Quantum walk on a graph of spins: magnetism and entanglement

Kevissen Sellapillay,1 and Alberto D. Verga1*

1 Aix-Marseille Université, CPT, Campus de Luminy, case 907, 13288 Marseille,
France

*<EMAIL_ADDRESS>
## Abstract

We introduce a model of a quantum walk on a graph in which a particle jumps
between neighboring nodes and interacts with independent spins sitting on the
edges. Entanglement propagates with the walker. We apply this model to the
case of a one dimensional lattice, to investigate its magnetic and
entanglement properties. In the continuum limit, we recover a Landau-Lifshitz
equation that describes the precession of spins. A rich dynamics is observed,
with regimes of particle propagation and localization, together with spin
oscillations and relaxation. Entanglement of the asymptotic states follows a
volume law for most parameters (the coin rotation angle and the particle-spin
coupling).

###### Contents

  1. 1 Introduction
  2. 2 Model
    1. 2.1 One dimensional lattice
    2. 2.2 Observables
  3. 3 Results
    1. 3.1 Weak coupling
    2. 3.2 Strong coupling and entanglement
    3. 3.3 Chaos
    4. 3.4 Topology
    5. 3.5 Parameter phase space
  4. 4 Discussion
    1. 4.1 Free walk
    2. 4.2 Homogeneous system
    3. 4.3 Small gradients
  5. 5 Conclusions
  6. A The one step operator $U$
  7. B $|V|=2$, full coupling

##  1 Introduction

The concept of information was introduced as a physical quantity measured in
bits by Leo Szilard in 1929 [1], and further explored by Rolf Landauer (1961)
[2] who establishes that erasing a bit is an irreversible process costing $\ln
2$ of entropy (in units of the Boltzmann constant). However, as the laws of
nature are quantum, information must ultimately be related to the quantum
state. The quantum extension of a bit was proposed by Benjamin Schumacher
(1995) [3] in order to demonstrate a quantum coding theorem analogous to the
Shannon one in classical information theory [4]. The quantum coding theorem
shows that the von Neumann entropy $S=-\operatorname{Tr}\rho\log\rho$ is a
measure of the information content of a general quantum system in a state
described by the density matrix $\rho$. In spite of the apparent similarity
between Shannon and von Neumann entropies, quantum information is essentially
different to its classical counterpart: general quantum states are entangled.

Quantum information [5, 6, 7] is a fundamental physical concept, now part of
the curriculum [8, 9], which has a profound impact in vast domains, well
beyond its obvious application to computing, as for instance in condensed
matter through the phenomenon of many-body entanglement [10, 11, 12]. An
interesting field of investigation opened at the frontier of matter and
information [13]. It is in this context that we propose here a model based on
a quantum walk interacting with a network of spins, in order to manipulate the
quantum state by local unitaries to create entanglement.

Discrete quantum walks [14, 15, 16], although initially proposed as an
extension of the classical random walk [17] with possible applications in
optics, were first investigated in relation with the theory of quantum
simulation by Meyer in 1996 [18]. In his seminal paper Meyer demonstrated that
a quantum system evolving by discrete steps on a lattice (cellular automaton),
cannot be homogeneous and local: in order to keep a unitary evolution one
needs to introduce an internal degree of freedom associated with the walking
particles and coupled to their motion. Applications of quantum walks range
from the implementation of the quantum circuit universal model of computation
[19, 20, 21, 22] to the study of topological phases of matter [23]. The
experimental realization [24] demonstrated the practical feasibility of
quantum walks, allowing the investigation of topological edge states [25] and
the measurement of topological invariants [26, 27, 28]. Generalizations from
the one particle case to two entangled particles, including possible
interactions, were also studied theoretically [29, 30, 31, 32] and
experimentally [33, 34]; besides, Anderson localization was observed in an
array of interferometers [35], and the effect of boson and fermion statistics
simulated using the symmetries of the photon’s state. Another important
extension of quantum walks from regular lattices to general graphs [36, 37,
38], led to interesting algorithms like the generalization of the Grover
search [39], showing that the square speedup over the classical algorithm
applies to structured data [40, 41, 42], or to algorithms aiming at
determining graph isomorphism [43, 44], although a solution to this problem is
still open [45]. Finally, quantum walks are a starting point to explore many-
body physics, once generalized to a quantum automaton [18, 46, 47, 48].

