text
stringlengths 448
13k
| label
int64 0
1
| doc_idx
stringlengths 8
14
|
|---|---|---|
---
abstract: 'The spin gauge field formalism has been used to explain the emergence of out of plane spin accumulation in two-dimensional spin orbit interaction (SOI) systems in the presence of an in-plane electric field. The adiabatic alignment of the charge carrier spins to the momentum dependent SOI field, which changes in time due to the electric field, can be mathematically captured by the addition of a gauge term in the Hamiltonian. This gauge term acts like an effective, electric field dependent magnetization. In this work we show that this effective magnetization can be generalized to systems which include additional discrete degrees of freedom to real spin, such as the pseudospin and/or valley degrees of freedom in emerging materials like molybdenum sulphide and silicene. We show that the generalized magnetization recovers key results from the Sundaram-Niu formalism as well as from the Kubo formula. We then use the generalized magnetization to study the exemplary system of a topological insulator thin film system where the presence of both a top as well as a bottom surface provides an additional discrete degree of freedom in addition to the real spin.'
author:
- Zhuo Bin Siu
- 'Mansoor B. A. Jalil'
- Seng | 1 | member_36 |
Ghee Tan
title: Gauge field in systems with spin orbit interactions and additional discrete degrees of freedom to real spin
---
Introduction
============
In the Spin Hall Effect (SHE) [@SHE1; @SHE2; @SHE3; @SHE4; @SHE5], the passage of an in-plane electric field in a two-dimensional electron gas (2DEG) with spin orbit interactions (SOIs) leads to the appearance of an out of plane spin accumulation.
Murakami [@Murakami] and Fujita [@Grp1; @Grp2; @Grp3; @Grp4; @SGTSciRep], and their respective coauthors, had independently studied the SHE. They showed that the out of plane spin accumulation can be understood as the response of the charge carriers as their spins align adiabatically with the momentum dependent SOI field. The direction of the SOI field changes in time due to the change in the momentum of the charge carriers as they accelerate under the electric field. Mathematically, the electric field gives to an effective magnetization term in the Hamiltonian which we shall, for short, call the Murakami-Fujita (MF) potential.
Many emerging material systems in interest in spintronics, for example silicene [@Sil1; @Sil2; @Sil3; @Sil4] and [@Mo1; @Mo2; @Mo3], possess discrete degrees of freedom (DoFs) such as the pseudospin and / or valley degrees of freedom, in addition to | 1 | member_36 |
their real spins. In this work, we show in the following sections that the MF potential can be readily extended to incorporate these additional degrees of freedom (DoF) which we shall for simplicity refer to collectively as pseudospin. To first order in the electric field, the MF potential accounts for the effects of a constant, in-plane electric field for the purposes of calculating spin / charge currents and spin accumulations to first order in the electric field.
We illustrate the application of the MF potential on a system with a spin$\otimes$pseudospin degrees of freedom through the example of the topological insulator (TI) thin film system [@PRB80_205401; @PRB81_041307; @PRB81_115407]. Unlike a semi-infinite TI slab, a TI thin film has both a top and a bottom surface which, due t the finite thickness, couple to each other. The low energy effective Hamiltonian for can be written as $$H = v(\vec{k}\times\vec{\sigma})\cdot\hat{z} \tau_z + \lambda \tau_x + \vec{M}\cdot\vec{\sigma} \label{TIham1}$$ Besides the real spin denoted as $\vec{\sigma}$ of the charge carriers there is another discrete degree of freedom $\vec{\tau}$ associated with whether the charge carriers are localized nearer the upper ( $|+\tau_z\rangle\langle +\tau_z|$ ) or lower ($|-\tau z \rangle\langle -\tau_z|$) surface of the film. The | 1 | member_36 |
$\tau_x$ term then represents the coupling between the two surfaces of the film due to the finite thickness.
This paper is organized as follows. We first revisit the emergence of the MF potential in a spin 1/2 SOI system. We then generalize the MF potential to include other discrete DoFs, and provide three evidences to support our claim that the MF potential accounts for the effects of the electric field in the sense that an effective Hamiltonian can be constructed by replacing the $\vec{E}\cdot\vec{r}$ term in the original Hamiltonian can be replaced by the position-*independent* MF potential.
We first show that taking the momentum derivative of the effective Hamiltonian in the Heisenberg equation of motion for the position operator reproduces the usual Berry curvature expression for the anomalous Hall velocity. As part of their paper on the microscopic origin of spin torque [@ChengRan], Cheng Ran and Niu Qian had extended the original Sundaram-Niu wavepacket formalism [@ChioGoo1], which gave only the time variation of the position and momentum expectation values, to now include the time variation of the spin expectation values. We show that Ran and Niu’s expressions for the time evolution of the spin expectation values can be readily extended | 1 | member_36 |
to incorporate the other discrete degrees of freedom present, and that the time evolution of these operators can be derived from applying the Heisenberg equation of motion to the MF potential. Finally, we show that the Kubo expression for non equilibrium expectation of spin$\otimes$pseudospin quantities can be interpreted as the first order time independent perturbation theory response to the MF potential.
We then move on to apply the MF potential formalism to study the emergence of a TI thin film system subjected to an in-plane magnetization and electric field. We first illustrate the effects of the interlayer coupling on the in-plane magnetization and the dispersion relations. We then show that the direction of the out of plane spin accumulation resulting from an in-plane electric field can be explained in terms of how the direction of the momentum dependent in-plane SOI field rotates with the change in momentum direction resulting from the electric field. The anti-symmetry of the out of plane spin accumulation in $k$ space can be broken with the application of an out of plane electric field in order to yield a finite spin accumulation after integrating over the Fermi surface.
Spin 1/2 systems {#spinhalf}
================
To familarize the | 1 | member_36 |
reader with the MF potential, we first review its appearance in spin 1/2 SOI systems without any additional discrete degrees of freedom. The Hamiltonian for a homogenous 2DEG with SOI and an electric field $E_x$ in the $x$ direction can be generically written as $$H = \frac{p^2}{2m} + \vec{B}(\vec{k})\cdot\vec{\sigma} + E_x x$$ where the $\vec{B}(\vec{k})$ represents a momentum dependent spin orbit interaction. We define a unitary transformation $U(\vec{k})$ which diagonalizes $\vec{B}.\vec{\sigma}$ in spin space so that after the unitary transformation, we have $$UHU^\dagger = \frac{p^2}{2m} + |\vec{B}|\sigma_z + E_x (x - i U \partial_{k_x} U^\dagger).$$
Mathematically, the effect of $U$ can be interpreted as rotating the spin space coordinates so that in the rotated frame, the spin $z$ axis points in the direction of the SOI field $\vec{B}(\vec{k})$. The non commutation between $x$ and the momentum dependent $U$ results in the appearance of the $-i E_x (U \partial_{k_x} U^\dagger)$ term which acts as an effective magnetization $M'_i \tilde{\sigma}_i$ in the rotated frame where the tilde on the $\tilde{\sigma}_i$ indicates that the index $i$ refers to the $i$th spin direction in the rotated frame. To determine what *lab* frame direction this effective magnetization points in, we perform the inverse unitary transformation | 1 | member_36 |
$U^\dagger (-i U \partial_{k_x}U^\dagger) U = -i (\partial_{k_x}U^\dagger)U$. This expression can be evaluated without an explicit form for $U$. To do this, we first note that by definition $U\hat{b}\cdot\vec{\sigma}U^\dagger = \sigma_z$ where $\hat{b} = \vec{B}/|\vec{B}|$. Thus, $$\begin{aligned}
&& U^\dagger\sigma_zU = \hat{b}\cdot\vec{\sigma} \\
&\Rightarrow& (\partial_{k_x}U^\dagger)\sigma_z U + U^\dagger\sigma_z(\partial_{k_x}U) = \partial_{k_x}\hat{b}\cdot\vec{\sigma} \\
&\Rightarrow& [(\partial_{k_x}U^\dagger)U, \hat{b}\cdot\vec{\sigma}] = \partial_{k_x}\hat{b}\cdot\vec{\sigma} \\\end{aligned}$$ In going from the first to second line, we differentiated the first line with respect to $k_x$ and then inserted $\mathbb{I}_\sigma=UU^\dagger$ in the appropriate places. From the last line, we use the fact that $[\vec{a}\cdot\vec{\sigma}, \vec{b}\cdot\vec{\sigma}] = i (\vec{a}\times\vec{b})\cdot\vec{\sigma}$ to conclude that $$-i E_x U\partial_{k_x}U^\dagger = E_x (\hat{b}\times\partial_{k_x}\hat{b})\cdot\vec{\sigma}.$$
This is the MF potential for a spin 1/2 system with an electric field in the $x$ direction.
Notice that although $U$ is not unique, the lab frame direction of $-i U\partial_{k_x}U^\dagger$ is independent of the specific choice of $U$. $E_x(-i U\partial_{k_x}U^\dagger)$ can then be thought as as an electric field dependent effective magnetization which confers a spin accumulation in the $(\hat{b}\times\partial_{k_x}\hat{b})\cdot\vec{\sigma}$ direction. Taking the specific example of the Rashba SOI where $\vec{B} = \alpha( p_y, -p_x)$ both $\vec{B}$ and $\partial_{k_x}\vec{B}$ lie on the $xy$ plane. $E_x(\hat{b}\times\partial_{k_x}\hat{b})$ thus points in the out of plane spin direction, | 1 | member_36 |
and confers an out of plane spin accumulation to the charge carriers
Physically, the origin of the $(\hat{b}\times(E_x \partial_{k_x}\hat{b})$ term can be explained by assuming that the spins of the charge carriers adiabatically follow the direction of the SOI field. As shown in Fig. \[gA2SOIfield\], $\vec{B}(\vec{k})\cdot\vec{\sigma}$ associates each point in $k$ space with a SOI field pointing in the $\hat{b}(\vec{k})$ direction. Assume that the electric field is initially switched off and consider a carrier with a definite $\vec{k}$. As the electric field is switched on, the field causes the charge carrier to accelerate in the direction of the field so that the momentum changes and the carrier traces out a trajectory along $k$ space. We assume that the electric field is weak enough so that the spin of the carrier rotates along with the direction of the SOI field as it successively moves through different $k$ points. The resulting rotation of the spin can be thought of as being due to an effective magnetic field pointing along the $\hat{b}\times\partial_t \hat{b} = E_x \hat{b}\times\partial_{k_x} \hat{b}$ direction which both provides the torque necessary to rotate the spin as well as confers a spin accumulation in the direction of the torque.
![ The | 1 | member_36 |
arrows at each point in $k$ space indicate the direction of the Rashba SOI field there. The application of the electric field causes the momentum of the charge carrier to trace out the trajectory in $k$ space indicated by the dotted line. The spin of the charge carrier adiabatically follows the direction of the SOI field at each point in $k$ space. The rotation of the spin can be thought of as being due to an effective magnetization which both creates the torque necessary to rotate the spin as shown in the inset, as well as confers an out of plane spin accumulation. []{data-label="gA2SOIfield"}](A2SOIfield.eps)
We now proceed to a general description of the MF potential generalized to include other discrete degrees of freedom.
The Murakami-Fujita potential
=============================
Consider now a generic Hamiltonian $$H_0 = b_i(\vec{k})\kappa_i \label{H0}$$ where the $\kappa_i$s are finite sized matrices representing the discrete degrees of freedom. For example, in a spin 1/2 system with SOC, the 4 $\kappa_i$s are the Pauli matrices and the identity matrix. In the TI thin film Hamiltonian Eq. \[TIham1\] the $\kappa_i$s represent the 16 $\vec{\sigma}\otimes\vec{\tau}$ matrices.
In order to write the Hamiltonian Eq. \[H0\] as a numerical matrix, we need to express | 1 | member_36 |
the matrix elements in terms of basis. For example, for spin 1/2 system it is common to adopt the usual representation of the Pauli matrices so that, for instance $H_0 = \vec{b}\cdot\vec{\sigma} \simeq \begin{pmatrix} b_z & b_x - i b_y \\ b_x + i b_y & -b_z \end{pmatrix}$. The numerical matrix on the rightmost side of the equal sign is written in the $|\pm z \rangle$ basis. We refer to the basis which $H_0$ as a ‘numerical matrix’ is in as the ‘laboratory frame’ with basis states $|\lambda_i\rangle$ ($\lambda$ for *l*aboratory. ) Label now the $i$th eigenstates of $H_0$ by $|\epsilon_i\rangle$. We assume that the laboratory basis is fixed, i.e. it has no dependence on any parameter in the Hamiltonian so that, for instance $\partial_{k_x} |\lambda_i\rangle = 0$. Instead of using the laboratory basis, we can also expand our states and operators in terms of the eigenbasis, and convert between the two basis through the unitary transformation $U$. Defining the $U$s so that $UH_0 U^\dagger$ is diagonal in the eigenbasis representation, we have $$U = \sum_{i,j} |\epsilon_i \rangle \langle \epsilon_i|\lambda _j\rangle \langle j|$$, i.e. the matrix $i,j$th elements in the numerical representation of $U$ is $\langle \epsilon_i|\lambda_j\rangle$. Notice that since | 1 | member_36 |
the phase factor $\exp(i\phi)$ can be introduced to $|\tilde{\epsilon}_a \rangle = |\epsilon_i\rangle\exp(i\phi_a)$ arbitrarily the values of the matrix elements $\langle \tilde{\epsilon}_i | \lambda_j \rangle$ will vary with the phase of $|\epsilon_i\rangle$s. Now consider adding a perturbative electric field . The Hamiltonian then becomes $H = H_0 + E_x x$, and we have $UHU^\dagger = UH_0U^\dagger + E_x ( x + i U\partial_{k_x} U^\dagger)$ where, in this rotated frame, $H_0$ is diagonal, and we have an additional $i U\partial_{k_x}U^\dagger$. In order to figure out the lab frame spin$\otimes$pseudospin ‘direction’ where this contribution points to, we transform the $i U\partial_{k_x}U^\dagger$ piece *without the diagonal elements* back to the laboratory frame . The reason for the removal of the diagonal elements will become apparent later. With the diagonal elements in place, we have $U^\dagger (i U\partial_k U^\dagger) U = -i U^\dagger\partial_k U$. (We have dropped the suffix $x$ from $k_x$ and $E_x$ for notational simplicity)
We stress that $-i U^\dagger\partial_k U$ has the same numerical matrix elements in the laboratory frame regardless of the phases of the $\langle \lambda_i | \epsilon_j \rangle$. This is because
$$\begin{aligned}
&& -i U^\dagger \partial_k U \\
&=& -i |\lambda_a \rangle \langle \lambda_a|\epsilon_b \rangle \langle \partial_k \epsilon_b|\lambda_c \rangle | 1 | member_36 |
\langle \lambda_c | \\
&=& -i |\epsilon_a \rangle \langle \partial_k \epsilon_a| \end{aligned}$$
The second line gives the numerical values of the laboratory frame $ac$th matrix elements, and the third line the simplification using a resolution of identity. Notice that we have the combination $|\epsilon_b \rangle \langle \partial_k \epsilon_b|$ with the same state index $b$ occurring together so that any phase factor $\exp(i \phi_b)$ introduced in $|\epsilon_b\rangle \rightarrow |\epsilon_b \rangle\exp(i \phi_b)$ cancels out.
Returning now to the diagonal elements of the rotated frame $i U\partial_k U^\dagger$, we see that they correspond to $i |\epsilon_i \rangle \langle \epsilon_i|\partial_k |\epsilon_i\rangle\langle \epsilon_i|$. Subtracting them off from $-i U^\dagger\partial_k U$ gives the MF potential $H_{MF}$ where $$H_{MF} = -i \sum_{a \neq b} |\epsilon_a \rangle \langle \partial_k \epsilon_a|\epsilon_b \rangle \langle \epsilon_b| E.$$
We argue that, at least for the purposes of calculating currents and spin$\otimes$pseudospin accumulations the effects of the electric field $E_i$ to the first order in $E$ can be incorporated by replacing $E_i x_i$ with $H_{MF}$ so that the effective Hamiltonian reads $$H' = H_0 + H_{MF} = b_i(\vec{k})\kappa_i - i \sum_{a \neq b} |\epsilon_a \rangle \langle \partial_{k_i} \epsilon_a|\epsilon_b \rangle \langle \epsilon_b| {E_i}. \label{Heff}$$
In order to support our claim, we list three examples | 1 | member_36 |
where the use of $H'$ recovers well-known results.
Anomalous velocity
------------------
In the presence of an electric field, charge carriers can acquire an anomalous velocity [@AV1; @AV2; @AV3] proportional to the Berry curvature [@AV4; @AV5; @ChioGoo1; @ChioGooRev]. We show that this result can be recovered via the Heisenberg equation of motion on Eq. \[Heff\]. Under the Heisenberg equation of motion, we have $\partial_t \vec{r} = -i ( [\vec{r}, H_0] + [\vec{r}, H_{MF}] = \nabla_{\vec{k}} (H_0 + H_{MF}) $ The first term is the usual velocity. We shall show that the second gives the usual Berry curvature anomalous contribution to the velocity. Taking the expectation value of the second with respect to the $i$th eigenstate of $H_0$, we have $$\begin{aligned}
&& -i \langle \epsilon_i | [x_b, H_{MF}] |\epsilon_i \rangle \\
&=& -i E_b \langle \epsilon_i| \partial_{k_b} \Big( \sum_{\substack{jk \\ j\neq k}} |\epsilon_j \rangle \langle \partial_{k_a}\epsilon_j| \epsilon_k \rangle\langle\epsilon_k| \Big) |\epsilon_i \rangle \\
&=&-i E_b \sum_j (\langle \epsilon_i |\partial_{k_b} \epsilon_j \rangle \langle \partial_{k_a}\epsilon_j|\epsilon_i\rangle + \langle \partial_{k_a} \epsilon_i | \epsilon_j\rangle\langle \partial_{k_b} \epsilon_j|\epsilon_i \rangle) \\
&=& 2 E_b \mathrm{Im} ( \langle \partial_{k_a} \epsilon_i|\partial_{k_b} \epsilon_i \rangle).\end{aligned}$$
The third line is simply the expansion of the $\partial_{k_b}$ differential. Notice that the requirement that $j \neq k$ stemming | 1 | member_36 |
from the removal of the diagonal terms of the rotated frame $-i U^\dagger \partial_k U$ results in the absence of terms $\langle \partial_{k_a}\partial_{k_b} \epsilon_i|\epsilon_i\rangle$ and $\langle \partial_{k_b} \epsilon_i|\partial_{k_a} \epsilon_i\rangle$ due to the $\partial_{k_b}$ acting on the second and third terms in the big bracket in the second line. The last line is the usual Berry curvature term for the anomalous velocity.
Spin and other discrete DoFs
----------------------------
As part of their paper on explaining the microscopic origin of spin torque, Cheng and Niu extended the original Sundaram-Niu formalism, which described only the spatial evolution of position and velocity, to now cover the time evolution of spin 1/2 as well. Their formalism can be easily extended to cover the time evolution of operators with finite discrete spectra. We describe the extension in the appendix, and simply state the end result here.
