. On associe à la sous-algèbre biparabolique $\mathfrak{q}^{C}_{n}(\underline{a}\mid \underline{b})$ de $\mathfrak{sp}(2n)$ le méandre $\Gamma^{C}_{n}(\underline{a}\mid\underline{b}):=\Gamma^{A}(\underline{a}^{'}\mid\underline{b}^{'})$, où $\underline{a}^{'}=(a_{1},\ldots,a_{k},2(n-|\underline{a}|),a_{k},\ldots,a_{1})$ et $\underline{b}^{'}=(b_{1},\ldots,b_{t},2(n-|\underline{b}|),b_{t},\ldots,b_{1})$. $\underline{a}^{'}$ et $\underline{b}^{'}$ sont deux compositions de $2n$, le nombre de sommets de $\Gamma^{C}_{n}(\underline{a}\mid\underline{b})$ est alors égal à $2n$. Soit $\sigma$ la symétrie par rapport \` a la droite verticale passant par le milieu des sommets $n$ et $n+1$. Par construction, le méandre $\Gamma^{C}_{n}(\underline{a}\mid\underline{b})$ est invariant par $\sigma$. Un cycle (resp. segment) X est dit invariant si $\sigma(X)=X$.
$\\$ $\Gamma^{C}_{8}(2,5\mid 1,4)={\setlength{\unitlength}{0.021in}
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\[thm2\][@meander.C] Soient $\mathfrak{q}^{C}_{n}(\underline{a}\mid\underline{b})$ une sous-algèbre biparabolique de $\mathfrak{sp}(2n)$ et $\Gamma^{C}_{n}(\underline{a}\mid\underline{b})$ son méandre, ona : $$\chi[\mathfrak{q}^{C}_{n}(\underline{a}\mid\underline{b})]=nombre\;de\;cycles\;+\frac{1}{2}\;(nombre\;de\;segments\;non\;invariants)$$
\[cor1\] Soient $\underline{a}=(a_{1},\dots, a_{k})$ et $\underline{b}=(b_{1},\ldots,b_{t})$ deux compositions telles que $|\underline{a}|\leq n$ et $|\underline{b}|\leq n$.
- $\chi[\mathfrak{q}^{C}_{n}(\underline{a}\mid\underline{b})]=\chi[\mathfrak{q}^{C}_{n}(\underline{b}\mid\underline{a})]$
- $\chi[\mathfrak{q}^{C}_{n}(\underline{a}\mid\underline{b})]=\chi[\mathfrak{q}^{C}_{\max(|\underline{a}|,|\underline{b}|)}(\underline{a}\mid\underline{b})]+n-\max(|\underline{a}|,|\underline{b}|)$
- $\chi[\mathfrak{q}^{C}_{n}(\underline{a}\mid\varnothing)]=\sum_{1\leq i\leq k}[\frac{a_{i}}{2}]+(n-|\underline{a}|)$
- S’il existe $1\leq i\leq k$ et $1\leq j\leq t$ tels que $a_{1}+\dots+a_{i}=b_{1}+\dots+b_{j}$, on a : $\chi[\mathfrak{q}^{C}_{n}(\underline{a}\mid\underline{b})]=\chi[\mathfrak{q}^{A}(a_{1},\ldots,a_{i}\mid b_{1},\ldots,b_{j})]+\chi[\mathfrak{q}^{C}_{n-(a_{1}+\dots+a_{i})}(a_{i+1},\ldots,a_{k}\mid b_{j+1},\ldots,b_{t})]$ où $\mathfrak{q}^{A}(a_{1},\ldots,a_{i}\mid b_{1},\ldots,b_{j})$ est la sous-algèbre biparabolique de $\mathfrak{gl}(a_{1}+\dots+a_{i})$ associée à la paire de compositions $((a_{1},\ldots,a_{i}),(b_{1},\ldots,b_{j}))$. En particulier, si $|\underline{a}|=|\underline{b}|$, alors\
$\chi[\mathfrak{q}^{C}_{n}(\underline{a}\mid\underline{b})]=\chi[\mathfrak{q}^{A}(\underline{a}\mid\underline{b})]+\chi[\mathfrak{q}^{C}_{n-|\underline{a}|}(\varnothing\mid\varnothing)]=\chi[\mathfrak{q}^{A}(\underline{a}\mid\underline{b})]+n-|\underline{a}|$
\[lem0’\] Soient $\underline{a}=(a_{1},\dots, a_{k})$ et $\underline{b}=(b_{1},\ldots,b_{t})$ deux compositions telles que $|\underline{b}|\leq|\underline{a}|\leq n$. On pose $\underline{a}^{-1}=(a_{k},\dots, a_{1})$. On a : $$\chi[\mathfrak{q}^{C}_{n}(\underline{a}\mid\underline{b})]=\chi[\mathfrak{q}^{C}_{n+|\underline{a}|}(2|\underline{a}|\mid\underline{a}^{-1},\underline{b})]$$
D’après le corollaire \[cor1\], on peut supposer $|\underline{a}|=n$. Soit $I=[1,n]$, $I^{'}=[1,|\underline{b}|]$ et $\underline{c}=(a_{k},\dots, a_{1},b_{1},\ldots,b_{t})$. On vérifie que $\theta_{a}$ (resp. $\theta_{b}$) est la restriction de $\theta_{n}\theta_{c}\theta_{n}$ (resp. $\theta_{n+|\underline{b}|}\theta_{c}\theta_{n+|\underline{b}|}$) à $I$ (resp. $I^{'}$). Puisque les méandres $\Gamma^{C}_{2n}(2n\mid\underline{a}^{-1},\underline{b})$ et $\Gamma^{C}_{n}(\underline{a}\mid\underline{b})$ sont invariants par la symétrie $\sigma$ définie au début de ce paragraphe, il existe alors une bijection entre les ensembles des composantes connexes des deux méandres conservant le nombre des cycles et le nombre des segments non invariants (voir exemple \[Ex1\]). Le résultat se déduit alors du théorème \[thm2\].
\[Ex1\] Considérons la paire de compositions $((2,3),(3,1))$, les arcs en bleu dans le méandre $\Gamma^{C}_{5}(2,3\mid 3,1)$ sont envoyés sur les arcs en bleu dans le méandre $\Gamma^{C}_{10}(10\mid 3,2,3,1)$.
$\\$ $\Gamma^{C}_{5}(2,3\mid 3,1)=
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$\\$ $\Gamma^{C}_{10}(10\mid 3,2,3,1)=
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Compte tenu du corollaire \[cor1\] et du lemme \[lem0’\], pour étudier l’indice des sous-algèbres biparaboliques de $\mathfrak{sp}(2n)$, on peut se ramener au cas des sous-algèbres biparaboliques de la forme $\mathfrak{q}^{C}_{n}(n\mid\underline{a})$, où $\underline{a}$ est une composition d’un entier inférieur ou égal à $n$.
Soient $\underline{a}=(a_{1},\ldots,a_{k})$ une composition vérifiant $|\underline{a}|\leq n$ et $s=n-|\underline{a}|$. Alors, pour tout $t\in\mathbb{N}$, on a $$\chi[\mathfrak{q}^{C}_{n+4ts}(n+4ts\mid \underbrace{2s,\ldots,2s}_{t},\underline{a},\underbrace{2s,\ldots,2s}_{t})]=\chi[\mathfrak{q}^{C}_{n}(n\mid\underline{a})]$$ En particulier, si $\mathfrak{q}^{C}_{n}(n\mid\underline{a})$ est une sous-algèbre de Frobenius de $\mathfrak{sp}(2n)$, alors, pour tout $t\in\mathbb{N}$, la sous-algèbre biparabolique $\mathfrak{q}^{C}_{n+4ts}(n+4ts\mid \underbrace{2s,\ldots,2s}_{t},\underline{a},\underbrace{2s,\ldots,2s}_{t})$ est une sous-algèbre de Frobenius de $\mathfrak{sp}(2(n+4ts))$.
La figure suivante montre que le résultat est vrai pour $t=1$. $\\$ $\Gamma^{C}_{n+4s}(n+4s\mid 2s,\underline{a},2s)=
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$(180,-50)[(0,1)[95]{}]{}$
$ $\\$ $\\$ $\\$ $\\$ $\\$ $\\$ Le résultat s’en déduit alors par récurrence sur $t$.
\[lemM\] Soient $\underline{a}=(a_{1},\ldots,a_{k})$ une composition vérifiant $|\underline{a}|\leq n$. Supposons qu’il existe $1\leq i\leq k$ tel que $|\underline{a}|_{i}:=a_{1}+\dots+a_{i}\leq n-|\underline{a}|$. Alors, on a $$\chi[\mathfrak{q}^{C}_{n}(n\mid\underline{a})]=\sum_{1\leq j\leq i}[\frac{a_{j}}{2}]+\chi[\mathfrak{q}^{C}_{n-2|\underline{a}|_{i}}(n-2|\underline{a}|_{i}\mid a_{i+1},\ldots,a_{k})]$$ En particulier, si $|\underline{a}|\leq [\frac{n}{2}]$, on a $$\chi[\mathfrak{q}^{C}_{n}(n\mid\underline{a})]=\sum_{1\leq j\leq k}[\frac{a_{j}}{2}]+[\frac{n-2|\underline{a}|}{2}]$$
Le méandre $\Gamma^{C}_{n}(n\mid\underline{a})$ est réunion disjointe des méandres $\Gamma^{C}_{2|\underline{a}|_{i}}(2|\underline{a}|_{i}\mid a_{1},\ldots,a_{i})$ et $\Gamma^{C}_{n-2|\underline{a}|_{i}}(n-2|\underline{a}|_{i}\mid a_{i+1},\ldots,a_{k})$. D’autre part, on vérifie que le méandre $\Gamma^{C}_{2|\underline{a}|_{i}}(2|\underline{a}|_{i}\mid a_{1},\ldots,a_{i})$ est composé de $\sum_{1\leq j\leq i}[\frac{a_{j}}{2}]$ cycles et $\sum_{1\leq j\leq i}[\frac{a_{j}+1}{2}]-\sum_{1\leq j\leq i}[\frac{a_{j}}{2}]$ segments qui sont tous invariants. Le résultat se déduit alors du théorème \[thm2\].
\[lem1\] Soient $\underline{a}=(a_{1},\ldots,a_{k})$ une composition vérifiant $|\underline{a}|\leq n$. On pose $a_{k+1}=n-|\underline{a}|$. Alors,
- Soient $a_{i,j}=(a_{1}+\dots +a_{i})-(a_{j}+\dots+a_{k+1})$ et $a^{i,j}=(a_{i+1}+\dots +a_{j-1})+|a_{i,j}|$, $1\leq i<j\leq k+1$. On a
$\chi[\mathfrak{q}^{C}_{n}(n\mid \underline{a})]=\begin{cases}\chi[\mathfrak{q}^{C}_{n+a^{i,j}}(n+a^{i,j}\mid a_{1},\ldots,a_{i},a^{i,j},a_{i+1},\ldots,a_{k})]\;si\;a_{i,j}< 0\\
\chi[\mathfrak{q}^{C}_{n+a^{i,j}}(n+a^{i,j}\mid a_{1},\ldots,a_{j-1},a^{i,j},a_{j},\ldots,a_{k})]\;si\;a_{i,j}\geq 0\\
\end{cases}$\
- Soit $d_{i}=(a_{1}+\dots +a_{i-1})-(a_{i+1}+\dots+a_{k+1}),\;1\leq i\leq k$. Supposons qu’il existe $1\leq i\leq k$ tel que $a_{i}\geq |d_{i}|$. On a
- $\chi[\mathfrak{q}^{C}_{n}(n\mid \underline{a})]=\begin{cases}\chi[\mathfrak{q}^{C}_{n+a_{i}+d_{i}}(n+a_{i}+d_{i}\mid a_{1},\ldots,a_{i},a_{i}+d_{i},a_{i+1},\ldots,a_{k})]\;si\;d_{i}< 0\\
\chi[\mathfrak{q}^{C}_{n+a_{i}-d_{i}}(n+a_{i}-d_{i}\mid a_{1},\ldots,a_{i-1},a_{i}-d_{i},a_{i},\ldots,a_{k})]\;si\;d_{i}\geq 0\\
\end{cases}$\
- $\chi[\mathfrak{q}^{C}_{n}(n\mid \underline{a})]=\chi[\mathfrak{q}^{C}_{n-|d_{i}|}(n-|d_{i}|\mid a_{1},\ldots,a_{i-1},a_{i}-|d_{i}|,a_{i+1},\ldots,a_{k})]$.
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- Supposons $a_{i,j}\leq 0$. Il suit de la démonstration du lemme \[lem0’\] qu’il existe une bijection entre les composantes connexes des méandres $\Gamma^{C}_{a^{i,j}}(a^{i,j}\mid a_{i+1},\ldots,a_{j-1})$ et $\Gamma^{C}_{2a^{i,j}}(2a^{i,j}\mid a^{i,j},a_{i+1},\ldots,a_{j-1})$ conservant le nombre de cycles et le nombre de segments invariants. Compte tenu de la symétrie $\sigma$, cette bijection se prolonge de manière naturelle en une bijection entre les composantes connexes des méandres $\Gamma^{C}_{n}(n\mid \underline{a})$ et $\Gamma^{C}_{n+a^{i,j}}(n+a^{i,j}\mid a_{1},\ldots,a_{i},a^{i,j},a_{i+1},\ldots,a_{k})$ qui conserve le nombre de cycles et le nombre de segments invariants. Le résultat découle du théorème \[lem0’\]. Le cas $a_{i,j}\geq 0$ se démontre de la même manière (voir exemple \[Ex2\]).
- - Il suffit de remarquer que $\begin{cases} a^{i,i+1}=a_{i,i+1}=a_{i}+d_{i}\;si\;d_{i}< 0\\
a^{i-1,i}=-a_{i-1,i}=a_{i}-d_{i}\;si\;d_{i}\geq 0\\
\end{cases}$ (voir exemple \[Ex3\]).
- Soit $\underline{b}=(b_{1},\ldots,b_{k})$ tel que $b_{j}=a_{j}$ si $j\neq i$ et $b_{i}=a_{i}-|d_{i}|$. On vérifie que $b_{i-1,i+1}=d_{i}$ et $b^{i-1,i+1}=a_{i}$, il suit de 1) que l’on a\
$\chi[\mathfrak{q}^{C}_{n}(\underline{b}\mid n)]=\begin{cases}\chi[\mathfrak{q}^{C}_{n+a_{i}+d_{i}}(n+a_{i}+d_{i}\mid a_{1},\ldots,a_{i},a_{i}+d_{i},a_{i+1},\ldots,a_{k})]\;si\;d_{i}< 0\\
\chi[\mathfrak{q}^{C}_{n+a_{i}-d_{i}}(n+a_{i}-d_{i}\mid a_{1},\ldots,a_{i-1},a_{i}-d_{i},a_{i},\ldots,a_{k})]\;si\;d_{i}\geq 0\\
\end{cases}$\
Le résultat se déduit de i)
\[Ex2\] Considérons la composition $\underline{a}=(a_{1},a_{2})=(3,2)$ et $n=7$, alors $|\underline{a}|=5$, $a_{3}=n-|\underline{a}|=2$, $a_{1,3}=a_{1}-a_{3}=1>0$ et $a^{1,3}=a_{2}+a_{1,3}=3$. Par suite, on a\
$\chi[ \mathfrak{q}^{C}_{7}(7\mid 3,2)]=\chi[\mathfrak{q}^{C}_{10}(10\mid 3,2,3)]$.
$\\$ $\Gamma^{C}_{7}(7\mid 3,2)=
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$\\$ $\Gamma^{C}_{10}(10\mid 3,2,3)=
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\[Ex3\] Considérons la composition $\underline{a}=(a_{1},a_{2})=(3,3)$ et $n=8$, alors $|\underline{a}|=6$, $a_{3}=n-|\underline{a}|=2$, $d_{2}=a_{1}-a_{3}=1>0$ et $a_{2}-d_{2}=2$. On a $\chi[ \mathfrak{q}_{8}(8\mid 3,3)]=\chi[\mathfrak{q}_{10}(10\mid 3,2,3)]$. Les arcs en bleu dans le méandre $\Gamma^{C}_{8}(8\mid 3,3)$ sont envoyés sur les arcs en bleu dans le méandre $\Gamma^{C}_{10}(10\mid 3,2,3)$.
$\\$ $\Gamma^{C}_{8}(8\mid 3,3)=
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$\\$ $\Gamma^{C}_{10}(10\mid 3,2,3)=
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Le théorème suivant est une conséquence imédiate du corollaire \[cor1\] et du lemme \[lem1\].
\[thm0’\] Soient $\underline{a}=(a_{1},\ldots,a_{k})$ une composition vérifiant $|\underline{a}|\leq n$. On pose $a_{k+1}=n-|\underline{a}|$ et $d_{i}=(a_{1}+\dots +a_{i-1})-(a_{i+1}+\dots+a_{k+1}),\;1\leq i\leq k$.
- Pour tout $1\leq i\leq k$ tel que $d_{i}\neq 0$ et tout $\alpha\in\mathbb{Z}$ tel que $a_{i}+\alpha|d_{i}|\geq 0$, on a $$\chi[\mathfrak{q}^{C}_{n}(n\mid\underline{a})]=\chi[\mathfrak{q}^{C}_{n+\alpha |d_{i}|}(n+\alpha |d_{i}|\mid a_{1},\ldots,a_{i-1},a_{i}+\alpha |d_{i}|,a_{i+1},\ldots,a_{k})]$$ En particulier, on a $$\chi[\mathfrak{q}^{C}_{n}(n\mid\underline{a})]=\chi[\mathfrak{q}^{C}_{n-a_{i}+a_{i}[|d_{i}|]}(n-a_{i}+a_{i}[|d_{i}|]\mid a_{1},\ldots,a_{i-1},a_{i}[|d_{i}|],a_{i+1},\ldots,a_{k})]$$
- Pour tout $1\leq i\leq k$ tel que $d_{i}=0$, on a $$\chi[\mathfrak{q}^{C}_{n}(n\mid\underline{a})]=a_{i}+\chi[\mathfrak{q}^{C}_{n-a_{i}}(a_{1},\ldots,a_{i-1},a_{i+1}\ldots,a_{k})]$$
\[rem\] Le lemme 2.4 de [@MB1] montre que si la composition $\underline{a}=(a_{1},\ldots,a_{k})$ et l’entier $n$ vérifient $|\underline{a}|\leq n$, alors il existe $\;1\leq i\leq k$ tel que $a_{i}\geq |d_{i}|$ ou $|\underline{a}|\leq [\frac{n}{2}]$.
Compte tenu du corollaire \[cor1\], des lemmes \[lem0’\] et \[lemM\] et de la remarque \[rem\], le théorème précédent donne un algorithme de réduction permettant le calcul de l’indice des sous-algèbres biparaboliques de $\mathfrak{sp}(2n)$.
Considérons la sous-algèbre biparabolique $\mathfrak{q}^{C}_{200}(15,185\mid 17,61,117)$ de $\mathfrak{sp}(400)$. Compte tenu du lemme \[lem0’\], on a $$\chi[\mathfrak{q}^{C}_{200}(15,185\mid 17,61,117)]=\chi[\mathfrak{q}^{C}_{400}(400\mid 185,15,17,61,117)]$$ Puis, en appliquant successivement le théorème \[thm0’\], on a $$\begin{aligned}
\chi[\mathfrak{q}^{C}_{400}(400\mid 185,15,17,61,117)]&=\chi[\mathfrak{q}^{C}_{385}(385\mid 185,17,61,117)]\\
&=\chi[\mathfrak{q}^{C}_{369}(369\mid 185,1,61,117)]\\
&=\chi[\mathfrak{q}^{C}_{185}(185\mid 1,1,61,117)]\\
&=\chi[\mathfrak{q}^{C}_{69}(69\mid 1,1,61,1)]\\
&=\chi[\mathfrak{q}^{C}_{9}(9\mid 1,1,1,1)]\\
\end{aligned}$$ Il suit du lemme \[lemM\] que l’on a $$\begin{aligned}
\chi[\mathfrak{q}^{C}_{400}(400\mid 185,15,17,61,117)]&=\chi[\mathfrak{q}^{C}_{9}(9\mid 1,1,1,1)]\\
&=0
\end{aligned}$$
\[lem F\]
Pour tout $k\in\mathbb{N}^{\times}$ et $(\alpha_{1},\ldots,\alpha_{k})\in\mathbb{N}^{k}$, soit $a_{k+1}=k$ et $\underline{a}=(a_{1},\ldots,a_{k})$ la composition définie par $a_{i}=1+\alpha_{i}(a_{i+1}+\dots+a_{k+1}-i+1),\;1\leq i\leq k$ et $r=|\underline{a}|+k$. Alors, $\mathfrak{q}^{C}_{r}(r\mid\underline{a})$ est une sous-algèbre de Frobenius de $\mathfrak{sp}(2r)$.
Soit $s_{i}=a_{i+1}+\dots+a_{k+1}-i+1, \;1\leq i\leq k$, alors $a_{i}[s_{i}]=1$. Utilisant la réduction du théorème \[thm0’\], on obtient $$\chi[\mathfrak{q}^{C}_{r}(r\mid\underline{a})]=\chi[\mathfrak{q}^{C}_{2k-1}(2k-1\mid \underbrace{1,\ldots,1}_{k-1})]$$ Il suit du lemme \[lemM\] que l’on a $$\chi[\mathfrak{q}^{C}_{r}(r\mid\underline{a})]=0$$
\[lem3\] Soient $\underline{a}=(a_{1},\ldots,a_{k})$ et $\underline{b}=(b_{1},\ldots,b_{t})$ deux compositions vérifiant $|\underline{b}|\leq|\underline{a}|\leq n$. Supposons $a_{1}>b_{1}$, on a : $$\chi[\mathfrak{q}^{C}_{n}(\underline{a}\mid\underline{b})]=\chi[\mathfrak{q}^{C}_{n-b_{1}}(a_{1}-b_{1}-a_{1}[a_{1}-b_{1}],a_{1}[a_{1}-b_{1}],a_{2},\ldots,a_{k}\mid b_{2}\ldots,b_{t})]$$.
D’après le corollaire \[cor1\], on peut supposer que $|\underline{a}|=n$. Utilisant le lemme \[lem0’\] et le théorème \[thm0’\], on a :\
$$\begin{aligned}
\chi[\mathfrak{q}^{C}_{n}(\underline{a}\mid\underline{b})]&=\chi[\mathfrak{q}^{C}_{2n}(2n\mid a_{k},\ldots,a_{1},\underline{b})]\\
&=\chi[\mathfrak{q}^{C}_{2n-b_{1}}(2n-b_{1}\mid a_{k},\ldots,a_{1},b_{2},\ldots,b_{t})]\\
&=\chi[\mathfrak{q}^{C}_{2n-b_{1}-a_{1}+a_{1}[a_{1}-b_{1}]}(2n-b_{1}-a_{1}+a_{1}[a_{1}-b_{1}]\mid a_{k},\ldots,a_{2},a_{1}[a_{1}-b_{1}],b_{2},\ldots,b_{t})]\\
&=\chi[\mathfrak{q}^{C}_{2n-2b_{1}}(2n-2b_{1}\mid a_{k},\ldots,a_{2},a_{1}[a_{1}-b_{1}],a_{1}-b_{1}-a_{1}[a_{1}-b_{1}],b_{2},\ldots,b_{t})]\\
&=\chi[\mathfrak{q}^{C}_{n-b_{1}}(a_{1}-b_{1}-a_{1}[a_{1}-b_{1}],a_{1}[a_{1}-b_{1}],a_{2},\ldots,a_{k}\mid b_{2}\ldots,b_{t})]\\\end{aligned}$$
\[lem3’\] Soient $\underline{a}=(a_{1},\ldots,a_{k})$ et $\underline{b}=(b_{1},\ldots,b_{t})$ deux compositions vérifiant $|\underline{b}|=|\underline{a}|=n$. Supposons $a_{1}>b_{1}$, on a : $$\chi[\mathfrak{q}^{A}(\underline{a}\mid\underline{b})]=\chi[\mathfrak{q}^{A}(a_{1}-b_{1}-a_{1}[a_{1}-b_{1}],a_{1}[a_{1}-b_{1}],a_{2},\ldots,a_{k}\mid b_{2}\ldots,b_{t})]$$.
Résulte du lemme précédent et du corollaire \[cor1\](4).
Soient $(a,b,n)\in(\mathbb{N}^{\times})^{3}$ tel que $b\leq a\leq n$. Alors, l’indice de $\mathfrak{q}^{C}_{n}(a\mid b)$ est donné par\
$$\chi[\mathfrak{q}^{C}_{n}(a\mid b)]=\begin{cases} n\;si\;a=b\\
[\frac{a[a-b]}{2}]+[\frac{a-b-a[a-b]}{2}]+n-a\;si\;a\neq b\\
\end{cases}$$
Supposons $a=b$, le résultat se déduit de l’assertion 4) du corollaire \[cor1\]. Supposons $a>b$, il suit du corolaire \[cor1\], qu’on peut supposer $a=n$. Utilisons le lemme \[lem3\], on a $$\chi[\mathfrak{q}^{C}_{n}(a\mid b)]=\chi[\mathfrak{q}^{C}_{a-b}(a-b-a[a-b],a[a-b]\mid\varnothing)$$ D’où le résultat.
\[lemP\] Soit $\mathfrak{q}^{C}_{n}(n\mid\underline{a})$ une sous-algèbre biparabolique de $\mathfrak{sp}(2n)$ où $\underline{a}=(a_{1},\ldots,a_{k})$ est une composition vérifiant $|\underline{a}|\leq n$. Soient $s=n-|\underline{a}|$ et $\underline{a}^{'}=(a_{1},\ldots,a_{k},s)$.Considérons $\mathfrak{q}^{A}(n\mid\underline{a}^{'})$ la sous-algèbre biparabolique de $\mathfrak{gl}(n)$ associée à la paire de compositions $(n,\underline{a}^{'})$. Alors, il existe $\alpha\in\mathbb{N}$ et une composition $\underline{c}=(c_{1},\ldots,c_{j})$ vérifiant $j\leq k$ et $|\underline{c}|\leq s$ tels que l’on a $$\chi[\mathfrak{q}^{C}_{n}(n\mid\underline{a})]=\alpha+\chi[\mathfrak{q}^{C}_{s+|\underline{c}|}(s+|\underline{c}|\mid\underline{c})]$$ $$\chi[\mathfrak{q}^{A}(n\mid\underline{a}^{'})]=\alpha+\chi[\mathfrak{q}^{A}(s+|\underline{c}|\mid\underline{c},s)]$$
Si $2|\underline{a}|\leq n$, il suffit de considérer $\alpha=0$ et $\underline{c}=\underline{a}$. Supposons que $2|\underline{a}|>n$, il suit du lemme 2.4 [@MB1] qu’il existe $1\leq i \leq k$ tel que $a_{i}\geq |d_{i}|$ (voir la définition des $d_{i}$ dans les théorèmes \[thm0\] et \[thm0’\]). Il résulte alors des théorèmes \[thm0\] et \[thm0’\] que le résultat s’obtient par récurrence sur $|\underline{a}|$.
Soient $\underline{a}=(a_{1},\ldots,a_{k})$ et $\underline{b}=(b_{1},\ldots,b_{t})$ deux compositions vérifiant $|\underline{b}|\leq|\underline{a}|=n$, et $s=|\underline{a}|-|\underline{b}|$. Supposons $k+t<s$, alors $\mathfrak{q}^{C}_{n}(\underline{a}\mid\underline{b})$ n’est pas une sous-algèbre de Frobenius de $\mathfrak{sp}(2n)$.
Utilisant le lemme \[lem0’\] et le théorème \[thm0’\], on a :\
$$\chi[\mathfrak{q}^{C}_{n}(\underline{a}\mid\underline{b})]=\chi[\mathfrak{q}^{C}_{2n-a_{1}}(2n-a_{1}\mid a_{k},\ldots,a_{2},\underline{b})]$$ Supposons que la sous-algèbre $\mathfrak{q}^{C}_{n}(\underline{a}\mid\underline{b})$ de $\mathfrak{sp}(2n)$ est de Frobenius. Il suit du lemme \[lemP\] qu’il existe une composition $\underline{c}=(c_{1},\ldots,c_{j})$ telle que $j\leq k+t-1<s-1$, $|\underline{c}|\leq s$ et $\chi[\mathfrak{q}^{C}_{s+|\underline{c}|}(s+|\underline{c}|\mid \underline{c})]=0$. Comme $|\underline{c}|\leq [\frac{s+|\underline{c}|}{2}]$, il suit du lemme \[lemM\] que $c_{i}=1,\;1\leq i\leq j$ et qu’il existe $\epsilon\in\{0,1\}$ tel que $j=|\underline{c}|=s-\epsilon$. On en déduit que $j\geq s-1$ : une contradiction.
\[I3\] Soient $(a,b,c)\in\mathbb{N}^{\times}$. On pose $\max(a+b,c)=n$, $p=(a+b)\wedge(b+c)$ et $r=|a+b-c|$. On a :
- Si $p> r$, alors $\chi(\mathfrak{q}^{C}_{n}(a,b\mid c))=p-r+[\frac{r}{2}]$.
- Si $p\leq r$, on a :\
$\chi[\mathfrak{q}^{C}_{n}(a,b\mid c)]=[\frac{r}{2}]$ si $p$ et $r$ de même parité.\
$\chi[\mathfrak{q}^{C}_{n}(a,b\mid c)]=[\frac{r}{2}]-1$ sinon.
Supposons $c\leq a+b=n$. Il suit du lemme \[lem0’\] et du théorème \[thm0’\] que l’on a
$$\chi[\mathfrak{q}^{C}_{n}(a,b\mid c)]=\chi[\mathfrak{q}^{C}_{2n}(2n\mid b,a,c)]=\chi[\mathfrak{q}^{C}_{b+c+r}(b+c+r\mid b,c)]$$ Supposons $r=0$, il suit du corollaire \[cor1\] (4) et du théorème \[ind\] que l’on a $$\chi[\mathfrak{q}^{C}_{b+c+r}(b+c+r\mid b,c)]=\chi[\mathfrak{q}^{A}(b+c\mid b,c)]=b\wedge c=p$$ Supposons dans la suite que $r\neq 0$ et rappelons les deux propriétés suivantes qui nous seront utiles après,
$\mathcal{P}:$ Tout triplet $(x,y,z)\in\mathbb{N}^{3}$ vérifie l’une des conditions suivantes
- $x+y\leq z$
- $x\geq y+z$
- $y>|x-z|$
$\mathcal{P}^{'}:$ Soit $z\in\mathbb{N}^{\times}$, pour toute paire $(x,y)\in\mathbb{N}^{2}$, il existe $(x^{'},y^{'})\in\mathbb{N}^{2}$ vérifiant $$\begin{cases}(x^{'}+y^{'})\wedge(y^{'}+z)=(x+y)\wedge(y+z)=:q\\
x^{'}=z\;et\;y^{'}=q-z\;si\;q> z\\
x^{'}+y^{'}\leq z\;si\;q\leq z\\
\end{cases}$$\
tel que l’on a $\chi[\mathfrak{q}^{C}_{x+y+z}(x+y+z\mid x,y)]=\chi[\mathfrak{q}^{C}_{x^{'}+y^{'}+z}(x^{'}+y^{'}+z\mid x^{'},y^{'})]$
La propriété $\mathcal{P}$ est évidente. Montrons la propriété $\mathcal{P}^{'}$ par récurrence sur la valeur de $x+y$.
Supposons $x+y\leq z$, en particulier $q\leq z$. Le résultat est vrai avec $x=x^{'}$ et $y=y^{'}$. Supposons $x+y>z$, il suit de la propriété $\mathcal{P}$ que l’on a $x\geq y+z$ ou $y> |x-z|$. Si $x=z$, en particulier $q=x+y=y+z>z$. Le résultat est encore vrai avec $x=x^{'}$ et $y=y^{'}$. Si $x\neq z$, il suit du théorème \[thm0’\] que l’on a $$\chi[\mathfrak{q}^{C}_{x+y+z}(x+y+z\mid x,y)]=\begin{cases}\chi[\mathfrak{q}^{C}_{x[|y+z|]+y+z}(x[|y+z|]+y+z\mid x[|y+z|],y)]\;si\;x\geq y+z\\
\chi[\mathfrak{q}^{C}_{x+y[|x-z|]+z}(x+y[|x-z|]+z\mid x,y[|x-z|])]\;si\;y>|x-z|
\end{cases}$$ Remarquons que $(x[|y+z|]+y)\wedge(y+z)=(x+y[|x-z|])\wedge(y[|x-z|]+z)=q$. Il suffit alors d’appliquer l’hypothèse de récurrence à la paire $$(x_{1},y_{1})=\begin{cases} (x[|y+z|],y)\;si\;x\geq y+z\\
(x,y[|x-z|])\;si\;y>|x-z|
\end{cases}$$ Il suit de ce qui précède qu’il existe $(b^{'},c^{'})\in\mathbb{N}^{2}$ vérifiant $$\begin{cases}(b^{'}+c^{'})\wedge(c^{'}+r)=(b+c)\wedge(c+r)=p\\
b^{'}=r\;et\;c^{'}=p-r\;si\;p> r\\
b^{'}+c^{'}\leq r\;si\;p\leq r\\
\end{cases}$$\
tel que l’on a $\chi[\mathfrak{q}^{C}_{n}(a,b\mid c)]=\chi[\mathfrak{q}^{C}_{b^{'}+c^{'}+r}(b^{'}+c^{'}+r\mid b^{'},c^{'})]$.\
Supposons $p> r$, en particulier $b'=r$ et $c^{'}=p-r$. Il suit du théorème \[thm0’\] et du lemme \[lemM\] que l’on a $$\chi[\mathfrak{q}^{C}_{n}(a,b\mid c)]=\chi[\mathfrak{q}^{C}_{b^{'}+c^{'}+r}(b^{'}+c^{'}+r\mid b^{'},c^{'})]=c^{'}+[\frac{r}{2}]=p-r+[\frac{r}{2}]$$ Supposons $p\leq r$, en particulier $b^{'}+c^{'}\leq r$. Il suit du lemme \[lemM\] que l’on a $$\chi[\mathfrak{q}^{C}_{n}(a,b\mid c)]=[\frac{b^{'}}{2}]+[\frac{c^{'}}{2}]+[\frac{r-b^{'}-c^{'}}{2}]$$ On distingue deux cas\
Si $p$ est pair, les entiers $c^{'},b^{'}\;et\;r$ sont alors de même parité. En particulier, on a $$\chi[\mathfrak{q}^{C}_{n}(a,b\mid c)]=\begin{cases}[\frac{r}{2}]\;si\;r\;est\;pair\\
[\frac{r}{2}]-1\;si\;r\;est\;impair\\
\end{cases}$$ Si $p$ est impair, alors parmi les entiers $c^{'},b^{'}\;et\;r-b^{'}-c^{'}$, il y a un entier impair et un autre pair. En particulier, on a $$\chi[\mathfrak{q}^{C}_{n}(a,b\mid c)]=\begin{cases}[\frac{r}{2}]-1\;si\;r\;est\;pair\\
[\frac{r}{2}]\;si\;r\;est\;impair\\
\end{cases}$$ Supposons $ a+b\leq c=n$. Il suit du corollaire \[cor1\] que l’on a $\chi[\mathfrak{q}^{C}_{n}(a,b\mid c)]=\chi[\mathfrak{q}^{C}_{n}(c\mid a,b)]$. Remarquons que $p=(a+b)\wedge (b+r)$, le résultat se déduit de ce qui précède.
\[J3\] Soit $(a,b,c)\in\mathbb{N}^{\times}$, $\mathfrak{q}^{C}_{n}(a,b\mid c)$ est une sous-algèbre de Frobenius de $\mathfrak{sp}(2n)$ si et seulement si $\max(a+b,c)=n$, et de plus, l’une des conditions suivantes est vérifiée :
- $r=1$ et $p=1$
- $r=2$ et $p=1$
- $r=3$ et $p=2$
Soient $\underline{a}=(a_{1},\ldots,a_{k})$ et $\underline{b}=(b_{1},\ldots,b_{t})$ deux compositions vérifiant $|\underline{b}|\leq|\underline{a}|=n$, et $s=|\underline{a}|-|\underline{b}|$. Considérons $\mathfrak{q}^{A}(\underline{a}\mid \underline{b},s)$ la sous-algèbre biparabolique de $\mathfrak{gl}(n)$ associée à la paire de compositions $(\underline{a},(b_{1},\ldots,b_{t},s))$. Alors,
- Il existe une composition $\underline{d}$ de $s$ qui vérifie :\
$\chi[\mathfrak{q}^{C}_{n}(\underline{a}\mid \underline{b})]-\chi[\mathfrak{q}^{A}(\underline{a}\mid \underline{b},s)]=\chi[\mathfrak{q}^{C}_{s}(\underline{d}\mid \varnothing)]-\chi[\mathfrak{q}^{A}(\underline{d}\mid s)]$
- - Supposons $\chi[\mathfrak{q}^{A}(\underline{a}\mid \underline{b},s)]=1$, alors $\chi[\mathfrak{q}^{C}_{n}(\underline{a}\mid \underline{b})]=\begin{cases} [\frac{s}{2}]-1\;si\;s\;est\;pair\\
[\frac{s}{2}]\;si\;s\;est\;impair\\
\end{cases}$
- Supposons $\chi[\mathfrak{q}^{C}_{n}(\underline{a}\mid \underline{b})]=0$, alors $\chi[\mathfrak{q}^{A}(\underline{a}\mid \underline{b},s)]=[\frac{s+1}{2}]$
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- Il suit des lemmes \[lem0\] et \[lem0’\] que l’on a $$\chi[\mathfrak{q}^{A}(\underline{a}\mid \underline{b},s)]=\chi[\mathfrak{q}^{A}(2n\mid \underline{a}^{-1}, \underline{b},s)]$$ $$\chi[\mathfrak{q}^{C}_{n}(\underline{a}\mid \underline{b})]=\chi[\mathfrak{q}^{C}_{2n}(2n\mid \underline{a}^{-1}, \underline{b})]$$ où $\underline{a}^{-1}=(a_{k},\ldots,a_{1})$. D’après le lemme \[lemP\], il existe $\alpha\in\mathbb{N}$ et une composition $\underline{c}=(c_{1},\ldots,c_{t})$ vérifiant $|\underline{c}|\leq s$ telle que l’on a $$\chi[\mathfrak{q}^{A}(2n\mid \underline{a}^{-1}, \underline{b},s)]=\alpha+\chi[\mathfrak{q}^{A}(s+|\underline{c}|\mid \underline{c},s)]$$ $$\chi[\mathfrak{q}^{C}_{2n}(2n\mid \underline{a}^{-1}, \underline{b})]=\alpha+\chi[\mathfrak{q}^{C}_{s+|\underline{c}|}(s+|\underline{c}|\mid \underline{c})]$$ Soit $\underline{d}=\begin{cases}(s-|\underline{c}|,c_{t},\ldots,c_{1})\;si\;|\underline{c}|<s\\
(c_{t},\ldots,c_{1})\;si\;|\underline{c}|=s\\
\end{cases} $. Puisque $|\underline{c}|\leq s$, alors $|\underline{c}|+s -2(c_{1}+\cdots+c_{i-1})\geq 2c_{i},\;1\leq i\leq t$. En appliquant les lemmes \[lem3\] et \[lem3’\] avec $a_{1}=|\underline{c}|+s -2(c_{1}+\cdots+c_{i-1})$ et $b_{1}=c_{i},\;i=1,\ldots,t$, on obtient $$\chi[\mathfrak{q}^{A}(s+|\underline{c}|\mid \underline{c},s)]=\chi[\mathfrak{q}^{A}(\underline{d}\mid s)]$$ $$\chi[\mathfrak{q}^{C}_{s+|\underline{c}|}(s+|\underline{c}|\mid \underline{c})]=\chi[\mathfrak{q}^{C}_{s}(\underline{d}\mid \varnothing)]$$
D’où le résultat.
- - Supposons $\chi[\mathfrak{q}^{A}(\underline{a}\mid \underline{b},s)]=1$, alors $\alpha=0$ et $\chi[\mathfrak{q}^{A}(\underline{d}\mid s)]=1$. Par suite $\chi[\mathfrak{q}^{C}_{n}(\underline{a}\mid \underline{b})]=\chi[\mathfrak{q}^{C}_{s}(\underline{d}\mid \varnothing)]=[\frac{c_{1}}{2}]+\dots+[\frac{c_{t}}{2}]+[\frac{s-|\underline{c}|}{2}]$. D’autre part, il suit du théorème \[thm1\] que le méandre $\Gamma^{A}(\underline{d}\mid s)$ de $\mathfrak{q}^{A}(\underline{d}\mid s)$ est un segment, ce qui implique que, parmi les entiers $c_{1},\ldots,c_{t},s-|\underline{c}|$ et $s$, il en existe uniquement deux qui sont impairs. En particulier, $$\chi[\mathfrak{q}^{C}_{n}(\underline{a}\mid \underline{b})]=\begin{cases} [\frac{s}{2}]-1\;si\;s\;est\;pair\\
[\frac{s}{2}]\;si\;s\;est\;impair\\
\end{cases}$$
- Supposons $\chi[\mathfrak{q}^{C}_{n}(\underline{a}\mid \underline{b})]=0$. Il suit du lemme \[lemM\] que $c_{i}=1,\;1\leq i\leq r$, $|\underline{c}|=s-\epsilon,\;\epsilon\in\{0,1\}$. Il en résulte que $\chi[\mathfrak{q}^{A}(\underline{a}\mid \underline{b},s)]=\chi[\mathfrak{q}^{A}(\underline{d}\mid s)]=[\frac{s+1}{2}]$.
Soient $\underline{a}=(a_{1},\ldots,a_{k})$ et $\underline{b}=(b_{1},\ldots,b_{t})$ deux compositions vérifiant $|\underline{b}|\leq|\underline{a}|= n$. Supposons que $s=|\underline{a}|-|\underline{b}|=1\;ou\;2$, alors $\mathfrak{q}^{C}_{n}(\underline{a}\mid \underline{b})$ est une sous-algèbre de Frobenius de $\mathfrak{sp}(2n)$ si et seulement si $\mathfrak{q}^{A}(\underline{a}\mid \underline{b},s)\cap \mathfrak{sl}(n)$ est une sous-algèbre de Frobenius de $\mathfrak{sl}(n)$.
sous-algèbres biparaboliques de $\mathfrak{so}(p)$
==================================================
Comme nous l’avons vu au numéro 2, toute sous-algèbre biparabolique de $\mathfrak{so}(2n+1)$ (resp. $\mathfrak{so}(2n)$) est conjuguée sous l’action du groupe adjoint connexe à l’une des $\mathfrak{q}^{B}_{n}(\underline{a}\mid \underline{b})$ (resp. $\mathfrak{q}^{D}_{n}(\underline{a}\mid \underline{b})$) où $\underline{a}$ et $\underline{b}$ sont deux compositions telles que $|\underline{a}|\leq n$ et $|\underline{b}|\leq n$. On associe à $\mathfrak{q}^{B}_{n}(\underline{a}\mid \underline{b})$ le même méandre associé à la sous-algèbre biparabolique $\mathfrak{q}_{n}^{C}(\underline{a}\mid \underline{b})$ de $\mathfrak{sp}(2n)$. De plus, l’indice de $\mathfrak{q}^{B}(\underline{a}\mid \underline{b})$ est égal à celui de $\mathfrak{q}_{n}^{C}(\underline{a}\mid \underline{b})$ (voir [@meander.C]). Ainsi, tous les résultats obtenus dans cet article pour les sous-algèbres biparaboliques de $\mathfrak{sp}(2n)$ sont encore vrais pour les sous-algèbres biparaboliques de $\mathfrak{so}(2n+1)$.\
Soit $\mathbb{C}^{2n}$ muni de la forme quadratique non dégénérée canonique. Comme expliqué dans [@DKT 5.24], un sous-espace isotrope de dimension $n-1$, $E$ de $\mathbb{C}^{2n}$, est contenu exactement dans deux sous-espaces isotropes de dimension $n$, chacun desquels est laissé invariant par le stabilisateur de $E$ dans $SO(2n)$. Il en résulte que dans le cas de type $D$, on peut supposer que si $|\underline{a}|=n$ (resp. $|\underline{b}|=n$), alors $a_{k}>1$ (resp. $b_{t}>1$), ce que nous faisons désormais.\
Soit $\Xi_{n}$ l’ensemble des paires de compositions $(\underline{a}=(a_{1},\ldots,a_{k}),\underline{b}=(b_{1},\ldots,b_{t}))$ qui vérifient : $|\underline{a}|=n$, $|\underline{b}|=n-1$ et $a_{k}>1$ ou $|\underline{b}|=n$, $|\underline{a}|=n-1$ et $b_{t}>1$.
Dans [@meander.D], Panyushev et Yakimova associent à chaque sous-algèbre biparabolique $\mathfrak{q}^{D}_{n}(\underline{a}\mid \underline{b})$ de $\mathfrak{so}(2n)$ un méandre de la manière suivante :\
\* Si $(\underline{a},\underline{b})\notin\Xi_{n}$, on associe à la sous-algèbre biparabolique $\mathfrak{q}^{D}_{n}(\underline{a}\mid \underline{b})$ le méandre $\Gamma^{D}_{n}(\underline{a}\mid \underline{b}):=\Gamma^{C}_{n}(\underline{a}\mid \underline{b})$, celui associé à la sous-algèbre biparabolique $\mathfrak{q}^{C}_{n}(\underline{a}\mid \underline{b})$ de $\mathfrak{sp}(2n)$.\
\* Si $(\underline{a}=(a_{1},\ldots,a_{k}),\underline{b}=(b_{1},\ldots,b_{t}))\in\Xi_{n}$ et $|\underline{a}|= n$, on pose $\underline{b}^{'}:=(b_{1},\ldots,b_{t-1},b_{t}+1)$ et on associe à la sous-algèbre biparabolique $\mathfrak{q}^{D}_{n}(\underline{a}\mid \underline{b})$ le méandre $\Gamma^{D}_{n}(\underline{a}\mid \underline{b})$ obtenu du méandre $\Gamma^{C}_{n}(\underline{a}\mid \underline{b}^{'})$ en remplaçant l’arc joignant les sommets $a_{1}+\cdots+a_{k-1}+1$ et $n$ par un arc joignant les sommets $a_{1}+\cdots+a_{k-1}+1$ et $n+1$, et l’arc joignant les sommets $n+1$ et $n+a_{k}$ par un arc joignant les sommets $n$ et $n+a_{k}$. Ces deux nouveaux arcs sont les seuls arcs de $\Gamma^{D}_{n}(\underline{a}\mid \underline{b})$ qui se croisent, on les appelle arcs croisés. De plus, ils sont soit dans un même cycle, soit chacun d’eux est dans un segment distinct du segment qui contient l’autre arc. Si $|\underline{b}|= n$, on associe à la sous-algèbre biparabolique $\mathfrak{q}^{D}_{n}(\underline{a}\mid \underline{b})$ le méandre $\Gamma^{D}_{n}(\underline{a}\mid \underline{b})$ symétrique de $\Gamma^{D}_{n}(\underline{b}\mid \underline{a})$ par rapport à la droite qui relie ses sommets.
$\\$
$\\$ $\\$ $\Gamma_{10}^{C}(1,6,3\mid 3,2,5)=
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$ $\\$\
\
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$\put(257,0){$\textcolor{blue}{\downarrow}$}$\
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$\\$ $\Gamma_{10}^{D}(1,6,3\mid 3,2,4)=
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$\\$
$\\$ $\\$ $\Gamma_{5}^{C}(5\mid 5)=
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\put(-96.5,-30){\line(0,1){65}}$ $\put(-20,0){$\textcolor{blue}{\rightarrow}$}$ $\Gamma_{5}^{D}(4\mid 5)=
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\[thm p\][@meander.D] Soit $\mathfrak{q}^{D}_{n}(\underline{a}\mid \underline{b})$ une sous-algèbre biparabolique de $\mathfrak{so}(2n)$, on a $$\chi[\mathfrak{q}^{D}_{n}(\underline{a}\mid \underline{b})]=nombre\;de\;cycles\;+\frac{1}{2}\;(nombre\;de\;segments\;non\;invariants)+\epsilon$$ où $\epsilon$ est donné de la manière suivante:$\\$ \* Si $(\underline{a},\underline{b})\notin\Xi_{n}$, alors\
$ \epsilon=\begin{cases} 0\;si\;|\underline{a}|-|\underline{b}|\;\text{est un entier pair} \\
1\;si\;|\underline{a}|-|\underline{b}|\;\text{ est un entier impair},\;\max(|\underline{a}|,|\underline{b}|)=n\;\text{et de plus l'arc joignant les sommets}\\
\;\;\;\text{n et n+1 est un arc d'un\ segment de }\Gamma^{D}_{n}(\underline{a}\mid \underline{b})\\
-1\;\text{dans les cas restants}\\
\end{cases}$\
\* Si $(\underline{a},\underline{b})\in\Xi_{n}$, alors\
$ \epsilon=\begin{cases} -1\;\text{si les arcs crois\'es sont dans un m\^eme cycle}\;de\;\Gamma_{n}^{D}(\underline{a}\mid\underline{b})\\
0\;sinon\\
\end{cases}$
Avec les notations du théorème \[thm p\], si $(\underline{a},\underline{b})\notin\Xi_{n}$, on a $$\chi[\mathfrak{q}^{D}_{n}(\underline{a}\mid \underline{b})]=\chi[\mathfrak{q}^{C}_{n}(\underline{a}\mid \underline{b})]+\epsilon$$
De manière analogue au cas $\mathfrak{sp}(2n)$, on a le lemme suivant,
\[lem r\]
Soient $\underline{a}=(a_{1},\dots, a_{k})$ et $\underline{b}=(b_{1},\ldots,b_{t})$ deux compositions telles que $|\underline{b}|\leq|\underline{a}|\leq n$. On pose $\underline{a}^{-1}=(a_{k},\dots, a_{1})$. On a :
- $\chi[\mathfrak{q}_{n}^{D}(\underline{a}\mid\underline{b})]=\chi[\mathfrak{q}_{n}^{D}(\underline{b}\mid\underline{a})]$
- $\chi[\mathfrak{q}_{n}^{D}(\underline{a}\mid\underline{b})]=\chi[\mathfrak{q}_{n+|\underline{a}|}^{D}(2|\underline{a}|\mid\underline{a}^{-1},\underline{b})]$
\[thm r1\] Soient $t\in\mathbb{N}^{\times}$ et $\underline{a}=(a_{1},\ldots,a_{k})$ une composition vérifiant $|\underline{a}|\leq t\leq n$ et $(t\mid\underline{a})\notin\Xi_{n}$. On pose $a_{k+1}=t-|\underline{a}|$ et $d_{i}=(a_{1}+\dots +a_{i-1})-(a_{i+1}+\dots+a_{k+1}),\;1\leq i\leq k$.
- Pour tout $1\leq i\leq k$ tel que $d_{i}\neq 0$ et tout $\alpha\in\mathbb{Z}$ tel que $a_{i}+\alpha|d_{i}|\geq 0$, on a $$\chi[\mathfrak{q}^{D}_{n}(t\mid\underline{a})]=\chi[\mathfrak{q}^{D}_{n+\alpha |d_{i}|}(t+\alpha |d_{i}|\mid a_{1},\ldots,a_{i-1},a_{i}+ \alpha |d_{i}|,a_{i+1},\ldots,a_{k})]$$ En particulier, on a $$\chi[\mathfrak{q}^{D}_{n}(t\mid\underline{a})]=\chi[\mathfrak{q}^{D}_{n-a_{i}+a_{i}[|d_{i}|]}(t-a_{i}+a_{i}[|d_{i}|]\mid a_{1},\ldots,a_{i-1},a_{i}[|d_{i}|],a_{i+1},\ldots,a_{k})]$$
- Pour tout $1\leq i\leq k$ tel que $d_{i}=0$, on a $$\chi[\mathfrak{q}^{D}_{n}(t\mid\underline{a})]=a_{i}+\chi[\mathfrak{q}^{D}_{n-a_{i}}(t-a_{i}\mid a_{1},\ldots,a_{i-1},a_{i+1}\ldots,a_{k})]$$
Rappelons que dans ce cas $\Gamma^{D}_{n}(t\mid\underline{a})=\Gamma^{C}_{n}(t\mid\underline{a})$, il suit du théorème \[thm p\] que l’on a $$\chi[\mathfrak{q}^{D}_{n}(t\mid\underline{a})]=\chi[\mathfrak{q}^{C}_{n}(t\mid\underline{a})]+\epsilon$$ où $\epsilon$ est donné par:$\\$ $ \epsilon=\begin{cases} 0\;\;\;\text{si $t-|\underline{a}|$ est un entier pair} \\
1\;\;\;\text{si $t-|\underline{a}|$ est un entier impair, $t=n$ et de plus l'arc de $\Gamma^{D}_{n}(t\mid \underline{a})$ joignant les sommets}\\
\text{$\;\;\;\;\;n$ et $n+1$ est un arc d'un segment de $\Gamma^{D}_{n}(t\mid \underline{a})$}\\
-1\;\text{dans les cas restants}\\
\end{cases}$\
Compte tenu du théorème \[thm0’\], il reste également à vérifier la condition sur l’arc joignant les sommets $n$ et $n+1$. Pour cela, on pose\
$(n^{'}\mid t^{'}\mid \underline{a}^{'}):=\begin{cases}(n+\alpha |d_{i}|\mid t+\alpha |d_{i}|\mid a_{1},\ldots,a_{i-1},a_{i}+ \alpha |d_{i}|,a_{i+1},\ldots,a_{k})\;si\;a_{i}+\alpha|d_{i}|\geq 0\\
(n-a_{i}\mid t-a_{i}\mid a_{1},\ldots,a_{i-1},a_{i+1}\ldots,a_{k})\;\text{si}\;d_{i}= 0
\end{cases}$,\
en particulier, $t^{'}-|\underline{a}^{'}|=t-|\underline{a}|$ et $(t^{'}\mid\underline{a}^{'})\notin\Xi_{n^{'}}$. Il s’ensuit que $\Gamma^{D}_{n^{'}}(t^{'}\mid \underline{a}^{'})=\Gamma^{C}_{n^{'}}(t^{'}\mid \underline{a}^{'})$. Remarquons que dans le cas $d_{i}= 0$, le méandre $\Gamma^{C}_{n}(t\mid \underline{a})$ est réunion disjointe des méandres $\Gamma^{C}_{n^{'}}(t^{'}\mid \underline{a}^{'})$ et $\Gamma^{C}_{a_{i}}(a_{i}\mid a_{i})$, et dans le cas $a_{i}+\alpha|d_{i}|\geq 0$, le méandre $\Gamma^{C}_{n^{'}}(t^{'}\mid \underline{a}^{'})$ est obtenu du méandre $\Gamma^{C}_{n}(t\mid \underline{a})$ comme expliqué dans la preuve du lemme \[lem1\]. Il en résulte que l’arc joignant les sommets $n$ et $n+1$ est un arc d’un segment de $\Gamma^{C}_{n}(t\mid \underline{a})$ si et seulement si l’arc joignant les sommets $n^{'}$ et $n^{'}+1$ est un arc d’un segment de $\Gamma^{C}_{n^{'}}(t^{'}\mid \underline{a}^{'})$.
Soient $(\underline{a}=(a_{1},\ldots,a_{k}),\underline{b}=(b_{1},\ldots,b_{t}))\in\Xi_{n}$ tel que $|\underline{b}|=n-1$. On pose $\underline{b}^{'}:=(b_{1},\ldots,b_{t-1},b_{t}+1)$ et on considère $\Gamma^{A}(\underline{a}\mid\underline{b}^{'})$ le méandre de la sous-algèbre biparabolique $\mathfrak{q}^{A}(\underline{a}\mid\underline{b}^{'})$ de $\mathfrak{gl}(n)$ dont les sommets sont les $n$ premiers sommets du méandre $\Gamma_{n}^{D}(\underline{a}\mid\underline{b})$. Alors, les arcs croisés du méandre $\Gamma_{n}^{D}(\underline{a}\mid\underline{b})$ sont dans un même cycle si et seulement si le dernier sommet $($le somment n$)$ du méandre $\Gamma^{A}(\underline{a}\mid\underline{b}^{'})$ appartient à un cycle de $\Gamma^{A}(\underline{a}\mid\underline{b}^{'})$.
\[corD\] Avec les notations précédentes, on a\
$\chi[\mathfrak{q}_{n}^{D}(\underline{a}\mid\underline{b})]=\begin{cases}\chi[\mathfrak{q}^{A}(\underline{a}\mid\underline{b}^{'})]\;si\;le\;somment\;n\;appartient\;\text{\`a un segment de }\Gamma^{A}(\underline{a}\mid\underline{b}^{'})\\
\chi[\mathfrak{q}^{A}(\underline{a}\mid\underline{b}^{'})]-2\;sinon\\
\end{cases}$\
En particulier, pour $n\geq 1$, on a $\chi[\mathfrak{q}_{n}^{D}(n\mid n-1)]=|n-2|$\
Pour une sous-algèbre biparabolique $\mathfrak{q}^{A}(\underline{a}\mid\underline{b})$ de $\mathfrak{gl}(n)$, on pose\
$\Psi[\mathfrak{q}^{A}(\underline{a}\mid\underline{b})]=\begin{cases}\chi[\mathfrak{q}^{A}(\underline{a}\mid\underline{b})]\;\text{si le somment $n$ appartient \`a un segment de }\Gamma^{A}(\underline{a}\mid\underline{b})\\
\chi[\mathfrak{q}^{A}(\underline{a}\mid\underline{b})]-2\;\text{sinon}\\
\end{cases}$\
\[thm r2\] Soient $\underline{a}=(a_{1},\ldots,a_{k})$ une composition vérifiant $1\leq |\underline{a}|=n-1$ $(i.e \;(n\mid\underline{a})\in\Xi_{n})$. On pose $\underline{a}^{'}=(a^{'}_{1},\ldots,a^{'}_{k})=(a_{1},\ldots,a_{k-1},a_{k}+1)$, $d_{k}=-(a^{'}_{1}+\cdots+a^{'}_{k-1})$ et $d_{i}=(a^{'}_{1}+\cdots+a^{'}_{i-1})-(a^{'}_{i+1}+\cdots+a^{'}_{k})$, $ 1\leq i\leq k-1$.
- Pour tout $1\leq i\leq k$ tel que $d_{i}\neq 0$ et tout $\alpha\in\mathbb{Z}$ tel que $a^{'}_{i}+\alpha|d_{i}|\geq 0$, on a $$\chi[\mathfrak{q}^{D}_{n}(n\mid a_{1},\ldots,a_{k})]=\Psi[\mathfrak{q}^{A}(n+\alpha|d_{i}|\mid a^{'}_{1},\ldots,a^{'}_{i-1},a^{'}_{i}+\alpha|d_{i}|,a^{'}_{i+1},\ldots,a^{'}_{k})]$$ En particulier, si on pose $t_{i}=a^{'}_{i}-a^{'}_{i}[|d_{i}|]$, on a $$\chi[\mathfrak{q}^{D}_{n}(n\mid a_{1},\ldots,a_{k})]=\Psi[\mathfrak{q}^{A}(n-t_{i}\mid a^{'}_{1},\ldots,a^{'}_{i-1},a^{'}_{i}[|d_{i}|],a^{'}_{i+1}\ldots,a^{'}_{k})]$$
- Pour tout $1\leq i\leq k$ tel que $d_{i}=0$, on a $$\chi[\mathfrak{q}^{D}_{n}(n\mid a_{1},\ldots,a_{k})]=a_{i}+\Psi[\mathfrak{q}^{A}(n-a_{i}\mid a_{1},\ldots,a_{i-1},a_{i+1}\ldots,a_{k})]$$
Soit $ 1\leq i\leq k$. On pose $(n^{'}\mid\underline{a}^{''})$ la paire de compositions donnée par\
$(n^{'}\mid\underline{a}^{''})=\begin{cases}(n+\alpha|d_{i}|\mid a^{'}_{1},\ldots,a^{'}_{i-1},a^{'}_{i}+\alpha|d_{i}|,a^{'}_{i+1},\ldots,a^{'}_{k})\;\text{si}\;a^{'}_{i}+\alpha|d_{i}|\geq 0\\
(n-a_{i}\mid a_{1},\ldots,a_{i-1},a_{i+1}\ldots,a_{k})\;\text{si}\;d_{i}=0\\
\end{cases}$\
Considérons $\Gamma^{A}(n\mid\underline{a}^{'})$ le méandre de $\mathfrak{q}^{A}(n\mid\underline{a}^{'})$ et $\Gamma^{A}(n^{'}\mid\underline{a}^{''})$ le méandre de $\mathfrak{q}^{A}(n^{'}\mid\underline{a}^{''})$ obtenu de $\Gamma^{A}(n\mid\underline{a}^{'})$ de la manière introduite dans [@MB1]. On vérifie que le dernier sommet de $\Gamma^{A}(n\mid\underline{a}^{'})$ appartient à un segment de $\Gamma^{A}(n\mid\underline{a}^{'})$ si et seulement si le dernier sommet de $\Gamma^{A}(n^{'}\mid\underline{a}^{''})$ appartient à un segment de $\Gamma^{A}(n^{'}\mid\underline{a}^{''})$. Il suit du théorème \[thm0\] que $\Psi[\mathfrak{q}^{A}(n\mid\underline{a}^{'})]=\Psi[\mathfrak{q}^{A}(n^{'}\mid\underline{a}^{''})]$. Le résultat se déduit imédiatement du corollaire précédent.
Compte tenu du lemme \[lem r\], les théorèmes \[thm r1\] et \[thm r2\] donnent un algorithme de réduction pour calculer l’indice des sous-algèbres biparaboliques de $\mathfrak{so}(2n)$.
Considérons la sous-algèbre biparabolique $\mathfrak{q}^{D}_{335}(218,15,102\mid 33,301)$ de $\mathfrak{so}(670)$. On vérifie que $(218,15,102\mid 33,301)\in\Xi_{335}$. Compte tenu du lemme \[lem r\], on a $$\mathfrak{q}^{D}_{335}(218,15,102\mid 33,301)=\chi[\mathfrak{q}^{D}_{670}(670\mid 102,15,218,33,301)]$$ Puis, on applique le théorème \[thm r1\], on a $$\begin{aligned}
\chi[\mathfrak{q}^{D}_{670}(670\mid 102,15,218,33,301)]&=\Psi[\mathfrak{q}^{A}(670\mid 102,15,218,33,302)]\\
&=\Psi[\mathfrak{q}^{A}(452\mid 102,15,33,302)]\\
&=\Psi[\mathfrak{q}^{A}(152\mid 102,15,33,2)]\\
&=\Psi[\mathfrak{q}^{A}(52\mid 2,15,33,2)]\\
&=\Psi[\mathfrak{q}^{A}(22\mid 2,15,3,2)]\\
&=\Psi[\mathfrak{q}^{A}(7\mid 2,3,2)]\\
&=3+\Psi[\mathfrak{q}^{A}(4\mid 2,2)]\\
&=3+\Psi[\mathfrak{q}^{A}(2\mid 2)]\\
&=3+2-2\\
&=3\\\end{aligned}$$
Considérons la famille des sous-algèbre biparaboliques de $\mathfrak{so}(2n)$ de la forme $\mathfrak{q}^{D}_{n}(a,b\mid c)$ où $(a,b,c)\in(\mathbb{N}^{\times})^{3}$. Dans le cas où $(a,b\mid c)\notin\Xi_{n}$, il est simple d’obtenir une formule pour l’indice de $\mathfrak{q}^{D}_{n}(a,b\mid c)$ à partir du théorème \[I3\] et du théorème \[thm p\]. Pour le cas $(a,b\mid c)\in\Xi_{n}$, il suit du lemme \[lem r\] et du corollaire \[corD\] qu’on peut supposer $(a,b\mid c)=(a,n-a-1\mid n)$. D’après le théorème \[ind\], on a $\chi[\mathfrak{q}^{A}(a,n-a\mid n)]=a\wedge n$. Il suit du lemme 3.3 de [@MB1] que le dernier sommet du méandre $\Gamma^{A}(a,n-a\mid n)$ appartient à un cycle de $\Gamma^{A}(a,n-a\mid n)$ si et seulement si $a\wedge n\geq 2$. Ainsi, on a le théorème suivant
\[I’3\] Soit $(a,n)\in(\mathbb{N}^{\times})^{2}$ tel que $a\leq n-2$, on a\
$\chi[\mathfrak{q}_{n}^{D}(a,n-a-1\mid n)]=\chi[\mathfrak{q}_{n}^{D}(a,n-a\mid n-1)]=|(a\wedge n)-2|$
\[J’3\] Soit $(a,b,c)\in(\mathbb{N}^{\times})^{3}$, $p=(a+b)\wedge(b+c)$, $r=|a+b-c|$ et $q=a\wedge n$. Alors $\mathfrak{q}^{D}_{n}(a,b\mid c)$ est une sous-algèbre de Frobenius de $\mathfrak{so}(2n)$ si et seulement si l’une des conditions suivantes est vérifiée :
- $r=1$, $q=2$ et $\max(a+b,c)=n$
- $r=1$, $p=1$ et $\max(a+b,c)=n-1$
- $r=2$, $p=1$ et $\max(a+b,c)=n$
- $r=3$, $p=2$ et $\max(a+b,c)=n-1$
\[th B\] On pose $\mathcal{F}^{A}_{n}:=\{\mathfrak{q}^{A}(\underline{a}\mid\underline{b})\subset\mathfrak{gl}(n)\;:\;\chi[\mathfrak{q}^{A}(\underline{a}\mid\underline{b})]=1\}$ et $\mathcal{F}^{D}_{n}:=\{\mathfrak{q}^{D}_{n}(\underline{a}\mid\underline{b})\subset\mathfrak{so}(2n)\;:\;(\underline{a}\mid\underline{b})\in\Xi_{n}\;et\;\chi[\mathfrak{q}^{D}_{n}(\underline{a}\mid\underline{b})]=0\}$. Soient $\underline{a}=(a_{1},\ldots,a_{m})$ et $\underline{b}=(b_{1},\ldots,b_{t})$ deux compositions de $n$ telles que $\mathfrak{q}^{A}(\underline{a}\mid\underline{b})\in\mathcal{F}^{A}_{n}$, alors les sous-algèbres $\mathfrak{q}^{D}_{2n}(2a_{1},\ldots,2a_{m}\mid 2b_{1},\ldots,2b_{t-1},2b_{t}-1)$ et $\mathfrak{q}^{D}_{2n}(2a_{1},\ldots,2a_{m-1},2a_{m}-1\mid 2b_{1},\ldots,2b_{t})$ appartiennent à $\mathcal{F}^{D}_{2n}$, et toutes les sous-algèbres de $\mathfrak{so}(2n)$ appartenant à $\mathcal{F}^{D}_{2n}$ sont ainsi obtenues.\
En particulier, pour tout $n\geq 1$, on a $\mathcal{F}^{D}_{2n+1}=\varnothing\;et\;\sharp\mathcal{F}^{D}_{2n}=2\sharp\mathcal{F}^{A}_{n}$
Soit $(\underline{c}=(c_{1},\ldots,c_{m}),\underline{d}=(d_{1},\ldots,d_{t}))\in\Xi_{n}$ tel que $|\underline{c}|=n$. On pose $\underline{d}^{'}=(d_{1},\ldots,d_{t-1},d_{t}+1)$, c’est une composition de $n$. Il résulte du corollaire \[corD\] que $\mathfrak{q}_{n}^{D}(\underline{c}\mid\underline{d})$ est une sous-algèbre de Frobenuis de $\mathfrak{so}(2n)$ si et seulement si $\Gamma^{A}(\underline{c}\mid\underline{d}^{'})$ est un cycle. D’autre part, il résulte du lemme 3.4 de [@MB1] que $\Gamma^{A}(\underline{c}\mid\underline{d}^{'})$ est un cycle si et seulement si $\chi[\mathfrak{q}^{A}(\underline{c}\mid\underline{d}^{'})]=c_{1}\wedge\ldots\wedge c_{m}\wedge d_{1}\wedge\ldots\wedge d_{t-1}\wedge(d_{t}+1)=2$. Il suit du théorème \[thm1\] que $\chi[\mathfrak{q}^{A}(\underline{a}\mid\underline{b})]=1$ si et seulement si $\Gamma^{A}(\underline{a}\mid\underline{b})$ est un segment. On déduit alors du lemme 2.6 de [@MB1] que l’application $\mathfrak{q}^{A}(\underline{a}\mid\underline{b})\longmapsto \mathfrak{q}^{A}(2a_{1},\ldots,2a_{m}\mid 2b_{1},\ldots,2b_{t})$ est une bijection de $\mathcal{F}^{A}_{n}$ sur l’ensemble des sous-algèbres biparaboliques $\mathfrak{q}^{A}(\underline{a}\mid\underline{b})$ de $\mathfrak{gl}(2n)$ tel que $\Gamma^{A}(\underline{a}\mid\underline{b})$ est un cycle. En particulier, $\mathfrak{q}^{D}_{2n}(2a_{1},\ldots,2a_{m}\mid 2b_{1},\ldots,2b_{t-1},2b_{t}-1)$ et $\mathfrak{q}^{D}_{2n}(2a_{1},\ldots,2a_{m-1},2a_{m}-1\mid 2b_{1},\ldots,2b_{t})$ appartiennent à $\mathcal{F}^{D}_{2n}$, et toutes les sous-algèbres de $\mathfrak{so}(2n)$ appartenant à $\mathcal{F}^{D}_{2n}$ sont ainsi obtenues. De plus, la condition $c_{1}\wedge\ldots\wedge c_{m}\wedge d_{1}\wedge\ldots\wedge d_{t-1}\wedge(d_{t}+1)=2$ montre que $\mathcal{F}^{D}_{n}=\varnothing$ si $n$ est impair.
Dans [@MB1], nous avons étudié la famille des sous-algèbres biparaboliques de $\mathfrak{gl}(n)$ de la forme $\mathfrak{q}^{A}(n\mid\underbrace{a,\ldots,a}_{m},b)$ où $(a,b,m)\in(\mathbb{N}^{\times})^{3}$. L’indice d’une telle sous-algèbre est donné par la formule suivante $$\chi[\mathfrak{q}^{A}(n\mid\underbrace{a,\ldots,a}_{m},b)]=(a\wedge b)\phi_{m}(\frac{a}{a\wedge b},\frac{b}{a\wedge b})$$ où $\phi_{m}$ la fonction définie sur $I:=\lbrace(a,b)\in\mathbb{N^{\times}}^{2}\mid \text{$a$ est impair ou b est impair}\rbrace$ par : $$\phi_{m}(a,b)=\begin{cases} [\frac{m}{2}]+1\;\text{si $a$ est impair et $b$ est impair}\\
[\frac{m+1}{2}]\;\text{si $a$ est impair et $b$ est pair}\\
1\;\text{si $a$ est pair et $b$ est impair}\\
\end{cases}$$ De plus, nous avons décrit explicitement le méandre $\Gamma^{A}(n\mid\underbrace{a,\ldots,a}_{m},b)$ (voir lemme 3.3 [@MB1]). Il en résulte que le dernier sommet de $\Gamma^{A}(n\mid\underbrace{a,\ldots,a}_{m},b)$ appartient à un segment de $\Gamma^{A}(n\mid\underbrace{a,\ldots,a}_{m},b)$ si et seulement si $(a\wedge b)=1$. On en déduit le théorème suivant :
\[thm a\] (Avec les notations précédentes), Soit $(a,b,m)\in(\mathbb{N}^{\times})^{3}$. On pose $n=ma+b+1$ et $p=a\wedge(b+1)$. Alors, $(n,(\underbrace{a,\ldots,a}_{m},b))\in\Xi_{n}$ et on a,
- $\chi[\mathfrak{q}^{D}_{n}(n\mid\underbrace{a,\ldots,a}_{m},b)]=\begin{cases}\phi_{m}(a,b+1)\;si\;p=1\\
p\phi_{m}(\frac{a}{p},\frac{b+1}{p})-2\;si\;p\geq 2\\
\end{cases}$
- $\mathfrak{q}^{D}_{n}(n\mid\underbrace{a,\ldots,a}_{m},b)$ est une sous-algèbre de Frobenius de $\mathfrak{so}(2n)$ si et seulement si $p=2$ et de plus l’une des conditions suivantes est vérifiée\
- m=1
- $\frac{a}{2}$ est pair et $\frac{b+1}{2}$ est impair
- $\frac{a}{2}$ est impair, $\frac{b+1}{2}$ est pair et $m=2$
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'This paper shows how one can directly apply natural language processing (NLP) methods to classification problems in cheminformatics. Connection between these seemingly separate fields is shown by considering standard textual representation of compound, SMILES. The problem of activity prediction against a target protein is considered, which is a crucial part of computer aided drug design process. Conducted experiments show that this way one can not only outrank state of the art results of hand crafted representations but also gets direct structural insights into the way decisions are made.'
author:
- |
Stanisław Jastrzębski, Damian Leśniak & Wojciech Marian Czarnecki\
Faculty of Mathematics and Computer Science\
Jagiellonian University\
Kraków, Poland\
`[email protected]`\
bibliography:
- 'paper.bib'
title: 'Learning to SMILE(S)'
---
Introduction
============
Computer aided drug design has become a very popular technique for speeding up the process of finding new biologically active compounds by drastically reducing number of compounds to be tested in laboratory. Crucial part of this process is virtual screening, where one considers a set of molecules and predicts whether the molecules will bind to a given protein. This research focuses on ligand-based virtual screening, where the problem is modelled as a supervised, binary classification task using only knowledge about ligands (drug candidates) rather than using information about the target (protein).
One of the most underrepresented application areas of deep learning (DL) is believed to be cheminformatics [@unterthiner2014deep; @bengio2012survey], mostly due the fact that data is naturally represented as graphs and there are little direct ways of applying DL in such setting [@DBLP:journals/corr/HenaffBL15]. Notable examples of DL successes in this domain are winning entry to Merck competition in 2012 [@DBLP:journals/corr/DahlJS14] and Convolutional Neural Network (CNN) used for improving data representation [@DBLP:journals/corr/DuvenaudMAGHAA15]. To the authors best knowledge all of the above methods use hand crafted representations (called fingerprints) or use DL methods in a limited fashion. The main contribution of the paper is showing that one can directly apply DL methods (*without* any customization) to the textual representation of compound (where characters are atoms and bonds). This is analogous to recent work showing that state of the art performance in language modelling can be achieved considering character-level representation of text [@DBLP:journals/corr/KimJSR15; @language_model].
Representing molecules
----------------------
Standard way of representing compound in any chemical database is called SMILES, which is just a string of atoms and bonds constructing the molecule (see Fig. \[fig:smiles\]) using a specific walk over the graph. Quite surprisingly, this representation is rarely used as a base of machine learning (ML) methods [@Worachartcheewan2014; @PMID:20570021].
Most of the classical ML models used in cheminformatics (such as Support Vector Machines or Random Forest) work with constant size vector representation through some predefined embedding (called *fingerprints*). As a result many such fingerprints have been proposed across the years [@fingerprint1; @fingerprint2]. One of the most common ones are the substructural ones - analogous of bag of word representation in NLP, where fingerprint is defined as a set of graph *templates* (SMARTS), which are then matched against the molecule to produce binary (set of words) or count (bag of words) representation. One could ask if this is really necessary, having at one’s disposal DL methods of feature learning.
Analogy to sentiment analysis
-----------------------------
The main contribution of this paper is identifying analogy to NLP and specifically sentiment analysis, which is tested by applying state of the art methods [@mesnil] directly to SMILES representation. The analogy is motivated by two facts. First, small local changes to structure can imply large overall activity change (see Fig. \[fig:SERT\_pivot\]), just like sentiment is a function of sentiments of different clauses and their connections, which is the main argument for effectiveness of DL methods in this task [@socher]. Second, perhaps surprisingly, compound graph is almost always nearly a tree. To confirm this claim we calculate molecules diameters, defined as a maximum over all atoms of minimum distance between given atom and the longest carbon chain in the molecule. It appears that in practise analyzed molecules have diameter between 1 and 6 with mean 4. Similarly, despite the way people write down text, human thoughts are not linear, and sentences can have complex clauses. Concluding, in organic chemistry one can make an analogy between longest carbon chain and sentence, where branches stemming out of the longest chain are treated as clauses in NLP.
![Visualization of CNN filters of size 5 for active (top row) and inactives molecules.[]{data-label="fig:cnn"}](fig/smiles_2.eps){height="0.07\textheight"}
![Visualization of CNN filters of size 5 for active (top row) and inactives molecules.[]{data-label="fig:cnn"}](fig/SERT_pivot_highlight.eps){height="0.11\textheight"}
![Visualization of CNN filters of size 5 for active (top row) and inactives molecules.[]{data-label="fig:cnn"}](fig/cnn.eps){height="0.11\textheight"}
\[fig:B\]
Experiments
===========
Five datasets are considered. Except SMILES, two baseline fingerprint compound representations are used, namely MACCS [@maccs] and Klekota–Roth [@klekota] (KR; considered state of the art in substructural representation [@czarnecki2015bayes]). Each dataset is fairly small (mean size is 3000) and most of the datasets are slightly imbalanced (with mean class ratio around 1:2). It is worth noting that chemical databases are usually fairly big (ChEMBL size is 1.5M compounds), which hints at possible gains by using semi-supervised learning techniques. Tested models include both traditional classifiers: Support Vector Machine (SVM) using Jaccard kernel, Naive Bayes (NB), Random Forest (RF) as well as neural network models: Recurrent Neural Network Language Model [@rnnlm] (RNNLM), Recurrent Neural Network (RNN) many to one classifier, Convolutional Neural Network (CNN) and Feed Forward Neural Network with ReLU activation. Models were selected to fit two criteria: span state of the art models in single target virtual screening [@czarnecki2015bayes; @smusz2013multidimensional] and also cover state of the art models in sentiment analysis. For CNN and RNN a form of data augmentation is used, where for each molecule random SMILES walks are computed and predictions are averaged (not doing so degrades strongly performance, mostly due to overfitting). For methods which are not designed to work on string representation (such as SVM, NB, RF, etc.) SMILES are embedded as n-gram models with simple tokenization (`[Na+]` becomes a single token). For all the remaining ones, SMILES are treated as strings composed of 2-chars symbols (thus capturing atom and its relation to the next one).
Using RNNLM, $p(\mbox{compound}|\mbox{active})$ and $p(\mbox{compound}|\mbox{inactive})$ are modelled separately and classification is done through logistic regression fitted on top. For CNN, purely supervised version of <span style="font-variant:small-caps;">context</span>, current state of the art in sentiment analysis [@DBLP:conf/naacl/Johnson015], is used. Notable feature of the model is working directly on one-hot representation of the data. Each model is evaluated using 5-fold stratified cross validation. Internal 5-fold grid is used for fitting hyperparameters (truncated in the case of deep models). We use log loss as an evaluation metric to include both classification results as well as uncertainty measure provided by models. Similar conclusions are true for accuracy.
Results
-------
Results are presented in Table \[tab:results\]. First, simple n-gram models (SVM, RF) performance is close to hand crafted state of the art representation, which suggests that potentially *any* NLP classifier working on n-gram representation might be applicable. Maybe even more interestingly, current state of the art model for sentiment analysis - CNN - despite small dataset size, outperforms (however by a small margin) traditional models.
model 5-HT$_\mathrm{1A}$ 5-HT$_\mathrm{2A}$ 5-HT$_7$ H1 SERT
-- ------- -------------------------- -------------------------- -------------------------- -------------------------- --------------------------
CNN $\mathbf{0.249\pm0.015}$ $\mathbf{0.284\pm0.026}$ $\mathbf{0.289\pm0.041}$ $\mathbf{0.182\pm0.030}$ $\mathbf{0.221\pm0.032}$
SVM $0.255\pm0.009$ $0.309\pm0.027$ $0.302\pm0.033$ $0.202\pm0.037$ $0.226\pm0.015$
GRU $0.274\pm0.016$ $0.340\pm0.035$ $0.347\pm0.045$ $0.222\pm0.042$ $0.269\pm0.032$
RNNLM $0.363\pm0.020$ $0.431\pm0.025$ $0.486\pm0.065$ $0.283\pm0.066$ $0.346\pm0.102$
SVM $0.262\pm0.016$ $0.311\pm0.021$ $0.326\pm0.035$ $0.188\pm0.022$ $0.226\pm0.014$
RF $0.264\pm0.029$ $0.297\pm0.012$ $0.322\pm0.038$ $0.210\pm0.015$ $0.228\pm0.022$
NN $0.285\pm0.026$ $0.331\pm0.015$ $0.375\pm0.072$ $0.232\pm0.034$ $0.240\pm0.024$
NB $0.634\pm0.045$ $0.788\pm0.073$ $1.201\pm0.315$ $0.986\pm0.331$ $0.726\pm0.066$
SVM $0.310\pm0.012$ $0.339\pm0.017$ $0.382\pm0.019$ $0.237\pm0.027$ $0.280\pm0.030$
RF $0.261\pm0.008$ $0.294\pm0.015$ $0.335\pm0.034$ $0.202\pm0.004$ $0.237\pm0.029$
NN $0.377\pm0.005$ $0.422\pm0.025$ $0.463\pm0.047$ $0.278\pm0.027$ $0.369\pm0.020$
NB $0.542\pm0.043$ $0.565\pm0.014$ $0.660\pm0.050$ $0.477\pm0.042$ $0.575\pm0.017$
: Log-loss ($\pm$ std) of each model for a given protein and representation.[]{data-label="results-table"}
\[tab:results\]
Hyperparameters selected for CNN (<span style="font-variant:small-caps;">context</span>) are similar to the parameters reported in [@DBLP:conf/naacl/Johnson015]. Especially the maximum pooling (as opposed to average pooling) and moderately sized regions (5 and 3) performed best (see Fig. \[fig:cnn\]). This effect for NLP is strongly correlated with the fact that small portion of sentence can contribute strongly to overall sentiment, thus confirming claimed molecule-sentiment analogy.
RNN classifier’s low performance can be attributed to small dataset sizes, as commonly RNN are applied to significantly larger volumes of data [@strategies]. One alternative is to consider semi-supervised version of RNN [@NIPS2015_5949]. Another problem is that compound activity prediction requires remembering very long interactions, especially that neighbouring atoms in SMILES walk are often disconnected in the original molecule.
Conclusions
===========
This work focuses on the problem of compounds activity prediction without hand crafted features used to represent complex molecules. Presented analogies with NLP problems, and in particular sentiment analysis, followed by experiments performed with the use of state of the art methods from both NLP and cheminformatics seem to confirm that one can actually learn directly from raw string representation of SMILES instead of currently used embedding. In particular, performed experiments show that despite being trained on relatively small datasets, CNN based solution can actually outperform state of the art methods based on structural fingerprints in ligand-based virtual screening task. At the same time it gives possibility to easily incorporate unsupervised and semi-supervised techniques into the models, making use of huge databases of chemical compounds. It appears that cheminformatics can strongly benefit from NLP and further research in this direction should be conducted.
Acknowledgments {#acknowledgments .unnumbered}
===============
First author was supported by Grant No. DI 2014/016644 from Ministry of Science and Higher Education, Poland.
|
{
"pile_set_name": "ArXiv"
}
|
---
author:
- |
Yafeng Zhang and Donatello Telesca\
Department of Biostatistics, University of California Los Angeles,\
Los Angeles, California, U.S.A.
bibliography:
- 'jointModel.bib'
title: |
**Joint Clustering and Registration\
of Functional Data**
---
Abstract
Curve registration and clustering are fundamental tools in the analysis of functional data. While several methods have been developed and explored for either task individually, limited work has been done to infer functional clusters and register curves simultaneously. We propose a hierarchical model for joint curve clustering and registration. Our proposal combines a Dirichlet process mixture model for clustering of common shapes, with a reproducing kernel representation of phase variability for registration. We show how inference can be carried out applying standard posterior simulation algorithms and compare our method to several alternatives in both engineered data and a benchmark analysis of the Berkeley growth data. We conclude our investigation with an application to time course gene expression.
.3in [*K*eywords:]{} Curve registration; Dirichlet process, Functional data clustering; Time course microarray data.
Introduction
============
Functional data is often characterized by both shape and phase variability. A typical example where these two sources of variation are clearly identified and interpreted are data arising from the study of human growth. Panel (a) and (b) of Figure \[growthVelocity\] shows growth velocity curves of 39 boys and 54 girls from the Berkeley Growth Study [@Tudden:Sny:1954]. An overall pattern is observed that growth velocity decelerates to zero from infancy to adulthood, with some subtle acceleration-deceleration pulses during late childhood and a prominent pubertal growth spurt. In this setting, phase variability is identified as variation in the timing of subject-specific growth. Explicit consideration of phase variability is necessary in order to obtain consistent estimation of typical growth patterns.
The formal analytical treatment of this problem has a long history in Statistics and Engineering. Initial contributions focused on curve alignment (registration) via dynamic time warping [@Sakoe:Chiba:1978; @Wang:Gass:alig:1997; @Wang:Gass:sync:1999] or landmark registration [@Gass:Knei:sear:1995]. Model-based alternatives represent subject-specific profiles as a parametric transformation of a common smooth regression function, evaluated over random functionals of time [@Lawt:Sylv:Magg:self:1972; @Knei:Gass:conv:1988]. Several of these methods involve a transformation of both the $x$ and $y$ axes, essentially defining the mean profile for curve $i$ as $f_i(x)=b_i + a_i m\big(\mu_i(x)\big) $, where $\mu_i(x)$ is a monotone transformation function accounting for phase variability. In longitudinal settings, @Brum:Lind:self:2004 introduced a mixed effect formulation of these models, formally accounting for dependence within subject. Similary, @Telesca:Inoue:2007 proposed a Bayesian hierarchical curve registration (BHCR) model allowing for posterior inference on both the shape function $m(\cdot)$ and transformation functions $\mu_i(x)$. Whereas these considerations are valid for any function argument $x$, it is most natural to think of $x$ as a time scale. In the following, we will therefore focus on the case of functional data observed over time.
Besides technical differences, these models of curve registration share a fundamental assumption, implying that all observed functional profiles are generated through semi-parametric transformations of a common shape $m(\cdot)$. While this assumption is likely to be warranted in standard applications, the increasing popularity of these methods for the analysis of more general data classes [@TelescaEtal2009; @TelescaErosheva:2012] motivates a methodological extension, conceiving the possible existence of shape-invariant subgroups, with group shapes $m_1(\cdot),\ldots,m_k(\cdot)$.
We are not the first to recognize a need for combing clustering and registration. Stepwise procedures, where first curves are registered and then clustered according to a chosen heuristic, have been explored in several applications [@LiuMuller2004; @TangMuller2009; @SlaetsEtal2012]. Joint clustering and registration procedures have been discussed by @GaffneySmyth2005 and @LiuYang2009. While stepwise procedures are likely to provide suboptimal estimation, available joint clustering and registration techniques have only been developed under the assumption of linear shape invariant time transformations, where $\mu_i(t) = \alpha_it + \beta_i$. Furthermore, model complexity, conceived as the number of clusters, is only treated as a nuisance parameter and fixed in a post-hock fashion via BIC or pBIC [@ChouReichl1999].
.1in We extend the BHCR model of @Telesca:Inoue:2007 to allow for shape-subgroups. Our proposal is based on a reproducing kernel (B-spline) representation of both shape and time transformation functions. To relax the homomorphic assumption, we define a non-parametric prior over shape functionals via a Dirichlet process (DP) mixture [@Ferguson1973; @Antoniak1974; @Quintana2006]. Clustering is achieved implicitly and is interpreted in terms of shape similarities. The number of clusters is subject to direct estimation and inferences account fully for this layer of uncertainty, without the need for post-hock adjustments. Furthermore, we show how posterior simulation remains straightforward via a simple extension to standard Metropolis within Gibbs MCMC transitions.
Following @LiuMuller2004, we show how this modeling approach is particularly useful for the analysis of time-course gene expression. While it is known that co-expressed genes are likely to be co-regulated, various regulation mechanisms, such as feedback loops and regulation cascades, may warp the timing of expression for genes involved in the same process or regulatory pathway [@Weber2007]. It is therefore desirable to have a model that can assign genes with similar, yet time-warped, expression profiles to the same cluster [@QianEtal2001; @Qin2006]. In other words, it is important to have a model that is phase-variation tolerant when defining curve subgroups.
The remainder of this article is organized as follows. In section \[modelFormulation\], we describe the the sampling model and priors. A posterior simulation strategy via MCMC is described in Section \[Posterior\]. In section \[simulation\], we apply the joint model to simulated datasets and compare it with single-purpose models: a clustering only model and a registration only model. In section \[growthData\] we apply the model to the Berkeley Growth Study data. In section \[fibroResponse\], we apply the model to time course microarray data of response of human fibroblasts to serum stimulation. Finally in section \[discussion\], we conclude the paper with a critical discussion.
Model Formulation {#modelFormulation}
=================
Sampling Model {#modelDescription}
--------------
Let $y_i(t)$ denote the observation of curve $i$ at time $t$, where $i=1, \ldots, N$ and $t \in T = [t_1, t_n]$. The sampling model is specified as follows: $$\label{modelStage1}
y_i(t)=c_i+a_i m_i\{\mu_i(t, {\mbox{\boldmath $\!\phi\!$ \unboldmath}}_i), {\mbox{\boldmath $\!\theta\!$ \unboldmath}}_i\} + \epsilon_{i}(t),$$ where $\epsilon_{i}(t)\sim \mathrm{N}(0, 1/\tau_i)$ and $\tau_i$ is the precision parameter.
In formula (\[modelStage1\]), $\mu_i()$ is the curve specific time transformation function, characterizing the latent time scale of curve $i$, and $m_i()$ is the curve specific shape function. The apparent lack of identifiability between $\mu_i()$ and $m_i()$ will be resolved in §\[Priors\] by specifying a random probability functional prior for $m_i()$, implicitly producing functional clusters.
To achieve flexible modeling of both time transformation and shape functions, we use B-splines [@deBoor:1978]. We model $\mu_i(t, {\mbox{\boldmath $\!\phi\!$ \unboldmath}}_i)={\mbox{\boldmath $\!B\!$ \unboldmath}}^T_{\mu}(t){\mbox{\boldmath $\!\phi\!$ \unboldmath}}_i$, where ${\mbox{\boldmath $\!B\!$ \unboldmath}}_{\mu}(t)$ is the B-spline basis vector at time $t$ and ${\mbox{\boldmath $\!\phi\!$ \unboldmath}}_i$ is the curve specific basis coefficient vector. $\mu_i()$ is a monotone function mapping the sampling time interval $T$ to the interval $\mathcal{T}=[t_1-\Delta, t_n+\Delta]$, with expansion constraint $\Delta \geq 0$ to allow the time scale to be transformed outside the observed sampling time interval $T$. To ensure monotonicity and function image boundaries, we impose the following constraints $$\label{phiConstraint}
(t_1-\Delta)\leq \phi_{i1} < \ldots < \phi_{iq} < \phi_{i(q+1)} < \ldots < \phi_{iQ} \leq (t_n+\Delta).$$ We model shape functions as $m_i\{\mu_i(t, {\mbox{\boldmath $\!\phi\!$ \unboldmath}}_i), {\mbox{\boldmath $\!\theta\!$ \unboldmath}}_i\}={\mbox{\boldmath $\!B\!$ \unboldmath}}^T_m\{\mu_i(t, {\mbox{\boldmath $\!\phi\!$ \unboldmath}}_i)\} {\mbox{\boldmath $\!\theta\!$ \unboldmath}}_i$ where ${\mbox{\boldmath $\!B\!$ \unboldmath}}_m(\cdot)$ is a B-spline basis vector and ${\mbox{\boldmath $\!\theta\!$ \unboldmath}}_i$ is the curve specific basis coefficient vector. No constraints are usually imposed on ${\mbox{\boldmath $\!\theta\!$ \unboldmath}}_i$, unless specific shapes are preferred a-priori (see for example, [@TelescaErosheva:2012]).
We note that the stochastic functionals $m_i()$ and $\mu_i()$ are now fully described by the distributions of ${\mbox{\boldmath $\!\theta\!$ \unboldmath}}_i$ and ${\mbox{\boldmath $\!\phi\!$ \unboldmath}}_i$ respectively. Identifiability of ${\mbox{\boldmath $\!\phi\!$ \unboldmath}}_i$ is ensured by modeling ${\mbox{\boldmath $\!\theta\!$ \unboldmath}}_i$ as a Dirichlet process mixture. In this setting, realizations of ${\mbox{\boldmath $\!\theta\!$ \unboldmath}}_i$ are discrete with probability one, with $K<N$ unique component vectors ${\mbox{\boldmath $\!\theta\!$ \unboldmath}}_k^*$, ($k=1,\ldots,K$). These component vectors, in turn, define cluster specific shape functions $m_k^*()$, to which member curves are aligned through $\mu_i(t)^{-1}$. Details are discussed in the following section.
Prior Model {#Priors}
-----------
We assume that shape function parameters ${\mbox{\boldmath $\!\theta\!$ \unboldmath}}_i$ and precisions $\tau_i$ to follow a Dirichlet process mixture prior. Let $G_0()$ be a base distribution absolutely continuous with respect to the Lebesque measure on $\mathbb{R}^p\cup \mathbb{R}^+$ and $\delta({\mbox{\boldmath $\!\theta\!$ \unboldmath}}_j, \tau_j)$ a Dirac mass at $({\mbox{\boldmath $\!\theta\!$ \unboldmath}}_j, \tau_j)$. Using a predictive Pòlya urn scheme [@BlackwellMacQueen1973], we specify the prior distribution as follows: $$\label{DPprior}
{\mbox{\boldmath $\!\theta\!$ \unboldmath}}_i, \tau_i|{\mbox{\boldmath $\!\theta\!$ \unboldmath}}_{-i}, \tau_{-i} \sim \frac{\alpha}{(\alpha+N-1)} G_0({\mbox{\boldmath $\!\theta\!$ \unboldmath}}_i, \tau_i)+\frac{1}{(\alpha+N-1)}\sum_{j\neq i}\delta({\mbox{\boldmath $\!\theta\!$ \unboldmath}}_j, \tau_j),$$ where $-i=\{j: j\neq i\}$ is the set all the indices other than $i$ and $\alpha$ is the weight parameter of the Dirichlet process model. This prior generates the shape ${\mbox{\boldmath $\!\theta\!$ \unboldmath}}_i$ and error precision $\tau_i$ for curve $i$, from a mixture involving a random draw from the base density $G_0()$ or the point mass $({\mbox{\boldmath $\!\theta\!$ \unboldmath}}_j, \tau_j)$’s, $j \neq i$.
Realizations from the prior in (\[DPprior\]) define a discrete distribution, implying ties among $({\mbox{\boldmath $\!\theta\!$ \unboldmath}}_i, \tau_i)$’s, $i=1,\ldots,N$. These ties are naturally interpreted as clusters among the $N$ curves, namely, curve $i$ and $j$ belong to the same cluster if $({\mbox{\boldmath $\!\theta\!$ \unboldmath}}_i, \tau_i)=({\mbox{\boldmath $\!\theta\!$ \unboldmath}}_j, \tau_j)$. As a result, only $K < N$ unique values are observed, each of which is associated with a cluster and is denoted by $({\mbox{\boldmath $\!\theta\!$ \unboldmath}}_k^*, \tau_k^*)$, $k=1, \ldots, K$. In this setting, we can re-express formula (\[DPprior\]) as: $$\label{DPprior2}
{\mbox{\boldmath $\!\theta\!$ \unboldmath}}_i, \tau_i|{\mbox{\boldmath $\!\theta\!$ \unboldmath}}_k^*, \tau_k^{*} \sim
\frac{\alpha}{(\alpha+N-1)} G_0({\mbox{\boldmath $\!\theta\!$ \unboldmath}}_i, \tau_i)+\frac{1}{(\alpha+N-1)}\sum_{k=1}^{K_{-i}}n_{k(-i)}\delta({\mbox{\boldmath $\!\theta\!$ \unboldmath}}_k^*, \tau_k^{*}),$$ where $n_{k(-i)}$ is the size of cluster $k$ and $K_{-i}$ is number of clusters when curve $i$ is excluded. The representation above implies that a complete sample of $({\mbox{\boldmath $\!\theta\!$ \unboldmath}}_i, \tau_i)$, $(i=1,\ldots,N)$ is in one to one correspondence with a set of unique values, $({\mbox{\boldmath $\!\theta\!$ \unboldmath}}_k^*, \tau_k^*)$, $(k=1,\ldots,K)$, through cluster labels ${\mbox{\boldmath $\!s\!$ \unboldmath}}=(s_1, \ldots, s_N)$. Specifically, $s_i=k$ if $({\mbox{\boldmath $\!\theta\!$ \unboldmath}}_i, \tau_i)=({\mbox{\boldmath $\!\theta\!$ \unboldmath}}_k^*, \tau_k^{*})$ and $s_i=K_{-i}+1$ if $({\mbox{\boldmath $\!\theta\!$ \unboldmath}}_i, \tau_i)$ is a new sample from $G_0({\mbox{\boldmath $\!\theta\!$ \unboldmath}}_i, \tau_i)$, which means curve $i$ forms a new cluster of its own. As a result, the number of clusters $K$ is also determined implicitly.
We note that if we omit the time transformation modeling with $\mu_i(t, {\mbox{\boldmath $\!\phi\!$ \unboldmath}}_i)$ and use time $t$ directly in the shape functions $m_i(t, {\mbox{\boldmath $\!\theta\!$ \unboldmath}}_i)$, our model reduces to standard functional clustering via Dirichlet process mixtures.
.2in We assume that the base DP mixture density factors as $G_0({\mbox{\boldmath $\!\theta\!$ \unboldmath}}_i, \tau_i)=p({\mbox{\boldmath $\!\theta\!$ \unboldmath}}_i \mid \tau_i)p(\tau_i)$, where ${\mbox{\boldmath $\!\theta\!$ \unboldmath}}_i \mid \tau_i \sim \mathrm{N}\left({\mbox{\boldmath $\!0\!$ \unboldmath}}, (\tau_{\theta}\tau_i{\mbox{\boldmath $\!\Sigma\!$ \unboldmath}})^{-1}\right)$ and $\tau_i \sim \mathrm{Ga}(a, b)$, a Gamma distribution with mean $a/b$. The specific form of the precision matrix $\Sigma$ is determined by a second-order shrinkage process: $\theta_{ip}-\theta_{i(p-1)}=$ $\theta_{i(p-1)}-\theta_{i(p-2)}+\xi_{ip}$ with $\xi_{ip} \sim \mathrm{N}\big(0, 1/(\tau_{\theta}\tau_i)\big)$ ($p=1,\ldots,P$) where $P$ is the dimension of ${\mbox{\boldmath $\!\theta\!$ \unboldmath}}_i$ and $\theta_{i0}=\theta_{i(-1)}=0$ [@Lang:Brez:baye:2004]. In this setting, the product $\tau_{\theta}\tau_i$ can be interpreted as a smoothing parameter for curve $i$.
Similarly, we also use a penalized B-spline prior on the time transformation function parameters ${\mbox{\boldmath $\!\phi\!$ \unboldmath}}_i$. In particular, letting ${\mbox{\boldmath $\!\phi\!$ \unboldmath}}_0$ be the vector associated with identity transformation so that $\mu(t, {\mbox{\boldmath $\!\phi\!$ \unboldmath}}_0)=t$, we assume $\phi_{iq}-\phi_{0q}=$ $\phi_{i(q-1)}-\phi_{0(q-1)}+\nu_{iq}$ with $\nu_{iq} \sim \mathrm{N}(0, 1/\tau_{\phi})$ ($q=1,\ldots, Q$) where $Q$ is dimension of ${\mbox{\boldmath $\!\phi\!$ \unboldmath}}_i$ and $\phi_{i0}=0$, implying ${\mbox{\boldmath $\!\phi\!$ \unboldmath}}_i \sim \mathrm{N}\left({\mbox{\boldmath $\!\phi\!$ \unboldmath}}_0, (\tau_{\phi}\Omega)^{-1}\right)$. In the foregoing, $\Omega$ is deterministic and $\tau_{\phi}$ is interpreted as a smoothing parameter.
Following @TelescaEtal2009, when modeling cluster specific common shape functions, we let the number of spline knots equal to the number of sampling time points. For the curve specific time transformation functions structural smoothness is imposed by their monotonicity (\[phiConstraint\]), suggesting parsimony in the choice of the number of knots. In many application contexts, $1$ to $4$ equally spaced interior knots allow for enough flexibility in the representation of time transformation.
.2in For ease of computation, we complete our model with priors and hyperpriors following principles of conditional conjugacy. Specifically, curve specific mean levels parameters are specified as $c_i \sim \mathrm{N}(c_0, 1/\tau_c)$ and curve specific amplitude parameters $a_i \sim \mathrm{N}(a_0, 1/\tau_a)I(a_i>0)$. The assumption of strictly positive amplitudes is appropriate if synchronous but negatively correlated curves are to be clustered separately. Removing positivity restrictions will imply clustering of synchronous profiles. We complete our prior specifications assuming $c_0 \sim \mathrm{N}(0, 1/\tau_{c_0})$, $a_0 \sim \mathrm{N}(1, 1/\tau_{a_0})$, $\tau_a \sim \mathrm{Ga}(a_a, b_a)$ and $\tau_c \sim \mathrm{Ga}(a_c, b_c)$. Smoothing parameters priors are specified as $\tau_{\theta} \sim \mathrm{Ga}(a_{\theta}, b_{\theta})$ and $\tau_{\phi} \sim \mathrm{Ga}(a_{\phi}, b_{\phi})$. Finally, the weight parameter of the Dirichlet process mixture is defined as $\alpha \sim \mathrm{Ga}(a_{\alpha}, b_{\alpha})$.
Posterior Inference {#Posterior}
===================
Posterior Simulation {#MCMC}
--------------------
Markov Chain Monte Carlo simulation from the posterior distribution is conceptually straightforward and obtained as a simple sequence of Metropolis-Hastings within Gibbs transitions.
For ease of notation, we let ${\mbox{\boldmath $\!\eta\!$ \unboldmath}}_i=({\mbox{\boldmath $\!\theta\!$ \unboldmath}}_i^t, \tau_i)^t$ and ${\mbox{\boldmath $\!y\!$ \unboldmath}}_i=(y_i(t_1), \ldots, y_i(t_n))^t$. Without loss of generality, we also assume that curves are of the same length $n$. The proposed Markov transition sequence is implemented by: (i) sampling $({\mbox{\boldmath $\!\phi\!$ \unboldmath}}_i,\,{\mbox{\boldmath $\!\eta\!$ \unboldmath}}_i,\,{\mbox{\boldmath $\!s\!$ \unboldmath}}_i)$ given all other parameters, (ii) resampling ${\mbox{\boldmath $\!\eta\!$ \unboldmath}}^*_k=({\mbox{\boldmath $\!\theta\!$ \unboldmath}}_k^{t*}, \tau_k^{*})^t$ given cluster indicators ${\mbox{\boldmath $\!s\!$ \unboldmath}}_i$ and all other parameters and (iii) sampling $\alpha$ and remaining parameters from their full conditional posteriors. We outline details as follows.
.1in (i) [*Sampling $({\mbox{\boldmath $\!\phi\!$ \unboldmath}}_i,\,{\mbox{\boldmath $\!\eta\!$ \unboldmath}}_i,\,{\mbox{\boldmath $\!s\!$ \unboldmath}}_i)$.*]{} The full conditional posterior of ${\mbox{\boldmath $\!\eta\!$ \unboldmath}}_i$ is a Dirichlet process mixture with updated mixing probabilities and components [@Escobar1994; @WestEtal1994]: $$\label{eta}
{\mbox{\boldmath $\!\eta\!$ \unboldmath}}_i\mid {\mbox{\boldmath $\!\eta\!$ \unboldmath}}_k^*, {\mbox{\boldmath $\!\phi\!$ \unboldmath}}_i, {\mbox{\boldmath $\!y\!$ \unboldmath}}_i
\sim \frac{q_{i0}G_i({\mbox{\boldmath $\!\eta\!$ \unboldmath}}_i \mid {\mbox{\boldmath $\!\phi\!$ \unboldmath}}_i, \,{\mbox{\boldmath $\!y\!$ \unboldmath}}_i) +\sum\limits_{k=1}^{K_{-i}}q_{ik}\delta({\mbox{\boldmath $\!\eta\!$ \unboldmath}}_k^*)} {q_{i0}+\sum\limits_{k=1}^{K_{-i}}q_{ik}},$$ where $q_{i0}=\alpha\int f({\mbox{\boldmath $\!y\!$ \unboldmath}}_i\mid {\mbox{\boldmath $\!\phi_i\!$ \unboldmath}}, {\mbox{\boldmath $\!\eta\!$ \unboldmath}}_i)\mathrm{d}G_0({\mbox{\boldmath $\!\eta\!$ \unboldmath}}_i)$ is $\alpha$ times the marginal likelihood of ${\mbox{\boldmath $\!y\!$ \unboldmath}}_i$, $G_i({\mbox{\boldmath $\!\eta\!$ \unboldmath}}_i \mid {\mbox{\boldmath $\!\phi\!$ \unboldmath}}_i, \,{\mbox{\boldmath $\!y\!$ \unboldmath}}_i) \propto f({\mbox{\boldmath $\!y\!$ \unboldmath}}_i\mid {\mbox{\boldmath $\!\phi_i\!$ \unboldmath}}, {\mbox{\boldmath $\!\eta\!$ \unboldmath}}_i)G_0({\mbox{\boldmath $\!\eta\!$ \unboldmath}}_i) $ is the full conditional density of ${\mbox{\boldmath $\!\eta\!$ \unboldmath}}_i$ and $q_{ik}=n_{k(-i)}f({\mbox{\boldmath $\!y\!$ \unboldmath}}_i \mid {\mbox{\boldmath $\!\phi\!$ \unboldmath}}_i, {\mbox{\boldmath $\!\eta\!$ \unboldmath}}_k^*)$ is the product of cluster size $n_{k(-i)}$ and the likelihood associated with ${\mbox{\boldmath $\!\eta\!$ \unboldmath}}_i={\mbox{\boldmath $\!\eta\!$ \unboldmath}}_k^*$.
To improve mixing rates, we combine the sampling of ${\mbox{\boldmath $\!s\!$ \unboldmath}}_i$, ${\mbox{\boldmath $\!\eta\!$ \unboldmath}}_i$ and ${\mbox{\boldmath $\!\phi\!$ \unboldmath}}_i$. Specifically, we use $K_{-i}$ copies of ${\mbox{\boldmath $\!\phi\!$ \unboldmath}}_i$, ${\mbox{\boldmath $\!\phi\!$ \unboldmath}}_i^1, \ldots, {\mbox{\boldmath $\!\phi\!$ \unboldmath}}_i^{K_{-i}}$, one for each cluster. We update ${\mbox{\boldmath $\!\phi\!$ \unboldmath}}_i^k$, assuming ${\mbox{\boldmath $\!\eta\!$ \unboldmath}}_i={\mbox{\boldmath $\!\eta\!$ \unboldmath}}_k^*$, with the Metropolis-Hastings algorithm as in @Telesca:Inoue:2007, so that the appropriate time transformation is found for curve $i$ to be registered with the common shape function of cluster $k$. We calculate each $q_{ik}$ $(k =1, \ldots, K_{-i})$ in (\[eta\]) using the corresponding ${\mbox{\boldmath $\!\eta\!$ \unboldmath}}_k^*$ and ${\mbox{\boldmath $\!\phi\!$ \unboldmath}}_i^k$. When calculating $q_{i0}$, we use the value of ${\mbox{\boldmath $\!\phi\!$ \unboldmath}}_i$ from the previous iteration of the Gibbs sampler.
Specifically, let ${\mbox{\boldmath $\!B\!$ \unboldmath}}_i={\mbox{\boldmath $\!B\!$ \unboldmath}}_m\{u_i({\mbox{\boldmath $\!t\!$ \unboldmath}}, {\mbox{\boldmath $\!\phi\!$ \unboldmath}}_i)\} = {\mbox{\boldmath $\!B\!$ \unboldmath}}_m\{{\mbox{\boldmath $\!B\!$ \unboldmath}}_{\mu}^T({\mbox{\boldmath $\!t\!$ \unboldmath}}){\mbox{\boldmath $\!\phi\!$ \unboldmath}}_i\}$ and ${\mbox{\boldmath $\!c\!$ \unboldmath}}_i=c_i{\mbox{\boldmath $\!1\!$ \unboldmath}}$, and define the following summaries: ${\mbox{\boldmath $\!E\!$ \unboldmath}}_i=a_i^2{\mbox{\boldmath $\!B\!$ \unboldmath}}_i^T{\mbox{\boldmath $\!B\!$ \unboldmath}}_i+\tau_{\theta}{\mbox{\boldmath $\!\Sigma\!$ \unboldmath}}$, ${\mbox{\boldmath $\!\mu\!$ \unboldmath}}_i=a_i{\mbox{\boldmath $\!B\!$ \unboldmath}}_i^T({\mbox{\boldmath $\!y\!$ \unboldmath}}_i-{\mbox{\boldmath $\!c\!$ \unboldmath}}_i)$, $a'_i=\frac{n}{2}+a$ and $b'_i=\frac{1}{2}({\mbox{\boldmath $\!y\!$ \unboldmath}}_i-{\mbox{\boldmath $\!c\!$ \unboldmath}}_i)^T ({\mbox{\boldmath $\!I\!$ \unboldmath}}-a_i^2{\mbox{\boldmath $\!B\!$ \unboldmath}}_i{\mbox{\boldmath $\!E\!$ \unboldmath}}_i^{-1}{\mbox{\boldmath $\!B\!$ \unboldmath}}_i^T)({\mbox{\boldmath $\!y\!$ \unboldmath}}_i-{\mbox{\boldmath $\!c\!$ \unboldmath}}_i) +b$. To sample ${\mbox{\boldmath $\!\eta\!$ \unboldmath}}_i$ from its full conditional (\[eta\]), we follow the procedure below:
1. Sample cluster membership $s_i$ which takes values on $K_{-i}+1, 1, \ldots, K_{-i}$ with probabilities proportional to $q_{i0}, q_{i1}, \ldots, q_{iK_{-i}}$.
2. If $s_i=K_{-i}+1$, we keep ${\mbox{\boldmath $\!\phi\!$ \unboldmath}}_i$ unchanged. Curve $i$ forms a new cluster, and a draw of ${\mbox{\boldmath $\!\eta\!$ \unboldmath}}_i$ from $G_i({\mbox{\boldmath $\!\eta\!$ \unboldmath}}_i \mid {\mbox{\boldmath $\!\phi\!$ \unboldmath}}_i,{\mbox{\boldmath $\!y\!$ \unboldmath}}_i)$ is obtained by first sampling $\tau_i \sim \mathrm{Ga}(a'_i, b'_i)$ and then sampling ${\mbox{\boldmath $\!\theta\!$ \unboldmath}}_i \mid \tau_i \sim \mathrm{N}({\mbox{\boldmath $\!E\!$ \unboldmath}}_i^{-1}{\mbox{\boldmath $\!\mu\!$ \unboldmath}}_i ,(\tau_i{\mbox{\boldmath $\!E\!$ \unboldmath}}_i)^{-1})$. If $s_i=k$, we use the corresponding ${\mbox{\boldmath $\!\phi\!$ \unboldmath}}_i^k$ as a draw of ${\mbox{\boldmath $\!\phi\!$ \unboldmath}}_i$ and let ${\mbox{\boldmath $\!\eta\!$ \unboldmath}}_i={\mbox{\boldmath $\!\eta\!$ \unboldmath}}_k^*$.
.2in (ii) [*Resampling ${\mbox{\boldmath $\!\eta\!$ \unboldmath}}^*_k$ given cluster indicators ${\mbox{\boldmath $\!s\!$ \unboldmath}}_i$.*]{} After a sample of ${\mbox{\boldmath $\!\eta\!$ \unboldmath}}^T=({\mbox{\boldmath $\!\eta\!$ \unboldmath}}_1^T, \ldots, {\mbox{\boldmath $\!\eta\!$ \unboldmath}}_N^T)$ and ${\mbox{\boldmath $\!s\!$ \unboldmath}}=(s_1, \ldots, s_N)^T$ is generated, to improve mixing rates, we update each ${\mbox{\boldmath $\!\eta\!$ \unboldmath}}_k^*$ from its full conditional $G_k({\mbox{\boldmath $\!\eta\!$ \unboldmath}}_k^*\mid {\mbox{\boldmath $\!y\!$ \unboldmath}}_{i\in S_k}) \propto \prod_{i\in S_k}\!f({\mbox{\boldmath $\!y\!$ \unboldmath}}_i\mid {\mbox{\boldmath $\!\phi_i\!$ \unboldmath}}, {\mbox{\boldmath $\!\eta\!$ \unboldmath}}_i)G_0({\mbox{\boldmath $\!\eta\!$ \unboldmath}}_k^*)$, where $S_k=\{i:s_i=k\}$ is the set of curves in cluster $k$ [@MacEachernMuller1998]. Furthermore, $G_k({\mbox{\boldmath $\!\eta\!$ \unboldmath}}_k^*\mid {\mbox{\boldmath $\!y\!$ \unboldmath}}_{i\in S_k}) \propto p({\mbox{\boldmath $\!\theta\!$ \unboldmath}}_k^*\mid\tau_k^*,{\mbox{\boldmath $\!y\!$ \unboldmath}}_{i\in S_k})p(\tau_k^*\mid{\mbox{\boldmath $\!y\!$ \unboldmath}}_{i\in S_k})$, s.t. $$\label{etak}
\begin{split}
&{\mbox{\boldmath $\!\eta\!$ \unboldmath}}_k^*\mid {\mbox{\boldmath $\!y\!$ \unboldmath}}_{i\in S_k} \sim \mathrm{N}({\mbox{\boldmath $\!E\!$ \unboldmath}}_k^{-1}{\mbox{\boldmath $\!\mu\!$ \unboldmath}}_k, (\tau_k^*{\mbox{\boldmath $\!E\!$ \unboldmath}}_k)^{-1})\\
&\times \mathrm{Ga}\left(\frac{1}{2}\sum_{i\in S_k}\!\!n_i+a, \frac{1}{2}\left(\sum_{i\in S_k}({\mbox{\boldmath $\!y\!$ \unboldmath}}_i-{\mbox{\boldmath $\!c\!$ \unboldmath}}_i)^T({\mbox{\boldmath $\!y\!$ \unboldmath}}_i-{\mbox{\boldmath $\!c\!$ \unboldmath}}_i) - {\mbox{\boldmath $\!\mu\!$ \unboldmath}}_k^T{\mbox{\boldmath $\!E\!$ \unboldmath}}_k^{-1}{\mbox{\boldmath $\!\mu\!$ \unboldmath}}_k\right)+b\right),
\end{split}$$ where ${\mbox{\boldmath $\!E\!$ \unboldmath}}_k=\sum_{i\in S_k} a_i^2{\mbox{\boldmath $\!B\!$ \unboldmath}}_i^T{\mbox{\boldmath $\!B\!$ \unboldmath}}_i + \tau_{\theta}{\mbox{\boldmath $\!\Sigma\!$ \unboldmath}}$ and ${\mbox{\boldmath $\!\mu\!$ \unboldmath}}_k=\sum_{i\in S_k}a_i{\mbox{\boldmath $\!B\!$ \unboldmath}}_i^T({\mbox{\boldmath $\!y\!$ \unboldmath}}_i-{\mbox{\boldmath $\!c\!$ \unboldmath}}_i)$.
.2in (iii) [*Sampling $\alpha$ and all hyper parameters.*]{} To develop the full conditional of $\alpha$, we note that $p(K|\alpha,N)\propto N!\alpha^K\frac{\Gamma(\alpha)}{\Gamma(\alpha+N)}$ [@Antoniak1974]. Following [@West1992], we define an auxiliary random quantity $x\mid \alpha \sim \mathrm{B}(\alpha+1, N)$ and a mixing probability $\pi_x$: $$\frac{\pi_x}{1-\pi_x}=\frac{a_{\alpha}+K-1}{N(b_{\alpha}-log(x))}.$$ Conditioning on $x$, it is easily shown that the full conditional distribution of $\alpha$ is a mixture of gamma densities. Specifically, $$\label{alpha3}
\begin{split}
&\alpha\mid x, K \sim\\
&\pi_x \mathrm{Ga}(a_{\alpha}+K, b_{\alpha}-log(x)) + (1-\pi_x)\mathrm{Ga}(a_{\alpha}+K-1, b_{\alpha}-log(x))
\end{split}$$ The rest of the model parameters are simulated directly from their full conditional posterior distributions. Detailed results are reported in Web Appendix A.
Posterior Inference {#inference}
-------------------
We base our inference on MCMC samples from the posterior distribution of the model parameters. Inference for functional quantities is obtained by post-processing these finite-dimensional posterior samples. To get a point estimate of the clustering structure we use the maximum a-posterior (MAP) clustering.
Given $M$ posterior samples of ${\mbox{\boldmath $\!\phi\!$ \unboldmath}}_i^{(j)}$, $(j=1, \ldots, M)$, posterior samples of the time transformation function $\mu_i(t)$ at any time point $t \in T$ can be calculated as: $$\label{muPosterior}
\mu_i^{(j)}(t)=\mu_i^{(j)}(t, {\mbox{\boldmath $\!\phi\!$ \unboldmath}}_i^{(j)})={\mbox{\boldmath $\!B\!$ \unboldmath}}_{\mu}^T(t){\mbox{\boldmath $\!\phi\!$ \unboldmath}}_i^{(j)}.$$ Here, the posterior mean function $\hat{\mu}_i(t)=\frac{1}{M}\sum_{j=1}^M\mu_i^{(j)}(t)$ provides an point estimate of $\mu_i(t)$, and curves are registered on the transformed time scales $\hat{\mu}_i(t)$ within each cluster.
Similar estimators are defined for cluster-specific shape functions: $$\label{clusterShapeFunction}
m_k(t)=c_0+a_0{\mbox{\boldmath $\!B\!$ \unboldmath}}^T_m(t){\mbox{\boldmath $\!\theta\!$ \unboldmath}}_k^*, \;\;\;\; (k=1,\ldots,K);$$ and curve specific profiles: $$\label{curveShapeFunction}
m_i(\mu_i(t))=c_i+a_i{\mbox{\boldmath $\!B\!$ \unboldmath}}^T_m({\mbox{\boldmath $\!B\!$ \unboldmath}}^T_{\mu}(t){\mbox{\boldmath $\!\phi\!$ \unboldmath}}_i){\mbox{\boldmath $\!\theta\!$ \unboldmath}}_i.$$ Point-wise credible intervals and functional bands are easily obtained as empirical quantiles. Alternatively, the simultaneous credible band for a function $f(\cdot)$ can be obtained as described in @CrainiceanuEtal2007 and @Telesca:Inoue:2007.
To assess the model fit, we use the Conditional Predictive Ordinate (CPO) [@geisser1979predictive; @Pettie1990]. The theoretical CPO for curve $i$ is defined as $$\label{CPO}
p({\mbox{\boldmath $\!y\!$ \unboldmath}}_i|{\mbox{\boldmath $\!y\!$ \unboldmath}}_{-i}) = \int p({\mbox{\boldmath $\!y\!$ \unboldmath}}_i|\Theta)p(\Theta|{\mbox{\boldmath $\!y\!$ \unboldmath}}_{-i})d\Theta,$$ where $\Theta$ denotes the collection of all model parameters. A Monte Carlo estimate, based on posterior draws is defined as: $$\label{CPOMCMC}
\mathrm{CPO}_i= \left\{M^{-1}\sum_{j=1}^M p({\mbox{\boldmath $\!y\!$ \unboldmath}}_i|\Theta^{(j)})^{-1}\right\}^{-1}.$$ Overall model fit is assessed using the log pseudo marginal likelihood (LPML), computed as: $$\label{LPML}
\mathrm{LPML}=\sum_{i=1}^N \log(\mathrm{CPO}_i).$$
A Monte Carlo Study of Engineered Data {#simulation}
======================================
We carry out a simulation study aimed at assessing the merits of joint clustering and registration and comparing the performance of our modeling strategy to common clustering techniques. We consider 100 datasets, each consisting of, $45$ curves in $4$ clusters. Each curve $i$ is generated as: $y_i(t)=c_i+a_if_k(\mu_i(t))+\epsilon_{it}$, (if $s_i=k$), with $c_i \sim \mathrm{N}(0, \sigma=0.3)$, $a_i \sim \mathrm{N}(1, \sigma=0.3)I(a_i>0)$ and $\epsilon_{it} \sim \mathrm{N}(0, \sigma=0.3)$. We simulate realizations at $21$ equidistant time points within interval $T=[0,20]$. We use the following cluster specific shape functions: $f_1(t)=\cos(t/4)+\sin(t/4)$, $f_2(t)=\cos(t/8)$, $f_3(t)=\sin(t/2)$ and $f_4(t)=0$. Cluster $4$ serves as a noise cluster with no signal. Finally, time transformations $\mu_i(t)$ are generated as a monotone linear combination of B-spline basis, defined by one interior knot at $t=10$.
We fit our model overparametrizing functional forms and fix $31$ equidistant interior knots between $-5$ to $25$ to model common shapes spline bases, and $3$ interior knots at $(5, 10, 15)$ to model time transformation spline bases. Precisions and the Dirichelet mixture weight $\alpha$ are assigned diffuse $\mathrm{Ga}(0.01,0.01)$ priors, (mean=1, variance=100).
.1in To assess the joint model’s ability to simultaneous cluster and register curves, we compare the model with a registration only (BHCR) model and the clustering only model as described in section \[Priors\]. For both the clustering only model and the registration only model, $20K$ iterations is run for the MCMC with the first $10K$ as burn-in. We also compare our model with model-based clustering (MCLUST) [@FraleyRaftery2002a] and functional clustering (FCM) [@JamesSugar2003].
Figure \[simulCurv\] shows the results from one of the simulations. Panel (a) shows $45$ curves color-coded by cluster membership. Panel (b) shows estimated individuals curves clustered and registered within each cluster, with superimposed cluster-specific shape functions (solid black). Panel (c) shows posterior expected cluster-specific shape functions (black) against the simulation truth (gray). The model is able to accurately recover cluster specific shapes. Panel (d)-(f) show results for three individual curves, each from one of the first three clusters. Posterior estimates of individual curves (solid black) are close to the simulation truth (solid gray) and $95\%$ simultaneous credible bands achieve calibrated coverage. Also shown are profile estimates from the the registration only model (dotdash) and the clustering only model (dotted). As expected, since the registration only model assumes all the curves share a common shape function, cluster-specific functional features are confounded and model fits tend to exhibit spurious features. The clustering only model is inherently highly flexible, as small sub-clusters are allowed to form and fit specific profiles. However, since there tend to be only few curves in each cluster, the loss of information results in noisier estimates.
![\[simulCurv\]**Simulation study: assessing model fit**. (a) Simulated unregistered curves in $4$ clusters shown in different colors from one sample dataset. (b) Estimated individuals curves clustered and registered within each cluster, superimposed by the posterior cluster specific common shape functions (solid black). (c) Estimated cluster specific common shape functions (black) and simulation truth (gray). (c)-(f) Three individual curves from cluster 1, 2 and 3, circles indicate the data points for each curve. Estimated individual curves and the true curves are shown in solid black and gray, respectively. $95\%$ simultaneous credible bands are shown as dashed lines. Estimated individual curves from the registration only model and the clustering only model are shown as the dotdash and dotted lines respectively.](simulCurv.eps){width="130mm"}
.1in We also compared our joint model with MCLUST and FCM in terms of both curve estimation accuracy and clustering accuracy. We apply MCLUST and FCM on the same 100 datasets, allowing for up to 10 clusters. Figure \[simulClust\] summarizes comparison results. Panel (a) shows boxplots of the log pseudo marginal likelihood (LPML) comparing the three Bayesian models. In panel (b) we show boxplots of the simulation mean squared error (MSE) between the estimated and true individual curves for all five models. The joint model exhibits best performance in terms of MSE, and the three Bayesian model perform better than MCLUST and FCM. To compare the clustering performance we used adjusted Rand index [@HubertArabie1985]. Panel (c) shows the boxplots of adjusted Rand indices for the four clustering models. The joint model leads to much higher indices, when compared to the other models considered. The clustering only model does as well as MCLUST and considerably better than FCM. Panel (d) shows a bar plot for the number of clusters identified by the four models. Out of the 100 datasets, the joint model identifies 4 clusters in 38 datasets and 5 clusters in 33 datasets. FCM also does well in identifying the correct cluster numbers, specifically it identifies 4 clusters in 46 datasets and 3 clusters in 27 datasets. The clustering only model and MCLUST tend to overestimate the number of clusters.
![\[simulClust\]**Simulation study: clustering comparison**. (a) Boxplots of log pseudo marginal likelihood (LPML) over the 100 datasets for the joint model (JM), the clustering only model (CO) and the registration only model (RO). (b) Boxplots of MSE over all the estimated curves by the five models, JM, CO, RO, MCLUST (MC) and FCM (FC). (c) Boxplots of adjusted Rand indices for the four clustering models, JM, CO, MC and FC. (d) Bar plots of cluster numbers identified by the four clustering models, JM (black), CO (dark gray), MC (light gray) and FC (white), in the 100 datasets. ](simulClust.eps){width="130mm"}
.1in We repeated the joint clustering and registration analysis under several prior specifications, in order to assess sensitivity. While the formal task is daunting, due to the large number of parameters in the model, we have found that reasonable variations in prior choice has little impact on final inference, detailed results are reported in Web Appendix A. Clearly, different considerations may apply under different sample size scenarios.
A Cluster Analysis of the Berkeley Growth Data {#growthData}
==============================================
We apply the proposed model to the well known Berkeley Growth data and compare it with the clustering only model, registration only model, MCLUST and FCM. As discussed in Section \[introduction\], the Berkeley Growth Study [@Tudden:Sny:1954] recorded the height of 39 boys and 54 girls for 27 time points between age 2 to 18, with one measurement a year before age 9 and two measurements a year after. To construct the growth velocity curves from the original growth curves, a smoothing spline model was fitted to each growth curve, and the first degree derivatives were obtained from the model and used in our comparisons. In Figure \[growthVelocity\](a) and (b), the growth velocity curves of boys (blue) and girls (pink) are plotted against age with superimposed cross-section means (black). Within each sex, curves have similarities in shape, while each curve shows individual time and amplitude variation. As pointed out by @Rams:Li:curv:1998 and @Gerv:Gass:self:2004, failing to account for time variability, produces inconsistent estimates of sex-specific growth velocities. Our analysis is non-standard, as we use sex as a hidden label to assess clustering performance. While illustrative, this exercise finds justification in the fact that sex is expected to explain a large portion of variation in adolescent growth patterns.
Shape functions basis are constructed fixing $\Delta=7$ and placing 27 equidistant interior knots between $-3$ to $23$. To model time transformation functions, we place four interior knots at $(5.2, 8.2, 11.6, 14.8)$ and partition the interval $T=[2, 18]$ into five subintervals. Priors on precisions and mixture weight are set as in Sec. \[simulation\]. Our inferences are based on $20K$ MCMC iterations, with $10K$ burn-in.
![\[growthVelocity\]**Growth velocity data analysis**. (a) and (b) Individual unregistered growth velocity curves for 39 boys (blue dashed) and 54 girls (pink dashed): cross-sectional means in solid-black. (c) Registered curves in the 1st cluster: 34 boys (blue dashed) and 11 girls (pick dashed). Estimated common shapes are indicated in (solid-black), MCLUST (red), FCM (green) and the cross sectional mean in (dashed-black). (d) Registered curves in the 2nd cluster: 43 girls (pink dashed) and 5 boys (blue dashed). Common shape functions as in (c). (e) and (f) Estimated curve specific time transformation functions for the two clusters: boys (dashed-blue) and girls (dashed-pink). ](growthVelocity.eps){width="130mm"}
The model identifies two clusters, seemingly discriminative according to sex.If we label the first cluster as the “boy” cluster and the second cluster as the “girl” cluster, then $43$ out of $54$ girls are clustered correctly and $34$ out of $39$ boys are clustered correctly. The overall classification accuracy is $83\%$. Estimated time transformation functions, common shape functions and registered curves are shown in Figure \[growthVelocity\]. Panel (c) and (d) show the registered curves for the $2$ clusters, superimposed by their corresponding common shape functions from the joint model (black solid), MCLUST (red), FCM (green) and the cross sectional mean curves (black dashed). Individual curves are colored by their true gender information, blue for boys and pink for girls. Therefore, pink curves in panel (c) and blue curves in panel (d) show the misclassified cases. Panel (e) and (f) show the estimated curve specific time transformation functions for the two clusters.
We compared the joint model with the clustering only model, the registration only model, MCLUST and FCM, and the results are shown in Figure \[growthCompPlot\]. Panel (a) shows the boxplots of CPO of the $93$ individual growth curves by the three Bayesian models. It shows that the joint model fits the data best, followed by the registration only model and the clustering only model. When the curves are not too dramatically different, the registration only model can fit the data accurately by finding a common shape function representing all the curves well.
As a comparable measure of model fit we compute the squared error (SE) between each curve and its fitted profiles.Panel (b) shows the boxplots of the SE over all the growth curves for the five models. The joint model gives the smallest SE, and the three Bayesian models fit the data better than MCLUST and FCM in terms of SE. Panel (c) and (d) show the model fitting results of two individual curves of a boy and a girl.
![\[growthCompPlot\]**Growth velocity data model comparison**. (a) Boxplots of log CPO of the $93$ individual growth curves by the joint model (JM), the clustering only model (CO) and the registration only model (RO). (b) Boxplots of the squared error (SE) over all the growth curves for JM, CO, RO, MCLUST (MC) and FCM (FC). (c) Model fitting results of the growth velocity curve of a boy (blue line with circles) by JM (black solid), CO (black dashed), RO (black dotdash), MCLUST (red) and FCM (green). (d)Model fitting results of the growth velocity curve of a girl (pick line with circles) by JM (black solid), CO (black dashed), RO (black dotdash), MCLUST (red) and FCM (green). ](growthCompPlot.eps){width="130mm"}
Interpreting sex as a clustering lable, we compare the joint model, the clustering only model, MCLUST and FCM using adjusted Rand indices (RIs). We find the following: FMC (RI = 0.61), MCLUST (RI = 0.47), JM (RI = 0.43) and CO (RI = 0.10). By this measure FCM and MCLUST seem to outperform our joint clustering and registrations model, with FCM giving the best clustering results. We note that when fitting MCLUST, we set the candidates of cluster numbers to be between 2 to 10, because when 1 is included as a candidate, MCLUST chooses it as the optimal cluster number, which leads to an adjusted Rand index of 0. On the other hand, as shown in panel (a) and (b) in Figure \[growthVelocity\], FMC and MCLUST seem to provide unsatisfactory estimates of cluster specific shape functions, which the joint clustering and registration model estimates consistently with the findings of @Rams:Gass:1995 and @Telesca:Inoue:2007, supporting the existence of the mid growth spurts.
Response of Human Fibroblasts to Serum {#fibroResponse}
======================================
In this section, we apply the joint model to time course expression data of the response of human fibroblasts to serum in a microarray experiment of $8613$ genes [@IyerEtal1999]. For human fibroblasts to proliferate in culture, they require growth factors provided by fetal bovine serum (FBS). In their study, after inducing primary cultured human fibroblasts to enter a quiescent state by serum deprivation for $48$ hours, the authors stimulated fibroblasts by adding medium containing $10\%$ FBS. A microarray experiment was then conducted to measure temporal gene expression levels at $12$ time points, from $15$ minutes to $24$ hours after serum stimulation. Furthermore, they selected $517$ genes with substantial time course expression change in response to serum and formed clusters using K-means clustering [@EisenEtal1998]. In our analysis, we consider a subset of $78$ genes, since they are associated with clear biological function categories as described in the original paper, and this provides a standard for us to validate the biological relevance of the clustered identified by our model.
We use the same prior setup as in previous sections. To model shape functions we use a maximum expansion constraint $\Delta=6$ and place interior knots at the sampling time points and $5$ equidistant points in two intervals from $-5$ to $-1$ and from $25$ to $29$ respectively. To estimate the time transformation functions, we place four interior knots at $(0.5, 2, 8, 16)$ in the sampling interval $T=[0, 24]$. Our inferences are based on $20K$ MCMC iterations, with $10K$ burn-in.
![\[geneCurves\]**Human fibroblast gene expression analysis**. (a) Unregistered time coures expression curves for 78 genes selected from a microarray experiment of Human fibroblasts’ response to serum. (b) Registered expression curves forming $4$ clusters colored by green, red, blue and pink. (c) Thirty five genes in cluster $1$ superimposed by the cluster specific common shape function (solid black). (d) Nine genes in cluster $2$. (e) Five genes in cluster $3$. (f) Twenty nice genes in cluster $4$.](geneClusters.eps){width="130mm"}
Panel (a) of Figure \[geneCurves\] shows the unregistered temporal expression curves of the 78 genes selected from the microarray experiment of human fibroblasts’ response to serum. Panel (b) shows the registered expression curves which are clustered into $4$ groups. Panel (c)-(f) show the $4$ clusters of registered expression curves separately, superimposed by their cluster specific common shape functions.
Cluster Size Typical Genes Functions
--------- ------ --------------------------- ------------------------------
1 35 PCNA, Cyclin A, Cyclin B1 Cell cycle and proliferation
CDC2, CDC28 kinase
2 9 LBR Cell cycle and proliferation
3 5 PAI1, PLAUR, ID3 Coagulation and hemostasis
Transcription factors
4 29 MINOR, JUNB, CPBP Signal transduction
TIGF, SGK, NET1 Transcriptional factors
: \[geneTable\] Clusters of genes and their biological functions
As shown in Figure \[geneCurves\] (c), genes in cluster $1$ are down-regulated at first and reach their lowest expression levels between $4$ and $12$ hours after serum stimulation. They begin to express about $16$ hours after the serum treatment, which is also the time when the stimulated fibroblasts replicate their DNA and reenter into the cell-division cycle. Several genes in cluster $1$ are known to be involved in mediating cell cycle and proliferation, for instance, PCNA, Cyclin A, Cyclin B1, CDC2 and CDC28 kinase, as shown in Table \[geneTable\]. Cluster $2$ in Figure \[geneCurves\](d) shows similar expression pattern to cluster $1$, except they expression level are lower than those in cluster $1$ through the time window. Genes in cluster $2$ are also involved in cell cycle and proliferation, such as LBR. Figure \[geneCurves\](e) shows that genes in cluster $3$ respond immediately to serum stimulation, reach their expression peaks around 10 hours later and remain induced towards the end. They are known to be transcription factors and involved in coagulation and hemostasis because of fibroblasts’ role in clot remodeling. Typical genes include PAI1, PLAUR and ID3. As shown in Figure \[geneCurves\](f), genes in cluster $4$ are also induced quickly by serum treatment, reach their peaks at about 2 hours, and then gradually return to a quiescent state. Several of the genes here are known to encode transcriptional factors and other proteins involved in signal transduction, such as MINOR, JUNB, CPBP, TIGF, SGK and NET1.
Discussion
==========
We propose a Bayesian hierarchical model for joint curve registration and clustering. Compared to previous methods, our proposal comes with several advantages. First, the model provides flexible nonlinear modeling for both components of variation. The Dirichlet process mixture prior over shape functionals strikes an automatic balance between complexity and parsimony. The implied posterior identifies subgroups of homomorphic curves, without the need to specify the number of clusters a priori. Finally, the increased model flexibility is still amenable to straightforward posterior simulation via MCMC, which provides exact inferences about a rich set of quantities of interest, without the need for simplifications or approximations.
The proposed B-spline representation of both shape and time transformation functions requires the a priori specification of the number and placement of spline knots. Our experiences suggests that a set of knots reproducing the original sampling time points works well for shape functions and 1 to 4 equidistant interior knots are enough for time transformation functions, as they carry smoothing properties through monotonicity constraints. Our simulation study shows that the model is robust to different prior choices. We however maintain, that different considerations may apply to ultra-sparse or, conversely, ultra-dense data settings.
The proposed modeling strategy has potentially broad applications to functional data analysis; especially when curve registration and clustering are of joint interest, as shown in our applications. In the first case study of the Berkeley Growth Data, our model is able to accurately separate growth curves into two clusters labelled by sex, and to correctly estimate the overall growth patterns for both sexes after registering curves in each cluster. In the second case study of time course expression data of human fibroblasts’ response to serum, our model identifies fours clusters of genes involved in distinct biological functions.
The proposed estimator of the clustering structure is the MAP clustering. Because Dirichlet process mixtures fully account for stochasticity in the potential alternative assignment of individual profiles to functional groups, it is possible, in principle, that the clustering structure with the second highest posterior probability is only a little less probable than the MAP clustering, yet it provides quite a different grouping structure.
We have not detected this type of phenomenon in our analyses. However, when it happens, some care is needed in summarizing complex posterior evidence. A possible alternative strategy to MAP is based on the estimation of a pairwise probability matrix whose elements are estimated probabilities that two curves are in the same cluster. Such a matrix can be easily generated by averaging the sampled association matrices from the MCMC output. Elements of an association matrix takes values $1$, if two corresponding curves are in the same cluster, and $0$ otherwise. Hierarchical clustering may be used subsequently as a way to explore grouping structures [@MedvedovicSivaganesan2002]. Alternatively, @Dahl2006 proposed a least squares clustering by selecting the sampled clustering which minimizes the sum of squared deviations of its association matrix from the pairwise probability matrix.
Finally, when covariate information is available, the proposed model is easily extended to include a dependent Dirichlet process prior, using covariates to inform clustering.
Supplementary Materials {#supplementary-materials .unnumbered}
=======================
Supplementary information is available from the authors.
\[lastpage\]
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'There exists a class of ultralight Dark Matter (DM) models which could form a Bose-Einstein condensate (BEC) in the early universe and behave as a single coherent wave instead of individual particles in galaxies. We show that a generic BEC-DM halo intervening along the line of sight of a gravitational wave (GW) signal could induce an observable change in the speed of GWs, with the effective refractive index depending only on the mass and self-interaction of the constituent DM particles and the GW frequency. Hence, we propose to use the deviation in the speed of GWs as a new probe of the BEC-DM parameter space. With a multi-messenger approach to GW astronomy and/or with extended sensitivity to lower GW frequencies, the entire BEC-DM parameter space can be effectively probed by our new method in the near future.'
address:
- '$^1$Max-Planck-Institut für Kernphysik, Saupfercheckweg 1, D-69117 Heidelberg, Germany'
- '$^2$Department of Physics and McDonnell Center for the Space Sciences, Washington University, St. Louis, MO 63130, USA'
author:
- 'P. S. Bhupal Dev$^{1,2}$, Manfred Lindner$^1$, and Sebastian Ohmer$^1$'
title: 'Gravitational Waves as a New Probe of Bose-Einstein Condensate Dark Matter'
---
Introduction {#sec:1}
============
Although the existence of Dark Matter (DM) constituting about 27% of the energy budget of our Universe [@Ade:2015xua] is by now well established through various cosmological and astrophysical observations, very little is known about its particle nature and interactions. While the standard $\Lambda$CDM model with collisionless cold DM (CDM) successfully explains the large-scale structure formation by the hierarchical clustering of DM fluctuations [@Blumenthal:1984bp; @Davis:1985rj], there are some unresolved issues on galactic and sub-galactic scales, such as the core-cusp [@Moore:1994yx; @Flores:1994gz; @Moore:1999gc; @deBlok:2009sp], missing satellite [@Kauffmann:1993gv; @Klypin:1999uc; @Moore:1999nt; @Bullock:2010uy], and too big to fail [@BoylanKolchin:2011de; @BoylanKolchin:2011dk; @Papastergis:2014aba] problems. All these small-scale structure anomalies can in principle be resolved if the DM is made up of ultralight bosons that form a Bose-Einstein condensate (BEC), i.e. a single coherent macroscopic wave function with long range correlation; for a review, see e.g., Ref. [@Suarez:2013iw].
There are two classes of BEC-DM, depending on whether DM self interactions are present or not. Without any self interactions, the quantum pressure of localized particles is sufficient to stabilize the DM halo against gravitational collapse only for a very light DM with mass $m\sim 10^{-22}$ eV [@Sahni:1999qe; @Hu:2000ke; @Sin:1992bg; @Matos:2008ag; @Lee:2008jp; @Lora:2011yc], whereas a small repulsive self-interaction can allow a much wider range of DM masses up to $m\lesssim 1$ eV [@Peebles:2000yy; @Goodman:2000tg; @Arbey:2003sj; @Eby:2015hsq; @Fan:2016rda].[^1] Concrete particle physics examples for BEC-DM are WISPs (Weakly Interacting Slim Particles) [@Ringwald:2012hr], which include the QCD axion or axion-like particles [@Sikivie:2009qn; @Mielke:2009zza; @Erken:2011dz; @Saikawa:2012uk; @Davidson:2013aba; @Berges:2014xea; @Davidson:2014hfa; @Guth:2014hsa; @Banik:2015sma] and hidden-sector gauge bosons [@Nelson:2011sf; @Arias:2012az; @Pires:2012yr; @Soni:2016gzf] ubiquitous in string theories, but our subsequent discussion will be generically applicable to any BEC-DM with a repulsive self-interaction, which is necessary to obtain long-range effects [@Guth:2014hsa].[^2]
The observational consequences on structure formation mentioned above cannot distinguish a BEC-DM from an ordinary self-interacting DM [@Spergel:1999mh]. Existing distinction methods include enhanced integrated Sachs-Wolfe effect [@Sikivie:2009qn], tidal torquing of galactic halos [@Banik:2015sma; @RindlerDaller:2011kx; @Banik:2013rxa], and effects on cosmic microwave background matter power spectrum [@Ferrer:2004xj; @Velten:2011ab]. We propose a new method to probe the BEC-DM parameter space using gravitational wave (GW) astronomy, inspired by the recent discovery of transient GW signals at LIGO [@Abbott:2016blz; @Abbott:2016nmj]. We show that if GWs pass through a BEC-DM halo on their way to Earth, the small spacetime distortions associated with them could produce phononic excitations in the BEC medium which in turn induce a small but potentially observable change in the speed of GWs, while the speed of light remains unchanged. This approach is very effective if any of the future multi-messenger searches for gamma-ray, optical, $X$-ray, or neutrino counterparts to the GW signal become successful. On the contrary, a lack of any observable deviation in the speed of GWs will put stringent constraints on the BEC-DM scenario. In fact, we find that even with the current LIGO sensitivity, it might be possible to partly rule out the BEC-DM parameter space otherwise preferred by existing cosmological data. Future GW detectors such as eLISA [@Seoane:2013qna] with extended sensitivity to lower GW frequencies will be able to completely rule out the cosmologically preferred region.
The rest of the paper is organized as follows: in Section \[sec:2\], we calculate the change in the speed of GWs due to energy loss inside the BEC medium. In Section \[sec:3\], we apply this result to derive constraints on the BEC-DM parameter space. In Section \[sec:4\], we discuss the effect of gravitational lensing. Our conclusions are given in Section \[sec:5\].
Speed of GW inside BEC medium {#sec:2}
=============================
The cosmological dynamics of BEC-DM can be described by a single classical scalar field $\phi$ [@Hertzberg:2016tal], with the effective Lagrangian $$\begin{aligned}
\mathcal{L} \ = \ \frac{1}{2} \partial_\mu \phi \partial^\mu \phi - \frac{1}{2} m^2\phi^2 - \lambda \phi^4 \, ,
\label{eq:lag}\end{aligned}$$ analogous to the Ginzburg-Landau free energy density in a neutral superfluid. A real scalar field will suffice for our discussion. In , we have considered a simple renormalizable scalar potential with only quadratic and quartic terms, the latter providing a repulsive self-interaction for the DM, as required in addition to the quantum pressure of localized particles to stabilize the DM halo core against gravitational collapse. For no self-interaction ($\lambda=0$), the quantum pressure is sufficient only if $m\sim 10^{-22}$ eV, a scenario known as fuzzy DM [@Hu:2000ke]. In principle, we could also have added a cubic term $-g \phi^3$ to ; however, for the self-interaction to be repulsive in the non-relativistic limit, we must have $\lambda> 5g^2/2$ [@Fan:2016rda]. Similarly, we do not include any higher-dimensional operators in .
Using , we calculate the stress-energy tensor $$\begin{aligned}
T^{\mu\nu} \ = \ \frac{\partial {\cal L}}{\partial(\partial _\mu \phi)}\partial^\nu \phi - g^{\mu\nu}{\cal L} \, ,
\label{eq:stress}\end{aligned}$$ where $g^{\mu\nu}$ is the spacetime metric. Far from the GW source, the linearized spacetime metric can be written as $g^{\mu\nu}= \eta^{\mu\nu}+h^{\mu\nu}$, where $\eta ={\rm diag}(1,-1,-1,-1)$ is the flat Minkowski metric (in particle physics conventions) and $h^{\mu\nu}$ is a small perturbation. To leading order, the background mean field values of the energy density $\rho_0 \equiv T^{00}$ and pressure $p_0 \equiv T^{ii}$ of the BEC medium are related by the equation of state (EoS) $$\begin{aligned}
p_0 \ = \ \frac{3}{2}\frac{\lambda}{m^4}\rho_0^2 \, .
\label{eq:eos1}\end{aligned}$$
Gravity is a long-range force. Because almost all particles in the BEC system are condensed into the lowest energy available state with very long de Broglie wavelength, the GWs can excite the massless phonon modes in the ground state of the BEC wave function [@Sabin:2014bua]. As a result, the GW undergoes enhanced coherent forward scattering inside a BEC-DM halo compared to an ordinary CDM halo. This effect is analogous to light traveling through an optically transparent medium (e.g. glass) with refractive index different from $1$. In general, there could be either refraction or absorption of the incident wave (apart from reflection), depending on the real or imaginary part of the refractive index, respectively. The refraction effect modifies the wavenumber and propagation speed of the wave in the medium (without change in its frequency and amplitude), while absorption results in the damping of the amplitude, and hence, attenuation of the wave in the medium. In the case of GWs incident on a BEC medium, the absorption effect is negligible, because it would require exciting the phonons from massless to massive modes, which in turn requires much larger energy comparable to the chemical potential of the BEC [@Sabin:2014bua]. Thus, only the propagation speed of the GW passing through a BEC-DM halo is reduced, but much more strongly than in a CDM halo, as we show below.
To estimate this effect, we first write down the effective metric of the BEC phononic excitations on the flat spacetime metric [@Sabin:2014bua; @Fagnocchi:2010sn; @Visser:2010xv] $$\begin{aligned}
g_{\rm eff} \ = \ \frac{n_0^2}{c_s(\rho_0+p_0)}{\rm diag}(c_s^2, -1, -1, -1) \, ,
\label{eq:geff}\end{aligned}$$ where $n_0 \equiv \rho_0/m$ is the number density of the background mean field and $$\begin{aligned}
c_s \ \equiv \ \left(\frac{\partial p_0}{\partial \rho_0}\right)^{1/2} \ = \ \left(\frac{3 \lambda \rho_0}{m^4}\right)^{1/2}\end{aligned}$$ is the speed of sound obtained from the background EoS . The solution to the Klein-Gordon equation with the metric in thus describes massless excitations propagating with the speed $c_s$. The frequency of the mode satisfies the linear dispersion relation $\omega_k=c_s|\mathbf{k}|$, where $\mathbf{k}$ is the 3-momentum of the mode.
The refractive index of a GW scattering off a gravitational potential was first calculated in [@Peters:1974gj] and was shown to be negligibly small for ordinary matter. Here, we provide an alternative derivation of the refractive index based on the optical theorem and argue that it could be relevant in our case due to the enhanced forward scattering rate in a BEC medium. The optical theorem links the refractive index $n_g$ to the forward scattering amplitude $f(0)$ of the incident wave with the scatterers inside the medium: $$\begin{aligned}
n_g = 1 + \frac{2 \pi n f(0)}{k^2}\,,\end{aligned}$$ with $n$ the number density of scatterers inside the medium and $k$ the wavenumber of the incident wave. We estimate the forward scattering $n f(0)$ of the GW in the BEC-DM halo by relating the energy density of the incident GW to the energy density of the massless phonon excitations in the ground state.
We assume for simplicity that the phonons can be described by a one-dimensional wave function with hard-wall boundary conditions. This approximation is also valid for a spherically-symmetric DM halo, such that only the radial component matters for the GW propagation through it. The energy spectrum of the massless modes is then given by $$\begin{aligned}
\omega_l \ = \ \frac{l \pi c_s}{\langle D_{\rm halo}\rangle} \,,
\label{eq:6}\end{aligned}$$ where $\langle D_{\rm halo}\rangle = 4R/\pi$ is the average distance the gravitational wave propagates through the spherically-symmetric DM halo with a radius $R$ and $l\in \{1,2,\cdots\}$. Therefore, the [*minimum*]{} energy density required to excite the massless phonon modes in the BEC medium is given by $$\begin{aligned}
\Delta \rho \ \equiv \ n \Delta \omega \ = \ \frac{n\pi^2 c_s}{4 R}\,,
\label{eq:7}\end{aligned}$$ where $n$ is the number density of phonons in the BEC and $\Delta \omega\equiv \omega_{l+1}-\omega_l$ is the energy difference between two adjacent massless modes \[cf. \].
The radius $R$ of the gravitationally bound BEC with a repulsive self-interaction only depends on the physical characteristics of the particles in the condensate [@Boehmer:2007um; @Chavanis:2011zi; @Chavanis:2011zm]: $$\begin{aligned}
R \ = \ \left(\frac{\pi^2 a_s}{Gm^3}\right)^{1/2} ,
\label{eq:R}\end{aligned}$$ where $a_s$ is the $s$-wave scattering length which in the low-energy limit is defined by $\displaystyle\lim_{k\to 0}\sigma(\phi \phi \to \phi \phi) = 4\pi a_s^2$. For the interaction Lagrangian given by , we find $$\begin{aligned}
\sigma \ = \ \frac{9\lambda^2}{\pi m^2} \, ,
\label{eq:sigma}\end{aligned}$$ and hence, from , we can extract $$\begin{aligned}
R \ = \ 2\pi \sqrt{3\lambda}\: \frac{M_{\rm Pl}}{m^2} \, .
\label{eq:R2}\end{aligned}$$
The number density of phonons in the BEC can be estimated in terms of the microscopic parameters from the expression for the critical temperature [@Bettoni:2013zma] $$\begin{aligned}
T_c \ = \ \frac{2\pi}{m}\left( \frac{n}{\zeta(3/2)} \right)^\frac{2}{3} \,,\end{aligned}$$ and equating it to the critical temperature for a self-interacting scalar gas: $T_c = (24m^2/\lambda)^{1/2}$ [@Dolan:1973qd; @Weinberg:1974hy; @Kapusta:1981aa]. We thus obtain $$\begin{aligned}
n \ = \ \left( \frac{6 m^4}{\pi^2 \lambda}\right)^\frac{3}{4} \zeta(3/2)\,.\end{aligned}$$
The typical energy density of a GW is given by [@Misner:1974qy] $$\begin{aligned}
\rho_{\rm{GW}} \ = \ \frac{1}{4}M^2_{\rm Pl}\omega_{\rm GW}^2 h^2 \,,\end{aligned}$$ where $\omega_{\rm GW}=2\pi f$ is the angular frequency ($f$ being the frequency) of the GW and $h$ is the amplitude (we assume $h_+ = h_\times = h/\sqrt{2}$ for the two polarization modes), and $M_{\rm Pl}$ is the Planck mass. Using a linear dispersion relation, we can calculate the relative change in the wavenumber of the GW due to its propagation in the BEC $$\begin{aligned}
\frac{\Delta \rho}{\rho_{\rm{GW}}} \ = \ 2\frac{\Delta k}{\omega_{\rm GW}}
\label{eq:14}
\,,\end{aligned}$$ where $\Delta \rho$ is given by and $\Delta k$ is the change in the wavenumber of GWs (which can be thought of as the induced mass of the graviton, if the GWs were quantized). Thus, the effective refractive index is given by $$\begin{aligned}
n_g \ =\ 1+\frac{\Delta k^2}{2\omega_{\rm GW}^2} \, .
\label{eq:ng}\end{aligned}$$ This effect is negligible for ordinary CDM (or for ordinary matter per se), since the number density in any given energy eigenstate is too small, which makes $\Delta k$ unobservable [@Elghozi:2016wzb]. However, the huge occupation number in the ground state and the long-range correlations of the BEC system could enhance this effect sizably, and we will exploit this key feature to derive constraints on the BEC-DM parameter space.
Thus, the change in refractive index experienced by the GWs inside the BEC medium is given by $$\begin{aligned}
\delta n_g \ \equiv \ n_g-1 \ = \ \sqrt{\frac{3}{2}} \frac{3 m^6 \rho_0 \zeta(3/2)^2}{8 \pi \lambda^{3/2} h^4 \omega_{\rm GW}^4 M^6_{\rm{Pl}} } \,,
\label{eq:dng}\end{aligned}$$ which depends on the average DM density $\rho_0$, the amplitude $h$ and angular frequency $\omega_{\rm GW}$ of the GW and the microscopic parameters $(m,\lambda)$. Fixing the macroscopic parameters $(\rho_0, h, \omega)$ we can directly translate a constraint on the speed of GWs into a constraint on the $(m,\lambda)$ parameter space of BEC-DM, as shown in Section \[sec:3\].
The density distribution of a static, spherically-symmetric BEC-DM halo can be obtained from the solution to the Lane-Emden equation [@chandra] in the weak field, Thomas-Fermi regime, given by the analytic form [@Boehmer:2007um] $$\begin{aligned}
\rho(r) \ = \ \rho_{\rm cr} \frac{\sin{\kappa r}}{\kappa r} \, ,
\label{eq:halo}\end{aligned}$$ where $\kappa=\sqrt{Gm^3/a_s}=\pi/R$ and $\rho_{\rm cr}$ is the central density of the condensate. The average density of a BEC-DM halo is thus given by $$\begin{aligned}
\rho_0 \ \equiv \ \langle \rho \rangle \ = \ \frac{3\rho_{\rm cr}}{\pi^2} \, ,\end{aligned}$$ which will be used in .
To calculate the change in the speed of the GWs due to the change in refractive index , let us assume that the GW is produced at a distance $D$ from Earth and encounters a spherical BEC-DM halo of radius $R$ en route.[^3] The average fraction of distance the GW propagates through the DM halo with a reduced speed $c_g=1/n_g$ is given by $$\begin{aligned}
x \ \equiv \ \frac{\langle D_{\rm halo}\rangle}{D} \ = \ \frac{4R}{\pi D}\, .
\label{eq:x}\end{aligned}$$ The effective speed of GWs is then given by $$\begin{aligned}
c_\text{eff} \ \equiv \ \frac{D}{\Delta \tau} \ = \ \frac{c_g}{x + (1-x)c_g} \, ,\end{aligned}$$ where $\Delta \tau = x D/c_g + (1-x) D$ is the proper time elapsed between the emission and detection of the GW signal. So the change in the speed of GWs from the speed of light in vacuum due to its encounter with the BEC-DM halo is given by $$\begin{aligned}
\delta c_g \ \equiv \ 1-c_{\rm eff} \ = \ \frac{x \delta n_g}{1+x \delta n_g} \, ,
\label{eq:deltacg}\end{aligned}$$ where $\delta n_g$ is given by . This is our key result that will be used to put new constraints on the BEC-DM properties.
GW Constraints on BEC-DM {#sec:3}
========================
Using , we numerically calculate the change in the speed of GWs $\delta c_g$ as a function of the microscopic BEC-DM parameters $m$ and $\lambda$ for given values of the source distance $D$, GW frequency $f$ and amplitude $h$. For illustration, we will fix the central core density at $\rho_{\rm cr}=0.04 M_{\odot}/{\rm pc}^3$ (where $M_\odot$ is the solar mass), which is within the range suggested by a recent $N$-body simulation of self-interacting DM [@Rocha:2012jg]. We also take $D=400$ Mpc, $f=35$ Hz and $h = 10^{-21}$ as representative values from the GW150914 event at LIGO [@Abbott:2016blz]. The DM particle mass is varied in the range $m\in [10^{-23},1]$ eV. The upper limit comes from the basic condition that the particle’s de Broglie wavelength, $\lambda_{\rm dB}=2\pi/mv$ (where $v\sim 10^{-3}$ is the virial velocity and we set $\hbar = 1$) should be larger than the inter-particle spacing, $d=(m/\rho)^{1/3}$, such that the wave functions of the individual particles in the system overlap with each other to form a BEC. The de Broglie wavelength also sets a natural lower limit to the core size of equilibrium BEC-DM halos that can form; taking $\lambda_{\rm dB}\lesssim 1$ kpc, the halo size of a typical dwarf spheroidal (DSph) galaxy, we get a lower limit of $m\gtrsim 5\times 10^{-23}$ eV, which is saturated for fuzzy DM [@Hu:2000ke].
With this choice of parameters, we find the minimum $\delta c_g$ that can rule out repulsive BEC-DM is at the level of $10^{-37}$ for macroscopic GW parameters which LIGO is sensitive to. Thus we need the experimental sensitivity of $\delta c_g^{\rm exp}$ at this level or below to be able to put constraints on the BEC-DM parameter space using our method. For comparison, the current best model-independent bound is $\delta c_g\leq 2\times 10^{-15}$ [@Moore:2001bv], deduced from the absence of gravitational Cherenkov radiation allowing for the unimpeded propagation of high-energy cosmic rays across our galaxy. Recently, assuming that the short gamma-ray burst above 50 keV detected by Fermi-GBM [@Connaughton:2016umz] just 0.4 seconds after the detection of GW150914 at LIGO [@Abbott:2016blz] originated from the same location, more stringent limits on $\delta c_g$ have been derived [@Li:2016iww; @Ellis:2016rrr; @Collett:2016dey; @Branchina:2016gad]. While a typical time-of-flight analysis [@Nishizawa:2014zna] gives $\delta c_g \lesssim 10^{-17}$ [@Li:2016iww; @Ellis:2016rrr; @Collett:2016dey], using modified energy dispersion relations (typical of many quantum gravity models) with the quantum gravity scale $E_G\geq M_{\rm Pl}$ yields a much stronger limit of $\delta c_g\lesssim 10^{-40}$ [@Branchina:2016gad]. However, whether the Fermi-GBM event originates from the same astrophysical source responsible for GW150914 is a controversial issue [@Loeb:2016fzn; @Lyutikov:2016mgv; @Janiuk:2016qpe; @Zhang:2016kyq; @Woosley:2016nnw; @Fraschetti:2016bpm; @Greiner:2016dsk] and according to a recent analysis [@Greiner:2016dsk], the GBM event is more likely a background fluctuation, which is consistent with the non-detection of similar gamma-ray events at SWIFT [@Evans:2016mta], INTEGRAL [@Savchenko:2016kiv], and AGILE [@Tavani:2016jrd]. Nevertheless, after the detection of the second LIGO event GW151226 [@Abbott:2016nmj], the multi-messenger searches have become more intense and now include searches for gamma-ray [@Racusin:2016fko; @Adriani:2016gdx; @Vianello:2016jzm], $X$-ray [@Evans:2016dgd; @Adriani:2016gdx], optical [@Cowperthwaite:2016shk; @Smartt:2016oeu] and neutrino [@Adrian-Martinez:2016xgn; @Gando:2016zhq; @Aab:2016ras; @Abe:2016jwn] counterparts. With more GW events expected from LIGO in the near future, these multi-messenger searches are likely to detect events coming from the same source and improve the limits on $\delta c_g$ significantly.
Since the change in refractive index in a BEC medium is inversely proportional to the fourth power of the GW frequency \[cf. \], a future space-based GW interferometer, such as eLISA [@Seoane:2013qna] with a lower operational frequency range of $0.1$–100 mHz, can further improve the sensitivity. For instance, for $f=1$ mHz, $D=3$ Gpc, $h=10^{-20}$ and $\rho_{\rm cr}$ same as above, the minimum $\delta c_g$ required to rule out BEC-DM is at the level of $10^{-24}$. Pulsar timing arrays, such as the ones united under IPTA [@IPTA:2013lea] and SKA [@Janssen:2014dka], probe much lower frequencies around 1 nHz and amplitudes at the level of $h=10^{-14}$ and are capable of detecting $\delta c_g\sim 10^{-23}$, similar to the eLISA sensitivity and within reach of not-too-far-distant multi-messenger searches.
{width="8.5cm"} {width="8.5cm"}
In Figure \[fig:1\] (left panel), we show the exclusion regions for $\delta c_g \leq 10^{-20}$ (red) and $\delta c_g \leq 10^{-37}$ (yellow) in the $(m,\lambda)$ plane using for a GW signal detected by LIGO with $D = 400$ Mpc, $f=35$ Hz and $h = 10^{-21}$. The dependence on the macroscopic GW parameters, and thus the possibility for a detection in future experiments, is demonstrated in Figure\[fig:1\] (right panel) where the exclusion regions for $\delta c_g \leq 10^{-17}$ (red) and $\delta c_g \leq 10^{-24}$ (orange) for a hypothetical GW event detected by eLISA with $f=1$ mHz, $D=3$ Gpc, $h=10^{-20}$ are shown. For comparison, we also show the region which gives $\sigma/m=(0.01-1)~{\rm cm}^2/g$ (blue shaded), as preferred by $N$-body simulations to explain the small-scale structure anomalies, while being consistent with all observational constraints from colliding galaxy clusters [@Markevitch:2003at; @Randall:2007ph; @Massey:2010nd; @Kahlhoefer:2015vua; @Harvey:2015hha] and halo shapes [@Vogelsberger:2012ku; @Rocha:2012jg; @Peter:2012jh; @Zavala:2012us; @Kaplinghat:2015aga]. Similarly, the viability of the BEC-DM halo model to fit the rotational curves of the most DM-dominated low surface brightness and DSph galaxies from different surveys implies $R\sim 0.5$–10 kpc [@Boehmer:2007um; @Arbey:2003sj; @Harko:2011xw; @Robles:2012uy; @Diez-Tejedor:2014naa], which can be translated to a preferred range of $m/\lambda^{1/4}\sim 4$–18 eV, as shown by the dark green shaded region in Figure \[fig:1\]. Note that the region of intersection between the blue and green shaded areas gives the physically preferred value of $(m,\lambda)\simeq (10^{-4}~{\rm eV}, 10^{-19})$, as shown by the yellow point. We find that for the LIGO frequency range $f=10$–350 Hz, the physical region can be completely excluded for $\delta c_g \leq 10^{-37}$. However, for the eLISA parameters the physically viable region can already be excluded for $\delta c_g \leq 10^{-24}$, which should be soon achievable in the multi-messenger approach. Further, eLISA will already be able to rule out complementary parameter space with current limits of $\delta c_g \leq 10^{-17}$.
The quantity $m^4/\lambda$ also gives a rough estimate of the total energy density of the DM field at the time of its transition from the radiation-like (when the scalar potential is dominated by the quartic term) to matter-like epoch (when the quadratic term in the scalar potential takes over the quartic term). The field density before this transition contributes to the extra relativistic species $\Delta N_{\rm eff}$ at Big Bang Nucleosynthesis (BBN) [@Arbey:2003sj; @Diez-Tejedor:2014naa; @Li:2013nal]. Using the latest Planck result on $\Delta N_{\rm eff}\lesssim 0.39$ [@Ade:2015xua], we obtain an additional constraint on $m/\lambda^{1/4}\gtrsim 8.5$ eV, as shown by the gray shaded region in Figure \[fig:1\], which disfavors part of the DSph-preferred region. Future constraints on the extra relativistic degrees of freedom at BBN with the precision of $\Delta N_{\rm eff} \lesssim 0.12$ could rule out the entire DSph-preferred region of BEC-DM with repulsive self-interaction.
Apart from the observational constraints, one should also satisfy important theoretical constraints from the BEC formation requirements. The orange shaded region in the lower right part of Figure \[fig:1\] is excluded as the relaxation time $t_{\rm relax}$ in the virialized DM clumps due to the scattering process $\phi\phi\to \phi \phi$ exceeds the age of the Universe $t_{\rm universe}$, which sets a lower limit on the self-interaction strength $\lambda \gtrsim 10^{-15}(m/{\rm eV})^{7/2}$ for the BEC to form [@Tkachev:1991ka; @Semikoz:1994zp; @Riotto:2000kh]. Similarly, the brown shaded region in the top left part of the parameter space is disfavored, as the critical temperature $T_c=(24m^2/\lambda)^{1/2}$ [@Dolan:1973qd; @Weinberg:1974hy; @Kapusta:1981aa] below which a BEC can form, falls below the temperature of the universe at the source redshift of $z=0.1$ (assuming that the system of DM particles has a temperature comparable with that of radiation), which means the BEC-DM halo could not have formed at the time the GW was emitted from the binary black hole merger event GW150914. However, such a scenario would imply that there must exist extra relativistic components, in addition to the DM, to ensure thermal equilibrium, which contribute to $\Delta N_{\rm eff}$ at BBN and the corresponding constraint is much stronger than the $T(z=0.1)<T_c$ requirement, as can be seen from Figure \[fig:1\]. Finally, an average halo size $\langle D_{\rm halo}\rangle \geq 1$ Mpc seems unrealistic for self-interacting DM halos and disfavored by simulations [@Boehmer:2007um; @Peter:2012jh], as shown by the lighter purple shaded region in Figure \[fig:1\].
Gravitational Lensing {#sec:4}
=====================
In the multi-messenger approach, one way to confirm the existence of a BEC-DM halo in the path of the GW is by studying the deflection of photons passing through the region where galactic rotation curves are flat. The deflection angle is given by the standard formula [@Misner:1974qy] $$\begin{aligned}
\delta \theta_{\rm def} \ = \ \frac{4GM}{b} \, ,
\label{eq:def}\end{aligned}$$ where $b$ is the impact parameter (i.e. distance of closest approach) for which we use the radius of the BEC-DM halo from and $M$ is the total mass of the DM halo, given by $$\begin{aligned}
M \ = \ 4\pi \int_0^R \rho(r)r^2 dr \ = \ \frac{4}{\pi}\rho_{\rm cr}R^3 \, ,
\label{eq:mass}\end{aligned}$$ using for the density profile. is valid in the limit $GM\ll b$ which is satisfied in our case as long as $R\ll {\cal O}(1~{\rm Mpc})$. Thus, for a BEC-DM halo, we can express completely in terms of the microscopic parameters: $$\begin{aligned}
\delta \theta_{\rm def} \ = \ \frac{24\lambda}{m^4}\rho_{\rm cr} \ =\ \frac{2R^2}{\pi^2 M^2_{\rm Pl}}\rho_{\rm cr} \, .
\label{eq:def2}\end{aligned}$$ The physically interesting yellow point in Figure \[fig:1\] corresponds to a halo radius of $\sim 1$ kpc where we find $\delta \theta_{\rm def}=10^{-7}$. For other values of $R$, the prediction for the deflection angle can be readily obtained from .
We should also clarify that the gravitational potential of the intervening DM halo along the line of sight will cause a Shapiro time delay [@Shapiro:1964uw] for the GW, as well as its multi-messenger counterparts. In the geometrical optics approximation, treating the total gravitating mass as a point source, the time delay is the same for GW, photons and neutrinos, given by the general formula [@Longo:1987gc; @Krauss:1987me] $$\begin{aligned}
\Delta t_{\rm Shapiro} \ = \ (1+\gamma)GM \ln{\left(\frac{D}{b}\right)} \, ,
\label{eq:shapiro}\end{aligned}$$ where $\gamma$ is a parametrized post-Newtonian parameter. However, this geometrical approximation breaks down for GWs with wavelengths larger than the size of the lensing object, which corresponds to lens masses $\lesssim 10^5M_\odot(f/{\rm Hz})^{-1}$. This can induce a differential Shapiro delay between the GW and photons/neutrinos of up to $0.1\:{\rm sec}(f/{\rm Hz})^{-1}$ [@Kahya:2016prx; @Takahashi:2016jom]. For $R\sim 1$ kpc, we estimate the mass of BEC-DM halo from to be $M\sim 10^8M_\odot$; so for the LIGO frequency range of 10-350 Hz, the geometrical optics approximation remains valid and there is no relative Shapiro time delay to be considered in the multi-messenger analysis. However, for smaller frequencies, such as those relevant for eLISA and IPTA, the additional time delay must be taken into account while deriving experimental bounds on $\delta c_g$.
Conclusion {#sec:5}
==========
We have proposed a new method to probe BEC-DM using GW astronomy. We have shown that GWs passing through a BEC-DM halo will get appreciably slowed down due to energy loss in collective phononic excitations. The effective refractive index depends only on the mass and quartic coupling of the DM particles, apart from the frequency and amplitude of the propagating GW. Thus, an observable deviation $\delta c_g$ in the speed of GW can be used to put stringent constraints on the BEC-DM parameter space, as demonstrated in Figure \[fig:1\]. The physically interesting region of BEC-DM parameter space satisfying all existing constraints can be completely probed by this new method for $\delta c_g \leq 10^{-37}$ in the LIGO frequency range and $\delta c_g \leq 10^{-24}$ in the eLISA frequency range, which is soon achievable in a multi-messenger approach to GW astronomy. This work of B.D. was supported in part by the DFG grant RO 2516/5-1. We thank the participants of the PhD Student Seminar on Gravitational Waves at MPIK for pedagogical talks and useful discussions. B.D. also acknowledges helpful discussions with Nick Mavromatos and the local hospitality at ECT\*, Trento where part of this work was done.
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[^1]: BEC configurations with heavier DM and/or an attractive self-interaction are usually unstable against gravity [@Khlopov:1985jw] and more likely to form local dense clumps such as Bose stars [@Tkachev:1986tr; @Colpi:1986ye; @Tkachev:1991ka; @Kolb:1993zz; @Lee:1995af], unless the thermalization rate is faster than the Hubble rate to overcome the Jeans instability.
[^2]: Although the simplest models, where the scalar potential has an approximate symmetry to ensure the radiative stability of the ultralight scalar, usually give rise to an attractive self-interaction in the non-relativistic limit, it is possible to have realistic models with repulsive self-interaction [@Fan:2016rda; @Berezhiani:2015bqa].
[^3]: For multiple DM halos along the line of slight, the calculation presented here can be extended in a straightforward manner, since it is only sensitive to the [*total*]{} distance traversed by the GWs through the DM halo(s).
|
{
"pile_set_name": "ArXiv"
}
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Quantum computing is an exciting field driven by the promise of exponential speedup of *a priori* arduous computational processes such as integer factoring [@Shor1994; @Nielsen2000]. Most proposals for experimentally implementing quantum computing call for the use of two-state quantum systems, or qubits [@Braunstein2001]. However, a quantum computer could very well use continuous quantum variables [@Lloyd1999; @Menicucci2006], such as position and momentum, or the quadrature amplitude operators of the quantized field [@Braunstein2005a; @Braunstein2003a; @Cerf2007]. Recently, some of us have proposed a new and extremely scalable method for building a quantum register by use of the set of quantum harmonic oscillators (“qumodes”) defined by a single optical resonator [@Menicucci2008; @Flammia2009]. In this proposal, the quantum correlations (entanglement) necessary for quantum computing will be implemented by a nonlinear medium placed inside the cavity, thereby realizing a sophisticated optical parametric oscillator (OPO). The sophistication stems from the fact that three different second-order nonlinear interactions must be simultaneously phasematched over the same set of cavity modes, i.e. must be concurrent. These interactions are parametric downconversion ($\lambda/2\mapsto\lambda$) of ZZZ (“type-0”), ZYY (type-I), and YYZ/YZY (type-II), where the first letter denotes the polarization of the pump field and the last two letters denote the polarization of the signal (entangled) beams. In previous work [@Pooser2005], we demonstrated the simultaneous quasi-phasematching (QPM) of this set of interactions at room temperature for $\lambda=1490$ nm in periodically poled $\rm KTiOPO_4$ (PPKTP) with a *single* period of $45.65\ \mu$m. This was a serendipitous discovery that relied upon a weak seventh-order QPM of the ZYY interaction (even though the final signal turned out to be much larger). Despite this result, designing concurrent nonlinear interactions remained difficult because the precision on the Sellmeier coefficients, as well known as they are, was still not high enough, in particular for $n_Y$.
In this Letter, we use Fourier engineering [@Fejer1992; @Lifshitz2005] to achieve and demonstrate a concurrent design with *low-order*, hence efficient, QPM at $\lambda=1560$ nm, close to the loss minimum of silica optical fibers. Recent advances in squeezing at and around this wavelength also make it a reasonable choice [@Feng2008; @Mehmet2009a]. This 1560 nm design required the use of three different poling periods. Two early iterations used published Sellmeier equations [@Fan1987; @Kato2002; @Emanueli2003]. In these initial versions, the ZZZ and ZYY QPM peaks overlapped well at 1560 nm at room temperature but the YZY interaction was quasi-phasematched for 1560 nm between average temperatures of $248.7^\circ$C (for a designed phase mismatch of $\rm 1.398 \times 10^5\ m^{-1}$ at room temperature) and $300.1^\circ$C (for a designed phase mismatch of $\rm 1.410 \times 10^5\ m^{-1}$ at room temperature). From these two measurements, and considering the corrections owing to the temperature expansion of the crystal [@Emanueli2003], we deduced that the phase mismatch value of the YZY process shifts with temperature with a slope of $\rm 22.34\ m^{-1}/K$. This enabled us to predict the expected phase mismatch of the YZY interaction at $40^\circ$C to be $\rm 1.348 \times 10^5\ m^{-1}$. However, owing to the uncertainty of this linear slope correction, we adopted a multi-section design for the crystal. The 10 mm $\times$ 6mm $\times$ 1 mm crystal was divided into two sections, lengthwise.
The first section, of length 5mm, was a Fourier-engineered ZZZ/ZYY concurrence grating created using the generalized dual grid method [@Lifshitz2005] which was previously shown to create nonlinear photonic quasicrystals whose reciprocal lattices contain an arbitrary set of desired wave vectors [@Bahabad2007; @Bahabad2008]. With corresponding mismatch values of $\Delta k_{ZZZ}=2.510 \times 10^5\ \rm m^{-1}$ and $\Delta k_{ZYY}=9.061\times 10^5\rm\ m^{-1}$, we designed a quasiperiodic structure with reciprocal base vectors $k_1=\Delta k_{ZZZ}+\Delta k_{ZYY}$ and $k_2=\Delta k_{ZYY}$ such that the desired orders for phase matching the two processes are $(1,-1)$ and $(0,1)$ in this basis. Feeding these values into the algorithm of the dual grid method, we got the two tiling vectors of the quasiperiodic structure to be of length $3.37\ \mu m$ and $2.64\ \mu m$. The duty cycles used for the two building blocks of the structure were $0\%$ and $100\%$ respectively. This means that the $3.37\ \mu m$ building block is fabricated with a positive value of the nonlinear susceptibility, and the $2.64\ \mu m$ building block with a negative value. The Fourier coefficients given by this structure for the ZZZ and ZYY processes are $0.112$ and $0.3855$ respectively. The reciprocal basis vectors and the duty cycles of the building blocks were chosen to both maximize the Fourier coefficients and to make the product of the Fourier coefficient and the material nonlinear coupling coefficient of the two processes approximately the same. We refer the interested reader to a detailed account of using the dual grid method for the design of quasiperiodic nonlinear photonic crystals able to phase match several different processes simultaneously [@Bahabad2007].
The second section of the crystal was composed of five parallel, 1 mm wide, gratings, of respective periods $45.9$, $46.3$, $46.7$, $47.2$, and $47.7\ \mu$m in an attempt to correctly sample the wider range of QPM variation for the YZY interaction. These periods are centered around the interpolated phase mismatch value of $\rm 1.348 \times 10^5\ m^{-1}=2\pi /46.6\ \mu$m that was obtained from the measurements with the two previous samples. The ZZZ/ZYY QPM section was as wide as the crystal and overlapped with all YZY channels.
The experimental study used second-harmonic generation (SHG) with the setup shown in Fig. \[setup\].
![Experimental setup. []{data-label="setup"}](1560setup.eps){width="3.25in"}
The input 1560 nm beam was emitted by a tunable fiber laser, amplified by an erbium-doped fiber amplifier (EDFA), and then collimated and sent through a chopper wheel that allowed us to easily observe the SHG signal, amplitude-modulated at 450 Hz, on a fast-Fourier-transform signal analyzer. After the chopper, the beam was sent through a half waveplate and polarizer, which allowed us to precisely control the polarization of the input beam. The input beam was then focused to a waist radius of approximately 30 $\mu$m in the crystal, which was temperature controlled to the nearest hundredth of a degree. Upon exiting the crystal, the input fundamental beam was filtered out by a pair of long-pass filters that reflected 99% of light in the 715–900 nm wavelength range while passing over 85% of light between 985 and 2000 nm. Before reaching the detector, the SHG light passed through a half-waveplate and polarizing beam splitter combination, which allowed us to choose the SHG polarization to be detected. Any residual fundamental light was filtered by the very low detection efficiency of our silicon photodiode at that wavelength.
![Temperature dependence of the YZY SHG signal for each of the 5 channels. The poling periods used from left to right were 45.9, 46.3, 46.7, 47.2, and 47.7 $\mu$m.[]{data-label="5channel"}](smaller5channel.eps){width="3.25in"}
-.2in
The detected light was measured by taking the average of ten measurements on the signal analyzer and recording the signal at 450 Hz. The efficiency of the various nonlinear interactions was controlled by adjusting both the crystal temperature and the wavelength of the input beam. The desired YZY poling period fell in between the 45.9 and 46.3 $\mu$m periods that were used to create our first two YZY channels. The other three YZY channels did not yield a significant SHG signal within the temperature range obtainable by our thermoelectric controller. Figure \[5channel\] shows the temperature dependence of the YZY SHG signal for each of the five YZY channels.
![Triply concurrent SHG as a function of temperature at 1560 nm in the $46.3\ \mu$m-period YZY channel. []{data-label="tripconctempdata"}](smallersigvtemp.eps){width="3.25in"}
-.2in
It appears that the $45.9\ \mu$m period corresponds to a QPM temperature larger than $65^\circ$C (which was the limit of our oven), while the $46.3\ \mu$m period is optimized just below $15^\circ$C. Despite the fact that none of our YZY channels used the exact poling period needed to put the SHG peak at 1560 nm at 40 degrees, the large temperature acceptance bandwidth of YZY phase-matching (approximately $30^\circ$ C $\times$ cm) yielded good overlap with the ZZZ and ZYY interactions, as can be seen from Fig. \[tripconctempdata\].
Figure \[tripconcdata\] shows results obtained using the YZY channel with the $46.3\ \mu$m period, at a temperature of $37^\circ$C.
![Triply concurrent SHG at $37^\circ$C in the $46.3\ \mu$m-period YZY channel.[]{data-label="tripconcdata"}](smallersigvwavelength.eps){width="3.25in"}
-.2in
A fit of the data shows the ZYY peak to occur at exactly 1560 nm at this temperature, while the ZZZ peak occurs at 1560.2 nm. The location of the beam waist in the crystal was adjusted so that the YZY SHG output matched that of ZZZ. When the waist was moved to maximize YZY, the YZY near-peak efficiency (at $15^\circ$C, see Fig. \[tripconctempdata\]) became approximately double that of the peak ZZZ efficiency at $37^\circ$C. Using the aforementioned Fourier coefficients and the values $d_{33}=15.4$ pm/V and $d_{32}=d_{24}=3.75$ pm/V [@Pack2004], we obtain a ZYY to ZZZ peak-efficiency ratio of $[(3.75\times 0.3855)/(15.4\times 0.112)]^2 = 2.09/2.97=0.70$, consistent with the experimental results of Figs. \[tripconctempdata\],\[tripconcdata\]. For the YZY to ZZZ peak-efficiency ratio, we obtain $[(3.75 \times 2/\pi )/(15.4\times 0.112)]^2=5.70/2.97=1.92$, again consistent with our experiment. This therefore confirms the values of Ref. . Note that the initial design used the different values $d_{33}=13.7$ pm/V and $d_{32}=5$ pm/V, which is why the ZYY interaction ends up weaker than ZZZ, but this can clearly be remedied.
In conclusion, we have designed and experimentally demonstrated a PPKTP crystal with three concurrent phase-matchings at 1560 nm. The knowledge gained about the YZY QPM period in this work can now be applied to generating a single Fourier-engineered grating for all three processes [@Lifshitz2005]. Having a triply concurrent crystal made with a single Fourier-engineered grating gives several advantages over a crystal containing three separate polings. In particular, the single grating would allow the crystal to be used in single-pass operations, such as those using a nonlinear waveguide, rather than an optical cavity. For example, just using the simultaneous ZZZ and ZYY phase-matchings in the Fourier-engineered crystal of this work could yield a useful source of collinear polarization-entangled photon pairs. Note that this method could also be used to make a crystal with four concurrent phase matchings in other materials, such as $\rm LiNbO_3$ and $\rm LiTaO_3$. Last but not least, and most importantly here, the crystal in this study represents the key component in the implementation of quantum computing over the optical frequency comb [@Menicucci2008; @Flammia2009], which is, in theory, extremely scalable.
MP, PP, and OP were supported by U.S. National Science Foundation grants Nos. PHY-0555522, CCF-062210, and PHY-0855632. AA was supported by the Israel Science Foundation, grant no. 960/05 and by the Israeli Ministry of Science, Culture and Sport.
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{
"pile_set_name": "ArXiv"
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---
abstract: 'The contribution of gravitational neutrino oscillations to the solar neutrino problem is studied by constructing the Dirac Hamiltonian and calculating the corresponding dynamical phase in the vicinity of the Sun in a non-Riemann background Kerr space-time with torsion and non-metricity. We show that certain components of non-metricity and the axial as well as non-axial components of torsion may contribute to neutrino oscillations. We also note that the rotation of the Sun may cause a suppression of transitions among neutrinos. However, the observed solar neutrino deficit could not be explained by any of these effects because they are of the order of Planck scales.'
author:
- |
M. Adak\
[Department of Physics, Pamukkale University,]{}\
[20100 Denizli, Turkey]{}\
[[email protected]]{}\
\
T. Dereli\
[Department of Physics, Koç University,]{}\
[34450 Sar[i]{}yer-İstanbul, Turkey ]{}\
[[email protected]]{}\
\
L. H. Ryder\
[Department of Physics, University of Kent,]{}\
[Canterbury, Kent CT2 7NF, UK]{}\
[[email protected]]{}
date:
title: 'Possible effects of space-time non-metricity on neutrino oscillations'
---
Introduction
============
Neutrinos always attracted a lot of attention in high energy physics [@hax]. A major problem of interest at present is the solar neutrino problem. The Sun is a strong source of electron neutrinos $\nu_e $ because of the thermonuclear reactions taking place in its core. According to the standard solar model, the number of $\nu_e$ to be emitted from the Sun can be predicted. At the same time, the flux of electron neutrinos coming from the Sun can be measured on earth. The measured amount of $\nu_e$ is approximately one third of the predicted amount. Essentially, this is the so-called [*solar neutrino problem*]{}. One well known solution to this problem is provided by the assumption of [*neutrino oscillations*]{} [@hax],[@bil2]. Briefly stated, the neutrino oscillations imply that the electron neutrinos coming out of the Sun may be converted to other neutrino species, muon $\nu_\mu $ and tau $\nu_\tau $ during their journey towards the earth, assuming neutrinos to have a mass whereas the standard electroweak model asserts zero mass for them. It should also be noted that all the above arguments have been cast in Minkowski space-time. However, we know that we live in a curved space-time – perhaps even in a curved space-time with torsion and non-metricity. Therefore, in more recent years, physicists have turned their attention to specifically gravitational contributions to neutrino oscillations – see [@ahl],[@wudka],[@lipkin],[@car],[@kon] and references therein. We recently investigated the effects of space-time torsion on neutrino oscillations [@muz1]- and see also [@gasperini],[@alimoh]. The essence of this work is to calculate the dynamical phase of neutrinos, by finding the form of the Hamiltonian, H, from the Dirac equation in a non-Riemannian space-time. The phase then follows from the formula i = H (t) = e\^[- ]{} (0) \[1\] where $\psi $ is a Dirac 4-spinor and $H$ is a $4\times 4$ matrix. The Hamiltonian $H$ will depend, for example, on momentum $\vec{p}$, and this is expressed not as a differential operator but simply as a vector.[^1] In this note we investigate within the same approach the possible effects of space-time non-metricity on neutrino oscillations.
Space-Time Geometry
===================
Space-time is denoted by the triple $ \{M,g,\nabla \} $ where M is a 4-dimensional differentiable manifold, equipped with a Lorentzian metric $ g $ which is a (0,2)-type covariant, symmetric, non-degenerate tensor and $ \nabla $ is a connection which defines parallel transport of vectors (or more generally tensors). We shall give a coordinate system set up at a point, $ p
\in M $, by coordinate functions (or independent variables) $ \{
x^\alpha (p) \} $, $ \alpha =\hat{0},\hat{1},\hat{2},\hat{3}$. This coordinate system forms a set of [*natural*]{} (or [*coordinate*]{}) [*reference frame*]{} at $ p $ as $ \{
\frac{\partial}{\partial x^\alpha }(p) \} $, with shorthand notation $ \partial_\alpha \equiv \frac{\partial }{\partial
x^\alpha } $. This natural reference frame is a basis vector set for the tangent space at $ p $, denoted by $ T_p(M) $. Similarly, differentials $ \{ dx^\alpha (p) \} $ of coordinate functions $ \{
x^\alpha (p) \} $ at $ p $, form a [*natural*]{} (or [*coordinate*]{}) [*reference co-frame*]{} in the co-tangent space at $
p $, denoted by $ T^*_p(M) $. Interior product of the basis vectors with the basis co-vectors is defined by the Kroenecker symbol: dx\^() \_[\_]{} dx\^= \^\_ . \[11\] In general, any set of linearly independent vectors in tangent space, $ T_p(M) $, can be taken as basis vectors and these vectors can be orthonormalized by, for example, the Gram-Schmidt process. We denote a set like this by $ \{ X_a \} $, $a = 0,1,2,3$ and call it an [*orthonormal reference frame*]{}. In this case the metric defined on M satisfies the relation g(X\_a,X\_b) = \_[ab]{} where $ \eta_{ab} $ is known as the Minkowski metric which is a matrix whose diagonal terms are -1,1,1,1 and off-diagonal terms are zero. The basis set dual to the orthonormal reference frame are denoted by $ \{ e^a \} $, a = 0,1,2,3, and called the [*orthonormal reference co-frame*]{}. $ \{ X_a \} $ and its dual $ \{
e^a \} $ satisfy the following set of equalities that is another manifestation of eqn.(\[11\]): e\^a(X\_b) \_[X\_b]{}(e\^a) = \^a\_b . \[12\] Here we adhere to the following conventions: indices denoted by Greek letters $\alpha$, $\beta , \;\; \cdots =
\hat{0},\hat{1},\hat{2},\hat{3}$ and $\mu$, $\nu , \;\; \cdots =
\hat{1},\hat{2},\hat{3}$ are holonomic or coordinate indices, $a$, $b, \;\; \cdots = 0,1,2,3$ and $i$, $j, \;\; \cdots = 1,2,3$ are anholonomic or frame indices. In terms of the local coordinate frame $ \partial_\alpha (p) $, the orthonormal frame $ X_a (p) $ can be expanded via the so-called vierbein (or tetrad) $
{h^\alpha}_a (p) $ as X\_a (p) =[h\^]{}\_a (p) \_(p) . In order for $ X_a $ to serve as an anholonomic basis, the $
{h^\alpha}_a (p) $ are required to be non-degenerate, i.e., $
\mbox{det} {h^\alpha}_a (p) \neq 0 $. In $ T^*_p (M) $ an orthonormal co-frame $ e^a (p)$ can be expanded in terms of the local coordinate co-frame $ dx^\alpha (p) $ as e\^b (p)= [h\^b]{}\_(p) dx\^(p) . The inverse vierbein $ {h^b}_\beta (p) $ have to be non-degenerate as well. Moreover, the duality of the frame and the co-frame requires for the vierbein and its inverse to satisfy \_[X\_a]{}e\^b = [h\^]{}\_a (p) [h\^b]{}\_(p) = \^b\_a . We set the space-time orientation by the choice $\epsilon_{0123}=
1$. The non-metricity 1-forms, torsion 2-forms and curvature 2-forms are defined by the Cartan structure equations 2Q\_[ab]{} &=& -D\_[ab]{} := \_[ab]{} + \_[ba]{} , \[nonmet\]\
T\^a &=& De\^a := de\^a + [\^a]{}\_b e\^b , \[torsion\]\
[R\^a]{}\_b &=& D[\^a]{}\_b := d[\^a]{}\_b +[\^a]{}\_c \_b .\[curva\] $ d,\quad D, \quad \imath_a,\quad *$ denote the exterior derivative, the covariant exterior derivative, the interior derivative and the Hodge star operator, respectively. The linear connection 1-forms can be decomposed in a unique way according to [@der1]: \_b = [\^a]{}\_b + [K\^a]{}\_b + [q\^a]{}\_b + [Q\^a]{}\_b \[connec\] where $ {\omega^a}_b $ are the Levi-Civita connection 1-forms \_b e\^b = -de\^a , $ {K^a}_b $ are the contortion 1-forms \_b e\^b = T\^a \[contor\] and $ {q^a}_b $ are the anti-symmetric tensor 1-forms q\_[ab]{} = -(\_a Q\_[bc]{}) e\^c + (\_b Q\_[ac]{}) e\^c .\[antisy\]
It is cumbersome to take into account all components of non-metricity and torsion in gravitational models. Therefore we will be content with dealing only with certain irreducible parts of them to gain physical insight. The irreducible decompositions of torsion and non-metricity invariant under the Lorentz group are summarily given below. For details one may consult Ref. [@heh]. The non-metricity 1-forms $ Q_{ab} $ can be split into their trace-free $ \overline{Q}_{ab} $ and the trace parts as Q\_[ab]{} = \_[ab]{} + \_[ab]{}Q \[118\] where the Weyl 1-form $Q={Q^a}_a$ and $ \eta^{ab}\overline{Q}_{ab}
= 0 $. Let us define \_b &:=& \_a [ \^a]{}\_b, := \_a e\^a,\
\_b &:=& \^\*(\_[ab]{} e\^a), := e\^b \_b, \_a := \_a -\_a\[119\] as to use them in the decomposition of $ Q_{ab} $ as Q\_[ab]{} = Q\_[ab]{}\^[(1)]{} + Q\_[ab]{}\^[(2)]{} + Q\_[ab]{}\^[(3)]{} +Q\_[ab]{}\^[(4)]{} \[120\] where Q\_[ab]{}\^[(2)]{} &=& \^\*(e\_a \_b +e\_b \_a)\
Q\_[ab]{}\^[(3)]{} &=& ( \_a e\_b +\_b e\_a - \_[ab]{} )\
Q\_[ab]{}\^[(4)]{} &=& \_[ab]{} Q\
Q\_[ab]{}\^[(1)]{} &=& Q\_[ab]{}-Q\_[ab]{}\^[(2)]{} - Q\_[ab]{}\^[(3)]{} -Q\_[ab]{}\^[(4)]{} . We have $ \imath^a Q_{ab}^{(1)} =\imath^a Q_{ab}^{(2)} =0,
\;\;\;\;\; \eta^{ab}Q_{ab}^{(1)} = \eta^{ab}Q_{ab}^{(2)}
=\eta^{ab}Q_{ab}^{(3)} = 0, \;\;$ and $ e^a \wedge Q_{ab}^{(1)} =0
$. In a similar way the irreducible decomposition of $ T^a $’s invariant under the Lorentz group are given in terms of =\_a T\^a , = e\_a T\^a so that T\^a = [T\^a]{}\^[(1)]{} +[T\^a]{}\^[(2)]{} +[T\^a]{}\^[(3)]{} \[121\] where \^[(2)]{} &=& e\^a ,\
[T\^a]{}\^[(3)]{} &=& \^a ,\
t\^a :=[T\^a]{}\^[(1)]{} &=& T\^a-[T\^a]{}\^[(2)]{} -[T\^a]{}\^[(3)]{} . Here $ \imath_a t^a = \imath_a {T^a}^{(3)} = 0, \;\;\; e_a \wedge
t^a = e_a \wedge {T^a}^{(2)} = 0 $. To give the contortion components in terms of the irreducible components of torsion, we firstly write 2K\_[ab]{} = \_a T\_b - \_b T\_a - (\_a \_b T\_c)e\^c \[122\] from (\[contor\]) and then substituting (\[121\]) into above we find 2K\_[ab]{} &=& \_a t\_b - \_b t\_a - (\_a \_b t\_c)e\^c\
&& + ( e\_a \_b - e\_b \_a )\
&& + ( \_a \_b ) - ( \_a \_b \_c )e\^c. \[123\] In components $ K_{ab} = K_{c,ab}e^c \; , \;\;\; t_a =
\frac{1}{2}t_{bc,a}e^{bc} \; , \;\;\; \alpha = F_a e^a \; , \;\;\;
\sigma = \frac{1}{3!} \sigma_{abc} e^{abc} $ this becomes K\_[c,ab]{} &=& ( t\_[ac,b]{} - t\_[bc,a]{} + t\_[ab,c]{} )\
&& +( F\_b \_[ac]{} - F\_a \_[bc]{}) - \_[abc]{}. \[D.15\]
Hamiltonian of a Dirac particle in arbitrary space-times {#gen-dir}
========================================================
The Dirac equation in a non-Riemannian space-time with torsion and non-metricity is written as [@kos],[@vandyck],[@muz2] \^[\*]{}D + M\^[\*]{}1 = 0 \[direqn\] in terms of the Clifford algebra ${\mathcal{C}}\ell_{3,1}$-valued 1-forms $ \gamma = \gamma^a e_a $ and $M =\frac{mc}{\hbar}$. We use the following Dirac matrices \^0 &=& i (
[cc]{} I & 0\
0 & -I
) , \^i = i (
[cc]{} 0 & \^i\
-\^i & 0
) \[dirmat\] where $\sigma^i $ are the Pauli matrices. $\psi$ is a 4-component complex valued Dirac spinor whose covariant exterior derivative is given explicitly by D= d+ \^[\[ab\]]{}\_[ab]{} + Q \[covder\] where \_[ab]{} = \[ \_a, \_b\] are the spin generators of the Lorentz group. We write it out explicitly as D&=& \_t dx\^ + \_dx\^+ e\^c \_[c,ab]{}\^[ab]{} + e\^c Q\_c\
&=& [h\^]{}\_c e\^c \_t + [h\^]{}\_c e\^c \_ + e\^c \_[c,ab]{}\^[ab]{} + e\^c Q\_c where $\Lambda_{[ab]} := \Omega_{ab} = \Omega_{c,ab}e^c$ anti-symmetric part of the full connection 1-form and $Q = Q_a e^a
$ and using ${ }^*\gamma = \gamma_a { }^*e^a$ and the identity ${
}^*e^a \wedge e^b = - \eta^{ab} { }^*1 $ calculate \^\*D= ( -[h\^]{}\_c \^c \_t - [h\^]{}\_c \^c \_ - \_[c,ab]{}\^c\^[ab]{} - Q\_c \^c ) [ ]{}\^\*1 . Putting this into (\[direqn\]) we obtain \_c \^c \_t = - [h\^]{}\_c \^c \_+ M - \_[c,ab]{}\^c\^[ab]{} - Q\_c \^c . We multiply this from left by i( [h\^]{}\_a \^a )\^[-1]{} = ( [h\^]{}\_a \^a ) where b\^2 := ( [h\^]{}\_0 )\^2 +[h\^]{}\_i h\^[i]{} . When we compare the result with the Schrödinger equation i = H , we deduce the Dirac Hamiltonian matrix [@kon; @muz1; @muz2; @ryder2; @ryder; @obukhov; @hehl2; @lammerzahl] H &=& \_a [h\^]{}\_b \^a \^b i\_ -\_a \^a\
& & + \_d \_[c,ab]{}\^d \^c \^[ab]{} + \_a Q\_b \^a \^b . \[hamilton\] The right hand side of (\[hamilton\]) need not be a hermitian matrix in general; e.g. if ${h^{\hat{0}}}_i \neq 0$, then the mass term contains an anti-hermitian part such as H= H\_0 +iH\_1 \[H0\] where $H_0^+ = H_0$ and $H_1^+ =H_1$. However, the decomposition (\[H0\]) is frame dependent. That is we can always find a local Lorentz frame in which Hamiltonian is fully hermitian [@ryder; @obukhov]. First we can get rid of the anti-hermitian part of the mass term by diagonalizing the matrix ${h^\alpha}_a$ via a frame transformation \_(x) & & \_(x) [[L\^[-1]{}]{}\^]{}\_(x)\
g\_(x) & & g\_(x) [[L\^[-1]{}]{}\^]{}\_\_. Thus[^2] \_a (x) f(x) \^\_a where $x$ stands for $x^\alpha$ and $f(x)$ is composed of ${h^\alpha}_a (x)$. Under this change (\[hamilton\]) goes over to H & & H = cf\_1(x) \^0 \^i i \_ -imc\^2 f\_2(x) \^0\
& & + ic f\_3(x) \_[c,ab]{}\^0 \^c \^[ab]{} + ic f\_4(x) Q\_b \^0 \^b where $f_i(x)$ are composed of ${h^\alpha}_a (x)$. Putting in the definition \_[c,ab]{} = \_[abcd]{}S\^d and using the identity \^a \^[bc]{} &=& \^[ab]{}\^c - \^[ac]{}\^b - \^[abcd]{} \_d \_5 where $\gamma_5=\gamma_0\gamma_1\gamma_2\gamma_3$, the Hamiltonian matrix becomes H = cf\_1(x) \^0 \^i i \_ -imc\^2 f\_2(x) \^0 + ic N\_a(x) \^0 \^a + ic f\_5(x) S\_a \^0 \^a \_5 \[hamil\] where we introduced N\_a := f\_3(x) [\^[b,]{}]{}\_[ba]{} + f\_4(x) Q\_a . If we now define the canonical momenta p\_i := -i( \_ + ) and assume p\_i\^+ = p\_i , (\[hamil\]) takes the form H = f\_1(x)cp\^i \_0 \_i + imc\^2 f\_2(x) \_0 + ic f\_5(x) S\^a \_0 \_a \_5 - ic N\_0(x) \[51\] . In order eliminate the last term in (\[51\]) one may further perform a locally unitary transformation (x)U\^+(x) (x) , H U\^+(x) H U(x) and obtain H & & H = f\_1(x)cp\^i U\^+(x)\_0 \_i U(x) + imc\^2 f\_2(x) U\^+(x) \_0 U(x)\
& & + ic f\_5(x) S\^a U\^+(x) \_0 \_a \_5 U(x)\
& &- ic \[ f\_1(x)U\^+(x)\_0 \_i \^U(x) + N\_0(x)\] . Under the following solvable matrix equation U\^+(x)\_0 \_i \^U(x) =- , we give the final form of our hermitian Hamiltonian matrix (up to a sign) by the expression H = f\_1(x)cp\^i \_0 \_i + imc\^2 f\_2(x) \_0 + ic f\_5(x) S\^a \_0 \_a \_5 .
Neutrino oscillations in the Kerr background
============================================
Here we construct the Hamiltonian matrix of a Dirac particle (i.e. a massive neutrino) of mass $m$ in the background space-time geometry of a heavy, slowly rotating body of mass $M$ such as the Sun. Its exterior gravitational field will be described by weak constant, uniform torsion and non-metricity fields, together with the Kerr metric [@kerr]: $$\begin{aligned}
ds^2 = -\left( 1- \frac{2MGr}{c^2 \rho^2} \right) c\, dt \otimes c\, dt + \frac{\rho^2}{\Delta} dr \otimes dr
+\rho^2 d\theta \otimes d\theta \nonumber \\
+ \left( r^2 +\frac{a^2}{c^2}+\frac{2MGa^2r}{c^4\rho^2} \sin^2\theta
\right) \sin^2\theta \, d\varphi \otimes d\varphi
-\frac{4MGar}{c^2\rho^2} \sin^2\theta \, dt \otimes d\varphi
\end{aligned}$$ where $\Delta = r^2 -\frac{2MG}{c^2}r + (\frac{a}{c})^2$ , $\rho^2
= r^2 + (\frac{a}{c})^2 \cos^2\theta $ , $ a \equiv \frac{J}{M}=
\frac{2}{5} R^2 \omega $. The Sun is assumed a uniform sphere of radius $R$. $M$, $J$ and $\omega$ are the mass, angular momentum and angular velocity of the Sun, respectively. We choose the orthonormal co-frame e\^0 &=& (cdt - \^[2]{}d) , e\^1 = dr\
e\^2 &=& d, e\^3 = ( ( r\^2 + ( )\^2 ) d- a dt ) and using the definitions de\^a + [\^a]{}\_b e\^b =0 \_[ab]{} = - \_a de\_b + \_b de\_a + (\_a \_b de\_c)e\^c , calculate the Levi-Civita connection 1-forms $$\begin{aligned}
\omega^{0}_{\,\,\,1} &=& \frac{MG[r^2 -
(\frac{a}{c})^2\cos^{2}\theta ] }{\rho^4 c}dt
+ \frac{[(\frac{MG}{c^2}-r)\rho^2 - \frac{2MGr^2}{c^2}]a\sin^{2}\theta}{\rho^4c} d\varphi ,
\nonumber \\
\omega^{2}_{\,\,\,3} &=& \frac{2MGra \cos \theta}{\rho^4c^2} dt +
\frac{\Delta (\frac{a}{c})^2 \sin^{2}\theta -( r^2 +
(\frac{a}{c})^2)^2 }{\rho^4} \cos \theta
d\varphi , \nonumber \\
\omega^{0}_{\,\,\,2} &=& - \frac{\sqrt{\Delta}a \sin \theta \cos
\theta}{\rho^2 c} d \varphi , \nonumber \\
\omega^{1}_{\,\,\,3} &=& - \frac{\sqrt{\Delta}r \sin
\theta}{\rho^2} d \varphi , \nonumber \\
\omega^{0}_{\,\,\,3} &=& \frac{\sqrt{\Delta}a \cos
\theta}{\rho^2c} d\theta
-\frac{ar \sin \theta}{\sqrt{\Delta}\rho^2c} dr , \nonumber \\
\omega^{1}_{\,\,\,2} &=&-\frac{a^2 \sin \theta \cos \theta}{\rho^2
\sqrt{\Delta}c^2} dr - \frac{r\sqrt{\Delta}}{\rho^2} d\theta .
\label{omega}\end{aligned}$$
To simplify the discussions, we consider only the motion of massive neutrinos restricted to the equatorial plane of the Sun. Thus we set $\theta = \pi / 2$ and $d\theta = 0$. Furthermore, since the Sun rotates very slowly \[$\omega \simeq 3 \times 10^{-6}
\; (rad/s)$\] we approximate the metric functions. Therefore, in reasonably far away distances from the Sun, the restricted line element will be taken as $$\begin{aligned}
ds^2 \simeq -\left( 1- \frac{2MG}{c^2 r} \right) c\, dt \otimes c\, dt
+ \left( 1- \frac{2MG}{c^2 r} \right)^{-1} dr \otimes dr \nonumber \\
+ r \, d\varphi \otimes r \, d\varphi
-4\frac{a}{c} \frac{MG}{c^2r^2} \, cdt \otimes r\, d\varphi \; .
\end{aligned}$$ We also write the orthonormal co-frame approximately up to $O \, (
\frac{a}{rc})$ as e\^0 = f cdt - d, e\^1 = dr , e\^2 =0 , e\^3 =- dt + r d where f\^2 1- . The inverses of these relations to the same order of approximation are cdt = e\^0 + e\^3 , dr = f e\^1 , d= 0 , d= e\^0 + e\^3 which give \_0 = , \_3 = , \_1 = f , \_0 = , \_3 = with all the other components neglected. To this order of approximation (\[omega\]) gives \_[01]{} f’ e\^0 + e\^3 , \_[03]{} e\^1, \_[31]{} e\^0 + e\^3 \[omega2\] with the remaining ones neglected. Then the Hamiltonian matrix (\[hamilton\]) reads H && f\^2 cp\_r \_0 \_1 + fc p\_ \_0 \_3 +ifmc\^2 \_0 -c ff’ \_0 \_1\
& & +ic f S\^a \_0 \_5\_a - ic f N\^a \_0 \_a\
& & - p\_r \_3\_1 + \_3 \_1 - \_3\
& & +\_0 \_2 \_5 + S\^a \_3 \_a\_5 +N\^a \_3 \_a where p\_r &:=& -i( +)\
p\_&:=& 0\
p\_&:=& -\
N\_a &:=& ([t\^b]{}\_[a,b]{} +F\_a + Q\_a -\_a) . \[Na\] Note that the contributions of axial components of torsion are given by $S^a$ while certain components of non-metricity and the non-axial components of torsion occur only in $N_a$ and the rotation effects are given in terms of the parameter $a$. We rewrite the Hamiltonian $4 \times 4$ matrix in terms of $2 \times
2$ matrices as follows: H = (
[cc]{} H\_[11]{} & H\_[12]{}\
H\_[21]{} & H\_[22]{}
) with H\_[11]{} &=& fmc\^2 +iA + B\_1 + (C-ip\_r)\_2 +D\_3\
H\_[22]{} &=&-fmc\^2 +iA + B\_1 + (C-ip\_r)\_2 +D\_3\
H\_[12]{} &=& F + (f\^2cp\_r +iG)\_1 + iH\_2 +(fcp\_++iK)\_3\
H\_[21]{} &=& F + (f\^2cp\_r +iG)\_1 + iH\_2 +(fcp\_-+iK)\_3 where we set A & & -cf N\_0 + N\_3\
B & & c S\_1 + N\_2\
C & & c S\_2 - (1+rf’ +rN\_1 )\
D & & c S\_3 - S\_0\
F & & c S\_0 - S\_3\
G & & -- cfN\_1 + S\_2\
H & & -cf N\_2 - S\_1\
K & & -cf N\_3 + N\_0 .
The way we approach the solar neutrino problem starts by writing down the Dirac equation in a rotating, axially symmetric background space-time geometry and finding phases corresponding to neutrino mass eigenstates, then finally calculating the phase differences among them. There are two cases of special interest: the azimuthal motion and the radial motion. The analysis of the azimuthal motion with $ \vec{p} =(p_r, p_\theta , p_\varphi) = (0,0,p) $ yields for ultrarelativistic neutrinos, for which $pc \simeq E$ and $cdt
\simeq Rd\varphi$, the phase for the spin up state \^ = ( fE + + +i(A+K) ) \[spinup\] and similarly for the phase of the spin down state \^ = ( fE + - +i(A+K) ) \[spindown\] where \_B\^2 +C\^2 +D\^2 +F\^2+G\^2+H\^2+ 2 (DF +BH -CG) . \[3.27\] These phases alone do not have an absolute meaning; the quantities relevant for the interference pattern at the observation point of the neutrinos are the phase differences $ \Delta \Phi = \Phi_2 -
\Phi_1 $ where $\Phi_1$ and $\Phi_2 $ are the absolute phases of the neutrino mass eigenstates $ \nu_1 $ and $ \nu_2 $. It is thus seen from equations (\[spinup\]) and (\[spindown\]) that the phase differences can have explicit dependence on non-metricity in the case of opposite spin polarizations of mass eigenstates for the azimuthal motion via (\[3.27\]): = \^\_2 - \^\_1 = ( -2 ) ,\
= \^\_2 - \^\_1 = ( +2 ) where $\Delta m^2 = {m_2}^2-{m_1}^2$.
The Hamiltonian for the radial motion on the other hand is obtained by the assumption $\vec{p} = (p,0,0) $. In this case with the further assumptions $ pc \simeq E$ and $cdt \simeq dr$, the phases appropriate to the spin up and spin down particles are, respectively, \^ &=& dr , \[eq24\]\
\^ &=& dr \[eq25\] where \_r && (D-H)\^2 +(B+F+)\^2 +(C+K-)\^2\
& &-(mc-fp)\^2 +(mc-fp)(C+K-) . \[Delta-r\] In this case the relevant phase differences depending on non-metricity via $N^a$ and rotation via $a$ come from the opposite spin polarization states &=& \^\_2 - \^\_1 = r - dr ,\[82\]\
&=& \^\_2 - \^\_1 = r + dr \[83\] . We point out that $\Delta_r = \mbox{Re}\Delta_r
+i\mbox{Im}\Delta_r$ implies $\sqrt{\Delta_r} = \alpha +i\beta$ and hence the rotation of the Sun would suppress the transitions among the neutrinos via the phase difference equations (\[82\]),(\[83\]) in opposite spin polarizations.
Conclusion
==========
We have here extended our recent study of gravitationally induced neutrino oscillations [@muz1] by including the effects of rotation of the Sun, space-time non-metricity and as well as components of torsion other than the axial ones. The rotation of the Sun implies a damping of neutrino oscillations. However, this result is frame dependent as we explained in Sect.\[gen-dir\] in general. We have shown that there are contributions coming from non-axial components of spacetime torsion and definite components of spacetime non-metricity depending on the polarizations of the spin states of the mass eigenstates. If we set the rotation parameter $a=0$, then (\[Delta-r\]) gives = c ( (S\_0 +S\_1)\^2 +(S\_2 -fN\_3)\^2 +(S\_3 +fN\_2)\^2 )\^[1/2]{} which means that there is no suppression among the neutrinos and only $N_2$ and $N_3$ components of $N^a$ contribute to the oscillations. If we further set $N^a=0$, we reach agreement with our previous results in [@muz1]. It should be clear that the above scheme only works if the neutrino masses are different from in each other and hence, in general, different from zero. This means there are right-handed neutrinos as well as left-handed ones which, however, must interact with matter very weakly as they have not yet been observed. Finally, we note that all the possible contributions discussed here so far would be of the order of Planck scales, and hence do not suffice to account for the observed solar neutrino deficit.
Acknowledgement
===============
We thank the referees for constructive criticisms. One of the authors (MA) acknowledges partial support through the research project BAP2002-FEF007 by Pamukkale University, Denizli.
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[^1]: The exponential of $ e^{-\frac{i
}{\hbar}\int{H dt}} $ is defined by its power series expansion.
[^2]: $L \in SO_+ (1,3)$ where $SO_+ (1,3)$ is special orthochronous Lorentz group.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Novelty search has shown to be a promising approach for the evolution of controllers for swarm robotics. In existing studies, however, the experimenter had to craft a domain dependent behaviour similarity measure to use novelty search in swarm robotics applications. The reliance on hand-crafted similarity measures places an additional burden to the experimenter and introduces a bias in the evolutionary process. In this paper, we propose and compare two task-independent, generic behaviour similarity measures: *combined state count* and *sampled average state*. The proposed measures use the values of sensors and effectors recorded for each individual robot of the swarm. The characterisation of the group-level behaviour is then obtained by combining the sensor-effector values from all the robots. We evaluate the proposed measures in an aggregation task and in a resource sharing task. We show that the generic measures match the performance of domain dependent measures in terms of solution quality. Our results indicate that the proposed generic measures operate as effective behaviour similarity measures, and that it is possible to leverage the benefits of novelty search without having to craft domain specific similarity measures.'
author:
- |
Jorge Gomes\
& LabMAg-FCUL\
\
Anders L. Christensen\
\
\
bibliography:
- 'geccobib.bib'
date: 31 January 2013
title: |
Generic Behaviour Similarity Measures\
for Evolutionary Swarm Robotics
---
=10000 = 10000
Introduction
============
Swarm robotics is a promising approach to collective robotics, where the group level behaviour emerges from the local interactions among agents, and from the interactions between the agents and the environment [@sahin05a]. This approach has the potential to incite several desirable properties in a group of agents, such as robustness, flexibility, and scalability [@sahin05a]. However, the complexity stemming from the intricate dynamics required to produce self-organised behaviour complicates the hand-design of control systems [@trianni03]. Artificial evolution has been shown capable of exploiting the intricate dynamics and synthesise self-organised behaviours (see for example [@trianni03; @trianni06a; @baldassarre03; @baldassarre07]), but the approach carries several issues. The most prominent issue associated with common evolutionary techniques is *deception* [@whitley91]. Deception occurs when the fitness function misguides the search towards local maxima that do not contain adequate solutions to the problem. As the complexity of a problem increases, the fitness landscape typically becomes more rugged and gains more local maxima [@nelson09]. As such, it becomes more difficult to craft a fitness function that can successfully guide the search towards the objective [@zaera96], i.e., the evolutionary process becomes more vulnerable to deception. Novelty search [@stan11a] is a distinctive evolutionary approach where candidate solutions are rewarded based solely on their behavioural novelty, with respect to previously evaluated solutions. In recent work [@gomes12a; @gomes12b], it was shown that novelty search can avoid deception in the evolution of swarm robotic systems. Besides not being affected by deception, it was also shown that novelty search is able to find a greater diversity of solutions, and the successful solutions were simpler in terms of neural network complexity, when compared to those found by fitness-based evolution. But these advantages come at a price: for novelty search to work, it is necessary to craft the domain dependent behaviour similarity measure, used to compute the novelty of the individuals. The results showed that the choice of the novelty metric has a significant impact in the performance of novelty search, and can introduce a significant bias in the evolutionary process.
Previous works have proposed behaviour similarity measures that are domain independent [@gomez09; @doncieux10; @mouret12]. These generic measures can potentially be used to overcome the aforementioned limitation of novelty search. Generic measures are typically based on the sensor and effector values of the agents exclusively, and do not rely on domain knowledge provided by the experimenter. However, the generic measures described in previous works are aimed at single robotic systems. In this paper, we study how generic measures can be adapted to swarm robotic systems.
This paper proposes generic behaviour similarity measures that use the sensor and effector values of the robots of the swarm to obtain a representation of the typical behaviour of the swarm as whole. The measures are evaluated in two swarm robotics tasks: (i) an aggregation task; and (ii) a task where robots must share an energy recharging station in order to survive. Following previous results [@gomes12b], novelty search is used in combination with the fitness function, through a linear scalarization. of novelty and fitness objectives. NEAT is used as the underlying neuroevolution method.
The results of our experiments suggest that novelty search with the proposed generic similarity measures can match the performance of novelty search with domain-dependent similarity measures, regarding the quality of the evolved solutions. We show that the documented advantages of novelty search, such as its capacity to bootstrap evolution and to circumvent deception [@gomes12a], are also present with the use of generic measures.
Related Work
============
In this section, we describe the novelty search algorithm, and how novelty search can be combined with fitness-based evolution to improve the effectiveness of the evolutionary process. We move on to discuss the previous applicatons of novelty search in evolutionary robotics. We conclude with a discussion of the generic behaviour similarity measures proposed in previous works.
Novelty Search
--------------
Novelty search [@stan11a] can be implemented over any evolutionary algorithm. The distinctive aspect of novelty search is how the individuals of the population are scored. Instead of being scored according to how well they perform a given task – which is typically measured by a fitness function, the individuals are scored based on their behavioural novelty – which is given by the novelty metric. This metric quantifies how different an individual is from the other, previously evaluated individuals with respect to behaviour.
To measure how far an individual is from others individuals in behaviour space, the novelty metric relies on the average distance of that individual to the $k$-nearest neighbours, among the current population and a sample of the previously seen behaviours (stored in an archive). The behaviour distance between each two individuals is given by a function $dist$ that should be provided by the experimenter. Candidates from sparse regions of the behaviour space thus tend to receive higher novelty scores, thereby creating a constant evolutionary pressure towards behavioural innovation.
The function $dist$ is typically defined with domain knowledge. Following this approach, the behaviour of each individual is characterised by a vector of real numbers. The experimenter should design the behaviour characterisation vector so that it captures behaviour features that are considered relevant to the problem or task. The behaviour distance between two individuals is then given as the Euclidian distance between the corresponding behaviour characterisation vectors of the individuals. A distinct approach is to use distance functions that do not rely on domain knowledge. This approach is the main focus of this paper and will be detailed in Section \[sec:generic\_measures\].
### Combining Novelty and Fitness {#sec:combination}
As novelty search is guided by behavioural innovation alone, its performance can be greatly affected by the size and shape of the behaviour space. In particular, behaviour spaces that are vast or contain dimensions not related with the task can negatively impact the performance of novelty search [@stan10a; @cuccu11b], because novelty search may spend most of its time exploring behaviours that are irrelevant for the goal task. To address this issue, several authors have proposed techniques that combine novelty with fitness in the evaluation of the individuals [@stan10a; @cuccu11b; @gomes12b; @mouret11; @mouret12].
In our experiments, we use a linear scalarization of the novelty and fitness objectives [@cuccu11b]. We chose this approach because it can be used together with NEAT without any further modifications, and has shown promising results in previous studies [@gomes12b]. Linear scalarization of the novelty and fitness objectives directs the search towards regions with high fitness in the behaviour space. An individual $i$ is evaluated to measure both fitness, $f\!it(i)$, and novelty, $nov(i)$, which after being normalised (Eq. \[eq:blendnorm\]) are combined according to Eq. \[eq:blend\]. $$\overline{f\!it}(i)=\frac{f\!it(i)-f\!it_{min}}{f\!it_{max}-f\!it_{min}} \; , \; \overline{nov}(i)=\frac{nov(i)-nov_{min}}{nov_{max}-nov_{min}}
\label{eq:blendnorm}$$ $$score(i)=(1-\rho)\cdot \overline{f\!it}(i)+\rho \cdot \overline{nov}(i)
\label{eq:blend}$$
The parameter $\rho$ controls the relative weight of novelty, and must be specified by the experimenter. $f\!it_{min}$ and $nov_{min}$ are the lowest fitness and lowest novelty score in the current population, and $f\!it_{max}$ and $nov_{max}$ are the highest fitness and highest novelty score, respectively.
### Novelty Search in Evolutionary Robotics
Novelty search, and other evolutionary techniques based on behavioural diversity, have been applied with success to single robotic systems. Some of these applications include body-brain co-evolution [@krcah10]; biped robot control [@stan11a]; robot navigation in deceptive mazes [@stan11a]; sequential light seeking [@mouret12]; and a robot ball-collecting task [@mouret12]. In [@mouret12], it is presented a comprehensive study of the use of diversity-based techniques in evolutionary robotics.
Gomes et al. [@gomes12a; @gomes12b] showed that novelty search can also provide a valuable contribution to the evolution of controllers for swarm robotics. In particular, the results showed that the use of novelty search circumvented deception and bootstrapping problems, and could unveil a broad diversity of solutions for the same problem. However, the same studies revealed that defining behaviour characterisations for this domain can be a delicate endeavour. Since there are infinitely many behavioural possibilities, many of these possibilities must be conflated in order to construct a viable search space. Excessive conflation, however, can hinder the evolution of certain types of solutions, and degrade the performance of novelty search. Furthermore, the definition of the behaviour characterisation adds a human bias to the process, which is an aspect that should be minimised in evolutionary robotics [@nelson09].
Generic Novelty Measures {#sec:generic_measures}
------------------------
Gomez [@gomez09] proposes the use of generic measures for assessing the behaviour similarity between individuals. The proposed approach consists of building state-action trajectories for the agent, i.e., the history of actions of the agent through time. These trajectories are then compared, obtaining a measure of behaviour similarity, without the need of providing domain-specific knowledge. To compare the sequences of actions, the author evaluates the use of Hamming distance, relative entropy, and normalised compression distance (NCD). The experimental setup is based on the Tartarus problem, and the results show that the NCD distance offers the best performance, followed closely by the Hamming distance. NCD is a similarity measure that exploits the algorithmic regularities of the sequences, but introduces a significant computational overhead.
To address the difficulty in designing behaviour characterisations for evolutionary robotics, Doncieux and Mouret [@doncieux10] proposed and compared generic behaviour similarity measures for evolutionary robotics. Any evolutionary robotics experiment involves robots with actuators and sensors, whose values reflect the microscopic behaviour of the robot. This notion led to the definition of the following generic measures [@doncieux10]:
Hamming distance
: A vector is built with the sensor and effector values of the robot, sampled throughout the simulation: $$\vartheta = [\{\mathbf{s}(t),\mathbf{e}(t)\},t\in [0,T]] \enspace ,$$ where $\mathbf{s}(t)$ and $\mathbf{e}(t)$ are the vectors of the sensor and effector values at time $t$, respectively, and $T$ is the simulation time. The vector $\vartheta$ is then binarised into $\vartheta_{bin}$, by transforming each value in either 0 or 1. The similarity measure is then given as the Hamming distance between the corresponding $\vartheta_{bin}$ vectors obtained for each individual.
Direct Fourier Transform
: The $\vartheta$ vectors are obtained for each individual, similar to the Hamming distance measure. But instead of using the complete vectors, a Discrete Fourier Transform (DFT) is used to reduce the dimensionality. The similarity measure is defined as the Euclidean distance between the first $n_{F}$ coefficients of the DFT.
Systematic State Count
: Perception-action states are defined based on the possible combinations of $\vartheta_{bin}$. Relying on the sensor-effector data, the number of times the robot was in a particular state is then evaluated, resulting in a vector of $n$ integers, $n$ being the number of such states. The similarity measure is then defined as the mean element-wise distance between the vectors.
These methods were evaluated in a ball collection task, where the robot had 9 sensors and 3 effectors. The novelty metric was combined with the fitness function through multi-objectivisation. The results showed that the Hamming distance measure was the most effective, being superior to the similarity measure defined with domain knowledge. The systematic state count and DFT measures displayed a significantly lower performance when compared to the Hamming distance.
The Hamming distance similarity measure was further tested in [@mouret12]. In these experiments, the measure was evaluated in three different tasks (deceptive maze, sequential light seeking, ball-collecting robot), and different diversity maintenance techniques. When using multi-objectivisation of novelty and fitness, the results showed that the generic Hamming distance was at least as good as the similarity measure manually defined with domain knowledge, regarding the quality of the evolved solutions.
Methods
=======
Combined State Count
--------------------
The proposed *Combined State Count* is an adaptation of *Systematic State Count* (see Section \[sec:generic\_measures\]). Despite the lower performance in the experiments of Doncieux & Mouret [@doncieux10], when compared to the other generic measures, the concept of this method can be directly adapted to swarms of robots. As such, it is the starting point of our study. The principle is to define states based on the values from the sensors and effectors recorded for each robot. Then, the number of times the robots of the swarm were in each state is computed. There is no discrimination in terms of which robot was in a particular state, i.e, the state counting at the swarm level is the sum of the state counts for each robot in the swarm.
The state counting approach is, however, prone to suffer from scalability issues, since the number of states grows exponentially with the number of sensors/effectors, and with the number of possible values for each sensor/effector. To address this issue, we propose modifications over the original *State count*. Scalability is achieved through the use of efficient structures for representing states and characterisations, and mechanisms for reducing the effective number of states.
### Efficient State Count Representation {#efficient-state-count-representation .unnumbered}
Representing each behaviour characterisation as a vector with one position for each state (as proposed in [@doncieux10]) can compromise the efficiency of the algorithm if there is a large number of states. However, the number of visited states in one simulation is only a small fraction of the total number of possible states. As such, we can represent each characterisation as a map from states to counts. The counting is normalised according to the size of the swarm, to allow fair comparisons between simulations with different swarm sizes. The behaviour similarity measure is then given by the difference between the state count maps. To calculate the characterisation map $m'$ for each individual, Algorithm \[alg:statecount\] is used:
$m \gets Map<Int, Float>$ $\vartheta_{r} \gets$ read-state($r$) $\vartheta'_{r} \gets$ discretise($\vartheta_{r}$) $h \gets$ hash($\vartheta'_{r}$) $m[h] \gets 0$ $m[h] \gets m[h] + 1/swarmsize$ $m' \gets$ filter($m$) $m'$
The function *read-state* retrieves the current sensor-effector state $\vartheta(r)$ for a particular robot $r$: $$\vartheta(r) = \{\mathbf{s}(r),\mathbf{e}(r)\} \enspace ,$$ where $\mathbf{s}(r)$ is the vector of size $n_{s}$, composed of the values coming from the $n_{s}$ sensors of the robot $r$; and $\mathbf{e}(r)$ is the vector composed of the effector values.
The discretised vector $\vartheta'(r)$ is obtained by independently normalising each element of $\vartheta(r)$ to the interval $[0,K-1]$, followed by an approximation to the nearest integer: $$\vartheta'_{i}(r) = \left \| \frac{\vartheta_{i}(r)-\vartheta_{i,max}}{\vartheta_{i,max}-\vartheta_{i,min}} \cdot (K-1)\right \| \enspace ,$$ where $\vartheta_{i,max}$ and $\vartheta_{i,min}$ are respectively the maximum and minimum values of the $i$-th sensor/effector, and $K$ is the number of target partitions. The parameter $K$ has direct implications in the number of possible states, and it should be empirically determined. A rule of thumb is to define it accordingly to the length of $\vartheta$. For most applications, $K$ values of 2 and 3 are adequate, categorising the value of each sensor in High/Low, or High/Medium/Low. However, if the robots have a small number of sensors ($\vartheta$ is relatively short), higher values of $K$ might be preferred, in order to operate with more detailed behaviour characterisations.
The function *hash* was implemented with the Jenkins’ one-at-a-time hash.[^1] The intent of hashing the vector $\vartheta'(r)$ is twofold. First, it allows lookups of the corresponding entry in the $m$ map in $\mathcal{O}(n)$ time, $n$ being the length of $\vartheta'(r)$. Second, as different vectors are hashed to different values (there is a very low probability of collisions), there is no need to store $\vartheta'$ vectors, which improves the space complexity of the algorithm.
### Reducing the Number of States {#reducing-the-number-of-states .unnumbered}
The function *filter* eliminates the least observed states, in order to improve the efficiency of the algorithm. Preliminary results revealed that robots tend to spend most of their time in a small subset of the state space. Most of the states are visited only in one or a few simulation steps. As such, eliminating these states from the behaviour characterisation can significantly improve the efficiency of the algorithm, practically without compromising the accuracy of the characterisation. The function *filter* removes from the characterisation the states where the robots spent less than $T\%$ of the time: $$m' = \left \{ (h,c) \in m : c > \sum_{i \in m} m[i] \cdot T \right \} \enspace .$$
The constant $T$ should be empirically determined. In our experiments, a value of only 1% was enough to drastically reduce the number of states in each characterisation. For instance, the preliminary results showed that on average 99% of the simulation time was spent on only 10% of the visited states.
### Distance Between Characterisations {#distance-between-characterisations .unnumbered}
To calculate the distance between two characterisations we chose to use the Bray-Curtis dissimilarity, a well-known measurement for quantifying the difference between samples of abundance data. Bray-Curtis is a modified Manhattan measure, where the summed differences between the variables are standardised by the summed variables of the samples. This measure is within the range of 0 to 1. A value of 0 indicates that the two samples have the same composition, while a value of 1 means the two samples do not share any element. Adapting the Bray-Curtis dissimilarity behaviour characterisations, the difference $b$ between two characterisations $m_{1}$ and $m_{2}$ is given by: $$\begin{multlined}
b(m_{1}, m_{2}) = \\
\frac{\displaystyle\sum_{i \, \in \, m_{1}\cap m_{2}} \hspace{-7pt} \left | m_{1}[i]-m_{2}[i] \right | + \hspace{-7pt} \sum_{i \, \in \, m_{1}\setminus m_{2}} \hspace{-7pt} m_{1}[i] + \hspace{-7pt} \sum_{i \, \in \, m_{2}\setminus m_{1}} \hspace{-7pt} m_{2}[i]}
{\displaystyle\sum_{i \, \in m_{1}} m_{1}[i] + \sum_{i \, \in m_{2}} m_{2}[i]} \enspace .
\end{multlined}$$
Sampled Average State
---------------------
The second similarity measure relies on the principles of the Hamming Distance measure (see Section \[sec:generic\_measures\]), which was one of the most successful generic similarity measures in previous works with single robotic systems [@doncieux10; @mouret12]. However, this measure relies on the full description of the sensor-effector states of the robot through time. As such, it can not be directly used with swarms of robots because (i) it would not scale with the number of robots, and (ii) the behaviour of an individual robot in a swarm often has a significant stochastic component. To overcome these issues, we propose the following modifications:
- The state of the swarm at a given instant is the average of the sensor-effector states of each robot. This allows scalability in respect to the size of the swarm.
- The state of the swarm is averaged over a certain time window. This reduces the sensitivity to the initial conditions, and to the stochastic nature of the individual robots behaviour.
The characterisation of an individual is given by: $$\vartheta = \left [\{ \overline{v_{0}}(w), \cdots, \overline{v_{n}}(w) \}, w \in [0,W[ \; \right ] \enspace ,$$ where $W$ is the number of time windows and $\overline{v_{i}}(p)$ is the average value of the $i$-th sensor/effector over the $w$-th time window: $$\overline{v_{i}}(w) = \frac{W}{T}\sum_{t=wT/P}^{(w+1)T/P} \frac{1}{R} \sum_{r=0}^{R-1}v'_{i,r}(t) \enspace ,$$ $T$ is the total simulation time, $R$ the number of robots, and $v'_{i,r}(t)$ is the normalised value of the $i$-th sensor/effector of the robot $r$, at instant $t$: $$v'_{i,r}(t) = \frac{v_{i,r}-v_{i,max}}{v_{i,max}-v_{i,min}} \enspace ,$$ $v_{i,max}$ and $v_{i,min}$ are the maximum and minimum values of the $i$-th sensor/effector, respectively.
The distance between two characterisations $\vartheta_{1}$ and $\vartheta_{2}$ is then given by the Manhattan distance between the vectors: $$d_{man}(\vartheta_{1},\vartheta_{2})= \sum |\vartheta_{1}[i]-\vartheta_{2}[i]| \enspace .$$
Experimental Setup
==================
The proposed generic similarity measures are evaluated over two swarm robotic tasks: aggregation and resource sharing. The generic measures are compared with domain dependent measures, and with fitness-based evolution.
Our experimental framework is based on Simbad 3d Robot Simulator [@hugues06] for the robotic simulations. In both tasks, the environment is a 3m by 3m square arena bounded by walls. The swarms are homogeneous. Each robot is modelled after the e-puck, but with modifications to the sensor setup. Each robot is circular with a diameter of 8cm, and is equipped with differential drive, capable of delivering speeds of up to 12cm/s. The local on-board controllers are recurrent neural networks. The inputs of the neural networks are the normalised values of the sensors of the robot, and there are three outputs: one to control each of the two motors, and one dedicated to completely halt the movement of the robot. Each simulation lasts for 2500 simulation steps, which corresponds to 250s of simulated time.
Aggregation Task
----------------
Aggregation is a commonly studied task in swarm robotics [@trianni03; @baldassarre03]. In this task, a dispersed robot swarm must form a single cluster in any point of the arena. The swarm has a fixed size of 7 robots. Each robot is equipped with (i) 8 IR sensors evenly distributed around its chassis for the detection of obstacles (walls or other robots) within a range of 10cm; (ii) 8 IR sensors dedicated to the detection of other robots within a range of 25cm; and (iii) a sensor that returns the percentage of nearby robots (within a radius of 25cm), relative to the swarm size.
The fitness function $F_{a}$ is defined as the average distance of the robots to the centre of mass of the swarm, measured at the last instant of the simulation: $$F_{a}=1-\sum_{i=1}^{N}\frac{dist(\mathbf{R}_T,\mathbf{r}_{i_T})}{N} \enspace ,$$ where $\mathbf{R}_T$ is the centre of mass in the last instant of simulation, and $\mathbf{r}_{i_{T}}$ is the position of robot $i$. The distance values are normalised to $[0,1]$.
The domain dependent behaviour characterisation, used as benchmark, is based on the average distance to the centre of mass of the swarm, and the number of clusters, sampled through the simulation time [@gomes12a]. Considering a simulation with $N$ robots and $T$ temporal samples, the characterisation $\mathbf{b_{a}}$ is given by: $$\begin{aligned}
\mathbf{b_{a}}&=\{\mathbf{cm},\mathbf{cl}\} \notag \\
\mathbf{cm}&=\frac{1}{N} \left [ \sum_{i=1}^{N}d(\mathbf{R}_1,\mathbf{r}_{i_1}),\cdots , \sum_{i=1}^{N}d(\mathbf{R}_T,\mathbf{r}_{i_T}) \right ] \\
\mathbf{cl}&=\frac{1}{N}\left [clusterCount(1), \cdots , clusterCount(T) \right ] \enspace ,\notag\end{aligned}$$ where $\mathbf{R}_t$ is the centre of mass at time $t$, and $\mathbf{r}_{i_{t}}$ is the position of robot $i$ at time $t$. The function $d$ gives the distance normalised in the range $[0,1]$. The function $clusterCount$ returns the number of robot clusters. Two robots belong to the same cluster if the distance between them is less than the robot IR sensor range (25cm).
Resource Sharing Task
---------------------
In this task, the swarm must coordinate in order to allow each member periodical access to a single battery charging station. The robots should first find the charging station, and then effectively share the station to ensure the survival of all the robots in the swarm. The charging station can only hold one robot at the time.
Our experiments use a group of 3 robots. Each robot has (i) 8 IR sensors for the detection of obstacles up to a range of 10cm; (ii) 8 sensors dedicated to the detection of other robots up to a range of 25cm; (iii) 8 sensors for the detection of the charging station up to a range of 1m; (iv) a binary sensor that indicates if the robot is over the charging station; and (v) a proprioceptive sensor that reads the current energy level of the robot.
Each robot starts with full energy (1000 units), and spends energy at a rate proportional to motor usage: a robot spends 5 units per second when motors are off, and 10 units of energy per second when motors propel the robot at its maximum speed. The charging station is placed in the centre of the arena, and charges a robot at a rate of 100 units of energy per second. The robots have to be completely stopped in order to charge.
The fitness function $F_{s}$ used to evaluate the controllers is a linear combination of the number of robots alive at the end of the simulation and the average energy of the robots throughout the entire simulation: $$F_{s} = 0.9\cdot\frac{|a_{T}|}{N}+0.1\cdot \sum_{t=1}^{T}\sum_{i=1}^{N}\frac{e_{i_{t}}}{TNe_{max}}\enspace ,$$ where $|a_{T}|$ is the number of robots still alive in the end of the simulation, $T$ is the length of the simulation, $N$ is the number of robots in the swarm, $e_{i_{t}}$ is the energy of the robot $i$ at time $t$, and $e_{max}$ is the maximum energy of a robot. The second term of $F_s$ concerning the average energy is included to differentiate solutions where the same number of robots survive.
The domain dependent behaviour characterisation is an extension of the characterisation used in previous experiments with this task [@gomes12b]. The characterisation is a vector of length four, composed by the following behavioural features that are related to the task: (i) The number of robots that reached the end of the simulation alive; (ii) the average energy of the alive robots throughout the simulation; (iii) the average movement of all alive robots; and (iv) the average distance of all alive robots to the charging station. Each of these elements is normalised to $[0,1]$.
Configuration of the Algorithms {#sec:treatments}
-------------------------------
NEAT [@stan02] is used as the underlying neuroevolution algorithm. NEAT is widely used, and one of the most successful neuroevolution approaches developed to date. We use the implementation provided in NEAT4J.[^2] The parameters for NEAT were the same in all experiments: recurrent links are allowed, crossover rate – 25%, mutation rate – 10%, population size – 200. The remaining parameters were assigned their default value in the NEAT4J implementation.
The implementation of novelty search follows the description in [@stan11a]. We used a $k$ value of 15 nearest neighbours, and the individuals are added to the novelty archive with a probability of 2% [@stan10b]. The size of the archive is bounded to 500 individuals. When the archive is full, individuals are randomly removed as needed.
Novelty search is combined with the fitness-function through a linear scalarization. of the novelty and fitness objectives (see Section \[sec:combination\]). In all novelty search experiments, the value of $\rho$ was set to 0.7, which means that the score of each individual is based on 70% of the novelty score and 30% of the fitness score. This value was empirically chosen, and in agreement with previous experiments [@gomes12b].
For the *combined state count* measure, the filter threshold $T$ was set to 1% in all experiments, and the discretisation level $K$ was set to 3. For the *sampled average state* measure, three values of $W$ were tested: 1, 10, and 50, which correspond to time windows of 250s, 25s, and 5s, respectively. In both generic similarity measures, the values coming from the sensor arrays (composed of 8 sensors for the detection of obstacles, other robots, or the charging station) were compressed in four values. These four values represent the closest distance measured at the front of the robot, left, right, and back. This compression was done to reduce the number of states (in the *combined state count* measure), and to reduce the length of the characterisation (in the *sampled average state* measure).
Each controller was evaluated in 10 simulations, randomly varying the initial positions and orientations of the robots. The fitness scores obtained in each simulation are combined to a single value using the harmonic mean as advocated in [@sahin05b]. The behaviour characterisations obtained in the multiple simulations are also merged in a single one through an element-wise average (in the domain dependent measures and *sampled average state*), and by summing the state counts (in *combined state count*). The best individuals found in each generation were post-evaluated with 50 simulations, in order to attain more reliable statistics.
Results
=======
The following treatments were applied to each task. Each evolutionary method was evaluated in 10 independent evolutionary runs. The parameters of each method were set as specified in Section \[sec:treatments\].
SC
: Combined state count
AS-1
: Sampled average state with $W=1$
AS-10
: Sampled average state with $W=10$
AS-50
: Sampled average state with $W=50$
DD
: Novelty with domain dependent similarity measure
Fit
: Fitness-based evolution
The quality of the solutions evolved with each evolutionary method is depicted in Figure \[fig:all\_boxplots\]. The boxplots represent the highest fitness score found until a given generation, in each evolutionary run of each treatment. The depicted results are further explained next.
![Performance comparison of the evolutionary treatments in both tasks, regarding the highest fitness score achieved at different stages of the evolutionary process. The boxplots represent the distribution of the fitness scores obtained in the 10 evolutionary runs of each treatment.[]{data-label="fig:all_boxplots"}](all_boxplots2.pdf){width="1\columnwidth"}
Aggregation
-----------
As the results show (Figure \[fig:all\_boxplots\] – Aggregation), the fitness function is not deceptive, as fitness-based evolution can almost always reach good quality solutions. The most notorious advantage of novelty search is its capacity of avoiding deception. However, previous work [@gomes12a] has shown that even in non-deceptive swarm robotic tasks, novelty search can offer a number of advantages. As such, it is still valuable to analyse the performance of novelty search with generic behaviour similarity measures in this non-deceptive task.
In early stages of evolution (at generation 20), novelty search has an advantage over fitness-based evolution, confirming that novelty search quickly bootstraps the evolutionary process [@gomes12a; @mouret09]. All similarity measures, except for *state count* were superior to fitness-based evolution ($p$-value $<$ 0.05, Mann-Whitney U test).
Around the middle of the evolution (generation 75), the differences between the multiple treatments are less pronounced. By the end of the evolution, the domain dependent similarity measure is only superior to the state count measure ($p$-value $<$ 0.05). This absence of significant difference between treatments is actually a promising result. Previous work [@gomes12a] has shown that when the behaviour similarity measure is poorly defined, the performance of novelty search tends to degrade significantly, regarding the quality of the solutions evolved. In our experiment, the generic measures yielded results similar to fitness-based evolution and to the domain dependent measure, which suggests that the generic measures are indeed acting as effective behaviour similarity measures.
Resource Sharing
----------------
As previous experiments have shown [@gomes12b], the resource sharing task is inherently deceptive. In particular, fitness-based evolution tends to get stuck in two local maxima: (i) The robots do not move at all in order to conserve energy and survive longer, and as a consequence, they can not find the charging station and all the robots run out of energy (fitness score around 0.04); and (ii) when a robot finds the charging station, it occupies it and never leaves, condemning the other robots (fitness score around 0.38). The deceptiveness of this task makes it especially suitable to solve using novelty search. As such, this task is a good benchmark to evaluate if the behaviour similarity measures are capable of avoiding deception and guiding evolution towards good solutions.
At the early stages of evolution (generation 50, see Figure \[fig:all\_boxplots\] – Resource sharing), almost all runs of fitness-based evolution are still stuck in the local maximum where the robots do not move. On the other hand, all treatments based on behaviour novelty could successfully bootstrap the evolution. At this early stage, there are still no significant differences between the novelty based treatments. By the middle of the evolutionary process, the domain dependent similarity measure stands out, being superior to all the treatments ($p$-value $<$ 0.05, Mann-Whitney U test), except for **AS-50**. There are no statistically significant differences between generic similarity measures at this stage.
Looking at the best fitness scores achieved in the whole evolution (generation 250), the superiority of the domain dependent similarity measure holds. However, all novelty based treatments were superior to fitness based-evolution ($p$-value $<$ 0.05), and more or less consistently, all reached high fitness scores. Regarding the generic similarity measures, the **AS-50** treatment stands out, being significantly superior to **SC** and **AS-1** ($p$-value $<$ 0.05).
Combined State Count
--------------------
In both tasks, the *combined state count* measure was the least effective generic measure. Nevertheless, the performance was close to the *sampled average state*, which contrasts with the results in [@doncieux10]. To shed some light on the inferior performance of *combined state count*, we analysed the sensor-effector states that are visited with each individual (Figure \[fig:statecount\]).
![Average number of sensor-effector states visited by each population individual (after the filtering step), compared with the average number of states that each individual shares with the current population and the novelty archive.[]{data-label="fig:statecount"}](statecount.pdf){width="1\columnwidth"}
The increasing average number of states depicts the increasing complexity of the solutions, throughout the evolution. However, the average number of common states do not follow this trend. Since the distance between two state count characterisations is essentially determined by the states they share, this distance can lose accuracy if the characterisations share few states. In the extreme case, if no states are shared, the distance value is always the same.
To overcome this issue, we suggest that the similarity between states should also be considered in the distance metric, besides the count of each state. This way, the distance will maintain its accuracy, regardless of the number of shared states. Further studies are required in order to assess the viability of this approach.
Sampled Average State
---------------------
Regarding the *sampled average state* technique, the most important factor to study is the influence of the parameter $W$. This parameter controls the length of the characterisation and how accurately it captures the temporal component of the robots behaviour. In the aggregation task, there was no significant difference among the treatments with different $W$ values ($p$-value $<$ 0.05). On the other hand, in the resource sharing task there is a trend in the results: the higher the $W$, the better the performance of the evolutionary process, regarding the quality of the solutions. The treatment with $W=50$ delivers significantly higher fitness scores than the treatment with $W=1$ ($p$-value $<$ 0.05).
The reason for the different impact of the $W$ value in different tasks is still not clear. Our hypothesis is that the difference is due to the degree of behaviour regularity necessary to solve each task. The aggregation task can be solved using a regular pattern of behaviour, almost a reactive approach. As such, a low $W$ value might be sufficient to adequately characterise the behaviour of the swarm. On the other hand, the resource sharing task requires a more sequential behaviour, which involves first finding the charging station, and then a different behaviour for sharing it with the other robots. As of consequence, higher $W$ values might be preferred, as they allow the sequential component of the behaviour to be adequately captured. Further experiments, with different tasks, are required to confirm or reject this hypothesis.
Conclusion
==========
We proposed two generic similarity measures for the domain of evolutionary swarm robotics, and used them to drive novelty search. The proposed measures rely on the principle that by analysing the microscopic behaviour of the robots of the swarm, it is possible to obtain a characterisation of the swarm behaviour as whole. The microscopic behaviour of each robot is exclusively based on the sensor and effector values of the robots, keeping the characterisation completely independent from the experimenter’s domain knowledge.
The proposed similarity measures were tested in two distinct tasks, and compared with carefully crafted domain dependent similarity measures. The results showed that the performance obtained with the generic measures is just slightly inferior to the performance obtained with the domain dependent measures, regarding the quality of the evolved solutions. In each task, the highest scoring generic measures were not significantly worse than the domain dependent measure. Furthermore, the results show that the advantages of novelty search identified in previous work [@gomes12a] hold with the generic measures: novelty search excelled at bootstrapping the evolutionary process, and was successful in circumventing deception.
In the comparison between the proposed generic similarity measures, we found that the *sampled average state* achieved the best results in both tasks. However, from a general perspective, this measure is associated with a number of limitations: (i) the characterisations can become too long if there is a high number of sensors/effectors and a high value of $W$ is necessary; (ii) it is not applicable to tasks where simulations can have different lengths; and (iii) in tasks where the robots of the swarm are performing different sub-tasks at the same time, averaging the sensor-effector states of all robots can result in a meaningless characterisation. On the other hand, the *combined state count* measure does not suffer from these limitations, despite the inferior performance verified in the two tasks presented in this paper. As such, we contend that the state count approach should not be discarded, and it should be further improved in future work. More experiments, with different tasks, are also needed in order to determine how well our results generalise, and clarify which measures are more suitable for each type of task.
The use of novelty search with generic behaviour similarity measures, in combination with traditional fitness-based evolution, opens interesting possibilities in the domain of evolutionary swarm robotics. First, it facilitates the use of straightforward fitness functions. There is no need to shape the fitness function in order to avoid local maxima, since novelty search circumvents that issue, without depending on additional information provided by the experimenter. It is a step towards evolving complex solutions with minimal intervention from the experimenter. Second, generic measures can potentially be used to unveil a true diversity of solutions based on self-organisation, with the evolved diversity not being conditioned by the experimenter.
[^1]: <http://www.burtleburtle.net/bob/hash/doobs.html>
[^2]: NeuroEvolution for Augmenting Topologies for Java – <http://neat4j.sourceforge.net>
|
{
"pile_set_name": "ArXiv"
}
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---
abstract: 'The classical and the quantum, spin $S=\frac{1}{2}$, versions of the uniaxially anisotropic Heisenberg antiferromagnet on a square lattice in a field parallel to the easy axis are studied using Monte Carlo techniques. For the classical version, attention is drawn to biconical structures and fluctuations at low temperatures in the transition region between the antiferromagnetic and spin-flop phases. For the quantum version, the previously proposed scenario of a first-order transition between the antiferromagnetic and spin-flop phases with a critical endpoint and a tricritical point is scrutinized.'
author:
- 'M. Holtschneider'
- 'S. Wessel'
- 'W. Selke'
title: 'Classical and quantum two-dimensional anisotropic Heisenberg antiferromagnets'
---
Introduction {#sec_in}
============
Uniaxially anisotropic Heisenberg antiferromagnets in an external field along the easy axis have attracted much interest, both theoretically and experimentally, due to their interesting structural and critical properties. In particular, they display a spin-flop phase, and multicritical behavior occurs at the triple point of the antiferromagnetic (AF), spin-flop (SF) and paramagnetic phases.[@fnk; @ro; @lb; @tk; @gj; @cab; @kst; @chl; @ou; @cu]
A prototypical model for such antiferromagnets is the XXZ model, with the Hamiltonian $$\mathcal{H} \; = \;
J \sum\limits_{(i,j)}\left[ \, \Delta (S_i^x S_j^x + S_i^y S_j^y) + S_i^z S_j^z \, \right] \; - \;
H \sum\limits_{i} S_i^z \quad \text{,}
\label{eq_ham}$$ where the sum runs over neighboring spins of a cubic, dimension $d=3$, or square lattice, $d=2$. The coupling constant $J$ and the field $H$ are positive; the anisotropy parameter $\Delta$ may range from zero to one. Furthermore, $S_i^x$, $S_i^y$, and $S_i^z$ denote the spin components at lattice site $i$.
For the three-dimensional case, early renormalization group arguments[@fnk] and Monte Carlo simulations[@lb] suggested that the triple point is a bicritical point with $O(3)$ symmetry. Only a few years ago, this scenario has been questioned, based on high-order perturbative renormalization group calculations.[@cbv] It has been predicted that there may be either a first order transition, or that the ’tetracritical biconical’ [@fnk] fixed point, due to an intervening ’mixed’ or ’biconical’ phase in between the AF and SF phases [@gor; @pt; @lf], may be stable.
In two dimensions, conflicting predictions on the nature of the triple point have been put forward recently [@ho; @zl; @pv; @st], when analyzing the classical version of the above model with spin vectors of unit length, and the quantum version with spin $S=\frac{1}{2}$.
Indeed, in the classical case, simulational evidence for a narrow (disordered) phase between the AF and SF phases has been presented [@ho], extending presumably down to zero temperature. [@zl] On the other hand, in the quantum case, based on simulations as well, a direct transition of first order between the AF and SF phases has been argued to occur at low temperatures. [@st; @koh; @yun]
Obviously, experimental data have to be viewed with care because deviations from the XXZ Hamiltonian, Eq. (\[eq\_ham\]), such as crystal field anisotropies or longer-range interactions, may affect relevantly the critical behavior of the triple point.[@gj; @lf; @lsk; @gor; @pt; @ba]
In the following, we present results from large-scale Monte Carlo simulations of the XXZ model on a square lattice for both the classical and the quantum variant. In the quantum Monte Carlo simulations, the method of the stochastic series expansion (SSE)[@sk] is used, and the standard Metropolis algorithm is applied for the classical case. The simulations are augmented by a ground-state analysis of the classical model, showing the significance of biconical structures. The outline of the paper is as follows: First we shall discuss our findings on the classical model, followed by a section on the quantum version of the XXZ model. A summary concludes the paper.
Classical model {#sec_cl}
===============
![Ground state configurations of the classical model sketched by the directions of spins on the two sublattices (i.e. at neighboring sites), from left to right: AF, SF, and biconical state. The circles denote the trivial degeneracy in the $xy$-plane.[]{data-label="fig_gstate"}](figure1)
The ground states of the classical model on a square lattice, see Hamiltonian (\[eq\_ham\]), can be determined exactly. The AF structure is stable for magnetic fields below the critical value $$H_{\text{c}1} \; = \; 4 J \sqrt{1-\Delta^2} \quad \text{,}
\label{eq_hc1}$$ while for larger fields the SF state is energetically favorable. At $H_{\text{c1}}$, the tilt angle $\theta_{\text{SF}}$ of the SF structures, see Fig. \[fig\_gstate\], is given by $$\theta_{\text{SF}} \; = \; \arccos \sqrt{\frac{1-\Delta}{1+\Delta}} \quad \text{.}
\label{eq_sfangle}$$ Increasing the field beyond , all spins perfectly align in the $z$-direction.
![Detail of the phase diagram of the XXZ model on a square lattice with $\Delta=\frac{4}{5}$, see Ref. . Squares refer to the boundary of the SF, circles to that of the AF phase. The solid line refers to the magnetic field $H/J=2.41$, where the probability distribution , depicted in Fig. \[fig\_2dhisto\], has been obtained. Here and in the following figures error bars are shown only if the errors are larger than the symbol size and dotted lines are guides to the eye.[]{data-label="fig_clpdiag"}](figure2)
At the critical field $H_{\text{c}1}$, see Eq. (\[eq\_hc1\]), the ground state is degenerate in the AF, the SF, and biconical structures, as illustrated in Fig. \[fig\_gstate\]. This degeneracy in the biconical configurations, following from straightforward energy considerations, seems to have been overlooked in the previous analyses. The structures may be described by the tilt angles, $\theta_1$ and $\theta_2$, formed between the directions of the spins on the two sublattices of the antiferromagnet and the easy axis. For a given value of $\theta_1$, the other angle $\theta_2$ is fixed by $$\theta_2 \; = \; \arccos
\left( \frac{ \sqrt{1-\Delta^2} \; - \; \cos\theta_1 }{ 1 \; - \; \sqrt{1-\Delta^2} \cos\theta_1 } \right)
\quad \text{.}
\label{eq_bic}$$ Obviously, the biconical configurations transform the AF into the SF state: The spins on the “up-sublattice” of the AF structure, with the spins pointing into the direction of the field, may be thought of to vary from $\theta_1=0$ to $\pm\theta_{\text{SF}}$, while the spins on the “down-sublattice” vary simultaneously from $\theta_2=\pi$ to $\mp\theta_{\text{SF}}$. Accordingly, $\theta_1$ determines uniquely $\theta_2$ and vice versa. Apart from this continuous degeneracy in $\theta_1$ (or $\theta_2$), there is an additional rotational degeneracy of the biconical configurations in the spin components perpendicular to the easy axis, the $xy$-components, as for the SF structure, see Fig. (\[fig\_gstate\]). These components are, of course, antiferromagnetically aligned at neighboring sites.
![Joint probability distribution showing the correlations between the tilt angles $\theta_m$ and $\theta_n$ on neighboring sites $m$ and $n$ for a system of size $L=80$ at $H/J=2.41$, $k_BT/J=0.255$, and $\Delta=\frac{4}{5}$. is proportional to the gray scale. The superimposed black line depicts the relation between the two angles in the biconical ground state, see Eq. (\[eq\_bic\]).[]{data-label="fig_2dhisto"}](figure3)
To study the possible thermal relevance of the biconical structures at $T>0$, we performed Monte Carlo simulations analyzing the joint probability distribution for having tilt angles $\theta_m$ and $\theta_n$ at neighboring sites, $m$ and $n$. For comparison with the previous studies[@lb; @ho; @zl] we set $\Delta=\frac{4}{5}$, leading to the phase diagram depicted in Fig. \[fig\_clpdiag\]. For example, fixing the field at $H=2.41J$, we observed at an Ising-type transition on approach from higher temperatures and a Kosterlitz-Thouless-type transition on approach from the low-temperature side, extending our corresponding previous findings[@ho] to even lower temperatures, and in agreement with recent results.[@zl] Indeed, as depicted in Fig. \[fig\_2dhisto\], in that part of the phase diagram, being in the vicinity of the very narrow intervening, supposedly disordered phase, the joint probability exhibits a line of local maxima following closely Eq. (\[eq\_bic\]), obtained for the ground state. That behavior is largely independent of the size of the lattices we studied. Similar signatures of the biconical structures are observed in the simulations at nearby temperatures, when fixing the field at $H=2.41J$, as well as in the vicinity of the entire transition region between the AF and SF phases, see Fig. \[fig\_clpdiag\], at higher fields and temperatures.
Accordingly, we tend to conclude that biconical fluctuations are dominating in the narrow intervening phase. Whether that phase exists as a disordered phase down to the ground state or whether there is a stable biconical phase in two dimensions, remain open questions, being beyond the scope of this article.
Note that our additional Monte Carlo simulations for the anisotropic XY antiferromagnet in a field on a square lattice show that the analogues of ’biconical’ structures (the orientation of the spins being now given by the two tilt angles only) and fluctuations play an important role near the transition regime between the AF and SF phases in that case as well. In fact, Eq. (\[eq\_bic\]) provides an excellent guidance for interpreting our simulational data similar to the ones presented in Fig. 3.
Quantum XXZ model {#sec_qu}
=================
![Phase diagram of the XXZ Heisenberg antiferromagnet with spin-$\frac{1}{2}$ and $\Delta=\frac{2}{3}$. The straight solid lines denote the choices of parameters where our very extensive simulations, discussed in the text, have been performed. The arrows mark the previously[@st] suggested locations of the tricritical point ($T_{\text{t}}$) and the critical endpoint ($T_{\text{ce}}$).[]{data-label="fig_phdg"}](figure4)
The aim of the study on the quantum version, $S=\frac{1}{2}$, of the XXZ model, Eq. (\[eq\_ham\]), has been to check the previously suggested scenario of a first-order phase transition between the AF and SF phases extending up to a critical endpoint and with a tricritical point on the AF phase boundary, see Fig. \[fig\_phdg\].
We performed quantum Monte Carlo (QMC) simulations in the framework of the stochastic series expansion (SSE)[@sk] using directed loop updates [@su]. We consider square lattices of $L\times L$ sites with the linear dimension $L$ ranging from $2$ to $150$, employing full periodic boundary conditions. Defining, as usual,[@sk] a single QMC step as one diagonal update followed by the construction of several operator-loops, each individual run typically consists of $10^6$ steps and is preceded by at least $2\cdot10^5$ steps for thermal equilibration. Averages and error bars are obtained by taking into account results of several, ranging from $8$ to $32$, Monte Carlo runs, choosing different initial configurations and random numbers. Especially for large systems and low temperatures we additionally utilize the technique of parallel tempering (or exchange Monte Carlo)[@pt1; @se] to enable the simulated systems to overcome the large energy barriers between configurations related to different phases more frequently. We typically work with a chain of $16$ to $32$ configurations in parallel which are simulated at different equally spaced temperature or magnetic field values allowing for an exchange of neighboring configurations after a constant number of QMC steps. The achieved reduction of the autocorrelation times, e.g. of the different magnetizations discussed below, amounts up to several orders of magnitude and therefore results in significantly smaller correlations between subsequent measurements which, in turn, allows for shorter simulation times.
To determine the phase diagram and to check against previous work[@st], we calculated various physical quantities including the $z$-component of the total magnetization, $$M^z \; = \; \frac{1}{L^2} \sum_i \langle S_i^z \rangle \quad \text{,}$$ and the square of the $z$-component of the staggered magnetization, $$(M^z_{\text{st}})^2 \; = \; \frac{1}{L^2}
\left[ \sum_{i_a} \langle S_{i_a}^z \rangle \; - \; \sum_{i_b}
\langle S_{i_b}^z \rangle \right]^2
\quad \text{,}$$ summing over all sites, $i_a$ and $i_b$, of the two sublattices of the antiferromagnet. A useful quantity in studying the SF phase is the spin-stiffness $\rho_s$ which is related to the change of the free-energy on imposing an infinitesimal twist on all bonds in one direction of the lattice. In QMC simulations the spin-stiffness can conveniently be measured by the fluctuations of the winding numbers $W_x$ and $W_y$,[@sk] $$\rho_s \; = \; \frac{k_B T}{2} \left( W_x^2 + W_y^2 \right) \quad \text{.}$$ The winding numbers themselves are given by $$W_{\alpha} \; = \; \frac{1}{L} \left( N^+_{\alpha} \; - \; N^-_{\alpha} \right),$$ where $N^+_{\alpha}$ and $N^-_{\alpha}$ denote the number of operators $S^+_iS^-_j$ and $S^-_iS^+_j$ in the SSE operator sequence with a bond $\langle i,j \rangle$ in the $\alpha$-direction, $\alpha\in\{x,y\}$.
All data for the quantum model presented here are obtained at an anisotropy parameter of $\Delta=\frac{2}{3}$ to allow for comparison with previous findings[@st; @ho]. The phase diagram in the region of interest, where all three phases, the AF, the SF, and the paramagnetic phase occur, is displayed in Fig. \[fig\_phdg\].
![Positions of the maxima of the magnetization histograms as a function of the inverse system size. The inset exemplifies two histograms for systems of size $L=32$ (circles) and $L=150$ (squares) at $k_BT/J=0.13$ and the coexistence fields $H/J=1.23075$ and $H/J=1.232245$.[]{data-label="fig_hist"}](figure5)
The earlier study[@st] asserted a phase diagram with a tricritical point at and a direct first-order transition between the SF and AF phases below the critical endpoint at , see Fig. \[fig\_phdg\]. In detail the authors identified a first-order AF to paramagnetic transition at $k_BT/J=0.13$ by means of an analysis of the magnetization histograms $p(M^z)$. We studied that case, improving the statistics and considering even larger lattice sizes. Indeed, as expected for a discontinuous change of the magnetization, the histograms of finite systems are confirmed to display two distinct maxima corresponding to the ordered and the disordered phase in the vicinity of the AF phase boundary (see inset of Fig. \[fig\_hist\]). Note however, that such a two-peak structure can also be found for small systems at a continuous transition, with a single peak in the thermodynamic limit. Thence, a careful finite-size analysis is needed to, possibly, discriminate the two different scenarios. We simulated lattice sizes with up to spins adjusting the magnetic field such that coexistence of the phases, i.e. equal weight of the two peaks, is provided. As depicted in Fig. \[fig\_hist\] the positions of the maxima as a function of the inverse system size exhibit a curvature, which becomes more pronounced for larger lattices. In contrast, in the previous analysis[@st] at the same temperature, linear dependences of the peak positions as a function of $1/L$ had been presumed, leading to distinct two peaks in the thermodynamic limit. We conclude, that the previous claims of a first-order transition at $k_BT/J=0.13$ and of the existence of a tricritical point at needs to be viewed with care. Indeed, the tricritical point seems, if it exists at all, to be shifted towards lower temperatures.
In the previous work[@st] a direct transition of first order between the AF and SF phases has been suggested to take place at lower temperatures, . To check this suggestion we studied the system at constant field $H/J = 1.225$, where such a direct transition would occur, see Fig. \[fig\_phdg\]. Calculating the expectation values of the different magnetizations as well as the corresponding histograms we obtain an estimate of the critical temperature of the AF phase, .
![Doubly logarithmic plot of the staggered magnetization $(M^z_{st})^2$ vs. the system size $L$ at $H/J=1.225$ for the temperatures $k_BT/J = 0.095$, $0.0955$, $0.096$, $0.0965$, $0.097$, and $0.0975$ (from bottom to top). The straight line proportional to $L^{\frac{1}{4}}$ illustrates the expected finite-size behavior close to a continuous transition of Ising type.[]{data-label="fig_stmz"}](figure6)
Surprisingly, approaching the transition from the AF phase, the finite-size behavior of the squared staggered magnetization $(M^z_{\text{st}})^2$, being the AF order parameter, is still consistent with a continuous transition in the Ising universality class: As illustrated in Fig. \[fig\_stmz\] the asymptotic region is very narrow, similar to the observations in the classical model.[@ho; @zl] The dependence on the system size seems to obey right at the transition, as expected for the Ising universality class.[@on]
Furthermore, approaching the transition from the SF phase, an analysis of the spin-stiffness $\rho_s$ at the same field value of $H/J=1.225$ results in about the same transition temperature, . Thence, there may be either a unique transition between the SF and AF phases, or, as observed in the classical case, an extremely narrow intervening phase, with phase boundaries of Ising and Kosterlitz-Thouless (KT) type.
