††thanks: The authors have contributed equally.††thanks: The authors have
contributed equally.††thanks: The authors have contributed equally.

#  Nonequilibrium symmetry-protected topological order:
emergence of semilocal Gibbs ensembles

Maurizio Fagotti  Université Paris-Saclay, CNRS, LPTMS, 91405, Orsay, France
Vanja Marić  Université Paris-Saclay, CNRS, LPTMS, 91405, Orsay, France
Lenart Zadnik  Université Paris-Saclay, CNRS, LPTMS, 91405, Orsay, France

###### Abstract

We consider nonequilibrium time evolution in quantum spin chains after a
global quench. We show that global symmetries can invalidate the standard
picture of local relaxation to a (generalised) Gibbs ensemble and provide a
solution to the problem. We introduce in particular a family of statistical
ensembles, which we dub “semilocal (generalised) Gibbs ensembles”. The issue
arises when the Hamiltonian possesses conservation laws that are not
(pseudo)local but act as such in the symmetry-restricted space where time
evolution occurs. Because of them, the stationary state emerging at infinite
time can exhibit exceptional features. We focus on a specific example with a
spin-flip symmetry, which is the commonest global symmetry encountered in
spin-$1/2$ chains. Among the exceptional properties, we find that, at late
times, the excess of entropy of a spin block triggered by a local perturbation
in the initial state grows logarithmically with the subsystem’s length. We
establish a connection with symmetry-protected topological order in
equilibrium at zero temperature and study the melting of the order induced
either by a (symmetry-breaking) rotation of the initial state or by an
increase of the temperature.

###### Contents

  1. I Introduction
  2. II Overview and results
    1. II.1 Local observables in spin chains
      1. II.1.1 Quasilocality
      2. II.1.2 Semilocality
      3. II.1.3 Cluster decomposition
      4. II.1.4 Semilocal _vs_. symmetry-protected topological order
    2. II.2 Conservation laws with semilocal densities
      1. II.2.1 Breakdown of maximum-entropy descriptions
      2. II.2.2 Semilocal (generalised) Gibbs ensemble
    3. II.3 Signatures of semilocal order
      1. II.3.1 Excess of entropy
      2. II.3.2 Melting of the order
  3. III Long time limit: state of the art
    1. III.1 Background
    2. III.2 Maximum-entropy descriptions
      1. III.2.1 Example: XY model
      2. III.2.2 Example: Dual XY model
    3. III.3 Failure of Gibbs and generalised Gibbs ensemble in symmetric systems
  4. IV Beyond (generalised) Gibbs ensembles
    1. IV.1 Kramers-Wannier duality
    2. IV.2 Semilocality and hidden symmetry breaking
    3. IV.3 Semilocal (generalised) Gibbs ensembles
      1. IV.3.1 Example: Dual XY model revisited
      2. IV.3.2 Example: generic model revisited
      3. IV.3.3 Canonical and non-canonical descriptions
      4. IV.3.4 Remark
  5. V Signatures of semilocal order
    1. V.1 Reduced density matrices across different theories of local observables
      1. V.1.1 Restricted duality transformation
      2. V.1.2 Dual reduced density matrices
    2. V.2 Excess of entropy
      1. V.2.1 Example: Dual generalised XY model
      2. V.2.2 Free-fermion techniques for the entropies
      3. V.2.3 Results
    3. V.3 Melting of the order
      1. V.3.1 Slow relaxation after weak symmetry breaking
      2. V.3.2 Melting at low finite temperatures
  6. VI Discussion
  7. A Convergence of the iTEBD data
  8. B $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ duality
  9. C Zero-temperature phases of the dual XY model
  10. D Tilted initial state
    1. D.1 Tilted state $\ket{\rotatebox[origin={c}]{45.0}{$\Rightarrow$}_{\theta}}_{\sigma}$
    2. D.2 Decay of correlations in the state $\ket{\rotatebox[origin={c}]{45.0}{$\Rightarrow$}_{\theta}}_{\sigma}$
  11. E Excess of entropy
    1. E.1 Bounds on Renyi entropies
    2. E.2 Correlation matrix after the quench
    3. E.3 Asymptotic results for Renyi entropies
    4. E.4 Exact Procedure
  12. F Melting of the order and scaling limits
    1. F.1 Effective density matrix
    2. F.2 Semilocal order parameter

##  I Introduction

There is no topological order in one-dimensional systems in equilibrium at
zero temperature; the ground states of gapped spin-chain Hamiltonians are all
equivalent to trivial product states [1]. When restricting to systems with a
given symmetry, however, it becomes possible to distinguish different phases.
On the one hand there are standard disordered and ordered Landau phases
characterised by symmetry breaking; on the other hand, there can be nontrivial
symmetry-protected topological phases, in which the order is manifested in
excitations, such as the celebrated edge modes [2, 3, 4, 5, 6, 7], or in
exceptional entanglement properties [8].

Focusing on symmetry-protected topological order, perhaps the most prominent
feature that is accessible in the bulk of the system is the so-called string
order [9, 10, 11]. It refers to the existence of sequences of bounded
operators with arbitrarily large support (e.g., strings of spins) whose
expectation values remain nonzero in the limit of infinite support. String
order was shown not to survive an increase in the temperature [12], and most
results pertaining to its fate when the system is not kept in equilibrium [13,
14, 15, 16, 17, 18, 19, 20, 12] seem to point at its melting.

This comes of no surprise for the physical community that has been working on
relaxation in nonequilibrium systems. Specifically, there has been a lot of
progress in predicting the behaviour of local observables in integrable and
generic systems composed of a macroscopic number of degrees of freedom [21,
22, 23, 24, 25, 26, 27, 28]. It is now well accepted that, in an isolated
many-body system, local relaxation occurs: finite regions of the full system
relax because the rest of the system acts as an effective bath for them. At
late times, local observables can be effectively described by statistical
ensembles incorporating conservation laws with certain locality properties
[29, 30], the simplest example being a conserved operator with a local
density. In the end the effective stationary state can be thought of as a
thermal state of an effective Hamiltonian. Since Ref. [12] ruled out string
order at finite temperature, there is little if any hope to keep string order
out of equilibrium.

In this work we reconsider symmetry-protected topological order from the
perspective of local relaxation after a global quench in a quantum spin chain.
We show that string order does not always melt down and explain a mechanism
behind its persistence. We study the corresponding exotic nonequilibrium
states emerging at late times. Embracing Occam’s razor for the description of
such novel states of matter forces us to a fundamental change of view. We
question the meaning of locality in a quantum spin chain and build on the
traditional representation of local observables; we reformulate the theory in
order for it to become consistent with the recent hints pointing at the
emergence of symmetry-protected topological order after global quantum
quenches [31].

The incriminated systems possess hidden symmetries that can not be associated
with local conservation laws. In infinite chains the corresponding charges lie
outside of the theory defined on the local operators and their quasilocal
completion, in turn invalidating standard descriptions in terms of maximum-
entropy statistical ensembles. This is observed both in generic and integrable
systems. In the generic case, the persistence of topological order impairs
local relaxation to an effective thermal state [32], together with related
results such as the celebrated eigenstate thermalisation hypothesis [33, 34,
35, 32]. In integrable systems, instead, we experience the failure of the
generalised Gibbs ensemble [36], the latter not being able to describe the
infinite-time limit even when defined in its more refined version that
includes every pseudolocal conservation law [29].

We overcome this problem by introducing two novel statistical ensembles: the
_$G$ -semilocal Gibbs ensemble_ and the _$G$ -semilocal generalised Gibbs
ensemble_, which live in an extension of the theory, characterised by the
symmetry $G$. These ensembles are able to capture string order in a setting
where the absence of string order is the norm: they are the defining feature
of a novel phase of nonequilibrium matter. This stands in stark contrast with
the situation in equilibrium, in which string order is not sufficient to
characterise a topological phase: even a trivial product state exhibits string
order.

Purely for the sake of simplicity, we focus on the commonest symmetry in
spin-$1/2$ chains — the invariance under a spin flip (a $\mathbbm{Z}_{2}$
symmetry). We show the failure of maximum-entropy statistical descriptions
both in generic and in integrable systems and report quantitative results for
the simplest model we know to exhibit a $\mathbbm{Z}_{2}$-semilocal
generalised Gibbs ensemble: the dual XY model, which, to the best of our
knowledge, has been introduced in Ref. [37]. We elaborate on the observations
of Ref. [31] about the macroscopic everlasting effects of a local perturbation
like a spin flip in the nonequilibrium time evolution of systems described by
Hamiltonians such as that of the dual XY model. While a similar phenomenology
had been observed before in the case of symmetry-breaking perturbations of
ground states [38, 39], and excited states in a jammed sector [40, 41], the
question of whether or not these situations are just different facets of the
same phenomenon remains open.

We take inspiration from Refs. [42, 43], on one side, and Ref. [44], on the
other, and investigate the late-time effect of the perturbation on the Rényi
entropies

$S_{\alpha}[A]=\frac{1}{1-\alpha}\log\mathrm{tr}(\rho_{A}^{\alpha})$ (1)

of spin blocks $A$ ($\rho_{A}$ is the reduced density matrix of $A$). We show
that a symmetry-breaking localised perturbation results in a logarithmic
correction to the standard extensive behaviour:

$S_{\alpha}[A]\xrightarrow{t\rightarrow\infty}a_{\alpha}|A|+b_{\alpha}\log|A|+\mathcal{O}(|A|^{0})\,,$
(2)

where $|A|$ is the subsystem’s length. In the examples considered the
prefactor $b_{\alpha}$ of the logarithm is computed analytically. Strictly
speaking, $b_{\alpha}$ is neither quantised nor universal, but it is
nevertheless nonzero and depends on only few system details.

Finally, we address the question of the instability of the symmetry-protected
topological phase from the constructive point of view of the separation of
time scales [45, 46, 47, 48]. Specifically, we consider two ways of breaking
the relevant symmetry: applying a weak symmetry-breaking transformation to the
initial state or heating it up to a finite low temperature. As expected,
string order melts down, but it does so over a time scale that is much longer
than the times that can be reached, for example, in numerical simulations
based on tensor networks.

##  II Overview and results

We consider time evolution after a global quench in an infinite spin-chain
system that, in almost the entire paper, is assumed to be translationally
invariant. That is to say, we investigate systems prepared in the ground state
or in a thermal state of a given pre-quench Hamiltonian $\boldsymbol{\bf
H}_{0}=\sum_{\ell}\boldsymbol{\bf h}_{0,\ell}$ and then time-evolved with a
different post-quench Hamiltonian $\boldsymbol{\bf
H}=\sum_{\ell}\boldsymbol{\bf h}_{\ell}$, e.g.,

$\displaystyle\ket{\Psi(t)}=e^{-i\boldsymbol{\bf H}t}\ket{\Psi(0)}\,,$ (3)
$\displaystyle(\boldsymbol{\bf H}_{0}\ket{\Psi(0)}=E_{\rm
GS}\ket{\Psi(0)})\,,$

where $E_{\rm GS}$ is the ground state energy of $\boldsymbol{\bf H}_{0}$. We
restrict ourselves to local Hamiltonian densities $\boldsymbol{\bf
h}_{0,\ell}$, $\boldsymbol{\bf h}_{\ell}$ (see discussion below). The focus is
on systems that are invariant under a global (sometimes called “on-site”)
symmetry characterised by some global unitary operator $\boldsymbol{\bf U}$,
such that $\boldsymbol{\bf U}\boldsymbol{\bf h}_{\ell}\boldsymbol{\bf
U}^{\dagger}=\boldsymbol{\bf h}_{\ell}$ and $\boldsymbol{\bf U}\boldsymbol{\bf
h}_{0,\ell}\boldsymbol{\bf U}^{\dagger}=\boldsymbol{\bf h}_{0,\ell}$.

We have three main goals:

  1. 1.

Show that the established picture of local relaxation to a Gibbs or a
generalised Gibbs ensemble can fail in symmetric systems;

  2. 2.

Uncover the reasons behind the failure of the standard descriptions and define
a novel family of statistical ensembles able to capture the stationary values
of local operators;

  3. 3.

Identify signatures of the exotic nonequilibrium phases.

We warn the reader that, in order to achieve our objectives, we need to
distinguish the notion of “local observable” from that of “local operator”.
The latter is for us only a representation of the former, as will be clarified
later.

We now provide an informal overview and contextualisation of our results.

###  II.1 Local observables in spin chains

In quantum mechanics the concept of locality is connected with the algebra of
the operators representing the observables. In practical terms, an observable
is local if it can be associated with a position in such a way that its
measurement does not affect the measurement of any other sufficiently distant
local observable. We wrote “sufficiently distant” because in a spin chain
local observables are not point-like objects but have a range corresponding to
the size of the finite subsystem affected by their measurement. Generically
local observables are represented by local operators of the form
$\boldsymbol{\bf O}_{A}\otimes\boldsymbol{\bf I}_{\bar{A}}$, where
$\boldsymbol{\bf O}_{A}$ has support in some finite connected subsystem $A$ of
the lattice, $\bar{A}$ denoting its complement. The range of the corresponding
local observable is then $|A|$.

####  II.1.1 Quasilocality

Time evolution does not preserve the locality of an operator: if the
Hamiltonian is not exceptionally simple, the Heisenberg representation of an
operator that is local at time $t=0$ does not have a finite range at any
$t\neq 0$. This problem can be resolved by weakening the definition of
locality so as to include also observables represented as limits of sequences
of local operators ordered by their range. For example
$\sum_{n=1}^{\infty}e^{-n}\boldsymbol{\bf\sigma}_{\ell-n}^{z}\boldsymbol{\bf\sigma}_{\ell+n}^{z}$
is not local but is quasilocal: it can be well approximated by the truncated
sums, and hence the limit is well defined. For local Hamiltonians it is
actually sufficient to restrict ourselves to the strong form of quasilocality
in which operators have exponentially decaying tails. A careful formalisation
of this intuitive idea eventually results in the definitions of Refs. [49, 50,
51, 30, 52].

Time evolution with local Hamiltonians preserves strong quasilocality [53,
54], but quantum mechanics is still a non-relativistic theory: causality,
usually expressed as a commutation of quasilocal operators at space-like
distances, does not hold. Nonetheless a weaker form of causality still
applies: Lieb and Robinson proved that the commutator of quasilocal operators
becomes exponentially small at space-like distances, providing a bound that
plays the role of the speed of light in relativistic systems [53].

####  II.1.2 Semilocality

Exponentially localised operators are usually regarded as the quintessential
representation of local observables in spin chains. Still, there are systems
with local observables that escape the boundaries of quasilocality. Let us
consider for example a quantum quench between two Hamiltonians with local
densities invariant under a spin flip $\mathcal{P}^{z}$:

$\displaystyle\begin{aligned} \boldsymbol{\bf
h}_{j}&=\mathcal{P}^{z}[\boldsymbol{\bf
h}_{j}]\equiv\lim_{n\rightarrow\infty}\Big{[}\prod_{\ell=-n}^{n}\boldsymbol{\bf\sigma}_{\ell}^{z}\Big{]}\boldsymbol{\bf
h}_{j}\Big{[}\prod_{\ell^{\prime}=-n}^{n}\boldsymbol{\bf\sigma}_{\ell^{\prime}}^{z}\Big{]}\,,\\\
\boldsymbol{\bf h}_{0,j}&=\mathcal{P}^{z}[\boldsymbol{\bf
h}_{0,j}]\,,\end{aligned}$ (4)

where $\boldsymbol{\bf\sigma}_{\ell}^{z}$ denotes the operator acting like the
Pauli $z$ matrix on site $\ell$ and like the identity elsewhere. Suppose that
the initial state is spin-flip symmetric. Thus, the expectation value of any
local observable $\boldsymbol{\bf O}_{\rm o}$ which is odd under spin flip,
i.e., $\mathcal{P}^{z}[\boldsymbol{\bf O}_{\rm o}]=-\boldsymbol{\bf O}_{\rm
o}$, vanishes. The symmetry of the Hamiltonian moreover implies that this
property holds true at any time. The system is completely characterised by
operators $\boldsymbol{\bf O}_{\rm e}$ that are even under spin flip, i.e.,
$\mathcal{P}^{z}[\boldsymbol{\bf O}_{\rm e}]=\boldsymbol{\bf O}_{\rm e}$;
without loss of information, we can restrict the space of operators to the
even ones.

Such a restriction has however a subtle by-product: in the restricted space
there are observables that are not represented by local operators but that
nevertheless behave as such. In this specific case the prototype of such an
observable is represented by an operator $\boldsymbol{\bf\Pi}^{z}(j)$ that
plays the role of a product of Pauli matrices $\boldsymbol{\bf\sigma}^{z}$
extending from site $j$ to infinity; see Section IV.1 for a proper definition.

$j$$\boldsymbol{\bf\Pi}^{z}(j)$$\boldsymbol{\bf O}_{\rm
o}(y)$$\boldsymbol{\bf-}\boldsymbol{\bf O}_{\rm o}(y)\,\,$$\boldsymbol{\bf
O}_{\rm o}(x)$$\boldsymbol{\bf O}_{\rm
o}(x)$(a)$j$$\boldsymbol{\bf\Pi}^{z}(j)$$\boldsymbol{\bf O}_{\rm
e}(y)$$\boldsymbol{\bf O}_{\rm e}(y)$$\boldsymbol{\bf O}_{\rm
e}(x)$$\boldsymbol{\bf O}_{\rm e}(x)$(b) Figure 1: Panel (a): odd operator
$\boldsymbol{\bf O}_{\rm o}(x)$ ($\boldsymbol{\bf O}_{\rm o}(y)$) with support
to the left (right) of the site $j$ commutes (anticommutes) with the string
$\boldsymbol{\bf\Pi}^{z}(j)$. For simplicity we assume $x\ll j\ll y$. Panel
(b): even operator with support that does not include the site $j$ commutes
with the string $\boldsymbol{\bf\Pi}^{z}(j)$. Restricting to the space of even
operators, the strings behave as local objects.

