# Entanglement dynamics and phase transitions of the Floquet cluster spin
chain

Alberto D. Verga<EMAIL_ADDRESS>CPT, Aix-Marseille Université,
Marseille, France

###### Abstract

Cluster states were introduced in the context of measurement based quantum
computing. In one dimension, the cluster Hamiltonian possesses topologically
protected states. We investigate the Floquet dynamics of the cluster spin
chain in an external field, interacting with a particle. We explore the
entanglement properties of the topological and magnetic phases, first in the
integrable spin lattice case, and then in the interacting quantum walk case.
We find, in addition to thermalization, dynamical phase transitions separating
low- and high-entanglement nonthermal states, reminiscent of the ones present
in the integrable case, but differing in their magnetic properties. The
nonergodic phases are characterized by the emergence of magnetic order,
persistent at long times.

##  I Introduction

One of the main trends in quantum information is the search for “computational
phases” of matter [1, 2]. Indeed, since the formulation of models of fault-
tolerant [3, 4], robust-against-error [5], quantum computation using logical
qubits encoded in the degenerate ground state of gapped Hamiltonians and their
topological excitations [6], or using persistent, highly entangled quantum
states [7], different models of quantum computation have been introduced. The
main idea has been to use the properties of topological phases of matter,
protected by some kind of symmetry to implement the qubit logical operations,
as in the topological model [8], or the measurement-based model [9] of quantum
computation.

One obstacle in the way of self-correcting fault-tolerant quantum computation
is the need for robust topologically ordered phases supporting highly
entangled states as a universal quantum resource [10, 11, 12, 13]. In fact,
closed quantum systems with short range, local interactions, conserving only
total energy, tend to thermalize: The expected value of the observables is
essentially given by their microcanonical value as derived from the
eigenvector thermalization [14].

As a result of thermalization, the states, although highly entangled, cannot
be a useful resource for quantum computation [15]. If, in addition, the system
is periodically driven by an applied field, breaking then the energy
conservation, it should evolve towards an infinite temperature ergodic state
[16, 17]. In systems without extrinsic disorder, protection against
thermalization can also be reached by ergodicity breaking, in analogy with
classical glasses, due to the existence of additional conservation laws [18,
19, 20], or by the emergence of a decoupled subspace of nonthermal states [21,
22].

In Floquet systems, in which we are interested, exceptions to relaxation
towards a thermal state are integrable systems whose large number of local
constants of motion prevent the emergence of an ergodic phase [23, 24], and
systems with dynamical constraints, as in arrays of Rydberg atoms [25]
supporting many-body scar states [26, 27, 28]. Another possibility for
suppression of unbounded energy absorption in strongly driven systems, is the
emergence of a robust (albeit not exact) conserved quantity leading to
dynamical freezing [29]. For a recent review see Ref. [30].

The search for nonergodic phases in Floquet systems is motivated by the
possibility of using them to engineer effective Hamiltonians in order to
describe, for instance, topological materials [31] or, from a more fundamental
perspective, to investigate nonequilibrium phases of matter [32, 33, 34]. One
interesting possibility is to create, using Floquet dynamics, nonthermal
states possessing useful entanglement properties, similar to the ones found in
symmetry-protected topological phases of gapped Hamiltonians [35], having the
potential to be a universal resource for quantum computation, such as for
instance the cluster state [36, 37].

Breaking of ergodicity in Floquet systems was recently demonstrated in the
case of a periodically perturbed ergodic Ising spin chain [38, 39], and in the
case of a quantum cellular automaton [40]. Although the two models are
unrelated, the mechanism of ergodicity breaking have in common the emergence
of approximate integrals of motion [29]: the magnetization in the first case
[38] and the conservation of the quasiparticle number in the second case [40].

For generic Ising chains, the emergence of an approximate conservation law,
absent in the undriven system, is related to the presence of resonances. These
resonances are a many-body generalization of the single-particle resonances
that, by interference at particular driven frequencies, coherently destroy
amplitude tunneling [41], leading to energy eigenstate localization. At strong
driving, these resonances effectively create constraints (similar to the ones
in scar models), eventually leading to the freezing of the system dynamics and
the breaking of the system’s ergodicity [39]. In contrast to the Ising chain,
ergodicity breaking in the automaton, is due to the persistence of weakly
interacting quasiparticles in the chaotic regime, even in the absence of
energy eigenstate localization, leading to the fragmentation of the Hilbert
space according to their content in quasiparticles [40].

Our aim in this paper is to investigate the fate of a topological phase
associated with the ground state of an error-correcting Hamiltonian, such as
the cluster phase [1, 42], first when externally driven by a periodic field,
and then when embedded in a larger Hilbert space by the introduction of an
interaction with a quantum walker [43]. More specifically, we generalize the
Floquet cluster model, which is integrable, by introducing an exchange
coupling between the chain spins and the walker spin (the coin internal degree
of freedom). We chose a coupling such that it preserves the original
symmetries, but destroys the local integrals of motion of the cluster chain.
The effect of the walker is to mediate, through its ballistic motion, the
interaction between spatially separated spins, effectively adding a nonlocal
interaction [44, 45].

We focus on a one dimensional spin chain whose Floquet dynamics can be solved
analytically (Sec. II), and demonstrate the existence of a topological phase
issued from the original static phase; in addition, computing the Loschmidt
rate [46] we show that the integrable model undergoes a dynamical phase
transition [47, 48] that can be characterized by the concomitant change in the
entanglement level [49, 50].

The extension of the model to an interacting quantum walk [44, 51] allows us
to determine the persistence of the two phases present in the integrable case,
the cluster phase and the topologically trivial paramagnetic phase (Sec. III).
In fact we find that, even if the dynamical phase transition is always present
in a range of parameters, new nonergodic phases appear characterized by a
finite magnetization and global entanglement. We discuss the mechanism of
ergodicity breaking in terms of the effective magnetic interaction mediated by
the particle between fixed spins. We show in particular, that a phase
transition between low- and high-entanglement phases, is possible even for the
case corresponding to the trivial phase in the original cluster chain.

##  II Floquet cluster model

We consider a system of $L$ spins in a one dimensional lattice. The system’s
Hamiltonian is [52]

$H_{C}=-\frac{J}{2}\sum_{x=0}^{L-1}Z_{x-1}X_{x}Z_{x+1},$ (1)

where $J$ is the coupling constant of the “cluster” interaction; we denote
$\bm{\sigma}_{x}=(X_{x},Y_{x},Z_{x})$ the vector of Pauli matrices at each
site $x=0,\ldots,L-1$ (we take the lattice constant as the unit of length). We
assume periodic boundary conditions, to ensure translation invariance. In
addition, an external field $B$ applies in the $x$ direction:

$H_{B}=-\frac{B}{2}\sum_{x=0}^{L-1}X_{x}.$ (2)

The spin Hilbert space is spanned by the basis states
$\ket{s}=\ket{s_{0}\cdots s_{L-1}}$, $s_{x}=\\{0,1\\}$, labeled by the integer
$s=0,\ldots,2^{L}-1$. The ground state of $H_{C}$, the cluster state, is the
eigenvector with eigenvalue $1$ common to every term in $H_{C}$; it can be
written in terms of the controlled $Z$ operator
$\operatorname{CZ}=\mathrm{diag}(1,1,1,-1)$:

$\ket{C}=\prod_{x}\operatorname{CZ}_{x,x+1}\ket{+}^{L}$ (3)

where $\ket{+}=(\ket{0}+\ket{1})/2^{1/2}$ (we use the tensor product notation
“$\otimes$” only in ambiguous cases, or to separate Hilbert subspaces). The
$H_{C}$ Hamiltonian is in fact a sum of stabilizer operators [5], whose terms
commute with each other and commute with the Hamiltonian, hence providing an
extensive number of integrals of motion.

The cluster state (3) is a highly entangled quantum state that can also be
defined over an arbitrary graph of qubits, used as an information resource for
measurement based quantum computing [7, 9]. Highly entangled means that its
Schmidt dimension, the minimal number of parameters needed for its
specification, grows exponentially with the number of qubits. It also has the
property of maximal connectedness: Any pair of qubits can be projected into a
Bell state by appropriately measuring the intermediate qubits joining them
along a path on the graph [53]. A consequence of the exponential Schmidt rank
and the inherent entanglement nonlocality, the cluster state is a genuine
quantum resource that goes beyond the possibilities of any classical resource:
A computation based on the cluster state cannot, in principle, be efficiently
simulated by a classical algorithm [54].

The dynamics of the system is governed by the Floquet operator

$\begin{split}F(J,B)&=\mathrm{e}^{-\mathrm{i}H_{C}}\mathrm{e}^{-\mathrm{i}H_{B}}\\\
&=\prod_{x=0}^{L-1}\mathrm{e}^{\mathrm{i}(J/2)Z_{x-1}X_{x}Z_{x+1}}\prod_{x=0}^{L-1}\mathrm{e}^{\mathrm{i}(B/2)X_{x}}\,\end{split}$
(4)

such that the state of the system $\ket{\psi(t)}$ at time $t$ changes in one
time step (our unit of time, with $\hbar=1$) according to

$\ket{\psi(t+1)}=F(J,B)\ket{\psi(t)}.$ (5)

The parity operators

$P_{o}=\prod_{x\in o}X_{x},\quad P_{e}=\prod_{x\in e}X_{x},\quad
P=P_{o}P_{e},$ (6)

where $o,e$ stand for odd and even sites, commute with $F$:

$[P_{e},F]=[P_{o},F]=[P,F].$ (7)

Therefore, $H_{C}$ is symmetric under the $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$
group generated by $P_{e,o}$, and its ground state belongs to a topological
phase protected by symmetry [42, 55]. This suggests the question about the
fate of this topological phase in the Floquet case.

Both the Hamiltonian $H_{C}+H_{B}$ and the Floquet $F$ models possess the same
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ symmetry; however $F$ do not commute with
$H_{C}+H_{B}$ breaking the energy conservation, which leads to essentially
different dynamical properties. The Hamiltonian model, often called the
transverse field cluster model, was extensively analyzed, starting with the
calculation by Suzuki [56] who used the Jordan-Wigner transformation to find
the eigenspectrum of $H_{C}+H_{B}$, and followed by the description of the
phase transition between the $J\rightarrow 0$ paramagnetic phase and the
$B\rightarrow 0$ topological phase [57, 1], and the discussion of the phase
diagram when magnetic interactions are added [58, 59, 42, 60].

