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abstract: 'The strong chromatic index of a graph $G$, denoted ${\chi_{s}''(G)}$, is the least number of colors needed to edge-color $G$ so that edges at distance at most two receive distinct colors. The strong list chromatic index, denoted ${\chi_{\ell,s}''(G)}$, is the least integer $k$ such that if arbitrary lists of size $k$ are assigned to each edge then $G$ can be edge-colored from those lists where edges at distance at most two receive distinct colors. We use the discharging method, the Combinatorial Nullstellensatz, and computation to show that if $G$ is a subcubic planar graph with $\operatorname{girth}(G) \geq 41$ then ${\chi_{\ell,s}''(G)}\leq 5$, answering a question of Borodin and Ivanova \[Precise upper bound for the strong edge chromatic number of sparse planar graphs, *Discuss. Math. Graph Theory*, 33(4), (2014) 759–770\]. We further show that if $G$ is a subcubic planar graph and $\operatorname{girth}(G) \geq 30$, then ${\chi_{s}''(G)}\leq 5$, improving a bound from the same paper. Finally, if $G$ is a planar graph with maximum degree at most four and $\operatorname{girth}(G) \geq 28$, then ${\chi_{s}''(G)}\leq 7$, improving a more general bound of Wang and Zhao from \[Odd graphs and its application on the strong edge coloring, `arXiv:1412.8358`\] in this case.'
author:
| 1 | member_28 |
- 'Philip DeOrsey$^{1,6}$'
- 'Jennifer Diemunsch$^{2,6}$'
- 'Michael Ferrara$^{2,6,7}$'
- 'Nathan Graber$^{2,6}$'
- 'Stephen G. Hartke$^{3,6,8}$'
- 'Sogol Jahanbekam$^{2,6}$'
- 'Bernard Lidický$^{4,6,9}$'
- 'Luke Nelsen$^{2,6}$'
- 'Derrick Stolee$^{4,5,6}$'
- 'Eric Sullivan$^{2,6}$'
title: On the Strong Chromatic Index of Sparse Graphs
---
Introduction
============
A [*proper edge-coloring*]{} of a graph $G$ is an assignment of colors to the edges so that incident edges receive distinct colors. A [*strong edge-coloring*]{} of a graph $G$ is an assignment of colors to the edges so that edges at distance at most two receive distinct colors. A proper edge-coloring is a decomposition of $G$ into matchings, while a strong edge-coloring is a decomposition of $G$ into *induced* matchings. Fouquet and Jolivet [@Fouquet83; @Fouquet84] defined the [*strong chromatic index*]{} of a graph $G$, denoted ${\chi_{s}'(G)}$, as the minimum integer $k$ such that $G$ has a strong edge-coloring using $k$ colors. Erdős and Nešetřil gave the following conjecture, which is still open, and provided an example to show that it would be sharp, if true.
\[Erdos\] For every graph $G$, $\chi'_s(G) \leq \dfrac{5}{4}\Delta(G)^2$ when $\Delta(G)$ is even, and $\chi'_s(G) \leq \dfrac{1}{4}(5\Delta(G)^2-2\Delta(G)+1)$ when $\Delta(G)$ is odd.
Towards this conjecture, Molloy and Reed [@MR] bounded ${\chi_{s}'(G)}$ away from the | 1 | member_28 |
trivial upper bound of $2\Delta(G)(\Delta(G)-1)+1$ by showing that every graph $G$ with sufficiently large maximum degree satisfies ${\chi_{s}'(G)}\leq1.998\Delta(G)^2$. Bruhn and Joos [@BJ] have announced an improvement, claiming ${\chi_{s}'(G)}\leq 1.93\Delta(G)^2$.
The focus of this paper is the study of strong edge-colorings of *subcubic graphs*, those with maximum degree at most three, and *subquartic graphs*, those with maximum degree at most four. Faudree, Gyárfas, Schelp, and Tuza [@Faudree90] studied ${\chi_{s}'(G)}$ in the class of subcubic graphs, and gave the following conjectures.
Let $G$ be a subcubic graph.
(1) ${\chi_{s}'(G)}\leq 10$.
(2) If $G$ is bipartite, then ${\chi_{s}'(G)}\leq 9$. \[con:bipartite\]
(3) If $G$ is planar, then ${\chi_{s}'(G)}\leq 9$. \[con:planar\]
(4) If $G$ is bipartite and for each edge $xy \in E(G)$, $d(x) + d(y) \leq 5$, then ${\chi_{s}'(G)}\leq 6$.
(5) If $G$ is bipartite and $C_4 \not \subset G$, then ${\chi_{s}'(G)}\leq 7$.
(6) If $G$ is bipartite and its girth is large, then ${\chi_{s}'(G)}\leq 5$. \[con:girth\]
\[con:Faud\]
Several of these conjectures have been verified, including (1) by Andersen [@Andersen] and (2) by Steger and Yu [@StegerYu]. Quite recently, Kostochka, Li, Ruksasakchai, Santana, Wang, and Yu [@KLRSWY2014] announced an affirmative resolution to (3). This result is best possible since the prism, shown in Figure | 1 | member_28 |
\[prism\], is a subcubic planar graph with ${\chi_{s}'(G)}= 9$.
(0,0)–(9,0) (0,0)–(0,4) (0,4)–(9,4) (9,4)–(9,0);
(0,0) node\[insep\] (0,4) node\[insep\] (9,4) node\[insep\] (9,0) node\[insep\] (2,2) node\[insep\] (7,2) node\[insep\];
(0,0)–(2,2) (0,4)–(2,2) (2,2)–(7,2) (7,2)–(9,4) (7,2)–(9,0);
Several papers prove sharper bounds on the strong chromatic index of planar graphs with additional structure [@Fouquet84; @Hocquard11; @Hocquard13; @Hudak], generally by introducing conditions on maximum average degree or girth to ensure that the target graph is sufficiently sparse. For a graph $G$, the [*maximum average degree*]{} of $G$, denoted $\operatorname{mad}(G)$, is the maximum of average degrees over all subgraphs of $G$. Hocquard, Montassier, Raspaud, and Valicov [@Hocquard11; @Hocquard13] proved the following.
\[thm:hocquard\] Let $G$ be a subcubic graph.
1. If $\operatorname{mad}(G) < \frac{7}{3}$, then ${\chi_{s}'(G)}\leq 6$. \[part1\]
2. If $\operatorname{mad}(G) < \frac{5}{2}$, then ${\chi_{s}'(G)}\leq 7$. \[part2\]
3. If $\operatorname{mad}(G) < \frac{8}{3}$, then ${\chi_{s}'(G)}\leq 8$.
\[thm:Hoc\]
(0,0)–(0,2) (0,2)–(2,2) (2,2)–(2,0) (2,0)–(0,0);
(0,0) node\[insep\] (0,2) node\[insep\] (2,2) node\[insep\] (2,0) node\[insep\] (1,2) node\[insep\] (1,3.5) node\[insep\] (0,2)–(1,3.5) (1,3.5)–(2,2);
\[fig:house\]
(-1.5,0)–(1.5,0) (-1.5,0) node\[insep\] (-.5,0) node\[insep\] (.5,0) node\[insep\] (1.5,0) node\[insep\] (90:1.5) node\[insep\] (270:1.5) node\[insep\] (-1.5,0)–(90:1.5) (90:1.5)–(1.5,0) (1.5,0)–(270:1.5) (270:1.5)–(-1.5,0) (90:1.5)–(2.5,1.5) (2.5,1.5)–(2.5,-1.5) (270:1.5)–(2.5,-1.5) (2.5,1.5) node\[insep\] (2.5,-1.5) node\[insep\] ;
\[fig:diamond\]
Parts (\[part1\]) and (\[part2\]) of Theorem \[thm:hocquard\] are sharp by the graphs shown in Figures \[fig:house\] and \[fig:diamond\], respectively. | 1 | member_28 |
An elementary application of Euler’s Formula (see [@west]) gives the following.
If $G$ is a planar graph with girth $g$ then $\operatorname{mad}(G) < \frac{2g}{g-2}$. \[prop:mad\]
Theorem \[thm:Hoc\] and Proposition \[prop:mad\] yield the following corollary.
Let $G$ be a subcubic planar graph with girth $g$.
1. If $g \geq 14$, then ${\chi_{s}'(G)}\leq 6$.
2. If $g \geq 10$, then ${\chi_{s}'(G)}\leq 7$.
3. If $g \geq 8$, then ${\chi_{s}'(G)}\leq 8$.
Note that no non-trivial sparsity condition on a graph $G$ with maximum degree $d$ will guarantee that ${\chi_{s}'(G)}< 2d - 1$ since any graph having two adjacent vertices of degree $d$ requires at least $2d-1$ colors to strongly edge-color the graph. We give sparsity conditions that imply a subcubic planar graph has strong chromatic index at most five and a subquartic planar graph has strong chromatic index at most seven. Previous work in this direction was initiated by Borodin and Ivanova [@BI], Chang, Montassier, Pěcher, and Raspaud [@CMPR], and most recently extended by Wang and Zhao [@WZ]. The current-best bounds are given by the following two results.
\[thm:bi\] Let $G$ be a subcubic graph.
1. If $G$ has girth at least $9$ and $\operatorname{mad}(G) < 2 + \frac{2}{23}$, then $\chi_s'(G) \leq 5$.
| 1 | member_28 |
2. If $G$ is planar and has girth at least $41$, then $\chi_s'(G) \leq 5$.
\[thm:wz\] Fix $d \geq 4$ and let $G$ be a graph with $\Delta(G) \leq d$.
1. If $G$ has girth at least $2d-1$ and $\operatorname{mad}(G) < 2 + \frac{2}{6d-7}$, then $\chi_s'(G) \leq 2d-1$.
2. If $G$ is planar and has girth at least $10d-4$, then $\chi_s'(G) \leq 2d-1$.
One barrier to proving sparsity conditions that imply ${\chi_{s}'(G)}\leq 5$ is that there exist graphs $G$ with $\operatorname{mad}(G) = 2$ and ${\chi_{s}'(G)}= 6$. Let $S_3$ be a triangle with pendant edges at each vertex, and let $S_4$ be a $4$-cycle with pendant edges at two adjacent vertices. For $k \geq 5$, let $S_k$ be a $k$-cycle with pendant edges at each vertex. Each of $S_3$, $S_4$ and $S_7$ have maximum average degree $2$ and strong chromatic index at least 6, see Figure \[fig:S3S4S5\]. However, these graphs are 6-critical with respect to ${\chi_{s}'(G)}$, as the removal of any edge from $S_3$, $S_4$ or $S_7$ results in a graph that has a strong edge-coloring using five colors.
Our main theorem demonstrates that if these few graphs are avoided, and the maximum average degree is not too large, then we | 1 | member_28 |
can find a strong 5-edge-coloring, improving Theorem \[thm:bi\].
\[thm:sparse\] Let $G$ be a subcubic graph.
1. If $G$ does not contain $S_3$, $S_4$, or $S_7$ and $\operatorname{mad}(G) < 2 + \frac{1}{7}$, then ${\chi_{s}'(G)}\leq 5$.
2. If $G$ is planar and has girth at least $30$, then ${\chi_{s}'(G)}\leq 5$.
The bound in Theorem \[thm:sparse\] is likely not sharp, but is close to optimal. The graph in Figure \[fig:thetaexample\] is subcubic, avoids $S_3$, $S_4$, and $S_7$, and satisfies both ${\chi_{s}'(G)}= 6$ and $\operatorname{mad}(G) = 2 + \frac{1}{6}$.
(0,0) node\[insep\](a) (5,0) node\[insep\](b)
\(a) to\[bend left=50\] node\[insep,pos=0.25\](x) node\[insep,pos=0.50\](y) node\[insep,pos=0.75\](z) (b) (x)– ++(90:0.5)node\[insep\] (y)– ++(90:0.5)node\[insep\] (z)– ++(90:0.5)node\[insep\]
\(a) to\[bend right=50\] node\[insep,pos=0.25\](x) node\[insep,pos=0.50\](y) node\[insep,pos=0.75\](z) (b) (x)– ++(-90:0.5)node\[insep\] (y)– ++(-90:0.5)node\[insep\] (z)– ++(-90:0.5)node\[insep\]
\(a) to node\[insep,pos=0.20\](x) node\[insep,pos=0.40\](y) node\[insep,pos=0.60\](z) node\[insep,pos=0.80\](w) (b) (x)– ++(90:0.5)node\[insep\] (y)– ++(90:0.5)node\[insep\] (z)– ++(90:0.5)node\[insep\] (w)– ++(90:0.5)node\[insep\] ;
;
Using similar methods, we improve the bounds in Theorem \[thm:wz\] when $d = 4$.
\[thm:sparse4\] Let $G$ be a subquartic graph.
1. If $G$ has girth at least $7$ and $\operatorname{mad}(G) < 2 + \frac{2}{13}$, then $\chi_s'(G) \leq 7$.
2. If $G$ is planar and has girth at least $28$, then $\chi_s'(G) \leq 7$.
We also consider a list variation of the strong chromatic index of $G$, first | 1 | member_28 |
introduced by Vu [@vu]. A [*strong list edge-coloring*]{} of a graph $G$ is an assignment of lists to $E(G)$ such that a strong edge-coloring can be chosen from the lists at each edge. The minimum $k$ such that a graph $G$ can be strongly list edge-colored using any lists of size at least $k$ on each edge is the [*strong list chromatic index*]{} of $G$, denoted ${\chi_{\ell,s}'(G)}$. Borodin and Ivanova [@BI] asked if there are sparsity conditions that imply ${\chi_{\ell,s}'(G)}\leq 2d-1$ for a planar graph $G$ with maximum degree $d$. We generalize the bounds in Theorem \[thm:bi\] to apply to list coloring.
\[thm:mainlist\] Let $G$ be a subcubic graph.
1. If $G$ has girth at least $9$ and $\operatorname{mad}(G) < 2 + \frac{2}{23}$, then ${\chi_{\ell,s}'(G)}\leq 5$.
2. If $G$ is planar and has girth at least $41$, then ${\chi_{\ell,s}'(G)}\leq 5$.
The proofs of Theorems \[thm:sparse\], \[thm:sparse4\], and \[thm:mainlist\] use the discharging method. We begin by proving Theorem \[thm:mainlist\] in Section \[sec:list\] as the proof is shorter and the one reducible configuration is used again in the proof of Theorem \[thm:sparse\] in Section \[sec:sparse\].
Preliminaries and Notation
--------------------------
Throughout this paper we will only consider simple, finite, undirected graphs. We refer | 1 | member_28 |
to [@west] for any undefined definitions and notation. A graph $G$ has vertex set $V(G)$, edge set $E(G)$, and maximum degree $\Delta(G)$. If a vertex $v$ has degree $j$ we refer to it as a [*$j$-vertex*]{}, and if $v$ has a neighbor that is a $j$-vertex, we say it is a [*$j$-neighbor*]{} of $v$. When $G$ is planar we let $F(G)$ denote the set of faces of $G$, and $\ell(f)$ denote the length of a face $f$. The [*girth*]{} of a graph $G$ is length of its shortest cycle. A graph $G$ is $\{a,b\}$-regular if for every $v$ in $G$, the degree of $v$ is either $a$ or $b$. Every graph $G$ with maximum degree $d$ is contained in a prescribed $\{1,d\}$-regular graph, denoted $\operatorname{ex}_d(G)$, the [*$d$-expansion*]{} of $G$. To construct $\operatorname{ex}_d(G)$, add $d-d(v)$ pendant edges to each vertex $v$ in $G$ where $d(v) \in \{2,\dots, d\}$. Additionally, let the [*contracted graph*]{} of $G$, denoted $\operatorname{ct}(G)$ be the graph obtained by deleting all 1-vertices of $G$. A vertex $v$ in $G$ is a *$2{{\ensuremath{^\perp}}}$-vertex* if $v$ is a 2-vertex in $\operatorname{ct}(G)$. Thus, for the remainder of the paper a vertex $v$ is a *$k^+$-vertex* in $G$ if it has degree | 1 | member_28 |
at least $k$ in $\operatorname{ct}(G)$.
We will make use of the discharging method for some of our results. For an introduction to this method, see the survey by Cranston and West [@CW]. We will directly use two standard results that can be proven using this method. Both of Theorems \[thm:bi\] and \[thm:wz\] rely on Lemmas \[lma:cw\] and \[lma:nrs\].
Let $G$ be a graph and $\operatorname{ct}(G$) be the contracted graph. An *$\ell$-thread* is a path $v_1\dots v_\ell$ in $\operatorname{ct}(G)$ where each $v_i$ is a $2{{\ensuremath{^\perp}}}$-vertex.
\[lma:cw\] If $G$ is a graph with girth at least $\ell+1$ and $\operatorname{mad}(G) < 2 + \frac{2}{3\ell - 1}$, then $\operatorname{ct}(G)$ contains a 1-vertex or an $\ell$-thread.
\[lma:nrs\] If $G$ is a planar graph with girth at least $5\ell + 1$, then $\operatorname{ct}(G)$ contains a 1-vertex or an $\ell$-thread.
Strong List Edge-Coloring of Subcubic Graphs {#sec:list}
============================================
In this section, we prove Theorem \[thm:mainlist\]. Our proof uses the discharging method, wherein we assign an initial charge to the vertices and faces of a theoretical minimal counterexample. This initial charge is then disbursed according to a set of discharging rules in order to draw a contradiction to the existence of such a minimal counterexample. We will often | 1 | member_28 |
make use of the following, which is another simple and well known application of Euler’s Formula.
\[prop:sum\] In a planar graph $G$,
$$\sum_{f \in F(G)} ( \ell(f) - 6 ) + \sum_{v \in V(G)} (2d(v) - 6) = -12.$$
We will also use the Combinatorial Nullstellensatz, which will be applied to show we can extend certain list colorings.
\[CN\] Let $f$ be a polynomial of degree $t$ in $m$ variables over a field $\mathbb{F}$. If there is a monomial $\prod x_i^{t_i}$ in $f$ with $\sum t_i=t$ whose coefficient is nonzero in $\mathbb{F}$, then $f$ is nonzero at some point of $\prod S_i$, where each $S_i$ is a set of $t_i+1$ distinct values in $\mathbb{F}$.
The first item of Theorem \[thm:mainlist\] follows from the following strengthened theorem.
Let $G$ be a planar $\{1,3\}$-regular graph of girth at least $41$, and let $p \in V(G)$. Assign distinct colors to the edges incident to $p$ and let $L$ be a $5$-list-assignment to the remaining edges of $G$. There exists a strong edge-coloring $c$ where $c(e) \in L(e)$ for all $e \in E(G)$.
For the sake of contradiction, select $G$, $p$, $c$, and $L$ as in the theorem statement, and assume there does not | 1 | member_28 |
exist a strong edge coloring of $E(G)$ using colors from $L$. In this selection, minimize $n(G)$. Note that $G$ is connected and $e(G) > 5$. We can further assume that $d(p) > 1$, since if $d(p)=1$ and $\{p'\}=N(p)$ then we can instead color the edges incident to $p'$.
\[lma:cutedge\] There does not exist a cut-edge $uv$ such that $d(u) = d(v) = 3$.
Suppose that $G$ contains a cut-edge $uv$ with $d(u) = d(v) = 3$. There are exactly two components in $G - uv$, call them $G_1$ and $G_2$, with $u \in V(G_1)$ and $v \in V(G_2)$. Without loss of generality, $p \in V(G_1)$. For each $i \in \{1,2\}$, let $G_i' = G_i + uv$.
Since $d(v) = 3$, $n(G_1') < n(G)$. Thus there is a strong edge-coloring of $G_1'$ using the 5-list-assignment $L$. Next, color the other two edges incident to $v$ using colors distinct from those on the edges incident to $u$. Now, $G_2'$ is a subcubic planar graph of girth at least 41 with distinctly colored edges about the vertex $v$ and $n(G_2') < n(G)$. Thus, there is an extension of the coloring to $G_2'$.
The colorings of $G_1'$ and $G_2'$ form a strong edge coloring | 1 | member_28 |
of $G$, a contradiction.
Define a *$k$-caterpillar* to be a $k$-thread $v_1,\dots,v_k$ in $G$ where $p \notin \{v_1,\dots,v_k\}$. Figure \[8cat\] is an $8$-caterpillar.
(0,0)–(9,0) (0,0) node\[insep\] (9,0) node\[insep\];
in [1,2,...,8]{}[ (,0)–(,1) (,0) node\[free\] (,1) node\[extra\]; at (,-0.35) [$v_{\x}$]{}; at (,1.5) [$v_{\x}'$]{}; ]{};
(9,0)–(10,.75) (9,0)–(10,-.75) (0,0)–(-1,.75) (0,0)–(-1,-.75);
(-1,-.75) node\[insep\]; (-1,.75) node\[insep\]; (10,-.75) node\[insep\]; (10,.75) node\[insep\]; at (-0.5,0) [$v_{0}$]{}; at (9.5,0) [$v_{9}$]{}; at (10.35,.75) [$v_9'$]{}; at (10.35,-.75) [$u_9'$]{}; at (-1.35,.75) [$v_0'$]{}; at (-1.35,-.75) [$u_0'$]{};
\[lma:caterpillar\] $G$ does not contain an $8$-caterpillar.
