# Microcanonical windows on quantum operators

Silvia Pappalardi<EMAIL_ADDRESS>Laboratoire de Physique de
l’École Normale Supérieure, ENS, Université PSL, CNRS, Sorbonne Université,
Université de Paris, F-75005 Paris, France  Institut für Theoretische Physik,
Universität zu Köln, Zülpicher Straße 77, 50937 Köln, Germany  Laura Foini
IPhT, CNRS, CEA, Université Paris Saclay, 91191 Gif-sur-Yvette, France  Jorge
Kurchan  Laboratoire de Physique de l’École Normale Supérieure, ENS,
Université PSL, CNRS, Sorbonne Université, Université de Paris, F-75005 Paris,
France

###### Abstract

We discuss a construction of a microcanonical projection WOW of a quantum
operator O induced by an energy window filter W, its spectrum, and the
retrieval of canonical many-time correlations from it.

###### Contents

  1. 1 Introduction
  2. 2 Why WOW: a summary of the results
    1. 2.1 Correlations on the energy shell
    2. 2.2 The problem
    3. 2.3 Results
  3. 3 The example of two-point functions
    1. 3.1 Window on operator
    2. 3.2 Not all windows are equal
      1. 3.2.1 Box window
      2. 3.2.2 Smooth and Gaussian window
  4. 4 Higher order correlations
  5. 5 The WOW spectrum and generating functions
    1. 5.1 Generating functions for WOW
    2. 5.2 Other generating functions
  6. 6 Discussion

##  1 Introduction

Quantum thermalization and dynamics at equilibrium are nowadays well
understood in terms of the matrix elements of physical operators in the energy
eigenbasis, i.e. $O_{ij}=\langle E_{i}|O|E_{j}\rangle$ [1, 2, 3]. These behave
as random matrices with correlated elements, whose smooth statistical
properties encode all the physical features [4]. Understanding the nature of
the randomness in $O_{ij}$ and how to extract from them or their spectrum
meaningful information is still an open, largely debated issue [5, 6, 7, 8, 9,
10, 11, 12, 13, 14, 15]. The purpose of this paper is to discuss the
construction of a microcanonical projection of an operator $O$, in the
thermodynamic limit:

$\tilde{O}\equiv W^{1/2}O\,W^{1/2}\ ,$ (1)

referred to as WOW, where $W$ is some ‘window’ operator which is centred
around an energy $E_{0}$, as illustrated in Fig.1.

The rules we set for this exercise are that (i) the projected operator
$\tilde{O}$ is such that its dynamical correlators among $n$-times allow one
to retrieve the corresponding canonical $n$-time correlators of $O$, and that
(ii) the spectrum of $\tilde{O}$ provides a meaningful large-deviation
function. For example, defining the commutator operator

${\cal{C}}\equiv[\tilde{O}(t),\tilde{O}]_{-}$

the spectrum of the operator ${\cal{C}}$ contains all the ‘multifractality’
properties of the quantum generalized Lyapunov exponent, through its moments
$\mathrm{Tr}\left\\{{\cal{C}}^{2n}\right\\}$, for $n>0$ [16].

It turns out that for this to work, as is to be expected, the width of the
energy window is not arbitrary, but, somewhat surprisingly, also its
smoothness properties have important consequences. We exemplify numerically
our findings on the Ising model with transverse and longitudinal fields.

Figure 1: Pictorial representation of the observable $O$ and of the
microcanonical projected WOW. (a) The matrix elements in the energy eigenbasis
$O_{ij}=\langle E_{i}|O|E_{j}\rangle$ look like structured random matrices
characterized by an intrinsic, observable dependent, energy scale
$\omega_{\max}$ after which they decay to zero. (b-c) Observable
$W^{1/2}OW^{1/2}$ projected in an energy window around $E_{0}$ with width
$\Delta$. (b) Box filter $W$: the microcanonical operator is given only matrix
elements with energies inside the box $|E_{i,j}-E_{0}|<\Delta/2$. (c) Gaussian
filter $W$: the observable is multiplied by a Gaussian function in both
directions $i,j$.

##  2 Why WOW: a summary of the results

We set ourselves to construct an operator which is a microcanonical
restriction of $O$ around some energy $E_{0}$. This operator shall not only
give the microcanonical expectation value, i.e. $\langle
E_{0}|O|E_{0}\rangle$, but should also reproduce all correlations _on the
energy shell_ of $E_{0}$.

###  2.1 Correlations on the energy shell

A way to introduce correlations on an energy shell is to look at the following
_canonical regularized correlator_

$M_{n}(\vec{t})=\mathrm{Tr}\left(\rho^{\frac{1}{n}}O(t_{1})\rho^{\frac{1}{n}}O(t_{2})\dots\rho^{\frac{1}{n}}O(t_{n})\right)\
,$ (2)

where $\rho=e^{-\beta H}/Z$ and $Z=\text{Tr}(e^{-\beta H})$ with $\beta$ the
inverse temperature, and $O(t)=e^{-iHt}Oe^{-iHt}$ for any (eventually complex)
$\vec{t}=(t_{1},t_{2},\dots t_{n})$ times 111We have absorbed here $\hbar$
into the $\vec{t}$, so that time has units of inverse energy. , where we set
$\hbar=1$. While familiar in high energy physics [17], these regulated
correlation functions may seem odd for a statistical physicist. To mitigate
the understandable prejudice, let us note that this correlator is related to
the more usual thermal average

$S_{n}(\vec{t})=\mathrm{Tr}\left(\rho O(t_{1})O(t_{2})\dots
O(t_{n-1})O(t_{n})\right)\ ,$ (3)

by a simple shift in imaginary time: Defining
$\vec{\ell}_{n}=(\frac{n-1}{n},\frac{n-2}{n},\dots,\frac{1}{n},0)$, one has

$S_{n}(\vec{t})=M_{n}(\vec{t}+i\beta\vec{\ell}_{n})\ ,$ (4)

which in frequency $\vec{\omega}=(\omega_{1},\dots,\omega_{n})$ reads

$S_{n}(\vec{\omega})=M_{n}(\vec{\omega})e^{-\beta\vec{\ell}_{n}\cdot\vec{\omega}}\
,$ (5)

thus making explicit the fluctuation-dissipation theorem [18] 222 With this
shift, one rephrases the usual Kubo-Martin Schwinger (KMS) relation as a a
time-reversal condition for Eq.(2):
$M_{n}(t_{1},...,t_{n})=\left\\{M_{n}(t_{n},...,t_{1})\right\\}^{*}$.. At
small frequencies, the two correlators coincide. Thus, the regulated
correlation function $M_{n}(\vec{t})$ may be expected to share the same
dynamics of the canonical one $S_{n}(\vec{t})$ at large times, exceptions may
be related to quantum bounds on time scales [17, 19, 20].

The _thermal projection_

$\rho^{\frac{1}{2n}}\,O(t)\,\rho^{\frac{1}{2n}}$ (6)

plays the role of constraining the support of each observable $O(t)$ on the
thermal energy shell: the correlator in Eq.(2) is evaluated on the shell
identified by the thermal energy $E_{\beta}=\mathrm{Tr}(He^{-\beta H})/Z$.
Reproducing this feature with a fixed, $n$-independent window is the focus of
this article.

Another way to discuss on-shell corrections is via the _quantum microcanonical
ensemble_ , nowadays well understood via the Eigenstate-Thermalization-
Hypothesis (ETH) [1, 2, 3, 4]. The general form of ETH states that observables
in the energy eigenbasis are pseudorandom matrices, whose statistical
properties are smooth functions. Specifically, products of $n$ matrix elements
with different indices read [4]

$\overline{O_{i_{1}i_{2}}O_{i_{2}i_{3}}\dots
O_{i_{n}i_{1}}}=e^{-(n-1)S(E^{+})}F_{e^{+}}^{(n)}(\vec{\omega})\ ,$ (7)

while products with repeated indices factorize in the large $N$ limit, in a
manner described by free probability [21]. Here,
$E^{+}=(E_{i_{1}}+\dots+E_{i_{n}})/n$ is the average energy,
$\vec{\omega}=(\omega_{i_{1}i_{2}},\dots,\omega_{i_{n-1}i_{n}})$ with
$\omega_{ij}=E_{i}-E_{j}$ are $n-1$ energy differences. The
$F^{(n)}_{e^{+}}(\vec{\omega})$ are smooth functions of the energy density
$e^{+}=E^{+}/N$ and $\vec{\omega}$ and correspond to the microcanonical (on-
shell) correlations of order $n$ at energy $e_{+}$.

The relation between the microcanonical definition via Eq.(7) and the
canonical one in Eq.(2) is known. As reviewed below, the ETH smooth functions
in Eq.(7) can be expressed as the Fourier transform of the following
correlator

$\displaystyle\overline{k}_{n}(\vec{t})$
$\displaystyle\equiv\mathrm{Tr}^{\prime}\left(\rho^{\frac{1}{n}}O(t_{1})\rho^{\frac{1}{n}}O(t_{2})\dots\rho^{\frac{1}{n}}O(t_{n})\right)=\int
d\vec{t}e^{i\vec{t}\cdot\vec{\omega}}\,\,F^{(n)}_{\beta}(\vec{\omega})$ (8)

where the notation $\mathrm{Tr}^{\prime}$ indicates the trace constrained to
matrix elements with different indices, e.g.
$\mathrm{Tr}^{\prime}(ABC)=\sum_{i\neq j\neq k}A_{ij}B_{jk}C_{ki}$. These
correlations are related to the thermal free cumulants [21] (a type of
connected correlation function defined in Free Probability [22]) and
constitute the building block of multi-time correlation functions. In Eq.(8)
and in what follows, we use $\beta$ intended as an implicit function of the
energy density $e_{\beta}=E_{\beta}/N$, e.g.
$F^{(n)}_{\beta}(\vec{\omega})=F^{(n)}_{e_{\beta}}(\vec{\omega})$.