For a state $$|\psi\rangle = \sum_i |\epsilon_i \rangle \eta_i$$ where the summation $i$ runs over the discrete DoFs (e.g spin up / down for spin 1/2, and the upper / lower surfaces for a TI thin film) and the continuous quantum numbers (e.g. $\vec{k}$ in SOI systems) and the $\eta_i$s are the weightages of the $i$th basis state, we show | 1 | member_36 |
in the appendix that for an operator $O$ in the discrete DoFs that
$$d_t \langle \psi | O | \psi \rangle = 2E_a \mathrm{Re} ( \eta_i^* \langle \epsilon_i|\partial_{k_a} \epsilon_j \rangle \langle \epsilon_j |O|\epsilon_k \rangle \eta_k$$
It is straightforward to show that this expression is $-i \langle \psi [ O, H_{MF}] |\psi\rangle$.
Recovery of the Kubo formula
----------------------------
Treating $H_{MF}$ as a perturbation to $H_0$ and applying the standard non-degenerate time-independent perturbation theory to the $i$th eigenstate of $H_0$, $|\epsilon_i\rangle$, the first order correction to $|\epsilon_i\rangle$ which we denote as $|\epsilon_i^{(1)}\rangle$ reads $$|\epsilon_i^{(1)}\rangle = \sum_j |\epsilon_j \rangle \frac{ \langle \epsilon_j|H_{MF}|\epsilon_i\rangle}{E_i - E_j}$$ so that to the correction to the expectation value of an observable $O$ for state $|\epsilon_i\rangle$ to first order in $\vec{E}$, $\delta \langle i| O | i \rangle$ is $$\begin{aligned}
&& \delta \langle i| O|i \rangle \\ \nonumber
&=& 2\mathrm{Re} (\langle \epsilon_i|O|\epsilon_i^{(1)}\rangle ) \\ \nonumber
&=& 2\mathrm{Re} \sum_j \frac{ \langle \epsilon_i |O| \epsilon_j \rangle \langle \epsilon_j |H_{MF}|\epsilon_i\rangle}{E_i - E_j}. \label{c1oe}\end{aligned}$$
However, since $$\begin{aligned}
&& \partial_k \langle \epsilon_i | H_0 |\epsilon_i \rangle = 0 \\
&\Rightarrow& \langle \partial_k \epsilon_i |\epsilon_j \rangle (E_i-E_j) = \langle \epsilon_i|\partial_k H_0 |\epsilon_j \rangle \\
&\Rightarrow& \langle \partial_k \epsilon_i|\epsilon_j \rangle = \frac{ \langle \epsilon_i| \partial_k H_0 | 1 | member_36 |
|\epsilon_j \rangle}{E_i - E_j},\end{aligned}$$ we can rewrite $$\begin{aligned}
H_{MF} &=& -i E_i \sum_{a \neq b} |\epsilon_a \rangle \langle \partial_{k_i} \epsilon_a|\epsilon_b \rangle \langle \epsilon_b|. \\
&=&-i \sum_{a \neq b} |\epsilon_a\rangle \frac{ \langle \epsilon_a| \partial_{k_i} H_0 |\epsilon_b \rangle}{E_a - E_b} \langle \epsilon_b|.\end{aligned}$$
A common form of the Kubo formula is $$\delta \langle O \rangle \propto \sum_{\vec{k}} \sum_{a= \neq b} \frac{n(E_a)-n(E_b)}{(E_a-E_b)^2} \mathrm{Im} ( \langle a|O|b\rangle \langle b| (\partial_k H_0)|a \rangle \label{kuboPaper}.$$
Substituting this back into Eq. \[c1oe\] gives a result similar to the Kubo expression for the change in an expectation value of $O$ under an electric field – $$\delta \langle i| O|i \rangle = 2 \mathrm{Im} \sum_j \vec{E}\cdot\frac{ \langle \epsilon_i |O| \epsilon_j \rangle \langle \epsilon_j |\nabla_{\vec{k}} H_0 |\epsilon_i\rangle}{(E_i - E_j)^2}. \label{c2oe}$$
Our result Eq. \[c2oe\] corresponds to Eq. \[kuboPaper\] with the occupancy factor $n$ set to 1 for the $i$th state we are interested in and 0 for the other states, and without a second summation over all states.
Having established the link between the MF potential and the Kubo formula, we now proceed to use Eq. \[c2oe\] to study the exemplary system of a topological insulator thin film system.
TI thin films
=============
The effective Hamiltonian for the surface states of | 1 | member_36 |
a TI thin film of infinite dimensions along the $x$ and $y$ directions, and small finite thickness along the $z$ direction, can be written as $$H = (\vec{k}\times\vec{\sigma})\cdot\hat{z}\tau_z + M_y\sigma_y + \lambda \tau_x. \label{tfHam0}$$
We use units where $e=\hbar=v_f=1$.
We first highlight the influence of the inter-surface coupling term $\lambda$ on the energy spectrum. Consider the limit where $\lambda \rightarrow 0$,$M_y \neq 0$. In this case, the upper and lower surfaces may be considered separately, and the energy spectrum consists of two Dirac cones. The states localized near the upper surface have $\langle \tau_z \rangle = +1$, while the state localized near the lower surface have the opposite sign of $\langle \tau_z \rangle$. The $M_y\sigma_y$ term, however, has the same sign for both the upper and lower surfaces. The Dirac points for the Dirac cones for the upper surface states and the lower surface states are hence displaced in opposite directions in $k$ space.
![ Panel (a) shows the dispersion relations for the two values of $\lambda$ indicated in the legend at $k_y = 0$ for $\vec{M} = 0.5\hat{y}$. The three horizontal dotted lines correspond to the values of energies at which the EECs in panels (b) to (d) at | 1 | member_36 |
$E = 0.5,\ 1$ and $1.5$ are plotted respectively. Panels (c) and (d) show the EECs and the in-plane spin accumulation directions at the two values of $\lambda$ indicated by different colors in panel (b). The inset of panel (d) shows a zoomed in view of the EECs near the intersection of the two Fermi ‘circles’ showing that the inter surface coupling leads to a breaking away of the lens shaped region where the two circles overlap into separate EEC curves.[]{data-label="glambdaComb2"}](lambdaComb2.eps)
We now turn on the inter-surface coupling. Fig. \[glambdaComb2\] shows the dispersion relations and equal energy contours (EECs) at two values of $\lambda < |\vec{M}|$. At these small (relative to $|\vec{M}|$) values of $\lambda$ the two Dirac cones corresponding to the surface states localized at the upper and lower surfaces of the thin film are still distinctly evident. At low values of energy where the two cones do not overlap (panel (b) ), the EECs consist of two almost circular curves that correspond to the cross sections of the two Dirac cones. As the energy increases and the two almost-circular cross sections begin to almost touch each other, the inter-surface coupling pushes the EECs outwards in $k$-space so that | 1 | member_36 |
the cross sections link up with each other and form a single curve ( panel (c) ). A further increase in energy causes the the two Dirac cones overlap with each other the anti crossing of the energy levels due to the inter surface coupling causes the $k$ space lens-shaped region where the Dirac cones overlap to break away from the outer perimeter of the overlapping ‘circles’ and form a second closed curve. Despite the distortions of the EECs from the perfectly circular profiles in the absence of inter-surface coupling, the directions of the in-plane spin accumulation along the EECs in the presence of inter-surface coupling still roughly follow those of the original Dirac cones. Returning now to panel (a) of the figure, it is evident that as the inter-surface coupling increases, the energy of the lowest energy particle (hole) band at $\vec{k}=0$ increases (decreases).
![ Panel (a) shows the dispersion relations for progressively larger values of $\lambda$ indicated by the colors in the legend relative to the fixed value of $\vec{M} = 0.5\hat{y}$. Panels (b) and (c) show the EECs (in solid lines) and in-plane spin accumulation directions for $\lambda = 0.7$ at the two values of energies (0.5 | 1 | member_36 |
and 1.5 respectively) indicated by the horizontal dotted liens in panel (a). The two dotted circles in panels (b) and (c) are indicative of the Fermi circles for $\pm +(\vec{k}\times\vec{\sigma})\cdot\hat{z}$ Dirac cones which provide rough indications of the in-plane spin accumulation directions at the $k$ space points on the EECs. []{data-label="gbigLambdaComb1"}](bigLambdaComb1.eps)
Fig. \[gbigLambdaComb1\] shows the dispersion relations and the EECs as $\lambda$ increases further relative to $|\vec{M}|$. As $\lambda$ is increased from 0, the energy of the lowest energy particle band at $\vec{k}=0$ is pushed downwards and that of the highest energy hole band pushed upwards until the two bands touch each other when $\lambda = |\vec{M}|$. At this point ($\lambda=0.5$ in panel (a)) we no longer have two the well-resolved Dirac cones with two separate Dirac points in $\lambda = 0.2$ in panel (a) of the figure. A further increase in $\lambda$ leads to a bandgap opening up between the particle and hole bands. Panels (b) and (c) of the figure show the EECs at two values of energy for $\lambda > |\vec{M}|$. The in-plane spin accumulations at various $k$ space points can still be roughly understood as the spin accumulations of two overlapping circular cross sections of perfect | 1 | member_36 |
Dirac cones.
![ The EECs, in-plane spin accumulation directions and out of plane spin accumulation at two representative values of energy in the $\lambda < |\vec{M}|$ regime ( (a) and (b) ) and $\lambda > |\vec{M}|$ regime ( (c) and (d) ) for electric fields applied in the $x$ ( (a) and (c) ) and $y$ ( (b) and (d) ) directions. The sizes on the circles on the EECs are indicative of the magnitudes of the out of plane spin accumulations due to the electric field (the sizes of the circles are *not* scaled linearly to the spin accumulation magnitudes) with green (red) circles indicating out of plane spin accumulations in the negative (positive) $z$ directions. The two dotted circles in the left panels are indicative of the Fermi circles for $\pm +(\vec{k}\times\vec{\sigma})\cdot\hat{z}$ Dirac cones which provide rough indications of the in-plane spin accumulation directions at the $k$ space points on the EECs. $E=1.5R$ for all panels; $\lambda = 0.1, \vec{M} = \hat{y}$ for (a) and (b) and $\lambda = 0.7, \vec{M} = 0.5 \hat{y}$ for (c) and (d). []{data-label="gspinZpoln"}](spinZpoln.eps)
We now turn our attention to the out of plane spin accumulation generated by an electrical field which we | 1 | member_36 |
calculate using Eq. \[c2oe\] . Fig. \[gspinZpoln\] shows the out of plane spin $z$ accumulation generated at various $k$ space points on the EECs of a TI thin film with $\lambda > |\vec{M}|$ (panels (a) and (b) ), and $\lambda > |\vec{M}|$ (panels (c) and (d) ) for electrical fields applied in the $x$ ( panels (a) and (c) ) direction perpendicular to the magnetization, and the $y$ direction ( panels (c) and (d) ) parallel to the magnetization. The sign of the resulting spin $z$ accumulation can be understood in terms of how the applied electric field changes the direction of the SOI field experienced by the charge carriers. We noted in our earlier discussion in Sect. \[spinhalf\] that each point on the EECs may be associated with the Fermi circle of either the $+(\vec{k}\times\vec{\sigma})\cdot\hat{z}$ Dirac cone, or the $-(\vec{k}\times\vec{\sigma})\cdot\hat{z}$ cone. This is also indicated on the left panels of Fig. \[gspinZpoln\] where the two Fermi circles are indicated by dotted circles of different colors. Consider now the $k$ space region denoted in the inset of panel (b). The inset shows the spin accumulations on two points in $k$ space with the red (blue) arrows denoting the spin accumulation | 1 | member_36 |
direction for a point on the + (-) Fermi circle. The passage of an electric field in the $y$ direction causes $\langle p_y \rangle$ to increase while $\langle p_x \rangle$ remains constant, so that the SOI field $\pm (\vec{k}\times\hat{z})$ as well as the spin accumulation rotates in opposite directions for the $\pm$ Fermi circles. Reminiscent of our earlier discussion on spin 1/2 systems, this rotation in turn indicates the existence of an out of plane effective magnetic field which in turn imparts an out of plane spin accumulation. Applying the same argument to most of the other $k$-space points on the EECs in the figure explains the *sign* of the out of plane spin accumulation there. The *magnitude* of the spin $z$ accumulation depends on how much relative change in the SOI field direction the application of the electric field leads to. For example, in the right panels of the figure, the largest spin $z$ accumulation are on those EEC segments where the in-plane spin accumulation are in the $\pm y$ directions so that the small increment in the SOI field in the $\pm x$ directions due to the $y$ electric field is a large increment compared to other $k$ | 1 | member_36 |
space points on the EECs where the spin accumulations already have large $x$ components.
The out of plane spin $z$ accumulations in the preceding figures are antisymetrically distributed in $k$ space on the EECs. This antisymmetry results there being no net out of plane spin accumulation in $k$ space after integrating over the entire Fermi surface. In order to break the antisymmetry, we now introduce a term $E_z \tau_z$ to the Hamiltonian Eq. \[tfHam0\] so that the Hamiltonian now reads $$H = (\vec{k}\times\vec{\sigma})\cdot\hat{z}\tau_z + \vec{M}\cdot\vec{\sigma} + \lambda \tau_x + E_z\tau_z. \label{tfHam1}$$
The $E_z\tau_z$ term introduces an asymmetry between the upper and lower surfaces. This asymmetry may physically result from the fact that in experimentally grown TI thin films the bottom surface of the film is in contact with the usually non-ferromagnetic substrate, and the upper surface either in contact with the vacuum (for $\vec{M}\cdot\vec{\sigma}$ being due to magnetic doping [@PRL102_156603; @Sci329_659; @NatPhy7_32] ) or with a FM layer (for $\vec{M}\cdot\vec{\sigma}$ being due to the proximity effect with a FM layer [@PRL104_146802; @PRL110_186807; @PRB88_081407] ) .
![ Panel (a) and (b) show the EECs at various energies in the (a) absence and (b) presence of $E_z$ for $\lambda < |\vec{M}|$. Panels | 1 | member_36 |
(c) and (d) show the EECs at various energies in the (c) absence and (d) presence of $E_z$ for $\lambda > |\vec{M}|$. ( $\vec{M} = 0.5\ \hat{y}$, $\lambda = 0.2$ in (a) and (b); $\vec{M} = 0.2 \hat{y}$ in (c) and (d). $E_z = 0.1$ in (b) and (d). ) []{data-label="gEZeec"}](EZeec.eps)
Fig. \[gEZeec\] compares the EECs in both the $\lambda < |\vec{M}|$ regime as well as the $\lambda > |\vec{M}|$ regimes in the presence and absence of the $E_z \tau_z$ term. The asymmetry between the upper and lower surfaces of the TI film due to the $E_z\tau_z$ term results in the states stemming from the Dirac cones corresponding to the two surfaces being shifted in opposite directions along the energy axis. The dispersion relations become ‘tilted’, and the EECs at a given value of energy becoming asymmetrical in $k$ space.
This asymmetry then results in a net out of plane spin accumulation after integrating over all the $k$ space points spanned by the EECs. Evidently, the spin accumulation increases with the magnitude of $\vec{M}$ and $E_z$. What is perhaps more interesting is the variation of $\langle \sigma_z (E) \rangle $, the out of plane spin accumulation integrated over the EECs | 1 | member_36 |
---
abstract: 'We introduce a symmetrization technique which can be used as an extra step in some continuous-variable quantum key distribution protocols. By randomizing the data in phase space, one can dramatically simplify the security analysis of the protocols, in particular in the case of collective attacks. The main application of this procedure concerns protocols with postselection, for which security was established only against Gaussian attacks until now. Here, we prove that under some experimentally verifiable conditions, Gaussian attacks are optimal among all collective attacks.'
author:
- Anthony Leverrier
title: 'A symmetrization technique for continuous-variable quantum key distribution'
---
Quantum key distribution (QKD) is the art of distilling a secret key among distant parties, Alice and Bob, in an untrusted environment. The remarkable feature of QKD is that it is secure in an information theoretic sense [@SBC08]. QKD protocols come in two flavors depending on the type of quantum measurement they use: either a photon counting measurement for *discrete-variable* protocols or a homodyne detection for *continuous-variable* (CV) QKD. While the security of the former is now rather well understood (with the notable exceptions of the differential phase shift [@IWY02] and coherent one-way [@SBG05] protocols), security of CV protocols has been | 1 | member_37 |
more elusive (see [@WPG11] for a recent review). This is mainly due to the fact that the infinite dimensional Hilbert space required to describe these protocols makes the analysis quite challenging.
Among all CVQKD schemes, the protocol GG02 is certainly the easiest one to analyze [@GG02]. In this protocol, Alice sends $n$ coherent states $|\alpha_k\rangle = |x_{2k} + i x_{2k+1}\rangle$ to Bob who measures the states he receives either with a homodyne detection (thus randomly choosing one quadrature to measure for each state) or a heterodyne detection (in which case, Bob measures both quadratures at the same time). Alice’s modulation is Gaussian, meaning that $x_k$ is a centered normal random variable with a given variance. In the case of a heterodyne detection for instance [@WLB04], Bob obtains a classical vector ${\bf y} = [y_1,\cdots, y_{2n}]$ which is correlated to Alice’s vector ${\bf x}= [x_1,\cdots, x_{2n}]$. In this paper, we use bold font to refer to vectors. Then, using parameter estimation, reconciliation and privacy amplification, they can extract a secret key. For this specific protocol, Gaussian attacks (where the action of the eavesdropper can be modeled by a Gaussian quantum channel between Alice and Bob) are known to be optimal among | 1 | member_37 |
collective attacks [@GC06; @NGA06; @PBL08]. Using de Finetti theorem and conditioned upon an extra verification step [@RC09], these collective attacks are actually optimal in general in the asymptotic limit. The only step which is currently missing in this security analysis is a tight reduction from coherent to collective attacks in the finite-size regime [@LGG10].
The security status of other CVQKD protocols is far less advanced. In particular, not much is known for protocols using postselection [@SRL02; @LKL04; @LSS05]. In these protocols, the idea is that Alice and Bob will only use some data to distill the secret key and discard the rest. More precisely, they only keep the data compatible with a positive key rate. This method, inspired by advantage distillation techniques, certainly makes the protocol more robust against imperfections such as losses or noise in the channel, and potentially gives the best practical CVQKD protocol (see Refs. [@LSS05; @LRH06] for experimental implementations). It is therefore of considerable importance to be able to assess its security. Unfortunately, there are currently no full security proof for this scheme, not even against collective attacks. In fact, the only result available so far is an analysis in the case of Gaussian attacks [@HL07; | 1 | member_37 |
@SAA07]. This is, however, far from being sufficient for two reasons: first, Gaussian attacks are not believed to be optimal against this protocol; second, one can never prove in practice that a given quantum channel is indeed Gaussian. The problem is that the only tool currently available to establish the security of a CV protocol against collective attacks, namely Gaussian optimality [@WGC06] does not seem to help much for protocols with postselection (see, however, a recent approach along those lines in Ref. [@WSR11]).