We will show that if $G-p$ contains an 8-caterpillar, then $G$ has a strong edge $L$-coloring. If $v_1,\dots,v_8$ form an 8-caterpillar, then let $v_i'$ be the 1-vertex adjacent to $v_i$, $v_0$ and $v_9$ be the other neighbors of $v_1$ and $v_8$. For $i \in \{0,9\}$, let $v_i'$ and $u_i'$ be the neighbors of $v_i$ other than $v_1$ or $v_8$. By removing all edges incident to $v_2,\dots, v_7$ and $u_1,\dots,u_8$, as well as any isolated vertices that are produced, we obtain a graph $G'$ with fewer vertices than $G$, so we can strongly edge-color $G'$ with 5 colors. We fix such a coloring of $G'$ and generate a contradiction by extending this coloring to a strong edge-coloring of $G$. Suppose that $c_1, | 1 | member_28 |
\dots, c_6$ are the colors of the edges incident to the vertices $v_0$ and $v_9$, and assign variables $y_1, \dots, y_8$ to the pendant edges, and variables $x_1, \dots, x_7$ to the interior edges as shown in Figure \[fig:lcat\].
(0,0)–(9,0) (0,0) node\[insep\] (9,0) node\[insep\];
in [1,2,...,8]{}[ (,0)–(,1) (,0) node\[free\] (,1) node\[extra\]; ]{};
(9,0)–(10,.75) (9,0)–(10,-.75) (0,0)–(-1,.75) (0,0)–(-1,-.75);
in [1,2,...,8]{}[ (-.2,.5) node[$y_{\x}$]{}; ]{}
in [1,2,...,7]{}[ (+.5, -.2) node[$x_{\x}$]{}; ]{}
(.5,-.2) node[$c_{3}$]{} (8.5,-.2) node[$c_{4}$]{} (-.25,.6) node[$c_1$]{} (-.25,-.6) node[$c_2$]{} (9.25,.6) node[$c_5$]{} (9.25,-.6) node[$c_6$]{} ;
(-1,-.75) node\[insep\]; (-1,.75) node\[insep\]; (10,-.75) node\[insep\]; (10,.75) node\[insep\];
Identifying the conflicts between variables and colors produces the following polynomial, $$\begin{aligned}
f(y_1, \dots, y_8,x_1,\dots, x_7) &= (y_2 - c_3)(x_2 - c_3)(y_7 - c_4)(x_6 - c_4) \\
&\quad\cdot
\prod_{i=1}^3(x_1 - c_i)\prod_{i=1}^3(y_1 - c_i) \prod_{i=4}^6 (x_7 - c_i) \prod_{i=4}^6 (y_8 - c_i)\\
&\quad\cdot \prod_{j -i \in \{1,2\}}(x_i-x_j) \prod_{j-i = 1}(y_i - y_j)\prod_{i - j \in \{-1,0,1,2\}} (y_i - x_j). \end{aligned}$$
We will use the Combinatorial Nullstellensatz to show that there is an assignment of colors $\hat{c}_1,\dots, \hat{c}_8$ and $c_1',\dots, c_7'$ such that $f(\hat{c}_1,\dots,\hat{c}_8,c_1',\dots,c_7')\ne 0$. Such an assignment of colors would extend the inductive coloring of $G-p$ to a strong edge-coloring of $G$. If the coefficient of $$(x_{1}~ x_{2}~ x_{3}~ x_{4} ~x_{5} ~x_{6} ~x_{7} | 1 | member_28 |
~y_{1} ~y_{2} ~y_{3} ~y_{4} ~y_{5} ~y_{6} ~y_{7} ~y_{8})^4$$ is nonzero, then there are values from $L$ for $x_1,\ldots,x_7,y_1,\ldots,y_8$ such that $f$ is nonzero by Theorem \[CN\]. Using the Magma algebra system [@magma], this monomial has coefficient $-2$, and thus there is a strong edge-coloring using the 5-list assignment[^1]. Thus, the 8-caterpillar does not exist in a vertex minimal counterexample.
Note that the proof in Lemma \[lma:caterpillar\] cannot be extended to exclude a 7-caterpillar in $G$, as there exists a 5-coloring of the external edges that does not extend to the caterpillar, even when the lists are all the same.
To complete the proof, we apply a discharging argument to $\operatorname{ct}(G)$. [^2]. First, observe that by Lemma \[lma:cutedge\], $\operatorname{ct}(G)$ is 2-connected and so every face is a simple cycle of length at least 41. Also observe that by Lemma \[lma:caterpillar\], $\operatorname{ct}(G)$ does not contain a path of length 8 where every vertex is of degree 2, unless one of those vertices is $p$.
Assign charge $2d(v)-6$ to every vertex $v \neq p$, charge $\ell(f) - 6$ to every face $f$, and charge $2d(p)+5$ to $p$. By Proposition \[prop:sum\], the total amount of charge on $\operatorname{ct}(G)$ is $-1$. Apply the following discharging | 1 | member_28 |
rules.
1. For every $v\in G-p$, if $v$ is a $2$–vertex, $v$ pulls charge 1 from each incident face.
2. If $p$ is a $2$–vertex, then $p$ gives charge $\frac{9}{2}$ to each incident face.
Observe that every vertex has nonnegative charge after this discharging process. It remains to show that every face has nonnegative charge.
Let $f$ be a face, and let $r_2$ be the number of 2–vertices on the boundary of $f$, not counting $p$, and consider two cases.
**Case 1**: $d(p)=3$ or $p$ is not adjacent to $f$.
In this case, $p$ does not give charge to $f$, and therefore $f$ has charge $\ell(f) - r_2 - 6$ after discharging. Also, the boundary of $f$ does not contain a path of length 8 containing only vertices of degree 2, thus $r_2 \leq \left\lfloor \frac{7}{8}\ell(f)\right\rfloor$. Since $\ell(f) \geq 41$, we have $$\ell(f) - r_2 - 6 \geq \ell(f) - \left\lfloor \frac{7}{8}\ell(f)\right\rfloor - 6 \geq 0.$$
**Case 2**: $d(p)=2$ and $p$ is adjacent to $f$.
By (R2), $p$ gives charge $\frac{9}{2}$ to $f$, so that $f$ has charge $\ell(f) - r_2 - \frac{3}{2}$ after discharging. The boundary of $f$ does not contain a path of length 8 containing only vertices | 1 | member_28 |
of degree 2, except when using $p$, so, $r_2 \leq \left\lfloor \frac{7}{8}\ell(f)\right\rfloor$. Since $\ell(f) \geq 41$, we have $$\ell(f) - r_2 - \frac{3}{2} \geq \ell(f) - \left\lfloor \frac{7}{8}\ell(f)\right\rfloor - \frac{3}{2} \geq 0.$$ Thus, all vertices and faces have nonnegative charge, contradicting Proposition \[prop:sum\].
The second item of Theorem \[thm:mainlist\] follows by similarly strengthening the statement to include a precolored vertex and using Lemmas \[lma:cw\], \[lma:cutedge\], and \[lma:caterpillar\].
Strong Edge-Coloring of Sparse Graphs {#sec:sparse}
=====================================
In this section, we prove Theorems \[thm:sparse\] and \[thm:sparse4\].
Let $G$ be a graph with maximum degree $\Delta(G) \leq d$. For a vertex $v$ in $\operatorname{ct}(G)$ denote by $N_3(v)$ the set of $3^+$-vertices $u$ where $\operatorname{ct}(G)$ contains a path $P$ from $u$ to $v$ where all internal vertices of $P$ are $2{{\ensuremath{^\perp}}}$-vertices. For $u \in N_3(v)$, let $\mu(v,u)$ be the number of paths from $v$ to $u$ whose internal vertices have degree 2 in $\operatorname{ct}(G)$. For a 3-vertex $v$, let the *responsibility set*, denoted $\operatorname{Resp}(v)$, be the set of $2{{\ensuremath{^\perp}}}$-vertices that appear on the paths between $v$ and the vertices in $N_3(v)$.
Let $D$ be a subgraph of $G$. We call $D$ a *$k$-reducible configuration* if there exists a subgraph $D'$ of $D$ such that any | 1 | member_28 |
strong $k$-edge-coloring of $G-D'$ can be extended to a strong $k$-edge-coloring of $G$. One necessary property for the selection of $D'$ is that no two edges that remain in $G-D'$ can have distance at most two in $G$ but distance strictly larger than two in $G - D'$. In the next subsection we describe several reducible configurations.
Reducible Configurations
------------------------
This subsection contains description of four types of reducible configurations. Each configuration is described in terms of how it appears within $\operatorname{ct}(G)$ where $G$ is a graph with maximum degree $\Delta(G) \leq d$ for some $d \geq 4$.
Let $t$ be a positive integer. The *$t$-caterpillar* is formed by two $3^+$-vertices $v_0$ and $v_{t+1}$ with a path $v_0v_1\dots v_tv_{t+1}$ where each $v_i$ is a $2{{\ensuremath{^\perp}}}$-vertex for every $i \in \{1,\dots,t\}$.
Let $t_1,\dots, t_k$ be nonnegative integers. A configuration $Y(t_1,\dots,t_k)$ is formed by a $k^+$-vertex $v$ and $k$ internally disjoint paths of lengths $t_1+1,\dots,t_k+1$ with $v$ as a common endpoint, where the internal vertices of the paths are $2{{\ensuremath{^\perp}}}$-vertices. We call such configuration a *$Y$-type configuration about $v$*, see Figure \[fig:Y\].
A configuration $H(t_1,t_2;r;s_1,s_2)$ is formed by two 3-vertices $u$ and $v$ and 5 internally disjoint paths of lengths $t_1+1$, $t_2+1$, | 1 | member_28 |
$r+1$, $s_1+1$, and $s_2+1$, where the internal vertices of the paths are $2{{\ensuremath{^\perp}}}$-vertices. The paths of lengths $t_1+1$ and $t_2+1$ have $v$ as an endpoint, the path of length $r+1$ has $u$ and $v$ as endpoints and the paths of lengths $s_1+1$ and $s_2+1$ have $u$ as an endpoint. We call such configuration an *$H$-type configuration about $v$ and $u$*, see Figure \[fig:H\].
A configuration $\Phi(t,a_1,a_2,s)$ is formed by two 3-vertices $u$ and $v$ and 4 internally disjoint paths of lengths $t+1$, $a_1+1$, $a_2+1$, and $s+1$, where the internal vertices of the paths are $2{{\ensuremath{^\perp}}}$-vertices. The path of length $t+1$ has $v$ as an endpoint, the paths of lengths $a_1+1$ and $a_2+1$ have $u$ and $v$ as endpoints and the path of length $s+1$ has $u$ as an endpoint. We call such configuration a *$\Phi$-type configuration about $v$ and $u$*, see Figure \[fig:Phi\].
The reducibility of these configurations was verified using computer[^3], and in addition the 8-caterpillar is addressed in Lemma \[lma:caterpillar\]. Given the definition of a $2{{\ensuremath{^\perp}}}$-vertex, the vertices of degree two in these configurations may, or may not, be adjacent to some 1-vertices in $G$. We demonstrate the reducibility of the instances of these configurations wherein each vertex | 1 | member_28 |
of degree 2 is adjacent to $d-2$ 1-vertices, as depicted in Figures \[fig:Y\]–\[fig:Phi\]. This suffices to address all other instances of these configurations that may occur.
\[claim:red\_cat\] The following caterpillars with maximum degree $d$ are reducible:
1. (Borodin and Ivanova [@BI]) For $d = 3$, the $8$-caterpillar is 5-reducible.
2. (Wang and Zhao [@WZ]) For $d \geq 4$, the $(2d-2)$-caterpillar is $(2d-1)$-reducible.
These caterpillars are likely the smallest that are reducible for each degree $d$. Thus, the bounds in Theorems \[thm:bi\] and \[thm:wz\] are best possible using only Lemma \[lma:nrs\]. To improve these bounds, we demonstrate larger reducible configurations and use a more complicated discharging argument.
\[clm:Reducible\] The following configurations with maximum degree 3 are 5-reducible:
1. $Y(1,6,7)$, $Y(2,5,6)$ and $Y(3,4,5)$.
2. $H(7,7;0;3,7),\, H(7,7;0;4,6),\, H(7,7;0;5,5),\, H(6,7;0;3,7),\, H(6,7;0;4,6),\\
H(6,7;0;5,5),\, H(6,6;1;2,7),\, H(6,6;1;3,6),\, H(6,6;1;4,5),\, H(5,7;1;2,7),\\
H(5,7;1;3,6),\, H(5,7;1;4,5),\, H(4,7;2;1,7),\, H(4,7;2;2,6),\, H(4,7;2;3,5),\\
H(4,7;2;4,4),\, H(3,7;3;1,6),\, H(3,7;3;2,5) \text{ and } H(3,7;3;3,4).$
3. $\Phi(7,0,7,1),\, \Phi(7,0,6,1),\, \Phi(6,0,7,1),\, \Phi(6,1,6,1),\, \Phi(7,1,5,1), \Phi(5,1,7,1),\\
\Phi(7,2,4,1),\, \Phi(4,2,7,1),\, \Phi(7,3,3,1),\, \Phi(3,3,7,1) \text{ and }\Phi(3,7,0,7).$
\[clm:Reducible4\] The following configurations with maximum degree 4 are $7$-reducible: $$Y(2,4,4),\ Y(1,5,5),\ Y(2,4,5),\ Y(3,4,4),\text{ and }Y(2,5,5).$$
Proof of Theorem \[thm:sparse\]
-------------------------------
Among graphs $G$ with $\operatorname{mad}(G) < 2 + \frac{1}{7}$ not containing $S_3$, $S_4$, or $S_7$, with ${\chi_{s}'(G)}> 5$, select $G$ while | 1 | member_28 |
minimizing the number of vertices in $\operatorname{ct}(G)$. Note that $e(G) > 5$ since ${\chi_{s}'(G)}> 5$, and let $n$ be the number of vertices in $\operatorname{ct}(G)$. By using the discharging method, we will show that $\operatorname{mad}(\operatorname{ct}(G)) \geq 2+\frac{1}{7}$, which is a contradiction, so no such minimal counterexample exists.
Observe that $G$ does not contain any of the reducible configurations addressed in Claim \[clm:Reducible\]. We also have the following additional structure on $\operatorname{ct}(G)$.
\[lma:cutedge2\] $\operatorname{ct}(G)$ is 2-connected.
Suppose that $\operatorname{ct}(G)$ contains a cut-edge $uv$. In $G$, the vertices $u$ and $v$ have degree at least two. There are exactly two components, $G_1$ and $G_2$, in $G - uv$, with $u \in V(G_1)$ and $v \in V(G_2)$. Let $u_1,u_2$ be neighbors of $u$ in $G_1$ and $v_1,v_2$ be neighbors of $v$ in $G_2$; let $u_1 = u_2$ only when $u$ has a unique neighbor in $G_1$, and $v_1 = v_2$ only when $v$ has a unique neighbor in $G_2$. Let $G_1' = G_1 + \{ uv, vv_1, vv_2\}$ and $G_2' = G_2 + \{ uv, uu_1, uu_2\}$.
If $G_1' = G$, then consider $G' = G - v_1 - v_2$. Since $n(G') < n(G)$ and $\operatorname{mad}(G') \leq \operatorname{mad}(G)$, there is a strong 5-edge-coloring | 1 | member_28 |
$c$ of $G'$. Extend the coloring $c$ to color $c(vv_1)$ and $c(vv_2)$ from the colors not in $\{ c(uv), c(uu_1), c(uu_2)\}$, a contradiction. We similarly reach a contradiction when $G_2' = G$.
Therefore, $n(G_i') < n(G)$ and $\operatorname{mad}(G_i') \leq \operatorname{mad}(G)$ for each $i \in \{1,2\}$. Thus, there exist strong 5-edge-colorings $c_1$ and $c_2$ of $G_1'$ and $G_2'$, respectively. For each coloring, the colors on the edges $uv, uu_1, uu_2, vv_1, vv_2$ are distinct. Let $\pi$ be a permutation of the five colors satisfying $\pi(c_2(e)) = c_1(e)$ for each edge $e \in \{uv, uu_1, uu_2, vv_1, vv_2\}$. Then, we extend the coloring $c_1$ of $G_1'$ to all of $G$ by assigning $c_1(e) = \pi(c_2(e))$ for all edges $e \in E(G_2')$. The coloring $c_1$ is a strong 5-edge-coloring of $G$, a contradiction.
If $\operatorname{ct}(G)$ does not have any $3$-vertices, then $\operatorname{ct}(G)$ must be isomorphic to cycle $C_n$. If $n \geq 9$, then $\operatorname{ex}_3(G)$ contains an 8-caterpillar. If $n \in \{5,6,8\}$, then $G$ is a subgraph of $S_5$, $S_6$, or $S_8$, which each has a strong edge-coloring using five colors, discovered by computer. When $n \in\{3,4,7\}$, $G$ does not contain $S_3$, $S_4$, or $S_7$, and any proper subgraph of these graphs is $5$ | 1 | member_28 |
strong edge-colorable, discovered by computer. Therefore, $\operatorname{ct}(G)$ is not isomorphic to a cycle, and hence for every $2{{\ensuremath{^\perp}}}$-vertex $u$ in $G$, $|N_3(u)| \geq 1$.
If $G$ has some vertex $v$ such that $|N_3(v)|=1$, then $G$ must be a subgraph of $\Theta(t_1,t_2,t_3)$, which is the graph consisting of three internally disjoint $x-y$ paths of length $t_1+1, t_2+1$ and $t_3+1$, for some $0 \leq t_1 \leq t_2 \leq t_3$.
If $t_3 \geq 8$, then $\operatorname{ex}_3(G)$ contains an 8-caterpillar, so we assume that $t_3 < 8$. Observe that if $\operatorname{mad}(\Theta(t_1,t_2,t_3)) < 2 + \frac{1}{7}$, then $t_1+t_2+t_3 \geq 13$. However, if $\Theta(t_1,t_2,t_3)$ does not contain a reducible $Y$-type configuration, then by Claim \[clm:Reducible\] the sequence $(t_1,t_2,t_3)$ is one of $(0,7,7)$, $(0,6,7)$, $(1,6,6)$, $(1,5,7)$, $(2,5,6)$, $(2,4,7)$, or $(3,3,4)$. In each of these cases, we have verified by computer that $\Theta(t_1,t_2,t_3)$ has a strong edge-coloring using five colors.
Therefore, $|N_3(v)|\geq 2$ for every $v\in\operatorname{ct}(G)$. We proceed using discharging. Assign each vertex initial charge $d(v)$. Note that the total charge on the graph is $2e(\operatorname{ct}(G))$, which is at most $\operatorname{mad}(G)n < (2+\frac{1}{7})n$. We shall distribute charge among the vertices of $\operatorname{ct}(G)$ and result with charge at least $2 + \frac{1}{7}$ on every vertex, giving a contradiction.
Distribute | 1 | member_28 |
charge among the vertices according to the following discharging rules, applied to each pair of vertices $u, v \in V(\operatorname{ct}(G))$:
1. \[2vtx\] If $u$ is a 2-vertex and $v \in N_3(u)$, then $v$ sends charge $\frac{1}{14}$ to $u$.
2. \[3vtx\] If $v$ is a 3-vertex with $|\operatorname{Resp}(v)|\leq 10$ and $u \in N_3(v)$, then
(a) \[adj\] if $d(u,v)=1$ and $|\operatorname{Resp}(u)| = 14$, then $v$ sends charge $\frac{1}{7}$ to $u$;
(b) \[near\] otherwise, if $d(u,v) \leq 4$, then $v$ sends charge $\frac{1}{14}$ to $u$.
We will now verify the assertion that each vertex has final charge at least $2 + \frac{1}{7}$. If $v$ is a 2-vertex, then since $|N_3(v)| = 2$ the final charge on $v$ is $2 + \frac{1}{7}$ after by Rule R\[2vtx\]. Let $v$ be a 3-vertex. If $u \in N_3(v)$, then $d(u,v) \leq 8$ by Lemma \[lma:caterpillar\]. Claim \[clm:Reducible\] implies that $|\operatorname{Resp}(v)|\leq 14$.
*Case : $|\operatorname{Resp}(v)| \in \{11, 12\}$.*
In this case, $v$ only loses charge by Rule R\[2vtx\], so the final charge is at least $3 - \frac{12}{14} = 2 + \frac{1}{7}$.
*Case : $|\operatorname{Resp}(v)| = 14$.*
By Claim \[clm:Reducible\], the $Y$-type configuration about $v$ is $Y(0,7,7)$. Thus, some vertex $u_1 \in N_3(v)$ is at distance one from | 1 | member_28 |
$v$. If $\mu(v,u_1) = 1$, then the $H$-type configuration about $v$ and $u_1$ is of the form $H(7,7;0;s_1,s_2)$; by Claim \[clm:Reducible\] $s_1+s_2 \leq 9$, $|\operatorname{Resp}(u_1)|\leq 9$, and $u_1$ sends charge $\frac{1}{7}$ to $v$ by Rule R\[adj\]. If $\mu(v,u_1) = 2$, then the $\Phi$-type configuration about $v$ and $u_1$ is of the form $\Phi(7,0,7,s)$; by Claim \[clm:Reducible\] $s = 0$, $|\operatorname{Resp}(u_1)|\leq 7$, and $u_1$ sends charge $\frac{1}{7}$ to $v$ by Rule R\[adj\].