As done in Ref. [31], we call $\boldsymbol{\bf\Pi}^{z}(j)$ semilocal to
remember that its action is local only in the restricted space of even
operators — see Fig. 1. This example with a $\mathbb{Z}_{2}$ (spin-flip)
symmetry will be our testing ground, but the idea can be applied to other
symmetries: if every subsystem has a density matrix that is written at any
time in terms of a restricted set of local operators,

> _an alternative representation of local observables can be obtained by
> supplementing the restricted set with operators that are not local but
> behave as such. As in the $\mathbb{Z}_{2}$ case, we call them “semilocal”._

##### Duality.

We expect quite generally that, once extended to include also semilocal
operators, the restricted space becomes isomorphic to the original space of
quasilocal operators. In the $\mathbb{Z}_{2}$ case, this is manifested in the
existence of a _duality transformation_ that preserves the locality of the
Hamiltonian and maps even semilocal operators into local ones that are odd
under a spin flip (see Section IV.1). Analogous duality transformations exist
also when the system exhibits other symmetries — see, e.g., Refs. [55, 56].

####  II.1.3 Cluster decomposition

Thermal states and ground states of local Hamiltonians have clustering
properties. That is to say, the expectation value of the product of two local
operators far away from each other factorises:

$\lim_{y\rightarrow\infty}\\!(\braket{\boldsymbol{\bf O}_{1}(x)\boldsymbol{\bf
O}_{2}(x+y)}\\!-\\!\braket{\boldsymbol{\bf
O}_{1}(x)}\\!\braket{\boldsymbol{\bf O}_{2}(x+y)})=0\,.$ (5)

Cluster decomposition is intimately connected with the phenomenon of symmetry
breaking in ordered phases: generic nonsymmetric perturbations break the
symmetry of the ground state in such a way that the state always ends up
satisfying clustering [57]. If our system is symmetric, semilocal operators
represent local observables as well, therefore we require that

> _a symmetric physical state satisfies clustering also for semilocal
> operators_.

This additional requirement allows us to completely break the barrier between
semilocality and locality: there is no physical property distinguishing
semilocal operators from local ones. In view of this, local, (strongly)
quasilocal, and semilocal operators can all represent local observables.
Incidentally, we mention that this equivalence becomes natural when taking the
continuum limit close to a critical point, where all the aforementioned
operators become represented by local fields.

In the previous example of the $\mathbb{Z}_{2}$ symmetry, clustering fixes the
expectation value of the semilocal operators up to an overall sign. Section
IV.2 will show that the auxiliary sign is determined by how the symmetry is
broken in the dual representation.

####  II.1.4 Semilocal _vs_. symmetry-protected topological order

We are considering noncritical spin-chain Hamiltonians with discrete global
symmetries. If, in equilibrium at zero temperature, some of the symmetries
remain unbroken, the state can exhibit so-called symmetry-protected
topological order [58]. Because of the symmetry of the Hamiltonian, we can
find string operators $\boldsymbol{\bf O}_{A}$ that act differently from the
identity everywhere within a connected region $A$ and commute with the energy
densities $\boldsymbol{\bf h}_{\ell}$ with support inside $A$. The order is
manifested in the fact that there are operators of that kind with a nonzero
expectation value in the limit $|A|\rightarrow\infty$. This is usually
referred to as “string order” [55].

> _We reinterpret string order as the existence of an alternative
> representation of local observables including semilocal operators with a
> nonzero expectation value._

As far as we can see, indeed, it is always possible to reinterpret
$\braket{\boldsymbol{\bf O}_{A}}$ as the correlation between two semilocal
operators positioned at the boundaries of $A$. Clustering of semilocal
operators together with the nonvanishing value of $\braket{\boldsymbol{\bf
O}_{A}}$ in the limit $|A|\rightarrow\infty$ imply that the expectation values
of those semilocal operators are different from zero. In analogy with the
local order parameters characterising a phase with a spontaneously broken
symmetry, we call them semilocal order parameters.

The reader can understand this as an alternative way of presenting the results
of Refs. [55, 56], which relate symmetry-protected topological order to a
standard Landau phase (with a broken symmetry) in a dual representation. For
example, in the $\mathbb{Z}_{2}$ case
$\lim_{n\rightarrow\infty}\braket{\prod_{j=\ell-n}^{\ell+n}\boldsymbol{\bf\sigma}_{j}^{z}}$
is the natural string-order parameter and we have
$\prod_{j=\ell-n}^{\ell+n}\boldsymbol{\bf\sigma}_{j}^{z}=\boldsymbol{\bf\Pi}^{z}(\ell-n)\boldsymbol{\bf\Pi}^{z}(\ell+n+1)$.

Since we have not carefully investigated the equivalence between semilocal
order and symmetry-protected topological order, we will refrain from using the
latter terminology when there is a risk that our claims could not hold in full
generality. We note however that the basic property of symmetry-protected
topological order is satisfied: semilocal order is preserved if the initial
state is perturbed without breaking the symmetry associated with the order.
For example, in the case of a $\mathbbm{Z}_{2}$-symmetric initial state
$\ket{\Psi(0)}$ the transformed state $e^{i\boldsymbol{\bf
W}}\\!\ket{\Psi(0)}$ retains the symmetry for any spin-flip invariant operator
$\boldsymbol{\bf W}$ with strongly quasilocal density. The collection of such
states obtained by all possible symmetry-preserving perturbations forms a
symmetry-protected phase [1]. All these states are characterised by a
nonvanishing expectation value of a semilocal operator.

###  II.2 Conservation laws with semilocal densities

Most spin-flip invariant Hamiltonians with local densities do not admit
dynamics in which the expectation values of semilocal operators remain nonzero
also at late times. This can be readily understood if, like in the
$\mathbb{Z}_{2}$ case, there is a duality transformation mapping semilocal
operators into local ones that break the symmetry. In the dual representation
their expectation value remains nonzero only if there are conserved charges
that break the symmetry of the dual Hamiltonian.

Conversely, for every symmetric Hamiltonian with at least one local charge
that doesn’t respect the symmetry, there is a dual Hamiltonian with at least
one conservation law with a semilocal density. In the specific case of spin-
flip symmetric Hamiltonians, the simplest examples of systems of this kind
have dual Hamiltonians that are spin-flip invariant in all the directions,
like the Hamiltonian of the Heisenberg model. In particular,

> _any local Hamiltonian with a spin-flip symmetry and a $U(1)$ charge that
> breaks spin flip is dual to a model with a semilocal charge._

The prototypical example of the latter has the following Hamiltonian

$\boldsymbol{\bf
H}=\sum_{\ell\in\mathbbm{Z}}\boldsymbol{\bf\sigma}_{\ell-1}^{x}(\boldsymbol{\bf
1}-\boldsymbol{\bf\sigma}_{\ell}^{z})\boldsymbol{\bf\sigma}_{\ell+1}^{x}+\boldsymbol{\bf
W}_{\ell}[\\{\boldsymbol{\bf\sigma}^{z}\\}]\,,$ (6)

where $\boldsymbol{\bf W}_{\ell}$ can be any local operator written only in
terms of $\boldsymbol{\bf\sigma}^{z}$. To the best of our knowledge, for
generic $\boldsymbol{\bf W}_{\ell}$ the model does not have conservation laws
with quasilocal densities. On the other hand, just using the algebra satisfied
by $\boldsymbol{\bf\Pi}^{z}(j)$, the reader can verify that $\boldsymbol{\bf
H}$ has the following conservation law with a semilocal density:

$\boldsymbol{\bf Q}=\sum_{\ell\in\mathbbm{Z}}\boldsymbol{\bf\Pi}^{z}(\ell)\,.$
(7)

Following Refs. [30, 52] literally, we would exclude this conservation law
from the description of the long-time behaviour of the expectation values of
local operators. Indeed $\boldsymbol{\bf\Pi}^{z}(\ell)$ cannot be written as a
limit of a sequence of local operators and does not even have a nonzero
infinite-temperature overlap with a local operator. Is this charge relevant
after a global quench?

####  II.2.1 Breakdown of maximum-entropy descriptions

After a quantum quench of a global parameter at zero temperature, the local
properties of the state approach those of an effective stationary state. The
standard picture is that the latter is completely characterised by
(pseudo)local integrals of motion carrying information about the initial
state, and it can be represented by the maximum-entropy ensemble satisfying
those constraints — see Section III.2. As such, the emerging state, which is
called Gibbs or generalised Gibbs ensemble, is expected to resemble an
equilibrium state at finite temperature. We remind the reader that finite-
temperature states in one dimension do not generally exhibit topological
order; this was proven, in particular, for global onsite symmetries [12].
Because of the similarity between (generalised) Gibbs ensembles and thermal
states, it is reasonable to expect that, in such maximum-entropy descriptions,
the two-point functions of semilocal operators approach zero at large
distances.

Let us now re-examine global quenches in symmetric systems in the light of our
previous considerations about semilocality. If the state before the quench is
not symmetric (i.e., its density matrix is not even), semilocal operators do
not represent local observables, exactly as in global quenches without global
symmetries. We expect, therefore, the emergence of a standard (generalised)
Gibbs ensemble; we will consider an explicit example in Section III.2.1.

The situation changes if the initial state exhibits semilocal order. It is
simple to show that

> _semilocal charges enable the possibility to keep memory of semilocal
> order,_

that is to say, they allow a symmetry-protected topological order phase to
survive the long-time limit after a quantum quench.

In the $\mathbb{Z}_{2}$ case, this can be readily proved considering the
fluctuations of a semilocal charge, let it be
$\sum_{j\in\mathbb{Z}}\boldsymbol{\bf q}^{\rm sl}_{j}$, with a nonzero
expectation value at the initial time. We indeed have

$\lim_{t\rightarrow\infty}\\!\frac{1}{|A|^{2}}\\!\sum_{j,n\in
A}\\!\braket{\Psi(t)}{\boldsymbol{\bf q}^{\rm sl}_{j}\boldsymbol{\bf q}^{\rm
sl}_{n}}{\Psi(t)}\\!\geq\\!\braket{\Psi(0)}{\boldsymbol{\bf q}^{\rm
sl}}{\Psi(0)}^{2}$ (8)

for every subsystem $A$, where we used the non-negativity of fluctuations and
$\braket{\Psi(t)}{\boldsymbol{\bf q}^{\rm
sl}}{\Psi(t)}=\braket{\Psi(0)}{\boldsymbol{\bf q}^{\rm sl}}{\Psi(0)}$. In the
limit of large $|A|$ the left-hand side of Eq. (8) can only be nonzero if the
state at infinite time exhibits string order, specifically, if
$\lim_{n\to\infty}\braket{\boldsymbol{\bf q}^{\rm sl}_{j}\boldsymbol{\bf
q}^{\rm sl}_{j+n}}\neq 0$. Thus, a nonzero initial expectation value
$\bra{\Psi(0)}\boldsymbol{\bf q}^{\rm sl}\ket{\Psi(0)}\neq 0$ implies survival
of the string order in the limit of infinite time. Being blind to a string
order, a (generalised) Gibbs ensemble constructed with the pseudolocal charges
of the model (see Section III.2 for the definition of pseudolocality) can not
describe such a limit. For an explicit example of the breakdown of the
maximum-entropy descriptions in integrable and generic systems we refer the
reader to Section III.3.

####  II.2.2 Semilocal (generalised) Gibbs ensemble

The charges responsible for the persistent order can be accommodated in a
minimal extension of the standard theory with quasilocal operators, which must
now include also semilocal operators. In such an extended space we define two
novel maximum-entropy statistical ensembles: the _$G$ -semilocal Gibbs
ensemble_ and the _$G$ -semilocal generalised Gibbs ensemble_, where $G$
refers to the symmetry exploited to enlarge the theory (e.g., the
$\mathbbm{Z}_{2}$ symmetry) — see Section IV.3. The former emerges in generic
systems, in which there is a finite number of local and semilocal conservation
laws; the latter incorporates instead the richer structure of integrable
systems, where there are infinitely many pseudolocal and semilocal charges.
They are supposed to capture the infinite-time limit as the corresponding
Gibbs and generalised Gibbs ensembles do in non-symmetric systems.

We posit an additional step: the semilocal ensemble should be projected back
onto a theory of local observables. Indeed, not all operators in the extended
theory can represent local observables (they do not satisfy the causality
principle sketched in Section II.1) and the choice of which of them can is
ambiguous: in the spin-flip example ($G=\mathbbm{Z}_{2}$), odd operators and
even operators with half-infinite strings do not commute at infinite distance
but they both, separately, can supplement the even quasilocal operators to
represent local observables — see Fig. 2.

$\mathcal{P}^{z}[\boldsymbol{\bf\sigma}^{z}_{\ell}]=\boldsymbol{\bf\sigma}^{z}_{\ell}$even
quasilocal$\mathcal{P}^{z}[\boldsymbol{\bf\sigma}^{x}_{\ell}]=-\boldsymbol{\bf\sigma}_{\ell}^{x}$odd
quasilocal$\mathcal{P}^{z}[\boldsymbol{\bf\Pi}^{z}(\ell)]=\boldsymbol{\bf\Pi}^{z}(\ell)$even
semilocal Figure 2: Quasilocal theory (red line) and semilocal theory (blue
dashed line). Each of them contains operators that are able to represent local
observables, provided that the operators of the other theory are excluded
(even semilocal and odd quasilocal operators can not simultaneously represent
local observables — see Fig. 1).

That said, local observables are typically represented by quasilocal operators
(excluding therefore, in the spin-flip case, operators with half-infinite
strings). In such a theory

> _persistent semilocal order manifests itself in the fact that the projected
> ensemble does not maximise the entropy constrained by the quasilocal
> integrals of motion._

Note however that, in all the cases we envisage, there is also an alternative
representation of local observables in which the projected ensemble is a
maximum-entropy state as in Ref. [59].

In conclusion, symmetric systems allow for multiple representations of local
observables and only a part of them, which we will refer to as “canonical”
theories, include the densities of all the operators that are conserved in the
extended space — see Section IV.3.3.

###  II.3 Signatures of semilocal order

The fact that the two-point function of semilocal operators does not vanish in
the limit of infinite distance is arguably the most evident signature of
semilocal order in nonequilibrium states (see Section V.3 for a study of a
semilocal order parameter in the dual XY model). In this respect, we only add
a remark:

> _the semilocal integrals of motion are semilocal order parameters with the
> additional property of being stationary._

In fact, this is only part of the story. That a symmetric system can be
described in alternative ways (e.g., by a quasilocal vs. by a semilocal
theory) can have striking consequences. In particular, perturbations that
generally have insignificant effects can trigger macroscopic changes. Two
phenomena of this kind are described in the following.

####  II.3.1 Excess of entropy

Ref. [31] has recently shown that, in a symmetric quench, a localised
perturbation to the initial state affects the stationary values of local
operators. Here we make the next step, studying the effect of the perturbation
on the entanglement properties of large subsystems. We parametrise the
perturbation by an odd transformation $\boldsymbol{\bf U}_{\ell}$ that
connects different symmetry sectors and is localised around some position
$\ell$ (in our specific example we will consider $\boldsymbol{\bf
U}_{\ell}=\boldsymbol{\bf\sigma}_{\ell}^{x}$):
$\ket{\Psi(0)}\rightarrow\boldsymbol{\bf U}_{\ell}\ket{\Psi(0)}$. The reader
can think of $\boldsymbol{\bf U}_{\ell}$ as of the result of a projective
measurement of a local operator — see also Ref. [40]. Since the perturbation
is local, it has a limited effect on the initial state. In particular, it does
not affect at all the entanglement entropies of subsystems that enclose the
support of $\boldsymbol{\bf U}_{\ell}$ completely.

After a global quench the entropies of spin blocks grow in time until reaching
an extensive value [60]. If there are no semilocal charges, the effect of
$\boldsymbol{\bf U}_{\ell}$ approaches zero in the limit of infinite time. On
the other hand, semilocal charges keep memory of the perturbation and even the
entanglement entropies of large subsystems remain affected. We consider in
particular the _excess of entropy_ , which was recently studied in states at
zero temperature in which a symmetry is spontaneously broken [43]. It is
defined as the increase in the entanglement entropies produced by the local
perturbation

$\displaystyle\begin{aligned} \Delta_{\boldsymbol{\bf
U}_{\ell}}S_{A}(t)=&S[\mathrm{tr}_{\bar{A}}(e^{-i\boldsymbol{\bf
H}t}\boldsymbol{\bf U}_{\ell}\ket{\Psi(0)}\bra{\Psi(0)}\boldsymbol{\bf
U}^{\dagger}_{\ell}e^{i\boldsymbol{\bf H}t})]\\\
&-S[\mathrm{tr}_{\bar{A}}(e^{-i\boldsymbol{\bf
H}t}\ket{\Psi(0)}\bra{\Psi(0)}e^{i\boldsymbol{\bf H}t})]\,,\end{aligned}$ (9)

where $S[\rho]$ can be any functional of the density matrix $\rho$ that
measures the entanglement between $A$ and the rest. We will focus on the Rényi
entropies $S[\rho]=[1/(1-\alpha)]\log\mathrm{tr}(\rho^{\alpha})$. Remarkably,

> _$\Delta_{\boldsymbol{\bf U}_{\ell}}S_{A}(t)$ does not approach zero in the
> limit of infinite time; it becomes proportional to the logarithm of the
> subsystem’s length_
>
> $\Delta_{\boldsymbol{\bf U}_{\ell}}S_{A}(t)\propto\log|A|\,.$ (10)

In Section V.2 we compute the prefactor analytically in systems that are dual
to noninteracting spin models. Despite appearing universal when considering
the dual XY model, which is our favourite testing ground, we argue that it is
not. The prefactor depends indeed on non-universal details that are
accidentally irrelevant in that model.

While the importance of the prefactor could be questioned, the logarithm
growth of the excess of entropy is, to the best of our understanding, a
striking exceptional property of systems with semilocal conservation laws.

####  II.3.2 Melting of the order

In generic systems after global quenches the energy is the only information
about the initial state that survives the limit of infinite time; the
stationary values of local operators are then captured by effective Gibbs
ensembles. Since finite-temperature phase transitions in spin chains described
by local Hamiltonians are exceptional if not forbidden 111Although a discrete
symmetry could in principle be broken, we are only aware of results pointing
at its impossibility in specific classes of models, the norm is that the
stationary expectation values of local observables are smooth functions of the
initial state.

The stationary expectation values of local operators are expected to be
described by an effective Gibbs ensemble even in the presence of semilocal
conservation laws, provided that the initial conditions are generic. Indeed,
semilocal charges are relevant only for symmetric initial states; one could
then argue that the systems we are considering require a fine-tuning and hence
are physically irrelevant.