The behaviour of all the $F_{\beta}^{(n)}(\vec{\omega})$ as functions of
frequencies encode all the physical properties, including some that are
neglected if elements are taken as independent [4, 21]. In Hamiltonian
systems, $F^{(n)}(\vec{\omega})$ decays rapidly at large frequencies,
reflecting the property that correlations between matrix elements with very
different energies shall be small. The case $n=2$ has been related to the
“operator growth hypothesis” which conjectures a universal exponential decay
for $F^{(2)}_{E}(\omega)$ in the case of chaotic systems [23, 24], see also
[25]. This structure has been discussed also for higher $n$ [26]: at large
frequencies, all $F^{(n)}(\vec{\omega})$ should fall at least as

$F^{(n)}(\vec{\omega})\sim e^{-|\omega_{i}|/\omega_{\max}}\quad\text{in all
directions }\omega_{i}\ ,$

where [26]

$\omega_{\max}\leq\frac{1}{\beta\hbar}\ .$ (9)

###  2.2 The problem

The issue with the definition in Eq.(2) is that the thermal projection (6)
depends on $n$: each $n$-th correlation function needs a different projection.
This poses the question: can we replace instead, for the purposes of computing
all $n$ on-shell correlation functions, the operator $O$ by a single projected
version?

In this work, we consider an _energy ‘window’ operator_

$W=W_{\Delta}(H-E_{0})$ (10)

centered around energy $E_{0}$ with width $\Delta$, such that $[W,H]=0$. The
window is a filter in energy around $E_{0}$ and on the basis of the
Hamiltonian reads

$W=\sum_{i}p_{i}|E_{i}\rangle\langle E_{i}|\ ,\qquad p_{i}=p(E_{i})\geq 0\ ,$
(11)

with a distribution $p_{i}$, which we leave unnormalized. Standard choices
include “box projectors” ($p_{i}=1$ for $|E_{i}-E_{0}|\leq\Delta$ and
vanishing otherwise), Gaussian ones
($p_{i}=e^{-(E_{i}-E_{0})^{2}/2\Delta^{2}}$), or more refined cosine filters
[27, 28]. These choices have an important impact on the results, as we shall
see. The window $W$ allows us to introduce the _microcanonical projection_

$\tilde{O}=W^{\frac{1}{2}}OW^{\frac{1}{2}}$ (12)

_whose eigenvalues and eigenvectors have information that apply to all $n$_,
unlike those of $\rho^{1/2n}O\rho^{1/2n}$. We will study dynamic correlations

$M^{\mbox{\tiny{W}}}_{n}(\vec{t})=\frac{1}{\mathrm{Tr}W}\mathrm{Tr}\left(\tilde{O}(t_{1})\tilde{O}(t_{2})\dots\tilde{O}(t_{n})\right)\
.$ (13)

###  2.3 Results

We show that the microcanonical operator $\tilde{O}$ can indeed be used to
compute on-shell correlations functions, namely

$\displaystyle\frac{1}{\mathrm{Tr}W}\mathrm{Tr}$
$\displaystyle\left(\tilde{O}(t_{1})\tilde{O}(t_{2})\dots\tilde{O}(t_{n})\right)\leftrightarrow\mathrm{Tr}\left(\rho^{\frac{1}{n}}O(t_{1})\rho^{\frac{1}{n}}O(t_{2})\dots\rho^{\frac{1}{n}}O(t_{n})\right)$
(14)

i.e. $M_{n}^{\mbox{\tiny{W}}}(\vec{t})\leftrightarrow M_{n}(\vec{t})$,
provided that the window $W$ at energy $E_{0}=E_{\beta}$ is chosen
appropriately. Our main findings below are summarized as follows:

  1. 1.

The smoothness properties of window filter $W$ are important. We discuss how
the box projector lead to dynamical effects introduced by the window itself, a
familiar fact in signal theory.

  2. 2.

The width $\Delta$ of the window should be _small_ enough to be a good
projector, but _larger_ than the characteristic energy scale $\omega_{\max}$
so as not to affect the physical correlations. One has

$\beta\omega_{\max}\ll\beta\Delta\ll\sqrt{N}\ ,$ (15)

($N$ is the number of degrees of freedom) a parametrically large choice of
$\Delta$ where the window is well defined.

  3. 3.

When the conditions 1. and 2. are met, the relation in Eq.(14) is given by

$\displaystyle M_{n}^{\mbox{\tiny{W}}}(\vec{t})$
$\displaystyle=\frac{\mathrm{Tr}(W^{n})}{\mathrm{Tr}(W)}M_{n}(\vec{t})\ ,$
(16)

where
$\frac{\mathrm{Tr}(W^{n})}{\mathrm{Tr}(W)}=e^{-\beta^{2}\Delta^{2}(1-1/n)}/\sqrt{n}$
in the case of a Gaussian window.

  4. 4.

Having a microcanonical operator $\tilde{O}$ leads us to consider the
associated _microcanonical operator spectral density_ defined as

$\rho(\alpha)\equiv\frac{1}{\mathrm{Tr}W}\sum_{i}\delta(\alpha-a_{i})\ ,$ (17)

where $a_{i}$ are the eigenvalues of $\tilde{O}$. Actually, we will focus on
$\alpha\rho(\alpha)$ which has a good limit for $\alpha\to 0$ and $\Delta$ as
in Eq.(15). We further discuss the generating functions of equal-time moments.

  5. 5.

On-shell correlations such as (13) are related to the usual microcanonical
average, e.g.

$S^{\mbox{\tiny{W}}}_{2}(t_{1}-t_{2})=\frac{1}{\mathrm{Tr}W}\mathrm{Tr}\left(W^{2}O(t_{1})O(t_{2})\right)$
(18)

in the frequency domain by

$S^{\mbox{\tiny{W}}}_{2}(\omega)=\frac{e^{S(E_{0}-\omega)}}{e^{S(E_{0}-\omega/2)}}M^{\mbox{\tiny{W}}}_{2}(\omega)\
,$ (19)

which plays the role of a _microcanonical fluctuation-dissipation theorem_ [3,
29].

We begin our paper by considering the more familiar case of two-time
correlation functions (Section 3). We then introduce free cumulants and
discuss results for $n>2$ (Section 4) and we study the spectral properties
(Section 5).

Our results are tested numerically on a paradigmatic model of many-body
quantum chaos: the Ising spin chain with longitudinal and transverse field,
described by the Hamiltonian

$H=\sum_{i=1}^{L}w\sigma_{i}^{x}+\sum_{i=1}^{L}h\sigma_{i}^{z}+\sum_{i=1}^{L}J\sigma_{i}^{z}\sigma_{i+1}^{z}\
,$ (20)

where $\sigma_{i}^{\alpha}$ are Pauli matrices on site $i$ in the direction
$\alpha=x,y,z$. We measure the energies in units of $J$ and set
$w=\sqrt{5}/2$, $h=(\sqrt{5}+5)/8$. With periodic boundary conditions, this
system is characterized by translational and inversion symmetry. We consider
the subsector at fixed $k=0$ momentum and the even inversion subsector.

##  3 The example of two-point functions

Most recent studies of quantum ensembles at equilibrium have focused on one-
time expectation values, both in the canonical
$M_{1}=\mathrm{Tr}(\rho))=S_{1}$ and in the microcanonical ensemble
$M^{\mbox{\tiny{W}}}_{1}=\mathrm{Tr}(WO)/\mathrm{Tr}W$ [3]. For example, the
issue of how to implement a smooth microcanonical filter $W$ to compute
efficiently $M^{\mbox{\tiny{W}}}_{1}$ has been posed lately in Refs.[27, 28].
In this section, we ask this question for two-time correlation functions.
These have been studied numerically using for instance microcanonical Lancsos
methods Refs.[30, 31, 32].

We start with the regularized and standard thermal canonical two-times
functions respectively:

$\displaystyle M_{2}(t_{1}-t_{2})$
$\displaystyle=\text{Tr}\left(\rho^{1/2}O(t_{1})\rho^{1/2}O(t_{2})\right)\ ,$
(21) $\displaystyle S_{2}(t_{1}-t_{2})$
$\displaystyle=\text{Tr}\left(\rho\,O(t_{1})O(t_{2})\right)\ .$ (22)

Since these depend only on the time differences $t=t_{1}-t_{2}$, we may write
everything in terms of independent time differences $t$, or set $t_{2}=0$,
without loss of generality. The same is true for the $n$-th correlation
function in Eqs.(2)-(3) and in what follows we will put $t_{n}=0$. The on-
shell function $M_{2}(t)$ is related to the canonical thermal average
$S_{2}(t)$ by a shift in imaginary time $S_{2}(t)=M_{2}(t+i\beta/2)$, namely

$\mathrm{Tr}\left(\rho
O(t)O(0)\right)=\text{Tr}\left(\rho^{1/2}O(t+i\beta/2)\rho^{1/2}A\right)\ .$

The two point functions can be expressed using ETH (7) for two matrix elements
[2]

$\displaystyle\overline{|O_{ii}|^{2}}$
$\displaystyle\simeq\overline{O_{ii}}\,^{2}\ ;\quad$ (23)
$\displaystyle\overline{|O_{ij}|^{2}}$
$\displaystyle=e^{-S(E_{ij}^{+})}\;F^{(2)}_{e_{ij}^{+}}(\omega_{ij})\quad
i\neq j\ ,$ (24)

where we recall that $\omega_{ij}=E_{i}-E_{j}$ and
$e_{ij}^{+}=\frac{E_{i}+E_{j}}{2N}$ the energy density and
$F^{(2)}_{e}(\omega)=|f_{O}(e,\omega)|^{2}$ with the standard notations [3].
By simple manipulations, one finds

$S_{2}(t)=[S_{1}]^{2}+k_{2}(t)\ ,$ (25)

where $S_{1}=\mathcal{O}(e_{\beta})$ is the one-point function and $k_{2}(t)$
is the connected part is the _second (free) cumulant_ , which encodes all the
information at the second order 333We recall that free cumulants coincide with
classical cumulants up to $n=3$ [22], so, in this case, the name is redundant,
but useful for later understanding.. Thanks to the smoothness properties of
ETH (23), it can be written as the constrained trace over different indices

$k_{2}(t)=\mathrm{Tr}^{\prime}\left(\rho O(t)O(0)\right)\equiv\sum_{i\neq
j}\frac{e^{-\beta E_{i}}}{Z}O_{ij}(t)O_{ij}\ ,$ (26)

where we recall that $\mathrm{Tr}^{\prime}(\bullet)$ denotes the trace of the
arguments with all different indices. Its Fourier transform reads

$k_{2}(\omega)=F^{(2)}_{\beta}(\omega)e^{-\beta\omega/2}\ .$ (27)

To isolate the on-shell correlation $F^{(2)}(\omega)$, one should look at the
following reduced trace of the thermally projected operator

$\displaystyle\begin{split}\overline{k}_{2}(t)\equiv\mathrm{Tr}^{\prime}\left(\rho^{1/2}O(t)\rho^{1/2}O(0)\right)\equiv&\sum_{i\neq
j}\frac{e^{-\beta(E_{i}+E_{j})/2}}{Z}O_{ij}(t)O_{ij}\end{split}$ (28)
$\displaystyle\to\overline{k}_{2}(\omega)$
$\displaystyle=F^{(2)}_{\beta}(\omega)\ ,$ (29)

which thus identifies the building block of two-point dynamical functions.
This expression shows that regularization of the correlation functions in
$M_{2}(t)$ (symmetrization in $\beta$) has the effect of removing the
exponential factor $e^{-\beta\omega/2}$ in the Fourier transform of the
connected trace $\overline{k}_{2}(\omega)$. This is why we referred to it as
correlation on the energy shell.