*Case : $|\operatorname{Resp}(v)| = 13$.*
By Claim \[clm:Reducible\], the $Y$-type configuration $Y(t_1,t_2,t_3)$ about $v$ is one of $Y(0,6,7)$, $Y(1,6,6)$, $Y(1,5,7)$, $Y(2,4,7)$, or $Y(3,3,7)$. We consider each case separately.
*Case [.]{}: $(t_1,t_2,t_3) = (0,6,7).$*
Let $u_1$ be the vertex in $N_3(v)$ at distance 1 from $v$. If $\mu(v,u_1) = 1$, then the $H$-type configuration about $v$ and $u_1$ is of the form $H(6,7;0;s_1,s_2)$; by Claim \[clm:Reducible\] $s_1+s_2 \leq 9$, $|\operatorname{Resp}(u_1)|\leq 9$, and $u_1$ sends charge $\frac{1}{14}$ to $v$ by Rule R\[near\]. If $\mu(v,u_1) = 2$, then the $\Phi$-type configuration about $v$ and $u_1$ is of the form $\Phi(6,0,7,s)$ or $\Phi(7,0,6,s)$; by Claim \[clm:Reducible\] $s = 0$, $|\operatorname{Resp}(u_1)|\leq 7$, and $u_1$ sends charge $\frac{1}{14}$ to $v$ by Rule R\[near\].
*Case [.]{}: $(t_1,t_2,t_3) = (1,6,6).$*
Let $u_1$ be the vertex in $N_3(v)$ | 1 | member_28 |
---
author:
- |
V.A. Baskov, A.V. Koltsov, A.I. L’vov, A.I. Lebedev, L.N. Pavlyuchenko, , E.V. Rzhanov, S.S. Sidorin, G.A. Sokol\
P.N. Lebedev Physical Institute, Leninsky prospect 53, Moscow 119991, Russia\
Email:
- |
S.V. Afanasiev, A.I. Malakhov\
Joint Institute for Nuclear Research, Joliot-Curie 6, Dubna 141980, Moscow region, Russia
- |
A.S. Ignatov, V.G. Nedorezov\
Institute for Nuclear Research, 60-letiya Oktyabrya prospekt 7a, Moscow 117312, Russia
title: 'Studies of eta-mesic nuclei at the LPI electron synchrotron'
---
Introduction: $\eta$-mesic nuclei {#introduction-eta-mesic-nuclei .unnumbered}
=================================
$\eta$-mesic nuclei, i.e. nuclear systems $_\eta A$ having the $\eta$-meson bound in a nuclear orbit by strong interaction with $A$ nucleons, have been predicted long ago [@haider86; @liu86] — soon after recognizing the attractive character of the $\eta N$ interaction at low energies [@bhalerao85]. Observations and investigations of these exotic systems would be very valuable for understanding meson-baryon interactions in free space and in nuclei and for studies of properties of hadrons in the dense nuclear matter.
[r]{}[0.5]{} {width="40.00000%"}
The $\eta$-meson, together with pions and kaons, belongs to the SU(3) octet of pseudoscalar mesons and has, therefore, a similar $q\bar q$ space structure. In contrast to the pion, however, the pseudoscalar coupling of $\eta$ to the | 1 | member_29 |
nucleon is empirically rather small [@tiator94]. Nevertheless the amplitude of $\eta N$ $s$-wave scattering is not as small as that for $\pi N$ scattering because of the contribution of the $s$-wave resonance $S_{11}(1535)$ which is actually a chiral partner of the nucleon — the lowest lying baryon with the opposite parity to the nucleon. This resonance has the mass slightly above the $\eta N$ threshold, $m_\eta+m_N = 1486$ MeV, and owing to its very strong coupling to the $\eta N$ channel \[with the branching ratio ${\rm Br}\;(S_{11}(1535)\to\eta N) \simeq 55\%$\] strongly enhances all interactions in this channel. A nice illustration of this feature is provided by Mainz data [@mcnicoll10] on the total cross section of $\eta$ photoproduction off protons. A huge near-threshold enhancement shown in Fig. \[eta-x-sect\] is just a manifestation of the $S_{11}(1535)$ resonance excited in the reaction $\gamma p\to
S_{11}(1535)\to\eta p$.
The $S_{11}(1535)$ resonance strongly contributes to the low-energy $\eta N$ scattering and, in particular, makes the threshold value of the $\eta N$ scattering amplitude (i.e. the $\eta N$ scattering length $a_{\eta N}$) positive. In the framework of a dynamical resonance model for the coupled channels $\pi N$, $\eta N$ and $\pi\pi
N$, Bhalerao and Liu [@bhalerao85] found | 1 | member_29 |
a\_[N]{} = 0.28 + i 0.19 fm. \[a-etaN-BL\] The positive value of ${\rm Re\,}a_{\eta N}$ means an effective attraction between $\eta$ and $N$, so that one can expect that several nucleons could jointly bind $\eta$ to a nuclear orbit. The first-order static-limit on-shell optical potential of $\eta$ in the nuclear matter at zero energy $E_\eta^{\rm kin}=0$ is equal to U(r) = -2 a\_[N]{}(r) ( + ), what gives \[together with Eq. (\[a-etaN-BL\])\] $U = -34 -i\;23$ MeV at normal nuclear matter density $\rho=\rho_0 = 0.17~\rm fm^{-3}$. The imaginary part of the potential describes a local absorption rate $\Gamma = -2\,{\rm Im}\,U$ of $\eta$ in the nuclear substance.
With the above strength of the $\eta A$ potential, $\eta$-mesic nuclei $_\eta A$ are expected to exist for all $A\ge 10$ [@haider02; @haider09]. Actually, due to a sharp (cusp) energy dependence of the $\eta N$ scattering amplitude near threshold, Fermi motion of nucleons and $\eta$ reduces the optical potential \[especially its imaginary part\], and this makes $\eta$-mesic nuclei to exist only for $A\ge 12$. For binding energies and widths of the lightest $\eta$-mesic nuclei Haider and Liu predicted [@haider02; @haider09] E\_= -1.19 [MeV]{}, && \_= 7.34 [MeV for\^[12]{}\_[ ]{}C]{}, E\_= -3.45 [MeV]{}, && | 1 | member_29 |
\_= 10.76 [MeV for\^[16]{}\_[ ]{}O]{}, E\_= -6.39 [MeV]{}, && \_= 13.20 [MeV for\^[26]{}\_[ ]{}Mg]{}. \[eq:EB-Liu\] Note, however, that a stronger $\eta N$ scattering amplitude was inferred in some other analyses. For example, using a $K$-matrix model for coupled channels $\pi N$, $\eta N$, $\gamma N$ and $\pi\pi N$, Green and Wycech [@green97; @green05] found from fit to available data a\_[N]{} = (0.910.06) + i (0.270.02) fm. With such a big strength of $\eta N$ interaction lighter $\eta$-mesic nuclei could also exist.
As an example of different predictions for binding energies and widths of $\eta$-mesic nuclei we mention very elaborated calculations [@oset02a; @oset02b; @oset02c], in which a model for meson-baryon interaction with dynamically generated resonances was build using a unitarized chiral perturbation theory for coupled channels $\pi N$, $\eta N$, $K\Lambda$, $K\Sigma$ and $\pi\pi N$ and then self-energies of all the particles in the nuclear matter were evaluated consistently. This approach leads to the $\eta N$ scattering length $a_{\eta N} =
0.264 + i\, 0.245~\rm fm$ close to that obtained in Eq. (\[a-etaN-BL\]). The resulting $\eta A$ potential is, however, found stronger owing to nonlinear dressing effects: $U = -54 -i\, 29$ MeV at normal nuclear density. Also stronger are $\eta$-meson | 1 | member_29 |
bindings found in [@oset02c]: E\_= -9.71 [MeV]{}, && \_= 35.0 [MeV for\^[12]{}\_[ ]{}C]{}, E\_= -12.57 [MeV]{}, && \_= 33.4 [MeV for\^[24]{}\_[ ]{}Mg]{}. \[eq:EB-Oset\]
Bindings with equally large widths arise also in calculations [@jido02; @nagahiro05; @jido08] that use a chiral doublet model and treat $\eta A$ and $S_{11}(1535)A$ attraction as a result of partial restoration of chiral symmetry in the dense nuclear matter leading to reduction of the $S_{11}(1535){-}N$ mass gap. It is clear that experimental data on energies and widths of $\eta$-mesic nuclei are needed to test these and many other models and calculations.
Signature for eta-mesic nuclei produced in photoreactions {#signature-for-eta-mesic-nuclei-produced-in-photoreactions .unnumbered}
=========================================================
A mechanism of $\eta$-mesic nuclei formation and decay in the photoreaction + A N’ + \_[ ]{}(A-1)N’ + + N + (A-2) \[reac:piN\] is shown in Fig. \[diagrams-piN\]a. A fast nucleon $N'$ ejected forward at the first stage of the reaction, i.e. in the subprocess + N’ N’ + \_[slow]{}, \[reac:eta\] escapes the nucleus, whereas a slow $\eta$ is captured by remaining $A-1$ nucleons to a bound state. At $E_\gamma \sim 800{-}900$ MeV, a minimal momentum transfer to $\eta$ in the reaction (\[reac:eta\]) is not large (less than $70~{\rm MeV}/c$). That is why the total cross | 1 | member_29 |
section of $\eta$-mesic nuclei formation off light nuclei (like carbon or oxygen implied in the following) turns out to be a few $\mu$b [@kohno89; @lebedev89; @lebedev91; @lebedev95; @tryasuchev99; @tryasuchev01], i.e. $\simeq 2{-}7\%$ of the total cross section $\sigma_{\gamma
A}^\eta$ of inclusive $\eta$ photoproduction, with the exact value strongly dependent on the assumed strength of the optical potential $U$.
![a) $\eta$-mesic nuclei formation and decay with the emission of back-to-back $\pi N$ pairs. b) Background creation of back-to-back $\pi N$ pairs by unbound $\eta$.[]{data-label="diagrams-piN"}](fig-diagram-piN.eps){width="80.00000%"}
Energies $E[{}_\eta(A-1)]$ of the produced $\eta$-mesic nuclei can, in principle, be determined through missing mass measurements in the reaction $(\gamma,p)$ using tagged photons $\gamma$ and a magnetic spectrometer for $N'=p$. Indirectly, the same energy E\[\_(A-1)\] = E\_+ E\_[A-1]{} = E\_[N]{} + E\_[A-2]{} \[eq:E-eta-(A-1)\] can also be found from the observed energy of a correlated back-to-back $\pi N$ pair produced at the second stage of the reaction (\[reac:piN\]) where the captured $\eta$ meson annihilates through the subprocess N N. \[etaNpiN\] The energy excitation of $(A-2)$ in (\[eq:E-eta-(A-1)\]) is not a fixed value. It rather depends on whether an $s$-shell or $p$-shell nucleon $N$ is knocked out in the process (\[etaNpiN\]). Therefore a distribution of the experimental observable $E_{\pi | 1 | member_29 |
N}$ has appropriately a bigger width than the width of the $\eta$-mesic nucleus.
Neglecting binding and Fermi motion of nucleons and $\eta$, we have the following kinematical characteristics of the ejected correlated $\pi N$ pairs (as for energies, momenta and velocities): && s = E\_+ E\_N = m\_+m\_N = 1486 MeV, && E\_\^[kin]{} = 313 [MeV]{},E\_N\^[kin]{} = 94 [MeV]{}, p\_= p\_N = 431 [MeV]{}/c, && \_= 0.95,\_N=0.42. \[kinema-piN\] A simple simulation that takes into account the Fermi motion of nucleons and $\eta$ as well as binding of these particles reveals that fluctuations around these ideal parameters are substantial (see Fig. \[simulation-piN\]) \[specifically, we used in this simulation the $\eta$-meson binding energy of 10 MeV with the width 25 MeV; for nucleons, we assumed a Fermi-gas distribution with binding energies distributed between 5 and 30 MeV\]. In particular, the angle $\theta_{\pi N}$ between the emitted pion and nucleon may not be so close to $180^\circ$, and a subtraction of background events with $\theta_{\pi N} \ne 180^\circ$ used sometimes in practice should be done cautiously. A shift of the peak down to 1486 MeV in the distribution of the total energy $E_{\pi N}=E_\pi+E_N$ seen in Fig. \[simulation-piN\] is related with binding of | 1 | member_29 |
both the $\eta$-meson (by 10 MeV) and the nucleon (by 15 MeV).
![Simulation of $\pi N$ pairs emitted in $\eta$-mesic nuclei decays. Shown are distributions over kinetic energies of the particles, their total energy, velocities, and the $\pi N$ relative angle.[]{data-label="simulation-piN"}](fig-Tpi.ps "fig:"){width="33.00000%"} ![Simulation of $\pi N$ pairs emitted in $\eta$-mesic nuclei decays. Shown are distributions over kinetic energies of the particles, their total energy, velocities, and the $\pi N$ relative angle.[]{data-label="simulation-piN"}](fig-TN.ps "fig:"){width="33.00000%"} ![Simulation of $\pi N$ pairs emitted in $\eta$-mesic nuclei decays. Shown are distributions over kinetic energies of the particles, their total energy, velocities, and the $\pi N$ relative angle.[]{data-label="simulation-piN"}](fig-Etot.ps "fig:"){width="33.00000%"}
![Simulation of $\pi N$ pairs emitted in $\eta$-mesic nuclei decays. Shown are distributions over kinetic energies of the particles, their total energy, velocities, and the $\pi N$ relative angle.[]{data-label="simulation-piN"}](fig-betapi.ps "fig:"){width="33.00000%"} ![Simulation of $\pi N$ pairs emitted in $\eta$-mesic nuclei decays. Shown are distributions over kinetic energies of the particles, their total energy, velocities, and the $\pi N$ relative angle.[]{data-label="simulation-piN"}](fig-betaN.ps "fig:"){width="33.00000%"} ![Simulation of $\pi N$ pairs emitted in $\eta$-mesic nuclei decays. Shown are distributions over kinetic energies of the particles, their total energy, velocities, and the $\pi N$ relative angle.[]{data-label="simulation-piN"}](fig-costheta.ps "fig:"){width="33.00000%"}
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Notice that $\pi N$ pairs with | 1 | member_29 |
the characteristics (\[kinema-piN\]) do not necessary originate from $\eta$-mesic nuclei decays. They can also be produced by slow etas in a background nonresonance process shown in Fig. \[diagrams-piN\]b. The resonance and nonresonance processes correspond to a resonance (Breit-Wigner) and nonresonance part of the full propagator \[i.e. the Green function $G({\bm r}_1,
{\bm r}_2; E_\eta)$\] of the $\eta$-meson moving in the optical potential $U(r)$. Jointly, these parts generate a complicated spectrum of $E_\eta$ similar to that obtained in a toy model with a square-well potential [@lvov98; @sokol99]. Shown in Fig. \[spectral-function\] is the spectral function in that model, S(E\_) = ([r]{}\_1) ([r]{}\_2) |G([r]{}\_1, [r]{}\_2; E\_)|\^2 d[r]{}\_1 d[r]{}\_2, that characterizes near-threshold energy distribution of the propagated etas as well as the near-threshold energy dependence of the yield of $\pi N$ pairs produced by these $\eta$. Bound states of the $\eta$-meson give pronounced peaks in the yield of the $\pi N$ pairs at subthreshold energies $E_\eta$. Generally, observation of a relatively narrow resonance peak in the spectrum of $E_\eta$ in the region $E_\eta < m_\eta$ is mandatory for claiming an observation of $\eta$-mesic nuclei at all. We refer to recent works by Haider and Liu [@haider10a; @haider10b] where a deeper and more elaborated | 1 | member_29 |
consideration is given in relation with a recent experiment.
Since $\eta$ is isoscalar, the $\pi N$ pairs produced in the subprocess (\[etaNpiN\]) have isospin $\frac12$ and hence the following isotopic contents \[for $\eta$-mesic nuclei with $A\gg 1$\]: (N) ={
[lll]{} 1/3 & [for]{} & \^+n,\
1/6 & [for]{} & \^0p,\
1/6 & [for]{} & \^0n,\
1/3 & [for]{} & \^-p.
. \[piN-modes\] From these, the channel $\pi^+n$ was chosen for detection in our experiment.
Previous searches for $\eta$-mesic nuclei {#previous-searches-for-eta-mesic-nuclei .unnumbered}
=========================================
Searches for $\eta$-mesic nuclei began very soon after their predictions [@haider86] followed by suggestions [@liu86; @kohno89; @lebedev89; @lebedev91; @kohno90] to seek these novel high-energy nuclear excitations in missing-mass experiments using the inclusive reactions $(\pi^+,p)$ and $(\gamma,p)$.
The first two experiments have been done along this line in 1988 at Brookhaven [@chrien88] and Los Alamos [@lieb88a; @lieb88b]. In both experiments, a $\pi^+$ beam was used and several targets (Li, C, O and Al) were examined. The inclusive $(\pi^+,p)$ reaction \^+ + A \_(A-1) + p was studied in [@chrien88] with a magnetic spectrometer, whereas the Los Alamos experiment had also an additional $4\pi$ BGO crystal ball for detecting charged paticles ejected in the subprocess (\[etaNpiN\]) of $\eta$-mesic nuclei decays | 1 | member_29 |
to $\pi N$ pairs in coincidence with the forward proton $p$.
The Brookhaven experiment did not find a theoretically expected signal [@liu86] — a relatively narrow peak of a predicted strength in the missing mass spectrum. The team working at Los Alamos did report a preliminary evidence for a wanted peak for the $^{16}$O target but this report was not confirmed (published) since then.
It was recognized in the following that the above obtained negative or incomplete results do not necessarily mean that the predicted $\eta$-mesic nuclei do not exist. It was possible that the binding energies and especially the widths of the $\eta$ bound states were theoretically underestimated. This point of view was supported by many-body calculations [@chiang91] taking into account some effects disregarded in the first theoretical works [@haider86; @liu86], in particular — dressing, binding and collisional decays of the $S_{11}(1535)$ resonance in the dense nuclear matter. The analysis of [@chiang91] was later extended and revised [@oset02a; @oset02b; @oset02c] (in particular, dressing of mesons was also included) with the main conclusion survived that $\eta$-mesic nuclei widths are bigger than those found in [@haider86; @liu86].
The next experiment has been performed at the Lebedev Physical Institute in Moscow/Troitsk [@sokol99; | 1 | member_29 |
@sokol00] (see also a summary in [@sokol08]). It was triggered [@sokol94; @lebedev95a] by a suggestion [@sokol91] to seek $\eta$-mesic nuclei through observing decay products of $\eta$-mesic nuclei, namely two correlated back-to-back particles, a pion and a nucleon, ejected in the process of annihilation of captured $\eta$-mesons in the nucleus, Eq. (\[etaNpiN\]). It was hoped that a background for the two very energetic particles (the pion and the nucleon) ejected in decays of $\eta$-mesic nuclei transversely to the beam would be lower than that for ejection of forward protons in the inclusive processes. Besides, it was hoped that background conditions in photon-induced reactions would be generally better than those in pion-induced ones.
Studies of the reaction + \^[12]{}[C]{} (\_[ ]{}\^[11]{}[Be]{} [ or ]{} \_[ ]{}\^[11]{}[C]{}) + N \^+ + n + X + N \[reac-C:piN\] done in the middle of 1990’s at the LPI electron synchrotron indeed showed a signal of an enhanced production of the correlated back-to-back $\pi^+n$ pairs ejected transversely to the photon beam when the photon energy exceeded the $\eta$-meson photoproduction threshold. Energy resolution of the experimental setup was, however, not sufficient to resolve a peak similar to that shown in Fig. \[spectral-function\] and to determine whether the | 1 | member_29 |
observed correlated pairs were produced by bound or unbound intermediate etas.
After the works [@sokol99; @sokol00] gaining and using information on the decay products became mandatory for experiment planning and data analysis in all further searches for $\eta$-mesic nuclei.
In 2004 an evidence for the $\eta$-mesic nucleus $_\eta^3$He formed in the reaction + \^3[He]{} \_\^3[He]{} \^0 + p + X has been reported from Mainz [@pfeiffer04]. A resonance-like structure was observed in a contribution to the cross section from back-to-back $\pi^0p$ pairs found after a background subtraction. A later study [@pheron12] revealed, however, that the background has a rather complicated structure, so that the conclusions of Ref. [@pfeiffer04] cannot be confirmed. At the moment their statement is that the existence of the $\eta$-mesic nucleus $_\eta^3$He is not yet established.
One more attempt to find $\eta$-mesic nuclei by detecting their $\pi^-p$ decay products has recently been done at the JINR nuclotron [@afanasiev11]. The reaction studied was d + \^[12]{}[C]{} (\_[ ]{}\^[11]{}[Be]{} [ or ]{} \_[ ]{}\^[11]{}[C]{}) + N\_1 + N\_2 \^- + p + X + N\_1 + N\_2. \[reac:Dubna\] The measured effective mass spectra of the correlated back-to-back $\pi^-p$ pairs show a presence of resonance-like peaks lying slightly below the | 1 | member_29 |
threshold energy $m_\eta+m_N=1486$ MeV. However, a consistent interpretation of these peaks was not yet obtained.