Loosely speaking, in quantum mechanics the state of a system can be modified
by a unitary transformation or by a projection over a smaller Hilbert space.
These two possibilities lead to two main models of quantum computing [49]: the
circuit model theorized by David Deutsch [20], and the measurement driven
computation [50, 51, 52, 49]. In the first model a universal set of one and
two qubits gates [53, 54, 55] are used to create a suitable entangled state,
while in the second model, a highly entangled state, a cluster state [56, 57],
is modified by one qubit measurements to imprint the logical operations. In
this fuzzy classification, quantum walks pertain to the ‘circuit’ category, in
which suitably local coin and motion operators entangle a state [58] in order
to coherently enhance the probability of certain configuration (as for example
in the search problem). Information processing requires in addition to
ressources, communication channels. A physical realization of a transmission
line is a spin chain [59, 60]. Therefore, very schematically a possible
computing system is a network of spin chains connecting logical gates,
implemented for example by interacting spins.

We present here a model in which the resource, the entangled state, is created
by a walker on a network of spins, as already mentioned. In its simplest form,
the spins are non interacting, and it is the motion of the particle that
induces spin correlations and a global entangled state. This is reminiscent to
the protocol used to entangle carbon nuclear spins by the electron of a
nitrogen vacancy in diamond; entanglement is produced by successive
interactions of the electron with the neighboring nuclear spins (two qubit
gates) [61]. This protocol was successfully applied to experimentally
demonstrate teleportation [62] and error correction [63].

One may consider a physical motivation to generalize the usual setup of a
quantum walk. Actually, in a simple discrete quantum walk the particle motion
is determined by an internal degree of freedom, however, the space geometry in
which the particle moves is imposed in the form of a lattice or more generally
a graph. At each time step, the position amplitudes in one node of the graph
is then distributed over the adjacent nodes. We can make an analogy with a
tight binding model, in which an electron jumps between sites of a crystal.
The main physical difference is that the crystal is itself a lattice of ions
whose intricate interaction with the electron gas (for instance in a
semiconductor) gives rise to both, the crystal structure and the
characteristic energies of the electron hopping. This observation motivates
our choice of assigning a material support to the graph in which the walker
moves. A natural way is to associate a spin degree of freedom to the graph
nodes [64], or, as we do here, to the links between nodes. At variance to the
model of Ref. [64, 65] the spins do not interact, are then located on the
edges of the graph, and, as a consequence, the particle-spin interaction is
independent of the local degree of the graph (which was the case in the node
spin model). The interaction of the walker with the geometry gives rise to an
extension of the usual model of a quantum walk to an interacting one. The main
consequence of this generalization is that we leave the simple world of one
particle to the complex many—body interacting quantum system.

Consequently the constituents of our system are a particle and a set of spins.
As usual in a discrete quantum walk, the particle’s motion between neighboring
nodes of the graph is controlled by a coin (an operator acting on the particle
internal degree of freedom); the coupling between the walker and the spin
network is defined by an exchange interaction, as in a Heisenberg magnet. This
model keeps some analogy, albeit with one free particle, with a magnetic
system where free electrons interact with local magnetic moments creating an
effective interaction between spins, like in the Ruderman-Kittel-Kasuya-Yosida
exchange interaction and the Kondo lattice [66, 67, 68]. The idea is that the
walker, which may spread ballistically over the graph, a defining feature of
the quantum walk already noted in the original paper by Aharonov et al. [17],
can create spin entanglement at a linear rate in a many-body system [69, 70,
71, 72].

Quantum walks can be used to simulate physical systems; see for instance the
recent experimental realization of a periodically driven Chern insulator [73].
Meyer [18] showed that the one particle case of his quantum automaton, in fact
a quantum walk, reduces to the Dirac equation in the continuum limit. Since
then, the continuum limit of quantum walks was extensively explored to
simulate neutrino oscillations [74], Weyl fermions [75], or the Dirac equation
in two dimensions [76], to give a few examples.

In the present model the dynamics of spins adds to the particle motion; we
investigate the continuum limit and show that, for a choice of parameters, the
dynamics can be modeled by a Landau-Lifshitz equation [77]. The relation with
the magnetization dynamics, described by the Landau-Lifshitz equation, can be
understood by observing that the exchange interaction of a moving spin with
the canted fixed spins should induce a torque in much the same way as a spin
polarized current induces a so-called spin-transfer torque [78, 79] (note
however that the current is here replaced by the current probability of one
quantum particle).