To determine, whether a KT transition describes the disordering of the SF phase, we check the theoretical prediction[@kt; @nk] that for the infinite system the spin-stiffness is finite within the SF phase, takes on a universal value at the KT transition related to $T_{\text{KT}}$ by $$\rho_s(T=T_{\text{KT}},L=\infty) \; = \; \frac{2}{\pi} \; k_B T_{\text{KT}} \quad \text{,}
\label{eq_ktstif}$$ and discontinuously vanishes in the disordered phase. As depicted in Fig. \[fig\_stif\], the spin-stiffness $\rho_s$ at $T = T_{\text{SF}}$ seems to be, at first sight, significantly larger than the KT-critical value given by Eq. (\[eq\_ktstif\]). Indeed, in the earlier study[@st] it has been argued, based on similar observations, that there is a direct first order AF to SF transition. However, the finite-size effects close to the transition deserve a careful analysis: For the KT scenario, renormalization group calculations[@wm; @wm1] predict the asymptotic size dependence at $T=T_{\text{KT}}$ to obey $$\begin{gathered}
\rho_s(T=T_{\text{KT}},L) \quad = \\
\quad \rho_s(T=T_{\text{KT}},L=\infty) \; \left( 1 \; + \; \frac{1}{2\ln L \; - \; C_0} \right) \quad \text{,}
\label{eq_stifffs}\end{gathered}$$ where $C_0$ denotes an apriorily unknown, non-universal, parameter. By studying the quantity[@hk] $$C(L) \; = \; -2 \left[ \left( \frac{\pi \rho_s}{k_B T} - 2 \right)^{-1} \; - \; \ln L \,\right] \quad \text{,}
\label{eq_stifcl}$$ which, according to Eqs. \[eq\_ktstif\] and \[eq\_stifffs\], converges for $L\rightarrow\infty$ and $T=T_{\text{KT}}$ to the value $C_0$ at a KT transition, we obtain a rough estimate of $C_0 \approx 5$. A prediction of the finite-size behavior at $T_{\text{SF}}$ is obtained by inserting this value, $C_0=5$, into Eq. (\[eq\_stifffs\]). Comparing the data of the spin-stiffness $\rho_s$ in the direct vicinity of the boundary of the SF phase with the prediction according to the KT theory, see Fig. \[fig\_stif\] b), one may conclude that the lattice sizes accessible by simulations, , seem to be too small to capture the asymptotic finite-size behavior. In any event, in case of a KT transition, the spin-stiffness $\rho_s$ drops asymptotically very rapidly to its universal critical value as a function of system size, being consistent with the relatively large values for the simulated finite lattices. Thus, a scenario with a KT transition between the SF and a narrow intervening disordered phase cannot be ruled out by the present large-scale simulations down to temperatures as low as.
![a) Spin stiffness $\rho_s/J$ vs. temperature $k_BT/J$ for the different system sizes $L=8$ (circles), $16$ (squares), $32$ (diamonds), and $64$ (triangles). The straight line denotes the critical value of the spin-stiffness according to the formula of Nelson and Kosterlitz[@nk], see Eq. (\[eq\_ktstif\]).\
b) Finite-size behavior of the spin-stiffness $\rho_s/J$ at $H/J=1.225$ as a function of the inverse system size, $1/L$ for the temperatures $k_BT/J=0.0955$, $0.096$, $0.0965$, $0.097$, and $0.0975$ (from top to bottom). The dashed curve illustrates the estimated asymptotic behavior according to Eq. (\[eq\_stifffs\]) with $k_BT_{\text{KT}}/J=0.09625$ and $C_0=5$, the corresponding critical value is marked by the filled circle.[]{data-label="fig_stif"}](figure7)
Of course, it is desirable to quantify the role of biconical fluctuations in the quantum case as well. However, accessing the probability distributions of the tilt angles studied in Sect. \[sec\_cl\] for the quantum case is beyond the scope of the present numerical analysis.
Summary {#sec_ds}
=======
We studied the classical and quantum, $S=\frac{1}{2}$, versions of the XXZ Heisenberg antiferromagnet on the square lattice in an external field along the easy axis. The model is known to display ordered AF and SF as well as disordered, paramagnetic phases. Here we focused attention to the region of the phase diagram near and below the temperature where the two boundary lines between the AF and the SF phases and the disordered phase approach each other, meeting eventually at a triple point. We performed Monte Carlo simulations, augmented, in the classical case, by a ground state analysis.
In the classical version, we presented first direct evidence for the importance of biconical structures in the XXZ model. Indeed, such configurations do exist already as ground states at the critical field $H_{\text{c}1}$, separating the AF and SF phases. The interdependence of the two tilt angles, characterizing the biconical ground states, persists at finite temperatures, in the region where the narrow phase between the AF and SF phases is expected to occur. Indeed, the joint probability distribution of the tilt angles at neighboring sites demonstrates the thermal significance of those configurations. Previous arguments on $O(3)$ symmetry in that region and down to zero temperature thus have to be viewed with care. The results of the present simulations suggest that, if the biconical configurations do not lead to a stable biconical phase in two dimensions, the narrow intervening phase is a disordered phase characterized by biconical fluctuations. In this sense the “hidden bicritical point” at $T=0$ may then be coined into a “hidden tetracritical point.”
In the quantum version, previous simulations suggested, on lowering the temperature, the existence of a tricritical point on the boundary line between the AF and disordered phases, followed by a critical endpoint being the triple point of the AF, SF and disordered phases, and eventually by a first-order transition between the AF and SF phases at sufficiently low temperatures. The present simulations, considering larger system sizes and improved statistics, provide evidence that this scenario, if it exists at all, has to be shifted to lower temperatures than proposed before. Of course, simulations on even larger lattices and lower temperatures would be desirable, but are extremely time consuming.
A clue on possible distinct phase diagrams for the classical and quantum versions may be obtained from an analysis of biconical fluctuations in the quantum case. Experimental studies on signatures of those fluctuations are also encouraged.
Financial support by the Deutsche Forschungsgemeinschaft under grant No. SE 324/4 is gratefully acknowledged. We thank A. Honecker, B. Kastening, R. Leidl, A. Pelissetto, M. Troyer, and E. Vicari for useful discussions and information.
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{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We study ionic liquids composed 1-alkyl-3-methylimidazolium cations and bis(trifluoromethyl-sulfonyl)imide anions (\[C$_n$MIm\]\[NTf$_2$\]) with varying chain-length $n\!=\!2, 4, 6, 8$ by using molecular dynamics simulations. We show that a reparametrization of the dihedral potentials as well as charges of the \[NTf$_2$\] anion leads to an improvment of the force field model introduced by Köddermann [*et al.*]{} \[ChemPhysChem, **8**, 2464 (2007)\] (KPL-force field). A crucial advantage of the new parameter set is that the minimum energy conformations of the anion ([*trans*]{} and [*gauche*]{}), as deduced from [*ab initio*]{} calculations and [Raman]{} experiments, are now both well represented by our model. In addition, the results for \[C$_n$MIm\]\[NTf$_2$\] show that this modification leads to an even better agreement between experiment and molecular dynamics simulation as demonstrated for densities, diffusion coefficients, vaporization enthalpies, reorientational correlation times, and viscosities. Even though we focused on a better representation of the anion conformation, also the alkyl chain-length dependence of the cation behaves closer to the experiment. We strongly encourage to use the new NGKPL force field for the \[NTf$_2$\] anion instead of the earlier KPL parameter set for computer simulations aiming to describe the thermodynamics, dynamics and also structure of imidazolium based ionic liquids.'
author:
- Benjamin Golub
- Jan Neumann
- 'Lisa-Marie Odebrecht'
- Ralf Ludwig
- Dietmar Paschek
title: 'Revisiting Imidazolium Based Ionic Liquids: Effect of the Conformation Bias of the \[NTf$_{2}$\] Anion Studied By Molecular Dynamics Simulations'
---
Introduction
============
Having a reliable force field available is one of the most important prerequisites for setting up a molecular dynamics simulation. Hence, a lot of effort has been put into the development of new as well as the improvement of existing force field models. There are essentially two different approaches on how to improve or optimize force fields:
One approach is trying to develop a “universal” force field parameter set which can be applied to a broad range of different molecules or ions, such as the force field parameters for ionic liquids introduced by Pádua et al.[@Padua:2004_1; @Padua:2004_2; @Padua:2006_1; @Padua:2006_2; @Shimizu:2008; @Ishiguro:2008_1; @Ishiguro:2008_2; @Padua:2009; @Lopes:2010; @Padua:2012]. These force fields are very popular in the ionic liquids molecular simulation community and yield in general good results in comparison with experimental data.
An alternative, less universal approach is to focus on a specific subset of molecules and ions, and to enhance the quality of the model by fitting the parameters of a system to a set of selected thermodynamical, dynamical and structural properties, which then can be accurately emulated by the force field. The most well-known example for the application of such a strategy is perhaps the water molecule. In 2002 Bertrand Guillot gave a comprehensive overview over (at the time) more than 40 different water models [@Guillot:2002], and the number has been increasing since then [@Baranyai:2012; @Bettens:2015; @Sadus:2015; @Paesani:2016]. Obviously, water is of great scientific interest. As a consequence, there exist a variety of force field models consisting mostly of three (SPC, TIP3P) to five (TIP5P, ST2) interaction sites, including (POL5) or without (SPC/E) polarizability an and even force fields optimized to best represent the solid phases of water (TIP4P/ICE) and their phase transitions.
The second strategy was employed by Köddermann [*et al.*]{} in 2007 to arrive at the KPL (Köddermann, Paschek, Ludwig) force field for a selected class of imidazolium based ionic liquids composed of 1-alkyl-3-methylimidazolium cations and bis(trifluoromethyl-sulfonyl)imide anions (\[C$_n$MIm\]\[NTf$_2$\]) [@Koeddermann:2007]. Aim of this work was to further optimize the force field of Pádua [*et al.*]{} to better represent dynamical properties like self-diffusion coefficients, reorientational correlation times, and viscosities. As shown in their original work from 2007 as well as in further works published by different groups, the KPL force field has been proven to yield reliable results for dynamical properties, but also thermodynamical properties, such as the free energies of solvation for light gases in ionic liquids [@Kerle:2009; @Kerle:2017], and is still used frequently to this date [@Daly:2017; @Stone:2017; @Kazemi:2017].
![Minimum energy conformations of the \[NTf$_2$\] anion taken from a MD simulation employing the KPL force field.[]{data-label="fig:kodd_conf"}](FIG01a.eps "fig:"){width="3.0cm"} ![Minimum energy conformations of the \[NTf$_2$\] anion taken from a MD simulation employing the KPL force field.[]{data-label="fig:kodd_conf"}](FIG01b.eps "fig:"){width="3.0cm"}
![Minimum energy conformations of the \[NTf$_2$\] anion obtained from [*ab initio*]{} calculations. The [*trans*]{} conformation (left) represents the global energy minimum, while the energy of the [*gauche*]{} conformation (right) is elevated by about $3\,\mbox{kJ}\,\mbox{mol}^{-1}$.[]{data-label="fig:minimum_conf"}](FIG02a.eps "fig:"){width="2.5cm"} ![Minimum energy conformations of the \[NTf$_2$\] anion obtained from [*ab initio*]{} calculations. The [*trans*]{} conformation (left) represents the global energy minimum, while the energy of the [*gauche*]{} conformation (right) is elevated by about $3\,\mbox{kJ}\,\mbox{mol}^{-1}$.[]{data-label="fig:minimum_conf"}](FIG02b.eps "fig:"){width="2.5cm"}
Here we want to present our take on further improving the KPL force field by revisiting the conformation-space explored by the \[NTf$_2$\] anion. Extensive studies of the conformation of the \[NTf$_2$\] anion using the KPL force field in comparison to experimental as well as quantum chemical calculations have revealed a significant mismatch of the energetically favored conformations. Therefore we feel the need for presenting a modified version of the force field, removing this conformation-bias. We are discussing the implications of this modification for a wealth of thermodynamical, dynamical, and structural quantities.
Conformation-space of the Anion
===============================
During MD simulations of ionic liquids of the type \[C$_n$MIm\]\[NTf$_2$\] with the force field of Köddermann [*et al.*]{} it became apparent that the favored \[NTf$_2$\] anion conformations observed in the simulation differ from what has been shown earlier from quantum chemical calculations [@Ishiguro:2008_1] as well as from [Raman]{} experiments [@Fujii:2006] (see Fig. \[fig:kodd\_conf\] and Fig. \[fig:minimum\_conf\]).
For locating the minimum energy conformations we performed extensive quantum chemical calculations with the [Gaussian 09]{} program [@Gaussian_09] following the approach of Pádua [*et al.*]{} [@Padua:2004_2]. We started by calculating the potential energy surface as a function of the two dihedral angles S1-N-S2-C2 ($\phi_{1}$) and S2-N-S1-C1 ($\phi_{2}$) on the HF level with a small basis set (6-31G\*). Subsequent to these optimizations we performed single point calculations on the MP2 level using the cc-pvtz basis set for all HF optimized conformations. In agreement with earlier calculations by Pádua [*et al.*]{} [@Ishiguro:2008_1], and [Raman]{} measurements of Fujii [*et al.*]{} [@Fujii:2006], we observe essentially two structurally distinct minimum energy conformations that can be identified as energy minima on the energy-landscape depicted in Fig. \[fig:abinitio\]. The [*trans*]{} conformations of the \[NTf$_2$\] anion are energetically preferred, followed by the gauche-conformations, which are elevated by about $3\,\mbox{kJ}\,\mbox{mol}^{-1}$ (see Fig. \[fig:minimum\_conf\]).
![[*Ab initio*]{} computation of the energy surface of the \[NTf$_{2}$\] anion as a function of the S1-N-S2-C2, and S2-N-S1-C1 dihedral angles $\phi_{1}$ and $\phi_{2}$. []{data-label="fig:abinitio"}](FIG03.eps){width="7.0cm"}
To compare these [*ab initio*]{} calculations with the KPL force field model, we employed the molecular dynamics package [Moscito]{} 4.180 and computed the same potential energy surface as a function of the two dihedral angles $\phi_{1}$ and $\phi_{2}$ (see Fig. \[fig:dihe\_sim\] top panel) by fixing the two dihedral angles and optimizing all other degrees of freedom. We would like to add that in the force field-optimizations all bond-lengths were kept fixed.
![Potential energy surface of the \[NTf$_{2}$\] anion computed for the two force field models. The new force field (bottom panel) provides a much better representation of the [*ab initio*]{} calculations shown in Fig. \[fig:abinitio\] than the the original KPL force field (top panel).[]{data-label="fig:dihe_sim"}](FIG04a.eps "fig:"){width="7.0cm"} ![Potential energy surface of the \[NTf$_{2}$\] anion computed for the two force field models. The new force field (bottom panel) provides a much better representation of the [*ab initio*]{} calculations shown in Fig. \[fig:abinitio\] than the the original KPL force field (top panel).[]{data-label="fig:dihe_sim"}](FIG04b.eps "fig:"){width="7.0cm"}
It is quite obvious that the KPL force field does not adequately reproduce the potential energy surface obtained from the quantum chemical calculations (compare the top panel in Fig. \[fig:dihe\_sim\] with FIG \[fig:abinitio\]). The minimum energy conformations of the KPL model reveals essentially two structurally distinct conformations illustrated in Fig. \[fig:kodd\_conf\]. However, both are somewhat similar, being positioned between the [*trans*]{} and [*gauche* ]{}conformations favoured in the [*ab initio*]{} calculations. The fact that the energy landscape does not reflect all the symmetry-features of the molecule, however, might be a lesser problem since energy barriers are rather large and the anion could explore similar conformations simply by rotation.
However, for arriving at a better representation of the [*ab initio*]{} energy surface, we reparameterized the charges as well as the two distinct independent dihedral potentials (S-N-S-C and F-C-S-N), while keeping the other parameters unchanged. From our quantum chemical calculations we yield the global minimum conformations at $\phi_{1}\!=\!\phi_{2}\!=\!90^\circ$ and $\phi_{1}\!=\!\phi_{2}\!=\!270^\circ$. Due to the symmetry of the \[NTf$_{2}$\] anion these two minima are conformationally identical. To calculate the parameters for the S-N-S-C dihedral angle, we fixed $\phi_{1}$ at $90^\circ$ and calculated the energy as function of the dihedral angle $\phi_{2}$ on the MP2 level using a cc-pvtz basis set (as shown in Fig. \[fig:snsc\_pot\]). The same procedure was applied using the KPL force field while switching of the dihedral potential, such that only the nonbonding (nb) interactions matter. We then subtracted the latter energy function from the energies obtained via the QM calculations, and arrive at the dihedral potential for the dihedral angle S-N-S-C, which should be reproduced by the torsion potential in our force field (see Fig. \[fig:snsc\_pot\] bottom panel).
In contrast to Köddermann [*et al.*]{}, we chose to fit a dihedral potential function obeying the conformational symmetry-features of the anion using $$V_{\kappa \lambda \omega \tau}^{\rm dp}=\sum_{n}k_{m}^{\rm dp}[1 + \cos({m_{n}\psi_{m}-\psi^{0}_{m})] }$$ (with $n\!=\!6$ and $\psi^{0}_{m}\!=\!0$) to the computed [*ab initio*]{} potential, leading to the proper minimum energy conformations of the \[NTf$_2$\] anion [@Padua:2004_2].
![Top panel: Potential energy of the entire \[NTf$_2$\] anion as a function of the of S2-N-S1-C1 dihedral angle $\phi_{2}$ with $\phi_{1}$ being fixed at $\phi_{1}\!=\!90^\circ$. Bottom panel: Torsion potential fitted to the difference between QM and force field model (with switched off torsion potential).[]{data-label="fig:snsc_pot"}](FIG05a.eps "fig:"){width="7.0cm"} ![Top panel: Potential energy of the entire \[NTf$_2$\] anion as a function of the of S2-N-S1-C1 dihedral angle $\phi_{2}$ with $\phi_{1}$ being fixed at $\phi_{1}\!=\!90^\circ$. Bottom panel: Torsion potential fitted to the difference between QM and force field model (with switched off torsion potential).[]{data-label="fig:snsc_pot"}](FIG05b.eps "fig:"){width="7.0cm"}
Similarly obtained were parameters for the F-C-S-N dihedral potential of the terminal $\mbox{C}\mbox{F}_3$-groups (see Fig. \[fig:fcsn\_pot\]). The complete set of new parameters for the NGKPL force field is given in Table \[tab:ntf2\_dieder\]. All charges were computed from the MP2-wavefunction using the method of Merz and Kollman as implemented in in the [Gaussian 09]{} programm. [@Singh:1984]. The refined charges are listed in Table \[tab:ntf2\_ljq\].
Finally, employing new refined parameters for the dihedral potentials and partial charges, we re-calculated the energy surface as a function of the two dihedral angles $\phi_{1}$ and $\phi_{2}$ (see Fig. \[fig:dihe\_sim\] bottom panel). The result is in much better agreement with the [*ab initio*]{} calculations and resolves the conformational mismatch issue for the force field of the \[NTf$_2$\] anion.
All parameters for the new \[NTf$_2$\] anion force field are listed in the Tables \[tab:ntf2\_ljq\]-\[tab:ntf2\_dieder\]. The original parameters as well as the parameters for the cations can be found in the publication of Köddermann [*et al.*]{} [@Koeddermann:2007].
![Potential energy of the \[NTf$_2$\] anion as function of the F-C-S-N dihedral angle.[]{data-label="fig:fcsn_pot"}](FIG06.eps){width="7.0cm"}
site $\sigma$ / Å $\epsilon$ / K $q$ / e
------ -------------- ---------------- ---------
F 2.6550 8.00 -0.189
C 3.1500 9.96 0.494
S 4.0825 37.73 1.076
O 3.4632 31.70 -0.579
N 3.2500 25.66 -0.690
: [Lennard-Jones]{} parameters $\sigma$, $\epsilon$ and charges $q$ for all interaction sites of the \[NTf${_2}$\] anion.
\[tab:ntf2\_ljq\]
-----------------------------------------------------------------------------------------------------------------------------------------
bond $r^0_{\kappa \lambda}$ / Å angle $\phi^0_{\kappa $k^{\rm a}_{\kappa \lambda \omega}$ / kJ mol$^{-1}$rad$^{-2}$
\lambda \omega}$ / $^\circ$
------ ---------------------------- ------- ----------------------------- ---------------------------------------------------------------
C-F 1.323 F-C-F 107.1 781.0
C-S 1.818 S-C-F 111.8 694.0
S-O 1.442 C-S-O 102.6 870.0
N-S 1.570 O-S-O 118.5 969.0
O-S-N 113.6 789.0
C-S-N 100.2 816.0
S-N-S 125.6 671.0
-----------------------------------------------------------------------------------------------------------------------------------------
: Bond length $r^0_{\kappa \lambda}$ and angle parameters $\phi^{0}_{\kappa
\lambda \omega}$ and $k^{\rm a}_{\kappa \lambda \omega}$ for the angle potential $V_{\kappa \lambda \omega}^{\rm a}\!=\! \tfrac{1}{2}
k^{\rm a}_{\kappa \lambda \omega}(\phi_{\kappa \lambda \omega} -
\phi^{0}_{\kappa \lambda \omega})^{2}$ in the force field of the \[NTf$_2$\] anions.
\[tab:ntf2\_bond\]
$n(\kappa \lambda \omega \tau)$ $m_{n}$ $k^{\rm dp}_{m}$ / kJ mol$^{-1}$ $\psi^{0}_{m}$ / $^\circ$
--------- --------------------------------- --------- ---------------------------------- ---------------------------
F-C-S-N 1 3 2.0401 0.0
S-N-S-C 1 1 23.7647 0.0
2 2 6.2081 0.0
3 3 -2.3684 0.0
4 4 -0.0298 0.0
5 5 0.6905 0.0
6 6 1.0165 0.0
: Parameters $k^{\rm dp}_{m}$ and $\psi^{0}_{m}$ for the torsion potential $V_{\kappa \lambda \omega \tau}^{\rm dp}
=
\sum_{n}k_{m}^{\rm dp}[1 + \cos({m_{n}\psi_{m}-\psi^{0}_{m})] }$ in the force field of the \[NTf$_2$\] anion.
\[tab:ntf2\_dieder\]
Molecular Dynamics Simulations
==============================
We performed MD simulations for the two force fields KPL and NGKPL with [Gromacs 5.0.6]{} [@Lindahl:2001; @Berendsen:1995; @Hess:2008; @Pronk:2013; @Gromacs_5.0.6] over a temperature range from $T\!=\!273$ – $483\,\mbox{K}$ to calculate thermodynamical and dynamical properties and compare them with the original KPL force field. All simulations were carried out in the $NpT$ ensemble. However, to compute viscosities, we performed additional $NVT$ simulations using starting configurations sampled along the $NpT$-trajectory. Periodic boundary conditions where applied using cubic simulation boxes containing 512 ion-pairs. We applied smooth particle mesh [Ewald]{} summation [@Essmann:1995] for the electrostatic interactions with a real space cutoff of $0.9\,\mbox{nm}$, a mesh spacing of $0.12\,\mbox{nm}$ and 4th order interpolation. The [Ewald]{} convergence factor $\alpha$ was set to $3.38\,\mbox{nm$^{-1}$}$ (corresponding to a relative accuracy of the [Ewald]{} sum of $10^{-5}$). All simulationa were carried out with a timestep of $2.0\,\mbox{fs}$, while keeping bond lengths fixed using the LINCS algorithm [@Hess:1997].
An initial equilibration was done for $2\,\mbox{ns}$ at $T\!=\!500\,\mbox{K}$ applying [Berendsen]{} thermostat as well as [Berendsen]{} barostat with coupling times $\tau_\textrm{T}\!=\!\tau_\textrm{p}\!=\!0.5\,\mbox{ps}$ [@Berendsen:1984]. After this another equilibration was done for $2\,\mbox{ns}$ at each of the desired temperatures. For each of the six temperatures 273K, 303K, 343K, 383K, 423K and 483K we performed production runs of $30\,\mbox{ns}$, keeping the the pressure fixed at $1\,\mbox{bar}$ applying [Nosé-Hoover]{} thermostats [@Nose:1984; @Hoover:1985] with $\tau_\textrm{T}\!=\!1\,\mbox{ps}$ and [Rahman-Parrinello]{} barostats [@Parrinello:1981; @Nose:1983] with $\tau_\textrm{p}\!=\!2\,\mbox{ps}$.
Results & Discussion
====================
Analogous to the publication of Köddermann [*et al.*]{} from 2007 [@Koeddermann:2007] we will compare densities, self-diffusion coefficients and vaporization enthalpies for \[C$_n$MIm\]\[NTf$_2$\] as function of temperature and alkyl chain-length as well as viscosities and reorientational correlation times for \[C$_2$MIm\]\[NTf$_2$\] as function of temperature. It is important to keep in mind, that the original force field was optimized to reproduce these properties and yields a good agreement between experiment and simulation. By resolving the mismatch of the favored conformations of the \[NTf$_2$\] anion we are able to describe these properties as good as the KPL force field or even better.
Structural Features
-------------------
Here we take a look at structural features of the liquid phase and in how they are influenced by changes in the conformation-population of the \[NTF$_2$\] anion. First we inspect the three distinct center of mass pair distribution functions between the different ions computed for \[C$_n$MIm\]\[NTf$_2$\] with $n=2$ at $T\!=\!303\,\mbox{K}$ (shown in Fig. \[fig:grcom\]). It is quite apparent that these distribution functions are only slightly affected by the alterations in the force field. Most notable are the differences observed in the anion-anion pair distribution function depicted in Fig. \[fig:grcom\]c with the first peak being significantly broadened. It is quite obvious to assume that this behavior is related to the more distinct conformational states ([*trans*]{} and [*gauche*]{}) that the reparameterized \[NTf$_2$\] anion is adopting as shown in Fig. \[fig:minimum\_conf\]. In the [*trans*]{} state the molecule is more elongated along the molecular axis and more compact perpendicular to it. In addition, the gauche-state is generally more compact than the minimum energy conformations adopted by the original KPL force field model shown in Fig. \[fig:kodd\_conf\]. This leads to an enhanced population of both, short and long anion-anion distances. This effect manifests itself also in the slight shift of the maximum of the first peak of the anion-cation pair distribution function towards smaller distances (see Fig. \[fig:grcom\]a).
{width="15cm"}
Another interesting distribution function is the pair distribution function of the anion-oxygens surrounding the C(2)-hydrogen site on the cation. The C(2)-position is deemed to act as a hydrogen-bond donor [@Fumino:2008; @Wulf:2010]. With changing conformations we expect an effect on the hydrogen bonding situation between the anion and cation. Here we observe that the NGKPL force field promotes hydrogen bonds between anions and cations as indicated by an increased first peak of the O-H pair distribution function shown in Fig. \[fig:groh\]. The computed number of hydrogen bonds increases throughout by about $4\,\%$, mostly unaffected by the alkyl chain-length and temperature (not shown). Taking into account the importance of more elongated [*trans*]{} configurations of the anion, it is also not surprising that the second peak is somewhat depleted, while the third peak is again enhanced (see Fig. \[fig:groh\]).
![Radial pair distribution function of the anion-oxygens around the C(2) hydrogen site on the cation for \[C$_2$MIm\]\[NTf$_2$\] at $T=303\,\mbox{K}$. The NGKPL force field is shown in the blue line and the KPL force field in the red dashed line.[]{data-label="fig:groh"}](FIG08.eps){width="7.0cm"}
We further investigate the hydrogen-bond situation by not just looking at the distance between the oxygen and hydrogen, but also at the angular distribution. Therefore we compute the probability density map of the anion-oxygens surrounding the C(2) hydrogen site on the cation. Again we focus on the C(2) hydrogen because its hydrogen-bond interaction with the anion is deemed the strongest and most important. To calculate this map we compute both, the O-H distance as well as angle between the C-H bond-vector on the cation and the intermolecular C-O vector, where C is the C(2)-position of the cation and O represents the oxygen-sites on the anions. In addition, the computed probabilities are weighted by $r^{-2}_\textrm{OH}$. It is revealed that the maximum of this probability density map does not quite represent a linear hydrogen bond at a distance of $2.3\,\mbox{\AA}$, but is tilted by about $25^\circ$, and is characterized by a rather broad angular distribution. (Fig. \[fig:histo\]).
![Probability density of the anion-oxygens around the C(2) hydrogen sites as function of the intermolecular distance $r_\textrm{OH}$ and the angle between the C(2)-H bond-vector and the intermolecular C(cation)-O(anion) vector. Shown for the NGKPL force field at $T=303\,\mbox{K}$.[]{data-label="fig:histo"}](FIG09.eps){width="8.0cm"}
Densities & Self-Diffusion Coefficients
---------------------------------------
To get an idea on how the changing conformation-populations influence the properties of the imidazolium based ionic liquids, we first take a look at the mass density of \[C$_2$MIm\]\[NTf$_2$\]. In molecular simulations the density has always been an important property for evaluating a force field. The enhanced conformational diversity of the \[NTf$_2$\] anion leads to a slight increase in the density over the whole temperature range (see Fig.\[fig:density\]). This overall increase is in better agreement with the experimental data from Tokuda [*et al.*]{} [@Tokuda:2005]. For lower temperatures the NGKPL force field even matches the experimental values. The thermal expansivity, however, is significantly overestimated, although at the highest temperatures the difference between experiment and simulation is still within about $5\;\%$. Despite the overall density increase from KPL to NGKPL, the thermal expansivities of both models are practically identical.
![Mass densities of \[C$_2$MIm\]\[NTf$_2$\] as function of temperature. The experimental data of Tokuda [*et al.*]{} is given according to their fitted temperature dependence (green dashed line) [@Tokuda:2005]. The results from our molecular dynamics simulation using the NGKPL (blue dots) and KPL (red squares) force fields were fitted with a linear function represented by the dashed lines. See also Table \[tab:density\_diffu\_c2mim\].[]{data-label="fig:density"}](FIG10.eps){width="7.0cm"}
With this increasing density, also slightly reduced self-diffusion coefficients for the \[NTf$_2$\] anion are observed (see Fig. \[fig:diffusion\]). We calculated the self-diffusion coefficient using the [Einstein]{} relation $$\begin{aligned}
D=\frac{1}{6}\lim\limits_{t \to \infty}\frac{d}{dt}\left\langle \left| \vec{r}{_i}(t) - \vec{r}{_i}(0)\right|^2 \right\rangle\end{aligned}$$ as function of the temperature for \[C$_2$MIm\]\[NTf$_2$\] (Fig. \[fig:diffusion\]) as well as as function of the alkyl chain-length of \[C$_n$MIm\]\[NTf$_2$\] at $T=303\,\mbox{K}$ (Fig. \[fig:diffusion\_2\]).
![Self-diffusion coefficients as function of the temperature for \[C$_{2}$MIm\]\[NTf$_2$\]. The experimental data of Tokuda [*et al.*]{} are represented according to their fitted temperature dependence (green dashed line) [@Tokuda:2005]. The red squares (KPL) and blue dots (NGKPL) represent the results from our molecular dynamics simulations. See also Table \[tab:density\_diffu\_c2mim\].[]{data-label="fig:diffusion"}](FIG11.eps){width="7.0cm"}
As shown in 2007 the KPL force field is able to yield self-diffusion coeffients in good agreement with the experimental data. Nevertheless, using the new NGKPL parameters we are able to describe the temperature dependence of the self-diffusion coefficient of the \[NTf$_2$\] anion in \[C$_2$MIm\]\[NTf$_2$\] even better (Fig. \[fig:diffusion\]).
![Self-diffusion coefficients as a function of the alkyl chain-length for \[C$_n$MIm\]\[NTf$_2$\] at $T=303\,\mbox{K}$. The experimental data are shown as green triangles, the KPL force field as red squares and the NGKPL force field as blue dots. The dashed lines are only guides for the eye. See also Table \[tab:diffu\_vap\].[]{data-label="fig:diffusion_2"}](FIG12.eps){width="7.0cm"}
Taking a look at the alkyl chain-length dependence we can support the findings for the $n=2$ imidazolium ionic liquid. The NGKPL force field is able to reproduce the dependence better, especially for $n\leq4$, for longer chains the KPL force field is closer to the experiment (see Fig. \[fig:diffusion\_2\]). As observed for the temperature dependence the general trend of the self-diffusion coefficient as function of the alkyl chain-length is identical for the KPL and NGKPL force field.
--------- ------ ------- ------ -------
$T$ / K KPL NGKPL KPL NGKPL
273 1525 1540 1.4 1.1
303 1485 1500 4.8 3.6
343 1433 1448 13.0 10.9
383 1383 1398 26.4 23.0
423 1335 1349 49.4 43.6
483 1265 1280 92.7 90.0
--------- ------ ------- ------ -------
: Temperature dependence of the density $\rho$ and the self-diffusion coefficients of the \[NTf$_{2}$\] anion $D_{-}$ in \[C$_{2}$MIm\]\[NTf$_{2}$\] according to the KPL and NGKPL force fields. See also Fig. \[fig:density\] and Fig. \[fig:diffusion\].
\[tab:density\_diffu\_c2mim\]
Vaporization Enthalpies
-----------------------
The magnitude of the vaporization enthalpy of ionic liquids was studied extensively over the last few years and has been sometimes discussed quite emotionally [@Zaitsau:2006; @Armstrong:2007; @Heintz:2007; @Santos:2007; @Luo:2008; @Heym:2011; @Rocha:2011; @Verevkin:2013; @Schroeder:2014]. For the purpose of this study we will compare our results with the more recent QCM data of imidazolium based ILs of type \[C$_n$MIm\]\[NTf$_2$\] from Verevkin [*et al.*]{} of 2013 [@Verevkin:2013] as shown in Fig. \[fig:vH\]. We would like to point out that an exhaustive overview of the huge amount of vaporization enthalpy data from different experiments as well as molecular simulation studies is provided in the supporting informations of Verevkin [*et al.*]{} [@Verevkin:2013] and in the COSMOS-RS study by Schröder and Coutinho [@Schroeder:2014]. The vaporization enthalpies per mol of \[C$_n$MIm\]\[NTf$_2$\] were here calculated by assuming ideal gas behavior with $$\Delta_{\rm v}H\approx\Delta_{\rm v}U + RT\;,
\label{eq:dvh1}$$ which is a well justified approximation, given the low vapor pressures of ILs at low temperatures. The energy difference between the liquid and gas phases were computed via $$\Delta_{\rm v}U=U'_{\rm g} - U'_{\rm l}
\label{eq:dvh2}$$ where $U'_{\rm l}$ and $U'_{\rm g}$ are the internal energies per mol ion-pairs of the liquid and gas phases, respectively. To determine $U'_{\rm g}$ we performed gas phase simulations of individual ion-pairs without periodic boundary conditions. It has been shown in the literature that the gas phase of ionic liquids consists mostly of ion-pairs [@Zaitsau:2016; @Verevkin:2011; @Verevkin:2013; @Zaitsau:2012; @Verevkin:2012; @Ahrenberg:2014; @Verevkin:2012_2; @Boeck:2014] tied together by strong long-range electrostatic forces. Hence, simulating an isolated ion-pair instead of separated ions is the most realistic approximation of the IL gas phase. As it is standard practice, during the simulation of of both, the liquid phase and also of the isolated ion-pair, the total linear momentum was set to zero, thus eliminating the systems center of mass translational motion. In addition, in the simulations of the isolated ion-pairs also the total angular momentum was set to zero. However, when comparing the internal energy of the gas phase and the liquid phase, we have to correct for differences in the kinetic energy stored in the translational/rotation motion of either system by adding $$\begin{aligned}
U'_{\rm g} &=&U_{\rm g} + \frac{6}{2}RT \\
U'_{\rm l} &=&U_{\rm l} + \frac{3}{2}RT\times \frac{1}{N_\textrm{IP}} \end{aligned}$$ per mole of ion-pairs, where $N_\textrm{IP}=512$ is the number of ion-pairs used in the liquid simulation, and $U_{\rm g}$ and $U_{\rm g}$ are the total energies per ion-pair as computed directly from the MD simulations. With these corrected molar internal energies $U'_{\rm g}$ and $U'_{\rm l}$ we compute the heat of vaporization $\Delta_{\rm v}H$ using Eq. \[eq:dvh1\] for a temperature of $T=303\,\mbox{K}$ shown in Fig. \[fig:vH\] and given in Table \[tab:diffu\_vap\].
Both the data computed from the KPL and from the NGKPL force field as a function of alkyl chain-length are rather close to the experimental data of of Verevkin [*et al.*]{} [@Verevkin:2013].
![Vaporization enthalpies as function of the alkyl chain-length for NGKPL (blue dots) KPL (red squares) at $T=303\,\mbox{K}$. For comparison we also show the QCM data of Verevkin [*et al.*]{} for $T=298\,\mbox{K}$ (green triangles) [@Verevkin:2013]. See also Table \[tab:diffu\_vap\].[]{data-label="fig:vH"}](FIG13.eps){width="7.0cm"}
----- ------ ------- ------- -------
$n$ KPL NGKPL KPL NGKPL
1 5.21 3.87 121.6 119.9
2 4.76 3.56 122.6 119.4
4 3.22 2.42 128.5 126.0
6 1.83 1.26 137.5 135.5
8 1.00 0.67 148.5 146.9
----- ------ ------- ------- -------
: MD simulated self-diffusion coefficients of the \[NTf$_{2}$\] anion $D_{-}$ as well as vaporization enthalpies $\Delta_{\rm v}H$ as a function of the alkyl chain-length $n$ in \[C$_{n}$MIm\]\[NTf$_{2}$\] for the KPL and the new NGKPL force field. See also Fig. \[fig:diffusion\_2\] and Fig. \[fig:vH\].
\[tab:diffu\_vap\]
However, we would like to point out, that the optimized NGKPL force field is in even better agreement with the QCM experiments, particularly for chain-lengths up to $n=4$. Not only are the data for $n=2$ now in quantitative agreement with the experimental data, but also the step from $n=1$ to $n=2$ is better captured by the new model, suggesting a significant influence of the enhanced conformational diversity of the \[NTf$_{2}$\] anion [@Fumino:2010]. Since the exact slope of $\Delta_{\rm v}H$ as a function of the alkyl chain-length has been shown to be controlled by the counterbalance of electrostatic and [van der Waals]{} forces [@Koddermann:2008], the increasing deviation for longer chain-length might indicate a slight misrepresentation of size of the dispersion interaction introduced by increasing the alkyl chain-length.
Viscosities & Reorientational Correlation Times
-----------------------------------------------
To further compare dynamical properties of the simulated ionic liquids with experimental data, the temperature dependence of the reorientational correlation times for the C(2)-H vector and viscosities for \[C$_2$MIm\]\[NTf$_2$\] where calculated. To compare with the quadrupolar relaxation experiments of Wulf [*et al.*]{} [@Wulf:2007] we computed reorientational correlation functions $R(t)$ of the C(2)-H bond-vector according to $$R(t) = \left< P_2\{\cos[\theta_\mathrm{CH}(t)]\} \right>,$$ where $P_2$ is the second [Legendre]{} polynomial and $$\cos[\theta_\mathrm{CH}(t)] =
\frac{\vec{r}_\mathrm{CH}(0) \cdot\vec{r}_\mathrm{CH}(t)}{|\vec{r}_\mathrm{CH}|^2}$$ represents the angle-cosine between the CH-bond vector at times “0” and $t$ and $|\vec{r}_\mathrm{CH}|$ is the CH-bond length, which is kept fixed during the simulation. The reorientational correlation times $\tau_\mathrm{c}$ are obtained as integral over the correlation function $$\tau_\mathrm{c} = \int\limits_0^\infty R(t)\,dt\,.$$ Here, the long-time behavior is fitted to a stretched exponential function and the total correlation time is determined by numerical integration.
![Reorientational correlation time of the C(2)-H vector in \[C$_{2}$MIm\]\[NTf$_{2}$\] as function of temperature. The experimental data of Wulf et al. [@Wulf:2007] are shown as green triangles, KPL-data as red squares, and NGKPL-data as blue dots. The data are summarized in Table \[tab:visc\_tau\_c2mim\].[]{data-label="fig:tau"}](FIG14.eps){width="7.0cm"}
Again we find that both force fields are in good agreement with the experimental values, albeit with the original KPL model being slightly closer to the experimental data (see Fig. \[fig:tau\]).
To determine the viscosities we used the approach of Zhang [*et al.*]{} [@Maginn:2015] to compute viscosities from equilibrium-fluctuations of the off-diagonal elements of the pressure tensor via the [Green-Kubo]{} relation $$\begin{aligned}
\eta = \frac{V}{k_{\rm B}T} \int^{\infty}_{0} \left< P_{\alpha\beta}(0)\cdot P_{\alpha\beta}(t)\right> dt.\end{aligned}$$ For each temperature we performed $15$ independent $NVT$ simulations, where the starting configurations where sampled from the earlier $NpT$ simulations with a constant time interval of $2\,\mbox{ns}$. After a $1\,\mbox{ns}$ equlibration we computed $8\,\mbox{ns}$ long productions runs for each of the sampled configurations storing the pressure tensor data for each time-step. Finally, the correlation function was calculated and integrated over a time-window of $1\,\mbox{ns}$ for each of the $15$ simulations. The average of the running integrals was calculated as well as standard deviation. The average over the running integrals as well as the standard deviation where handled as suggested by Zhang [*et al.*]{} [@Maginn:2015] with a fitting cut off $t_{\rm cut}$ at the point where $\sigma(t)$ is $40\,\%$ of the calculated average viscosity.
We find that the differences between the KPL and NGKPL models to be rather small. Both are basically lying within the statistical errors of this method. However, both force field model yield viscosities very close to the experiment (Fig. \[fig:vis\]).
![Viscosities as function of temperature for \[C$_{\rm 2}$MIm\]\[NTf$_2$\] for NGKPL (blue dots) and KPL (red squares). The experimental data was taken from Tokuda [*et al.*]{} (green dashed line) [@Tokuda:2005]. See also Table \[tab:visc\_tau\_c2mim\].[]{data-label="fig:vis"}](FIG15.eps){width="7.0cm"}
--------- --------------- ----------------- ------- -------
$T$ / K KPL NGKPL KPL NGKPL
273 67 $\pm$ 24 82 $\pm$ 22 173.1 114.7
303 26 $\pm$ 6 25 $\pm$ 4 51.6 38.2
343 8.6 $\pm$ 2.0 9.6 $\pm$ 2.4 17.3 14.1
383 4.4 $\pm$ 0.9 4.9 $\pm$ 0.9 8.1 7.0
423 2.9 $\pm$ 1.0 2.9 $\pm$ 0.6 4.7 4.1
483 1.6 $\pm$ 0.3 1.48 $\pm$ 0.21 2.6 2.3
--------- --------------- ----------------- ------- -------
: Viscosities $\eta$ and reorientational correlation times of the C(2)-H vector $\tau_{\rm c}$ as a function of temperature calculated from MD simulations of \[C$_{2}$MIm\]\[NTf$_{2}$\] employing the KPL and the NGKPL force fields. See also Fig. \[fig:tau\] and Fig. \[fig:vis\].
\[tab:visc\_tau\_c2mim\]
Conclusions
===========
We showed that the reparametrization of the dihedral potentials as well as charges of the \[NTf$_2$\] anion leads to an improvment of the force field model of Köddermann [*et al.*]{} for imidazolium based ionic liquids from 2007. The most prominent advantage of the new parameter set is that the minimum energy conformations ([*trans*]{} and [*gauche*]{}) of the anion, as demonstrated from [*ab initio*]{} calculations and [Raman]{} experiments, are now well reproduced.
The results obtained for \[C$_n$MIm\]\[NTf$_2$\] show that this correction leads to a slightly better agreement between experiment and molecular dynamics simulation for a variety of properties, such as densities, diffusion coefficients, vaporization enthalpies, reorientational correlation times, and viscosities. Even though we focused on optimizing the anion parameters, the alkyl chain-length dependence is found to be general also closer to the experiment.
With this work we want to point out that it is important to re-examine established force field and, if necessary, to improve those. We highly recommend to use the new NGKPL force field for the \[NTf$_2$\] anion instead of the original KPL force field. Especially for simulation aiming to describe the thermodynamics, dynamics and also structure of imidazolium based ionic liquids.
Acknowlegements
===============
B.G. is thankful for financial support provided by COST Action CM 1206 (ÒEXIL - Exchange on Ionic LiquidsÓ).
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{
"pile_set_name": "ArXiv"
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---
abstract: 'We report on high resolution spectroscopy of four giant stars in the Galactic old open clusters Berkeley 22 and Berkeley 66 obtained with HIRES at the Keck telescope. We find that $[Fe/H]=-0.32\pm0.19$ and $[Fe/H]=-0.48\pm0.24$ for Berkeley 22 and Berkeley 66, respectively. Based on these data, we first revise the fundamental parameters of the clusters, and then discuss them in the context of the Galactic disk radial abundance gradient. We found that both clusters nicely obey the most updated estimate of the slope of the gradient from @fri02 and are genuine Galactic disk objects.'
author:
- Sandro Villanova
- Giovanni Carraro
- Fabio Bresolin
- Ferdinando Patat
title: |
Metal abundances in extremely distant Galactic old open clusters.\
II. Berkeley 22 and Berkeley 66$^1$
---
Introduction
============
This paper is the second of a series dedicated at obtaining high resolution spectroscopy of distant Galactic old open cluster giant stars to derive new or improved estimates of their metal content. In @car04 we presented results for Berkeley 29 and Saurer 1, the two old open clusters possessing the largest galactocentric distance known, and showed that they do not belong to the disk, but to the Monoceros feature (see also the discussion in @fri05). Here we present high-resolution spectra of four giant stars in the old open clusters Berkeley 22 and Berkeley 66. The latter cluster, in particular, is of particular interest, since its heliocentric distance is among the largest currently known.\
With this paper we aim to enlarge the sample of old open clusters with metallicity obtained from high resolution spectra, and to test the common assumption of axisymmetry made in chemical evolution models about the structure of the Milky Way disk. For this purpose we selected two open clusters with about the same age located one (Berkeley 66) in the second Galactic Quadrant, and the other (Berkeley 22) in the third Galactic Quadrant. Significant differences in metal abundance for clusters located in different disk zones would hopefully provide useful clues about the chemical evolution of the disk and about the role of accretion and infall phenomena.\
The layout of the paper is as follows. Sections 2 and 3 illustrate the observations and the data reduction strategies, while Section 4 deals with radial velocity determinations. In Section 5 we derive the stellar abundances and in Section 6 we revise the cluster fundamental parameters. The radial abundance gradient and the abundance ratios are discussed in Section 7 and 8, respectively. The results of this paper are finally discussed in Section 9
Observations
============
The observations were carried out on the night of November 30, 2004 at the W.M. Keck Observatory under photometric conditions and typical seeing of 11 arcsec. The HIRES spectrograph [@vog94] on the Keck I telescope was used with a 11 x 7 slit to provide a spectral resolution R = 34,000 in the wavelength range 5200$-$8900 Å on the three 2048$\times$4096 CCDs of the mosaic detector. A blocking filter was used to remove second-order contamination from blue wavelengths. Three exposures of 1500–1800 seconds were obtained for the stars Berkeley 22$-$400 and Berkeley 22$-$579. For both Berkeley 66$-$785 and Berkeley 66$-$934 we took four exposures of 2700 seconds each. During the first part of the night, when the telescope pointed to Berkeley 66, the observing conditions were far from optimal, due to the presence of thick clouds. For this reason our abundance analysis in this first cluster is limited to Berkeley 66$-$785. For the wavelength calibration, spectra of a thorium-argon lamp were secured after the set of exposures for each star was completed. The radial velocity standard HD 82106 was observed at the end of the night.\
In Fig. 1 we show a finding chart for the two clusters where the four observed stars are indicated, while in Fig. 2 we show the position of the stars in the Color$-$Magnitude Diagram (CMD), based on published photometry (@kal94, @phe96).
Data Reduction
==============
Images were reduced using IRAF, including bias subtraction, flat-field correction, frame combination, extraction of spectral orders, wavelength calibration, sky subtraction and spectral rectification. The single orders were merged into a single spectrum. As an example, we show in Fig. 3 a portion of the reduced, normalized spectrum of Be 22-400.
Radial Velocities
=================
No radial velocity estimates were previously available for Berkeley 22 and Berkeley 66. The radial velocities of the target stars were measured using the IRAF FXCOR task, which cross-correlates the object spectrum with the template (HD 82106). The peak of the cross-correlation was fitted with a Gaussian curve after rejecting the spectral regions contaminated by telluric lines ($\lambda > 6850$ Å). In order to check our wavelength calibration we also measured the radial velocity of HD 82106 itself, by cross-correlation with a solar-spectrum template. We obtained a radial heliocentric velocity of 29.8$\pm$0.1 km s$^{-1}$, which perfectly matches the published value (29.7 km/sec; @Udr99). The final error in the radial velocities was typically about 0.2 km s$^{-1}$. The two stars we measured in each clusters have compatible radial velocities (see Table 1), and are considered, therefore, [*bona fide*]{} cluster members.
ABUNDANCE ANALYSIS
==================
Atomic parameters and equivalent widths
---------------------------------------
We derived equivalent widths of spectral lines by using the standard IRAF routine [*SPLOT*]{}. Repeated measurements show a typical error of about 5$-$10 mÅ, also for the weakest lines. The line list (FeI, FeII, Mg, Si, Ca, Al,, Na, Ni and Ti, see Table 3) was taken from @fri03, who considered only lines with equivalent widths narrower than 150mÅ, in order to avoid non-linear effects in the LTE analysis of the spectral features. log(gf) parameters of these lines were re-determinated using equivalent widths from the solar-spectrum template, solar abundances from @and89 and standard solar parameters ($T_{eff}=5777~K, log(g)=4.44, v_t=0.8~Km\,
s^{-1}$).
Atmospheric parameters
----------------------
Initial estimates of the atmospheric parameter $T_{eff}$ were obtained from photometric observations in the optical. BVI data were available for Berkeley 22 [@kal94], while VI photometry for Berkeley 66 has been taken from [@phe96]. Reddening values are E(B$-$V)= 0.62 (E(V$-$I)= 0.74), and E(V$-$I)= 1.60, respectively. First guess effective temperatures were derived from the (V$-$I)–$T_{eff}$ and (B$-$V)–$T_{eff}$ relations, the former from @alo99 and the latter from @gra96. We then adjusted the effective temperature by minimizing the slope of the abundances obtained from Fe I lines with respect to the excitation potential in the curve of growth analysis. For both clusters the derived temperature yields a reddening consistent with the photometric one.\
Initial guesses for the gravity $\log$(g) were derived from:
$$log(\frac{g}{g_{\odot}}) = log(\frac{M}{M_{\odot}}) + 4 \times
log(\frac{T_{eff}}{T_{\odot}}) - log(\frac{L}{L_{\odot}})$$
taken from @carre97. In this equation the mass $\frac{M}{M_{\odot}}$ was derived from the comparison between the position of the star in the Hertzsprung$-$Russell diagram and the Padova Isochrones [@gir00]. The luminosity $\frac{L}{L_{\odot}}$ was derived from the the absolute magnitude $M_V$, assuming the literature distance moduli of 15.9 for Berkeley 22 [@kal94] and 17.4 for Berkeley 66 [@phe96]. The bolometric correction (BC) was derived from the relation BC–$T_{eff}$ from @alo99. The input $\log$(g) values were then adjusted in order to satisfy the ionization equilibrium of Fe I and Fe II during the abundance analysis. Finally, the micro$-$turbulence velocity is given by the following relation [@gra96]:
$$v_t[km\,s^{-1}] = 1.19 \times 10^{-3} \times T_{eff} - 0.90 \times
log(g) - 2$$
The final adopted parameters are listed in Table 2.
Abundance determination
-----------------------
The LTE abundance program MOOG (freely distributed by Chris Sneden, University of Texas, Austin) was used to determine the metal abundances. Model atmospheres were interpolated from the grid of Kurucz (1992) models by using the values of $T_{eff}$ and $\log$(g) determined as explained in the previous section. During the abundance analysis $T_{eff}$, $\log$(g) and $v_t$ were adjusted to remove trends in excitation potential, ionization equilibrium and equivalent width for Fe I and Fe II lines. Table 3 contains the atomic parameters and equivalent widths for the lines used. The first column contains the name of the element, the second the wavelength in Å, the third the excitation potential, the fourth the oscillator strength $\log$ ([*gf*]{}), and the remaining ones the equivalent widths of the lines for the observed stars.\
The derived abundances are listed in Table 4, together with their uncertainties. The measured iron abundances are \[Fe/H\]=$-$0.32$\pm0.19$ and \[Fe/H\]=$-$0.48$\pm0.24$ for Berkeley 22 and Berkeley 66, respectively. The reported errors are derived from the uncertainties on the single star abundance determination (see Table 4). For Berkeley 66$-$934 no abundance determination was possible because the S/N ratio in our spectrum was too low to perform any equivalent width determination.\
Finally, using the stellar parameters (colors, T$_{eff}$ and log(g)) and the absolute calibration of the MK system (@str81), we derived the stellar spectral classification, which we provide in Table 1.
REVISION OF CLUSTER PROPERTIES
==============================
Our study is the first to provide spectral abundance determinations of stars in Berkeley 22 and Berkeley 66. Here we briefly discuss the revision of the properties of these two clusters which follow from our measured chemical abundances (see Figs. 4 and 5). We use the isochrone fitting method and adopt for this purpose the Padova models from [@gir00].
Berkeley 22
-----------
Berkeley 22 is an old open cluster located in the third Galactic quadrant, first studied by @kal94. On the basis of deep VI photometry he derived an age of 3 Gyr, a distance of 6.0 Kpc and a reddening E(V$-$I)=0.74. The author suggests that the probable metal content of the cluster is lower than solar. Here we obtained \[Fe/H\]=$-$0.32, which corresponds to Z=0.008 and roughly half the solar metal content. Very recently @dif05 presented new BVI photometry, on the basis of which they derive a younger age (2.0-2.5 Gyr), but similar reddening and distance, also suggesting that the cluster posseses solar metal abundance. The two photometric studies are compatible in the VI filters, being the difference in the V and I zeropoints less than 0.03 mag. Both the studies show that the cluster turnoff is located at V = 19 and a clump at V = 16.65.\
According to @car94, with a $\Delta V \approx 2.35$ mag ($\Delta V$ being the magnitude difference between the turnoff and the clump) and for the derived metallicity, one would expect an age around 3.5 Gyr. This preliminary estimate of the age is in fact confirmed by the isochrone fitting method. In Fig. 4 the solid line is a 3.3 Gyr isochrone, which provides a new age estimate of 3.3$\pm$0.3 Gyrs for the same photometric reddening and Galactocentric distance derived by [@kal94]. Both the turnoff and the clump location are nicely reproduced. The uncertainty reflects the range of isochrones which produce an acceptable fit.\
For comparison, in the same figure we over-impose a solar metallicity isochrone for the age of 2.25 Gyr (dashed line in Fig. 4), and for the same reddening and distance reported by @dif05. This isochrone clearly does not provide a comparable good fit. When trying to fit the turnoff, both the color of the Red Giant Branch and the position of the clump cannot be reproduced.
Berkeley 66
-----------
Berkeley 66 was studied by @gua97 and @phe96, who suggested a reddening E(V$-$I)=1.60 and a metallicity in the range $-0.23 \leq [Fe/H] \leq 0.0$. By assuming these values, @phe96 derived a galactocentric distance of 12.9 Kpc and an age of 3.5 Gyrs. We obtain a significantly smaller abundance value \[Fe/H\]=$-$0.48$\pm$0.24 for a spectroscopic reddening of E(V$-$I)=1.60, which is consistent with the @phe96 estimate. By looking at the CMD, the turnoff is situated at V = 20.75, whereas the clump is at V = 18.25, implying a $\Delta V$ of 2.5. For this $\Delta V$ and the derived \[Fe/H\], the @car94 calibration yields an age of about 4.7 Gry, significantly larger than previous estimate. The value \[Fe/H\]=$-$0.48 tanslates into Z=0.006, and we generated a few isochrones for this exact metal abundance from [@gir00]. The result is shown in Fig 5, where a 4 and 5 Gyr isochrones (dashed and solid line, respectively) are overimposed to the cluster CMD. Both the isochrones reasonably reproduce the turnoff shape but the 5 Gyr one (solid line) clearly presents a too bright clump. On the other hand, the 4 Gyr isochrone reproduces well all the CMD feature. This new age estimate provides a reddening E(V-I) = 1.60 and a heliocentric distance of 5 Kpc.
THE RADIAL ABUNDANCE GRADIENT
=============================
In Fig. 6 we plot the open cluster Galactic radial abundance gradient, as derived from @fri02, which is at present the most updated version of the gradient itself. The clusters included in their work (open squares) define an overall slope of $-$0.06$\pm$0.01 dex Kpc$^{-1}$ (solid line). The filled circles represent the two clusters analyzed here, and it can be seen that they clearly follow very nicely the general trend. In fact the dashed line, which represents the radial abundance gradient determined by including Berkeley 22 and Berkeley 66, basically coincides with the @fri02 one.\
We note that the gradient exhibits quite a significant scatter. One may wonder whether this solely depends on observational errors, or whether this scatter reflects a true chemical inhomogeneity in the Galactic disk.\
It must be noted however that between 10 and 14 kpc the scatter increases and the distribution of the cluster abundances is compatible also with a flat gradient. Again, it is extremely difficult to conclude whether this behaviour of the gradient is significant, or whether the distribution of abundances is mostly affected by the size of the observational errors and the small number of clusters involved.
ABUNDANCE RATIOS
================
Abundance ratios constitute a powerfull tool to assign a cluster to a stellar population @fri03. In PaperI (@car04) we found that Saurer 1 and Berkeley 29 exhibit enhanced abundance ratio with respect to the Sun, and we concluded that they probably do not belong to the Galactic disk, since all the old open cluster for which detailed abundance analysis is available show solar scaled abundance ratios.\
In Table 5 we list the abundance ratios for the observed stars in Berkeley 22 and Berkeley 66. Our program clusters have ages around 3-4 Gyrs and iron metal content \[Fe/H\] $\approx$ $-$0.3-0.4. They are therefore easily comparable with similar clusters from the literature, such as Tombaugh 2 and Melotte 66 (see @fri03, Tab. 7). We note that the latter two clusters and our program clusters have scaled solar abundances. Indeed, at a similar \[Fe/H\] and age , Berkeley 22 and Melotte 66 have similar values for all the abundance ratios.\
Similar conclusions can be drawn for Berkeley 66, when compared with Tombaugh 2, an old open cluster of similar age and \[Fe/H\]. All the abundance ratios we could measure are comparable in these two clusters.\
These results therefore indicate that Berkeley 22 and Berkeley 66 are two genuine old disk clusters, which well fit in the overall Galactic radial abundance gradient (see previous section).
DISCUSSION and CONCLUSIONS
==========================
Berkeley 22 and Berkeley 66 are two similar age open clusters located roughly at symmetric positions with respect to the virtual line connecting the Sun to the Galactic center. In fact for Berkeley 22 we derive +5.5, $-$2.0 and $-$0.8 Kpc for the rectangular Galactic coordinate X, Y and Z, respectively, while for Berkeley 66 we obtain +4, +2.5 and 0.01 Kpc. The corresponding Galactocentric distances $R_{GC}$ are 12.7 and 14.2 Kpc for Berkeley 66 and Berkeley 22, respectively.\
Within the errors the two clusters possess the same metal abundance, suggesting that at the distance of 12-14 Kpc from the Galactic Center the metal distribution over the second and third quadrant of the Galaxy is basically the same.\
It is worthwhile to point out here that only 3 clusters are insofar known to lie outside the 14 Kpc-radius ring from the Galactic center: Berkeley 20, Berkeley 29 and Saurer 1. @car04 showed that both Berkeley 29 and Saurer 1 do not belong to the disk, and therefore we are probably sampling here the real outskirts of the Galactic stellar disk.\
The axisymmetric homogeneity is confirmed when we add a few more old open clusters located in this strip, like NGC 1193, NGC 2158, NGC 2141, Berkeley 31, Tombaugh 2 and Berkeley 21 (@fri02). All these clusters are located in the second and third quadrant, have metallicities $ -0.62 \leq [Fe/H] \leq -0.25$ and probably belong all to the same generation (ages between 2 and 4 Gyr). Therefore, although with a significant spread, we conclude that at 12-14 Kpc the disk is chemically homogeneous in \[Fe/H\] within the observational errors. The scatter, if real, may be due to local inhomogeneities.
The work of GC is supported by [*Fundacion Andes*]{}. We thank the anonymous referee for his/her report which helped a lot to improve on the paper presentation.
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[lccccccccc]{} Be 22-400 & 05:58:30.97 & +07:46:15.3 & 16.70 & 1.58 & 1.78 & 93.3$\pm$0.2 & 25 & G8III & @kal94\
Be 22-579 & 05:58:25.78 & +07:45:31.2 & 16.88 & 1.66 & 1.80 & 97.3$\pm$0.2 & 20 & K0III & @kal94\
Be 66-785 & 03:04:02.90 & +58:43:57.0 & 18.23 && 2.64 & -50.7$\pm$0.1 & 15 & K1III & @phe96\
Be 66-934 & 03:04:06.41 & +58:43:31.0 & 18.23 && 2.64 & -50.6$\pm$0.3 & 5 & K1III & @phe96\
[lccc]{} Be 22-400 & 4790$\pm$100 & 2.8$\pm$0.1 & 1.3\
Be 22-579 & 4690$\pm$50 & 2.8$\pm$0.1 & 1.2\
Be 66-785 & 4640$\pm$100 & 2.7$\pm$0.1 & 1.2\
Be 66-934 & & &\
[lcccccc]{} Fe I & 5379.570 & 3.680 & -1.57 & 83 & 102 &\
Fe I & 5417.033 & 4.415 & -1.45 & 45 & 44 & 54\
Fe I & 5466.988 & 3.573 & -2.24 & 69 &&\
Fe I & 5633.946 & 4.990 & -0.23 & 68 & 86 &\
Fe I & 5662.520 & 4.160 & -0.59 & 106 & 111 &\
Fe I & 5701.550 & 2.560 & -2.19 & 125 & 113 & 146\
Fe I & 5753.120 & 4.240 & -0.76 & 102 & 108 & 94\
Fe I & 5775.081 & 4.220 & -1.17 & 96 & 77 & 82\
Fe I & 5809.218 & 3.883 & -1.67 & 80 & 83 &\
Fe I & 6024.058 & 4.548 & +0.03 & 124 & 108 & 104\
Fe I & 6034.036 & 4.310 & -2.35 & 58 &&\
Fe I & 6056.005 & 4.733 & -0.48 & 84 & 82 & 71\
Fe I & 6082.72 & 2.22 & -3.62 & 77 & 76 &\
Fe I & 6093.645 & 4.607 & -1.38 & 47 &&\
Fe I & 6096.666 & 3.984 & -1.82 & 47 & 46 &\
Fe I & 6151.620 & 2.180 & -3.34 & 63 & 103 & 81\
Fe I & 6165.360 & 4.143 & -1.53 & 47 & 66 &\
Fe I & 6173.340 & 2.220 & -2.90 & 115 & 101 & 104\
Fe I & 6200.313 & 2.608 & -2.38 & 124 & 102 & 81\
Fe I & 6229.230 & 2.845 & -2.93 & 82 &&\
Fe I & 6246.320 & 3.590 & -0.77 & 119 &&\
Fe I & 6344.15 & 2.43 & -2.90 & 116 & 107 & 78\
Fe I & 6481.880 & 2.280 & -2.95 & 121 & 109 & 97\
Fe I & 6574.229 & 0.990 & -5.11 & 97 & 92 & 90\
Fe I & 6609.120 & 2.560 & -2.67 & 108 & 93 & 125\
Fe I & 6703.570 & 2.758 & -3.08 & 78 & 71 & 52\
Fe I & 6705.103 & 4.607 & -1.07 & 57 & 60 &\
Fe I & 6733.151 & 4.638 & -1.48 &&& 37\
Fe I & 6810.263 & 4.607 & -0.99 & 64 & 72 &\
Fe I & 6820.372 & 4.638 & -1.14 & 50 & 56 &\
Fe I & 6839.831 & 2.559 & -3.42 & 70 & 67 & 74\
Fe I & 7540.430 & 2.730 & -3.87 & 56 &&\
Fe I & 7568.900 & 4.280 & -0.85 & 80 & 92 & 91\
Fe II& 5414.080 & 3.22 & -3.60 & 27 &&\
Fe II& 6084.100 & 3.20 & -3.78 & 21 &&\
Fe II& 6149.250 & 3.89 & -2.67 & 36 &&\
Fe II& 6247.560 & 3.89 & -2.31 & 63 & 48 &\
Fe II& 6369.463 & 2.89 & -4.18 & 28 &&\
Fe II& 6456.390 & 3.90 & -2.05 & 50 & 57 &\
Fe II& 6516.080 & 2.89 & -3.24 && 56 &\
Al I & 6696.03 & 3.14 & -1.56 & 71 && 43\
Al I & 6698.67 & 3.13 & && 44 &\
Ca I & 5581.97 & 2.52 & -0.62 & 114 & 119 & 107\
Ca I & 5590.12 & 2.52 & -0.82 & 105 & 112 & 100\
Ca I & 5867.57 & 2.93 & -1.65 & 45 &&\
Ca I & 6161.30 & 2.52 & -1.27 & 93 & 109 & 92\
Ca I & 6166.44 & 2.52 & -1.12 & 94 & 83 & 103\
Ca I & 6455.60 & 2.52 & -1.41 & 82 & 70 & 72\
Ca I & 6499.65 & 2.52 & -0.91 & 117 & 111 & 85\
Mg I & 5711.09 & 4.33 & -1.71 & 117 & 109 &\
Mg I & 7387.70 & 5.75 & -1.09 & 105 & 67 &\
Na I & 5682.65 & 2.10 & -0.75 & 126 & 128 &\
Na I & 5688.21 & 2.10 & -0.72 & 136 & 138 & 160\
Na I & 6154.23 & 2.10 & -1.61 & 56 & 45 & 51\
Na I & 6160.75 & 2.10 & -1.38 & 73 & 73 & 64\
Ni I & 6175.37 & 4.09 & -0.52 & 74 & 64 & 62\
Ni I & 6176.81 & 4.09 & -0.19 & 116 & 78 & 80\
Ni I & 6177.25 & 1.83 & -3.60 & 36 & 42 &\
Ni I & 6223.99 & 4.10 & &&& 64\
Si I & 5665.60 & 4.90 & -1.98 && 53 &\
Si I & 5684.52 & 4.93 & -1.63 & 56 &&\
Si I & 5701.12 & 4.93 & -1.99 && 41 &\
Si I & 5793.08 & 4.93 & -1.89 & 38 & 49 &\
Si I & 6142.49 & 5.62 & -1.47 & 27 &&\
Si I & 6145.02 & 5.61 & &&&\
Si I & 6243.82 & 5.61 & -1.30 & 39 & 54 &\
Si I & 7034.91 & 5.87 & -0.74 & 55 & 103 &\
Ti I & 5978.54 & 1.87 & -0.65 & 71 & 67 & 67\
Ti II& 5418.77 & 1.58 & -2.12 & 74 & 76 & 103\
[lccccccccc]{} Be 22-400 & -0.29$\pm$0.21 & -0.32$\pm$0.19 & +0.05$\pm$0.20 & -0.35$\pm$0.11 & -0.37$\pm$0.20 & -0.23$\pm$0.05 & -0.26$\pm$0.20 & -0.37$\pm$0.08 & -0.19$\pm$0.11\
Be 22-579 & -0.35$\pm$0.17 & -0.28$\pm$0.04 & -0.12$\pm$0.20 & -0.45$\pm$0.11 & -0.41$\pm$0.14 & -0.33$\pm$0.12 & -0.29$\pm$0.07 & -0.20$\pm$0.11 & -0.22$\pm$0.04\
Be 66-785 & -0.48$\pm$0.24 & & -0.48$\pm$0.20 & -0.53$\pm$0.21 & & -0.33$\pm$0.18 & -0.24$\pm$0.25 & & -0.05$\pm$0.23\
[lcccccccc]{} Be 22-400 & -0.29 & -0.06 & -0.08 & -0.08 & +0.10 & +0.06 & +0.34 & +0.03\
Be 22-579 & -0.35 & -0.10 & -0.06 & +0.15 & +0.13 & +0.02 & +0.23 & +0.06\
Be 66-785 & -0.48 & -0.05 & & & +0.43 & +0.15 & +0.00 & +0.24\
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Using train tracks on a nonexceptional oriented surface $S$ of finite type in a systematic way we give a proof that the complex of curves $\mathcal C (S)$ of $S$ is a hyperbolic geodesic metric space. We also discuss the relation between the geometry of the complex of curves and the geometry of Teichmüller space.'
author:
- Ursula Hamenstädt
title: Geometry of the complex of curves and of Teichmüller space
---
[Partially supported by DFG-SPP 1154 and DFG-SFB 611.]{}
Introduction
============
Consider a compact oriented surface $S$ of genus $g\geq 0$ from which $m\geq 0$ points, so-called *punctures*, have been deleted. Such a surface is called *of finite type*. We assume that $S$ is *non-exceptional*, i.e. that $3g-3+m\geq 2$; this rules out a sphere with at most four punctures and a torus with at most one puncture.
In [@Ha], Harvey associates to such a surface the following simplical complex.
The *complex of curves* ${{\mathcal}C}(S)$ for the surface $S$ is the simplicial complex whose vertices are the free homotopy classes of essential simple closed curves on $S$ and whose simplices are spanned by collections of such curves which can be realized disjointly.
Here we mean by an *essential* simple closed curve a simple closed curve which is not contractible nor homotopic into a puncture. Since $3g-3+m$ is the number of curves in a *pants decomposition* of $S$, i.e. a maximal collection of disjoint mutually not freely homotopic essential simple closed curves which decompose $S$ into $2g-2+m$ open subsurfaces homeomorphic to a thrice punctured sphere, the dimension of ${{\mathcal}C}(S)$ equals $3g-4+m$.
In the sequel we restrict our attention to the one-skeleton of the complex of curves which is usually called the *curve graph*; by abuse of notation, we denote it again by ${{\mathcal}C}(S)$. Since $3g-3+m\geq 2$ by assumption, ${{\mathcal}C}(S)$ is a nontrivial graph which moreover is connected [@Ha]. However, this graph is locally infinite. Namely, for every simple closed curve $\alpha$ on $S$ the surface $S-\alpha$ which we obtain by cutting $S$ open along $\alpha$ contains at least one connected component of Euler characteristic at most $-2$, and such a component contains infinitely many distinct free homotopy classes of simple closed curves which viewed as curves in $S$ are disjoint from $\alpha$.
Providing each edge in ${{\mathcal}C}(S)$ with the standard euclidean metric of diameter 1 equips the curve graph with the structure of a geodesic metric space. Since ${{\mathcal}C}(S)$ is not locally finite, this metric space $({{\mathcal}C}(S),d)$ is not locally compact. Masur and Minsky [@MM1] showed that nevertheless its geometry can be understood quite explicitly. Namely, ${{\mathcal}C}(S)$ is hyperbolic of infinite diameter. Here for some $\delta
>0$ a geodesic metric space is called *$\delta$-hyperbolic in the sense of Gromov* if it satisfies the *$\delta$-thin triangle condition*: For every geodesic triangle with sides $a,b,c$ the side $c$ is contained in the $\delta$-neighborhood of $a\cup b$. Later Bowditch [@B] gave a simplified proof of the result of Masur and Minsky which can also be used to compute explicit bounds for the hyperbolicity constant $\delta$.
Since the Euler characteristic of $S$ is negative, the surface $S$ admits a complete hyperbolic metric of finite volume. The group of diffeomorphisms of $S$ which are isotopic to the identity acts on the space of such metrics. The quotient space under this action is the *Teichmüller space* ${{\mathcal}T}_{g,m}$ for $S$ of all *marked* isometry classes of complete hyperbolic metrics on $S$ of finite volume, or, equivalently, the space of all marked complex structures on $S$ of finite type. The Teichmüller space can be equipped with a natural topology, and with this topology it is homeomorphic to $\mathbb{R}^{6g-6+2m}$. The *mapping class group* ${{\mathcal}M}_{g,m}$ of all isotopy classes of orientation preserving diffeomorphisms of $S$ acts properly discontinuously as a group of diffeomorphisms of Teichmüller space preserving a complete Finsler metric, the so-called *Teichmüller metric*. The quotient orbifold is the *moduli space* ${\rm Mod}(S)$ of $S$ of all isometry classes of complete hyperbolic metrics of finite volume on $S$ (for all this see [@IT]).
The significance of the curve graph for the geometry of Teichmüller space comes from the obvious fact that the mapping class group acts on ${{\mathcal}C}(S)$ as a group of simplicial isometries. Even more is true: If $S$ is not a twice punctured torus or a closed surface of genus $2$, then the *extended mapping class group* of isotopy classes of *all* diffeomorphisms of $S$ coincides precisely with the group of simplicial isometries of ${{\mathcal}C}(S)$; for a closed surface of genus 2, the group of simplicial isometries of ${{\mathcal}C}(S)$ is the quotient of the extended mapping class group under the hyperelliptic involution which acts trivially on ${{\mathcal}C}(S)$ (see [@I] for an overview on this and related results). Moreover, there is a natural map $\Psi:{{\mathcal}T}_{g,m}\to
{{\mathcal}C}(S)$ which is *coarsely ${{\mathcal}M}_{g,m}$-equivariant* and *coarsely Lipschitz* with respect to the Teichmüller metric on ${{\mathcal}T}_{g,m}$. By this we mean that there is a number $a>1$ such that $d(\Psi (\phi h),\phi(\Psi h))\leq a$ for all $h\in {{\mathcal}T}_{g,m}$ and all $\phi\in {{\mathcal}M}_{g,m}$ and that moreover $d(\Psi h,\Psi h^\prime)\leq a d_T(h,h^\prime)+a$ for all $h,h^\prime\in {{\mathcal}T}_{g,m}$ where $d_T$ denotes the distance function on ${{\mathcal}T}_{g,m}$ induced by the Teichmüller metric (see Section 4).
As a consequence, the geometry of ${{\mathcal}C}(S)$ is related to the large-scale geometry of the Teichmüller space and the mapping class group. We discuss this relation in Section 4. In Section 3 we give a proof of the hyperbolicity of the curve graph using *train tracks* and splitting sequences of train tracks in a consistent way as the main tool. Section 2 introduces train tracks, geodesic laminations and quadratic differentials and summarizes some of their properties.
Train tracks and geodesic laminations
=====================================
Let $S$ be a nonexceptional surface of finite type and choose a complete hyperbolic metric on $S$ of finite volume. With respect to this metric, every essential free homotopy class of loops can be represented by a closed geodesic which is unique up to parametrization. This geodesic is *simple*, i.e. without self-intersection, if and only if the free homotopy class has a simple representative (see [@Bu]). In other words, there is a one-to-one correspondence between vertices of the curve graph and simple closed geodesics on $S$. Moreover, there is a fixed *compact* subset $S_0$ of $S$ containing all simple closed geodesics.
The *Hausdorff distance* between two closed bounded subsets $A,B$ of a metric space $X$ is defined to be the infimum of all numbers $\epsilon >0$ such that $A$ is contained in the $\epsilon$-neighborhood of $B$ and $B$ is contained in the $\epsilon$-neighborhood of $A$. This defines indeed a distance and hence a topology on the space of closed bounded subsets of $X$; this topology is called the *Hausdorff topology*. If $X$ is compact then the space of closed subsets of $X$ is compact as well. In particular, for the distance on $S$ induced by a complete hyperbolic metric of finite volume, the space of closed subsets of the compact set $S_0\subset S$ is compact with respect to the Hausdorff topology.
The collection of all simple closed geodesics on $S$ is *not* a closed set with respect to the Hausdorff topology, but a point in its closure can be described as follows.
A *geodesic lamination* for a complete hyperbolic structure of finite volume on $S$ is a compact subset of $S$ which is foliated into simple geodesics.
Thus every simple closed geodesic is a geodesic lamination which consists of a single leaf. The space of geodesic laminations on $S$ equipped with the Hausdorff topology is compact, and it contains the closure of the set of simple closed geodesics as a *proper* subset. Note that every lamination in this closure is necessarily connected.
To describe the structure of the space of geodesic laminations more explicitly we introduce some more terminology.
A geodesic lamination $\lambda$ is called *minimal* if each of its half-leaves is dense in $\lambda$. A geodesic lamination $\lambda$ is *maximal* if all its complementary components are ideal triangles or once punctured monogons. A geodesic lamination is called *complete* if it is maximal and can be approximated in the Hausdorff topology for compact subsets of $S$ by simple closed geodesics.
As an example, a simple closed geodesic is a minimal geodesic lamination. A minimal geodesic lamination with more than one leaf has uncountably many leaves. Every minimal geodesic lamination can be approximated in the Hausdorff topology by simple closed geodesics [@CEG]. Moreover, a minimal geodesic lamination $\lambda$ is a *sublamination* of a complete geodesic lamination [@H1], i.e. there is a complete geodesic lamination which contains $\lambda$ as a closed subset. In particular, every simple closed geodesic on $S$ is a sublamination of a complete geodesic lamination. *Every* geodesic lamination $\lambda$ is a disjoint union of finitely many minimal components and a finite number of isolated leaves. Each of the isolated leaves of $\lambda$ either is an isolated closed geodesic and hence a minimal component, or it *spirals* about one or two minimal components [@Bo; @CEG; @O]. This means that the set of accumulation points of an isolated half-leaf of $\lambda$ is a minimal component of $\lambda$.
Geodesic laminations which are disjoint unions of minimal components can be equipped with the following additional structure.
A *measured geodesic lamination* is a geodesic lamination together with a translation invariant *transverse measure*.
A transverse measure for a geodesic lamination $\lambda$ assigns to every smooth compact arc $c$ on $S$ with endpoints in the complement of $\lambda$ and which intersects $\lambda$ transversely a finite Borel measure on $c$ supported in $c\cap \lambda$. These measures transform in the natural way under homotopies of $c$ by smooth arcs transverse to $\lambda$ which move the endpoints of the arc $c$ within fixed complementary components of $\lambda$. The support of the measure is the smallest sublamination $\nu$ of $\lambda$ such that the measure on any such arc $c$ which does not intersect $\nu$ is trivial. This support is necessarily a union of minimal components of $\lambda$. An example for a measured geodesic lamination is a *weighted simple closed geodesic* which consists of a simple closed geodesic $\alpha$ and a positive weight $a>0$. The measure disposed on a transverse arc $c$ is then the sum of the Dirac masses on the intersection points between $c$ and $\alpha$ multiplied with the weight $a$.
The space ${{\mathcal}M{\mathcal}L}$ of measured geodesic laminations on $S$ can naturally be equipped with the *weak$^*$-topology*. This topology restricts to the weak$^*$-topology on the space of measures on a given arc $c$ which is transverse to each lamination from an open subset of lamination space. The natural action of the group $(0,\infty)$ by scaling is continuous with respect to this topology, and the quotient is the space ${{\mathcal}P{\mathcal}M{\mathcal}L}$ of *projective measured geodesic laminations*. This space is homeomorphic to a sphere of dimension $6g-7+2m$ (see [@CEG; @FLP; @PH]).
The *intersection number* $i(\gamma,\delta)$ between two simple closed curves $\gamma,\delta\in {{\mathcal}C}(S)$ equals the minimal number of intersection points between representatives of the free homotopy classes of $\gamma,\delta$. This intersection function extends to a continuous pairing $i:{{\mathcal}M{\mathcal}L}\times{{\mathcal}M{\mathcal}L}\to [0,\infty)$, called the *intersection form*.
Measured geodesic laminations are intimately related to more classical objects associated to Riemann surfaces, namely *holomorphic quadratic differentials.* A holomorphic quadratic differential $q$ on a Riemann surface $S$ assigns to each complex coordinate $z$ an expression of the form $q(z)dz^2$ where $q(z)$ is a holomorphic function on the domain of the coordinate system, and $q(z)(dz/dw)^2=q(w)$ for overlapping coordinates $z,w$. We require that $q$ has at most a simple pole at each puncture of $S$. If $q$ does not vanish identically, then its zeros are isolated and independent of the choice of a complex coordinate. If $p\in S$ is not a zero for $q$ then there is a coordinate $z$ near $p$, unique up to multiplication with $\pm 1$, such that $p$ corresponds to the origin and that $q(z)\equiv 1$. Writing $z=x+iy$ for this coordinate, the euclidean metric $dx^2+dy^2$ is uniquely determined by $q$. The arcs parallel to the $x$-axis (or $y$-axis, respectively) define a foliation ${{\mathcal}F}_h$ (or ${{\mathcal}F}_v$) on the set of regular points of $q$ called the *horizontal* (or *vertical*) foliation. The vertical length $\vert dy\vert$ defines a *transverse measure* for the horizontal foliation, and the horizontal length $\vert dx\vert$ defines a transverse measure for the vertical foliation. The foliations ${{\mathcal}F}_h,{{\mathcal}F}_v$ have singularities of the same type at the zeros of $q$ and at the punctures of $S$ (see [@S] for more on quadratic differentials and measured foliations).
There is a one-to-one correspondence between measured geodesic laminations and (equivalence classes of) measured foliations on $S$ (see [@L] for a precise statement). The pair of measured foliations defined by a quadratic differential $q$ corresponds under this identification to a pair of measured geodesic laminations $\lambda\not=\mu\in {{\mathcal}M{\mathcal}L}$ which *jointly fill up $S$*. This means that for every $\eta\in {{\mathcal}M{\mathcal}L}$ we have $i(\lambda,\eta)+i(\mu,\eta)>0$.
Vice versa, every pair of measured geodesic laminations $\lambda\not= \mu\in {{\mathcal}M{\mathcal}L}$ which jointly fill up $S$ defines a unique complex structure of finite type on $S$ together with a holomorphic quadratic differential $q(\lambda,\mu)$ (see [@Ke] and the references given there) whose *area*, i.e. the area of the singular euclidean metric defined by $q(\lambda,\mu)$, equals $i(\lambda,\mu)$. If $\alpha,\beta$ are *simple multi-curves* on $S$, which means that $\alpha$ and $\beta$ consist of collections $c=c_1\cup \dots \cup c_\ell\subset
{{\mathcal}C}(S)$ of free homotopy classes of simple closed curves which can be realized disjointly, and if $\alpha,\beta$ jointly fill up $S$, then for all $a>0,b>0$ the quadratic differential $q(a\alpha,b\beta)$ defined by the measured geodesic laminations $a\alpha,b\beta$ can explicitly be constructed as follows. Choose smooth representatives of $\alpha,\beta$, again denoted by $\alpha,\beta$, which intersect transversely in precisely $i(\alpha,\beta)$ points; for example, the geodesic representatives of $\alpha,\beta$ with respect to any complete hyperbolic metric on $S$ of finite volume have this property. For each intersection point between $\alpha,\beta$ choose a closed rectangle in $S$ with piecewise smooth boundary containing this point in its interior and which does not contain any other intersection point between $\alpha,\beta$. We allow that some of the vertices of such a rectangle are punctures of $S$. These rectangles can be chosen in such a way that they provide $S$ with the structure of a cubical complex: The boundary of each component $D$ of $S-\alpha-\beta$ is a polygon with an even number of sides which are subarcs of $\alpha,\beta$ in alternating order. If $D$ does not contain a puncture, then its boundary has at least four sides. Thus we can construct the rectangles in such a way that their union is all of $S$ and that the intersection between any two distinct such rectangles either is a common side or a common vertex. Each rectangle from this cubical complex has two sides which are “parallel” to $\alpha$ and two sides “parallel” to $\beta$ (see Section 4 of [@Ke] for a detailed discussion of this construction).
Equip each rectangle with an euclidean metric such that the sides parallel to $\alpha$ are of length $b$, the sides parallel to $\beta$ are of length $a$ and such that the metrics on two rectangles coincide on a common boundary arc. These metrics define a piecewise euclidean metric on $S$ with a singularity of cone angle $k\pi\geq 3\pi$ in the interior of each disc component of $S-\alpha-\beta$ whose boundary consists of $2k\geq 6$ sides. The metric also has a singularity of cone angle $\pi$ at each puncture of $S$ which is contained in a punctured disc component with two sides. Since there are precisely $i(\alpha,\beta)$ rectangles, the area of this singular euclidean metric on $S$ equals $ab i(\alpha,\beta)$. The line segments of this singular euclidean metric which are parallel to $\alpha$ and $\beta$ define singular foliations ${{\mathcal}F}_{\alpha},{{\mathcal}F}_{\beta}$ on $S$ with transverse measures induced by the singular metric. The metric defines a complex structure on $S$ and a quadratic differential $q(a\alpha,b\beta)$ which is holomorphic for this structure and whose horizontal and vertical foliations are just ${{\mathcal}F}_\alpha,{{\mathcal}F}_{\beta}$, with transverse measures determined by the weights $a$ and $b$. The assignment which associates to $a>0$ the Riemann surface structure determined by $q(a\alpha,\beta)$ is up to parametrization the geodesic in the Teichmüller space with respect to the *Teichmüller metric* whose cotangent bundle contains the differentials $q(a\alpha,\beta)$ (compare [@IT; @Ke]).
A *geodesic segment* for a quadratic differential $q$ is a path in $S$ not containing any singularities in its interior and which is a geodesic in the local euclidean structure defined by $q$. A closed geodesic is composed of a finite number of such geodesic segments which meet at singular points of $q$ and make an angle at least $\pi$ on either side. Every essential closed curve $c$ on $S$ is freely homotopic to a closed geodesic with respect to $q$, and the length of such a geodesic $\eta$ is the infimum of the $q$-lengths of any curve freely homotopic to $\eta$ (compare [@R] for a detailed discussion of the technical difficulties caused by the punctures of $S$) and will be called the *$q$-length* of our closed curve $c$. If $q=q(\lambda,\nu)$ for $\lambda,\nu\in {{\mathcal}M{\mathcal}L}$ then this $q$-length is bounded from above by $2i(\lambda,c)+2i(\mu,c)$ (see [@R]).
Thurston invented a way to understand the structure of the space of geodesic laminations by squeezing almost parallel strands of such a lamination to a simple arc and analyzing the resulting graph. The structure of such a graph is as follows.
A *train track* on the surface $S$ is an embedded 1-complex $\tau\subset S$ whose edges (called *branches*) are smooth arcs with well-defined tangent vectors at the endpoints. At any vertex (called a *switch*) the incident edges are mutually tangent. Through each switch there is a path of class $C^1$ which is embedded in $\tau$ and contains the switch in its interior. In particular, the branches which are incident on a fixed switch are divided into “incoming” and “outgoing” branches according to their inward pointing tangent at the switch. Each closed curve component of $\tau$ has a unique bivalent switch, and all other switches are at least trivalent. The complementary regions of the train track have negative Euler characteristic, which means that they are different from discs with $0,1$ or $2$ cusps at the boundary and different from annuli and once-punctured discs with no cusps at the boundary. A train track is called *generic* if all switches are at most trivalent.
In the sequel we only consider generic train tracks. For such a train track $\tau$, every complementary component is a bordered subsurface of $S$ whose boundary consists of a finite number of arcs of class $C^1$ which come together at a finite number of *cusps*. Moreover, for every switch of $\tau$ there is precisely one complementary component containing the switch in its closure which has a cusp at the switch. A detailed account on train tracks can be found in [@PH] and [@M].
A geodesic lamination or a train track $\lambda$ is *carried* by a train track $\tau$ if there is a map $F:S\to S$ of class $C^1$ which is isotopic to the identity and maps $\lambda$ to $\tau$ in such a way that the restriction of its differential $dF$ to every tangent line of $\lambda$ is non-singular. Note that this makes sense since a train track has a tangent line everywhere.
If $c$ is a simple closed curve carried by $\tau$ with carrying map $F:c\to \tau$ then $c$ defines a *counting measure* $\mu_c$ on $\tau$. This counting measure is the non-negative weight function on the branches of $\tau$ which associates to an open branch $b$ of $\tau$ the number of connected components of $F^{-1}(b)$. A counting measure is an example for a *transverse measure* on $\tau$ which is defined to be a nonnegative weight function $\mu$ on the branches of $\tau$ satisfying the *switch condition*: for every switch $s$ of $\tau$, the sum of the weights over all incoming branches at $s$ is required to coincide with the sum of the weights over all outgoing branches at $s$. The set $V(\tau)$ of all transverse measures on $\tau$ is a closed convex cone in a linear space and hence topologically it is a closed cell. More generally, every measured geodesic lamination $\lambda$ on $S$ which is carried by $\tau$ via a carrying map $F:\lambda\to \tau$ defines a transverse measure on $\tau$ by assigning to a branch $b$ the total mass of the pre-image of $b$ under $F$; the resulting weight function is independent of the particular choice of $F$. Moreover, every transverse measure for $\tau$ can be obtained in this way (see [@PH]).
A train track is called *recurrent* if it admits a transverse measure which is positive on every branch. A train track $\tau$ is called *transversely recurrent* if every branch $b$ of $\tau$ is intersected by an embedded simple closed curve $c=c(b)\subset S$ which intersects $\tau$ transversely and is such that $S-\tau-c$ does not contain an embedded *bigon*, i.e. a disc with two corners at the boundary. A recurrent and transversely recurrent train track is called *birecurrent*. A generic transversely recurrent train track which carries a complete geodesic lamination is called *complete*.
For every recurrent train track $\tau$, measures which are positive on every branch define the interior of the convex cone $V(\tau)$ of all transverse measures. A complete train track is birecurrent [@H1].
A half-branch $\tilde b$ in a generic train track $\tau$ incident on a switch $v$ is called *large* if the switch $v$ is trivalent and if every arc $\rho:(-\epsilon,\epsilon)\to
\tau$ of class $C^1$ which passes through $v$ meets the interior of $\tilde b$. A branch $b$ in $\tau$ is called *large* if each of its two half-branches is large; in this case $b$ is necessarily incident on two distinct switches (for all this, see [@PH]).
There is a simple way to modify a transversely recurrent train track $\tau$ to another transversely recurrent train track. Namely, if $e$ is a large branch of $\tau$ then we can perform a right or left *split* of $\tau$ at $e$ as shown in Figure A below. The split $\tau^\prime$ of a train track $\tau$ is carried by $\tau$. If $\tau$ is complete and if the complete geodesic lamination $\lambda$ is carried by $\tau$, then for every large branch $e$ of $\tau$ there is a unique choice of a right or left split of $\tau$ at $e$ with the property that the split track $\tau^\prime$ carries $\lambda$, and $\tau^\prime$ is complete. In particular, a complete train track $\tau$ can always be split at any large branch $e$ to a complete train track $\tau^\prime$; however there may be a choice of a right or left split at $e$ such that the resulting train track is not complete any more (compare p.120 in [@PH]).
In the sequel we denote by ${{\mathcal}T}T$ the collection of all isotopy classes of complete train tracks on $S$. A sequence $(\tau_i)\subset {{\mathcal}T}T$ of complete train tracks is called a *splitting sequence* if $\tau_{i+1}$ can be obtained from $\tau_i$ by a single split at some large branch $e$.
Hyperbolicity of the complex of curves
======================================
In this section we present a proof of hyperbolicity of the curve graph using the main strategy of Masur and Minsky [@MM1] and Bowditch [@B] in a modified form. The first step consists in guessing a family of uniform *quasi-geodesics* in the curve graph connecting any two points. Here a $p$-quasi-geodesic for some $p>1$ is a curve $c:[a,b]\to {{\mathcal}C}(S)$ which satisfies $$d(c(s),c(t))/p-p\leq \vert s-t\vert \leq p d(c(s),c(t))+p
\quad \hbox{for all}\, s,t\in [a,b].$$ Note that a quasi-geodesic does not have to be continuous. In a hyperbolic geodesic metric space, every $p$-quasi-geodesic is contained in a fixed tubular neighborhood of any geodesic joining the same endpoints, so the $\delta$-thin triangle condition also holds for triangles whose sides are uniform quasi-geodesics [@BH]. As a consequence, for every triangle in a hyperbolic geodesic metric space with uniform quasi-geodesic sides there is a “midpoint” whose distance to each side of the triangle is bounded from above by a universal constant. The second step of the proof consists in finding such a midpoint for triangles whose sides are curves of the distinguished curve family. This is then used in a third step to establish the $\delta$-thin triangle condition for the distinguished family of curves and derive from this hyperbolicity of ${{\mathcal}C}(S)$. By abuse of notation, in the sequel we simply write $\alpha\in {{\mathcal}C}(S)$ if $\alpha$ is a free homotopy class of an essential simple closed curve on $S$, i.e. if $\alpha$ is a vertex of ${{\mathcal}C}(S)$.
We begin with defining a map from the set ${{\mathcal}T}T$ of complete train tracks on $S$ into ${{\mathcal}C}(S)$. For this we call a transverse measure $\mu$ for a complete train track $\tau$ a *vertex cycle* [@MM1] if $\mu$ spans an extreme ray in the convex cone $V(\tau)$ of all transverse measures on $\tau$.
Up to scaling, every vertex cycle $\mu$ is a counting measure of a simple closed curve $c$ which is carried by $\tau$ [@MM1]. Namely, the switch conditions are a family of linear equations with integer coefficients for the transverse measures on $\tau$. Thus an extreme ray is spanned by a nonnegative *rational* solution which can be scaled to a nonnegative integral solution. From every integral transverse measure $\mu$ for $\tau$ we can construct a unique *simple weighted multi-curve*, i.e. a simple multi-curve together with a family of weights for each of its components, which is carried by $\tau$ and whose counting measure coincides with $\mu$ as follows. For each branch $b$ of $\tau$ draw $\mu(b)$ disjoint arcs parallel to $b$. By the switch condition, the endpoints of these arcs can be connected near the switches in a unique way so that the resulting family of arcs does not have self-intersections. Let $c$ be the simple multi-curve consisting of the free homotopy classes of the connected components of the resulting curve $\tilde c$. To each such homotopy class associate the number of components of $\tilde c$ in this class as a weight. The resulting simple weighted multi-curve is carried by $\tau$, and its counting measure equals $\mu$. Thus if there are at least two components of $\tilde c$ which are not freely homotopic then the weighted counting measures of these components determine a decomposition of $\mu$ into transverse measures for $\tau$ which are not multiples of $\mu$. This is impossible if $\mu$ is a vertex cycle. Hence $c$ consists of a single component and up to scaling, $\mu$ is the counting measure of a simple closed curve on $S$.
A simple closed curve which is carried by $\tau$, with carrying map $F:c\to \tau$, defines a vertex cycle for $\tau$ only if $F(c)$ passes through every branch of $\tau$ at most twice, with different orientation (Lemma 2.2 of [@H2]). In particular, the counting measure $\mu_c$ of a simple closed curve $c$ which defines a vertex cycle for $\tau$ satisfies $\mu_c(b)\leq 2$ for every branch $b$ of $\tau$.
In the sequel we mean by a vertex cycle of a complete train track $\tau$ an *integral* transverse measure on $\tau$ which is the counting measure of a simple closed curve $c$ on $S$ carried by $\tau$ and which spans an extreme ray of $V(\tau)$; we also use the notion vertex cycle for the simple closed curve $c$. Since the number of branches of a complete train track on $S$ only depends on the topological type of $S$, the number of vertex cycles for a complete train track on $S$ is bounded by a universal constant (see [@MM1] and [@H2]).
The following observation of Penner and Harer [@PH] is essential for all what follows. Denote by ${{\mathcal}M{\mathcal}C}(S)$ the space of all simple multi-curves on $S$. Let $P=\cup_{i=1}^{3g-3+m}\gamma_i\in {{\mathcal}M{\mathcal}C}(S)$ be a pants decomposition for $S$, i.e. a simple multi-curve with the maximal number of components. Then there is a special family of complete train tracks with the property that each pants curve $\gamma_i$ admits a closed neighborhood $A$ diffeomorphic to an annulus and such that $\tau\cap A$ is diffeomorphic to a *standard twist connector* depicted in Figure B.
Such a train track clearly carries each pants curve from the pants decomposition $P$ as a vertex cycle; we call it *adapted* to $P$. For every complete geodesic lamination $\lambda$ there is a train track $\tau$ adapted to $P$ which carries $\lambda$ ([@PH], see also [@H1],[@H2]).
Since every simple multi-curve is a subset of a pants decomposition of $S$, we can conclude.
For every pair $(\alpha,\beta)\in {{\mathcal}M{\mathcal}C}(S)\times {{\mathcal}M{\mathcal}C}(S)$ there is a splitting sequence $(\tau_i)_{0\leq i\leq m}\subset {{\mathcal}T}T$ of complete train tracks with the property that $\tau_0$ is adapted to a pants decomposition $P_\alpha\supset \alpha$ and that each component of $\beta$ is a vertex cycle for $\tau_m$.
We call a splitting sequence as in the lemma an $\alpha\to
\beta$-splitting sequence. Note that such a sequence is by no means unique.
The distance in ${{\mathcal}C}(S)$ between two simple closed curves $\alpha,\beta$ is bounded from above by $i(\alpha,\beta)+1$ (Lemma 1.1 of [@B] and Lemma 2.1 of [@MM1]). In particular, there is a number $D_0>0$ with the following property. Let $\tau,\tau^\prime\in {{\mathcal}T}T$ and assume that $\tau^\prime$ is obtained from $\tau$ by at most one split. Then the distance in ${{\mathcal}C}(S)$ between any vertex cycle of $\tau$ and any vertex cycle of $\tau^\prime$ is at most $D_0$ (see [@MM1] and the discussion following Corollary 2.3 in [@H2]).
Define a map $\Phi:{{\mathcal}T}T\to {{\mathcal}C}(S)$ by assigning to a train track $\tau\in {{\mathcal}T}T$ a vertex cycle $\Phi(\tau)$ for $\tau$. By our above discussion, for any two choices $\Phi,\Phi^\prime$ of such a map we have $d(\Phi(\tau),\Phi^\prime(\tau))
\leq D_0$ for all $\tau\in {{\mathcal}T}T$. Images under the map $\Phi$ of splitting sequences then define a family of curves in ${{\mathcal}C}(S)$ which connect any pair of points in a $D_0$-dense subset of ${{\mathcal}C}(S)\times {{\mathcal}C}(S)$, equipped with the product metric. As a consequence, we can use such images of splitting sequences as our guesses for uniform quasi-geodesics. It turns out that up to parametrization, these curves are indeed $p$-quasi-geodesics in ${{\mathcal}C}(S)$ for a universal number $p>0$ only depending on the topological type of the surface $S$ ([@MM3], see also [@H2]).
To explain this fact we use the following construction of Bowditch [@B]. For multi-curves $\alpha,\beta \in
{{\mathcal}M{\mathcal}C}(S)$ which jointly fill up $S$, i.e. which cut $S$ into components which are homeomorphic to discs and once punctured discs, and for a number $a>0$ let $q(a\alpha,\beta/ai(\alpha,\beta))$ be the area one quadratic differential whose horizontal foliation corresponds to the measured geodesic lamination $a\alpha$ and whose vertical measured foliation corresponds to the measured geodesic lamination $\beta/ai(\alpha,\beta)$. For $r>0$ define $$L_a(\alpha,\beta,r)=\{\gamma\in {{\mathcal}C}(S)\mid
\max\{ai(\gamma,\alpha),i(\gamma,\beta)/ai(\alpha,\beta)\}\leq r\}.$$ Then $L_a(\alpha,\beta,r)$ is contained in the set of all all simple closed curves on $S$ whose $q(a\alpha,\beta/ai(\alpha,\beta))$-length does not exceed $2r$. Note that we have $L_a(\alpha,\beta,r)=L_{1/ai(\alpha,\beta)}(\beta,\alpha,r)$ for all $r>0$, moreover $\alpha^\prime\in L_a(\alpha,\beta,r)$ for every component $\alpha^\prime$ of $\alpha$ and every sufficiently large $a>0$, and $\beta^\prime\in L_a(\alpha,\beta,r)$ for every component $\beta^\prime$ of $\beta$ and every sufficiently small $a>0$. Thus for fixed $r>0$ we can think of a suitably chosen assignment which associates to a number $s>0$ a point in $L_s(\alpha,\beta,r)$ as a curve in ${{\mathcal}C}(S)$ connecting a component of $\beta$ to a component of $\alpha$ (provided, of course, that the sets $L_s(\alpha,\beta,r)$ are non-empty). Lemma 2.5 of [@H2] links such curves to splitting sequences.
There is a number $k_0\geq 1$ with the following property. Let $P$ be a pants decomposition of $S$, let $\alpha\in {{\mathcal}M{\mathcal}C}(S)$ be such that $\alpha$ and $P$ jointly fill up $S$ and let $(\tau_i)_{0\leq i\leq m}\subset {{\mathcal}T}T$ be a $P\to \alpha$-splitting sequence. Then there is a non-decreasing surjective function $\kappa:(0,\infty)\to \{0,\dots,m\}$ such that $\kappa(s)=0$ for all sufficiently small $s>0, \kappa(s)=m$ for all sufficiently large $s>0$ and that for all $s\in (0,\infty)$ there is a vertex cycle of $\tau_{\kappa(s)}$ which is contained in $L_s(\alpha,P,k_0)$.
Since for every multi-curve $\alpha\in {{\mathcal}M{\mathcal}C}(S)$ and every pants decomposition $P$ of $S$ there is a $P\to \alpha$-splitting sequence, we conclude that for every $k\geq k_0$ and every $s>0$ the set $L_s(\alpha,P,k)$ is non-empty. To obtain a control of the size of these sets, Bowditch [@B] uses the following observation (Lemma 4.1 in [@B]) whose first part was earlier shown by Masur and Minsky (Lemma 5.1 of [@MM1]).
There is a number $k_1\geq k_0$ with the following property. For all $\alpha,\beta\in {{\mathcal}M{\mathcal}C}(S)$ which jointly full up $S$ and every $a\in
(0,\infty)$ there is some $\delta\in L_a(\alpha,\beta,k_1)$ such that for every $\gamma\in {{\mathcal}M{\mathcal}C}(S)$ we have $$i(\delta,\gamma)\leq
k_1\max\{ai(\alpha,\gamma),i(\gamma,\beta)/ai(\alpha,\beta)\}.$$ In particular, for every $R>0$, for all $\alpha,\beta\in
{{\mathcal}M {\mathcal}C}(S)$ and for every $a>0$ the diameter of the set $L_a(\alpha,\beta,R)$ is not bigger than $2k_1R+1$.
In [@MM1; @B] it is shown that there is a number $\nu >0$ only depending on the topological type of $S$ and there is an embedded essential annulus in $S$ whose width with respect to the piecewise euclidean metric defined by the quadratic differential $q(a\alpha,\beta/ai(\alpha,\beta))$ is at least $\nu$. This means that the distance between the boundary circles of the annulus is at least $\nu$. Assuming the existence of such an annulus, let $\delta$ be its core-curve. Then for every simple closed curve $\gamma$ on $S$ and for every essential intersection of $\gamma$ with $\delta$ there is a subarc of $\gamma$ which crosses through this annulus and hence whose length is at least $\nu$; moreover, different subarcs of $\gamma$ corresponding to different essential intersections between $\gamma$ and $\delta$ are disjoint. Thus the length with respect to the singular euclidean metric on $S$ of any simple closed curve $\gamma$ on $S$ is at least $\nu i(\gamma,\delta)$. On the other hand, by construction the minimal length with respect to this metric of a curve in the free homotopy class of $\gamma$ is bounded from above by $2\max\{ai(\alpha,\gamma),i(\beta,\gamma)/ai(\alpha,\beta)\}$ and therefore the core curve $\delta$ of the annulus has the properties stated in the first part of our lemma (see [@B]).
The second part of the lemma is immediate from the first. Namely, let $\alpha,\beta\in {{\mathcal}M {\mathcal}C}(S)$ and let $a>0$. Choose $\delta\in L_a(\alpha,\beta,k_1)$ which satisfies the properties stated in the first part of the lemma. If $\gamma\in L_a(\alpha,\beta,R)$ for some $R>0$ then we have $i(\gamma,\delta)\leq k_1 R$ and hence $d(\gamma,\delta)\leq
k_1R+1$.
As an immediate consequence of Lemma 3.2 and Lemma 3.3 we observe that there is a universal number $D_1>0$ with the following property. Let $P$ be a pants decomposition for $S$, let $\beta\in
{{\mathcal}M{\mathcal}C}(S)$ and let $(\tau_i)_{0\leq i\leq m}\subset
{{\mathcal}T}T$ be any $P\to \beta$-splitting sequence; then the Hausdorff distance in ${{\mathcal}C}(S)$ between the sets $\{\Phi(\tau_i)\mid 0\leq i\leq m\}$ and $\cup_{a>0}L_a(\beta,P,k_1)$ is at most $D_1/16$. If $c>0$ and if $j\leq m$ is such that there is a vertex cycle $\gamma$ for $\tau(j)$ which is contained in $L_c(\beta,P,k_1)$ then the splitting sequence $(\tau_i)_{0\leq i\leq j}$ is a $P\to \gamma$-splitting sequence and hence the Hausdorff distance between $\cup_{a>0}L_a(\gamma,P,k_1)$ and $\cup_{a\geq c}L_a(\beta,P,k_1)$ is at most $D_1/8$. Moreover, for every $\beta\in {{\mathcal}C}(S)$ and every simple multi-curve $Q$ containing $\beta$ as a component the Hausdorff-distance between the sets $\cup_{a>0}L_a(\beta,P,k_1)$ and $\cup_{a>0}L_a(Q,P,k_1)$ is at most $D_1/8$. Thus if $Q,Q^\prime$ are pants decompositions for $S$ containing a common curve $\beta\in {{\mathcal}C}(S)$ then the Hausdorff distance between $\cup_aL_a(Q,P,k_1)$ and $\cup_aL_a(Q^\prime,P,k_1)$ is at most $D_1/4$.
On the other hand, for multi-curves $P,Q\in {{\mathcal}M{\mathcal}C}(S)$ we have $$\cup_{a>0}L_a(P,Q,k_1)=\cup_{a>0}L_a(Q,P,k_1).$$ Therefore from two applications of our above consideration we obtain the following. Let $\alpha,\beta\in {{\mathcal}C}(S)$ and let $P,P^\prime,Q,Q^\prime$ be any pants decompositions for $S$ containing $\alpha,\beta$; then the Hausdorff distance between $\cup_{a>0}L_a(P,Q,k_1)$ and $\cup_{a>0}L_a(P^\prime,Q^\prime,k_1)$ is not bigger than $D_1/2$. By our choice of $D_1$ this implies that the Hausdorff distance between the images under $\Phi$ of *any* $\alpha\to\beta$- or $\beta\to\alpha$-splitting sequences is bounded from above by $D_1$.
Now let $\alpha,\beta,\gamma\in {{\mathcal}C}(S)$ be such that their pairwise distance in ${{\mathcal}C}(S)$ is at least 3; then any two of these curves jointly fill up $S$. Choose pants decompositions $P_\alpha,P_\beta,P_\gamma$ containing $\alpha,\beta,\gamma$. Then there are unique numbers $a,b,c>0$ such that $abi(P_\alpha,P_\beta)=bci(P_\beta,P_\gamma)=aci(P_\gamma,P_\alpha)=1$. By construction, we have $$L_a(P_\alpha,P_\beta,k_1)=L_b(P_\beta,P_\alpha,k_1),\,
L_b(P_\beta,P_\gamma,k_1)=L_c(P_\gamma,P_\beta,k_1)$$ and $L_c(P_\gamma,P_\alpha,k_1)=L_a(P_\alpha,P_\gamma,k_1)$. Choose a point $\delta\in L_a(P_\alpha,P_\beta,k_1)$ such that for every $\zeta$ in ${{\mathcal}M{\mathcal}C}(S)$ we have $$i(\delta,\zeta)\leq k_1\max\{ai(P_\alpha,\zeta),
i(\zeta,P_\beta)/ai(P_\alpha,P_\beta)\};$$ such a point exists by Lemma 3.3. Applying this inequality to $\zeta=P_\gamma$ yields $ci(\delta,P_\gamma)\leq k_1$. For $\zeta=P_\beta$ we obtain $$i(\delta,P_\beta)/ci(P_\gamma,P_\beta)\leq
ai(P_\alpha,P_\beta)/ci(P_\gamma,P_\beta)=1,$$ and for $\zeta=P_\alpha$ we obtain $$i(\delta,P_\alpha)/ci(P_\gamma,P_\alpha)
\leq 1/aci(P_\gamma,P_\alpha)=1.$$ Therefore we have $\delta\in
L_c(P_\gamma,P_\beta,k_1)\cap L_a(P_\alpha,P_\beta,k_1)
\cap L_c(P_\gamma,P_\alpha,k_1)$. Together with Lemma 3.2 and our above remark we conclude that there is a universal constant $D_2>0$ such that the distance between $\phi(\alpha,\beta,\gamma)=\delta$ and the image under $\Phi$ of any $\alpha\to\beta$-splitting sequence, any $\alpha\to\gamma$-splitting sequence and any $\gamma\to\beta$-splitting sequence is bounded from above by $D_2$.
We use the map $\phi$ to derive the $\delta$-thin triangle condition for triangles whose sides are images under the map $\Phi$ of splitting sequences in ${{\mathcal}T}T$.
There is a number $D_3>0$ with the following property. Let $\alpha,\beta,\gamma\in {{\mathcal}C}(S)$ and let $a,b,c$ be the image under $\Phi$ of a $\beta\to\gamma$, $\gamma\to \alpha$, $\alpha\to \beta$-splitting sequence. Then the $D_3$-neighborhood of $a\cup b$ contains $c$.
Let $\alpha,\beta,\gamma\in {{\mathcal}C}(S)$ and assume that $d(\beta,\gamma)\leq p$ for some $p>0$. Let $(\tau_i)_{0\leq i\leq m}$ be an $\alpha\to \beta$-splitting sequence and let $(\eta_j)_{0\leq j\leq \ell}$ be an $\alpha\to\gamma$-splitting sequence; if $D_1>0$ is as above then the Hausdorff distance between $\{\Phi(\tau_i)\mid 0\leq i\leq m\}$ and $\{\Phi(\eta_j)\mid 0\leq j\leq \ell\}$ is at most $2pD_1$. Namely, we observed that the Hausdorff distance between the image under $\Phi$ of any two $\alpha\to\beta$-splitting sequences is bounded from above by $D_1$. Moreover, if $d(\beta,\gamma)=1$ then $\beta\cup\gamma\in
{{\mathcal}M{\mathcal}C}(S)$ and hence there is an $\alpha\to\beta$-splitting sequence which also is an $\alpha\to\gamma$-splitting sequence. Thus the statement of the corollary holds for $p=1$, and the general case follows from a successive application of this fact for the points on a geodesic in ${{\mathcal}C}(S)$ connecting $\beta$ to $\gamma$.
Now let $\alpha,\beta,\gamma\in {{\mathcal}C}(S)$ be arbitrary points whose pairwise distance is at least 3. Let again $(\tau_i)_{0\leq i\leq m}$ be an $\alpha\to\beta$-splitting sequence. By the definition of $\phi$ and the choice of the constant $D_2>0$ above there is some $i_0\leq m$ such that the distance betweeen $\Phi(\tau_{i_0})$ and $\phi(\alpha,\beta,\gamma)$ is at most $D_2$. Let $(\eta_j)_{0\leq j\leq \ell}$ be any $\alpha\to\phi(\alpha,\beta,\gamma)$-splitting sequence. By our above consideration, the Hausdorff distance between $\{\Phi(\tau_i)\mid 0\leq i\leq i_0\}$ and $\{\Phi(\eta_j)\mid 0\leq j\leq \ell\}$ is at most $2D_1D_2$. Similarly, let $(\zeta_j)_{0\leq j\leq n}$ be an $\alpha\to\gamma$-splitting sequence. Then there is some $j_0>0$ such that $d(\Phi(\zeta_{j_0}),
\phi(\alpha,\beta,\gamma))\leq D_2$. By our above argument, the Hausdorff distance between the sets $\{\Phi(\tau_i)\mid 0\leq i\leq i_0\}$ and $\{\Phi(\zeta_j)\mid 0\leq j\leq j_0\}$ is at most $4D_1D_2$.
As a consequence, there are numbers $a(\alpha,\beta)>0,a(\alpha,\gamma)>0$ such that $$\phi(\alpha,\beta,\gamma)\in
L_{a(\alpha,\beta)}(\beta,\alpha,k_1)\cap
L_{a(\alpha,\gamma)}(\gamma,\alpha,k_1)$$ and that the Hausdorff distance between $\cup_{a\geq a(\alpha,\beta)}L_a(\beta,\alpha,k_1)$ and\
$\cup_{a\geq a(\alpha,\gamma)}L_a(\gamma,\alpha,k_1)$ is at most $6D_1D_2$. The same argument, applied to a $\beta\to \alpha$-splitting sequence and a $\beta\to\gamma$-splitting sequence, shows that $\cup_{a\leq a(\alpha,\beta)}L_a(\alpha,\beta,k_1)$ is contained in the $6D_1D_2$-neighborhood of $\cup_aL_a(\beta,\gamma,k_1)$. Then $\{\Phi(\tau_i)\mid 0\leq i\leq m\}$ is contained in the $12D_1D_2$-neighborhood of the union of the image under $\Phi$ of a $\gamma\to\alpha$-splitting sequence and a $\beta\to\gamma$-splitting sequence. This shows the lemma.
Hyperbolicity of the curve graph now follows from Lemma 3.4 and the following criterion.
Let $(X,d)$ be a geodesic metric space. Assume that there is a number $D>0$ and for every pair of points $x,y\in X$ there is an arc $\eta(x,y):[0,1]\to X$ connecting $\eta(x,y)(0)=x$ to $\eta(x,y)(1)=y$ so that the following conditions are satisfied.
1. If $d(x,y)\leq 1$ then the diameter of $\eta(x,y)[0,1]$ is at most $D$.
2. For $x,y\in X$ and $0\leq s\leq t\leq 1$, the Hausdorff distance between $\eta(x,y)[s,t]$ and $\eta(\eta(x,y)(s),\eta(x,y)(t))[0,1]$ is at most $D$.
3. For any $x,y,z\in X$ the set $\eta(x,y)[0,1]$ is contained in the $D$-neighborhood of $\eta(x,z)[0,1]\cup \eta(z,y)[0,1].$
Then $(X,d)$ is $\delta$-hyperbolic for a number $\delta>0$ only depending on $D$.
Let $(X,d)$ be a geodesic metric space. Assume that there is a number $D>0$ and there is a family of paths $\eta(x,y):[0,1]\to X$, one for every pair of points $x,y\in X$, which satisfy the hypotheses in the statement of the proposition. To show hyperbolicity for $X$ it is then enough to show the existence of a constant $\kappa >0$ such that for all $x,y\in X$ and every geodesic $\nu:[0,\ell]\to X$ connecting $x$ to $y$, the Hausdorff-distance between $\nu[0,\ell]$ and $\eta(x,y)[0,1]$ is at most $\kappa$. Namely, if this is the case then for every geodesic triangle with sides $a,b,c$ the side $a$ is contained in the $3\kappa+D$-neighborhood of $b\cup c$.
To show the existence of such a constant $\kappa >0$, let $x,y\in X$ and let $c:[0,2^k]\to X$ be *any* path of length $\ell(c)=2^k$ parametrized by arc length connecting $x$ to $y$. Write $\eta_1=\eta(c(0),c(2^{k-1}))$ and write $\eta_2=
\eta(c(2^{k-1}),c(2^k)).$ By our assumption, the $D$-neighborhood of $\eta_1\cup \eta_2$ contains $\eta(c(0),c(2^k))$. Repeat this construction with the points $c(2^{k-2}),c(3\cdot 2^{k-2})$ and the arcs $\eta_1,\eta_2$. Inductively we conclude that the path $\eta(c(0),c(2^k))$ is contained in the $(\log_2\ell(c))D$-neighborhood of a path $\tilde c:[0,2^k]\to {{\mathcal}C}(S)$ whose restriction to each interval $[m-1,m]$ $(m\leq 2^k)$ equals up to parametrization the arc $\eta(c(m-1),c(m))$. Since $d(c(m-1),c(m))\leq 1$, by assumption the diameter of each of the sets $\eta(c(m-1),c(m))[0,1]$ is bounded from above by $D$ and therefore the arc $\eta(c(0),c(2^k))$ is contained in the $(\log_2\ell(c))D+D$-neighborhood of $c[0,2^k]$.
Now let $c:[0,k]\to X$ be a geodesic connecting $c(0)=x$ to $c(k)=y$ which is parametrized by arc length. Let $t>0$ be such that $\eta(x,y)(t)$ has maximal distance to $c[0,k]$, say that this distance equals $\chi$. Choose some $s>0$ such that $d(c(s),\eta(x,y)(t))=\chi$ and let $t_1<t<t_2$ be such that $d(\eta(x,y)(t),\eta(x,y)(t_u))= 2\chi$ $(u=1,2)$. In the case that there is no $t_1\in [0,t)$ (or $t_2\in (t,1]$) with $d(\eta(x,y)(t),\eta(x,y)(t_1))\geq 2\chi$ (or $d(\eta(x,y)(t),\eta(x,y)(t_2))\geq 2\chi$) we choose $t_1=0$ (or $t_2=1$). By our choice of $\chi$, there are numbers $s_u\in
[0,k]$ such that $d(c(s_u),\eta(x,y)(t_u))\leq \chi
\,(u=1,2)$. Then the distance between $c(s_1)$ and $c(s_2)$ is at most $6\chi$. Compose the subarc $c[s_1,s_2]$ of $c$ with a geodesic connecting $\eta(x,y)(t_1)$ to $c(s_1)$ and a geodesic connecting $c(s_2)$ to $\eta(x,y)(t_2)$. We obtain a curve $\nu$ of length at most $8\chi$. By our above observation, the $(\log_2(8\chi))D+D$-neighborhood of this curve contains the arc $\eta(\eta(x,y)(t_1),\eta(x,y)(t_2))$. However, the Hausdorff distance between $\eta(x,y)[t_1,t_2]$ and $\eta(\eta(x,y)(t_1),
\eta(x,t)(t_2))$ is at most $D$ and therefore the $(\log_2(8\chi))D+2D$-neighborhood of the arc $\nu$ contains $\eta(x,y)[t_1,t_2]$. But the distance between $\eta(t)$ and our curve $\nu$ equals $\chi$ by construction and hence we have $\chi\leq (\log_2(8\chi))D+2D$. In other words, $\chi$ is bounded from above by a universal constant $\kappa_1 >0$, and $\eta(x,y)$ is contained in the $\kappa_1$-neighborhood of the geodesic $c$.
A similar argument also shows that the $3\kappa_1$-neighborhood of $\eta(x,y)$ contains $c[0,k]$. Namely, by the above consideration, for every $t\leq 1$ the set $A(t)=\{s\in [0,k]\mid
d(c(s),\eta(x,y)(t))\leq \kappa_1\}$ is a non-empty closed subset of $[0,k]$. The diameter of the sets $A(t)$ is bounded from above by $2\kappa_1$. Assume to the contrary that $c[0,k]$ is not contained in the $3\kappa_1$-neighborhood of $\eta(x,y)$. Then there is a subinterval $[a_1,a_2]\subset [0,k]$ of length $a_2-a_1\geq 4\kappa_1$ such that $d(c(s),\eta(x,y)[0,1])>\kappa_1$ for every $s\in (a_1,a_2)$. Since $d(c(s),c(t))=\vert s-t\vert$ for all $s,t$ we conclude that for every $t\in [0,1]$ the set $A(t)$ either is entirely contained in $[0,a_1]$ or it is entirely contained in $[a_2,k]$. Define $C_1=\{t\in [0,1]\mid
A(t)\subset [0,a_1]\}$ and $C_2=\{t\in [0,1]\mid
A(t)\subset [a_2,1]\}$. Then the sets $C_1,C_2$ are disjoint and their union equals $[0,1]$; moreover, we have $0\in C_1$ and $1\in C_2$. On the other hand, the sets $C_i$ are closed. Namely, let $(t_i)\subset C_1$ be a sequence converging to some $t\in [0,1]$. Let $s_i\in A(t_i)$ and assume after passing to a subsequence that $s_i\to s\in [0,a_1]$. Now $\kappa_1\geq d(c(s_i),\eta(x,y)(t_i))\to
d(c(s),\eta(x,y)(t))$ and therefore $s\in A(t)$ and hence $t\in C_1$. However, $[0,1]$ is connected and hence we arrive at a contradiction. In other words, the geodesic $c$ is contained in the $3\kappa_1$-neighborhood of $\eta(x,y)$. This completes the proof of the proposition.
As an immediate corollary, we obtain.
The curve graph is hyperbolic.
Let $\Phi:{{\mathcal}T}T\to {{\mathcal}C}(S)$ be as before. For $\alpha,\beta\in {{\mathcal}C}(S)$ choose an $\alpha\to\beta$-splitting sequence $(\tau_i)_{0\leq i\leq m}$. Define an arc $\eta(\alpha,\beta):[0,1]\to
{{\mathcal}C}(S)$ by requiring that for $1\leq i\leq m$ the restriction of $\eta(\alpha,\beta)$ to the interval $[\frac{i}{m+2},\frac{i+1}{m+2}]$ is a geodesic connecting $\Phi(\tau_{i-1})$ to $\Phi(\tau_i)$ and that the restriction of $\eta(\alpha,\beta)$ to $[0,\frac{1}{m+2}]$ (or $[\frac{m-1}{m-2},1]$) is a geodesic connecting $\alpha$ to $\Phi(\tau_0)$ (or $\Phi(\tau_m)$ to $\beta$).
We claim that this family of arcs satisfy the assumptions in Proposition 3.5. Namely, we observed before that for all $\alpha,\beta\in {{\mathcal}C}(S)$, the Hausdorff distance between the image under $\Phi$ of *any* two $\alpha\to\beta$-splitting sequences is bounded from above by a universal constant. Now if $(\tau_i)_{0\leq i\leq m}$ is any splitting sequence, then for all $0\leq k\leq \ell\leq m$ the sequence $(\tau_i)_{k\leq i\leq \ell}$ is a $\Phi(\tau_k)\to\Phi(\tau_\ell)$-splitting sequence and hence our curve system satisfies the second condition in Proposition 3.5.
Moreover, curves $\alpha,\beta\in {{\mathcal}C}(S)$ with $d(\alpha,\beta)=1$ can be realized disjointly, and $\alpha\cup \beta$ is a multi-curve. For such a pair of curves we can choose a *constant* $\alpha\to\beta$-splitting sequence; hence our curve system also satisfies the first condition stated in Proposition 3.5. Finally, the third condition was shown to hold in Lemma 3.4.
Now hyperbolicity of the curve graph follows from Proposition 3.5.
A curve $c:[0,m]\to {{\mathcal}C}(S)$ is called an *unparametrized $p$-quasi-geodesic* for some $p>1$ if there is a homeomorphism $\rho:[0,u]\to [0,m]$ for some $u>0$ such that $$d(c(\rho(s)),c(\rho(t)))/p-p\leq \vert s-t\vert \leq
pd(c(\rho(s)),c(\rho(t)))+p$$ for all $s,t\in [0,u]$. We define a map $c:\{0,\dots,m\}\to {{\mathcal}C}(S)$ to be an unparametrized $q$-quasi-geodesic if this is the case for the curve $\tilde c$ whose restriction to each interval $[i,i+1)$ coincides with $c(i)$. The following observation is immediate from Proposition 3.5 and its proof.
There is a number $p>0$ such that the image under $\Phi$ of an arbitrary splitting sequence is an unparametrized $p$-quasi-geodesic.
By Proposition 3.5, Theorem 3.6 and their proofs, there is a universal number $D>0$ with the property that for every splitting sequence $(\tau_i)_{0\leq i\leq m}$ and every geodesic $c:[0,s]\to {{\mathcal}C}(S)$ connecting $c(0)=\Phi(\tau_0)$ to $c(s)=\Phi(\tau_m)$, the Hausdorff distance between the sets $\{\Phi(\tau_i)\mid 0\leq i\leq m\}$ and $c[0,s]$ is at most $D$. From this the corollary is immediate.
Geometry of Teichmüller space
=============================
In this section, we relate the geometry of the curve graph to the geometry of Teichmüller space equipped with the Teichmüller metric. For this we first define a map $\Psi:{{\mathcal}T}_{g,m}\to {{\mathcal}C}(S)$ as follows. By a well-known result of Bers (see [@Bu]) there is a number $\chi>0$ only depending on the topological type of $S$ such that for every complete hyperbolic metric on $S$ of finite volume there is a pants decomposition $P$ for $S$ which consists of simple closed geodesics of length at most $\chi$. Since the distance between any two points $\alpha,\beta
\in {{\mathcal}C}(S)$ is bounded from above by $i(\alpha,\beta)+1$, the collar lemma for hyperbolic surfaces (see [@Bu]) implies that the diameter in ${{\mathcal}C}(S)$ of the set of simple closed curves whose length with respect to the fixed metric is at most $\chi$ is bounded from above by a universal constant $D>0$. Define $\Psi:{{\mathcal}T}_{g,m} \to
{{\mathcal}C}(S)$ by assigning to a finite volume hyperbolic metric $h$ on $S$ a simple closed curve $\Psi(h)$ whose $h$-length is at most $\chi$. Then for any two maps $\Psi,\Psi^\prime$ with this property and every $h\in {{\mathcal}T}_{g,m}$ the distance in ${{\mathcal}C}(S)$ between $\Psi(h)$ and $\Psi^\prime(h)$ is at most $D$. Moreover, the map $\Psi$ is coarsely equivariant with respect to the action of the mapping class group ${{\mathcal}M}_{g,m}$ on ${{\mathcal}T}_{g,m}$ and ${{\mathcal}C}(S)$: For every $h\in {{\mathcal}T}_{g,m}$ and every $\phi\in {{\mathcal}M}_{g,m}$ we have $d(\Psi(\phi(h)),\phi(\Psi(h)))\leq D$.
The following result is due to Masur and Minsky (Theorem 2.6 and Theorem 2.3 of [@MM1]). For its formulation, let $d_T$ be the distance function on ${{\mathcal}T}_{g,m}$ induced by the Teichmüller metric.
1. There is a number $a>0$ such that $d(\Psi h,\Psi h^\prime)\leq ad_T(h,h^\prime)+a$ for all $h,h^\prime\in {{\mathcal}T}_{g,m}$.
2. There is a number $\tilde p>0$ with the following property. Let $\gamma:(-\infty,\infty)\to {{\mathcal}T}_{g,m}$ be any Teichmüller geodesic; then the assignment $t\to \Psi(\gamma(t))$ is an unparametrized $\tilde p$-quasi-geodesic in ${{\mathcal}C}(S)$.
Let $\gamma:(-\infty,\infty)\to {{\mathcal}T}_{g,m}$ be any Teichmüller geodesic parametrized by arc length. Then the cotangent of $\gamma$ at $t=0$ is a quadratic differential $q$ of area one defined by a pair $(\lambda,\mu)\in {{\mathcal}M{\mathcal}L}\times {{\mathcal}M{\mathcal}L}$ of measured geodesic laminations which jointly fill up $S$. The cotangent of $\gamma$ at $t$ is given by the quadratic differential $q(t)$ defined by the pair $(e^t\lambda,e^{-t}\mu)$. For $k_1>0$ as in Lemma 3.3 and $t\in \mathbb{R}$ let $\zeta(t)\in {{\mathcal}C}(S)$ be a curve whose $q(t)$-length is at most $2k_1$. For every $\beta\in [0,1]$ the $q(t+\beta)$-length of $\zeta(t)$ is bounded from above by $2ek_1$ and therefore by Lemma 3.3, for every $t$ the distance in ${{\mathcal}C}(S)$ between $\zeta(t)$ and $\zeta(t+\beta)$ is bounded from above by a universal constant $k_2>0$. In particular, the assignment $t\to \zeta(t)$ satisfies $d(\zeta(s),\zeta(t))\leq k_2\vert s-t\vert
+k_2$. Hence for the proof of our lemma, we only have to show that there is a constant $k_3>0$ such that for every $h\in {{\mathcal}T}_{g,m}$ and every holomorphic quadratic differential $q$ of area one for $h$, the distance between $\Psi(h)$ and a curve on $S$ whose $q$-length is bounded from above by $2k_1$ is uniformly bounded.
Thus let $h$ be a complete hyperbolic metric of finite volume and let $q$ be a holomorphic quadratic differential for $h$ of area one. By the collar lemma of hyperbolic geometry, a simple closed geodesic $c$ for $h$ whose length is bounded from above by $\chi$ is the core curve of an embedded annulus $A$ whose *modulus* is bounded from below by a universal constant $\epsilon >0$; we refer to [@S] for a definition of the modulus of an annulus and its properties. Then the *extremal length* of the core curve of $A$ is bounded from above by a universal constant $m >0$. Now the area of $q$ equals one and therefore the $q$-length of the core curve $c$ does not exceed $\sqrt{m}$ by the definition of extremal length (see e.g. [@Mi]). In other words, the $q$-length of the curve $\Psi(h)$ is uniformly bounded which together with Lemma 3.3 implies our claim. The theorem follows.
There are also Teichmüller geodesics in Teichmüller space which are mapped by $\Psi$ to *parametrized* quasi-geodesics in ${{\mathcal}C}(S)$. For their characterization, denote for $\epsilon >0$ by ${{\mathcal}T}_{g,m}^\epsilon$ the subset of Teichmüller space consisting of all marked hyperbolic metrics for which the length of the shortest closed geodesic is at least $\epsilon$. The set ${{\mathcal}T}_{g,m}^\epsilon$ is invariant under the action of the mapping class group and projects to a *compact* subset of moduli space. Moreover, every compact subset of moduli space is contained in the projection of ${{\mathcal}T}_{g,m}^\epsilon$ for some $\epsilon >0$.
*Cobounded* Teichmüller geodesics. i.e. Teichmüller geodesic which project to a compact subset of moduli space, relate the geometry of Teichmüller space to the geometry of the curve graph. We have.
The image under $\Psi$ of a Teichmüller geodesic $\gamma:\mathbb{R}\to {{\mathcal}T}_{g,m}$ is a parametrized quasi-geodesic in ${{\mathcal}C}(S)$ if and only if there is some $\epsilon >0$ such that $\gamma(\mathbb{R})\subset {{\mathcal}T}_{g,m}^\epsilon$.
Minsky [@Mi96] discovered earlier that the Teichmüller metric near a cobounded geodesic line has properties similar to properties of a hyperbolic geodesic metric space. Namely, for a Teichmüller geodesic $\gamma:\mathbb{R}\to {{\mathcal}T}_{g,m}^\epsilon$ the map which associates to a point $h\in {{\mathcal}T}_{g,m}$ a point on $\gamma(\mathbb{R})$ which minimizes the Teichmüller distance is coarsely Lipschitz and contracts distances in a way which is similar to the contraction property of the closest point projection from a $\delta$-hyperbolic geodesic metric space to any of its bi-infinite geodesics.
A hyperbolic geodesic metric space $X$ admits a *Gromov boundary* which is defined as follows. Fix a point $p\in X$ and for two points $x,y\in X$ define the *Gromov product* $(x,y)_p=\frac{1}{2}(d(x,p)+d(y,p)-d(x,y))$. Call a sequence $(x_i)\subset X$ *admissible* if $(x_i,x_j)_p\to \infty$ $(i,j\to \infty)$. We define two admissible sequences $(x_i),(y_i)\subset X$ to be *equivalent* if $(x_i,y_i)_p\to
\infty$. Since $X$ is hyperbolic, this defines indeed an equivalence relation (see [@BH]). The Gromov boundary $\partial X$ of $X$ is the set of equivalence classes of admissible sequences $(x_i)\subset X$. It carries a natural Hausdorff topology. For the curve graph, the Gromov boundary was determined by Klarreich [@K] (see also [@H2]).
For the formulation of Klarreich’s result, we say that a minimal geodesic lamination $\lambda$ *fills up $S$* if every simple closed geodesic on $S$ intersects $\lambda$ transversely, i.e. if every complementary component of $\lambda$ is an ideal polygon or a once punctured ideal polygon with geodesic boundary [@CEG]. For any minimal geodesic lamination $\lambda$ which fills up $S$, the number of geodesic laminations $\mu$ which contain $\lambda$ as a sublamination is bounded by a universal constant only depending on the topological type of the surface $S$. Namely, each such lamination $\mu$ can be obtained from $\lambda$ by successively subdividing complementary components $P$ of $\lambda$ which are different from an ideal triangle or a once punctured monogon by adding a simple geodesic line which either connects two non-adjacent cusps or goes around a puncture. Note that every leaf of $\mu$ which is not contained in $\lambda$ is necessarily isolated in $\mu$.
Recall that the space ${{\mathcal}L}$ of geodesic laminations on $S$ equipped with the restriction of the Hausdorff topology for compact subsets of $S$ is compact and metrizable. It contains the set ${{\mathcal}B}$ of minimal geodesic laminations which fill up $S$ as a subset which is neither closed nor dense. We define on ${{\mathcal}B}$ a new topology which is coarser than the restriction of the Hausdorff topology as follows. Say that a sequence $(\lambda_i)\subset {{\mathcal}L}$ *converges in the coarse Hausdorff topology* to a minimal geodesic lamination $\mu$ which fills up $S$ if every accumulation point of $(\lambda_i)$ with respect to the Hausdorff topology contains $\mu$ as a sublamination. Define a subset $A$ of ${{\mathcal}B}$ to be closed if and only if for every sequence $(\lambda_i)\subset A$ which converges in the coarse Hausdorff topology to a lamination $\lambda\in {{\mathcal}B}$ we have $\lambda\in A$. We call the resulting topology on ${{\mathcal}B}$ the *coarse Hausdorff topology*. The space ${{\mathcal}B}$ is not locally compact. Using this terminology, Klarreich’s result [@K] can be formulated as follows.
1. There is a natural homeomorphism $\Lambda$ of ${{\mathcal}B}$\
equipped with the coarse Hausdorff topology onto the Gromov boundary $\partial {{\mathcal}C}(S)$ of the complex of curves ${{\mathcal}C}(S)$ for $S$.
2. For $\mu\in {{\mathcal}B}$ a sequence $(c_i)\subset {{\mathcal}C}(S)$ is admissible and defines the point $\Lambda(\mu)\in \partial {{\mathcal}C}(S)$ if and only if $(c_i)$ converges in the coarse Hausdorff topology to $\mu$.
Recall that every Teichmüller geodesic in ${{\mathcal}T}_{g,m}$ is uniquely determined by a pair of projective measured laminations which jointly fill up $S$. The following corollary is immediate from Theorem 4.1 and Theorem 4.3 with $\tilde p>0$ as in Theorem 4.1.
Let $\lambda,\mu\in {{\mathcal}P{\mathcal}M{\mathcal}L}$ be such that their supports are minimal and fill up $S$. Then the image under $\Psi$ of the unique Teichmüller geodesic in ${{\mathcal}T}_{g,m}$ determined by $\lambda$ and $\mu$ is a biinfinite unparametrized $\tilde p$-quasigeodesic in ${{\mathcal}C}(S)$, and every biinfinite unparametrized $\tilde p$-quasi-geodesic in ${{\mathcal}C}(S)$ is contained in a uniformly bounded neighborhood of a curve of this form.
Our above discussion also gives information on images under $\Psi$ of a convergent sequence of geodesic lines in Teichmüller space. Namely, if $(\gamma_i)$ is such a sequence of Teichmüller geodesic lines converging to a Teichmüller geodesic which is determined by a pair of projective measured geodesic laminations $(\alpha,\beta)$ so that the support of $\alpha$ does not fill up $S$ then there is a curve $\zeta\in {{\mathcal}C}(S)$, a number $m>0$ and a sequence $j(i)\to \infty$ such that $d(\Psi(\gamma_i[0,j(i)]),\zeta)\leq m$.
On the other hand, the image under $\Psi$ of “most” Teichmüller geodesics are unparametrized quasi-geodesics of infinite diameter which are not *parametrized* quasi-geodesics. For example, let $\lambda\in {{\mathcal}P{\mathcal}M{\mathcal}L}$ be a projective measured geodesic lamination whose support $\lambda_0$ is minimal and fills up $S$ but is not *uniquely ergodic*. This means that the dimension of the space of transverse measures supported in $\lambda_0$ is at least 2. Let $\gamma:[0,\infty)\to
{{\mathcal}T}_{g,m}$ be a Teichmüller geodesic ray determined by a quadratic differential whose horizontal foliation corresponds to $\lambda$. By a result of Masur [@Ma82a], the projection of $\gamma$ to moduli space eventually leaves every compact set. On the other hand, since $\lambda_0$ is minimal and fills up $S$ the points $\gamma(t)$ converge as $t\to \infty$ to $\lambda$ viewed as a point in the *Thurston boundary* of the Thurston compactification of Teichmüller space [@Ma82b] (compare also [@FLP] for the construction of the Thurston compactification). By the definition of the map $\Psi$, the projective measured geodesic laminations defined by the curves $\Psi\gamma(t)$ converge as $t\to\infty$ to $\lambda$ and therefore the curves $\Psi\gamma(t)$ converge in the coarse Hausdorff topology to $\lambda_0$. By Theorem 4.3, this implies that the diameter of $\Psi\gamma[0,\infty)$ is infinite.
[CEG99]{}
\[Bo\] F. Bonahon, Geodesic laminations on surfaces, in [*Laminations and foliations in dynamics, geometry and topology*]{}, (Stony Brook, NY, 1998), 1–37, [*Contemp. Math.*]{}, 269, Amer. Math. Soc., Providence, RI, 2001.
\[B\] B. Bowditch, Intersection numbers and the hyperbolicity of the curve complex, [*J. reine angew. Math.*]{} 598 (2006), 105–129.
\[BH\] M. Bridson, A. Haefliger, [*Metric spaces of non-positive curvature*]{}, Springer Grundlehren 319, Springer, Berlin 1999.
\[Bu\] P. Buser, [*Geometry and spectra of compact Riemann surfaces*]{}, Birkhäuser, Boston 1992.
\[CEG\] R. Canary, D. Epstein, P. Green, Notes on notes of Thurston, in [*Analytical and geometric aspects of hyperbolic space*]{}, edited by D. Epstein, London Math. Soc. Lecture Notes 111, Cambridge University Press, Cambridge 1987.
\[FLP\] A. Fathi, F. Laudenbach, V. Poénaru, [*Travaux de Thurston sur les surfaces*]{}, Astérisque 66-67, Soc. Math. Fr. 1991/1979.
\[H2\] U. Hamenstädt, Train tracks and the Gromov boundary of the complex of curves. In [*Spaces of Kleinian groups*]{} (Y. Minsky, M. Sakuma, C. Series, eds.), London Math. Soc. Lec. Notes 329, Cambridge University Press 2006, 187–207.
\[H05\] U. Hamenstädt, Word hyperbolic extensions of surface groups, [math.GT/0505244]{}.
\[H1\] U. Hamenstädt, Geometry of the mapping class groups I: Boundary amenability, [math.GR/0510116]{}.
\[H4\] U. Hamenstädt, Geometry of the mapping class groups II: (Quasi)-geodesics, [math.GR/0511349]{}.
\[Ha\] W. J. Harvey, Boundary structure of the modular group, in [*Riemann Surfaces and Related topics: Proceedings of the 1978 Stony Brook Conference*]{}, edited by I. Kra and B. Maskit, Ann. Math. Stud. 97, Princeton, 1981.
\[IT\] Y. Imayoshi, M. Taniguchi, [*An introduction to Teichmüller spaces*]{}, Springer,\
Tokyo 1992.
\[I\] N. V. Ivanov, Mapping class groups, Chapter 12 in [*Handbook of geometric topology*]{}, edited by R. J. Daverman and R. B. Sher, Elsevier Science 2002, 523–633.
\[Ke\] S. Kerckhoff, Lines of minima in Teichmüller space, [*Duke Math. J.*]{} 65 (1992), 187–213.
\[K\] E. Klarreich, The boundary at infinity of the curve complex and the relative Teichmüller space, unpublished manuscript, 1999.
\[L\] G. Levitt, Foliations and laminations on hyperbolic surfaces, [*Topology*]{} 22 (1983), 119–135.
\[Ma82a\] H. Masur, Interval exchange transformations and measured foliations, [*Ann. Math. 155*]{} (1982), 169–200.
\[Ma82b\] H. Masur, Two boundaries of Teichmüller space, [*Duke Math. J.*]{} 49 (1982), 183–190.
\[MM1\] H. Masur, Y. Minsky, Geometry of the complex of curves I: Hyperbolicity, [*Invent. Math.*]{} 138 (1999), 103–149.
\[MM3\] H. Masur, Y. Minsky, Quasiconvexity in the curve complex, [*In the tradition of Ahlfors and Bers,*]{} III, 309–320, [*Contemp. Math.*]{}, 355, Amer. Math. Soc., Providence, RI, 2004.
\[Mi96\] Y. Minsky, Quasi-projections in Teichmüller space, [*J. Reine Angew. Math.*]{} 473 (1996), 121-136.
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{
"pile_set_name": "ArXiv"
}
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---
address: |
$^1$Image Processing Laboratory, Universitat de València, València, Spain\
$^2$Department of Statistics, University of Oxford, Oxford, UK\
$^3$German Aerospace Center, Institute of Data Science, Jena, Germany\
$^4$Max Planck Institute for Biogeochemistry, Jena, Germany
author:
- '[^1]'
bibliography:
- 'nsr.bib'
title: A Perspective on Gaussian Processes for Earth Observation
---
Introduction {#sec1 .unnumbered}
============
Earth observation (EO) by airborne and satellite remote sensing and in-situ observations play a fundamental role in monitoring our planet. In the last decade, machine learning has attained outstanding results in the estimation of bio-geo-physical variables from the acquired images at local and global scales in a time-resolved manner. Gaussian processes (GPs) [@Rasmussen06], as flexible nonparametric models to find functional relationships, have excelled in EO problems in recent years, mainly introduced for model inversion and emulation of complex codes [@CampsValls16grsm]. GPs provide not only accurate estimates but also principled uncertainty estimates for the predictions. Besides, GPs can easily accommodate multimodal data coming from different sensors and from multitemporal acquisitions. Due to their solid Bayesian formalism, GPs can include prior physical knowledge about the problem, and allow for a formal treatment of uncertainty quantification and error propagation.
In remote sensing, we often deal with [*radiative transfer models*]{} (RTMs) which implement the equations of energy transfer. These codes are needed for modelling, understanding, and predicting some variables of interest related to the state of the land cover, water bodies and atmosphere. An RTM $\f$ operating in [*forward mode*]{} generates a multidimensional [*radiance observation*]{} $\y\in\mathbb{R}^p$ seen by the sensor given a multidimensional [*parameter state vector*]{} $\x\in\mathbb{R}^d$, see Fig. \[forward\_inverse\]. Running forward simulations yields a look-up-table (LUT) of input-output pairs, $\dataset=\{(\x_i,\y_i)\}_{i=1}^n$. Solving the [*inverse problem*]{} implies learning the function $\g$ using $\dataset$ to return an estimate $\x_*$ each time a new satellite observation $\y_*$ is acquired. GPs have been used to learn both the often costly forward model $\f$ as well as the inverse model $\g$. Learning the forward model allows for faster simulations, while learning an inverse model has allowed to provide physically-meaningful, spatially-explicit, and temporally-resolved maps of variables of interest.
![Forward (solid lines) and inverse (dashed lines) modelling in Earth observation.[]{data-label="forward_inverse"}](figure){width="7.5cm"}
Despite great advances in forward and inverse modelling, GP models still have to face important challenges, such as the high computational cost involved or the derivation of faithful confidence intervals. More importantly, we posit that GP models should evolve towards [*data-driven physics-aware models*]{} that respect signal characteristics, be consistent with elementary laws of physics, and move from pure regression to observational [*causal inference*]{}.
Advances in GP inverse modelling {#advances-in-gp-inverse-modelling .unnumbered}
================================
The most important shortcoming of GPs is their high computational cost and the memory requirements, which grows cubically and quadratically with the number of training points, respectively. Recently, a great progress has been made in constructing scalable versions of GPs, demonstrating their utility in big data regimes [@hensman2013gaussian].
An important challenge in Earth observation relates to the fact that data comes with complex nonlinearities, levels and sources of noise, and non-stationarities. Standard GPs often assume homoscedastic noise and use stationary kernels though. The current state-of-the-art GP to deal with heteroscedastic noise makes use of a marginalized variational approximation [@CampsValls16grsm]. The method has resulted in excellent performance in estimating biophysical parameters (chlorophyll-a content in plants and water bodies) from acquired reflectances. In many EO applications one transforms the observed variable to linearize or Gaussianize the data via parametric transforms. A [*warped*]{} GP model has allowed learning a non-parametric optimal transformation from data, and has shown very good results in predicting vegetation parameters (chlorophyll, leaf area index, and fractional vegetation cover) from hyperspectral images [@Mateo18wgp]. Another common problem in remote sensing is that of ensuring consistency across products: estimating several related variables simultaneously can incorporate their relations in a single model. A recent latent force model (LFM) GP can encode ordinary/partial differential equations governing the system, and has allowed to monitoring crops, estimate multiple vegetation covariates simultaneously, and deal with missing observations due to the presence of clouds or sensor acquisition problems [@CampsValls18sciasi].
Making inferences with GPs is not only about obtaining point-wise estimates but also faithful uncertainty estimates, essential to perform error propagation. Inference should also contemplate [*extrapolation*]{} analysis as an ambitious far-end goal. Besides, note that we ultimately aim to characterize model error by comparing simulators to reality, calibrate models by proper estimation of (hyper)parameters, and make uncertainty statements about the world that combine models, data, and their corresponding errors. We think that the Bayesian formalism of GPs is the natural framework to tackle these yet unresolved problems.
Advances in GP forward modelling {#advances-in-gp-forward-modelling .unnumbered}
================================
Surrogate modelling, also known as [*emulation*]{}, based on GPs is gaining popularity in remote sensing. Emulators are essentially statistical models that learn to mimic the RTM code using a representative dataset $\dataset$. GPs have largely dominated the field for decades and have provided excellent accuracy and physical consistency as studied via sensitivity analysis in the context of vegetation and atmosphere models in [@CampsValls16grsm]. Once the GP model is trained, one can readily perform fast forward simulations, which in turn allows improved inversion. However, replacing an RTM with a GP model requires running expensive evaluations of $\f$ first. Recent more efficient alternatives construct an approximation to $\f$ starting with a set of support points selected iteratively [@CampsValls18sciasi]. This topic is related to active learning and Bayesian optimization, which might push results further in accuracy and sparsity, especially when modelling complex codes.
RTMs are the result of many decades of scientific research and continuous development, so they often include [*ad hoc*]{} rules, heuristics, and non-differentiable links that hamper analytic treatment. Emulation allows to account for input errors, derive predictive variance estimates, infer sensitivity values of parameters, calculate Jacobians, and perform uncertainty propagation and quantification analytically. Besides, a lot of physical knowledge used for designing RTMs could be translated in designing priors (e.g. physically plausible parameter values). These excellent capabilities have not been widely exploited in EO applications though.
Towards physics-aware GP modelling {#sec3 .unnumbered}
==================================
The GP framework allows us to include constraints and priors adapted to [*signal features*]{} such as non-stationarity, circularity, spatial-temporal relations, coloured-noise processes, and non-i.i.d. relations. Nevertheless, data-driven GP models should be further constrained to provide physically-plausible predictions. Recent approaches consider designing joint observation-simulation cross-covariances [@CampsValls18sciasi]. Recently we suggested a full framework for hybrid modelling with machine learning [@reichstein18nat], which could be formalized within the GP probabilistic framework too.
Learning dynamical physical systems is very challenging. Recent regression approaches have learned the governing equations of nonlinear dynamical systems from data, such as the Lorenz, Navier-Stokes and Schrödinger equations. Models typically impose sparsity and hierarchical modelling, but also a GP probabilistic approach has excelled in discovering ordinary and partial differential, integro-differential, and fractional order operators [@Raissi17].
The integration of physics into GP models does not only achieve improved generalization but, more importantly, endorses these grey-box models with [*consistency*]{} and [*faithfulness*]{}. As a by-product, the hybridization process has an interesting regularization effect, as physics discards implausible models and promotes simpler structures.
From regression to causation {#from-regression-to-causation .unnumbered}
============================
Understanding is more challenging than predicting, especially when no interventional studies can be conducted, as in the Earth sciences. Causal inference from observational data to estimate causal graphical models has become a mature science with effective machine learning methods to deal with both time series and non-time ordered data, see [@Peters18; @ZhaSchSpiGly17] and references therein. Causal inference methods can be classified roughly into conditional independence or constraint-based approaches and structural causal models. Constraint-based causal discovery algorithms iteratively infer graphical models utilizing conditional independence testing. In [@Runge2018d] a GP-based conditional independence test is combined with a scalable causal discovery algorithm allowing to infer high-dimensional graphical models from time series data. Constraint-based algorithms only allow to infer causal graphical models up to a Markov equivalence class. Utilizing additional assumptions, such as on the noise distribution or functional dependence, the class of structural causal models [@Peters18] allows to infer causal directionality in such undecidable Markov equivalent cases. Further GP-based causal discovery methods include [@HuaZhaSch15] where a GP model was used as a prior to capture the time-varying causal association in a non-parametric manner, while in [@FlaxmanNS16] GPs were exploited as an efficient pre-whitening step to deal with non-iid observations so common in remote sensing. Recently, [@Mateo18wgp; @CampsValls16grsm] introduced the WGP regression in additive noise models to account for post-nonlinear effects and heteroscedastic noise respectively, and applied it successfully to a set of geoscience and remote sensing bivariate problems. Some important challenges in causal inference for the Earth science are still to be solved: how to scale GP models to deal with millions of points, missing data and time aggregation as well as time sub-sampling, and complex spatial-temporal dependency structures. Testing scientific hypotheses, comparing model-vs-data causal graphs, and assessing the impacts of extreme events, are just some exciting further avenues of research.
Funding {#funding .unnumbered}
=======
GCV would like to acknowledge the support from the European Research Council (ERC) under the ERC Consolidator Grant 2014 project SEDAL (grant agreement 647423).
[^1]: Published in National Science Review 6 (4):616-618, 2019 DOI: https://academic.oup.com/nsr/article/6/4/616/5369430.
|
{
"pile_set_name": "ArXiv"
}
|
---
author:
- Nathaniel Eldredge
bibliography:
- 'allpapers.bib'
title: 'Analysis and Probability on Infinite-Dimensional Spaces'
---
Preface {#preface .unnumbered}
=======
I wrote these lecture notes for a graduate topics course I taught at Cornell University in Fall 2011 (Math 7770). The ostensible primary goal of the course was for the students to learn some of the fundamental results and techniques in the study of probability on infinite-dimensional spaces, particularly Gaussian measures on Banach spaces (also known as abstract Wiener spaces). As others who have taught such courses will understand, a nontrivial secondary goal of the course was for the instructor (i.e., me) to do the same. These notes only scratch the very surface of the subject, but I tried to use them to work through some of the basics and see how they fit together into a bigger picture. In addition to theorems and proofs, I’ve left in some more informal discussions that attempt to develop intuition.
Most of the material here comes from the books [@kuo-gaussian-book; @nualart; @bogachev-gaussian-book], and the lecture notes prepared by Bruce Driver for the 2010 Cornell Probability Summer School [@driver-cpss-notes; @driver-probability]. If you are looking to learn more, these are great places to look.[^1]
Any text marked **Question N** is something that I found myself wondering while writing this, but didn’t ever resolve. I’m not proposing them as open problems; the answers could be well-known, just not by me. If you know the answer to any of them, I’d be happy to hear about it! There are also still a few places where proofs are rough or have some gaps that I never got around to filling in.
On the other hand, something marked **Exercise N** is really meant as an exercise.
I would like to take this opportunity to thank the graduate students who attended the course. These notes were much improved by their questions and contributions. I’d also like to thank several colleagues who sat in on the course or otherwise contributed to these notes, particularly Clinton Conley, Bruce Driver, Leonard Gross, Ambar Sengupta, and Benjamin Steinhurst. Obviously, the many deficiencies in these notes are my responsibility and not theirs.
Questions and comments on these notes are most welcome. I am now at the University of Northern Colorado, and you can email me at `[email protected]`.
Introduction
============
Why analysis and probability on $\mathbb{R}^n$ is nice
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Classically, real analysis is usually based on the study of real-valued functions on finite-dimensional Euclidean space $\R^n$, and operations on those functions involving limits, differentiation, and integration. Why is $\R^n$ such a nice space for this theory?
- $\R^n$ is a nice topological space, so limits behave well. Specifically, it is a complete separable metric space, and it’s locally compact.
- $\R^n$ has a nice algebraic structure: it’s a vector space, so translation and scaling make sense. This is where differentiation comes in: the derivative of a function just measures how it changes under infinitesimal translation.
- $\R^n$ has a natural measure space structure; namely, Lebesgue measure $m$ on the Borel $\sigma$-algebra. The most important property of Lebesgue measure is that it is invariant under translation. This leads to nice interactions between differentiation and integration, such as integration by parts, and it gives nice functional-analytic properties to differentiation operators: for instance, the Laplacian $\Delta$ is a self-adjoint operator on the Hilbert space $L^2(\R^n, m)$.
Of course, a lot of analysis only involves local properties, and so it can be done on spaces that are locally like $\R^n$: e.g. manifolds. Let’s set this idea aside for now.
Why infinite-dimensional spaces might be less nice
--------------------------------------------------
The fundamental idea in this course will be: how can we do analysis when we replace $\R^n$ by an infinite dimensional space? First we should ask: what sort of space should we use? Separable Banach spaces seem to be nice. They have a nice topology (complete separable metric spaces) and are vector spaces. But what’s missing is Lebesgue measure. Specifically:
“There is no infinite-dimensional Lebesgue measure.” Let $W$ be an infinite-dimensional separable Banach space. There does not exist a translation-invariant Borel measure on $W$ which assigns positive finite measure to open balls. In fact, any translation-invariant Borel measure $m$ on $W$ is either the zero measure or assigns infinite measure to every open set.
Essentially, the problem is that inside any ball $B(x,r)$, one can find infinitely many disjoint balls $B(x_i, s)$ of some fixed smaller radius $s$. By translation invariance, all the $B(x_i, s)$ have the same measure. If that measure is positive, then $m(B(x,r)) =
\infty$. If that measure is zero, then we observe that $W$ can be covered by countably many balls of radius $s$ (by separability) and so $m$ is the zero measure.
The first sentence is essentially Riesz’s lemma: given any proper closed subspace $E$, one can find a point $x$ with $||x|| \le 1$ and $d(x,E) > 1/2$. (Start by picking any $y \notin E$, so that $d(y,E)
> 0$; then by definition there is an $z \in E$ with $d(y,z) < 2d(y,E)$. Now look at $y-z$ and rescale as needed.) Now let’s look at $B(0,2)$ for concreteness. Construct $x_1, x_2, \dots$ inductively by letting $E_n = \spanop\{x_1, \dots, x_n\}$ (which is closed) and choosing $x_{n+1}$ as in Riesz’s lemma with $||x_{n+1}|| \le 1$ and $d(x_{n+1}, E_n) > 1/2$. In particular, $d(x_{n+1}, x_i) > 1/2$ for $i \le n$. Since our space is infinite dimensional, the finite-dimensional subspaces $E_n$ are always proper and the induction can continue, producing a sequence $\{x_i\}$ with $d(x_i,
x_j) > 1/2$ for $i \ne j$, and thus the balls $B(x_i, 1/4)$ are pairwise disjoint.
Prove the above theorem for $W$ an infinite-dimensional Hausdorff topological vector space. (Do we need separability?)
A more intuitive idea why infinite-dimensional Lebesgue measure can’t exist comes from considering the effect of scaling. In $\R^n$, the measure of $B(0,2)$ is $2^n$ times larger than $B(0,1)$. When $n =
\infty$ this suggests that one cannot get sensible numbers for the measures of balls.
There are nontrivial translation-invariant Borel measures on infinite-dimensional spaces: for instance, counting measure. But these measures are useless for analysis since they cannot say anything helpful about open sets.
So we are going to have to give up on translation invariance, at least for now. Later, as it turns out, we will study some measures that recover a little bit of this: they are *quasi*-invariant under *some* translations. This will be explained in due course.
Probability measures in infinite dimensions
-------------------------------------------
If we just wanted to think about Borel measures on infinite-dimensional topological vector spaces, we actually have lots of examples from probability, that we deal with every day.
\[product-example\] Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space and $X_1, X_2,
\dots$ a sequence of random variables. Consider the infinite product space $\R^\infty$, thought of as the space of all sequences $\{x(i)\}_{i=1}^\infty$ of real numbers. This is a topological vector space when equipped with its product topology. We can equip $\R^\infty$ with its Borel $\sigma$-algebra, which is the same as the product Borel $\sigma$-algebra (verify). Then the map from $\Omega$ to $\R^\infty$ which sends $\omega$ to the sequence $x(i) =
X_i(\omega)$ is measurable. The pushforward of $\mathbb{P}$ under this map gives a Borel probability measure $\mu$ on $\R^\infty$.
The Kolmogorov extension theorem guarantees lots of choices for the joint distribution of the $X_i$, and hence lots of probability measures $\mu$. Perhaps the simplest interesting case is when the $X_i$ are iid with distribution $\nu$, in which case $\mu$ is the infinite product measure $\mu = \prod_{i=1}^\infty \nu$. Note that in general one can only take the infinite product of *probability* measures (essentially because the only number $a$ with $0 < \prod_{i=1}^\infty a < \infty$ is $a=1$).
Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space, and $\{X_t : 0 \le t \le 1\}$ be any stochastic process. We could play the same game as before, getting a probability measure on $\R^{[0,1]}$ (with its product $\sigma$-algebra). This case is not as pleasant because nothing is countable. In particular, the Borel $\sigma$-algebra generated by the product topology is not the same as the product $\sigma$-algebra (exercise: verify this, perhaps by showing that the latter does not contain singleton sets.) Also, the product topology on $\R^{[0,1]}$ is rather nasty; for example it is not first countable. (In contrast, $\R^\infty$ with its product topology is actually a Polish space.) So we will avoid examples like this one.
As before, but now assume $\{X_t : 0 \le t \le 1\}$ is a *continuous* stochastic process. We can then map $\Omega$ into the Banach space $C([0,1])$ in the natural way, by sending $\omega$ to the continuous function $X_\cdot(\omega)$. One can check that this map is measurable when $C([0,1])$ is equipped with its Borel $\sigma$-algebra. (Hint: $||x|| \le 1$ if and only if $|x(t)| \le
1$ for all $t$ in a countable dense subset of $[0,1]$.) So by pushing forward $\mathbb{P}$ we get a Borel probability measure on $C([0,1])$. For example, if $X_t$ is Brownian motion, this is the classical Wiener measure.
So probability measures seem more promising. We are going to concentrate on Gaussian probability measures. Let’s start by looking at them in finite dimensions.
Gaussian measures in finite dimensions
--------------------------------------
In one dimension everyone knows what Gaussian means. We are going to require our measures / random variables to be centered (mean zero) to have fewer letters floating around. However we are going to include the degenerate case of zero variance.
A Borel probability measure $\mu$ on $\R$ is **Gaussian** with variance $\sigma^2$ iff $$\mu(B) = \int_B \frac{1}{\sqrt{2 \pi} \sigma} e^{-x^2/2\sigma^2}\,dx$$ for all Borel sets $B \subset \R$. We also want to allow the case $\sigma = 0$, which corresponds to $\mu = \delta_0$ being a Dirac mass at 0.
We could also specify $\mu$ in terms of its Fourier transform (or characteristic function). The above condition is equivalent to having $$\int_\R e^{i \lambda x} \mu(dx) = e^{-\sigma^2 \lambda^2/2}$$ for all $\lambda \in \R$. (Note $\sigma = 0$ is naturally included in this formulation.)
A random variable $X$ on some probability space $(\Omega,
\mathcal{F}, \mathbb{P})$ is Gaussian with variance $\sigma^2$ if its distribution measure is Gaussian, i.e. $$\begin{aligned}
\mathbb{P}(X \in B) = \int_B \frac{1}{\sqrt{2 \pi} \sigma} e^{-x^2/2\sigma^2}\,dx.
\end{aligned}$$ for all Borel sets $B$. For $\sigma = 0$ we have the constant random variable $X = 0$. Equivalently $$\mathbb{E}[e^{i \lambda X}] = e^{- \sigma^2 \lambda^2/2}$$ for all $\lambda \in \R$.
Let’s make a trivial observation: $\mu$ is not translation invariant. However, translation doesn’t mess it up completely.
Let $\mu$ be a measure on a vector space $W$. For $y \in W$, we denote by $\mu_y$ the translated measure defined by $\mu_y(A) =
\mu(A-y)$. In other words, $\int_W f(x) \mu_y(dx) = \int_W f(x+y)$.
Check that I didn’t screw up the signs in the previous paragraph.
A measure $\mu$ on a vector space $W$ is said to be **quasi-invariant** under translation by $y \in W$ if the measures $\mu, \mu_y$ are mutually absolutely continuous (or **equivalent**); that is, if $\mu(A)=0 \Leftrightarrow \mu_y(A)
= 0$ for measurable sets $A \subset W$.
Intuitively, quasi-invariance means that translation can change the measure of a set, but it doesn’t change whether or not the measure is zero.
One way I like to think about equivalent measures is with the following baby example. Suppose I have two dice which look identical on the surface, but one of them is fair, and the other produces numbers according to the distribution $(0.1,0.1,0.1,0.1,0.1,0.5)$ (i.e. it comes up $6$ half the time). (Note that they induce equivalent measures on $\{1,2,3,4,5,6\}$: in both cases the only set of measure zero is the empty set.) I pick one of the dice and ask you to determine which one it is. If you roll a lot of 6s, you will have a strong suspicion that it’s the unfair die, but you can’t absolutely rule out the possibility that it’s the fair die and you are just unlucky.
On the other hand, suppose one of my dice always comes up even, and the other always comes up odd. In this case the induced measures are mutually singular: there is a set (namely $\{1,3,5\}$) to which one gives measure 0 and the other gives measure 1. If I give you one of these dice, then all you have to do is roll it once and see whether the number is even or odd, and you can be (almost) sure which die you have.
For Gaussian measures on $\R$, note that if $\sigma \ne 0$, then $\mu$ is quasi-invariant under translation by any $y \in \R$. This is a trivial fact: both $\mu$ and $\mu_y$ have positive densities with respect to Lebesgue measure $m$, so $\mu(A) = 0$ iff $\mu_y(A) = 0$ iff $m(A) = 0$. We’ll also note that, as absolutely continuous measures, they have a Radon-Nikodym derivative, which we can compute just by dividing the densities: $$\begin{aligned}
\frac{d \mu_y}{d \mu}(x) &= \frac{\frac{1}{\sqrt{2 \pi} \sigma} e^{-(x-y)^2/2\sigma}}
{\frac{1}{\sqrt{2 \pi} \sigma} e^{-x^2/2\sigma^2}} \\
&= e^{-\frac{y^2}{2\sigma^2} + \frac{xy}{\sigma^2}}.\end{aligned}$$ Just pay attention to the form of this expression, as we will see it again later.
On the other hand, in the degenerate case $\sigma = 0$, then $\mu =
\delta_0$ and $\mu_y = \delta_y$ are mutually singular.
Now let’s look at the $n$-dimensional case.
An $n$-dimensional random vector $\mathbf{X} = (X_1, \dots, X_n)$ is **Gaussian** if and only if $\mathbf{\lambda} \cdot \mathbf{X} := \sum
\lambda_i X_i$ is a Gaussian random variable for all $\mathbf{\lambda} = (\lambda_1, \dots, \lambda_n) \in \R^n$.
Or in terms of measures:
Let $\mu$ be a Borel probability measure on $\R^n$. For each $\mathbf{\lambda} \in \R^n$, we can think of the map $\R^n \ni \mathbf{x}
\mapsto \mathbf{\lambda} \cdot \mathbf{x} \in \R$ as a random variable on the probability space $(\R^n, \mathcal{B}_{\R^n},
\mu)$. $\mu$ is **Gaussian** if and only if this random variable is Gaussian for each $\mathbf{\lambda}$.
Of course, we know that the distribution of $\mathbf{X}$ is uniquely determined by its $n \times n$ covariance matrix $\Sigma$, where $\Sigma_{ij} = \Cov(X_i, X_j)$. Note that $\Sigma$ is clearly symmetric and positive semidefinite. Furthermore, any symmetric, positive semidefinite matrix $\Sigma$ can arise as a covariance matrix: let $\mathbf{Z} = (Z_1, \dots, Z_n)$ where $Z_i$ are iid Gaussian with variance $1$, and set $\mathbf{X} = \Sigma^{1/2} \mathbf{Z}$.
A consequence of this is that if $(X_1, \dots, X_n)$ has a joint Gaussian distribution, then the $X_i$ are independent if and only if they are uncorrelated (i.e. $\Cov(X_i, X_j) = 0$ for $i \ne j$, i.e. $\Sigma$ is diagonal). Note that this *fails* if all we know is that each $X_i$ has a Gaussian distribution.
$\mathbf{X}$ is Gaussian if and only if it has characteristic function $$\mathbb{E}[e^{i \mathbf{\lambda} \cdot \mathbf{X}}] = e^{- \frac{1}{2} \mathbf{\lambda} \cdot \Sigma \mathbf{\lambda}}$$ where $\Sigma$ is the covariance matrix of $\mathbf{X}$.
Or, in terms of measures:
A probability measure $\mu$ on $\R^n$ is Gaussian if and only if $$\int_{\R^n} e^{i \mathbf{\lambda} \cdot \mathbf{x}} \mu(d\mathbf{x}) = e^{-
\frac{1}{2}\mathbf{\lambda} \cdot \Sigma \mathbf{\lambda}}$$ for some $n \times n$ matrix $\Sigma$, which is necessarily positive semidefinite and can be chosen symmetric.
We would like to work more abstractly and basis-free, in preparation for the move to infinite dimensions. The map $\mathbf{x} \mapsto
\mathbf{\lambda} \cdot \mathbf{x}$ is really just a linear functional on $\R^n$. So let’s write:
\[finite-gaussian-good\] Let $\mu$ be a Borel probability measure on a finite-dimensional topological vector space $W$. Then each $f \in W^*$ can be seen as a random variable on the probability space $(W, \mathcal{B}_W, \mu)$. $\mu$ is **Gaussian** if and only if, for each $f \in W^*$, this random variable is Gaussian.
Equivalently, $\mu$ is Gaussian iff the pushforward $\mu \circ f^{-1}$ is a Gaussian measure on $\R$ for each $f \in W^*$.
Of course we have not done anything here because a finite-dimensional topological vector space $W$ is just some $\R^n$ with its usual topology. Every linear functional $f \in W^*$ is of the form $f(\mathbf{x}) = \mathbf{\lambda} \cdot \mathbf{x}$, and all such linear functionals are continuous, hence measurable.
If $f, g \in W^*$ are thought of as Gaussian random variables on $(W,
\mu)$, then $q(f,g) = \Cov(f,g)$ is a symmetric, positive semidefinite, bilinear form on $W^*$. We’ll also write $q(f) = q(f,f) = \Var(f)$. $q$ is the basis-free analogue of the covariance matrix; we could call it the covariance form. As we argued above, in this finite-dimensional case, any such bilinear form can arise as a covariance form.
Another way to think about this is that since each $f \in W^*$ is Gaussian, it is certainly square-integrable, i.e. $\int_W |f(x)|^2
\mu(dx) = E|f|^2 = \Var(f) < \infty$. So $V^*$ can be thought of as a subspace of $L^2(V, \mu)$. Then $q$ is nothing but the restriction of the $L^2$ inner product to the subspace $W^*$.
(Technically, $q$ may be degenerate, in which case it is actually the quotient of $W^*$ by the kernel of $q$ that we identify as a subspace of $L^2(W, \mu)$.)
The support of the measure $\mu$ is given by $$\supp \mu = \bigcap_{q(f,f) = 0} \ker f.$$ One could write $\supp \mu = (\ker q)^\perp$. In particular, if $q$ is positive definite, then $\mu$ has full support. (Recall that the support of a measure $\mu$ is defined as the smallest closed set with full measure.)
The restriction of $\mu$ to its support is a nondegenerate Gaussian measure (i.e. the covariance form is positive definite).
$\mu$ is quasi-invariant under translation by $y$ if and only if $y
\in \supp \mu$. If $y \notin \supp \mu$, then $\mu$ and $\mu_y$ are mutually singular. (We’ll see that in infinite dimensions, the situation is more complex.)
In terms of characteristic functions, then, we have
A Borel probability measure $\mu$ on a finite-dimensional topological vector space $W$ is Gaussian if and only if, for each $f
\in W^*$, we have $$\int_W e^{i f(x)} \mu(dx) = e^{-\frac{1}{2}q(f,f)}$$ where $q$ is some positive semidefinite symmetric bilinear form on $W^*$.
Infinite-dimensional Gaussian measures
======================================
Definition \[finite-gaussian-good\] will generalize pretty immediately to infinite-dimensional topological vector spaces. There is just one problem. An arbitrary linear functional on a topological vector space can be nasty; in particular, it need not be Borel measurable, in which case it doesn’t represent a random variable. But continuous linear functionals are much nicer, and are Borel measurable for sure, so we’ll restrict our attention to them.[^2]
As usual, $W^*$ will denote the continuous dual of $W$.
Let $W$ be a topological vector space, and $\mu$ a Borel probability measure on $W$. $\mu$ is **Gaussian** iff, for each continuous linear functional $f \in W^*$, the pushforward $\mu \circ f^{-1}$ is a Gaussian measure on $\R$, i.e. $f$ is a Gaussian random variable on $(W, \mathcal{B}_W, \mu)$.
As before, we get a covariance form $q$ on $W^*$ where $q(f,g) =
\Cov(f,g)$. Again, $W^*$ can be identified as a subspace of $L^2(W,
\mu)$, and $q$ is the restriction of the $L^2$ inner product.
A Borel probability measure $\mu$ on a topological vector space $W$ is Gaussian if and only if, for each $f \in W^*$, we have $$\int_W e^{i f(x)} \mu(dx) = e^{-\frac{1}{2}q(f,f)}$$ where $q$ is some positive semidefinite symmetric bilinear form on $W^*$.
If $f_1, \dots, f_n$ in $W^*$, then $(f_1, \dots, f_n)$ has a joint Gaussian distribution.
Any $q$-orthogonal subset of $W^*$ is an independent set of random variables on $(W, \mu)$.
For the most part, we will concentrate on the case that $W$ is a separable Banach space. But as motivation, we first want to look at a single case where it isn’t. If you want more detail on the general theory for topological vector spaces, see Bogachev [@bogachev-gaussian-book].
Motivating example: $\R^\infty$ with product Gaussian measure
=============================================================
As in Example \[product-example\], let’s take $W = \R^\infty$ with its product topology. Let’s record some basic facts about this topological vector space.
$W$ is a Fréchet space (its topology is generated by a countable family of seminorms).
$W$ is a Polish space. (So we are justified in doing all our topological arguments with sequences.)
The topology of $W$ does not come from any norm.
The Borel $\sigma$-algebra of $W$ is the same as the product $\sigma$-algebra.
Every continuous linear functional $f \in W^*$ is of the form $$f(x) = \sum_{i=1}^n a_i x(i)$$ for some $a_1, \dots, a_n \in \R$. Thus $W^*$ can be identified with $c_{00}$, the set of all real sequences which are eventually zero.
Let’s write $e_i$ for the element of $W$ with $e_i(j) = \delta_{ij}$, and $\pi_i$ for the projection onto the $i$’th coordinate $\pi_i(x) =
x(i)$. (Note $\pi_i \in W^*$; indeed they form a basis.)
As in Example \[product-example\], we choose $\mu$ to be an infinite product of Gaussian measures with variance $1$. Equivalently, $\mu$ is the distribution of an iid sequence of standard Gaussian random variables. So the random variables $\pi_i$ are iid standard Gaussian.
$\mu$ is a Gaussian measure. The covariance form of $\mu$ is given by $$q(f,g) = \sum_{i=1}^\infty f(e_i) g(e_i).$$ (Note that the sum is actually finite.)
$\mu$ has full support.
$q$ is actually positive definite: the only $f \in W^*$ with $q(f,f) =
0$ is $f=0$. So $W^*$ is an honest subspace of $L^2(W, \mu)$. It is not a closed subspace, though; that is, $W^*$ is not complete in the $q$ inner product. Let $K$ denote the $L^2(W,\mu)$-closure of $W^*$.
\[Kexercise\] Show that $K$ consists of all functions $f : W \to \R$ of the form $$\label{Kdef}
f(x) = \sum_{i=1}^\infty a_i x(i)$$ where $\sum_{i=1}^\infty |a_i|^2 < \infty$. This formula requires some explanation. For an arbitrary $x \in W$, the sum in (\[Kdef\]) may not converge. However, show that it does converge for $\mu$-a.e. $x \in W$. (Hint: Sums of independent random variables converge a.s. as soon as they converge in $L^2$; see Theorem 2.5.3 of Durrett [@durrett].) Note well that the measure-1 set on which (\[Kdef\]) converges depends on $f$, and there will not be a single measure-1 set where convergence holds for every $f$. Moreover, show each $f \in K$ is a Gaussian random variable.
($K$ is isomorphic to $\ell^2$; this should make sense, since it is the completion of $W^* = c_{00}$ in the $q$ inner product, which is really the $\ell^2$ inner product.)
Now let’s think about how $\mu$ behaves under translation. A first guess, by analogy with the case of product Gaussian measure on $\R^n$, is that it is quasi-invariant under all translations. But let’s look closer at the finite-dimensional case. If $\nu$ is standard Gaussian measure on $\R^n$, i.e. $$d\nu = \frac{1}{(2 \pi)^{n/2}} e^{-|x|^2/2} dx$$ then a simple calculation shows $$\label{radon-finite}
\frac{d \nu_y}{d \nu}(x) = e^{-\frac{1}{2} |y|^2 + x \cdot y}.$$ Note that the Euclidean norm of $y$ appears. Sending $n \to \infty$, the Euclidean norm becomes the $\ell^2$ norm. This suggests that $\ell^2$ should play a special role. In particular, translation by $y$ is not going to produce a reasonable positive Radon-Nikodym derivative if $\norm{y}_{\ell^2} = \infty$.
Let’s denote $\ell^2$, considered as a subset of $W$, by $H$. $H$ has a Hilbert space structure coming from the $\ell^2$ norm, which we’ll denote by ${\norm{\cdot}}_H$. We note that, as shown in Exercise \[Kexercise\], that for fixed $h \in H$, $(h,x)_H$ makes sense not only for $x \in H$ but for $\mu$-a.e. $x \in W$, and $(h,\cdot)_H$ is a Gaussian random variable on $(W,\mu)$ with variance $\norm{h}_H^2$.
\[Special case of the Cameron-Martin theorem\] If $h \in H$, then $\mu$ is quasi-invariant under translation by $h$, and $$\label{radon-infinite}
\frac{d \mu_h}{d \mu}(x) = e^{-\frac{1}{2} \norm{h}_H^2 + (h,x)_H}.$$ Conversely, if $y \notin H$, then $\mu, \mu_y$ are mutually singular.
We are trying to show that $$\label{radon2}
\mu_h(B) = \int_B e^{-\frac{1}{2} \norm{h}_H^2 + (h,x)_H} \mu(dx)$$ for all Borel sets $B \subset W$. It is sufficient to consider the case where $B$ is a “cylinder set” of the form $B = B_1 \times
\dots \times B_n \times \R \times \dots$, since the collection of all cylinder sets is a $\pi$-system which generates the Borel $\sigma$-algebra. But this effectively takes us back to the $n$-dimensional setting, and unwinding notation will show that in this case (\[radon-infinite\]) is the same as (\[radon-finite\]).
This is a bit messy to write out; here is an attempt. If you don’t like it I encourage you to try to just work it out yourself.
Since $W$ is a product space, let us decompose it as $W = \R^n \times R^\infty$, writing $x \in W$ as $(x_n, x_\infty)$ with $x_n = (x(1), \dots, x(n))$ and $x_\infty =
(x(n+1), \dots)$. Then $\mu$ factors as $\mu^n \times \mu^\infty$, where $\mu^n$ is standard Gaussian measure on $\R^n$ and $\mu^\infty$ is again product Gaussian measure on $\R^\infty$. $\mu_h$ factors as $\mu^n_{h_n} \times \mu^\infty_{h_\infty}$. Also, the integrand in (\[radon2\]) factors as $$e^{-\frac{1}{2} |h_n|^2 + h_n \cdot x_n} e^{-\frac{1}{2}
||h_\infty||_{\ell_\infty}^2 + (h_\infty, x_\infty)_{\ell^2}}.$$ So by Tonelli’s theorem the right side of (\[radon2\]) equals $$\int_{B_1 \times \dots \times B_n} e^{-\frac{1}{2} |h_n|^2 + h_n
\cdot x_n} \mu^n(dx_n) \int_{\R^\infty} e^{-\frac{1}{2}
||h_\infty||_{\ell_\infty}^2 + (h_\infty, x_\infty)_{\ell^2}} \mu^\infty(dx_\infty).$$ The first factor is equal to $\mu^n_{h_n}(B_1 \times \dots \times
B_n)$ as shown in (\[radon-finite\]). Since $(h_\infty,
\cdot)_{\ell^2}$ is a Gaussian random variable on $(\R^\infty,
\mu^\infty)$ with variance $||h_\infty||^2_{\ell^2}$, the second factor is of the form $E[e^{X - \sigma^2/2}]$ for $X \sim N(0,
\sigma^2)$, which is easily computed to be $1$. Since $\mu^\infty_{h_\infty}(\R^\infty) = 1$ also (it is a probability measure), we are done with the forward direction.
For the converse direction, suppose $h \notin H$. Then, by the contrapositive of Lemma \[ell2\], there exists $g \in \ell^2$ such that $\sum h(i) g(i)$ diverges. Let $A = \{ x \in W : \sum x(i)
g(i) \text{ converges}\}$; this set is clearly Borel, and we know $\mu(A) = 1$ by Exercise \[Kexercise\]. But if $\sum x(i) g(i)$ converges, then $\sum (x-h)(i)g(i)$ diverges, so $A-h$ is disjoint from $A$ and $\mu_h(A) = \mu(A-h) = 0$.
We call $H$ the **Cameron–Martin space** associated to $(W, \mu)$.
$H$ is dense in $W$, and the inclusion map $H \hookrightarrow W$ is continuous (with respect to the $\ell^2$ topology on $H$ and the product topology on $W$).
Although $H$ is dense, there are several senses in which it is small.
$\mu(H) = 0$.
For $x \in W$, $x \in H$ iff $\sum_i |\pi_i(x)|^2 < \infty$. Note that the $\pi_i$ are iid $N(0,1)$ random variables on $(W, \mu)$. So by the strong law of large numbers, for $\mu$-a.e. $x \in W$ we have $\frac{1}{n} \sum_{i=1}^n
|\pi_i(x)|^2 \to 1$; in particular $\sum_i |\pi_i(x)|^2 = \infty$.
Any bounded subset of $H$ is precompact and nowhere dense in $W$. In particular, $H$ is meager in $W$.
So $\mu$ is quasi-invariant only under translation by elements of the small subset $H$.
Abstract Wiener space
=====================
Much of this section comes from Bruce Driver’s notes [@driver-probability] and from Kuo’s book [@kuo-gaussian-book].
An **abstract Wiener space** is a pair $(W,\mu)$ consisting of a separable Banach space $W$ and a Gaussian measure $\mu$ on $W$.
Later we will write an abstract Wiener space as $(W,H,\mu)$ where $H$ is the Cameron–Martin space. Technically this is redundant because $H$ will be completely determined by $W$ and $\mu$. Len Gross’s original development [@gross-measurable-hilbert-1962; @gross-abstract-wiener] went the other way, starting with $H$ and choosing a $(W,\mu)$ to match it, and this choice is not unique. We’ll discuss this more later.
$(W,\mu)$ is **non-degenerate** if the covariance form $q$ on $W^*$ is positive definite.
\[full-support-implies-nondegenerate\] If $\mu$ has full support (i.e. $\mu(U) > 0$ for every nonempty open $U$) then $(W,\mu)$ is non-degenerate. (For the converse, see Exercise \[nondegenerate-implies-full-support\] below.)
From now on, we will assume $(W,\mu)$ is non-degenerate unless otherwise specified. (This assumption is really harmless, as will be justified in Remark \[rk-non-degenerate\] below.) So $W^*$ is honestly (injectively) embedded into $L^2(\mu)$, and $q$ is the restriction to $W^*$ of the $L^2(\mu)$ inner product. As before, we let $K$ denote the closure of $W^*$ in $L^2(\mu)$.
Note that we now have two different topologies on $W^*$: the operator norm topology (under which it is complete), and the topology induced by the $q$ or $L^2$ inner product (under which, as we shall see, it is not complete). The interplay between them will be a big part of what we do here.
Measure-theoretic technicalities
--------------------------------
The main point of this subsection is that the continuous linear functionals $f \in W^*$, and other functions you can easily build from them, are the only functions on $W$ that you really have to care about.
Let $\mathcal{B}$ denote the Borel $\sigma$-algebra on $W$.
\[weak-sigma-field\] Let $\sigma(W^*)$ be the $\sigma$-field on $W$ generated by $W^*$, i.e. the smallest $\sigma$-field that makes every $f \in W^*$ measurable. Then $\sigma(W^*) = \mathcal{B}$.
Note that the *topology* generated by $W^*$ is not the same as the original topology on $W$; instead it’s the weak topology.
Since each $f \in B^*$ is Borel measurable, $\sigma(W^*) \subset
\mathcal{B}$ is automatic.
Let $B$ be the closed unit ball of $W$; we will show $B \in
\sigma(W^*)$. Let $\{x_n\}$ be a countable dense subset of $W$. By the Hahn-Banach theorem, for each $x_n$ there exists $f_n \in W^*$ with $||f_n||_{W^*} =1$ and $f_n(x_n) = ||x_n||$. I claim that $$\label{B-cap}
B = \bigcap_{n = 1}^\infty \{x : |f_n(x)| \le 1\}.$$ The $\subset$ direction is clear because for $x \in B$, $|f_n(x)|
\le ||f_n||\cdot ||x|| = ||x|| \le 1$. For the reverse direction, suppose $|f_n(x)| \le 1$ for all $n$. Choose a sequence $x_{n_k} \to x$; in particular $f_{n_k}(x_{n_k}) = ||x_{n_k}|| \to ||x||$. But since $||f_{n_k}|| = 1$, we have $||f_{n_k}(x_{n_k}) - f_{n_k}(x)|| \le ||x_{n_k} - x|| \to 0$, so $||x|| = \lim f_{n_k}(x_{n_k}) = \lim f_{n_k}(x) \le 1$. We have thus shown $B \in \sigma(W^*)$, since the right side of (\[B-cap\]) is a countable intersection of sets from $W^*$.
If you want to show $B(y,r) \in \sigma(W^*)$, we have $$B(y,r) = \bigcap_{n = 1}^\infty \{x : |f_n(x) - f_n(y))| < r\}.$$
Now note that any open subset $U$ of $W$ is a *countable* union of closed balls (by separability) so $U \in \sigma(W^*)$ also. Thus $\mathcal{B} \subset \sigma(W^*)$ and we are done.
Note that we used the separability of $W$, but we did not assume that $W^*$ is separable.
If $W$ is not separable, Lemma \[weak-sigma-field\] may be false. For a counterexample, consider $W = \ell^2(E)$ for some uncountable set $E$. One can show that $\sigma(W^*)$ consists only of sets that depend on countably many coordinates. More precisely, for $A \subset E$ let $\pi_A : \ell^2(E) \to \ell^2(A)$ be the restriction map. Show that $\sigma(W^*)$ is exactly the set of all $\pi_A^{-1}(B)$, where $A$ is countable and $B \subset \ell^2(A)$ is Borel. In particular, $\sigma(W^*)$ doesn’t contain any singletons (in fact, every nonempty subset of $\sigma(W^*)$ is non-separable).
Is Lemma \[weak-sigma-field\] *always* false for non-separable $W$?
Functionals $f \in W^*$ are good for approximation in several senses. We are just going to quote the following results. The proofs can be found in [@driver-probability], and are mostly along the same lines that you prove approximation theorems in basic measure theory. For this subsection, assume $\mu$ is a Borel probability measure on $W$ (not necessarily Gaussian).
Let $\mathcal{F} C_c^\infty(W)$ denote the “smooth cylinder functions” on $W$: those functions $F : W \to \R$ of the form $F(x)
= \varphi(f_1(x), \dots, f_n(x))$ for some $f_1, \dots, f_n \in W^*$ and $\varphi \in C_c^\infty(\R)$. (Note despite the notation that $F$ is not compactly supported; in fact there are no nontrivial continuous functions on $W$ with compact support.)
Let $\mathcal{T}$ be the “trigonometric polynomials” on $W$: those functions $F : W \to \R$ of the form $F(x) = a_1 e^{i f_1(x)} +
\dots + a_n e^{i f_n(x)}$ for $a_1, \dots, a_n \in \R$, $f_1, \dots,
f_n \in W^*$.
$\mathcal{F} C^\infty_c$ and $\mathcal{T}$ are each dense in $L^p(W,
\mu)$ for any $1 \le p < \infty$.
A nice way to prove this is via Dynkin’s multiplicative system theorem (a functional version of the $\pi$-$\lambda$ theorem).
\[fourier-unique\] Let $\mu$, $\nu$ be two Borel probability measures on $W$. If $\int
e^{i f(x)} \mu(dx) = \int e^{i f(x)} \nu(dx)$ for all $f \in W^*$, then $\mu = \nu$.
We could think of the Fourier transform of $\mu$ as the map $\hat{\mu}
: W^* \to \R$ defined by $\hat{\mu}(f) = \int e^{i f(x)} \mu(dx)$. The previous theorem says that $\hat{\mu}$ completely determines $\mu$.
Fernique’s theorem
------------------
The first result we want to prove is Fernique’s theorem [@fernique70], which in some sense says that a Gaussian measure has Gaussian tails: the probability of a randomly sampled point being at least a distance $t$ from the origin decays like $e^{-t^2}$. In one dimension this is easy to prove: if $\mu$ is a Gaussian measure on $\R$ with, say, variance 1, we have $$\label{gaussian-tail-1d}
\begin{split}
\mu(\{x : |x| > t\}) &= 2 \int_t^\infty \frac{1}{\sqrt{2 \pi}}
e^{-x^2/2}\,dx \\
&\le \frac{2}{\sqrt{2\pi}} \int_t^\infty \frac{x}{t} e^{-x^2/2}\,dx
\\
&= \frac{2}{t \sqrt{2 \pi}} e^{-t^2/2}
\end{split}$$ where the second line uses the fact that $\frac{x}{t} \ge 1$ for $x
\ge t$, and the third line computes the integral directly.
This is sort of like a heat kernel estimate.
\[fernique-theorem\] Let $(W,\mu)$ be an abstract Wiener space. There exist $\epsilon >
0$, $C > 0$ such that for all $t$, $$\mu(\{x : ||x||_W \ge t\}) \le C e^{-\epsilon t^2}.$$
The proof is surprisingly elementary and quite ingenious.
Let’s prove Fernique’s theorem. We follow Driver’s proof [@driver-probability Section 43.1]. Some of the details will be sketched; refer to Driver to see them filled in.
The key idea is that products of Gaussian measures are “rotation-invariant.”
Let $(W,\mu)$ be an abstract Wiener space. Then the product measure $\mu^2 = \mu \times \mu$ is a Gaussian measure on $W^2$.
If you’re worried about technicalities, you can check the following: $W^2$ is a Banach space under the norm $||(x,y)||_{W^2} := ||x||_W + ||y||_W$; the norm topology is the same as the product topology; the Borel $\sigma$-field on $W^2$ is the same as the product of the Borel $\sigma$-fields on $W$.
Let $F \in (W^2)^*$. If we set $f(x) = F(x,0)$, $g(y) = F(0,y)$, we see that $f,g \in W^*$ and $F(x,y) =f(x) + g(y)$. Now when we compute the Fourier transform of $F$, we find $$\begin{aligned}
\int_{W^2} e^{i \lambda F(x,y)} \mu^2(dx,dy) &= \int_W e^{i \lambda
f(x)} \mu(dx)
\int_W e^{i \lambda g(y)} \mu(dy) \\
&= e^{-\frac{1}{2}\lambda^2 (q(f,f) + q(g,g))}.
\end{aligned}$$
For $\theta \in \R$, define the “rotation” $R_\theta$ on $W^2$ by $$R_\theta(x,y) = (x \cos\theta - y \sin\theta, x \sin \theta + y
\cos \theta).$$ (We are actually only going to use $R_{\pi/4}(x,y) =
\frac{1}{\sqrt{2}} (x-y, x+y)$.) If $\mu$ is Gaussian, $\mu^2$ is invariant under $R_\theta$.
Invariance of $\mu^2$ under $R_{\pi/4}$ is the only hypothesis we need in order to prove Fernique’s theorem. You might think this is a very weak hypothesis, and hence Fernique’s theorem should apply to many other classes of measures. However, it can actually be shown that any measure $\mu$ satisfying this condition must in fact be Gaussian, so no generality is really gained.
Let $\nu = \mu^2 \circ R_\theta^{-1}$; we must show that $\nu = \mu^2$. It is enough to compare their Fourier transforms. Let $F \in (W^2)^*$, so $W(x,y) = f(x) + g(y)$, and then $$\begin{aligned}
\int_{W^2} e^{i (f(x) + g(y))} \nu(dx,dy) &= \int_{W^2} e^{i (f(Tx) +
g(Ty))} \mu^2(dx,dy) \\
&= \int_{W^2} e^{i (\cos \theta f(x) - \sin \theta f(y) +
\sin\theta g(x) + \cos\theta g(y))}\, \mu^2(dx,dy) \\
&= \int_{W} e^{i (\cos \theta f(x) +\sin\theta g(x))}\, \mu(dx)
\int_{W} e^{i (- \sin \theta f(y) + \cos\theta g(y))}\, \mu(dy)
\\
&= e^{-\frac{1}{2}(\cancel{(\sin^2 \theta + \cos^2 \theta)}q(f,f)
+ \cancel{(\sin^2 \theta + \cos^2 \theta)}q(g,g))} \\
&= \int_{W^2} e^{i (f(x) + g(y))} \mu^2(dx,dy).\end{aligned}$$
We can now really prove Fernique’s theorem.
In this proof we shall write $\mu(\norm{x} \le t)$ as shorthand for $\mu(\{x : \norm{x} \le t\})$, etc.
Let $0 \le s \le t$, and consider $$\begin{aligned}
\mu(\norm{x} \le s) \mu(\norm{x} \ge t) &= \mu^2(\{(x,y) : \norm{x} \le s,
\norm{y} \ge t\}) \\
&= \mu^2\left(\norm{\frac{1}{\sqrt{2}}(x-y)} \le s,
\norm{\frac{1}{\sqrt{2}}(x+y)} \ge t\right)
\end{aligned}$$ by $R_{\pi/4}$ invariance of $\mu^2$. Now some gymnastics with the triangle inequality shows that if we have $\norm{\frac{1}{\sqrt{2}}(x-y)}
\le s$ and $\norm{\frac{1}{\sqrt{2}}(x+y)} \ge t$, then $\norm{x},
\norm{y} \ge \frac{t-s}{\sqrt{2}}$. So we have $$\begin{aligned}
\mu(\norm{x} \le s) \mu(\norm{x} \ge t) &\le \mu^2\left(\norm{x} \ge
\frac{t-s}{\sqrt{2}}, \norm{y} \ge \frac{t-s}{\sqrt{2}}\right)\\
&= \left(\mu\left(\norm{x} \ge \frac{t-s}{\sqrt{2}}\right)\right)^2.
\end{aligned}$$ If we rearrange and let $a(t) = \frac{\mu(\norm{x} \ge
t)}{\mu(\norm{x} \le s)}$, this gives $$\label{a-relation}
a(t) \le a\left(\frac{t-s}{\sqrt{2}}\right)^2.$$ Once and for all, fix an $s$ large enough that $\mu(\norm{x} \ge s)
< \mu(\norm{x} \le s)$ (so that $a(s) < 1$). Now we’ll iterate (\[a-relation\]). Set $t_0 = s$, $t_{n+1} = \sqrt{2}(t_n + s)$, so that (\[a-relation\]) reads $a(t_{n+1}) \le a(t_n)^2$, which by iteration implies $a(t_n) \le a(s)^{2^n}$.
Since $t_n \uparrow \infty$, for any $r \ge s$ we have $t_n \le r
\le t_{n+1}$ for some $n$. Note that $$t_{n+1} = s \sum_{k=0}^{n+1} 2^{k/2} \le C 2^{n/2}$$ since the largest term dominates. $a$ is decreasing so we have $$\begin{aligned}
a(r) \le a(t_n) \le a(s)^{2^n} \le a(s)^{r^2/C^2}
\end{aligned}$$ so that $a(r) \le e^{-\epsilon r^2}$, taking $\epsilon = -\log(a(s))
/C^2$. Since $a(r) = \mu(\norm{x} \le s) \mu(\norm{x} \ge r)$ we are done.
If $\epsilon$ is as provided by Fernique’s theorem, for $\epsilon' <
\epsilon$ we have $\int_W
e^{\epsilon' ||x||^2} \mu(dx) < \infty$.
Standard trick: for a nonnegative random variable $X$, $EX =
\int_0^\infty P(X > t)\,dt$. So $$\begin{aligned}
\int_W e^{\epsilon' ||x||^2}\mu(dx) &= \int_0^\infty \mu(\{x :
e^{\epsilon' ||x||^2} > t\})dt \\
&= \int_0^\infty \mu\left(\left\{x :
||x|| > \sqrt{\frac{\log t}{\epsilon'}}\right\}\right)dt \\
&\le \int_0^\infty t^{-\epsilon/\epsilon'}\,dt < \infty.
\end{aligned}$$
The following corollary is very convenient for dominated convergence arguments.
For any $p > 0$, $\int_W ||x||_W^p \mu(dx) <\infty$.
$t^p$ grows more slowly than $e^{\epsilon t^2}$.
The inclusion $W^* \hookrightarrow L^2(\mu)$ is bounded. In particular, the $L^2$ norm on $W^*$ is weaker than the operator norm.
For $f \in W^*$, $||f||_{L^2}^2 = \int_W |f(x)|^2 \mu(dx) \le
||f||_{W^*}^2 \int_W ||x||^2 \mu(dx) \le C ||f||_{W^*}^2$ by the previous corollary.
(This would be a good time to look at Exercises \[topologies-first\]–\[topologies-last\] to get some practice working with different topologies on a set.)
Actually we can say more than the previous corollary. Recall that an operator $T : X \to Y$ on normed spaces is said to be **compact** if it maps bounded sets to precompact sets, or equivalently if for every bounded sequence $\{x_n\} \subset X$, the sequence $\{T x_n\}
\subset Y$ has a convergent subsequence.
The inclusion $W^* \hookrightarrow L^2(\mu)$ is compact.
Suppose $\{f_n\}$ is a bounded sequence in $W^*$; say $\norm{f_n}_{W^*} \le 1$ for all $n$. By Alaoglu’s theorem there is a weak-\* convergent subsequence $f_{n_k}$, which is to say that $f_{n_k}$ converges pointwise to some $f \in W^*$. Note also that $|f_{n_k}(x)| \le \norm{x}_W$ for all $k$, and $\int_W \norm{x}_W^2
\mu(dx) < \infty$ as we showed. So by dominated convergence, $f_{n_k} \to f$ in $L^2(W,\mu)$, and we found an $L^2$-convergent subsequence.
This fact is rather significant: since compact maps on infinite-dimensional spaces can’t have continuous inverses, this shows that the $W^*$ and $L^2$ topologies on $W^*$ must be quite different. In particular:
$W^*$ is not complete in the $q$ inner product (i.e. in the $L^2(\mu)$ inner product), except in the trivial case that $W$ is finite dimensional.
We’ve shown the identity map $(W^*, \norm{\cdot}_{W^*}) \to (W^*, q)$ is continuous and bijective. If $(W^*, q)$ is complete, then by the open mapping theorem, this identity map is a homeomorphism, i.e. the $W^*$ and $q$ norms are equivalent. But the identity map is also compact, which means that bounded sets, such as the unit ball, are precompact (under either topology). This means that $W^*$ is locally compact and hence finite dimensional.
So the closure $K$ of $W^*$ in $L^2(W,\mu)$ is a *proper* superset of $W^*$.
The Cameron–Martin space {#sec-cameron-martin}
------------------------
Our goal is to find a Hilbert space $H \subset W$ which will play the same role that $\ell^2$ played for $\R^\infty$. The key is that, for $h \in H$, the map $W^* \ni f \mapsto f(h)$ should be continuous with respect to the $q$ inner product on $W^*$.
As before, let $K$ be the closure of $W^*$ in $L^2(W,\mu)$. We’ll continue to denote the covariance form on $K$ (and on $W^*$) by $q$. We’ll also use $m$ to denote the inclusion map $m : W^* \to K$. Recall that we previously argued that $m$ is compact.
Every $k \in K$ is a Gaussian random variable on $(W,\mu)$.
Since $W^*$ is dense in $K$, there is a sequence $f_n \in W^*$ converging to $k$ in $L^2(W,\mu)$. In particular, they converge in distribution. By Lemma \[limit-of-gaussian\], $k$ is Gaussian.
The **Cameron–Martin space** $H$ of $(W,\mu)$ consists of those $h \in W$ such that the evaluation functional $f \mapsto f(h)$ on $W^*$ is continuous with respect to the $q$ inner product.
$H$ is obviously a vector space.
For $h \in H$, the map $W^* \ni f \mapsto f(h)$ extends by continuity to a continuous linear functional on $K$. Since $K$ is a Hilbert space this may be identified with an element of $K$ itself. Thus we have a mapping $T : H \to K$ such that for $f \in W^*$, $$q(Th, f) = f(h).$$ A natural norm on $H$ is defined by $$\norm{h}_H = \sup\left\{\frac{|f(h)|}{\sqrt{q(f,f)}} : f \in W^*, f \ne 0\right\}.$$ This makes $T$ into an isometry, so $\norm{\cdot}_H$ is in fact induced by an inner product ${\langle \cdot , \cdot \rangle}_H$ on $H$.
Next, we note that $H$ is continuously embedded into $W$. We have previously shown (using Fernique) that the embedding of $W^*$ into $K$ is continuous, i.e. $q(f,f) \le C^2 \norm{f}_{W^*}^2$. So for $h \in
H$ and $f \in W^*$, we have $$\frac{|f(h)|}{\norm{f}_{W^*}} \le C \frac{|f(h)|}{\sqrt{q(f,f)}}.$$ When we take the supremum over all nonzero $f \in W^*$, the left side becomes $\norm{h}_W$ (by Hahn–Banach) and the right side becomes $C
\norm{h}_H$. So we have $\norm{h}_W \le C \norm{h}_H$ and the inclusion $i : H \hookrightarrow W$ is continuous.
(Redundant given the next paragraph.) Next, we check that $(H, \norm{\cdot}_H)$ is complete. Suppose $h_n$ is Cauchy in $H$-norm. In particular, it is bounded in $H$ norm, so say $\norm{h_n}_H \le M$ for all $n$. Since the inclusion of $H$ into $W$ is bounded, $h_n$ is also Cauchy in $W$-norm, hence converges in $W$-norm to some $x \in W$. Now fix $\epsilon > 0$, and choose $n$ so large that $||h_n - h_m||_H < \epsilon$ for all $m \ge n$. Given a nonzero $f \in W^*$, we can choose $m \ge n$ so large that $|f(h_m -
x)| \le \epsilon \sqrt{q(f,f)}$. Then $$\frac{f(h_n - x)}{\sqrt{q(f,f)}} \le \frac{|f(h_n -
h_m)}{\sqrt{q(f,f)}} + \frac{|f(h_m - x)|}{\sqrt{q(f,f)}} <
\norm{h_n - h_m}_H + \epsilon < 2\epsilon.$$ We can then take the supremum over $f$ to find that $\norm{h_n - x}_H
< 2 \epsilon$, so $h_n \to x$ in $H$-norm.
Next, we claim the inverse of $T$ is given by $$J k = \int_W x k(x) \mu(dx)$$ where the integral is in the sense of Bochner. (To see that the integral exists, note that by Fernique $\norm{\cdot} \in L^2(W,
\mu)$.) For $f \in W^*, k \in K$, we have $$\abs{f(Jk)} = \abs{\int_W f(x)
k(x)\mu(dx)} = \abs{q(f,k)} \le \sqrt{q(f,f) q(k,k)}$$ whence $\norm{\int_W x k(x) \mu(dx)}_H \le \sqrt{q(k,k)}$. So $J$ is a continuous operator from $K$ to $H$. Next, for $f \in W^*$ we have $$q(TJk, f) = f(Jk) = q(k,f)$$ as we just argued. Since $W^*$ is dense in $K$, we have $TJk = k$. In particular, $T$ is surjective, and hence unitary.
Could we have done this without the Bochner integral?
We previously showed that the inclusion map $i : H \to W$ is continuous, and it’s clearly 1-1. It has an adjoint operator $i^* :
W^* \to H$. We note that for $f \in W^*$ and $h \in H$, we have $$\begin{aligned}
q(f, Th) = f(h) = {\langle i^* f , h \rangle}_H = q(Ti^*f, Th).\end{aligned}$$ Since $T$ is surjective we have $q(f,k) = q(T i^* f,k)$ for all $k \in
K$; thus $T i^*$ is precisely the inclusion map $m : W^* \to K$. Since $m$ is compact and 1-1 and $T$ is unitary, it follows that $i^*$ is compact and 1-1.
Since $i^*$ is 1-1, it follows that $H$ is dense in $W$: if $f \in
W^*$ vanishes on $H$, it means that for all $h \in H$, $0 = f(h) =
{\langle i^* f , h \rangle}_H$, so $i^* f = 0$ and $f = 0$. The Hahn–Banach theorem then implies $H$ is dense in $W$. Moreover, Schauder’s theorem from functional analysis (see for example [@conway Theorem VI.3.4]) states that an operator between Banach spaces is compact iff its adjoint is compact, so $i$ is compact as well. In particular, $H$ is not equal to $W$, and is not complete in the $W$ norm.
We can sum up all these results with a diagram.
The following diagram commutes. $$\xymatrix{
& & W^* \ar@{.>}[ddll]_{i^*} \ar@{.>}[rr]^m & & K \ar@{->}@/^/[ddllll]^J \\ \\
H \ar@{.>}[rr]^i \ar@{->}@/^/[uurrrr]^T & & W
}$$ All spaces are complete in their own norms. All dotted arrows are compact, 1-1, and have dense image. All solid arrows are unitary.
Sometimes it’s convenient to work things out with a basis.
\[ek-basis\] There exists a sequence $\{e_k\}_{k=1}^\infty \subset W^*$ which is an orthonormal basis for $K$. $e_k$ are iid $N(0,1)$ random variables under $\mu$. For $h \in H$, we have $\norm{h}_H^2
= \sum_{k=1}^\infty |e_k(h)|^2$, and the sum is infinite for $h \in
W \backslash H$.
The existence of $\{e_k\}$ is proved in Lemma \[dense-subspace-basis\]. They are jointly Gaussian random variables since $\mu$ is a Gaussian measure. Orthonormality means they each have variance 1 and are uncorrelated, so are iid.
If $h \in H$, then $\sum_k |e_k(h)|^2 = \sum_k |q(e_k, Th)|^2 =
\norm{Th}_K^2 = \norm{h}_H^2$ since $T$ is an isometry. Conversely, suppose $x \in W$ and $M := \sum_k |e_k(x)|^2 < \infty$. Let $E
\subset X^*$ be the linear span of $\{e_k\}$, i.e. the set of all $f \in
W^*$ of the form $f = \sum_{k=1}^n a_k e_k$. For such $f$ we have $$\begin{aligned}
|f(x)|^2 &= \abs{\sum_{k=1}^n a_k e_k(x)}^2 \\
&\le \left(\sum_{k=1}^n |a_k|^2\right) \left(\sum_{k=1}^n
|e_k(x)|^2\right) && \text{(Cauchy--Schwarz)} \\
\le M q(f,f)
\end{aligned}$$ Thus $x \mapsto f(x)$ is a bounded linear functional on $(E,
q)$. $(E,q)$ is dense in $(W^*, q)$ so the same bound holds for all $f \in X^*$. Thus by definition we have $x \in H$.
$\mu(H) = 0$.
For $h \in H$ we have $\sum |e_k(h)|^2 < \infty$. But since $e_k$ are iid, by the strong law of large numbers we have that $\sum
|e_k(x)|^2 = +\infty$ for $\mu$-a.e. $x$.
Fix $h \in H$. Then ${\langle h , x \rangle}_H$ is unambiguous for all $x \in
H$. If we interpret ${\langle h , x \rangle}_H$ as $(Th)(x)$, it is also well-defined for almost every $x$, and so ${\langle h , \cdot \rangle}_H$ is a Gaussian random variable on $(W,\mu)$ with variance $\norm{h}_H^2$.
For $h \in H$, $\mu_h$ is absolutely continuous with respect to $\mu$, and $$\frac{d \mu_h}{d \mu}(x) = e^{- \frac{1}{2} \norm{h}_H^2 +
{\langle h , x \rangle}_H}.$$ For $x \in W \backslash H$, $\mu_x$ and $\mu$ are singular.
Suppose $h \in H$. We have to show $\mu_h(dx) = e^{- \frac{1}{2}
\norm{h}_H^2 + {\langle h , x \rangle}_H} \mu(dx)$. It is enough to show their Fourier transforms are the same (Theorem \[fourier-unique\]). For $f \in W^*$ we have $$\int_W e^{i f(x)} \mu_h(dx) = \int_W e^{i f(x+h)} \mu(dx) = e^{i
f(h) - \frac{1}{2} q(f,f)}.$$ On the other hand, $$\begin{aligned}
\int_W e^{i f(x)} e^{- \frac{1}{2}
\norm{h}_H^2 + {\langle h , x \rangle}_H} \mu(dx) &= e^{- \frac{1}{2}
\norm{h}_H^2} \int_W e^{i (f-iTh)(x)} \mu(dx) \\
&= e^{- \frac{1}{2}
\norm{h}_H^2} e^{-\frac{1}{2} q(f-iTh, f-iTh)}
\end{aligned}$$ since $f - iTh$ is a complex Gaussian random variable (we will let the reader check that everything works fine with complex numbers here). But we have $$\begin{aligned}
q(f-iTh, f-iTh) = q(f,f) - 2 i q(f, Th) - q(Th, Th) = q(f,f) -
2i f(h) - \norm{h}_H^2
\end{aligned}$$ by properties of $T$, and so in fact the Fourier transforms are equal.
Conversely, if $x \in W \backslash H$, by Lemma \[ek-basis\] we have $\sum_k |e_k(x)|^2 = \infty$. By Lemma \[ell2\] there exists $a \in \ell^2$ such that $\sum a_k e_k(x)$ diverges. Set $A = \{y \in W : \sum a_k e_k(y) \text{ converges}\}$. We know that $\sum_k a_k e_k$ converges in $L^2(W,\mu)$, and is a sum of independent random variables (under $\mu$), hence it converges $\mu$-a.s. Thus $\mu(A) = 1$. However, if $y \in A$, then $\sum
a_k e_k(y-x)$ diverges, so $A-x$ is disjoint from $A$, and thus $\mu_x(A) = \mu(A-x) = 0$.
\[nondegenerate-implies-full-support\] $\mu$ has full support, i.e. for any nonempty open set $U$, $\mu(U)
> 0$. This is the converse of Exercise \[full-support-implies-nondegenerate\]. (Hint: First show this for $U \ni 0$. Then note any nonempty open $U$ contains a neighborhood of some $h \in H$. Translate.) (Question: Can we prove this without needing the Cameron–Martin hammer? I think yes, look for references.)
\[rk-non-degenerate\] There really isn’t any generality lost by assuming that $(W,\mu)$ is non-degenerate. If you want to study the degenerate case, let $F =
\{f \in W^* : q(f,f) = 0\}$ be the kernel of $q$, and consider the closed subspace $$W_0 := \bigcap_{f \in F} \ker f \subset W.$$ We claim that $\mu(W_0) = 1$. For each $f \in F$, the condition $q(f,f) = \int f^2\,d\mu = 0$ implies that $f = 0$ $\mu$-almost everywhere, so $\mu(\ker f) = 1$, but as written, $W_0$ is an uncountable intersection of such sets. To fix that, note that since $W$ is separable, the unit ball $B^*$ of $W^*$ is weak-\* compact metrizable, hence weak-\* separable metrizable, hence so is its subset $F \cap B^*$. So we can choose a countable weak-\* sequence $\{f_n\} \subset F \cap B^*$. Then I claim $$W_0 = \bigcap_n \ker f_n.$$ The $\subset$ inclusion is obvious. To see the other direction, suppose $x \in \bigcap_n \ker f_n$ and $f \in F$; we will show $f(x)
= 0$. By rescaling, we can assume without loss of generality that $f \in B^*$. Now choose a subsequence $f_{n_k}$ converging weak-\* to $f$; since $f_{n_k}(x) = 0$ by assumption, we have $f(x) = 0$ also. Now $W_0$ is written as a *countable* intersection of measure-$1$ subsets, so $\mu(W_0) = 1$.
We can now work on the abstract Wiener space $(W_0, \mu|_{W_0})$. Note that the covariance form $q_0$ defined on $W_0^*$ by $q_0(f_0,
f_0) = \int_{W_0} f_0^2\,d\mu$ agrees with $q$, since given any extension $f \in W^*$ of $f_0$ will satisfy $$q(f,f) = \int_W
f^2\,d\mu = \int_{W_0} f^2\,d\mu = \int_{W_0} f_0^2\,d\mu = q_0(f_0,
f_0).$$
This makes it easy to see that that $q_0$ is positive definite on $W_0^*$. Suppose $q_0(f_0, f_0) = 0$ and use Hahn–Banach to choose an extension $f \in W^*$ of $f_0$. Then $q(f,f) = 0$, so by definition of $W_0$, we have $W_0 \subset \ker f$; that is, $f$ vanishes on $W_0$, so the restriction $f_0 = f|_{W_0}$ is the zero functional.
It now follows, from the previous exercise, that the support of $\mu$ is precisely $W_0$. So $(W_0, \mu|_{W_0})$ is a non-degenerate abstract Wiener space, and we can do all our work on this smaller space.
I’d like to thank Philipp Wacker for suggesting this remark and sorting out some of the details.
Example: Gaussian processes
---------------------------
Recall that a one-dimensional stochastic process $X_t, 0 \le t \le 1$ is said to be **Gaussian** if, for any $t_1, \dots, t_n \ge 0$, the random vector $(X_{t_1}, \dots, X_{t_n})$ has a joint Gaussian distribution. If the process is continuous, its distribution gives a probability measure $\mu$ on $W = C([0,1])$. If there is any good in the world, this ought to be an example of a Gaussian measure.
By the Riesz representation theorem, we know exactly what $W^*$ is: it’s the set of all finite signed Borel measures $\nu$ on $[0,1]$. We don’t yet know that all of these measures represent Gaussian random variables, but we know that some of them do. Let $\delta_t$ denote the measure putting unit mass at $t$, so $\delta_t(\omega) = \int_0^1
\omega(t)\,d\delta_t = \omega(t)$. We know that $\{\delta_t\}_{t \in
[0,1]}$ are jointly Gaussian. If we let $E \subset W^*$ be their linear span, i.e. the set of all finitely supported signed measures, i.e. the set of measures $\nu = \sum_{i=1}^n a_i \delta_{t_i}$, then all measures in $E$ are Gaussian random variables.
$E$ is weak-\* dense in $W^*$, and dense in $K$.
Suppose $\nu \in W^*$. Given a partition $\mathcal{P} = \{0 = t_0 < \dots < t_n
= 1\}$ of $[0,1]$, set $\nu_{\mathcal{P}} = \sum_{j=1}^n \int
1_{(t_{j-1}, t_j]} d\nu \delta_{t_j}$. Then for each $\omega \in
C([0,1])$, $\int \omega \,d\nu_{\mathcal{P}} = \int
\omega_{\mathcal{P}} \,d\nu$, where $$\omega_{\mathcal{P}} =
\sum_{j=1}^n \omega(t_j) 1_{(t_{j-1}, t_j]}.$$ But by uniform continuity, as the mesh size of $\mathcal{P}$ goes to 0, we have $\omega_{\mathcal{P}} \to \omega$ uniformly, and so $\int
\omega_{\mathcal{P}} d\nu \to \int
\omega \, d\nu$. Thus $\nu_{\mathcal{P}} \to \nu$ weakly-\*.
$\mu$ is a Gaussian measure.
Every $\nu \in W^*$ is a pointwise limit of a sequence of Gaussian random variables, hence Gaussian.
$E$ is dense in $K$.
$\{\nu_{\mathcal{P}}\}$ is bounded in total variation (in fact $\norm{\nu_{\mathcal{P}}} \le \norm{\nu}$). So by Fernique’s theorem and dominated convergence, $\nu_{\mathcal{P}} \to \nu$ in $L^2(X,\mu)$. Thus $E$ is $L^2$-dense in $W^*$. Since $W^*$ is $L^2$-dense in $K$, $E$ is dense in $K$.
Note that in order to get $\mu$ to be non-degenerate, it may be necessary to replace $W$ by a smaller space. For example, if $X_t$ is Brownian motion started at 0, the linear functional $\omega \mapsto
\omega(0)$ is a.s. zero. So we should take $W = \{ \omega \in
C([0,1]) : \omega(0)=0\}$. One might write this as $C_0((0,1])$.
Recall that a Gaussian process is determined by its covariance function $a(s,t) = E[X_s X_t] = q(\delta_s, \delta_t)$. Some examples:
1. Standard Brownian motion $X_t = B_t$ started at 0: $a(s,t) =
s$ for $s < t$. Markov, martingale, independent increments, stationary increments.
2. Ornstein–Uhlenbeck process defined by $dX_t = -X_t \,dt +
\sigma \,dB_t$: $a(s,t) = \frac{\sigma^2}{2} (e^{-(t-s)} -
e^{-(t+s)})$, $s < t$. Markov, not a martingale.
3. Fractional Brownian motion with Hurst parameter $H \in (0,1)$: $a(s,t) = \frac{1}{2}(t^{2H} + s^{2H} - (t-s)^{2H})$, $s < t$. Not Markov.
4. Brownian bridge $X_t = B_t - t B_1$: $a(s,t) = s(1-t)$, $s < t$. (Here $W$ should be taken as $\{\omega \in C([0,1]) : \omega(0) =
\omega(1) = 0\} = C_0((0,1))$, the so-called pinned loop space.)
The covariance form $q$ for a Gaussian process is defined by $$q(\nu_1, \nu_2) = \int_0^1 \int_0^1 a(s,t) \nu_1(ds) \nu_2(dt)$$ for $\nu_1, \nu_2 \in W^*$.
Fubini’s theorem, justified with the help of Fernique.
$J : K \to H$ is defined by $Jk(t) = q(k, \delta_t)$. For $k = \nu
\in W^*$ this gives $J\nu(t) = i^* \nu(t) = \int_0^1 a(s,t)
\nu(ds)$. In particular $J \delta_s(t) = a(s,t)$.
$Jk(t) = \delta_t(Jk) = q(k, \delta_t)$.
Observe that $a$ plays the role of a reproducing kernel in $H$: we have $$\begin{aligned}
{\langle h , a(s, \cdot) \rangle}_H = {\langle h , J \delta_s \rangle} = \delta_s(h) = h(s).\end{aligned}$$ This is why $H$ is sometimes called the “reproducing kernel Hilbert space” or “RKHS” for $W$.
Classical Wiener space
----------------------
Let $\mu$ be Wiener measure on classical Wiener space $W$, so $a(s,t)
= s \wedge t$.
\[classical-cameron-martin\] The Cameron–Martin space $H \subset W$ is given by the set of all $h \in W$ which are absolutely continuous and have $\dot{h} \in
L^2([0,1], m)$. The Cameron-Martin inner product is given by ${\langle h_1 , h_2 \rangle}_H = \int_0^1 \dot{h_1}(t) \dot{h_2}(t)\,dt$.
Let $\tilde{H}$ be the candidate space with the candidate norm ${\norm{\cdot}}_{\tilde{H}}$. It’s easy to see that $\tilde{H}$ is a Hilbert space.
Note that $J \delta_s(t) = s \wedge t \in \tilde{H}$, so by linearity $J$ maps $E$ into $\tilde{H}$. Note $\dot{J \delta_s} =
1_{[0,s]}$. Moreover, $$\begin{aligned}
{\langle J \delta_s , J \delta_r \rangle}_{\tilde{H}} = \int_0^1 1_{[0,s]}
1_{[0,r]} dm = s \wedge r = q(\delta_s, \delta_r)
\end{aligned}$$ so $J$ is an isometry from $(E,q)$ to $\tilde{H}$. Hence it extends to an isometry of $K$ to $\tilde{H}$. Since $J$ is already an isometry from $K$ to $H$ we have $H = \tilde{H}$ isometrically.
Now what can we say about $T$? It’s a map that takes a continuous function from $H$ and returns a random variable. Working informally, we would say that $$\label{T-informal}
Th(\omega) = {\langle h , \omega \rangle}_H = \int_0^1
\dot{h}(t) \dot{\omega}(t)\,dt.$$ This formula is absurd because $\dot{\omega}$ is nonexistent for $\mu$-a.e. $\omega$ (Brownian motion sample paths are nowhere differentiable). However, it is actually the right answer if interpreted correctly.
Let’s suppose that $h$ is piecewise linear: then its derivative is a step function $\dot{h} = \sum_{i=1}^n b_i 1_{[c_i, d_i]}$. Note that the reproducing kernel $a(s, \cdot)$ has as its derivative the step function $1_{[0,s]}$. So by integrating, we see that we can write $$h(t) = \sum_{i=1}^n b_i (a(d_i, t) - a(c_i, t)).$$ Now we know that $T[a(s,\cdot)] = \delta_s$, i.e. the random variable $B_s$. So we have $$Th = \sum_{i=1}^n b_i (B_{d_i} - B_{c_i}).$$ We can recognize this as the stochastic integral of the step function $\dot{h} = \sum_{i=1}^n b_i 1_{[c_i, d_i]}$: $$\label{T-integral}
Th = \int_0^1
\dot{h}(t)\,dB_t.$$ Moreover, by the Itô isometry we know that $$\norm{ \int_0^1
\dot{h}(t)\,dB_t}_{L^2(W,\mu)}^2 = \norm{\dot{h}}_{L^2([0,1])}^2 = \norm{h}_H^2.$$ Thus both sides of (\[T-integral\]) are isometries on $H$, and they are equal for all piecewise linear $H$. Since the step functions are dense in $L^2([0,1])$, the piecewise linear functions are dense in $H$ (take derivatives), so in fact (\[T-integral\]) holds for all $h \in
H$. We have rediscovered the stochastic integral, at least for deterministic integrands. This is sometimes called the Wiener integral. Of course, the Itô integral also works for stochastic integrands, as long as they are adapted to the filtration of the Brownian motion. Later we shall use our machinery to produce the Skorohod integral, which will generalize the Itô integral to integrands which need not be adapted, giving us an “anticipating stochastic calculus.”
For the Ornstein–Uhlenbeck process, show that $H$ is again the set of absolutely continuous functions $h$ with $\dot{h} \in
L^2([0,1])$, and the Cameron–Martin inner product is given by $${\langle h_1 , h_2 \rangle}_H = \frac{1}{\sigma^2}\int_0^1 \dot{h_1}(t) \dot{h_2}(t) + h_1(t) h_2(t)\,dt.$$
For the Brownian bridge, show that $H$ is again the set of absolutely continuous functions $h$ with $\dot{h} \in L^2([0,1])$, and the Cameron–Martin inner product is given by ${\langle h_1 , h_2 \rangle}_H = \int_0^1 \hat{h_1}(t) \hat{h_2}(t)\,dt$, where $$\hat{h}(t) = \dot{h}(t) + \frac{h(t)}{1-t}.$$
Perhaps later when we look at some stochastic differential equations, we will see where these formulas come from.
Note that in this case the Cameron–Martin theorem is a special case of Girsanov’s theorem: it says that a Brownian motion with a “smooth” drift becomes a Brownian motion without drift under an equivalent measure. Indeed, suppose $h \in H$. If we write $B_t(\omega) = \omega(t)$, so that $\{B_t\}$ is a Brownian motion on $(W,\mu)$, then $B_t + h(t)$ is certainly a Brownian motion (without drift!) on $(W, \mu_h)$. The Cameron-Martin theorem says that $\mu_h$ is an equivalent measure to $\mu$. Anything that $B_t$ can’t do, $B_t + h(t)$ can’t do either (since the $\mu$-null and $\mu_h$-null sets are the same). This fact has many useful applications. For example, in mathematical finance, one might model the price of an asset by a geometric Brownian motion with a drift indicating its average rate of return (as in the Black–Scholes model). The Cameron–Martin/Girsanov theorem provides an equivalent measure under which this process is a martingale, which makes it possible to compute the arbitrage-free price for options involving the asset. The equivalence of the measures is important because it guarantees that changing the measure didn’t allow arbitrage opportunities to creep in.
Construction of $(W,\mu)$ from $H$
----------------------------------
This section originates in [@gross-abstract-wiener] via Bruce Driver’s notes [@driver-probability].
When $W = \R^n$ is finite-dimensional and $\mu$ is non-degenerate, the Cameron–Martin space $H$ is all of $W$ (since $H$ is known to be dense in $W$), and one can check that the Cameron–Martin norm is $${\langle x , y \rangle}_H = x \cdot \Sigma^{-1} y$$ where $\Sigma$ is the covariance matrix. We also know that $\mu$ has a density with respect to Lebesgue measure $dx$, which we can write as $$\mu(dx) = \frac{1}{Z} e^{-\frac{1}{2} \norm{x}_H^2} dx$$ where $Z = \int_{\R^n} e^{-\frac{1}{2} \norm{x}_H^2} dx$ is a normalizing constant chosen to make $\mu$ a probability measure. Informally, we can think of $\mu$ as being given by a similar formula in infinite dimensions: $$\label{nonsense-abstract}
\mu(dx) {\text{`` $ = $ ''}} \frac{1}{\mathcal{Z}} e^{-\frac{1}{2} \norm{x}_H^2} \mathcal{D}x$$ where $\mathcal{Z}$ is an appropriate normalizing constant, and $\mathcal{D}x$ is infinite-dimensional Lebesgue measure. Of course this is nonsense in at least three different ways, but that doesn’t stop physicists, for instance.
For classical Wiener measure this reads $$\label{nonsense-classical}
\mu(dx) {\text{`` $ = $ ''}} \frac{1}{\mathcal{Z}} e^{-\frac{1}{2} \int_0^1
|\dot{\omega}(t)|^2 dt} \mathcal{D}\omega.$$
Since the only meaningful object appearing on the right side of (\[nonsense-abstract\]) is $\norm{\cdot}_H$, it is reasonable to ask if we can start with a Hilbert space $H$ and produce an abstract Wiener space $(W,\mu)$ for which $H$ is the Cameron–Martin space.
### Cylinder sets
Let $(H, \norm{\cdot}_H)$ be a separable Hilbert space.
A **cylinder set** is a subset $C \subset H$ of the form $$\label{cylinder-set}
C = \{h \in H : ({\langle h , k_1 \rangle}_H, \dots, {\langle h , k_n \rangle}_H) \in A\}$$ for some $n \ge 1$, orthonormal $k_1, \dots, k_n$, and $A \subset
\R^n$ Borel.
Let $\mathcal{R}$ denote the collection of all cylinder sets in $H$. $\mathcal{R}$ is an algebra: we have $\emptyset \in
\mathcal{R}$ and $\mathcal{R}$ is closed under complements and *finite* unions (and intersections). However, if $H$ is infinite dimensional then $\mathcal{R}$ is not a $\sigma$-algebra.
Note by Lemma \[weak-sigma-field\] that $\sigma(\mathcal{R}) =
\mathcal{B}_H$, the Borel $\sigma$-algebra.
We are going to try to construct a Gaussian measure $\tilde{\mu}$ on $H$ with covariance form given by ${\langle \cdot , \cdot \rangle}_H$. Obviously we can only get so far, since we know of several obstructions to completing the task. At some point we will have to do something different. But by analogy with finite dimensions, we know what value $\tilde{\mu}$ should give to a cylinder set of the form (\[cylinder-set\]): since $k_1,
\dots, k_n$ are orthonormal, they should be iid standard normal with respect to $\tilde{\mu}$, so we should have $$\label{mu-cylinder-def}
\tilde{\mu}(C) = \mu_n(A)$$ where $d \mu_n = \frac{1}{(2 \pi)^{n/2}} e^{-|x|^2/2}dx$ is standard Gaussian measure on $\R^n$.
The expression for $\tilde{\mu}(C)$ in (\[mu-cylinder-def\]) is well-defined, and $\tilde{\mu}$ is a finitely additive probability measure on $\mathcal{R}$.
To check that $\tilde{\mu}(C)$ is well-defined, suppose that $$\label{Ceq}
C = \{h \in H : ({\langle h , k_1 \rangle}_H, \dots, {\langle h , k_n \rangle}_H) \in
A \subset \R^n\} = \{h \in H : ({\langle h , k'_1 \rangle}_H, \dots,
{\langle h , k'_{n'} \rangle}_H) \in A' \subset \R^{n'}\}.$$ Let $E$ be the span in $H$ of $\{k_1, \dots, k_n, k'_1, \dots,
k'_{n'}\}$, and let $m = \dim E$. Since $\{k_1, \dots, k_n\}$ is orthonormal in $E$, we can extend it to an orthonormal basis $\{k_1,
\dots, k_m\}$ for $E$, and then we have $$C = \{h \in H : ({\langle h , k_1 \rangle}_H, \dots, {\langle h , k_m \rangle}_H) \in
A \times \R^{m-n}\}.$$ Since $\mu_m$ is a product measure, we have $\mu_m(A \times
\R^{m-n}) = \mu_n(A)$. So by playing the same game for $\{k'_1,
\dots, k'_{n'}\}$, there is no loss of generality in assuming that in (\[Ceq\]) we have $n = n' = m$, and that $\{k_1,
\dots, k_m\}$ and $\{k'_1, \dots, k'_m\}$ are two orthonormal bases for the same $E \subset H$. We then have to show that $\mu_m(A) = \mu_m(A')$.
We have two orthonormal bases for $E$, so there is a unitary $T : E
\to E$ such that $T k_i = k'_i$. Let $P : H \to E$ be orthogonal projection, and define $S : E \to \R^m$ by $Sx = ((x, k_1), \dots,
(x, k_m))$. Then $S$ is unitary. If we define $S'$ analogously, then $S' = ST^* = ST$, and we have $$C = P^{-1} S^{-1} A = P^{-1} S'^{-1} A' = P^{-1} T^{-1} S^{-1} A'.$$ Since $P : H \to E$ is surjective, we must have $S^{-1} A = T^{-1}
S^{-1} A'$; since $S,T$ are bijective this says $A' = STS^{-1} A$, so $A'$ is the image of $A$ under a unitary map. But standard Gaussian measure on $\R^m$ is invariant under unitary transformations, so indeed $\mu_m(A) = \mu_m(A')$, and the expression (\[mu-cylinder-def\]) is well defined.
It is obvious that $\tilde{\mu}(\emptyset) = 0$ and $\tilde{\mu}(H) = 1$. For finite additivity, suppose $C_1, \dots, C_n \in \mathcal{R}$ are disjoint. By playing the same game as above, we can write $C_i =
P^{-1} (A_i)$ for some common $P : H \to \R^m$, where the $A_i
\subset \R^m$ are necessarily disjoint, and then $\tilde{\mu}(C_i) =
\mu_m(A_i)$. Since $\bigcup_i C_i = P^{-1}\left(\bigcup_i
A_i\right)$, the additivity of $\mu_m$ gives us that $\tilde{\mu}\left(\bigcup_i C_i\right) = \sum_i \tilde{\mu}(C_i)$.
We will call $\tilde{\mu}$ the canonical Gaussian measure on $H$. As we see in the next proposition, we’re using the term “measure” loosely.
If $H$ is infinite dimensional, $\tilde{\mu}$ is not countably additive on $\mathcal{R}$. In particular, it does not extend to a countably additive measure on $\sigma(\mathcal{R}) = \mathcal{B}_H$.
Fix an orthonormal sequence $\{e_i\}$ in $H$. Let $$A_{n,k} = \{ x \in H : |{\langle x , e_i \rangle}| \le k, i=1,\dots, n\}.$$ $A_{n,k}$ is a cylinder set, and we have $B(0,k) \subset A_{n,k}$ for any $n$. Also, we have $\tilde{\mu}(A_{n,k}) =
\mu_n([-k,k]^n) = \mu_1([-k,k])^n$ since $\mu_n$ is a product measure. Since $\mu_1([-k,k]) < 1$, for each $k$ we can choose an $n_k$ so large that $\tilde{\mu}(A_{n_k,k}) = \mu_1([-k,k])^{n_k} <
2^{-k}$. Thus $\sum_{k=1}^\infty \tilde{\mu}(A_{n_k, k}) < 1$, but since $B(0,k) \subset A_{n_k, k}$ we have $\bigcup_{k=1}^\infty A_{n_k,k}
= H$ and $\tilde{\mu}(H)=1$. So countable additivity does not hold.
Of course we already knew that this construction cannot produce a genuine Gaussian measure on $H$, since any Gaussian measure has to assign measure 0 to its Cameron–Martin space. The genuine measure has to live on some larger space $W$, so we have to find a way to produce $W$. We’ll produce it by producing a new norm $\norm{\cdot}_W$ on $H$ which is not complete, and set $W$ to be the completion of $H$ under $\norm{\cdot}_W$. Then we will be able to extend $\tilde{\mu}$, in a certain sense, to an honest Borel measure $\mu$ on $W$.
It’s common to make an analogy here with Lebesgue measure. Suppose we were trying to construct Lebesgue measure $m$ on $\Q$. We could define the measure of an interval $(a,b) \subset \Q$ to be $b-a$, and this would give a finitely additive measure on the algebra of sets generated by such intervals. But it could not be countably additive. If we want a countably additive measure, it has to live on $\R$, which we can obtain as the completion of $\Q$ under the Euclidean metric.
### Measurable norms
By a **finite rank projection** we mean a map $P : H \to H$ which is orthogonal projection onto its image $PH$ with $PH$ finite dimensional. We will sometimes abuse notation and identify $P$ with the finite-dimensional subspace $PH$, since they are in 1-1 correspondence. We will write things like $P_1 \perp P_2$, $P_1
\subset P_2$, etc.
We are going to obtain $W$ as the completion of $H$ under some norm $\norm{\cdot}_W$. Here is the condition that this norm has to satisfy.
A norm ${\norm{\cdot}}_{W}$ on $H$ is said to be **measurable** if for every $\epsilon > 0$ there exists a finite rank projection $P_0$ such that $$\label{measurable-condition}
\tilde{\mu}(\{h : \norm{Ph}_W > \epsilon\}) < \epsilon \text{ for
all $P \perp P_0$ of finite rank}$$ where $\tilde{\mu}$ is the canonical Gaussian “measure” on $H$. (Note that $\{x : \norm{Ph}_W > \epsilon\}$ is a cylinder set.)
A quick remark: if $P_0$ satisfies (\[measurable-condition\]) for some $\epsilon$, and $P_0 \subset P_0'$, then $P_0'$ also satisfies (\[measurable-condition\]) for the same $\epsilon$. This is because any $P \perp P_0'$ also has $P \perp P_0$.
In words, this definition requires that $\tilde{\mu}$ puts most of its mass in “tubular neighborhoods” of $P_0 H$. Saying $\norm{Ph}_W >
\epsilon$ means that $x$ is more than distance $\epsilon$ (in $W$-norm) from $P_0 H$ along one of the directions from $PH$.
As usual, doing the simplest possible thing doesn’t work.
${\norm{\cdot}}_{H}$ is not a measurable norm on $H$.
For any finite-rank projection $P$ of some rank $n$, we can find an orthonormal basis $\{h_1, \dots, h_n\}$ for $PH$. Then it’s clear that $Ph = \sum_{i=1}^n {\langle h , h_i \rangle}_H h_i$, so $\{ h :
\norm{Ph}_H > \epsilon\} = P^{-1}(\closure{B_{PH}(0,\epsilon)}^C)$, where $B_{PH}(0,\epsilon)$ is a ball in $PH$. By definition of $\tilde{\mu}$ we can see that $$\begin{aligned}
\tilde{\mu}(\{ h : \norm{Ph}_H >
\epsilon\}) &= \mu_n(\closure{B_{\R^n}(0,\epsilon)}^c) \\
&\ge \mu_n(([-\epsilon, \epsilon]^n)^C) \intertext{(since the ball
is contained in the cube)}
&= 1 - \mu_1([-\epsilon,\epsilon])^n.
\end{aligned}$$ Thus for any $\epsilon > 0$ and any finite-rank projection $P_0$, if we choose $n$ so large that $1 - \mu_1([-\epsilon,\epsilon])^n >
\epsilon$, then for any projection $P$ of rank $n$ which is orthogonal to $P_0$ (of which there are lots), we have $\tilde{\mu}(\{h : \norm{Ph}_H > \epsilon\}) > \epsilon$. So ${\norm{\cdot}}_{H}$ is not measurable.
As a diversion, let’s explicitly verify this for the classical example.
Let $H$ be the classical Cameron–Martin space of Theorem \[classical-cameron-martin\]. The supremum norm $\norm{h}_W = \sup_{t \in
[0,1]} h(t)$ is a measurable norm on $H$.
Together with Gross’s theorem (Theorem \[gross-theorem\] below), this proposition constitutes a construction of Brownian motion: the completion $W$ of $H$ under ${\norm{\cdot}}_W$ is precisely $C([0,1])$ (since $H$ is dense in $C([0,1])$), and the measure $\mu$ on $W$ is Wiener measure (having $H$ as its Cameron–Martin space, we can check that its covariance function is $a(s,t) = s \wedge t$ as it ought to be).
With the proof we will give, however, it will not be an essentially new construction. Indeed, we are going to steal the key ideas from a construction which is apparently due to Lévy and can be found in [@karatzas-shreve Section 2.3], which one might benefit from reading in conjunction with this proof. In some sense, Gross’s theorem is simply an abstract version of an essential step of that construction.
Observe up front that by Cauchy–Schwarz $$\abs{h(t)} = \abs{\int_0^t \dot{h}(t)\,dt} \le t \norm{h}_H$$ so taking the supremum over $t \in [0,1]$, we have $\norm{h}_W \le
\norm{h}_H$.
We want to choose a good orthonormal basis for $H$. We use the so-called “Schauder functions” which correspond to the “Haar functions” in $L^2([0,1])$. The Haar functions are given by $$f^n_k(t) :=
\begin{cases}
2^{(n-1)/2}, & \frac{k-1}{2^n} \le t < \frac{k}{2^n} \\
-2^{(n-1)/2}, & \frac{k}{2^n} \le t \le \frac{k+1}{2^n} \\
0, & \text{else}
\end{cases}$$ where $f_0^1 = 1$. Here $k$ should be taken to range over the set $I(n)$ consisting of all odd integers between $0$ and $2^n$. (This somewhat peculiar indexing is from Karatzas and Shreve’s proof. It may or may not be optimal.) We note that for $n \ge 1$, we have $\int_0^1 f^n_k(t)\,dt = 0$; that for $n > m$, $f^m_j$ is constant on the support of $f^n_k$; and that for fixed $n$, $\{f^n_k : k \in
I(n)\}$ have disjoint supports. From this it is not hard to check that $\{f^n_k : k \in I(n), n \ge 0\}$ are an orthonormal set in $L^2(0,1)$. Indeed, the set forms an orthonormal basis.
The Schauder functions are defined by $h^n_k(t) := \int_0^t f^n_k(s)\,ds$; since $f \mapsto
\int_0^\cdot f\,dt$ is an isometric isomorphism from $L^2([0,1])$ to $H$, we have that $\{h^n_k : k \in I(n), n \ge 0\}$ is an orthonormal basis for $H$. (We can check easily that it is a basis: if ${\langle h , h^n_k \rangle}_H = 0$ then $h(\frac{k-1}{2^n}) =
h(\frac{k+1}{2^n})$. If this holds for all $n,k$, then $h(t)=0$ for all dyadic rationals $t$, whence by continuity $h=0$.) Stealing Karatzas and Shreve’s phrase, $h^n_k$ is a “little tent” of height $2^{-(n+1)/2}$ supported in $[\frac{k-1}{2^n}, \frac{k+1}{2^n}]$; in particular, for each $n$, $\{h^n_k : k \in I(n)\}$ have disjoint supports.
Let $P_m$ be orthogonal projection onto the span of $\{h^n_k : k \in
I(n), n < m\}$, and suppose $P$ is a projection of finite rank which is orthogonal to $P_m$. Then for any $h \in H$, we can write $Ph$ in terms of the Schauder functions $$Ph = \sum_{n=m}^\infty \sum_{k \in I(n)} h^n_k {\langle Ph , h^n_k \rangle}_H.$$ where the sum converges in $H$ and hence also in $W$-norm, i.e. uniformly. Since for fixed $n$ the $h^n_k$ have disjoint supports, we can say $$\label{W-norm-estimate}
\begin{split}
\norm{Ph}_W &\le \sum_{n=m}^\infty \norm{\sum_{k \in I(n)} h^n_k
{\langle Ph , h^n_k \rangle}_H}_W \quad \text{(Triangle inequality)}
\\
&= \sum_{n=m}^\infty \max_{k \in I(n)} \norm{h^n_k}_W \abs{
{\langle Ph , h^n_k \rangle}_H} \quad \text{(since $h^n_k$ have disjoint
support)}
\\
&= \sum_{n=m}^\infty 2^{-(n+1)/2} \max_{k \in I(n)} \abs{
{\langle Ph , h^n_k \rangle}_H}.
\end{split}$$
To forestall any nervousness, let us point out that all the following appearances of $\tilde{\mu}$ will be to measure sets of the form $P^{-1} B$ for our single, fixed $P$, and on such sets $\tilde{\mu}$ is an honest, countably additive measure (since it is just standard Gaussian measure on the finite-dimensional Hilbert space $PH$). Under $\tilde{\mu}$, each ${\langle Ph , h^n_k \rangle}_H$ is a centered Gaussian random variable of variance $\norm{P h^n_k}_H^2
\le 1$ (note that ${\langle Ph , h^n_k \rangle}_H = {\langle Ph , P h^n_k \rangle}_H$, and that $P$ is a contraction). These random variables will be correlated in some way, but that will not bother us since we are not going to use anything fancier than union bounds.
We recall the standard Gaussian tail estimate: if $N$ is a Gaussian random variable with variance $\sigma^2 \le 1$, then $P(\abs{N} \ge
t) \le C e^{-t^2/2}$ for some universal constant $C$. (See (\[gaussian-tail-1d\], or for overkill, Fernique’s theorem.) Thus we have for each $n,k$ $$\tilde{\mu}(\{h : \abs{{\langle Ph , h^n_k \rangle}_H} \ge n\}) \le C e^{-n^2/2}$$ and so by union bound $$\tilde{\mu}\left(\left\{ h : \max_{k \in I(n)} \abs{{\langle Ph , h^n_k \rangle}_H} \ge n\right\}\right) =
\tilde{\mu}\left(\bigcup_{k \in I(n)} \{h : \abs{{\langle Ph , h^n_k \rangle}_H} \ge n\}\right)
\le C 2^n e^{-n^2/2}$$ since, being crude, $\abs{I(n)} \le 2^n$. By another union bound, $$\tilde{\mu}\left(\bigcup_{n=m}^\infty \left\{ h: \max_{k \in I(n)}
\abs{{\langle Ph , h^n_k \rangle}_H} \ge n\right\}\right) \le C \sum_{n=m}^\infty 2^n e^{-n^2/2}.$$ On the complement of this event, we have $\max_{k \in I(n)}
\abs{{\langle Ph , h^n_k \rangle}_H} < n$ for every $n$, and so using (\[W-norm-estimate\]) we have $\norm{Ph}_W < \sum_{n=m}^\infty
n 2^{-(n+1)/2}$. Thus we have shown $$\tilde{\mu}\left(\left\{ h: \norm{Ph}_W \ge \sum_{n=m}^\infty n
2^{-(n+1)/2}\right\}\right) \le C \sum_{n=m}^\infty 2^n e^{-n^2/2}.$$ Since $\sum n 2^{-(n+1)/2}$ and $\sum
2^n e^{-n^2/2}$ both converge, for any given $\epsilon > 0$ we may choose $m$ so large that $\sum_{n=m}^\infty n 2^{-(n+1)/2} <
\epsilon$ and $C \sum_{n=m}^\infty
2^n e^{-n^2/2} < \epsilon$. Then for any finite-rank projection $P$ orthogonal to $P_m$, we have $$\tilde{\mu}(\{ h : \norm{Ph}_W > \epsilon \}) \le \epsilon$$ which is to say that ${\norm{\cdot}}_W$ is a measurable norm.
The name “measurable” is perhaps a bit misleading on its face: we are not talking about whether $h \mapsto \norm{h}_W$ is a measurable function on $H$. It just means that ${\norm{\cdot}}_{W}$ interacts nicely with the “measure” $\tilde{\mu}$. However, ${\norm{\cdot}}_{W}$ is in fact a measurable function on $H$, in fact a continuous function, so that it is a weaker norm than ${\norm{\cdot}}_{H}$.
\[cont-embed\] If ${\norm{\cdot}}_{W}$ is a measurable norm on $H$, then $\norm{h}_W \le C
\norm{h}_H$ for some constant $C$.
Choose a $P_0$ such that (\[measurable-condition\]) holds with $\epsilon = 1/2$. Pick some vector $k \in (P_0 H)^\perp$ with $\norm{k}_H = 1$. Then $Ph = {\langle h , k \rangle} k$ is a (rank-one) projection orthogonal to $P_0$, so $$\begin{aligned}
\frac{1}{2} &> \tilde{\mu}(\{h : \norm{Ph}_W > \frac{1}{2}\}) \\
&= \tilde{\mu}(\{h : |{\langle h , k \rangle}| > \frac{1}{2 \norm{k}_W}\}) \\
&= \mu_1\left(\left[-\frac{1}{2 \norm{k}_W}, \frac{1}{2 \norm{k}_W}\right]^C\right).
\end{aligned}$$ Since $\mu_1([-t,t]^C) = 1 - \mu_1([-t,t])$ is a decreasing function in $t$, the last line is an increasing function in $\norm{k}_W$, so it follows that $\norm{k}_W \le M$ for some $M$. $k \in (P_0
H)^\perp$ was arbitrary, and so by scaling we have that $\norm{k}_W
\le M \norm{k}_H$ for all $k \in (P_0 H)^\perp$. On the other hand, $P_0 H$ is finite-dimensional, so by equivalence of norms we also have $\norm{k}_W \le M\norm{k}_H$ for all $k \in P_0 H$, taking $M$ larger if needed. Then for any $k \in H$, we can decompose $k$ orthogonally as $(k - P_0 k) +
P_0 k$ and obtain $$\begin{aligned}
\norm{k}_W^2 &= \norm{(k - P_0 k) + P_0 k}_W^2 \\
&\le (\norm{k - P_0 k}_W + \norm{P_0 k}_W)^2 &&
\text{Triangle inequality} \\
&\le 2 (\norm{k - P_0 k}_W^2 + \norm{P_0 k}_W^2) &&
\text{since $(a+b)^2 \le 2(a^2 + b^2)$, follows from AM-GM}\\
&\le 2 M^2 (\norm{k - P_0 k}_H^2 + \norm{P_0 k}_H^2) \\
&= 2 M^2 \norm{k}_H^2 && \text{Pythagorean theorem}
\end{aligned}$$ and so the desired statement holds with $C = \sqrt{2} M$.
\[gross-theorem\] Suppose $H$ is a separable Hilbert space and ${\norm{\cdot}}_W$ is a measurable norm on $H$. Let $W$ be the completion of $H$ under ${\norm{\cdot}}_{W}$. There exists a Gaussian measure $\mu$ on $(W, {\norm{\cdot}}_{W})$ whose Cameron-Martin space is $(H, {\norm{\cdot}}_{H})$.
We start by constructing a sequence of finite-rank projections $P_n$ inductively. First, pick a countable dense sequence $\{v_n\}$ of $H$. Let $P_0 = 0$. Then suppose that $P_{n-1}$ has been given. By the measurability of ${\norm{\cdot}}_{W}$, for each $n$ we can find a finite-rank projection $P_n$ such that for all finite-rank projections $P \perp P_n$, we have $$\tilde{\mu}(\{h \in H : \norm{Ph}_W > 2^{-n}\}) < 2^{-n}.$$ As we remarked earlier, we can always choose $P_n$ to be larger, so we can also assume that $P_{n-1} \subset P_n$ and also $v_n \in P_n
H$. The latter condition ensures that $\bigcup_n P_n H$ is dense in $H$, from which it follows that $P_n h \to h$ in $H$-norm for all $h \in H$, i.e. $P_n \uparrow I$ strongly. Let us also note that $R_n := P_n - P_{n-1}$ is a finite-rank projection which is orthogonal to $P_{n-1}$ (in fact, it is projection onto the orthogonal complement of $P_{n-1}$ in $P_n$), and in fact we have the orthogonal decomposition $H = \bigoplus_{n=1}^\infty R_n H$.
Given an orthonormal basis for $P_n H$, we can extend it to an orthonormal basis for $P_{n+1} H$. Repeating this process, we can find a sequence $\{h_j\}_{j=1}^\infty$ such that $\{h_1, \dots,
h_{k_n}\}$ is an orthonormal basis for $P_n H$. Since $\bigcup P_n
H$ is dense in $H$, it follows that the entire sequence $\{h_j\}$ is an orthonormal basis for $H$.
Let $\{X_n\}$ be a sequence of iid standard normal random variables defined on some unrelated probability space $(\Omega,
\mathcal{F}, \mathbb{P})$. Consider the $W$-valued random variable $$S_n = \sum_{j=1}^{k_n} X_j h_j.$$ Note that $S_n - S_{n-1}$ has a standard normal distribution on the finite-dimensional Hilbert space $R_n H$, so by definition of $\tilde{\mu}$ we have $$\begin{aligned}
\mathbb{P}(\norm{S_n - S_{n-1}}_W > 2^{-n}) = \tilde{\mu}(\{h \in H
: \norm{R_n h}_W > 2^{-n}\}) < 2^{-n}.
\end{aligned}$$ Thus $\sum_n \mathbb{P}(\norm{S_n - S_{n-1}}_W > 2^{-n}) < \infty$, and by Borel–Cantelli, we have that, $\mathbb{P}$-almost surely, $\norm{S_n
- S_{n-1}}_W \le 2^{-n}$ for all but finitely many $n$. In particular, $\mathbb{P}$-a.s., $S_n$ is Cauchy in $W$-norm, and hence convergent to some $W$-valued random variable $S$.
Let $\mu = \mathbb{P} \circ S^{-1}$ be the law of $S$; $\mu$ is a Borel measure on $W$. If $f \in W^*$, then $f(S) = \lim_{n \to
\infty} f(S_n) = \lim_{n \to \infty} \sum_{j=1}^{k_n} f(h_j) X_j$ is a limit of Gaussian random variables. Hence by Lemma \[limit-of-gaussian\] $f(S)$ is Gaussian, and moreover we have $$\infty > \Var(f(S)) = \lim_{n \to \infty} \Var(f(S_n)) = \lim_{n \to
\infty} \sum_{j=1}^{k_n} |f(h_j)|^2 = \sum_{j=1}^\infty |f(h_j)|^2.$$ Pushing forward, we have that $f$ is a Gaussian random variable on $(W,\mu)$ with variance $q(f,f) = \sum_{j=1}^\infty
|f(h_j)|^2 < \infty$. So $\mu$ is a Gaussian measure and $q$ is its covariance form. Let $H_\mu$ be the Cameron–Martin space associated to $(W, \mu)$. We want to show that $H = H_\mu$ isometrically. This is basically just another diagram chase.
Let $i$ denote the inclusion map $i : H \hookrightarrow W$. We know by Lemma \[cont-embed\] that $i$ is 1-1, continuous and has dense range, so its adjoint $i^* : W^* \to H$ is also 1-1 and continuous with dense range (Exercise \[adjoint-exercise\]). Also, we have $$\norm{i^* f}_H^2 = \sum_{j=1}^\infty |{\langle i^* f , h_j \rangle}_H|^2 =
\sum_{j=1}^\infty |f(h_j)|^2 = q(f,f)$$ so that $i^* : (W^*,q) \to H$ is an isometry.
Next, for any $h \in H$ and any $f \in W^*$, Cauchy–Schwarz gives $$|f(h)|^2 = |{\langle i^* f , h \rangle}_H|^2 \le \norm{i^*f}_H^2 \norm{h}_H^2
= q(f,f) \norm{h}_H^2$$ so that $f \mapsto f(h)$ is a continuous linear functional on $(W^*,q)$. That is, $h \in H_\mu$, and rearranging and taking the supremum over $f$ shows $\norm{h}_{H_\mu} \le \norm{h}_H$. On the other hand, if $f_n \in W^*$ with $i^* f_n \to h$ in $H$, we have by definition $$\frac{|f_n(h)|}{\sqrt{q(f_n,f_n)}} \le \norm{h}_{H_\mu}.$$ As $n \to \infty$, $f_n(h) = {\langle i^* f_n , h \rangle}_H \to \norm{h}_H^2$, and since $i^*$ is an isometry, $q(f_n, f_n) = \norm{i^* f_n}_H^2
\to \norm{h}_H^2$, so the left side tends to $\norm{h}_H$. Thus $\norm{h}_H = \norm{h}_{h_\mu}$.
We have shown $H \subset H_\mu$ isometrically; we want equality. Note that $H$ is closed in $H_\mu$, since $H$ is complete in $H$-norm and hence also in $H_\mu$-norm. So it suffices to show $H$ is dense in $H_\mu$. Suppose there exists $g
\in H_\mu$ with ${\langle g , h \rangle}_{H_\mu} = 0$ for all $h \in H$. If $i_\mu : H_\mu \hookrightarrow W$ is the inclusion map, we know that $i_\mu^* : (W^*,q) \to H_\mu$ has dense image and is an isometry. So choose $f_n \in W^*$ with $i_\mu^* f_n \to g$ in $H_\mu$. Then $f_n$ is $q$-Cauchy, and so $i^* f_n$ converges in $H$-norm to some $k \in H$. But for $h \in H$, $${\langle k , h \rangle}_H = \lim {\langle i^* f_n , h \rangle}_H = \lim f_n(h) = \lim
{\langle i_\mu^* f_n , h \rangle}_{H_\mu} = {\langle g , h \rangle}_{H_\mu} = 0$$ so that $k = 0$. Then $$\norm{g}_{H_\mu}^2 = \lim \norm{i_\mu^* f_n}_{H_\mu}^2 = \lim
q(f_n, f_n) = \lim \norm{i^* f_n}_H^2 = 0$$ so $g = 0$ and we are done.
Here is one way to describe what is going on here. If $h_j$ is an orthonormal basis for $H$, then $S = \sum X_j h_j$ should be a random variable with law $\mu$. However, this sum diverges in $H$ almost surely (since $\sum |X_j|^2 = \infty$ a.s.). So if we want it to converge, we have to choose a weaker norm.
The condition of measurability is not only sufficient but also necessary.
Let $(W, \mu)$ be an abstract Wiener space with Cameron–Martin space $H$. Then ${\norm{\cdot}}_{W}$ is a measurable norm on $H$.
The first proof of this statement, in this generality, seems to have appeared in [@dudley-feldman-lecam]. For a nice proof due to Daniel Stroock, see Bruce Driver’s notes [@driver-probability].
The mere existence of a measurable norm on a given Hilbert space $H$ is trivial. Indeed, since all infinite-dimensional separable Hilbert spaces are isomorphic, as soon as we have found a measurable norm for one Hilbert space, we have found one for any Hilbert space.
One might wonder if the completion $W$ has any restrictions on its structure. Equivalently, which separable Banach spaces $W$ admit Gaussian measures? This is a reasonable question, since Banach spaces can have strange “geometry.”[^3] However, the answer is that there are no restrictions.
If $W$ is any separable Banach space, there exists a separable Hilbert space densely embedded in $W$, on which the $W$-norm is measurable. Equivalently, there exists a non-degenerate Gaussian measure on $W$.
The finite-dimensional case is trivial, so we suppose $W$ is infinite dimensional. We start with the case of Hilbert spaces.
First, there exists a separable (infinite-dimensional) Hilbert space $W$ with a densely embedded separable Hilbert space $H$ on which the $W$-norm is measurable. Proposition \[hs-implies-measurable\] tells us that, given any separable Hilbert space $H$, we can construct a measurable norm ${\norm{\cdot}}_W$ on $H$ by letting $\norm{h}_W =
\norm{Ah}_H$, where $A$ is a Hilbert–Schmidt operator on $H$. Note that ${\norm{\cdot}}_W$ is induced by the inner product ${\langle h , k \rangle}_W =
{\langle Ah , Ak \rangle}_H$, so if we let $W$ be the completion of $H$ under ${\norm{\cdot}}_W$, then $W$ is a separable Hilbert space with $H$ densely embedded. (We should take $A$ to be injective. An example of such an operator is given by taking an orthonormal basis $\{e_n\}$ and letting $A e_n = \frac{1}{n} e_n$.)
Now, since all infinite-dimensional separable Hilbert spaces are isomorphic, this shows that the theorem holds for any separable Hilbert space $W$.
Suppose now that $W$ is a separable Banach space. By the following lemma, there exists a separable Hilbert space $H_1$ densely embedded in $W$. In turn, there is a separable Hilbert space $H$ densely embedded in $H_1$, on which the $H_1$-norm is measurable. The $W$-norm on $H$ is weaker than the $H_1$-norm, so it is measurable as well. (Exercise: check the details.)
Alternatively, there exists a non-degenerate Gaussian measure $\mu_1$ on $H_1$. Push it forward under the inclusion map. As an exercise, verify that this gives a non-degenerate Gaussian measure on $W$.
If $W$ is a separable Banach space, there exists a separable Hilbert space $H$ densely embedded in $W$.
We repeat a construction of Gross [@gross-abstract-wiener]. Since $W$ is separable, we may find a countable set $\{z_i\} \subset
W$ whose linear span is dense in $W$; without loss of generality, we can take $\{z_n\}$ to be linearly independent. We will construct an inner product ${\langle \cdot , \cdot \rangle}_K$ on $K = \spanop\{z_i\}$ such that $\norm{x}_W \le \norm{x}_K$ for all $x \in K$; thus $K$ will be an inner product space densely embedded in $W$.
We inductively construct a sequence $\{a_i\}$ such that $a_i \ne 0$ for any real numbers $b_1, \dots, b_n$ with $\sum_{i=1}^n b_i^2 \le
1$, we have $\norm{\sum_{i=1}^n a_i b_i z_i}_W < 1$. To begin, choose $a_1$ with $0 < |a_1| < \norm{z_1}_W^{-1}$. Suppose now that $a_1, \dots,
a_{n-1}$ have been appropriately chosen. Let $D^n = \{(b_1, \dots,
b_n) : \sum_{i=1}^n b_n^2 \le 1$ be the closed Euclidean unit disk of $\R^n$ and consider the map $f : D^n \times \R \to W$ defined by $$f(b_1, \dots, b_n, a) = \sum_{i=1}^{n-1} a_i b_i z_i + a b_n z_n.$$ Now $f$ is obviously continuous, and by the induction hypothesis we have $f(D^n \times \{0\}) \subset S$, where $S$ is the open unit ball of $W$. So by continuity, $f^{-1}(S)$ is an open set containing $D^n \times \{0\}$; hence $f^{-1}(S)$ contains some set of the form $D^n \times (-\epsilon, \epsilon)$. Thus if we choose any $a_n$ with $0 \le |a_n| < \epsilon$, we have the desired property for $a_1, \dots, a_n$.
Set $y_i = a_i z_i$; since the $a_i$ are nonzero, the $y_i$ span $K$ and are linearly independent. Let ${\langle \cdot , \cdot \rangle}_K$ be the inner product on $K$ which makes the $y_i$ orthonormal; then we have $\norm{\sum_{i=1}^n b_i y_i}_K^2 = \sum_{i=1}^n b_i^2$. By our construction, we have that any $x \in K$ with $\norm{x}_K^2 \le 1$ has $\norm{x}_W < 1$ as well, so $K$ is continuously and densely embedded in $W$. That is to say, the inclusion map $i : (K, {\norm{\cdot}}_K)
\to (W, {\norm{\cdot}}_W)$ is continuous.
Let $\bar{K}$ be the abstract completion of $K$, so $\bar{K}$ is a Hilbert space. Since $W$ is Banach, the continuous map $i : K \to
W$ extends to a continuous map $\bar{i} : \bar{K} \to W$ whose image is dense (as it contains $K$). It is possible that $\bar{i}$ is not injective, so let $H = (\ker \bar{i})^\perp$ be the orthogonal complement in $\bar{K}$ of its kernel. $H$ is a closed subspace of $\bar{K}$, hence a Hilbert space in its own right, and the restriction $\bar{i}|_H : H \to W$ is continuous and injective, and its range is the same as that of $\bar{i}$, hence still dense in $W$.
The final step of the previous proof (passing to $(\ker
\bar{i})^\perp$) is missing from Gross’s original proof, as was noticed by Ambar Sengupta, who asked if it is actually necessary. Here is an example to show that it is.
Let $W$ be a separable Hilbert space with orthonormal basis $\{e_n\}_{n=1}^\infty$, and let $S$ be the left shift operator defined by $S e_1
= 0$, $S e_n = e_{n-1}$ for $n \ge 2$. Note that the kernel of $S$ is one-dimensional and spanned by $e_1$. Let $E$ be the subspace of $W$ spanned by the vectors $h_n = e_n - e_{n+1}$, $n = 1, 2, \dots$. It is easy to check that $e_1 \notin E$, so the restriction of $S$ to $E$ is injective. On the other hand, $E$ is dense in $W$: for suppose $x \in E^\perp$. Since ${\langle x , h_n \rangle}_W = 0$, we have ${\langle x , e_n \rangle}_W = {\langle x , e_{n+1} \rangle}_W$, so in fact there is a constant $c$ with ${\langle x , e_n \rangle}_W = c$ for all $n$. But Parseval’s identity says $\sum_{n=1}^\infty
|{\langle x , e_n \rangle}_W|^2 = \norm{x}_W^2 < \infty$ so we must have $c=0$ and thus $x=0$.
We also remark that $SE$ is also dense in $W$: since $S h_n =
h_{n-1}$, we actually have $E \subset SE$.
So we have a separable inner product space $E$, a separable Hilbert space $W$, and a continuous injective map $S|_E : E \to W$ with dense image, such that the continuous extension of $S|_E$ to the completion of $E$ (namely $W$) is not injective (since the extension is just $S$ again).
To make this look more like Gross’s construction, we just rename things. Set $K = SE$ and define an inner product on $K$ by ${\langle Sx , Sy \rangle}_K = {\langle x , y \rangle}_W$ (this is well defined because $S$ is injective on $E$). Now $K$ is an inner product space, continuously and densely embedded in $W$, but the completion of $K$ does not embed in $W$ (the continuous extension of the inclusion map is not injective, since it is really $S$ in disguise).
The inner product space $(K, {\langle \cdot , \cdot \rangle}_K)$ could actually be produced by Gross’s construction. By applying the Gram–Schmidt algorithm to $\{h_n\}$, we get an orthonormal set $\{g_n\}$ (with respect to ${\langle \cdot , \cdot \rangle}_W$) which still spans $E$. (In fact, $\{g_n\}$ is also an orthonormal basis for $W$.) Take $z_n = S g_n$; the $z_n$s are linearly independent and span $K$, which is dense in $W$. If $\sum_{i=1}^n
b_i^2 \le 1$, then $\norm{\sum_{i=1}^n b_i z_i}_W = \norm{S
\sum_{i=1}^n b_i g_i}_W \le 1$ because $S$ is a contraction and the $g_i$ are orthonormal. So we can take $a_i = 1$ in the induction.[^4] Then of course the inner product which makes the $z_n$ orthonormal is just ${\langle \cdot , \cdot \rangle}_K$.
We can make the issue even more explicit: consider the series $\sum_{n=1}^\infty {\langle g_n , e_1 \rangle}_W z_n$. Under ${\norm{\cdot}}_K$, this series is Cauchy, since $z_n$ is orthonormal and $\sum_n
|{\langle g_n , e_1 \rangle}_W|^2 = \norm{e_1}_W^2 = 1$; and its limit is not zero, since there must be some $g_k$ with ${\langle g_k , e_1 \rangle}_W \ne
0$, and then we have ${\langle z_k , \sum_{n=1}^m {\langle g_n , e_1 \rangle}_W
z_n \rangle} = {\langle g_k , e_1 \rangle}_W$ for all $m \ge k$. So the series corresponds to some nonzero element of the completion $\bar{K}$. However, under ${\norm{\cdot}}_H$, the series converges to zero, since $\sum_{n=1}^m {\langle g_n , e_1 \rangle}_W z_n = S \sum_{n=1}^m
{\langle g_n , e_1 \rangle}_W g_n \to S e_1 = 0$, using the continuity of $S$ and the fact that $g_n$ is an orthonormal basis for $W$.
The following theorem points out that measurable norms are far from unique.
Suppose ${\norm{\cdot}}_W$ is a measurable norm on a Hilbert space $(H,
{\norm{\cdot}}_H)$. Then there exists another measurable norm ${\norm{\cdot}}_{W'}$ which is stronger than ${\norm{\cdot}}_W$, and if we write $W, W'$ for the corresponding completions, the inclusions $H \hookrightarrow W'
\hookrightarrow W$ are compact.
See [@kuo-gaussian-book Lemma 4.5].
Gaussian measures on Hilbert spaces
-----------------------------------
We have been discussing Gaussian measures on separable Banach spaces $W$. This includes the possibility that $W$ is a separable Hilbert space. In this case, there is more that can be said about the relationship between $W$ and its Cameron–Martin space $H$.
Let $H,K$ be separable Hilbert spaces.
Let $A : H \to K$ be a bounded operator, $A^*$ its adjoint. Let $\{h_n\}$, $\{k_m\}$ be orthonormal bases for $H,K$ respectively. Then $$\sum_{n=1}^\infty \norm{A h_n}_K^2 = \sum_{m=1}^\infty \norm{A^* k_m}_H^2.$$
\[hs-def\] A bounded operator $A : H \to K$ is said to be **Hilbert–Schmidt** if $$\norm{A}_{HS}^2 = \sum_{i=1}^\infty \norm{A e_n}_K^2 < \infty$$ for some orthonormal basis $\{e_n\}$ of $H$. By the previous exercise, this does not depend on the choice of basis, and $\norm{A}_{HS} = \norm{A^*}_{HS}$.
If $\norm{A}_{L(H,K)}$ denotes the operator norm of $A$, then $\norm{A}_{L(H)} \le \norm{A}_{HS}$.
${\norm{\cdot}}_{HS}$ is induced by the inner product ${\langle A , B \rangle}_{HS} =
\sum_{n=1}^\infty {\langle A e_n , B e_n \rangle}_K$ and makes the set of all Hilbert–Schmidt operators from $H$ to $K$ into a Hilbert space.