To date the strongest evidence for the existence of $\eta$-mesic nuclei came from the precision COSY-GEM experiment [@budzanowski09]. Following ideas of the work [@hayano99] borrowed in turn from previous experience in studying deeply-bound pionic states in nuclei, the reaction p + \^[27]{} \^3[He]{} + \^[25]{}\_[ ]{}[Mg]{} \^3[He]{} + p + \^- + X \[reac:COSY-GEM\] of a recoilless formation of the $\eta$-mesic nuclei $^{25}_{~\eta}\rm
Mg$ was explored and the mass of this $\eta$-mesic nucleus was determined through precision missing-mass measurements in $(p,
{}^3{\rm He})$. A clear peak was found in the missing mass spectrum that corresponds to the binding energy $-13.13\pm 1.64$ MeV and the width $10.22\pm 2.98$ MeV of the formed $\eta$-mesic nucleus. An upper limit of $\approx 0.5$ nb was found for the cross section of the $\eta$-mesic nucleus formation.
Recently Haider and Liu argued [@haider10a; @haider10b] that the observed peak in (\[reac:COSY-GEM\]) is shifted down from the genuine binding energy of $\eta$ because of interference of the resonance and nonresonance mechanisms of the reaction (similar to those shown in Fig. \[diagrams-piN\]). This very interesting effect signifies that the genuine $\eta$ binding in ${}_{~\eta}^{25}{\rm | 1 | member_29 |
Mg}$ is $\approx -8$ MeV with the width $\approx 19$ MeV.
On the two-nucleon decay mode of $\eta$-mesic nuclei {#on-the-two-nucleon-decay-mode-of-eta-mesic-nuclei .unnumbered}
====================================================
[r]{}[0.4]{} {width="35.00000%"}
{width="40.00000%"}
The main novelty in our present research is exploring a new possibility for searching for $\eta$-mesic nuclei, namely through observation of their two-nucleon decay mode arising owing to the two-nucleon absorption of the captured $\eta$ in the nucleus, NN NN, \[2N-absorption\] see Fig. \[diagram-NN\]. Ejected in this process correlated back-to-back nucleons of the $NN$ pairs have very high energies ($E_N^{\rm kin}\simeq \frac12 m_\eta = 274$ MeV) and momenta ($p_N\simeq 770~{\rm MeV}/c$), so that they are to be visible (especially in coincidence) at the background of other particles emitted in photoreactions at $E_\gamma\sim 800$ MeV and thus should provide a bright signature for the $\eta$-mesic nucleus formation.
The $NN$ pair production in decays of $\eta$ in the nuclear matter was considered among other channels by Chiang, Oset and Liu [@chiang91] in terms of the self-energy of $S_{11}(1535)$ that includes a contribution of $S_{11}(1535)N\to NN$. A more direct and rather transparent evaluation of this process has been done by Kulpa and Wycech [@kulpa98b] who used available experimental data on the inverse reactions $pp\to pp\eta$, $pn\to pn\eta$ and | 1 | member_29 |
$pn\to d\eta$ and then converted them into the rate of (\[2N-absorption\]). In terms of the imaginary part $W_{NN}$ of the optical potential $U$, this rate was found to be proportional to $\rho^2$, being $W_{NN}=3.4$ MeV at central nuclear density. This is only about 15% of $W_N \sim 23$ MeV related with the absorption of $\eta$ by one nucleon. Nevertheless such a small fraction of $NN$ can be quite visible experimentally because of a specific isotopic contents of the $\pi N$ and $NN$ pairs.
The matter is that $\gtrsim 90\%$ of these $NN$ pairs are proton plus neutron because the cross section of $pp\to pp\eta$ (and $nn\to
nn\eta$) is by order or magnitude less than that of $pn\to pn\eta$ (plus $pn\to d\eta$), see Fig. \[fig-etaNN\] where pertinent Uppsala-Celsius [@calen96; @calen97; @calen98a; @calen98b] and COSY [@smyrski00; @moskal09] data are shown (and see also, e.g., [@baru03] for theoretical explanations). This difference can be traced to isospin factors and Fermi statistics signs in the dominating pion-exchange mechanism of the reaction $NN\to NN\eta$ shown in Fig. \[diagrams-NNeta\]. If the experimental setup detects one charged and one neutral particle from the pairs, it detects $\sim90\%$ of $NN$ and only $\sim33\%$ of $\pi N$. Then count rates | 1 | member_29 |
of the setup would not be so different for $pn$ and $\pi^+n$ pairs. That seems to be exactly what we see in our experiment.
![Pion-exchange mechanism of $NN\to NN\eta$. Isospin factors, which accompany the $\pi NN$ coupling $g$ and the $\pi N\to\eta N$ amplitude $T$, and the Fermi-statistics signs (both shown in this Figure) jointly determine the big difference between the cross sections of $pp\to pp\eta$ and $pn\to pn\eta$ (plus $pn\to d\eta$). Antisymmetrization of the initial state and initial/final state interactions are not shown.[]{data-label="diagrams-NNeta"}](fig_pp.ps "fig:"){width="50.00000%" height="13ex"} ![Pion-exchange mechanism of $NN\to NN\eta$. Isospin factors, which accompany the $\pi NN$ coupling $g$ and the $\pi N\to\eta N$ amplitude $T$, and the Fermi-statistics signs (both shown in this Figure) jointly determine the big difference between the cross sections of $pp\to pp\eta$ and $pn\to pn\eta$ (plus $pn\to d\eta$). Antisymmetrization of the initial state and initial/final state interactions are not shown.[]{data-label="diagrams-NNeta"}](fig_pn.ps "fig:"){width="50.00000%" height="13ex"}
![Simulation of $NN$ pairs emitted in decays of $\eta$-mesic nuclei. Shown are distributions over the kinetic energy and velocity of one of the nucleons, the total energy of the pair and the relative angle.[]{data-label="simulation-NN"}](figNN-TN.ps "fig:"){width="36.00000%"} ![Simulation of $NN$ pairs emitted in decays of $\eta$-mesic nuclei. Shown are distributions over the kinetic energy | 1 | member_29 |
and velocity of one of the nucleons, the total energy of the pair and the relative angle.[]{data-label="simulation-NN"}](figNN-Etot.ps "fig:"){width="36.00000%"}\
![Simulation of $NN$ pairs emitted in decays of $\eta$-mesic nuclei. Shown are distributions over the kinetic energy and velocity of one of the nucleons, the total energy of the pair and the relative angle.[]{data-label="simulation-NN"}](figNN-bN.ps "fig:"){width="36.00000%"} ![Simulation of $NN$ pairs emitted in decays of $\eta$-mesic nuclei. Shown are distributions over the kinetic energy and velocity of one of the nucleons, the total energy of the pair and the relative angle.[]{data-label="simulation-NN"}](figNN-costheta.ps "fig:"){width="36.00000%"}
Neglecting binding effects and effects of Fermi motion of nucleons and $\eta$, we have the following kinematical characteristics of the correlated $NN$ pairs (i.e. energies, momenta, velocities) ejected in $\eta$-mesic nuclei decays: E\_[N\_1]{}\^[kin]{} = E\_[N\_2]{}\^[kin]{} = 12 m\_= 274 [MeV]{}, p\_[N\_1]{} = p\_[N\_2]{} = 767 [MeV]{}/c, \_[N\_1]{} = \_[N\_2]{} = 0.63. \[kinema-NN\] Actually, the Fermi motion and binding leads to fluctuations around these ideal parameters as a simple simulation reveals, see Fig. \[simulation-NN\]. Note that the angular correlation in the $NN$ pairs is stronger than that in the $\pi N$ pairs — owing to higher momenta of particles in $NN$.
The first studies of the photoreaction + \^[12]{}[C]{} (\_[ ]{}\^[11]{}[Be]{} [ or | 1 | member_29 |
]{} \_[ ]{}\^[11]{}[C]{}) + N p + n + X + N \[reac:NN\] have recently been done at the LPI synchrotron and we report below on the obtained results.
Experimental setup at LPI {#experimental-setup-at-lpi .unnumbered}
=========================
[r]{}[0.3]{} {width="30.00000%"}
Our experiment was performed at the bremsstrahlung photon beam of the 1.2-GeV electron synchrotron of the Lebedev Physical Institute. Photons were produced with an electron beam of intensity $I_e \simeq 10^{12}
~\rm s^{-1}$ and the duty factor $\simeq 10\%$. The energy of the beam was usually $E_e = E_{\gamma\;\rm max} = 850~\rm MeV$ (i.e. above the $\eta$ photoproduction threshold off free nucleons, $E_{\eta\;\rm
thr}=708~\rm MeV$); additional measurements of subthreshold backgrounds have been done at $E_e = E_{\gamma\;\rm max} = 650~\rm
MeV$.
The experimental setup included two time-of-flight arms (two scintillation telescopes — C and N arms) for detecting in coincidence charged and neutral particles (back-to-back pairs), see Fig. \[exp-setup\]. These arms were both positioned at $90^\circ{-}90^\circ$ with respect to the beam axis in order to minimize background.
The C-arm used for detection of charged particles is a plastic TOF spectrometer for charged pions and protons. It consists of a start detector T1 ($20\times 20\times 2~\rm cm^3$), a stop detector T2 ($50\times 50\times | 1 | member_29 |
5~\rm cm^3$) and three energy losses detectors $\Delta E_1$, $\Delta E_2$ and $\Delta E_3$ ($40\times 40\times 2~\rm
cm^3$ each). A 4 mm lead (Pb) plate was used in some runs for TOF calibrations with ultrarelativistic electrons/positrons produced in the lead plate with high-energy photons emitted from the target owing to production and decays of neutral pions.
The N-arm is a plastic TOF spectrometer for neutrons. It consists of a veto counter A ($50\times 50\times 2~\rm cm^3$) and four plastic blocks — the neutron detectors N1, N2, N3 and N4 ($50\times 50\times
10~\rm cm^3$ each). Again, a 4 mm lead plate was used in some runs for TOF calibrations. The efficiency of the N-arm for neutrons of energies above 50 MeV was $\approx 30\%$.
In both arms each volume of scintillator counters/blocks was viewed from corners with 4 phototubes. The time-of-flight bases in the C and N arms were 1.4 m and the time resolution was $\simeq
200$ ps ($1\sigma$). The target was a carbon cylinder of the 10 cm length along the beam axis. Its diameter was 4 cm, i.e. slightly more than the diameter of the collimated photon beam (3 cm). The distance between the target and the start | 1 | member_29 |
detector T1 was 0.7 m.
Mostly, the setup was the same as in our previous work [@sokol00; @sokol08] but a few useful changes have been made:
- $\Delta E_i$ detectors have been placed after the time-of-flight interval T1-T2. This enabled us to have a better $\pi^\pm/p$ separation and time resolution.
- A transverse size of the start detector T1 was cut off according to required geometry. This reduced a background load of the C-arm.
- A thickness of the start detector was also reduced in order to improve time resolution.
- All unnecessary layers of absorbers used previously to suppress radiative backgrounds have been removed from the time-of-flight interval, with the effect of reducing the $e^+/e^-$ background created by photons from $\pi^0$ decays.
[r]{}[0.4]{} {width="35.00000%"}
General tests of the setup, including preliminary time calibrations of the arms, have been done in special runs, in which the quasifree reaction $\gamma p\to\pi^+n$ inside carbon nuclei was observed. In such runs the two arms of the setup have been moved to the angles $50^\circ{-}50^\circ$ where the high count rate enabled one to do the calibrations quickly. Lead convertors used in these runs provided reliable ultrarelativistic reference points $\beta=1$ for particle’s velocities $\beta_C$ and | 1 | member_29 |
$\beta_N$ measured in the C- and N-arms. A two-dimensional $\beta_C{-}\beta_N$ plot on Fig. \[2-dim-bb\] illustrates this procedure.
The calibration done provided a linear scale of velocities in the range $\beta = 0.6{-}1$ with errors of about 3% ($1\sigma$). We have checked the linearity of the scale by using cosmic rays and setting different distances between detectors.
Results and comparison with simulations {#results-and-comparison-with-simulations .unnumbered}
=======================================
Measurement runs have mostly been done in 2009 at two maximal beam energies: $E_{\gamma\;\rm max} = 650$ MeV and 850 MeV. The on-line trigger was a coincidence of particles in the C- and N-arms within a time gate of 50 ns.
For further off-line analysis events were selected with an additional condition of sufficiently long ranges of the charged particles, E\_i > E\_i\^[thr]{} \[eq:selection\] with experimentally adjusted thresholds $E_i^{\rm thr}$. In this way low-energy particles in the C-arm were rejected.
A two-dimensional histogram in the variables $\Delta E{-}\beta_C$, where $\Delta E$ is the minimal energy loss in the $\Delta E_i$ detectors, E=\_i(E\_i), is shown in Fig. \[2-dim-bE\] for the beam energy $E_{\gamma\;\rm
max} = 850$ MeV. Results of simulations using the Intra Nuclear Cascade (INC) model [@pschenichnov97] in the GEANT-3 framework are shown in Fig. \[2-dim-bE-INC\] | 1 | member_29 |
for comparison. The INC model takes into account production of various mesons and baryon resonances, their free propagation in the nuclear matter, and then various $2\to 2$ collisional reactions including $\eta N\to\pi N$. This model successfully describes many photoreactions in wide kinematical ranges as was demonstrated, beyond [@pschenichnov97], in simulations of the GRAAL experiment at energies 500–1500 MeV [@ignatov08]. Binding effects for $\eta$ and reactions like $\eta NN\to NN$ were not included into the model, so one can try to find effects arising due to formation and decay of $\eta$-mesic nuclei through characteristic deviations of the model predictions from the experimental data.
The simulation shows that the selection (\[eq:selection\]) of particles with sufficiently long ranges distinguishes very well protons (as particles with $\beta_C \leq 0.7$) and pions (as particles with $\beta_C \geq 0.7$): the overlap is less than 1%.
![Two-dimensional $\Delta E{-}\beta_C$ distribution, the INC model.[]{data-label="2-dim-bE-INC"}](fig-2dim-bE.eps){width="60.00000%"}
![Two-dimensional $\Delta E{-}\beta_C$ distribution, the INC model.[]{data-label="2-dim-bE-INC"}](fig-2dim-bE-INC.eps){width="60.00000%"}
Considering one-dimensional spectra over $\beta_C$ of events selected according to the condition (\[eq:selection\]) of sufficiently long ranges and imposing the additional cut-off $0.3 <\beta_N < 0.7$ for neutron velocities, we find rather interesting structures in the spectra. Shown in Fig. \[bC\] are experimental data (blue areas) together | 1 | member_29 |
with results of the INC simulation (pink hatched areas). Separately shown are INC predictions for the number of protons and charged pions in the C-arm. There is a qualitative agreement of the INC simulation with the experimental data for the case of the subthreshold beam energy, $E_e=650$ MeV. Meanwhile, in the case of $E_e=850$ MeV there is a clear excess of the experimentally observed events over the simulation results in two velocity regions closely corresponding to the kinematics of $\eta$-mesic nuclei decays with emission of $\pi N$ and $NN$ correlated pairs, Eqs. (\[kinema-piN\]) and (\[kinema-NN\]).
Knowing from the INC simulations that the ”normal” (without $\eta$-mesic nuclei) dynamics of the considered reaction does not yield a sufficient amount of protons and pions with the velocities of about $\beta_C\sim 0.7$, we interpret the found anomaly at $\beta_C\sim 0.7$ as a result of production of low-energy $\eta$-mesons followed by their two-nucleon annihilation.
The energy resolution of the experimental setup is not sufficient to say whether an essential part of these $\eta$-mesons is produced in the bound state, but theoretical arguments discussed in above make such a statement plausible.
Concerning the excess of pions with $\beta_C\simeq 0.95$, this feature is in agreement with our | 1 | member_29 |
measurements reported earlier [@sokol99; @sokol00; @sokol08]. It can be interpreted as an evidence for one-nucleon annihilation of produced low-energy $\eta$-mesons (bound or unbound).
Electron/positron peaks shown in Fig. \[bC\] originate from calibration runs with the lead plate inserted. They were not included into simulations made.
![Velocity distribution of charged particles selected according to the criterion $\Delta E_i > 0$ (for all $i=1,2,3$) at $E_e=650$ and 850 MeV. A well visible excess of events over the INC simulation is seen at the right panel — in the case of the beam energy exceeding the $\eta$-photoproduction threshold — in both velocity regions corresponding to the expected velocities of the $\pi N$ and $NN$ decay products of $\eta$-mesic nuclei.[]{data-label="bC"}](fig-bC-650.eps){width="45.00000%"}
The observed proton peak in the $\beta_C$ distribution is very unusual because it corresponds to $pn$ pairs with very high kinetic energies $T_p\sim T_n\sim 200{-}300$ MeV and transverse momenta $p_p\sim
p_n\sim 400{-}800$ MeV/c. One should keep in mind that photons which produce such pairs have quite a modest energy $650~{\rm MeV} < E_\gamma
< 850$ MeV. Ordinary photoproduction reactions do not give nucleons with such a high energy and momentum. Creation and annihilation of intermediate low-energy $\eta$-mesons seems to be the only explanation to | 1 | member_29 |
---
abstract: 'Dissipationless collapses in Modified Newtonian Dynamics (MOND) are studied by using a new particle-mesh N-body code based on our numerical MOND potential solver. We found that low surface-density end-products have shallower inner density profile, flatter radial velocity-dispersion profile, and more radially anisotropic orbital distribution than high surface-density end-products. The projected density profiles of the final virialized systems are well described by Sersic profiles with index $m \lsim 4$, down to $m\sim 2$ for a deep-MOND collapse. Consistently with observations of elliptical galaxies, the MOND end-products, if interpreted in the context of Newtonian gravity, would appear to have little or no dark matter within the effective radius. However, we found impossible (under the assumption of constant mass-to-light ratio) to simultaneously place the resulting systems on the observed Kormendy, Faber-Jackson and Fundamental Plane relations of elliptical galaxies. Finally, the simulations provide strong evidence that phase mixing is less effective in MOND than in Newtonian gravity.'
author:
- Carlo Nipoti
- Pasquale Londrillo
- Luca Ciotti
title: Dissipationless collapses in MOND
---
Introduction
============
In Bekenstein & Milgrom’s (1984, hereafter BM) Lagrangian formulation of Milgrom’s (1983) Modified Newtonian Dynamics (MOND), the Poisson equation $$\nabla^2{\phi_{\rm N}}=4\pi G\rho
\label{eqPoisson}$$ for the Newtonian gravitational | 1 | member_30 |
potential ${\phi_{\rm N}}$ is replaced by the field equation $$\nabla\cdot\left[\mu\left({\Vert\nabla\phi\Vert\over{a_0}}\right)
\nabla\phi\right] = 4\pi G \rho,
\label{eqMOND}$$ where ${a_0}\simeq 1.2 \times 10^{-10} {\rm m\, s^{-2}}$ is a characteristic acceleration, $\Vert ...\Vert$ is the standard Euclidean norm, $\phi$ is the MOND gravitational potential produced by the density distribution $\rho$, and in finite mass systems $\nabla\phi\to 0$ for $\Vert{{\bf x}}\Vert\to\infty$. The MOND gravitational field ${{\bf g}}$ experienced by a test particle is $${{\bf g}}=-\nabla\phi,
\label{eqgv}$$ and the function $\mu$ is such that $$\mu(y)\sim\cases{y&for $y\ll 1$,\cr 1&for $y\gg 1$;}
\label{eqmulim}$$ throughout this paper we use $$\mu (y)={y\over\sqrt{1+y^2}}.
\label{eqmu}$$ In the so-called ‘deep MOND regime’ (hereafter dMOND), describing low-acceleration systems ($\Vert\nabla\phi\Vert \ll{a_0}$), $\mu(y)=y$ and so equation (\[eqMOND\]) simplifies to $$\nabla\cdot\left({\Vert\nabla\phi\Vert}\nabla\phi\right) = 4\pi G {a_0}\rho.
\label{eqdMOND}$$ The source term in equation (\[eqMOND\]) can be eliminated by using equation (\[eqPoisson\]), giving $$\mu\left(\Vert\nabla\phi\Vert\over{a_0}\right)\nabla\phi=\nabla{\phi_{\rm N}}+{{\bf S}},
\label{eqcurl}$$ where ${{\bf S}}={\rm curl\,}{{\bf h}}$ is a solenoidal field dependent on $\rho$ and in general different from zero. When ${{\bf S}}=0$ equation (\[eqcurl\]) reduces to Milgrom’s (1983) formulation and can be solved explicitly. Such reduction is possible for configurations with spherical, cylindrical or planar symmetry, which are special cases of a more general family of stratifications (BM; Brada & Milgrom 1995). Though | 1 | member_30 |
the solenoidal field ${{\bf S}}$ has been shown to be small for some configurations (Brada & Milgrom 1995; Ciotti, Londrillo & Nipoti 2006, hereafter CLN), neglecting it when simulating time-dependent dynamical processes has dramatic effects such as non-conservation of total linear momentum (e.g. Felten 1984; see also Section \[secic\]).