This unremarkable picture is however a consequence of a naive physical
interpretation of asymptotic behaviour. In reality the limit of infinite time
is an effective description of times sufficiently larger than the relaxation
time 222There is no standard definition of “relaxation time”, but our
considerations are quite independent of the details of the definition.. The
latter diverges in the limit in which the symmetry in the initial state
becomes exact but it is still finite for an exactly symmetric initial state.
This happens because the limit of exact symmetry does not commute with the
limit of infinite time. Consequently, for any given arbitrarily large time,
the discrepancy between the expectation value of local observable and its
infinite time limit becomes larger and larger the closer the system is to the
symmetric point (see, e.g., Fig. 5 in Section III.3). Such a slow relaxation
can be understood only taking into account the existence of semilocal charges
at the symmetric point.

In Section V.3 we consider two simple ways of breaking the symmetry in the
initial state: a global rotation and an increase of the temperature. In both
cases we find a

> _nontrivial scaling behaviour in the limit where the time is large and
> comparable with the correlation length of semilocal operators._

The latter, indeed, diverges in the limit of exact symmetry (while being
finite at the symmetric point): the limit corresponds to recovering the
symmetry without allowing semilocal operators to have a nonzero expectation
value, finally leading to the breakdown of cluster decomposition.

##### Organisation of the rest of the paper:

Section III

It reviews some established results on relaxation after global quenches in
quantum spin chains. Section III.1 presents the general picture in a
descriptive way, whereas Section III.2 goes into more details about the late-
time description. Two simple examples supporting the validity of the picture
are exhibited. Section III.3 points out when the standard description fails,
exhibiting also a suspected counterexample in a generic system.

Section IV

It is the core of the paper, where the failure of the standard picture is
explained and a theory able to overcome those problems is developed from
scratch. Since the issue has deep roots in the mathematical structure
underlying the representation of local observables in infinite spin chains,
the section is more abstract than the rest of the paper. Section IV.1
describes a Kramers-Wannier duality transformation introducing ingredients
that will be used in the following. Section IV.2 explains how a state can have
distinct representations in different theories of local observables. Section
IV.3 formalises the structure revealed in the previous section and defines the
semilocal statistical ensembles able to capture the infinite time limit.

Section V

It describes some effects of persistent semilocal order. Section V.1 shows
that the notion of subsystem changes across different theories of local
observables, culminating in a formula that expresses a reduced density matrix
in a theory in terms of reduced density matrices of a different theory.
Section V.2 studies the excess of Rényi entropy induced by a local
perturbation. The asymptotic behaviour of the excess of entropy is computed
analytically in an exactly solvable model, exhibiting an unusual log behaviour
that is interpreted as a consequence of semilocal order. Section V.3 shows
that, despite semilocal order being fragile under symmetry-breaking
perturbations, it leaves clear marks in the time evolution of the expectation
value of local operators.

Section VI

It collects additional comments and a list of open problems.

##  III Long time limit: state of the art

This section presents some established results on quench dynamics in
translationally invariant systems described by Hamiltonians with densities
$\boldsymbol{\bf h}_{\ell}$ that are _local_. As reviewed in Section III.2,
this means that each density $\boldsymbol{\bf h}_{\ell}$ acts non-trivially on
a finite connected subsystem $A_{\ell}$ of the spin chain, while on the rest
of the system $\bar{A_{\ell}}$ it acts as an identity.

If the reader is familiar with relaxation in isolated quantum many-body
systems, they can skip Section III.1. If they are also familiar with the
concept of pseudolocality, they can also skip Section III.2. We advise however
to read Section III.3, as it shows when the standard descriptions of
relaxation fail.

###  III.1 Background

Generally only a tiny part of the excited states of the Hamiltonian gives a
significant contribution to the expectation values of local observables after
a global quench. Even if tiny with respect to the total Hilbert space, this
part is however still exponentially large with respect to the volume of the
system (this can be a-posteriori understood by observing that the entanglement
entropy of subsystems becomes extensive after global quenches [63]).
Consequently, studies of global quenches are challenging both numerically and
analytically.

Quantum quenches in translationally invariant many-body systems have been
intensively investigated in the last two decades, leading to the following
picture:

  * –

In finite systems, consistently with the quantum recurrence theorem [64],
there is no relaxation. The distributions of the expectation values over time,
however, are highly peaked at values that can be described by the so-called
_diagonal ensemble_ [32].

  * –

In the thermodynamic limit, if the initial state has clustering properties and
the Hamiltonian is local, relaxation is the norm, and the expectation values
of local observables can be described by _Gibbs_ or _generalised Gibbs
ensembles_ (GGE) [25].

  * –

Generally, the diagonal ensemble and the (generalised) Gibbs ensemble are
locally equivalent [28]. They belong to a family of stationary states with the
same local properties, the (generalised) Gibbs ensemble being the state
keeping the least amount of information about the initial state. This
equivalence class is sometimes called _macrostate_ [25].

  * –

Eigenstate thermalisation hypothesis [33, 34, 35, 32] and its generalisation
to integrable systems [65] point at the fact that the aforementioned family of
stationary states includes excited states, usually referred to as
_representative states_ [66].

The diagonal ensemble can be obtained by killing the non-stationary part of
the time-evolving density matrix in a basis diagonalising the Hamiltonian
(i.e., only the (block)-diagonal part of the density matrix remains). In
practice this can only be done in sufficiently small systems. (Generalised)
Gibbs ensembles and representative states are more suitable for analytical
investigations. In integrable systems, in particular, they can be directly
used to compute expectation values. Generalised Gibbs ensembles rely on the
identification of the smallest set of conserved operators characterising the
stationary properties of local observables. Representative states instead
require the knowledge of the overlaps between the initial state and a generic
excited state.

Contrary to classical systems, every quantum system has a number of
conservation laws in involution equal to its dimension (namely, the projectors
on the eigenstates). In the thermodynamic limit, however, relaxation is a
property of a restricted class of observables, therefore that exponentially
large (in volume) set of conservation laws is redundant, like in turn is
redundant the description in terms of the diagonal ensemble.

The first studies on local relaxation after global quenches focused on non-
interacting systems of fermions and bosons [36]. In those cases the Wick’s
theorem is sufficient to conclude that the mode occupation numbers provide a
sufficiently large set of conserved operators. Having in mind interacting
models, the attention had then moved to reinterpreting the known results in a
way that could be easily generalised in the presence of interactions.

The importance of locality of the conservation laws [67] was then pointed out.
It sets a clear (in translationally invariant systems) division between
integrable and generic systems. Such a locality principle was strongly
questioned after some discrepancies between theory and numerics in the
Heisenberg XXZ spin-$1/2$ chain were observed. Using the so-called “quench
action” method [27], based on representative states, it was shown that the
generalised Gibbs ensemble constructed with local charges in the XXZ
spin-$1/2$ chain was inadequate [68, 69]. These problems have been finally
resolved with the discovery of new families of conservation laws [70, 51, 71,
72] which had been previously overlooked. Their peculiarity is that their
densities are not strictly local but exhibit exponential tails. Locality, now
including also the new kind of operators, called pseudolocal [30], had
resurrected. Some technical discussions apart [73, 74, 75], the new picture
was positively received, and after the axiomatic definition of generalised
Gibbs ensemble put forward in Ref. [52] the question of the relevance of
conservation laws has been practically archived.

###  III.2 Maximum-entropy descriptions

Conserved quantities constrain the dynamics of the system and prevent the loss
of information about the initial state. An important question of the past
decade has been, which of these quantities enter in the ensemble
$\boldsymbol{\bf\rho}_{\infty}$ that describes local relaxation according to

$\displaystyle\lim_{t\to\infty}\bra{\Psi(t)}\boldsymbol{\bf
O}\ket{\Psi(t)}={\rm tr}(\boldsymbol{\bf\rho}_{\infty}\boldsymbol{\bf
O}),\quad\forall\boldsymbol{\bf O}\text{ local}.$ (11)

We remind the reader again that by “local” we mean that $\boldsymbol{\bf O}$
acts non-trivially only on a finite connected subsystem $A$, while on its
complement $\bar{A}$ it acts as an identity. We focus here on representations
of $\boldsymbol{\bf\rho}_{\infty}$ through maximum-entropy ensembles [59]. In
our specific case they are called _Gibbs_ or _generalised Gibbs_ , depending
on whether the system is generic or integrable.

The first assumption behind such a description is that, at late times, the
entanglement entropy

$S[\rho]=-{\rm tr}[\rho\log\rho]$ (12)

of subsystems becomes an extensive quantity. One can then look for a
representation of $\boldsymbol{\bf\rho}_{\infty}$ in terms of a state with an
extensive entropy. Thermal ensembles are the typical examples of such states.
Imagine then to use a thermal ensemble as a testing description of the
stationary expectation values (cf. Ref. [76]). Specifically, we define it in
such a way as to capture the energy of the state. We wonder whether the
presence of an additional conservation law $\boldsymbol{\bf
K}=\sum_{j}\boldsymbol{\bf k}_{j}$ could change the expectation value of local
operators. Guided by statistical physics, we use the Ansatz

$\displaystyle\boldsymbol{\bf\rho}_{\infty}=\frac{e^{\boldsymbol{\bf Q}}}{{\rm
tr}(e^{\boldsymbol{\bf Q}})}$ (13)

where $\boldsymbol{\bf Q}$ was originally assumed to be proportional to
$\boldsymbol{\bf H}$ and now is questioned whether to include or not an
additional linear dependence on $\boldsymbol{\bf K}$. In the absence of
$\boldsymbol{\bf K}$, $\boldsymbol{\bf Q}=-\beta\boldsymbol{\bf H}$ maximises
the entropy under the constraint given by the energy per site
$\braket{\boldsymbol{\bf h}_{j}}$ in the initial state. In order to be
relevant, the introduction of $\boldsymbol{\bf K}$ should result in

  1. (i)

a change in the expectation value of local operators;

  2. (ii)

an extensive reduction of the entropy.

While i is self-explanatory, ii is subtler and can be seen as a condition
allowing us to use the principle of maximum entropy [59].

To be more quantitative, the effect of including $\boldsymbol{\bf K}$ in the
ensemble can be interpreted as the result of a smooth variation
$\boldsymbol{\bf Q}=-\beta\boldsymbol{\bf H}\mapsto\boldsymbol{\bf
Q}(\lambda)=-\beta(\lambda)\boldsymbol{\bf H}+\lambda\boldsymbol{\bf K}$ that
brings the original ensemble into the new one. The variation of the
expectation value of a local operator $\boldsymbol{\bf O}$ under a small
change of $\lambda$ is

$\delta\braket{\boldsymbol{\bf O}}_{\lambda}=\braket{\delta\boldsymbol{\bf
Q}(\lambda),\boldsymbol{\bf O}}_{\lambda}\,,$ (14)

where $\delta\boldsymbol{\bf Q}(\lambda)=\boldsymbol{\bf
Q}^{\prime}(\lambda)\delta\lambda$, and the right-hand side is the Kubo-Mori
inner product. In this specific case ($[\delta\boldsymbol{\bf
Q},\boldsymbol{\bf Q}]=0$) the latter is reduced to the connected correlation
function $\braket{\boldsymbol{\bf A},\boldsymbol{\bf
B}}=\braket{\boldsymbol{\bf A}\boldsymbol{\bf B}}-\braket{\boldsymbol{\bf
A}}\braket{\boldsymbol{\bf B}}$; the subscript $\lambda$ indicates that the
expectation values should be taken in the state
$\boldsymbol{\bf\rho}_{\infty}(\lambda)$.

Since the energy density is fixed by the initial state, the variation should
preserve it, hence

$\displaystyle\braket{\delta\boldsymbol{\bf Q}(\lambda),\boldsymbol{\bf
h}_{j}}_{\lambda}=\delta\lambda\braket{-\beta^{\prime}(\lambda)\boldsymbol{\bf
H}+\boldsymbol{\bf K},\boldsymbol{\bf h}_{j}}_{\lambda}=0\,.$ (15)

Condition i then requires the existence of a local operator $\boldsymbol{\bf
O}$ satisfying 333Suppose instead that $\braket{\boldsymbol{\bf
K},\boldsymbol{\bf O}}_{\lambda}=0$ for all local operators $\bf O$ (including
$\boldsymbol{\bf O}=\boldsymbol{\bf h}_{j}$). Then condition
$\braket{-\beta^{\prime}(\lambda)\boldsymbol{\bf H}+\boldsymbol{\bf
K},\boldsymbol{\bf h}_{j}}_{\lambda}=0$ would imply either
$\braket{\boldsymbol{\bf H},\boldsymbol{\bf h}_{j}}=0$, which is clearly
false, or $\beta^{\prime}(\lambda)=0$, the latter leading to
$\delta\braket{\boldsymbol{\bf O}}_{\lambda}=0$ in Eq. (14).

$\displaystyle 0<\braket{\boldsymbol{\bf K},\boldsymbol{\bf
O}}_{\lambda}<\infty\,.$ (16)

On the other hand, the variation of the entropy per unit site is
$\delta\mathfrak{s}[\boldsymbol{\bf\rho}_{\infty}(\lambda)]=-\braket{\delta\boldsymbol{\bf
Q}(\lambda),\boldsymbol{\bf q}_{j}(\lambda)}_{\lambda}$, where
$\boldsymbol{\bf q}_{j}(\lambda)$ is the density of $\boldsymbol{\bf
Q}(\lambda)$ 444We consider the entropy per site rather than the entropy
because the latter is infinite in the thermodynamic limit.. Imposing the
energy constraint (15) we then obtain

$\delta\mathfrak{s}[\boldsymbol{\bf\rho}_{\infty}(\lambda)]=-\frac{1}{2}\delta(\lambda^{2})\braket{\boldsymbol{\bf
Q}^{\prime}(\lambda),\boldsymbol{\bf q}_{j}^{\prime}(\lambda)}_{\lambda}\,,$
(17)

which is non-positive along a path with increasing $|\lambda|$. Explicitly it
reads
$\delta\mathfrak{s}[\boldsymbol{\bf\rho}_{\infty}(\lambda)]=-\frac{1}{2}\delta(\lambda^{2})[\braket{\boldsymbol{\bf
K},\boldsymbol{\bf
k}_{j}}_{\lambda}-(\beta^{\prime}(\lambda))^{2}\braket{\boldsymbol{\bf
H},\boldsymbol{\bf h}_{j}}_{\lambda}]$, where $\boldsymbol{\bf k}_{j}$ is the
density of $\boldsymbol{\bf K}$. In order to apply the maximum entropy
principle, we need the entropy to be a smooth function of the Lagrange
multiplier $\lambda$, which requires 555If $\braket{\boldsymbol{\bf
K},\boldsymbol{\bf k}_{j}}$ was infinite $\beta^{\prime}(\lambda)$ would have
to compensate it in order for
$\delta\mathfrak{s}[\boldsymbol{\bf\rho}_{\infty}(\lambda)]$ to remain finite.
The entropy density would then not be a smooth function of $\lambda$.

$0<\braket{\boldsymbol{\bf K},\boldsymbol{\bf k}_{j}}_{\lambda}<\infty\,.$
(18)

This is usually referred to as “extensivity condition” and represents ii. The
generalisation to systems with more conservation laws is straightforward and
results in the same equations.

As a matter of fact, in the most formal works dealing with the emergence of
maximum-entropy ensembles after global quenches (see, e.g., Ref. [52]), an
additional assumption is made:

  1. ($\ast$)

the density $\boldsymbol{\bf k}_{j}$ can be obtained as a limit of a sequence
of local operators.

From our simplified explanation, it might not be evident where this extra
condition comes from. Explaining it goes beyond the mathematical rigour of
this work; we only mention that 1 has a role in making sense to the
thermodynamic limit. We will see however that, in order to describe the
stationary values in symmetric systems with maximum-entropy ensembles, we will
be forced to relax 1.

Conditions (16), (18), and 1 define what is now known as _pseudolocality_.
Charges that satisfy them include translationally invariant sums
$\boldsymbol{\bf Q}=\sum_{\ell}\boldsymbol{\bf q}_{\ell}$, where the density
has the form $\boldsymbol{\bf q}_{\ell}=\sum_{r=1}^{\infty}\boldsymbol{\bf
q}_{\ell}(r)$ with $\\{\boldsymbol{\bf q}_{\ell}(r)\\}$ being a sequence of
local operators that act on subsystems of increasing size $r$ and are
exponentially suppressed in $r$, e.g., $\langle\boldsymbol{\bf
Q},\boldsymbol{\bf q}_{\ell}(r)\rangle\leq Ce^{-r/\xi}$, for some $C,\xi>0$
[51, 30].

In the following we consider two examples of maximum-entropy descriptions in
integrable exactly solvable models. Note that, since the systems are
integrable, the corresponding maximum-entropy ensembles will have to
incorporate the constraints of infinitely many pseudolocal integrals of
motion.

####  III.2.1 Example: XY model

To illustrate the validity of the generalised Gibbs ensemble description we
consider quench dynamics in the XY model, whose time evolution is generated by
the Hamiltonian

$\displaystyle\boldsymbol{\bf
H}=\sum_{\ell\in\mathbbm{Z}}J_{x}\boldsymbol{\bf\sigma}_{\ell}^{x}\boldsymbol{\bf\sigma}_{\ell+1}^{x}+J_{y}\boldsymbol{\bf\sigma}_{\ell}^{y}\boldsymbol{\bf\sigma}_{\ell+1}^{y}\,.$
(19)

This is a very well known model [80]. We mention that the ground state is
noncritical for $|J_{x}|\neq|J_{y}|$. The critical line separates two ordered
phases where spin flip symmetry is broken (for $J_{x}>J_{y}$ the spins acquire
a nonzero $x$ component, whereas for $J_{y}>J_{x}$ they acquire a nonzero $y$
component).

Let the system be prepared in the initial state

$\displaystyle\ket{\Psi(0)}=\ket{\rotatebox[origin={c}]{45.0}{$\Rightarrow$}_{\theta}}:=\bigotimes_{\ell\in\mathbbm{Z}}\begin{pmatrix}\cos\tfrac{\theta}{2}\\\
\sin\tfrac{\theta}{2}\end{pmatrix}\,,$ (20)

which is the ground state of the Hamiltonian

$\displaystyle\boldsymbol{\bf
H}_{0}=-\sum_{\ell\in\mathbbm{Z}}\cos\left(\tfrac{\theta}{2}\right)\boldsymbol{\bf\sigma}_{\ell}^{z}+\sin\left(\tfrac{\theta}{2}\right)\boldsymbol{\bf\sigma}_{\ell}^{x}\,.$
(21)

For $\theta\notin\\{0,\pi\\}$ the initial state breaks the $\mathbbm{Z}_{2}$
symmetry $\mathcal{P}^{z}$ of the post-quench Hamiltonian $\boldsymbol{\bf H}$
(see Eq. (4)). The latter can be conveniently rewritten in terms of Majorana
fermions $\boldsymbol{\bf
a}^{x,y}_{\ell}=(\Pi_{j<\ell}\boldsymbol{\bf\sigma}^{z}_{j})\boldsymbol{\bf\sigma}^{x,y}_{\ell}$
as 666We remind the reader that the half-infinite string of Pauli matrices is
not well defined in an infinite chain; this is however irrelevant as long as
only operators quadratic in Majorana fermions are considered. For a rigorous
definition of Jordan-Wigner transformation see instead Ref. [91]. We will use
the latter definition in Section IV.1 where we discuss duality transformation
and semilocal operators.