Every Hilbert–Schmidt operator is compact. In particular, Hilbert–Schmidt operators do not have bounded inverses if $H$ is infinite-dimensional. The identity operator is not Hilbert–Schmidt.
If $A$ is Hilbert–Schmidt and $B$ is bounded, then $BA$ and $AB$ are Hilbert–Schmidt. So the Hilbert–Schmidt operators form a two-sided ideal inside the ring of bounded operators.
If $A$ is a bounded operator on $H$, $H_0$ is a closed subspace of $H$, and $A|_{H_0}$ is the restriction of $A$ to $H_0$, then $\norm{A|_{H_0}}_{HS} \le \norm{A}_{HS}$.
If $H$ is a finite-dimensional Hilbert space, $A : H \to K$ is linear, and $\mathbf{X}$ has a standard normal distribution on $H$, then $\mathbb{E} \norm{A\mathbf{X}}_K^2 = \norm{A}_{HS}^2$.
If $\mathbf{Z}$ has a normal distribution on $\R^n$ with covariance matrix $\Sigma$, then clearly $E |\mathbf{Z}|^2 = {\operatorname{tr}}\Sigma$. The covariance matrix of $A \mathbf{X}$ is $A^* A$, and ${\operatorname{tr}}(A^* A) =
\norm{A}_{HS}^2$.
\[hs-implies-measurable\] Let $A$ be a Hilbert–Schmidt operator on $H$. Then $\norm{h}_{W} =
\norm{Ah}_H$ is a measurable norm on $H$.
Fix an orthonormal basis $\{e_n\}$ for $H$, and suppose $\epsilon > 0$. Since $\sum_{n=1}^\infty \norm{A e_n}_H^2 < \infty$, we can choose $N$ so large that $\sum_{n=N}^\infty \norm{A e_n}_H^2 < \epsilon^3$. Let $P_0$ be orthogonal projection onto the span of $\{e_1, \dots,
e_{N-1}\}$. Note in particular that $\norm{A|_{(P_0
H)^\perp}}_{HS}^2 < \epsilon^3$. Now suppose $P \perp P_0$ is a finite rank projection. Then $$\tilde{\mu}(\{h : \norm{Ph}_{W} > \epsilon\}) =
\tilde{\mu}(\{h : \norm{APh}_{H} > \epsilon\}) =
\mathbb{P}(\norm{A \mathbf{X}}_H > \epsilon)$$ where $\mathbf{X}$ has a standard normal distribution on $PH$. By the previous lemma, $$\mathbb{E} \norm{A \mathbf{X}}_H^2 = \norm{A|_{PH}}_{HS}^2 \le
\norm{A|_{(P_0 H)^\perp}}_{HS}^2 < \epsilon^3$$ so Chebyshev’s inequality gives $\mathbb{P}(\norm{A \mathbf{X}}_H >
\epsilon) < \epsilon$ as desired.
Since the norm $\norm{h}_W = \norm{Ah}_H$ is induced by an inner product (namely ${\langle h , k \rangle}_W = {\langle Ah , Ak \rangle}_H$), the completion $W$ is a Hilbert space. Actually, this is the only way to get $W$ to be a Hilbert space. Here is a more general result, due to Kuo.
Let $W$ be a separable Banach space with Gaussian measure $\mu$ and Cameron–Martin space $H$, $i : H \to W$ the inclusion map, and let $Y$ be some other separable Hilbert space. Suppose $A : W \to Y$ is a bounded operator. Then $Ai : H \to Y$ (i.e. the restriction of $A$ to $H \subset W$) is Hilbert–Schmidt, and $\norm{Ai}_{HS} \le C \norm{A}_{L(W,Y)}$ for some constant $C$ depending only on $(W,\mu)$.
We consider instead the adjoint $(Ai)^* = i^* A^* : Y \to H$. Note $A^* : Y \to W^*$ is bounded, and $\norm{i^* A^* y}_H^2 = q(A^* y,
A^* y)$. So if we fix an orthonormal basis $\{e_n\}$ for $Y$, we have $$\begin{aligned}
\norm{i^* A^*}_{HS}^2 &= \sum_{n=1}^\infty q(A^* e_n, A^* e_n) \\
&= \sum_{n=1}^\infty \int_W |(A^* e_n)(x)|^2 \mu(dx) \\
&= \int_W \sum_{n=1}^\infty |(A^* e_n)(x)|^2 \mu(dx) &&
\text{(Tonelli)} \\
&= \int_W \sum_{n=1}^\infty |{\langle A x , e_n \rangle}_Y|^2
\mu(dx) \\
&= \int_W \norm{Ax}_Y^2 \mu(dx) \\
&\le \norm{A}_{L(W,Y)}^2 \int_W \norm{x}_W^2 \mu(dx).
\end{aligned}$$ By Fernique’s theorem we are done.
\[inclusion-HS\] If $W$ is a separable Hilbert space with a Gaussian measure $\mu$ and Cameron–Martin space $H$, then the inclusion $i : H \to W$ is Hilbert–Schmidt, as is the inclusion $m : W^* \to K$.
Take $Y = W$ and $A = I$ in the above lemma to see that $i$ is Hilbert–Schmidt. To see $m$ is, chase the diagram.
\[hilbert-in-hilbert\] Let ${\norm{\cdot}}_W$ be a norm on a separable Hilbert space $H$. Then the following are equivalent:
1. \[cor-measurable\] ${\norm{\cdot}}_W$ is measurable and induced by an inner product ${\langle \cdot , \cdot \rangle}_W$;
2. \[cor-hs\] $\norm{h}_W = \norm{Ah}_H$ for some Hermitian, positive definite, Hilbert–Schmidt operator $A$ on $H$.
Suppose \[cor-measurable\] holds. Then by Gross’s theorem (Theorem \[gross-theorem\]) the completion $W$, which is a Hilbert space, admits a Gaussian measure with Cameron–Martin space $H$. Let $i :
H \to W$ be the inclusion; by Corollary \[inclusion-HS\] $i$ is Hilbert–Schmidt, and so is its adjoint $i^* : W \to H$. Then $i^*
i : H \to H$ is continuous, Hermitian, and positive semidefinite. It is also positive definite because $i$ and $i^*$ are both injective. Take $A = (i^* i)^{1/2}$. $A$ is also continuous, Hermitian, and positive definite, and we have $\norm{Ah}_H^2 =
{\langle i^* i h , h \rangle}_H = {\langle ih , ih \rangle}_W = \norm{h}_W^2$. $A$ is also Hilbert–Schmidt since $\sum \norm{A e_n}_H^2 = \sum \norm{i
e_n}_W^2$ and $i$ is Hilbert–Schmidt.
The converse is Lemma \[hs-implies-measurable\].
Brownian motion on abstract Wiener space
========================================
Let $(W,H,\mu)$ be an abstract Wiener space.
For $t \ge 0$, let $\mu_t$ be the rescaled measure $\mu_t(A) =
\mu(t^{-1/2} A)$ (with $\mu_0 = \delta_0$). It is easy to check that $\mu_t$ is a Gaussian measure on $W$ with covariance form $q_t(f,g) = t q(f,g)$. For short, we could call $\mu_t$ **Gaussian measure with variance $t$**.
If $W$ is finite dimensional, then $\mu_s \sim \mu_t$ for all $s,t$. If $W$ is infinite dimensional, then $\mu_s \perp \mu_t$ for $s \ne t$.
$\mu_s * \mu_t = \mu_{s+t}$, where $*$ denotes convolution: $\mu *
\nu(E) = \iint_{W^2} 1_E(x+y) \mu(dx) \nu(dy)$. In other words, $\{\mu_t : t \ge 0\}$ is a convolution semigroup.
Compute Fourier transforms: if $f \in W^*$, then $$\begin{aligned}
\widehat{\mu_s * \mu_t}(f) &= \int_W \int_W e^{i f(x+y)} \mu_s(dx)
\mu_t(dy) \\
&= \int_W e^{i f(x)} \mu_s(dx) \int_W e^{i f(y)} \mu_t(dy) \\
&= e^{- \frac{1}{2} sq(f,f)} e^{- \frac{1}{2} tq(f,f)} \\
&= e^{- \frac{1}{2} (s+t) q(f,f)} \\
&= \widehat{\mu_{s+t}}(f).
\end{aligned}$$
There exists a stochastic process $\{B_t, t \ge 0\}$ with values in $W$ which is a.s. continuous in $t$ (with respect to the norm topology on $W$), has independent increments, and for $t > s$ has $B_t - B_s \sim \mu_{t-s}$, with $B_0 = 0$ a.s. $B_t$ is called **standard Brownian motion on $(W,\mu)$**.
Your favorite proof of the existence of one-dimensional Brownian motion should work. For instance, one can use the Kolmogorov extension theorem to construct a countable set of $W$-valued random variables $\{B_t : t \in E \}$, indexed by the dyadic rationals $E$, with independent increments and $B_t - B_s \sim \mu_{t-s}$. (The consistency of the relevant family of measures comes from the property $\mu_t * \mu_s = \mu_{t+s}$, just as in the one-dimensional case.) If you are worried that you only know the Kolmogorov extension theorem for $\R$-valued random variables, you can use the fact that any Polish space can be measurably embedded into $[0,1]$. Then the Kolmogorov continuity theorem (replacing absolute values with ${\norm{\cdot}}_W$) can be used to show that, almost surely, $B_t$ is Hölder continuous as a function between the metric spaces $E$ and $W$. Use the fact that $$\mathbb{E} \norm{B_t - B_s}_W^\beta = \int_W \norm{x}_W^\beta
\mu_{t-s}(dx) = (t-s)^{\beta/2} \int_W \norm{x}_W^\beta \mu(dx) \le
C (t-s)^{\beta/2}$$ by Fernique. In particular $B_t$ is, almost surely, uniformly continuous and so extends to a continuous function on $[0,\infty)$.
For any $f \in W^*$, $f(B_t)$ is a one-dimensional Brownian motion with variance $q(f,f)$. If $f_1, f_2, \dots$ are $q$-orthogonal, then the Brownian motions $f_1(B_t), f_2(B_t), \dots$ are independent.
If $h_j$ is an orthonormal basis for $H$, and $B_t^j$ is an iid sequence of one-dimensional standard Brownian motions, does $\sum_{j=1}^\infty B_t^j h_j$ converge uniformly in $W$, almost surely, to a Brownian motion on $W$? That would be an even easier construction.
Let $W' = C([0,1], W)$ (which is a separable Banach space) and consider the measure $\mu'$ on $W'$ induced by $\{B_t, 0 \le t \le
1\}$. Show that $\mu'$ is a Gaussian measure. For extra credit, find a nice way to write its covariance form.
Suppose $W = C([0,1])$ and $\mu$ is the law of a one-dimensional continuous Gaussian process $X_s$ with covariance function $a(s_1,s_2)$. Let $Y_{s,t} = B_t(s)$ be the corresponding two-parameter process (note $B_t$ is a random element of $C([0,1])$ so $B_t(s)$ is a random variable). Show $Y_{s,t}$ is a continuous Gaussian process whose covariance function is $$\mathbb{E}[Y_{s_1,t_1} Y_{s_2, t_2}] = (t_1 \wedge t_2) a(s_1, s_2).$$ If $X_s$ is one-dimensional Brownian motion, then $Y_{s,t}$ is called the **Brownian sheet**.
$B_t$ has essentially all the properties you would expect a Brownian motion to have. You can open your favorite textbook on Brownian motion and pick most any theorem that applies to $d$-dimensional Brownian motion, and the proof should go through with minimal changes. We note a few important properties here.
$B_t$ is a Markov process, with transition probabilities $\mathbb{P}^x (B_t \in
A) = \mu_t^x(A) := \mu(t^{-1/2}(A - x))$.
The Markov property is immediate, because $B_t$ has independent increments. Computing the transition probabilities is also very simple.
$B_t$ is a martingale.
Obvious, because it has independent mean-zero increments.
$B_t$ obeys the Blumenthal $0$-$1$ law: let $\mathcal{F}_t =
\sigma(B_s : 0 \le s \le t)$ and $\mathcal{F}_t^+ = \bigcap_{s > t}
\mathcal{F}_s$. Then $\mathcal{F}_0^+$ is $\mathbb{P}^x$-almost trivial, i.e. for any $A \in \mathcal{F}_0^+$ and any $x \in W$, $\mathbb{P}^x(A) = 0$ or $1$.
This holds for any continuous Markov process.
The transition semigroup of $B_t$ is $$P_t F(x) = E_x F(B_t) = \int F \mu^x_t = \int F(x + t^{1/2}y) \mu(dy)$$ which makes sense for any bounded measurable function. Clearly $P_t$ is Markovian (positivity-preserving and a contraction with respect to the uniform norm).
Let $C_b(W)$ denote the space of bounded continuous functions $F : W
\to \R$. Let $C_u(W)$ denote the subspace of bounded *uniformly continuous* functions.
$C_b(W)$ and $C_u(W)$ are Banach spaces.
$P_t$ is a Feller semigroup: if $F$ is continuous, so is $P_t F$.
Fix $x \in W$, and $\epsilon > 0$. Since $\mu_t$ is Radon (see Section \[radon\]), there exists a compact $K$ with $\mu_t(K^C) < \epsilon$.
For any $z \in K$, the function $F(\cdot + z)$ is continuous at $x$, so there exists $\delta_z$ such that for any $u$ with $\norm{u}_W <
\delta_z$, we have $|F(x+z) -F(x+u+z)| < \epsilon$. The balls $B(z,
\delta_z/2)$ cover $K$ so we can take a finite subcover $B(z_i,
\delta_i/2)$. Now suppose $\norm{x-y}_W < \min \delta_i/2$. For any $z
\in K$, we may choose $z_i$ with $z \in B(z_i, \delta_i/2)$. We then have $$\begin{aligned}
|F(x+z) - F(y+z)| &\le |F(x+z) - F(x+z_i)| + |F(x+z_i) - F(y+z)| \\
&= |F((x + (z - z_i)) + z_i) - F(x + z_i)| \\
&\quad + |F(x + z_i) - F((x +
(y-x) + (z-z_i)) + z_i)|
\end{aligned}$$ Each term is of the form $|F(x+z_i) - F(x+u+z_i)|$ for some $u$ with $\norm{u}_W < \delta_i$, and hence is bounded by $\epsilon$.
Now we have $$\begin{aligned}
|P_t F(x) - P_t F(y)| &\le \int_W |F(x+z)-F(y+z)| \mu_t(dz) \\
&= \int_K |F(x+z)-F(y+z)| \mu_t(dz) + \int_{K^C} |F(x+z)-F(y+z)|
\mu_t(dz) \\
&\le \epsilon + 2 \epsilon \norm{F}_\infty.
\end{aligned}$$
This shows that $P_t$ is a contraction semigroup on $C_b(W)$. We would really like to have a strongly continuous contraction semigroup. However, $P_t$ is not in general strongly continuous on $C_b(W)$. Indeed, take the one-dimensional case $W = \R$, and let $f(x) = \cos(x^2)$, so that $f$ is continuous and bounded but not uniformly continuous. One can check that $P_t f$ vanishes at infinity, for any $t$. (For instance, take Fourier transforms, so the convolution in $P_t f$ becomes multiplication. The Fourier transform $\hat{f}$ is just a scaled and shifted version of $f$; in particular it is bounded, so $\widehat{P_t f}$ is integrable. Then the Riemann–Lebesgue lemma implies that $P_t f \in C_0(\R)$. One can also compute directly, perhaps by writing $f$ as the real part of $e^{i x^2}$.) Thus if $P_t f$ were to converge uniformly as $t \to
0$, the limit would also vanish at infinity, and so could not be $f$. (In fact, $P_t f \to f$ pointwise as $t \to 0$, so $P_t f$ does not converge uniformly.)
One should note that the term “Feller semigroup” has several different and incompatible definitions in the literature, so caution is required when invoking results from other sources. One other common definition assumes the state space $X$ is locally compact, and requires that $P_t$ be a strongly continuous contraction semigroup on $C_0(X)$, the space of continuous functions “vanishing at infinity”, i.e. the uniform closure of the continuous functions with compact support. In our non-locally-compact setting this condition is meaningless, since $C_0(W) = 0$.
$B_t$ has the strong Markov property.
This should hold for any Feller process. The proof in Durrett looks like it would work.
$P_t$ is a strongly continuous contraction semigroup on $C_u(W)$.
Let $F \in C_u(W)$. We first check that $P_t F \in C_u(W)$. It is clear that $P_t F$ is bounded; indeed, $\norm{P_t F}_\infty \le \norm{F}_\infty$. Fix $\epsilon > 0$. There exists $\delta > 0$ such that $|F(x) - F(y)|
< \epsilon$ whenever $\norm{x-y}_W < \delta$. For such $x,y$ we have $$|P_t F(x) - P_t F(y)| \le \int |F(x+z)-F(y+z)| \mu_t(dz) \le
\epsilon.$$ Thus $P_t F$ is uniformly continuous.
Next, we have $$\begin{aligned}
|P_t F(x) - F(x)| &\le \int |F(x + t^{1/2}y) - F(x)| \mu(dy) \\
&= \int_{\norm{t^{1/2} y}_W < \delta} |F(x + t^{1/2}y) - F(x)|
\mu(dy) + \int_{\norm{t^{1/2} y}_W \ge \delta} |F(x + t^{1/2}y) -
F(x)| \mu(dy) \\
&\le \epsilon + 2 \norm{F}_\infty \mu(\{y : \norm{t^{1/2}y}_W \ge \delta\}).
\end{aligned}$$ But $\mu(\{y : \norm{t^{1/2}y}_W \ge \delta\}) =
\mu(B(0,t^{-1/2})^C) \to 0$ as $t \to 0$. So for small enough $t$ we have $|P_t F(x) - F(x)| \le 2 \epsilon$ independent of $x$.
$C_u(X)$ is not the nicest Banach space to work with here; in particular it is not separable, and its dual space is hard to describe. However, the usual nice choices that work in finite dimensions don’t help us here. In $\R^n$, $P_t$ is a strongly continuous semigroup on $C_0(\R^n)$; but in infinite dimensions $C_0(W) = 0$.
In $\R^n$, $P_t$ is also a strongly continuous symmetric semigroup on $L^2(\R^n, m)$, where $m$ is Lebesgue measure. In infinite dimensions we don’t have Lebesgue measure, but we might wonder whether $\mu$ could stand in: is $P_t$ a reasonable semigroup on $L^2(W,\mu)$? The answer is emphatically no; it is not even a well-defined operator. First note that $P_t 1 = 1$ for all $t$. Let $e_i \in W^*$ be $q$-orthonormal, so that under $\mu$ the $e_i$ are iid $N(0,1)$. Under $\mu_t$ they are iid $N(0,t)$. Set $s_n(x) = \frac{1}{n} \sum_{i=1}^n |e_i(x)|^2$; by the strong law of large numbers, $s_n \to t$, $\mu_t$-a.e. Let $A = \{x : s_n(x) \to 1\}$, so $1_A = 1$ $\mu$-a.e. On the other hand, for any $t > 0$, $$\int_W P_t 1_A(x)\,\mu(dx) = \int_W \int_W 1_A(x+y) \mu_t(dy) \mu(dx)
= (\mu_t * \mu)(A) = \mu_{1+t}(A) = 0.$$ Since $P_t 1_A \ge 0$, it must be that $P_t 1_A = 0$, $\mu$-a.e. Thus $1$ and $1_A$ are the same element of $L^2(W,\mu)$, but $P_t 1$ and $P_t 1_A$ are not.
We knew that $H$ was “thin” in $W$ in the sense that $\mu(H) = 0$. In particular, this means that for any $t$, $\mathbb{P}(B_t \in H) = 0$. Actually more is true.
Let $\sigma_H = \inf\{t > 0 : B_t \in H\}$. Then for any $x \in W$, $\mathbb{P}^x(\sigma_H = \infty) = 1$. That is, from any starting point, with probability one, $B_t$ *never* hits $H$. In other words, $H$ is polar for $B_t$.
Fix $0 < t_1 < t_2 < \infty$. If $b_t$ is a standard one-dimensional Brownian motion started at $0$, let $c = P(\inf\{b_t
: t_1 \le t \le t_2\} \ge 1)$, i.e. the probability that $b_t$ is above $1$ for all times between $t_1$ and $t_2$. Clearly $c > 0$. By the strong Markov property it is clear that for $x_0 > 0$, $P(\inf\{b_t + x_0 : t_1 \le t \le t_2\} \ge 1) > c$, and by symmetry $P(\sup\{b_t - x_0 : t_1 \le t \le t_2\} \le -1) > c$ also. So for any $x_0 \in \R$, we have $$P(\inf\{|b_t + x_0|^2 : t_1 \le t \le t_2\} \ge 1) > c.$$
Fix a $q$-orthonormal basis $\{e_k\} \subset W^*$ as in Lemma \[ek-basis\], so that $B_t \in H$ iff $\norm{B_t}_H^2
= \sum_k |e_k(B_t)|^2 < \infty$. Under $\mathbb{P}^x$, $e_k(B_t)$ are independent one-dimensional Brownian motions with variance 1 and starting points $e_k(x)$. So if we let $A_k$ be the event $A_k =
\{\inf\{|e_k(B_t)|^2 : t_1 \le t \le t_2\} \ge 1\}$, by the above computation we have $\mathbb{P}^x(A_k) > c$. Since the $A_k$ are independent we have $\mathbb{P}^x(A_k \text{ i.o.}) = 1$. But on the event $\{A_k \text{ i.o}\}$ we have $\norm{B_t}_H^2 = \sum_{k=1}^\infty
|e_k(B_t)|^2 = \infty$ for all $t \in [t_1, t_2]$. Thus $\mathbb{P}^x$-a.s. $B_t$ does not hit $H$ between times $t_1$ and $t_2$. Now let $t_1 \downarrow 0$ and $t_2 \uparrow \infty$ along sequences to get the conclusion.
A priori it is not obvious that $\sigma_H : \Omega \to [0,\infty]$ is even measurable (its measurability it is defined by an uncountable infimum, and there is no apparent way to reduce it to a countable infimum), or that $\{\sigma_H = \infty\}$ is a measurable subset of $\Omega$. What we really showed is that there is a (measurable) event $A = \{A_k \text{ i.o.}\}$ with $\mathbb{P}^x(A)
= 1$ and $\sigma_H = \infty$ on $A$. If we complete the measurable space $(\Omega, \mathcal{F})$ by throwing in all the sets which are $\mathbb{P}^x$-null for every $x$, then $\{\sigma_h = \infty\}$ will be measurable and so will $\sigma_H$.
In the general theory of Markov processes one shows that under some mild assumptions, including the above completion technique, $\sigma_B$ is indeed measurable for any Borel (or even analytic) set $B$.
Calculus on abstract Wiener space
=================================
The strongly continuous semigroup $P_t$ on $C_u(W)$ has a generator $L$, defined by $$Lf = \lim_{t \downarrow 0} \frac{1}{t} (P_t f - f).$$ This is an unbounded operator on $C_u(W)$ whose domain $D(L)$ is the set of all $f$ for which the limit converges in $C_u(W)$. It is a general fact that $L$ is densely defined and closed.
In the classical setting where $W = \R^n$ and $\mu$ is standard Gaussian measure (i.e. $q = {\langle \cdot , \cdot \rangle}_{\R^n}$), so that $B_t$ is standard Brownian motion, we know that $L = - \frac{1}{2} \Delta$ is the Laplace operator, which sums the second partial derivatives in all orthogonal directions. Note that “orthogonal” is with respect to the Euclidean inner product, which is also the Cameron–Martin inner product in this case.
We should expect that in the setting of abstract Wiener space, $L$ should again be a second-order differential operator that should play the same role as the Laplacian. So we need to investigate differentiation on $W$.
Let $W$ be a Banach space, and $F : W \to \R$ a function. We say $F$ is **Fréchet differentiable** at $x \in W$ if there exists $g_x \in W^*$ such that, for any sequence $W \ni y_n \to 0$ in $W$-norm, $$\frac{F(x+y_n) - F(x) - g_x(y_n)}{\norm{y_n}_W} \to 0.$$ $g_x$ is the Fréchet derivative of $F$ at $x$. One could write $F'(x) = g_x$. It may be helpful to think in terms of directional derivatives and write $\partial_y F(x) = F'(x)y = g_x(y)$.
Suppose $F(x) = \phi(f_1(x), \dots, f_n(x))$ is a cylinder function. Then $$F'(x) = \sum_{i=1}^n \partial_i \phi(f_1(x), \dots, f_n(x)) f_i.$$
As it turns out, we will be most interested in differentiating in directions $h
\in H$, since in some sense that is really what the usual Laplacian does. Also, Fréchet differentiability seems to be too much to ask for; according to references in Kuo [@kuo-gaussian-book], the set of continuously Fréchet differentiable functions is not dense in $C_u(W)$.
$F : W \to \R$ is **$H$-differentiable** at $x \in W$ if there exists $g_x \in H$ such that for any sequence $H \ni h_n \to 0$ in $H$-norm, $$\frac{F(x+h_n) - F(x) - {\langle g_x, h_n , _ \rangle}H}{\norm{h_n}_H} \to 0.$$ We will denote the element $g_x$ by $DF(x)$, and we have $\partial_h
F(x) = {\langle DF(x) , h \rangle}_H$. $DF : W \to H$ is sometimes called the **Malliavin derivative** or **gradient** of $F$.
For a cylinder function $F(x) = \phi(f_1(x), \dots, f_n(x))$, we have $${\langle DF(x) , h \rangle}_H = \sum_{i=1}^n \partial_i \phi(f_1(x), \dots, f_n(x)) f_i(h)$$ or alternatively $$\label{DF-cylinder}
DF(x) = \sum_{i=1}^n \partial_i \phi(f_1(x), \dots, f_n(x)) J f_i$$
We know that for $F \in C_u(W)$, $P_t F$ should belong to the domain of the generator $L$, for any $t > 0$. The next proposition shows that we are on the right track with $H$-differentiability.
For $F \in C_u(W)$ and $t > 0$, $P_t F$ is $H$-differentiable, and $${\langle D P_t F(x) , h \rangle}_H = \frac{1}{t} \int_W F(x+y) {\langle h , y \rangle}_H\, \mu_t(dy).$$
It is sufficient to show that $$P_t F(x+h) - P_t F(x) = \frac{1}{t} \int_W F(x+y) {\langle h , y \rangle}_H\,
\mu_t(dy) + o(\norm{h}_H)$$ since the first term on the right side is a bounded linear functional of $h$ (by Fernique’s theorem). The Cameron–Martin theorem gives us $$P_t F(x+h) = \int_W F(x+y)\, \mu_t^h(dy) = \int_W F(x+y) J_t(h,y)\, \mu_t(dy)$$ where $$J_t(h,y) = \exp\left(-\frac{1}{2t} \norm{h}_H^2 + \frac{1}{t} {\langle h , y \rangle}_H\right)$$ is the Radon–Nikodym derivative, or in other words the “Jacobian determinant.” Then we have $$\begin{aligned}
P_t F(x+h) - P_t F(x) = \int_W F(x+y) (J_t(h,y) - 1) \,\mu_t(dy).
\end{aligned}$$ Since $J_t(0,y) = 1$, we can write $J_t(h,y) - 1 = \int_0^1
\frac{d}{ds} J_t(sh,y)\,ds$ by the fundamental theorem of calculus. Now we can easily compute that $\frac{d}{ds} J_t(sh,y) = \frac{1}{t}
({\langle h , y \rangle}_H - s \norm{h}_H^2) J_t(sh,y)$, so we have $$\begin{aligned}
P_t F(x+h) - P_t F(x) &= \frac{1}{t} \int_W F(x+y) \int_0^1
({\langle h , y \rangle}_H - s \norm{h}_H^2) J_t(sh,y)\,ds \,\mu_t(dy) \\
&= \frac{1}{t} \int_W F(x+y) {\langle h , y \rangle}_H \mu_t(dy) \\
&\quad +
\frac{1}{t} \int_W
F(x+y) \int_0^1
{\langle h , y \rangle}_H (J_t(sh,y)-1)\,ds \,\mu_t(dy) && (\alpha) \\
&\quad - \frac{1}{t} \int_W F(x+y) \int_0^1 s \norm{h}_H^2
J_t(sh,y) \,ds\,\mu_t(dy) && (\beta).
\end{aligned}$$ So it remains to show that the remainder terms $\alpha, \beta$ are $o(\norm{h}_H)$.
To estimate $\alpha$, we crash through with absolute values and use Tonelli’s theorem and Cauchy–Schwarz to obtain $$\begin{aligned}
\abs{\alpha} &\le \frac{\norm{F}_\infty}{t} \int_0^1 \int_W
\abs{{\langle h , y \rangle}_H} \abs{J_t(sh,y) - 1}\,\mu_t(dy)\,ds \\
&\le \frac{\norm{F}_\infty}{t} \int_0^1 \sqrt{\int_W
\abs{{\langle h , y \rangle}_H}^2\,\mu_t(dy) \int_W \abs{J_t(sh,y) - 1}^2\,\mu_t(dy)}\,ds.
\end{aligned}$$ But $\int_W
\abs{{\langle h , y \rangle}_H}^2\,mu_t(dy) = t \norm{h}_H^2$ (since under $\mu_t$, ${\langle h , \cdot \rangle}_H \sim N(0, t \norm{h}_H^2)$). Thus $$\abs{\alpha} \le \frac{\norm{F}_\infty}{\sqrt{t}} \norm{h}_H
\int_0^1 \sqrt{\int_W \abs{J_t(sh,y) - 1}^2\,\mu_t(dy)}\,ds.$$
Now, a quick computation shows $$\begin{aligned}
\abs{J_t(sh,y)-1}^2 = e^{s^2 \norm{h}_H^2/t} J_t(2sh, y) - 2
J_t(sh,y) + 1.
\end{aligned}$$ But $J_t(g,y) \mu_t(dy) = \mu_t^g(dy)$ is a probability measure for any $g \in H$, so integrating with respect to $\mu_t(dy)$ gives $$\begin{aligned}
\int_W \abs{J_t(sh,y) - 1}^2\,\mu_t(dy) = e^{s^2 \norm{h}_H^2/t}
- 1.
\end{aligned}$$ So we have $$\int_0^1 \sqrt{\int_W \abs{J_t(sh,y) - 1}^2\,\mu_t(dy)}\,ds =
\int_0^1 \sqrt{e^{s^2 \norm{h}_H^2/t}
- 1}\,ds = o(1)$$ as $\norm{h}_H \to 0$, by dominated convergence. Thus we have shown $\alpha = o(\norm{h}_H)$.
The $\beta$ term is easier: crashing through with absolute values and using Tonelli’s theorem (and the fact that $J_t \ge 0$), we have $$\begin{aligned}
\abs{\beta} &\le \frac{1}{t} \norm{F}_\infty \norm{h}_H^2
\int_0^1 s \int_W J_t(sh,y) \,\mu_t(dy)\,ds \\
&= \frac{1}{t} \norm{F}_\infty \norm{h}_H^2
\int_0^1 s \cancelto{1}{\int_W \,\mu_t^{sh}(dy)}\,ds \\
&= \frac{1}{2t} \norm{F}_\infty \norm{h}_H^2 = o(\norm{h}_H).
\end{aligned}$$
With more work it can be shown that $P_t F$ is in fact infinitely $H$-differentiable.
Kuo claims that the second derivative of $P_t F$ is given by $${\langle D^2 P_t F h , k \rangle}_H = \frac{1}{t} \int_W F(x+y)
\left(\frac{{\langle h , y \rangle}_H {\langle k , y \rangle}_H}{t} -
{\langle h , k \rangle}\right) \mu_t(dy).$$ If the generator $L$ is really the Laplacian $\Delta$ defined below, then in particular $D^2 P_t F$ should be trace class. But this doesn’t seem to be obvious from this formula. In particular, if we let $h = k = h_n$ and sum over an orthonormal basis $h_n$, the obvious approach of interchanging the integral and sum doesn’t work, because we get an integrand of the form $\sum_n (\xi_n^2 - 1)$ where $\xi_n$ are iid $N(0,1)$, which diverges almost surely.
The (infinitely) $H$-differentiable functions are dense in $C_u(W)$.
$P_t$ is a strongly continuous semigroup on $C_u(W)$, so for any $F
\in C_u(W)$ and any sequence $t_n \downarrow 0$, we have $P_{t_n} F \to F$ uniformly, and we just showed that $P_{t_n} F$ is $H$-differentiable.
Now, on the premise that the $H$ inner product should play the same role as the Euclidean inner product on $\R^n$, we define the Laplacian as follows.
The **Laplacian** of a function $F : W \to \R$ is $$\Delta F(x) = \sum_{k=1}^\infty \partial_{h_k} \partial_{h_k} F(x)$$ if it exists, where $\{h_k\}$ is an orthonormal basis for $H$. (This assumes that $F$ is $H$-differentiable, as well as each $\partial_h F$.)
If $F : W \to \R$ is a cylinder function as above, then $$\Delta F(x) = \sum_{i,j=1}^n \partial_i \partial_j \phi(f_1(x),
\dots, f_n(x)) q(f_i, f_j).$$
If $F$ is a cylinder function, then $F \in D(L)$, and $LF = -
\frac{1}{2} \Delta F$.
We have to show that $$\frac{P_t F(x) - F(x)}{t} \to \frac{1}{2}\Delta F(x)$$ uniformly in $x \in W$. As shorthand, write $$\begin{aligned}
G_i(x) &= \partial_i
\phi(f_1(x), \dots, f_n(x)) \\
G_{ij}(x) &= \partial_i
\partial_j \phi(f_1(x), \dots, f_n(x))
\end{aligned}$$ so that $\Delta F(x) =
\sum_{i,j=1}^n G_{ij}(x) q(f_i, f_j)$. Note that $G_{ij}$ is Lipschitz.
First note the following identity for $\alpha \in C^2([0,1])$, which is easily checked by integration by parts: $$\alpha(1) - \alpha(0) = \alpha'(0) + \int_0^1 (1-s) \alpha''(s)\,ds$$ Using $\alpha(s) = F(x+sy)$, we have $$\begin{aligned}
F(x+y) - F(x) = \sum_{i=1}^n G_i(x) f_i(y) + \int_0^1 (1-s)
\sum_{i,j=1}^n G_{ij}(x+sy) f_i(y) f_j(y)\,ds.
\end{aligned}$$ Integrating with respect to $\mu_t(dy)$, we have $$\begin{aligned}
P_t F(x) - F(x) &= \sum_{i=1}^n G_i(x) \cancelto{0}{\int_W f_i(y)
\,\mu_t(dy)} + \sum_{i,j=1}^n \int_W f_i(y) f_j(y) \int_0^1 (1-s)
G_{ij}(x+sy)\,ds \,\mu_t(dy) \\
&= t \sum_{i,j=1}^n \int_W f_i(y) f_j(y) \int_0^1 (1-s)
G_{ij}(x+st^{1/2}y)\,ds \,\mu(dy)
\end{aligned}$$ rewriting the $\mu_t$ integral in terms of $\mu$ and using the linearity of $f_i, f_j$. Now if we add and subtract $G_{ij}(x)$ from $G_{ij}(x+st^{1/2}y)$, we have $$\begin{aligned}
\int_W f_i(y) f_j(y)
G_{ij}(x) \,\mu(dy) \int_0^1 (1-s) \,ds = \frac{1}{2} G_{ij}(x)
\int_W f_i(y) f_j(y)\,\mu(dy) = \frac{1}{2} G_{ij}(x) q(f_i, f_j)
\end{aligned}$$ and, if $C$ is the Lipschitz constant of $G_{ij}$, $$\begin{aligned}
& \abs{\int_W f_i(y) f_j(y) \int_0^1 (1-s)
(G_{ij}(x+st^{1/2}y) - G_{ij}(x))\,ds \,\mu(dy)} \\
&\le
\norm{f_i}_{W^*} \norm{f_j}_{W^*} \int_W \norm{y}_W^2 \int_0^1
(1-s) (Cst^{1/2}\norm{y}_W)\,ds\,\mu(dy) \\
&\le
C t^{1/2} \norm{f_i}_{W^*} \norm{f_j}_{W^*} \int_0^1 (s-s^2)\,ds
\int_0^1 \norm{y}_W^3\,\mu(dy).
\end{aligned}$$ The $\mu$ integral is finite by Fernique’s theorem, so this goes to $0$ as $t \to 0$, independent of $x$. Thus we have shown $$P_t F(x) - F(x) = \frac{t}{2}\left(\sum_{i,j=1}^n G_{ij}(x) q(f_i,
f_j) + o(1)\right)$$ uniformly in $x$, which is what we wanted.
Kuo [@kuo-gaussian-book] proves the stronger statement that this holds for $F$ which are (more or less) Fréchet-$C^2$. Even this is not quite satisfactory, because, as claimed by Kuo’s references, these functions are not dense in $C_u(W)$. In particular, they are not a core for $L$. Can we produce a Laplacian-like formula for $L$ which makes sense and holds on a core of $L$?
Some $L^p$ theory
-----------------
Here we will follow Nualart [@nualart] for a while.
It is worth mentioning that Nualart’s approach (and notation) are a bit different from ours. His setting is a probability space $(\Omega,
\mathcal{F}, P)$ and a “process” $W$, i.e. a family $\{W(h) : h \in
H\}$ of jointly Gaussian random variables indexed by a Hilbert space $H$, with the property that $E[W(h) W(k)] = {\langle h , k \rangle}_H$. This includes our setting: take an abstract Wiener space $(B, H, \mu)$, set $\Omega = B$, $P = \mu$, and $W(h) = {\langle h , \cdot \rangle}_H$. We will stick to our notation.
We want to study the properties of the Malliavin derivative $D$ as an unbounded operator on $L^p(W,\mu)$, $p \ge 1$. If we take the domain of $D$ to be the smooth cylinder functions $\mathcal{F}
C_c^\infty(W)$, we have a densely defined unbounded operator from $L^p(W,\mu)$ into the vector-valued space $L^p(W,\mu; H)$. (Note that $\norm{DF(x)}_H$ is bounded as a function of $x$, so there is no question that $DF \in L^p(W; H)$.)
Let $F \in \mathcal{F} C_c^\infty(W)$ be a cylinder function, and $h
\in H$. Then $$\label{byparts-eq}
\int_W {\langle DF(x) , h \rangle}_H \mu(dx) = \int_W F(x) {\langle h , x \rangle}_H \mu(dx).$$
It is easy to see that both sides of (\[byparts-eq\]) are bounded linear functionals with respect to $h$, so it suffices to show that (\[byparts-eq\]) holds for all $h$ in a dense subset: hence suppose $h = i^* f$ for some $f \in
W^*$.
Now basically the proof is to reduce to the finite dimensional case. By adjusting $\phi$ as needed, there is no loss of generality in writing $F(x) = \phi(f_1(x), \dots, f_n(x))$ where $f_1 = f$. We can also apply Gram–Schmidt and assume that all the $f_i$ are $q$-orthonormal. Then ${\langle DF(x) , h \rangle}_H = \partial_1 \phi(f_1(x),
\dots, f_n(x))$. The $f_i$ are iid $N(0,1)$ random variables under $\mu$, so we have $$\begin{aligned}
\int_W {\langle DF(x) , h \rangle}_H \,\mu(dx) &= \int_{\R^n} \partial_1
\phi(x_1, \dots, x_n) \frac{1}{(2\pi)^{n/2}} e^{-|x|^2/2} \,dx \\
&= \int_{\R^n} \phi(x_1, \dots, x_n) x_1 \frac{1}{(2\pi)^{n/2}}
e^{-|x|^2/2}\,dx \\
&= \int_{W} \phi(f_1(x), \dots, f_n(x)) f_1(x) \,\mu(dx) \\
&= \int_W F(x) {\langle h , x \rangle}_H \,\mu(dx)
\end{aligned}$$ so we are done.
We also observe that $D$ obeys the product rule: $D(F \cdot G) = F
\cdot DG + DF \cdot G$. (This is easy to check by assuming, without loss of generality, that we have written $F,G$ in terms of the same functionals $f_1, \dots, f_n \in W^*$.) Applying the previous lemma to this identity gives:
$$\label{product-byparts}
\int_W G(x) {\langle DF(x) , h \rangle}_H \,\mu(dx) = \int_W (-F(x)
{\langle DG(x) , h \rangle}_H + F(x) G(x) {\langle h , x \rangle}_H) \mu(dx).$$
We can use this to prove:
The operator $D : L^p(W,\mu) \to L^p(W,\mu; H)$ is closable.
Suppose that $F_n \in \mathcal{F} C_c^\infty(W)$ are converging to 0 in $L^p(W,\mu)$, and that $D F_n \to \eta$ in $L^p(W;H)$. We have to show $\eta = 0$. It is sufficient to show that $\int_W
{\langle \eta(x) , h \rangle} G(x)\,mu(dx) = 0$ for all $h \in H$ and $G \in
\mathcal{F} C_c^\infty(W)$. (Why?) Now applying (\[product-byparts\]) we have $$\int_W G(x) {\langle DF_n(x) , h \rangle}_H \mu(dx) = -\int_W F_n(x)
{\langle DG(x) , h \rangle}_H + \int_W F_n(x) G(x) {\langle h , x \rangle}_H \mu(dx).$$ As $n \to \infty$, the left side goes to $\int_W G(x)
{\langle \eta(x) , h \rangle}_H \mu(dx)$. The first term on the right side goes to 0 (since $DG \in L^q(W;H)$ and hence ${\langle DG , h \rangle} \in
L^q(W)$) as does the second term (since $G$ is bounded and ${\langle h , \cdot \rangle} \in L^q(W)$ because it is a Gaussian random variable).
Now we can define the Sobolev space $\mathbb{D}^{1,p}$ as the completion of $\mathcal{F} C_c^\infty(W)$ under the norm $$\norm{F}_{\mathbb{D}^{1,p}}^p = \int_W |F(x)|^p + \norm{DF}_H^p \mu(dx).$$ Since $D$ was closable, $\mathbb{D}^{1,p} \subset L^p(W,\mu)$. We can also iterate this process to define higher derivatives $D^k$ and higher order Sobolev spaces.
If $\phi \in C^\infty(\R^n)$ and $\phi$ and its first partials have polynomial growth, then $F(x) = \phi(f_1(x), \dots, f_n(x)) \in \mathbb{D}^{1,p}$.
Cutoff functions.
\[polynomials-dense\] The set of functions $F(x) = p(f_1(x), \dots, f_n(x))$ where $p$ is a polynomial in $n$ variables and $f_i \in W^*$, is dense in $L^p(W,\mu)$.
See [@driver-probability Theorem 39.8].
Let $H_n(s)$ be the $n$’th **Hermite polynomial** defined by $$H_n(s) = \frac{(-1)^n}{n!} e^{s^2/2} \frac{d^n}{ds^n} e^{-s^2/2}.$$ $H_n$ is a polynomial of degree $n$. Fact: $H_n' = H_{n-1}$, and $(n+1) H_{n+1}(s) = s H_n(s) - H_{n-1}(s)$. In particular, $H_n$ is an eigenfunction of the one-dimensional Ornstein–Uhlenbeck operator $Af = f'' - xf'$ with eigenvalue $n$.
Also, we have the property that if $X,Y$ are jointly Gaussian with variance $1$, then $$E[H_n(X) H_m(Y)] =
\begin{cases}
0, & n \ne m \\
\frac{1}{n!} E[XY]^n, & n=m.
\end{cases}$$ (See Nualart for the proof, it’s simple.)
This implies:
If $\{e_i\} \subset W^*$ is a $q$-orthonormal basis, then the functions $$F_{n_1, \dots, n_k}(x) = \prod_i \sqrt{n_i!} H_{n_i}(e_i(x))$$ are an orthonormal basis for $L^2(W,\mu)$.
If $\mathcal{H}_n$ is the closed span of all $F_{n_1, \dots, n_k}$ with $n_1
+ \dots + n_k = n$ (i.e. multivariable Hermite polynomials of degree $n$) then we have an orthogonal decomposition of $L^2$. Let $J_n$ be orthogonal projection onto $\mathcal{H}_n$. This decomposition is called **Wiener chaos**. Note $\mathcal{H}_0$ is the constants, and $\mathcal{H}_1 = K$. For $n \ge 2$, the random variables in $\mathcal{H}_n$ are not normally distributed.
We can decompose $L^2(W;H)$ in a similar way: if $h_j$ is an orthonormal basis for $H$, then $\{ F_{n_1, \dots, n_k} h_j \}$ is an orthonormal basis for $L^2(W;H)$, and if $\mathcal{H}_n(H)$ is the closed span of functions of the form $F h$ where $F \in
\mathcal{H}_n$, $h \in H$, then $L^2(W;H) = \bigoplus
\mathcal{H}_n(H)$ is an orthogonal decomposition, and we again use $J_n$ to denote the orthogonal projections. We could do the same for $L^2(W; H^{\otimes m})$.
Note that $$\begin{aligned}
D F_{n_1, \dots, n_k}(x) &= \sum_{j=1}^k \sqrt{n_j!}
H_{n_j-1}(e_j(x)) \prod_{i \ne j} \sqrt{n_i!} H_{n_i}(e_i(x)) J e_j
\\
&= \sum_{j=1}^k \sqrt{n_j} F_{n_1, \dots, n_j-1, \dots, n_k}(x) J e_j.\end{aligned}$$
We can see from this that $\norm{D F_{n_1, \dots, n_k}}_{L^2(W; H)}^2
= n_1 + \dots + n_k$, or for short, $\norm{D F_\alpha}_{L^2(W;H)}^2 =
|\alpha|$. Also, if $\alpha \ne \beta$, ${\langle D F_\alpha , D
F_\beta \rangle}_{L^2(W;H)} = 0$.
For each $n$, $\mathcal{H}_n \subset \mathbb{D}^{1,2}$. Moreover, $\{F_\alpha : |\alpha| = n\}$ are an orthogonal basis for $\mathcal{H}_n$ with respect to the $\mathbb{D}^{1,2}$ inner product, and $\norm{F_\alpha}^2 = 1+n$.
Since $\{F_{\alpha} : |\alpha| = n\}$ are an orthonormal basis for $\mathcal{H}_n$, we can write $F = \sum_{i=1}^\infty a_i
F_{\alpha_i}$ where $\sum a_i^2 < \infty$ and the $\alpha_i$ are distinct. Let $F_m = \sum_{i=1}^m a_i F_{\alpha_i}$, so that $F_m
\to F$ in $L^2(W)$. Clearly $F_m \in \mathbb{D}^{1,2}$ and $D F_m =
\sum_{i=1}^m a_i D F_{\alpha_i}$. Now we have ${\langle D
F_{\alpha_i} , D F_{\alpha_j} \rangle}_{L^2(W;H)} = n \delta_{ij}$, so for $k
\le m$ we have $$\norm{D F_m - D F_k}_{L^2(W;H)}^2 = \norm{\sum_{i=k}^m a_i D
F_{\alpha_i}}_{L^2(W;H)} = n \sum_{i=k}^m a_i^2$$ which goes to $0$ as $m,k \to \infty$. So we have $F_m \to F$ in $L^2(W)$ and $D F_m$ Cauchy in $L^2(W;H)$. Since $D$ is closed, we have $F \in \mathbb{D}^{1,2}$.
In fact, we have shown that $F$ is a $\mathbb{D}^{1,2}$-limit of elements of the span of $\{F_\alpha\}$, and we know the $F_\alpha$ are $\mathbb{D}^{1,2}$-orthogonal.
Note that $\mathcal{H}_n$ are thus pairwise orthogonal closed subsets of $\mathbb{D}^{1,2}$. Furthermore, $D$ maps $\mathcal{H}_n$ into $\mathcal{H}_{n-1}(H)$.
The span of all $\mathcal{H}_n$ is dense in $\mathbb{D}^{1,2}$, so we can write $\mathbb{D}^{1,2} = \bigoplus \mathcal{H}_n$.
This proof is taken from [@schmuland92].
We begin with the finite dimensional case. Let $\mu_k$ be standard Gaussian measure on $\R^k$, and let $\phi \in C_c^\infty(\R^k)$. We will show that there is a sequence of polynomials $p_m$ such that $p_m \to \phi$ and $\partial_i p_m \to \partial_i \phi$ in $L^2(\R^k, \mu_k)$ for all $k$.
For continuous $\psi : \R^k \to \R$, let $$I_i \psi(x_1, \dots, x_k) = \int_0^{x_i} \psi(x_1, \dots, x_{i-1},
y, x_{i+1}, \dots, x_k)\,dy.$$ By Fubini’s theorem, all operators $I_i, 1 \le i \le k$ commute. If $\psi \in L^2(\mu_k)$ is continuous, then $I_i \psi$ is also continuous, and $\partial_i I_i \psi = \psi$. Moreover, $$\begin{aligned}
\int_{0}^\infty |I_i \psi (x_1, \dots, x_k)|^2 e^{-x_i^2/2} dx_i &=
\int_{0}^\infty \abs{\int_0^{x_i} \psi(\dots, y,\dots)\,dy}^2
e^{-x_i^2/2} dx_i \\
&\le \int_0^\infty \int_0^{x_i} |\psi(\dots, y, \dots)|^2 \,dy x_i
e^{-x_i^2/2}\,dx_i && \text{Cauchy--Schwarz} \\
&= \int_0^\infty |\psi(\dots, x_i, \dots)|^2 e^{-x_i^2/2} dx_i
\end{aligned}$$ where in the last line we integrated by parts. We can make the same argument for the integral from $-\infty$ to $0$, adjusting signs as needed, so we have $$\int_\R |I_i \psi(x)|^2 e^{-x_i^2/2} dx_i \le \int_\R |\psi_i(x)|^2
e^{-x_i^2/2} dx_i.$$ Integrating out the remaining $x_j$ with respect to $e^{-x_j^2/2}$ shows $$\norm{I_i \psi}_{L^2(\mu_k)}^2 \le \norm{\psi}_{L^2(\mu_k)}^2,$$ i.e. $I_i$ is a contraction on $L^2(\mu_k)$.
Now for $\phi \in C_c^\infty(\R^k)$, we can approximate $\partial_1
\dots \partial_k \phi$ in $L^2(\mu_k)$ norm by polynomials $q_n$ (by the finite-dimensional case of Proposition \[polynomials-dense\]). If we let $p_n = I_1 \dots I_k q_n$, then $p_n$ is again a polynomial, and $p_n \to I_1 \dots I_k \partial_1 \dots \partial_k
\phi = \phi$ in $L^2(\mu_k)$. Moreover, $\partial_i p_n = I_1 \dots I_{i-1} I_{i+1} \dots I_k q_n \to I_1
\dots I_{i-1} I_{i+1} \dots I_k \partial_1 \dots \partial_k \phi =
\partial_i \phi$ in $L^2(\mu_k)$ also.
Now back to the infinite-dimensional case. Let $F \in \mathcal{F}
C^\infty_c(W)$, so we can write $F(x) = \phi(e_1(x), \dots, e_k(x))$ where $e_i$ are $q$-orthonormal. Choose polynomials $p_n \to \phi$ in $L^2(\R^k, \mu_k)$ with $\partial_i p_n \to \partial_i \phi$ in $L^2(\mu_k)$ also, and set $P_n(x) = p_n(e_1(x), \dots, e_k(x))$. Note $P_n \in \mathcal{H}_m$ for some $m = m_n$. Then $$\int_W |F(x) - P_n(x)|^2 \mu(dx) = \int_{\R^k} |\phi(y) -
p_n(y)|^2 \mu_k(dy) \to 0.$$ Exercise: write out $\norm{DF - DP_n}_{L^2(W;H)}$ and show that it goes to 0 also.
\[DJ-commute\] For $F \in \mathbb{D}^{1,2}$, $D J_n F = J_{n-1} DF$.
If $F = F_\alpha$ where $|\alpha| = n$, then $J_n F_\alpha =
F_\alpha$ and $DF_\alpha \in \mathcal{H}_{n-1}(H)$ so $J_{n-1} D
F_\alpha = D F_\alpha$, so this is trivial. If $|\alpha| \ne n$ then both sides are zero.
Now for general $F \in \mathbb{D}^{1,2}$, by the previous proposition we can approximate $F$ in $\mathbb{D}^{1,2}$-norm by functions $F_m$ which are finite linear combinations of $F_\alpha$. In particular, $F_m \to F$ in $L^2(W)$. Since $J_n$ is continuous on $L^2(W)$, $J_n F_m \to J_n F$ in $L^2(W)$. Also, $D F_m \to DF$ in $L^2(W;H)$, so $J_{n-1} D F_m \to J_{n-1} DF$. But $J_{n-1} D
F_m = D J_n F_m$.
We have shown $J_n F_m \to J_n F$ and $D J_n F_m \to J_{n-1} DF$. By closedness of $D$, we have $D J_n F = J_{n-1} DF$.
$J_n$ is a continuous operator on $\mathbb{D}^{1,2}$.
$F_m \to F$ in $\mathbb{D}^{1,2}$ means $F_m \to F$ in $L^2(W)$ and $D F_m \to DF$ in $L^2(W;H)$. When this happens, we have $J_n F_m
\to J_n F$ since $J_n$ is continuous on $L^2(W)$, and $D J_n F_m =
J_{n-1} D F_m \to J_{n-1} D F = D J_n F$.
$J_n$ is orthogonal projection onto $\mathcal{H}_n$ with respect to the $\mathbb{D}^{1,2}$ inner product.
$J_n$ is the identity on $\mathcal{H}_n$, and vanishes on any $\mathcal{H}_m$ for $m \ne n$. Thus by continuity it vanishes on $\bigoplus_{m \ne n} \mathcal{H}_m$ which is the $\mathbb{D}^{1,2}$-orthogonal complement of $\mathcal{H}_n$.
For $F \in \mathbb{D}^{1,2}$, $F = \sum_{n=0}^\infty J_n F$ where the sum converges in $\mathbb{D}^{1,2}$.
For $F \in \mathbb{D}^{1,2}$, $DF = \sum_{n=0}^\infty D J_n F =
\sum_{n=1}^\infty J_{n-1} DF$ where the sums converge in $L^2(W;H)$.
The first equality follows from the previous corollary, since $D :
\mathbb{D}^{1,2} \to L^2(W;H)$ is continuous. The second equality is Lemma \[DJ-commute\].
\[D-chaos\] $F \in \mathbb{D}^{1,2}$ if and only if $\sum_n n \norm{J_n
F}_{L^2(W)}^2 < \infty$, in which case $\norm{DF}_{L^2(W;H)}^2 =
\sum_n n \norm{J_n F}_{L^2(W)}^2$.
If $f \in \mathbb{D}^{1,2}$, we have $DF = \sum_{n=0}^\infty D J_n
F$. Since the terms of this sum are orthogonal in $L^2(W;H)$, we have $$\infty > \norm{DF}^2_{L^2(W;H)} = \sum_{n=0}^\infty \norm{D J_n
F}^2_{L^2(W;H)} = \sum_{n=0}^\infty n \norm{J_n F}^2_{L^2(W;H)}.$$ Conversely, if $\sum_n n \norm{J_n F}^2 = \sum_n \norm{D J_n F}^2 < \infty$ then $\sum J_n F$ converges to $F$ and $\sum D J_n F$ converges, therefore by closedness of $D$, $F \in \mathbb{D}^{1,2}$.
If $F \in \mathbb{D}^{1,2}$ and $DF = 0$ then $F$ is constant.
\[chain-rule\] If $\psi \in C^\infty_c(\R)$, $F \in \mathbb{D}^{1,2}$, then $\psi(F)
\in \mathbb{D}^{1,2}$ and $$D \psi(F) = \psi'(F) DF.$$
For $F \in \mathcal{F} C_c^\infty(W)$ this is just the regular chain rule. For general $F \in \mathbb{D}^{1,2}$, choose $F_n \in
C_c^\infty(W)$ with $F_n \to F$, $D F_n \to DF$. Then use dominated convergence.
Actually the chain rule also holds for any $\psi \in C^1$ with bounded first derivative. Exercise: prove.
\[zero-one-law\] If $A \subset W$ is Borel and $1_A \in \mathbb{D}^{1,2}$ then $\mu(A)$ is 0 or 1.
Let $\psi \in C^\infty_c(\R)$ with $\psi(s) = s^2$ on $[0,1]$. Then $$D 1_A = D \psi(1_A) = 2 1_A D 1_A$$ so by considering whether $x \in A$ or $x \in A^c$ we have $D 1_A =
0$ a.e. Then by an above lemma, $1_A$ is (a.e.) equal to a constant.
As a closed densely defined operator between Hilbert spaces $L^2(W)$ and $L^2(W;H)$, $D$ has an adjoint operator $\delta$, which is a closed densely defined operator from $L^2(W;H)$ to $L^2(W)$.
To get an idea what $\delta$ does, let’s start by evaluating it on some simple functions.
\[delta-simple\] If $u(x) = G(x) h$ where $G \in \mathbb{D}^{1,2}$ and $h \in H$, then $u \in {\operatorname{dom}}(\delta)$ and $$\delta u(x) = G(x) {\langle h , x \rangle}_H - {\langle DG(x) , h \rangle}_H.$$
If $G \in \mathcal{F} C_c^\infty$, use (\[product-byparts\]). Otherwise, approximate. (Hmm, maybe we actually need $G \in
L^{2+\epsilon}(W)$ for this to work completely.)
Recall in the special case of Brownian motion, where $W = C([0,1])$ and $H = H^1_0([0,1])$, we had found that ${\langle h , \omega \rangle}_H =
\int_0^1 \dot{h}(s) dB_s(\omega)$, i.e. ${\langle h , \cdot \rangle}_H$ produces the Wiener integral, a special case of the Itô integral with a deterministic integrand. So it appears that $\delta$ is also some sort of integral. We call it the Skorohod integral. In fact, the Itô integral is a special case of it!
Let $A(t, \omega)$ be an adapted process in $L^2([0,1] \times
W)$. Set $u(t, \omega) = \int_0^t A(\tau,\omega)\,d\tau$, so $u(\cdot,
\omega) \in H$ for each $\omega$. Then $u \in {\operatorname{dom}}(\delta)$ and $\delta u = \int_0^t A_\tau dB_\tau$.
First suppose $A$ is of the form $A(t, \omega) = 1_{(r,s]}(t)
F(\omega)$ where $F(\omega) = \phi(B_{t_1}(\omega), \dots,
B_{t_n}(\omega))$ for some $0 \le t_1,
\dots, t_n \le r$ and $\phi \in C_c^\infty(\R^n)$. (Recall $B_t(\omega) = \omega(t)$ is just the evaluation map, a continuous linear functional of $\omega$.) We have $F \in \mathcal{F}_r$ so $A$ is adapted. Then $u(t, \omega) = h(t)
F(\omega)$ where $h(t) = \int_0^t 1_{(r,s)}(\tau) d\tau$. In particular $h(t) = 0$ for $t \le r$, so $${\langle DF(\omega) , h \rangle}_H = \sum_i \partial_i \phi(B_{t_1}(\omega),
\dots, B_{t_n}(\omega)) h(t_i) = 0.$$ Thus $$\begin{aligned}
\delta u(\omega) &= F(\omega) {\langle h , \omega \rangle}_H -
\cancel{{\langle DF(\omega) , h \rangle}_H} \\
&= F(\omega) (B_r(\omega)-B_s(\omega)) \\
&= \int_0^1 A_\tau \,dB_\tau (\omega).
\end{aligned}$$ Now we just need to do some approximation. If $F \in L^2(W,
\mathcal{F}_r)$, we can approximate $F$ in $L^2$ by cylinder functions $F_n$ of the above form. Then defining $u_n$, $u$ accordingly we have $u_n \to u$ in $L^2(W;H)$ and $\delta u_n \to F
\cdot (B_r - B_s) = \int_0^1 A_\tau \,dB_\tau$ in $L^2(W)$, so the conclusion holds for $A = F(\omega) 1_{(r,s]}(t)$. By linearity it also holds for any linear combination of such processes. The set of such linear combinations is dense in the adapted processes in $L^2([0,1] \times W)$, so choose such $A^{(n)} \to A$. Note that $A(t,\omega) \mapsto \int_0^t A(\tau, \omega)\,d\tau$ is an isometry of $L^2([0,1] \times W)$ into $L^2(W; H)$ so the corresponding $u_n$ converge to $u$ in $L^2(W;H)$, and by the Itô isometry, $\delta
u_n = \int_0^1 A^{(n)}_\tau dB_\tau \to \int_0^1 A_\tau dB_\tau$. Since $\delta$ is closed we are done.
This is neat because defining an integral in terms of $\delta$ lets us integrate a lot more processes.
Let’s compute $\int_0^1 B_1 dB_s$ (Skorohod). We have $u(t,\omega)
= t B_1 = h(t) G(\omega)$ where $h(t)=t$, and $G(\omega) =
\phi(f(\omega))$ where $\phi(x) = x$, and $f(\omega) = \omega(1)$. So $\delta u (\omega) = G(\omega)
{\langle h , \omega \rangle}_H - {\langle DG(\omega) , h \rangle}_H$. But ${\langle h , \omega \rangle} = \int_0^1 \dot{h}(t) dB_t = B_1$. And $DG(\omega) = Jf$ so ${\langle DG(\omega) , h \rangle}_H = f(h) = h(1) = 1$. So $\int_0^1 B_1 dB_s = B_1^2 - 1$.
Note that $B_1$, although it doesn’t depend on $t$, is not adapted. Indeed, a random variable is an adapted process iff it is in $\mathcal{F}_0$, which means it has to be constant.
All the time derivatives and integrals here are sort of a red herring; they just come from the fact that the Cameron–Martin inner product has a time derivative in it.
Another cool fact is that we can use this machinery to construct integration with respect to other continuous Gaussian processes. Again let $W =
C([0,1])$ (or an appropriate subspace), $\mu$ the law of a continuous centered Gaussian process $X_t$ with covariance function $a(s,t) = E[X_s X_t]$. Since $\delta$ applies to elements of $L^2(W; H)$, and we want to integrate honest processes (such as elements of $L^2([0,1] \times
W)$), we need some way to map processes to elements of $L^2(W;H)$. For Brownian motion it was $A_t \mapsto \int_0^t A_s ds$, an isometry of $L^2([0,1]
\times W) = L^2(W; L^2([0,1])) = L^2(W) \otimes L^2([0,1])$ into $L^2(W; H)$. To construct such an map $\Phi$ in this case, we start with the idea that we want $\int_0^t dX_s = X_t$, so we should have $\Phi 1_{[0,T]} = J \delta_s \in H$. We can extend this map linearly to $\mathcal{E}$, the set of all step functions on $[0,1]$. To make it an isometry, equip $\mathcal{E}$ with the inner product defined by $${\langle 1_{[0,s]} , 1_{[0,t]} \rangle}_{\mathcal{E}} = {\langle J \delta_s , J
\delta_t \rangle}_H = a(s,t)$$ again extended by bilinearity. It extends isometrically to the completion of $\mathcal{E}$ under this inner product, whatever that may be. So the processes we can integrate are we can integrate processes from $L^2(W; \bar{\mathcal{E}}) = L^2(W)
\otimes \bar{\mathcal{E}}$ just by taking $\int_0^1 A\,dX = \delta
u$ where $u(\omega) = \Phi(A(\omega))$.
Exactly what are the elements of $L^2(W; \bar{\mathcal{E}})$ is a little hard to say. The elements of $\bar{\mathcal{E}}$ can’t necessarily be identified as functions on $[0,1]$; they might be distributions, for instance. But we for sure know it contains step functions, so $L^2(W; \bar{\mathcal{E}})$ at least contains “simple processes” of the form $\sum Y_i 1_{[a_i, b_i]}(t)$. In the case of fractional Brownian motion, one can show that $\bar{\mathcal{E}}$ contains $L^2([0,1])$, so in particular $\Phi$ makes sense for any process in $L^2(W \times [0,1])$. Of course, there is still the question of whether $\Phi(A) \in {\operatorname{dom}}\delta$.
There’s a chapter in Nualart which works out a lot of this in the context of fractional Brownian motion. Being able to integrate with respect to fBM is a big deal, because fBM is not a semimartingale and so it is not covered by any version of Itô integration.
A couple of properties of $\delta$ in terms of the Wiener chaos:
1. $\mathcal{H}_n(H) \subset {\operatorname{dom}}\delta$ for each $n$. (Follows from Lemma \[delta-simple\].)
2. For $u \in {\operatorname{dom}}\delta$, $J_n \delta u = \delta J_{n-1} u$.
Using Lemma \[DJ-commute\] and the fact that the $J_n$, being orthogonal projections, are self-adjoint, we have for any $F \in
\mathbb{D}^{1,2}$, $$\begin{aligned}
{\langle J_n \delta u , F \rangle}_{L^2(W)} &= {\langle \delta u , J_n
F \rangle}_{L^2(W)} \\ &= {\langle u , D J_n F \rangle}_{L^2(W;H)} \\ &= {\langle u , J_{n-1}
D F \rangle}_{L^2(W;H)} \\ &= {\langle \delta J_{n-1} u , F \rangle}_{L^2(W)}.
\end{aligned}$$ $\mathbb{D}^{1,2}$ is dense in $L^2(W)$ so we are done.
3. $J_0 \delta u = 0$. For if $F \in L^2(W)$, then ${\langle J_0 \delta u , F \rangle} = {\langle \delta u , J_0 F \rangle} = {\langle u , D J_0
F \rangle}$. But $J_0 F$ is a constant so $D J_0 F = 0$.
The Clark–Ocone formula
-----------------------
Until further notice, we are working on classical Wiener space, $W =
C_0([0,1])$, with $\mu$ being Wiener measure.
A standard result in stochastic calculus is the **Itô representation theorem**, which in its classical form says:
Let $\{B_t\}$ be a Brownian motion on $\R^d$, let $\{\mathcal{F}_t\}$ be the filtration it generates, and let $Z$ be an $L^2$ random variable which is $\mathcal{F}_1$-measurable (sometimes called a **Brownian functional**). Then there exists an adapted $L^2$ process $Y_t$ such that $$Z = \mathbb{E}[Z] + \int_0^1 Y_t \,dB_t, \quad \text{a.s.}$$
This is claiming that the range of the Itô integral contains all the $L^2$ random variables with mean zero (which we’ll denote $\mathcal{H}_0^\perp$). Since the Itô integral is an isometry, its range is automatically closed, so it suffices to show it is dense in $\mathcal{H}_0^\perp$. One can explicitly produce a dense set.
Look up a proof.
An important application of this theorem is in finance. Suppose we have a stochastic process $\{X_t\}$ which gives the price of a stock (call it Acme) at time $t$. (Temporarily you can think $X_t = B_t$ is Brownian motion, though this is not a good model and we might improve it later.) We may want to study an **option** or **contingent claim**, some contract whose ultimate value $Z$ is determined by the behavior of the stock. For example:
- A **European call option** is a contract which gives you the right, but not the obligation, to buy one share of Acme at time 1 for a pre-agreed **strike price** $K$. So if the price $X_1$ at time 1 is greater than $K$, you will **exercise** your option, buy a share for $K$ dollars, and then you can immediately sell it for $X_1$ dollars, turning a quick profit of $Z = X_1 - K$ dollars. If $X_1 < K$, then you should not exercise the option; it is worthless, and $Z=0$. Thus we can write $Z = (X_1 - K)^+$.
- A **European put option** gives the right to sell one share of Acme at a price $K$. Similarly we have $Z = (K - X_1)^+$.
- A **floating lookback put option** gives one the right, at time 1, to sell one share of Acme at the highest price it ever attained between times 0 and 1. So $Z = \sup_{t \in [0,1]} X_t - X_1$.
- There are many more.
You can’t lose money with these contracts, because you can always just not exercise it, and you could gain a profit. Conversely, your counterparty can only lose money. So you are going to have to pay your counterparty some money up front to get them to enter into such a contract. How much should you pay? A “fair” price would be $\mathbb{E}[Z]$. But it may be that the contract would be worth more or less to you than that, depending on your appetite for risk. (Say more about this.)
Here the Itô representation theorem comes to the rescue. If $X_t =
B_t$ is a Brownian motion, it says that $Z = E[Z] + \int_0^1
Y_t\,dB_t$. This represents a **hedging strategy**. Consider a trading strategy where at time $t$ we want to own $Y_t$ shares of Acme (where we can hold or borrow cash as needed to achieve this; negative shares are also okay because we can sell short). $Y_t$ is adapted, meaning the number of shares to own can be determined by what the stock has already done. A moment’s thought shows that the net value of your portfolio at time $1$ is $\int_0^1 Y_t \,dB_t$. Thus, if we start with $\mathbb{E}[Z]$ dollars in the bank and then follow the strategy $Y_t$, at the end we will have exactly $Z$ dollars, almost surely. We can **replicate** the option $Z$ for $\mathbb{E}[Z]$ dollars (not counting transaction costs, which we assume to be negligible). So anybody that wants more than $\mathbb{E}[Z]$ dollars is ripping us off, and we shouldn’t pay it even if we would be willing to.
So a key question is whether we can explicitly find $Y_t$.
In Wiener space notation, $Z$ is an element of $L^2(W,\mu)$, which we had usually called $F$. Also, now $\mathcal{F}_t$ is the $\sigma$-algebra generated by the linear functionals $\{\delta_s : s
\le t\}$; since these span a weak-\* dense subset of $W^*$ we have $\mathcal{F}_1 = \sigma(W^*) = \mathcal{B}_W$, the Borel $\sigma$-algebra of $W$.
Let $L^2_a([0,1] \times W)$ be the space of adapted processes.
$L^2_a([0,1] \times W)$ is a closed subspace of $L^2([0,1] \times W)$.
$Y_t \mapsto \mathbb{E}[Y_t | \mathcal{F}_t]$ is orthogonal projection from $L^2([0,1] \times W)$ onto $L^2_a([0,1] \times W)$.
This section is tangled up a bit by some derivatives coming and going. Remember that $H$ is naturally isomorphic to $L^2([0,1])$ via the map $\Phi : L^2([0,1]) \to H$ given by $\Phi f(t) = \int_0^t
f(s) ds$ (its inverse is simply $\frac{d}{dt}$). Thus $L^2(W;H)$ is naturally isomorphic to $L^2([0,1] \times W)$. Under this identification, we can identify $D : \mathbb{D}^{1,2} \to L^2(W;H)$ with a map that takes an element $F \in \mathbb{D}^{1,2}$ to a process $D_t F \in
L^2([0,1] \times W)$; namely, $D_t F(\omega) = \frac{d}{dt}
DF(\omega)(t) = \frac{d}{dt} {\langle DF(\omega) , J \delta_t \rangle}_H =
\frac{d}{dt} {\langle DF(\omega) , \cdot \wedge t \rangle}_H$. So $D_t F =
\Phi^{-1} D F$.
The Clark–Ocone theorem states:
For $F \in \mathbb{D}^{1,2}$, $$F = \int F\,d\mu + \int_0^1 \mathbb{E}[D_t F | \mathcal{F}_t]\,dB_t.$$
To prove this, we want to reduce everything to Skorohod integrals. Let $E \subset L^2(W;H)$ be the image of $L^2_a([0,1] \times W)$ under the isomorphism $\Phi$. Then, since the Skorohod integral extends the Itô integral, we know that $E \subset {\operatorname{dom}}\delta$, and $\delta : E
\to L^2(W)$ is an isometry. Moreover, by the Itô representation theorem, the image $\delta(E)$ is exactly $\mathcal{H}_0^\perp$, i.e. the orthogonal complement of the constants, i.e. functions with zero mean.
Let $P$ denote orthogonal projection onto $E$, so that $\mathbb{E}[\cdot | \mathcal{F}_t] = \Phi^{-1} P \Phi$.
We summarize this discussion by saying that the following diagram commutes. $$\xymatrix{
& L^2([0,1] \times W) \ar@{->}[rr]^{\mathbb{E}[\,\cdot\, |
\mathcal{F}_t]}
\ar@{<=>}[dd]_{\Phi}
&& L^2_a([0,1] \times W) \ar@{<=>}[dd]_{\Phi} \ar@{->}[dr]^{\int_0^1
\cdot \,dB_t}
\\
\mathbb{D}^{1,2} \ar@{->}[ur]^{D_t} \ar@{->}[dr]_{D} & & & & L^2(W) \\
& L^2(W;H) \ar@{->}[rr]^{P}
& & E \ar@{->}[ur]_{\delta}
}$$
From this diagram, we see that the Clark–Ocone theorem reads: $$F = \int F d\mu + \delta P D F.$$
Now the proof is basically just a diagram chase.
Suppose without loss of generality that $\int F \,d\mu = 0$, so that $F \in \mathcal{H}_0^\perp$. Let $u \in E$. Then $$\begin{aligned}
{\langle F , \delta u \rangle}_{L^2(W)} &= {\langle DF , u \rangle}_{L^2(W;H)} \\
&= {\langle DF , Pu \rangle}_{L^2(W;H)} && \text{since $u \in E$} \\
&= {\langle PDF , u \rangle}_{L^2(W;H)}
\intertext{(since orthogonal projections are self-adjoint)}
&= {\langle \delta P D F , \delta u \rangle}_{L^2(W)}
\end{aligned}$$ since $PDF \in E$, $u \in E$, and $\delta$ is an isometry on $E$. As $u$ ranges over $E$, $\delta u$ ranges over $\mathcal{H}_0^\perp$, so we must have $F = \delta P D F$.
If the stock price is Brownian motion ($X_t = B_t$), compute the hedging strategy $Y_t$ for a European call option $Z = (X_1 - K)^+$.
Again take $X_t = B_t$. Compute the hedging strategy for a floating lookback call option $Z
= M - X_1$, where $M = \sup_{t \in [0,1]} X_t$. (Show that $D_t M =
1_{\{t \le T\}}$ where $T = \arg\max X_t$, which is a.s. unique, by approximating $M$ by the maximum over a finite set.)
Let $X_t$ be a **geometric Brownian motion** $X_t =
\exp\left(B_t - \frac{t}{2}\right)$. Compute the hedging strategy for a European call option $Z = (X_1 - K)^+$. (Note by Itô’s formula that $dX_t = X_t dB_t$.)
Ornstein–Uhlenbeck process
==========================
We’ve constructed one canonical process on $W$, namely Brownian motion $B_t$, defined by having independent increments distributed according to $\mu$ (appropriately scaled). In finite dimensions, another canonical process related to Gaussian measure is the **Ornstein–Uhlenbeck process**. This is a Gaussian process $X_t$ which can be defined by the SDE $dX_t = \sqrt{2} dB_t - X_t dt$. Intuitively, $X_t$ tries to move like a Brownian motion, but it experiences a “restoring force” that always pulls it back toward the origin. Imagine a Brownian particle on a spring. A key relationship between $X_t$ and standard Gaussian measure $\mu$ is that $X_t$ has $\mu$ as its stationary distribution: if we start $X_t$ in a random position chosen according to $\mu$, then $X_t$ itself is also distributed according to $\mu$ at all later times. This also means that, from any starting distribution, the distribution of $X_t$ converges to $\mu$ as $t \to \infty$.
One way to get a handle on the Ornstein–Uhlenbeck process, in finite or infinite dimensions, is via its **Dirichlet form**. Here are some basics on the subject.
Crash course on Dirichlet forms
-------------------------------
Suppose $X_t$ is a symmetric Markov process on some topological space $X$ equipped with a Borel measure $m$. This means that its transition semigroup $T_t f(x) = E_x[f(X_t)]$ is a Hermitian operator on $L^2(X,m)$. If we add a few extra mild conditions (e.g. cádlág, strong Markov) and make $X_t$ a **Hunt process**, the semigroup $T_t$ will be strongly continuous. It is also **Markovian**, i.e. if $0 \le f \le 1$, then $0 \le T_t f \le 1$. For example, if $X = \R^n$, $m$ is Lebesgue measure, and $X_t$ is Brownian motion, then $T_t f(x) = \frac{1}{(2 \pi t)^{n/2}} \int_{\R^n} f(y)
e^{-|x-y|^2/2t}\,m(dy)$ is the usual heat semigroup.
A strongly continuous contraction semigroup has an associated **generator**, a nonnegative-definite self-adjoint operator $(L, D(L))$ which in general is unbounded, such that $T_t = e^{-tL}$. For Brownian motion it is $L = -\Delta/2$ with $D(L) = H^2(\R^n)$.
Associated to a nonnegative self-adjoint operator is an unbounded bilinear symmetric form $\mathcal{E}$ with domain $\mathbb{D}$, such that $\mathcal{E}(f,g) = (f,Lg)$ for every $f \in \mathbb{D}$ and $g
\in D(L)$. We can take $(\mathcal{E},\mathbb{D})$ to be a closed form, which essentially says that $\mathcal{E}_1(f,g) =
\mathcal{E}(f,g) + (f,g)$ is a Hilbert inner product on $\mathbb{D}$. Note that $\mathbb{D}$ is generally larger than $D(L)$. For Brownian motion, $\mathcal{E}(f,g) = \int_{\R^n} {\nabla}f \cdot
{\nabla}g\,dm$ and $\mathbb{D} = H^1(\R^n)$. Note $\mathcal{E}_1$ is the usual Sobolev inner product on $H^1(\R^n)$. $\mathcal{E}(f,f)$ can be interpreted as the amount of “energy” contained in the distribution $f dm$. Letting this distribution evolve under the process will tend to reduce the amount of energy as quickly as possible.
When $T_t$ is Markovian, $(\mathcal{E},\mathbb{D})$ has a corresponding property, also called Markovian. Namely, if $f \in
\mathbb{D}$, let $\bar{f} = f \wedge 1 \vee 0$ be a “truncated” version of $f$. The Markovian property asserts that $\bar{f} \in
\mathbb{D}$ and $\mathcal{E}(\bar{f}, \bar{f}) \le \mathcal{E}(f,f)$. A bilinear, symmetric, closed, Markovian form on $L^2(X,m)$ is called a **Dirichlet form**.
So far this is nice but not terribly interesting. What’s neat is that this game can be played backwards. Under certain conditions, one can start with a Dirichlet form and recover a Hunt process with which it is associated. This is great, because constructing a process is usually a lot of work, but one can often just write down a Dirichlet form. Moreover, one finds that properties of the process often have corresponding properties for the Dirichlet form.
For example, if the process $X_t$ has continuous sample paths, the form $(\mathcal{E},\mathbb{D})$ will be **local**: namely, if $f=0$ on the support of $g$, then $\mathcal{E}(f,g) = 0$. Conversely, if the form is local, then the associated process will have continuous sample paths. If additionally the process is not killed inside $X$, the form is **strongly local**: if $f$ is constant on the support of $g$, then $\mathcal{E}(f,g) = 0$; and the converse is also true.
So you might ask: under what conditions must a Dirichlet form be associated with a process? One sufficient condition is that $(\mathcal{E},\mathbb{D})$ be **regular**: that $\mathbb{D} \cap
C_c(X)$ is $\mathcal{E}_1$-dense in $\mathbb{D}$ and uniformly dense in $C_c(X)$. We also have to assume that $X$, as a topological space, is locally compact. The main purpose of this condition is to exclude the possibility that $X$ contains “holes” that the process would have to pass through. Unfortunately, this condition is useless in infinite dimensions, since if $X = W$ is, say, an infinite-dimensional Banach space, then $C_c(W) = 0$.
There is a more general condition called **quasi-regular**, which is actually necessary and sufficient for the existence of a process. It is sufficiently complicated that I won’t describe it here; see Ma and Röckner’s book for the complete treatment.
The Ornstein–Uhlenbeck Dirichlet form
-------------------------------------
We are going to define the Ornstein–Uhlenbeck process via its Dirichlet form. For $F,G \in \mathbb{D}^{1,2}$, let $\mathcal{E}(F,G)
= {\langle DF , DG \rangle}_{L^2(W;H)}$. This form is obviously bilinear, symmetric, and positive semidefinite. With the domain $\mathbb{D}^{1,2}$, $\mathcal{E}$ is also a closed form (in fact, $\mathcal{E}_1$ is exactly the Sobolev inner product on $\mathbb{D}^{1,2}$, which we know is complete).
$(\mathcal{E}, \mathbb{D}^{1,2})$ is Markovian.
Fix $\epsilon > 0$. Let $\varphi_n \in C^\infty(\R)$ be a sequence of smooth functions with $0 \le \varphi_n \le 1$, $|\varphi_n'| \le
1+\epsilon$, and $\varphi_n(x) \to x \wedge 1 \vee 0$ pointwise. (Draw a picture to convince yourself this is possible.) Then $\varphi_n(F) \to F \wedge 1 \vee 0$ in $L^2(W)$ by dominated convergence. Then, by the chain rule, for $F \in \mathbb{D}^{1,2}$, we have $\norm{D \varphi_n(F)}_{L^2(W;H)} = \norm{\varphi_n'(F)
DF}_{L^2(W;H)} \le (1+\epsilon) \norm{DF}_{L^2(W;H)}$. It follows from Alaoglu’s theorem that $F \wedge 1 \vee 0 \in
\mathbb{D}^{1,2}$, and moreover, $\norm{D[F \wedge 1 \vee
0]}_{L^2(W;H)} \le (1+\epsilon) \norm{DF}_{L^2(W;H)}$. Letting $\epsilon \to 0$ we are done.
Fill in the details in the preceding proof.
$(\mathcal{E}, \mathbb{D}^{1,2})$ is quasi-regular. Therefore, there exists a Hunt process $X_t$ whose transition semigroup is $T_t$, the semigroup corresponding to $(\mathcal{E}, \mathbb{D}^{1,2})$.
See [@ma-rockner-book IV.4.b].
The operator $D$ is **local** in the sense that for any $F \in
\mathbb{D}^{1,2}$, $DF = 0$ $\mu$-a.e. on $\{F = 0 \}$.
Let $\varphi_n \in C^\infty_c(\R)$ have $\varphi_n(0) = 1$, $0 \le
\varphi_n \le 1$, and $\varphi_n$ supported inside $\left[-\frac{1}{n}, \frac{1}{n}\right]$; note that $\varphi_n \to
1_{\{0\}}$ pointwise and boundedly. Then as $n \to \infty$, $\varphi_n(F) DF \to 1_{\{F = 0\}} DF$ in $L^2(W;H)$. Let $\psi_n(t) =
\int_{-\infty}^t \varphi_n(s)\,ds$, so that $\varphi_n = \psi_n'$; then $\psi_n \to 0$ uniformly. By the chain rule we have $D(\psi_n(F)) = \varphi_n(F) DF$. Now if we fix $u \in {\operatorname{dom}}\delta$, we have $$\begin{aligned}
{\langle 1_{\{F=0\}} DF , u \rangle}_{L^2(W;H)} &= \lim_{n \to \infty}
{\langle \varphi_n(F) DF , u \rangle}_{L^2(W;H)}\\
&= \lim_{n \to \infty} {\langle D (\psi_n(F)) , u \rangle}_{L^2(W;H)} \\
&= \lim_{n \to \infty} {\langle \psi_n(F) , \delta u \rangle}_{L^2(W)} = 0
\end{aligned}$$ since $\psi_n(F) \to 0$ uniformly and hence in $L^2(W)$. Since ${\operatorname{dom}}\delta$ is dense in $L^2(W;H)$, we have $1_{\{F=0\}} DF = 0$ $\mu$-a.e., which is the desired statement.
The Ornstein–Uhlenbeck Dirichlet form $(\mathcal{E},
\mathbb{D}^{1,2})$ is strongly local.
Let $F,G \in \mathbb{D}^{1,2}$. Suppose first that $F = 0$ on the support of $G$. By the previous lemma we have (up to $\mu$-null sets) $\{DF
= 0\} \supset \{F = 0\} \supset \{G \ne 0\} \supset \{DG \ne 0\}$. Thus, for a.e. $x$ either $DF(x) = 0$ or $DG(x) = 0$. So $\mathcal{E}(F,G) = \int_X {\langle DF(x) , DG(x) \rangle}_H \,\mu(dx) = 0$.
If $F = 1$ on the support of $G$, write $\mathcal{E}(F,G) =
\mathcal{E}(F-1,G) + \mathcal{E}(1,G)$. The first term vanishes by the previous step, while the second term vanishes since $D1 = 0$.
We now want to investigate the generator $N$ associated to $(\mathcal{E},\mathbb{D})$.
For $F \in L^2(W)$, $J_0 F = \int F d\mu$, where $J_0$ is the orthogonal projection onto $\mathcal{H}_0$, the constant functions in $L^2(W)$.
This holds over any probability space. Write $EF = \int F d\mu$. Clearly $E$ is continuous, $E$ is the identity on the constants $\mathcal{H}_0$, and if $F \perp \mathcal{H}_0$, then we have $EF =
{\langle F , 1 \rangle}_{L^2(W)} = 0$ since $1 \in \mathcal{H}_0$. So $E$ must be orthogonal projection onto $\mathcal{H}_0$.
\[poincare\]\[A Poincaré inequality\] For $F \in \mathbb{D}^{1,2}$, we have $$\norm{F - \int F d\mu}_{L^2(W)} \le \norm{DF}_{L^2(W;H)}.$$
Set $G = F - \int F d\mu$, so that $J_0 G = \int G d\mu = 0$. Note that $DF
= DG$ since $D1 = 0$. Then by Lemma \[D-chaos\], $$\norm{DG}_{L^2(W;H)}^2 = \sum_{n=0}^\infty n \norm{J_n
G}_{L^2(W)}^2 \ge \sum_{n=1}^\infty \norm{J_n
G}_{L^2(W)}^2 = \sum_{n=0}^\infty \norm{J_n
G}_{L^2(W)}^2 = \norm{G}_{L^2(W)}^2.$$
Note that by taking $F(x) = f(x)$ for $f \in W^*$, we can see that the Poincaré inequality is sharp.
$N = \delta D$. More precisely, if we set $${\operatorname{dom}}N = {\operatorname{dom}}\delta D = \{ F \in \mathbb{D}^{1,2} : DF \in {\operatorname{dom}}\delta \}$$ and $NF = \delta D F$ for $F \in {\operatorname{dom}}N$, then $(N, {\operatorname{dom}}N)$ is the unique self-adjoint operator satisfying ${\operatorname{dom}}N \subset
\mathbb{D}^{1,2}$ and $$\label{Ndef} \mathcal{E}(F,G) = {\langle F , NG \rangle}_{L^2(W)}
\text{ for all } F \in \mathbb{D}^{1,2}, G \in {\operatorname{dom}}N.$$
It is clear that ${\operatorname{dom}}N \subset \mathbb{D}^{1,2}$ and that (\[Ndef\]) holds. Moreover, it is known there is a unique self-adjoint operator with this property (reference?). We have to check that $N$ as defined above is in fact self-adjoint. (Should fill this in?)
$N F_\alpha = |\alpha| F_\alpha$. That is, the Hermite polynomials $F_\alpha$ are eigenfunctions for $N$, with eigenvalues $|\alpha|$. So the $\mathcal{H}_n$ are eigenspaces.
Since $F_\alpha$ is a cylinder function, it is easy to see it is in the domain of $N$. Then ${\langle N F_\alpha , F_\beta \rangle}_{L^2(W)} =
{\langle D F_\alpha , D F_\beta \rangle}_{L^2(W;H)} = |\alpha| \delta_{\alpha
\beta}$. Since the $\{F_\beta\}$ are an orthonormal basis for $L^2(W)$, we are done.
There is a natural identification of $\mathcal{H}_n$ with $H^{\otimes
n}$, which gives an identification of $L^2(W)$ with Fock space $\bigoplus_n H^{\otimes n}$. In quantum mechanics this is the state space for a system with an arbitrary number of particles, $H^{\otimes n}$ corresponding to those states with exactly $n$ particles. $N$ is thus called the number operator because ${\langle N
F , F \rangle}$ gives the (expected) number of particles in the state $F$.
$NF = \sum_{n=0}^\infty n J_n F$, where the sum on the right converges iff $F \in {\operatorname{dom}}N$.
For each $m$, we have $$N \sum_{n=0}^m J_n F = \sum_{n=0}^m N J_n F =
\sum_{n=0}^m n J_n F.$$ Since $\sum_{n=0}^m J_n F \to F$ as $m \to \infty$ and $N$ is closed, if the right side converges then $F \in {\operatorname{dom}}N$ and $NF$ equals the limit of the right side.
Conversely, if $F \in {\operatorname{dom}}N$, we have $\infty >
\norm{NF}^2_{L^2(W)} = \sum_{n=0}^\infty \norm{J_n N F}^2$. But, repeatedly using the self-adjointness of $J_n$ and $N$ and the relationships $J_n = J_n^2$ and $N J_n = n J_n$, $$\begin{aligned}
\norm{J_n N F}^2 = {\langle F , N J_n N F \rangle} = n {\langle F , J_n N F \rangle} = n
{\langle N J_n F , F \rangle} = n^2 {\langle J_n F , F \rangle} = n^2 \norm{J_n F}^2.
\end{aligned}$$ Thus $\sum n^2 \norm{J_n F}^2 < \infty$, so $\sum n J_n F$ converges.
Let $T_t = e^{-tN}$ be the semigroup generated by $N$. Note that each $T_t$ is a contraction on $L^2(W)$, and $T_t$ is strongly continuous in $t$.
\[Tt-chaos\] For any $F \in L^2(W)$, $$\label{Tt-chaos-eqn}
T_t F = \sum_{n=0}^\infty e^{-tn} J_n F.$$
Since $J_n F$ is an eigenfunction of $N$, we must have $$\frac{d}{dt} T_t J_n F = T_t N J_n F = n T_t J_n f.$$ Since $T_0 J_n F = J_n F$, the only solution of this ODE is $T_t J_n
F = e^{-tn} J_n F$. Now sum over $n$.
\[Tt-exponential-decay\] $\norm{T_t F - \int F d\mu}_{L^2(W)} \le e^{-t} \norm{F - \int F d\mu}$.
Let $G = F - \int F d\mu$; in particular $J_0 G = 0$. Then $$\begin{aligned}
\norm{T_t G}^2 = \sum_{n=1}^\infty e^{-2tn} \norm{J_n G}^2 \le
e^{-2t} \sum_{n=1}^\infty \norm{J_n G}^2 = e^{-2t} \norm{G}^2.
\end{aligned}$$
This is also a consequence of the Poincaré inequality (Lemma \[poincare\]) via the spectral theorem.
$T_t$ is the transition semigroup of the Ornstein–Uhlenbeck process $X_t$, i.e. $T_t F(x) = \mathbb{E}_x[F(X_t)]$ for $\mu$-a.e. $x \in
X$. To get a better understanding of this process, we’ll study $T_t$ and $N$ some more.
The finite-dimensional Ornstein–Uhlenbeck operator is given by $$\tilde{N} \phi (x) = \Delta \phi (x) - x \cdot {\nabla}\phi(x).$$
The same formula essentially works in infinite dimensions.
For $F \in \mathcal{F} C^\infty_c(W)$ of the form $F(x) =
\phi(e_1(x), \dots, e_n(x))$ with $e_i$ $q$-orthonormal, we have $$N F(x) = (\tilde{N} \phi)(e_1(x), \dots, e_n(x)).$$
This follows from the formula $N = \delta D$ and (\[DF-cylinder\]) and Proposition \[delta-simple\], and the fact that $J : (W^*, q)
\to H$ is an isometry. Note for finite dimensions, if we take $e_1,
\dots, e_n$ to be the coordinate functions on $\R^n$, this shows that $\tilde{N}$ really is the Ornstein–Uhlenbeck operator.
The Ornstein–Uhlenbeck semigroup $T_t$ is given by $$\label{Tt-formula}
T_t F(x) = \int_W F\left(e^{-t} x + \sqrt{1-e^{-2t}} y\right)\,\mu(dy).$$
Since this is mostly computation, I’ll just sketch it.
Let $R_t$ denote the right side. We’ll show that $R_t$ is another semigroup with the same generator.
Showing that $R_t$ is a semigroup is easy once you remember that $\mu_t$ is a convolution semigroup, or in other words $$\int_W \int_W G(ax+by) \,\mu(dy)\,\mu(dx) = \int_W
G\left(\sqrt{a^2+b^2} z\right) \,\mu(dz).$$
To check the generator is right, start with the finite dimensional case. If $\phi$ is a nice smooth function on $\R^n$, and $p(y) dy$ is standard Gaussian measure, then show that $$\frac{d}{dt}|_{t=0} \int_{\R^n} \phi\left(e^{-t} x + \sqrt{1-e^{-2t}}y\right)
p(y) dy = \tilde{N} \phi(x).$$ (First differentiate under the integral sign. For the term with the $x$, evaluate at $t=0$. For the term with $y$, integrate by parts, remembering that $y p(y) = -{\nabla}p(y)$. If in doubt, assign it as homework to a Math 2220 class.)
Now if $F$ is a smooth cylinder function on $W$, do the same and use the previous lemma, noting that $(e_1, \dots, e_n)$ have a standard normal distribution under $\mu$.
There is probably some annoying density argument as the last step. The interested reader can work it out and let me know how it went.
This shows that at time $t$, $X_t$ started at $x$ has a Gaussian distribution (derived from $\mu$) with mean $e^{-t} x$ and variance $1-e^{-2t}$.
Here is a general property of Markovian semigroups that we will use later:
\[Tt-cauchy-schwarz\] For bounded nonnegative functions $F,G$, we have $$\label{Tt-cauchy-schwarz-eqn}
|T_t (F G)(x)|^2 \le T_t (F^2)(x) T_t (G^2)(x).$$
Note the following identity: for $a, b \ge 0$, $$ab = \frac{1}{2} \inf_{r > 0} \left(r a^2 + \frac{1}{r} b^2\right).$$ (One direction is the AM-GM inequality, and the other comes from taking $r = b/a$.) So $$\begin{aligned}
T_t (FG) &= \frac{1}{2} T_t \left(\inf_{r > 0} \left(r F^2 +
\frac{1}{r} G^2\right)\right) \\
&\le \frac{1}{2} \inf_{r > 0} \left( r T_t (F^2) + \frac{1}{r} T_t
(G^2) \right) \\
&= \sqrt{T_t(F^2) T_t(G^2)}
\end{aligned}$$ where in the second line we used the fact that $T_t$ is linear and Markovian (i.e. if $f \le g$ then $T_t f \le T_t g$).
As a special case, taking $G=1$, we have $|T_t F(x)|^2 \le T_t (F^2)(x)$.
Alternative proof: use (\[Tt-formula\]), or the fact that $T_t F(x)
= E_x[F(X_t)]$, and Cauchy–Schwarz.
Log Sobolev inequality
----------------------
Recall that in finite dimensions, the classical Sobolev embedding theorem says that for $\phi \in C^\infty_c(\R^n)$ (or more generally $\phi \in W^{1,p}(\R^n)$), $$\norm{\phi}_{L^{p^*}(\R^n,m)} \le C_{n,p}(\norm{\phi}_{L^p(\R^n, m)} +
\norm{{\nabla}\phi}_{L^p(\R^n, m)})$$ where $\frac{1}{p^*} = \frac{1}{p} - \frac{1}{n}$. Note everything is with respect to Lebesgue measure. In particular, this says that if $\phi$ and ${\nabla}\phi$ are both in $L^p$, then the integrability of $\phi$ is actually better: we have $\phi \in L^{p^*}$. So $$W^{1,p} \subset L^{p^*}$$ and the inclusion is continuous (actually, if the inclusion holds at all it has to be continuous, by the closed graph theorem).
This theorem is useless in infinite dimensions in two different ways. First, it involves Lebesgue measure, which doesn’t exist. Second, when $n = \infty$ we get $p^* = p$ so the conclusion is a triviality anyway.
In 1975, Len Gross discovered the logarithmic Sobolev inequality [@gross75] which fixes both of these defects by using Gaussian measure and being dimension-independent. Thus it has a chance of holding in infinite dimensions. In fact, it does.
The log-Sobolev inequality says that in an abstract Wiener space, for $F \in \mathbb{D}^{1,2}$ with $$\label{log-sobolev-eqn}
\int |F|^2 \ln |F| \,d\mu \le \norm{F}_{L^2(W,\mu)}^2 \ln
\norm{F}_{L^2(W,\mu)} + \norm{DF}_{L^2(W;H)}^2.$$
If you are worried what happens for $F$ near 0:
$g(x) = x^2 \ln x$ is bounded below on $(0, \infty)$, and $g(x) \to
0$ as $x \downarrow 0$.
So if we define “$0^2 \ln 0 = 0$”, there is no concern about the existence of the integral on the left side (however, what is not obvious is that it is finite). What’s really of interest are the places where $|F|$ is large, since then $|F|^2 \ln |F|$ is bigger than $|F|^2$.
It’s worth noting that (\[log-sobolev-eqn\]) also holds in finite dimensions, but there are no dimension-dependent constants appearing in it.
A concise way of stating the log Sobolev inequality is to say that $$\mathbb{D}^{1,2} \subset L^2 \ln L$$ where $L^2 \ln L$, by analogy with $L^p$, represents the set of measurable functions $F$ with $\int |F|^2 \ln |F| < \infty$. This is called an Orlicz space; one can play this game to define $\phi(L)$ spaces for a variety of reasonable functions $\phi$.
Our proof of the log Sobolev inequality hinges on the following completely innocuous looking commutation relation.
\[Tt-D-commute\] For $F \in \mathbb{D}^{1,2}$, $D T_t F = e^{-t} T_t DF$.
You may object that on the right side we are applying $T_t$, an operator on the real-valued function space $L^2(W)$, to the $H$-valued function $DF$. Okay then: we can define $T_t$ on $L^2(W;H)$ in any of the following ways:
1. Componentwise: $T_t u =
\sum_i (T_t {\langle u(\cdot) , h_i \rangle}_H)(x) h_i$ where $h_i$ is an orthonormal basis for $H$.
2. Via (\[Tt-formula\]), replacing the Lebesgue integral with Bochner.
3. Via (\[Tt-chaos-eqn\]): set $T_t u = \sum_{n=0}^\infty e^{-tn}
J_n u$ where $J_n$ is orthogonal projection onto $\mathcal{H}_n(H)
\subset L^2(W;H)$.
Verify that these are all the same. Also verify the inequality $$\label{Tt-H-markov}
\norm{T_t u(x)}_H \le T_t \norm{u}_H (x).$$
It’s worth noting that for any $F \in L^2$, $T_t F \in
\mathbb{D}^{1,2}$. This follows either from the spectral theorem, or from the observation that for any $t$, the sequence $\{n e^{-2tn}\}$ is bounded, so $\sum_n n \norm{J_n T_t F}^2 = \sum_n n e^{2tn}
\norm{J_n F}^2 \le C \sum \norm{J_n F}^2 \le C \norm{F}^2$. In fact, more is true: we have $T_t F \in {\operatorname{dom}}N$, and indeed $T_t F \in {\operatorname{dom}}N^\infty$.
$$\begin{aligned}
D T_t F &= D \sum_{n=0}^\infty e^{-tn} J_n F \\
&= \sum_{n=1}^\infty e^{-tn} D J_n F && \text{(recall $D J_0 =
0$)} \\
&= \sum_{n=1}^\infty e^{-tn} J_{n-1} DF \\
&= \sum_{k=0}^\infty e^{-t(k+1)} J_k DF = e^{-t} T_t DF
\end{aligned}$$
where we re-indexed by letting $k=n-1$. We’ve extended to $L^2(W;H)$ some Wiener chaos identities that we only really proved for $L^2(W)$; as an exercise you can check the details.
There’s also an infinitesimal version of this commutation:
For $F \in \mathcal{F} C^\infty_c(W)$, $D N F = (N+1) DF$.
Differentiate the previous lemma at $t=0$. Or, use Wiener chaos expansion.
(Not necessarily very interesting) Characterize the set of $F$ for which the foregoing identity makes sense and is true.
We can now prove the log Sobolev inequality (\[log-sobolev-eqn\]). This proof is taken from [@ustunel2010] which actually contains several proofs.
First, let $F$ be a smooth cylinder function which is bounded above and bounded below away from 0: $0 < a \le F \le b < \infty$. Take $G = F^2$; $G$ has the same properties. Note in particular that $G \in {\operatorname{dom}}N$. We have $$\label{ls1}
Q := 2 \left(\int F^2 \ln F d\mu - \norm{F}^2 \ln \norm{F}\right) =
\int G \ln G d\mu - \int G d\mu \ln \int G d\mu.$$ and we want to bound this quantity $Q$ by $2 \norm{DF}_{L^2(W;H)}^2$.
Note that for any $G \in L^2(W)$ we have $\lim_{t \to \infty} T_t G
= J_0 G = \int G d\mu$. (Use Lemma \[Tt-chaos\] and monotone convergence.) So we can think of $T_t G$ as a continuous function from $[0, \infty]$ to $L^2(W)$. It is continuously differentiable on $(0,\infty)$ and has derivative $-N T_t G = -T_t N G$. So define $A : [0,\infty] \to L^2(W)$ by $A(t) = (T_t G)\cdot(\ln T_t G)$ (noting that as $T_t$ is Markovian, $T_t G$ is bounded above and below, so $(T_t G)\cdot(\ln T_t G)$ is also bounded and hence in $L^2$). Then $Q = \int_W (A(0) - A(\infty)) d\mu$. Since we want to use the fundamental theorem of calculus, we use the chain rule to see that $$A'(t) = -(N T_t G) (1 + \ln T_t G).$$ So by the fundamental theorem of calculus, we have $$\begin{aligned}
Q &= - \int_W \int_0^\infty A'(t)\,dt d\mu \\
&= \int_W \int_0^\infty (N T_t G)(1 + \ln T_t G)\,dt\,d\mu.
\end{aligned}$$
There are two integrals in this expression, so of course we want to interchange them. To justify this, we note that $1 + \ln T_t G$ is bounded (since $0 < a^2 \le G \le b^2$ and $T_t$ is Markovian, we also have $a^2 \le T_t G \le b^2)$), and so it is enough to bound $$\begin{aligned}
\int_W \int_0^\infty |N T_t G| \,dt\,d\mu &= \int_0^\infty \norm{N
T_t G}_{L^1(W,\mu)} dt \\
&\le \int_0^\infty \norm{N T_t G}_{L^2(W,\mu)} dt
\end{aligned}$$ since $\norm{\cdot}_{L^1} \le \norm{\cdot}_{L^2}$ over a probability measure (Cauchy–Schwarz or Jensen). Note that $N T_t G = T_t N G$ is continuous from $[0,\infty]$ to $L^2(W,\mu)$, so $\norm{N T_t G}_{L^2(W)}$ is continuous in $t$ and hence bounded on compact sets. So we only have to worry about what happens for large $t$. But Corollary \[Tt-exponential-decay\] says that it decays exponentially, and so is integrable. (Note that $\int NG d\mu = {\langle NG , 1 \rangle}_{L^2(W)} = {\langle DG , D1 \rangle}_{L^2(W)} =
0$.)
So after applying Fubini’s theorem, we get $$\begin{aligned}
Q &= \int_0^\infty \int_W (N T_t G)(1 + \ln T_t G)\,d\mu \,dt \\
&= \int_0^\infty {\langle N T_t G , 1 + \ln T_t G \rangle}_{L^2(W)}\,dt.
\end{aligned}$$
Now since $N = \delta D$ we have, using the chain rule, $$\begin{aligned}
{\langle N T_t G , 1 + \ln T_t G \rangle}_{L^2(W)} &= {\langle D T_t G , \cancel{D1} + D
\ln T_t G \rangle}_{L^2(W;H)} \\
&= {\langle D T_t G , \frac{D T_t G}{T_t G} \rangle}_{L^2(W;H)} \\
&= \int_W \frac{1}{T_t G} \norm{D T_t G}_H^2 d\mu \\
&= e^{-2t} \int_W \frac{1}{T_t G} \norm{T_t D G}_H^2 d\mu
\end{aligned}$$ where we have just used the commutation $D T_t = e^{-t} T_t D$.
Let’s look at $\norm{T_t DG}_H^2$. Noting that $DG = 2 F DF$, we have $$\begin{aligned}
\norm{T_t DG}_H^2 &\le (T_t \norm{DG}_H)^2 && \text{by
(\ref{Tt-H-markov})} \\
&= 4 (T_t (F \norm{DF}_H))^2 \\
&\le 4 (T_t(F^2)) (T_t \norm{DF}_H^2) && \text{by (\ref{Tt-cauchy-schwarz-eqn})}.
\end{aligned}$$
Thus we have reached $$\begin{aligned}
\int_W \frac{1}{T_t G} \norm{T_t D G}_H^2 d\mu \le 4 \int_W T_t \norm{DF}_H^2\,d\mu.
\end{aligned}$$ But since $T_t$ is self-adjoint and $T_t 1 = 1$ (or, if you like, the fact that $T_t$ commutes with $J_0$, we have $\int_W T_t f d\mu
= \int f d\mu$ for any $t$. Thus $\int_W T_t \norm{DF}_H^2 d\mu =
\int_W \norm{DF}_H^2 \,d\mu = \norm{DF}_{L^2(W;H)}^2$. So we have $$\begin{aligned}
Q \le \left(4 \int_0^\infty e^{-2t}\,dt \right) \norm{DF}_{L^2(W;H)}^2
\end{aligned}$$ The parenthesized constant equals 2 (consult a Math 1120 student if in doubt). This is what we wanted.
To extend this to all $F \in \mathbb{D}^{1,2}$, we need some density arguments. Suppose now that $F$ is a smooth cylinder function which is bounded, say $|F| \le M$. Fix $\epsilon > 0$, and for each $n$ let $\varphi_n \in C^\infty(\R)$ be a positive smooth function, such that:
1. $\varphi_n$ is bounded away from 0;
2. $\varphi_n \le M$;
3. $\varphi_n'| \le 1+\epsilon$;
4. $\varphi_n(x) \to |x|$ pointwise on $[-M,M]$.
Thus $\varphi_n(F)$ is a smooth cylinder function, bounded away from 0 and bounded above, so it satisfies the log Sobolev inequality. Since $\varphi_n(F) \to |F|$ pointwise and boundedly, we have $\norm{\varphi_n(F)}_{L^2(W)} \to \norm{F}_{L^2(W)}$ by dominated convergence. We also have, by the chain rule, $\norm{D
\varphi_n(F)}_{L^2(W;H)} \le (1+\epsilon) \norm{DF}_{L^2(W;H)}$. Thus $$\limsup_{n \to \infty} \int_W \varphi_n(F)^2 \ln \varphi_n(F)\,d\mu \le
\norm{F}^2 \ln \norm{F} + (1+\epsilon) \norm{DF}^2.$$ Now since $x^2 \ln x$ is continuous, we have $\varphi_n(F)^2 \ln
\varphi_n(F) \to |F|^2 \ln |F|$ pointwise. Since $x^2 \ln x$ is bounded below, Fatou’s lemma gives $$\int_W |F|^2 \ln |F| \,d\mu \le \liminf_{n \to \infty} \int_W \varphi_n(F)^2 \ln \varphi_n(F)\,d\mu$$ and so this case is done after we send $\epsilon \to 0$. (Dominated convergence could also have been used, which would give equality in the last line.)
Finally, let $F \in \mathbb{D}^{1,2}$. We can find a sequence of bounded cylinder functions $F_n$ such that $F_n \to F$ in $L^2(W)$ and $DF_n \to DF$ in $L^2(W;H)$. Passing to a subsequence, we can also assume that $F_n \to F$ $\mu$-a.e., and we use Fatou’s lemma as before to see that the log Sobolev inequality holds in the limit.
Note that we mostly just used properties that are true for any Markovian semigroup $T_t$ that is conservative ($T_t 1 = 1$). The only exception was the commutation $D T_t = e^{-t} T_t D$. In fact, an inequality like $\norm{D T_t F}_H \le C(t) T_t \norm{DF}_H$ would have been good enough, provided that $C(t)$ is appropriately integrable. (One of the main results in my thesis was to prove an inequality like this for a certain finite-dimensional Lie group, in order to obtain a log-Sobolev inequality by precisely this method.)
Also, you might wonder: since the statement of the log-Sobolev inequality only involved $D$ and $\mu$, why did we drag the Ornstein–Uhlenbeck semigroup into it? Really the only reason was the fact that $T_\infty F = \int F d\mu$, which is just saying that $T_t$ is the semigroup of a Markov process whose distribution at a certain time $t_0$ (we took $t_0=\infty$) is the measure $\mu$ we want to use. If we want to prove this theorem in finite dimensions, we could instead use the heat semigroup $P_t$ (which is symmetric with respect to Lebesgue measure) and take $t=1$, beak Brownian motion at time 1 also has a standard Gaussian distribution.
Absolute continuity and smoothness of distributions
===================================================
This section will just hint at some of the very important applications of Malliavin calculus to proving absolute continuity results.
When presented with a random variable (or random vector) $X$, a very basic question is “What is its distribution?”, i.e. what is $\nu(A) := P(X \in A)$ for Borel sets $A$? A more basic question is “Does $X$ has a continuous distribution?”, i.e. is $\nu$ absolutely continuous to Lebesgue measure? If so, it has a Radon–Nikodym derivative $f \in L^1(m)$, which is a density function for $X$. It may happen that $f$ is continuous or $C^k$ or $C^\infty$, in which case so much the better.
Given a Brownian motion $B_t$ or similar process, one can cook up lots of complicated random variables whose distributions may be very hard to work out. For example:
- $X = f(B_t)$ for some fixed $t$ (this is not so hard)
- $X = f(B_T)$ for some stopping time $T$
- $X = \sup_{t \in [0,1]} B_t$
- $X = \int_0^1 Y_t \,dB_t$
- $X = Z_t$, where $Z$ is the solution to some SDE $dZ_t = f(Z_t) dB_t$.
Malliavin calculus gives us some tools to learn something about the absolute continuity of such random variables, and the smoothness of their densities.
Let $(W, H, \mu)$ be an abstract Wiener space. A measurable function $F : W \to \R$ is then a random variable, and we can ask about its distribution. If we’re going to use Malliavin calculus, we’d better concentrate on $F \in \mathbb{D}^{1,p}$. An obvious obstruction to absolute continuity would be if $F$ is constant on some set $A$ of positive $\mu$-measure; in this case, as we have previously shown, $DF = 0$ on $A$. The following theorem says if we ensure that $DF$ doesn’t vanish, then $F$ must be absolutely continuous.
Let $F \in \mathbb{D}^{1,1}$, and suppose that $DF$ is nonzero $\mu$-a.e. Then the law of $F$ is absolutely continuous to Lebesgue measure.
Let $\nu = \mu \circ F^{-1}$ be the law of $F$; our goal is to show $\nu \ll m$.
By replacing $F$ with something like $\arctan(F)$, we can assume that $F$ is bounded; say $0 \le F \le 1$. So we want to show that $\nu$ is absolutely continuous to Lebesgue measure $m$ on $[0,1]$. Let $A \subset [0,1]$ be Borel with $m(A) = 0$; we want to show $\nu(A) = 0$.
Choose a sequence $g_n \in C^\infty([0,1])$ such that $g_n \to 1_A$ $m+\nu$-a.e., and such that the $g_n$ are uniformly bounded (say $|g_n| \le 2$). Set $\psi_n(t) = \int_0^t
g_n(s) ds$. Then $\psi_n \in C^\infty$, $|\psi_n| \le 2$, and $\psi_n \to 0$ pointwise (everywhere).
In particular $\psi_n(F) \to
0$ $\mu$-a.e. (in fact everywhere), and thus also in $L^1(W, \mu)$ by bounded convergence. On the other hand, by the chain rule, $D \psi_n (F) = g_n(F) DF$. Now since $g_n \to 1_A$ $\nu$-a.e., we have $g_n(F) \to 1_A(F)$ $\mu$-a.e., and boundedly. Thus $g_n(F) DF \to 1_A(F) DF$ in $L^1(W; H)$. Now $D$ is a closed operator, so we must have $1_A(F) DF = D0 = 0$. But by assumption $DF \ne 0$ $\mu$-a.e., so we have to have $1_A F = 0$ $\mu$-a.e., that is, $\nu(A) = 0$.
So knowing that the derivative $DF$ “never” vanishes guarantees that the law of $F$ has a density. If $DF$ mostly stays away from zero in the sense that $\norm{DF}_H^{-1} \in L^p(W)$ for some $p$, then this gives more smoothness (e.g. differentiability) for the density. See Nualart for precise statements.
In higher dimensions, if we have a function $F = (F^1, \dots, F^n) : W
\to \R^n$, the object to look at is the “Jacobian,” the matrix-valued function $\gamma_F : W \to \R^{n \times n}$ defined by $\gamma_F(x)_{ij} = {\langle DF^i(x) , DF^j(x) \rangle}_H$. If $\gamma_F$ is almost everywhere nonsingular, then the law of $F$ has a density. If we have $(\det \gamma_F)^{-1} \in L^p(W)$ for some $p$, then we get more smoothness.
Here’s another interesting fact. Recall that the **support** of a Borel measure $\nu$ on a topological space $\Omega$ is by definition the set of all $x \in \Omega$ such that every neighborhood of $x$ has nonzero $\nu$ measure. This set is closed.
If $F \in \mathbb{D}^{1,2}$, then the support of the law of $F$ is connected, i.e. is a closed interval in $\R$.
Let $\nu = \mu \circ F^{-1}$. Suppose $\supp \nu$ is not connected. Then there exists $a \in \R$ such that there are points of $\supp \nu$ to the left and right of $a$. Since $\supp \nu$ is closed, there is an open interval $(a,b)$ in the complement of $\supp \nu$. That is, we have $\mu(a < F < b) = 0$ but $0 < \mu(F
\le a) < 1$. Let $\psi \in C^\infty(\R)$ have $\psi(t) = 1$ for $t
\le a$ and $\psi(t) = 0$ for $t \ge b$, and moreover take $\psi$ and all its derivatives to be bounded. Then $\psi(F) = 1_{(-\infty,
a]}(F) = 1_{\{F \le a\}}$. Since $\psi$ is smooth, $1_{\{F \le a\}}
= \psi(F) \in
\mathbb{D}^{1,2}$ by the chain rule (Lemma \[chain-rule\]). By the zero-one law of Proposition \[zero-one-law\], $\mu(F \le a)$ is either 0 or 1, a contradiction.
As an example, let’s look at the maximum of a continuous process.
Let $(W,H,\mu)$ be an abstract Wiener space. Suppose we have a process $\{X_t : t \in [0,1]\}$ defined on $W$, i.e. a measurable map $X : [0,1] \times W \to \R$, which is a.s. continuous in $t$. (If we take $W = C([0,1])$ and $\mu$ the law of some continuous Gaussian process $Y_t$, then $X_t = Y_t$, in other words $X_t(\omega)=\omega(t)$, would be an example. Another natural example would be to take classical Wiener space and let $X_t$ be the solution of some SDE.) Let $M = \sup_{t \in [0,1]} X_t$. We will show that under certain conditions, $M$ has an absolutely continuous law.
(Note you can also index $\{X_t\}$ by any other compact metric space $S$ and the below proofs will go through just fine. If you take $S$ finite, the results are trivial. You can take $S = [0,1]^2$ and prove things about Brownian sheet. You can even take $S$ to be Cantor space if you really want (hi Clinton!).)
Suppose $F_n \in \mathbb{D}^{1,2}$, $F_n \to F$ in $L^2(W)$, and $\sup_n \norm{DF_n}_{L^2(W;H)} < \infty$. Then $F \in
\mathbb{D}^{1,2}$ and $D F_n \to DF$ weakly in $L^2(W;H)$.
This is really a general fact about closed operators on Hilbert space. Since $\{DF_n\}$ is a bounded sequence in $L^2(W;H)$, by Alaoglu’s theorem we can pass to a subsequence and assume that $DF_n$ converges weakly in $L^2(W;H)$, to some element $u$. Suppose $v \in {\operatorname{dom}}\delta$. Then ${\langle DF_n , v \rangle}_{L^2(W;H)} =
{\langle F_n , \delta v \rangle}_{L^2(W)}$. The left side converges to ${\langle u , v \rangle}_{L^2(W;H)}$ and the right side to ${\langle F , \delta
v \rangle}_{L^2(W)}$. Since the left side is continuous in $v$, we have $F
\in {\operatorname{dom}}\delta^* = {\operatorname{dom}}D = \mathbb{D}^{1,2}$. Moreover, since we have ${\langle D F_n , v \rangle} \to {\langle DF , v \rangle}$ for all $v$ in a dense subset of $L^2(W;H)$, and $\{D F_n\}$ is bounded, it follows from the triangle inequality that $D F_n \to DF$ weakly. Since we get the same limit no matter which weakly convergent subsequence we passed to, it must be that the original sequence $D F_n$ also converges weakly to $DF$.
Recall, as we’ve previously argued, that if $F \in \mathbb{D}^{1,2}$, then $|F| \in \mathbb{D}^{1,2}$ also, and $\norm{D|F|}_H \le
\norm{DF}_H$ a.e. (Approximate $|t|$ by smooth functions with uniformly bounded derivatives.) It follows that if $F_1, F_2 \in \mathbb{D}^{1,2}$, then $F_1 \wedge F_2$, $F_1 \vee F_2
\in \mathbb{D}^{1,2}$ also. ($F_1 \wedge F_2 = F_1 + F_2 - |F_1 -
F_2|$, and $F_1 \vee F_2 = F_1 + F_2 + |F_1 - F_2|$.) Then by iteration, if $F_1, \dots, F_n \in \mathbb{D}^{1,2}$, then $\min_k
F_k, \max_k F_k \in \mathbb{D}^{1,2}$ as well.
\[Md12\] Suppose $X, M$ are as above, and:
1. $\int_W \sup_{t \in [0,1]} |X_t(\omega)|^2 \,\mu(d\omega) <
\infty$;
2. For any $t \in [0,1]$, $X_t \in \mathbb{D}^{1,2}$;
3. The $H$-valued process $DX_t$ has an a.s. continuous version (which we henceforth fix);
4. $\int_W \sup_{t \in [0,1]} \norm{DX_t(\omega)}_H^2
\,\mu(d\omega) < \infty$.
Then $M \in \mathbb{D}^{1,2}$.
The first property guarantees $M \in L^2(W)$. Enumerate the rationals in $[0,1]$ as $\{q_n\}$. Set $M_n = \max\{X_{q_1}, \dots,
X_{q_n}\}$. Then $M_n \in \mathbb{D}^{1,2}$ (using item 2). Clearly $M_n \uparrow M$ so by monotone convergence $M_n \to M$ in $L^2(W)$. It suffices now to show that $\sup_n
\norm{DM_n}_{L^2(W;H)} < \infty$. Fix $n$, and for $k = 1, \dots,
n$ let $A_k$ be the set of all $\omega$ where the maximum in $M_n$ is achieved by $X_{q_k}$, with ties going to the smaller $k$. That is, $$\begin{aligned}
A_1 &= \{ \omega : X_{q_1}(\omega) = M_n(\omega)\} \\
A_2 &= \{ \omega : X_{q_1}(\omega) \ne M_n(\omega),
X_{q_2}(\omega) = M_n(\omega)\} \\
&\vdots \\
A_n &= \{ \omega : X_{q_1}(\omega) \ne M_n(\omega), \dots,
X_{q_{n-1}}(\omega) \ne M_n(\omega), X_{q_n}(\omega) = M_n(\omega)\}
\end{aligned}$$ Clearly the $A_k$ are Borel and partition $W$, and $M_n = X_{q_k}$ on $A_k$. By the local property of $D$, we have $D M_n = D X_{q_k}$ a.e. on $A_k$. In particular, $\norm{D M_n}_H \le \sup_{t \in
[0,1]} \norm{D X_t}_H$ a.e. Squaring and integrating both sides, we are done by the last assumption.
\[gpex\] Let $\{X_t, t \in [0,1]\}$ be a continuous centered Gaussian process. Then we can take $W = C([0,1])$ (or a closed subspace thereof) and $\mu$ to be the law of the process, and define $X_t$ on $W$ by $X_t(\omega) = \omega(t)$. Verify that the hypotheses of Proposition \[Md12\] are satisfied.
\[M-ac\] Suppose $X_t$ satisfies the hypotheses of the previous theorem, and moreover $$\mu(\{\omega : X_t(\omega) = M(\omega) \implies DX_t(\omega) \ne
0\}) = 1.$$ (Note we are fixing continuous versions of $X_t$ and $DX_t$ so the above expression makes sense.) Then $DM \ne 0$ a.e. and $M$ has an absolutely continuous law.
It is enough to show $$\mu(\{\omega : X_t(\omega) = M(\omega) \implies DX_t(\omega) =
DM(\omega)\}) = 1.$$ Call the above set $A$. (Note that for every fixed $\omega$, $M(\omega) = X_t(\omega)$ for some $t$.)
Let $E$ be a countable dense subset of $H$. For fixed $r,s \in \Q$, $h \in E$, $k > 0$, let $$G_{r,s,h,k} = \{ \omega : \sup_{t \in (r,s)} X_t(\omega) =
M(\omega), {{\langle DX_t(\omega) - DM(\omega) , h \rangle}_H} \ge
\frac{1}{n} \text{ for all $r < t < s$} \}.$$ Enumerate the rationals in $(r,s)$ as $\{q_i\}$. If we let $M' =
\sup_{t \in (r,s)} X_t$, $M_n' = \max \{ X_{q_1}, \dots, X_{q_n}\}$, then as we argued before, $M_n' \to M'$ in $L^2(W)$, and $D M_n' \to
DM'$ weakly in $L^2(W;H)$. On the other hand, by the local property used before, for every $\omega$ there is some $t_i$ with $D M_n' = D
X_{t_i}$. Thus for $\omega \in G_{r,s,h,k}$ we have ${{\langle DM'_n(\omega) - DM'(\omega) , h \rangle}_H} \ge \frac{1}{n}$ for all $r < t < s$. Integrating this inequality, we have ${\langle D
M'_n - D M' , h 1_{G_{r,s,h,k}} \rangle}_{L^2(W;H)} \ge \frac{1}{n} \mu(G_{r,s,h,k})$ for all $n$. The left side goes to 0 by weak convergence, so it must be that $\mu(G_{r,s,h,k}) = 0$.
However, $A^c = \bigcup G_{r,s,h,k}$ which is a countable union. (If $\omega \in A^c$, there exists $t$ such that $X_t(\omega) =
M(\omega)$ but $DX_t(\omega) \ne DM(\omega)$. As such, there must exist $h \in E$ with ${\langle DX_t(\omega) - DM(\omega) , h \rangle}_H \ne 0$; by replacing $h$ by $-h$ or something very close to it, we can assume ${\langle DX_t(\omega) - DM(\omega) , h \rangle}_H > 0$. As $DX_t$ is assumed continuous, there exists $(r,s) \in \Q$ and $k > 0$ such that ${\langle DX_t(\omega) - DM(\omega) , h \rangle}_H > \frac{1}{k}$ for all $t \in (r,s)$. So we have $\omega \in G_{r,s,h,k}$.)
Again let $X_t$ be a centered Gaussian process as in Exercise \[gpex\] above. Give an example of a process for which $M$ does not have an absolutely continuous law. However, show that if $P(M=0)=0$, then the hypothesis of Proposition \[M-ac\] is satisfied. (Can we show this always holds whenever $X_t$ is strong Markov?)
Miscellaneous lemmas
====================
\[ell2\] Let $y \in \R^\infty$, and suppose that $\sum y(i) g(i)$ converges for every $g \in \ell^2$. Then $y \in \ell^2$.
For each $n$, let $H_n \in
(\ell^2)^*$ be the bounded linear functional $H_n(g) = \sum_{i=1}^n
y(i) g(i)$. By assumption, for each $g \in \ell^2$, the sequence $\{H_n(g)\}$ converges; in particular $\sup_n |H_n(g)| < \infty$. So by the uniform boundedness principle, $\sup_n
||H_n||_{(\ell^2)^*} < \infty$. But $||H_n||_{(\ell^2)^*}^2 =
\sum_{i=1}^n |y(i)|^2$, so $\sum_{i=1}^\infty |y(i)|^2 = \sup_n
||H_n||_{(\ell^2)^*}^2 < \infty$ and $y \in \ell^2$.
For an elementary, constructive proof, see also [@piau-ell2-argument].
\[dense-subspace-basis\] Let $H$ be a separable Hilbert space and $E \subset H$ a dense subspace. There exists an orthonormal basis $\{e_i\}$ for $H$ with $\{e_i\} \subset E$.
Choose a sequence $\{x_i\} \subset E$ which is dense in $H$. (To see that this is possible, let $\{y_k\}$ be a countable dense subset of $H$, and choose one $x_i$ inside each ball $B(y_k, 1/m)$.) Then apply Gram-Schmidt to $x_i$ to get an orthonormal sequence $\{e_i\}
\subset E$ with $x_n \in \spanop\{e_1, \dots, e_n\}$. Then since $\{x_i\} \subset \spanop\{e_i\}$, $\spanop\{e_i\}$ is dense in $H$, so $\{e_i\}$ is an orthonormal basis for $H$.
\[limit-of-gaussian\] Let $X_n \sim N(0, \sigma_n^2)$ be a sequence of mean-zero Gaussian random variables converging in distribution to a finite random variable $X$. Then $X$ is also Gaussian, with mean zero and variance $\sigma^2 = \lim
\sigma_n^2$ (and the limit exists).
Suppose $\sigma_{n_k}^2$ is a subsequence of $\sigma_n^2$ converging to some $\sigma^2 \in [0,+\infty]$. (By compactness, such a subsequence must exist.) Now taking Fourier transforms, we have $e^{- \lambda^2 \sigma_n^2/2} = E[e^{i
\lambda X_n}] \to E[e^{i \lambda X}]$ for each $X$, so $E[e^{i
\lambda X}] = e^{-\lambda^2 \sigma^2/2}$. Moreover, the Fourier transform of $X$ must be continuous and equal 1 at $\lambda = 0$, which rules out the case $\sigma^2 = +\infty$. So $X \sim N(0,
\sigma^2)$. Since we get the same $\sigma^2$ no matter which convergent subsequence of $\sigma_n^2$ we start with, $\sigma_n^2$ must converge to $\sigma^2$.
\[smooth-dense\] Let $\mu$ be any finite Borel measure on $[0,1]$. Then $C^\infty([0,1])$ is dense in $L^p([0,1], \mu)$.
Use Dynkin’s multiplicative system theorem. Let $M$ consist of all $\mu$-versions of all bounded measurable functions in the closure of $C^\infty$ in $L^p(\mu)$. Then $M$ is a vector space closed under bounded convergence (since bounded convergence implies $L^p(\mu)$ convergence) and it contains $C^\infty([0,1])$. By Dynkin’s theorem, $M$ contains all bounded $\mathcal{F}$-measurable functions, where $\mathcal{F}$ is the smallest $\sigma$-algebra that makes all functions from $C^\infty([0,1])$ measurable. But the identity function $f(x) = x$ is in $C^\infty$. So for any Borel set $B$, we must have $B = f^{-1}(B) \in \mathcal{F}$. Thus $\mathcal{F}$ is actually the Borel $\sigma$-algebra, and $M$ contains all bounded measurable functions. Since the bounded functions are certainly dense in $L^p(\mu)$ (by dominated convergence), we are done.
Radon measures {#radon}
==============
A finite Borel measure $\mu$ on a topological space $W$ is said to be **Radon** if for every Borel set $B$, we have $$\label{inner-regular}
\mu(B) = \sup\{\mu(K) : K \subset B, K \text{ compact}\}$$ (we say that such a set $B$ is **inner regular**). Equivalently, $\mu$ is Radon if for every Borel set $B$ and every $\epsilon > 0$, there exists a compact $K \subset B$ with $\mu(B
\backslash K) < \epsilon$.
If $X$ is a compact metric space, every finite Borel measure on $X$ is Radon.
Let $(X,d)$ be a compact metric space, and $\mu$ a Borel measure. Let $\mathcal{F}$ denote the collection of all Borel sets $B$ such that $B$ and $B^C$ are both inner regular. I claim $\mathcal{F}$ is a $\sigma$-algebra. Clearly $\emptyset \in
\mathcal{F}$ and $\mathcal{F}$ is also closed under complements. If $B_1, B_2, \dots \in \mathcal{F}$ are disjoint, and $B =
\bigcup_n B_n$ then since $\sum_n
\mu(B_n) = \mu(B) < \infty$, there exists $n$ so large that $\sum_{n=N}^\infty \mu(B_n) < \epsilon$. For $n = 1, \dots, N$, choose a compact $K_n \subset B_n$ with $\mu(B_n \backslash K_n) <
\epsilon/N$. Then if $K = \bigcup_{n=1}^N K_n$, $K$ is compact, $K
\subset B$, and $\mu(B \backslash K) < 2 \epsilon$. So $B$ is inner regular.
Next, $\mathcal{F}$ contains all open sets $U$. For any open set $U$ may be written as a countable union of compact sets $K_n$. (For every $x \in U$ there is an open ball $B(x,r_x)$ contained in $U$, hence $\closure{B(x,r_x/2)} \subset U$ also. Since $X$ is second countable we can find a basic open set $V_x$ with $x \in V_x \subset
B(x, r_x/2)$, so $\closure{V_x} \subset U$. Then $U = \bigcup_{x
\in U} \closure{V_x}$. But this union actually contains only countably many distinct sets.) Thus by countable additivity, $U$ is inner regular. $U^C$ is compact and so obviously inner regular. Thus $U \in \mathcal{F}$. Since $\mathcal{F}$ is a $\sigma$-algebra and contains all open sets, it contains all Borel sets.
Every complete separable metric space $(X,d)$ is homeomorphic to a Borel subset of the compact metric space $[0,1]^\infty$.
Without loss of generality, assume $d \le 1$. Fix a dense sequence $x_1, x_2, \dots$ in $X$ and for each $x \in X$, set $F(x) =
(d(x,x_1), d(x,x_2), \dots) \in [0,1]^\infty$. It is easy to check that $F$ is continuous. $F$ is also injective: for any $x \in X$ we can choose a subsequence $x_{n_k} \to x$, so that $d(x_{n_k}, x) \to
0$. Then if $F(x) = F(y)$, then $d(x_n, x) = d(x_n, y)$ for all $n$, so $x_{n_k} \to y$ as well, and $x=y$. Finally, $F$ has a continuous inverse. Suppose $F(y_m) \to F(y)$. Choose an $x_n$ such that $d(x_n, y) < \epsilon$. We have $F(y_m)_n = d(x_n, y_m)
\to d(x_n, y) = F(y)_n$, so for sufficiently large $m$, $d(y_m, x_n)
< \epsilon$, and by the triangle inequality $d(y_m, y) < 2\epsilon$.
Lastly, we check $F(X)$ is Borel. Well, this theorem is standard and I’m too lazy to write it out. See, e.g. Srivastava’s *A course on Borel sets*, section 2.2.
Any finite Borel measure $\mu$ on a complete separable metric space $X$ is Radon.
Let $F$ be the above embedding of $X$ into $[0,1]^\infty$. Then $\mu \circ F^{-1}$ defines a Borel measure on $F(X)$. We can extend it to a Borel measure on $[0,1]^\infty$ by setting $\tilde{\mu}(B) = \mu(F^{-1}(B
\cap F(X)))$, i.e. $\tilde{\mu}$ assigns measure zero to all sets outside $F(X)$. Then we know that $\tilde{\mu}$ is Radon and hence so is $\mu$.
As a corollary of this, for any Borel probability measure on a Polish space, there is a sequence of compact sets $K_n$ such that $\mu(\bigcup K_n) = 1$. This is perhaps surprising because compact sets in an infinite dimensional Banach space are very thin; in particular they are nowhere dense. For classical Wiener space with Wiener measure, find explicit sets $K_n$ with this property. (Hint: Think of some well-known sample path properties of Brownian motion.)
Miscellaneous Exercises
=======================
\[topologies-first\] Let $X$ be a set, and let $\tau_s$ and $\tau_w$ be two topologies on $X$ such that $\tau_w \subset \tau_s$. $\tau_w$ is said to be “weaker” or “coarser,” while $\tau_s$ is “stronger” or “finer.”
Fill in the following chart. Here $A \subset X$, and $Y,Z$ are some other topological spaces. All terms such as “more,” “less,” “larger,” “smaller” should be understood in the sense of implication or containment. For instance, since every set which is open in $\tau_w$ is also open in $\tau_s$, we might say $\tau_s$ has “more” open sets and $\tau_w$ has “fewer.”
---------------------------------------------------------------------------------------------------
Property $\tau_w$ $\tau_s$ Choices
-------------------------------- ---------- ------------------------------------ ------------------
Open sets More / fewer
Closed sets More / fewer
Dense sets More / fewer
Compact sets More / fewer
Connected sets More / fewer
Closure $\bar{A}$ Larger / smaller
Interior $A^\circ$ Larger / smaller
Precompact sets More / fewer
Separable sets More / fewer
Continuous functions $X \to Y$ More / fewer
Continuous functions $Z \to X$ More / fewer
Identity map continuous $(X, \tau_s)
\to (X, \tau_w)$ or vice versa
Convergent sequences More / fewer
---------------------------------------------------------------------------------------------------
Now suppose that $X$ is a vector space, and $\tau_w \subset \tau_s$ are generated by two norms ${\norm{\cdot}}_w, {\norm{\cdot}}_s$. Also let $Y,Z$ be other normed spaces.
Property ${\norm{\cdot}}_w$ ${\norm{\cdot}}_s$ Choices
---------------------------------------- -------------------- -------------------- ---------------------------------------------------------------------
Size of norm ${\norm{\cdot}}_s \le C {\norm{\cdot}}_w$ or vice versa, or neither
Closed (unbounded) operators $X \to Y$ More / fewer
Closed (unbounded) operators $Z \to X$ More / fewer
Cauchy sequences More / fewer
Give an example where $X$ is complete in ${\norm{\cdot}}_s$ but not in ${\norm{\cdot}}_w$.
Give an example where $X$ is complete in ${\norm{\cdot}}_w$ but not in ${\norm{\cdot}}_s$. (This exercise is “abstract nonsense,” i.e. it uses the axiom of choice.)
If $X$ is complete in both ${\norm{\cdot}}_s$ and ${\norm{\cdot}}_w$, show that the two norms are equivalent, i.e. $c {\norm{\cdot}}_s \le {\norm{\cdot}}_w \le C {\norm{\cdot}}_s$ (and in particular $\tau_s = \tau_w$).
\[topologies-last\] In the previous problem, the assumption that $\tau_w \subset
\tau_s$ was necessary. Give an example of a vector space $X$ and complete norms ${\norm{\cdot}}_1$, ${\norm{\cdot}}_2$ which are not equivalent. (Abstract nonsense.)
\[adjoint-exercise\] Let $X,Y$ be Banach spaces with $X$ reflexive, $T : X \to Y$ a bounded operator, and $T^* : Y^* \to X^*$ its adjoint.
1. If $T$ is injective, then $T^*$ has dense range.
2. If $T$ has dense range, then $T^*$ is injective.
For classical Wiener space $(W, \mu)$, find an explicit sequence of compact sets $K_n \subset W$ with $\mu\left(\bigcup_n K_n\right) = 1$.
Questions for Nate
==================
1. Is a Gaussian Borel measure on a separable Banach space always Radon? (Yes, a finite Borel measure on a Polish space is always Radon. See Bogachev Theorem A.3.11.)
2. Compute the Cameron-Martin space $H$ for various continuous Gaussian processes (Ornstein–Uhlenbeck, fractional Brownian motion).
3. Why should Brownian motion “live” in the space $C([0,1])$ instead of the smaller Hölder space $C^{0,\alpha}([0,1])$ for $\alpha < 1/2$?
4. What’s the relationship between Brownian motion on classical Wiener space and various other 2-parameter Gaussian processes (e.g. Brownian sheet)? (Compute covariances.)
[^1]: In reading these notes in conjunction with [@nualart], you should identify Nualart’s abstract probability space $(\Omega, \mathcal{F}, P)$ with our Banach space $(W, \mathcal{B}, \mu)$. His “Gaussian process” $h \mapsto W(h)$ should be viewed as corresponding to our map $T$ defined in Section \[sec-cameron-martin\]; his indexing Hilbert space $H$ may be identified with the Cameron–Martin space, and his $W(h)$ is the random variable, defined on the Banach space, that we have denoted by $Th$ or $\langle h, \cdot \rangle$. There’s a general principle in this area that all the “action” takes place on the Cameron–Martin space, so one doesn’t really lose much by dropping the Banach space structure on the space $W$ and replacing it with an generic $\Omega$ (and moreover generality is gained). Nonetheless, I found it helpful in building intuition to work on a concrete space $W$; this also gives one the opportunity to explore how the topologies of $W$ and $H$ interact.
[^2]: In the case of separable Banach spaces, or more generally Polish topological vector spaces, this sufficient condition is also necessary: a linear functional is Borel measurable if and only if it is continuous. Even the weaker assumption of so-called Baire measurability is sufficient, in fact. See 9.C of [@kechris]. So we are not giving up anything by requiring continuity. Thanks to Clinton Conley for explaining this to me and providing a reference. This sort of goes to show that a linear functional on a separable Banach space is either continuous or really really nasty.
[^3]: A particularly bizarre example was given recently in [@argyros-haydon]: a separable Banach space $X$ such that every bounded operator $T$ on $X$ is of the form $T =
\lambda I + K$, where $K$ is a compact operator. In some sense, $X$ has almost the minimum possible number of bounded operators.
[^4]: Technically, since we were supposed to have $\norm{\sum a_i b_i z_i}_W < 1$ with a strict inequality, we should take $a_i = c < 1$, and this argument will actually produce $c
{\norm{\cdot}}_K$ instead of ${\norm{\cdot}}_K$, which of course makes no difference.
|
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"pile_set_name": "ArXiv"
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abstract: 'We use a single trapped-ion qutrit to demonstrate the violation of input-state-independent non-contextuality inequalities using a sequence of randomly chosen quantum non-demolition projective measurements. We concatenate 53 million sequential measurements of 13 observables, and violate an optimal non-contextual bound by 214 standard deviations. We use the same dataset to characterize imperfections including signaling and repeatability of the measurements. The experimental sequence was generated in real time with a quantum random number generator integrated into our control system to select the subsequent observable with a latency below $\SI{50}{\micro\second}$, which can be used to constrain hidden-variable models that might describe our results. The state-recycling experimental procedure is resilient to noise, self-correcting and independent of the qutrit state, substantiating the fact that the contextual nature of quantum physics is connected to measurements as opposed to designated states. The use of extended sequences of quantum non-demolition measurements finds applications in the fields of sensing and quantum information.'
author:
- 'F. M. Leupold'
- 'M. Malinowski'
- 'C. Zhang'
- 'V. Negnevitsky'
- 'J. Alonso'
- 'J. P. Home'
- 'A. Cabello'
title: 'Sustained state-independent quantum contextual correlations from a single ion'
---
Two measurements are said to be compatible when the outcome statistics of each of them individually is independent of whether the other is carried out or not. In classical theories, outcomes of measurements are consistent with each measurement result having a pre-existing value, independent of which other compatible measurements are performed. However, correlations between the outcomes of compatible observables in Quantum Mechanics (QM) can be stronger than in classical theories. This feature, which is known as contextuality, has been linked to the power of quantum computation [@14Howard; @16Bermejo; @17Bermejo] and its most famous manifestation is Bell non-locality [@64Bell]. In this sense, the violation of a Bell inequality demonstrates contextuality. However, non-locality requires composite systems in entangled states. A more general result is that of Kochen and Specker [@67Kochen], who showed that any state of any quantum system in a Hilbert space of dimension greater than 2 can be used to reveal contextuality.
In a similar sense to a Bell-inequality, the contextuality of QM can be shown through the violation of a number of inequalities, which have been derived for systems of various Hilbert space dimension. Such inequalities can be split into those which are violated for a given input state [@69Clauser; @08Klyachko], and those for which the violation is input-state independent [@08Cabello; @12Yu]. We will refer to the latter as State-Independent-Contextuality (SIC) tests. SIC tests have been performed using a number of systems [@09Kirchmair2; @09Amselem; @10Moussa; @12Zu; @13Zhang; @13DAmbrosio; @13Huang; @14Canas; @14Canas2], but thus far they all used the following approach: i) prepare an input states, ii) measure multiple observables. This was repeated for each of a finite number of input states, and using all combinations of observables required for the test. Measurements on each observable can either be carried out simultaneously or sequentially [@09Guhne], with the sequential approach being the most popular.
An alternative proposal [@16Wajs] is to perform a SIC test using sequences of ideal Quantum Non-Demolition (QND) projective measurements (which in the context of general probabilistic theories are known as sharp measurements [@14Chiribella]). Each measurement is performed on the state into which the system was projected by the previous measurement. When executed in this manner, contextuality tests intrinsically stabilize the generation of quantum correlations and are self-correcting, which can be used to generate and certify continuous strings of random numbers [@ThColbeck; @13Um].
In this Letter, we demonstrate SIC sustainable in time using state-recycling over a sequence of 53 million measurements. To that end we have adopted: i) the simplest system featuring SIC, a three-level quantum system or qutrit [@66Bell; @67Kochen], ii) the smallest set of elementary quantum measurements needed for SIC, namely, the Yu-Oh set with 13 observables [@12Yu; @15Cabello; @16Cabello], and iii) the original Yu-Oh and an optimal witness of SIC [@12Kleinmann]. Our results violate the bounds imposed by non-contextual hidden-variable models. We use a commercial Quantum Random Number Generator (QRNG) to create the sequence of measured observables in real time. This places constraints on contextual hidden-variable models attempting to explain our results, which must cover the behavior not only of the qutrit but also of the QRNG [@88Peres]. We quantify the sharpness and compatibility of our measurements by extracting high-order correlators from the dataset.
The 13 dichotomic (“yes-no”) observables or “rays” in the Yu-Oh set [@12Yu] are of the form $A_v=I-2P_v$, where $I$ is the identity, $P_v$ is the normalized projection operator onto a vector $\ket*{v}=a\ket*{0}+b\ket*{1}+c\ket*{2}$, and $\Bqty{\ket*{0},\ket*{1},\ket*{2}}$ form a qutrit basis. Since the eigenvalues of $P_v$ are 0 and 1, ray measurements result in values $+1$ and $-1$. The 13 vectors $\ket*{v}$ with real-valued coefficients $\qty(a,b,c)$ are defined by points on the surface of a $3\times 3$ cube in a three-dimensional Hilbert space (FIG. \[fig:YuOhRays\]a, TABLE \[tab:RayAngles\]). Two rays are compatible if the corresponding vectors are orthogonal. This can be visualized in an orthogonality graph (FIG. \[fig:YuOhRays\]b) by drawing all vectors from the set $V=\qty\big{y_k^\sigma,h_\alpha,z_k|k=1,2,3;\sigma=\pm;\alpha=0,1,2,3}$ as vertices and linking vertices of compatible rays. In this notation, $z_k$ are the basis states, $y_k$ are superpositions of two basis states, and $h_k$ are superpositions of all three. In total, there are 24 edges in the graph, representing the 24 compatible pairs $(u,v)\in E$ with $P_uP_v=0$ (each edge is counted only once).
Besides the original Yu-Oh witness [@12Yu]
\[eq:SICWitnesses\] $$\begin{aligned}
\expval*{\chi_\text{YO}}=\sum_{v\in V} \expval*{A_v} - \sum_{(u,v)\in E} \frac{1}{2}\,\expval*{A_u A_v},\end{aligned}$$ we use the optimal SIC witness opt3 for which the QM and classical predictions differ maximally [@12Kleinmann] $$\begin{aligned}
\expval*{\chi_\text{opt3}}=&\sum_{v\in V_h}2\,\expval*{A_v}+\sum_{v\in V\setminus V_h} \expval*{A_v} \notag\\
&-\sum_{(u,v)\in E\setminus C_2}2\,\expval*{A_u A_v}-\sum_{(u,v)\in C_2} \expval*{A_u A_v} \notag\\
&-\sum_{(u,v,w)\in C_3}3\,\expval*{A_u A_v A_w}.\end{aligned}$$
Here $V_h=\qty\big{h_\alpha}$, $C_2=\qty\big{\pqty\big{z_k,y_k^+},\pqty\big{z_k,y_k^-},\pqty\big{y_k^+,y_k^-}}$ and $C_3=\qty\big{\pqty\big{z_k,y_k^+,y_k^-}}$, with indices $k$ and $\alpha$ running as for $V$. A necessary condition for a set of correlations to be non-contextual is $$\begin{aligned}
\label{eq:NoncontextualInequalities}
\expval*{\chi_\text{YO}}\leq 8 \qq{and}
\expval*{\chi_\text{opt3}}\leq 25,\end{aligned}$$ and any violation of these inequalities demonstrates contextuality. The prediction of quantum theory is that, for any qutrit state and under ideal conditions, $$\begin{aligned}
\expval*{\chi_\text{YO}}=\frac{25}{3}\approx 8.333 \qq{and}
\expval*{\chi_\text{opt3}}=\frac{83}{3}\approx 27.667.\end{aligned}$$
[LRRRRRc]{} v & (a,b,c) & \^[(1)]{}\_v & \^[(1)]{}\_v & \^[(2)]{}\_v & \^[(2)]{}\_v & QRNG\
y\_1\^- & (0,1,|[1]{}) & & 3/2 & /2 & /2 & 0001\
y\_2\^- & (|[1]{},0,1) & 0 & [0]{} & 3/2 & 3/2 & 0010\
y\_3\^- & (1,|[1]{},0) & /2 & /2 & 0 & [0]{} & 0011\
y\_1\^+ & (0,1,1) & & 3/2 & /2 & 3/2 & 0100\
y\_2\^+ & (1,0,1) & 0 & [0]{} & /2 & 3/2 & 0101\
y\_3\^+ & (1,1,0) & /2 & 3/2 & 0 & [0]{} & 0110\
h\_1 & (|[1]{},1,1) & 3/2 & 3/2 & \^[(2)]{}\_[h]{} & 3/2 & 0111\
h\_2 & (1,|[1]{},1) & /2 & /2 & \^[(2)]{}\_[h]{} & 3/2 & 1000\
h\_3 & (1,1,|[1]{}) & /2 & 3/2 & \^[(2)]{}\_[h]{} & /2 & 1001\
h\_0 & (1,1,1) & /2 & 3/2 & \^[(2)]{}\_[h]{} & 3/2 & 1010\
z\_1 & (1,0,0) & 0 & [0]{} & 0 & [0]{} & 1011\
z\_2 & (0,1,0) & & 3/2 & 0 & [0]{} & 1100\
z\_3 & (0,0,1) & 0 & [0]{} & & 3/2 & 1101
Our experimental platform to test these witnesses uses a single ${}^{40}\text{Ca}^+$ ion confined in a surface-electrode radio-frequency trap in the setup described in [@16Alonso]. The qutrit basis states are represented by three fine-structure levels in a ${}^{40}\text{Ca}^+$ ion: $\ket*{0}=\ket*{S_{1/2}(m_j=-1/2)}$ in the ground-state manifold, and $\ket*{1}=\ket*{D_{5/2}(m_j=-3/2)}$ and $\ket*{2}=\ket*{D_{5/2}(m_j=-1/2)}$ in the metastable $D_{5/2}$ manifold (FIG. \[fig:LevelDiagram\]). The two metastable states have a Zeeman-shifted energy difference $\hbar(\omega_2-\omega_1)=(2\pi\hbar)\,\SI{6.47}{\mega\hertz}$ in an external magnetic field of $B=\SI{0.385}{\milli\tesla}$.
Every experimental sequence starts with of Doppler cooling using a laser red-detuned approximately half a natural linewidth from resonance with the cycling transition between the $S_{1/2}$ and $P_{1/2}$ manifolds, and with close to one saturation intensity [@SuppMat; @16Alonso]. This is followed by of optical pumping to initialize the qutrit to the $\ket*{0}$ state. Subsequently, measurements of the observables $\{A_v\}$ are performed, which consist of coherent rotations between the qutrit states and projective measurements. Coherent rotations are achieved using laser pulses resonant with the transitions between $\ket*{0}$ and $\ket*{1}$ (at $\omega_1$), and between $\ket*{0}$ and $\ket*{2}$ (at $\omega_2$). Matrix representations of the rotations in the Hilbert space spanned by the basis $\qty\big{\ket*{0},\ket*{1},\ket*{2}}$ are given by
\[eq:RotationOperators\] $$\begin{aligned}
R^{(1)}(\theta,\phi)&=\left(\begin{array}{ccc}
\cos(\frac{\theta}{2}) & -{\ensuremath{\mathrm{i}}}{\mathrm{e}^{-{\ensuremath{\mathrm{i}}}\phi}} \sin(\frac{\theta}{2}) & 0 \\[0.5ex]
-{\ensuremath{\mathrm{i}}}{\mathrm{e}^{{\ensuremath{\mathrm{i}}}\phi}} \sin(\frac{\theta}{2}) & \cos(\frac{\theta}{2}) & 0 \\[0.5ex]
0 & 0 & 1
\end{array}\right), \\
R^{(2)}(\theta,\phi)&=\left(\begin{array}{ccc}
\cos(\frac{\theta}{2}) & 0 & -{\ensuremath{\mathrm{i}}}{\mathrm{e}^{-{\ensuremath{\mathrm{i}}}\phi}} \sin(\frac{\theta}{2}) \\[0.5ex]
0 & 1 & 0 \\[0.5ex]
-{\ensuremath{\mathrm{i}}}{\mathrm{e}^{{\ensuremath{\mathrm{i}}}\phi}} \sin(\frac{\theta}{2}) & 0 & \cos(\frac{\theta}{2})
\end{array}\right).\end{aligned}$$
The angles $\theta$ and $\phi$ for a certain rotation (TABLE \[tab:RayAngles\]) are controlled via the duration and phase of the corresponding laser pulse using an acousto-optic modulator. Projective measurements are realized by illuminating the ion for with the same settings used for Doppler cooling [@SuppMat]. If photons are scattered, the qutrit state is projected onto $\ket*{0}$ (“bright state”); if not, the qutrit is projected onto the $D_{5/2}$ manifold (“dark states”), preserving the coherence between $\ket*{1}$ and $\ket*{2}$ (FIG. \[fig:LevelDiagram\]). For the bright / dark states, we register on average / photons through a high-numerical aperture objective on a photomultiplier tube. Thresholding single-shot photon counts at for the detection window allows us to distinguish bright from dark states with an estimated mean detection-error of $<\num{2e-4}$ [@SuppMat].
Testing the SIC inequalities on the Yu-Oh set [@12Yu] requires projective measurements along all 13 rays (FIG. \[fig:YuOhRays\]). By design, the fluorescence detection projects onto either the qutrit state $\ket*{0}$ itself, i.e. the $z_1$ ray, or the plane orthogonal to it, spanned by $\ket*{1}$ and $\ket*{2}$. For any other observable $A_v$, we apply first a unitary rotation $U_v=R^{(2)}\pqty\big{\theta^{(2)}_v,\phi^{(2)}_v}\,R^{(1)}\pqty\big{\theta^{(1)}_v,\phi^{(1)}_v}$, which rotates $v$ onto $z_1$, then fluorescence detection (followed by optical pumping of the $S_{1/2}$ population to $\ket*{0}$), and finally the reverse rotation $U_v^\dagger$ (FIG. \[fig:MeasurementSequence\]). Every measurement of an observable is thus uniquely determined by $v$ and is independent of the context.
Ideally, we would perform a single long series of measurements of randomly chosen observables. In practice, we interrupt the sequence to save collected data and periodically calibrate laser frequencies and pulse times. To sustain the sequence, we take subsequences containing a minimum of 1,000 measurements, which we interrupt when the last detection projected the qutrit onto $\ket*{0}$. The next subsequence then starts by initializing the qutrit to $\ket*{0}$ and applying the rotation $U_{v_0}^\dagger$, with $v_0=v_l$ the last ray from the previous sequence. In this way, all performed measurement sequences can be concatenated up to the 53 million in the present dataset [@SuppMat].
We randomize the sequence of measured observables using a QRNG (model Quantis from ID Quantique SA). It delivers a constant stream of random bits, from which we take groups of four and assign rays $v$ to them (TABLE \[tab:RayAngles\]). The random bits for an observable are created after the detection event of the previous observable (FIG. \[fig:MeasurementSequence\]). In this way, if we acknowledge the randomness of the QRNG, we prevent a hypothetically conspiring ion from knowing what the context of a measurement will be [@88Peres]. Everything from the QRNG output to the pulse sequence programmed in the computer-control system is updated in real time within a time window between unitary rotations.
In a typical sequence of 1 million measurements, we observe between two and five subsequences containing more than dark measurements in a row. In a random sequence of 55 ideal measurements, we would, however, only expect such a set to occur with a probability of $(2/3)^{55}\approx\num{2e-10}$, which corresponds to a probability for it to appear once in the full set of 53 million measurements. We attribute this anomalous effect to off-resonant leakage into the states $\ket*{D_{5/2},m_j=-5/2}$ and $\ket*{D_{5/2},m_j=+1/2}$, which are long-lived dark states outside our computational Hilbert space [@SuppMat]. The control system for the experiment spots these events in real time and breaks, purging the subsequence and starting a new subsequence from the same $v_0$ as was used for the purged subsequence.
Every data point measured for an observable $A_v$ consists of the measurement ray $v$ and an outcome $a=\pm 1$. From the full data set, we collect the numbers $N(A_v{=}a_1)$, $N(A_u{=}a_1,A_v{=}a_2)$, and $N(A_u{=}a_1,A_v{=}a_2,A_w{=}a_3)$, where $A_u$, $A_v$, and $A_w$ are successive measurements in that order, for all $u,v,w \in V$ and all $a_1,a_2,a_3 \in \Bqty{1,-1}$. Based on these numbers, we compute the expectation values
$$\begin{aligned}
\expval*{A_v}&=\frac{\sum_{a_1} a_1 N(A_v{=}a_1)}
{\sum_{a_1} N(A_v{=}a_1)}, \\
\expval*{A_u A_v}&=\frac{\sum'_{a_1,a_2} a_1 a_2 N(A_u{=}a_1,A_v{=}a_2)}
{\sum'_{a_1,a_2} N(A_u{=}a_1,A_v{=}a_2)}, \\
\expval*{A_u A_v A_w}&=\frac{\sum'_{a_1,a_2,a_3} a_1 a_2 a_3 N(A_u{=}a_1,A_v{=}a_2,A_w{=}a_3)}
{\sum'_{a_1,a_2,a_3} N(A_u{=}a_1,A_v{=}a_2,A_w{=}a_3)},\end{aligned}$$
where $\sum'$ additionally sums over all permutations of the argument list of $N$, i.e. the measurement order. Substituting the obtained values (FIG. \[fig:Correlators\]) into the SIC witnesses in Equations , we find $$\begin{aligned}
\expval*{\chi_\text{YO}}=\num{8.279(4)} \qq{and} \expval*{\chi_\text{opt3}}=\num{27.357(11)}.\end{aligned}$$ Our results thus violate Inequalities by 69 and 214 standard deviations. These deviations are solely based on statistical uncertainties, which are small due to the large number of measurements in the complete dataset. We, however, believe that significance of these violations should be penalized according to experimental imperfections and systematic errors [@SuppMat], and elaborate on this issue below.
Our dataset additionally allows for evaluation of the SIC witnesses in Eqs. (\[eq:SICWitnesses\]) based on the “standard approach”, where measurements are repeatedly performed on specifically prepared states of the system. For this, we calculate the averages conditioned on a preceding projection onto one of the states $i \in V$. We do this for all 13 input states and observe violations of the SIC inequalities by at least and standard deviations, respectively [@SuppMat].
Inequalities are satisfied by any theory assuming non-contextuality and their violation indicates contextuality if certain underlying assumptions are satisfied. There are some such assumptions that are untestable, e.g., the assumption that observers have free will for choosing which measurement to make at any time (here implemented with a QRNG). Nevertheless, there are underlying assumptions that are (at least partially) testable. One is the assumption that measurements are sharp, i.e. they are minimally disturbing [@14Chiribella] and their outcomes are the same if performed repeatedly. Note that sharpness implies that measurements are repeatable even when other sharp compatible measurements are performed in between two successive realizations of the same measurement. In quantum theory, sharp measurements are represented by self-adjoint operators; the “ideal measurements” as defined by von Neumann [@32vonNeumann] are sharp measurements. While perfect sharpness can never be fulfilled in a real experiment, we find the repeatability of our measurements (including rotations and projections) to be above $\SI{99.6}{\percent}$ [@SuppMat]. Another assumption is compatibility between the 24 pairs of observables in $E$ (FIG. \[fig:YuOhRays\]b). While compatibility is among the untestable assumptions, one of its consequences are testable: given compatibility, there should be no context signaling in the data. We indeed find no signature of signaling backward in time, and attribute detectable, yet small traces of signaling forward in time to imperfect coherent rotations [@SuppMat]. The large number of measurements comprising our dataset render statistical uncertainties very small and we are able to resolve small systematic deviations from the ideal case in both these measures. We believe these should reflect on our results for the SIC witnesses (Eqs. (\[eq:SICWitnesses\])). But whereas there exist analytical methods to take into account such imperfections for non-contextuality inequalities for scenarios with cyclic systems in which dichotomic observables are measured in only two contexts [@15Kujala], we are not aware of any standard method to account for these imperfections when evaluating SIC witnesses.
In [@SuppMat], we characterize the quantum-vs-classical advantage of this experiment based on the fact that QM predictions for this system cannot be simulated with a classical trit as this would require a classical system with a substantially larger memory. Furthermore, we show that the compatibility structure between observables need not be assumed a priori, but can be inferred from the resulting statistics without invoking QM.
Beyond addressing fundamental aspects of QM, this work demonstrates a system capable of autonomously generating quantum operations, a feature desirable for a prospective quantum computer. The system concatenates hundreds of millions of coherent rotations and projective measurements, rather than repeating a finite sequence which starts with a pre-defined quantum state and consumes a resource at the end of the computation. Such long sequences of QND measurements are interesting in a range of areas including sensing and quantum computing [@13Haroche]. Furthermore, the methods presented in this paper might be generalized to multi-particle quantum systems, providing more powerful tests of fundamental physics [@12Cabello; @16Zhan] and addressing the question of how to optimally generate, certify and make use of quantum contextual correlations.
We thank IdQuantique for the QRNG, and Matt Grau for partial checking of our data analysis and comments on the manuscript. AC thanks Matthias Kleinmann for discussions, and Mile Gu and Zhen-Peng Xu for contributions to the memory estimation in [@SuppMat]. We acknowledge support from the Swiss National Science Foundation under grant no. 200021 134776, ETH Research Grant under grant no. ETH-18 12-2, and from the Swiss National Science Foundation through the National Centre of Competence in Research for Quantum Science and Technology (QSIT). The research is partly based upon work supported by the Office of the Director of National Intelligence (ODNI), Intelligence Advanced Research Projects Activity (IARPA), via the U.S. Army Research Office grant W911NF-16-1-0070. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of the ODNI, IARPA, or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright annotation thereon. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the view of the U.S. Army Research Office. AC acknowledges support from Project No. FIS2014-60843-P, “Advanced Quantum Information” (MINECO, Spain), with FEDER funds, the FQXi Large Grant “The Observer Observed: A Bayesian Route to the Reconstruction of Quantum Theory,” and the project “Photonic Quantum Information” (Knut and Alice Wallenberg Foundation, Sweden).
**Author Contributions:** Experimental data were taken by FML, MM and JA, using an apparatus primarily built up by FML and JA, and with significant contributions from MM, CZ and VN. Data analysis was performed by FML, JA, JPH and AC. The paper was written by FML, JA, AC and JPH, with input from all authors. Experiments were conceived by AC, FML, JA and JPH. The authors declare that they have no competing financial interests.
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{
"pile_set_name": "ArXiv"
}
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abstract: 'We show the occurrence of Dirac, Triple point, Weyl semimetal and topological insulating phase in a single ternary compound using specific symmetry preserving perturbations. Based on [*first principle*]{} calculations, ***k.p*** model and symmetry analysis, we show that alloying induced precise symmetry breaking in SrAgAs (space group P6$_3/mmc$) leads to tune various low energy excitonic phases transforming from Dirac to topological insulating via intermediate triple point and Weyl semimetal phase. We also consider the effect of external magnetic field, causing time reversal symmetry (TRS) breaking, and analyze the effect of TRS towards the realization of Weyl state. Importantly, in this material, the Fermi level lies extremely close to the nodal point with no extra Fermi pockets which further, make this compound as an ideal platform for topological study. The multi fold band degeneracies in these topological phases are analyzed based on point group representation theory. Topological insulating phase is further confirmed by calculating *Z$_2$* index. Furthermore, the topologically protected surface states and Fermi arcs are investigated in some detail.'
author:
- Chiranjit Mondal
- 'C. K. Barman'
- Aftab Alam
- Biswarup Pathak
title: 'Broken symmetry driven topological semi-metal to gapped phase transitions in SrAgAs'
---
@matrix\[1\]\[\*@MaxMatrixCols c\][ - ifnextchar@ifnextchar ]{}
[^1]
[^2]
[***Introduction***]{}:The degrees of band degeneracy near the Fermi level classify the topological gapless phases into three distinct categories; namely Dirac Semimetal (DSM),[@DS-4; @DS-5; @DS-6; @DS-7; @DS-8; @DS-9; @DS-10] Triple point semimetal (TPSM),[@MoP2017; @QWUPRX2016; @ZrTe2016; @MgTa2N32018; @pureMgTa2N3; @InAsSb2016; @HgTe2013; @NexusFermion; @Jianfeng2017; @HWeng2016; @XZhang2017; @GangLi2017] and Weyl Semimetal (WSM).[@WS-1; @WS-2; @WS-3; @WS-4; @WS-5; @WS-6; @WS-7] The DSM and WSM are the low energy excitations of relativistic Dirac and Weyl fermions having four and two fold band degeneracies, respectively. However, the TPSM does not have any high energy analogue in the quantum field theory. The TPSM is considered to be the intermediate state of the other two phases. All these three semi-metals have been theoretically predicted and experimentally verified in various condensed matter systems (crystal) holding appropriate crystalline symmetries. On the other hand, topological gapped states have been studied extensively because of its unique bulk to surface correspondence.[@DS-1; @DS-2; @DS-3] Gradual reduction of the crystalline symmetries removes the band degeneracies which drives the system from one semimetal phase to other semimetal phase and also to gapped states.[@MgTa2N32018; @pureMgTa2N3] For an example, crystals having inversion symmetry with C$_{3v}$ little group along some high symmetry line (HSL) in Brillouin zone (BZ) could provide four fold Dirac node if there is an accidental band crossing on that HSL. Two triply degenerate nodes (TDN) can be formed by simply breaking the inversion symmetry (keeping C$_{3v}$ intact). These pairs of TDN can further be splitted into four Weyl points (WP) when lowering C$_{3v}$ to C$_3$ (breaking vertical mirror plane). Finally, topological insulator (TI) phase could be realized by further breaking the C$_3$ symmetry. Although the theoretical pathway is utterly straightforward but the realization of such phase in a single realistic material is non-trivial. Breaking of crystalline symmetry is usually associated with doping, alloying, strain, pressure or some other external perturbations which often destroy the local chemical environment causing the lifting of degeneracies. As a result, the accidental degeneracies are quite fragile under such perturbations.
Doping or alloying has been proven to be an efficient mechanism to tune the materials electronic properties. The advantage of doping or alloying is that one can break the structural symmetry and remove the band degeneracies easily without much affecting the overall electronic structure with proper choice of dopant and doping site. Also from experimental point of view, these approaches are well established to tailor the properties of the materials. Very recently, alloying mechanism has been applied effectively to tune the topological properties of MgTa$_2$N$_3$.[@MgTa2N32018; @pureMgTa2N3]
In this letter, we report the emergence of various topological semi-metals and topological insulating phase in Cu-doped SrAgAs. We have doped Cu in place of Ag in a precise way such that the doped compound SrAg$_{1-x}$Cu$_x$As holds the required crystalline symmetry to induce different topological phases (DSM, TPSM, TI) for different values of $x$. Unlike previous study[@MgTa2N32018] on a similar theme, we have also discussed the effect of time reversal symmetry (TRS) breaking and curbs of doping towards the realization of WSM phase. The details about the broken symmetry structure are given in supplement (SM)[@supp] (Section I). Another key advantage of the present system over the previously reported one is that, all the Dirac and triple point nodes in the current system lie either close or just below the Fermi level. This can greatly help probes such as photo-emission spectroscopy to locate them easily. The possibility of the experimental realization of the proposed compounds have been discussed in SM [@supp](Section II). We confirm the feasibility of the experimental synthesis of the proposed alloys via phonon calculation. Moreover, we confirm the chemical stability of these alloys by simulating their formation energies (which are $-$ve in sign).
SrAgAs was first synthesized by Albrecht Mewis in 1978[@expt] and it belongs to prototype ZrBeSi-type [@ZrBeSi] hexagonal family which crystallizes in P6$_3/mmc$ (\# 194) space group. The crystal structure (Fig. \[fig1\](a,b)) can be viewed as staffed graphene layers $-$ the Sr$^{2+}$ cations are staffed between \[Ag$^{1+}$As$^{3-}$\]$^{2-}$ honeycomb network. Such structural arrangements further hold the space inversion symmetry. The presence of time reversal symmetry (TRS), center of inversion symmetry and C$_{3v}$ little group along the k$_z$ axis leads to generate four fold Dirac nodes in SrAgAs. Further, in the present work, we replace 50$\%$ Ag atoms by Cu to break the inversion symmetry and hence to realize three component TPSM state. Moreover, 25$\%$ Cu doping at the Ag sites breaks the C$_3$ rotational symmetry transforming to C$_{2h}$ symmetry which, in turn, allows to develop a topological insulating phase in SrAg$_{0.75}$Cu$_{0.25}$As alloy. Furthermore, for SrAg$_{0.75}$Cu$_{0.25}$As, we simulated all possible configurations of Cu dopant sites (see Sec. II of SM[@supp]). Out of them, the structure with the lowest energy configuration (which shows C$_{2h}$ point group symmetry) is chosen to present further results in the manuscript. Nonetheless, all other possible configurations also lead to open up a topologically non-trivial band gap along $\Gamma$-A direction which is ensured by its structural point group symmetry (see Sec. II of SM[@supp] for further detail discussions). Subsequently, we induce a Weyl semimetal phase in the parent SrAgAs by breaking TRS with the application of external magnetic field. The summary of the broken symmetries and the corresponding topological phases have been listed in SM [@supp](Table-S1). It is important to note that both Ag and As holds the equivalent wyckoff positions in the unit cell. So the above symmetry lowering mechanism equivalently holds for the doping at As site as well. We chose to present the detailed results for Cu alloying @ Ag site here. A similar realization of various topological phases by alloying antimony (Sb) @ As site is discussed in detail in SM [@supp](Section III).
We have performed [*first principle*]{} calculation using Vienna Ab Initio Simulation Package (VASP).[@PEBLOCHL1994; @GKRESSE1993; @JOUBERT1999] Other details of the calculations including results based on more accurate exchange correlation functionals are given in SM [@supp](Section IV).
![(Color online) (a) Crystal structure of SrAgAs with (b) Hexagonal Bulk Brillouin zone (BZ) and surface BZ. The high symmetry points are shown in BZ. Bulk band structure of SrAgAs (c) without SOC and (d) with SOC. The red (blue) symbols in (c) represents s (p$_{x}$,p$_y$)-like orbital contribution. $\Gamma_i$s in (d) indicate different irreducible representations of bands. (e) band structure in k$_x$-k$_y$ plane surrounding the Dirac point (K$_\text{D}$ (0, 0, 0.194$\frac{\pi}{c}$)).[]{data-label="fig1"}](Fig1.png){width="\linewidth"}
[***Results and Discussions:***]{} We start with a question; what exactly defines the states of a topological material ? Is it DSM, TPSM, TI or WSM ? The answer lies in its point group symmetry. The central concept is that two bands which are composed with atomic orbitals will cross each other at any k-points in the BZ and the degeneracy of that crossing points will be protected by the site-symmetry group of that points and the band hybridization will be restricted by the group orthogonality relations.[@WS-1] The dimension on the irreducible representation (IR) of the corresponding bands define the particular state of the compound.
Now, coming to our parent compound, SrAgAs has D$_{6h}$ point group which immediately suggests that there is an inversion center along with the C$_{6v}$ little group along $\Gamma$-A HSL. The electronic structure of SrAgAs in the absence of spin-orbit coupling (SOC) is shown Fig. \[fig1\](c). The red and blue symbols in the band structure represent the s- and (p$_{x}$,p$_{y}$)-like orbital character respectively. The topological non-triviality of SrAgAs is dictated by the presence of an inverted band ordering (between s and p$_{xy}$ orbitals) at $\Gamma$ point near the Fermi level (E$_F$), as observed from Fig. \[fig1\](c). Further, inclusion of SOC splits the band degeneracy and open up a gap at $\Gamma$ point as shown in Fig. \[fig1\](d). Under the double group representations in the presence of SOC, the conduction band minima (CBM) at $\Gamma$-point holds $\Gamma_9^-$ (J$_z$= $\pm$ $\frac{3}{2}$) irreducible representations (IRs). On the other hand, highest occupied valence band maxima (VBM) and second highest VBM posses $\Gamma_{7}^-$ (J$_z$= $\pm$ $\frac{1}{2}$) and $\Gamma_{7}^+$ (J$_z$= $\pm$ $\frac{1}{2}$) IRs respectively. Note that, the $\Gamma_7^+$ bands are composed of As-(s)-like orbitals. While $\Gamma_7^-$ and $\Gamma_9^-$ bands are majorly are contributed by As-($p_{x}$,p$_{y}$) orbitals and a very small contribution of Ag-d states. The position of s-like $\Gamma_7^+$ and p-like $\Gamma_7^-$ bands at $\Gamma$ indicate the band inversion in the presence of SOC, hence the non-trivial band order of SrAgAs. Now, along $\Gamma$-A direction $\Gamma_7^{+,-}$ and $\Gamma_9^-$ bands transformed into $\Gamma_7$ and $\Gamma_9$ IRs according to the linking rule of bands. The $\Gamma_7$ and $\Gamma_9$ are the representations of C$_{6v}$ little group. Furthermore, $\Gamma_9$ and $\Gamma_7$ bands near the Fermi level, have similar slope up to K$_D$ along $\Gamma$-A direction. However, the $\Gamma_7$ band (which lies near the E$_F$) suddenly changes its slope and disperse towards the higher energy due to the band repulsion with another $\Gamma_7$ band which is originated from $\Gamma_7^+$. Therefore, $\Gamma_7$ and $\Gamma_9$ bands cross each other at a large momenta K$_D$(0, 0, $\pm$0.194$\frac{\pi}{c}$) and hence a Dirac node has been formed at K$_D$ as shown in Fig. \[fig1\](d). Figure \[fig1\](e) shows the in-plane (k$_x$-k$_y$ plane) band structure with a k$_z$=0.194$\frac{\pi}{c}$. Interestingly, the Dirac nodes in SrAgAs enjoy the double protection against external perturbations. As the material SrAgAs holds both inversion and time reversal symmetry, all the bands are doubly degenerate over the BZ. Besides, all bands along $\Gamma$-A direction are two dimensional IRs, which is ensured by double group representation of C$_{6v}$. Therefore, any perturbations which breaks the inversion symmetry but do not interrupt C$_{6v}$ site-symmetry, will not be able to break the Dirac nodes. The breaking of inversion symmetry only removes the band degeneracies and changes the in-plane velocity of the bands away from the C$_{6v}$-axis. Such a scenario has been realized in non-centrosymmetric P6$_3mc$ (\# 186) space group materials.[@LiZnBi] We have discussed these story-line of breaking various symmetries and their consequences more explicitly using DFT calculation (see Section V of SM[@supp]) as well as model Hamiltonian consideration later in the manuscript.
![ (Color online) (a) Bulk band structure of TPSM phase in SrAg$_{0.5}$Cu$_{0.5}$As alloy. Inset in (a) shows the triply degenerate nodal point TP1(TP2) formed by crossings of $\Lambda_{4}$( $\Lambda_{5}$) and $\Lambda_{6}$ IRs at around K$_{\text{TP}}$. (b,d) shows the three fold band degeneracy at K$_{\text{TP1}}$ (0, 0, 0.180$\frac{\pi}{c}$) and K$_{\text{TP2}}$ (0, 0, 0.183$\frac{\pi}{c}$) for TP1 and TP2 respectively. (d) Band structure of topological insulating SrAg$_{0.75}$Cu$_{0.25}$As. $\Delta_{3,4}$ are the band representations along $\Gamma$-A under C$_S$ point group. Two doubly degenerate bands belong to same IRs ($\Delta_{3}$+$\Delta_{4}$) hybridize at nodal point (K$_{\text{TP}}$) and open up a non-trivial band gap along $\Gamma$-A direction. []{data-label="fig2"}](Fig2.png){width="\linewidth"}
Next, we induce a triple point semi-metallic (TPSM) phase in SrAg$_{1-x}$Cu$_x$As via symmetry allowed Copper (Cu) alloying. The parent compound SrAgAs possess two equivalent positions of Ag-atoms in the unit cell. We replace one of the Ag with a Cu (i.e 50% alloying) to realize TPSM state in SrAgAs. Such alloying breaks the center of inversion, three dihedral mirror planes ($\sigma_d$) and C$_6$ rotational symmetries which in turn deduces the D$_{6h}$ point group symmetry to D$_{3h}$. The little group along the $\Gamma$-A path is now C$_{3v}$ in SrAg$_{1-0.5}$Cu$_{0.5}$As alloy. The C$_{3v}$ point group allows two one-dimensional ($\Lambda_4$ and $\Lambda_5$) and one two-dimensional ($\Lambda_6$) representations in the 2$\pi$ spin rotational sub-space. The electronic structure of SrAg$_{0.5}$Cu$_{0.5}$As alloy with SOC effect is shown in Fig. \[fig2\](a). As we go from parent SrAgAs to SrAg$_{0.5}$Cu$_{0.5}$As compound, the alloying transforms the $\Gamma_7$ bands to $\Lambda_6$ band and $\Gamma_9$ splits into $\Lambda_4$ and $\Lambda_5$ ($\Gamma_9$ $\rightarrow$ $\Lambda_4 \oplus \Lambda_5$) under C$_{3v}$. The intersection of $\Lambda_4$ ($\Lambda_5$) and $\Lambda_6$ leads to form a pair of triply degenerate nodal points on k$_z$-axis as shown in Fig. \[fig2\](a) inset. The in-plane (k$_x$-$k_y$ plane) band structure around the triple points TP1 and TP2 are shown in Fig. \[fig2\](b) and \[fig2\](c) respectively, which further confirm the presence of 3-fold degeneracies as has been predicted via group symmetry analysis (inset of Fig. \[fig2\](a))
Next, we discuss the possible symmetry breaking mechanism to decompose the four fold Dirac nodes into two Weyl nodes. Further breaking of vertical mirror ($\sigma_v$) symmetry in SrAg$_{0.5}$Cu$_{0.5}$As alloy transforms the C$_{3v}$ into C$_3$ point group. Since the IRs of C$_3$ point group are one dimensional, the band crossing of two different IRs is two fold degenerate and hence C$_3$ is an allowed symmetry environment to form a WSM phase. But unfortunately, it is not possible to get such a symmetry environment (C$_3$) via controlled doping engineering in our material SrAgAs. Another allowed symmetry group for Weyl phase is C$_s$ which can be easily achieved under precise doping concentrations. For example, replacing one Ag by Cu atom in a 2$\times$2$\times$2 supercell ensures C$_s$ group in SrAgAs. Yet another route to observe WSM state is to break TRS symmetry. In facts, from the experimental point of view, breaking of TRS is quite easy (compared to breaking point group symmetry) as it can be done by using an external magnetic field. We, therefore, introduce a Zeeman field along the z-direction in the low energy Dirac Hamiltonian which directly transforms the DSM state into WSM (explained in Fig. \[fig3\]).
A (0,0,$\pi$) $\Gamma$ (0,0,0) 3M ($\pi$,0,0) 3L ($\pi$,0,$\pi$) product
-- --------------- ------------------ ---------------- -------------------- ---------
$+$ $-$ $+$ $+$ $-$
: Parity eigenvalues of occupied bands at TRIM points for non-trivial SrAg$_{0.75}$Cu$_{0.25}$As. []{data-label="Table1"}
Further, breaking of three fold rotational symmetry will allow to open up a gap along $\Gamma$-A line at the nodal point. 25% (x=0.25) Cu alloying in SrAgAs opens up a band gap throughout the BZ. It is important to mention here that breaking of C$_3$ rotation can convert the system into either gapped TI phase (for C$_{2v}$ and C$_{2h}$) or WSM (C$_{2}$ & C$_{s}$) state. In our case, 25% Cu alloying restores the structural inversion symmetry with C$_{2h}$ point group. In such symmetry environment, the alloy SrAg$_{0.75}$Cu$_{0.25}$As ensures it’s strong topological insulating phase. To observe the evolution of TI phase, we have taken a 2$\times$2$\times$2 supercell where four out of 16 Ag atoms replaced by Cu atoms. For further study, we considered the energetically most favorable structure of SrAg$_{0.75}$Cu$_{0.25}$As alloy. Most stable crystal structure and the dopant positions are shown in supplement (see Fig. S1(c)).[@supp] The electronic structure of SrAg$_{0.75}$Cu$_{0.25}$As alloy is shown in Fig. \[fig2\](d) which indeed indicates an insulating phase. To further confirm the topological insulating behavior of SrAg$_{0.75}$Cu$_{0.25}$As, we compute the *Z$_2$* index by counting the parity eigen values over the occupied bands in eight time reversal invariant momenta (TRIM) points as given in Table \[Table1\]. The $-$ve products of all parity eigenvalues at eight TRIM points confirm the TI state in SrAg$_{0.75}$Cu$_{0.25}$As alloy with topological index *Z$_2$*=1.
[***Minimal Hamiltonian:***]{} To get a better understanding of the broken symmetry driven various topological phases, we demonstrated a low energy ***k.p*** model Hamiltonian around the $\Gamma$ point. The low energy ***k.p*** Hamiltonian can be derived using method of invariants similar to those used in Na$_3$Bi[@A3Bi2012] and Cd$_3$As$_2$.[@Cd3As22013] Since our [*ab-initio*]{} calculations show that the low energy states are mostly contributed by Sr-s, Ag-s and As-p orbitals, we choose $|S^+_\frac{1}{2},\frac{1}{2}\rangle$, $|P^-_\frac{3}{2},\frac{3}{2}\rangle$, $|S^+_\frac{1}{2},-\frac{1}{2}\rangle$, $|P^-_\frac{3}{2},-\frac{3}{2}\rangle$ as basis sets considering the above atomic like orbitals under inversion, time reversal and D$_{6h}$ symmetry. The superscript $\pm$ in the basis set represents the parity of the states. The 4$\times$4 minimal Hamiltonian around $\Gamma$ using these basis for D$_{6h}$ point group is given by,
$$H(\bf k) = \epsilon_{0}(\bf k) \mathbb{1} + \begin{pmatrix}
M(\bf k) & Ak_{+} & Dk_{-} & -B^{*}(\bf k) \\
Ak_{-} & -M(\bf k) & B^{*}(\bf k) & 0 \\
Dk_{+} & B(\bf k) & M(\bf k) & Ak_{-} \\
-B(\bf k) & 0 & Ak_{+} & - M(\bf k) \\
\end{pmatrix}$$
where $ \epsilon_{0}({\bf k}) = C_0 + C_1k^{2}_{z} + C_2(k^{2}_{x}+k^{2}_{y}),\quad k_{\pm} = k_{x} \pm ik_{y},\quad M({\bf k}) = -M_0 + M_{1}k^{2}_{z} + M_2(k^{2}_{x}+k^{2}_{y})$ with $M_0$, $M_1$, $M_2$ $\textgreater$ 0 to confirm the band inversion. Finite value of parameter D introduces broken inversion symmetry. Therefore, D$=$0 for centro-symmetric Dirac semimetal SrAgAs. The eigenvalues of the above Hamiltonian are,
$$E({\bf k}) = \epsilon_{0}({\bf k}) \pm \sqrt{M({\bf k})^2 + A^2k_+k_- + |B({\bf k})|^2}$$
![(Color online) Low energy band dispersion near the nodal points using [*first principles*]{} based fitting parameters from ***k.p*** model Hamiltonian. (a) Dirac nodes (b) DSM state via breaking of inversion symmetry keeping C$_{6v}$ intact (c) TI states by breaking C$_3$ rotation symmetry and (d) WSM state via breaking of TRS.[]{data-label="fig3"}](Fig3.png){width="\linewidth"}
This eigen value equation gives two gapless solution at [**k**]{}$_d$ = (0, 0, $\pm\sqrt{M_0/M_1}$), which are nothing but the two Dirac nodes on k$_z$ axis. Under three fold rotational symmetry, the off diagonal term B([**k**]{}) takes only the higher order form of B$_3$k$_z$k$_+^{2}$. So, in the vicinity of the Dirac nodes, the higher order terms vanish, i.e., B([**k**]{}) = 0. We have fitted the energy spectrum of the above model Hamiltonian with [*ab-initio*]{} band structure of SrAgAs in the vicinity of Dirac node (Fig. \[fig3\](a)). The fitting parameters are $C_0$=$-$0.047 eV, $C_1$=0.942 eV $\AA^2$, $C_2$=179.209 eV $\AA^2$, $M_0$=0.134 eV, $M_1$=3.534 eV $\AA^2$, $M_2$=228.122 eV $\AA^2$, $A$=5.790 eV $\AA$. As mentioned earlier, the breaking of inversion symmetry still keeps the Dirac node intact on k$_z$ axis because all the bands along the $\Gamma$-A have two dimensional IRs of C$_{6v}$ double group. The breaking of inversion center only removes the band degeneracies and changes the effective mass of bands away from the C$_{6v}$ axis. Figure \[fig3\](b) shows the Dirac node for the inversion breaking term $D$ = 1.0 eV $\AA$. For the case of 25% alloying (SrAg$_{0.75}$Cu$_{0.25}$As), broken C$_3$ rotational symmetry further introduces an additional linear leading order term of B([**k**]{}) = B$_1$k$_z$ into the above Hamiltonian. Now restoring the inversion symmetry (i.e, $D$ = 0) in the Hamiltonian introduces a topological insulating state by opening up a gap at Dirac nodes as shown in Fig. \[fig3\](c).
![(Color online) Projected (100) surface density of states and corresponding Fermi arcs at Fermi level for (a,b) DSM SrAgAs, (d,e) TPSM SrAg$_{0.5}$Cu$_{0.5}$As. (c) Schematic position of Weyl nodes with their chirality. (f) TI surface state projected on (001) surface. Location of Dirac cone is indicated within a box.[]{data-label="fig4"}](Fig4.png){width="\linewidth"}
Now, concentrating at the neighborhood of the Dirac nodes, we can neglect the higher oder terms (i.e B([**k**]{}) = 0) which transform the Hamiltonian into block-diagonal form of the 4$\times$4 Dirac Hamiltonian. The block-diagonal form allows the Dirac Hamiltonian to de-couple it into two 2$\times$2 Weyl Hamiltonian. Further addition of magnetic field breaks the degeneracy of the Weyl nodes and separate them in momentum space. Application of an external magnetic field along z-direction changes the Dirac Hamiltonian as:
H$_{mag}$($\bf k$) = H($\bf k$) + $h \sigma_z \tau_z$
The external magnetic field ($h=0.008$ eV) decouples the Dirac nodes into two Weyl nodes and they appear at [**k**]{}$_{WSM}$ = (0,0, $\pm \sqrt{M_0 \pm h /M_1}$) points on k$_z$-axis as shown in Fig. \[fig3\](d). Schematic diagram of Weyl nodes with their chirality is shown in Fig. \[fig4\](c). Detailed calculation of chiral charges of the Weyl nodes is presented in Sec. VI of SM.[@supp]
[***Surface states:***]{} Now,we address the surface properties of the compounds on their bulk projected surfaces. Fig. \[fig4\](a,b) shows the momentum resolved surface density of states and the Fermi arc of SrAgAs on the (100) surface. Two Fermi arcs originate and terminate into two Dirac nodes on the k$_z$-axis forming a continuous close arc loop on $\bar{\Gamma}$-$\bar{X}$-$\bar{U}$-$\bar{Z}$ plane. Fermi arcs of SrAgAs are somewhat similar to (two super-imposed Weyl arcs arising out of two Weyl nodes coming together and forming a Dirac node in the presence of particular rotational symmetry) of Na$_3$Bi and Cd$_3$As$_2$. Under specific symmetry preserving perturbations, a Dirac node of SrAgAs splits into two triple point nodes in SrAg$_{0.5}$Cu$_{0.5}$As. Therefore, the corresponding two surface states shrink into two isolated triple point nodes as shown Fig. \[fig4\](d). Figure \[fig4\](e) shows the typical TPSM arc generated from the TPSM surface states. Further analysis of the Fermi arc topology for DSM and TPSM are presented in Sec. VII of SM.[@supp] We have also simulated the TI surface states on (001) surface, as shown in Fig. \[fig4\](f). In this case, a non-trivial band gap is opened up using a strain along crystallographic [*a-axis*]{}, mimicking the corresponding symmetry breaking. It is important to note that, in the (001) surface BZ, the $\Gamma$-A high symmetry line falls on $\overline{\Gamma}$ (see supplement Sec. VIII[@supp] for more details).
In general, robustness of the Fermi arcs come from either symmetry or topology or both. Dirac Fermi arcs are the “doubly degenerate Fermi arcs” and its formation mechanism can be understood from the super-imposition of two Weyl nodes of opposite chirality in momentum space. In that sense, four fold Dirac nodes give rise to the “doubly degenerate Fermi arcs”. However, these Fermi arcs are not topologically protected because the Dirac nodes do not have any topological invariant protection.[@PNS2018; @PNS2016; @Mehdi2018] Similar to DSM, TPSM Fermi arcs whose protection comes from the crystalline symmetry, are also not topologically protected. In contrast, Weyl nodes always have a non-trivial chern number associated with them and do not need any crystalline symmetry to appear. As such, WSM arcs are more robust than DSM arcs. Considering Dirac state as the parent state of Weyl and topological insulator, one can deform the close Dirac Fermi arc to a Weyl type open Fermi arcs which have non-trivial chern number. Similarly, it can also be converted to a gapped topological phase which owns a Z$_2$ invariant surface state. Another example is the nodal line semi-metal state, where the nodal loop is protected by crystalline symmetry (in general, mirror reflection) and they have a non-trivial berry phase associated with them. So, a system having a topological invariance is more robust against the external perturbations where as symmetry protected system requires extra care to get its surface signatures.
[***Conclusion***]{}: In conclusion, we predict a single material SrAgAs which can host topologically distinct phases (DSM, TPSM, WSM and TI). Using the appropriate symmetry analysis and [*first principles*]{} calculations, we show that all these distinct topological phases can be realized in SrAgAs via proper alloy engineering. We systematically explain the multi dimensional band degeneracies and phase transition from one to another, within the concept of group theoretical analysis. Further, the non-trivial bulk band signatures of DSM and TPSM have been projected onto the rectangular (100) surface. Our surface analysis indeed show the existence of topological surface states and Fermi arc, originating from the nodal points of DSM and TPSM. We believe that the stoichiometric SrAgAs can serve as a host for all distinct topological phases and hence pave a path for experimentalists to verify the outcomes with appropriate probe. Such discovery of new promising topological candidates using alloy engineering is extremely useful to guide the further design of topological materials.
[***Acknowledgements***]{}: This work is financially supported by DST SERB (EMR/2015/002057), India. We thank IIT Indore for the lab and computing facilities. CKB and CM acknowledge MHRD-India for financial support.
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We consider the convergence of the eigenvalues to the support of the equilibrium measure in the $\beta$ matrix models at criticality. We show a phase transition phenomenon, namely that, with probability one, all eigenvalues will fall in the support of the limiting spectral measure when $\beta>1$, whereas this fails when $\beta<1$.'
author:
- 'C. Fan $^\ddagger$'
- 'A. Guionnet $^\dagger$'
- 'Y. Song $^\S$'
- 'A. Wang$^\sharp$'
bibliography:
- 'BG.bib'
title: 'Convergence of eigenvalues to the support of the limiting measure in critical $\beta$ matrix models'
---
[^1]
**introduction and statement of the result**
============================================
Definitions and Known Results
-----------------------------
Let $\boldsymbol{B}$ be a subset of the real line. $\boldsymbol{B}$ can be chosen as the whole real line, an interval, or the union of finitely many disjoint intervals. For now, let $V:\boldsymbol{B}\rightarrow\mathbb{R}$ be an arbitrary function, and let $\beta>0$ be a positive real number. In this paper, we consider the $\beta$ ensemble, i.e a sequence of $N$ random variables $(\lambda_{1}, \ldots, \lambda_{N})$ with law $\mu_{N, \beta}^{V;\boldsymbol{B}}$ defined as the probability measure on $\boldsymbol{B}^N$ given by $$\label{defi1}
d\mu_{N, \beta}^{V;\boldsymbol{B}}(\lambda)=\frac{1}{Z_{N, \beta}^{V;\boldsymbol{B}}}\prod_{i=1}^{N}d\lambda_{i}e^{-\frac{N\beta}{2}V(\lambda_{i})}\mathbf{1}_{\boldsymbol{B}}(\lambda_{i})\prod_{1\leq i<j\leq N}\left|\lambda_{i}-\lambda_{j}\right|^{\beta},$$ where $Z_{N, \beta}^{V;\boldsymbol{B}}$ is the partition function $$\label{defi2}
Z_{N, \beta}^{V;\boldsymbol{B}}=\int_{\mathbb{R}}\cdots\int_{\mathbb{R}}\prod_{i=1}^{N}d\lambda_{i}e^{-\frac{N\beta}{2}V(\lambda_{i})}\mathbf{1}_{\boldsymbol{B}}(\lambda_{i})\prod_{1\leq i<j\leq N}\left|\lambda_{i}-\lambda_{j}\right|^{\beta}.$$
If $\beta$ is equal to 1, 2, or 4, $\mu_{N, \beta}^{V;\mathbb{R}}$ is the probability measure induced on the eigenvalues of $\Omega$ by the probability measure $d\Omega e^{-\frac{N\beta}{2}\mathrm{Tr}(V(\Omega))}$ on a vector space of real symmetric, Hermitian, and self-dual quaternionic $N\times N$ matrices respectively, see [@meh].\
Therefore, the $\beta$ models can be viewed as the natural generalization of these matrix models and we will refer to $\lambda_{i}$ as “eigenvalue” of a “matrix model”. For a quadratic potential, the $\beta$ ensembles can also be realized as the eigenvalues of tridiagonal matrices [@DE02]. Even though such a construction is not known for general potentials, $\beta$ matrix models are natural Coulomb interaction probability measures which appear in many different settings. These laws have been intensively studied, both in physics and in mathematics. In particular, the convergence of the empirical measure of the $\lambda_i$’s (which we will call hereafter the spectral measure) was proved [@SaffTotik; @Defcours; @AGZ], and its fluctuations analyzed [@Johansson98; @Pasturs; @Shch]. Moreover, the partition functions as well as the mean Stieltjes transforms can be expanded as a function of the dimension to all orders [@bp78; @ACKMe; @ACM92; @Ake96; @CE06; @Ch06; @BG1; @BG2]. It turns out that both central limit theorems and all order expansions depend heavily on whether the limiting spectral measure has a connected support. Indeed, when the limiting spectral measure has a disconnected support, it turns out that even though most eigenvalues will stick into one of its connected components, some eigenvalues will randomly switch from one to the other connected components of the support even at the large dimension limit. This phenomenon can invalidate the central limit theorem, see e.g. [@Pasturs; @Shch], and results with the presence of a Theta function in the large dimension expansion of the partition function [@BG2]. In the case where the limiting measure has a connected support $\boldsymbol{S}$, and that the eigenvalues are assumed to belong asymptotically to $\boldsymbol{S}$, even more refined information could be derived. Indeed, in this case, local fluctuations of the $\lambda_i$’s could first be established in the case corresponding to Gaussian random matrices, $\beta=1, 2$ or $4$ and $V(x)=x^2$ [@meh], then to tridiagonal ensembles (all $\beta\ge 0$ but $V(x)=x^2$) [@RRV06] and more recently for general potentials and $\beta\ge 0$ [@BEY1; @BEY2; @BEY3; @Shch; @BFG]. However, all these articles consider non-critical potentials. We shall below define more precisely the later case but let us say already that a non-critical potential prevents the eigenvalues to deviate from the support of the limiting spectral measure as the dimension goes to infinity. We study in this article $\beta$ models with critical potentials and whether the eigenvalues stay confined in the limiting support. In fact, we exhibit an interesting phase transition: we show that if $\beta>1$ the eigenvalues stay confined whereas if $\beta<1$ some deviate towards the critical point with probability one. We postpone the study of the critical case $\beta=1$ to further research. Let us finally point out that the case where the potential is critical, but with critical parameters tuned with the dimension so that new phenomena occur, was studied in [@tb; @eynardbirth]. We restrict ourselves to potentials independent of the dimension.
We next describe more precisely the definition of criticality and our results.
Consider the *spectral measure* $L_{N}:=\frac{1}{N}\sum_{i=1}^{N}\delta_{\lambda_{i}}$, where $\delta_{\lambda_{i}}$ is the Dirac measure centered on $\lambda_{i}$. $L_N$ belongs to the set $M_1(\boldsymbol{B})$ of probability measures on the real line. We endow this space with the weak topology. Then, $L_N$ converges almost surely. This convergence can be derived from the following large deviation result (see [@BAG], and [@AGZ Theorem 2.6.1]) :
\[large-deviation\] Assume that $V$ is continuous and goes to infinity faster than $2\log|x|$ (if $\boldsymbol{B}$ is not bounded). The law of $L_{N}$ under $\mu_{N,\beta}^{V; \boldsymbol{B}}$ satisfies a large deviation principle with speed $N^{2}$ and good rate function $\tilde{\mathcal{E}}$, where $ \tilde{\mathcal{E}}={\mathcal {E}}-\inf\{ {\mathcal {E}}(\mu), \mu\in M_1(\boldsymbol{B})\}$ with $${\mathcal{E}}[\mu]=\frac{\beta}{4} \iint \left( V(\xi)+V(\eta) -2\log\left|\xi-\eta\right|\right)d\mu(\xi)d\mu(\eta)\,.$$ In other words,
1. $$\lim_{N\rightarrow\infty}\frac{1}{N^2}\log Z_{N,\beta}^{V;\boldsymbol{B}}=-\inf_{\mu\in M_1(\boldsymbol{B})} \mathcal E[\mu]\,.$$
2. $\tilde{\mathcal{E}}:M_{1}(\mathbb{R})\rightarrow[0, \infty]$ possesses compact level sets $\{v:\tilde{\mathcal{E}}(v)\leq M\}$ for all $M\in\mathbb{R}^{+}$.
3. for any open set $O\subset M_{1}(\boldsymbol{B})$, $$\liminf_{N\rightarrow\infty}\frac{1}{N^{2}}\log\mu_{N, \beta}^{V;\boldsymbol{B}}\left(L_{N}\in O\right)\geq-\inf_{O}\tilde{\mathcal{E}}.$$
4. for any closed set $F\subset M_{1}(\boldsymbol{B})$, $$\liminf_{N\rightarrow\infty}\frac{1}{N^{2}}\log\mu_{N, \beta}^{V;\boldsymbol{B}}\left(L_{N}\in F\right)\leq-\inf_{F}\tilde{\mathcal{E}}.$$
The minimizers of $\mathcal{E}$ are described as follows (see [@AGZ Lemma 2.6.2]):
\[t2\] $\mathcal{E}$ achieves its minimal value at a unique minimizer $\mu_{\mathrm{eq}}$. Moreover, $\mu_{\rm eq}$ has a compact support $\boldsymbol{S}$. In addition, there exists a constant $C_V$ such that: $$\label{theeq}
\begin{cases}
\mathrm{for}\; x\in \boldsymbol{S} & 2\int_{\mathbb{R}}d\mu_{\mathrm{eq}}(\xi)\ln\left|x-\xi\right|-V(x)=C_V\\
\mathrm{for}\; x\;\mathrm{ Lebesgue\; almost\; everywhere\; in\;} \boldsymbol{S}^c & 2\int_{\mathbb{R}}d\mu_{\mathrm{eq}}(\xi)\ln\left|x-\xi\right|-V(x)<C_V.
\end{cases}$$
We will refer to $\mu_{\rm eq}$, which is compactly supported, as the equilibrium measure.
\[weak convergence\] Theorem \[large-deviation\] and Theorem \[t2\] imply that under $\mu_{N, \beta}^{V;\boldsymbol{B}}$, $L_{N}$ converges to the equilibrium measure $\mu_{\rm eq}$ almost surely.
Once the existence of the equilibrium measure is established, one may explore the convergence of the eigenvalues to the support of the equilibrium measure $\mu_{eq}$. It is shown in [@BG1; @BG2] that the probability that eigenvalues escape this limiting support is governed by a large deviation principle with rate function given by $$\mathcal{\tilde{\mathcal{J}}}^{V;\boldsymbol{B}}(x)=\mathcal{J}^{V;\boldsymbol{B}}(x)+C_V$$ with $$\label{jjj}
\mathcal{J}^{V;\boldsymbol{B}}(x)=\begin{cases}
V(x)-2\int d\mu_{\mathrm{eq}}(\xi)\ln\left|x-\xi\right| & x\in\boldsymbol{B}\backslash \boldsymbol{S}\\
-C_{V} & \mathrm{otherwise}.
\end{cases}$$ The large deviation principle states as follows:
\[original-thm\] Assume $V$ continuous and going to infinity faster than $2\log|x|$ (in the case where $\boldsymbol{B}$ is not bounded). Then
1. $\tilde{\mathcal{J}}^{V;\boldsymbol{B}}$ is a good rate function.
2. We have large deviation estimates: for any $\mathsf{F} \subseteq \overline{\boldsymbol{B}\backslash \boldsymbol{S}}$ closed and $\mathsf{O} \subseteq \boldsymbol{B}\backslash \boldsymbol{S}$ open, $$\begin{aligned}
\limsup_{N\rightarrow\infty}\frac{1}{N}\ln \mu^{V;\mathsf{B}}_{N,\beta}\left[\exists i\quad\lambda_i \in \mathsf{F}\right] & \le & -\frac{\beta}{2}\,\inf_{x \in \mathsf{F}} \tilde{ \mathcal{J}}^{V;\boldsymbol{B}}(x), \\
\liminf_{N\rightarrow\infty}\frac{1}{N}\ln \mu^{V;\mathsf{B}}_{N,\beta}\left[\exists i\quad\lambda_i \in \mathsf{O}\right] & \ge & -\frac{\beta}{2}\,\inf_{x \in \mathsf{O}} \tilde {\mathcal{J}}^{V;\boldsymbol{B}}(x).\end{aligned}$$
The last theorem shows that the support of the spectrum is governed by the minimizers of $\tilde{\mathcal{J}}^{V;\boldsymbol{B}}$ .
Assume $V$ is continuous. We say that $V$ is [*non-critical*]{} iff [$\tilde{\mathcal{J}}^{V;\boldsymbol{B}}$]{} is positive everywhere outside of the support of $\mu_{\rm eq}$.
Theorem \[t2\] only ensures [$\tilde{\mathcal{J}}^{V;\boldsymbol{B}}$]{} is positive **almost** everywhere outside of the support of $\mu_{\rm eq}$.
In the literature, the potential $V$ is also said to be critical when the density of the equilibrium measure vanishes at an interior point or at an edge point at a fast rate. We will assume that our potential is *[not]{} critical in this sense (see the fourth point in Assumption \[main-assume\]).*
A consequence of the second part of the aforementioned theorem is the following:
\[original-cor\] Let the assumptions in Theorem \[original-thm\] hold. Assume that $V$ is non-critical. Then $$\label{re}
\lim_{N\rightarrow\infty}\mu_{N, \beta}^{V;\boldsymbol{B}}\left(\exists \lambda_{i}\notin \boldsymbol{A}\right)=0\,,$$ for any open set $\boldsymbol{A}$ containing the support of $\mu_{\rm eq}$.
\[rem3\] Since the law of the eigenvalues satisfies a large deviation principle with rate N, the eigenvalues actually converge to the support exponentially fast (or more precisely, $\exists c >0, $ s.t, $\mu_{N, \beta}^{V;\boldsymbol{B}}\left(\exists \lambda_{i}\notin \boldsymbol{A}\right)\leq e^{-c N}$ for any open set $\boldsymbol{A}$ containing the support of $\mu_{\rm eq}$ ).
\[imprem\] By the definition of the partition function, $1-\mu_{N, \beta}^{V;\boldsymbol{B}}\left(\exists \lambda_{i}\notin \boldsymbol{A}\right)=\frac{Z_{N, \beta}^{V;\boldsymbol{A}}}{Z_{N, \beta}^{V;\boldsymbol{B}}}$, thus, $$(\ref{re})\Leftrightarrow \lim_{N\rightarrow \infty}\frac{Z_{N, \beta}^{V;\boldsymbol{A}}}{Z_{N, \beta}^{V;\boldsymbol{B}}}=1.$$
In the rest of this article we investigate what happens in the case where $V$ is critical. This investigation will require the uses of precise estimates on $\beta$ models partitions functions derived in [@BG1; @BG2] and to apply these results we shall make the following assumption :
\[main-assume\]
- $V:\boldsymbol{B}\rightarrow\mathbb{R}$ is a continuous function independent of $N$.
- If $\pm\infty\in\boldsymbol{B}$, $$\liminf_{x\rightarrow\pm\infty}\frac{V(x)}{2\ln\left|x\right|}>1.$$
- $\mathrm{supp}\left(\mu_{\mathrm{eq}}\right)$ is a finite union of disjoint intervals, i.e. $\mathrm{supp}\left(\mu_{\mathrm{eq}}\right)$ of the form $\boldsymbol{S}=\bigcup_{h=1}^{g}\boldsymbol{S}_{h}$, where $\boldsymbol{S}_{h}=[\alpha_{h}^{-}, \alpha_{h}^{+}]$.
- Let $\boldsymbol{B}=\cup_h [b_h^-,b_h^+]$ with $b_h^-\le\alpha_h^-\le\alpha_h^+\le b_h^+$ and set $ \mathrm{Hard}=\{a\in\cup\{\alpha_h^-,\alpha_h^+\}:\alpha_h^\pm=b_h^\pm\}$ and $\mathrm{Soft}=\cup\{\alpha_h^-,\alpha_h^+\}\backslash \mathrm{Hard}$. Then we assume that $$S(x)=\pi\frac{d\mu_{\mathrm{eq}}}{dx}\sqrt{\left|\frac{\prod_{a\in\mathrm{Hard}}(x-a)}{\prod_{a\in\mathrm{Soft}}(x-a)}\right|}.$$ is strictly positive whenever $x\in\boldsymbol{S}$.
- V is a real analytic function in some open neighborhood ${\boldsymbol{A}}$ of $\boldsymbol{S}$ : ${\boldsymbol{A}}= \cup_{h=1}^g \boldsymbol{A}_h$, $\boldsymbol{A}_h=(a_h^-,a_h^+)$ for some $a_h^-<\alpha_h^-<\alpha_h^+<a_h^+$, and $A_{h},A_{h'}$ are disjoint for any $h\neq h'$.
When $V$ is analytic in a neighborhood of the real line, the third point of our assumption is automatically satisfied. Here, we assume analyticity only in a neighborhood of $\boldsymbol{S}$.
Hereafter the neighborhood ${\boldsymbol{A}}$ will be fixed, but clearly can be chosen as small as wished, being given it is open and containing $\boldsymbol{S}$.
We want to investigate whether (\[re\]) still holds when the restriction on $\tilde{\mathcal{J}}^{V;\boldsymbol{B}}$ is weakened so that it vanishes outside the support $\boldsymbol{S}$. Our working hypothesis will be the following:
\[as\] With ${\boldsymbol{A}}$ given in Assumption \[main-assume\], assume that $\tilde {\mathcal{J}}^{V;\boldsymbol{B}}$ vanishes only on the support of the equilibrium measure $\boldsymbol{S}$ and at one point $c_{0}$ in $\overline{{\boldsymbol{A}}}^c$. We also require that $V$ (and therefore $ \tilde {\mathcal{J}}^{V;\boldsymbol{B}}$) extends as a twice continuously differentiable function in a open neighborhood $(c_0-\varepsilon, c_0+\varepsilon)$ of $c_0$, for some $\varepsilon>0$ and that $\frac{d^{2}}{dx^{2}} \tilde {\mathcal{J}}^{V;\boldsymbol{B}}(c_{0})>0$. Moreover, for technical reason, we require $\frac{d^{2}}{dx^{2}} V\geq\sigma>0$ on $\boldsymbol{A}$.
\[remsup\] Because $\tilde{\mathcal{J}}^{V; \boldsymbol{B}}$ is a good rate function which is positive outside $\boldsymbol{A}\cup (c_{0}-\epsilon, c_{0}+\epsilon)$ for any $\epsilon>0$ small enough under our assumptions, Theorem \[original-thm\] implies that $\mu_{N, \beta}^{V;\boldsymbol{B}}\left(\exists \lambda_{i}\notin \boldsymbol{A}\cup(c_{0}-\epsilon,c_{0}+\epsilon)\right) $ goes to zero exponentially fast. In other words, it is bounded above by $e^{-Nc_{\epsilon}}$ for some $c_{\epsilon}>0$.
[ Potentials $V$ satisfying [Assumptions \[main-assume\] and \[as\]]{} are easy to build. Indeed, being given a probability measure $\mu$ so that $x\rightarrow f(x)=2\int\log|x-y|d\mu(y)$ is well defined and continuous on the whole real line, we simply choose $V$ to be equal to $f$ on the support $S$ of $\mu$ and Lebesgue almost surely strictly greater than $f$ outside $\boldsymbol{S}$. This insures that $\mu$ is the equilibrium measure of our $\beta$-model with potential $V$ as it satisfies . We can then choose $V-f$ strictly positive outside $\boldsymbol{S}$ except at the point $c_0$ where it is strictly convex. To make sure that $V$ also satisfies Assumption \[main-assume\], we can take $\mu$ to be the equilibrium measure for a potential $W$ which is real-analytic, going to infinity faster than $2\ln |x|$, strictly convex in a neighborhood of $\boldsymbol{S}$, and non-critical (in the one cut case, we can take $W$ strictly convex everywhere). We let ${\boldsymbol{A}}=\cup_{h=1}^g (a_h^-,a_h^+)$ be a bounded, open neighborhood of the support of $\mu$. We choose $V=W+C_{W}$ on $ (-\infty,a_g^+)$. We may assume without loss of generality that $\tilde{\mathcal J}^{V;\boldsymbol{B}} (a_g^+)$ is strictly positive. We then choose for $x>a_g^+$, $V(x)-f(x)=d(x-c_0)^2$ for some $d>0$ and $c_0>a_g^+$ so that $d(a_g^+-c_0)^2=\tilde{\mathcal J}^{W;\boldsymbol{B}}(a_g^+)$. We have constructed an equilibrium measure $\mu$ for a potential $V$ satisfying Assumptions \[main-assume\] and \[as\]. ]{}
Main Results
------------
\[main\] Given Assumptions \[main-assume\] and \[as\], and with ${\boldsymbol{A}}$ as in assumption \[main-assume\], $c_0,\varepsilon$ as in Assumption \[as\], we have the following alternative :
- when $\beta>1$, $$\lim_{N\rightarrow\infty}\mu_{N, \beta}^{V;\boldsymbol{B}}\left(\exists \lambda_{i}\notin \boldsymbol{A}\right)=0,$$
- when $\beta<1$, $$\lim_{N\rightarrow\infty}\mu_{N, \beta}^{V;\boldsymbol{B}}\left(\exists \lambda_{i}\notin \boldsymbol{A}\right)=1.$$
[Equivalently, for any $\epsilon\in (0,\varepsilon)$, the probability that there exists an eigenvalue in $(c_0-\epsilon,c_0+\epsilon)$ goes to zero when $\beta>1$ and to one when $\beta<1$. ]{}
The behavior below $\beta=1$ can be illustrated with the case $\beta=0$ where one would consider a potential $V$ vanishing on a support $\boldsymbol{S}$ and at a point $c_0$ (where its second derivative is positive), being strictly positive everywhere else. This corresponds to $N$ independent variables with probability of order $N^{-1/2}$ to belong to a small neighborhood of $c_0$ (where the latter probability can be estimated by Laplace method). However, we will show in this article that this weight has to be corrected by a term $N^{-\frac{\beta}{2}}$ by studying the precise estimates derived in [@BG1; @BG2]. Essentially, digging back into the latter estimates, one can realize that these corrections come from Selberg integral and in fact is due to the Coulomb gas interaction. In this case, it is clear that some eigenvalues will lie in the neighborhood of $c_0$ with positive probability as soon as $\beta+1<2$. The existence of a phase transition for this phenomenon at $\beta=1$ is new to our knowledge. It suggests that the support of the eigenvalues of matrices with real coefficients corresponding to $\beta=1$ matrix models might be more sensible to perturbations of the potential than matrices with complex coefficients (corresponding to $\beta=2$). This is however not supported by finite dimensional perturbations of the Wigner matrices since the BBP transition [@BBP] does not vary much between these two cases. Let us observe that our arguments could be carried similarly with several critical points similar to $c_0$ without changing the phase transition. However, if the second derivative of $ {\mathcal{J}^{V;\boldsymbol{B}}}$ at these critical points could vanish so that $ {\mathcal{J}}^{V;\boldsymbol{B}}$ behaves as $|x-c_0|^q$ in the vicinity of $c_0$ for some $q> 2$, the phase transition would occur at a threshold $\beta_q$ depending on $q$ (see Remark \[beha\]).
Structure of the paper
----------------------
In Section 2 we reduce the problem to the analysis of the probability that M eigenvalues are contained in a small neighborhood of $c_{0}$ while the rest of the $N-M$ eigenvalues are contained in $\boldsymbol{A}$ and state the main proposition, Proposition \[mainprop\], which give precise estimates of this probability. We deduce our main result Theorem \[main\] in the case $\beta>1$ in Section \[betamore\] and the case $\beta<1$ in Section \[betaless\]. Section \[proofprop1\] is devoted to the proof of Proposition \[mainprop\], which we first give in the case where the equilibrium measure has a connected support and then extend to the general case. The appendix contains precise concentration of measures results which are key to our estimates.
Notation
--------
We use the notation $X\lesssim Y$ (resp. $X\gtrsim Y$) to denote $X\leq CY $ (resp. $X\geq CY$) for some universal constant $C$ . $X\approx Y$ when both $X\lesssim Y$ and $X\gtrsim Y$ hold. We sometimes use $a\ll 1$ to denote that $a$ is smaller than any universal constant involved in the proof.
**Preliminary and Basic analysis**
==================================
The probability that a specific subset of $M$ eigenvalues are contained in a small neighborhood $(c_0-\epsilon,c_0+\epsilon)$ of $c_0$ while the other $N-M$ eigenvalues are contained in $\boldsymbol{A}$ shall be denoted by $\Phi_{N, M, \beta}^{V;{\epsilon}}$ : $$\label{main3}
\Phi_{N, M, \beta}^{V;{\epsilon}}:=\mu_{N, \beta}^{V;\boldsymbol{B}}\left(\lambda_{N-M+1}, ..., \lambda_{N}\in (c_{0}-\epsilon, c_{0}+\epsilon), \lambda_{1}, ...\lambda_{N-M}\in\boldsymbol{A}\right).$$ [[ $\Phi_{N, M, \beta}^{V;{\epsilon}}$ depends also on ${\boldsymbol{A}}$ and $\boldsymbol{B}$ but it will be fixed hereafter as in Assumption \[main-assume\] so that we do not stress this dependency. $\epsilon$ will later be chosen small enough, but notice that the conclusion will not depend on this choice since the probability that eigenvalues go in $[c_0-\varepsilon,c_0+\varepsilon]\backslash [c_0-\epsilon,c_0+\epsilon]$ goes to zero as $N$ goes to infinity for any $\epsilon>0$ by the large deviation principle Theorem \[original-thm\] and Assumption \[as\].]{}]{} The key to prove our main result is to compute the speed at which $\Phi_{N, M, \beta}^{V;{\epsilon}}$ goes to zero as $N$ goes to +$\infty$. Indeed, note that $$\begin{aligned}
\quad\mu_{N, \beta}^{V;\boldsymbol{B}}\left(\exists \lambda_{i} \notin \boldsymbol{A}\right)
&=& \mu_{N, \beta}^{V;\boldsymbol{B}}\left(\exists \lambda_{i} \notin \boldsymbol{A}\cup (c_{0}-\epsilon, c_{0}+\epsilon)\right) \nonumber\\
&+&\sum_{M>\delta N} \left(\begin{array}{c}
N\\
M
\end{array}\right){\Phi}^{V;{\epsilon}}_{N, M, \beta}\nonumber\\
&+&\sum_{1\le M \leq \delta N}\left(\begin{array}{c}
N\\
M
\end{array}\right){\Phi}^{V;{\epsilon}}_{N, M, \beta}\nonumber\\
&=:&P_{1}+P_{2}+P_{3}.\label{finalstep}\end{aligned}$$ Here $\delta>0$ is a small fixed constant [which will be chosen later]{}.
Since $\tilde{\mathcal{J}}^{V; \boldsymbol{B}}$ is a good rate function which is positive outside $\boldsymbol{A}\cup [c_{0}-\epsilon, c_{0}+\epsilon]$, Theorem \[original-thm\] implies that for any fixed $\epsilon>0$, $P_{1}$ approaches 0 exponentially fast. In other words, it is controlled by $e^{-Nc_{\epsilon}}$ for some $c_{\epsilon}>0$.
$P_2$ is bounded above by the probability that there are at least $\delta N$ eigenvalues close to $c_0$. This implies that the empirical measure $L_{N}$ must put a mass $\delta$ in $(c_0-\epsilon,c_0+\epsilon)$, so that it must be at a positive distance of $\mu_{\rm eq}$ since $(c_0-\epsilon,c_0+\epsilon)\cap\boldsymbol{S}=\emptyset$. By the large deviation principle for the law of $L_{N}$ described in Theorem \[large-deviation\], $P_{2}$ is bounded above by $e^{-{c_{\delta}N^{2}}}$ for some $c_{\delta}>0$. Therefore we deduce that for any $\delta,\varepsilon >0$, there exists $c(\delta,\epsilon)>0$ such that $$\label{finalstep2}
\mu_{N, \beta}^{V;\boldsymbol{B}}\left(\exists \lambda_{i} \notin \boldsymbol{A}\right)= P_{3} +O(e^{-c(\delta,\epsilon)N}).$$ Our goal, therefore, is to control the third term $P_{3}$.
Since $\mathcal{J}^{V;\boldsymbol{B}}$ goes to infinity at infinity, Theorem \[original-thm\] also shows that the probability to have an eigenvalue above some finite threshold goes to zero exponentially fast. Therefore, we may assume without loss of generality that $\boldsymbol{B}$ is a bounded set.
Thus, we are left to analyze ${\Phi}^{V;{\epsilon}}_{N, M, \beta}$ for a bounded set $\boldsymbol{B}$.
We prove the following bounds in section \[proofprop\]:
\[mainprop\] \[multi-eig-thm\] Let Assumptions \[main-assume\] and \[as\] hold. Then, there exist $c, \delta_0,\epsilon_0>0$ so that for $\delta\in (0,\delta_0)$, $\epsilon\in (0,\epsilon_0\wedge \varepsilon)$, we have uniformly in $M\le \delta N$ $$\label{ub}
\Phi_{N, M, \beta}^{V;{\epsilon}}\lesssim \frac{1}{N^{\frac{M(\beta+1)}{2}}} +O(e^{-c N^2}).$$ On the other hand, $$\label{lb}
\frac{1}{N^{\frac{(\beta+1)}{2}}}\frac{Z_{N,\beta}^{V; \boldsymbol{ A}}}{Z_{N,\beta}^{V;\boldsymbol{B}}}\lesssim \Phi_{N, 1, \beta}^{V;{\epsilon}}+{O(e^{-c N^{2}})}.$$
[If one assumes $\boldsymbol{A}$ is connected, Proposition \[mainprop\] follows from the calculation from [@BG1] where the one-cut case in considered. In the general multi-cut case, the proof of Proposition \[mainprop\] is based on the precise estimate derived in [@BG2] for the partition function and correlators for fixed filling fraction measure, that is with given number of eigenvalues in each connected part of the support $\boldsymbol{S}$. One could also consider the more general multi-cut case where the potential $V$ is replaced by a non-linear statistic and then use similar estimates derived in [@BGK].]{}
We next give the proof of our main result.
Convergence of the eigenvalues to the support $\boldsymbol{S}$ when $\beta>1$ {#betamore}
=============================================================================
We next prove the first half of our main Theorem \[main\]. To this end, we use the upper bound provided by Proposition \[mainprop\] to find $$P_{3}\leq \sum_{1\le M\le \delta N}\left(\begin{array}{c}
N\\
M
\end{array}\right)\Phi_{N,M, \beta}^{V;{\epsilon}} \lesssim \left( 1+N^{-\frac{(\beta+1)}{2}}\right)^N-1\,.$$ where we used that any error of order $e^{-cN^2}$ in $\Phi_{N, M, \beta}^{V;{\epsilon}}, 1\le M\le \delta N,$ is neglectable in the above sum. Hence, when $\beta>1$, $P_{3}$ goes to $0$ as $N$ goes to $+\infty$. We deduce by that $$\lim_{N\to\infty} \mu_{N, \beta}^{V;\boldsymbol{B}}\left(\exists \lambda_{i} \notin \boldsymbol{A}\right)=0\,.$$ Moreover, this is equivalent to the fact that $ \mu_{N, \beta}^{V;\boldsymbol{B}}\left(\exists \lambda_{i} \in (c_0-\epsilon, c_0+\epsilon)\right)$ goes to zero by Remark \[remsup\].
Escaping eigenvalues when $\beta<1$ {#betaless}
====================================
We prove that when $\beta<1$ the probability that no eigenvalues lies in the neighborhood of $c_0$ goes to zero, that is by Remark \[imprem\], that we have: $$\label{eod}
\lim_{N\rightarrow \infty}\frac{Z_{N, \beta}^{V;\boldsymbol{A}}}{Z_{N, \beta}^{V; \boldsymbol{B}}}=0.$$ This is done by lower bounding the probability $p_{N,\beta}^{V}$ that one eigenvalue exactly lies in the neighborhood of $c_0$. Indeed, as $A_i=\{\lambda_i\in (c_0-\epsilon,c_0+\epsilon), \lambda_j\in \boldsymbol{A}, j\neq i\}$ are disjoint as soon as $(c_0-\epsilon,c_0+\epsilon)\cap \boldsymbol{ A}=\emptyset$, we have by symmetry $$p_{N,\beta}^V=N\Phi_{N, 1, \beta}^{V;{\epsilon}}.$$ Since $p_{N,\beta}^V\le 1$, we deduce from that $$\frac{N}{N^{\frac{(\beta+1)}{2}}}\frac{Z_{N,\beta}^{V; \boldsymbol{A}}}{Z_{N,\beta}^{V;\boldsymbol{B}}}
\lesssim 1+N\times {O(e^{-c N^{2}})}\,,$$ which results with $$\frac{Z_{N,\beta}^{V; \boldsymbol{A}}}{Z_{N,\beta}^{V;\boldsymbol{B}}}\lesssim N^{\frac{\beta-1}{2}}\,,$$ so that , and therefore the second part of our main Theorem \[main\], follows.
We first prove Proposition \[mainprop\] in the case where $\mu_{\rm eq}$ has a connected support (the one cut case) where our proof is based on the expansion of partition functions obtained in the one cut case in [@BG1], and then turn to the more delicate general case (multi-cut case) which is based from estimates from [@BG2].
[Proof of Proposition \[mainprop\] ]{} {#proofprop1}
======================================
We start with the explicit formula for $\Phi_{N, M, \beta}^{V; {\epsilon}}$ which can be written as follows : $$\begin{aligned}
\Phi_{N,M,\beta}^{V;{\epsilon}}
&=&\frac{Z_{N-M,\beta}^{\frac{N}{N-M} V; \boldsymbol{A}}}{Z_{N, \beta}^{V;\boldsymbol{B}}}
\int_{[c_0-\epsilon,c_0+\epsilon]^M} \Xi(\eta_1,\ldots, \eta_M) \prod_{1\le k<l\le M} |{\eta_k-\eta_l}|^\beta
\prod_{j=1}^M e^{-\frac{\beta M}{2} V(\eta_j)} d\eta_j \nonumber\end{aligned}$$ where
$${\Xi(\eta_1, \cdots, \eta_M):=\mu^{\frac{N}{N-M}V;\boldsymbol{A}}_{N-M, \beta}\left(\prod_{j=1}^{M}e^{\beta\sum_{i=1}^{N-M}\ln|\eta_{j}-\lambda_{i}|-\frac{\beta}{2}(N-M)V(\eta_{j})}\right)}$$ The term $\prod_{1\leq k<l\leq M}|\eta_{k}-\eta_{l}|^{\beta}$ is bounded by $(2\epsilon)^{\frac{\beta M(M-1)}{2}}$. Thus, we have: $$\label{main4}
\Phi_{N, M, \beta}^{V;{\epsilon}}\leq (2\epsilon)^{\frac{\beta M(M-1)}{2}}
Y_{N, M} L_{N,M}\,.$$ with $$\begin{aligned}
\label{defLY}
L_{N,M}&:=&\int_{[c_{0}-\epsilon, c_{0}+\epsilon]^{M}}\Xi(\eta_{1}, \cdots, \eta_{M})\prod_{j=1}^{M}e^{-\frac{M\beta}{2}V(\eta_{j})}d\eta_{j}.\nonumber\\
Y_{N, M}&:=&\frac{Z^{\frac{N}{N-M}V;\boldsymbol{A}}_{N-M, \beta}}{Z^{V;\boldsymbol{B}}_{N, \beta}},\nonumber\\\end{aligned}$$ We wish to split $Y_{N, M}$ into components for further analysis. We make the decomposition: $$\label{main5}
Y_{N, M}=\frac{Z^{V,\boldsymbol{A}}_{N,\beta}}{Z^{V,\boldsymbol{B}}_{N,\beta}}
\tilde{Y}_{N, M}\,, \qquad
\tilde{Y}_{N, M}
=F_{N,M}G_{N,M}\,,$$ where $$\label{defFG}
F_{N, M}=\frac{Z_{N-M, \beta}^{\frac{N}{N-M}V;\boldsymbol{A}}}{Z_{N-M, \beta}^{V;\boldsymbol{A}}}\,,\qquad G_{N, M}=\frac{Z_{N-M, \beta}^{V;\boldsymbol{A}}}{Z_{N, \beta}^{V;\boldsymbol{A}}}\,.$$ [To get the upper bound in Proposition \[mainprop\], we will use the trivial estimate $Y_{N,M}\leq \tilde{Y}_{N,M}$ since $Z^{V,\boldsymbol{A}}_{N,\beta}\le Z^{V,\boldsymbol{B}}_{N,\beta}$.]{} Thus one finally rewrites formula (\[main4\]) as: $$\label{clean}
\Phi_{N, M, \beta}^{V;{\epsilon}}\leq (2\epsilon)^{\frac{\beta M(M-1)}{2}}
G_{N, M}F_{N, M} L_{N,M}\,.$$ We remark here that when M=1, the inequality (\[main4\]) becomes an equality: $$\label{counter}
\Phi_{N, 1, \beta}^{V; {\epsilon}}=\frac{Z^{V;\boldsymbol{A}}_{N,\beta}}{Z^{V;\boldsymbol{B}}_{N,\beta}}
G_{N, 1}F_{N,1} L_{N,1}\,.$$ ($\ref{counter}$) will be used to prove .\
We next estimate $G_{N,M}$ and $F_{N,M}\times L_{N,M}$. We shall prove that
\[order0\] Under Assumptions \[main-assume\] and \[as\], there exists a small $\delta>0$, such that uniformly for $M\le \delta N$, we have $$G_{N, M}\approx C_{M}\frac{1}{N^{\frac{M\beta}{2}}}e^{NM(\frac{\beta}{2}\inf_{\xi\in\boldsymbol{B}}\mathcal{J}^{V;\boldsymbol{B}}(\xi)+\frac{\beta}{2}\int V(\eta)d\mu_{\mathrm{eq}}(\eta))}(\frac{N-M}{N})^{\frac{\beta}{2}(N-M)} \,.$$ Furthermore, there exists a finite positive constant $C$ such that for all $1\le M\le \delta N$, $$\frac{1}{C}e^{-CM^{2}}\le C_{M}\le C e^{CM^{2}}.$$
For the term $F_{N,M}\times L_{N,M}$ we have the estimate
\[o3\] Under Assumptions \[main-assume\] and \[as\], there exists a positive constant $c$ and a finite constant $C$ so that for $\delta>0$ small enough, such that uniformly on $M\le \delta N$, $$F_{N,M} L_{N,M}\lesssim e^{CM^2} e^{-\frac{\beta}{2}NM(\int V(\eta)d\mu_{\rm eq}(\eta)+\inf \mathcal J^{V;\boldsymbol{B}})} \frac{1}{N^{\frac{M}{2}}} +O(e^{-c N^2})$$ and $$F_{N,1} L_{N,1}\gtrsim e^{-\frac{\beta}{2}N(\int V(\eta)d\mu_{\rm eq}(\eta)+ \inf \mathcal J^{V;\boldsymbol{B}})} \frac{1}{N^{\frac{1}{2}}}+O(e^{-c N^2})\,.$$
Clearly, Propositions \[order0\] and \[o3\] give Proposition \[mainprop\]. First, as the constant $c$ in Proposition \[o3\] is independent from $\delta$, we can choose $\delta$ small enough so that $O(e^{-c N^2})$ is negligible. Indeed, since $G_{N,M}^{-1}$ is at most of order $e^{C\delta N^2}$, for $\delta\le c/2C$ $$G_{N,M} e^{-c N^2}\le e^{-c N^2/2}$$ is negligible compared to $$G_{N,M} e^{CM^2} e^{-\frac{\beta}{2}NM(\int V(\eta)d\mu_{\rm eq}(\eta)+\inf \mathcal J^{V;\boldsymbol{B}})} \frac{1}{N^{\frac{M}{2}}}$$ which is decaying only polynomially by Proposition \[order0\]. We get by using , Propositions \[order0\] and \[o3\], we choose $\epsilon$ small enough so that $2\epsilon e^{4C}\le 1$ so that the terms in $e^{CM^2}$ disappear, and we observe that $(N-M)/N\le 1$. To derive the lower bound , we use together with Propositions \[order0\] and \[o3\]. in this case the term $((N-1)/N)^{\frac{\beta}{2} (N-1)}$ is of order one.
The proof of Proposition \[o3\] is based on the following proposition.
\[o2\] Under Assumption \[main-assume\] and \[as\], there exists positive constants $c, C, \delta_0$ so that for $\delta\in (0,\delta_0)$, $M\le \delta N$, for any $\eta_{1}, \cdots, \eta_{M}$ belonging to $[c_{0}-\varepsilon,c_0+\varepsilon]$, we have the following uniform estimate: $$F_{N, M}\Xi\left(\eta_{1}, \cdots, \eta_{M}\right)\lesssim e^{CM^{2}}e^{-\frac{\beta NM}{2}\int V(\eta)d\mu_{eq}(\eta)}e^{-\frac{\beta}{2}N\sum_{j=1}^{M}\mathcal{J}^{V;\boldsymbol{B}}(\eta_{j})}+O(e^{-c N^2}).$$ Moreover, for $\eta\in [c_{0}-\varepsilon,c_0+\varepsilon]$ $$\label{lkj}F_{N, 1}\Xi\left(\eta\right)\gtrsim e^{-\frac{\beta N}{2}( \int V(x)d\mu_{eq}(x)+\mathcal{J}^{V;\boldsymbol{B}}(\eta))}+O(e^{-c N^2}).$$
Let us first deduce Proposition \[o3\] from Proposition \[o2\]. The proof is straightforward since by Proposition \[o2\] $$\begin{aligned}
F_{N,M}L_{N,M}&\lesssim & e^{CM^2} e^{-\frac{\beta }{2}NM\int V(\eta)d\mu_{\rm eq}(\eta) -NM\frac{\beta}{2}\inf \mathcal J^{V;\boldsymbol{B}} }\left(\int_{c_0-\epsilon}^{c_0+\epsilon}e^{-N \tilde{\mathcal J}^{V;\boldsymbol{B}}(x)} dx\right)^M\\
&&+O(e^{-c N^2})\end{aligned}$$ where we can use Laplace method (recall we assume $\tilde{\mathcal{J}}^{V;\boldsymbol{B}}(c_0)=0, \frac{d}{dx} \tilde{\mathcal{J}}^{V;\boldsymbol{B}}(c_0)=0,\frac{d^{2}}{dx^{2}} \tilde{\mathcal{J}}^{V;\boldsymbol{B}}(c_0)>0$, see [@AGZ section 3.5.3] for details) to get $$\int_{c_0-\epsilon}^{c_0+\epsilon} e^{-N\tilde {\mathcal J}^{V;\boldsymbol{B}}(\lambda)}d\lambda\approx \frac{1}{\sqrt{N}}\,.$$ The proof of the lower bound is similarly deduced from .
\[beha\]Note that if we would have assumed instead of $\frac{d^{2}}{dx^{2}} \tilde{\mathcal{J}}^{V;\boldsymbol{B}}(c_0)>0$ that for some $q>2$, $\tilde{\mathcal{J}}^{V;\boldsymbol{B}}(x)\simeq |x-c_0|^q$ in a neighborhood of $c_0$, we would have obtained $$F_{N,M} L_{N,M}\lesssim Ce^{CM^2} e^{-\frac{\beta}{2}NM\int V(\eta)d\mu_{\rm eq}(\eta)} \frac{1}{N^{\frac{M}{q}}} +O(e^{-c N^2})$$ and criticality would have occurred at $\beta=2/q$.
Proof of Proposition \[order0\] and \[o2\] in the one cut case.
---------------------------------------------------------------
To estimate $G_{N,M}$ and $F_{N,M}L_{N,M}$ in the one cut case, we shall rely on the following results from [@BG1]:
\[art0\][@BG1 Propositions 1.1 and 1.2] If $V$ satisfies Assumption \[main-assume\] and \[as\] on $\boldsymbol{A}$, there exists a universal constant $e$, and constants $F_\beta^{\{k\}}$ so that we have for $N$ large enough: $$\label{expZ}
Z_{N, \beta}^{V;\mathsf{A}} = N^{(\frac{\beta}{2})N + e}\exp\Big(\sum_{k =-2}^K N^{-k}\, F^{\{k\}}_{ \beta} + o(N^{-K})\Big)\,.$$ Moreover, define the *correlators* given for $x\in \mathbb C\backslash \boldsymbol{A}$ by: $$W_{1} (x):=\mu_{N, \beta}^{V;\boldsymbol{A}}(\sum \frac{1}{x-\lambda_{i}}), \quad
W_1^{\{-1\}}(x):={\mu_{\mathrm{eq}}}(\frac{1}{x-\lambda}).$$ Then $$\label{expco0}W_{1}(x) = NW_1^{\{-1\}}(x) +O(1).$$ holds uniformly for $x$ in compact regions outside $\boldsymbol{A}$(in particular near the critical point $c_{0}$).
[**Proof of Proposition \[order0\].**]{} The fact that $$\label{GNM0}
G_{N, M}\approx C_{M}e^{MNA_{\beta}}\frac{1}{N^{\frac{M\beta}{2}}}(\frac{N-M}{N})^{\frac{\beta}{2}(N-M)},$$ where $A_{\beta}=-2F^{\{-2\}}_\beta$ does not depend on $N$ or $M$ and $C_{M}\lesssim e^{CM^{2}}$ is a direct consequence of . Moreover, the large deviation principle of Theorem \[large-deviation\](1) yields $$-F^{\{-2\}}_\beta=\inf \mathcal E= \frac{\beta}{2}( \int V(x)d\mu_{\rm eq}(x)-\int\int\log|x-y|d\mu_{\rm eq}(x)d\mu_{\rm eq}(y))\,.$$ Also, by definition, since the effective potential is constant on the support of the equilibrium measure, $$\inf \mathcal J^{V;\boldsymbol{B}}=-C_V= \int V(x)d\mu_{\rm eq}(x)-2\int\int\log|x-y|d\mu_{\rm eq}(x)d\mu_{\rm eq}(y)\,.$$ As a consequence $$A_{\beta}=\frac{\beta}{2}\inf_{\xi\in\boldsymbol{A}}\mathcal{J}^{V;\boldsymbol{A}}(\xi)+\frac{\beta}{2}\int V(\eta)d\mu_{\mathrm{eq}}(\eta)\,.$$
[**Proof of Proposition \[o2\].**]{} In the one-cut case we can prove this proposition without the error terms $O(e^{-c N^2})$ which we will need to deal with the several cut case. The main tool is to use concentration of measure. In fact we can write $$F_{N,M}\Xi(\eta_1,\ldots,\eta_M)=\mu_{N-M,\beta}^{ V; \boldsymbol{A}}\left( e^{\sum_{i=1}^{N-M} h_\eta (\lambda_i)}\right)$$ with $$h_\eta (x)=\beta \sum_{i=1}^M (\ln(\eta_i-x)-\frac{1}{2} V(\eta_i)-\frac{1}{2} V(x))\,.$$ Eventhough this function depends on $M$, concentration inequalities will allow a uniform control. We develop the necessary estimates in the appendix, see Lemma \[central\]. $\|f\|_{\mathcal L}$ denotes the Lipschitz norm and $\|f\|_\infty$ the uniform bound on a neighborhood of $\boldsymbol{A}$. We apply Lemma \[central\] with $h=h_\eta$. As the $\eta_i$ are away from $\boldsymbol{A}$, [$\|h_\eta\|_{\mathcal L}^2$]{} is uniformly bounded by $CM^2$ for some finite constant $C$, whereas $\|h_\eta\|_\infty$ is of order $M$. Hence, we deduce from Lemma \[central\] that $$\label{polk} e^{-CM} e^{(N-M)\mu_{\rm eq }(h_\eta)}\le F_{N,M} \Xi(\eta_1,\ldots,\eta_M)\le e^{C M^2} e^{ (N-M) \mu_{\rm eq}( h_\eta) }\,.$$ Note that we can replace above $(N-M)\mu_{\rm eq }(h_\eta)$ by $N\mu_{\rm eq }(h_\eta)$ up to an error of order $ M^2$ which amounts to change the constant $C$. This completes the proof of Proposition \[o2\] since $$\begin{aligned}
\mu_{\rm eq}( h_\eta)&=&\frac{\beta}{2}( \sum_{i=1}^M 2 \int \log |\eta_i-x|d\mu_{\rm eq}(x)-V(\eta_i))-\frac{\beta}{2} M \int V(x)d\mu_{\rm eq}(x)\nonumber\\
&=&-\frac{\beta}{2}\sum_{i=1}^M \mathcal J^{V;\boldsymbol{B}}(\eta_i) -\frac{\beta}{2} M \int V(x)d\mu_{\rm eq}(x)\,.\label{eqr}\end{aligned}$$
[Proof of Propositions \[order0\] and \[o2\] in the general multi-cut case]{} {#proofprop}
-----------------------------------------------------------------------------
In the multi cut case, we have to be more careful since the number of eigenvalues that are in each connected component of the support of $\mu_{\rm eq}$ is not a priori fixed. The idea is therefore that we will have to sum over all possible number of eigenvalues in these components. Our proof is based on estimates from [@BG2] on the fixed filling fraction measure [(see Definition \[defi333\] below)]{}. We introduce some new notation below.
[From the viewpoint of Large Deviation Principle on $L_{N}$, the number of eigenvalues in the interval $A_{h}$, which we will call the filling fractions, should in principle be proportionnal to $\mu_{eq}(S_{h})$, which is the mass of equilibrium measure accumulated on $A_{h}$.]{}
[Thus, let’s define $\boldsymbol{f}_{\star}$ as the $g$-tuple denoting the mass of the equilibrium measure in each of the intervals that comprise the support of the equilibrium measure, i.e. $$\label{eps} \boldsymbol{f}_{\star}:=\left(\mu_{\mathrm{eq}}(S_{1}), \cdots, \mu_{\mathrm{eq}}(S_{g})\right)$$ ]{} [ In order to describe how the $N$ eigenvalues are distributed in the $g$ intervals ${\boldsymbol{A}}_{h}$, we let $$\label{gtuple}
\mathcal{E}_g:=\left\lbrace(f_{1}, \cdots , f_{g})|\sum_{h=1}^{g} f_{h}=1, f_{1}, \ldots , f_{g}\geq 0\right\rbrace .$$]{}
Now we can define the fixed filling fraction probability measure:
\[defi333\] For any $\overrightarrow{\boldsymbol{N}}=(N_1,\ldots,N_g)$ so that $\overrightarrow{\boldsymbol{N}}/N\in \mathcal{E}_g$, let the fixed filling fraction probability measure $d\mu_{N, \frac{\overrightarrow{\boldsymbol{N}}}{N}, \beta}^{V;\boldsymbol{A}}$ be given by: $$\label{defi3}
\begin{aligned}
d\mu_{N, \frac{\overrightarrow{\boldsymbol{N}}}{N}, \beta}^{V;\boldsymbol{A}}(\boldsymbol{\lambda})
:= & \frac{1}{Z_{N, \boldsymbol{f}, \beta}^{V;\boldsymbol{A}}}\prod_{h = 1}^g \Big[\prod_{i = 1}^{N_h} \mathrm{d}\lambda_{h, i}\, \mathbf{1}_{\boldsymbol{A}_{h}}(\lambda_{h, i})\, e^{-\frac{\beta N}{2}\, V(\lambda_{h, i})}\, \prod_{1 \leq i < j \leq N} |\lambda_{h, i} - \lambda_{h, j}|^{\beta}\Big] \nonumber \\
& \times \prod_{1 \leq h < h' \leq g} \prod_{\substack{1 \leq i \leq N_h \\ 1 \leq j \leq N_{h'}}} |\lambda_{h, i} - \lambda_{h', j}|^{\beta},
\end{aligned}$$ where $Z_{N, \boldsymbol{f}, \beta}^{V;\boldsymbol{A}}$ is the partition function.
The following precise estimate of the fixed filling fraction measure from [@BG2 Theorem 1.4] will be essential in our proof. It extends Theorem \[art0\].
\[part\] If $V$ satisfies Assumption \[main-assume\] and \[as\] on $\boldsymbol{A}$, there exists $t > 0$ such that, uniformly for $\frac{\overrightarrow{\boldsymbol{N}}}{N} \in \mathcal{E}_g$ that satisfies $|\frac{\overrightarrow{\boldsymbol{N}}}{N} - \boldsymbol{f}_{\star}| < t$, we have: $$\label{sqiqq}\frac{N!}{\prod_{h = 1}^{g} (N_h)!}\, Z_{N, \frac{\overrightarrow{\boldsymbol{N}}}{N}, \beta}^{V;\mathsf{A}} = N^{(\frac{\beta}{2})N + e}\exp\Big(\sum_{k =-2}^K N^{-k}\, F^{\{k\}}_{\frac{\overrightarrow{\boldsymbol{N}}}{N}, \beta} + o(N^{-K})\Big)\,.$$ $e$ is some universal constant, $F^{\{k\}}_{\boldsymbol{f}, \beta}$ extends as a smooth function for $\boldsymbol{f}$ close enough to $\boldsymbol{f}_{\star}$, and at the value $\boldsymbol{f} = \boldsymbol{f}_{\star}$, the derivative of $F^{\{-2\}}_{\boldsymbol{f}, \beta}$ vanishes and its Hessian is negative definite. Assume that $\overrightarrow{\boldsymbol{N}}/N$ converges towards $\boldsymbol{f}$. Then, the law of the empirical measure $L_N$ under $\mu_{N, \boldsymbol{f}, \beta}^{V;\boldsymbol{A}}$ satisfies a large deviation principle with speed $N^2$ and good rate function $\tilde{\mathcal E}_{\boldsymbol{f}}$ which is minimized at a unique probability measure $\mu_{{\rm eq}, \boldsymbol{f}}$, which is also the minimum of $\mathcal E$ under the constraint that $\mu(\boldsymbol{A}_h)=f_h$. In particular $L_N$ converges $\mu_{N, \frac{\overrightarrow{\boldsymbol{N}}}{N}, \beta}^{V;\boldsymbol{A}}$ almost surely to $\mu_{{\rm eq},\boldsymbol{f}}$. Moreover, for $x\in \mathbb C\backslash \boldsymbol{A}$ let $$W_{\frac{\overrightarrow{\boldsymbol{N}}}{N}}(x):=\mu_{N, \frac{\overrightarrow{\boldsymbol{N}}}{N}, \beta}^{V;\boldsymbol{A}}(\sum \frac{1}{x-\lambda_{i}}), \quad
W_{\boldsymbol{f}}^{\{-1\}}(x):=\mu_{\mathrm{eq}, \boldsymbol{f}}(\frac{1}{x-\lambda}).$$ Then, there exists $t > 0$ such that, uniformly for $\boldsymbol{f} \in \mathcal{E}_g$ and $|\boldsymbol{f} - \boldsymbol{f}_{\star}| < t$, we have an expansion for the correlators: $$\label{expco}W_{\frac{\overrightarrow{\boldsymbol{N}}}{N}}(x) = NW_{\frac{\overrightarrow{\boldsymbol{N}}}{N}}^{\{-1\}}(x) +O(1).$$ holds uniformly for $x$ in compact regions outside $\boldsymbol{A}$(in our case in particular near the critical point $c_{0}$). $\boldsymbol{f}\rightarrow
W_{\boldsymbol{f}}^{\{-1\}}(x)$ extends as a smooth function in a neighborhood of $\boldsymbol{f}_{\star}$ for any $x\in \mathbb{C}\backslash \boldsymbol{S}$.
[**Proof of Proposition \[order0\].**]{} By the partition function estimate from Theorem \[part\], for a sufficiently small $\kappa$, we have $$\label{G1}
\begin{aligned}
&Z_{N-M, \beta}^{V; \boldsymbol{A}}=\sum_{N_{1}+\cdots+N_{g}=N-M}\frac{(N-M)!}{N_{1}!\cdots N_{g}!}\cdot Z^{V; \boldsymbol{A}}_{N-M, \frac{\overrightarrow{\boldsymbol{N}}}{N-M}, \beta}, \\
&\approx
\sum_{
\begin{subarray}
\quad N_{1}+\cdots+N_{g}=N-M, \\
|\frac{\overrightarrow{\boldsymbol{N}}}{N-M}-\boldsymbol{f_{\star}}|<\kappa
\end{subarray}
}(N-M)^{(\frac{\beta}{2})(N-M)+e}\exp\left((N-M)^{2}F^{\{- 2\}}_{\frac{\overrightarrow{\boldsymbol{N}}}{N-M}, \beta}+(N-M)F^{\{-1\}}_{\frac{\overrightarrow{\boldsymbol{N}}}{N-M}, \beta}\right)\\
&
+O(e^{-C_\kappa N^2})Z_{N-M, \beta}^{V; \boldsymbol{A}}\end{aligned}$$ with some $C_\kappa>0$. In the last step we applied the large deviation principle for the empirical measure $L_{N-M}$; in other words, the sum over all $\overrightarrow{N}$ such that $|\frac{\overrightarrow{\boldsymbol{N}}}{N-M}-\boldsymbol{f}_{\star}|\geq\kappa$ divided by $Z_{N-M, \beta}^{V; \boldsymbol{A}}$ is negligible since it is the probability that the filling fractions are away from the equilibrium ones, a set on which the distance between the empirical measure $L_{N-M}$ and the equilibrium measure is positive. We used Theorem \[part\] to estimate each partition functions in the remaining sum. Similarly, $$\label{G2}
\begin{aligned}
Z_{N, \beta}^{V; \boldsymbol{A}}&=\sum_{N_{1}+\cdots+N_{g}=N}\frac{N!}{N_{1}!\cdots N_{g}!}\cdot Z^{\frac{N}{N}; \boldsymbol{A}}_{N, \frac{\overrightarrow{\boldsymbol{N}}}{N}, \beta}, \\
&\approx \sum_{\begin{subarray}
\quad
N_{1}+\cdots+N_{g}=N, \\
|\frac{\overrightarrow{\boldsymbol{N}}}{N}-\boldsymbol{f_{\star}}|<\kappa\end{subarray}}(N)^{(\frac{\beta}{2})(N)+e}\exp\left((N)^{2}F^{\{-2\}}_{\frac{\overrightarrow{\boldsymbol{N}}}{N},\beta}+(N)F^{\{-1\}}_{\frac{\overrightarrow{\boldsymbol{N}}}{N},\beta}\right).
\end{aligned}$$\
All that is left to do is to analyze the limiting behavior of : $$\label{G3}
L_{{K}}:= \sum_{\begin{subarray}
\quad
N_{1}+\cdots+N_{g}=K, \\
|\frac{\overrightarrow{\boldsymbol{N}}}{K}-\boldsymbol{f_{\star}}|<\kappa\end{subarray}} \exp((K)^{2}F^{\{-2\}}_{\frac{\overrightarrow{\boldsymbol{N}}}{K}, \beta}+(K)F^{\{-1\}}_{\frac{\overrightarrow{\boldsymbol{N}}}{K}, \beta}) .$$ Here $\overrightarrow{\boldsymbol{N}}=(N_{1}, \cdots, N_{g})$ with $\sum {N_{i}}=K$. [This is done in Lemma \[lemG4\] below, from where the rest of the argument is exactly as in the one-cut case:]{}
\[lemG4\] $$\label{gfd}
L_{{K}}\approx \exp(K^{2}F^{\{-2\}}_{\boldsymbol{f_{\star}},\beta}+KF^{\{-1\}}_{\boldsymbol{f_{\star}},\beta})$$
According to Theorem \[part\], [ for ${\boldsymbol{f}}$ sufficiently close to $\boldsymbol{f}_{\star}$, $F_{\boldsymbol{f}}^{\{-1\}}$ and $F_{\boldsymbol{f}}^{\{-2\}}$ are smooth, and the Hessian of $F^{\{-2\}}_{{\boldsymbol{f}},\beta}$ is negative definite at $\boldsymbol{f_{\star}}$.]{} Thus we can find constant c, C such that for $|\boldsymbol{f}-\boldsymbol{f_{\star}}|\le \kappa <t$, we have $$\label{lem1}|F^{\{-1\}}_{\boldsymbol{f}, \beta}-F^{\{-1\}}_{\boldsymbol{\boldsymbol{f_{\star}}}, \beta}|\leq C|\boldsymbol{f}-\boldsymbol{f_{\star}}|,$$ $$\label{lem2}
F^{\{-2\}}_{\boldsymbol{f}, \beta}-F^{\{-2\}}_{\boldsymbol{\boldsymbol{f_{\star}}}, \beta}\leq-c|\boldsymbol{f}-\boldsymbol{f_{\star}}|^{2},$$ $$\label{lem3}|F^{\{-2\}}_{\boldsymbol{f}, \beta}-F^{\{-2\}}_{\boldsymbol{\boldsymbol{f_{\star}}}, \beta}|\leq C|\boldsymbol{f}-\boldsymbol{f_{\star}}|^{2}.$$
[We first derive the lower bound of $L_{K}$. Indeed, there exists at least one $\overrightarrow{\boldsymbol{N_{1}}}:=(N_{1},\cdots, N_{g})$, such that $|\frac{\overrightarrow{\boldsymbol{N_{1}}}}{K}-\boldsymbol{f_{\star}}|{\lesssim}\frac{1}{K}$ and $N_{1}+\cdots+ N_{g}=K$. Thus we can easily get the following lower bound of $L_{k}$ with (\[lem1\]), (\[lem3\]):]{} $$L_{{K}}{\geq \exp(K^{2}F^{\{-2\}}_{\frac{\overrightarrow{\boldsymbol{N}}_{1}}{K}, \beta}+KF^{\{-1\}}_{\frac{\overrightarrow{\boldsymbol{N}}_{1}}{K}, \beta})\gtrsim} \exp(K^{2}F^{\{-2\}}_{\boldsymbol{f_{\star}, \beta} }+KF^{\{-1\}}_{\boldsymbol{f_{\star}}, \beta}).$$ Next, we compute the upper bound of $L_{K}$. Direct computation leads to $$\label{tempupper}
\begin{aligned}
&\frac{L_{K}}{\exp(K^{2}F^{\{-2\}}_{\boldsymbol{f_{\star}}, \beta}+KF^{\{-1\}}_{\boldsymbol{f_{\star}}, \beta})}\\
&=\sum_{\begin{subarray}
\quad
N_{1}+\cdots+N_{g}=K, \\
|\frac{\overrightarrow{\boldsymbol{N}}}{K}-\boldsymbol{f_{\star}}|<\kappa\end{subarray}} \exp(K^{2}(F^{\{-2\}}_{\frac{\overrightarrow{\boldsymbol{N}}}{K}, \beta}-F^{\{-2\}}_{{\boldsymbol{f}}_{\star}, \beta})+K(F^{\{-1\}}_{\frac{\overrightarrow{\boldsymbol{K}}}{K}, \beta}-F^{\{-2\}}_{{\boldsymbol{f}}_{\star}, \beta}))\\
&\leq \sum_{|\frac{\overrightarrow{\boldsymbol{N}}}{K}-\boldsymbol{f_{\star}}|\le \kappa}\exp(-cK^{2}|\frac{\overrightarrow{\boldsymbol{N}}}{K}-\boldsymbol{f_{\star}}|^2+CK|\frac{\overrightarrow{\boldsymbol{N}}}{K}-\boldsymbol{f_{\star}}|).
\end{aligned}$$ In the last step of , we use and . It is clear that the above right hand side is bounded from which the claim follows.
Having established Lemma \[lemG4\], we complete the proof of Proposition \[order0\] as in the one-cut case.
To prove Proposition \[o2\], let us notice first that we can replace $\Xi$ by taking the integral only on filling fractions close to that of the equilibrium measure since the error will be otherwise of order $e^{-cN^2}$: for a given configuration $\boldsymbol{\lambda}=(\lambda_1,\ldots, \lambda_{N-M-1})$ denote ${\overrightarrow{\boldsymbol{N(\lambda)}}}= (N_1(\lambda),\ldots, N_g(\lambda))$ with $N_h(\lambda)=\#\{i: \lambda_i\in \boldsymbol{A}_h\}$. Then hereafter we replace $\Xi$ by its localized version : $$\begin{aligned}
\label{defLY}
{\Xi(\eta_1, \cdots, \eta_M):=\mu^{\frac{N}{N-M}V;\boldsymbol{A}}_{N-M, \beta}(\prod_{j=1}^{M}e^{\beta\sum_{i=1}^{N-M}\ln|\eta_{j}-\lambda_{i}|-\frac{\beta}{2}(N-M)V(\eta_{j})}\mathbf{1}_{|\frac{{\overrightarrow{\boldsymbol{N(\lambda)}}}}{N-M} -{\boldsymbol{f}}_{\star}|<\kappa})}\,.\end{aligned}$$ For any $\kappa>0$, the part we cut-off can be controlled by large deviation principle of the empirical measure $L_{N}$, which is of order $e^{-c_{\kappa}N^{2}}$. All our estimates on $F_{N,M}\Xi$ below will be made up to this error that we will not write done to simplify the exposition. The proof goes again through an expansion of terms where the filling fractions are fixed.
$$\label{keykey}
F_{N, M}\Xi (\eta_{1}, \cdots, \eta_{M}){=}
\sum_{
\begin{subarray}
\quad N-M=N_{1}+\cdots+N_{g}, \\|\frac{\overrightarrow{\boldsymbol{N}}}{N-M}-\boldsymbol{f_{\star}}|<\kappa
\end{subarray}
}
c_{\overrightarrow{\boldsymbol{N}}}d_{\overrightarrow{\boldsymbol{N}}},$$
where
- $$\label{def c}
c_{\overrightarrow{\boldsymbol{N}}}:=\frac{1}{Z_{N-M, \beta}^{V;\boldsymbol{A}}}
\frac{(N-M)!}{N_{1}!\cdots
N_{g}!}\cdot Z^{V; \boldsymbol{A}}_{N-M, \frac{\overrightarrow{\boldsymbol{N}}}{N-M}, \beta},$$
- $$\label{def d} d_{\overrightarrow{\boldsymbol{N}}}:=\mu^{V;\boldsymbol{A}}_{N-M, \frac{\overrightarrow{\boldsymbol{N}}}{N-M}\beta}(\prod_{i=1}^{M}e^{\sum_{j=1}^{N-M}(-\frac{\beta}{2}V(\eta_{i})+\beta \ln|\lambda_{j}-\eta_{i}|-\frac{\beta}{2}V(\lambda_{j}))}).$$
We use the concentration Lemma \[central\] with $$h(x)=\sum_{i=1}^M h_{\eta_i} (x)\,, \quad h_\eta(x)=
-\frac{\beta}{2}V(x)+\beta \ln|\eta_i-x |\, .$$ Note that $\|h\|_\La^2$ is of order $M^2$ for [$\eta_{1},\cdots,\eta_M$]{} close to $c_0$ and $\|h\|_\infty$ is of order $M$. We first estimate $d_{\overrightarrow{\boldsymbol{N}}}$ and then substitute the estimate into (\[keykey\]).
$$\begin{aligned}
d_{\overrightarrow{\boldsymbol{N}}}
&=&\mu^{V;\boldsymbol{A}}_{N-M, \frac{\overrightarrow{\boldsymbol{N}}}{N-M};\beta}(e^{\sum_{i=1}^{N-M} h(\lambda_i)})\prod_{j=1}^{M}e^{-\frac{\beta}{2}(N-M)V(\eta_{j})} \nonumber\\
&\leq &Ce^{CM^{2}} e^{(N-M) \mu_{\rm{eq}, \frac{\overrightarrow{\boldsymbol{N}}}{N-M}}(h)}\prod_{j=1}^{M}e^{-\frac{\beta}{2}(N-M)V(\eta_{j})}.\label{d}\end{aligned}$$
Next, we want to substitute $\mu_{\rm eq, \frac{\overrightarrow{\boldsymbol{N}}}{N-M}}$ by $\mu_{\rm eq}$. By Appendix A.1 in [@BG2], $\boldsymbol{f}\to \mu_{\rm eq,\boldsymbol{f}} (h_\eta)$ is Lipschitz in a neighborhood of $\boldsymbol{f_{\star}}$, uniformly in $\eta\in [c_0-\epsilon,c_0+\epsilon]$ so that $$\label{k3}
|\mu_{\rm{eq}}(h)-
\mu_{\rm{eq}, \frac{\overrightarrow{\boldsymbol{N}}}{N-M}}(h)|=|\mu_{\rm{eq}, \boldsymbol{f_{\star}}}(h)-
\mu_{\rm{eq}, \frac{\overrightarrow{\boldsymbol{N}}}{N-M}}(h)|\leq CM|\frac{\overrightarrow{\boldsymbol{N}}}{N-M}-\boldsymbol{f_{\star}}|.$$ Combining (\[d\]), , (\[k3\]) gives $$\label{k4}
\begin{aligned}
&F_{N, M}\Xi\left(\eta_{1}, \cdots, \eta_{M} \right)
\lesssim e^{CM^{2}}(\prod_{j=1}^{M}e^{-\frac{\beta}{2}NV(\eta_{j})})\\
&\times \sum_{|\frac{\overrightarrow{\boldsymbol{N}}}{N-M}-\boldsymbol{f_{\star}}|<\kappa}
c_{\overrightarrow{\boldsymbol{N}}}\exp\left((N-M)(\mu_{{\rm eq}}^{V;\boldsymbol{A}}(h)+CM|\frac{\overrightarrow{\boldsymbol{N}}}{N-M}-\boldsymbol{f_{\star}}|)+O(M)\right).
\end{aligned}$$ Finally, observe as in that $c_{\overrightarrow{\boldsymbol{N}}}$ has a sub-Gaussian tail, that is: $$\label{k7}
c_{\overrightarrow{\boldsymbol{N}}}\leq Ce^{-c(N-M)^{2}|\frac{\overrightarrow{\boldsymbol{N}}}{N-M}-\boldsymbol{f_{\star}}|^{2}+C(N-M)|\frac{\overrightarrow{\boldsymbol{N}}}{N-M}-\boldsymbol{f_{\star}}|}.$$ so that we deduce that $$\label{k6}
\sum_{|\frac{\overrightarrow{\boldsymbol{N}}}{N-M}-\boldsymbol{f_{\star}}|<\kappa}c_{\overrightarrow{\boldsymbol{N}}}e^{CM(N-M)|\frac{\overrightarrow{\boldsymbol{N}}}{N-M}-\boldsymbol{f_{\star}}|}\leq Ce^{CM^{2}},$$ [Indeed, $$CM(N-M)|\frac{\overrightarrow{\boldsymbol{N}}}{N-M}-\boldsymbol{f_{\star}}|\leq
{\frac{2}{c} C^2M^{2}+\frac{c}{2}|N-M|^{2}|\frac{\overrightarrow{\boldsymbol{N}}}{N-M}-\boldsymbol{f_{\star}}|^{2}}$$ so that the term $c_{\overrightarrow{\boldsymbol{N}}}e^{CM(N-M)|\frac{\overrightarrow{\boldsymbol{N}}}{N-M}-\boldsymbol{f_{\star}}|}$ has also a sub-Gaussian tail up to multiplying it by $e^{CM^{2}}$, thus follows from . ]{} Therefore yields the desired upper bound, as in :
$$\label{k5}
\begin{aligned}
F_{N.M}\Xi({\eta_{1}, \cdots, \eta_{M}})&\lesssim& e^{CM^{2}}(\prod_{j=1}^{M}e^{-\frac{\beta}{2}(N-M)V(\eta_{j})})e^{(N-M)\mu_{{\rm eq}}(h)} \\
&\le& Ce^{CM^2}e^{-\frac{\beta NM}{2}\int V(\eta)d\mu_{{\rm eq}}(\eta)}e^{-N\frac{\beta}{2}\sum_{j=1}^{M}{\mathcal{J}}^{V;\boldsymbol{B}}(\eta_{j})}\,.
\end{aligned}$$
The proof of is similar to the one cut case. From , choosing $\overrightarrow{\boldsymbol{N}}$ so that $|\overrightarrow{\boldsymbol{N}}- N\boldsymbol{f}_\star|\le 1$, we get $$F_{N,1}\Xi(\eta)\ge c_{\overrightarrow{\boldsymbol{N}}} d_{\overrightarrow{\boldsymbol{N}}}\,.$$ We use Lemma \[central\] to lower bound the term in $d_{\overrightarrow{\boldsymbol{N}}}$: $$d_{\overrightarrow{\boldsymbol{N}}} \ge e^{-cM^2} e^{ -\frac{\beta}{2}N( \mathcal J^{V;\boldsymbol{B}}(\eta) +\int V(x)d\mu_{\rm eq}(x))}\,.$$ The $c_{\overrightarrow{\boldsymbol{N}}}$ term is bounded from below by the partition function estimate from Theorem \[part\] as well as the upper bound for $ Z_{N, \beta}^{V; \boldsymbol{A}}$ provided by and Lemma \[lemG4\].
Acknowledgments
===============
Part of this work was completed as part of the MIT SPUR program over the summer of 2013. AG was partially supported by the Simons Foundation and by NSF Grant DMS-1307704. The authors are very grateful to a anonymous referee for his helpful comments.
Concentration lemmas
====================
Moments estimate
----------------
Here we deduce the following lemma by using the estimate for correlator, by a direct application of Cauchy’s integral formula.
\[core\] Let Assumption \[main-assume\] and \[as\] hold, let $\overrightarrow{\boldsymbol{N}}=(N_1,\ldots,N_g)$ so that $\sum N_i=N$ and let $\mu_{N, \frac{\overrightarrow{\boldsymbol{N}}}{N}, \beta}^{V;\boldsymbol{A}}$be the fixed filling fractions measure, and $\mu_{{\rm eq}, \frac{\overrightarrow{\boldsymbol{N}}}{N},}$ be its limiting measure. Let $h$ be a function that is holomorphic in a open neighborhood $\boldsymbol{U}$ of $\boldsymbol{A}$. Then, $$|\mu_{N, \frac{\overrightarrow{\boldsymbol{N}}}{N}, \beta}^{V;\boldsymbol{A}}(\sum h(\lambda_{i}))
-N\mu_{{\rm eq}, \frac{\overrightarrow{\boldsymbol{N}}}{N}}(h)|\lesssim C\|h\|_\infty.$$ where $\|h\|_\infty$ is the supremum norm of $h$ on a contour around $\boldsymbol{A}$ inside $\boldsymbol{U}$.\
This result in particular hold in the one cut case where the fraction is always set to $\overrightarrow{\boldsymbol{N}}=N$.
As $h$ is holomorphic, we can write by Cauchy formula, for a contour $\mathcal C$ around $\boldsymbol{A}$, $$\mu_{N, \boldsymbol{f}, \beta}^{V;\boldsymbol{A}}(\sum h(\lambda_{i}))
=\frac{1}{2i\pi}\int_{\mathcal C} h(\xi) W_{ \boldsymbol{f}}(\xi) d\xi$$ from which the estimate follows from (and in the one cut case).
Concentration estimates
-----------------------
We assumed in Assumption \[as\] that $V$ is strictly convex in a neighborhood of $\boldsymbol{A}$ in order to use the following concentration inequality, see sections 2.3.2, 4.4.17, 4.4.26 in [@AGZ] for more details. Note here that we fix the component in which each eigenvalue is living so that indeed they only see a convex potential.
\[comi\] Let $\boldsymbol{f}\in \mathcal E_g$ be given. Let $V$ be a smooth function such that $V''(x)\geq C>0$ for all $x\in\boldsymbol{A}$. Let $h$ be a function that is class $C^{1}$ on $\mathbb{R}^{N}$. Let $\overrightarrow{\boldsymbol{N}}=(N_1,\ldots,N_g)$ so that $\sum N_i=N$. Then $$\mu_{N, \frac{\overrightarrow{\boldsymbol{N}}}{N},
\beta}^{V;\boldsymbol{A}}\left[\exp\{\left(f-\mu_{N, \frac{\overrightarrow{\boldsymbol{N}}}{N}, \beta}^{V;\boldsymbol{A}}(h)\right)\}\right]\lesssim e^{\frac{1}{NC}\|h\|_{\mathcal{L}}^{2}},$$ where $$\|h\|_{\mathcal{L}}:=\sqrt{\sum_{i=1}^{N}\sup_{x\in\boldsymbol{A}^N}|\partial_{\lambda_{i}}h(x)|^{2}}.$$
Note that this lemma applies in particular in the one cut case.
\[central\] Let Assumption \[main-assume\] and \[as\] hold and let $\overrightarrow{\boldsymbol{N}}=(N_1,\ldots,N_g)$ so that $\sum N_i=N$. Then, there exists a finite constant $C$ so that for any h holomorphic in an open neighborhood of $\boldsymbol{A}$, we have $$\label{eq1}e^{-C\|h\|_\infty} \lesssim
\mu_{N,\frac{\overrightarrow{\boldsymbol{N}}}{N},
\beta}^{V;\boldsymbol{A}}(\exp(\sum_{i=1}^N ( h(\lambda_{i})-\int h(\eta)d\mu_{{\rm eq}, \frac{\overrightarrow{\boldsymbol{N}}}{N}}(\eta)))) \lesssim e^ {C(\|h\|_{\mathcal{L}}^{2}+\|h\|_\infty)}\,.$$
By Jensen’s Inequality: $$\label{eq3}
\begin{aligned}
\mu_{N, \frac{\overrightarrow{\boldsymbol{N}}}{N}, \beta}^{V;\boldsymbol{A}}(\exp(\sum h(\lambda_i)))
\geq \exp(\mu_{N, \frac{\overrightarrow{\boldsymbol{N}}}{N}, \beta}^{V;\boldsymbol{A}}(\sum h(\lambda_{i}))).