Summarizing, the dynamic features of two-point functions are encoded in the
second cumulants. Their structure is given by the trace $\mathrm{Tr}^{\prime}$
of the regularized correlator, reduced to different indices, whose Fourier
transform yields ETH on-shell correlations $F^{(2)}_{\beta}(\omega)$
[cf.Eq.(28)].

As mentioned above, ETH correlation functions are expected to decay fast at
large frequencies $\omega\gg 1$. Based on the growth of operators for chaotic
systems, a universal exponential decay has been conjectured [24] 444 In the
case of interacting integrable systems, there is evidence that
$F^{(2)}(\omega)$ has instead a Gaussian decay [24]. :

$F^{(2)}_{\beta}(\omega)\sim e^{-|\omega|/\omega^{(2)}_{\max}}\ .$ (30)

Using that $k_{2}(0)$ is finite, in Ref.[26] it was shown that

$\omega^{(2)}_{\max}\leq\frac{2}{\beta\hbar}\ .$ (31)

Therefore, throughout this paper, we shall assume that the frequency
$\omega\ll\omega^{(2)}_{\max}$ encompass all the interesting physical
properties of a system.

In summary, ETH tells us that the physical operators have, on the basis of the
Hamiltonian, a ‘band’ structure: the width of the band being $O(1)$ in energy,
while the variations parallel to the diagonal are much smoother, depending
continuously on energy density.

###  3.1 Window on operator

We consider the following microcanonical projection

$\tilde{O}=W^{\frac{1}{2}}OW^{\frac{1}{2}}$ (32)

where $W=W_{\Delta}(H-E_{0})$ is the generic window filter defined in Eq.(10),
well-peaked around energy $E_{0}$ and with a small width $\Delta/E_{0}\ll 1$.
The wish that filtered two-point correlator

$M_{2}^{\mbox{\tiny{W}}}(t)=\frac{1}{\mathrm{Tr}W}\text{Tr}\left(WO(t)WO\right)\
,$ (33)

reproduces the thermal regularized one [cf. (21)] and in particular the
connected part:
$\overline{k}_{2}(t)\equiv\mathrm{Tr}^{\prime}\left(\rho^{1/2}O(t)\rho^{1/2}O(0)\right)$.
Hence, we look at

$\overline{k}_{2}^{\mbox{\tiny{W}}}(t)\equiv\frac{1}{\mathrm{Tr}W}\text{Tr}^{\prime}\left(WO(t)WO\right)\
,$ (34)

where the trace is constrained to different indices. The size $\Delta$ of the
window should obey

$\Delta\gg\omega^{(2)}_{\max}\ ,$ (35)

in order not to interfere with the band structure of the operator.

Using standard ETH manipulations [33] (substituting sums with integrals and
expanding
$S(E+\omega/2)+S(E-\omega/2)-S(E)=S(E)-\frac{\omega^{2}}{4}S^{\prime\prime}(E)+\dots$),
one may thus re-write Eq.(34) to leading order in $N$:

$\displaystyle\overline{k}_{2}^{\mbox{\tiny{W}}}(t)$
$\displaystyle=\frac{1}{\mathrm{Tr}W}\sum_{i\neq
j}p_{i}p_{j}e^{-S(E^{+}_{ij})}F^{(2)}_{e^{+}_{ij}}(\omega_{ij})e^{i\omega_{ij}t}$
(36) $\displaystyle\simeq\frac{1}{\mathrm{Tr}W}\int
dEe^{S(E)}\;p(E+\omega/2)p(E-\omega/2)\int d\omega\;e^{i\omega
t}F^{(2)}_{E}(\omega)\ .$ (37)

where $p_{i}=p(E_{i})$ is the window function [cf. Eq.(11)].

The integral over $E$ in Eq.(37) is solved by a saddle point, explicit
examples will be discussed below. Since $F_{E}^{(2)}(\omega)$ is of order one,
one can evaluate in on the maximum, leading to

$\overline{k}^{\mbox{\tiny{W}}}_{2}(\omega)=F^{(2)}_{E}(\omega)\Big{|}_{\text{max}}\,P_{\Delta}(\omega)$
(38)

with

$P_{\Delta}(\omega)=\frac{1}{\mathrm{Tr}W}\int
dEe^{S(E)}\;p(E+\omega/2)p(E-\omega/2)\ .$ (39)

Eqs.(38)-(39) represent the link between thermal on-shell correlations
$F^{(2)}_{E}(\omega)$ and the ones obtained from the window and are the main
result of this section.

To summarize, we have shown that the microcanonical projected observable $WOW$
yields the same ETH 2-point on-shell correlations but multiplied by a function
$P_{\Delta}(\omega)$ [cf. Eq.(38)], which depends on the different energy
windows. We now discuss if and how one can retrieve dynamical correlations.

###  3.2 Not all windows are equal

####  3.2.1 Box window

A natural choice for the window is to consider flat “box projectors”, namely

$p(E)=\theta\left(\frac{\Delta}{2}-|E-E_{0}|\right)\ ,$ (40)

where $\theta(x)$ is a step function. Flat windows have the following property

$\displaystyle\mathrm{Tr}\left[W^{n}\right]$
$\displaystyle=\mathrm{Tr}\left[W\right]=\int dE\;e^{S(E)}p(E)$
$\displaystyle=e^{S(E_{0})}\,\frac{2}{\beta}\sinh(\beta\Delta/2)\ ,$ (41)

where integral is solved by saddle point by expanding the entropic factor as
$e^{S(E)}=e^{S(E_{0})+\beta x}$ around $E_{0}$ with $\beta=S^{\prime}(E_{0})$
and integrating by $x$.

To see _if_ we can retrieve information about two-point dynamical correlators,
we need to compute $P_{\Delta}(\omega)$ in Eq.(39). A short calculation shows
that the box projector constraints both the average energy and frequency
inside a box, namely

$\displaystyle\begin{split}p(E+\omega/2)&\,p(E-\omega/2)=\theta(\Delta-|\omega|)\,\theta\left(\frac{\Delta-|\omega|}{2}-|E-E_{0}|\right)\
.\end{split}$ (42)

Hence, we perform the integral over energy $E$ in Eq.(39) solving again by a
saddle point. By expanding the entropic factor as $e^{S(E)}=e^{S(E_{0})+\beta
x}$ and integrating by $x$, we get

$\displaystyle P_{\Delta}(\omega)$
$\displaystyle=\frac{e^{S(E_{0})}}{\mathrm{Tr}W}\int_{-\frac{\Delta-|\omega|}{2}}^{\frac{\Delta-|\omega|}{2}}e^{\beta
x}dx\,\theta(\Delta-|\omega|)=\frac{\sinh(\beta(\Delta-|\omega|)/2)}{\sinh(\beta\Delta/2)}\theta(\Delta-|\omega|)\
.$ (43)

The effect of the window does not become negligible for any $\Delta$. This is
most striking in the time domain, where Eq.(38) becomes a convolution between
the two-point correlation and a Lorentzian. In fact, the Fourier transform of
Eq.(43) reads

$P_{\Delta}(t)=\frac{4\beta}{\sinh(\beta\Delta/2)}\frac{\cosh(\beta\Delta/2)-\cos(\Delta
t)}{\beta^{2}+4t^{2}}\ .$ (44)

Therefore, the box projector has the effect of inducing by itself a slow
(Planckian) decay in the dynamical correlator at a time-scale $1/\beta$, see
e.g. [34]. This can be traced back to the non-analytic factor
$\theta(\Delta-|\omega|)$ in Eq.(42) 555We note that this problem is well-
recognized in signal processing, where it is known that nonanalytic windows
turn out to have a slowly decaying and oscillating Fourier transform.. Such
behaviour could overcome the intrinsic decay of the correlation functions,
which is typically exponentially decaying in time. On the other hand, the box
projector can still be used for studying the dynamics of correlators with
slower decay at long times, such as the long hydrodynamic tails of _diffusive
observables_ [35].

These predictions are consistent with the numerical evaluation for the Ising
model, described by the Hamiltonian (20). We focus here on the collective
observable $O=\sum\sigma_{i}^{z}/\sqrt{L}$. In Fig.2 (left panel), we compare
the thermally regulated dynamical correlator (red line) at temperature $T=5$,
with the result obtained by WOW obtained with a box window (33). There is no
value of $\Delta$ for which one can reproduce the thermal result. We also
contrast the results with the behavior obtained for infinite temperature,
which should be retrieved for very large $\Delta$ independently of where the
window is centered. Even for very large $\Delta$, one is not able to recover
the infinite temperature dynamics (dashed black), which is obtained in the
absence of a window filter. See for comparison Fig.3 below for a smooth
window. This result is induced by the non-analiticities of the window. On the
right panel of Fig.2, we plot the ratio between the Fourier transform of the
box energy window and the thermally regulated one for different $\Delta$. The
result matches the exact prediction in Eq.(43) with no fitting parameter.

Figure 2: Numerical evaluation of the connected on-shell correlation functions
in Eq.(21),(33) with the box projector (40) for
$O=\frac{1}{\sqrt{L}}\sum_{i}\sigma_{i}^{z}$ with $L=14$. (Left) The thermally
regulated result (red line) is contrasted with one obtained with window
projections with different $\Delta=0.5,4,10$ (blue lines). When the width is
too large $\Delta=2L$ (blue dashed line) one still can not retrieve the result
at infinite temperature which, for very large $\Delta$, should hold
independently of where the window is centered (black line). (Right) The ratio
between the thermal correlation and the regulated thermal ones (dots) compared
with Eq.(43) without fitting parameters (dashed line).