Nowadays several astronomical observational data appear consistent with the MOND hypothesis (see, e.g., Milgrom 2002; Sanders & McGaugh 2002). In addition, Bekenstein (2004) recently proposed a relativistic version of MOND (Tensor-Vector-Scalar theory, TeVeS), making it an interesting alternative to the cold dark matter paradigm. However, dynamical processes in MOND have been investigated very little so far, mainly due to difficulties posed by the non-linearity of equation (\[eqMOND\]). Here we recall the spherically symmetric simulations (in which ${{\bf S}}=0$) of gaseous collapse in MOND by Stachniewicz & Kutschera (2005) and Nusser & Pointecouteau (2006). The only genuine three-dimensional MOND N-body simulations (in which equation \[\[eqMOND\]\] is solved exactly) are those by Brada & Milgrom (1999, 2000), who studied the stability of disk galaxies and the external field effect, and those of Tiret & Combes (2007). Other attempts to study MOND dynamical processes have been conducted using three-dimensional N-body codes by arbitrarily setting | 1 | member_30 |
${{\bf S}}=0$: Christodoulou (1991) investigated disk stability, while Nusser (2002) and Knebe & Gibson (2004) explored cosmological structure formation[^1].
In this paper we present results of N-body simulations of dissipationless collapse in MOND. The simulations were performed with an original three-dimensional particle-mesh N-body code, based on the numerical MOND potential solver presented in CLN, which solves equation (\[eqMOND\]) exactly. These numerical experiments are interesting both from a purely dynamical point of view, allowing for the first time to explore the relaxation processes in MOND, and in the context of elliptical galaxy formation. In fact, the ability of dissipationless collapse at producing systems strikingly similar to real ellipticals is a remarkable success of Newtonian dynamics (e.g., van Albada 1982; Aguilar & Merritt 1990; Londrillo, Messina & Stiavelli 1991; Udry 1993; Trenti, Bertin & van Albada 2005; Nipoti, Londrillo, & Ciotti 2006, hereafter NLC06), while there have been no indications so far that MOND can work as well in this respect. Here we study the structural and kinematical properties of the end-products of MOND simulations, and we compare them with the observed scaling relations of elliptical galaxies: the Faber–Jackson (FJ) relation (Faber & Jackson 1976), the Kormendy (1977) relation, and the Fundamental | 1 | member_30 |
Plane (FP) relation (Djorgovski & Davis 1987, Dressler et al. 1987).
The paper is organized as follows. The main features of the new N-body code are presented in Section \[seccod\], while Section \[secsim\] describes the set-up and the analysis of the numerical simulations. The results are presented in Section \[secres\] and discussed in Section \[secdis\].
The N-body code {#seccod}
===============
While most N-body codes for simulations in Newtonian gravity are based on the gridless multipole expansion treecode scheme (Barnes & Hut 1986; see also Dehnen 2002), the non-linearity of the MOND field equation (\[eqMOND\]) forces one to resort to other methods, such as the particle-mesh technique (see Hockney & Eastwood 1988). In this approach, particles are moved under the action of a gravitational field which is computed on a grid, with particle-mesh interpolation providing the link between the two representations. In our MOND particle-mesh N-body code, we adopt a spherical grid of coordinates ($r$, $\vartheta$, $\varphi$), made of $\Nr\times\Nth\times\Nph$ points, on which the MOND field equation is solved as in CLN. Particle-mesh interpolations are obtained with a quadratic spline in each coordinate, while time stepping is given by a classical leap-frog scheme (Hockney & Eastwood 1988). The time-step $\Dt$ is | 1 | member_30 |
the same for all particles and is allowed to vary adaptively in time. In particular, according to the stability criterion for the leap-frog time integration, we adopt $\Dt=\eta/\sqrt{\max{|\nabla^2
\phi|}}$, where $\eta\lsim0.3$ is a dimensionless parameter. We found that $\eta=0.1$ assures good conservation of the total energy in the Newtonian cases (see Section \[secic\]). In the present version of the code, all the computations on the particles and the particle-mesh interpolations can be split among different processors, while the computations relative to the potential solver are not performed in parallel. The solution of equation (\[eqMOND\]) over the grid is then the bottleneck of the simulations: however, the iterative procedure on which the potential solver is based (see CLN) allows to adopt as seed solution at each time step the potential previously determined.
The MOND potential solver can also solve the Poisson equation (obtained by imposing $\mu=1$ in equation \[eqMOND\]), so Newtonian simulations can be run with the same code. We exploited this property to test the code by running several Newtonian simulations of both equilibrium distributions and collapses, comparing the results with those of simulations (starting from the same initial conditions) performed with the FVFPS treecode (Londrillo, Nipoti & Ciotti 2003; | 1 | member_30 |
Nipoti, Londrillo & Ciotti 2003). One of these tests is described in Section \[seckin\].
We also verified that the code reproduces the Newtonian and MOND conservation laws (see Section \[secic\]): note that the conservation laws in MOND present some peculiarities with respect to the Newtonian case, so we give here a brief discussion of the subject. As already stressed by BM, equation (\[eqMOND\]) is obtained from a variational principle applied to a Lagrangian with all the required symmetries, so energy, linear and angular momentum are conserved. Unfortunately, as also shown by BM, the total energy diverges even for finite mass systems, thus posing a computational challenge to code validation. We solved this problem by checking the volume-limited energy balance equation $$\begin{aligned}
\label{eqbaltext}
{d \over d t}\int_{V_0}\left[k+\rho\phi+{{a_0}^2\over 8 \pi
G}\mathcal{F}\left({||\nabla\phi||\over
{a_0}}\right)\right]\dxcube=\nonumber\\{1\over 4\pi G}\int_{\partial
V_0}\mu{\partial \phi \over \partial t} <\nabla \phi,\hat{\bf n}> d a,\end{aligned}$$ which is derived in Appendix \[appetot\]. In equation (\[eqbaltext\]) $V_0$ is an arbitrary (but fixed) volume enclosing all the system mass, $k$ is the kinetic energy per unit volume, and $$\mathcal{F}(y)\equiv2\int_{y_0}^{y} \mu(\xi) \xi d \xi,
\label{eqeffe}$$ where $y_0$ is an arbitrary constant; note that only finite quantities are involved. Another important relation between global quantities for a system | 1 | member_30 |
at equilibrium (in MOND as in Newtonian gravity) is the virial theorem $$\label{eqvirtheo}
2 K + W=0,$$ where $K$ is the total kinetic energy and $W=\Tr W_{ij}$ is the trace of the Chandrasekhar potential energy tensor $$\label{eqwij}
W_{ij}\equiv-\int \rho({{\bf x}}) x_i {\partial \phi({{\bf x}}) \over \partial x_j} \dxcube$$ (e.g., Binney & Tremaine 1987). Note that in MOND $K+W$ is [*not*]{} the total energy, and is not conserved. However, [*$W$ is conserved in the limit of dMOND*]{}, being $W=-(2/3)\sqrt{G{a_0}M_*^3}$ for [*all*]{} systems of finite total mass ${{M_*}}$ (see Appendix \[appw\] for the proof). As a consequence, in dMOND the virial theorem writes simply $\sgv^4=4G{{M_*}}{a_0}/9$, where $\sgv \equiv \sqrt{2 K / {{M_*}}}$ is the system virial velocity dispersion (this relation was proved for dMOND spherical systems by Gerhard & Spergel 1992; see also Milgrom 1984). In our simulations we also tested that equation (\[eqvirtheo\]) is satisfied at equilibrium, and that $W$ is conserved in the dMOND case (see Sections \[secic\] and \[secres\]).
Numerical simulations {#secsim}
=====================
----------------------------------------------- ---------------------------------------------
${t_{\rm *n}}={r_*}^{3/2} (G {{M_*}})^{-1/2}$ ${t_{\rm *d}}={r_*}(G {{M_*}}{a_0})^{-1/4}$
${v_{\rm *n}}=(G {{M_*}})^{1/2}{r_*}^{-1/2}$ ${v_{\rm *d}}=(G {{M_*}}{a_0})^{1/4}$
${E_{\rm *n}}=G {{M_*}}^2{r_*}^{-1}$ ${E_{\rm *d}}=(G{a_0})^{1/2}{{M_*}}^{3/2}$
----------------------------------------------- ---------------------------------------------
: Time, velocity, and energy units for Newtonian and MOND (subscript n), and dMOND | 1 | member_30 |
(subscript d) N-body simulations.
The choice of appropriate scaling physical units is an important aspect of N-body simulations. This is especially true in the present case, in which we want to compare MOND and Newtonian simulations having the same initial conditions. As well known, due to the scale-free nature of Newtonian gravity, a Newtonian $N$-body simulation starting from a given initial condition describes in practice $\infty^2$ systems of arbitrary mass and size. Each of them is obtained by assigning specific values to the length and mass units, ${r_*}$ and ${{M_*}}$, in which the initial conditions are expressed. Also dMOND gravity is scale free, because ${a_0}$ appears only as a multiplicative factor in equation (\[eqdMOND\]), and so a simulation in dMOND gravity represents systems with arbitrary mass and size (though in principle the results apply only to systems with accelerations much smaller than ${a_0}$). MOND simulations can also be rescaled, but, due to the presence of the characteristic acceleration ${a_0}$ in the non-linear function $\mu$, each simulation describes only $\infty^1$ systems, because ${r_*}$ and ${{M_*}}$ cannot be chosen independently of each other.
On the basis of the above discussion, we fix the physical units as follows (see Appendix \[appscal\] for a | 1 | member_30 |
detailed description of the scaling procedure). Let the initial density distribution be characterized by a total mass ${{M_*}}$ and a characteristic radius ${r_*}$. We rescale the field equations so that the dimensionless source term is the same in Newtonian, MOND and dMOND simulations. We also require that the Second Law of Dynamics, when cast in dimensionless form, is independent of the specific force law considered, and this leads to fix the time unit. As a result, Newtonian and MOND simulations have the same time unit ${t_{\rm *n}}={r_*}^{3/2} (G {{M_*}})^{-1/2}$, while the natural time unit in dMOND simulations is ${t_{\rm *d}}={r_*}(G {{M_*}}{a_0})^{-1/4}$. Note that MOND simulations are characterized by the dimensionless parameter $\kappa=G {{M_*}}/{r_*}^2{a_0}$, and scaling of a specific simulation is allowed provided the value of $\kappa$ is maintained constant. So, simulations with lower $\kappa$ values describe lower surface-density, weaker acceleration systems; dMOND simulations represent the limit case $\kappa \ll 1$, while Newtonian ones describe the regime with $\kappa \gg 1$. With the time units fixed, the corresponding velocity and energy units are ${v_{\rm *n}}\equiv{r_*}/{t_{\rm *n}}$, ${v_{\rm *d}}\equiv{r_*}/{t_{\rm *d}}$, ${E_{\rm *n}}={{M_*}}{v_{\rm *n}}^2$, and ${E_{\rm *d}}={{M_*}}{v_{\rm *d}}^2$ (see Table 1 for a summary).
Initial conditions and analysis of the simulations {#secic}
--------------------------------------------------
| 1 | member_30 |
We performed a set of five dissipationless-collapse N-body simulations, starting from the same phase-space configuration: the initial particle distribution follows the Plummer (1911) spherically symmetric density distribution $$\label{eqplum}
\rho(r)={{3 {{M_*}}{r_*}^2 }\over 4 {\pi (r^2 +{r_*}^2)^{5/2}}},$$ where ${{M_*}}$ is the total mass and ${r_*}$ a characteristic radius. The choice of a Plummer sphere as initial condition is quite artificial, and not necessarily the most realistic to reproduce initial conditions in the cosmological context (e.g., Gunn & Gott 1972). We adopt such a distribution to adhere to other papers dealing with collisionless collapse (e.g., Londrillo et al. 1991; NLC06; see also Section \[secdis\], in which we present the results of a set of simulations starting from different initial conditions). The particles are at rest, so the initial virial ratio $2K/|W|=0$. What is different in each simulation is the adopted gravitational potential, which is Newtonian in simulation N, dMOND in simulation D, and MOND with acceleration ratio $\kappa$ in simulations M$\kappa$ ($\kappa$=1, 2, 4). For each simulation we define the dynamical time ${t_{\rm dyn}}$ as the time at which the virial ratio $2K/|W|$ reaches its maximum value. In particular, we find ${t_{\rm dyn}}\sim2{t_{\rm *d}}$ in simulation D, and ${t_{\rm dyn}}\sim2{t_{\rm *n}}$ in simulations | 1 | member_30 |
N, M1, M2 and M4. We note that ${t_{\rm dyn}}\sim{G {{M_*}}^{5/2}(2|K+W|)^{-3/2}}$ in simulation N.
Following NLC06, the particles are spatially distributed according to equation (\[eqplum\]) and then randomly shifted in position (up to ${r_*}/5$ in modulus). This artificial, small-scale “noise” is introduced to enhance the phase mixing at the beginning of the collapse, because the numerical noise is small, and the velocity dispersion is zero (see also Section 4.2). As such, these fluctuations are not intended to reproduce any physical clumpiness.
All the simulations (realized with $N=10^6$ particles, and a grid with $\Nr=64$, $\Nth=16$ and $\Nph=32$) are evolved up $t=150{t_{\rm dyn}}$. In all cases the modulus of the center of mass position oscillates around zero with r.m.s $\lsim 0.1{r_*}$; similarly, the modulus of the total angular momentum oscillates around zero[^2] with r.m.s. $\lsim0.02$, in units of ${r_*}{{M_*}}{v_{\rm *n}}$ (simulations M$\kappa$ and N) and of ${r_*}{{M_*}}{v_{\rm *d}}$ (simulation D). $K+W$ in the Newtonian simulation and $W$ in the dMOND simulation are conserved to within $2\%$ and $0.6\%$, respectively. The volume-limited energy balance equation (\[eqbaltext\]) is conserved with an accuracy of $1\%$ in MOND simulations, independently of the adopted $V_0$. To estimate possible numerical effects, we reran one of the MOND collapse | 1 | member_30 |
simulations (M1) using $N=2\times 10^6$, $\Nr=80$, $\Nth=24$, and $\Nph=48$: we found that the end-products of these two simulations do not differ significantly, as far as the properties relevant to the present work are concerned.
The intrinsic and projected properties of the collapse end-products are determined as in NLC06. In particular, the position of the center of the system is determined using the iterative technique described by Power et al. (2003). Following Nipoti et al. (2002), we measure the axis ratios $c/a$ and $b/a$ of the inertia ellipsoid (where $a$, $b$ and $c$ are the major, intermediate and minor axis) of the final density distributions, their angle-averaged profile and half-mass radius $\rhalf$. We fitted the final angle-averaged density profiles with the $\gamma$-model (Dehnen 1993; Tremaine et al. 1994) $$\label{eqgamma}
\rho (r)= {\rho_0 \rc^4 \over r^{\gamma} (\rc +r)^{4-\gamma}},$$ where the inner slope $\gamma$ and the break radius $\rc$ are free parameters, and the reference density $\rho_0$ is fixed by the total mass ${{M_*}}$. The fitting radial range is $0.06\,\lsim\,
r/\rhalf\,\lsim\, 10$. In order to estimate the importance of projection effects, for each end-product we consider three orthogonal projections along the principal axes of the inertia tensor, measuring the ellipticity $\epsilon=1-\sbe/\sae$, the circularized | 1 | member_30 |
projected density profile and the circularized effective radius [$\Re\equiv\sqrt{\sae\sbe}$]{} (where [$\sae$]{} and [$\sbe$]{} are the major and minor semi-axis of the effective isodensity ellipse). We fit (over the radial range $0.1 \, \lsim \, R/\Re \, \lsim \, 10$) the circularized projected density profiles of the end-products with the $R^{1/m}$ Sersic (1968) law: $$\label{eqser}
I(R)=\Ie \,
\exp\left\{-b(m)\left[ \left( \frac{R}{\Re} \right)^{1/m} -1 \right]\right\},$$ where $\Ie\equiv I(\Re)$ and $b(m)\simeq 2m-1/3+4/405m$ (Ciotti & Bertin 1999). In the fitting procedure $m$ is the only free parameter, because $\Re$ and $\Ie$ are determined by their measured values obtained by particle count. In addition, we measure the central velocity dispersion $\sgz$, obtained by averaging the projected velocity dispersion over the circularized surface density profile within an aperture of $\Re/8$. Some of these structural parameters are reported in Table 2 for the five simulations described above, as well as for three additional simulations, which start from different initial conditions (see Section \[secdis\]).
Results {#secres}
=======
In Newtonian gravity, collisionless systems reach virialization through violent relaxation in few dynamical times, as predicted by the theory (Lynden-Bell 1967) and confirmed by numerical simulations (e.g. van Albada 1982). On the other hand, due to the non linearity of the theory | 1 | member_30 |
and the lack of numerical simulations, the details of relaxation processes and virialization in MOND are much less known. Thus, before discussing the specific properties of the collapse end-products we present a general overview of the time evolution of the virial quantities in our simulations, postponing to Section \[secphsp\] a more detailed description of the phase-space evolution. In particular, in Fig. \[figtime\] we show the time evolution of $2 K/|W|$, $K$, $W$, and $K+W$ for simulations D, M1, and N. In the diagrams time is normalized to ${t_{\rm dyn}}$, so plots referring to different simulations are directly comparable (the values of ${t_{\rm dyn}}$ in time units for the five simulations are given in Section \[secic\]). In simulation N (right column) we find the well known behavior of Newtonian dissipationless collapses: $2 K/|W|$ has a peak, then oscillates, and eventually converges to the equilibrium value $2 K/|W|=1$; the total energy $K+W$ is nicely conserved during the collapse, though it presents a secular drift, a well known feature of time integration in N-body codes. The time evolution of the same quantities is significantly different in a dMOND simulation (left column). In particular, the virial ratio $2 K/|W|$ quickly becomes close to one, | 1 | member_30 |
but is still oscillating at very late times because of the oscillations of $K$, while $W$ is constant as expected. As we show in Section \[secphsp\], these oscillations are related to a peculiar behavior of the system in phase space. Finally, simulation M1 (central column) represents an intermediate case between models N and D: the system starts as dMOND, but soon its core becomes concentrated enough to enter the Newtonian regime. After the initial phases of the collapse, Newtonian gravity acts effectively in damping the oscillations of the virial ratio. Overall, it is apparent how the system is in a “mixed” state, neither Newtonian ($K+W$ is not conserved) nor dMOND ($W$ is not constant).
Properties of the collapse end-products
---------------------------------------
### Spatial and projected density profiles
We found that all the simulated systems, once virialized, are not spherically symmetric. However, while the dMOND collapse end-product is triaxial ($c/a\sim0.2$, $b/a\sim0.4$), MOND and Newtonian end-products are oblate ($c/a \sim c/b \sim 0.5$). The ellipticity $\epsilon$ of the projected density distributions (measured for each of the principal projections) is found in the range $0.5-0.8$ in D, and $0 - 0.5$ in M1, M2, M4 and N. These values are consistent with those observed | 1 | member_30 |
in real ellipticals, with the exception of $\epsilon_b$ in model D (see Table 2), which would correspond - if taken at face value - to an E8 galaxy. These result could be just due to the procedure adopted to measure the ellipticity (see Section \[secic\]), however we find interesting that dMOND gravity could be able to produce some system that would be unstable in Newtonian gravity. We remark that a similar result, in the different context of disk stability in MOND, has been obtained by Brada & Milgrom (1999).
In order to describe the radial mass distribution of the final virialized systems, we fitted their angle-averaged density profiles with the $\gamma$-model (\[eqgamma\]) over the radial range $0.06\,\lsim\, r/\rhalf\,\lsim\, 10$. The best-fit $\gamma$ and $\rc$ for the final distribution of each simulation are reported in Table 2 together with their $1\sigma$ uncertainties (calculated from $\Delta
\chi^2=2.30$ contours in the space $\gamma-\rc$). As also apparent from Fig. \[fig3d\] (bottom), the Newtonian collapse produced the system with the steepest inner profile ($\gamma\sim1.7$), the dMOND end-product has inner logarithmic slope close to zero, while MOND collapses led to intermediate cases, with $\gamma$ ranging from $\sim
1.2$ ($\kappa=1$) to $\sim1.5$ ($\kappa=4$). We also note that | 1 | member_30 |
the ratio $\rc/\rhalf$ (indicating the position of the knee in the density profile) increases systematically from dMOND to Newtonian simulations.