To find the spectrum of $F$ we follow the usual method [56, 61] and introduce
the Jordan-Wigner transformation [62, 63] of the spin operators in terms of
the fermion operators,

$\\{f_{x},f_{y}\\}=\\{f_{x}^{\dagger},f_{y}^{\dagger}\\},\quad\\{f_{x},f_{y}^{\dagger}\\}=\delta_{xy}.$
(8)

It is defined by

$\begin{split}X_{x}&=1-2f^{\dagger}_{x}f_{x},\\\
Y_{x}&=\mathrm{i}K_{x}(f^{\dagger}_{x}-f_{x}),\\\
Z_{x}&=-K_{x}(f^{\dagger}_{x}+f_{x}),\end{split}$ (9)

where

$K_{x}=\prod_{y=0}^{x-1}\mathrm{e}^{\mathrm{i}\pi
f^{\dagger}_{y}f_{y}}=\prod_{y=0}^{x-1}(1-2f^{\dagger}_{y}f_{y}).$ (10)

Once introduced into (1), the Jordan-Wigner transformation leads to the
bilinear fermion Hamiltonian

$\displaystyle H_{C}$
$\displaystyle=\frac{J}{2}\sum_{x=0}^{L-1}(f^{\dagger}_{x-1}-f_{x-1})(f^{\dagger}_{x+1}+f_{x+1}),$
(11) $\displaystyle H_{B}$
$\displaystyle=\frac{B}{2}\sum_{x=0}^{L-1}(2f^{\dagger}_{x}f_{x}-1).$ (12)

The translation invariance allows the use of the Fourier transform

$f_{x}=\frac{\mathrm{e}^{\mathrm{i}\pi/4}}{\sqrt{L}}\sum_{k}\mathrm{e}^{\mathrm{i}kx}f_{k},$
(13)

where the set of wavenumbers in the Brillouin zone $k\in(-\pi,\pi]$, is
divided into “even” and “odd” sectors,

$\begin{split}e&=\\{k=\pm\frac{\pi n}{L}\mid n=1,3,\ldots,L-1\\}\\\
&=e_{+}\cup e_{-},\\\ o&=\\{k=\frac{\pi n}{L}\mid n=0,\pm
2,\ldots,\pm(L-2),L\\}\\\ &=\\{0\\}\cup o_{+}\cup
o_{-}\cup\\{\pi\\},\end{split}$ (14)

for an even and odd number of fermions, respectively (taking $L$ to be even).
Indeed, the sign of the parity operator $P$ depends on the total number of
fermions $N_{F}$,

$P=(-1)^{N_{F}},\quad N_{F}=\sum_{x=0}^{L-1}f_{x}^{\dagger}f_{x},$ (15)

for even $N_{F}$, we impose anti-periodic boundary conditions
$f_{L+1}=-f_{1}$, and we impose periodic boundary conditions $f_{L+1}=f_{1}$,
for $N_{F}$ odd. The Fourier transformed cluster Hamiltonian is, in the even
sector,

$H_{C}=J\sum_{k\in e_{+}}\left[\cos
2k(f^{\dagger}_{k}f_{k}-f_{-k}f^{\dagger}_{-k})\right.\\\ \left.+\sin
2k(f^{\dagger}_{k}f^{\dagger}_{-k}+f_{-k}f_{k})\right],$ (16)

and the field Hamiltonian is,

$H_{B}=B\sum_{k\in e_{+}}(f^{\dagger}_{k}f_{k}-f_{-k}f^{\dagger}_{-k}).$ (17)

Similar expressions hold for the odd sector.

The unitary map $F$ is, in the even sector

$F^{(e)}=\prod_{k\in e_{+}}V_{k}\prod_{k\in e_{+}}W_{k},$ (18)

where

$V_{k}=\mathrm{e}^{\mathrm{i}JC_{k}^{\dagger}(\cos 2kZ+\sin 2kX)C_{k}},\quad
W_{k}=\mathrm{e}^{\mathrm{i}BC_{k}^{\dagger}ZC_{k}},$ (19)

and

$C_{k}=\begin{pmatrix}f_{k}\\\ f^{\dagger}_{-k}\end{pmatrix},\quad
C^{\dagger}_{k}=\begin{pmatrix}f^{\dagger}_{k}&f_{-k}\end{pmatrix}.$ (20)

In the odd sector we add the terms $k=0,\pi$:

$F^{(o)}=F_{0}\prod_{k\in o_{+}}V_{k}\prod_{k\in o_{+}}W_{k}F_{\pi},$ (21)

where

$F_{0}=\mathrm{e}^{\mathrm{i}(J+B)(2f^{\dagger}_{0}f_{0}-1)},\quad
F_{\pi}=\mathrm{e}^{\mathrm{i}(J+B)(2f^{\dagger}_{\pi}f_{\pi}-1)}.$ (22)

Combining the two rotations we obtain the Floquet operator in terms of an
effective Hamiltonian $H_{F}$:

$F^{(e)}=\mathrm{e}^{-\mathrm{i}H_{F}},\;F^{(o)}=F_{0}\mathrm{e}^{-\mathrm{i}H_{F}}F_{\pi},$
(23)

where,

$H_{F}=-\sum_{k}C^{\dagger}h_{k}C_{k}$ (24)

and

$h_{k}=\epsilon_{k}\bm{n}_{k}\cdot\bm{\sigma}=\bm{d}_{k}\cdot\bm{\sigma},$
(25)

with $\bm{d}_{k}=\epsilon_{k}\bm{n}_{k}$ and

$\cos(\epsilon_{k})=\cos J\cos B-\sin J\sin B\cos 2k;$ (26)
$\bm{n}_{k}=\frac{1}{\big{|}\sin\epsilon_{k}\big{|}}\begin{pmatrix}\sin J\cos
B\sin 2k\\\ \sin J\sin B\sin 2k\\\ \sin J\cos B\cos 2k+\cos J\sin
B\end{pmatrix}$ (27)

is a unit vector of $(n_{x},n_{y},n_{z})$ components. Note that similar
effective Hamiltonians appear in the Floquet transverse field Ising model
[61], and topological quantum walks [43].

Figure 1: Family of quasienergy bands $\epsilon_{k}$ of the effective
Hamiltonian, as a function of the quasimomentum $k$ and $J$, for $B=0.7\pi/2$,
fixed. The gap between positive and negative quasienergy bands closes on the
line $J=B$, at $\epsilon_{k}=0$ when $k=\pm\pi/2$, and at $\epsilon_{k}=\pi$
when $k=0,\pi$.

###  II.1 Winding number

The dispersion relation $\epsilon_{k}$ (26), is represented in Fig. 1 as a
function of $J$ for fixed $B$; two Dirac points appear in the Brillouin zone
$k\in(-\pi,\pi]$ when $J=B$ for $\epsilon_{k}=0,\pi$. We demonstrate now that
the change in the band structure through the line in parameter space $J=B$
separates two topological distinct phases, related to a chiral symmetry of the
effective Hamiltonian.

We show in Fig. 2 the locus of $\bm{n}_{k}$ for two values of $(J,B)$. When
the spin coupling constant is small with respect to the applied field $J<B$,
the vector describes an arc, and in the opposite case it describes a complete
circle; both run twice around the center.

We note that the vector $\bm{A}$

$\bm{A}=\bm{A}(B)=\begin{pmatrix}\sin(B)\\\ -\cos(B)\\\ 0\end{pmatrix}$ (28)

is perpendicular to $\bm{n}_{\bm{k}}$, independently of $k$. The existence of
such vector is related to the symmetry

$\mathrm{e}^{\mathrm{i}\pi\bm{A}\cdot\bm{\sigma}/2}h\mathrm{e}^{-\mathrm{i}\pi\bm{A}\cdot\bm{\sigma}/2}=-h,$
(29)

indicative of a chiral invariance of the effective Hamiltonian. Indeed,
applying the rotation

$\mathrm{e}^{-\mathrm{i}\pi
X/4}\mathrm{e}^{\mathrm{i}BY/2}h\mathrm{e}^{-\mathrm{i}BY/2}\mathrm{e}^{\mathrm{i}\pi
X/4}=h_{c},$ (30)

we can off-diagonalize the effective Hamiltonian, explicitly exhibiting its
chiral symmetry:

$h_{c}=\begin{pmatrix}0&g(k)\\\
\bar{g}(k)&0\end{pmatrix}=\bm{d}_{c}(k)\cdot\bm{\sigma},$ (31)

where $g(k)=d_{cx}(k)-\mathrm{i}d_{cy}(k)$, and where

$\bm{d}_{c}=\frac{\epsilon_{k}}{\sin\epsilon_{k}}\begin{pmatrix}\sin J\sin
2k\\\ \sin J\cos B\cos 2k+\cos J\sin B\\\ 0\end{pmatrix}$ (32)

lies on the $(x,y)$ plane. Therefore, the winding number can be easily
computed using the standard formula [64]

$\nu=\frac{1}{2\pi\mathrm{i}}\int_{-\pi}^{\pi}\mathrm{d}k\,\frac{\mathrm{d}}{\mathrm{d}k}\ln
g(k).$ (33)

(Note that the wavenumber can be extended to the whole Brillouin zone
$k\in(-\pi,\pi]$, since $\epsilon_{k}$ is even.)

Noting that the coordinates of $\bm{d}_{c}$ define the parametric equations of
an ellipse centered at $(0,\cos J\sin B)$, we obtain

$\nu=\begin{cases}2\operatorname{sgn}(\sin J)&\text{if }\tan J>\tan B\\\
0&\text{if }\tan J<\tan B\end{cases},$ (34)

the critical points locate at the lines $J=B\mod\pi/2$, splitting the $(J,B)$
plane into sectors with winding number $\nu=0,\pm 2$ (Fig. 3).

Figure 2: The rotation axis vector of the effective Hamiltonian accumulates a
zero phase for $J<B$ and a $4\pi$ phase for $J>B$, when $k$ spans the
Brillouin zone. (The two circles are shifted for clarity; the dot marks the
origin of coordinates.) Parameters: $J>B$ case, $J=0.9\pi/2$, $B=0.7\pi/2$,
and $J<B$ case, $J=0.5\pi/2$, $B=0.7\pi/2$.

We remark that the use of (33) to characterize the topological phases is based
on the symmetries of the effective Hamiltonian, in particular the winding
number is invariant under the unitary transformation (30) leading to the
manifestly chiral form $h_{c}$ (31). However it is possible to generalize
$\nu$ from the symmetries of the Floquet operator instead of those related to
the effective Hamiltonian [65]. More specifically, one may take into account
the fact that $\epsilon_{k}$ is a quasienergy and define a pair of winding
numbers associated with the closing of band gaps at $\epsilon_{k}(J,B)=0,\pi$
[66, 67]. The result (34) shows that the Floquet generalization of the cluster
model does not destroy the cluster nontrivial topology [68, 55].