####  3.2.2 Smooth and Gaussian window

The long-time tail induced by the box window can be avoided by using a smooth
one. As an illustrative example, we consider here Gaussian energy filters:

$W=e^{-\frac{(\hat{H}-E_{0})^{2}}{2\Delta^{2}}}\ ,\quad\text{i.e.}\quad
p(E)=e^{-\frac{(E-E_{0})^{2}}{2\Delta^{2}}}\ .$ (45)

Let us recall, that we want that the entropic term $e^{S(E)}$ to be overridden
by $e^{-(E-E_{0})^{2}/\Delta^{2}}$ so that the Gaussian dominates in the
location of the window. At the same time, the width has to be large enough so
as not to spoil the physical correlations of the two-point functions (30).
Hence we need

$\beta\omega^{(2)}_{\max}\ll\beta\Delta\ll\sqrt{N}$ (46)

which implies Eq.(15).

To see _how_ to retrieve on-shell correlations from the Gaussian window, one
needs to evaluate Eq.(39). First of all, one has

$p(E+\omega/2)p(E-\omega/2)=p^{2}(E)\,e^{-\frac{\omega^{2}\,\,}{2\cdot
2\Delta^{2}}}\ ,$ (47)

that back in Eq.(39) leads to

$\displaystyle P_{\Delta}(\omega)$
$\displaystyle=e^{-\frac{\omega^{2}\,\,}{2\cdot
2\Delta^{2}}}\frac{\mathrm{Tr}[W^{2}]}{\mathrm{Tr}W}\ .$ (48)

To simplify this equation, let us compute more generally $\mathrm{Tr}[W^{n}]$.
By saddle point we have

$\displaystyle\mathrm{Tr}[W^{n}]$ $\displaystyle=\int
dEe^{S(E)}e^{-n\frac{(E-E_{0})^{2}}{2\Delta^{2}}}=e^{S(E_{0})}\int
dxe^{-\frac{x^{2}}{2(\Delta/\sqrt{n})^{2}}+\beta
x}=e^{S(E_{0})}\frac{\Delta\sqrt{2\pi}}{\sqrt{n}}e^{\beta^{2}\Delta^{2}/n}$
(49)

where from the left to the right of the first line we have expanded the
entropic contribution $e^{S(E)}=e^{S(E_{0})+\beta x}$ and then solved the
resulting Gaussian integral in $x$.

All in all, we have shown that, whenever $\Delta$ is large enough [cf. (15)],
the frequency dependence behaviour of $P_{\Delta}(\omega)$ can be neglected.
We conclude that

$\mathrm{Tr}(\rho^{1/2}O(t)\rho^{1/2}O)\leftrightarrow\mathrm{Tr}(WO(t)WO)$
(50)

where the proportionality constant is
$\frac{\mathrm{Tr}[W^{2}]}{\mathrm{Tr}[W]}\simeq\frac{e^{-\beta^{2}\Delta^{2}/2}}{\sqrt{2}}$,
which is finite, since $\beta\Delta$ is of order one, even if large.

Figure 3: Numerical evaluation of the connected on-shell correlation functions
in Eq.(21)-(33) with the gaussian Window (45) for
$A=\frac{1}{N}\sum_{i}\sigma_{i}^{z}$ and $L=16$. (Left) The thermally
regulated result (red line) is contrasted with one obtained with window
projections with different $\Delta=0.5,1,3.5$ (blue lines). When the width is
too large $\Delta=2\,L$ (dark blue) one can retrieve the result at infinite
temperature (black dashed line). (Right) The ratio between the thermal
correlation and the regulated thermal ones (dots) compared with Eq.(48)
without fitting parameters.

##  4 Higher order correlations

We wish now to access higher order on-shell correlation functions
$F^{(n)}_{e}(\vec{\omega})$ defined via Eq.(7). In Ref.[4], it was shown that
$F^{(n)}_{\beta}(\vec{\omega})$ are given by the Fourier Transform of the
following constrained trace

$\displaystyle\begin{split}&\overline{k}_{n}(\vec{t})=\mathrm{Tr}^{\prime}\left(\rho^{\frac{1}{n}}O(t_{1})\rho^{\frac{1}{n}}O(t_{2})\dots\rho^{\frac{1}{n}}O(0)\right)\\\
&=\frac{1}{Z}\sum_{i_{1}\neq\hbox
to7.00002pt{.\hss.\hss.}i_{n}}e^{-\frac{\beta}{q}(E_{i_{1}}+\dots
E_{i_{n}})}[O(t_{1})]_{i_{1}i_{2}}\hbox
to10.00002pt{.\hss.\hss.}[O(0)]_{i_{n}i_{1}}=\int
d\vec{\omega}e^{i\vec{\omega}\cdot\vec{t}}F^{(n)}_{\beta}(\vec{\omega})\
,\end{split}$ (51)

which corresponds to the connected part of the regularized thermal function in
Eq.(2), i.e.
$\mathrm{Tr}\left(\rho^{\frac{1}{n}}O(t_{1})\rho^{\frac{1}{n}}O(t_{2})\dots\rho^{\frac{1}{n}}O(t_{n})\right)$.

These correlations are in fact related to the _thermal free cumulants_ [36], a
specific form of connected correlation functions defined from the thermal
moments as [21]

$\displaystyle S_{n}(\vec{t})$ $\displaystyle=\text{Tr}(\rho O(t_{1})\dots
O(t_{n}))=\sum_{\pi\in NC(n)}k_{\pi}(O(t_{1})\dots O(t_{n}))\ ,$ (52)

where $NC(n)$ is the set of the non-crossing partitions of $\\{1,2,\dots,n\\}$
and $k_{\pi}$ is product of cumulants, one for each block of $\pi$. This
decomposition generalizes the implicit definition of connected cumulant
$S_{2}(t)=[k_{1}]^{2}+k_{2}(t)$ [cf. Eq.(25)] to higher-order 666For instance
$\displaystyle k_{1}$ $\displaystyle\equiv S_{1}$ $\displaystyle k_{2}(t)$
$\displaystyle\equiv S_{2}(t)-(S_{1})^{2}$ $\displaystyle k_{3}(t_{1},t_{2})$
$\displaystyle\equiv
S_{3}(t_{1},t_{2})-S_{1}S_{2}(t_{1}-t_{2})-S_{1}S_{2}(t_{1})-S_{1}S_{2}(-t_{2})+2(S_{1})^{3}$
$\displaystyle k_{4}(t_{1},t_{2},t_{3})$ $\displaystyle\equiv
S_{4}(t_{1},t_{2},t_{3})-S_{2}(t_{1}-t_{2})S_{2}(t_{3})-S_{2}(t_{2}-t_{3})S_{2}(t_{1})\quad\text{for}\quad
S_{1}=0\ .$ . See also Refs.[37, 38] for other appearances of free cumulants
in many-body quantum systems. The smoothness of the ETH ansatz (7) implies
that the thermal free cumulants acquire a simple form, given only by the
constrained trace over different indices

$\displaystyle k_{n}(\vec{t})$ $\displaystyle=\mathrm{Tr}^{\prime}(\rho
O(t_{1})\dots O(t_{n}))=\int
d\vec{\omega}e^{i\vec{\omega}\cdot\vec{t}}F^{(n)}_{\beta}(\vec{\omega})e^{-\beta\vec{\omega}\cdot\vec{\ell}_{n}},$
(53)

where the right-hand side is the Fourier transform obtained using the
smoothness of ETH ansatz. The factor
$e^{-\beta\vec{\omega}\cdot\vec{\ell}_{n}}$ is the thermal shift implementing
KMS. This expression thus generalizes the $n=2$ case of Eqs.(26)-(27). To
isolate $F^{(n)}_{\beta}(\vec{\omega})$, this result naturally leads us to
study the symmetrized correlator $\overline{k}_{n}(\vec{t})$ in Eq.(51), which
thus identifies the building block of multi-time functions at all orders $n$.

We demand now that WOW contains all such correlations. We compute the
corresponding object using the window filter:

$\displaystyle\overline{k}_{n}^{\mbox{\tiny{W}}}(\vec{t})$
$\displaystyle\equiv\frac{1}{\mathrm{Tr}W}\sum_{i_{1}\neq\hbox
to7.00002pt{.\hss.\hss.}\neq i_{n}}W_{i_{1},i_{1}}\dots
W_{i_{n},i_{n}}[O(t_{1})]_{i_{1}i_{2}}\hbox
to10.00002pt{.\hss.\hss.}[O(0)]_{i_{n-1}i_{n}}$ (54)
$\displaystyle=\frac{1}{\mathrm{Tr}W}\int
dEe^{S(E)}d\vec{\omega}\;p(E_{1})\dots
p(E_{n})\;F^{(n)}_{e^{+}}(\vec{\omega})e^{i\vec{\omega}\cdot\vec{t}}\ .$ (55)

The same factorization of the probability discussed for $n=2$ [cf. Eq.(47)]
applies for generic $n$, namely

$\displaystyle W_{i_{1},i_{1}}$ $\displaystyle\dots
W_{i_{n},i_{n}}=\;p(E_{1})\dots
p(E_{n})\;=e^{-{n}(E-E_{0})^{2}/2\Delta^{2}}g_{\Delta}(\vec{\omega})\ ,$ (56)

where

$g_{\Delta}(\vec{\omega})=e^{-\frac{1}{2\Delta^{2}}\sum_{i-1}^{n}\left(\sum_{j=1}^{n-1}c_{j,i}\omega_{j}\right)^{2}}\
,$

with $c_{j<i}=\frac{j}{n}$ and $c_{j\geq i}=\frac{j-n}{n}$ depends only on
energy differences, as before. If we now take into account that
$\beta\Delta\ll\sqrt{N}$, the first factor in (56) dominates over $S(E)$, and
the saddle is $E_{0}$.