$\displaystyle\boldsymbol{\bf
H}=\frac{1}{4}\sum_{\ell,n\in\mathbbm{Z}}\begin{pmatrix}\boldsymbol{\bf
a}^{x}_{\ell}&\boldsymbol{\bf
a}^{y}_{\ell}\end{pmatrix}\mathcal{H}_{\ell,n}\begin{pmatrix}\boldsymbol{\bf
a}^{x}_{n}\\\ \boldsymbol{\bf a}^{y}_{n}\end{pmatrix},$ (22)

where $\mathcal{H}_{\ell,n}=\int\tfrac{{\rm
d}k}{2\pi}e^{i(\ell-n)k}\mathcal{H}(k)$ is the Fourier transform of the
$2\times 2$ matrix

$\displaystyle\mathcal{H}(k)=2(J_{x}-J_{y})\sin
k\,\sigma^{x}-2(J_{x}+J_{y})\cos k\,\sigma^{y}.$ (23)

On account of the Toeplitz structure of $\mathcal{H}$, $\mathcal{H}(k)$ is
also referred to as the symbol of the Hamiltonian (see, e.g., Ref. [82]). The
symbol generates the evolution of the two-point correlations of Majorana
fermions according to

$\displaystyle\Gamma_{t}(k)=e^{-i\mathcal{H}(k)t}\Gamma(k)e^{i\mathcal{H}(k)t},$
(24)

where $\Gamma(k)$ is the symbol of the two-point correlation matrix

$\displaystyle\Gamma_{\ell,n}=\delta_{\ell,n}I-\langle\begin{pmatrix}\boldsymbol{\bf
a}^{x}_{\ell}\\\ \boldsymbol{\bf
a}^{y}_{\ell}\end{pmatrix}\begin{pmatrix}\boldsymbol{\bf
a}^{x}_{n}&\boldsymbol{\bf a}^{y}_{n}\end{pmatrix}\rangle\,,$ (25)

in the initial state. For example, in the ferromagnetic initial state (20)
with $\theta=0$ one has $\Gamma_{\ell,n}=\delta_{\ell,n}\sigma^{y}$ and
$\Gamma(k)=\sigma^{y}$.

Under some very mild assumptions on the dispersion relation which are
practically always satisfied (for example, the dispersion relation should not
be flat), at large time the symbol of the correlation matrix relaxes to its
time-averaged value
$\overline{\Gamma}(k)=\lim_{T\to\infty}\frac{1}{T}\int_{0}^{T}{\rm
d}t\,\Gamma_{t}(k)$. In our specific case we find

$\displaystyle\begin{aligned}
\overline{\Gamma}(k)=\frac{(\cos^{2}\\!\theta\\!+\\!1)\\!\cos^{2}\\!k\\!-\\!2\\!\cos\\!\theta\cos\\!k\\!+\\!\frac{J_{x}\\!-\\!J_{y}}{J_{x}\\!+\\!J_{y}}\\!\sin^{2}\\!\theta\sin^{2}\\!k}{2\cos\\!\theta\cos^{2}\\!k\\!-\\!(\cos^{2}\\!\theta\\!+\\!1)\cos\\!k}\begin{pmatrix}0&\tfrac{\\!-\\!i(J_{x}\\!+\\!J_{y})\cos\\!k}{(J_{x}\\!+\\!J_{y})\cos\\!k+i(J_{x}\\!-\\!J_{y})\sin\\!k}\\\
\tfrac{i(J_{x}\\!+\\!J_{y})\cos\\!k}{(J_{x}\\!+\\!J_{y})\cos\\!k-i(J_{x}\\!-\\!J_{y})\sin\\!k}&0\\\
\end{pmatrix}\,.\end{aligned}$ (26)

In a non-interacting model, Ref. [83] established that (26) is actually
equivalent to a symbol of the two-point correlation matrix in a generalised
Gibbs ensemble, i.e., $\overline{\Gamma}(k)\equiv\Gamma_{\rm GGE}(k)$. Under
some assumptions it was later proven [84, 85] that, if the Hamiltonian
generating the time evolution is translationally invariant, and if the initial
state has clustering properties, the asymptotic state of the system is
Gaussian, whence higher-order correlations can be accessed using Wick’s
theorem. These conditions are satisfied by Hamiltonian (19) and our state
(20). The time-averaged correlation matrix (26) thus determines not only the
two-point correlations of Majorana fermions at late times (e.g.
$\braket{\boldsymbol{\bf\sigma}^{z}_{\ell}}$, shown in Fig. 3), but also
higher-order correlations, as demonstrated in Fig. 4, which shows the
relaxation of $\langle\boldsymbol{\bf\sigma}^{x}_{\ell}\rangle$ (a string of
Majorana fermions).

Figure 3: Relaxation of $\langle\boldsymbol{\bf\sigma}_{\ell}^{z}\rangle$
towards $\langle\boldsymbol{\bf\sigma}_{\ell}^{z}\rangle_{\rm GGE}\approx
0.466\,(0.717)$ (dashed lines) in the XY model, computed from Eq. (26). The
dots represent the results of the iTEBD simulation, while the solid curve is
the exact time evolution using Eq. (24). Parameters in Eqs. (19) and (20) are
$J_{x}=1$, $J_{y}=2$, and $\theta=0.9\,(0.3)$. The iTEBD evolution uses a
second order Trotter scheme with two-site quantum gates, time step $\delta
t=0.01$, Schmidt values cutoff $10^{-6}$ and maximal allowed bond dimension
$M_{\rm max}=800$. Figure 4: Relaxation of
$\langle\boldsymbol{\bf\sigma}_{\ell}^{x}\rangle$ towards
$\langle\boldsymbol{\bf\sigma}_{\ell}^{x}\rangle_{\rm GGE}\approx 0$ (dashed
lines) in the XY model. Parameters are the same as in Fig. 3.

####  III.2.2 Example: Dual XY model

We now consider a global quench from the same initial state
$\ket{\rotatebox[origin={c}]{45.0}{$\Rightarrow$}_{\theta}}$ in the model
described by the Hamiltonian

$\displaystyle\boldsymbol{\bf
H}=\sum_{\ell\in\mathbbm{Z}}\boldsymbol{\bf\sigma}^{x}_{\ell-1}(J_{x}\boldsymbol{\bf
I}-J_{y}\boldsymbol{\bf\sigma}^{z}_{\ell})\boldsymbol{\bf\sigma}^{x}_{\ell+1}.$
(27)

To the best of our knowledge, this model has not been studied much. We mention
that the ground state is noncritical for $|J_{x}|\neq|J_{y}|$. As shown in
Appendix C, for $|J_{y}|<|J_{x}|$ the ground state is in a Landau phase where
both spin flip symmetries over odd and even sites are broken, for
$|J_{y}|>|J_{x}|$, instead, it is in a nontrivial
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ protected topological phase (see, e.g.,
Ref. [86] for $J_{x}=0$).

Hamiltonian (27) can be mapped into the one of the quantum XY model, given in
Eq. (19), by means of a Kramers-Wannier duality transformation, which will be
described in detail in Section IV.1. After the duality transformation the
Hamiltonian (27) is therefore quadratic in terms of Majorana fermions: the
time evolution of the $n$-point correlation functions is determined only by
the $n$-point correlation functions at the initial time. As reviewed in the
previous section, only the $2$-point functions survive the limit of infinite
time (the asymptotic state is Gaussian). The correlation matrix of the
generalised Gibbs ensemble can thus easily be determined from the initial
correlation matrix (which consists of the initial expectation values of all
local operators mapped by the duality transformation into operators that are
quadratic in Majorana fermions). This allows one to obtain the stationary
values of local operators with minimal effort. Since however we are
considering this model as a representative of a larger class of systems for
which this procedure would not apply, we describe also an alternative
approach.

Specifically, one could infer that this is an integrable model from the fact
that a boost operator $\boldsymbol{\bf B}$ exists and takes the standard form

$\boldsymbol{\bf B}=\sum_{\ell\in\mathbb{Z}}\ell\,\boldsymbol{\bf h}_{\ell},$
(28)

where $\boldsymbol{\bf
h}_{\ell}=\boldsymbol{\bf\sigma}^{x}_{\ell-1}(J_{x}\boldsymbol{\bf
I}-J_{y}\boldsymbol{\bf\sigma}^{z}_{\ell})\boldsymbol{\bf\sigma}^{x}_{\ell+1}$.
That is to say, we can construct a tower of conserved operators
$\boldsymbol{\bf Q}_{n}$ in the following manner:

$\boldsymbol{\bf Q}_{n+1}=i[\boldsymbol{\bf B},\boldsymbol{\bf
Q}_{n}],\qquad\boldsymbol{\bf Q}_{1}=\boldsymbol{\bf H}.$ (29)

We are then in a position to either guess the corresponding Lax operator
(which, in this model, was obtained in Ref. [87]) and exploit the integrable
structure to compute the GGE expectation values [88], or to construct a
truncated generalised Gibbs ensemble [89] using the most local charges. For
generic $\theta$ all these methods converge to the same values.

Relaxation of local observables in the dual XY model is shown in Figs. 5 and
6. For $\theta\notin\\{0,\pi\\}$ all local observables eventually relax to
their respective GGE predictions. Note, however, that for some local
observables the time scale on which relaxation happens tends to increase when
$\theta$ decreases — see Fig. 5.

Figure 5: Numerical (iTEBD) time evolution of
$\braket{\boldsymbol{\bf\sigma}_{\ell}^{z}}$ and GGE predictions
($\braket{\boldsymbol{\bf\sigma}_{\ell}^{z}}_{\rm GGE}\approx 0.038$,
$0.0012$, and $0.000078$ for $\theta=0.9$, $0.3$, and $0.15$, respectively) in
the dual XY model. Relaxation towards the GGE prediction is slower for smaller
$\theta$. Parameters of the Hamiltonian (27) are $J_{x}=1$, $J_{y}=2$. The
iTEBD uses second order Trotter scheme with four-site quantum gates, $\delta
t=0.01$, Schmidt values cutoff $10^{-6}$, and maximum bond dimension $M_{\rm
max}=1000$. The data converges up to times $t\sim 4.5$ (see Appendix A).
Figure 6: Relaxation of
$\boldsymbol{\bf\sigma}^{x}_{\ell-1}\boldsymbol{\bf\sigma}^{x}_{\ell+1}$
towards the GGE prediction
($\braket{\boldsymbol{\bf\sigma}_{\ell-1}^{x}\boldsymbol{\bf\sigma}_{\ell+1}^{x}}_{\rm
GGE}\approx 0.022808\,(0.005646)$ for $\theta=0.3\,(0.15)$, respectively) in
the dual XY model. Parameters are the same as in Fig. 5, except for the
maximal bond dimension: here, $M_{\rm max}=1000$ ($\theta=0.3$), $M_{\rm
max}=600$ ($\theta=0.15$).

###  III.3 Failure of Gibbs and generalised Gibbs ensemble in symmetric
systems

The Hamiltonians of the XY and the dual XY model have a $\mathbbm{Z}_{2}$
symmetry: $\mathcal{P}^{z}[\boldsymbol{\bf H}]=0$. If also the initial state
is even under $\mathcal{P}^{z}$ (e.g.,
$\ket{\rotatebox[origin={c}]{45.0}{$\Rightarrow$}_{\theta}}\\!\bra{\rotatebox[origin={c}]{45.0}{$\Rightarrow$}_{\theta}}$
for $\theta\in\\{0,\pi\\}$), the entire system is symmetric. Hence, only the
even conservation laws are expected to contribute to the stationary behaviour
of the local observables.

Figure 7: Relaxation of $\braket{\boldsymbol{\bf\sigma}_{\ell}^{z}}$ and
$\braket{\boldsymbol{\bf\sigma}_{\ell}^{z}\boldsymbol{\bf\sigma}_{\ell+1}^{z}}$
towards the respective GGE predictions (dashed lines) in the XY model.
Parameters are $J_{x}=1$, $J_{y}=2$, and $\theta=0$. The colored solid lines
correspond to exact calculation using Eq. (24), while the dots represent the
iTEBD results. The inset shows the $1/\sqrt{t}$ decay of oscillations (the
differences between the local maxima of oscillations and the GGE predictions
are plotted versus $1/\sqrt{t}$). The iTEBD uses second order Trotter scheme
with a two-site update rule, $\delta t=0.01$, Schmidt values cutoff $10^{-5}$,
and maximal bond dimension $M_{\rm max}=400$. Figure 8: Numerical evolution of
$\boldsymbol{\bf\sigma}^{z}_{\ell}$ (points) and the GGE prediction
$\braket{\boldsymbol{\bf\sigma}^{z}_{\ell}}_{\rm GGE}=0$ (blue line) in the
dual XY model, for $\theta=0$, $J_{x}=1$ and $J_{y}=2$. The time evolution
relaxes towards the value
$\overline{\braket{\boldsymbol{\bf\sigma}^{z}_{\ell}}}\approx 0.5625$. The
iTEBD parameters are the same as in Fig. 5.

Using the even-parity local charges in each model reproduces the generalised
Gibbs ensembles described in the previous examples. As demonstrated in Fig. 7,
the GGE description is correct in the XY model for $\theta=0$. In contrast, in
the dual XY model the prediction for the expectation value of
$\boldsymbol{\bf\sigma}^{z}_{\ell}$ is zero, while its time evolution
approaches a finite value
$\braket{\boldsymbol{\bf\sigma}^{z}_{\ell}}_{\infty}\approx 0.5625$ — see Fig.
8 777In fact, it will later become clear that the time evolution of
$\braket{\boldsymbol{\bf\sigma}^{z}_{\ell}}$ in the dual XY model is exactly
the same as that of
$\braket{\boldsymbol{\bf\sigma}^{z}_{\ell}\boldsymbol{\bf\sigma}^{z}_{\ell+1}}$
in the XY model shown in Fig. 7.. The generalised Gibbs ensemble using the
even local charges of the dual XY model thus _fails_ to capture the asymptotic
behaviour of $\boldsymbol{\bf\sigma}^{z}_{\ell}$.

Normally this would be indicative of having overlooked some pseudolocal
integrals of motion. Here, however, the discrepancy has a different origin
(which will be explained in Section IV) and the failure of the generalised
Gibbs ensemble acquires a more fundamental meaning.

Finally, as a concrete example of a generic system, we consider Hamiltonian
(6) with

$\displaystyle\boldsymbol{\bf
W}_{\ell}[\\{\boldsymbol{\bf\sigma}^{z}\\}]=w_{1}\boldsymbol{\bf\sigma}_{\ell}^{z}+\sum_{n=1}^{\infty}w_{2,n}\boldsymbol{\bf\sigma}_{\ell}^{z}\boldsymbol{\bf\sigma}_{\ell+n}^{z}$
(30)

and initial state

$\displaystyle\ket{\Psi(0)}=e^{i\frac{\varphi}{2}\sum_{\ell}\boldsymbol{\bf\sigma}_{\ell}^{x}\boldsymbol{\bf\sigma}_{\ell+1}^{x}}\ket{\Uparrow}\,,$
(31)

where $\ket{\Uparrow}$ means that all spins are up. To the best of our
knowledge, there is no (pseudo)local operator commuting with this Hamiltonian.
The expectation value of the energy density reads $\braket{\boldsymbol{\bf
h}_{\ell}}_{t}=1+(w_{1}+w_{2,1}-1)\cos^{2}\varphi+(\sum_{n=2}^{\infty}w_{2,n})\cos^{4}\varphi$.
We now provide numerical evidence that the state does not thermalise by
comparing the time evolution of $\braket{\boldsymbol{\bf\sigma}_{\ell}^{z}}$
starting from two initial states with the same energy. To that aim, we choose
the Hamiltonian’s parameters in such a way that
$0<(1-w_{1}-w_{2,1})/(\sum_{n=2}^{\infty}w_{2,n})<1$, so that the energy is
the same for initial states with either $\cos^{2}\varphi=0$ or
$\cos^{2}\varphi=(1-w_{1}-w_{2,1})/(\sum_{n=2}^{\infty}w_{2,n})$. Fig. 9 shows
quite clearly that the infinite-time limit of
$\braket{\boldsymbol{\bf\sigma}_{\ell}^{z}}$ depends on the initial state (at
the same energy), in contrast with the eigenstate thermalisation hypothesis
and the conjecture of local thermalisation.

Figure 9: Numerical evolution of $\boldsymbol{\bf\sigma}^{z}_{\ell}$ in the
generic model given by Eqs. (6) and (30). Parameters are
$J_{x}=J_{y}=w_{2,2}=1$, and $w_{1}=0.6$. The iTEBD parameters are the same as
in Fig. 5, except for the maximum bond dimension $M_{\rm max}$ (here various
bond dimensions are used, yielding comparable results).

##  IV Beyond (generalised) Gibbs ensembles

We aim at obtaining a canonical description of the macrostate describing the
infinite-time limit after symmetric quenches in systems such as the dual XY
model in which the standard (generalised) Gibbs ensemble fails — see Fig. 8.
Since the limit of infinite time makes sense only in the thermodynamic limit,
we make an effort to consider infinite chains directly.

Bearing in mind the duality correspondence highlighted in Refs. [55, 56]
between symmetry-protected topological phases and Landau phases, we start this
section describing a duality transformation. The most important property that
we want to stress since the beginning is that duality transformations can be
used to map algebras of operators representing local observables into one
another. This will prove essential in unveiling the importance of semilocal
integrals of motion on the relaxation of local observables.

###  IV.1 Kramers-Wannier duality

Let us denote by $\mathfrak{A}_{\rm ql}$ the ($C^{\star}$) algebra of
quasilocal operators in an infinite spin-$1/2$ chain. This algebra includes
local operators and limits of their sequences that converge in the operator
norm. It is generated by the local spin operators, which act like Pauli
matrices on site $j$ and like the identity elsewhere; we will denote them
either by $\boldsymbol{\bf\sigma}_{j}^{\alpha}$ or by
$\boldsymbol{\bf\tau}_{j}^{\alpha}$.