In order to process information the physical system must be able to transfer
the information through the network and also to stock it at some location.
This means that it is desirable to find regimes of the quantum walk in which
the quantum state can spread over the available space or instead, can remain
localized. We focus here on the simplest case, in which the quantum walk is
defined on a one dimensional lattice with spin interaction absent, to show
that even in this geometry, and with only the local particle-spin coupling
present, a rich variety of dynamical regimes exists. We observe, both
propagation and localization of the particle distribution, together with spin
oscillations, irregular dynamics and relaxation to some uniform state. As in
condensed matter topological phases allow the control of the conduction to
insulator transition, quantum walks exhibit analogous topological properties
[15] in which the interaction may play an important role [32]. We investigate
the topological properties of the particle-spin quantum walk to identify the
effect of the interaction on an otherwise single step Dirac walk, which is
known to support edge states at the interface between two non equivalent
phases. (We call “Dirac” the walk with a $SU(2)$ coin that converges to the
Dirac equation in the continuum limit [80, 81].)

What is special about the quantum state is that we need an exponentially large
number of parameters to specify it; this is a straightforward consequence of
the superposition principle: while a classical state is defined by a point in
phase space, the quantum one is a superposition of all the basis states [82].
The amount of information encoded in the quantum state is potentially
exponentially larger than in a classical state, as shown by the existence of
superdense coding [83], polynomial algorithm to factorize large numbers [84],
and fast methods to solve linear equations [85]. These algorithms exploit the
informational resources of an entangled state, the question that naturally
arises is how to build useful quantum states. We propose, in the sequel of
[64], to use a walker on a graph to construct an entangled state of spins.
Instead of using single qubit measurements on a previously prepared generic
entangled state as in one-way computing, we let the particle to explore the
graph creating through its interaction with the spins, the entangled quantum
state. The resulting state depends, in addition to the initial condition and
parameters, on the graph connectivity and topology. It is worth noting that
the analogy with the graph state is nevertheless limited because of the
different structure of the Hilbert space; the walk build state possesses
besides the spin degrees of freedom, which are localized on the network edges,
the walker position and internal degrees of freedom, which spread generally
over the network: this structure changes the entanglement characteristics,
impeding, for instance, to spatially partition the graph as can be done in
pure spin systems (the walker wave function can be delocalized on the whole
graph). This difference is exploited here to investigate the case where
entanglement of spins is indirectly induced by a moving particle, in
particular we can analyze the role of local and nonlocal effect in the
multipartite entanglement of spins.

Our aim in this work is to demonstrate that a simple quantum system governed
by local unitary rules leads to complex dynamics suitable to investigate the
links between entanglement, topological, and magnetic properties. The paper
starts with the presentation of the model, we show in particular how to handle
the spins lying on the graph edges and their coupling with the itinerant
particle. We specializes the model to the simple Dirac walk on the line and
explore its phenomenology using exact numerical computations. We discuss the
different dynamical regimes and their relation with the entanglement of the
quantum state. We observe that the many-body interaction is essential in the
setting of well defined magnetic properties and in the evolution towards a
stationary regime. These magnetic properties can be described by a
semiclassical theory and the corresponding Landau-Lifshitz equation derived.
We end by a discussion of the results and a conclusion.

##  2 Model

We implement a quantum walk on a simple, undirected, simply connected graph
$G(V,E)$, where $V$ is the set of nodes and $E$ the set of edges. The number
of nodes is the cardinal $|V|$ (we write $|\cdot|$ for the number of elements
of a set). Nodes are denoted $x,y,\ldots\in V$ and edges are pairs of linked
nodes $(x,y)\in E$. The set

$V_{x}=\\{y\mid(x,y)\in E\\}$ (1)

is the set of nodes $y\in V$ neighbors of $x$, and $d_{x}=|V_{x}|$ the degree
of node $x$. A particle jumps between neighboring nodes ($y\in V_{x},\;\forall
x\in V$) and interacts with spins located on the edges $(x,y)\in E$. The
motion between adjacent vertices depends on an internal degree of freedom, the
“color” $c$, taking $d_{x}$ values. The color decorates the graph nodes with a
label ‘$c$’, the “subnode”, for each incident edge [86]. In addition, the
color degree of freedom is coupled with the particle motion, and with the edge
spin through a spin-particle interaction. This model simulates a kind of
itinerant magnetism in which the motion of a particle determines the
correlations between localized spins.