In order to choose the window appropriately, we recall that, just as the two-
point functions [cf.(30)], also $n$-point on-shell correlators are supposed to
decay fast at large frequencies, at least as fast as

$F^{(n)}(\vec{\omega})=e^{-|\omega_{i}|/\omega_{\max}^{(n)}}$ (57)

in each direction $\omega_{i}$ with
$\omega^{(n)}_{\max}\leq\frac{n-1}{n}\frac{1}{\beta\hbar}$ [26, 21]. From
this, we conclude that the largest energy scale characterizing the
correlations of the operator $O$ obey

$\omega_{\max}=\max_{n}\omega^{(n)}_{\max}\leq\frac{1}{\beta\hbar}\ .$ (58)

In the time domain, we thus have

$\displaystyle\overline{k}_{n}^{\mbox{\tiny{W}}}(\vec{t})$
$\displaystyle=\frac{1}{\mathrm{Tr}W}\int dEe^{S(E)}p^{n}(E)\int
d\vec{\omega}\,g_{\Delta}(\vec{\omega})\,\;F_{E}^{(n)}(\vec{\omega})e^{i\vec{\omega}\cdot\vec{t}}$
(59) $\displaystyle=\frac{\mathrm{Tr}[W^{n}]}{\mathrm{Tr}[W]}\int
d\vec{\omega}\,g_{\Delta}(\vec{\omega})\,F_{E}^{(n)}(\vec{\omega})e^{i\vec{\omega}\cdot\vec{t}}\
,$ (60)

where the proportionality constant in the case of the Gaussian can be computed
using Eq.(49), leading to

$\displaystyle\frac{\mathrm{Tr}[W^{n}]}{\mathrm{Tr}[W]}=\frac{e^{-\beta^{2}\Delta^{2}\frac{n-1}{n}}}{n^{1/2}}\
.$ (61)

Thus, by choosing $\beta\omega_{\max}\ll\Delta\beta\ll\sqrt{N}$ [cf. Eq.(15)],
we may neglect the contribution of the filter and set
$g_{\Delta}(\vec{\omega})\sim 1$. All in all, we have obtained a relation
between the time-dependent on-shell correlators and the ones obtained by WOW,
i.e.

$\overline{k}_{n}^{\mbox{\tiny{W}}}(\vec{t})=\frac{\mathrm{Tr}[W^{n}]}{\mathrm{Tr}[W]}\overline{k}_{n}(\vec{t})\
.$ (62)

Thus, to compute dynamical correlation functions, all the on-shell
correlations are encoded in the filtered observable WOW. As a result, the
canonical multi-point correlations can be computed directly by looking at
their correlations as

$\displaystyle\mathrm{Tr}$ $\displaystyle(\rho^{1/n}O(t_{1})\rho^{1/n}\dots
O(t_{n-1})\rho^{1/n}O(0)\rho^{1/n})\leftrightarrow\frac{1}{\mathrm{Tr}W}\mathrm{Tr}(WO(t_{1})\dots
WO(t_{n-1})WO(0))$ (63)

where the proportionality constant is given by Eq.(61).

##  5 The WOW spectrum and generating functions

In the previous sections, we have shown that all dynamical on-shell
correlations can be retrieved by looking at the structured microcanonical
operator WOW. We now explore its spectrum. The spectrum of the
microcanonically truncated operator was introduced and studied by Richter et
al. in Ref.[13], to determine the emergence of random matrix description for
small windows, i.e. when $\Delta$ is small with the system size $L$. Our
approach has a different goal because it aims to access the physical
properties of the finite-frequency operator, exactly those that go beyond
random matrix theory.
The _WOW spectral density_ is defined as

$\rho(\alpha)\equiv\frac{1}{\text{tr}W}\sum_{i}\delta(a_{i}-\alpha)\ ,$ (64)

where $a_{i}$ are the eigenvalues of $\tilde{O}$. Note that with this
definitions $\rho(\alpha)$ is not normalized, i.e. $\int
d\alpha\,\rho(\alpha)=\frac{\text{tr}1}{\text{tr}W}=e^{S(0)-S(e_{0})}$. In
fact, depending on the different choices of the filter W, the distribution
$\rho(\alpha)$ could develop a divergent behavior at $\alpha\to 0$. Imagine we
consider a Gaussian filter as in Eq.(45): its effect is to project almost all
matrix elements to zero, besides the ones in the energy shall of $E_{0}$ with
variance $\Delta$. Correspondingly, also many eigenvalues will become zero,
with the effect of an accumulation in $\rho(\alpha\to 0)$. On the other hand,
this effect is not present by choosing a box filter function as in Eq.(40). In
this case, one can restrain the analysis to the submatrix of $O$ with finite
matrix elements, and all the zeros are automatically avoided. Near zero
eigenvalues are in any case irrelevant as far as the integer, positive moments
of $\tilde{O}$ are concerned.
In Fig.(4), we compare the numerical histogram of the eigenvalues of
$\tilde{O}$, obtained with a Gaussian filter (left) and the boxed one (right).
We consider a single site observable $O=\sigma^{z}_{L/2}$, which is
characterized by $+1,-1$ eigenvalues in the absence of the microcanonical
projection. As is evident from the plot for the Gaussian filter (Fig.(4)a), a
diverging peak develops in the limit $\alpha\to 0$. This is absent in the case
of the box projector (Fig.(4)c), which displays a regular distribution
$\rho(\alpha)$.

Figure 4: Numerical evaluation of the spectrum of $WOW$ with Gaussian (left)
and boxed (right) window filter for different sizes $L=12,14,16,18$. In the
bottom panels we plot $\alpha\rho(\alpha)$.

To avoid the pitfalls of the energy filtering for $\alpha\to 0$, we look at

$\alpha\rho(\alpha)$ (65)

which will appear in the generating function of the finite moments, discussed
below. This function of the eigenvalues is a well-defined object, which shall
yield a universal result independently of the choice of the window Function
$W$. This is shown in Fig.(4)bd, where we plot $\alpha\rho(\alpha)$ for the
Gaussian and the filter functions respectively, which shows a good agreement
between the two functions.

###  5.1 Generating functions for WOW

The spectral properties of WOW lead us to the definition of some generating
functions to directly access equal-time moments and free cumulants,
generalizing known concepts in random matrix theory [39]. The equal-time
moments are defined as

$\overline{m}_{n}=\frac{1}{\mathrm{Tr}[W^{n}]}\mathrm{Tr}[\tilde{O}^{n}]\simeq\mathrm{Tr}\left(\rho^{1/n}O\dots\rho^{1/n}O\right)\
,$ (66)

where on the right-hand side we have used the Eq.(63) derived in the previous
section valid for $\omega_{\max}\beta\ll\Delta\beta\ll\sqrt{N}$, cf.(15).
Thus, the information of the equal-times moments implies knowledge of the
following function

$\bar{G}(z)=\sum_{n=0}^{\infty}\frac{\bar{m}_{n}}{z^{n+1}}=\sum_{n=0}^{\infty}\frac{1}{\mathrm{Tr}[W^{n}]}\frac{{\mathrm{Tr}[\tilde{O}^{n}]}}{z^{n+1}}\
,$ (67)

related to the _moment generating function_. This expression can always be re-
written in terms of the spectral density discussed above, namely

$\bar{G}(z)\equiv\frac{1}{z}+\frac{1}{z}\int
d\alpha\,\alpha\rho(\alpha)\sum_{n=1}^{\infty}\frac{1}{\mathrm{Tr}[W^{n}]}\frac{\alpha^{n-1}}{z^{n}}\
,$ (68)

from which is clear that $\rho$ only appears through $\alpha\rho(\alpha)$ in
the expression for the moment $\bar{m}_{n}$ generating functions.
In the case of a box window as in Eq.(40), then one has
$\mathrm{Tr}[W^{n}]=\mathrm{Tr}[W]$ and the generating function $\bar{G}(z)$
corresponds to the Stieljes (Cauchy) transform [39] of the operator WOW:

$\displaystyle\bar{G}(z)$
$\displaystyle=G(z)\equiv\frac{1}{\text{tr}W}\text{tr}\left(\frac{1}{z-W^{1/2}OW^{1/2}}\right)=\int
d\alpha\frac{\rho(\alpha)}{z-\alpha}\ .$ (69)

Here we use $G(z)$ to denote the standard definition of the Stieljes transform
[39], as opposed to $\bar{G}(z)$ defined in Eq.(68). The two correspond only
in the case of a box filter, where one can consider a matrix of reduced shape.
In this case, all the known properties of $G(z)$ can be used. For instance,
the spectral density $\rho(\alpha)$ can be retrieve from it by 777Here one
uses $\lim_{\epsilon\to 0^{+}}1/(x\pm i\epsilon)=\text{p.p.}\frac{1}{x}+\mp
i\pi\delta(x)$, where $p.p$ stands for the Cauchy principal value.

$\rho(\alpha)=\frac{1}{\pi}\lim_{\epsilon\to
0^{+}}\text{Im}[G(\alpha-i\epsilon)]\ .$ (70)

###  5.2 Other generating functions

Before concluding, let us add a brief discussion of an alternative generating
function defined from the thermal moments of the observable $O$. Standard
moments at equal times are defined with respect to the average over the
density matrix:

$S_{n}^{\beta}(0)=\text{Tr}(\,\rho_{\beta}\,O^{n})\ .$ (71)

One can use a standard thermal density matrix
$\rho_{\beta}=\frac{1}{Z}e^{-\beta H}$ or a projector over an eigenstate
$\rho_{\beta}=|E_{\beta}\rangle\langle E_{\beta}|$, the two are equivalent in
describing $S_{n}^{\beta}$ according to ETH. Associated with this one can
define a Stieltjes transform (resolvent):

$G^{S}_{\beta}(z)=\text{Tr}\,\left(\rho_{\beta}\,\frac{1}{z-O}\right)\ .$ (72)

Via the Stieltjes one can define an effective probability or density of
eigenvalues:

$\displaystyle\rho^{S}(x)$ $\displaystyle=\frac{1}{\pi}\lim_{\epsilon\to
0^{+}}\text{Im}[G^{S}_{\beta}(x-i\epsilon)]$
$\displaystyle=\frac{1}{\pi}\lim_{\epsilon\to
0}\sum_{\alpha}\langle\alpha|\rho_{\beta}|\alpha\rangle\frac{\epsilon}{(x-O_{\alpha})^{2}+\epsilon^{2}}\
,$ (73)

where $|\alpha\rangle$ are the eigenvectors of $O$ and $O_{\alpha}$ its
eigenvalues. One then recovers that in the infinite temperature limit
$\rho\propto I$ this density is simply the density of eigenvalues of $O$,
while at finite temperature it gets reweighed with the probability
$\langle\alpha|\rho_{\beta}|\alpha\rangle$. The eigenvalues of an operator can
be highly degenerate and in this case, this probability is enhanced by summing
over all degenerate eigenvectors.

##  6 Discussion

In this paper, we proposed the construction of a microcanonical observable,
which encodes the dynamical correlation functions, and allows us to explore
its spectral properties.

In order to fix ideas, in this paper we have assumed that the Hamiltonian is
sufficiently chaotic, so that the matrix elements of a local operator satisfy
the randomicity properties encompassed in the Eigenstate Thermalization
Hypothesis. It is however very likely that this hypothesis may be relaxed and
that one can generalize this construction to generic microcanonical ensembles
which include integrable systems, where one only fixes the energy.