The circularized projected density profiles of the end-products are analyzed as described in Section \[secic\]. The best-fit Sersic indices $m_a$, $m_b$ and $m_c$ (for projections along the axes $a$, $b$, and $c$, respectively) are reported in Table 2, together with the $1\sigma$ uncertainties corresponding to $\Delta \chi^2=1$; the relative uncertainties on the best-fit Sersic indices are in all cases smaller than 5 per cent and the average residuals between the data and the fits are typically $0.05 \lsim\langle\Delta{SB}\rangle \lsim 0.2$, where $SB\equiv-2.5 \log [I(R)/\Ie]$. The fitting radial range $0.1\,\lsim\, R/\Re\,\lsim\, 10$ is comparable with or larger than the typical ranges spanned by observations (e.g., see Bertin, Ciotti & Del Principe 2002). In agreement with previous investigations, we found that the Newtonian collapse produced a system well fitted by the de Vaucouleurs (1948) law. MOND collapses led to systems with Sersic index $m<4$, down to $m\sim2$ in the case of the dMOND collapse. Figure \[fig2d\] (bottom) shows the circularized (major-axis) projected density profiles for the end-products of simulations D, M1 and N together with their best-fit Sersic laws ($m=2.87$, $m=3.20$, and | 1 | member_30 |
$m=4.21$, respectively), and the corresponding residuals. Curiously, NLC06 found that low-$m$ systems can be also obtained in Newtonian dissipationless collapses in the presence of a pre-existing dark-matter halo, with Sersic index value decreasing for increasing dark-to-luminous mass ratio.
### Kinematics {#seckin}
We quantify the internal kinematics of the collapse end-products by measuring the angle-averaged radial and tangential components ($\sigma_r$ and $\sigma_{\rm t}$) of their velocity-dispersion tensor, and the anisotropy parameter $\beta(r) \equiv 1 -0.5\sigma^2_{\rm
t}/\sigma^2_r$. These quantities are shown in Fig. \[fig3d\] for simulations D, M1, and N. We note that the $\sigma_r$ profile decreases more steeply in the Newtonian than in the MOND end-products, while it presents a hole in the inner regions of the dMOND system. In addition, the dMOND galaxy is radially anisotropic ($\beta\sim0.4$) even in the central regions, where models N and M1 are approximately isotropic ($\beta \sim 0.1$). All systems are strongly radially anisotropic for $r\gsim\rhalf$. For each model projection we computed the line-of-sight velocity dispersion $\sglos$, considering particles in a strip of width $\Re/4$ centered on the semi-major axis of the isophotal ellipse. The line-of-sight velocity-dispersion profiles (for the major-axis projection) are plotted in the top panels of Fig. \[fig2d\]. The Newtonian profile is | 1 | member_30 |
very steep within $\Re$, while MOND and dMOND profiles are significantly flatter there. As well as $\sigma_r$, $\sglos$ decreases for decreasing radius in the inner region of model D. The kinematical properties of M2 and M4 are intermediate between those of M1 and of N: overall we find only weak dynamical non-homology among MOND end-products. The empty symbols in Fig. \[fig3d\] (right column) refer to a test Newtonian simulation run with the FVFPS treecode (with $4\times10^5$ particles). The structural and kinematical properties of the end-product of this simulation are clearly in good agreement with those of the end-product of simulation N (solid lines), which started from the same initial conditions.
Phase-space properties of MOND collapses {#secphsp}
----------------------------------------
To explore the phase-space evolution of the systems during the collapse and the following relaxation we consider time snapshots of the particles radial velocity ($\vr$) vs. radius as in Londrillo et al. (1991). In Fig. \[figphsp\] we plot five of these diagrams for simulations D, M1 and N: each plot shows the phase-space coordinates of 32000 particles randomly extracted from the corresponding simulation, and, as in Fig. \[figtime\], times are normalized to the dynamical time ${t_{\rm dyn}}$ (see Section \[secic\]). At time $t=0.5{t_{\rm | 1 | member_30 |
dyn}}$ all particles are still collapsing in simulation N, while in MOND simulations a minority of particles have already crossed the center of mass, as revealed by the vertical distribution of points at $r\sim0$ in panels D and M1. At $t={t_{\rm dyn}}$ (time of the peak of $2
K/|W|$ in the three models), sharp shells in phase space are present, indicating that particles are moving in and out collectively and phase mixing has not taken place yet. At $t=4{t_{\rm dyn}}$ is already apparent that phase mixing is operating more efficiently in simulation N than in simulation M1, while there is very little phase mixing in the dMOND collapse. At significantly late times ($t=44{t_{\rm dyn}}$), when the three systems are almost virialized ($2 K/|W|\sim 1$; see Fig. \[figtime\]), phase mixing is complete in simulation N, but phase-space shells still survive in models M1 and D. Finally, the bottom panels show the phase-space diagrams at equilibrium ($t=150{t_{\rm dyn}}$), when phase mixing is completed also in the MOND and dMOND galaxies: note that the populated region in the ($r$,$\vr$) space is significantly different in MOND and in Newtonian gravity, consistently with the sharper decline of radial velocity dispersion in the Newtonian system.
Thus, | 1 | member_30 |
our results indicate that phase mixing is more effective in Newtonian gravity than in MOND[^3]. It is then interesting to estimate in physical units the phase-mixing timescales of MOND systems. From Table 1 it follows that ${t_{\rm *n}}\simeq4.7\,({r_*}/\kpc)^{3/2}({{M_*}}/10^{10}
\Msun)^{-1/2}\Myr=29.8\kappa^{-3/4}({{M_*}}/10^{10} \Msun)^{1/4}\Myr$ for ${a_0}=1.2\times 10^{-10} {\rm m}\,{\rm s}^{-2}$. For example, in the case of model M1, adopting ${{M_*}}=10^{12} \Msun$ (and ${r_*}=\sqrt{G{{M_*}}/{a_0}}\simeq34\kpc$), shells in phase space are still apparent after $\sim8.3\Gyr$ ($\simeq44{t_{\rm dyn}}$). Simulation M1 might also be interpreted as representing a dwarf elliptical galaxy of, say, ${{M_*}}= 10^9 \Msun$ (and ${r_*}=\sqrt{G{{M_*}}/{a_0}}\simeq
1.1\kpc$). In this case $44{t_{\rm dyn}}\sim 1.5 \Gyr$. We conclude that in some MOND systems substructures in phase space can survive for significantly long times.
In addition to the ($r$,$\vr$) diagram, another useful diagnostic to investigate phase-space properties of gravitational systems is the energy distribution $N(E)$ (i.e. the number of particles with energy per unit mass between $E$ and $E+dE$; e.g., Binney & Tremaine 1987; Trenti & Bertin 2005). Independently of the force law, the energy per unit mass of a particle orbiting at ${{\bf x}}$ with speed $v$ in a gravitational potential $\phi({{\bf x}})$ is $E=v^2/2+\phi({{\bf x}})$, and $E$ is constant if $\phi$ is time-independent. In Newtonian gravity $\phi$ is | 1 | member_30 |
usually set to zero at infinity for finite-mass systems, so $E<0$ for bound particles; in MOND all particles are bound, independently of their velocity, because $\phi$ is confining, and all energies are admissible. This difference is reflected in Fig. \[fignde\], which plots the initial (top) and final (bottom) differential energy distributions for simulations D, M1, and N. Given that the particles are at rest at $t=0$, the initial $N(E)$ depends only on the structure of the gravitational potential, and is significantly different in the Newtonian and MOND cases. We also note that $N(E)$ is basically the same in models D and M1 at $t=0$, because model M1 is initially in dMOND regime. In accordance with previous studies, in the Newtonian case the final differential $N(E)$ is well represented by an exponential function over most of the populated energy range (Binney 1982; van Albada 1982; Ciotti 1991; Londrillo et al. 1991; NLC06). In contrast, in model D the final $N(E)$ decreases for increasing energy, qualitatively preserving its initial shape. In the case of simulation M1 it is apparent a dichotomy between a Newtonian part at lower energies (more bound particles), where $N(E)$ is exponential, and a dMOND part at higher energies, | 1 | member_30 |
where the final $N(E)$ resembles the initial one. We interpret this result as another manifestation of a less effective phase-space reorganization in MOND than in Newtonian collapses.
Comparison with the observed scaling relations of elliptical galaxies {#secsca}
---------------------------------------------------------------------
It is not surprising that galaxy scaling relations represent an even stronger test for MOND than for Newtonian gravity, due to the absence of dark matter and the existence of the critical acceleration ${a_0}$ with a universal value in the former theory (e.g., see Milgrom 1984; Sanders 2000). For example, when interpreting the FP tilt in Newtonian gravity one can invoke a systematic and fine-tuned increase of the galaxy dark-to-luminous mass ratio with luminosity (e.g., Bender, Burstein & Faber 1992; Renzini & Ciotti 1993; Ciotti, Lanzoni & Renzini 1996), while in MOND the tilt should be related to the characteristic acceleration ${a_0}$. Note, however, that in MOND as well as in Newtonian gravity other important physical properties may help to explain the FP tilt, such as a systematic increase of radial orbital anisotropy with mass or a systematic structural weak homology (Bertin et al. 2002). Due to the relevance of the subject, we attempt here to derive some preliminary hints. In particular, | 1 | member_30 |
for the first time, we can compare with the scaling relations of elliptical galaxies MOND models produced by a formation mechanism, yet as simple as the dissipationless collapse.
In this Section we consider the end-products of simulations M1, M2, and M4. As already discussed in Section \[secsim\], each of the three systems corresponds to a family with constant ${{M_*}}/{r_*}^2$. This degeneracy is represented by the straight dotted lines in Fig. \[figfp\]a: all galaxies on the same dotted line have the same $\kappa$ value. This behavior is very different from the Newtonian case, in which the result of a N-body simulation can be placed anywhere in the space $\Re-{{M_*}}$, by arbitrarily choosing ${{M_*}}$ and ${r_*}$. For comparison with observations, the specific scaling laws represented in Fig. \[figfp\] (thick solid lines) are the near-infrared $z^*$-band Kormendy relation $\Re \propto
{{M_*}}^{0.63}$ and FJ relation ${{M_*}}\propto\sigma_0^{3.92}$ (Bernardi et al. 2003a), and the edge-on FP relation in the same band $\log \Re = A \log \sigma_0 + B \log ({{M_*}}/ \Re^2) +const$ (with $A=1.49$, $B=- 0.75$; Bernardi et al. 2003b), under the assumption of luminosity-independent mass-to-light ratio.
The physical properties of each model are determined as follows. First, for each model (identified by a value | 1 | member_30 |
---
abstract: |
We formulate the Hubbard model for the simple cubic lattice in the representation of interacting dimers applying the exact solution of the dimer problem. By eliminating from the considerations unoccupied dimer energy levels in the large $U$ limit (it is the only assumption) we analytically derive the Hubbard Hamiltonian for the dimer (analogous to the well-known $t-J$ model), as well as, the Hubbard Hamiltonian for the crystal as a whole by means of the projection technique. Using this approach we can better visualize the complexity of the model, so deeply hidden in its original form. The resulting Hamiltonian is a mixture of many multiple ferromagnetic, antiferromagnetic and more exotic interactions competing one with another. The interplay between different competitive interactions has a decisive influence on the resulting thermodynamic properties of the model, depending on temperature, model parameters and assumed average number of electrons per lattice site. A simplified form of the derived Hamiltonian can be obtained using additionally Taylor expansion with respect to $x=\frac{t}{U}$ ($t$-hopping integral between nearest neighbours, $U$-Coulomb repulsion). As an example, we present the expansion including all terms proportional to $t$ and to $\frac{t^2}U$ and we reproduce the exact form of the Hubbard Hamiltonian | 1 | member_31 |
in the limit $U\rightarrow \infty $.
The nonperturbative approach, presented in this paper, can, in principle, be applied to clusters of any size, as well as, to another types of model Hamiltonians.
author:
- |
M.Matlak[^1], J.Aksamit, B.Grabiec\
Institute of Physics, Silesian University,\
4 Uniwersytecka, PL-40-007 Katowice,Poland\
and\
W.Nolting\
Institute of Physics, Humboldt University,\
110 Invalidenstr., D-10115 Berlin, Germany
title: 'Hubbard Hamiltonian in the dimer representation. Large $U$ limit'
---
Introduction
============
The single-band Hubbard model, Ref. \[1\], plays in the solid state physics a similar principal role as the hydrogen atom in the atomic physics. This explains a continuous interest in its properties. The Hubbard Hamiltonian reads $$H=\sum\limits_{i,j,\sigma }t_{i,j}c_{i,\sigma }^{+}c_{j,\sigma
}+U\sum\limits_in_{i,\uparrow }n_{i,\downarrow }.$$ Here $c_{i,\sigma }$ $(c_{i,\sigma }^{+})$ are annihilation (creation) operators of an electron with spin $\sigma =\uparrow $, $\downarrow $ in the Wannier representation at the lattice site $\mathbf{R}_i$ and $n_{i,\sigma
}=c_{i,\sigma }^{+}c_{i,\sigma }$. Moreover, $t_{i,j}$ is the hopping integral between different lattice sites $i$ and $j$ ($t_{i,i}=0)$ and $U$ is the intrasite Coulomb repulsion. The Bloch conduction band energy $%
\varepsilon _{\mathbf{k}}$ is given by
$$\varepsilon _{\mathbf{k}}=\sum\limits_{i-j}t_{i,j}e^{-i\mathbf{k\cdot (R}_i-%
\mathbf{R}_j)}.$$
In the following we restrict ourselves to the simple cubic (sc) lattice and assume that
$$t_{i,j}=\left\{
\begin{array}{ll}
| 1 | member_31 |
-t & i,j\mbox{-nearest neighbours} \\
0 & \mbox{otherwise}.
\end{array}
\right.$$ Then
$$\varepsilon _{\mathbf{k}}=-2t(\cos k_xa+\cos k_ya+\cos k_za).$$
The hopping parameter $t$ is simply related to the bandwidth $W$ of the Bloch band (2), e. g. $W=12t$ for the sc lattice$.$ The interplay between the two model parameters, $W$ and $U,$ is decisive for the properties of the model resulting in strong electron correlations, leading to band magnetism (see e.g. Refs \[2\], \[3\] for a review), insulator-to-metal transition (see e.g. Refs \[3\], \[4\] and papers cited therein) and high-$T_c$ superconductivity (negative $U$-model, see e.g. Ref. \[5\]). The Hubbard model very often plays also a role of a submodel for many other more complicated models (as e.g. Anderson model (see Ref. \[6\]), s-f model (see Refs \[7,8\]) and so on). Especially interesting, but difficult to handle are the properties of the Hubbard model in such a regime of the model parameters where the bandwidth $W$ is comparable to the Coulomb repulsion $U$. In a large number of papers \[9-30\] the authors tried to solve this model using many sophisticated methods. The exact solution, however, does not exist till now. Many authors tried to change the situation in this field by introducing the expansion | 1 | member_31 |
parameter $x=\frac{t}{U}$ $(x\ll 1)$. This idea (cf Refs \[31,32\]) consists in replacing ”difficult physics” connected with the model by ”difficult mathematics” obtained by a laborious expansion with respect to $x.$ Different methods connected with this problem have been applied as e.g. the perturbation expansion (see Refs \[18\], \[21\]), canonical transformation (see Refs \[9\], \[12\], \[17\], \[22\], \[23\]) or ab initio derivations (see Refs \[25\], \[28-30\]). Most of the methods, leading to the $%
t-J$ model (or generalized $t-J$ model) are also summarized in Refs \[27\], \[4\].
The goal of the present paper is just to show that we can take another, nonperturbative way. In the first step we divide the crystal lattice into a set of interacting dimers. In other words, we can rewrite the Hubbard Hamiltonian (1) for the sc lattice (see Fig. 1) in the equivalent form [^2]:
$$\begin{array}{ll}
H= & \sum\limits_IH_I^d-t\sum\limits_{I,\sigma }(c_{I,2,\sigma
}^{+}c_{I+1,1,\sigma }+c_{I+1,1,\sigma }^{+}c_{I,2,\sigma }) \\
& -t\sum\limits_{I\neq J,\sigma }(c_{I,1,\sigma }^{+}c_{J,1,\sigma
}+c_{I,2,\sigma }^{+}c_{J,2,\sigma })
\end{array}$$
where
$$H_I^d=-t\sum\limits_{\sigma} (c_{I,1,\sigma }^{+}c_{I,2,\sigma
}+c_{I,2,\sigma }^{+}c_{I,1,\sigma })+U(n_{I,1,\uparrow }n_{I,1,\downarrow
}+n_{I,2,\uparrow }n_{I,2,\downarrow }).$$
The indices $I$ and $J$ enumerate the dimers and $H_I^d$ is the dimer Hamiltonian. The second term in (4) describes the hopping between nearest dimers in the $z$-direction, the third | 1 | member_31 |
one represents the hopping between nearest dimers (($y,z$)-plane) and between different dimer planes (see Fig. 1).
The equivalent form of the Hubbard Hamiltonian (4) is especially suitable because we can apply in the following the exact solution of the dimer problem (5). In the next step we express the construction operators $%
a_{\sigma}$ ($a_{\sigma}^{+}$) as linear combinations of the transition operators between different dimer states. This, in turn, allows to find the exact dimer representation of the whole Hamiltonian (4). The space of the dimer eigenvectors consists, however, of two subspaces. One of them corresponds to the lowest lying energy levels, the second one contains the levels with energies which in the large $U$ limit take on large, positive values. These levels in a reasonable temperature range cannot be occupied by electrons and therefore we can exclude them from further considerations. It is interesting to note that this approch, without any additional assumptions, applied to the dimer Hamiltonian (5) produces the analogy of the well known $t-J$ model (cf Refs \[31\], \[32\]). A similar approach can be applied to the Hamiltonian (4), describing the crystal as a whole. By eliminating from the considerations the unoccupied dimer energy levels in the | 1 | member_31 |
large $U$ limit (it is the only assumption) with the use of the projection technique we can find in a straightforward way the final form of the Hamiltonian in this limit without using perturbation expansion or canonical transformation. The resulting Hamiltonian, obtained in this way, is very complicated. It, however, explicitly shows all possible magnetic, nonmagnetic or more complex competitive interaction processes very deeply hidden in the original Hamiltonian written in the site representation (1). It is the aim of this paper just to reveal these important but normally invisible elementary interactions. One important advantage might be that a given approach to the unsolvable Hubbard problem can be tested with respect to the types of neglected interaction processes. Besides, the new and straightforward method, presented in this paper, can easily be adopted to clusters of any size and also to another types of model Hamiltonians.
The paper is organized as follows. In Sec. 2 we find the exact solution of the Hubbard dimer (5) and give the exact expressions for the annihilation operators $c_{I,1(2),\sigma }$ in the dimer representation. In Sec. 3 we derive the Hubbard dimer Hamilonian (5) in the large $U$ limit. With the use of the projection | 1 | member_31 |
technique onto the lowest lying dimer states we derive in Sec. 4 the Hubbard Hamiltonian for the crystal in this limit (central formula of this paper). A simplified version of the derived Hamiltonian with the use of the Taylor expansion with respect to $x=\frac{t}{U}$ $(x\ll 1)$ and the case $U\rightarrow \infty $ is discussed in Sec. 5.
Exact solution of the Hubbard dimer
===================================
The eigenvalues and eigenvectors of the dimer Hamiltonian (5) can be found by using a standard procedure (cf Refs \[33-35\]). We start with the vectors $%
|n_{1,\uparrow },n_{1,\downarrow };n_{2,\uparrow },n_{2,\downarrow }\rangle $ $(n_{i,\sigma }=0,1;$ $i=1,2$; $\sigma =\uparrow ,\downarrow )$ which constitute the Fock basis of the single-dimer space of states:
$$\begin{array}{lll}
\begin{array}{l}
\begin{array}{l}
|0\rangle =|0,0;0,0\rangle ,
\end{array}
\\
\\
\begin{array}{l}
|11\rangle =|1,0;0,0\rangle , \\
|12\rangle =|0,1;0,0\rangle , \\
|13\rangle =|0,0;1,0\rangle , \\
|14\rangle =|0,0;0,1\rangle ,
\end{array}
\end{array}
&
\begin{array}{l}
|21\rangle =|1,1;0,0\rangle , \\
|22\rangle =|1,0;1,0\rangle , \\
|23\rangle =|1,0;0,1\rangle , \\
|24\rangle =|0,1;1,0\rangle , \\
|25\rangle =|0,1;0,1\rangle , \\
|26\rangle =|0,0;1,1\rangle ,
\end{array}
&
\begin{array}{l}
\begin{array}{l}
|31\rangle =|0,1;1,1\rangle , \\
|32\rangle =|1,0;1,1\rangle , \\
|33\rangle =|1,1;0,1\rangle , \\
|34\rangle =|1,1;1,0\rangle ,
\end{array}
\\
\\
\begin{array}{l}
|4\rangle =|1,1;1,1\rangle .