The basis of the graph Hilbert space $\mathcal{H}_{G}$ is the set of kets of
the form

$\ket{xcs}=\ket{x}\otimes\ket{c}\otimes\ket{s_{0}\ldots
s_{|E|-1}}\in\mathcal{H}_{G}\,,$ (2)

(the canonical basis) where the labels ‘$x,c,s$’ stand for:

  * •

the position $x$, spanning the nodes $x=0,\ldots,|V|-1\in V$;

  * •

the color $c=c(x)$, spanning the incoming edges of each node $\forall x\in
V,\;c=0,\ldots,d_{x}-1$;

  * •

and the set of spins $s=s_{0}s_{1}\ldots s_{|E|-1}$, represented by a string
of binary numbers $s_{e}=0,1,\;e=0,\ldots,|E|-1$ ($e$ labels the edges) for
the up and down states at each edge, and spanning the configurations
$s\in\\{0,\ldots,2^{|E|}-1\\}$.

The Hilbert space dimension is

$\mathrm{dim}\mathcal{H}_{G}=\left(\sum_{x\in V}d_{x}\right)\times
2^{|E|}\leq|V|\times d_{\text{max}}\times 2^{|E|}$ (3)

($d_{\text{max}}=\max_{x}|V_{x}|$ is the maximum degree). A node state is a
superposition of $\ket{xcs}$ vectors with fixed $x$, and the edge state is
given by a superposition of pairs of node vectors of the form,

$\left\\{\ket{xc_{y}s_{e}},\ket{yc_{x}s_{e}}\right\\},\quad(x,y)\in E\,,$ (4)

where $c_{e}=(c_{y},c_{x})$, formed by the subnodes of the corresponding
neighbors, is defined as the edge color (Fig. 1).

$x$$y$$c_{y}$$s_{e}$$c_{x}$120 Figure 1: Graph showing the labeling of
subnodes (color, $0,1,\ldots$) between nodes (position $x,y,\ldots$) and edges
(spin $s_{e}$).

The interacting quantum walk is defined by a unitary operator,

$U=V(J)W\,,$ (5)

which splits into two parts, the walk $W$ part and the interaction $V(J)$ part
that depends on a particle-spin coupling constant $J$:

  * •

The walk part $W=MC$, includes a motion operator ($M$) that permutes the
amplitudes at a node $x$ with the amplitudes of its neighbors $y\in V_{x}$,
according to the edge color state, and a coin operator ($C$) that actualizes
the color state at each node; the motion step $M$ is,

$M\ket{xc_{y}s}=\ket{yc_{x}s},\;\forall y\in V_{x}\,,$ (6)

where the set of values of $c=c_{y}$ with $y\in V_{x}$, corresponding to the
subnodes of $x$, determines the direction of the particle motion; and the coin
step $C$,

$\ket{xcs}\rightarrow C\ket{xcs},\quad C=\mathbb{I}^{|V|}\otimes
C_{c}\otimes\mathbb{I}^{2^{|E|}}$ (7)

($\mathbb{I}^{D}$ is the $D$-dimensional identity matrix) where $C_{c}$ can be
any operator of dimension $d_{x}$ (conveniently completed with zeros up to the
total dimension $d_{\text{max}}$). The composite operator $W$ defines a simple
quantum walk in a graph [36, 37, 87, 43]. The choice of the coin $C_{c}$ is
somewhat arbitrary but the Grover and Fourier operators possess complementary
properties [14]. The Grover operator,

$G(d)=\frac{2}{d}\mathbb{J}^{d}-\mathbb{I}^{d}\,,$ (8)

where $\mathbb{J}^{d}$ is the $d$-dimensional matrix filled with 1, is not
balanced but distributes the walker amplitudes at each node preserving the
graph symmetries; the Fourier operator is the quantum version of a balanced
coin,

$F(d)=\frac{1}{\sqrt{d}}\mathrm{e}^{2\mathrm{i}\pi\bm{c}\bm{c}^{\textsc{T}}/d}\,,$
(9)

where $\bm{c}$ is the vector whose coordinates are the colors at each node.