We consider a duality transformation $\mathcal{D}_{\mathbb{Z}_{2}}$ that
differs from the standard Kramers-Wannier duality, responsible for the self-
duality of the transverse-field Ising model, just in an additional rotation:

$\boldsymbol{\bf\tau}_{j}^{x}=\boldsymbol{\bf\sigma}_{j-1}^{x}\boldsymbol{\bf\sigma}_{j}^{x}\,,\qquad\boldsymbol{\bf\tau}_{j}^{z}\boldsymbol{\bf\tau}_{j+1}^{z}=\boldsymbol{\bf\sigma}_{j}^{z}\,.$
(32)

In order to define it properly in the infinite chain, we exploit the same
trick as Ref. [91], which needed to define a Jordan-Wigner transformation
directly in infinite systems.

In particular, we define $\boldsymbol{\bf T}^{\alpha}_{\sigma,s}$, where
$\alpha\in\\{x,z\\}$, $s=\pm$, and $\sigma\leftrightarrow\tau$, as the
Hermitian operators satisfying

$\displaystyle[\boldsymbol{\bf T}_{\sigma,s}^{\alpha}]^{2}=\boldsymbol{\bf
I},\quad\boldsymbol{\bf T}_{\sigma,s}^{\alpha}\boldsymbol{\bf
O}\boldsymbol{\bf
T}_{\sigma,s}^{\alpha}=\lim_{n\to\infty}\Big{[}\prod_{\ell=0}^{n}\boldsymbol{\bf\sigma}_{s\ell}^{\alpha}\Big{]}\boldsymbol{\bf
O}\Big{[}\prod_{\ell=0}^{n}\boldsymbol{\bf\sigma}_{s\ell}^{\alpha}\Big{]}$
(33)

for all quasilocal operators $\boldsymbol{\bf O}\in\mathfrak{A}_{\rm ql}$,
extended then by linearity as $\boldsymbol{\bf
T}_{\sigma,s}^{\alpha}(\boldsymbol{\bf O}_{1}+\boldsymbol{\bf
O}_{2}\boldsymbol{\bf T}_{\sigma,s}^{\alpha})\boldsymbol{\bf
T}_{\sigma,s}^{\alpha}=\boldsymbol{\bf T}_{\sigma,s}^{\alpha}\boldsymbol{\bf
O}_{1}\boldsymbol{\bf T}_{\sigma,s}^{\alpha}+(\boldsymbol{\bf
T}_{\sigma,s}^{\alpha}\boldsymbol{\bf O}_{2}\boldsymbol{\bf
T}_{\sigma,s}^{\alpha})\boldsymbol{\bf T}_{\sigma,s}^{\alpha}$ for
$\boldsymbol{\bf O}_{1},\boldsymbol{\bf O}_{2}\in\mathfrak{A}_{\rm ql}$ (the
role of such an extension will be discussed in Section IV.3). We also define
the auxiliary operators

$\displaystyle\boldsymbol{\bf\Pi}^{z}_{\sigma,+}(j)=\boldsymbol{\bf
T}_{\sigma,+}^{z}\cdot\begin{cases}\boldsymbol{\bf\sigma}^{z}_{j}\boldsymbol{\bf\sigma}^{z}_{j+1}\cdots\boldsymbol{\bf\sigma}^{z}_{-1}&j<0\\\
\boldsymbol{\bf I}&j=0\\\
\boldsymbol{\bf\sigma}_{0}^{z}\boldsymbol{\bf\sigma}^{z}_{1}\cdots\boldsymbol{\bf\sigma}^{z}_{j-1}&j>0.\end{cases}$
(34)

and

$\displaystyle\boldsymbol{\bf\Pi}^{x}_{\tau,-}(j)=\boldsymbol{\bf
T}_{\tau,-}^{x}\cdot\begin{cases}\boldsymbol{\bf\tau}^{x}_{j+1}\boldsymbol{\bf\tau}^{x}_{j+2}\cdots\boldsymbol{\bf\tau}^{x}_{0}&j<0\\\
\boldsymbol{\bf I}&j=0\\\
\boldsymbol{\bf\tau}_{1}^{x}\boldsymbol{\bf\tau}^{x}_{2}\cdots\boldsymbol{\bf\tau}^{x}_{j}&j>0,\end{cases}$
(35)

These operators act like a spin flip on the right or left of site $j$ included
and like the identity elsewhere. The rotated Kramers-Wannier duality is then
an algebra homomorphism mapping the algebra of operators generated by
$\\{\boldsymbol{\bf\tau},\boldsymbol{\bf T}^{x}_{\tau,-}\\}$ to the one
generated by $\\{\boldsymbol{\bf\sigma},\boldsymbol{\bf T}^{z}_{\sigma,+}\\}$,
both algebras being extensions of $\mathfrak{A}_{\rm ql}$. It reads

$\displaystyle\boldsymbol{\bf\Pi}^{x}_{\tau,-}(j)=$
$\displaystyle\boldsymbol{\bf\sigma}_{j}^{x}\,,$ (36)
$\displaystyle\boldsymbol{\bf\tau}_{j}^{y}=$
$\displaystyle\boldsymbol{\bf\sigma}_{j-1}^{x}\boldsymbol{\bf\sigma}_{j}^{y}\boldsymbol{\bf\Pi}_{\sigma,+}^{z}(j+1)\,,$
$\displaystyle\boldsymbol{\bf\tau}_{j}^{z}=$
$\displaystyle\boldsymbol{\bf\Pi}_{\sigma,+}^{z}(j)\,.$

We stress that, while $\boldsymbol{\bf T}^{z}_{\sigma,\pm}$ resembles a string
of $\boldsymbol{\bf\sigma}^{z}_{j}$ with $j$ extending up to $\pm\infty$, the
sequence $\\{\prod_{\ell=0}^{n}\boldsymbol{\bf\sigma}^{z}_{\pm\ell}\\}$ does
not converge in the operator norm as $n\to\infty$. This is why we could not
avoid extending the algebra of the quasilocal operators.

The duality transformation (36) mixes the local space with some inaccessible
degrees of freedom and only a subset of the operators remain local under the
mapping — see Table 1.

$\mathcal{D}$ | $LE_{z}$ | $LO_{z}$ | $S_{z}E_{z}$ | $S_{z}O_{z}$
---|---|---|---|---
$LE_{x}$ | $\bullet$ |  |  |
[dashed] $LO_{x}$ |  |  | $\bullet$ |
$S_{x}E_{x}$ |  | $\bullet$ |  |
[dashed] $S_{x}O_{x}$ |  |  |  | $\bullet$
Table 1: The duality transformation $\mathcal{D}$. $L$ stands for “local”,
$S_{\alpha}$ for “semilocal” with a string along the axis $\alpha$,
$E_{\alpha}$ and $O_{\alpha}$ for “even” and “odd”, respectively, with respect
to spin flip along the axis $\alpha$. The transformation preserves only even
locality and odd semilocality.

In the table we also consider parities under spin flips. Specifically, even
(odd) quasilocal operators obey $\mathcal{P}^{\alpha}_{\nu}[\boldsymbol{\bf
O}]=\boldsymbol{\bf O}$ ($\mathcal{P}^{\alpha}_{\nu}[\boldsymbol{\bf
O}]=-\boldsymbol{\bf O}$), where $(\alpha,\nu)$ corresponds to either
$(x,\tau)$ or $(z,\sigma)$ — cf. Eq. (4)

$\displaystyle\mathcal{P}^{\alpha}_{\nu}\\![\boldsymbol{\bf
O}]\\!=\\!\lim_{n\rightarrow\infty}\\!\boldsymbol{\bf\Pi}^{\alpha}_{\nu,s_{\nu}}\\!(-n)\boldsymbol{\bf\Pi}^{\alpha}_{\nu,s_{\nu}}\\!(n)\boldsymbol{\bf
O}\boldsymbol{\bf\Pi}^{\alpha}_{\nu,s_{\nu}}\\!(-n)\boldsymbol{\bf\Pi}^{\alpha}_{\nu,s_{\nu}}\\!(n)\,,$
(37)

and $s_{\sigma(\tau)}=+(-)$. The transformations $\mathcal{P}^{\alpha}_{\nu}$
can be extended by linearity to operators $\boldsymbol{\bf
O}_{1}+\boldsymbol{\bf O}_{2}\boldsymbol{\bf T}^{\alpha}_{\nu,s_{\nu}}$.

Finally, we report the inverse transformation
$\mathcal{D}^{-1}_{\mathbb{Z}_{2}}$, which maps the algebra of observables
generated by $\\{\boldsymbol{\bf\sigma},\boldsymbol{\bf T}^{z}_{\sigma,+}\\}$
to the one generated by $\\{\boldsymbol{\bf\tau},\boldsymbol{\bf
T}^{x}_{\tau,-}\\}$:

$\displaystyle\boldsymbol{\bf\sigma}_{j}^{x}=$
$\displaystyle\boldsymbol{\bf\Pi}^{x}_{\tau,-}(j)\,,$ (38)
$\displaystyle\boldsymbol{\bf\sigma}_{j}^{y}=$
$\displaystyle\boldsymbol{\bf\Pi}^{x}_{\tau,-}(j-1)\boldsymbol{\bf\tau}_{j}^{y}\boldsymbol{\bf\tau}_{j+1}^{z}\,,$
$\displaystyle\boldsymbol{\bf\Pi}_{\sigma,+}^{z}(j)=$
$\displaystyle\boldsymbol{\bf\tau}_{j}^{z}\,.$

###  IV.2 Semilocality and hidden symmetry breaking

When presenting the duality transformation $\mathcal{D}_{\mathbb{Z}_{2}}$ we
were forced to extend the algebra of operators. Not all operators of the
extended algebra can however be associated with local observables. A theory of
local observables, indeed, requires some basic notion of locality:

  1. (a)

the theory is generated by operators that, in the limit of infinite distance,
commute with one another

$\lim_{|x-y|\rightarrow\infty}[\boldsymbol{\bf O}_{1}(x),\boldsymbol{\bf
O}_{2}(y)]=0\,;$ (39)

  2. (b)

the state has clustering properties for the aforementioned operators

$\braket{\boldsymbol{\bf O}_{1}(x)\boldsymbol{\bf
O}_{2}(y)}\xrightarrow{|x-y|\rightarrow\infty}\braket{\boldsymbol{\bf
O}_{1}(x)}\braket{\boldsymbol{\bf O}_{2}(y)}\,.$ (40)

After an inspection of Eq. (36), we realise that condition a forces us to
exclude either operators with strings or operators that are odd under spin
flip (that is, under $\mathcal{P}^{x}_{\tau}$ or $\mathcal{P}^{z}_{\sigma}$).
In either of the cases the remaining operators can represent local
observables. This can be understood by noting that a state can not
simultaneously have clustering properties, i.e., condition b, and a nonzero
expectation value of two operators that anticommute at an infinite distance:

$\braket{\boldsymbol{\bf O}_{1}(x)}\\!\braket{\boldsymbol{\bf
O}_{2}(y)}\\!\xleftarrow{|x-y|\rightarrow\infty}\\!\braket{\boldsymbol{\bf
O}_{1}(x)\boldsymbol{\bf O}_{2}(y)}\\!=\\!\\\ =\\!-\\!\braket{\boldsymbol{\bf
O}_{2}(y)\boldsymbol{\bf
O}_{1}(x)}\\!\xrightarrow{|x-y|\rightarrow\infty}\\!-\\!\braket{\boldsymbol{\bf
O}_{1}(x)}\\!\braket{\boldsymbol{\bf O}_{2}(y)}.$ (41)

Duality transformations such as $\mathcal{D}_{\mathbb{Z}_{2}}$ help identify
theories in which local observables are not necessarily represented by local
operators. When that happens we call the corresponding operators “semilocal”.
In the following we will refer to the theory consisting of only quasilocal
operators as “quasilocal theory”, in contrast to “semilocal theories”, which
include also semilocal operators.

Specifically, in the $\mathbb{Z}_{2}$ case we identify two theories of local
observables:

  * •

Quasilocal Theory: It consists of quasilocal operators;

  * •

Even $\mathbb{Z}_{2}$-Semilocal Theory: It consists of even quasilocal
operators and operators with even quasilocal “heads” and half-infinite “tails”
(see (IV.3.1) for an example).

Semilocal operators represent information that was lost in the thermodynamic
limit, e.g., anything related to the boundaries of the system. This becomes
evident when considering their expectation values. For example, let us take
the equilibrium state

$\boldsymbol{\bf\rho}^{(\sigma)}(\beta)=\frac{e^{-\beta\boldsymbol{\bf
H}^{(\sigma)}}}{Z_{\beta}}\,,$ (42)

where $\boldsymbol{\bf
H}^{(\sigma)}=\sum_{\ell}\boldsymbol{\bf\sigma}_{\ell}^{z}$, the upper index
referring to the representation of spins through
$\boldsymbol{\bf\sigma}_{\ell}^{\alpha}$. Being even, i.e.,
$\mathcal{P}^{z}_{\sigma}[\boldsymbol{\bf\rho}^{(\sigma)}(\beta)]=\boldsymbol{\bf\rho}^{(\sigma)}(\beta)$,
this state can be described both by the quasilocal theory and by the even
$\mathbb{Z}_{2}$-semilocal one, the latter being generated by the even local
operators and the semilocal operator $\boldsymbol{\bf T}^{z}_{\sigma,+}$.
Indeed, the Hamiltonian $\boldsymbol{\bf H}^{(\sigma)}$ through which the
state is defined belongs to both theories — see Fig. 2. Since it is gapped
with a nondegenerate ground state, one would be tempted to say that the limits
$\beta\to\pm\infty$ of Eq. (42) are given by the states $\ket{\Uparrow}$ and
$\ket{\Downarrow}$, respectively; one would also conclude that they are one-
site shift invariant. Such a conclusion is correct in the quasilocal theory,
in which we have

$\displaystyle\lim_{\beta\to\infty}{\rm
tr}[\boldsymbol{\bf\rho}(\beta)\boldsymbol{\bf
O}]=\bra{\Uparrow}\boldsymbol{\bf
O}\ket{\Uparrow}_{\sigma},\quad\forall\boldsymbol{\bf
O}\in\mathfrak{A}^{(\sigma)}_{\rm ql}\,.$ (43)

We added the subscript $\sigma$ in $\ket{\Uparrow}_{\sigma}$ to stress that
the state can be represented by all spins up when considering operators in
$\mathfrak{A}^{(\sigma)}_{\rm ql}$.

Instead, in the even $\mathbb{Z}_{2}$-semilocal theory semilocal operators,
such as $\boldsymbol{\bf\Pi}_{\sigma,+}^{z}(\ell)$, do not satisfy Eq. (43):
they are affected by what is not in the bulk of the system. We denote this
uncertainty by $\ket{\Uparrow\bullet}_{\sigma,\mathrm{sl}}$ and
$\ket{\Downarrow\bullet}_{\sigma,\mathrm{sl}}$, where $\bullet$ represents our
ignorance of what is not in the bulk. Since we are already in the
thermodynamic limit, this independent degree of freedom risks to appear very
abstract. In order to partially overcome this problem, it is convenient to map
semilocal operators into local ones through a duality transformation. In our
specific case this is achieved by $\mathcal{D}_{\mathbb{Z}_{2}}^{-1}$. As in
Section IV.1, we denote the transformed spins by
$\boldsymbol{\bf\tau}_{\ell}^{\alpha}$; the reader can think of the $\tau$
representation as of a shortcut for operators that could be semilocal in the
$\sigma$ representation (the system has not changed). We then have — see Eq.
(38) —

$\boldsymbol{\bf\rho}^{(\tau)}(\beta)=\frac{e^{-\beta\boldsymbol{\bf
H}^{(\tau)}}}{Z_{\beta}}\,,$ (44)

with $\boldsymbol{\bf
H}^{(\tau)}=\sum_{\ell}\boldsymbol{\bf\tau}_{\ell}^{z}\boldsymbol{\bf\tau}_{\ell+1}^{z}$,
which is even under $\mathcal{P}^{x}_{\tau}$. The limits
$\beta\rightarrow\pm\infty$ now exhibit a completely different phenomenology
that uncovers the role of $\bullet$. Specifically, cluster decomposition
requires spin-flip symmetry $\mathcal{P}_{\tau}^{x}$ to be spontaneously
broken [92, 57]:

$\displaystyle\lim_{\beta\to\infty}\\!{\rm
tr}[\boldsymbol{\bf\rho}(\beta)\boldsymbol{\bf
O}]\\!=\\!\begin{cases}\\!\bra{\Uparrow}\boldsymbol{\bf
O}\ket{\Uparrow}_{\tau}\\\ \\!\bra{\Downarrow}\boldsymbol{\bf
O}\ket{\Downarrow}_{\tau}\end{cases}\\!\\!\forall\boldsymbol{\bf
O}\\!\in\\!\mathfrak{A}^{(\tau)}_{\rm ql}\,.$ (45)

(Note that, in the limit $\beta\rightarrow-\infty$, the state breaks also one-
site shift invariance and becomes Néel or anti-Néel.) Semilocal operators,
which have vanishing expectation value at any finite temperature (they are odd
under $\mathcal{P}_{\tau}^{x}$ while the state is even), acquire a nonzero
expectation value at zero temperature due to spontaneous hidden symmetry
breaking.

We are now in a position to quantify $\bullet$. We show, in particular, that
the ambiguity hidden behind $\bullet$ is just a global sign. To that aim, let
us consider a triplet of semilocal operators $\boldsymbol{\bf
O}^{(n_{j})}(x_{j})$ ($j\in\\{1,2,3\\}$) representing local observables at
arbitrarily large distances $|x_{2}-x_{1}|$, $|x_{3}-x_{2}|$ and
$|x_{3}-x_{1}|$. Using clustering we have

$\braket{\boldsymbol{\bf
O}^{(n_{2})}(x_{2})}=s^{(n_{2})}(x_{2})\lim_{|x_{j}-x_{k}|\rightarrow\infty}\\\
\sqrt{\tfrac{\braket{\boldsymbol{\bf O}^{(n_{2})}(x_{2})\boldsymbol{\bf
O}^{(n_{3})}(x_{3})}\braket{\boldsymbol{\bf O}^{(n_{1})}(x_{1})\boldsymbol{\bf
O}^{(n_{2})}(x_{2})}}{\braket{\boldsymbol{\bf
O}^{(n_{1})}(x_{1})\boldsymbol{\bf O}^{(n_{3})}(x_{3})}}}\,,$ (46)

which tells us that we can determine the expectation value of $\boldsymbol{\bf
O}^{(n_{2})}(x_{2})$ up to a sign $s^{(n_{2})}(x_{2})$ from the expectation
values of local operators ($\boldsymbol{\bf O}^{(n_{j})}(x_{j})\boldsymbol{\bf
O}^{(n_{k})}(x_{k})$ are indeed local).