\end{aligned}$$ The upper bound is based on the concentration equality for fixed filling fractions of Lemma \[comi\] with $\|\sum h(\lambda_i)\|_{\mathcal L}^2=N\|h\|_{\mathcal L}^2$ so that $$\label{eq4}
\begin{aligned}
\mu_{N, \frac{\overrightarrow{\boldsymbol{N}}}{N}, \beta}^{V;\boldsymbol{A}}(\exp(\sum h(\lambda_i)))
\leq \exp\{\frac{1}{C}\|h\|^2_{\mathcal L}\} \exp(\mu_{N, \frac{\overrightarrow{\boldsymbol{N}}}{N}, \beta}^{V;\boldsymbol{A}}(\sum h(\lambda_{i}))).
\end{aligned}$$ Lemma \[comi\] completes the proof.
[^1]: \
$^{\ddagger} $Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Ave, Cambridge, MA 02139-4307 USA. email: [email protected].\
$^{\dagger}$ Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Ave, Cambridge, MA 02139-4307 USA. email: [email protected].\
$^{\S}$Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Ave, Cambridge, MA 02139-4307 USA. email: [email protected].\
$^{\sharp}$Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Ave, Cambridge, MA 02139-4307 USA. email: [email protected].
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'A lower bound on the secrecy capacity of the wiretap channel with state information available causally at both the encoder and decoder is established. The lower bound is shown to be strictly larger than that for the noncausal case by Liu and Chen. Achievability is proved using block Markov coding, Shannon strategy, and key generation from common state information. The state sequence available at the end of each block is used to generate a key, which is used to enhance the transmission rate of the confidential message in the following block. An upper bound on the secrecy capacity when the state is available noncausally at the encoder and decoder is established and is shown to coincide with the lower bound for several classes of wiretap channels with state.'
author:
- '\'
bibliography:
- 'secrecy.bib'
title: Wiretap Channel with Causal State Information
---
Introduction
============
Consider the 2-receiver wiretap channel with state depicted in Figure \[fig1\]. The sender $X$ wishes to send a message to the legitimate receiver $Y$ while keeping it asymptotically secret from the eavesdropper $Z$. The secrecy capacity for this channel can be defined under various scenarios of state information availability at the encoder and decoder. When the state information is not available at either party, the problem reduces to the classical wiretap channel for the channel averaged over the state and the secrecy capacity is known [@Wyner], [@Csiszar]. When the state is available only at the decoder, the problem reduces to the wiretap channel with augmented receiver $(Y,S)$.
\[c\][$M$]{} \[c\][$$]{} \[c\][$X_i$]{} \[l\][$S_i$]{} \[c\] \[c\][$Y_{i}$]{} \[c\][$Z_{i}$]{} \[c\][$\Mh$]{} \[c\] \[c\][Encoder]{} \[c\][Decoder]{} \[c\][[Eavesdropper]{}]{} \[c\][$p(s)$]{} \[c\][$p(y,z|x,s)$]{} \[b\] ![Wiretap channel with State[]{data-label="fig1"}](figs/CSI-BC_c.eps "fig:"){width="0.8\linewidth"}
The interesting scenarios to consider therefore are when the state information is available at the encoder and may or may not be available at the decoder. This raises the question of how the encoder and decoder can make use of the state information to increase the secrecy rate. In [@Chen--Vinck2006], Chen and Vinck established a lower bound on the secrecy capacity when the state information is available noncausally only at the encoder. The lower bound is established using a combination of Gelfand–Pinsker coding and Wyner wiretap coding. Subsequently, Liu and Chen [@Liu--Chen2007] used the same techniques to establish a lower bound on the secrecy capacity when the state information is available noncausally at both the encoder and decoder. In a related direction, Khisti, Diggavi, and Wornell [@Khisti--Diggavi--Wornell2009] considered the problem of secret key agreement first studied in [@Maurer1993] and [@Ahlswede--Csiszar1993] for the wiretap channel with state and established the secret key capacity when the state is available causally or noncausally at the encoder and decoder. The key is generated in two parts; the first using a wiretap channel code while treating the state sequence as a time-sharing sequence, and the second part is generated from the state itself.
In this paper, we consider the wiretap channel with state information available [*causally*]{} at the encoder and decoder. We show that the lower bound for the noncausal case in [@Liu--Chen2007] is achievable when only causal state information is available. Our achievability scheme, however, is quite different from that for the noncausal case. We use block Markov coding, Shannon strategy for channels with state [@Shannon1958a], and secret key agreement from state information, which builds on the work in [@Khisti--Diggavi--Wornell2009]. However, unlike [@Khisti--Diggavi--Wornell2009], we are not directly interested in the size of the secret key, but rather in using the secret key generated from the state sequence in one transmission block to increase the secrecy rate in the following block. This block Markov scheme causes additional information leakage through the correlation between the secret key generated in a block and the received sequences at the eavesdropper in subsequent blocks. Although a similar block Markov coding scheme was used in [@Ardestanizadeh--Franceschetti--Javidi--Kim2008] to establish the secrecy capacity of the degraded wiretap channel with rate limited secure feedback, in their setup no information about the key is leaked to the eavesdropper because the feedback link is assumed to be secure.
We also establish an upper bound on the secrecy capacity of the wiretap channel with state information available noncausally at the encoder and decoder. We show that the upper bound coincides with the aforementioned lower bound for several classes of channels. Thus, the secrecy capacity for these classes does not depend on whether the state information is known causally or noncausally at the encoder.
The rest of the paper is organized as follows. In Section \[sect:2\], we provide the needed definitions. In Section \[sect:3\], we summarize and discuss the main results in the paper. The proofs of the lower and upper bounds are detailed in Sections \[sect:4\] and \[sect:5\], respectively.
Problem Definition {#sect:2}
==================
Consider a discrete memoryless wiretap channel (DM-WTC) with discrete memoryless state (DM)\
$(\Xc\times\Sc, p(y,z|x,s)p(s), \Yc, \Zc)$ consisting of a finite input alphabet $\Xc$, finite output alphabets $\Yc$, $\Zc$, a finite [*state*]{} alphabet $\Sc$, a collection of conditional pmfs $p(y,z|x,s)$ on $\Yc\times \Zc$, and a pmf $p(s)$ on the state alphabet $\Sc$. The sender $X$ wishes to send a confidential message $M \in [1:2^{nR}]$ to the receiver $Y$ while keeping it secret from the eavesdropper $Z$ with either causal or noncausal state information available at both the encoder and decoder.
A $(2^{nR}, n)$ code for the DM-WTC with causal state information at the encoder and decoder consists of: (i) a message set $[1:2^{nR}]$, (ii) an encoder that generates a symbol $X_i(m)$ according to a conditional pmf $p(x_i|m, s^{i}, x^{i-1})$ for $i\in[1:n]$; and a decoder that assigns an estimate $\hat{M}$ or an error message to each received sequence pair $(y^n,s^n)$. We assume throughout that the message $M$ is uniformly distributed over the message set. The probability of error is defined as $P_{e}^{(n)} = \P\{\Mh \neq M\}$. The information leakage rate at the eavesdropper $Z$, which measures the amount of information about $M$ that leaks out to the eavesdropper, is defined as $R_L = \frac{1}{n} I(M;Z^n)$. A secrecy rate $R$ is said to be achievable if there exists a sequence of codes with $P_{e}^{(n)} \to 0$ and $R_L \to 0$ as $n\to \infty$. The secrecy capacity $C_{\rm S-CSI}$ is the supremum of the set of achievable rates.
We also consider the case when the state information is available noncausally at the encoder. The only change in the above definitions is that the encoder now generates a codeword $X^n(m)$ according to the conditional pmf $p(x^n|m, s^{n})$, i.e., the stochastic mapping is allowed to depend on the entire state sequence instead of just the past and present state sequence. The secrecy capacity for this scenario is denoted by $C_{\rm S-NCSI}$.
The notation used in this paper will follow that of El Gamal–Kim Lectures on Network Information Theory [@El-Gamal--Kim2010].
Summary of Main Results {#sect:3}
=======================
We summarize the results in this paper. Proofs are given in the following two sections and in the Appendix.
Lower Bound {#lower-bound .unnumbered}
-----------
The main result in this paper is the following lower bound on the secrecy capacity of the DM-WTC with causal state information available causally at both the encoder and decoder.
\[thm:1\] The secrecy capacity of the DM-WTC with state information available causally at the encoder and decoder is lower bounded as $$\begin{aligned}
C_{\rm S-CSI} & \ge \max \{ \max_{p(v|s)p(x|v,s)} \min \{I(V;Y|S) - I(V;Z|S)+ H(S|Z), I(V;Y|S)\}, \nonumber \\
& \left. \qquad \qquad \max_{p(v)p(x|v,s)} \min \{H(S|Z,V), I(V;Y|S)\}\right\}. \label{eqn:rate1}\end{aligned}$$
Note that if $S = \emptyset$, the above lower bound reduces to the secrecy capacity for the wiretap channel. Define $$\begin{aligned}
R_{\rm S-CSI-1} &= \max_{p(v|s)p(x|v,s)}\min\{I(V;Y|S) - I(V;Z|S) +H(S|Z), I(V;Y|S)\}, \\
R_{\rm S-CSI-2} &= \max_{p(v)p(x|v,s)}\min\{H(S|Z,V), I(V;Y|S)\}.\end{aligned}$$ Then, (\[eqn:rate1\]) can be expressed as $$\begin{aligned}
C_{\rm S-CSI} \ge \max\{R_{\rm S-CSI-1}, R_{\rm S-CSI-2}\}.\end{aligned}$$ The proof of this theorem is detailed in Section \[sect:4\].
In [@Liu--Chen2007], the authors established the following lower bound for the noncausal case $$\begin{aligned}
C_{\rm S-NCSI} &\ge \max_{p(u|s)p(x|u,s)} (I(U;Y,S)-\max\{I(U;Z), I(U;S)\}) \nonumber \\
& = \max_{p(u|s)p(x|u,s)}\min \left \{I(U;Y|S)-I(U;Z|S) + I(S;U|Z), I(U;Y|S)\right\}.\label{eqn:rate2}\end{aligned}$$ Clearly, $R_{\rm S-CSI-1}$ is at least as large as this lower bound. Hence, our lower bound (\[eqn:rate1\]) is at least as large as this lower bound (\[eqn:rate2\]). We now show that the lower bound (\[eqn:rate2\]) is as large as $R_{\rm S-CSI-1}$.
Fix $V\in [0:|\Vc|-1],$ $p(v|s),$ and $p(x|v,s)$ in $R_{\rm S-CSI-1}$. Let $U \in [0:|\Vc||\Sc|-1]$ in bound (\[eqn:rate2\]). Define the conditional probability mass functions: For $u = v + s|\Vc|$, let $p(u|s) = p(v|s), \, p(x|u,s) = p(x|v,s),$ and let $p(u|s) = p(x|u,s) =0$ otherwise. Under this mapping, it is easy to see that $H(S|Z,U) = 0$ and the other terms in (\[eqn:rate2\]) reduce to those in $R_{\rm S-CSI-1}$.
We now show that our lower bound (\[eqn:rate1\]) can be strictly larger than that for the noncausal case (\[eqn:rate2\])). This is done via an example for which $R_{\rm S-CSI-2}> R_{\rm S-CSI-1}$.
Consider the channel in Figure \[fig:0\], where $\Xc, \Yc, \Zc, \Sc \in \{0,1\}$ and $p(y,z|x,s) = p(y,z|x)$ with channel transition probabilities as defined in the Figure. The state $S$ is an i.i.d process that is observed by $X$ and $Y$ with $H(S) = 1 - H(0.1)$.
\[c\][$0.1$]{} \[c\][$0$]{} \[c\][$1$]{} \[c\][$X$]{} \[c\][$Z$]{} \[c\][$Y$]{} \[c\][$S$]{}
By setting $V = X$ independent of $S$ and $\P\{X = 1\} = \P\{X = 0\} = 0.5$ in $R_{\rm S-CSI-2}$, we obtain $R_{\rm S-CSI-2} = 1- H(0.1)$.
We now show that $R_{\rm S-CSI-1}$ is strictly smaller than $1 - H(0.1)$. First, note that $$\begin{aligned}
I(V;Y|S) &= H(Y|S) - H(Y|V,S) \\
& \le H(Y) - H(Y|X) \\
& = I(X;Y) \le 1-H(0.1). \end{aligned}$$ However, for $R_{\rm S-CSI-1} \ge 1-H(0.1)$, we must have $I(V;Y|S) \ge 1- H(0.1)$. Hence, we must have $I(V;Y|S) = 1- H(0.1)$. Next, consider $$\begin{aligned}
I(V;Y|S) &= H(Y|S) - H(Y|V,S)\\
& \stackrel{(a)}{\le} 1 - H(Y|V,S) \\
& \stackrel{(b)}{\le} 1 - H(Y|V,S,X) \\
& = 1-H(0.1)\end{aligned}$$ Step $(a)$ holds with equality iff $p(y|s) = 0.5$ for all $y,s \in \{0,1\}$. From the structure of the channel, this implies that $p(x|s) = 0.5$ for all $x,s \in \{0,1\}$. Step $(b)$ holds with equality iff $H(Y|X,V,S) = H(Y|V,S)$, or equivalently $I(X;Y|V,S) = 0$. This implies that given $V,S$, $X$ and $Y$ are independent, $p(x,y|v,s) = p(x|v,s)p(y|v,s)$. But since $p(x,y|v,s) = p(x|v,s)p(y|x)$, either (i) $p(x|v,s) = 0$ or (ii) $p(y|v,s) = p(y|x)$ must hold. Now, consider the pair $x=1, y=1$. Then, we must have either (i) $p(x=1|v,s) = 0$ or (ii) $p(y=1|v,s) = p(y=1|x=1) = 0.9$. In (i), $X$ is a function of $V$ and $S$. In (ii), we have $$\begin{aligned}
p(y=1|v,s) &= p(x=1|v,s)p(y=1|x=1) + (1-p(x=1|v,s))p(y=1|x=0) \\
& = 0.9p(x=1|v,s) +0.1 -0.1p(x=1|v,s) \\
& = 0.8p(x=1|v,s) +0.1. \end{aligned}$$ Using the fact that $p(y=1|v,s) = 0.9$, we have $0.8p(x=1|v,s) +0.1 = 0.9 \Rightarrow p(x=1|v,s) = 1$. This implies again that $X$ is a function of $V,S$. In both cases (i) and (ii), we see that $X$ is necessarily a function of $V$ and $S$, which implies that $Z =X$ is also a function of $V$ and $S$. Using the fact that $p(x|s) = p(z|s) =0.5$ for all $x,s$, we have $$\begin{aligned}
I(V;Z|S) &= H(Z|S) - H(Z|V,S) = H(X|S) =1.\end{aligned}$$ The first expression in $R_{\rm S-CSI-1}$ is then upper bounded by $$\begin{aligned}
I(V;Y|S) - I(V;Z|S) + H(S|Z) & \le I(V;Y|S) - I(V;Z|S) + H(S) \\
& = 1-H(0.1) -1 + 1-H(0.1) \\
& = 1-2H(0.1) <1-H(0.1).\end{aligned}$$ This shows that $R_{\rm S-CSI-1} < R_{\rm S-CSI-2}$, which completes the example.
To illustrate the main ideas of the achievability proof of Theorem 1, we provide an outline for part of the proof of the rate expression $R_{\rm S-CSI-1}$. Using the functional representation lemma [@Willems--Meulen1985], we can show that it suffices to perform the maximization in $R_{\rm S-CSI-1}$ over $p(u), p(x|v,s)$, and functions $v(u,s)$. Thus, we prove achievability for the equivalent characterization of $R_{\rm S-CSI-1}$ $$\begin{aligned}
R_{\rm S-CSI-1} \ge \max_{p(u), v(u,s),p(x|v,s)}\min\{I(U;Y,S) - I(U;Z,S) + H(S|Z), I(U;Y,S)\}. \label{eqn:1}\end{aligned}$$ We will outline the proof for the case where $I(U;Y,S) - I(U;Z,S)> 0$. Our coding scheme involves the transmission of $b-1$ independent messages over $b$ $n$-transmission blocks. We split the message $M_j$, $j \in [2:b]$, into two independent messages $M_{j0} \in [1:2^{nR_0}]$ and $M_{j1} \in [1:2^{nR_1}]$, where $R_0 + R_1 = R$. The codebook generation consists of two steps. The first step is the generation of the [*message codebook*]{}. We randomly generate $2^{nI(U;Y,S)}$ $u^n(l)$ sequences and partition them into $2^{nR_0}$ equal size bins. The codewords in each bin are further partitioned into $2^{nR_1}$ equal size sub-bins $\Cc(m_0,m_1)$. The second step is to generate the key codebook. We randomly bin the set of state sequences $s^n$ into $2^{nR_K}$ bins $\Bc(k)$. The key $K_{j-1}$ used in block $j$ is the bin index of the state sequence $\Sv(j-1)$ in block $j-1$.
To send message $M_j$, $M_{j1}$ is encrypted with the key $K_{j-1}$ to obtain the index $M'_{j1}=M_{j1}\oplus K_{j-1}$. A codeword $u^n(L)$ is selected uniformly at random from sub-bin $\Cc(M_{j0}, M_{j1}\oplus K_{j-1})$ and transmitted using Shannon’s strategy as depicted in Figure \[fig:1\]. The decoder uses joint typicality decoding together with its knowledge of the key to decode message $M_j$ at the end of block $j$. Finally, at the end of block $j$, the encoder and decoder declare the bin index $K_j$ of the state sequence $\sv(j)$ as the key to be used in block $j+1$.
\[r\][$M_j$]{} \[c\][$M_{j1}$]{} \[c\][$M_{j0}$]{} \[c\][$M'_{j1}$]{} \[b\][$K_{j-1}$]{} \[c\][$U_i$]{} \[c\][$V_i$]{} \[l\][$X_i$]{} \[c\][$p(s)$]{} \[c\][$v(u,s)$]{} \[c\][$p(x|v,s)$]{} \[l\][$S_i$]{} \[b\][$S_i$]{} ![Encoding in block $j$.[]{data-label="fig:1"}](figs/shannon.eps "fig:"){width="0.85\linewidth"}
To show that the messages can be kept asymptotically secret from the eavesdropper, note that $M_{j0}$ is transmitted using Wyner wiretap coding. Hence, it can be kept secret from eavesdropper provided $I(U;Y,S) - I(U;Z,S) >0$. The key part of the proof is to show that the second part of the message $M_{j1}$, which is encrypted with the key $K_{j-1}$, can be kept secret from the eavesdropper. This involves showing that the eavesdropper has negligible information about $K_{j-1}$. However, the fact that $K_{j-1}$ is generated from the state sequence in block $j-1$ and used in block $j$ results in correlation between it and all received sequences at the eavesdropper from subsequent blocks. We show that the eavesdropper has negligible information about $K_{j-1}$ given all its received sequences provided $R_K < H(S|Z)$.
Upper Bound {#upper-bound .unnumbered}
-----------
We establish the following upper bound on the secrecy capacity of the wiretap channel with noncausal state information available at both the encoder and decoder (which holds also for the causal case).
\[thm:2\] The following is an upper bound to the secrecy capacity of the DM-WTC with state noncausally available at the encoder and decoder $$\begin{aligned}
C_{\rm S-NCSI} \le \min\left \{I(V_1;Y|U,S) - I(V_1;Z|U,S)+ H(S|Z,U), I(V_2;Y|S)\right\}.\end{aligned}$$ for some $U,$ $V_1$ and $V_2$ such that $p(u,v_1,v_2,x|s) = p(u|s)p(v_1|u,s)p(v_2|v_1,s)p(x|v_2,s).$
The proof of this theorem is given in Section \[sect:5\].
Secrecy Capacity Results {#secrecy-capacity-results .unnumbered}
------------------------
1\. Following the lines of [@Chen--Vinck2006], we can show that Theorems \[thm:1\] and \[thm:2\] are tight for the following two special cases.
- If there exists a $V^*$ such that $\max_{p(v|s)p(x|v,s)}(I(V;Y|S) - I(V;Z|S) + H(S|Z)) = I(V^*;Y|S) - I(V^*;Z|S) + H(S|Z)$ and $I(V^*;Y|S) - I(V^*;Z|S) + H(S|Z) \le I(V^*;Y|S)$, then the secrecy capacity is $C_{\rm S-CSI}=C_{\rm S-NCSI} = I(V^*;Y|S) - I(V^*;Z|S) + H(S|Z)$.
- If there exists a $V'$ such that $\max_{p(v|s)p(x|v,s)}I(V;Y|S)= I(V';Y|S)$ and $I(V';Y|S) \le I(V';Y|S) - I(V';Z|S) + H(S|Z)$, then the secrecy capacity is $C_{\rm S-CSI}=C_{\rm S-NCSI} = I(V';Y|S)$.
2\. We show that Theorems \[thm:1\] and \[thm:2\] are also tight when $I(U;Y|S)\ge I(U;Z|S)$ for $U$ such that $(U,S) \to (X,S) \to (Y,Z)$ form a Markov chain, i.e., when $Y$ is [*less noisy*]{} than $Z$ for every state $s\in\Sc$ [@Korner--Marton].
\[thm:3\] The secrecy capacity for the DM-WTC with the state information available causally or noncausally at the encoder and decoder when $Y$ is [less noisy]{} than $Z$ is $$\begin{aligned}
C_{\rm S-CSI}& =C_{\rm S-NCSI} = \max_{p(x|s)}\min\{I(X;Y|S)- I(X;Z|S) + H(S|Z), I(X;Y|S)\}.\end{aligned}$$
Consider the special case when $p(y,z|x,s) = p(y,z|x)$ and $Z$ is a degraded version of $Y$, then Theorem \[thm:3\] specializes to the secrecy capacity for the wiretap channel with a key [@Yamamoto1997] $$\begin{aligned}
& C_{\rm S-CSI}=C_{\rm S-NCSI} \\
& = \max_{p(x)} \min\{I(X;Y) - I(X;Z) + H(S), I(X;Y)\}.\end{aligned}$$
Achievability for Theorem \[thm:3\] follows directly from Theorem \[thm:1\] by setting $V= X$ and observing that the expression reduces to $R_{\rm S-CSI-1}$ since $Y$ is less noisy than $Z$. To establish the converse, we use the less noisy assumption to strengthen the first inequality in Theorem \[thm:2\] as follows $$\begin{aligned}
I(V_1;Y|U,S) - I(V_1;Z|U,S) + H(S|Z,U) &\le I(V_1;Y|U,S) - I(V_1;Z|U,S) + H(S|Z) \\
&\stackrel{(a)}{\le} I(V_1;Y|S) - I(V_1;Z|S) + H(S|Z) \\
& \stackrel{(b)}{\le} I(X;Y|S) - I(X;Z|S) + H(S|Z),\end{aligned}$$ where $(a), (b)$ follow from the less noisy assumption. The proof of the second inequality follows by the data processing inequality: $I(V_2;Y|S) \le I(X;Y|S)$.
3\. Next, consider the case where $p(y,z|x,s) = p(y,z|x)$ and the eavesdropper $Z$ is [*less noisy*]{} [@Korner--Marton] than $Y.$ That is, $I(U;Z) \ge I(U;Y)$ for every $U$ such that $U \to X \to (Y,Z).$ Then, the capacity of this special class of channels is $$\begin{aligned}
C_{\rm S-CSI} = C_{\rm S-NCSI}= \max_{p(x)}\min\{H(S), I(X;Y)\}.\end{aligned}$$ Achievability follows by setting $V=X$ independent of $S$. The converse follows from Theorem \[thm:2\] and the observation that since $Z$ is less noisy than $Y$ and $p(y,z|x,s) = p(y,z|x),$ $$\begin{aligned}
I(V_1;Y|U,S) - I(V_1;Z|U,S) + H(S|Z, U) &\le H(S|Z,U) \\
& \le H(S),\end{aligned}$$ and $I(V_2;Y|S) \le I(X;Y)$.
Proof of Theorem \[thm:1\] {#sect:4}
==========================
We will prove the achievability of $R_{\rm S-CSI-1}$ and $R_{\rm S-CSI-2}$ separately. For $R_{\rm S-CSI-1}$, we will prove the equivalent expression stated in equation \[eqn:1\].
The proof of achievability for $R_{\rm S-CSI-1}$ is split into two cases (Cases 1 and 2) while $R_{\rm S-CSI-2}$ is proved in Case 3.
Case 1: $R_{\rm S-CSI-1}$ with $I(U;Y,S) > I(U;Z,S)$ {#case-1-r_rm-s-csi-1-with-iuys-iuzs .unnumbered}
----------------------------------------------------
### Codebook generation {#codebook-generation .unnumbered}
Split message $M_j$ into two independent messages $M_{j0}\in [1:2^{nR_0}]$ and $M_{j1} \in [1:2^{nR_1}]$, thus $R=R_0+R_1$. Let $\Rt \ge R$. The codebook generation consists of two steps.
[*Message codeword generation*]{}: We randomly and independently generate $2^{n\Rt}$ sequences $u^n(l)$, $l \in [1:2^{n\Rt}]$, each according to $\prod_{i=1}^n p(u_i)$ and partition them into $2^{nR_0}$ equal-size bins $\Cc(m_0)$, $m_0 \in [1:2^{nR_0}]$. We further partition the sequences within each bin $\Cc(m_0)$ into $2^{nR_K}$ equal size sub-bins, $\Cc(m_0, m_1)$, $m_1 \in [1:2^{nR_1}]$.
[*Key codebook generation*]{}: We randomly and uniformly partition the set of $s^n$ sequences into $2^{nR_K}$ bins $\Bc(k)$, $k\in [1:2^{nR_K}]$.
Both codebooks are revealed to all parties.
### Encoding {#encoding .unnumbered}
We send $b-1$ messages over $b$ $n$-transmission blocks. In the first block, we randomly select a sequence $u^n(L) \in \Cc(m_{10},m'_{11})$. The encoder then computes $v_i= v(u_i(L), s_i)$ and transmits a randomly generated symbol $X_i \sim p(x_i|s_i,v_i)$ for $i \in [1:n]$. At the end of the first block, the encoder and decoder declare $k_1 \in [1:2^{nR_K}]$ such that $\sv(1) \in \Bc(k_1)$ as the key to be used in block 2.
Encoding in block $j \in [2:b]$ proceeds as follows. To send message $m_j=(m_{j0},m_{j1})$ and given key $k_{j-1}$, the encoder computes $m'_{j1} = m_{j1}\oplus k_{j-1}$. To ensure secrecy, we must have $R_1 \le R_K$ [@Shannon1949]. The encoder then randomly selects a sequence $u^n(L) \in \Cc(m_{j0}, m'_{j1})$. It then computes $v_i= v(u_i(L), s_i)$ and transmits a randomly generated symbol $X_i \sim p(x_i|s_i,v_i)$ for $i \in [(j-1)n+1:jn]$.
### Decoding and analysis of the probability of error {#decoding-and-analysis-of-the-probability-of-error .unnumbered}
At the end of block $j$, the decoder declares that $\lh$ is sent if it is the unique index such that $(u^n(\lh), \Yv(j), \Sv(j))\in \aep$, otherwise it declares an error. It then finds the indices $(\mh_{j0},\mh'_{j1})$ such that $u^n(l) \in \Cc(\mh_{j0},\mh'_{j1})$. Finally, it recovers $\mh_{j1}$ by computing $\mh_{j1} = \mh_{j1}'\oplus k_{j-1}$.
To analyze the probability of error, let $\e'' > \e' > \e>0$ and define the following events for every $j\in [2:b]$: $$\begin{aligned}
\Ec(j) &= \{\Mh_j \neq M_j\}, \\
%\Ec(j) &= \{\Sv(j) \notin \aep \text{ or } \Mh_j \neq M_j\}, \\
%\Ec_1(j) & = \{\Sv(j) \notin \aep\}, \\
\Ec_1(j) & = \{(U^n(L), \Sv(j)) \notin \mathcal{T}^n_{\e'}\}, \\
\Ec_2(j) & = \{(U^n(L),\Sv(j), \Yv(j)) \notin \mathcal{T}^n_{\e''}\}, \\
\Ec_3(j) & = \{(U^n(\lh),\Sv(j), \Yv(j)) \in \mathcal{T}^n_{\e''} \text{ for some } \lh \neq L\}.\end{aligned}$$ The probability of error is upper bounded as $$\begin{aligned}
\P(\Ec) & = \P\{\cup_{j=2}^b \Ec(j)\} \le \sum_{j=2}^b \P(\Ec(j)).\end{aligned}$$ Each probability of error term can be upper bounded as $$\begin{aligned}
\P(\Ec(j)) &\le \P(\Ec_1(j)) + \P(\Ec_2(j)\cap\Ec^c_1(j)) + \P(\Ec_3(j)\cap\Ec^c_2(j)). %+\P(\Ec_4\cap\Ec_3(j)^c).\end{aligned}$$ Now, $\P(\Ec_1(j))\to 0$ as $n\to \infty$ by Law of Large Numbers (LLN) since $\P\{(U^n(L) \in \aep)\} \to 1$ as $n \to \infty$ and $\Sv(j)\sim\prod_{i=1}^n p(s_i) = \prod_{i=1}^n p(s_i|u_i)$ by independence. The term $\P(\Ec_2(j)\cap\Ec^c_1(j))\to 0$ as $n \to \infty$ by LLN since $(U^n(L), \Sv(j) \in \mathcal{T}_{e'}^n$ and $Y^n\sim\prod_{i=1}^n p(y_i|u_i, s_i)$. For the last term, consider [$$\begin{aligned}
\P(\Ec_3\cap\Ec^c_2(j)) &\le \sum_{l}p(l) \sum_{\lh\neq l} \P\{(U^n(\lh),\Sv(j), \Yv(j)) \in \mathcal{T}^n_{\e''}|\Ec_2^c(j), L=l\} \\
&\stackrel{(a)}{\le} \sum_{\lh \neq l} 2^{-n(I(U;Y,S) - \d(\e''))} \le 2^{n(\Rt - I(U;Y,S) + \d(\e''))},\end{aligned}$$ where $(a)$ follows from: (i) $L$ is independent of the transmission codebook sequences $U^n$ and the current state sequence $\Sv(j)$; and (ii) the conditional joint typicality lemma [@El-Gamal--Kim2010 Lecture 2]. Hence, $\P(\Ec_3\cap\Ec^c_2(j))\to 0$ as $n\to \infty$ if $\Rt < I(U;Y,S) - \d(\e'')$.]{}
### Analysis of the information leakage rate {#analysis-of-the-information-leakage-rate .unnumbered}
We use $\Zv^j$ to denote the eavesdropper’s received sequence from blocks 1 to $j$ and $\Zv(j)$ to denote the received sequence in block $j$. We will need the following two results.
\[prop1\] If $R_{K} < H(S|Z) - 4\d(\e)$ and $\Rt \ge I(U;Z,S)$, then the following holds for every $j\in [1:b]$.
1. $H(K_{j}|\Cc) \ge n(R_K - \d(\e))$.
2. $I(K_j; \Zv(j)|\Cc) \le 2n\d(\e)$.
3. $I(K_j;\Zv^{j}|\Cc) \le n\d'(\e)$, where $\d(\e)\to 0$ and $\d'(\e)\to 0$ as $\e \to 0$.
The proof of this proposition is given in Appendix \[appen1\].
[ [@Chia--El-Gamal_a]]{} \[lem1\] Let $(U,V,Z) \sim p(u,v,z)$, $\bar{R} \ge 0$ and $\e >0$. Let $U^n$ be a random sequence distributed according to $\prod_{i=1}^np(u_i)$. Let $V^n(l)$, $l \in [1:2^{n\bar{R}}]$, be a set of random sequences that are conditionally independent given $U^n$ and each distributed according to $\prod_{i=1}^n p(v_i|u_i)$. Let $L$ be a random index with an arbitrary distribution over $[1:2^{n\bar{R}}]$ independent of $(U^n, V^n(l)), l \in [1:2^{n\bar{R}}]$. Then, if $\P\{(U^n,V^n(L), Z^n)\in \aep\}\to 1$ as $n\to \infty$ and $\bar{R} \geq I(V;Z|U)$, there exists a $\d(\e)>0$, where $\d(\e) \to 0$ as $\e \to 0$, such that $H(L|Z^n,U^n) \leq n(\bar{R} - I(V;Z|U)) +n\d(\e)$.
We are now ready to upper bound the leakage rate averaged over codes. Consider $$\begin{aligned}
I(M_2,M_3, \ldots, M_b;\Zv^b|\Cc)& = \sum_{j=2}^b I(M_{j};\Zv^b|\Cc, M_{j+1}^b) \\
& \stackrel{(a)}{\le} \sum_{j=2}^b I(M_{j};\Zv^b|\Cc, \Sv(j), M_{j+1}^b) \\
& \stackrel{(b)}{=} \sum_{j=2}^b I(M_{j};\Zv^j|\Cc, \Sv(j)),\end{aligned}$$ where $(a)$ follows by the independence of $M_{j}$ and $(\Sv(j), M_{j+1}^b)$, and $(b)$ follows by the Markov Chain relation $(\Zv_{j+1}^b, M_{j+1}^b,\Cc) \to (\Zv^j,\Sv(j),\Cc) \to (M_j, \Cc)$. Hence, it suffices to upper bound each individual term $I(M_{j};\Zv^j|\Cc, \Sv(j))$. Consider $$\begin{aligned}
I(M_{j};\Zv^j|\Cc, \Sv(j)) & = I(M_{j0}, M_{j1};\Zv^j|\Cc, \Sv(j)) \\
& = I(M_{j0}, M_{j1};\Zv^{j-1}|\Cc, \Sv(j)) + I(M_{j0}, M_{j1};\Zv(j)|\Cc, \Sv(j), \Zv^{j-1}).\end{aligned}$$ Note that the first term is equal to zero by the independence of $M_j$ and past transmissions, the codebook, and state sequence. For the second term, we have $$\begin{aligned}
I(M_{j0}, M_{j1};\Zv(j)|\Cc, \Sv(j), \Zv^{j-1}) &= I(M_{j0};\Zv(j)|\Cc, \Sv(j), \Zv^{j-1}) + I(M_{j1};\Zv(j)|\Cc, M_{j0}, \Sv(j), \Zv^{j-1}).\end{aligned}$$ We now bound the each term separately. Consider the first term
$$\begin{aligned}
I(M_{j0};\Zv(j)|\Cc, \Sv(j), \Zv^{j-1}) & = I(M_{j0}, L; \Zv(j)|\Cc, \Sv(j), \Zv^{j-1}) - I(L; \Zv(j)|\Cc, M_{j0}, \Sv(j), \Zv^{j-1}) \\
& \le I(U^n;\Zv(j)|\Cc, \Sv(j), \Zv^{j-1}) -H(L|\Cc, M_{j0}, \Sv(j), \Zv^{j-1}) \\
& \qquad + H(L|\Zv(j), M_{j0}, \Sv(j)) \\
& \le \sum_{i=1}^n(H(\Zv_i(j)|\Cc, \Sv_i(j)) - H(\Zv_i(j)|\Cc, U_i, \Sv_i(j))) \\
& \qquad -H(L|\Cc, M_{j0}, \Sv(j), \Zv^{j-1})+ H(L|\Zv(j), M_{j0}, \Sv(j)) \\
& \stackrel{(a)}{\le} nI(U;Z|S)-H(L|\Cc, M_{j0}, \Sv(j), \Zv^{j-1}) + H(L|\Zv(j), M_{j0}, \Sv(j)) \\
& \stackrel{(b)}{\le} nI(U;Z|S) -H(L|\Cc, M_{j0}, \Sv(j), \Zv^{j-1}) \\ & \quad + n(\Rt - R_0 - I(U;Z,S) + \d(\e)) \\
& \stackrel{(c)}{=} n(\Rt-R_0) - H(L|\Cc, M_{j0}, \Sv(j), \Zv^{j-1}) +n\d(\e) \\
& = n(\Rt-R_0) - H(M_{j1}\oplus K_{j-1}|\Cc, M_{j0}, \Sv(j), \Zv^{j-1}) \\
& \qquad - H(L|\Cc, M_{j0}, \Sv(j), \Zv^{j-1}, M_{j1}\oplus K_{j-1}) +n\d(\e) \\
& \le n(\Rt-R_0) - H(M_{j1}\oplus K_{j-1}|\Cc, M_{j0}, \Sv(j), K_{j-1}, \Zv^{j-1}) \\
& \qquad - n(\Rt-R_0 - R_K) +n\d(\e) \\
& \stackrel{(d)}{=} nR_K - H(M_{j1}\oplus K_{j-1}|\Cc, M_{j0}, \Sv(j), K_{j-1}) + n\d(\e)\\
& = nR_K - H(M_{j1}|\Cc, M_{j0}, \Sv(j), K_{j-1}) + n\d(\e) = n\d(\e),\end{aligned}$$ where $(a)$ follows from the fact that $H(\Zv_i(j)|\Cc, \Sv_i(j)) \le H(\Zv_i(j)|\Sv_i(j))= H(Z|S)$ and $H(\Zv_i(j)|\Cc, U_i, \Sv_i(j)) = H(Z|U,S)$. Step $(b)$ follows by Lemma 1 which requires that (i) $\P\{(U^n(L),\Sv(j), \Zv(j)) \in \aep\}\to 1$ as $n\to \infty$, and (ii) $\Rt - R_0 \ge I(U;Z,S)$; where (i) can be shown using the same steps as in the analysis of probability of error. Step $(c)$ follows by the independence of $U$ and $S$. Step $(d)$ follows from the Markov Chain relation $(\Zv^{j-1}, M_{j0}, \Sv(j)) \to (K_{j-1}, M_{j0}, \Sv(j)) \to (M_{j1}\oplus K_{j-1}, M_{j0}, \Sv(j))$. The last step follows by the fact that $M_{j1}$ is independent of $(\Cc, M_{j0}, \Sv(j), K_{j-1})$ and uniformly distributed over $[1:2^{nR_K}]$.
Next, consider the second term $$\begin{aligned}
I(M_{j1};\Zv(j)|\Cc, M_{j0}, \Sv(j), \Zv^{j-1}) &\le I(M_{j1},L;\Zv(j)|\Cc, M_{j0}, \Sv(j), \Zv^{j-1}) \\
& \qquad - I(L;\Zv(j)|\Cc, M_{j0}, M_{j1}, \Sv(j), \Zv^{j-1}) \\
& \le I(U^n;\Zv(j)|\Cc, M_{j0}, \Sv(j), \Zv^{j-1}) - H(L|\Cc, M_{j0}, M_{j1}, \Sv(j), \Zv^{j-1}) \\
& \qquad + H(L|\Cc, M_{j0}, M_{j1}, \Sv(j), \Zv^{j}) \\
& \stackrel{(a)}{\le} nI(U;Z|S) - H(L|\Cc, M_{j0}, M_{j1}, \Sv(j), \Zv^{j-1}) \\
& \qquad + H(L|\Cc, M_{j0}, M_{j1}, \Sv(j), \Zv^{j}) \\
& \le nI(U;Z|S) - H(L|\Cc, M_{j0}, M_{j1}, \Sv(j), \Zv^{j-1}) + H(L|M_{j0}, \Sv(j), \Zv(j)) \\
& \stackrel{(b)}{\le} nI(U;Z|S) - H(L|\Cc, M_{j0}, M_{j1}, \Sv(j), \Zv^{j-1}) \\
& \qquad + n(\Rt-R_0) - nI(U;Z,S)+n\d(\e) \\
& = n(\Rt-R_0) - H(L|\Cc, M_{j0}, M_{j1}, \Sv(j), \Zv^{j-1}) + n\d(\e),\end{aligned}$$ where $(a)$ follows from the same steps used in bounding $I(M_{j0};\Zv(j)|\Cc, \Sv(j), \Zv^{j-1})$; $(b)$ follows from Lemma 1. Next consider $$\begin{aligned}
H(L|\Cc, M_{j0}, M_{j1}, \Sv(j), \Zv^{j-1}) &= H(M_{j1}\oplus K_{j-1}|\Cc, M_{j0}, M_{j1}, \Sv(j), \Zv^{j-1}) \\
& \quad + H(L|\Cc, M_{j0}, M_{j1}, M_{j1}\oplus K_{j-1}, \Sv(j), \Zv^{j-1}) \\
& = H(K_{j-1}|\Cc, M_{j0}, M_{j1}, \Sv(j), \Zv^{j-1}) + n(\Rt - R_0 - R_K) \\
& = H(K_{j-1}|\Cc, \Zv^{j-1}) + n(\Rt - R_0 - R_K).\end{aligned}$$
From Proposition \[prop1\], $H(K_{j-1}|\Cc, \Zv^{j-1}) \ge n(R_K -\d(\e)-\d'(\e))$, which implies that $$\begin{aligned}
I(M_{j1};\Zv(j)|\Cc, M_{j0}, \Sv(j), \Zv^{j-1}) &\le n(\d'(\e) + 2\d(\e)).\end{aligned}$$ This completes the analysis of information leakage rate.
### Rate analysis {#rate-analysis .unnumbered}
From the analysis of probability of error and information leakage rate, we see that the rate constraints are $$\begin{aligned}
\Rt &< I(U;Y,S) -\d(\e), \\
\Rt -R_0 &\ge I(U;Z,S), \\
R_K &< H(S|Z) -4\d(\e), \\
R_0+R_1 &\le \Rt, \\
R_1 &\le R_K, \\
R &= R_0 + R_1.\end{aligned}$$ Using Fourier-Motzkin elimination (see for e.g. Lecture 6 of [@El-Gamal--Kim2010]), we obtain $$\begin{aligned}
R &< \max_{p(u), v(u,s), x(u,s)}\min \{I(U;Y,S) - I(U;Z,S) + H(S|Z), I(U;Y,S)\} \\
& \stackrel{(a)}{=} \max_{p(u), v(u,s), p(x|s,v)}\min \{I(V;Y|S) - I(V;Z|S) + H(S|Z), I(V;Y|S)\},
%& \stackrel{(b)}{=} \max_{p(v|s)p(x|s,v)}\min \{I(V;Y|S) - I(V;Z|S) + H(S|Z), I(V;Y|S)\},\end{aligned}$$ where $(a)$ follows by the independence of $U$ and $S$ and the fact that $V$ is a function of $U$ and $S$.
Case 2: $R_{\rm S-CSI-1}$ with $I(U;Y,S) \le I(U;Z,S)$ {#case-2-r_rm-s-csi-1-with-iuys-le-iuzs .unnumbered}
------------------------------------------------------
Under this condition, the decoder cannot rely on the wiretap channel to send a confidential message. Therefore, only the key is used to encrypt the message and transmit it securely. Note that we only need to consider the case where $H(S|Z) - (I(U;Z,S)- I(U;Y,S)) >0$.
### Codebook generation {#codebook-generation-1 .unnumbered}
Codebook generation again consists of two steps.
[*Message codebook generation*]{}: Let $\Rt \ge R_d$ and $R \le \Rt-R_d$. Randomly and independently generate $2^{n\Rt}$ sequences $u^n(l)$, $l \in [1:2^{n\Rt}]$, each according to $\prod_{i=1}^n p(u_i)$ and partition them into $2^{nR_d}$ equal-size bins $\Cc(m_d)$, $m_d \in [1:2^{nR_d}]$. We further partition the set of sequences in each bin $\Cc(m_d)$ into sub-bins, $\Cc(m_d, m)$, $m \in [1:2^{nR}]$.
[*Key codebook generation*]{}: We randomly bin the set of $s^n \in \Sc^n$ sequences into $2^{nR_K}$ bins $\Bc(k)$, $k\in [1:2^{nR_K}]$.
### Encoding {#encoding-1 .unnumbered}
We send $b-1$ messages over $b$ $n$-transmission blocks. In the first block, we randomly select a $u^n(L)$ sequence. The encoder then computes $v_i= v(u_i(L), s_i)$, $i \in [1:n]$, and transmits a randomly generates sequence $X^n$ according to $\prod_{i=1}^n p(x_i|s_i,v_i)$. At the end of the first block, the encoder and decoder declare $k_1 \in [1:2^{nR_K}]$ such that $\sv(1) \in \Bc(k_1)$ as the key to be used in block 2.
Encoding in block $j \in [2:b]$ is as follows. We split the key $k_{j-1}$ into two independent parts, $K_{j-1,d}$ and $K_{j-1, m}$ at rates $R_d$ and $R$, respectively. To send message $m_j$, the encoder computes $m' = m_j\oplus k_{(j-1)m}$. This requires that $R_K \ge R + R_d$. The encoder then randomly selects a sequence $u^n(L) \in \Cc(k_{(j-1)d}, m')$. At time $i \in [(j-1)n+1:jn]$, it computes $v_i= v(u_i(L), s_i)$, and transmits a randomly generated symbol $X_i$ according to $p(x_i|s_i,v_i)$.
### Decoding and analysis of the probability of error {#decoding-and-analysis-of-the-probability-of-error-1 .unnumbered}
At the end of block $j$, the decoder declares that $\lh$ is sent if it is the unique index such that $(u^n(\lh), \Yv(j), \Sv(j))\in \aep$ and $u^n(\lh) \in \Cc(k_{(j-1)d})$. Otherwise, it declares an error. It then finds the index $\mh'$ such that $u^n(\lh) \in \Cc(k_{(j-1)d}, \mh')$. Finally, it recovers $\mh_j$ by computing $\mh_j = \mh' \oplus k_{(j-1)m}$. Following similar steps to the analysis for Case 1, it can be shown that $\P_e \to 0$ as $n \to \infty$ if $\Rt - R_d < I(U;Y,S) - \d(\e)$.
### Analysis of the information leakage rate {#analysis-of-the-information-leakage-rate-1 .unnumbered}
Following the same steps as for Case 1, we can show that it suffices to upper bound the terms $I(M_j;\Zv(j)|\Cc, \Sv(j), \Zv^{j-1})$ for $j \in [2:b]$. Consider $$\begin{aligned}
I(M_j;\Zv(j)|\Cc, \Sv(j), \Zv^{j-1}) &= H(M_j) - H(M_j|\Cc, \Sv(j), \Zv^{j}) \\
&\le H(M_j) - H(M_j|\Cc, \Sv(j), K_{(j-1)d}, M_{j}\oplus K_{(j-1)m},\Zv^{j}) \\
&= H(M_j) - H(M_j|\Cc, K_{(j-1)d}, M_{j}\oplus K_{(j-1)m},\Zv^{j-1}) \\
&= H(M_j) - H(M_j|\Cc, \Zv^{j-1}, K_{(j-1)d})- H(M_{j}\oplus K_{(j-1)m}|\Cc, \Zv^{j-1},K_{(j-1)d},M_j)\\
&\qquad +H(M_{j}\oplus K_{(j-1)m}|\Cc, \Zv^{j-1}, K_{(j-1)d}) \\
& = nR -H(M_j) +H(M_{j}\oplus K_{(j-1)m}|\Cc, \Zv^{j-1}, K_{(j-1)d}) \\
& \quad - H(M_{j}\oplus K_{(j-1)m}|\Cc, \Zv^{j-1},K_{(j-1)d},M_j)\\
& \le nR - H(K_{(j-1)m}|\Cc,\Zv^{j-1}, K_{(j-1)d}).\end{aligned}$$ Thus, showing that $$\begin{aligned}
I(K_{(j-1)m};\Zv^{j-1}|\Cc, K_{(j-1)d}) &\le n\d'(\e), \label{case2_1}\\
H(K_{(j-1)m}|\Cc, K_{(j-1)d}) &\ge n(R_K-R_d - \d(\e)) \label{case2_2}\end{aligned}$$ implies $$\begin{aligned}
I(M_j;\Zv(j)|\Cc, \Sv(j), \Zv^{j-1}) \le nR - n(R_K - R_d) + n(\d'(\e)+\d(\e)).\end{aligned}$$ Hence, the rate of information leakage approaches zero as $n\to \infty$ if $R \le R_K - R_d$. To prove (\[case2\_1\]) and (\[case2\_2\]), we need the following Proposition.
\[prop2\] If $\Rt \ge I(U;Z,S)$ and $R_K < H(S|Z) -4\d(\e)$, then for all $j\in [1:b]$,
1. $H(K_{j}|\Cc) \ge n(R_K - \d(\e))$.
2. $I(K_j; \Zv(j)|\Cc) \le 3n\d(\e)$.
3. $I(K_{j};\Zv^{j}|\Cc) \le n\d'(\e)$, where $\d(\e)\to 0$ and $\d'(\e)\to 0$ as $\e \to 0$.
The proof of this Proposition is given in Appendix \[appen2\].
Part 3 of Proposition \[prop2\] implies (\[case2\_1\]), since $$\begin{aligned}
I(K_{j-1};\Zv^{j-1}|\Cc) &= I(K_{(j-1)d}, K_{(j-1)m};\Zv^{j-1}|\Cc) \\
& = I(K_{(j-1)d};\Zv^{j-1}|\Cc) + I(K_{(j-1)m};\Zv^{j-1}|\Cc, K_{(j-1)d}).\end{aligned}$$
Part 1 of Proposition 2 implies (\[case2\_2\]), since $H(K_{(j-1)}|\Cc) = H(K_{(j-1)m}, K_{(j-1)d}|\Cc)\ge n(R_K - \d(\e))$, which implies that $H(K_{(j-1)m}|\Cc, K_{(j-1)d}) \ge n(R_K-R_d - \d(\e))$.
### Rate analysis {#rate-analysis-1 .unnumbered}
The following rate constraints are necessary for Case 2.[$$\begin{aligned}
\Rt &\ge I(U;Z,S), \\
\Rt - R_d & < I(U;Y,S) - \d(\e), \\
R &\le \Rt - R_d, \\
R_K &< H(S|Z) -4\d(\e), \\
R &\le R_K -R_d.\end{aligned}$$ ]{} Using Fourier Motzkin elimination, we obtain $$\begin{aligned}
R &< \max_{p(u), v(u,s)}\min \{I(U;Y,S) - I(U;Z,S) + H(S|Z), I(U;Y,S)\} \\
& = \max_{p(u), v(u,s), p(x|s,v)}\min \{I(V;Y|S) - I(V;Z|S) + H(S|Z), I(V;Y|S)\}.
%& = \max_{p(v|s)p(x|s,v)}\min \{I(V;Y|S) - I(V;Z|S) + H(S|Z), I(V;Y|S)\}.\end{aligned}$$
Case 3: $R_{\rm S-CSI-2}$ {#case-3-r_rm-s-csi-2 .unnumbered}
-------------------------
For $R_{\rm S-CSI-2}$, the key generated in a block is used purely to encrypt the message in the following block. This implies that there is a possibility that the eavesdropper can decode the codeword transmitted in the current block, which reduces the key rate that can be generated at the current block. This is compensated for by the fact that the entire key is used for message transmission. The codebook generation, encoding and analysis of probability of error and equivocation are therefore similar to that in Case 2.
### Codebook generation {#codebook-generation-2 .unnumbered}
Codebook generation again consists of two steps.
[*Message codebook generation*]{}: Randomly and independently generate $2^{nR}$ sequences $v^n(l)$, $l \in [1:2^{nR}]$, each according to $\prod_{i=1}^n p(v_i).$
[*Key codebook generation*]{}: Set $R_K = R.$ We randomly bin the set of $s^n \in \Sc^n$ sequences into $2^{nR_K}$ bins $\Bc(k)$, $k\in [1:2^{nR_K}]$.
### Encoding {#encoding-2 .unnumbered}
We send $b-1$ messages over $b$ $n$-transmission blocks. In the first block, we randomly select a $v^n(L)$ sequence. The encoder then transmits a randomly generated sequence $X^n$ according to $\prod_{i=1}^n p(x_i|s_i,v_i)$. At the end of the first block, the encoder and decoder declare $k_1 \in [1:2^{nR_K}]$ such that $\sv(1) \in \Bc(k_1)$ as the key to be used in block 2.
Encoding in block $j \in [2:b]$ is as follows. To send message $m_j$, the encoder computes $m' = m_j\oplus k_{j-1}$. The encoder then selects the sequence $v^n(m')$. At time $i \in [(j-1)n+1:jn]$, it transmits a randomly generated symbol $X_i$ according to $p(x_i|s_i,v_i)$.
### Decoding and analysis of the probability of error {#decoding-and-analysis-of-the-probability-of-error-2 .unnumbered}
At the end of block $j$, the decoder declares that $\mh'$ is sent if it is the unique index such that $(v^n(\mh'), \Yv(j), \Sv(j))\in \aep$. Otherwise, it declares an error. It then recovers $\mh_j$ by computing $\mh_j = \mh' \oplus k_{j-1}$. Following similar steps to the analysis for Case 1, it can be shown that $\P_e \to 0$ as $n \to \infty$ if $R < I(V;Y,S) - \d(\e)$.
### Analysis of the information leakage rate {#analysis-of-the-information-leakage-rate-2 .unnumbered}
Following the same steps as for Case 1, we can show that it suffices to upper bound the terms $I(M_j;\Zv(j)|\Cc, \Sv(j), \Zv^{j-1})$ for $j \in [2:b]$. Consider $$\begin{aligned}
I(M_j;\Zv(j)|\Cc, \Sv(j), \Zv^{j-1}) &= H(M_j) - H(M_j|\Cc, \Sv(j), \Zv^{j}) \\
&\le H(M_j) - H(M_j|\Cc, \Sv(j), M_{j}\oplus K_{j-1},\Zv^{j}) \\
& = H(M_j) - H(M_j|\Cc, M_{j}\oplus K_{j-1},\Zv^{j-1}) \\
& = H(M_j) - H(M_{j}\oplus K_{j-1}, M_{j}|\Cc, \Zv^{j-1}) + H(M_j\oplus K_{j-1}|\Cc, \Zv^{j-1}) \\
& \le nR - H(M_{j}|\Cc, \Zv^{j-1}) - H(M_j\oplus K_{j-1}|\Cc,\Zv^{j-1}, M_j) + nR \\
& = nR - H(K_{j-1}|\Cc,\Zv^{j-1}).\end{aligned}$$ Thus, showing that $$\begin{aligned}
I(K_{j-1};\Zv^{j-1}|\Cc) &\le n\d'(\e), \label{case3_1}\\
H(K_{j-1}|\Cc) &\ge n(R_K - \d(\e)) \label{case3_2}\end{aligned}$$ implies $$\begin{aligned}
I(M_j;\Zv(j)|\Cc, \Sv(j), \Zv^{j-1}) \le n(\d'(\e)+\d(\e)).\end{aligned}$$ To prove (\[case3\_1\]) and (\[case3\_2\]), we will use the following Proposition
\[prop3\] If $R_K < H(S|Z, V) -4\d(\e)$, then for all $j\in [1:b]$,
1. $H(K_{j}|\Cc) \ge n(R_K - \d(\e))$.
2. $I(K_j; \Zv(j)|\Cc) \le 3n\d(\e)$.
3. $I(K_{j};\Zv^{j}|\Cc) \le n\d'(\e)$, where $\d(\e)\to 0$ and $\d'(\e)\to 0$ as $\e \to 0$.
The proof of this Proposition is given in Appendix \[appen3\]. It is clear that equations (\[case3\_1\]) and (\[case3\_2\]) are implied by Proposition \[prop3\], which completes the analysis of information leakage rate.
### Rate analysis {#rate-analysis-2 .unnumbered}
The following rate constraints are necessary for Case 3.[$$\begin{aligned}
R &= R_K, \\
R &< I(V;Y,S) - \d(\e), \\
R_K &< H(S|Z,V) -4\d(\e).\end{aligned}$$ ]{} These constraints imply the achievability of $$\begin{aligned}
R &< \max_{p(v)p(x|s,v)}\min \{H(S|Z,V), I(V;Y|S)\}.\end{aligned}$$
Proof of Theorem \[thm:2\] {#sect:5}
==========================
For any sequence of codes with probability of error and leakage rate that approach zero as $n \to \infty$, consider [$$\begin{aligned}
nR & = H(M) \stackrel{(a)}{\le} I(M;Y^n,S^n) + n\e_n \\
& \stackrel{(b)}{\le} I(M;Y^n,S^n) - I(M;Z^n) + 2n\e_n \\
& = \sum_{i=1}^n(I(M;Y_i,S_i|Y_{i+1}^n, S_{i+1}^n) - I(M;Z_i|Z^{i-1})) + 2n\e_n\\
& \stackrel{(c)}{=}\sum_{i=1}^n(I(M, Z^{i-1};Y_i,S_i|Y_{i+1}^n, S_{i+1}^n) - I(M, Y_{i+1}^n, S_{i+1}^n;Z_i|Z^{i-1})) + 2n\e_n\\
& \stackrel{(d)}{=}\sum_{i=1}^n(I(M;Y_i,S_i|Y_{i+1}^n, S_{i+1}^n, Z^{i-1}) - I(M;Z_i|Y_{i+1}^n, S_{i+1}^n, Z^{i-1})) + 2n\e_n\\
& \stackrel{(e)}{=}\sum_{i=1}^n(I(V_{1i};Y_i,S_i|U_i) - I(V_{1i};Z_i|U_i)) + 2n\e_n\\
%& = \sum_{i=1}^n(I(V_{1i};Y_i,S_i|U_i) - I(V_{1i};Z_i|U_i) + 2n\e_n\\
& = \sum_{i=1}^n(I(V_{1i};Y_i,S_i|U_i) - I(V_{1i};Z_i,S_i|U_i) + I(V_{1i};S_i|Z_i,U_i))+ 2n\e_n\\
& \le \sum_{i=1}^n(I(V_{1i};Y_i,S_i|U_i) - I(V_{1i};Z_i,S_i|U_i) + H(S_i|Z_i,U_i))+ 2n\e_n\\
& \le \sum_{i=1}^n(I(V_{1i};Y_i|U_i, S_i) - I(V_{1i};Z_i,S_i|U_i,S_i) + H(S_i|Z_i,U_i))+ 2n\e_n\\
& \stackrel{(f)}{=}n(I(V_1;Y|U,S) - I(V_1;Z|U,S) + H(S|Z,U))+ 2n\e_n,\end{aligned}$$ where $(a)$ follows by Fano’s inequality; $(b)$ follows from the secrecy condition; $(c)$ and $(d)$ follows the Csiszár sum identity; $(e)$ follows from defining $U_i = (Y_{i+1}^n, S_{i+1}^n, Z^{i-1})$ and $V_{1i} = (M,Y_{i+1}^n, S_{i+1}^n, Z^{i-1})$; and $(f)$ follows from setting $Q$ to be a uniform random variable over $[1:n]$, independent of all other variables, and defining $U = (U_Q,Q)$, $V_{1} = (V_{1Q}, Q)$, $S = S_Q$, $Y = Y_Q$ and $Z = Z_Q$. ]{}
For the second upper bound, we have[$$\begin{aligned}
nR &\le I(M;Y^n,S^n) + n\e_n \\
& \stackrel{(a)}{=} I(M;Y^n|S^n) + n\e_n \\
& = \sum_{i=1}^n I(M;Y_i|S^n, Y_{i+1}^n) \\
& \le \sum_{i=1}^n I(M, Y_{i+1}^n, Z^{i-1}, S_{i+1}^n, S^{i-1};Y_i|S_i) \\
& \stackrel{(b)}{=} \sum_{i=1}^n I(V_{2i}; Y_i|S_i) \\
& = nI(V_{2Q};Y|S,Q) \\
& \stackrel{(c)}{\le} nI(V_2;Y|S), \end{aligned}$$ where $(a)$ follows from the independence between $M$ and $S^n$; $(b)$ follows from defining\
$V_{2i} = (M, Y_{i+1}^n, Z^{i-1}, S_{i+1}^n, S^{i-1})$ and $(c)$ follows from defining $V_2 = (V_{2Q}, Q)$.]{}
Conclusion {#sect:6}
==========
We established bounds on the secrecy capacity of the wiretap channel with state information causally available at the encoder and decoder. We showed that our lower bound can be strictly larger than the best known lower bound for the noncausal state information case. The upper bound holds when the state information is available noncausally at the encoder and decoder. We showed that the bounds are tight for several classes of wiretap channels.
We used key generation from state information to improve the message transmission rate. It may be possible to extend this idea to the case when state information is available only at the encoder. This case, however, is not straightforward to analyze since it would be necessary for the encoder to reveal some state information to the decoder (and hence partially to the eavesdropper) in order to agree on a secret key. This may reduce the wiretap coding part of the rate.
Appendix: Proof of Proposition \[prop1\] {#appen1}
========================================
1\. The proof of this result follows largely from Lemma 2 in Lecture 23 of Lectures on Network Information Theory by El Gamal and Kim [@El-Gamal--Kim2010]. For completeness, we give the proof here. Consider $$\begin{aligned}
H(K_j|\Cc) &\ge \P\{S^n \in \aep \} H(K_j |\Cc, \Sv(j) \in \aep)\\
&\ge (1-\e'_n) H(K_j|\Cc, \Sv(j) \in \aep).\end{aligned}$$ Let $P(k_j)$ be the [*random*]{} pmf of $K_j$ given $\{\Sv(j) \in \aep\}$, where the randomness is induced by the random bin assignment (codebook) $\Cc$.
By symmetry, $P(k_j)$, $k_j \in [1:2^{nR_K}]$, are identically distributed. We express $P(1)$ in terms of a weighted sum of indicator functions as $$P(1) = \sum_{s^n \in \aep} \frac{p(s^n)}{\P\{S^n \in \aep \}} \cdot {I}_{\{s^n \in \Bc(1)\}}.$$ It can be easily shown that $$\begin{aligned}
\E_{\Cc}(P(1)) &= 2^{-nR_K},\\
\var(P(1)) &= 2^{-nR_K}(1- 2^{-nR_K}) \sum_{x^n \in \aep} \left(\frac{p(s^n)}{\P\{\Sv(j) \in \aep\}}\right)^2 \\
&\le 2^{-nR_K} 2^{n(H(S) + \d(\e))} \frac{2^{-2n(H(S) - \d(\e)) }} {(1- \e'_n)^2} \\
&\le 2^{-n(R_K+H(S) - 4\d(\e))}\end{aligned}$$ for sufficiently large $n$.
By the Chebyshev inequality, $$\begin{aligned}
\P\{ |P(1) - \E(P(1))|
\ge \e \E(P(1)) \} &\le \frac{\var(P(1))}{(\e \E(P(1)))^2}\\
&\le \frac{2^{-n(H(S)-R_K - 4 \d(\e)) }}{ \e^2}.\end{aligned}$$ Note that if $R_K< H(S)-4 \d(\e)$, this probability $\to 0$ as $n \to \infty$. [Now, by symmetry $$\begin{aligned}
& H(K_1|\Cc, \Sv(j) \in \aep) \\
& = 2^{nR_K} \E(P(1)) \log (1/P(1))) \\
& \ge 2^{nR_K} \P\{|P(1) - \E(P(1))| < \e 2^{-nR_K} \}
\E\bigl(P(1)\log (1/P(1)) \,\big|\,
|P(1) - \E(P(1))| < \e 2^{-nR_K} \bigr)\\
& \ge \left(1- \frac{2^{-n(H(S)-R_K - 4\d(\e))}}{\e^2} \right)
\cdot (nR_K (1-\e)-(1-\e)\log (1+\e)) \\
& \ge n(R_K - \d(\e))\end{aligned}$$ ]{}for sufficiently large $n$ and $R_K< H(S)-4 \d(\e)$.
Thus, we have shown that if $R_K< H(S)-4 \d(\e)$, $H(K_j|\Cc) \ge n(R_K - \d(\e))$ for $n$ sufficiently large. This completes the proof of part 1 of Proposition \[prop1\]. Note now that since $H(S|Z) \le H(S)$, the same results also holds if $R_K \le H(S|Z) -4\d(\e)$.
2\. We need to show that if $R_K < H(S|Z) - 3\d(\e)$, then $I(K_j; \Zv(j)|\Cc) \le 2n\d(\e)$ for every $j\in [1:b]$. We have $$\begin{aligned}
I(K_j;\Zv(j)|\Cc) &= I(\Sv(j); \Zv(j)|\Cc) - I(\Sv(j);\Zv(j)|K_j, \Cc).\end{aligned}$$ We analyze the terms separately. For the first term, we have $$\begin{aligned}
I(\Sv(j); \Zv(j)|\Cc) & = I(\Sv(j),L;\Zv(j)|\Cc) - I(L;\Zv(j)|\Sv(j), \Cc) \\
& \le I(U^n,\Sv(j);Z|\Cc) - H(L|\Sv(j), \Cc) + H(L|\Sv(j), Z^n) \\
& \le nI(U,S;Z) - H(L|\Sv(j), \Cc) + H(L|\Sv(j), Z^n) \\
& \stackrel{(a)}{\le} nI(U,S;Z) - H(L|\Sv(j), \Cc) + n(\Rt - I(U;Z,S) + \d(\e)) \\
& = n\Rt - H(M_{j0}|\Cc) -H(M_{j1}\oplus K_{j-1}|\Cc, M_{j0}) \\
& \quad -H(L|M_{j0}, M_{j1}\oplus K_{j-1}, \Cc) + nI(S;Z) +n\d(\e) \\
& \le n\Rt - nR_0 -H(M_{j1}\oplus K_{j-1}|\Cc, M_{j0}, K_{j-1}) - n(\Rt-R_0-R_K) + nI(S;Z) +n\d(\e) \\
& = nR_K - H(M_{j1}|\Cc, M_{j0}, K_{j-1}) + nI(S;Z) +n\d(\e) \\
& = n(I(S;Z)+\d(\e)), \end{aligned}$$ where step $(a)$ follows from application of Lemma 1 which holds since $\Rt-R_0 \ge I(U;Z,S)$. For the second term we have $$\begin{aligned}
I(\Sv(j);\Zv(j)|K_j, \Cc) & = H(\Sv(j)|K_j, \Cc) - H(\Sv(j)|\Zv(j), K_j, \Cc) \\
& = H(\Sv(j), K_j|\Cc) -H(K_j|\Cc) - H(\Sv(j)|\Zv(j), K_j, \Cc) \\
& \ge nH(S) -nR_K -H(\Sv(j)|\Zv(j), K_j, \Cc) \\
& \ge n(H(S) - R_K) -H(\Sv(j)|\Zv(j), K_j) \\
& \stackrel{(b)}{\ge} n(H(S) - R_K) - n(H(S|Z) - R_K + \d'(\e)) \\
& = nI(S;Z) -n\d(\e),\end{aligned}$$ where $(b)$ follows from showing that $H(\Sv(j)|\Zv(j), K_j) \le n(H(S|Z) - R_K + \d(\e))$. This requires the condition $R_K < H(S|Z) - 3\d(\e)$. Combining the bounds for the 2 expressions gives $I(K_j;\Zv(j)|\Cc) \le 2n\d(\e)$.
*Proof of step $(b)$*: Give an arbitrary ordering to the set of all state sequences $s^n$ with $\Sv(j) = s^n(T)$ for some $T\in [1:2^{n\log|S|}]$. Hence, $H(\Sv(j)|\Zv(j), K) = H(T|K,\Zv(j))$.
From the coding scheme, we know that $\P\{(s^n(T), \Zv(j))\in \aep\}\to 1$ as $n \to \infty$. Note here that $T$ is random and corresponds to the realization of $S^n$.
Now, fix $T = t$, $\Zv(j) = z^n$, $K = k$ and define $N(z^n, k, t) := |\lt \in [1: |\aep(S)|]: (s^n(\lt),z^n)\in \aep, \; \lt \neq t, \; s^n(\lt) \in \Bc(k)|$. For $z^n \notin \aep$, $N(z^n, k, t) = 0$. For $z^n\in \aep$, it is easy to show that $$\begin{aligned}
\frac{|\aep(S|Z)|-1}{2^{nR_K}}\le \E(N(z^n, k, t)) &\le \frac{|\aep(S|Z)|}{2^{nR_K}}, \\
\var(N(z^n, k, t)) &\le \frac{|\aep(S|Z)|}{2^{nR_K}}.\end{aligned}$$
By the Chebyshev inequality, $$\begin{aligned}
\P\{ N(z^n, k,t)\ge (1+\e) \E(N(z^n, k,t)) \} &\le \frac{\var(N(z^n, k,t))}{(\e \E(N(z^n, k,t)))^2}\\
&\le \frac{2^{-n(H(S|Z) - 3\d(\e)-R_K)}}{ \e^2}.\end{aligned}$$ Note that $\P\{ N(z^n, k,t)\ge (1+\e) \E(N(z^n, k,t)) \}\to 0$ as $n \to \infty$ if $R< H(S|Z) - 3\d(\e)$. Now, define the following events $$\begin{aligned}
\Ec_1 &:= \{(\Sv(j), \Zv(j)) \notin \aep\}, \\
\Ec_2 & := \{N(\Zv(j), K, T)\ge (1+\e)\E(N(\Zv(j), K, T))\}.\end{aligned}$$
Let $E = 0$ if $\Ec_1^c \cap \Ec_2^c$ occurs and $1$ otherwise. We have $$\begin{aligned}
\P(E = 1) &\le \P(\Ec_1) + \P(\Ec_2) \\
& \le \P(\Ec_1) + \sum_{(z^n,s^n(t))\in \aep,\; k} p(z^n,t,k)\P\{N(z^n,k,t)\ge (1+\e)\E(N(z^n,k,t))\}\\
& \quad + \P\{(s^n(T),\Zv(j))\notin \aep\}. \end{aligned}$$ $\P\{(s^n(T),\Zv(j))\notin \aep\} = \P(\Ec_1)$ and $\P(\Ec_1) \to 0$ as $n\to \infty$ by the coding scheme. For the second term, $\P\{N(z^n,k,t)\ge (1+\e)\E(N(z^n,k,t))\} \to 0$ as $n \to \infty$ if $R< H(S|Z) - 3\d(\e)$. Hence, $\P(E=1) \to 0$ as $n \to \infty$ if if $R< H(S|Z) - 3\d(\e)$.
We can now bound $H(T|K,Z^n)$ by $$\begin{aligned}
H(T|K,Z^n) &\le 1 + \P(E = 1)H(T|K,Z^n, E= 1) + H(T|K,Z^n, E= 0) \\
&\le n(H(S|Z) -R_K + \d(\e)).\end{aligned}$$
3\. To upper bound $I(K_j;\Zv^{j}|\Cc)$, we use an induction argument assuming that $I(K_{j-1};\Zv^{j-1}|\Cc) \le n\d_{j-1}(\e),$ where $\d_{j-1}(\e)\to 0$ as $\e \to 0$. Note that the proof for $j=2$ follows from part 2. Consider[$$\begin{aligned}
I(K_j; \Zv^{j}|\Cc) & = I(K_j; \Zv(j)|\Cc) + I(K_j; \Zv^{j-1}|\Cc, \Zv(j)) \\
& \stackrel{(a)}{\le} 2n\d(\e) + I(K_j; \Zv^{j-1}|\Cc, \Zv(j)) \\
& = H(\Zv^{j-1}|\Cc, \Zv(j)) - H(\Zv^{j-1}|\Cc, \Zv(j), K_j) +2n\d(\e)\\
& \le H(\Zv^{j-1}|\Cc) - H(\Zv^{j-1}|\Cc, K_{j-1}, \Zv(j), K_j) +2n\d(\e) \\
& \stackrel{(b)}{=} H(\Zv^{j-1}|\Cc) - H(\Zv^{j-1}|\Cc, K_{j-1}) +2n\d(\e)\\
& = I(K_{j-1}; \Zv^{j-1}|\Cc) + 2n\d(\e) \\
& \stackrel{(c)}{\le} n\d_{j-1}(\e) + 2n\d(\e),\end{aligned}$$ ]{}where $(a)$ follows from part 2 of the Proposition; $(b)$ follows from the Markov Chain relation $\Zv^{j-1} \to K_{j-1} \to (\Zv(j), K_{j})$; $(c)$ follows from the induction hypothesis. This completes the proof since the last line implies that there exists a $\d'(\e),$ where $\d'(\e) \to 0$ as $\e \to 0,$ that upper bounds $I(K_j; \Zv^{j}|\Cc)$ for $j\in [1:b].$
Proof of Proposition 2
======================
\[appen2\] 1. We first show that if $R_K < H(S) - 4\d(\e)$, then $H(K_j|\Cc) \ge n(R_K - \d(\e))$. This is done in the same manner as 1 of Proposition 1. The proof is therefore omitted.
2\. We need to show that if $R_K < H(S|Z) - 3\d(\e)$, then $I(K_j; \Zv(j)|\Cc) \le 2n\d(\e)$ for every $j\in [1:b]$. We have $$\begin{aligned}
I(K_j;\Zv(j)|\Cc) &= I(\Sv(j); \Zv(j)|\Cc) - I(\Sv(j);\Zv(j)|K_j, \Cc).\end{aligned}$$ We analyze the terms separately. For the first term, we have $$\begin{aligned}
I(\Sv(j); \Zv(j)|\Cc) & = I(\Sv(j),L;\Zv(j)|\Cc) - I(L;\Zv(j)|\Sv(j), \Cc) \\
& \le I(U^n,\Sv(j);Z|\Cc) - H(L|\Sv(j), \Cc) + H(L|\Sv(j), Z^n) \\
& \le nI(U,S;Z) - H(L|\Sv(j), \Cc) + H(L|\Sv(j), Z^n) \\
& \stackrel{(a)}{\le} nI(U,S;Z) - H(L|\Cc) + n(\Rt - I(U;Z,S) + \d(\e)) \\
& = n\Rt - H(K_{(j-1)d}|\Cc) - H(K_{(j-1)m}\oplus M_j|\Cc) + nI(S;Z) \\
& \quad - H(L|K_{(j-1)m}\oplus M_j, K_{(j-1)d}) + n\d(\e) \\
& \stackrel{(b)}{\le} n(\Rt -R_d - R - \Rt + R_d +R + 2\d(\e))+nI(S;Z)\\
& = n(I(S;Z)+2\d(\e)), \end{aligned}$$ where step $(a)$ follows from application of Lemma 1, which holds from the condition that $\Rt \ge I(U;Z,S)$, and the fact that $\Sv(j)$ is independent of $L$. Step $(b)$ follows from part 1 of Proposition \[prop2\]: $H(K_{j-1}|\Cc) \ge n(R_K - \d(\e))$, which implies that $H(K_{(j-1)d}|\Cc) \ge n(R_d - \d(\e))$. Note we implicitly assumed $j\ge 2$. The case of $j=1$ is straightforward, since $H(L|\Cc) = n\Rt$ by the fact that we transmit a codeword picked uniformly at random.
The proof that $I(\Sv(j);\Zv(j)|K_j, \Cc)\ge nI(S;Z) - n\d(\e)$ follows the same steps as the proof of part 2 of Proposition \[prop1\] and requires the same condition that $R_K < H(S|Z) - 3\d(\e).$
3\. Part 3 of the Proposition is proved in the same manner as part 3 of Proposition \[prop1\].
Proof of Proposition 3
======================
\[appen3\] 1. We first show that if $R_K < H(S) - 4\d(\e)$, then $H(K_j|\Cc) \ge n(R_K - \d(\e))$. This is done in the same manner as 1 of Proposition 1. The proof is therefore omitted.
2\. We need to show that if $R_K < H(S|Z,V) - 3\d(\e)$, then $I(K_j; \Zv(j)|\Cc) \le n\d(\e)$ for every $j\in [1:b]$. We have $$\begin{aligned}
I(K_j;\Zv(j)|\Cc) &\le I(K_j;\Zv(j), U^n|\Cc) \\
& =I(\Sv(j); \Zv(j), U^n|\Cc) - I(\Sv(j);\Zv(j),U^n|K_j, \Cc).\end{aligned}$$ We analyze the terms separately. For the first term, we have $$\begin{aligned}
I(\Sv(j); \Zv(j), V^n|\Cc) &= I(\Sv(j); \Zv(j)|V^n, \Cc) \\
& = \sum_{i=1}^n (H(\Zv_i(j)|\Cc, V^n, \Zv^{i-1}(j)) - H(\Zv_i(j)|\Cc, V^n, \Sv(j), \Zv^{i-1}(j))) \\
& \le \sum_{i=1}^{n}(H(\Zv_i(j)|\Cc, V_i) - H(\Zv_i(j)|\Cc, V_i, \Sv_i(j))) \\
& \le n(H(Z|V) - H(Z|V,S)) \\
& = nI(Z;S|V) = nI(Z,V;S).\end{aligned}$$
For the second term, we have $$\begin{aligned}
I(\Sv(j);\Zv(j), V^n|K_j, \Cc) & = H(\Sv(j)|K_j, \Cc) - H(\Sv(j)|\Zv(j), V^n, K_j, \Cc) \\
& = H(\Sv(j), K_j|\Cc) -H(K_j|\Cc) - H(\Sv(j)|\Zv(j), V^n, K_j, \Cc) \\
& \ge nH(S) -nR_K -H(\Sv(j)|\Zv(j), V^n, K_j, \Cc) \\
& \ge n(H(S) - R_K) -H(\Sv(j)|\Zv(j),V^n, K_j) \\
& \stackrel{(b)}{\ge} n(H(S) - R_K) - n(H(S|Z,V) - R_K + \d'(\e)) \\
& = nI(S;Z,V) -n\d(\e),\end{aligned}$$
The proof of step $(b)$ follows the same steps as in the proof of part 2 of Proposition \[prop1\]. We can show that step $(b)$ holds if $R_K < H(S|Z,V) - 3\d(\e).$
Combining the two terms then give the required upper bound which completes the proof of Part 2.
3\. Part 3 of the Proposition is proved in the same manner as part 3 of Proposition \[prop1\].
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We extend, to include the effects of finite temperature, our earlier study of the interband dynamics of electrons with Markoffian dephasing under the influence of uniform static electric fields. We use a simple two-band tight-binding model and study the electric current response as a function of field strength and the model parameters. In addition to the Esaki-Tsu peak, near where the Bloch frequency equals the damping rate, we find current peaks near the Zener resonances, at equally spaced values of the inverse electric field. These become more prominenent and numerous with increasing bandwidth (in units of the temperature, with other parameters fixed). As expected, they broaden with increasing damping (dephasing).'
address:
- |
CCAST (World Laboratory) P.O. Box 8730, Beijing 100080, China\
Institute of Applied Physics and Computational Mathematics,\
P.O. Box 8009, Beijing 100088, China\
- 'Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106 '
author:
- 'Xian-Geng Zhao'
- 'Daniel W. Hone'
---
Introduction
============
In a previous paper[@yanzhao] (henceforth referred to as I) we began a study of the effects on interband transitions of the scattering of electrons from static imperfections in a semiconductor superlattice. A uniform static electric field was applied. It has long been recognized that scattering destroys the coherence necessary to sustain the Bloch oscillations predicted in such a field, and in practice this delayed the observation of these oscillations until the development of semiconductor superlattices[@esaki; @expt]. But the scattering destroys other types of interesting coherent motion, as well. Dunlap and Kenkre[@dunken; @dunkenpla], in particular, looked at the effects on dynamic localization of electrons in time periodic electric fields. We were interested in I in a multiband effect, the Rabi oscillations[@xgniu2; @rjs; @rjs2] of electron population between bands of a crystal in a uniform static electric field of appropriate magnitude — near an avoided crossing of the two interpenetrating Wannier-Stark Ladders (WSL) arising essentially from different bands[@rjs]. These are the so-called Zener resonances. Within a simple two-band tight-binding model we demonstrated the destruction of localization (of occupied electron states) to a single miniband by the dephasing associated with the scattering. The decay rate for the approach to steady state band populations exhibits sharp peaks at values of the static electric field which give Zener resonances in the absence of scattering. The specific approximations in I, however, effectively limited the results to the case of infinite temperature. In particular, the steady state was assumed to be equal population of the two bands. In this paper we remove that restriction, to discuss these effects at finite temperature. The necessary modification was described in Ref. (), namely relaxation of band populations toward values set by the Boltzmann factors describing thermal equilibrium. This will allow us, in particular, to look at the electric current, a quantity of obvious experimental interest which vanishes in the infinite temperature limit.
Model
=====
We consider the same model that we treated in I, a standard simple tight-binding model[@fukuyama] of a two-band system in a static electric field E. The Hamiltonian can be written as $$\begin{aligned}
H =
&&\sum_n\bigg[(\Delta_a + n\omega_B)a_n^\dagger a_n + (\Delta_b +
n\omega_B)b_n^\dagger b_n \nonumber\\ &&- (W_a/4)(a^\dagger_{n+1} a_n +
h.c.) + (W_b/4)(b^\dagger_{n+1} b_n + h.c.)\nonumber\\ && + eER(a^\dagger_n
b_n + b^\dagger_n a_n)\bigg].
\label{HAM} \end{aligned}$$ Here the subscripts label the lattice sites and the lower and upper minibands are designated by symbols $a$ and $b$, respectively. We have introduced the notation $\omega_B\equiv eEd$, where $d$ is the lattice constant, for the Bloch frequency, which will appear often below. The first two terms describe the site energies of the Wannier states in the presence of the electric field, and $W_{a,b}$ are the widths of the isolated ($ E=0 $) minibands induced by nearest neighbor hopping: $\epsilon_{a,b}(k) = \Delta_{a,b} \mp (W_{a,b}/2) \cos k $, where the dimensionless wave vector $k$ is in units of the inverse lattice constant $d$. The last term is the on-site electric dipole coupling between minibands; $eR$ is the corresponding dipole moment. This Hamiltonian does neglect Coulomb interactions and electric dipole elements between Wannier states on different sites, but it contains the essential physics for the problem[@rjs; @rjs2; @fukuyama; @honexg]. Note that the hopping parameters $W_{a,b}$ are written here with opposite signs, so that with both parameters positive the band structure at $E=0$ is of the standard nearly free electron character, with direct band gaps at the zone boundary. But the calculation to follow is valid for arbitrary signs of the parameters.
It is easily shown[@fukuyama; @honexg] that the exact spectrum of $H$ is two interpenetrating Wannier-Stark Ladders. But what do the corresponding states represent in terms of the occupation of the original bands as a function of time, and what is the influence of scattering? For vanishing dipole matrix element between bands, $R=0$, there is no interband mixing. Each of the two bands gives rise to a single WSL. Clearly, when the electric field amplitude is such that the ladders become degenerate, even small values of $R$ lead to strong interband mixing. The crossing of the ladders is “avoided" by any finite $R$, and the behavior near those avoided crossings (the Zener resonances) is of particular importance and interest. In general there are peaks in the current response at those values of the electric field, but the peaks are broadened by increasing temperature, as well as by decreasing bandwidth relative to band separation.
We start by defining the density matrix in the representation of the two bands, $$\rho(t)=\sum_{ijmn}\rho_{mn}^{ij}\xi_{m}^{i\dagger}\xi_{n}^{j},$$ where $i,j = 1$ or $2$ are band indices: $\xi_{m}^{1\dagger}$ ($\xi_{m}^{1}$) and $\xi_{m}^{2\dagger}$ ($\xi_{m}^{2}$) designate $a_m^{\dagger}$ ($a_m$) and $b_m^\dagger$ ($b_m$), respectively. Since we are interested in the dynamics of occupation of various band states, it is convenient to work in a wave vector basis, by Fourier transforming the density matrix. In general, since $\rho_{mn}$ is not translationally invariant (a function only of $m-n$), we have a full set $\rho^{ij}_{kq} = \sum_{mn} \rho_{mn}^{ij}(t)\exp[-ikm+iqn]$ of Fourier components. But we will be interested in the wave vector diagonal band occupation numbers $\rho^{ij}_{kk}(t) \equiv \rho^{ij}(k,t)$. Then at finite temperature $T$ we insist that the wave vector and band diagonal occupation number relax to the thermal equilibrium value, $$\rho^{ii}_{kk} \rightarrow \rho^i_T(k) \equiv e^{-\beta\epsilon_i(k)} \Big/
\sum e^{-\beta\epsilon_j(q)}, \label{rhot}$$ where $\beta = 1/k_BT$, the sum in the partition function is over $j=1,2$ and over all wave vectors $q$, and the band energies are those given above, $$\epsilon_{1,2}(k) = \Delta _{a,b} \mp (W_{a,b}/2)\cos k ~.$$
Within a constant relaxation rate approximation[@dunken], the density matrix $\rho(k,t)$ satisfies the following stochastic Liouville equation (SLE) (we set $\hbar=1$ throughout this paper), $$i\frac{d\rho}{dt}=[H, \rho(t)]-i\Gamma \left[\rho(t) - \rho_T\right].
\label{SLE}$$ Here each of $H$, $\rho$, and $\rho_T$ is labelled by (the same) wave vector $k$. The operator $\Gamma $ describes the relaxation of the off-diagonal elements of $\rho$ through dephasing: $$\Gamma [\rho - \rho_T] = \sum_{ij}\alpha_{ij}\left[ \rho^{ij}(k,t)
-\delta_{ij} \rho_T(k) \right] \xi^{i\dagger}(k)\xi^j(k).$$ The utility of this simplest form of the SLE has been discussed by Kenkre and collaborators (see Ref. and references therein). The parameters $\alpha_{ij}$ measure the loss of phase coherence between sites, or the scattering lifetime of band states labeled by quasimomentum.
As in I it is convenient to introduce the linear combinations of density matrix elements: $$\begin{aligned}
\rho_+(k,t)=\rho^{11}(k,t)+\rho^{22}(k,t)~,\\
\rho_-(k,t)=\rho^{11}(k,t)-\rho^{22}(k,t)~,\\
\rho_{+-}(k,t)=\rho^{12}(k,t)+\rho^{21}(k,t)~,\\
\rho_{-+}(k,t)=i[\rho^{21}(k,t)-\rho^{12}(k,t)]~. \end{aligned}$$ For simplicity, we also take $\alpha_{11}=\alpha_{22}=
\alpha_{12}=\alpha_{21}=\alpha$ to reduce the number of parameters in the theory. Then the SLE (\[SLE\]) has the explicit components $$\frac{\partial}{\partial
t}\rho_+(k,t)-\omega_B\frac{\partial}{\partial k} \rho_+(k,t)=
- -\alpha\left[\rho_+(k,t)-\rho^+_T(k)\right]~,
\label{rplus}$$ $$\frac{\partial}{\partial
t}\rho_-(k,t)-\omega_B\frac{\partial}{\partial k} \rho_-(k,t)= -2eER
\rho_{-+}(k,t)-\alpha\left[\rho_-(k,t)-\rho^-_T(k)\right]~,$$ $$\frac{\partial}{\partial
t}\rho_{+-}(k,t)-\omega_B\frac{\partial} {\partial k} \rho_{+-}(k,t)=
\Bigl(\Delta - W\cos k\Bigr) \rho_{-+}(k,t) -\alpha\rho_{+-}(k,t)~,$$ $$\frac{\partial}{\partial
t}\rho_{-+}(k,t)-\omega_B\frac{\partial} {\partial k} \rho_{-+}(k,t)=
- -\Bigl(\Delta - W\cos k\Bigr) \rho_{+-}(k,t) +2eER
\rho_-(k,t)-\alpha\rho_{-+}(k,t)~.$$ Here we have used the simplified notation $\Delta\equiv\Delta_a - \Delta_b$ and $W \equiv
(W_a+W_b)/2$. The equation (\[rplus\]) for $\rho_+$ is decoupled from the others, and is readily integrated to give $$\rho_+(k,t) =
e^{-\alpha t}\left\{\rho^+_T(k+\omega_Bt) + \alpha\int^t_0 dt'\, e^{\alpha
t'} \rho^+_T[k+\omega_B(t-t')]\right\}.
\label{rhoplus}$$ The equations for $\rho_-(k,t),~~\rho_{+-}(k,t),$ and $\rho_{-+}(k,t)$ can be reduced to the following ordinary differential equations in an accelerated basis[@kria], $k(t) = k-\omega_B t$ or, equivalently, in the transverse or vector gauge discussed in Ref. , $$\frac{d}{dt}X(k,t)=-2eER~Z(k,t)-\alpha[X(k,t)-\rho^-_T(k-\omega_Bt)],$$ $$\frac{d}{dt}Y(k,t)= [\Delta -
W\cos(k-\omega_B t)]Z(k,t) -\alpha Y(k,t),$$ $$\frac{d}{dt}Z(k,t)= -[\Delta - W\cos(k-\omega_B t)]Y(k,t)
+2eER~X(k,t)-\alpha Z(k,t)~.$$
Here $X(k,t)=\rho_-(k-\omega_B t,t)$, $Y(k,t)=\rho_{+-}(k-\omega_B t,t)$, and $Z(k,t)=\rho_{-+}(k-\omega_B t,t)$. The structure is exactly the same as Eqs. (17 - 19) in I, except for the $k$-dependent relaxation of $X$ here. As we did there, we can integrate these equations analytically as a perturbation series in the parameter $\mu\equiv 2eER$, which characterizes the electric dipole coupling between bands. To lowest nontrivial (second) order in $\mu$ we find $$\begin{aligned}
\rho_-(k,t) = e^{-\alpha t} \bigg\{\rho^-_T(k+\omega_Bt)
&+& \alpha \int_0^t dt'\,e^{\alpha t'}\rho^-_T[k+\omega_B(t-t')] -\mu^2
\int_0^t dt'\int_0^{t'} dt''\, \Big[\rho^-_T(k+\omega_Bt)\nonumber\\
&+&\alpha \int_0^{t''} d\tau\,e^{\alpha \tau}\rho^-_T[k+\omega_B(t-\tau)]
\bigg] \cos \int_{t''}^{t'} d\tau '\left[\Delta
- -W\cos\left(k+\omega_B(t-\tau ') \right)\right]\Big\}~.
\label{rhominus} \end{aligned}$$
Current
=======
We turn now to the calculation of the interesting physical quantity, the current $j(t)$ along the superlattice direction. In the band $i=a,b$ the instantaneous current is given by the sum over wave vectors of the relevant electron velocity $v_i(k) = (W_i d/2)\sin k$ times the number density in that band at that wave vector, $n_0\rho_i(k,t)$, where $n_0$ is the number of carriers per unit area in each cell of the superlattice. In terms of the convenient quantities $\rho_{\pm}(k,t)$ we then have $$j(t) = \int_0^{2\pi}\frac{dk}{8\pi}\left[(W_a-W_b)\rho_+(k,t) +
2W\rho_-(k,t)\right] n_0 d\sin k.$$ Of particular interest is the steady state long time average of this, $$\langle j\rangle = \lim_{T_0\rightarrow\infty} \frac{1}{T_0}
\int_0^{T_0} dt \, j(t).
\label{jav}$$ To second order in $\mu$ we have the probability densities $\rho_{\pm}(k,t)$, in Eqs. (\[rhoplus\]) and (\[rhominus\]), and we find $$\begin{aligned}
\frac{\langle j\rangle}{n_0 d} &=& \left(\frac{W_a}{2}\right)\left(\frac
{\alpha\omega_B}{\alpha^2+\omega_B^2}\right)[C_1+(W_b/W_a)C_2] \nonumber\\
&-& \left(\frac{W}{4}\right)\left(\frac{\alpha\mu}{\alpha^2+
\omega_B^2}\right)^2 [C_1+C_2] \sum_{\ell=-\infty}^{\infty} \left[
D_{\ell}^- + D_{\ell}^+\right]J_{\ell}^2(W/\omega_B) ,
\label{javg}\end{aligned}$$ with $$C_1 \equiv
\frac{e^{-\beta\Delta}I_1(\beta W_a/2)} {e^{-\beta\Delta}I_0(\beta
W_a/2)+ I_0(\beta W_b/2)} ~,$$ $$C_2 \equiv \frac{I_1(\beta W_b/2)} {e^{-\beta\Delta}I_0(\beta W_a/2)
+ I_0(\beta W_b/2)}~,$$ $$D_{\ell}^{\pm} \equiv \frac{\alpha^2-\omega_B^2-2\omega_B
[(\ell+1)\omega_B\pm\Delta]}{\alpha^2+[(\ell+1)\omega_B\pm\Delta]^2}~,$$ where $I_n$ is the modified Bessel function and $J_n$ the ordinary Bessel function of order $n$. The interband effects are all contained in the second term on the right hand side of (\[javg\]), proportional to $\mu^2$. The Zener resonances, near $\Delta = n\omega_B$, with $n$ an integer, exhibit themselves as peaks in the factors $D_{\ell}^-$.
We will look at the nonlinear conductance, the current as a function of increasing electric field, which is conveniently parameterized in the dimensionless form $\omega_B/\Delta$. At sufficiently high fields, as the Wannier-Stark functions become increasingly localized in space, the current falls off inversely with $\omega_B$. To the extent that the interband contribution can be neglected (small $\mu$) the field dependence of the response is given by the factor $\omega_B/(\alpha^2+\omega_B^2)$, with an “Esaki-Tsu" peak[@esaki] near $\omega_B = \alpha$.
There are two sources of temperature dependence in (\[javg\]). The first is the explicit appearance of $\beta \equiv 1/k_BT$ in the factors $C_1$ and $C_2$. The other is implicit; the relaxation rate $\alpha$ is ordinarily also temperature dependent. Though that can be modeled for a specific relaxation mechanism, we simply take it to be constant below. At high temperatures, with $\beta\Delta$, $\beta W_i \ll 1$, we have $C_1$ and $C_2$ approximately linear in $\beta$; the current $\langle j\rangle$ also therefore falls off as $1/T$. This is the well known effect of approaching uniform thermal population of the states throughout each band as the temperature rises; at complete uniformity the current vanishes.
All of these effects are seen in Fig. 1, where the interband matrix element $\mu$ has been set equal to zero, and $\alpha/\Delta=0.1$. The second term on the right hand side of Eq. (\[javg\]) is then absent, and the single remaining peak arises from the first term in that equation. To avoid a multiplicity of parameters we have taken the two bandwidths to be equal, $W_a=W_b=W = 0.8\Delta$. In the remaining figures we have set the interband coupling to be $\mu/\Delta = 0.4$. Again $W_a=W_b=W$, with increasing values $W/\Delta =$ 0.4 and 0.8 in Figs. 2 and 3, respectively. Each figure gives results for three values of the parameter $\beta\Delta$, namely 1, 10 and 100. The value of unity is representative of a typical superlattice, of lattice parameter of order 100 Å and carrier effective mass $m^*/m\approx 0.1$, at room temperature. From Eq. (\[javg\]) we see that the height in $\langle j\rangle$ of the $n$th peak (for values of $n$ up to 5 or 6, where $\alpha^2\ll\omega_B^2$), at $\omega_B=\Delta/n$ is approximately proportional to $[n/(1+.01n^2)]^2J_{n-1}^2(nW/\Delta)$ . The Bessel function overlap factors increase rapidly with increasing bandwidth $W$ over the range we have chosen for $W$, and so we see more well-defined Zener resonance peaks with increasing bandwidth , going from Fig. 2 to Fig. 3. The major effect of increasing the damping rate $\alpha$ is to broaden the peaks in the current. Thus, for the particular choice of $W/\Delta =$ 0.4 and $\mu/\Delta = 0.4$, we can compare the results for $\alpha/\Delta = 0.05$ in Fig. 4 and $\alpha/\Delta = 0.2$ in Fig. 5 with the previous curves for $\alpha/\Delta = 0.1$ in Fig. 2.
Conclusions
===========
Within a simple two-band tight binding model we have studied the electric current response of a semiconductor superlattice subject to a finite uniform electric field. Relaxation processes have been assumed Markoffian, and they have been described by a single parameter characterizing the rate at which a nonequilibrium density is restored to thermal equilibrium, using a stochastic Liouville equation. We have been particularly interested in the interband transitions, which lead to current peaks near the values of external field where Zener resonances occur. We have exhibited the variation in height and width of these peaks, as well as in the overall current magnitude with temperature and with the various parameters of the theory, including bandwidth, band separation, interband dipole coupling, and relaxation rate.
The results we have obtained are perturbational in $\mu$; we do require weak interband coupling. Moreover, we have limited the discussion to a single pair of bands, assuming the impact of all other bands to be negligible on the Zener tunneling between these two. This can be realized in practice by using a dimerized semiconductor superlattice with, for example, uniform wells but alternating thick and thin barriers between them. With weak alternation of barrier thickness one can adjust a pair of bands resulting from the doubling of the unit cell to be well isolated from all other bands, and the predictions of this paper can be studied in such a system by varying the external uniform electric field. Another way to observe the effects predicted here is to use ultracold atoms in accelerating optical potentials. Recently, Rabi oscillations were observed in such a system, where the interband coupling was generated by a small phase modulation [@Raizen]. Very low temperatures can be realized in these systems, and it should be possible to observe the resonant structures in current that we have predicted in this paper.
This work was supported in part by the National Natural Science Foundations of China under Grant No. 19725417, and in part by the U.S. National Science Foundation Grant No. PHY94-07194.
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'The estimation and utilization of photometric redshift probability density functions (photo-$z$ PDFs) has become increasingly important over the last few years and currently there exist a wide variety of algorithms to compute photo-$z$’s, each with their own strengths and weaknesses. In this paper, we present a novel and efficient Bayesian framework that combines the results from different photo-$z$ techniques into a more powerful and robust estimate by maximizing the information from the photometric data. To demonstrate this we use a supervised machine learning technique based on random forest, an unsupervised method based on self-organizing maps, and a standard template fitting method but can be easily extend to other existing techniques. We use data from the DEEP2 and the SDSS surveys to explore different methods for combining the predictions from these techniques. By using different performance metrics, we demonstrate that we can improve the accuracy of our final photo-$z$ estimate over the best input technique, that the fraction of outliers is reduced, and that the identification of outliers is significantly improved when we apply a Naïve Bayes Classifier to this combined information. Our more robust and accurate photo-$z$ PDFs will allow even more precise cosmological constraints to be made by using current and future photometric surveys. These improvements are crucial as we move to analyze photometric data that push to or even past the limits of the available training data, which will be the case with the Large Synoptic Survey Telescope.'
bibliography:
- 'combine\_bayes\_final.bib'
title: 'Exhausting the Information: Novel Bayesian Combination of Photometric Redshift PDFs'
---
\[firstpage\]
methods: data analysis – methods: statistical – surveys – galaxies: distances and redshifts – galaxies: statistics.
Introduction
============
Spectroscopic galaxy surveys have played an important role in understanding the origin, composition, and evolution of our Universe. Surveys like the Sloan Digital Sky Survey (SDSS; @York2000), WiggleZ [@Drinkwater2010], and BOSS [@Dawson2013] have imposed important constraints on the allowed parameter values of the standard cosmological model [[[e.g., ]{}]{} @Percival2010; @Blake2011; @Sanchez2013]. However, spectroscopic measurements are considerable more expensive to obtain than photometric data, they are more likely to suffer from selection effects, and they provide much smaller galaxy samples per unit telescope time. As a consequence, current ongoing and future galaxy surveys like the Dark Energy Survey (DES[^1]) and the Large Synoptic Survey Telescope (LSST[^2]) are pure photometric surveys. These surveys will enable cosmological measurements on galaxy samples that are currently at least a hundred times larger than comparable spectroscopic samples, that have relatively simple and uniform selection functions, that extend to fainter flux limits and larger angular scales, thereby probing much larger cosmic volumes and will photometrically detect galaxies that are too faint to be spectroscopically observed.
With the growth of these large photometric surveys, the estimation of galaxy redshifts by using multi band photometry has grown significantly over the last two decades. As a result, a variety of different algorithms for estimating [photo-$z$ ]{}’s based on statistical techniques have been developed [see, [[e.g., ]{}]{} @Hildebrandt2010; @Abdalla2011; @Sanchez2014 for a review of current [photo-$z$ ]{}techniques]. Over the last several years, particular attention has been focused on techniques that compute a full probability density function (PDF) for each galaxy in the sample. A [photo-$z$ ]{}PDF contains more information than a single [photo-$z$ ]{}estimate, and the use of [photo-$z$ ]{}PDFs has been shown to improve the accuracy of cosmological measurements [[[e.g., ]{}]{} @Mandelbaum2008; @Myers2009; @Jee2013].
[Photo-$z$ ]{}techniques can be broadly divided into two categories: spectral energy distribution (SED) fitting, and training based algorithms. Template fitting approaches [see [[e.g., ]{}]{} @Benitez2000; @Bolzonella2000; @Feldmann2006; @Ilbert2006; @Assef2010] estimate [photo-$z$]{}s by finding the best match between the observed set of magnitudes or colors, and the synthetic magnitudes or colors taken from the suite of templates that are sampled across the expected redshift range of the photometric observations. This method is often preferred over empirical techniques as they can be applied without obtaining a high-quality spectroscopic training sample. However, these techniques do require a representative sample of template galaxy spectra, and they are not exempt from uncertainties due to measurement errors on the survey filter transmission curves, mismatches when fitting the observed magnitudes or colors to template SEDs, and color–redshift degeneracies. The use of training data that include known redshifts can also improve these predictions [[[e.g., ]{}]{} @Ilbert2006; @Newman2013b]. On the other hand, machine learning methods have been shown to have similar or even better performance [[[e.g., ]{}]{} @Collister2004; @CarrascoKind2013a] when the spectroscopic training sample is populated by representative galaxies from the photometric sample.
Machine learning methods have the advantage that it is easier to include extra information, such as galaxy profiles, concentrations, or different modeled magnitudes within the algorithm. However, they are only reliable within the limits of the training data, and one must exercise sufficient caution when extrapolating these algorithms. These techniques can be sub-categorized into supervised and unsupervised machine learning approaches. For supervised techniques [[[e.g., ]{}]{} @Connolly1995; @Brunner1997; @Collister2004; @Wadadekar2005; @Ball2008; @Lima2008; @Freeman2009; @Gerdes2010; @CarrascoKind2013a], the input attributes (e.g., magnitudes or colors) are provided along with the desired output (e.g., redshift). This training information is directly used by the algorithm during the learning process. In this case, the redshift information from the training set *supervises* the learning process and decisions are made by using this information. On the other hand, unsupervised machine learning [photo-$z$ ]{}techniques [[[e.g., ]{}]{} @Geach2012; @Way2012; @CarrascoKind2014a] are less common as they do not use the desired output value (e.g., redshifts from the spectroscopic sample) during the training process. Only the input attributes are processed during the training, leaving aside the redshift information until the evaluation phase.
Given the importance of these [photo-$z$ ]{}PDFs, there is a present demand to compute them as efficiently and accurately as possible. Additional requirements include the need to understand the impact of systematics from the spectroscopic sample on the estimation of these PDFs [[[e.g., ]{}]{} @Oyaizu2008; @Cunha2012a; @Cunha2012b], and to maximally reduce the fraction of catastrophic outliers [[[e.g., ]{}]{} @Gorecki2014]. Considerable effort has, therefore, been put into both the development of different techniques and the exploration of new approaches in order to maximize the efficacy of [photo-$z$ ]{}PDF estimation. Yet, the combination of multiple, independent [photo-$z$ ]{}PDF techniques has remained under explored [[[e.g., ]{}]{} @CarrascoKind2013b; @Dahlen2013].
In this paper we extend our previous exploratory work in combining machine learning techniques with template fitting methods [@CarrascoKind2013b] to explicitly address this issue by presenting a novel Bayesian framework to combine and fully exploit different [photo-$z$ ]{}PDF techniques. In particular, we show that the combination of a standard template fitting technique with both a supervised and an unsupervised machine learning method can improve the overall accuracy over any individual method. We also demonstrate how this combined approach can both reduce the number of outliers and improve the identification of catastrophic outliers when compared to the individual techniques. Finally, we show that this methodology can be easily extended to include additional, independent techniques and that we can maximize the complex information contained within a photometric galaxy sample.
This paper is organized as follows. In Section 2 we present the algorithms used in this work to generate the individual [photo-$z$ ]{}PDF estimates and we provide a brief description on their individual functionality. We describe, in Section 3, the different Bayesian approaches by which different [photo-$z$ ]{}techniques are combined. Section 4 introduces the data sets employed to test this Bayesian approach taken from the SDSS and DEEP2 surveys. In Section 5 we present the main results of our combination approach and compare these results to those from the individual [photo-$z$ ]{}PDF methods. In Section 6 we discuss the application of a Naïve Bayes combination technique for outlier detection. In Section 7 we conclude with a summary of our main points and a more general discussion of this new approach.
Photo-z methods {#pz_methods}
===============
To develop and test our combination framework, we consider three, distinct [photo-$z$ ]{}PDF estimation techniques; we briefly discuss each one of them in this section. We make the reasonable assumption that these three techniques are independent in their nature where two of these methods implement machine learning algorithms. The first method is a supervised machine learning technique we have published called [`TPZ` ]{} [Trees for Photo-Z, @CarrascoKind2013a hereafter CB13], which uses prediction trees and a random forest to produce probability density functions. The second method is an unsupervised technique we have published called [`SOMz` ]{} [@CarrascoKind2014a hereafter CB14], which uses self organizing maps (SOM) and a random atlas to produce a probability density function. We have recently incorporated these two implementations into a new, publicly available and growing [photo-$z$ ]{}PDF prediction framework called `MLZ`[^3] (Machine Learning for photo-Z).
The third method is a Bayesian template fitting technique based on [`BPZ` ]{} [Bayesian Photometric Redshifts; @Benitez2000], which fits spectral energy density templates from a preselected library to an observed set of measured flux values. Taken together, these three methods span the three standard published approaches in computing [photo-$z$]{}s in the literature. Any new method would, very likely, be functionally similar to one of these three methods; therefore, any of these three methods could in principle be replaced by a similar method to avoid redundancy. This can be most easily demonstrated for template fitting methods, where an additional set of [photo-$z$ ]{}estimations can be utilized by adopting a different template library [e.g., @Dahlen2013]. In this particular case, the underlying code is essentially unchanged, but the [photo-$z$ ]{}results will change as different spectral libraries are adopted.
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TPZ
---
[`TPZ` ]{}(CB13) is a parallel, supervised algorithm that uses prediction trees and random forest techniques [@Breiman1984; @Breiman2001] to produce [photo-$z$ ]{}PDFs and ancillary information for a sample of galaxies. Among the different non-linear methods that are used to compute photometric redshifts, prediction trees and random forests are one of the simplest yet most accurate techniques. Furthermore, they have been shown to be one of the most accurate algorithms for low as well as high multi-dimensional data [@Caruana2008].
Prediction trees are built by asking a sequence of questions that recursively split the data into two branches until a terminal leaf is created that meets a pre-defined stopping criterion ([[e.g., ]{}]{}a minimum leaf size or a maximum rms within that leaf). The small region bounding the data in the terminal leaf node represents a specific subsample of the entire data that all share similar characteristics. A comprehensive predictive model is applied to the data within each leaf that enables predictions to be rapidly computed in situations where many variables might exist that possibly interact in a nonlinear manner, which is often the case with [photo-$z$ ]{}estimation. A visualization of an example tree generated by [`TPZ` ]{}is shown in the left panel of Figure \[fig:mlz\]. In this figure, the plotting colors represent the magnitudes (or source colors) in which the data are recursively divided. In practice, however, the prediction trees are generally both denser and deeper than the sample tree shown in the Figure.
To compute [photo-$z$ ]{}PDFs in this study, we have used regression trees, which are a specific type of prediction trees. Regression trees are built by first starting with a single node that encompasses the entire data, and subsequently splitting the data within a node recursively into two branches along the dimension that provides the most information about the desired output. The procedure used to select the optimal split dimension is based on the minimization of the sum of the squared errors, which for a specific ${\rm node}$ is given by $$\label{S_RT1}
S({\rm node}) = \sum\limits_{m \in values(M)} \sum\limits_{i \in m} (z_i - \hat{z}_m)^2$$ where $m$ are the possible values (bins) of the dimension $M$, $z_i$ are the values of the target variable on each branch, and $\hat{z}_m$ is the specific prediction model used. In the case of the *arithmetic mean*, for example, we would have that $\hat{z}_m = \frac{1}{n_m}\sum_{i \in m} z_i$, where $n_m$ are the members on branch $m$. This allows us to rewrite Equation \[S\_RT1\] as $$\label{S_RT2}
S({\rm node}) = \sum\limits_{m \in values(M)} n_m V_m$$ where $V_m$ is the variance of the estimator $\hat{z}_m$.
At each node in our tree, we scan all dimensions to identify the split point that minimizes the function $S({\rm node})$. We choose the dimension that minimizes $S({\rm node})$ as the splitting direction, and this process is recursively repeated until either a predefined threshold in $S({\rm node})$ is reached or any new child nodes would contain less than the predefined minimum leaf size. When constructed, each terminal leaf within the prediction tree *contains* spectroscopic data with different redshift values; the final prediction value for a given leaf node is determined from a regression model that covers these spectroscopic data. The simplest model is to simply return the mean value of the set of spectroscopic training redshifts contained within the leaf node, which provides a single estimate of a continuous variable. Alternatively, all of the spectroscopic training redshifts can be retained and subsequently combined with data from the matching leaf nodes in other prediction trees to form an aggregate, final prediction.
We create bootstrap samples from the input training data by sampling repeatedly from the magnitude using the magnitude errors. We use these bootstrap samples to construct multiple, uncorrelated prediction trees whose individual predictions are aggregated to construct a [photo-$z$ ]{}PDF for each individual galaxy by using a technique called a random forest. We also use a cross validation technique called Out-of-Bag [@Breiman1984 CB13] within [`TPZ` ]{}to provide extra information about the galaxy sample. This information includes an unbiased estimation of the errors and a ranking of the relative importance of the individual input attributes used for the prediction. This extra information can prove extremely valuable when calibrating the algorithm, when deciding what attributes to incorporate in the construction of the forest, and when combining this approach with other techniques.
[`TPZ` ]{}has been tested extensively on different datasets, including the SDSS, DEEP2, and DES. In all tests, [`TPZ` ]{}has performed comparable to if not better than other machine learning approaches. When high quality training data are available, [`TPZ` ]{}has been shown to actually outperform other comparable techniques, both training and template based. [@CarrascoKind2013a] provides a more detailed discussion of the [`TPZ` ]{}algorithm and its application to different datasets.
SOM$z$
------
A Self Organized Map (SOM): [@Kohonen1990; @Kohonen2001] is an unsupervised, artificial neural network algorithm that is capable of projecting high-dimensional input data onto a low-dimensional map through a process of competitive learning. In our case, the high dimensional input data can be galaxy magnitudes, colors, or some other photometric attributes, and two dimensions are generally sufficient for the output map. A SOM differs from other neural network based-algorithms in that a SOM is unsupervised (the redshift information is not used during training), there are no hidden layers and therefore no extra parameters, and it produces a direct mapping between the training set and the output network. In fact, a SOM can be viewed as a non-linear generalization of a principal component analysis (PCA).
The key characteristic of the self organization is that it retains the *topology* of the input training set, revealing correlations between inputs that are not obvious. The method is unsupervised since the user is not required to specify the desired output during the creation of the low-dimensional map, as the *mapping* of the components from the input vectors is a natural outcome of the competitive learning process. Another important characteristic of a SOM when applied to [photo-$z$ ]{}estimation is the creation of a structured ordering of the spectroscopic training data, since similar galaxies in the training sample are mapped to neighboring neural nodes in the trained feature map (CB14).
We demonstrate the construction of a self-organizing map in the right-hand panel of Figure \[fig:mlz\]. During this phase, each node on the two-dimensional map is represented by weight vectors of the same dimension as the number of attributes used to create the map itself. In an iterative process, each galaxy in the input sample is individually used to correct these weight vectors. This correction is determined so that the specific neuron (or node), which at a given moment best represents the input galaxy, is modified along with the weight vectors of that node’s neighboring neurons. As a result, this *sector* within the map becomes a better representation of the current input galaxy.
This process is repeated for every galaxy in the train sample, and this entire process is repeated for several iterations. Eventually the SOM converges to its final form where the training data is separated into *groups* of similar features, which is illustrated in Figure \[fig:mlz\] by the different cell colors within the output map. The result of this direct mapping procedure is an approximation of the galaxy training probability density function, and the map itself can be considered a simplified representation of the full attribute space of the input galaxy sample.
Building on our experience in creating [`TPZ`]{}, we have developed a similar approach, named [`SOMz` ]{}(CB14), where prediction trees are replaced by SOMs to create what we called a *random atlas*. The random atlas is constructed from multiple maps that are each constructed from different bootstrap samples selected from the input training data by perturbing the input attributes using their measured error, where each one of these maps are built using a random subsample of the attribute space. The multiple, uncorrelated maps are aggregated to generate a [photo-$z$ ]{}PDF, in a similar manner as described earlier for the random forest.
As described previously, our SOM implementation not only updates the best-matching node but also the topologically closest nodes to it. This functionality ensures that the entire region surrounding the best-matching node is identified as being similar to the current input galaxy. As a result, similar nodes within the map are co-located, which naturally mimics how the input galaxies that have similar properties tend to be co-located in the higher dimensional input parameter space. We apply this procedure iteratively to all input galaxies, which are processed randomly during each iteration to avoid any biases that might arise if galaxies are processed in a specific order.
When running [`SOMz`]{}, there are few different parameters that must be determined, including the map resolution ([[i.e., ]{}]{}the number of pixels in the map), the number of iterations required to build the map, and, most importantly, the underlying two-dimensional topology used for the maps. In this paper we follow the guidelines we presented in CB14 for these parameters, and use a spherical topology for the map, which are constructed by using `HEALPIX` [@Gorski2005], where each pixel in our maps has the same area. This topology was shown to be more accurate in many cases when compared to other topologies like a rectangular or hexagonal grid. In addition, a spherical topology has natural periodic boundary conditions which avoids possible edge effects.
In analogy with [`TPZ`]{}, we use cross validation, or OOB data, to estimate unbiased errors and to determine the relative importance of the different input attributes for this technique. These are both key pieces of information that will be used during the combination process, as we need to ensure that the same process is uniformly applied to each [photo-$z$ ]{}estimation technique. By doing this, we will enable a robust analysis of the final results from the combination of the different techniques. [@CarrascoKind2014a] (CB14) provides a complete description of the [`SOMz` ]{}implementation, the performance of this technique when applied to real data, and an exploration of specific parameter configurations.
Template fitting approach {#template}
-------------------------
Using spectral templates to estimate galaxy [photo-$z$s ]{}from broadband photometry has a long history [@Baum1962]; and this approach is, not surprisingly, one of the most utilized techniques. A primary advantage of this technique is the fact that a training sample is not required, thus this approach can be considered unsupervised. On the other hand, this technique has the disadvantage that a complete and representative library of spectral energy distributions (SEDs) are required. Thus any incompleteness in our knowledge of the template SEDs that fully span the input galaxy photometry will lead to inaccuracies or misestimates in the computation of a galaxy [photo-$z$]{}.
A number of different groups have published template fitting [photo-$z$ ]{}estimation methods, all of which are roughly similar in nature. In this work, we have modified and parallelized one of the most popular, publicly available template fitting algorithms, [`BPZ` ]{} [@Benitez2000]. [`BPZ` ]{}uses Bayesian inference to quantify the relative probability that each template matches the galaxy input photometry and determines a [photo-$z$ ]{}PDF by computing the posterior probability that a given galaxy is at a particular redshift. We can write this probability as $P(z\mid{\mathbf{x}})$ for a specific template $t$, where ${\mathbf{x}}$ represents a given set of magnitudes (or colors). If the identification of a specific template is not required, we can later marginalize over the entire set of templates ${\mathbf{T}}$.
By using Bayes theorem, we have: $$P(z \mid {\mathbf{x}}) = \sum\limits_{t \in {\mathbf{T}}} P(z,t\mid{\mathbf{x}}) \propto \sum\limits_{t \in {\mathbf{T}}} \mathcal{L}({\mathbf{x}} \mid z,t) P(z,t) .$$ $\mathcal{L} ({\mathbf{x}} \mid z,t) $ is the likelihood that, for a given redshift $z$ and spectral template $t$, a specific galaxy has the set of magnitudes (or colors) ${\mathbf{x}}$. $P(z,t)$ is the prior probability of a specific galaxy is at redshift $z$ and has spectral type $t$, this prior probability can be computed from a spectroscopic sample if one is available. The [photo-$z$ ]{}PDF is, therefore, either the posterior probability, if a prior is used, or the likelihood itself if no prior is used. This last point arises since the likelihood only depends on the collection of template SEDs; and, if this collection is representative of the overall galaxy sample, the likelihood can be used by itself as a [photo-$z$ ]{}PDF even without a spectroscopic training sample.
\[fig:filters\] ![ An Elliptical galaxy spectrum at z=0 and redshifted to z = 0.4 overlaid by the eight photometric filters from the DEEP2 galaxy survey (3 from the original survey and $ugriz$ from a matched catalog [@Matthews2013]).[]{data-label="fig:bpz_example"}](Figures/deep2_filt.png "fig:"){width="44.00000%"}
The use of a prior in a Bayesian analysis, however, is recommended. In this case, the prior probability can be computed directly from physical assumptions, from an empirical function calibrated by using a spectroscopic training sample [[[e.g., ]{}]{} @Benitez2000], or from an empirical function calibrated by using machine learning techniques [see [[e.g., ]{}]{} @CarrascoKind2013b where we used Random Naïve Bayesian methods to compute the prior probabilities]. For example, [@Benitez2000] propose the following function for a single magnitude $m_0$: $$\begin{gathered}
P(z,t\mid m_0) = P(t\mid m_0)P(z\mid t,m_0) \\
\propto f_T e^{-k_t (m-m_0)} \times z^{\alpha_t} \exp\left( -\left[\frac{z}{z_{mt}(m)} \right]^{\alpha_t}\right).\end{gathered}$$ where $z_{mt}(m) = z_0t + k_{mt} (m-m_0)$. The five parameters of this function: $f_T$, $m_0$, $\alpha_t$, $z_{mt}$, and $k_{mt}$ can be constrained either by using direct fitting routines, or by using Markov Chain Monte Carlo methods to sample these parameters. These five parameters are dependent on the template $t$ and can be quantified independently. For additional details on the underlying Bayesian approach, we refer the reader to the original paper by [@Benitez2000].
As the goal of a template fitting method is to minimize the difference between observed and theoretical magnitudes (or colors), this approach is heavily dependent on both the library of galaxy SED templates that are used for the computation and the accuracy of the transmission functions for the filters used for particular survey. SED libraries are generally built from a base set of SED templates. These base templates broadly cover the Elliptical, Spiral, and Irregular categories, and a template library can be constructed by interpolating between the base spectral templates to create new spectra. One of the most widely used set of base templates are the four CWW spectra [@Coleman1980], which include an Elliptical, an Sba, an Sbb, and an Irregular galaxy template. When extending an analysis to higher redshift, these temples are often augmented with two star bursting galaxy templates published by [@Kinney1996]. One additional effect some template approaches consider is the presence of interstellar dust, which will introduce artificial reddening.
Once the library of galaxy SED templates has been constructed, the templates are convolved with the transmission functions for a particular survey to generate synthetic magnitudes as a function of redshift for each galaxy template. For the most accurate results, these transmission functions should include the effects of the Earth’s atmosphere (if the observations are ground-based), as well as all telescope and instrument effects. This convolution process is demonstrated visually in Figure \[fig:bpz\_example\], which presents an example Elliptical galaxy spectral template at redshift zero and at a redshift 0.4. Overplotted on this figure is the filter set ($B$, $R$, and $I$) used by the DEEP2 survey, which is the data analyzed in this paper, along with the five extra filters: $u, g, r, i, z$ presented in the DEEP2 photometry catalog compiled by [@Matthews2013].
Photo-$Z$ PDF Combination Methods {#combine}
=================================
We now turn our attention to the different methods with which we can combine distinct [photo-$z$ ]{}PDF estimation techniques [see [[e.g., ]{}]{} @CarrascoKind2013b where we first discussed combining Bayesian and machine learning predictions]. In the statistics and machine learning communities, this topic is known as *ensemble learning* [@Rokach2010]. Recently, [@Dahlen2013] have demonstrated that, on average, an improved [photo-$z$ ]{}estimate can be realized by combining the results from multiple template fitting methods. In this section, we build on this previous work to identify how Bayesian techniques can be used to construct a combined [photo-$z$ ]{}PDF estimator.
We can frame the problem mathematically by writing the set of [photo-$z$ ]{}PDFs for a given galaxy as a set of models ${\mathbf{M}}$, where each individual model $M_k$ ([[e.g., ]{}]{}[`TPZ`]{}, [`SOMz`]{}, or modified [`BPZ`]{}) provides a distinct [photo-$z$ ]{}PDF or posterior probability. A [photo-$z$ ]{}PDF can be written as $P(z \mid {\mathbf{x}}, {\mathbf{D}}, M_k)$, where ${\mathbf{x}}$ is the set of magnitudes or colors (note that without loss of generality we can use other attributes in this process) used to make the prediction and ${\mathbf{D}}$ corresponds to the training set which consists of $N_d$ galaxies. We can also abbreviate this [photo-$z$ ]{}PDF as $P_k(z)$. These [photo-$z$ ]{}PDFs are each subject to the following constraint: $$\label{pz1}
\int_{z_1}^{z_2} P_k(z) dz = 1$$ for every model $M_k$, where $z_1$ and $z_2$ are the lower and upper limits, respectively, for the redshift range spanned by the galaxy sample. In the following subsections, we introduce different methods to aggregate these [photo-$z$ ]{}PDFs and show the results of these different methods in §\[App\].
Given the variety of [photo-$z$ ]{}PDF estimation methods we are using ([[i.e., ]{}]{}supervised, unsupervised, and model-based), we fully expect the relative performance of the individual techniques to vary across the parameter space spanned by the data. For example, supervised methods should perform the best in areas populated by high quality training data, while unsupervised or model-based methods should perform better where we have little or no training data. As a result, we can bin a specific subspace of our multi-dimensional parameter space and apply an individual combination method to each bin separately. This technique is demonstrated later in more detail with the Bayesian Model Averaging method (although it is more generally applicable).
Weighted Average {#addition}
----------------
The simplest approach to combine different [photo-$z$ ]{}PDF techniques is to simply add the individual PDFs and renormalize the sum. In this case the final [photo-$z$ ]{}PDF is given by: $$P(z \mid {\mathbf{x}},{\mathbf{M}})=\sum\limits_{k}P(z \mid {\mathbf{x}},M_k) .$$ We can improve on this simple approach by including weights in the previous equation: $$P(z \mid {\mathbf{x}},{\mathbf{M}})=\sum\limits_{k} \omega_k P(z \mid {\mathbf{x}},M_k) .
\label{swa}$$ These weights, $\omega_k$, can be estimated for each input method by using the cross validation or OOB data, or from an intrinsic characteristic of the [photo-$z$ ]{}PDF, such as $zConf$ that we introduced in CB13. In this work we use three weight schemes in addition to the uniform case:
### PDF shape weights {#pdf-shape-weights .unnumbered}
In this case, $\omega_k$ is given by the the $zConf$ parameter, which is similar to the *odds* parameter presented in [@Benitez2000] $zConf$ is defined as the integrated probability between $z_{\rm phot} \pm \sigma_{k}(1+z_{\rm phot})$, where $z_{\rm phot}$ is a single estimated value for the [photo-$z$ ]{}PDF. This single [photo-$z$ ]{}estimate can be either the mean or the mode of the [photo-$z$ ]{}PDF. Likewise, we can estimate $\sigma_k$ for each input method either by using the OOB data, by selecting a constant value across all input methods, or by selecting these values separately so that all [photo-$z$ ]{}PDFs have the same cumulative $zConf$ distributions. $zConf$ quantifies the sharpness of the PDF and can take values from zero to one. In CB13 and CB14, we demonstrated that there is a correlation between this value and the accuracy of the overall [photo-$z$]{}. Specifically, we observed that, on average, galaxies with higher $zConf$ have more accurate [photo-$z$ ]{}PDFs than galaxies with lower $zConf$ values.
### Best fit weights {#best-fit-weights .unnumbered}
An alternative method to compute the values of $\omega_k$ is to use the cross-validation data to first determine the weight values that minimize the difference between $z_{\rm phot}$ and $z_{\rm spec}$; and, second to apply these best fit values to the test data. This method seeks the optimal linear combination of each individual PDF, thus it allows the values of $\omega_k$ to be negative. After the combination is completed, we renormalize according to Equation \[pz1\]. This method can be applied to a binned sub-sample to take advantages of the performance of each method in different areas of the attribute space.
### Oracle scheme {#oracle-scheme .unnumbered}
As mentioned, when the input, multi-dimensional data have been binned (c.f. Figure \[fig:combined\_map\_cfh\]), we can use the cross-validation data to select only one model from among all available input models to only be used with the test data located within that specific bin. Since we are allowed to only select one input model, this will result in an assigned weight value of one for the chosen model and zero otherwise, however the chosen model is allowed to vary between bins.\
The primary disadvantage of these simple, additive models is that incorrect estimates for the errors for the selected input model can bias the final result. On the one hand, if a technique has underestimated errors, the final result will be biased towards this one input method. On the other hand, overestimation of the errors will bias the final result away from this particular method. One approach to address this issue, as discussed by [@Dahlen2013], is to either smooth or sharpen the [photo-$z$ ]{}PDFs estimated by each method by using the OOB data until their error distributions are approximately Gaussian with unit variance. We can generalize this approach to transform a [photo-$z$ ]{}PDF as $P_k(z) = P_k(z)^{\alpha_k}$, where we adjust the value of $\alpha_k$ by using either the cross validation data when errors are over estimated or use a Gaussian smoothing filter when they are under estimated.
Bayesian Model Averaging {#bma}
------------------------
Bayesian Model Averaging (BMA) is an ensemble technique that combines different models within a Bayesian framework. BMA accounts for any uncertainty in the correctness of a given model by integrating over the model space and weighting each model by the estimated probability of being the *correct* model. As a result, BMA acts as a model selection procedure that handles the uncertainty in selecting the best model by using a combination of models instead. This is because BMA considers the uncertainty in selecting the best model while working under the assumption that only one model is actually the best [@Monteith2011]. BMA has been used for astrophysical problems [see [[e.g., ]{}]{} @Gregory1992; @Trotta2007; @Debosscher2007] in, for example, the determination of cosmological parameters and variable star classification [see, @Parkinson2013 for a review on using BMA in astronomy].
When using BMA, the training data are used to characterize each of the models that will be combined. For each galaxy, the final PDF, $P(z \mid {\mathbf{x}}, {\mathbf{D}}, {\mathbf{M}})$, is given by: $$\label{BMA1}
P(z \mid {\mathbf{x}},{\mathbf{D}},{\mathbf{M}}) = \sum\limits_{k}P(z \mid {\mathbf{x}},M_k)P(M_k \mid {\mathbf{D}}) .$$ $P(M_k \mid {\mathbf{D}})$ is the probability of the model $M_k$ given the training data ${\mathbf{D}}$, which can be viewed as a simple, model dependent weighting scheme. This probability can be computed by using Bayes’ Theorem: $$\begin{gathered}
\label{BMA2}
P(M_k \mid {\mathbf{D}}) = \frac{P(M_k)}{P({\mathbf{D}})} P({\mathbf{D}} \mid M_k) \\
\propto P(M_k) \prod\limits_{i=1}^{N_d} P(d_i \mid M_k) .\end{gathered}$$ We have omitted the $P({\mathbf{D}})$ term as it is merely a normalization factor and we use the same data for all models. $d_i$ is the $i^{\textrm{th}}$ element from the training data ${\mathbf{D}}$, which are assumed to be independent.
For each model, we assign the value $\epsilon_k$ as an average error for the estimation process. $\epsilon_k$ can be computed as the fraction $N^{(b)}_k/N_d$, where $N^{(b)}_k$ is the number of galaxies considered to be misestimated or *bad* for the particular [photo-$z$ ]{}PDF method $k$. To quantify when a specific galaxy is a bad prediction we compute $$\label{BMA3}
N^{(b)}_{k,i} =
\left\{
\begin{array}{ll}
1 & \mbox{if } \int_{z_s-\delta_z}^{z_s+\delta_z} P(z \mid {\mathbf{x}},d_i) dz \leq \pi_z ,\\
0 & \mbox{otherwise} .
\end{array}
\right.$$ In this equation, $z_s$ is the spectroscopic redshift for the $i^{\textrm{th}}$ training set galaxy. The first parameter, $\delta_z$, controls the width of a window centered on $z_s$ within which we accumulate [photo-$z$ ]{}probability for the $i^{\textrm{th}}$ training galaxy around the true redshift. The second parameter, $\pi_z$, is the minimum probability within this window for which we consider the model prediction to be good. We find that $\pi_z = 0.5$ and $\delta_z = 0.05$ provides a good discriminant between good and bad [photo-$z$ ]{}model estimates.
Given the individual good/bad predictions for each training set galaxy, we can compute the total number of bad predictions, $N^{(b)}_k$, by summing over the individual predictions, $N^{(b)}_{k,i}$, for the entire training data, ${\mathbf{D}}$. The total number of good prediction will naturally be $N_d-N^{(b)}_k$. As a result, we can rewrite Equation \[BMA2\]: $$P(M_k \mid {\mathbf{D}}) \propto P(M_k) (1-\epsilon_k)^{N_d-N^{(b)}_k}(\epsilon_k)^{N_k^{(b)}},$$ where $P(M_k)$ is the probability of each model $k$, which we can assume to be unity for all models. Therefore, the final PDF for each galaxy is given by $$\begin{gathered}
\label{final_P}
P(z \mid {\mathbf{x}},{\mathbf{D}}, {\mathbf{M}}) \propto \sum\limits_{k}P(z \mid {\mathbf{x}},M_k) P(M_k) \times \\
(1-\epsilon_k)^{N_d-N^{(b)}_k}(\epsilon_k)^{N_k^{(b)}} .\end{gathered}$$ We applied the BMA technique to individual bins within the multi-dimensional parameter space occupied by a given data set. We demonstrate this binned BMA technique in Figure \[fig:combined\_map\_cfh\], where we use a Self Organized Map to project our entire input parameter space to a two-dimensional map. In this manner, all magnitudes or colors are used to form the binned regions within which the parameters of the ensemble learning approach can vary. After computing [photo-$z$ ]{}PDFs for all galaxies with each method, we use BMA to determine the relative weights for these input techniques within each bin; we can visualize these weights as different colors across the two-dimensional map, as shown in Figure \[fig:combined\_map\_cfh\]. This figure graphically displays how the *accuracy* of each [photo-$z$ ]{}PDF estimation varies across the parameter space, and thus how the different weights themselves vary.
Bayesian Model Combination {#bmc}
--------------------------
As discussed, Bayesian Model Averaging tries to select the best model among the ones introduced to the algorithm. Alternatively, we can modify BMA to produce an more optimal model combination technique [@Monteith2011] known as Bayesian Model Combination (BMC). With BMC, instead of directly combining the three different [photo-$z$ ]{}PDF estimates as was the case with BMA, the Bayesian process is used to explore different combinations of the individual [photo-$z$ ]{}PDF techniques. Thus, an ensemble of different [photo-$z$ ]{}PDF combinations are generated and we directly compare different model combinations.
As a simple example, we could first generate hundreds different random weights for all three of our [photo-$z$ ]{}PDF estimation techniques, and second use these to compute hundreds of new *sets* of PDFs by computing a simple weighted average by using Equation \[swa\]. Finally, we could apply BMA to this PDF ensemble to determine the final PDF. In this case, we could write Equation \[BMA1\]: $$\label{BMC1}
P(z \mid {\mathbf{x}},{\mathbf{D}},{\mathbf{M}}, {\mathbf{E}}) = \sum\limits_{e \in {\mathbf{E}}}P(z \mid {\mathbf{x}},{\mathbf{M}},e)P(e \mid {\mathbf{D}}) ,$$ where $e$ is an element from the set ${\mathbf{E}}$ of these hundreds combined models. Here we need to compute the performance of each combination $e$ and apply the BMA formulation, shown in Equations \[BMA2\] and \[BMA3\], to those models by using the model $e$ instead of $M_k$, i.e., $$P(e \mid {\mathbf{D}}) \propto P(e) \prod\limits_{i=1}^{N_d} P(d_i \mid e) .$$ Fundamentally, with BMC we are marginalizing over the uncertainty in the correct model combination, where in BMA we marginalized over the uncertainty in identifying the correct model from the entire ensemble.
The number of model combinations ${\mathbf{E}}$ is, in principle, infinite, and in practice can be very large. To overcome this, we can use sampling techniques over a reasonable, finite number of models. Naively we might use randomly generated weights, however, this approach can be costly to fully span the allowed range of weights and convergence towards a satisfactory solution might be slow. Thus, instead of assigning weights randomly or using incremental steps within a regular grid, we sample the weights from a Dirichlet distribution where the *concentration* parameters are modified until they converge to stable values. We require that the set of weights, $w_k$, for each of the three models, $M_k$, satisfy $\sum w_k = 1$ and also $w_k > 0$.
For a concentration parameter $\boldsymbol{\alpha}$ of the same dimension as ${\mathbf{w}}$, we have that the probability distribution for ${\mathbf{w}}$ is given by: $$P({\mathbf{w}}) \sim \mathcal{D}{\rm ir}(\boldsymbol{\alpha}) = \frac{\Gamma(\sum_k \alpha_k)}{\prod_k \Gamma(\alpha_k)}\prod\limits_k w_k^{\alpha_k -1} ,$$ where $\mathcal{D}{\rm ir}(\boldsymbol{\alpha})$ is the *Dirichlet* distribution, $\Gamma(\alpha_k)$ is the *gamma function* and $k$ are the base models, which in this paper are [`TPZ`]{}, [`SOMz`]{}, and our modified [`BPZ`]{}. In order to generate a set ${\mathbf{E}}$ of combined models, we first set $\alpha_k$ to unity for all values of $k$. Second, we sample from this distribution $n_s$ times ($n_s$ is a fixed number, generally between 2 and 5, which we fixed at 3) to get a set of $n_s$ weights and $n_s$ new model combinations. Next, we compute $P(e \mid D)$ by using Equations \[BMA2\] and \[BMA3\] for each model in the set of $n_s$ models. We, temporarily, select the best model among the set $n_s$, i.e, the one with highest $P(e \mid {\mathbf{D}})$, and update the $\alpha_k$ parameters by simply adding the weights from the corresponding model to the current values of $\boldsymbol{\alpha}$, $$\boldsymbol{\alpha}^{(t+1)} = \boldsymbol{\alpha}^{t} + \max_{{\mathbf{w}}_e \in n_s} P(e \mid {\mathbf{D}})$$ where $t$ is just a symbolic reference to the fact that $\boldsymbol{\alpha}$ is being updated every 3 steps.
We use the latest values for $\boldsymbol{\alpha}$ to continue the sampling process to obtain the next set $n_s$ of model combinations. As a result, we continually (by adding $n_s$ new models at each step) extend our set of model combinations ${\mathbf{E}}$. As the chain of models in this set is constructed iteratively, the process can be terminated either when a predefined number of model combinations has been reached or when new model combinations have started to converge. This process behaves similarly to a Markov Chain Monte Carlo process, and we have an analogous phase to the *burn in* step, where we can omit some number of model combinations at the start of our set ${\mathbf{E}}$ of model combinations. Thus, our final [photo-$z$ ]{}PDF prediction is the application of BMA over the remaining elements in ${\mathbf{E}}$, we have set for this work the size of $E$ to be 800. Finally, we note that, as was the case with BMA, we can develop a binned version of our BMC technique, where we develop different model combinations for different region of the magnitude (color) space by using a SOM.
Hierarchical Bayes
------------------
A Hierarchical Bayesian (HB) method provides a different approach to combine the individual [photo-$z$ ]{}PDFs. In a manner similar to BMA, we include the uncertainty that a given [photo-$z$ ]{}PDF for a specific galaxy might be incorrectly predicted as a set of nuisance parameters over which we later marginalize.
Adopting our previous notation, we follow a similar approach to [@Fadely2012] and [@Dahlen2013], and we write the [photo-$z$ ]{}PDF for an individual galaxy for each base method $k$: $$\begin{gathered}
\label{HB1}
P(z \mid {\mathbf{x}},{\mathbf{D}},M_k, \theta_k) = \sum\limits_j P(z \mid {\mathbf{x}}, {\mathbf{D}}, M_k, \theta_{k j}) \times \\
P(\theta_{k j} \mid {\mathbf{D}}, M_k) ,
$$ where we have introduced the *hyperparameter* $\theta_{k}$, a nuisance parameter that characterizes our uncertainty in the prior distribution of model $k$. The parameter $\theta_{k}$ can be quantified in different forms, but essentially is the misclassification probability of the $k^{\textrm{th}}$ method. Thus, we quantify this mis-prediction probability with $P(\theta_k)$; and we drop the dependence on ${\mathbf{x}}$, the measured galaxy attributes, as it does not directly affect the parameter $\theta_k$. Since we will marginalize over $\theta$, we keep the term ${\mathbf{D}}$ as we can use the training data to place limits on $\theta_k$ by using the cross-validation data. We note that these probabilities are subject to: $$\sum\limits_{j} P(\theta_{k j} \mid {\mathbf{D}}, M_k) = 1 .$$
If we consider the case where galaxies are predicted correctly or are outliers, $j$ is a binary state. In this model, if we assume that $\gamma_k$ is the fraction of galaxies that are mispredictions or are labeled as outliers for method $k$, we have: $P(\theta_{k 0} \mid {\mathbf{D}}, M_k) = \gamma_k$ and $P(\theta_{k 1} \mid {\mathbf{D}}, M_k) = (1 - \gamma_k)$. In this case, Equation \[HB1\] becomes: $$\begin{gathered}
P(z \mid {\mathbf{x}},{\mathbf{D}},M_k, \theta_k) = P_{def}(z \mid M_k, \theta_k)\gamma_k + \\
P(z \mid {\mathbf{x}}, {\mathbf{D}}, M_k, \theta_k)(1-\gamma_k) ,
$$ where $P_{def}(z \mid M_k, \theta_k)$ is the default PDF that should be used for the $k^{\textrm{th}}$ method when the original PDF for that method has been determined to be mis-predicted or wrong. In the second term, we use the original PDF for the method $k$, which is multiplied by the fraction of well predicted objects $1-\gamma_k$.
The final PDF after we combine the different [photo-$z$ ]{}PDFs from our base methods in the HB approach is given by: $$P(z \mid {\mathbf{x}},{\mathbf{D}},\theta) = \prod\limits_k P(z \mid {\mathbf{x}},{\mathbf{D}},M_k, \theta_k)^{1/\beta} .$$ Here, following [@Dahlen2013], we have introduced an extra parameter $\beta$, which is a constant value that quantifies the degree of covariance between the different base methods. $\beta =1$ corresponds to complete independence between the base methods, while $\beta=3$ (or, more generally, the total number of methods) would correspond to full covariance between them. We can compute $\beta$ from the OOB sample in such way the final error distribution follows a normal distribution with zero mean and unit variance, as we have done in this paper. Alternatively, we can marginalize over all possibles values of $\beta$ when no cross validation data is available and we can integrate over the uncertainty of this parameter.
Finally, by marginalizing over $\theta$ we have our final PDF: $P(z \mid {\mathbf{x}}, {\mathbf{D}} )$, or simply $P(z)$ given by: $$P(z) = \int_0^1 P(z \mid {\mathbf{x}},{\mathbf{D}},\theta)P(\theta)d\theta ,$$ where $P(\theta)$ is a constant which in the simple case is equal to unity. If OOB data is available, we can narrow down the range of allowed values for $\theta$ (or effectively $\gamma_k$), so we can set up a limited range for $\gamma_k$ based on the performance of each method $k$ on this data. In this case, $P(\theta)$ will act as a top-hat window function. In any case, the final $P(z)$ is subject to Equation \[pz1\]. As discussed before, we can either apply the HB approach to the entire data set, or we can partition the input space and apply the HB approach independently to the binned regions of the parameter space.
Method Weights[^4] Abbreviation
---------------------------- ------------------ -------------------------
Weighted Average Uniform ${\rm WA}_{\rm flat}$
Weighted Average $zConf$ ${\rm WA}_{\rm shape}$
Weighted Average best fit ${\rm WA}_{\rm fit}$
Weighted Average oracle predictor ${\rm WA}_{\rm oracle}$
Bayesian Model Averaging ${\rm BMA}$
Bayesian Model Combination ${\rm BMC}$
Hierarchical Bayes ${\rm HB}$
: The [photo-$z$ ]{}PDF combination methods, their weights and abbreviations presented in this paper.[]{data-label="tab:methods"}
DATA {#deep2data}
====
To explore different configurations and to demonstrate the capabilities and the efficacy of these [photo-$z$ ]{}combination techniques, we follow the approach we presented in CB13 and CB14, but in this paper we restrict our analysis to data obtained by the Deep Extragalactic Evolutionary Probe (DEEP) survey and the Sloan Digital Sky Survey (SDSS). In the rest of this section we provide a summary of these data and detail how we extracted the data sets from these surveys that we use in the analysis presented in §\[App\].
Deep Extragalactic Evolutionary Probe
-------------------------------------
The DEEP survey is a multi-phase, deep spectroscopic survey that was performed with the Keck telescope. Phase I used the Low Resolution Imaging Spectrometer (LIRS) instrument [@Oke1995], while phase II used the DEep Imaging Multi-Object Spectrograph (DEIMOS) [@Faber2003]. The DEEP2 Galaxy Redshift Survey is a magnitude limited spectroscopic survey of objects with $R_{AB} < 24.1$ [@Davis2003; @Newman2013a]. The survey includes photometry in three bands from the Canada-France-Hawaii Telescope (CFHT) 12K: $B$, $R$, and $I$ and it was recently extended by cross-matching the data to other photometric data sets. In this work, we use Data Release 4 [@Matthews2013], the latest DEEP2 release that includes secure and accurate spectroscopy for over 38,000 sources. The original input photometry for the sources in this catalog was supplemented by using two $u$, $g$, $r$, $i$, and $z$ surveys: the Canada-France-Hawaii Legacy Survey [CFHTLS; @Gwyn2012], and the SDSS. For additional details about the photometric extension of the DEEP2 catalog, see [@Matthews2013].
To use the DEEP2 data with our implementation, we have selected sources with secure redshifts (ZQUALITY$\geq 3$), which were securely classified as galaxies, have no bad flags, and have full photometry. Even though the filter responses are similar, the $u$, $g$, $r$, $i$, and $z$ photometry originates from two different surveys and are thus not identical. We therefore only present the results from those galaxies that lie within field 1 that have CFHTLS photometry. Furthermore, we have corrected these observed magnitudes by using the extinction maps from [@Schlegel1998]. In the end, this leaves us with a total of 10,210 galaxies each with eight band photometry and redshifts. From this data set, we randomly select 5,000 galaxies for training and hold the remainder out for testing. The computation of [photo-$z$ ]{}PDFs was completed by using the magnitudes in the bands $B$, $R$, $I$, $u$, $g$, $r$, $i$, and $z$ and their corresponding colors $B-R$, $R-I$, $u - g$, $g - r$, $r - i$, and $i - z$, providing a total of fourteen dimensions.
Sloan Digital Sky Survey
------------------------
The Sloan Digital Sky Survey [SDSS; @York2000] phases I, II and III conducted a photometric survey in the optical bands: $u$, $g$, $r$, $i$, $z$ that covered more than 14,000 square degrees, more then one-quarter of the entire sky. The resultant photometric catalog contains photometry for over $10^8$ galaxies, making the SDSS one of the largest sky surveys ever completed. The SDSS also conducted a spectroscopic survey of targets selected from the SDSS photometric catalog. In this paper, we use a subset of the spectroscopic data contained within the Data Release 10 catalog [@Ahn2013 SDSS-DR10], which includes over two million spectra of galaxies and quasars which include those taken as apart as the Baryonic Oscillation Spectroscopic Survey (BOSS) program [@Dawson2013].
Specifically, we selected galaxies by using the online CasJobs website[^5] and the following query from the DR10 data base:
SELECT spec.specObjID,
gal.dered_u, gal.dered_g, gal.dered_r,
gal.dered_i, gal.dered_z,
gal.err_u, gal.err_g, gal.err_r,
gal.err_i, gal.err_z,
spec.z AS zs
INTO mydb.DR10_spec_clean_phot
FROM SpecObj AS spec
JOIN Galaxy AS gal
ON spec.specobjid = gal.specobjid,
PhotoObj AS phot
WHERE spec.class = `GALAXY' -- Spectroscopic class
-- (GALAXY, QSO, or STAR)
AND gal.objId = phot.ObjID
AND phot.CLEAN=1 -- Clean photometry flag
-- (1=clean, 0=unclean)
AND spec.zWarning = 0 -- Bitmask of warning
-- vaules; 0 means all
-- is well
We also removed some additional bad photometric observations, ensured the redshift values were positive, and compute colors for the final catalog, which contains 1,147,397 galaxies. The spectroscopic data range from $z \approx 0.02$ up to $z \approx 0.8$; the full spectroscopic redshift distribution of these galaxies is shown in the gray shaded histogram presented in Figure \[fig:N\_z\_sdss\]. These data are dominated by the Main Galaxy Sample (MGS) at low redshifts, with mean redshift of $z \sim 0.1$, and by luminous red galaxies (LRG) at higher redshifts, with mean redshift of $z \sim 0.5$.
From this sample, we randomly selected 50,000 galaxies for training and hold the remaining 1,097,397 for testing. This training set corresponds to approximately 4.5% of the test set. We note that this is a blind test, as the testing data are not used in any way to train or calibrate the algorithms. Of all the measured attributes in the SDSS photometric catalog, we have only used the nine dimensions corresponding to the five galaxy, extinction corrected, model magnitudes and the four colors derived from these five magnitudes: $u$, $g$, $r$, $i$, $z$, $u - g$, $g - r$, $r - i$, and $i - z$.
results/discussion {#App}
==================
We now turn to the actual application of the ensemble learning approaches described in §\[combine\] to the data introduced in §\[deep2data\]. We present the seven combination methodologies we use in this section in Table \[tab:methods\], which also includes an abbreviated name that we will use to refer to a specific technique. We follow a similar approach to CB14 in order to compare different combination methods, and define the bias to be $\Delta z' = |z_{\rm phot}-z_{\rm spec}|/(1+z_{\rm spec})$. We also present the standard metrics we use to compare the performance of the different combination techniques in Table \[tab:def\_metrics\]. As shown in this table, we define five metrics to address the bias and the variance of the results (the first five rows) and we present three values to characterize the outlier fraction.
We also use the $KS$ metric, which represents the results of a Kolmogorov–Smirnov test that quantifies the likelihood that the predicted [photo-$z$ ]{}distribution and the spectroscopic redshift distribution $N(z)$ are drawn from the same underlying population. This metric provides a single, robust value to compare both distributions that does not depend on how the results are binned by redshift, and it is defined as the maximum distance between both empirical distributions.
To determine this statistic, we compute the empirical cumulative distribution function (ECDF) for both distributions. For the spectroscopic sample, the ECDF is defined as: $$F_{\rm spec} (z) = \sum\limits_{i=1}^{N} \Omega_{z_{\rm spec}^i < z}$$ where N is the number of galaxies in the redshift sample, and $$\Omega_{z_{\rm spec}^i < z} = \begin{cases} 1, & \mbox{if } z_{{\rm spec},i} < z \\ 0 , & \mbox{otherwise } \end{cases}$$ The ECDF for the [photo-$z$ ]{}distribution is simply the accumulation of the probability presented in the [photo-$z$ ]{}PDF. The summation is carried out over all galaxies in the sample. Given the ECDF for both the [photo-$z$ ]{}and spectroscopic distributions, we compute the KS statistic as: $${\rm KS} = \max_z \left(\lvert\lvert F_{\rm phot} (z) - F_{\rm spec} (z)\rvert\rvert \right)$$ Thus, as the KS statistic decreases, the two distributions become more similar.
All of the metrics listed in Table \[tab:def\_metrics\] are positive and characterized by the fact that lower metric values indicate a more accurate [photo-$z$ ]{}PDF. In CB14 we defined a new, meta-statistic called $I$-score (symbolically represented by $I_{\Delta z'}$) that provides a single statistic to simplify the comparison of different [photo-$z$ ]{}techniques. To compute this metric, we first normalize each set of metrics across all different [photo-$z$ ]{}estimation techniques so that we are not biased by different dynamic ranges. Thus, for example, we first compute the mean and standard deviation for $<\Delta z'>$ for each combination technique, and subsequently rescale all individual $<\Delta z'>$ values so that this set of values has zero mean and unit variance.
We continue this process for all nine statistics listed in Table \[tab:def\_metrics\], and compute their weighted sum to obtain the total $I$-score: $$I_{\Delta z'} = \sum w_i M_i,$$ where $M_i$ is the rescaled metric and weight value for metric $i$ out of the nine available. For simplicity, we use equal weights in the remainder of this paper (and thus the $I$-score is simply the average of the nine rescaled metrics for each technique). As a result, the [photo-$z$ ]{}method (or parameter configuration) with the lowest $I$-score will be the optimal estimation technique. On the other hand, if we were looking for the technique or the specific parameter configuration with, for instance, the lower outlier fraction, we could assign higher weights accordingly to select the best technique. In this way, we can efficiently select the best method or configuration for specific research requirement.
Metric Meaning
----------------------- -------------------------------------------------------------------------------
$<\Delta z'>$ mean of $\Delta z'$
$|\Delta z'|_{50}$ median of $\Delta z'$
$\sigma_{\Delta z'}$ Standard deviation of $\Delta z'$
$\sigma_{68}$ Sigma value at which 68% of $\Delta z'$ is enclosed
$\sigma_{\rm MAD}$ Median absolute deviation = ${\rm median}(||\Delta z' - |\Delta z'|_{50}||)$
${\rm KS}$ Kolmogorov - Smirnov statistic for $N(z)$
${\rm out}_{0.1}$ Fraction of outliers where $\Delta z' > 0.1$
${\rm out}_{2\sigma}$ Fraction of outliers where $|\Delta z' - <\Delta z'> | > 2\sigma_{\Delta z'}$
${\rm out}_{3\sigma}$ Fraction of outliers where $|\Delta z' - <\Delta z'> | > 3\sigma_{\Delta z'}$
$I_{\Delta z'}$ $I$-score, a weighted combination of all other metrics.
: The definition of the metrics used to compare different [photo-$z$ ]{}combination methods.[]{data-label="tab:def_metrics"}
Cross validation data
---------------------
In CB13, we introduced OOB data and demonstrated its use as a cross-validation data set that provided error quantification and overall performance similar to what could be expected when applying an algorithm directly to the test data set. When building a tree with [`TPZ` ]{}or a map with [`SOMz`]{}, a fraction of the overall training data, usually one-third, is extracted and not used during the tree/map construction process. The resultant tree/map is subsequently applied to this unused data to make a [photo-$z$ ]{}prediction, and this process is repeated for every tree/map. These [photo-$z$ ]{}predications are aggregated for each galaxy to make a [photo-$z$ ]{}PDF; and by construction a galaxy can never be used to train any tree/map that is subsequently used to predict that galaxy’s [photo-$z$]{}. Thus, as long as the OOB data remains similar to the final testing data, the OOB data provide results that will be similar to the final test data results and can be used to guide expectations when applied blindly to other data.
As an illustration of this process, Figure \[fig:true\_both\_deep\] compares the photometric (as computed by using [`SOMz`]{}) and spectroscopic redshifts for galaxies in the training (5,000 in total) and testing (5,210) samples as selected from field 1 of the DEEP2 data set. As shown in this Figure, the performance on both the OOB and the testing data are visually similar and there is no indication of overfitting. In addition, general features in the result, like the spread of the data or the slight tilt of the distribution of points relative to the diagonal, are observed in both samples.
A similar conclusion is observed with the SDSS data, as shown in Figure \[fig:true\_both\_sdss\] where the photometric (as computed by using [`TPZ`]{}) and spectroscopic redshifts for 50,000 galaxies from the training set are compared to 50,000 randomly selected galaxies from the test set. Both distributions show similar behavior and global trends, thus we conclude that, as expected, the OOB data can be used to predict the performance of an PDF combination algorithm on real data.
![A comparison of the photometric (computed by using [`SOMz`]{}) and spectroscopic redshifts for training set (left) and test set (right) galaxies from field 1 of the DEEP2 survey.[]{data-label="fig:true_both_deep"}](Figures/true_test_oob_deep.png){width="46.00000%"}
![A comparison of the photometric (computed by using [`TPZ`]{}) and the spectroscopic redshift from the SDSS-DR10 for the 50,000 training set galaxies (left) and 50,000 galaxies randomly subsampled from the 1,097,397 galaxies in the test set (right). []{data-label="fig:true_both_sdss"}](Figures/true_test_oob_sdss.png){width="46.00000%"}
Another method to contrast the results from these data is to compute the correlation between each of the three [photo-$z$ ]{}estimation techniques discussed earlier as a function of redshift. For this, we use the [photo-$z$ ]{}PDFs for all galaxies, and we calculate the Pearson correlation coefficient $R_{ik}$ within each redshift bin. Even if the three input methods are completely independent, we should expect a positive correlation between them if their predictions are similar. In fact, we desire a positive correlation (but not necessarily a perfect correlation) between the techniques as this will indicate the different techniques are all performing well.
We present the Pearson correlation coefficient for the three [photo-$z$ ]{}PDF estimation techniques for the DEEP2 data (top panel) and the SDSS data (bottom panel) in Figure \[fig:correlation\]. In this figure we display these correlation coefficient computed from the cross-validation (OOB) data (dashed line) and the test data (solid line). The global agreement between these lines further demonstrates the importance of the OOB data as a predictor of the performance of a given technique. This figure also demonstrates a tighter correlation between the two machine learning algorithms than between any machine learning algorithm and the template technique, which is not surprising given the similarities in the methods. While not shown, the shape of the covariance matrices resemble the spectroscopic $N(z)$ distributions presented in Figures \[fig:N\_z\_deep\] and \[fig:N\_z\_sdss\]. We conclude that this is expected since a larger number of galaxies can naturally produce a greater chance for divergent [photo-$z$ ]{}estimates.
![The Pearson correlation coefficient between the individual [photo-$z$ ]{}PDF estimation methods as a function of redshift for the DEEP2 (top) and SDSS (bottom) data. The coefficients measured from the cross-validation (OOB) data (dashed line) and from the test data (solid line) are nearly identical, indicating the utility of the OOB data in predicting the performance of an algorithm on blind test data. Note that a positive correlation is beneficial since this measures the relative performance of different techniques in predicting redshifts.[]{data-label="fig:correlation"}](Figures/cor_deep2.png "fig:"){width="46.00000%"} ![The Pearson correlation coefficient between the individual [photo-$z$ ]{}PDF estimation methods as a function of redshift for the DEEP2 (top) and SDSS (bottom) data. The coefficients measured from the cross-validation (OOB) data (dashed line) and from the test data (solid line) are nearly identical, indicating the utility of the OOB data in predicting the performance of an algorithm on blind test data. Note that a positive correlation is beneficial since this measures the relative performance of different techniques in predicting redshifts.[]{data-label="fig:correlation"}](Figures/cor_sdss.png "fig:"){width="46.00000%"}
As mentioned previously, a concern when combining [photo-$z$ ]{}PDFs from different methods is to reduce the likelihood of being biased by methods that might under- or overestimate their errors. To further demonstrate the importance of the cross-validation data, we compare the normalized error distribution between the cross-validation (OOB) and test data in Figure \[fig:err\_test\_oob\] for both DEEP2 (top panel) and SDSS (bottom panel) data, where the [photo-$z$ ]{}PDFs were generated by [`TPZ` ]{}. In both cases, the two curves are nearly identical, and we confirmed the same result with both [`SOMz` ]{}and [`BPZ`]{}. Thus we can use the OOB data error estimate to rescale the PDF for the test data by using the results computed from the OOB data.
![The normalized error distributions for galaxies in DEEP2 (top) and SDSS (bottom). The error distribution computed from the test data is shown in red, while the error distribution for the cross-validation (OOB data) is shown in black. The excellent agreement highlights the importance of the OOB data in predicting the results of blind test data predictions.[]{data-label="fig:err_test_oob"}](Figures/err_test_oob.png){width="44.00000%"}
[Photo-$z$ ]{}PDF Combination for DEEP2
---------------------------------------
To combine the three [photo-$z$ ]{}PDF techniques discussed in §\[pz\_methods\], we employ a binning strategy to allow different method combinations to be used in different parts of parameter space. We first create a two dimensional, $10 \times 10$ SOM representation of the full 14-dimensional space (eight magnitudes and six colors, note that we do not compute a color between the two different photometric input surveys) by using a rectangular topology to facilitate visualization. With this map we can perform an analysis of all galaxies that lie within the same cell, in a similar process to that described in CB14, but now instead of predicting a [photo-$z$]{}, we are computing the optimal model combination. We apply all seven combination methods presented in Table \[tab:methods\] to all galaxies within each cell by using the OOB data that are also contained within the same cell. We note that the ${\rm WA}_{\rm flat}$ and ${\rm WA}_{\rm shape}$ methods do not depend on this binning, and can, therefore, be used without OOB data. We also could employ the ${\rm HB}$ approach without using this map, but in this case we would need to define $P_{def}(z \mid M_k, \theta_k)$ and perform the marginalization over the entire range of $\theta_k$ without any prior on this value.
![A comparison of the average performance for the three individual [photo-$z$ ]{}PDF estimation methods and the seven different [photo-$z$ ]{}PDF combination approaches for five different metrics as defined in Table \[tab:def\_metrics\] for the DEEP2 data. The horizontal dashed line indicates the best result for a given statistic among the three individual methods (note, [`BPZ` ]{}is not always shown at the provided scale), and the shaded area separates the individual methods from the combined approaches. All values are presented in Table \[tab:big\_metrics\_cfh\].[]{data-label="fig:metrics_cfh_nocut"}](Figures/metrics_deep.pdf){width="44.00000%"}
Combination method $<\Delta z'>$ $|\Delta z'|_{50}$ $\sigma_{\Delta z'}$ $\sigma_{68}$ $\sigma_{\rm MAD}$ ${\rm KS}$ ${\rm out}_{0.1}$ ${\rm out}_{2\sigma}$ ${\rm out}_{3\sigma}$ $I_{\Delta z'}$
------------------------- --------------- -------------------- ---------------------- --------------- -------------------- ------------ ------------------- ----------------------- ----------------------- -----------------
${\rm TPZ}$ 0.0361 0.0205 0.0561 0.0257 0.0139 0.0235 0.0647 0.0307 0.0184 -0.3021
${\rm SOM}$ 0.0431 0.0291 0.0547 0.0325 0.0188 0.0350 0.0862 **0.0284** **0.0150** -0.2035
${\rm BPZ}$ 0.0635 0.0476 0.0679 0.0428 0.0273 0.1342 0.1636 0.0338 0.0170 2.3255
${\rm WA_{\rm flat}}$ 0.0386 0.0231 0.0573 0.0285 0.0155 0.0537 0.0691 0.0313 0.0192 0.1409
${\rm WA_{\rm oracle}}$ 0.0364 0.0206 0.0563 0.0260 0.0139 0.0245 0.0659 0.0313 0.0184 -0.2385
${\rm WA_{\rm shape}}$ 0.0366 0.0217 0.0556 0.0268 0.0146 0.0450 0.0614 0.0297 0.0186 -0.2392
${\rm WA_{\rm fit}}$ 0.0359 0.0208 0.0551 **0.0253** **0.0137** **0.0227** 0.0616 0.0318 0.0178 -0.3404
${\rm BMA}$ 0.0355 0.0211 0.0549 0.0257 0.0140 0.0245 0.0584 0.0289 0.0178 -0.5339
${\rm BMC}$ **0.0350** 0.0208 **0.0531** 0.0255 0.0140 0.0233 **0.0570** 0.0297 0.0176 **-0.5734**
${\rm HB}$ 0.0359 **0.0199** 0.0568 0.0259 **0.0137** 0.0244 0.0641 0.0329 0.0196 -0.0354
We present a summary of the results obtained by applying the seven different combination techniques to all the galaxies within the DEEP2 data in Table \[tab:big\_metrics\_cfh\]. The bold entries in this Table highlight the best technique for any particular metric. The first three rows in this Table show the individual [photo-$z$ ]{}PDF estimation techniques, of which [`TPZ` ]{}generally performs the best and is thus shown in the first row as the benchmark. This Table also clearly indicates that the seven different combination techniques generally have a similar performance, and, as shown in the last four rows, often perform better than [`TPZ`]{}.
We observe that the last four methods: ${\rm WA_{\rm fit}}$, ${\rm BMA}$, ${\rm BMC}$, and ${\rm HB}$ all use the binned model combination approach, and thus can take advantage of the different performance characteristics of individual codes. In this case, ${\rm BMC}$ provides the best performance as measured by the $I$-score $I_{\Delta z'}$, the bias $<\Delta z'>$, the scatter $\sigma_{\Delta z'}$, and the outlier fraction ${\rm out}_{0.1}$. Overall, the differences are close to 5% for many of the metrics, which, while small, are still significant since these are averaged metrics over the full test galaxy sample.
In Figure \[fig:metrics\_cfh\_nocut\], we present a visual comparison between the ten different [photo-$z$ ]{}estimation techniques for five different metrics: bias, scatter, outlier fraction, KS test, and the $I$-score. In each panel, the horizontal dashed line shows the best value from the individual [photo-$z$ ]{}PDF estimation methods and the shaded area separates the individual from the combined methods. This Figure demonstrates that the Bayesian modeling techniques provide better performance than the best individual method over all five metrics, and also that by employing the binning scheme to optimize the combination approach we achieve better performance than for the best individual technique.
We compare the actual [photo-$z$ ]{}PDF for a single galaxy selected from the DEEP2 survey as estimated by the three individual techniques with the [photo-$z$ ]{}PDF estimated by the ${\rm BMC}$ method in Figure \[fig:combined\_pdf\_cfh\]. This Figure clearly shows how the re-normalized combined PDF from the three individual [photo-$z$ ]{}PDF estimation techniques has been improved as the ${\rm BMC}$ result is closer to the true galaxy redshift, shown by the vertical line. These combination techniques identify which individual method works best in different cells, and can use that information to either weight the individual [photo-$z$ ]{}PDFs accordingly, or in the case of ${\rm BMC}$ to marginalize over the uncertainty in the correct weights to produce the best combination.
![An comparison between the three individual [photo-$z$ ]{}PDF estimation techniques and a combined PDF computed by using ${\rm BMC}$ and Equation \[final\_P\] for a single example galaxy taken from the DEEP2. The vertical line indicates the true source redshift.[]{data-label="fig:combined_pdf_cfh"}](Figures/example_2527.png){width="44.00000%"}
![A two-dimensional SOM showing the relative weights for the BMA combination scheme applied to the three individual methods for the DEEP2 field 1 data ([`TPZ` ]{}is top left, [`BPZ` ]{}is top right, and [`SOMz` ]{}is bottom left). In each panel, the color map indicates the value of the weight relative to the other cells in the map. The bottom right panel shows the same cells colored by the mean $R$-band magnitude for the cross validation galaxies.[]{data-label="fig:combined_map_cfh"}](Figures/bma_w.png "fig:"){width="48.00000%"} ![A two-dimensional SOM showing the relative weights for the BMA combination scheme applied to the three individual methods for the DEEP2 field 1 data ([`TPZ` ]{}is top left, [`BPZ` ]{}is top right, and [`SOMz` ]{}is bottom left). In each panel, the color map indicates the value of the weight relative to the other cells in the map. The bottom right panel shows the same cells colored by the mean $R$-band magnitude for the cross validation galaxies.[]{data-label="fig:combined_map_cfh"}](Figures/bma_w_bar.png "fig:"){width="48.00000%"}
We apply a SOM to the DEEP2 field 1 data in order to construct a two-dimensional, binned combination of the three individual [photo-$z$ ]{}PDF estimation methods. We use this SOM to determine the weights for the three individual methods for each cell, and present the results in Figure \[fig:combined\_map\_cfh\] when using the BMA approach as it is easy to interpret. We also show the mean DEEP2 $R$-band magnitude for all galaxies in a given cell in the lower right panel, which clearly indicates the ability of the SOM to preserve relationships between galaxies when projecting from the higher dimensional space to the two-dimensional map. Of course, the SOM mapping is a non-linear representation of all magnitudes and colors, thus the DEEP2 $R$-band map should only be used to provide guidance.
In the three weight maps, a redder color indicates a higher weight, or equivalently that the corresponding method performs better in that region. These weight maps demonstrate the variation in the performance of the individual techniques across the two-dimensional parameter space defined by the SOM. For example, [`BPZ` ]{}performs the best, as expected, in the upper left corner of the map, which is approximately where the faintest galaxies, at least in the DEEP2 $R$-band, are stored. On the other hand, [`TPZ` ]{}performs better in the lower sections of the map, which approximates to brighter DEEP2 $R$-band magnitudes. Interestingly, [`SOMz` ]{}performs relatively better in the upper middle of the map, which corresponds to the middle range $21 \la R \la 23$. The overall variation in weights across the map reflects the performance differences between the individual methods, which are exploited by the combination algorithms in order to identify the optimal combined performance.
We can also compare the global performance of the ${\rm BMC}$ method with the three individual [photo-$z$ ]{}PDF methods as a function of the spectroscopic redshift as shown in Figure \[fig:combined\_true\_deep\]. In this Figure, the photometric redshifts are the computed as the mean of each PDF, and the median is shown as black points along with the tenth and ninetieth percentiles as vertical error bars, enclosing 80% of the distribution on each redshift bin. The performance of the ${\rm BMC}$ method is generally more accurate, resulting in a tighter distribution that suffers fewer outliers when compared to the benchmark [`TPZ` ]{}method. Interestingly, the [`SOMz` ]{}performance is similar to [`TPZ`]{}, while [`BPZ` ]{}is worse, with wider spread and several discontinuities. Nevertheless, the combined method still uses [`BPZ`]{}, as shown in the weight maps, as appropriate to generate an overall improved performance, especially for the faintest galaxies as discussed previously. We note, however, that the number counts in the last few bins are very low for the DEEP2 training and testing sets as shown in Figure \[fig:N\_z\_deep\]. Therefore, although on average [`BPZ` ]{}has better performance statistics over those bins (with large error bars), the [photo-$z$ ]{}results remain subject to Poissonian fluctuations (which is important when constructing a SOM to subdivide the galaxies when applying the combination models), thus the BMC results do not emphasize the [`BPZ` ]{}results in the highest redshift bins.
{width="95.00000%"}
Of all of the ten different metrics presented in Table \[tab:big\_metrics\_cfh\], only the $KS$ test does not show a marked improvement over the benchmark [`TPZ` ]{}method. This metric does not depend on the redshift binning and it is computed by using the stacked PDF for each method. As a result, this metric is expected to be less sensitive to a combination approach, since stacking the PDF smooths out little discrepancies between the models. After integrating over a large number of galaxies PDFs, the individual methods will not differ significantly from one another and the final $N(z)$ distribution will resemble the one from the benchmark method.
Figure \[fig:N\_z\_deep\] shows the final $N(z)$ produced by stacking the PDFs from the ${\rm BMC}$ technique for galaxies from the DEEP2 (in solid black) and the corresponding DEEP2 spectroscopic $N(z)$ for the same galaxies (in gray). As also seen in CB13 and CB14 for [`TPZ` ]{}and [`SOMz` ]{}respectively, both distributions match exceedingly well.
![Top panel: The $N(z)$ for the DEEP2 sample computed directly from the spectroscopic redshifts (gray) and by stacking the [photo-$z$ ]{}PDF estimates from the $BMC$ method (black). Bottom Panel: The absolute difference between these two $N(z)$ distributions.[]{data-label="fig:N_z_deep"}](Figures/N_z_deep2.pdf){width="48.00000%"}
[Photo-$z$ ]{}PDF Combination for the SDSS
------------------------------------------
We now change our focus to the analysis of the SDSS galaxy sample, which consists of 1,097,397 galaxies taken from the SDSS-DR10 data; we now retain 50,000 galaxies for training purposes. We apply the same three [photo-$z$ ]{}PDF estimation methods and seven different combination methods. We construct a SOM-defined, $10 \times 10$ two-dimensional map to subdivide the multi-dimensional magnitude and color space by using a rectangular topology to facilitate visualization. As before, we use cross-validation data to identify the best set of model parameters within each individual cell in our two-dimensional map. As shown in Figures \[fig:correlation\] and \[fig:err\_test\_oob\], the [photo-$z$ ]{}PDFs computed by using the cross-validation and testing data sets are comparable and unbiased.
We present in Table \[tab:big\_metrics\_sdss\] the same ten metrics for each method, and in bold we highlight the best method for each metric. Overall, the results obtained for this data set are remarkable, especially for the outlier fraction and the dispersion. We once again treat [`TPZ` ]{}as the benchmark method; but note that, interestingly enough, in two cases, including the $KS$ metric, [`TPZ` ]{}does provide the best result. In addition, both ${\rm BMA}$ and ${\rm BMC}$ have very similar results, with the latter being slightly better.
After these two models, ${\rm WA_{\rm shape}}$, which is OOB data independent, shows good performance, especially when looking at the $I_{\Delta z'}$ score. For any given individual metric, however, it does not perform better than other combination methods. For this data, [`BPZ` ]{}provides good results; thus we expect that the set of template described in §\[template\] are a good representation of the galaxies in the SDSS photometric data. In particular, this seems true of the LRGs that dominate this sample for $z \ga 0.3$.
Combination method $<\Delta z'>$ $|\Delta z'|_{50}$ $\sigma_{\Delta z'}$ $\sigma_{68}$ $\sigma_{\rm MAD}$ ${\rm KS}$ ${\rm out}_{0.1}$ ${\rm out}_{2\sigma}$ ${\rm out}_{3\sigma}$ $I_{\Delta z'}$
------------------------- --------------- -------------------- ---------------------- --------------- -------------------- ------------ ------------------- ----------------------- ----------------------- -----------------
${\rm TPZ}$ 0.0188 0.0137 0.0219 0.0139 **0.0082** **0.0260** 0.0078 0.0297 0.0121 -0.2875
${\rm SOM}$ 0.0201 0.0149 0.0209 0.0152 0.0094 0.0381 0.0070 0.0334 0.0125 0.7836
${\rm BPZ}$ 0.0230 0.0164 0.0289 0.0167 0.0103 0.0367 0.0134 **0.0228** 0.0111 1.7143
${\rm WA_{\rm flat}}$ 0.0195 0.0139 0.0235 0.0145 0.0088 0.0292 0.0082 0.0251 0.0104 -0.2507
${\rm WA_{\rm oracle}}$ 0.0193 0.0141 0.0220 0.0145 0.0089 0.0373 0.0067 0.0266 **0.0100** -0.1495
${\rm WA_{\rm shape}}$ 0.0192 0.0136 0.0236 0.0143 0.0086 0.0297 0.0081 0.0243 0.0102 -0.4114
${\rm WA_{\rm fit}}$ 0.0200 0.0141 0.0242 0.0149 0.0090 0.0274 0.0090 0.0255 0.0107 0.0244
${\rm BMA}$ 0.0183 0.0133 0.0209 0.0139 0.0084 0.0261 0.0060 0.0296 0.0110 -0.6384
${\rm BMC}$ **0.0183** **0.0133** **0.0203** **0.0138** 0.0084 0.0267 **0.0059** 0.0296 0.0109 **-0.6873**
${\rm HB}$ 0.0198 0.0143 0.0237 0.0147 0.0090 0.0271 0.0084 0.0251 0.0106 -0.0975
We present the performance of the three individual and seven combination methods when applied to the SDSS data for five of the most common metrics in Figure \[fig:metrics\_sdss\]. As was the case with the DEEP2 data, the Bayesian combination methods provide good performance. We also see the same variation in the $KS$ metric, especially when comparing the combination methods to [`TPZ`]{}. However, [`TPZ` ]{}is not always the best performer among the individual techniques, for example [`SOMz` ]{}displays the best performance as measured by $\sigma_{\Delta z'}$ and ${\rm out}_{0.1}$.
As we discussed in CB14, [`SOMz` ]{}performs quite well when using a spherical topology; in the current application to the SDSS data, we have used a random atlas containing 300 maps that use spherical topology each with 3072 total cells. Interestingly, the ${\rm WA_{\rm oracle}}$ method, which selects the best method within each binned cell, often selects the [`SOMz` ]{}result as we can infer from Figure \[fig:metrics\_sdss\]. Although in general the *oracle* combination method is not the best possible combination, as shown by the overall performance of the ${\rm BMA}$ and ${\rm BMC}$ combination methods on this data.
![A comparison of the average performance for the three individual [photo-$z$ ]{}PDF estimation methods and the seven different [photo-$z$ ]{}PDF combination approaches for five different metrics as defined in Table \[tab:def\_metrics\] for the SDSS data. The horizontal dashed line indicates the best result for a given statistic among the three individual methods, and the shaded area separates the individual methods from the combined approaches. All values are presented in Table \[tab:big\_metrics\_sdss\]. []{data-label="fig:metrics_sdss"}](Figures/metrics_sdss.pdf){width="44.00000%"}
We also display the SOM-defined, $10 \times 10$ two-dimensional map used to determine the weights for the three individual methods for each cell in Figure \[fig:combined\_map\_sdss\]. In this map, we identify galaxies within the OOB and test data to determine the parameters for the combination models. One of the benefits of using an unsupervised learning method for this mapping is that we can use any property from the galaxies within this map to construct a representation, such as the mean SDSS $r$-band magnitude map shown in the bottom right panel of Figure \[fig:combined\_map\_sdss\]. In this panel the brighter galaxies are generally on the right while the fainter galaxies are on the left, even though all five magnitudes and four colors were used to construct the SOM-defined, two-dimensional map.
The weighting for the three individual methods show interesting patterns, and [`TPZ` ]{}and [`SOMz` ]{}seem complimentary in that [`TPZ` ]{}is weighted most strongly at fainter $r$-band magnitudes (the left side of the map) while [`SOMz` ]{}is weighted most strongly at brighter $r$-band magnitudes (the right side of the map). This result is most likely an artifact from the bi-modality of the training data, which is dominated at low redshift by the SDSS main galaxy sample and at high redshifts by the SDSS-III LRG sample. At brighter magnitudes and lower redshifts, the [`SOMz` ]{}approach where a high-dimensional space is projected to two-dimensions does a better job of maintaining complex relationships within the data. At fainter magnitudes and higher redshifts, however, the data are dominated by the homogeneous LRG sample. The [`TPZ` ]{}approach performs better for this sample, since the high-dimensional space is recursively sub-divided by [`TPZ` ]{}to maximize the information gain, which may only require one or two dimensions.
![A two-dimensional SOM showing the relative weights for the BMA combination scheme applied to the three individual methods for the SDSS data ([`TPZ` ]{}is top left, [`BPZ` ]{}is top right, and [`SOMz` ]{}is bottom left). In each panel, the color map indicates the value of the weight relative to the other cells in the map. The bottom right panel shows the same cells colored by the mean SDSS $r$-band magnitude for the cross validation galaxies.[]{data-label="fig:combined_map_sdss"}](Figures/bma_w_sdss.png "fig:"){width="48.00000%"} ![A two-dimensional SOM showing the relative weights for the BMA combination scheme applied to the three individual methods for the SDSS data ([`TPZ` ]{}is top left, [`BPZ` ]{}is top right, and [`SOMz` ]{}is bottom left). In each panel, the color map indicates the value of the weight relative to the other cells in the map. The bottom right panel shows the same cells colored by the mean SDSS $r$-band magnitude for the cross validation galaxies.[]{data-label="fig:combined_map_sdss"}](Figures/bma_w_bar_sdss.png "fig:"){width="48.00000%"}
{width="95.00000%"}
Another interesting observation from these weight maps is that [`BPZ` ]{}performs well over much of the parameter space, with a particular strong weighting in a narrow vertical band on the extreme left of the map and again in the center of the map. Given the nature of the input galaxy sample, it seems reasonable to expect that these areas of the map are dominated by Elliptical galaxies. Another interesting observation is that there are six cells in the second column from the left that all have the same value in each weight map (pink for [`TPZ`]{}, white for [`BPZ`]{}, and light blue for [`SOMz`]{}). These cells are primarily empty, [[i.e., ]{}]{}they contain weights and training data but they lack test galaxies and thus have a constant value, which illustrates how strongly the galaxies ([[i.e., ]{}]{}MGS or LRG) are concentrated in this SOM-defined, two-dimensional topology.
The number of galaxies, either for training or testing, within each cell can vary significantly, which is simply due to the fact that we used a fixed number of cells (in this case 100) to represent the higher dimensional space when fewer cells would have been sufficient. However, the empty cells do not affect the performance of the [photo-$z$ ]{}combination methods, they are simply not used during the analysis. It is the fact that these individual methods perform differently across these cells that makes the combination approach a powerful technique to maximally extract information from the available data.
We next provide a comparison between the [photo-$z$ ]{}PDFs computed by the three individual techniques and the ${\rm BMC}$ technique and the SDSS spectroscopic redshift for all 1,097,397 galaxies in Figure \[fig:combined\_true\_sdss\]. The first observation from the figure is the bi-modality of the sample, which is the result of the two primary sub-populations ([[i.e., ]{}]{}MGS and LRGs). Overall, the results are quite good with a very tight correlation, especially in areas of high source density areas. The main exception is at the highest redshifts where there is a slight underestimation; and, as seen before, we can observe how these different approaches provide similar results, which are therefore correlated, while still differing in other areas where one method may outperform the others. The most right panel is the ${\rm BMC}$ which shows a slightly tighter distribution in comparison to the others.
Finally, in Figure \[fig:N\_z\_sdss\] we present the galaxy redshift distribution for both the spectroscopic sample (in gray) and the photometric redshift distribution, computed by stacking the individual galaxy PDFs (in black). This Figure highlights that the underestimation of the [photo-$z$ ]{}at high redshifts in Figure \[fig:combined\_true\_sdss\] coincides with the strong decline in the number of galaxies after $z = 0.75$. More importantly, however, this $N(z)$ figure shows the excellent agreement between the photometric and spectroscopic galaxy redshift distributions. Given the fact that the SDSS galaxy sample contains two distinct populations, this agreement is remarkable.
![Top panel: The $N(z)$ computed directly from the spectroscopic redshifts (gray) and by stacking the [photo-$z$ ]{}PDF estimates from the $BMC$ method (black). Bottom Panel: The absolute difference between these two $N(z)$ distributions.[]{data-label="fig:N_z_sdss"}](Figures/N_z_sdss.pdf){width="48.00000%"}
Outliers identification
=======================
As we have discussed previously, aggregating information from multiple [photo-$z$ ]{}PDFs estimation techniques can improve the overall [photo-$z$ ]{}solution. In this section, however, we explore how this information can be combined to improve the identification of outliers within the test data. In particular, we attempt to use all possible information in order to identify these objects, from the shape of each [photo-$z$ ]{}PDF as computed by all individual methods to the differences in their predicted [photo-$z$]{}. We adopt a Naïve Bayes Classifier (NBC) [@Zhang2004] to identify these two groups, a technique that has found widespread adoption to identify spam email messages. The advantage of this approach is that it is easy to implement, is fast and efficient for large dimensional data, and can be very competitive with other classifiers [@Domingos1997; @Frank2000].
Let $\boldsymbol{\theta}$ be the set of $N_{\theta}$ parameters, $\theta_i$, we will use to identify the outliers. By using the Bayes Theorem, we can compute the probability for an object to be an outlier, given $\boldsymbol{\theta}$ as: $$\label{NBC1}
P({\rm out} \mid \boldsymbol{\theta} ) = \frac{P({\rm out}) P(\boldsymbol{\theta} \mid {\rm out})}{P(\boldsymbol{\theta})}$$ where the *evidence*, $P(\boldsymbol{\theta})$ is given by $$P(\boldsymbol{\theta}) = P(\boldsymbol{\theta} \mid {\rm out}) + P(\boldsymbol{\theta} \mid {\rm in})$$ and *out* refers to outliers and *in* refers to inliers, the only two classes we identify in this analysis. The Naïve Bayes Classifier assumes that all $\theta_i$ variables are independent, even if their independence is weak or even if there is a strong dependence between any of them. Each variable provides information about these two classes, and this information can be combined to make a stronger classifier [@Zhang2004]. For instance, in CB13 we showed that outliers tend to have a broader (larger values of $zConf$) and multi-peaked PDFs, and herein we treat these values as independent data even though multi-peaked PDFs are indeed generally broader.
By using this assumption, we can write: $$P(\boldsymbol{\theta} \mid {\rm out}) = P(\theta_1, \theta_2, \dots, \theta_{N_\theta} \mid {\rm out}) = \prod\limits_{i=1}^{N_\theta} P(\theta_i \mid {\rm out})$$ and similarly, $$P(\boldsymbol{\theta} \mid {\rm in}) = \prod\limits_{i=1}^{N_\theta} P(\theta_i \mid {\rm in})$$ We can now rewrite Equation \[NBC1\]: $$\label{NBC2}
P({\rm out} \mid \boldsymbol{\theta} ) = \frac{ P({\rm out}) \prod P(\theta_i \mid {\rm out})}{\prod P(\theta_i \mid {\rm out}) + \prod P(\theta_i \mid {\rm in}) } ,$$ which is similar to the method used by [@Gorecki2014], who demonstrated the potential of this approach to identify [photo-$z$ ]{}outliers. Here, however, we use a different set of variables that are generated for all three individual [photo-$z$ ]{}PDF methods.
In our case we use $N_{peak}$, the number of peaks in each [photo-$z$ ]{}PDF; $r_{peak}$, the logarithm of the ratio between the height of the first peak and the height of the second peak; $z_{mean}$, the mean of each [photo-$z$ ]{}PDF; $z_{mode}$, the mode of each PDF;$zConf$, measured with respect to the mean and the mode of the [photo-$z$ ]{}PDF; and the difference in the [photo-$z$ ]{}, as enumerated by the mean and the mode between each of the three methods. Thus, we have six metrics computed individually for each of our three [photo-$z$ ]{}PDF estimation techniques, and an additional six metrics for the difference in [photo-$z$ ]{}mean and mode between the three techniques. As a result, we have a total of twenty-four metrics, to which we can add the input data for each survey.
We, therefore, have a total of thirty-eight variables for the DEEP2 survey, while for the SDSS we have a total of thirty-three variables to use for outlier detection. For convenience, we rescale each of these variables to lie between zero and one. $P(\theta_i \mid {\rm in})$ and $P(\theta_i \mid {\rm out})$ are evaluated by using the OOB or cross-validation data, which we have shown can reliably predict the results on the test data. Once computed, these distributions are evaluated for the test data, where $ P({\rm out} \mid \boldsymbol{\theta} ) $ is evaluated separately for each galaxy in the test data.
![The normalized distributions of four of the set of thirty-eight (rescaled) $\boldsymbol{\theta} $ variables from the DEEP2 data that are used for outlier detection. The variables are binned as outliers (black line histograms) or inliers (gray histogram). From the top left and following in a clockwise direction: $N_{peak}$, the number of peaks in the [`TPZ` ]{}PDF; $zConf$, as computed from [`TPZ`]{}, the $R$-band magnitude, and the difference between the [photo-$z$ ]{}computed by using the mean of the [`TPZ` ]{}and [`BPZ` ]{}PDFs. \[fig:P\_theta\]](Figures/P_theta.png){width="48.00000%"}
![The normalized distributions of four of the set of thirty-three (rescaled) $\boldsymbol{\theta} $ variables from the SDSS data that are used for outlier detection. The variables are binned as outliers (black line histograms) or inliers (gray histogram). From the top left and following in a clockwise direction: $r_{peak}$, the logarithmic ratio of the first two peaks in the [`TPZ` ]{}PDF; $zConf$, as computed from [`SOMz`]{}, the SDSS $z$-band magnitude, and the difference between the [photo-$z$ ]{}computed by using the mode of the [`SOMz` ]{}and [`BPZ` ]{}PDFs. \[fig:P\_theta\_sdss\]](Figures/P_theta_sdss.png){width="48.00000%"}
Figure \[fig:P\_theta\] presents the normalized distributions of four rescaled variables ([[i.e., ]{}]{}$\theta_i$) taken from the DEEP2 test data. Note that the inlier and outlier distributions are normalized to have unit area, thus these distributions illustrate how these two populations differ and not how the relative numbers between the inlier and outlier populations vary. The four variables shown in this Figure include the number of peaks in the [`TPZ` ]{}PDFs, $zConf$ computed by [`TPZ`]{}, the $R$-band magnitude, and the difference between the mean of the [`TPZ` ]{}and [`BPZ` ]{}[photo-$z$ ]{}PDFs. In just these four distributions, there is clear separation between the galaxies labeled as outliers (black line) and inliers (gray shaded area), where the outlier identification metrics are defined by using Table \[tab:def\_metrics\]. In particular, for this Figure we use ${\rm out}_{0.1}$, [[i.e., ]{}]{}galaxies for which $\Delta z' > 0.1$. While not shown, a similar result is seen for the other distributions. The result that outliers and inliers follow distinct distributions is what makes this a powerful approach. In effect, all information is assumed to be independent, and when combined allows an efficient identification of catastrophic outliers.
We see a similar trend in Figure \[fig:P\_theta\_sdss\], but now for galaxies in the SDSS test data. In this Figure, we have selected four different rescaled variables; namely, the logarithmic ratio between the first and the second peaks of the [`TPZ` ]{}PDF (note that if the PDF has one peak, we fix this value to be four), the $zConf$ computed from [`SOMz`]{}, the SDSS $z$-band magnitude, and the difference between the mode of the [`SOMz` ]{}and [`BPZ` ]{}[photo-$z$ ]{}PDFs. Once again, this Figure highlights that in each of these distributions there is a separation between the outliers and inliers, and that in combination we obtain an even better discriminant between these two classes.
By using Equation \[NBC2\], we can combine the values of all of the rescaled variables ([[i.e., ]{}]{}$\theta_i$) to compute $P({\rm out} \mid \boldsymbol{\theta} )$ for each galaxy in the DEEP2 and SDSS, both for the OOB and the test data. We present these $P({\rm out} \mid \boldsymbol{\theta} )$ distributions for the DEEP2 in Figure \[fig:P\_outlier\_deep\] and for the SDSS in Figure \[fig:P\_outlier\_sdss\]. Both Figures are similar, showing a clear separation between the outliers and inliers in both data sets. The probability ranges between zero and one, and the outliers are generally concentrated near one, while the inliers are concentrated near zero. While some mis-classifications remain, the contamination has been greatly reduced, meaning we can successfully identify a majority of the outlier population. Lastly, while there are a few galaxies with probabilities lying somewhere between zero and one, these distributions are highly bimodal, which reinforces the belief that this method provides a remarkably good discriminant between these two populations.
![The count distribution of $P({\rm out} \mid \boldsymbol{\theta} )$ for the DEEP2 OOB data (top) and test data (bottom) showing both the outliers (orange) and inliers (gray). \[fig:P\_outlier\_deep\]](Figures/outliers_deep2.png){width="48.00000%"}
![The count distribution of $P({\rm out} \mid \boldsymbol{\theta} )$ for the SDSS OOB data (top) and test data (bottom) showing both the outliers (orange) and inliers (gray ). \[fig:P\_outlier\_sdss\]](Figures/outliers_sdss.png){width="48.00000%"}
Once again, in both Figures \[fig:P\_outlier\_deep\] and \[fig:P\_outlier\_sdss\], the OOB and test data distributions show strong similarities. As a result, we can expect that any cut we make on the OOB data will produce similar results in the test data, allowing us to make a robust classification of outliers in potentially blind test data. To quantify this, we show in Table \[tab:outliers\_deep\] the effects of selecting outliers by using this NBC approach and by using the $zConf$ approach we initially presented in CB13 for the DEEP2 data. To simplify the comparison, we first select inlier galaxies by using the $P({\rm out} \mid \boldsymbol{\theta} )$ to cut the test data sample, and subsequently choosing those galaxies in the test data that have the highest $zConf$ so that we have the same number of galaxies selected via both techniques.
Method Criteria Fraction $<\Delta z'>$ $\sigma_{\Delta z'}$ ${\rm out}_{0.1}$
--------- ----------- ---------- --------------- ---------------------- -------------------
NBC $<$ 0.998 83.0 % 0.02819 0.03948 0.0362
$zConf$ $>$ 0.854 83.0 % 0.02868 0.04186 0.0371
NBC $<$ 0.894 72.0 % 0.02616 0.03548 0.0304
$zConf$ $>$ 0.893 72.0 % 0.02721 0.03895 0.0330
NBC $<$ 0.174 56.0 % 0.02565 0.03470 0.0251
$zConf$ $>$ 0.918 56.0 % 0.02595 0.03575 0.0289
: The effect of removing outliers from the DEEP2 test data on several, select performance metrics by using the Naïve Bayes Classifier and the $zConf$ cut approach. The two techniques are applied to ensure equal numbers of galaxies are selected, which is indicated by the *Fraction* column. \[tab:outliers\_deep\]
The information in this Table demonstrates that the NBC approach produces a sample of galaxies that have a smaller spread in $\Delta z'$ along with a smaller number of outliers than the $zConf$ method, which was previously shown to be beneficial in this regard (CB13). We interpret this result as suggesting that a $zConf$ cut can potentially remove *good* galaxies whose [photo-$z$ ]{}PDF happens top be broad, while retaining some *bad* galaxies that have a well-localized [photo-$z$ ]{}PDF. By using a Naïve Bayes approach, we collect all information from [photo-$z$ ]{}PDFs predicted by using different, semi-independent methods, allowing a more robust discriminant between outliers and inliers. Finally, we notice that as always there is a trade-off between completeness, whereby we try to retain as many *good* galaxies, and contamination, whereby we try to minimize the inclusion of *bad* galaxies. The final choice in this conflict should be determined by the scientific application, but by producing a probabilistic value, subsequent researchers can make these cuts more easily.
We performed a similar analysis on the SDSS galaxy sample and present the results in Table \[tab:outliers\_sdss\]. As was the case with the DEEP2 galaxies, we see that the NBC approach once again does better in identifying outliers within the sample, as the NBC cuts have a smaller scatter and the fraction of remaining outliers is remarkably small. We also notice that the mean bias is similar between the two approaches, but the number of outliers, defined as $\Delta z' > 0.1$, is significantly reduced when we adopt the Bayesian approach. This is yet another piece of evidence supporting the benefits of aggregating information to make decisions.
Method Criteria Fraction $<\Delta z'>$ $\sigma_{\Delta z'}$ ${\rm out}_{0.1}$
--------- ------------ ---------- --------------- ---------------------- -------------------
NBC $<$ 0.999 83.0 % 0.01560 0.01533 0.0022
$zConf$ $>$ 0.7018 83.0 % 0.01589 0.01704 0.0035
NBC $<$ 0.802 72.0 % 0.01473 0.01411 0.0012
$zConf$ $>$ 0.755 72.0 % 0.01475 0.01549 0.0026
NBC $<$ 0.001 56.0 % 0.01387 0.01309 0.0006
$zConf$ $>$ 0.807 56.0 % 0.01366 0.01410 0.0020
: The effect of removing outliers from the SDSS test data on several, select performance metrics by using the Naïve Bayes Classifier and the $zConf$ cut approach. The two techniques are applied to ensure equal numbers of galaxies are selected, which is indicated by the *Fraction* column. \[tab:outliers\_sdss\]
We can also test how the definition of an outlier affects this approach. Previously we identified an outlier as a galaxy that had $\Delta z' > 0.1$; but for the purpose of this test, we apply a much more restrictive cut of $\Delta z' > 0.05$. We apply the NBC cut and produce a matched sample by imposing a $zConf$ cut to both the DEEP2 and the SDSS galaxies, presenting the information in Table \[tab:outliers\_both\]. We find, once again, that even for this more restrictive approach we produce a cleaner catalog (of the same size) as compared to using only the $zConf$ parameter. Interestingly, even after removing almost 30% of the galaxies from the DEEP2 galaxy sample, we still have over a 10% outlier contamination. On the other hand, this tight cut applied to the SDSS galaxies produces a very small contamination of $\sim$ 2%, for both methods, albeit the NBC approach is still slightly better.
Method Criteria Fraction $<\Delta z'>$ $\sigma_{\Delta z'}$ ${\rm out}_{0.05}$
--------- ----------- ---------- --------------- ---------------------- --------------------
DEEP2
NBC $<$ 0.996 72.0 % 0.02780 0.03934 0.138
$zConf$ $>$ 0.878 72.0 % 0.02809 0.04244 0.141
SDSS
NBC $<$ 0.85 72.0 % 0.01461 0.01407 0.0247
$zConf$ $>$ 0.75 72.0 % 0.01479 0.01554 0.0278
: The effect of removing outliers, defined as $\Delta z' > 0.05$, from the DEEP2 and SDSS test data on several, select performance metrics by using the Naïve Bayes Classifier and the $zConf$ cut approach. For each data set, the two techniques are applied to ensure equal numbers of galaxies are selected, which is indicated by the *Fraction* column. \[tab:outliers\_both\]
While producing galaxy samples that are less affected by outliers than competing techniques, the NBC approach has an additional advantage in that it can easily be extended to other variables and to other [photo-$z$ ]{}algorithms. In effect, any information that might increase the efficacy of outlier identification can be included in order to improve this discriminant while still maximizing the overall galaxy sample size.
Conclusions
===========
We have presented and analyzed different techniques for combining [photo-$z$ ]{}PDF estimations on galaxy samples from the DEEP2 and SDSS projects. In particular, we use three independent [photo-$z$ ]{}PDF estimation methods: [`TPZ`]{}, a supervised machine learning technique based on prediction trees and a random forest; [`SOMz`]{}, an unsupervised machine learning approach based on self organizing maps and a random atlas; and [`BPZ`]{}, a standard template-fitting method that we have slightly modified to parallelize the implementation. Both [`TPZ` ]{}and [`SOMz` ]{}are currently available within a new software package entitled `MLZ`[^6].
We developed seven different combination methods that employ ensemble learning with cross-validation data to maximize the information extracted. Of these seven methods, four employ a weighted average where the weights can either be selected to be uniform across the input methods, to be determined from the shape of the [photo-$z$ ]{}PDF (e.g., by using the $zConf$ parameter), to be determined by an *oracle* estimator where one (ideally the best) method is preferentially selected, and where the weights are obtained by a fitting procedure applied to the OOB data. Three of the combination methods were Bayesian techniques: Bayesian Model Averaging (BMA), Bayesian Model Combination (BMC), and Hierarchical Bayes (HB).
We expect the individual [photo-$z$ ]{}PDF estimation techniques to perform differently across the parameter space spanned by our galaxy samples; for example, template-fitting techniques are expected to work better at higher redshifts than machine learning methods, which perform optimally when provided high-quality, representative training data. Thus we construct a two-dimensional, $10 \times 10$ self-organizing map (SOM) to subdivide the high-dimensional parameter space occupied by the galaxy samples. We apply different [photo-$z$ ]{}PDF estimation techniques within each cell in this map, since each cell should contain galaxies with similar properties. A visual inspection of these maps indicates that the two machine learning methods: [`TPZ` ]{}and [`SOMz` ]{}are generally complementary, and that in combination with a model based technique such as [`BPZ` ]{}we are able to maximize the coverage of this multidimensional space efficiently.
We also verified that by using the OOB data, as introduced in CB13, we can an obtain an accurate, unbiased and *honest* estimation of the performance of a [photo-$z$ ]{}PDF estimation technique on the test data. We also computed the correlation coefficient and the error distribution and showed they also behave similarly for the cross-validation ([[i.e., ]{}]{}the OOB data) and the test data. These computations are extremely important when combining [photo-$z$ ]{}PDF techniques as we can learn from the OOB data the optimal parameters needed for a specific ensemble learning approach, and thereby maximize the performance of that combination technique when applied to *blind* test data.
Overall, we found that the BMA and BMC are the best [photo-$z$ ]{}PDF combination techniques as they have better performance metrics when compared to the individual [photo-$z$ ]{}PDF estimation techniques, especially when unbiased cross-validation data is available. This result is true for both the DEEP2 and the SDSS data. When OOB data is not available, we can instead use the $zConf$ parameter as a weight for each method after first renormalizing the individual [photo-$z$ ]{}PDFs. We can also use the Hierarchical Bayes method to combine these predictions, which we demonstrated can also lead to better results.
Within this Bayesian Framework, we also developed a novel, Naïve Bayesian Classifier (NBC) that efficiently identifies outliers within the galaxy sample. The approach we present gathers all available information from the different [photo-$z$ ]{}PDF estimation techniques regarding the shape of the PDF, the location of the mean and mode, and the magnitudes and colors, which are all *naively* assumed to be independent, in order to compute a Bayesian posterior probability that a certain galaxy is an outlier. The distribution of these probabilities for an entire galaxy sample indicate that this is a very powerful method to separate outliers from inliers ([[i.e., ]{}]{}*good* galaxies), and we further demonstrated that this approach can produce a more accurate and cleaner sample of galaxies than competing techniques, such as the use of the $zConf$ parameter. An important takeaway point is that all information provided by the catalogs and the [photo-$z$ ]{}PDF methods, no matter how redundant the information might appear, helps in building this discriminant probability. Given the probabilistic nature of this computation, the final application of this technique can be chosen to maximize the scientific utility of the resulting galaxy data for a specific application.
The computational cost to apply these Bayesian models to galaxy samples will depend directly on the size of the data set, the number of [photo-$z$ ]{}estimation techniques used, and the resolution of the given [photo-$z$ ]{}PDFs. In [@CarrascoKind2014b] we demonstrate how a sparse basis representation can reduce the storage significantly and that manipulation of these PDFs can be improved within the bases framework thereby reducing computational costs. We plan to adopt this representation framework to compute the combination models, which will allow fast and accurate combination of multiple [photo-$z$ ]{}PDFs.
Finally, we have demonstrated that even when a [photo-$z$ ]{}PDF technique is very accurate, we can still make improvements by extracting additional information about the distribution of galaxies in the higher dimensional parameter space and the individual performance of the [photo-$z$ ]{}PDF algorithms. There are currently a large number of published algorithms to compute [photo-$z$ ]{}’s, many of which also compute [photo-$z$ ]{}PDFs. Even if their performance is similar, these techniques will all have their own advantages and disadvantages. Thus we believe the combination of different techniques is the future of [photo-$z$ ]{}research, and we expect additional research to be forthcoming in this area. Overall, the combination of [photo-$z$ ]{}PDFs is a powerful, new approach that can be easily extended to incorporate new techniques in order to generate a meta-predictor that accelerate our knowledge and understanding of the Universe.
Acknowledgements {#acknowledgements .unnumbered}
================
The authors thank the referee for a careful reading of the manuscript and for comments that improved this work. RJB and MCK acknowledge support from the National Science Foundation Grant No. AST-1313415. MCK has been supported by the Computational Science and Engineering (CSE) fellowship at the University of Illinois at Urbana-Champaign. RJB has been supported in part by the Institute for Advanced Computing Applications and Technologies faculty fellowship at the University of Illinois.
The authors gratefully acknowledge the use of the parallel computing resource provided by the Computational Science and Engineering Program at the University of Illinois. The CSE computing resource, provided as part of the Taub cluster, is devoted to high performance computing in engineering and science. This work also used resources from the Extreme Science and Engineering Discovery Environment (XSEDE), which is supported by National Science Foundation grant number OCI-1053575.
Funding for the DEEP2 Galaxy Redshift Survey has been provided by NSF grants AST-95-09298, AST-0071048, AST-0507428, and AST-0507483 as well as NASA LTSA grant NNG04GC89G.
Funding for SDSS-III has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, and the U.S. Department of Energy Office of Science. The SDSS-III web site is http://www.sdss3.org/.
SDSS-III is managed by the Astrophysical Research Consortium for the Participating Institutions of the SDSS-III Collaboration including the University of Arizona, the Brazilian Participation Group, Brookhaven National Laboratory, Carnegie Mellon University, University of Florida, the French Participation Group, the German Participation Group, Harvard University, the Instituto de Astrofisica de Canarias, the Michigan State/Notre Dame/JINA Participation Group, Johns Hopkins University, Lawrence Berkeley National Laboratory, Max Planck Institute for Astrophysics, Max Planck Institute for Extraterrestrial Physics, New Mexico State University, New York University, Ohio State University, Pennsylvania State University, University of Portsmouth, Princeton University, the Spanish Participation Group, University of Tokyo, University of Utah, Vanderbilt University, University of Virginia, University of Washington, and Yale University.
\[lastpage\]
[^1]: http://www.darkenergysurvey.org/
[^2]: http://www.lsst.org/lsst/
[^3]: \[fnote\] http://lcdm.astro.illinois.edu/code/mlz.html
[^4]: if applicable
[^5]: http://skyserver.sdss3.org/CasJobs/
[^6]: http://lcdm.astro.illinois.edu/code/mlz.html
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Using inelastic neutron scattering, we show that the onset of superconductivity in underdoped Ba(Fe$_{1-x}$Co$_{x}$)$_{2}$As$_{2}$ coincides with a crossover from well-defined spin waves to overdamped and diffusive spin excitations. This crossover occurs despite the presence of long-range stripe antiferromagnetic order for samples in a compositional range from $x=0.04-0.055$, and is a consequence of the shrinking spin-density wave gap and a corresponding increase in the particle-hole (Landau) damping. The latter effect is captured by a simple itinerant model relating Co doping to changes in the hot spots of the Fermi surface. We argue that the overdamped spin fluctuations provide a pairing mechanism for superconductivity in these materials.'
author:
- 'G. S. Tucker'
- 'R. M. Fernandes'
- 'D. K. Pratt'
- 'A. Thaler'
- 'N. Ni'
- 'K. Marty'
- 'A. D. Christianson'
- 'M. D. Lumsden'
- 'B. C. Sales'
- 'A. S. Sefat'
- 'S. L. Bud’ko'
- 'P. C. Canfield'
- 'A. Kreyssig'
- 'A. I. Goldman'
- 'R. J. McQueeney'
title: 'Crossover from spin-waves to diffusive spin excitations in underdoped '
---
The key to many unconventional superconductors lies in their proximity to an ordered antiferromagnetic (AFM) phase [@Pines92; @Chubukov03; @Scalapino12]. As a ground state that competes with superconductivity (SC), the suppression of AFM order (by chemical tuning or applied pressure) is required for the SC state to appear. However, the vestiges of AFM order that remain as correlated spin fluctuations have been proposed to provide the glue that pairs electrons in the SC state [@magnetic]. It is, therefore, very important to understand how the spin excitations evolve from collective spin waves in the AFM ordered state to the overdamped correlated spin fluctuations characteristic of the SC state. In the iron pnictide , the suppression of AFM ordering upon Co substitution of a few percent allows a SC ground state to appear [@Ni08] in the presence of substantial spin fluctuations at the AFM wavevector, ${\ensuremath{\mathbf{Q}{\ensuremath{_{\mbox{\scriptsize AFM}}}}}}$. Unlike some unconventional superconductors, e.g., Ba$_{1-x}$K$_x$Fe$_2$As$_2$, the competing AFM ordered and SC states actually coexist microscopically in a limited compositional range from $x\approx0.04-0.06$, the so-called underdoped compositions [@FernandesPRB10]. This allows one to investigate how the normal state spin fluctuations provide the conditions for SC to emerge, even in the presence of weak AFM order.
Given the important connection between superconductivity and magnetism, extensive studies of the magnetic dynamics have been performed in these compounds as a function of composition. The magnetic dynamics of electron doped compounds, ($M=$ Co, Ni) have been studied in some detail by inelastic neutron scattering (INS) [@Lester10; @Li10; @Harriger11; @Dai12; @Tucker12; @Liu12]. These investigations found that the high-energy spin dynamics ($E>50$ meV) are relatively insensitive to electron doping whereas the low-energy spin dynamics show a strong dependence on electron doping. Deep within the AFM ordered state of the parent [BaFe$_2$As$_2$]{} compound (Néel transition temperature, $T{\ensuremath{_{\mbox{\scriptsize N}}}}=136$ K), the low-energy spin dynamics are dominated by a large spin gap $\Delta\approx10$ meV that characterizes the ordered AFM state [@Sato09]. Above the spin gap, very steep spin wave excitations propagate within the Fe layer, while much lower spin wave velocities are found for modes propagating perpendicular to the layers, indicative of quasi-two-dimensional magnetism. In [BaFe$_2$As$_2$]{} [@Ewings08], as well as [CaFe$_2$As$_2$]{} [@McQueeney08] and [SrFe$_2$As$_2$]{} [@Zhao08], the low-energy spin waves have very long lifetimes (no substantial energy-dependent damping). The large spin gap and small damping of the collective spin wave modes highlight the robust AFM state of the parent compounds. Within an itinerant spin-density wave picture for the AFM order in the iron pnictides, such behavior indicates that the electronic spin-density wave (SDW) gap is large, estimated to be $\Delta_{\mathrm{SDW}}>50$ meV via optical conductivity measurements [@Hu08], thereby gapping out particle-hole (Landau) damping mechanisms. Note that while the spin gap $\Delta$ is related to anisotropies in spin space (such as single-ion anisotropy), the SDW gap $\Delta_{\mathrm{SDW}}$ is proportional to the magnetization and, therefore, to the energy gain in the magnetically ordered state.
At the opposite extreme, those compositions without long-range AFM order ($x>0.06$ for Co substitutions) display low-energy spin excitations that are diffusive (overdamped) in nature, and typical of systems close to a critical point [@Inosov10; @Li10]. The low-energy spin fluctuations are still centered at $\textbf{Q}_{\mathrm{AFM}}$, but appear gapless and are characterized by a finite spin-spin correlation length ($\xi$) and relaxational energy scale ($\Gamma$) related to the Landau damping. The presence of substantial magnetic spectral weight at low energies (as obtained from a gapless spectrum with $\Gamma\sim k{\ensuremath{_{\mbox{\scriptsize B}}}}T{\ensuremath{_{\mbox{\scriptsize c}}}}$, with superconducting transition temperature $T{\ensuremath{_{\mbox{\scriptsize c}}}}$) is considered an important ingredient for magnetically-mediated SC [@Chubukov03].
Here, we study the evolution of the normal state spin dynamics between these two extremes. Most of the compositions are underdoped, possessing both weak AFM ordering and superconductivity at low temperatures (i.e. small SDW and SC gaps). In the normal state of the underdoped compounds, we find clear signatures of diffusive behavior in the low-energy spin dynamics (spatial disorder and a gapless spectrum with overdamped dynamics) despite the AFM ordering. The crossover of the spin dynamics is associated both with the collapse of the spin-density wave gap, see Ref. , and the subsequent development of strong Landau (particle-hole) damping. This crossover coincides with the appearance of SC in underdoped samples.
The INS measurements were carried out on the HB3 spectrometer at the High Flux Isotope Reactor at the Oak Ridge National Laboratory. Samples were grown and characterized as outlined in Ref. and were mounted in the $[1,1,0]$–$[0,0,1]$ scattering plane. We define $\mathbf{Q}$ = $\frac{2\pi}{a}H\hat{\mathrm{i}}+\frac{2\pi}{a}K\hat{\mathrm{j}}+\frac{2\pi}{c}L\hat{\mathrm{k}}$ = $(H,K,L)$ in reciprocal lattice units as referenced to the tetragonal $I4/mmm$ unit cell. Details about the instrumental configuration and resolution are given elsewhere.
![ Temperature dependence of INS data for with $x=0.015$ \[(a,c,e)\] and $x=0.033$ \[(b,d,f)\] plus fits to the spinwave model. (a,b) Energy scans at ${\ensuremath{\mathbf{Q}{\ensuremath{_{\mbox{\scriptsize AFM}}}}}}=(0.5,0.5,1)$ performed at the indicated temperatures are offset vertically. (c,d) Reduced temperature dependence of spinwave model parameters $\alpha$ (open symbols) and $\Delta$ (filled symbols). (e,f) Reduced temperature dependence of the ordered magnetic moment, $\mu$, normalized to its low-temperature value. Light gray symbols in panels (a) and (b) represent measured intensity which was excluded from fitting due to concerns with the validity of background estimates at those points. \[fig:tempdep\] ](Ba122CoSeries_temperature0)
Figure \[fig:tempdep\] shows the ${\ensuremath{\mathbf{Q}{\ensuremath{_{\mbox{\scriptsize AFM}}}}}}=(0.5,0.5,1)$ spectrum of at several different temperatures for lightly doped and nonsuperconducting $x=0.015$ and $0.033$. The spectra are dominated by a large spin gap at $\sim$ 10 meV for both compositions. These data can be fit to a damped spin wave form for ${\ensuremath{\chi''{\ensuremath{_{\mbox{\scriptsize }}}}\!\!\left({\ensuremath{\mathbf{Q}{\ensuremath{_{\mbox{\scriptsize }}}}}},E\right)}}$, $${\ensuremath{\chi''{\ensuremath{_{\mbox{\scriptsize s}}}}\!\!\left({\ensuremath{\mathbf{Q}{\ensuremath{_{\mbox{\scriptsize }}}}}},E\right)}} \propto \frac{E}{{\left(\Delta^2+c^2q^2-E^2\right)^2+\alpha^2E^2}},
\label{eqn1}$$ where $\Delta$ is a spin gap, $c$ is the spin wave velocity, $\alpha$ is a damping rate, and ${\ensuremath{\mathbf{q}{\ensuremath{_{\mbox{\scriptsize }}}}}}\equiv{\ensuremath{\mathbf{Q}{\ensuremath{_{\mbox{\scriptsize }}}}}}-{\ensuremath{\mathbf{Q}{\ensuremath{_{\mbox{\scriptsize AFM}}}}}}$ is the reduced momentum transfer. The full anisotropic form for the damped spin wave susceptibility is given elsewhere. For $x=0.015$ at $11$ K, $\alpha=3.6(4)$ meV is small in comparison to other energy scales and, in principle, can arise from a combination of different damping processes (such as Landau damping for energy scales larger than the SDW gap, or magnon-magnon interactions). The fit to the $x=0.015$ $11$ K data shows a large spin gap $\Delta=9.73(14)$ meV characteristic of the parent AFM ordered state.
The solid lines in Fig. \[fig:tempdep\] (a) and (b) represent independent fits to the damped spin wave model where the gap and damping rate are allowed to vary freely. The magnitude of the spin gap is determined to be nearly constant with temperature up to our closest approach of $T/T{\ensuremath{_{\mbox{\scriptsize N}}}}=0.95$ where the ordered magnetic moment $\mu(T)/\mu(11\text{ K})\approx0.5$. Similar to the results described for NaFeAs [@Park12], [BaFe$_2$As$_2$]{} [@Park12], and LaFeAsO [@Ramazanoglu13], we find that the spin gap energy, $\Delta$, is roughly 9 meV in the ordered state, regardless of size of the ordered moment or the concentration, $x$, and the dynamics become overdamped as $T{\ensuremath{_{\mbox{\scriptsize N}}}}$ is approached.
![ Background subtracted INS intensity of corrected for the Bose thermal population factor and the Fe$^{2+}$ single-ion magnetic form factor plus best fit lines to the diffusive (light green lines) and the damped spin-wave (black lines) models. (a-e) Constant-[$\mathbf{Q}{\ensuremath{_{\mbox{\scriptsize }}}}$]{} energy scans at ${\ensuremath{\mathbf{Q}{\ensuremath{_{\mbox{\scriptsize AFM}}}}}}=(0.5,0.5,1)$ for five compositions. (f-h) Constant-$E$ scans in the $[h,h,0]$-direction across ${\ensuremath{\mathbf{Q}{\ensuremath{_{\mbox{\scriptsize AFM}}}}}}=(0.5,\,0.5,\,1)$ at $E=7$ meV. (i-j) Constant-$E$ scans in the $[h,h,0]$-direction across ${\ensuremath{\mathbf{Q}{\ensuremath{_{\mbox{\scriptsize AFM}}}}}}=(0.5,\,0.5,\,3)$ at $E=10$ meV. (k-m) Constant-$E$ scans in the $[0,0,l]$-direction, perpendicular to the Fe layer, across ${\ensuremath{\mathbf{Q}{\ensuremath{_{\mbox{\scriptsize AFM}}}}}}=(0.5,0.5,1)$ at $E=7$ meV. (n-o) Constant-$E$ scans in the $[0,0,l]$-direction across ${\ensuremath{\mathbf{Q}{\ensuremath{_{\mbox{\scriptsize AFM}}}}}}=(0.5,\,0.5,\,1)$ at $E=10$ meV. Light gray symbols represent measured intensity which was excluded from fitting due to concerns with the validity of background estimates at those points. \[fig:scans\] ](Ba122CoSeries_overview1)
Figure \[fig:scans\] shows a series of representative low-energy INS scans taken in the AFM ordered and normal state ($T{\ensuremath{_{\mbox{\scriptsize c}}}}<T<T{\ensuremath{_{\mbox{\scriptsize N}}}}$) for each composition. Upon increased Co substitution, the spin gap appears to gradually close \[[Figs. \[fig:scans\]]{}(a)-(e)\] and is completely absent at $x=0.055$. One can also observe a gradual reciprocal space broadening of the longitudinal cut \[[Figs. \[fig:scans\]]{}(f)-(j)\] with increasing Co composition. Finally, the modulations along $(1/2,1/2,l)$ \[[Figs. \[fig:scans\]]{}(k)-(o)\] are reduced, signaling a gradual evolution to two-dimensional spin dynamics. Within the damped spin wave model of [Eq. (\[eqn1\])]{}, the data at all compositions have been successfully fit by assuming that: in accordance with our temperature-dependent results, the spin gap remains constant; the damping increases dramatically with $x$; and both the in-plane and inter-plane spin wave velocities are reduced with $x$. However, it is clear from high-energy INS investigations that the in-plane spin velocities are independent of composition (see the discussion in Ref. ) and constraining the in-plane velocity to this value leads to poorer and poorer agreement of the low-energy data with the damped spin wave model (as shown by the black lines in [Fig. \[fig:scans\]]{}).
One major assumption of our data analysis using the spin wave model is that the spin gap is independent of composition. If the spin gap is due to single-ion anisotropy, then its magnitude should be proportional to some power of $\mu$ [@Fishman98]. Data fitting in which the spin gap was allowed to freely vary resulted in an *increase* of the gap with composition, and fits in which the spin gap was constrained to be proportional to $\mu$ gave worse results.
The increased reciprocal space broadening suggests that another length scale must be introduced for low-energy magnetic fluctuations, such as a spin-spin correlation length. Considering also the gapless form of the magnetic excitations, the data at higher compositions resemble the diffusive response that has been used to describe the optimal and overdoped samples.[@Li10; @Matan10] This diffusive response has the form $$\begin{aligned}
{\ensuremath{\chi''{\ensuremath{_{\mbox{\scriptsize d}}}}\!\!\left({\ensuremath{\mathbf{Q}{\ensuremath{_{\mbox{\scriptsize }}}}}},E\right)}} &\propto \frac{E}{a^4\left(q^2+\xi^{-2}\right)^2+\gamma^2E^2} \\
&=\frac{E}{\Gamma^2\left(1+q^2\xi^{2}\right)^2+E^2 },
\label{eqn2}\end{aligned}$$ where $\xi$ is the spin-spin correlation length, $a$ is the tetragonal lattice constant, and $\gamma$ is the Landau damping coefficient. One can also define $\Gamma\equiv a^{2}/\gamma\xi^{2}$ as the spin relaxation rate. Fits to the diffusive form for ${\ensuremath{\chi''{\ensuremath{_{\mbox{\scriptsize }}}}\!\!\left({\ensuremath{\mathbf{Q}{\ensuremath{_{\mbox{\scriptsize }}}}}},E\right)}}$ are shown as light green lines in [Fig. \[fig:scans\]]{}. While the diffusive form does a poor job at the lowest compositions where the spin gap is sharp, it works exceptionally well at the higher compositions where the spectrum appears gapless and the increased reciprocal space broadening for longitudinal scans shown in [Fig. \[fig:scans\]]{}(f)-(j) is captured by a smaller correlation length.
![ (a) Phase diagram of showing regions of AFM, SC, and their coexistence; colored symbols show the locations in phase-space of the measurements performed in this study. (b-f) Select model parameters as a function of composition for the diffusive (green diamonds) and spin wave (black circles) models. All data points shown in (b-h) were determined at the lowest temperature indicated in (a), and the lightly shaded background indicates compositions which exhibit SC at low temperature. Spin-wave model: (c) spin gap $\Delta$ (filled), damping $\alpha$ (open), and $\Gamma{\ensuremath{_{\mbox{\scriptsize s}}}}=\Delta^2/\alpha$ (diamonds); (e) Inter-plane spin wave velocity. Diffusive model: (b) Landau damping $\gamma$ and the corresponding theoretical prediction; (d) Spin relaxation characteristic energy $\Gamma$ (filled) and the effective magnetic energy $E{\ensuremath{_{\mbox{\scriptsize SF}}}}=1/\gamma$ (open); (f) correlation length $\xi$. In (c-d) an estimate for the SDW gap – derived from $\mu(x)$ from Ref. and $\Delta{\ensuremath{_{\mbox{\scriptsize SDW}}}}(x=0)$ from Ref. – is also shown (tan line). (g) Residual $\chi^{2}$ for each fit model. (h) Spectral weight of the $(0.5,0.5,1)$ excitation. The spectral weight is the ${\ensuremath{\mathbf{Q}{\ensuremath{_{\mbox{\scriptsize }}}}}}$-averaged energy integration of the trace of the imaginary component of the magnetic susceptibility tensor. The averaging-range in ${\ensuremath{\mathbf{Q}{\ensuremath{_{\mbox{\scriptsize }}}}}}$ here is $0\!<\!H\!<\!1$, $0\!<\!K\!<\!1$, $0\!<\!L\!<\!2$; the energy integration is over the range $0\!<\!E\!<\!35$ meV. All errorbars represent the combined errors for all function parameters. The solid green and black lines in (c-h) are guides to the eye. \[fig:params\] ](Ba122CoSeries_overview2)
Figure \[fig:params\] shows the locations of our measurements in a phase-space diagram, the fitting parameters for both the damped spin wave and diffusive models in [Eqs. (\[eqn1\])]{} and [(\[eqn2\])]{}, a $\chi^{2}$ measure of the goodness-of-fit for the constant-[$\mathbf{Q}{\ensuremath{_{\mbox{\scriptsize }}}}$]{} and constant-$E$ scans presented in Fig. \[fig:scans\] for these two models, and the composition-dependence of the low-energy spectral weight. For $x=0.015$, the damped spin wave model is the best and $\alpha/\Delta=0.37(8)$ is consistent with underdamped dynamics. For intermediate composition, $x=0.033$, both models are of comparable quality. As shown in [Fig. \[fig:params\]]{}(c), within the damped spin wave model $\alpha/\Delta>1$ and the dynamics have become overdamped causing the spin gap to disappear. In the limit where $\alpha/\Delta\gg1$, the overdamped spin wave model also takes on a relaxational form with $\Gamma{\ensuremath{_{\mbox{\scriptsize s}}}}=\Delta^2/\alpha$; as shown in [Figs. \[fig:params\]]{}(c) and (d) $\Gamma{\ensuremath{_{\mbox{\scriptsize s}}}}$, $\Gamma$, and the effective magnetic energy, $E{\ensuremath{_{\mbox{\scriptsize SF}}}}=1/\gamma$, decrease as the critical concentration for which the AFM order is fully suppressed is approached, as indicated by vanishing $\Delta{\ensuremath{_{\mbox{\scriptsize SDW}}}}$. As seen in [Figs. \[fig:scans\]]{}(k-o) the excitation becomes increasingly two-dimensional with $x$ as captured by the damped spin-wave model parameter $c{\ensuremath{_{\mbox{\scriptsize z}}}}$, [Fig. \[fig:params\]]{}(e). For $x=0.040$, $0.047$, and $0.055$, the diffusive model becomes the better fit, as the smaller correlation length \[[Fig. \[fig:params\]]{}(f)\] is able to capture the reciprocal space broadening of the in-plane spin fluctuations. A comparison of the residual for each fit model in [Fig. \[fig:params\]]{}(g) clearly shows the crossover from spin-wave- to diffusive-like excitations. In [Fig. \[fig:params\]]{}(h), a sharp increase in the low-energy spectral weight ($<35$ meV) coincides with the appearance of SC.
From [Fig. \[fig:params\]]{}, regardless of the model used to fit our data, it is clear that upon approaching the optimally-doped composition, damping becomes stronger, the spin fluctuations acquire a more two-dimensional character, and the energy scale associated with these fluctuations ($\Gamma$ or $\Gamma{\ensuremath{_{\mbox{\scriptsize s}}}}$) become smaller. These features, as well as the crossover from spin-wave to diffusive excitations, are consistent with a suppression of the SDW gap $\Delta_{\mathrm{SDW}}$ upon doping. In [Figs. \[fig:params\]]{}(c-d), we show the experimentally determined suppression of $\Delta_{\mathrm{SDW}}$ obtained by combining the doping evolution of the zero-temperature ordered magnetic moment, $\mu(x)$, from Ref. with the optical conductivity derived value of $\Delta{\ensuremath{_{\mbox{\scriptsize SDW}}}}(x=0)$ from Ref. , using the fact that $\Delta{\ensuremath{_{\mbox{\scriptsize SDW}}}}\propto \mu$ [@FernandesPRB10; @Fernandes_Schmalian; @Vorontsov10; @Eremin10].
Based on this information, we can conclude that the presence of sub-gap spectral weight which appears with either an increase in temperature or Co composition is driven entirely by damping. For the temperature-driven transition, we find an increase of damping close to $T{\ensuremath{_{\mbox{\scriptsize N}}}}$. Given the similarities between the spin fluctuations above and below $T_{N}$ near optimal doping and the smallness of the spin-wave gap $\Delta_{\mathrm{SDW}}$ in this regime \[see [Fig. \[fig:params\]]{}(c)\], we compare the fitted damping rate $\gamma$ with the calculated Landau damping $\gamma_{\mathrm{calc}}$ due to the decay of spin excitations into particle-hole pairs near the Fermi level in [Fig. \[fig:params\]]{}(b). Using a simplified two-band model, which was previously shown to successfully capture the coexistence of SC and AFM [@Fernandes_Schmalian; @Vorontsov10; @Eremin10], the Landau damping is given by $\gamma_{\mathrm{calc}}^{-1}\propto\left|\mathbf{v}_{e}\times\mathbf{v}_{h}\right|$ [@Sachdev_book], where $\mathbf{v}_{e}$ and $\mathbf{v}_{h}$ are respectively the Fermi velocities of the electron and hole pockets at the hot spots (i.e. points connected by the AFM ordering vector ${\ensuremath{\mathbf{Q}{\ensuremath{_{\mbox{\scriptsize AFM}}}}}}$). Upon Co substitution, electrons are introduced into the system, making the hole pocket shrink and the electron pocket expand. As revealed by ARPES [@Liu10], this moves the hot spots, making their Fermi velocities become nearly parallel around optimal doping. As a result, $\gamma_{\mathrm{calc}}^{-1}\rightarrow0$, as seen experimentally. Note that $\gamma_{\mathrm{calc}}$ describes well the data only in compositions near optimal doping, indicating that in slightly-doped compositions the damping comes from another mechanism, such as magnon-magnon interactions.
In summary, we have shown that in the low-energy spin dynamics, which are most strongly tied to excitations in close proximity to the Fermi surface, display a crossover from gapped spin waves to a regime of strong damping and short correlation length, even though weak AFM order persists. The appearance of strong Landau damping near $x=0.03$ – $0.04$ coincides with the appearance of superconductivity, suggesting that the corresponding increase of low energy spectral weight below the spin gap is a key ingredient for superconductivity to develop. In theories where pairing is mediated by spin fluctuations, their energy scale ($E{\ensuremath{_{\mbox{\scriptsize SF}}}}$) is usually positively correlated to $T{\ensuremath{_{\mbox{\scriptsize c}}}}$ [@Pines92; @Chubukov03; @Scalapino12]. We instead observe that $E{\ensuremath{_{\mbox{\scriptsize SF}}}}$ decreases with increasing $x$ (and $T{\ensuremath{_{\mbox{\scriptsize c}}}}$), see [Fig. \[fig:params\]]{}(d). To avoid the apparent contradiction one must also consider that AFM and SC order compete [@Pratt09Christianson09] thereby effectively decreasing $T{\ensuremath{_{\mbox{\scriptsize c}}}}$ for underdoped samples (and eliminating SC for the parent compound). Indeed, when the long-range AFM order is suppresed by pressure, $T{\ensuremath{_{\mbox{\scriptsize c}}}}$ for lower $x$ samples is enhanced beyond that for optimal doping[@Colombier10Ni10]. Therefore, $T{\ensuremath{_{\mbox{\scriptsize c}}}}$ and $E{\ensuremath{_{\mbox{\scriptsize SF}}}}$ are infact positively correlated. Iron pnictide compositions on either side of the SC region — such as with $x=0.015$ or $x=0.14$ [@Sato11], or [@Wang13] — lack overdamped spin fluctuations, in contrast to the underdoped SC compositions presented here; this provides further evidence that overdamped spin fluctuations are a necessary component in the paring mechanism for superconductivity in the iron pnictides.
The authors would like to thank D. C. Johnston and A. Kaminski for useful discussions. Work at the Ames Laboratory was supported by the Department of Energy – Basic Energy Sciences, Division of Materials Sciences and Engineering under Contract No. DE-AC02-07CH11358. Part of the research conducted at ORNL’s High Flux Isotope Reactor was sponsored by the Scientific User Facilities Division, Office of Basic Energy Sciences, U.S. Department of Energy. Some work at Oak Ridge (BCS, ASS) was supported by the Department of Energy, Basic Energy Sciences, Materials Sciences and Engineering Division.
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'The Legendre transformation method, applied in 1987 to deal with purely metric gravitational Lagrangians with nonlinear dependence on the Ricci tensor, is extended to metric–affine models and is shown to provide a concise and insightful comparison of the dynamical content of the two variational frameworks.'
author:
- |
Guido Magnano\
Università di Torino, Dipartimento di Matematica “G. Peano”\
via Carlo Alberto 10, I-10123 Torino, Italy\
[email protected]
date: December 2015
title: Nonlinear Gravitational Lagrangians revisited
---
Thirty years after
==================
In june 1987, the article *Nonlinear Gravitational Lagrangians*[@NGL] was published in *General Relativity and Gravitation*. The core of the article was the application of a (generalized) Legendre transformation to models of gravity in which the Lagrangian depended in a nonlinear way on the Ricci tensor of the space-time metric. The method turned out to work for both Lagrangians depending only on the scalar curvature and Lagrangians depending on the full Ricci tensor, although the appropriate Legendre transformation is different in the two cases. A parallel, independent work by A. Jakubiec and J. Kijowski discussing the application of the Legendre transformation to gravitational theories [@JK] was published in the same volume of the GRG journal. Both papers followed an earlier proposal by Kijowski [@K79]; a mathematical framework for the Legendre transformation in higher–order field theories, based on Poincaré-Cartan forms rather that on symplectic geometry, was introduced in [@MFF90].
The common reception of these results was that any nonlinear gravitational Lagrangian is equivalent to the well-known Einstein-Hilbert Lagrangian (with additional, minimally coupled terms for some auxiliary field variable) *up to a redefinition of the metric.* Whenever the Lagrangian depends only on the scalar curvature (which is universally referred to as the $f(R)$ case), the redefinition of the metric consists in a conformal rescaling: this was in line with previous observations made by different authors (for a comprehensive list of early references, see [@MS1]), which led most researchers, in the subsequent years, to regard the Legendre transformation and the metric redefinition as a single operation that can be performed to exhibit a dynamical equivalence between “nonlinear gravity” and General Relativity. In more recent years, this has been repeatedly rephrased as the existence of two “frames”: the *Jordan frame*, defined by the original metric obeying a fourth–order field equation, and the *Einstein frame* defined by the new metric and leading to the familiar Einstein equation.
Many authors have then addressed the question whether the “physical frame” should be assumed to be the Jordan or the Einstein one. Actually, the question had previously arisen for the Jordan-Brans-Dicke scalar-tensor theory, whereby a conformal rescaling was known to recast the Lagrangian in the usual Einstein-Hilbert form: this is the origin of the widespread “Jordan/Einstein frame” terminology (although the reference to a “frame” is potentially misleading: the two pictures, in fact, are not related by a change of coordinates or of reference frame). The analogous debate for the $f(R)$ models was started by Carl Brans in a letter [@Brans] in response to [@NGL]: Brans pointed out that, after redefining the metric, the gravitational mass of bodies becomes dependent on the point in space and time, and stress–energy conservation laws are broken. In reply to [@Brans], it was observed that assuming minimal coupling between matter fields and the original metric, in the construction of a gravity model, entails that one has already postulated that the original metric is the “physical” one: in that case, indeed, one cannot maintain that the new metric is physical as well, and it is not surprising that the interaction of the new metric with matter becomes somehow “unphysical”. But the argument can be reversed (stress–energy conservation, in particular, holds in Einstein frame if the covariant derivatives are taken with respect to the new metric) and therefore it cannot be invoked to claim that either metric is unphysical [@rBrans].
In other terms, the question of which tensor field has to be identified with the physical spacetime metric should be addressed *before* devising the additional terms which describe the gravitational interaction of matter fields. The equivalence of $f(R)$ models with General Relativity plus a scalar field, holding in vacuum, is indeed broken if matter fields are coupled to the original metric tensor as if the latter were the physical one.
A criterion to decide which metric is the physical one, based on structural properties on the model (positivity of energy), has been proposed in [@MS1], and we shall recall it below; but this has not settled the dispute, seemingly, even in cases where the criterion would be applicable. In fact, the attitude of the various authors depends on their purposes for introducing a $f(R)$ gravity model. Some authors regarded such models as a nice way to make a “quintessential” scalar field appear not as an additional field, but rather as an additional degree of freedom which is already contained in the original metric and reveals itself as a separate field upon switching to the Einstein frame. In that case, the original (Jordan–frame) metric is regarded as a sort of *unifying* field variable, while the role of physical metric is attributed to the Einstein–frame metric. In the last decade, the prevailing viewpoint seems to be the opposite one: many authors explore the cosmological effects of adding ordinary cosmic matter terms to $f(R)$ Lagrangians, without considering any metric redefinition. Meanwhile, most authors also turned to what appears to be a deep change in the basic assumptions of the model: namely, the affine connection $\Gamma$ whose curvature enters the Lagrangian is no longer assumed to be the Levi–Civita connection of the metric $g$, but instead becomes an independent field. This is the metric–affine, or Palatini, version of the action principle [@Cap].
Even in the Palatini setup it is found that, upon suitable assumptions, a metric redefinition (which in the vacuum reduces to the identity) leads to a dynamically equivalent picture where the dynamical connection *is* the Levi–Civita connection of the new metric, which in turn obeys Einstein equations; however, it is exactly the possibility of coupling cosmic matter to a metric describing a *different* spacetime geometry (with respect to the connection) which seems appealing, in order to fit the observational data. The fact that $\Gamma$ coincides with the Levi–Civita connection for a conformally-rescaled metric (as a consequence of the field equations), on the other hand, ensures that light propagation under the original metric $g$ is physically compatible (in the sense of Ehlers–Pirani–Schild [@EPS]) with the free fall of massive bodies described by the geodesics of $\Gamma$.
The Palatini versions of $f(R)$ models are now often referred to as “extended gravity theories” [@Cap2]. Although a detailed comparison between the equations of Palatini f(R) models, purely metric f(R) models and scalar-tensor models has been presented by various authors, the involvement of field redefinitions tends to obscure the exact extent of the differences between the dynamical contents of these models.
Palatini $f(R)$ models were not considered in [@NGL], and one may suspect that the approach of that article is intrinsically confined to the purely metric setup. On the contrary, in the sequel we show that the (distinct) ideas of Legendre transformation and of metric redefinition, both introduced in [@NGL], provide a nice way to
(i)
: obtain a systematic comparison between Palatini and purely metric models; such a comparison, through a suitable scalar-tensor reformulation of both models, has been already discussed in the previous literature (see e.g. [@Cap]), but our approach reveals in a more direct way that the difference between the purely metric variation and the Palatini variation of a given $f(R)$ Lagrangian merely amounts to swapping (with a relevant change of sign) the dynamical term of a scalar field from the “Einstein frame” to the “Jordan frame” Lagrangian;
(ii)
: generalize this result to Lagrangians depending on the (symmetric part) of the Ricci tensor; the Palatini versions of such models have been studied by some authors ([@Bor],[@LBM]), but an insightful comparison with their purely metric counterparts is still lacking.
The following discussion is thus aimed at exploiting the Legendre transformation to put the Palatini and the purely metric gravitational models on the same footing, so that the actual differences between them can be clearly seen already at the level of the action principle.
Purely metric theories
======================
Let $g_{\mu\nu}$ be a metric (of the appropriate signature) on the spacetime manifold $M$ (we shall assume that $\dim(M)=4$, but this is not crucial for most of the subsequent discussion). We shall denote by $\left\lbrace{}^\lambda_{\mu\nu}\right\rbrace_g$ the Christoffel symbols, i.e. the components of the Levi–Civita connection, for the metric $g_{\mu\nu}$; the scalar density $\sqrt{|g|}$ will be the component of the associate volume form; $R^{\alpha}_{\mu\beta\nu} $, $R_{\mu\nu}\equiv R^{\alpha}_{\mu\alpha\nu}$ and $R\equiv R_{\mu\nu}g^{\mu\nu}$ will denote, respectively, the Riemann tensor, the Ricci tensor and the curvature scalar of the metric.
We recall that a covariant Lagrangian cannot depend only on first derivatives of the metric, unless a separate affine connection is used (either as a fixed background or as an independent field variable): thus, in purely metric theories the gravitational Lagrangian should depend at least on second derivatives of the metric, and only through the components of the Riemann tensor. To the vacuum Lagrangian one adds interaction terms for the other physical fields (that we collectively denote by $\Phi$), possibly depending on first derivatives of the metric. In the sequel, we shall use the jet notation $j^k x$ to indicate that a function depends on some field $x$ and on its spacetime derivatives up to the $k$-th order.
Although Lagrangians depending on the full Riemann tensor have sometimes been considered in the literature, here we shall focus on models depending only on the Ricci tensor, as was done in [@NGL]: $$L=f(R_{\mu\nu},g_{\mu\nu})\sqrt{|g|}+L_{\mathrm{mat}}(j^1 \Phi,j^1 g).\label{HOG}$$
A particular case is $$L=f(R)\sqrt{|g|}+L_{\mathrm{mat}}(j^1 \Phi,j^1 g)\label{NL},$$ whereby the Lagrangian depends solely on the curvature scalar. The Einstein–Hilbert Lagrangian belongs to this class, with $f(R)\equiv R$, and is the only one which generates second–order Euler-Lagrange equations for the metric: for any nonlinear function $f(R)$ (or, more generally, $f(R_{\mu\nu},g_{\mu\nu})$) the equations for the gravitational metric will be of order four. The function $f$ is generally assumed to be differentiable, but a class of models have been proposed in recent years [@CCT] where $f$ has a pole at $R=0$.
Variation of (\[HOG\]) with respect to the metric produces the equation $$J^{\alpha\beta}+\frac{1}{2}\left(P^{\beta\nu}{}_{;\mu\nu}g^{\alpha\mu}+P^{\alpha\nu}{}_{;\mu\nu}g^{\beta\mu}-P^{\mu\nu}{}_{;\mu\nu}g^{\alpha\beta}-P^{\alpha\beta}{}_{;\mu\nu}g^{\mu\nu}\right)=\kappa T^{\alpha\beta}\sqrt{|g|}\label{HOGeq}$$ where $$\begin{aligned}
P^{\alpha\beta} &\equiv& \frac{\partial f}{\partial R_{\alpha\beta}}\sqrt{|g|}, \\
J^{\alpha\beta} &\equiv& \left(\frac{\partial f}{\partial g_{\alpha\beta}}+\frac{1}{2}g^{\alpha\beta}f\right)\sqrt{|g|},\end{aligned}$$ $T^{\alpha\beta}\sqrt{|g|}$ is (up to a constant factor) the variational derivative of the scalar density $L_{\mathrm{mat}}$ with respect to the metric and the semicolon denotes covariant differentiation under the connection $\left\lbrace{}^\lambda_{\mu\nu}\right\rbrace_g$.
For the Lagrangian (\[NL\]), equation (\[HOGeq\]) reduces to $$f'(R)R^{\alpha\beta}-\frac{1}{2}f(R)g^{\alpha\beta}+f'(R)_{;\mu\nu}\left(g^{\mu\nu}g^{\alpha\beta}-g^{\mu\alpha}g^{\nu\beta}\right)=\kappa T^{\alpha\beta},\label{NLeq}$$ where $f'(R)\equiv\frac{df}{dR}$.
The idea of performing a Legendre transformation of $L$ with respect to the Ricci tensor is due to J. Kijowski [@K79]: he proposed that, in a gravity theory, the fields which jointly describe gravity (i.e. the metric, defining infinitesimal spacetime separation, and the affine connection, defining the free–fall worldlines and the inertial reference frames) should be regarded as *mutually conjugate variables* in the sense of symplectic geometry. More precisely, the metric is conjugate to the connection, whereby the components of the (symmetrized) Ricci tensor of the connection play the role of the “velocity components” in the Legendre transformation.
Originally, this idea was exploited to introduce a metric tensor in purely affine models, whereby the Lagrangian contains only an affine connection $\Gamma$ (e.g. in the Einstein-Eddington model, see [@FK] and references therein): a tensor density $\pi^{\mu\nu}$ is then introduced through the Legendre map $$\pi^{\mu\nu}=\frac{\partial L}{\partial R_{(\mu\nu)}}$$ and subsequently converted into a symmetric tensorfield through multiplying $\pi^{\mu\nu}$ by the square root of (the absolute value of) its determinant. For relevant Lagrangians the resulting tensorfield is generically nondegererate, and the appropriate signature can be imposed, for instance, on the Cauchy data.
In metric-affine models and purely metric models, a metric $g_{\mu\nu}$ is already present among the dynamical variables: not only there is no apparent reason to introduce a new field conjugate to the connection, but one should expect that such a new field would *not* coincide with $g_{\mu\nu}$ (the only exception being the Einstein-Hilbert Lagrangian, as is easy to check). In spite of that, for the Lagrangian (\[HOG\]) we do introduce the new tensorfield and call it $\tilde{g}^{\mu\nu}$: $$\tilde{g}^{\mu\nu}\sqrt{|\tilde{g}|}=\frac{\partial L}{\partial R_{\mu\nu}}=\frac{\partial f}{\partial R_{\mu\nu}}\sqrt{|g|}.
\label{LM1}$$ (the covariant components $\tilde{g}_{\mu\nu}$ are defined as the entries of the inverse matrix, $\tilde{g}_{\alpha\nu}\tilde{g}^{\nu\beta}\equiv\delta^{\beta}_{\alpha}$, *not* by lowering the indices with the original metric). In order to perform a Legendre transformation, however, the map (\[LM1\]) is useful only if it can be inverted, i.e. if a tensor function $r_{\mu\nu}(g_{\mu\nu},\tilde{g}_{\mu\nu})$ exists such that $$\sqrt{|g|}\left.\frac{\partial f}{\partial R_{\mu\nu}}\right|_{R_{\alpha\beta}=r_{\alpha\beta}}\equiv \tilde{g}^{\mu\nu}\sqrt{|\tilde{g}|}
\label{iLM1}$$ It is easy to see, for instance, that this *cannot* be done if $L$ has the form (\[NL\]), while it works if the Lagrangian depends, for instance, on the squared Ricci tensor $R_{\mu\nu}R^{\mu\nu}$. For the moment, let us assume that the Lagrangian is “Ricci–regular”, i.e. that the inverse Legendre map $r_{\mu\nu}$ exists. Then, we apply a suitable generalization of a procedure of analytical mechanics that is completely equivalent to the usual Legendre transformation (although seldom described in current textbooks). Suppose that $L(q^{\lambda},\dot{q}^{\lambda})$ is the Lagrangian for some holonomic mechanical system on a configuration space $Q$; let $p_{\lambda}=\frac{\partial L}{\partial q^{\lambda}}$ be the Legendre map which associates to each vector in $TQ$ its conjugate momentum in $T^*Q$ and let $u^{\mu}(q^{\lambda},p_{\lambda})$ be the inverse Legendre map, such that $$\left.\frac{\partial L}{\partial \dot{q}^{\lambda}}\right|_{\dot{q}^{\mu}=u^{\mu}(q^{\lambda},p_{\lambda})}\equiv p_{\lambda};\label{LM2}$$ one can now introduce the *Helmholtz Lagrangian*, which defines a holonomic system in $T^*Q$: $$L_H(q^{\lambda},p_{\lambda},\dot{q}^{\lambda})=p_{\mu}(\dot{q}^{\mu}-u^{\mu})+L(q^{\lambda},u^{\nu}(q^{\lambda},p_{\lambda}))$$ (technically speaking, $L_Hdt$ is the pull-back on a curve in $T^*Q\times\mathbb{R}$ of the Poincaré-Cartan one-form for the Lagrangian $L$). $L_H$ is a degenerate Lagrangian, because it does not depend on $\dot{p}_{\lambda}$ and depends linearly on $\dot{q}_{\lambda}$: the resulting Euler-Lagrange equations, instead of being second-order equations (in $T^*Q$), are first order and are easily found to reproduce the dynamics of the original system: $\dfrac{\delta L_H}{\delta p_{\lambda}}=0 \Leftrightarrow \dot{q}^{\lambda}=u^{\lambda}$ and $\dfrac{\delta L_H}{\delta q^{\lambda}}=0 \Leftrightarrow \dot{p}_{\lambda}=\frac{\partial L}{\partial q^{\lambda}}$. These equations, in fact, reduce either to the Lagrange equations for $L$, taking into account (\[LM2\]) and eliminating the conjugate momenta $p_{\mu}$, or to the Hamilton equations, upon introducing $H(q^{\lambda},p_{\lambda})=p_{\mu}u^{\mu}-L(q^{\lambda},u^{\nu}(q^{\lambda},p_{\lambda}))$ and observing that $u^{\lambda}\equiv \frac{\partial H}{\partial p_{\lambda}}$ and $\left.\frac{\partial L}{\partial q^{\lambda}}\right|_{\dot{q}^{\mu}=u^{\mu}}\equiv\frac{\partial H}{\partial q^{\lambda}}$.
Going back to the gravitational Lagrangian (\[HOG\]), following the same steps one finds that the Helmholtz Lagrangian $$L_{H}=\tilde{g}^{\alpha\beta}(R_{\alpha\beta}-r_{\alpha\beta})\sqrt{|\tilde{g}|}+f(r_{\mu\nu},g_{\mu\nu})\sqrt{|g|}+L_{\mathrm{mat}}(j^1 \Phi,j^1 g) \label{LH1}$$ produces Euler-Lagrange equations which are exactly equivalent to (\[HOGeq\]). Notice that, so far, we have performed a Legendre transformation by introducing an independent “conjugate field”, *not* redefined the metric. The new tensor $\tilde{g}_{\mu\nu}$ is symmetric by definition and is generically nondegenerate, so one can rightfully regard the Lagrangian (\[LH1\]) as describing a particular *bimetric* theory. The two metrics play an unequal role, because $L_H$ depends linearly on the Ricci tensor of $g_{\mu\nu}$ and does not depend at all on derivatives of $\tilde{g}_{\mu\nu}$. However, subtracting a full divergence (which can be done without affecting the Euler-Lagrange equations), we can almost exchange the two roles. It is sufficient to use a well-known identity for the difference between the Ricci tensors of two symmetric affine connections on the same manifold: $$R_{\beta\nu}-\tilde{R}_{\beta\nu}\equiv\tilde{\nabla}_{\alpha}Q^{\alpha}{}_{\beta\nu}-
\tilde{\nabla}_{\nu}Q^{\alpha}{}_{\alpha\beta}+Q^{\alpha}{}_{\alpha\sigma}Q^{\sigma}{}_{\beta\nu}-
Q^{\alpha}{}_{\beta\sigma}Q^{\sigma}{}_{\alpha\nu}\label{Riccidiff}$$ where $\tilde{\nabla}$ denotes covariant differentiation with respect to the second connection, and the tensor $Q^{\alpha}{}_{\beta\nu}$ is the difference between the two affine connections; in our case, these are the Levi-Civita connections of the two metrics, i.e. $$Q^{\alpha}{}_{\beta\nu}=\left\lbrace{}^\alpha_{\beta\nu}\right\rbrace_g-\left\lbrace{}^\alpha_{\beta\nu}\right\rbrace_{\tilde{g}}=\frac{1}{2}g^{\alpha\sigma}(\tilde{\nabla}_{\nu}g_{\sigma\beta}+\tilde{\nabla}_{\beta}g_{\sigma\nu}-\tilde{\nabla}_{\sigma}g_{\nu\beta})\label{Qmet}$$ Once multiplied by $\sqrt{|\tilde{g}|}$, the two terms with the covariant derivatives of $Q^{\alpha}{}_{\beta\nu}$ become a full divergence. Therefore, the Helmholtz Lagrangian (\[LH1\]) is dynamically equivalent to the following Lagrangian: $$\begin{aligned}
L_{E}=&\ \tilde{R}_{\mu\nu}\tilde{g}^{\mu\nu}\sqrt{|\tilde{g}|}+\tilde{g}^{\alpha\beta}\left(Q^{\rho}_{\sigma\alpha}Q^{\sigma}_{\rho\beta}-Q^{\rho}_{\alpha\beta}Q^{\sigma}_{\sigma\rho}-r_{\alpha\beta}\right)\sqrt{|\tilde{g}|}\ +\nonumber\\&+f(r_{\mu\nu},g_{\mu\nu})\sqrt{|g|}+L_{\mathrm{mat}}(j^1 \Phi,j^1 g)\end{aligned}$$ The dynamical term for $\tilde{g}_{\mu\nu}$ is now an Einstein–Hilbert term, so we immediately know that the variation of $L_{E}$ with respect to $\tilde{g}_{\mu\nu}$ produces an Einstein equation for this metric, with a complicated stress-energy tensor describing the interaction with $g_{\mu\nu}$; the full equations and their detailed analysis can be found in [@MS2]. However, it turns out that also the metric $g_{\mu\nu}$ obeys an Einsten equation: we know, in fact, that the variation of $L_H$ (\[LH1\]) automatically reproduces the inverse Legendre map, i.e. (in our case) $R_{\mu\nu}=r_{\mu\nu}(g_{\alpha\beta},\tilde{g}_{\alpha\beta})$, which can easily be recast in Einstein form (by taking its trace with $g^{\mu\nu}$). The Einstein equations for the two metrics, however, are not mutually independent: they are, in fact, two versions of the *same equation*, derived from the variation with respect to $\tilde{g}_{\mu\nu}$. One could equally pass from one form to the other using (\[Riccidiff\]), without relying on the existence of two equivalent Lagrangians.
In this picture, there is no direct interaction between $\tilde{g}_{\mu\nu}$ and the matterfields $\Phi$: this is a mere consequence of the fact that $\Phi$ has been coupled from the very beginning (\[HOG\]) to the metric $g_{\mu\nu}$.
In short, the introduction of the “conjugate” tensorfield $\tilde{g}_{\mu\nu}$ leads to an equivalent formulation of the original fourth–order equation (\[HOGeq\]): the latter splits into two second–order equations (in much the same way as the second–order Lagrange equation, in classical mechanics, splits into a pair of first–order Hamilton equations). One of these second–order equations can be put in Einstein form, either for the original metric $g_{\mu\nu}$ or for the new tensorfield $\tilde{g}_{\mu\nu}$; only in the second case (i.e. for the metric $\tilde{g}_{\mu\nu}$), the Einstein tensor is equated to the variational stress tensor. The latter, however, does not contain the true matterfields $\Phi$ which are instead coupled to $g_{\mu\nu}$. This situation may seem strange from the physical viewpoint, and is indeed *not* physically equivalent to ordinary General Relativity with the interaction term $L_{\mathrm{mat}}$, whichever metric is considered: yet, this is nothing but an exactly equivalent representation of the dynamics described by the original Lagrangian (\[HOG\]).
Instead of starting from (\[HOG\]), one could first consider the *vacuum* Lagrangian (deleting $L_{\mathrm{mat}}$ from (\[HOG\])), so to keep the freedom of deciding *after* the Legendre trasformation whether matter should be coupled (in the ordinary sense) either to $g_{\mu\nu}$ or to $\tilde{g}_{\mu\nu}$. This would amount to decide – on independent grounds – which metric should play the physical role assigned by General Relativity. Assuming that one of the two metrics defines the causal structure of spacetime, and the other, through its Levi-Civita connection, defines the free-fall geodesics, would be physically untenable: generically, the two metric are not conformal to each other, and the inertial structure of space-time would then become incompatible with the causal structure (in the sense of Ehlers–Pirani–Schild). Thus, either $g_{\mu\nu}$ and $\tilde{g}_{\mu\nu}$ should be regarded as providing both causal and inertial structure. The other tensorfield, then, could be assimilated to a gravitating massive field, and might be conveniently split into a scalar field (the trace $g_{\mu\nu}\tilde{g}^{\mu\nu}$) and a traceless symmetric tensor, corresponding to a spin-0 and a spin-2 field respectively (see [@MS2] for a complete discussion). The dynamical term for the spin-2 field, in both cases, has unphysical signature (still, it is the only known consistent model of a spin-2 field interacting with gravity).
Now, let us revert to the restricted case (\[NL\]), which enjoyed a much larger popularity in cosmology. As we have already remarked, from the viewpoint of the Legendre transformation this is *not* a particular case of the setup described in the previous part of this section. The Lagrangian (\[NL\]), in fact, is not “Ricci-regular”, meaning that the Legendre map (\[LM1\]) cannot be inverted to re-express the components of the Ricci tensor as functions of $g_{\mu\nu}$ and $\tilde{g}_{\mu\nu}$. This is simply because the r.h.s. of (\[LM1\]) does not contain now the full Ricci tensor, but merely its trace. According to the mathematical framework which is explained in [@MFF90], since the Lagrangian (\[NL\]) depends only on the curvature scalar $R$, then it is the (scalar) conjugate field to $R$ which should be introduced by a suitable Legendre map: $$p=\frac{1}{\sqrt{|g|}}\frac{\partial L}{\partial R}=f'(R).\label{p1}$$ Then, whenever $f''(R)\neq 0$ (we may say that the Lagrangian is “R-regular” in this case), one can invert (\[p1\]) and get a function $r(p)$ (inverse Legendre map) such that $$\left.f'(R)\right|_{R=r(p)}.\equiv p\label{ip1}$$ The Helmholtz Lagrangian for this case is then $$L_{H}=p(R-r)\sqrt{-g}+f(r)\sqrt{|g|}+L_{\mathrm{mat}}(j^1\Phi,j^1g).\label{LH}$$ Instead of a bimetric theory, the Legendre transformation now led us to a scalar-tensor theory. The scalar field $p$, however, has no separate dynamical term. As is well known, at this point one can perform a conformal rescaling of the metric (whenever $p>0$), so that the coupling of $p$ with the Ricci tensor is replaced by an ordinary quadratic first-order term. Setting $$\tilde{g}_{\mu\nu}=pg_{\mu\nu}\label{cr}$$ and subtracting a full divergence, the Helmholtz Lagrangian transforms into $$\begin{aligned}
L_{E}=&\ \tilde{R}\sqrt{|\tilde{g}|}+\frac{1}{p^{2}}\left(-\frac{3}{2}\tilde{g}^{\mu\nu}\partial_{\mu} p\, \partial_{\nu} p+
f(r)-p\cdot r(p)\right)\sqrt{|\tilde{g}|}\ +\nonumber\\ &+L_{\mathrm{mat}}(j^1\Phi,j^1g),\end{aligned}$$ which can be put in more familiar form by redefining also the scalar field by $p=e^{\sqrt{\frac{3}{2}}\varphi}$: $$L_{E}=\left(\tilde{R}-\tilde{g}^{\mu\nu}\varphi_{,\mu}\varphi_{,\nu}-2V(\varphi)\right)\sqrt{|\tilde{g}|}+L_{\mathrm{mat}}(j^1\Phi,j^1g).$$ The last term can, in principle, rewritten so that matter fields $\Phi$ appear to interact with $\varphi$ and $\tilde{g}_{\mu\nu}$. The equation for the gravitational field can be written as an Einstein equation for either the “Jordan” metric $g_{\mu\nu}$ or the rescaled “Einstein” metric $\tilde{g}_{\mu\nu}$. Only in the second case, however, the Einstein tensor is equated to a variational stress–energy tensor, which is covariantly conserved along the connection $\lbrace{}^\alpha_{\beta\nu}\rbrace_{\tilde{g}}$. The r.h.s. of the Einstein equation for $g_{\mu\nu}$ can be viewed as an “effective energy–momentum tensor”, and is covariantly conserved along the connection $\lbrace{}^\alpha_{\beta\nu}\rbrace_g$, but does not coincide with the variational derivative of the interaction terms of the Lagrangian. Furthermore, whenever $f''(R)>0$ at $R=0$, *in the Einstein frame* the stress–energy tensor for the scalar field fulfills the dominant energy condition [@MS1]: this provides a possible criterion (that we mentioned in Section 1) to consider the Einstein metric $\tilde{g}_{\mu\nu}$ to be the physical one. If this criterion is adopted, then one should ensure that the coupling with matter fields $\Phi$ becomes the physically appropriate one when expressed in terms of the Einstein metric.
It is striking to remark that the rescaled metric $\tilde{g}_{\mu\nu}$, even in this case, meets the Kijowski prescription: since for $f(R)$ one has , we see that (\[LM1\]) holds true. The idea of defining a metric as the conjugate momentum to the Ricci tensor is somehow independent of which Legendre map is actually invertible (and should therefore be used to perform a Legendre transformation): this fact is at the core of the “universality of the Einstein equation" repeatedly advocated by J. Kijowski himself and by many other authors after him.
On the other hand, it is worthwhile to stress again the differences between the dynamics generated by Ricci-regular Lagrangians (\[HOG\]) and by R-regular Lagrangians (\[NL\]). In the Ricci-regular case, the model is equivalent to a bimetric theory, while in the R-regular case the gravitational degrees of freedom are represented by a metric and a scalar field. In the first case, the two metrics coexist as independent dynamical variables in the second-order formulation of the model. In the $f(R)$ case, instead, the Jordan metric and the Einstein metric are conformally related, and only either of them can appear in the field equations (jointly with the scalar field); the Einstein metric does not pop up from the Legendre transformation, but from the subsequent (and independent) conformal rescaling.
Metric–affine (Palatini) theories
=================================
In the previous section we have basically rephrased the results presented in [@NGL] (taking into account the further insight given by the more detailed analysis in [@MS1] and [@MS2]). Let us now consider the metric–affine version of these models, adopting the same approach. Here, the affine connection $\Gamma^{\lambda}_{\mu\nu}$ is not assumed to be metric, and becomes and independent dynamical field.
In a lecture given in Turin a few years ago, Mauro Francaviglia quoted a celebrated passage that can be found in Galileo Galilei’s *Saggiatore*:
*“Philosophy is written in this grand book, which stands continually open before our eyes (I say the Universe), but cannot be understood without first learning to comprehend the language and know the characters in which it is written. It is written in mathematical language, and its characters are triangles, circles and other geometric figures, without which it is impossible for humans to understand a word; without these, one is wandering in a dark labyrinth.”*
Mauro remarked that Galileo’s reference to “triangles”, geometric figures whose definition implies the notion of *straight line*, i.e. geodesic, and to “circles” – figures which encapsulate the *metric* structure of space – may sound as a striking anticipation of the modern view of the structure of the Universe. He confronted this idea with the Palatini framework (which is actually due to Einstein), where the geodesic and the causal structures are kept independent in principle, and become related only by the dynamical equations. We had no opportunity of learning from Mauro whether this fascinating and daring parallel to Galileo’s text was due to himself or to another author, but it is definitely representative of Mauro’s penetrating view on research in mathematical physics.
We shall denote by $\mathcal{R}_{\mu\nu}$ the Ricci tensor of the connection $\Gamma^{\lambda}_{\mu\nu}$, and by $\mathcal{R}$ its trace with the metric, $\mathcal{R}=\mathcal{R}_{\alpha\beta}g^{\alpha\beta}$. We shall restrict to what is commonly considered the “Palatini setup” (in contrast to the most general metric–affine setup), by the following three relevant assumptions:
(A)
: $\Gamma^{\lambda}_{\mu\nu}$ is symmetric: the torsion of the connection (i.e. its antisymmetric part, which does not enter the geodesic equation) would have a non–gravitational physical interpretation, which is beyond the scope of our discussion;
(B)
: the Lagrangian depends only on the symmetric part of the Ricci tensor; the possible role of the antisymmetric part $\mathcal{R}_{[\mu\nu]}$ (which vanishes identically if the connection is metric) has been studied many years ago [@FK] and is known to produce the appearance of a spin-1, massive (Proca) field in the model, of doubtful physical interpretation;
(C)
: the matterfields $\Phi$ interact only with the metric: the matter Lagrangian, which in principle could be expected to be of the form $L_{\mathrm{mat}}(j^1 \Phi,g,\Gamma)$, is instead $L_{\mathrm{mat}}(j^1 \Phi,j^1 g)$. In other terms, possible covariant derivatives of the matterfields are defined using the Levi-Civita connection of $g$ instead of $\Gamma$. A sound physical motivation for this assumption has never been given, to our knowledge, but it turns out that a direct coupling of matter with $\Gamma$, as will become clear in the next paragraphs, would radically change the situation and make it much more distant from General Relativity: hence, some authors purposely reserve the name “Palatini theories” only for models where the matter is not coupled to the independent connection $\Gamma$ [@Cap].
With the above assumptions, the Palatini counterpart of Lagrangian (\[HOG\]) becomes $$L=f(\mathcal{R}_{(\mu\nu)},g_{\mu\nu})\sqrt{|g|}+L_{\mathrm{mat}}(j^1 \Phi,j^1 g),\label{PalRic}$$ and the counterpart of (\[NL\]) is $$L=f(\mathcal{R})\sqrt{|g|}+L_{\mathrm{mat}}(j^1 \Phi,j^1 g)\label{PalR}.$$ Contrary to the previous Section, for the reader’s convenience we shall first consider the $f(\mathcal{R})$ case (\[PalR\]), since it is much more popular in the current literature and the resulting picture will be somehow easier to interpret.
In the vacuum case ($L_{\mathrm{mat}}\equiv 0$), the Euler-Lagrange equations read $$\begin{aligned}
&\bar{\nabla}_{\lambda}\left(f'(\mathcal{R})g^{\mu\nu}\sqrt{|g|}\right) = 0 \label{met}\\
&f'(\mathcal{R})\mathcal{R}_{\mu\nu}-\frac{1}{2}f(\mathcal{R})g_{\mu\nu} = 0,\label{ME0}\end{aligned}$$ where $\bar{\nabla}$ denotes covariant derivation with the connection $\Gamma$. As is well known, taking the trace of the second equation (\[ME0\]) with $g^{\mu\nu}$ one gets an algebraic equation for $\mathcal{R}$, $$f'(\tilde{R})\tilde{R}-2f(\tilde{R})=0\quad\Rightarrow\quad \tilde{R}=c_k$$ (in dimension four) where the roots $c_k$ (and their number) only depend on the function $f$ in the Lagrangian. $\mathcal{R}$ being constant, $f'(\mathcal{R}$ should be constant, too, and equation (\[met\]) implies that $\Gamma^{\lambda}_{\mu\nu}$ (under the assumption (A) above) should coincide with the Levi-Civita connection $\left\lbrace{}^\alpha_{\beta\nu}\right\rbrace_g$. Therefore, $\mathcal{R}_{\mu\nu}$ coincides with $R_{\mu\nu}$, the Ricci tensor of the metric. Equation (\[ME0\]) then becomes an Einstein equation with cosmological constant $\Lambda$ (whose value depends on the function $f$ and on the particular root $c_k$ considered: $\Lambda_k=\frac{1}{2}\left(\frac{f(c_k)}{f'(c_k)}-c_k\right)$). Hence, the set of solutions of a vacuum Palatini $f(\mathcal{R})$ is exactly the union of the solutions of vacuum General Relativity over the set of possible values $\Lambda_k$ of the cosmological constant [@FFV].