Figure 3: Winding number as a function of the coupling $J$ and field $B$. The
two dots correspond to the trivial and nontrivial phases of Fig. 2.

###  II.2 Eigenstates

The diagonalization of the effective Hamiltonian can be obtained by a unitary
Bogoliubov transformation

$R_{k}=\frac{1}{\sqrt{2}\sqrt{1-n_{z}(k)}}\begin{pmatrix}1-n_{z}(k)&n_{-}(k)\\\
-n_{+}(k)&1-n_{z}(k)\end{pmatrix},$ (35)

where $n_{\pm}=n_{x}\pm\mathrm{i}n_{y}$ and the columns of $R_{k}$ are the
eigenvectors of $h_{k}$ corresponding to the eigenvalues $\mp\epsilon_{k}$,
respectively. This transformation maps the Fermi operators $C_{k}$ into the
new operators

$A_{k}=\begin{pmatrix}a_{k}\\\
a^{\dagger}_{-k}\end{pmatrix}=R_{k}^{\dagger}\begin{pmatrix}f_{k}\\\
f^{\dagger}_{-k}\end{pmatrix}$ (36)

preserving the commutation relations. In the new basis the effective
Hamiltonian reads,

$H_{F}=\sum_{k>0}A^{\dagger}_{k}\begin{pmatrix}\epsilon_{k}&0\\\
0&-\epsilon_{k}\end{pmatrix}A_{k}.$ (37)

The corresponding evolution operator factorizes as

$F=\left\\{\begin{aligned} &\prod_{k\in e_{+}}F_{k},\quad\text{even}\\\
&F_{0}\left(\prod_{k\in
o_{+}}F_{k}\right)F_{\pi},\quad\text{odd}\end{aligned}\right.$ (38)

where,

$F_{k}=\mathrm{e}^{\mathrm{i}C^{\dagger}_{k}h_{k}C_{k}}.$ (39)

Its eigenstates span a four dimensional space for each $k$, corresponding to
the four eigenvalues of the fermion number operators $f^{\dagger}_{k}f_{k}$
and $f^{\dagger}_{-k}f_{-k}$. The basis of this four dimensional space is

$\big{\\{}\ket{0},f^{\dagger}_{k}\ket{0},f^{\dagger}_{-k}\ket{0},f^{\dagger}_{-k}f^{\dagger}_{k}\ket{0}\big{\\}}$
(40)

where $f_{k}\ket{0}=f_{-k}\ket{0}=0$. The vacuum state $\ket{0}$ corresponds
to the completely polarized state $\ket{+}^{L}$ in the original spin
configuration basis [cf. (3)]. In this basis, the eigenstates of $F_{k}$ are,

$\displaystyle F_{k}\ket{\pm k}=\ket{\pm k},$ $\displaystyle F_{k}\ket{\pm
kk}=\mathrm{e}^{\pm\mathrm{i}\epsilon_{k}}\ket{\pm kk},$ (41)

where

$\displaystyle\ket{\pm k}=f^{\dagger}_{\pm k}\ket{0},$
$\displaystyle\ket{+kk}=\frac{1-n_{z}(k)+n_{-}(k)f^{\dagger}_{-k}f^{\dagger}_{k}}{\sqrt{2(1-n_{z}(k))}}\ket{0},$
$\displaystyle\ket{-kk}=\frac{n_{+}(k)-(1-n_{z}(k))f^{\dagger}_{-k}f^{\dagger}_{k}}{\sqrt{2(1-n_{z}(k))}}\ket{0},$
(42)

(see Appendix A). The $k=\\{0,\pi\\}$ subspace is spanned by the eigenvectors

$\big{\\{}\ket{0},f^{\dagger}_{0}\ket{0},f^{\dagger}_{\pi}\ket{0},f^{\dagger}_{0}f^{\dagger}_{\pi}\ket{0}\big{\\}},$
(43)

which also are the eigenvectors of $F_{0}F_{\pi}$ with eigenvalues:

$\big{\\{}\mathrm{e}^{-2\mathrm{i}(J+B)},\mathrm{e}^{-\mathrm{i}(J+B)},\mathrm{e}^{-\mathrm{i}(J+B)},1\big{\\}},$
(44)

respectively. Therefore, these terms contribute with a constant phase and do
not play any dynamical role.

###  II.3 Time evolution

The discrete time evolution of an arbitrary state
$\ket{\psi(t)}=F(t)\ket{\psi(0)}$, for integer $t$, is governed by,

$F(t)=\prod_{k}F_{k}(t)=\prod_{k}\mathrm{e}^{\mathrm{i}t\epsilon_{k}C_{k}^{\dagger}\bm{n}_{k}\cdot\bm{\sigma}C_{k}},$
(45)

where the product is over the relevant set of wavenumbers [cf. (14) and (38)].
For the vacuum state $\ket{0}$, a simple calculation using the eigenstates
(42), gives (see Appendix A)

$\ket{t}=\prod_{k}\big{[}\cos(\epsilon_{k}t)+\mathrm{i}n_{z}(k)\sin(\epsilon_{k}t)\\\
+\mathrm{i}n_{-}(k)\sin(\epsilon_{k}t)f_{-k}^{\dagger}f_{k}^{\dagger}\big{]}\ket{0}.$
(46)

We are interested in the time evolution of the entanglement, when the system
is initially in the product state $\ket{0}$ (or equivalently $\ket{+}^{L}$). A
computable measure of the global entanglement can be defined in terms of the
spin purity $\operatorname{tr}\rho_{x}^{2}$, where $\rho_{x}$ is the reduced
density matrix of the spin located at $x$, assuming that the system is in an
arbitrary pure state $\ket{\psi(t)}$ [69]:

$\mathcal{Q}(t)=2-\frac{2}{L}\sum_{x=0}^{L-1}\operatorname{tr}\rho^{2}_{x}(t).$
(47)

This formula can easily be written as

$\mathcal{Q}(t)=1-\frac{1}{L}\sum_{x}\braket{\psi(t)}{\bm{\sigma}_{x}}{\psi(t)}^{2},$
(48)

where, from the expression of the density matrix,

$\rho_{x}=\frac{1+\braket{\bm{\sigma}_{x}}\cdot\bm{\sigma}}{2},$ (49)

we derived the purity in terms of the expected value of the spin at site $x$,
$\braket{\bm{\sigma}_{x}}$ [61]. The $\mathcal{Q}$ measure, first introduced
in Ref. [70], quantifies the multipartite entanglement of a given state as an
average over the bipartite entanglement of each spin with the rest of the
system [69]. $\mathcal{Q}$ vanishes only if the global state is a product
state, and is maximum for a globally entangled state. It distinguishes
localized and chaotic random states [71] and is useful to characterize the
quantum phase transition of the XY model [72]. The time evolution of
$\mathcal{Q}$ in the kicked Ising model, extensively investigated by
Lakshminarayan and Subrahmanyam in Ref. [61], shows recurrences to an initial
unentangled state and smooth variations for zero field, but at nonzero
transverse field, its dynamics becomes very complex due to the large energy
spectrum (time scales) of the Floquet operator (even if the system is
completely integrable).

Figure 4: Mean global entanglement $\mathcal{Q}_{0}$, averaged over the last
50 time steps (over 200), as a function of $J$ and $B$. Computed from (51)
with $1000$ values of $k$, $L=1000$.

Equations (9) and (46) imply that the expected values in the evolved vacuum
state of $\braket{Y_{x}}(t)$ and $\braket{Z_{x}}(t)$, vanish, therefore

$\begin{split}\mathcal{Q}_{0}(t)&=1-\frac{1}{L}\sum_{x}\braket{t}{X_{x}}{t}^{2}\\\
&=\frac{4}{L}\sum_{x}\braket{t}{f_{x}^{\dagger}f_{x}}{t}\big{(}1-\braket{t}{f_{x}^{\dagger}f_{x}}{t}\big{)},\end{split}$
(50)

which gives (see Appendix A),

$\mathcal{Q}_{0}(t)=\frac{4}{L}\sum_{k\in\mathcal{B}}N_{k}\left[1-\frac{1}{L}\sum_{k\in\mathcal{B}}N_{k}\right],$
(51)

where

$N_{k}=\braket{t}{f^{\dagger}_{k}f_{k}}{t}=(1-n_{z}^{2})\sin^{2}(\epsilon_{k}t).$
(52)

In the case $J=B=\pi/2$, $n_{z}(k)=0$ and $\epsilon_{k}=2k$ (Dirac dispersion)
the global entanglement for the initial $\ket{0}$ state, reduces to

$\mathcal{Q}_{0}(t;J=B)=1-\delta_{t,mL/4},$ (53)

where we used the identity

$\frac{1}{L}\sum_{k\in\mathcal{B}}\sin^{2}(2kt)=\frac{1-\delta_{t,mL/4}}{2},$
(54)

with $m$ being an integer ($t=0,1,\ldots$, and $L$ is even). We find that the
entanglement present revivals with a period proportional to the system’s size,
in which it is maximal during one step. This result is similar to the one
obtained for the transverse Ising model [61]. In the $J=0$ case,
$\mathcal{Q}_{0}$ vanishes, while for $B=0$ it becomes,

$\mathcal{Q}_{0}(t;J=0)=0,\quad\mathcal{Q}_{0}(t;B=0)=1-\cos^{4}(Jt),$ (55)

and tends to $1/2$ at large times

$\lim_{t\rightarrow\infty}\mathcal{Q}_{0}(t;B=0^{+})=5/8.$ (56)

The global entanglement as a function of $J$ and $B$ is represented in Fig. 4.
On the critical lines $J=B$, $\mathcal{Q}_{0}$ is maximum, reaching the
absolute maximum at $J=B=\pi/2$. Therefore, $\mathcal{Q}_{0}$ is a good
indicator of the symmetry-breaking phase transition, although it cannot detect
the distinct topological phases.

###  II.4 Loschmidt echo

Figure 5: Loschmidt echo rate $\lambda(t)$ [(a) and (b)] and global
entanglement $\mathcal{Q}_{0}(t)$ [(c) and (d)] of the full polarized state.
(a) Near $J,B=0$ we find the behavior of the Hamiltonian model; (b) for larger
values of $J$ and $B$, $\lambda(t)$ loses regularity. In both (a) and (b), the
topological phase ($J>B$, black upper line) is clearly distinguished from the
trivial one ($J<B$, red bottom line). The global entanglement [(c) and (d)]
follows a similar pattern with higher values for $J>B$. Parameters: $L=600$,
for (a) and (c) ($J=0.03$, $B=0.01$) and ($J=0.01$, $B=0.03$), and for (b) and
(d), ($J=0.6$, $B=0.2$) and ($J=0.2$, $B=0.6$).