  * •

And the color-spin interaction part, which superposes the color state of the
edge $c_{e}=(c_{y},c_{x})$ with the edge spin $s_{e}$

$\forall e\in E,\;\ket{c_{e}}\otimes\ket{s_{e}}\rightarrow
V(J)\ket{c_{e}}\otimes\ket{s_{e}}\,.$ (10)

This operator acts then on the set of edges $E\in G$,

$V_{xy}(J)\begin{pmatrix}xc_{y}0_{e}\\\ xc_{y}1_{e}\\\ yc_{x}0_{e}\\\
yc_{x}1_{e}\end{pmatrix}\,,\;\forall(x,y)\in E\,,$ (11)

in an obvious notation, for instance $0_{e}$ denotes a configuration
$s=s_{0}\ldots s_{e}\ldots$ such that $s_{e}=0$. The choice of $V(J)$
determines the physics of the system. In the framework of an itinerant
electron we can implement an analogous to the so-called $sd$ exchange
interaction $H_{J}$ [88, 79]:

$V_{xy}(J)=\exp(-\mathrm{i}H_{J}),\;H_{J}=-\frac{J}{4}\bm{\tau}\cdot\bm{\sigma}\,,$
(12)

with $\bm{\tau},\bm{\sigma}=(X,Y,Z)$ vectors of Pauli matrices acting on the
color and spin spaces, respectively; $J$ is the color-spin coupling constant.
(Remark that, for the sake of simplicity, we define the matrix $V(J)$ by its
action on the edges $(x,y)$, which are not in the basis of the Hilbert space
$\mathcal{H}_{G}$.) This interaction

$V_{xy}(J)=\mathrm{e}^{-\mathrm{i}J/4}\begin{pmatrix}\mathrm{e}^{\mathrm{i}J/2}&0&0&0\\\
0&\cos(J/2)&\mathrm{i}\sin(J/2)&0\\\ 0&\mathrm{i}\sin(J/2)&\cos(J/2)&0\\\
0&0&0&\mathrm{e}^{\mathrm{i}J/2}\end{pmatrix}\,,$ (13)

is related to the swap gate for the particular choice of the coupling $J=\pi$.
The interaction $V(J)$ is able to entangle the color and spin degrees of
freedom, while $W$ may entangle the color and particle position; as a result,
distant spins generically get entangled through the particle motion (for
standard choices of the coin operator and parameters).

Therefore, the unitary evolution of the quantum state $\ket{\psi}$ is governed
by the operator $U$ such that,

$\ket{\psi(t+1)}=U\ket{\psi(t)},\quad\ket{\psi(t)}\in\mathcal{H}_{G}\,,$ (14)

which advances the quantum state by one time step (we chose units such that
$\hbar=1$ and the time step $\Delta t=1$). In summary, this equation describes
the quantum walk on the nodes of an arbitrary graph of a colored particle
interacting with the spins on the corresponding edges. In Appendix A we study
in more detail the tensor structure of $U$, to explicitly show its local
action on the canonical basis of $\mathcal{H}_{G}$ in the specific case of a
line graph.

###  2.1 One dimensional lattice

In the following we will investigate the one dimensional lattice. The graph
reduces to the set of nodes $x\in V\subset\mathbb{Z}$ with edges $e\in E$
($e=0,1,\ldots,|E|-1$) simply given by $(x,x+1)$. The set of neighbors (1) of
node $x$ is $V_{x}=\\{x-1,x+1\\}$. The Hilbert space dimension (3) is
$|V|\times 2^{|E|+1}$. We consider periodic ($|E|=|V|$), and finite lattices
($|E|=|V|-1$), for different initial states.

We choose a rotation matrix of angle $\theta$ for the coin operator $C_{c}$,

$R(\theta)=\begin{pmatrix}\cos\theta&-\sin\theta\\\
\sin\theta&\cos\theta\end{pmatrix}\,,$ (15)

which, for the special value $\theta=\pi/2$, exchanges amplitudes like a two
dimensional Grover matrix $G(2)$, and for $\theta=\pi/4$, distributes the
amplitudes like the balanced Hadamard coin $F(2)$, thus conveniently
interpolating between Grover and Fourier coins. The motion operator reduces to
a translation such that the swapping of neighbor amplitudes can be written as,

$M=\sum_{x}\left(\ket{x+1}\bra{x}\otimes\ket{1}\bra{0}+\ket{x-1}\bra{x}\otimes\ket{0}\bra{1}\right)\,,$
(16)

so that the $0$ color amplitude moves to the right, while its state flips to
$1$. The one step evolution operator is then given by,

$U(J,\theta)=V(J)MR(\theta)\,,$ (17)

which depends on two parameters taking values in the relevant ranges
$\theta\in(0,\pi)$ and $J\in(0,\pi)$. Appendix A shows the action of $U$ on
the basis vectors arranged in edges related amplitudes, and Appendix B
illustrates the case $|V|=2$.