Let us now switch on matter interaction. Equation (\[met\]), coming from the variation of (\[PalR\]) with respect to the connection, is not affected (it is here that assumption (C) plays a crucial role); (\[ME0\]), instead, becomes $$f'(\mathcal{R})\mathcal{R}_{\mu\nu}-\frac{1}{2}f(\mathcal{R})g_{\mu\nu} =\kappa T_{\mu\nu}.\label{ME1}$$ Its trace is again an algebraic equation for the scalar $\mathcal{R}$, but now the r.h.s. is, in general, a function of $j^1\Phi$ (and possibly of $j^2 g$). Hence, the roots will be non–constant functions in spacetime, through the trace $T$ of the stress-energy tensor. Therefore, $f'(\mathcal{R})$ will no longer be constant: equation (\[met\]), then, implies that $\Gamma^{\lambda}_{\mu\nu}$ coincides with the Levi-Civita connection $\lbrace{}^\alpha_{\beta\nu}\rbrace_{\tilde{g}}$ for a different metric tensor, namely the conformally rescaled metric fulfilling $$\tilde{g}^{\mu\nu}\sqrt{|\tilde{g}|}=f'(c_k(T))g^{\mu\nu}\sqrt{|g|}.$$ Let us reinterpret these well–known facts, by applying a Legendre transformation: analogously to (\[p1\]) we set $$p=\frac{1}{\sqrt{|g|}}\frac{\partial L}{\partial \mathcal{R}}=f'(\mathcal{R}),\label{p2}$$ we introduce the inverse Legendre map $r=r(p)$ and we obtain the (Palatini) Helmholtz Lagrangian $$L_{H}=p(\mathcal{R}-r)\sqrt{|g|}+f(r)\sqrt{-g}+L_{\mathrm{mat}}(j^1 \Phi,j^1 g).\label{LHP}$$ The difference, with respect to (\[LH\]), is that the dynamical fields are now the connection $\Gamma$, the scalar field $p$ and the metric $g$. The variations with respect to these three fields, in this order, yield the following equations, which are manifestly equivalent to (\[met\]) and (\[ME1\]): $$\begin{aligned}
& \bar{\nabla}_{\lambda}\left(p g^{\mu\nu}\sqrt{|g|}\right)=0\label{met2} \\[6pt]
& \mathcal{R}_{\alpha\beta}g^{\alpha\beta}=r(p)\label{uf1}\\
& p\mathcal{R}_{\mu\nu}-\frac{1}{2}f(r(p))g_{\mu\nu}=T_{\mu\nu}.\label{uf2}\end{aligned}$$ We know that eq. (\[met2\]) is equivalent to $$\Gamma^\lambda_{\mu\nu}=\left\lbrace{}^\lambda_{\mu\nu}\right\rbrace_{\tilde{g}},\quad\text{where}\quad \tilde{g}_{\mu\nu}=pg_{\mu\nu};\label{vin}$$ this means that the dynamics can be completely described by one metric and one scalar field. One has, as a matter of fact, two possibilities: either one chooses to represent the dynamics in terms of the scalar field $p$ and the metric $g_{\mu\nu}$, or one chooses the pair $(p, \tilde{g}_{\mu\nu})$. The steps are as follows: first, one observes that the lower–order relations (\[vin\]), being *identically satisfied for all solutions* of the field equations, can be safely plugged into in the Lagrangian itself. This amounts to substituting the Ricci tensor $\mathcal{R}_{\mu\nu}$ with the Ricci tensor of $\tilde{g}_{\mu\nu}$, obtaining $$L_{H}=p(\tilde{R}_{\mu\nu}g^{\mu\nu}-r)\sqrt{|g|}+f(r)\sqrt{-g}+L_{\mathrm{mat}}(j^1 \Phi,j^1 g).\label{LHP2}$$ On account of the relation between the two metrics (\[vin\]), one can immediately get rid of the metric $g_{\mu\nu}$, and in dimension four the Lagrangian (in what can be called the Einstein frame) becomes $$L_{E}=\tilde{R}\sqrt{|\tilde{g}|}+\left(p^{-2}f(r)-p^{-1}r\right)\sqrt{|\tilde{g}|}+L_{\mathrm{mat}}(j^1\Phi,j^1\tilde{g},j^1p).\label{LEH2}$$ To keep the Jordan–frame metric, instead, one can exploit once more the identity for the difference of the two Ricci tensors; in the case of two conformally–related metrics, it is known that the difference of the two Levi–Civita connections can be expressed in terms of the first derivatives of the conformal factor. The resulting Jordan–frame Lagrangian ([@Cap], [@FF]) is $$L_{J}=pR\sqrt{|g|}+\left(\frac{3}{2p}g^{\mu\nu}\partial_\mu p\, \partial_\nu p-p\cdot r+f(r)\right)\sqrt{|g|}+L_{\mathrm{mat}}(j^1\Phi,j^1g).$$ For the reader’s convenience, let us put alongside the Legendre–transformed versions of purely metric and Palatini $f(R)$ theories:
In both cases, $r=r(p)$ is the map such that $f'(x)|_{x=r(p)}\equiv p$. Notice that the coupling between $\Phi$ and the scalar field $p$ in the matter Lagrangian in the Einstein frame is not unavoidable. In some relevant cases (e.g. scalar fields, electromagnetic field, cosmic dust), the matter Lagrangian contains only ordinary derivatives of $\Phi$, not covariant ones, so it would be $L_{\mathrm{mat}}(j^1\Phi,\tilde{g},p)$; then, it may be possible to devise a suitable non–minimal coupling in the original Lagrangian (which should then be more appropriately referred to as a $f(R,\Phi)$ Lagrangian), so that the dependence of $L_{\mathrm{mat}}$ on $p$ disappears in the Einstein frame picture (various examples are given in [@MS1]).
Direct comparison of the boxed expressions shows that the difference between the purely metric models and the corresponding Palatini models entirely consists in the fact that the dynamical term for the scalar field, which in the purely metric setup is absent in the Jordan frame and appears in the Einstein frame, for the Palatini models is instead found (with opposite sign) in the Jordan-frame scalar-tensor Lagrangian, but clears away in the Einstein frame.
This difference has, indeed, relevant consequences. For instance, in vacuum the scalar field $p$ has still a nontrivial dynamics in the purely metric setup, while in the Palatini setup $p$ is forced to be constant in spacetime and can assume a set of possible values being the roots of the equation $p\cdot r(p)-2f(r(p))=0$, which is the trace of eq. (\[uf2\]) combined with (\[uf1\]). It is easy to see that these are nothing but the particular solutions of the purely metric model for which $p$ is constant. Thus, we see that in vacuum the set of solutions for the Palatini model are a *proper subset* of the solutions on the purely metric model (this fact was first observed in [@mio]). In the presence of matter, this is no longer true. The field $p$ cannot be constant unless $T$, the trace of the matter stress-energy tensor, is constant as well: in general the dynamical term for $p$, which makes the difference between the two models, cannot vanish. It is still true, however, that in the purely metric theory $p$ behaves (in the Einstein frame) as a true independent dynamical scalar field, while in the Palatini framework it becomes a mere function of the metric and of the matter/energy distribution, without independent propagation.
Let us eventually turn to the Ricci-regular case. For brevity, in the sequel we drop the matter interaction term and consider only the vacuum Lagrangian. As usual, we introduce the Legendre map for the Lagrangian (\[PalRic\]), and produce a symmetric tensor from the conjugate momentum (which is a tensor density): $$\tilde{g}^{\mu\nu}\sqrt{|\tilde{g}|}=\frac{\partial L}{\partial \mathcal{R}_{(\mu\nu)}}=\frac{\partial f}{\partial\mathcal{R_{(\mu\nu)}}}\sqrt{|g|}.
\label{LM3}$$ Setting $r_{\mu\nu}=r_{\mu\nu}(g_{\alpha\beta},\tilde{g}_{\alpha\beta})$ to be the symmetric tensor-valued function being the inverse of the Legendre map (notice that by construction $r_{\mu\nu}$ does not depend on the connection $\Gamma$), the Helmholtz Lagrangian becomes $$L_{H}=\tilde{g}^{\alpha\beta}(\mathcal{R}_{(\alpha\beta)}-r_{\alpha\beta})\sqrt{|\tilde{g}|}+f(r_{\mu\nu},g_{\mu\nu})\sqrt{|g|}.$$ Taking the variation of the original Lagrangian with respect to the connection $\Gamma^{\lambda}_{\mu\nu}$, one finds the equation $$\bar{\nabla}_{\lambda}\left(\dfrac{\partial f}{\partial\mathcal{R}_{(\mu\nu)}}\sqrt{-g}\right)=0,$$ which in the Legendre–transformed picture (i.e., taking the corresponding variation of $L_H$) becomes $$\bar{\nabla}_{\lambda}\left(\tilde{g}^{\mu\nu}\sqrt{|\tilde{g}|}\right)=0,$$ which means that the (symmetric) connection $\Gamma^{\lambda}_{\mu\nu}$ should be the Levi–Civita connection of the metric $\tilde{g}^{\mu\nu}$. This holds true even in the presence of matter, *provided* the matter Lagrangian does not contain $\Gamma^{\lambda}_{\mu\nu}$ (assumption (C) above). Hence, such a metric-affine theory is equivalent to a bimetric theory where the (independent) dynamical fields are $g^{\mu\nu}$ and $\tilde{g}^{\mu\nu}$, with the Lagrangian $$L_{E}=\tilde{g}^{\alpha\beta}\tilde{R}_{\alpha\beta}+\left(f(r_{\mu\nu},g_{\mu\nu})\sqrt{|g|}-r_{\alpha\beta}\tilde{g}^{\alpha\beta}\sqrt{|\tilde{g}|}\right).\label{uf3}$$ One therefore finds directly an Einstein–frame Lagrangian, in contrast to (\[LH1\]). Here, it is the Ricci tensor of the metric $\tilde{g}_{\mu\nu}$ which enters the Helmholtz Lagrangian, and therefore the dynamical term for $g_{\mu\nu}$, which in the purely metric setup appears in the Einstein frame after replacing $R_{\alpha\beta}$ with $\tilde{R}_{\alpha\beta}$, is absent from (\[uf3\]). Once again, we summarize the outcome of the discussion in the following box:
Again, we see that the difference lies in the dynamical term for the metric $g_{\mu\nu}$, which appears in the Einstein frame Lagrangian for the purely metric framework, and does not occur in the Einstein frame Lagrangian for the Palatini model. This term can be written as a quadratic term in the covariant derivatives $\tilde{\nabla}_{\lambda}g_{\mu\nu}$, and can be further decomposed into a standard dynamical term for a scalar field and a dynamical term for a spin-two field.
In the Palatini case, instead, the tensor $g_{\mu\nu}$ does not propagate independently: its configuration depends on $\tilde{g}_{\mu\nu}$ and on the possible coupling with matter in the original Lagrangian. There are particular cases, e.g. Palatini Lagrangians depending only on the square of the Ricci tensor, $f(\mathcal{R}_{\alpha\beta}\mathcal{R}^{\alpha\beta})$ where (in vacuum) the equations completely reduce to Einstein equations for the original metric $g_{\mu\nu}$ alone [@Bor]. Otherwise, the two metrics are not even conformally related, and in general one should not expect that the causal structure defined by $g_{\mu\nu}$ and the geodesic worldlines associated to $\tilde{g}_{\mu\nu}$ (and therefore to the original connection $\Gamma^{\lambda}_{\mu\nu}$) can be compatible in the EPS sense. This somehow undermines the physical consistency of generic Ricci-regular Palatini models, unless one is willing to assume that $\tilde{g}_{\mu\nu}$ is the physical metric (or, alternatively, that the inertial structure is defined by the Levi-Civita connection of $g_{\mu\nu}$, rather than by $\Gamma^{\lambda}_{\mu\nu}$).
Acknowledgments {#acknowledgments .unnumbered}
===============
The author wishes to thank Marco Ferraris, Leszek and Lorenzo for innumerable fruitful discussions in the past years; Jerzy , Andrzej and Demeter Krupka for sound remarks during the workshop.
This article has been written as a tribute to the memory of Mauro , who initiated the author into scientific research.
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"pile_set_name": "ArXiv"
}
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---
bibliography:
- 'CharacteristicClasses.bib'
---
#### Abstract.
We propose a definition of persistent Stiefel-Whitney classes of vector bundle filtrations. It relies on seeing vector bundles as subsets of some Euclidean spaces. The usual Čech filtration of such a subset can be endowed with a vector bundle structure, that we call a Čech bundle filtration. We show that this construction is stable and consistent. When the dataset is a finite sample of a line bundle, we implement an effective algorithm to compute its persistent Stiefel-Whitney classes. In order to use simplicial approximation techniques in practice, we develop a notion of weak simplicial approximation. As a theoretical example, we give an in-depth study of the normal bundle of the circle, which reduces to understanding the persistent cohomology of the torus knot (1,2).
#### Numerical experiments.
A Python notebook containing illustrations can be found at <https://github.com/raphaeltinarrage/PersistentCharacteristicClasses/blob/master/Demo.ipynb>.
#### Acknowledgements.
The author want to thank Frédéric Chazal and Marc Glisse for fruitful discussions and corrections.
Introduction {#sec:intro}
============
Statement of the problem
------------------------
Let $\MMo$ and $\MMo'$ denote the torus and the Klein bottle. Only one of them is orientable, hence these two manifolds are not homeomorphic. Let $\Zd$ be the field with two elements. Observe that the cohomology groups of $\MMo$ and $\MMo'$ over $\Zd$ are equal: $$\begin{aligned}
H^0(\MMo) = H^0(\MMo') &= \Zd, \\
H^1(\MMo) = H^1(\MMo') &= \Zd\times\Zd, \\
H^2(\MMo) = H^2(\MMo') &= \Zd.\end{aligned}$$ Therefore, the cohomology groups alone do not permit to differenciate the manifolds $\MMo$ and $\MMo'$. To do so, several refinements from algebraic topology may be used. For example, the first cohomology groups $H^1(\MMo)$ and $H^1(\MMo')$, or the second ones $H^2(\MMo)$ and $H^2(\MMo')$ are distinct when computed over the rings $\Z$ or $\Zp$, $p \geq 3$. Also, the cup product structures on the cohomology rings $H^*(\MMo)$ and $H^*(\MMo')$ are distinct, even over $\Zd$. In this paper, we will consider another invariant associated to $\MMo$ and $\MMo'$: the characteristic classes of their vector bundles. For instance, if we equip $\MMo$ and $\MMo'$ with their tangent bundles, their first Stiefel-Whitney classes are distinct: only one of them is zero. Hence we are able to differenciate these two manifolds.
{width="0.6\linewidth"} $$\begin{aligned}
H^*(\MMo) &= \Zd[x,y] / \langle x^2, y^2 \rangle \\
w_1(\tau_{\MMo}) &= 0\end{aligned}$$
{width="0.5\linewidth"} $$\begin{aligned}
H^*(\MMo') &= \Zd[x,y] / \langle x^3, x^2y^{-2}, xy \rangle \\
w_1(\tau_{\MMo'}) &= x\end{aligned}$$
In general, if $X$ is a topological space endowed with a vector bundle $\xi$ of dimension $d$, there exists a collection of cohomology classes $w_1(\xi), ..., w_d(\xi)$, the Stiefel-Whitney classes, such that $w_i(\xi)$ is an element of the cohomology group $H^i(\MMo)$ over $\Zd$ for $i \in [1, d]$. We discuss in Subsection \[subsec:background\_stiefelwhitney\] the interpretation of these classes. As we explain in Subsection \[subsec:background\_vectorbundles\], defining a vector bundle over a compact space $\MMo$ is equivalent to defining a continuous map $p \colon \MMo \rightarrow \Grass{d}{\R^m}$ for $m$ large enough, where $\Grass{d}{\R^m}$ is the Grassmann manifold of $d$-planes in $\R^m$. Such a map is called a classifying map for $\xi$. It is closely related to the Gauss map of submanifolds of $\R^3$.
{width="0.85\linewidth"}
Given a classifying map $p\colon X \rightarrow \Grass{d}{\R^m}$ of a vector bundle $\xi$, the Stiefel-Whitney classes $w_1(\xi), ..., w_d(\xi)$ can be defined by pushing forward some particular classes of the Grassmannian via the induced map in cohomology $p^*\colon H^*(X) \leftarrow H^*(\Grass{d}{\R^m})$.
In order to translate these considerations in a persistent-theoretic setting, suppose that we are given a dataset of the form $(X, p)$, where $X$ is a finite subset of $\R^n$, and $p$ is a map $p\colon X \rightarrow \Grass{d}{\R^m}$. Denote by $(X^t)_{t \geq 0}$ the Čech filtration of $X$, which is the collection of the $t$-thickenings $X^t$ of $X$ in the ambient space $\R^n$. In order to define some persistent Stiefel-Whitney classes, one would try to extend the map $p\colon X \rightarrow \Grass{d}{\R^m}$ to $p^t\colon X^t \rightarrow \Grass{d}{\R^m}$. However, the author did not find any interesting way to extend this map.
To adopt another point of view, $(X, p)$ can be seen as a subset of $\R^n \times \Grass{d}{\R^m}$, via $\checkX = \left\{\left(x, p(x)\right), x \in X\right\}$. The Grassmann manifold $\Grass{d}{\R^m}$ can be naturally embedded in the matrix space $\matrixspace{\R^m}$, hence $\checkX$ can be seen as a subset $\R^n \times \matrixspace{\R^m}$. If $(\checkX^t)_{t \geq 0}$ denotes the Čech filtration of $\checkX$ in the ambient space $\R^n \times \matrixspace{\R^m}$, then a natural map $p^t\colon \checkX^t \rightarrow \Grass{d}{\R^m}$ can be defined: map a point $(x, A) \in \checkX^t$ to the projection of $A$ on $\Grass{d}{\R^m}$, seen as a subset of $\matrixspace{\R^m}$. Using the extended maps $p^t\colon \checkX^t \rightarrow \Grass{d}{\R^m}$, we are able to define a notion of persistent Stiefel-Whitney classes (Definition \[def:persistent\_SF\_classes\]). The nullity of a persistent Stiefel-Whitney class is summarized in a diagram that we call a lifebar.
As an example, consider the embedding of the torus $\imm\colon \MMo \rightarrow \MM \subset \R^3$ depicted in Figure \[fig:intro\_torus\_embedding\]. Denote $P_{x}$ the tangent space of $\MM$ at $x$. The set $\MMcheck = \{ (x, P_x), x \in \MM \}$ can be seen as a subset of $\R^3 \times \matrixspace{\R^3}$.
![The submanifold $\MM \subset \R^3$, and the submanifold $\MMcheck \subset \R^3 \times \matrixspace{\R^3} \simeq \R^{12}$ projected in a 3-dimensional subspace via PCA.[]{data-label="fig:intro_torus_embedding"}](intro_torus.png){width="0.5\linewidth"}
![The submanifold $\MM \subset \R^3$, and the submanifold $\MMcheck \subset \R^3 \times \matrixspace{\R^3} \simeq \R^{12}$ projected in a 3-dimensional subspace via PCA.[]{data-label="fig:intro_torus_embedding"}](intro_torus_pca.png){width="0.5\linewidth"}
The lifebar of the first persistent Stiefel-Whitney class of this torus is depicted in Figure \[fig:intro\_torus\_lifespan\]. The bar is hatched, which means that the class is zero all along the filtration. This is coherent with the actual first Stiefel-Whitney class of the normal bundle of the torus, which is zero.
![Lifebar of the first persistent Stiefel-Whitney class of $\MMcheck$. It is only defined on the interval $\left[0, \frac{\sqrt{2}}{2}\right)$ (see Definition \[def:filtered\_cech\_bundle\]).[]{data-label="fig:intro_torus_lifespan"}](intro_torus_lifespan.png){width="0.5\linewidth"}
To continue, consider the immersion of the Klein bottle $\imm'\colon \MMo' \rightarrow \MM' \subset \R^3$ depicted in Figure \[fig:intro\_klein\_embedding\]. For $x_0 \in \MMo'$, denote $P_{x_0}$ the tangent space of $\MMo'$ at $x_0$, seen in $\R^3$. The set $\MMcheck' = \left\{ \left(u(x_0), P_{x_0}\right), x_0 \in \MMo' \right\}$ can be seen as a subset of $\R^3 \times \matrixspace{\R^3}$. Note that $\MMcheck'$ is a submanifold (diffeomorphic to the Klein bottle), while $\MM'$ is not.
![The set $\MM' \subset \R^3$, and the submanifold $\MMcheck' \subset \R^3 \times \matrixspace{\R^3} \simeq \R^{12}$ projected in a 3-dimensional subspace via PCA.[]{data-label="fig:intro_klein_embedding"}](intro_klein.png){width="0.5\linewidth"}
![The set $\MM' \subset \R^3$, and the submanifold $\MMcheck' \subset \R^3 \times \matrixspace{\R^3} \simeq \R^{12}$ projected in a 3-dimensional subspace via PCA.[]{data-label="fig:intro_klein_embedding"}](intro_klein_pca.png){width="0.5\linewidth"}
Just as before, we can define persistent Stiefel-Whitney classes over the Čech filtration of $\MMcheck'$. Figure \[fig:intro\_klein\_lifespan\] represents the lifebar of the first Stiefel-Whitney class of this filtration. The bar is filled, which means that the class is nonzero all along the filtration. This is coherent with the first Stiefel-Whitney class of the normal bundle of the Klein bottle, which is nonzero.
![Lifebar of the first persistent Stiefel-Whitney class of $\MMcheck'$.[]{data-label="fig:intro_klein_lifespan"}](intro_klein_lifespan.png){width="0.5\linewidth"}
The construction we propose is defined for any subset $X \subset \R^n \times \matrixspace{\R^m}$. In particular, it can be applied to finite samples of $\MMcheck$ and $\MMcheck'$. We prove that it is stable and consistent (Theorems \[thm:stability\] and \[thm:consistency\]). As an illustration, Figure \[fig:intro\_samples\] represents the lifebars of the first persistent Stiefel-Whitney classes of samples $X$ and $X'$ of $\MMcheck$ and $\MMcheck'$. Observe that they are close to the original ones.
![Left: a sample of $\MMcheck \subset \R^3 \times \matrixspace{\R^3}$, seen in $\R^3$, and the lifebar of its first persistent Stiefel-Whitney class. Right: same for $\MMcheck'$.[]{data-label="fig:intro_samples"}](intro_torus_sample.png "fig:"){width="0.5\linewidth"} ![Left: a sample of $\MMcheck \subset \R^3 \times \matrixspace{\R^3}$, seen in $\R^3$, and the lifebar of its first persistent Stiefel-Whitney class. Right: same for $\MMcheck'$.[]{data-label="fig:intro_samples"}](intro_bundle_torus_barcode_lifespan.png "fig:"){width=".8\linewidth"}
![Left: a sample of $\MMcheck \subset \R^3 \times \matrixspace{\R^3}$, seen in $\R^3$, and the lifebar of its first persistent Stiefel-Whitney class. Right: same for $\MMcheck'$.[]{data-label="fig:intro_samples"}](intro_klein_sample.png "fig:"){width="0.5\linewidth"} ![Left: a sample of $\MMcheck \subset \R^3 \times \matrixspace{\R^3}$, seen in $\R^3$, and the lifebar of its first persistent Stiefel-Whitney class. Right: same for $\MMcheck'$.[]{data-label="fig:intro_samples"}](intro_bundle_klein_barcode_lifespan.png "fig:"){width=".8\linewidth"}
Notations
---------
We adopt the following notations:
- $I$ denotes a set, $\card{I}$ its cardinal and $\complementaire{I}$ its complement.
- $\R^n$ and $\R^m$ denotes the Euclidean spaces of dimension $n$ and $m$, $E$ denotes a Euclidean space.
- $\matrixspace{\R^m}$ the vector space of $m \times m$ matrices, $\Grass{d}{\R^m}$ the Grassmannian of $d$-subspaces of $\R^m$, and $\S_k \subset \R^{k+1}$ the unit $k$-sphere.
- $\eucN{\cdot}$ the usual Euclidean norm on $\R^n$, $\frobN{\cdot}$ the Frobenius norm on $\matrixspace{\R^m}$, $\gammaN{\cdot}$ the norm on $\R^n \times \matrixspace{\R^m}$ defined as $\gammaN{(x,A)}^2 = \eucN{x}^2 + \gamma^2 \frobN{A}^2$ where $\gamma>0$ is a parameter.
- $\X = (X^t)_{t\in T}$ denotes a set filtration. $\V[\X]$ denotes the corresponding persistent cohomology module. If $X$ is a subset of $E$, then $\X = (X^t)_{t\in T}$ denotes the Čech set filtration of $X$.
- $(\V, \vbb)$ denotes a persistence module over $T$, with $\V = (V^t)_{t \in T}$ a family of vector spaces, and $\vbb = (v_s^t \colon X^s \leftarrow X^t)_{s\leq t \in T}$ a family of linear maps.
- $\UU$ denotes a cover of a topological space, and $\NN(\UU)$ its nerve. $\S = (S^t)_{t \in T}$ denotes a simplicial filtration.
- $(\X, \p)$ denotes a vector bundle filtration, with $\X$ a set filtration, and $\p = (p^t)_{t \in T}$ a family of maps $p^t \colon X^t \rightarrow \Grass{d}{\R^m}$. If $X$ is a subset of $\R^n \times \matrixspace{\R^m}$, then $(\X, \p)$ denotes the Čech bundle filtration associated to $X$.
- If $X$ is a topological space, $H^*(X)$ denotes its cohomology ring, and $H^i(X)$ the $i$th cohomology group. If $f\colon X\rightarrow Y$ is a continuous map, $f^*\colon H^*(X)\leftarrow H^*(Y)$ is the map induced in cohomology.
- If $\xi$ is a vector bundle, $w_i(\xi)$ denotes its $i$th Stiefel-Whitney class. If $(\X, \p)$ is a vector bundle filtration, $w_i(\p)$ denotes the $i$th persistent Stiefel-Whitney class, with $w_i(\p) = (w_i^t(\p))_{t \in T}$ (see Definition \[def:persistent\_SF\_classes\]).
- If $A$ is a subset of $E$, then $\med{A}$ denotes its medial axis, $\reach{A}$ its reach, $\dist{\cdot}{A}$ the distance to $A$ (see Subsection \[subsec:background\_persistentcohomology\]). The projection on $A$ is denoted $\proj{\cdot}{A}$ or $\projj{\cdot}{A}$. $\Hdist{\cdot}{\cdot}$ denotes the Hausdorff distance between two sets of $E$.
- If $K$ is a simplicial complex, $\skeleton{K}{i}$ denotes its $i$-skeleton. For every vertex $v \in \skeleton{K}{0}$, $\Star{v}$ and $\closedStar{v}$ denote its open and closed star. The topological realization of $K$ is denoted $\topreal{K}$, and the topological realization of a simplex $\sigma \in K$ is $\topreal{\sigma}$. The face map is denoted $\facemapK{K} \colon \topreal{K} \rightarrow K$ (see Subsection \[subsec:background\_simplicialcomplexes\]).
- If $f\colon K \rightarrow L$ is a simplicial map, $\topreal{f} \colon \topreal{K} \rightarrow \topreal{L}$ denotes its topological realization. The $i$th barycentric subdivision of the simplicial complex $K$ is denoted $\subdiv{K}{i}$ (see Subsection \[subsec:simplicial\_approximation\]).
Background on persistent cohomology {#subsec:background_persistentcohomology}
-----------------------------------
In the following, we consider interleavings of filtrations, interleavings of persistence modules and their associated pseudo-distances. Their definitions, in the context of cohomology, are recalled in this subsection. Compared to the standard definitions of persistent homology, the arrows go backward. Let $T \subseteq [0, +\infty)$ be an interval that contains 0, and let $E$ be a Euclidean space.
#### Filtrations of sets and simplicial complexes.
A family of subsets $\X=(X^t)_{t \in T}$ of $E$ is a [*filtration*]{} if it is non-decreasing for the inclusion, i.e. for any $s, t \in T$, if $s \leq t$ then $X^s \subseteq X^t$. Given $\epsilon\geq 0$, two filtrations $\X=(X^t)_{t \in T}$ and $\Y=(Y^t)_{t \in T}$ of $E$ are [*$\epsilon$-interleaved*]{} if, for every $t \in T$, $X^t \subseteq Y^{t+\epsilon}$ and $Y^t \subseteq X^{t+\epsilon}$. The interleaving pseudo-distance between $\X$ and $\Y$ is defined as the infimum of such $\epsilon$: $$d_i(\X, \Y) = \inf \left\{ \epsilon, ~\X \ \mbox{\rm and} \ \Y \ \mbox{\rm are $\epsilon$-interleaved} \right\}.$$ Filtrations of simplicial complexes and their interleaving distance are similarly defined: given an abstract simplex $S$, a [*filtration of $S$*]{} is a non-decreasing family $\S = (S^t)_{t \in T}$ of subcomplexes of $S$. The interleaving pseudo-distance between two filtrations $(S_1^t)_{t \in T}$ and $(S_2^t)_{t \in T}$ of $S$ is the infimum of the $\epsilon \geq 0$ such that they are $\epsilon$-interleaved, i.e., for any $t \in T$, we have $S_1^{t} \subseteq S_2^{t+\epsilon}$ and $S_2^{t} \subseteq S_1^{t+\epsilon}$.
#### Persistence modules and interleavings.
Let $k$ be a field. A [*persistence module*]{} over $T$ is a pair $(\V, \vbb)$ where $\V = (V^t)_{t\in T}$ is a family of $k$-vector spaces, and $\vbb = (v_s^t)_{s\leq t \in T}$ is a family of linear maps $v_s^t\colon V^s \leftarrow V^t$ such that:
- for every $t\in T$, $v_t^t\colon V^t \leftarrow V^t$ is the identity map,
- for every $r, s,t\in T$ such that $r\leq s\leq t$, $v_r^s \circ v_s^t = v_r^t$.
When there is no risk of confusion, we may denote a persistence module by $\V$ instead of $(\V, \vbb)$. Given $\epsilon \geq 0$, an [*$\epsilon$-morphism*]{} between two persistence modules $(\V, \vbb)$ and $(\W, \w)$ is a family of linear maps $(\phi_t\colon V^t \rightarrow W^{t-\epsilon})_{t \geq \epsilon}$ such that the following diagrams commute for every $\epsilon \leq s \leq t$:
V\^s &V\^t\
W\^[s-]{} &W\^[t-]{}
If $\epsilon = 0$ and each $\phi_t$ is an isomorphism, the family $(\phi_t)_{t \in T}$ is an [*isomorphism*]{} of persistence modules. An [*$\epsilon$-interleaving*]{} between two persistence modules $(\V, \vbb)$ and $(\W, \w)$ is a pair of $\epsilon$-morphisms $(\phi_t\colon V^t \rightarrow W^{t-\epsilon})_{t \geq \epsilon}$ and $(\psi_t\colon W^t \rightarrow V^{t-\epsilon})_{t \geq \epsilon}$ such that the following diagrams commute for every $t \geq 2 \epsilon$:
V\^[t-2]{} & & V\^[t]{}\
& W\^[t-]{} &
& V\^[t-]{} &\
W\^[t-2]{} & & W\^t
The interleaving pseudo-distance between $(\V, \vbb)$ and $(\W, \w)$ is defined as $$d_i(\V, \W) = \inf \{\epsilon \geq 0, ~\V \text{ and } \W \text{ are } \epsilon \text{-interleaved}\}.$$ In some cases, the proximity between persistence modules is expressed with a function. Let $T'\subseteq T$ and $\eta\colon T' \rightarrow T$ be a non-increasing function such that for any $t \in T'$, $\eta(t) \leq t$. A $\eta$-interleaving between two persistence modules $(\V, \vbb)$ and $(\W, \w)$ is a pair of families of linear maps $(\phi_t\colon V^t \rightarrow W^{\eta(t) })_{t \in T'}$ and $(\psi_t\colon W^t \rightarrow V^{\eta(t)})_{t \in T'}$ such that the following diagrams commute for every $t \in T'$:
V\^[((t))]{} & & V\^[t]{}\
& W\^[(t)]{} &
& V\^[(t)]{} &\
W\^[((t))]{} & & W\^t
When $\eta$ is $t \mapsto t-c$ for some $c > 0$, it is called an *additive $c$-interleaving* and corresponds with the previous definition. When $\eta$ is $t \mapsto ct$ for some $0 < c < 1$, it is called a *multiplicative $c$-interleaving*.
#### Persistence diagrams.
A persistence module $(\V, \vbb)$ is said to be [*pointwise finite-dimensional*]{} if for every $t \in T$, $V^t$ is finite-dimensional. This implies that we can define a notion of persistence diagram [@botnan2018decomposition Theorem 1.2]. It is based on the algebraic decomposition of the persistence module into interval modules. Moreover, given two pointwise finite-dimensional persistence modules $\V, \W$ with persistence diagrams $D(\V), D(\W)$, the so-called isometry theorem states that $d_b(D(\V), D(\W)) = d_i(\V, \W)$ where $d_i(\cdot,\cdot)$ denotes the interleaving distance, and $d_b(\cdot,\cdot)$ denotes the bottleneck distance between diagrams.
More generally, the persistence module $(\V, \vbb)$ is said to be [*$q$-tame*]{} if for every $s,t \in T$ such that $s < t$, the map $v_s^t$ is of finite rank. The $q$-tameness of a persistence module ensures that we can still define a notion of persistence diagram, even though the module may not be decomposable into interval modules. Moreover, the isometry theorem still holds [@Chazal_Persistencemodules Theorem 4.11].
#### Relation between filtrations and persistence modules.
Applying the singular cohomology functor to a set filtration gives rise to a persistence module whose linear maps between cohomology groups are induced by the inclusion maps between sets. As a consequence, if two filtrations are $\epsilon$-interleaved, then their associated cohomology persistence modules are also $\epsilon$-interleaved, the interleaving homomorphisms being induced by the interleaving inclusion maps. As a consequence of the isometry theorem, if the modules are $q$-tame, then the bottleneck distance between their persistence diagrams is upperbounded by $\epsilon$.
The same remarks hold when applying the simplicial cohomology functor to simplicial filtrations.
#### Reach of subsets of $E$.
Let $X$ be any subset of $E$. Following [@federer1959curvature Definition 4.1], the function *distance to $X$* is the map $\dist{\cdot}{X} \colon y\in E \mapsto \inf\{ \eucN{y-x}, x \in X \}$. A projection of $y$ on $X$ is a point $x\in X$ which attains this infimum. The medial axis of $X$ is the subset $\med{X} \subset E$ which consists of points $y\in E$ that admits at least two projections: $$\begin{aligned}
\med{X} = \left\{ y \in E, \exists x,x' \in X, x \neq x', \eucN{y-x}=\eucN{y-x}=\dist{y}{X} \right\}.\end{aligned}$$ The *reach* of $X$ is $$\reach{X} = \inf\left\{ \eucN{x-y}, x \in X, y \in \med{X} \right\}.$$ Alternatively, let $X^t$ denote the $t$-thickening of $X$, i.e. the subset of points of $E$ at distance at most $t$ from $X$. Then the reach of $X$ can be defined as the supremum of $t\geq 0$ such that $X^t$ does not intersect $\med{X}$.
Suppose that $X$ is closed and let $\reach{X}$ be the reach of $X$. One shows that each $X^t$ deform retracts onto $X$ for $0 \leq t< \reach{X}$. Besides, if $Y$ is any other subset of $E$ with Hausdorff distance $\Hdist{X}{Y}\leq\epsilon$, then for any $t \in [4\epsilon, \reach{X} -3 \epsilon)$, $Y^t$ deform retracts on $X$ [@chazal2009sampling Theorem 4.6, case $\mu=1$].
#### Weak feature size of compact subsets of $E$.
Let $X$ be any compact subset of $E$, and denote by $d_X$ the distance function to $X$. It is not differentiable in general. However, one can define a generalized gradient vector field $\nabla_X\colon E \rightarrow E$, as in [@boissonnat2018geometric Section 9.2]. A point $x \in E$ is called a critical point of $d_X$ if $\nabla_X(x) = 0$. One shows that $x$ is a critical point of $d_X$ if it lies in the convex hull of its projections on $X$. The *weak feature size* of $X$ is defined as $$\begin{aligned}
\wfs{X} = \inf\left\{\dist{x}{X}, x \text{ is a critical point of } d_X\right\}.\end{aligned}$$ The Isotopy Lemma [@boissonnat2018geometric Theorem 9.5] states that for every $s,t \in \R$ such that $0<s\leq t < \wfs{X}$, the thickening $X^t$ is isotopic to $X^s$. This isotopy can be chosen to be a deformation retraction. If $X$ admits a positive reach, we deduce that the thickenings $X^t$ deform retracts on $X$. The weak feature size and reach of $X$ satisfy the inequality $\reach{X} \leq \wfs{X}$.
#### Čech set filtrations.
Let $X$ denote any subset of $E$. The *Čech set filtration* associated to $X$ is the filtration of $E$ defined as the collection of subsets $\X = (X^t)_{t \geq 0}$, where $X^t$ denotes the $t$-thickening of $X$ in $E$, that is, $X^t = \{x \in E, \dist{x}{X} \leq t\}$.
If $X$ is a compact submanifold, then according to the previous considerations about the reach, for every $t \in [0, \reach{X})$, $X^t$ deform retracts on $X$. Therefore, the corresponding cohomology persistence module is constant on the interval $[0, \tau)$, and is equal to the cohomology of $X$. Moreover, if $Y$ is any other subset of $E$ with Hausdorff distance $\Hdist{X}{Y}\leq\epsilon$, then the cohomology persistence module of the Čech filtration associated to $Y$ is constant on the interval $[4\epsilon, \tau -3 \epsilon)$ and is equal to the cohomology of $X$.
#### Čech simplicial filtrations.
Let $X$ denote a finite subset of $E$ and $\X = (X^t)_{t\geq 0}$ its associated Čech set filtration. For all $t \geq 0$, $X^t$ is a union of closed balls of radius $t$: $X^t = \bigcup_{x \in X} \closedball{x}{t}$. Consider the simplicial filtration $\S = (S^t)$, where $S^t$ is the nerve of the cover $\UU^t$ defined as $\UU^t = \{\closedball{x}{t}, x \in X\}$. It is called the *Čech simplicial filtration* associated to $X$. The persistent nerve lemma [@Chazal_Towards Lemma 3.4] states that the persistence (singular) cohomology module associated to $\X$ and the persistent (simplicial) cohomology module associated to $\S$ are isomorphic.
Background on vector bundles {#subsec:background_vectorbundles}
----------------------------
This subsection and the next one follow the presentation of [@Milnor_Characteristic].
#### Vector bundles.
Let $X$ be a topological space. A *vector bundle $\xi$* of dimension $d$ consists of a topological space $A = A(\xi)$, the *total space*, a continuous map $\pi = \pi(\xi) \colon A \rightarrow X$, the *projection map*, and for every $x \in X$, a structure of $d$-dimensional vector space on $\pi^{-1}(\{ x \})$. Moreover, $\xi$ must satisfies the local triviality condition: for every $x \in X$, there exists a neighborhood $U \subseteq X$ of $x$ and a homeomorphism $h \colon U \times \R^d \rightarrow \pi^{-1}(U)$ such that for every $y \in U$, the map $z \mapsto h(y,z)$ defines an isomorphism between the vector spaces $\R^d$ and $\pi^{-1}(\{ y \})$.
A()\
X
\^[-1]{}(U) &\[1em\] U \^d\
U &
In this subsection, the fibers $\pi^{-1}(\{ x \})$ will be denoted $F_x(\xi)$.
#### Isomorphisms of vector bundles.
An *isomorphism of vector bundles* $\xi, \eta$ with common base space $X$ is a homeomorphism $f\colon A(\xi) \rightarrow A(\eta)$ which sends each fiber $F_{x}(\xi)$ isomorphically into $F_{f(x)}(\eta)$. We obtain a commutative diagram
A() & & A()\
& X &
The *trivial bundle* of dimension $d$ over $X$, denoted $\epsilon = \epsilon_X^d$, is defined with the total space $A(\epsilon) = X \times \R^d$, with the projection map $\pi$ being the projection on the first coordinate, and where each fiber is endowed with the usual vector space structure of $\R^d$. A vector bundle $\xi$ over $X$ is said trivial if it is isomorphic to $\epsilon$.
#### Operations on vector bundles.
If $\xi, \eta$ are two vector bundles on $X$, we define their *Whitney sum* $\xi \oplus \eta$ by $$\begin{aligned}
A(\xi \oplus \eta) = \{(x,a,b), x\in X, a \in F_x(\xi), b \in F_x(\eta) \},\end{aligned}$$ where the projection map is given by the projection on the first coordinate, and where the vector space structures are the product structures. If $\eta$ is a vector bundle on $Y$ and $g\colon X \rightarrow Y$ a continuous map, the *pullback bundle* $g^* \xi$ is the vector bundle on $X$ defined by $$\begin{aligned}
A(g^* \xi) = \{(x, a), x \in X, a \in F_{g(x)}(\xi)\},\end{aligned}$$ where the projection map is given by the projection on the first coordinate.
#### Bundle maps.
A *bundle map* between two vector bundles $\xi, \eta$ with base spaces $X$ and $Y$ is a contiuous map $f\colon A(\xi) \rightarrow A(\eta)$ which sends each fiber $F_x(\xi)$ isomorphically into another fiber $F_{x'}(\eta)$. If such a map exists, there exist a unique map $\overline f$ which makes the following diagram commute:
A() &\[1em\] A()\
X & Y
In this case, $\xi$ is isomorphic to the pullback bundle $\overline{f}^* \eta$ [@Milnor_Characteristic Lemma 3.1]. We say that the map $\overline f$ *covers* $f$.
#### Universal bundles.
Let $0<d\leq m$. The Grassmann manifold $\Grass{d}{\R^m}$ is a set which consists of all $d$-dimensional linear subspaces of $\R^m$. It can be given a smooth manifold structure. When $d=1$, $\Grass{1}{\R^m}$ corresponds to the real projective space $\P_n(\R)$. On $\Grass{d}{\R^m}$, there exists a canonical vector bundle of dimension $d$, denoted $\gamma_d^m$. It consists in the total space $$\begin{aligned}
A(\gamma_d^m) = \left\{ (V, v), V \in \Grass{d}{\R^m}, v \in V \right\} \subset \Grass{d}{\R^m} \times \R^m,\end{aligned}$$ with the projection map on the first coordinate, and the linear structure inherited from $\R^m$.
Let $\xi$ be vector bundle of dimension $d$ over a compact space $X$. Then for $m$ large enough, there exists a bundle map from $\xi$ to $\gamma_d^m$. \[lem:bundlemap\_universal\]
If such a bundle map $f \colon \xi \rightarrow \gamma_d^m$ exists, then $\xi$ is isomorphic to the pullback $\overline f^* \gamma_d^m$, where $\overline f$ denotes the map that $f$ covers.
In order to avoid mentionning $m$, it is convenient to consider the infinite Grassmannian. The infinite Grassmann manifold $\Grass{d}{\R^{\infty}}$ is the set of all $d$-dimensional linear subspaces of $\R^{\infty}$, where $\R^{\infty}$ is the vector space of series with a finite number of nonzero terms. The infinite Grassmannian is topologized as the direct limit of the sequence $\Grass{d}{\R^{d}} \subset \Grass{d}{\R^{d+1}} \subset \Grass{d}{\R^{d+2}} \subset \cdots$. Just as before, there exists on $\Grass{d}{\R^{\infty}}$ a canonical bundle $\gamma_d^\infty$. It is called a *universal bundle*, for the following reason:
if $\xi$ is vector bundle of dimension $d$ over a paracompact space $X$, then there exists a bundle map from $\xi \rightarrow \gamma_d^\infty$. \[lem:bundlemap\_universal\_infinite\]
Such a bundle map is denoted $f_\xi \colon A(\xi) \rightarrow A(\gamma_d^\infty)$. The underlying map between base spaces, denoted $\overline f_\xi : X \rightarrow \Grass{d}{\R^{\infty}}$, is called a *classifying map for $\xi$*. As before, $\xi$ is isomorphic to the pullback $(\overline f_\xi)^* \gamma_d^{\infty}$. Note that if $f$ is a bundle map given by Lemma \[lem:bundlemap\_universal\], then the following composition is a classifying map for $\xi$:
X & & .
#### A correspondance.
Let $\xi, \eta$ be bundles over $X$, and let $\overline{f_\xi}, \overline{f_\eta}$ be classifying maps. If these maps are homotopic, one shows that the bundles $\xi$ and $\eta$ are isomorphic. The following theorem states that the converse is also true.
Let $X$ be a paracompact space. There exists a bijection between the vector bundles over $X$ (up to isomorphism) and the continuous maps $X \rightarrow \Grass{d}{\R^\infty}$ (up to homotopy). It is given by $\xi \mapsto \overline f_\xi$, where $\overline f_\xi$ denotes a the classifying map for $\xi$. \[th:correspondance\_classifyingmap\]
This result leads to the following convention:
Background on Stiefel-Whitney classes {#subsec:background_stiefelwhitney}
-------------------------------------
The Stiefel-Whitney classes are a particular instance of the theory of characteristic classes, with coefficient group being $\Zd$. We first define them axiomatically, and then describe their construction.
#### Axioms for Stiefel-Whitney classes.
To each vector bundle $\xi$ over a paracompact base space $X$, one associates a sequence of cohomology classes $$\begin{aligned}
w_i(\xi) \in H^i(X, \Z_2), ~~~~~i \in \N,\end{aligned}$$ called the *Stiefel-Whitney classes of $\xi$*. These classes satisfy:
- **Axiom 1:** $w_0$ is equal to $1 \in H^0(X, \Zd)$, and if $\xi$ is of dimension $d$ then $w_i(\xi) = 0$ for $i>d$.
- **Axiom 2:** if $f\colon \xi \rightarrow \eta$ is a bundle map, then $w_i(\xi) = \overline f^* w_i(\eta)$, where $\overline f^*$ is the map in cohomology induced by the underlying map $\overline f$.
- **Axiom 3:** if $\xi, \eta$ are bundles over the same base space $X$, then for all $k \in \N$, $w_k(\xi \oplus \eta) = \sum_{i=0}^k w_i(\xi) \cupp w_{k-i}(\eta)$, where $\cupp$ denotes the cup product.
- **Axiom 4:** $w_1(\gamma_1^1) \neq 0$, where $\gamma_1^1$ denotes the universal bundle of the projective line $\Grass{1}{\R^2}$.
The Stiefel-Whitney classes are invariants of vector bundles, and carry topological information. For instance, the following lemma shows that the first Stefel-Whitney class detects orientability.
\[prop:SF\_orientability\] If $X$ is a compact manifold and $\tau$ its tangent bundle, then $X$ is orientable if and only if $w_1(\tau) = 0$.
#### Construction of the Stiefel-Whitney classes.
The cohomology ring of the Grassmann manifolds admits a simple description: $H^*(G_d(\R^\infty), \Z_2)$ is the free abelian ring generated by $d$ elements $w_1, ..., w_d$. As a graded algebra, the degree of these elements are $|w_1| = 1,..., |w_d| = d$ [@Milnor_Characteristic Theorem 7.1]. Hence we can write $$\begin{aligned}
H^*(G_d(\R^\infty), \Z_2) \simeq \Z_2[w_1, ..., w_d].\end{aligned}$$ In particular, the infinite projective space $\P_\infty = G_1(\R^\infty)$ space has cohomology $H^*(\P_\infty, \Z_2) = \Z_2[w_1]$, the polynomial ring.
The generators $w_1, ..., w_d$ can be seen as the Stiefel-Whitney classes of the universal bundle $\gamma_d^\infty$ on $\Grass{d}{\R^\infty}$. Now, for any vector bundle $\xi$, define $$\begin{aligned}
w_i(\xi) = \overline f_\xi^* (w_i),\end{aligned}$$ where $\overline f_\xi \colon X \rightarrow \Grass{d}{\R^\infty}$ is a classifying map for $\xi$ (as in Theorem \[th:correspondance\_classifyingmap\]) and $\overline f_\xi^* \colon H^*(X )\leftarrow H^*(\Grass{d}{\R^\infty})$ the induced map in cohomology. This construction yields the Stiefel-Whitney classes:
Defined this way, the classes satisfy the four axioms. And they are unique. \[th:def\_SF\_classes\]
Persistent Stiefel-Whitney classes {#sec:persistentSWclasses}
==================================
Definition
----------
Let $E$ be a Euclidean space, and $\X = (X^t)_{t\in T}$ a set filtration of $E$ (see Subsection \[subsec:background\_persistentcohomology\]). Let us denote by $i_s^t$ the inclusion map from $X^s$ to $X^t$. In order to define persistent Stiefel-Whitney classes, we have to give such a filtration a vector bundle structure.
A vector bundle filtration of dimension $d$ on $E$ is a couple $(\X,\p)$ where $\X = (X^t)_{t \in T}$ is a set filtration of $E$ and $\p = (p^t)_{t \in T}$ a family of continuous maps $p^t \colon X^t \rightarrow \Grass{d}{\R^\infty}$ such that, for every $s,t \in T$ with $s \leq t$, we have $p^t \circ i_s^t = p^s$. In other words, the following diagram commutes:
X\^s & & X\^t\
& &
Let us fix a $t \in T$. The map $p^t \colon X^t \rightarrow \Grass{d}{\R^\infty}$ gives the topological space $X^t$ a vector bundle structure, as discussed in Subsection \[subsec:background\_vectorbundles\]. Following Subsection \[subsec:background\_stiefelwhitney\], the induced map in cohomology, $(p^t)^*$, allows to define the Stiefel-Whitney classes of this vector bundle. Let us introduce some notations. The Stiefel-Whitney classes of $\Grass{d}{\R^\infty}$ are denoted $w_1, ..., w_d$. The Stiefel-Whitney classes of the vector bundle $(X^t, p^t)$ are denoted $w_1^t(\p), ..., w_d^t(\p)$, and can be defined as $w_i^t(\p) = (p^t)^*(w_i)$ (as in Theorem \[th:def\_SF\_classes\]).
(p\^t)\^\* &\
w\_1\^t() & w\_1\
& \
w\_d\^t() & w\_d
Let $(\V, \vbb)$ denote the persistence module corresponding to the filtration $\X$, with $\V = (V^t)_{t \in T}$ and $\vbb = (v_s^t)_{s\leq t \in T}$. For every $t \in T$, the classes $w_1^t(\p), \cdots, w_d^t(\p)$ belong to the vector space $V^t$. The persistent Stiefel-Whitney classes are defined to be the collection of such classes over $t$.
\[def:persistent\_SF\_classes\] Let $(\X, \p)$ be a vector bundle filtration. The persistent Stiefel-Whitney classes of $(\X, \p)$ are the families of classes $$\begin{aligned}
w_1(\p) &= \big(w_1^t(\p)\big)_{t \in T} \\
&\vdots \\
w_d(\p)& = \big(w_d^t(\p)\big)_{t \in T}.\end{aligned}$$
Let $i \in [1,d]$, and consider a persistent Stiefel-Whitney class $w_i(\p)$. Note that it satisfies the following property: for all $s,t \in T$ such that $s\leq t$, we have $w_i^s(\p) = v_s^t\big( w_i^t(\p) \big)$. As a consequence, if a class $w_i^t(\p)$ is given for a $t \in T$, one obtains all the others $w_i^s(\p)$, with $s \leq t$, by applying the maps $v_s^t$. In particular, if $w_i^t(\p) = 0$, then $w_i^s(\p) = 0$ for all $s \in T$ such that $s \leq t$.
#### Lifebar.
In order to visualize the evolution of a persistent Stiefel-Whitney class through the persistence module $(\V, \vbb)$, we propose the following bar representation: the lifebar of $w_i(\p)$ is the set $$\begin{aligned}
\left\{ t \in T, w_i^t(\p) \neq 0 \right\}.\end{aligned}$$ According to the last paragraph, the lifebar of a persistent class is an interval of $T$, of the form $[\tdeatho, \sup(T))$ or $(\tdeatho, \sup(T))$, where $$\begin{aligned}
\tdeatho = \inf \left\{t \in T, w_i^t(\p) \neq 0\right\},\end{aligned}$$ with the convention $\inf (\emptyset) = \inf (T)$. In order to distinguish the lifebar of a persistent Stiefel-Whitney class from the bars of the persistence barcodes, we draw the rest of the interval hatched.
{width="0.6\linewidth"}
Čech bundle filtration {#subsec:filtered_cech_bundle}
----------------------
In this subsection, we propose a particular construction of vector bundle filtration, called the Čech bundle filtration. We will work in the ambient space $E = \R^n \times \matrixspace{\R^m}$. Let $\eucN{\cdot}$ be the usual Euclidean norm on the space $\R^n$, and $\frobN{\cdot}$ the Frobenius norm on the matrix space $\matrixspace{\R^m}$. Let $\gamma > 0$. We endow the vector space $E$ with the Euclidean norm $\gammaN{\cdot}$ defined for every $(x,A) \in E$ as $$\gammaN{(x,A)}^2 = \eucN{x}^2 + \gamma^2 \frobN{A}^2.
\label{eq:gammaN}$$ See Subsection \[subsec:choice\_of\_gamma\] for a discussion about the parameter $\gamma$.
In order to define the Čech bundle filtration, we will first study the usual embedding of the Grassmann manifold $\Grass{d}{\R^m}$ into the matrix space $\matrixspace{\R^m}$.
#### Embedding of $\Grass{d}{\R^m}$.
We embed the Grassmannian $\Grass{d}{\R^m}$ into $\matrixspace{\R^m}$ via the application which sends a $d$-dimensional subspace $T \subset \R^m$ to its orthogonal projection matrix $\projmatrix{T}$. We can now see $\Grass{d}{\R^m}$ as a submanifold of $\matrixspace{\R^m}$. Recall that $\matrixspace{\R^m}$ is endowed with the Frobenius norm. According to this metric, $\Grass{d}{\R^m}$ is included in the sphere of center 0 and radius $\sqrt{d}$ of $\matrixspace{\R^m}$.
In the metric space $(\matrixspace{\R^m}, \frobN{\cdot})$, consider the distance function to $\Grass{d}{\R^m}$, denoted $\dist{\cdot}{\Grass{d}{\R^m}}$. Let $\med{\Grass{d}{\R^m}}$ denote the medial axis of $\Grass{d}{\R^m}$. It consists in the points $A \in \matrixspace{\R^m}$ which admit at least two projections on $\Grass{d}{\R^m}$: $$\begin{aligned}
\med{\Grass{d}{\R^m}} = \{A \in \matrixspace{\R^m}, \exists P, P' \in \Grass{d}{\R^m}&, P \neq P', \\
&\frobN{A - P} = \frobN{A - P} = \dist{A}{\Grass{d}{\R^m}} \}.\end{aligned}$$
{width="0.6\linewidth"}
On the set $\matrixspace{\R^m} \setminus \med{\Grass{d}{\R^m}}$, the projection on $\Grass{d}{\R^m}$ is well-defined: $$\begin{aligned}
\proj{\cdot}{\Grass{d}{\R^m}} \colon \matrixspace{\R^m} \setminus \med{\Grass{d}{\R^m}} &\longrightarrow \Grass{d}{\R^m} \subset \matrixspace{\R^m} \\
A &\longmapsto P \text{ such that } \frobN{P-A} = \dist{A}{\Grass{d}{\R^m}}.\end{aligned}$$ The following lemma describes this projection explicitly. We defer its proof to Appendix \[sec:appendix\_persistentSFclasses\].
\[lem:projongrass\] For any $A \in \matrixspace{\R^m}$, let $A^s$ denote the matrix $A^s = \frac{1}{2}(A + \transp{A})$, and let $\lambda_1(A^s), ..., \lambda_n(A^s)$ be the eigenvalues of $A^s$ in decreasing order. The distance from $A$ to $\med{\Grass{d}{\R^m}}$ is $$\begin{aligned}
\dist{A}{\med{\Grass{d}{\R^m}}} = \frac{\sqrt{2}}{2} \big|\lambda_d(A^s) - \lambda_{d+1}(A^s)\big|.\end{aligned}$$ If this distance is positive, the projection of $A$ on $\Grass{d}{\R^m}$ can be described as follows: consider the symmetric matrix $A^s$, and let $A^s = O D \transp{O}$, with $O$ an orthogonal matrix, and $D$ the diagonal matrix containing the eigenvalues of $A^s$ in decreasing order. Let $J_d$ be the diagonal matrix whose first $d$ terms are 1, and the other ones are zero. We have $$\begin{aligned}
\proj{A}{\Grass{d}{\R^m}} = O J_d \transp{O}.\end{aligned}$$
Observe that, as a consequence of this lemma, every point of $\Grass{d}{\R^m}$ is at equal distance from $\med{\Grass{d}{\R^m}}$, and this distance is equal to $\frac{\sqrt{2}}{2}$. Therefore the reach of the subset $\Grass{d}{\R^m} \subset \matrixspace{\R^m}$ is $$\begin{aligned}
\reach{\Grass{d}{\R^m}} = \frac{\sqrt{2}}{2}. \end{aligned}$$
#### Čech bundle filtration.
Let $X$ be a subset of $E= \R^n \times \matrixspace{\R^m}$. Consider the usual Čech filtration $\X = (X^t)_{t \geq 0}$, where $X^t$ denotes the $t$-thickening of $\check{X}$ in the metric space $(E, \gammaN{\cdot})$. In order to give this filtration a vector bundle structure, consider the map $p^t$ defined as the composition $$\begin{tikzcd}[baseline=(current bounding box.center), column sep = 6em]
X^t \subset \R^n \times \matrixspace{\R^m} \arrow[r, "\mathrm{proj}_2"] & \matrixspace{\R^m} \setminus \med{\Grass{d}{\R^m}} \arrow[r, "\proj{\cdot}{\Grass{d}{\R^m}}"]
& \Grass{d}{\R^m},
\end{tikzcd}
\label{eq:def_cech_bundle_proj}$$ where $\mathrm{proj}_2$ represents the projection on the second coordinate of $\R^n \times \matrixspace{\R^m}$, and $\proj{\cdot}{\Grass{d}{\R^m}}$ the projection on $\Grass{d}{\R^m} \subset \matrixspace{\R^m}$. Note that $p^t$ is well-defined only when $X^t$ does not intersect $\R^n \times \med{\Grass{d}{\R^m}}$. The supremum of such $t$’s is denoted $\tmaxgamma{X}$. We have $$\label{eq:tmaxgamma}
\tmaxgamma{X} = \inf \left\{ \distgamma{x}{\R^n \times \med{ \Grass{d}{\R^m} }}, x \in X \right\},$$ where $\distgamma{x}{\R^n \times \med{ \Grass{d}{\R^m} }}$ is the distance between the point $x \in \R^n \times \matrixspace{\R^m}$ and the subspace $\R^n \times \med{ \Grass{d}{\R^m}}$, with respect to the norm $\gammaN{\cdot}$. By definition of $\gammaN{\cdot}$, Equation \[eq:tmaxgamma\] rewrites as $$\tmaxgamma{X} = \gamma \cdot \inf \{ \dist{A}{\med{ \Grass{d}{\R^m} }}, (y,A) \in X \},$$ where $\dist{A}{\med{ \Grass{d}{\R^m} }}$ represents the distance between the matrix $A$ and the subspace $\med{ \Grass{d}{\R^m}}$ with respect to the Frobenius norm $\frobN{\cdot}$. Denoting $\tmax{X}$ the value $\tmaxgamma{X}$ for $\gamma = 1$, we obtain $$\begin{aligned}
\label{eq:tmax}
\begin{split}
\tmaxgamma{X} &= \gamma \cdot \tmax{X} \\
\mathrm{and}~~~~~~~~~~~~~\tmax{X} &=\inf \{ \dist{A}{\med{ \Grass{d}{\R^m} }}, (y,A) \in X \}.
\end{split}\end{aligned}$$
Note that the values $\tmax{X}$ can be computed explicitly thanks to Lemma \[lem:projongrass\]. In particular, if $X$ is a subset of $\R^n \times \Grass{d}{\R^m}$, then $\tmax{X} = \frac{\sqrt{2}}{2}$. Accordingly, $$\label{eq:tmax_subset_grass}
\tmaxgamma{X} = \frac{\sqrt{2}}{2} \gamma.$$
\[def:filtered\_cech\_bundle\] Consider a subset $X$ of $E=\R^n \times \matrixspace{\R^m}$, and suppose that $\tmax{X} > 0$. The Čech bundle filtration associated to $X$ in the ambient space $(E, \gammaN{\cdot})$ is the vector bundle filtration $(\X, \p)$ consisting of the Čech filtration $\X = (X^t)_{t \in T}$, and the maps $\p = (p^t)_{t \in T}$ as defined in Equation \[eq:def\_cech\_bundle\_proj\]. This vector bundle filtration is defined on the index set $T = \left[0, \tmaxgamma{X} \right)$, where $\tmaxgamma{X}$ is defined in Equation \[eq:tmax\].
The $i$th persistent Stiefel-Whitney class of the Čech bundle filtration $(\X, \p)$, as in Definition \[def:persistent\_SF\_classes\], will be denoted $w_i(X)$ instead of $w_i(\p)$.
\[ex:normal\_mobius\] Let $E = \R^2 \times \matrixspace{\R^2}$. Let $X$ and $Y$ be the subsets of $E$ defined as: $$\begin{aligned}
&X = \bigg\{
\bigg(
\begin{pmatrix}
\cos(\theta) \\
\sin(\theta)
\end{pmatrix}
,
\begin{pmatrix}
\cos(\theta)^2 & \cos(\theta) \sin(\theta) \\
\cos(\theta) \sin(\theta) & \sin(\theta)^2
\end{pmatrix}
\bigg),
\theta \in [0, 2\pi) \bigg\} \\
&Y = \bigg\{
\bigg(
\begin{pmatrix}
\cos(\theta) \\
\sin(\theta)
\end{pmatrix}
,
\begin{pmatrix}
\cos(\frac{\theta}{2})^2 & \cos(\frac{\theta}{2}) \sin(\frac{\theta}{2}) \\
\cos(\frac{\theta}{2}) \sin(\frac{\theta}{2}) & \sin(\frac{\theta}{2})^2
\end{pmatrix}
\bigg),
\theta \in [0, 2\pi) \bigg\}\end{aligned}$$ The set $X$ is to be seen as the normal bundle of the circle, and $Y$ as the universal bundle of the circle, known as the Mobius band. We have $\tmax{X} = \tmax{Y} = \frac{\sqrt{2}}{2}$ as in Lemma \[lem:projongrass\]. Let $\gamma = 1$.
{width=".5\linewidth"}
{width=".5\linewidth"}
{width=".5\linewidth"}
{width=".5\linewidth"}
We now compute the persistence diagrams of the Čech filtrations of $X$ and $Y$ in the ambient space $E$.
![$H^0$ and $H^1$ persistence barcode of the Čech filtration of $X$ (left) and $Y$ (right).[]{data-label="fig:example_bundles_barcode_0"}](bundle_normal_barcode_H0.png "fig:"){width=".9\linewidth"} ![$H^0$ and $H^1$ persistence barcode of the Čech filtration of $X$ (left) and $Y$ (right).[]{data-label="fig:example_bundles_barcode_0"}](bundle_normal_barcode_H1.png "fig:"){width=".9\linewidth"}
![$H^0$ and $H^1$ persistence barcode of the Čech filtration of $X$ (left) and $Y$ (right).[]{data-label="fig:example_bundles_barcode_0"}](bundle_mobius_barcode_H0.png "fig:"){width=".9\linewidth"} ![$H^0$ and $H^1$ persistence barcode of the Čech filtration of $X$ (left) and $Y$ (right).[]{data-label="fig:example_bundles_barcode_0"}](bundle_mobius_barcode_H1.png "fig:"){width=".9\linewidth"}
Consider the first persistent Stiefel-Whitney classes $w_1(X)$ and $w_1(Y)$ of the corresponding Čech bundle filtrations. We compute that their lifebars are $\emptyset$ for $w_1(X)$, and $\left[0, \tmax{Y}\right)$ for $w_1(Y)$. This is illustrated in Figure \[fig:example\_bundles\_lifespan\]. One reads these bars as follows: $w_1^t(X)$ is zero for every $t \in \left[0, \frac{\sqrt{2}}{2}\right)$, while $w_1^t(Y)$ is nonzero.
![Lifebars of the first persistent Stiefel-Whitney classes $w_1(X)$ and $w_1(Y)$.[]{data-label="fig:example_bundles_lifespan"}](bundle_normal_lifespan.png){width=".9\linewidth"}
![Lifebars of the first persistent Stiefel-Whitney classes $w_1(X)$ and $w_1(Y)$.[]{data-label="fig:example_bundles_lifespan"}](bundle_mobius_lifespan.png){width=".9\linewidth"}
Stability
---------
In this subsection we derive a straigthforward stability result for persistent Stiefel-Whitney classes. We start by defining a notion of interleavings for vector bundle filtrations, in the same vein as the usual interleavings of set filtrations.
\[def:interleavings\_bundles\] Let $\epsilon \geq 0$, and consider two vector bundle filtrations $(\X, \p)$, $(\Y,\q)$ of dimension $d$ on $E$ with respective index sets $T$ and $U$. They are $\epsilon$-interleaved if the underlying filtrations $\X = (X^t)_{t \in T}$ and $\Y = (Y^t)_{t \in U}$ are $\epsilon$-interleaved, and if the following diagrams commute for every $t \in T \cap (U - \epsilon)$ and $s \in U \cap (T - \epsilon)$:
X\^t & & Y\^[t+]{}\
& &
Y\^s & & X\^[s+]{}\
& &
The following theorem shows that interleavings of vector bundle filtrations give rise to interleavings of persistence modules which respect the persistent Stiefel-Whitney classes.
\[thm:stability\] Consider two vector bundle filtrations $(\X, \p)$, $(\Y,\q)$ of dimension $d$ with respective index sets $T$ and $U$. Suppose that they are $\epsilon$-interleaved. Then there exists an $\epsilon$-interleaving $(\phi, \psi)$ between their corresponding persistent cohomology modules which sends persistent Stiefel-Whitney classes on persistent Stiefel-Whitney classes. In other words, for every $i \in [1,d]$, and for every $t \in (T + \epsilon) \cap U$ and $s \in U \cap (T + \epsilon)$, we have $$\begin{aligned}
&\phi^t( w_i^t(\p) ) = w_i^{t-\epsilon}(\q) \\
\text{ and ~~ } &\psi^s( w_i^s(\p) ) = w_i^{s-\epsilon}(\q).\end{aligned}$$
Define $(\phi, \psi)$ to be the $\epsilon$-interleaving between the cohomology persistence modules $\V(\X)$ and $\V(\Y)$ given by the $\epsilon$-interleaving between the filtrations $\X$ and $\Y$. Explicitly, if $i_t^{t+\epsilon}$ denotes the inclusion $X^t \hookrightarrow Y^{t+\epsilon}$ and $j_s^{s+\epsilon}$ denotes the inclusion $Y^s \hookrightarrow X^{s+\epsilon}$, then $\phi = (\phi^t)_{t \in (T+\epsilon) \cap U}$ is given by the induced maps in cohomology $\phi^t = (i_{t-\epsilon}^{t})^*$, and $\psi = (\psi^s)_{s \in (U+\epsilon) \cap T}$ is given by $\psi^s = (j_{s-\epsilon}^{s})^*$. Now, by fonctoriality, the diagrams of Definition \[def:interleavings\_bundles\] give rise to commutative diagrams in cohomology:
H\^\*(X\^[t-]{}) & & H\^\*(Y\^[t]{})\
& H\^\*() &
H\^\*(Y\^[s-]{}) & & H\^\*(X\^[s]{})\
& H\^\*() &
Let $i \in [1,d]$. By definition, the persistent Stiefel-Whitney classes $w_i(\p) = (w_i^t(\p))_{t \in T}$ and $w_i(\q) = (w_i^s(\q))_{s \in U}$ are $w_i^t(\p) = (p^t)^*(w_i)$ and $w_i^s(\q) = (q^s)^*(w_i)$, where $w_i$ is the $i$th Stiefel-Whitney class of $\Grass{d}{\R^\infty}$. The previous commutative diagrams then tranlates as $\phi^t( w_i^t(\p) ) = w_i^{t-\epsilon}(\q)$ and $\psi^s( w_i^s(\p) ) = w_i^{s-\epsilon}(\q)$, as wanted.
Consider two vector bundle filtrations $(\X, \p)$, $(\Y,\q)$ such that there exists an $\epsilon$-interleaving $(\phi, \psi)$ between their persistent cohomology modules $\V(\X)$, $\V(\Y)$ which sends persistent Stiefel-Whitney classes on persistent Stiefel-Whitney classes. Let $i \in [1,d]$. Then the lifebars of their $i$th persistent Stiefel-Whitney classes $w_i(\p)$ and $w_i(\q)$ are $\epsilon$-close in the following sense: if we denote $\tdeatho(\p) = \inf \{t \in T, w_i^t(\p) \neq 0\}$ and $\tdeatho(\q) = \inf \{t \in T, w_i^t(\q) \neq 0\}$, then $|\tdeatho(\p) - \tdeatho(\q)| \leq \epsilon$.
{width=".5\linewidth"} {width=".5\linewidth"}
Let us apply this result to the Čech bundle filtrations. Let $X$ and $Y$ be two subsets of $E = \R^n \times \matrixspace{\R^m}$. Suppose that the Hausdorff distance $\Hdist{X}{Y}$, with respect to the norm $\gammaN{\cdot}$, is not greater than $\epsilon$, meaning that the $\epsilon$-thickenings $X^\epsilon$ and $Y^\epsilon$ satisfiy $Y \subseteq X^\epsilon$ and $X \subseteq Y^\epsilon$. It is then clear that the vector bundle filtrations are $\epsilon$-interleaved, and we can apply Theorem \[thm:stability\] to obtain the following result.
\[cor:stability\] If two subsets $X, Y \subset E$ satisfy $\Hdist{X}{Y} \leq \epsilon$, then there exists an $\epsilon$-interleaving between the persistent cohomology modules of their corresponding Čech bundle filtrations which sends persistent Stiefel-Whitney classes on persistent Stiefel-Whitney classes.
In order to illustrate Corollary \[cor:stability\], consider the sets $X'$ and $Y'$ represented in Figure \[fig:stability\_X\_Y\_prime\]. They are noisy samples of the sets $X$ and $Y$ defined in Example \[ex:normal\_mobius\]. They contain 50 points each.
![Representation of the sets $X', Y' \subset \R^2 \times \matrixspace{\R^2}$.[]{data-label="fig:stability_X_Y_prime"}](bundle_normal_X_prime.png){width=".6\linewidth"}
![Representation of the sets $X', Y' \subset \R^2 \times \matrixspace{\R^2}$.[]{data-label="fig:stability_X_Y_prime"}](bundle_mobius_X_prime.png){width=".6\linewidth"}
We choose the parameter $\gamma=1$. Figure \[fig:stability\_X\_Y\_prime\_barcode\] represents the barcodes of the Čech filtrations of the sets $X'$ and $Y'$, together with the lifebar of the first persistent Stiefel-Whitney class of their corresponding Čech bundle filtrations. Observe that they are close to the original descriptors of $X$ and $Y$ (Figures \[fig:example\_bundles\_barcode\_0\] and \[fig:example\_bundles\_lifespan\]).
Experimentally, we computed that the Hausdorff distances between $X,X'$ and $Y,Y'$ are approximately $\Hdist{X}{X'} \approx 0{,}43$ and $\Hdist{Y}{Y'} \approx 0{,}39$. This is coherent with the lifebar of $w_1(Y')$, which is $\epsilon$-close to the lifebar of $w_1(Y)$ with $\epsilon \approx 0{,}3 \leq 0{,}39$.
![Left: $H^0$ and $H^1$ barcodes of $X'$ and lifebar of $w_1(X')$. Right: same for $Y'$.[]{data-label="fig:stability_X_Y_prime_barcode"}](bundle_normal_prime_barcode_H0.png "fig:"){width=".9\linewidth"} ![Left: $H^0$ and $H^1$ barcodes of $X'$ and lifebar of $w_1(X')$. Right: same for $Y'$.[]{data-label="fig:stability_X_Y_prime_barcode"}](bundle_normal_prime_barcode_H1.png "fig:"){width=".9\linewidth"} ![Left: $H^0$ and $H^1$ barcodes of $X'$ and lifebar of $w_1(X')$. Right: same for $Y'$.[]{data-label="fig:stability_X_Y_prime_barcode"}](bundle_normal_prime_lifespan.png "fig:"){width=".9\linewidth"}
![Left: $H^0$ and $H^1$ barcodes of $X'$ and lifebar of $w_1(X')$. Right: same for $Y'$.[]{data-label="fig:stability_X_Y_prime_barcode"}](bundle_mobius_prime_barcode_H0.png "fig:"){width=".9\linewidth"} ![Left: $H^0$ and $H^1$ barcodes of $X'$ and lifebar of $w_1(X')$. Right: same for $Y'$.[]{data-label="fig:stability_X_Y_prime_barcode"}](bundle_mobius_prime_barcode_H1.png "fig:"){width=".9\linewidth"} ![Left: $H^0$ and $H^1$ barcodes of $X'$ and lifebar of $w_1(X')$. Right: same for $Y'$.[]{data-label="fig:stability_X_Y_prime_barcode"}](bundle_mobius_prime_lifespan.png "fig:"){width=".9\linewidth"}
Consistency
-----------
In this subsection we describe a setting where the persistent Stiefel-Whitney classes $w_i(X)$ of the Čech bundle filtration of a set $X$ can be seen as consistent estimators of the Stiefel-Whitney classes of some underlying vector bundle.
Let $\MMo$ be a compact $\CC^2
$-manifold, and $u_0 \colon \MMo \rightarrow \R^n$ an immersion. Suppose that $\MMo$ is given a $d$-dimensional vector bundle structure $p \colon \MMo \rightarrow \Grass{d}{\R^m}$. Let $E = \R^n \times \matrixspace{\R^m}$, and consider the set $$\label{eq:lifted_set}
\MM = \left\{ \left(\immo(x_0), \projmatrix{p(x_0)}\right), x_0 \in \MMo \right\} \subset E,$$ where $\projmatrix{p(x_0)}$ denotes the orthogonal projection matrix onto the subspace $p(x_0) \subset \R^m$. The set $\MM$ is called the *lift* of $\MMo$. Consider the *lifting map* defined as $$\begin{aligned}
\label{eq:lifting_map}
\begin{split}
\imm\colon \MMo & \longrightarrow \MM \subset E \\
x_0 & \longmapsto \left(\immo(x_0), \projmatrix{p(x_0)}\right).
\end{split}\end{aligned}$$ We make the following assumption: $\imm$ is an embedding. As a consequence, $\MM$ is a submanifold of $E$, and $\MMo$ and $\MM$ are diffeomorphic. An extensive study of this setting can be found in [@tinarrage2019recovering].
The persistent cohomology of $\MM$ can be used to recover the cohomology of $\MMo$. To see this, select $\gamma>0$, and denote by $\reach{\MM}$ the reach of $\MM$, where $E$ is endowed with the norm $\gammaN{\cdot}$. Note that $\reach{\MM}$ is positive and depends on $\gamma$. Let $\M = (\MM^t)_{t\geq 0}$ be the Čech set filtration of $\MM$ in the ambient space $(E, \gammaN{\cdot})$, and let $\V(\MM)$ be the corresponding persistent cohomology module. For every $s, t \in [0, \reach{\MM})$ such that $s \leq t$, we know that the inclusion maps $i_s^t\colon \MM^s \hookrightarrow \MM^t$ are homotopy equivalences (see Subsection \[subsec:background\_persistentcohomology\]). Hence the persistence module $\V(\MM)$ is constant on the interval $[0, \reach{\MM})$, and is equal to the cohomology $H^*(\MM) = H^*(\MMo)$.
Consider the Čech bundle filtration $(\M, \p)$ of $\MM$. The following theorem shows that the persistent Stiefel-Whitney classes $w_i^t(\MM)$ are also equal to the usual Stiefel-Whitney classes of the vector bundle $(\MMo, p)$.
\[thm:consistency\] Let $\MMo$ be a compact $\CC^2$-manifold, $u_0 \colon \MMo \rightarrow \R^n$ an immersion and $p \colon \MMo \rightarrow \Grass{d}{\R^m}$ a continuous map. Let $\MM$ be the lift of $\MMo$ (Equation \[eq:lifted\_set\]) and $u$ the lifting map (Equation \[eq:lifting\_map\]). Suppose that $u$ is an embedding.
Let $\gamma > 0$ and consider the Čech bundle filtration $(\M, \p)$ of $\MM$. Its maximal filtration value is $\tmaxgamma{\MM} = \frac{\sqrt{2}}{2} \gamma$. Denote by $\pSF{i}{\mathbbm{p}} = (\pSFt{i}{\mathbbm{p}}{t})_{t \in T}$ its persistent Stiefel-Whitney classes, $i \in [1,d]$. Denote also by $i_0^t$ the inclusion $\MM \rightarrow \MM^t$, for $t \in [0, \reach{\MM})$.
Let $t\geq0$ be such that $t < \min\left(\reach{\MM}, \tmaxgamma{\MM}\right)$. Then the map $i_0^t \circ \imm\colon \MMo \rightarrow \MM^t$ induces an isomorphism $H^*(\MMo) \leftarrow H^*(\MM^t)$ which maps the $i$th persistent Stiefel-Whitney class $\pSFt{i}{\p}{t}$ of $(\M, \p)$ to the $i$th Stiefel-Whitney class of $(\MMo, p)$.
Consider the following commutative diagram, defined for every $t < \tmaxgamma{\MM}$: $$\begin{tikzcd}
\MMo \arrow[r, "u"] \arrow[dr, "p", swap] & \MM \arrow[r, hook, "i_0^t"] & \MM^t \arrow[dl, "p^t" ] \\
& \Grass{d}{\R^m} & &
\end{tikzcd}$$ We obtain a commutative diagram in cohomology: $$\begin{tikzcd}
H^*(\MMo) & H^*(\MM) \arrow[l, "u^*", swap] & H^*(\MM^t) \arrow[l, "(i_0^t)^*", swap] \\ & H^*(\Grass{d}{\R^m}) \arrow[ur, "(p^t)^*", swap] \arrow[ul, "p^*"] & &
\end{tikzcd}$$ Since $t < \reach{\MM}$, the map $(i_0^t)^*$ is an isomorphism (see Subsection \[subsec:background\_persistentcohomology\]). So is $u^*$ since $u$ is an embedding. As a consequence, the map $i_0^t\circ u$ induces an isomorphism $H^*(\MMo) \simeq H^*(\MM^t)$.
Let $w_i$ denotes the $i$th Stiefel-Whitney class of $\Grass{d}{\R^m}$. By definition, the $i$th Stiefel-Whitney class of $(\MMo, p)$ is $p^*(w_i)$, and the $i$th persistent Stiefel-Whitney class of $(\M, \p)$ is $\pSFt{i}{\p}{t} = (p^t)^*(w_i)$. By commutativity of the diagram, we obtain $p^*(w_i) = (p^t)^*(w_i)$, under the identification $H^*(\MMo) \simeq H^*(\MM^t)$.
Applying Theorems \[thm:stability\], \[thm:consistency\] and the considerations of Subsection \[subsec:background\_persistentcohomology\] yield an estimation result.
\[cor:consistency\_stability\] Let $X \subset E$ be any subset such that $\Hdist{X}{\MM} \leq \epsilon$. Then for every $t\in [4\epsilon, \reach{\MM}-3\epsilon)$, the composition of inclusions $\MMo \hookrightarrow \MM \hookrightarrow X^t$ induces an isomorphism $H^*(\MMo) \leftarrow H^*(X^t)$ which sends the $i$th persistent Stiefel-Whitney class $\pSFt{i}{X}{t}$ of the Čech bundle filtration of $X$ to the $i$th Stiefel-Whitney class of $(\MMo, p)$.
As a consequence of this corollary, on the set $[4\epsilon, \reach{\MM}-3\epsilon)$, the $i$th persistent Stiefel-Whitney class of the Čech bundle filtration of $X$ is zero if and only if the $i$th Stiefel-Whitney class of $(\MMo, p)$ is.
In order to illustrate Corollary \[cor:consistency\_stability\], consider the torus and the Klein bottle, immersed in $\R^3$ as in Figure \[fig:torus\_and\_klein\].
![Immersion of the torus and the Klein bottle in $\R^3$.[]{data-label="fig:torus_and_klein"}](intro_torus.png){width=".5\linewidth"}
![Immersion of the torus and the Klein bottle in $\R^3$.[]{data-label="fig:torus_and_klein"}](intro_klein.png){width=".5\linewidth"}
Let them be endowed with their normal bundles. They can be seen as submanifolds $\MM, \MM'$ of $\R^3 \times \matrixspace{\R^3}$. We consider two samples $X, X'$ of $\MM, \MM'$, represented in Figure \[fig:torus\_and\_klein\_sample\]. They contain respectively 346 and 1489 points. We computed experimentally the Hausdorff distances $\Hdist{X}{\MM} \approx 0{,}6$ and $\Hdist{X'}{\MM'} \approx 0{,}45$, with respect to the norm $\gammaN{\cdot}$ where $\gamma=1$.
![Samples $X$ and $X'$ of $\MM$ and $\MM'$. The black points corresponds to the $\R^3$-coordinate, and the pink arrows over them correspond to the orientation of the $\matrixspace{\R^3}$-coordinate.[]{data-label="fig:torus_and_klein_sample"}](bundle_torus.png){width=".6\linewidth"}
![Samples $X$ and $X'$ of $\MM$ and $\MM'$. The black points corresponds to the $\R^3$-coordinate, and the pink arrows over them correspond to the orientation of the $\matrixspace{\R^3}$-coordinate.[]{data-label="fig:torus_and_klein_sample"}](bundle_klein.png){width=".6\linewidth"}
Figure \[fig:torus\_klein\_barcodes\] represents the barcodes of the persistent cohomology of $X$ and $X'$, and the lifebars of their first persistent Stiefel-Whitney classes $w_1(X)$ and $w_1(X')$. Observe that $w_1(X)$ is always zero, while $w_1(X')$ is nonzero for $t\geq 0{,}3$. This is an indication that $\MM$, the underlying manifold of $X$, is orientable, while $\MM'$ is not. To see this, recall Proposition \[prop:SF\_orientability\]: the first Stiefel-Whitney class of the tangent bundle of a manifold is zero if and only if the manifold is orientable. One can deduce the following fact: the first Stiefel-Whitney class of the normal bundle of an immersed manifold is zero if and only if the manifold is orientable (see the following lemma). Therefore, one interprets these lifebars as follows: $X$ is sampled on an orientable manifold, while $X'$ is sampled on a non-orientable one.
![Left: $H^0$, $H^1$ and $H^2$ barcodes of $X$ and lifebar of $w_1(X)$. Right: same for $X'$.[]{data-label="fig:torus_klein_barcodes"}](bundle_torus_barcode_H0.png "fig:"){width=".9\linewidth"} ![Left: $H^0$, $H^1$ and $H^2$ barcodes of $X$ and lifebar of $w_1(X)$. Right: same for $X'$.[]{data-label="fig:torus_klein_barcodes"}](bundle_torus_barcode_H1.png "fig:"){width=".9\linewidth"} ![Left: $H^0$, $H^1$ and $H^2$ barcodes of $X$ and lifebar of $w_1(X)$. Right: same for $X'$.[]{data-label="fig:torus_klein_barcodes"}](bundle_torus_barcode_H2.png "fig:"){width=".9\linewidth"} ![Left: $H^0$, $H^1$ and $H^2$ barcodes of $X$ and lifebar of $w_1(X)$. Right: same for $X'$.[]{data-label="fig:torus_klein_barcodes"}](bundle_torus_barcode_lifespan.png "fig:"){width=".9\linewidth"}
![Left: $H^0$, $H^1$ and $H^2$ barcodes of $X$ and lifebar of $w_1(X)$. Right: same for $X'$.[]{data-label="fig:torus_klein_barcodes"}](bundle_klein_barcode_H0.png "fig:"){width=".9\linewidth"} ![Left: $H^0$, $H^1$ and $H^2$ barcodes of $X$ and lifebar of $w_1(X)$. Right: same for $X'$.[]{data-label="fig:torus_klein_barcodes"}](bundle_klein_barcode_H1.png "fig:"){width=".9\linewidth"} ![Left: $H^0$, $H^1$ and $H^2$ barcodes of $X$ and lifebar of $w_1(X)$. Right: same for $X'$.[]{data-label="fig:torus_klein_barcodes"}](bundle_klein_barcode_H2.png "fig:"){width=".9\linewidth"} ![Left: $H^0$, $H^1$ and $H^2$ barcodes of $X$ and lifebar of $w_1(X)$. Right: same for $X'$.[]{data-label="fig:torus_klein_barcodes"}](bundle_klein_barcode_lifespan.png "fig:"){width=".9\linewidth"}
Let $\MMo \rightarrow \MM$ be an immersion of a manifold $\MMo$ in a Euclidean space. Then $\MMo$ is orientable if and only if the first Stiefel-Whitney class of its normal bundle is zero.
Let $\tau$ and $\nu$ denote the tangent and normal bundles of $\MMo$. The Whitney sum $\tau \oplus \nu$ is a trivial bundle, hence its first Stiefel-Whitney class is $w_1(\tau \oplus \nu) = 0$. Using Axioms 1 and 3 of the Stiefel-Whitney classes, we obtain $$\begin{aligned}
w_1(\tau \oplus \nu)
&= w_1(\tau)\cupp w_0(\nu) + w_0(\tau) \cupp w_1(\nu) \\
&= w_1(\tau) \cupp 1 + 1 \cupp w_1(\nu) \\
&= w_1(\tau) + w_1(\nu).\end{aligned}$$ Therefore, $w_1(\tau) = w_1(\nu)$. To conclude, $w_1(\tau)$ is zero if and only if $w_1(\nu)$ is zero, and Proposition \[prop:SF\_orientability\] yields the result.
Simplicial approximation of Čech bundle filtrations
===================================================
In order to build an effective algorithm to compute the persistent Stiefel-Whitney classes, we have to find an equivalent formulation in terms of simplicial cohomology. We start by reviewing the usual technique of simplicial approximation, and then apply it to the particular case of Čech bundle filtrations.
Background on simplicial complexes {#subsec:background_simplicialcomplexes}
----------------------------------
To start, we recall some elements of combinatorics and topology of simplicial complexes.
#### (Combinatorial) simplicial complexes.
Let $K$ be a simplicial complex. It means that there exists a set $V$, the set of *vertices*, such that $K \subseteq \mathcal{P}(V)$, and $K$ satisfies the following condition: for every $\sigma \in K$ and every subset $\nu \subseteq \sigma$, $\nu$ is in $K$. The elements of $K$ are called *faces* or *simplices* of the simplicial complex $K$.
For every simplex $\sigma \in K$, we define its dimension $\dim(\sigma) = \card{\sigma} -1$. The dimension of $K$, denoted $\dim (K)$, is the maximal dimension of its simplices. For every $i\geq 0$, the $i$-skeleton $\skeleton{K}{i}$ is defined as the subset of $K$ consisting of simplices of dimension at most $i$. Note that $\skeleton{K}{0}$ corresponds to the underlying vertex set $V$, and $\skeleton{K}{1}$ is a graph.
Given a simplex $\sigma \in K$, its (open) star $\Star{\sigma}$ is the set of all the simplices $\nu \in K$ that contain $\sigma$. The open star is not a simplicial complex in general. We also define its closed star $\closedStar{\sigma}$ as the smallest simplicial subcomplex of $K$ which contains $\Star{\sigma}$.
{width=".6\linewidth"} $K$
{width=".6\linewidth"} $\Star{v}$ in red and pink
{width=".6\linewidth"} $\closedStar{v}$ in red and pink
Given a graph $G$, the corresponding clique complex is the simplicial complex whose simplices are the sets of vertices of the cliques of $G$. We say that a simplicial complex $K$ is a flag complex if it is the clique complex of its 1-skeleton $\skeleton{K}{1}$.
#### Topological realizations.
For every $p\geq 0$, the standard $p$-simplex $\Delta^p$ is a topological space defined as the convex hull of the canonical basis vectors $e_1, ..., e_{p+1}$ of $\R^{p+1}$, endowed with the subspace topology. We now describe the construction of the *topological realization* of the simplicial complex $K$, denoted $\topreal{K}$. It is a particular case of the construction CW-complexes [@Hatcher_Algebraic Appendix].
1. Start with the discrete topological space $\topreal{\skeleton{K}{0}}$ consisting of the vertices of $K$.
2. Inductively, form the $p$-skeleton $\topreal{\skeleton{K}{p}}$ from $\topreal{\skeleton{K}{p-1}}$ by attaching $p$-dimensional simplices to $\topreal{\skeleton{K}{p-1}}$. More precisely, for each $\sigma \in K$ of dimension $p$, take a copy of the standard $p$-simplex $\Delta^p$. Denote this simplex by $\Delta \sigma$. Label its vertices with the elements of $\sigma$. Whenever $\tau \subset \sigma \in K$, identify $\Delta \tau$ with a subset of $\Delta \sigma$, via the face inclusion which sends the elements of $\tau$ to the corresponding elements of $\sigma$. Give $\topreal{\skeleton{K}{p}}$ the quotient topology.
3. \[enum:topological\_realization\] Endow $\topreal{K} = \bigcup_{p \geq 0} \topreal{\skeleton{K}{p}}$ with the weak topology: a set $A \subset \topreal{K}$ is open if and only if $A \cap \topreal{\skeleton{K}{p}}$ is open in $\topreal{\skeleton{K}{p}}$ for each $p\geq0$.
Alternatively, the topology on $\topreal{K}$ can be described as follows: a subset $A \subset \topreal{K}$ is open (or closed) if and only if for every $\sigma \in K$, the set $A \cap \Delta\sigma$ is open (or closed) in $\Delta\sigma$. Note that condition \[enum:topological\_realization\] is superfluous when $K$ is finite dimensional.
If $\sigma = [v]$ is a vertex of $K$, we will denote by $\topreal{\sigma}$ the singleton $\{v\}$, seen as a subset of $\topreal{K}$. If $\sigma$ is a face of $K$ of dimension at least 1, we will denote by $\topreal{\sigma}$ the open subset of $\topreal{K}$ which corresponds to the interior of the face $\Delta \sigma \subset \topreal{K}$. We denote by $\overline{\topreal{\sigma}}$ its closure in $\topreal{K}$. Observe that if $\overline{\sigma}$ denotes the smallest simplicial subcomplex of $K$ that contains $\sigma$, then $\topreal{\overline{\sigma}} = \Delta \sigma = \overline{\topreal{\sigma}}$. The following set is a partition of $\topreal{K}$: $$\left\{ \topreal{\sigma}, \sigma \in K \right\}.$$ This allows to define the *face map* of $K$. It is the unique map $\facemapK{K} \colon \topreal{K} \rightarrow K$ that satisfies $x \in \topreal{\facemapK{K}(x)}$ for every $x \in \topreal{K}$.
If $L$ is a subset of $K$, we define its topological realization as $\topreal{L} = \bigcup_{\sigma\in L} \topreal{\sigma}$. For every simplex $\sigma \in K$, the topological realization of its open star, $\topreal{\Star{\sigma}}$, is open in $\topreal{K}$. Besides, the topological realization of its closed star, $\topreal{\closedStar{\sigma}}$, is equal to $\overline{\topreal{\Star{\sigma}}}$, hence is closed with respect to the weak topology.
If $\sigma$ is a face of $K$ of dimension at least 1, the subset $\topreal{\sigma}$ of $\topreal{K}$ is canonically homeomorphic to the interior of the standard $p$-simplex $\Delta^p$, where $p = \dim(\sigma)$. This allows to define on $\topreal{K}$ the barycentric coordinates: for every face $\sigma = [v_0, ..., v_p] \in K$, the points $x \in \topreal{\sigma}$ can be written as $$x = \sum_{i=0}^p \lambda_i v_i$$ with $\lambda_0, ..., \lambda_p > 0$ and $\sum_{i=0}^p \lambda_i = 1$.
#### Triangulation and geometric realizations.
Let $X$ be a subset of $E$. A triangulation of $X$ consists of a simplicial complex $K$ together with a homeomorphism $h\colon X \rightarrow \topreal{K}$. The set $X$ is called a *geometric realization* of $K$, and $K$ is called a *triangulation* of $X$.
Simplicial approximation {#subsec:simplicial_approximation}
------------------------
This subsection is based on [@Hatcher_Algebraic Section 2.C]. In the following, $K$ and $L$ are two simplicial complexes. We recall the reader that $\topreal{K}$ denotes the topological realization of $K$, and $\Star{v}, \closedStar{v}$ denote the open and closed star of a vertex $v \in \skeleton{K}{0}$.
#### Simplicial maps.
A *simplicial map* between simplicial complexes $K$ and $L$ is a map $g \colon \topreal{K} \rightarrow \topreal{L}$ which sends vertices on vertices and is linear on every simplices. In other words, for every $\sigma = [v_0, ..., v_p] \in K$, the map $g$ restricted to $\topreal{\sigma} \subset \topreal{K}$ can be written in barycentric coordinates as $$\label{eq:simplicial_map_1}
\sum_{i=0}^p \lambda_i v_i ~ \longmapsto ~ \sum_{i=0}^p \lambda_i g(v_i).$$ A simplicial map $g \colon \topreal{K} \rightarrow \topreal{L}$ is uniquely determined by its restriction to the vertex sets $g_{| \skeleton{K}{0}} \colon \skeleton{K}{0} \rightarrow \skeleton{L}{0}$. Reciprocally, let $f \colon \skeleton{K}{0} \rightarrow \skeleton{L}{0}$ be a map between vertex sets which satisfies the following condition: $$\label{eq:simplicial_map_2}
\forall \sigma \in K, f(\sigma) \in L.$$ Then $f$ induces a simplicial map via barycentric coordinates, denoted $|f| \colon \topreal{K} \rightarrow \topreal{L}$. In the rest of the paper, a simplicial map will either refer to a map $g \colon \topreal{K} \rightarrow \topreal{L}$ which satisfies Equation \[eq:simplicial\_map\_1\], to a map $f\colon \skeleton{K}{0} \rightarrow \skeleton{L}{0}$ which satisfies Equation \[eq:simplicial\_map\_2\], or to the induced map $f \colon K \rightarrow L$.
#### Simplicial approximation.
Let $g \colon \topreal{K} \rightarrow \topreal{L}$ be any continuous map. The problem of the simplicial approximation consists in finding a simplicial map $f \colon K \rightarrow L$ with topological realization $\topreal{f} \colon \topreal{K} \rightarrow \topreal{L}$ homotopic to $g$. A way to solve this problem is to consider the following property: we say that the map $g$ satisfies the *star condition* if for every vertex $v$ of $K$, there exists a vertex $w$ of $L$ such that $$\begin{aligned}
g\left(\topreal{ \closedStar{v} }\right) \subseteq \topreal{ \Star{w} }.\end{aligned}$$ If this is the case, let $f \colon \skeleton{K}{0} \rightarrow \skeleton{L}{0}$ be any map between vertex sets such that for every vertex $v$ of $K$, we have $g\left(\topreal{ \closedStar{v}} \right) \subseteq \topreal{ \Star{f(v)} }$. Equivalently, $f$ satisfies $$g\left(\closedStar{v} \right) \subseteq \Star{f(v)}.$$ Such a map is called a *simplicial approximation to $g$*. One shows that it is a simplicial map, and that its topological realization $\topreal{f}$ is homotopic to $g$ [@Hatcher_Algebraic Theorem 2C.1].
![The map $f$ (in red) is a simplicial approximation to $g$.[]{data-label="fig:simplicial_approximation"}](simplicial_approx_K.png "fig:"){width=".9\linewidth"} $K$
![The map $f$ (in red) is a simplicial approximation to $g$.[]{data-label="fig:simplicial_approximation"}](simplicial_approx_L.png "fig:"){width=".9\linewidth"} $L$
![The map $f$ (in red) is a simplicial approximation to $g$.[]{data-label="fig:simplicial_approximation"}](simplicial_approx_g.png "fig:"){width=".9\linewidth"} $g$
![The map $f$ (in red) is a simplicial approximation to $g$.[]{data-label="fig:simplicial_approximation"}](simplicial_approx_f.png "fig:"){width=".9\linewidth"} $f$
In general, a map $g$ may not satisfy the star condition. However, there is always a way to subdivise the simplicial complex $K$ in order to obtain an induced map which does (see Theorem \[th:starcondition\]). We describe this construction in the following paragraph.
#### Barycentric subdivisions.
Let us describe briefly the process of barycentric subdivision of a simplicial complex. A more extensive description can be found in [@Hatcher_Algebraic Proof of Proposition 2.21]. Let $\Delta^p$ denote the standard $p$-simplex, with vertices denoted $v_0, ..., v_p$. The barycentric subdivision of $\Delta^p$ consists in decomposing $\Delta^p$ into $(p+1)!$ simplices of dimension $p$. It is a simplicial complex, whose vertex set corresponds to the points $\sum_{i=0}^p \lambda_i v_i$ for which some $\lambda_i$ are zero and the other ones are equal. Equivalently, one can see these this new set of vertices as a the power set of the set of vertices of $\Delta^p$.
More generally, if $K$ is a simplicial complex, its barycentric subdivision $\text{sub}(K)$ is the simplicial complex obtained by subdivising each of its faces. The set of vertices of $\text{sub}(K)$ can be seen as a subset of the power set of the set of vertices of $K$.
{width=".8\linewidth"}
By subdivising $K$, we shrink its faces. More precisely, if $h\colon X \rightarrow \topreal{K}$ is a geometric realization of $K$, with $X \subset \R^n$, and if $D$ is the diameter of a face $\sigma \in K$ seen in $X$, then the faces of the barycentric subdivision of $\sigma$ are of diameter at most $\frac{\dim(\sigma)}{\dim(\sigma) +1} D$. Therefore one can repeat the subdivision to obtain arbitrarly small faces. Applying barycentric subdivisions $n$ times will be denoted $\subdiv{K}{n}$.
Consider two simplicial complexes $K, L$ with $K$ finite, and let $g \colon \topreal{K} \rightarrow \topreal{L}$ be a continuous map. Then there exists $n \geq 0$ such that $g \colon \topreal{\subdiv{K}{n}} \rightarrow \topreal{L}$ satisfies the star condition. \[th:starcondition\]
As a consequence, such a map $g \colon \topreal{\subdiv{K}{n}} \rightarrow \topreal{L}$ admits a simplicial approximation. This is known as the simplical approximation theorem. As an illustration, Figure \[fig:barycentric\_subdiv\_approx\] represents a map $g \colon \topreal{K} \rightarrow \topreal{L}$ which does not satisfies the star condition, but whose first barycentric subdivision does.
![The map $g\colon \topreal{K} \rightarrow \topreal{L}$ does not satisfy the star condition, but its first barycentric subdivision does (see Figure \[fig:simplicial\_approximation\]).[]{data-label="fig:barycentric_subdiv_approx"}](simplicial_approx_counterexample.png){width=".3\linewidth"}
Application to Čech bundle filtrations {#subsec:simplicial_approx_cech}
--------------------------------------
In this subsection, we apply the principle of simplicial approximation to the particular case of persistent Stiefel-Whitney classes of Čech bundle filtrations.
Let $X$ be a subset of $E = \R^n \times \matrixspace{\R^m}$. Let us recall Definition \[def:filtered\_cech\_bundle\]: the Čech bundle filtration associated to $X$ is the vector bundle filtration $(\X, \p)$ whose underlying filtration is the Čech filtration $\X = (X^t)_{t \in T}$, with $T = [0, \tmaxgamma{X})$, and whose maps $\p = (p^t)_{t \in T}$ are given by the following composition, as in Equation \[eq:def\_cech\_bundle\_proj\]: $$\begin{tikzcd}[baseline=(current bounding box.center), column sep = 6em]
X^t \arrow[r, "\mathrm{proj}_2"] \arrow[rr, bend right = 20, "p^t", swap]
& \matrixspace{\R^m} \setminus \med{\Grass{d}{\R^m}} \arrow[r, "\proj{\cdot}{\Grass{d}{\R^m}}"]
& \Grass{d}{\R^m}.
\end{tikzcd}$$ Let $t \in T$. The aim of this subsection is to describe a simplicial approximation to $p^t \colon X^t \rightarrow \Grass{d}{\R^m}$. To do so, let us fix a triangulation $L$ of $\Grass{d}{\R^m}$. It comes with a homeomorphism $h \colon \Grass{d}{\R^m} \rightarrow \topreal{L}$. We will now triangulate the Čech set filtration $X^t$, as described in Subsection \[subsec:background\_persistentcohomology\]. The thickening $X^t$ is a subset of the metric space $(E, \gammaN{\cdot})$ which consists in a union of closed balls centered around points of $X$: $$\begin{aligned}
X^t = \bigcup_{x \in X} \closedballM{x}{t}{\gamma},\end{aligned}$$ where $\closedballM{x}{t}{\gamma}$ denotes the closed ball of center $x$ and radius $t$ for the norm $\gammaN{\cdot}$. Let $\UU^t$ denote the cover $\{ \closedballM{x}{t}{\gamma}, x \in X\}$ of $X^t$, and let $\NN(\UU^t)$ be its nerve. By the nerve theorem for convex closed covers [@boissonnat2018geometric Theorem 2.9], the simplicial complex $\NN(\UU^t)$ is homotopy equivalent to its underlying set $X^t$. That is to say, there exists a continuous map $g^t \colon \topreal{\NN(\UU^t)} \rightarrow X^t$ which is a homotopy equivalence.
As a consequence, in cohomological terms, the map $p^t \colon X^t \rightarrow \Grass{d}{E}$ is equivalent to the map $q^t$ defined as $q^t = h \circ p^t \circ g^t$. $$\begin{tikzcd}
X^t \arrow[r, "p^t"]
&[1em] \Grass{d}{\R^m} \arrow[d, "h"] \\
\topreal{\NN(\UU^t)} \arrow[u, "g^t"] \arrow[r, "q^t", dashed]
& \topreal{L}
\end{tikzcd}
\label{eq:map_approx_cech}$$ This gives a way to compute the induced map $(p^t)^* \colon \coring{X^t} \leftarrow \coring{\Grass{d}{\R^m}}$ algorithmically:
- Subdivise $\NN(\UU^t)$ until $q^t$ satisfies the star condition (as in Theorem \[th:starcondition\]),
- Choose a simplicial approximation $f^t$ to $q^t$,
- Compute the induced map between simplicial cohomology groups $(f^t)^* \colon H^*(\NN(\UU^t)) \leftarrow H^*(L)$.
By correspondance between simplicial and singular cohomology, the map $(f^t)^*$ corresponds to $(p^t)^*$. Hence the problem of computing $(p^t)^*$ is solved, if it were not for the following issue: in practice, the map $g^t\colon \topreal{\NN(\UU^t)} \rightarrow X^t$ given by the nerve theorem is not explicit. The rest of this subsection is devoted to showing that $g^t$ can be chosen canonically as the *shadow map*.
#### Shadow map.
We still consider the thickening $X^t$, the corresponding cover $\UU^t$ and its nerve $\NN(\UU^t)$. The underlying vertex set of the simplicial complex $\NN(\UU^t)$ is the set $X$ itself. The shadow map $g^t \colon \topreal{ \NN(\UU^t) } \rightarrow X^t$ is defined as follows: for every simplex $\sigma = [x_0, ..., x_p] \in \NN(\UU^t)$ and every point $\sum_{i=0}^p \lambda_i x_i$ of $\topreal{\sigma}$ written in barycentric coordinates, associate the point $\sum_{i=0}^p \lambda_i x_i$ of $E$: $$\begin{aligned}
g^t \colon \sum_{i=0}^p \lambda_i x_i \in \topreal{\sigma} \longmapsto \sum_{i=0}^p \lambda_i x_i \in E.\end{aligned}$$ The author is not aware if the shadow map is indeed a homotopy equivalence from $|\NN(\UU^t)|$ to $X^t$. Nevertheless, the following result will be enough for our purposes: the shadow map induces an isomorphism at cohomology level.
Suppose that $X$ is finite and in general position. Then the shadow map $g^t \colon |\NN(\UU^t)| \rightarrow X^t$ induces an isomorphism $(g^t)^* \colon H^*(|\NN(\UU^t)|) \leftarrow H^*(X^t)$.
Recall that $\UU^t = \left\{ \closedballM{x}{t}{\gamma}, x \in X \right \}$. Let us consider a smaller cover. For every $x \in X$, let $\vor{x}$ denote the Voronoi cell of $x$ in the ambient metric space $(E, \gammaN{\cdot})$, and define $$\begin{aligned}
\VV^t = \left\{\closedball{x}{t} \cap \vor{x} , x \in X\right \}.\end{aligned}$$ The set $\VV^t$ is a cover of $X^t$, and its nerve $\NN(\VV^t)$ is known as the Delaunay complex (see [@bauer2017morse]). Let $h^t \colon \topreal{\NN(\VV^t)} \rightarrow X^t$ denote the shadow map of $\NN(\VV^t)$. The Delaunay complex is a subcomplex of the Čech complex, hence we can consider the following diagram:
|(\^t)| &|(\^t)| &X\^t .
This yields the following commutative diagram between cohomology rings:
H\^\*(|(\^t)|) & H\^\*(|(\^t)|) & H\^\*(X\^t) .
Now, it is proven in [@edelsbrunner1993union Theorem 3.2] that the shadow map $h^t \colon |\NN(\VV^t)| \rightarrow X^t$ is a homotopy equivalence (it is required here that $X$ is in general position). Therefore the map $(h^t)^* \colon H^*(|\NN(\VV^t)|) \leftarrow H^*(X^t)$ is an isomorphism. Moreover, we know from [@bauer2017morse Theorem 5.10] that $\NN(\UU^t)$ collapses to $\NN(\VV^t)$. Therefore the inclusion $\topreal{\NN(\VV^t)} \hookrightarrow \topreal{\NN(\UU^t)}$ also is a homotopy equivalence, hence the induced map $H^*(|\NN(\VV^t)|) \leftarrow H^*(|\NN(\UU^t)|)$ is an isomorphism. We conclude from the last diagram that $(g^t)^*$ is an isomorphism.
A sketch of algorithm
---------------------
Suppose that we are given a finite set $X \subset E = \R^n \times \matrixspace{\R^m}$. Choose $d \in [1, n-1]$ and $\gamma > 0$. Consider the Čech bundle filtration of dimension $d$ of $X$. Let $T = \left[0, \tmaxgamma{X}\right)$, $t \in T$ and $i \in [1,d]$. From the previous discussion we can infer an algorithm to solve the following problem:
Denote:
- $\X = (X^t)_{t\geq0}$ the Čech set filtration of $X$,
- $\S$ the Čech simplicial filtration of $X$, and $g^t \colon \topreal{S^t} \rightarrow X^t$ the shadow map,
- $L$ a triangulation of $\Grass{d}{\R^n}$ and $h \colon \Grass{d}{\R^n} \rightarrow \topreal{L}$ a homeomorphism,
- $(\X, \p)$ the Čech bundle filtration of $X$,
- $(\V, \vbb)$ the persistent cohomology module of $\X$,
- $w_i \in H^i(\Grass{d}{\R^n})$ the $i$th Stiefel-Whitney class of the Grassmannian.
Let $t\in T$ and consider the map $q^t$, as defined in Equation \[eq:map\_approx\_cech\]: $$\begin{tikzcd}[baseline=(current bounding box.center), column sep = 4em]
\topreal{S^t} \arrow[r, "g^t", swap]
\arrow[rrr, bend left = 18, "q^t"]
& X^t \arrow[r, "p^t", swap]
&\Grass{d}{\R^m} \arrow[r, "h", swap]
& \topreal{L}.
\end{tikzcd}$$ We propose the following algorithm:
- Subdivise barycentrically $S^t$ until $q^t$ satisfies the star condition. Denote $k$ the number of subdivisions needed.
- Consider a simplicial approximation $f^t \colon \subdiv{S^t}{k} \rightarrow L$ to $q^t$.
- Compute the class $(f^t)^*(w_i)$.
The output $(f^t)^*(w_i)$ is equal to the persistent Stiefel-Whitney class $\pSFt{i}{X}{t}$ at time $t$, seen in the simplicial cohomology group $H^i(S^t) = H^i(\subdiv{S^t}{k})$. In the following section, we gather some technical details needed to implement this algorithm in practice.
Note that this also gives a way to compute the lifebar of $\pSF{i}{X}$. This bar is determined by the value $\tdeatho = \inf \{t \in T, \pSF{i}{X} \neq 0\}$. This quantity can be approximated by computing the classes $\pSFt{i}{X}{t}$ for several values of $t$. We point out that, in order to compute the value $\tdeatho$, there may exist a better algorithm than evaluating the class $\pSFt{i}{X}{t}$ several times.
An algorithm when $d=1$ {#sec:algorithm}
=======================
Even though the last sections described a theoretical way to compute the persistent Stiefel-Whitney classes, some concrete issues are still to be discussed:
- verifying that the star condition is satisfied,
- the Grassmann manifold has to be triangulated,
- in practice, the Vietoris-Rips filtration is prefered to the Čech filtration,
- the parameter $\gamma$ has to be tuned.
The following subsections will elucidate these points. Concerning the first one, the author is not aware of a computational-explicit process to triangulate the Grassmann manifolds $\Grass{d}{\R^m}$, except when $d=1$, which corresponds to the projective spaces $\Grass{1}{\R^m}$. We will then restrict to the case $d=1$.
The star condition in practice {#subsec:combinatorial_star_condition}
------------------------------
Let us get back to the context of Subsection \[subsec:simplicial\_approximation\]: $K, L$ are two simplicial complexes, $K$ is finite, and $g \colon \topreal{K} \rightarrow \topreal{L}$ is a continuous map. We have seen that finding a simplicial approximation to $g$ reduces to finding a small enough barycentric subdivision $\subdiv{K}{n}$ of $K$ such that $g \colon \topreal{\subdiv{K}{n}} \rightarrow \topreal{L}$ satisfies the star condition, that is, for every vertex $v$ of $\subdiv{K}{n}$, there exists a vertex $w$ of $L$ such that $$g\left(\topreal{\closedStar{v}}\right) \subseteq \topreal{\Star{w}}.$$ In practice, one can compute the closed star $\closedStar{v}$ from the finite simplicial complex $\subdiv{K}{n}$. However, computing $g\left(\topreal{\closedStar{v}}\right)$ requires to evaluate $g$ on the infinite set $\topreal{\closedStar{v}}$. In order to reduce the problem to a finite number of evaluations of $g$, we will consider a related property that we call the *weak star condition*.
A map $g\colon \topreal{K} \rightarrow \topreal{L}$ between topological realizations of simplicial complexes $K$ and $L$ satisfies the *weak star condition* if for every vertex $v$ of $\subdiv{K}{n}$, there exists a vertex $w$ of $L$ such that $$\topreal{ g \left( \skeleton{\closedStar{v}}{0} \right) } \subseteq \topreal{\Star{w}},$$ where $\skeleton{\closedStar{v}}{0}$ denotes the 0-skeleton of $\closedStar{v}$, i.e. its vertices.
Observe that the practical verification of the condition $\topreal{ g \left( \skeleton{\closedStar{v}}{0} \right) } \subseteq \topreal{\Star{w}}$ requires only a finite number of computations. Indeed, one just has to check whether every neighbor $v'$ of $v$ in the graph $\skeleton{K}{1}$, $v$ included, satisfies $g(v') \in \topreal{\Star{w}}$. The following lemma rephrases this condition by using the face map $\facemapK{L}\colon \topreal{L} \rightarrow L$ defined in Subsection \[subsec:background\_simplicialcomplexes\]. We remind the reader that the face map is defined by the relation $x \in \facemapK{L}(x)$ for all $x \in \topreal{L}$.
\[lem:weak\_star\] The map $g$ satisfies the weak star condition if and only if for every vertex $v$ of $K$, there exists a vertex $w$ of $L$ such that for every neighbor $v'$ of $v$ in $\skeleton{K}{1}$, we have $$w \in \facemapK{L}(g(v')).$$
Let us show that the assertion “$w \in \facemapK{L}(g(v'))$" is equivalent to “$g(v') \in \topreal{\Star{w}}$". Remind that the open star $\Star{w}$ consists of simplices of $L$ that contain $w$. Moreover, the topological realization $\topreal{\Star{w}}$ is the union of $\topreal{\sigma}$ for $\sigma \in \Star{w}$. As a consequence, $g(v')$ belongs to $\topreal{\Star{w}}$ if and only if it belongs to $\topreal{\sigma}$ for some simplex $\sigma \in L$ that contains $w$. Equivalently, the face map $\facemapK{L}(g(v'))$ contains $w$.
Suppose that $g$ satisfies the weak star condition. Let $f\colon \skeleton{K}{0} \rightarrow \skeleton{L}{0}$ be a map between vertex sets such that for every $v \in \skeleton{K}{0}$, $$\topreal{ g \left( \skeleton{\closedStar{v}}{0} \right) } \subseteq \topreal{\Star{f(v)}}.$$ According to the proof of Lemma \[lem:weak\_star\], an equivalent formulation of this condition is: for all neighbor $v'$ of $v$ in $\skeleton{K}{1}$, $$\label{eq:face_map_weakstarcondition}
f(v) \in \facemapK{L}(g(v')).$$ Such a map is called a *weak simplicial approximation to $g$*. It plays a similar role as the simplicial approximations to $g$.
\[lem:combinatorial\_is\_simplicial\] If $f\colon \skeleton{K}{0} \rightarrow \skeleton{L}{0}$ is a weak simplicial approximation to $g\colon \topreal{K} \rightarrow \topreal{L}$, then $f$ is a simplicial map.
Let $\sigma = [v_0, ..., v_n]$ be a simplex of $K$. We have to show that $f(\sigma ) = [f(v_0), ..., f(v_n)]$ is a simplex of $L$. Note that each closed star $\closedStar{v_i}$ contains $\sigma$. Therefore each $\topreal{ g \left( \skeleton{\closedStar{v_i}}{0} \right) }$ contains $\topreal{ g \left( \skeleton{\sigma}{0} \right) } = \{g(v_0), ..., g(v_n)\}$. Using the weak simplicial approximation property of $f$, we deduce that each $\topreal{\Star{f(v_i)}}$ contains $\{g(v_0), ..., g(v_n)\}$. Using Lemma \[lem:intersection\_stars\] stated below, we obtain that $[f(v_0), ..., f(v_n)]$ is a simplex of $L$.
\[lem:intersection\_stars\] Let $w_0, ... w_n$ be vertices of a simplicial complex $L$. Then $\bigcap_{i=0}^n \Star{w_i} \neq \emptyset$ if and only if $[w_0, ..., w_n]$ is a simplex of $L$.
As one can see from the definitions, the weak star condition is weaker than the star condition. Consequently, the simplicial approximation theorem admits the following corollary.
Consider two simplicial complexes $K, L$ with $K$ finite, and let $g\colon \topreal{K} \rightarrow \topreal{L}$ be a continuous map. Then there exists $n \geq 0$ such that $g\colon \topreal{\subdiv{K}{n}} \rightarrow \topreal{L}$ satisfies the weak star condition.
However, some weak simplicial approximations to $g$ may not be simplicial approximations, and may not even be homotopic to $g$. Figure \[fig:combinatorial\_not\_simplicial\_approx\] gives such an example.
![The map $g$ admits a weak simplicial approximation which is constant.[]{data-label="fig:combinatorial_not_simplicial_approx"}](combinatorial_approx_K.png "fig:"){width=".6\linewidth"} $K$
![The map $g$ admits a weak simplicial approximation which is constant.[]{data-label="fig:combinatorial_not_simplicial_approx"}](combinatorial_approx_L.png "fig:"){width=".7\linewidth"} $L$
![The map $g$ admits a weak simplicial approximation which is constant.[]{data-label="fig:combinatorial_not_simplicial_approx"}](combinatorial_approx_g.png "fig:"){width=".7\linewidth"} $g\colon \topreal{K} \rightarrow \topreal{L}$
Fortunately, these two notions coincides under the star condition assumption:
\[prop:weak\_star\_strong\_star\] Suppose that $g$ satisfies the star condition. Then every weak simplicial approximation to $g$ is a simplicial approximation.
Let $f$ be a weak simplicial approximation to $g$, and $f'$ any simplicial approximation. Let us show that $f$ and $f'$ are contiguous simplicial maps. Let $\sigma = [v_0, ..., v_n]$ be a simplex of $K$. We have to show that $[f(v_0), ..., f(v_n), f'(v_0), ..., f'(v_n)]$ is a simplex of $L$. As we have seen in the proof of Lemma \[lem:combinatorial\_is\_simplicial\], each $\topreal{ g \left( \skeleton{\closedStar{v_i}}{0} \right) }$ contains $\{g(v_0), ..., g(v_n)\}$. Therefore, by definition of weak simplicial approximations and simplicial approximations, each $\topreal{\Star{f(v_i)}}$ and $\topreal{\Star{f'(v_i)}}$ contains $\{g(v_0), ..., g(v_n)\}$. We conclude by applying Lemma \[lem:intersection\_stars\].
Remark that the proof of this proposition can be adapted to obtain the following fact: any two weak simplicial approximations are equivalent—as well as any two simplicial approximations.
Let us comment Proposition \[prop:weak\_star\_strong\_star\]. If $K$ is subdivised enough, then every weak simplicial approximation to $g$ is homotopic to $g$. But in practice, the number of subdivisions needed by the star condition is not known. We propose to subdivise the complex $K$ until it satisfies the weak star condition, and then use a weak simplicial approximation to $g$. However, such a weak simplicial approximation may not be homotopic to $g$, and our algorithm would output a wrong result. To close this subsection, we state a lemma that gives a quantitative idea of the number of subdivisions needed by the star condition. We say that a Lebesgue number for an open cover $\UU$ of a compact metric space $X$ is a positive number $\epsilon$ such that every subset of $X$ with diameter less than $\epsilon$ is included in some member of the cover $\UU$.
Let $\topreal{K}, \topreal{L}$ be endowed with metrics. Suppose that $g\colon \topreal{K} \rightarrow \topreal{L}$ is $l$-Lipschitz with respect to these metrics. Let $\epsilon$ be a Lebesgue number for the open cover $\left \{ \topreal{\Star{w}}, w \in L \right\}$ of $\topreal{L}$. Let $p$ be the dimension of $K$ and $D$ an upper bound on the diameter of its faces. Then for $n > \log(\frac{Dl}{\epsilon})\big/\log(\frac{p+1}{p})$, the map $g\colon \topreal{\subdiv{K}{n}}\rightarrow \topreal{L}$ satisfies the star condition.
The map $g$ satisfies the star condition if for every vertex $v$ of $K$, there exists a vertex $w$ of $L$ such that $g(\topreal{\closedStar{v}}) \subseteq \topreal{\Star{w}}$. Since the cover $\left \{ \topreal{\Star{w}}, w \in L \right\}$ admits $\epsilon$ as a Lebesgue number, it is enough for $v$ to satisfy the following inequality: $$\label{eq:proof_number_subdivision}
\diam{ g(\topreal{\closedStar{v}}) } < \epsilon.$$ Since $g$ is $l$-Lipschitz, we have $\diam{ g\left(\topreal{\closedStar{v}} \right) } \leq l\cdot \diam{ \topreal{\closedStar{v}} }$. Using the hypothesis $\diam{ \topreal{\closedStar{v}} } \leq D$, Equation \[eq:proof\_number\_subdivision\] leads to the condition $D l < \epsilon$. Now, we use the fact that a barycentric subdivision reduces the diameter of each face by a factor $\frac{p}{p+1}$. After $n$ barycentric subdivision, the last inequality rewrites $\left( \frac{p}{p+1} \right)^n D l < \epsilon$. It admits $n > \log(\frac{Dl}{\epsilon})\big/\log(\frac{p+1}{p})$ as a solution.
Triangulating the projective spaces
-----------------------------------
As we described in Subsection \[subsec:combinatorial\_star\_condition\], the algorithm we propose rests on a triangulation $L$ of the Grassmannian $\Grass{1}{\R^m}$, together the map $\facemapK{L} \circ h \colon \Grass{1}{\R^m} \rightarrow L$, where $h \colon \Grass{1}{\R^m} \rightarrow \topreal{L}$ is a homeomorphism and $\facemapK{L} \colon \Grass{1}{\R^m} \rightarrow L$ is the face map. In the following, we also refer to $\facemap:= \facemapK{L} \circ h$ as the face map. We will use the following folklore triangulation of the projective space $\Grass{1}{\R^m}$. It uses the fact that the quotient of the sphere $\S_{m-1}$ by the antipodal relation gives $\Grass{1}{\R^m}$. Let $\Delta^{m}$ denote the standard $m$-simplex, $v_0, ..., v_m$ its vertices, and $\partial \Delta^m$ its boundary. The simplicial complex $\partial \Delta^m$ is a triangulation of the sphere $\mathbb{S}_{m-1}$. Denote its first barycentric subdivision as $\subdiv{\partial \Delta^m}{1}$. The vertices of $\subdiv{\partial \Delta^m}{1}$ are in bijection with the non-empty proper subsets of $\{v_0, ..., v_m\}$ (see Subsection \[subsec:simplicial\_approximation\]). Consider the equivalence relation on these vertices which associates a vertex to its complement. The quotient simplicial complex under this relation, $L$, is a triangulation of $\Grass{1}{\R^m}$.
{width=".9\linewidth"} $\partial \Delta^2$
{width=".9\linewidth"} $\subdiv{\partial \Delta^2}{1}$
{width=".9\linewidth"} Equivalence relation
{width=".9\linewidth"} Quotient complex $L$
Let us now describe how to define the homeomorphism $h\colon \Grass{1}{\R^m} \rightarrow \topreal{L}$. First, embed $\Delta^m$ in $\R^{m+1}$ via $v_i \mapsto (0, ..., 0, 1, 0, ...)$, where $1$ sits at the $i$th coordinate. Its image lies on a $m$-dimensional affine subspace $P$, with origin being the barycenter of $v_0, ..., v_m$. Seen in $P$, the vertices of $\Delta^m$ now belong to the sphere centered at the origin and of radius $\sqrt{\frac{m}{m+1}}$ (see Figure \[fig:triangulation\_homeomorphism\]). Let us denote this sphere as $\mathbb{S}_{m-1}$. Next, subdivise barycentrically $\partial \Delta^m$ once, and project each vertex of $\subdiv{\partial \Delta^m}{1}$ on $\mathbb{S}_{m-1}$. By taking the convex hulls of its faces, we now see $\topreal{\subdiv{\partial \Delta^m}{1}}$ as a subset of $P$. Define an application $p\colon \mathbb{S}_{m-1} \rightarrow \topreal{\subdiv{\partial \Delta^m}{1}}$ as follows: for every $x \in \mathbb{S}_{m-1}$, the image $p(x)$ is the unique intersection point between the segment $[0, x]$ and the set $\topreal{\subdiv{\partial \Delta^m}{1}}$. The application $p$ can also be seen as the inverse function of the projection on $\S_{m-1}$, written $\projj{ \cdot }{\S_{m-1}}\colon \topreal{\subdiv{\partial \Delta^m}{1}} \rightarrow \mathbb{S}_{m-1}$.
![Triangulating $\Grass{1}{\R^3}$.[]{data-label="fig:triangulation_homeomorphism"}](triangulate_plane_1.png "fig:"){width=".7\linewidth"} $\partial \Delta^3$ is included in $\mathbb{S}_{m-1}$
![Triangulating $\Grass{1}{\R^3}$.[]{data-label="fig:triangulation_homeomorphism"}](triangulate_plane_2.png "fig:"){width=".7\linewidth"} $\subdiv{\partial \Delta^3}{1}$ and $\mathbb{S}_{m-1}$
![Triangulating $\Grass{1}{\R^3}$.[]{data-label="fig:triangulation_homeomorphism"}](triangulate_plane_3.png "fig:"){width=".7\linewidth"} $L$
The next lemma shows that the antipodal relation on $\mathbb{S}_{m-1}$ can be pulled-back to $\topreal{\subdiv{\partial \Delta^m}{1}}$ via $p$, and it corresponds to the equivalence relation we defined on $\subdiv{\partial \Delta^m}{1}$. As a consequence, we can factorize $p\colon \mathbb{S}_{m-1} \rightarrow \topreal{\subdiv{\partial \Delta^m}{1}}$ as $$\begin{aligned}
h\colon \left(\mathbb{S}_{m-1} / {\sim} \right) \rightarrow \left(\topreal{\subdiv{\partial \Delta^m}{1}} / {\sim} \right),\end{aligned}$$ and we can identify these spaces with $$\begin{aligned}
h\colon \Grass{1}{\R^m} \rightarrow \topreal{L},\end{aligned}$$ giving the desired homeomorphism.
For any vertex $x \in \subdiv{\partial \Delta^m}{1}$, denote by $\topreal{x}$ its embedding in $P$. Let $-\topreal{x}$ denote the image of $\topreal{x}$ by the antipodal relation on $\S_{m-1}$. Denote by $y$ the image of $x$ by the relation on $\subdiv{\partial \Delta^m}{1}$. Then $y = -\topreal{x}$.
More generally, pulling back the antipodal relation onto $\topreal{\subdiv{\partial \Delta^m}{1}}$ via $p$ gives the relation we defined on $\subdiv{\partial \Delta^m}{1}$.
Pick a vertex $x$ of $\subdiv{\partial \Delta^m}{1}$. It can be described as a proper subset $\{v_i, i \in I\}$ of the vertex set $\skeleton{(\partial \Delta^m)}{0} = \{v_0, ..., v_m\}$, where $I \subset [0,m]$. According to the relation on $(\partial \Delta^m)$, the vertex $x$ is in relation with the vertex $y$ described by the proper subset $\{v_i, i \in \complementaire{I}\}$. The point $x$ can be written in barycentric coordinates as $\frac{1}{\card{I}} \sum_{i \in I} \topreal{v_i}$. Seen in $P$, $\topreal{x}$ can be written $\topreal{x} = \projj{\sum_{i \in I} v_i}{\S_{m-1}}$. Similarly, $\topreal{y}$ can be written $\topreal{y} = \projj{\sum_{i \in \complementaire{I}} v_i}{\S_{m-1}}$.
Now, denote by 0 the origin of the hyperplane $P$, and embed the vertices $v_0, ..., v_m$ in $P$. Observe that $$\begin{aligned}
0
= \sum_{i \leq 0} v_i
= \sum_{i \in I} v_i + \sum_{i \in \complementaire{I}} v_i.\end{aligned}$$ Hence $- \sum_{i \in I} v_i = \sum_{i \in \complementaire{I}} v_i$, and we deduce that $$\begin{aligned}
-\topreal{x}
= \projj{- \sum_{i \in I} v_i}{\S_{m-1}}
= \projj{\sum_{i \in \complementaire{I}} v_i}{\S_{m-1}}
= \topreal{y}.\end{aligned}$$
Applying the same reasoning, one obtains the following result: for every simplex $\sigma$ of $\subdiv{\partial \Delta^m}{1}$, if $\nu$ denotes the image of $\sigma$ by the relation of $\subdiv{\partial \Delta^m}{1}$, then the image of $\topreal{\sigma}$ by the antipodal relation is also $\topreal{\nu}$. As a consequence, these two relations coincide.
At a computational level, let us describe how to compute the face map $\facemap \colon \Grass{1}{\R^m} \rightarrow L$. Since $\facemap$ can be obtained as a quotient, it is enough to compute the face map of the sphere, $\facemap' \colon \mathbb{S}_{m-1} \rightarrow \subdiv{\partial \Delta^m}{1}$, which corresponds to the homeomorphism $p\colon \S_{m-1} \rightarrow \topreal{\subdiv{\partial \Delta^m}{1}}$. It is given by the following lemma, which can be used in practice.
For every $x \in \S_{m-1}$, the image of $x$ by the face map $\facemap'$ is equal to the intersection of all maximal faces $\sigma = [w_0, ..., w_m]$ of $\subdiv{\partial \Delta^m}{1}$ that satisfies the following conditions: denoting by $x_0$ any point of the affine hyperplane spanned by $\{w_0, ..., w_m\}$, and by $h$ a vector orthogonal to the corresponding linear hyperplane,
- the inner product $\eucP{x}{h}$ has the same sign as $\eucP{x_0}{h}$,
- the point $\frac{\eucP{x_0}{h} }{\eucP{x}{h}} x$, which is included in the affine hyperplane spanned by $\{w_0, ..., w_m\}$, has nonnegative barycentric coordinates.
Recall that for every $x \in \mathbb{S}_{m-1}$, the image $p(x)$ is defined as the unique intersection point between the segment $[0, x]$ and the set $\topreal{\subdiv{\partial \Delta^m}{1}}$. Besides, the face map $\facemap'(x)$ is the unique simplex $\sigma \in \subdiv{\partial \Delta^m}{1}$ such that $p(x) \in \topreal{\sigma}$. Equivalently, $\facemap'(x)$ is equal to the intersection of all maximal faces $\sigma \in \subdiv{\partial \Delta^m}{1}$ such that $p(x)$ belongs to the closure $\overline{ \topreal{\sigma} }$.
Consider any maximal face $\sigma = [w_0, ..., w_m]$ of $\subdiv{\partial \Delta^m}{1}$. The first condition of the lemma ensures that the segment $[0,x]$ intersects the affine hyperplane spanned by $\{w_0, ..., w_m\}$. In this case, a computation shows that this intersection consists of the point $\frac{\eucP{x_0}{h} }{\eucP{x}{h}} x$. Then, the second condition of the lemma tests whether this point belongs to the convex hull of $\{w_0, ..., w_k\}$. In conclusion, if $\sigma$ satisfies these two conditions, then $p(x) \in \overline{ \topreal{\sigma} }$.
As a remark, let us point out that the verification of the conditions of this lemma is subject to numerical errors. In particular, the point $\frac{\eucP{x_0}{h} }{\eucP{x}{h}} x$ may have nonnegative coordinates, yet mathematical softwares may return (small) negative values. Consequently, the algorithm may recognize less maximal faces that satisfy these conditions, hence return a simplex that strictly contains the wanted simplex $\facemap'(x)$. Nonetheless, such an error will not affect the output of the algorithm. Indeed, if we denote by $\widetilde{\facemap'}$ the face map computed by the algorithm, we have that $\facemap'(x) \subseteq \widetilde{\facemap'}(x)$ for all $x \in \S_{m-1}$. As a consequence of Lemma \[lem:weak\_star\], $\widetilde{\facemap'}$ satisfies the weak star condition if $\facemap'$ does, and Equation \[eq:face\_map\_weakstarcondition\] shows that every weak simplicial approximations for $\facemap'$ are weak simplicial approximations for $\widetilde{\facemap'}$. Since every weak simplicial approximations are homotopic, we obtain that the induced maps in cohomology are equal, therefore the output of the algorithm is unchanged.
Vietoris-Rips version of the Čech bundle filtration
---------------------------------------------------
We still consider a subset $X \subset \R^n \times \matrixspace{\R^m}$. Denote by $\X$ the corresponding Čech set filtration, and by $\S = (S^t)_{t\geq0}$ the simplicial Čech filtration. For every $t \geq 0$, let $R^t$ be the flag complex of $S^t$, i.e. the clique complex of the 1-skeleton $\skeleton{(S^t)}{1}$ of $S^t$. It is known as the *Vietoris-Rips complex* of $X$ at time $t$. The collection $\R = (R^t)_{t \geq 0}$ is called the *Vietoris-Rips filtration* of $X$. The simplicial filtrations $\S$ and $\R$ are multiplicatively $\sqrt{2}$-interleaved [@bell2017weighted Theorem 3.1]. In other words, for every $t \geq 0$, we have $$\begin{aligned}
S^t \subseteq R^t \subseteq S^{\sqrt{2} t}.\end{aligned}$$
Let $\gamma>0$ and consider the Čech bundle filtration $(\X, \p)$ of $X$. Suppose that its maximal filtration value $\tmaxgamma{X}$ is positive. Let $\topreal{\R} = (\topreal{R^t})_{t\geq 0}$ denote the topological realization of the Vietoris-Rips filtration. We can give $\topreal{\R}$ a vector bundle filtration structure with $(p')^t \colon \topreal{R^t} \rightarrow \Grass{d}{\R^m}$ defined as $$(p')^t = p^{\sqrt{2}t} \circ i^t,$$ where $p^{\sqrt{2}t}$ denotes the maps of the Čech bundle filtration $(\X, \p)$, and $i^t$ denotes the inclusion $\topreal{R^t} \hookrightarrow \topreal{S^{\sqrt{2}t}}$. These maps fit in the following diagram: $$\begin{tikzcd}
\topreal{R^t} \arrow[r, hook, "i^t"] \arrow[drr, "(p')^t", dashed, swap]
&[0em] \topreal{S^{\sqrt{2} t}} \arrow[r, equal, shorten <= 1em, shorten >= 1em]
& X^{\sqrt{2} t} \arrow[d, "p^{\sqrt{2} t}"] \\
&& \Grass{d}{\R^m}
\end{tikzcd}$$ This new vector bundle filtration is defined on the index set $T' = \left[0, \frac{1}{\sqrt{2}}\tmaxgamma{X}\right)$.
It is clear from the construction that the vector bundle filtrations $(\X, \p)$ and $(\topreal{\R}, \p')$ are multiplicatively $\sqrt{2}$-interleaved, with an interleaving that preserves the persistent Stiefel-Whitney classes. This property is a multiplicative equivalent of Theorem \[thm:stability\].
We recall the reader that, if $X$ is a subset of $\R^n \times \Grass{d}{\R^m}$, then the maximal filtration value of the Čech bundle filtration on $X$ is $\tmaxgamma{X} = \frac{\sqrt{2}}{2} \gamma$ (see Equation \[eq:tmax\_subset\_grass\]). Consequently, the maximal filtration value of its Vietoris-Rips version is $\frac{1}{2}\gamma$.
From an application perspective, we choose to work with the Vietoris-Rips filtration since it is easier to compute. Indeed, its construction only relies on computing pairwise distances and finding cliques in graphs.
Choice of the parameter $\gamma$ {#subsec:choice_of_gamma}
--------------------------------
This subsection is devoted to discussing the influence of the parameter $\gamma>0$. Recall that $\gamma$ affects the norm $\gammaN{\cdot}$ we chose on $\R^n \times \matrixspace{\R^m}$: $$\begin{aligned}
\gammaN{(x,A)}^2 = \eucN{x}^2 + \gamma^2 \frobN{A}^2.\end{aligned}$$ Let $X \subset \R^n \times \matrixspace{\R^m}$. If $\gamma_1 \leq \gamma_2$ are two positive real numbers, the corresponding Čech filtrations $\X_1$ and $\X_2$, as well as the Čech bundle filtrations $(\X_1, \p_1)$ and $(\X_2, \p_2)$, are $\frac{\gamma_2}{\gamma_1}$-interleaved multiplicatively. This comes from the straightforward inequality $$\begin{aligned}
\|\cdot\|_{\gamma_1}
~\leq~ \|\cdot\|_{\gamma_2}
~\leq~ \frac{\gamma_2}{\gamma_1} \|\cdot\|_{\gamma_1}.\end{aligned}$$ Note that we also have the additive inequality $$\begin{aligned}
\| (x,A) \|_{\gamma_1}
~\leq~ \| (x,A) \|_{\gamma_2}
~\leq~ \| (x,A) \|_{\gamma_1} + \sqrt{\gamma_2^2-\gamma_1^2} \frobN{A}.\end{aligned}$$ One deduces that the Čech bundle filtrations $(\X_1, \p_1)$ and $(\X_2, \p_2)$ are $\sqrt{\gamma_2^2 - \gamma_1^2} \cdot \tmax{X}$-interleaved additively, where $\tmax{X}$ is the maximal filtration value when $\gamma=1$. As a consequence of these interleavings, when the values $\gamma_1$ and $\gamma_2$ are close, the persistence diagrams and the lifebars of the persistent Stiefel-Whitney classes are close (see Theorem \[thm:stability\]). As a general principle, one would choose the parameter $\gamma$ to be large, since it would lead to large filtrations. More precisely, if $t_{\gamma_1}^{\mathrm{max}}\left( X \right)$ and $t_{\gamma_2}^{\mathrm{max}}\left( X \right)$ denote repectively the maximal filtration values of $(\X_1,\p_1)$ and $(\X_2,\p_2)$, then $t_{\gamma_1}^{\mathrm{max}}\left( X \right) = \gamma_1 \cdot \tmax{X}$ and $t_{\gamma_2}^{\mathrm{max}}\left( X \right) = \gamma_2 \cdot \tmax{X}$, as in Equation \[eq:tmax\]. Moreover, we have the following inclusion: $$\begin{aligned}
X_1^{t_{\gamma_1}^{\mathrm{max}}\left( X \right)}
\subseteq X_2^{t_{\gamma_2}^{\mathrm{max}}\left( X \right)},\end{aligned}$$ where $X_1^{t_{\gamma_1}^{\mathrm{max}}\left( X \right)}$ denotes the thickening of $X$ with respect to the norm $\|\cdot\|_{\gamma_1}$, and $X_2^{t_{\gamma_2}^{\mathrm{max}}\left( X \right)}$ with respect to $\|\cdot\|_{\gamma_2}$. This inclusion can be proven from the following fact, valid for every $x \in \R^n$ and $A \in \matrixspace{\R^m}$ such that $\frobN{A} \leq \tmax{X}$: $$\begin{aligned}
\| (x,A) \|_{\gamma_1} \leq t_{\gamma_1}^{\mathrm{max}}\left( X \right)
\implies
\| (x,A) \|_{\gamma_2} \leq t_{\gamma_2}^{\mathrm{max}}\left( X \right).\end{aligned}$$
Hence larger parameters $\gamma$ lead to larger maximal filtration values and larger filtrations.
However, as we show in the following examples, different values of $\gamma$ may result in different behaviours of the persistent Stiefel-Whitney classes. In Example \[ex:gamma\_normal\], large values of $\gamma$ highlight properties of the dataset that are not consistent with the underlying vector bundle: the persistent Stiefel-Whitney class is nonzero, yet the vector bundle is orientable. Notice that, so far, we always picked the value $\gamma = 1$, for it seemed experimentally relevant with the datasets we chose.
\[ex:gamma\_mobius\] Consider the set $Y \subset \R^2 \times \matrixspace{\R^2}$ representing the Mobius band, as in Example \[ex:normal\_mobius\] of Subsection \[subsec:filtered\_cech\_bundle\]: $$\begin{aligned}
&Y = \bigg\{
\bigg(
\begin{pmatrix}
\cos(\theta) \\
\sin(\theta)
\end{pmatrix}
,
\begin{pmatrix}
\cos(\frac{\theta}{2})^2 & \cos(\frac{\theta}{2}) \sin(\frac{\theta}{2}) \\
\cos(\frac{\theta}{2}) \sin(\frac{\theta}{2}) & \sin(\frac{\theta}{2})^2
\end{pmatrix}
\bigg),
\theta \in [0, 2\pi) \bigg\}.\end{aligned}$$ As we show in Appendix \[subsec:gamma\_mobius\], $Y$ is a circle, included in a 2-dimensional affine subspace of $\R^2 \times \matrixspace{\R^2}$. Its radius is $\sqrt{1 + \frac{\gamma^2}{2}}$. As a consequence, the persistence of the Čech filtration of $Y$ consists of two bars: one $H^0$-feature, the bar $[0, +\infty)$, and one $H^1$-feature, the bar $\left[0, \sqrt{1 + \frac{\gamma^2}{2}}\right)$. For any $\gamma>0$, the maximal filtration value of the Čech bundle filtration of $Y$ is $\tmaxgamma{Y} = \frac{\sqrt{2}}{2} \gamma$. Moreover, the persistent Stiefel-Whitney class $w_1^t(Y)$ is nonzero all along the filtration.
In this example, we see that the parameter $\gamma$ does not influence the qualitative interpretation of the persistent Stiefel-Whitney class. It is always nonzero where it is defined. The following example shows a case where $\gamma$ does influence the persistent Stiefel-Whitney class.
\[ex:gamma\_normal\] Consider the set $X \subset \R^2 \times \matrixspace{\R^2}$ representing the normal bundle of the circle $\S_1$, as in Example \[ex:normal\_mobius\]: $$\begin{aligned}
&X = \bigg\{
\bigg(
\begin{pmatrix}
\cos(\theta) \\
\sin(\theta)
\end{pmatrix}
,
\begin{pmatrix}
\cos(\theta)^2 & \cos(\theta) \sin(\theta) \\
\cos(\theta) \sin(\theta) & \sin(\theta)^2
\end{pmatrix}
\bigg),
\theta \in [0, 2\pi) \bigg\}.\end{aligned}$$ As we show in Appendix \[subsec:gamma\_normal\], $X$ is a subset of a 2-dimensional flat torus embedded in $\R^2 \times \matrixspace{\R^2}$, hence can be seen as a torus knot.
Before studying the Čech bundle filtration of $X$, we discuss the Čech filtration $\X$. Its behaviour depends on $\gamma$:
- if $\gamma \leq \frac{\sqrt{2}}{2}$, then $X^t$ retracts on a circle for $t \in [0,1)$, $X^t$ retracts on a 3-sphere for $t \in \left [ 1, \sqrt{1+\frac{1}{2}\gamma^2} \right)$, and $X^t$ retracts on a point for $t \geq \sqrt{1+\frac{1}{2}\gamma^2}$.
- if $\gamma \geq \frac{\sqrt{2}}{2}$, then $X^t$ retracts on a circle for $t \in [0,1)$, $X^t$ retracts on another circle for $t \in \left[1,\frac{\sqrt{2} }{2}\sqrt{1 + \gamma^2 + \frac{1}{4 \gamma^2}}\right)$, $X^t$ retracts on a 3-sphere for $t \in \left[ \frac{\sqrt{2} }{2}\sqrt{1 + \gamma^2 + \frac{1}{4 \gamma^2}}, \sqrt{1+\frac{1}{2}\gamma^2}\right)$, and $X^t$ has the homotopy type of a point for $t \geq \sqrt{1+\frac{1}{2}\gamma^2}$.
Let us interpret these facts. If $\gamma \leq \frac{\sqrt{2}}{2}$, then the persistent cohomology of $X$ looks similar to the persistent cohomology of the underlying set $\left\{\begin{psmallmatrix} \cos(\theta) \\ \sin(\theta) \end{psmallmatrix}, \theta \in [0, 2\pi)\right\} \subset \R^2$, but with a $H^3$ cohomology feature added. Besides, if $\gamma \geq \frac{\sqrt{2}}{2}$, a new topological feature appears in the $H^1$-barcode: the bar $\left[1,\sqrt{2} \sqrt{1 + \gamma^2 + \frac{1}{4 \gamma^2}}\right)$. These barcodes are depicted in Figures \[fig:parameter\_gamma\] and \[fig:parameter\_gamma2\].
![$H^0$-, $H^1$-, $H^3$-barcodes and lifebar of the first persistent Stiefel-Whitney class of $X$ with $\gamma=\frac{1}{2}$ (left) and $\gamma=\frac{\sqrt{2}}{2}$ (right).[]{data-label="fig:parameter_gamma"}](gamma_1_2_H0.png "fig:"){width=".9\linewidth"} ![$H^0$-, $H^1$-, $H^3$-barcodes and lifebar of the first persistent Stiefel-Whitney class of $X$ with $\gamma=\frac{1}{2}$ (left) and $\gamma=\frac{\sqrt{2}}{2}$ (right).[]{data-label="fig:parameter_gamma"}](gamma_1_2_H1.png "fig:"){width=".9\linewidth"} ![$H^0$-, $H^1$-, $H^3$-barcodes and lifebar of the first persistent Stiefel-Whitney class of $X$ with $\gamma=\frac{1}{2}$ (left) and $\gamma=\frac{\sqrt{2}}{2}$ (right).[]{data-label="fig:parameter_gamma"}](gamma_1_2_H3.png "fig:"){width=".9\linewidth"} ![$H^0$-, $H^1$-, $H^3$-barcodes and lifebar of the first persistent Stiefel-Whitney class of $X$ with $\gamma=\frac{1}{2}$ (left) and $\gamma=\frac{\sqrt{2}}{2}$ (right).[]{data-label="fig:parameter_gamma"}](gamma_1_2_lifespan.png "fig:"){width=".9\linewidth"}
![$H^0$-, $H^1$-, $H^3$-barcodes and lifebar of the first persistent Stiefel-Whitney class of $X$ with $\gamma=\frac{1}{2}$ (left) and $\gamma=\frac{\sqrt{2}}{2}$ (right).[]{data-label="fig:parameter_gamma"}](gamma_sqrt2_2_H0.png "fig:"){width=".9\linewidth"} ![$H^0$-, $H^1$-, $H^3$-barcodes and lifebar of the first persistent Stiefel-Whitney class of $X$ with $\gamma=\frac{1}{2}$ (left) and $\gamma=\frac{\sqrt{2}}{2}$ (right).[]{data-label="fig:parameter_gamma"}](gamma_sqrt2_2_H1.png "fig:"){width=".9\linewidth"} ![$H^0$-, $H^1$-, $H^3$-barcodes and lifebar of the first persistent Stiefel-Whitney class of $X$ with $\gamma=\frac{1}{2}$ (left) and $\gamma=\frac{\sqrt{2}}{2}$ (right).[]{data-label="fig:parameter_gamma"}](gamma_sqrt2_2_H3.png "fig:"){width=".9\linewidth"} ![$H^0$-, $H^1$-, $H^3$-barcodes and lifebar of the first persistent Stiefel-Whitney class of $X$ with $\gamma=\frac{1}{2}$ (left) and $\gamma=\frac{\sqrt{2}}{2}$ (right).[]{data-label="fig:parameter_gamma"}](gamma_sqrt2_2_lifespan.png "fig:"){width=".9\linewidth"}
![$H^0$-, $H^1$-, $H^3$-barcodes and lifebar of the first persistent Stiefel-Whitney class of $X$ with $\gamma=1$ (left) and $\gamma=2$ (right).[]{data-label="fig:parameter_gamma2"}](gamma_1_H0.png "fig:"){width=".9\linewidth"} ![$H^0$-, $H^1$-, $H^3$-barcodes and lifebar of the first persistent Stiefel-Whitney class of $X$ with $\gamma=1$ (left) and $\gamma=2$ (right).[]{data-label="fig:parameter_gamma2"}](gamma_1_H1.png "fig:"){width=".9\linewidth"} ![$H^0$-, $H^1$-, $H^3$-barcodes and lifebar of the first persistent Stiefel-Whitney class of $X$ with $\gamma=1$ (left) and $\gamma=2$ (right).[]{data-label="fig:parameter_gamma2"}](gamma_1_H3.png "fig:"){width=".9\linewidth"} ![$H^0$-, $H^1$-, $H^3$-barcodes and lifebar of the first persistent Stiefel-Whitney class of $X$ with $\gamma=1$ (left) and $\gamma=2$ (right).[]{data-label="fig:parameter_gamma2"}](gamma_1_lifespan.png "fig:"){width=".9\linewidth"}
![$H^0$-, $H^1$-, $H^3$-barcodes and lifebar of the first persistent Stiefel-Whitney class of $X$ with $\gamma=1$ (left) and $\gamma=2$ (right).[]{data-label="fig:parameter_gamma2"}](gamma_2_H0.png "fig:"){width=".9\linewidth"} ![$H^0$-, $H^1$-, $H^3$-barcodes and lifebar of the first persistent Stiefel-Whitney class of $X$ with $\gamma=1$ (left) and $\gamma=2$ (right).[]{data-label="fig:parameter_gamma2"}](gamma_2_H1.png "fig:"){width=".9\linewidth"} ![$H^0$-, $H^1$-, $H^3$-barcodes and lifebar of the first persistent Stiefel-Whitney class of $X$ with $\gamma=1$ (left) and $\gamma=2$ (right).[]{data-label="fig:parameter_gamma2"}](gamma_2_H3.png "fig:"){width=".9\linewidth"} ![$H^0$-, $H^1$-, $H^3$-barcodes and lifebar of the first persistent Stiefel-Whitney class of $X$ with $\gamma=1$ (left) and $\gamma=2$ (right).[]{data-label="fig:parameter_gamma2"}](gamma_2_lifespan.png "fig:"){width=".9\linewidth"}
Let us now discuss the corresponding Čech bundle filtrations. For any $\gamma>0$, the maximal filtration value of the Čech bundle filtration of $X$ is $\tmaxgamma{X} = \frac{\sqrt{2}}{2} \gamma$. We observe two behaviours:
- if $\gamma \leq \frac{\sqrt{2}}{2}$, then $w_1^t(X)$ is zero all along the filtration,
- if $\gamma > \frac{\sqrt{2}}{2}$, then $w_1^t(X)$ is nonzero from $t^\dagger = 1$.
This in proven in Appendix \[subsec:gamma\_normal\]. To conclude, this persistent Stiefel-Whitney class is consistent with the underlying bunlde—the normal bundle of the circle, which is trivial—only for $t\leq 1$.
Conclusion
==========
In this paper we defined the persistent Stiefel-Whitney classes of vector bundle filtrations. We proved that they are stable with respect to the interleaving distance between vector bundle filtrations. We studied the particular case of Čech bundle filtrations of subsets of $\R^n \times \matrixspace{\R^m}$, and showed that they yield consistent estimators of the usual Stiefel-Whitney classes of some underlying vector bundle. Moreover, when the dimension of the bundle is 1 and $X$ is finite, we proposed an algorithm to compute the persistent Stiefel-Whitney classes.
Our algorithm is limited to the bundles of dimension 1, since we only implemented triangulations of the Grassmannian $\Grass{d}{\R^m}$ when $d=1$. However, any other triangulation of $\Grass{d}{\R^m}$, with a computable face map, could be included in the algorithm without any modification. We also described a way to compute the lifebar of the persistent Stiefel-Whitney classes, by evaluating the class for several values of $t$.
Supplementary material for Section \[sec:persistentSWclasses\] {#sec:appendix_persistentSFclasses}
==============================================================
We prove Lemma \[lem:projongrass\], stated page .
Note that $\Grass{d}{\R^m}$ is contained in the linear subspace $\SS$ of symmetric matrices. Therefore, to project a matrix $A \in \matrixspace{\R^m}$ onto $\Grass{d}{\R^m}$, we may project on $\SS$ first. It is well known that the projection of $A$ onto $\SS$ is the matrix $A^s = \frac{1}{2}(A + \transp{A})$.
Suppose now that we are given a symmetric matrix $B$. Let it be diagonalized as $B = O D \transp{O}$. A projection of $B$ onto $\Grass{d}{\R^m}$ is a matrix $P$ which minimizes the following quantity: $$\label{eq:min_grass}
\min_{P \in \Grass{d}{E}} \frobN{ B - P }.$$ This problem is equivalent to $$\begin{aligned}
\min_{P \in \Grass{d}{E}} \frobN{ D - P }\end{aligned}$$ via $P \mapsto \transp{O} P O$. Now, let $e_1, \cdots, e_n$ denote the canonical basis of $\R^m$. We have $$\begin{aligned}
\frobN{ D - P}^2
&= \frobN{D}^2 + \frobN{P}^2 - 2\frobP{D}{P} \\
&= \frobN{D}^2 + \frobN{P}^2 - 2\sum \eucP{\lambda_i e_i}{P(e_i)},\end{aligned}$$ where $\frobP{\cdot}{\cdot}$ is the Frobenius inner product, and $\eucP{\cdot}{\cdot}$ the usual inner product on $\R^m$. Therefore, Equation \[eq:min\_grass\] is a problem equivalent to $$\begin{aligned}
\max_{P\in \Grass{d}{E}} \sum \lambda_i \eucP{e_i}{P(e_i)}.\end{aligned}$$ Since $P$ is an orthogonal projection, we have $\eucP{e_i}{P(e_i)} = \eucP{P(e_i)}{P(e_i)} = \eucN{P(e_i)}^2$ for all $i \in [1,n]$. Moreover, $d = \frobN{P}^2 = \sum \eucN{P(e_i)}^2$. Denoting $p_i = \eucN{P(e_i)}^2 \in [0,1]$, we finally obtain the following alternative formulation of Equation \[eq:min\_grass\]: $$\begin{aligned}
\max_{\substack{p_1,...p_n \in [0,1] \\ p_1+...+p_n = d}} \sum \lambda_i p_i.\end{aligned}$$ Using that $\lambda_1 \geq ... \geq \lambda_n$, we see that this maximum is attained when $p_0 = ... = p_d=1$ and $p_{d+1}=...=p_n = 0$. Consequently, a minimizer of Equation \[eq:min\_grass\] is $P = J_d$, where $J_d$ is the diagonal matrix whose first $d$ terms are 1, and the other ones are zero. Moreover, it is unique if $\lambda_d \neq \lambda_{d+1}$.
As a consequence of these considerations, we obtain the following characterization: for every $B\in \matrixspace{\R^m}$, $$\label{eq:med_Gd}
B \in \med{\Grass{d}{\R^m}} \iff \lambda_d(B^s) = \lambda_{d+1}(B^s).$$
Let us now show that for every matrix $A\in \matrixspace{\R^m}$, we have $$\begin{aligned}
\dist{A}{\med{\Grass{d}{\R^m}}} = \frac{\sqrt{2}}{2} \big|\lambda_d(A^s) - \lambda_{d+1}(A^s)\big|.\end{aligned}$$ First, remark that $$\label{eq:proj_on_med}
\dist{A}{\med{\Grass{d}{\R^m}}} = \dist{A^s}{\med{\Grass{d}{\R^m}}}.$$ Indeed, if $B$ is a projection of $A$ on $\med{\Grass{d}{\R^m}}$, then $B^s$ is still in $\med{\Grass{d}{\R^m}}$ according to Equation \[eq:med\_Gd\], and $$\begin{aligned}
\dist{A}{\med{\Grass{d}{\R^m}}} = \frobN{A-B} \geq \frobN{A^s - B^s} \geq \dist{A^s}{\med{\Grass{d}{\R^m}}}.\end{aligned}$$ Conversely, if $B$ is a projection of $A^s$ on $\med{\Grass{d}{\R^m}}$, then $\hat B = B + A - A^s$ is still in $\med{\Grass{d}{\R^m}}$, and $$\begin{aligned}
\dist{A}{\med{\Grass{d}{\R^m}}} \leq \frobN{A - \hat B} = \frobN{A^s - B} = \dist{A^s}{\med{\Grass{d}{\R^m}}}.\end{aligned}$$ We deduce Equation \[eq:proj\_on\_med\].
Now, let $A \in \SS$ and $B \in \med{\Grass{d}{\R^m}}$. Let $e_1,...,e_n$ be a basis of $\R^m$ that diagonalizes $A$. Writing $\frobN{A-B} = \sum \eucN{A(e_i) - B(e_i)}^2 = \sum \eucN{\lambda_i(A) e_i - B(e_i)}^2$, it is clear that the closest matrix $B$ must satisfy $B(e_i) = \lambda_i(B) e_i$, with
- $\lambda_i(B) = \lambda_i(A)$ for $i \notin \{d,d+1\}$,
- $\lambda_d(B) = \lambda_{d+1}(B) = \frac{1}{2}(\lambda_d(A)+\lambda_{d+1}(A))$.
We finally compute $$\begin{aligned}
\frobN{A-B}^2
&= \sum \eucN{\lambda_i(A) e_i - \lambda_i(B) e_i}^2 \\
&= \left\lvert\lambda_d(A) - \lambda_d(B) \right\rvert^2 + \left \lvert\lambda_{d+1}(A) - \lambda_{d+1}(B) \right\rvert^2 \\
&= \frac{1}{2} \left\lvert \lambda_d(A) - \lambda_{d+1}(A)\right\rvert^2.
\qedhere\end{aligned}$$
Supplementary material for Section \[sec:algorithm\] {#sec:appendix_algorithm}
====================================================
Study of Example \[ex:gamma\_mobius\] {#subsec:gamma_mobius}
-------------------------------------
We consider the set $$\begin{aligned}
&X = \bigg\{
\bigg(
\begin{pmatrix}
\cos(\theta) \\
\sin(\theta)
\end{pmatrix}
,
\begin{pmatrix}
\cos(\frac{\theta}{2})^2 & \cos(\frac{\theta}{2}) \sin(\frac{\theta}{2}) \\
\cos(\frac{\theta}{2}) \sin(\frac{\theta}{2}) & \sin(\frac{\theta}{2})^2
\end{pmatrix}
\bigg),
\theta \in [0, 2\pi) \bigg\}.\end{aligned}$$
To study the Čech filtration of $X$, we will apply the following affine transformation: let $Y$ be the subset of $\R^2 \times \matrixspace{\R^2}$ defined as $$\begin{aligned}
&Y = \bigg\{
\bigg(
\begin{pmatrix}
\cos(\theta) \\
\sin(\theta)
\end{pmatrix}
,
\gamma \begin{pmatrix}
\cos(\frac{\theta}{2})^2 & \cos(\frac{\theta}{2}) \sin(\frac{\theta}{2}) \\
\cos(\frac{\theta}{2}) \sin(\frac{\theta}{2}) & \sin(\frac{\theta}{2})^2
\end{pmatrix}
\bigg),
\theta \in [0, 2\pi) \bigg\}.\end{aligned}$$ and let $\Y = (Y^t)_{t\geq 0}$ be the Čech filtration of $Y$ in $\R^2 \times \matrixspace{\R^2}$ endowed with the usual norm $\gammaNun{(x,A)} = \sqrt{\eucN{x}^2 + \frobN{A}^2}$. We recall that the Čech filtration of $X$, denoted $\X = (X^t)_{t\geq 0}$, is defined with respect to the norm $\gammaN{\cdot}$. It is clear that, for every $t \geq 0$, the thickenings $X^t$ and $Y^t$ are homeomorphic via the application $$\begin{aligned}
h \colon \R^2 \times \matrixspace{\R^2} &\longrightarrow \R^2 \times \matrixspace{\R^2} \\
(x, A) &\longmapsto O + (x, \gamma A).\end{aligned}$$ As a consequence, the persistence cohomology modules associated to $\X$ and $\Y$ are isomorphic.
Next, notice that $Y$ is a subset of the affine subspace of dimension 2 of $\R^2 \times \matrixspace{\R^2}$ with origin $O$ and spanned by the vectors $e_1$ and $e_2$, where $$\begin{aligned}
O &= \left( \begin{pmatrix} 0 \\ 0 \end{pmatrix}, \frac{\gamma}{2}\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \right),
~~~~~~
e_1 = \left( \begin{pmatrix} 1 \\ 0 \end{pmatrix},
\frac{\gamma}{2} \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} \right),
~~~~~~
e_2 = \left( \begin{pmatrix} 0 \\ 1 \end{pmatrix},
\frac{\gamma}{2} \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \right).\end{aligned}$$ Indeed, using the equality $$\begin{aligned}
\begin{pmatrix}
\cos(\frac{\theta}{2})^2 & \cos(\frac{\theta}{2}) \sin(\frac{\theta}{2}) \\
\cos(\frac{\theta}{2}) \sin(\frac{\theta}{2}) & \sin(\frac{\theta}{2})^2
\end{pmatrix}
\bigg)
=
\frac{1}{2}\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}
+
\frac{1}{2}
\begin{pmatrix}
\cos(\theta) & \sin(\theta) \\
\sin(\theta) & -\cos(\theta)
\end{pmatrix},\end{aligned}$$ we obtain $$\begin{aligned}
Y =
O
+
\left\{
\cos(\theta) e_1 + \sin(\theta) e_2,
~~\theta \in [0, 2\pi)
\right \}.\end{aligned}$$ We see that $Y$ is a circle, of radius $\eucN{e_1} = \eucN{e_2} = \sqrt{1 + \frac{\gamma^2}{2}}$.
Let $E$ denotes the affine space with origin $O$ and spanned by the vectors $e_1$ and $e_2$. Lemma \[lem:persistence\_lift\], stated below, shows that the persistent cohomology of $Y$, seen in the ambient space $\R^2 \times \matrixspace{\R^2}$, is the same as the persistent cohomology of $Y$ restricted to the subspace $E$. As a consequence, $Y$ has the same persistence as a circle of radius $\sqrt{1 + \frac{\gamma^2}{2}}$ in the plane. Its barcode can be described as follows:
- one $H^0$-feature: the bar $[0, +\infty)$,
- one $H^1$-feature: the bar $\left[0, \sqrt{1 + \frac{\gamma^2}{2}}\right)$.
\[lem:persistence\_lift\] Let $Y \subset \R^n$ be any subset, and define $\checkY = Y \times\{(0, ..., 0)\} \subset \R^n \times \R^m$. Let these spaces be endowed with the usual Euclidean norms. Then the Čech filtrations of $Y$ and $\checkY$ yields isomorphic persistence modules.
Let $\mathrm{proj}_n\colon \R^n \times \R^m \rightarrow \R^n$ be the projection on the first $n$ coordinates. One verifies that, for every $t \geq 0$, the map $\mathrm{proj}_n \colon\checkY^t \rightarrow Y^t$ is a homotopy equivalence. At cohomology level, these maps induce an isomorphism of persistence modules.
Let us now study the Čech bundle filtration of $Y$, denoted $(\Y, \p)$. According to Equation \[eq:tmax\_subset\_grass\], its filtration maximal value is $\tmax{Y} = \tmaxgamma{X} = \frac{\gamma}{\sqrt{2}}$. Note that $\frac{\gamma}{\sqrt{2}}$ is lower than $\sqrt{1+ \frac{\gamma^2}{2}}$, which is the radius of the circle $Y$. Hence, for $t < \tmax{Y}$, the inclusion $Y \hookrightarrow Y^t$ is a homotopy equivalence. Consider the following commutative diagram:
Y & & Y\^[t]{}\
& &
It induces the following diagram in cohomology:
H\^\*(Y) & & H\^\*(Y\^[t]{})\
& H\^\*() &
The horizontal arrow is an isomorphism. Hence the map $(p^t)^*\colon H^*(Y^t) \leftarrow H^*(\Grass{1}{\R^2})$ is equal to $(p^0)^*$. We only have to understand $(p^0)^*$.
Remark that the map $p^0\colon Y \rightarrow \Grass{1}{\R^2}$ can be seen as the universal bundle of the circle. Therefore $(p^0)^*\colon H^*(Y) \leftarrow H^*(\Grass{1}{\R^m})$ is nontrivial. Alternatively, $p^0$ can be seen as a map between two circles. It is injective, hence its degree (modulo 2) is one. We still deduce that $(p^0)^*$ is nontrivial. As a consequence, the persistent Stiefel-Whitney class $w_1^t(X)$ is nonzero for every $t< \tmax{Y}$.
Study of Example \[ex:gamma\_normal\] {#subsec:gamma_normal}
-------------------------------------
We consider the set $$\begin{aligned}
&X = \bigg\{
\bigg(
\begin{pmatrix}
\cos(\theta) \\
\sin(\theta)
\end{pmatrix}
,
\begin{pmatrix}
\cos(\theta)^2 & \cos(\theta) \sin(\theta) \\
\cos(\theta) \sin(\theta) & \sin(\theta)^2
\end{pmatrix}
\bigg),
\theta \in [0, 2\pi) \bigg\}.\end{aligned}$$ As we explained in the previous subsection, the Čech filtration of $X$ with respect to the norm $\gammaN{\cdot}$ yields the same persistence as the Čech filtration of $Y$ with respect to the usual norm $\eucN{\cdot}$, where $$\begin{aligned}
&Y = \bigg\{
\bigg(
\begin{pmatrix}
\cos(\theta) \\
\sin(\theta)
\end{pmatrix}
,
\gamma \begin{pmatrix}
\cos(\theta)^2 & \cos(\theta) \sin(\theta) \\
\cos(\theta) \sin(\theta) & \sin(\theta)^2
\end{pmatrix}
\bigg),
\theta \in [0, 2\pi) \bigg\}.\end{aligned}$$
Notice that $Y$ is a subset of the affine subspace of dimension 4 of $\R^2 \times \matrixspace{\R^2}$ with origin $O = \left( \begin{psmallmatrix} 0 \\ 0 \end{psmallmatrix}, \frac{1}{2}\begin{psmallmatrix} 1 & 0 \\ 0 & 1 \end{psmallmatrix} \right)$ and spanned by the vectors $e_1, e_2, e_3$ and $e_4$, where $$\begin{aligned}
e_1 = &\left( \begin{pmatrix} 1 \\ 0 \end{pmatrix},
\begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix} \right),
~~~~~~~~~~~~~~e_2 = \left( \begin{pmatrix} 0 \\ 1 \end{pmatrix},
\begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix} \right), \\
e_3 = \frac{1}{\sqrt{2}} &\left( \begin{pmatrix} 0 \\ 0 \end{pmatrix},
\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} \right),
~~~~~~~e_4 = \frac{1}{\sqrt{2}} \left( \begin{pmatrix} 0 \\ 0 \end{pmatrix},
\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \right).\end{aligned}$$ Indeed, $Y$ can be written as $$\begin{aligned}
Y = O +
\left\{
\cos(\theta) e_1 + \sin(\theta) e_2 + \frac{\gamma}{\sqrt{2}}\cos(2\theta) e_3 + \frac{\gamma}{\sqrt{2}}\sin(2\theta) e_4,
~~\theta \in [0, 2\pi)
\right \}.\end{aligned}$$ This comes from the equality $$\begin{aligned}
\begin{pmatrix}
\cos(\theta)^2 & \cos(\theta) \sin(\theta) \\
\cos(\theta) \sin(\theta) & \sin(\theta)^2
\end{pmatrix}
=
\frac{1}{2}\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}
+
\frac{1}{2}
\begin{pmatrix}
\cos(2\theta) & \sin(2\theta) \\
\sin(2\theta) & -\cos(2\theta)
\end{pmatrix}.\end{aligned}$$ Observe that $Y$ is a torus knot, i.e. a simple closed curve included in the torus $\T$, defined as $$\begin{aligned}
\T =
O
+
\left\{
\cos(\theta) e_1 + \sin(\theta) e_2 + \frac{\gamma}{\sqrt{2}}\cos(\nu) e_3 + \frac{\gamma}{\sqrt{2}}\sin(\nu) e_4,
~~\theta, \nu \in [0, 2\pi)
\right \}.\end{aligned}$$ The curve $Y$ winds one time around the first circle of the torus, and two times around the second one. It is known as the torus knot $(1,2)$.
Let $E$ denotes the affine subspace with origin $O$ and spanned by $e_1, e_2, e_3, e_4$. Since $Y$ is a subset of $E$, it is equivalent to study the Čech filtration of $Y$ restricted to this subset (as in Lemma \[lem:persistence\_lift\]). We will denote the coordinates of points $x \in E$ with respect to the orthonormal basis $(e_1, e_2, e_3, e_4)$. That is, a tuple $(x_1, x_2, x_3, x_4)$ will refer to the point $O+x_1 e_1 + x_2e_2 + x_3e_3 + x_4 e_4$ of $E$. Seen in $E$, the set $Y$ can be written as $$\begin{aligned}
Y =
\left\{\left(\cos(\theta), \sin(\theta),
\frac{\gamma}{\sqrt{2}}\cos(2\theta), \frac{\gamma}{\sqrt{2}}\sin(2\theta)\right), \theta \in [0, 2\pi)\right\}.\end{aligned}$$ For every $\theta \in [0, 2\pi)$, we will denote $y_\theta = \left(\cos(\theta), \sin(\theta), \frac{\gamma}{\sqrt{2}} \cos(2\theta), \frac{\gamma}{\sqrt{2}} \sin(2\theta)\right)$.
{width=".5\linewidth"}
{width=".6\linewidth"}
We now state two lemmas that will be useful in what follows.
\[lem:torusknot\_diameter\] For every $\theta \in [0, 2\pi)$, the map $\theta' \mapsto \eucN{y_{\theta} - y_{\theta'}}$ admits the following critical points:
- $\theta'-\theta = 0$ and $\theta'-\theta = \pi$ if $\gamma \leq \frac{1}{\sqrt{2}}$,
- $\theta'-\theta = 0$, $\pi$, $\arccos(-\frac{1}{2\gamma^2})$ and $-\arccos(-\frac{1}{2\gamma^2})$ if $\gamma \geq \frac{1}{\sqrt{2}}$.
They correspond to the values
- $\eucN{y_{\theta} - y_{\theta'}} = 0$ if $\theta'-\theta = 0$,
- $\eucN{y_{\theta} - y_{\theta'}} = 2$ if $\theta'-\theta = \pi$,
- $\eucN{y_{\theta} - y_{\theta'}} = \sqrt{2}\sqrt{1 + \gamma^2 + \frac{1}{4 \gamma^2}}$ if $\theta'-\theta = \pm \arccos(-\frac{1}{2\gamma^2})$.
Moreover, we have $\sqrt{2}\sqrt{1 + \gamma^2 + \frac{1}{4 \gamma^2}} \geq 2$ when $\gamma \geq \frac{1}{\sqrt{2}}$.
Let $\theta, \theta' \in [0, 2\pi)$. One computes that $$\eucN{y_{\theta} - y_{\theta'}}^2
= 4 \sin^2\left(\frac{\theta-\theta'}{2}\right) + 2 \gamma^2 \sin^2(\theta-\theta').$$ Consider the map $f\colon x \in [0, 2\pi) \mapsto 4 \sin^2\left(\frac{x}{2}\right) + 2 \gamma^2 \sin^2(x)$. Its derivative is $$\begin{aligned}
f'(x)
&= 4 \cos\left(\frac{x}{2}\right) \sin\left(\frac{x}{2}\right) + 4 \gamma^2 \cos(x) \sin(x) \\
&= 2 \sin(x)\left(1+2\gamma^2 \cos(x)\right).\end{aligned}$$ It vanishes when $x = 0$, $x = \pi$, or $x = \pm \arccos(-\frac{1}{2\gamma^2})$ if $\gamma \geq \frac{1}{\sqrt{2}}$. A computation shows that $f(0) = 0$, $f(\pi) = 4$ and $f\left(\pm \arccos\left(-\frac{1}{2\gamma^2}\right) \right) = 2\left( 1 + \gamma^2 + \frac{1}{4\gamma^2}\right)$.
\[lem:torusknot\_distance\] For every $x \in E$ such that $x \neq 0$, the map $\theta \mapsto \eucN{x-y_\theta}$ admits at most two local maxima and two local minima.
Consider the map $g\colon \theta \in [0, 2\pi) \mapsto \eucN{x-y_\theta}^2$. It can be written as $$\begin{aligned}
g(\theta)
&= \eucN{x}^2 + \eucN{y_\theta}^2 - 2\eucP{x}{y_\theta} \\
&= \eucN{x}^2 + 1 + \frac{\gamma^2}{2} - 2\eucP{x}{y_\theta}.\end{aligned}$$ Let us show that its derivative $g'$ vanishes at most four times on $[0,2\pi)$, which would show the result. Using the expression of $y_\theta$, we see that $g'$ can be written as $$\begin{aligned}
g'(\theta)
= a \cos(\theta) + b \sin(\theta)+ c \cos(2\theta)+ d \sin (2\theta),\end{aligned}$$ where $a,b,c,d \in \R$ are not all zero. Denoting $\omega = \cos(\theta)$ and $\xi = \sin(\theta)$, we have $\xi^2 = 1-\omega^2$, $\cos(2 \theta) = \cos^2(\theta) - \sin^2(\theta) = 2\omega^2 -1$ and $\sin(2\theta) = 2 \cos(\theta) \sin(\theta) = 2\omega\xi$. Hence $$\begin{aligned}
g'(\theta)
= a \omega + b \xi + 2 c \omega^2 + 2d \omega \xi.\end{aligned}$$ Now, if $g'(\theta) = 0$, we get $$\label{eq:torusknot_distance}
a \omega + 2 c \omega^2 = -( b + 2d \omega) \xi$$ Squaring this equality yields $\left(a \omega + 2 c \omega^2\right)^2
= \left( b + 2d \omega \right)^2 (1-\omega^2)$. This degree four equation, with variable $\omega$, admits at most four roots. To each of these $w$, there exists a unique $\xi = \pm \sqrt{1-w^2}$ that satisfies Equation \[eq:torusknot\_distance\]. In other words, the corresponding $\theta \in [0,2\pi)$ such that $\omega = \cos(\theta)$ is unique. We deduce the result.
Before studying the Čech filtration of $Y$, let us describe some geometric quantities associated to it. Using a symbolic computation software, we see that the curvature of $Y$ is contant and equal to $$\begin{aligned}
\rho = \frac{\sqrt{1+8\gamma^2}}{1+2\gamma^2}.\end{aligned}$$ In particular, we have $\rho \geq 1$ if $\gamma \leq 1$, and $\rho < 1$ if $\gamma > 1$. We also have an expression for the diameter of $Y$: $$\begin{aligned}
\frac{1}{2}\diam{Y}
= \left\{
\begin{array}{ll}
1 & \mbox{ if } \gamma \leq \frac{1}{\sqrt{2}}, \\
\frac{1}{\sqrt{2}} \sqrt{1 + \gamma^2 + \frac{1}{4 \gamma^2}} & \mbox{ if } \gamma \geq \frac{1}{\sqrt{2}}.
\end{array}
\right.\end{aligned}$$ It is a consequence of Lemma \[lem:torusknot\_diameter\]. We now describe the reach of $Y$: $$\label{eq:reach_Y}
\reach{Y}
= \left\{
\begin{array}{ll}
\frac{1+2\gamma^2}{\sqrt{1+8\gamma^2}} & \mbox{ if } \gamma \leq 1, \\
1 & \mbox{ if } \gamma \geq 1.
\end{array}
\right.$$ Let us prove this by using [@aamari:hal-01521955 Theorem 3.4]. We define a bottleneck of $Y$ as pair of distinct points $(y,y') \in Y^2$ such that the open ball $\openball{\frac{1}{2}(y+y')}{\frac{1}{2}\eucN{y-y'}}$ does not intersect $Y$. Its length is defined as $\frac{1}{2}\eucN{y-y'}$. Then the reach of $Y$ is equal to $$\begin{aligned}
\reach{Y} = \min\left\{\frac{1}{\rho}, \delta\right\},\end{aligned}$$ where $\frac{1}{\rho}$ is the inverse curvature of $Y$, and $\delta$ is the minimal length of bottlenecks of $Y$. As we computed, $\frac{1}{\rho}$ is equal to $\frac{1+2\gamma^2}{\sqrt{1+8\gamma^2}} $. Besides, according to Lemma \[lem:torusknot\_diameter\], a bottleneck $(y_\theta, y_{\theta'})$ has to satisfy $\theta'-\theta = \pi$ or $\pm\arccos(-\frac{1}{2\gamma^2})$. The smallest length is attained when $\theta'-\theta = \pi$, for which $\frac{1}{2}\eucN{y_\theta - y_{\theta'}}=1$. It is straightforward to verify that the pair $(y_\theta, y_{\theta'})$ is indeed a bottleneck. Therefore we have $\delta=1$, and we deduce the expression of $\reach{Y}$.
Last, the weak feature size of $Y$ does not depend on $\gamma$ and is equal to 1: $$\label{eq:torusknot_wfs}
\wfs{Y} = 1.$$ We will prove it by using the characterization of Subsection \[subsec:background\_persistentcohomology\]: $\wfs{Y}$ is the infimum of distances $\dist{x}{Y}$, where $x \in E$ is a critical point of the distance function $d_Y$. In this context, $x$ is a critical point if it lies in the convex hull of its projections on $Y$. Remark that, if $x \neq 0$, then $x$ admits at most two projections on $Y$. This follows from Lemma \[lem:torusknot\_distance\]. As a consequence, if $x$ is a critical point, then there exists $y, y' \in Y$ such that $x$ lies in the middle of the segment $[y,y']$, and the open ball $\openball{x}{\dist{x}{Y}}$ does not intersect $Y$. Therefore $y'$ is a critical point of $y' \mapsto \eucN{y-y'}$, hence Lemma \[lem:torusknot\_diameter\] gives that $\eucN{y-y'} \geq 2$. We deduce the result.
We now describe the thickenings $Y^t$. They present four different behaviours:
- $0\leq t<1$: $Y^t$ is homotopy equivalent to a circle,
- $1 \leq t < \frac{1}{2}\diam{Y}$: $Y^t$ is homotopy equivalent to a circle,
- $ \frac{1}{2}\diam{Y} \leq t < \sqrt{1+\frac{\gamma^2}{2}}$: $Y^t$ is homotopy equivalent to a 3-sphere,
- $ t \geq \sqrt{1+\frac{\gamma^2}{2}}$: $Y^t$ is homotopy equivalent to a point.
Recall that, in the case where $\gamma \leq \frac{1}{\sqrt{2}}$, we have $\frac{1}{2}\diam{Y} = 1$. Consequently, the interval $\left[1,\frac{1}{2}\diam{Y}\right)$ is empty, and the second point does not appear in this case.
#### Study of the case $0\leq t<1$.
For $t \in [0,1)$, let us show that $Y^t$ deform retracts on $Y$. According to Equation \[eq:torusknot\_wfs\], we have $\wfs{Y} = 1$. Moreover, Equation \[eq:reach\_Y\] gives that $\reach{Y} > 0$. Using the results of Subsection \[subsec:background\_persistentcohomology\], we deduce that $Y^t$ is isotopic to $Y$.
#### Study of the case $1 \leq t < \frac{1}{2}\diam{Y}$.
Denote $z_\theta = \left(0,0,\frac{\gamma}{\sqrt{2}}\cos(2\theta), \frac{\gamma}{\sqrt{2}}\sin(2\theta)\right)$, and define the circle $Z = \left\{ z_\theta, \theta \in [0,\pi) \right\}$.
{width=".25\linewidth"}
We claim that $Y^t$ deform retracts on $Z$. To prove so, we will define a continuous application $f\colon Y^t \rightarrow Z$ such that, for every $y\in Y^t$, the segment $[y, f(y)]$ is included in $Y^t$. This would lead to a deformation retraction of $Y^t$ onto $Z$, via $$\begin{aligned}
(s,y) \in [0,1]\times Y^t \mapsto (1-s)y + s f(y).\end{aligned}$$ Equivalently, we will define an application $\Theta\colon Y^t \rightarrow [0,\pi)$ such that the segment $[y, z_{\Theta(y)}]$ is included in $Y^t$.
Let $y \in Y^t$. According to Lemma \[lem:torusknot\_distance\], $y$ admits at most two projection on $Y$. We start with the case where $y$ admits only one projection, namely $y_{\theta}$ with $\theta \in [0, 2\pi)$. Let $\overline{\theta} \in [0, \pi)$ be the reduction of $\theta$ modulo $\pi$, and consider the point $z_{\overline{\theta}}$ of $Z$. A computation shows that the distance $\eucN{y_\theta - z_{\overline{\theta}}}$ is equal to 1. Besides, since $y \in Y^t$, the distance $\eucN{y_\theta - y}$ is at most $t$. By convexity, the segment $\left[y, z_{\overline{\theta}}\right]$ is included in the ball $\closedball{y_\theta}{t}$, which is a subset of $Y^t$. We then define $\Theta(y)= \overline{\theta}$.
Now suppose that $y$ admits exactly two projection $y_\theta$ and $y_{\theta'}$. According to Lemma \[lem:torusknot\_diameter\], these angles must satisfy $\theta'-\theta = \pi$. Indeed, the case $\eucN{y_{\theta} - y_{\theta'}} = \sqrt{2}\sqrt{1 + \gamma^2 + \frac{1}{4 \gamma^2}}$ does not occur since we chose $t < \frac{1}{2}\diam{Y} = \frac{\sqrt{2}}{2}\sqrt{1 + \gamma^2 + \frac{1}{4 \gamma^2}}$. The angles $\theta$ and $\theta'$ correspond to the same reduction modulo $\pi$, denoted $\overline{\theta}$, and we also define $\Theta(y)= \overline{\theta}$.
#### Study of the case $t\in \left[ \frac{1}{2}\diam{X}, \sqrt{1+\frac{\gamma^2}{2}}\right)$.
Let $\S_3$ denotes the unit sphere of $E$. For every $v = (v_1,v_2,v_3,v_4) \in \S_3$, we will denote by $\langle v \rangle$ the linear subspace spanned by $v$, and by $\langle v \rangle_+$ the cone $\{\lambda v, \lambda \geq 0\}$. Moreover, we define the quantity $$\begin{aligned}
\delta(v) = \min_{y \in Y} \dist{y}{\langle v \rangle_+}.\end{aligned}$$ and the set $$\begin{aligned}
S = \left\{ \delta(v) v, v \in \S_3 \right\}.\end{aligned}$$ We claim that $S$ is a subset of $Y^t$, and that $Y^t$ deform retracts on it. This follows from the two following facts: for every $v \in \S_3$,
1. $\delta(v)$ is not greater than $\frac{1}{2}\diam{Y}$,
2. $\langle v \rangle_+ \cap Y^t$ consists of one connected component: an interval centered on $\delta(v)v$, that does not contains the point 0.
Suppose that these assertions are true. Then one defines a deformation retraction of $Y^t$ on $S$ by retracting each fiber $\langle v \rangle_+ \cap Y^t$ linearly on the singleton $\{ \delta(v)v \}$. We will now prove the two items.
{width=".3\linewidth"}
*Item 1.* Note that Item 1 can be reformulated as follows: $$\label{eq:torusknot_maxmin_plus}
\max_{v \in \S_3} \min_{y \in Y} \dist{y}{\langle v \rangle_+}
\leq \frac{1}{2} \diam{Y}.$$ Let us justify that the pairs $(v,y)$ that attain this maximum-minimum are the same as in $$\label{eq:torusknot_maxmin_norm}
\max_{v \in \S_3} \min_{y \in Y} \eucN{y-v}.$$ From the definition of $Y = \{y_\theta, \theta \in [0,2\pi)\}$, we see that $\min_{y \in Y} \dist{y}{\langle v \rangle_+} = \min_{y \in Y} \dist{y}{\langle v \rangle}$. A vector $v \in \S_3$ being fixed, let us show that $y\mapsto \dist{y}{\langle v \rangle}$ is minimized when $y \mapsto \eucN{v-y}$ is. Let $y\in Y$. Since $v$ is a unit vector, the projection of $y$ on $\langle v \rangle$ can be written as $\eucP{y}{v}v$. Hence $\dist{y}{\langle v \rangle}^2 = \eucN{\eucP{y}{v}v - y}^2$, and expanding this norm yields $$\begin{aligned}
\dist{y}{\langle v \rangle}^2 = \eucN{y}^2-\eucP{y}{v}^2.\end{aligned}$$ Expanding the norm $\eucN{y-v}^2$ and using that $\eucN{y}^2 = 1 + \frac{\gamma^2}{2}$, we get $\eucP{y}{v} = \frac{1}{2}\left( 2 + \frac{\gamma^2}{2} - \eucN{y-v}^2\right)$. We inject this relation in the preceding equation to obtain $$\begin{aligned}
\dist{y}{\langle v \rangle}^2
= -\left(\frac{\gamma}{2}\right)^4+\gamma^2 + \frac{1}{4}\eucN{y-v}^2\left(4+\gamma^2 -\eucN{y-v}^2\right).
$$ Now we can deduce that $y \mapsto \dist{y}{\langle v \rangle}^2$ is minimized when $y \mapsto \eucN{y-v}$ is minimized. Indeed, the map $\eucN{y-v}\mapsto \frac{1}{4}\eucN{y-v}^2\left(4+\gamma^2 -\eucN{y-v}^2\right)$ is increasing on $\left[0, \frac{1}{2}(4+\gamma^2)\right]$. But $\eucN{y-v} \leq \eucN{y} + \eucN{v} = \frac{1}{2}(4+\gamma^2)$. We deduce that that studying the left hand term of Equation \[eq:torusknot\_maxmin\_plus\] is equivalent to studying Equation \[eq:torusknot\_maxmin\_norm\]. We will denote by $g\colon \S_3 \rightarrow \R$ the map $$\label{eq:torusknot_g}
g(v) = \min_{y \in Y} \eucN{y-v}.$$
Let $v \in \S_3$ that attains the maximum of $g$, and let $y$ be a corresponding point that attains the minimum of $\eucN{y-v}$. The points $v$ and $y$ attains the quantity in Equation \[eq:torusknot\_maxmin\_plus\]. In order to prove that $\dist{y}{\langle v \rangle} \leq \frac{1}{2}\diam{Y}$, let $p(y)$ denotes the projection of $y$ on $\langle v \rangle$. We will show that there exists another point $y' \in Y$ such that $p(y)$ is equal to $\frac{1}{2}(y+y')$ Consequently, we would have $\eucN{y - p(y)} = \frac{1}{2}\eucN{y' - y} \leq \frac{1}{2}\diam{Y}$, i.e. $$\begin{aligned}
\dist{y}{\langle v \rangle} \leq \frac{1}{2}\diam{Y}.\end{aligned}$$
Remark the following fact: if $w \in \S_3$ is a unit vector such that $\eucP{p(y)-y}{w} >0$, then for $\epsilon>0$ small enough, we have $$\begin{aligned}
\dist{y}{\langle v+\epsilon w \rangle} > \dist{y}{\langle v \rangle}. \end{aligned}$$ Equivalently, this statement reformulates as $0 \leq \eucP{y}{\frac{1}{\eucN{v+\epsilon w}}(v+\epsilon w)} < \eucP{y}{v}$. Let us show that $$\label{eq:torusknot_eucP}
\eucP{y}{\frac{1}{\eucN{v+\epsilon w}}(v+\epsilon w)}
= \eucP{y}{v} - \epsilon \kappa + \petito{\epsilon},$$ where $\kappa = \eucP{p(y)-y}{w} > 0$, and where $\petito{\epsilon}$ is the little-o notation. Note that $\frac{1}{\eucN{v+\epsilon w}} = 1-\epsilon \eucP{v}{w} + \petito{\epsilon}$. We also have $$\begin{aligned}
\frac{1}{\eucN{v+\epsilon w}}(v+\epsilon w)
&= v + \epsilon \left(w-\eucP{v}{w}v \right) + \petito{\epsilon}.
$$ Expanding the inner product in Equation \[eq:torusknot\_eucP\] gives $$\begin{aligned}
\eucP{y}{\frac{1}{\eucN{v+\epsilon w}}(v+\epsilon w)}
&= \eucP{y}{v} + \epsilon \left( \eucP{y}{w} - \epsilon\eucP{v}{w} \eucP{y}{v} \right) + \petito{\epsilon} \\
&= \eucP{y}{v} + \epsilon \eucP{ y-\eucP{y}{v}v }{w} +\petito{\epsilon} \\
&= \eucP{y}{v} + \epsilon \eucP{ y-p(y) }{w} +\petito{\epsilon},\end{aligned}$$ and we obtain the result.
Next, let us prove that $y$ is not the only point of $Y$ that attains the minimum in Equation \[eq:torusknot\_g\]. Suppose that it is the case by contradiction. Let $w \in \S_3$ be a unit vector such that $\eucP{p(y)-y}{w} >0$. For $\epsilon$ small enough, let us prove that the vector $v' = \frac{1}{\eucN{v+\epsilon w}}(v+\epsilon w)$ of $\S_3$ contradicts the maximality of $v$. That is, let us prove that $g(v') > g(v)$. Let $y' \in Y$ be a minimizer $\eucN{y'-v'}$. We have to show that $\eucN{y'-v'} > \eucN{y-v}$. This would lead to $g(v') > g(v)$, hence the contradiction. Expanding the norm yields $$\begin{aligned}
\eucN{v'-y'}^2
= \eucN{v'-v+v-y'}^2
&\geq \eucN{v'-v}^2 + \eucN{v-y'}^2 -2\eucP{v'-v}{v-y'}.\end{aligned}$$ Using $\eucN{v'-v}^2 \geq 0$ and $\eucN{v-y'}^2 \geq \eucN{v-y}^2$ by definition of $y$, we obtain $$\begin{aligned}
\eucN{v'-y'}^2
\geq \eucN{v-y}^2 -2\eucP{v'-v}{v-y'}.\end{aligned}$$ We have to show that $\eucP{v'-v}{y-y'}$ is positive for $\epsilon$ small enough. By writing $v-y' = v-y+(y-y')$ we get $$\begin{aligned}
\eucP{v'-v}{v-y'}
&= \eucP{v'-v}{v} - \eucP{v'-v}{y} + \eucP{v'-v}{y-y'}\end{aligned}$$ According to Equation \[eq:torusknot\_eucP\], $- \eucP{v'-v}{y} = \epsilon \kappa + \petito{\epsilon}$. Besides, using $v' - v = \epsilon(w-\eucP{v}{w}v) + \petito{\epsilon}$, we get $\eucP{v'-v}{v} = \petito{\epsilon}$. Last, Cauchy-Schwarz inequality gives $|\eucP{v'-v}{y-y'}| \leq \eucN{v'-v}\eucN{y-y'}$. Therefore, $\eucP{v'-v}{y-y'} = \grando{\epsilon}\eucN{y-y'}$, where $\grando{\epsilon}$ is the big-o notation. Gathering these three equalities, we obtain $$\begin{aligned}
\eucP{v'-v}{v-y'}
&= \petito{\epsilon} + \epsilon \kappa + \grando{\epsilon}\eucN{y-y'}.\end{aligned}$$ As we can read from this equation, if $\eucN{y-y'}$ goes to zero as $\epsilon$ does, then $\eucP{v'-v}{v-y'}$ is positive for $\epsilon$ small enough. Observe that $v'$ goes to $v$ when $\epsilon$ goes to 0. By assumption $y$ is the only minimizer in Equation \[eq:torusknot\_g\]. By continuity of $g$, we deduce that $y'$ goes to $y$.
By contradiction, we deduce that there exists another point $y'$ which attains the minimum in $g(v)$. Note that it is the only other one, according to Lemma \[lem:torusknot\_distance\]. Let us show that $p(y)$ lies in the middle of the segment $[y,y']$. Suppose that it is not the case. Then $p(y)-y$ is not equal to $-(p(y')-y')$, where $p(y')$ denotes the projection of $y'$ on $\langle v \rangle$. Consequently, the half-spaces $\{w \in E, \eucP{p(y)-y}{w} > 0\}$ and $\{w \in E, \eucP{p(y')-y'}{w} > 0\}$ intersects. Let $w$ be any vector in the intersection. For $\epsilon>0$, denote $v' = \frac{1}{\eucN{v+\epsilon w}}(1+\epsilon w)$. If $\epsilon$ is small enough, the same reasoning as before shows that $v'$ contradicts the maximality of $v$.
![Left: Representation of the situation where $y$ and $y'$ are minimizers of Equation \[eq:torusknot\_g\]. Right: Representation in the plane passing through the points $y$, $y'$ and $p(y)$. The dashed area corresponds to the intersection of the half-spaces $\{w \in E, \eucP{p(y)-y}{w} > 0\}$ and $\{w \in E, \eucP{p(y')-y'}{w} > 0\}$.](example2_2.png){width=".6\linewidth"}
![Left: Representation of the situation where $y$ and $y'$ are minimizers of Equation \[eq:torusknot\_g\]. Right: Representation in the plane passing through the points $y$, $y'$ and $p(y)$. The dashed area corresponds to the intersection of the half-spaces $\{w \in E, \eucP{p(y)-y}{w} > 0\}$ and $\{w \in E, \eucP{p(y')-y'}{w} > 0\}$.](example2_3.png){width=".7\linewidth"}
*Item 2.* Let $v \in \S_3$. The set $\langle v \rangle_+ \cap Y^t$ can be described as $$\begin{aligned}
\langle v \rangle_+ \cap \bigcup_{y \in Y} \closedball{y}{t}.\end{aligned}$$ Let $y \in Y$ such that $\langle v \rangle_+ \cap \closedball{y}{t} \neq \emptyset$. Denote by $p(y)$ the projection of $y$ on $\langle v \rangle_+$. It is equal to $\eucP{y}{v}v$. Using Pythagoras’ theorem, we obtain that the set $\langle v \rangle_+ \cap \closedball{y}{t}$ is equal to the interval $$\begin{aligned}
\left[ p(y) \pm \sqrt{t^2 - \dist{y}{\langle v \rangle}^2} v \right].\end{aligned}$$ Using the identity $\dist{y}{\langle v \rangle}^2 = \eucN{y} - \eucP{y}{v}^2 = 1 + \frac{\gamma^2}{2} - \eucP{y}{v}^2$, we can write this interval as $$\begin{aligned}
\big[I_1(y) \cdot v, ~ I_2(y) \cdot v\big],\end{aligned}$$ where $I_1(y) = \eucP{y}{v} - \sqrt{ \eucP{y}{v}^2 - (1 + \frac{\gamma^2}{2} -t^2)}$ and $I_2(y) = \eucP{y}{v} + \sqrt{ \eucP{y}{v}^2 - (1 + \frac{\gamma^2}{2} -t^2)}$. Seen as functions of $\eucP{y}{v}$, the map $I_1$ is decreasing, and the map $I_2$ is increasing (see Figure \[fig:I1andI2\]). Let $y^* \in Y$ that minimizes $\dist{y}{\langle v \rangle}$. Equivalently, $y^*$ maximizes $\langle y,v\rangle$. It follows that the corresponding interval $\big[I_1(y^*) \cdot v, ~ I_2(y^*) \cdot v\big]$ contains all the others. We deduce that the set $\langle v \rangle_+ \cap Y^t$ is equal to this interval.
![Left: Representation of two intervals $\langle v \rangle_+ \cup \closedball{y}{t}$ and $\langle v \rangle_+ \cup \closedball{y'}{t}$. Right: Representation of the maps $x\mapsto x \pm \sqrt{x^2-1}$.[]{data-label="fig:I1andI2"}](example2_4.png){width=".6\linewidth"}
![Left: Representation of two intervals $\langle v \rangle_+ \cup \closedball{y}{t}$ and $\langle v \rangle_+ \cup \closedball{y'}{t}$. Right: Representation of the maps $x\mapsto x \pm \sqrt{x^2-1}$.[]{data-label="fig:I1andI2"}](example2_5.png){width=".8\linewidth"}
#### Study of the case $t \geq \sqrt{1+\frac{1}{2}\gamma^2}$.
For every $y \in Y$, we have $\eucN{y} = \sqrt{1+\frac{1}{2}\gamma^2}$. Therefore, if $t \geq \sqrt{1+\frac{1}{2}\gamma^2}$, then $Y^t$ is star shaped around the point 0, hence it deform retracts on it.
#### Čech bundle filtration of $Y$.
To close this subsection, let us study the Čech bundle filtration $(\Y, \p)$ of $Y$. According to Equation \[eq:tmax\_subset\_grass\], its filtration maximal value is $\tmax{Y} = \tmaxgamma{X} = \frac{\gamma}{\sqrt{2}}$. Note that $\frac{\gamma}{\sqrt{2}}$ is lower than $\frac{1}{2}\diam{Y}$. Consequently, only two cases are to be studied: $t \in [0,1)$, and $t \in \left[1, \frac{1}{2}\diam{Y} \right)$.
The same argument as in Subsection \[subsec:gamma\_normal\] yields that for every $t \in [0,1)$, the persistent Stiefel-Whitney class $w_1^t(Y)$ is equal to $w_1^0(Y)$. Accordingly, for every $t \in \left[1, \frac{1}{2}\diam{Y} \right)$, the class $w_1^t(Y)$ is equal to $w_1^1(Y)$. Let us show that $w_1^0(Y)$ is zero, and that $w_1^1(Y)$ is not.
First, remark that the map $p^0\colon Y \rightarrow \Grass{1}{\R^2}$ can be seen as the normal bundle of the circle. Hence $(p^0)^*\colon H^*(Y) \leftarrow H^*(\Grass{1}{\R^2})$ is nontrivial, and we deduce that $w_1^0(Y)=0$. As a consequence, the persistent Stiefel-Whitney class $w_1^t(X)$ is nonzero for every $t < 1$.
Next, consider $p^1\colon Y^1 \rightarrow \Grass{1}{\R^2}$. Recall that $Y^1$ deform retracts on the circle $$\begin{aligned}
Z = \left\{ \left(0,0,\frac{\gamma}{\sqrt{2}}\cos(2\theta), \frac{\gamma}{\sqrt{2}}\sin(2\theta)\right), \theta \in [0,\pi) \right\}.\end{aligned}$$ Seen in $\R^2 \times \matrixspace{\R^2}$, we have $$\begin{aligned}
Z = \bigg\{
\bigg(
\begin{pmatrix}
0 \\
0
\end{pmatrix}
,
\gamma \begin{pmatrix}
\cos(\theta)^2 & \cos(\theta) \sin(\theta) \\
\cos(\theta) \sin(\theta) & \sin(\theta)^2
\end{pmatrix}
\bigg),
\theta \in [0, \pi) \bigg\}.\end{aligned}$$ Notice that the map $q\colon Z \rightarrow \Grass{1}{\R^2}$, the projection on $\Grass{1}{\R^2}$, is injective. Seen as a map between two circles, it has degree (modulo 2) equal to 1. We deduce that $q^*\colon H^*(Z) \leftarrow H^*(\Grass{1}{\R^2})$ is nontrivial. Now, remark that the map $q$ factorizes through $p^1$:
Z & & Y\^[1]{}\
& &
It induces the following diagram in cohomology:
H\^\*(Z) & & H\^\*(Y\^[1]{})\
& H\^\*() &
Since $q^*$ is nontrivial, this commutative diagram yields that the persistent Stiefel-Whitney class $w_1^1(Y)$ is nonzero. As a consequence, the persistent Stiefel-Whitney class $w_1^t(Y)$ is nonzero for every $t \geq 1$.
|
{
"pile_set_name": "ArXiv"
}
|
---
author:
- |
Tristan Dennen[^1], andYu-tin Huang[^2]\
\
\
*Department of Physics and Astronomy,\
UCLA,\
Los Angeles, CA 90095-1547, USA*
title: 'Dual Conformal Properties of Six-Dimensional Maximal Super Yang-Mills Amplitudes'
---
Introduction
============
Dual superconformal symmetry [@DualConformal; @DualConformal2] has played an important role in understanding the structure of planar four-dimensional $\mathcal{N}=4$ super-Yang-Mills (sYM) theory at both strong [@Strong] and weak coupling [@Weak; @anomaly]. In particular, the closure of the original and dual superconformal symmetries forms an infinite-dimensional Yangian symmetry [@Drummond:2009fd], which has been extremely useful in determining the planar amplitudes of four-dimensional $\mathcal{N}=4$ sYM [@Drummond:2008cr; @Bern:2006ew; @Korchemsky:2010ut; @ArkaniHamed:2010kv].
Because the realization of this symmetry relies heavily on four-dimensional twistor variables [@Hodges:2009hk; @ArkaniHamed:2009vw; @Mason:2009qx], it is not immediately apparent how the symmetry behaves away from four dimensions. This is an important question because the loop amplitudes are infrared divergent and require regularization in four dimensions, and the dimensional regulator breaks the symmetry [@DualConformal2; @anomaly; @anomaly2]. Generically, any regularization scheme will result in either altering the dimensionality or the massless condition of the external momenta, both of which are essential to the definition of twistors. While one can modify the dual symmetry generators to account for massive regulators [@Alday:2009zm], thus making the symmetry exact, it is a priori not apparent that such a symmetry should exist without explicit calculation of the loop amplitudes, although it is expected to exist.
To clarify these issues, six-dimensional four-point sYM multiloop amplitudes were recently set up [@Bern:2010qa] using the six-dimensional spinor helicity formalism and on-shell superspace of refs. [@CheungOConnell; @DHS]. If one restricts the external momenta to a four-dimensional subspace, these should correspond to four-dimensional $\mathcal{N}=4$ sYM amplitudes with loop momenta continued to six dimensions. Interestingly, four-dimensional dual conformal symmetry can be used to restrict the form of the multiloop planar integrand, and at four points, this integrand can be straightforwardly extended to six dimensions. Furthermore, the four-dimensional dual conformal boost generator can be extended to incorporate a massive regulator [@Drummond:2008cr], which can be interpreted as extra-dimensional momenta.
In ref. [@Bern:2010qa] it was conjectured that the six-dimensional maximal sYM $n$-point tree amplitude, when stripped of the momentum and supermomentum delta functions, transforms covariantly under dual conformal inversion. More precisely, the delta-function-independent part of the amplitude inverts with the same inversion weight on all external lines. The delta functions then introduce extra inversion weight due to the mismatch of mass dimensions of the momentum and supermomentum delta functions. This conjecture was checked explicitly against the simple four-point tree amplitude.
In this paper, we will show that the conjecture holds for all $n\geq4$–point tree amplitudes. We will establish the proof by induction; assuming that the $(n-1)$–point amplitude inverts covariantly, via BCFW recursion relations [@BCFW], the $n$–point amplitude will invert in the same way. This proof follows a similar line given for the four-dimensional $\mathcal{N}=4$ sYM theory in ref. [@Brandhuber:2008pf]. In addition, while this paper was in preparation, a tree-level proof of dual conformal symmetry of ten-dimensional sYM was given in ref. [@CaronHuot:2010rj]
At loop level, while it is expected that the six-dimensional loop integration measure spoils any dual conformal properties present at tree level, we can recover good behavior by restricting our attention to the integrand. Using the tree-level result, we will demonstrate that the multiloop planar integrands invert in the same fashion as in four dimensions; they are covariant with equal weight on all external lines, and with extra weight for the dual loop variables. We proceed by combining the tree-level result with the generalized unitarity method [@UnitarityMethod] to show that all planar cuts, after restoring the cut propagators, invert uniformly, and thus the planar multiloop integrand inverts in the same way.
By restricting the loop integration to a four-dimensional subspace, the six-dimensional maximal sYM amplitudes can be interpreted as four-dimensional massively regulated $\NeqFour$ sYM amplitudes. Furthermore, the four-dimensional loop integration measure inverts with the precise weight to cancel the extra weight of dual loop variables in the integrand. Because ultraviolet divergences are absent in four dimensions, and the massive regulator does not break the six-dimensional dual conformal symmetry, one concludes that the regulated $\NeqFour$ amplitude will obey the exact symmetry. Assuming cut constructability of the loop amplitudes, which is expected for maximally supersymmetric Yang-Mills, this demonstrates that the dual conformal symmetry is an exact symmetry of the planar amplitude of massively regulated $\NeqFour$ theory.
This paper is organized as follows: In , we give a brief review of the six-dimensional spinor helicity formalism, which provides a convenient set of on-shell variables for the representation of amplitudes. In , we introduce constraint equations which define dual coordinates in terms of the original on-shell coordinates. The dual conformal symmetry is then defined on these dual coordinates. Through the constraint equations, we are also able to define how the on-shell variables transform under dual conformal inversion. In , we prove the covariance of the tree-level amplitudes via induction using BCFW recursion. In , we use the generalized unitarity method [@UnitarityMethod] and the tree-level covariance to extend the result to loop level.
Review of spinor helicity in six dimensions {#SixDimHelicitySubSection}
===========================================
The six-dimensional spinor helicity formalism laid out in refs. [@CheungOConnell; @DHS] provides a convenient set of variables to represent six-dimensional massless theories. For a discussion of the spinor helicity formalism in general dimensions see ref. [@Boels:2009bv]. This formalism has been successfully applied to computations of loop amplitudes of the six-dimensional $\mathcal{N}=(1,1)$ sYM theory [@Bern:2010qa; @Brandhuber:2010mm].[^3]
The on-shell degrees of freedom of each external particle are described by the variables $$\left(\lambda_{i}^{Aa},\tilde{\lambda}_{iA{{\dot{a}}}},\eta_{ia},\tilde{\eta}_{i}^{{{\dot{a}}}}\right) \,,$$ subject to the constraint $$\lambda_i^{Aa}\lambda_{ia}^B = \frac{1}{2}\epsilon^{ABCD}\tilde\lambda_{iC{{\dot{a}}}}\tilde\lambda_{iD}^{{{\dot{a}}}} \,.
\label{lambdaconstraint}$$ The indices used here and throughout this paper represent various transformation properties, summarized in the following: $$\begin{aligned}
\hbox{SU$^*$(4) Lorentz group labels:}&& \quad A,B,C, \cdots = 1,2,3,4 \nn\\
\hbox{SU(2)$\times$SU(2) little group labels:} && \quad a,b,c, \cdots =1,2 \,
\hskip .3 cm \hbox{and} \hskip .3 cm {{\dot{a}}}, {{\dot{b}}}, {{\dot{c}}}, \cdots = 1,2 \,.\nn\\
\hbox{SO(5,1) vector labels:} && \quad \mu, \nu, \rho, \cdot\cdot\cdot = 0,1,2,\dots5\, \nn\\
\hbox{Particle/region labels:} && \quad i,j,k,r,s,l_i \,.\end{aligned}$$ The bosonic variables $\left(\lambda_{i}^{Aa},\tilde{\lambda}_{iA{{\dot{a}}}}\right)$ are related to the momentum via $$\begin{aligned}
& p_i^{AB}=\lambda_i^{A a}\,\epsilon_{ab}\lambda_i^{B b} \,, &
& p_{iAB}=\tilde{\lambda}_{iA{{\dot{a}}}}\,\epsilon^{{{\dot{a}}}{{\dot{b}}}}\tilde{\lambda}_{iB{{\dot{b}}}} \,,
\label{VectorSpinor}\end{aligned}$$ where the matrices ${\epsilon}_{ab}$ and ${\epsilon}^{{{\dot{a}}}{{\dot{b}}}}$ are the SU(2) little group metric, and the lowering and raising of the spinor variables are defined as $$\begin{aligned}
& \lambda_a = {\epsilon}_{ab}\lambda^b\,,
& &\tilde\lambda^{\dot{a}} = {\epsilon}^{\dot{a}\dot{b}}\tilde\lambda_{\dot{b}}\,,\end{aligned}$$ with ${\epsilon}_{12}=-1$, ${\epsilon}^{12}=1$. One can see that solves the massless condition $$p^2_i \propto {\epsilon}_{ABCD}p^{AB}_ip^{CD}_i = 0.$$ We represent the contraction between chiral and anti-chiral spinors as $$\lambda^{Aa}_{i}\tilde{\lambda}_{jA{{\dot{b}}}}=\langle i^a|j_{{{\dot{b}}}}].$$
The fermionic variables $\eta_{ia},\tilde{\eta}_{i}^{{{\dot{a}}}}$ carry the information of the on-shell states of the maximal sYM theory. More explicitly, the on-shell states correspond to the coefficients of the $\eta_{ia},\tilde{\eta}_{i}^{{{\dot{a}}}}$ expansion of the scalar superfield, $$\begin{aligned}
\Phi(\eta,\tilde{\eta}) &=&
\phi
+ \chi^a \eta_a
+ \phi'(\eta)^2
+ \tilde{\chi}_{{{\dot{a}}}}\tilde{\eta}^{{{\dot{a}}}}
+ g^a\,_{{{\dot{a}}}}\eta_a\tilde{\eta}^{{{\dot{a}}}}
+ \tilde{\psi}_{{{\dot{a}}}}(\eta)^2\tilde{\eta}^{{{\dot{a}}}} \nn\\
&& \null
+ \phi''(\tilde{\eta})^2
+ \psi^a\eta_a(\tilde{\eta})^2
+ \phi'''(\eta)^2(\tilde{\eta})^2 \,,\end{aligned}$$ where $(\eta)^2 \equiv\tfrac{1}{2}\epsilon^{ab}\eta_b\eta_a$ and $(\tilde{\eta})^2 \equiv\tfrac{1}{2}\epsilon_{\dot{a}\dot{b}}
\tilde{\eta}^{{{\dot{b}}}}\tilde{\eta}^{{{\dot{a}}}}$. Similar to the relationship between the spinor variables and the momenta $p_i$, one can solve the on-shell condition for supermomenta $q_i,\tilde{q}_i$ as $$\begin{aligned}
& q_i^A= \lambda^{Aa}_i\eta_{ia}\,,
&& \tilde{q}_{iA} = \tilde{\lambda}_{iA{{\dot{a}}}}\tilde{\eta}_{i}^{{{\dot{a}}}} \,.\end{aligned}$$ For $n\ge 4$, the superamplitude can be written as a function of $(p_i,q_i,\tilde q_i)$, $$\mathcal{A}_n=
\overalldelta{\E}
f_n(p_i,q_i,\tilde{q}_i) \,,
\label{factorization}$$ where here and throughout this paper, we use $\mathcal{E}$ to indicate the set of external legs, and the fermionic delta function is defined as $$\delta^4 \left(\sum_{i\in\mathcal{E}} q_i^A\right) \equiv
\frac{1}{4!} \, {\epsilon}_{BCDE} \,
\left(\sum_{i\in\mathcal{E}} q_i^B\right)
\left(\sum_{i\in\mathcal{E}} q_i^C\right)
\left(\sum_{i\in\mathcal{E}} q_i^D\right)
\left(\sum_{i\in\mathcal{E}}q_i^E\right) \,,$$ and similarly for the antichiral $\tilde{q}_A$.
Due to the special kinematics of the three-point amplitude, one introduces additional SU(2) variables which are related to the usual spinor variables as [@CheungOConnell] $$\begin{aligned}
&\langle i^a| i+1_{{{\dot{a}}}}]=u_{i}^{a}\tilde{u}_{i+1{{\dot{a}}}}, && \langle i^a| i-1_{{{\dot{a}}}}]=-u_{i}^{a}\tilde{u}_{i-1{{\dot{a}}}} \,.
\label{udefine}\end{aligned}$$ One also defines the pseudoinverse of $u$ as $$u_{ia}w_{ib}-u_{ib}w_{ia}=\epsilon_{ab} \,.
\label{wdefine}$$ With these new variables, it can be shown that the three-point superamplitude is given by $$\begin{aligned}
\null \hskip -1.3 cm
\mathcal{A}_{3}^{\tree}(1,2,3) &=&
-i \bigl({\mathbf{u}}_{1}{\mathbf{u}}_{2}+
{\mathbf{u}}_{2}{\mathbf{u}}_{3}+
{\mathbf{u}}_{3}{\mathbf{u}}_{1}\bigr)
\biggl(\sum_{i=1}^3 {\mathbf{w}}_i\biggr)
\bigl({\tilde{\mathbf{u}}}_{1}{\tilde{\mathbf{u}}}_{2}+
{\tilde{\mathbf{u}}}_{2}{\tilde{\mathbf{u}}}_{3}+
{\tilde{\mathbf{u}}}_{3}{\tilde{\mathbf{u}}}_{1}\bigr)
\biggl(\sum_{i=1}^3 {\tilde{\mathbf{w}}}_i\biggr) \,,
\label{ThreePointSuperAmplitude}\end{aligned}$$ where ${\mathbf{u}}_i$ and ${\mathbf{w}}_i$ are defined in terms of the $u_i^a$ and $w_i^a$ as $$\begin{aligned}
{\mathbf{u}}_i = u_i^a\eta_{ia}, \quad
{\tilde{\mathbf{u}}}_i = \tilde{u}_{i{{\dot{a}}}}\tilde{\eta}_{i}^{{{\dot{a}}}}, \quad
{\mathbf{w}}_i=w_i^a\eta_{ia}, \quad
{\tilde{\mathbf{w}}}_i = \tilde{w}_{i{{\dot{a}}}}\tilde{\eta}_{i}^{{{\dot{a}}}} \,.\end{aligned}$$
Dual conformal symmetry {#OnShellInversion}
=======================
Dual conformal symmetry is a symmetry of the superamplitude that is made manifest by introducing dual (or region) variables subject to the following constraints [@DualConformal2]: $$\begin{aligned}
\nonumber
&&(x_i-x_j)^{AB}=\lambda_{\{ij\}}^{Aa}\lambda^{B}_{\{ij\}a} \,, \hskip 1.0cm
(x_i-x_j)_{AB}=\tilde{\lambda}_{\{ij\}A\dot{a}}\tilde{\lambda}_{\{ij\}B}^{\dot{a}} \,, \\
&&(\theta_i-\theta_j)^{A}=\lambda_{\{ij\}}^{Aa}\eta_{\{ij\}a} \,, \hskip 1.55cm
(\tilde{\theta}_i-\tilde{\theta}_j)_{A}=\tilde{\lambda}_{\{ij\}A\dot{a}}\tilde{\eta}_{\{ij\}}^{\dot{a}},
\label{constraints}\end{aligned}$$ where each leg is labeled by the indices $\{ij\}$ of the two adjacent regions, the order of which indicates the direction of momentum flow along the leg (for example, $p^\mu_{\{ij\}} = -p^\mu_{\{ji\}}$). For tree amplitudes, this notation is redundant since $j$ can always be chosen as $i+1$. However this prescription does not generalize to loop level, and thus we use a more general notation in anticipation of the multiloop discussion in . We will go back and forth between using indices $(i,j,\ldots)$ to label regions and to label legs; the meaning of the indices should be clear from the context. The superamplitude is viewed as a distribution on the full space $(x,\theta,\tilde{\theta},\lambda,\tilde{\lambda},\eta,\tilde{\eta})$, with delta function support on the constraint equations (\[constraints\]). The cyclic nature of the region variables then automatically enforces momentum and supermomentum conservation, and the first two equations in (\[constraints\]) also imply .
To obtain the four-dimensional massive amplitudes, we break the six-dimensional spinors up into four-dimensional representations. Explicit details can be found in refs. [@Bern:2010qa; @Boels:2009bv]. Here we just note that the dual variables should also be broken into four-dimensional pieces and the fifth and sixth dimensional components. With $p_{\{ij\}}=(\check{p}_{\{ij\}},m_{\{ij\}},\tilde{m}_{\{ij\}})$, we have: $$\check{x}_i-\check{x}_{j}=\check{p}_{\{ij\}},\;\;\;n_i-n_{j}=m_{\{ij\}},\;\;\;\tilde{n}_i-\tilde{n}_{j}=\tilde{m}_{\{ij\}}\,,$$ where we use a check mark over a variable to indicate the components in the four-dimensional subspace. The physical mass squared is then $m_{\{ij\}}^2+\tilde{m}^2_{\{ij\}}$.
The dual conformal boost generator can be expressed as a composition of dual conformal inversions and translations, $$K^{\mu}=I\, P_{\mu}\, I\,,
\label{KPrelate}$$ so we begin our discussion with the dual conformal inversion operator $I$. The inversion is defined on the Clifford algebra as $$\begin{aligned}
I[(\sigma^\mu)_{AB}] \equiv (\tilde{\sigma}_\mu)^{BA}\,, \hskip 1.0cm I[(\tilde{\sigma}^\mu)^{AB}] \equiv (\sigma_\mu)_{BA}\,,\end{aligned}$$ and on the region variables as $$I[x_i^\mu] \equiv (x_i^{-1})_\mu = \frac{x_{i\mu}}{x_i^2}\,,\hskip 1.0cm
I[\theta_i^A] \equiv (x_i^{-1})_{AB} \theta_i^B \,, \hskip 1.0cm
I[\tilde{\theta}_{iA}] \equiv (x_i^{-1})^{AB}\tilde{\theta}_{iB}\,.
\label{variableinversion}$$ From the inversion of $x^\mu$, we also see that $$\begin{aligned}
I[(x_i-x_j)^{AB}] &=& (x_i^{-1})_{AC}(x_i-x_j)^{CD}(x_j^{-1})_{DB} \nn\\
&=& (x_j^{-1})_{AC}(x_i-x_j)^{CD}(x_i^{-1})_{DB}\,,\end{aligned}$$ and integration measures invert as $$\begin{aligned}
& I[d^6x_i] = (x_i^2)^{-6}d^6x_i \,,
&& I[d^4\theta_i] = (x_i^2)^2 d^4\theta_i \,,
&& I[d^4\tilde\theta_i] = (x_i^2)^2 d^4\tilde\theta_i\,.\end{aligned}$$ With these definitions in hand, we can deduce the inversion properties of all of the other variables by requiring the invariance of the constraint equations (\[constraints\]) and the definitions of the $u$ and $w$ variables in . We leave the proofs of these properties to and collect the results here: $$\begin{aligned}
&I[\lambda_{\{ij\}a}^A] =
\frac{x_{iAB}\lambda_{\{ij\}}^{Ba}}{\sqrt{x_i^2x_j^2}} =
\frac{x_{jAB}\lambda_{\{ij\}}^{Ba}}{\sqrt{x_i^2x_j^2}} \,,
&&I[\eta_{\{ij\}a}] = -\sqrt{\frac{x^2_i}{x^2_j}}
\Bigl(\eta^a_{\{ij\}}+(x^{-1}_i)_{AB} \, \theta^A_i\lambda_{\{ij\}}^{Ba}\Bigr) \,, \nn\\
&I[\tilde{\lambda}_{\{ij\}A{{\dot{a}}}}] =
\frac{x_i^{AB}\tilde{\lambda}_{\{ij\}B}^{{{\dot{a}}}}}{\sqrt{x_i^2x_j^2}} =
\frac{x_j^{AB}\tilde{\lambda}_{\{ij\}B}^{{{\dot{a}}}}}{\sqrt{x_i^2x_j^2}} \,,
&&I[\tilde{\eta}_{\{ij\}}^{{{\dot{a}}}}] = -\sqrt{\frac{x^2_i}{x^2_j}}
\Bigl( \tilde{\eta}_{\{ij\}{{\dot{a}}}} + (x_i^{-1})^{AB} \, \tilde{\theta}_{iA}\tilde{\lambda}_{\{ij\}B{{\dot{a}}}}\Bigr) \,, \nn\\
&I[u_{ia}]=\frac{\beta u_i^a}{\sqrt{x^2_{i-1}}} \,,
&&I[w_{ia}]=-\frac{1}{\beta}\sqrt{x^2_{i-1}}w_i^{a} \,, \nn\\
&I[\tilde{u}_{i{{\dot{a}}}}]=\frac{\tilde{u}_i^{{{\dot{a}}}}}{\beta\sqrt{x^2_{i-1}}} \,,
&&I[\tilde{w}_{i{{\dot{a}}}}]=-\beta\sqrt{x^2_{i-1}}\tilde{w}_i^{{{\dot{a}}}} \,,
\label{moreinversions}\end{aligned}$$ where $\beta$ is an unfixed parameter that is irrelevant in our calculations.
Given these inversion rules, one can immediately deduce via how each variable transforms under the dual conformal boost generator $K^\mu$. Alternatively, one can deduce the same information by requiring that the dual conformal boost generator respects all of the constraints in . If we were to use the usual dual conformal boost generator in $x$ space, $$\begin{aligned}
K^{\mu}=\sum_{i}\left(2\,x_i^{\mu}x_i^{\nu} - x_i^2\,\eta^{\mu\nu}\right)\frac{\partial}{\partial x_i^{\nu}}\,,\end{aligned}$$ the LHS of the definition of the $x_i$ in would be nonzero under boosts, while the RHS would vanish. To correct this, we must add derivatives with respect to $\lambda$ and $\tilde\lambda$ to $K^\mu$. These new derivatives in turn would not be compatible with the definition of $\theta_i$, so we must also add $\theta$ and $\eta$ derivatives. Requiring that all of the constraints in are consistent with $K^\mu$ then yields $$\begin{aligned}
K^\mu&=&\sum_{i}\left[
\left(2 \, x^\mu_i x^\nu_i-x_i^2 \, \eta^{\mu\nu}\right)\frac{\partial}{\partial x_{i}^{\nu}}
+\theta^A_i(\sigma^{\mu})_{AB}x_i^{BC}\frac{\partial}{\partial \theta^C_i}
+\tilde{\theta}_{iA}(\tilde{\sigma}^{\mu})^{AB}x_{iBC}\frac{\partial}{\partial \tilde{\theta}_{iC}}\right] \nn\\
&+&\frac{1}{2}\sum_{\{jk\}} \left[\lambda^{Aa}_{\{jk\}}(\sigma^\mu)_{AB}(x_j+x_k)^{BC}\frac{\partial}{\partial \lambda^{Ca}_{\{jk\}}}
-(\theta_j+\theta_k)^A(\sigma^\mu)_{AB}\lambda_{\{jk\}a}^{B}\frac{\partial}{\partial \eta_{\{jk\}a}} \right.\nn\\
&&+\left.\tilde{\lambda}_{\{jk\}A{{\dot{a}}}}(\tilde{\sigma}^\mu)^{AB}(x_j+x_k)_{BC}\frac{\partial}{\partial \tilde{\lambda}_{\{jk\}C{{\dot{a}}}}}
-(\tilde{\theta}_j+\tilde{\theta}_k)_{A}(\tilde{\sigma}^{\mu})^{AB}\tilde{\lambda}^{{{\dot{a}}}}_{\{jk\}B}\frac{\partial}{\partial \tilde{\eta}_{\{jk\}}^{{{\dot{a}}}}}\right] \,,\nn\\\end{aligned}$$ where $i$ runs over all regions, and $\{jk\}$ runs over all legs. The bosonic part of this generator was given in ref. [@Bern:2010qa]. One can explicitly check that the infinitesimal transformations generated by this dual conformal boost generator match with those generated by .
Dual conformal properties of tree-level amplitudes {#TreeLevelProof}
==================================================
In this section, we show that the tree-level amplitudes of six-dimensional maximal sYM exhibit dual conformal covariance. In ref. [@Bern:2010qa], the four-point tree-level amplitude was shown to be covariant under dual conformal inversion, $$I[\mathcal{A}_4^{\tree}]=(x^2_1)^2(x^2_1x^2_2x^2_3x^2_4)\mathcal{A}_4^{\tree}.$$ Note that the extra factor $(x^2_1)^2$ relative to the four-dimensional result comes from the mismatch of the degrees of the momentum and supermomentum delta functions in six dimensions. In six dimensions, the momentum conservation delta function is of degree six instead of degree four as in four dimensions. Since the fermionic delta function is still of degree eight, there will be a mismatch in inversion weights of degree two in $(x^2_1)$. After separating out the delta functions from the rest of the amplitude, $$\mathcal{A}_n^{\tree}=\overalldelta{\E} f_n\,,$$ it was conjectured that the function $f_n$, for $n\ge4$, transforms as $$I[f_n]=\left(\prod_{i\in\E}x_i^2\right) f_n
\label{conjecture}$$ under dual conformal inversion. We prove this by induction, utilizing the BCFW recursion relations [@BCFW]; assuming that all $f_{m}$ transform as in for $4\le m<n$, each term in the BCFW recursive construction of $f_n$ will respect , and hence so will $f_n$. For the three-point amplitude, due to special kinematics, it is possible to consider the external momenta in a four-dimensional subspace. It is then conceivable that the four-dimensional dual conformal properties carry over to higher dimensions via covariance. However, closer inspection is warranted, because the polarization vectors of the gluons could point outside of the subspace. Furthermore, the six-dimensional three-point amplitude is not proportional to the supermomentum delta function, and hence $f_3$ cannot be defined.
Given that the function $f_n$ inverts as , acting with the dual conformal boost generator then gives $$K^{\mu}[f_n] =\left(\sum_{i\in\E} 2x^{\mu}_i\right)f_n.$$ The above results can be rewritten for the massive amplitudes. In four-dimensional notation, the conformal inversion acts as $$\begin{aligned}
&I\left[\check{x}^\mu\right]=\frac{\check{x}_\mu}{x^2}\,,
&&I\left[n\right]=-\frac{n}{x^2}\,,
&&I\left[\tilde{n}\right]=-\frac{\tilde{n}}{x^2}\,,\end{aligned}$$ where $x^2=\check{x}^2-n^2-\tilde{n}^2$. The massive amplitude then transforms under the dual conformal boost generators as $$\begin{aligned}
& \check{K}^\mu [f_n] =\left(\sum_{i\in\E} 2\check{x}^{\hat{\mu}}_i\right)f_n \,,
&& K^{n} [f_n] =\left(\sum_{i\in\E} 2n_i\right)f_n \,,
&& K^{\tilde{n}} [f_n] =\left(\sum_{i\in\E} 2\tilde{n}_i\right)f_n \,.\end{aligned}$$ The generator $\check{K}^\mu$ is closely related to the dual generator for the massively regulated amplitude [@Alday:2009zm]. The bosonic dual variable part is $$\check{K}^\mu=\sum_{i}\left[2\,\check{x}_i^{\mu}\left(\check{x}_i^{\nu}\frac{\partial}{\partial \check{x}_i^{\nu}}
+n_i\frac{\partial}{\partial n_i}
+\tilde{n}_i\frac{\partial}{\partial \tilde{n}_i}\right)
- x_i^2\frac{\partial}{\partial \check{x}_{i\mu}}\,\right]\,,$$ while the bosonic part of the fifth and sixth components of $K^\mu$ is $$\begin{aligned}
\nonumber K^n&=&\sum_{i}\left[2\,n_i\left(\check{x}_i^{\nu}\frac{\partial}{\partial \check{x}_i^{\nu}}
+n_i\frac{\partial}{\partial n_i}
+\tilde{n}_i\frac{\partial}{\partial \tilde{n}_i}\right)
+ x_i^2\frac{\partial}{\partial n_{i}}\,\right] \,,\\
K^{\tilde{n}}&=&\sum_{i}\left[2\,\tilde{n}_i\left(\check{x}_i^{\nu}\frac{\partial}{\partial \check{x}_i^{\nu}}
+n_i\frac{\partial}{\partial n_i}
+\tilde{n}_i\frac{\partial}{\partial \tilde{n}_i}\right)
+ x_i^2\frac{\partial}{\partial \tilde{n}_{i}}\,\right]\,.\end{aligned}$$ Since the massive formulation is obtained straightforwardly from the six-dimensional formalism, from now on we will work with manifest six-dimensional covariance.
The BCFW shift in dual coordinates.
-----------------------------------
Taking the BCFW shift to be on legs $1$ and $n$, we have $$\begin{aligned}
p_{1}(z)&=p_1+zr \,,
&q_{1}(z)&=q_1+zs \,,
&\tilde{q}_{1}(z)&=\tilde{q}_1+z\tilde{s} \,,\nn\\
p_{n}(z)&=p_{n}-zr \,,
& q_{n}(z)&=q_{n}-zs \,,
& \tilde{q}_n(z)&=\tilde{q}_n-z\tilde{s} \,.\end{aligned}$$ The precise forms of $r$, $s$ and $\tilde{s}$ are given in refs. [@CheungOConnell; @DHS]. For our purposes, it is sufficient to note that this implies a shift in only the dual coordinates $x_{1}$, $\theta_{1}$ and $\tilde{\theta}_{1}$, $$\begin{aligned}
p_{1}(z)&=x_{1}(z)-x_{2} \,,
& q_{1}(z)&=\theta_{1}(z)-\theta_{2} \,,
& \tilde{q}_1(z)&=\tilde{\theta}_1(z)-\tilde{\theta}_2 \,,\nn\\
p_{n}(z)&=x_{n}-x_{1}(z) \,,
& q_{n}(z)&=\theta_{n}-\theta_{1}(z) \,,
& \tilde{\theta}_n(z)&=\tilde{\theta}_n-\tilde{\theta}_1(z)\,,\end{aligned}$$ where $$\begin{aligned}
x_{1}(z)&=x_{1}+zr\,, & \theta_{1}(z)&=\theta_{1}+zs\,, & \tilde{\theta}_1(z)&=\tilde{\theta}_1+z\tilde{s}\,.\end{aligned}$$ Thus each BCFW term can be defined in a dual graph with just one shifted dual coordinate. We will denote the legs with shifted momentum by placing hats over the leg labels, while a hat over $x$ and $\theta$ is used for shifted regions.
There are two types of BCFW diagrams, characterized by the presence or absence of a three-point subamplitude. We must consider each case separately, due to the fact that we cannot pull out an overall supermomentum conservation delta function from the three-point amplitude, and thus the three-point amplitude does not have the straightforward inversion of .
BCFW diagrams without three-point subamplitudes {#BCFWNoThree}
-----------------------------------------------
We first consider the case where there is no three-point subamplitude, as in .
![A BCFW diagram without three-point subamplitudes.[]{data-label="BCFW"}](BCFW.eps)
The amplitudes on the left and right can be written as $$\begin{aligned}
\nonumber\mathcal{A}_L&=&\overalldelta{L}f_L(\hat{1},\cdots,j,\widehat{P}) \,,\\
\mathcal{A}_R&=&\overalldelta{R} f_R(-\widehat{P},j+1,\cdots,\hat{n}) \,.\end{aligned}$$ Each term in the BCFW recursion can then be written as $$\overalldelta{\E} f_{n}^{(j)}\,,$$ where $f_{n}^{(j)}$ is the contribution to $f_n$ from the BCFW diagram labeled by $j$, $$f_{n}^{(j)} = \frac{i}{P^2}\int d^2\eta_Pd^2\tilde{\eta}_P
\,\delta^4\left(\sum_{i\in L} q_i\right)\delta^4\left(\sum_{i\in L} \tilde{q}_i\right) f_Lf_R\,.$$ From the induction step, the functions $f_L$ and $f_R$ invert as $$\begin{aligned}
I\left[f_L\right]&=&\Bigl(\widehat{x}_1^2 x_2^2 \cdots x_{j+1}^2\Bigr)f_L\,, \nn\\
I\left[f_R\right]&=&\Bigl(x_{j+1}^2 \cdots x_n^2 \widehat{x}_1^2\Bigr)f_R\,.
\label{finversions}\end{aligned}$$ The propagator in $f_{n}^{(j)}$ has a simple inversion, given by $$I\left[\frac{1}{P^2}\right]=I\left[\frac{1}{x^2_{1,j+1}}\right]=\frac{x^2_{1}x^2_{j+1}}{x^2_{1,j+1}}\,,
\label{propagatorinversion}$$ so the only remaining piece of $f_{n}^{(j)}$ is the fermionic integral. Since the fermionic delta function is of degree eight, the fermionic integral can be completely localized by the delta functions, and the $\eta_P,\tilde{\eta}_P$s in $f_L,f_R$ will be replaced by the solution of the delta functions. The replacement does not affect the inversion properties of $f_L,f_R$ because it simply amounts to the use of supermomentum conservation. The integral has been shown previously [@Bern:2010qa] to give $$\begin{aligned}
\nonumber \int d^2\eta_Pd^2\tilde{\eta}_P \,
\delta^4\left(\sum_{i\in L} q_i\right)\delta^4\left(\sum_{i\in L} \tilde{q}_i\right)
&=&\left(\widehat{\theta}_1-\theta_{j+1}\right)^A \tilde{\lambda}_{\widehat{P}A{{\dot{a}}}}\tilde{\lambda}_{\widehat{P}B}^{{{\dot{a}}}} \left(\widehat{\theta}_1-\theta_{j+1}\right)^B \\
&\phantom{=}&\times\biggl(\widehat{\tilde{\theta}}_1-\tilde{\theta}_{j+1}\biggr)_C \lambda_{\widehat{P}}^{Ca}\lambda_{\widehat{P}a}^D \biggl(\widehat{\tilde{\theta}}_1-\tilde{\theta}_{j+1}\biggr)_D\,.\end{aligned}$$ Note that we do not write $f_L$ and $f_R$ in the integral because they are independent of $\eta_P,\tilde{\eta}_P$ after the replacement. To see how this expression inverts, we use on each factor, such as $$\begin{aligned}
\nonumber I\left[\left(\widehat\theta_{1}-\theta_{j+1}\right)^A\tilde\lambda_{\widehat{P}A{{\dot{a}}}}\right]&=&
-\frac{1}{\sqrt{\widehat{x}^2_{1}x^2_{j+1}}} \left(\widehat\theta_1^B(\widehat{x}^{-1}_1)_{BA}\widehat{x}_1^{AC}\tilde\lambda^{{{\dot{a}}}}_{\widehat{P}C} - \theta_{j+1}^B(x^{-1}_{j+1})_{BA}x^{AC}_{j+1}\tilde\lambda^{{{\dot{a}}}}_{\widehat{P}C}\right)\\
&=&-\frac{1}{\sqrt{\widehat{x}^2_1 x^2_{j+1}}} \left(\widehat\theta_1-\theta_{j+1}\right)^A\tilde\lambda^{{{\dot{a}}}}_{\widehat{P}A}\,.\end{aligned}$$ Doing the same for the other factors, we find $$\begin{aligned}
&& I\left[\int d^2\eta_Pd^2\tilde{\eta}_P \, \delta^4\left(\sum_{i\in L} q_i\right)\delta^4\left(\sum_{i\in L} \tilde{q}_i\right)\right] \nn\\
&&\hskip 2.0cm =\frac{1}{(\widehat{x}^2_1 x^2_{j+1})^2}\int d^2\eta_Pd^2\tilde{\eta}_P \, \delta^4\left(\sum_{i\in L} q_i\right)\delta^4\left(\sum_{i\in L} \tilde{q}_i\right)
\label{integralinversionA}\end{aligned}$$ Combining equations (\[finversions\]), (\[propagatorinversion\]) and (\[integralinversionA\]), we arrive at the desired result $$\begin{aligned}
I\left[f_{n}^{(j)}\right]=\left(\prod_{i\in \mathcal{E}}x^2_{i}\right)f_{n}^{(j)} \,.\end{aligned}$$
BCFW diagrams with a three-point subamplitude
---------------------------------------------
To make a statement about the inversion weight of the entire $n$-point amplitude, we must also consider the BCFW terms which contain a three-point subamplitude, as shown in .
![A BCFW diagram with a three-point subamplitude.[]{data-label="BCFW2"}](BCFW2.eps)
It was shown in ref. [@Bern:2010qa] that the contribution of such a diagram is given as, $$\begin{aligned}
&&\int d^2\eta_P d^2\tilde{\eta}_P \, \mathcal{A}_3\frac{i}{P^2}\mathcal{A}_{n-1}\\
\nonumber&=&-\overalldelta{\E}
\left({\mathbf{u}}_{2}-{\mathbf{u}}_{\hat1}\right)\left({\tilde{\mathbf{u}}}_2-{\tilde{\mathbf{u}}}_{\hat1}\right)\frac{1}{P^2}f_{n-1}\,,\end{aligned}$$ where $f_{n-1}$ has been rewritten completely in terms of external leg variables by using the substitutions $q_{\widehat{P}}=-q_{\hat{1}}-q_{2}$ etc. Hence, $$f_{n}^{(2)}=-\left({\mathbf{u}}_{2}-{\mathbf{u}}_{\hat1}\right)\left({\tilde{\mathbf{u}}}_{2}-{\tilde{\mathbf{u}}}_{\hat1}\right) \frac{1}{P^2} f_{n-1}\,.$$ The inversion of $P^2$ and $f_{n-1}$ here are straightforward, and we are left with the remaining factors involving ${\mathbf{u}}$ and ${\tilde{\mathbf{u}}}$. We consider the inversion of $\left({\mathbf{u}}_{2}-{\mathbf{u}}_{\hat1}\right)$ in detail. After applying , we get $$\begin{aligned}
I\left[{\mathbf{u}}_{2}-{\mathbf{u}}_{\hat1}\right]&=&
-\sqrt{\frac{x^2_{2}}{\widehat{x}^2_{1}x^2_{3}}} \, \beta u_{2a}\left(\eta^a_{2}+(x^{-1}_{2})_{AB}\theta^A_{2}\lambda_{2}^{Ba}\right) \nn\\
&&+\sqrt{\frac{\widehat{x}^2_1}{x^2_2 x^2_3}} \, \beta u_{\hat1a}\left(\eta^a_{\hat1}+(\widehat{x}^{-1}_{1})_{AB}\widehat\theta^A_{1}\lambda_{\hat{1}}^{Ba}\right)\,.
\label{invertuterm}\end{aligned}$$ We can combine the $\theta$-dependent terms in the above equation as $$\begin{aligned}
&&\frac{\beta}{\sqrt{\widehat{x}^2_1 x^2_2 x^2_3}}
\left(-u_{2a}\,x_{2AB}\,\theta^A_2\lambda_2^{Ba}
+u_{\hat1a}\,\widehat{x}_{1AB}\,\widehat\theta^A_1\lambda_{\hat1}^{Ba}\right) \nn\\
&&\hskip2.0cm =\frac{-\beta}{2\sqrt{\widehat{x}^2_1 x^2_2 x^2_3}}
\, u_{\hat1a}(\widehat{x}_1+x_2)_{AB}(\theta^A_{2}-\widehat\theta^A_{1})\lambda_{\hat1}^{Ba} \nn\\
&&\hskip2.0cm =\frac{\beta}{2\sqrt{\widehat{x}^2_1 x^2_2 x^2_3}}
\, u_{\hat1a}(\widehat{x}_1+x_2)_{AB}\lambda^{Ab}_{\hat1}\lambda_{\hat1}^{Ba}\eta_{\hat1b} \nn\\
&&\hskip2.0cm =\frac{-\beta}{4\sqrt{\widehat{x}^2_1 x^2_2 x^2_3}}
\, (\widehat{x}_1-x_2)^{AB} \, (\widehat{x}_1+x_2)_{AB} \, {\mathbf{u}}_{\hat1} \nn\\
&&\hskip2.0cm =\beta \, {\mathbf{u}}_{\hat1}
\left(\sqrt{\frac{\widehat{x}^2_1}{x^2_2 x^2_3}} - \sqrt{\frac{x^2_2}{\widehat{x}^2_1 x^2_3}}\right) \,,\end{aligned}$$ where in the second line we have used $\widehat{x}_{1AB}\,\lambda_{\hat1}^{Ba} = x_{2AB}\,\lambda_{\hat1}^{Ba}$ and $u_{\hat1a}\lambda_{\hat1}^{Ba} = u_{2a}\lambda_{2}^{Ba}$. Putting this back into , we arrive at $$I\left[({\mathbf{u}}_{2}-{\mathbf{u}}_{\hat1})\right]= \beta
\sqrt{\frac{x^2_2}{\widehat{x}^2_1 x^2_3}} ({\mathbf{u}}_2-{\mathbf{u}}_{\hat1}) \,.$$ The inversion of the antichiral factor $({\tilde{\mathbf{u}}}_2-{\tilde{\mathbf{u}}}_{\hat1})$ behaves in the same way, except that $\beta$ appears in the denominator. Thus, putting everything together, we have $$I\left[f_{n}^{(2)}\right]=\biggl(\frac{x^2_2}{\widehat{x}^2_1 x^2_3}\biggr) \left(x^2_1 x^2_3\right)
\left(\widehat{x}^2_1 x^2_3 \cdots x^2_n \right) f_n^{(2)}
=\left(\prod_{i\in\E}x^2_{i}\right)f_{n}^{(2)}.
\label{f2invert}$$ This completes the proof of . In the next section, we turn our attention to planar multiloop amplitudes.
Loop amplitudes through unitarity cuts {#LoopProof}
======================================
In this section, we will demonstrate to all loop orders that the $L$–loop planar integrand is covariant under inversion in the following way: $$I\left[\mathcal{I}_n^{L}\right]= \left(\prod_{i\in\E} x_i^2\right) \left(\prod_{i=1}^L (x_{l_i}^2)^4\right) \mathcal{I}_n^{L} \,,
\label{loopinvert}$$ where the integrand is defined with respect to the amplitude as $$\begin{aligned}
\mathcal{A}_n^{L} = \overalldelta{\E}
\int \Biggl( \prod_{i=1}^{L}d^6 x_{l_i} \Biggr) \mathcal{I}_n^{L}\,.
\label{integranddefinition}\end{aligned}$$ Because we are focusing on the integrand itself, there are extra loop region weights $(x_{l_i}^2)^4$. This is the same result as in four dimensions, although in six dimensions the loop integration measure inverts with weight $(x_{l_i}^2)^{-6}$, which does not exactly cancel the weight of the integrand. Therefore, the amplitude after integration will not be covariant unless the integral is restricted to four dimensions, which, as we have discussed, is the case when interpreting the extra two dimensions as a massive regulator [@Alday:2009zm].
Our approach to is to study the inversion properties of unitarity cuts of the amplitude. In the unitarity method, we are required to perform state sums across the cut propagators, which is achieved by integrating the Grassmann variables $\eta_{l_i},\tilde{\eta}_{l_i}$ of the cut lines. Since the tree amplitudes contributing to the cuts have definite inversion properties, we only need to understand how the $\eta_{l_i},\tilde{\eta}_{l_i}$ integration modifies the inversion weight.
To make statements about inversion properties, it is more natural to express everything in terms of dual variables than in terms of $\eta$ and $\lambda$. We therefore trade the supersum $\eta$ integrals for $\theta$ integrals. Suppose a cut not containing any three-point subamplitudes has an internal line between regions $i$ and $j$. The supersum across this line is expressed as an integral with measure $d^2\eta_{\{ij\}} d^2\tilde{\eta}_{\{ij\}}$. The transformation to dual coordinates is achieved by inserting 1 into the cut in a particular way, given by $$\begin{aligned}
\mathcal{A}_n^{L}\Bigr|_{\hbox{\footnotesize{cut}}} &=& \int
\prod_{\{ij\}} d^2\eta_{\{ij\}}d^2\tilde\eta_{\{ij\}} \times
\mathcal{A}^{\tree}_{(1)}\mathcal{A}^{\tree}_{(2)}\mathcal{A}^{\tree}_{(3)}\ldots\mathcal{A}^{\tree}_{(m)} \nn\\
&=& \int
\prod_{\{ij\}} d^2\eta_{\{ij\}}d^2\tilde\eta_{\{ij\}} \times
\prod_{\alpha} \delta^4\left(\sum_{k \in\alpha} q_k\right) \delta^4 \left(\sum_{k\in\alpha} \tilde q_k\right) f_{\alpha} \nn\\
&=& \int
\prod_{\{ij\}} d^2\eta_{\{ij\}}d^2\tilde\eta_{\{ij\}} \times
\prod_k d^4\theta_kd^4\tilde\theta_k \times
\prod_\alpha f_\alpha \nn\\
&&\times\prod_{\{rs\}}\delta^4\left(\theta_r^A-\theta_s^A-\lambda_{\{rs\}}^{Aa}\eta_{\{rs\}a}\right)
\delta^4\left(\tilde\theta_{rB}-\tilde\theta_{sB}-\tilde\lambda_{\{rs\}B{{\dot{a}}}} \tilde\eta_{\{rs\}}^{{{\dot{a}}}}\right) \,,
\label{loopamplitude}\end{aligned}$$ where the product over $\{ij\}$ runs over all internal cut lines, the product over $k$ runs over all regions, the product over $\{rs\}$ runs over all lines, and the product over $\alpha$ runs over the tree subamplitudes. The first two lines of this equality are the definition of the cut, where we have ignored the momentum conservation delta functions on the subamplitudes, because they combine straightforwardly into an overall momentum conservation when cut conditions are relaxed and loop integrals are replaced. Because the integrand in the third and fourth lines has a shift symmetry in the $\theta$ variables, the measure $\prod d^4\theta$ is understood to include only $(F-1)$ of the regions, where $F=n+L$ is the total number of regions in the graph. An explicit example for the two-loop four-point amplitude is given schematically in . It does not matter how we fix the symmetry in the measure; our choice will only affect the overall supermomentum delta function, which does not contribute to the conjectured transformation . We therefore leave this detail implicit.
![A cut of the two-loop four-point amplitude. (a) In the usual expression of the cut, this diagram is dressed with a tree-level amplitude for each blob and a state sum over each internal line. (b) As discussed in the text, for planar cuts this is equivalent to dressing the diagram with an $f$ function for each blob, introducing the dual variable constraints for every line, and integrating over the dual $\theta$ variables of every region. Finally, a state sum over each internal line is performed. One can check that the dressing of (b) contains $8\times 8=64$ fermionic delta functions and $5\times8=40$ integrations over $\theta$ (because one of the six regions is fixed by the shift symmetry), leaving $24$ unintegrated fermionic delta functions, which are exactly the supermomentum conservation of the subamplitudes in dressing (a).[]{data-label="loopexample"}](loopexample.eps)
To see the equality of , note that we can pull the subamplitude supermomentum delta functions out of the $\theta$ delta functions in the fourth line, leaving behind $(P-V)$ delta functions to be used for localizing the $\theta$ integrals, where $P$ is the number of lines in the graph, and $V$ is the number of subamplitudes. Because there are $(F-1)$ of the $\theta$ integrals, the leftover delta functions saturate the integral when $F-1=P-V$, which is indeed the case for planar graphs.
We can now use the $\theta$ delta functions to eliminate all explicit $\eta$ dependence from each $f_\alpha$, so that the entire $\eta$ dependence of the cut appears in the form $$\begin{aligned}
\int d^2\eta_{\{ij\}} \delta^4\left(\theta_i^A-\theta_j^A-\lambda_{\{ij\}}^{Aa}\eta_{\{ij\}a}\right) \,.\end{aligned}$$ This performs the chiral half of the supersum across the line between regions $i$ and $j$. The antichiral half of the supersum is completely analogous, so we leave it out. The integration over $\eta_{\{ij\}}$ thus contributes $$\begin{aligned}
\theta_{ij}\cdot x_{ij}\cdot \theta_{ij} \equiv (\theta_i-\theta_j)^A (x_i-x_j)_{AB} (\theta_i-\theta_j)^B \,.\end{aligned}$$ We demonstrated in that this factor inverts with weight $(x_i^2x_j^2)^{-1}$.
Returning to the cut in , the result of doing the $\eta$ integrals is $$\begin{aligned}
\mathcal{A}_n^{L}\Bigr|_{\hbox{\footnotesize{cut}}} &=& \int
\prod_k d^4\theta_kd^4\tilde\theta_k \times
\prod_\alpha f_\alpha^0 \times
\prod_{\{ij\}} (\theta_{ij}\cdot x_{ij} \cdot \theta_{ij})
(\tilde\theta_{ij} \cdot x_{ij} \cdot \tilde\theta_{ij})\nn\\
&&\times\prod_{\{rs\}}\delta^4\left(\theta_r^A-\theta_s^A-\lambda_{\{rs\}}^{Aa}\eta_{\{rs\}a}\right)
\delta^4\left(\tilde\theta_{rA}-\tilde\theta_{sA}-\tilde\lambda_{\{rs\}A{{\dot{a}}}} \tilde\eta_{\{rs\}}^{{{\dot{a}}}}\right) \,,\end{aligned}$$ where now $\{rs\}$ only runs over the external lines. An overall supermomentum delta function pulls out, leaving $(n-1)$ delta functions of each chirality, which completely saturate the $\theta$ integrations over the external regions (this also takes care of the shift symmetry detail). We are finally left with $$\begin{aligned}
\mathcal{A}_n^{L}\Bigr|_{\hbox{\footnotesize{cut}}} &=& \overalldelta{\E} \nn\\
&&\times \int \left(\prod_k d^4 \theta_k d^4 \tilde\theta_k \right)
\left( \prod_{\{ij\}} (\theta_{ij}\cdot x_{ij}\cdot \theta_{ij}) (\tilde\theta_{ij}\cdot x_{ij} \cdot \tilde\theta_{ij})\right)
\prod_\alpha f_\alpha \,,
\label{cutintegrand}\end{aligned}$$ where the product over $k$ now runs only over the internal regions, and we have replaced the overall momentum conservation.
We are now in a position to formulate a set of diagrammatic rules for inverting the cut, after restoring the cut propagators. Because each piece of the second line of inverts covariantly, the cut inverts to itself multiplied by an overall prefactor (not considering the inversion of the overall delta functions). To calculate the prefactor for a given cut, we have the following rules:
- For every loop region $k$, the $\theta_k,\tilde\theta_k$ measure contributes a factor $(x_k^2)^4$.
- Each internal leg $\{ij\}$ contributes $(x_i^2x_j^2)^{-1}$, where a factor of $x_i^2 x_j^2$ comes from the cut propagator, and a factor of $(x_i^2x_j^2)^{-2}$ comes from $(\theta_{ij}\cdot x_{ij}\cdot\theta_{ij})(\tilde\theta_{ij}\cdot x_{ij}\cdot\tilde\theta_{ij})$
- Each tree-level subamplitude contributes $\prod_i x_i^2$, where $i$ runs over all regions adjacent to the tree.
Given a region $i$, it is straightforward to invert these rules to figure out what power of $x_i^2$ appears in the prefactor. If $i$ is an external region, $x_i^2$ must appear to the power $(\rho_i-\sigma_i)$, where $\rho_i$ and $\sigma_i$ are the number of tree-level subamplitudes and the number of internal propagators, respectively, adjacent to region $i$. Each external region necessarily borders one fewer of the internal propagators than the subamplitudes, so the external regions each give $x_i^2$. If, on the other hand, $i$ is an internal region, then $x_i^2$ appears to the power $(\rho_i-\sigma_i+4)$. All internal regions necessarily border the same number of internal propagators as subamplitudes, so the internal regions each give $(x_i^2)^4$. Therefore, we have reached the result that each planar cut with no three-point subamplitudes inverts with the prefactor $$\left(\prod_{i\in\E} x_i^2\right)\left(\prod_{i=1}^{L} (x_{l_i}^2)^4\right)\,,$$ after the cut propagators have been restored, and not including the overall momentum and supermomentum conservation. It is not difficult to extend this result to cuts involving three-point subamplitudes. The supersum between a three-point subamplitude and another subamplitude in a cut proceeds in the same way as sewing a three-point tree in BCFW. The resulting merged subamplitudes then invert as in .
Because all cuts invert in exactly the same way, and the correct amplitude must satisfy all generalized unitarity cuts, we conclude that the $L$–loop integrand inverts as $$I\left[\mathcal{I}_n^L\right] = \left(\prod_{i\in\E} x_i^2\right)\left(\prod_{i=1}^{L} (x_{l_i}^2)^4\right) \mathcal{I}_n^L\,.
\label{conclusion}$$ For a recent discussion of the transition from cuts to the amplitude, see ref. [@Bern:2010tq]. Note that bubbles on external lines are not cut detectable, so they potentially violate . However, because this is the maximally supersymmetric theory, we do not expect these contributions to appear [@Bern:2010tq]. If we restrict the loop integration measure in to a four-dimensional subspace, as when interpreting the two extra dimensions as a massive regulator, the measure will provide an extra inversion weight of $\prod_i (x_{l_i}^2)^{-4}$, which exactly cancels the extra weight of the integrand. The inversion then commutes with the integration, since the infrared singularities have been regulated, and the amplitude obeys an exact dual conformal symmetry to all loops, which we may write as $$I\left[\int \Biggl( \prod_{i=1}^{L}d^4 x_{l_i} \Biggr) \mathcal{I}_n^{L}\right] =
\left(\prod_{i\in\E} x_i^2\right) \int \Biggl( \prod_{i=1}^{L}d^4 x_{l_i} \Biggr) \mathcal{I}_n^{L}
\,, \hskip 1.0cm \hbox{(massively regulated $\NeqFour$)} \,.$$
Conclusion
==========
In this paper we demonstrated that the six-dimensional maximal sYM tree-level amplitudes and multiloop integrands exhibit dual conformal covariance. While dual conformal symmetry has been shown to exist for theories in $D\neq4$ [@Huang:2010qy], it is noteworthy that such a symmetry can be defined for theories which are not invariant under ordinary conformal symmetry [@Bern:2010qa; @CaronHuot:2010rj]. Also, because a massless on-shell particle in six dimensions is equivalent to a massive particle in four dimensions, our six-dimensional result then naturally gives the dual conformal properties of the massively regulated four-dimensional $\NeqFour$ theory [@Alday:2009zm].
Covariance under dual conformal symmetry facilitated the construction of four-dimensional $\NeqFour$ sYM tree-level amplitudes by expressing the amplitudes in terms of dual conformal invariant “$R$" functions [@Drummond:2008cr]. Therefore, an obvious task is to formulate the corresponding “$R$" covariants for the six-dimensional theory and construct the general $n$-point tree amplitude. This would serve as an efficient way to compute massive amplitudes in four dimensions.
One of the important new ingredients in utilizing the four-dimensional dual conformal symmetry to determine amplitudes is the notion of momentum twistors [@Hodges:2009hk]. These are twistor variables whose incidence relations are defined in the dual momentum space instead of the ordinary spacetime. Similarly, one can now hope to express six-dimensional amplitudes in terms of momentum twistors defined in six dimensions. It would be interesting to see if such a construction leads to alternative representations of planar maximal sYM amplitudes.
Acknowledgements
================
We thank Zvi Bern for suggesting this problem and for many stimulating discussions. We would also like to thank Donal O’Connell and Emery Sokatchev for their useful communications. This research was supported by the US Department of Energy under contract DE–FG03–91ER40662. T.D. gratefully acknowledges the financial support of a Robert Finkelstein grant.
Clifford algebra conventions
============================
Here we follow the conventions of [@CheungOConnell]. The Clifford algebra is given as, $$\sigma^\mu_{AB}\tilde{\sigma}^{\nu BC}+\sigma^\nu_{AB}\tilde{\sigma}^{\mu BC}=2\eta^{\mu\nu}\delta^C_A\,.$$ The explicit forms of the matrices $\sigma, \tilde{\sigma}$ are given in [@CheungOConnell]. They satisfy the following identities: $$\begin{aligned}
\nonumber&&\sigma^\mu_{AB}\sigma_{\mu CD}=-2\epsilon_{ABCD} \,,\\
\nonumber&&\tilde{\sigma}^{\mu AB}\tilde{\sigma}_{\mu}^{CD}=-2\epsilon^{ABCD} \,,\\
\nonumber&&\tilde{\sigma}^{\mu AB}\sigma_{\mu CD}=-2\left(\delta_C^{[A}\delta_{D}^{B]}\right) \,,\\
&&tr(\sigma^\mu\tilde{\sigma}_\nu)=\sigma^{\mu}_{AB}\tilde{\sigma}^{\nu BA}=4\eta^{\mu\nu}\,.\end{aligned}$$ From the above, one can deduce, $$\begin{aligned}
&&x^\mu=\frac{1}{4}(\tilde{\sigma}^\mu)^{BA}x_{AB} = \frac{1}{4}(\sigma^\mu)_{BA}x^{AB} \,,\nn\\
&&x^{AB}=\frac{1}{2}\epsilon^{ABCD}x_{CD}\,, \nn\\
&&x^{AB}x_{BE}=\frac{1}{2}\epsilon^{ABCD}x_{CD}x_{BE}=x^2\delta^A_E\,,\nn\\
&&x^2=x^\nu x_\nu=-\frac{1}{8}\epsilon_{ABCD}x^{AB}x^{CD}\,.\end{aligned}$$ Some useful formulæ: $$\begin{aligned}
\nonumber X_{[ab]}&=&\epsilon_{ab}X^c\,_c,\;X^{[ab]}=-\epsilon^{ab}X^c\,_c\\
\nonumber\epsilon^{ABCD}\epsilon_{AEFG}&=&\left(\delta^B_E\delta^C_F\delta^D_G+\delta^C_E\delta^D_F\delta^B_G+\delta^D_E\delta^B_F\delta^C_G\right.\label{econtraction}\\
&&\left.-\delta^B_F\delta^C_E\delta^D_G-\delta^C_F\delta^D_E\delta^B_G-\delta^D_F\delta^B_E\delta^C_G\right)\end{aligned}$$
Proof of variable inversion formulæ {#InversionAppendix}
===================================
In this appendix, we derive the inversion properties in .
- $I[\lambda_{\{ij\}a}^A] = \frac{x_{iAB}\lambda_{\{ij\}}^{Ba}}{\sqrt{x_i^2x_j^2}} = \frac{x_{jAB}\lambda_{\{ij\}}^{Ba}}{\sqrt{x_i^2x_j^2}}$
Our starting point is the constraint equation $(x_i-x_j)_{AB} = \tilde\lambda_{\{ij\}A{{\dot{a}}}}\tilde\lambda_{\{ij\}B}^{{{\dot{a}}}}\,$. Contracting both sides with $\lambda_{\{ij\}}^{Ba}$ provides an equation linear in $\lambda$, but loses normalization information. We then proceed with the inversion $$\begin{aligned}
0 &=& I[(x_i-x_j)_{AB}\lambda_{\{ij\}}^{Ba}] \nn\\
&=& (x_j^{-1})^{AC}(x_i-x_j)_{CD}(x_i^{-1})^{DB} I[\lambda_{\{ij\}}^{Ba}].\end{aligned}$$ This implies that $(x_{i}^{-1})^{DB} I[\lambda_{\{ij\}}^{Ba}]$ is in the null space of $(x_i-x_j)_{CD}$, so $$\begin{aligned}
(x_{i}^{-1})^{DB} I[\lambda_{\{ij\}}^{Ba}] &=& M_{ab}\lambda_{\{ij\}}^{Db} \nn\\
\Rightarrow \quad I[\lambda_{\{ij\}}^{Aa}] &=& x_{iAB} M_{ab} \lambda_{\{ij\}}^{Bb} \nn\\
&=& x_{jAB}M_{ab}\lambda_{\{ij\}}^{Bb}\,,\end{aligned}$$ where $M_{ab}$ is a normalization matrix, which we partially fix by inverting the original constraint equation $$\begin{aligned}
&&I[(x_i-x_j)^{AB}] = I[\lambda_{\{ij\}}^{Aa}] I[\lambda_{\{ij\}a}^B] \nn\\
\Rightarrow&& (x_i^{-1})_{AC}(x_i-x_j)^{CD}(x_j^{-1})_{DB} = x_{iAC} M_{ab}\lambda_{\{ij\}}^{Cb} \, x_{jBD}M^{ac}\lambda_{\{ij\}c}^D \nn\\
\Rightarrow&& (x_i-x_j)^{CD} = -x_i^2 x_j^2 M_{ab}M^{ac} \lambda_{\{ij\}}^{Cb} \lambda_{\{ij\}c}^{D} \nn\\
\Rightarrow&& M_{ab}M^{ac} = -\frac{\delta_b^c}{x_i^2x_j^2}\,.\end{aligned}$$ This is the only constraint on $M$. Without loss of generality, we choose $M_{ab}=\epsilon_{ab}(x_i^2x_j^2)^{-1/2}$.
- $I[\eta_{\{ij\}a}] = -\sqrt{\frac{x^2_i}{x^2_j}} \Bigl(\eta^a_{\{ij\}}+(x^{-1}_i)_{AB}\theta^A_i\lambda_{\{ij\}}^{Ba}\Bigr)$
To invert $\eta$, we begin with the constraint equation $(\theta_i-\theta_j)^A = \lambda_{\{ij\}}^{Aa}\eta_{\{ij\}a}\,$. Inverting, we have $$\begin{aligned}
&&I[(\theta_i-\theta_j)^A] = I[\lambda_{\{ij\}}^{Aa}]I[\eta_{\{ij\}a}] \nn\\
\Rightarrow&& (x_i^{-1})_{AB}\,\theta_i^B - (x_j^{-1})_{AB}\,\theta_j^B =
\frac{x_{iAB}}{\sqrt{x_i^2x_j^2}}\lambda_{\{ij\}a}^{B}I[\eta_{\{ij\}a}] \nn\\
&& \phantom{ (x_i^{-1})_{AB}\,\theta_i^B - (x_j^{-1})_{AB}\,\theta_j^B} =
\frac{x_{jAB}}{\sqrt{x_i^2x_j^2}}\lambda_{\{ij\}a}^{B}I[\eta_{\{ij\}a}]\,.\end{aligned}$$ Multiplying the above equations by $x_i$ and $x_j$, respectively, we get $$\begin{aligned}
\theta_i^A - (x_ix_j^{-1})^A_{\phantom{A}B}\,\theta_j^B &=& \sqrt{\frac{x_i^2}{x_j^2}} \lambda_{\{ij\}a}^{A}I[\eta_{\{ij\}a}]\,, \nn\\
(x_jx_i^{-1})^A_{\phantom{A}B}\,\theta_i^B - \theta_j^A &=& \sqrt{\frac{x_j^2}{x_i^2}} \lambda_{\{ij\}a}^{A}I[\eta_{\{ij\}a}]\,.\end{aligned}$$ Adding these two equations gives $$\begin{aligned}
(\theta_i-\theta_j)^A - (x_ix_j^{-1})^A_{\phantom{A}B}\,\theta_j^B + (x_jx_i^{-1})^A_{\phantom{A}B}\,\theta_i^B
= \frac{x_i^2+x_j^2}{\sqrt{x_i^2x_j^2}}\lambda_{\{ij\}a}^{A}I[\eta_{\{ij\}a}]\,.\end{aligned}$$ We rewrite the LHS as $$\begin{aligned}
&&\hskip -2.0cm -\lambda_{\{ij\}a}^A\eta_{\{ij\}}^{a} - \frac{(x_ix_j)^A_{\phantom{A}B}\,\theta_j^B}{x_j^2}
+\frac{(x_jx_i)^A_{\phantom{A}B}\,\theta_{i}^B}{x_i^2} \nn\\
&& =-\lambda_{\{ij\}a}^A\eta_{\{ij\}}^{a}
- \frac{(x_ix_j)^A_{\phantom{A}B}\,\theta_{i}^B - x_i^2(\theta_i-\theta_j)^A}{x_j^2}
+ \frac{(x_jx_i)^A_{\phantom{A}B}\,\theta_{i}^B}{x_i^2} \nn\\
&& = -\frac{x_i^2+x_j^2}{x_j^2}\left(\lambda_{\{ij\}a}^A\eta_{\{ij\}}^{a}
+ \theta_i^A - (x_jx_i^{-1})^A_{\phantom{A}B}\,\theta_{i}^B \right) \nn\\
&& = -\frac{x_i^2+x_j^2}{x_j^2}\left(\lambda_{\{ij\}a}^A\eta_{\{ij\}}^{a}
+(x_i-x_j)^{AB}(x_i^{-1})_{BC}\,\theta_i^C\right) \nn\\
&& = -\frac{x_i^2+x_j^2}{x_j^2}\lambda_{\{ij\}a}^A\left(\eta_{\{ij\}}^{a}
-\lambda^{Ba}_{\{ij\}}(x_i^{-1})_{BC}\,\theta_i^C\right) \,.\end{aligned}$$ where in the second line we used $(x_i-x_j)_{AB}(\theta_i-\theta_j)^B=0\,$, and in the third line we used $(x_i-x_j)^{AB}(x_i-x_j)_{BC} = 0\,$. We can now read off the solution.
- $I[u_{ia}]=\frac{\beta u_i^a}{\sqrt{x^2_{i+2}}}\,, \quad I[\tilde{u}_{i{{\dot{a}}}}]=\frac{\tilde{u}_i^{{{\dot{a}}}}}{\beta\sqrt{x^2_{i+2}}}$
Here, we begin with the definition $\langle i_a |i+1_{{{\dot{b}}}}] = u_{ia}\tilde{u}_{i+1{{\dot{b}}}}$. Contracting both sides with $u_i^a$ and inverting, we have $$\begin{aligned}
I[u_i^a] I\left[\langle i_a|i+1_{{{\dot{b}}}}]\right] = -I[u_i^a] \frac{\langle i^a|i+1^{{{\dot{b}}}}]}{\sqrt{x_i^2x_{i+2}^2}} = 0\,.\end{aligned}$$ Since $u_{ia}$ is the only vector annihilated by the matrix $\langle i|i+1]$, we conclude that $$I[u_i^a] = \alpha_i u_{ia}\,,$$ for some $\alpha_i$. Returning to the original equation defining $u$ and $\tilde{u}$, we get a set of constraints on $\alpha_i$, $$\begin{aligned}
\alpha_1 \tilde\alpha_2 &= -(x_1^2 x_3^2)^{-1/2}\,, & \tilde\alpha_1\alpha_2 =& -(x_1^2 x_3^2)^{-1/2}\,, \nn\\
\alpha_2 \tilde\alpha_3 &= -(x_2^2 x_1^2)^{-1/2}\,, & \tilde\alpha_2\alpha_3 =& -(x_2^2 x_1^2)^{-1/2}\,, \nn\\
\alpha_3 \tilde\alpha_1 &= -(x_3^2 x_2^2)^{-1/2}\,, & \tilde\alpha_3\alpha_1 =& -(x_3^2 x_2^2)^{-1/2} \,.\end{aligned}$$ The solution to these equations is $$\begin{aligned}
\alpha_i &= \frac{\beta}{\sqrt{x_{i+2}^2}}\,, & \tilde\alpha_i = \frac{1}{\beta\sqrt{x_{i+2}^2}}\,.\end{aligned}$$
- $I[w_{ia}]=-\frac{1}{\beta}\sqrt{x^2_{i+2}}w_i^{a}\,, \quad
I[\tilde{w}_{i\dot{a}}]=-\beta\sqrt{x^2_{i+2}}\tilde{w}_i^{\dot{a}}$
Because $w$ is defined as the pseudoinverse of $u$ via $u_{ia}w_{ib}-u_{ib}w_{ia}=\epsilon_{ab}$, its inversion is straightforward. The definition inverts as $$\begin{aligned}
\epsilon^{ba} &=& I[u_{ia}]I[w_{ib}]-I[u_{ib}]I[w_{ia}] \,, \nn\\
&=& \frac{\beta}{\sqrt{x_{i+2}^2}} \left(u_i^aI[w_{ib}]-u_i^bI[w_{ia}]\right) \,.\end{aligned}$$ This is again the definition of $w$ as the psuedoinverse of $u$, $$\frac{\beta}{\sqrt{x_{i+2}^2}}I[w_{ia}] = -w_i^a \,,$$ whence the result follows.