In this paper, we introduce a new proof technique based on a symmetrization procedure that allows us to make some progress concerning the security analysis of CVQKD with postselection. In particular, we will show how this symmetrization allows us, under some verifiable conditions, to consider that the quantum channel is indeed Gaussian, even though the physical channel may actually be non-Gaussian. This then means that checking the security against Gaussian attacks (which can be done with present tools) is indeed sufficient to get full security against collective attacks, and in fact against arbitrary attacks in the asymptotic limit thanks to de Finetti theorem [@RC09].
A symmetrized protocol
======================
The usual technique to prove the security of a | 1 | member_37 |
Prepare and Measure (PM) protocol such as GG02 where Alice sends coherent states to Bob who measures them, is to consider an equivalent Entanglement-Based (EB) protocol. In the latter, Alice prepares two-mode squeezed vacuum states of which she measures one mode with a heterodyne detection and sends the second mode to Bob. Interestingly, before Alice and Bob measure their respective modes, their share a bipartite state $\rho_{AB}$. In this paper, we will restrict our attention to collective attacks, meaning that at the end of the protocol, Alice and Bob share $n$ copies of that state, that is, $\rho_{AB}^{\otimes n}$.
In general, one cannot perform a perfect tomography of this state, simply because it lives in an infinite dimensional Hilbert space. In the case of protocols without postselection, this is not a problem since the secret key rate can be safely computed from the the Gaussian state with the same first two moments as $\rho_{AB}$ [@GC06; @NGA06]. This is remarkable because one only needs to compute the covariance matrix of the state instead of its whole density matrix.
Unfortunately, this approach fails in the case of protocols with postselection. Indeed, one would then need to compute the covariance matrix of the | 1 | member_37 |
state *given it was postselected*. In principle, one could do this analysis with the experimental data obtained from the EB version of the protocol; but one cannot directly reconstruct this covariance matrix from the data observed in the actual PM version of the protocol. Indeed, the probabilistic map corresponding to a successful postselection is too complicated and one cannot expect to analyze its effect on general non-Gaussian states. For this reason, our only hope for a security proof seems to be to somehow enforce the Gaussianity of the state $\rho_{AB}$ Alice and Bob will use in their protocol. The idea is therefore to add an extra step to the usual protocol, that will make the state $\rho_{AB}^{\otimes n}$ more Gaussian. Let us note $\mathcal{S}$ the quantum map induced by this symmetrization.
It is now clear that if one had $\mathcal{S}\left(\rho_{AB}^{\otimes n}\right) = \rho_G^{\otimes n}$ where $\rho_G$ is the bipartite Gaussian state with the same covariance matrix as $\rho_{AB}$, then the security of the symmetrized protocol against general collective attacks would be identical to the security of the original protocol against Gaussian attacks. One could then compute the secret key rate, simply from the transmission and excess noise of the quantum | 1 | member_37 |
channel, exactly as in the case of protocols without postselection. The symmetrization we introduce below will not induce an exact Gaussification, but only an approximate one. However, the quality of the Gaussification, characterized by the fidelity between $\mathcal{S}\left(\rho_{AB}^{\otimes n}\right)$ and $\rho_G^{\otimes n}$ will increase with $n$, and tend to 1 if some experimentally verifiable conditions (on the moment of order 4 of $\rho_{AB}$) are met.
The symmetrization we consider here was introduced in [@LKG09] where it was argued that it corresponds to the natural symmetry for protocols using a Gaussian modulation of coherent states. In the EB scenario, before they both perform their heterodyne measurements, Alice and Bob would apply random conjugate passive linear transformations over their $n$ modes. Once this is done, they apply the usual postselection protocol. This symmetrization can also be used in the PM scenario, and crucially, one can simply apply it to the *classical* data ${\bf x}$ and ${\bf y}$ of Alice and Bob. More concretely, in the PM scenario, Alice and Bob follow the standard scenario of sending coherent states and performing a heterodyne detection for Bob. Then, Alice draws a random transformation with the Haar measure on the group $K(n):=O(2n,\mathbb{R}) \cap Sp(2n,\mathbb{R})$ (isomorphic | 1 | member_37 |
to the unitary group $U(n)$), that is the transformations corresponding to linear passive transformations in phase-space. She informs Bob of her choice of transformation (over the authenticated classical channel), and both parties apply this transformation to their respective $2n$-vectors ${\bf x}$ and ${\bf y}$.
Equivalence between the EB and PM symmetrized protocols
=======================================================
In order to study the security of the symmetrized PM protocol, one needs to show that its equivalent EB protocol corresponds to the one symmetrized through the application of random conjugate passive transformations in phase-space. It is useful to introduce three different distributions that can be used to describe the two scenarios. In order to simplify the exposition, let us first consider the analysis of a generic CVQKD protocol in the case of collective attacks, meaning that the protocol is entirely described by a single use of the quantum channel. First, the PM protocol is naturally described by a joint probability distribution $P(x_1, x_{2}, y_1, y_{2})$ where $x_1,x_2$ (resp. $y_1,y_2$) refers to Alice’s (resp. Bob’s) measurement results. The EB scenario is characterized by the bipartite state $\rho$ shared by Alice and Bob before their respective measurements. In the context of a CV protocol, it is natural to | 1 | member_37 |
describe this state by its Wigner function $W(x_1,x_2,y_1,y_2)$ where index 1 (resp. 2) refers to the first (resp. second) quadrature of Alice or Bob’s mode. Alternatively, since we restrict ourselves to protocols where both Alice and Bob perform a heterodyne detection, we can also consider the convenient characterization in terms of the $Q$-function $Q(x_1,x_2,y_1,y_2)$ of the state $\rho$. This $Q$-function corresponds to the probability distribution sampled from when measuring the state with heterodyne detection [@leo97] and is given by: $$Q(x_1, x_2, y_1,y_2) = \frac{1}{\pi^2} \langle \alpha_A, \alpha_B |\rho|\alpha_A, \alpha_B \rangle,$$ where $|\alpha_{A (B)}\rangle$ is a coherent state centered on $\alpha_{A(B)} = x_{A (B)} + i p_{A (B)}$ in Alice’s (Bob’s) phase space. Interestingly, if we denote $W_0$ the Wigner function of the vacuum, then $Q$ simply corresponds to the convolution of $W$ and $W_0$: $Q= W \star W_0$.
The relation between the two probability distributions $P$ and $Q$ can also be made explicit: if Alice measures one mode of the two-mode squeezed vacuum with a heterodyne detection and obtains outcomes $x_1$, $x_2$, she projects the second mode on the coherent state $|\gamma (x_1 - i x_2)\rangle$ where the factor $\gamma=\sqrt{2(V-1)/(V+1)}$ depends on the variance $V$ of the two-mode squeezed vacuum | 1 | member_37 |
[@GCW03][^1]. This means that we have the one-to-one relation: $$Q(x_1, x_2, y_1,y_2) = P(\gamma x_1, -\gamma x_2, y_1, y_2).$$
We now consider general $n$-mode states. Because of the correspondence above, applying a random transformation $R\otimes R$ with $R$ drawn with the Haar measure on $K(n)$ on the classical data represented by the distribution $P$ is equivalent to a symmetrization in phase-space corresponding to the application of random conjugate passive linear transformations over the $n$ modes of Alice and Bob. Noting $\mathcal{G}$ the group of passive linear transformations in phase-space, the state obtained after the symmetrization of $n$ copies of the state $\rho$ (i.e., for a collective attack) is: $$\label{symm_state}
\mathcal{S}(\rho^{\otimes n}) := \int_{U \in \mathcal{G}} (U \otimes U^*) \, \rho^{\otimes n} \, (U \otimes U^*)^\dagger \mathrm{d}U,$$ where $\mathrm{d} U$ is the Haar measure over the group $\mathcal{G}$.
Sketch of the security proof
============================
The rest of the paper consists in analyzing the state $\mathcal{S}(\rho^{\otimes n})$, and in particular proving that it becomes approximately Gaussian under conditions on the second and fourth moment of $P(x_1,x_2,y_1,y_2)$ which are usually met in practical implementations of a CVQKD protocol. Since the three distributions $W$, $Q$ and $P$, equivalently describe the protocol, we choose here | 1 | member_37 |
to work with $P$, which is the one directly observable in the practical implementation of the protocol.
Our proof will consist of two steps. First, we show that the distribution $P$ describing the state after the symmetrization tends to an explicit limiting function, where the convergence speed is $O(1/\sqrt{n})$. While one could in principle stop at this point and directly compute the secret key rate that can be extracted from this state, we will focus on the experimentally relevant scenario where the quantum channel behaves approximately as a Gaussian channel. In this case, we can bound the distance between the limiting distribution describing the whole protocol and a Gaussian identically and independently distributed (i.i.d.) function (thus corresponding to a collective Gaussian attack for which the key rate is already known) with an error term of order $O(1/\sqrt{n})$. In such a practical scenario, one can therefore bound the distance between the actual state and the state corresponding to a Gaussian attack by an arbitrary small quantity. Taking $n$ large enough, the secret key rate of the symmetrized protocol is therefore identical to the secret key rate against collective Gaussian attacks.
Convergence to a limiting distribution
======================================
To simplify the notations, we | 1 | member_37 |
write $\tilde{P}$ the distribution corresponding to the state $\mathcal{S}(\rho^{\otimes n})$. The following lemma from [@LG10b] (proven in Appendix \[proof-three-variables\] for completeness) shows that this function only depends on 3 variables, instead of $4n$ for the non-symmetrized scenario.
\[three-variables\] For $n \geq 2$, the symmetrized distribution $$\tilde{P}({\bf x}, {\bf y}) = \int_{K(n)} P(R{\bf x}, R{\bf y}) \mathrm{d} R,$$ where $\mathrm{d}R$ refers to the Haar measure on $K(n)$, only depends on $||{\bf x}||^2, ||{\bf y}||^2, {\bf x} \cdot {\bf y}$.
Let us note $X_i = x_i^2, Y_i = y_i^2, Z_i = x_i y_i$ and $X^n = \sum_{i=1}^n X_i, Y^n = \sum_{i=1}^n Y_i, Z^n = \sum_{i=1}^n Z_i$. Because $\tilde{P}({\bf x}, {\bf y})$ only depends on $X^n=||{\bf x}||^2$, $Y^n=||{\bf y}||^2$ and $Z^n={\bf x} \cdot {\bf y}$, it actually corresponds to the probability distribution $P_v$ of the vector $V^n=[X^n, Y^n, Z^n]$: $$\tilde{P}({\bf x}, {\bf y}) \mathrm{d}{\bf x} \mathrm{d}{\bf y} = P_v(V^n) \mathrm{d} V^n.$$
According to the central limit theorem, since the vectors $V_i$ are i.i.d. (which follows from the collective attack assumption), the distribution $P_v$ converges to a Gaussian distribution as $n$ tends to infinity, and one can use a multidimensional version of Berry-Esseen theorem to bound the distance between the two distributions. Noting $P_G$ | 1 | member_37 |
the Gaussian distribution with the same first two moments as $P_v$, one can prove that the variational distance $\Delta$ between the two distributions is of the form (see Appendix \[BE\] for a more precise bound): $$\Delta := \iiint_{\mathbb{R}^3} \left|P_v(V)-P_G(V)) \right| \mathrm{d}V= O\left(\frac{1}{\sqrt{n}}\right).$$ The scaling law in $O(1/\sqrt{n})$ is generic for Berry-Esseen theorem and the constant factor depends on the covariance matrix of $[X,Y,Z]$, that is, on the moment of order 4 of the measurement results $(x,y)$.
In general, one could compute the secret key rate corresponding to a state described by the distribution $P_G$. Here, we will restrict ourselves to a very concrete scenario, that of a quantum channel acting like a Gaussian channel (like an optical fiber typically [^2]). We insist that this is not a new assumption since one can always compute the covariance matrix $\Sigma$ of the distribution $P_G$ above. However, in general, the quantum channel will be such that $\Sigma$ will be close (up to sampling errors) to the covariance matrix obtained for a Gaussian channel.
The limiting distribution $P_G$ is close to an i.i.d. Gaussian distribution in typical implementations
======================================================================================================
In practice, the coherent states are sent through an optical fiber acting as a Gaussian | 1 | member_37 |
quantum channel, meaning that the data obtained by Alice and Bob follow a Gaussian distribution. In general, this does not simplify the security analysis since observing variables that *look* Gaussian does not mean that they indeed *are* Gaussian. Here, we will show that *looking* Gaussian is sufficient for the bound obtained through Berry-Esseen theorem to be useful. Let us consider the case where $$(x_i, y_i) \sim \mathcal{N}\left(
\begin{bmatrix}
0\\
0\\
\end{bmatrix},
\begin{bmatrix}
\langle x_i^2 \rangle & \langle x_i y_i \rangle \\
\langle x_i y_i \rangle & \langle y_i^2 \rangle \\
\end{bmatrix}
\right).$$ Then, applying the symmetrization and using the results of Berry-Esseen theorem, one obtains that the symmetrized (normalized) distribution $P_v$ tends to a Gaussian distribution with covariance matrix $\Sigma_G$: $$\Sigma_G = \left[
\begin{smallmatrix}
3\langle x_i^2 \rangle^2 & \langle x_i^2 \rangle \langle y_i^2 \rangle + 2 \langle x_i y_i \rangle^2 & 3 \langle x_i^2 \rangle \langle x_i y_i \rangle \\
\langle x_i^2 \rangle \langle y_i^2 \rangle + 2 \langle x_i y_i \rangle^2 & 3\langle y_i^2 \rangle^2& 3 \langle y_i^2 \rangle \langle x_i y_i \rangle\\
3 \langle x_i^2 \rangle \langle x_i y_i \rangle & 3 \langle y_i^2 \rangle \langle x_i y_i \rangle & \langle x_i^2 y_i^2 \rangle\\
\end{smallmatrix}\right].$$ Unfortunately, however, in | 1 | member_37 |
a practical protocol, Alice and Bob only have access to a finite-precision estimation of the covariance matrix, and the one they measure, $\Sigma_{\mathrm{est}}$, and that they use in Berry-Esseen theorem will slightly differ from the ideal one above $\Sigma_G$. The typical estimation error is of the order of $1/\sqrt{m}$ if $m$ samples are used in the procedure. Assuming that the estimation is performed with a (small) constant fraction of the total samples $n$, the typical error will be on the order of $1/\sqrt{n}$, which is comparable to the error term of the Berry-Esseen theorem. This then implies that the variational distance between the two final distributions is also of that order (see Appendix \[finite\] for details).
Security analysis of a CV QKD protocol with postselection
=========================================================
Using the results above, the distribution $P$ (or equivalently the $Q$-function) of the state describing the symmetrized version of the QKD protocol is $1/\sqrt{n}$-close in variational distance to a Gaussian distribution. Moreover, in a practical scenario, this Gaussian distribution corresponds to that of an i.i.d. Gaussian state. If one can make the error $1/\sqrt{n}$ small enough, then the security of the symmetrized protocol against collective attacks is equivalent to that of the usual (non-symmetrized) | 1 | member_37 |
protocol against Gaussian attacks. In particular, the secret key rate for the symmetrized protocol is equal to the secret key rate against Gaussian attacks [@HL07; @SAA07].
Although the variational distance between the $Q$-functions is a weaker criterion than the usual trace distance between the states, one can argue that this distance makes sense when considering CV QKD protocols. Indeed, if two states have $2\epsilon$-close $Q$-functions (for the variational distance), it means that the probability to successfully distinguish them using heterodyne detection is bounded by $1/2+\epsilon$.
Sampling from the Haar measure on $K(n)$ for $n$ large might be quite unpractical. Methods to achieve it in complexity $O(n^2)$ are known [@LG11; @JKL11]. Fortunately, for our purpose, it is sufficient to sample from the different measure on $K(n)$ provided that the symmetrized state can be made arbitrary close to the state $\mathcal{S}\left( \rho^{\otimes n}\right)$ of Eq. \[symm\_state\]. This can be achieved by means of quantum $k$-designs as presented in Appendix \[design\]. In particular, it is reasonable to conjecture that this can be done in complexity $O(n \log n)$, which would be compatible with a practical implementation.
One could wonder why the symmetrization has to be *active* here in contrast to protocols such as | 1 | member_37 |
GG02. The difference is that the postselection, performed along the quadrature axes, introduces some privileged directions in phase space. Consequently one needs to actively symmetrize the protocol to make it invariant under rotations in phase space.
Conclusion
==========
In this paper, we provided a first step towards a general security proof of CVQKD protocols with postselection. Until now, its security was only established in the very restricted case of Gaussian attacks, which are very unlikely to be optimal. Thanks to an active symmetrization of the protocol (performed on the classical data of Alice and Bob), one can show that collective attacks and actually arbitrary attacks in the asymptotic limit basically reduce to Gaussian attacks. The present solution is still not very practical since one would need to randomly sample from the unitary group in a very large dimension. Two possible approaches should be considered: either looking at a much smaller set of transformations for which the sampling can be performed efficiently, or improving the bounds derived here, possibly combining the symmetrization technique with some de Finetti-type arguments. It seems almost clear, at any rate, that the symmetrization technique introduced here will be required for any further advance in the study | 1 | member_37 |
of the security of CV QKD with postselection.
Acknowledgments
===============
I thank Antonio Acín, Frédéric Grosshans and Philippe Grangier for fruitful discussions. I acknowledge support from the European Union under the ERC Starting grant PERCENT.
Proof of Lemma $[1]$ (Lemma \[1\] from [@LG10b]) {#proof-three-variables}
================================================
Since the probability distribution $P$ is being randomized under the action of the group $K(n) = O(2n,\mathbb{R}) \cap Sp(2n,\mathbb{R})$ to give $\tilde{P}$ defined as $$\tilde{P}({\bf x}, {\bf y}) = \int_{K(n)} P(R{\bf x}, R{\bf y}) \mathrm{d} R$$ where $\mathrm{d}R$ refers to the Haar measure on $K(n)$, one has: $$\tilde{P}(R{\bf x}, R{\bf y}) = \tilde{P}({\bf x}, {\bf y})$$ for any ${\bf x}, {\bf y} \in \mathbb{R}^{2n}$ and $R \in K(n)$.
We want to show that any function $\tilde{P}: \mathbb{R}^{2n} \times \mathbb{R}^{2n}$, such that $\tilde{P}(R{\bf x}, R{\bf y}) = \tilde{P}({\bf x}, {\bf y})$ for any transformation $R \in K(n)$ only depends on the three following parameters: $||{\bf x}||^2, ||{\bf y}||^2, {\bf x} \cdot {\bf y}$.
Given any four vectors ${\bf x}, {\bf x'}, {\bf y}, {\bf y'} \in \mathbb{R}^{2n}$ such that $||{\bf x}||^2=||{\bf x'}||^2, ||{\bf y}||^2=||{\bf y'}||^2, {\bf x} \cdot {\bf y}={\bf x'} \cdot {\bf y'}$, it is sufficient to exhibit a transformation $R\in K(n)$ such that ${\bf | 1 | member_37 |
x'}=R {\bf x}$ and ${\bf y'}= R {\bf y}$ to prove Lemma $[1]$.