To characterize the topological phases we investigate the Loschmidt echo [46],

$\mathcal{L}(t)=|\braket{0}{t}|^{2},$ (57)

here defined for the initial completely polarized pure state $\ket{0}$, and
$\ket{t}$ is given by (46):

$\mathcal{L}(t)=\prod_{k\in\mathcal{B}}\big{\\{}1-[1-n_{z}(k)^{2}]\sin^{2}(\epsilon_{k}t)\big{\\}}.$
(58)

A more suitable quantity is the intensive variable, well defined in the large
system limit $L\rightarrow\infty$, the Loschmidt ratio

$\lambda(t)=-\frac{1}{L}\ln\mathcal{L}(t),$ (59)

which we represent in Fig. 5(a) and 5(b). We observe that the evolution of
$\lambda(t)$ well discriminates the trivial $J<B$ and nontrivial $J>B$ phases;
however, only near the Hamiltonian limit (in the sense of the Trotter
approximation of the evolution operator) is $\lambda(t)$ smooth, with isolated
singularities (cusps) signaling the presence of a dynamical phase transition
[Fig. 5(a)], similar to the Ising transverse field model [46, 48]. For larger
values of $(J,B)$, it becomes irregular, with an increase in the frequency of
the singularities, and displaying intermittent large fluctuations at long
times (revivals) [73, 60, 61].

We find that, in Fig. 5c, the first peaks of $\mathcal{Q}_{0}$ are well
correlated with the period $t=\pi/2J,3\pi/2J,\ldots\approx 52,157,\ldots$,
predicted by formula (55), valid for vanishing field. Moreover, the long time
behavior of both $\lambda(t)$ and $\mathcal{Q}_{0}(t)$ essentially depends on
the topological phase rather than the actual values of $J$ and $B$.

More precisely, in the small coupling limit $J,B\rightarrow 0$ with $J/B\sim
O(1)$ one can introduce a small parameter $\Delta t=t/n$ with
$n\rightarrow\infty$ such that $J=\bar{J}\Delta t$ and $B=\bar{B}\Delta t$,
where the barred constants are order 1 $\bar{J},\bar{B}\sim O(1)$. As a
consequence, the Floquet evolution (4), $F(t)=F(J,B)^{t}$, can be approximated
by [74]

$F(t)\approx\lim_{\Delta t\rightarrow
0}\left(\mathrm{e}^{-\mathrm{i}\bar{H}_{C}\Delta
t}\mathrm{e}^{-\mathrm{i}\bar{H}_{B}\Delta t}\right)^{\frac{t}{\Delta
t}}=\mathrm{e}^{-\mathrm{i}(\bar{H}_{C}+\bar{H}_{B})t},$ (60)

the evolution operator of the transverse cluster model, where, in the barred
Hamiltonians, we substituted the barred couplings, $(\bar{J},\bar{B})$. The
behavior of $\lambda(t)$ [Fig. 5(a)] is compatible with the known
phenomenology of the model [60]. In contrast, when both couplings are order 1,
the stroboscopic dynamics appears to be irregular [Fig. 5(b), even if the
underlying system is integrable]. Note however that the asymptotic levels of
the Loschmidt rate (as well as the global entanglement) are comparable in the
two regimes.

It is worth noting that the global entanglement given by (51) follows
qualitatively the same pattern as the Loschmidt ratio (58) [Fig. 5(c) and
5(d)]: The $J>B$ case corresponds to a high global entanglement with a maximum
at the first singular point, and the other case ($J<B$) corresponds to a
featureless, low-entanglement evolution. However, the $\mathcal{Q}$ measure
appears to follow a smooth evolution for small $(J,B)$, in contrast to the
appearance of a cusp in $\lambda$ [Fig. 5(a) and 5(c)], showing that the
global entanglement is a poor indicator of the dynamical phase transition.

In conclusion the integrable Floquet cluster model display a dynamical phase
transition between low- and high-entanglement states, extending the phases of
the static Hamiltonian’s ground state to the driven nonequilibrium regime. The
existence of symmetry-protected topological phases in Floquet models is well
documented in integrable or near integrable models, for example the ones
related to quantum cellular automata [23, 24] or nonthermal states in
constrained systems [26, 75, 40]; yet it is of interest to investigate Floquet
nonergodic states in noiseless interacting systems.

##  III Cluster quantum walk

To investigate nonergodic behavior beyond the integrable case, we extend the
Floquet cluster model to consider the interaction of the chain spins with a
moving particle. We introduce then a quantum walk, which in the continuous
limit represents a Dirac particle, coupled with the lattice spins by an
exchange interaction characterized by the parameter $J_{w}$. Related models of
interacting quantum walks were used in the study of thermal relaxation [44,
45] and spin dynamics [76].

The system’s total Hilbert space $\mathcal{H}$ is the tensor product of the
walker and spin chain Hilbert spaces; it is then spanned by the basis vectors

$\ket{xcs}=\ket{x}\otimes\ket{c}\otimes\ket{s}\in\mathcal{H},\quad
x\in\\{0,\ldots,L-1\\},\\\ c\in\\{0,1\\},\;s\in\\{0,\ldots,2^{L}-1\\},$ (61)

where $x$ is the particle position, $c$ is the particle spin, which we call
the coin state (heads or tails) as usual for quantum walks, and $s$ is the
spin configuration ($\ket{s}=\ket{s_{0}\cdots s_{L-1}}$; cf. Sec. II). An
arbitrary state of the interacting quantum walk can be written in the
canonical basis (61),

$\ket{\psi}=\sum_{xcs}\psi_{xcs}\ket{xcs},\quad\sum_{xcs}|\psi_{xcs}|^{2}=1.$
(62)

We choose the particle-spin exchange interaction in the form

$W_{x}(J_{w})=\exp\big{(}\mathrm{i}J_{w}\tau^{(x)}X_{x}\big{)},$ (63)

where $\tau^{(x)}$ is the particle coin operator, the Pauli matrix in the
$x$-direction (it flips the coin and acts as the identity on the position and
spin spaces). This operator acts at each site $x$, on the local coin-spin four
dimensional Hilbert space, spanned by vectors of the form

$\begin{pmatrix}x00_{x}\\\ x01_{x}\\\ x10_{x}\\\ x11_{x}\end{pmatrix},$ (64)

where $0_{x},1_{x}$ denote a spin configuration with a spin up or down at site
$x$, respectively. Note that $W$ acts as the identity on the position Hilbert
subspace. Therefore, $W_{x}(J_{w})$ couples the coin and the local spin
degrees of freedom, entangling the walker with the spin chain.

The motion operator $M$ of the walker is controlled by its coin degree of
freedom $c$. At each time step we modify the coin state applying a rotation of
angle $\theta$:

$R(\theta)=\exp\big{(}-\mathrm{i}\theta\tau^{(y)}\big{)},$ (65)

whatever the particle position ($\tau^{(y)}$ is the $y$ Pauli matrix),
followed by a switch of the particle’s position between neighbors
$x+1\rightarrow x$ if the coin state is heads and $x\rightarrow x+1$ if it is
tails:

$M\ket{x1s}=\ket{x+10s},\quad M\ket{x+11s}=\ket{x0s}.$ (66)

The sequence $MC$, where

$C(\theta)=1_{L}\otimes R(\theta)\otimes 1_{2^{L}},$ (67)

is the usual definition of a quantum walk [43]; with our choice of coin it
belongs to the class of Dirac walks, which tend in the continuous limit to the
Dirac evolution operator [77, 78, 79, 76]. Schematically, the walker’s
behavior in the absence of coupling with the spins ($J_{w}=0$), depends on the
rotation angle; when $\theta\approx\pi/2$ the particle propagates as a chiral
excitation walking to the left (if $c=1$) or to the right (if $c=0$); when
$\theta\approx\pi/4$ the particle propagates ballistically in the two
directions.

Finally, we apply the interaction operators, particle-spin $W$ and spin-spin
$F$. In summary, the one time step operator is

$F_{\text{QW}}=F(J,B)W(J_{w})MC(\theta),$ (68)

The coupling of the particle with the spins ($J_{w}\neq 0$), through $W$ and
$F$, introduces an entanglement mechanism between the two parties. It also
modifies the system’s dispersion properties leading to strong effects on the
walker’s motion. In Appendix B we show the explicit formulas of the different
operators in (68).

It is important to note that the introduction of the coupling (63) preserves
the $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ symmetry of the cluster model, Eq.
(7). However, it deeply modifies the properties of the system, which cannot be
consider any more as “integrable” [80]. One may identify an integrable or
quasi-integrable quantum system by the presence of weakly interacting
quasiparticles, related to the existence of a set of local conservation laws.
The introduction of the walker changes the nature of the degrees of freedom:
local and attached to the lattice in the case of the spins, and spread over
the whole lattice in the case of the particle. (Appendix B shows the complex
structure of the Floquet operator, which cannot be associated with a local
effective Hamiltonian).

The nonlocality of the particle wave function allows the spins to indirectly
interact at large distance [81, 76]. This fact do not forbids the emergence of
dynamical approximated conserved quantities [38, 39], and might facilitate the
creation of long range entanglement through the transfer of information
carried by the walker to separated locations. Qualitatively, using a naive
mean-field reasoning, the spin-particle interaction adds to the applied field
$B$, it might thus suppress its action and restore for example a high
entangled phase even if $J<B$. This is the point we want to study.

In contrast to the effect of an external field, the presence of a self-
consistent spin-particle interaction enlarge the cluster model Hilbert space;
this means that the state of the spin subsystem is generally mixed. The spin
subsystem is then described by the reduced density matrix

$\rho_{s}(t)=\operatorname{tr}_{xc}\rho(t),\;\rho(t)=\ket{\psi(t)}\bra{\psi(t)},$
(69)

where $\ket{\psi(t)}$ is the state that has evolved from the initial
$\ket{\psi(0)}$ state, usually a simple product state,

$\ket{\psi(t)}=F_{\text{QW}}^{t}\ket{\psi(0)}.$ (70)

In (69) we took the partial trace over the particle and coin degrees of
freedom $\\{x,c\\}$ of the total system density matrix $\rho(t)$.