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[^1]: Email: [email protected]
[^2]: Email: [email protected]
[^3]: Besides (super) Yang-Mills amplitudes, these variables have also been used to analyze the $\mathcal{N}=(2,0)$ theory in ref. [@Huang:2010rn].
|
{
"pile_set_name": "ArXiv"
}
|
---
address:
- 'Institut f[ü]{}r Theoretische Physik und Astrophysik, Universit[ä]{}t W[ü]{}rzburg, 97074 W[ü]{}rzburg, Germany,'
- 'Institut f[ü]{}r Theoretische Physik und Astrophysik, Universit[ä]{}t W[ü]{}rzburg, 97074 W[ü]{}rzburg, Germany,'
author:
- 'Pin-Jui Hsu'
- 'Jens K[ü]{}gel'
- Jeannette Kemmer
- Francesco Parisen Toldin
- Tobias Mauerer
- Matthias Vogt
- Fakher Assaad
- Matthias Bode
bibliography:
- 'article.bib'
title: 'Coexistence of Charge- and Ferromagnetic-Order in fcc Fe'
---
[ **Phase coexistence phenomena have been intensively studied in strongly correlated materials where several ordered states simultaneously occur or compete. Material properties critically depend on external parameters and boundary conditions, where tiny changes result in qualitatively different ground states. However, up to date, phase coexistence phenomena have exclusively been reported for complex compounds composed of multiple elements. Here we show that charge- and magnetically ordered states coexist in double-layer Fe/Rh(001). Scanning tunneling microscopy and spectroscopy measurements reveal periodic charge-order stripes below $T_{\rm P} = 130$K. At $T = 6$K they are superimposed by ferromagnetic domains as observed by spin-polarized scanning tunneling microscopy. Temperature-dependent measurements reveal a pronounced cross-talk between charge- and spin-order at the ferromagnetic ordering temperature $T_{\rm C} \approx 70$K, which is successfully modeled within an effective Ginzburg-Landau ansatz including sixth-order terms. Our results show that subtle balance between structural modifications can lead to competing ordering phenomena.** ]{} In the recent past competing order phenomena, such as the interplay between spin- and charge-order in copper- and iron-based superconductors [@JHoffman; @JDavis; @PCai], the magnetic modulation–induced emergence of spontaneous polarization in multiferroics [@PRL85_3720; @TKimura; @MKenzelmann; @SCheong], or the coexistence of magnetism and superconductivity at the interface of oxide heterostructures [@DADikin; @LLi; @JABert] have intensively been investigated. In these materials subtle changes of the chemical composition or external stimuli may eventually lead to nontrivial emergent excitations at quantum critical points between lowest energy states [@Sachdev_Science].
Although iron (Fe) is usually considered the prototypical ferromagnetic material, it exhibits strong correlations between atomic, orbital, and magnetic spin structure, that result in a rich variety of interesting magnetic properties. Bulk Fe crystallizes in a body-center-cubic (bcc) crystal structure and shows robust ferromagnetism (FM) with a Curie temperature $T_{\rm C} = 1043$K [@Kittel]. For low-dimensional Fe ultra-thin films and nanostructures, however, various magnetic ground states and nontrivial spin textures have been theoretically predicted, including non-magnetic [@CSWang], non-collinear antiferromagnetic (AFM) ordering [@FJPinski; @TAsada], incommensurate spin-density wave (SDW) [@DSpisak], helical spin spiral (SS) [@KKnopfle], and magnetic skyrmions [@SHeinze].
Recent advanced experimental studies [@DQian; @AKubetzka; @SMeckler; @NRomming] indicate, that these apparently contradicting reports are caused by the fact that the magnetic ground state of Fe is highly sensitive to the interplay of electronic hybridization and structural instabilities in reduced dimensions. In particular, the magnetism of ultra-thin pseudomorphic Fe films on fcc Rh(001), which has been subject of several investigations [@PhysRevB.64.054417; @Hayashi_JPSJap; @DSpisak2; @AlZubi; @MTakada; @KWK2015], appears to be strongly influenced by the competition between ferromagnetic order in bcc $\alpha$- and antiferromagnetism in fcc $\gamma$-Fe as well as electronic hybridization of the film’s $3d$ with substrate’s $4d$ states [@DSpisak2]. While the monolayer exhibits an antiferromagnetic c$(2 \times 2)$ spin structure [@DSpisak2; @AlZubi; @KWK2015], films with a local thickness of 2 and 3atomic layers (AL) are ferromagnetically ordered with the easy axis of magnetization along the surface normal [@MTakada; @KWK2015]. It has been speculated that the competition between antiferromagnetic and ferromagnetic order in tetragonally distorted films may lead to low-energy excitations or competing phase transitions [@DSpisak2].
In this study, we report on the observation of a phase coexistence phenomenon in a pseudomorphic Fe double-layer films grown on Rh(001). Our scanning tunneling microscopy (STM) and spectroscopy (STS) measurements reveal that different order phenomena, i.e. charge- and ferromagnetic spin-order, coexist at low temperatures for below the respective phase transitions. Interestingly, we observe observe a remitted reduction of the charge-order parameter $\phi$ at the ferromagnetic Curie temperature, indicating a crosstalk between charge- and spin-order. This behavior is successfully be modeled by Ginzburg-Landau (GL) calculations. We speculate that this crosstalk may be mediated by electronic states at or very close to the Fermi level. Since the system investigated here is structurally much more simple than other materials with coexisting order phenomena, it may become a model system and allow for a better understandings of competing order phenomena.
Results
=======
**Coexistence of charge- and ferromagnetic spin-order.** Figures\[fig:Magn\_and\_stripes\]a and b show the topography and the differential conductivity $\mathrm{d}I/\mathrm{d}U$, respectively, of a $(1.95 \pm 0.02)$AL Fe film on Rh(001) resolved by using spin-polarized scanning tunneling microscopy (SP-STM) at $T = 5$K. An almost perfectly closed double-layer is obtained, with a few holes and some tiny triple-layer islands as the only imperfections. Both data sets were measured simultaneously using a Cr-coated probe tip with out-of-plane magnetic sensitivity [@MBode1]. The magnetic contrast is particularly well visible in Fig.\[fig:Magn\_and\_stripes\]b which shows the tunneling magneto-resistance (TMR) contrast of two domains with opposite perpendicular magnetization as signaled by the dark and bright regions in the lower left and upper right of the image.
The Curie temperature $T_{\rm C}$ of this double-layer turns out to be very low. While $T_{\rm C}$ of a 3.0AL Fe film on Rh(001) amounts to approximately 320K, a steep linear decrease has been observed towards thinner films [@Hayashi_JPSJap]. For the ferromagnetic double-layer no magnetic signal could be detected down to $T = 97$K [@Hayashi_JPSJap]. Linear extrapolation to an Fe coverage of 2.0AL [@Hayashi_JPSJap] suggests a Curie temperature below 80K. In fact, our temperature-dependent SP-STM measurements confirm this finding (see Supplementary Figure 1). It has been speculated that the surprisingly low Curie temperature may be related to the above-mentioned FM–AFM competition which potentially results in a very low Curie temperature and/or excited magnetic states at relatively low excitation energies [@DSpisak2]. As we will show below the situation appears to be even more complex, with different competing ordering phenomena at work.
![[**Coexistence of magnetic domains and a striped phase.**]{} ([**a**]{}) Topography and ([**b**]{}) the simultaneously measured spin-resolved $\mathrm{d}I/\mathrm{d}U$ map of $(1.95 \pm 0.02)$AL Fe/Rh(001) showing out-of-plane magnetic domains (scan parameters: $U = -0.7$V, $I = 500$pA, $T = 5$K). Scale bar is 15 nm. The white and black boxes mark regions where the stripe superstructure is oriented parallel (${\mathbf{q}_{1}}$) or perpendicular (${\mathbf{q}_{2}}$) to the magnetic domain wall, respectively. ([**c**]{}) Schematic drawing of the magnetic domain and stripe patterns observed in [**b**]{}. ([**d**]{}),([**e**]{}) Zoomed-in topographic and $\mathrm{d}I/\mathrm{d}U$ image of a location similar to the one shown in the black box in [**b**]{} ($U = -1.0$V, $I = 500$pA). Scale bars are 3 nm. ([**f**]{}),([**g**]{}) Line sections obtained from [**e**]{} and [**d**]{}, showing spin-resolved $\mathrm{d}I/\mathrm{d}U$ signal together with topographic corrugation. Fitting the domain wall (red line) gives a wall width $w = (4.32 \pm 0.35)$nm. Note, that the presence of the domain wall does not significantly influence the modulation of the stripe superstructure, as indicated by the equally spaced dashed red lines. \[fig:Magn\_and\_stripes\] ](Fig1.pdf){width="0.99\columnwidth"}
Interestingly, Fig.\[fig:Magn\_and\_stripes\]a and b also reveal that the out-of-plane FM spin-order of the Fe double-layer on Rh(001) coexists with a periodic one-dimensional superstructure that consists of stripes along the \[100\] and \[010\] directions of the substrate. For clarity the coexisting magnetic domain and stripe patterns observed in Fig.\[fig:Magn\_and\_stripes\]b is schematically represented by dark/bright background and differently oriented periodic lines in Fig.\[fig:Magn\_and\_stripes\]c, respectively. The periodicity of the stripes amounts to $(1.48 \pm 0.22)$nm, corresponding to a superstructures with wave vectors ${\mathbf{q}_{1}} = (2 \pi / a)(0.26 \pm 0.03,0,0)$ and $\mathbf{q_{2}} = (2 \pi / a)(0,0.25 \pm 0.03,0)$, where $a$ is the Rh(001) atomic lattice constant of 3.80[Å]{}.
Detailed analysis indicates that at the measurement temperature of 5K there is no significant correlation between the stripe pattern and the magnetic domain structure. For example, the black and white boxes in Fig.\[fig:Magn\_and\_stripes\]b mark surface areas where the stripe superstructure is oriented parallel (${\mathbf{q}_{2}}$) or perpendicular (${\mathbf{q}_{1}}$) to this domain wall, respectively. In neither case the presence of the domain wall seems to have any significant influence on the stripes.
This impression is also confirmed by the analysis of an area with a configuration similar to the black box in Fig.\[fig:Magn\_and\_stripes\]b, i.e. with stripes along the \[010\] directions (${\mathbf{q}_{2}}$) across a magnetic domain wall. Fig.\[fig:Magn\_and\_stripes\]d and e show the topography and the $\mathrm{d}I/\mathrm{d}U$ map, respectively. An individual defect is marked by circles in both images. Fig.\[fig:Magn\_and\_stripes\]f presents line sections drawn along the long axes of these images. Comparison of the peak position of the stripe superstructure in both the topographic as well as the spin-resolved $\mathrm{d}I/\mathrm{d}U$ channel shows no indication for any changes across the domain wall (see dashed red vertical lines in Fig.\[fig:Magn\_and\_stripes\]e).
![[**Scanning tunneling spectroscopy and bias voltage–dependent corrugation.**]{} ([**a**]{}) Topography of the Fe double-layer on Rh(001) and ([**b**]{})-([**e**]{}) d*I*/d*U* maps taken at the indicated bias voltages from constant-separation STS data (setpoint parameters: $U = +1.0$V, $I = 500$pA). The stripe pattern is only visible in the occupied energy range. Image sizes are 15 x 15 nm$^{2}$. ([**f**]{}) Tunneling spectra measured along the box in [**d**]{}. The peak at $-0.2$V is more intensive on the bright stripes than between them. ([**g**]{}) Plot of the electronic asymmetry as a function of bias voltages. While the electronic asymmetry is negligible in the empty states, it becomes maximal in the occupied states at about $-0.2$V. ([**h**]{}),([**i**]{}) Topographic images taken at $U = +0.6$V and $-0.6$V. Scale bars are 5 nm. ([**j**]{}) Averaged line sections measured along the red and black line. ([**k**]{}) Bias-dependence of the corrugation. Error bars are given by standard deviation of corrugation peak heights. \[fig:Spec\] ](Fig2.pdf){width="0.99\columnwidth"}
**Electronic structure of charge-ordered phase.** In order to unravel the physical origin of the stripes we have performed STS measurements to probe the local density-of-states (LDOS) of the region shown in Fig.\[fig:Spec\]a (topography). Fig.\[fig:Spec\]b-e shows a series of $\mathrm{d}I/\mathrm{d}U$ maps extracted from this data set at some representative bias voltages $U$. Obviously, the appearance of stripes strongly depends on $U$. While the stripes cannot be detected within the signal-to-noise ratio at positive bias voltages, i.e. when tunneling into empty sample states, they are clearly visible at negative bias (occupied states). As can be seen in curves 1 through 15 of Fig.\[fig:Spec\]f (obtained within the box in Fig.\[fig:Spec\]d), the spectra are characterized by a pronounced peak at $-0.2$V, a weaker peak at $-0.6$V, and a dip at $-0.8$V (which are all absent in the relatively featureless spectrum of the Fe monolayer; not shown here). The sequence of spectra reveals that the peak intensity varies periodically. Obviously, the main peak at $-0.2$V appears much more intense when the tip positioned above a bright stripe (see, e.g., spectrum 2) than above a dark one (spectrum 5).
The sign and intensity of the bias-dependent contrast of the striped superstructure can be analyzed more systematically by calculating the energy-dependent asymmetry which is defined as the difference of the differential conductance measured on and off a bright stripe at a particular energy, $E - E_{\rm F}$, divided by their sum, i.e. ${\left( \mathrm{d}I/\mathrm{d}U \right)_{\rm on}}
\mathbin{/}
{\left( \mathrm{d}I/\mathrm{d}U \right)_{\rm off}}$. The asymmetry curve calculated from tunneling spectra is plotted in Fig.\[fig:Spec\]g. While the asymmetry is negligible at positive sample bias several features can be recognized at negative bias voltages, with a pronounced asymmetry maximum at $-0.2$V.
Fig.\[fig:Spec\]h and i exemplarily show two topographic STM images obtained at the same sample location. While only subtle modulations can be recognized at $U = +0.6$V (Fig.\[fig:Spec\]h) the stripes are clearly resolved at $U = -0.6$V (Fig.\[fig:Spec\]i). The corresponding line profiles in Fig.\[fig:Spec\]j reveal a corrugation of about 14pm at $U = -0.6$V, whereas it is below $2$pm at $U = +0.6$V. Fig.\[fig:Spec\]k summarizes the observed bias-dependence of the corrugation. Apparently the corrugation is extremely low at positive bias, rises up to a maximum value of about 17pm at $U \approx -0.2 ... 0.4$V—a value that is in line with the above-mentioned energy-dependent asymmetry (cf. Fig.\[fig:Spec\]g)—and then slowly decreases to $\approx 12$pm at $U = -1$V.
[**Temperature-dependent electronic reconstruction.**]{} While the measurements presented so far suggest the existence of two apparently independent ordering phenomena, i.e. ferromagnetism and the formation of stripes with a pronounced LDOS modulation, the following data indicate that the two phenomena are coupled and influence each other. Fig.\[fig:VT\]a-c show three STM topographic images of a 2 AL Fe film on Rh(001) taken at $T = 33$K, 49K, and 67K. In order to exclude any potential influence of local fluctuations of sample quality all data were taken at the same location as emphasized by white arrows pointing at one particular island. Since the rather small corrugation of the stripes indicative for charge-order is difficult to recognize, the corresponding rendered perspective images of Fig.\[fig:VT\]a-c (viewing direction indicated by a black arrow) are displayed in Fig. \[fig:VT\]d. While stripes are clearly visible at 33K (Fig.\[fig:VT\]a), a much lower corrugation amplitude can be recognized at 49K (Fig.\[fig:VT\]b). Surprisingly, the intensity of the stripes increases again as the temperature is further increased to $T = 67$K (Fig.\[fig:VT\]c).
![[**Temperature-dependent charge ordering.**]{} ([**a**]{})-([**c**]{}) STM topographic showing the very same location of the sample surface at $T = 33$K, $49$K, and $67$K. The scale bars are 20nm long. One island is marked by white arrows. ([**d**]{}) Rendered perspective images as seen along the viewing direction marked by a black arrow in (a). While the stripes are clearly visible at low and high temperatures (see, e.g., the area in front of the island marked by a white arrow), the intensity is strongly reduced at the intermediate temperature. ([**e**]{}) Plot of the temperature-dependent normalized intensity of the spots indicative for the stripe pattern (see arrow). A dip starting at around $T = 70$K can be recognized. An example of a Fourier-transformed STM image taken at $T = 85$K is shown in the inset. Error bars represent the spot’s full width at hals maximum after subtraction of the background intensity. Scale bar is 0.5nm$^{-1}$. \[fig:VT\] ](Fig3.pdf){width="0.7\columnwidth"}
Figure\[fig:VT\]e presents a summary of several temperature-dependent data sets that were obtained through fast Fourier transformation (FFT) of constant-current STM images. As can be seen in the inset of Fig.\[fig:VT\]e this results in four spots—one of which is marked by an arrow—indicative of the above-mentioned superstructures with ${\mathbf{q}_{1,2}}$. Starting at the lowest temperature accessible with our variable-temperature STM, i.e. 35K, the normalized intensity of these FFT spots first decreases with increasing temperature up to $T \approx 50$K. When increasing the temperature further, however, an unexpected upturn of the FFT intensity is observed until approximately the initial value is reached at $T \approx 70$K. Raising the temperature beyond this value leads to another reduction of the FFT intensity until it eventually becomes indistinguishable from the background above the charge-order transition temperature $T_{\rm P} = (128 \pm 12)$K. We note that this temperature dependence is continuous and reversible, i.e. the periodic modulations as well as the spots in the FFT image reappear as the temperature is lowered, consistent with a second-order phase transition and excluding any potential aging or contamination effects.
Our experimental observations indicate that there are two competing ordering phenomena at work: (i) Charge-order sets in at about 130K and results in stripes visible at negative bias voltages and (ii) ferromagnetic order which can be observed by spin-polarized STM below about 75K. Since both ordering phenomena can be explained by Fermi surface instabilities, some degree of cross-correlation can readily be expected. It should be noted that in many cases the onset of charge-order coincides with electronic structure changes which are not largest directly at the Fermi level but at slightly different binding energies. For example, when cooling through the charge-density wave phase transition temperatures of 1$T$-TiSe$_2$ [@Claessen1990; @Rossnagel2002] or 1$H$-TaSe$_2$ [@Valla2000] in temperature-dependent photoemission spectroscopy experiments the strongest variation in the energy distribution curves was observed at $E - E_{\rm F} = -0.2$eV, i.e. at a binding energy similar to what we present in Fig.\[fig:Spec\](f). We can only speculate why the peak that is indicative of charge-order in our STS spectra doesn’t appear directly at but 200meV below the Fermi level. One possibility would be that the responsible bands somewhat disperse and exhibit tunneling matrix elements that are higher for states which are still involved in the phase transition but further away from the Fermi level.
[**Ginzburg-Landau theory.**]{} Indeed, the observed phenomenology can be modeled within a Ginzburg-Landau (GL) theory. Its formulation is dictated by the universal properties of the system, such as the number of components of the order parameter and the symmetries of the system. In the present case there are two order parameters describing the charge and magnetic order. Pinning of the charge order and magnetic anisotropy allow to consider two scalar order parameters, the charge order parameter $\phi$, which can be identified with the intensity of the FFT spots, and the Ising-like magnetization $m$. The systems exhibits a ${\mathbb Z}_2$ symmetry on both order parameters, $\phi\rightarrow -\phi$, $m\rightarrow -m$, so that the global symmetry group is ${\mathbb Z}_2\oplus{\mathbb Z}_2$. A GL free energy is obtained by expanding the free energy $F$ in powers of $\phi$ and $m$, retaining only the terms which respect the given symmetry group (see, e.g., Ref. [@critbook]): $$\begin{gathered}
F = F_0 + \frac{a}{2}\left(T-T_{\rm P}\right)\phi^2 + \frac{b}{4}\phi^4
+ \frac{a'}{2}\left(T-T_{\rm C'}\right)m^2+\frac{b'}4m^4+\frac{\gamma}{2}\phi^2m^2,
\label{LandauF}\end{gathered}$$ where $\gamma$ is the coupling constant between $\phi$ and $m$, and we have already encoded the expected temperature dependence of the quadratic terms close to the onset of non-zero order parameters.
The minimization of the free energy $F$ determines thermal equilibrium, whose stability requires $b$, $b' > 0$. Depending on its coefficients, one finds in general four possible solutions to the minimization of $F$: a solution where both order parameters vanish, two solutions where one of the order parameter is vanishing, and a solution with a coexistence of both order parameters. The observed charge-order in the absence of magnetization implies that $T_{\rm C'} < T_{\rm P}$, and $a$, $a' > 0$, so that $\phi$ orders at $T=T_{\rm P}$. When the coupling constant $\gamma$ satisfies the constraints $ab'/a' < \gamma < a'b/a$ and $\gamma^2<bb'$, $\phi$ exhibits a maximum at the magnetic critical temperature $T_{\rm C} = (a'bT_{\rm C'}-\gamma aT_{\rm P}) / (a'b-\gamma a)$.
![[**Expansion of the temperature-dependent Ginzburg-Landau free energy.**]{} Plots showing the results of an expansion of the GL free energy $F$ to the forth (([**a**]{}) parameters: $a = 0.9$, $a' = 1$, $T_{\rm P} = 130$, $T_{\rm C'} = 100$, $b = 2.4$, $b' = 0.9$, $\gamma = 1.4$, $c = c' = 0$), and to the sixth power (([**b**]{}) same parameters except for $c = c' = 0.015$), as described in Eqs. (\[LandauF\]) and (\[LandauF6\]), respectively.[]{data-label="phim"}](Fig4.pdf){width="0.99\columnwidth"}
An example of the resulting order parameters is shown in Fig. \[phim\]a. We observe that this solution displays an interval of temperature where $\phi$ vanishes while $m > 0$. This behavior is a result of the competition between the $\frac{a}{2}\left(T-T_{\rm P}\right)\phi^2$ term, which is negative for $T<T_{\rm P}$, and the positive coupling with $m$, $\frac{\gamma}{2}\phi^2m^2$: upon decreasing the temperature, $m$ grows and a strong enough coupling $\gamma$ pushes down the value of $\phi$ which minimizes $F$, eventually leading to $\phi = 0$. However, this zero of the charge-order parameter is unstable with respect to the inclusion of higher-order terms in Eq. (\[LandauF\]). Although such corrections are irrelevant close to the phase transitions of $\phi$ and $m$, they nevertheless influence their growth in a wider range of temperatures. In fact, the expansion of Eq. (\[LandauF\]) up to the fourth order predicts the order parameters to grow indefinitely below the critical temperature, whereas in real materials a saturation effect is expected. This suggests to study an improved GL free energy expansion, including next-to-leading powers:
$$\begin{gathered}
F = F_0 + \frac{a}{2}\left(T-T_{\rm P}\right)\phi^2 + \frac{b}{4}\phi^4 + \frac{c}{6}\phi^6
+ \frac{a'}{2}\left(T-T_{\rm C'}\right)m^2+\frac{b'}4m^4+\frac{\gamma}{2}\phi^2m^2+\frac{c'}{6}m^6,
\label{LandauF6}\end{gathered}$$
where stability requires $c$, $c' > 0$. The inclusion of the sixth power qualitatively changes the behavior of $\phi$, giving rise to a dip qualitatively similar to the experimentally observed behavior. In other words, the solution with a vanishing charge-order and non-zero magnetization requires a fine-tuning of the higher-order terms $c = c' = 0$, whereas their inclusion naturally explains the observed dip in $\phi$. The sixth term effectively damps the growth of $m$ away from $T=T_{\rm C}$, such that $\phi$ is not pushed down to $0$, but instead displays a minimum.
Discussion
==========
Our experiments show that a sample system that is conceptually as simple as the pseudomorphic Fe double-layer of Fe/Rh(001) may possess different ordering phenomena. Even more importantly, our data reveal that the ordering phenomena at play here, i.e. charge-order and ferromagnetism, compete with each other as evidenced by the intermediate reduction of the charge-order parameter $\phi$ observed experimentally and in GL calculations. We speculate that this cross-talk is caused by the fact that the electronic structure close to the Fermi level, which is responsible for charge- and magnetic-order phenomena through Fermi surface nesting and the exchange interaction, respectively, drastically changes at the respective critical temperatures. We expect that details of the temperature-dependent evolution of the electronic structure will be subject of future experiments, such as angular-resolved photoemission spectroscopy, to shed light on the subtle balance between nested and exchange-split electronic states.
Methods
=======
[**STM, STS, and SPSTM measurements.**]{} Scanning tunneling microscopy (STM) measurements were performed under ultrahigh vacuum ($p \leq 5.0 \times 10^{-11}$mbar) with a home-built low-temperature (LT)-STM and a commercial variable-temperature (VT)-STM at sample temperatures of $T_{\rm LT} = 5$K and $T_{\rm VT} = 30 ... 300$ K, respectively. STM tips were prepared from electro-chemically etched tungsten (W) wires which were flashed under ultrahigh vacuum conditions. For spin-resolved measurements the W tips were coated with $\approx 20$ atomic layers (AL) of Cr, resulting in out-of-plane magnetic sensitivity, as verified by test measurements on samples with well-known magnetization directions, i.e. Fe/W(110) or Mn/W(110) [@MBode1]. For scanning tunneling spectroscopy (STS) measurements a small modulation was added to the sample bias voltage $U$ (frequency $\nu = 5.777$kHz; amplitude 5 to 15mV), such that tunneling differential conductance d$I/$d$U$ spectra as well as d$I/$d$U$ maps can be acquired by detecting the first harmonic signal with a lock-in amplifier.
[**Sample preparation of Fe/Rh(001).**]{} The Rh(001) surface was prepared by cycles that consist of about 10min Ar ion sputtering at room temperature ($p_{\rm Ar} = 5 \times 10^{-6}$mbar, $E_{\rm ion} = 1$keV), followed by 150sec annealing at $T_{\rm an} = 1300$K in an oxygen atmosphere, and a final flash (duration about 30sec) without oxygen at the same temperature. It has been shown that this procedure reliably removes carbon impurities from the surface [@KWK2015]. Subsequently, Fe films were grown on the Rh(001) surface at $T = 315$K by means of the e-beam evaporation.
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Acknowledgments
===============
This work has been funded by Deutsche Forschungsgemeinschaft within BO 1468/22-1 and through SFB 1170 “ToCoTronics”.
Author contributions
====================
P.-J.H., J.K., J.K., T.B., and M.V. performed the STM experiment and jointly with M.B. analyzed the experimental data. F.P.T. and F.A. performed Ginzburg-Landau modeling. All authors contributed to the scientific discussion. P.-J.H., F.P.T. and M.B. wrote the manuscript with input and comments from all co-authors.
Additional information
======================
Supplementary information is included in this submission.
Competing financial interests
=============================
The authors declare no competing financial interests.
|
{
"pile_set_name": "ArXiv"
}
|
[Variation formulas for transversally harmonic and bi-harmonic maps ]{}
[^1][^2]
[**Abstract.**]{} In this paper, we study variation formulas for transversally harmonic maps and bi-harmonic maps, respectively. We also study the transversal Jacobi field along a map and give several relations with infinitesimal automorphisms.
Introduction
============
Let $(M,\mathcal F)$ and $(M',\mathcal F')$ be two foliated Riemannian manifolds and let $\phi:M\to M'$ be a smooth foliated map, i.e., $\phi$ is a leaf-preserving map. Then $\phi$ is transversally harmonic if $\phi$ is a critical point of the transversal energy functional on any compact domain of $M$, which is defined in Section 3 (cf. \[\[JJ2\],\[KW1\],\[KW2\]\]). Equivalently, it is a solution of $\tau_b(\phi)=0$, where $\tau_b(\phi)$ is a transversal tension field, which is given by $\tau_b(\phi)={\rm tr}_Q\tilde\nabla d_T\phi$ (see \[\[KW1\]\] for more details). That is, transversally harmonic maps are considered as harmonic maps between the leaf spaces \[\[KW1\],\[KW2\]\]. For harmonic maps, see \[\[ES\],\[XI\]\]. Also, we study the transversally bi-harmonic map as the critical point of the transversal bi-energy functional on any compact domain of $M$ (Section 6). In this paper, we study the second variation formulas for the transversal energy and transversal bi-energy of $\phi$. And we give some applications. This paper is organized as follows. In Section 2, we recall the basic facts on foliated manifolds. In Section 3, we review transversally harmonic maps and the first variation formula. In Section 4, we give the second variation formula for the transversal energy. In Section 5, we define the transversal Jacobi operator along the foliated map and study its realtion with infinitesimal automorphisms. In Section 6, we study transversally bi-harmonic maps and their applications. In Section 7, we give the second variation formula for the transversal bi-energy. Note that some results in Section 6 and Section 7 of the present paper can be found in \[\[CW\]\], but the approach is different in a technical sense. Throughout this paper, $(M,\mathcal F)$ is considered as a foliated Riemannian manifold, i.e., a Riemannian manifold with a Riemannian foliation, and all leaves of $\mathcal F$ are compact.
Preliminaries
==============
Let $(M,g,\mathcal F)$ be a $(p+q)$-dimensional foliated Riemannian manifold with foliation $\mathcal F$ of codimension $q$ and a bundle-like metric $g$ with respect to $\mathcal F$. Let $TM$ be the tangent bundle of $M$, $L$ the tangent bundle of $\mathcal F$, and $Q=TM/L$ the corresponding normal bundle of $\mathcal F$. Let $g_Q$ be the holonomy invariant metric on $Q$ induced by $g$. We denote by $\nabla^Q$ the transverse Levi-Civita connection on the normal bundle $Q$ \[\[TO1\],\[TO2\]\]. Let $R^Q, K^Q,{\rm Ric}^Q $ and $\sigma^Q$ be the transversal curvature tensor, transversal sectional curvature, transversal Ricci operator and transversal scalar curvature with respect to $\nabla^Q\equiv\nabla$, respectively. Let $\Omega_B^r(\mathcal F)$ be the space of all [*basic $r$-forms*]{}, i.e., $\omega\in\Omega_B^r(\mathcal F)$ if and only if $i(X)\omega=0=i(X)d\omega$ for any $X\in\Gamma L$, where $i(X)$ is the interior product. Then $\Omega^r(M)=\Omega_B^r(\mathcal F)\oplus \Omega_B^r(\mathcal F)^\perp$ \[\[LO\]\]. Let $\kappa_B$ be the basic part of $\kappa$, the mean curvature form of $\mathcal F$. Then $\kappa_B$ is closed, i.e., $d\kappa_B=0$ \[\[LO\]\]. The [*basic Laplacian*]{} $\Delta_B$ acting on $\Omega_B^*(\mathcal F)$ is defined by $$\label{2-1}
\Delta_B=d_B\delta_B+\delta_B d_B,$$ where $\delta_B$ is the formal adjoint of $d_B=d|_{\Omega_B^*(\mathcal F)}$\[\[PR\],\[TO2\]\]. Let $V(\mathcal F)$ be the space of all transversal infinitesimal automorphisms $Y$ of $\mathcal F$, i.e., $[Y,Z]\in \Gamma L$ for all $Z\in \Gamma L$. Let $\bar V(\mathcal F)=\{\bar Y=\pi(Y)| Y\in V(\mathcal F)\}$, where $\pi:TM\to Q$ is a projection. Trivially, $\bar V(\mathcal F)\cong \Omega_B^1(\mathcal F)$ \[\[MO\]\]. For later use, we recall the transversal divergence theorem \[\[YO\]\] on a foliated Riemannian manifold.
\[thm1-1\] Let $(M,g_M,\mathcal F)$ be a closed, oriented Riemannian manifold with a transversally oriented foliation $\mathcal F$ and a bundle-like metric $g_M$ with respect to $\mathcal F$. Then $$\label{2-4}
\int_M \operatorname{div_\nabla}\bar X = \int_M g_Q(\bar
X,\kappa_B^\sharp)$$ for all $X\in V(\mathcal F)$, where $\operatorname{div_\nabla}\bar X$ denotes the transversal divergence of $\bar X$ with respect to the connection $\nabla^Q$.
Now, we define the bundle map $A_Y:\Lambda^r
Q^*\to\Lambda^r Q^*$ for any $Y\in V(\mathcal F)$ \[\[KT\]\] by $$\begin{aligned}
\label{2-5}
A_Y\omega =\theta(Y)\omega-\nabla_Y\omega \quad\forall\omega\in\Lambda^r Q^*,\end{aligned}$$ where $\theta(Y)$ is the transverse Lie derivative. It is well-known \[\[KT\]\] that, on $\Gamma Q$ $$\begin{aligned}
\label{2-5-1}
A_Y s = -\nabla_{Y_s}\bar Y \quad\forall s\in\Gamma Q,\end{aligned}$$ where $Y_s$ is the vector field such that $\pi(Y_s)=s$. So $A_Y$ depends only on $\bar Y=\pi(Y)$. Since $\theta(X)\omega=\nabla_X\omega$ for any $X\in\Gamma L$, $A_Y$ preserves the basic forms and depends only on $\bar Y$. Let $E\to M$ be a vector bundle over $M$ and $\Omega_B^r (E)\equiv \Omega_B^r(\mathcal F)\otimes E$ be the space of all $E$-valued basic $r$-forms. Let $\nabla$ be also the connection on $E$. Then the operator $A_X$ is extended to $\Omega_B^r(E)$ \[\[JJ2\]\]. Now we define $d_\nabla :\Omega_B^r(E)\to \Omega_B^{r+1}(E)$ by $$\begin{aligned}
\label{2-6}
d_\nabla( \omega\otimes s)= d_B \omega\otimes s+ (-1)^r\omega\wedge \nabla s\end{aligned}$$ for any $\omega\in\Omega_B^r(\mathcal F)$ and $s\in E$. Let $\delta_\nabla$ be the formal adjoint of $d_\nabla$. Then we define the Laplacian $\Delta$ on $\Omega_B^r(E)$ by $$\begin{aligned}
\label{2-6-1}
\Delta=d_\nabla \delta_\nabla +\delta_\nabla d_\nabla.
\end{aligned}$$ From now on, let $\{E_a\}(a=1,\cdots,q)$ be a local orthonormal frame on $Q$ and $\theta^a$ be the $g_Q$-dual 1-form to $E_a$. Then the generalized Weitzenböck type formula on $\Omega_B^r(E)$ is given by \[\[JJ2\]\] $$\begin{aligned}
\label{2-7}
\Delta \Phi = \nabla_{\rm tr}^*\nabla_{\rm tr} \Phi
+ F(\Phi) + A_{\kappa_{B}^\sharp} \Phi,\quad\forall \Phi\in\Omega_B^r(E),\end{aligned}$$ where $\nabla_{\rm tr}^*\nabla_{\rm tr} =-\sum_a \nabla^2_{E_a,E_a}
+\nabla_{\kappa_B^\sharp}$ and $F=\sum_{a,b=1}^{q}\theta^a\wedge i(E_b) R^\nabla(E_b,E_a)$. From (\[2-7\]), we also have $$\begin{aligned}
\label{2-8}
\frac12\Delta_B| \Phi |^2 &= \langle\Delta \Phi, \Phi\rangle -
|\nabla_{\rm tr} \Phi|^2 -
\langle A_{\kappa_{B}^\sharp}\Phi, \Phi\rangle -\langle F(\Phi),\Phi\rangle,\end{aligned}$$ where $\langle\cdot,\cdot\rangle$ is an inner product on $\Omega_B^r(E)$.
Now, we recall the following generalized maximum principles.
$[\ref{JLK}]$ Let $\mathcal F$ be a Riemannian foliation on a closed, oriented Riemannian manifold $(M,g_M)$. If $(\Delta_B -\kappa_B^\sharp)f\geq 0$ $($or $\leq 0)$ for any basic function $f$, then $f$ is constant.
Transversally harmonic maps
===========================
Let $(M, g,\mathcal F)$ and $(M', g',\mathcal F')$ be two foliated Riemannian manifolds and all leaves of $\mathcal F$ are compact. Let $\nabla^{M}$ and $\nabla^{M'}$ be the Levi-Civita connections on $M$ and $M'$, respectively. And $\nabla$ and $\nabla'$ be the transverse Levi-Civita connections on $Q$ and $Q'$, respectively. Let $\phi:(M,g,\mathcal
F)\to (M', g',\mathcal F')$ be a smooth foliated map, i.e., $d\phi(L)\subset L'$. We define $d_T\phi:Q \to Q'$ by $$\begin{aligned}
\label{3-1}
d_T\phi := \pi' \circ d \phi \circ \sigma,\end{aligned}$$ where $\sigma : Q \to L^\perp$ is a bundle map satisfying $\pi\circ\sigma={\rm id}$. Then $d_T\phi$ is a section in $ Q^*\otimes
\phi^{-1}Q'$. Let $\nabla^\phi$ and $\tilde \nabla$ be the connections on $\phi^{-1}Q'$ and $Q^*\otimes \phi^{-1}Q'$, respectively. Then $\phi:(M,g,\mathcal F)\to (M',g',\mathcal F')$ is called [*transversally totally geodesic*]{} if it satisfies $$\begin{aligned}
\label{3-2}
\tilde\nabla_{\rm tr}d_T\phi=0,\end{aligned}$$ where $(\tilde\nabla_{\rm tr}d_T\phi)(X,Y)=(\tilde\nabla_X d_T\phi)(Y)$ for any $X,Y\in \Gamma Q$. And the [*transversal tension field*]{} of $\phi$ is defined by $$\begin{aligned}
\label{3-3}
\tau_b(\phi)={\rm tr}_{Q}\tilde \nabla d_T
\phi=\sum_{a=1}^{q}(\tilde\nabla_{E_a} d_T\phi)(E_a),\end{aligned}$$ where $\{E_a\}(a=1,\cdots,q)$ is a local orthonormal frame on $Q$. Trivially, the transversal tension field $\tau_b(\phi)$ is a section of $\phi^{-1}Q'$.
[Let $\phi: (M, g,\mathcal F) \to (M', g',\mathcal F')$ be a smooth foliated map. Then $\phi$ is said to be ]{} transversally harmonic [if the transversal tension field vanishes]{}, i.e., $\tau_b(\phi)=0$.
Let ${ vol}_L:M \to [0,\infty]$ be the volume map for which ${
vol}_L (x)$ is the volume of the leaf passing through $x\in
M$. It is trivial that $vol_L$ is a basic function. And it holds \[\[JJ2\]\] that $$\begin{aligned}
d_B vol_L + (vol_L) \kappa_B =0.\end{aligned}$$ The transversal energy of $\phi$ on a compact domain $\Omega\subset
M$ is defined by $$\begin{aligned}
\label{eq2-4}
E_B(\phi;\Omega)={1\over 2}\int_{\Omega} | d_T \phi|^2 {1\over vol_L}
\mu_{M},\end{aligned}$$\[3-4\] where $\mu_{M}$ is the volume element of $M$. Let $V\in\phi^{-1}Q'$. Obviously, $V$ may be considered as a vector field on $Q'$ along $\phi$. Then there is a 1-parameter family of foliated maps $\phi_t$ with $\phi_0=\phi$ and ${d\phi_t\over
dt}|_{t=0}=V$. The family $\{\phi_t\}$ is said to be a [*foliated variation*]{} of $\phi$ with the normal variation vector field $V$. Then we have the first variation formula.
$[\ref{JJ2}]$ \[Thm3-2\] [(The first variation formula)]{} Let $\phi:(M,g,\mathcal F)\to (M',g',\mathcal F')$ be a smooth foliated map, and all leaves of $\mathcal F$ be compact. Let $\{\phi_t\}$ be a smooth foliated variation of $\phi$ supported in a compact domain $\Omega$. Then $$\begin{aligned}
\label{eq2-5}
{d\over dt}E_B(\phi_t,\Omega)|_{t=0}=-\int_{\Omega} \langle
V,\tau_b(\phi)\rangle {1\over vol_L}\mu_{M},\end{aligned}$$ where $V(x)={d\phi_t\over dt}(x)|_{t=0}$ is the normal variation vector field of $\{\phi_t\}$.
[Let $\phi:(M,g,\mathcal F)\to (M',g',\mathcal F')$ be a smooth foliated map. Then the]{} transversal stress-energy tensor $S_T(\phi)$ [of $\phi$ is defined by]{} $$\begin{aligned}
\label{3-6}
S_T(\phi)={1\over 2}|d_T\phi|^2 g_{Q}-\phi^* g_{Q'},
\end{aligned}$$ where $\phi^*$ is the pull-back of $\phi$.
Trivially, $S_T(\phi)\in\otimes^2 Q^*$ is the symmetric 2-covariant normal tensor field on $M$.
$[\ref{CW}]$ Let $\phi:(M,g,\mathcal F)\to (M',g',\mathcal F')$ be a smooth foliated map. Then, for any vector field $X\in \Gamma Q$, $$\begin{aligned}
({\rm div}_\nabla S_T(\phi))(X)=-\langle\tau_b(\phi),d_T\phi(X)\rangle,
\end{aligned}$$ where $({\rm div}_\nabla S_T(\phi))(\cdot)=\sum_{a=1}^q (\nabla_{E_a}S_T(\phi))(E_a,\cdot)$.
[**Proof.**]{} Note that $[d_T\phi(X),d_T\phi(Y)]=d_T\phi([X,Y])$ for any $X,Y\in\Gamma Q$. So, by direct calculation, the proof follows. $\Box$
If ${\rm div}_\nabla S_T(\phi)=0$, then we say that $\phi$ satisfies the [*transverse conservation law*]{} \[\[CW\]\]. The foliated map satisfying the transverse conservation law is said to be [*transversally relatively harmonic*]{}. Then we have the following.
Any transversally harmonic map is transversally relatively harmonic.
The converse of Corollary 3.5 does not hold. For the converse, see Theorem 7.4 below.
[**Remark.**]{} $[\ref{JJ2}]$ Let $\phi:(M,g,\mathcal F)\to (M',g',\mathcal F')$ be a smooth foliated map. Then $$\begin{aligned}
\label{3-8}
d_\nabla d_T\phi=0,\quad \tilde\delta d_T\phi=-\tau_b(\phi),\end{aligned}$$ where $\tilde\delta = \delta_\nabla -i(\kappa_B^\sharp)$.
The second variation formula for the transversal energy
=======================================================
Let $(M, g,\mathcal F)$ and $(M', g',\mathcal F')$ be two foliated Riemannian manifolds, and all leaves of $\mathcal F$ be compact. Let $\phi:(M,g,\mathcal F)\to (M',g',\mathcal F')$ be a transversally harmonic map. For any $V,W\in \phi^{-1}Q'$, there exists a family of foliated maps $\phi_{t,s} (-\epsilon<s,t<\epsilon)$ satisfying $$\label{4-1}
\left\{\begin{split}&V={\partial\phi_{t,s}\over \partial t}\Big|_{(t,s)=(0,0)},\\
&W={\partial\phi_{t,s}\over \partial s}\Big|_{(t,s)=(0,0)},\\
&\phi_{0,0}=\phi.\qquad\qquad\qquad{}
\end{split}
\right.$$ Then $\{\phi_{t,s}\}$ is said to be the [*foliated variation*]{} of $\phi$ with the [*normal variation vector fields*]{} $V$ and $W$. Then we have the second variation formula for the transversal energy.
[(The second variation formula)]{} Let $\phi:(M,g,\mathcal F)\to (M',g',\mathcal F')$ be a transversally harmonic map with $M$ compact without boundary, and all leaves of $\mathcal F$ be compact. Let $\{\phi_{t,s}\}$ be the foliated variation of $\phi$ with the normal variation vector fields $V$ and $W$. Then $$\begin{aligned}
\label{4-2}
&{\partial^2\over\partial t\partial s}E_B(\phi_{t,s})\Big|_{(t,s)=(0,0)}\\
&=\int_M \langle (\nabla_{\rm tr}^\phi)^*(\nabla_{\rm tr}^\phi) V-\nabla_{\kappa_B^\sharp}^\phi V-{\rm tr}_Q R^{Q'}(V,d_T\phi)d_T\phi,W\rangle {1\over vol_L}\mu_M,\notag\end{aligned}$$ where ${\rm tr}_Q R^{Q'}(V,d_T\phi)d_T\phi=\sum_{a=1}^q R^{Q'}(V,d_T\phi(E_a))d_T\phi(E_a)$.
[**Proof.**]{} Let $\Phi:M \times (-\epsilon,\epsilon)\times(-\epsilon,\epsilon)\to M'$ be a smooth map, which is defined by $\Phi(x,t,s)=\phi_{t,s}(x)$. Let $\nabla^\Phi$ be the pull-back connection on $\Phi^{-1}Q'$. It is trivial that $[X,{\partial\over\partial t}]=[X,{\partial\over\partial s}]=0$ for any vector field $X\in TM$. From the first normal variation formula (Theorem \[Thm3-2\]), we have $$\begin{aligned}
\label{4-3}
{\partial\over\partial s}E_B(\phi_{t,s})=-\int_M\langle {\partial\Phi\over \partial s},\tau_b(\phi_{t,s})\rangle {1\over vol_L}\mu_M.\end{aligned}$$ By differentiating (\[4-3\]) with respect to $t$, we have $$\begin{aligned}
{\partial^2\over\partial t\partial s}E_B(\phi_{t,s})=-\int_M\{\langle {\partial^2\Phi\over\partial t\partial s},\tau_b(\phi_{t,s})\rangle +\langle{\partial\Phi\over\partial s},\nabla_{\partial\over\partial t}^\Phi\tau_b(\phi_{t,s})\rangle\}{1\over vol_L}\mu_M.\end{aligned}$$ At $(t,s)=(0,0)$, the first term vanishes since $\tau_b(\phi)=0$. Hence we have $$\begin{aligned}
\label{4-4}
{\partial^2\over\partial t\partial s}E_B(\phi_{t,s})\Big |_{(t,s)=(0,0)}=-\int_M \langle W,\nabla^\Phi_{\partial\over\partial t}\tau_b(\phi_{t,s})\Big|_{(t,s)=(0,0)}\rangle{1\over vol_L}\mu_M.\end{aligned}$$ We choose a local orthonormal basic frame field $\{E_a\}$ with $(\nabla E_a)(x)=0$. Then, at $x\in M$, $$\begin{aligned}
\nabla^\Phi_{\partial\over\partial t}\tau_b(\phi_{t,s})
&=\sum_a \{\nabla^\Phi_{\partial\over\partial t}\nabla^\Phi_{E_a}d_T\Phi(E_a)-\nabla^\Phi_{\partial\over\partial t} d_T\Phi(\nabla_{E_a}E_a)\}\\
&=\sum_a\{\nabla^\Phi_{E_a}\nabla^\Phi_{\partial\over\partial t} d_T\Phi(E_a)+R^\Phi({\partial\over\partial t},E_a)d_T\Phi(E_a)-\nabla^\Phi_{\partial\over\partial t} d_T\Phi(\nabla_{E_a}E_a)\}\\
&=\sum_a\{\nabla^\Phi_{E_a}\nabla^\Phi_{E_a}{\partial \Phi\over\partial t}-\nabla^\Phi_{\nabla_{E_a}E_a}{\partial \Phi\over\partial t}+R^{Q'}({\partial \Phi\over\partial t},d_T\Phi(E_a))d_T\Phi(E_a)\}.\end{aligned}$$ Hence, at $(t,s)=(0,0)$, we have $$\begin{aligned}
\label{4-5}
&\nabla^\Phi_{\partial\over\partial t}\tau_b(\phi_{t,s})\Big|_{(t,s)=(0,0)}\notag\\
&= \sum_a\{\nabla^\phi_{E_a}\nabla^\phi_{E_a}V-\nabla^\phi_{\nabla_{E_a}E_a}V +R^{Q'}(V,d_T\phi(E_a))d_T\phi(E_a)\}.
\end{aligned}$$ Hence the proof is complete. $\Box$
Let $\phi:(M,g,\mathcal F)\to (M',g',\mathcal F')$ be a foliated map with $M$ compact. Then we define the [*transversal Hessian*]{} $THess_\phi$ of $\phi$ by $$\begin{aligned}
\label{4-6}
THess_\phi(V,W)={\partial^2 \over\partial t \partial s}E_B(\phi_{t,s})\Big|_{(t,s)=(0,0)},\end{aligned}$$ where $\{\phi_{t,s}\}$ is a foliated variation of $\phi$ with the normal variation vector fields $V$ and $W$. Then we have the following corollary.
Let $\phi:(M,g,\mathcal F)\to (M',g',\mathcal F')$ be a transversally harmonic map with $M$ compact without boundary, and all leaves of $\mathcal F$ be compact. Then, for any $V,W\in\phi^{-1}Q'$, $$\begin{aligned}
\label{4-7}
THess_\phi(V,W)&=\int_M\langle\nabla_{\rm tr}^\phi V,\nabla_{\rm tr}^\phi W\rangle{1\over vol_L}\mu_M\\
&-\int_M\langle {\rm tr}_QR^{Q'}(V,d_T\phi)d_T\phi,W\rangle{1\over vol_L}\mu_M\notag
\end{aligned}$$ and $THess_\phi$ is symmetric, i.e., $THess_\phi(V,W)=THess_\phi(W,V)$ for any normal vector fields $V$ and $W$ along $\phi$.
[**Proof.**]{} From (\[4-2\]) and (\[4-6\]), we have $$\begin{aligned}
& THess_\phi(V,W)\\
&=\sum_a\int_M\langle \nabla_{E_a}^\phi V,\nabla_{E_a}^\phi({1\over vol_L}W)\rangle\mu_M
-\int_M \langle\nabla_{(vol_L^{-1})\kappa_B^\sharp}^\phi V,W\rangle\mu_M \\
&-\int_M\langle{\rm tr}_QR^{Q'}(V,d_T\phi)d_T\phi,W\rangle{1\over vol_L}\mu_M\\
&=\int_M \sum_a\langle \nabla_{E_a}^\phi V,\nabla_{E_a}^\phi W\rangle{1\over vol_L}\mu_M -\int_M\langle{\rm tr}_QR^{Q'}(V,d_T\phi)d_T\phi,W\rangle{1\over vol_L}\mu_M\\
&-\int_M \langle \nabla^\phi_{d_B vol_L^\sharp + (vol_L)\kappa_B^\sharp}V,W\rangle {1\over vol_L^2}\mu_M.
\end{aligned}$$ By (3.4), the last term in the last equality above vanishes. So the proof is completed. $\Box$
If the transversal Hessian of $\phi:(M,g,\mathcal F)\to (M',g',\mathcal F')$ is positive semi-definite, i.e., $THess_\phi(V,V)\geq 0$ for any normal vector field $V$ along $\phi$, then $\phi$ is said to be [*transversally stable*]{}. From (\[4-7\]), we have the following corollary.
[(Stability)]{} Any transversally harmonic map from a compact(without boundary) foliated Riemannian manifold to a foliated Riemannian manifold of non-positive transversal sectional curvature is transversally stable.
Transversal Jacobi operator along a map
=======================================
Let $(M,g,\mathcal F)$ be a compact foliated Riemannian manifold, and all leaves of $\mathcal F$ are compact.
[Let $\phi:(M,g,\mathcal F)\to (M',g',\mathcal F')$ be a foliated map. Then the]{} transversal Jacobi operator $J_\phi^T:\Gamma \phi^{-1}Q'\to \Gamma \phi^{-1}Q'$ [along $\phi$ is defined by $$\begin{aligned}
\label{5-1}
J^T_\phi(V)=(\nabla_{\rm tr}^\phi)^*(\nabla_{\rm tr}^\phi)V-\nabla_{\kappa_B^\sharp}^\phi V-{\rm tr}_Q R^{Q'}(V,d_T\phi)d_T\phi.\end{aligned}$$ Any $V\in {\rm Ker}J^T_\phi$ is called a]{} transversal Jacobi field [along $\phi$ for the transversal energy.]{}
From (\[2-5-1\]), the transversal Jacobi operator $J_{\nabla}^T\equiv J_{\rm id}^T$ along the identity map is given by $$\begin{aligned}
\label{5-1-1}
J_{\nabla}^T (\bar Y)=\nabla_{\rm tr}^*\nabla_{\rm tr}\bar Y -\rho^\nabla(\bar Y)+A_Y \kappa_B^\sharp\end{aligned}$$ for any $Y\in V(\mathcal F)$, which is called to [*generalized Jacobi operator*]{} of $\mathcal F$ on $M$. From (\[4-2\]) and (\[4-6\]), if $M$ is compact without boundary, then we have $$\begin{aligned}
\label{5-2}
THess_\phi(V,W)=\int_M\langle J^T_\phi(V),W\rangle {1\over vol_L}\mu_M.\end{aligned}$$ Let $\{\phi_t\}$ be a smooth foliated variation of $\phi$ with the normal variation vector field $V$. From (4.4) and (5.3), we have $$\begin{aligned}
\label{5-3}
J^T_\phi(V)=-\nabla_{d\over dt} \tau_b(\phi_t)\Big|_{t=0}.\end{aligned}$$ Hence we have the following proposition.
Let $\phi:(M,g,\mathcal F)\to (M',g',\mathcal F')$ be a transversally harmonic map and $\{\phi_t\}$ be a smooth foliated variation of $\phi$ with the normal variation vector field $V$. Then $J^T_\phi(V)=0$, i.e., $V$ is a transversal Jacobi field along $\phi$ for the energy.
Let $Y\in V(\mathcal F)$ be an infinitesimal automorphism on $(M,\mathcal F)$. It is well-known \[\[JJ1\]\] that, if $\bar Y$ is a transversal affine field, i.e., $\theta(Y)\nabla=0$, then $$\begin{aligned}
\label{5-4}
\nabla_{\rm tr}^*\nabla_{\rm tr}\bar Y-\rho^\nabla(\bar Y)+A_Y\kappa_B^\sharp=0.\end{aligned}$$ Hence we have the following proposition.
On $(M,\mathcal F)$, any transversal affine field is a generalized Jacobi field for $\mathcal F$.
[**Remark.**]{} Since any transversal Killing field is transversal affine \[\[KT\]\], any transversal Killing field is also generalized Jacobi field of $\mathcal F$. And the converse in Proposition 5.3 does not hold unless $\mathcal F$ is harmonic.
$(cf. [\ref{JJ1}])$ On $(M,\mathcal F)$ with $M$ compact without boundary, the followings are equivalent: for any vector field $Y\in V(\mathcal F)$,
$(1)$ $\bar Y$ is a transversal Killing field, that is, $\theta(Y)g_Q=0$;
$(2)$ $\bar Y$ is a generalized Jacobi field for $\mathcal F$ satisfying $(i)$ ${\rm div}_\nabla(\bar Y)=0$ and $(ii)$ $\int_M g_Q(B_Y\bar Y,\kappa_B^\sharp)\geq 0$, where $B_Y=A_Y +A_Y^t$.
Now, we recall the [*transversal Jacobi operator*]{} $J_\nabla:\Gamma Q\to \Gamma Q$ of $\mathcal F$ \[\[KT\]\] by $$\begin{aligned}
J_\nabla =\nabla_{\rm tr}^*\nabla_{\rm tr} -\rho^\nabla.\end{aligned}$$ Then $\bar Y\in {\rm Ker}J_\nabla$ is called a [*transversal Jacobi field for $\mathcal F$*]{} \[\[KT\]\]. Note that two operators $J_\nabla$ and $J_{\nabla}^T$ are related by $$\begin{aligned}
J_{\nabla}^T(\bar Y)=J_\nabla(\bar Y)+A_Y\kappa_B^\sharp.
\end{aligned}$$ On a harmonic foliation $\mathcal F$, $J_{\nabla}^T =J_\nabla$. Hence, from Proposition 5.3 and Corollary 5.4, we have the following corollary.
$[\ref{KTT}]$ Let $\mathcal F$ be a harmonic foliation on a compact Riemannian manifold $(M,g)$. Then the following are equivalent:
$(1)$ $\bar Y$ is a transversal Killing field, i.e., $\theta(Y)g_Q=0$;
$(2)$ $\bar Y$ is a transversal Jacobi field of $\mathcal F$ and ${\rm div}_\nabla(\bar Y)=0$;
$(3)$ $\bar Y$ is transversal affine field, i.e., $\theta(Y)\nabla=0$.
Now, we have the vanishing theorem about the transversal Jacobi field along the map.
Let $(M,g,\mathcal F)$ be a closed,connected Riemannian manifold with a foliation $\mathcal F$ and a bundle-like metric $g$. Assume the transversal Ricci operator is non-positive and negative at some point. Then any generalized Jacobi field $\bar Y$ for $\mathcal F$ is trivial, i.e, $Y$ is tangential to $\mathcal F$.
[**Proof.**]{} It is well-known \[\[JJ1\]\] that $$\begin{aligned}
\label{5-6}
\frac12(\Delta_B-\kappa_B^\sharp)|\bar Y|^2 = g_Q(J_{\nabla}^T(\bar Y)+g_Q(\rho^\nabla(\bar Y),\bar Y)-|\nabla_{\rm tr}\bar Y|^2.\end{aligned}$$ Let $\bar Y$ be a generalized Jacobi field for $\mathcal F$. Then we have $$\begin{aligned}
\label{5-7}
\frac12(\Delta_B-\kappa_B^\sharp)|\bar Y|^2 = g_Q(\rho^\nabla(\bar Y),\bar Y)-|\nabla_{\rm tr}\bar Y|^2.\end{aligned}$$ Since the transversal Ricci curvature is non-positive, we have $$\begin{aligned}
\label{5-8}
(\Delta_B-\kappa_B^\sharp)|\bar Y|^2\leq 0.\end{aligned}$$ Hence, by the generalized maximum principle (Lemma 2.2), $|\bar Y|$ is constant. Again, from (\[5-7\]), $\bar Y$ is parallel. Moreover, since $\rho^\nabla$ is negative at some point, $\bar Y$ is trivial. Equivalently, $Y$ is tangential to $\mathcal F$. $\Box$
If $\bar Y$ is a transversal Jacobi field of $\mathcal F$, i.e., $J_\nabla(\bar Y)=0$, then $J_{\nabla}^T(\bar Y)=-\nabla_{\kappa_B^\sharp}\bar Y$. Therefore, from (\[5-6\]) we have $$\begin{aligned}
\label{5-12}
\frac12 \Delta_B |\bar Y|^2 =g_Q(\rho^\nabla(\bar Y),\bar Y) -|\nabla_{\rm tr}\bar Y|^2.
\end{aligned}$$ From (\[5-12\]), we have the following corollary \[\[KT\], p535\].
$[\ref{KT}]$ Let $(M,g,\mathcal F)$ be as in Theorem 5.6. Assume the transversal Ricci operator is non-positive and negative at some point. Then any transversal Jacobi field of $\mathcal F$ is trivial, i.e., $Y$ is tangential to $\mathcal F$.
[**Proof.**]{} Let $\bar Y$ be a transversal Jacobi field of $\mathcal F$. Since $\rho^\nabla$ is non-positive, from (\[5-12\]), $\Delta_B |\bar Y|^2\leq 0$. Since $\Delta=\Delta_B$ on a basic function, by the maximum principle, $|\bar Y|$ is constant. Since $\rho^\nabla$ is negative at some point, from (5.11), $\bar Y$ is trivial. $\Box$
Transversally bi-harmonic maps
===============================
Let $(M, g,\mathcal F)$ and $(M', g',\mathcal F')$ be two foliated Riemannian manifolds. Let $\phi:(M,g,\mathcal
F)\to (M', g',\mathcal F')$ be a smooth foliated map. Now we define the [*transversal bi-tension field $(\tau_2)_b(\phi)$* ]{} of $\phi$ by $$\begin{aligned}
\label{6-9}
(\tau_2)_b(\phi)=J_\phi^T(\tau_b(\phi)).\end{aligned}$$
[Let $\phi:(M,g,\mathcal F)\to (M',g',\mathcal F')$ be a smooth foliated map. Then $\phi$ is said to be]{} transversally bi-harmonic [if the transversal bi-tension field vanishes, i.e., $(\tau_2)_b(\phi)=0$.]{}
Trivially, $\phi$ is a transversally bi-harmonic map if and only if the transversal tension field $\tau_b(\phi)$ is a transversal Jacobi field along $\phi$. Moreover, any transversal harmonic map is a transversal bi-harmonic map. Now, we define the [*transversal bi-energy*]{} of $\phi$ supported in a compact domain $\Omega$ by $$\begin{aligned}
\label{6-12}
(E_2)_B(\phi,\Omega)=\frac12\int_\Omega |\tilde\delta d_T\phi|^2{1\over vol_L}\mu_M.\end{aligned}$$ Then we have the following theorem.
$($[The first variation formula for the transversal bi-energy]{}$)$ Let $\phi:(M,g,\mathcal F)\to (M',g',\mathcal F')$ be a smooth foliated map, and all leaves of $\mathcal F$ be compact. Let $\{\phi_t\}$ be a foliated variation of $\phi$ with the variation vector field $V$ in a compact domain $\Omega$. Then we have $$\begin{aligned}
\label{6-13}
{d\over dt}(E_2)_B(\phi_t,\Omega)\Big|_{t=0}=-\int_\Omega \langle (\tau_2)_b(\phi),V\rangle{1\over vol_L}\mu_M.\end{aligned}$$
[**Proof.**]{} Let $\{\phi_t\}$ be a foliated variation of $\phi$ such that ${d\phi_t\over dt}\Big|_{t=0}=V$ and $\phi_0=\phi$. Choose a local orthonormal basic frame $\{E_a\}$ with $(\nabla E_a)(x)=0$. Define $\Phi:M\times (-\epsilon,\epsilon)\to M'$ by $\Phi(x,t)=\phi_t(x)$. Let $\nabla^\Phi$ be the pull-back connection on $\Phi^{-1}Q'$. Obviously, $d_T\Phi(E_a)=d_T\phi(E_a)$ and $d\Phi({d\over dt})={d\phi_t\over dt}$. Moreover, it is trivial that $\nabla_{d\over dt}{d\over dt}=\nabla_{d\over dt}E_a =\nabla_{E_a}{d\over dt}=0$. Hence, from (\[3-8\]), we have $$\begin{aligned}
\label{4-4-1}
{d\over dt}(E_2)_B(\phi_t,\Omega)&=\int_\Omega \langle \nabla_{d\over dt}^\Phi \tau_b(\phi_t),\tau_b(\phi_t)\rangle{1\over vol_L}\mu_M.\end{aligned}$$ From (\[5-3\]), it follows that $$\begin{aligned}
{d\over dt}(E_2)_B(\phi_t,\Omega)|_{t=0}&=-\int_\Omega \langle J_\phi^T(V),\tau_b(\phi)\rangle{1\over vol_L}\mu_M\\
&=-\int_\Omega\langle J_\phi^T(\tau_b(\phi)),V\rangle{1\over vol_L}\mu_M.\end{aligned}$$ The last equality above follows from (5.3) and the symmetry of the transversal Hessian $THess$ of $\phi$. From (\[6-9\]), the proof is complete. $\Box$
Let $\phi:(M,g,\mathcal F)\to (M',g',\mathcal F')$ be a smooth foliated map, and all leaves of $\mathcal F$ be compact. Then $\phi$ is transversally bi-harmonic if and only if it is a critical point of the transversal bi-energy $(E_2)_B(\phi)$ of $\phi$ on any compact domain.
Then we have the following (cf. \[\[CW\]\]).
Let $\phi:(M,g,\mathcal F)\to (M',g',\mathcal F')$ be a transversally bi-harmonic map with $M$ compact without boundary, and all leaves of $\mathcal F$ be compact. Assume that the transversal sectional curvature $K^{Q'}$ of $\mathcal F'$ is non-positive. Then $\phi$ is transversally harmonic.
[**Proof.**]{} Let $\{E_a\}$ be a local orthonormal basic frame of $Q$. Then $(\tau_2)_b(\phi)=0$ implies that $$\begin{aligned}
\label{6-14}
(\nabla_{\rm tr}^\phi)^*\nabla_{\rm tr}^\phi \tau_b(\phi)-\nabla_{\kappa_B^\sharp}^\phi \tau_b(\phi)-\sum_a R^{Q'}(\tau_b(\phi),d_T\phi(E_a))d_T\phi(E_a)=0.\end{aligned}$$ From (\[2-7\]) and (\[2-8\]), we have $\Delta_B |\tau_b(\phi)|^2 =2\langle (\nabla_{\rm tr}^\phi)^*\nabla_{\rm tr}^\phi \tau_b(\phi),\tau_b(\phi)\rangle -2|\nabla_{\rm tr}\tau_b(\phi)|^2$. Hence from (\[6-14\]), we have $$\begin{aligned}
\label{6-15}
{1\over 2}(\Delta_B-\kappa_B^\sharp)|\tau_b(\phi)|^2 &=-|\nabla_{\rm tr}\tau_b(\phi)|^2 \notag\\&
+ \sum_a\langle R^{Q'}(\tau_b(\phi),d_T\phi(E_a))d_T\phi(E_a),\tau_b(\phi)\rangle.
\end{aligned}$$ Since the transversal sectional curvature $K^{Q'}$ of $\mathcal F'$ is non-positive, we have $$\begin{aligned}
(\Delta_B-\kappa_B^\sharp)|\tau_b(\phi)|^2\leq 0.
\end{aligned}$$ Hence, by the generalized maximum principle (Lemma 2.3), $|\tau_b(\phi)|$ is constant. Again, from (\[6-15\]), we have that for all $a$, $$\begin{aligned}
\label{eq5-7}
\nabla_{E_a}\tau_b(\phi)=0.\end{aligned}$$ Now, we define the normal vector bundle $X$ by $$\begin{aligned}
X={1\over vol_L}\sum_a\langle d_T\phi(E_a),\tau_b(\phi)\rangle E_a.\end{aligned}$$ Then we have $$\begin{aligned}
{\rm div}_\nabla (X)&=-{1\over {vol_L}^{2}}\langle d_T\phi(d_B vol_L^\sharp),\tau_b(\phi)\rangle +{1\over vol_L}
|\tau_b(\phi)|^2\\
&={1\over vol_L}\langle d_T\phi(\kappa_B^\sharp),\tau_b(\phi)\rangle +{1\over vol_L}|\tau_b(\phi)|^2.\end{aligned}$$ The last equality above follows from Lemma 2.2. By integrating and by using the transversal divergence theorem (Theorem 2.1), we have $$\begin{aligned}
\label{6-16}
\int_M |\tau_b(\phi)|^2{1\over vol_L}\mu_M=0,\end{aligned}$$ which implies that $\tau_b(\phi)=0$. So $\phi$ is transversally harmonic. $\Box$
The second variation formula for the transversal bi-energy
==========================================================
Let $(M,g,\mathcal F)$ and $(M',g',\mathcal F')$ be two foliated Riemannian manifolds. Let $\phi:(M,g,\mathcal F)\to (M',g',\mathcal F')$ be a transversally bi-harmonic map. Then we have the second variational formula of the transversal bi-energy as follows.
$($ [The second variation formula for the transversal bi-energy]{}$)$ Let $\phi:(M,g,\mathcal F)\to (M',g',\mathcal F')$ be a smooth foliated map with $M$ compact without boundary and all leaves be compact. Let $\{\phi_t\}$ be a foliated variation of $\phi$ with the normal variation vector field $V$. Then $$\begin{aligned}
&{d^2\over dt^2} (E_2)_B (\phi_t)\Big|_{t=0}\\
&=\int_M \{|(\tau_2)_b(\phi))|^2 -\langle (\tau_2)_b(\phi)),\nabla_VV\rangle-\langle R^{Q'}(V,\tau_b(\phi))\tau_b(\phi),V\rangle\}{1\over vol_L}\mu_M\\
&-2\sum_a\int_M\langle(\nabla_{E_a}R^{Q'})(V,d_T\phi(E_a))\tau_b(\phi),V\rangle{1\over vol_L}\mu_M\\
&+\sum_a\int_M\langle (\nabla_{\tau_b(\phi)}R^{Q'})(V,d_T\phi(E_a))d_T\phi(E_a),V\rangle{1\over vol_L}\mu_M\\
&-4\sum_a\int_M\langle R^{Q'}(\nabla^\phi_{E_a}V,\tau_b(\phi))d_T\phi(E_a),V\rangle{1\over vol_L}\mu_M.\end{aligned}$$
[**Proof.**]{} Let $V\in \phi^{-1}Q'$ and let $\{\phi_t\}$ be a foliated variation of $\phi$ such that ${d\phi_t\over dt}\Big|_{t=0}=V$ and $\phi_0=\phi$. We choose a local orthonormal basic frame $\{E_a\}$ such that $(\nabla E_a)(x)=0$ at point $x$. Define $\Phi:M\times (-\epsilon,\epsilon)\to M'$ by $\Phi(x,t)=\phi_t(x)$. Let $\nabla^\Phi$ be the pull-back connection on $\Phi^{-1}Q'$. Obviously, $d_T\Phi(E_a)=d_T\phi(E_a)$ and ${\partial\Phi\over \partial t}={d\phi_t\over dt}$. Trivially, $\nabla_{d\over dt}{d\over dt}=\nabla_{d\over dt}E_a =\nabla_{E_a}{d\over dt}=0$. Hence, from (\[4-4-1\]) we have $$\begin{aligned}
\label{7-1}
{d^2\over dt^2}(E_2)_B(\phi_t)&=\int_M \Big(\langle\nabla^\Phi_{d\over dt} \nabla^\Phi_{d\over dt}\tau_b(\phi_t),\tau_b(\phi_t)\rangle+|\nabla^\Phi_{d\over dt} \tau_b(\phi_t)|^2\Big){1\over vol_L}\mu_M.\end{aligned}$$ From (\[4-5\]), we have $$\begin{aligned}
\nabla^\Phi_{d\over dt} \tau_b(\phi_t)=\sum_a(\nabla^\Phi)^2_{E_a,E_a}{\partial\Phi\over \partial t} + \sum_aR^{Q'}({\partial\Phi\over \partial t},d_T\phi_t(E_a))d_T\phi_t(E_a),\end{aligned}$$ and then $$\begin{aligned}
\nabla^\Phi_{d\over dt}\nabla^\Phi_{d\over dt} \tau_b(\phi_t)=\sum_a\nabla^\Phi_{d\over dt}\Big((\nabla^\Phi)^2_{E_a,E_a} {\partial\Phi\over \partial t} + R^{Q'}({\partial\Phi\over \partial t},d_T\phi_t(E_a))d_T\phi_t(E_a)\Big).\end{aligned}$$ By a long calculation, we get $$\begin{aligned}
\nabla_{d\over dt}^\Phi(\nabla^\Phi)^2_{E_a,E_a}{\partial\Phi\over \partial t}&=(\nabla^\Phi)^2_{E_a,E_a}\nabla^\Phi_{d\over dt}{\partial\Phi\over \partial t}+\nabla^\Phi_{E_a}R^\Phi({d\over dt},E_a){\partial\Phi\over \partial t}\\
&+R^\Phi({d\over dt},E_a)\nabla^\Phi_{E_a}{\partial\Phi\over \partial t} +R^\Phi(\nabla_{E_a}E_a,{d\over dt}){\partial\Phi\over \partial t}\end{aligned}$$ and $$\begin{aligned}
&\nabla^\Phi_{d\over dt} R^{Q'}({\partial\Phi\over \partial t},d_T\phi_t(E_a))d_T\phi_t(E_a)\\
&=(\nabla_{\partial\Phi\over \partial t}R^{Q'})({\partial\Phi\over \partial t},d_T\phi_t(E_a))d_T\phi_t(E_a)+R^{Q'}(\nabla^\Phi_{d\over dt}{\partial\Phi\over \partial t},d_T\phi_t(E_a))d_T\phi_t(E_a)\\
&+R^{Q'}({\partial\Phi\over \partial t},\nabla^\Phi_{d\over dt}d_T\phi_t(E_a))d_T\phi_t(E_a)+R^{Q'}({\partial\Phi\over\partial t},d_T\phi_t(E_a))\nabla^\Phi_{d\over dt}d_T\phi_t(E_a).\end{aligned}$$ Since $[{d\over dt},E_a]=0$, we have $\nabla_V d_T\phi(E_a)=\nabla_{d_T\phi(E_a)}V=\nabla^\phi_{E_a}V$. Hence from the equations above, we have $$\begin{aligned}
&\nabla^\Phi_{d\over dt}\nabla^\Phi_{d\over dt} \tau_b(\phi_t)|_{t=0}\\
&=-J^T_\phi(\nabla_VV)+\sum_a\nabla^\phi_{E_a}R^{Q'}(V,d_T\phi(E_a))V\\
&+2\sum_aR^{Q'}(V,d_T\phi(E_a))\nabla^\phi_{E_a}V +\sum_aR^{Q'}(d_T\phi(\nabla_{E_a}E_a),V)V\\
&+\sum_a(\nabla_V R^{Q'})(V,d_T\phi(E_a))d_T\phi(E_a)+\sum_aR^{Q'}(V,\nabla^\phi_{E_a}V)d_T\phi(E_a).\end{aligned}$$ So, by the first and second Bianchi identites, we have $$\begin{aligned}
\langle\nabla^\Phi_{d\over dt} \nabla_{d\over dt}^\Phi \tau_b(\phi_t),\tau_b(\phi_t)\rangle|_{t=0}
&=-\langle J_\phi^T(\nabla_VV),\tau_b(\phi)\rangle +\langle R^{Q'}(V,\tau_b(\phi))V,\tau_b(\phi)\rangle \\
&+\sum_a\langle (\nabla_V R^{Q'})(V,d_T\phi(E_a))d_T\phi(E_a),\tau_b(\phi)\rangle\\
&+\sum_a\langle (\nabla_{E_a}R^{Q'})(V,d_T\phi(E_a))V,\tau_b(\phi)\rangle\\
& +4\sum_a\langle R^{Q'}(V,d_T\phi(E_a))\nabla^\phi_{E_a}V,\tau_b(\phi)\rangle.\end{aligned}$$ By integrating together with (\[5-3\]), we have $$\begin{aligned}
&\int_M\langle\nabla^\Phi_{d\over dt}\nabla^\Phi_{d\over dt} \tau_b(\phi_t)|_{t=0},\tau_b(\phi)\rangle{1\over vol_L}\mu_M\\
&=-\int_M\langle J_\phi^T(\tau_b(\phi)),\nabla_VV\rangle{1\over vol_L}\mu_M-\int_M\langle R^{Q'}(V,\tau_b(\phi))\tau_b(\phi),V\rangle{1\over vol_L}\mu_M\notag\\
&-2\sum_a\int_M\langle(\nabla_{E_a}R^{Q'})(V,d_T\phi(E_a))\tau_b(\phi),V\rangle{1\over vol_L}\mu_M \notag\\
&+\sum_a\int_M\langle(\nabla_{\tau_b(\phi)} R^{Q'})(V,d_T\phi(E_a))d_T\phi(E_a),V\rangle{1\over vol_L}\mu_M\notag\\
& -4\sum_a\int_M\langle R^{Q'}(\nabla^\phi_{E_a}V,\tau_b(\phi))d_T\phi(E_a),V\rangle{1\over vol_L}\mu_M.\end{aligned}$$ From (\[6-9\]) and (\[7-1\]), the proof follows. $\Box$
Then we have the following corollary (cf. \[\[CW\]\]).
Let $\phi:(M,g,\mathcal F)\to (M',g',\mathcal F')$ be a transversally bi-harmonic map with $M$ compact without boundary. Let $\{\phi_t\}$ be a foliated variation of $\phi$ with the normal variation vector field $V$. Then $$\begin{aligned}
{d^2\over dt^2} (E_2)_B (\phi_t)\Big|_{t=0}=&-\int_M\langle R^{Q'}(V,\tau_b(\phi))\tau_b(\phi),V\rangle{1\over vol_L}\mu_M\\
&-2\sum_a\int_M\langle(\nabla_{E_a}R^{Q'})(V,d_T\phi(E_a))\tau_b(\phi),V\rangle{1\over vol_L}\mu_M\\
&+\sum_a\int_M\langle (\nabla_{\tau_b(\phi)}R^{Q'})(V,d_T\phi(E_a))d_T\phi(E_a),V\rangle{1\over vol_L}\mu_M\\
&-4\sum_a\int_M\langle R^{Q'}(\nabla_{E_a}^\phi V,\tau_b(\phi))d_T\phi(E_a),V\rangle{1\over vol_L}\mu_M.\end{aligned}$$
[If the transversally bi-harmonic map $\phi:(M,g,\mathcal F)\to (M',g',\mathcal F')$ satisfies ${d^2\over dt^2}(E_2)_B(\phi_t)\Big|_{t=0}\geq 0$, then $\phi$ is said to be]{} stable.
Note that any transversally harmonic map can be considered as a stable transversally bi-harmonic map. We also have the following theorem (\[\[CW\]\]).
Let $(M,\mathcal F)$ be a closed Riemannian manifold with a foliation $\mathcal F$ and let $(M',\mathcal F')$ be a Riemannian manifold with a constant transversal sectional curvature $C>0$. If a foliated map $\phi:(M,\mathcal F)\to (M',\mathcal F')$ is stable transversally bi-harmonic and relatively harmonic, then $\phi$ is transversally harmonic.
[**Proof.**]{} Since the transversal sectionsal curvature $K^{Q'}$ of $\mathcal F'$ is constant $C>0$, from Corollary 7.2, we have $$\begin{aligned}
{d^2\over dt^2} (E_2)_B (\phi_t)\Big|_{t=0}=&-\int_M\langle R^{Q'}(V,\tau_b(\phi))\tau_b(\phi),V\rangle{1\over vol_L}\mu_M\\
&-4\sum_a\int_M\langle R^{Q'}(\nabla_{E_a}^\phi V,\tau_b(\phi))d_T\phi(E_a),V\rangle{1\over vol_L}\mu_M.\end{aligned}$$ Let $V=\tau_b(\phi)$. Since $\phi$ is relatively harmonic, i.e., $\langle \tau_b(\phi),d_T\phi(X)\rangle$ for any vector field $X\in \Gamma Q'$, we have $$\begin{aligned}
{d^2\over dt^2} (E_2)_B (\phi_t)\Big|_{t=0}&=-4\sum_a\int_M\langle R^{Q'}(\nabla_{E_a}^\phi \tau_b(\phi),\tau_b(\phi))d_T\phi(E_a),\tau_b(\phi)\rangle{1\over vol_L}\mu_M\\
&=4C\sum_a\int_M \langle \nabla_{E_a}^\phi \tau_b(\phi),d_B\phi(E_a)\rangle |\tau_b(\phi)|^2{1\over vol_L}\mu_M\\
&=-4C\int_M|\tau_b(\phi)|^4{1\over vol_L}\mu_M.\end{aligned}$$ This stability implies that $\tau_b(\phi)=0$, i.e., $\phi$ is transversally harmonic. $\Box$
[**Acknowledgements.**]{} This research was supported by the Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education, Science and Technology (2010-0021005) and NRF-2011-616-C00040.
[\[9\]]{}
\[LO\] J. A. Alvarez [López]{}, *The basic component of the mean curvature of Riemannian foliations*, Ann. Global Anal. Geom. 10(1992), 179-194.
\[CW\] Y.-J. Chiang and R. Wolak, *Transversally biharmonic maps between foliated Riemannian manifolds*, International J. Math. 19(2008), 981-996.
\[ES\] J. Eells and J. H. Sampson, *Harmonic mappings of Riemannian manifolds*, Amer. J. Math. 86(1964), 106-160.
\[JJ1\] M. J. Jung and S. D. Jung, *Riemannian foliations admitting transversal conformal fields*, Geometriae Dedicata 133(2008), 155-168.
\[JJ2\] M. J. Jung and S. D. Jung, *On transversally harmonic maps of foliated Riemannian manifolds*, to be appeared in J. Korean Math. Soc.
\[JLK\] S. D. Jung, K. R. Lee and K. Richardson, *Generalized Obata theorem and its applications on foliations*, J. Math. Anal. Appl. 376(2011), 129-135.
\[KT\] F. W. Kamber and Ph. Tondeur, *Infinitesimal automorphisms and second variation of the energy for harmonic foliations*, Tôhoku Math. J. 34(1982), 525-538.
\[KTT\] F. W. Kamber, Ph. Tondeur and G. Toth, *Transversal Jacobi fields for harmonic foliations*, Michigan Math. J. 34(1987), 261-266.
\[KW1\] J. Konderak and R. Wolak, *On transversally harmonic maps between manifolds with Riemannian foliations*, Quart. J. Math. Oxford Ser.(2) 54(2003), 335-354.
\[KW2\] J. Konderak and R. Wolak, *Some remarks on transversally harmonic maps*, Glasgow Math. J. 50(2008), 1-16.
\[MO\] P. Molino, *Riemannian foliations*, translated from the French by Grant Cairns, Boston: Birkhäser, 1988.
\[PR\] E. Park and K. Richardson, *The basic Laplacian of a Riemannian foliation*, Amer. J. Math. 118(1996), 1249-1275.
\[TO1\] Ph. Tondeur, *Foliations on Riemannian manifolds*, New-York: Springer-Verlag, 1988.
\[TO2\] Ph. Tondeur, *Geometry of foliations*, Basel: Birkhäuser Verlag, 1997.
\[XI\] Y.L. Xin, *Geometry of harmonic maps*, Birkhäuser, Boston, 1996.
\[YO\] S. Yorozu and T. Tanemura, *Green’s theorem on a foliated Riemannian manifold and its applications*, Acta Math. Hungar. 56 (1990), 239-245.
Seoung Dal Jung
Department of Mathematics and Research Institute for Basic Sciences, Jeju National University, Jeju 690-756, Korea
[*E-mail address*]{} : [email protected]
[^1]: 2000 *Mathematics Subject Classification.* 53C12, 58E20
[^2]: *Key words and phrases.* Transversally harmonic map, transversally bi-harmonic map, trasnversal Jacobi operator, variation formulas.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'A typical quantum state obeying the area law for entanglement on an infinite 2D lattice can be represented by a tensor network ansatz – known as an infinite projected entangled pair state (iPEPS) – with a finite bond dimension $D$. Its real/imaginary time evolution can be split into small time steps. An application of a time step generates a new iPEPS with a bond dimension $k$ times the original one. The new iPEPS does not make optimal use of its enlarged bond dimension $kD$, hence in principle it can be represented accurately by a more compact ansatz, favourably with the original $D$. In this work we show how the more compact iPEPS can be optimized variationally to maximize its overlap with the new iPEPS. To compute the overlap we use the corner transfer matrix renormalization group (CTMRG). By simulating sudden quench of the transverse field in the 2D quantum Ising model with the proposed algorithm, we provide a proof of principle that real time evolution can be simulated with iPEPS. A similar proof is provided in the same model for imaginary time evolution of purification of its thermal states.'
author:
- Piotr Czarnik
- Jacek Dziarmaga
title: |
Time Evolution of an Infinite Projected Entangled Pair State:\
an Algorithm from First Principles
---
Introduction {#sec:introduction}
============
Tensor networks are a natural language to represent quantum states of strongly correlated systems[@Verstraete_review_08; @Orus_review_14]. Among them the most widely used ansatze are a matrix product states (MPS) [@Fannes_MPS_92] and its 2D generalization: pair-entangled projected state (PEPS) [@Verstraete_PEPS_04] also known as a tensor product state. Both obey the area law for entanglement entropy. In 1D matrix product states are efficient parameterizations of ground states of gapped local Hamiltonians [@Verstraete_review_08; @Hastings_GSarealaw_07; @Schuch_MPSapprox_08] and purifications of thermal states of 1D local Hamiltonians [@Barthel_1DTMPSapprox_17]. MPS is the ansatz optimized by the density matrix renormalization group (DMRG) [@White_DMRG_92; @White_DMRG_93] which is one of the most powerful methods to simulate not only ground states of 1D systems but also theirs exited states, thermal states or dynamic properties [@Schollwock_review_05; @Schollwock_review_11].
PEPS are expected to be an efficient parametrization of ground states of 2D gapped local Hamiltonians [@Verstraete_review_08; @Orus_review_14] and were shown to be an efficient representation of thermal states of 2D local Hamiltonians [@Molnar_TPEPSapprox_15], though in 2D there are limitations to the assumed representability of area-law states by tensor networks [@Eisert_TNapprox_16]. Furthermore tensor networks can be used to represent efficiently systems with fermionic degrees of freedom [@Eisert_fMERA_09; @Corboz_fMERA_09; @Barthel_fTN_09; @Gu_fTN_10] as was demonstrated for both finite [@Cirac_fPEPS_10] and infinite PEPS [@Corboz_fiPEPS_10; @Corboz_stripes_11].
PEPS was originally proposed as a varaitional ansatz for ground states of 2D finite systems [@Verstraete_PEPS_04; @Murg_finitePEPS_07] generalizing earlier attempts to construct trial wave-functions for specific 2D models using 2D tensor networks [@Nishino_2DvarTN_04]. Efficient numerical methods enabling optimisation and controlled approximate contraction of infinite PEPS (iPEPS) [@Cirac_iPEPS_08; @Xiang_SU_08; @Gu_TERG_08; @Orus_CTM_09] became basis for promising new methods for strongly correlated systems. Among recent achievements of those methods are solution of a long standing magnetization plateaus problem in highly frustrated compound $\textrm{SrCu}_2(\textrm{BO}_3)_2$ [@Matsuda_SS_13; @Corboz_SS_14] and obtaining coexistence of superconductivity and striped order in the underdoped regime of the Hubbard model – a result which is corroborated by other numerical methods (among them another tensor network approach - DMRG simulations of finite-width cylinders) – apparently settling one of long standing controversies concerning that model [@Simons_Hubb_17]. Another example of a recent contribution of iPEPS-based methods to condensed matter physics is a problem of existence and nature of spin liquid phase in kagome Heisenberg antiferromagnet for which new evidence in support of gapless spin liquid was obtained [@Xinag_kagome_17]. This progress was accompanied and partly made possible by new developments in iPEPS optimization [@Corboz_varopt_16; @Vanderstraeten_varopt_16], iPEPS contraction [@Fishman_FPCTM_17; @Xie_PEPScontr_17; @Czarnik_fVTNR_16], energy extrapolations [@Corboz_Eextrap_16], and universality class estimation [@Corboz_FCLS_18; @Rader_FCLS_18; @Rams_xiD_18]. These achievements encourage attempts to use iPEPS to simulate broad class of states obeying 2D area law like thermal states [@Czarnik_evproj_12; @Czarnik_fevproj_14; @Czarnik_SCevproj_15; @Czarnik_compass_16; @Czarnik_VTNR_15; @Czarnik_fVTNR_16; @Czarnik_eg_17; @Dai_fidelity_17], states of dissipative systems [@Kshetrimayum_diss_17] or exited states [@Vanderstraeten_tangentPEPS_15].
Among alternative tensor network approaches to strongly correlated systems are methods of direct contraction and renormalization of a 3D tensor network representing a density operator of a 2D thermal state [@Li_LTRG_11; @Xie_HOSRG_12; @Ran_ODTNS_12; @Ran_NCD_13; @Ran_THAFstar_18; @Su_THAFoctakagome_17; @Su_THAFkagome_17] and, technically challenging yet able to represent critical states with subleading logarithmic corrections to the area law, multi-scale entanglement renormalization ansatz (MERA) [@Vidal_MERA_07; @Vidal_MERA_08] and its generalization branching MERA [@Evenbly_branchMERA_14; @Evenbly_branchMERAarea_14]. Recent years brought also progress in using DMRG to simulate cylinders with finite width. Such simulations are routinely used alongside iPEPS to investigate 2D systems ground states (see e.g. Ref. ) and were applied recently also to thermal states [@Stoudenmire_2DMETTS_17; @Weichselbaum_Tdec_18].
In this work we test an algorithm to simulate either real or imaginary time evolution with iPEPS. The algorithm uses second order Suzuki-Trotter decomposition of the evolution operator into small time steps [@Trotter_59; @Suzuki_66; @Suzuki_76]. A straightforward application of a time step creates a new iPEPS with a bond dimension $k$ times the original bond dimension $D$. If not truncated, the evolution would result in an exponential growth of the bond dimension. Therefore, the new iPEPS is approximated variationally by an iPEPS with the original $D$. The algorithm is a straightforward construction directly from first principles with a minimal number of approximations controlled by the iPEPS bond dimension $D$ and the environmental bond dimension $\chi$ in CTMRG. It uses CTMRG [@Baxter_CTM_78; @Nishino_CTMRG_96; @Orus_CTM_09; @Corboz_CTM_14] to compute fidelity between the new iPEPS and its variational approximation. The very calculation of fidelity between two close iPEPS was shown to be tractable only very recently [@Orus_GSfidelity_17]. In this work we go further and demonstrate that the fidelity can be optimized variationally effectively enough for time evolution.
A challenging application of the method is real time evolution after a sudden quench. A sudden quench of a parameter in a Hamiltonian excites entangled pairs of quasiparticles with opposite quasimomenta that run away from each other crossing the boundary of the subsystem. Consequently, the number of pairs that are entangled across the boundary (proportional to the entanglement entropy) grows linearly with time requiring an exponential growth of the bond dimension. Therefore, a tensor network is doomed to fail after a finite evolution time. Nevertheless, matrix product states proved to be useful for simulating time evolution after sudden quenches in 1D [@Zaunerstauber_DPT_17]. As a proof of principle that the same can be attempted with iPEPS in 2D, in this work we simulate a sudden quench in the transverse field quantum Ising model.
Moreover, there are other – easier from the entanglement point of view – potential applications of the real time variational evolution. For instance, a smooth ramp of a parameter in a Hamiltonian across a quantum critical point generates the entanglement entropy proportional to the area of the boundary times a logarithm of the Kibble-Zurek correlation length $\hat\xi$ that in turn is a power of the ramp time [@Cincio_KZ_07]. Thanks to this dynamical area law, the required $D$ instead of growing exponentially with time saturates becoming a power of the ramp time. Even stronger limitations may apply in many-body localization (MBL), where localized excitations are not able to spread the entanglement. Tensor networks have already been applied to 2D MBL phenomena [@Wahl_MBL_17]. Finally, after vectorization of the density matrix, the unitary time evolution can be generalized to a Markovian master equation with a Lindblad superoperator, where local decoherence limits the entanglement making the time evolution with a tensor network feasible [@Montangero_master_16; @Kshetrimayum_diss_17].
Another promising application is imaginary time evolution generating thermal states of a quantum Hamiltonian. By definition, a thermal Gibbs state maximizes entropy for a given average energy. As this maximal entropy is the entropy of entanglement of the system with the rest of the universe, then – by the monogamy of entanglement – there is little entanglement left inside the system. In more quantitative terms, both thermal states of local Hamiltonians and iPEPS representations of density operators obey area law for mutual information making an iPEPS a good ansatz for thermal states [@Wolf_Tarealaw_08]. In this paper we evolve a purification of thermal states in the quantum Ising model obtaining results convergent to the variational tensor network renormalization (VTNR) introduced and applied to a number of models in [@Czarnik_VTNR_15; @Czarnik_compass_16; @Czarnik_fVTNR_16; @Czarnik_eg_17]. This test is a proof of principle that thermal states can be obtained with the variational imaginary time evolution.
![ In a, an elementary rank-6 tensor $A$ of a purification. The top (orange) index numbers ancilla states $a=0,1$, the bottom (red) index numbers spins states $s=0,1$, the four (black) bond indices have a bond dimension $D$. In b, an iPEPS representation of the purification. Here pairs of elementary tensors at NN lattice sites were contracted through their connecting bond indices. The whole network is an amplitude for a joint spins’ and ancillas’ state labelled by the open spin and ancilla indices. Reducing the dimension of ancilla indices to 1 (or simply ignoring the ancilla lines) we obtain a well known iPEPS representation of a pure state. []{data-label="fig:A"}](A.pdf){width="0.9999\columnwidth"}
The paper is organized as follows. In section \[sec:purification\] we introduce purification of a thermal state to be evolved in imaginary time. In section \[sec:algorithm\] we introduce the algorithm in the more general case of imaginary time evolution of a thermal state purification. A modification to real time evolution of a pure state is straightforward. In subsection \[sec:ST\] we make Suzuki-Trotter decomposition of a small time step and represent it by a tensor network. In subsection \[sec:step\] we outline the algorithm whose further details are refined in subsections \[sec:fom\],\[sec:lu\], and appendix \[sec:2site\]. In section \[sec:im\] the algorithm is applied to simulate imaginary time evolution generating thermal states. Its results are compared with VTNR. In section \[sec:re\] the real time version of the algorithm is tested in the challenging problem of time evolution after a sudden quench. Finally, we conclude in section \[sec:conclusion\].
Purification of thermal states {#sec:purification}
==============================
We will exemplify the general idea with the transverse field quantum Ising model on an infinite square lattice H = - \_[j,j’]{}Z\_jZ\_[j’]{} - \_j ( h\_x X\_j + h\_z Z\_j ). \[calH\] Here $Z,X$ are Pauli matrices. At zero longitudinal bias, $h_z=0$, the model has a ferromagnetic phase with a non-zero spontaneous magnetization $\langle Z \rangle$ for sufficiently small transverse field $h_x$ and sufficiently large inverse temperature $\beta$. At $h_x=0$ the critical $\beta$ is $\beta_0=-\ln(\sqrt{2}-1)/2\approx 0.441$ and at zero temperature the quantum critical point is $h_0=3.04438(2)$ [@Deng_QIshc_02].
In an enlarged Hilbert space, every spin with states $s=0,1$ is accompanied by an ancilla with states $a=0,1$. The space is spanned by states $\prod_j |s_j,a_j\rangle$, where $j$ is a lattice site. The Gibbs operator at an inverse temperature $\beta$ is obtained from its purification $|\psi(\beta)\rangle$ (defined in the enlarged space) by tracing out the ancillas, () e\^[-H]{} = [Tr]{}\_[ancillas]{}|()()|. \[rhobeta\] At $\beta=0$ we choose a product over lattice sites, |(0)= \_j \_[s=0,1]{} |s\_j,s\_j, \[psi0\] to initialize the imaginary time evolution |() = e\^[-12H]{}|(0) = U(-i/2)|(0). \[psibeta\] The evolution operator $U(\tau)=e^{-i\tau H}$ acts in the Hilbert space of spins. With the initial state (\[psi0\]) Eq. (\[rhobeta\]) becomes () U(-i/2) U\^(-i/2). \[UU\] Just like a pure state of spins, the purification can be represented by a iPEPS, see Fig. \[fig:A\].
![ In a, an elementary rank-6 Trotter tensor $T$ with two (red) spin indices and four (black) bond indices, each of dimension 2. In b, a layer of Trotter tensors representing a small time step $U(d\tau)$. In c, the time step $U(d\tau)$ is applied to spin indices of the purification. In d, the tensors $T$ and $A$ can be contracted into a single new tensor $A'$. A layer of $A'$ makes a new iPEPS that looks like the original one in Fig. \[fig:A\]b but has a doubled bond dimension $2D$. []{data-label="fig:AT"}](AT.pdf){width="0.9999\columnwidth"}
The Method {#sec:algorithm}
==========
We introduce the algorithm in the more general case of thermal states simulation by imaginary time evolution of their purification. To be more specific, we use the example of the quantum Ising model. Modification to real time evolution amounts to ignoring any ancilla lines in the diagrams. For the sake of clarity, in the main text we fully employ the symmetry of the Ising model but we do our numerical simulations with a more efficient algorithm, described in Appendix \[sec:2site\], that breaks the symmetry by applying 2-site nearest-neighbor gates. That algorithm can be generalized to less symmetric models in a straightforward manner.
Suzuki-Trotter decomposition {#sec:ST}
----------------------------
In the second-order Suzuki-Trotter decomposition a small time step is U(d) &=& U\_[h]{} (d/2) U\_[ZZ]{}(d) U\_[h]{} (d/2), \[Udbeta\] where U\_[ZZ]{}(d) = \_[j,j’]{}e\^[i dZ\_jZ\_[j’]{}]{}, U\_[h]{}(d) = \_j e\^[i dh\_j]{} \[UZZ\] are elementary gates and $h_j = h_x X_j + h_z Z_j$.
In order to rearrange $U(d\tau)$ as a tensor network, we use singular value decomposition to rewrite a 2-site term $e^{id\tau Z_jZ_{j'}}$ acting on a NN bond as a contraction of 2 smaller tensors acting on single sites: e\^[idZ\_jZ\_[j’]{}]{} &=& \_[=0,1]{} z\_[j,]{} z\_[j’,]{}. \[svdgate\] Here $\mu$ is a bond index and $z_{j,\mu}\equiv\sqrt{\Lambda_\mu}\,(Z_j)^\mu$ and $\Lambda_0=\cos d\tau$ and $\Lambda_1=i\sin d\tau$. Now we can write U(d) &=& \_[{}]{} \_j . \[Tx\] Here $\mu_{\langle j,j'\rangle}$ is a bond index on the NN bond $\langle j,j'\rangle$ and $\{\mu\}$ is a collection of all such bond indices. The square brackets enclose a Trotter tensor $T(d\tau)$ at site $j$, see Fig. \[fig:AT\]a. It is a spin operator depending on the bond indices connecting its site with its four NNs. A contraction of these Trotter tensors is the gate $U(d\tau)$ in Fig. \[fig:AT\]b. The evolution operator is a product of such time steps, $U(Nd\tau)=\left[U(d\tau)\right]^N$.
![ In a, tensor $A'$ is contracted with a complex conjugate of $A''$ into a transfer tensor $t'$ with a bond dimension $d=2D^2$. In b, tensor $A''$ is contracted with its own complex conjugate into a transfer tensor $t''$ with a bond dimension $d=D^2$. In c, an infinite layer of tensors $t'$ ($t''$) represents the overlap $\langle\psi''|\psi'\rangle$ ($\langle\psi''|\psi''\rangle$). []{data-label="fig:t"}](kuku.pdf){width="0.9999\columnwidth"}
Variational truncation {#sec:step}
----------------------
The time step $U(d\tau)$ applied to the state $|\psi\rangle$ yields a new state |’=U(d)|, see Figs. \[fig:AT\]c and d. If $|\psi\rangle$ has a bond dimension $D$, then the new iPEPS has twice the original bond dimension $2D$.
In order to prevent exponential growth of the dimension in time, the new iPEPS has to be approximated by a more compact one, $|\psi''\rangle$, made of tensors $A''$ with the original bond dimension $D$. The best $|\psi''\rangle$ minimizes the norm | |”- |’|\^2. \[norm\] Equivalently – up to normalization of $|\psi''\rangle$ – the quality of the approximation can be measured by a global fidelity F=. \[F\] After a rearrangement in section \[sec:fom\] below, it becomes an efficient figure of merit.
The iPEPS tensor $A''$ – the same at all sites – has to be optimized globally. However, the first step towards this global optimum is a local pre-update. We choose a site $j$ and label the tensor at this site as $A''_j$. This tensor is optimized while all other tensors are kept fixed as $A''$. With the last constraint the norm (\[norm\]) becomes a quadratic form in $A''_j$. The quadratic form is minimized with respect to $A''_j$ by $\tilde{A}$ that solves the linear equation GA=V. \[GA=V\] Here G= , V= \[GV\] are, respectively, a metric tensor and a gradient. Further details on the local pre-update can be found in section \[sec:lu\] below.
The global fidelity (\[F\]) is not warranted to increase when the local optimum $\tilde A$ is substituted globally, i.e., in place of every $A''$ at every lattice site. However, $\tilde A$ can be used as an estimate of the most desired direction of the change of $A''$. In this vein, we attempt an update A”=A+A, \[Aepsilon\] with an adjustable parameter $\epsilon\in[-\pi/2,\pi/2]$ using an algorithm proposed in Ref. which simplified version was introduced in Refs. . This update was successfully used in a similar variational problem of minimizing energy of an iPEPS as a function of $A$ [@Corboz_varopt_16], where we refer for its detailed account. Here we just sketch the general idea.
To begin with, the global fidelity $F_0$ is calculated for the “old” tensor $A''=A$ with $\epsilon=0$. For small $\epsilon$ the optimization is prone to get trapped in a local optimum. This is why large $\epsilon=\pi/2$ is tried first and if $F>F_0$ then $A''=\tilde A$ is accepted. Otherwise, $\epsilon$ is halved as many times as necessary for $F$ to increase above $F_0$ and then $A''=\tilde A$ is accepted. Negative $\epsilon$ are also considered in case the global $F$ does not increase for a positive $\epsilon$.
Once $A''$ in (\[Aepsilon\]) is accepted, the whole procedure beginning with a solution of (\[GA=V\]) is iterated until $F$ is converged. The final converged $A''$ is accepted as a global optimum.
![ Left, planar version of Fig. \[fig:t\]c. Right, its approximate representation with corner tensors $C$ and edge tensors $E$. Here $C$ effectively represents a corner of the infinite graph on the left and $E$ its semi-infinite edge. Environmental bond dimension $\chi$ controls accuracy of the approximation. Tensors $C$ and $E$ are obtained with corner transfer matrix renormalization group [@Baxter_CTM_78; @Nishino_CTMRG_96; @Orus_CTM_09; @Corboz_CTM_14]. []{data-label="fig:CMR"}](CMR.pdf){width="0.9999\columnwidth"}
![ The environmental tensors introduced in Fig. \[fig:CMR\] can be used to calculate the figure of merit (\[f\]). This diagram shows a fourth power of a factor $q$ by which the diagram in Fig. \[fig:t\]c (or, equivalently, the left panel of Fig. \[fig:CMR\]) is multiplied when $4$ sites are added to the network. Depending on the overlap in question – either $\langle\psi''|\psi'\rangle$ or $\langle\psi''|\psi''\rangle$, see Fig. \[fig:t\] – the factor is either $n=q$ or $d=q$, respectively. The diagram is equivalent to Fig.13.9 in R. J. Baxter’s textbook [@Baxter_Textbook_82]. []{data-label="fig:Baxter"}](Baxter.pdf){width="0.8\columnwidth"}
Efficient fidelity computation {#sec:fom}
------------------------------
In the limit of infinite lattice, the overlaps in the fidelity (\[F\]) become ”|’= \_[N]{} n\^N, ”|”= \_[N]{} d\^N, where $N$ is the number of lattice sites. Consequently, the fidelity becomes $
F = \lim_{N\to\infty} f^{N},
$ where f= \[f\] is a figure of merit per site.
The factors $n$ and $d$ can be computed by CTMRG[@Orus_GSfidelity_17] generalizing the CTMRG approach to compute a partition function per site for 2D statistical models [@Baxter_CTM_78; @Baxter_Textbook_82; @Chan_fpsCTMRG_12; @Chan_fpsCTMRG_13]. First of all, each overlap – either $\langle\psi''|\psi'\rangle$ or $\langle\psi''|\psi''\rangle$ – can be represented by a planar network in Fig. \[fig:t\]c. With the help of CTMRG [@Baxter_CTM_78; @Nishino_CTMRG_96; @Orus_CTM_09; @Corboz_CTM_14], this infinite network can be effectively replaced by a finite one, as shown in Fig. \[fig:CMR\]. Figure \[fig:Baxter\] shows how to obtain $n$ and $d$ with the effective environmental tensors introduced in Fig. \[fig:CMR\].
![ In a, tensor environment for $t'$ ($t''$). It is obtained by removing one tensor $t'$ ($t''$) from the overlap in Fig. \[fig:t\]c or equivalently from the right diagram in Fig. \[fig:CMR\]. The environment represents a derivative of the overlap $\langle\psi''|\psi'\rangle$ ($\langle\psi''|\psi''\rangle$) with respect to $t'$ ($t''$) (\[G2\],\[V2\]). This rank-4 tensor has 4 indices with dimension $D\times 2D$ ($D\times D$), respectively. In b, in case of $t''$ (\[G2\]) each of the 4 indices in (a) is decomposed back into two indices, each of dimension $D$. The diagram represents the metric tensor $G$. The open (red) spin line is a Kronecker delta for spin states and the open (orange) ancilla line is a delta for ancillas. Therefore, the metric can be decomposed as $G=g\otimes 1_s\otimes 1_a$, where $g$ is the tensor environment for $t$. In c, in case of $t'$ (\[V2\]) each of the 4 indices in (a) is decomposed back into two indices of dimension $2D$ (upper) and $D$ (lower). After contracting the upper indices with $A'$ the diagram becomes the gradient $V$. []{data-label="fig:GV"}](GV.pdf){width="0.9999\columnwidth"}
Local pre-update {#sec:lu}
----------------
In order to construct $G$ and $V$ from the effective environmental tensors $C$ and $T$, it is useful to note first that a derivative of a contraction of two rank-$n$ tensors $f=\sum_{i_1,...,i_n} A_{i_1,...,i_n} B_{i_1,...,i_n}$ with respect to one of them gives the other one: $\partial f/\partial A_{i_1,...,i_n} = B_{i_1,...,i_n}$. Futhermore, we note that both the optimized tensor $A''_j$ and its conjugate $\left(A''_j\right)^*$ are located at the same site $j$ and they enter the overlap $\langle\psi''|\psi''\rangle$ ($\langle\psi''|\psi'\rangle$) only through the tensor $t''$ ($t'$)defined in Fig. \[fig:t\] (\[fig:t\]a), located at this site. We distinguish this tensor $t''$ ($t'$) by an index $j$ and call it $t''_j$ ($t'_j$). Therefore, the derivatives in Eq. (\[GV\]) decompose into a tensor contraction of derivatives G &=& , \[G2\]\
V &=& \[V2\] The derivatives of the overlaps with respect to $t'_j$ ($t''_j$) are represented by Fig. \[fig:GV\]a, where one tensor $t'_j$ ($t''_j$) at site $j$ was removed from the overlap shown in Figs. \[fig:t\]c, \[fig:CMR\]. Indeed, a contraction of the missing $t'_j$ ($t''_j$) with its environment in Fig. \[fig:GV\]a through corresponding indices gives back the overlap. Diagramatically, this contraction amounts to filling the hole in Fig. \[fig:GV\]a with the missing $t'_j$ ($t''_j$). In numerical calculations, the infinite diagram in Fig. \[fig:GV\]a is approximated by a equivalent finite one in a similar way as in Fig. \[fig:CMR\].
The rank-4 tensor in Fig. \[fig:GV\]a is a tensor environment for $t'_j$ ($t''_j$). Each of its 4 indices is a concatenation of two iPEPS bond indices, one from the ket and one from the bra iPEPS layer and has a dimension equal to ($2D\times D$) $D\times D$. After splitting each index back into ket and bra indices, this environment can be used to calculate ($V$) $G$, as shown in Fig. \[fig:GV\]c (Fig. \[fig:GV\]b). In Fig. \[fig:GV\]b the hole in Fig. \[fig:GV\]a (with split ket and bra indices) is filled with the second derivative of $t''_j$ with respect to $A''_j$ and $\left(A''_j\right)^*$. Similarly as the derivative of an overlap with respect to $t''_j$, this derivative is obtained from the tensor $t''_j$ in Fig. \[fig:t\]b by removing both $A''_j$ and $\left(A''_j\right)^*$ from the diagram. In Fig. \[fig:GV\]c the hole in Fig. \[fig:GV\]a is filled by the derivative of $t'_j$ with respect to $\left(A''_j\right)^*$. This derivative is obtained from the tensor $t'_j$ in Fig. \[fig:t\]a by removing $\left(A''_j\right)^*$ from the diagram.
We have to keep in mind that the evironmental tensors are converged with limited precision that is usually set by demanding that local observables are converged with precision $\simeq 10^{-8}$. This precision limits the accuracy to which the matrix $G$ is Hermitean and positive definite. In order to avoid numerical instabilities this error has to filtered out by elliminating the anti-Hermitean part of $G$ and then truncating its eigenvalues that are less than a fraction of its maximal eigenvalue. The fraction is usually set at $10^{-8}$. To this end we solve the linear equation (\[GA=V\]) using the Moore-Penrose pseudo-inverse =[pinv]{}(G)V, \[tildeA\] where the truncation is implemented by setting an appropriate tolerance in the pseudo-inverse procedure.
Another advantage of the pseudo-inverse solution is that it does not contain any zero modes of $G$. By definition, these zero modes do not matter for the local optimization problem but they can make futile the attempt in (\[Aepsilon\]) to use $\tilde{A}$ as a significant part of the global solution.
A possibility of further simplification occurs in Fig. \[fig:GV\]b, where the open spin and ancilla lines represent two Kronecker symbols. The symbols are identities in the spin and ancilla subspace and, therefore, the metric $G$ has a convenient tensor-product structure $G=g\otimes 1_s\otimes 1_a$, where $g$ is a reshaped tensor environment for $t''_j$ and $1_s$ and $1_a$ are identities for spins and ancillas, respectively. Therefore – after appropriate reshaping of tensors – Eq. (\[tildeA\]) can be reduced to A = [pinv]{}(g) V, where only the small tensor environment $g$ has to be pseudo-inverted.
![ Thermal states for a transverse field $h_x=2.5$ with a longitudinal bias $h_z=0.01$. The stars are results from variational tensor network renormalization (VTNR) and the solid lines from the imaginary time evolution. With increasing bond dimension $D$ the two methods converge to each other. In a, longitudinal magnetization $\langle Z\rangle$ in function of inverse temperature. In b, energy per site $E$ in function of inverse temperature. []{data-label="fig:imag25"}](gx2p5.pdf){width="0.9999\columnwidth"}
![ Thermal states for a transverse field $h_x=2.9$ with a longitudinal bias $h_z=0.01$. The stars are results from variational tensor network renormalization (VTNR) and the solid lines from the imaginary time evolution. With increasing bond dimension $D$ the two methods converge to each other. In a, longitudinal magnetization $\langle Z\rangle$ in function of inverse temperature. In b, energy per site $E$ in function of inverse temperature. []{data-label="fig:imag29"}](gx2p9.pdf){width="0.9999\columnwidth"}
Thermal states from imaginary time evolution {#sec:im}
============================================
In this section we present results obtained by imaginary time evolution for two values of the transverse field $h_x=2.5$ and $h_x=2.9$, see Figures \[fig:imag25\] and \[fig:imag29\], corresponding to critical temperatures $\beta_c=0.7851(4)$ and $\beta_c=1.643(2)$, respectively [@Hesselmann_TIsingQMC_16]. We show data with $D=2,3,4,5$. The stronger field is closer to the quantum critical point at $h_0$, hence quantum fluctuations are stronger and a bigger bond dimension $D$ is required to converge. For the evolution to run smoothly across the critical point we added a small longitudinal bias $h_z=0.01$.
Figures \[fig:imag25\]a and \[fig:imag25\]b show the longitudinal magnetization $\langle Z\rangle$ and energy $E$ for the two transverse fields. The data from the evolution are compared to results obtained with the variational tensor network renormalization (VTNR) [@Czarnik_VTNR_15; @Czarnik_compass_16; @Czarnik_fVTNR_16; @Czarnik_eg_17]. With increasing $D$ each of the two methods converges and they converge to each other. This is a proof of principle that the variational time evolution can be applied to thermal states.
The data at hand suggest that with increasing $D$ the evolution converges faster than VTNR. However, at least for the Ising benchmark, numerical effort necessary to obtain results of similar accuracy is roughly the same. In both methods the bottleneck is the corner transfer matrix renormalization procedure. In the case of VTNR larger D is necessary but in the case of the evolution the environmental tensors need to be computed more times.
The advantage of VTNR is that it targets the desired temperature directly, there is no need to evolve from $\beta=0$ and thus no evolution errors are accumulated. In order to minimize the accumulation when evolving across the critical regime a small longitudinal bias has to be applied. The critical singularity is recovered in the limit of small bias that requires large $D$. However, one big advantage of the variational evolution is that – unlike VTNR targeting the accuracy of the partition function – it aims directly at an accurate thermal state. In some models this may prove to be a major advantage.
![ Transverse magnetization $\langle X \rangle$ (left column) and energy per site (right column) after a sudden quench from a ground state in a strong transverse field, $h_x\gg h_0$, with all spins pointing along $x$ down to a finite $h_x=2h_0$ (top row), $h_x=h_0$ (middle row), and $h_x=h_0/10$ (bottom row). The quench is, respectively, within the same phase, to the quantum critical point, and to a different phase. Energy conservation shows systematic improvement with increasing bond dimension $D=2,3,4$. We see that for sufficiently small times seemingly converged results for transverse magnetization can be obtained. While approaching the limit of small entanglement ($h_x=h_0/10$) we see that the “convergence time” is growing longer as expected. []{data-label="fig:quench"}](time-crop.pdf){width="0.9999\columnwidth"}
Time evolution after a sudden quench {#sec:re}
====================================
Next we move to simulation of a real time evolution after a quench in an unbiased model (\[calH\]) with $h_z=0$. The initial state is the ground state for $h_x\gg h_0$ with all spins pointing along $x$. At $t=0$ the Hamiltonian is suddenly quenched down to a finite $h_x=2h_0,h_0,h_0/10$ that is, respectively, above, at, and below the quantum critical point $h_0$.
Figure \[fig:quench\] shows a time evolution of the magnetization $\langle X\rangle$ and energy per site $E$ after the sudden quench for bond dimensions $D=2,3,4$. With increasing $D$ the energy becomes conserved more accurately for a longer time. This is an indication of the general convergence of the algorithm.
Not quite surprisingly, the results are most accurate for $h_x=h_0/10$. This weak transverse field is close to $h_x=0$ when the Hamiltonian is classical and the time evolution can be represented exactly with $D=2$. At $h_x=0$ quasiparticles have flat dispersion relation and do not propagate, hence – even though they are excited as entangled pairs with opposite quasi-momenta – they do not spread entanglement across the system. For any $h_x>0$, however, the entanglement grows with time and any bond dimension is bound to become insufficient after a finite evolution time. However, as discussed in Sec. \[sec:introduction\], there are potential applications where this effect is of limited importance.
Conclusion {#sec:conclusion}
==========
We tested a straightforward algorithm to simulate real and imaginary time evolution with infinite iPEPS. The algorithm is based on variational maximization of a fidelity between a new iPEPS obtained after a direct application of a time step and its approximation by an iPEPS with the original bond dimension.
The main result is simulation of real time evolution after a sudden quench of a Hamiltonian. With increasing bond dimension the results converge over increasing evolution time. This is a proof of principle demonstration that simulation of a real time evolution with a 2D tensor network is feasible.
We also apply the same algorithm to evolve purification of thermal states. These results converge to the established VTNR method providing a proof of principle that the algorithm can be applied to 2D strongly correlated systems at finite temperature.
P. C. acknowledges inspiring discussions with Philippe Corboz on application of CTMRG to calculation of partition function per site and simulations of thermal states. We thank Stefan Wessel for numerical values of data publised in Ref. . Simulations were done with extensive use of ncon function [@ncon]. This research was funded by National Science Center, Poland under project 2016/23/B/ST3/00830 (PC) and QuantERA program 2017/25/Z/ST2/03028 (JD).
\[sec:appendix\]
![ In a, the infinite square lattice is divided into two sublattices with tensors $A$ (lighter green) and $B$ (darker green). In b, SVD decomposition of a NN gate is applied to every pair $A$ and $B$ of NN tensors. In c, when the tensors $A$ and $B$ are contracted with their respective $z$’s, then they become new tensors $A'$ and $B'$ with a doubled bond dimension $2D$ on their common NN bond. By variational optimization the iPEPS made of $A'$ and $B'$ is approximated by a new iPEPS made of $A''$ and $B''$ with the original bond dimension $D$. []{data-label="fig:2site"}](2site.pdf){width="0.9999\columnwidth"}
2-site gates {#sec:2site}
============
For the sake of clarity, the main text presents a straightforward single-site version of the algorithm. In practice it is more efficient to implement the gate $U_{ZZ}(d\tau)$ as a product of two-site gates. To this end the infinite square lattice is divided into two sublattices $A$ and $B$, see Fig. \[fig:2site\]a. On the checkerboard the gate becomes a product && U\_[ZZ]{}(d) = U\^a\_0(d) U\^a\_1(d) U\^b\_0(d) U\^b\_1(d). \[UZZ2s\] Here $a$ and $b$ are the Cartesian lattice directions spanned by $e_a$ and $e_b$, U\^a\_s(d) &=& \_[mn]{} e\^[i dZ\_[2m+s-1,n]{}Z\_[2m+s,n]{}]{},\[Ua\]\
U\^b\_s(d) &=& \_[mn]{} e\^[i dZ\_[m,2n+s-1]{}Z\_[m,2n+s]{}]{},\[Ub\] and $Z_{m,n}$ is an operator at a site $me_a+ne_b$.
Every NN gate in (\[Ua\],\[Ub\]) is decomposed as in (\[svdgate\]). Consequently, when a gate, say, $U^a_0(d\tau)$ is applied to the checkerboard $AB$-iPEPS in Fig. \[fig:2site\]a, then every pair of tensors $A$ and $B$ at every pair of NN sites $(2m-1)e_a+ne_b$ and $2me_a+ne_b$ is applied with the NN-gate’s decomposition as in Fig. \[fig:2site\]b. When the tensors $A$ and $B$ are fused with their respective $z$’s, they become $A'$ and $B'$, respectively, that are connected by an index with a bond dimension $2D$, see Fig. \[fig:2site\]c. The action of the gate $U^a_0(d\tau)$ is completed when the $A'B'$-iPEPS is approximated by a (variationally optimized) new $A''B''$-iPEPS with the original bond dimension $D$ at every bond.
Apart from the opportunity to use reduced tensors in the variational optimization, the main advantage of the 2-site gates is that the enlarged bond dimension $2D$ appears only on a minority of bonds. This speeds up the CTMRG for the overlap $\langle\psi'|\psi''\rangle$ that is the most time-consuming part of the algorithm. The decomposition into 2-site gates breaks the symmetry of the lattice. Therefore we use the efficient non-symmetric version of CTMRG [@Corboz_CTM_14] for checkerboard lattice.
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---
abstract: |
Belgium is amongst few artificial countries, established on purpose, when Dutch and French speaking parts were joined in a single unit. This makes Belgium a particularly interesting testbed for studying bio-inspired techniques for simulation and analysis of vehicular transport networks. We imitate growth and formation of a transport network between major urban areas in Belgium using the acellular slime mould *Physarum polycephalum*. We represent the urban areas with the sources of nutrients. The slime mould spans the sources of nutrients with a network of protoplasmic tubes. The protoplasmic tubes represent the motorways. In an experimental laboratory analysis we compare the motorway network approximated by *P. polycephalum* and the man-made motorway network of Belgium. We evaluate the efficiency of the slime mould network and the motorway network using proximity graphs.
*Keywords: transport networks, unconventional computing, slime mould*
address:
- 'Unconventional Computing Centre, University of the West of England, Bristol, United Kingdom'
- 'Department of Mathematical Modelling, Statistics and Bioinformatics, Ghent University, Gent 9000, Belgium'
- 'Scientific Institute of Public Health, Brussels B1050, Belgium'
author:
- Andrew Adamatzky
- Bernard De Baets
- Wesley Van Dessel
title: |
Slime mould imitation of\
Belgian transport networks:\
redundancy, bio-essential motorways, and dissolution
---
Introduction
============
Plasmodium is a vegetative stage of the acellular slime mould *Physarum polycephalum*. This is a single cell with many nuclei. The plasmodium feeds on microscopic particles [@stephenson_2000]. During its foraging behaviour the plasmodium spans scattered sources of nutrients with a network of protoplasmic tubes. The protoplasmic network is optimised to cover all sources of food and to provide a robust and speedy transportation of nutrients and metabolites in the plasmodium body. The plasmodium’s foraging behaviour can be interpreted as computation. Data are represented by spatial configurations of attractants and repellents, and results of computation by structures of protoplasmic network formed by the plasmodium on the data sites [@nakagaki_2000; @nakagaki_2001a; @PhysarumMachines]. The problems solved by plasmodium of *P. polycephalum* include shortest path [@nakagaki_2000; @nakagaki_2001a], implementation of storage modification machines [@PhysarumMachines], Voronoi diagram [@shirakawa], Delaunay triangulation [@PhysarumMachines], logical computing [@tsuda_2004; @adamatzky_gates], and process algebra [@schumann_adamatzky_2009]; see an overview in [@PhysarumMachines]. Previously [@adamatzky_UC07] we have evaluated the road-modelling potential of *P. polycephalum*, however no conclusive results were presented back in 2007. A step towards biological approximation, or evaluation, of man-made road networks was done in our previous papers on the approximation of motorways/highways in the United Kingdom [@adamatzky_jones_2009], Mexico [@adamatzky_Mexico] and Australia [@adamatzky_Australia] by plasmodium of *P. polycephalum*. For these countries we found that, in principle, the network of protoplasmic tubes developed by plasmodium matches, at least partly, the network of man-made transport networks. However a country’s shape and spatial configuration of urban areas, which are experimentally represented by sources of nutrients, might play a key role in determining the exact structure of plasmodium networks. Also we suspect that the degree of matching between the Physarum networks and the motorway networks is determined by original government designs of motorways in any particular country. This is why it is so important to collect data on the development of plasmodium networks in all major countries, and then undertake a comparative analysis.
Belgium is a good testbed for the evaluation of slime-mould approximation of motorways because
- Belgium is an artificial country created relatively recently, in 1830.
- It is amongst the most populated area in Europe.
- There is a density misbalance between two major communities: Flanders is more densely populated than Wallonia.
- the Belgian economy is centred around Brussels, by far biggest city, with hundreds of thousands of workers commuting to Brussels every day.
In the early days the Belgian highways were constructed to provide a solution against the overcharged national and local roads, caused by the expanding number of cars. In the North of the country, construction was generally based on growing demands from the economic and touristic sectors. The first highway was the one between Brussels and Ostend. Another ‘early’ highway was the one between Antwerp and Liège (E313) to open up the port of Antwerp’s access to the ‘hinterland’. At the end of 1972 the most important cities were connected by highways. However, from the point of view of transport economics, only two of them were answering to an economic demand, a need for construction based on increasingly busy roads: the one between Brussels and Antwerp (E19), and the one between Brussels and Liège (E40). The others were intended as an investment trigger. The Autoroute de Wallonie (E42) was aimed at the economic reconversion of the old industrial axis in Wallonia (steel and coal industry). The E17 and E34 motorways provided an additional connection between the port of Antwerp and the French and German inner lands. The purpose of the E314 was to open up the province of Limburg (coal mining industry), and to provide a shortcut between Antwerp and the German Rhineland (Ruhrgebiet). Highway construction has been the result of political negotiations and the desire or need of the Northern and Southern partners to balance large investments in both parts of the country [@WegenRoutes].
The paper is structured as follows. In Sect. \[methods\] we give an overview of the experimental setup employed. Analysis of protoplasmic networks produced by slime mould *P. polycephalum* in laboratory experiments is provided in Sect. \[experimentalresults\]. We compare slime mould generated and man-built motorways in Sect. \[comparingPhysarumMotorways\]. Section \[proximity\] considers protoplasmic networks and Belgian motorways in the context of planar proximity graphs. Relations between the experimental results and the administrative subdivision of Belgium are discussed in Sect. \[adminstrativesubdivision\]. In Sect. \[contamination\] we study outcomes of large-scale contamination and resulting reconfiguration of slime mould transport networks.
Experimental {#methods}
============
Plasmodium of *P. polycephalum* is cultivated in plastic container, on paper kitchen towels moistened with still water, and fed with oat flakes. For experiments we use $120 \times 120$ mm polystyrene square Petri dishes and 2% agar gel (Select agar, by Sigma Aldrich) as a substrate. Agar plates, about 2-3 mm in depth, are cut in the shape of Belgium.
We consider the twenty-one most[^1] populous urban areas in Belgium $\mathbf U$ (Fig. \[urbanareas\]a), shown below in descending order of population size:
[2]{}
1. Brussels area, including Dilbeek and Vilvoorde
2. Antwerp area, including Beveren and Brasschaat
3. Gent
4. Charleroi area, including La Louvière and Chatelet
5. Liège area, including Seraing, Verviers and Herstal
6. Brugge
7. Namur
8. Leuven
9. Mons
10. Aalst
11. Mechelen
12. Kortrijk area, including Mouscron and Waregem
13. Hasselt
14. Oostende
15. Sint-Niklaas
16. Tournai
17. Genk area, including Maasmechelen
18. Roeselare
19. Turnhout
20. Arlon
21. Sankt-Vith
To represent areas of $\mathbf{U}$ we place oat flakes in the positions of agar plate corresponding to the areas. At the beginning of each experiment an oat flake colonised by plasmodium is placed in the Brussels area (Fig. \[mappopulation\]). Our choice of inoculation site does not reflect the historical development of transport routes in Belgium (where inoculation should start in Aalst, Brugge, Kortrijk, or Gent) however it conveys the overwhelming economic power of the capital. We undertook 28 experiments. The Petri dishes with plasmodium are kept in darkness, at temperature 22-25$^\text{o}$C, except for observation and image recording. Periodically, the dishes are scanned with an Epson Perfection 4490 scanner.
Slime mould transport networks: bio-essential motorways {#experimentalresults}
=======================================================
Plasmodium is inoculated in the Brussels area. In the first 24 h it propagates towards and occupies Leuven and Mechelen, and then propagates from Mechelen to Antwerp, from Antwerp to Sint-Niklaas (Fig. \[exampleA11\]ab). In the next 24 h slime mould propagates from Sint-Niklaas to Aalst and Gent, from Gent to Brugge and from Aalst to Kortrijk. Links from Brugge to Oostende and Roeselare are built during the same time interval (Fig. \[exampleA11\]cd). Westward development of plasmodium is somehow stopped. Despite an attempted propagation from Turnhout towards the Hasselt and Genk areas the slime mould never actually reaches these areas in the first 48 h (Fig. \[exampleA11\]cd).
By the 72nd hour after being inoculated in Brussels almost all urban areas but Hasselt and Genk are colonised by slime mould. Namely in the time interval 48-72 h plasmodium grows from Aalst to Tournai, from Tournai to Mons, and from Mons to Charleroi. Slime mould branches at Charleroi and grows in parallel to Namur and Arlon. It propagates from Namur to Sankt-Vith, from Sankt-Vith to Liège. (Fig. \[exampleA11\]ef). By the 96th h after being inoculated the slime mould propagates from Liège to Genk and Hasselt areas, and from Hasselt to Leuven. The plasmodium’s explorative activities in the Brussels area are ’resumed’ when the slime propagates from Leuven and re-occupies the Brussels area (Fig. \[exampleA11continuation\] and Fig. \[A11\_12\_schemes\]a.).
Colonisation of Belgium by slime mould shown in Fig. \[exampleA12\] develops initially according to the scenario described above with the following deviations. Plasmodium propagates from Gent to Brugge, Roeselare and Kortrijk at the same time (Figs. \[exampleA12\]a–d and \[A11\_12\_schemes\]b). Slime mould grows from Leuven to Namur and then to Charleroi. From Charleroi plasmodium propagates to Mons and from Mons to Tournai (Figs. \[exampleA12\]ab). Arlon and Sinkt-Vith are reached by slime mould via Leuven, Hasselt and Liège (Figs. \[exampleA12\]cd and \[exampleA12\]ef). Antwerp is never colonised by slime mould in this particular experiment (Fig. \[A11\_12\_schemes\]).
Two examples discussed above show that the dynamics of colonisation varies from experiment to experiment. Therefore from experimental data we extract a generalised Physarum graph. To generalise our experimental results we constructed a Physarum graph with weighted edges. A Physarum graph is a tuple ${\mathbf P} = \langle {\mathbf U}, {\mathbf E}, w \rangle$, where $\mathbf U$ is a set of urban areas, $\mathbf E$ is a set of edges, and $w: {\mathbf E} \rightarrow [0,1]$ associates to each edge of $\mathbf{E}$ a probability (or weights). For every two regions $a$ and $b$ from $\mathbf U$ there is an edge connecting $a$ and $b$ if a plasmodium’s protoplasmic link is recorded at least in one of $k$ experiments, and the edge $(a,b)$ has a probability calculated as the ratio of experiments where protoplasmic link $(a,b)$ occurred in the total number of experiments ($k=23$). For example, if we observed a protoplasmic tube connecting areas $a$ and $b$ in 7 experiments, the weight of edge $(a,b)$ will be $w(a,b)=\frac{7}{23}$. We do not take into account the exact configuration of the protoplasmic tubes but merely their existence. Further we will be dealing with threshold Physarum graphs $\mathbf{P}(\theta) = \langle {\mathbf U}, T({\mathbf E}), w, \theta \rangle$. A threshold Physarum graph is obtained from Physarum graph by the transformation: $T({\mathbf E})=\{ e \in {\mathbf E}: w(e) geq \theta \}$. That is, all edges with weight less than $\theta$ are removed. Examples of threshold Physarum graphs for various values of $\theta$ are shown in Fig. \[allphysarumgraphs\].
The ’raw’ Physarum graph $\mathbf{P}(\frac{1}{28})$ is a non-planar[^2] acyclic graph (Fig. \[selectedphysarumgraphs\]a). In a raw graph each edge appears at least in one laboratory experiment. We call an edge of Physarum *credible* if this edge is represented by a protoplasmic tube in over 20% of laboratory experiments. Such graph $\mathbf{P}(\frac{6}{28})$ is shown in Fig. \[selectedphysarumgraphs\]b. It is still non-planar, however, the only intersecting edges are the links Liège area — Arlon and Namur — Sankt-Vith.
The Physarum graph with all credible edges is a non-planar cyclic graph.
The graph remains connected while $\theta$ grows up to $\frac{10}{28}$ (Fig. \[selectedphysarumgraphs\]c). The Turnhout urban area becomes an isolated vertex when $\theta=\frac{11}{28}$ (Fig. \[selectedphysarumgraphs\]d). For this value of $\theta$ the Physarum graph becomes planar because the link (Liège area – Arlon) is represented in more experiments than the link (Namur – Sankt-Vith). The graph $\mathbf{P}(\frac{11}{28})$ has the largest (among all Physarum graphs studied here) empty, i.e. not having any edges inside, circle. Clockwise, starting in the Brussels area it spans Leuven, Hasselt, Liège, Namur, Charleroi, Mons, Tournai, Kortrijk, Gent, Aalst and finishes in Brussels.
The Physarum graph $\mathbf{P}(\theta)$ splits into three components when $\theta=\frac{16}{28}$. The smallest component is the isolated vertex Turnhout. The medium size component is a cycle Liège — Hasselt — Genk — Liège attached to a segment Arlon — Sank-Vith (Fig. \[selectedphysarumgraphs\]e). The largest component is a proximity graph spanning all remaining urban areas.
The following slime mould transport chains appear in almost all experiments (Fig. \[selectedphysarumgraphs\]f):
- chain Roeselare — Kortrijk — Mons — Charleroi — Namur
- chain Oostende — Brugge — Gent — Aalst — Brussels — Leuven — Mechelen — Antwerp — Sint-Niklaas
- chain Liège — Sankt-Vith — Arlon
- one-link chain Hasselt — Genk.
Slime mould versus Belgian motorways {#comparingPhysarumMotorways}
====================================
As we can see in the examples of the experimental configurations in Fig. \[PhysarumOnMotorwayMap\], plasmodium networks are polymorphic and no two networks are exactly the same. Some motorways are matched by protoplasmic tubes well, others just approximated, and some do not have a slime mould representation at all.
For example, Fig. \[PhysarumOnMotorwayMap\]a demonstrates that slime mould developed protoplasmic tubes corresponding to motorway A10/E40 between Gent and Brugge, E17 between Gent and Kortrijk, E17 and E403/A17 between Gent and Tournai, and E42 connecting Mons to Charleroi to Namur to Liège. Motorways E42 (Tournai – Mons), E46 (Antwerp – Turnhout) and E40 (Leuven – Liège) are represented by slime mould in the experiment illustrated in Fig. \[PhysarumOnMotorwayMap\]b. At the same time, the transport link E411 from Brussels to Namur to Arlon is not represented in the configurations in Fig. \[PhysarumOnMotorwayMap\].
Due to the variability of the slime mould networks it would be unreasonable to attempt comparing the exact topology of slime mould and man-made transport links. Instead we will compare them at the level of graph-theoretical representations.
![Graph $\mathbf H$ of Belgian motorway network.[]{data-label="motorways"}](figs/BelgianMotorways){width="70.00000%"}
The graph $\mathbf H$ of the Belgian motorway network (Fig. \[motorways\]) is constructed as follows. Let $\mathbf U$ be a set of urban regions/cities; for any two regions $a$ and $b$ from $\mathbf U$, the nodes $a$ and $b$ are connected by an edge $(a,b)$ if there is a motorway starting in the vicinity of $a$, passing in the vicinity of $b$, and not passing in the vicinity of any other urban area $c \in \mathbf U$. In the case of branching – that is, a motorway starts in $a$, goes in the direction of $b$ and $c$, and at some point branches towards $b$ and $c$ – we then add two separate edges $(a,b)$ and $(a,c)$ to the graph $\mathbf H$. The motorway graph is planar (Fig. \[motorways\]).
Physarum polycephalum almost completely approximates the Belgian motorway network.
Namely, $\mathbf{H} \nsubseteq \mathbf{P}(\frac{1}{28})$ but 25 of 28 edges of $\mathbf{H}$ are edges of $\mathbf{P}(\frac{1}{28})$. The following motorway links are never represented by protoplasmic tubes, not in a single experiment: Brussels to Tournai, Brussels to Antwerp, Antwerp to Hasselt (Fig. \[IntersectPhysarumMotorways\]a and Fig. \[motorways\]).
**On redundancy**. Motorway links connecting Brussels with Antwerp, Tournai, Mons, Charleroi, Namur, and links connecting Leuven with Liège and Antwerp with Genk and Turnhout are proved to be redundant components of the Belgian transport system in the slime mould experiments.
The skeletal motorway network represented by the plasmodium network in laboratory experiments is shown in Fig. \[IntersectPhysarumMotorways\]b. This network is represented by protoplasmic tubes in over 35% of experiments; and in almost 40% of experiments without the Antwerp — Turnhout link (Fig. \[IntersectPhysarumMotorways\]cd). By increasing the level of slime mould’s “confidence" to almost 60% we loose motorway links connecting Leuven with Hasselt, Namur with Liège and Arlon, and Liège with Arlon (Fig. \[IntersectPhysarumMotorways\]cd).
Motorway segments A17 (Antwerp — Sint-Niklaas), A19 (Antwerp — Mechelen), E42 (Liège — Sankt-Vith), A10/E40 (Oostende — Gent — Aalst — Brussels — Leuven), A17/E403 (Roeselare — Kortrijk — Tournai), E42/E19 (Tournai — Mons), E42 (Mons — Charleroi — Namur) are essential transport links from the slime mould’s point of view.
As illustrated in Fig. \[IntersectPhysarumMotorways\]f these transport links are represented by the protoplasmic network in almost 80% of laboratory experiments.
Proximity graphs {#proximity}
================
A planar proximity graph is a planar graph where two points are connected by an edge if they are close in some sense. A pair of points is assigned a certain neighbourhood, and points of the pair are connected by an edge if their neighbourhood is empty. Here we consider the most common proximity graph as follows.
- $\mathbf{GG}$: Points $a$ and $b$ are connected by an edge in the Gabriel Graph $\mathbf{GG}$ if the disc with diameter $dist(a,b)$ centred in middle of the segment $ab$ is empty [@gabriel_sokal_1969; @matula_sokal_1984] (Fig. \[proximity\]a).
- $\mathbf{RNG}$: Points $a$ and $b$ are connected by an edge in the Relative Neighbourhood Graph $\mathbf{RNG}$ if no other point $c$ is closer to $a$ and $b$ than $dist(a,b)$ [@toussaint_1980] (Fig. \[proximity\]b).
- $\mathbf{MST}$: The Euclidean minimum spanning tree (MST) [@nesetril] is a connected acyclic graph which has the minimum possible sum of edges’ lengths (Fig. \[proximity\]b).
In general, the graphs relate as $\mathbf{MST} \subseteq \mathbf{RNG} \subseteq\mathbf{GG}$ [@jaromczyk_toussaint_1992; @matula_sokal_1984; @toussaint_1980]; this is called the Toussaint hierarchy.
The Belgian motorway graph is approximated by the Gabriel graph with 80% accuracy and by the relative neighbourhood graph with 70% accuracy.
The following motorways links from $\mathbf{H}$ are not presented in the Gabriel graph $\mathbf{GG}$: (Brussels area — Antwerp area), ( Brussels area — Tournai), (Brussels area — Namur), (Antwerp area — Hasselt), (Leuven — Liège area), (Liège area — Arlon) (Fig. \[motorwaysintersection\]a). With regards to the relative neighbourhood graph, a few more transport links from $\mathbf{H}$ are not a part of $\mathbf{RNG}$: (Brussels area — Mons), (Brussels area — Leuven) and (Namur — Arlon) (Fig. \[motorwaysintersection\]b). Thus $\mathbf{GG}$ represents 23 of 29 edges of the motorway graph $\mathbf{H}$ and $\mathbf{RNG}$ 20 of 29 edges of $\mathbf{H}$.
The minimum spanning tree rooted in Brussels is not a subgraph of the Belgian motorways graph, $\mathbf{MST} \nsubseteq \mathbf{H}$.
This is because the links (Mechelen — Sint-Niklaas) and (Mechelen — Leuven) exist in $\mathbf{MST}$ but not are not present in $\mathbf{H}$ (Fig. \[motorwaysintersection\]c). The minimum spanning tree is considered to be an optimal, in a sense of minimality of edge lengths, acyclic planar graph. The fact that two of the spanning tree edges are not represented by man-made motorway links allows us to suggest that the Belgian motorway network is not optimal, at least not optimal in spanning of major urban areas.
Considering Physarum graph $\mathbf{P}(\frac{11}{28})$ is credible because its edges are represented by protoplasmic tubes in at least 40% of laboratory experiments, we can compare it with three basic proximity graphs (Fig. \[physarum11\_intersectproximitygraphs\]).
If slime mould represented A7/A54 Brussels — Charleroi, non-existing in reality motorways between Leuven — Namur and Turnhout — Hasselt its credible transport network would be a super-graph of the relative neighbourhood graph.
This is a direct outcome of comparing Fig. \[physarum11\_intersectproximitygraphs\]a and Fig. \[proximitygraphs\]a. Relative neighbourhood graphs are considered to be optimal cyclic graphs in terms of total edge length and travel distance, and are known to be a good approximation of road networks [@watanabe_2005; @watanabe_2008]. The Leuven — Namur and Brussels — Charleroi links are represented by slime mould in less than 20% of laboratory experiments (Fig. \[selectedphysarumgraphs\]b). The intersection of the Physarum graph $\mathbf{P}(\frac{11}{28})$ (Fig. \[selectedphysarumgraphs\]d) with the minimum spanning tree (Fig. \[proximitygraphs\]c) comprises two disconnected components: one lies in Flanders and another in Wallonia (Fig. \[physarum11\_intersectproximitygraphs\]c).
Dissolution: Transport networks and administrative subdivision {#adminstrativesubdivision}
==============================================================
The province Brabant Wallone has no transport routes originating in and passing through it for $\theta=\frac{10}{28}$. When $\theta$ increases to $\frac{11}{28}$, the Antwerp province becomes isolated from the other provinces of Belgium. For $\theta=\frac{22}{28}$ we have isolated Antwerp and Limburg provinces, and clusters of interconnected regions:
1. West-Vlaanderen, Hainaut and Namur provinces,
2. West-Vlaanderen, Oost-Vlaanderen, Vlaams Brabant, Oost-Vlaanderen provinces,
3. Liège and Luxembourg provinces.
![Intersection of Physarum graph $\mathbf{P}(\frac{11}{28})$ with the minimum spanning tree rooted in Brussels.[]{data-label="walloniaflanders"}](figs/WalloniaFlandersScheme){width="49.00000%"}
In terms of spanning tree (Fig. \[proximitygraphs\]) the transport link between the Brussels area and the Charleroi area is the only means of keeping the Dutch and French speaking communities together.
If the two parts of Belgium were separated with Brussels in Flanders, the Walloon region of the Belgian transport network would be represented by a single chain from Tournai in the north-west to the Liège area in the north-east and down to southmost Arlon.
This transport link is not part of the spanning tree rooted in Brussels, and when omitted the two Belgian communities become isolated from each other (Fig. \[walloniaflanders\]).
Response of slime mould networks to imitated disasters {#contamination}
======================================================
To study the reaction of Physarum-grown transport networks on major disasters, we placed crystals of sodium chloride in the approximate positions of Doel Nuclear Power Plant, near Antwerpen (seven experiments) and Tihange Nuclear Power Station, near Liège (nine experiments). Sodium chloride is a strong repellent for *P. polycephalum*. The salt diffuses in the substrate outwards its original application site (epicentre of disaster). It can therefore imitate radioactive and/or chemical pollution and subsequent disturbance spreading along Belgian transport networks. Images of protoplasmic networks reconfigured 24 h after start of contamination are shown in Figs. \[saltat2\] and \[saltat5\].
![Illustration of the slime mould’s response to contamination. (a) epicentre of contamination. (b) abandoned and not repositioned transport link. (c) abandoned and repositioned transport link. (d) new location of the abandoned transport link. (e) explorative activity. (f) emergency preparations, initial stage of sclerotinisation. (g) enhanced transport links. Image is taken 24 h after initiation of contamination in Liège area.[]{data-label="saltexample"}](scheme3){width="95.00000%"}
A typical response to propagating contamination is dissected in Fig. \[saltexample\]. In this example we place a salt grain on the oat flake representing the Liège area (Fig. \[saltexample\], ’a’). In 24 h after the imitated accident, the contamination spreads as far Namur on the west and Sankt-Vith on the east and Hasselt and Genk in the south. Transport links in proximity of the contamination epicentre become destroyed and abandoned. An example of abandoned transport link is a protoplasmic tube (marked ’b’ Fig. \[saltexample\]) representing E42/A27 (Liège to Sankt-Vikt motorway). No alternative routes for such destroyed transport links are offered by the slime mould.
Transport links being at a significant distance from the epicentre but yet inside the contamination zone are shifted further away from the contamination. For example, slime mould abandons protoplasmic tube representing motorway E411, marked ’c’ in Fig. \[saltexample\] but grows another tube, marked ’d’ in Fig. \[saltexample\] slightly westwardly. Urban areas not directly affected by contamination show signs of increased explorative and scouting activity, for example, Antwerp and Turnhout areas, marked ’e’ in Fig. \[saltexample\]. Also transport links not affected by contamination became visibly enhanced (Fig. \[saltexample\], ’g’). In the situation of continuing contamination plasmodium ’considers’ the last opportunity to survive by forming sclerotium (hardened body of the ’hibernating’ slime mould), initial stage of the sclerotium formation is labelled ’f’ in Fig. \[saltexample\].
Based on outcomes of our scoping experiments (Figs. \[saltat2\] and \[saltat5\]) we can propose the following scenarios of response to contaminations:
- Epicentre of contamination is at the Tihange Nuclear Power Station near Liège. Segments of transport network connecting Namur, Liège, Hasselt, Genk, Sankt-Vith and Arlon are destroyed or abandoned. Hyperactivity is observed in domains surrounding Sinkt-Niklaas, Antwerp and Turnhout. Preparations for emergency hibernation take place at Aalst, Brussels, Leuven and Mechelen. Transport links between Gent, Roeselare, Kortijk, Tournai, Mons, Charleroi, Namur are hypertrophied (Fig. \[saltat5\]).
- Epicentre of contamination is at the Doel Nuclear Power Plant near Antwerpen. Domains surrounding Antwerp, Sinklaas, Mechelen, Turnhout become depopulated and transport links are abandoned. Explorative activity is observed in areas of Oostende, Brugge, Roeselare, and Arlon and Sankt-Vith. Attempted mass-migration is recorded into Northern France, South of the Netherlands and Luxemburg and West Germany. Transport links in the chain Brugge — Roeselare — Kortijk — Tournai — Mons — Charleroi — Namur — Liège — Sankt-Vith/Arlon are significantly hypertrophied (Fig. \[saltat2\]). Preparations for emergency hibernation take place at Brussels and Leuven, and at Hasselt and Genk areas.
Conclusions
===========
To evaluate how good Belgian motorways are from an amorphous living creature point of view, we conducted the following laboratory experiments with slime mould *P. polycephalum*. We represented major urban areas with oat flakes and inoculated slime mould in the oat flake corresponding to Brussels. We waited till the slime mould colonised all oat flakes and then analysed the slime mould’s protoplasmic network structure and compared it with the man-made motorway network and basic planar proximity graphs. We found that *P. polycephalum* almost but not completely approximates the Belgian motorway network. Transport links Roeselare — Kortrijk — Mons — Charleroi — Namur, Oostende — Brugge — Gent — Aalst — Brussels — Leuven — Mechelen — Antwerp — Sint-Niklaas, Liège — Sankt-Vith — Arlon, and Hasselt — Genk appear as protoplasmic tubes in almost all laboratory experiments. Motorway links connecting Brussels with Antwerp, Tournai, Mons, Charleroi, Namur, and links connecting Leuven with Liège and Antwerp with Genk and Turnhout are “considered” by slime mould to be redundant and thus almost never appear in our experiments with the slime mould. If slime mould represented A7/A54 Brussels — Charleroi, non-existing in reality, motorways between Leuven — Namur and Turnhout — Hasselt its credible transport network would be a super-graph of the relative neighbourhood graph.
Motorway segments A17 (Antwerp — Sint-Niklaas), A19 (Antwerp — Mechelen), E42 (Liège — Sankt-Vith), A10/E40 (Oostende — Gent — Aalst — Brussels — Leuven), A17/E403 (Roeselare — Kortrijk — Tournai), E42/E19 (Tournai — Mons), E42 (Mons — Charleroi — Namur) are essential transport links from the slime mould’s point of view. If the two parts of Belgium were separated with Brussels in Flanders, the Walloon region of the Belgian transport network would be represented by a single chain from Tournai in the north-west to Liège area in the north-east and down to southmost Arlon.
While imitating major disasters leading to contamination propagating from Liège and Antwerp areas we found several scenarios of major restructuring of transport networks and possible routes of mass-migration.
We believe the results of our scoping experiments and analysis can be used not only in bio-inspired unconventional computing but also in more classical fields of urban and transport planning. Possible applications include but are not limited to restructuring of rural landscape [@froment:1987], novel approaches towards mapping vehicular accessibility [@Vandenbulcke:2009], simulation of traffic dynamics in Belgian motorways [@Boel:2006], development of Trans-European transport networks [@De; @Lathauwer:1995], bottom up landscape planning [@Sevenant:2010] and modelling relations between urban sprawl and growing transport networks [@Poelmans:2009].
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<http://wegen-routes.be/hist/histn.html>
Appendix A: Physarum graphs for all values of $\theta$
======================================================
$\frac{\includegraphics[width=0.23\textwidth]{figs/PhysarumGraphs/physarum_01}}{\theta=\frac{1}{28}}$ $\frac{\includegraphics[width=0.23\textwidth]{figs/PhysarumGraphs/physarum_02}}{\theta=\frac{2}{28}}$ $\frac{\includegraphics[width=0.23\textwidth]{figs/PhysarumGraphs/physarum_03}}{\theta=\frac{3}{28}}$ $\frac{\includegraphics[width=0.23\textwidth]{figs/PhysarumGraphs/physarum_04}}{\theta=\frac{4}{28}}$ $\frac{\includegraphics[width=0.23\textwidth]{figs/PhysarumGraphs/physarum_05}}{\theta=\frac{5}{28}}$ $\frac{\includegraphics[width=0.23\textwidth]{figs/PhysarumGraphs/physarum_06}}{\theta=\frac{6}{28}}$ $\frac{\includegraphics[width=0.23\textwidth]{figs/PhysarumGraphs/physarum_07}}{\theta=\frac{7}{28}}$ $\frac{\includegraphics[width=0.23\textwidth]{figs/PhysarumGraphs/physarum_08}}{\theta=\frac{8}{28}}$ $\frac{\includegraphics[width=0.23\textwidth]{figs/PhysarumGraphs/physarum_09}}{\theta=\frac{9}{28}}$ $\frac{\includegraphics[width=0.23\textwidth]{figs/PhysarumGraphs/physarum_10}}{\theta=\frac{10}{28}}$ $\frac{\includegraphics[width=0.23\textwidth]{figs/PhysarumGraphs/physarum_11}}{\theta=\frac{11}{28}}$ $\frac{\includegraphics[width=0.23\textwidth]{figs/PhysarumGraphs/physarum_12}}{\theta=\frac{12}{28}}$ $\frac{\includegraphics[width=0.23\textwidth]{figs/PhysarumGraphs/physarum_13}}{\theta=\frac{13}{28}}$ $\frac{\includegraphics[width=0.23\textwidth]{figs/PhysarumGraphs/physarum_14}}{\theta=\frac{14}{28}}$ $\frac{\includegraphics[width=0.23\textwidth]{figs/PhysarumGraphs/physarum_15}}{\theta=\frac{15}{28}}$ $\frac{\includegraphics[width=0.23\textwidth]{figs/PhysarumGraphs/physarum_16}}{\theta=\frac{16}{28}}$ $\frac{\includegraphics[width=0.23\textwidth]{figs/PhysarumGraphs/physarum_17}}{\theta=\frac{17}{28}}$ $\frac{\includegraphics[width=0.23\textwidth]{figs/PhysarumGraphs/physarum_18}}{\theta=\frac{18}{28}}$ $\frac{\includegraphics[width=0.23\textwidth]{figs/PhysarumGraphs/physarum_19}}{\theta=\frac{19}{28}}$ $\frac{\includegraphics[width=0.23\textwidth]{figs/PhysarumGraphs/physarum_20}}{\theta=\frac{20}{28}}$ $\frac{\includegraphics[width=0.23\textwidth]{figs/PhysarumGraphs/physarum_21}}{\theta=\frac{21}{28}}$ $\frac{\includegraphics[width=0.23\textwidth]{figs/PhysarumGraphs/physarum_22}}{\theta=\frac{22}{28}}$ $\frac{\includegraphics[width=0.23\textwidth]{figs/PhysarumGraphs/physarum_23}}{\theta=\frac{23}{28}}$ $\frac{\includegraphics[width=0.23\textwidth]{figs/PhysarumGraphs/physarum_24}}{\theta=\frac{24}{28}}$ $\frac{\includegraphics[width=0.23\textwidth]{figs/PhysarumGraphs/physarum_25}}{\theta=\frac{25}{28}}$ $\frac{\includegraphics[width=0.23\textwidth]{figs/PhysarumGraphs/physarum_26}}{\theta=\frac{26}{28}}$ $\frac{\includegraphics[width=0.23\textwidth]{figs/PhysarumGraphs/physarum_27}}{\theta=\frac{27}{28}}$ $\frac{\includegraphics[width=0.23\textwidth]{figs/PhysarumGraphs/physarum_28}}{\theta=\frac{28}{28}}$
[^1]: Arlon and Sankt-Vith are not amongst the most populated areas but we added them for completeness, to allows the slime mould propagating towards Luxembourg and Germany
[^2]: A planar graph consists of nodes which are points of the Euclidean plane and edges which are straight segments connecting the points.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'The quantum mechanical analysis of the canonical hamiltonian description of the effective action of a SD$p$-brane in bosonic ten dimensional Type II supergravity in a homogeneous background is given. We find exact solutions for the corresponding quantum theory by solving the Wheeler-deWitt equation in the late-time limit of the rolling tachyon. The probability densities for several values of $p$ are shown and their possible interpretation is discussed. In the process the effects of electromagnetic fields are also incorporated and it is shown that in this case the interpretation of tachyon regarded as “matter clock” is modified.'
author:
- 'H. García-Compeán'
- 'G. García-Jiménez'
- 'O. Obregón'
- 'C. Ramírez'
title: 'Towards SD$p-$brane quantization'
---
-1truecm
-1.3truecm
Introduction
============
In recent years a great deal of attention has been paid to open string tachyon states, which arise in unstable D$p$-branes or brane-antibrane systems. These tachyon states have a symmetric potential $V(T)$, with a central maximum and two symmetric minima, and to it D$(p-1)$ branes are associated, which arise as a kink interpolating states between these minima. If the boundary conditions on the tachyon are spacelike, then usual D$(p-1)$-branes arise (for a review, see [@sennonbps]). However if one of these conditions is timelike, then the tachyon rolls down and time-dependent, spacelike SD$(p-1)$-branes arise [@gs]. These branes are localized in time, i.e. they exist for a short time and, due to the coupling of the tachyon with Ramond-Ramond (RR) fields, they carry the same type of charge as D-branes [@gs]. Moreover, the study of the gravitational backreaction of the tachyon matter has been done.
As soon as the tachyon field rolls down from the top of $V(T)$ towards one of its minima, it starts to excite open and closed string modes in such a way that the energy of the unstable D-brane is radiated away. When the tachyon arrives to its minimum, the radiation is in the form of only closed strings because open strings cannot exist in the bulk. This has been computed explicitly, see [@malda; @rastelli] and references therein. Actually in this context, a *dual* correspondence between open and closed string modes has been conjectured, which can be very helpful in the computation of the effects when tachyon condensates [@conjetura]. Such a conjecture states that the tree level open string theory provides a description of the rolling tachyon system in terms of the closed string emission [@conjetura]. Moreover this conjecture can be generalized to include quantum corrections and the full tachyon dynamics [@senconj].
On the other hand, based in previous work [@tdynamics], Sen proposed a field theory describing the dynamics of the rolling tachyon [@senthree; @senfour; @senfive]. In this context, he found that the tachyon field can be interpreted as the [*time*]{} in quantum cosmology [@sentime]. This was done by coupling the “tachyon matter” to a gravitational field and then performing its canonical quantization. From it, a Wheeler-deWitt equation turns out, which can be regarded as a time-dependent Schrödinger equation for this gravity-tachyon matter system. The coupling of the tachyon to gravity has been studied in connection with classical cosmological evolution [@gibb; @fquevedo; @revsen]. In particular its role related to inflation has been discussed, see [@revgibb] and references therein.
The classical solutions to supergravity including S-branes have been worked out in some cases, see e.g. [@galtsov; @martin; @fernando]. In Ref. [@buchelone] solutions of the Einstein-Maxwell effective description, in four dimensions, of the rolling tachyon of the S0-brane proposed in Ref. [@gs], have been found.
Further, in [@peettwo; @bucheltwo; @peetone] the bosonic sector of the effective ten dimensional supergravity action, coupled to tachyonic matter, under a maximal symmetric ansatz $ISO(p+1)\times SO(8-p,1)$, has been considered. There, the time-dependent models were extensively studied and some classical solutions to supergravity with SD-branes have been worked out. For recent developments in this direction see Ref. [@leblond],
In the present work, we will consider the canonical quantization of the above mentioned effective action. In quantizing the classical field theory in [@peettwo; @bucheltwo; @peetone], we do not expect to describe rigorously quantum aspects of string theory. Nevertheless, the quantum properties of the considered field theory seem to be an interesting problem by itself, as already pointed out by Sen in Ref. [@sentime], where he considers a quantum cosmology model coupled to the tachyon matter. The SD$p-$brane model [@peettwo; @bucheltwo; @peetone] we are going to consider can also be understood as cosmology with dilaton and RR fields, driven by the tachyon matter. We show that the proposal by Sen, concerning the interpretation of the tachyon as time, in the late ‘time’ decoupling limit, is valid for the model under consideration. We find an exact wave function, finite and continuous everywhere for the corresponding Schrödinger equation. The associated probability density shows an infinity of continuous degenerated maxima describing a path in minisuperspace. Its behavior with respect to some interesting values of $p$ of the SD$p$-brane is also shown. Moreover, we will show that even for the next order approximation from the late-time decoupling limit (still with $T$ large but with nonvanishing $V(T)$ and $f(T)$, see Ref. [@senfour; @senfive] and Sec. 2 in Ref. [@sentime]), the RR coupling allows an interpretation of the tachyon as time, in this case with a Schrödinger equation with a time-dependent potential.
It should be remarked however that in the presence of a uniform electric field, the interpretation of the tachyon as time seems to be spoiled. In the late-time limit, the tachyon does not decouple from the electric field. This electric field has been considered, for example in connection with what has been called the carrollian confinement mechanism for open string states [@yi; @revgibb].
This paper is organized as follows: in section 2 we briefly discuss the model proposed in Refs. [@peettwo; @bucheltwo] for the effective action of a SD$p$-brane. In section 3 we find the hamiltonian constraint for the SD-brane. Section 4 is devoted to the study of quantum solutions with the rolling tachyon approximation in the decoupling limit. We also comment about a possible extension of the interpretation of tachyon as time to a first approximation around the limit at $T\rightarrow\infty$. In section 5, we include electric and magnetic fields and exhibit the relevant part of the Hamiltonian in the late-time limit. Our conclusions are finally presented in section 6.
The SD$p$-brane Action
======================
The case we analyze here is that of the low energy effective action of the closed string interaction with the rolling tachyon matter. This can be done by means of an action $S_{brane}$ given by the Dirac-Born-Infeld action of the tachyon plus a Wess-Zumino term describing its coupling to the RR fields. To this, the action $S_{bulk}$ of the background ten dimensional supergravity is added, from which we will consider only the bosonic sector. The action proposed in [@peettwo; @bucheltwo] for this theory is: $$\begin{aligned}
S & =S_{bulk}+S_{brane},\\
S_{bulk} & =\frac{1}{16\pi G_{10}}\int d^{10}x\sqrt{-g}\left( R-\frac{1}
{2}(\partial\phi)^{2}-\frac{e^{a\phi}}{2(p+2)!}F_{p+2}^{2}\right) ,\\
S_{brane} & =\frac{\Lambda}{16\pi G_{10}}\int d^{p+2}x_{\parallel}%
\widehat{\varrho}_{\perp}\left( -V(T)e^{-\phi}\sqrt{-{\cal A}}\text{ }\right)
+\frac{\Lambda}{16\pi G_{10}}\int\widehat{\varrho}_{\perp}\mathcal{F(}%
T\mathcal{)}dT\wedge C_{p+1},\label{accion}%\end{aligned}$$ where $G_{10}$ is the Newton’s constant in the ten-dimensional theory, $a\equiv(3-p)/2$ is the dilaton coupling, ${\cal A}=\det {\cal A}_{\alpha\beta},$ ${\cal A}_{\alpha\beta}=g_{\alpha\beta}e^{\phi/2}+\partial_{\alpha}T\partial_{\beta
}T$ is the tachyon metric, $\mathcal{F(}T\mathcal{)}$ is the factor of coupling between the tachyon and the RR fields $C_{p+1}$, and $V(T)$ is the tachyon potential. $\widehat{\varrho}_{\perp}$ is the “density of branes”, which does not depend on the parallel coordinates of the brane $x_{\parallel
\text{ }}.$ Greek indices $\alpha,\beta=0,1,\dots,p+1,$ label the time and parallel coordinates (denoted by $\parallel$) to the SD$p$-brane. Latin indices $i,j=1,..,8-p$ label the perpendicular coordinates of the brane (denoted by $\perp$), and capital letters $A,B, \dots,$ etc. stand for space-time coordinates of the bulk.
Following Refs. [@peettwo; @bucheltwo], the simplest model that we can study is assuming the homogeneous (but non-isotropic) FRW metric. Making the space decomposition into maximal symmetric direct product $ISO(p+1)\times SO(8-p,1)$ we have the metric $$ds^{2}=-N^{2}(t)dt^{2}+a_{\parallel}^{2}(t)dx_{\parallel}^{2}+a_{\perp}%
^{2}(t)dx_{\perp}^{2}, \label{hmetric}%$$ where $a_{\parallel}(t)$ and $a_{\perp}(t)$ are the parallel and perpendicular scaling factors of the brane and $N(t)$ is the lapse function. In [@buchelone; @bucheltwo] it was noticed that the SD$p$-brane is not suitable to be localized by means of a delta function, because it could break at short scales the $R$-symmetry present in $SO(8-p,1)$. In order to preserve this symmetry, it was proposed to “smear out” the localization of the brane by a homogeneous distribution along $x_{\perp}.$ Thus the density $\widehat
{\varrho}_{\perp}$ is given by $$\widehat{\varrho}_{\perp}=\rho_{\perp}d^{8-p}x_{\perp},$$ where $\rho_{\perp}=\rho_{0}\sqrt{g_{H_{8-p}}}=\rho_{0}a_{\perp}^{8-p}$, $\rho_{0}$ is a constant and $g_{H_{8-p}}$ is the determinant of the metric of the hyperbolic space perpendicular to the brane. The $\left( p+2\right)
-$form field strength $F_{p+2}$ is given in terms of the $\left( p+1\right)
-$form RR potential $C_{p+1}$, which is chosen in a gauge in which the only nonvanishing component is $C_{12\cdots p+1}=C(t),$ $$F_{p+2}^{2}=-N^{-2}\dot{C}_{p+1}^{2}=-N^{-2}\dot{C}^{2}.$$ In order to preserve homogeneity, the tachyon field is function only of time $T=T(t)$. Hence the tachyon couples to RR fields in the following form$$dT\wedge C_{p+1}=\dot{T}Cd^{p+2}x_{\parallel}.$$ In order to simplify the Lagrangian we can introduce the coordinates $\beta_{1},\beta_{2}$ defined as, $$\beta_{1}=\frac{1}{9}\left[ (p+1)\beta_{\parallel}+(8-p)\beta_{\perp}\right],$$$$\beta_{2}=\beta_{\parallel}-\beta_{\perp},$$ where $\beta_{\parallel}=\ln a_{\parallel}$ and $\beta_{\perp}=\ln a_{\perp}.$ Also the space volume is given by $V_{S}=\frac{1}{16\pi G_{10}}\int
d^{p+1}x_{\parallel}d^{8-p}x_{\perp}$, so $S=\int d^{10}x\mathcal{L},$ with $\mathcal{L=}V_{S}\int dt\ L.$ In these coordinates and with the ansatz (\[hmetric\]) we have the Lagrangian $$\begin{aligned}
L & =-\frac{e^{9\beta_{1}}}{N}\left[ 72\dot{\beta}_{1}^{2}-\frac
{(p+1)(8-p)}{9}\dot{\beta}_{2}^{2}-\frac{1}{2}\dot{\phi}^{2}-\frac{e^{a\phi}%
}{2(p+2)!}\dot{C}^{2}\right] -\lambda e^{9\beta_{1}-a\phi/2}V(T)\sqrt
{N^{2}e^{\phi/2}-\dot{T}^{2}}\nonumber\\
& +\lambda e^{(8-p)\left[ \beta_{1}-\frac{1}{9}(p+1)\beta_{2}\right]
}\mathcal{F(}T\mathcal{)}\dot{T}C. \label{lagrangiano}%\end{aligned}$$ In order to manage the square root part of the Lagrangian, we introduce a Lagrange multiplier $\Omega$ [@tseytlin] into the Lagrangian (\[lagrangiano\]) as follows, $$\begin{aligned}
L & =-\frac{e^{9\beta_{1}}}{N}\left[ 72\dot{\beta}_{1}^{2}-\frac
{(p+1)(8-p)}{9}\dot{\beta}_{2}^{2}-\frac{1}{2}\dot{\phi}^{2}-\frac{e^{a\phi}%
}{2(p+2)!}\dot{C}^{2}\right] -\frac{1}{2}\Omega^{-1}\left( N^{2}e^{\phi
/2}-\dot{T}^{2}\right) \nonumber\\
& -\frac{1}{2}\lambda^{2}e^{18\beta_{1}-a\phi}V^{2}(T)\Omega+\lambda
e^{(8-p)\left[ \beta_{1}-\frac{1}{9}(p+1)\beta_{2}\right] }\mathcal{F}%
(T)\dot{T}C,\label{lagrados}%\end{aligned}$$ where $\lambda=\Lambda\rho_{0}.$ As usual, variating this action with respect to $\Omega$, ${\frac{\partial L}{\partial\Omega}}=0$, and substituting $\Omega$ from it into Lagrangian (\[lagrados\]) the Lagrangian (\[lagrangiano\]) follows.
The SD$p$-brane Hamiltonian
===========================
In this section we discuss the canonical hamiltonian formalism of the Lagrangian (\[lagrados\]). The resulting hamiltonian constraint will be used in the next section to give the corresponding Wheeler-deWitt equation. The canonical momenta obtained from the Lagrangian (\[lagrados\]) are given by,$$P_{1}=\frac{\partial L}{\partial\dot{\beta}_{1}}=-\frac{144}{N}e^{9\beta_{1}%
}\dot{\beta}_{1},$$$$P_{2}=\frac{\partial L}{\partial\dot{\beta}_{2}}=\frac{2}{9}\frac
{(p+1)(8-p)}{N}e^{9\beta_{1}}\dot{\beta}_{2},$$$$P_{\phi}=\frac{\partial L}{\partial\dot{\phi}}=\frac{e^{9\beta_{1}}}{N}%
\dot{\phi},$$$$P_{C}=\frac{\partial L}{\partial\dot{C}}=\frac{e^{9\beta_{1}+a\phi}}%
{N(p+2)!}\dot{C},$$ $$P_{T}=\frac{\partial L}{\partial\dot{T}}=\Omega^{-1}\dot{T}+\lambda
e^{(8-p)\left[ \beta_{1}-\frac{1}{9}(p+1)\beta_{2}\right] }\mathcal{F(}%
T\mathcal{)}C.$$
With the constraints $P_{\Omega}=P_{N}=0$ implemented, the Hamiltonian is given by $$H=\dot{\beta}_{1}P_{1}+\dot{\beta}_{2}P_{2}+\dot{\phi}P_{\phi}+\dot{C}%
P_{C}+\dot{T}P_{T}-L$$ $$\begin{aligned}
& =\frac{N}{2}\left\{ -\frac{1}{144}e^{-9\beta_{1}}P_{1}^{2}+\frac
{9e^{-9\beta_{1}}}{2(p+1)(8-p)}P_{2}^{2}+e^{-9\beta_{1}}P_{\phi}%
^{2}+(p+2)!e^{-(9\beta_{1}+a\phi)}P_{C}^{2}\right\} \nonumber\\
& +\frac{\lambda^{2}}{2}V^{2}(T)e^{18\beta_{1}-a\phi}\Omega+\frac{N^{2}%
\Omega^{-1}}{2}e^{\phi/2}+\frac{\Omega}{2}\bigg[P_{T}-\lambda e^{(8-p)\left[
\beta_{1}-\frac{1}{9}(p+1)\beta_{2}\right] }\mathcal{F(}T\mathcal{)}%
C\bigg]^{2}.\end{aligned}$$ After elimination of $\Omega$ by its equation of motion $\partial
H/\partial\Omega=0,$ the Hamiltonian gets the form $H=NH_{0}$, where, $$\begin{aligned}
H_{0} & =-\frac{1}{144}e^{-9\beta_{1}}P_{1}^{2}+\frac{9e^{-9\beta_{1}}%
}{2(p+1)(8-p)}P_{2}^{2}+e^{-9\beta_{1}}P_{\phi}^{2}+(p+2)!e^{-(9\beta
_{1}+a\phi)}P_{C}^{2}\nonumber\\
& +2e^{\phi/4}\left\{ \lambda^{2}V^{2}(T)e^{18\beta_{1}-a\phi}%
+[P_{T}-\lambda e^{(8-p)\left[ \beta_{1}-\frac{1}{9}(p+1)\beta_{2}\right]
}\mathcal{F(}T\mathcal{)}C]^{2}\right\} ^{1/2}=0. \label{hamiltonn}%\end{aligned}$$ is the hamiltonian constraint.
It is worth to notice that when this constraint is applied at the quantum level, the resulting Wheeler-deWitt equation does not provide a time evolution of the system, and the corresponding wave function is not normalizable. This is known as the “time problem” [@kuchar].
Canonical Quantization
======================
Exact expressions for the potential $V(T)$ and the coupling factor $\mathcal{F(}T\mathcal{)}$ are not known. However, their asymptotic form $V(T)=e^{-\alpha\left\vert T\right\vert /2}$ and $\mathcal{F(}T\mathcal{)=}%
{\rm sign}(T)e^{-\alpha\left\vert T\right\vert /2}$ as $\left\vert T\right\vert
\rightarrow\infty,$ is known from string theory [@senthree; @senfour; @senfive; @sentime]. Thus we only assume that $V(T)$ has a maximum at $T=0$ and a minimum at $\left\vert
T\right\vert \rightarrow\infty$, where $V(T)=0$. Also, we see that in this limit the tachyon decouples also from the RR fields as $\mathcal{F}(T)\rightarrow0$. The canonical hamiltonian (\[hamiltonn\]) takes in this limit the form $$H_{0}=-\frac{1}{144}e^{-9\beta_{1}}P_{1}^{2}+\frac{9}{2}\frac{e^{-9\beta_{1}}%
}{(p+1)(8-p)}P_{2}^{2}+e^{-9\beta_{1}}P_{\phi}^{2}+(p+2)!e^{-\left(
9\beta_{1}+a\phi\right) }P_{C}^{2}+2eç^{\phi/4}P_{T}=0. \label{hamilzero}%$$ The resulting equation is the Wheeler-deWitt equation, $$\widehat{H}_{0}\Psi=0, \label{qconstra}%$$ where $\widehat{H}_{0}$ is given by (\[hamilzero\]), with $P_{1}%
=-i\frac{\partial}{\partial\beta_{1}},P_{2}=-i\frac{\partial}{\partial
\beta_{2}},P_{C}=-i\frac{\partial}{\partial C}$ and $P_{T}=-i\frac{\partial
}{\partial T}.$ Assuming that the dilaton field is given by its vacuum expectation value, i.e. $g_{s}=e^{\left\langle \phi\right\rangle }$, where $g_s$ is the string coupling constant, then $P_{\phi}=0$, and we have (with a particular factor ordering), $$e^{-9\beta_{1}}\left[ C_{1}\frac{\partial^{2}\Psi}{\partial\beta_{1}^{2}%
}-C_{2}\frac{\partial^{2}\Psi}{\partial\beta_{2}^{2}}-C_{3}\frac{\partial
^{2}\Psi}{\partial C^{2}}\right] =iC_{4}\frac{\partial\Psi}{\partial T},
\label{wdw}%$$ where $C_{1}=\frac{1}{144},C_{2}=\frac{9}{2(p+1)(8-p)},C_{3}=(p+2)!g_{s}%
^{-a},C_{4}=2g_{s}^{1/4}.$ Now, we see that the Wheeler-deWitt equation (\[qconstra\]) leads to a Schrödinger-like equation.
Thus in this limit, the tachyon is a scalar field which provides a useful parametrization of time, because the tachyon momentum enters linearly in (\[hamilzero\]). This can be interpreted as a “matter clock”[^1]. In string theory, the corresponding low energy effective action contains the action of the brane, in which the tachyon arises. This matter accompanies gravitation ($S_{bulk}$) in a natural and consistent manner. On the other hand, as mentioned, the tachyon momentum appears linearly in (\[wdw\]). So it seems that at least some of the criticisms and problems related to a “matter clock” can be in this case avoided. Moreover, Sen [@senfive] showed that for large values of time $x_{0}$, the classical tachyon solution goes as $T\simeq x_{0}+\mathcal{O}(e^{-\alpha x_{0}%
})$ thus, this result provide us another way to recognize $T$ as a time.
The solution of (\[wdw\]) is straightforward. Assuming separation of variables for $\Psi$ is of the form: $\Psi=$ $\psi_{\beta_{1}}(\beta_{1}%
)\psi_{\beta_{2}}(\beta_{2})\psi_{T}(T)\psi_{C}(C)$ we can rewrite the equation (\[wdw\]) as $$e^{-9\beta_{1}}\left[ C_{1}\frac{\psi_{\beta_{1}}^{\prime\prime}}{\psi
_{\beta_{1}}}-C_{2}\frac{\psi_{\beta_{2}}^{\prime\prime}}{\psi_{\beta_{2}}%
}-C_{3}\frac{\psi_{C}^{\prime\prime}}{\psi_{C}}\right] =iC_{4}\frac{\psi
_{T}^{\prime}}{\psi_{T}}=-\mu,$$ where $\mu$ is a separation constant, which we take to be real. Thus, the tachyon wave function is given by $$\psi_{T}(T)=e^{i\left( \mu/C_{4}\right) T}.$$ Similarly, we find for the other field components $$\begin{aligned}
\psi_{\beta_{2}}(\beta_{2}) & =e^{\pm i\sqrt{\frac{\sigma}{C_{2}}}\beta_{2}%
},\\
\psi_{C}(C) & =e^{\pm i\sqrt{\frac{\xi}{C_{3}}}C},\end{aligned}$$ with $\lambda=\xi+\sigma\geq0$. Thus the remaining equation is given by $$C_{1}\frac{\psi_{\beta_{1}}^{\prime\prime}}{\psi_{\beta_{1}}}+\mu
e^{9\beta_{1}}+\lambda=0.$$ This equation has as solution the modified Bessel function $$\psi_{\beta_{1}}(\beta_{1})=K_{i\nu}\bigg(\frac{8}{3}\sqrt{\mu}e^{\frac{9}{2}
\beta_{1}}\bigg),$$ where $\nu=\frac{2}{9}\sqrt{\frac{\lambda}{C_{1}}}.$ The general solutions are then, $$\Psi^{\pm}=\mathcal{N}e^{i\left( \mu/C_{4}\right) T}e^{\pm i\sqrt{\frac{\xi
}{C_{3}}}C}e^{\pm i\sqrt{\frac{\sigma}{C_{2}}}\beta_{2}}K_{i\nu}\bigg(\frac
{8}{3}\sqrt{\mu}e^{\frac{9}{2}\beta_{1}}\bigg),\label{solgen}%$$ where $\mathcal{N}$ is a normalization constant. This is a plane wave, that represents a free particle, with respect to the variables $C$ and $\beta_{2}$ and with $T$ playing the role of time. In terms of the radii $a_{\parallel}$ and $a_{\perp}$ we have, $$\Psi^{\pm}=\mathcal{N}e^{i(\mu g_{s}^{-1/4})\,T}e^{\pm i\sqrt{\xi/(p+2)!}%
C}\left( \frac{a_{\parallel}}{a_{\perp}}\right) ^{\pm\frac{i}{3}%
\sqrt{2(p+1)(8-p)\sigma}}K_{i\nu}\left( \frac{8}{3}\sqrt{\mu a_{\parallel
}^{p+1}a_{\perp}^{8-p}}\right) .$$ If we compute the expectation value of $a_{\parallel}$, for a certain constant value of $a_{\perp}$, we have $$\langle a_{\parallel}\rangle=\mathcal{N}\int_{0}^{\infty}\Phi^{\ast
}a_{\parallel}\Phi da_{\parallel}=\mathcal{N}\int_{0}^{\infty}\left[ K_{i\nu
}\left( \frac{8}{3}\sqrt{\mu a_{\parallel}^{p+1}a_{\perp}^{8-p}}\right)
\right] ^{2}a_{\parallel}da_{\parallel}.$$ From which we get $$\langle a_{\parallel}\rangle=\mathcal{N}\sqrt{\pi}\frac{\Gamma\left( \frac
{2}{p+1}\right) \Gamma\left( \frac{2}{p+1}+i\nu\right) \Gamma\left(
\frac{2}{p+1}-i\nu\right) }{(3-p)\Gamma\left( \frac{3-p}{2(p+1)}\right)
}\left( \frac{9}{64\mu\langle a_{\perp}\rangle^{8-p}}\right) ^{\frac{2}%
{p+1}}. \label{apar}%$$ This relation can also be written as a sort of uncertainty relation between the two radii $\langle a_{\parallel}\rangle\sim\langle a_{\perp}%
\rangle^{-2\frac{8-p}{p+1}}$, where the proportionality factor is, for $\nu=1$, of the order of $10^{-2}$ $\mathcal{N}$ and decreases exponentially as $\nu$ increases. Note that the denominator in (\[apar\]) does not diverge at $p=3$ due to the properties of the Gamma function, in fact $(3-p)\Gamma\left(
\frac{3-p}{2(p+1)}\right) =2(p+1)\Gamma\left( \frac{5+p}{2(p+1)}\right) $.
In Figure 1, we plotted the probability density $|\Psi|^2$ for the physical (‘realistic’) case of $p=3$, where it is shown a continuum of maxima in the $a_{\perp}-a_{\parallel}$ plane, following a path in minisuperspace and showing an inverse relation between the two radii given by Eq. (\[apar\]). Figure 2 and Figure 3 show two extreme cases with $p=1$ and $p=8$, respectively. These figures also shown that, for $p=1$, the probability density is almost projected on the $a_{\perp}$-axis, that is, for large values of $a_{\parallel}$, it is almost independent on it. Figure 3, is the case for $p=8$ and it shows that the probability density $|\Psi|^2$ is independent on $a_{\perp}$ and therefore is projected on the $a_{\parallel}$-axis. All these figures are plotted for the specific values of the parameters given by $\mu=0.1$ and $\nu = 0.7$.
![The figure shows the probability density $|\Psi|^2$ For the ’physical’ case of a SD3-brane ($p=3$) (with $\mu = 0.1$ and $\nu =0.7$) and its variation with respect to the radii $a_{\perp}$ and $a_{\parallel}$. The maxima of the quantum solution $\Psi$ determines a trajectory in the $a_{\perp}-a_{\parallel}$ plane. These maxima satisfy an inverse relation between both radii as shown in Eq. (27).[]{data-label="wavef3"}](f3.eps "fig:"){width="10"}\
![The probability density $|\Psi|^2$ (also with $\mu = 0.1$ and $\nu =0.7$) for one extreme case with $p=1$. The solution shows that for this case, the maxima of $|\Psi|^2$ determines an evolution which is almost projected on the axis $a_{\perp}$, that is, for large values of $a_{\perp}$ it is almost independent on $a_{\parallel}$.[]{data-label="wavef1"}](f1.eps "fig:"){width="10"}\
![The figure corresponds with the other extreme case with $p=8$ which shows that the probability density $|\Psi|^2$ is independent on $a_{\perp}$. This corresponds with a wave function $\Psi$ describing an evolution whose maxima are localized around fixed values of $a_{\parallel}$.[]{data-label="wavef8"}](f8.eps "fig:"){width="10"}\
Let us now consider the next leading order of the approximation of the hamiltonian (\[hamiltonn\]), in which $V^{2}(T)$ is neglected with respect to the $V(T)$ or $\mathcal{F}(T)$. As we mentioned in the introduction, this approximation corresponds to the first order correction from the late-time decoupling limit with $T$ still large but nonvanishing $V(T)$ and $f(T)$. This configuration was considered previously by Sen in Refs. [@senfour; @senfive; @sentime]. In this case we have, $$\begin{aligned}
H_{0} & =2e^{\phi/4}P_{T}-\frac{1}{144}e^{-9\beta_{1}}P_{1}^{2}%
+\frac{9e^{-9\beta_{1}}}{2(p+1)(8-p)}P_{2}^{2}+e^{-9\beta_{1}}P_{\phi}%
^{2}+(p+2)!e^{-(9\beta_{1}+a\phi)}P_{C}^{2}\nonumber\\
& -2\lambda e^{\phi /4} e^{(8-p)\left[ \beta_{1}-\frac{1}{9}(p+1)\beta_{2}\right]
-\frac{\alpha T}{2}}C=0. \label{hamiltonnt}%\end{aligned}$$ After quantization, we obtain from this hamiltonian again a Schrödinger equation, now with a time dependent potential for the RR field. It is interesting to note that in this case, the term coming from the RR coupling still allows to interpret the tachyon field as time, because its moment still appears linearly, but in this case Eq. (\[hamiltonnt\]) leads to a Schrödinger equation with a time-dependent potential. Of course in the absence of RR and dilaton fields, the tachyon field is coupled only to gravity and we recover the situation discussed by Sen in Refs. [@senfour; @senfive; @sentime].
One way to solve equation (\[hamiltonnt\]), could be by traying the time-dependent term as a perturbation. In this case we could look for a solution of the form, $$\Psi(T,\beta_1,\beta_2,C)=\psi(T,\beta_1,\beta_2,C)+
e^{-\frac{\alpha T}{2}+\frac{i}{2}\mu g_S^{1/4}T}\psi_1(\beta_1,\beta_2,C).
\label{pert}$$ However, when substituted into (\[hamiltonnt\]), it gives an equation for $\psi_1$ too complicated for an exact solution. The probability density obtained from (\[pert\]) $|\Psi|^2\simeq|\psi|^2+2{\rm Re}[\,e^{-\frac{1}{2}(\alpha-i\mu g_S^{1/4})T}\overline\psi\psi_1]$, contains time dependent interference terms corresponding to interactions of the tachyon matter (open strings) with background fields (closed strings). This interference represents a manifestation of the quantum backreaction of the tachyon field by the background.
Inclusion of electromagnetic fields
===================================
We want to see in this section how the tachyon dynamics is modified in the presence of electromagnetic fields. Let us consider the case in which electric and magnetic fields $f_{\alpha\beta}$ are included. The brane action $S_{Brane}$ from Eq. (\[accion\]) is modified as follows [@senmuk; @roy; @rey],
$$S_{Brane}=\frac{\Lambda}{16\pi G_{10}}\int d^{p+2}x_{\parallel}\widehat
{\varrho}_{\perp}\left( -V(T)e^{-\phi}\sqrt{-{\cal A}}\text{ }\right)
+\frac{\Lambda}{16\pi G_{10}}\int\widehat{\varrho}_{\perp}\mathcal{F(}%
T\mathcal{)}dT\wedge C_{p+1}\wedge e^{f},\label{accionb}%$$
where now the tachyon metric is ${\cal A}_{\alpha\beta}=g_{\alpha\beta}e^{\phi
/2}+\partial_{\alpha}T\partial_{\beta}T+f_{\alpha\beta}$ and $f=f_{\alpha
\beta}dx^{\alpha}\wedge dx^{\beta}$. For simplicity, we will consider only one nonvanishing component for the electric and magnetic fields, $E=f_{01}%
=\partial_{0}A_{1}$ and $B=f_{12}=-\partial_{2}A_{1}$. With this choice, the exponential $e^{f}$ in the last term of action (\[accionb\]) contributes only with a factor one. This can be obtained by direct calculation or following [@peetone], taking into account the ansatz $ISO(p+1)\times
SO(8-p,1)$. After integration of the space coordinates, we get the Lagrangian $$\begin{aligned}
& L_{brane}=\lambda e^{(8-p)\left[ \beta_{1}-\frac{1}{9}(p+1)\beta
_{2}\right] }\mathcal{F}(T)\dot{T}C\nonumber\\
& -\lambda\,e^{9\beta_{1}-a\phi/2}\,V(T)\left[ \bigg(N^{2}e^{\phi/2}-\dot{T}%
^{2}\bigg)\bigg(1+e^{-2\left( \beta_{1}+\frac{8-p}{9}\beta_{2}+\phi/4\right) }%
B^{2}\bigg)-E^{2}\right] ^{1/2}.\end{aligned}$$ Making the same procedure of introducing a Lagrange multiplier $\Omega$ we found in the late-time limit ($V(T)\rightarrow0$ as $\left\vert T\right\vert
\rightarrow\infty$) that the relevant part of the hamiltonian turns out to be, $$H_{\substack{brane\\\left\vert T\right\vert \rightarrow\infty}}=2e^{\phi
/4}\left[ \left( 1+e^{-2\left( \beta_{1}+\frac{8-p}{9}\beta_{2}%
+\phi/4\right) }B^{2}\right) P_{T}^{2}+\Pi^{2}\right] ^{1/2},\label{spoil}%$$ where $\Pi$ is the momentum conjugated to $A_{1}$. From this expression, we see that the tachyon would decouple only if $\Pi$ vanishes. Thus, under the presence of electromagnetic fields, the tachyon cannot be identified with time in the sense of a Schrödinger-type equation even in the late-time limit.
Conclusions
===========
In this work, we have provided an exact solution to the canonical quantization of the SD$p$-brane model [@gs; @buchelone; @peettwo; @bucheltwo; @peetone]. For this effective action, a Wheeler-deWitt equation has been obtained from the hamiltonian analysis. Following Ref. [@tseytlin], the square root in the tachyonic matter action (\[lagrangiano\]) was eliminated by the introduction of a Lagrange multiplier $\Omega$. From the resulting action the Hamiltonian (\[hamiltonn\]) has been computed and the decoupling late-time limit ($\left\vert T\right\vert \rightarrow\infty$) has been done. Even though we have considered the canonical quantization of the effective action with a maximally symmetric metric (\[hmetric\]), the quantum version of this field theory and in particular of the model under consideration is interesting on its own right [@sentime]. Moreover, it could provide some insight on string theory beyond the classical limit.
Further we show that the proposal by Sen, concerning the interpretation of the tachyon as time, in the late-time decoupling limit, is valid for this model. In this limit we find an exact wave function for the corresponding Schrödinger equation. The associated probability density is a finite and continuous function of the radii $a_{\parallel}$ and $a_{\perp}$, it shows (a non-singular) continuum of maxima along a definite trajectory, in such a way that if the mean value of one of the radii increases, the mean value of the other one decreases, as shown in Figure 1 for $p=3$ and in Figure 2 and Figure 3 for the extreme cases of $p=1$ and $p=8$, respectively. We have also considered the situation beyond the late-time decoupling limit in which still $T$ is large but $V(T)$ and $f(T)$ are nonvanishing. The coupling of the tachyon with the RR fields allows us still to interpret the tachyon as time. However in this case the Wheeler-deWitt equation (\[hamiltonnt\]) leads to a Schrödinger equation with a time-dependent potential. This situation has been already discussed in Refs. [@senfour; @senfive; @sentime] at the classical level. If quantum corrections of the string theory have to be taken into account and if the open-closed duality holds (see remarks of review, [@revsen]), it would be very interesting to explore if solutions of the Schrödinger equation (\[hamiltonnt\]), or its generalizations (representing open-closed states), correspond to a description (at the lowest level) of the physics of the quantum string theory associated to SD$p$-branes.
Finally, we have also shown that in the presence of electromagnetic fields, the interpretation of the tachyon as time seems to be spoiled. Indeed, as can be seen from Eq. (\[spoil\]) that even in the late-time limit the tachyon does not decouple from the electric field.
2truecm
**Acknowledgments**
This work was supported in part by CONACyT México Grants Nos. 37851E and 41993F, PROMEP and Gto. University Projects.
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A. Sen, “Non-BPS States and Branes in String Theory", Published in Cargese 1999, [*Progress in string theory and M-theory*]{}, pp 187-234, hep-th/9904207. M. Gutperle and A. Strominger, “Spacelike Branes” JHEP [**0204**]{} (2002) 018, hep-th/0202210. N. Lambert, H. Liu and J. Maldacena, hep-th/0303139. D. Gaiotto, N. Itzhaki and L. Rastelli, Nucl. Phys. B [**688**]{} (2004) 70, hep-th/0304192. A. Sen, Phys. Rev. D [**68**]{} (2003) 106003, hep-th/0305011; Phys. Rev. Lett. [**91**]{} (2003) 181601, hep-th/0306137. A. Sen, “Open Closed Duality: Lessons from Matrix Model”, Mod. Phys. Lett. A [**19**]{} (2004) 841, hep-th/0308068. M.R. Garousi, Nucl. Phys. B [**584**]{} (2000) 284; E.A. Bergshoeff, M. de Roo, T.C. de Wit, E. Eyras and S. Panda, JHEP [**0005**]{} (2000) 009; J. Kluson, Phys. Rev. D [**62**]{} (2000) 126003; G.W. Gibbons, K. Hori and P. Yi, Nucl. Phys. B [**596**]{} (2001) 136. A. Sen, “Rolling Tachyon”, JHEP [**0204**]{} (2002) 048, hep-th/0203211. A. Sen, “Tachyon Matter”, JHEP [**0207**]{} (2002) 065, hep-th/0203265. A. Sen, “Field Theory of Tachyon Matter”, Mod. Phys. Lett. A [**17**]{} (2002) 1797, hep-th/0204143. A. Sen, “Time and Tachyon”, Int. J. Mod. Phys. A [**18**]{} (2003) 4869, hep-th/0209122. F. Quevedo, “Lectures on String/Brane Cosmology”, Class. Quant. Grav. [**19**]{} (2002) 5721, hep-th/0210292. A. Sen, “Remarks on Tachyon Driven Cosmology”, Talk at Nobel Symposium on Cosmology and String Theory and IIT Kanpur workshop on String Theory, hep-th/0312153. G.W. Gibbons, “Cosmological Evolution of the Rolling Tachyon”, Phys. Lett. B [**537**]{} (2002) 1, hep-th/0204008. G.W. Gibbons, “Thoughts of Tachyon Cosmology”, Class. Quant. Grav. [**20**]{} (2003) S321, hep-th/0301117. C.-M. Chen, D.V. Galtsov and M. Gutperle, “S-branes Solutions in Supergravity Theories´´, Phys. Rev. D [**66**]{} (2002) 024043, hep-th/0204071. M. Kruczenski, R.C. Myers and A.W. Peet, JHEP [**0205**]{} (2002) 039, hep-th/0204144. F. Quevedo, G. Tasinato and I. Zavala C., “$S$-branes, Negative Tension Branes and Cosmology”, hep-th/0211031. A. Buchel, P. Langfelder and J. Walcher, “Does the Tachyon Matter?”, Annals Phys. [**302**]{} (2002) 78, hep-th/0207235; A. Buchel and J. Walcher, “The Tachyon does Matter”, Fortsch. Phys. [**51**]{} (2003) 885, hep-th/0212150. F. Leblond and A.W. Peet, “SD-brane Gravity Fields and Rolling Tachyons”, JHEP [**0304**]{} (2003) 048, hep-th/0303035. A. Buchel and J. Walcher, “Comments on Supergravity Description of S-branes”, JHEP [**0305**]{} (2003) 069, hep-th/0305055. F. Leblond and A.W. Peet, “A Note on the Singularity Theorem for Supergravity SD-branes”, JHEP [**0404**]{} (2004) 022, hep-th/0305059. F. Leblond, “Mirage Resolution of Cosmological Singularities”, hep-th/0403221. G.W. Gibbons, K. Hashimoto and P. Yi, “Tachyon Condensates, Carrollian Contraction of Lorentz Group, and Fundamental Strings”, JHEP [**0209**]{} (2002) 061, hep-th/0209034. A.A. Tseytlin, Nucl. Phys. B [**469**]{} (1996) 51. K.V. Kuchar, “Time and Interpretations in Quantum Gravity”, in proceedings of the 4th conference on General Relativity and Relativistic Astrophysics, eds G. Kunstatter, D. Vincent and J. Williams (World Scientific, Singapore 1992). P. Mukhopadhyay and A. Sen, JHEP [**0211**]{} (2002) 047, hep-th/0208142. S. Bhattacharya, S. Mukherji and S. Roy, “On the Effective Action of Space-like Brane”, Phys. Lett. B [**584**]{} (2004) 163, hep-th/0308069. S.-J. Rey and S. Sugimoto, “Rolling Tachyon with Electric and Magnetic Fields- T-duality Approach-”, Phys. Rev. D [**67**]{} (2003) 086008, hep-th/0301049.
[^1]: The analysis and criticism of different proposals as time, in particular the introduction of matter clocks in quantum gravity and quantum cosmology, has been nicely done in Ref. [@kuchar].
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Every $d \times d$ bipartite system is shown to have a large family of undistillable states with nonpositive partial transpose (NPPT). This family subsumes the family of conjectured NPPT bound entangled Werner states. In particular, all one-copy undistillable NPPT Werner states are shown to be bound entangled.'
author:
- Rajiah Simon
date: 'August 30, 2006'
title: NPPT Bound Entanglement Exists
---
[**Introduction:**]{} Maximally entangled bipartite states form an indispensable resource in several quantum information processing situations[@ekert; @cleve; @teleport; @commit]. In reality, however, readily available states may be mixed and less than maximally entangled due to a variety of reasons including influence of the environment. Thus distillation, the process by which some copies of (nearly) maximally entangled pure states are extracted from several copies of partially entangled mixed states using local quantum operations and classical communication (LOCC), is of singular importance.
Rapid progress has been achieved in this regard, beginning with the works of Popescu[@distil1], Bennett [*et al.*]{}[@distil2], Deutsch [*et al.*]{}[@distil3], and Gisin[@distil4; @distil5]. An important question that presented itself at an early stage of this development was this: Can every entangled state be distilled? The Horodecki family answered this question affirmatively in a significant particular case: Every inseparable state of a pair of qubits can, given sufficiently many copies, be distilled into a singlet[@allqubits].
Subsequently they proved a result that applies to arbitrary $m
\times n$ bipartite systems[@distil-nc]: A state that does not violate the Peres-Horodecki PPT (positive partial transpose) criterion[@peres; @peres-horo] can not be distilled. Since inseparable PPT states can not be distilled, they are said to possess bound ([*i.e.*]{}, undistillable) entanglement. The first examples of such undistillably entangled PPT states were constructed by Horodecki[@range], and several families of PPT bound-entangled states have been presented since then. The role PPT bound-entanglement could play as a quantum resource has also been studied.
[**The Problem of NPPT Bound Entanglement:**]{} Given a bipartite state $\rho$, let $({\rho}^{T_B})^{\otimes n} =
({\rho}^{\otimes n})^{T_B}$ denote the partial transpose of the state of $n$ identical copies. Then the necessary and sufficient condition for distillability of $\rho$ is that the inequality[@distil-nc] $$\label{eqn1}
\left\langle{\psi}\right|\left({\rho}^{T_B}\right)^{\otimes
n}\left|{\psi}\right\rangle \ge 0,$$ be violated by a Schmidt rank two vector $|\psi\rangle$ in the $n$-copy Hilbert space, for some $n$. Thus in order to be distillable, the state should be NPPT ([*i.e.*]{}, the state should have non-positive partial transpose). What has remained an open problem is this: Are there undistillable NPPT states? Does undistillability imply PPT[@open]?
Clearly, violation of the above inequality for a particular $n =
n_0$ implies violation for all $n > n_0$. Given an NPPT state $\rho$, if it satisfies this inequality for all Schmidt rank two states, for a particular $n$, we say that the state is $n$-copy undistillable or, equivalently, that the partial transpose ${\rho}^{T_B}$ is $n$-copy $2$-positive. Thus $\rho$ is undistillable if ${\rho}^{T_B}$ is $n$-copy $2$-positive for all $n$.
While Horodecki [*et al.*]{} pointed out in a subsequent work[@reduction] that NPPT bound entangled states, if they exist in nature, should be found among the one-copy undistillable NPPT Werner states[@wernerstate], it may be fair to say that this problem attained its present state of fame only when two leading groups produced independently, and about the same time, analytical and numerical evidence[@nppt-divincenzo; @nppt-dur] for its existence in the context of the Werner family of states \[More recent numerical attempt has been undertaken in Ref.[@nppt-vianna]\]. As the analytic part of the evidence it was shown, for every $n$, that there is a corresponding range of parameter values over which the NPPT Werner state remains provably $n$-copy undistillable. But the range itself becomes rapidly smaller with increasing $n$, and was not proved to remain nonzero as $n \rightarrow \infty$. It should be added that no one-copy undistillable NPPT Werner state was shown to be $n$-copy distillable either. Similar evidence in the multipartite case has been considered by explicit construction of $n$-copy undistillable NPPT states, for every $n \ge 1$[@nppt-som].
It should be noted for completeness that some systems are known not to support NPPT bound entangled states at all. These include $2 \times n$ dimensional systems[@allqubits; @nppt-dur] and all bipartite Gaussian states[@giedke].
Shor [*et al.*]{}[@nppt-shor] have used the conjectured existence of NPPT bound entanglement to prove nonadditivity of bipartite distillable entanglement. And Eggeling [*et al.*]{}[@nppt-eggeling] have related the existence of NPPT bound entanglement to the connection between the sets of separable superoperators and PPT-preserving channels. Further, Vollbrecht and Wolf[@VW] have shown that an additional resource in the form of infinitesimal amount of PPT bound-entanglement can render any one-copy undistillable NPPT state one-copy distillable.
In a more recent paper Watrous[@nppt-watrous] has constructed, for every $n \ge 1$, states which are $n$-copy undistillable, and yet are distillable. This is a surprising and important result, for it shows that $n$-copy undistillability, even if $n$ is very large, does not by itself prove undistillability. The burden this finding places on numerical evidence for NPPT bound entanglement is evident.
We conclude this brief summary of the present status of the conjectured existence of NPPT bound entanglement by noting that the conjecture itself seems to enjoy the confidence of researchers in quantum information theory, even though the evidence presented so far has been assessed differently by different authors[@nppt-shor; @nppt-eggeling; @nppt-watrous].
In this Letter we prove that any $d \times d$ bipartite system with $d
\ge 3$ has a fairly large family of NPPT states which are undistillable. As will be seen, this family is much larger than the Werner family of conjectured NPPT bound entangled states, but it turns out to be convenient to begin our proof with the Werner family.
[**Proof of Existence of NPPT Bound Entanglement:**]{} We may define the one-parameter family of Werner state ${\rho}_{\alpha}$ in $d \times
d$ dimensions through the partial transpose of $\rho_\alpha$: $${\rho}^{\rm T_B}_{\alpha} = {\rm Id} - d \alpha P.$$ Here $P$ is the projection on the standard maximally entangled state: $$P= |\Psi\rangle \langle \Psi |, \,\,\,~~|\Psi\rangle =
\frac{1}{\sqrt{d}} \sum_{k=1}^{d} | k, k \rangle.$$ These states as defined are not normalized to unit trace, but this does in no way affect our considerations below.
Nonnegativity of ${\rho}_{\alpha}$ forces on $\alpha$ the restriction $-1 \leq \alpha \leq 1$. This allowed range for $\alpha$ divides into three interesting regions[@nppt-divincenzo; @nppt-dur]: $$\begin{aligned}
-1 \leq \alpha \leq \frac{1}{d}, & \,\,{\rm PPT},\,\,& {\rm separable},
\nonumber \\
\frac{1}{2} < \alpha \leq 1,& \,\,{\rm NPPT},\,\,& {\rm one}{\mathrm -}
{\rm copy\,\,\, distillable}, \nonumber \\
\frac{1}{d} < \alpha \leq \frac{1}{2}, &
\,\,{\rm NPPT},\,\, & {\rm one}{\mathrm -}{\rm copy\,\,\, undistillable}.\end{aligned}$$ Clearly, it is the last mentioned range which is of interest for the issue on hand. In this range ${\rho}_{\alpha}$ is NPPT, yet ${{\rho}_{\alpha}}^{T_B}$ is $2$-positive. Since ${\rho}_{\alpha}$ being NPPT implies that ${\rho}_{\alpha}^{\otimes n}$ is NPPT, for all $n$, the issue really is whether $({{{\rho}_{\alpha}}^{T_B}})^{\otimes n}$ is $2$-positive as well. We show that it indeed is, for all $n$.
Let us denote by ${{\cal S}^{(1)}}$ the collection of all pure and mixed states of Schmidt rank (number) $\leq 2$[@terhal]. This is a convex set whose extremals are all pure states of Schmidt rank $1$ or $2$. Let $\theta =
({\theta}_{1},{\theta}_{2},\cdots,{\theta}_{d})$ be a $d$-tuple of angles, and consider the subgroup of $U(d)$, the $d$-dimensional unitary group, consisting of diagonal matrices $U_{\theta}$: $${(U_{\theta})}_{kl} =
{\delta}_{kl} e^{i {\theta}_{k}}.$$ This is the standard maximal abelian subgroup $U_{A}$ of $U(d)$. Now average each element $\sigma$ of ${{\cal S}^{(1)}}$ over the local group $U_{A} \otimes {U}_{A}^{*}$: $$\sigma \in {\cal S}^{(1)}\,\rightarrow\, {\sigma}^{'} =
\int d \theta\, U_{\theta} \otimes {U}_{\theta}^{*}
\,\sigma\, {U}^{\dagger}_{\theta} \otimes
{U}_{\theta}^{T}\,.$$ Here $d \theta = d{\theta}_{1}d{\theta}_{2} \cdots d{\theta}_{d}$, with ${\theta}_{k}$ running over the interval $[0,2\pi]$, independently for every $k$. This group-averaging process is analogous to the diagonal twirl operation[@nppt-divincenzo]. The resulting images of elements of ${\cal
S}^{(1)}$ constitute a new convex set ${\Omega}^{(1)}$, where the superscript on ${\cal S}$ and $\Omega$ reflects our intention to extend these considerations to several copies of the Werner state.
The reduction in complexity achieved by going from ${{\cal S}^{(1)}}$ to ${\Omega}^{(1)}$ should be appreciated. For instance, whereas ${{\cal S}^{(1)}}$ has a $C{P}^{d-1} \times C{P}^{d-1}$ worth of product states among its extremals, the product states among the extremals of ${\Omega}^{(1)}$ are precisely $d^2$ in number; these are the standard computational basis states $|k,l \rangle$. Similarly the pure states of Schmidt rank = $2$ among the extremals of ${\Omega}^{(1)}$ necessarily have one of the forms ${\alpha}_1|11\rangle + {\alpha}_2|22\rangle$, ${\beta}_2|22\rangle + {\beta}_3|33\rangle$, and ${\gamma}_1|11\rangle + {\gamma}_3|33\rangle$; each of these three sets is a Bloch-sphere in worth. And, therefore the maximally entangled rank-two states are precisely three circles worth in number, being of the form $\frac{1}{\sqrt{2}} (|11\rangle + e^{i {\delta}_{1}}
|22\rangle)$, $\frac{1}{\sqrt{2}} (|22\rangle + e^{i
{\delta}_{2}} |33\rangle)$, or $\frac{1}{\sqrt{2}} (|33\rangle +
e^{i {\delta}_{3}} |11\rangle)$.
Now $2$-positivity of ${\rho}_{\alpha}^{T_B}$ is equivalent to the demand that ${\rm Tr}({\rho}^{T_B} \sigma) \geq 0$, for all $\sigma
\in {{\cal S}^{(1)}}$. In view of the $U_{A} \otimes {U}_{A}^{*}$ symmetry of ${\rho}^{T_B}$, this is equivalent to the demand that ${\rm Tr}({\rho}^{T_B} \sigma) \geq 0$, for all $\sigma \in
{\Omega}^{(1)}$ and in view of the convexity of ${\Omega}^{(1)}$, it is sufficient for the extremals of ${\Omega}^{(1)}$ to meet this demand. This convexity argment is in essence a recognition of the fact that the real-valued expression ${\rm Tr} (\rho^{T_B}_\alpha sigma)$ is linear in $\sigma$.
We thus find that the minimum of ${\rm Tr}({\rho}^{T_B}
\sigma)$ over ${\Omega}^{(1)}$ equals $(1-2\alpha)$, showing that ${\rho}^{T_B}$ is $2$-positive for all $\alpha \leq \frac{1}{2}$. The minimum is achieved by $\frac{1}{\sqrt{2}} (|11\rangle +
|22\rangle)$, $\frac{1}{\sqrt{2}} (|22\rangle + |33\rangle)$, and $\frac{1}{\sqrt{2}} (|33\rangle + |11\rangle)$, and by no other states.
We may note in passing that the local symmetry of ${\rho}^{T_B}$ is not $U_{A} \otimes {U}_{A}^{*}$, but the full group $U(d) \otimes {U(d)}^{*}$. We wish to go beyond the Werner states later in this Letter, and for this reason we have based our analysis on the subgroup $U_{A} \otimes
{U}_{A}^{*}$, rather than on the full group.
We now move on to consider ${\rho}_{\alpha} \otimes
{\rho}_{\alpha}$, two copies of the Werner state ${\rho}_{\alpha}$. The set ${{\cal S}}^{(2)}$ is constructed as the collection of $(d^2 \times d^2)$-dimensional states of Schmidt number $\leq
2$. From the convex set ${{\cal S}}^{(2)}$ we obtain ${\Omega}^{(2)}$ by averaging each $\sigma \in{{\cal S}}^{(2)}$ over the local group $(U_{A} \otimes {U}_{A}^{*}) \otimes (U_{A}
\otimes {U}_{A}^{*})$. It is to be understood that the first $(U_{A} \otimes {U}_{A}^{*})$ factor acts on the Hilbert space of the first copy and the second on that of the second copy, independently.
The extremals of ${\Omega}^{(2)}$ are readily enumerated. The rank-$1$ extremals of ${\Omega}^{(2)}$ are necessarily of the form $|k,l\rangle_1 \otimes |i,j\rangle_2$. They are $d^4$ in number. The rank two states are of two types. $$\begin{aligned}
{\rm Type}\,\,{\rm I}:&&~~ (\alpha |kk\rangle_1 + \beta |ll\rangle_1 )
\otimes |ij\rangle_2, \nonumber \\
&&~~ |kl\rangle_1 \otimes
(\alpha |ii\rangle_2 + \beta |jj\rangle_2 ); \nonumber \\
{\rm Type}\,\,{\rm II}:&&~~ \alpha |kk\rangle_1 \otimes |ii\rangle_2
+ \beta |ll\rangle_1 \otimes |jj\rangle_2.\end{aligned}$$ At the risk of sounding repetitive, we emphasize that this is a complete enumeration of the extremals of ${\Omega}^{(2)}$.
As with the single copy case $2$-positivity of ${\rho}_{\alpha} \otimes {\rho}_{\alpha}$ is equivalent to the demand ${\rm Tr}[({\rho}_{\alpha}^{T_B}
\otimes {\rho}_{\alpha}^{T_B}) \sigma] \geq 0$, for all $\sigma
\in {\Omega}^{(2)}$, which in turn is equivalent to the demand that this condition be met by all the extremal states of ${\Omega}^{(2)}$. The type-I rank-$2$ states being products across the two copies, cannot bring out genuinely two-copy properties if any, and thus we are left with only the type-II states to examine: $$\begin{aligned}
\langle \Psi_{II} |{\rho}_{\alpha}^{T_B} \otimes {\rho}_{\alpha}^{T_B}
| \Psi_{II} \rangle & \nonumber \\
= &|\alpha|^2 \langle kk | {\rho}_{\alpha}^{T_B} |kk \rangle
\langle ii | {\rho}_{\alpha}^{T_B} |ii \rangle \nonumber \\
&+ |\beta|^2 \langle ll | {\rho}_{\alpha}^{T_B} |ll \rangle
\langle jj | {\rho}_{\alpha}^{T_B} |jj \rangle \nonumber \\
&+ {\alpha}^{*} \beta \langle kk | {\rho}_{\alpha}^{T_B} |ll \rangle
\langle ii | {\rho}_{\alpha}^{T_B} |jj \rangle \nonumber \\
&+ \alpha {\beta}^{*} \langle ll | {\rho}_{\alpha}^{T_B} |kk \rangle
\langle jj | {\rho}_{\alpha}^{T_B} |ii \rangle.\end{aligned}$$ Now, $2$-positivity of ${\rho}^{T_B}$ is equivalent to the Schwartz inequality $$\langle kk | {\rho}_{\alpha}^{T_B} |ll \rangle \leq
[\langle kk | {\rho}_{\alpha}^{T_B} |kk \rangle \langle ll |
{\rho}_{\alpha}^{T_B} |ll \rangle]^{\frac{1}{2}},$$ for rank-1 states. Use of this inequality in Eq.(8) proves $$\langle \Psi_{II} |{\rho}_{\alpha}^{T_B} \otimes {\rho}_{\alpha}^{T_B}
| \Psi_{II} \rangle \geq 0.$$ That is one-copy $2$-positivity of the $(U_{A} \otimes {U}_{A}^{*})$ invariant operator ${\rho}^{T_B}$ implies its two-copy $2$-positivity. And we have proved
[**Theorem 1**]{}: All one-copy undistillable Werner states, that is all ${\rho}_{\alpha}$’s in the entire range $\frac{1}{d} \le \alpha \leq \frac{1}{2}$, are two-copy undistillable.
To move on to the $n$-copy case, assume that ${\rho}_{\alpha}^{T_B}$ is $(n-1)$-copy $2$-positive. That is $\langle \psi {({\rho}_{\alpha}^{T_B})}^{\otimes (n-1)}|\psi \rangle
\geq 0$ for all rank-$2$ states of the $(n-1)$-copy Hilbert space. We wish to prove that this implies ${({\rho}_{\alpha}^{T_B})}^{\otimes
(n)}$ is $2$-positive.
To this end, form ${\Omega}^{(n)}$ by averaging each $\sigma \in {\cal S}^{(n)}$ over the local group ${(U_{A} \otimes {U}_{A}^{*})}^{\otimes n}$, with one $(U_{A} \otimes {U}_{A}^{*})$ factor acting on the Hilbert space of each copy independently. Again, the extremals of ${\Omega}^{(n)}$ consist of rank-$1$ and rank-$2$ pure states. The rank-$1$ states are of the form $|i_1,j_1\rangle \otimes
|i_2,j_2\rangle\otimes ....|i_n,j_n\rangle$. That is, these are tensor products of computational basis states, one picked from each copy, and thus are $d^{\,2n}$ in number.
The rank-$2$ extremal (or pure) states in ${\Omega}^{(n)}$ are of two types, as in the two-copy case: $$\begin{aligned}
{\rm Type}\,\,{\rm I}:&&~~ |\Psi_{I}\rangle = |\psi \rangle \otimes
|i,j\rangle,\nonumber\\
{\rm Type}\,\,{\rm II}:&&~~ |\Psi_{II} \rangle =|{\phi}_{1} \rangle
\otimes |i,i\rangle + |{\phi}_{2} \rangle \otimes |j,j\rangle,\end{aligned}$$ where $|\psi \rangle$ is a $(n-1)$-copy rank-$2$ state, and $|{\phi}_{1} \rangle$, $|{\phi}_{2} \rangle$ are $(n-1)$-copy rank-$1$ states. Since type-I states have a product structure across the copies, nonnegative expectation values of $({{\rho}_{\alpha}^{T_B}})^{\otimes n}$ in respect of type-I rank-$2$ states follows directly from the assumed $(n-1)$-copy $2$-positivity, the additional copy offering nothing new in the type-I case. The same $(n-1)$-copy $2$-positivity is equivalent to the validity of the Schwartz inequality for $(n-1)$-copy rank-$1$ states, and this in turn implies the nonnegativity of the expectation values of $({{\rho}_{\alpha}^{T_B}})^{\otimes n}$ for type-II states. We have thus proved, by induction,
[**Theorem 2**]{}: The one-copy $2$-positive $({{\rho}_{\alpha}^{T_B}})^{\otimes n}$’s in the entire parameter range $\frac{1}{d} \le \alpha \leq \frac{1}{2}$ are $n$-copy $2$-positive, for all $n$. That is, these one-copy undistillable NPPT Werner states ${\rho}_{\alpha}$ are $n$-copy undistillable, for all $n$, and hence are bound entangled.
Finally, it should be evident that the only property of ${\rho}_{\alpha}$, apart from its NPPT and one-copy $2$-positivity properties, used in our analysis is its $(U_{A}\otimes {U}_{A})$ invariance or, equivalently, the $(U_{A}\otimes {U}_{A}^{*})$ invariance of its partial transpose. It follows that our conclusions apply to all states with these properties. That is,
[**Theorem 3**]{}: Every one-copy undistillable NPPT state in $d \times d$ dimensions is bound entangled if it possesses $(U_{A} \otimes {U}_{A}^{})$ symmetry.
[*This is the main result of this Letter*]{}. It shows that the family of NPPT bound entangled states in $d \times d$ dimensions, for any $d \geq 3$, is much larger than what might have been anticipated. This point is worth illustrating.
It is easily seen that in $3 \times 3$ dimensions the most general $(U_{A} \otimes {U}_{A}^{*})$ invariant ${\rho}^{T_B}$ has the form $$\begin{aligned}
{\rho}^{T_B}=
\left[
\begin{array}{ccccccccc}
{\rho}_{11} & 0 & 0 & 0 & -z_{12} & 0 &0 & 0 & -{z}^{*}_{31} \\
0 & {\rho}_{12} & 0 &0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0& {\rho}_{13}& 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & {\rho}_{21}& 0 & 0 & 0 & 0 & 0 \\
{-z}^{*}_{12}& 0 & 0 & 0 & {\rho}_{22} & 0 & 0 & 0 & -z_{23}\\
0 & 0 & 0 & 0 & 0 & {\rho}_{23} & 0 & 0 & 0 \\
0 & 0 & 0 &0 & 0 & 0 & {\rho}_{31} & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & {\rho}_{32} & 0 \\
{-z}_{31} & 0 & 0 & 0 & {-z}^{*}_{23} &
0 & 0& 0& {\rho}_{33}
\end{array} \right] \end{aligned}$$ Clearly, $\rho$ has to be positive semidefinite in order to be a valid density matrix. This demand is equivalent to the conditions: $(i)$ all the diagonal elements ${\rho}_{ij} \geq 0$, and ${\rho}_{ij} {\rho}_{ji} \geq {|z_{ij}|}^2$ for all $i \neq j$. The NPPT requirement demands that the $3 \times 3$ submatrix $$\begin{aligned}
\left[ \begin{array}{ccc}
{\rho}_{11} & -z_{12} & -{z}^{*}_{31} \\
{-z}^{*}_{12} & {\rho}_{22} & -z_{23}\\
{-z}_{31} & {-z}^{*}_{23} &{\rho}_{33}
\end{array} \right] \end{aligned}$$ should be nonpositive, and the $2$-positivity demand is equivalent to the three inequalities ${\rho}_{11}{\rho}_{22} \geq {|z_{12}|}^2$, ${\rho}_{22}{\rho}_{33}\geq {|z_{23}|}^2$, and ${\rho}_{33}{\rho}_{11}\geq {|z_{31}|}^2$. The NPPT demand and the $2$-positivity demands thus involve only the six parameters ${\rho}_{11}$, ${\rho}_{22}$, ${\rho}_{33}$ and $z_{12}$, $z_{23}$, $z_{31}$. The phases of the complex z-parameters can be tuned by (local) change of phases of the basis vectors on the $A$ and $B$ sides (to be precise it is sufficient to carry out the changes on one side only), but the argument of $z_{12}$ $z_{23}$ $z_{31}$ is invariant under such gauge transformations.
Thus our family of NPPT bound entangled states in $3 \times 3$ involves, when normalized to unit trace, $12$ parameters; eight coming from the diagonals ${\rho}_{ij}$, three coming from the magnitudes of the $z$-parameters, and one gauge-invariant phase. These are canonical parameters, and do not take into consideration parameters arising from local unitary transformations.
[**Acknowledgement**]{}: I would like to thank Sibasish Ghosh, Solomon Ivan, Guruprasad Kar, and Anirban Roy for many discussions on the problem of NPPT bound entanglement.
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Using Talagrand’s concentration inequality on the discrete cube $\{0,1\}^m$ we show that given a real-valued function $Z(x)$ on $\{0,1\}^m$ that satisfies certain monotonicity conditions one can control the deviations of $Z(x)$ above its median by a local Lipschitz norm of $Z$ at the point $x.$ As one application, we give a simple proof of a nearly optimal deviation inequality for the number of $k$-cycles in a random graph.'
author:
- |
Dmitry Panchenko\
[*Massachusetts Institute of Technology*]{}\
title: ' Deviation inequality for monotonic Boolean functions with application to a number of $k$-cycles in a random graph.'
---
Introduction and main results.
==============================
In this paper we suggest a new way to use Talagrand’s concentration inequality on the cube to control the deviations of Boolean functions that satisfy certain monotonicity conditions. As one application we prove a suboptimal deviation inequality for the count of $k$-cycles in a random graph.
Let $\X=\{0,1\}$ and define a probability measure $\mu$ on $\X$ by $\mu(\{1\})=p, \mu(\{0\})=1-p.$ Consider a product space $\X^m$ with a product probability measure ${{\Bbbb P}}=\mu^m.$ Given a function $Z: \X^m \to{\mathbb{R}}$ and a point $x=(x_1,\ldots,x_m)\in \X^m$ we define $$V_i(x)=Z(x) - Z(x_1,\ldots,x_{i-1},0,x_{i+1},\ldots,x_m),$$ and $$V(x)=\sum_{i=1}^m V_i^2(x).$$ Note that $V_i(x)=0$ if $x_i=0.$
Let us state the main result of this paper.
If $Z(x)$ and $V_i(x), i\leq m$ are non-decreasing in each coordinate then for any $a\in {\mathbb{R}}$ and $t>0,$ $${{\Bbbb P}}\bigl(Z(x)\geq a +\sqrt{V(x) t}\bigr)
{{\Bbbb P}}\bigl(Z(x)\leq a\bigr) \leq e^{-t/2}.
\label{selfnorm}$$
To understand the statement of Theorem 1, we notice that a function $(V(x))^{1/2}$ can be interpreted as a kind of discrete Lipschitz norm of $Z$ locally at the point $x.$ For example, since $Z(x)$ is defined only on the vertices of $m$-dimensional cube, if one extends $Z(x)$ linearly from the point $x$ to its neighbours only along the coordinates where $x_i=1,$ then $(V(x))^{1/2}$ is the norm of that linear map. Indeed, if we denote the map by $L: {\mathbb{R}}^m \to {\mathbb{R}},$ then $$L(z)=\sum_{i=1}^m V_i(x)(z_i-x_i) + Z(x),$$ and $\|L\|=(V(x))^{1/2}.$
The proof of Theorem1 is based on Talagrand’s concentration inequality on $\X^m.$ In order to give more clear interpretation of (\[selfnorm\]), let us compare it with some typical ways of using Talagrand’s inequality. One common application is the following. Given a convex function $f:[0,1]^m\to{\mathbb{R}}$ with a Lipschitz norm $$\|f\|_{L}=\sup_{x,y \in [0,1]^m} \frac{|f(x)-f(y)|}{|x-y|}< \infty,$$ where the supremum is taken over $x\not = y,$ the following inequality holds: $${{\Bbbb P}}\bigl(f\geq a + \|f\|_{L} \sqrt{t}\bigr)
{{\Bbbb P}}\bigl(f\leq a\bigr) \leq e^{-t/2}.
\label{convin}$$ If the function $f$ is defined only on the vertices of the cube $\X^m,$ then it is possible to state a similar result where one has to use a discrete analog of the Lipschitz norm. For example, the following deviation inequality holds (see [@Bobkov]). For $i\leq m$ we define $x^{i}\in \X^m$ such that $x_j^i = x_j$ for $j\not = i$ and $x_i^i \not = x_i,$ and define $$\|f\|_{d}=
\sup_{x\in\X^m}\bigl(\sum_{i=1}^m (f(x) - f(x^i))^2\bigr)^{1/2}.$$ Then $${{\Bbbb P}}\bigl(f\geq {{\Bbbb E}}f + \|f\|_{d} \sqrt{t}\bigr)
\leq e^{-t/4}.
\label{bob}$$ Both inequalities (\[convin\]) and (\[bob\]) use global Lipschitz condition to control the deviation of $f(x).$ Theorem 1 suggests a possibility of using a local Lipschitz norm $V(x)^{1/2}$ at the point $x,$ provided that the monotonicity conditions are satisfied. The reason why we compute the Lipschitz norm only in the direction of decreasing $Z$ is because we control the deviation of $Z$ [*above*]{} level $a.$ Theorem 1 is similar in spirit to the ideas in [@Kimvu], [@Vu1],[@Vu2] (see also references therein), where the authors describe a way of using average Lipschitz norm of $Z$ to control its deviations.
One example when the monotonicity conditions are satisfied is the following. Let us consider a set of indices ${\cal M}=\{1,2,\ldots,m\}$ and a set of nonnegative numbers $\alpha_{\cal C}\geq 0$ indexed by the subsets ${\cal C}\subseteq {\cal M}.$ Consider the function $Z(x)$ defined by $$Z(x)=\sum_{{\cal C}\subseteq {\cal M}}\alpha_{\cal C}
\prod_{i\in {\cal C}} x_i.
\label{pos}$$ In this case the fact that $\alpha_{\cal C}$’s are non-negative implies that functions $Z(x)$ and $V_i(x), i\leq m$ are non-decreasing in each coordinate. Below we will consider the example of counting the number of $k$-cycles in a random graph which can be represented in the form (\[pos\]) and, thus, Theorem 1 is applicable.
Consider a standard Erdös-Rényi model of a random graph $G(n,p).$ Let $V$ be a set of $n$ vertices, $m={n\choose 2}$ and let $E=\{e_1,\ldots,e_m\}$ denote a set of edges of a complete graph $K_n$ on $n$ vertices. Given $x=(x_1,\ldots,x_m)\in\X^m,$ the fact that $x_i=1$ or $0$ describes that the edge $e_i$ is present or not present in the graph $G(n,p)$ respectively. Let $$C_k=\Bigl\{\{e_{i_1},\ldots,e_{i_k}\} : e_{i_j}\in E, \mbox{ and
$\{ e_{i_1},\ldots,e_{i_k} \}$
form a $k$-cycle}
\Bigr\}$$ be a collection of all $k$-cycles, and for $e\in E$ let $$C_k(e)=\{c\in C_k : e\in c\}$$ be a set of all $k$-cycles containing the edge $e.$ We consider the following function on $\X^m$ $$Z(x)= \sum_{c\in C_k}\prod_{e\in c}x_e,$$ which is the number of $k$-cycles in a random graph $G(n,p)$. In this case $V(x)$ can be clearly written as $$V(x)=\sum_{e\in E} x_e \Bigl(
\sum_{c\in C_k(e)}\prod_{e'\not = e}x_{e'}
\Bigr)^2.
\label{Vee}$$ In this case, in order to use Theorem 1 to control the deviation of $Z(x)$ above its median $M(Z)$ (or its expectation ${{\Bbbb E}}Z$) we will proceed by showing how to control $V(x)$ in terms of $Z(x).$
We assume that for some large enough $C(k)>0,$ $$np \geq C(k) \log n.
\label{pee}$$ The following theorem holds.
If (\[pee\]) holds then there exists a constant $C(k)>0$ that depends on $k$ only such that $${{\Bbbb P}}\Bigl(
V(x)\geq C(k)
\bigl( (np)^{k-2}Z(x) + (np)^{2(k-1)} \bigr)
\Bigr) \leq \exp\Bigl(-\frac{(np)^2}{C(k)\log\log np}\Bigr).$$
Theorems 1 and 2 will readily imply the following theorem.
If (\[pee\]) holds then,
\(1) For any $\eps>0$ there exists a constant $C(k,\eps)$ that depends on $k$ and $\eps$ only such that the following holds $$\mbox{If }\,\,\,
{{\Bbbb E}}Z\geq C(k,\eps)
\,\,\,\mbox{ then }\,\,\,
M(Z)\leq (1+\eps){{\Bbbb E}}Z.$$
\(2) There exists a constant $C(k)>0$ such that if ${{\Bbbb E}}Z \geq C(k)$ then $${{\Bbbb P}}(Z\geq 2{{\Bbbb E}}Z)\leq \exp\Bigl(- \frac{(np)^2}{C(k)\log\log np}\Bigr).
\label{dev}$$
Recently the authors of [@Jan2] proved a more general result describing the deviations of the count of any subgraph in a random graph. In the case of $k$-cycles their bound gives $${{\Bbbb P}}(Z\geq 2{{\Bbbb E}}Z)\leq \exp(-(np)^2/C(k)).$$ This shows that the factor $\log\log np$ in (\[dev\]) is unnecessary, but at the moment we don’t see how to get rid of it using our approach. This has nothing to do with Theorem 1, since the factor $\log\log np$ comes directly from Theorem 2 which, probably, can be improved. In the case of triangles ($k=3$) the bound ${{\Bbbb P}}(Z\geq 2{{\Bbbb E}}Z)\leq \exp(-(np)^2/C(k))$ was also proved in [@Kimvu].
Proof of Theorem 1.
===================
Talagrand’s concentration inequality on the discrete cube is the main tool in the proof of Theorem 1. Let us recall it first.
Given a point $x\in\X^m = \{0,1\}^m$ and a set ${\cal A}\subseteq \X^m,$ let us denote $$U_{\cal A}(x)=\{(s_i)_{i\leq m}\in\{0,1\}^m ,
\exists y\in{\cal A} , s_i=0 \Rightarrow y_i=x_i\}.$$ The “convex hull” distance between the point $x$ and the set $\cal A$ is defined as $$f_c({\cal A},x)=\inf\{|s| : s\in\mbox{conv}U_{\cal A}(x)\},$$ where $|s|$ is the Euclidean norm of $s.$ The concentration inequality of Talagrand (Theorem 4.3.1 in [@Ta1]) states the following.
For any $t>0,$ $${{\Bbbb P}}({\cal A})
{{\Bbbb P}}\Bigl(x\in\X^m : f_{c}^{2}({\cal A},x)\geq t\Bigr)\leq
e^{-t/2}.
\label{T1}$$
The main feature of this distance is that (Theorem 4.1.2 in [@Ta1]) $$\forall (\lambda_i)_{i\leq m}\,\,\,\,\,
\exists y\in{\cal A}\,\,\,\,\,\,\,\,
\sum_{i=1}^{m}\lambda_i I(y_i\not = x_i)\leq
f_c({\cal A},x)\Bigl(\sum_{i=1}^{m}\lambda_i^2\Bigr)^{1/2}.
\label{T2}$$ [**Proof of Theorem 1.**]{} For a fixed number $a\in {\mathbb{R}}$ consider a set $$\A=\{y\in \X^m : Z(y)\leq a\}.$$ For a fixed $x\in \X^m$ and an arbitrary $y\in\A,$ since $Z(y)\leq a,$ we can write $Z(x)-a\leq Z(x) - Z(y).$ Consider three sets of indices $$I_1 =\{i: x_i=1, y_i=0\},\,\,\,\,
I_2 = \{i: x_i = 0, y_i = 1\}, \,\,\,\,
I_3 = \{i: x_i = y_i\}.$$ Without loss of generality we will assume that $I_1=\{1,\ldots,k\}$ and $I_2 = \{k+1,\ldots, l\}.$ Define a sequence $$z^i = (y_1,\ldots,y_i,x_{i+1},\ldots, x_m), \,\,\,
i=0,\ldots,m.$$ We have $$Z(x) - Z(y)=\sum_{i=1}^{m} (Z(z^{i-1})-Z(z^i))=
\sum_{i=1}^{m} (Z(z^{i-1})-Z(z^i))I(x_i\not = y_i),$$ since $x_i=y_i$ (i.e. $i>l$) implies that $Z(z^{i-1})-Z(z^i)=0.$ We have $$Z(z^{i-1})-Z(z^i)=0\leq V_i(x)\,\,\,
\mbox{ for }\,\,\,
i=l+1,\ldots,m,$$ since for this range of indices $z^{i-1} = z^i,$ $$Z(z^{i-1})-Z(z^i)\leq 0\leq V_i(x)\,\,\,
\mbox{ for }\,\,\,
i=k+1,\ldots,l,$$ since the function $Z$ is non-decreasing in each coordinate and for $i\in I_2,$ $z^{i-1}_i = 0,$ $z_i^i = 1$ and all other coordinates of $z^{i-1}$ and $z^i$ coincide, and $$Z(z^{i-1})-Z(z^i) = V_i(z^{i-1})\leq V_i(x)\,\,\,
\mbox{ for }\,\,\,
i=1,\ldots,k,$$ since for $i\in I_1$ each coordinate of $z^{i-1}$ is smaller than the corresponding coordinate of $x.$ Thus we proved that for any $y\in {\cal A}$ $$Z(x) - a \leq \sum_{i=1}^m V_i(x) I(x_i\not =y_i).$$ By (\[T2\]) there exists $y\in {\cal A}$ such that the last expression can be bounded $$\sum_{i=1}^m V_i(x) I(x_i\not =y_i) \leq
f_c({\cal A},x)\Bigl(\sum_{i=1}^m V_i^2(x)\Bigr)^{1/2} =
f_c({\cal A},x)\sqrt{V(x)}$$ Talagrand’s inequality (\[T1\]) states that $${{\Bbbb P}}(f_c(\A,x)\geq \sqrt{t}) {{\Bbbb P}}(\A)
\leq e^{-t/2},$$ and, therefore, we finally get $${{\Bbbb P}}(Z(x)\geq a + \sqrt{V(x)t}){{\Bbbb P}}(Z(x)\leq a) \leq e^{-t/2}.$$
Proof of Theorem 2.
====================
We will denote by $d_v$ the degree of a vertex $v.$ Consider the sequence of sets $$V_1=\{v: d_v < 16 np\},\,\,\,\,
V_j=\{v : d_v\in [2^{j+2} np, 2^{j+3} np)\},\,\,\,
j\geq 2.
\label{sets}$$ We will start by stating several basic facts that will be used in the proof of Theorem 2.
If (\[pee\]) holds then, $${{\Bbbb P}}\Bigl(\exists v: d_v \geq (np)^2\Bigr)\leq e^{-(np)^2/2}.
\label{Basic1}$$
[**Proof.**]{} For a fixed vertex $v$ its degree $d_v$ is a sum of $(n-1)$ independent variables with the distribution $\mu.$ Using Bernstein’s inequality one can easily check that for $np\geq 4$ $${{\Bbbb P}}\Bigl(d_v \geq (np)^2\Bigr)\leq e^{-(np)^2}.$$ The union bound will produce a factor $n$ and, therefore, using (\[pee\]) implies (\[Basic1\]).
Thus, with high probability we can assume that the degree of each vertex is bounded by $(np)^2$ and, therefore, we can only consider the sets $V_j$ in (\[sets\]) such that $2^{j+2} \leq np$ and, therefore, $j\leq \log np.$
Next we will bound the cardinality of each $V_j.$
For $C(k)>0$ large enough we have, $${{\Bbbb P}}\Bigl(
\exists 2 \leq j\leq \log np\,\,\,\,\,
\mbox{\rm card} V_j \geq \frac{np}{j 2^j \log\log np}
\Bigr) \leq
\exp\Bigl(-\frac{(np)^2}{C(k)\log\log np} \Bigr).
\label{Basic2}$$
[**Proof.**]{} We copy the proof from [@Kimvu] (see equation (12) in section 4.2 there). For a fixed $j\geq 2,$ assume that $$\mbox{card} V_j \geq r = \frac{np}{j 2^j \log\log np}.$$ In this case, there exists a set of $r$ vertices each with degree at least $2^{j+2}np.$ It implies that the number of edges containing exactly one of these vertices exceeds $$(2^{j+2}np - r)r \geq 2^{j+1} np r,$$ The probability that such a set of edges exists is bounded by $$\begin{aligned}
& &
{n\choose r}{r(n-r) \choose 2^{j+1}npr} p^{2^{j+1}npr}\leq
\exp\Bigl(
r\log\frac{en}{r} + 2^{j+1}npr \log\frac{er(n-r)}{2^{j+1}npr} +
2^{j+1}npr \log p
\Bigr)
\\
& &
\leq
\exp\Bigl(
r\log\frac{en}{r} + 2^{j+1}npr \log\frac{e}{2^{j+1}}
\Bigr)\leq
\exp\Bigl(
-\frac{(np)^2}{C(k)\log\log np}
\Bigr),\end{aligned}$$ where in the last inequality we used the estimate $$2^{j+1}npr \log\frac{e}{2^{j+1}}\leq \frac{(np)^2}{2j\log\log np}
\log\frac{1}{2^{j-1}}\leq
-\frac{(np)^2}{C(k)\log\log np},$$ and the first term $r\log(en/r)$ was negligible compared to the second term. Taking the union bound over $j\leq \log np,$ we get a factor $\log np$ in front of the exponent that can be ignored by increasing $C(k).$
Before we will state our next lemma, we need to make one remark about the proof of Theorem 2. Multiplying out the right-hand side of (\[Vee\]) we observe that $V(x)$ can be written as a sum of terms $$x_e \prod_{e'\in c}x_{e'}\prod_{e'\in c'}x_{e'},\,
\mbox{ where } e\in E, c,c'\in C_k(e).$$ Each of these terms may appear several times, but, clearly, the number of appearances will be bounded by $C(k)$ that depends on $k$ only. Each of these term represents two cycles that have at least one edge in common. There are many different isometric configurations of such two cycles but, clearly, the number of them is bounded by a constant that depends on $k$ only. Hence, $V(x)$ can be decomposed into the sum of the counts of such pairs of cycles over different configuration. With minor modifications it is possible to prove the statement of the theorem for each of these configuration.
We will only look at the pairs of cycles that have exactly one edge in common. Let us denote the number of such pair by $W(x).$ We will identify each pair of cycles with an injection $\sigma:\{1,\ldots,2k-2\}\to V(G),$ such that $(\sigma(1),\sigma(2)\ldots,\sigma(k))$ and $(\sigma(k),\sigma(k+1),\ldots,\sigma(1))$ are the ordered vertices of these two cycles, and $\sigma(1)\sigma(k)$ is their only common edge. Let us denote the set of these injections by $\Sigma_0.$
There is a partition of vertices $V(G)=F_1\cup\ldots\cup F_{2k-2}$ such that $$W(x)\leq C(k)\mbox{\rm card}
\{\sigma\in\Sigma_0 :\forall i\,\,\, \sigma(i)\in F_i\}.
\label{Basic3}$$
[**Proof.**]{} See Proposition 1.3 in [@Friedgut].
Let us denote the set in the statement of Lemma 3 by $$\Sigma=\{\sigma\in\Sigma_0 :\forall i\,\,\, \sigma(i)\in F_i\}.
\label{sigma}$$ [**Proof of Theorem 2.**]{} By Lemma 3 and the discussion preceeding Lemma 3, all we need to do is to estimate the cardinality of $\Sigma$ in (\[sigma\]). Let us consider the event $${\cal E}=\Bigl\{
\forall j\geq 2\,\,\,\,
\mbox{\rm card} V_j \leq \frac{np}{j 2^j \log\log np}
\Bigr\}
\bigcup
\bigl\{
\forall v: d_v \leq (np)^2
\bigr\}.$$ By Lemma 1 and Lemma 2 this event holds with probability at least $$1-\exp\Bigl(-\frac{(np)^2}{C(k)\log\log np}\Bigr),$$ for some $C(k)$ large enough. From now on we assume that this event occurs. For each vertex $v\in F_1$ let us denote $$S_l(v)=\{(\sigma(1),\ldots,\sigma(l)) : \sigma\in\Sigma,
\sigma(1)=v\},\,\,\,
l\geq 1.$$ Let us denote $d_v^{+}=\max(d_v,np).$ We will prove that if ${\cal E}$ occurs than for $l\geq 2$ $$\mbox{card}S_l(v) \leq C(k)d_v^{+} (np)^{l-2}.
\label{path}$$ For $l=2,$ this obviously holds with $C(k)=1.$ We proceed by induction over $l.$ Let us decompose $$S_l(v)=\bigcup_{j\geq 1} S_l^j(v),$$ where $$S_l^j(v)=\{(\sigma(1),\ldots,\sigma(l))\in S_l(v):
\sigma(l-1)\in V_j\}.$$
To bound the cardinality of $\mbox{card}S_l^1(v)$ we use the induction hypothesis and the fact that the degree of each vertex in the set $V_1$ is bounded by $16np.$ We get $$\mbox{card}S_l^1(v)\leq (16np)\ \mbox{card}{S_{l-1}(v)}\leq
C(k)d_v^{+} (np)^{l-2}.$$ To bound the cardinality of $S_l^j(v)$ for $j\geq 2$ we notice that for each $(\sigma(1),\ldots,\sigma(l))\in S_l^j(v),$ we have $(\sigma(1),\ldots,\sigma(l-2))\in S_{l-2}(v)$ and $\sigma(l-1)\in V_j.$ On the event $\cal E$ we can control the cardinality of $V_j$ and, moreover, the degree $d_{\sigma(l-1)}\leq 2^{j+3}np.$ For $l=3,$ since $\mbox{card}S_1(v)=1,$ we get $$\mbox{card}S_3^j(v)\leq
(\mbox{card}V_j)\ (2^{j+3}np)
\leq
\frac{np}{2^{j}j\log\log np} 2^{j+3}np
\leq 8 (np)^{2}\frac{1}{j\log\log np}$$ and since on the event ${\cal E}$ we can assume that $2^{j+2}\leq np,$ which implies that $j\leq \log np,$ we get $$\sum_{j=2}^{\log np} \mbox{card}S_3^j(v)\leq
8 (np)^{2}
\sum_{j=2}^{\log np}\frac{1}{j\log\log np}\leq
C(k) (np)^{2}\leq C(k)d_v^{+}(np),$$ and this proves the induction step for $l=3.$ The last inequality explains the appearance of the factor $\log\log np$ in Theorem 3. Similarly, for $l\geq 4$ we get $$\begin{aligned}
&&
\mbox{card}S_l^j(v)\leq
\mbox{card}S_{l-2}(v)\
(\mbox{card}V_j)\ (2^{j+3}np)
\\
&&
\leq
C(k)d_v^{+} (np)^{l-4}\frac{np}{2^{j}j\log\log np} 2^{j+3}np
\leq C(k) d_v^{+} (np)^{l-2}\frac{1}{j\log\log np} \end{aligned}$$ and $$\sum_{j=2}^{\log np} \mbox{card}S_l^j(v)\leq
C(k)d_v^{+} (np)^{l-2}
\sum_{j=2}^{\log np}\frac{1}{j\log\log np}\leq
C(k) d_v^{+} (np)^{l-2}.$$ This completes the proof of the induction step and (\[path\]).
To estimate the cardinality of $\Sigma$ we will decompose it into $\Sigma=\Sigma_1\cup\Sigma_2,$ where $$\Sigma_1=\{\sigma\in\Sigma : \sigma(1)\in V_1\},\,\,\,\,
\Sigma_2=\{\sigma\in\Sigma : \sigma(1)\in V_j, 2\leq j\}.$$ We will estimate the cardinality of $\Sigma_1$ and $\Sigma_2$ differently (the idea is illustrated in Figure 1). First of all, since we can control the cardinality of $V_j$ for $j\geq 2,$ we will simply use (\[path\]) for $l=2k-2$ to compute the number of different paths from $v\in F_1$ to $F_{2k-2},$ and then add them up. This will give us the bound on cardinality of $\Sigma_2.$ On the other hand, for $\sigma\in\Sigma_1$ we can represent it as a cycle on $F_1, F_k,\ldots,F_{2k-2}$ and a path from $F_1$ to $F_{k-1}.$ In this case, the number of cycles is bounded by $Z(x),$ and to bound the number of paths we again use (\[path\]).
Let first estimate the cardinality of $\Sigma_2.$ First of all by (\[path\]) for each vertex $v\in V_j \cap F_1$ $$\mbox{card} S_{2k-2}(v)\leq C(k) 2^{j+3}(np) (np)^{2k-4},$$ and, therefore, on the event ${\cal E},$ $$\mbox{card}\{\sigma\in\Sigma : \sigma(1)\in V_j\} \leq
C(k) 2^{j+3}(np)^{2k-3}
(\mbox{card} V_j)\leq
C(k) 2^{j+3}(np)^{2k-3}\frac{np}{j 2^j\log\log np}.$$ When we add up these injections over $j\geq 2$ we get $$\sum_{j=2}^{\log np}
C(k) 2^{j+3}(np)^{2k-3}\frac{np}{j 2^j\log\log np}
\leq
C(k) (np)^{2k-2}.$$ This accounts for the second term in the bound of the theorem.
Now consider all injections $\sigma$ such that $\sigma(1)\in V_1$. Consider the trace of the set of images of the injections from $\Sigma$ (in other words, pairs of cycles) on the set $$(V_1\cap F_1) \cup F_k \cup F_{k+1}\cup \ldots \cup F_{2k-2},$$ i.e. $${\cal P}=
\{(\sigma(1),\sigma(k),\sigma(k+1),\ldots,\sigma(2k-2))
: \sigma\in \Sigma, \sigma(1)\in V_1\}.$$ First of all, the cardinality of ${\cal P}$ is bounded by $Z(x),$ since ${\cal P}$ can be identified with the subset of all cycles in the random graph. Moreover, for each $(v_1,v_k,v_{k+1},\ldots,v_{2k-2})\in{\cal P},$ the number of injections $\sigma\in\Sigma$ such that $\sigma(1)=v_1, \sigma(k) = v_k,\ldots, \sigma(2k-2)=v_{2k-2}$ is bounded by $\mbox{card}S_{k-1}(v_1),$ since all values of the injection are fixed except for $\sigma(2),\ldots,\sigma(k-1).$ But since $v_1\in V_1$ implies that the degree $d_{v_1}\leq 16 np$ and, thus, $d_{v_1}^{+}\leq 16 np,$ we have by (\[path\]) $$\mbox{card}S_{k-1}(v_1)\leq C(k)(np)(np)^{k-3}\leq C(k)(np)^{k-2}.$$ Therefore, the cardinality of all injections such that $\sigma(1)\in V_1$ is bounded by $$C(k) (\mbox{card}{\cal P}) (np)^{k-2} \leq C(k) Z(x) (np)^{k-2},$$ which accounts for the first term in the statement of the theorem.
Proof of Theorem 3.
===================
Theorem 2 implies that for any $a\in {\mathbb{R}}$ and $t>0,$ $$\begin{aligned}
&&
{{\Bbbb P}}\Bigl(
Z\leq a +\sqrt{Vt}
\Bigr)
\leq
{{\Bbbb P}}\Bigl(
Z\leq a +\Bigl(C(k)
\bigl((np)^{k-2} Z +(np)^{2(k-1)}\bigr)t\Bigr)^{1/2}
\Bigr)
\\
&&
+
{{\Bbbb P}}\Bigl(
V \geq C(k)((np)^k Z +(np)^{2k})
\Bigr)
\\
&&
\leq
{{\Bbbb P}}\Bigl(
Z\leq a +\Bigl(C(k)
\bigl((np)^{k-2} Z +(np)^{2(k-1)}\bigr)t\Bigr)^{1/2}
\Bigr)
+
\exp\Bigl(
-\frac{(np)^2}{C(k)\log\log np}
\Bigr),\end{aligned}$$ which implies that $$\begin{aligned}
{{\Bbbb P}}\Bigl(
Z\geq a +\Bigl(C(k)
\bigl((np)^{k-2} Z +(np)^{2(k-1)}\bigr)t\Bigr)^{1/2}
\Bigr)
&\leq &
{{\Bbbb P}}\Bigl(
Z\geq a +\sqrt{V t}
\Bigr)
\\
& + &
\exp\Bigl(
-\frac{(np)^2}{C(k)\log\log np}
\Bigr).\end{aligned}$$ Multiplying both sides by ${{\Bbbb P}}(Z\leq a)$ and using Theorem 1 with $t=2\eps (np)^2$ we get $$\begin{aligned}
&&
{{\Bbbb P}}\Bigl(
Z\geq a +\Bigl(C(k) \eps
\bigl((np)^k Z +(np)^{2k}\bigr)\Bigr)^{1/2}
\Bigr)
{{\Bbbb P}}\Bigl(
Z\leq a
\Bigr)
\nonumber
\\
&&
\leq
{{\Bbbb P}}(Z\leq a)\exp\Bigl(
-\frac{(np)^2}{C(k)\log\log np}
\Bigr)
+
e^{-\eps(np)^2}.
\label{final}\end{aligned}$$ If we take $$a=M - \sqrt{C(k) \eps ((np)^k M +(np)^{2k})}
\label{a}$$ then, clearly, the following two events are equal $$\{Z\geq a +\sqrt{C(k) \eps ((np)^k Z +(np)^{2k})}\} =
\{ Z\geq M\},$$ and, therefore, (\[final\]) implies that for $np\geq C(k)$ for large enough $C(k)>0,$ $${{\Bbbb P}}\Bigl(
Z\leq M - \sqrt{C(k) \eps ((np)^k M +(np)^{2k})}
\Bigr) \leq
3e^{-\eps(np)^2}.$$ Since ${{\Bbbb E}}Z \geq a{{\Bbbb P}}(Z\geq a),$ the choice of $a$ as in (\[a\]) gives $$M - \sqrt{C(k) \eps ((np)^k M +(np)^{2k})}\leq
{{\Bbbb E}}Z \Bigl(1-3e^{-\eps(np)^2}\Bigr)^{-1}.$$ This, clearly, implies the first statement of Theorem 3.
To prove the second statement we use (\[final\]) with $a=M,$ and assume that $np$ is large enough, so that $M\leq (1+\eps){{\Bbbb E}}Z.$ Then with probability at least $$1-2e^{-\eps (np)^2} -
\exp\Bigl(
-\frac{(np)^2}{C(k)\log\log np}
\Bigr)$$ we have $$Z\leq (1+\eps){{\Bbbb E}}Z +\sqrt{C(k) \eps ((np)^k Z +(np)^{2k})}.$$ Since ${{\Bbbb E}}Z \sim (np)^k,$ for small enough $\eps$ this implies that $Z\leq 2{{\Bbbb E}}Z,$ which completes the proof of the second statement of Theorem 3.
[**Acknowledgment.**]{} We want to thank anonymous referee for helpful comments, especially, for suggesting the present formulation of Theorem 1.
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S. Boucheron, G. Lugosi, P. Massart, Concentration inequalities using the entropy method, To appear in [*Ann. Probab.*]{} (2002).
E. Friedgut, J. Kahn, On the number of copies of one hypergraph in another, [*Israel Journal of Mathematics*]{} [**105**]{} (1998), 251 - 256.
S. Janson, A. Ruciński, The infamous upper tail. Probabilistic methods in combinatorics, [*Random Structures Algorithms*]{} [**20**]{} (2002), no.3, 317 - 342.
S. Janson, K. Oleszkiewicz, A. Ruciński, Upper tail for subgraph counts in random graphs, preprint (2002).
J.H. Kim, V.H. Vu, Divide and conquer martingales and the number of triangles in a random graph, preprint (2002).
M. Talagrand, Concentration of measure and isoperimetric inequalities in product spaces, *Publications Mathématiques de l’I.H.E.S. [**81**]{} (1995), 73-205.*
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Department of MathematicsMassachusetts Institute of Technology77 Massachusetts Avenue, Room 2-181Cambridge, MA, 02139-4307 URL: http://www-math.mit.edu/\~panchenke-mail: [email protected]
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We present a comprehensive theory of critical spaces for the broad class of quasilinear parabolic evolution equations. The approach is based on maximal $L_p$-regularity in time-weighted function spaces. It is shown that our notion of critical spaces coincides with the concept of scaling invariant spaces in case that the underlying partial differential equation enjoys a scaling invariance. Applications to the vorticity equations for the Navier-Stokes problem, convection-diffusion equations, the Nernst-Planck-Poisson equations in electro-chemistry, chemotaxis equations, the MHD equations, and some other well-known parabolic equations are given.'
address:
- |
Martin-Luther-Universität Halle-Wittenberg\
Institut für Mathematik\
Theodor-Lieser-Strasse 5\
D-06120 Halle, Germany
- |
Department of Mathematics\
Vanderbilt University\
Nashville, Tennessee\
USA
- |
Universität Regensburg\
Fakultät für Mathematik\
D-93040 Regensburg, Germany
author:
- Jan Prüss
- Gieri Simonett
- Mathias Wilke
title: Critical Spaces For Quasilinear Parabolic Evolution Equations and Applications
---
[^1]
Introduction
============
In the last decades there has been an increasing interest in finding critical spaces for nonlinear parabolic partial differential equations. There is an extensive literature on this program, but so far a general unified approach seems to be lacking and each equation seems to require its own theory.
As a matter of fact, there is no generally accepted definition in the mathematical literature concerning the notion of critical spaces. One possible definition may be based on the idea of a ‘largest space of initial data such that the given PDE is well-posed.’ However, this is a rather vague concept that requires additional clarifying information. Critical spaces are often introduced as ‘scaling invariant spaces,’ provided the underlying PDE enjoys a scaling invariance. A prototype example is given by the Navier-Stokes problem on ${{\mathbb R}}^d$, $$\begin{aligned}
\partial_t u + u\cdot \nabla u -\Delta u +\nabla\pi =0,\quad
{\rm div}\, u =0, \quad
u(0)= u_0,
\end{aligned}$$ which is invariant under the scaling $$(u_\lambda (t,x),\pi_\lambda(t,x)):=(\lambda u(\lambda^2 t,\lambda x),\lambda^2 \pi(\lambda^2 t,\lambda x)).$$ In this case one shows that the spaces $L_d({{\mathbb R}}^d)$ and $\dot B^{d/q-1}_{qp}({{\mathbb R}}^d)$ are scaling invariant for $u$, and thus are ‘critical spaces.’
Clearly, this latter concept of ‘critical space’ breaks down as soon as a given equation fails to have a scaling invariance.
In this paper we present a comprehensive theory of critical spaces for the broad class of quasilinear parabolic evolution equations. Our approach is based on the concept of maximal $L_p$-regularity in time-weighted function spaces. In this framework, we introduce the notion of a ‘critical weight’ $\mu_c$ and a corresponding ‘critical space’ $X_c=X_{\gamma,\mu_c}$. We will show that
1. $X_{c}$ is, in a generic sense, the largest space of initial data for which the given equation is well-posed.
2. $X_{c}$ is scaling invariant, provided the given equation admits a scaling.
The spaces $X_{c}$, thus, encompass and combine the properties mentioned above. We shall also show that this definition of ‘critical space’ awards us with considerable flexibility in choosing an appropriate setting for analyzing a given equation. For instance, it turns out that the critical spaces $X_{c}$ are independent within the scale of interpolation-extrapolation spaces associated with a given partial differential equation (in a sense to be made more precise below).
With our approach, we are able to recover many known results in a unified way, and on the other side, we will be able to add a variety of new results for some well-known partial differential equations.
The concept of ‘critical weight’ was first introduced by Prüss and Wilke in [@PrWi17] and was then applied to the Navier-Stokes equations by Prüss and Wilke [@PrWi17; @PrWi17a], and to the quasi-geostrophic equations by Prüss [@Pru17].
In this paper, we elaborate on the properties of the critical spaces $X_{c}$ alluded to above. In addition, we include applications to the Cahn-Hilliard equations, the vorticity equations for the Navier-Stokes problem, convection-diffusion equations, the Nernst-Planck-Poisson equations in electro-chemistry, chemotaxis equations, and the MHD equations.
For the reader’s convenience we now state and explain the basic underlying result on quasilinear parabolic evolution equations obtained recently in Prüss and Wilke [@PrWi17].
Let $X_0,X_1$ be Banach spaces such that $X_1$ embeds densely in $X_0$, let $p\in(1,\infty)$ and $1/p<\mu\leq1$. We consider the following quasilinear parabolic evolution equation $$\label{qpp}
\dot{u} +A(u)u = F_1(u)+F_2(u),\; t>0,\quad u(0)=u_1.$$ The space of initial data will be the real interpolation space $X_{\gamma,\mu} =(X_0,X_1)_{\mu-1/p,p}$, and the state space of the problem is $X_\gamma=X_{\gamma,1}$. Let $ V_\mu\subset X_{\gamma,\mu}$ be open and $u_1\in V_\mu$. Furthermore, let $X_\beta=(X_0,X_1)_\beta$, $\beta\in (0,1)$, denote the complex interpolation spaces. We will impose the following assumptions.
[**(H1)**]{} $(A,F_1)\in C^{1-}(V_\mu; {{\mathcal B}}(X_1,X_0)\times X_0)$.
[**(H2)**]{} $F_2: V_\mu\cap X_\beta \to X_0$ satisfies the estimate $$|F_2(u_1)-F_2(u_2)|_{X_0} \leq C \sum_{j=1}^m (1+|u_1|_{X_\beta}^{\rho_j}+|u_2|_{X_\beta}^{\rho_j})|u_1-u_2|_{X_{\beta_j}},$$ $ u_1, u_2\in V_\mu\cap X_\beta$, for some numbers $m\in{{\mathbb N}}$, $\rho_j\geq 0$, $\beta\in (\mu-1/p,1)$, $\beta_j\in [\mu-1/p, \beta]$, where $C$ denotes a constant which may depend on $|u_i|_{X_{\gamma,\mu}}$. The case $\beta_j=\mu-1/p$ is only admissible if [**(H2)**]{} holds with $X_{\beta_j}$ replaced by $X_{\gamma,\mu}$.
[**(H3)**]{} For all $j=1,\ldots,m$ we have $$\rho_j( \beta-(\mu-1/p)) + (\beta_j -(\mu-1/p)) \leq 1 -(\mu-1/p).$$
Allowing for equality in [**(H3)**]{} is not for free and we additionally need to impose the following structural [**Condition (S)**]{} on the Banach spaces $X_0$ and $X_1$.
[**(S)**]{} The space $X_0$ is of class UMD. The embedding $${H}^{1}_p({{\mathbb R}};X_0)\cap L_{p}({{\mathbb R}};X_1)\hookrightarrow {H}^{1-\beta}_{p}({{\mathbb R}};X_\beta),$$ is valid for each $\beta\in [0,1]$.
By the [mixed derivative theorem]{}, Condition [**(S)**]{} is valid if $X_0$ is of class UMD and if there is an operator $A_{\#}\in {{\mathcal H}}^\infty(X_0)$, with domain ${\sf D}(A_\#)=X_1$, and ${{\mathcal H}}^\infty$-angle $\phi_{A_\#}^\infty<\pi/2$. We refer to Prüss and Simonett [@PrSi16 Chapter 4].
The usual [*solution spaces*]{} in the framework of [*maximal $L_p$*]{}-regularity are $$u\in H^1_p((0,a);X_0)\cap L_p((0,a),X_1)\hookrightarrow C([0,a];X_{\gamma,1}),$$ where the [*state space*]{} $X_{\gamma,1}$ is defined by $$X_{\gamma,\mu}=(X_0,X_1)_{\mu-1/p,p},$$ with $\mu=1$. Here we want to advertise for [*time-weighted spaces*]{}, defined by $$u\in L_{p,\mu}((0,a);Y) \quad \Leftrightarrow \quad t^{1-\mu} u \in L_p((0,a);Y)),\quad 1\geq\mu>1/p.$$ The corresponding solution classes in the time weighted case are $$u\in H^1_{p,\mu}((0,a);X_0)\cap L_{p,\mu}((0,a),X_1)\hookrightarrow C([0,a];X_{\gamma,\mu}).$$ There are several compelling reasons for time weights, among them the following:
- Reduced initial regularity,
- Instantaneous gain of regularity,
- Compactness properties of orbits.
Important is the fact that maximal regularity is independent of $\mu\in (1/p,1]$. In the $L_p$-framework, this was first observed by Prüss and Simonett [@PrSi04].
In Prüss and Wilke [@PrWi17] the following extension of Theorem 2.1 in LeCrone, Prüss and Wilke [@LPW14] was obtained.
\[main\] Suppose that the structural assumption [**(S)**]{} holds, and assume that hypotheses [**(H1), (H2), (H3)**]{} are valid. Fix any $u_0\in V_\mu$ such that $A_0:= A(u_0)$ has maximal $L_p$-regularity. Then there is $a=a(u_0)>0$ and $\varepsilon =\varepsilon(u_0)>0$ with $\bar{B}_{X_{\gamma,\mu}}(u_0,\varepsilon) \subset V_\mu$ such that problem admits a unique solution $$u(\cdot, u_1)\in H^1_{p,\mu}((0,a);X_0)\cap L_{p,\mu}((0,a); X_1) \cap C([0,a]; V_\mu),$$ for each initial value $u_1\in \bar{B}_{X_{\gamma,\mu}}(u_0,\varepsilon).$ Furthermore, there is a constant $c= c(u_0)>0$ such that $$|u(\cdot,u_1)-u(\cdot,u_2)|_{{{\mathbb E}}_{1,\mu}(0,a)} \leq c|u_1-u_2|_{X_{\gamma,\mu}}$$ holds for all $u_1,u_2\in \bar{B}_{X_{\gamma,\mu}}(u_0,\varepsilon)$.
We call $j$ subcritical if strict inequality holds in [**(H3)**]{}, and critical otherwise. As $\beta_j\leq \beta<1$, any $j$ with $\rho_j=0$ is subcritical. Furthermore, [**(H3)**]{} is equivalent to $ \rho_j\beta+\beta_j -1\leq \rho_j(\mu-1/p)$, hence the minimal value of $\mu$ is given by $$\mu_{c} = \frac{1}{p} + \beta -\min_j(1-\beta_j)/\rho_j.$$ We call this value the [*critical weight*]{}. Thus Theorem \[main\] shows that we have local well-posedness of for initial values in the space $X_{\gamma,\mu_{}}$. Therefore, it is meaningful to name this space the [*critical space*]{} for .
Note that the critical space $X_{\gamma,\mu_c}$ is given by the real interpolation space $$X_{\gamma,\mu_c}= (X_0,X_1)_{\mu_c-1/p,p},$$ and $\mu_{c} -\frac{1}{p} =\beta -\min_j(1-\beta_j)/\rho_j$ is independent of $p$. Therefore, the exponent $p$ only shows up as a microscopic parameter.
Recall the embeddings $$\begin{aligned}
&(X_0,X_1)_{\alpha,p_1}\hookrightarrow (X_0,X_1)_{\alpha,p_2}\hookrightarrow (X_0,X_1)_{\beta,1},
\quad \mbox{for } p_1\leq p_2,\; 0<\alpha<\beta<1,\\
&(X_0,X_1)_{\beta,1}\hookrightarrow (X_0,X_1)_\beta\hookrightarrow (X_0,X_1)_{\beta,\infty}.
\end{aligned}$$ The philosophy is to choose $p$ large, say $1/p< 1-\beta$. We then have $$X_{\gamma,1} = (X_0,X_1)_{1-1/p,p}\hookrightarrow X_\beta\hookrightarrow X_{\gamma,\mu_c},$$ as $\mu_c-1/p<\beta <1-1/p$. As a consequence, the qualitative theory of Köhne, Prüss and Wilke [@KPW10] and Prüss, Simonett and Zacher [@PSZ09] is available; see also Prüss and Simonett [@PrSi16], Chapter 5.
In Section 2 we give an example which shows that Theorem \[main\] is optimal in a generic sense (but might not be if additional structural properties hold). This means in specific applications that our theory yields generic lower bounds for the critical space. Nevertheless, we find in all PDE applications considered so far that the critical spaces obtained by our theory coincide with the known ones, which in most cases come from (local) scaling invariance. Also, a PDE equation can often be considered in a scale of function spaces, and then it turns out that the critical spaces are widely independent of this scale.
Semilinear parabolic evolution equations with bilinear nonlinearities
=====================================================================
This is a special case of the [quasilinear theory]{} presented above, and it encompasses many important differential equations in fluid dynamics, physics, and chemistry; for instance, the Navier-Stokes equations, vorticity equations for the Navier-Stokes problem, quasi-geostrophic (subcritical) equations, convection-diffusion equations, Nernst-Planck-Poisson equations, magneto-hydrodynamics, and many more.
The section is organized as follows. In Subsection 2.1 we first formulate a result on local well-posedness for equation and give a sketch of the proof for the critical case. The ingredients of the proof for the critical case rely in an essential way on the mixed derivative theorem and sharp embedding results for time-weighted Sobolev spaces. Corollaries \[cor2\] and \[cor1\] describe conditions for global existence, while Theorem \[thm5\] contains a result of Serrin type which states that global existence is equivalent to an integral a priori bound. In Subsection 2.2 we show by means of a counterexample that the critical spaces $X_{\gamma,\mu_c}$ identified in Theorem \[thm1\] are - in a generic sense - the largest spaces of initial data for which equation is well-posed. In Subsection 2.3 it is shown that the (homogeneous versions of) critical spaces are scaling invariant, provided equation admits a scaling. Finally, in Subsection 2.4 we show that the critical spaces are invariant with respect to the extrapolation-interpolation scale associated with the operator $A$ in equation . Here we would like to mention that similar results are also true (with appropriate modifications) for semilinar parabolic equations with multilinear nonlinearities.
Local and global existence of solutions
---------------------------------------
Let $X_0$ be a UMD-space, $X_1\hookrightarrow X_0$ densely, $A:X_1\to X_0$ bounded and such that $A\in\mathcal{BIP}(X_0)$ with power angle $\theta_A<\pi/2$, and let $p\in(1,\infty)$. Consider the semilinear parabolic evolution equation $$\label{3.1}
\partial_t u +Au = G(u,u),\; t>0,\quad u(0)=u_0.$$ Here $G:X_\beta\times X_\beta \to X_0$ is bilinear and bounded, with $$X_\beta = (X_0,X_1)_\beta = {\sf D}(A^\beta),$$ for some $\beta\in [0,1)$.
Our main result for reads as follows.
\[thm1\] Assume $p\in (1,\infty)$, $\mu\in (1/p,1]$, $\beta\in (\mu-1/p,1)$ and $$\label{3.2}
2\beta-1 \leq \mu-1/p.$$ Then for each initial value $u_0\in X_{\gamma,\mu}$ there is $a=a(u_0)>0$ and a unique solution of (\[3.1\]) in the class $$u\in H^1_{p,\mu}((0,a);X_0)\cap L_{p,\mu}((0,a),X_1)\hookrightarrow C([0,a];X_{\gamma,\mu}).$$ The solution exists on a maximal time interval $[0,t_+(u_0))$ and depends continuously on the data and moreover satisfies $$u\in H^1_{p,loc}((0,t_+);X_0)\cap L_{p,loc}((0,t_+),X_1)\hookrightarrow C((0,t_+);X_{\gamma,1}).$$ Hence it regularizes instantly, provided $\mu<1$.
Hence, $\mu$ is [*subcritical*]{} for if strict inequality holds in , and [*critical*]{} otherwise. The case $\beta\leq 1/2$ is always subcritical, and if $\beta>1/2$ then $\mu_c:= 2\beta -1+1/p$ is the [critical weight]{} and $$X_{\gamma,\mu_c} = D_A(2\beta-1,p)$$ is the [critical space]{} for (\[3.1\]). We observe again that $p$ appears only as a microscopic parameter.
[**Main idea for the proof of Theorem \[thm1\]**]{}. The semilinear case with bilinear nonlinearity is considerably simpler than the quasilinear case , and we give here an outline of the proof for this simpler case. The arguments then are rather short and employ the contraction mapping principle and several sharp embeddings. We focus on the critical case, i.e. we assume that $\beta>1/2$ and we choose $\mu=\mu_c=2\beta -1 +1/p$.
The fixed point equation reads $$v = Tv:=e^{-At}* G(u,u), \quad v=u-u_*, \quad u_*= e^{-At}u_0,$$ in a ball ${{\mathbb B}}_r:= \bar{B}_{{_0{{\mathbb E}}}_\mu(a)}(0,r)$ in the space $${_0{{\mathbb E}}}_\mu(a) = {_0H}^1_{p,\mu}((0,a);X_0)\cap L_{p,\mu}((0,a);X_1),$$ where $r$ and $a\in (0,a_0)$, for some fixed $a_0$, are at our disposal. Here ${_0H}^1_{p,\mu}((0,a);X_0)$ denotes the functions $v\in H^1_{p,\mu}((0,a);X_0)$ with $v(0)=0$.