A transformation $R \in K(n)$ can be described as a symplectic map: $$\label{symplectic}
R = R(X,Y) \equiv
\begin{bmatrix}
X & Y \\
-Y & X \\
\end{bmatrix}$$ where the matrices $X$ and $Y$ are such that [@ADMS95]: $$\begin{aligned}
X^T X+Y^T Y &=& X X^T + Y Y^T =1\\
X^T Y &,& X Y^T \quad \mathrm{symmetric}.\end{aligned}$$ Note that this matrix is written for reordered vectors of the form $[x_1, x_3, \cdots, x_{2n-1},x_2, x_4, \cdots, x_{2n}]$, that is, one first writes the $n$ $q$-quadratures then the $n$ $p$-quadratures for all the vectors.
Let us introduce the following vectors ${\bf a}, {\bf a'}, {\bf b}, {\bf b'} \in \mathbb{C}^n$ defined as $$\begin{aligned}
a_k = x_{2k-1} + i x_{2k} &,& a'_k = x'_{2k-1}+ i x'_{2k}\\
b_k = y_{2k-1} + i y_{2k} &,& b'_k = y'_{2k-1}+ i y'_{2k}.\end{aligned}$$ Then, the conditions read: $$\label{condition2}
\left\{
\begin{array}{ccc}
||{\bf a}||^2=||{\bf a'}||^2\\
||{\bf b}||^2=||{\bf b'}||^2\\
\mathrm{Re}\langle {\bf a} | {\bf b} \rangle = \mathrm{Re}\langle {\bf a'} | {\bf b'} \rangle
\end{array}
\right. ,$$ where $\mathrm{Re}(x)$ refers to the real part of $x$.
For our purpose, it is therefore sufficient to prove that there exists an unitary | 1 | member_37 |
transformation $U \in U(n)$ such that $U {\bf a} ={\bf a'}$ and $U {\bf b} = {\bf b'}$. Indeed, one can split $U$ into real and imaginary parts: $U = X - iY$, and it is easy to check that $R=R(X,Y)$ is such that $R{\bf x} ={\bf x'}$ and $R{\bf y} = {\bf y'}$.
Let us introduce the following notations: $A \equiv ||{\bf a}||^2=||{\bf a'}||^2, B \equiv ||{\bf b}||^2=||{\bf b'}||^2$ and $C \equiv \mathrm{Re}\langle {\bf a}| {\bf b} \rangle = \mathrm{Re}\langle {\bf a'}| {\bf b'} \rangle$.
Consider first the case where ${\bf a}$ and ${\bf b}$ are colinear. This means that ${\bf b} = C/A {\bf a}$ and $C = \pm \sqrt{AB}$. Using the Cauchy-Schwarz inequality, $|C| = |{\bf a'} \cdot {\bf b'}| \leq ||{\bf a'}|| \cdot||{\bf b'}|| = \sqrt{AB}$ with equality if and only if ${\bf a'}$ and ${\bf b'}$ are colinear. This means that ${\bf a'}$ and ${\bf b'}$ are colinear and that ${\bf b'} = (C/A) \, {\bf a'}$. Because $||{\bf a}|| = ||{\bf a'}||$, the reflexion $U$ across the mediator hyperplane of ${\bf a}$ and ${\bf a'}$ is a unitary transformation that maps ${\bf a}$ to ${\bf a'}$. This reflexion also maps ${\bf b}$ to ${\bf | 1 | member_37 |
b'}$. This ends the proof in the case where ${\bf a}$ and ${\bf b}$ are colinear.
Let us now consider the general case where ${\bf a}$ and ${\bf b}$ are not colinear. It is clear that ${\bf a'}$ and ${\bf b'}$ cannot be colinear either. We take two bases $({\bf a}, {\bf b}, {\bf f_3}, \cdots, {\bf f_n})$ and $({\bf a'}, {\bf b'}, {\bf f_3'}, \cdots, {\bf f_n'})$ of $\mathbbm{C}^n$ and use the Gram-Schmidt process to obtain two orthonormal bases $\mathcal{B}=({\bf e_1}, \cdots, {\bf e_n})$ and $\mathcal{B}'=({\bf e_1'}, \cdots, {\bf e_n'})$. Note that vectors ${\bf e_1}, {\bf e_2}, {\bf e_1'}$ and $e{\bf _2'}$ are given by: $$\begin{aligned}
{\bf e_1} = \frac{{\bf a}}{\sqrt{A}} &,& {\bf e_2} = \frac{{\bf b} - \langle {\bf e_1}|{\bf b}\rangle {\bf e_1}}{||{\bf b} - \langle {\bf e_1}|{\bf b}\rangle {\bf e_1} ||}\\
{\bf e_1'} = \frac{{\bf a'}}{\sqrt{A}} &,& {\bf e_2'} = \frac{{\bf b'} - \langle {\bf e_1'}|{\bf b'}\rangle {\bf e_1'}}{||{\bf b'} - \langle {\bf e_1'}|{\bf b}\rangle {\bf e_1'} ||}.\end{aligned}$$ Let us call $U$ the unitary operator mapping $\mathcal{B}$ to $\mathcal{B}'$. It is easy to see that $U$ maps ${\bf a}$ and ${\bf b}$ to ${\bf a'}$ and ${\bf b'}$, respectively. This concludes our proof.
Distance between the symmetrized | 1 | member_37 |
distribution and a Gaussian distribution: Berry-Esseen theorem {#BE}
===============================================================================================
We use a multidimensional local version of the Berry-Esseen theorem due to Zitikis. More precisely, Theorem 1.2 of [@Zit93] reads:
\[berry-esseen\] Let $V_1, \cdots, V_n$ be a sequence of independent and identically distributed $d$-variate random vectors, let $S_n = \tfrac1n \sum_{i=1}^n V_i$. Let ${\bf \mu}$ and $\Sigma$ be the first two moments of $V_1$, and let $\lambda_{\mathrm{min}}$ be the least eigenvalue of $\Sigma$. Let $G$ be a Gaussian random vector with zero mean and covariance matrix $\mathbbm{1}_d$. Let $\mathcal{C}$ be the class of all convex Borel sets. Then there exists a universal constant $c\geq 0$ such that $$\begin{gathered}
\sup_{C \in \mathcal{C}} \left|P(\sqrt{n} (S_n-\mu)\Sigma^{-1/2}\in \mathcal{C}) - P(G \in \mathcal{C}) \right|\\
\leq c \sqrt{d} \lambda_{\mathrm{min}}^{-3/2}\mathbb{E}\left(||V_1||^3\right)/\sqrt{n},\end{gathered}$$ where $L_n(1)=1/n \sum_i V_i$, $V_i$ is a $d$-dimensional vector and $||.||$ refers to the Euclidean norm.
First, we note that the quantity $$\sup_{C \in \mathcal{C}} \left|P({\bf x} \in \mathcal{C}) - P({\bf y} \in \mathcal{C}) \right|$$ corresponds to the variational distance between the distributions of ${\bf x}$ and ${\bf y}$.
In the case of CVQKD, we need to consider tridimensional random vectors $V_i=[X_i,Y_i,Z_i]$ and one can immediately estimate ${\bf \mu}$ and $\Sigma$ from the experimental data. Using the notations | 1 | member_37 |
of the main text and applying Theorem \[berry-esseen\] gives $$\begin{gathered}
\label{bound}
\iiint_{\mathbb{R}^3} \left|P_v(V)-P_G(V)) \right| \mathrm{d}V\\
\leq c \sqrt{3} \lambda_{\mathrm{min}}^{-3/2}\mathbb{E}\left(||X_1^2+Y_1^2+Z_1^2||^{3/2}\right)/\sqrt{n},\end{gathered}$$ where $P_G$ corresponds to the multivariate Gaussian distribution with the same first two moments as $P_v$.
Error due to the finite estimation sample {#finite}
=========================================
Let us assume that the variables $(x_k,y_k)$ follow a centered bivariate normal distribution, as typical in experimental implementations of CVQKD.
During the parameter estimation, Alice and Bob need to estimate the fourth moments of $(x_k,y_k)$ given by the covariance matrix $\Sigma$: $$\Sigma = \left[
\begin{smallmatrix}
\langle x_i^4 \rangle & \langle x_i^2 y_i^2 \rangle & \langle x_i^3 y_i \rangle \\
\langle x_i^2 y_i^2 \rangle & \langle y_i^4 \rangle & \langle x_i y_i^3 \rangle\\
\langle x_i^3 y_i \rangle & \langle x_i y_i^3 \rangle & \langle x_i^2 y_i^2 \rangle\\
\end{smallmatrix}\right]$$ Alternatively, one can describe this matrix by the vector $V_i = [\langle x_i^4 \rangle, \langle x_i^3 y_i \rangle, \langle x_i^2 y_i^2 \rangle, \langle x_i y_i^3 \rangle, \langle y_i^4 \rangle]$. In order to estimate this vector, one uses the estimator $\bar{V}^m$ defined as $$\bar{V}^m:= \frac{1}{\sqrt{m}} \sum_{i_1,\cdots, i_m} V_i,$$ where $i_1,\cdots, i_m$ are $m$ randomly chosen indices among $\{1,\cdots,n\}$. Using the mutlivariate version of the Central Limit Theorem, one obtains that | 1 | member_37 |
the estimator $\bar{V}^m$ converges to the true value and that the error follows a normal distribution.
More precisely, let us note $\Sigma_G$ the true covariance matrix and $\Sigma_{\mathrm{est}}$ the estimated matrix, i.e., $$\Sigma_G = \left[
\begin{smallmatrix}
3\langle x_i^2 \rangle^2 & \langle x_i^2 \rangle \langle y_i^2 \rangle + 2 \langle x_i y_i \rangle^2 & 3 \langle x_i^2 \rangle \langle x_i y_i \rangle \\
\langle x_i^2 \rangle \langle y_i^2 \rangle + 2 \langle x_i y_i \rangle^2 & 3\langle y_i^2 \rangle^2& 3 \langle y_i^2 \rangle \langle x_i y_i \rangle\\
3 \langle x_i^2 \rangle \langle x_i y_i \rangle & 3 \langle y_i^2 \rangle \langle x_i y_i \rangle & \langle x_i^2 y_i^2 \rangle\\
\end{smallmatrix}\right],$$ and $$\Sigma_\mathrm{est} = \frac{1}{m} \sum_{i_1, \cdots, i_m}\left[
\begin{smallmatrix}
x_i^4 & x_i^2 y_i^2 & x_i^3 y_i \\
x_i^2 y_i^2 & y_i^4 & x_i y_i^3 \\
x_i^3 y_i & x_i y_i^3 & x_i^2 y_i^2\\
\end{smallmatrix}\right].$$ Then, the Central Limit Theorem asserts that the random matrix $\sqrt{m} (\Sigma_\mathrm{est}-\Sigma_G)$ converges in distribution to a centered multivariate normal distribution with a covariance matrix depending on moments of order 8 of $(x_k,y_k)$. We do not explicitate this matrix here as it is rather cumbersome, but it is straightforward to compute it.
The result of this analysis is | 1 | member_37 |
that the error in estimating the correct covariance matrix $\Sigma_g$ scales as $1/\sqrt{m}$ where $m$ is the number of samples used. In order to obtain an error of the same order of magnitude as the one due to Berry-Esseen theorem, one should use a constant fraction of the data for the parameter estimation. In that case, the covariance matrix would be estimated with a precision $1/\sqrt{n}$.
We now prove that this error for the covariance matrices translates into an error (computed with respect to the variational distance) of the same magnitude for the probability distributions.
Let us first consider the case of univariate normal distributions, that is two normal distributions $\mathcal{N}(0,\sigma_1^2)$ and $\mathcal{N}(0,\sigma_2^2)$ with $\sigma_2 > \sigma_1$. Let us note $g_1$ and $g_2$ their respective density function. One has: $$\begin{aligned}
&& \int_{-\infty}^{\infty} |g_1(x) - g_2(x)| \mathrm{d}x \nonumber \\
&&= 2 \mathrm{erf} \left(\sigma_2 \sqrt{\frac{\ln (\sigma_2/\sigma_1)}{\sigma_2^2-\sigma_1^2}} \right)-2 \mathrm{erf} \left(\sigma_1 \sqrt{\frac{\ln (\sigma_2/\sigma_1)}{\sigma_2^2-\sigma_1^2}} \right) \nonumber\end{aligned}$$ where $\mathrm{erf}(x) = 2/\sqrt{\pi} \int_0^x e^{-t^2}\mathrm{d}t$ is the error function. In particular, if one has $\sigma_2 = \sigma_1 + \delta$ where $\delta=O(1/\sqrt{n})$ is a small error, then a first order expansion of the expression above gives $$\int_{-\infty}^{\infty} |g_1(x) - g_2(x)| \mathrm{d}x = \delta \sqrt{\frac{8}{e \pi \sigma_1^2}} + o\left(\frac{1}{\sqrt{n}}\right).$$
One | 1 | member_37 |
---
abstract: 'Let $X$ be a Calabi–Yau threefold fibred over ${{\mathbb P}}^1$ by non-constant semi-stable K3 surfaces and reaching the Arakelov–Yau bound. In \[STZ\], X. Sun, Sh.-L. Tan, and K. Zuo proved that $X$ is modular in a certain sense. In particular, the base curve is a modular curve. In their result they distinguish the rigid and the non-rigid cases. In \[SY\] and \[Y\] rigid examples were constructed. In this paper we construct explicit examples in non-rigid cases. Moreover, we prove for our threefolds that the “interesting” part of their $L$-series is attached to an automorphic form, and hence that they are modular in yet another sense.'
address:
- 'Institute of Mathematics, Hebrew University of Jerusalem, Givat-Ram, Jerusalem 91904, Israel'
- 'Department of Mathematics, Queen’s University, Kingston. Ontario Canada K7L 3N6'
author:
- Ron Livné
- Noriko Yui
date: 'Version of June 16, 2005'
title: 'The modularity of certain non-rigid Calabi–Yau threefolds'
---
\[section\] \[section\] \[section\] \[section\] \[section\] \[section\] \[section\] \[section\] \[section\] \[section\] \[section\]
[^1]
[^2]
Introduction
============
Let $X$ be an algebraic threefold and let $f:X\to
{{\mathbb P}}^1$ be a non-isotrivial morphism whose fibers are semi-stable K3 surfaces. Let $S\subset {{\mathbb P}}^1$ be the finite set of points above | 1 | member_38 |
which $f$ is non-smooth, and assume that the monodromy at each point of $S$ is non-trivial. Jost and Zuo \[JZ\] proved the Arakelov–Yau type inequality: $$\mbox{deg} f_* \omega_{X/{{\mathbb P}}^1}\leq \mbox{deg}\,
\Omega_{{{\mathbb P}}^1}^1 (\mbox{log} S).$$
Let $\Delta\subset X$ be the pull-back of $S$. Let $\omega_{X/{{{\mathbb P}}}^1}$ be the canonical sheaf. The Kodaira–Spencer maps $\theta^{2,0}$ and $\theta^{1,1}$ are maps of sheaves fitting into the following diagram: $$f_*\Omega^2_{X/{{{\mathbb P}}}^1}(\mbox{log}\,\Delta)
\overset{\theta^{2,0}}\to
R^1f_*\Omega^1_{X/{{{\mathbb P}}}^1}(\mbox{log}\,\Delta)
\otimes \Omega^1_{{{{\mathbb P}}}^1}(\mbox{log}\,S)
\overset{\theta^{1,1}}\to R^2\,
f_*{{\mathcal O}}_{X/{{{\mathbb P}}}^1}\otimes\Omega^1_{{{{\mathbb P}}}^1}
(\mbox{log}\,S)^{{\otimes}2}.$$ The iterated Kodaira–Spencer map of $f$ is defined to be the map $\theta^{1,1}\theta^{2,0}$.
It is known (see \[STZ\]) that when the (iterated) Kodaira–Spencer map is $0$, one actually has the stronger inequality $$\mbox{deg} f_* \omega_{X/{{\mathbb P}}^1}\leq
\frac{1}{2}\mbox{deg}\,
\Omega_{{{\mathbb P}}^1}^1 (\mbox{log} S).$$
Assume from now on that $X$ is a Calabi–Yau threefold. Then the triviality of the canonical bundle implies that $\mbox{deg}\, f_*
\omega_{X/{{\mathbb P}}^1}=2$ (see (\[deg\]) below).
Recently X. Sun, S-L. Tan and K. Zuo \[STZ\] considered Calabi–Yau threefolds for which the Arakelov–Yau inequality becomes equality. Thus $S$ consists of $6$ points when the Kodaira–Spencer map is $0$ and of $4$ points otherwise.
As a consequence of the main theorem of \[STZ\], the following results were obtained.
[**Theorem 1**]{} (Corollary 0.4 | 1 | member_38 |
in \[STZ\]):
*(i) If the iterated Kodaira–Spencer map $\theta^{1,1}\theta^{2,0}$ of $f$ is non-zero, then $X$ is rigid (i.e., $h^{2,1}=0$) and birational to the Nikulin–Kummer construction of a symmetric square of a family of elliptic curves $f: E\to {{\mathbb P}}^1$. After passing to a double cover $E^{\prime}\to E$ (if necessary), the family $g^{\prime}: E^{\prime}\to
{{\mathbb P}}^1$ is one of the six modular families of elliptic curves on the Beauville’s list (\[B\]).*
\(ii) If the iterated Kodaira–Spencer map $\theta^{1,1}\theta^{2,0}$ of $f$ is zero, then $X$ is a non-rigid Calabi–Yau threefold (i.e., $h^{2,1}\neq 0$), the general fibers have Picard number at least $18$, and ${{\mathbb P}}^1\setminus
S\simeq \mathfrak H/\Gamma$ where $\Gamma$ is a congruence subgroup of $PSL(2,{{\mathbb Z}})$ of index $24$.
[**Remark**]{}: The base curve ${{\mathbb P}}^1\setminus S$ is a modular variety of genus zero, i.e., $\mathfrak H/\Gamma$ where $\Gamma$ is a torsion-free genus zero congruence subgroup of $PSL(2,{{\mathbb Z}})$ of index $12$ in case (i), and of index $24$ in case (ii). In the paper of Sun, Tan and Zuo \[STZ\], the word “modularity” refers to this fact.
The third cohomology of each of the six rigid Calabi–Yau threefolds in Theorem 1 (i) arises from a weight $4$ modular form. In the | 1 | member_38 |
articles of Saito and Yui \[SY\] and of Verrill in Yui \[Y\], these forms were explicitly determined. Saito and Yui use geometric structures; while Verrill uses point counting method, to obtain the results.