Table 1:  Numerical parameters used in cases (a)-(d) of Fig. 6: size $L=14$, particle-spin coupling $J_{w}=1.6$; $J>B$ [cases (a) and (b)], $J<B$ [cases (c) and (d)]; $\theta\approx\pi/2$ [cases (a) and (c)], $\theta=\pi/4$ [cases (b) and (d)]. Case | $J$ | $B$ | $\theta$ | $\mathcal{Q}_{s}$ | thermal
---|---|---|---|---|---
(a) | 0.03 | 0.01 | 1.6 | high | no
(b) | 0.03 | 0.01 | $\pi/4$ | low | no
(c) | 0.2 | 0.6 | 1.6 | low | no
(d) | 0.2 | 0.6 | $\pi/4$ | high | yes

Motivated by the formula proposed by Peres [82], we introduce a generalization
of the Loschmidt overlap to mixed systems in terms of reduced density
matrices, here the one of the spin subsystem,

$\mathcal{L}_{s}(t)=\frac{\operatorname{tr}\rho_{s}(0)\rho_{s}(t)}{\operatorname{tr}\rho_{s}(0)^{2}},\quad\lambda_{s}(t)=-\frac{1}{L}\ln\mathcal{L}_{s}(t),$
(71)

which, for pure systems and initially mixed systems, reduces to the usual
definitions [82]. Note that this definition does not use the subsystem
evolution operator, which in our case is not necessarily a local Floquet
Hamiltonian: Formula (71) generalizes the overlap of two pure states with the
one built from the forward and backward time evolution of the partial density
matrices of a larger pure system. Alternative definitions do use the subsystem
Hamiltonian to evolve an initial mixed state [83, 84].

Figure 6: Influence of the spin-particle interaction on the dynamical
topological transition. Rows: particle distribution $p$, tangle $\tau$, global
spin entanglement $\mathcal{Q}_{s}(t)$ and magnetization per site
$\braket{X}(t)$ , and spin Loschmidt ratio $\lambda_{s}(t)$; the initial state
is $\ket{L/2,0,+}$. For parameters see Table 1. Columns (a), (b) and (d) show
evolution towards a highly entangled state, while column (c) show a low-
entanglement one. (c) and (d) display a transition between low- and high-
entanglement phases induced by the particle (change in the coin angle), even
if $J<B$.

Entanglement of the spin subsystem can be measured with

$\mathcal{Q}_{s}(t)=\frac{1}{L}\sum_{x=1}^{L}\tau_{s}(x,t),\quad\tau_{s}(x,t)=4\det\rho_{s}(x,t),$
(72)

where $\tau_{s}$ is the so called (one) tangle [85], and

$\rho_{s}(x,t)=\operatorname{tr}_{\bar{s}_{x}}\rho_{s}(t),\quad\bar{s}_{x}=s_{1}\ldots
s_{x-1}s_{x+1}\ldots s_{L}$

is the density matrix of the spin at site $x$, which is also a straightforward
generalization of the pure state case. For a single spin reduced density
matrix, the tangle is useful to visualize the entanglement distribution along
the chain. In addition, the walker space density is characterized by the wave
function amplitudes $\psi_{xcs}(t)=\braket{xcs}{\psi(t)}$:

$p(x,t)=\operatorname{tr}_{cs}\rho(t)=\sum_{cs}|\psi_{xcs}(t)|^{2}.$ (73)

It simply gives the probability of finding the walker at position $x$ and time
$t$, whatever its coin state and surrounding spin configuration. The
magnetization distribution per site is defined by

$\braket{\bm{\sigma}}(x,t)=\operatorname{tr}\rho_{s}(x,t)\bm{\sigma},$ (74)

its mean value over the sites, in contrast to the pure state case, does not
entirely determines the global entanglement, although in our model where the
number of degrees of freedom in the spin subsystem overwhelms the number in
the particle subsystem, its behavior is well correlated with the entanglement
measures. In particular, in a mixed state, the norm of $\braket{\bm{\sigma}}$
is smaller than 1.

We compute the time evolution of an initial state in which all spins are
polarized in the $+$ direction, and the particle is located at $x=L/2$ with a
heads coin ($c=0$)

$\ket{\psi(0)}=\ket{L/2,0}\ket{+}^{L}.$ (75)

In the uncoupled case $J_{w}=0$ this state would evolve into a quantum Dirac
walk for the particle subspace [78, 86] and into the Floquet cluster for the
spin subspace (cf. Sec. II). Instead, when $J_{w}\neq 0$, the different
degrees of freedom, position, coin and spins, get entangled. We compare the
results of four numerical computations labeled as cases (a)-(d) in Fig. 6,
using the parameters of Table 1. The four cases use a strong particle-spin
coupling $J_{w}=1.6\approx\pi/2$, corresponding to the exchange between the
coin and local spin states. In addition, we distinguish the $J>B$ case (a,b),
and the $J<B$ case (c,d). Finally, for the same values of $(J,B)$, we compare
the weak dispersive case (a,c), $\theta\approx\pi/2$, with the strong
dispersive one (b,d), $\theta=\pi/4$.

Case (a) keeps similarities with the integrable case shown in Fig. 5(a) and
5(c), in which a dynamical phase transition appeared. The coupling with the
walker results in an increase in the entanglement after the transition, as can
be inferred form the behavior of $\mathcal{Q}_{s}(t)$ and $\lambda_{s}(t)$
[Fig. 6(a), rows 3 and 4]. One remarkable effect of the particle-spin
interaction is the localization of the walker state around its initial
position [Fig. 6(a)], row 1). This effect contrasts with the fast dispersion
of the $\theta=\pi/4$ case (b). The asymptotic state reached in both case (a)
and case (b), which only differ in the particle dispersion, possesses
essentially identical entanglement and magnetization properties. However, the
transition dynamics between the initial low- entanglement state and the final
highly entangled state is singular in the case where the walker is localized
and smooth when the particle-spin interaction is important. It is worth noting
that in case (a), the initial growth of the spin entanglement is faster in the
region where the particle density is low, and is depleted in the central
region as measured by $\tau_{s}(x,t)$ [Fig. 6(a), row 2].

The effect of the particle on the propagation of the spin’s entanglement is
also present in cases (c) and (d). In case (c) the walker is localized and the
spin entanglement remains inhomogeneous for long times, while its mean value
$\mathcal{Q}_{s}$ rapidly reaches its saturation value at a low level. The
persistence of inhomogeneities in both spin entanglement and magnetization
$\braket{X}$, is typical of nonergodic chaotic states: The evolution of the
system and the long time stationary state depend on the initial configuration.
Case (d), in which the walker spreads over the whole space, shows entanglement
growth and homogenization, with a very small value of the Loschmidt ratio,
indicating that the system evolved into a chaotic (thermal) state, in which
the magnetization tends to zero after an exponential relaxation. In this $J<B$
case the transition between the low-entanglement state (c) and high
entanglement state (d) is driven by the particle, and controlled by $\theta$.

A significant difference exists between the integrable case and the
interacting quantum walk case. For $J_{w}=0$ the topological phase is
magnetically disordered with a vanishing mean magnetization
$\braket{\bm{\sigma}_{x}}$; for $J_{w}\neq 0$, the topological phase coexists
with a magnetic order, as can be verified from the results of Fig. 6, where we
plot $\braket{X_{x}}(t)$ [cf. Eq. (74), where $\rho_{s}$ replaces
$\ket{\psi}$]. In the integrable case $Q_{s}=1$ implies $\braket{X}=0$, but in
the interacting case the spin subsystem can be maximally entangled even in the
presence of a finite value of the magnetization, due to the interaction with
the particle [Fig. 6(a) and 6(b)]. We further discuss the magnetic order in
Appendix C.

In summary (see Table 1 last two columns), case (a) shows a dynamical
entanglement transition reminiscent to the one present in the integrable case
for $J>B$; case (b) shows a smooth evolution of the entanglement towards a
nonthermal high entangled state; cases (c) and (d) illustrate the transition
between a nonthermal chaotic regime and a thermal phase, respectively, induced
by the interaction with the particle, and controlled by the quantum walk
parameter $\theta$.

##  IV Conclusion

We investigated the entanglement properties of the Floquet cluster spin chain
coupled with a particle via an exchange interaction. The system is invariant
with respect to a global $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ symmetry.
Already in the integrable case, when the coupling vanishes, the periodically
driven spin chain exhibits phase transitions between a low-entanglement phase
and a topological highly entangled phase, as demonstrated by explicit analytic
computations.

More interestingly, we found that the (interacting) combination of the chiral
particle motion with the spin chain leads to nonthermal states and dynamical
phase transitions between low and high entanglement regimes. In fact, not only
can the particle inhibit the topological phase, but also it can be localized
by its interaction with the spins; in this case a dynamical phase transition
can arise, allowing the initial product state to evolve into a strongly
entangled one.

We identified different regimes, some of them extending the properties of the
topological ordered cluster phase of the integrable model; however, there is a
difference: the emergence of an approximately conserved magnetization. The
transition between the initial product state and the high, albeit nonthermal,
entangled state could be reached through a dynamical phase transition
reminiscent of the one present in the integrable case, or directly following a
path of nonexponential relaxation. Exponential relaxation to a paramagnetic
regime was also observed, typical of the infinite-temperature phase.

We note that the magnetization vanishes in this paramagnetic regime, even if
the applied field is nonzero, and that in the nonthermal phases the presence
of a long-time finite magnetization generally correlates with the enhancement
of the entanglement with respect to the integrable case. These effects are
related to the complex magnetic interactions mediated by the particle
scattering off the fixed spins, which can generate effective magnetic (Ising)
spin-spin couplings. These magnetic interactions compete with the external
field, modifying the original regimes of the effective noninteracting case,
and allowing the emergence of other regimes, notably affecting the
entanglement dynamics.

In our model the driven frequency is fixed (the unit of time), however it
could be of interest to study the behavior of the system as a function of the
driving strength, in particular the robustness of the emerging conserved law
and the corresponding dynamical phases [31, 87, 30]. Another interesting
generalization would be to consider an arbitrary graph of spins, to probe the
topological properties of the corresponding cluster state together with those
of the underlying graph topology.

The experimental realization of an interacting quantum walk of the type
defined by (68) is certainly challenging. However, the basic ingredients (the
cluster coupling and the particle walk), can probably be implemented in
lattices of Rydberg excited atoms and combined to obtain an effective system
approaching $F_{\text{QW}}$. Cluster states were recently created using highly
selective interactions of atoms in a Rydberg array [88]. Moreover, it was
proposed that with a similar experimental setup, it is possible to simulate a
discrete time quantum walk possessing nontrivial topological properties [89].

In conclusion, ergodicity can then be broken in a periodically driven system
by the interplay of qualitatively different interacting degrees of freedom,
blocking the evolution towards an infinite temperature state.