We estimate as follows, with maximal regularity constant $M=M(a_0)\geq1$ and $1-\mu= 2(1-\tau)$, $$\begin{aligned}
|Tv|_{{_0{{\mathbb E}}_\mu(a)}} &\leq M |G(u,u)|_{L_{p,\mu}(X_0)} \leq MC | |u|^2_{X_\beta}|_{L_{p,\mu}}\\
& =MC |u|^2_{L_{2p,\tau}(X_\beta)}\leq 2MC [ |u_*|^2_{L_{2p,\tau}(X_\beta)}+|v|^2_{L_{2p,\tau}(X_\beta)}].
\end{aligned}$$ As $u_0\in D_A(2\beta-1,p)\subset D_A(2\beta-1,2p)=D_A(\beta -1 + \tau -1/2p, 2p)$ we obtain $$u_*\in L_{2p,\tau}((0,a);X_\beta),$$ see [@PrSi16 Proposition 3.4.3]. The first term $2MC|u_*|^2$ can be made small, say smaller than $r/2$, by choosing $a\in (0,a_0)$ small. Next we have the embeddings $$\begin{aligned}
{_0{{\mathbb E}}}_{\mu}(a) &\hookrightarrow {_0H}^{1-\beta}_{p,\mu}((0,a); X_\beta) \quad \mbox{(mixed derivative theorem)}\\
& \hookrightarrow L_{2p,\tau}((0,a);X_\beta)\quad \mbox{(Sobolev embedding)},
\end{aligned}$$ as the Sobolev indices for these spaces are the same, $$1-\beta -1/p -(1-\mu) = -1/2p-(1-\tau) \quad \mbox{\Black{as}} \quad 2\beta-1= \mu -1/p.$$ [We refer to [@MeVe12 Corollary 1.4] for embedding results in weighted Bessel-potential spaces.]{} Note that the embedding constants do not depend on $a>0$. So choosing $r>0$ small enough, the remaining term $2MC|v|^2$ will also be small, say smaller than $r/2$. This shows that $T:{{\mathbb B}}_r\to {{\mathbb B}}_r$ is a self-map. The contraction property is proved in a similar way, by the estimate $$\begin{aligned}
|Tv_1-Tv_2|_{{_0{{\mathbb E}}}(a)}&\leq M|G(u_1,u_1)-G(u_2,u_2)|_{L_{p,\mu}(X_0)}\\
&\le MC\Big(|u_1|_{L_{2p,\tau}(X_\beta)}+|u_2|_{L_{2p,\tau}(X_\beta)}\Big) |v_1-v_2|_{L_{2p,\tau}(X_\beta)}.
\quad\square
\end{aligned}$$
Instead of choosing $a>0$ small, we may instead, for a given $a>0$, choose $r>0$ small enough to obtain a unique solution on $(0,a)$. If $0\in\rho(A)$ then the maximal regularity constant is independent of $a$ and $r>0$ may be chosen uniformly in $a>0$, to obtain global existence and exponential stability.
\[cor2\] Let the assumptions of Theorem \[thm1\] hold. Then
1. For any given $a>0$ there is $r=r(a)>0$ such that the solution of (\[3.1\]) exists on $[0,a]$, whenever $|u_0|_{X_{\gamma,\mu}}\leq r$.
2. If $0\in\rho(A)$ then $r>0$ is independent of $a$.
3. If $0\in\rho(A)$ and $1/p<1-\beta$, then the trivial solution of (\[3.1\]) is exponentially stable in the state space $X_{\gamma,1}$. Moreover, there is $r_0>0$ such that the solution $u(t)$ of (\[3.1\]) converges exponentially to zero in $X_{\gamma,1}$, provided $|u_0|_{X_{\gamma,\mu}}\leq r_0$.
Concerning [*conditional global existence*]{} we can prove the following result.
\[cor1\] The local solution of Theorem \[thm1\] exists globally, provided
1. $u([0,t_+))\subset X_{\gamma,\mu}$ is bounded in the subcritical case; or
2. $u([0,t_+))\subset X_{\gamma,\mu_c}$ is relatively compact in the critical case.
[**(i)**]{} Suppose $\mu$ is subcritical, i.e. $\mu-1/p>2\beta-1$, and let $$\alpha:=\frac{\beta -(\mu-1/p)}{1-(\mu-1/p)}.$$ By interpolation theory we obtain $(X_{\gamma,\mu}, X_1)_{\alpha,1}=(X_0,X_1)_{\beta,1}\hookrightarrow X_\beta.$ Hence, $$|G(u,u)|_{X_0}\le C |u|^2_{X_\beta}\le C |u|^{2\alpha}_{X_1} |u|^{2(1-\alpha)}_{X_{\gamma,\mu}}.$$ To establish global existence we may assume w.l.o.g that $u_0\in X_{\gamma,1}$. Let $u$ be the unique solution of with initial value $u_0\in X_{\gamma,1}$, defined on its maximal existence interval $[0,t_+(u_0))$ and suppose that $t_+:=t_+(u_0)<\infty$. Then we obtain for any $a\in (0,t_+)$ by means of maximal regularity $$\begin{aligned}
|u|_{{{\mathbb E}}_1(a)}
&\le C_1 |u_0|_{X_{\gamma,1}} + M |G(u,u)|_{L_p((0,a);X_0)}\\
& \le C_1 |u_0|_{X_{\gamma,1}} + MC |u|^{2(1-\alpha)}_{L_\infty((0,t_+); X_{\gamma,\mu})} |u|^{2\alpha}_{{{\mathbb E}}_1(a)} \\
&\le C_1 |u_0|_{X_{\gamma,1}} + C_2 |u|^{2\alpha}_{{{\mathbb E}}_1(a)},
\end{aligned}$$ where $M=M(t_+)$ is the constant of maximal regularity. As $2\alpha<1$, we conclude that $|u|_{{{\mathbb E}}_1(a)}$ is bounded, uniformly in $a\in (0,t_+)$. Therefore, $|u|_{{{\mathbb E}}_1(t_+)}$ is bounded, implying that $u\in C([0,t_+]; X_{\gamma,1})$. Hence, the solution can be continued beyond $t_+$, a contradiction.\
[**(ii)**]{} Let $u_0\in X_{\gamma,\mu_c}$ be given and suppose that $t_+=t_+(u_0)<\infty$. Then the set $\Gamma:= \overline{u([0,t_+))}$ is compact in $X_{\gamma,\mu_c}$. It follows from Theorem \[thm1\] and a covering argument that there is $\delta>0$ such that equation has for each $v_0\in \Gamma$ a unique solution $v\in {{\mathbb E}}_{1,\mu_c}(\delta)$. But this implies that the solution $u$ can be continued beyond $t_+$, leading once more to a contradiction.
Next we prove a result of Serrin type which states that global existence is equivalent to an integral a priori bound.
\[thm5\] Let $p\in (1,\infty)$, $\beta>1/2$ and $\mu :=2\beta -1 +1/p\leq 1$ the critical weight. Assume $u_0\in X_{\gamma,\mu}$, and let $u$ denote the unique solution of defined by Theorem \[thm1\], with maximal interval of existence $[0,t_+)$. Then
1. $u\in L_p((0,a);X_\mu)$, for each $a<t_+$.
2. If $t_+<\infty$ then $ u\not\in L_p((0,t_+);X_\mu)$.
In particular, the solution exists globally if $u\in L_p((0,a);X_\mu)$ for any finite number $a$ with $a\le t_+$.
We remind that $X_\mu=(X_0,X_1)_\mu$ denote the complex interpolation spaces. It is interesting to observe that the Sobolev indices of the spaces ${{\mathbb E}}_\mu(a)$, $L_p((0,a);X_\mu)$ and $C([0,a];X_{\gamma,\mu})$ are all the same, given by $\mu-1/p$.
[**(i)**]{} Let $a\in (0,t_+)$ be fixed. By the mixed derivative theorem of Sobolevskii and [Sobolev embedding in weighted spaces, see for instance [@MeVe12 Corollary 1.4],]{} we have $${{\mathbb E}}_\mu(a)= H^1_{p,\mu}((0,a); X_0)\cap L_{p,\mu}((0,a);X_1)\hookrightarrow H^{1-\mu}_{p,\mu}((0,a);X_\mu)\hookrightarrow L_p((0,a);X_\mu).$$
[**(ii)**]{} From the mixed derivative theorem, Proposition \[interpol\] in the appendix, and [Sobolev embedding in weighted spaces]{} follows $$\begin{aligned}
(L_{p}((0,a);X_\mu), {{\mathbb E}}_\mu(a))_{1/2} &\hookrightarrow (L_{p}((0,a);X_\mu), H^{1-\alpha}_{p,\mu}((0,a);X_\alpha))_{1/2}\\
&= H^{(1-\alpha)/2}_{p,(1+\mu)/2}((0,a);X_{(\mu+\alpha)/2}) \hookrightarrow L_{2p,\tau}((0,a); X_\beta),\end{aligned}$$ where $\alpha=1-1/p$ and $2(1-\tau)=1-\mu$, i.e. $ \tau =(1+\mu)/2$.\
Suppose $t_+<\infty$ and let $a_0\in (0,t_+)$ be fixed. Employing the interpolation inequality, the quadratic estimate $|G(u,u)|_{X_0}\leq C_1 |u|_{X_\beta}^2$ implies $$|G(u,u)|_{L_{p,\mu}((a_0,a);X_0)} \leq C_1 |u|^2_{L_{2p,\tau}((a_0,a);X_\beta)}
\leq C_2 |u|_{L_p((a_0,a);X_\mu)} |u|_{{{\mathbb E}}_\mu(a_0,a)},$$ where the constant $C_2$ is independent of $a\in (a_0,t_+)$. Let $M$ be the constant of maximal regularity for the interval $[0,t_+)$, and let $\eta=1/(2MC_2)$. We choose $t_0\in (a_0,t_+)$ sufficiently close to $t_+$ such that $|u|_{L_p((t_0,t_+);X_\mu)}\le \eta$. Then by maximal regularity we obtain $$|u|_{{{\mathbb E}}_{\mu}(t_0,a)} \leq M\big(|u(t_0)|_{X_{\gamma,\mu}} + C_2\eta |u|_{{{\mathbb E}}_{\mu}(t_0,a)}\big).$$ By the definition of $\eta$, this yields $$|u|_{{{\mathbb E}}_{\mu}(t_0,a)} \leq 2M|u(t_0)|_{X_{\gamma,\mu}}$$ for any $a\in (t_0,t_+)$. Therefore, $u|_{(t_0,t_+)}\in {{\mathbb E}}_1(t_0,t_+)\hookrightarrow C([t_0,t_+],X_\mu)$. This implies that the solution $u$ can be continued beyond $t_+$, leading to a contradiction.
A Counterexample
----------------
We want to show that Theorem \[thm1\], and hence also Theorem \[main\], are optimal in the sense that, generically, is not well-posed in spaces strictly larger than the critical space $X_{\gamma,\mu_c}$.
[**(i)**]{} We choose a sequence $a_k>0$ with $a_k\to\infty$, and set $$X_0 =l_2({{\mathbb N}}),\quad (Au)_k= a_k u_k,\quad X_1={\sf D}(A)= l_2({{\mathbb N}};a_k).$$ This operator is selfadjoint and positive definite in the Hilbert space $X_0$. Furthermore, [by complex interpolation in weighted $l_2$-spaces]{} we have $X_\beta = l_2({{\mathbb N}};a_k^\beta)={\sf D}(A^\beta)$, for all $\beta\geq0$. Next we define a symmetric bilinear operator $G$ by means of $$G:X_\beta\times X_\beta \to X_0,\quad G(u,v)_k := a_k^{2\beta} u_k v_k.$$ Obviously, $G$ is bilinear, and it also bounded by the Cauchy-Schwarz inequality, as $l_2({{\mathbb N}})\hookrightarrow l_4({{\mathbb N}})$.
Consider the evolution equation $$\label{1}
\partial_t u +Au =G(u,u),\; t>0,\quad u(0)=u_0,$$ in $X_0$. Then we are in the situation of Theorem \[thm1\] hence we have local well-posedness in $L_{p,\mu}$ for initial data in $X_{\gamma,\mu}=D_A(\mu-1/p,p)$, for all $\mu\geq\mu_c$, where the critical weight $\mu_c$ is defined by $$2\beta-1 = \mu_c-1/p.$$ Below we require $\beta>1/2$ and $\beta \leq 1-1/2p$. By an appropriate choice of the coefficients $a_k$, we want to show that this problem is ill-posed for any weight $1/p<\mu<\mu_c$, showing that Theorem \[thm1\] cannot be improved, and hence the condition is sharp.
[**(ii)**]{} In components, problem reads $$\begin{aligned}
\partial_t u_k +a_k u_k &= a_k^{2\beta} u_k^2,\quad t>0,\; k\in{{\mathbb N}},\\
u_k(0) &= u^0_k.\end{aligned}$$ This system can be solved explicitly. In fact, by means of the scaling $v_k(t)= a_k^{2\beta-1} u_k(t/a_k)$, the system transforms to $$\begin{aligned}
\partial_t v_k +v_k &= v_k^2,\quad t>0,\; k\in{{\mathbb N}},\\
v_k(0) &= v^0_k= a_k^{2\beta-1}u_k^0.\end{aligned}$$ Solving the bi-stable equation $$\partial_t w +w = w^2,\; t>0,\quad w(0)=w_0,$$ we get $$w(t)= \frac{w_0 e^{-t}}{ 1-w_0 +e^{-t}w_0} = \frac{w_0 e^{-t}}{ 1-w_0(1- e^{-t})}.$$ So $w(t)$ exists globally to the right if $w_0\leq1$ and a blow up occurs if $w_0>1$. This yields for $v_k(t)$ the formula $$v_k(t) = \frac{v_k^0 e^{-t}}{1-v_k^0(1-e^{-t})},$$ hence inverting the scaling $$u_k(t) = \frac{ u_k^0 e^{-a_k t}}{1-a_k^{2\beta-1} u_k^0(1-e^{-a_kt})},\quad t>0,\; k\in{{\mathbb N}}.$$ This implies that whenever a solution of class $L_{p,\mu}$, i.e. $$u\in H^1_{p,\mu}((0,a); X_0)\cap L_{p,\mu}((0,a);X_1)$$ exists, then the initial value $u_0$ must satisfy $$\label{nec-cond}
\overline{\lim}_{k\to\infty} a_k^{2\beta-1}u_k^0 \leq 1.$$ [**(iii)**]{} Now suppose that $u_k^0\geq 0$ for all $k$, and set $a_k = 2^k$. If holds, then $|u_k^0|\leq c 2^{k(1-2\beta)}$, for some constant $c>1$, hence $$|A^s u_0|^2_{X_0} = \sum_{k\geq1} 2^{2ks}|u_k^0|^2 \leq c\sum_{k\geq1} 2^{2k(s-2\beta +1)} <\infty,$$ for $s<2\beta-1$. This implies that whenever a solution of class $L_{p,\mu}$ exists, then $u_0\in {\sf D}(A^s)$ for all $s<2\beta-1$.
We thus find ourselves in the following situation: If there is some $\mu\in (1/p,\mu_c)$ and some initial value $u_0\in X_{\gamma,\mu}=(X_0,X_1)_{\mu-1/p,p}$ such that has a solution in the class $H^1_{p,\mu}((0,a);X_1)\cap L_{p,\mu}((0,a);X_0)$ then $u_0$ must be in $\bigcap_{s<2\beta-1}{\sf D}(A^s)$, which is a space strictly contained in $(X_0,X_1)_{\mu-1/p,p}$. Hence the assumption that the problem is well-posed in the class $L_{p,\mu}$ for $\mu<\mu_c$ leads to a contradiction.
[**(iv)**]{} Note that in this example we have global existence in $L_{p,\mu}$ for any $1\geq\mu >1/p$ if the initial data are non-positive. In that case we have the estimates $$|u_k(t)|\leq |u_k^0|e^{-a_kt},\; t\geq0,\; k\in{{\mathbb N}},$$ hence the solution of the nonlinear problem is dominated by the semigroup $e^{-At}$. This shows that in this case we have well-posedness for all initial data, even for $u_0\in X_0$.
We may modify the example in several ways. Replacing the nonlinearity by $-w^3$ we obtain global $L_{p,\mu}$-solutions for all initial values $u_0\in D_A(\mu-1/p,p)$, as well as global exponential stability. And if we replace the nonlinearity by $w|w|$ the signs of the initial conditions are not important.
Such examples show that it may very well happen that is still well-posed in spaces of initial data larger than $X_{\gamma,\mu_c}$, but this involves more structural properties of the equation under consideration.
Scaling Invariance
------------------
Let $A\in\mathcal{BIP}(X_0)$ and $G:\dot{X}_\beta\times \dot{X}_\beta\to X_0$ bounded bilinear. Here $\dot{X}_\beta$ means the completion of ${\sf D}(A^\beta)$ in the homogeneous norm $|A^\beta\cdot|$. In a similar way we define the spaces $\dot{D}_A(\alpha,p)$ as the completions of $D_A(\alpha,p)$ in the homogeneous norms $$|x|_{\dot{D}_A(\alpha,p)} = [\int_0^\infty |r^\alpha A(r+A)^{-1}x|^pdr/r]^{1/p}.$$ We begin with a definition.
A family of operators $\{T_\lambda\}_{\lambda>0}\subset {{\mathcal B}}(X_0)\cap {{\mathcal B}}(X_1)$ is a called a [**scaling** ]{} for if the following conditions hold.\
[**(i)**]{} $AT_\lambda =\lambda T_\lambda A$, for all $\lambda>0$;\
[**(ii)**]{} $\lambda T_\lambda G(x,x) = G(T_\lambda x, T_\lambda x)$, for all $\lambda>0, \; x\in \dot{X}_\beta$;\
[**(iii)**]{} there are constants $c>0$ and $\delta\in {{\mathbb R}}$ such that $$c^{-1}\lambda^{-\delta} |x|\leq |T_\lambda x|\leq c\lambda^{-\delta}|x|,\quad \lambda>0,\; x\in X_0.$$
It is easy to see that if $u(t)$ is a solution of , then $u_\lambda(t) = T_\lambda u(\lambda t)$ is again a solution of with initial value $u_\lambda(0) = T_\lambda u_0$.
From [**(i)**]{} we obtain $$(z-A) T_\lambda = T_\lambda z - \lambda T_\lambda A = \lambda T_\lambda (z/\lambda -A),$$ hence $$(z-A)^{-1} T_\lambda = \frac{1}{\lambda} T_\lambda(z/\lambda -A)^{-1},$$ and so again by [**(i)**]{} $$\label{AT}
A(z-A)^{-1}T_\lambda = T_\lambda A(z/\lambda -A)^{-1}.$$ This implies $$\begin{aligned} |T_\lambda x|_{\dot{D}_A(\alpha,p)} &= [ \int_0^\infty |r^\alpha A(r+A)^{-1}T_\lambda x|^pdr/r]^{1/p} \\
&= \lambda^\alpha [ \int_0^\infty |r^\alpha T_\lambda A(r+A)^{-1} x|^pdr/r]^{1/p},
\end{aligned}$$ which by (iii) yields $$c^{-1}\lambda^{\alpha-\delta} |x|_{\dot{D}_A(\alpha,p)}\leq |T_\lambda x|_{\dot{D}_A(\alpha,p)}\leq c\lambda^{\alpha-\delta} |x|_{\dot{D}_A(\alpha,p)}.$$ In particular, the norm of $T_\lambda x$ in $\dot{D}_A(\alpha,p)$ is, up to constants, independent of $\lambda>0$ if and only if $\alpha=\delta$. Thus the spaces $\dot{D}_A(\delta,p)$ are [*scaling invariant*]{} for the scaling $T_\lambda$. From we also obtain, with an appropriate contour $\Gamma$ $$A^\beta T_\lambda x = \frac{1}{2\pi i}\int_\Gamma z^\beta A(z-A)^{-1} T_\lambda x dz/z= \lambda^\beta T_\lambda A^\beta x.$$ Next we employ [**(ii)**]{} and [**(iii)**]{} to obtain $$\begin{aligned}
c^{-1} \lambda^{1-\delta}|G(x,x)|&\leq |\lambda T_\lambda G(x,x)|= |G(T_\lambda x, T_\lambda x)|\\
&\leq C |A^\beta T_\lambda x|^2 = C |\lambda^\beta T_\lambda A^\beta x|^2 \leq Cc^2 \lambda^{2\beta-2\delta}|A^\beta x|^2.\end{aligned}$$ This implies the inequality $$|G(x,x)|\leq Cc^3 \lambda^{2\beta-1-\delta}|A^\beta x|^2, \quad \lambda>0,\; x\in X_\beta.$$ If $G$ is nontrivial, i.e. $G(x_0,x_0)\neq 0$ for some $x_0\in \dot{X}_\beta$, this implies $\delta = 2\beta-1$, otherwise we obtain a contradiction by letting either $\lambda\to\infty$ or $\lambda\to 0$.
This shows that the critical weight $\mu_c$ satisfies $ \mu_c -1/p =2\beta-1 = \delta,$ thereby proving that the critical spaces for are scaling invariant.
[**Example.**]{} [*The Navier-Stokes equations in ${{\mathbb R}}^d$.*]{}\
Consider the Navier-Sokes problem in ${{\mathbb R}}^d$, which reads $$\begin{aligned}
\dot{u} -\Delta u +\nabla \pi&= -u\cdot\nabla u && \mbox{in }\; {{\mathbb R}}^d,\\
{\rm div}\, u &=0 && \mbox{in }\; {{\mathbb R}}^d,\\
u(0)&=u_0 && \mbox{in }\; {{\mathbb R}}^d.
\end{aligned}$$ This problem has the scaling invariance given by $$u_\lambda(t,x) = \sqrt\lambda u(\lambda t,\sqrt\lambda x),\quad \pi_\lambda(t,x) = \lambda \pi(\lambda t, \sqrt\lambda x),$$ i.e. we have $T_\lambda u(x) =\sqrt\lambda u(\sqrt\lambda x)$. In the $L_q$-setting, it is easy to compute the scaling number $\delta$ in [**(iii)**]{} to the result $\delta= (d/q-1)/2$. Therefore, with $$\dot{X_\beta}= \dot{H}^{2\beta}_{q,\sigma}({{\mathbb R}}^d),\quad \dot{D}_A(\alpha,p) =\dot{B}_{qp,\sigma}^{2\alpha}({{\mathbb R}}^d)$$ as well as $\beta = (d/q + 1)/4$ we obtain that the critical spaces are the scaling invariant spaces $\dot{B}^{d/q-1}_{qp,\sigma}({{\mathbb R}}^d)$.
Independence of the Scale
-------------------------
Consider the complex interpolation-extrapolation scale $(X_s,A_s)$ generated by $A_0:=A\in \mathcal{BIP}(X_0)$ with $X_1={\sf D}(A)$, where we assume w.l.o.g that $A$ is invertible. Suppose that for some $s_0>0$ we have $$G:X_{\beta-s/2}\times X_{\beta-s/2} \to X_{-s}\quad \mbox{bilinear bounded}, \quad s\in[0,s_0]$$ We claim that the critical spaces for are independent of $s$. Assume first that $\beta>1/2$. Then we find $$\mu_c^0 -1/p= 2\beta -1$$ for the critical weight in case $s=0$ and solutions in the class $$u\in H^1_{p,\mu_c^0}(J;X_0) \cap L_{p,\mu_c^0}(J;X_1),$$ for initial values in the critical space $ (X_0, X_1)_{2\beta-1,p}$.
Next fix any $s\in (0,s_0]$ and set $X_0^{\sf w}=X_{-s}$, $A^{\sf w}=A_{-s}$ and $X_1^{\sf w} =X_{1-s}$. Then we have with $\beta^{\sf w}=\beta + s/2$ $$G:X_{\beta^{\sf w}}^{\sf w}\times X_{\beta^{\sf w}}^{\sf w}\to X^{\sf w}_0\quad \mbox{is bilinear and bounded},$$ hence with the critical weight $$\mu_c^s -1/p= 2\beta^{\sf w} -1 = 2\beta -1 +s =\mu_c^0-1/p+s$$ we obtain solutions $$u\in H^1_{p,\mu_c^s}(J;X_0^{\sf w}) \cap L_{p,\mu_c^s}(J;X_1^{\sf w}),$$ for initial values $u_0 \in (X_0^{\sf w}, X_1^{\sf w})_{\mu_c^{\sf w}-1/p,p}$. But by $$(X_0^{\sf w}, X_1^{\sf w})_{\mu_c^{\sf w}-1/p,p}= (X_0^{\sf w}, X_1^{\sf w})_{2\beta^{\sf w}-1,p}= (X_{-s},X_{1-s})_{2\beta -1+s,p}=(X_0,X_1)_{2\beta-1,p},$$ which shows the invariance of the critical spaces w.r.t. $s\in[0,s_0]$.
[**Remark.**]{} There is the restriction $\beta^{\sf w}<1$, which means $s<2-2\beta$. This yields an upper bound for $s$. On the other hand, if $s=0$ is subcritical, i.e if $\beta\leq 1/2$, the problem will be critical in $X_{-s}$ provided $2\beta-1+s>0$ which means $ s> 1-2\beta$. Thus we have the window $$1-2\beta <s<2-2\beta$$ for the best choice of $s$.
Examples of scalar parabolic equations
======================================
In this section we consider three scalar parabolic equations. We begin with a very classical one, namely with a famous problem studied by Fujita and later on by Weissler.
[**Example 1.**]{} Let $\Omega$ be a bounded domain of class $C^{2}$ in ${{\mathbb R}}^d$, and consider the Dirichlet problem $$\label{eq:Ex1}
\begin{aligned}
\partial_t u -\Delta u &= |u|^{\kappa-1} u &&\mbox{in}\;\; \Omega, \\
u&=0 && \mbox{on}\;\; \partial\Omega,\\
u&=u_0 && \mbox{in}\;\; \Omega,\
\end{aligned}$$ where $\kappa>1$.
We consider this example in strong and weak functional analytic settings, to be made precise below. Note that the (local) scaling invariance is given by $$u_\lambda(t,x)= \lambda^{\frac{1}{\kappa-1}}u(\lambda t,\lambda^{\frac{1}{2}}x).$$ *Strong setting:* Let $X_0=L_q(\Omega)$, $1<q<\infty$, $$X_1:=\{u\in H_q^2(\Omega):u|_{\partial\Omega}=0\}$$ and define an operator $A:X_1\to X_0$ by $Au:=-\Delta u$. With this choice, it holds that $$X_\beta=(X_0,X_1)_{\beta}
=
\left\{
\begin{aligned}
& \{u\in H_q^{2\beta}(\Omega):u|_{\partial\Omega}=0\} &&\text{for} && \beta\in (1/2q,1],\\
& H_q^{2\beta}(\Omega) &&\text{for} &&\beta\in [0,1/2q).
\end{aligned}
\right.$$ Define $F(u)=|u|^{\kappa-1}u$ for $u\in X_\beta$. Then $$|F(u)|_{X_0} = |u|_{\kappa q}^\kappa\le C|u|_{X_\beta}^\kappa,$$ and by the fundamental theorem of calculus $$|F(u)-F(\bar{u})|_{X_0}\le C(|u|_{X_\beta}^{\kappa-1}+|\bar{u}|_{X_\beta}^{\kappa-1})|u-\bar{u}|_{X_\beta},$$ provided $H_{q}^{2\beta}(\Omega)\hookrightarrow L_{\kappa q}(\Omega)$, which is the case for $$\beta=\frac{d}{2q}\left(1-\frac{1}{\kappa}\right),$$ hence $q>\frac{d}{2}(1-\frac{1}{\kappa})$, by the constraint $\beta<1$. Setting $\rho_1=\kappa-1$, $\beta_1=\beta$ in **(H2)**, the critical weight $\mu_c$ is given by $$\mu_c=\frac{1}{p}+\frac{\kappa\beta-1}{\kappa-1}=\frac{1}{p}+\frac{d}{2q}-\frac{1}{\kappa-1},$$ which results from **(H3)**. The condition $\mu_c>1/p$ is then equivalent to $\beta>1/\kappa$, hence $$\label{eq:Ex1q}
\frac{d(\kappa-1)}{2\kappa}<q<\frac{d(\kappa-1)}{2}.$$ As $q>1$, this means $d(\kappa-1)/2>1$, hence $\kappa>1+2/d$. Since $\mu_c\le 1$, we obtain the additional relation $$\frac{2}{p}+\frac{d}{q}\le \frac{2\kappa}{\kappa-1}$$ for $p,q\in (1,\infty)$ satisfying . In the sequel, for $s\in (0,1)\backslash\{1/2q\}$, let $${_0}B_{qp}^{2s}(\Omega)=(X_0,X_1)_{s,p}=
\left\{
\begin{aligned}
& \{u\in B_{qp}^{2s}(\Omega):u|_{\partial\Omega}=0\} &&\text{for} && s\in(1/2q,1),\\
& B_{qp}^{2s}(\Omega) &&\text{for} &&s\in (0,1/2q).
\end{aligned}
\right.$$ Then, the critical space is given by $$X_{\gamma,\mu_c}=(X_0,X_1)_{\mu_c-1/p,p}={_0}B_{qp}^{d/q-2/(\kappa-1)}(\Omega).$$ Applying Theorem \[main\], we obtain the following result.
\[thm:Ex1strong\] Let $\kappa>1+2/d$, $p\in (1,\infty)$ and let $q\in (1,\infty)$ satisfy such that $2/p+d/q\le 2\kappa/(\kappa-1)$.
Then, for each [$u_0\in {_0B}^{d/q-2/(\kappa-1)}_{qp}(\Omega)$]{}, problem admits a unique solution $$u \in H^1_{p,\mu_c}((0,a);L_q(\Omega))\cap L_{p,\mu_c}((0,a); H_q^2(\Omega)),$$ for some [$a>0$]{}, with critical weight [$\mu_c = 1/p+ d/2q-1/(\kappa-1)$]{}. The solution exists on a maximal interval [$(0,t_+(u_0))$]{} and depends continuously on [$u_0$]{}. In addition, $$u \in C([0,t_+); {_0B}^{d/q-2/(\kappa-1)}_{qp}(\Omega))\cap C((0,t_+);{_0 B}^{2(1-1/p)}_{qp}(\Omega)),$$ i.e. the solutions regularize instantly if [$2/p +d/q<2\kappa/(\kappa-1)$]{}.
*Weak setting:* Here, we employ the theory of interpolation-extrapolation scales, see the Appendix. Let $A_0:=A$ with domain $X_1$ and $(X_\alpha,A_\alpha)$, $\alpha\in{{\mathbb R}}$, the interpolation-extrapolation scale $(X_\alpha,A_\alpha)$, $\alpha\in{{\mathbb R}}$, generated by $(X_0,A_0)$ with respect to the complex interpolation functor. Let further $1/q+1/q'=1$, $X_0^\sharp=L_{q'}(\Omega)$, $A_0^\sharp=-\Delta|_{X_0^\sharp}$ with domain $$X_1^\sharp=\{u\in H_{q'}^2(\Omega):u|_{\partial\Omega}=0\}.$$ Then $(X_0^\sharp,A_0^\sharp)$ generates the dual scale $(X_\alpha^\sharp,A_\alpha^\sharp)$. In particular, it holds $$X_{1/2}=(X_0,X_1)_{1/2}=\{u\in H_q^1(\Omega):u|_{\partial\Omega}=0\},$$ $$X_{1/2}^\sharp=(X_0^\sharp,X_1^\sharp)_{1/2}=\{u\in H_{q'}^1(\Omega):u|_{\partial\Omega}=0\}$$ and $X_{-1/2}=(X_{1/2}^\sharp)'$, see the Appendix. Moreover, the operator $A_{-1/2}:X_{1/2}\to X_{-1/2}$ is given by $$\langle A_{-1/2}u,\phi\rangle=\int_\Omega\nabla u\cdot \nabla \phi\ dx$$ for all $(u,\phi)\in X_{1/2}\times X_{1/2}^\sharp$, which follows from integration by parts and the density of $X_1$ in $X_{1/2}$.
So in the week setting, we choose $X_0^{\sf w}=X_{-1/2}$, $X_1^{\sf w}=X_{1/2}$ and $A^{\sf w}=A_{-1/2}$ with domain $X_{1/2}$ to rewrite as the semilinear evolution equation $$\label{eq:Ex1weak}
\partial_tu+A^{\sf w} u=F^{\sf w}(u),$$ where $\langle F^{\sf w}(u)|\phi\rangle:=(F(u)|\phi)_{L_2(\Omega)}$. In the sequel let $${_0 H}_q^{r}(\Omega):=
\left\{
\begin{aligned}
&\{u\in H_q^{r}(\Omega):u|_{\partial\Omega}=0\} && \text{for} && r\in (1/q,1],\\
&H_q^{r}(\Omega) && \text{for} && r\in [0,1/q),\\
\end{aligned}
\right.$$ and ${_0}H_q^{-r}(\Omega):=({_0}H_{q'}^{r}(\Omega))'$ if $r\in [0,1]\backslash\{1-1/q\}$. It follows readily from the reiteration property that $$X_\beta^{\sf w}=(X_0^{\sf w},X_1^{\sf w})_{\beta}={_0}H_q^{2\beta-1}(\Omega).$$ For $u\in X_\beta^{\sf w}$ and $\phi\in{_0}H_{q'}^1(\Omega)$, by Hölder’s inequality, we therefore obtain $$|\langle F(u)|\phi \rangle | \le |u|_{L_{r\kappa}(\Omega)}^\kappa|\phi|_{L_{r'}(\Omega)},$$ where $1/r+1/r'=1$. We choose $q,r,q'$ and $r'$ such that $$H_{q}^{2\beta-1}(\Omega)\hookrightarrow L_{r\kappa}(\Omega)\quad\text{and}\quad H_{q'}^1(\Omega)\hookrightarrow L_{r'}(\Omega),$$ to be precise $$2\beta-1-\frac{d}{q}= -\frac{d}{r\kappa}\quad\text{and}\quad 1-\frac{d}{q'}= -\frac{d}{r'}.$$ The last equality is equivalent to $1+d/q= d/r$. This yields $$-\kappa\left(2\beta-1-\frac{d}{q}\right)= 1+\frac{d}{q}$$ or equivalently $$\beta=\frac{1}{2}\left(1+d/q\right)\left(1-1/\kappa\right).$$ Note that the condition $\frac{1}{d}+\frac{1}{q}=\frac{1}{r}<1$ is equivalent to $q>\frac{d}{d-1}$.
Furthermore, the constraint $\beta<1$ leads to $q>d(\kappa-1)/(\kappa+1)$. Under these assumptions we obtain the estimates $$|F^{\sf w}(u)|_{X_0^{\sf w}}\le C|u|_{X_\beta^{\sf w}}^\kappa,$$ and $$|F^{\sf w}(u)-F^{\sf w}(\bar{u})|_{X_0^{\sf w}}\le C(|u|_{X_\beta^{\sf w}}^{\kappa-1}+|\bar{u}|_{X_\beta^{\sf w}}^{\kappa-1})|u-\bar{u}|_{X_\beta^{\sf w}}.$$ From **(H3)** with $\rho_1=\kappa-1$ and $\beta_1=\beta$, we obtain the critical weight $$\mu^{\sf w}_c=\frac{1}{p}+\frac{1}{2}\left(1+\frac{d}{q}\right)-\frac{1}{\kappa-1}=\frac{1}{p}+\frac{d}{2q}+\frac{\kappa-3}{2(\kappa-1)}.$$ which in turn yields the restriction $$\frac{2}{p}+\frac{d}{q}\le \frac{\kappa+1}{(\kappa-1)}$$ as $\mu_c\le 1$. Note that the requirement $\mu_c>1/p$ leads to $$\frac{3-\kappa}{\kappa-1}<\frac{d}{q}$$ which is always satisfied provided $\kappa\ge 3$. We assume this in the sequel.
For $s\in (0,1)\backslash\{1/q\}$, we define the spaces $${_0}B_{qp}^s(\Omega):=(X_0,X_{1/2})_{s,p}=
\left\{
\begin{aligned}
& \{u\in B_{qp}^{s}(\Omega):u|_{\partial\Omega}=0\} && \text{for} && s\in (1/q,1),\\
& B_{qp}^{s}(\Omega) && \text{for} && s\in (0,1/q)
\end{aligned}
\right.$$ and set ${_0}B_{qp}^{-s}(\Omega):=({_0}B_{q'p'}^s(\Omega))'$ for $s\in (0,1)\backslash\{1-1/q\}$. It follows from that $$(X_0^{\sf w},X_1^{\sf w})_{s,p}=(X_{-1/2},X_{1/2})_{s,p}={_0}B_{qp}^{2s-1}(\Omega),$$ for all $s\in (0,1)$, where, by reiteration $${_0}B_{qp}^0(\Omega):=(X_{-1/2},X_{1/2})_{1/2,p}
=(X_{-\alpha},X_{\alpha})_{1/2,p}=B_{qp}^0(\Omega)$$ for any $\alpha\in (0,1/2q)$. Therefore, the critical space in the weak setting is given by $$X_{\gamma,\mu_c}^{\sf w}=(X_0^{\sf w},X_1^{\sf w})_{\mu^{\sf w}_c-1/p,p}={_0}B_{qp}^{d/q-2/(\kappa-1)}(\Omega).$$ Then we have the following result.
\[thm:Ex1weak\] Let $\kappa\ge3$, $p\in (1,\infty)$ and let $q\in \left(d\cdot\max\{\frac{1}{d-1},\frac{\kappa-1}{\kappa+1}\},\infty\right)$ satisfy $2/p+d/q\le (\kappa+1)/(\kappa-1)$.
Then, for each [$u_0\in {_0B}^{d/q-2/(\kappa-1)}_{qp}(\Omega)$]{}, equation admits a unique solution $$u \in H^1_{p,\mu_c}((0,a);{_0}H_q^{-1}(\Omega))\cap L_{p,\mu_c}((0,a); {_0}H_q^1(\Omega)),$$ for some [$a>0$]{}, with critical weight $$\mu^{\sf w}_c = \frac{1}{p}+\frac{1}{2}\left(1+\frac{d}{q}\right)-\frac{1}{\kappa-1}.$$ The solution exists on a maximal interval [$(0,t_+(u_0))$]{} and depends continuously on [$u_0$]{}. In addition, $$u \in C([0,t_+); {_0B}^{d/q-1/(\kappa-1)}_{qp}(\Omega))\cap C((0,t_+);{_0 B}^{1-2/p}_{qp}(\Omega)),$$ i.e. the solutions regularize instantly if [$2/p +d/q<(\kappa+1)/(\kappa-1)$]{}.
Equation has been considered by many authors in the last four decades, see for instance [@CDW09; @Fuj66; @QuSo07; @Wei81; @Wei86]. To the best of our knowledge, the results in Theorems \[thm:Ex1strong\] and \[thm:Ex1weak\] are new.
[**Example 2.**]{} Let $\Omega$ be a bounded domain of class $C^{2}$ in ${{\mathbb R}}^d$, and consider the Neumann problem $$\label{eq:ex2}
\begin{aligned}
\partial_t u -{\rm div}(a(u)\nabla u) &= |\nabla u|^\kappa && \mbox{in}\;\; \Omega, \\
\partial_\nu u&=0 && \mbox{on}\;\; \partial\Omega,\\
u&=u_0 && \mbox{in} \;\;\Omega,
\end{aligned}$$ where $\kappa>2$. Observe that this problem is a scaling invariant with respect to $$u_\lambda(t,x)=\lambda^{-\frac{\kappa-2}{2(\kappa-1)}}u(\lambda t,\lambda^{\frac{1}{2}}x)$$ if e.g. $a(u)\equiv a_0=const$. In the sequel, we assume that $a\in C^{1}({{\mathbb R}})$, $a(s)>0$ for all $s\in{{\mathbb R}}$, and there exists $C>0$ such that $$\label{eq:a_cond}
|a'(s_1)-a'(s_2)|\le C|s_1-s_2|,\quad s_1,s_2\in{{\mathbb R}}.$$ For sufficiently smooth $u$, this yields $${\rm div}(a(u)\nabla u)=a(u)\Delta u+a'(u)|\nabla u|^2,$$ hence we may rewrite $\eqref{eq:ex2}_1$ as $$\partial_t u -a(u)\Delta u = |\nabla u|^\kappa+a'(u)|\nabla u|^2\quad \mbox{in } \Omega.$$ Let $X_0=L_q(\Omega)$, $1<q<\infty$, and $$X_1=\{u\in H_q^2(\Omega):\partial_\nu u=0\ \text{on}\ \partial\Omega\}.$$ This in turn implies $$X_\beta=(X_0,X_1)_{\beta}=
\left\{
\begin{aligned}
& \{u\in H_q^{2\beta}(\Omega):\partial_\nu u=0\} &&\text{for} && \beta\in (1/2+1/2q,1],\\
& H_q^{2\beta}(\Omega) && \text{for} &&\beta\in [0,1/2+1/2q).
\end{aligned}
\right.$$ Define $F_\kappa,F_a:X_\beta\to X_0$ by $$F_\kappa(u)=|\nabla u|^\kappa\quad\text{and}\quad F_a(u)=a'(u)|\nabla u|^2.$$ Note that if $H^{2\beta}_q(\Omega\hookrightarrow H_{q\kappa}^1(\Omega)$, i.e. $$\beta=\frac{1}{2}+\frac{d}{2q}\left(1-\frac{1}{\kappa}\right),$$ the nonlinearities $F_\kappa$ and $F_a$ are well-defined, since $a'\in C({{\mathbb R}})$, $\kappa>2$ and $H_q^{2\beta}\hookrightarrow C(\overline{\Omega})$ for any $q\in (1,\infty)$. For $u_1,u_2\in X_\beta$, we obtain as in Example 1 $$|F_\kappa(u_1)-F_\kappa(u_2)|_{X_0}\le C(|u_1|_{X_\beta}^{\kappa-1}+|u_2|_{X_\beta}^{\kappa-1})|u_1-u_2|_{X_\beta}.$$ For this nonlinearity, the smallest possible value for $\mu$ may be computed from **(H3)** with $\rho_1=\kappa-1$ and $\beta_1=\beta$ to the result $$\mu_c=\frac{1}{p}+\frac{\kappa\beta-1}{\kappa-1}=\frac{1}{p}+\frac{d}{2q}+\frac{\kappa-2}{2(\kappa-1)}.$$ Setting $${_\nu}B_{qp}^{2s}(\Omega):=(X_0,X_1)_{s,p}=
\left\{
\begin{aligned}
& \{u\in B_{qp}^{2s}(\Omega):\partial_\nu u=0\} &&\text{for} &&s\in(1/2+1/2q,1),\\
& B_{qp}^{2s}(\Omega) && \text{for} && s\in (0,1/2+1/2q),
\end{aligned}
\right.$$ the critical space is $$X_{\gamma,\mu_c}=(X_0,X_1)_{\mu_c-1/p,p}={_\nu B}_{qp}^{d/q+(\kappa-2)/(\kappa-1)}(\Omega),$$ which is embedded into $C(\overline{\Omega})$ for all $p,q\in (1,\infty)$, $\kappa>2$.
Concerning $F_a$, we write $$F_a(u_1)-F_a(u_2)=a'(u_1)(|\nabla u_1|^2-|\nabla u_2|^2)+|\nabla u_2|^2(a'(u_1)-a'(u_2)),$$ for $u_1,u_2\in X_\beta$. It follows from Hölder’s inequality and $$\begin{aligned}
|a'(u_1)(|\nabla u_1|^2-|\nabla u_2|^2)|_{X_0}&\le C(1+|u_1|_{X_{\gamma,\mu_c}})(|\nabla u_1|_{L_{2q}}+|\nabla u_2|_{L_{2q}})|\nabla u_1-\nabla u_2|_{L_{2q}}\\
&\le C_\kappa(|u_1|_{X_{\gamma,\mu_c}})(1+| u_1|_{X_\beta}^{\kappa-1}+| u_2|_{X_\beta}^{\kappa-1})|u_1- u_2|_{X_\beta},\end{aligned}$$ since $\kappa-1>1$, where we also applied Young’s inequality. So for this part we may set as before $\rho_2=\kappa-1$ and $\beta_2=\beta$. For the remaining part, we obtain $$|\nabla u_2|^2(a'(u_1)-a'(u_2))|_{X_0}\le C|u_1-u_2|_{L_\infty}|u_2|_{H_{2q}^1}^2\le C|u_2|_{X_\beta}^2|u_1-u_2|_{X_{\gamma,\mu_c}}.$$ A straightforward calculation shows that there is strict inequality in **(H3)** with $\rho_3=2$ and $\beta_3=\mu_c-1/p$ if and only if $\beta<1$.
In other words, this part of the nonlinearity $F_a$ is always subcritical and so $\mu_c$ defined above is the critical weight for the nonlinearity $F(u):=F_\kappa(u)+F_a(u)$. Note that the condition $\mu_c>1/p$ is always satisfied as $\kappa>2$. The restrictions $\beta<1$ and $\mu_c\le 1$ then lead to $$\begin{aligned}
& \frac{d}{q} < \frac{\kappa}{(\kappa-1)}\quad\text{and}\quad
&\frac{1}{p}+\frac{d}{2q}+\frac{\kappa-2}{2(\kappa-1)}\le 1\Leftrightarrow
\frac{2}{p}+\frac{d}{q}\le \frac{\kappa}{(\kappa-1)},
\end{aligned}$$ respectively.
For $v\in X_{\gamma,\mu_c}$ we define an operator $A(v):X_1\to X_0$ by $[A(v)u](x):=a(v(x))\Delta u(x)$, $x\in\Omega$. By compactness, there exists $a_0>0$ such that $a(v(x))\ge a_0>0$ for all $x\in\Omega$, since $x\mapsto a(v(x))$ is continuous. Furthermore, it follows from [@DDHPV04] that $A_\#:=A(v)\in\mathcal{H}^\infty(X_0)$ with $\mathcal{H}^\infty$-angle $\phi_{A_\#}^\infty<\pi/2$.
\[thm:Ex2\] Let $a\in C^1({{\mathbb R}}) $, $a(s)>0$ for all $s\in{{\mathbb R}}$ and assume . Suppose that $\kappa>2$, $p\in (1,\infty)$, and $2/p+d/q\le \kappa/(\kappa-1)$.
Then, for each $u_0\in {_\nu B}^{d/q+(\kappa-2)/(\kappa-1)}_{qp}(\Omega)$, problem admits a unique solution $$u \in H^1_{p,\mu_c}((0,a);L_q(\Omega))\cap L_{p,\mu_c}((0,a); H_q^2(\Omega)),$$ for some $a>0$, with critical weight $\mu_c = 1/p+ d/2q-(\kappa-2)/2(\kappa-1)$. The solution exists on a maximal interval $(0,t_+(u_0))$ and depends continuously on $u_0$. In addition, $$u \in C([0,t_+); {_\nu B}^{d/q+(\kappa-2)/(\kappa-1)}_{qp}(\Omega))\cap C((0,t_+);{_\nu B}^{2(1-1/p)}_{qp}(\Omega)),$$ i.e. the solutions regularize instantly if $2/p +d/q<\kappa/(\kappa-1)$.
[The results contained in Theorem \[thm:Ex2\] seem to be new. We refer to the monograph [@QuSo07] for additional results concerning equation .]{}\
[**Example 3.**]{} Let $\Omega\subset{{\mathbb R}}^d$ be a bounded domain with boundary $\partial\Omega\in C^{4-}$. Consider the *Cahn-Hilliard equation* $$\label{eq:CH}
\begin{aligned}
\partial_t u-\Delta v&=0 && \text{in} \;\;\Omega,\\
v+\Delta u-\Phi'(u)&=0 && \text{in}\;\; \Omega,\\
\partial_\nu u=\partial_\nu v&=0 &&\text{on}\;\; \partial\Omega,\\
u(0)&=u_0 &&\text{in}\;\;\Omega.
\end{aligned}$$ Here $u$ is an order parameter, $v$ is the chemical potential and $\Phi$ denotes the physical potential, which is often assumed to be of double-well type, i.e. $\Phi(s)=(s^2-1)^2$ for $s\in{{\mathbb R}}$. Note that in view of the homogeneous Neumann boundary conditions, the elliptic-parabolic problem is equivalent to the purely parabolic problem $$\label{eq:CH2}
\begin{aligned}
\partial_t u+\Delta^2 u-\Delta\Phi'(u)&=0 &&\text{in}\;\; \Omega,\\
\partial_\nu u=\partial_\nu \Delta u&=0 &&\text{on}\;\; \partial\Omega,\\
u(0)&=u_0 && \text{in}\;\;\Omega.
\end{aligned}$$ Let $X_0:=L_q(\Omega)$, $1<q<\infty$, $$X_1:=\{u\in H_q^4(\Omega):\partial_\nu u=\partial_\nu\Delta u=0\}$$ and define an operator $A_0:X_1\to X_0$ by $A_0u:=\Delta^2 u$. By [@DDHPV04], $A_0\in\mathcal{H}^\infty(X_0)$ with $\mathcal{H}^\infty$-angle $\phi_{A_0}^\infty<\pi/2$. Let $(X_\alpha,A_\alpha)$, $\alpha\in{{\mathbb R}}$, denote the interpolation-extrapolation scale with respect to the complex interpolation functor (see the Appendix). Then $X_{-1/2}=(X_{1/2}^\sharp)'$ and $$X_{1/2}=\{u\in H_{q}^2(\Omega):\partial_\nu u=0\},$$ as well as $$X_{1/2}^\sharp=\{u\in H_{q'}^2(\Omega):\partial_\nu u=0\}.$$ The operator $A_{-1/2}:X_{1/2}\to X_{-1/2}$ may be represented as $$\langle A_{-1/2}u,v\rangle=\int_\Omega \Delta u\,\Delta v\ dx$$ for all $(u,v)\in X_{1/2}\times X_{1/2}^\sharp$, which follows from integration by parts and density of $X_1$ in $X_{1/2}$. We consider the weak formulation $$\label{eq:CHweak}
\partial_tu+A^{\sf w} u=F^{\sf w}(u),\ t>0,\quad u(0)=u_0,$$ of in $X_0^{\sf w}:=X_{-1/2}$, where $A^{\sf w}=A_{-1/2}$ with domain $X_1^{\sf w}:=X_{1/2}$ and $$\langle F^{\sf w}(u),v\rangle:=(\Phi'(u)|\Delta v)_{L_2(\Omega)}.$$ In the sequel, we assume $\Phi\in C^1({{\mathbb R}})$ and there are constants $C>0$ and $\kappa>0$ such that $$\label{eq:condPhi}
|\Phi'(s)-\Phi'(\bar{s})|\le C(1+|s|^\kappa+|\bar{s}|^\kappa)|s-\bar{s}|,\quad s,\bar s\in{{\mathbb R}}.$$ Hölder’s inequality in combination with yields $$|F^{\sf w}(u)|_{X_{0}^{\sf w}}\le |\Phi'(u)|_{L_q}\le C(1+|u|_{L_{(\kappa+1)q}}^{\kappa+1}),$$ for all $u\in X_\beta^{\sf w}=(X_0^{\sf w},X_1^{\sf w})_\beta={_\nu H}_q^{4\beta-2}(\Omega)$, $\beta\in (0,1)$ (see the Appendix for details). Here $${_\nu H}_q^{r}(\Omega):=
\left\{
\begin{aligned}
& \{u\in H_q^{r}(\Omega):\partial_\nu u=0\} && \text{for} && r\in (1+1/q,2],\\
& H_q^{r}(\Omega) &&\text{for} && r\in [0,1+1/q),\\
\end{aligned}
\right.$$
and ${_\nu}H_q^{-r}(\Omega):=({_\nu}H_{q'}^{r}(\Omega))'$ if $r\in [0,2]\backslash\{2-1/q\}$.
Thus, $F^{\sf w}:X_\beta^{\sf w}\to X_0^{\sf w}$ is well-defined, provided $H_q^{4\beta-2}\hookrightarrow L_{(\kappa+1)q}$, hence $$4\beta-2-\frac{d}{q}=-\frac{d}{(\kappa+1)q}\Leftrightarrow \beta=\frac{1}{2}+\frac{d\kappa}{4q(\kappa+1)}.$$ The condition $\beta<1$ is then equivalent to $q>\frac{d\kappa}{2(\kappa+1)}$ and the estimate $$|F^{\sf w}(u)-F^{\sf w}(\bar{u})|_{X_0^{\sf w}}\le C(1+|u|_{X_\beta^{\sf w}}^{\kappa}+|\bar{u}|_{X_\beta^{\sf w}}^{\kappa})|u-\bar{u}|_{X_\beta^{\sf w}}.$$ holds for all $u,\bar{u}\in X_\beta^{\sf w}$. From **(H3)** with $\rho_1=\kappa$ and $\beta_1=\beta$, we obtain the critical weight $$\mu_c=\frac{1}{p}+\frac{d}{4q}+\frac{\kappa-1}{2\kappa}.$$ Hence $\mu_c\le 1$ iff $$\frac{1}{p}+\frac{d}{4q}\le \frac{\kappa+1}{2\kappa},$$ and in case $\kappa\in (0,1)$ it is $\mu_c>1/p$ if and only if $$q<\frac{d\kappa}{2(1-\kappa)}.$$ As $q>1$, this implies the lower bound $\kappa>2/(d+2)$. The critical space is then given by $$X_{\gamma,\mu_c}^{\sf w}={_\nu}B_{qp}^{4(\mu_c-1/p)-2}(\Omega)={_\nu}B_{qp}^{d/q-2/\kappa}(\Omega),$$ where $${_\nu B}_{qp}^{r}(\Omega):=
\left\{
\begin{aligned}
& \{u\in B_{qp}^{r}(\Omega):\partial_\nu u=0\} &&\text{for} && r\in (1+1/q,2],\\
& B_{qp}^{r}(\Omega) &&\text{for} && r\in [0,1+1/q),\\
\end{aligned}
\right.$$ and ${_\nu}B_{qp}^{-r}(\Omega):=({_\nu}B_{q'p'}^{r}(\Omega))'$ if $r\in [0,2]\backslash\{2-1/q\}$. In case of the double-well potential $\Phi(s)=(s^2-1)^2$ we may set $\kappa=2$ in , hence the critical space for this case reads $${_\nu}B_{qp}^{d/q-1}(\Omega).$$
\[thm:Ex3\] Let $\Phi\in C^1({{\mathbb R}}) $ and assume . Suppose that $p\in (1,\infty)$ and $2/p+d/2q\le (\kappa+1)/\kappa$, where $q\in (1,\infty)$, $\kappa>2/(d+2)$ and $q<d\kappa/(2(1-\kappa))$ in case $\kappa\in (2/(d+2),1)$. Then, for each [$u_0\in {_\nu B}^{d/q-\kappa/2}_{qp}(\Omega)$]{}, problem admits a unique solution $$u \in H^1_{p,\mu_c}((0,a);{_\nu}H_q^{-2}(\Omega))\cap L_{p,\mu_c}((0,a); {_\nu}H_q^2(\Omega)),$$ for some [$a>0$]{}, with critical weight [$\mu_c = 1/p+ d/4q+(\kappa-1)/2\kappa$]{}. The solution exists on a maximal interval [$(0,t_+(u_0))$]{} and depends continuously on [$u_0$]{}. In addition, $$u \in C([0,t_+); {_\nu B}^{d/q-\kappa/2}_{qp}(\Omega))\cap C((0,t_+);{_\nu B}^{2-4/p}_{qp}(\Omega)),$$ i.e. the solutions regularize instantly if [$2/p +d/2q<(\kappa+1)/\kappa$]{}.
[The Cahn-Hilliard equation has been proposed in the pioneering work [@CaHi], to model the separation of phases of a binary fluid. It has been subject of intensive research during the last decades, see for instance [@AbWi; @ElZhe; @HR99] and the references therein. So far, there seem to be no results on critical spaces for the Cahn-Hilliard equation.]{}
Vorticity Equation
==================
Let $\Omega$ be a bounded, simply connected domain in ${{\mathbb R}}^3$ with boundary $\Sigma:=\partial\Omega$ of class $C^{3}$. We consider the Navier-Stokes equation with boundary conditions of Navier type. $$\label{NS}
\begin{aligned}
\partial_t u -\upmu\, \Delta u +u\cdot\nabla u +\nabla \pi&=0 && \mbox{in}\;\; \Omega,\\
{\rm div}\, u &=0 &&\mbox{in}\;\; \Omega,\\
u\cdot\nu =0,\; 2\upmu\, {\sf P}_\Sigma D(u) \nu +\alpha {\sf P}_\Sigma u &=0 && \mbox{on}\;\; \Sigma,\\
u(0)&=u_0 && \mbox{in}\;\; \Omega.
\end{aligned}$$ Here $\upmu>0$ and $\alpha\geq0$ are constants, ${\sf P}_\Sigma = I-\nu\otimes\nu$ is the orthogonal projection onto the tangent bundle $T\Sigma$, $D(u) = (\nabla u +[\nabla u]^{\sf T})/2$ denotes the symmetric velocity gradient and $R(u)= (\nabla u -[\nabla u]^{\sf T})/2$ its asymmetric part, for future reference. The parameter $\alpha\ge 0$ takes friction on the boundary $\Sigma$ into consideration. If $\alpha=0$, we are in the case of pure-slip boundary conditions, whereas $\alpha>0$ corresponds to the case of partial-slip. We want to study this problem in terms of its vorticity and its stream function. By proper scaling we may assume $\upmu=1$.
The Navier Condition.
---------------------
We first reformulate the Navier boundary condition in a way that is more convenient for our analysis. For this purpose, we make use of the splitting $$w=(I- \nu\otimes\nu)w + (w\cdot\nu)\nu=: w_\parallel +w_\nu\nu,$$ where $w_\parallel$ and $w_\nu$ denote the tangential and the normal part, respectively, of a vector field $w$ defined on $\Sigma$. It is important to note that we can extend the unit normal $\nu$, defined on $\Sigma$, to a tubular neighborhood $U$ of $\Sigma$ according to $\tilde\nu(x) = \nu(\Pi_\Sigma(x))$, where $\Pi_\Sigma$ denotes the metric projection onto $\Sigma$; see for instance Prüss and Simonett [@PrSi16 Chapter 2] for more details. We can then also extend the above decomposition of $w$ to the tubular neighborhood $U$. In the following we always assume that a given vector field $w$ defined on $\bar\Omega \cap U$ is decomposed in a tangential and normal component according to $$\label{splitting-extended}
w=(I- \tilde\nu\otimes\tilde\nu)w + (w\cdot\tilde\nu)\tilde\nu=: w_\parallel +w_\nu\tilde\nu.$$ In order to not overburden the notation, we will drop the tilde in the sequel. Note that with this convention we have $$\label{derivatives}
\partial_\nu \nu=0, \quad \partial_\nu w_\parallel\cdot \nu=0 \quad\mbox{on}\;\; \Sigma,$$ for any vector field $w$ defined on $\bar\Omega \cap U$. An easy computation then yields $$\begin{aligned}
2{\sf P}_\Sigma D(u) \nu &= \partial_\nu u_\parallel +{\sf L}_\Sigma u_\parallel +\nabla_\Sigma u_\nu &\text{on}\;\;\Sigma ,\\
2{\sf P}_\Sigma R(u) \nu &= -\partial_\nu u_\parallel +{\sf L}_\Sigma u_\parallel +\nabla_\Sigma u_\nu &\text{on}\;\;\Sigma ,
\end{aligned}$$ where ${\sf L}_\Sigma=-\nabla_\Sigma \nu$ is the Weingarten tensor and $\nabla_\Sigma$ denotes the surface gradient on $\Sigma$, see also [@PrWi17a Section 5.4]. For dimension 3, we obtain $$2{\sf P}_\Sigma R(u) \nu= \nu\times {\rm rot}\, u.$$ As $u_\nu=0$ by the first boundary condition, with ${\sf B}_\Sigma = -2 {\sf L}_\Sigma -\alpha {\sf P}_\Sigma$ the second implies $$\begin{aligned}
0&= 2{\sf P}_\Sigma D(u) \nu +\alpha {\sf P}_\Sigma u
= \partial_\nu u_\parallel +{\sf L}_\Sigma u_\parallel +\alpha u_\parallel\\
&= \partial_\nu u_\parallel -{\sf L}_\Sigma u_\parallel -{\sf B}_\Sigma u_\parallel
= {\rm rot}\, u \times \nu - {\sf B}_\Sigma u_\parallel,\end{aligned}$$ hence the Navier boundary conditions are equivalent to $$\label{nbc}
u\cdot\nu=0,\quad {\rm rot}\, u \times \nu = {\sf B}_\Sigma u\quad \mbox{on } \Sigma.$$ Note that this is a lower order perturbation of the so-called [*perfect slip*]{} boundary conditions $$u\cdot\nu=0,\quad {\rm rot}\,u \times \nu =0.$$
The Stream Function.
--------------------
As ${\rm div}\, u=0$ in $\Omega$, $u\cdot\nu=0$ on $\Sigma=\partial\Omega$, and $\Omega$ is simply connected by assumption, there is a unique solution $v$ of the problem $$\begin{aligned}
\label{SF}
{\rm rot}\, v&=u && \mbox{in}\;\; \Omega,\\
{\rm div}\, v&=0 && \mbox{in}\;\; \Omega,\\
v_\parallel &=0 && \mbox{on}\;\; \Sigma.
\end{aligned}$$ We call $v$ the [*stream function*]{}, below.
[**(i)**]{} We shall first show uniqueness. Suppose $u=0$ and $v$ is a solution of . As ${\rm rot}\,v=0$ and $\Omega$ is simply connected, there exists a potential function $\phi$, i.e. we have $v=\nabla \phi$. From the second line follows $\Delta\phi=0$ in $\Omega$, while the boundary condition $v_\parallel=0$ implies $\nabla_\Sigma \phi=0$ on $\Sigma$. Hence, there is a constant $c$ such that $\phi\equiv c$ on $\Sigma$ and the function $\phi-c$ then solves the elliptic problem$$\Delta (\phi-c)=0\;\;\text{in}\;\;\Omega,
\quad \phi-c=0 \;\;\text{on}\;\;\Sigma.$$ Therefore, $\phi$ is constant in all of $\Omega$ and $v=\nabla\phi=0$.
[**(ii)**]{} In order to show existence, we consider the elliptic problem $$\begin{aligned}
\label{LP}
-\Delta v &= {\rm rot}\, u && \mbox{in}\;\; \Omega,\\
v_\parallel &=0 && \mbox{on}\;\; \Sigma,\\
\partial_\nu v_\nu - \kappa_\Sigma v_\nu &=0 && \mbox{on}\;\; \Sigma,
\end{aligned}$$ where $\kappa_\Sigma$ is the mean curvature (more precisely, the sum of the principal curvatures) of $\Sigma$. For later use we record the important relationship $$\label{div-boundary}
{\rm div}\,v = {\rm div}_\Sigma v_\parallel + \partial_\nu v_\nu -\kappa_\Sigma v_\nu\;\;\text{on}\;\;\Sigma.$$ Hence the boundary conditions of imply ${\rm div}\,v=0$ on $\Sigma$.
Problem is uniquely solvable, with $v=A^{-1}{\rm rot}\, u$, where the operator $A$ is defined in Section 4.4 below.
[**(iii)**]{} It remains to show that ${\rm div}\,v=0$ and ${\rm rot}\, v=u$ in $\Omega$. The first assertion readily follows from the observation that the solution $v$ of satisfies $$\begin{aligned}
\Delta\, {\rm div}\,v &=0 &&\mbox{in}\;\; \Omega,\\
{\rm div}\,v &=0 &&\mbox{on}\;\; \Sigma,
\end{aligned}$$ which only admits the trivial solution. Hence, ${\rm rot}\,({\rm rot}\, v -u)=0$ in $\Omega$, and by simple connectedness of $\Omega$ this yields ${(\rm rot}\,v -u )=\nabla\phi$ for some harmonic function $\phi$. We claim that $({\rm rot}\,v -u)\cdot\nu=0$ on $\Sigma$. By assumption, $u\cdot\nu=0$. Hence $$({\rm rot}\,v-u)\cdot\nu
=\big((\nabla_\Sigma +\nu\partial_\nu)\times (v_\parallel + v_\nu\nu)\big)\cdot\nu
=\big((\nabla_\Sigma v_\nu-\partial_\nu v_\parallel)\times \nu\big)\cdot\nu =0
\;\;\mbox{on}\;\;\Sigma.$$ Here we used the properties that $v_\parallel=0$, $\partial_\nu\nu=0$, and $\nabla_\Sigma\times\nu=0$ on $\Sigma$. The latter assertion can be verified by means of local coordinates, see for instance Section 2.1 in [@PrSi16], $$\nabla_\Sigma\times\nu
=\tau^j\partial_j\times\nu =\tau^j \times \partial_j\nu
=\tau^j \times l_{jk}\tau^k =\tau^1\times l_{12}\tau^2 + \tau^2\times l_{21}\tau^1=0,$$ as $l_{12}=l_{21}$. Noting that $\phi$ solves the elliptic problem $$\Delta \phi =0 \;\;\mbox{in}\;\; \Omega, \quad
\partial_\nu\phi =({\rm rot}\,v -u )\cdot\nu=0\;\;\mbox{on}\;\; \Sigma,$$ we conclude that $\phi$ is constant on $\Omega$, and hence ${(\rm rot}\,v -u )=\nabla\phi=0$ in $\Omega$, i.e. ${\rm rot}\,v = u$.
For $q\in (1,\infty)$ and $s\in [-1,1]\setminus\{1/q, 1/q-1\}$ one can show that $$A^{-1}{\rm rot}: {_{\sf N} H}^{s}_{q}(\Omega) \to {_\parallel H}^{s+1}_q(\Omega),$$ is linear and bounded, where ${_{\sf N}H}^s_q(\Omega)$ is the complex interpolation-extrapolation scale associated to problem , and where the spaces ${_\parallel H}^s_q(\Omega)$ are defined in Section 4.4 below.
The Vorticity Equation.
------------------------
We define the [*vorticity*]{} by means of $w:={\rm rot}\, u$. As a consequence of the above considerations, we then have $$u ={\rm rot}\, v ={\rm rot}\, A^{-1} {\rm rot}\, u ={\rm rot}\, A^{-1}w =: L_0 w.$$ Observe that $ L_0$ is an operator of order $-1$. This way $u$ is determined uniquely by the stream function $v$, or equivalently by the vorticity $w$. The latter property is usually referred to as the Biot-Savart law. Note that $$\begin{aligned}
\label{vortex-relation}
{\rm rot}\,({\rm div}\,(u\otimes u))
&={\rm rot}\,(u\cdot\nabla u)
= (u\cdot\nabla)\, {\rm rot}\,u -({\rm rot}\, u\cdot\nabla) u\\
&= u\cdot\nabla w-w\cdot \nabla u
={\rm div}\,(u\otimes w)-{\rm div}\,(w\otimes u),
\end{aligned}$$ as ${\rm div}\,u = {\rm div}\,w=0$. The vorticity equation now reads $$\begin{aligned}
\label{vorteq}
\partial_t w + u\cdot\nabla w -w\cdot\nabla u-\Delta w &=0 && \mbox{in}\;\; \Omega,\\
u &= L_0 w && \mbox{in } \Omega, \\
w\times \nu =B_\Sigma u &=0 && \mbox{on}\;\; \Sigma,\\
{\rm div}_\Sigma\, w_\parallel + \partial_\nu w_\nu -\kappa_\Sigma w_\nu &=0 && \mbox{on}\;\; \Sigma,\\
w(0)&=w_0 && \mbox{in}\;\; \Omega.
\end{aligned}$$ Here we note that the second boundary condition ensures ${\rm div}\, w=0$, as soon as ${\rm div}\, w_0=0$, which is natural as $w_0={\rm rot }\, u_0$. In fact, for $\phi={\rm div}\, w$ we then obtain, formally at least, $$\begin{aligned}
\partial_t \phi -\Delta \phi&=0 && \mbox{in}\;\; \Omega, \\
\phi &=0 && \mbox{on}\;\; \Sigma,\\
\phi(0)&=0 && \mbox{in}\;\; \Omega,
\end{aligned}$$ hence ${\rm div}\, w =\phi=0$. We also observe that with $$\nu\times(w\times\nu) = w(\nu\cdot\nu) -\nu(w\cdot\nu) = w_\parallel,$$ the first boundary condition for $w$ can be rephrased as $$\label{w-parallel}
w_\parallel =\nu\times {\sf B}_\Sigma u =: L_1w.$$ Observe that $L_1= \nu\times {\sf B}_\Sigma L_0$ is an operator of order $-1$, hence a lower order perturbation. We also recall that we have $$u\cdot\nabla u = {\rm div}(u\otimes u),$$ as ${\rm div}\, u=0$. This will be useful for the very weak formulation below.
The Scale of the Principal Operator $A$.
-----------------------------------------
We define the principal operator $A$ in $L_q(\Omega)^3$ by means of $$\label{princop}
Aw=-\Delta w, \quad w\in {\sf D}(A) =\{ w\in H^2_q(\Omega)^3:
\, w_\parallel =0,\;\; \partial_\nu w_\nu-\kappa_\Sigma w_\nu=0\}.$$ The operator $A$ has some beautiful properties. Firstly, it admits an ${{\mathcal H}}^\infty$-calculus with ${{\mathcal H}}^\infty$-angle $\phi_A^\infty=0$ in $L_q(\Omega)^3$. Next we note that $A$ is positive definite in $L_2(\Omega)^3$. In order to see this, we employ the relation $- \Delta w = {\rm rot}\,{\rm rot}\, w - \nabla {\rm div}\, w$ and integrate by parts. This yields $$\label{Aww}
\begin{aligned}
-(\Delta w|w)_{\Omega} &= |{\rm rot}\, w|^2_{\Omega} + (\nu\times {\rm rot}\, w|w)_{\Sigma}
+ |{\rm div}\, w|^2_{\Omega} - ({\rm div}\,w|w_\nu)_{\Sigma} \\
&=|{\rm rot}\, w|^2_{\Omega} + |{\rm div}\, w|^2_{\Omega},
\end{aligned}$$ where $(u|v)_\Omega:=(u|v)_{L_2(\Omega)}$ and $(u|v)_\Sigma:=(u|v)_{L_2(\Sigma)}$. Here we used $w_\parallel=0$, and . This shows that $A$ is positive semi-definite. Suppose $Aw=0.$ Then implies ${\rm rot}\,w=0$ and ${\rm div}\,w=0$ in $\Omega$. By uniqueness of problem , see Step (i) in Section 4.2, $w=0$. Hence $A$ is injective in $L_2(\Omega)^3$. As $A$ has compact resolvent in $L_2(\Omega)^3$, its spectrum consists of eigenvalues of finite multiplicity. Therefore, $0$ lies in the resolvent set of $A$ in $L_2(\Omega)^3$. Since the spectrum of $A$ is independent of $q\in (1,\infty)$, we conclude that $A$ is invertible in $L_q(\Omega)^3$ for all $q\in (1,\infty)$. In summary, $A$ is invertible, and the spectrum of $A$ in $L_q(\Omega)^3$ consists only of eigenvalues of finite multiplicity which are all positive.
The pair $(L_q(\Omega), A)$ generates an interpolation-extrapolation scale, see the Appendix, and we have explicit expressions for the extrapolation-interpolation spaces, i.e. we have for ${_\parallel H}^{2s}_q(\Omega):=(L_q(\Omega),{\sf D}(A))_s$
$${_\parallel H}^{2s}_q(\Omega)=
\left\{
\begin{aligned}
& \{w\in H^{2s}_q(\Omega)^3:\, w_\parallel=0,\, \partial_\nu w_\nu -{\kappa_\Sigma w_\nu} =0\}, && \!\!\! s\in (1/2+1/2q,1),\\
& \{w\in H^{2s}_q(\Omega)^3:\, w_\parallel=0\}, && \!\!\! s\in (1/2q,1/2+1/2q),\\
& H^{2s}_q(\Omega)^3, && \!\!\! s\in (0,1/2q),
\end{aligned}
\right.$$
and for ${_\parallel B}^{2s}_{qp}(\Omega):=(L_q,{\sf D}(A))_{s,p}$, $${_\parallel B}^{2s}_{qp}(\Omega)=
\left\{
\begin{aligned}
& \{w\in B^{2s}_{qp}(\Omega)^3:\,w_\parallel=0,\, \partial_\nu w_\nu -{\kappa_\Sigma w_\nu} =0\}, && \!\!\! s\in (1/2+1/2q,1),\\
& \{w\in B^{2s}_{qp}(\Omega)^3:\, w_\parallel=0\}, && \!\!\! s\in (1/2q,1/2+1/2q),\\
& B^{2s}_{qp}(\Omega)^3, && \!\!\! s\in (0,1/2q).
\end{aligned}
\right.$$ Moreover, $${_\parallel H}^{-2s}_q(\Omega):=\big({_\parallel H}^{2s}_{q^\prime}(\Omega)\big)^\prime,\quad
{_\parallel B}^{-2s}_{qp}(\Omega) := \big({_\parallel B}^{2s}_{q^\prime p^\prime}(\Omega)\big)^\prime,$$ for $s\in [0,1]\setminus\{1/2-1/2q, 1-1/2q\}.$
Moreover, $A$ commutes with the [*Weyl projection*]{} ${{\mathbb P}}_W$ defined by $w= {{\mathbb P}}_W w+\nabla\varphi$, where $$\Delta \varphi ={\rm div}\, w \quad \mbox{in } \Omega,\quad \varphi=0 \quad \mbox{on } \Sigma.$$ Therefore, its restriction $A_0$ to $X_0 :={{\mathbb P}}_W L_q(\Omega)^3=: {_\parallel L}_{q,\sigma}(\Omega)$ with domain $$\begin{aligned}
&X_1:={\sf D}(A_0)= {{\mathbb P}}_W{\sf D}(A), \\
&X_1=\{ w\in H^2_q(\Omega)^3: {\rm div}\, w =0 \; \mbox{in}\; \Omega, w_\parallel = 0\, \mbox{on}\; \Sigma\}
=: {_\parallel H}^2_{q,\sigma}(\Omega),
\end{aligned}$$ has the same properties as $A$. We note on the go that the conditions ${\rm div}\,w=0$ and $w_\parallel =0$ imply [$\partial_\nu w_\nu - \kappa_\Sigma w_\nu=0$.]{}
Hence, the pair $(X_0,A_0)$ generates the complex interpolation scale $(X_\alpha,A_\alpha)$, $\alpha\in {{\mathbb R}}$, see the Appendix. Here we are particularly interested in the cases $\alpha=-1/2$ for the weak formulation and $\alpha= -1$ for the very weak setting. Observe that all these spaces are of class $U\!M\!D$, and all these operators admit an ${{\mathcal H}}^\infty$-calculus with ${{\mathcal H}}^\infty$-angle $\phi_A^\infty=0$. The corresponding complex interpolation spaces are given by $$X_\alpha = {\sf D}(A_0^\alpha)={{\mathbb P}}_W{\sf D}(A^\alpha),\quad D_{A_0}(\alpha,p) = {{\mathbb P}}_W D_A(\alpha,p),\quad \mbox{for} \; \alpha>0,$$ and $$X_{\alpha}= \big({\sf D}([A_0^\#]^{-\alpha})\big)^\prime,
\quad D_{A_0}(\alpha,p)= \big(D_{A^\#_0}(-\alpha,p^\prime)\big)^\prime,\quad \mbox{for} \; \alpha<0.$$ Here $A_0^\#$ means $A_0$ considered in $X_0^\sharp =L_{q^\prime}(\Omega)$, i.e. in the dual scale. In the sequel, we set $${_\parallel H}^s_{q,\sigma}(\Omega):= X_{s/2}
\quad \mbox{and}\quad {_\parallel B}^s_{qp,\sigma}(\Omega):= D_{A_0}(s/2,p):=(X_0,{\sf D}(A_0))_{s/2,p}.$$
Very Weak Formulation.
----------------------
In the very weak formulation we define $$X_0^{\sf vw}= X_{-1}= {_\parallel H}^{-2}_{q,\sigma}(\Omega), \quad X_1^{\sf vw}= X_{0}=:{_\parallel L}_{q,\sigma}(\Omega).$$ Then for $\phi \in {_\parallel H}^2_{q^\prime,\sigma}(\Omega)$ we obtain with two integrations by parts $$\begin{aligned}
0&= (\partial_t w -\Delta w +{\rm rot}\,(u\cdot\nabla u) |\phi)_\Omega\\
&= (\partial_t w|\phi)_\Omega -(w|\Delta\phi)_\Omega - (u\otimes u:{\nabla\rm rot}\, \phi)_\Omega
-(\partial_\nu w|\phi)_\Sigma +(w|\partial_\nu\phi)_\Sigma\\
&= \langle\partial_t w + A_{-1} w|\phi\rangle - \langle (B^{\sf vw} w+F^{\sf vw}(w)|\phi\rangle,\end{aligned}$$ with $$\langle F^{\sf vw}(w)|\phi\rangle= (u\otimes u|\nabla{\rm rot}\, \phi)_\Omega,\quad
\langle B^{\sf vw} w|\phi\rangle =
(\nu\times{\sf B}_\Sigma u|\partial_\nu \phi_\parallel -\nabla_\Sigma \phi_\nu)_\Sigma.$$ Here the expression for $B^{\sf vw}$ was derived as follows. From , , the surface divergence theorem, and $\phi_\parallel=0$, [$\partial_\nu\phi_\nu-\kappa_\Sigma\phi_\nu =0$]{} on $\Sigma$ we obtain $$\begin{aligned}
(\partial_\nu w|\phi)_\Sigma
&=(\partial_\nu w_\parallel +(\partial_\nu w_\nu)\nu|\phi_\nu\nu)_\Sigma
=((\partial_\nu w_\nu)\nu|\phi_\nu\nu)_\Sigma \\
&= (-{\rm div}_\Sigma w_\parallel {+\kappa_\Sigma w_\nu} |\phi_\nu )_\Sigma
=(w_\parallel |\nabla_\Sigma \phi_\nu)_\Sigma {+ ( \kappa_\Sigma w_\nu |\phi_\nu )_\Sigma }
\end{aligned}$$ and $$(w|\partial_\nu\phi)_\Sigma
= {(w_\parallel | \partial_{\nu}\phi_\parallel)_\Sigma + (w_\nu |\kappa_\Sigma \phi_\nu)_\Sigma.}$$ This shows that the very weak formulation of the vorticity equation reads $$\label{vorteq-vw}
\partial_t w + A_{-1} w = B^{\sf vw} w + F^{\sf vw}(w),\; t>0,\quad w(0)=w_0.$$ Here $B^{\sf vw}$ is a linear lower order perturbation and $F^{\sf vw}$ is bilinear.
To show that $B^{\sf vw}$ is lower order, we estimate as follows. $$\begin{aligned}
|\langle B^{\sf vw}w|\phi\rangle|&\leq |\nu \times {\sf B}_\Sigma u|_{L_q(\Sigma)}|\partial_\nu \phi_\parallel -\nabla_\Sigma \phi_\nu|_{L_{q^\prime}(\Sigma)}\\
&\leq |u|_{H^{1/q+\varepsilon}_q(\Omega)} |\phi|_{H^{1+1/q^\prime+\varepsilon}_{q^\prime}(\Omega)}\\
& \leq |w|_{{_\parallel H}^{1/q+\varepsilon -1}_q(\Omega)}|\phi|_{H^{2-1/q+\varepsilon}_{q^\prime}(\Omega)}.\end{aligned}$$ Here we have employed the mapping properties of ${\rm rot}$ and $A^{-1}$ for $u={\rm rot}\, A^{-1}w$. This shows that $$B^{\sf vw} : {_\parallel H}^{1/q+\varepsilon -1}_{q,\sigma}(\Omega) \to {_\parallel H}^{1/q-\varepsilon -2}_{q,\sigma}(\Omega),$$ hence $B^{\sf vw}$ is a lower order perturbation of $A_{-1}$. Below we set $A^{\sf vw}=A_{-1}-B^{\sf vw}$, and observe that $A^{\sf vw}$ also admits a bounded ${{\mathcal H}}^\infty$-calculus with ${{\mathcal H}}^\infty$-angle 0.
The bilinearity $F^{\sf vw}(w)=G^{\sf vw}(w,w)$ can be estimated as follows $$|\langle G^{\sf vw}(w_1,w_2)|\phi\rangle|= |(u_1\otimes u_2|\nabla{\rm rot}\, \phi)_\Omega|\leq |u_1|_{L_{2q}(\Omega)}|u_2|_{L_{2q}(\Omega)}|\phi|_{H^2_{q^\prime}(\Omega)},$$ hence $G^{\sf vw}:X_{\beta^{\sf vw}}^{\sf vw}\times X_{\beta^{\sf vw}}^{\sf vw}\to X_0^{\sf vw}$ is bounded, with $$\beta^{\sf vw}= 1/2+3/4q<1\quad \mbox{and}\quad \mu_c^{\sf vw}-1/p = 2\beta^{\sf vw}-1= 3/2q.$$ So here we require $2/p+3/q\leq2$, to have $\mu_c^{\sf vw}\leq1$. Now we are in position to apply the results from Section 2 to this very weak setting of the vorticity equation, useful to cover the range $q>3/2$, with critical space $X_{\gamma,\mu_c^{\sf vw}}^{\sf vw} = {_\parallel B}^{3/q-2}_{qp,\sigma}(\Omega)$.
\[thm:vwvort\] Let $q\in (3/2,\infty)$, $p\in (1,\infty)$ such that $2/p+3/q\leq2$.
Then, for each $w_0\in {_\parallel B}^{3/q-2}_{qp,\sigma}(\Omega)$, the vorticity equation admits a unique very weak solution $$w \in H^1_{p,\mu^{\sf vw}_c}((0,a); {_\parallel H}^{-2}_{q,\sigma}(\Omega)))\cap L_{p,\mu^{\sf vw}_c}((0,a); L_q(\Omega)),$$ for some $a>0$, with critical weight $\mu^{\sf vw}_c = 1/p+ 3/2q$. The solution exists on a maximal interval $(0,t_+(w_0))$ and depends continuously on $w_0$. In addition, $$w \in C([0,t_+); {_\parallel B}^{3/q-2}_{qp,\sigma}(\Omega))
\cap C((0,t_+);{_\parallel B}^{2(1-1/p)-2}_{qp,\sigma}(\Omega)),$$ i.e. the solutions regularize instantly if $2/p +3/q<2$.
Weak Formulation.
------------------
In the weak formulation of the vorticity equation we choose $X_0^{\sf w}= X_{-1/2}= {_\parallel H}^{-1}_{q,\sigma}(\Omega)$. Then for $\phi \in {_\parallel H}^1_{q^\prime,\sigma}(\Omega)$ we obtain with an integration by parts $$\begin{aligned}
0&= (\partial_t w -\Delta w +{\rm rot}\,(u\cdot\nabla u) |\phi)_\Omega\\
&= (\partial_t w|\phi)_\Omega +(\nabla w|\nabla \phi)_\Omega - (u\cdot\nabla u|{\rm rot}\, \phi)_\Omega
-(\partial_\nu w|\phi)_\Sigma \\
&= \langle\partial_t w + A^{\sf w} w|\phi\rangle - \langle F^{\sf w}(w)|\phi\rangle,\end{aligned}$$ with $$\langle F^{\sf w}(w)|\phi\rangle= (u\cdot\nabla u|{\rm rot}\, \phi)_\Omega,\quad \langle A^{\sf w} w|\phi\rangle
=(\nabla w|\nabla \phi)_\Omega +
({\rm div}_\Sigma w_\parallel -\kappa_\Sigma w_\nu |\phi_\nu )_\Sigma,$$ and we keep the boundary condition $$w_\parallel = \nu\times{\sf B}_\Sigma u \quad \mbox{on } \Sigma.$$ This means $$X_1^{\sf w}={\sf D}(A^{\sf w})=\{ w\in H^1_q(\Omega)^3:\, {\rm div}\, w=0 \;\mbox{in}\; \Omega, \; w_\parallel = \nu\times {\sf B}_\Sigma u \; \mbox{on} \; \Sigma\}.$$ Then the weak formulation of the vorticity equation reads $$\label{vorteq-w}
\partial_t w + A^{\sf w} w = F^{\sf w}(w),\; t>0,\quad w(0)=w_0.$$ The operator $A^{\sf w}$ generates its own scale, which differs from that of $A_0$ through the boundary condition $w_\parallel =\nu\times B_\Sigma u$. By definition of $A^{\sf w}$ and an integration by parts it follows that $A^{\sf w}=A^{\sf vw}_{1/2}$. In particular, $A^{\sf w}$ admits a bounded ${{\mathcal H}}^\infty$-calculus with angle 0 as well.
Next, we estimate the bilinearity $F^{\sf w}(w) = G^{\sf w}(w,w)$ as follows. $$|\langle G^{\sf w}(w_1,w_2)|\phi\rangle|= |(u_1\cdot \nabla u_2|{\rm rot}\, \phi)_\Omega|\leq |u_1|_{L_{qr^\prime}(\Omega)}|u_2|_{H^1_{qr}(\Omega)}|\phi|_{H^1_{q^\prime}(\Omega)},$$ where we choose $r>1$ in such a way that the Sobolev indices of $L_{qr^\prime}$ and $H^1_{qr}$ are equal, which means $$1-3/qr = -3/qr^\prime = -3/q +3/qr,\quad \mbox{i.e.} \quad 3/qr = (1+3/q)/2.$$ This is feasible if $q<3$. Then we have with $X_{\beta^{\sf w}}^{\sf w} = {_\parallel H}^{2\beta^{\sf w}-1}_{q,\sigma}(\Omega)$ $$G^{\sf w} : X_{\beta^{\sf w}}^{\sf w} \times X_{\beta^{\sf w}}^{\sf w} \to X_0^{\sf w} \quad \mbox{bounded},$$ provided $$\beta^{\sf w} = (1+3/q)/4, \quad \mu^{\sf w}_c -1/p = 2\beta^{\sf w} -1 = (3/q-1)/2.$$ Obviously, $\beta^{\sf w}<1$ and for $\mu_c^{\sf w}\leq1$ we require $ 2/p+ 3/q \leq3$. As a consequence, the results of Section 2 apply to the vorticity equation in the weak setting for $q<3$, with critical space $ X^{\sf w}_{\gamma,\mu_c^{\sf w}} = {_\parallel B}_{qp,\sigma}^{3/q-2}(\Omega)$, the same spaces as for the very weak formulation in case $3/2<q< 3$. We observe that the Sobolev indices of these critical spaces equal $-2$, i.e. it is independent of $q$.
\[thm:wvort\] Let $q\in (1,3)$, $p\in (1,\infty)$ such that $2/p + 3/q\leq3$.
Then, for each $w_0\in {_\parallel B}^{3/q-2}_{qp,\sigma}(\Omega)$, the vorticity equation admits a unique weak solution $$w \in H^1_{p,\mu^{\sf w}_c}((0,a); {_\parallel H}^{-1}_{q,\sigma}(\Omega))\cap L_{p,\mu^{\sf w}_c}((0,a); H^1_q(\Omega)),$$ for some $a>0$, with critical weight $\mu^{\sf w}_c = 1/p+ 3/2q -1/2$. The solution exists on a maximal interval $(0,t_+(w_0))$ and depends continuously on $w_0$. In addition, $$w \in C([0,t_+); {_\parallel B}^{3/q-2}_{qp,\sigma}(\Omega))\cap C((0,t_+); B^{2(1-1/p)-1}_{qp}(\Omega)^3),$$ i.e. the solutions regularize instantly if $2/p +3/q<3$.
Conditional Global Existence
----------------------------
Next we employ the abstract Serrin condition to characterize global existence. For this purpose we only need to compute the spaces $X_{\mu_c^{\sf w}}^{\sf w}$ and $X_{\mu_c^{\sf vw}}^{\sf vw}$. We have $$X_{\mu_c^{\sf w}}^{\sf w} = {_\parallel H}^{2\mu_c^{\sf w}-1}_{q,\sigma}(\Omega) = {_\parallel H}^{2/p+3/q-2}_{q,\sigma}(\Omega)={_\parallel H}^{2\mu_c^{\sf vw}-2}_{q,\sigma}(\Omega) = X_{\mu_c^{\sf vw}}^{\sf vw},$$ a surprise? This yields with Theorem \[thm5\] the following result.
Let $p\in (1,\infty)$, $q\in (1,\infty)$ such that $s:=2/p+3/q\leq 2$, and $s\leq 3$ in case $q<3$. Assume
$w_0\in {_\parallel B}^{3/q-2}_{qp,\sigma}(\Omega)$, and let $w$ denote the unique weak or very weak solution of according to Theorems \[thm:wvort\] or \[thm:vwvort\], with maximal interval of existence $[0,t_+)$. Then
1. $w\in L_p((0,a);{_\parallel H}^{s-2}_{q,\sigma}(\Omega))$, for each $a<t_+$.
2. If $t_+<\infty$ then $ w\not\in L_p((0,t_+);{_\parallel H}^{s-2}_{q,\sigma}(\Omega))$.
In particular, the solution exists globally if $w\in L_p((0,a);{_\parallel H}^{s-2}_{q,\sigma}(\Omega))$
for any finite number $a$ such that $a\leq t_+$.
We emphasize the case $s=2$, i.e. $2/p+3/q=2$. Then ${_\parallel H}^{s-2}_{q,\sigma}(\Omega)={_\parallel L}_{q,\sigma}(\Omega)$. So we have e.g. global existence if $w$ stays bounded in $L_2(J;L_3(\Omega)^3)$ or in $L_4(J;L_2(\Omega)^3)$.
Small Data
----------
In case $p>2, q\geq 3$ we may continue the very weak solution instantly to a weak solution. In fact, for $q\geq 3$ we have the estimate $$|G^{\sf w}(w_1,w_2)|_{{_\parallel H}^{-1}_{q,\sigma}} \leq |u_1|_{L_\infty}|u_2|_{H^1_q}
\le C |u_1|_{H^{2\beta}_q} |u_2|_{H^{2\beta}_q} \leq C |w_1|_{H^{2\beta-1}_q} |w_2|_{H^{2\beta-1}_q},$$ for any $2\beta>1$. This shows that any $\mu>1/p$ is admissible, we are in the subcritical case. For sufficiently small $\mu>1/p$ we have $$X_\gamma^{\sf vw}={_\parallel B}^{-2/p}_{qp,\sigma}(\Omega)\hookrightarrow {_\parallel B}^{2(\mu-1/p)-1}_{qp,\sigma}(\Omega)=X^{\sf w}_{\gamma,\mu},$$ hence we obtain the following result.
Let $q\in[3,\infty)$ and $p\in(2,\infty)$.\
Then, for each $w_0\in {_\parallel B}^{3/q-2}_{qp,\sigma}(\Omega)$, the vorticity equation admits a unique weak solution $$u \in H^1_{p,loc}((0,t_+); {_\parallel H}^{-1}_{q,\sigma}(\Omega)))\cap L_{p,loc}((0,t_+); {H}^1_q(\Omega)^3).$$ on a maximal time interval $(0,t_+)$.
Now we want to consider data which are small in the critical spaces ${_\parallel B}^{3/q-2}_{qp,\sigma}$ where $p,q\in (1,\infty)$, with $2/p+3/q<2$. To apply Corollary \[cor2\] in Section 2, we need to study the spectrum of the weak operator $A^{\sf w}$, which consists only of eigenvalues and by elliptic regularity is independent of $q$. So it is sufficient to consider the case $q=2$.
Suppose $\lambda\in {{\mathbb C}}$ is an eigenvalue of $A^{\sf w}$ with eigenfunction $w\neq 0$. Then setting $\phi=v$, with $v$ being the stream function, in the definition of the operator $A^{\sf w}$ we obtain $$\lambda (w|v)_\Omega = \langle A^{\sf w}w|v\rangle,\quad w_\parallel= \nu\times {\sf B}_\Sigma u.$$ Integrating by parts on the left hand side, with $u={\rm rot}\, v$, $w={\rm rot }\,u$, we obtain after some calculations involving the boundary conditions as well as ${\rm div}\, w ={\rm div}\, u={\rm div}\, v =0$ the identity $$\lambda|u|_\Omega^2 = 2|D(u)|_\Omega^2 + \alpha |u|_\Sigma^2.$$ In order to verify the assertion on the right hand side, we first use a partial integration to obtain $\langle A^{\sf w}w|v\rangle= -(\Delta w|v)_\Omega$ (assuming for the moment that all functions be sufficiently smooth). Employing $-\Delta w={\rm rot}\,{\rm rot}\,w$ (as ${\rm div}\,w=0$) and the relationships between $u,v,w$, an integration by parts yields $(\Delta w|v)_\Omega=(\Delta u|u)_\Omega.$ Yet another integration by parts in conjunction with the Navier boundary conditions and the relation $\Delta u = 2\,{\rm div}D(u)$ (as ${\rm div}\,u=0$) implies the assertion.
This shows that the eigenvalues of $A^{\sf w}$ are real and nonnegative. In addition, if $\alpha>0$ then Korn’s inequality shows that $0$ is not an eigenvalue. Therefore, the analytic $C_0$-semigroup generated by $A^{\sf w}$ is exponentially stable. Corollary \[cor2\] then yields the following result.
\[thm:ex-stable\] Suppose $p,q\in (1,\infty)$ such that $2/p+3/q<2$ and $p\ge2$.\
Then the trivial solution of the vorticity equation is globally exponentially stable in $X_{\gamma}^{\sf w} \subset B^{1-2/p}_{qp}(\Omega)^3.$ Moreover, there is $r_0>0$ such that every very weak solution $w$ with initial value $w_0\in {_\parallel B}^{3/q-2}_{qp,\sigma}(\Omega)$ with norm $|w_0|_{{_\parallel B}^{3/q-2}_{qp,\sigma}}\leq r_0$ exists globally and converges exponentially to zero in the norm of $B^{1-2/p}_{qp}(\Omega)^3$.
The Navier boundary conditions considered in were first introduced by Navier in [@Na27] and later derived by Maxwell in [@Ma79] from the kinetic theory of gases. Problem has been considered by several authors, see for instance [@PrWi17a] and the references therein. The construction of a stream function in Subsection 4.2, based on solvability of the elliptic problem , seems to be new. Moreover, to the best of our knowledge, the results contained in Theorems \[thm:vwvort\]-\[thm:ex-stable\] are new.
We refer to [@Za05] for results concerning the vorticity equations (corresponding to the Naveier-Stokes system with slip boundary conditions) in a cylindrical domain, and to [@GiMi89] concerning the vorticity equations in ${{\mathbb R}}^3$.
Further Applications
====================
In this section, we apply our theory to a variety of well-known problems. Applying Theorem \[thm1\], as well as Corollaries \[cor2\] and \[cor1\], we obtain results in critical spaces which seem to be new in the case of bounded domains. We would like to emphasize that our proofs are rather simple, as they do not involve the microscopic structure of Besov spaces. Corresponding results in the literature seem restricted to the case of ${{\mathbb R}}^d$, where techniques from harmonic analysis are applied.
Convection-Diffusion
--------------------
Let $\Omega\subset{{\mathbb R}}^d$ be a bounded domain of class $C^{4}$ and consider the following non-local convection-diffusion problem. $$\begin{aligned}
\label{C-D}
\partial_t u -\Delta u &=-{\rm div}(u\nabla w) &&\mbox{in} \;\; \Omega,\\
-\Delta w &= \pm u &&\mbox{in} \;\; \Omega,\\
\partial_\nu u =\partial_\nu w &=0 && \mbox{on} \;\; \partial\Omega,\\
u(0)&=u_0 && \mbox{in} \;\; \Omega.
\end{aligned}$$ Here $u$ means a scalar variable, such as a density or a concentration, and $w$ a potential. Observe that the mean value of $u$ vanishes identically if that of $u_0$ does. We assume this throughout. Then $w$ is uniquely defined with mean zero. Without loss of generality, we may assume $\Delta w=u$. If not, we replace $u$ by $-u$. Note that this system is (locally) scaling invariant w.r.t the scaling $$\big(u_\lambda,w_\lambda \big)(t,x):=\big(\lambda^2 u,w\big)(\lambda^2t,\lambda x).$$ For $q\in (1,\infty)$, define $$X_0:=L_{q,(0)}(\Omega):=\{u\in L_q(\Omega):\int_\Omega u\, dx=0\}$$ and an operator $A:X_1\to X_0$ by $Au=-\Delta u$ with domain $$X_1=\{u\in \,H_q^2(\Omega)\cap L_{q,(0)}(\Omega):\partial_\nu u=0\ \text{on}\ \partial\Omega\}.$$ By [@DDHPV04], it holds that $A\in \mathcal{H}^\infty(X_0)$ with angle $\phi_{A}^\infty=0$. Moreover, $X_\beta=(X_0,X_1)_\beta={_\nu H_{q,(0)}^{2\beta}(\Omega)}$, $\beta\in [0,1]$, where $${_\nu H_{q,(0)}^{2\beta}(\Omega)}:=L_{q,(0)}(\Omega)\cap
\left\{
\begin{aligned}
& \{u\in H_q^{2\beta}(\Omega):\partial_\nu u=0\ \text{on}\ \partial\Omega\} && \text{for} && 2\beta\in (1+1/q,2],\\
& H_q^{2\beta}(\Omega) &&\text{for} && 2\beta\in [0,1+1/q).
\end{aligned}
\right.$$ For $s\in (1,\infty)$ and $\tau\in [0,2]$, we denote by $S: H_{s,(0)}^{\tau}(\Omega)\to H_s^{\tau+2}(\Omega)$ the linear solution map $u\mapsto w$ for the elliptic problem $$\begin{aligned}
\Delta w&=u&&\mbox{in} \;\; \Omega,\\
\partial_\nu w&=0&&\mbox{on} \;\; \partial\Omega,
\end{aligned}$$ which is well-defined thanks to [@Tri78 Theorem 5.5.1] and the fact that $u$ has mean value zero. We note on the go that there exists a constant $C>0$ such that for all $u\in H_{s,(0)}^\tau(\Omega)$ $$|Su|_{H_s^{\tau+2}(\Omega)}\le C|u|_{H_s^\tau(\Omega)}.$$ With the operator $S$ at hand, we may reduce to the single equation $$\label{eq:C-D-u}
\partial_t u+Au=F(u),\ t>0,\quad u(0)=u_0,$$ where $F(u)=G(u,u)$ and (since $\Delta Sv=v$) $$G(u,v)= -{\rm div}(u\nabla Sv)= -\nabla u\cdot\nabla Sv - u\Delta Sv= -\nabla u\cdot\nabla Sv - uv$$ is bilinear in $(u,v)\in X_\beta\times X_\beta$. For $u\in X_\beta$ and by Hölder’s inequality, we obtain $$|F(u)|_{L_q}\le |u|_{H_{qr}^1}|Su|_{H_{qr'}^1}+|u|_{L_{2q}}^2.$$ Choose $\beta\in (0,1)$ such that $$H_q^{2\beta}(\Omega)\hookrightarrow H_{qr}^1(\Omega),\ H_q^{2\beta+2}(\Omega)\hookrightarrow H_{qr'}^1(\Omega)\quad\text{and}\quad H_{q}^{2\beta}(\Omega)\hookrightarrow L_{2q}(\Omega).$$ The first two embeddings hold if $$2\beta-\frac{d}{q}=1-\frac{d}{qr}\quad\text{and}\quad 2\beta+1-\frac{d}{q}=-\frac{d}{qr'}.$$ It turns out that this can always be achieved if $q\in (1,d/2)$, hence $d\ge 3$ is necessary. The number $\beta$ can then be computed to the result $\beta=d/4q$. In particular, the restriction $\beta<1$ is satisfied if $q\in (d/4,d/2)$. Note that for the above value of $\beta$, we have $H_q^{2\beta}(\Omega)\hookrightarrow L_{2q}(\Omega)$.
It is now easy to see that the estimate $$|F(u)-F(\bar{u})|_{X_0}\le C(|u|_{X_\beta}+|\bar{u}|_{X_\beta})|u-\bar{u}|_{X_\beta}$$ holds for some constant $C>0$ and all $u,\bar{u}\in X_\beta$, hence **(H2)** is satisfied with $m=\rho=1$ and $\beta_1=\beta$. Thus, the critical weight $\mu_c$ is given by $$\mu_c=\frac{1}{p}+\frac{d}{2q}-1,$$ which results from **(H3)**.
It holds that $\mu_c>1/p$ if $q\in (1,d/2)$ and $\mu_c\le 1$ if $1/p+d/2q\le 2$. The critical space reads $$X_{\gamma,\mu_c}=(X_0,X_1)_{\mu_c-1/p,p}={_\nu}B_{qp,(0)}^{2(\mu_c-1/p)}(\Omega)={_\nu}B_{qp,(0)}^{d/q-2}(\Omega),$$ where $$\label{thm:C-D}
{_\nu B}_{qp,(0)}^{r}(\Omega):=L_{q,(0)}(\Omega)\cap
\left\{
\begin{aligned}
& \{u\in B_{qp}^{r}(\Omega):\partial_\nu u=0\} &&\text{for} && r\in (1+1/q,2],\\
& B_{qp}^{r}(\Omega) &&\text{for} && r\in (0,1+1/q).\\
\end{aligned}
\right.$$ Here we assume that $d\ge 3$, $p\in (1,\infty)$ $q\in (d/4,d/2)$ and $2/p+d/q\le 4$.
Choosing $X_0=H_q^{-1}(\Omega):=(H_{q'}^1(\Omega))'$ as a base space in the weak setting, one obtains $$\mu_c^{\sf w}=\frac{1}{p}+\frac{d}{2q}-\frac{1}{2}$$ as the critical weight, hence $q<d$, by the condition $\mu_c^{\sf w}>1/p$. Furthermore, $\mu_c^{\sf w}\le 1$ if and only if $$\frac{1}{p}+\frac{d}{2q}\le \frac{3}{2},$$ hence, in particular, $q>d/3$. This shows that we may consider space dimensions $d\ge 2$ in the weak setting.
Electro-Chemistry {#subsec:EC}
-----------------
Let $\Omega\subset{{\mathbb R}}^d$ be a bounded domain of class $C^{3}$ and consider the following problem of Nernst-Planck-Poisson type. $$\begin{aligned}
\label{NPP}
\partial_t u -\upmu_u\,\Delta u &={\rm div}(u\nabla w) && \mbox{in} \;\; \Omega,\\
\partial_t v -\upmu_v\,\Delta v &=-{\rm div}(v\nabla w) && \mbox{in} \;\; \Omega,\\
\partial_t w-\Delta w &= u-v && \mbox{in} \;\; \Omega,\\
\partial_\nu u =\partial_\nu v=\partial_\nu w &=0 && \mbox{on} \;\; \partial\Omega,\\
u(0)=u_0,\; v(0)&=v_0 && \mbox{in} \;\; \Omega.
\end{aligned}$$ The variables $u$ and $v$ denote concentrations of oppositely charged ions, and $w$ the induced electrical potential. Here $ \upmu_u,\upmu_v>0$ are assumed to be constant. In the following, we set $ \upmu_u=\upmu_v=1$. Note that this system is scaling invariant w.r.t the scaling $$\big(u_\lambda,v_\lambda, w_\lambda\big)(t,x) :=\big(\lambda^2 u,\lambda^2 v,w\big)(\lambda^2t,\lambda x).$$ Let $B_N=-\Delta$ in $L_q(\Omega)$, $1<q<\infty$, with domain $${\sf D}(B_N)=\{u\in H_q^2(\Omega):\partial_\nu u=0\ \text{on}\ \Omega\}.$$ It is well-known that for each $\omega>0$, $\omega+B_N\in \mathcal{H}^\infty(L_q(\Omega))$ with angle $\phi_{B_N}^\infty=0$, see e.g. [@DDHPV04]. The pair $(Y_0,B_0)=(L_q(\Omega),B)$ generates the complex extrapolation-interpolation scale $(Y_\alpha,B_\alpha)$, $\alpha\in{{\mathbb R}}$, see the Appendix. Consider the operator $$B_N^{\sf w}:=B_{-1/2}:H_q^1(\Omega)\to H_q^{-1}(\Omega):=(H_{q'}^1(\Omega))',$$ which has the explicit representation $$\langle B_N^{\sf w}u|\phi\rangle=(\nabla u|\nabla\phi)_{L_2(\Omega)}$$ for all $(u,\phi)\in H_q^1(\Omega)\times H_{q'}^1(\Omega)$. Then $B_{N}^{\sf w}\in\mathcal{H}^\infty(H_q^{-1}(\Omega))$ with the same angle as $B_N$.
As a base space for the system variable $z=(u,v,w)$ we take $$\begin{aligned}
X_0 &= H^{-1}_q(\Omega)\times H^{-1}_q(\Omega)\times H^1_q(\Omega)\end{aligned}$$ and we define $Az:= {\rm diag }\big(B_{N}^{\sf w}, B_{N}^{\sf w},B_{N}|_{H_q^3}\big)z+(0,0,v-u)$, with domain $$X_1:={\sf D}(A) = \{ z=(u,v,w)\in H^1_q(\Omega)^2\times H^3_q(\Omega):\; \partial_\nu w=0 \mbox{ on } \partial\Omega\}.$$ By the triangular structure of $A$, it follows readily that $A\in\mathcal{H}^\infty(X_0)$ with $\phi^{\infty}_A=0$. For the complex interpolation spaces $X_\beta=(X_0,X_1)_\beta$, $\beta\in (0,1)$, we obtain $$X_\beta = \{ z=(u,v,w)\in H^{2\beta-1}_q(\Omega)^2\times H^{2\beta+1}_q(\Omega):\; \partial_\nu w=0 \mbox{ on } \partial\Omega\},$$ if $2\beta>1/q$ and $$X_\beta=H^{2\beta-1}_q(\Omega)^2\times H^{2\beta+1}_q(\Omega)$$ if $2\beta<1/q$, where $H_q^{-r}(\Omega):=(H_{q'}^r(\Omega))'$ for $r\in [0,1]$. Define $F=(F_1,F_2,F_3):X_\beta\to X_0$ by $$\langle (F_1(z),F_2(z))|(\phi_1,\phi_2)\rangle := \left(-(u\nabla w|\nabla\phi_1)_{L_2},(v\nabla w|\nabla\phi_2)_{L_2}\right),$$ for all $(\phi_1,\phi_2)\in H_{q'}^1(\Omega)^2$ and $F_3(z):=0$. It follows that $$|F(z)|_{X_0}\le C (|u|_{L_{qr}(\Omega)}+|v|_{L_{qr}(\Omega)})|w|_{H_{qr'}^1(\Omega)},$$ hence there exists a positive number $C$ such that $$|F(z_1)-F(z_2)|_{X_0}\le C(|z_1|_{X_\beta}+|z_2|_{X_\beta})|z_1-z_2|_{X_\beta},\quad z_1,z_2\in X_\beta,$$ provided $\beta = \frac{1}{4}( 1+\frac{d}{q})$, $q\in (d/3,d)$, which yields the critical space $$X_{\gamma,\mu_c} = B^{d/q-2}_{qp}(\Omega)^2\times
{_\nu}B^{d/q}_{qp}(\Omega)
,$$ where $${_\nu B}_{qp}^{r}(\Omega):=
\left\{
\begin{aligned}
& \{w\in B_{qp}^{r}(\Omega):\partial_\nu w=0\} &&\text{for} && r\in (1+1/q,3),\\
& B_{qp}^{r}(\Omega) &&\text{for} && r\in (0,1+1/q)\\
\end{aligned}\right.$$ and $B_{qp}^{-s}(\Omega):=\left(B_{q'p'}^s(\Omega)\right)'$ for $s\in (0,1)$. Here we assume $ 2/p+d/q\le 3$, to make sure that $\mu_c\le1$.
Critical Besov spaces for system in the case $\Omega={{\mathbb R}}^d$ and with the parabolic equation $\partial_t w-\Delta w = u-v$ is replaced by the corresponding elliptic problem have been studied in [@Zhao17].
Chemotaxis equations
--------------------
Let $\Omega\subset{{\mathbb R}}^d$ be a bounded domain of class $C^{3}$. Then we consider the system $$\begin{aligned}
\label{CNS}
\partial_t u + u\cdot\nabla u-\upmu_u\, \Delta u +\nabla\pi &=0 && \mbox{in} \;\; \Omega,\\
{\rm div}\,u &=0 && \mbox{in} \;\; \Omega, \\
\partial_t v + u\cdot\nabla v- \upmu_v\,\Delta v &= -{\rm div}(v\nabla w) && \mbox{in} \;\; \Omega,\\
\partial_t w + u\cdot\nabla w -\upmu_w\,\Delta w &=-v && \mbox{in} \;\; \Omega,\\
u=0,\; \partial_\nu v=\partial_\nu w &=0 && \mbox{on} \;\; \partial\Omega,\\
u(0)=u_0,\; v(0)=v_0,\; w(0) &= w_0 && \mbox{in} \;\; \Omega.
\end{aligned}$$ Here $u$ is the velocity field, $w$ denotes the cell density, and $v$ a chemical potential. For simplicity, we choose all constants $\upmu_j$, $j\in\{u,w,v\}$, equal to one, as this does not affect the analysis. Note that the system is (locally) scaling invariant w.r.t. the scaling $$\big(u_\lambda ,\pi_\lambda, v_\lambda ,w_\lambda\big)(t,x) :=\big(\lambda u,\lambda^2\pi,\lambda^2 v, w\big)(\lambda^2 t,\lambda x).$$ Denote by $\mathbb{P}$ the Helmholtz projection in $L_q$ and let $L_{q,\sigma}(\Omega):=\mathbb{P}L_q(\Omega)^d$. We choose $X_0=L_{q,\sigma}(\Omega)\times H_q^{-1}(\Omega)\times H_q^1(\Omega)$ as a base space for $z=(u,v,w)$, where $H_q^{-1}(\Omega):=(H_{q'}^1(\Omega))'$. Define a linear operator $A:X_1\to X_0$ by $$Az={\rm diag }\left(B_S, B_N^{\sf w},B_{N}\right)z-(0,0,v)$$ with domain $$X_1=\{z=(u,v,w)\in H_{q,\sigma}^2(\Omega)\times H_q^1(\Omega)\times H_q^3(\Omega):u=0,\;\partial_\nu w=0\ \text{on}\ \partial\Omega\}.$$ The operators $B_{N}^{\sf w}$, $B_N$ are defined as in Subsection \[subsec:EC\] and $B_Su:=-\mathbb{P}\Delta u.$ Furthermore, $A\in \mathcal{H}^\infty(X_0)$ with angle $\phi_A^\infty=0$. The complex interpolation spaces $X_\beta=(X_0,X_1)_\beta$, $\beta\in (0,1)$, then read $$X_\beta = \{(u,v,w)\in H^{2\beta}_{q,\sigma}(\Omega)\times H^{2\beta-1}_q(\Omega)\times H^{2\beta+1}_q(\Omega):\; u=0,\,\partial_\nu w=0 \mbox{ on } \partial\Omega\}$$ if $2\beta\in (1/q,2]$ and $$X_\beta =H^{2\beta}_{q,\sigma}(\Omega)\times H^{2\beta-1}_q(\Omega)\times H^{2\beta+1}_q(\Omega)$$ if $2\beta\in [0,1/q)$, where $H_{q,\sigma}^r(\Omega):=H_q^r(\Omega)^d\cap L_{q,\sigma}(\Omega)$ and $H_q^{-s}(\Omega):=(H_{q'}^s(\Omega))'$ for $r\in[0,2]$ and $s\in [0,1]$.
For $\beta\ge 1/2$, we define $F=(F_1,F_2,F_3):X_\beta\to X_0$ by $F_1(z):=-\mathbb{P}(u\cdot\nabla u)$ $$\langle F_2(z)|\phi\rangle:=(v(u+\nabla w)|\nabla\phi)_{L_2(\Omega)},\quad\phi\in H_{q'}^1(\Omega),$$ and $F_3(z):=-u\cdot \nabla w$. By similar arguments as in Subsection \[subsec:EC\], it follows that there is a constant $C>0$ such that $$|F(z_1)-F(z_2)|_{X_0}\le C(|z_1|_{X_\beta}+|z_2|_{X_\beta})|z_1-z_2|_{X_\beta},\quad z_1,z_2\in X_\beta,$$ provided $\beta=\frac{1}{4}(1+\frac{d}{q})$ and $q\in (d/3,d)$. This in turn yields the critical space $$X_{\gamma,\mu_c} = {_0B}^{d/q-1}_{qp,\sigma}(\Omega)\times B^{d/q-2}_{qp}(\Omega) \times
{_\nu}B^{d/q}_{qp}(\Omega)
,$$ where $B_{qp}^{-r}(\Omega):=\left(B_{q'p'}^r(\Omega)\right)'$ for $r\in (0,1)$, $${_0 B}_{qp,\sigma}^{r}(\Omega):=L_{q,\sigma}\cap
\left\{
\begin{aligned}
& \{u\in B_{qp}^{r}(\Omega):u=0\} &&\text{for} && r\in (1/q,2),\\
& B_{qp}^{r}(\Omega) &&\text{for} && r\in (0,1/q),\\
\end{aligned}\right.$$ and $${_\nu B}_{qp}^{r}(\Omega):=
\left\{
\begin{aligned}
& \{w\in B_{qp}^{r}(\Omega):\partial_\nu w=0\} &&\text{for} && r\in (1+1/q,3),\\
& B_{qp}^{r}(\Omega) &&\text{for} && r\in (0,1+1/q).\\
\end{aligned}\right.$$
Magneto-Hydrodynamics
---------------------
In this last subsection, we consider the equations of magneto-hydrodynamics which read $$\label{MHD1}
\begin{aligned}
\varrho (\partial_t +u \cdot\nabla) u -\upnu\, \Delta u +\nabla \pi &= \frac{1}{\upmu_0} {\rm rot}\, B \times B && \mbox{in}\;\; \Omega,\\
\varrho (\partial_t +u\cdot\nabla) B -\frac{1}{\upmu_0\,\sigma} \Delta B &= B\cdot\nabla u && \mbox{in}\;\; \Omega,\\[2pt]
{\rm div}\, u = {\rm div}\,B &=0 && \mbox{in}\;\; \Omega,\\
u=0,\quad B\cdot\nu =0,\quad \nu\times {\rm rot}\, B &=0 && \mbox{on}\;\; \partial\Omega,\\
u(0)=u_0,\quad B(0)&=B_0 && \mbox{in}\;\; \Omega.
\end{aligned}$$ Here $u$ means the velocity field, $\pi$ the pressure, and $B$ the magnetic field. The parameters $\varrho,\upnu,\upmu_0,\sigma>0$ denote physical constants, which we set identical to one in the sequel. Note that the system is (locally) scaling invariant w.r.t. the scaling $$\big(u_\lambda,\pi_\lambda ,B_\lambda\big)(t,x) := \big(\lambda u, \lambda^2 \pi, \lambda B\big)(\lambda^2 t, \lambda x).$$ Next, observe that ${\rm rot}\, B\times B = B\cdot \nabla B -\frac{1}{2}\nabla |B|^2.$ As ${\rm div}\, u = {\rm div}\, B=0$ we may rewrite system in the following way. $$\label{MHD2}
\begin{aligned}
\partial_t u +{\rm div}\, (u\otimes u) - \Delta u +\nabla \tilde\pi &= {\rm div}\, (B\otimes B) && \mbox{in}\;\; \Omega,\\
\partial_t B + {\rm div}\,(u\otimes B) -\Delta B &= {\rm div}\, (B\otimes u) && \mbox{in}\;\; \Omega,\\
{\rm div}\, u = {\rm div}\,B &=0 && \mbox{in}\;\; \Omega,\\
u=0,\quad B\cdot\nu =0,\quad \nu\times {\rm rot}\, B &=0 && \mbox{on}\;\; \partial\Omega,\\
u(0)=u_0,\quad B(0)&=B_0 && \mbox{in}\;\; \Omega,
\end{aligned}$$ with $\tilde \pi =\pi + (1/2)|B|^2$. Let $\mathbb{P}$ denote the Helmholtz projection in $L_{q}(\Omega)^3$ and define $A_0(u,B):=(-\mathbb{P}\Delta u,-\mathbb{P}\Delta B)$ in $X_0:=L_{q,\sigma}(\Omega)^2$ with domain $$X_1:={\sf D}(A_0):=\{(u,B)\in {H}^2_{q,\sigma}(\Omega)^2:u=0,\;\nu\times{\rm rot}\,B=0\ \text{on}\ \partial\Omega\}.$$ Note that by the properties of the Helmholtz projection, $B\cdot \nu=0$ on $\partial\Omega$ for $B\in H^{r}_{q,\sigma}(\Omega)$ with $r\in (1/q,2]$. The pair $(X_0,A_0)$ generates the complex extrapolation-interpolation scale $(X_\alpha,A_\alpha)$, $\alpha\in{{\mathbb R}}$, see the Appendix. In the sequel, we choose the weak setting for $u$ and $B$, i.e. $\alpha=-1/2$. This yields $$X_0^{\sf w}:=X_{-1/2}={_0H}_{q,\sigma}^{-1}(\Omega)\times {H}_{q,\sigma}^{-1}(\Omega),$$ where ${_0H}_{q,\sigma}^{-1}(\Omega):=({_0H}_{q,\sigma}^{1}(\Omega))'$, ${H}_{q,\sigma}^{-1}(\Omega):=({H}_{q,\sigma}^{1}(\Omega))'$, $${H}_{q,\sigma}^{1}(\Omega):=H_q^1(\Omega)^3\cap L_{q,\sigma}(\Omega)\quad\text{and}\quad{_0H}_{q,\sigma}^{1}(\Omega):=\{u\in {H}_{q,\sigma}^{1}(\Omega):u=0\ \text{on}\ \partial\Omega\}.$$ Denote by $A^{\sf w}$ the operator $A_{-1/2}$ with domain $X_1^{\sf w}:=X_{1/2}={_0H}_{q,\sigma}^{1}(\Omega)\times {H}_{q,\sigma}^{1}(\Omega)$. This way, we may rewrite as the bilinear evolution equation $$\partial_t z+A^{\sf w}z=F^{\sf w}(z),\ t>0,\quad z(0)=z_0,$$ in $X_0^{\sf w}$ with $z:=(u,B)$ and $z_0:=(u_0,B_0)$. The nonlinearity $F$ is defined by $$\langle F^{\sf w}(z)|\phi\rangle:=\left((u\otimes u-B\otimes B|\nabla\phi_1)_{L_2},(u\otimes B-B\otimes u|\nabla\phi_2)_{L_2}\right)$$ for $\phi=(\phi_1,\phi_2)\in {_0H}_{q,\sigma}^{1}(\Omega)\times {H}_{q,\sigma}^{1}(\Omega)$ and $z=(u,B)\in X_\beta$, with $$X_\beta={_0\mathbb{H}}_{q,\sigma}^{2\beta-1}(\Omega):=
\left\{
\begin{aligned}
& {_0H}_{q,\sigma}^{2\beta-1}(\Omega)\times {H}_{q,\sigma}^{2\beta-1}(\Omega) &&\text{for} && 2\beta-1\in (1/q,1],\\
& {H}_{q,\sigma}^{2\beta-1}(\Omega)\times {H}_{q,\sigma}^{2\beta-1}(\Omega) &&\text{for} && 2\beta-1\in [0,1/q),
\end{aligned}
\right.$$ and ${_0\mathbb{H}}_{q,\sigma}^{-r}(\Omega):=\left({_0\mathbb{H}}_{q',\sigma}^{r}(\Omega)\right)'$ if $r\in [0,1]$. By similar arguments as in [@PrWi17a Section 5] one shows that the critical space is given by $$X_{\gamma, \mu_c}={_0 B}^{3/q-1}_{q,\sigma}(\Omega)\times {B}^{3/q-1}_{q,\sigma}(\Omega),$$ where we assume $2/p + 3/q\le 2$ for $p>1$ and $q>3/2$. We refer to [@ZLY17] for corresponding results in the case $\Omega={{\mathbb R}}^3$.
Multilinear Nonlinearities
==========================
In this last section, we consider (1.1) with multilinear nonlinearities of the form $F_2(u)=G(u,\ldots,u)$, where $$G:\Pi_{k=1}^m X_{\beta_k} \to X_0$$ is multilinear and bounded, with $\beta_k\in(0,1)$ and $m\geq 2$. Although the results derived here are not used in this publication, they are, nevertheless, relevant for applications. As before, $X_{\beta_k}=(X_0,X_1)_{\beta_k}$ are complex interpolation spaces. Here we show how to find the critical weight for this nonlinearity.
For this purpose, we may assume that the sequence $\beta_k$ is non-increasing, i.e. $$1>\beta_1 \geq \beta_2\geq \ldots \geq \beta_m>0,$$ and suppose that $\sum_1^m \beta_k>1$; otherwise we are in the subcritical case as will turn out below. Define a sequence $\mu_j$ by means of $$\mu_j = \frac{1}{p} + \frac{1}{j-1}\big(\sum_{k=1}^j \beta_k -1\big),\quad j=2,\ldots,m .$$ For the sake of definiteness, we set $\mu_1=-1$. We have $$\begin{aligned}
\mu_{j+1}\geq\mu_j \quad & \Leftrightarrow \quad\frac{1}{j}\big(\sum_{k=1}^{j+1} \beta_k -1\big) \geq \frac{1}{j-1}\big(\sum_{k=1}^j\beta_k-1\big) \\
&\Leftrightarrow \quad(j-1)\big(\sum_{k=1}^{j+1} \beta_k -1\big) \geq j\big(\sum_{k=1}^{j+1} \beta_k -1\big)-j\beta_{j+1}\\
&\Leftrightarrow \quad \beta_{j+1} \geq \frac{1}{j}\big(\sum_{k=1}^{j+1}\beta_k-1\big) = \mu_{j+1}-\frac{1}{p}.
\end{aligned}$$ We now assume that there is a unique number $l\in\{2,\ldots,m\}$ such that $\mu_l=\max_j \mu_j$. Then the critical weight is given by $$\mu_c := \mu_l = \frac{1}{p} + \frac{1}{l-1}\big(\sum_{k=1}^l \beta_k -1\big).$$ We observe that $\mu_{l}>\mu_{l-1}$ implies $\beta_l>\mu_l-1/p$, while $\mu_{l+1}<\mu_{l}$ implies $\beta_{l+1}<\mu_{l+1}-1/p<\mu_l-1/p.$ Hence, $$\beta_j>\mu_c-1/p\;\text{ for $j\le l$,\quad $\beta_j<\mu_c-1/p\;$ for $j>l$}.$$ The assumption $\sum_{k=1}^m \beta_k>1$ yields $\mu_c-1/p\ge \mu_m-1/p >0$. On the other side we have $\mu_c-1/p<1$, hence $\mu_c\le 1$ if $p$ is large enough.
We show that with this choice of $\mu_c$, Conditions [**(H2)**]{} and [**(H3)**]{} are valid. In fact, the identity $$F(u)-F(\bar{u}) = \sum_{j=1}^m G(\bar{u},\ldots,\bar{u}, u-\bar{u}, u,\ldots, u)$$ implies $$|F(u)-F(\bar{u})|_{X_0} \leq C \sum_{j=1}^m \Pi_{k=1}^{j-1}|\bar{u}|_{X_{\beta_k}}|u-\bar{u}|_{X_{\beta_j}} \Pi_{j+1}^m |u|_{X_{\beta_k}}.$$ Note that $X_{\gamma,\mu_c}\hookrightarrow X_{\beta_k}$ for $k>l$ and $ X_\beta\hookrightarrow X_{\beta_k}\hookrightarrow X_{\gamma,\mu_c}$ for $k\le l$, where $\beta=\max\beta_k =\beta_1$. Setting $$\alpha_k :=\frac{\beta_k-(\mu_c-1/p)}{\beta-(\mu_c-1/p)}$$ we obtain by interpolation $$|u|_{X_{\beta_k}}\leq c |u|_{X_{\gamma,\mu_c}}^{1-\alpha_k} |u|_{X_\beta}^{\alpha_k},\quad k\le l.$$ Setting $\alpha_k=0$ for $k>l$, this yields with $\rho_j = \sum_{k\neq j} \alpha_k$ and Young’s inequality $$\begin{aligned}
\Pi_{k=1}^{j-1}|\bar{u}|_{X_{\beta_k}} \Pi_{k=j+1}^m |u|_{X_{\beta_k}}
&\leq C(|u|_{X_{\gamma,\mu_c}},|\bar{u}|_{X_{\gamma,\mu_c}}) |\bar{u}|_{X_\beta}^{\sum_{k=1}^{j-1} \alpha_k}|u|_{X_\beta}^{\sum_{k=j+1}^{m} \alpha_k}\\
&\leq C(|u|_{X_{\gamma,\mu_c}},|\bar{u}|_{X_{\gamma,\mu_c}})\big( |u|_{X_\beta}^{\rho_j}+|\bar{u}|_{X_\beta}^{\rho_j}\big),\end{aligned}$$ and thus $$\Pi_{k=1}^{j-1}|\bar{u}|_{X_{\beta_k}}\Pi_{j+1}^m |u|_{X_{\beta_k}} |u-\bar{u}|_{X_{\beta_j}}
\le C(|u|_{X_{\gamma,\mu_c}},|\bar{u}|_{X_{\gamma,\mu_c}})\big( |u|_{X_\beta}^{\rho_j}+|\bar{u}|_{X_\beta}^{\rho_j}\big)
|u-\bar u|_{X_{\beta_j}}.$$
This shows that Condition [**(H2)**]{} holds. In order to verify Condition [**(H3)**]{} we observe that $\rho_j$ is given by $$\rho_j=\frac{1}{\beta-(\mu_c-1/p)}\sum_{1\le k\le l,k\neq j }(\beta_k-(\mu_c-1/p)).$$ This yields for $j\le l$ $$\begin{aligned}
\rho_j(\beta-(\mu_c-1/p))+ (\beta_j-(\mu_c-1/p) = \big(\sum_{k=1}^l\beta_k\big)-l(\mu_c-1/p))
= 1-(\mu_c-1/p),
\end{aligned}$$ showing that $\mu_c$ is critical. For $j>l$ we obtain $$\begin{aligned}
\rho_j(\beta-(\mu_c-1/p))+ (\beta_j-(\mu_c-1/p))
&=(1-(\mu_c-1/p))+(\beta_j-(\mu_c-1/p)) \\
& < (1-(\mu_c-1/p)).
\end{aligned}$$ Hence $\mu_c$ is subcritical in this case and we may use the estimate $|u-\bar u|_{X_{\beta_j}}\le c|u-\bar u|_{X_{\gamma,\mu}}$. Combining, we see that $\mu_c$ defined above is the critical weight for multilinear maps.
\[rem-multi\] Some special cases should be kept in mind.\
[**(i)**]{} $m=2$. Then $l=2$ and $\mu_c -1/p= \beta_1+\beta_2-1$.\
[**(ii)**]{} $m=3$. Then $\mu_c -1/p= \beta_1+\beta_2-1$ if $\beta_3< \beta_1+\beta_2-1$, and $\mu_c -1/p= (\beta_1+\beta_2+\beta_3-1)/2$ if $\beta_3> \beta_1+\beta_2-1$.\
[**(iii)**]{} If $\beta_k=\beta$ for all $k$ then $\mu_c = 1/p + (m\beta-1)/(m-1)$.
Appendix
========
Interpolation-extrapolation scale
---------------------------------
Here we collect some basic facts from the theory of Banach scales. Let $X_0$ a reflexive Banach space and $A_0\in\mathcal{BIP}(X_0)$ a linear operator with dense domain $X_1\hookrightarrow X_0$ and $0\in\rho(A_0)$. By [@Ama95 Theorem V.1.5.1] the pair $(X_0,A_0)$ generates an interpolation-extrapolation scale $(X_\alpha,A_\alpha)$, $\alpha\in{{\mathbb R}}$, with respect to the complex interpolation functor $(\cdot,\cdot)_\theta$, $\theta\in (0,1)$. In particular, for any $\alpha\in{{\mathbb R}}$, the operator $A_\alpha:X_{\alpha+1}\to X_\alpha$ is a linear isomorphism. If $0\notin \rho(A_0)$, then choose $\omega>0$ such that $0\in\rho(\omega+A_0)$ and replace $A_0$ by $\omega+A_0$.
For $\alpha<\beta$, $\rho(A_\alpha)=\rho(A_\beta)$ and the scale is densely injected, meaning that the embedding $X_\beta\hookrightarrow X_\alpha$ is dense. If $\alpha\ge 0$, $A_\alpha$ is the maximal restriction of $A_0$ to $X_\alpha$ and if $\alpha<0$, then $A_\alpha$ is the closure of $A_0$ in $X_\alpha$, hence $$A_\alpha u=A_0 u,\quad\text{if}\ u\in X_{1+\max\{\alpha,0\}},\ \alpha\in{{\mathbb R}}.$$
By [@Ama95 Theorem V.1.5.4] the scale $(X_\alpha,A_\alpha)$, $\alpha\in{{\mathbb R}}$, is equivalent to the fractional power scale, generated by $(X_0,A_0)$. In particular, for $\alpha>0$ it holds that $$X_\alpha=(D(A^\alpha),|A^\alpha\cdot |)$$ up to equivalent norms and the reiteration property $$\label{eq:ReitProp}
(X_\alpha,X_\beta)_\theta=X_{(1-\theta)\alpha+\theta\beta},\ \alpha<\beta,\ \theta\in (0,1)$$ holds.
Since $X_0$ is reflexive, [@Ama95 Theorem V.1.5.12] yields that $X_\alpha$ is reflexive, $$(X_\alpha)'=X_{-\alpha}^\sharp\quad\text{and}\quad (A_\alpha)'=A_{-\alpha}^\sharp$$ for any $\alpha\in{{\mathbb R}}$, where $(X_\alpha^\sharp,A_\alpha^\sharp)$, $\alpha\in{{\mathbb R}}$, denotes the dual interpolation-extrapolation scale generated by $(X_0^\sharp,A_0^\sharp)$. Here $X_0^\sharp$ is the dual space of $X_0$ and $A_0^\sharp$ denotes the dual operator of $A_0$ in $X_0^\sharp$ with domain $X_1^\sharp$. Furthermore, by [@Ama95 Proposition V.1.5.5], $A_\alpha\in\mathcal{BIP}(X_\alpha)$ for any $\alpha\in{{\mathbb R}}$, in particular, the reiteration property holds for the dual scale as well.
Concerning real interpolation of the spaces $X_\alpha$, we note that for all $\alpha,\beta\in [0,1]$ with $(\alpha,\beta)\neq (0,0)$ and $\theta\in (0,1)$, it follows from the reiteration theorem $$(X_{0},X_{\beta})_{\theta,p}=((X_{-\alpha},X_\beta)_{\frac{\alpha}{\alpha+\beta}},X_\beta)_{\theta,p}=(X_{-\alpha},X_{\beta})_{\frac{\alpha+\theta\beta}{\alpha+\beta},p}$$ and $$(X_{0},X_{\beta})_{\theta,p}=(X_0,(X_0,X_1)_\beta)_{\theta,p}=(X_{0},X_{1})_{\beta\theta,p},$$ where we made also use of . In summary, we obtain $$(X_{-\alpha},X_\beta)_{\tau,p}=(X_0,X_1)_{\tau(\alpha+\beta)-\alpha,p},$$ provided $\tau>\frac{\alpha}{\alpha+\beta}$ and $\tau<1$. For $\tau\in (0,\frac{\alpha}{\alpha+\beta})$ we make use of duality properties to derive $$(X_{-\alpha},X_\beta)_{\tau,p}=\left((X_{-\beta}^\sharp,X_{\alpha}^\sharp)
_{1-\tau,p'}\right)'=\left((X_{0}^\sharp,X_{1}^\sharp)
_{\alpha-\tau(\alpha+\beta),p'}\right)',$$ provided $1-\tau>\frac{\beta}{\alpha+\beta}$ or equivalently $\tau<\frac{\alpha}{\alpha+\beta}$.
In particular, if $\alpha=s$ and $\beta=1-s$ for some $s\in [0,1]$, this yields $$\label{re-int}
(X_{-s},X_{1-s})_{\tau,p}=
\left\{
\begin{aligned}
& (X_0,X_1)_{\tau-s,p} &&\text{for} && \tau\in (s,1)\\
& \left((X_0^\sharp,X_1^\sharp)_{s-\tau,p'}\right)' && \text{for} &&\tau\in (0,s).
\end{aligned}
\right.$$
An interpolation result
-----------------------
The following interpolation result, which seems to be new, was used in the proof of Theorem \[thm5\]. For the function spaces ${{\mathcal F}}_j$ appearing in the next proposition, the reader should think of $L_{p,\mu}$, $H^s_{p,\mu}$.
\[interpol\] Suppose $X_1$ is densely embedded in $X_0$, $A:X_1\to X_0$ is bounded and $A\in \mathcal{BIP}(X_0)$. Let ${{\mathcal F}}_j$, $j=0,1,$ be complete function spaces over an interval $J=(0,a)$ and let $\theta\in (0,1)$. Then $$({{\mathcal F}}_0(J;X_{\beta_0}),{{\mathcal F}}_1(J;X_{\beta_1}))_\theta \cong {{\mathcal F}}_\theta(J;X_\beta),
\quad \beta = (1-\theta)\beta_0 + \theta \beta_1,$$ where $(\cdot,\cdot)_\theta$ means complex interpolation, ${{\mathcal F}}_\theta =({{\mathcal F}}_0,{{\mathcal F}}_1)_\theta$, and $X_\alpha=(X_0,X_1)_\alpha$ for $\alpha\in (0,1).$
As $A$ has bounded imaginary powers, we know that $X_\alpha ={\sf D}(A^\alpha)$, $\alpha\in (0,1).$ We may assume w.l.o.g that $A$ is invertible.
[**(i)**]{} Let $x\in ({{\mathcal F}}_0(J;X_{\beta_0}),{{\mathcal F}}_1(J;X_{\beta_1}))_\theta$ be given. By definition of the complex interpolation method, there exists a bounded and continuous function $$h:\bar S\to {{\mathcal F}}_0(J;X_{\beta_0})+{{\mathcal F}}_1(J;X_{\beta_1}),$$ where $S:=[0<{\rm Re}\,z<1]$, such that $h$ is holomorphic on $S$, $$h(i\cdot)\in C_0({{\mathbb R}}; {{\mathcal F}}_0(J;X_{\beta_0})),\quad
h(1+i\cdot)\in C_0({{\mathbb R}}; {{\mathcal F}}_1(J;X_{\beta_1})),\quad \text{and}\;\; x=h(\theta).$$ Here, $C_0$ denotes the space of continuous functions vanishing at infinity. The norm of $x$ in $({{\mathcal F}}_0(J;X_{\beta_0}),{{\mathcal F}}_1(J;X_{\beta_1}))_\theta$ is given by the infimum of $$|h(i\cdot)|_{L_\infty({{\mathbb R}}; {{\mathcal F}}_0(J;X_{\beta_0}))} + |h(1+i\cdot)|_{L_\infty({{\mathbb R}}; {{\mathcal F}}_1(J;X_{\beta_1}))},$$ taken over all such functions $h$ with $h(\theta)=x$. Let $g(z):=e^{z^2-\theta^2} A^{\beta_0 + (\beta_1-\beta_0)z}h(z)$ for $z\in\bar S$. Using the fact that $A\in\mathcal{BIP}(X_0)$ one shows that $$g(i\cdot)\in C_0({{\mathbb R}};{{\mathcal F}}_0(J;X_0)),\quad g(1+i\cdot)\in C_0({{\mathbb R}};{{\mathcal F}}_1(J;X_0)).$$ This implies $h(\theta)=A^\beta x\in {{\mathcal F}}_\theta(J;X_0)$ by definition of the complex interpolation method and we can now conclude that $x\in {{\mathcal F}}_\theta(J;X_\beta)$. The argument also shows that $({{\mathcal F}}_0(J;X_{\beta_0}),{{\mathcal F}}_1(J;X_{\beta_1}))_\theta$ is embedded in ${{\mathcal F}}_\theta(J;X_\beta)$.
[**(ii)**]{} Suppose $x\in {{\mathcal F}}_\theta(J;X_\beta)$. Then there exists a bounded and continuous function $$g:\bar S\to {{\mathcal F}}_0(J;X_{\beta})+{{\mathcal F}}_1(J;X_{\beta})$$ such that $h$ is holomorphic on $S$, and $$g(i\cdot)\in C_0({{\mathbb R}}; {{\mathcal F}}_0(J;X_{\beta})),\quad
g(1+i\cdot)\in C_0({{\mathbb R}}; {{\mathcal F}}_1(J;X_{\beta})),\quad \text{and}\;\; x=g(\theta).$$ Let $h(z)=e^{z^2-\theta^2} A^{\beta-\beta_0 - (\beta_1-\beta_0)z}g(z)$ for $z\in\bar S$. Using once more the property that $A$ has bounded imaginary powers one shows that $$h(i\cdot)\in C_0({{\mathbb R}};{{\mathcal F}}_0(J,X_{\beta_0})),
\quad h(1+i\cdot)\in C_0({{\mathbb R}};{{\mathcal F}}_1(J,X_{\beta_1})).$$ Noting that $h(\theta)=x$ we can now conclude that $x\in ({{\mathcal F}}_0(J;X_{\beta_0}),{{\mathcal F}}_1(J;X_{\beta_1}))_\theta$, with continuous embedding.
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[^1]: This work was supported by a grant from the Simons Foundation (\#426729, Gieri Simonett).
|
{
"pile_set_name": "ArXiv"
}
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---
author:
- |
<span style="font-variant:small-caps;">Gerhard Kramm</span>\
\
University of Alaska Fairbanks, Geophysical Institute\
903 Koyokuk Drive, P.O. Box 757320 Fairbanks, AK 99775-7320, USA\
Email: [email protected]\
Phone: + 1 907 474 5992
title: '**Comment to “Recent Climate Observations Compared to Projections” by Rahmstorf et al.**'
---
With great interest I read this article of Rahmstorf et al. [@Ra07]. It is surprising to me that the authors only consider a period from the beginning of the seventies to recent years. I think that this is, clearly, a source of misinterpretation.\
The Mauna Loa observation of the atmospheric carbon dioxide ($CO_2$) concentration (probably the best $CO_2$ data we have) started in 1958. Therefore, one should consider the whole period of these observations. As illustrated in Figures \[Figure\_1\] and \[Figure\_2\], the correlations for the period 1958 to 2004 show a somewhat different picture as presented by Rahmstorf et al. [@Ra07] in their Figure 1. The results of my figures are based on the Mauna Loa $CO_2$ data (monthly and annual averages) and the mean near surface temperature anomalies of the Hadley Centre for Climate Prediction and Research, MetOffice, UK, for the northern hemisphere (also monthly and annual averages), too.\
If we do not consider the whole period of available data, then we might run in the wrong direction. Figures \[Figure\_3\] and \[Figure\_4\], for instance, illustrate results from correlation calculations for the period ranging from 1958 to 1988. Remember that in 1988 the Intergovernmental Panel of Climate Change (IPCC) of the United Nations and the World Meteorological Organization (WMO) was established. As shown in the figures attached, during 1988 there was certainly no correlation between $CO_2$ and the temperature anomalies, neither on the annual time scale (Figure \[Figure\_3\]) nor on the monthly time scale (Figure \[Figure\_4\]). Consequently, I wonder why the IPCC was established during that time.
[label]{} S. Rahmstorf, A. Cazenave, J.A. Church, J.E. Hansen, R.F. Keeling, D.E. Parker, R.C.J. Somerville. Recent climate observations compared to projections. Science 316 (4 May 2007), p. 709.
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{
"pile_set_name": "ArXiv"
}
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---
abstract: 'In this work a simple method to enforce the positivity-preserving property for general high-order conservative schemes is proposed. The method keeps the original scheme unchanged and detects critical numerical fluxes which may lead to negative density and pressure, and then imposes a cut-off flux limiter to satisfy a sufficient condition for preserving positivity. Though an extra time-step size condition is required to maintain the formal order of accuracy, it is less restrictive than those in previous works. A number of numerical examples suggest that this method, when applied on an essentially non-oscillatory base scheme, can be used to prevent positivity failure when the flow involves vacuum or near vacuum and very strong discontinuities.'
address:
- 'Lehrstuhl für Aerodynamik und Strömungsmechanik, Technische Universität München, 85748 Garching, Germany'
- 'Division of Applied Mathematics, Brown University, Providence, RI 02912, USA'
author:
- 'X. Y. Hu,'
- 'N. A. Adams'
- 'C.-W. Shu'
title: 'Positivity-preserving flux limiters for high-order conservative schemes'
---
numerical method, compressible flow, high-order conservative scheme, positivity-preserving
Introduction {#sec:intro}
============
Compressible flow problems are usually solved by conservative schemes. High-order conservative schemes are suitable for simulating flows with both shock waves and rich flow features (acoustic waves, turbulence) since they are capable of handling flow discontinuities and accurately resolve a broad range of length scales. One important issue of high-order conservative schemes is that non-physical negative density or pressure (failure of positivity) can lead to an ill-posed system, which may cause blow-ups of the numerical solution. While for some first-order schemes negative density or pressure can occur when a vacuum or near vacuum is reached, for higher-order conservative schemes positivity failure can also occur due to interpolation errors at or near very strong discontinuities even though the flow physically is far away from vacuum.
It is known that many first order Godunov-type schemes [@einfeldt1991godunov; @toro2009riemann; @gressier1999positivity] have the so called positivity-preserving property and can maintain positive density and pressure. It has been also proved that some second-order conservative schemes [@tao1999gas; @hu2004kinetic] are positivity-preserving with or without a more restrictive Courant-Friedrichs-Lewy (CFL) condition. For even higher-order conservative schemes, Perthame and Shu [@perthame1996positivity] proved that, given a first-order positivity-preserving scheme, such as Godunov-type schemes, one can always build a higher-order positivity-preserving finite volume scheme under the following constraints: (a) the cell-face values for the numerical flux calculation have positive density and pressure, (b) additional limits on the interpolation under a more restrictive CFL-like condition. With a different interpretation of these constraints based on certain Gauss-Lobatto quadratures, positivity-preserving methods have been successfully developed for high-order discontinuous Galerkin (DG) methods [@zhang2010positivity] and weighted essentially non-oscillatory (WENO) finite volume and finite difference schemes [@zhang2011maximum; @zhang2011positivity].
In this paper, we propose an alternative method to enforce the positivity-preserving property with a simple cut-off flux limiter. The flux limiter first detects critical numerical fluxes which may lead to negative density and pressure, then limits these fluxes to satisfy a sufficient condition for preserving positivity. Unlike the approaches in [@zhang2010positivity; @zhang2011maximum; @zhang2011positivity], in which positivity-preserving and the maintenance of high order accuracy are considered simultaneously when designing the limiter, here we design the cut-off flux limiter to satisfy positivity only, and then prove [*a posteriori*]{} the maintenance of high order accuracy under a time step restriction. It appears that, in our numerical experiments, a much less restrictive time-step size condition is sufficient for preserving positivity without destroying overall accuracy. An advantage of the approach in this paper is that the cut-off limiter is directly applied to the numerical flux and it can be applied to arbitrary high-order conservative schemes.
Method {#sec:method}
======
For presentation of the positivity-preserving flux limiters we assume that the fluid is inviscid and compressible, described by the one-dimensional Euler equations as $$\label{governing-equation}
\frac{\partial \mathbf{U}}{\partial t} + \frac{\partial \mathbf{F}(\mathbf{U})}{\partial x} = 0,$$ where $\mathbf{U} = (\rho, m, E)^{T}$, and $\mathbf{F}(\mathbf{U}) = [m, \rho u^2 + p, (E + p)u]^{T}$. This set of equations describes the conservation laws for mass density $\rho$, momentum density $m \equiv \rho u$ and total energy density $E=\rho e + \rho u^2/2$, where $e$ is the internal energy per unit mass. To close this set of equations, the ideal-gas equation of state $p = (\gamma -1)\rho e$ with a constant $\gamma$ is used. Note that the density and pressure have the relations with the conservative variables as $$\label{density-pressure}
\rho(\mathbf{U}) = \rho, \quad p(\mathbf{U}) = (\gamma - 1)\left(E - \frac{1}{2}\frac{m^2}{\rho}\right).$$ It is easy to find that they are locally Lipschitz continuous, i.e. $$\begin{aligned}
|\rho(\mathbf{U}_2) - \rho(\mathbf{U}_1)| & \leq & L_{\rho}||\mathbf{U}_2 - \mathbf{U}_1||, \label{lipschitz-density} \\ |p(\mathbf{U}_2) - p(\mathbf{U}_1)| & \leq & L_{p}||\mathbf{U}_2 - \mathbf{U}_1||, ~{\rm if} ~\rho(\mathbf{U}_1) > 0, ~\rho(\mathbf{U}_2)
> 0, \label{lipschitz-pressure}\end{aligned}$$ where $L_{\rho}$ and $L_{p}$ are Lipschitz constants. For $1 \geq \theta \geq 0$, $\rho(\mathbf{U})$ and $p(\mathbf{U})$ have the properties $$\begin{aligned}
\label{inequality}
\rho\left[(1 - \theta)\mathbf{U}_1 + \theta \mathbf{U}_2\right] & = & (1 - \theta)\rho(\mathbf{U}_1) + \theta \rho(\mathbf{U}_2), \label{inequality-rho} \\
p\left[(1 - \theta)\mathbf{U}_1 + \theta \mathbf{U}_2\right] & \geq & (1 - \theta)p(\mathbf{U}_1) + \theta p(\mathbf{U}_2),
~{\rm if} ~\rho(\mathbf{U}_1), \, ~\rho(\mathbf{U}_2) > 0,
\label{inequality-p}\end{aligned}$$ where Eq. (\[inequality-rho\]) is straightforward and Eq. (\[inequality-p\]) is implied by the Jensen’s inequality since $p(\mathbf{U})$ is a concave function.
Finite-volume and finite-difference conservative schemes
--------------------------------------------------------
When Eq. (\[governing-equation\]) is discretized within the spatial domain such that $x_i =
i\Delta x$, $i=0, ..., N$, where $\Delta x$ is the spatial step, a general explicit $k$th-order conservative scheme with Euler-forward time integration can be written as $$\label{conservative-scheme}
\mathbf{U}^{n+1}_i = \mathbf{U}^{n}_i + \lambda \left(\hat{\mathbf{F}}_{i-1/2} - \hat{\mathbf{F}}_{i+1/2}\right),$$ where the superscript $n$ and $n+1$ represent the old and new time steps, respectively, and $\lambda = \Delta t/\Delta x$, where $\Delta t$ is time-step size. Note that with the CFL condition $$\label{CFL}
\Delta t = \frac{{\rm CFL} \cdot \Delta x}{(|u| + c)_{\max}},$$ where $c=\sqrt{\gamma p /\rho}$ is the sound speed and the CFL number $0<{\rm CFL}<1$, one has the relation $$\label{lambda0}
\lambda = \frac{\rm CFL}{(|u| + c)_{\max}}.$$ For a finite-volume scheme, $\mathbf{U}^{n}_i$ and $\mathbf{U}^{n+1}_i$ are the cell averaged conservative variables on the cell $i$ defined on the computational cell between $(i-1/2)\Delta x$ and $(i+1/2)\Delta x$, i.e. $I_i = [x_{i-1/2}, x_{i+1/2}]$, $\hat{\mathbf{F}}_{i\pm1/2} = \mathbf{F}_{i\pm1/2} + \mathbf{O}(\Delta x^{k+1})$ are the numerical fluxes, which are based on the cell-face values $\mathbf{U}_{i\pm1/2}$ reconstructed from the cell averages $\{ \mathbf{U}_j \}$ and $\mathbf{F}_{i\pm1/2} = \mathbf{F}(\mathbf{U}_{i\pm1/2})$.
For a finite-difference scheme, $\mathbf{U}^{n}_i$ and $\mathbf{U}^{n+1}_i$ are the nodal values, and $(\hat{\mathbf{F}}_{i+1/2} - \hat{\mathbf{F}}_{i-1/2})/\Delta x$ is a $k$th order approximation to $\partial \mathbf{F}(\mathbf{U})/\partial x$ at $x = x_i$. Assume there exists a function $\mathbf{H}(x)$ depending on $\Delta x$ such that $$\label{reconstruction-pair}
\mathbf{F}\left[\mathbf{U}(x)\right] = \frac{1}{\Delta x}\int^{x + \Delta x /2}_{x - \Delta x /2} \mathbf{H}(\xi) d \xi,$$ then the same reconstruction procedure in a finite-volume scheme can be used to obtain the numerical fluxes $\hat{\mathbf{F}}_{i\pm1/2} = \mathbf{H}_{i\pm1/2} + O(\Delta x^{k+1})$ based on the cell-face values of $\mathbf{H}(x)$ reconstructed from its cell-average values $\mathbf{F}\left[\mathbf{U}_j\right] = \int^{x_j
+ \Delta x /2}_{x_j - \Delta x /2} \mathbf{H}(\xi) d \xi /\Delta x $. We refer to [@SO2] for the discussion of this formulation of conservative finite difference schemes.
Positivity preserving cut-off flux limiter
------------------------------------------
The positivity-preserving property for the scheme Eq. (\[conservative-scheme\]) refers to the property that the density and pressure are positive for $\mathbf{U}^{n+1}_i$ when $\mathbf{U}^{n}_i$ has positive density and pressure. Since Eq. (\[conservative-scheme\]) can be rewritten as a convex combination $$\begin{aligned}
\label{rewritten-scheme}
\mathbf{U}^{n+1}_i & = & \frac{1}{2}\left(\mathbf{U}^{n}_i + 2\lambda \hat{\mathbf{F}}_{i-1/2} \right) + \frac{1}{2}\left(\mathbf{U}^{n}_i - 2\lambda \hat{\mathbf{F}}_{i+1/2}\right) \nonumber \\
& = & \frac{1}{2}\mathbf{U}^{-}_i + \frac{1}{2}\mathbf{U}^{+}_i, \end{aligned}$$ a sufficient condition for preserving positivity is that $\mathbf{U}^{\pm}_i$ have positive density and pressure, i.e. $g(\mathbf{U}^{\pm}_i)>0$, where $g$ represents $\rho$ and $p$. Since the first-order Lax-Friedrichs flux $$\label{Lax-Friedrichs}
\hat{\mathbf{F}}^{LF}_{i+1/2} = \frac{1}{2}\left[\mathbf{F}_{i} + \mathbf{F}_{i+1} + (|u| + c)_{max}(\mathbf{U}^{n}_i - \mathbf{U}^{n}_{i+1})\right]$$ has the property $g(\mathbf{U}^{LF,\pm}_i) = g(\mathbf{U}^{n}_i \mp 2\lambda \hat{\mathbf{F}}^{LF}_{i \pm 1/2})>0$, under an additional CFL condition $$\label{CFL-condition-1}
{\rm CFL} \le \frac{1}{2}$$ (see [@zhang2010positivity]), a straightforward way to ensure positivity is to limit the magnitude of $\hat{\mathbf{F}}_{i+1/2}$ by utilizing the properties in Eqs. (\[inequality-rho\]) and (\[inequality-p\]). The positive density is first enforced by:
Cut-off flux limiter for positive density
:
1.
: For all $i$: initialize $\theta^{+}_{i+1/2} = 1$, $\theta^{-}_{i+1/2} = 1$.
2.
: If $ \rho(\mathbf{U}^{+}_i)<\epsilon_{\rho}$ , solve $\theta^{+}_{i+1/2}$ from $(1 - \theta^{+}_{i+1/2})\rho(\mathbf{U}^{LF,+}_i) + \theta^{+}_{i+1/2} \rho(\mathbf{U}^{+}_i) = \epsilon_{\rho}$.
3.
: If $ \rho(\mathbf{U}^{-}_{i+1})<\epsilon_{\rho}$, solve $\theta^{-}_{i+1/2}$ from $(1 - \theta^{-}_{i+1/2})\rho(\mathbf{U}^{LF,-}_{i+1}) + \theta^{-}_{i+1/2} \rho(\mathbf{U}^{-}_{i+1}) = \epsilon_{\rho}$.
4.
: Set $\theta_{\rho,i+1/2} = \min(\theta^{+}_{i+1/2}, \theta^{-}_{i+1/2})$, $\hat{\mathbf{F}}^{*}_{i+1/2} = (1-\theta_{\rho,i+1/2})\hat{\mathbf{F}}^{LF}_{i+1/2} + \theta_{\rho,i+1/2} \hat{\mathbf{F}}_{i+1/2}$.
Here, $\epsilon_{\rho} = \min\left\{10^{-13}, \rho^{0}_{min}\right\}$, where $\rho^{0}_{min}$ is the minimum density in the initial condition, $\hat{\mathbf{F}}^{*}_{i+1/2}$ is the limited flux, $0 \leq \theta^{\pm}_{i+1/2} \leq 1$ are the limiting factors corresponding to the two neighboring cells, which share the same flux $\hat{\mathbf{F}}_{i+1/2}$. After applying this flux limiter, Eq. (\[rewritten-scheme\]) becomes $$\begin{aligned}
\label{rewritten-scheme-1}
\mathbf{U}^{n+1}_i & = & \frac{1}{2}\left(\mathbf{U}^{n}_i + 2\lambda \hat{\mathbf{F}}^{*}_{i-1/2} \right) + \frac{1}{2}\left(\mathbf{U}^{n}_i - 2\lambda \hat{\mathbf{F}}^{*}_{i+1/2}\right) \nonumber \\
& = & \frac{1}{2}\mathbf{U}^{*,-}_i + \frac{1}{2}\mathbf{U}^{*,+}_i. \end{aligned}$$ Clearly, by Eq. (\[inequality-rho\]), both $\mathbf{U}^{*,-}_i$ and $\mathbf{U}^{*,+}_i$ have positive density, so does $\mathbf{U}^{n+1}_i$. The positive pressure is further enforced by:
Cut-off flux limiter for positive pressure
:
1.
: For all $i$: initialize $\theta^{+}_{i+1/2} = 1$, $\theta^{-}_{i+1/2} = 1$.
2.
: If $ p(\mathbf{U}^{*,+}_i)<\epsilon_{p}$ , solve $\theta^{+}_{i+1/2}$ from $(1 - \theta^{+}_{i+1/2})p(\mathbf{U}^{LF,+}_i) + \theta^{+}_{i+1/2} p(\mathbf{U}^{*,+}_i) = \epsilon_{p}$.
3.
: If $ p(\mathbf{U}^{*,-}_{i+1})<\epsilon_{p}$, solve $\theta^{-}_{i+1/2}$ from $(1 - \theta^{-}_{i+1/2})p(\mathbf{U}^{LF,-}_{i+1}) + \theta^{-}_{i+1/2} p(\mathbf{U}^{*,-}_{i+1}) = \epsilon_{p}$.
4.
: Set $\theta_{p,i+1/2} = \min(\theta^{+}_{i+1/2}, \theta^{-}_{i+1/2})$, $\hat{\mathbf{F}}^{**}_{i+1/2} = (1-\theta_{p,i+1/2})\hat{\mathbf{F}}^{LF}_{i+1/2} + \theta_{p,i+1/2} \hat{\mathbf{F}}^{*}_{i+1/2}$.
Again, $\epsilon_{p} = \min\left\{10^{-13}, p^{0}_{min}\right\}$, where $p^{0}_{min}$ is the minimum pressure in the initial condition, and $\hat{\mathbf{F}}^{**}_{i+1/2}$ is the further limited flux. After applying this flux limiter, Eq. (\[rewritten-scheme-1\]) becomes $$\begin{aligned}
\label{conservative-scheme-limited}
\mathbf{U}^{n+1}_i & = &
\frac 12 \left(\mathbf{U}^{n}_i + 2\lambda
\hat{\mathbf{F}}^{**}_{i-1/2} \right) + \frac 12 \left(\mathbf{U}^{n}_i -
2\lambda \hat{\mathbf{F}}^{**}_{i+1/2}\right) \nonumber \\
& = & \frac 12 \left( \mathbf{U}^{**,-}_i + \mathbf{U}^{**,+}_i
\right).\end{aligned}$$ Clearly, by Eqs. (\[inequality-rho\]) and (\[inequality-p\]), both $\mathbf{U}^{**,-}_i$ and $\mathbf{U}^{**,+}_i$ have positive density and pressure, so does $\mathbf{U}^{n+1}_i$. Note that these limiters can be applied at each sub-stage of a TVD Runge-Kutta [@shu1988efficient] method, which is a convex combination of Euler-forward time steps.
Consistency and accuracy
------------------------
Now we address two important issues for the cut-off flux limiter. First, the limited flux is a consistent flux since it is the convex combination of two consistent fluxes, i.e. the first-order Lax-Friedrichs flux $\mathbf{U}^{LF}_{i+1/2}$ and the original high-order numerical flux $\hat{\mathbf{F}}^{o}_{i+1/2}$, which represents $\hat{\mathbf{F}}_{i+1/2}$ and $\hat{\mathbf{F}}^{*}_{i+1/2}$. Second, when the limiter is active, the difference between the original flux $\hat{\mathbf{F}}^{o}_{i+1/2}$ and the limited flux $\hat{\mathbf{F}}^{lim}_{i+1/2}$ representing $\hat{\mathbf{F}}^{*}_{i+1/2}$ and $\hat{\mathbf{F}}^{**}_{i+1/2}$ is $$\label{correction}
||\hat{\mathbf{F}}^{lim}_{i+1/2} - \hat{\mathbf{F}}^{o}_{i+1/2}|| = (1 - \theta_{g,i+1/2})||\hat{\mathbf{F}}^{o}_{i+1/2} - \hat{\mathbf{F}}^{LF}_{i+1/2}||.$$ We only need to consider accuracy maintenance when $\theta_{g,i+1/2} <1$, for otherwise the limiter does not take any effect. Without loss of generality we may assume $\theta_{g,i+1/2} = \theta^+_{g,i+1/2}$. In this situation we have $g(\mathbf{U}^{o,+}_i) < \epsilon_{g}$, in which $\mathbf{U}^{o,+}_i$ represents $\mathbf{U}^{+}_i$ and $\mathbf{U}^{*,+}_i$, and $\epsilon_{g}$ is negligibly small, and $$\label{accuracy-condition}
1 - \theta_{g,i+1/2} = \frac{\epsilon_{g} - g(\mathbf{U}^{o,+}_i)}
{g(\mathbf{U}^{LF,+}_i) - g(\mathbf{U}^{o,+}_i)}
\approx \frac{ - g(\mathbf{U}^{o,+}_i)}
{g(\mathbf{U}^{LF,+}_i) - g(\mathbf{U}^{o,+}_i)}
\leq
\frac{|g(\mathbf{U}^{o,+}_i)|}
{g(\mathbf{U}^{LF,+}_i)} .$$ Since $\hat{\mathbf{F}}^{o}_{i+1/2}$ and $\hat{\mathbf{F}}^{LF}_{i+1/2}$ are both bounded in smooth regions, it is sufficient to show that the accuracy is not destroyed if the limiting factor satisfies $$\label{accuracy-condition2}
1 - \theta_{g,i+1/2} = O(\Delta x^{k+1}),$$ a sufficient condition for which would be $|g(\mathbf{U}^{o,+}_i)| = O(\Delta x^{k+1})$ and $g(\mathbf{U}^{LF,+}_i)$ is bounded away from zero.
Similar to Zhang and Shu [@zhang2010positivity], we assume the exact solution $\mathbf{U}(x)$ is smooth and $g(\widetilde{\mathbf{U}}_i) \geq M$, where $\widetilde{\mathbf{U}}_i$ is either the cell-average (for the finite-volume scheme) or the nodal value (for the finite-different scheme) of the exact solution $\mathbf{U}(x)$ and $M>0$ is a constant. Since $g(\mathbf{U}_i)$ is obtained from a $k$th order approximation, one has $g(\mathbf{U}_i)\geq M - O(\Delta x^{k+1}) > M/2$ if $\Delta x$ is sufficiently small, therefore $$\begin{aligned}
\label{bounding}
g(\mathbf{U}^{LF,+}_i) & = & g\left[(1-\hat{w})\mathbf{U}_i + \hat{w}\left(\mathbf{U}_i- \frac{2\lambda}{\hat{w}} \hat{\mathbf{F}}^{LF}_{i+1/2}\right)\right] \nonumber \\
& \geq & (1-\hat{w})g(\mathbf{U}_i) + \hat{w}g\left(\mathbf{U}_i- \frac{2\lambda}{\hat{w}} \hat{\mathbf{F}}^{LF}_{i+1/2}\right) \\
& \geq & \frac{(1-\hat{w})}{2} M >0, \nonumber\end{aligned}$$ where $1>\hat{w}>0$ is a constant, under an extra CFL condition $$\label{CFL-condition-2}
{\rm CFL} \le \frac{\hat{w}}{2}.$$ Furthermore, one has $$\begin{aligned}
\label{rewritten-positivity}
\mathbf{U}^{o,+}_i & = & \mathbf{U}^{n}_i -
2\lambda \hat{\mathbf{F}}^{o}_{i+1/2}\nonumber \\
& = & \mathbf{U}^{LF,+}_i + 2\lambda
\left(\hat{\mathbf{F}}^{LF}_{i+1/2} -
\hat{\mathbf{F}}^{o}_{i+1/2} \right) \nonumber \\
& = & \mathbf{U}^{LF,+}_i + 2\lambda
\left(\hat{\mathbf{F}}^{LF}_{i+1/2} -
\widetilde{\mathbf{F}}_{i+1/2}\right)
+ \mathbf{O}(\Delta x^{k+1}), \end{aligned}$$ where $\widetilde{\mathbf{F}}_{i+1/2} = \mathbf{F}_{i+1/2}$ for the finite-volume scheme, and $\widetilde{\mathbf{F}}_{i+1/2} = \mathbf{H}_{i+1/2}$ for the finite-difference scheme. Let $\mathbf{U}^{s}_i = \mathbf{U}^{LF,+}_i + 2\lambda
\left(\hat{\mathbf{F}}^{LF}_{i+1/2} -
\widetilde{\mathbf{F}}_{i+1/2}\right)$, and with Eqs. (\[lipschitz-density\]) and (\[lipschitz-pressure\]), one has $$\label{rewritten-positivity-1}
|g(\mathbf{U}^{s}_i) - g(\mathbf{U}^{o,+}_i)|
\leq L_{g}||\mathbf{U}^{o,+}_i - \mathbf{U}^{s}_i|| = O(\Delta x^{k+1}),$$ where $L_{g}$ is the Lipschitz constant. Note that the first term of $\mathbf{U}^{s}_i$ has positive density and pressure. For the second term, since the first-order Lax-Friedrichs flux $\hat{\mathbf{F}}^{LF}_{i+1/2}$ is a first order approximation to the exact flux $\widetilde{\mathbf{F}}_{i+1/2}$, that is $||\hat{\mathbf{F}}^{LF}_{i+1/2} -
\widetilde{\mathbf{F}}_{i+1/2}|| = O(\Delta x) $. With bounded $g(\mathbf{U}^{LF,+}_i)$ from Eq. (\[bounding\]), one has $$\rho(\mathbf{U}^{s}_i) \geq \frac{(1-\hat{w})}{2} M - O(\Delta x)
\geq \frac{(1-\hat{w})}{4} M >0$$ for sufficiently small $\Delta x$, according to Eq. (\[lipschitz-density\]), and furthermore $p(\mathbf{U}^{s}_i) > \epsilon_{p}$ according to Eq. (\[lipschitz-pressure\]). Since $g(\mathbf{U}^{s}_i) > \epsilon_{g}$ and $g(\mathbf{U}^{o,+}_i) < \epsilon_{g}$, i.e. $\rho(\mathbf{U}^{+}_i) < \epsilon_{\rho}$ while enforcing positive density and $p(\mathbf{U}^{*,+}_i) < \epsilon_{p}$ but $\rho(\mathbf{U}^{*,+}_i) > \epsilon_{\rho}$ while enforcing positive pressure, Eq. (\[rewritten-positivity-1\]) leads to $|g(\mathbf{U}^{o,+}_i)|= O(\Delta x^{k+1})$. Hence, we have proved that the cut-off flux limiter preserves high-order accuracy.
Note that, for given values of $M$ and grid size, Eqs. (\[bounding\]) and (\[rewritten-positivity\]) suggest that the errors introduced by the the cut-off flux limiter decrease with the time-step sizes. Also note that, the condition Eq. (\[CFL-condition-2\]) is less restrictive than the time-step size conditions in Refs. [@zhang2010positivity; @zhang2011maximum; @zhang2011positivity], and is desirable for higher computational efficiency.
Assessment of accuracy
----------------------
As a simple way to test the accuracy of the present flux limiters, we consider the one-dimensional linear advection equation $$\frac{\partial u}{\partial t} + \frac{\partial u}{\partial x}
= 0 \label{1d-linear-advection}$$ with initial condition $u(x)>0$. Applying the cut-off flux limiter to preserve positivity results in the limiter (denoted as HAS) $$\begin{aligned}
f^{*}_{i+1/2} & = & \theta(u_{i+1/2}^- - u^{n}_i) +
u^{n}_i, \quad \theta = \min \left\{\frac{u^{n}_i}{u^{n}_i - u_{\min}},
1\right\}, \nonumber\\ u_{\min} & = &\min \left\{u^{n}_i -
2\lambda u_{i+1/2}^-, u^{n}_{i+1} + 2\lambda u_{i+1/2}^-,
10^{-13}\right\}. \label{1d-linear-advection-limiter-1}\end{aligned}$$ Here $u_{i+1/2}^-$ is the approximated upwind flux at the cell face ${i+1/2}$. Note that only one of $u^{n}_i - 2\lambda u_{i+1/2}^-$ or $u^{n}_{i+1} + 2\lambda u_{i+1/2}^-$ being negative will activate the limiter. The limiter of Zhang and Shu [@zhang2011maximum] (denoted as ZS) for Eq. (\[1d-linear-advection\]) can be written as $$\begin{aligned}
f^{*}_{i+1/2} & = & \theta(u^{-}_{i+1/2} - u^{n}_i) + u^{n}_i,
\quad \theta = \min \left\{\frac{u^{n}_i}{u^{n}_i
- u_{\min}}, 1\right\}, \nonumber\\ u_{\min} &
= & \min \left\{\frac{u^{n}_i - \hat{w}_1 (u^{+}_{i-1/2}
+ u^{-}_{i+1/2})}{1-2\hat{w}_1}, u^{+}_{i-1/2}, u^{-}_{i+1/2},
10^{-13}\right\}, \label{1d-linear-advection-limiter-shu}\end{aligned}$$ where $u^{+}_{i-1/2}$ and $u^{-}_{i+1/2}$ are the high-order approximations of the cell-face values at $u(x_{i-1/2})$ and $u(x_{i+1/2})$ within cell $i$. For Eq. (\[1d-linear-advection\]), one has $u^{+}_{i-1/2} = u_{i+1/2}$. Comparing $u_{\min}$ in Eqs. (\[1d-linear-advection-limiter-1\]) and (\[1d-linear-advection-limiter-shu\]), it can be observed that the HAS limiter does not directly constrain the cell-face values to be non-negative.
To further illustrate the accuracy of the HAS limiter and its relation to the ZS limiter, we compute the advection of a function $u = 1 + 10^{-6} + \cos(2\pi x)$ in domain \[0, 1\] with a fifth-order conservative finite difference WENO-5 scheme [@jiang1996cient] with third-order TVD Runge-Kutta time integration [@shu1988efficient]. A periodic boundary condition is applied at $x=0$ and $x=1$. The final time is $t=1$, which corresponds to one period.
![Linear advection problem at $t=1$: (a) Error distribution vs. time-step sizes on 200 grid points; (b) Evolution of the $L_\infty$ error with decreasing grid size..[]{data-label="1d-advection-weno5"}](1d-advection-weno5.eps){width="120.00000%"}
This problem is computed on different grids with $N = 50$, 100, 200, 400 and 800 grid points. Figure \[1d-advection-weno5\]a shows the error distributions for the results on 200 grid points. It can be observed that if the maximum admissible CFL number of 0.5 is used, the HAS limiter produces larger errors than the ZS limiter. However, the HAS limiter is already as accurate as the ZS limiter when a smaller CFL number of $1/12$, which corresponds to the maximum admissible value for the latter, is used. If the time-step size is decreased further, errors produced by the ZS limiter do not change considerably, whereas the errors produced by the HAS limiter decrease further. This behavior is also shown in Fig. \[1d-advection-weno5\]b for the evolution of the $L_\infty$ error with decreasing grid size. Here, the time-step size $\Delta t = 0.5\Delta x^{5/3}$ is used to keep the spatial errors dominant. Note that Fig. \[1d-advection-weno5\]b clearly shows that the theoretical order of accuracy is achieved.
Extension to multiple dimensions
--------------------------------
To present the extension of the positivity-preserving flux limiters to multiple dimensions we consider the two-dimensional Euler equation $$\label{governing-equation-2d}
\frac{\partial \mathbf{U}}{\partial t} + \frac{\partial
\mathbf{F}(\mathbf{U})}{\partial x}
+ \frac{\partial \mathbf{G}(\mathbf{U})}{\partial y} = 0.$$ where $\mathbf{U} = (\rho, \rho u, \rho v, E)^{T}$, $\mathbf{F}(\mathbf{U}) = [\rho u, \rho u^2 + p, \rho u v,
(E + p)u]^{T}$ and $\mathbf{G}(\mathbf{U}) =
[\rho v, \rho v u, \rho v^2 + p, (E + p)v]^{T}$. Compared to the one-dimensional equation Eq. (\[governing-equation\]), the momentum density is $\rho\mathbf{v} = (\rho u, \rho v)$, where $u$ and $v$ are velocities in the $x$ and $y$ directions, respectively, and the total energy density is $E=\rho e +
\rho \mathbf{v}^2/2$. As an extension of Eq. (\[rewritten-scheme\]), the conservative scheme for Eq. (\[governing-equation-2d\]) can be rewritten as a convex combination $$\begin{aligned}
\label{rewritten-scheme-multi}
\mathbf{U}^{n+1}_{i,j} & = & \frac{\alpha_x}{2}\left(\mathbf{U}^{n}_{i,j}
+ 2\lambda_x \hat{\mathbf{F}}_{i-1/2, j}\right)
+ \frac{\alpha_x}{2}\left(\mathbf{U}^{n}_{i,j} -
2\lambda_x\hat{\mathbf{\mathbf{F}}}_{i+1/2, j}\right) \nonumber \\
& + & \frac{\alpha_y}{2}\left(\mathbf{U}^{n}_{i,j}
+ 2\lambda_y \hat{\mathbf{G}}_{i, j-1/2}\right)
+ \frac{\alpha_y}{2}\left(\mathbf{U}^{n}_{i,j} -
2\lambda_y \hat{\mathbf{G}}_{i, j+1/2}\right),\end{aligned}$$ where $\lambda_x = \Delta t/\Delta x \alpha_x$ and $\lambda_y = \Delta t/\Delta y \alpha_y$, $\alpha_x + \alpha_y = 1$, with $\alpha_x>0$ and $\alpha_y>0$ being partitions of the contribution in the $x$ and $y$ directions. A simple way to obtain this partition is to set $\alpha_x = \alpha_y = 1/2$ as in Zhang and Shu [@zhang2010positivity; @zhang2011positivity]. Another straightforward way to determine $\alpha_x$ and $\alpha_y = 1- \alpha_x$ is $$\label{partition}
\alpha_x = \frac{\tau_x}{\tau_x + \tau_y},
\quad \tau_x = \frac{(|u| + c)_{\max}}{\Delta x}, \quad
\tau_y = \frac{(|v| + c)_{\max}}{\Delta y}.$$ Note that, since the time-step size for integrating Eq. (\[rewritten-scheme-multi\]) is given by $$\label{CFL-2d}
\Delta t = \frac{\rm CFL}{\tau_x + \tau_y},$$ one has the relation $$\label{lambda-2d}
\lambda_x = \frac{\rm CFL}{(|u| + c)_{\max}} \quad {\rm and}
\quad \lambda_y = \frac{\rm CFL}{(|v| + c)_{\max}},$$ which gives an extended form from Eq. (\[lambda0\]). Also note that, since the components in Eq. (\[rewritten-scheme-multi\]) and Eq. (\[rewritten-scheme\]) have the same form, it is straightforward to implement the positivity-preserving flux limiters in a dimension-by-dimension fashion.
Test cases
==========
In the following, we illustrate that a number of typical numerical test cases, where the original high-order conservative schemes fail, can be simulated by using the proposed positivity-preserving flux limiters. For the first type of cases involving vacuum or near vacuum, the flux limiters are combined with the finite difference WENO-5 scheme [@jiang1996cient], which is a shock-capturing scheme with fifth-order accuracy for smooth solutions. For the second type of cases involving very strong discontinuities, the flux limiters are combined with the WENO-CU6-M1 scheme [@hu2011scale], which can be used for implicit large eddy simulation (LES) of turbulent flow and has sixth-order accuracy for smooth solutions. For both variants of the WENO schemes the Roe approximation is used for the characteristic decomposition at the cell faces, the Lax-Friedrichs formulation is used for the numerical fluxes, and the third-order TVD Runge-Kutta scheme is used for time integration [@shu1988efficient]. If not mentioned otherwise, the computations are carried out with a CFL number of 0.5.
One-dimensional problems involving vacuum or near vacuum
--------------------------------------------------------
Here we show that the proposed method passes two one-dimensional test problems involving vacuum or near vacuum: the double rarefaction problem [@hu2004kinetic], where a vacuum occurs, and the planar Sedov blast-wave problem [@sedov1959similarity; @zhang2011positivity], where a point-blast wave propagates. For the first problem, the initial condition is $$(\rho, u, p)=\cases{(1, -2, 0.1) & if $0<x<0.5$ \\
\cr (1, 2, 0.1) & if $1>x>0.5$ \\},$$ $\Delta x = 2.5\times 10^{-3}$ and the final time is $t=0.1$. For the second problem, the initial condition is $$(\rho, u, p)=\cases{(1, 0, 4\times 10^{-13}) & if $0<x<2-0.5\Delta x, \quad 2 + 0.5\Delta x < x <4$ \\
\cr (1, 0, 2.56\times 10^{8}) & if $2-0.5\Delta x<x<2 + 0.5\Delta x$ \\},$$ $\Delta x = 5\times 10^{-3}$ and the final time is $t=10^{-3}$.
Figure \[1d\] gives the computed pressure, density and velocity distributions, which show good agreement with the exact solutions.
![One-dimensional problems involving vacuum or near vacuum: (left) double rarefaction problem; (right) planar Sedov blast-wave problem.[]{data-label="1d"}](1d.eps){width="120.00000%"}
Although a vacuum occurs in the solution of the double rarefaction problem, the results still exhibit accurate density and pressure profiles in the rarefaction-wave regions. As a vacuum occurs, the solution at the center of the domain strictly speaking has no physical meaning. Note that compared to Zhang and Shu [@zhang2011positivity] (see their Fig. 5.1 (right)) for the planar Sedov blast-wave problem a slightly sharper blast wave is obtained in the present results. This may be due to the fact that Zhang and Shu [@zhang2011positivity] have modified the original Lax-Friedrichs flux to use a single maximum signal speed other than the respective maximum eigenvalues.
Two-dimensional problems involving vacuum or near vacuum
--------------------------------------------------------
We consider two two-dimensional problems involving vacuum or near vacuum. The first problem is the two-dimensional Sedov problem which has been studied in Zhang and Shu [@zhang2010positivity; @zhang2011positivity]. The computation is performed on the domain $[0,0]\times[1.1, 1.1]$, where a high pressure region occupies the computation cell at the lower-left corner. The initial condition is given by $$(\rho, u, v, p)=\cases{(1, 0, 0, 4\times 10^{-13}) & if $x > \Delta x, \quad y > \Delta y$ \\
\cr (1, 0, 0, \frac{9.79264}{\Delta x \Delta y}\times 10^{4}) & else \\},$$ where $\Delta x = \Delta y = 1.1/160$. The final time is $t = 1.0 \times 10^{-3}$. A reflective boundary condition is applied at the lower and left boundaries, and an outflow condition is applied at the right and upper boundaries.
![Two-dimensional Sedov problem: (left) 10 density contours from 0 to 6; (right) density profile along $y = 0$.[]{data-label="2d-sedov"}](2d-sedov.eps){width="120.00000%"}
Figure \[2d-sedov\] gives the computed density profiles. One can observe that these results are in very good agreement with the exact solution.
The second problem is the Mach-2000 jet problem, which has been computed in Zhang and Shu [@zhang2010positivity; @zhang2011maximum; @zhang2011positivity]. The computation is performed on the domain $[0,1]\times[0, 0.25]$, Initially, the entire domain is filled with ambient gas with $(\rho, u, v, p) = (0.5, 0, 0, 0.4127)$. A reflective condition is applied at the lower boundary, an outflow condition is applied at the right and upper boundaries, and an inflow condition is applied at the left boundary with states $(\rho, u, v, p) = (5, 800, 0, 0.4127)$ if $y<0.05$ and $(\rho, u, v, p) = (0.5, 0, 0, 0.4127)$ otherwise. A CFL number of 0.25 is used and the final time is 0.001. Since $\gamma = 5/3$ is used, the speed of the jet 800 gives about Mach 2100 with respect to the sound speed in the jet gas. Figure \[2d-mach-2000\] gives the computed density and pressure profiles in logarithmic scale.
![Mach-2000 jet problem: (upper) 30 density contours of logarithmic scale from -4 to 4; (lower) 30 pressure contours of logarithmic scale from -1 to 13.[]{data-label="2d-mach-2000"}](2d-mach-2000.eps){width="120.00000%"}
One can observe that these results are in very good agreement with those in Zhang and Shu [@zhang2010positivity] (their Fig. 4.6) computed with the same resolution.
One-dimensional problems involving very strong discontinuities
--------------------------------------------------------------
We show that, combined with the proposed flux limiters, the WENO-CU6-M1 scheme passes two one-dimensional test problems, which cannot be computed with the original scheme without limiting, involving very strong discontinuities: the two blast-wave interaction problem [@woodward1984numerical], and the Le Blanc problem [@loubcre2005subcell]. The latter is an extreme shock-tube problem. For the first problem, the initial condition is $$(\rho, u, p)=\cases{(1, 0, 1000) & if $0<x<0.1$ \\ \cr
(1, 0, 0.01) & if $0.1<x<0.9$ \\
\cr (1, 0, 100) & if $1>x>0.9$
\\},$$ $\Delta x = 2.5\times 10^{-3}$, and the final time is $t=0.038$. Reflective boundary conditions are applied at both $x=0$ and $x=1$. The reference “exact" solution is a high-resolution numerical solution on $3200$ grid points calculated by the WENO-CU6 scheme [@hu2010adaptive]. For the second problem, the initial condition is $$(\rho, u, p)=\cases{(1, 0, \frac{2}{3}\times 10^{-1}) & if $0<x<3$ \\
\cr (10^{-3}, 0, \frac{2}{3}\times 10^{-10}) & if $3<x<9$},$$ $\gamma = 5/3$, $\Delta x = 9/800$ and the final time is $t=6$.
Figure \[1d-strong\] gives the computed pressure, density and velocity distributions, although at relatively low resolution, which show a good agreement with the exact or reference solutions.
![One-dimensional problems involving very strong discontinuities: (left) two blast wave problem; (right) Le Blanc shock-tube problem.[]{data-label="1d-strong"}](1d-strong.eps){width="120.00000%"}
The magnitudes of the small over-shoots (see Fig. \[1d-strong\](left)) and the small errors at the shock position (see Fig. \[1d-strong\](right)) decrease when the resolution is increased (not shown here). For the two blast-wave interaction problem the present results are comparable to those obtained by the WENO-CU6-M2 scheme [@hu2011scale] at the same resolution (see their Fig. 3). Note that the WENO-CU6-M2 scheme stabilizes for very strong discontinuities in a different way, but still cannot compute the Le Blanc problem.
Two-dimensional problems involving very strong discontinuities
--------------------------------------------------------------
We first consider the problem from Woodward and Colella [@woodward1984numerical] on the double Mach reflection of a strong shock. A Mach 10 shock in air is reflected from the wall with incidence angle of $60^{\circ}$. The initial condition is $$(\rho, u, v, p)=\cases{(1.4, 0, 0, 1) & if $y < 1.732(x - 0.1667)$ \\
\cr (8, 7.145, -4.125, 116.8333) & else \\
},$$ and the final time is $t = 0.2$. The computational domain for this problem is $[0,0]\times[4,1]$. Initially, the shock extends from the point $x = 0.1667$ at the bottom to the top of the computational domain. Along the bottom boundary, at $y = 0$, from $x = 0$ to $x = 0.1667$ the post-shock conditions are imposed, whereas a reflective condition is set from $x = 0.1667$ to $x = 4$. Inflow and outflow conditions are applied at the left and right boundaries, respectively. The states at the top boundary are set to describe the exact motion of a Mach 10 shock.
![Double-Mach reflection of a Mach 10 shock wave at $t = 0.2$: (upper) 30 pressure contours from 0.92 to 520; (lower) 30 density contours from 1.73 to 21.[]{data-label="double-mach"}](double-mach.eps){width="120.00000%"}
Figure \[double-mach\] shows the pressure and density contours of the solution on a $240\times 60$ grid. Note that compared to the results obtained by WENO-CU6-M2 [@hu2011scale] (their Fig. 4) a good agreement is observed. Especially, both predict a strong near-wall jet, which is usually smeared in the previous computations with the same resolution [@kim2005high; @kawai2008localized; @hu2010adaptive].
We then consider a shock-bubble interaction problem, when a Mach 6 shock wave in air impacts on a cylindrical helium bubble. Air and helium are treated as the same ideal gas fluid for simplicity. Numerical computations for this problem can be found in Bagabir and Drikakis [@bagabir2001mach]. The initial conditions are $$\cases{(\rho=1, u=-3, v=0, p=1) & pre-shocked air \\
\cr (\rho=5.268, u=2.752, v=0, p=41.83) & post-shocked air \\
\cr (\rho=0.138, u=-3, v=0, p=1) & helium bubble \\
},\label{eq:sgn}$$ and the final time is $t = 0.15$. The computational domain for this problem is $[0,0]\times[1,0.5]$. Initially, the shock wave is at $x=0.05$, and the half helium bubble of radius 0.15 is at (0,0.25). Note that a frame velocity $u = -3$ is applied to keep the bubble approximately in the center of the computational domain. Reflective conditions are applied at the lower and upper boundaries, an outflow condition is applied at the right boundary, and an inflow condition is applied to the left boundary with the post-shocked state. Figure \[mach-6\] shows the pressure and density contours of the solution on a $200\times 100$ grid.
![Shock-bubble interaction problem at $t = 0.15$: (left) 30 pressure from 0.9 to 62; (right) 30 density contours from 0.5 to 8.[]{data-label="mach-6"}](mach-6.eps){width="75.00000%"}
These results show a fairly good agreement with those in Bagabir and Drikakis [@bagabir2001mach] (their Fig. 6) at the same resolution. The secondary reflected shock wave and triple-wave configurations are calculated with good resolution. Note that since the WENO-CU6-M1 scheme has smaller numerical dissipation than the MUSCL scheme used in Bagabir and Drikakis [@bagabir2001mach], the present results show a less smeared bubble interface and more detailed structures near the triple-wave region.
Concluding remarks
==================
In this paper we have proposed a very simple method to enforce the positivity-preserving property for general high-order conservative schemes. The method first detects critical numerical fluxes which may lead to negative density and pressure, then limits the fluxes to satisfy a sufficient condition for preserving positivity. Though an extra time-step size condition is required to maintain the formal order of accuracy, it is less restrictive than those in previous works. In addition, since the method uses the general form of a conservative scheme, similarly as the approaches of Zhang and Shu [@zhang2011positivity], it can be applied to flows with a general equation of state and source terms in a straightforward way.
Acknowledgment {#acknowledgment .unnumbered}
==============
We thank Dr. Xiangxiong Zhang for inspirational discussions. The research of the third author is supported by AFOSR grant F49550-12-1-0399 and NSF grant DMS-1112700.
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: |
Using the two-fluid formalism with the polytropic approximation, we examine the axisymmetric stability criterion for a composite system of gravitationally coupled stellar and gaseous singular isothermal discs (SIDs). Both SIDs are taken to be infinitely thin and are in a self-consistent steady background rotational equilibrium with power-law surface mass densities ($\propto
r^{-1}$) and flat rotation curves. Recently, Lou & Shen (2003) derived exact solutions for both axisymmetric and nonaxisymmetric stationary perturbations in such a composite SID system and proposed the $D_s-$criterion of stability for axisymmetric perturbations. Here for axisymmetric perturbations, we derive and analyze the time-dependent WKBJ dispersion relation to study stability properties. By introducing a dimensionless stellar SID rotation parameter $D_s$, defined as the ratio of the constant stellar rotation speed $V_s$ to the constant stellar velocity dispersion $a_s$, one can readily determine the axisymmetric stability $D_s-$criterion numerically by identifying a stable range of $D_s$. Those systems which rotate too slow (collapse) or too fast (ring fragmentation) are unstable. We found that the stable range of $D_s^2$ depends on the mass ratio $\delta$ of the gaseous SID to the stellar SID and on the square of the ratio $\beta$ of the stellar velocity dispersion (which mimics the sound speed) to the gaseous isothermal sound speed. Increment of either $\delta$ or $\beta$ or both will diminish the stable range of $D_s^2$. The WKBJ results of instabilities provide physical explanations for the stationary configurations derived by Lou & Shen. It is feasible to introduce an effective $Q$ parameter for a composite SID system. The closely relevant theoretical studies of Elmegreen (1995), Jog (1996) and Shu et al. (2000) are discussed. A study of composite partial SID system reveals that an axisymmetric dark matter halo will promote stability of composite SID system against axisymmetric disturbances. Potential applications to disc galaxies, circumnucleus discs around nuclei of galaxies, and protostellar discs are briefly discussed.
author:
- |
Yue Shen$^1$ and Yu-Qing Lou$^{1,2,3}$\
$^1$Physics Department,The Tsinghua Center for Astrophysics, Tsinghua University, Beijing 100084,China\
$^2$Department of Astronomy and Astrophysics, The University of Chicago, 5640 S. Ellis Ave., Chicago, IL 60637 USA\
$^3$National Astronomical Observatories, Chinese Academy of Sciences, A20, Datun Road, Beijing, 100012 China.
date: 'Accepted 2003 ... Received 2003 ...; in original form 2003 ... '
title: Axisymmetric stability criterion for two gravitationally coupled singular isothermal discs
---
stars: formation—ISM: general—galaxies: kinematics and dynamics—galaxies: spiral—galaxies: structure.
Introduction
============
In the context of galactic disc dynamics, there have been numerous investigations on criteria of axisymmetric and non-axisymmetric disc instabilities (e.g., Binney & Tremaine 1987). For a single disc of either gaseous or stellar content, Safronov (1960) and Toomre (1964) originally derived a dimensionless $Q$ parameter to determine the local stability (i.e., $Q>1$) against axisymmetric ring-like disturbances. Besides a massive dark matter halo, a more realistic disc galaxy involves both gas and stars. It is thus sensible to consider a composite system of one gas disc and one stellar disc under the gravitational control of the dark matter halo. Theoretical studies on this type of two-component disc systems were extensive in the past (Lin & Shu 1966, 1968; Kato 1972; Jog & Solomon 1984a, b; Bertin & Romeo 1988; Romeo 1992; Elmegreen 1995; Jog 1996; Lou & Fan 1998b, 2000a, b). In particular, it has been attempted to introduce proper definitions of an effective $Q$ parameter relevant to a composite disc system for the criterion of local axisymmetric instabilities (Elmegreen 1995; Jog 1996; Lou & Fan 1998b). The results of such a stability analysis may provide a basis for understanding composite disc dynamics (e.g. Lou & Fan 2000a, b; Lou & Shen 2003) and for estimating global star formation rate in a disc galaxy (e.g., Kennicutt 1989; Silk 1997).
There are several reasons that lead us to once again look into this axisymmetric stability problem. A few years ago, Shu, Laughlin, Lizano & Galli (2000; see also Galli et al. 2001) studied stationary coplanar perturbation structures in an isopedically magnetized singular isothermal disc (SID) without using the usual WKBJ approximation. They found exact analytical solutions for stationary (i.e., zero pattern speed) configurations of axisymmetric and non-axisymmetric logarithmic spiral perturbations. According to their analysis, for axisymmetric perturbations with radial propagation, a SID with sufficiently slow rotation speed will collapse for sufficiently large radial perturbation scales, while a SID with sufficiently fast rotation speed will be unstable to ring fragmentations for relatively small radial perturbation scales (see Fig. 2 of Shu et al. 2000). For axisymmetric perturbations, the stable regime of SID rotation speed may be characterized by a dimensionless rotation parameter $D$, which is the ratio of the constant SID rotation speed $V$ to the isothermal sound speed $a$. The critical values of the lowest and the highest $D$ for axisymmetric stability can be derived from their marginal stability curves. Shu et al. (2000) supported their interpretations by invoking the familiar Toomre $Q$ parameter, implying that a local WKBJ analysis may still have some relevance or validity to global SID perturbation solutions. In their proposed scheme, Shu et al. noted that the two critical values of $D$ are fairly close to $Q=1$ for neutral stability. As they did not invoke the WKBJ approximation in deriving the analytic perturbation solutions, the $Q-$criterion and the $D-$criterion correspond to each other well with large wavenumber (ring fragmentation), while the rough correspondence of the $Q-$criterion and the $D-$criterion for small wavenumbers (collapse) is unexpected where the WKBJ requirement is poorly met.
Regarding astrophysical applications, these basic properties of $D-$criterion should be properly applied in contexts even at qualitative or conceptual levels. For example, for majority of currently observed disc galaxies, the typical rotation parameter $D$ may be sufficiently large to avoid the collapse regime, that is, these disc galaxies are rotationally supported. Besides other instabilities, the ring fragmentation instability is conceptually pertinent to disc galaxies such as estimates of global star formation rates (e.g., Jog & Solomon 1984a; Kennicutt 1989; Silk 1997; Lou & Fan 1998b, 2002a, b). Having said these, we should add that the collapse regime of $D-$criterion might be relevant in early stages of galactic disc evolution. That is, slow and fast disc rotations of proto-galaxies may be responsible for eventual bifurcations into different morphologies of galactic systems. Furthermore, for central discs (with radii less than a few kiloparsecs) surrounding galactic nuclei with supermassive black holes (e.g., Lynden-Bell 1969) or for protostellar discs around central collapsed cores (e.g., Shu 1977), this $D$ parameter may be sufficiently small to induce disc collapses. Therefore, both the collapse and ring-fragmentation regimes can be of considerable interest in various astrophysical applications.
Recently, we investigated coplanar perturbation configurations in a composite system of two gravitationally coupled stellar and gaseous SIDs, both taken to be razor thin (Lou & Shen 2003). In the ideal two-fluid formalism, we derived analytical solutions for stationary axisymmetric and logarithmic spiral configurations in a composite SID system in terms of a dimensionless rotation parameter $D_s$ for the stellar SID[^1]. By the analogy of a single SID (Shu et al. 2000), these stationary configurations should form marginal stability curves (see Fig. 2 and more extensive results of Lou & Shen 2003) that separate the stable region from unstable ’ regions for axisymmetric disturbances. Compared with the single SID case, the stable range of $D_s$ is reduced owing to the presence of an additional gaseous SID and can vary for different parameters.
In order to make convincing arguments for the above analogy and our physical interpretations, we perform here a time-dependent local stability analysis using the WKBJ approximation for the composite SID system (Lou & Shen 2003). In a proper parameter regime, we demonstrate the correctness of interpretations by Shu et al. (2000), and in general, we clearly show the validity of using $D_s$ parameter to demark stable and unstable regimes. To place our analysis in relevant contexts, we also discuss how the two effective $Q$ parameters of Elmegreen (1995) and of Jog (1996) are related to our $D_s$ parameter when other pertinent parameters are specified.
In Section 2, we derive the time-dependent WKBJ dispersion relation for the composite SID system and define several useful dimensionless parameters. In Section 3, we present the criterion for the SID system being stable against all axisymmetric disturbances. Finally, our results are discussed and summarized in Section 4.
Two-fluid composite SID system
==============================
Here, we go through the basic fluid equations and derive the local dispersion relation in the WKBJ approximation (Lin & Shu 1964). For physical variables, we use either superscript or subscript $s$ for the stellar SID and $g$ for the gaseous SID. As both SIDs are assumed to have flat rotation curves with different speeds $V_s$ and $V_g$, we express the mean angular rotation speeds of the two SIDs in the forms of $$\Omega_s=V_s/r=a_sD_s/r,\ \qquad\ \Omega_g=V_g/r=a_gD_g/r\ ,$$ where $a_s$ and $a_g$ are the velocity dispersion of the stellar SID and the isothermal sound speed in the gaseous SID, respectively. Dimensionless rotation parameters $D_s$ and $D_g$ are defined as the ratios of stellar SID rotation speed to velocity dispersion and gaseous SID rotation speed to sound speed, respectively. The two epicyclic frequencies are $$\begin{split}
\kappa_s\equiv\{(2\Omega_s/r)[d(r^2\Omega_s)/dr]\}^{1/2}
=\sqrt{2}\Omega_s\ ,\\
\kappa_g\equiv\{(2\Omega_g/r)[d(r^2\Omega_g)/dr]\}^{1/2}
=\sqrt{2}\Omega_g\ .
\end{split}$$ Note that $\kappa_s$ and $\kappa_g$ are different in general. By the polytropic assumption for the surface mass density and the two-dimensional pressure (e.g., Binney & Tremaine 1987), the fluid equations for the stellar disc, in the cylindrical coordinates $(r,\varphi,z)$ at the $z=0$ plane, are $$\frac{\partial \Sigma^{s}}{\partial t} +\frac{1}{r}\frac{\partial
}{\partial r} (r\Sigma^{s}u^{s})+\frac{1}{r^{2}}\frac{\partial
}{\partial \varphi } (\Sigma^{s}j^{s})=0\ ,$$ $$\frac{\partial u^{s}}{\partial t} +u^{s}\frac{\partial
u^{s}}{\partial r} +\frac{j^{s}}{r^{2}}\frac{\partial u^{s}}
{\partial \varphi }-\frac{j^{s2}}{r^{3}}
=-\frac{1}{\Sigma^{s}}\frac{\partial }{\partial
r}(a_{s}^{2}\Sigma^{s}) -\frac{\partial \phi }{\partial r}\ ,$$ $$\frac{\partial j^{s}}{\partial t}+u^{s}\frac{\partial
j^{s}}{\partial r} +\frac{j^{s}}{r^{2}}\frac{\partial
j^{s}}{\partial \varphi } =-\frac{1}{\Sigma^{s}}\frac{\partial }
{\partial \varphi }(a_{s}^{2}\Sigma^{s}) -\frac{\partial \phi
}{\partial \varphi }\ ,$$ where $\Sigma^{s}$ is the stellar surface mass density, $u^{s}$ is the radial component of the bulk fluid velocity, $j^{s}$ is the $z-$component of the specific angular momentum, $a_{s}$ is the stellar velocity dispersion (or an effective “isothermal sound speed"), $a_{s}^{2}\Sigma^{s}$ stands for an effective two-dimensional pressure in the polytropic approximation, and $\phi$ is the total gravitational potential perturbation. For the corresponding fluid equations in the gaseous disc, we simply replace the relevant superscript or subscript $s$ with $g$. The two sets of fluid equations are coupled by the total gravitational potential $\phi$ through the Poisson integral $$\begin{aligned}
%& &\!\!\!\!
%\phi (r,\varphi ,t)=
%\nonumber \\ && \quad
\phi(r,\varphi,t)=
\oint\!d\psi\!\!\int_0^{\infty}\!\!\frac{-G\Sigma (r^{\prime
},\psi ,t)r^{\prime }dr^{\prime }}{\left[ r^{\prime
2}+r^{2}-2rr^{\prime }\cos (\psi -\varphi )\right]^{1/2}}\ ,
%\nonumber\end{aligned}$$ where $\Sigma =\Sigma^{s}+\Sigma^{g}$ is the total surface mass density. Here the stellar and gaseous SIDs interact mainly through the mutual gravity on large scales (Jog & Solomon 1984a,b; Bertin & Romeo 1988; Romeo 1992; Elmegreen 1995; Jog 1996; Lou & Fan 1998b, 2000a,b; Lou & Shen 2003).
Using the above equations for the steady background rotational equilibrium (indicated by an explicit subscript $0$) with $u_0^s=u_0^g=0$, $\Omega_s=j_0^s/r^2$, and $\Omega_g=j_0^g/r^2$, we obtain $$\begin{split}
\Sigma_0^s=a_s^2(1+D_s^2)/[2\pi Gr(1+\delta)]\ ,\\
\Sigma_0^g=a_g^2(1+D_g^2)\delta/[2\pi Gr(1+\delta)]\ ,
\end{split}$$ and an important relation[^2] $$\label{ad}
a_s^2(D_s^2+1)=a_g^2(D_g^2+1)\ ,$$ where $\delta\equiv\Sigma_0^g/\Sigma_0^s$ is the SID surface mass density ratio, $\beta\equiv a_s^2/a_g^2$ stands for the square of the ratio of the stellar velocity dispersion to the gas sound speed. In our analysis, we specify values $\delta$ and $\beta$ to characterize different composite SID systems. For late-type spiral galaxies, gas materials are less than the stellar mass with $\delta<1$. For young proto disc galaxies, gas materials may exceed stellar mass in general with $\delta>1$. Thus, in our analysis and computations, both cases of $\delta<1$ and $\delta\geq 1$ are considered. As the stellar velocity dispersion is usually larger than the gas sound speed ($a_s^2>a_g^2$), we then have $\beta>1$. When $\beta=1$, we have $D_s=D_g$ by condition (\[ad\]), which means the two SIDs may be treated as a single SID (Lou & Shen 2003).
For small perturbations denoted by subscripts $1$, we obtain the following linearized equations $$\frac{\partial \Sigma_{1}^{s}}{\partial
t}+\frac{1}{r}\frac{\partial } {\partial
r}(r\Sigma_{0}^{s}u_{1}^{s}) +\Omega_s \frac{\partial \Sigma
{}_{1}^{s} } {\partial \varphi
}+\frac{\Sigma_{0}^{s}}{r^{2}}\frac{\partial j_{1}^{s}} {\partial
\varphi }=0\ ,$$ $$\frac{\partial u_{1}^{s}}{\partial t}+\Omega_s \frac{\partial
u_{1}^{s}} {\partial \varphi }-2\Omega_s \frac{j_{1}^{s}}{r}
=-\frac{\partial }{\partial r}
\bigg(a_{s}^{2}\frac{\Sigma_{1}^{s}}{\Sigma_{0}^{s}}+\phi_{1}\bigg)\
,$$ $$\frac{\partial j_{1}^{s}}{\partial t}+r\frac{\kappa_s
^{2}}{2\Omega_s } u_{1}^{s}+\Omega_s \frac{\partial
j_{1}^{s}}{\partial \varphi } =-\frac{\partial }{\partial \varphi
}\bigg(a_{s}^{2}\frac{\Sigma_{1}^{s}} {\Sigma_{0}^{s} }+\phi
_{1}\bigg)$$ for the stellar SID, and the corresponding equations for the gaseous SID with subscript or superscript $s$ replaced by $g$ in equations (9)$-$(11). The total gravitational potential perturbation is $$\begin{aligned}
%& &\!\!\!\!\phi _{1}(r,\varphi ,t)=
%\nonumber \\ & &\quad
\phi_1(r,\varphi,t)= \oint\!\! d\psi\!\!\int_0^{\infty}\!\!\!
\frac{-G(\Sigma_1^s +\Sigma_1^g)r^{\prime }dr^{\prime
}}{\left[r^{\prime 2}+r^{2}-2rr^{\prime }\cos (\psi -\varphi
)\right]^{1/2}}\ .\end{aligned}$$ In the usual WKBJ approximation, we write the coplanar axisymmetric perturbations with the periodic dependence of $\exp(ikr+i\omega t)$ where $k$ is the radial wavenumber and $\omega$ is the angular frequency. With this dependence in the perturbation equations, we obtain the local WKBJ dispersion relation for the composite SID system in the form of $$\label{dis}
\begin{split}
(\omega^2-\kappa_s^2&-k^2a_s^2+2\pi G|k|\Sigma_0^s)\\
&\times
(\omega^2-\kappa_g^2-k^2a_g^2+2\pi G|k|\Sigma_0^g)\\
&-(2\pi G|k|\Sigma_0^s)(2\pi G|k|\Sigma_0^g)=0\ .
\end{split}$$ Jog & Solomon (1984a) have previously derived the similar dispersion relation with $V_s=V_g$ and thus $\kappa_s=\kappa_g$. The stability analyses of Elmegreen (1995) and Jog (1996) start with the same dispersion relation of Jog & Solomon (1984a). The WKBJ approach of Jog & Solomon (1984a) is generally applicable to any locally prescribed properties of a two-fluid disc system. In earlier theoretical studies including that of Jog & Solomon (1984a), it is usually taken that the rotation speeds of the two fluid discs are equal. Our model of two coupled SIDs here is self consistent globally with a central singularity and with different SID rotation speeds in general. When one applies the local WKBJ analysis to our composite SID system, the procedure is identical to that of Jog & Solomon (1984a) but with different local SID speeds $V_s$ and $V_g$ and hence different epicyclic frequencies $\kappa_s$ and $\kappa_g$. Therefore, our dispersion (13) appears strikingly similar to that of Jog & Solomon (1984a) yet with different $\kappa_s$ and $\kappa_g$. In short, if one prescribes two different disc speeds and thus two different epicyclic frequencies in the analysis of Jog & Solomon (1984a), the resulting dispersion relation should be our equation (13).
In our analysis, the rotation speeds of the stellar and the gaseous SIDs differ in general. In particular, since $a_s^2$ is usually higher than $a_g^2$, it follows from equations (1) and (8) that $\Omega_s<\Omega_g$. That is, the stellar disc rotates somewhat slower than the gaseous disc does. Such phenomena have been observed in the solar neighborhood as well as in external galaxies (K. C. Freeman, 2003, private communications). This is related to the so-called [*asymmetric drift*]{} phenomena (e.g., Mihalas & Binney 1981) and can be understood in terms of the Jeans equations (Jeans 1919) originally derived by Maxwell from the collisionless Boltzmann equation. In essence, random stellar velocity dispersions produce a pressure-like effect such that mean circular motions become slower (Binney & Tremaine 1987). In our treatment, this effect of stellar velocity dispersions is modelled in the polytropic approximation (see eqns (2) and (8)). Biermann & Davis (1960) discussed the possibility that the mean rotation speeds of the stars and gas in the Galaxy may be different and reached the opposite conclusion that a gas disc should rotate slower than a stellar disc does. However, their analysis based on a virial theorem that excludes the effect of stellar velocity dispersions. In other words, the stress tensor due to stellar velocity dispersions in the Jeans equation has been ignored in their considerations.
Axisymmetric stability analysis
===============================
We study the axisymmetric stability problem based on the WKBJ dispersion relation. Similar to previous analysis (Toomre 1964; Elmegreen 1995; Jog 1996; Lou & Fan 1998b), we also derive an effective $Q$ parameter. This $Q_{\hbox{eff}}$ is expected to depend on $D_s^2$ once $\delta$ and $\beta$ are known. For the convenience of analysis, we define notations $$\label{1}
\begin{split}
H_1\equiv\kappa_s^2+k^2a_s^2&-2\pi G|k|\Sigma_0^s\ ,\\
H_2\equiv\kappa_g^2+k^2a_g^2&-2\pi G|k|\Sigma_0^g\ ,\\
G_1\equiv 2\pi &G|k|\Sigma_0^s\ ,\\
G_2\equiv 2\pi &G|k|\Sigma_0^g\ ,\\
\end{split}$$ to express dispersion relation (\[dis\]) as an explicit quadratic equation in terms of $\omega^2$, namely $$\label{2}
\omega^4-(H_1+H_2)\omega^2+(H_1H_2-G_1G_2)=0\ .$$ The two roots[^3] $\omega_{+}^2$ and $\omega_{-}^2$ of equation (\[2\]) are $$\begin{split}
&\omega_{\pm}^2(k)=
\frac{1}{2}\{(H_1+H_2)\\
&\qquad\qquad\pm[(H_1+H_2)^2-4(H_1H_2-G_1G_2)]^{1/2}\}\ .
\end{split}$$ The $\omega_{+}^2$ root is always positive, which can be readily proven.[^4] We naturally focus on the $\omega_{-}^2$ root, namely $$\label{3}
\begin{split}
&\omega_{-}^2(k)=\frac{1}{2}\{(H_1+H_2)\\
&\qquad\qquad -[(H_1+H_2)^2-4(H_1H_2-G_1G_2)]^{1/2}\}\ ,
\end{split}$$ to search for stable conditions that make the minimum of $\omega_{-}^2(k)\ge 0$ for all $k$.
By expanding the $\omega_{-}^2(k)$ root in terms of definition (\[1\]), we obtain an expression involving parameters $\kappa_s^2$, $\kappa_g^2$, $a_s^2$, $a_g^2$, $\Sigma_0^s$, $\Sigma_0^g$, and $k$. For a composite SID system with power-law surface mass densities, flat rotation curves, and the equilibrium properties of $\kappa_s$, $\kappa_g$, $\Sigma_0^s$ and $\Sigma_0^g$, equation (\[3\]) can also be expressed in terms of four parameters $a_s^2$, $D_s^2$, $\delta$ and $\beta$, namely $$\label{4}
\begin{split}
\omega_{-}^2(K)=\frac{a_s^2}{2r^2}[A_2K^2+A_1K+A_0-\wp^{1/2}]\ ,
\end{split}$$ where $K\equiv |k|r$ is the dimensionless local radial wavenumber and the relevant coefficients are defined by $$\begin{split}
A_2&\equiv 1+1/\beta\ ,\\
A_1&\equiv -(1+D_s^2)\ ,\\
A_0&\equiv 4D_s^2+2(1-1/\beta)\ ,
\end{split}$$ $$\begin{split}
\wp\equiv B_4K^4+B_3K^3+B_2K^2+B_1K+B_0\ ,
\end{split}$$ with $$\begin{split}
B_4&\equiv (1-1/\beta)^2\ ,\\
B_3&\equiv 2(1+D_s^2)(1-1/\beta)(\delta-1)/(1+\delta)\ ,\\
B_2&\equiv (D_s^2-1+2/\beta)(D_s^2+3-2/\beta)\ ,\\
B_1&\equiv 4(1+D_s^2)(1-1/\beta)(1-\delta)/(1+\delta)\ ,\\
B_0&\equiv 4(1-1/\beta)^2\ .
\end{split}$$ As $a_s^2$ has been taken out as a common factor of equation (\[4\]), the most relevant part is simply $$\label{5}
A_2K^2+A_1K+A_0-\wp^{1/2}$$ that involves the stellar rotation parameter $D_s^2$ and the two ratios $\delta$ and $\beta$. We need to determine the value of $K_{\hbox{min}}$ at which $\omega_{-}^2(K)$ takes the minimum value, in terms of $D_s^2$. We then substitute this $K_{\hbox{min}}$ into expression (\[5\]) and derive the condition for $D_s^2$ that makes this minimal $\omega_{\hbox{min}}^2\geq 0$.
The $D_s-$criterion in the WKBJ regime
--------------------------------------
To demonstrate the stability properties unambiguously, and to confirm our previous interpretations for the marginal stability curves in a composite SID system (Lou & Shen 2003), we first present some numerical results that contain the same physics and are simple enough to be understood.
According to equation (\[4\]), $\omega_{-}^2$ is a function of rotation parameter $D_s^2$ and wavenumber $K$. We then use $K$ as the horizontal axis and $D_s^2$ as the vertical axis to plot contours of $\omega_{-}^2$ numerically. A specific example of $\delta=0.2$ and $\beta=10$ is shown in Fig. 1. The shaded regions are where $\omega_{-}^2$ takes on negative values and the blank region is where $\omega_{-}^2$ takes on positive values. In this manner, we clearly obtain marginal stability curves (solid lines) along which $\omega_{-}^2=0$. Physically, the shaded regions of negative $\omega_{-}^2$ values are unstable, while the blank region of positive $\omega_{-}^2$ values is stable. From such contour plots of $\omega_{-}^2$, one can readily determine the specific range of stellar rotation parameter $D_s^2$ such that the composite SID system is stable against axisymmetric disturbances. We therefore confirm our previous interpretations for the marginal stability curves for a composite SID system (Lou & Shen 2003). For a comparison, Fig. 2 shows the corresponding marginal stability curves with $\delta=0.2$ and $\beta=10$ that we derived previously (Lou & Shen 2003) as the stationary axisymmetric perturbation configuration where $\alpha$ is a dimensionless effective radial wavenumber (see also Shu et al. 2000). The apparent difference in Figs. 1 and 2, mainly in the collapse regions, arises because the results of Fig. 2 are exact perturbation solutions without using the WKBJ approximation.
Now in the WKBJ approximation, the stable range of $D_s^2$ can be read off from Fig. 1. For the case of $\delta=0.2$ and $\beta=10$, the composite SID system is stable from $D_s^2=0.1193$ at $K=0.5121$ to $D_s^2=4.1500$ at $K=3.9237$. In comparison with the single SID case (see Fig. 2 of Shu et al. 2000), this stable range of $D_s^2$ diminishes. With other sets of $\delta$ and $\beta$ parameters, we can derive similar diagrams of Fig. 2 as have been done in our recent work, for examples, Figs. $7-10$ of Lou & Shen (2003). In the WKBJ approximation, we show variations with $\delta$ and $\beta$ in Figs. 3 and 4. When $\delta$ is larger, the system tends more likely to be unstable for small $\beta$. In reference to Fig. 10 of Lou & Shen (2003) with $\delta=1$ and $\beta=30$, we plot contours of $\omega_{-}^2$ in Fig. 5 in the WKBJ approximation. In this case, the stable range of $D_s^2$ disappears, that is, the composite SID system cannot be stable against axisymmetric disturbances. We note again that the collapse region with small wavenumber deviates from the exact result of Fig. 10 of Lou & Shen (2003), due to the obvious limitation of the WKBJ approximation. For the unstable region of large wavenumber, which we labelled as ring fragmentation previously (Lou & Shen 2003), Fig. 5 here and Fig. 10 of Lou & Shen (2003) show good correspondence as expected. For the purpose of comparison, Table 1 lists several sets of $\delta$ and $\beta$ and the corresponding stable range of $D_s^2$, selected from our stationary perturbation configurations studied earlier (see Figs. 5-10 of Lou & Shen 2003), using both the WKBJ approximation here and the marginal stability curves of Lou & Shen (2003).
It is not surprising that there exist one lower limit and one upper limit to bracket the stable range of $D_s^2$. We now provide interpretations for the single SID case. In the usual WKBJ analyses, the surface density $\Sigma$ is prescribed and hence independent of the epicyclic frequency $\kappa$. In our global SID formalism, the surface mass density and the epicyclic frequency are coupled through the rotation parameter $D$. Both $\Sigma$ and $\kappa$ decrease with decreasing $D$ but with different rates. Specifically, the surface mass density $\Sigma$ is proportional to $1+D^2$ (see eq. (7)) while $\kappa$ is proportional to $D$ (see eq. (2)). For either large or small enough $D$, the ratio of $\kappa$ to $\Sigma$ may become less than the critical value (sound speed is constant by the isothermal assumption), that is, the Q parameter becomes less than unity for instabilities.
We stress that ring fragmentation instabilities occurring at large wavenumbers are familiar in the usual local WKBJ analysis (Safronov 1960; Toomre 1964). While collapse instabilities at small wavenumbers are known recently for the SID system (Shu et al. 2000; Lou 2002; Lou & Shen 2003). The physical nature of the new “collapse” regime is fundamentally due to the Jeans instability when the radial perturbation scale becomes sufficiently large.[^5] By the conservation of angular momentum, SID rotation plays a stabilizing role to modify the onset of Jeans collapse (e.g., Chandrasekhar 1961). Hence, there appears a critical $D_s^2$ below which exists the collapse regime. Of course, too large a $D_s^2$ causes the other type of ring fragmentation instabilities to set in. For most disc galaxies currently observed, the rotation parameter $D_s^2$ is typically large enough to avoid such collapses.
Elsewhere, the collapse regime studied here may have relevant astrophysical applications, e.g., in circumnuclear discs around nuclei of galaxies and protostellar discs around protostellar cores etc., where $D_s^2$ may be small enough (either a high velocity dispersion $a_s$ or a low rotation speed $V_s$) to initiate collapses during certain phases of disc system evolution. In fact, for a low rotation speed $V_s$, we note from the perspective of evolution, that those proto disc galaxies of low rotation speeds in earlier epochs, will Jeans collapse and lead to other morphologies, while those proto disc galaxies of sufficiently fast rotation speeds in earlier epochs will gradually evolve into disc galaxies we observe today. Meanwhile, ring fragmentation instabilities are thought to be relevant to global star formation in a disc galaxy – another important aspect of galactic evolution.
The addition of a gaseous SID to a stellar SID will decrease the overall stability against axisymmetric disturbances at any wavelengths in general, that is, the stable range of $D_s^2$ shrinks. However, for the two types of instabilities, namely, the collapse instability at small wavenumbers and the ring-fragmentation instability at relatively large wavenumbers, the gravitational coupling of the two SIDs play different roles. For instance, a composite SID system tends more likely to fall into ring fragmentations but less likely to collapse. For either larger $\delta$ or larger $\beta$, this tendency becomes more apparent.
This trend can be understood physically. For the ring-fragmentation regime, suppose the stellar SID by itself alone is initially stable and a gaseous SID component is added to the stellar SID system in a dynamically consistent manner. The gaseous SID may be either stable or unstable by itself alone. By itself, the gaseous SID tends to be more unstable if the sound speed $a_g$ becomes smaller or $\Sigma_0^g$ becomes larger according to the definition of the $Q$ parameter (valid for the ring fragmentation instability), which means larger $\beta$ or larger $\delta$, respectively. Thus, with either $\beta$ or $\delta$ or both being larger, a composite SID system tends to be more vulnerable to ring fragmentations. The interpretation for the collapse regime is associated with the dynamical coupling of surface mass density $\Sigma$ and epicyclic frequency $\kappa$ that leads to the following conclusion, namely, a lower velocity dispersion of the gas component seems to prevent collapse. This should be understood from condition (8): for larger $\beta$, the rotation speed $V_g$ will exceeds $V_s$ by a larger margin and helps to prevent an overall SID collapse.
As already noted, the WKBJ approximation becomes worse in the quantitative sense when dealing with small wavenumbers where the collapse regime exists. But the stability properties of a SID can be qualitatively understood through the WKBJ analysis. The exact perturbation analysis of our recent work (Lou & Shen 2003) has shown that the collapse regime diminishes with increasing values of $\delta$ and $\beta$. Therefore, once the interpretations are justified physically, the exact procedure of Lou & Shen (2003) should be adopted to identify the relevant stable range of $D_s^2$ of a composite SID. As much as Shu et al. (2000) have done for a single SID analytically, Lou & Shen (2003) did the same for a composite SID system with exact solutions to the Poisson integral. The stationary axisymmetric perturbation configurations were derived to determine the marginal stability curves. These marginal stability curves have been proven to have the same physical interpretation as the WKBJ marginal curves obtained here.
---------- --------- ---------------- ----------------
$\delta$ $\beta$ lower limit higher limit
of $D_s^2$ of $D_s^2$
- 1 0.1716(0.9261) 5.8284(5.6259)
0.2 1.5 0.1350(0.8406) 5.3974(5.2117)
5 0.1207(0.7705) 4.4782(4.3558)
10 0.1193(0.7602) 4.1500(4.0611)
30 0.1185(0.7539) 3.8062(3.7670)
1 1.5 0.0596(0.6556) 4.5945(4.4352)
5 0.0411(0.4478) 2.1275(2.0853)
10 0.0401(0.4221) 1.0947(1.0834)
30 -(-) -(-)
5 3 0.0039(0.1418) 1.7397(1.6718)
10 5 0.0011(-) 0.5696(0.5317)
---------- --------- ---------------- ----------------
: The overall stable ranges of $D_s^2$ against axisymmetric disturbances at any wavelengths for different sets of $\delta$ and $\beta$. The values in parentheses are derived by Lou & Shen (2003) without the WKBJ approximation and are preferred to describe the collapse region. The overall stable range of $D_s^2$ appears to decrease when either $\delta$ or $\beta$ increases. The stable range of $D_s^2$ may disappear for a case like $\delta=1$ and $\beta=30$. When $\beta=1$, the problem is independent of $\delta$, that is, $\delta$ can be arbitrary (Lou & Shen 2003).
Effective $Q$ parameters
------------------------
It is natural to extend the concept of the $Q$ parameter for a single disc to an effective $Q$ parameter for a composite disc system for the purpose of understanding axisymmetric stability properties. In the following, we discuss effective $Q$ parameters for a composite SID system in reference to the works of Elmegreen (1995) and Jog (1996). There are several points worth noting in our formulation. First, $\kappa_s$ and $\kappa_g$ are related to each other but are allowed to be different in general. Secondly, the background surface mass densities are related to SID rotation speeds through the polytropic approximation and the steady background rotational equilibrium. For the convenience of notations, the effective $Q$ parameters introduced by Elmegreen (1995) and by Jog (1996) are denoted by $Q_E$ and $Q_J$, respectively.
### The $Q_E$ parameter of Elmegreen
Starting from equation (18) or (22), we define an effective $Q_E$ parameter in a composite SID system following the procedure of Elmegreen (1995) but with two different epicyclic frequencies $\kappa_s$ and $\kappa_g$. The minimum of $\omega^2_{-}$ is given by $$\label{6}
\begin{split}
\omega^2_{-min}&=\frac{a_s^2}{2r^2}[A_2K_{min}^2+A_1K_{min}+A_0
-\wp^{1/2}]\\
&=\frac{a_s^2}{2r^2}(\wp^{1/2}-A_2K_{min}^2-A_1K_{min})(Q_E^2-1)\
,
\end{split}$$ where $$\label{Qeff}
Q_E^2\equiv\frac{A_0}{\wp^{1/2}-A_2K_{min}^2-A_1K_{min}}\$$ and $\wp$ takes on the value at $K=K_{min}$. As $K_{min}$ depends on $D_s^2$, $\delta$ and $\beta$, so does $Q_E^2$. For specified parameters $\delta$ and $\beta$ of a composite SID system, the parameter $Q_E^2$, giving the criterion of axisymmetric stability (i.e. stable when $Q_E^2>1$), corresponds to the rotation parameter $D_s^2$ of the stellar disc. One purpose of the present analysis is to establish the correspondence between the axisymmetric stability condition and the marginal stability curves of axisymmetric stationary perturbation configuration derived recently by Lou & Shen (2003) for a composite SID system. The pertinent results for a single SID can be found in Shu et al. (2000).
Our task now is to find the solution $K_{min}$. As $\kappa_s\neq\kappa_g$ in general, our computation turns out to be somewhat more complicated or involved than that of Elmegreen (1995) who took $\kappa_s=\kappa_g$. It becomes more difficult to solve this algebraic problem of order higher than the third analytically. However, if one multiplies both sides of equation (17) by the positive root $\omega_{+}^2$, a simple expression appears $${\cal W}\equiv\omega_{+}^2\omega_{-}^2=H_1H_2-G_1G_2\ .$$ Instead of minimizing $\omega_{-}^2$, we solve for $K_{c}$ in terms of $D_s^2$ at which ${\cal W}\equiv\omega_{+}^2\omega_{-}^2$ attains the minimum value. This $K_{c}$ must satisfy the following equation $$\begin{split}
\frac{d{\cal W}}{dK}=\frac{d(H_1H_2-G_1G_2)}{dK}=0\ .
\end{split}$$ By substituting $H_1$, $H_2$, $G_1$ and $G_2$ into equation (26) and regarding $\delta$ and $\beta$ as specified parameters, equation (26) can be reduced to a cubic equation of $K$ involving $D_s^2$, namely, $$\label{K3}
\begin{split}
K^3+aK^2+bK+c=0\ ,
%\frac{1}{4}K^4+\frac{a}{3}K^3+\frac{b}{2}K^2+cK+d=0\ ,
\end{split}$$ where $$a=-3(D_s^2+1)(1+\beta\delta)/[4(1+\delta)]\ ,$$ $$b=D_s^2(\beta+1)+\beta-1\ ,$$ $$c=-(D_s^2+1)[\beta D_s^2(1+\delta)+\beta-1]/[2(1+\delta)]\ .$$ In most cases [^6], equation (27) has only one real solution in the form of $$K_{c}=(x-q/2)^{1/3}+(-x-q/2)^{1/3}-a/3\ ,$$ where $x=(q^2/4+p^3/27)^{1/2}$, $p=b-a^2/3$ and $q=2a^3/27-ab/3+c$. For a single real root, it is required that $q^2/4+p^3/27>0$ . Through numerical computations, we find that this condition is met for most pairs of $\delta$ and $\beta$.
We then regard this $K_c$ as an estimator[^7] for $K_{min}$ to determine $Q_E^2$. As already known in our analysis (Lou & Shen 2003), when the ratio $\beta$ is equal to 1, the properties of a composite SID system have something in common with those of a single SID. Moreover, the stationary axisymmetric configuration becomes the same as a single SID. We expect that the present problem should reduce to a single SID case when $\beta=1$. We consider below the special case of $\beta=1$.
With $\beta=1$ in equation (\[K3\]), $K_c$ can be determined in terms of $D_s^2$ without involving $\delta$ at all. We then use $K_c$ as $K_{min}$ in the definition of $Q_E^2$, namely, condition (\[Qeff\]). One can readily verify that $Q_E^2>1$ for $3-2\sqrt{2}<D_s^2<3+2\sqrt{2}$; $Q_E^2<1$ for $D_s^2>3+2\sqrt{2}$ or $D_s^2<3-2\sqrt{2}$; and $Q_E^2=1$ for $D_s^2=3\pm2\sqrt{2}$. The range of $D_s^2$ that makes $Q_E^2>1$ is $0.1716<D_s^2<5.8284$, which is exactly the same as the $Q$ estimator for the single SID of Shu et al. (2000). Based on estimates of $Q$ parameter alone, Shu et al. (2000) made physical interpretations for their axisymmetric stationary perturbation solution curves for a single SID. According to their interpretation, the maximum $D^2=0.9320$ of the collapse branch and the minimum $D^2=5.410$ of the ring-fragmentation branch (see Fig. 2 of Shu et al. 2000) encompass the required range of $D^2$ for stability against axisymmetric disturbances. As the formulation of Shu et al. (2000) is exact in contrast to the WKBJ approximation, the stable range of $D^2$ slightly deviates from that determined by the effective $Q_E$ parameter. And for the upper bound of $D^2$, with a larger wavenumber (or a smaller wavelength), the two approaches lead to consistent results as expected.
In our recent analysis (Lou & Shen 2003), we obtained curves for stationary axisymmetric perturbations given various combinations of $\delta$ and $\beta$ for a composite SID system. These curves differ from those for a single SID and the stable range of $D_s^2$ varies for different combinations of $\delta$ and $\beta$. It is found that this stable range of $D_s^2$ may be significantly reduced for certain values of $\delta$ and $\beta$, as shown in Fig. $7-9$ of Lou & Shen (2003) and in the analysis of Section 3.1. Moreover, when $\delta$ and $\beta$ are sufficiently large, the stable range of $D_s^2$ may no longer exist, as shown in Fig. 10 of Lou & Shen (2003) and Fig. 5 here. For these cases, a composite SID system becomes vulnerable to axisymmetric instabilities.
For $\delta=0.2$ and $\beta=10$, as shown in Fig. 6 of Lou & Shen (2003) and also Fig. 2 of this paper, we search for the approximate stable range of $D_s^2$ with the WKBJ dispersion relation (\[4\]) for a composite SID system. By using the $Q_E^2$ defined by equation (\[Qeff\]), one can directly plot the curve of $Q_E^2$ versus $D_s^2$ as shown in Fig. 6, from which one can identify the approximate stable range of $D_s^2$ from $\sim 0.1193$ to $\sim 4.1500$ where $Q_E^2=1$. Several other curves with different $\delta$ and $\beta$ are also shown in Fig. 6.
One thing that should be emphasized is that the $Q_E^2$ parameter as defined by equation (\[Qeff\]) must be positive, otherwise it may occasionally lead to incorrect conclusions regarding the sign of $\omega_{-}^2$ as defined by equation (23). More detailed investigations indicate that it is only applicable to cases where a composite SID system can be stabilized in a proper range of $D_s^2$. For special cases that cannot be stabilized at all, the $Q_E^2$ thus defined may not be relevant.
### The $Q_J$ parameter of Jog
The $Q_E$ parameter can be derived by solving cubic equation (\[K3\]) for $K$. It may happen that equation (\[K3\]) gives three real solutions. In order to obtain an analytical form of $K_{min}$, one must identify the absolute minimum of $\omega_{-}^2$. This procedure can be cumbersome. Alternatively, Jog (1996) used a seminumerical way to define another effective $Q$ parameter, referred to as $Q_J$ here, that is different from $Q_E$ defined by equation (\[Qeff\]).
According to solution (17), $\omega_{-}^2$ will be positive or negative if $H_1H_2-G_1G_2>0$ or $<0$. The critical condition of neutral stability is thus $H_1H_2-G_1G_2=0$, which can be cast into the form of $$\frac{2\pi Gk\Sigma_0^s}{\kappa_s^2+k^2a_s^2}+\frac{2\pi
Gk\Sigma_0^g}{\kappa_g^2+k^2a_g^2}=1\ .$$ One can now define a function ${\cal F}$ such that $$\begin{split}
{\cal F}&\equiv\frac{2\pi Gk\Sigma_0^s}{\kappa_s^2+k^2a_s^2}
+\frac{2\pi Gk\Sigma_0^g}{\kappa_g^2+k^2a_g^2}\\
&=\frac{K(D_s^2+1)/(1+\delta)}{2D_s^2+K^2}+
\frac{K\beta(D_s^2+1)\delta/(1+\delta)}{2[\beta(D_s^2+1)-1]+K^2}\
,
\end{split}$$ where expressions (2), (7), (8) and $K\equiv |k|r$ are used. We then search for $K_{min}$ at which a composite SID system becomes hardest to be stabilized, that is, when $\omega_{-}^2$ reaches the minimum. The effective $Q$ parameter, $Q_J$, can thus be defined as $$\label{QJ2}
\begin{split}
\frac{2}{1+Q_J^2}&\equiv
\frac{K_{min}(D_s^2+1)/(1+\delta)}{2D_s^2+K^2_{min}}\\
&\qquad +\frac{K_{min}\beta(D_s^2+1)\delta/(1+\delta)}
{2[\beta(D_s^2+1)-1]+K^2_{min}}\
\end{split}$$ (see eqns. (5) and (6) of Jog 1996). It follows that $Q_J^2<1$ or $>1$ correspond to instability or stability, respectively. Therefore, for a given set of $\delta$, $\beta$ and $D_s^2$, one can numerically determine the value of $K_{min}$. By inserting this $K_{min}$ into equation (\[QJ2\]), one obtains the value of $Q_J^2$ for any given set of $\{\delta,\beta,D_s^2\}$. It is then possible to explore relevant parameter regimes of interest. Jog (1996) introduced three parameters, namely, the two Toomre $Q$ parameters for each disc and the surface mass density ratio of the two discs, to determine $Q_J^2$. Equivalently, we use three dimensionless parameters $\delta$, $\beta$ and $D_s^2$ instead. Figs. 7 and 8 show variations of $Q^2_J$ with $D_s^2$. Similar to $Q_E^2$, the value of $Q_J^2>1$ when $D_s^2$ falls in the same ranges. When $\delta$ is fixed and $\beta$ increases, the left and right bounds of the stable $D_s^2$ range move towards left together, while the width of the stable range appears to decrease. We have revealed the same trend of variations in our recent work (Lou & Shen 2003). It is possible for the left bound to disappear when $\beta$ is sufficiently large, and it is easier to achieve this when $\delta$ becomes larger. For a sufficiently large $\delta$, the increase of $\beta$ may completely suppress the stable range of $D_s^2$ as shown in Fig. 5 for $\delta=1$ and $\beta=30$.
In comparison to $Q_E$, the $Q_J$ definition (\[QJ2\]) is valid for all parameter regimes and avoids improper situations for unusual $\delta$ and $\beta$ values. However, to find the value of $Q_J^2$ one must perform numerical exploration for each given $D_s^2$, while for $Q_E^2$, the procedure is analytical as long as there exists a stable range of $D_s^2$. Regardless, the critical values of $D_s^2$ found by $Q_E^2$ and $Q_J^2$ are equivalent.
Composite partial SID system
----------------------------
In disc galaxies, there are overwhelming evidence for the existence of massive dark matter halos as inferred from more or less flat rotation curves. To mimic the gravitational effect of a dark matter halo, we include gravity terms $\partial\Phi/\partial r$ and $\partial\Phi/\partial\varphi$ in the radial and azimuthal momentum equations (4) and (5), respectively, where $\Phi$ is an axisymmetric gravitational potential due to the dark matter halo. It is convenient to introduce a dimensionless parameter $F\equiv\phi/(\phi+\Phi)$ for the fraction of the disc potential relative to the total potential (e.g., Syer & Tremaine 1996; Shu et al. 2000; Lou 2002). The background rotational equilibrium is thus modified by this additional $\Phi$ accordingly. As before, we write $\Omega_s=a_sD_s/r$, $\Omega_g=a_gD_g/r$, and $\kappa_s=\sqrt{2}\Omega_s$, $\kappa_g=\sqrt{2}\Omega_g$. The equilibrium surface mass densities now become $$\Sigma_0^s=\frac{a_s^2(1+D_s^2)F}{2\pi Gr(1+\delta)},
\qquad\qquad
\Sigma_0^g=\frac{a_g^2(1+D_g^2)F\delta}{2\pi Gr(1+\delta)}\ ,$$ where $0\leq F\leq 1$. In our perturbation analysis, the dynamic response of this axisymmetric massive dark matter halo to coplanar perturbations in a composite SID system is ignored[^8] (Syer & Tremaine 1996; Shu et al. 2000; Lou 2002). We thus derive similar perturbation equations as those of a full SID case ($F=1$) and consequently the similar dispersion relation as equation (13) yet with modified equilibrium properties.
Now following the same procedure of WKBJ perturbation analysis described in Section 3.1, we can plot contours of $\omega_{-}^2$ as a function of rotation parameter $D_s^2$ and wavenumber $K$ for a composite partial SID system. The case of $F=1$ corresponds to a composite full SID system that has been studied in Section 3. When $F$ becomes less than 1 (i.e., a composite partial SID system), which means the fraction of the dark matter halo increases, the stable range of $D_s^2$ becomes enlarged, as shown in Figs. 9 and 10 for $\delta=0.2$ and $\beta=10$. From these contour plots, it is clear that the introduction of a dark matter halo tends to stabilize a composite partial SID system. For late-type disc galaxies, one may take $F=0.1$ or smaller. Such composite partial SID systems are stable against axisymmetric disturbances in a wide range of $D_s^2$. Moreover, those composite full SID systems that are unstable may be stabilized by the presence of a dark matter halo, as shown by the example of Fig. 11 for $\delta=1$, $\beta=30$ and $F=0.5$ (this case is unstable for a composite full SID system).
Discussions and summary
=======================
Our model of a composite (full or partial) SID system is an idealization of any actual disc systems, such as disc galaxies, circumnuclear discs surrounding nuclei of galaxies, or protostellar discs. One key feature of the SID model is the flat rotation curve. One major deficiency in galactic application is the central singularity that sometimes even gives rise to conflicting theoretical interpretations (see Shu et al. 2000). The isothermality is yet another simplifying approximation. In galactic applications, the common wisdom is to introduce a bulge around the center to avoid the central singularity. By a local WKBJ stability analysis of axisymmetric perturbations, this model can be utilized as usual with qualifications. We now discuss the WKBJ results in the context of our own Galaxy, the Milky Way, for which the necessary observational data are available, such as the gas fraction, the stellar and the gaseous velocity dispersions and the epicyclic frequency, and so forth.
There are several inadequacies of our model. First, the isothermal assumption implying constant $a_s$ and $a_g$ throughout the disc system is a gross simplification; the velocity dispersion $a_s$ in the Milky Way decreases with increasing radius (e.g., Lewis & Freeman 1989). Secondly, by the polytropic approximation, the surface mass densities are characterized by power-law $\propto
r^{-1}$ profiles, which appear not to be the case of the Milky Way (e.g., Caldwell & Ostriker 1981). Nevertheless, to estimate local stability properties of a disc portion, we may take $a_s=50\hbox{
km s}^{-1}$, $V_s=220\hbox{ km s}^{-1}$, $\delta=0.1$ and flat rotation speed $V_s=220\hbox{ km s}^{-1}$ around $4\hbox{ kpc}$ from the center of our Galaxy. For a massive dark matter halo such that $F=0.1$, the relevant parameters to determine the effective $Q$ parameter for a composite partial SID system are $\delta=0.1$, $\beta=50$, $D_s^2=20$ and $F=0.1$. Following the definition of Jog (1996), the value of $Q_J\gg1$. While for full SIDs with $F=1$, we get $Q_J=0.17$ which means local instability. Obviously, the dark matter halo plays the key role in stabilizing the system.
The composite partial SID model with power-law $\propto r^{-1}$ surface mass densities may not describe the Milky Way well. But for those disc galaxies with flat rotation curves and approximate power-law $\propto r^{-1}$ surface mass densities, our model should offer a sensible local criterion for axisymmetric instability. For example, given radial variations of velocity dispersions and gas mass fraction, we may deduce locally unstable zones.
For circumnuclear discs around nuclei of galaxies, the stellar velocity dispersion $a_s$ can be as high as several hundred kilometers per second along the line of sight (e.g., Whitmore et al. 1985; McElroy 1994); this may then lead to small $D_s^2$. Within such a parameter regime, the collapse instability may set in during a certain phase of system evolution. For instance, in the case of Fig. 2 with $\delta=0.2$ and $\beta=10$, we choose $V_s=200\hbox{ km s}^{-1}$ and $a_s=300\hbox{ km s}^{-1}$ in a circumnuclear disc region of a disc galaxy. The resulting $D_s^2=0.44$ would give rise to collapse and the composite SID system might eventually evolve into a bulge of high velocity dispersion.
Another possible collapse situation of interest corresponds to small $D_s^2$ for sufficiently low disc rotation speed $V$. Potential applications include but not limited to protostellar discs in the context of star formation. We here briefly discuss several qualitative aspects. For a protostellar disc system, the two disc components here may be identified with the relatively “hot" gas disc and the relatively “cool" dust disc. Interactions of radiation fields from the central protostar with cool dusts in the disc lead to infrared emissions. In our idealized treatment, the gas and dust discs are treated as two gravitationally coupled thin SIDs. This simple model treatment may need to be complemented by other processes of coupling such as collisions between gas and dust particles. When such a composite disc system rotates at a low speed (e.g., Shu et al. 1987), the rotation parameter $D$ could be low enough to initiate large-scale collapse. The subsequent dynamical processes will drive the composite SID system to more violent star formation activities.
So far we have investigated the axisymmetric stability problem for a composite system of gravitationally coupled stellar and gaseous SIDs using the WKBJ approximation and we now summarize the results of our analysis.
First, because the single SID case studied by Shu et al. (2000) may be regarded as the special case of a composite SID system, the results of our WKBJ analysis clearly support the physical interpretations of Shu et al. (2000) for the marginal stability curves of axisymmetric perturbations. The “ring fragmentation regime" is related to the familiar $Q$ parameter (Safronov 1960; Toomre 1964). The presence of the Jeans “collapse regime" can be traced to the SID model where self-gravity, effective pressure, and SID rotation compete with each other.
Secondly, the recent study of Lou & Shen (2003) generalizes that of Shu et al. (2000) and describes exact perturbation solutions in a composite SID system. We obtained marginal stability curves for axisymmetric perturbations that are qualitatively similar to those of Shu et al. (2000) and that depend on additional dimensionless parameters. It is natural to extend the interpretations of Shu et al. (2000). Through the WKBJ analysis here, we now firmly establish the presence of two regimes of “ring fragmentation" and “collapse" in a composite SID system. It is fairly straightforward to apply our exact $D-$criterion for a composite SID system, not only for disc galaxies, but also for other disc systems including circumnuclear discs and protostellar discs etc.
Thirdly, in the WKBJ approximation, it is also possible to relate our $D-$criterion to the $Q_E$ criterion similar to that of Elmegreen (1995). Because $\kappa_s$ and $\kappa_g$ are different in general, our WKBJ treatment is not the same as that of Elmegreen (1995).
Fourthly, also in the WKBJ approximation, we relate our $D-$criterion to the $Q_J$ criterion as defined by Jog (1996) but with $\kappa_s\neq\kappa_g$ in general. We find that the $Q_J$ criterion is fairly robust in the WKBJ regime.
Finally, we further consider the axisymmetric stability of a composite partial SID system to include the gravitational effect from an axisymmetric dark matter halo. The stabilizing effect of the dark matter halo is apparent.
Acknowledgments {#acknowledgments .unnumbered}
===============
We thank the referee C. J. Jog for comments and suggestions. This research has been supported in part by the ASCI Center for Astrophysical Thermonuclear Flashes at the University of Chicago under Department of Energy contract B341495, by the Special Funds for Major State Basic Science Research Projects of China, by the Tsinghua Center for Astrophysics, by the Collaborative Research Fund from the NSF of China (NSFC) for Young Outstanding Overseas Chinese Scholars (NSFC 10028306) at the National Astronomical Observatory, Chinese Academy of Sciences, and by the Yangtze Endowment from the Ministry of Education through the Tsinghua University. Affiliated institutions of Y.Q.L. share the contribution.
[9]{}
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[^1]: Subscript $_s$ indicates the relevance to the stellar SID.
[^2]: This implies a close relation among the two SID rotation speeds, the stellar velocity dispersion and the gas sound speed. In this aspect, it is different from the usual local prescription of a composite disc system (e.g., Jog & Solomon 1984a, b; Bertin & Romeo 1988; Elmegreen 1995; Jog 1996; Lou & Fan 1998b). Once the ratio of $a_s^2$ to $a_g^2$ is known, the rotation speeds of the composite SID system can be determined through that of the stellar disc, i.e. $D_s^2$, as the two rotation speeds are coupled dynamically. In the limiting regime of $a\ll V$ and $\beta\gg 1$, $V_s$ and $V_g$ are comparable with $V_g$ slightly larger by an amount of $\sim (a_s^2-a_g^2)/(2V_s)$. In this sense, the assumption $\kappa_s\approx\kappa_g$ is reasonable.
[^3]: The determinant $\Delta\equiv (H_1-H_2)^2+4G_1G_2\geq0$.
[^4]: This conclusion holds true if $H_1+H_2\geq0$. If $H_1+H_2<0$, then $H_1H_2-G_1G_2<0$ and the conclusion still holds.
[^5]: One might wonder how a large-scale disc Jeans instability can be revealed in a local WKBJ analysis although somewhat crudely. Perhaps, the best analogy is the WKBJ dispersion relation for spiral density waves in a thin rotating disc (Lin & Shu 1966, 1968). This WKBJ dispersion relation is quadratic in the radial wavenumber $|k|$ and can be solved for a larger $|k|$ and a smaller $|k|$ corresponding to short- and long-wave branches respectively (e.g., Binney & Tremaine 1987). The former is naturally consistent with the WKBJ approximation, while the latter, inconsistent with the WKBJ approximation, turns out to be a necessary key ingredient in the swing process (Goldreich & Lynden-Bell 1965; Toomre 1981; Fan & Lou 1997) as revealed by analytical and numerical analyses. In the end, one needs to perform global analyses to justify the WKBJ hint for instabilities of relatively large scales.
[^6]: When $\kappa_s=\kappa_g$, Bertin & Romeo (1988) give the proof that as long as $1/\beta>0.0294$ and $\delta>0.172$, there exists only one minimum of $\omega_{-}^2$. When there are more than one local minima for $\omega_{-}^2$, one should choose $K_c$ for the smallest minimum of $\omega_{-}^2$.
[^7]: While $K_c$ is obtained as the value at which $\omega_{-}^2$ multiplying $\omega_{+}^2$ reaches the minimum. It is valid to use $K_{c}$ to determine whether $Q_E^2>1$ or $Q_E^2<1$ by realizing that $\omega_{+}^2$ remains positive and the critical condition $Q_E^2=1$ is equivalent to $\omega_{+}^2\omega_{-}^2=0$.
[^8]: Strictly speaking, gravitational perturbation coupling between a thin, less massive, rotating disc and a more massive dark matter halo together with proper matching and boundary conditions should lead to an increasing number of modes that may or may not be stable. We simplify by noting numerical simulations have indicated that a typical galactic dark matter halo has a relatively high velocity dispersion of the order of several hundred kilometers per second. The rationale of the simplification is that perturbation coupling between those in the SID system and those in the dark matter halo becomes weak for a high velocity dispersion in the dark matter halo. For example, for a non-rotating spherical dark matter halo, effective “acoustic" and “gravity" modes (i.e., $p-$modes and $g-$modes in the parlance of stellar oscillations; Lou 1995) can exist within; these $p-$ and $g-$modes are expected to be Landau damped in a collisionless system. However, the stability of “interface modes" between a thin rotating disc and a dark matter halo remains to be investigated.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: '[ We investigate the hydrodynamic recovery of Lattice Boltzmann Method (LBM) by analyzing exact balance relations for energy and enstrophy derived from averaging the equations of motion on sub-volumes of different sizes.]{} In the context of 2D isotropic homogeneous turbulence, we first validate this approach on decaying turbulence by comparing the hydrodynamic recovery of an ensemble of LBM simulations against the one of an ensemble of Pseudo-Spectral (PS) simulations. We then conduct a benchmark of LBM simulations of forced turbulence with increasing Reynolds number by varying the input relaxation times of LBM. This approach can be extended to the study of implicit subgrid-scale (SGS) models, thus offering a promising route to quantify the implicit SGS models implied by existing stabilization techniques within the LBM framework.'
address:
- 'Dipartimento di Fisica and INFN, Università di Roma “Tor Vergata", Via della Ricerca Scientifica 1, 00133 Roma, Italy'
- 'Department of Applied Physics and Department of Mathematics and Computer Science, Eindhoven University of Technology, 5612 AZ Eindhoven, Netherlands'
- 'Chair of Applied Mathematics and Numerical Analysis, Bergische Universität Wuppertal, Gaußstrasse 20, 42119 Wuppertal, Germany'
author:
- Guillaume Tauzin
- Luca Biferale
- Mauro Sbragaglia
- Abhineet Gupta
- Federico Toschi
- Andreas Bartel
- Matthias Ehrhardt
bibliography:
- 'references.bib'
title: 'A numerical tool for the study of the hydrodynamic recovery of the Lattice Boltzmann Method [^1] '
---
Lattice Boltzmann Method ,Hydrodynamics ,Turbulence modeling
Introduction
============
The simulation of turbulent flows pertains to a vast diversity of applications in engineering [@Galperin2010]. The high Reynolds number associated with the phenomenon of turbulence requires solving a wide range of scales on a high resolution computational grid, making their Direct Numerical Simulation (DNS) typically out of reach [@Pope2000; @Davidson2015]. Large-Eddy Simulation (LES) is a workaround which allows a reduction of the number of degrees of freedom. LES is acknowledged in the engineering community as a cost-effective alternative to DNS [@Pitsch2006; @Wagner2007; @Sullivan1994]. The principle of LES is to solve flow scales up to a cut-off and to filter the small scales out. As large scales and smaller scales are coupled, unresolved small scales need to be modeled using a so-called subgrid-scale (SGS) model. A large number of filtering techniques and SGS models have been proposed in the Navier-Stokes framework [@Sagaut2002].\
The Lattice Boltzmann Method (LBM) is a meso-scale flow solver that has been gaining popularity because of its intrinsic scalability, as well as its ability to deal with multiple physics and complex boundary conditions [@Succi2001; @Wolf-Gladrow2000; @Kruger2017]. The LBM equation describes the streaming and collision of distribution functions $f_{\ell}(\vec{x},t)$ on a lattice with a finite set of kinetic velocities $\vec{c}_{\ell},\,\, {\ell}=0 \ldots q-1$. The collision operator is popularly modeled by the Bhatnagar-Gross-Krook (BGK) [@Bhatnagar1954] relaxation towards a local equilibrium with a dimensionless relaxation time $\tau$ $$\label{eq:LBM}
f_{\ell}(\vec{x}+\vec{c}_{\ell} \Delta t,t + \Delta t) - f_{\ell}(\vec{x},t)= -\frac{1}{\tau}\left[ f_{\ell}(\vec{x},t)-f^{eq}_{\ell}(\vec{x},t)\right]+F_{\ell}$$ where $F_{\ell}$ is a suitable forcing term designed to reproduce a macroscopic forcing [@Succi2001; @Wolf-Gladrow2000; @Kruger2017]. From a theoretical point of view, the use of a multi-scale Chapman-Enskog (CE) perturbative expansion allows to recover hydrodynamic equations. In brief, one expands the distribution function in a power-series: $f_{\ell}=f^{(eq)}_{\ell}+K_n f^{(1)}_{\ell}+K^2_n f^{(2)}_{\ell}+...$, where $K_n=\lambda/L \ll 1$ is the Knudsen number, giving the ratio between the particles mean free path $\lambda$ and the macroscopic scale $L$. Furthermore, space and time are rescaled, *i.e.* $\vec{x}^{(1)}=K_n \vec{x}$, $t^{(1)}=K_n t$, $t^{(2)}=K_n^2 t$ by introducing separate time scales for the effect of advection ($t^{(1)}$) and dissipation ($t^{(2)}$) [@Succi2001; @Wolf-Gladrow2000]. Performing this procedure for a local equilibrium distribution chosen as (repeated indices are meant summed upon)
$$\label{eq:feq}
f^{eq}_{\ell} ( \vec{x}, \, t)
= f^{eq}_{\ell} \left( \rho (\vec{x}, \, t), \, \vec{u} (\vec{x}, \, t) \right)
= t_{\ell} \, \rho \left[ 1 + \frac{c_{\ell, \, i} u_i}{c^2_s} + \frac{\left( c_{\ell, \, i} u_i \right)^2}{2 c^4_s} - \frac{u_i u_i}{2 c^2_s} \right],$$
where $t_{\ell}$ is a set of lattice-dependent weighting factors and $c_s$ the speed of sound in the lattice, one can recover the athermal weekly compressible Navier-Stokes hydrodynamic equations for the density field $\rho(\vec{x}, \, t)=\sum^{q-1}_{\ell=0} f_{\ell}(\vec{x}, \,t)$ and velocity field $\vec{u}(\vec{x}, \, t)=\sum^{q-1}_{\ell=0} f_i(\vec{x}, \, t) \, \vec{c}_{\ell}/\rho(\vec{x}, \,t)$
$$\label{eq:mass_LBM}
\partial_t \rho + \partial_j (\rho u_j)=0 + \mathcal{O} (K_n^2)$$
$$\label{eq:N-S_LBM}
\partial_t \left( \rho u_i \right) + \partial_j \left( \rho u_i u_j \right)
= - \partial_i p + \partial_j \left( \rho \nu \left( \partial_j u_i + \partial_i u_j \right) \right) + F_i
+ \mathcal{O} (K_n^2) + \mathcal{O} (Ma^3).$$
Beyond the higher order corrections in the Knudsen number, in the recovery of the momentum equations one usually neglects terms which are cubic in the velocity [@Viggen2009], hence we find the term $\mathcal{O} (Ma^3)$, where the Mach number $Ma = \frac{U_{RMS}}{c_s}$ represents the ratio of the root mean square velocity $U_{RMS}$ to $c_s$. The term $p = c_s^2 \rho$ is the fluid pressure and the viscosity $\nu$ is linearly dependent on the relaxation time $\tau$ in and vanishes as $\tau \rightarrow 0.5$: $$\label{eq:N-S_LBM_nu}
\nu=c_s^2 \left( \tau - \frac{1}{2} \right) \Delta t.$$ The LBM community has been keenly proposing Navier-Stokes inspired LES techniques to combine the intrinsic scalability of LBM with turbulence SGS models. The majority of them are eddy viscosities models implemented by locally modifying the relaxation time $\tau$, i.e. assuming that Eq. holds and that an effective relaxation time $\tau_{\text{eff}} (\vec{x}, \, t)$ results in an effective viscosity $\nu_{\text{eff}} (\vec{x}, \, t)$ [@Filippova2001a; @Dong2008a; @Dong2008b; @Chen2009]. Malaspinas & Sagaut have shown that this method is only valid in the athermal weakly compressible limit and proposed a consistent eddy viscosity closure extension for compressible thermal flows [@Malaspinas2012]. Instabilities of the LBM with a BGK collision operator (LBGK) arising for an input relaxation time $\tau_0 \rightarrow 0.5$, *i.e.* for an input viscosity $\nu_0 \rightarrow 0$, along with the low $Ma$, which is required to remain in a good approximation of Navier-Stokes, significantly limit the range of Reynolds number reachable in practice. Some eddy viscosity methods have been shown to extend the range of stability to relaxation times $\tau_0 \rightarrow 0.5$, making it possible to simulate higher Reynolds number flows for a fixed grid resolution [@Premnath2009]. Stabilization of LBGK has been linked to the existence of an underlying Lyapunov functional in the form of a discrete Boltzmann H-functional [@Boghosian2001]. Karlin *et al.* [@Karlin1999] introduced the Entropic Lattice Boltzmann (ELBM): an LBGK ensuring the monotonicity of a convex H-functional commonly chosen as $$H \left( \mathbf{f} \right) = \sum^{q-1}_{\ell=0} f_{\ell} \log \left( \frac{f_{\ell}}{t_{\ell}} \right), \,\, {\bf f} = \left\{ f_{\ell} \right\}^{q-1}_{\ell=0}.$$ To equip a LBGK with an H-theorem, ELBM implements a collisional process with an effective relaxation time $\tau_{\text{eff}} = \frac{2 \tau_0}{\alpha}$ to a local equilibrium distribution $\mathbf{f^{eq}}$ defined as the extremum of the H-functional under the constraints of mass and momentum conservation. The parameter $\alpha$ is calculated locally (in space and time) and has a non-linear dependency on the distribution functions $f_{\ell}$. While the result is an unconditionally stable LBGK for $\tau_0 \rightarrow 0.5$ ($\nu_0 \rightarrow 0$), we are also left with a side-effect effective viscosity $\nu_{\text{eff}}$. Unfortunately, the non-linear dependency of the effective relaxation time on the distribution functions does not allow this effective viscosity to be expressed in terms of macroscopic quantities and therefore the physics behind it remains hidden. In 2008, Malaspinas *et al.* [@Malaspinas2008] proposed an approximate formulation of the effective viscosity $\nu_{\text{eff}} (\vec{x}, \, t) = \nu_0 + \nu_t (\vec{x}, \, t)$ using CE expansion assuming $\alpha \approx 2$ ($\tau_{\text{eff}} \approx \tau_0$). The resulting turbulent viscosity $\nu_\text{t}$ is $$\label{eq:viscosity_malaspinas}
\nu_{\text{t}} = - \frac{c^{2}_{s}}{3} \tau_0^2 \Delta t^2
\frac{S_{\theta \kappa} S_{\kappa \gamma} S_{\gamma \theta}}{S_{\lambda \mu} S_{\lambda \mu}}
\propto \frac{Tr(S^3)}{Tr(S^2)}$$ where $S_{ij} = \frac{1}{2} (\partial_i u_j + \partial_j u_i )$ is the strain-rate tensor. The above formula suggests a similarity with the Smagorinsky SGS model [@Smagorinsky1963] $\nu_{\text{t}} = C_{smago} \Delta x^2 \sqrt{ S_{\theta \kappa} S_{\theta \kappa} } \propto \sqrt{ Tr(S^2)}$ while allowing back-scatter as it can change sign.\
In order to quantify the validity of the ELBM methodology as a LES turbulence SGS model, one needs to be able to evaluate and understand the physics it implies. Firstly, one needs to control the hydrodynamic recovery and [ determine to which accuracy the Navier-Stokes equations are recovered as a function of the analyzing sub-volume size [@Biferale2010]]{}. This is an unquestionable prerequisite. Secondly, one needs to further study the subgrid-scale model implied by the ELBM. Based on this philosophy, in this paper we propose a tool to numerically evaluate the Navier-Stokes hydrodynamic recovery of fluid flow simulations in the context of isotropic homogeneous turbulence. This tool is based on the systematic calculation of each term of the kinetic energy and enstrophy balance equations averaged over a suitable ensemble of sub-volumes of the computational grid. [A similar approach to characterize LBM hydrodynamics was successfully used in [@Bosch; @Dorschner2016] by estimating the input viscosity $\nu_0$ from the incompressible energy and enstrophy equations averaged over the whole volume. Here, we define an error with respect to an exact balance of the equation of motion and conduct a statistical analysis over sub-volumes of different sizes to assess the locality of the hydrodynamic recovery.]{} The paper is organized as follows: in section \[sec:2\] we introduce the balance equations, their averaged counterparts over a sub-volume $V$ and we define balancing errors as a measure of the hydrodynamic recovery; in section \[sec:3\] we present the numerical set-up for the simulations of 2D isotropic homogeneous turbulence and for the statistical analysis of the balancing errors; in section \[sec:4\] we present a validation of the tool by comparing the hydrodynamic recovery of an ensemble of LBGK simulations to an ensemble of Pseudo-Spectral (PS) simulations in the case of decaying flows; in section \[sec:5\] we benchmark the tool on LBGK simulations of forced turbulence for a range of [ increasing Reynolds numbers]{}, [ while linking the results to the corresponding statistics of the Mach number;]{} some concluding remarks will follow in section \[sec:6\].
Hydrodynamic recovery for energy and enstrophy balance in 2D {#sec:2}
============================================================
In order to characterize the hydrodynamic recovery of a simulation, we calculate the average over sub-volumes of the terms in both the kinetic energy and the enstrophy balance equations. Starting from the formulation of the macroscopic LBM momentum conservation (see Eq. ) and mass conservation (see Eq. ), one can obtain the kinetic energy ($E = \frac{\rho u_i u_i}{2}$) balance equation and the enstrophy ($\Omega = \frac{\omega_i \omega_i}{2}$, with $\omega_i$ the component of the vorticity $\vec{\omega} = \vec{\nabla} \times \vec{u}$ along $\vec{e}_i$) balance equation $$\begin{aligned}
\label{eq:E_balance}
\begin{split}
\partial_t \left(\frac{\rho u_i u_i}{2}\right)
=& - u_i \partial_i p - \nu \rho \left( \partial_j u_i
+ \partial_i u_j \right) \partial_j u_i + u_i F_i \\
& - \partial_j \left(\frac{\rho u_i u_i}{2} u_j\right)
+ \partial_j \left(\nu \rho u_i \left( \partial_j u_i + \partial_i u_j \right)\right)
\end{split}\end{aligned}$$ $$\begin{aligned}
\label{eq:Z_balance}
\begin{split}
\partial_t \left(\frac{\omega_i \omega_i}{2}\right)
=&- \partial_j \left(\frac{\omega_i \omega_i}{2} u_j\right)
+ \omega_i \omega_j \partial_j u_i
+ H_i (\nu) \epsilon_{i j k} \partial_j \omega_k
+ \omega_i \epsilon_{i j k} \partial_j \left(\frac{1}{\rho} F_k \right)\\
& - \partial_j \left(\frac{\omega_i \omega_i}{2} u_j\right)
+ \partial_j \left(\epsilon_{i j k} \omega_i H_k (\nu)\right)
\end{split}\end{aligned}$$ where $\epsilon$ is the Levi-Civita symbol and $H_i (\nu)= \frac{1}{\rho} \partial_j \nu \rho$ $\left( \partial_i u_j + \partial_j u_i \right)$. Equations and are locally valid. The next step is to calculate the average of each term of the balance equations over [ a sub-volume $V$]{} $$\begin{aligned}
\label{eq:E_balance_average}
\begin{split}
LHS_V^E =& \, \partial_t \big \langle \frac{\rho u_i u_i}{2} \big \rangle_V\\
=& - \big \langle \partial_j \left(\frac{\rho u_i u_i}{2} u_j\right) \big \rangle_V
- \big \langle u_i \partial_i p \big \rangle_V
+ \big \langle u_i F_i \big \rangle_V\\
&- \big \langle \nu \rho \left( \partial_j u_i + \partial_i u_j \right) \partial_j u_i \big \rangle_V
+ \big \langle \partial_j \left(\nu \rho u_i \left( \partial_j u_i + \partial_i u_j \right)\right) \big \rangle_V
\\
=& \, RHS^{E, \, 1}_V + RHS^{E, \, 2}_V + RHS^{E, \, 3}_V+ RHS^{E, \, 4}_V + RHS^{E, \, 5}_V\\
=& \, RHS_V^E
\end{split}\end{aligned}$$ $$\begin{aligned}
\label{eq:Z_balance_average}
\begin{split}
LHS_V^{\Omega} =& \, \partial_t \big \langle \frac{\omega_i \omega_i}{2} \big \rangle_V\\
=& - \big \langle \partial_j \left(\frac{\omega_i \omega_i}{2} u_j\right) \big \rangle_V
- \big \langle \frac{\omega_i \omega_i}{2} \partial_j u_j \big \rangle_V
+ \big \langle \omega_i \epsilon_{i j k} \partial_j \left(\frac{1}{\rho} F_k \right) \big \rangle_V\\
&+ \big \langle H_i (\nu) \epsilon_{i j k} \partial_j \omega_k \big \rangle_V
+ \big \langle \partial_j \left(\epsilon_{i j k} \omega_i H_k(\nu)\right) \big \rangle_V
+ \big \langle \omega_i \omega_j \partial_j u_i \big \rangle_V\\
=& \, RHS^{\Omega, \, 1}_V + RHS^{\Omega, \, 2}_V + RHS^{\Omega, \, 3}_V + RHS^{\Omega, \, 4}_V
+ RHS^{\Omega, \, 5}_V + RHS^{\Omega, \, 6}_V\\
=& \, RHS^{\Omega}_V
\end{split}\end{aligned}$$ where $ \big \langle \cdots \big \rangle_V$ denotes the average over a generic volume $V$. Equations and describe the physical balance between the time derivative of the averaged energy and enstrophy ($LHS^{E, \, \Omega}_V$) and the right-hand side ($RHS^{E, \, \Omega}_V$) comprising all the physical contributions responsible for their evolution: the effect of compressibility, dissipation, input, and the transport and diffusive fluxes. It is worth pointing out that equations and remain valid for a viscosity changing in space and time $\nu = \nu_{\text{eff}} (\vec{x}, \, t)= \nu_0 + \nu_{\text{t}} (\vec{x}, \, t)$. Notice that in 3D, the enstrophy balance must include another additional term stemming from vortex stretching [@Davidson2015].\
To measure the accuracy of the hydrodynamic recovery over a sub-volume $V$, we define a balancing error for the kinetic energy and enstrophy balance, $\delta_V^{E}$ and $\delta_V^{\Omega}$ respectively. At a time $t$, $\delta^{E, \, \Omega}_V(t)$ is obtained by dividing the absolute difference between the $RHS^{E, \, \Omega}_V(t)$ and the $LHS^{E, \, \Omega}_V(t)$ terms by the term of the right-hand side with the maximum absolute value [*i.e.*]{} $$\label{eq:delta_E}
\delta_{V}^{E}(t)=\frac{\left| RHS_{V}^{E}(t)-LHS_{V}^{E}(t) \right|}{\max_i \left| RHS_{V}^{E, \, i}(t) \right|}$$ and $$\label{eq:delta_Z}
\delta_{V}^{\Omega}(t)=\frac{\left| RHS_{V}^{\Omega}(t)-LHS_{V}^{\Omega}(t) \right|}{\max_i \left| RHS_{V}^{\Omega, \, i}(t) \right|}.$$ If for a sub-volume $V$ at a time $t$ the balance equations are perfectly respected on average, we must have $\delta_{V}^{E}(t) \equiv \delta_{V}^{\Omega}(t) \equiv 0$.
Numerical set-up for the statistical analysis of 2D homogeneous isotropic turbulence hydrodynamics {#sec:3}
==================================================================================================
To validate this hydrodynamic recovery check tool, we apply it to configurations obtained from simulations conducted on a periodic two-dimensional $256 \times 256$ computational grid. Turbulence is triggered by a homogeneous isotropic forcing with a constant phase $\phi$ on a shell of (dimensionless) wavenumbers $\vec{k}$ of magnitude from 5 to 7 given in a stream-function formulation $$\label{eq:forcing_turbulence_streamfunction}
F^{T}_{\Psi} (\vec{x})=F^{T}_{0}\sum_{5 \leq \|\vec{k}\| \leq 7}\cos\left(\frac{2\,\pi}{256}\vec{k} \cdot \vec{x}+\phi\right).$$ The corresponding force is then obtained by taking $$\label{eq:forcing_turbulence}
F^{T}_x = \partial_y F^{T}_{\Psi} \qquad \text{and} \qquad F^{T}_y = -\partial_x F^{T}_{\Psi},$$ which ensures that it does not input any incompressibility in the system as $\vec{\nabla} \cdot \vec{F}^{T} \equiv 0$. We use this forcing to define a time scale $T_f = \sqrt{ \frac{2 \pi}{k_f F^T_{0}} }$, where $k_f$ is taken equal to six. To have some control on the Mach number and limit the effect of the backward energy cascade, characteristic of 2D turbulence [@Boffetta2012; @Frisch1995], we introduce a spectral forcing to damp large-scale energy $$\label{eq:forcing_damping}
\vec{F}^{R} \left( \vec{x}, \, t \right) = - F^{R}_{0} \sum_{1 \leq \|\vec{k}\| \leq 2} \vec{\hat{u}} (\vec{k}, \, t) \, e^{\frac{2\,\pi}{256}\vec{k} \cdot \vec{x} }$$ where $\vec{\hat{u}} (\vec{k}, \, t)$ is the Fourier transform of $\vec{u} (\vec{x}, \, t)$. The forcing amplitudes are fixed for all simulations to $F^{T}_{0} = 0.0008$ and $F^{R}_{0}=0.00001$. LBGK simulations are conducted on a 2D lattice with 9 discrete velocities, the D2Q9 [@Succi2001; @Wolf-Gladrow2000; @Kruger2017], on which forcings are implemented using the exact-difference method forcing scheme [@Kuperstokh2004]. The sub-volume averaged terms are calculated offline based on the [ output]{} configuration fields.