More precisely, the following was proved for the natural models over ${{\mathbb Q}}$ of these six rigid threefolds:
[**Theorem 2**]{} (Saito and Yui \[SY\] and Verrill in Yui \[Y\]): [*For each of the six rigid Calabi–Yau threefold over ${{\mathbb Q}}$, the L-series of the third cohomology coincides with the L-series arising from the cusp form of weight $4$ of one variable on the modular group in the Beauville’s list. Beauville’s list and the corresponding cusp forms are given in Table 1.*]{} $$\begin{array}{|c|c|c|} \hline \hline
\mbox{Group} & \mbox{Number of components} &
\mbox{Cusp forms} \\
\Gamma & \mbox{of singular fibers} &
\mbox{of weight $4$} \\ \hline
\Gamma(3) & 3,3,3,3 &
\eta(q^3)^8 \\ \hline
\Gamma_1(4)\cap \Gamma(2) & 4,4,2,2 &
\eta(q^2)^4\eta(q^4)^4 \\ \hline
\Gamma_1(5) & 5,5,1,1 &
\eta(q)^4\eta(q^5)^4 \\ \hline
\Gamma_1(6) & 6,3,2,1 &
\eta(q)^2\eta(q^2)^2\eta(q^3)^2\eta(q^6)^2 \\
\hline
\Gamma_0(8)\cap\Gamma_1(4) & 8,2,1,1&
\eta(q^4)^{16}\eta(q^2)^{-4}\eta(q^8)^{-4} \\
\hline
\Gamma_0(9)\cap\Gamma_1(3) & 9,1,1,1 &
\eta(q^3)^8 \\ \hline \hline
\end{array}$$
Table 1: Rigid Calabi–Yau threefolds and cusp forms
Here $\eta(q)$ denotes the Dedekind eta-function: $\eta(q)=q^{1/24}\prod_{n\geq 1}(1-q^n)$ with $q=e^{2\pi i\tau}$.
| 1 | member_38 |
It might be helpful to recall the six rigid Calabi–Yau threefolds in Theorem 2. These six rigid Calabi–Yau threefolds are obtained (by Schoen \[S\]; see also Beauville \[B\]) as the self-fiber products of stable families of elliptic curves admitting exactly four singular fibers. The base curve is a rational modular curve and correspond to the torsion-free genus zero congruence subgroups $\Gamma$ of $PSL(2,{{\mathbb Z}})$ in Table 1. Note that the $4$-tuples of natural numbers appearing in the second column add up to $12$, which is the index of the modular group $\Gamma$ in $PSL(2,{{\mathbb Z}})$.
In \[STZ\] the authors indicate one example for the non-rigid extremal case. It is related to $\Gamma(4)$, which is a torsion-free congruence subgroup of genus $0$ and index $24$ in $PSL(2,{{\mathbb Z}})$. The list of torsion-free congruence subgroups of genus $0$ and index 24 in $PSL(2,{{\mathbb Z}})$ is known (see Sebbar \[Se\], and Table 2 below). In this paper we will show that most of them give rise to a similar collection of examples. In each of these cases we will compute the interesting part of the $L$-series of the third cohomology of an appropriate natural model over ${{\mathbb Q}}$ in terms of automorphic forms.
| 1 | member_38 |
This paper is organized as follows. In Section 2, we use work of Sebbar \[Se\] to determine the groups $\Gamma$ corresponding to case (ii) of Theorem 2. These are subgroups of $PSL(2,{{\mathbb Z}})$, and associated to each $PSL(2,{{\mathbb Z}})$-conjugacy class there is a natural elliptic fibration over the base curve, defined over ${{\mathbb Q}}$. The total spaces of these fibrations are elliptic modular surfaces in the sense of Shioda \[Sh1\]. Moreover each is an extremal K3 surface (namely their Picard number is 20, the maximum possible). We explain the relation between the motive of their transcendental cycles, and specific CM forms of weight $3$ using a result of Livné on orthogonal rank 2 motives in \[L2\].
Section 3 contains our main results: we construct our examples, verify the required properties, and analyze the interesting part of their cohomology. (See the final Remarks 8 (2) for the other parts.) Then in Section 4 we give explicit formulas for the weight $3$ cusp forms and defining equations for the elliptic fibrations of Section 2.
The paper is supplemented by the article of Hulek and Verrill \[HV\] where they treat Kummar varieties, one of which is the case associated to the modular group | 1 | member_38 |
$\Gamma_1(7)$. This case differs from the examples considered in our paper with the main difference being the fact that the $2$-torsion points do not decompose into four sections, leading to non-semi-stable fibrations. But it still gives rise to a Calabi–Yau threefold (Theorem 2.2 of Hulek and Verrill \[HV\]), and the modularity question can still be considered, and this is exactly what Hulek and Verrill deals with in the supplement \[HV\] to this article.
Extremal congruence K3 surfaces
===============================
The torsion-free genus zero congruence subgroups of $PSL(2,{{\mathbb Z}})$ of index $24$ were classified in Sebbar \[Se\]. There are precisely nine conjugacy classes of such congruence subgroups.
The second column in the following Table 2 gives the complete list of the torsion-free congruence subgroups of $PSL(2,{{\mathbb Z}})$ of index $24$ up to conjugacy. Each has precisely 6 cusps. The third column in the table gives the widths of these cusps.
$$\begin{array}{|c|c|c|} \hline\hline
\#& \mbox{The group $\Gamma$} &
\mbox{Widths of the cusps} \\
\hline
1& \Gamma(4) & 4,4,4,4,4,4 \\ \hline
2& \Gamma_0(3)\cap \Gamma(2) & 6,6,6,2,2,2 \\
\hline
3& \Gamma_1(7) & 7,7,7,1,1,1 \\ \hline
4& \Gamma_1(8) & 8,8,4,2,1,1 \\ \hline
5& \Gamma_0(8)\cap \Gamma(2) & 8,8,2,2,2,2 \\
\hline
6& \Gamma_1(8;4,1,2) & 8,4,4,4,2,2 \\ \hline
7& | 1 | member_38 |
\Gamma_0(12) & 12,4,3,3,1,1 \\ \hline
8& \Gamma_0(16) & 16,4,1,1,1,1 \\ \hline
9& \Gamma_1(16;16,2,2) & 16,2,2,2,1,1 \\
\hline \hline
\end{array}$$
Table 2: Torsion-free congruence subgroups of index 24.
Here $$\Gamma_1(8;4,1,2):=\{\pm
\begin{pmatrix} 1+4a & 2b \\ 4c & 1+4d
\end{pmatrix},\, a\equiv c\pmod 2\}$$ and $$\Gamma_1(16;16,2,2):=\{\pm \begin{pmatrix}
1+4a & b \\
8c & 1+4d \end{pmatrix}, \, a\equiv c\pmod 2\}.$$ : [*If we are interested in conjugacy as Fuchsian groups (in $PSL(2,{{\mathbb R}})$), Examples \#1,\#5, and \#8 are conjugate (use the matrix $\begin{pmatrix} 1 & 0
\\ 0 & 2
\end{pmatrix})$, Examples \#2 and \#7 are conjugate (use the same matrix), and Examples \#4, \#6, and \#9 are conjugate (use the same matrix as well as $\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix})$.*]{}
[**Proposition 4**]{}:
*Let $\Gamma$ be one of the congruence subgroups in Table 2. Then $\Gamma$ has an explicit congruence lift $\tilde{\Gamma}$ to $SL(2,{{\mathbb Z}})$ with the following properties:*
\(1) $\tilde{\Gamma}$ has no elliptic elements. In particular $-{\rm Id}$ is not in $\tilde{\Gamma}$.
\(2) $\tilde{\Gamma}$ contains no element of trace $-2$.
[**Proof**]{}: We let $\tilde{\Gamma}$ be the subgroup of $SL(2,{{\mathbb Z}})$ consisting of the elements above $\Gamma$. Indeed, the lifts $\tilde{\Gamma}$ are sometimes the same as the groups $\Gamma$ themselves. | 1 | member_38 |
In fact, for the cases \#1,\#2,\#3, \#4 and \#5, the lifts are the same and respectively given by: $\Gamma(4)$, $\Gamma_0(3)\cap\Gamma(2)$,$\Gamma_1(7)$,$\Gamma_1(8)$ and $\Gamma_0(8)\cap\Gamma(2)$. In the cases \#6,\#7,\#8 and \#9, the lifts are not unique, and we choose respectively the following lifts: $\Gamma_1^{\prime}(8;4,1,2)$, $\Gamma_0(12)\cap\Gamma_1(3)$, $\Gamma_0(16)\cap\Gamma_1(4)$ and $\Gamma_1^{\prime}(16;16,2,2)$. Here $\Gamma_1^{\prime}(8;4,1,2)$ and $\Gamma_1^{\prime}(16;16,2,2)$ are defined in the same way as $\Gamma_1(8;4,1,2)$ and $\Gamma_1(16;16,2,2)$ but without the $\pm$. Note that the widths of the cusps are not affected by taking a lift as $-{\rm Id}$ is the only difference.
(We should remark that the lifts are not unique; for instacne, in the case \#7, there are four lifts, but only one has no elements of trace $-2$, which is the one given above.)
In fact, Proposition 4 has also been obtained by A. Sebbar in his unpublished note.
Proposition 4 will pave a way to the definition of elliptic modular surfaces, which we will discuss next.
[**Elliptic Modular Surfaces**]{}: In \[Sh1\] Shioda has shown how to associate to any subgroup $G$ of $SL(2,{{\mathbb Z}})$ of finite index and not containing $-$Id an elliptic fibration $E(G)$, called the elliptic modular surface associated to $G$, over the modular curve $X(G)=\overline{G\backslash{\frak{H}}}$.
(Shioda’s construction requires that modular groups ought | 1 | member_38 |
to be a subgroup of $SL(2,{{\mathbb Z}})$ (rather than a subgroup of $PSL(2,{{\mathbb Z}})$) that contains no element of order $2$. This is a reason we consider a lift $\tilde{\Gamma}$ of $\Gamma$ in our discussion.)
[**Remark 5 **]{}: [*It follows from Kodaira’s theory that when $G$ contains no elliptic elements and no elements of trace $-$2, all the singular fibers are above the cusps and are of type $I_n$, where $n$ is the width of the cusp. On the other hand, elements of trace $-2$ give rise to $I_n^*$-fibers above the cusps.*]{}
For the $\tilde{\Gamma}$’s of Proposition 4 the modular curve $X(\tilde{\Gamma})$ has genus $0$, and, since the sum of the widths of the cusps in Table 2 is always 24, each $E(\tilde{\Gamma})$ is an extremal K3 surface. The space $S_3(\tilde{\Gamma})$ of cusp forms of weight 3 for $\tilde{\Gamma}$ is therefore one-dimensional. Up to a square, the discriminant of the intersection form on the rank 2 motive of the transcendental cycles $T: =
T(E(\tilde{\Gamma})) =
H^2(E(\tilde{\Gamma}),{{\mathbb Q}})/{\rm NS}
(E(\tilde{\Gamma}))$ is given by
$$\label{delta}
\delta = \delta_k = -1,-3,-7,-2,-1,-2,-3,-1,-2$$
in cases $k = 1,\dots,9$ respectively. To see this one computes the discriminant of the (known) lattice $NS(E(\tilde{\Gamma}))$ and passes to the | 1 | member_38 |
orthogonal complement in $H^2(E(\tilde{\Gamma}),{{\mathbb Q}})$. For details, see e. g. Besser-Livné \[BS\]. By \[L2\] it follows that the normalized newform $g_{3,\Gamma}$ generating $S_3(\tilde{\Gamma})$ has CM by ${{\mathbb Q}}(\sqrt{\delta}\,)$.
To each of our 9 examples there is a naturally associated moduli problem of classifying (generalized) elliptic curves with a given level structure (Katz and Mazur \[KM\]). Each of these moduli problems refines the respective moduli problem $Y_1(M)$, of classifying elliptic curves with a point of order $M$, where $M$ is as above. Since $M\geq 4$, these moduli problems are all represented by universal families $E(\tilde{\Gamma})/X(\tilde{\Gamma})/{{\mathbb Z}}[1/M]$ (see Katz–Mazur \[KM\]). The geometric fibers are geometrically connected in all these examples, and their compactified fibers over ${{\mathbb C}}$ are the corresponding elliptic modular surfaces above.
We shall now compute the $L$-series $L(T,s)$ of the transcendental cycles. By the Eichler–Shimura Isomorphism, this is the parabolic cohomology $$\tilde{H}: =
\tilde{H}{\vphantom{H}}^1(
X(\tilde{\Gamma})\times_{{{\mathbb Z}}[1/M]}\overline{{{\mathbb Q}}},
R^1(E(\tilde{\Gamma})\rightarrow
X(\tilde{\Gamma}))).$$ Moreover, Deligne proved (\[D\]) the Eichler–Shimura congruence relation $$\text{Frob}_p + \text{Frob}'_p = T_p
\qquad\text{for any $p\not|M$},$$ where $T_p$ is the $p$-th Hecke operator on $S_3(\tilde{\Gamma})$. This is the same as the $p$-th Fourier coefficient of the normalized newform $g_{3,\Gamma}$. Summarizing, we proved $$L(T(X(\tilde{\Gamma}),s) = L(g_{3,\Gamma},s).$$
Explicit Weierstrass equations for the | 1 | member_38 |
elliptic fibrations $E(\tilde{\Gamma})/X(\tilde{\Gamma})/{{\mathbb Z}}[1/M]$ will be given in Section 4 below.
[**Remarks**]{}:
*The list in Table 2 exhausts all of the families of semi-stable elliptic $K3$ surfaces with exactly six singular fibers, which correspond to torsion-free genus zero [*congruence*]{} subgroups of $PSL_2({{\mathbb Z}})$ of index $24$. The $6$-tuples of natural numbers appearing in the third column add up to $24$. Therefore, the number of such $6$-tuples is a priori finite. That this list is complete was proved by Sebbar \[Se\].*
\(2) Miranda and Persson \[MP\] studied all possible configurations of $I_n$ fibers on elliptic $K3$ surfaces. In the case of exactly six singular fibers, they obtained $112$ possible configurations including the above nine cases. All these $K3$ surfaces have the maximal possible Picard number $20$. It should be emphasized that the exactly nine configurations correspond to genus zero congruence subgroups of $PSL_2({{\mathbb Z}})$ of index $24$.
\(3) The theory of Miranda and Persson had been extended to prove the uniqueness (over ${{\mathbb C}}$) of K3 surfaces having each of these types of singular fibers by Artal-Bartolo, Tokunaga and Zhang \[BHZ\]. Confer also the article of Shimada and Zhang \[SZ\] for a useful table of extremal elliptic $K3$ surfaces.
The non-rigid | 1 | member_38 |
examples
======================
Let $Y=E(\tilde{\Gamma})$ be one of the K3 surfaces of the previous Section, and let $g_Y =
g_{3,\Gamma}$ denote the corresponding cusp form of level $3$. If $\tilde{\Gamma}$ contains $\Gamma_1(M)$ (in Table 2 this happens in cases \#3, \#4, \#7, \#8, and \#9), and $M = M_Y$ is the maximal possible, then $M$ is the [*level*]{} of $g_Y$, and the [*newtype*]{} of $g_Y$ is the Dirichlet character $$\label{new}
\epsilon = \epsilon_Y,$$ of conductor $M$, so that $g_Y$ is in $S_3(\Gamma_0(M), \epsilon_Y)$. Notice that $\epsilon$ is odd (namely $\epsilon(-1) = -1$). Moreover, since the coefficients of $g_Y$ are integers, $\epsilon$ must be quadratic. (We will determine $\epsilon$ for our examples in Proposition 10 below.)
Let $E$ be an elliptic curve. We view the product $Y\times E$ as a family of abelian surfaces over $X(\tilde{\Gamma})$. The fiber $A_t
= A_{\Gamma,t}$ over each point $t\in
X(\tilde{\Gamma})$ is the product of the fiber $E_{\Gamma,t}$ of $E(\tilde{\Gamma})$ with $E$. Then we have the following
[**Theorem 6**]{}:
*(1) The product $Y\times E$ has the Hodge numbers $$h^{0,3}(Y\times E)=1,\, h^{1,0}(Y\times E) =
1\,\,\mbox{and}\,\, B_3(Y\times E)=44$$ (so that $Y\times E$ is not a Calabi–Yau threefold).*
\(2) The motive $T(Y\times E)=T(Y)\times H^1(E)$ is a submotive of $H^3(Y\times E)$. | 1 | member_38 |
If $E$ and $Y$ are defined over ${{\mathbb Q}}$, this submotive is modular, in the sense that its $L$-series is associated to a cusp form $g_Y$ on $GL(2,{{\mathbb Q}}(\sqrt{\delta}\,))$. Let $g_E$ be the cusp form of weight $2$ associated to $E$ by Wiles et. al. (\[W\]). Let $A(p)$ (respectively $B(p)$) be the $p$th Fourier coefficient of $g_E$ (respectively of $g_Y$), and let $\epsilon_Y$ be the newtype character of $g_Y$ (see Section 3). Then for any good prime $p$, the local Euler factor $L_p(s)$ of the $L$-series $L(T(Y\times E),s) = L(g_E\otimes g_Y,s)$ is $$1 - A(p)B(p)p^{-s} + (B(p)^2+\epsilon_Y(p)pA(p)^2-2p^2
\epsilon_Y(p))p^{1-2s}
- A(p)B(p)\epsilon_Y(p)p^{3-3s} + p^{6-4s}.$$
[**Proof**]{}: The statements about the Hodge and Betti numbers follow from the Künneth formula. Since $T(Y)$ is a factor of $H^2(Y)$, it follows again from the Künneth formula that $T(Y)\times H^1(E)$ is a factor of $H^3(Y\times E)$.
For the second part, we know that $g_Y$ is a CM form. Hence it is induced from an algebraic Hecke character $\chi = \chi_Y$ of the imaginary quadratic field $F = K_i$. Let $\chi_G$ be the compatible system of $1$-dimensional $\ell$-adic representations of $G_F =
\text{Gal}(\overline{{\mathbb Q}}/F)$ corresponding to $\chi$. Then the $2$-dimensional Galois representation associated to $T(Y)$ is ind$_{G_F}^{G_{{\mathbb Q}}}\chi_G$. | 1 | member_38 |
Hence we obtain the $4$-dimensional Galois representation $$\rho_E \otimes \text{ind}_{G_F}^{G_{{\mathbb Q}}} \chi_G
\simeq \text{ind}_{G_F}^{G_{{\mathbb Q}}}(\chi_G \otimes
\text{Res}^{G_{{\mathbb Q}}}_{G_F}\rho_E),$$ where $\rho_E$ is the Galois representation associated to $H^1(E)$. The operation of restricting $\rho_E$ to $G_F$ and of twisting by characters have automorphic analogs. Let $\pi_E$ be the automorphic representation associated to $E$. Then $\pi' = \chi\otimes$ Res$^{{\mathbb Q}}_F\pi_E$ makes sense as an automorphic cuspidal irreducible representation of $GL(2,F)$, and we have the characterizing relationship $$L(\pi',s) = L(\text{ind}_F^{{\mathbb Q}}\pi',s) =
L(\pi_E \otimes \text{ind}_F^{{\mathbb Q}}\chi,s) =
L(g_E\otimes g_Y,s).$$
For the last part, write the $p$th Euler factors of $g_E$ and $g_Y$ respectively as $$(1-\alpha_p p^{-s})(1-\alpha'_p p^{-s}) =
1-A(p) p^{-s} + p^{1-2s} \qquad\text{and}$$ $$(1-\beta_p p^{-s})(1-\beta'_p p^{-s}) =
1 - B(p) p^{-s} +\epsilon_Y(p) p^{2-2s}.$$ Then the Euler factor $L_p$ is defined as $$L_p(s) =(1-\alpha_p\beta_p p^{-s})
(1-\alpha'_p\beta_p p^{-s})
(1-\alpha_p\beta_p' p^{-s})
(1-\alpha_p'\beta_p' p^{-s}),$$ and the claim follows by a direct calculation.