##  Appendix A Eigenvectors and evolution

In this appendix we give some details about the calculations of Sec. II. In
terms of the fermion operators $f_{k}$, the Fourier components of the
Hamiltonian (24) can be written as

$H_{k}=\epsilon_{k}\big{[}n_{z}(f^{\dagger}_{k}f_{k}-f_{-k}f^{\dagger}_{-k})\\\
+n_{-}f^{\dagger}_{k}f^{\dagger}_{-k}+n_{+}f_{-k}f_{k}\big{]},$ (76)

where $n_{\pm}=n_{x}\pm\mathrm{i}n_{y}$ and $n_{z}$ are given by (27). The
eigenvectors (42) are obtained from the following identities:

$\displaystyle
H_{k}\ket{0}=\epsilon_{k}\big{(}-n_{z}\ket{0}+n_{-}f^{\dagger}_{k}f^{\dagger}_{-k}\big{)},$
(77) $\displaystyle H_{k}f^{\dagger}_{\pm k}\ket{0}=0,$ (78) $\displaystyle
H_{k}f^{\dagger}_{k}f^{\dagger}_{-k}\ket{0}=\epsilon_{k}\big{(}n_{z}f^{\dagger}_{-k}\ket{0}\ket{0}+n_{+}\ket{0}\big{)}.$
(79)

For example, we have

$\begin{split}H_{k}&\left(n_{+}\ket{0}-(1-n_{z})f^{\dagger}_{k}f^{\dagger}_{-k}\ket{0}\right)\\\
&=\epsilon_{k}\left[-n_{z}n_{+}+n_{-}n_{+}f^{\dagger}_{k}f^{\dagger}_{-k}-(1-n_{z})n_{z}f^{\dagger}_{k}f^{\dagger}_{-k}-(1-n_{z})n_{+}\right]\ket{0}\\\
&=\epsilon_{k}\left[-n_{+}+(n_{x}^{2}+n_{y}^{2}-n_{z}+n_{z}^{2})f^{\dagger}_{k}f^{\dagger}_{-k}\right]\ket{0},\end{split}$
(80)

or

$H_{k}\left(n_{+}-(1-n_{z})f^{\dagger}_{k}f^{\dagger}_{-k}\right)\ket{0}=-\epsilon_{k}\left(n_{+}-(1-n_{z})f^{\dagger}_{k}f^{\dagger}_{-k}\right)\ket{0},$
(81)

which leads to the eigenvector $\ket{-kk}$.

The evolution of the vacuum state is

$\ket{t}=F(t)\ket{0}=\prod_{k}F_{k}(t)\ket{0}.$ (82)

We verify that $\ket{0}$ can be written as a superposition,

$\sqrt{\frac{1-n_{z}}{2}}\ket{kk}+\frac{n_{-}}{\sqrt{2(1-n_{z})}}\ket{-kk}=\left(\frac{1-n_{z}}{2}+\frac{1+n_{z}}{2}\right)\ket{0}=\ket{0}$
(83)

of the $\pm\epsilon_{k}$ eigenvectors. Therefore,

$\begin{split}F_{k}(t)\ket{0}&=\mathrm{e}^{\mathrm{i}\epsilon_{k}t}\sqrt{\frac{1-n_{z}}{2}}\ket{kk}+\mathrm{e}^{-\mathrm{i}\epsilon_{k}t}\frac{n_{-}}{\sqrt{2(1-n_{z})}}\ket{-kk}\\\
&=\mathrm{e}^{\mathrm{i}\epsilon_{k}t}\left(\frac{1-n_{z}}{2}+\frac{n_{-}}{2}f^{\dagger}_{-k}f^{\dagger}_{k}\right)\ket{0}+\mathrm{e}^{-\mathrm{i}\epsilon_{k}t}\left(\frac{1+n_{z}}{2}-\frac{n_{-}}{2}f^{\dagger}_{-k}f^{\dagger}_{k}\right)\ket{0}\\\
&=\left[\cos(\epsilon_{k}t)-\mathrm{i}n_{z}\sin(\epsilon_{k}t)+\mathrm{i}n_{-}\sin(\epsilon_{k}t)f^{\dagger}_{-k}f^{\dagger}_{k}\right]\ket{0}\end{split}$
(84)

which leads to (46). Knowing the explicit expression of $\ket{t}$ one deduces
straightforwardly $Q_{0}(t)$ from its definition (51), as well as $\lambda(t)$
from (57) and (59).

A useful formula, used to compute the global entanglement (51), is the
expected value:

$\begin{split}\braket{t}{f_{k}^{\dagger}f_{k}}{t}&=\bra{0}(c+\mathrm{i}n_{z}s-\mathrm{i}n_{+}sf_{-k}f_{k})f^{\dagger}_{k}f_{k}(c-\mathrm{i}n_{z}s+\mathrm{i}n_{-}sf_{k}^{\dagger}f_{-k}^{\dagger})\ket{0}\\\
&=(1-n_{z})^{2}\sin^{2}(\epsilon_{k}t).\end{split}$ (85)

where $s=\sin(\epsilon_{k}t)$ and $c=\cos(\epsilon_{k}t)$. In the previous
equation we used the fact that

$f_{k}\ket{t}=\mathrm{i}n_{-}\sin(\epsilon_{k}t)f^{\dagger}_{-k}\ket{0}.$ (86)

Moreover, the expected value of the spin $X_{x}$ does not depend on $x$ in the
state $\ket{t}$:

$\braket{t}{X_{x}}{t}=1-\frac{2}{L}\sum_{k_{1},k_{2}}\mathrm{e}^{\mathrm{i}(k_{1}-k_{2})x}\braket{t}{f^{\dagger}_{k_{2}}f_{k_{1}}}{t},$
(87)

where we used the Fourier transform (13), because the expected value of the
fermion product vanishes for $k_{1}\neq k_{2}$, leading to the simple relation

$\braket{t}{X_{x}}{t}=1-\frac{2}{L}\sum_{k}\braket{t}{f^{\dagger}_{k}f_{k}}{t}.$
(88)

This formula together with (85) allow us to obtain the expression of
$\mathcal{Q}_{0}$ (51).

##  Appendix B Interacting quantum walk operator

We show the explicit form of the operators $MC(\theta)$, $F(J,B)$, and
$W(j_{w})$ which define the interacting quantum walk evolution operator
$F_{\text{QW}}$, (68).

The combined motion and coin operators governing the particle quantum walk can
be written as

$MC(\theta)=\sum_{x}\left[\ket{x+1}\bra{x}\otimes\ket{0}\bra{1}R(\theta)\right.\\\
+\left.\ket{x}\bra{x+1}\otimes\ket{1}\bra{0}R(\theta)\right]\otimes
1_{2^{L}}.$ (89)

It acts trivially on the spin Hilbert subspace, and exchange the amplitudes of
neighboring sites according to their coin state; the coin state is rotated by
an angle $\theta$.

The cluster Floquet operator (4) is readily extended to the whole Hilbert
space:

$F(J,B)\rightarrow 1_{L}\otimes 1_{2}\otimes F(J,B).$ (90)

Finally, the coin-spin interaction is given by

$W(J_{w})=1_{L}\otimes\prod_{x}\left[\cos J_{w}1_{2}\otimes
1_{2^{L}}+\mathrm{i}\sin J_{w}\tau^{(x)}\otimes X_{x}\right],$ (91)

where the operator

$X_{x}=1_{2^{x-1}}\otimes X\otimes 1_{2^{L-x}},$ (92)

belongs to the spin Hilbert subspace and $\tau^{(x)}$ belongs to the coin
subspace. It is worth noting that the global structure of $F_{\text{QW}}$, in
particular due to the sum over $x$ in (89), cannot be reduced to a tensor
product of local unitary operators, as is usual for near-neighbor interacting
Hamiltonian systems (as is indeed the case for the driven cluster model).
Indeed, if in principle it is possible to embed the quantum walk into a local
quantum automaton, it can be done, using a Fourier space representation of the
motion operator, at the cost of an exponentially large internal particle space
[90]; however, in our case, the interaction of the walker, this construction
would give a highly nonlocal spin Floquet operator.

Figure 7: Negativity between two 3-site connected and disconnected sets of
spins. Parameters are as in Fig. 6 (vertical lines at $\theta=\pi/4,1.6$): (a)
$J=0.03$, $B=0.01$; (b) $J=0.2$, $B=0.6$; (a) and (b) $J_{w}=1.6$, $L=12$.

##  Appendix C Magnetic order

We showed in the main text (Sec. III) that a transition was possible even in
the $J<B$ case, between different entanglement regimes, characterized by low-
entanglement non ergodic and highly entangled thermal states, whereas in the
three-spin interaction dominant case $J>B$ we observed two types of relaxation
toward a highly entangled state, with and without dynamical phase transition.
We complement the characterization of these regimes with the computation of
the entanglement negativity:

$\mathcal{N}_{s}(A)=\ln|\rho_{s}^{T_{B}}|,\quad|\rho_{s}^{T_{B}}|=1+2\sum_{n}|\lambda_{n}|$
(93)

where AB is a bipartition of the spin subsystem and $T_{B}$ denotes the
partial transpose over B; the norm of the density matrix is computed from its
negative eigenvalues $\lambda_{n}<0$ [91]. The negativity is a measure of the
entanglement of mixed states, here the spin subsystem, sensitive to the range
of entanglement.

Figure 8: Entanglement in the magnetic ordered phase. (a) Subsystem spin $s$,
position $x$, and coin $c$ von Neumann entropies; (b) Loschmidt rate; (c)
half-chain entanglement entropy normalized to its maximum value
$\mathcal{S}_{0}=L/2$; and (d) global entanglement. Parameters: $L=12$,
$J=0.2$, $B=0.6$, $J_{w}=0.5$, and $\theta=\pi/4$ [compare with the parameters
of Fig. 6(d)].

We computed $\mathcal{N}_{s}$ as a function the rotation angle $\theta$, using
a connected A set of six spins and a disconnected $3+3$ set ($L=12$), for the
two regimes of Fig. 6, $J>B$ and $J<B$. The result is shown in Fig, 7. For
$J>B$ the system remains in its high entanglement phase, while for $J<B$ it
displays a transition for values of $\theta\sim 1.3$ (only the qualitative
behavior can be inferred from such small system sizes, here $L=12$). A slight
difference between the connected and disconnected sets is observed for low
entanglement, the disconnected set entanglement being slightly larger than the
connected set one.

The parameter $J_{w}$ controls the strength of the particle-spin coupling, by
tuning its value a near-adiabatic regime in which the fixed spins follow the
particle dynamics can be set in. It differs from the topological and
paramagnetic phases by its magnetic order. Even in the case where $J=0$, a
two-spin interaction can be mediated by the successive particle scattering off
the fixed spins, in much the same way as the Ruderman–Kittel–Kasuya–Yosida
(RKKY) interaction of magnetic impurities in a metal([92]. In this case the
orientation of the spins is essentially determined by their indirect
interaction through the particle’s coin degree of freedom.