Let us now consider whatever semilocal operator $\boldsymbol{\bf O}^{(n)}(x)$
with $|x-x_{2}|\rightarrow\infty$. Using again clustering we immediately
obtain $s^{(n)}(x)=s^{(n_{2})}(x_{2})\mathrm{sgn}(\braket{\boldsymbol{\bf
O}^{(n)}(x)\boldsymbol{\bf O}^{(n_{2})}(x_{2})})$, and hence a single global
sign $s^{(n_{2})}(x_{2})$ fixes the expectation value of every semilocal
operator. We can choose, for example, $s^{(n_{2})}(x_{2})$ to be the sign of
the expectation value of the fundamental semilocal operator $\boldsymbol{\bf
T}^{z}_{\sigma,+}$. We can then indicate the state in the
$\mathbb{Z}_{2}$-semilocal theory by

$\ket{\Uparrow;\mathrm{sgn}[\braket{\boldsymbol{\bf
T}^{z}_{\sigma,+}}]}_{\sigma,\mathrm{sl}}\quad\text{or}\quad\ket{\Downarrow;\mathrm{sgn}[\braket{\boldsymbol{\bf
T}^{z}_{\sigma,+}}]}_{\sigma,\mathrm{sl}}\,.$ (47)

Note that the symmetry breaking states
$\ket{\Uparrow,\pm}_{\sigma,\mathrm{sl}}$ and
$\ket{\Downarrow,\pm}_{\sigma,\mathrm{sl}}$ in the semilocal theory are stable
under symmetry-breaking perturbations, which can now also be semilocal. This
is the hidden symmetry breaking studied in Refs. [93, 94, 55]. It is hidden
because no local operator can distinguish between
$\ket{\Uparrow,+}_{\sigma,\rm sl}$ and $\ket{\Uparrow,-}_{\sigma,\rm sl}$ (or
$\ket{\Downarrow,+}_{\sigma,\rm sl}$ and $\ket{\Downarrow,-}_{\sigma,\rm
sl}$).

More generally, a pure state that can be written as

$\displaystyle\ket{\Psi_{0}}^{(\sigma)}=\boldsymbol{\bf
V}(1)\ket{\Uparrow;s}_{\sigma,\mathrm{sl}}\,\,\text{or}\quad\ket{\Psi_{0}}^{(\sigma)}=\boldsymbol{\bf
V}(1)\ket{\Downarrow;s}_{\sigma,\mathrm{sl}}\,,$ (48) $\displaystyle
i\partial_{\tau}\boldsymbol{\bf V}(\tau)=\boldsymbol{\bf
W}(\tau)\boldsymbol{\bf V}(\tau),\qquad\boldsymbol{\bf V}(0)=\boldsymbol{\bf
I}\,,$

where $s$ is the sign ambiguity and $\boldsymbol{\bf W}(\tau)$ an extensive
translationally invariant Hermitian operator with a
$\mathcal{P}^{z}_{\sigma}$-invariant local density 888As far as we can see,
semilocal terms in $\boldsymbol{\bf W}(\tau)$, can always be replaced by local
terms. can be described by the even $\mathbb{Z}_{2}$-semilocal theory.

###  IV.3 Semilocal (generalised) Gibbs ensembles

In equilibrium at zero temperature the interplay between the symmetries of the
(ground) state and of the Hamiltonian is sufficient to discriminate the
symmetry-protected topological phases [1]. After a global quench that is not
sufficient anymore: conserved operators constrain the dynamics as well as the
Hamiltonian, therefore it is reasonable to expect that also their group of
symmetry becomes important.

We have already reviewed that not every conservation law affects the late-time
behaviour of local operators. In generic systems it was shown that the
discriminating criterion is pseudolocality, which, in view of condition 1 in
Section III.2, is related to the fact that the operators we are interested in
form the algebra of quasilocal operators $\mathfrak{A}_{\rm ql}$. In the
presence of a symmetry of the system (i.e., initial state and Hamiltonian),
such as the $\mathbb{Z}_{2}$ symmetry taken as an example in this paper, only
a subset of operators is relevant: by symmetry the rest of them have zero
expectation values. In the $\mathbb{Z}_{2}$ case, the relevant operators form
the subalgebra $\mathfrak{A}^{+}_{\rm ql}\subset\mathfrak{A}_{\rm ql}$ of the
quasilocal operators that are even under spin flip. We remark that the full
algebra $\mathfrak{A}_{\rm ql}$ can be obtained as an extension of the even
subalgebra, generated by multiplying the latter by a single element, which can
be whatever invertible local odd operator. Specifically, denoting the latter
by $\boldsymbol{\bf O}$, one has

$\displaystyle\mathfrak{A}_{\rm ql}=\mathfrak{A}^{+}_{\rm
ql}\oplus(\mathfrak{A}^{+}_{\rm ql}\boldsymbol{\bf O})\,.$ (49)

Because of the symmetry, however, this is not the only extension giving rise
to an algebra of operators that represent local observables. We have indeed
shown that also the semilocal operator $\boldsymbol{\bf T}^{z}_{+}$ represents
a local observable (which can be shifted by means of local operators to become
$\boldsymbol{\bf\Pi}_{+}^{z}(\ell)$, for any $\ell$ — Eq. (34)), therefore we
can use it to extend $\mathfrak{A}_{\rm ql}^{+}$ into another algebra
associated with local observables, say

$\displaystyle\mathfrak{A}_{\mathbbm{Z}_{2}\text{-sl}}^{+}=\mathfrak{A}^{+}_{\rm
ql}\oplus(\mathfrak{A}^{+}_{\rm ql}\boldsymbol{\bf T}_{+}^{z})\,.$ (50)

This provides a more abstract definition of the semilocal theories informally
introduced in the previous section: they are associated with different
extensions of the symmetric subalgebra of quasilocal operators through
operators that are not local but still represent local observables.

We are now in a position to generalise the requisite of pseudolocality (i.e.
Eqs. (16), (18)) for the relevance of a conservation law to semilocal
theories. To that aim, let us call $\boldsymbol{\bf T}_{i}$, with $i=1,\dots
n$, the operators that, in a given semilocal theory, are added to the
symmetric subalgebra $\mathfrak{A}_{\rm ql}^{+}$ to form a representation of
local observables. Such an algebra is supposed to be closed both under time
evolution and under shifts of lattice sites. Without loss of generality we can
assume that $\boldsymbol{\bf T}_{i}$ commute with every local operator in the
symmetric subalgebra $\mathfrak{A}_{\rm ql}^{+}$ as long as the latter’s
support is far enough from position $0$. A conservation law is then relevant
if its density around position $0$ reads $\boldsymbol{\bf
q}_{0}^{(0)}+\sum_{i=1}^{n}\boldsymbol{\bf q}_{0}^{(i)}\boldsymbol{\bf
T}_{i}$, with $\boldsymbol{\bf q}_{0}^{(i)}\in\mathfrak{A}^{+}_{\rm ql}$ being
even (strongly) quasilocal operators. We will refer to it as a “semilocal
conservation law”.

If there are several semilocal theories, it could not be obvious which
theories include all the relevant conservation laws. We propose then to extend
the entire algebra of quasilocal operators by adding all the semilocal
operators that generate the various semilocal theories. In the
$\mathbb{Z}_{2}$ case this corresponds to considering the algebra

$\mathfrak{A}_{\mathbbm{Z}_{2}\text{-sl}}=\mathfrak{A}_{\mathrm{ql}}\oplus(\mathfrak{A}_{\mathrm{ql}}\boldsymbol{\bf
T}^{z}_{+})\,,$ (51)

which we have already encountered when we defined the rotated Kramers-Wannier
transformation (36).

_We conjecture that in the extended space there exists a maximum-entropy
representation of the macrostate emerging at infinite time after the global
quench. We call it $G$-semilocal (generalised) Gibbs ensemble, where $G$ is
the symmetry used to extend the algebra_ (e.g., $G=\mathbbm{Z}_{2}$ in most
cases considered here). Extending the algebra allows us to recover a simple
description of the late-time stationary values.

There is however an inconvenience: in the extended theory there are operators
that do not represent local observables. In fact, except for the operators in
the subalgebra that is common to all theories, there is no unambiguous way to
associate local observables to operators. For example, in the $\mathbb{Z}_{2}$
case, both $\mathfrak{A}_{\mathrm{ql}}$ and
$\mathfrak{A}_{\mathbb{Z}_{2}\text{-sl}}^{+}$, which are subalgebras of
$\mathfrak{A}_{\mathbb{Z}_{2}\text{-sl}}$, are generated by operators
representing local observables, but the operators do not coincide. While the
even local operators form a common subset $\mathfrak{A}_{\rm ql}^{+}$ of both
algebras (see Fig. 10) and thus enter the description of the local observables
in both theories, it is less clear how to choose between the even semilocal or
odd local operators ($\mathfrak{A}_{\rm ql}^{+}\boldsymbol{\bf T}_{+}^{z}$ or
$\mathfrak{A}_{\rm ql}^{+}\boldsymbol{\bf\sigma}^{x}_{0}$, respectively). We
will expand on this in Section IV.3.3. Before that, we provide an explicit
example of a model with semilocal conservation laws.

quasilocal$\boldsymbol{\cdot}\boldsymbol{\bf
T}^{z}_{+}$evenodd$\mathfrak{A}_{\rm
ql}$$\mathfrak{A}^{+}_{\mathbbm{Z}_{2}\text{-sl}}$$\mathfrak{A}_{\mathbbm{Z}_{2}\text{-sl}}$$\mathfrak{A}_{\rm
ql}^{+}$$\mathfrak{A}_{\rm ql}^{+}\boldsymbol{\bf
T}^{z}_{+}$$\mathfrak{A}_{\rm
ql}^{+}\boldsymbol{\bf\sigma}^{x}_{0}$$(\mathfrak{A}_{\rm
ql}^{+}\boldsymbol{\bf\sigma}^{x}_{0})\boldsymbol{\bf T}^{z}_{+}$ Figure 10:
Representation of local observables in a $\mathbbm{Z}_{2}$-symmetric theory.
Quasilocal operators (left column, red line) can be extended to semilocal ones
through multiplication by $\boldsymbol{\bf T}^{z}_{+}$. The full semilocal
theory (black frame) can be projected either onto the quasilocal theory (red
frame), or even semilocal theory (blue dashed frame): both can represent local
observables.

####  IV.3.1 Example: Dual XY model revisited

Contrary to the transverse-field Ising model, the quantum XY model

$\displaystyle\boldsymbol{\bf
H}^{(\tau)}=\sum_{\ell\in\mathbbm{Z}}J_{x}\boldsymbol{\bf\tau}_{\ell}^{x}\boldsymbol{\bf\tau}_{\ell+1}^{x}+J_{y}\boldsymbol{\bf\tau}_{\ell}^{y}\boldsymbol{\bf\tau}_{\ell+1}^{y}$
(52)

is not self-dual under the Kramers-Wannier transformation. The duality
transformation (36) maps its Hamiltonian into the one of the dual XY model

$\displaystyle\boldsymbol{\bf
H}^{(\sigma)}=\sum_{\ell\in\mathbbm{Z}}\boldsymbol{\bf\sigma}_{\ell-1}^{x}(J_{x}\boldsymbol{\bf
I}-J_{y}\boldsymbol{\bf\sigma}^{z}_{\ell})\boldsymbol{\bf\sigma}^{x}_{\ell+1}\,.$
(53)

We denote the corresponding local Hamiltonian densities by $\boldsymbol{\bf
h}^{(\tau)}_{\ell}=J_{x}\boldsymbol{\bf\tau}_{\ell}^{x}\boldsymbol{\bf\tau}_{\ell+1}^{x}+J_{y}\boldsymbol{\bf\tau}_{\ell}^{y}\boldsymbol{\bf\tau}_{\ell+1}^{y}$,
and $\boldsymbol{\bf
h}^{(\sigma)}_{\ell}=\boldsymbol{\bf\sigma}_{\ell-1}^{x}(J_{x}\boldsymbol{\bf
I}-J_{y}\boldsymbol{\bf\sigma}^{z}_{\ell})\boldsymbol{\bf\sigma}^{x}_{\ell+1}$,
respectively.

Under the duality transformation $\mathcal{D}_{\mathbb{Z}_{2}}$, the spin-flip
symmetry $\mathcal{P}^{x}_{\tau}[\boldsymbol{\bf
h}^{(\tau)}_{\ell}]=\boldsymbol{\bf h}^{(\tau)}_{\ell}$, for any $\ell$,
becomes the following invariance of the dual XY model’s local densities:

$\displaystyle\lim_{n\to\infty}[\boldsymbol{\bf\sigma}^{x}_{-n}\boldsymbol{\bf\sigma}^{x}_{n}]\boldsymbol{\bf
h}^{(\sigma)}_{\ell}[\boldsymbol{\bf\sigma}^{x}_{-n}\boldsymbol{\bf\sigma}^{x}_{n}]=\boldsymbol{\bf
h}^{(\sigma)}_{\ell},\quad\forall\ell.$ (54)

While this invariance holds trivially for (quasi)local densities, the same is
not true for the semilocal operators. Considering, for example, half-infinite
strings $\boldsymbol{\bf\Pi}^{z}_{\sigma,+}(\ell)$, we have

$\displaystyle\lim_{n\to\infty}[\boldsymbol{\bf\sigma}^{x}_{-n}\boldsymbol{\bf\sigma}^{x}_{n}]\boldsymbol{\bf\Pi}^{z}_{\sigma,+}(\ell)[\boldsymbol{\bf\sigma}^{x}_{-n}\boldsymbol{\bf\sigma}^{x}_{n}]=-\boldsymbol{\bf\Pi}^{z}_{\sigma,+}(\ell),$
(55)

i.e., $\boldsymbol{\bf\Pi}^{z}_{\sigma,+}(\ell)$ are odd under the operation
that is dual to $\mathcal{P}^{x}_{\tau}$. Indeed, recall that
$\mathcal{D}_{\mathbb{Z}_{2}}$ maps all operators that are odd under
$\mathcal{P}^{x}_{\tau}$ into semilocal operators, as described in Table 1.
With this in mind, let us now consider the charges of the XY model and its
dual counterpart.

The quantum XY model is non-abelian integrable [82]. That is to say, the
Hamiltonian commutes with infinitely many pseudolocal operators, not
necessarily commuting with one another. Its translationally invariant local
charges are

$\displaystyle\boldsymbol{\bf
Q}^{(n,\pm;\tau)}=\sum_{\ell\in\mathbbm{Z}}\boldsymbol{\bf
q}^{(n,\pm;\tau)}_{\ell}\,,$ (56)

where the local densities (including the Hamiltonian’s one $\boldsymbol{\bf
h}^{(\tau)}_{\ell}\equiv\boldsymbol{\bf q}^{(2,+;\tau)}_{\ell}$) read (see,
e.g., [96])

$\displaystyle\begin{aligned} \boldsymbol{\bf
q}^{(2,+;\tau)}_{\ell}=&J_{x}\boldsymbol{\bf\tau}^{x}_{\ell}\boldsymbol{\bf\tau}^{x}_{\ell+1}+J_{y}\boldsymbol{\bf\tau}^{y}_{\ell}\boldsymbol{\bf\tau}^{y}_{\ell+1}\,,\qquad\boldsymbol{\bf
q}^{(3,+;\tau)}_{\ell}=(J_{x}\boldsymbol{\bf\tau}^{x}_{\ell}\boldsymbol{\bf\tau}^{x}_{\ell+2}+J_{y}\boldsymbol{\bf\tau}^{y}_{\ell}\boldsymbol{\bf\tau}^{y}_{\ell+2})\boldsymbol{\bf\tau}^{z}_{\ell+1}-(J_{x}+J_{y})\boldsymbol{\bf\tau}^{z}_{\ell+1}\,,\\\
\boldsymbol{\bf
q}^{(n,+;\tau)}_{\ell}=&(J_{x}\boldsymbol{\bf\tau}^{x}_{\ell}\boldsymbol{\bf\tau}^{x}_{\ell+n-1}+J_{y}\boldsymbol{\bf\tau}^{y}_{\ell}\boldsymbol{\bf\tau}^{y}_{\ell+n-1})\prod_{j=1}^{n-2}\boldsymbol{\bf\tau}^{z}_{\ell+j}+(J_{x}\boldsymbol{\bf\tau}^{y}_{\ell}\boldsymbol{\bf\tau}^{y}_{\ell+n-3}+J_{y}\boldsymbol{\bf\tau}^{x}_{\ell}\boldsymbol{\bf\tau}^{x}_{\ell+n-3})\prod_{j=1}^{n-4}\boldsymbol{\bf\tau}^{z}_{\ell+j},\quad\text{for
}n>3\,,\\\ \boldsymbol{\bf
q}^{(n,-;\tau)}_{\ell}=&(\boldsymbol{\bf\tau}^{x}_{\ell}\boldsymbol{\bf\tau}^{y}_{\ell+n-1}\\!-\\!\boldsymbol{\bf\tau}^{y}_{\ell}\boldsymbol{\bf\tau}^{x}_{\ell+n-1})\prod_{j=1}^{n-2}\boldsymbol{\bf\tau}_{\ell+j}^{z},\quad\text{for
}n\geq 2\,.\end{aligned}$ (57)

For the product we use the standard convention with
$\prod_{j=\ell}^{\ell-1}\boldsymbol{\bf\tau}^{\alpha}_{j}=\boldsymbol{\bf I}$.
The upper indices $n$ and $\pm$ denote, respectively, the number of sites the
charge’s local density acts upon and the charge’s parity under spatial
reflection. Note that the reflection-odd charges $\boldsymbol{\bf
Q}^{(n,-;\tau)}$ do not depend on the coupling constants $J_{x}$ and $J_{y}$.