[**Remark**]{}: [*If a K3 surface has the Picard number $19$ or $18$, the modularity question for the product $Y\times E$ is still open. However, if the Picard number is 19, one knows at least that the rank $3$ motive $T(Y)$ of the transcendental cycles is self dual orthogonal via the cup product. (For explicit examples of | 1 | member_38 |
K3 surfaces with Picard number 19, see e. g. Besser and Livné \[BL\].) Thus one can use a result of Tate to lift each $\ell$-adic representation to the associated spin cover, which is the multiplicative group of some quaternion algebra over ${{\mathbb Q}}$. If the spin representation is modular (which should always be the case), then it is associated to a cusp form $h$ of weight $2$ on $GL(2)$, so that Symm$^2 h$ realizes $T(Y)$. Let $g_E$ be again the weight $2$ cusp form associated with $E$. It follows, by work of Gelbart and Jacquet, that $T(Y)$ is realized by an automorphic representation on $GL(3,{{\mathbb Q}})$. Hence, by the work of Kim and Shahidi (\[KS\]), $T(Y)\times E$ is realized by an automorphic form on $GL(6,{{\mathbb Q}})$. In particular, $L(\text{Symm}^2h \otimes g_E,s)$ has the expected analytic properties.*]{}
To construct our promised examples, let $X=
X_\Gamma{\rightarrow}X(\tilde{\Gamma})$ be the associated Kummer family, in which we divide each fiber $A_t$ of $Y(\tilde{\Gamma})\times E$ by $\pm 1$ and then blow up the locus of points of order $2$. We now have the following
[**Theorem 7 **]{}: [*In the Examples \#1, \#2, \#5, and \#6 of Table 2 the resulting $X$ is a smooth Calabi–Yau threefold. | 1 | member_38 |
It is non-rigid, and the given fibration $f:X{\rightarrow}X(\tilde{\Gamma})$ is semi-stable, with vanishing (iterated) Kodaira–Spencer mapping. We have $$\deg{f_*\omega_{X/{{\mathbb P}}^1}} = 2 =
\frac{1}{2}\,\mbox{deg}\,\Omega^1_{{{\mathbb P}}^1}
(\mbox{log}\,S),$$ In other words, $X$ reaches the (stronger) Arakelov–Yau bound.* ]{}
[**Remark**]{}: [*For the first case in Table 2 ($\tilde{\Gamma} = \Gamma(4)$) this example is indicated in \[STZ\].*]{}
[**Proof**]{}: The Examples we chose are those in which $\tilde{\Gamma}$ is a subgroup of $\Gamma(2)$. (This is because otherwise, the points of order $2$ of $X(\Gamma)$ coincide (over the cusps).) Thus the $2$-torsion points (of $E_t$ and hence of $A_t$) are distinct for [*all*]{} $t\in
X(\tilde{\Gamma})$. It follows that the locus $A[2]$ of $2$-torsion points is smooth, and hence so is the blow-up $X$. We have $H^i(X) =
H^i(Y(\tilde{\Gamma})\times E)^{<\pm 1>}$. But $\pm 1$ acts as $\pm 1$ on both the non-trivial holomorphic $1$-form $\omega_1$ of $E$ and on the non-trivial holomorphic $2$-form $\omega_2$ of $Y(\tilde{\Gamma})$. Hence $\omega_1\wedge
\omega_2$ descends to a holomorphic $3$-form $\omega_3$ on $X$. Its divisor can only be supported on the proper transform ${\mathcal{F}}$ of $A[2]$; however $\mathcal{F}$ intersects each fiber $f^{-1}(t)$ in sixteen $(-2)$-curves, which do not contribute to the canonical class, so that $\omega_3$ is indeed nowhere-vanishing. The Künneth formula gives that | 1 | member_38 |
$$H^1(X) = H^1(Y)^{<\pm 1>} =
H^1(E)^{<\pm 1> } = 0, \qquad\text{and}$$ $$H^{2,0}(X) = H^{2,0}(Y \times E)^{<\pm 1>} =
H^2(Y)^{<\pm 1>} = 0.$$ Thus $X$ is indeed a smooth Calabi–Yau threefold. It is non-rigid, because $T(Y\times
E)$ descends to $X$ and each of its Hodge pieces $H^{p,q}(T(Y\times E))$ is $1$-dimensional.
To compute the monodromy around each singular fiber, we notice that for a generic fiber $X_t =
f^{-1}(t)$ the Kummer structure gives a canonical decomposition $$H^2(X_t,{{\mathbb Q}}) =
(H^2(A_t,{{\mathbb Q}}) \oplus {{\mathbb Q}}A_t[2])^{<\pm 1>}.$$ Our examples were chosen so that the action of $\pm 1$ on $A_t[2]= E_{\Gamma,t}[2] \times
E_t[2]$ is trivial. Moreover, in the Künneth decomposition $$H^2(A_t) = H^2(E) \oplus (H^1(E)\otimes
H^1(E_{\Gamma,t})) \oplus H^2(E_{\Gamma,t})$$ the $\pm 1$ action is trivial on the first and last factors, is trivial on $H^1(E)$ and is unipotent on $H^1(E_\Gamma,t)$ around each singular fiber of $f$ (namely the cusps of $\Gamma$). Thus the monodromy of the fibration $f$ is unipotent as well.
To compute the Kodaira–Spencer map $\Theta(f)$ for our $f$ we embed it into the Kodaira–Spencer map for $Y\times E \rightarrow
X(\tilde{\Gamma})$. This map is the cup product with the Kodaira–Spencer class $\Theta$ which itself is $\Theta_{Y/X(\tilde{\Gamma})} \otimes
\Theta_{X(\tilde{\Gamma}) \times
E/X(\tilde{\Gamma})}$. Since the Kodaira–Spencer | 1 | member_38 |
class of a trivial fibration vanishes, it follows that $\Theta(f) = 0$.
Our examples all have $6$ singular fibers. Hence $$\frac{1}{2}\deg \Omega^1_{{{\mathbb P}}^1}(\log S) =
\frac{1}{2} \deg{{\mathcal O}}_{{{\mathbb P}}^1}(-2+6) = 2.$$
On the other hand, since $X$ is a Calabi–Yau variety we have $$\omega_{X/{{\mathbb P}}^1} = \omega_X\otimes
(f^*\omega_{{{\mathbb P}}^1})^{-1} =
(f^*\omega_{{{\mathbb P}}^1})^{-1}.$$ Hence $$\label{deg}
f_*\omega_{X/{{\mathbb P}}^1} =
f_*f^*(\omega_{{{\mathbb P}}^1})^{-1}=
(f_*f^*\omega_{{{\mathbb P}}^1})^{-1}=
\omega_{{{\mathbb P}}^1}^{-1},$$ whose degree is $2$ as well, concluding the proof of Theorem 7.
[**Remarks 8**]{}:
*(1) In the other cases in Table 2 the monodromy on the points of order $2$ of $A_t[2]$ is non-trivial, and the calculation gives that the monodromy of $f$ around the cusps is not unipotent. We know by Remark 3, the groups \#1, \#5 and \#8 are in the same $PSL(2,{{\mathbb R}})$-conjugacy class. However, this group theoretic property does not guarantee isomorphisms of the corresponding Calabi–Yau threefolds, since the fiber structures are not preserved. Similarly, the groups \#4, \#6 and \#9 are $PSL(2,{{\mathbb R}})$-conjugate, but geometric structures are different (as the fibers over the cusps are different). The same applies to Examples \#2 and \#7. Therefore, Examples \#4, \#7, \#8 and \#9 are not covered by our examples. Also we do not know | 1 | member_38 |
how to construct examples corresponding to Example \#3 in Table 2, for which $\tilde{\Gamma} = \Gamma_1(7)$. We also do not know whether our examples are the only ones.*
\(2) It is an interesting exercise to compute the full $L$-series of our examples. The results are as follows: Let $N_+$ (respectively $N_-$) be the motive of algebraic cycles on $Y$ invariant (respectively anti-invariant) by $\pm
1$ acting on the elliptic fibrations of $Y$. Let $n_{\pm}$ be the rank of $N_\pm$. Then $n_+ + n_-
= 20$, and if we let $\chi_{\delta^{\prime}}$ denote the quadratic character cut by ${{\mathbb Q}}(\sqrt{\delta^{\prime}}\,)$ (not necessarily the same quadratic field pre-determined by the modular group corresponding to the surface), then $N_+= {{\mathbb Z}}(1)^{n^{\prime}_+}\oplus{{\mathbb Z}}(\chi_{\delta^{\prime}}(1))^{n^{\prime\prime}_+}$ and $N_- = {{\mathbb Z}}(1)^{n^{\prime}_-}\oplus {{\mathbb Z}}(\chi_\delta^{\prime}(1))^{n^{\prime\prime}_-}$. Here $n^{\prime}_{+}$ (resp. $n^{\prime\prime}_+$) denotes the number of cycles defined over ${{\mathbb Q}}$ (resp. ${{\mathbb Q}}(\delta^{\prime})$) and similarly for $n^{\prime}_-$ (resp. $n^{\prime\prime}_-$). We have $n_{\pm}=
(n^{\prime}+n^{\prime\prime})_{\pm}$. Then we have
$$\begin{aligned}
L(H^0,s) & = & L({{\mathbb Z}},s) = \zeta(s) \\
L(H^1,s) & = & 1 \\
L(H^2,s) &= & L(H^2({{{\mathbb P}}}^1\times{{{\mathbb P}}}^1),s)L({{\mathbb Z}}(1),s)L(N_+,s)\\
&=& \zeta(s-1)^{16}\zeta(s-1)^{1+n_+^{\prime}}
L({{\mathbb Q}}\otimes\chi_{\delta^{\prime}},s-1)^{n_+^{\prime\prime}}\\
L(H^3,s) & = & L(T(Y)\otimes H^1(E),s)L(N_-\times H^1(E),s) \\
\quad & = & L(g_3\otimes g_2,s)L(E, s-1)^{n_-^{\prime}}
\prod_{\delta^{\prime}}L(E\otimes\chi_{{\delta}^{\prime}},s-1)^{n_-^{\prime\prime}}\end{aligned}$$
(The higher cohomologies | 1 | member_38 |
are determined by Poincaré duality.)
[**Lemma 9 **]{}: In cases \#1, \#2, \#5, and \#6 in Table 2, we have $n_+=14$ and $n_-=6$ (so $n_+-n_-=2+6=8$). Furthermore, we have $$\begin{aligned}
(n_+^{\prime},n_+^{\prime\prime})=\begin{cases}
(12,2)\quad &\mbox{for \#1} \\
(14,0)\quad &\mbox{for \#2} \\
(13,1)\quad &\mbox{for \#5,\, \#6}\end{cases}\end{aligned}$$ and $$\begin{aligned}
(n_-^{\prime},n_-^{\prime\prime})=\begin{cases}
(3,3)\quad &\mbox{for \#1} \\
(6,0)\quad &\mbox{for \#2} \\
(5,1)\quad &\mbox{for \#5,\,\#6}\end{cases}\end{aligned}$$ In case \#3, $n_+= 11$ and $n_-=9$. (For the last case, confer the article of Hulek and Verrill \[HV\] for more detailed discussion.)
For the computations of $n_+$ and $n_-$, confer Proposition 2.4 of Hulek and Verrill \[HV\]. The Proof of Lemma 9 will be given at the end of Section 4.
Explicit Formulas
=================
We shall now give explicit formulas for the weight $3$ forms $g_Y = g_{3,\Gamma}$ for the examples in Table 2. We will denote the weight $3$ form in the $i$th case by $h_i$. By Remark 3 it suffices to compute $h_i$ for $i=8,7,3,4$, and then $h_8(\tau) = h_5(\tau/2) = h_1(\tau/4)
$, $h_7(\tau) = h_2(\tau/2)$, and $h_6(\tau) =
h_9(\tau/2) = h_4(\tau/2 -1/2)$. Two kinds of formulas suggest themselves for the $h_i$’s: as a product of $\eta$-functions or as inverse Mellin transforms of the Dirichlet series attached to Hecke characters. The | 1 | member_38 |
second method is always possible since the $g_Y$’s are CM forms. In \[M\] Martin determined which modular forms on $\Gamma_1(N)$ can be expressed as a product of $\eta$-functions. This applies to cases \#8, \#7, \#3, and \#4 in Table 2. Hence the same is also true for the \#6 and \#9 cases. For cases \#3, \#7, and \#8 the corresponding spaces of cusp forms of weight $3$ are $1$-dimensional, hence the conditions in \[M\] are satisfied and Martin gives the corresponding forms as $h_3=
\eta(q)^3\eta(q^7)^3$ and $h_7=
\eta(q^2)^3\eta(q^6)^3$. The modular form in case \#1 is classically known to be $h_1=
\eta(q)^6$, which implies $h_8= \eta(q^4)^6$. Lastly $h_2= \eta(q)^3\eta(q^3)^3$, and $h_5=
\eta(q^2)^6$. For \#4, we have $h_4(q)=\eta(q)^2\eta(q^2)\eta(q^4)\eta(q^8)^2$. We will prove the following
[**Proposition 10**]{}:
*Let $\chi_i$ be the Hecke character for which $L(h_i,s) =
L(\chi_i,s)$ (so that the inverse Mellin transform of $L(\chi_i,s)$ is $h_i$). Let $a_p(h_i)$ be the $p$th Fourier coefficient of $h_i$, and let $K_i = {{\mathbb Q}}(\sqrt{\delta_i})$. Then we have the following:*
\(1) The infinite component of $\chi_i:{{\mathbb A}}_{K_i}^\times \rightarrow {{\mathbb C}}$ is given by $\chi_{i,\infty}(z) = z^{-2}$. Moreover $\chi_i$ is the unique such Hecke character of conductor $c_i{{\mathcal O}}_{K_i}$, where $c_i = 2,2,1,1
\in {{\mathcal O}}_{K_i}$ for | 1 | member_38 |
$i = 8,7,3,4$ respectively.
\(2) For each rational prime $p$ which is prime to the level of the corresponding $\Gamma$, we have $a_p(h_i) = 0$ if $p$ is inert in $K_i$. Otherwise, there are $a$, $b$, which are integers in case \#8 and half integers in the three other cases, so that $p = a^2+d_i b^2$, where $d_i = 4,
3, 7, 2$ for $i = 8,7,3,4$ respectively. Then $a$ and $b$ are unique up to signs, and $a_p(h_i) =
(a^2-d_ib^2)/2$.
\(3) The newtype of $h_i$ (see (\[new\])) is the character defining $K_i$, namely $p\mapsto
\left(\frac{\delta_i}{p}\right)$.
[**Proof: **]{} See e.g. \[L2\] for the generalities (in particular regarding the $\infty$-component of $\chi_i$), as well as the following formula: the conductors of $\chi_i$ and of $h_i$ are related by $$\text{cond}(h_i) = \text{Nm}^{K_i}_{{{\mathbb Q}}}
\text{cond}(\chi_i) \text{Disc}(K_i).$$ Since the level of $h_i$ is respectively $M=16$, $12$, $7$, and $8$ in cases \#8, \#7, \#3, and \#4 of Table 2, we get the asserted value of the $c_i$’s, and since all the fields $K_i$ involved have class number $1$ we have $$\label{dec}
{{\mathbb A}}_{K_i}^\times = (K_i^\times \times U_i \times
{{\mathbb C}}^\times)/\mu(K_i)$$ where $U_i$ is the maximal compact subgroup $\hat{{{\mathcal O}}}\vphantom{{{\mathcal O}}}_{K_i}^\times$ of the finite idèles | 1 | member_38 |
of $K_i$, and $\mu(K_i)$ is the group of roots of unity of $K_i$, acting diagonally (we view ${{\mathbb C}}$ as the infinite completion of $K_i$). The existence and the uniqueness of $\chi_i$ are then verified in each case by a straightforward calculation (compare \[L1\]).
For the second part, the vanishing of $a_p(h_i)$ for $p$ inert in $K_i$ is a general property of CM forms. For a split $p$ (prime to $\text{cond}(h_i)$), write $p = a^2 + d_i b^2 =
\text{Nm}^{K_i}_{{{\mathbb Q}}} \pi$. Here $\pi$ is a prime element of ${{\mathcal O}}_{K_i}$, so $a$ and $b$ are half integers. We verify that, up to multiplying $\pi$ by a unit, we can guarantee that $a$ and $b$ are integers for $i=8$. In all cases, the $a$’s and the $b$’s are unique up to signs. Next one verifies that $\pi\equiv \pm 1
\pmod{c_i{{\mathcal O}}_{K_i}}$. Let $\wp$ be the ideal generated by $\pi$ (notice that changing the sign of $b$ replaces $\wp$ by its conjugate). Let ${\operatorname{tr}}$ denote the trace from $K_i$ to ${{\mathbb Q}}$. By the general theory, we have that $$a_p(h_i) = {\operatorname{tr}}\chi_\wp(\pi) =
{\operatorname{tr}}\chi_\infty(\pi)^{-1} = {\operatorname{tr}}\pi^2 =
2(a^2-d_ib^2),$$ where the second equality holds since the finite components of $\chi$ other than | 1 | member_38 |
$\pi$ are now $1$.
For the third part we notice that the restriction of $\chi_i$ to $U_i$ in the decomposition (\[dec\]) above gives a Dirichlet character $\epsilon'_i$ on ${{\mathcal O}}_{K_i}$ of conductor $c_i{{\mathcal O}}_{K_i}$. The newtype Dirichlet character $\epsilon_i$ (on ${{\mathbb Z}}$) is then the product $\chi_{K_i}$ by the restriction of $\epsilon'_i$ to ${{\mathbb Z}}$. However, the conductors $c_i{{\mathcal O}}_{K_i}$ of $\chi_i$ are all $1$ or $2$, and the only character of ${{\mathbb Z}}$ of conductor $1$ or $2$ is trivial. Hence the newtype character of $h_i$ is $K_i$, concluding the proof of Proposition 10.
[**Defining equations for extremal K3 surfaces**]{}: We now discuss how to determine defining equations for the extremal K3 surfaces in Theorem 7. This problem has been getting a considerable attention lately, for instance, Shioda \[Sh3\] and (independently and by a different method by Y. Iron \[I\]) have determined a defining equation for the semi-stable elliptic K3 surface with singular fibers of type $I_1,I_1,I_1,I_1,I_1,I_{19}$ whose existence was established in Miranda and Persson \[MP\] (this is given as $\#1$ in their list). As we shall see, our examples can be determined by a more classical method.