Initially the particle can be located on one node, or uniformly distributed on
the line (‘i’), in different color superpositions, and spin states: all spins
$z$-polarized (‘z’), $x$-polarized (‘x’), or a single spin up in a background
of ‘x’ spins. We label the initial state as follows,

  * •

particle at $x=x_{0}$ and spins up $\ket{0}$,

$\text{`z':
}\ket{\text{z}}=\ket{x_{0}00}=\ket{x_{0}0}\otimes\ket{0}^{\otimes|E|}\,,$

  * •

particle at $x=x_{0}$ and spin ‘right’
$\ket{+}=(1/\sqrt{2})(\ket{0}+\ket{1})$,

$\text{`x':
}\ket{\text{x}}=\frac{1}{2^{|E|/2}}\sum_{s=0}^{2^{|E|}-1}\ket{x_{0}0s}=\ket{x_{0}0}\otimes\ket{+}^{\otimes|E|}\,,$

  * •

particle at $x=x_{0}$ spin up at $e=(x_{0},x_{0}+1)$, and the other spins in
$\ket{+}$,

$\text{`zx': }\ket{\text{zx}}=\frac{1}{\sqrt{2^{|E|-1}}}\sum_{\\{\forall s\mid
s_{e}=0\\}}\ket{x_{0}0s}\,,$

  * •

(entangled spins) the particle at $x=x_{0}$ is in a superposition of up and
down spins,

$\text{`e':
}\ket{\text{e}}=\frac{1}{\sqrt{2}}\left(\ket{x_{0}00}+\ket{x_{0}02^{|E|-1}}\right)\,.$

If initially the particle is uniformly distributed over the position states we
add a prefix ‘i’ to the initial state label (‘iz’, ‘ix’, etc.).

We also studied different boundary conditions: (‘p’) periodic (‘b’) reflective
and (‘t’) with an interface. With reflection of the walker at the line
boundary the translation symmetry preserved by the periodicity is broken. The
interface is introduced to investigate topological properties of the quantum
walk, using different coins on the two sides.

In summary, the system is determined by the boundary and initial conditions
and the couple of parameters $(\theta,J)$: an example is $(\pi/3,1)$, ‘p’,
‘z’.

###  2.2 Observables

After the $t$ step, the quantum walk (pure) state is given by the density
matrix,

$\rho(t)=\rho(xcs,t)=\ket{\psi(t)}\bra{\psi(t)},\;\ket{\psi(t)}=\sum_{xcs}\psi_{xcs}(t)\ket{xcs}\,,$
(18)

where we explicitly wrote the functional dependency of $\rho$ on the basis
labels $xcs$. This state is numerically computed iteratively multiplying by
$U$, and then it is exactly known. The knowledge of the quantum state allows
us to monitor the physical properties of the system; we focus on the
probabilities distribution of position and spin, as well as different measures
of the entanglement entropy. We denote $l\in\\{x,c,s\\}$ one of the basis
state labels and $\bar{l}$ the complement set of the label $l$ (i.e. if $l=x$,
$\bar{l}=\\{c,s\\}$), and

$\braket{O}(l,t)=\operatorname{Tr}O\rho(l,t),\;\rho(l,t)=\operatorname{Tr}_{\bar{l}}O\rho(xcs,t)$
(19)

the expected value of the magnitude $O$ as a function of the variables
($l,t$), obtained from the partial density matrix $\rho(l,t)$, which is the
partial trace $\operatorname{Tr}_{\bar{l}}$ over the complement $\bar{l}$ of
the total matrix $\rho(xcs,t)$.

The particle distribution probability, function of the nodes $x$ and step $t$,
is given by

$p(x,t)=\operatorname{Tr}_{\bar{x}}\rho(xcs,t)=\operatorname{Tr}\rho(x,t)\,,$
(20)

where we used the notation (19). The expected value of the spin $\bm{s}(e,t)$
located at edge $e$ in the state $\rho(t)$, is

$\bm{s}(e,t)=\operatorname{Tr}\bm{\sigma}\rho(s_{e},t)\,,$ (21)

where, as noted in the preceding section, $s_{e}$ labels the spin at edge $e$
in the configuration string $s=s_{0}\ldots s_{e}\dots s_{|E|-1}$. We can label
the edges simply using the node label and write $\bm{s}(x,t)$ for the edge
$e=(x,x+1)$. The spatial mean is then given by,

$\bm{s}(t)=\frac{1}{|E|}\sum_{e}\bm{s}(e,t)\,.$ (22)

The expected value $\bm{s}$ corresponds to the graph magnetization
distribution (here the magnetization on the line).