Besides the abelian charges with densities (57) the XY model possesses also
local charges of a staggered form $\sum_{\ell}(-1)^{\ell}\boldsymbol{\bf
w}_{\ell}$ that do not commute with $\boldsymbol{\bf Q}^{(n,\pm;\tau)}$ [82].
Since the expectation value of any staggered operator is zero in a
translationally invariant state, these additional nonabelian charges are
irrelevant for our discussion and we will therefore not report their explicit
form here.

Consider now the densities (57) after the duality transformation. According to
Table 1 the local densities $\\{\boldsymbol{\bf
q}^{(2n,+;\tau)}_{\ell},\boldsymbol{\bf
q}^{(2n+1,-;\tau)}_{\ell}\\}_{n\in\mathbbm{N}}$ remain local even in the
$\sigma$ representation, since they are even under $\mathcal{P}^{x}_{\tau}$.
Instead $\\{\boldsymbol{\bf q}^{(2n+1,+;\tau)}_{\ell},\boldsymbol{\bf
q}^{(2n,-;\tau)}_{\ell}\\}_{n\in\mathbbm{N}}$ are odd under
$\mathcal{P}^{x}_{\tau}$ and are mapped into operators with half-infinite
strings. Hence, the set $\\{\boldsymbol{\bf Q}^{(2n+1,+;\tau)},\boldsymbol{\bf
Q}^{(2n,-;\tau)}\\}_{n\in\mathbbm{N}}$ is mapped into a set of semilocal
charges in the dual XY model. For example, the density of the charge
$\boldsymbol{\bf Q}^{(3,+;\tau)}$ becomes

$\displaystyle\boldsymbol{\bf q}^{(3,+;\sigma)}_{\ell}\\!=$
$\displaystyle[\boldsymbol{\bf\sigma}^{x}_{\ell-1}(J_{x}\boldsymbol{\bf\sigma}^{x}_{\ell}\boldsymbol{\bf\sigma}^{x}_{\ell+1}\\!+\\!J_{y}\boldsymbol{\bf\sigma}^{y}_{\ell}\boldsymbol{\bf\sigma}^{y}_{\ell+1})\boldsymbol{\bf\sigma}^{x}_{\ell+2}\\!-\\!(J_{x}\\!+\\!J_{y})\boldsymbol{\bf
I}]$ $\displaystyle\times\boldsymbol{\bf\Pi}^{z}_{\sigma,+}(\ell+1)\,.$ (58)

In the symmetric quench we considered in Section III.3 this is a special
charge: it is the only one with a nonzero expectation value in the state
$\ket{\Uparrow}_{\sigma}$. This is because its density is the only one among
$\boldsymbol{\bf q}^{(n,\pm;\sigma)}_{\ell}$ containing a term that consists
solely of $\boldsymbol{\bf\sigma}_{j}^{z}$ matrices.

####  IV.3.2 Example: generic model revisited

Let us reconsider the generic model of Section III.3 — cf. Eqs (6) and (30) —
which is described by the Hamiltonian

$\boldsymbol{\bf
H}=\sum_{\ell\in\mathbbm{Z}}\boldsymbol{\bf\sigma}_{\ell-1}^{x}(\boldsymbol{\bf
1}-\boldsymbol{\bf\sigma}_{\ell}^{z})\boldsymbol{\bf\sigma}_{\ell+1}^{x}+w_{1}\boldsymbol{\bf\sigma}_{\ell}^{z}+\sum_{n=1}^{\infty}w_{2,n}\boldsymbol{\bf\sigma}_{\ell}^{z}\boldsymbol{\bf\sigma}_{\ell+n}^{z}\,.$
(59)

As anticipated in Section II.2, this Hamiltonian has a semilocal conservation
law, namely

$\boldsymbol{\bf Q}=\sum_{\ell\in\mathbbm{Z}}\boldsymbol{\bf\Pi}^{z}(\ell)\,.$
(60)

The expectation value of this semilocal charge is defined through clustering
(i.e., using Eq. (5)), which yields
$\braket{\boldsymbol{\bf\Pi}^{z}(j)}^{2}_{t}=\cos^{2}\varphi$ for its density.
The latter provides a positive lower bound for the string order parameter
$\lim_{n\rightarrow\infty}\braket{\prod_{j=-n}^{n}\boldsymbol{\bf\sigma}_{n}^{z}}$
at infinite time through Eq. (8). According to Ref. [12], a thermal state can
not exhibit string order, therefore we can immediately conclude that, for
$\cos\varphi\neq 0$, the state at infinite time is not thermal. This is
consistent with the behaviour shown in Fig. 9.

####  IV.3.3 Canonical and non-canonical descriptions

A priori we do not see any reason to choose one theory of local observables
over the other. If however we imagine the theoretical system as an
idealisation of an experiment and the experimental apparatus as something that
goes beyond the system under investigation, a theory could be somehow selected
by how the apparatus was designed. For example, in the $\mathbb{Z}_{2}$ case,
if the experimental apparatus is able to preserve spin-flip symmetry with high
accuracy, the even $\mathbb{Z}_{2}$ semilocal theory could become a better
framework where to study the effect of noise (even under spin flip) or other
blind spots of the experiment. The question then becomes, how the
$G$-semilocal (generalised) Gibbs ensemble is represented within the chosen
theory. To answer it, the ensemble should be projected onto the subalgebra
associated with that theory. There are two possibilities:

  1. 1.

the $G$-semilocal (generalised) Gibbs ensemble is part of the subalgebra,
hence the projected ensemble coincides with the maximum-entropy ensemble in
the theory. We call such theories “canonical”;

  2. 2.

the $G$-semilocal (generalised) Gibbs ensemble includes also conserved
operators outside the theory; the theory is termed “non-canonical”.

For example, a $\mathbbm{Z}_{2}$-semilocal (generalised) Gibbs ensemble
$\boldsymbol{\bf\rho}_{\mathbbm{Z}_{2}\text{-sl}}$ constructed from an even
local charge $\boldsymbol{\bf Q}_{\rm ql}^{+}$ and an even semilocal charge
$\boldsymbol{\bf Q}^{+}_{\mathbbm{Z}_{2}\text{-sl}}$ can be written as

$\displaystyle\boldsymbol{\bf\rho}_{\mathbb{Z}_{2}\text{-sl}}\propto
e^{-\boldsymbol{\bf Q}^{+}_{\rm ql}-\boldsymbol{\bf
Q}^{+}_{\mathbb{Z}_{2}\text{-}\mathrm{sl}}}\,.$ (61)

It coincides with its projection onto the theory represented by
$\mathfrak{A}^{+}_{\mathbbm{Z}_{2}\text{-sl}}$, i.e.,

$\displaystyle\boldsymbol{\bf\rho}_{\mathbb{Z}_{2}\text{-sl}}=\boldsymbol{\bf\rho}_{\mathbb{Z}_{2}\text{-sl}}\bigr{|}_{\mathfrak{A}^{+}_{\mathbbm{Z}_{2}\text{-sl}}}\,.$
(62)

In the quasilocal theory associated with $\mathfrak{A}_{\rm ql}$ it instead
takes a different form. Specifically, only the first term in

$\displaystyle\boldsymbol{\bf\rho}_{\mathbbm{Z}_{2}\text{-sl}}\\!\propto\\!e^{-\boldsymbol{\bf
Q}^{+}_{\rm ql}}\\!\cosh\\!\boldsymbol{\bf
Q}^{+}_{\mathbbm{Z}_{2}\text{-sl}}\\!-\\!e^{-\boldsymbol{\bf Q}^{+}_{\rm
ql}}\sinh\\!\boldsymbol{\bf Q}^{+}_{\mathbbm{Z}_{2}\text{-sl}}$ (63)

survives the projection onto the quasilocal algebra (the second one consists
of odd powers of $\boldsymbol{\bf Q}_{\mathbbm{Z}_{2}\text{-sl}}$ and thus
contains strings). The projected ensemble

$\displaystyle\boldsymbol{\bf\rho}_{\mathbb{Z}_{2}\text{-sl}}\bigr{|}_{\mathfrak{A}_{\rm
ql}}\propto e^{-\boldsymbol{\bf Q}^{+}_{\rm ql}}\cosh\boldsymbol{\bf
Q}^{+}_{\mathbbm{Z}_{2}\text{-sl}}$ (64)

does not maximize the von Neumann entropy (12) constrained by the pseudolocal
charges, making the theory non-canonical 999Note that the projected ensemble
(64) gives correct predictions for the stationary expectation values of
quasilocal operators, as can be seen by mapping it into the $\tau$
representation..

Returning to the problem of defining and classifying nonequilibrium phases
after global quenches, the way local observables are represented in a
canonical theory constitutes a fundamental distinction, which goes much beyond
the differences associated with the multiplicity and the symmetries of the
conservation laws. It was observed in Ref. [65] that the generalised
eigenstate thermalisation hypothesis (gETH), according to which all
eigenstates with the same local integrals of motion are locally equivalent,
could be sufficient to prove that a maximum-entropy ensemble description is
possible (the criticism raised in Ref. [98] is resolved once pseudolocal
integrals of motion are taken into account). In non-canonical theories we
claim that gETH fails. As a matter of fact, the following much weaker
assumption fails: in the limit of infinite time the only information retained
from the initial state is encoded in the (pseudo)local conserved operators.

We have already shown an example of the breakdown of such a key property in
the previous section, when we tried to describe the infinite-time limit after
a symmetric quench in the dual XY model through the maximum-entropy ensemble
of a non-canonical theory, which in the specific case was the (quasilocal)
generalised Gibbs ensemble. That ensemble was unable to describe even a local
observable such as $\boldsymbol{\bf\sigma}_{\ell}^{z}$ — Fig. 8. As expected,
instead, the stationary values are described by the $\mathbb{Z}_{2}$-semilocal
generalised Gibbs ensemble (61). In that specific case, the relaxation to the
$\mathbb{Z}_{2}$-semilocal GGE is a trivial consequence of the established
result that noninteracting systems relax to generalised Gibbs ensembles. More
generally, if a duality transformation between a canonical theory and the
quasilocal theory is known, proving relaxation to the (generalised) Gibbs
ensemble in the canonical theory becomes equivalent to proving relaxation of
non-symmetric states in the standard quasilocal one.

##### Non-symmetric states.

So far we have assumed that the entire system (initial state and Hamiltonian)
is symmetric. On the other hand, we defined the semilocal charges in an
extended algebra, which includes also non-symmetric operators. It is then
natural to wonder whether a semilocal charge could make sense also with non-
symmetric initial states. To that aim, we consider again the state
$\ket{\Psi}^{(\sigma)}=\ket{\rotatebox[origin={c}]{45.0}{$\Rightarrow$}_{\theta}}_{\sigma}$
and $\theta\notin\\{0,\pi\\}$. As discussed in Section IV.2, we must be
careful about the interpretation of $\ket{\Psi}$ when we extend the theory so
as to include also semilocal operators. It is again convenient to map even
semilocality into odd locality. Appendix D.1 proves

$\ket{\Psi}^{(\tau)}=\ket{\beta_{\theta}}_{\tau}:=\frac{\sum_{\underline{s}}e^{-\frac{\beta_{\theta}}{2}E(\underline{s})}\ket{\underline{s}}_{\tau}}{\sum_{\underline{s}}e^{-\beta_{\theta}E(\underline{s})}}\,,$
(65)

where $\beta_{\theta}=-\log\tan(\theta/2)$,
$\underline{s}=(\ldots,s_{-1},s_{0},s_{1},\ldots)$, with $s_{j}\in\\{-1,1\\}$
denoting the eigenvalues of the spin operator $\boldsymbol{\bf\tau}_{j}^{z}$,
and $E(\underline{s})$ is the energy of the classical Ising model
$E(\underline{s})=-\sum_{j}s_{j}s_{j+1}$.

One can show that $\ket{\beta}_{\tau}$ has clustering properties for any
finite $\beta$, i.e., for $\theta\notin\\{0,\pi\\}$ (see Appendix D.2).
Importantly, despite the tilted state not being symmetric, the corresponding
state in the dual representation, i.e., $\ket{\beta_{\theta}}_{\tau}$, is even
under $\mathcal{P}^{x}_{\tau}$. More generally we can understand this by
interpreting $\ket{\Psi}$ as the ground state of a symmetric Hamiltonian in
which the symmetry is spontaneously broken at zero temperature in the $\sigma$
representation (a simple Hamiltonian with this symmetry-breaking ground state
is known [99]: the quantum XY model in a transverse field, with density
$\boldsymbol{\bf
h}_{\ell}=J_{x}\boldsymbol{\bf\sigma}_{\ell}^{x}\boldsymbol{\bf\sigma}_{\ell+1}^{x}+J_{y}\boldsymbol{\bf\sigma}_{\ell}^{y}\boldsymbol{\bf\sigma}_{\ell+1}^{y}-2\sqrt{J_{x}J_{y}}\boldsymbol{\bf\sigma}_{\ell}^{z}$,
with $J_{y}/J_{x}=\cos^{2}\theta$). It turns out that the symmetry in the dual
Hamiltonian remains unbroken and the ground state is symmetric. This implies
that even semilocal operators have zero expectation values in the original
system, and hence the expectation values of all semilocal charges vanish in
$\ket{\Psi}$. On the one hand, this justifies taking into account semilocal
charges also with non-symmetric initial states; on the other hands, it shows
their irrelevance (as the corresponding integrals of motion vanish).

As a matter of fact, the naive approach of interpreting half-infinite strings
as infinite products of Pauli matrices gives, in this case, the correct
result: since $\ket{\rotatebox[origin={c}]{45.0}{$\Rightarrow$}_{\theta}}$ is
a product state, for $\theta\notin\\{0,\pi\\}$ one has
$|\bra{\rotatebox[origin={c}]{45.0}{$\Rightarrow$}_{\theta}}\boldsymbol{\bf\sigma}_{j}^{z}\ket{\rotatebox[origin={c}]{45.0}{$\Rightarrow$}_{\theta}}|<1$
for all $j$, so the strings are exponentially suppressed. The two approaches
give the same result because the sequence of the expectation values of
$\\{\boldsymbol{\bf
O}_{0}\Pi_{\ell=0}^{n}\boldsymbol{\bf\sigma}^{z}_{\ell}\\}$ converges (to
zero) as $n\rightarrow\infty$ for any operator $\boldsymbol{\bf O}_{0}$
quasilocalised around $0$. This actually allows one to go even further and
conclude that also the expectation values of odd semilocal operators vanish.

####  IV.3.4 Remark

We conclude with a clarification. So far we have presented (27) as a
Hamiltonian with a $\mathbb{Z}_{2}$ symmetry, but, in fact, it exhibits a
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ symmetry. Indeed, the energy density is
also even under

$\mathcal{P}^{z}_{\mathrm{e}}[\boldsymbol{\bf
h}_{j}]=\lim_{n\rightarrow\infty}\Bigl{[}\prod_{\ell=-n}^{n}\boldsymbol{\bf\sigma}_{2\ell}^{z}\Bigr{]}\boldsymbol{\bf
h}_{j}\Bigl{[}\prod_{\ell=-n}^{n}\boldsymbol{\bf\sigma}_{2\ell}^{z}\Bigr{]}\,.$
(66)

The same comment applies to the symmetric initial state studied in the example
of Section III.3, namely, $\ket{\Uparrow}$. The extended algebra is now
obtained by supplementing the local operators that are even under both spin
flips by two (commuting) semilocal operators $\boldsymbol{\bf T}^{z}_{2,0}$
and $\boldsymbol{\bf T}^{z}_{2,1}$, which play the role of products of Pauli
matrices $\boldsymbol{\bf\sigma}^{z}$ over even or odd sites, respectively,
extending from site $0$ or $1$ to infinity (see Appendix B for a proper
definition).

Let $\mathfrak{A}_{\mathrm{ql}}^{+\bullet}$,
$\mathfrak{A}_{\mathrm{ql}}^{\bullet+}$,
$\mathfrak{A}_{\mathrm{ql}}^{\pm\pm}$, $\mathfrak{A}_{\mathrm{ql}}^{++}$ be
the subalgebras of quasilocal operators even under $\mathcal{P}^{z}_{\rm e}$,
$\mathcal{P}^{z}_{\rm o}:=\mathcal{P}^{z}_{\rm e}\circ\mathcal{P}^{z}$,
$\mathcal{P}^{z}(=\mathcal{P}^{z}_{\rm e}\circ\mathcal{P}^{z}_{\rm o})$, both
$\mathcal{P}^{z}$ and $\mathcal{P}^{z}_{\rm e}$, respectively
($\mathfrak{A}_{\mathrm{ql}}^{\pm\pm}$ was previously denoted by
$\mathfrak{A}_{\mathrm{ql}}^{+}$, since only the symmetry $\mathcal{P}^{z}$
was relevant). We identify five subalgebras generated by operators
representing local observables:

$\mathfrak{A}_{\rm ql}$

this is the standard algebra of quasilocal operators, given by
$\mathfrak{A}_{\mathrm{ql}}^{\pm\pm}\oplus(\mathfrak{A}_{\mathrm{ql}}^{\pm\pm}\boldsymbol{\bf
O})$, with $\boldsymbol{\bf O}$ an invertible local odd operator;

$\mathfrak{A}^{+\bullet}_{\mathbb{Z}_{2}\text{-sl}}$

this is the $\mathbb{Z}_{2}$-semilocal algebra even under spin flip on even
sites, and it is given by
$\mathfrak{A}_{\mathrm{ql}}^{+\bullet}\oplus(\mathfrak{A}_{\mathrm{ql}}^{+\bullet}\boldsymbol{\bf
T}^{z}_{2,0})$;

$\mathfrak{A}^{\bullet+}_{\mathbb{Z}_{2}\text{-sl}}$

this is the $\mathbb{Z}_{2}$-semilocal algebra even under spin flip on odd
sites, and it is given by
$\mathfrak{A}_{\mathrm{ql}}^{\bullet+}\oplus(\mathfrak{A}_{\mathrm{ql}}^{\bullet+}\boldsymbol{\bf
T}^{z}_{2,1})$;

$\mathfrak{A}^{\pm\pm}_{\mathbb{Z}_{2}\text{-sl}}$

this is the $\mathbb{Z}_{2}$-semilocal algebra even under spin flip, and it is
given by
$\mathfrak{A}_{\mathrm{ql}}^{\pm\pm}\oplus(\mathfrak{A}_{\mathrm{ql}}^{\pm\pm}\boldsymbol{\bf
T}^{z}_{2,0}\boldsymbol{\bf T}^{z}_{2,1})$;

$\mathfrak{A}^{++}_{\mathbb{Z}_{2}\times\mathbb{Z}_{2}\text{-sl}}$

this is the $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$-semilocal algebra even both
under spin flip on odd sites and under spin flip on even sites, and it is
given by
$\mathfrak{A}_{\mathrm{ql}}^{++}\oplus(\mathfrak{A}_{\mathrm{ql}}^{++}\boldsymbol{\bf
T}^{z}_{2,0})\oplus(\mathfrak{A}_{\mathrm{ql}}^{++}\boldsymbol{\bf
T}^{z}_{2,1})\oplus(\mathfrak{A}_{\mathrm{ql}}^{++}\boldsymbol{\bf
T}^{z}_{2,0}\boldsymbol{\bf T}^{z}_{2,1})$.