Indeed, a simple perturbation expansion argument, in analogy with the above
mentioned RKKY interaction [93], show that the effective spin Hamiltonian
should contain a long range $XX$ coupling. This coupling is at the origin of
the magnetic phase we illustrate in Fig. 8, which is qualitatively similar to
the one in Fig. 6(c). To characterize this regime we measured the von Neumann
entropy

$\mathcal{S}_{l}(t)=\operatorname{tr}_{\bar{l}}\ket{\psi(t)}\bra{\psi(t)},\quad
l=\\{x,c,s\\},$ (94)

of the $x,c,s$ subsystems, as well as the half-chain entanglement
$\mathcal{S}(t)$, the global spin entanglement $\mathcal{Q}_{s}(t)$, and the
expected value of the magnetization vector $\braket{\bm{\sigma}_{x}}$,
averaged over $x$.

Figure 9: Distribution of the one site magnetization, showing the persistent
spatio-temporal fluctuations. Parameters are the sames as in Fig. 8.

We observe that the spin entanglement follows the particle, as measured by the
von Neumann entropy [Fig. 8(a)]; the Loschmidt ratio is small, characteristic
of a chaotic state [Fig. 8(b)]; and the half-chain entropy and the global
entanglement [Fig. 8(c) and 8(d)] saturate at levels well below their maximum
values, following the same pattern as the von Neumann entropy, and they also
are well correlated with the stochastic variations of the Loschmidt ratio. One
important point here is that the stationary state possesses a finite
magnetization, signaling a magnetic order (Fig. 9). This asymptotic
magnetization appears as an emergent conserved quantity [38]. We deduce that
the magnetic interaction between the chain spins mediated by the walker, in a
near adiabatic regime, can establish a high entanglement regime with magnetic
order, counterbalancing the paramagnetic effect of the external field ($B>J$).

In conclusion, the magnetic order can be attributed to the spin-spin
interaction mediated by the walker. It emerges for a range of parameters
characterizing the walker motion and coupling with the spins, and its
entanglement behavior essentially differs with the paramagnetic
(noninteracting) phase.