There are several cases where defining equations can be found in | 1 | member_38 |
---
abstract: 'We provide complete characterizations, on Banach spaces with cotype 2, of those linear operators which happen to be weakly mixing or strongly mixing transformations with respect to some nondegenerate Gaussian measure. These characterizations involve two families of small subsets of the circle: the countable sets, and the so-called *sets of uniqueness* for Fourier-Stieltjes series. The most interesting part, the sufficient conditions for weak and strong mixing, is valid on an arbitrary (complex, separable) Fréchet space.'
address:
- 'Clermont Université, Université Blaise Pascal, Laboratoire de Mathématiques, BP 10448, F-63000 Clermont-Ferrand - CNRS, UMR 6620, Laboratoire de Mathématiques, F-63177 Aubière.'
- 'Laboratoire de Mathématiques de Lens, Université d’Artois, Rue Jean Souvraz S. P. 18, 62307 Lens.'
author:
- Frédéric Bayart
- Étienne Matheron
title: |
Mixing operators\
and small subsets of the circle
---
Introduction
============
A basic problem in topological dynamics is to determine whether a given continuous map $T:X\to X$ acting on a topological space $X$ admits an ergodic probability measure. One may also ask for stronger ergodicity properties such as weak mixing or strong mixing, and put additional constraints on the measure $\mu$, for example that $\mu$ should have no discrete part, or that it should belong | 1 | member_39 |
to some natural class of measures related to the structure of the underlying space $X$. Especially significant is the requirement that $\mu$ should have *full support* ( $\mu (V)>0$ for every open set $V\neq\emptyset$) since in this case any ergodicity property implies its topological counterpart. There is, of course, a huge literature on these matters since the classical work of Oxtoby and Ulam ([@OU]).
In recent years, the above problem has received a lot of attention in the specific setting of *linear* dynamics, when the transformation $T$ is a continuous linear operator acting on a topological vector space $X$ ([@F], [@BG2], [@BG3], [@BoGE], [@Sophie]). The main reason is that people working in linear dynamics are mostly interested in studying *hypercyclic* operators, operators having dense orbits. When the space $X$ is second-countable, it is very easy to see that if a continuous map $T:X\to X$ happens to be ergodic with respect to some Borel probability measure $\mu$ with full support, then almost every $x\in X$ (relative to $\mu$) has a dense $T$-orbit. (In fact, one can say more: it follows from Birkhoff’s ergodic theorem that almost all $T$-orbits visit every non-empty open set along a set of integers having positive lower | 1 | member_39 |
density. In the linear setting, an operator having at least one orbit with that property is said to be *frequently hypercyclic*. This notion was introduced in [@BG3] and extensively studied since then; see the books [@BM] and [@GP] for more information). Hence, to find an ergodic measure with full support is an efficient way of showing that a given operator is hypercyclic, which comes as a measure-theoretic counterpart to the more traditional Baire category approach.
Throughout the paper, we shall restrict ourselves to the best understood infinite-dimensional measures, the so-called *Gaussian* measures. Moreover, the underlying topological vector space $X$ will always be a complex separable Fréchet space. (The reason for considering *complex* spaces only will become clear in the next few lines). In this setting, a Borel probability measure on $X$ is Gaussian if and only if it is the distribution of an almost surely convergent random series of the form $\xi=\sum_0^\infty g_n x_n$, where $(x_n)\subset X$ and $(g_n)$ is a sequence of independent, standard complex Gaussian variables. Given any property (P) relative to measure-preserving transformations, we shall say that an operator $T\in\mathfrak L(X)$ has property (P) *in the Gaussian sense* if there exists some Gaussian probability measure $\mu$ on | 1 | member_39 |
$X$ with full support with respect to which $T$ has (P). The problem of determining which operators are ergodic in the Gaussian sense was investigated by E. Flytzanis ([@F]), in a Hilbert space setting. The fundamental idea of [@F] is that one has to look at the *$\TT$-eigenvectors* of the operator, the eigenvectors associated with eigenvalues of modulus $1$: roughly speaking, ergodicity is equivalent to the existence of “sufficiently many $\TT$-eigenvectors and eigenvalues". This is of course to be compared with the now classical eigenvalue criterion for hypercyclicity found by G. Godefroy and J. Shapiro ([@GS]), which says in essence that an operator having enough eigenvalues inside and outside the unit circle must be hypercyclic.
The importance of the $\TT$-eigenvectors is easy to explain. Indeed, it is almost trivial that if $T\in\mathfrak L(X)$ is an operator whose $\TT$-eigenvectors span a dense subspace of $X$, then $T$ admits an invariant Gaussian measure with full support: choose a sequence of $\TT$-eigenvectors $(x_n)_{n\geq 0}$ (say $T(x_n)=\lambda_n x_n$) with dense linear span such that $\sum_0^\infty\Vert x_n\Vert<\infty$ for every continuous semi-norm $\Vert\,\cdot\,\Vert$ on $X$, and let $\mu$ be the distribution of the random variable $\xi=\sum_0^\infty g_n x_n$. That $\mu$ is $T$-invariant follows from the linearity | 1 | member_39 |
of $T$ and the rotational invariance of the Gaussian variables $g_n$ ($\mu\circ T^{-1}\sim\sum_0^\infty g_n T(x_n)=\sum_0^\infty (\lambda_ng_n)\, x_n\sim\sum_0^\infty g_n x_n=\mu$). However, this particular measure $\mu$ cannot be ergodic ([@Sophie]).
Building on Flytzanis’ ideas, the first named author and S. Grivaux came rather close to characterizing the weak and strong mixing properties for Banach space operators in terms of the $\TT$-eigenvectors ([@BG3], [@BG2]). However, this was not quite the end of the story because the sufficient conditions for weak or strong mixing found in [@BG3] and [@BG2] depend on some geometrical property of the underlying Banach space, or on some “regularity" property of the $\TT$-eigenvectors (see the remark just after Corollary \[eigvectfield\]).
In the present paper, our aim is to show that in fact, these assumptions can be completely removed. Thus, we intend to establish “optimal" sufficient conditions for weak and strong mixing in terms of the $\TT$-eigenvectors which are valid on an arbitrary Fréchet space. These conditions turn out to be also necessary when the underlying space $X$ is a Banach space with *cotype 2*, and hence we get complete characterizations of weak and strong mixing in this case. We shall in fact consider some more general notions of “mixing", but | 1 | member_39 |
our main concerns are really the weak and strong mixing properties.
At this point, we should recall the definitions. A measure-preserving transformation $T:(X,\mathfrak B,\mu)\to(X,\mathfrak B,\mu)$ is *weakly mixing* (with respect to $\mu$) if $$\frac{1}{N}\sum_{n=0}^{N-1} \vert \mu(A\cap T^{-n}(B))-\mu (A)\mu(B)\vert\xrightarrow{N\to\infty} 0$$ for any measurable sets $A,B\subset X$; and $T$ is *strongly mixing* if $$\mu(A\cap T^{-n}(B))\xrightarrow{n\to\infty} \mu(A)\mu(B)$$ for any $A,B\in\mathfrak B$. (Ergodicity can be defined exactly as weak mixing, but removing the absolute value in the Cesàro mean).
According to the “spectral viewpoint" on ergodic theory, weakly mixing transformations are closely related to *continuous* measures on the circle $\TT$, and strongly mixing transformations are related to *Rajchman* measures, i.e. measures whose Fourier coefficients vanish at infinity. Without going into any detail at this point, we just recall that, by a classical result of Wiener (see [@Ktz]), continuous measures on $\TT$ are characterized by the behaviour of their Fourier coefficients: a measure $\sigma$ is continuous if and only if $$\frac{1}{N}\sum_{n=0}^{N-1} \vert \widehat\sigma (n)\vert\xrightarrow{N\to\infty} 0\, .$$ Wiener’s lemma is usually stated with symmetric Cesàro means, but this turns out to be equivalent. Likewise, by the so-called *Rajchman’s lemma*, a measure $\sigma$ is Rajchman if and only if $\widehat\sigma (n)\to 0$ as $n\to +\infty$ (that is, | 1 | member_39 |
a one-sided limit is enough). Especially important for us will be the corresponding families of “small" sets of the circle; that is, the sets which are annihilated by every positive measure in the family under consideration (continuous measures, or Rajchman measures). Obviously, a Borel set $D\subset\TT$ is small for continuous measures if and only if it is countable. The small sets for Rajchman measures are the so-called *sets of extended uniqueness* or *sets of uniqueness for Fourier-Stieltjes series*, which have been extensively studied since the beginning of the 20th century (see [@KL]). The family of all sets of extended uniqueness is usually denoted by $\mathcal U_0$.
Our main results can now be summarized as follows.
\[WS\] Let $X$ be a complex separable Fréchet space, and let $T\in\mathfrak L(X)$.
1. Assume that the $\TT$-eigenvectors are *perfectly spanning*, in the following sense: for any countable set $D\subset \TT$, the linear span of $\bigcup_{\lambda\in\TT\setminus D}\ker (T-\lambda)$ is dense in $X$. Then $T$ is weakly mixing in the Gaussian sense.
2. Assume that the $\TT$-eigenvectors are *$\mathcal U_0$-perfectly spanning*, in the following sense: for any Borel set of extended uniqueness $D\subset \TT$, the linear span of $\bigcup_{\lambda\in\TT\setminus D}\ker (T-\lambda)$ is dense in $X$. Then | 1 | member_39 |
$T$ is strongly mixing in the Gaussian sense.
3. In [(1)]{} and [(2)]{}, the converse implications are true if $X$ is a Banach space with [cotype 2]{}.
Some remarks are in order regarding the scope and the “history" of these results.
1. When $X$ is a Hilbert space, (1) is stated in [@F] (with some additional assumptions on the operator $T$) and a detailed proof is given in [@BG3] (without these additional assumptions). The definition of “perfectly spanning" used in [@BG3] is formally stronger than the above one, but the two notions are in fact equivalent ([@Sophie]).
2. It is shown in [@Sophie] that under the assumption of (1), the operator $T$ is frequently hypercyclic. The proof is rather complicated, and it is not clear that it could be modified to get weak mixing in the Gaussian sense. However, some of the ideas of [@Sophie] will be crucial for us. In particular, sub-section \[Sophiesection\] owes a lot to [@Sophie].
3. In the weak mixing case, (3) is proved in [@BG2 Theorem 4.1].
4. It seems unnecessary to recall here the definition of cotype (see any book on Banach space theory, [@AK]). Suffices it to say that this is a geometrical | 1 | member_39 |
property of the space, and that Hilbert space has cotype $2$ as well as $L^p$ spaces for $p\in [1,2]$ (but *not* for $p>2$).
5. As observed in [@BG2 Example 4.2], (3) does not hold on an arbitrary Banach space $X$. Indeed, let $X:=\mathcal C_0([0,2\pi])=\{ f\in\mathcal C([0,2\pi]);\; f(0)=0\}$ and let $V:L^2(0,2\pi)\to X$ be the Volterra operator, $Vf(t)=\int_0^t f(s)\, ds$. There is a unique operator $T:X\to X$ such that $TV=VM_\phi$, where $M_\phi:L^2(0,2\pi)\to L^2(0,2\pi)$ is the multiplication operator associated with the function $\phi (t)=e^{it}$. The operator $T$ is given by the formula $$\label{Kal}
Tf(t)=\phi(t)f(t)-\int_0^t \phi'(s)f(s)\, ds\, .$$ It is easy to check that $T$ has no eigenvalues. On the other hand, $T$ is strongly mixing with respect to the Wiener measure on $\mathcal C_0([0,2\pi])$.
As it turns out, ergodicity and weak mixing in the Gaussian sense are in fact equivalent (see e.g. [@G], or [@BG2 Theorem 4.1]). Hence, from Theorem \[WS\] we immediately get the following result. (A Gaussian measure $\mu$ is *nontrivial* if $\mu\neq\delta_0$).
\[characexistergod\] For a linear operator $T$ acting on a Banach space $X$ with cotype 2, the following are equivalent:
- $T$ admits a nontrivial ergodic Gaussian measure;
- there exists a closed, $T$-invariant subspace $Z\neq \{ 0\}$ such | 1 | member_39 |
that $$\overline{\rm span}\, \bigcup_{\lambda\in\TT\setminus D}\ker (T_{\vert Z}-\lambda)=Z$$ for every countable set $D\subset\TT$.
In this case, $T$ admits an ergodic Gaussian measure with support $Z$, for any such subspace $Z$.
If $T$ admits an ergodic Gaussian measure $\mu\neq\delta_0$, then $Z:={\rm supp}(\mu)$ is a non-zero $T$-invariant subspace, and $Z$ satisfies (b) by Theorem \[WS\] (3). The converse follows from Theorem \[WS\] (1).
For concrete applications, it is useful to formulate Theorem \[WS\] in terms of *$\TT$-eigenvector fields* for the operator $T$. A $\TT$-eigenvector field for $T$ is a map $E:\Lambda\to X$ defined on some set $\Lambda\subset\TT$, such that $$TE(\lambda)=\lambda E(\lambda)$$ for every $\lambda\in\Lambda$. (The terminology is not perfectly accurate: strictly speaking, $E(\lambda)$ is perhaps not a $\TT$-eigenvector because it is allowed to be $0$). Recall also that a closed set $\Lambda\subset\TT$ is *perfect* if it has no isolated points or, equivalently, if $V\cap\Lambda$ is uncountable for any open set $V\subset\TT$ such that $V\cap\Lambda\neq\emptyset$. Analogously, a closed set $\Lambda\subset \TT$ is said to be *$\mathcal U_0$-perfect* if $V\cap \Lambda$ is not a set of extended uniqueness for any open set $V$ such that $V\cap\Lambda\neq\emptyset$. (For example, any nontrivial closed arc is $\mathcal U_0$-perfect).
\[eigvectfield\] Let $X$ be a separable complex Fréchet space, and | 1 | member_39 |
let $T\in\mathfrak L(X)$. Assume that one has at hand a family of *continuous* $\TT$-eigenvector fields $(E_i)_{i\in I}$ for $T$, where $E_i:\Lambda_i\to X$ is defined on some closed set $\Lambda_i\subset\TT$, such that ${\rm span}\left(\bigcup_{i\in I} E_i(\Lambda_i)\right)$ is dense in $X$.
1. If each $\Lambda_i$ is a perfect set, then $T$ is weakly mixing in the Gaussian sense.
2. If each $\Lambda_i$ is $\mathcal U_0$-perfect, then $T$ is strongly mixing in the Gaussian sense.
This follows immediately from Theorem \[WS\]. Indeed, if $\Lambda\subset\TT$ is a perfect set then $\Lambda\setminus D$ is dense in $\Lambda$ for any countable set $D$, whereas if $\Lambda$ is $\mathcal U_0$-perfect then $\Lambda\setminus D$ is dense in $\Lambda$ for any $\mathcal U_0$-set $D$. Since the $\TT$-eigenvector fields $E_i$ are assumed to be continuous, it follows that the $\TT$-eigenvectors of $T$ are perfectly spanning in case (i), and $\mathcal U_0$-perfectly spanning in case (ii).
Several results of this kind are proved in [@BG2] and in [@BM Chapter 5], all of them being based on an interplay between the geometry of the (Banach) space $X$ and the regularity of the $\TT$-eigenvector fields $E_i$. For example, it is shown that if $X$ has *type 2*, then continuity of the $E_i$ is | 1 | member_39 |
enough, whereas if the $E_i$ are Lipschitz and defined on (nontrivial) closed arcs then no assumption on $X$ is needed. “Intermediate" cases involve the [type]{} of the Banach space $X$ and H" older conditions on the $E_i$. What Corollary \[eigvectfield\] says is that continuity of the $E_i$ is *always* enough, regardless of the underlying space $X$.
We also point out that the assumption in (i), i.e. the existence of $\TT$-eigenvector fields with the required spanning property defined on perfect sets, is in fact equivalent to the perfect spanning property ([@Sophie]). Likewise, the assumption in (ii) is equivalent to the $\mathcal U_0$-perfect spanning property (see Proposition \[perfect\]).
In order to illustrate our results, two examples are worth presenting immediately. Other examples will be reviewed in section \[final\].
Let $\mathbf w=(w_n)_{n\geq 1}$ be a bounded sequence of nonzero complex numbers, and let $B_{\bf w}$ be the associated *weighted backward shift* acting on $X_p=\ell^p(\NN)$, $1\leq p<\infty$ or $X_\infty=c_0(\NN)$; that is, $B_{\bf w}(x_0,x_1,x_2,\dots )=(w_1x_1,w_2x_2,\dots ).$ Solving the equation $B_{\mathbf w}(x)=\lambda x$, it is easy to check that $B_{\mathbf w}$ has eigenvalues of modulus 1 if and only if $$\label{shift}
\hbox{the sequence $\displaystyle\left(\frac{1}{w_0\cdots w_n}\right)_{n\geq 0}$ is in $X_p$}$$ (we have put $w_0:=1$). In this case | 1 | member_39 |
the formula $$E(\lambda):=\sum_{n=0}^\infty \frac{\lambda^n}{w_0\cdots w_n}\, e_n$$ defines a continuous $\TT$-eigenvector field $E:\TT\to X_p$ such that $\overline{\rm span}\, E(\TT)=X_p$. Hence $B_{\mathbf w}$ is strongly mixing in the Gaussian sense. This is known since [@BG2] if $p<\infty$, but it appears to be new for weighted shifts on $c_0(\NN)$.
The converse is true if $p\leq 2$ ( (\[shift\]) is satisfied if $B_{\mathbf w}$ is strongly mixing in the Gaussian sense) since in this case $X_p$ has cotype 2, but the case $p>2$ is not covered by Theorem \[WS\]. However, it turns out that the converse does hold true for any $p<\infty$. In fact, (\[shift\]) is satisfied as soon as the weighted shift $B_{\mathbf w}$ is frequently hypercyclic ([@BR]). As shown in [@BG2], this breaks down completely when $p=\infty$: there is a frequently hypercyclic weighted shift $B_{\mathbf w}$ on $c_0(\NN)$ whose weight sequence satisfies $w_1\cdots w_n=1$ for infinitely many $n$. Such a weighted shift does not admit any (nontrivial) invariant Gaussian measure.
Let us also recall that, in contrast with the ergodic properties, the hypercyclicity of $B_{\bf w}$ does not depend on $p$: by a well known result of H. Salas ([@Sal]), $B_{\mathbf w}$ is hypercyclic on $X_p$ for any $p$ if and only | 1 | member_39 |
if $\sup_{n\geq 1} \vert w_1\cdots w_n\vert=\infty$. Likewise, $B_{\mathbf w}$ is strongly mixing in the topological sense (on any $X_p$) iff $\vert w_1\cdots w_n\vert\to\infty$. Hence, we see that strong mixing in the topological sense turns out to be equivalent to strong mixing in the Gaussian sense for weighted shifts on $c_0(\NN)$.