\end{array}
\end{array}
\end{array}$$
Starting with the vectors (6) | 1 | member_31 |
we easily get the eigenvalues $E_\alpha $ and the eigenvectors $|E_{\alpha} \rangle $ of the Hubbard dimer (5). Here, we only mention that the space of 16 eigenvectors $|E_{\alpha} \rangle $ can be devided into some subspaces numbered by $n=\sum\limits_{i,\sigma
}n_{i,\sigma }$. The subspace, belonging to $n=0$ and $n=4$ is 1-dimensional ($E_0=0,$ $|E_0\rangle =|0\rangle ;$ $E_4=2U,$ $|E_4\rangle =|4\rangle )$. There are, however, two 2-dimensional subspaces corresponding to $n=1$ (as e.g. $E_{11}=-t,$ $|E_{11}\rangle =\frac 1{\sqrt{2}}(|11\rangle +|13\rangle
),$ etc.) and two 2-dimensional subspaces corresponding to $n=3$ (as e.g. $%
E_{31}=t+U,$ $|E_{31}\rangle =\frac 1{\sqrt{2}}(|31\rangle +|33\rangle ),$ etc.). The subspace, belonging to $n=2$ consists of one 4-dimensional subspace (as e.g. $E_{21}=0,$ $|E_{21}\rangle =\frac 1{\sqrt{2}}(|23\rangle
+|24\rangle ),$ etc.) and two 1-dimensional subspaces ($E_{25}=E_{26}=0,$ $%
|E_{25}\rangle =|22\rangle ,$ $|E_{26}\rangle =|25\rangle $). The complete set of the eigenvalues $E_{\alpha} $ and eigenvectors $|E_{\alpha} \rangle $ of the Hubbard dimer is given in the Appendix A. It allows to express the dimer Hamiltonian (5) in the equivalent form
$$H^d=\sum\limits_{\alpha} E_{\alpha} |E_{\alpha} \rangle \langle E_{\alpha} |.$$
The next important step in our calculations is the possibility to express the annihilation operators $c_{1(2),\sigma }$ as linear combinations of transition operators between dimer states $P_{\alpha ,\beta }=|E_{\alpha}
\rangle \langle E_{\beta} |.$ | 1 | member_31 |
This procedure can easily be performed when acting with the annihilation operators on the basis vectors (6) and using the reciprocal relations to (A.1). In this way we can obtain the dimer representation of the annihilation (creation) operators, given in Appedix B. After this operation we can insert so prepared $c_{I,1(2),\sigma }$ $%
(c_{I,1(2),\sigma }^{+})$ into the Hubbard Hamiltonian (4) to obtain the Hubbard model in the dimer representation for the sc lattice. This representation will be used later to derive the Hubbard Hamiltonian in the large $U$ limit (see Sec. 4).
Hubbard dimer for large $U$
===========================
Looking at the eigenvalues (A.1) of the Hubbard dimer it is easy to see that for large $U$ $(U\gg t)$ the energies $E_{\alpha \text{ }%
}=E_{22},E_{23},E_{31},E_{32},E_{33},E_{34}$ and $E_4$ take on large, positive values, producing in the partition function the terms which can practically be neglected. It means that the mentioned energies are not occupied in the reasonable temperature range ($1$ $eV\sim 11604.5$ $K$) and can be excluded from our considerations. Therefore the dimer Hamiltonian, given by (7), reduces to
$$\overline{H}^d=-tP_{11,11}+tP_{12,12}-tP_{13,13}+tP_{14,14}+\left( -C+\frac
U2\right) P_{24,24}.$$
To bring this expression into a compact (second quanization) form we introduce the Hubbard operators
$$\begin{aligned}
a_{i,\sigma } &=&c_{i,\sigma | 1 | member_31 |
}(1-n_{i,-\sigma }), \\
b_{i,\sigma } &=&c_{i,\sigma }n_{i,-\sigma }\end{aligned}$$
and spin operators
$$\begin{aligned}
S_i^z &=&\frac 12(n_{i,\uparrow }-n_{i,\downarrow })=\frac 12(n_{i,\uparrow
}^a-n_{i,\downarrow }^a), \\
S_i^{+} &=&c_{i,\uparrow }^{+}c_{i,\downarrow }=a_{i,\uparrow
}^{+}a_{i,\downarrow }, \\
S_i^{-} &=&c_{i,\downarrow }^{+}c_{i,\uparrow }=a_{i,\downarrow
}^{+}a_{i,\uparrow }\end{aligned}$$
where $n_{i,\sigma }^a=a_{i,\sigma }^{+}a_{i,\sigma }$ $(i=1,2;$ $\sigma
=\uparrow $, $\downarrow ).$
With the use of (9)-(13) the Hamiltonian of the Hubbard dimer (8) for large $%
U$ can be presented in the form (we introduce the omitted earlier dimer index $I$)
$$\begin{aligned}
\overline{H}_I^d &=&-t\sum\limits_{\sigma} [a_{I,1,\sigma }^{+}a_{I,2,\sigma
}+a_{I,2,\sigma }^{+}a_{I,1,\sigma }] \nonumber \\
&& \nonumber \\
&&+\frac{4t^2}{U\sqrt{1+(\frac{4t}U)^2}}[\overrightarrow{S}_{I,1}\cdot
\overrightarrow{S}_{I,2}-\frac{n_{I,1}^an_{I,2}^a}4] \nonumber \\
&& \nonumber \\
&&+\frac{t^2(1-\sqrt{1+(\frac{4t}U)^2})}{U(1+\sqrt{1+(\frac{4t}U)^2})\sqrt{%
1+(\frac{4t}U)^2}}[2(b_{I,1,\uparrow }^{+}a_{I,1,\downarrow
}^{+}a_{I,2,\downarrow }b_{I,2,\uparrow } \nonumber \\
&& \nonumber \\
&&+b_{I,2,\uparrow }^{+}a_{I,2,\downarrow }^{+}a_{I,1,\downarrow
}b_{I,1,\uparrow }) \\
&& \nonumber \\
&&+(1-n_{I,1}^a)n_{I,2}^b+(1-n_{I,2}^a)n_{I,1}^b-n_{I,1}^bn_{I,2}^b]
\nonumber \\
&& \nonumber \\
&&+\frac{t(1-\sqrt{1+(\frac{4t}U})^2)}{2\sqrt{1+(\frac{4t}U)^2}}%
\sum\limits_{\sigma} \sum\limits_{\alpha =1,2}[a_{I,\alpha ,\sigma }^{+}b_{I,%
\overline{\alpha },\sigma }+b_{I,\alpha ,\sigma }^{+}a_{I,\overline{\alpha }%
,\sigma }] \nonumber\end{aligned}$$
where $n_{I,\alpha }^{a,b}=\sum\limits_{\sigma} n_{I,\alpha ,\sigma }^{a,b},$ $n_{I,\alpha ,\sigma }^b=b_{I,\alpha ,\sigma }^{+}b_{I,\alpha ,\sigma
}=n_{I,\alpha ,\sigma }n_{I,\alpha ,-\sigma }$ $(\alpha =1,2)$, $\overline{%
\alpha }=1$ when $\alpha =2$ and $\overline{\alpha }=2$ when $\alpha =1$. It is very important to stress that the formula (14) has been obtained in a nonperturbative way, starting from the exact form of the dimer Hamiltonian (see (5) or (7)), and excluding from | 1 | member_31 |
the considerations unoccupied dimer energy levels in the large $U$ limit. Let us note that using the Taylor expansion in (14) with respect to $x=\frac{t}{U}$ and retaining the terms proportional to $\frac{t^2}U$ we obtain
$$\overline{H}_I^d=-t\sum\limits_\sigma [a_{I,1,\sigma }^{+}a_{I,2,\sigma
}+a_{I,2,\sigma }^{+}a_{I,1,\sigma }]+\frac{4t^2}U[\overrightarrow{S}%
_{I,1}\cdot \overrightarrow{S}_{I,2}-\frac{n_{I,1}^an_{I,2}^a}4].$$
The formula (15) is the well-known $t-J$ model for the Hubbard dimer where the first part in (15), similarly to (14), represents the exact form of the dimer Hamiltonian (5) in the limit $U\rightarrow \infty $.
The same result (14) can also be obtained applying a more general approach which will be used later to derive the Hubbard Hamiltonian for large $U$ in the case of a crystal. Let us note that after elimination of the unoccupied levels the subspace of the eigenvectors of the Hubbard dimer (A.1) for large $U$ consists of the following eigenvectors: $|E_0\rangle $, $|E_{11}\rangle $, $|E_{12}\rangle $, $|E_{13}\rangle $, $|E_{14}\rangle $, $|E_{21}\rangle $, $|E_{24}\rangle $, $|E_{25}\rangle $, and $|E_{26}\rangle $. It means that we can define a projection operator onto this subspace
$$\begin{array}{ll}
P_I= &
P_{0,0}^{(I)}+P_{11,11}^{(I)}+P_{12,12}^{(I)}+P_{13,13}^{(I)}+P_{14,14}^{(I)}
\\
& +P_{21,21}^{(I)}+P_{24,24}^{(I)}+P_{25,25}^{(I)}+P_{26,26}^{(I)}
\end{array}$$
which, in the second quantization form, reads
$$\begin{array}{ll}
P_I= & 1-\frac 12(n_{I,1}^b+n_{I,2}^b)+\frac 14n_{I,1}^bn_{I,2}^b \\
& \\
&
\begin{array}{ll}
& +\frac 18(1-\frac 1{\sqrt{1+(\frac{4t}U)^2}})[4(\overrightarrow{S}%
_{I,1}\cdot | 1 | member_31 |
\overrightarrow{S}_{I,2}-\frac 14n_{I,1}^an_{I,2}^a) \\
& \\
& +2(b_{I,1,\uparrow }^{+}a_{I,1,\downarrow }^{+}a_{I,2,\downarrow
}b_{I,2,\uparrow }+b_{I,2,\uparrow }^{+}a_{I,2,\downarrow
}^{+}a_{I,1,\downarrow }b_{I,1,\uparrow }) \\
& \\
& +(1-n_{I,1}^a)n_{I,2}^b+(1-n_{I,2}^a)n_{I,1}^b-n_{I,1}^bn_{I,2}^b]
\end{array}
\\
& \\
& +\frac t{U\sqrt{1+(\frac{4t}U)^2}}\sum\limits_{\sigma} \sum\limits_{\alpha
=1,2}(a_{I,\alpha ,\sigma }^{+}b_{I,\overline{\alpha },\sigma }+b_{I,\alpha
,\sigma }^{+}a_{I,\overline{\alpha },\sigma }).
\end{array}$$
Now, the Hamiltonian (14) can also be obtained from (5) (or (7)) with the use of the projection operator (16) or (17) using the relation
$$\overline{H}_I^d=P_IH_I^dP_I$$
and applying a straightforward but laborious algebraic calculation.
Hubbard model for large $U$
===========================
The Hamiltonian of the whole crystal (4) can be expressed (similar to (7)) in the form $$H=\sum\limits_{\gamma} \overline{E}_{\gamma} |\overline{E}_{\gamma} \rangle
\langle \overline{E}_{\gamma} |$$ with unknown energies $\overline{E}_{\gamma} $ and eigenvectors $|\overline{E%
}_{\gamma} \rangle $. We can, however, expand the eigenvectors $|\overline{E}%
_{\gamma} \rangle $ in the series of the dimer eigenvectors (see (A.1))
$$|\overline{E}_{\gamma} \rangle =\sum\limits_{\gamma _1,..,\gamma
_M}c_{\gamma _{1,}.._{,}\gamma _M}^{\gamma} |E_{\gamma _1}\rangle
..|E_{\gamma _M}\rangle$$
assuming that the crystal consists of $M$ dimers. Using (19) and (20) we obtain
$$H=\sum\limits_{\gamma} \overline{E}_{\gamma} \sum\limits_{\gamma
_1,..,\gamma _M}\sum\limits_{\gamma _1^{^{,}},..,\gamma _M^{,}}c_{\gamma
_{1,}.._{,}\gamma _M}^{\gamma} c_{\gamma _{1,}^{,}.._{,}\gamma
_M^{,}}^{\gamma *}|E_{\gamma _1}\rangle ..|E_{\gamma _M}\rangle \langle
E_{\gamma _1^{,}}|..\langle E_{\gamma _M^{,}}|.$$
It is clear that to obtain the Hubbard Hamiltonian for large $U$ we have to project (21) onto the subspace of the lowest | 1 | member_31 |
lying dimer states with the use of the projection operator
$$P=P_1P_2...P_M$$
where $P_{I}$ is given by (16) or (17). In analogy to (14) we denote the Hubbard Hamiltonian in the large $U$ limit by $\overline{H}$. Similar to (18) we write
$$\overline{H}=PHP$$
and instead of the form (21) for the Hamiltonian $H$ we can use (4). Taking into account that $P^2=P$ $(P_I^2=P_I,$ $\left[ P_I,P_J\right] =0)$ we obtain
$$\begin{array}{ll}
\overline{H}= & P[\sum\limits_I\overline{H}_I^d-t\sum\limits_{I,\sigma
}\left( \overline{c}_{I,2,\sigma }^{+}\overline{c}_{I+1,1,\sigma }+\overline{%
c}_{I+1,1,\sigma }^{+}\overline{c}_{I,2,\sigma }\right) \\
& -t\sum\limits_{I\neq J,\sigma }\left( \overline{c}_{I,1,\sigma }^{+}%
\overline{c}_{J,1\sigma }+\overline{c}_{I,2,\sigma }^{+}\overline{c}%
_{J,2,\sigma }\right) ]\equiv P\overline{\overline{H}}
\end{array}$$
where $\overline{H}_I^d$ is given by (14) and $(\alpha =1,2)$
$$\overline{c}_{I,\alpha ,\sigma }=c_{I,\alpha ,\sigma }P_I,$$
$$\overline{c}_{I,\alpha ,\sigma }^{+}=P_Ic_{I,\alpha ,\sigma }^{+}.$$
Applying the projection operator (16) or (17) to (B.1) - (B.4) and introducing Hubbard- and spin operators (9) - (13) we obtain
$$\overline{c}_{I,1,\uparrow }=\underline{\underline{a}}_{I,1,\uparrow }+\beta
\left[ S_{I,2}^za_{I,1,\uparrow }+S_{I,2}^{-}a_{I,1,\downarrow }\right]
-\delta \left[ S_{I,1}^za_{I,2,\uparrow }+S_{I,1}^{-}a_{I,2,\downarrow
}\right] ,$$
$$\overline{c}_{I,1,\downarrow }=\underline{\underline{a}}_{I,1,\downarrow
}-\beta \left[ S_{I,2}^za_{I,1,\downarrow }-S_{I,2}^{+}a_{I,1,\uparrow
}\right] +\delta \left[ S_{I,1}^za_{I,2,\downarrow
}-S_{I,1}^{+}a_{I,2,\uparrow }\right]$$
where the corresponding expressions for $\overline{c}_{I,2,\sigma }$ ($%
\sigma =\uparrow ,\downarrow $) can easily be obtained by changing the internal dimer index $1\Leftrightarrow 2$ in (27) and (28).
The new operators $\underline{\underline{a}}_{I,\alpha ,\sigma }\left(
\alpha =1,2;\sigma =\uparrow ,\downarrow \right) $ are introduced | 1 | member_31 |
to obtain a relatively compact form of (27) and (28). They are defined as follows
$$\begin{array}{ll}
\underline{\underline{a}}_{I,\alpha ,\sigma }= & \underline{a}_{I,\alpha
,\sigma }+\beta \left( \underline{b}_{I,\alpha ,\sigma }+a_{I,\alpha
,-\sigma }^{+}a_{I,\overline{\alpha },-\sigma }b_{I,\overline{\alpha }%
,\sigma }\right) \\
& +\delta \left[ \underline{b}_{I,\overline{\alpha },\sigma }+a_{I,\overline{%
\alpha },-\sigma }^{+}a_{I,\alpha ,-\sigma }b_{I,\alpha ,\sigma }+a_{I,%
\overline{\alpha },\sigma }\frac{n_{I,\alpha }^a}2\right]
\end{array}$$
where
$$\underline{a}_{I,\alpha ,\sigma }=a_{I,\alpha ,\sigma }\left( 1-\frac \beta
2n_{I,\overline{\alpha }}^a-\frac 12n_{I,\overline{\alpha }}^b\right) ,$$
$$\underline{b}_{I,\alpha ,\sigma }=b_{I,\alpha ,\sigma }\left( 1-n_{I,%
\overline{\alpha }}^a-\frac 12n_{I,\overline{\alpha }}^b\right)$$
and
$$\beta =\frac 14\left( 1-\frac 1{\sqrt{1+\left( \frac{4t}U\right) ^2}}\right)
,$$
$$\delta =\frac{t}{U}\frac 1{\sqrt{1+\left( \frac{4t}U\right) ^2}}.$$
To write down the explicit form of the Hubbard Hamiltonian $\overline{%
\overline{H}}$ (see (24)) we have to insert (27) and (28) into $\overline{%
\overline{H}}$. This operation leads, however, to a very complicated form of $\overline{\overline{H}},$ given in the Appendix C (a simplified form of this Hamiltonian is discussed in the next Section). Here again the only assumption made to derive $\overline{\overline{H}}$ was the reduction of the whole dimer space (A.1) to the subspace of the dimer eigenvectors $\left|
E_0\right\rangle $, $\left| E_{11}\right\rangle $, $\left|
E_{12}\right\rangle $, $\left| E_{13}\right\rangle $, $\left|
E_{14}\right\rangle $, $\left| E_{21}\right\rangle $, $\left|
E_{24}\right\rangle $, $\left| E_{25}\right\rangle $ and $\left|
E_{26}\right\rangle $, corresponding to the lowest lying | 1 | member_31 |
dimer energy levels because in the large $U$ limit only these levels can be occupied. The Hamiltonian $\overline{\overline{H}}$, obtained in this way, contains many competing, magnetic, nonmagnetic and more complex interactions. Among them we can find a direct antiferromagnetic interaction generated by the term $%
\stackrel{\rightarrow }{S}_{I,1}\cdot \stackrel{\rightarrow }{S}_{I,2}$ (Heisenberg exchange interaction) multiplied by the positive coupling constant. Such a term appears in $\overline{H}_I^d$ (see (C.6) and (14)). Inside of $\overline{\overline{H}}$ (see (C.6) and (C.2) - (C.5)) a kind of ferromagnetic interactions between spins from different dimers, represented by (C.3), appears within the terms proportional to $\beta ^2$ and $\delta ^2$ (negative coupling constants). The antiferromagnetic interactions, however, appear again in terms proportional to $\beta \delta $. There are also many other magnetic, more exotic interactions, represented by (C.2), (C.4) and (C.5), entering into (C.6). The situation is, however, much more complicated when we consider the total Hamiltonian $\overline{H}$ (see (24)) in the large $U$ limit. $\overline{H}$ differs from $\overline{\overline{H}}$ by the multiplicative factor $P$ (a product of the projection operators $P_I$ (see (22) and (17)), standing on the left. Inside of each $P_I$ the mentioned antiferromagnetic interaction also appears (see (17)). In other words, the total Hamiltonian $\overline{H}$, we | 1 | member_31 |
are interested in, is actually a sum of the products of many competitive ferromagnetic, antiferromagnetic and more exotic interactions. The thermodynamic properties of the system, described by the Hamiltoniam $\overline{H}$ (24), are then a result of the competition between all of them. Which interaction wins in such a competition it certainly depends on temperature, model parameters ($%
t,U$) and on the average number of electrons per lattice site, determining the chemical potential of the system.
The formalism presented in this paper is also applicable to a more complicated decomposition of the Hubbard Hamiltonian (1) into a set od interacting clusters consisting e.g. of one central atom and $z$ its nearest neighbours. We, however, know (see e.g. Refs \[36\]-\[39\] and papers cited therein) that, unfortunately, the mathematical problems in this case exponentially grows up with the size of the cluster.
Taylor expansion
================
The complicated form of the Hubbard Hamiltonian $\overline{H}$ in the large $%
U$ limit (see (24), (C.6)) where $P$ is given by (22) (see also (17)) can essentially be reduced when applying the Taylor expansion with respect to the parameter $x=\frac{t}{U}\ll 1$. To do it we have to expand all the coefficients in (14), (17) and (C.6) including | 1 | member_31 |
also $\beta $ and $\delta $ (see (32), (33)). Such an expansion can be performed to any power of $x$, however, the most simple form we obtain when we restrict ourselves to the linear approximation, resulting in the terms proportional to $t$ and $\frac{%
t^2}U.$ The accuracy of this expansion can easily be verified assumming e.g. a typical value of the ratio $\frac{W}{U}=\frac 15$ (or less). Because the bandwidth of the conduction band for the sc lattice is $W=12t,$ it results in a small value of the expansion parameter $x=\frac tU=\frac 1{60}$ in this case. It, however, means that the linear approximation is quite reasonable because all higher terms in the expansion, proportional to $x^n$ ($%
n=2,3,...),$ produce $60$ times smaller contribution.
To present the results of the Taylor expansion including all the terms proportional to $t$ and $\frac{t^2}U$ let us first define several auxiliary quantities
$$P^{(1)}=\prod\limits_{I=1}^MP_I^{(1)},$$
$$P^{(2)}=\frac tU\sum\limits_{I=1}^MP_1^{(1)}\cdot ...\cdot
P_{I-1}^{(1)}P_I^{(2)}P_{I+1}^{(1)}\cdot ...\cdot P_M^{(1)}$$
where
$$P_I^{(1)}=1-\frac 12\left( n_{I,1}^b+n_{I,2}^b\right) +\frac
14n_{I,1}^bn_{I,2}^b,$$
$$P_I^{(2)}=\sum\limits_{\sigma} \sum\limits_{\alpha =1}^2\left( a_{I,\alpha
,\sigma }^{+}b_{I,\overline{\alpha },\sigma }+b_{I,\alpha ,\sigma }^{+}a_{I,%
\overline{\alpha },\sigma }\right)$$
and ($\alpha =1,$ $2$)
$$\widetilde{a}_{I,\alpha ,\sigma }=a_{I,\alpha ,\sigma }\left( 1-\frac 12n_{I,%
\overline{\alpha }}^b\right) ,$$
$$\widetilde{\widetilde{a}}_{I,\alpha ,\sigma }=\underline{b}_{I,\overline{%
\alpha },\sigma }+a_{I,\overline{\alpha },-\sigma }^{+}a_{I,\alpha ,-\sigma
}b_{I,\alpha ,\sigma }+a_{I,\overline{\alpha },\sigma }\frac{n_{I,\alpha | 1 | member_31 |
}^a}%
2$$
where $\underline{b}_{I,\alpha ,\sigma }$ is given by (31).