In order to measure the von Neumann entropy we partition the Hilbert space
into two parties, selecting the relevant degrees of freedom. The entropies
related to the basis degrees of freedom are defined by,

$S_{l}(t)=-\operatorname{Tr}\rho(l,t)\log\rho(l,t)$ (23)

(we use throughout base 2 logarithms denoted $\log$). It is also interesting
to measure the entanglement of a subset $A$ of spins. We take,

$A=\\{s=0\ldots s_{e}\ldots\mid e=e_{0},e_{1},\dots\in
E\\},\;\bar{A}=\\{xcs\mid s\notin A\\}\,,$ (24)

where here $e$ is a list of selected edges. With this definition, the
entanglement entropy of $A$ spins is,

$S_{A}(t)=-\operatorname{Tr}\rho(A,t)\log\rho(A,t),\quad\rho(A,t)=\operatorname{Tr}_{\bar{A}}\rho(xcs,t)\,.$
(25)

For a binary partition of the system the von Neumann entropy measures the
entanglement of the corresponding reduced state [3, 89, 90]. It is important
to note that (23) is a global quantity defined with respect to the Hilbert’s
space degrees of freedom, while $S_{A}$ is much a local quantity in the sense
that the set $A$ is spatially defined by the set of edges; as a consequence we
may have in general that

$S_{A}>S_{s}\,,$ (26)

for a large enough $|A|$ (the number of spins in $A$). In particular $S_{A}$
can be used to test entanglement for spatially separated spins, using a
disconnected set $A$.

Figure 2: Spatiotemporal plot of the particle density $p(x,t)$, colors in
logarithmic scale. Parameters (a) $(\pi/8,0.2)$ ‘x’, (b) $(\pi/2,0.2)$ ‘x’.
(a) Dispersive ballistic propagation; (b) ballistic propagation with weak
dispersion.

Figure 3: Spin and entanglement. Parameters as in the previous figure, (a,b)
$(\pi/8,0.2)$ ‘x’, (c,d) $(\pi/2,0.2)$ ‘x’. (a,c) Average spin
$\bar{\bm{s}}(t)$; (b,d) entanglement entropy $S_{l}(t)$, $l=x,c,s$. (a)
oscillations, (c) oscillations with a weak damping; (b) recursive behavior of
the particle and spin entropies, (b) growth of the entanglement.

To establish the entanglement between two spins at edges $(e_{1},e_{2})$ we
use the standard notion of concurrence [91, 92], defined by,

$C(e_{1},e_{2})=C[\rho_{2}]=\max\\{0,\sqrt{\lambda_{1}}-\sqrt{\lambda_{2}}-\sqrt{\lambda_{3}}-\sqrt{\lambda_{4}}\\},$
(27)

where $\rho_{2}$ is the reduced matrix of the couple of spins, and
$\lambda_{i}$ are (in decreasing order, $i=1,\ldots,4$) the eigenvalues of the
matrix,

$\rho_{2}(Y\otimes Y)\rho_{2}^{\star}(Y\otimes Y)\,,$ (28)

where $\rho^{\star}$ is the conjugate matrix in the canonical basis (the same
as the $\rho_{2}$ basis). The concurrence measures the entanglement of
formation of a two qubits state, it is zero for a separable state and one for
a maximally entangled one; in particular, if $\rho_{2}$ is diagonal it
vanishes, and hence no two-spins entanglement would be present. See Appendix B
for an application to the case $|V|=2$ of some of these formulas.

##  3 Results

Figure 4: Spin relaxation for strong coupling. Parameters (a) $(3\pi/8,1.2)$,
$|V|=13$, ‘z’, (b) $(\pi/8,1.2)$, $|V|=19$, ‘z’.

Figure 5: Position density and entanglement. Parameters as in Fig. 4, (a,b)
$(3\pi/8,1.2)$, $|V|=13$, ‘z’, (c,d) $(\pi/8,1.2)$, $|V|=19$, ‘z’. Propagation
(a) and localization (c) regimes. (b,d) Entanglement of a set of spins. (b)
The one spin entropy saturates to its maximum value ($S_{1}=1$); the $|A|=6$
curve shows a linear range. (d) The one spin entropy saturates; $S_{A}(t)$ for
$|A|=10$ slowly increases following a curve that can be fitted by a stretched
exponential with exponent $1/2$ (see text).