It turns out that the $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$-semilocal
generalised Gibbs ensemble emerging after a quench from $\ket{\Uparrow}$
belongs to the intersection of
$\mathfrak{A}^{\pm\pm}_{\mathbb{Z}_{2}\text{-sl}}$ and
$\mathfrak{A}^{++}_{\mathbb{Z}_{2}\times\mathbb{Z}_{2}\text{-sl}}$ 101010A
$\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$-semilocal charge would be mapped by
$\mathcal{D}_{\mathbbm{Z}_{2}}^{-1}$ into a $\mathbbm{Z}_{2}$-semilocal charge
in the $\tau$ representation, but there are no such charges in the latter —
see Section IV.3.1. The latter two algebras represent therefore canonical
theories. On the other hand, $\mathfrak{A}_{\mathrm{ql}}$,
$\mathfrak{A}^{+\bullet}_{\mathbb{Z}_{2}\text{-sl}}$, and
$\mathfrak{A}^{\bullet+}_{\mathbb{Z}_{2}\text{-sl}}$ are associated with non-
canonical theories. There, the stationary state capturing the infinite time
expectation values is not a maximum-entropy statistical ensemble. Once
projected onto the quasilocal theory, the
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}$-semilocal generalised Gibbs ensemble has
the form (64) with $\boldsymbol{\bf Q}_{\mathrm{ql}}=0$ (the form of the
ensemble is specific to the initial state considered, i.e., $\ket{\Uparrow}$).
Since it is even both under $\mathcal{P}^{z}_{\rm e}$ and
$\mathcal{P}^{z}_{\rm o}$, the $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$-semilocal
generalised Gibbs ensemble belongs to the intersection of
$\mathfrak{A}_{\mathrm{ql}}$,
$\mathfrak{A}^{+\bullet}_{\mathbb{Z}_{2}\text{-sl}}$, and
$\mathfrak{A}^{\bullet+}_{\mathbb{Z}_{2}\text{-sl}}$: the three theories share
the same projected ensemble. This makes the algebraic structure associated
with the $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ symmetry of the dual XY model
redundant, and the reader can now understand why we have completely overlooked
this larger symmetry group when describing the model and its conservation
laws.

The remaining question is: can we realise that the state does not locally
relax to the maximum-entropy state without comparing the predictions from the
maximum-entropy state? This is the subject of the next two sections.

##  V Signatures of semilocal order

A way to assess whether a theory is canonical or not is by perturbing the
initial state with an operator that represents a local observable but does not
belong to the common subalgebra. Specifically, we consider a unitary
transformation with finite support connecting different symmetry sectors, such
as $\boldsymbol{\bf\sigma}_{r}^{x}$ in the spin-flip case [31]. Such
perturbations are semilocal in the eyes of charges belonging to a different
theory, which will therefore be affected in a nonlocal way. If the quasilocal
theory is not canonical, we find it reasonable to expect signatures of such a
nonlocality in the entanglement of subsystems.

Being the first investigation of this kind in nonequilibrium symmetry-
protected topological order phases, we focus on the entanglement entropies of
connected blocks of spins $A$, which in the rest of the paper will be
identified with the following set of sites

$A\equiv\\{1,\ldots,\ell\\}\,.$ (67)

To that aim, we need to construct reduced density matrices of finite
subsystems. Reduced density matrices describe the expectation values of local
observables represented by local operators with support in the subsystem. In a
spin-$1/2$ chain they can be expanded in an orthogonal basis of Hermitian
matrices $O_{A}$
($\mathrm{tr}[O_{A}O^{\prime}_{A}]=\delta_{O_{A},O^{\prime}_{A}}\mathrm{tr}[\mathbb{I}_{A}]$)
representing local operators $\boldsymbol{\bf O}_{A}$ with support in the
subsystem $A$

$\rho_{A}(\Psi)=\frac{1}{\mathrm{tr}[\mathbb{I}_{A}]}\sum_{O_{A}}\braket{\Psi}{\boldsymbol{\bf
O}_{A}}{\Psi}O_{A}\,,$ (68)

where $\boldsymbol{\bf O}_{A}$ is the operator that acts like $O_{A}$ in the
subsystem and like the identity elsewhere. Reduced density matrices are
therefore embedded in $\mathfrak{A}_{\mathrm{ql}}$: investigating reduced
density matrices implicitly selects the quasilocal theory of local
observables. The Rényi entanglement entropies are then defined as

$S_{\alpha}(\ell,\Psi)\equiv\frac{1}{1-\alpha}\log\mathrm{tr}[\rho_{A}^{\alpha}(\Psi)]\,,$
(69)

which include the von Neumann entropy “$S_{1}$”, also known as entanglement
entropy, as the limit $\alpha\rightarrow 1^{+}$ (this limit makes sense
because the support of the distribution of eigenvalues of the density matrix
is bounded, and hence the Rènyi entropies with $\alpha=2,3,\ldots$
characterise the distribution completely [101]).

###  V.1 Reduced density matrices across different theories of local
observables

In a symmetric system $\rho_{A}(\Psi)$ is embedded in the subalgebra common to
all the theories of local observables, which in the $\mathbb{Z}_{2}$ case is
$\mathfrak{A}^{+}_{\mathrm{ql}}$ (the sum in Eq. (68) can be restricted to
matrices associated with operators that are even under spin flip). Thus,
$\rho_{A}(\Psi)$ is also represented in the other theories of local
observables, although, there, its interpretation as a subsystem’s reduced
density matrix breaks down. The simplest way to understand the representation
of a reduced density matrix in a semilocal theory is by mapping semilocal
operators into local operators through a duality transformation. This is
discussed below, where we specify the action of the duality transformation on
the reduced density matrices and show how to estimate the entanglement entropy
in the semilocal theory using the dual reduced density matrices.

####  V.1.1 Restricted duality transformation

We remind the reader that, when restricted to the even quasilocal subalgebra,
the duality transformation defined in Eq. (36) maps local operators into local
operators (see Table 1). This allows us to use the rotated Kramer-Wannier
duality transformation also in finite subsystems. For a subsystem $\tilde{A}$
of $\ell+1$ sites

$\tilde{A}=\\{1,\ldots,\ell+1\\}\,,$ (70)

we define $\mathcal{D}^{-1}_{\mathbb{Z}_{2}}(\ell)$ as follows

$\displaystyle\mathbb{I}^{\otimes(j-1)}\otimes\sigma^{z}\otimes\mathbb{I}^{\otimes(\ell+1-j)}\mapsto$
$\displaystyle\mathbb{I}^{\otimes(j-1)}\otimes(\tau^{z})^{\otimes
2}\otimes\mathbb{I}^{\otimes(\ell-j)}$ (71)
$\displaystyle\mathbb{I}^{\otimes(j^{\prime}-1)}\otimes(\sigma^{x})^{\otimes
2}\otimes\mathbb{I}^{\otimes(\ell-j^{\prime})}\mapsto$
$\displaystyle\mathbb{I}^{\otimes
j^{\prime}}\otimes\tau^{x}\otimes\mathbb{I}^{\otimes(\ell-j^{\prime})}\,,$

where $j=1,\ldots\ell$ and $j^{\prime}=1,\ldots\ell-1$. Note that we have
given up the bold notation because $\sigma^{\alpha}$ are not spin operators in
an infinite system but Pauli matrices spanning the local 2-dimensional Hilbert
space.

Since the matrices on the left hand side of Eq. (71) generate all matrices
representing even operators with support in $A$ — (67) (acting therefore as
the identity on site $\ell+1$), it is convenient to consider an alternative
representation of $S_{\alpha}(\ell,\Psi)$ (cf. Eq. (69)):

$S_{\alpha}(\ell,\Psi)=\frac{\log\mathrm{tr}[(\rho_{A}(\Psi)\otimes\tfrac{\mathbb{I}}{2})^{\alpha}]}{1-\alpha}-\log
2\,.$ (72)

##### Remark.

Similarly to $\mathcal{D}^{-1}_{\mathbb{Z}_{2}}$, also
$\mathcal{D}^{-1}_{\mathbb{Z}_{2}}(\ell)$ is an algebra homomorphism, although
it is defined only in a subspace of the full matrix algebra associated with
the subsystem. Specifically, let us introduce the following:

$\mathfrak{A}_{A}^{(\sigma)}$

the full matrix algebra generated by the tensor products of Pauli matrices
$\otimes_{j\in A}\sigma_{j}^{\alpha_{j}}$, with $\alpha_{j}=0,x,y,z$, in the
subsystem consisting of the sites $j\in A$ (and analogously for
$\mathfrak{A}_{\tilde{A}}^{(\tau)}$);

$\mathfrak{A}_{A}^{+(\sigma)}$

the restriction of $\mathfrak{A}_{A}^{(\sigma)}$ to matrices commuting with
$\bigotimes_{j\in A}\sigma_{j}^{z}$, which we refer to as even;

$\mathfrak{A}_{\tilde{A}}^{+(\tau)}$

the restriction of $\mathfrak{A}_{\tilde{A}}^{(\tau)}$ to matrices commuting
with $\bigotimes_{j\in\tilde{A}}\tau_{j}^{x}$, which we refer to as even;

$\mathfrak{A}_{B}\otimes\mathbb{I}_{C}$

(with $B\cap C=\emptyset$) the matrix subalgebra of $\mathfrak{A}_{B\cup C}$
consisting of the matrices in $\mathfrak{A}_{B}$ extended with the identity in
$C$, to act on sites $j\in B\cup C$.

We recognise $\mathcal{D}^{-1}_{\mathbb{Z}_{2}}(\ell)$, defined in Eq. (71),
as the _trace-preserving_ isomorphism that acts like (32) in a restricted
space of matrices with support on $\tilde{A}$:

$\mathcal{D}^{-1}_{\mathbb{Z}_{2}}(\ell):\mathfrak{A}^{+(\sigma)}_{A}\otimes\mathbb{I}_{\ell+1}\rightarrow\tilde{\mathfrak{A}}^{+(\tau)}_{\tilde{A}}\subset\mathfrak{A}^{+(\tau)}_{\tilde{A}}\,.$
(73)

Here the image $\tilde{\mathfrak{A}}^{+(\tau)}_{\tilde{A}}$, which is a
subalgebra of $\mathfrak{A}^{+(\tau)}_{\tilde{A}}$, is isomorphic to
${\mathfrak{A}}^{+(\sigma)}_{A}$ despite consisting also of operators whose
range extends up to $\ell+1$.

Since we are considering a mapping that spoils the notion of a spatial
subsystem, it is convenient to consider a generalisation of reduced density
matrices which, instead of describing the expectation values of operators with
support in a given region, describes the expectation values of operators that
are represented by matrices belonging to a given algebra. Specifically, we
introduce the following notation

$\hat{P}_{\mathfrak{A}}(\Psi)=\frac{1}{\mathrm{tr}[\mathbb{I}_{\mathfrak{A}}]}\sum_{O\in\mathfrak{A}}\braket{\Psi}{\boldsymbol{\bf
O}}{\Psi}O\,,$ (74)

where $\mathbb{I}_{\mathfrak{A}}$ is the identity in $\mathfrak{A}$ and the
sum is over an orthogonal basis of Hermitian matrices, such that
$\mathrm{tr}[OO^{\prime}]=\delta_{OO^{\prime}}\mathrm{tr}[\mathbb{I}_{\mathfrak{A}}]$
(we have always in mind operators written as tensor products of Pauli
matrices).

If the reduced density matrix $\rho_{A}(\Psi)$ — (68) of the subsystem is
even, the matrix $\rho_{A}(\Psi)\otimes\tfrac{\mathbb{I}}{2}$, which appears
in Eq. (72), belongs to
$\mathfrak{A}^{+(\sigma)}_{A}\otimes\mathbb{I}_{\ell+1}$,

$\rho_{A}(\Psi)\otimes\frac{\mathbb{I}}{2}=\hat{P}_{\mathfrak{A}^{+(\sigma)}_{A}\otimes\mathbb{I}_{\ell+1}}(\Psi)\,,$
(75)

and hence it is in the domain of $\mathcal{D}^{-1}_{\mathbb{Z}_{2}}(\ell)$.

Since this mapping preserves the trace,
$\mathcal{D}^{-1}_{\mathbb{Z}_{2}}(\ell)$ does not affect the value of
functionals of $\rho_{A}(\Psi)$ such as the Rényi entropies (72). In light of
Eq. (73), $\mathcal{D}^{-1}_{\mathbb{Z}_{2}}(\ell)$ maps
$\rho^{(\sigma)}_{A}(\Psi)\otimes\frac{\mathbb{I}}{2}$ into the projection of
$\rho^{(\tau)}_{A}(\Psi)$ on the subalgebra
$\tilde{\mathfrak{A}}^{+(\tau)}_{\tilde{A}}$

$\mathcal{D}^{-1}_{\mathbb{Z}_{2}}(\ell):\rho^{(\sigma)}_{A}(\Psi)\otimes\frac{\mathbb{I}}{2}\mapsto\rho^{(\tau)}_{\tilde{A}}(\Psi)\bigr{|}_{\tilde{\mathfrak{A}}^{+(\tau)}_{\tilde{A}}}\,.$
(76)

We will see in a moment how to carry out this projection.

####  V.1.2 Dual reduced density matrices

By construction, the operators $\boldsymbol{\bf O}^{(\sigma)}$ that have
support in $A$ and are represented by matrices $O^{(\sigma)}\otimes\mathbb{I}$
in the domain of $\mathcal{D}_{\mathbb{Z}_{2}}^{-1}(\ell)$ are mapped by the
rotated Kramers-Wannier duality transformation (36) into operators
$\tilde{\boldsymbol{\bf O}}$ that have support in $\tilde{A}$ and are
represented by
$\tilde{O}=\mathcal{D}_{\mathbb{Z}_{2}}^{-1}(\ell)[O^{(\sigma)}\otimes\mathbb{I}]$.
The following equality holds for them:

$\braket{\Psi}{\boldsymbol{\bf
O}}{\Psi}^{(\sigma)}=\braket{\Psi}{\tilde{\boldsymbol{\bf
O}}}{\Psi}^{(\tau)}\,.$ (77)

Thus, the (extended) reduced density matrix
$\rho^{(\sigma)}_{A}(\Psi)\otimes\frac{\mathbb{I}}{2}$ is mapped into

$\rho^{(\sigma)}_{A}(\Psi)\otimes\frac{\mathbb{I}}{2}\mapsto
2^{-\ell-1}\kern-10.00002pt\sum_{O\in\mathfrak{A}^{+(\sigma)}_{A}\atop\tilde{O}=\mathcal{D}_{\mathbb{Z}_{2}}^{-1}(\ell)[O\otimes\mathbb{I}]}\kern-10.00002pt\bra{\Psi}\boldsymbol{\bf\tilde{O}}\ket{\Psi}^{(\tau)}\tilde{O}\,,$
(78)

which can be compactly written as

$\hat{P}_{\mathfrak{A}^{+(\sigma)}_{A}\otimes\mathbb{I}_{\ell+1}}\mapsto\hat{P}_{\mathfrak{\tilde{A}}^{+(\tau)}_{\tilde{A}}}\,.$
(79)

While the right-hand side of Eq. (78) resembles

$\rho^{(\tau)}_{\tilde{A}}(\Psi):=\hat{P}_{\mathfrak{A}^{(\tau)}_{\tilde{A}}}=2^{-\ell-1}\sum_{\tilde{O}\in\mathfrak{A}^{(\tau)}_{\tilde{A}}}\kern-10.00002pt\bra{\Psi}\boldsymbol{\bf\tilde{O}}\ket{\Psi}^{(\tau)}\tilde{O}\,,$
(80)

the sum in Eq. (78) does not cover the full matrix algebra of the subsystem
($\mathfrak{\tilde{A}}^{+(\tau)}_{\tilde{A}}$ is strictly contained in
$\mathfrak{A}^{(\tau)}_{\tilde{A}}$). In order to obtain Eq. (78) from
$\rho^{(\tau)}_{\tilde{A}}(\Psi)$ we must therefore project out all the
operators that are not in the image of
$\mathcal{D}_{\mathbb{Z}_{2}}^{-1}(\ell)$. This can be done by identifying the
symmetries characterising $\tilde{\mathfrak{A}}^{+(\tau)}_{\tilde{A}}$:

  1. 1.

Only operators even under spin flip $\mathcal{P}^{x}_{\tau}$ are represented
in $\tilde{\mathfrak{A}}^{+(\tau)}_{\tilde{A}}$;

  2. 2.

Operators $\boldsymbol{\bf\tilde{O}}$ that are represented on the right hand
side of (78) commute both with $\boldsymbol{\bf\tau}_{1}^{z}$ and
$\boldsymbol{\bf\tau}_{\ell+1}^{z}$ — see Eq. (71).

These $\mathbb{Z}_{2}$ symmetries can be enforced by averaging over the
corresponding (Hermitian) flip matrices $P_{i}$ as follows

$\displaystyle\hat{P}_{\mathfrak{\tilde{A}}_{\tilde{A}}^{+(\tau)}}(\Psi)\\!=\\!\bar{\rho}^{(\tau)}_{\tilde{A}}(\Psi)\\!:=\\!\sum_{j_{1,2,3}=0}^{1}\kern-5.0pt\tfrac{P_{3}^{j_{3}}P_{2}^{j_{2}}P_{1}^{j_{1}}\rho^{(\tau)}_{\tilde{A}}(\Psi)P_{1}^{j_{1}}P_{2}^{j_{2}}P_{3}^{j_{3}}}{8},$
(81) $\displaystyle P_{1}=(\tau^{x})^{\otimes(\ell+1)},\quad
P_{2}=\tau^{z}\otimes\mathbb{I}^{\otimes\ell},\quad
P_{3}=\mathbb{I}^{\otimes\ell}\otimes\tau^{z}$

(see Fig. 11 for a graphical representation).