It would also be interesting to understand how to generalize to Floquet
systems. On one side, one can certainly make a smooth window periodic in the
$2\pi$ period Floquet spectrum. On the other hand, Floquet systems differ from
Hamiltonian ones in that their correlations do not decay exponentially [40,
41], making the $\omega_{\max}$ ill-defined and the filter function definition
ambiguous.

## Acknowledgements

We acknowledge useful discussions with T. Prosen and M. Srednicki, as well as
useful suggestions from X. Turkeshi and L. Cugliandolo. This work was
completed during the workshop “Dynamical Foundation of Many-Body quantum
chaos” held at the Institut Pascal at Université Paris-Saclay made possible
with the support of the program “Investissements d’avenir”
ANR-11-IDEX-0003-01. S.P. acknowledges support by the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s
Excellence Strategy - Cluster of Excellence Matter and Light for Quantum
Computing (ML4Q) EXC 2004/1 -390534769.

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# Spontaneous superconducting diode effect in non-magnetic Nb/Ru/Sr2RuO4
topological junctions

M. S. Anwar<EMAIL_ADDRESS>Department of Physics, Kyoto University,
Kyoto 606-8502, Japan  Department of Materials Science & Metallurgy,
University of Cambridge, CB3 0FS Cambridge, United Kingdom  T. Nakamura
Department of Physics, Kyoto University, Kyoto 606-8502, Japan  Fukuoka
Institute of Technology, 3-30-1 Wajiro-higashi, Higashi-ku, Fukuoka, 811-0295
Japan  R. Ishiguro  Department of Mathematical and Physical Sciences, Faculty
of Science, Japan Women’s University, Tokyo 112-8681, Japan  Department of
Applied Physics, Faculty of Science, Tokyo University of Science, Katsushika
Tokyo 162-8601, Japan  S. Arif  Department of Physics, Lahore College for
Women University, Lahore, 54000, Pakistan  J. W. A. Robinson  Department of
Materials Science & Metallurgy, University of Cambridge, CB3 0FS Cambridge,
United Kingdom  S. Yonezawa  Department of Physics, Kyoto University, Kyoto
606-8502, Japan  M. Sigrist  Theoretische Physik, ETH Zurich, CH-8093 Zurich,
Switzerland  Y. Maeno  Department of Physics, Kyoto University, Kyoto
606-8502, Japan  Toyota Riken–-Kyoto University Research Center (TRiKUC),
Kyoto 606-8501, Japan

Non-reciprocal electronic transport in a material occurs if both time reversal
and inversion symmetries are broken. The superconducting diode effect (SDE) is
an exotic manifestation of this type of behavior where the critical current
for positive and negative currents are mismatched, as recently observed in
some non-centrosymmetric superconductors with a magnetic field. Here, we
demonstrate a SDE in non-magnetic Nb/Ru/Sr2RuO4 Josephson junctions without
applying an external magnetic field. The cooling history dependence of the SDE
suggests that time-reversal symmetry is intrinsically broken by the
superconducting phase of Sr2RuO4. Applied magnetic fields modify the SDE
dynamically by randomly changing the sign of the non-reciprocity. We propose a
model for such a topological junction with a conventional superconductor
surrounded by a chiral superconductor with broken time reversal symmetry.

Diodes are one-way electronic switches for charge flow, and are a fundamental
component in modern circuits. Recently a superconducting diode effect (SDE)
(i.e. non-reciprocal flow of supercurrents) was observed in superconducting
superlattices Ando2020 ; Narita2022 , 2-dimensional materials Ye2022 ;
Bauriedl2022 ; Wakatsuki2017 , multilayers Hou2022  and Josephson junctions
Baumgartner2022 ; Wu2022 ; Pal2021 . Such a dissipationless SDE paves-the-way
to the development of energy-efficient electronics and computation.

Non-reciprocal electron transport is realized in materials in which both
inversion and time-reversal symmetries are broken. On a microscopic level the
lack of inversion symmetry induces specific forms of spin-orbit coupling
(SOC). For instance, magnetochiral anisotropy (MCA) in a chiral conductor is
described through the non-linear dependence of the resistivity $R(H)=R_{\rm
o}(1-\gamma HI)$ for specific current directions Rikken1997 ; Rikken2001 ,
where $\gamma$ is the MCA coefficient originating from SOC, $H$ is magnetic
field applied perpendicular to the current flow and $R_{\rm o}$ is magnetic
field independent resistance.

Non-reciprocity in superconductors is seen as a difference in critical current
for positive and negative flow of the current, constituting a SDE. A recently
discussed example are non-centrosymmetric superconductors, where a lack of an
inversion center in the crystal lattice induces antisymmetric SOC and
influences the superconducting phase. Applying a magnetic field to a non-
centrosymmetric superconductor can give rise to SDE. This effect is seen for
systems that exhibit Rashba-type SOC due to the absence of specific mirror
symmetries in crystal lattice. A magnetic field directed along such planes
establishes a helical phase with non-reciprocal in-plane supercurrents
perpendicular to the magnetic field Daido2022 ; Yuan2022 ; Ilic2022 ;
Scammell2022 ; He2022 . An example includes Nb/Ta/V superlattices with
antisymmetric SOC; the SDE in such structures is observed when the magnetic
field is applied perpendicular to the current flow, such that both inversion
and time reversal symmetries are broken. Insertion of ferromagnetic layers is
reported to serve a similar role as external magnetic field Narita2022 . In
case of a magnetic field-free SDE Baumgartner2022 ; Wu2022 ; Pal2021 , broken
inversion symmetry at the interfaces may produce required SOC and presence of
a magnetic layer in the devices breaks time-reversal symmetry. This may also
be true for EuS/Nb/Pt multilayers Hou2022 .

The SDE can also occur without an applied magnetic field in a superconductor
with spontaneously broken time-reversal symmetry and, as noted in Zinkl2022 ,
without SOC if inversion symmetry is violated. One of the first observations
of this type of “spontaneous” SDE was reported for Sr2RuO4 (SRO) but above the
bulk superconducting transition temperature where a filamentary
superconducting phase so-called 3-Kalvin (3-K) phase exists Hooper2004 .

This article investigates Josephson junctions based on SRO, which exhibit an
unconventional superconducting order parameter with spontaneous time-reversal
symmetry violation Maeno2012 ; Mackenzie2017 ; Anwar2021 . A eutectic single
crystal of SRO with Ru metal inclusions shows two phases of superconductivity,
bulk with a transition temperature ($T_{\rm c}$) of 1.5 K (1.5-K phase) and a
filamentary superconducting phase at the Ru/SRO interface Ghosh2017  below 3 K
(the so-called 3-K phase Maeno1998 ; Haka2009 ). It has been speculated that
strain at the interface may be responsible for the locally enhanced onset of
superconductivity in the 3-K phase Hicks2014 ; Taniguchi2015 ; Hicks2017 .
Bulk superconductivity in SRO is sensitive to impurities, consistent with an
unconventional order parameter Mackenzie1998 . Evidence for broken time-
reversal symmetry in the superconducting state originates from the observation
of an intrinsic magnetic field below $T_{\rm c}$ by zero-field muon spin
relaxation Luke1998 ; Grinenko2021  and Kerr rotation Xia2006  experiments.
Furthermore, electronic transport measurements of SRO junctions indicate two-
fold degenerate domains Kindwingira2006 ; Nakamura2011 ; Nakamura2012 ;
Anwar2013 ; Anwar2017 . Among other experiments, these observations suggest a
two-component order parameter with chiral symmetry, while the symmetry of the
electron pairing is still under debate Hassinger2017 ; Romer2019 ;
Pustogow2019 ; Ishida2020 ; Petsch2020 ; Kivelson2020 ; Suh2020 ; Li2021 ;
Grinenko2021 .

In this article, we report a magnetic field-free (spontaneous) SDE in
Nb/Ru/SRO Josephson junctions. Josephson currents are observed below about 2 K
with the SDE appearing below 1.4 K when bulk superconductivity is established
in SRO. The difference between the critical current for opposite current
directions increases below this onset and passes through a maximum around 0.5
K, below which the Ru-metal inclusion becomes intrinsically superconducting
Nago2014 . This SDE depends on thermal cycles and is affected by applying
magnetic fields parallel to the basal plane of SRO.