The Hamiltonian $\overline{\overline{H}}$ (C.6), including all the terms proportional to $t$ and $\frac{t^2}U$, takes on the form
$$\overline{\overline{H}}=\overline{\overline{H}}^{(1)}+\overline{\overline{H}}%
^{(2)}$$
where $$\begin{aligned}
\overline{\overline{H}}^{(1)} &=&-t\sum\limits_{I,\sigma }\left[
a_{I,1,\sigma }^{+}a_{I,2,\sigma }+a_{I,2,\sigma }^{+}a_{I,1,\sigma }\right]
\nonumber \\[0.2cm]
&&-t\sum\limits_{I,\sigma }\left[ \widetilde{a}_{I,2,\sigma }^{+}\widetilde{%
a}_{I+1,1,\sigma }+\widetilde{a}_{I+1,1,\sigma }^{+}\widetilde{a}%
_{I,2,\sigma }\right] \\[0.2cm]
&&-t\sum\limits_{I\neq J,\sigma }\sum\limits_{\alpha =1}^2\widetilde{a}%
_{I,\alpha ,\sigma }^{+}\widetilde{a}_{J,\alpha ,\sigma } \nonumber\end{aligned}$$
and $$\begin{aligned}
\overline{\overline{H}}^{(2)} &=&\frac{4t^2}U\sum\limits_I[\overrightarrow{S%
}_{I,1}\cdot \overrightarrow{S}_{I,2}-\frac 14n_{I,1}^an_{I,2}^a] \nonumber
\\
&&-\frac{t^2}U\sum\limits_{I,\sigma }[\widetilde{a}_{I,2,\sigma }^{+}%
\widetilde{\widetilde{a}}_{I+1,1,\sigma }+\widetilde{\widetilde{a}}%
_{I,2,\sigma }^{+}\widetilde{a}_{I+1,1,\sigma } \nonumber \\
&&+\widetilde{a}_{I+1,1,\sigma }^{+}\widetilde{\widetilde{a}}_{I,2,\sigma }+%
\widetilde{\widetilde{a}}_{I+1,1,\sigma }^{+}\widetilde{a}_{I,2,\sigma }]
\nonumber \\
&&-\frac{t^2}U\sum\limits_{I\neq J,\sigma }\sum\limits_{\alpha =1}^2[%
\widetilde{a}_{I,\alpha ,\sigma }^{+}\widetilde{\widetilde{a}}_{J,\alpha
,\sigma }+\widetilde{\widetilde{a}}_{I,\alpha ,\sigma }^{+}\widetilde{a}%
_{J,\alpha ,\sigma }] \\
&&+\frac{2t^2}U\sum\limits_I[\overrightarrow{S}_{I,2}\cdot (\overrightarrow{%
\underline{\underline{s}}}_{I,1;I+1,1}+\overrightarrow{\underline{s}}%
_{I+1,1;I,1}) \nonumber \\
&&+\overrightarrow{S}_{I+1,1}\cdot (\overrightarrow{\underline{\underline{s}}%
}_{I+1,2;I,2}+\overrightarrow{\underline{s}}_{I,2;I+1,2})] \nonumber \\
&&+\frac{2t^2}U\sum\limits_{I\neq J}[\overrightarrow{S}_{I,1}\cdot (%
\overrightarrow{\underline{\underline{s}}}_{I,2;J,1}+\overrightarrow{%
\underline{s}}_{J,1;I,2}) \nonumber \\
&&+\overrightarrow{S}_{I,2}\cdot (\overrightarrow{\underline{\underline{s}}}%
_{I,1;J,2}+\overrightarrow{\underline{s}}_{J,2;I,1})]. \nonumber\end{aligned}$$
The operators $\underline{\underline{s}}_{I,\mu ;J,\nu }^{z,\pm }$ and $%
\underline{s}_{I,\mu ;J,\nu }^{z,\pm }$ in (42) retain their forms introduced in (C.1) but $\underline{\underline{a}}_{I,\mu ,\sigma }(%
\underline{\underline{a}}_{I,\mu ,\sigma }^{+})$ in (C.1) should actually be replaced by $\widetilde{a}_{I,\mu ,\sigma }(\widetilde{a}_{I,\mu ,\sigma
}^{+})$, defined by (38). The total Hamiltonian $\overline{H}$ in the large $%
U$ limit (24) including all terms proportional to $t$ and $\frac{t^2}U$ can thus be written in the form
$$\overline{H}=P^{(1)}(\overline{\overline{H}}^{(1)}+\overline{\overline{H}}%
^{(2)})+P^{(2)}\overline{\overline{H}}^{(1)}=P^{(1)}\overline{\overline{H}}%
^{(1)}+(P^{(1)}\overline{\overline{H}}^{(2)}+P^{(2)}\overline{\overline{H}}%
^{(1)}).$$
The first part, | 1 | member_31 |
$P^{(1)}\overline{\overline{H}}^{(1)}$, contains the terms proportional to $t$ whereas the term in the parentheses is proportional to $%
\frac{t^2}U$ (see (35) and (42)). In the expression for $\overline{\overline{%
H}}^{(2)}$(see (42)) the first term describes the antiferromagnetic, Heisenberg intradimer interaction (see also the second term in (15)). The apearence of this term may suggest that such an interaction should also arise between different dimers. This is of course the case. However, because of applied procedure such terms do not appear explicitly. In the last analysis our method treats all interactions within and between dimers, on the same quality level, i.e. all interactions are taken into account. The magnetic interdimer interactions, represented by the fourth and fifth term in (42), have formally the same structure as the Heisenberg interactions but instead of the scalar products of spin operators there are the products of spin operators and ”hopping spin” operators (defined in C.1). All the terms presented in (42) are correct because they originate from the exact decomposition of the Hubbard Hamiltonian (1) into a set of interacting dimers (4) where each dimer problem has been exactly solved (exact dimer representation of the construction operators after applying the projection procedure, given by (27) and | 1 | member_31 |
(28)). The total Hamiltonian $\overline{H}$ (43), we are interested in, is much more complicated than $\overline{%
\overline{H}}^{(1)}$ and $\overline{\overline{H}}^{(2)}$alone (see (41)-(43)) because of the presence of the projection operators $P^{(1)}$ and $P^{(2)}$ (see (34)-(37)) in the expression (43).
It is also interesting to see what happens in the special case when taking the limit $U\rightarrow \infty $. The second term in the parentheses of Eq.(43) vanishes in this limit (cf (35) and (42)). Besides, each lattice site cannot be at the same time occupied by two electrons what is equivalent to the assumption ($\alpha =1,2)$
$$n_{I,\alpha ,\sigma }^b=b_{I,\alpha ,\sigma }^{+}b_{I,\alpha ,\sigma
}=n_{I,\alpha ,\sigma }n_{I,\alpha ,-\sigma }=0,$$
$$n_{I,\alpha }^b=\sum\limits_{\sigma} n_{I,\alpha ,\sigma }^b=0$$
and (cf (38))
$$\widetilde{a}_{I,\alpha ,\sigma }=a_{I,\alpha ,\sigma }.$$
The total Hamiltonian (43) is thus given by a simple formula
$$\begin{array}{ll}
\overline{H}=P^{(1)}\overline{\overline{H}}= & -t\sum\limits_{I,\sigma
}\left[ a_{I,1,\sigma }^{+}a_{I,2,\sigma }+a_{I,2,\sigma }^{+}a_{I,1,\sigma
}\right] \\
& \\
& -t\sum\limits_{I,\sigma }\left[ a_{I,2,\sigma }^{+}a_{I+1,1,\sigma
}+a_{I+1,1,\sigma }^{+}a_{I,2,\sigma }\right] \\
& \\
& -t\sum\limits_{I\neq J,\sigma }\sum\limits_{\alpha =1}^2a_{I,\alpha
,\sigma }^{+}a_{J,\alpha ,\sigma }.
\end{array}$$
Going back to the original lattice (cf (4),(5) and (1)) it is easy to see that the exact Hubbard Hamiltonian in the limit $U\rightarrow \infty $ takes on the form
$$\overline{H}=-t\sum\limits_{i,j,\sigma }a_{i,\sigma }^{+}a_{j,\sigma }$$
where $i$ | 1 | member_31 |
and $j$ (as before) number the lattice points and $a_{i,\sigma }$ $%
(a_{i,\sigma }^{+})$ are the Hubbard operators, defined by (9).
Conclusions
===========
Using a new, nonperturbative approach, basing on the equivalent form of the Hubbard Hamiltonian, represented by the collection of interacting dimers (4) where each dimer problem has been exactly solved, we have expressed the annihilation (creation) operators (B.1)-(B.4) as linear combinations of transition operators between different dimer states (dimer representation). This method made it possible to exclude from the considerations the unoccupied dimer energy levels in the large $U$ limit by means of the projection technique resulting in the final Hamiltonian for the dimer itself (14), as well as, for the crystal as a whole (see (23), (24) and (C.6)). It is important to stress that the elimination of the unoccupied dimer energy levels was the only assumption to derive the Hubbard Hamiltonian (24) in the large $U$ limit. Therefore we can be sure that expanding (24) with respect to $x=\frac tU$ (Taylor expansion) we obtain absolutely all terms proportional to $x^n$ $(n=1,2,...).$ In other words all the coefficients proportional to $x^n$ are easy to control what is not always the case when using another methods. The | 1 | member_31 |
final form of the obtained Hamiltonian in the large $U$ limit (see (23), (24) and (C.6)) visualizes high degree of complexity of the model, deeply hidden in its original version, forming a mixture of many multiple ferromagnetic, antiferromagnetic and more complex interactions competing one with another. This fact seems to be decisive for our final conclusion. Because we are still dealing with the approximate solutions of the model (the exact solution does not exist till now) it may happen that we underestimate in this way some important interactions and overestimate the others. It is the reason why the resulting thermodynamic properties of the Hubbard model, obtained in an approximate way, so strongly depend on the quality of applied approximations.
We note in passing that the exact dimer solution may serve as a novel alloy analogy for the Hubbard model, which could be treated by coherent potential approximation (CPA). It is well-known that the standard alloy analogy, based on the atomic limit, does not allow for ferromagnetism in the Hubbard model. This may change by application of the dimer solution which already accounts for a restricted hopping of the band electrons. A corresponding study is in preparation.
Another example of a | 1 | member_31 |
dimer approach to Hubbard-like models is the bond operator theory as an extension of the slave bosonic and fermionic operators (see Refs \[40\], \[41\] and papers cited therein).
[Appendix A]{}
The exact solution of the dimer eigenvalue problem (5) reads:
$$\begin{array}{ll}
E_0=0; & |E_0\rangle =|0\rangle ,
\end{array}$$
$$\begin{array}{ll}
E_{11}=-t; & |E_{11}\rangle =\frac{1}{\sqrt{2}}(|11\rangle +|13\rangle ), \\
E_{12}=t; & |E_{12}\rangle =\frac{1}{\sqrt{2}}(|11\rangle -|13\rangle ), \\
E_{13}=-t; & |E_{13}\rangle =\frac{1}{\sqrt{2}}(|12\rangle +|14\rangle ), \\
E_{14}=t; & |E_{14}\rangle =\frac{1}{\sqrt{2}}(|12\rangle -|14\rangle ),
\end{array}$$
$$\begin{array}{ll}
E_{21}=0; & |E_{21}\rangle =\frac 1{\sqrt{2}}(|23\rangle +|24\rangle ), \\
E_{22}=U; & |E_{22}\rangle =\frac 1{\sqrt{2}}(|21\rangle -|26\rangle ), \\
E_{23}=C+\frac U2; & |E_{23}\rangle =a_1(|21\rangle +|26\rangle
)-a_2(|23\rangle -|24\rangle ), \\
E_{24}=-C+\frac U2; & |E_{24}\rangle =a_2(|21\rangle +|26\rangle
)+a_1(|23\rangle -|24\rangle ), \\
E_{25}=0; & |E_{25}\rangle =|22\rangle , \\
E_{26}=0; & |E_{26}\rangle =|25\rangle ,
\end{array}
\tag{A.1}$$
$$\begin{array}{ll}
E_{31}=t+U; & |E_{31}\rangle =\frac 1{\sqrt{2}}(|31\rangle +|33\rangle ), \\
E_{32}=-t+U; & |E_{32}\rangle =\frac 1{\sqrt{2}}(|31\rangle -|33\rangle ),
\\
E_{33}=t+U; & |E_{33}\rangle =\frac 1{\sqrt{2}}(|32\rangle +|34\rangle ), \\
E_{34}=-t+U; & |E_{34}\rangle =\frac 1{\sqrt{2}}(|32\rangle -|34\rangle ),
\end{array}$$
$$\begin{array}{ll}
E_4=2U; & |E_4\rangle =|4\rangle
\end{array}$$ where
$$C=\sqrt{{\left( \frac U2\right) }^2+4t^2}, \tag{A.2}$$
$$a_1=\frac 12\sqrt{1+\frac U{2C}}, \tag{A.3}$$
$$a_2=\frac 12\sqrt{1-\frac U{2C}}. \tag{A.4}$$
[Appendix B]{}
The exact dimer representation of the construction operators is given by the following expressions:
$$\begin{array}[t]{ll}
c_{1,\uparrow }= & \frac | 1 | member_31 |
1{\sqrt{2}}\left( P_{0,11}+P_{0,12}\right) +\frac 1{%
\sqrt{2}}P_{11,25}-\frac 1{\sqrt{2}}P_{12,25} \\
& +\frac 12\left( P_{13,21}+P_{13,22}\right) +\frac 1{\sqrt{2}}\left(
bP_{13,23}+aP_{13,24}\right) \\
& -\frac 12\left( P_{14,21}-P_{14,22}\right) +\frac 1{\sqrt{2}}\left(
aP_{14,23}-bP_{14,24}\right) \\
& +\frac 12\left( P_{21,33}-P_{21,34}\right) -\frac 12\left(
P_{22,33}+P_{22,34}\right) \\
& +\frac 1{\sqrt{2}}\left( aP_{23,33}+bP_{23,34}\right) -\frac 1{\sqrt{2}%
}\left( bP_{24,33}-aP_{24,34}\right) \\
& +\frac 1{\sqrt{2}}\left( P_{26,31}-P_{26,32}\right) +\frac 1{\sqrt{2}%
}P_{31,4}+\frac 1{\sqrt{2}}P_{32,4},
\end{array}
\tag{B.1}$$
$$\begin{array}[t]{ll}
c_{1,\downarrow }= & \frac 1{\sqrt{2}}\left( P_{0,13}+P_{0,14}\right) +\frac
1{\sqrt{2}}P_{13,26}-\frac 1{\sqrt{2}}P_{14,26} \\
& +\frac 12\left( P_{11,21}-P_{11,22}\right) -\frac 1{\sqrt{2}}\left(
bP_{11,23}+aP_{11,24}\right) \\
& -\frac 12\left( P_{12,21}+P_{12,22}\right) -\frac 1{\sqrt{2}}\left(
aP_{12,23}-bP_{12,24}\right) \\
& -\frac 12\left( P_{21,31}-P_{21,32}\right) -\frac 12\left(
P_{22,31}+P_{22,32}\right) \\
& +\frac 1{\sqrt{2}}\left( aP_{23,31}+bP_{23,32}\right) -\frac 1{\sqrt{2}%
}\left( bP_{24,31}-aP_{24,32}\right) \\
& -\frac 1{\sqrt{2}}\left( P_{25,33}-P_{25,34}\right) -\frac 1{\sqrt{2}%
}P_{33,4}-\frac 1{\sqrt{2}}P_{34,4},
\end{array}
\tag{B.2}$$
$$\begin{array}[t]{ll}
c_{2,\uparrow }= & \frac 1{\sqrt{2}}\left( P_{0,11}-P_{0,12}\right) -\frac 1{%
\sqrt{2}}P_{11,25}-\frac 1{\sqrt{2}}P_{12,25} \\
& -\frac 12\left( P_{13,21}+P_{13,22}\right) +\frac 1{\sqrt{2}}\left(
bP_{13,23}+aP_{13,24}\right) \\
& -\frac 12\left( P_{14,21}-P_{14,22}\right) -\frac 1{\sqrt{2}}\left(
aP_{14,23}-bP_{14,24}\right) \\
& -\frac 12\left( P_{21,33}+P_{21,34}\right) +\frac 12\left(
P_{22,33}-P_{22,34}\right) \\
& +\frac 1{\sqrt{2}}\left( aP_{23,33}-bP_{23,34}\right) -\frac 1{\sqrt{2}%
}\left( bP_{24,33}+aP_{24,34}\right) \\
& -\frac 1{\sqrt{2}}\left( P_{26,31}+P_{26,32}\right) +\frac 1{\sqrt{2}%
}P_{31,4}-\frac 1{\sqrt{2}}P_{32,4},
\end{array}
\tag{B.3}$$
$$\begin{array}[t]{ll}
c_{2,\downarrow }= & \frac 1{\sqrt{2}}\left( P_{0,13}-P_{0,14}\right) -\frac
1{\sqrt{2}}P_{13,26}-\frac 1{\sqrt{2}}P_{14,26} \\
& -\frac 12\left( P_{11,21}-P_{11,22}\right) -\frac 1{\sqrt{2}}\left(
bP_{11,23}+aP_{11,24}\right) \\
& -\frac 12\left( P_{12,21}+P_{12,22}\right) +\frac 1{\sqrt{2}}\left(
aP_{12,23}-bP_{12,24}\right) \\
& +\frac 12\left( P_{21,31}+P_{21,32}\right) +\frac 12\left(
P_{22,31}-P_{22,32}\right) \\
& +\frac 1{\sqrt{2}}\left( aP_{23,31}-bP_{23,32}\right) -\frac 1{\sqrt{2}%
| 1 | member_31 |
}\left( bP_{24,31}+aP_{24,32}\right) \\
& +\frac 1{\sqrt{2}}\left( P_{25,33}+P_{25,34}\right) -\frac 1{\sqrt{2}%
}P_{33,4}+\frac 1{\sqrt{2}}P_{34,4}
\end{array}
\tag{B.4}$$
where
$$P_{\alpha ,\beta }=|E_{\alpha} \rangle \langle E_{\beta} | \tag{B.5}$$
and
$$\begin{array}{l}
a=a_1+a_2 \\
b=a_1-a_2.
\end{array}
\tag{B.6}$$
To obtain the annihilation operators of the $I-$th dimer, the index $I$ should be added $(c_{1(2),\sigma }\longrightarrow c_{I,1(2),\sigma }$, $%
P_{\alpha ,\beta }\rightarrow P_{\alpha ,\beta }^{(I)})$ in (B.1)-(B.4).
[Appendix C]{}
To present the Hamiltonian $\overline{\overline{H}}$ in a compact form we first introduce the following operators $\left( \mu ,\nu =1,2\right) $ :
$$\begin{array}{ll}
s_{I,\mu ;J,\nu }^{+}= & a_{I,\mu ,\uparrow }^{+}a_{J,\nu ,\downarrow }, \\
& \\
s_{I,\mu ;J,\nu }^{-}= & a_{J,\nu ,\downarrow }^{+}a_{I,\mu ,\uparrow }, \\
& \\
n_{I,\mu ;J,\nu ;\sigma }= & a_{I,\mu ,\sigma }^{+}a_{J,\nu ,\sigma }, \\
& \\
n_{I,\mu ;J,\nu }= & \sum\limits_{\sigma} n_{I,\mu ;J,\nu ;\sigma }, \\
& \\
s_{I,\mu ;J,\nu }^z= & \frac 12\left( n_{I,\mu ;J,\nu ;\uparrow }-n_{I,\mu
;J,\nu ;\downarrow }\right) ; \\
&
\end{array}$$
$$\begin{array}{ll}
\underline{s}_{I,\mu ;J,\nu }^{+}= & \underline{\underline{a}}_{I,\mu
,\uparrow }^{+}a_{J,\nu ,\downarrow }, \\
& \\
\underline{s}_{I,\mu ;J,\nu }^{-}= & a_{J,\nu ,\downarrow }^{+}\underline{%
\underline{a}}_{I,\mu ,\uparrow }, \\
& \\
\underline{n}_{I,\mu ;J,\nu ;\sigma }= & \underline{\underline{a}}_{I,\mu
,\sigma }^{+}a_{J,\nu ,\sigma }, \\
& \\
\underline{n}_{I,\mu ;J,\nu }= & \sum\limits_{\sigma} \underline{n}_{I,\mu
;J,\nu ;\sigma }, \\
& \\
\underline{s}_{I,\mu ;J,\nu | 1 | member_31 |