## References

  * Doherty and Bartlett [2009] A. C. Doherty and S. D. Bartlett, Identifying Phases of Quantum Many-Body Systems That Are Universal for Quantum Computation, Phys. Rev. Lett. 103, 020506 (2009).
  * Stephen  _et al._ [2019] D. T. Stephen, H. P. Nautrup, J. Bermejo-Vega, J. Eisert, and R. Raussendorf, Subsystem symmetries, quantum cellular automata, and computational phases of quantum matter, Quantum 3, 142 (2019).
  * Shor [1996] P. W. Shor, Fault-tolerant quantum computation, in  _Proc. 37th Conf. Found. Comput. Sci._ (IEEE, 1996) pp. 56–65.
  * Preskill [1998] J. Preskill, Fault-tolerant quantum computation, in  _Introduction to Quantum Computation and Information_ (World Scientific, Singapore, 1998) pp. 213–269.
  * Gottesman [1997] D. Gottesman, Stabilizer Codes and Quantum Error Correction, ArXivquant-Ph9705052 (1997), arXiv:quant-ph/9705052 .
  * Kitaev [2003] A. Y. Kitaev, Fault-tolerant quantum computation by anyons, Ann. Phys. 303, 2 (2003).
  * Briegel and Raussendorf [2001] H. J. Briegel and R. Raussendorf, Persistent Entanglement in Arrays of Interacting Particles, Phys. Rev. Lett. 86, 910 (2001).
  * Freedman  _et al._ [2003] M. Freedman, A. Kitaev, M. Larsen, and Z. Wang, Topological quantum computation, Bull. Amer. Math. Soc. 40, 31 (2003).
  * Raussendorf and Briegel [2001] R. Raussendorf and H. J. Briegel, A One-Way Quantum Computer, Phys. Rev. Lett. 86, 5188 (2001).
  * Dennis  _et al._ [2002] E. Dennis, A. Kitaev, A. Landahl, and J. Preskill, Topological quantum memory, Journal of Mathematical Physics 43, 4452 (2002).
  * Brown  _et al._ [2016] B. J. Brown, D. Loss, J. K. Pachos, C. N. Self, and J. R. Wootton, Quantum memories at finite temperature, Rev. Mod. Phys. 88, 045005 (2016).
  * Roberts [2019] S. Roberts,  _Symmetry-Protected Topological Phases for Robust Quantum Computation_ , Thesis (2019).
  * Wildeboer  _et al._ [2022] J. Wildeboer, T. Iadecola, and D. J. Williamson, Symmetry-Protected Infinite-Temperature Quantum Memory from Subsystem Codes, PRX Quantum 3, 020330 (2022).
  * D’Alessio  _et al._ [2016] L. D’Alessio, Y. Kafri, A. Polkovnikov, and M. Rigol, From quantum chaos and eigenstate thermalization to statistical mechanics and thermodynamics, Adv. Phys. 65, 239 (2016).
  * Gross  _et al._ [2009] D. Gross, S. T. Flammia, and J. Eisert, Most Quantum States Are Too Entangled To Be Useful As Computational Resources, Phys. Rev. Lett. 102, 190501 (2009).
  * D’Alessio and Rigol [2014] L. D’Alessio and M. Rigol, Long-time Behavior of Isolated Periodically Driven Interacting Lattice Systems, Phys. Rev. X 4, 041048 (2014).
  * Lazarides  _et al._ [2014] A. Lazarides, A. Das, and R. Moessner, Equilibrium states of generic quantum systems subject to periodic driving, Phys. Rev. E 90, 012110 (2014).
  * Chamon [2005] C. Chamon, Quantum Glassiness in Strongly Correlated Clean Systems: An Example of Topological Overprotection, Phys. Rev. Lett. 94, 040402 (2005).
  * Sala  _et al._ [2020] P. Sala, T. Rakovszky, R. Verresen, M. Knap, and F. Pollmann, Ergodicity Breaking Arising from Hilbert Space Fragmentation in Dipole-Conserving Hamiltonians, Phys. Rev. X 10, 011047 (2020).
  * Scherg  _et al._ [2021] S. Scherg, T. Kohlert, P. Sala, F. Pollmann, B. Hebbe Madhusudhana, I. Bloch, and M. Aidelsburger, Observing non-ergodicity due to kinetic constraints in tilted Fermi-Hubbard chains, Nat Commun 12, 4490 (2021).
  * Shiraishi and Mori [2017] N. Shiraishi and T. Mori, Systematic Construction of Counterexamples to the Eigenstate Thermalization Hypothesis, Phys. Rev. Lett. 119, 030601 (2017).
  * Papić [2022] Z. Papić, Weak Ergodicity Breaking Through the Lens of Quantum Entanglement, in  _Entanglement in Spin Chains: From Theory to Quantum Technology Applications_, Quantum Science and Technology, edited by A. Bayat, S. Bose, and H. Johannesson (Springer, Cham, Switzerland, 2022) pp. 341–395.
  * Gopalakrishnan and Zakirov [2018] S. Gopalakrishnan and B. Zakirov, Facilitated quantum cellular automata as simple models with non-thermal eigenstates and dynamics, Quantum Sci. Technol. 3, 044004 (2018).
  * Friedman  _et al._ [2019] A. J. Friedman, S. Gopalakrishnan, and R. Vasseur, Integrable Many-Body Quantum Floquet-Thouless Pumps, Phys. Rev. Lett. 123, 170603 (2019).
  * Bernien  _et al._ [2017] H. Bernien, S. Schwartz, A. Keesling, H. Levine, A. Omran, H. Pichler, S. Choi, A. S. Zibrov, M. Endres, M. Greiner, V. Vuletić, and M. D. Lukin, Probing many-body dynamics on a 51-atom quantum simulator, Nature 551, 579 (2017).
  * Iadecola and Vijay [2020] T. Iadecola and S. Vijay, Nonergodic quantum dynamics from deformations of classical cellular automata, Phys. Rev. B 102, 180302 (2020).
  * Mizuta  _et al._ [2020] K. Mizuta, K. Takasan, and N. Kawakami, Exact Floquet quantum many-body scars under Rydberg blockade, Phys. Rev. Research 2, 033284 (2020).
  * Sugiura  _et al._ [2021] S. Sugiura, T. Kuwahara, and K. Saito, Many-body scar state intrinsic to periodically driven system, Phys. Rev. Research 3, L012010 (2021).
  * Das [2010] A. Das, Exotic freezing of response in a quantum many-body system, Phys. Rev. B 82, 172402 (2010).
  * Haldar and Das [2022] A. Haldar and A. Das, Statistical mechanics of Floquet quantum matter: Exact and emergent conservation laws, J. Phys.: Condens. Matter 34, 234001 (2022).
  * Oka and Kitamura [2019] T. Oka and S. Kitamura, Floquet Engineering of Quantum Materials, Annu. Rev. Condens. Matter Phys. 10, 387 (2019).
  * Harper  _et al._ [2020] F. Harper, R. Roy, M. S. Rudner, and S. Sondhi, Topology and Broken Symmetry in Floquet Systems, Annu. Rev. Condens. Matter Phys. 11, 345 (2020).
  * Yates  _et al._ [2022] D. J. Yates, A. G. Abanov, and A. Mitra, Long-lived period-doubled edge modes of interacting and disorder-free Floquet spin chains, Commun Phys 5, 43 (2022).
  * Wybo  _et al._ [2021] E. Wybo, F. Pollmann, S. L. Sondhi, and Y. You, Visualizing quasiparticles from quantum entanglement for general one-dimensional phases, Phys. Rev. B 103, 115120 (2021).
  * Zeng  _et al._ [2019] B. Zeng, X. Chen, D.-L. Zhou, and X.-G. Wen,  _Quantum Information Meets Quantum Matter_ (Springer, New York, NY, 2019) arXiv:1508.02595 .
  * Nielsen [2006] M. A. Nielsen, Cluster-state quantum computation, Rep. Math. Phys. 57, 147 (2006).
  * Raussendorf and Wei [2012] R. Raussendorf and T.-C. Wei, Quantum Computation by Local Measurement, Annu. Rev. Condens. Matter Phys. 3, 239 (2012).
  * Haldar  _et al._ [2018] A. Haldar, R. Moessner, and A. Das, Onset of Floquet thermalization, Phys. Rev. B 97, 245122 (2018).
  * Haldar  _et al._ [2021] A. Haldar, D. Sen, R. Moessner, and A. Das, Dynamical Freezing and Scar Points in Strongly Driven Floquet Matter: Resonance vs Emergent Conservation Laws, Phys. Rev. X 11, 021008 (2021).
  * Sellapillay  _et al._ [2022a] K. Sellapillay, A. D. Verga, and G. Di Molfetta, Entanglement dynamics and ergodicity breaking in a quantum cellular automaton, Phys. Rev. B 106, 104309 (2022a).
  * Grossmann  _et al._ [1991] F. Grossmann, T. Dittrich, P. Jung, and P. Hänggi, Coherent destruction of tunneling, Phys. Rev. Lett. 67, 516 (1991).
  * Son  _et al._ [2011] W. Son, L. Amico, R. Fazio, A. Hamma, S. Pascazio, and V. Vedral, Quantum phase transition between cluster and antiferromagnetic states, EPL 95, 50001 (2011).
  * Kitagawa [2012] T. Kitagawa, Topological phenomena in quantum walks: Elementary introduction to the physics of topological phases, Quantum Inf. Process. 11, 1107 (2012).
  * Verga [2019] A. D. Verga, Interacting quantum walk on a graph, Phys. Rev. E 99, 012127 (2019).
  * Verga and Elías [2019] A. D. Verga and R. G. Elías, Thermal state entanglement entropy on a quantum graph, Phys. Rev. E 100, 062137 (2019).
  * Heyl [2015] M. Heyl, Scaling and Universality at Dynamical Quantum Phase Transitions, Phys. Rev. Lett. 115, 140602 (2015).
  * Heyl  _et al._ [2013] M. Heyl, A. Polkovnikov, and S. Kehrein, Dynamical Quantum Phase Transitions in the Transverse-Field Ising Model, Phys. Rev. Lett. 110, 135704 (2013).
  * Heyl [2018] M. Heyl, Dynamical quantum phase transitions: A review, Rep. Prog. Phys. 81, 054001 (2018).
  * De Nicola  _et al._ [2021] S. De Nicola, A. A. Michailidis, and M. Serbyn, Entanglement View of Dynamical Quantum Phase Transitions, Phys. Rev. Lett. 126, 040602 (2021).
  * Jafari and Akbari [2021] R. Jafari and A. Akbari, Floquet dynamical phase transition and entanglement spectrum, Phys. Rev. A 103, 012204 (2021).
  * Sellapillay  _et al._ [2022b] K. Sellapillay, P. Arrighi, and G. Di Molfetta, A discrete relativistic spacetime formalism for 1 + 1-QED with continuum limits, Sci Rep 12, 2198 (2022b).
  * Raussendorf  _et al._ [2005] R. Raussendorf, S. Bravyi, and J. Harrington, Long-range quantum entanglement in noisy cluster states, Phys. Rev. A 71, 062313 (2005).
  * Hein  _et al._ [2006] M. Hein, W. Dür, J. Eisert, R. Raussendorf, M. Nest, and H.-J. Briegel, Entanglement in graph states and its applications, ArXiv Prepr. 10.48550/arXiv.quant-ph/0602096 (2006).
  * Vidal [2003] G. Vidal, Efficient Classical Simulation of Slightly Entangled Quantum Computations, Phys. Rev. Lett. 91, 147902 (2003).
  * Verresen  _et al._ [2017] R. Verresen, R. Moessner, and F. Pollmann, One-dimensional symmetry protected topological phases and their transitions, Phys. Rev. B 96, 165124 (2017).
  * Suzuki [1971] M. Suzuki, Relationship among Exactly Soluble Models of Critical Phenomena. I: 2D Ising Model, Dimer Problem and the Generalized XY-Model, Prog. Theor. Phys. 46, 1337 (1971).
  * Pachos and Plenio [2004] J. K. Pachos and M. B. Plenio, Three-Spin Interactions in Optical Lattices and Criticality in Cluster Hamiltonians, Phys. Rev. Lett. 93, 056402 (2004).
  * Skrøvseth and Bartlett [2009] S. O. Skrøvseth and S. D. Bartlett, Phase transitions and localizable entanglement in cluster-state spin chains with Ising couplings and local fields, Phys. Rev. A 80, 022316 (2009).
  * Smacchia  _et al._ [2011] P. Smacchia, L. Amico, P. Facchi, R. Fazio, G. Florio, S. Pascazio, and V. Vedral, Statistical mechanics of the cluster Ising model, Phys. Rev. A 84, 022304 (2011).
  * Montes and Hamma [2012] S. Montes and A. Hamma, Phase diagram and quench dynamics of the cluster-XY spin chain, Phys. Rev. E 86, 021101 (2012).
  * Lakshminarayan and Subrahmanyam [2005] A. Lakshminarayan and V. Subrahmanyam, Multipartite entanglement in a one-dimensional time-dependent Ising model, Phys. Rev. A 71, 062334 (2005).
  * Jordan and Wigner [1928] P. Jordan and E. Wigner, Über das Paulische Äquivalenzverbot, Z. Physik 47, 631 (1928).
  * Mbeng  _et al._ [2020] G. B. Mbeng, A. Russomanno, and G. E. Santoro, The quantum Ising chain for beginners, arXiv:2009.09208 10.48550/arXiv.2009.09208 (2020), arXiv:2009.09208 .
  * Asbóth  _et al._ [2016] J. K. Asbóth, L. Oroszlány, and A. Pályi,  _A Short Course on Topological Insulators_, Lecture Notes in Physics, Vol. 919 (Springer, 2016).
  * Roy and Harper [2017] R. Roy and F. Harper, Periodic table for Floquet topological insulators, Phys. Rev. B 96, 155118 (2017).
  * Asbóth and Obuse [2013] J. K. Asbóth and H. Obuse, Bulk-boundary correspondence for chiral symmetric quantum walks, Phys. Rev. B 88, 121406 (2013).
  * Zhou and Gong [2018] L. Zhou and J. Gong, Floquet topological phases in a spin-1/2 double kicked rotor, Phys. Rev. A 97, 063603 (2018).
  * Ohta  _et al._ [2016] T. Ohta, S. Tanaka, I. Danshita, and K. Totsuka, Topological and dynamical properties of a generalized cluster model in one dimension, Phys. Rev. B 93, 165423 (2016).
  * Brennen [2003] G. K. Brennen, An observable measure of entanglement for pure states of multi-qubit systems, Quantum Info. Comput. 3, 619 (2003).
  * Meyer and Wallach [2002] D. A. Meyer and N. R. Wallach, Global entanglement in multiparticle systems, J. Math. Phys. 43, 4273 (2002).
  * Giraud  _et al._ [2007] O. Giraud, J. Martin, and B. Georgeot, Entanglement of localized states, Phys. Rev. A 76, 042333 (2007).
  * Radgohar and Montakhab [2018] R. Radgohar and A. Montakhab, Global entanglement and quantum phase transitions in the transverse XY Heisenberg chain, Phys. Rev. B 97, 024434 (2018).
  * Häppölä  _et al._ [2012] J. Häppölä, G. B. Halász, and A. Hamma, Universality and robustness of revivals in the transverse field XY model, Phys. Rev. A 85, 032114 (2012).
  * Vanicat  _et al._ [2018] M. Vanicat, L. Zadnik, and T. Prosen, Integrable Trotterization: Local Conservation Laws and Boundary Driving, Phys. Rev. Lett. 121, 030606 (2018).
  * Pai and Pretko [2019] S. Pai and M. Pretko, Dynamical Scar States in Driven Fracton Systems, Phys. Rev. Lett. 123, 136401 (2019).
  * Sellapillay and Verga [2021] K. Sellapillay and A. D. Verga, Quantum walk on a graph of spins: Magnetism and entanglement, Phys. Rev. E 103, 032123 (2021).
  * Meyer [1996] D. A. Meyer, From quantum cellular automata to quantum lattice gases, J. Stat. Phys. 85, 551 (1996).
  * Strauch [2006] F. W. Strauch, Relativistic quantum walks, Phys. Rev. A 73, 054302 (2006).
  * Di Molfetta and Debbasch [2012] G. Di Molfetta and F. Debbasch, Discrete-time quantum walks: Continuous limit and symmetries, J. Math. Phys. 53, 123302 (2012).
  * Caux and Mossel [2011] J.-S. Caux and J. Mossel, Remarks on the notion of quantum integrability, J. Stat. Mech. 2011, P02023 (2011).
  * Klinovaja  _et al._ [2013] J. Klinovaja, P. Stano, A. Yazdani, and D. Loss, Topological Superconductivity and Majorana Fermions in RKKY Systems, Phys. Rev. Lett. 111, 186805 (2013).
  * Peres [1984] A. Peres, Stability of quantum motion in chaotic and regular systems, Phys. Rev. A 30, 1610 (1984).
  * Heyl and Budich [2017] M. Heyl and J. C. Budich, Dynamical topological quantum phase transitions for mixed states, Phys. Rev. B 96, 180304 (2017).
  * Bhattacharya  _et al._ [2017] U. Bhattacharya, S. Bandyopadhyay, and A. Dutta, Mixed state dynamical quantum phase transitions, Phys. Rev. B 96, 180303 (2017).
  * Coffman  _et al._ [2000] V. Coffman, J. Kundu, and W. K. Wootters, Distributed entanglement, Phys. Rev. A 61, 052306 (2000).
  * Asbóth [2012] J. K. Asbóth, Symmetries, topological phases, and bound states in the one-dimensional quantum walk, Phys. Rev. B 86, 195414 (2012).
  * Bukov  _et al._ [2015] M. Bukov, L. D’Alessio, and A. Polkovnikov, Universal high-frequency behavior of periodically driven systems: From dynamical stabilization to Floquet engineering, Adv. Phys. 64, 139 (2015).
  * Hollerith  _et al._ [2022] S. Hollerith, K. Srakaew, D. Wei, A. Rubio-Abadal, D. Adler, P. Weckesser, A. Kruckenhauser, V. Walther, R. van Bijnen, J. Rui, C. Gross, I. Bloch, and J. Zeiher, Realizing Distance-Selective Interactions in a Rydberg-Dressed Atom Array, Phys. Rev. Lett. 128, 113602 (2022).
  * Khazali [2022] M. Khazali, Discrete-Time Quantum-Walk & Floquet Topological Insulators via Distance-Selective Rydberg-Interaction, Quantum 6, 664 (2022).
  * Brun and Mlodinow [2020] T. A. Brun and L. Mlodinow, Quantum cellular automata and quantum field theory in two spatial dimensions, Phys. Rev. A 102, 062222 (2020).
  * Lee and Vidal [2013] Y. A. Lee and G. Vidal, Entanglement negativity and topological order, Phys. Rev. A 88, 042318 (2013).
  * Kittel [2018] C. Kittel,  _Introduction to Solid State Physics_ , 9th ed. (Wiley-VCH, 2018).
  * Ruderman and Kittel [1954] M. A. Ruderman and C. Kittel, Indirect Exchange Coupling of Nuclear Magnetic Moments by Conduction Electrons, Phys. Rev. 96, 99 (1954).