Figure 1: Junction geometry and electronic transport properties. a A scanning
electron micrograph showing two junctions A (Left) and B (right). The brighter
areas are the two top Nb electrodes and the darker area is the SiO2/SRO
substrate. b The SiO2 insulating layer is removed in small area covering the
top of a single Ru inclusion as shown in the inset (junction A). An analogous
setup is realized for the junction B. c A schematic view of a Nb/Ru/SRO
junction fabricated by depositing 600 nm thick Nb layer on top of a SiO2 layer
with a hole allowing direct contact of Nb to a single Ru inclusion of SRO-Ru
eutectic crystal (layer orientation with normal vector along the $c$-axis of
SRO). d Resistance as a function of temperature of junction A. The first
transition at 9.2 K and the transition at 2.2 K correspond to the onset of
superconductivity in the Nb electrode and in the SRO-Ru eutectic crystal,
respectively. Zero resistance is achieved at 1.7 K well above 1.5 K that
indicates the Josephson coupling develops first through 3-K phase. There are
two more transition at 8.5 K and 6.5 K, which corresponds to the geometry of
the junction. Inset shows the $R$($T$) at lower temperature. e A non-
hysteretic $V$($I$) curve measured at 0.3 K with current scans in the sequence
of i) zero to positive (0 - p), ii) positive to zero (p - 0), iii) zero to
negative (0 - n) and iv) negative to zero (n - 0). The comparison with a
symmetric theoretical $V$($I$) curve (black dotted line) clearly shows non-
reciprocity of the experimental junction: $I_{\rm c+}\neq|I_{\rm c-}|$. f
$V$($I$) curves for the absolute values of current and voltage at different
temperatures. Solid and dotted lines mark the branches belonging to positive
and negative current direction, respectively. The difference between
magnitudes of $I_{\rm c+}$ and $I_{\rm c-}$ decreases with increasing
temperature and disappears above 1.4 K (see also inset with an $V$($I$) curve
at 1.5 K). Figure 2: Temperature dependence of the superconducting diode
effect. a Critical currents $I_{\rm c+}$ and $|I_{\rm c-}|$ vs $T$. The two
critical currents are non-vanishing below 2 K as a result of the 3 K-phase and
start to differ from each other below $T_{\rm c}\approx$ 1.4 K when SRO shows
bulk superconductivity. Above $T_{\rm c}$ there is no difference in the
critical currents. $\Delta I_{\rm c}=I_{\rm c+}-|I_{\rm c-}|$ and $Q$ defined
as $(I_{\rm c+}-|I_{\rm c-}|)/(I_{\rm c+}+|I_{\rm c-}|)$ are plotted against
$T$ in panel b and c, respectively. The insets in both panels show a distinct
$T$-dependence close to the onset of the bulk $T_{\rm c}$, which is $(T_{\rm
c}-T)^{2}$ for $\Delta I_{\rm c}(T)$ and $(T_{\rm c}-T)^{1/2}$ for $Q(T)$.
Note that the standard deviation in $I_{\rm c}$ for all temperatures is
smaller than the data points. Figure 3: Effect of thermal cycling on the
superconducting diode effect. Three different $V$($I$) curves with absolute
values of current and voltage measured for different cooling cycles; the black
(red) curve shows the positive (negative) branch. a Junction A exhibits a
forward diode effect with $I_{\rm c+}>I_{\rm c-}$, b standard junction
behavior with $I_{\rm c+}=|I_{\rm c-}|$ and c reverse diode effect with
$I_{\rm c+}<|I_{\rm c-}|$. Insets show $V$($I$) curves for junction B for
equivalent thermal cycles. Figure 4: Effect of in-plane magnetic fields on the
superconducting diode effect. The effect of magnetic fields applied
perpendicular to the $c$-axis is investigated for junction-A using the
following sequence: i) the magnetic field is applied in the superconducting
state (zero-field cooled state), ii) the magnetic field is switched off at 0.3
K after measuring some $V$($I$) curves and iii) $V$($I$) curves are measured
in the zero field. The $V$($I$) characteristic behave unstable as long as the
magnetic field is switched on and does not allow to determine critical
currents reliably (see Supplementary Fig. S1c and S1d). After switching off
the magnetic field $V$($I$) curves show relatively stable critical currents,
but the non-reciprocal component $\Delta I_{\rm c}$ is “random”; $V$($I$)
scans at different times sometimes show the behavior of a forward diode
$I_{\rm c+}>|I_{\rm c-}|$, b standard junction $(I_{\rm c+}\approx|I_{\rm
c-}|)$ and c reverse diode $(I_{\rm c+}<|I_{\rm c-}|)$. d Histograms of
$\Delta I_{\rm c}$ for 34 sequential $V$($I$) scans showing quantitative
scatter of the SDE around $\Delta I_{\rm c}=\pm$20 $\mu$A. Blue, black and red
bars indicate forward diode effect ($\Delta I_{\rm c}>0$), standard junction
($\Delta I_{\rm c}=0$) and reverse diode effect ($\Delta I_{\rm c}<0$). Note
that $V$($I$) curves are measured successively at 0.3 K with a negligible time
delay between measurements.

Results

Junction fabrication and electronic characteristics Nb/Ru/SRO junctions were
fabricated by depositing a 600-nm-thick Nb film on the $ab$-surface of the
SRO-Ru eutectic crystal covered by an insulating mask of SiO2 with a hole such
that the Nb film is in contact with a single Ru-inclusion only (see Fig. 1
a-c); details are given in the Methods Section. The $\mu$m-sized Ru-metal
inclusion is embedded in a single crystal of SRO such that the interface
between the two forms a Josephson junction. Assuming a conventional $s$-wave
superconducting phase within Ru (proximity-induced by Nb and intrinsic below
0.5 K) and a chiral superconducting phase in SRO, this Josephson junction
behaves as a topological junction as it involves a winding of the Josephson
phase Nakamura2011 ; Nakamura2012 ; Anwar2013 ; Anwar2017 ; Kaneyasu2010-1 ;
Kaneyasu2010-2 ; Etter2014 . We have measured electronic transport properties
through Nb/Ru/SRO junctions down to 0.3 K (electronic setup is described in
the Methods Section). Figure 1d shows the resistance vs temperature $R$($T$)
measured with a 100 $\mu$A dc current. Note that the $R$ was measured in delta
mode and the average value of $R$ was measured with step-like current
variations (positive and negative) with a frequency of 5 Hz. The drop of
resistance coincides with the onset of superconductivity in Nb at $\approx$
9.2 K, followed by a further transition around 2.2 K corresponding to the 3-K
phase at the Ru/SRO interface. Zero-resistance is achieved below 1.7 K which
is above the bulk $T_{\rm c}$ of 1.4 K of SRO. Note additional transitions
between 9.2 K and 2.2 K that may relate with geometry of the junctions (narrow
parts at the corners of the junction).

Temperature effect on non-reciprocity

To investigate non-reciprocity of the critical current we measured current
voltage $V$($I$) characteristics between 0.3 and 2 K. Figure 1e shows a set of
$V$($I$) curves at 0.3 K using the following current cycles: i) zero to
positive (0 - p), positive to zero (p - 0), zero to negative (0 - n) and
negative to zero (n - 0). As there is no hysteretic behavior visible in the
$V$($I$) curves, the junction is in the overdamped regime. A fit based on
$V=R(I^{2}-I_{\rm c}^{2})^{1/2}$ (dotted line) shows that the critical
currents $I_{c+}$ (positive direction) is higher than $|I_{\rm c-}|$ (negative
direction). To compare opposite current directions for different temperatures,
in Fig. 1f we have plotted $V$($I$) curves for absolute currents and voltages,
where solid (dotted) lines indicate the positive (negative) branch of the
curve. At $T=1.5$ K there is no difference between $I_{\rm c+}$ and $|I_{c-}|$
(see also the inset of Fig. 1f), $\Delta I_{c}=I_{\rm c+}-|I_{\rm c-}|$ is
non-vanishing at lower temperatures. Similar behavior is observed for
junction-B (see Supplementary Fig. S2).

In Fig. 2a-c we have plotted the $T$-dependence of the critical current for
both directions and their difference. Both $I_{c+}$ (black squares) and
$|I_{\rm c-}|$ (red circles) decrease for increasing temperature and merge
above 1.4 K (Fig. 2a). The SDE displayed in $\Delta I_{\rm c}$ grows
continuously below 1.4 K and shows a maximum around 0.5 K (Fig. 2b). The
$T$-dependence below $T_{\rm c}$ follows $\Delta I_{\rm c}(T)\propto(T_{\rm
c}-T)^{2}$, similar to the theoretical expectations for non-centrosymmetric
superconductors Daido2022 ; He2022 .

The efficiency of the diode effect can be scaled with the quality parameter
$Q$, defined as $Q=\frac{I_{c+}-|I_{c-}|}{I_{c+}+|I_{c-}|}$. For our
junctions, we find an increase in the $Q$ with decreasing temperature,
following $\sqrt{T_{\rm c}-T}$ (Fig. 2c), which is an expected behavior
Daido2022 ; He2022 . The $Q$ reaches a maximum of 0.34 (34$\%$) at 0.5 K,
which is the highest value achieved so far in a Josephson junction
Baumgartner2022 ; Wu2022 ; Pal2021 ; Scammell2022  but lower than the value
observed in EuS/Nb/Pt Hou2022  and NbSe2 Bauriedl2022  .

Effect of thermal cycle on non-reciprocity

After cooling to 0.3 K, the junctions are warmed to 10 K to reach the normal
state. The junctions are then cooled to $\approx$ 3 K and maintained at this
temperature for 30 min before cooling to 0.3 K. We observe that with each
cooling cycle our junctions randomly change the SDE state among one of the
three states, forward diode ($\Delta I_{\rm c}=+$; see Fig. 3a), standard
junction ($\Delta I_{\rm c}=0$; see Fig. 3b) and reverse diode ($\Delta I_{\rm
c}=-$; see Fig. 3c). The junction remains in its state persistently down to
the lowest temperature and at a fixed temperature within one thermal cycle
$\Delta{I_{\rm c}}$ remains almost constant (see Supplementary Fig. S3). Note
that the standard junction behaviour ($\Delta I_{\rm c}=0$) occurs rarely.
Thus, we conclude that the SDE is dependent on the cooling cycle, a feature
not seen in non-centrosymmetric superconductors where the sign of non-
reciprocity depends on the direction of the applied magnetic field.

Effect of magnetic fields on non-reciprocity

We note that the SDE in our junctions does not require an external magnetic
field nor the addition of a magnetic layer. Nevertheless, we have investigated
the effect of magnetic fields oriented parallel to the $ab$-surface of SRO on
the SDE. Such magnetic fields cause $I_{\rm c}$ of the junctions to become
unstable (see Supplementary Fig. S1). After switching off the magnetic field
the junctions $I_{\rm c}$ are partially re-stabilized but the magnitude and
sign of the SDE is randomly changing. Figure 4a-c shows the $V$($I$) curves in
zero magnetic field measured after an in-plane field cycle ($\pm$20mT) at 0.3
K, showing that all three states (defined above) are reached. Considering a
number of field sweeps, Fig. 4d displays histograms of $\Delta I_{\rm c}$ at
zero magnetic field. This demonstrates a dynamical behavior stimulated by the
external magnetic field. We would like to emphasize the $ab$-plane direction
of the magnetic field would not couple to the chirality of the superconducting
order parameter in SRO, thus, would unlikely reverse chirality. Similar
behavior has been observed in junction B (see Supplementary Fig. S4).

Figure 5: Spontaneously induced vortex at the Ru/SRO interface. a A schematic
illustration of the concentrated flux line (vortex) at the interface between a
non-chiral superconductor (Sc) and chiral Sc, which is Ru/SRO interface for
our junctions. This appears below bulk superconducting transition temperature
of SRO. The magnetic flux of this vortex is compensated by a magnetic flux in
opposite direction spread out over the remaining interface (here for a
cylindrically shaped Ru-inclusion embedded in SRO). b Flux distribution, where
$\theta$ is the angle indicating the position on the interface around the
inclusion. c Illustration of pinning of the flux line at scattered pinning
sites in zero field and d the flux line tilted under applied in-plane magnetic
field. e Schematic picture of Lorentz force on the flux line (red dot) for
negative and positive current direction. Note that the force direction
reverses for the superconducting phase of opposite chirality in SRO. f
Asymmetric pinning potential for the vortex which allows for easier depinning
for motion to the right, leading to forward bias diode effect. g Symmetric
pinning potential that has no preferred depinning direction. There is no diode
effect.