A 2^nd^ order explicit Euler scheme is used to evaluate time derivatives, while a 8^th^ order centered scheme is applied for the space-derivatives, respectively $$\left. \frac{\partial \mathbf{A}}{\partial t}\right |^n_{i,j}
\sim
\frac{3 A_{i,j}^{n}-4 A_{i,j}^{n-1}+A_{i,j}^{n-2}}{2 \,\Delta t}
\label{eq:discretisation_t2} \text{, and }$$
$$\begin{aligned}
\begin{split}
\left. \frac{\partial \mathbf{A}}{\partial x} \right|^n_{i,j}
\sim &
\frac{- \frac{1}{56} A^n_{i+4,j} + \frac{4}{21} A^n_{i+3,j} - A^n_{i+2,j} + 4 A^n_{i+1,j}
- 4 A^n_{i-1,j} + A^n_{i-2,j} - \frac{4}{21} A^n_{i-3,j} + \frac{1}{56} A^n_{i-4,j}} {5 \,\Delta x}\\
\text{\& } \left. \frac{\partial \mathbf{A}}{\partial y} \right|^n_{i,j}
\sim &
\frac{- \frac{1}{56} A^n_{i,j+4} + \frac{4}{21} A^n_{i,j+3} - A^n_{i,j+2} + 4 A^n_{i,j+1}
- 4 A^n_{i,j-1} + A^n_{i,j-2} - \frac{4}{21} A^n_{i,j-3} + \frac{1}{56} A^n_{i,j-4}} {5 \,\Delta y} \text{.}
\end{split}
\label{eq:dscretisation_s8}\end{aligned}$$
Examples of the balancing of the terms of the energy and enstrophy equations are illustrated in Figs. \[fig:example\_E\_balance\] and \[fig:example\_Z\_balance\] respectively. [ In both cases, the matching between the left-hand side ($LHS^{E, \, \Omega}_V$) and the right-hand side ($RHS^{E, \, \Omega}_V$) highlights very small discrepancies observed. Typically, the total $RHS^{E, \, \Omega}_V$ terms are the result of the sum of significantly higher amplitude terms. Eventually, the resulting balancing errors $\delta^{E, \, \Omega}_V$ is of the order ${\cal O} (10^{-3})$ for both the kinetic energy balancing and the enstrophy balancing, resulting in an excellent hydrodynamic recovery.]{}
![Typical time-evolution of the kinetic energy balancing over a single sub-volume of size [ $181 \times 181$ shown for a forced LBGK simulation with $\tau_0 = 0.60$ ($Re \approx 90$)]{} on a $256 \times 256$ grid. The top figure shows the matching between the $LHS^{E}_V$ and the $RHS^{E}_V$, the middle figure shows the contribution of each $RHS^{E, \, i}_V$ term and their sum $RHS^{E}_V$, and the bottom figure shows the balancing error $\delta^{E}_V$.[]{data-label="fig:example_E_balance"}](balance_example_energy_LBGK){width="14cm"}
![Typical time-evolution of the enstrophy balancing over a single sub-volume of size [ $181 \times 181$ shown for a forced LBGK simulation with $\tau_0 = 0.60$ ($Re \approx 90$)]{} on a $256 \times 256$ grid. The top figure shows the matching between the $LHS^{\Omega}_V$ and the $RHS^{\Omega}_V$, the middle figure shows the contribution of each $RHS^{\Omega, \, i}_V$ term and their sum $RHS^{\Omega}_V$, and the bottom figure shows the balancing error $\delta^{\Omega}_V$.[]{data-label="fig:example_Z_balance"}](balance_example_enstrophy_LBGK){width="14cm"}
[In order to gather statistics of both balancing errors $\delta_{V}^{E, \, \Omega}(t)$ for a given sub-volume size $L$, we calculate them over squared sub-volumes $V= L \times L$ randomly chosen in space as illustrated in Fig. \[fig:sub-volume\_example\]]{}.
![Illustration on a snapshot of the vorticity field of three random squared sub-volumes $V_1=L_1 \times L_1$, $V_2=L_2 \times L_2$, and $V_3=L_3 \times L_3$ [ corresponding to the sub-volume size $L_1$, $L_2$, and $L_3$ respectively.]{}[]{data-label="fig:sub-volume_example"}](sub-volume_example.jpg){width="7cm"}
[ To present the results, we introduce the normalized sub-volume size $l=\frac{L}{L_0}$ with $L_0 = 256$ the size of the squared computational domain, and we group together the balancing errors $\delta^{E, \, \Omega}_{l}(t) = \delta^{E, \, \Omega}_{V = L \times L}(t)$ obtained for all sub-volumes of the same normalized sub-volume size $l$ on the same configuration at time $t$. We conduct a statistical analysis and define their mean $\mu^{E, \, \Omega}_{l}(t)$ and their standard deviation $\sigma^{E, \, \Omega}_{l}(t)$. The number of sub-volumes processed for a normalized sub-volume size $l$ is shown in Table \[tab:number\_subvolumes\]]{}.
Sub-volume size $L$ [Corresponding normalized sub-volume size $l$]{} Number of sub-volumes processed
--------------------- -------------------------------------------------- ---------------------------------
$L = 256$ [$l=1$]{} 1
$100 \leq L < 256$ [$0.4 \leq l < 1$]{} 1000
$10 \leq L < 100$ [$0.04 \leq l < 0.4$]{} 5000
$L < 10$ [$l < 0.04$]{} 10000
: [ Number of sub-volumes processed per sub-volume size $L$]{}[]{data-label="tab:number_subvolumes"}
Validation: LBGK against Pseudo-Spectral on an ensemble of decaying flow simulations {#sec:4}
====================================================================================
To understand how LBGK recovers hydrodynamics, we compare the statistics of the balancing errors obtained from LBGK simulations to the one obtained from PS simulations, which are used as a reference. To this aim we generate ensembles of LBGK and PS simulations: we conduct a statistically [ stationary forced LBGK $Re \approx 1200$ ($\tau_0 = 0.52$)]{} simulation that we sample into 25 configurations as shown in Fig. \[fig:sampling\_PS\], the number 25 being chosen in order to recover smooth statistics. [ Each of those configurations is then used to restart a LBGK simulation and to compute the corresponding vector potential $\vec{b}$ such as $\vec{u} = \vec{\nabla} \times \vec{b}$ to initialize an incompressible PS simulation]{} at the same Reynolds number, thus ensuring that they solve the same physics. Specifically, we set $$\label{eq:reynolds_PS-LBGK}
Re=\frac{U^{LBGK}_{RMS} L^{LBGK}}{\nu^{LBGK}_0}=\frac{U^{PS}_{RMS} L^{PS}}{\nu^{PS}_0}$$ with $U^{PS}_{RMS} = U^{LBGK}_{RMS}\frac{\Delta x^{LBGK}}{\Delta t^{LBGK}}$, $L^{PS} = 2 \pi = L^{LBGK} \Delta x^{LBGK}$, and $\nu_0^{PS} = \nu_0^{LBGK} \frac{(\Delta x^{LBGK})^2}{\Delta t^{LBGK}}$ and where $\nu_0^{LBGK} = c_s^2 (\tau_0 - 0.5)$ with $\tau_0 = 0.52$ in all simulations. Having fixed $\Delta x^{LBGK} = \frac{2 \pi}{256}$, $\tau_0 = 0.52$, and $\Delta t^{LBGK} = 0.001$, we obtain $\nu_0^{PS} \approx 0.004 $. We set $\Delta t^{PS} = 0.0005$ in order to be able to dump configurations of PS and LBGK simulations at the same physical time ($\Delta t^{LBGK} \propto \Delta t^{PS}$), while ensuring the stability of the PS simulations. Moreover, the velocity fields generated by the forced LBGK simulation have to be normalized by a factor $\frac{\Delta x^{LBGK}}{\Delta t^{LBGK}}$ before they are used to initialize the PS simulations. After initialization, the simulations are then left with no forcing to decay for a duration of $450 \, T_f$, where $T_f$ is the time scale based on the forcing as discussed in section \[sec:3\]. Eventually, the superposed ensemble-averaged energy spectrum for both ensemble at three selected times $t_1= 0$, $t_2 = 225 T_f$, and $t_3 = 450 T_f$ are in very good agreement (Fig. \[fig:spectra\_PS\]). The pressure field for the PS simulations is obtained by solving, for each configuration, the Poisson equation for pressure, while the pressure field for the LBGK simulations is obtained directly from the density field $p = c_s^2 \rho$.\
![Evolution of the kinetic energy (a) and of the enstrophy (b) of the forced LBGK simulation. The 25 vertical lines highlight the sampled configurations used to initialize the 25 decaying flow simulations of the PS and the LBGK ensembles.[]{data-label="fig:sampling_PS"}](energy_sampling_PS-LBGK "fig:"){height="5cm"} ![Evolution of the kinetic energy (a) and of the enstrophy (b) of the forced LBGK simulation. The 25 vertical lines highlight the sampled configurations used to initialize the 25 decaying flow simulations of the PS and the LBGK ensembles.[]{data-label="fig:sampling_PS"}](enstrophy_sampling_PS-LBGK "fig:"){height="5cm"}
![Superposed ensemble-averaged energy spectrum shown for three selected time instances for the PS and the LBGK simulations.[]{data-label="fig:spectra_PS"}](spectra_decaying_PS-LBGK){width="10cm"}
We show the results of the statistical analysis of the kinetic energy balancing error [ $\delta_l^{E}$]{} and enstrophy balancing error [ $\delta_l^{\Omega}$]{} in Figs. \[fig:results\_E\_PS-LBGK\] and \[fig:results\_Z\_PS-LBGK\] respectively. As expected, the PS method recovers hydrodynamics with a significant higher accuracy than the LBGK, with a clear improvement with time as the Reynolds number decreases and the simulations become increasingly resolved. [ This improvement with time cannot be well appreciated in the LBGK simulations, as it appears to be sub-leading in both the energy balance statistics $\mu_l^{E}$ and $\sigma_l^{E}$ (Fig. \[fig:results\_E\_PS-LBGK\], Panels (c)-(d)) and the the enstrophy balance statistics $\mu_l^{\Omega}$ and $\sigma_l^{\Omega}$ (Fig. \[fig:results\_Z\_PS-LBGK\], Panels (c)-(d)).]{} Taken all together, the statistical analysis of the balancing errors [$\delta_l^{E}$ and $\delta_l^{\Omega}$ show that hydrodynamic recovery is excellent on large sub-volumes and two orders of magnitude larger on small sub-volumes (see Figs. \[fig:results\_E\_PS-LBGK\] and \[fig:results\_Z\_PS-LBGK\], Panels (a)-(b)), the errors remaining however of order ${\cal O} (10^{-1})$.]{}\
To understand if the range of Mach numbers simulated affects the hydrodynamic recovery, we plot the statistics on the Mach number at [ the normalized sub-volume size $l$, [*i.e.*]{} $$\label{eq:MaL}
Ma_l = \big \langle \frac{U_{RMS}}{c_s} \big \rangle_{V=L \times L} \text{, } l = \frac{L}{L_0}$$]{} as shown in Fig. \[fig:mach\_decaying\_LBGK\]. We observe a steady mean (Fig. \[fig:mach\_decaying\_LBGK\]-(c)) going from about 0.55 to 0.4, and a steady standard deviation (Fig. \[fig:mach\_decaying\_LBGK\]-(d)) up to $L \approx 20$. As expected for decaying flows, the Mach number gradually decreases in time [ for all sub-volume sizes]{}. [ The statistical analysis of the decaying LBGK simulations is quite helpful to further assess the importance of the terms proportional to $Ma^3$ neglected in the momentum equation (see Eq. ). Indeed, if we look at the statistics of the energy and enstrophy balancing errors in Figs. \[fig:results\_E\_PS-LBGK\] and \[fig:results\_Z\_PS-LBGK\], we notice that if the Mach number was impacting the balancing errors, we would have observed a statistics that varies in time as the Mach number decays.]{} Thus, we can conclude that for the range of simulated Mach numbers the LBGK is a trustworthy Navier-Stokes solver, *i.e.* the Mach number is low enough so that all higher order Mach number terms that were neglected in the momentum equation do not affect the hydrodynamics.
![Statistics of the balancing error obtained from the kinetic energy balance [$\delta_l^{E}$ (see Eq. ) against the normalized size of the sub-volume $l$ shown for the PS and LBGK ensemble of 25 decaying simulations for three selected times. Top figures are PDF of the balancing error for sub-volumes corresponding to $l \approx 0.01$ (Panel (a)) and $l \approx 0.707$ (Panel (b)) and insets shows the PDFs of the balancing error for the PS ensemble alone]{}. Bottom figures are the mean (Panel (c)) and the standard deviation (Panel (d)) of the balancing error.\[fig:results\_E\_PS-LBGK\]](pdf_delta_energy_3x3_decaying_PS-LBGK "fig:"){width="49.00000%"} ![Statistics of the balancing error obtained from the kinetic energy balance [$\delta_l^{E}$ (see Eq. ) against the normalized size of the sub-volume $l$ shown for the PS and LBGK ensemble of 25 decaying simulations for three selected times. Top figures are PDF of the balancing error for sub-volumes corresponding to $l \approx 0.01$ (Panel (a)) and $l \approx 0.707$ (Panel (b)) and insets shows the PDFs of the balancing error for the PS ensemble alone]{}. Bottom figures are the mean (Panel (c)) and the standard deviation (Panel (d)) of the balancing error.\[fig:results\_E\_PS-LBGK\]](pdf_delta_energy_181x181_decaying_PS-LBGK "fig:"){width="49.00000%"} ![Statistics of the balancing error obtained from the kinetic energy balance [$\delta_l^{E}$ (see Eq. ) against the normalized size of the sub-volume $l$ shown for the PS and LBGK ensemble of 25 decaying simulations for three selected times. Top figures are PDF of the balancing error for sub-volumes corresponding to $l \approx 0.01$ (Panel (a)) and $l \approx 0.707$ (Panel (b)) and insets shows the PDFs of the balancing error for the PS ensemble alone]{}. Bottom figures are the mean (Panel (c)) and the standard deviation (Panel (d)) of the balancing error.\[fig:results\_E\_PS-LBGK\]](mean_energy_decaying_PS-LBGK "fig:"){width="49.00000%"} ![Statistics of the balancing error obtained from the kinetic energy balance [$\delta_l^{E}$ (see Eq. ) against the normalized size of the sub-volume $l$ shown for the PS and LBGK ensemble of 25 decaying simulations for three selected times. Top figures are PDF of the balancing error for sub-volumes corresponding to $l \approx 0.01$ (Panel (a)) and $l \approx 0.707$ (Panel (b)) and insets shows the PDFs of the balancing error for the PS ensemble alone]{}. Bottom figures are the mean (Panel (c)) and the standard deviation (Panel (d)) of the balancing error.\[fig:results\_E\_PS-LBGK\]](std_energy_decaying_PS-LBGK "fig:"){width="49.00000%"}
![Statistics of the balancing error obtained from the kinetic energy balance [$\delta_l^{E}$ (see Eq. ) against the normalized size of the sub-volume $l$ shown for the PS and LBGK ensemble of 25 decaying simulations for three selected times. Top figures are PDF of the balancing error for sub-volumes corresponding to $l \approx 0.01$ (Panel (a)) and $l \approx 0.707$ (Panel (b)) and insets shows the PDFs of the balancing error for the PS ensemble alone]{}. Bottom figures are the mean (Panel (c)) and the standard deviation (Panel (d)) of the balancing error.\[fig:results\_E\_PS-LBGK\]](legend_PS-LBGK){width="90.00000%"}
![Statistics of the balancing error obtained from the enstrophy balance [$\delta_l^{\Omega}$ (see Eq. ) against the normalized size of the sub-volume $l$ shown for the PS and LBGK ensemble of 25 decaying simulations for three selected times. Top figures are PDFs of the balancing error for sub-volumes corresponding to $l \approx 0.01$ (Panel (a)) and $l \approx 0.707$ (Panel (b)) and insets shows the PDFs of the balancing error for the PS ensemble alone]{}. Bottom figures are the mean (Panel (c)) and the standard deviation (Panel (d)) of the balancing error.\[fig:results\_Z\_PS-LBGK\]](pdf_delta_enstrophy_3x3_decaying_PS-LBGK "fig:"){width="49.00000%"} ![Statistics of the balancing error obtained from the enstrophy balance [$\delta_l^{\Omega}$ (see Eq. ) against the normalized size of the sub-volume $l$ shown for the PS and LBGK ensemble of 25 decaying simulations for three selected times. Top figures are PDFs of the balancing error for sub-volumes corresponding to $l \approx 0.01$ (Panel (a)) and $l \approx 0.707$ (Panel (b)) and insets shows the PDFs of the balancing error for the PS ensemble alone]{}. Bottom figures are the mean (Panel (c)) and the standard deviation (Panel (d)) of the balancing error.\[fig:results\_Z\_PS-LBGK\]](pdf_delta_enstrophy_181x181_decaying_PS-LBGK "fig:"){width="49.00000%"} ![Statistics of the balancing error obtained from the enstrophy balance [$\delta_l^{\Omega}$ (see Eq. ) against the normalized size of the sub-volume $l$ shown for the PS and LBGK ensemble of 25 decaying simulations for three selected times. Top figures are PDFs of the balancing error for sub-volumes corresponding to $l \approx 0.01$ (Panel (a)) and $l \approx 0.707$ (Panel (b)) and insets shows the PDFs of the balancing error for the PS ensemble alone]{}. Bottom figures are the mean (Panel (c)) and the standard deviation (Panel (d)) of the balancing error.\[fig:results\_Z\_PS-LBGK\]](mean_enstrophy_decaying_PS-LBGK "fig:"){width="49.00000%"} ![Statistics of the balancing error obtained from the enstrophy balance [$\delta_l^{\Omega}$ (see Eq. ) against the normalized size of the sub-volume $l$ shown for the PS and LBGK ensemble of 25 decaying simulations for three selected times. Top figures are PDFs of the balancing error for sub-volumes corresponding to $l \approx 0.01$ (Panel (a)) and $l \approx 0.707$ (Panel (b)) and insets shows the PDFs of the balancing error for the PS ensemble alone]{}. Bottom figures are the mean (Panel (c)) and the standard deviation (Panel (d)) of the balancing error.\[fig:results\_Z\_PS-LBGK\]](std_enstrophy_decaying_PS-LBGK "fig:"){width="49.00000%"}
![Statistics of the balancing error obtained from the enstrophy balance [$\delta_l^{\Omega}$ (see Eq. ) against the normalized size of the sub-volume $l$ shown for the PS and LBGK ensemble of 25 decaying simulations for three selected times. Top figures are PDFs of the balancing error for sub-volumes corresponding to $l \approx 0.01$ (Panel (a)) and $l \approx 0.707$ (Panel (b)) and insets shows the PDFs of the balancing error for the PS ensemble alone]{}. Bottom figures are the mean (Panel (c)) and the standard deviation (Panel (d)) of the balancing error.\[fig:results\_Z\_PS-LBGK\]](legend_PS-LBGK){width="90.00000%"}
![Statistics of the Mach number at [ normalized sub-volume size $l$ (see Eq. ) $Ma_l$ against the normalized size of the sub-volume $l$ shown for the LBGK ensemble of 25 decaying simulations for three selected times. Top figures are PDFs of $Ma_l$ for sub-volumes corresponding to $l\approx 0.01$ (Panel (a)) and $l \approx 0.707$ (Panel (b)). Bottom figures are the mean (Panel (c)) and the standard deviation (Panel (d)) of $Ma_l$]{}.\[fig:mach\_decaying\_LBGK\]](pdf_mach_3x3_decaying_LBGK "fig:"){width="49.00000%"} ![Statistics of the Mach number at [ normalized sub-volume size $l$ (see Eq. ) $Ma_l$ against the normalized size of the sub-volume $l$ shown for the LBGK ensemble of 25 decaying simulations for three selected times. Top figures are PDFs of $Ma_l$ for sub-volumes corresponding to $l\approx 0.01$ (Panel (a)) and $l \approx 0.707$ (Panel (b)). Bottom figures are the mean (Panel (c)) and the standard deviation (Panel (d)) of $Ma_l$]{}.\[fig:mach\_decaying\_LBGK\]](pdf_mach_181x181_decaying_LBGK "fig:"){width="49.00000%"} ![Statistics of the Mach number at [ normalized sub-volume size $l$ (see Eq. ) $Ma_l$ against the normalized size of the sub-volume $l$ shown for the LBGK ensemble of 25 decaying simulations for three selected times. Top figures are PDFs of $Ma_l$ for sub-volumes corresponding to $l\approx 0.01$ (Panel (a)) and $l \approx 0.707$ (Panel (b)). Bottom figures are the mean (Panel (c)) and the standard deviation (Panel (d)) of $Ma_l$]{}.\[fig:mach\_decaying\_LBGK\]](mean_mach_decaying_LBGK "fig:"){width="49.00000%"} ![Statistics of the Mach number at [ normalized sub-volume size $l$ (see Eq. ) $Ma_l$ against the normalized size of the sub-volume $l$ shown for the LBGK ensemble of 25 decaying simulations for three selected times. Top figures are PDFs of $Ma_l$ for sub-volumes corresponding to $l\approx 0.01$ (Panel (a)) and $l \approx 0.707$ (Panel (b)). Bottom figures are the mean (Panel (c)) and the standard deviation (Panel (d)) of $Ma_l$]{}.\[fig:mach\_decaying\_LBGK\]](std_mach_decaying_LBGK "fig:"){width="49.00000%"}
Forced LBGK hydrodynamics {#sec:5}
=========================
Setting up the forcings as described in section \[sec:3\], [ we analyze configurations of statistically stationary simulations at five different Reynolds numbers $Re \approx 90$, $390$, $640$, $1200$ and $1800$ respectively corresponding to relaxation times $\tau_0 = 0.60$, $0.54$, $0.53$, $0.52$ and $\tau_0^{last}=0.515$, beyond which LBGK is no longer stable]{}. We then obtain statistics of the balancing errors by averaging both in space and in time on 25 different configurations (see Fig. \[fig:sampling\_LBGK\]). We show in Fig. \[fig:spectra\_LBGK\] the superposed time-averaged spectrum for the conducted simulations. At large scales, we can see the effect of the energy removal preventing the energy to accumulate and maintaining the large-scale slope over the backward energy cascade slope of $-\frac{5}{3}$. On the other hand, at small scales, we observe that when we decrease $\tau_0$ ([ that is, increasing $Re$]{}) the flow becomes more turbulent and the slope gets increasingly closer to the forward enstrophy cascade slope of $-3$ [@Boffetta2012; @Frisch1995].
![Evolution of the kinetic energy (a) and of the enstrophy (b) of LBGK simulations for five different relaxation times. The 25 vertical lines highlight the time when configurations were processed to gather statistics in space and time of the balancing errors.[]{data-label="fig:sampling_LBGK"}](energy_sampling_LBGK "fig:"){height="5cm"} ![Evolution of the kinetic energy (a) and of the enstrophy (b) of LBGK simulations for five different relaxation times. The 25 vertical lines highlight the time when configurations were processed to gather statistics in space and time of the balancing errors.[]{data-label="fig:sampling_LBGK"}](enstrophy_sampling_LBGK "fig:"){height="5cm"} ![Evolution of the kinetic energy (a) and of the enstrophy (b) of LBGK simulations for five different relaxation times. The 25 vertical lines highlight the time when configurations were processed to gather statistics in space and time of the balancing errors.[]{data-label="fig:sampling_LBGK"}](legend_LBGK "fig:"){width="90.00000%"}
![Superposed time-averaged spectrum of LBGK simulations for five different relaxation times.[]{data-label="fig:spectra_LBGK"}](spectra_LBGK){width="10cm"}
We present the results of the statistical analysis of the kinetic energy balancing error [$\delta_l^{E}$]{} and the enstrophy balancing error [ $\delta_l^{\Omega}$]{} in Figs. \[fig:results\_E\_LBGK\] and \[fig:results\_Z\_LBGK\] respectively. As expected from the LBGK-PS validation results, [the hydrodynamic recovery largely depends on the size of the sub-volume it is measured on. Indeed, hydrodynamic recovery is again excellent on large sub-volumes with an order of magnitude of up to ${\cal O} (10^{-3})$, than on small sub-volumes, where we obtain an error that is of orders of magnitude ${\cal O} (10^{-1})$ (see dashed lines in Figs. \[fig:results\_E\_LBGK\] and \[fig:results\_Z\_LBGK\], Panels (c)-(d)). For the energy balancing error presented Fig. \[fig:results\_E\_LBGK\], we observe a small dependence on the Reynolds number. However, as shown on Fig. \[fig:results\_Z\_LBGK\], the enstrophy balance becomes better by decreasing Reynolds number, as it is expected for a quantity that is strongly sensitive to the small-scales resolution.]{}\
Having forced with fixed forcing amplitudes, the Mach number of the conducted simulations also varies as a [ function of the Reynolds number]{}. To highlight potential high Mach number effects, we plot again the statistics on the Mach number at [ sub-volume size $l$, $Ma_l$ (Eq. )]{} as shown in Fig. \[fig:mach\_LBGK\]. We observe that we are working with Mach number that are qualitatively and quantitatively similar to the ones studied in the previous section (see Fig. \[fig:mach\_decaying\_LBGK\]), hence we conclude again that we work on a range of Mach number that does not impact the hydrodynamics.
![Statistics of the balancing error obtained from the kinetic energy balance [$\delta_l^{E}$]{} (see Eq. ) against the [size of the sub-volume $l$ for 5 forced LBGK simulation of different Reynolds numbers. Top figures are PDF of the balancing error for sub-volumes corresponding to $l \approx 0.01$ (Panel (a)) and $l \approx 0.707$ (Panel (b))]{}. Bottom figures are the mean (Panel (c)) and the standard deviation (Panel (d)) of the balancing error.[]{data-label="fig:results_E_LBGK"}](pdf_delta_energy_3x3_LBGK "fig:"){width="49.00000%"} ![Statistics of the balancing error obtained from the kinetic energy balance [$\delta_l^{E}$]{} (see Eq. ) against the [size of the sub-volume $l$ for 5 forced LBGK simulation of different Reynolds numbers. Top figures are PDF of the balancing error for sub-volumes corresponding to $l \approx 0.01$ (Panel (a)) and $l \approx 0.707$ (Panel (b))]{}. Bottom figures are the mean (Panel (c)) and the standard deviation (Panel (d)) of the balancing error.[]{data-label="fig:results_E_LBGK"}](pdf_delta_energy_181x181_LBGK "fig:"){width="49.00000%"} ![Statistics of the balancing error obtained from the kinetic energy balance [$\delta_l^{E}$]{} (see Eq. ) against the [size of the sub-volume $l$ for 5 forced LBGK simulation of different Reynolds numbers. Top figures are PDF of the balancing error for sub-volumes corresponding to $l \approx 0.01$ (Panel (a)) and $l \approx 0.707$ (Panel (b))]{}. Bottom figures are the mean (Panel (c)) and the standard deviation (Panel (d)) of the balancing error.[]{data-label="fig:results_E_LBGK"}](mean_energy_LBGK "fig:"){width="49.00000%"} ![Statistics of the balancing error obtained from the kinetic energy balance [$\delta_l^{E}$]{} (see Eq. ) against the [size of the sub-volume $l$ for 5 forced LBGK simulation of different Reynolds numbers. Top figures are PDF of the balancing error for sub-volumes corresponding to $l \approx 0.01$ (Panel (a)) and $l \approx 0.707$ (Panel (b))]{}. Bottom figures are the mean (Panel (c)) and the standard deviation (Panel (d)) of the balancing error.[]{data-label="fig:results_E_LBGK"}](std_energy_LBGK "fig:"){width="49.00000%"}
![Statistics of the balancing error obtained from the enstrophy balance [ $\delta_l^{\Omega}$]{} (see Eq. ) against the [ size of the sub-volume $l$ shown for 5 forced LBGK simulation of different Reynolds numbers. Top figures are PDF of the balancing error for sub-volumes corresponding to $l \approx 0.01$ (Panel (a)) and $l \approx 0.707$ (Panel (b))]{}. Bottom figures are the mean (Panel (c)) and the standard deviation (Panel (d)) of the balancing error.[]{data-label="fig:results_Z_LBGK"}](pdf_delta_enstrophy_3x3_LBGK "fig:"){width="49.00000%"} ![Statistics of the balancing error obtained from the enstrophy balance [ $\delta_l^{\Omega}$]{} (see Eq. ) against the [ size of the sub-volume $l$ shown for 5 forced LBGK simulation of different Reynolds numbers. Top figures are PDF of the balancing error for sub-volumes corresponding to $l \approx 0.01$ (Panel (a)) and $l \approx 0.707$ (Panel (b))]{}. Bottom figures are the mean (Panel (c)) and the standard deviation (Panel (d)) of the balancing error.[]{data-label="fig:results_Z_LBGK"}](pdf_delta_enstrophy_181x181_LBGK "fig:"){width="49.00000%"} ![Statistics of the balancing error obtained from the enstrophy balance [ $\delta_l^{\Omega}$]{} (see Eq. ) against the [ size of the sub-volume $l$ shown for 5 forced LBGK simulation of different Reynolds numbers. Top figures are PDF of the balancing error for sub-volumes corresponding to $l \approx 0.01$ (Panel (a)) and $l \approx 0.707$ (Panel (b))]{}. Bottom figures are the mean (Panel (c)) and the standard deviation (Panel (d)) of the balancing error.[]{data-label="fig:results_Z_LBGK"}](mean_enstrophy_LBGK "fig:"){width="49.00000%"} ![Statistics of the balancing error obtained from the enstrophy balance [ $\delta_l^{\Omega}$]{} (see Eq. ) against the [ size of the sub-volume $l$ shown for 5 forced LBGK simulation of different Reynolds numbers. Top figures are PDF of the balancing error for sub-volumes corresponding to $l \approx 0.01$ (Panel (a)) and $l \approx 0.707$ (Panel (b))]{}. Bottom figures are the mean (Panel (c)) and the standard deviation (Panel (d)) of the balancing error.[]{data-label="fig:results_Z_LBGK"}](std_enstrophy_LBGK "fig:"){width="49.00000%"}
![Statistics of the Mach number at [ $Ma_l$ normalized sub-volume size $l$ (see Eq. ) $Ma_l$ against the normalized size of the sub-volume $l$ shown for 5 forced LBGK simulation of different Reynolds numbers. Top figures are PDF of the balancing error for sub-volumes corresponding to $l \approx 0.01$ (Panel (a)) and $l \approx 0.707$ (Panel (b)). Bottom figures are the mean (Panel (c)) and the standard deviation (Panel (d)) of $Ma_l$]{}.[]{data-label="fig:mach_LBGK"}](pdf_mach_3x3_LBGK "fig:"){width="49.00000%"} ![Statistics of the Mach number at [ $Ma_l$ normalized sub-volume size $l$ (see Eq. ) $Ma_l$ against the normalized size of the sub-volume $l$ shown for 5 forced LBGK simulation of different Reynolds numbers. Top figures are PDF of the balancing error for sub-volumes corresponding to $l \approx 0.01$ (Panel (a)) and $l \approx 0.707$ (Panel (b)). Bottom figures are the mean (Panel (c)) and the standard deviation (Panel (d)) of $Ma_l$]{}.[]{data-label="fig:mach_LBGK"}](pdf_mach_181x181_LBGK "fig:"){width="49.00000%"} ![Statistics of the Mach number at [ $Ma_l$ normalized sub-volume size $l$ (see Eq. ) $Ma_l$ against the normalized size of the sub-volume $l$ shown for 5 forced LBGK simulation of different Reynolds numbers. Top figures are PDF of the balancing error for sub-volumes corresponding to $l \approx 0.01$ (Panel (a)) and $l \approx 0.707$ (Panel (b)). Bottom figures are the mean (Panel (c)) and the standard deviation (Panel (d)) of $Ma_l$]{}.[]{data-label="fig:mach_LBGK"}](mean_mach_LBGK "fig:"){width="49.00000%"} ![Statistics of the Mach number at [ $Ma_l$ normalized sub-volume size $l$ (see Eq. ) $Ma_l$ against the normalized size of the sub-volume $l$ shown for 5 forced LBGK simulation of different Reynolds numbers. Top figures are PDF of the balancing error for sub-volumes corresponding to $l \approx 0.01$ (Panel (a)) and $l \approx 0.707$ (Panel (b)). Bottom figures are the mean (Panel (c)) and the standard deviation (Panel (d)) of $Ma_l$]{}.[]{data-label="fig:mach_LBGK"}](std_mach_LBGK "fig:"){width="49.00000%"}
Concluding remarks {#sec:6}
==================
We have proposed a general tool to check the generated hydrodynamics of fluid flow simulations. The tool hinges on the calculation of the kinetic energy and the enstrophy balance equation terms averaged over randomly chosen [ sub-volumes of different size]{}. We have defined balancing errors, representing the accuracy of the hydrodynamic recovery across [ sub-volume sizes]{} and conducted a statistical analysis in the context of 2D homogeneous isotropic turbulence. Firstly, we validated this tool on decaying 2D turbulence by systematically comparing an ensemble of LBGK simulations with an ensemble of PS simulations, both initialized with the same configurations. [ The PS simulations hydrodynamic recovery accuracy is two to six orders of magnitudes higher than the LBGK simulations’. Moreover, in all cases hydrodynamic recovery is better verified by looking at larger and larger sub-volumes]{}. [Besides, although the enstrophy balance involves higher order derivatives than those present in the kinetic energy equation [@Biferale2010], the associated extra discretization error was shown to be negligible as both statistics of the energy and enstrophy balancing errors shows similar order of magnitudes]{}. Secondly, we have applied this tool to check LBGK hydrodynamic in the context of forced 2D turbulence at increasing Reynolds number. All in all, we have observed statistics of the balancing errors both from kinetic energy balance and enstrophy balance that are very similar to the validation LBGK ensemble’s results. In both the validation and benchmark, the Mach number was maintained low enough for its effect to be sub-leading in the hydrodynamic recovery.\
The ideal continuation of this work is the study of hydrodynamic recovery with LBM in presence of SGS models of eddy viscosity. To this aim, the developed tool is particularly useful, since it allows to quantitatively describe the effects of under-resolution and the possible improvements led by the SGS model. [ An expansion of this tool to 3D turbulence is also being developed. Indeed, 3D turbulence is of interest, as it exhibits a direct cascade of energy with a Kolmogorov-predicted slope of $k^{\frac{5}{3}}$, which does not ensure that the flow remains differentiable.]{}
Acknowledgement {#acknowledgement .unnumbered}
===============
The authors would like to thank Fabio Bonaccorso and Michele Buzzicotti at the University of Rome “Tor Vergata” for their support in conducting the PS simulations. This work was supported by the European Unions Framework Programme for Research and Innovation Horizon 2020 (2014-2020) under the Marie Skłodowska-Curie grant \[grant number 642069\] [ for the High Performance Computing in Life sciences, Engineering and Physics (HPC-LEAP) project]{} and by the European Research Council under the ERC grant \[grant number 339032\]. It is also part of the research programme CSER \[project number 12CS034\], which is (partly) financed by the Netherlands Organisation for Scientific Research (NWO).
References {#references .unnumbered}
==========
[^1]: Postprint version of the article published on Computers & Fluids 172 (2018) 241-250
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We present a detailed analysis of the role of native point defects in the antiferromagnetic (AFM) phases of bulk chromium nitride (CrN). We perform first-principles calculations using local spin-density approximation, including local interaction effects (LSDA+U), to study the two lowest energy AFM models expected to describe the low-temperature phase of the material. We study the formation energies, lattice deformations and electronic and magnetic structure introduced by native point defects. We find that, as expected, nitrogen vacancies are the most likely defect present in the material at low temperatures. Nitrogen vacancies present different charged states in the cubic AFM model, exhibiting two transition energies, which could be measurable by thermometry experiments and could help identify the AFM structure in a sample. These vacancies also result in partial spin polarization of the induced impurity band, which would have interesting consequences in transport experiments. Other point defects have also signature electronic and magnetic structure that could be identified in scanning probe experiments.'
author:
- Tomas Rojas
- 'Sergio E. Ulloa'
bibliography:
- 'Mendeley.bib'
title: Native Point Defects in Antiferromagnetic Phases of CrN
---
Introduction
============
Antiferromagnetic (AFM) materials exhibit an interesting long-range order that sets in below a characteristic temperature known as the Néel scale, $T_N$. In this regime, there is an alternating pattern for the magnetic moment orientation throughout the crystal that results in a zero net macroscopic magnetization and is robust to external magnetic and electric fields. [@Coey2009] In recent years, AFM materials have been considered in spintronics applications, as the development of methods to control the microscopic AFM ordering by applying electrical pulses has proven successful. [@Jungwirth2016; @Baltz2018] Deeper understanding of these materials, including the role that pervasive defects play, would be of considerable interest, especially as possible applications appear on the horizon.
Transition metal nitrides (TMNs) represent a large family of materials that play a significant role in many technological applications. They are characterized by high hardness, corrosion resistance and unusual electrical properties that make them useful, [@Navinsek2001a] and in some cases they also exhibit interesting magnetic behavior. [@Zilske2017] Among them, chromium nitride (CrN) possesses unusual electronic and magnetic properties. Specifically, CrN has been shown to exhibit a phase transition in both crystalline structure and magnetic ordering, as the material transitions from a paramagnet with cubic rock-salt structure at high temperature to an antiferromagnet with orthorhombic $P_{nma}$ structure at $T_{N}\simeq 280$K [@Gall2002].
The details of the AFM and structural ordering of the CrN crystal have however been subject to some controversy. Several first-principles calculations have shown that an orthorhombic $P_{nma}$ AFM model, in which the spin changes every two layers along the $[110]$ direction (and denoted as $AFM^2_{[110]}$), is the most energetically favorable [@Corliss1960AntiferromagneticCrN; @Botana2012; @Zhou2014; @Filippetti1999a]. However, as discussed by Zhou [*et al.*]{} [@Zhou2014], the set of [*ad hoc*]{} parameters used in LSDA+U calculations affects the energetic difference between the $AFM^2_{[110]}$ model and a competing cubic arrangement (shown in Fig. \[fig:afm\]) in which local magnetic moments change at every layer in the $[010]$ direction (and denoted as $AFM^1_{[010]}$). The energetic difference between these two configurations is small and it is therefore likely that both phases would appear in thin film experiments. A recent study of thin films of CrN grown by molecular beam epitaxy (MBE) detected the phase transition between the cubic paramagnet and the orthorhombic $AFM^2_{[110]}$. However, they also find experimental evidence of a low temperature cubic phase, perhaps stabilized near the surface of the thin film crystal. [@Alam2017]
![(Color online) Unit cells of CrN for a) cubic $AFM^1_{[010]}$, and b) orthorhombic $AFM^2_{[110]}$ structures. Silver/red spheres indicate alternating magnetic moment direction on Cr sites. Blue spheres are N atoms. Black arrows indicate different lattice directions.[]{data-label="fig:afm"}](fig_afm_new2.png)
It is also important to mention that thin film experiments disagree on the intrinsic character of the electronic resistivity $\rho$ [@Constantin2004; @Inumaru2007], with some experiments showing metallic behavior as $\rho$ decreases with lower temperatures,[@Gall2002; @SubramanyaHerle1997a] while others show semiconducting behavior, with increasing $\rho$ as the temperature drops. This disagreement has been associated with the presence of nitrogen vacancies or other dopants that tend to complicate transport measurements on CrN thin films.[@Quintela2009]
A related issue is determining the presence and/or size of an energy bandgap in the material: an optical gap $\approx 0.7$ eV was reported in single crystals of CrN [@Gall2002]; however, a much smaller band gap ($\approx 90$ meV) was reported from resistivity measurements in powder samples. From first principles calculations, the inclusion of a Hubbard correction (LSDA+U) results in reported band gaps from $0.2$ to $2.13$ eV, [@Botana2012; @Zhou2014] depending on the magnitude of the U correction, although a moderate gap ($\lesssim 0.8$ eV) is believed to be the best theoretical estimate. Moreover, strain also induces significant changes in the electronic structure, [@Rojas2017; @Filippetti2000] which may affect different observations. For instance, a moderate strain of $\approx 1.3\%$ is predicted to close the gap, as well as to strongly modify the effective masses of both conduction and valence bands. [@Rojas2017] Interestingly, the masses are changed anisotropically, with principal axes heavily influenced by the magnetic ordering.[@Rojas2017] This points to the strong connection between structural and magnetic ordering at low temperatures, and response of the system to external fields.
As mentioned, the presence of N vacancies and the accompanying carrier doping may have important roles in understanding the disagreements in resistivity measurements, and previous studies have indeed focused on this effect in CrN. Zhang [*et al*]{}.[@Zhang2013a] analyze structural changes and core-level shifts due to a high concentration of defects in high-resolution transmission electron microscopy experiments. They conclude that high N-vacancy concentration ($\gtrsim 10\%$) leads to lattice distortions and overall reduction of the lattice volume. Mozafari [*et al*]{}.[@Mozafari2015] studied vacancies in CrN at high temperatures (the paramagnetic phase), and conclude that both N vacancies and interstitials act as donors in the system. As entropy facilitates the appearance of such defects, it is expected that they would naturally contribute to self-doping of films and the overall resistivity behavior away from carrier freeze out.
A related issue that has not received much attention is the effect of vacancies and other point defects on the magnetic structure and response of CrN crystals. As the atomic arrangement and magnetic structure are closely intertwined in this material, it would be of interest to study how defects modify not only the charge but magnetic moment profiles. Such effects would not only be observable by local probes, such as STM and spin-polarized STM, but are also likely to affect the electromagnetic response that is now being used to control the distribution of magnetic moments in AFM materials. [@Jungwirth2016; @Baltz2018]
It is with these effects in mind that we have carried out a detailed theoretical analysis of native point defects, focusing on their effect on the electronic and magnetic properties of CrN. Utilizing the LSDA+U approach we study the formation energies of four native defects in different expected AFM phases. We calculate band structures and determine orbital weight shifts near the Fermi level produced by defects. We analyze the structural changes and its consequences on the electronic structure, charge, and magnetic moment distributions created as a consequence. We find that nitrogen vacancies with different charge states are realistically created in different structures, as function of doping, and result in distortions of charge and spin profiles that are rather local. Different point defects have different remnant spin structures and associated partially polarized impurity bands which could be identified in transport and in experiments using local scanning probes.
Computational Approach
======================
We carry out density functional theory (DFT) calculations in the local spin density approximation as included in the Quantum Espresso package[@Giannozzi2009a]. We include the LSDA+U correction on the $3d$ orbitals of Cr using the rotationally invariant formulation of Liechtenstein [*et al.*]{}[@Liechtenstein1995b] Identification of the appropriate $ad$ $hoc$ constants for this system is assisted by previous work; Herwadkar [*et al.*]{}[@Herwadkar2009] estimated $U \simeq 3$ to 5 eV, and $J=0.94$ eV, using the Cococcoine and Gironcoli algorithm[@Cococcioni2005]. Similar values were estimated by Zhou [*et al.*]{}[@Zhou2014] In our calculations we selected $J=0.94$ eV, with $U=3$ eV for the $AFM^2_{[110]}$ structural model, and $U=5$ eV for the $AFM^1_{[010]}$ arrangement. These choices represent low values of $U$ that achieve similar energy gaps in both models.
We have performed calculations in a 64 atom unit cell ($2\times2\times2$ cubic cells with 8 atoms each), with Brillouin zone sampling using $4\times4\times4$ k-point meshes and with an energy cutoff of 410 eV. The defects are created by either adding or removing the appropriate atom in the supercell, and performing full structural relaxation with a force convergence threshold of $10^{-4}$ a.u. (atomic units). This allows us to analyze the accompanying deformations of the lattice structure as well as the energetics and spatial distribution of the associated charge and spin distribution of the defect state.
We have specifically studied different native point defects, with the simplest being the nitrogen ($V_N$) and chromium vacancies ($V_{Cr}$). We also consider antisite defects, replacing the atom of one species with the other, which we denote as anti-N ($A_N$, where N is in a Cr site) and anti-Cr ($A_{Cr}$, for Cr in an N site). Lastly, we also consider different interstitial defects, $I_N$ or $I_{Cr}$, with the corresponding impurity placed in a previously hollow location. We have attempted to obtain all these point defects for both AFM models; however, it was not possible to reach convergence for $A_{Cr}$ in the $AFM^1_{[010]}$ structure. A similar situation was found for $I_N$ in the $AFM^2_{[110]}$ structure, reflecting the high energy cost of the drastic change in atomic coordination for such defects.
Finding equilibrium structures of different defects indicates the possibility of generating such configurations in real crystals, as the energetics are important in determining abundance under different growth conditions. Knowing the formation energy of neutral species is crucial, although charged states are possible. For the latter, theoretical evaluation must also consider the electrostatic contributions, with obvious different energy costs for different charged states. To this end, Freysoldt [*et al*]{}.[@Freysoldt2011] have proposed a successful scheme to include the charge density of the defect to find the corresponding short range potential generated. The method has been used in several studies to examine defects in ternary oxides, among other materials.[@Hautier2013a] In this approach, the formation energy is defined as [@Freysoldt2011; @Freysoldt2014]
$$\begin{split}
E_{\rm defect} (q) =E^{DFT}_{{\rm def+bulk}} - E^{DFT}_{{\rm bulk}} -E^{latt}[q^{model}] \\ + \,q\Delta V - \sum n_s \mu_s + \, (E_{F}+\varepsilon^{\rm vbm} )q ,
\end{split}
\label{EqFormation}$$
where the total energy of the bulk supercell, $E^{DFT}_{{\rm bulk}}$, is subtracted from the cell containing the point defect $E^{DFT}_{{\rm def+bulk}}$. The next term includes the electrostatic interaction of the lattice of charged defects generated by the periodic boundary conditions. $\Delta V$ represents a constant alignment potential added to correct the long and short-range potential between defects of charge $q$. In the next term, $\mu_s$ represents the chemical potential for each reservoir for the species involved. For CrN we use the ground state energy of a $N_2$ molecule, and an AFM bulk Cr crystal as the individual points of reference. Likewise, the Fermi energy $E_{F}$ (measured from the valence band maximum $\varepsilon^{\rm vbm}$) is the electronic chemical potential. Notice the approach requires knowledge of the dielectric constant $\epsilon$, to properly screen the charged defects. We estimate $\epsilon$ by applying a sawtooth potential to a slab of material and calculating the ratio between external and internal potential slopes [@Freysoldt2011]; for CrN we have estimated $\epsilon=1.43$.
As we will see below, the nature of the point defect changes the electronic structure in a variety of ways and the entire set of properties of the material.
Results
=======
Formation energies
------------------
The formation energies were evaluated using the scheme developed by Freysoldt [*et al.*]{},[@Freysoldt2011; @Freysoldt2014] as discussed above, and shown in Fig. \[fig:ForEn1\] for the $AFM^1_{[010]}$ structure. Our calculations indicate that only the $V_N$ vacancies in this cubic model have charged states within a reasonable range of the Fermi level. Figure \[fig:ForEn1\] shows that at zero Fermi energy ($E_F=0$) the formation energy of the neutral species for the nitrogen vacancy is $V_N^{q=0}= 2.48$ eV, while its charged states are at $V_N^{q=-1}=2.63$ eV and $V_N^{q=1}=2.94 $ eV. Considering a Fermi energy range from $-0.2$ to 0.8 eV, we find two transition energies between charged states, with $\varepsilon_{-1,0} = -0.147 $ eV, and $\varepsilon_{0,1} = 0.46 $ eV. Both transitions are likely reachable through doping, and may be in the energy gap, as indicated by two different gap estimates in the figure. \[A vertical line at 0.79 eV indicates the gap reported by Botana [*et al*]{}. [@Botana2012] using the TB-mBJLDA functional [@Tran2009].\] In the cubic AFM structure we also obtain $V_{Cr}$ at 6.90 eV, $A_{Cr}$ at 10.05 eV, and the interstitial $I_N$ at 12.01 eV. These high formation energy values suggest that their appearance is most unlikely in well relaxed crystals, although thermodynamic analysis under different growth conditions would be required to confirm this conclusion. [@Freysoldt2014]
Similarly, in the orthorhombic $AFM^2_{[110]}$ model, the neutral $V_N$ was also found to have the lowest formation energy at 2.13 eV, followed by $V_{Cr}= 5.72 $ eV, $A_{Cr}= 9.7 $ eV, and $A_N =10.14 $eV. Charged states of any of these defects are at least 2 eV away, making them uninteresting. For this structure, we also found that it was not possible to obtain convergence for interstitial point defects. This is likely related to the orthorhombic distortion present in this model, which having lower symmetry makes for difficult force balance on the interstitial sites.
![(Color online) Formation energies for the different $V_N$ nitrogen vacancy different charged states in the $AFM^1_{[010]}$ model structure. $E_F=0$ is set at the edge of the valence band. Over the $E_F$ range in the figure, there are two transition energies at $\varepsilon_{-1,0}= -0.147$ eV and $\varepsilon_{0,1}= 0.46 $ eV. Vertical lines at $E_{\rm gap}^{\rm LDA+U}$ and $E_{\rm gap}^{\rm TB-mBJLDA}$ (see Ref. ) indicate corresponding estimates of the energy gap in this phase.[]{data-label="fig:ForEn1"}](AFM1_energies.png)
Geometry of relaxed structures
------------------------------
Table 1 shows a summary of the distortions in the relaxed structures for different point defects and AFM models. Positive and negative displacement values represent atomic displacements away and towards the defect, respectively. Except for the interstitials and antisites, most values are relatively small, as one would expect.
[lcccc]{}\
\
-------
Model
-------
: Atom displacement (in ${\textup{\AA}}$) of nearest neighbor atoms with respect of the defect site in the ideal crystal. Positive (negative) values indicate motion away (towards) the point defect.[]{data-label="table1"}
& Defect & $Cr (\uparrow)$ & $Cr(\downarrow)$ & N\
\
\
& $V_N$ & 0.005 & –0.001 & –0.002\
& $V_{Cr(\uparrow)}$ & 0.00 & –0.010 & –0.082\
& $A_N$ & –0.03 & –0.02 & –0.171\
& $I_{N}$ & 0.032 & 0.032 & 0.221\
\
& $V_N$ & 0.01 & –0.002;0.03 & –0.01\
& $V_{Cr}$ & –0.07 & –0.07 & 0.07;0.08\
& $A_N$ & –0.298;0.055 & –0.031 & –0.022\
& $A_{Cr}$ & 0.309 & 0.085 & –0.11\
For the $AFM^1_{[010]}$ structure, the nitrogen vacancy produces significantly smaller displacements ($\lesssim 0.2\%$), reflecting the low energies of formation, with slight differences among Cr atoms, depending on their local magnetic moment projection. The Cr$(\uparrow)$ neighbors move away from the defect site, while the Cr$(\downarrow)$ atoms move inwards slightly. Meanwhile, the nearest N atoms move closer to the defect by $0.002~{\textup{\AA}}$. Chromium vacancies produce larger deformations, with inward displacements of all neighboring N atoms and opposite spin Cr sites.
The $A_N$ defects produce motion of all neighbor Cr atoms towards the defect, as well as significant displacement of the neighbor N atoms. In contrast, the $I_N$ defect pushes the neighbors away, with especially large shifts for the nearest N atoms. It is interesting to see that in nearly all defects the magnetic moments on the Cr atoms seem to play a role on the cell distortions, reflecting the coupling of magnetic and lattice structure in this material.
In the $AFM^2_{[110]}$ model, the $V_N$ defects produce similar distortions as in the $AFM^1_{[010]}$ structure. However, $V_{Cr}$ vacancies produce equal distortions towards the defect for both orientations of the magnetic moment, and a slight shift in the opposite direction for the neighbor N atoms. The anti-N defect produces a large shift ($\approx 10\%$ of the unperturbed separation) of the spin-up Cr atoms towards it, arising from the fact that the Cr atom removed was in that direction. A related situation occurs for the $A_{Cr}$ defect, as Cr atoms of the same spin direction as the atom added show significant shifts towards the defect, and only a slight shift away for opposite-spin Cr atoms.
As we will illustrate below, even the relatively small shifts produced by the $V_N$ and $V_{Cr}$ defects in these crystals result in significant charge and spinful redistributions with unique characteristics.
Electronic structure of defects
-------------------------------
In this section we analyze changes in the electronic structure introduced by the various defects in different AFM model structures. For this purpose we calculate the partial density of states (PDOS) for the atoms close to the point defect. We emphasize that the results here are for a single defect in a 64 atom supercell, corresponding to 3.1% concentration. We have verified, however, that results are fully converged, and that especially for the neutral vacancy species, the defect interactions across supercells is negligible.
As the $AFM^1_{[010]}$ model is characterized by a cubic symmetric structure, its PDOS in the pristine case exhibits spin-symmetric results–even as the magnetic arrangement reduces that somewhat–see Fig. \[fig:PDOS1\], top panel. Notice, for example, that $d_{xy}$ and $d_{zx}$ have identical PDOS, as in this AFM model two spatial directions show isotropy with respect of the magnetic ordering.
Introduction of a nitrogen vacancy in the supercell has the lowest energy of formation, as discussed above. As seen in Fig. \[fig:PDOS1\], second panel, the vacancy gives rise to a redistribution of $d$-orbitals in the neighbors, with a peak of $d_{z^2}$ states just below the Fermi level for one of the spins. This shallow orbital state exhibits partial spin polarization, and would contribute significantly to mobile carriers in a typical crystal, impart a polarization to the response, and affect the measured resistivity. The spin polarization seen in this PDOS is also present in the overall band structure, and no gap appears for the minority component. Similarly, a large spin polarization in the $d_{zy}$ orbital is seen, peaking at $\approx-1.5$ eV, with opposite direction to that of the $d_{z^2}$ peak, as the nearest Cr atoms rearrange. A noticeable rearrangement of the $p$-orbitals in nitrogen is also seen for this $V_N$ defect.
![(Color online) Partial density of states in the $AFM^1_{[010]}$ structure for the nearest neighbor atoms (Cr($\uparrow$), Cr($\downarrow$) and N) surrounding each specific point defect, for up (positive) and down (negative) spin projections. Individual $d$-orbitals of the Cr and $p$-orbitals of N are shown with different colors. The vertical line at zero energy represents the Fermi level.[]{data-label="fig:PDOS1"}](afm1_pdos2.png)
In the case of $V_{Cr}$, symmetrically surrounded by nearest neighbors nitrogen atoms, the vacancy produces an enhancement of $p$-orbitals near the Fermi level with no net spin polarization (as shown in Fig. \[fig:PDOS1\], third panel). The defect, however, alters the AFM order as evidenced by the asymmetrical changes in the $t_{2g}$ orbitals of the nearby Cr atoms, especially $d_{zy}$.
The $A_{N}$ defect strongly suppresses $p$-orbitals near the Fermi level, as additional $p$ bonds are created around the defect. In contrast, $I_N$ represents an interesting case, being the only point defect that distorts the orbital distribution on the neighbors while preserving spin-polarization symmetry. It also shows symmetry among $d_{zy}$ and $d_{dxy}$ orbitals.
The pristine $AFM^2_{[110]}$ model displays lower orbital symmetry overall, as seen in Fig. \[fig:PDOS2\], product of the orthorhombic structure. The N vacancy produces an enhancement of $d_{z^2}$ states, and just like in the cubic model, it is partially spin polarized. The $V_{Cr}$ in this model shows also significantly enhanced PDOS at the Fermi level, but with a more complex rearrangement that has larger $d_{x^2-y^2}$ amplitude in one spin projection and $d_{zx}$ on the opposite. Both antisite defects, $A_N$ and $A_{Cr}$ maintain an open bandgap, although with strong orbital rearrangement and only slight local polarization.
![(Color online) Partial density of states for the $AFM^2_{[110]}$ structure for nearest neighbor atoms surrounding each specific point defect. Spin up (down) projections are shown as positive (negative) values. Individual Cr $d$-orbitals and N $p$-orbitals shown with different colors. Vertical line at zero energy represents the Fermi level.[]{data-label="fig:PDOS2"}](afm2_pdos2.png)
![(Color online) Charge density near chromium and nitrogen vacancies for the $AFM^1_{[010]}$ and $AFM^2_{[110]}$ models. Red and green represent charge deficiency and excess, respectively. Blue spheres represent N atoms; gray represent Cr. The isosurface value was taken as 0.075 electron/Bohr$^3$[]{data-label="fig:chargeclouds"}](ChargeDensity.png)
To complement the PDOS information, Fig. \[fig:chargeclouds\] shows 3D representations of the charge densities near the nitrogen and chromium vacancies in the two different structures. In all cases, the local character of the induced charge is evident.
Magnetic structure of defects
-----------------------------
In addition to the changes seen in the PDOS, all point defects are seen to disturb the ideal AFM ordering of the pristine crystal in either of the two models we explored. The presence of different defects results often in a net magnetic moment for the supercell even when the added or removed atom has no magnetic moment. Moreover, defects are seen to induce different spatial patters of local spin polarization in the unit cell.
![(Color online) Two dimensional map of the spin polarization on (001) plane, for point defects in the $AFM^1_{[010]}$ structure. Red and blue amplitudes represent extremal spin projections (up and down respectively), as per color bars. Notice the very different scale in the case of $V_N$. $x$, $y$ and spin projections given in atomic units. Images in the background represent the atom positions obtained afterfull structure relaxations. Blue and grey spheres represent N and Cr atoms, respectively.[]{data-label="fig:spinpol1"}](afm1_sp.png)
In Fig. \[fig:spinpol1\] we present a 2D cross section of the spin polarization on the $(001)$ plane for the cubic $AFM^1_{[010]}$ structure, centered at the different point defects. The results for $V_{N}$ show strong spin polarization associated with the extra charge in the neighboring Cr atoms, product of the broken bonds, and clearly dominated by the $d_{z^2}$ orbital. It is also interesting to see the axial symmetry of the additional spin on the neighboring Cr atoms. The net magnetic moment of the $V_N$ supercell is found to be 0.69$\mu_B$, reflecting the spin polarization of the orbitals seen in Fig. \[fig:PDOS1\]. The chromium vacancy in this model shows also spin accumulation at the neighboring Cr atoms, having amplitudes in both spin directions with complex spatial patterns in nodules pointing towards the point defect. Notice also a small remnant spin density on the vacancy site, all contributing to the net supercell magnetic moment of 1.66$\mu_B$ (nearly half of the Cr magnetic moment in the pristine crystal of 2.90$\mu_B$). The $A_N$ defect is also accompanied by a slight spin accumulation at the defect location, as well as clear dipolar patterns in the neighboring Cr sites. Finally, the interstitial defect in this model, $I_N$, shows a strong dipolar spin accumulation at the defect. Notice also the alternating slight polarization that extends throughout the supercell, an indication of the strong local distortion that has not fully healed over the supercell volume (and yet has zero net magnetic moment).
For the orthorhombic cell, $AFM^2_{[110]}$, we show the corresponding spin polarization on the $(001)$ plane in Fig. \[fig:spinpol2\]. In general, one finds more structured spin distributions, reflecting the lower symmetry of this structure. The $V_N$ defect shows spin concentrations in the nearest Cr atoms with amplitudes that are rather asymmetric among different Cr neighbors, contributing to a small magnetic moment of 0.3$\mu_B$ in the supercell. In the $V_{Cr}$ defect we see a high spin concentration at the defect site, as well as in the neighboring Cr atoms. For $V_{Cr}$, the net magnetic moment of the supercell is 2.98$\mu_B$, nearly the moment of the Cr sites in the pristine lattice. The antisites, with strong structural deformations, produce spin polarization not only slightly at the $A_N$ defect site, but strongly in the neighboring Cr sites for $A_{Cr}$.
We should comment that although we are analyzing point defects in the bulk, similar charge and spin polarization would likely be found near crystal surfaces. As such, some of these spatial characteristics and spin polarization could be identified by scanning probes, such as a spin polarized scanning tunneling microscope. Analysis of the defect symmetries and spin polarizations they induce would help identify the appropriate atomic AFM model in a specific sample.
![(Color online) Two dimensional spin polarization map on $(001)$ plane for point defects in the $AFM^2_{[110]}$ model. Red and blue represent extremal spin projections (up and down respectively), as per color bars. Notice different scale for $V_N$. $x$, $y$ and spin projections given in atomic units. Images in the background show atom positions of fully relaxed structure. Blue/gray spheres represent N/Cr atoms, respectively. []{data-label="fig:spinpol2"}](afm2_2sq.pdf)
Conclusions
===========
We have analyzed the presence of native point defects on two microscopic AFM models of CrN, widely believed to describe the low temperature phase of the crystal. The formation energies, including corrections to avoid supercell artifacts, have been evaluated for the different defects. We have found, as suspected from general considerations, that the N vacancy defects are indeed those with the lowest formation energies, and therefore likely to appear in crystals in either of the AFM models. We further determine that only the N vacancies in the cubic AFM model should exhibit charged states, showing two possible transitions (to +1 and $-1$ charge states) within a reasonable doping range. We have found that such vacancies produce orbital restructuring near the Fermi level, providing significant self-doping which would affect the resistivity in real samples and may exhibit partial spin polarization of the impurity-created band. The formation energy values for neutral and charged species provide predictions that could be verified from thermometry experiments. Identification of these transitions would further assist in the identification of the AFM model present in a specific thin film sample. We trust this would contribute to clarify disagreements among several studies.
The changes in PDOS induced by defects have also been shown to carry interesting local structure in magnetic moment polarizations. The symmetries of these spin clouds reflect the underlying crystal symmetries and should be detectable with the use of spin polarized scanning tunneling spectroscopy. We trust our results would motivate further theoretical and experimental explorations of the microscopic details of point defects in this interesting system. We also hope that these results would contribute towards clarifying some of the controversial characteristics of CrN thin films.
\[sec:ack\] Acknowledgments {#secack-acknowledgments .unnumbered}
===========================
The work was supported by National Science Foundation grant DMR-1508325. Most calculations were performed at the Ohio Supercomputing Center under project PHS0219.
|
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"pile_set_name": "ArXiv"
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[**A** ]{}[**DETAILED RE**]{}[**CALCULATION OF THE SPECTRUM OF THE RADIATION EMITTED DURING GRAVITATIONAL COLLAPSE OF A SPHERICALLYSYMMETRIC STAR**]{}
[**A Calogeracos**]{}(\*)
[*Division of Theoretical Mechanics, Hellenic Air Force Academy TG1010, Dekelia Air Force Base, Greece*]{}
July 2003
(\*) [email protected]
[**Abstract**]{}
We address the question of radiation emission from a collapsing star. We consider the simple model of a spherical star consisting of pressure-free dust and we derive the emission spectrum via a systematic asymptotic expansion of the complete Bogolubov amplitude. Inconsistencies in derivations of the black body spectrum are pointed out.
Introduction
============
More than a quarter of a century has passed since Hawking’s remarkable suggestion that a star collapsing to a black hole gives rise to radiation emission at a steady rate characterized by the black body spectrum (Hawking 1975). That particle creation takes place is not in itself surprising. Let us take the case of a spherical star with radius initially larger than its Schwarzschild radius that eventually collapses contracting to a point (according to classical gravitation). If we consider a quantized scalar photon field the quantum spaces of the $in$ states (before the initiation of collapse) and of the $out$ states (after collapse is completed) are certainly different. Hence particle creation clearly takes place and is determined by the Bogolubov $\alpha (\omega ,\omega ^{\prime })$ and $\beta
(\omega ,\omega ^{\prime })$ amplitudes. The fact that the spectrum is that of a black body is indeed noticeable, and ties up with the somewhat earlier results on black hole thermodynamics. From the mathematical point of view the black body result is due to the special behaviour of the photon modes near the stellar surface just before the horizon is formed. Hawking in his derivation makes heavy use of the asymptotic (i.e. near the horizon) form of the modes. The singular behaviour of the modes in this regime has in turn given rise to various statements in the literature that are not strictly correct. For example there have been references to ”parts of $\alpha
(\omega ,\omega ^{\prime })$ and $\beta (\omega ,\omega ^{\prime })$ that relate to the steady-state regime at late times” ((DeWitt 1975), p. 327). However the Bogolubov amplitudes are global constructs and the distinction between early and late times does not make strict sense. Hawking too talks about particle production that depends on the details of the collapse and that such particles ”will disperse”, thus leaving only the thermal part at late times (Hawking (1975), p. 207). The point of view we are adopting in this paper is the one dictated by quantum-mechanical orthodoxy, namely that particle production is a global process ((Hawking 1975) p. 216) and that one should start with the full standard expression for $\beta (\omega ,\omega
^{\prime })$. Of course at some point in the mathematical handling of the amplitude $\beta (\omega ,\omega ^{\prime })$ the special role of the horizon will show up. The above remarks are certainly not meant to imply that one cannot calculate [*local*]{} quantities like $\left\langle T_{\mu
\nu }\right\rangle $; such quantities certainly behave very differently during the various phases of the collapse.
Shortly after Hawking’s work on black hole evaporation two classic papers (Fulling and Davies 1976) and (Davies and Fulling 1977) were published (the latter shall be quoted as DF in what follows). The authors demonstrated an illuminating analogy, physical as well as mathematical, between gravitational collapse and the seemingly rather different problem of a perfect mirror starting from rest and accelerating for an infinite time. The renormalized matrix element of the $T_{uu}$ component of the energy momentum tensor, defined as $$T_{uu}=\left( \partial _{u}\phi \right) ^{2}$$ is calculated in DF. The result is that [*asymptotically for* ]{}$%
t\rightarrow \infty $$$\left\langle T_{uu}\right\rangle \rightarrow \frac{\kappa ^{2}}{48\pi }
\label{bb1a}$$ (see e.g. equation (4.16) of (Birrell and Davies 1982) where a comprehensive review of particle production from both mirrors and black holes is presented; $\kappa $ is a constant characterizing the trajectory.) Equation (\[bb1a\]) shows that there is a constant energy flux at late times, analogous to the thermal energy flux found in (Hawking 1975). Davies and Fulling (1977) also calculated the Bogolubov amplitude following Hawking’s lines and arrived at the black body spectrum relevant to an accelerating mirror $$\left| \beta (\omega ,\omega ^{\prime })\right| ^{2}=\frac{1}{2\pi \omega
^{\prime }\kappa }\frac{1}{e^{2\pi \omega /\kappa }-1} \label{bb1c}$$
$$n(\omega )=\int_{0}^{\infty }d\omega ^{\prime }\left| \beta (\omega ,\omega
^{\prime })\right| ^{2} \label{bb2}$$
Certain technicalities in the latter calculation have been elucidated in a previous paper by the author (Calogeracos 2002, hereafter referred to as I). In the present paper we intend to show that similar points may be raised in connection to the standard black hole literature (see (Birrell and Davies 1982), chapter 8 and references therein). There are steps in the derivation of the black body spectrum that both obscure the mathematics and create confusion regarding the physics of the problem. Our findings, summarized in the concluding section, in no way do they diminish the astounding intuition shown by the early workers on the subject.
In section 2 we examine the simple model of a collapsing spherical star consisting of pressure-free dust and we review the fundamental features of the collapse. The problem is widely treated in the literature and some of the results are included so that the paper be self-contained. In section 3 we write down the photon modes and the full expression for the Bogolubov amplitude $\beta (\omega ,\omega ^{\prime })$. In section 4 we show that the asymptotic behaviour of the latter for large $\omega ^{\prime }$ is $$\beta (\omega ,\omega ^{\prime })\approx \left( \omega ^{\prime }\right) ^{-%
\frac{1}{2}}+O\left( \left( \omega ^{\prime }\right) ^{-N}\right) \left(
N>1\right) \label{bb2a}$$
The $1/\omega ^{\prime }$ in (\[bb1c\]) leads to a logarithmic divergence in (\[bb2\]) and this signals the production of particles at a finite rate for an infinite time. One often refers to the ultraviolet divergence by saying that large $\omega ^{\prime }$ frequencies dominate. Our analysis emphasizes two points, stressed also in I: (a) the thermal result depends crucially on the behaviour of the photon modes near the horizon, (b) for a consistent derivation one must consider the [*whole* ]{}collapse and [*not* ]{}just the late phase. The truth of statement (a) is usually taken as common knowledge. However the importance of statement (b) is often not appreciated. Comparison of our approach with the somewhat more conventional one is presented in Appendix A.
The ultraviolet divergence previously mentioned is widely taken to signify that one cannot fully resolve the problem within the context of classical gravitation. It is well known that the constants we have at our disposal are $c=3\times 10^{10}$ cm$\cdot $s$^{-1}$, $\not{h}=1.05\times 10^{-27}$ g$%
\cdot $cm$^{2}\cdot $s$^{-1}$, $G=6.67\times 10^{-8}$ cm$^{3}\cdot $s$%
^{-2}\cdot $g$^{-1\text{ }}$and that we can form three quantities with dimensions of mass, length, and time respectively: $M=\left( \not%
{h}c/G\right) ^{1/2}=2\times 10^{-5}$ g, $L=\left( \not{h}G/c^{3}\right)
^{1/2}=1.6\times 10^{-33}$ cm, $T=\left( \not{h}G/c^{5}\right)
^{1/2}=5\times 10^{-44}$ s. It is clear that when the radius of the contracting star becomes of order $L$ then quantum gravitational effects have to come into play. Such matters are not discussed here. Nor do we touch upon the profound problem of loss of information during collapse (see e.g. Preskill 1992, Helfer 2003).
Note: in what follows $\not{h}=G=c=1$.
A collapsing sphere of dust
===========================
We consider a collapsing spherical star consisting of pressure-free dust and follow the approach of (Weinberg 1972), chapter 11, sections 8 and 9. Since each dust particle falls freely the spacetime geometry inside the star is most appropriately described in a comoving frame. Let $M$ be the mass of the star. The metric reads $$ds^{2}=dt^{2}-R^{2}(t)\left[ \frac{dr^{2}}{1-kr^{2}}+r^{2}d\theta
^{2}+r^{2}\sin ^{2}\theta d\varphi ^{2}\right] \label{c1}$$
In the comoving frame each dust particle is labelled by a unique set of $(r,\theta ,\varphi )$ and the time $t$ stands for the proper time registered by the particle in question. The radius of the star is specified in the comoving frame by $r=a$ where $a$ is by definition constant. The quantities $k$ and $R(t)$ refer to details of the model and their meaning shall be explained shortly. The collapse is initiated at time $t=0$, and we normalize $R(t)$ by requiring $$R(0)=1 \label{co4}$$ Outside the star spacetime is described by the Schwarzschild metric $$ds^{2}=\left( 1-\frac{2M}{\bar{r}}\right) d\bar{t}^{2}-\left( 1-\frac{2M}{%
\bar{r}}\right) ^{-1}d\bar{r}^{2}-\bar{r}^{2}d\bar{\theta}^{2}-\bar{r}%
^{2}\sin ^{2}\bar{\theta}d\bar{\varphi}^{2} \label{c2}$$
The match of the two metrics at the surface implies (Weinberg 1972) $$\bar{r}=rR(t),\theta =\bar{\theta},\varphi =\bar{\varphi} \label{co3}$$
Relation (\[co4\]) together with the first of (\[co3\]) imply that the quantity $a$ stands for the radius of the star in Schwarzschild spacetime at the moment when collapse starts. Thus the radius of the star in Schwarzschild coordinates is given by $\bar{R}=aR$. The constant $k$ is proportional to the star’s density and is related to the mass via $$2M=ka^{3} \label{co5}$$
At times we shall consider the case of an initially very large star $a>>M$, which implies via (\[co5\]) $$ka^{2}<<1 \label{co31}$$ The relation between $\bar{t}$ and $t$ is a rather more intricate one and is treated in detail by Weinberg [*op. cit.* ]{}(see the remark after (\[co8\]) below). It has already been mentioned that since the pressure inside the star vanishes the motion of the surface corresponds to that of a free falling particle in the gravitational field of a body of mass $M$. The proper time $t$ registered by an observer located on the surface of the collapsing star and the quantity $\bar{R}$ are best given parametrically in terms of the cycloidal variable $\eta $ (Weinberg 1972): $$\bar{R}(\eta )=aR(\eta ),R(\eta )=\frac{1}{2}\left( 1+\cos \eta \right)
\label{co6}$$
$$t(\eta )=\frac{1}{2\sqrt{k}}\left( \eta +\sin \eta \right) \label{co7}$$
where $0\leq \eta \leq \pi $. Collapse starts at $\eta =0$ and is completed at $\eta =\pi $. Thus according to (\[co7\]) the total time of the collapse as measured by an observer comoving with the star’s surface is given by $\pi /\left( 2\sqrt{k}\right) $. The black hole is formed when $%
\bar{R}=2M$. According to (\[co6\]) this happens at a value $\eta _{0}$ of the parameter given by $$\frac{\eta _{0}}{2}=\cos ^{-1}a\sqrt{k} \label{co9}$$ The Schwarzschild time $\bar{t}$ for a particle on the stellar surface is given in terms of $\eta $ by the expression (see (Chandrasekhar 1983) chapter 3, section 19) : $$\bar{t}=2a\sqrt{1-ka^{2}}\left[ \frac{1}{2}\left( \eta +\sin \eta \right)
+ka^{2}\eta \right] +2M\ln \left[ \frac{\tan \frac{\eta _{0}}{2}+\tan \frac{%
\eta }{2}}{\tan \frac{\eta _{0}}{2}-\tan \frac{\eta }{2}}\right] \label{co8}$$
Relations (\[co7\]) and (\[co8\]) parametrically connect the time $t$ in the comoving frame to the Schwarzschild time $\bar{t}$ on the stellar surface. Note that as $\eta \rightarrow \eta _{0}$ from the left (i.e. before the black hole is formed) $t$ diverges. This reflects the well-known fact that collapse appears to the Schwarzschild observer to take an infinitely long time.
Radial null geodesics in Schwarzschild spacetime satisfy (see (\[c2\])) $$dt^{2}=\frac{1}{\left( 1-\frac{2M}{\bar{r}}\right) ^{2}}d\bar{r}^{2}
\label{c10}$$
This leads to the definition of the Regge-Wheeler radial coordinate $r^{*}$$$r^{*}=\bar{r}+2M\ln \left( \frac{\bar{r}}{2M}-1\right) \label{co11}$$
so that (\[c10\]) may be written as $d\bar{t}^{2}-dr^{*2}=0$, and to the introduction of the incoming and outgoing Eddington-Finkelstein coordinates $$v=\bar{t}+r^{*},u=\bar{t}-r^{*} \label{co12}$$
The quantities $v$ and $u$ label photon trajectories in Schwarzschild spacetime.
In order not to confuse the reader with the proliferation of radial variables let us clarify: $\bar{R}$ stands for the stellar radius in Schwarzschild coordinates, $R^{*}$ is the corresponding Regge-Wheeler coordinate, $R$ is the auxiliary function given by the second of (\[co6\]) and appearing in (\[c1\]), $\bar{r}$ is the radial Schwarzschild coordinate of an arbitrary point outside the star, $r^{*}$ the corresponding Regge-Wheeler coordinate, and $r$ the radial coordinate inside the star in the comoving frame.
A property that one must establish is that as the star approaches collapse $$\frac{\bar{R}}{2M}-1\simeq Ae^{-\bar{t}/2M} \label{co14}$$
where $A$ is a calculable constant. This expression is of course widely known. It agrees with the remark following (\[co8\]) that as far as the Schwarzschild observer is concerned the collapse takes an infinitely long time. A similar exponential appears in the calculation of the red shift as observed by the Schwarzschild observer; see (Weinberg 1972), p. 348 (the asymptotic form of the mirror trajectory given by DF, equation (2.3), has a form identical to that of (\[co14\])). To establish the validity of (\[co14\]) and also calculate $A$ one takes logarithms of both sides and uses (\[co6\]), (\[co8\]). The logarithmic divergences cancel and $A$ is expressed in terms of $\eta _{0}$ (see Appendix A). The quantity $A$ depends on the parameters $a$ and $M$ of the collapse and does not turn up in the expression for the spectrum. Note that relations (\[co12\]) define parametrically (via $\eta $) a function $$u=f_{st}(v) \label{co14b}$$
and its inverse $$v=p_{st}(u) \label{co14c}$$
which give the trajectory of the stellar surface in terms of $u,v$ coordinates.
In the limit $\eta \rightarrow \eta _{0}$ both $R^{*}$ and $\bar{t}$ are singular (as is obvious from (\[co8\]) and (\[co11\])), however their combination in the Eddington-Finkelstein coordinate $v$ is analytic. The cancellation of the logarithmic singularities is hardly surprising since $%
v_{0}$ must have a well-defined value labelling a null line; for details see Appendix A. For given values of $M$ and $a$ the value of $\eta _{0}$ is readily calculable via (\[co9\]) and then so is $v_{0}$ after some algebra (setting $\eta =\eta _{0}$ in (\[a3\])).Thus $v_{0}-v$ admits a Taylor expansion near $\eta _{0}$ and of course so does $2M-\bar{R}$ which is analytic for all $0<\eta <\pi $. Writing $$2M-\bar{R}\simeq C_{1}\left( \eta _{0}-\eta \right) ,v_{0}-v\simeq
C_{2}\left( \eta _{0}-\eta \right) \label{co33}$$ we can read the positive constants $C_{1},C_{2}$ off (\[a5\]), (\[a4\]). Then $$\bar{R}-2M\simeq C\left( v_{0}-v\right) ,C=C_{1}/C_{2} \label{co34}$$ Combining the above with (\[co14\]) we get $$v\simeq v_{0}-\frac{2MA}{C}e^{-\bar{t}/2M} \label{co35}$$ For $\eta \rightarrow \eta _{0}$ we can combine the two relations (\[co12\]) and write $$\bar{t}\simeq \frac{u+v_{0}}{2} \label{co18}$$ Then (\[co35\]) reads $$v\simeq v_{0}-Be^{-u/4M},B\equiv \frac{2MA}{C} \label{co36}$$
This defines $p_{st}(u)$ in (\[co14c\]).
In what follows we shall need the metric (\[c2\]) expressed in terms of Kruskal coordinates; see e.g. (Townsend 1997), (Misner, Thorne, Wheeler 1973). We introduce $${\it U}=-Ee^{-u/4M},{\it V}=\frac{e^{v/4M}}{E} \label{co50}$$
and we transform the metric to $$ds^{2}=\frac{32M^{3}}{\bar{r}}e^{-\bar{r}/2M}d{\it U}d{\it V}-\bar{r}%
^{2}d\Omega \label{co51}$$
where $r$ is given in terms of ${\it U,V}$ implicitly via $${\it UV}=\frac{\bar{r}-2M}{2M}e^{\bar{r}/2M} \label{co52}$$
At early times spacetime is taken to be essentially flat (cf remark preceding (\[co21\])) and the metric is simply expressed in terms of advanced and retarded Eddington-Finkelstein coordinates $$ds^{2}=dudv-\bar{r}^{2}d\Omega \label{co53}$$
We now turn to ray-tracing and consider light rays obeying the condition that they be reflected at the centre of the star $r=0$. The rationale for the boundary condition at $r=0$ is reviewed in the next section. An incident ray corresponding to a certain $v$ at early times upon reflection becomes an outgoing ray corresponding to a certain $u$ at late times, thus defining a function $u=f(v)$. The inverse function is given by $v=p(u)$. Note that the function $u=f(v)$ is by construction quite distinct from the function $%
u=f_{st}(v)$ (\[co14b\]) (the former involves reflection of the ray at the centre of the star whereas the latter at the surface). The importance of ray tracing lies in the fact that once we determine the function $f(v)$ (or its inverse) we can immediately construct the photon modes via (\[e3\]), (\[e5\]). Note also that these expressions involve the Eddington-Finkelstein coordinates rather than the Kruskal ones. It is convenient to exhibit collapse via a Kruskal diagram, where radial null lines are at $45^{0}$ on the plane of the paper. The line $l(1)$ represents a ray that crosses the star’s surface at point B, is reflected at the centre of the star at spacetime point C, and after reflection crosses the star’s surface at A (reflected ray labelled by $l^{\prime }(1)$). By definition ray $l(1)$ is the last incoming ray that manages to escape before the black hole is formed, and corresponds to an incoming Eddington-Finkelstein coordinate $%
v_{H}$. At A the star’s surface crosses the Schwarzschild radius, and the coordinates of A correspond to the value $\eta _{0}$ of the cycloidal parameter $\eta $ in (\[co6\]), (\[co7\]). The extension of the line CA is the future horizon and corresponds to the value ${\it U}=0$ of the Kruskal coordinate. Let $l^{\prime }(i)$ be an outgoing ray slightly preceding $l(1)$, corresponding to an incoming ray $l(i)$. We also trace the incoming light ray $m(1)$ which becomes $l(1)$ upon reflection on the star’s surface at A without entering the star (this is an auxiliary ray and the reflection mechanism is an entirely hypothetical one, not connected to partial reflection that may take place on the star’s surface). Similarly let $m(i)$ be an incoming ray slightly preceding $m(1)$. The problem is to relate the $u$ of the outgoing $l^{\prime }(i)$ to the $v$ of the incoming $%
l(i)$, thus determining the function $u=f(v).$ We review the argument in (Hawking 1975); see also (Townsend 1997), p. 125. Recall ((Townsend 1997), p. 29 for proof) that ${\it U}$ is an affine parameter for the null geodesic ${\it U}=$[*constant*]{}. (This means that ${\bf L\cdot D}L^{\mu }=0$ rather than ${\bf L\cdot D}L^{\mu }$ ${\bf \varpropto }L^{\mu }$.) Instead of ${\it %
U}=-e^{-u/4M}$ one may also use any parameter $\lambda =-Ee^{-u/4M}$, $E$ being some constant; we exploit this freedom and will comment on the value of $E$ later on. Let ${\bf T}$ be a null vector on ${\it U}=0$ parallel to the horizon and pointing along the radial spatial direction, and let ${\bf N}
$ be a null vector pointing to the future and along the radial spatial direction and satisfying ${\bf N\cdot T}=-1$. The above requirements are consistent (see (Townsend 1997), p. 109) if ${\bf N}$ is parallely transported along the geodesic defined by ${\bf T}$. Consider a neighbouring null geodesic labelled by ${\it U}=-\varepsilon $ where is small. This corresponds to constant $u$ according to (\[co50\]): $$\varepsilon =Ee^{-u/4M}\Rightarrow \ln \varepsilon =\ln E-\frac{u}{4M}
\label{co54}$$
The two geodesics are thus connected by the displacement vector $%
-\varepsilon {\bf N}$. We parallely transport the pair ${\bf N}$, ${\bf T}$ to the point where the ${\it U}=0$ geodesic cuts the trajectory of centre of the star. We then reflect ${\bf N}$ and ${\bf T}$ as in figure 2 and parallely transport the new pair to past infinity where the stellar radius satisfies (\[co21\]). Then spacetime becomes essentially flat, the metric is given by (\[co53\]) and the affine parameter is given by $v$ for a general geodesic $l(i)$ (figures 1, 2) and by $v_{H}$ for the special geodesic $l(1)$. We thus have $$\varepsilon =v_{H}-v \label{co55}$$
Comparing (\[co54\]) and (\[co55\]) we obtain $$v_{H}-v=Ee^{-u/4M}$$
or $$v\simeq v_{H}-Ee^{-u/4M} \label{co37}$$
(the $\simeq $ symbol above refers to the fact that this approximation is valid near the horizon). Note that the preceding argument involving ${\bf N}$, ${\bf T}$ can be equally well applied to the incoming rays $m(i),$ $m(1)$ which upon reflection on the surface of the star coincide with $l^{\prime }(i),l^{\prime }(1)$. In particular the affine distance between $l(1)$ and $l(i)$ equals the distance between $m(1)$ and $%
m(i)$. This implies that the quantity $E$ appearing in (\[co37\]) is equal to the $B$ appearing in (\[co36\]). (The identification of $E$ with $B$ does not affect the derivation of the black body spectrum and we shall keep $%
E$ in notation.) Note also that the quantity
$$d\equiv v_{0}-v_{H} \label{co13}$$
is calculable; see appendix A, the remark following (\[co56\]). Inverting (\[co37\]) we get $$u=f(v)\simeq -\frac{1}{4M}\ln \left( \frac{v_{H}-v}{E}\right) \label{co38}$$
We now turn to the early stage of the collapse (figure 3). We assume that the collapse is initiated at $\bar{t}=0$ (and that for $\bar{t}<0$ the star is held stable by some external means). Let us assume large initial stellar radius $$a>>M \label{co21}$$ Expanding (\[co6\]), (\[co7\]) for early times, i.e. small $%
\eta $, we obtain $$\bar{t}\simeq \frac{\eta }{\sqrt{k}},R\simeq 1-\frac{\eta ^{2}}{4}$$
Hence the radius as a function of time at the early stage is given by $$\bar{R}=a\left( 1-\frac{k\bar{t}^{2}}{4}\right) \label{co20}$$
Recalling the definition (\[co5\]) we observe that the star’s surface initially contracts according to non-relativistic kinematics with acceleration $M/a^{2}$ (in accordance with Newton’s law of gravity). Given the assumption (\[co21\]) the initial gravitational field is sufficiently weak so the light rays in the Schwarzschild geometry are straight lines. Let us consider a $\bar{t}-\bar{R}$ diagram ( $\bar{t}$, $\bar{R}$ being Schwarzschild coordinates; at early times when (\[co31\]) is satisfied there is little difference between radial and Regge-Wheeler coordinates) and take an incoming light ray $q$ which has $v=0$. The reflected ray $q^{\prime
}$ has $u=0$ and takes time $a$ to travel from the centre of the star to the surface. During this time the gravitational field is still weak. This may be seen by setting $t=a$ in (\[co20\]); then (\[co31\]) implies that by the time the ray emerges from the star’s surface $\bar{R}$ still is almost equal to $a$. To summarize the function $u=f(v)$ has the properties $$f(v)=v=0,initially \label{co23}$$ $$f(v)\simeq -4M\ln \left( \frac{v_{H}-v}{E}\right) ,v\rightarrow v_{H}
\label{co41}$$
Construction of the [*in*]{} and [*out*]{} states
=================================================
We follow (Birrell and Davies 1982), (Brout et al. 1995). A scalar mode corresponding to angular momentum $l$ and satisfying the Klein-Gordon equation is written in the form $$\psi _{l}=\left( \sqrt{4\pi }r\right) ^{-1}\phi _{l}(r)Y_{l}^{m}(\theta
,\varphi ) \label{co25}$$
where $$\left( \frac{\partial ^{2}}{\partial \bar{t}^{2}}-\frac{\partial ^{2}}{%
\partial r^{*2}}-V_{l}(r)\right) \phi _{l}=0 \label{co26}$$
with $$V_{l}(r)=\left( 1-\frac{2M}{r}\right) \left( \frac{2M}{r^{3}}+\frac{l(l+1)}{%
r^{2}}\right) \label{co27}$$
As far as conceptual purposes and technical details are concerned it suffices to restrict ourselves to $s$ waves (following the references in the beginning of this section). Note that according to (\[co27\]) there is a centrifugal barrier even for $s$ waves which does not affect either the high frequency modes or the modes that are present at the early stages of the collapse (when the gravitational field is weak). Again following the standard treatments of the problem we are going to neglect the barrier. For a discussion of its effect on the modes see (DeWitt 1975), section 5.2. Thus (\[co26\]) becomes $$\left( \frac{\partial ^{2}}{\partial \bar{t}^{2}}-\frac{\partial ^{2}}{%
\partial r^{*2}}\right) \phi =0 \label{co29}$$ (in what follows the index $0$ in $\phi $ is suppressed). Since $%
\psi _{0}(r=0)$ must be finite it follows from (\[co25\]) that $$\phi (r=0)=0 \label{co28}$$
Relations (\[co29\]) and (\[co28\]) define the problem.
Relation (\[co29\]) can be written in the form $$%TCIMACRO{\dfrac{\partial ^{2}\phi }{\partial u\partial v} }
%BeginExpansion
{\displaystyle {\partial ^{2}\phi \over \partial u\partial v}}%
%EndExpansion
=0 \label{e1}$$
Hence any function that depends only on $u$ or $v$ (or the sum of two such functions) is a solution of (\[e1\]). We can now make contact with the accelerating mirror problem (DF,I). One set of modes satisfying (\[e1\]) and the boundary condition (\[co28\]) is given by $$\varphi _{\omega }(u,v)=%
%TCIMACRO{\dfrac{i}{2\sqrt{\pi \omega }} }
%BeginExpansion
{\displaystyle {i \over 2\sqrt{\pi \omega }}}%
%EndExpansion
\left( \exp (-i\omega v)-\exp \left( -i\omega p(u)\right) \right) \label{e3}$$
Another set of modes satisfying the boundary condition is immediately obtained from (\[e3\]) $$\bar{\varphi}_{\omega }(u,v)=%
%TCIMACRO{\dfrac{i}{2\sqrt{\pi \omega }} }
%BeginExpansion
{\displaystyle {i \over 2\sqrt{\pi \omega }}}%
%EndExpansion
\left( \exp \left( -i\omega f(v)\right) -\exp \left( -i\omega u\right)
\right) \label{e5}$$
The modes $\varphi _{\omega }(u,v)$ of (\[e3\]) describe sinusoidal waves incident from $r^{*}=\infty $ as it is clear from the sign of the exponential in the first term; the second term represents the outgoing part which has a rather complicated behaviour depending on the motion of the mirror. These modes constitute the [*in* ]{}space and should obviously be absent before collapse, $$a(\omega ^{\prime })\left| 0in\right\rangle =0 \label{bb4}$$ Similarly the modes $\bar{\varphi}_{\omega }(u,v)$ describe sinusoidal outgoing waves travelling towards $r^{*}=\infty $ (emitted by the star) as can be seen from the exponential of the second term. Correspondingly the first term is complicated.
Recall that apart from the modes $\bar{\varphi}_{\omega }(u,v)$ we should include modes $\bar{q}_{\omega }(u,v)$ that contain no outgoing component; instead they are confined inside the black hole (the letter $q$ complies with (Hawking 1975, 1976)). The modes $\bar{q}_{\omega }(u,v)$ are undetectable by an outside observer, but they are needed to make the set $%
\bar{\varphi}_{\omega },\bar{q}_{\omega }$ complete. Correlations between the two sets is part of the information problem alluded to in the Introduction.
The $\bar{\varphi}_{\omega }(u,v)$ modes should be absent from the [*out* ]{}vacuum $$\bar{a}(\omega )\left| 0out\right\rangle =0 \label{bb6}$$ The state $\left| 0out\right\rangle $ corresponds to the state where there are no outgoing particles detectable by an outside observer. The two representations are connected by the Bogolubov transformation $$\bar{a}\left( \omega \right) =\int_{0}^{\infty }d\omega \left( \alpha
(\omega ,\omega ^{\prime })a(\omega ^{\prime })+\beta ^{*}(\omega ,\omega
^{\prime })a^{\dagger }(\omega ^{\prime })\right) \label{e011}$$
Using (\[e011\]) and its hermitean conjugate we may immediately verify that the expectation value of the number of excitations of the mode $%
\varphi _{\omega }$ in the $\left| 0in\right\rangle $ vacuum $$\left\langle 0in\right| N\left( \omega \right) \left| 0in\right\rangle
=\int_{0}^{\infty }d\omega ^{\prime }\left| \beta (\omega ,\omega ^{\prime
})\right| ^{2} \label{e0011}$$
The matrix element $\beta (\omega ,\omega ^{\prime })$ is given by (21) of I (we use $z$ for either $r$ or $r^{*}$ as the case may be to ease up on notation and make contact with I) $$\beta (\omega ,\omega ^{\prime })=-i\int dz\varphi _{\omega ^{\prime }}(z,0)%
%TCIMACRO{\dfrac{\partial }{\partial t} }
%BeginExpansion
{\displaystyle {\partial \over \partial t}}%
%EndExpansion
\bar{\varphi}_{\omega }(z,0)+i\int dz\left(
%TCIMACRO{\dfrac{\partial }{\partial t} }
%BeginExpansion
{\displaystyle {\partial \over \partial t}}%
%EndExpansion
\varphi _{\omega ^{\prime }}(z,0)\right) \bar{\varphi}_{\omega }(z,0)
\label{e11}$$
The integration in (\[e11\]) can be over any spacelike hypersurface. Since collapse starts at $t=0$ the choice $t=0$ for the hypersurface is convenient. The $in$ modes evaluated at $t=0$ are given by the simple expression (\[e3\]) (i.e. $p(u)=u)$$$\varphi _{\omega }(u,v)=%
%TCIMACRO{\dfrac{i}{2\sqrt{\pi \omega }} }
%BeginExpansion
{\displaystyle {i \over 2\sqrt{\pi \omega }}}%
%EndExpansion
\left( \exp (-i\omega z)-\exp \left( i\omega z\right) \right) \theta \left(
z\right) \label{e99}$$ where the presence of $\theta \left( z\right) $ emphasizes the fact that $z$ is a radial coordinate. The $\bar{\varphi}$ modes are given by (\[e5\]) with $f$ depending on the history of the collapse. Relation (\[e11\]) is rewritten in the form (the endpoints of integration shall be stated presently) $$\begin{aligned}
\beta (\omega ,\omega ^{\prime }) &=&-i\int dz%
%TCIMACRO{\dfrac{i}{2\sqrt{\pi \omega ^{\prime }}} }
%BeginExpansion
{\displaystyle {i \over 2\sqrt{\pi \omega ^{\prime }}}}%
%EndExpansion
\left\{ e^{-i\omega ^{\prime }z}-e^{i\omega ^{\prime }z}\right\} \theta
\left( z\right) \frac{\omega }{2\sqrt{\pi \omega }}\left\{ f^{\prime
}e^{-i\omega f}-e^{i\omega z}\right\} + \label{e100} \\
&&+i\int dz%
%TCIMACRO{\dfrac{\omega ^{\prime }}{2\sqrt{\pi \omega ^{\prime }}} }
%BeginExpansion
{\displaystyle {\omega ^{\prime } \over 2\sqrt{\pi \omega ^{\prime }}}}%
%EndExpansion
\left\{ e^{-i\omega ^{\prime }z}-e^{i\omega ^{\prime }z}\right\} \theta
\left( z\right) \frac{i}{2\sqrt{\pi \omega }}\left\{ e^{-i\omega
f}-e^{i\omega z}\right\} \nonumber\end{aligned}$$
The above expression may be rearranged in the form $$\beta (\omega ,\omega ^{\prime })=%
%TCIMACRO{\dfrac{1}{4\pi \sqrt{\omega \omega ^{\prime }}} }
%BeginExpansion
{\displaystyle {1 \over 4\pi \sqrt{\omega \omega ^{\prime }}}}%
%EndExpansion
\int_{0}^{v_{H}}dz\left\{ e^{i\omega ^{\prime }z}-e^{-i\omega ^{\prime
}z}\right\} \theta \left( z\right) \left\{ \omega ^{\prime }e^{-i\omega
f}-\omega f^{\prime }e^{-i\omega f}\right\} +
%TCIMACRO{
%\dfrac{\left( \omega -\omega ^{\prime }\right) }{4\pi \sqrt{\omega \omega ^{\prime }}} }
%BeginExpansion
{\displaystyle {\left( \omega -\omega ^{\prime }\right) \over 4\pi \sqrt{\omega \omega ^{\prime }}}}%
%EndExpansion
\int_{0}^{\infty }dz\left\{ e^{i\omega ^{\prime }z}-e^{-i\omega ^{\prime
}z}\right\} \theta \left( z\right) e^{i\omega z} \label{bb9}$$
where the limits of integration are displayed (we took inro account the fact that the argument of $f$ runs up to $v_{H}$). The rays with $v<0$ do not affect the amplitude due to the presence of the $\theta $ function. The first and second integral in the above relation will be denoted by $\beta _{1}(\omega ,\omega ^{\prime })$ and $\beta _{2}(\omega
,\omega ^{\prime })$ respectively. Note that the second integration is of kinematic origin and totally independent of the collapse. We also quote the expression for the $\alpha (\omega ,\omega ^{\prime })$ amplitude $$\alpha (\omega ,\omega ^{\prime })=-i\int_{0}^{\infty }dz\varphi _{\omega
^{\prime }}(z,0)%
%TCIMACRO{\dfrac{\partial }{\partial t}}
%BeginExpansion
{\displaystyle {\partial \over \partial t}}%
%EndExpansion
\bar{\varphi}_{\omega }^{*}(z,0)+i\int_{0}^{\infty }dz\left(
%TCIMACRO{\dfrac{\partial }{\partial t}}
%BeginExpansion
{\displaystyle {\partial \over \partial t}}%
%EndExpansion
\varphi _{\omega ^{\prime }}(z,0)\right) \bar{\varphi}_{\omega }^{*}(z,0)
\label{91}$$
Observe the unitarity condition $$\int_{0}^{\infty }d\widetilde{\omega }\left( \alpha \left( \omega _{1},%
\widetilde{\omega }\right) \alpha ^{*}\left( \omega _{2},\widetilde{\omega }%
\right) -\beta \left( \omega _{1},\widetilde{\omega }\right) \beta
^{*}\left( \omega _{2},\widetilde{\omega }\right) \right) =\delta \left(
\omega _{1}-\omega _{2}\right) \label{92}$$
which is a consequence of the fact that the set of [*in* ]{}states is complete.
Recall from I that we can introduce quantities $A(\omega ,\omega ^{\prime
}),B(\omega ,\omega ^{\prime })$ that are analytic functions of the frequencies via $$\alpha (\omega ,\omega ^{\prime })=\frac{A(\omega ,\omega ^{\prime })}{\sqrt{%
\omega \omega ^{\prime }}},\beta (\omega ,\omega ^{\prime })=\frac{B(\omega
,\omega ^{\prime })}{\sqrt{\omega \omega ^{\prime }}} \label{e106}$$
The quantity $B(\omega ,\omega ^{\prime })$ is read off (\[e100\]) (and $A(\omega ,\omega ^{\prime })$ from the corresponding expression for $%
\alpha (\omega ,\omega ^{\prime })$). From the definitions of the Bogolubov coefficients, the explicit form (\[e100\]) of the overlap integral and expressions (\[e3\]) and (\[e5\]) for the field modes one can deduce that $$B^{*}(\omega ,\omega ^{\prime })=A(-\omega ,\omega ^{\prime }),A^{*}(\omega
,\omega ^{\prime })=B(-\omega ,\omega ^{\prime }) \label{e107}$$
The above relations allow the calculation of $\alpha (\omega
,\omega ^{\prime })$ once $\beta (\omega ,\omega ^{\prime })$ is determined.
Calculation of the Bogolubov amplitudes
=======================================
The strategy we adopt in handling (\[bb9\]) is as follows. The first integral will be evaluated via an asymptotic expansion in negative powers of $\omega ^{\prime }$, which will in fact show that the $\omega ^{\prime }$ integration in (\[bb2\]) is logarithmically divergent. We start with the second integral in (\[bb9\]): $$\beta _{2}(\omega ,\omega ^{\prime })=
%TCIMACRO{
%\dfrac{\left( \omega -\omega ^{\prime }\right) }{4\pi \sqrt{\omega \omega ^{\prime }}} }
%BeginExpansion
{\displaystyle {\left( \omega -\omega ^{\prime }\right) \over 4\pi \sqrt{\omega \omega ^{\prime }}}}%
%EndExpansion
\left\{ \int_{0}^{\infty }dze^{i\left( \omega +\omega ^{\prime }\right)
z}-\int_{0}^{\infty }dze^{i\left( \omega -\omega ^{\prime }\right) z}\right\}
\label{co80}$$ The integrals in (\[co80\]) are readily evaluated in terms of the function $\zeta $ and its complex conjugate $\zeta ^{*}$(see e.g. Heitler (1954), pages 66-71): $$\zeta (x)\equiv -i\int_{0}^{\infty }e^{i\kappa x}d\kappa =P\frac{1}{x}-i\pi
\delta (x) \label{delta}$$
Since we are interested in the asymptotic limit $\omega ^{\prime
}\rightarrow \infty $ the argument of the $\delta $ function in (\[delta\]) never vanishes so it is only the first term in (\[delta\]) that is operative as far as the calculation of the $\beta _{2}(\omega ,\omega
^{\prime })$ goes. (The $\delta $ proportional term is relevant in the calculation of the $\alpha (\omega ,\omega ^{\prime })$ amplitude via relations (\[e106\]), (\[e107\]).) Thus asymptotically in the said limit $$\beta _{2}(\omega ,\omega ^{\prime })\simeq \frac{1}{2\pi i\sqrt{\omega
\omega ^{\prime }}} \label{bb10}$$
We turn to the first integral $\beta _{1}(\omega ,\omega ^{\prime })$ in (\[bb9\]). Rather than splitting it to four integrals we perform an integration by parts to get (\[bb9\]) in the form $$\beta _{1}(\omega ,\omega ^{\prime })=-\frac{1}{2\pi }\sqrt{\frac{\omega
^{\prime }}{\omega }}\int_{0}^{v_{H}}dze^{-i\omega f(z)-i\omega ^{\prime }z}+%
\frac{1}{2\pi }\frac{1}{\sqrt{\omega \omega ^{\prime }}}\sin \left( \omega
^{\prime }v_{H}\right) e^{-i\omega f\left( v_{H}\right) } \label{bb22}$$ This is the same integration by parts that was used to obtain (35) of I and also the one that is used in DF to go from their (2.10a) to (2.10b). The lower limit contribution of the integration by parts vanishes (being proportional to $\sin \omega ^{\prime }z$), quite irrespectively of the value of $f(0)$. The second term in (\[bb22\]) (originating from the upper limit) oscillates rapidly since the exponent tends to infinity. Thus the term tends distributionally to zero it may be neglected as in DF (also it is one power of $\omega ^{\prime }$ down compared to the first term). So asymptotically in $\omega ^{\prime }$ we are entitled to write $$\beta _{1}(\omega ,\omega ^{\prime })\simeq -\frac{1}{2\pi }\sqrt{\frac{%
\omega ^{\prime }}{\omega }}\int_{0}^{v_{H}}dze^{-i\omega f(z)-i\omega
^{\prime }z} \label{bb23}$$ To bring the singularity in the integral to zero we make the change of variable
$$z=v_{H}-x \label{e22a}$$
and rewrite $\beta _{1}(\omega ,\omega ^{\prime })$ in the form $$\beta _{1}(\omega ,\omega ^{\prime })=-\frac{e^{-i\omega v_{H}}}{2\pi }\sqrt{%
\frac{\omega ^{\prime }}{\omega }}\int_{0}^{v_{H}}dxe^{-i\omega
g(x)}e^{i\omega ^{\prime }x} \label{bb24}$$
where the function $g(x)\equiv f\left(
v_{H}-z\right) $ is defined in the range $0<x<v_{H}$ and has the properties that follow from (\[co23\]), (\[co41\]): $$g(x)\simeq -4M\ln \left( \frac{x}{E}\right) ,x\rightarrow 0 \label{co42}$$ $$g(v_{H})=f(0) \label{co43}$$ We isolate the integral
$$I\equiv \int_{0}^{v_{H}}dxe^{-i\omega g(x)}e^{i\omega ^{\prime }x}
\label{e24b}$$
To obtain the asymptotic behaviour of (\[e24b\]) for $\omega ^{\prime }$ large we adopt the standard technique of deforming the integration path to a contour in the complex plane; see (Bender and Orszag 1978), chapter 6; (Ablowitz and Fokas 1997), chapter 6; (Morse and Feshbach 1953), p. 610 where a very similar contour is used in the study of the asymptotic expansion of the confluent hypergeometric. The deformed contour runs from 0 up the imaginary axis till $iT$ (we eventually take $T\rightarrow \infty $), then parallel to the real axis from $iT$ to $iT+v_{H}$, and then down again parallel to the imaginary axis from $iT+v_{H}$ to $v_{H}$. The contribution of the segment parallel to the real axis vanishes exponentially in the limit $T\rightarrow \infty $. We thus get $$I=i\int_{0}^{\infty }dse^{-\omega ^{\prime }s}e^{-i\omega
g(is)}-i\int_{0}^{\infty }dse^{i\omega ^{\prime }\left( v_{H}+is\right)
}e^{-i\omega g(v_{H}+is)} \label{co44}$$ In both integrations the dominant contribution comes from the region where $%
s\simeq 0$. The second integral (including the minus sign in front) takes the form using (\[co43\]) $$-ie^{i\omega ^{\prime }v_{H}}e^{-i\omega f(0)}\int_{0}^{\infty }dse^{-\omega
^{\prime }s}=-i\frac{e^{i\omega ^{\prime }v_{H}}e^{-i\omega f(0)}}{\omega
^{\prime }} \label{co45}$$
In the first integral we use the asymptotic form (\[co42\]) valid for small $x$ and write $$\exp \left( -i\omega g\left( is\right) \right) =\exp \left( i4M\omega \ln
\left( \frac{is}{E}\right) \right) =\left( \frac{is}{E}\right) ^{i4M\omega
}=\exp \left( -\frac{\pi }{2}4M\omega \right) \left( \frac{s}{E}\right)
^{i4M\omega }$$
where we took the branch cut of the function $x^{i\omega }$ to run from zero along the negative $x$ axis, wrote $x^{i\omega }=\exp \left(
i\omega \left( \ln x+i2N\pi \right) \right) $ and chose the branch $N=0$. Thus $$i\int_{0}^{\infty }dse^{-\omega ^{\prime }s}e^{-i\omega g(is)}=i\exp \left( -%
\frac{\pi }{2}4M\omega \right) E^{-i4M\omega }\int_{0}^{\infty }dse^{-\omega
^{\prime }s}\left( s\right) ^{i4M\omega }=i\exp \left( -\frac{\pi }{2}%
4M\omega \right) E^{-i4M\omega }\frac{\Gamma \left( 1+i4M\omega \right) }{%
\left( \omega ^{\prime }\right) ^{1+i4M\omega }} \label{co46}$$
We substitute (\[co45\]) and (\[co46\]) in (\[co44\]) and then in (\[bb24\]) to get
$$\beta _{1}(\omega ,\omega ^{\prime })=-i\frac{e^{-i\omega v_{H}}}{2\pi \sqrt{%
\omega \omega ^{\prime }}}\exp \left( -\frac{\pi }{2}4M\omega \right)
E^{-i4M\omega }\frac{\Gamma \left( 1+i4M\omega \right) }{\left( \omega
^{\prime }\right) ^{i4M\omega }}-\frac{e^{-i\omega f(0)}}{i2\pi \sqrt{\omega
\omega ^{\prime }}} \label{co47}$$
Collecting (\[bb10\]) and (\[co47\]) we get $$\beta \left( \omega ,\omega ^{\prime }\right) =-i\frac{e^{-i\omega v_{H}}}{%
2\pi \sqrt{\omega \omega ^{\prime }}}\exp \left( -\frac{\pi }{2}4M\omega
\right) E^{-i4M\omega }\frac{\Gamma \left( 1+i4M\omega \right) }{\left(
\omega ^{\prime }\right) ^{i4M\omega }}-\frac{e^{-i\omega f(0)}}{i2\pi \sqrt{%
\omega \omega ^{\prime }}}+\frac{1}{2\pi i\sqrt{\omega \omega ^{\prime }}}
\label{co48}$$
The day is saved by (\[co23\]) which causes an exact cancellation of the last two terms. The first term on its own immediately leads to the black body spectrum. Taking its modulus, squaring, and using the property $$\left| \Gamma \left( 1+iy\right) \right| ^{2}=\pi y/\sinh \left( \pi
y\right)$$
we get $$\left| \beta \left( \omega ,\omega ^{\prime }\right) \right| ^{2}=\frac{4M}{%
2\pi \omega ^{\prime }}\frac{1}{e^{8\pi \omega M}-1}$$
Conclusion
==========
The objective of the paper was to prove that the Bogolubov amplitude $\beta
(\omega ,\omega ^{\prime })$ has the asymptotic form (\[bb2a\]) and that the radiation emitted has the spectrum of a black body. Standard quantum mechanics dictate that we should specify the initial and final states before calculating the transition amplitude, and to this end we considered the photon [*in* ]{}states before the collapse and[* *]{}the [*out* ]{}states after collapse has taken place (the final state does not have a $\bar{q}%
_{\omega }(u,v)$ component; cf the remark following (\[bb4\])). We consider the case where the gravitational field is initially weak so that we may use the simple modes (\[e99\]) as [*in* ]{}modes; this is standard practice and leads to expression (\[bb24\]) for the amplitude. Unfortunately the term (\[co80\]) and the consequent (\[bb10\]) are missing from the standard treatments. Term (\[co80\]) does not depend on the kinematics of the collapse and is certainly there in order to give a diagonal [*S* ]{}matrix in the trivial case where collapse is never initiated and nothing is produced. A correct treatment of the large $\omega
^{\prime }$ asymptotics of the amplitude (\[bb24\]) yields one contribution that leads to the black body spectrum and a second contribution (which again is missed in the standard treatments; see Appendix A) that precisely cancels (\[bb10\]). However this second contribution [*does* ]{}depend on the collapse and the exact cancellation takes place only in the case of an initially weak gravitational field. In the light of the above remarks the black body result is indeed idependent of the details of the collapse (as often asserted) but does depend on the assumption of an initially weak gravitational field. The question as to what would happen in the case of an initial photon state in a gravitational background corresponding to an advanced state of collapse is not answered either here or in the standard treatments of Hawking radiation. The question may be of academic interest in the context of gravity, but it may be relevant in other cases where the analog of Hawking radiation is expected to occur.
The derivation of the black body spectrum presented in this note is based on the calculation of the Bogolubov $\beta \left( \omega ,\omega ^{\prime
}\right) $ amplitude. As emphasized in the Introduction this quantity is by definition time-independent, and thus the question as to where and when the photons are produced simply does not arise. It is certainly true that were it not for the singularity on the horizon the thermal spectrum would not arise. There are of course arguments based on calculation of [*local* ]{}field quantities to the effect that the regime in question is the important one; or even simpler classical arguments related to the red shift during collapse ((Weinberg 1972), p. 347). However such statements may be misleading in connection to the quantum mechanical calculation of global ([*time independent)* ]{}quantities. Similarly attempts to distinguish between ”transient” and ”steady state” radiation at the level of the $%
\alpha $ and $\beta $ amplitudes are bound to fail; the emphasis in the literature on the behaviour of the amplitude near the horizon has unfortunately led to such statements. One of the main conclusions of this note is that the correct derivation of the thermal result requires the consideration of the function $f(v)$ throughout its range and not just of its asymptotic part.
[**Acknowledgment**]{}
The author is greatly indebted to Professor S A Fulling for correspondence.
[**Appendix A: Comparison with previous derivations**]{}
The black body result is often obtained via a sequence of somewhat peculiar mathematical steps. One often starts (see e.g. (Birrell and Davies 1982) p. 108 and also references cited therein) with expression (\[bb24\]) thus wrongly disregarding (\[bb10\]). Following that step one unaccountably uses (\[co42\]) throughout the range of integration and not just asymptotically near the horizon where it is valid. One thus (incorrectly) gets dropping constant prefactors $$\beta _{1}(\omega ,\omega ^{\prime })\approx \int_{0}^{v_{H}}dx\left( \frac{x%
}{E}\right) ^{i\omega 4M}e^{i\omega ^{\prime }x} \label{a6}$$ That (\[a6\]) is a wrong approximation to the original integral may easily be seen by the fact that the use of (\[co42\]) has changed the behaviour of the integrand at $x=v_{H}$ (the non-singular end) and an asymptotic estimate similar to that given in section 4 would not lead to the cancellation that took place between the last two terms in (\[co48\]). One then proceeds to rewrite (\[a6\]) by rescaling $\omega ^{\prime
}x\rightarrow x$$$\beta _{1}(\omega ,\omega ^{\prime })\approx E^{-i\omega 4M}\left( \omega
^{\prime }\right) ^{-i\omega 4M-1}\int_{0}^{\omega ^{\prime
}v_{H}}dxx^{i\omega 4M}e^{ix} \label{680a}$$
Since one is chasing the ultraviolet divergence one simply sets $%
\omega ^{\prime }v_{H}=\infty $, changes variable $\rho =i\sigma $ and rotates in the complex plane to get (\[680a\]) in the form $$\beta _{1}(\omega ,\omega ^{\prime })\approx E^{-i\omega 4M}\left( \omega
^{\prime }\right) ^{-i\omega 4M-1}e^{-\frac{\pi }{2}\omega
4M}\int_{0}^{\infty }d\sigma e^{-\sigma }\sigma ^{i\omega } \label{e18}$$
Note that setting $\omega ^{\prime }v_{H}=\infty $ certainly does [*not* ]{}amount to a systematic expansion in $\left( \omega ^{\prime
}\right) ^{-1}$. The $\sigma $ integration yields $\Gamma (1+i\omega )$ and one thus obtains the form for the $\beta _{1}$ amplitude leading to the black body spectrum. On the other hand the step from (\[680a\]) to (\[e18\]) is again incorrect. Integral (\[a6\]) can be performed exactly in terms of the confluent hypergeometric function and the asymptotic estimate for large $\omega ^{\prime }$ may be examined afterwards. Indeed let us rescale the variable in (\[a6\]) $x\rightarrow x/v_{H}$ and rewrite $$\begin{aligned}
\beta _{1}(\omega ,\omega ^{\prime }) &\approx &v_{H}^{i\omega
4M+1}E^{-i\omega 4M}\int_{0}^{1}dxe^{i\omega ^{\prime }v_{H}x}x^{i\omega 4M}=
\label{bb18} \\
&=&v_{H}^{i\omega 4M+1}E^{-i\omega 4M}\frac{1}{i\omega +1}M\left( 1+i\omega
4M,2+i\omega 4M,i\omega ^{\prime }v_{H}\right) \nonumber\end{aligned}$$ where $M$ is the confluent hypergeometric. We can now examine the asymptotic limit of (\[bb18\]) for large $\omega ^{\prime }$. The asymptotic limit of the confluent $M(a,b,i\left| z\right| )$ for large values of $\left|
z\right| $ is given by item 13.5.1 of (Abramowitz and Stegun 1972) ($z\equiv
i\omega ^{\prime }v_{H}$). In the case $b=a+1$ some simplifications occur and we get $$M\left( 1+i\omega ,2+i\omega ,i\left| z\right| \right) \simeq -\left(
1+i\omega \right) e^{i\left| z\right| }\frac{i}{\left| z\right| }+i\Gamma
\left( 2+i\omega \right) \frac{e^{-\frac{\pi \omega }{2}}}{\left| z\right|
^{1+i\omega }} \label{bb20}$$ (other terms are down by higher powers of $1/\left| z\right| $). The second term of the above relation combined with the prefactors in (\[bb18\]) does feature the $\Gamma \left( 1+i\omega \right) e^{-\frac{\pi \omega }{2}}$ factor characteristic of the black body spectrum. The reason for the discrepancy between (\[e18\]) and (\[bb20\]) lies in the fact that one should first evaluate the integral in terms of the confluent and then take the $\omega ^{\prime }\rightarrow \infty $ limit rather than take the limit first. The rotation in the complex plane stumbles upon the Stokes phenomenon for the confluent (different limits for $\left| z\right| \rightarrow \infty $ depending on $\arg z)$). In short the black body spectrum (\[e18\]) is obtained by (a) incorrectly dropping (\[bb10\]), (b) incorrectly approximating (\[bb24\]) by (\[a6\]), (c) wrongly estimating the $\omega
^{\prime }$ asymptotics of the latter.
[**Appendix B: Technical remarks on the collapse of a sphere of dust**]{}
We write down the expression for the Regge-Wheeler coordinate of the stellar radius as given by (\[co6\]), (\[co11\]): $$R^{*}\left( \eta \right) =\frac{a}{2}\left( 1+\cos \eta \right) +2M\ln
\left[ \frac{a}{4M}\left( 1+\cos \eta \right) -1\right]$$
It takes a sequence of trivial trigonometric transformations to bring this to the form $$R^{*}\left( \eta \right) =\frac{a}{2}\left( 1+\cos \eta \right) +2M\ln
\left[ \frac{\sin \frac{\eta _{0}+\eta }{2}\sin \frac{\eta _{0}-\eta }{2}}{%
\cos ^{2}\frac{\eta _{0}}{2}}\right] \label{a1}$$
We similarly transform the argument of the logarithm in (\[co8\]): $$\bar{t}\left( \eta \right) =2a\sqrt{1-ka^{2}}\left[ \frac{1}{2}\left( \eta
+\sin \eta \right) +ka^{2}\eta \right] +2M\ln \left[ \frac{\sin \frac{\eta
_{0}+\eta }{2}}{\sin \frac{\eta _{0}-\eta }{2}}\right] \label{a2}$$
Both $R^{*}\left( \eta \right) $ and $\bar{t}\left( \eta \right) $ diverge as $\eta \rightarrow \eta _{0}$ (i.e. as the stellar radius approaches the Schwarzschild radius), but the combination $v=\bar{t}+R^{*}$ does not: $$v=2a\sqrt{1-ka^{2}}\left[ \frac{1}{2}\left( \eta +\sin \eta \right)
+ka^{2}\eta \right] +a\cos ^{2}\frac{\eta }{2}+2M\ln \left[ \frac{\sin
^{2}\left( \frac{\eta _{0}+\eta }{2}\right) }{\cos ^{2}\frac{\eta _{0}}{2}}%
\right] \label{a3}$$
The quantity $C_{2}$ in (\[co33\]) is the derivative of the above evaluated at $\eta _{0}$: $$C_{2}=2a\sqrt{1-ka^{2}}\left( \frac{1}{2}+\frac{1}{2}\cos \eta _{0}\right) -%
\frac{a}{2}\sin \eta _{0}+2M\cot \eta _{0} \label{a4}$$
The quantity $C_{1}$ in (\[co33\]) is trivially obtained by differentiating (\[co7\]): $$C_{1}=\frac{a}{2}\sin \eta _{0} \label{a5}$$
To calculate $A$ in (\[co14\]) we rewrite the latter in the form $$\frac{\bar{R}-2M}{2M}\simeq Ae^{-\bar{t}/2M} \label{a7}$$
We take logarithms of both sides of (\[a7\]) to rewrite it in the form $$\ln \left( \frac{\bar{R}-2M}{2M}\right) \simeq \ln A-\frac{\bar{t}}{2M}
\label{a9}$$ We now use (\[co5\]), (\[co9\]), (\[a9\]) and a sequence of trigonometric identities to obtain $$\ln \left[ \frac{2}{ka^{2}}\sin \eta _{0}\right] =\ln A-\frac{a}{M}\sqrt{%
1-ka^{2}}\left[ \frac{1}{2}\left( \eta _{0}+\sin \eta _{0}\right)
+ka^{2}\eta _{0}\right] \label{a8}$$
thus determining $A$. The crucial step in the calculation of $A$ lies in the fact that although both sides of (\[a9\]) diverge in the limit $\eta \rightarrow \eta _{0}$, there is precise cancellation of a term $$\ln \left( \sin \frac{\eta _{0}-\eta }{2}\right)$$
on each side thus leading to the finite result (\[a8\]).
For a given value of $\eta _{0}$ the corresponding value $t(\eta _{0})$ given by (\[co7\]). We take advantage of the fact that in the comoving frame the endpoints B and A of the ray BCA lie at $r=a$. Thus the unknown is the time $t_{H}$ in the comoving frame where ray $l1$ hits the star’s surface. The latter may be determined via the equation of a null geodesic inside the star obtained through (\[c1\]). The path of a light ray BCA propagating inside the star is given by $$\int_{t_{H}}^{t_{0}}\frac{dt}{R(t)}=2\int_{0}^{a}\frac{dr}{\sqrt{1-kr^{2}}}
\label{co15}$$
To evaluate the left hand side of (\[co15\]) we change variable according to (\[co7\]) $$\frac{d\tau }{d\eta }=\frac{1}{2\sqrt{k}}\left( 1+\cos \eta \right)$$ Using (\[co6\]) for $R(t)$ the left hand side of (\[co15\]) immediately yields $$\frac{1}{4\sqrt{k}}\left( \eta _{0}-\eta _{H}\right) \label{co30}$$ The right hand side of (\[co15\]) yields $$\frac{2}{\sqrt{k}}\arcsin \left( \sqrt{k}a\right)$$ which in the limit (\[co31\]) reduces to $2a$. Combining with (\[co30\]) we get $$\eta _{0}-\eta _{H}=8\sqrt{k}a \label{co56}$$ In the limit (\[co31\]) we get that $v_{H}$ lies very close to $v_{0}$.
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: |
The production of a stochastic background of relic gravitational waves is well known in various works in the literature, where, by using the so called adiabatically-amplified zero-point fluctuations process, it has been shown how the standard inflationary scenario for the early universe can in principle provide a distinctive spectrum of relic gravitational waves. In this paper, it is shown that, in general, f(R) theories of gravity produce a third massive polarization of gravitational waves and the primordial production of this polarization is analysed adapting the adiabatically-amplified zero-point fluctuations process at this case. In this way, previous results, where only particular cases of f(R) theories have been analysed, will be generalized.
The presence of the mass could also have important applications in cosmology, because the fact that gravitational waves can have mass could give a contribution to the dark matter of the Universe.
An upper bound for these relic gravitational waves, which arises from the WMAP constrains, is also released.
At the end of the paper, the potential detection of such massive gravitational waves using interferometers like Virgo and LIGO is discussed.
author:
- '**Christian Corda**'
title: '**Massive relic gravitational waves from f(R) theories of gravity: production and potential detection**'
---
Associazione Galileo Galilei, Via Pier Cironi 16 - 59100 PRATO, Italy
*E-mail address:*
Introduction
============
Recently, the data analysis of interferometric gravitational waves (GWs) detectors has been started (for the current status of GWs interferometers see [@key-1; @key-2; @key-3; @key-4; @key-5; @key-6; @key-7; @key-8]) and the scientific community aims in a first direct detection of GWs in next years.
Detectors for GWs will be important for a better knowledge of the Universe and also to confirm or to rule out the physical consistency of General Relativity or of any other theory of gravitation [@key-9; @key-10; @key-11; @key-12; @key-13; @key-14]. In fact, in the context of Extended Theories of Gravity, some differences between General Relativity and the others theories can be pointed out starting by the linearized theory of gravity [@key-9; @key-10; @key-12; @key-14].
In this picture, detectors for GWs are in principle sensitive also to a hypothetical *scalar* component of gravitational radiation, that appears in extended theories of gravity like scalar-tensor gravity and high order theories [@key-12; @key-15; @key-16; @key-17; @key-18; @key-19; @key-20; @key-21; @key-22], Brans-Dicke theory [@key-23] and string theory [@key-24].
A possible target of these experiments is the so called stochastic background of gravitational waves [@key-25; @key-26; @key-27; @key-28; @key-29; @key-30].
The production of the primordial part of this stochastic background (relic GWs) is well known in the literature starting by the works of [@key-25; @key-26] and [@key-27; @key-28], that, using the so called adiabatically-amplified zero-point fluctuations process, have shown in two different ways how the standard inflationary scenario for the early universe can in principle provide a distinctive spectrum of relic gravitational waves. In [@key-29; @key-30] the primordial production has been analysed for the scalar component admitted from scalar-tensor gravity.
In this paper, it is shown that, in general, f(R) theories of gravity produce a third massive polarization of gravitational waves and the primordial production of this polarization is analysed adapting the adiabatically-amplified zero-point fluctuations process at this case. In a recent paper [@key-21], such a process has been applied to the same class of theories, i.e. the $f(R)$ ones, which will be discussed in the present work. But, in [@key-21] a different point of view has been considered. In that case, by using a conform analysis, the authors discussed such a process in respect to the two standard polarizations which arises from standard General Relativity. In the present paper the analysis is focused to the third massive polarization. In this way, we generalize previous results where only particular cases of f(R) theories have been analysed (see [@key-40] for example).
Regarding f(R) theories, even if such theories have to be dismissed either because they contradict the current cosmology and solar system tests or generate ‘"ghosts‘" in what would be a quantum version of the finite range gravity[@key-41], a number of authors manage to avoid both of such problems [@key-41].
Concerning the presence of the graviton mass, it was considered by many authors since and Pauli (1939) [@key-42; @key-43]. Important contributions in this sense are the ones in [@key-44; @key-45; @key-46]. In these papers, it is emphasized that the presence of the mass modifies General Relativity on various scales, which are defined by the Compton wavelength of gravitons. This fact leads to dispersion of waves and Yukawa-like potentials in the linearized theory. On the other hand, it also leads to other modifications in General Relativity solutions in high-order (i.e. non-linear) regimes. The fact that gravitational waves can have mass could also have important applications in cosmology because such masses could give a contribution to the dark matter of the Universe.
An upper bound for these relic gravitational waves, which arises from the WMAP constrains, is also released. About this point, a different interesting treatment of the additional polarizations and impact on Cosmic Microwave Background has been recently discussed in the good paper [@key-47].
At the end of the paper the potential detection of such massive gravitational waves using interferometers like Virgo and LIGO is discussed.
f(R) theories of gravity
========================
Let us consider the action$$S=\int d^{4}x\sqrt{-g}f(R)+\mathcal{L}_{m},\label{eq: high order 1}$$
where $R$ is the Ricci curvature scalar.
Equation (\[eq: high order 1\]) represents the action of the so called $f(R)$ theories of gravity [@key-9; @key-10; @key-11; @key-19; @key-21; @key-29; @key-33; @key-34; @key-35] in respect to the well known canonical one of General Relativity (the Einstein - Hilbert action [@key-31; @key-32]) which is
$$S=\int d^{4}x\sqrt{-g}R+\mathcal{L}_{m}.\label{eq: EH}$$
The action (\[eq: high order 1\]) has been analysed in [@key-33; @key-34; @key-35] in cosmological contexts.
As we will interact with gravitational waves, i.e. the linearized theory in vacuum, $\mathcal{L}_{m}=0$ will be put and the pure curvature action $$S=\int d^{4}x\sqrt{-g}f(R)\label{eq: high order 12}$$ will be considered.
Field equations and linearized theory
=====================================
Following [@key-32] (note that in this paper we work with $8\pi G=1$, $c=1$ and $\hbar=1$), the variational principle
$$\delta\int d^{4}x\sqrt{-g}f(R)=0\label{eq: high order 2}$$
in a local Lorentz frame can be used, obtaining:
$$f'(R)R_{\mu\nu}-\frac{1}{2}f(R)g_{\mu\nu}-f'(R)_{;\mu;\nu}+g_{\mu\nu}\square f'(R)=0\label{eq: einstein-general}$$
which are the modified Einstein field equations. $f'(R)$ is the derivative of $f$ in respect to the Ricci scalar. Writing down, explicitly, the Einstein tensor, eqs. (\[eq: einstein-general\]) become$$G_{\mu\nu}=\frac{1}{f'(R)}\{\frac{1}{2}g_{\mu\nu}[f(R)-f'(R)R]+f'(R)_{;\mu;\nu}-g_{\mu\nu}\square f'(R)\}.\label{eq: einstein 2}$$
The trace of the field equations (\[eq: einstein 2\]) gives
$$3\square f'(R)+Rf'(R)-2f(R)=0,\label{eq: KG}$$
and, with the identifications [@key-40]
$$\begin{array}{ccccc}
\Phi\rightarrow f'(R) & & \textrm{and } & & \frac{dV}{d\Phi}\rightarrow\frac{2f(R)-Rf'(R)}{3}\end{array}\label{eq: identifica}$$
a Klein - Gordon equation for the effective $\Phi$ scalar field is obtained:
[@key-23]$$\square\Phi=\frac{dV}{d\Phi}.\label{eq: KG2}$$
To study gravitational waves, the linearized theory has to be analysed, with a little perturbation of the background, which is assumed given by a near Minkowskian background, i.e. a Minkowskian background plus $\Phi=\Phi_{0}$ (the Ricci scalar is assumed constant in the background) [@key-9; @key-19]. $\Phi_{0}$ is also assumed to be a steady minimum for the effective potential $V$, that called $V_{0}$. This assumption is vital for the further calculations [@key-41] and its physical justification arises from the fact that the effective $\Phi$ scalar field is a function of the Ricci curvature and here the linearized theory is developed, i.e. only weak perturbations near a *fixed* curvature are considered. Thus, such a minimum has to be *steady*. This is an analysis *totally equivalent* to the case of the linearization process for scalar-tensor gravity, see for example [@key-16]. In Section 4 of [@key-16] the authors claim that *“we linearize the equations near the background ($\eta_{\mu\nu}$, $\varphi_{0}$ ), where $\varphi_{0}$ is a minimum of $V$”*. The difference with the present analysis is that in [@key-16] the scalar field which is a minimum of the potential arises directly from the Brans-Dicke theory [@key-23], while in the present analysis an effective scalar field and an effective potential arise directly from spacetime curvature (the effective scalar field is the prime derivative $f'(R)$, see eq. (\[eq: identifica\])).
Thus, the potential presents a square (i.e. parabolic) trend, in function of the effective scalar field, near the minimum [@key-41], i.e.
$$V\simeq V_{0}+\frac{1}{2}\alpha\delta\Phi^{2}\Rightarrow\frac{dV}{d\Phi}\simeq m^{2}\delta\Phi,\label{eq: minimo}$$
and the constant $m$ has mass dimension.
Putting
$$\begin{array}{c}
g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}\\
\\\Phi=\Phi_{0}+\delta\Phi.\end{array}\label{eq: linearizza}$$
to first order in $h_{\mu\nu}$ and $\delta\Phi$, calling $\widetilde{R}_{\mu\nu\rho\sigma}$ , $\widetilde{R}_{\mu\nu}$ and $\widetilde{R}$ the linearized quantities which correspond to $R_{\mu\nu\rho\sigma}$ , $R_{\mu\nu}$ and $R$, the linearized field equations are obtained [@key-12; @key-19; @key-31]:
$$\begin{array}{c}
\widetilde{R}_{\mu\nu}-\frac{\widetilde{R}}{2}\eta_{\mu\nu}=(\partial_{\mu}\partial_{\nu}h_{f}-\eta_{\mu\nu}\square h_{f})\\
\\{}\square h_{f}=m^{2}h_{f},\end{array}\label{eq: linearizzate1}$$
where
$$h_{f}\equiv\frac{\delta\Phi}{\Phi_{0}}.\label{eq: definizione}$$
Then, from the second of eqs. (\[eq: linearizzate1\]), the mass can be defined like
$$m\equiv\sqrt{\frac{\square h_{f}}{h_{f}}}=\sqrt{\frac{\square\delta\Phi}{\delta\Phi}}.\label{eq: massa}$$
Thus, as the mass is generated by variation of a function of the Ricci scalar, in a certain sense, it is generated by variation of spacetime curvature. In this way, the theory is totally generalized for an arbitrary $f$ function of the Ricci scalar, improving the results in [@key-9; @key-19; @key-40], where only particular theories have been discussed..
$\widetilde{R}_{\mu\nu\rho\sigma}$ and eqs. (\[eq: linearizzate1\]) are invariants for gauge transformations [@key-9; @key-12; @key-19; @key-40]
$$\begin{array}{c}
h_{\mu\nu}\rightarrow h'_{\mu\nu}=h_{\mu\nu}-\partial_{(\mu}\epsilon_{\nu)}\\
\\\delta\Phi\rightarrow\delta\Phi'=\delta\Phi;\end{array}\label{eq: gauge}$$
then
$$\bar{h}_{\mu\nu}\equiv h_{\mu\nu}-\frac{h}{2}\eta_{\mu\nu}+\eta_{\mu\nu}h_{f}\label{eq: ridefiniz}$$
can be defined, and, considering the transform for the parameter $\epsilon^{\mu}$
$$\square\epsilon_{\nu}=\partial^{\mu}\bar{h}_{\mu\nu},\label{eq:lorentziana}$$
a gauge parallel to the Lorenz one of electromagnetic waves can be chosen:
$$\partial^{\mu}\bar{h}_{\mu\nu}=0.\label{eq: cond lorentz}$$
In this way,the field equations read like
$$\square\bar{h}_{\mu\nu}=0\label{eq: onda T}$$
$$\square h_{f}=m^{2}h_{f}\label{eq: onda S}$$
Solutions of eqs. (\[eq: onda T\]) and (\[eq: onda S\]) are plan waves [@key-12; @key-19]:
$$\bar{h}_{\mu\nu}=A_{\mu\nu}(\overrightarrow{p})\exp(ip^{\alpha}x_{\alpha})+c.c.\label{eq: sol T}$$
$$h_{f}=a(\overrightarrow{p})\exp(iq^{\alpha}x_{\alpha})+c.c.\label{eq: sol S}$$
where
$$\begin{array}{ccc}
k^{\alpha}\equiv(\omega,\overrightarrow{p}) & & \omega=p\equiv|\overrightarrow{p}|\\
\\q^{\alpha}\equiv(\omega_{m},\overrightarrow{p}) & & \omega_{m}=\sqrt{m^{2}+p^{2}}.\end{array}\label{eq: k e q}$$
In eqs. (\[eq: onda T\]) and (\[eq: sol T\]) the equation and the solution for the standard waves of General Relativity [@key-31; @key-32] have been obtained, while eqs. (\[eq: onda S\]) and (\[eq: sol S\]) are respectively the equation and the solution for the massive mode (see also [@key-9; @key-12; @key-19; @key-40]).
The fact that the dispersion law for the modes of the massive field $h_{f}$ is not linear has to be emphasized. The velocity of every “ordinary” (i.e. which arises from General Relativity) mode $\bar{h}_{\mu\nu}$ is the light speed $c$, but the dispersion law (the second of eq. (\[eq: k e q\])) for the modes of $h_{f}$ is that of a massive field which can be discussed like a wave-packet [@key-9; @key-12; @key-19; @key-40]. Also, the group-velocity of a wave-packet of $h_{f}$ centred in $\overrightarrow{p}$ is
$$\overrightarrow{v_{G}}=\frac{\overrightarrow{p}}{\omega},\label{eq: velocita' di gruppo}$$
which is exactly the velocity of a massive particle with mass $m$ and momentum $\overrightarrow{p}$.
From the second of eqs. (\[eq: k e q\]) and eq. (\[eq: velocita’ di gruppo\]) it is simple to obtain:
$$v_{G}=\frac{\sqrt{\omega^{2}-m^{2}}}{\omega}.\label{eq: velocita' di gruppo 2}$$
Then, as one wants a constant speed of the wave-packet, it has to be [@key-9; @key-12; @key-19]
$$m=\sqrt{(1-v_{G}^{2})}\omega.\label{eq: relazione massa-frequenza}$$
Now, the analysis can remain in the Lorenz gauge with transformations of the type $\square\epsilon_{\nu}=0$; this gauge gives a condition of transverse effect for the ordinary part of the field: $k^{\mu}A_{\mu\nu}=0$, but does not give the transverse effect for the total field $h_{\mu\nu}$. From eq. (\[eq: ridefiniz\]) it is
$$h_{\mu\nu}=\bar{h}_{\mu\nu}-\frac{\bar{h}}{2}\eta_{\mu\nu}+\eta_{\mu\nu}h_{f}.\label{eq: ridefiniz 2}$$
At this point, if being in the massless case [@key-9; @key-12; @key-19], one puts
$$\begin{array}{c}
\square\epsilon^{\mu}=0\\
\\\partial_{\mu}\epsilon^{\mu}=-\frac{\bar{h}}{2}+h_{f},\end{array}\label{eq: gauge2}$$
which gives the total transverse effect of the field. But in the massive case this is impossible. In fact, by applying the Dalembertian operator to the second of eqs. (\[eq: gauge2\]) and by using the field equations (\[eq: onda T\]) and (\[eq: onda S\]) it is
$$\square\epsilon^{\mu}=m^{2}h_{f},\label{eq: contrasto}$$
which is in contrast with the first of eqs. (\[eq: gauge2\]). In the same way, it is possible to show that it does not exist any linear relation between the tensor field $\bar{h}_{\mu\nu}$ and the massive field $h_{f}$. Thus, a gauge in which $h_{\mu\nu}$ is purely spatial cannot be chosen (i.e. $h_{\mu0}=0$ cannot be put, see eq. (\[eq: ridefiniz 2\])) . But the traceless condition to the field $\bar{h}_{\mu\nu}$ can be put :
$$\begin{array}{c}
\square\epsilon^{\mu}=0\\
\\\partial_{\mu}\epsilon^{\mu}=-\frac{\bar{h}}{2}.\end{array}\label{eq: gauge traceless}$$
These equations imply
$$\partial^{\mu}\bar{h}_{\mu\nu}=0.\label{eq: vincolo}$$
To save the conditions $\partial_{\mu}\bar{h}^{\mu\nu}$ and $\bar{h}=0$ transformations like
$$\begin{array}{c}
\square\epsilon^{\mu}=0\\
\\\partial_{\mu}\epsilon^{\mu}=0\end{array}\label{eq: gauge 3}$$
can be used and, taking $\overrightarrow{p}$ in the $z$ direction, a gauge in which only $A_{11}$, $A_{22}$, and $A_{12}=A_{21}$ are different to zero can be chosen. The condition $\bar{h}=0$ gives $A_{11}=-A_{22}$. Now, putting these equations in eq. (\[eq: ridefiniz 2\]), it is
$$h_{\mu\nu}(t,z)=A^{+}(t-z)e_{\mu\nu}^{(+)}+A^{\times}(t-z)e_{\mu\nu}^{(\times)}+h_{f}(t-v_{G}z)\eta_{\mu\nu}.\label{eq: perturbazione totale}$$
The term $A^{+}(t-z)e_{\mu\nu}^{(+)}+A^{\times}(t-z)e_{\mu\nu}^{(\times)}$ describes the two standard polarizations of gravitational waves which arise from General Relativity, while the term $h_{f}(t-v_{G}z)\eta_{\mu\nu}$ is the massive field arising from the high order theory. In other words, the function $f'$ of the Ricci scalar generates a third massive polarization for gravitational waves which is not present in standard General Relativity.
The primordial production of the third polarization
===================================================
Now, let us consider the primordial physical process, which gave rise to a characteristic spectrum $\Omega_{gw}$ for relic GWs. Such physical process has been analysed in different ways: respectively in refs. [@key-25; @key-26] and [@key-27; @key-28] but only for the components of eq. (\[eq: perturbazione totale\]) which arises from General Relativity. In [@key-29] the process has been extended to scalar-tensor gravity. Actually, the process can be further improved showing the primordial production of the third polarization of eq. (\[eq: perturbazione totale\]).
Before starting with the analysis, let us recall that, considering a stochastic background of GWs, it can be characterized by a dimensionless spectrum [@key-25; @key-26; @key-27; @key-28; @key-29]$$\Omega_{gw}(f)\equiv\frac{1}{\rho_{c}}\frac{d\rho_{gw}}{d\ln f},\label{eq: spettro}$$
where $$\rho_{c}\equiv\frac{3H_{0}^{2}}{8G}\label{eq: densita' critica}$$
is the (actual) critical density energy, $\rho_{c}$ of the Universe, $H_{0}$ the actual value of the Hubble expansion rate and $d\rho_{gw}$ the energy density of relic GWs in the frequency range $f$ to $f+df$.
The existence of a relic stochastic background of GWs arises from generals assumptions, i.e. from a mixing between basic principles of classical theories of gravity and of quantum field theory. The strong variations of the gravitational field in the early universe amplify the zero-point quantum oscillations and produce relic GWs. It is well known that the detection of relic GWs is the only way to learn about the evolution of the very early universe, up to the bounds of the Planck epoch and the initial singularity [@key-21; @key-25; @key-26; @key-27; @key-28; @key-29]. It is very important to stress the unavoidable and fundamental character of this mechanism. The model derives from the inflationary scenario for the early universe [@key-36; @key-37], which is tuned in a good way with the WMAP data on the Cosmic Background Radiation (CBR) (in particular exponential inflation and spectral index $\approx1$ [@key-38; @key-39]). Inflationary models of the early Universe were analysed in the early and middles 1980’s (see [@key-36] for a review ), starting from an idea of A. Guth [@key-37]. These are cosmological models in which the Universe undergoes a brief phase of a very rapid expansion in early times. In this context, the expansion could be power-law or exponential in time. Inflationary models provide solutions to the horizon and flatness problems and contain a mechanism which creates perturbations in all fields. Important for our goals is that this mechanism also provides a distinctive spectrum of relic GWs. The GWs perturbations arise from the uncertainty principle and the spectrum of relic GWs is generated from the adiabatically-amplified zero-point fluctuations [@key-21; @key-25; @key-26; @key-27; @key-28; @key-29].
Now, the calculation for a simple inflationary model will be shown for the third polarization of eq. (\[eq: perturbazione totale\]), following the works of Allen [@key-25; @key-26] that performed the calculation in the case of standard General Relativity and Corda, Capozziello and De Laurentis [@key-29; @key-30] that extended the process to scalar GWs. In a recent paper [@key-21], such a process has been applied to the $f(R)$ theories arising from the action (\[eq: high order 1\]). But, in [@key-21] a different point of view has been considered. In that case, using a conform analysis, the authors discussed such a process in respect to the two standard polarizations which arises from standard General Relativity. In the following the analysis is focused to the third massive polarization. Thus, in a certain sense, one can say that the present analysis is an integration of the analysis in [@key-21].
It will be assumed that the universe is described by a simple cosmology in two stages, an inflationary De Sitter phase and a radiation dominated phase [@key-21; @key-25; @key-26; @key-27; @key-28; @key-29]. The line element of the spacetime is given by
$$ds^{2}=a^{2}(\eta)[-d\eta^{2}+d\overrightarrow{x}^{2}+h_{\mu\nu}(\eta,\overrightarrow{x})dx^{\mu}dx^{\nu}].\label{eq: metrica}$$
(\[eq: metrica\]) has to be a solution of the general field equations (6). In fact, even if such a form is allowed, it could have absolutely different behavior (see [@key-44; @key-45; @key-46]). For instance one needs to show that inflation is present in the proposed model. But, in the case of f(R) theories, which are the ones that we are treating here, both of the two conditions are, in general, satisfied. In fact, we recall that the original inflation was proposed by Starobinsky in the classical papers [@key-48; @key-49] by using the simplest f(R) theory, i.e. the $R^{2}$ one. On the other hand, the potential presence and the importance of standard De Sitter inflation in the general framework of the primordial production of relic gravitational waves has been recently shown in [@key-50] considering a different point of view. In that case, using a conform analysis, the authors discussed such a process in respect to the two standard polarizations which arises from standard General Relativity. In the following, the analysis is focused to the third massive polarization. Thus, in a certain sense, the present analysis is an integration of the analysis in [@key-50]. Remaining in the tapestry of f(R) theories, there are lots of examples in the literature where the line element (\[eq: metrica\]), which is the perturbed form of the standard conformally flat Robertson-Walker one, results a solution of the general field equations (6). One can see for example the recent reviews [@key-51; @key-52; @key-53].
In the line element (\[eq: metrica\]), by considering only the third polarization, the metric perturbation (\[eq: perturbazione totale\]) reduces to
$$h_{\mu\nu}=h_{f}I_{\mu\nu},\label{eq: perturbazione scalare}$$
where $$I_{\mu\nu}\equiv\begin{array}{cccc}
1 & 0 & 0 & 0\\
0 & 1 & 0 & 0\\
0 & 0 & 1 & 0\\
0 & 0 & 0 & 1.\end{array}\label{eq: identica}$$
In the De Sitter phase ($\eta<\eta_{1}$) the equation of state is $P=-\rho=const$, the scale factor is $a(\eta)=\eta_{1}^{2}\eta_{0}^{-1}(2\eta_{1}-\eta)^{-1}$ and the Hubble constant is given by $H(\eta)=H_{ds}=c\eta_{0}/\eta_{1}^{2}$.
In the radiation dominated phase $(\eta>\eta_{1})$ the equation of state is $P=\rho/3$, the scale factor is $a(\eta)=\eta/\eta_{0}$ and the Hubble constant is given by $H(\eta)=c\eta_{0}/\eta^{2}$ [@key-21; @key-25; @key-26; @key-29; @key-30].
Expressing the scale factor in terms of comoving time defined by
$$cdt=a(t)d\eta\label{eq: tempo conforme}$$
one gets
$$a(t)\propto\exp(H_{ds}t)\label{eq: inflazione}$$
during the De Sitter phase and
$$a(t)\propto\sqrt{t}\label{eq: dominio radiazione}$$
during the radiation dominated phase. The horizon and flatness problems are solved if [@key-36; @key-37]
$\frac{a(\eta_{0})}{a(\eta_{1})}>10^{27}$
The third polarization generates weak perturbations $h_{\mu\nu}(\eta,\overrightarrow{x})$ of the metric (\[eq: perturbazione scalare\]) that can be written, in terms of the conformal time $\eta$, in the form
$$h_{\mu\nu}=I_{\mu\nu}(\hat{k})X(\eta)\exp(\overrightarrow{k}\cdot\overrightarrow{x}),\label{eq: relic gravity-waves}$$
where $\overrightarrow{k}$ is a constant wavevector and
$$h_{f}(\eta,\overrightarrow{k},\overrightarrow{x})=X(\eta)\exp(\overrightarrow{k}\cdot\overrightarrow{x}).\label{eq: phi}$$
By putting $Y(\eta)=a(\eta)X(\eta)$ one performs the standard linearized calculation in which the connections (i.e. the Cristoffel coefficients), the Riemann tensor, the Ricci tensor and the Ricci scalar curvature are computed. Then, from the Friedman linearized equations, the function $Y(\eta)$ satisfies the equation
$$Y''+(|\overrightarrow{k}|^{2}-\frac{a''}{a})Y=0\label{eq: Klein-Gordon}$$
where $'$ denotes derivative with respect to the conformal time. Clearly, this is the equation for a parametrically disturbed oscillator.
The solutions of eq. (\[eq: Klein-Gordon\]) give the solutions for the function $X(\eta)$, that can be expressed in terms of elementary functions simple cases of half integer Bessel or Hankel functions [@key-21; @key-25; @key-26; @key-29; @key-30] in both of the inflationary and radiation dominated eras:
For $\eta<\eta_{1}$ $$X(\eta)=\frac{a(\eta_{1})}{a(\eta)}[1+H_{ds}\omega^{-1}]\exp-ik(\eta-\eta_{1}),\label{eq: ampiezza inflaz.}$$
for $\eta>\eta_{1}$ $$X(\eta)=\frac{a(\eta_{1})}{a(\eta)}[\alpha\exp-ik(\eta-\eta_{1})+\beta\exp ik(\eta-\eta_{1}),\label{eq: ampiezza rad.}$$
where $\omega=ck/a$ is the angular frequency of the wave (that is function of the time because of the constance of $k=|\overrightarrow{k}|$), $\alpha$ and $\beta$ are time-independent constants which can be obtained demanding that both $X$ and $dX/d\eta$ are continuous at the boundary $\eta=\eta_{1}$ between the inflationary and the radiation dominated eras of the cosmological expansion. With this constrain it is
$$\alpha=1+i\frac{\sqrt{H_{ds}H_{0}}}{\omega}-\frac{H_{ds}H_{0}}{2\omega^{2}}\label{eq: alfa}$$
$$\beta=\frac{H_{ds}H_{0}}{2\omega^{2}}\label{eq: beta}$$
In eqs. (\[eq: alfa\]), (\[eq: beta\]) $\omega=ck/a(\eta_{0})$ is the angular frequency that would be observed today. Calculations like this are referred in the literature as Bogoliubov coefficient methods [@key-21; @key-25; @key-26; @key-29; @key-30].
As inflation damps out any classical or macroscopic perturbations, the minimum allowed level of fluctuations is that required by the uncertainty principle. The solution (\[eq: ampiezza inflaz.\]) corresponds precisely to this De Sitter vacuum state [@key-21; @key-25; @key-26; @key-29; @key-30]. Then, if the period of inflation was long enough, the observable properties of the Universe today should be the same properties of a Universe started in the De Sitter vacuum state.
In the radiation dominated phase the coefficients of $\alpha$ are the eigenmodes which describe particles while the coefficients of $\beta$ are the eigenmodes which describe antiparticles. Thus, the number of created particles of angular frequency $\omega$ in this phase is
$$N_{\omega}=|\beta_{\omega}|^{2}=(\frac{H_{ds}H_{0}}{2\omega^{2}})^{2}.\label{eq: numero quanti}$$
Now, one can write an expression for the energy spectrum of the relic gravitational waves background in the frequency interval $(\omega,\omega+d\omega)$ as
$$d\rho_{gw}=2\hbar\omega(\frac{\omega^{2}d\omega}{2\pi^{2}c^{3}})N_{\omega}=\frac{\hbar H_{ds}^{2}H_{0}^{2}}{4\pi^{2}c^{3}}\frac{d\omega}{\omega}=\frac{\hbar H_{ds}^{2}H_{0}^{2}}{4\pi^{2}c^{3}}\frac{df}{f}.\label{eq: de energia}$$
Eq. (\[eq: de energia\]) can be rewritten in terms of the present day and the De Sitter energy-density of the Universe. The Hubble expansion rates is
$H_{0}^{2}=\frac{8\pi G\rho_{c}}{3}$, $H_{ds}^{2}=\frac{8\pi G\rho_{ds}}{3}$.
Then, introducing the Planck density $$\rho_{Planck}\equiv\frac{c^{7}}{\hbar G^{2}}\label{eq: Shoooortyyyyyy}$$ the spectrum is
$$\Omega_{gw}(f)=\frac{1}{\rho_{c}}\frac{d\rho_{sgw}}{d\ln f}=\frac{f}{\rho_{c}}\frac{d\rho_{gw}}{df}=\frac{16}{9}\frac{\rho_{ds}}{\rho_{Planck}}.\label{eq: spettro gravitoni}$$
Some comments are needed. The computation works for a very simplified model that does not include the matter dominated era. Including this era, the redshift has to be considered. An enlightening computation parallel to the one in [@key-26] gives
$$\Omega_{gw}(f)=\frac{16}{9}\frac{\rho_{ds}}{\rho_{Planck}}(1+z_{eq})^{-1},\label{eq: spettro gravitoni redshiftato}$$
for the waves which at the time in which the Universe was becoming matter dominated had a frequency higher than $H_{eq}$, the Hubble constant at that time. This corresponds to frequencies $f>(1+z_{eq})^{1/2}H_{0}$, where $z_{eq}$ is the redshift of the Universe when the matter and radiation energy density were equal. The redshift correction in equation (\[eq: spettro gravitoni redshiftato\]) is needed because the Hubble parameter, which is governed by Friedman equations, should be different from the observed one $H_{0}$ for a Universe without matter dominated era.
At lower frequencies the spectrum is [@key-21; @key-25; @key-26; @key-29; @key-30]
$$\Omega_{gw}(f)\propto f^{-2}.\label{eq: spettro basse frequenze}$$
Moreover, the results (\[eq: spettro gravitoni\]) and (\[eq: spettro gravitoni redshiftato\]), which are not frequency dependent, cannot be applied to all the frequencies. For waves with frequencies less than $H_{0}$ today, the energy density cannot be defined, because the wavelength becomes longer than the Hubble radius. In the same way, at high frequencies there is a maximum frequency above which the spectrum drops to zero rapidly. In the above computation it has been implicitly assumed that the phase transition from the inflationary to the radiation dominated epoch is instantaneous. In the real Universe this phase transition occurs over some finite time $\Delta\tau$, and above a frequency
$$f_{max}=\frac{a(t_{1})}{a(t_{0})}\frac{1}{\Delta\tau},\label{eq: freq. max}$$
which is the redshifted rate of the transition, $\Omega_{gw}$ drops rapidly. These two cutoffs, at low and high frequencies, to the spectrum force the total energy density of the relic gravitational waves to be finite. For GUT energy-scale inflation it is [@key-21; @key-25; @key-26; @key-29; @key-30].
$$\frac{\rho_{ds}}{\rho_{Planck}}\approx10^{-12}.\label{eq: rapporto densita' primordiali}$$
Tuning with WMAP data
=====================
It is well known that WMAP observations put strongly severe restrictions on the spectrum of relic gravitational waves. In fig. 1 the spectrum $\Omega_{gw}$is mapped following [@key-20]: the amplitude is chosen (determined by the ratio $\frac{\rho_{ds}}{\rho_{Planck}}$) to be *as large as possible, consistent with the WMAP constraints* o*n tensor perturbations.* Nevertheless, because the spectrum falls off $\propto f^{-2}$ at low frequencies, this means that today, at LIGO-Virgo and LISA frequencies (indicate by the lines in fig. 1) [@key-20], it is
$$\Omega_{gw}(f)h_{100}^{2}<9*10^{-13}.\label{eq: limite spettro WMAP}$$
Let us calculate the correspondent strain at $\approx100Hz$, where interferometers like Virgo and LIGO have a maximum in sensitivity. The well known equation for the characteristic amplitude, adapted for the third component of GWs can be used [@key-20]:
$$h_{fc}(f)\simeq1.26*10^{-18}(\frac{1Hz}{f})\sqrt{h_{100}^{2}\Omega_{gw}(f)},\label{eq: legame ampiezza-spettro}$$
obtaining [@key-20]
$$h_{fc}(100Hz)<1.7*10^{-26}.\label{eq: limite per lo strain}$$
Then, as we expect a sensitivity order of $10^{-22}$ for interferometers at $\approx100Hz$, four order of magnitude have to be gained. Let us analyse smaller frequencies too. The sensitivity of the Virgo interferometer is of the order of $10^{-21}$ at $\approx10Hz$ and in that case it is [@key-20]
$$h_{fc}(10Hz)<1.7*10^{-25}.\label{eq: limite per lo strain2}$$
The sensitivity of the LISA interferometer will be of the order of $10^{-22}$ at $10^{-3}\approx Hz$ and in that case it is [@key-20]
$$h_{fc}(100Hz)<1.7*10^{-21}.\label{eq: limite per lo strain3}$$
Then, a stochastic background of relic gravitational waves could be, in principle, detected by the LISA interferometer.
We emphasize the sentence *in principle*. Actually, one has to take into account the Galactic confusion background which will dominate over the instrumental noise. Thus, it is still questionable whether the relic gravitons produced in GR could be detected by LISA [@key-41].
On the other hand, the assumption that all the tensor perturbations in the Universe are due to a stochastic background of GWs is quit strong, but the results (\[eq: limite spettro WMAP\]), (\[eq: limite per lo strain\]), (\[eq: limite per lo strain2\]) and (\[eq: limite per lo strain3\]) can be considered like upper bounds.
Figure 1, which is adapted from ref. [@key-20], shows that the spectrum of relic SGWs in inflationary models is flat over a wide range of frequencies. The horizontal axis is $\log_{10}$ of frequency, in Hz. The vertical axis is $\log_{10}\Omega_{gsw}$. The inflationary spectrum rises quickly at low frequencies (wave which re-entered in the Hubble sphere after the Universe became matter dominated) and falls off above the (appropriately redshifted) frequency scale $f_{max}$ associated with the fastest characteristic time of the phase transition at the end of inflation. The amplitude of the flat region depends only on the energy density during the inflationary stage; we have chosen the largest amplitude consistent with the WMAP constrains on tensor perturbations. This means that at LIGO and LISA frequencies, $\Omega_{gw}(f)h_{100}^{2}<9*10^{-13}.$
Potential detection with interferometers
========================================
Before starting the discussion of the potential interferometric detection of the massive GWs polarization, there is another point which has to be clarified. The attentive reader [@key-41] asks what would happen to Hulse-Taylor pulsar [@key-54]. As it is clear from eq. (\[eq: perturbazione totale\]), by fixing the massless component of the metric to GR value, the third term should carry the additional energy which should affect evolution of the binary system [@key-41]. This problem has been discussed by Shibata, Nakao and Nakamura in [@key-15]. In such a work, it has been shown that the energy associated to the monopole mode of the scalar particle is quite lower in respect to the energy carried by ordinary quadrupole modes arising from the linearization of standard General Relativity. Thus, in these conditions, the evolution of the binary system should be not affected by the third polarization if one remains into observing error bars.
On the other and, even if the ratio of differential mode to common mode is large in the case of the binary pulsar, it could be, in principle, similar and potentially small in cosmology. In fact, in the case of $f(R)$ theories such a mode arises directly from a function of the Ricci scalar, see the first identification (\[eq: identifica\]), i.e. *it arises from spacetime curvature*, and it is well known that, being the production of relic GWs near the bounds of the Planck epoch and the initial singularity, in this early Era both of spacetime curvature and its variations were very high. Thus, in this cosmological case, the amplitude of the common mode could be even dominant in respect to differential modes.
After these clarifications, let us start the discussion regarding the potential detection.
Even if eqs. (\[eq: limite per lo strain\]) and (\[eq: limite per lo strain2\]) show that the detection of relic GWs is quite difficult, people hope in better sensitivity of advanced projects. In this case, it is interesting to discuss the interaction between interferometers and massive GWs.
Considering only the third polarization $h_{f}(t-v_{G}z)\eta_{\mu\nu}$ the line element associated to eq. (\[eq: perturbazione totale\]) becomes the conformally flat one
$$ds^{2}=[1+h_{f}(t,z)](-dt^{2}+dz^{2}+dx^{2}+dy^{2}).\label{eq: metrica puramente scalare}$$
As the analysis on the motion of test masses is performed in a laboratory environment on Earth, the coordinate system in which the space-time is locally flat is typically used and the distance between any two points is given simply by the difference in their coordinates in the sense of Newtonian physics [@key-12; @key-13; @key-14; @key-16; @key-31]. This frame is the proper reference frame of a local observer, located for example in the position of the beam splitter of an interferometer. In this frame GWs manifest themself by exerting tidal forces on the masses (the mirror and the beam-splitter in the case of an interferometer). A detailed analysis of the frame of the local observer is given in ref. [@key-31], sect. 13.6. Here only the more important features of this coordinate system are recalled:
the time coordinate $x_{0}$ is the proper time of the observer O;
spatial axes are centred in O;
in the special case of zero acceleration and zero rotation the spatial coordinates $x_{j}$ are the proper distances along the axes and the frame of the local observer reduces to a local Lorentz frame: in this case the line element reads [@key-31]
$$ds^{2}=-(dx^{0})^{2}+\delta_{ij}dx^{i}dx^{j}+O(|x^{j}|^{2})dx^{\alpha}dx^{\beta}.\label{eq: metrica local lorentz}$$
The effect of the gravitational wave on test masses is described by the equation
$$\ddot{x^{i}}=-\widetilde{R}_{0k0}^{i}x^{k},\label{eq: deviazione geodetiche}$$
which is the equation for geodesic deviation in this frame.
Thus, to study the effect of the massive gravitational wave on test masses, $\widetilde{R}_{0k0}^{i}$ have to be computed in the proper reference frame of the local observer. But, because the linearized Riemann tensor $\widetilde{R}_{\mu\nu\rho\sigma}$ is invariant under gauge transformations [@key-9; @key-12; @key-13; @key-14; @key-31], it can be directly computed from eq. (\[eq: perturbazione scalare\]).
From [@key-31] it is:
$$\widetilde{R}_{\mu\nu\rho\sigma}=\frac{1}{2}\{\partial_{\mu}\partial_{\beta}h_{\alpha\nu}+\partial_{\nu}\partial_{\alpha}h_{\mu\beta}-\partial_{\alpha}\partial_{\beta}h_{\mu\nu}-\partial_{\mu}\partial_{\nu}h_{\alpha\beta}\},\label{eq: riemann lineare}$$
that, in the case eq. (\[eq: perturbazione scalare\]), begins
$$\widetilde{R}_{0\gamma0}^{\alpha}=\frac{1}{2}\{\partial^{\alpha}\partial_{0}h_{f}\eta_{0\gamma}+\partial_{0}\partial_{\gamma}h_{f}\delta_{0}^{\alpha}-\partial^{\alpha}\partial_{\gamma}h_{f}\eta_{00}-\partial_{0}\partial_{0}h_{f}\delta_{\gamma}^{\alpha}\};\label{eq: riemann lin scalare}$$
the different elements are (only the non zero ones will be written):
$$\partial^{\alpha}\partial_{0}h_{f}\eta_{0\gamma}=\left\{ \begin{array}{ccc}
\partial_{t}^{2}h_{f} & for & \alpha=\gamma=0\\
\\-\partial_{z}\partial_{t}h_{f} & for & \alpha=3;\gamma=0\end{array}\right\} \label{eq: calcoli}$$
$$\partial_{0}\partial_{\gamma}h_{f}\delta_{0}^{\alpha}=\left\{ \begin{array}{ccc}
\partial_{t}^{2}h_{f} & for & \alpha=\gamma=0\\
\\\partial_{t}\partial_{z}h_{f} & for & \alpha=0;\gamma=3\end{array}\right\} \label{eq: calcoli2}$$
$$-\partial^{\alpha}\partial_{\gamma}h_{f}\eta_{00}=\partial^{\alpha}\partial_{\gamma}h_{f}=\left\{ \begin{array}{ccc}
-\partial_{t}^{2}h_{f} & for & \alpha=\gamma=0\\
\\\partial_{z}^{2}h_{f} & for & \alpha=\gamma=3\\
\\-\partial_{t}\partial_{z}h_{f} & for & \alpha=0;\gamma=3\\
\\\partial_{z}\partial_{t}h_{f} & for & \alpha=3;\gamma=0\end{array}\right\} \label{eq: calcoli3}$$
$$-\partial_{0}\partial_{0}h_{f}\delta_{\gamma}^{\alpha}=\begin{array}{ccc}
-\partial_{z}^{2}h_{f} & for & \alpha=\gamma\end{array}.\label{eq: calcoli4}$$
Now, putting these results in eq. (\[eq: riemann lin scalare\]) one obtains:
$$\begin{array}{c}
\widetilde{R}_{010}^{1}=-\frac{1}{2}\ddot{h}_{f}\\
\\\widetilde{R}_{010}^{2}=-\frac{1}{2}\ddot{h}_{f}\\
\\\widetilde{R}_{030}^{3}=\frac{1}{2}\square h_{f}.\end{array}\label{eq: componenti riemann}$$
But, putting the second of equations (\[eq: linearizzate1\]) in the third of eqs. (\[eq: componenti riemann\]), it is
$$\widetilde{R}_{030}^{3}=\frac{1}{2}m^{2}h_{f},\label{eq: terza riemann}$$
which shows that the field is not transversal.
In fact, using eq. (\[eq: deviazione geodetiche\]) it results
$$\ddot{x}=\frac{1}{2}\ddot{h}_{f}x,\label{eq: accelerazione mareale lungo x}$$
$$\ddot{y}=\frac{1}{2}\ddot{h}_{f}y\label{eq: accelerazione mareale lungo y}$$
and
$$\ddot{z}=-\frac{1}{2}m^{2}h_{f}(t,z)z.\label{eq: accelerazione mareale lungo z}$$
Then, the effect of the mass is the generation of a *longitudinal* force (in addition to the transverse one).
For a better understanding of this longitudinal force, let us analyse the effect on test masses in the context of the geodesic deviation.
Following [@key-14] one puts$$\widetilde{R}_{0j0}^{i}=\frac{1}{2}\left(\begin{array}{ccc}
-\partial_{t}^{2} & 0 & 0\\
0 & -\partial_{t}^{2} & 0\\
0 & 0 & m^{2}\end{array}\right)h_{f}(t,z)=-\frac{1}{2}T_{ij}\partial_{t}^{2}h_{f}+\frac{1}{2}L_{ij}m^{2}h_{f}.\label{eq: eqs}$$
Here the transverse projector with respect to the direction of propagation of the GW $\widehat{n}$, defined by
$$T_{ij}=\delta_{ij}-\widehat{n}_{i}\widehat{n}_{j},\label{eq: Tij}$$
and the longitudinal projector defined by
$$L_{ij}=\widehat{n}_{i}\widehat{n}_{j}\label{eq: Lij}$$
have been used. In this way, the geodesic deviation equation (\[eq: deviazione geodetiche\]) can be rewritten like
$$\frac{d^{2}}{dt^{2}}x_{i}=\frac{1}{2}\partial_{t}^{2}h_{f}T_{ij}x_{j}-\frac{1}{2}m^{2}h_{f}L_{ij}x_{j}.\label{eq: TL}$$
Thus, it appears clear that the effect of the mass present in the GW generates a longitudinal force proportional to $m^{2}$ which is in addition to the transverse one. But if $v(\omega)\rightarrow1$ in eq. (\[eq: velocita’ di gruppo 2\]) one gets $m\rightarrow0$, and the longitudinal force vanishes. Then, it is clear that the longitudinal mode arises from the fact that the GW does no propagate at the speed of light.
Now, let us analyse the detectability of the third polarization computing the pattern function of a detector to this massive component. One has to recall that it is possible to associate to a detector a *detector tensor* that, for an interferometer with arms along the $\hat{u}$ e $\hat{v}$ directions in respect to the propagating gravitational wave (see figure 2), is defined by [@key-14]$$D^{ij}\equiv\frac{1}{2}(\hat{v}^{i}\hat{v}^{j}-\hat{u}^{i}\hat{u}^{j}).\label{eq: definizione D}$$
If the detector is an interferometer [@key-1; @key-2; @key-3; @key-4; @key-5; @key-6; @key-7; @key-8; @key-21], the signal induced by a GW of a generic polarization, here labelled with $s(t),$ is the phase shift, which is proportional to [@key-14]
$$s(t)\sim D^{ij}\widetilde{R}_{i0j0}\label{eq: legame onda-output}$$
and, using equations (\[eq: eqs\]), one gets
$$s(t)\sim-\sin^{2}\theta\cos2\phi.\label{eq: legame onda-output 2}$$
The angular dependence (\[eq: legame onda-output 2\]), is different from the two well known standard ones arising from general relativity which are, respectively $(1+\cos^{2}\theta)\cos2\phi$ for the $+$ polarization and $-\cos\theta\sin2\vartheta$ for the $\times$ polarization. Thus, in principle, the angular dependence (\[eq: legame onda-output 2\]) could be used to discriminate among $f(R)$ theories and general relativity, if present or future detectors will achieve a high sensitivity. The third angular dependence is shown in figure.
![The (dimensionless) angular dependence (\[eq: legame onda-output 2\])](AngoliVirgo_gr1)
For a sake of completeness, let us recall that there is one more problem with the potential detection of the common mode. This is because the sensitivity curve drawn for different detectors is for the differential mode. The common mode is also registered during the GW experiment but it is much noisier, therefore it has different sensitivity level [@key-41]. Actually, this is correct for studies on potential detection in the case of wavelength of the wave much larger than the linear dimension of the interferometer (low frequencies approximation). In previous analysis, we have implicitly assumed to be in such an approximation, and in particular, eqs. (\[eq: definizione D\]), (\[eq: legame onda-output\]) and (\[eq: legame onda-output 2\]) are strictly valid only in this approximation. A recent analysis [@key-55] has shown that, in the case in which the differential mode is massive, the response function of an interferometer *increases* with increasing frequency differently from the case of massless modes where the response function *decreases* with increasing frequency. This opens important perspectives for a potential detection of the massive mode at high frequency. In [@key-55] the response function has only been computed in the case of a massive mode propagating parallel to one arm of the interferometer, thus, further studies in this direction are needed. For example, it will be quite interesting to generalize the frequency dependence of the angular pattern (\[eq: legame onda-output 2\]).
Conclusion remarks
==================
It has been shown that, in general, $f(R)$ theories produce a third massive polarization of gravitational waves and the primordial production of this polarization has been analysed adapting the adiabatically-amplified zero-point fluctuations process at this case and generalizing previous results in which only particular cases have been discussed.
The presence of the mass could also have important applications in cosmology because the fact that gravitational waves can have mass could give a contribution to the dark matter of the Universe.
An upper bound for these relic gravitational waves, which arises from the WMAP constrains, has been also released and at the end of the paper and the potential detection of such massive GWs using interferometers like Virgo and LIGO has been discussed.
Acknowledgements {#acknowledgements .unnumbered}
================
I would like to thank Francesco Rubanu for helpful advices during my work. I have to strongly thank the referees for useful advices and precious comments which permitted in improving the paper.
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'In this paper we show how the method of Lie algebra expansions may be used to obtain, in a simple way, both the extended Bargmann Lie superalgebra and the Chern-Simons action associated to it in three dimensions, starting from $D=3$, $\mathcal{N}=2$ superPoincaré and its corresponding Chern-Simons supergravity.'
author:
- |
J.A. de Azcárraga,\
Departamento de Física Teórica and IFIC (CSIC-UVEG),\
46100-Burjassot (Valencia), Spain\
D. Gútiez,\
Department of Physics, Universidad de Oviedo,\
33007-Oviedo, Spain\
J. M. Izquierdo,\
Departamento de Física Teórica, Universidad de Valladolid,\
47011-Valladolid, Spain
date: 'Jul. 12, 2019'
title: '[**Extended $D=3$ Bargmann supergravity from a Lie algebra expansion**]{}'
---
Introduction
============
In recent years, the supersymmetric version of Newtonian Gravity, [*i.e.*]{} Newtonian supergravity, has received some attention in the context of a non-relativistic version of the AdS/CFT correspondence (see, for instance, [@BG:09; @CHOR:14]). However, certain problems need to be solved, and progress is still being made.
The natural way to address the problem requires using a Galilean superalgebra as a starting point, plus a gauging procedure that, in the bosonic case, should recover Newtonian gravity. In [@ABPR:11] this was done in $D=3,4$ and in the absence of fermions by starting from the centrally extended Galilei algebra or Bargmann algebra, and imposing certain conditions on the curvatures associated to its gauging. These conditions allowed the authors to obtain Newtonian gravity in the Newton-Cartan (NC) formalism [@C:23], which formulates Newtonian gravity in a way that resembles general relativity. Subsequently, the supersymmetric case was studied [@ABRS:13; @BRZ:15] in $D=3$ for a superalgebra that contains two fermionic generators and such that the bosonic part is the Bargmann algebra. In this way, the $D=3$ NC supergravity was obtained.
The solution to the problem mentioned above, however, has some limitations: spatial geometry is fixed to be flat, and there is no satisfactory action principle associated to it. An action was obtained in [@BR:16] that overcomes these difficulties, which was called the extended Bargmann supergravity action. In contrast with the NC supergravity case, the bosonic part of the supersymmetry algebra is a further extension of the Bargmann algebra, and the action itself is a Chern-Simons (CS) action, as it is also the case of $D=3$ Poincaré supergravity [@AT:86]. In fact, it was shown in [@BR:16] that both the Galilei action and algebra may be obtained from the Poincaré ones as a limit that, although it looks like a contraction, it is not so since it does not preserve the dimension of the algebra.
In this paper, we point out that the Galilean superalgebra and the CS action mentioned above may be found alternatively by using the method of Lie algebra expansions, which has its origin in the work of [@HS:03] and was formulated and studied in general in [@AIPV:03; @AIPV:07] (see also [@AI:12; @AILW:13] for other applications and [@IRS:06] for a generalization involving semigroups[^1]). In addition to the other three ways of obtaining new algebras from given ones, namely contractions and deformations (both preserving the dimension of the original Lie algebra) and extensions (which require two algebras), the method of expansions provides a way to obtain in general larger Lie algebras from a given one (see [@AIPV:07]). Presumably, all algebras obtained by the expansion method may be obtained by a combination of extensions and contractions, but from the computational point of view expansions are more interesting, as some calculations simplify considerably.
More precisely, we show here that, starting from $D=3$, $\mathcal{N}=2$ Poincaré supergravity and a CS action associated to it, the method of expansions applied to the algebra leads to the extended Bargmann superalgebra of [@BR:16]. Further, when the superPoincaré action is expanded, the result is also the action of [@BR:16]. As it will become apparent, the calculations involved are very simple.
The plan of the paper is the following. In Sec. 2 we will briefly review the method of Lie algebra expansions [@AIPV:03; @AIPV:07] in the particular case of interest to us. In section 3 it will be applied to the $D=3$ Poincaré gravity to obtain a bosonic Galilean CS action. Section 4 is devoted to the expansions of the $D=3$, $\mathcal{N}=2$ superPoincaré based CS model that leads to the algebra and action obtained in [@BR:16]. In the conclusions, we will comment on the possible future applications of the method in the context of Galilean gravity and supergravity.
Lie Algebra expansions
======================
In a nutshell, Lie algebra expansions consists of three steps. Given a Lie (super)algebra $\mathcal{G}$,
1. Write a formal series expansion in $\lambda$ of the Maurer-Cartan (MC) one-forms associated with the Lie algebra,
2. Insert the expansions into the MC equations of $\mathcal{G}$ and identify equal powers of $\lambda$ to obtain a consistent infinite set of MC equations, and
3. Cut the infinite expansions in a consistent way so that a finite Lie algebra, the expanded algebra, is obtained in terms of its MC equations.
We give the details of the construction in the case of interest in this paper, namely when $\mathcal{G}$ has a symmetric coset structure. Let $\mathcal{G}=V_0 \oplus V_1$, where $V_1$ is a symmetric coset, $$\label{simcos}
[V_0,V_0]\subset V_0 \ , \quad [V_0,V_1]\subset V_1\ ,\quad [V_1,V_1]\subset V_0\ .$$ Let $\omega^{i_p}$, be the MC forms valued on the spaces $V_p$, $p=0,1$ and $i_p=1,\dots , \mathrm{dim} V_p$. Let us write the MC equations of $\mathcal{G}$ in the form $$\
\label{MCsup} d \omega^{k_s} = - \frac{1}{2} c^{k_s}_{i_p j_q}
\omega^{i_p} \wedge \omega^{j_q} \quad (p,q,s= 0,1) \ .$$ Condition implies that the structure constants of the algebra satisfy $$\label{simcosb}
c^{k_1}_{i_0 j_0}= 0 \ , \quad c^{k_0}_{i_0 j_1} = 0 \ , \quad c^{k_1}_{i_1 j_1}= 0 \ .$$ Then, it is consistent to expand the MC forms of $V_0$ in terms of even powers of $\lambda$ and those of $V_1$ in terms of odd powers of $\lambda$ as $$\begin{aligned}
\label{expthinf}
\omega^{i_0} &=& \sum^{\infty}_{\alpha_0=0,\ \alpha_0 \, even}
\lambda^{\alpha_0} \omega^{i_0,\alpha_0} \nonumber \\
\omega^{i_1} &=& \sum^{\infty}_{\alpha_1=1,\ \alpha_1 \, odd}
\lambda^{\alpha_1} \omega^{i_1,\alpha_1}
\ .\end{aligned}$$ When these expansions are inserted in the MC equations , and the equal powers of $\lambda$ in both sides are identified, we obtain a consistent infinite number of MC one-forms and equations. To obtain finite Lie algebras, the expansions must be cut in a consistent way, so that they correspond to the MC equations of a Lie (super)algebra. It can be shown that this is achieved when $$\begin{aligned}
\label{expth}
\omega^{i_0} &=& \sum^{N_0}_{\alpha_0=0,\ \alpha_0 \, even}
\lambda^{\alpha_0} \omega^{i_0,\alpha_0} \nonumber \\
\omega^{i_1} &=& \sum^{N_1}_{\alpha_1=1,\ \alpha_1 \, odd}
\lambda^{\alpha_1} \omega^{i_1,\alpha_1}
\ ,\end{aligned}$$ provided that the $N_0$ and $N_1$ integers satisfy one of the two conditions below $$\begin{aligned}
\label{conditions}
& & N_0= N_1+1 \, \nonumber\\
& & N_0= N_1-1 \ ,\end{aligned}$$ (see [@AIPV:03] for the proof). This leads to a series of finite-dimensional superalgebras, which are denoted by $\mathcal{G}(N_0,N_1)$, with structure constants given by $$\label{expsc}
C^{k_s,\alpha_s}_{i_p,\beta_p\, j_q,\gamma_q} =
\left\{\begin{array}{cc}
0 & \mathrm{if}\; \beta_p+\gamma_q \neq \alpha_s \\
c^{k_s}_{i_p j_q} & \mathrm{if}\; \beta_p+\gamma_q = \alpha_s\\
\end{array}\right. \quad .$$
From the MC equations we may obtain the gauge curvatures of the same Lie algebra by noticing that the latter may be viewed as an equation that expresses the failure of the MC equations. So if $0=d \theta
+\theta \wedge \theta$ are the MC equations, then the curvatures are given by $F= dA + A \wedge A$, and by taking $F=0$ we recover the MC equations[^2]. Then, making the replacement $\omega^{i_s,\alpha_s} \rightarrow A^{i_s,\alpha_s}$, the MC forms and MC equations are replaced by gauge one-forms and by the equations defining of the curvatures, $$\label{Curvatures}
F^{k_s,\alpha_s} = dA^{k_s,\alpha_s}+ \frac{1}{2}
C^{k_s,\alpha_s}_{i_p,\beta_p\, j_q,\gamma_q} A^{i_p,\beta_p}
\wedge
A^{j_q,\gamma_q} \ ,$$ so that the MC equations may be recovered by setting $F^{k_s,\alpha_s}=0$.The Bianchi identities and gauge variations (of infinitesimal parameters $\varphi^{i_s,\alpha_s}$) are given by $$\begin{aligned}
\label{expFDA}
d F^{k_s,\alpha_s} &=& C^{k_s,\alpha_s}_{i_p,\beta_p\,
j_q,\gamma_q} F^{i_p,\beta_p} \wedge
A^{j_q,\gamma_q} \ , \nonumber\\
\delta A^{k_s,\alpha_s} &=& d \varphi^{k_s,\alpha_s} -
C^{k_s,\alpha_s}_{i_p,\beta_p\, j_q,\gamma_q}
\varphi^{i_p,\beta_p} A^{j_q,\gamma_q} \ .\end{aligned}$$ It is crucial for our construction to note that, alternatively, these equations may be obtained by substituting, in the equations that would correspond to the original $\mathcal{G}$, that is ([*cf.*]{} ) $$\begin{aligned}
\label{gaugeorig}
F^{k_s} &=& dA^{k_s}+ \frac{1}{2}
c^{k_s}_{i_p\, j_q} A^{i_p}
\wedge
A^{j_q} \ , \nonumber \\
\delta A^{k_s} &=& d \varphi^{k_s} -
C^{k_s}_{i_p\, j_q}
\varphi^{i_p} \wedge A^{j_q} \ ,\end{aligned}$$ the expansions of $A^{k_s}$, $F^{k_s}$ and $\varphi^{k_s}$ with exactly the same structure as the of $\omega^{k_s}$ in and then identifying equal powers of $\lambda$ (see [@AIPV:03]).
Expanded CS actions
-------------------
We can use the expansions of the gauge one-forms and curvature two-forms to obtain, in some cases, new actions from a given one. As an example, we consider now the important case of the CS actions.
Let $\mathcal{G}$ be a Lie superalgebra, and let $k_{I_1,\dots I_l}$ be the coordinates of a symmetric invariant $l$-tensor of $\mathcal{G}$. Then, the $2l$-form $$\label{2lform}
H= k_{I_1,\dots I_l} F^{I_1} \wedge \dots \wedge F^{I_l}$$ is closed and invariant under gauge transformations. Since the gauge FDAs (given by the definition of the curvatures plus the Bianchi identities) are contractible, this defines a $(2l-1)$-form $B$ (the CS form, see [*e.g.*]{} [@AI:95]), such that $dB=H$, and if $B$ is integrated over a $(2l-1)$-dimensional manifold $\mathcal{M}^{2l-1}$, a CS model is obtained through the action $$\label{CSaction}
I[A] = \int_{\mathcal{M}^{2l-1}} B(A) \ ,$$ where ${\mathcal{M}^{2l-1}}$ is the $(2l-1)$-dimensional spacetime.
New CS actions for the expanded algebras may be obtained by inserting the expansions of $A^I$ and $F^I$ in the CS action for $\mathcal{G}$, $$\label{CSexpansion}
I[A,\lambda] = \int_{\mathcal{M}^{2l-1}} B(A,\lambda) =
\int_{\mathcal{M}^{2l-1}} \sum_{N=0}^\infty \lambda^N B_N(A) =
\sum_{N=0}^\infty \lambda^N I_N[A] \ .$$ The same expansion, when applied to , leads to $$\label{Hexp}
H(F,\lambda) = \sum_{N=0}^\infty \lambda^N H_N \ ,
\quad
H_N = d B_N(A) \ .$$ This means that the actions given by $$\label{NCS}
I_N = \int_{\mathcal{M}^{2l-1}} B_N(A)$$ define CS models that have been obtained by expanding of the original $\mathcal{G}$-based one. The corresponding Lie algebra is the smallest one that contains [*all*]{} the fields appearing in $I_N$. Not keeping all the fields may result in a lack of gauge invariance of the actions, which is otherwise guaranteed if the expansion is kept infinite. Then, in practice, the power of $\lambda$ in the expansion of the action selects the corresponding finite expanded algebra. In general, the expanded actions and algebras ‘remember’ the structure of the original ones (see Eq. ) a fact that simplifies the calculations.
One computational advantage of expansions is the fact that the equation of motion for $A^{k_s,\alpha_s}$ in $I_N$, which may be represented by $E(A^{k_s,\alpha_s})=0$, satisfies $$E(A^{k_s,\alpha_s})= E(A^{k_s})|_{N-\alpha_s} ,$$ where $E(A^{k_s})|_{N-\alpha_s}$ is the coefficient of $\lambda^{N-\alpha_s}$ in the expansion of $E(A^{k_s})$.
Galilei expansion of arbitrary $D$ poincaré and $D=3$ gravity.
==============================================================
Before going to the supersymmetric case, we consider in this section the bosonic expanded algebras and action to illustrate the method. Although the subject of this paper is $D=3$, we will keep $D$ arbitrary for the expansion of the gauge fields and curvatures, and fix $D=3$ when constructing the action (gravity in $D>3$ is not CS; see, however, the Outlook).
Poincaré algebra and space-time splitting
-----------------------------------------
Our starting algebra $\mathcal{G}$ will be the Poincaré algebra in arbitrary dimensions, which in a certain basis can be described by the MC equations $$\begin{aligned}
\label{MCPoin}
d \tilde{e}^A &=& - \tilde{\omega}^A{}_B \wedge \tilde{e}^B \nonumber\\
d \tilde{\omega}^{AB} &=& -\tilde{\omega}^A{}_C \wedge \tilde{\omega}^{CB} \ ,\end{aligned}$$ where $A,B,C=0,\dots , D-1$. We will use a ‘mostly plus’ $(1,D-1)$ signature for the Minkowski metric $\eta_{AB}$. In order to perform an expansion leading to an extension of the Galilei algebra, we split the Poincaré algebra generators as follows: $$\begin{aligned}
\label{bosonicspl}
\tilde{e}^A & \rightarrow & ( \tilde{e}^a, \, \tilde{e}^0=\tilde{\phi}) \ , \nonumber\\
\tilde{\omega}^{AB} & \rightarrow & ( \tilde{\omega}^{ab},\, \tilde{\omega}^a{}_0= \tilde{\omega}^a)\ ,\end{aligned}$$ where $a=1,\dots , D-1$. In terms of these one-forms, the MC equations read $$\begin{aligned}
\label{MCPoinSp}
d \tilde{e}^a &=& - \tilde{\omega}^a{}_b \wedge \tilde{e}^b - \tilde{\omega}^a \wedge \tilde{\phi} \nonumber\\
d \tilde{\phi} &=& - \tilde{\omega}_a \wedge \tilde{e}^a \nonumber\\
d \tilde{\omega}^{ab} &=& -\tilde{\omega}^a{}_c \wedge \tilde{\omega}^{cb} - \tilde{\omega}^a \wedge \tilde{\omega}^b \nonumber\\
d \tilde{\omega}^a &=& -\tilde{\omega}^a{}_b \wedge \tilde{\omega}^b \ .\end{aligned}$$
As mentioned earlier, the gauge curvatures can be viewed as the two-forms that express the failure of the MC equations; then, the MC one-forms become the gauge one-form fields (again, denoted by the same symbols). The gauge curvatures of the Poincaré algebra are the two-forms $\widetilde{T}^A$, $\widetilde{R}^{AB}$ given by $$\begin{aligned}
\label{GaugePoin}
\widetilde{T}^A &=& d \tilde{e}^A + \tilde{\omega}^A{}_B \wedge \tilde{e}^B \\
\widetilde{R}^{AB} &=& d \tilde{\omega}^{AB} + \tilde{\omega}^A{}_C \wedge \tilde{\omega}^{CB}\ .\end{aligned}$$ By using the space-time splitting for the curvatures, $$\begin{aligned}
\label{bosonicsplG}
\widetilde{T}^A & \rightarrow & ( \widetilde{T}^a, \, \widetilde{T}^0=\widetilde{\Omega}) \ , \nonumber\\
\widetilde{R}^{AB} & \rightarrow & ( \widetilde{R}^{ab},\, \widetilde{R}^a{}_0= \widetilde{R}^a)\ ,\end{aligned}$$ the gauge curvatures $\widetilde{T}^a$, $\widetilde{\Omega}$, $\widetilde{R}^{ab}$ and $\widetilde{R}^a$ are given in terms of the gauge fields by $$\begin{aligned}
\label{GaugePoinSp}
\widetilde{T}^a &=& d \tilde{e}^a + \tilde{\omega}^a{}_b \wedge \tilde{e}^b + \tilde{\omega}^a \wedge \tilde{\phi} \nonumber\\
\widetilde{\Omega} &=& d \tilde{\phi} - \tilde{\omega}_a \wedge \tilde{e}^a \nonumber\\
\widetilde{R}^{ab} &=& d \tilde{\omega}^{ab} +\tilde{\omega}^a{}_c \wedge \tilde{\omega}^{cb} + \tilde{\omega}^a \wedge \tilde{\omega}^b \nonumber\\
\widetilde{R}^a &=& d \tilde{\omega}^a +\tilde{\omega}^a{}_b \wedge \tilde{\omega}^b \ .\end{aligned}$$ It is seen that the MC equations and are recovered when the curvatures are set to zero in and .
Expansion of the algebra and the $D=3$ action
---------------------------------------------
If we choose $V_0^*$ as the vector space generated by $\tilde{\omega}^{ab}, \tilde{\phi}$, and $V_1^*$ as the one generated by $\tilde{e}^a, \tilde{\omega}^a$, we have precisely the structure . Thus, we may perform the following consistent expansion in terms of a parameter $\lambda$: $$\label{BosExp}
\begin{array}{ll}
\tilde{e}^a= \lambda e^a + \sum_{k=1}^\infty \lambda^{2k+1} \tilde{e}^a_{(2k+1)}\, , &
\widetilde{T}^a =\lambda T^a + \sum_{k=1}^\infty \lambda^{2k+1} \widetilde{T}^a_{(2k+1)} \\
\tilde{\phi} = \phi +\lambda^2 \varphi + \sum_{k=2}^\infty \lambda^{2k} \tilde{\phi}_{(2k)}\, , & \widetilde{\Omega} = \Omega + \lambda^2 \Lambda + \sum_{k=2}^\infty \lambda^{2k} \widetilde{\Omega}_{(2k)} \\
\tilde{\omega}^{ab} = {\omega}^{ab}+ \lambda^2 \ell^{ab} + \sum_{k=2}^\infty \lambda^{2k} \tilde{\omega}^{ab}_{(2k)} \, , & \widetilde{R}^{ab} = R^{ab} + \lambda^2 L^{ab} + \sum_{k=2}^\infty \lambda^{2k} \widetilde{R}^{ab}_{(2k)}\\
\tilde{\omega}^a = \lambda \omega^a + \sum_{k=1}^\infty \lambda^{2k+1} \tilde{\omega}^a_{(2k+1)}\, , &
\widetilde{R}^a= \lambda R^a
+ \sum_{k=1}^\infty \lambda^{2k+1} \widetilde{R}^a_{(2k+1)}
\end{array}\ .$$ The expansion is infinite, but it may be cut in a consistent manner (see eq. ). As argued before, we will consider the finite algebra that contains all the fields that appear in a suitable term of the expanded action. More explicitly, let us start from the four-form $\widetilde{H}$ given by $$\begin{aligned}
\label{tildeHB}
\widetilde{H} = \epsilon_{ABC} \widetilde{R}^{AB} \wedge \widetilde{T}^{C} ,\end{aligned}$$ with $A,B,C = 0,1,2$. This form is closed, $d\widetilde{H} = 0$, so there exists a three-form $\widetilde{B}$ such that $d\widetilde{B}=\widetilde{H}$. The integral over three-dimensional spacetime gives an action that describes general relativity in three dimensions and in the absence of matter. We will however use $\widetilde{H}$, because it is much simpler to derive the field equations from it, and also exhibits the CS character of the action. Let us now rewrite $\widetilde{H}$ using the space-time splitting and : $$\label{tildeHBspl}
\widetilde{H} = 2 \epsilon_{ab} \widetilde{R}^{a} \wedge \widetilde{T}^{b} + \epsilon_{ab} \widetilde{R}^{ab}\wedge \widetilde{\Omega}\ ,$$ where $\epsilon_{ab}$ is the Levy-Civita symbol in $2$ dimensions, $\epsilon_{0 ab} = \epsilon_{ab}$.
Let us now replace the fields in by their expansions . This leads to an expansion of $\widetilde{H}$, $$\label{expH}
\widetilde{H} = \sum_{k=0}^{\infty} \lambda^k \widetilde{H}|_k \ ,$$ where the terms $\widetilde{H}|_k$ depend on the fields of the expansion and, since they are closed, define actions on these fields. The gauge algebra corresponding to a particular term $\widetilde{H}|_k$ will be the consistent truncation of the infinite expansion that has the gauge fields corresponding to the curvatures that it contains.
The lowest order term in $\lambda$ of the expansion of the first term of is $\lambda^{2} \epsilon_{ab} R^{a} \wedge T^{b} $. We need to keep this term if the resulting action has to be related with gravity, because we need $T^a=0$ and the contribution for the $\omega^a$ equation of this term will be of this sort. This means that our model corresponds to the term $\widetilde{H}|_{2}$ in . We now have to find out which curvatures appear in the $4$-form $\widetilde{H}$ of . By selecting the $\lambda^2$ in the expansion of $\widetilde{H}$, we obtain the four form $H$ given by $$\label{HD3B}
H= \widetilde{H}|_2= 2\epsilon_{a_1 a_2} R^{a_1} \wedge T^{a_2} + \epsilon_{a_1 a_2} L^{a_1a_2} \wedge \Omega + \epsilon_{a_1 a_2} R^{a_1a_2} \wedge \Lambda \ .$$ which means that the gauge curvatures for this model are $$\begin{aligned}
\label{gaugeB2}
T^a &=& de^a +\omega^a{}_b\wedge e^b \nonumber + \omega^a \wedge \phi \\ \Omega &=& d\phi \nonumber \\
\Lambda &=& d\varphi + \omega_a \wedge e^a \nonumber \\
R^{ab} &=& d\omega^{ab} +\omega^a{}_c \wedge \omega^{cb} \nonumber \\
L^{ab} &=& d\ell^{ab} +\omega^a{}_c \wedge \ell^{cb} + \ell^a{}_c \wedge \omega^{cb} +\omega^a\wedge \omega^b\nonumber \\
R^a &=& d\omega^a+ \omega^a{}_b \wedge \omega^b \ ,\end{aligned}$$ expressions that are valid for any $D$. For $D=3$, $R^{ab}$ reduces to $d\omega^{ab}$ and the second and third terms in the expression of $L^{ab}$ cancel each other. The corresponding Lie algebra (remember that the MC equations may be recovered by setting the curvatures to zero) is precisely the extension of the Bargmann algebra studied by Bergshoeff et al. in [@BR:16]. If we set the curvatures equal to zero, the MC forms $e^a$ are dual to the generators of space translations, $\phi$ is dual to the generator of the time translations, $\omega_a$ correspond to the Galilean boosts, and $\omega_{ab}$ to the rotations, while $\varphi$ and $\ell^{ab}$ are dual to commuting extension generators that determine, respectively, the Bargmann and the extended Bargmann algebra. Note that is closed and only depends on the curvatures. Hence $H$ is invariant under the gauge transformations of the algebra corresponding to , and therefore defines a CS action, which coincides with the bosonic sector of the one obtained in [@BR:16].
### On the physical dimensions of $\lambda$
We now comment on the issue of the physical dimensions of the expansion parameter $\lambda$. Although the expansion in terms of powers of the parameter $\lambda$ is formal, the gauge fields ultimately involved have physical dimensions. This is achieved in general by assigning a suitable dimension to the parameter $\lambda$. In [@AIPV:03], $D=3$ Poincaré supergravity was obtained by expanding a CS action based on a simple superalgebra. The generators of a simple algebra are dimensionless, and those of the superPoincaré algebra have to be dimensionful if they are to be associated with Poincaré supergravity, so $\lambda$ has to have dimensions. In our case, the starting Poincaré fields do have dimensions, but these are different from the dimensions of the fields in the expansion .
Let us start with the fields in the Poincaré action. We may choose $[\tilde{e}^A]=T$, while $\omega^{AB}$ has to be dimensionless. Since the metric is given in terms of the dreibein by$$\label{eg}
g_{\mu\nu} = e^A_\mu e_{A\nu}\ ,$$ where $e^A= e^A_\mu dx^\mu$ ($e^A_\mu$ are the coordinates of $e^A$ in the basis $dx^\mu$). If we take $[x^0]=T$, $[x^a]=L$, then $g_{00}$ is dimensionless and $[g_{ij}]=T^2L^{-2}$. This is compatible with the flat spacetime expression $$\label{mink}
\left(
\begin{array}{ccc}
-1 & 0 & 0 \\
0 & \frac{1}{c^2} & 0 \\
0 & 0 & \frac{1}{c^2} \\
\end{array}
\right) \ .$$ Now, consider the algebra obtained by setting the curvatures equal to zero in . Since $\omega^a$ correspond to the Galilean boosts, it makes sense to set $[\omega^a]=L T^{-1}$. Also, we would like to have $[e]=L^{-1}T$ as they are dual to the space translations. From we deduce that $[\lambda]= L^{-1}T$, that is, the inverse of a velocity. This argument, of course, is valid in every dimension $D$. This means that it is consistent, although not necessary in this context, to assume that the parameter $\lambda$ is equal to $c^{-1}$. For a construction that does include a $c^{-1}$ expansion, see [@HHO:19].
From $N=2$ superPoincaré to the extended superGalilei in $D=3$
==============================================================
We now start from the superPoincaré algebra in $D=3$, which in terms of its MC forms is given by $$\begin{aligned}
\label{sPD3}
d \widetilde{e}^A &=& -\widetilde{\omega}^A{}_B \wedge \widetilde{e}^B -i \overline{\widetilde{\psi}}
\gamma^A \wedge \widetilde{\psi} \nonumber\\
d \widetilde{\omega}^{AB} &=& -\widetilde{\omega}^A{}_C \wedge \widetilde{\omega}^{CB} \nonumber\\
d \widetilde{\psi} &=& - \frac{1}{4} \gamma^{AB} \widetilde{\omega}_{AB} \wedge \widetilde{\psi} \ ,\end{aligned}$$ where $A,B,C=0,1,2$ and we are using the $(-++)$ metric (we use the convention that complex conjugation reorders the product of Grassmann-odd symbols).
When the fermion one-forms $\widetilde{\psi}$ are complex, the algebra is that of $\mathcal{N}=2$ superPoincaré, and it is $\mathcal{N}=1$ when they are Majorana spinors. We are interested in obtaining a superGalilei algebra by expanding superPoincaré, with anticommutators of the type $\{ Q, Q\} \propto H$, where $Q$ is a $so(2)$ spinor supersymmetry generator and $H$ generates the time translations. Looking at the most general expansion it turns out that this requires starting from $\mathcal{N}=2$ superPoincaré. Additionally, this fact may be justified by noticing that we will need to split the original $so(1,2)$ spinor into two $so(2)$ spinors as suggested by the results of [@ABRS:13], but a real $so(1,2)$ spinor has two real components, so the $so(2)$ spinors must have one real component each. But this would correspond to Majorana-Weyl spinors, which do not exist in $D=2$ with signature $(++)$ (although they do exist when the signature is $(-+)$). So we are forced to consider the case $\mathcal{N}=2$. In what follows our spinors will be complex, with no reality condition assumed.
Let us now perform the space-time splitting including the fermions. First, we take $\gamma^A$ real for convenience; for instance, $$\label{D3realgamma}
\gamma^0=i\sigma^2\ ,\quad \gamma^1=\sigma^1 \ ,\quad \gamma^2=\sigma^3 \ .$$ We then define the following one-forms: $$\begin{aligned}
\label{fermionicspl}
\tilde{e}^A & \rightarrow & ( \tilde{e}^a, \, \tilde{e}^0=\tilde{\phi}) \ , \nonumber\\
\tilde{\omega}^{AB} & \rightarrow & ( \tilde{\omega}^{ab},\, \tilde{\omega}^a{}_0= \tilde{\omega}^a)\nonumber\\
\tilde{\psi} &=& P_+ \tilde{\xi}_+ + P_- \tilde{\xi}_- \ \quad \left( \overline{\tilde{\psi}} = \overline{\tilde{\xi}_+} P_+ + \overline{\tilde{\xi}_-} P_-\right)\ ,\end{aligned}$$ where $P_\pm= \frac{1}{2}(1\pm i\gamma_0)$, and $\tilde{\xi}_\pm$ are real, as can be seen from $$\label{lambdapsi}
\tilde{\xi}_\pm = \textrm{Re}\, \tilde{\psi} \pm \gamma^0 \textrm{Im}\, \tilde{\psi} \ .$$ In terms of these forms, the MC equations read $$\begin{aligned}
\label{MCspPoinSp}
d \tilde{e}^a &=& - \tilde{\omega}^a{}_b \wedge \tilde{e}^b - \tilde{\omega}^a \wedge \tilde{\phi}-i \overline{\tilde{\xi}}_+ \gamma^a \wedge \tilde{\xi}_- \nonumber\\
d \tilde{\phi} &=& - \tilde{\omega}_a \wedge \tilde{e}^a + \frac{i}{2} \tilde{\xi}^t_+ \wedge \tilde{\xi}_+ + \frac{i}{2} \tilde{\xi}^t_- \wedge \tilde{\xi}_-\nonumber\\
d \tilde{\omega}^{ab} &=& -\tilde{\omega}^a{}_c \wedge \tilde{\omega}^{cb} - \tilde{\omega}^a \wedge \tilde{\omega}^b \nonumber\\
d \tilde{\omega}^a &=& -\tilde{\omega}^a{}_b \wedge \tilde{\omega}^b\nonumber\\
d \tilde{\xi}_\pm &=& -\frac{1}{4} \omega_{ab} \gamma^{ab}\wedge \tilde{\xi}_\pm -\frac{1}{2}
\gamma^a \tilde{\omega}_a \gamma^0 \wedge \tilde{\xi}_\mp \ .\end{aligned}$$ As before, the MC one-forms become the gauge one-forms (denoted by the same letters), and the gauge curvatures of the Poincaré algebra are the two-forms $\widetilde{T}^A$, $\widetilde{R}^{AB}$, $\tilde{\rho}$ given by $$\begin{aligned}
\label{GaugesPoin}
\widetilde{T}^A &=& d \tilde{e}^A + \tilde{\omega}^A{}_B \wedge \tilde{e}^B +i \overline{\tilde{\psi}} \gamma^A \wedge \tilde{\psi} \nonumber \\
\widetilde{R}^{AB} &=& d \tilde{\omega}^{AB} + \tilde{\omega}^A{}_C \wedge \tilde{\omega}^{CB}\nonumber
\\
\tilde{\rho} &=& d\tilde{\psi} +\frac{1}{4} \tilde{\omega}_{AB}\gamma^{AB}\wedge \tilde{\psi} \ .\end{aligned}$$ By using the space-time splitting for the curvatures, $$\begin{aligned}
\label{fermionicsplG}
\widetilde{T}^A & \rightarrow & ( \widetilde{T}^a, \, \widetilde{T}^0=\widetilde{\Omega}) \ , \nonumber\\
\widetilde{R}^{AB} & \rightarrow & ( \widetilde{R}^{ab},\, \widetilde{R}^a{}_0= \widetilde{R}^a)\ ,\nonumber\\
\tilde{\rho}&=& P_+ \widetilde{\Xi}_+ + P_- \widetilde{\Xi}_- \ ,\end{aligned}$$ the gauge curvatures $\widetilde{T}^a$, $\widetilde{\Omega}$, $\widetilde{R}^{ab}$, $\widetilde{R}^a$ and $\widetilde{\Xi}_\pm$ are given in terms of the gauge fields by $$\begin{aligned}
\label{GaugePoinSp1}
\widetilde{T}^a &=& d \tilde{e}^a + \tilde{\omega}^a{}_b \wedge \tilde{e}^b + \tilde{\omega}^a \wedge \tilde{\phi} + i \overline{\tilde{\xi}}_+ \gamma^a \wedge \tilde{\xi}_- \nonumber\\
\widetilde{\Omega}&=& d \tilde{\phi} + \tilde{\omega}_a \wedge \tilde{e}^a - \frac{i}{2} \tilde{\xi}^t_+ \wedge \tilde{\xi}_+ - \frac{i}{2} \tilde{\xi}^t_- \wedge \tilde{\xi}_-\nonumber\\
\widetilde{R}^{ab} &=& d \tilde{\omega}^{ab} +\tilde{\omega}^a{}_c \wedge \tilde{\omega}^{cb} + \tilde{\omega}^a \wedge \tilde{\omega}^b \nonumber\\
\widetilde{R}^a &=& d \tilde{\omega}^a +\tilde{\omega}^a{}_b \wedge \tilde{\omega}^b\nonumber\\
\widetilde{\Xi}_\pm &=& d\tilde{\xi}_\pm +\frac{1}{4} \omega_{ab} \gamma^{ab}\wedge \tilde{\xi}_\pm +\frac{1}{2}
\gamma^a \tilde{\omega}_a \gamma^0 \wedge \tilde{\xi}_\mp \ .\end{aligned}$$
We now expand the gauge one-forms and curvature two-forms contained in the gauge algebra . To do this, we notice that if we make the choice $V_0^*=\{\tilde{\omega}^{ab},\tilde{\phi},\tilde{\xi}_+\}$ and $V_1^*=\{\tilde{e}^a, \tilde{\omega}^a, \tilde{\xi}_-\}$, then the superalgebra has the structure . So we may write $$\label{FermExp}
\begin{array}{ll}
\tilde{e}^a= \lambda e^a + \sum_{k=1}^\infty \lambda^{2k+1} \tilde{e}^a_{(2k+1)}\, , &
\widetilde{T}^a =\lambda T^a + \sum_{k=1}^\infty \lambda^{2k+1} \widetilde{T}^a_{(2k+1)} \\
\tilde{\phi} = \phi +\lambda^2 \varphi + \sum_{k=2}^\infty \lambda^{2k} \tilde{\phi}_{(2k)}\, , & \widetilde{\Omega} = \Omega + \lambda^2 \Lambda + \sum_{k=2}^\infty \lambda^{2k} \widetilde{\Omega}_{(2k)} \\
\tilde{\omega}^{ab} = {\omega}^{ab}+ \lambda^2 \ell^{ab} + \sum_{k=2}^\infty \lambda^{2k} \tilde{\omega}^{ab}_{(2k)} \, , & \widetilde{R}^{ab} = R^{ab} + \lambda^2 L^{ab} + \sum_{k=2}^\infty \lambda^{2k} \widetilde{R}^{ab}_{(2k)}\\
\tilde{\omega}^a = \lambda \omega^a + \sum_{k=1}^\infty \lambda^{2k+1} \tilde{\omega}^a_{(2k+1)}\, , &
\widetilde{R}^a= R^a
+ \sum_{k=1}^\infty \lambda^{2k+1} \widetilde{R}^a_{(2k+1)}\\
\tilde{\xi}_+= \psi +\lambda^2 \xi + \sum_{k=2}^\infty \lambda^{2k} \tilde{\xi}_+{}_{(2k)}\, , &
\widetilde{\Xi}_+ = \rho + \lambda^2 \Xi + \sum_{k=2}^\infty \lambda^{2k} \widetilde{\Xi}_+{}_{(2k)}\\
\tilde{\xi}_-= \lambda \pi + \sum_{k=1}^\infty \lambda^{2k+1} \tilde{\xi}_-{}_{(2k+1)}\, , &
\widetilde{\Xi}_- = \lambda \Pi + \sum_{k=1}^\infty \lambda^{2k+1} \widetilde{\Xi}_-{}_{(2k)}
\end{array}\ .$$ Since we need to select the $\lambda^2$ term in the expansion of the $\mathcal{N}=2$, $D=3$ supergravity action, we shall consistently cut the expansion at the power $\lambda^2$. The resulting gauge algebra is given by $$\begin{aligned}
\label{gaugeF2}
T^a &=& de^a +\omega^a{}_b\wedge e^b+ \omega^a \wedge \phi +i \bar{\psi} \gamma^a\wedge \pi\nonumber \\
\Omega &=& d\phi -\frac{i}{2} \psi^t \wedge \psi \nonumber \\
\Lambda &=& d\varphi + \omega_a \wedge e^a -i\psi^t\wedge \xi - \frac{i}{2} \pi^t \wedge \pi\nonumber \\
R^{ab} &=& d\omega^{ab} +\omega^a{}_c \wedge \omega^{cb} \nonumber \\
L^{ab} &=& d\ell^{ab} +\omega^a{}_c \wedge \ell^{cb} + \ell^a{}_c \wedge \omega^{cb} +\omega^a\wedge \omega^b\nonumber \\
R^a &=& d\omega^a+ \omega^a{}_b \wedge \omega^b \nonumber\\
\rho &=& d\psi + \frac{1}{4} \omega_{ab}\gamma^{ab} \wedge \psi\nonumber\\
\Xi &=& d\xi + \frac{1}{4} \omega_{ab}\gamma^{ab} \wedge \xi + \frac{1}{4} \ell_{ab}\gamma^{ab} \wedge \psi +
\frac{1}{2} \gamma^a \omega_a\gamma^0\wedge \pi\nonumber\\
\Pi &=& d\pi + \frac{1}{4} \omega_{ab}\gamma^{ab} \wedge \pi + \frac{1}{2} \gamma^a \omega_a\gamma^0\wedge \psi\ .\end{aligned}$$ Again, the MC equations of the algebra are recovered setting all curvatures equal to zero. The bosonic subalgebra is precisely the extended bosonic Bargmann algebra of . Eqs, contain also three real $so(2)$ odd spinor gauge one-forms $\psi$, $\xi$ and $\pi$ and three fermionic curvatures $\rho, \Xi$ and $\Pi$.
Dual version of the algebra
---------------------------
Let us find the (anti)commutators of the generators dual to the MC forms of the algebra obtained from , when the curvatures vanish. Since in our $D=3$ case $a=1,2$, in order to make contact with [@ABRS:13] we write $\omega_{ab}=\epsilon_{ab}\omega$, $\ell_{ab}=\epsilon_{ab}q$ so the MC equations read $$\begin{aligned}
\label{MCF2}
de^a &=& -\epsilon^a{}_b\omega \wedge e^b- \omega^a \wedge \phi -i \bar{\psi} \gamma^a\wedge \pi\nonumber \\
d\phi &=&- \frac{i}{2} \psi^t \wedge \psi \nonumber \\
d\varphi &=& -\omega_a \wedge e^a +i\psi^t\wedge \xi + \frac{i}{2} \pi^t \wedge \pi\nonumber \\
d\omega &=& 0 \nonumber \\
dq &=& -\frac{1}{2}\epsilon_{ab} \omega^a\wedge \omega^b\nonumber \\
d\omega^a&=& -\epsilon^a{}_b \omega\wedge \omega^b \nonumber\\
d\psi &=& - \frac{1}{2} \omega \gamma^{0} \wedge \psi\nonumber\\
d\xi &=& - \frac{1}{2} \omega\gamma^{0} \wedge \xi - \frac{1}{2} q\gamma^{0} \wedge \psi
-\frac{1}{2} \gamma^a \omega_a\gamma^0\wedge \pi\nonumber\\
d\pi &=& \frac{1}{2} \omega\gamma^{0} \wedge \pi + \frac{1}{2} \gamma^a \omega_a\gamma^0\wedge \psi\ .\end{aligned}$$ Now we call the generators dual to $e^a$, $\phi$, $\varphi$, $\omega$, $q$, $\omega^a$, $\psi$, $\xi$ y $\pi$, $P_a$, $H$, $M$, $J$, $S$, $G^a$, $Q^+$, $U$ and $Q^-$ respectively. A convenient way of finding the commutators is using of the canonical one-form $$\label{canonical1}
\theta = e^a P_a +\phi H + \varphi M+ \omega J+ q S + \omega^a G_a+ \psi^\alpha Q^+_\alpha + \xi^\alpha U_\alpha + \pi^\alpha Q^+_\alpha\ .$$ In terms of $\theta$ the MC equations $d\theta= -\theta \wedge \theta$ lead immediately to the superalgebra commutators simply by inserting $\theta= \omega^I X_I$. In this way, the following non-zero (anti)-commutators are obtained: $$\begin{aligned}
\label{explie}
& & \left[G_a,H \right] =P_a \ ,\quad \left[G_a, P_b\right]= \delta_{ab} M \ ,\quad
\left[G_a, G_b\right] =\epsilon_{ab} S\ ,\nonumber\\
& &\left\{Q_\alpha^+, Q_\beta^+ \right\}= i\delta_{\alpha\beta}H \ ,\quad
\left\{Q^+_\alpha, Q^-_\beta\right\}= -i(\gamma^0\gamma^a)_{\alpha\beta} P_a \ ,\nonumber\\
& & \{ Q^-_\alpha, Q^-_\beta\}= -i \delta_{\alpha\beta}M \ ,\quad \left\{Q^+_\alpha, U_\beta \right\} =-i \delta_{\alpha\beta}M \ ,\nonumber\\
& & \left[S, Q^+_\alpha\right] = \frac{1}{2}(\gamma^0)^\beta{}_\alpha U_\beta\ ,\quad
\left[G_a, Q^+_\alpha\right] = \frac{1}{2}(\gamma^0\gamma^a)^\beta{}_\alpha Q^-_\beta \ ,\nonumber\\
& & \left[G_a, Q^-_\alpha\right] = \frac{1}{2}(\gamma^0\gamma^a)^\beta{}_\alpha U_\beta \ ,\nonumber\\
& & \left[J, P_a\right]=-\epsilon_{ab} P^b \ ,\quad \left[ J, G_a\right]= -\epsilon_{ab} G^b \ ,\nonumber\\
& & \left[J, Q^{\pm}_\alpha\right]= \frac{1}{2}(\gamma^0)^\beta{}_\alpha Q_\beta^{\pm} \ , \quad
\left[J, U_\alpha\right] = \frac{1}{2}(\gamma^0)^\beta{}_\alpha U_\beta\ .\end{aligned}$$
The last two lines exhibit the semidirect structure of the algebra, $J$ being the generator of the two-dimensional rotations. The first line is the extended Bargmann algebra (omitting rotations), where $S$ is the generator of the central extension; the second and third lines contain the anticommutators of the fermionic generators and the fourth and the fifth one give the commutators of bosonic and fermionic generators, excluding the rotations. Apart from minor redefinitions, these commutators coincide with those of [@BR:16].
Expansion of the action
-----------------------
The next step is to expand the action, or equivalently $\widetilde{H}$, of $D=3$, $\mathcal{N}=2$ supergravity and select the coefficient of the $\lambda^2$ term. To this end, we need to start with the action of D=3 Poincaré supergravity. It is given by $$\label{Hdef}
\widetilde{H}= \epsilon^{ABC} \widetilde{R}_{AB}\wedge \widetilde{T}_C-4i \overline{\tilde{\rho}}\wedge \tilde{\rho}\ ,$$ where $\overline{\tilde{\rho}}$ is the Dirac adjoint of $\tilde{\rho}$. In order to check that the four-form $\widetilde{H}$, which depends only on the curvatures is closed and thus defines a CS action, we have used that, with the choice of gamma matrices, $\gamma^0\gamma^1\gamma^2= i\sigma^2\sigma^1\sigma^3=I_3$. Thus, if we define $\epsilon^{012}=1$, then $\gamma^{ABC}=\epsilon^{ABC}$. This in turn gives $$\label{gammaabc}
\gamma^{AB}= \epsilon^{ABC} \gamma_C \ .$$ When calculating the exterior differential of $\widetilde{H}$, we have used the expression of the differentials of the curvatures $\widetilde{R}_{AB}$, $\widetilde{T}_C$ and $\tilde{\rho}$, $$\begin{aligned}
\label{cFPoin}
d\widetilde{R}_{AB} &=& \widetilde{R}_{AC}\wedge \tilde{\omega}^C{}_A-
\tilde{\omega}_{AC}\wedge \widetilde{R}^C{}_A\ ,\quad (D\widetilde{R}_{AB} =0) \nonumber\\
d\widetilde{T}^A &=& \widetilde{R}^A{}_B \wedge \tilde{e}^B - \tilde{\omega} ^A{}_B \wedge \widetilde{T}^B + i \overline{\tilde{\rho}} \gamma^A\wedge \tilde{\psi}- i\overline{\tilde{\psi}} \gamma^A\wedge \tilde{\rho} \ ,\nonumber\\
& & \quad\quad\quad (D\widetilde{T}^A = \widetilde{R}^A{}_B \wedge e^B + i \overline{\tilde{\rho}} \gamma^A\wedge \tilde{\psi}- i\overline{\tilde{\psi}} \gamma^A\wedge \tilde{\rho})\nonumber\\
d \tilde{\rho} &=& \frac{1}{4} \widetilde{R}_{AB} \gamma^{AB}\wedge \tilde{\psi} - \frac{1}{4} \tilde{\omega}_{AB} \gamma^{AB}\wedge \tilde{\rho}\ ,\nonumber\\
&& \quad\quad\quad (D \tilde{\rho} = \frac{1}{4} \widetilde{R}_{AB} \gamma^{AB}\wedge \tilde{\psi}\ ,\ D \overline{\tilde{\rho}} = \frac{1}{4} \overline{\tilde{\psi}} \wedge \widetilde{R}_{AB} \gamma^{AB}) \ ,\end{aligned}$$ where $D$ is the Lorentz covariant exterior differential. Using Eqs. and the gamma matrix identity , the differential of $\widetilde{H}$ in is seen to vanish.
We have to rewrite in terms of the spacetime splitting to perform the expansion. The result is (we write $\epsilon_{0ab}=\epsilon_{ab}$) $$\label{Hdefspl}
\widetilde{H}=2 \epsilon_{ab} \widetilde{R}^a\wedge \widetilde{T}^b + \epsilon_{ab} \widetilde{R}^{ab}
\wedge \widetilde{\Omega} -2i \overline{\tilde{\Xi}}_+\wedge \tilde{\Xi}_+
-2i\overline{\tilde{\Xi}}_-\wedge \tilde{\Xi}_- \ .$$ We now expand the gauge one-forms and curvature two-forms as in and insert the expansion into . Then, we select the $\lambda^2$ term, which is given by $$\begin{aligned}
\label{H2ferm}
H=\widetilde{H}|_2 & =& 2\epsilon_{ab} R^a\wedge T^b + \epsilon_{ab} R^{ab} \wedge \Lambda +\epsilon_{ab} L^{ab} \wedge \Omega \nonumber\\
& & -4i \bar{\rho} \wedge \Xi -2i \widetilde{\Pi} \wedge \Pi \ .\end{aligned}$$ The equations of the action $I=\int_{\mathcal{M}^{3}} B$, where $dB=H$ and $\mathcal{M}^{3}$ denotes the two-dimensional space and time, are given by the vanishing of all the curvatures included in , and, since $H$ is gauge invariant under the gauge transformations that correspond to the gauge algebra , then $I$ will be invariant too, up to topological effects. The action obtained coincides with that of [@BR:16]
Outlook
=======
In this paper we have shown how to obtain the Galilean (super)gravity action in $D=3$ by using the Lie (super)algebras expansion method. Although this method may give less physical insight than the procedures based on limits, it has the advantage of being systematic and involving simpler calculations.
We have applied here our method to the $D=3$ case, but in principle it could be applied to higher dimensions, provided the starting action is one that can be expressed as the integral over spacetime of a differential form constructed from the fields and curvatures of a certain Lie (super)algebra. Since for $D>4$ the starting (super)gravity action will no longer be gauge invariant, a question to be answered is to what extent the actions obtained by expansion are invariant under some symmetries of the expanded algebra.
The expansion method has recently been used in [@BIOR:19] to derive general actions for any $D$ and $p$-brane[^3], thus recovering some known examples of actions existing in the literature, and providing a way of reproducing others, such as Carrol gravity [@BGRRV:17; @H:15]. This indicates that our method, at least in the bosonic case, can be applied when $D>3$. So it is natural to think that maybe this is also possible in the supersymmetric case. One potential problem is the fact that, in general, Poincaré supergravity with $N=2$ is required as the starting point. In $D=4$, for instance, the first order supergravity action contains not only the gauge fields of a centrally extended $N=2$ superPoincaré algebra, but also some auxiliary zero-forms that are needed to write in first order form the kinetic term of the gauge field associated with the central extension generator (see [@CDF:91]). So this problem may presumably be overcome by applying the expansion method to general free differential algebras, the gauge algebra being just an example.
Another difficulty is the local supersymmetry of the actions obtained by expansion. In [@BIOR:19], it was shown that the actions do possess local symmetries corresponding to the generators of their algebras, but the argument given there will not be applicable in the case of supersymmetry. Also, it is well known that in the case of Poincaré supergravity the supersymmetry variations realize the supersymmetry algebra up to field equations. It is not clear whether this is going to be the case when applying the expansion procedure. One possible approach is to take as the starting point the action with auxiliary fields that ensure the closure of the supersymmetry algebra off-shell but, also here, the auxiliary fields do not correspond to gauge fields of a Lie algebra, so they should be treated as zero forms of a free differential algebra.
Acknowledgements
================
This work has been partially supported by the grants MTM2014-57129-C2-1-P from the MINECO (Spain), VA137G18 Spanish Junta de Castilla y León and FEDER BU229P18. Useful conversations with E. Bergshoeff, T. Ortín and L. Romano are also appreciated.
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[^1]: For D=3 constructions based on the Nappi-Witten and the AdS-Lorentz algebras see [@PS:19] and [@CR:19], and [@GRSS:16] for D = 5 starting from a CS gravity.
[^2]: To avoid complicating the notation, in this paper we will use the same symbols to denote the MC one-forms and the corresponding gauge one-form fields.
[^3]: An example of string ($p=1$) action was found in [@GO:01]
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{
"pile_set_name": "ArXiv"
}
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\#1[Contribution ]{} \#1[Chapter \[Chapter\#1\]]{}
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{
"pile_set_name": "ArXiv"
}
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---
abstract: 'By means of first principles calculations, we computed the effective electron-phonon coupling constant $G_0$ governing the electron cooling in photoexcited bismuth. $G_0$ strongly increases as a function of electron temperature, which can be traced back to the semi-metallic nature of bismuth. We also used a thermodynamical model to compute the time evolution of both electron and lattice temperatures following laser excitation. Thereby, we simulated the time evolution of (1 -1 0), (-2 1 1) and (2 -2 0) Bragg peak intensities measured by Sciaini [*[ et al]{}*]{} \[Nature [**458**]{}, 56 (2009)\] in femtosecond electron diffraction experiments. The effect of the electron temperature on the Debye-Waller factors through the softening of all optical modes across the whole Brillouin zone turns out to be crucial to reproduce the time evolution of these Bragg peak intensities.'
author:
- 'B. Arnaud$^{1}$ and Y. Giret$^{2,3}$'
date:
-
-
title: 'Electron cooling and Debye-Waller effect in photoexcited bismuth '
---
Femtosecond time resolved X-ray or electron diffraction emerges as a unique tool to study laser-induced structural dynamics with a picometer spatial and femtosecond temporal resolution. A large class of systems including organic compounds[@collet_2003], correlated materials[@cavalleri_2001; @baum_2007] and semi-metallic systems like graphite[@carbone_2008; @raman_2008] or bismuth[@sokolowski_2003; @fritz_2007; @beaud_2007; @johnson_2008; @johnson_2009; @sciaini_2009] have been investigated, providing enlightening information about atomic motions following laser excitation. However, it is often difficult to interpret the time evolution of all Bragg peak intensities and to understand the different time scales showing up in these experiments because models or calculations describing laser-induced electron-phonon processes are lacking.
A first step towards a better understanding of time resolved X-ray diffraction experiments performed on bismuth[@fritz_2007] was recently achieved[@giret_2011]. Indeed, a thermodynamical model based on the assumption that an electron temperature $T_e$ and a lattice temperature $T_l$ can be defined at any time delay following the laser pulse arrival, was shown to reproduce the (111) bragg peak intensities measured by Fritz [*[et al]{}*]{}[@fritz_2007] for time delays up to $\sim 2.5$ ps. While most of the parameters encompassed in the model were obtained from first principles calculations, the effective electron-phonon coupling constant $G_0$ governing the electron cooling was taken as a temperature independent parameter fitted to reproduce the measured Bragg intensities. Obviously, knowledge of the temperature dependence of $G_0$ is mandatory to perform simulations over longer time scales.
In this letter, we computed the phonon spectrum of Bi for a large range of electron temperatures and calculated the effective electron-phonon coupling constant $G_0(T_e)$ using the theory developped by Allen[@allen_1987] which is applied for the first time to a material showing a strongly varying electron density of states at the Fermi level. We found that all the optical modes are affected by an increase of $T_e$ and that $G_0$ strongly increases as $T_e$ increases. Then, we solved the model proposed in Ref.[@giret_2011] and simulated the intensity decay of Bragg peaks measured by Sciaini [*[ et al]{}*]{}[@sciaini_2009] in femtosecond electron diffraction (FED) experiments for time delays up to 14 ps. Our calculations firmly establish that the softening of all optical modes is crucial to achieve a good agreement between experiment and theory. This finding parallels that of Johnson [*[et al]{}*]{}[@johnson_2009] who scale all phonon modes by an empirical constant factor smaller than one to reproduce their normalized diffracted intensities.
All the ab-initio calculations were performed within the framework of the local density approximation (LDA) using the ABINIT code[@gonze_2009]. Spin-orbit coupling was included and an energy cut-off of 15 Hartree in the planewave expansion of wavefunctions as well as a $16\times 16\times 16$ kpoint grid for the Brillouin zone integration were used. We computed the evolution of the phonon spectrum as a function of $T_e$ using density functional perturbation theory (DFPT)[@gonze_1997]. Our calculations show that the phonon spectrum is not affected by $T_e$ below 600 K. However, an increase of $T_e$ above this critical temperature leads to a redshift of all optical phonon modes. Figure 1(a) shows the calculated phonon density of states for $T_e=0$ K and $T_e=2100$ K. All the optical modes are softened for $T_e=2100$ K whereas the acoustic modes are practically unaffected. This effect has already been highlighted in other theoretical works[@murray_2007; @zijlstra_2010] and should be contrasted with the bond hardening predicted theoretically[@recoules_2006] and observed experimentally in gold[@ernstorfer_2009].
Knowledge of the phonon spectrum is essential to compute the decay of electron energy due to electron-phonon interaction. A key dimensionless quantity is the so called generalized Eliashberg function[@allen_1987] defined by $$\begin{aligned}
\label{spectral_function1}
\alpha^2F(\epsilon, \epsilon^\prime, \omega)&=&\frac{2}{\hbar N_k N_q g(\epsilon_F)}
\sum_{{\bf{k}}, {\bf{q}}, n, m, \lambda} |g_{{\bf q}\lambda}^{{\bf{k}} n m}|^2
\delta[\omega-\omega_{\bf{q}\lambda}] \nonumber \\
& &\times \delta[\epsilon-\epsilon_{{\bf{k}}n}]
\delta[\epsilon^\prime-\epsilon_{{\bf{k+q}}m}],\end{aligned}$$ where the sum is performed on $N_k$ ($N_q$) electron (phonon) wavevectors over the Brillouin zone, $g(\epsilon_F)$ is the density of states per unit cell at the Fermi level, $\epsilon_{{\bf{k}}n}$ are Kohn-Sham energies for band $n$ and wavevector ${\bf{k}}$ and where the electron-phonon matrix elements read $$\label{matrix_elements}
g_{{\bf q}\lambda}^{{\bf{k}} n m}=\sum_{p,\alpha} \sqrt{\frac{\hbar}{2 M \omega_{\bf{q}\lambda}}}
e_p^\alpha({\bf q}, \lambda) \langle {\bf{k+q}}m |\frac{\delta V^{SCF}}{\delta u_p^\alpha({\bf q})} |{\bf{k}}n\rangle$$ where $e_p^\alpha({\bf q}, \lambda)$ is the displacement of the p$^{th}$ atom of mass $M$ in the direction $\alpha$ for the mode $({\bf q}, \lambda)$ of frequency $\omega_{\bf{q}\lambda}$. By solving the Boltzmann equation for both the electron and phonon systems, one can show that the variation of electron energy per unit volume is given by $$\label{electronic_energy1}
\frac{\partial E_e}{\partial t}=G_0(T_e)\times(T_l-T_e)$$ provided that the lattice temperature $T_l$ is larger than the Debye temperature $\theta_D\simeq 119$ K of Bi. The effective electron-phonon coupling constant $G_0(T_e)$ reads[@allen_1987] $$\begin{aligned}
\label{electronic_energy2}
G_0(T_e)& = &\frac{2\pi g(\epsilon_F) k_B}{v} \int d\omega~
\int d\epsilon~ \alpha^2F(\epsilon, \epsilon+\hbar\omega, \omega) \nonumber \\
& & \times [f(\epsilon)-f(\epsilon+\hbar\omega)],\end{aligned}$$ where $v$ is the unit cell volume, $k_B$ is the Boltzmann constant, and $f(\epsilon)$ is the Fermi-Dirac distribution at temperature $T_e$. This expression shows that the integration over $\epsilon$ should be done in a range of energy around $\epsilon_F$ whose width increases with $T_e$. The usual approximation is tantamount to neglecting the energy dependence of $\alpha^2F(\epsilon, \epsilon+\hbar\omega, \omega)$ which is then approximated by $\alpha^2F(\epsilon_F, \epsilon_F, \omega)\equiv \alpha^2F(\omega)$ when the phonon energies are also neglected. Thus, the electron-phonon coupling constant becomes independent of $T_e$ and is given by $$\label{electronic_energy3}
G_0= \frac{\pi}{v\hbar} k_B \lambda\langle \omega^2\rangle g(\epsilon_F),$$ where $\lambda\langle \omega^2\rangle$ is the second moment of the Eliashberg function $\alpha^2F(\omega)$. Eq. \[electronic\_energy3\] is justified for low electron temperatures or for all electron temperatures when the electron density of states (DOS) is only weakly energy dependent. The last assumption is far from being satisfied since Bi is a semi-metallic material with a strongly varying DOS near the Fermi level. Therefore, a temperature dependent $G_0$ is expected.
1.0truecm ![\[phonon-curves\] (color online) a/ Calculated phonon density of states in meV$^{-1}$ per cell for $T_e=0$ K (thin line) and $T_e=2100$ K (thick line). The inset shows the fraction of acoustic (TA+LA) and optical (TO and LO) phonons contributing to the effective electron-phonon coupling $G_0$ for electron temperature $T_e$ ranging from 294 K to 4000 K. b/ Calculated effective electron-phonon coupling constant $G_0$ (in W.m$^{-3}$.K$^{-1}$) as a function of $T_e$ (in K) compared to the values fitted in Ref.[@giret_2011] to reproduce the time evolution of (111) Bragg peak intensities measured by Fritz [*et al*]{}[@fritz_2007]. The inset shows the calculated bandstructure of Bi along high symmetry lines for $T_e=0$ K. ](figure1.eps "fig:"){width="8.5cm"}
The ab-initio calculated electron-phonon coupling constant $G_0$ is shown in Fig. \[phonon-curves\](b). The low temperature value is well described by Eq. \[electronic\_energy3\] which explains why $G_0$ is so small since $g(\epsilon_F)\sim 1.94\times 10^{-2}$ eV$^{-1}$ per cell. The huge increase of $G_0$ as $T_e$ increases can be easily interpreted by looking at the electronic structure of Bi depicted in the inset of Fig. \[phonon-curves\](b). At low $T_e$, only intraband scattering processes involving nearly zone center phonons or interband scattering processes involving phonons connecting the L pocket to the T pockets participate to the energy exchange between the electrons and the lattice. Thus, the energy is transferred to only a few phonon modes in this regime. When $T_e$ increases, the number of phonon modes participating in the intraband scattering processes increases tremendously because of larger ${\bf{q}}$-wavectors allowed by momentum and energy conservation. Consequently, $G_0$ increases by four orders of magnitude when $T_e$ increases from 100 K to 4000 K. The inset in Fig. \[phonon-curves\](a) shows that both optical and acoustic phonons contribute to electron cooling. At 2100 K, transverse optical phonons (TO), longitudinal optical phonons (LO) and acoustic phonons (TA+LA) respectively contribute for 51.6, 27.1 and 21.3 % to the total electron-phonon coupling constant $G_0$. In Ref.[@giret_2011], the electron-phonon coupling constants were fitted to reproduce the time evolution of (111) Bragg peak intensities measured by Fritz [*[et al]{}*]{}[@fritz_2007] for four different pump laser fluences. All the fitted values denoted as stars in Fig. \[phonon-curves\](b) are underestimated by $\sim$ 20% with respect to the calculated values. However, the (111) Bragg peak intensities are still perfectly reproduced by using our calculated $G_0(T_e)$, which was also succesfully used to reproduce the time evolution of the electron temperature inferred from time-resolved photoemission experiments[@papalazarou_2012]. We next calculate the mean square displacement of atoms with respect to their equilibrium positions as a function of both lattice temperature $T_l$ and electron temperature $T_e$. Defining the displacement of the p$^{th}$ atom in the unit cell in the direction $\alpha$ as $u_p^\alpha$, one can show that $$\label{mean_squared_displacement}
\langle u_p^\alpha u_p^\beta\rangle= \frac{\hbar}{2 N_q M }\sum_{{\bf q}, \lambda}
\frac{1}{\omega_{{\bf q}\lambda}} e_p^\alpha({\bf q}, \lambda) e_p^\beta(-{\bf q}, \lambda)
\left[1+ 2 n_{{\bf q}, \lambda } \right]
%\left[1+ 2 \langle n_{{\bf q}, \lambda } \rangle \right]$$ where $n_{{\bf q}, \lambda } $ is the Bose occupation factor at temperature $T_l$ for a phonon with frequency $\omega_{{\bf q}\lambda}$. The mean values defined by Eq. \[mean\_squared\_displacement\] have been calculated using the method introduced by Lee and Gonze[@lee_1995]. Our calculations show that the anisotropy in the mean square displacements is negligible. Figure \[mean-squared-displacment\](a) displays the mean square displacement of one Bismuth atom (in Å$^2$) for $T_e=$ 294 K as a function of $T_l$ (thin solid line). The root mean square displacement increases from 0.070 Å at $T_l$=0K to 0.241 Å at $T_l$=294 K. The room temperature value is slightly overestimated with respect to the experimental value of 0.21 Å extracted from LEED experiments[@monig_2005].
1.0truecm ![\[mean-squared-displacment\] (color online) a/Mean square displacements (in Å$^2$) of one Bi atom as a function of lattice temperature T$_l$ (in K) from [*ab initio*]{} calculations for $T_e$=294 K and $T_e$=2100 K (thin and thick solid lines). The vertical line indicates the melting temperature of Bi. b/Calculated normalized intensities defined by $\textrm{I}_{\textrm{hkl}}(T_l, T_e=T_l)/
\textrm{I}_{\textrm{hkl}}(T_l=294 ~{\textrm K}, T_e=294 ~{\textrm K})$ for (1 -1 0), (-2 1 1) and (2 -2 0) Bragg peaks as a function of lattice temperature $T_l$ in K. The horizontal lines are the experimental intensities obtained in FED experiments[@sciaini_2009] 14 ps after the arrival of the 200-fs pump pulse of fluence 0.84 mJ.cm$^{-2}$. The experimental data are shown in Fig. \[time-evolution\](c). ](figure2.eps "fig:"){width="8.5cm"}
The softening of the optical phonon modes over the whole Brillouin zone sketched in Fig.\[phonon-curves\](a) has a strong impact on the mean square displacements which are as higher as the phonon frequencies are lower (see Eq. \[mean\_squared\_displacement\]). Figure \[mean-squared-displacment\](a) shows the increase of the mean square displacements upon increasing $T_e$ from 294 K (thin solid line) to 2100 K (thick solid line) and suggests that the Bragg peak intensities are affected by $T_e$.
The diffracted intensity for a given scattering vector defined by the Miller indices (h,k,l) is given by $$\label{intensity-definition}
I_{h,k,l}(T_l, T_e)=I_{h,k,l}^0 \exp[-2W(T_l, T_e)]$$ where the intensity $I_{h,k,l}^0$, which can be affected by the coherent A$_{1g}$ phonon coordinate[@giret_2011], is reduced by the so-called Debye-Waller factor. Here $W(T_l, T_e)$ reads $$\label{def_W}
W(T_l, T_e)=
\frac{1}{2} \sum_{\alpha,\beta} G_\alpha \langle u_p^\alpha u_p^\beta \rangle G_\beta$$ where $\langle u_p^\alpha u_p^\beta\rangle$ is obtained from equation \[mean\_squared\_displacement\]. Figure \[mean-squared-displacment\](b) displays the calculated ratios $\textrm{I}_{\textrm{hkl}}(T_l, T_e=T_l)/
\textrm{I}_{\textrm{hkl}}(T_l=294 ~{\textrm K}, T_e=294 ~{\textrm K})$ for (1 -1 0), (-2 1 1) and (2 -2 0) Bragg peaks as a function of $T_l$. The horizontal lines correspond to the normalized intensities obtained by Sciaini [*[ et al]{}*]{}[@sciaini_2009] in their FED studies 14 ps after the arrival of the laser pulse. As can be seen from Fig.\[time-evolution\](c), the experimental normalized diffracted peak intensities are stationary for this time delay. Assuming that the lattice is at equilibrium with the electron system for larger time delays, we can deduce from our calculated normalized intensities three different lattice temperatures shown by arrows in Fig.\[mean-squared-displacment\](b) which are close to each other. The temperature $T_{l,eq}$ reached by the lattice, when averaging these three values, is found to be 408 K and is smaller than the temperature $T_{l,eq}\simeq 460$ K obtained by Sciaini [*[ et al]{}*]{}[@sciaini_2009] on the basis of a parametrized Debye-Waller model[@gao_1999].
1.0truecm ![\[time-evolution\] (color online) a/ and b/ Lattice $T_l$ and electron temperature $T_e$ in K, as a function of time delay $t$ in ps obtained from the model proposed in Ref.[@giret_2011] for a 30-nm-thick Bi film excited by a 200-fs laser pulse of incident fluence $F_{inc}$=1.11 mJ.cm$^{-2}$ at a wavelength of 775 nm. The inset shows the function $g$ in nm$^{-1}$ as a function of $z$ in nm. c/ Normalized intensity decays of (1 -1 0) , (-2 1 1) and (2 -2 0) Bragg peaks measured by Sciaini [*[ et al]{}*]{}[@sciaini_2009] as a function of $t$ compared to the theoretical calculations. The thin lines represent $I_{h,k,l}(T_l, T_e=T_l)/I_{h,k,l}(T_l=294 ~{\textrm K}, T_e=294 ~{\textrm K})$ while the thick lines represent $I_{h,k,l}(T_l, T_e)/I_{h,k,l}(T_l=294 ~{\textrm K}, T_e=294 ~{\textrm K})$. ](figure3.eps "fig:"){width="8.5cm"}
In order to simulate the time-resolved FED experiments performed by Sciaini [*[ et al]{}*]{}[@sciaini_2009] on a free standing Bi film of thickness L=30 nm excited by a near-infrared laser pulse whose FWHM is $t_w=200$ fs, we used the model proposed in Ref.[@giret_2011] where the source term $P(z,t)$ describing the energy deposited per unit volume and time by the laser pulse at time $t$ and at position $z$ reads $$\label{source-term}
P(z,t)=\frac{F_{inc}}{t_w} \sqrt{\frac{4\ln 2}{\pi}}
g(z) \exp\left[-4 \ln 2\,\frac{t^2}{t_w^2}\right]$$ where $F_{inc}$ is the incident fluence and where $g(z)$ is a function plotted in the inset of Fig.\[time-evolution\](a) and calculated by using both the imaginary and real part of the optical index of Bi for a wavelength $\lambda=775$ nm. We note in passing that the absorption coefficient $A$ defined by $\int_0^L g(z) dz$ is slightly larger for a 30 nm Bi film ($A=0.36$) than for a semi-infinite film ($A=0.31$). The incident fluence $F_{inc}$ needed to reproduce the lattice temperature rise extracted from Fig. \[mean-squared-displacment\](b) is given by $F_{inc}=L\times C_l\times (T_{l,eq}-T_{l,0})/A$ where $T_{l,0}$ is the initial lattice temperature and $C_l$ is the Dulong-Petit lattice specific heat. Thus, we estimate an incident fluence of 1.11 mJ.cm$^{-2}$. This value is overestimated by 27 % with respect to the measured incident fluence of 0.84 mJ.cm$^{-2}$, which might be traced back to the problem of making an accurate measurement of the laser fluence.
Using our calculated effective electron-phonon coupling constant $G_0$ shown in Fig. \[phonon-curves\](b), we numerically solved the three coupled differential equations underlying the model proposed in Ref.[@giret_2011]. Thereby, we obtained the spatial and temporal evolution of the coherent phonon coordinate $u$ and of both electron $T_e$ and lattice temperature $T_l$ following the arrival of the 200 fs laser pulse on the sample. As shown in Fig. \[time-evolution\](b), the electron temperature reaches its maximum value $T_{e,max}\sim 2257$ K only 0.3 ps after the arrival of the laser pulse. After a strong overheating of the electron system, the electron temperature decreases whereas the lattice temperature increases. At $t=$ 14 ps, the lattice is not at equilibrium with the electron system due to the slow down of electron cooling for low electron temperatures. However, the lattice temperature almost reaches $T_{l,eq}\sim 408$ K because the energy stored in the electron system is very small.
The normalized intensities of (1 -1 0), (-2 1 1) and (2 -2 0) bragg peaks measured by Sciaini [*et al*]{} [@sciaini_2009] in FED experiments are shown in Fig. \[time-evolution\](c). These intensities are compared to the theoretical intensities (thick solid lines) defined by $I_{h,k,l}(T_l, T_e)/I_{h,k,l}(T_l=294 ~{\textrm K}, T_e=294 ~{\textrm K})$. The agreement between theory and experiment is noteworthy given the fact that all parameters of the model have been obtained from [*[ab-initio]{}*]{} calculations. In order to point out the role played by $T_e$, we have also displayed $I_{h,k,l}(T_l, T_e=T_l)/I_{h,k,l}(T_l=294 ~{\textrm K}, T_e=294 ~{\textrm K})$ (thin solid lines) in Fig. \[time-evolution\](c). Our results show that it is not possible to reproduce the normalized intensity decays if we assume that the electron system is at equilibrium with the lattice. The Bragg intensities are systematically overestimated with respect to experiments. From a physical point of view, the softening of the optical phonon modes across the whole brillouin zone when $T_e$ is larger than $600$ K leads to a strong decrease of the diffracted intensities and brings the theoretical calculations in close agreement with the experimental results. In conclusion, we computed the effective electron-phonon coupling constant using first-principles density functionnal theory and found that it strongly depends on the electron temperature. In addition, we reproduced the time evolution of Bragg peak intensities measured by Sciaini [*[ et al]{}*]{}[@sciaini_2009] in FED experiments and found that the decay of these intensities is not only due to the increase of the lattice temperature but also to the redshift of all optical modes arising from an increase of the electron temperature.
Calculations were performed using HPC resources from GENCI-CINES (project 095096). We aknowledge G. Sciaini and R.J.D. Miller for providing us with their experimental results and also for enlightening discussions.
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|
---
abstract: 'Norms are behavioral expectations in communities. Online communities are also expected to abide by the established practices that are expressed in the code of conduct of a system. Even though community authorities continuously prompt their users to follow the regulations, it is observed that hate speech and abusive language usage are on the rise. In this paper, we quantify and analyze the patterns of violations of normative behaviour among the users of Stack Overflow (SO) a well-known technical question-answer site for professionals and enthusiast programmers, while posting a comment. Even though the site has been dedicated to technical problem solving and debugging, hate speech as well as posting offensive comments make the community “toxic". By identifying and minimising various patterns of norm violations in different SO communities, the community would become less toxic and thereby the community can engage more effectively in its goal of knowledge sharing. Moreover, through automatic detection of such comments, the authors can be warned by the moderators, so that it is less likely to be repeated, thereby the reputation of the site and community can be improved. Based on the comments extracted from two different data sources on SO, this work first presents a taxonomy of norms that are violated. Second, it demonstrates the sanctions for certain norm violations. Third, it proposes a recommendation system that can be used to warn users that they are about to violate a norm. This can help achieve norm adherence in online communities.'
author:
- Jithin Cheriyan
- Bastin Tony Roy Savarimuthu
- Stephen Cranefield
bibliography:
- 'biblio\_norms.bib'
title: Norm violation in online communities A study of Stack Overflow comments
---
Introduction
============
Online social media platforms have enabled users to express their viewpoints and hence have become a place for information sharing. Applications like Facebook and Twitter are the forerunners in this arena along with multitudes of other applications. Interactions among users of these applications are generally observed as positive, inclusive and creative. SO, a technological division of Stack Exchange — a network of question-answer websites on topics in diverse fields, is a platform for beginners to get free technical support from professionals and well-versed programmers. Those who are interested can join the site for free and post questions they may have about programming to get the best possible theoretical and practical support. Anyone can ask questions in any joined communities and anyone can post answers or comments to that question, making the site dynamic and inclusive. The intent of SO is to give power back to the community [@noauthor_who_nodate] so that a new way of knowledge sharing can be created.
This work is inspired by the post by the Executive Vice President of SO, regarding the alarming transformation of SO as an unwelcoming place [@culture_stack_2018]. Even though millions of comments are generated by the users of various communities day-by-day, a considerable amount of them were found to violate the Code of Conduct (CoC) of the site [@noauthor_code_nodate], which advocates for friendliness and inclusiveness. To monitor the proper usage of the site, site authorities have selected reputed community members as moderators to monitor and review all the posts [@culture_stack_2018]. The reputation score in SO decides a user’s future as a moderator. Reputation score comes from a range of activities including the up-votes of all the answers that one makes and it reflects how much a person has been accepted as a valued resource in that community [@noauthor_what_nodate]. If a comment is found to breach the CoC of the site, moderators would either remove that comments or may contact the author to remove it. Some examples of comments that have been deleted by the moderator are given below.
[*“shut up sir....."*]{}\
[*“I just hate this answer oh downvote you senseless clods <profanity >."*]{}\
[*“Its called your brain. If you can’t review your code ask someone else to do it."*]{}\
[*“you can convert it into seconds, then compare. I think you learned that in your school."*]{}
In addition to these human moderators, SO brings into play an automatic bot [@noauthor_app_nodate] which helps to identify comments containing certain triggering keywords reflecting toxic contents, and will report that to the moderators if they exceed a certain threshold. These norm violating comments, bearing highly toxic contents, would be flagged ‘red’ and those will be moderated by the human moderators immediately [@co-founder_raising_2009]. Moderators who are online will be sent a notification to deal with this. Thereby an important set of norms are enforced on the site. Figure 1 shows the process of moderation in SO. However, no prior work has investigated the types of norms and their violations pertaining to SO and this work bridges this gap.
{width="\textwidth"}
Even though SO has been offered in myriads of languages like Spanish, Portuguese, Russian and Japanese [@noauthor_posts_nodate6], this work intends to study the norm violations in comment posting in English only [@co-founder_non-english_2009]. In this proposed work, by analysing comments on SO, we address three objectives: 1) propose a classification of the types of norms and also quantify norm violations, 2) quantify the different types of punishments for norm violations and 3) propose a recommendation system to minimise norm violations in SO in terms of comment posting.
The paper is structured as follows. Section 2 provides background and related work in norms, norm identification and norm violations. Also, this section establishes the purpose of identification of norm violation in online discussion forums. Section 3 describes the context of norm investigation process in this study. The methodology is explained in Section 4 and Section 5 presents our results. Finally, Section 6 presents the implications of the study in the context of a normative recommendation system and Section 7 concludes the paper.
Background and related work
===========================
In many areas of human life, rules, conventions and norms play a vital role in the smooth functioning of the system [@Savarimuthu1; @sen2007emergence]. Even though rules are enforced by law or authorities, norms are considered as the set of expected practices of interpersonal behaviour in societies, and may only be socially monitored [@Savarimuthu1]. While the breaching of rules results in punishment, usually norm violations may not get punished all the time [@Savarimuthu1; @Savarimuthu2]. As norms are not been imposed by those in power, it is the mindset of the people and the extent to which sanctions are perceived to be applied which make members of society follow or violate the norms. Since norms are specific to a society, identifying and following a certain norm in an unknown community is challenging for people [@Gao; @Savarimuthu1; @savarimuthu2017developers; @sen2007emergence; @singh2013uses]. A system that is able to recognise potential norm violations and warn users about them would be ideal. However, such a system must be able to identify what are the norms that exist in the society.
Many researchers have investigated the norm creation, norm learning and norm emergence processes in agent systems, especially in multi-agent systems (MAS) [@Savarimuthu1; @savarimuthu2013social; @sen2007emergence]. Other than research on agent based systems, many others have focused on norm adherence and violation by the members of various social media platforms like Facebook and Reddit [@chandrasekharan2019hybrid]. For example, Chandrasekharan et al. have tried to identify and classify the macro and meso norms being violated in Reddit comments [@Chandrasekharan2]. Nowadays, to identify and moderate online hate speech, for an example, a meaningful amalgamation of both these streams has been utilized by all leading social networking sites [@park_one-step_2017; @razavi2010offensive]. We examine both of these streams of norms identification in the following sub-sections.
Norms in multi-agent systems
----------------------------
MAS may contain both artificial and human agents. Therefore, it is expected that these communities also would follow certain behavioural norms inside the community. Usually, agents follow the norm life cycle — norm creation, spreading, learning, enforcement and emergence [@Gao; @Savarimuthu1; @Savarimuthu4]. Norms are created by the norm-leaders and inferred by agents by observing the patterns of actions of other agents of the society. Thus, that predominant practice would become a norm to be followed by the community.
In a MAS, the learning process can be either offline or online [@Savarimuthu1; @singh2013uses]. The offline mode of learning, certain rules would be embodied into the agents and they would follow these regulations in a top-down model. But, in a MAS, the norms may be changed dynamically, requiring an ability of an agent to learn the updated norm from the behaviour of other agents. Therefore, in the online fashion of learning, multiple agents interact with others simultaneously and at the same time they would learn the etiquette through the interactions. This is a bottom-up approach and it is usually expected that the agents should learn from their own experiences to fit in a dynamic community. Like human beings, agents also may communicate to achieve coordination and cooperation [@Xuan]. Usually, as part of online learning, the agents may exchange information regarding their present state or the resources they hold, thereby gaining a better understanding of themselves and others’ expectations of its behaviour [@anastassacos_understanding_2019].
Norm enforcement refers to the process of discouraging norm violation either by sanctions or punishments [@Savarimuthu1]. Punishment could be monetary or blacklisting an agent. As per Singh et al. [@singh2013uses] and Savarimuthu et al. [@Savarimuthu1], categories of norms are a) obligation norms 2) prohibition norms and 3) permission norms. Obligation norms are the set of patterns of actions which an agent is supposed to do like tipping in a restaurant. Likewise, an agent system is not expected to perform an action that has been prohibited by the society such as littering the park. Violating the above mentioned two classes of norms would invite sanctions or punishments while the third one, permission norms, usually may not get sanctioned as it refers to the set of actions an agent is permitted to do. This work concerns prohibition norms.
Mining norms from online social media platforms
-----------------------------------------------
Online social networking sites like Facebook, Twitter and Reddit provide a digital world where interactions happen. Even though there are centralised rules regarding the effective and constructive usage of these applications, there exist some users who do not abide by these rules, resulting in abuse and hate speech. When people get upset, especially in online media, they are likely to overlook the norms and may start bullying others for even small mistakes, which has found a surge in last decade [@noauthor_alw_nodate]. Also, it is observed that aggressive behaviour is more common in online spaces than face-to-face [@jones_online_2013]. This hostile nature has long lasting detrimental effects, e.g. the CEO of Twitter has admitted that Twitter loses their users because of online hate speech [@noauthor_alw_nodate]. The European Union has passed a law that all leading social media platforms must delete abusive contents within 24 hours [@noauthor_alw_nodate]. This justifies the requirement of a scalable computational system to systematically locate and remove online hate speech in social media platforms.
As a result of social and legislative pressure, all prominent social media applications are forced to employ human moderators from within the community to identify and moderate provoking, abusive and unnecessary contents [@corr2018legal]. These community moderators monitor the posts that have violated the norms of the community, and potentially delete these posts. Along with these human moderators, computational tools like chatbots (agents) are used to identify abusive or offensive terms and notify the human moderators for possible moderation. Abusive language detection is an interdisciplinary domain which integrates technical components like natural language processing and machine learning [@gupta_natural_2019]. Even though the human moderation process has been found effective, this tedious task could be minimised if all the community members follow the norms. Unfortunately, norm violation in social media is becoming commonplace and there is an increased need for understanding the types of norms and their violations in online communications. In this work we investigate the norms of SO on a posted comment.
The goal of the paper is to identify the norms that a comment is expected to follow and the nature of violations. We provide the context of our study in the next section.
Norm investigation in Stack Overflow
====================================
Having learned about the toxicity through the aforementioned blog post of SO top officials, we investigated the nature of comments by collecting the comments from one particular day with the goal of checking the number of comments that are deleted in subsequent days. We found that around 9.3% of total comments of that day disappeared within one month and this increased to 14% within two months, and this process still continues. Comments were collected between 2 December 2019 and 13 February 2020 for posts with tags ‘Java’ and ‘Python’. Between these two dates, 3221 comments that originally appeared on 2 December were deleted. There are two possible reasons for a comment’s disappearance: either voluntarily deleted by the author or removed by the moderator as part of moderation (i.e. deleted because it violates a norm). Therefore, the set of deleted comments comprise those that were voluntarily removed by the author and deleted by the moderator. The reason for moderation could be the possible violation of generic rules of a society or, some rules or norms specific to SO. Based on prior work [@al-hassan_detection_2019] and a bottom-up analysis of deleted comments, in this work we investigate two major classifications of norms and patterns of norm violations that can be observed from textual comments: (1) Generic norms and (2) SO specific norms.
#### **Generic norms and their violations -**
Generally, it is expected that in the community we live in, we are not supposed to use rude or offensive words that may hurt others. So, the norm is to be polite, and also respect individuals and their unique attributes such as gender and ethnicity. While interacting in online communities we are also expected to follow these rules to keep the propriety. SO is such a forum where the CoC requires its users to respect fellow community members irrespective of their knowledge level, ethnicity or gender. Therefore, generic norms refer to a set of universally accepted norms pertaining to the use of refined language. These norms are likely to be similar across online communities. The norms under this category of prohibition norms are norm against a) personal harassment, b) use of racial slurs, c) use of swear words, and d) use of unwelcoming language. A fine grained description along with examples of these norms are given in Table 1.
#### **SO specific norms and their violations -**
It is a common courtesy to express gratitude for any kind of help, especially in online media. People will acknowledge and apologize for small mistakes to keep the space amiable. However, as aforementioned, the CoC of SO has the policy of keeping the site useful to everyone by containing only relevant knowledge and reducing noise. As a part of that, even though users usually post thanksgiving or apologising comments, these are usually removed by moderator as those comments may distract the users from gaining knowledge, or the newly joining members may consider the site to be too chatty [@rahman_cleaning_2019]. Also, if someone asks an irrelevant question as a follow up of a posted question, this may also get removed. These norms of not accepting or not promoting pleasantries are unique to SO as these comments distract from the intended purpose of knowledge sharing. Therefore, the very specific norms of SO is to keep the site less noisy (free of clutter). Details about four SO-specific norms are shown in the lower half of Table 1.
Methodology
===========
There are two parts of this study: identifying violations of [*generic*]{} and [*specific*]{} norms in SO. To study the pattern of generic norms, we collected the data from the SO heat detector bot [@noauthor_search_nodate] that identifies violations based on regular expressions and various machine learning algorithms. We collected all comments that were flagged by the bot between 16 May 2016 and 31 January 2020. There were a total of 56382 comments. Since these comments are related to violations of generic norms (norm type 1) and did not contain violations of SO specific norms (norm type 2), we created a deleted comments dataset, by collecting comments that were posted on a particular day (2 December 2019), and subsequently checking which of those were deleted from that set in the next two months. We discuss these in detail in the following sub-sections.
#### **Identifying generic norms and their violations -**
In SO, a comment flagged by the bot may violate the abuse norms of SO. A sample entry from the bot is [*“SCORE: 7 (Regex:(?i)/bullshit NaiveBayes:1.00 OpenNLP:0.75 Perspective:0.90)"*]{}. Here the Regex attribute shows the nature of the violation (i.e. the use of the word bullshit). While the entry has other attributes such as the SCORE indicating the extent to which a comment might be a violation (i.e. 7 out of 10) and the output from other machine learning algorithms such as NaiveBayes, we consider only those comments that have regular expressions. Only these can be classified into one of the four groups, as the actual comments have been deleted by SO. We only have the regular expression that was matched. After the bot flags a comment, the moderators review the nature of the violation. The outcome of the review process may be any one of the following: (1) the moderators may deem the comment to be appropriate and permit the comment to be available in the site, (2) the comment may be deleted from the site but the reason is not disclosed, (3) the author may remove the comment voluntarily, after the moderator contacts the author because of its inappropriateness, (4) the comment may be deleted by the moderator because of its inappropriateness. The classification of comments based on the outcome from the moderation process is shown in Table 2. Outcomes A and M can be considered as punishments for norm violation.
Number Type Description
-------- ------ -------------------------------------------------------------------------- --
1 E Contains matching keywords, but not deleted (i.e. they **E**xist in SO).
2 U Deleted ( **U**nknown reason).
3 A Deleted voluntarily by the **A**uthor.
4 M Deleted by the **M**oderator.
: Table describing outcomes of moderator review []{data-label="tab2"}
The dataset contained 56382 comments of which 673 comments do not have the links to locate the comment. In the remaining 55709 comments we found that only 19872 have Regex values to consider for norm classification. After collecting the set of Regex keywords used in SO from the GitHub repository of the heat detector bot[^1], we manually classified the Regex collection into four norm groups — personal harassment, racial, swearing and other unwelcoming comments as listed in Table 1. The full list of Regexs that correspond to each group can be found online[^2]. Then a Python program was used to identify the outcome of each violation by clicking on the link for each comment and extracting the nature of the outcome from the resulting page (e.g. moderator deleted the comment). This extracted data was then grouped based on different outcomes.
#### **Identifying SO specific norms and their violations -**
To identify SO specific norm violations, we collected comments from one particular day - 2 December 2019, for two months. And we identified that 3221 comments were deleted within that period. We examined the reason for moderation of the comments. The reasons fall under two major categories. The first one is pleasantries. The reason for moderation is that, being a technical question-answering site, SO discourages the formalities of expressing gratitude, apology and welcoming someone new to the site or community. The second measure is the scale of essentiality of the comment. The reason of moderation is that the moderators may find those comments no longer needed towards knowledge sharing goal of the site.
Results
=======
This section presents results of norm violation for the two categories of norms. The discussion of these results is presented in the subsequent section.
#### **Violation of generic norms -**
Figure 2 shows the occurrence of the top fifty Regexs in the dataset. Out of the top ten Regex keywords, both swearing and personal targeting keywords appear thrice each and unwelcoming keywords appear four times. This shows that even in SO, a technical discussion forum, people may violate the norms of politeness and may tend to use abusive words so frequently. Moreover, the presence of keywords representing other unwelcoming comments show that people may accuse others or others’ comments as being a spam, sarcastic or rude, which has been traditionally been considered as the duty of moderators. Racial abuse related keywords are found in small numbers (i.e. 377 comments). Complete Regex occurrence details can be found online[^3].
{width="\textwidth" height="6cm"}
Figure 3 shows the percentage of comments that violated norms belonging to the four categories. We have observed that personal harassment keywords and swearing terms are the two most common reasons for norm violation in the examined dataset (33.3% and 33.2% respectively). 66.5% of the total violations come from both these two groups. However, another 31.5% comes from other unwelcoming comments category. Only just 2% of the violations were found for racial norms.
{width="\textwidth" height="6cm"}
Figure 4 shows the outcome of the flagging process of the bot. As presented earlier, the outcome is one of the four options shown in Table 2. We observed that in all four categories of norm violations, the type U outcome occurs the most, showing that the comments have been deleted from the site for an unknown reason. It is likely that the authors of the comments (without any prompting) realized the issue with their own posts and removed them. Out of 19872 comments evaluated, 45% of comments belong to this category. The next category of outcome is those comments deleted by the moderator (M) with 28%. Also, in 9% of comments, the author voluntarily removed them (type A). The rest of the comments (18%) are still present on the site (type E) showing that despite possessing certain objectionable words, the human tolerance for these comments have not been exceeded. On the other hand, strict punishments are imposed on norm-violating comments. This is evident from the moderation process which removed 37% of total comments which fell in categories A and M.
{width="\textwidth" height="6cm"}
Figure 5 shows the outcome of the flagging process by the bot for the four norm categories. It is evident that in all four categories, most of the comments are deleted from the site for an unknown reason (type U). Followed by this is type M where the moderator has deleted all these comments. Voluntary removal of the comment by the author (type A) is the outcome with the smallest count. It can be observed that the type E (comments still exist) outcome happens more than the type A outcome. Therefore, the general trend is U >M >E >A in all four norm categories.
{width="\textwidth" height="6cm"}
#### **Violation of SO specific norms -**
Figure 6 presents the percentage of various SO specific norm violations in the dataset. Out of 3221 comments evaluated, 84% of the comments were in the ‘no longer needed’ category. Since these are no longer needed for the site, those are deleted. 2% are apologies that were removed. 2% are welcome messages and 12% are gratitude messages. It is interesting that pleasantries (apologies, welcome and gratitude) account for 16% of the comments and these are in fact deemed to be not useful to the knowledge creation process. In addition to these, we also observed 9 comments in the dataset that were personal harassment messages indicating the violation of generic norms (not shown in Figure 6). This shows that the moderation process is not instantaneous (i.e., removing harassment messages takes time) since these are possibly milder (borderline) offenses and may require multiple users flagging them before a decision could be made.
{width="\textwidth" height="5.5cm"}
Discussion
==========
In this section we provide a discussion of the results presented in Section 5, particularly discussing their implications for developing a recommendation system to prompt the users to follow the norms of SO when posting a comment.
#### **Generic norms -**
From the results shown in Section 5, it can be inferred that nearly equal contribution for generic norms violation come from the three categories of personal, swearing and other unwelcoming comments. Also, these three together contribute substantially towards total generic norm violation of 98%. Therefore, if we can restrict the abusive language usage, we would be able to improve the quality of the system for everyone. This provides an opportunity to develop an online norm recommendation system for comments which may violate the generic norms of SO. Figure 7 displays the workflow of the application which provides norm recommendation.
{width="\textwidth" height="8cm"}
In the recommendation system, when a new comment has been entered, the generic norm violations can be detected. If the comment violates norms, the system would alert the user regarding the abusive content and the nature of norm violation. Moreover, the system would provide certain rephrased options for the same comment using deep reinforcement learning techniques such as the ones that suggest code auto-complete [@synced_deep_2019]. If the user accepts the proposal, the rephrased comment is posted. If the user does not rephrase or accept the options presented, the post would be allowed, however, a notification will be sent to the moderators regarding the norm violation. Then it would be the discretion of the moderator to review and decide the destiny of the comment. The user also has the option to abort the comment in which case the comment will not be posted. Figure 8 shows an example of the options available to the user for rephrasing a bad comment.
#### **SO specific norms violations -**
From the results shown in Section 5, it is evident that more than three-fourths of the comments in SO are in the category ‘no longer needed’ which are abiding by the specific norms of SO and are neutral in nature. However, these comments are not contributing anything productive to the community, which is likely to be the reason for moderation. In addition to these set of comments, pleasantries also became a substantial reason for norm violations. Therefore, if we could limit the usage of these two categories of comments, SO specific norm adherence can be enhanced. Figure 7 shows that if a comment violates SO specific norms, the system would notify the user about the nature of norm violation. The user can abstain from posting the comment. If not, the user will be allowed to post the comment and the moderator would be notified regarding norm violation.
{width="\textwidth" height="6cm"}
In the future, we intend to build the norm recommendation system proposed in Figure 8 using deep learning techniques. Also, we plan to extend the proposed system to improve the reputation of users in SO community by abiding by the norms pertaining to the site. Thereby, better knowledge sharing without clutter can be facilitated, trust can be guaranteed and gentler treatment can be expected among community members.
Conclusion
==========
The type of norms and their violations in SO are seldom addressed by prior work and that formed the focus of the current work. Our objectives are to identify and quantify the patterns of norm violations in SO comments and to propose a norm recommendation system for SO comments. We have identified two categories of norms, the generic norms and SO specific norms. We found that a significant proportion of violations in the first category has been contributed by the violation of three norms: personal harassment, swearing and other unwelcoming comments. In the second category, the main violations are ‘no longer needed’ and pleasantries. We have proposed an approach that can identify and alert the user regarding the presence of violations in comments which would potentially limit norm violations in SO.
[^1]: <https://github.com/SOBotics/HeatDetector>
[^2]: <http://www.mediafire.com/file/o39ypolha0v0ik0/Regex_classification.pdf/file>
[^3]: <http://www.mediafire.com/file/mez6z6lcszz6ybi/Regex_occurrence.pdf/file>
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We address the problem of complying with the GDPR while processing and transferring personal data on the web. For this purpose we introduce an extensible profile of OWL2 for representing data protection policies. With this language, a company’s data usage policy can be checked for compliance with data subjects’ consent and with a formalized fragment of the GDPR by means of subsumption queries. The outer structure of the policies is restricted in order to make compliance checking highly scalable, as required when processing high-frequency data streams or large data volumes. However, the vocabularies for specifying policy properties can be chosen rather freely from expressive Horn fragments of OWL2. We exploit IBQ reasoning to integrate specialized reasoners for the policy language and the vocabulary’s language. Our experiments show that this approach significantly improves performance.'
author:
- 'Piero A. Bonatti'
- Luca Ioffredo
- 'Iliana M. Petrova'
- Luigi Sauro
bibliography:
- 'biblio.bib'
- 'oracle.bib'
- 'bib-prelim.bib'
- 'newbiblio.bib'
title: 'Fast Compliance Checking with General Vocabularies[^1]'
---
Introduction {#sec:intro}
============
Preliminaries {#sec:prelim}
=============
Supporting General Vocabularies with IBQ Reasoning {#sec:IBQ}
==================================================
Experimental Evaluation {#sec:experiments}
=======================
Related Work {#sec:related}
============
Conclusions {#sec:conclusions}
===========
[^1]: This research is funded by the European Union’s Horizon 2020 research and innovation programme under grant agreement N. 731601.
|
{
"pile_set_name": "ArXiv"
}
|
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