Let $T$ be the operator defined by formula (\[Kal\]), but acting on $L^2(0,2\pi)$. It is straightforward to check that for any $t\in (0,2\pi)$, the function $f_t=\mathbf 1_{(0,t)}$ is an eigenvector for $T$ with associated eigenvalue $\lambda =e^{it}$. Moreover the map $E:\TT\setminus\{ \mathbf 1\}\to L^2(0,2\pi)$ defined by $E(e^{it})=f_t$ is clearly continuous. Now, let $\Lambda$ be an arbitrary compact subset of $\TT\setminus\{ \mathbf 1\}$. Let us denote by $H_\Lambda$ the closed linear span of $E(\Lambda)$, and let $T_\Lambda$ be the restriction of $T$ to $H_\Lambda$. By a result of G. Kalisch ([@Kal], see also [@BG1 Lemma 2.12]), the point spectrum of $T_\Lambda$ is exactly equal to $\Lambda$. By Corollary \[eigvectfield\] and Theorem \[WS\] (3), it follows that the operator $T_\Lambda$ is weakly mixing in the Gaussian sense if and only if $\Lambda$ is a perfect set, and strongly mixing iff $\Lambda$ is $\mathcal U_0$-perfect. Hence, any perfect $\mathcal U_0$-set $\Lambda$ gives rise | 1 | member_39 |
to a very simple example of a weakly mixing transformation which is not strongly mixing. This could be of some interest since the classical concrete examples are arguably more complicated (see the one given in [@P section 4.5]).
Regarding the difference between weak and strong mixing, it is also worth pointing out that there exist Hilbert space operators which are weakly mixing in the Gaussian sense but not even strongly mixing in the [topological]{} sense. Indeed, in the beautiful paper [@BadG], C. Badea and S. Grivaux are able to construct a weakly mixing operator (in the Gaussian sense) which is *partially power-bounded*, $\sup_{n\in I} \Vert T^n\Vert<\infty$ for some infinite set $I\subset\NN$. This line of investigations was pursued even much further in [@BadG2] and [@EG].
We have deliberately stressed the formal analogy between weak and strong mixing in the statement of Theorem \[WS\]. In view of this analogy, it should not come as a surprise that Theorem \[WS\] can be deduced from some more general results dealing with abstract notions of “mixing". (In order not to make this introduction exceedingly long, these results will be described in the next section). In particular, (1) and (2) are formal consequences of Theorem \[abstract\] | 1 | member_39 |
below. However, even though the proof of Theorem \[abstract\] is “conceptually" simple, the technical details make it rather long. This would be exactly the same for the strong mixing case (i.e. part (2) of Theorem \[WS\]), but in the weak mixing case it is possible to give a technically much simpler and hence much shorter proof. For the sake of readability, it seems desirable to present this proof separately. But since there is no point in repeating identical arguments, we shall follow the abstract approach as long as this does not appear to be artificial.
The paper is organized as follows. In section 2, we present our abstract results. In section 3, we review some basic facts concerning Gaussian measures and we outline the strategy for proving the abstract results and hence Theorem \[WS\]. Apart from some details in the presentation and the level of generality, this follows the scheme described in [@BG3], [@BG2] and [@BM]. In section 4, we prove part (1) of Theorem \[WS\] (the sufficient condition for weak mixing). The abstract results are proved in sections \[proofabstract1\] and \[proofabstract2\]. Section \[final\] contains some additional examples and miscellaneous remarks. In particular, we briefly discuss the “continous" analogues of | 1 | member_39 |
our results (i.e. the case of $1$-parameter semigroups), and we show that for a large class of strongly mixing weighted shifts, the set of hypercyclic vectors turns out to be rather small, namely Haar-null in the sense of Christensen. We conclude the paper with some possibly interesting questions.
[**Notation and conventions.**]{} The set of natural numbers is denoted either by $\NN$ or by $\ZZ_+$. We denote by $\mathcal M(\TT)$ the space of all complex measures on $\TT$, endowed with total variation norm. The Fourier transform of a measure $\sigma\in\mathcal M(\TT)$ is denoted either by $\widehat\sigma$ or by $\mathcal F(\sigma)$. As a rule, all measurable spaces $(\Omega,\mathfrak A)$ are standard Borel, and all measure spaces $(\Omega,\mathfrak A, m)$ are sigma-finite. All Hilbert spaces $\mathcal H$ are (complex) separable and infinite-dimensional. The scalar product $\langle u,v\rangle_{\mathcal H}$ is linear with respect to $u$ and conjugate-linear with respect to $v$.
Abstract results
================
$\mathbf S$-mixing
------------------
It is well known (and easy to check) that the definitions of ergodicity, weak and strong mixing can be reformulated as follows. Let $(X,\mathfrak B,\mu)$ be a probability space, and set $$L^2_0(\mu):=\left\{ f\in L^2(\mu);\; \int_X f\, d\mu=0\right\} .$$ Then, a measure-preserving transformation $T:(X,\mathfrak B,\mu)\to(X,\mathfrak B,\mu)$ is ergodic | 1 | member_39 |
with respect to $\mu$ if and only if $$\frac{1}{N}\sum_{n=0}^{N-1} \langle f\circ T^n,g\rangle_{L^2(\mu)}\xrightarrow{N\to\infty} 0$$ for any $f,g\in L^2_0(\mu)$. The transformation $T$ is weakly mixing iff $$\frac{1}{N}\sum_{n=0}^{N-1} \left\vert \langle f\circ T^n,g\rangle_{L^2(\mu)}\right\vert\xrightarrow{N\to\infty} 0$$ for any $f,g\in L^2_0(\mu)$, and $T$ is strongly mixing iff $$\langle f\circ T^n,g\rangle_{L^2(\mu)}\xrightarrow{n\to\infty} 0\, .$$
Now, let us denote by $V_T:L^2(\mu)\to L^2(\mu)$ the Koopman operator associated with a measure-preserving transformation $T:(X,\mathfrak B,\mu)\to(X,\mathfrak B,\mu)$, the isometry defined by $$V_Tf=f\circ T\, .$$
For any $f,g\in L^2(\mu)$, there is a uniquely defined complex measure $\sigma_{f,g}=\sigma_{f,g}^T$ on $\TT$ such that $$\widehat\sigma_{f,g} (n)=\left\{
\begin{matrix} \langle V_T^n f,g\rangle_{L^2(\mu)}&{\rm if}&n\geq 0\\
\langle V_T^{*\vert n\vert} f,g\rangle_{L^2(\mu)}&{\rm if}&n< 0
\end{matrix}
\right.$$ (When $f=g$, this follows from Bochner’s theorem because in this case the right-hand side defines a positive-definite function on $\ZZ$; and then one can use a “polarization" argument). We denote by $\Sigma(T,\mu)$ the collection of all measures $\sigma_{f,g}$, $f,g\in L^2_0(\mu)$, and forgetting the measure $\mu$ we refer to $\Sigma(T,\mu)$ as “the spectral measure of $T$".
With these notations, we see that $T$ is weakly mixing with respect to $\mu$ iff all measures $\sigma\in \Sigma(T,\mu)$ are continuous (by Wiener’s lemma), and that $T$ is strongly mixing iff all measures $\sigma\in \Sigma(T,\mu)$ are Rajchman (by Rajchman’s lemma). Likewise, $T$ is | 1 | member_39 |
ergodic iff $\sigma (\{ \mathbf 1\})=0$ for every $\sigma\in \Sigma(T,\mu)$.
More generally, given any family of measures $\mathcal B\subset\mathcal M(\TT)$, one may say that $T$ is *$\mathcal B$-mixing* with respect to $\mu$ if the spectral measure of $T$ lies in $\mathcal B$, all measures $\sigma\in\Sigma(T,\mu)$ are in $\mathcal B$. We shall in fact consider a more specific case which seems to be the most natural one for our concerns. Let us denote by $\mathcal F_+:\mathcal M(\TT)\to \ell^\infty(\ZZ_+)$ the positive part of the Fourier transformation, i.e. $\mathcal F_+(\sigma)= \widehat\sigma_{\vert \ZZ_+}$.
Given any family $\mathbf S\subset\ell^\infty(\ZZ_+)$, we say that a measure $\sigma\in\mathcal M(\TT)$ is *$\mathbf S$-continuous* if $\mathcal F_+(\sigma)\in\mathbf S$. A measure-preserving transformation $T:(X,\mu)\to (X,\mu)$ is *$\mathbf S$-mixing* with respect to $\mu$ if every measure $\sigma \in\Sigma (T,\mu)$ is $\mathbf S$-continuous.
Thus, strong mixing is just $\mathbf S$-mixing for the family $\mathbf S=c_0(\ZZ_+)$, weak mixing is $\mathbf S$-mixing for the the family $\mathbf S$ of all sequences $(a_n)\in\ell^\infty(\ZZ_+)$ such that $\vert a_n\vert\to 0$ in the Cesàro sense, and ergodicity corresponds to the family $\mathbf S$ of all $a\in\ell^\infty (\ZZ_+)$ tending to $0$ in the Cesàro sense. In what follows, these families will be denoted by $\mathbf S_{\rm strong}$, $\mathbf S_{\rm weak}$ and | 1 | member_39 |
$\mathbf S_{\rm erg}$, respectively.
Small subsets of the circle
---------------------------
Given a family of measures $\mathcal B\subset\mathcal M(\TT)$, it is quite natural in harmonic analysis to try to say something about the *$\mathcal B$-small* subsets of $\TT$, i.e. the sets $D\subset\TT$ that are annihilated by all positive measures $\sigma\in\mathcal B$. By this we mean that for any such measure $\sigma$, one can find a Borel set $\widetilde D$ (possibly depending on $\sigma$) such that $D\subset\widetilde D$ and $\sigma (\widetilde D)=0$. When the family $\mathcal B$ has the form $\mathcal B=\mathcal F_+^{-1}(\mathbf S)$ for some $\mathbf S\subset\ell^\infty (\ZZ_+)$, we call these sets *$\mathbf S$-small*.
To avoid trivialities concerning $\mathcal B$-small sets, the family $\mathcal B$ under consideration should contain nonzero [positive]{} measures, and in fact it is desirable that it should be *hereditary* with respect to absolute continuity; that is, any measure absolutely continuous with respect to some $\sigma\in\mathcal B$ is again in $\mathcal B$. The following simple lemma shows how to achieve this for families of the form $\mathcal F_{+}^{-1}(\mathbf S)$. Let us say that a family $\mathbf S\subset\ell^\infty(\ZZ_+)$ is *translation-invariant* if it is invariant under both the forward and the backward shift on $\ell^\infty(\ZZ_+)$.
\[hereditary\] If $\mathbf S$ is | 1 | member_39 |
a translation-invariant linear subspace of $\ell^\infty(\ZZ_+)$ such that $\mathcal F_{+}^{-1}(\mathbf S)$ is norm-closed in $\mathcal M(\TT)$, then $\mathcal F_{+}^{-1}(\mathbf S)$ is hereditary with respect to absolute continuity.
If $\sigma\in\mathcal F_+^{-1}(\mathbf S)$ then $P\sigma$ is in $\mathcal F_+^{-1}(\mathbf S)$ for any trigonometric polynomial $P$, by translation-invariance. So the result follows by approximation.
We shall also make use of the following well known result concerning *$\mathcal B$-perfect* sets. By definition, a set $\Lambda\subset \TT$ is $\mathcal B$-perfect if $V\cap\Lambda$ is not $\mathcal B$-small for any open set $V\subset\TT$ such that $V\cap\Lambda\neq\emptyset$.
\[Bperfect\] Let $\mathcal B$ be a norm-closed linear subspace of $\mathcal M(\TT)$, and assume that $\mathcal B$ is hereditary with respect to absolute continuity. For a closed set $\Lambda\subset\TT$, the following are equivalent:
- $\Lambda$ is $\mathcal B$-perfect;
- $\Lambda$ is the support of some probability measure $\sigma\in\mathcal B$.
That (b) implies (a) is clear (without any assumption on $\mathcal B$). Conversely, assume that $\Lambda$ is $\mathcal B$-perfect. Let $(W_n)_{n\geq 1}$ be a countable basis of open sets for $\Lambda$, with $W_n\neq \emptyset$. Since $\mathcal B$ is hereditary, one can find for each $n$ a probability measure $\sigma_n\in\mathcal B$ such that ${\rm supp}(\sigma_n)\subset\Lambda$ and $\sigma_n (W_n)>0$. Then the probability measure $\sigma=\sum_1^\infty | 1 | member_39 |
2^{-n}\sigma_n$ is in $\mathcal B$ and ${\rm supp}(\sigma)=\Lambda$.
The results
-----------
To state our results we need two more definitions.
\[defBperfspan\] Let $T$ be an operator acting on a complex separable Fréchet space $X$, and let $\mathcal B\subset\mathcal M(\TT)$. We say that the $\TT$-eigenvectors of $T$ are *$\mathcal B$-perfectly spanning* if, for any Borel $\mathcal B$-small set $D\subset \TT$, the linear span of $\bigcup_{\lambda\in\TT\setminus D}\ker (T-\lambda)$ is dense in $X$. When $\mathcal B$ has the form $\mathcal F_+^{-1}(\mathbf S)$, the terminology *$\mathbf S$-perfectly spanning* is used.
Thus, perfect spanning is $\mathcal B$-perfect spanning for the family of continuous measures, and $\mathcal U_0$-perfect spanning is $\mathcal B$-perfect spanning for the family of Rajchman measures.
At some places, we will encounter sets which are *analytic* but perhaps non Borel. Recall that a set $D$ in some Polish space $Z$ is [analytic]{} if one can find a Borel relation $B\subset Z\times E$ (where $E$ is Polish) such that $z\in D\Leftrightarrow\exists u\;:\; B(z,u)$. If the spanning property of the above definition holds for every analytic $\mathcal B$-small set $D$, we say that the $\TT$-eigenvectors of $T$ are [$\mathcal B$-perfectly spanning]{} *for analytic sets*.
We shall say that a family $\mathbf S\subset\ell^\infty(\ZZ_+)$ is *$c_0$-like* if | 1 | member_39 |
it has the form $$\mathbf S=\{ a\in\ell^\infty(\ZZ_+);\; \lim_{n\to\infty} \Phi_n(a)=0\}\,$$ where $(\Phi_n)_{n\geq 0}$ is a uniformly bounded sequence of $w^*$-$\,$continuous semi-norms on $\ell^\infty(\ZZ_+)$. (By “uniformly bounded", we mean that $\Phi_n(a)\leq C\, \Vert a\Vert_\infty$ for all $n$ and some finite constant $C$).
For example, the families $\mathbf S_{\rm strong}$, $\mathbf S_{\rm weak}$ and $\mathbf S_{\rm erg}$ are $c_0$-like: just put $\Phi_n(a)=\vert a_n\vert$ in the strong mixing case, $\Phi_n(a)=\frac{1}{n}\sum_{k=0}^{n-1}\vert a_k\vert$ in the weak mixing case, and $\Phi_n(a)=\left\vert \frac{1}{n}\sum_{k=0}^{n-1} a_k\right\vert$ in the ergodic case.
Our main result is the following theorem, from which (1) and (2) in Theorem \[WS\] follow immediately. Recall that a family $\mathbf S\subset \ell^\infty(\ZZ_+)$ is an *ideal* if it is a linear subspace and $ua\in \mathbf S$ for any $(u,a)\in\ell^\infty(\ZZ_+)\times\mathbf S$.
\[abstract\] Let $X$ be a separable complex Fréchet space, and let $T\in\mathfrak L(X)$. Let also $\mathbf S\subset \ell^\infty (\ZZ_+)$. Assume that $\mathbf S$ is a translation-invariant and $c_0$-like [ideal]{}, and that any $\mathbf S$-continuous measure is continuous. If the $\TT$-eigenvectors of $T$ are $\mathbf S$-perfectly spanning, then $T$ is $\mathbf S$-mixing in the Gaussian sense.
This theorem cannot be applied to the ergodic case, for two reasons: the family $\mathbf S_{\rm erg}$ is not an ideal of $\ell^\infty (\ZZ_+)$, | 1 | member_39 |
and $\mathbf S_{\rm erg}$-continuous measures need not be continuous (a measure $\sigma$ is $\mathbf S_{\rm erg}$-continuous iff $\sigma(\{ \mathbf 1\})=0$).
Theorem \[abstract\] will be proved in section \[proofabstract1\]. The following much simpler converse result (which corresponds to (3) in Theorem \[WS\]) will be proved in section \[proofabstract2\].
\[converse\] Let $X$ be a separable complex Banach space, and assume that $X$ has cotype 2. Let also $\mathbf S$ be an arbitrary subset of $\ell^\infty(\ZZ_+)$. If $T\in\mathfrak L(X)$ is $\mathbf S$-mixing in the Gaussian sense, then the $\TT$-eigenvectors of $T$ are $\mathbf S$-perfectly spanning for analytic sets.
Two “trivial" cases are worth mentioning. If we take $\mathbf S=\mathbf S_{\rm erg}$, then $\mathcal F_+^{-1}(\mathbf S)=\{ \sigma\in\mathcal M(\TT);\; \sigma (\{\mathbf 1\})=0\}$ and hence a set $D\subset\TT$ is $\mathbf S$-small if and only if $D\subset\{\mathbf 1\}$. If we take $\mathbf S=\ell^\infty (\ZZ_+)$, then $\mathcal F_+^{-1}(\mathbf S)=\mathcal M(\TT)$ and hence the only $\mathbf S$-small set is $D=\emptyset$. So, assuming that $X$ has cotype $2$, we get the following (known) results: if $T$ admits an invariant Gaussian measure with full support, then the $\TT$-eigenvectors of $T$ span a dense subspace of $X$; and if $T$ is ergodic in the Gaussian sense, then ${\overline{\rm span}}\,\left(\bigcup_{\lambda\in\TT\setminus\{ \mathbf 1\}}\ker(T-\lambda)\right)=X$. As | 1 | member_39 |
already pointed out, much more is true in the ergodic case: it is proved in [@BG2 Theorem 4.1] that if $T$ is ergodic in the Gaussian sense, then in fact the $\TT$-eigenvectors are perfectly spanning (provided that $X$ has cotype 2).
Our last result (to be proved also in section \[proofabstract2\]) says that when the space $X$ is well-behaved, the assumptions in Theorem \[abstract\] can be relaxed: it is no longer necessary to assume that the family $\mathbf S$ is $c_0$-like, nor that $\mathbf S$-continuous measures are continuous. However, the price to pay is that one has to use the strengthened version of $\mathbf S$-perfect spanning.
\[abstracteasy\] Let $X$ be a separable complex Banach space, and assume that $X$ has *type 2*. Let also $\mathbf S$ be a norm-closed, translation-invariant ideal of $\ell^\infty(\ZZ_+)$, and let $T\in\mathfrak L(X)$. If the $\TT$-eigenvectors of $T$ are $\mathbf S$-perfectly spanning for analytic sets, then $T$ is $\mathbf S$-mixing in the Gaussian sense.
Since Hilbert space has both type 2 and cotype 2, we immediately deduce
\[Smix=span\] Let $\mathbf S$ be a norm-closed, translation-invariant ideal of $\ell^\infty(\ZZ_+)$. For Hilbert space operators, $\mathbf S$-mixing in the Gaussian sense is equivalent to the $\mathbf S$-spanning property of | 1 | member_39 |