Discussion

First we discuss whether our results are intrinsic to the junctions or related
to the junction geometry. One might suspect that the observed SDE is spurious
due to structural artefacts in the junction such as defects that may lead to
anisotropic $V$($I$) curves mimicking a SDE. However, effects triggered in
this way are unlikely to display a history dependence with cooling cycles, but
would rather remain persistent. This is clearly not the case for the SDE
observed in our junctions. The temperature effect and dynamical behavior of
SDE under magnetic fields suggest that the SDE is a feature connected with the
non-trivial superconducting state of SRO.

Our junctions consist of non-magnetic materials, except surface magnetism of
SRO associated with an in-plane rotation of the RuO6 Matzdorf2000 ; Angelo2021
, which may break time reversal symmetry. According to the geometry of our
junctions, the SDE is mainly observed for the in-plane (along the $ab$-plane)
electronic transport for which surface magnetism may not play any important
role. If non-reciprocity of our junctions relates with the surface magnetism,
the SDE should also emerge in the 3-K phase as well. But, in our junctions, we
only observe SDE below the bulk $T_{\rm c}$ (1.5-K phase). Surface magnetism
may only contribute in case of a $c$-axis oriented junction Anwar2015 ;
Anwar2016 .

We believe that the understanding of the non-reciprocal behavior of these
junctions is connected with the mechanism limiting the Josephson critical
current between the Ru-inclusion embedded in bulk SRO and the bulk SRO. The
Ru-SRO interface has been found to be atomically sharp and providing good
conditions for Josephson coupling between two superconductors with order
parameters of different symmetry Ghosh2017 .

Through the Nb top electrode the Ru inclusion acquires conventional $s$-wave
superconductivity due to the proximity effect and becomes intrinsically
superconducting below $T=$ 0.5 K. We assume that the bulk superconductivity
(1.5-K phase) of SRO has a chiral order parameter whose phase varies under
rotation around the crystal $c$-axis. Under this condition, the in-plane
Josephson coupling between Ru and SRO is phase frustrated on a closed
interface like for an inclusion of cylindrical geometry (closed surface),
because the $s$-wave order parameter of Ru would interact with a different
phase for the chiral order parameter for different junction orientation
Kaneyasu2010-1 ; Etter2014 . Around the closed interface the Josephson phase
picks up a full $2\pi$-phase winding going once around the interface between
Ru-SRO enclosing the $c$-axis. It has been shown that this configuration
induces a spontaneous magnetic flux distribution ($c$-axis oriented) along the
interface with one rather well localized vortex-like flux line in order to
relax that phase frustration Kaneyasu2010-1 , as illustrated in Fig. 5a-c.
Note that the total magnetic flux of the interface vanishes, compensated by
widely spread flux of opposite sign (Fig. 5a and b). When a current flows
through the interface a Lorentz force acts on the vortex pushing it along the
interface (Fig. 5e and f). Thus, such a current only flows without
dissipation, as long as this vortex remains pinned and does not move. It was
shown theoretically that the depinning of this spontaneous vortex would
determine the limiting of the Josephson current and defines the critical
current Etter2014 . In case of a perfectly symmetric pinning potential (Fig.
5g) to move the vortex left or right the depinning force has the same
magnitude for both directions of the current. In reality pinning potentials
around a Ru inclusion are generally asymmetric simply by the nature of pinning
defects and the irregular geometry of the inclusion. Consequently, we find
different limiting (critical) current for flow direction into and out of the
Ru-inclusion (Fig. 5e and f). Thus, the origin of the non-reciprocal behavior
in this case relies on broken time-reversal symmetry (chiral order parameter
in SRO) responsible for existence of the spontaneous flux line and the broken
inversion symmetry for the asymmetric pinning potential. Unlike in non-
centrosymmetric superconductors the lack of inversion symmetry is not a
microscopic feature and, thus, not related to SOC, but is connected with the
lack of certain reflection symmetries of the junction Zinkl2022 .

It has been demonstrated that the $T_{\rm c}$ of bulk SRO is only enhanced for
the uniaxial strain along the [100] direction Hicks2014 . This indicates that
$T_{\rm c}$ of the emerged 3-K phase around a Ru in SRO-Ru eutectic can be
distributed non-uniformly around the Ru inclusion. This non-uniform local
$T_{\rm c}$ around the Ru-inclusion do not effect our model as DSE is observed
only below bulk superconducting transition temperature (in the 1.5-K phase).
In our model, we mainly consider the interaction of bulk superconductivity of
SRO and induced $s$-wave superconductivity in a Ru inclusion. Therefore, this
model is valid for our experimental results.

The mechanism proposed here yields a behavior which can depend on thermal
cycling, because chirality is a feature of spontaneous symmetry breaking and
can have either sign equally likely. For opposite signs the flux line would
have opposite magnetic flux reversing the Lorentz force and the sign of the
difference of the critical currents, i.e. $\Delta I_{\rm c}\lessgtr 0$.
Because there may be several pinning positions along the interface, there can
also be a history dependence of the magnitude of $\Delta I_{\rm c}$ besides
its sign.

In this context it is also not surprising that pinning conditions change, when
the superconductivity in Ru becomes intrinsic below 0.5 K. It is, however, not
obvious why this would always lead to a reduction of non-reciprocity as
observed in the experiment. In-plane magnetic fields would tilt or twist the
spontaneous flux lines, which likely leads to a change of the pinning
conditions as well for the flux line (Fig. 5d). The pronounced influence of
the in-plane magnetic field is not astonishing. A tilted flux line may remain
trapped in a metastable situation once the magnetic field is turned off. Thus,
the relaxation behavior of the flux line returning towards a more $c$-axis
oriented position would naturally impact the depinning properties for a
certain time after removing the magnetic field.

As a control experiment to test our model of vortex pinning potential, we
fabricated additional junctions in different configurations. A parallel
junction, where two parallel Ru inclusions are used to create Nb/Ru/SRO/Ru/Nb
Josephson junction (see Supplementary Fig. S5a). This junction shows a SDE.
However, the SDE is not observed in a tip junction where Nb is deposited only
on a tip of the Ru inclusion as shown in Supplementary Fig. S5b. For a tip
junction, higher curvature of the tip may provide stronger pinning sites,
which also justifies our model.

In summary, we have demonstrated the existence of a field-free superconducting
diode effect in non-magnetic Nb/Ru/SRO topological Josephson junctions
prepared by using a SRO-Ru eutectic single crystal. Non-reciprocal component
$\Delta I_{\rm c}$, the difference of the critical current in opposite
directions, is observed below 1.4 K, when the bulk superconducting phase of
SRO is established. $\Delta I_{\rm c}$ rises with lowering temperature and
passes through a maximum at around 0.5 K. The sign of $\Delta I_{\rm c}$
varies with different cooling cycles and applied in-plane magnetic field
induces an increased variability in $\Delta I_{\rm c}$. These effects are
attributed to the chiral superconductivity in the bulk SRO. The observed non-
reciprocal behavior is explained by a limiting mechanism of the supercurrent
in the topological Josephson junction with a phase frustration. Such a
junction is established between the two superconductors with incompatible
order parameters: a conventional (non-chiral) in Nb/Ru and a chiral in bulk
SRO. The fact that the chiral superconducting phase spontaneously breaks time
reversal symmetry and the absence of inversion symmetry due to the geometry of
the interface are sufficient conditions to enable the SDE in this type of
junctions. This means that no magnetic fields need to be applied and the SDE
is a signature of spontaneously broken time reversal symmetry in at least one
of the two superconductors. For this reason, the observation of a spontaneous
SDE may be a diagnostic tool to detect time reversal symmetry violating
superconductors. It remains open whether this type of phenomenon might become
part of a building block for future dissipationless electronics.

Methods

Fabrication of junctions: Nb/Ru/SRO Josephson junctions were fabricated from
SRO-Ru eutectic single crystal using the floating zone method Bobowski2019 .
The junctions were prepared by the following protocol. i) A 300-nm-thick SiO2
layer was deposited by RF sputtering with a backing pressure of $\approx$10-7
mbar on a polished $ab$-surface of a rectangular piece of SRO-Ru eutectic
crystal with dimensions of 3$\times$3$\times$0.5 mm3. ii) Using laser
lithography and etching with CHF3 gas, a window was opened over a single Ru-
inclusion (see Fig. 1b and Supplementary Fig. S1a). During this etching
process, a fluoride thin film may be generated on the surface of the sample.
Therefore, before removing the photo resist, O2 plasma cleaning was preformed
to etch away the fluoride film. iii) A 600-nm-thick top Nb electrode was
deposited using a lift-off technique. The Nb was deposited by magnetron
sputtering with a base pressure of $\approx$ 10-7 mbar. Note that the Nb top
electrode deposited by this protocol is not only in contact with Ru but also
the $ab$-surface of SRO. The $ab$-surface of SRO develops a poor contact with
other normal metals. It may happen because of atomic reconstruction at the
surface Veenstra2013  that avoids to develop superconducting coupling at
Nb/SRO and no flow of critical current through the interface between Nb and
$ab$-surface of SRO. Therefore, superconducting coupling mainly develops
through Ru metal.

Electrical Measurements Electronic transport properties were investigated by
measuring $V$($I$) curves using a four-point current-bias technique with two
contacts on the Nb electrode and two contacts on the side of the SRO crystal
as shown in the schematic diagram in Fig. 1c. The current is applied using a
Yokogawa current source (GS200) and the voltage is measured using a Keithley
2082. The measurements were performed with a He-3 cryostat down to 300 mK. The
cryostat was magnetically shielded with high-permeability material (Hamamatsu
Photonics, mu-metal). Inside the shield, we placed a superconducting magnet
(with maximum field range of 1 T) to apply magnetic fields.

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Acknowledments

We are grateful to B. Zinkl and G. Mattoni for helpful discussion. This work
is supported by JSPS KAKENHI (Nos. JP15H05852, JP15K21717, JP22H01168 and
JP26287078), JSPS-EPSRC Core-to-Core Programme (No. JPJSCCA20170002), and the
Swiss National Science Foundation (SNSF) through Division II (No. 184739). M.
S. A. and J. W. A. R. acknowledges funding from the EPSRC International
Network Grant “Oxide Superspin” (No. EP/P026311/1).

Authors Contributions

M.S.A. devised the concept and performed the measurements. T.N., R.I. and
M.S.A. prepared the devices. S.Y. was also involved in the measurements.
M.S.A., S.A. J.W.A.R. and Y.M. analysed the data. M.S. provided theoretical
input. M.S.A. wrote the paper. M.S., J.W.A.R, S.Y. and Y.M. contributed in
writing and reviewed the manuscript. All the authors were involved in the
discussion.

Competing interests

The authors declare no competing interests.