We investigate now the dynamical evolution of the system in the parameter
phase space $(\theta,J)$. We consider systems of different sizes, the typical
one being $|V|=13$, whose Hilbert space dimension is about $2\,10^{5}$; some
simulations were performed for $|V|=19$ (dimension about $2\,10^{7}$).

In the absence of interaction $J=0$ we identify two extreme behaviors: for
$\theta=0$ the particle remains confined in the neighborhood of its initial
position; in contrast, for $\theta=\pi/2$ the walker translates at speed 1
without spreading (the motion reduces to an amplitude swap to the left or to
the right). Intermediate values give the usual ballistic spreading along the
line of the Dirac quantum walk [93]. In the interval $\theta\in(\pi/2,\pi)$
the system’s behavior symmetrically reverses. However, it should be noted that
interactions may in principle break this symmetry.

In the interacting case we may classify the different regimes according to the
particle motion (using $p(x,t)$), which can be localized or propagating, and
to the spin dynamics (using $\bm{s}(x,t)$), which may exhibit oscillations,
relaxation or be chaotic. In addition, each of these regimes may display a
variety of entanglement behaviors that can be characterized by the von Neumann
entropies $S_{l}$ associated to each class of degrees of freedom, position,
color and spin, and also by the entanglement of a connected or unconnected
subset of spins $S_{A}$.

###  3.1 Weak coupling

The motion of the walker for weak coupling $J=0.2$ is illustrated in Fig. 2,
where we represent in a logarithmic scale the probability $\ln p(x,t)$ for the
particle to be at step $t$ at node $x$. Between the case $\theta=\pi/8$ (Fig.
2a) and the case $\theta=\pi/2$ (Fig. 2b) we note that not only the
propagation speed differs, with values of about $1$ for $\theta=\pi/2$ and
about $1/3$ for $\theta=\pi/8$, but that the probability spreading is much
stronger for small rotation angles. It is worth mentioning that, at variance
to the case ‘x’ ($\theta=\pi/2$), the initial condition ‘z’ is a proper state
of $U(J,\pi/2)$ leading to a trivial translation. A slight dispersion of the
position probability appears in the case $\theta=\pi/2$ for larger times
(visible for $t\gtrsim 150$). This is an interesting effect of the
interaction, it induces a dispersion on the underlying Dirac quantum walk.

The difference in the spreading of the particle reflects in the spin dynamics.
We show in Fig. 3ac, for the same parameters ($\theta=\pi/8,\pi/2,J=0.2$), the
mean spin vector $\bm{s}(t)$ as a function of time and the von Neumann spin
entropy $S_{l}(t)$ ($l=x,c,s$). Both cases exhibit spin oscillations, yet for
$\theta=\pi/8$ an almost recurrence to the initial state appears (relaxation
is very weak: the peaks are $1,0.98,0.97$, at $t=0,839,1648$), while for
$\theta=\pi/2$ oscillations are damped (relaxation is stronger: the peaks are
$1,0.74,0.39$, at $t=0,834,1679$). We observe that in spite of the irregular
and fine grained features of the particle motion, the mean magnetization is
smooth and behaves as if it would obey to a simple dynamical system.

The oscillation period can be empirically estimated to be,

$T\approx 4\pi|V|/J\,,$ (29)

which is $T=817$ for $|V|=13$ and $J=0.2$, and depends essentially on $J$.
However, the amplitude of the oscillations is strongly dependent on the
rotation angle $\theta$: for $\pi/8$ the amplitudes are much smaller than for
$\pi/2$.

Figure 6: Simulation with closed boundary conditions ‘b’. Spatiotemporal
particle and $z$-spin density. Parameters $(3\pi/8,1)$, ‘z’. The arrows in (a)
show the set A of selected spins.

Figure 7: Entanglement with closed boundary conditions ‘b’. Parameters
$(3\pi/8,1)$, ‘z’. (a) The $(xcs)$ entropies saturate to their maximum values.
(b) Spin entropy of a disconnected set A, shown in Fig. 6a.

The motion and spin distributions for weak coupling display some qualitative
differences depending on the values of the rotation angle, these differences
also appear in the walk entanglement properties (Fig. 3bd). The recursive
behavior found in the case $\pi/8$ is also present in the entropy $S_{l}(t)$.
In the case $\pi/2$ we have instead a steady increase of the entanglement
towards saturation. The spin entropy is smaller than the position one
$S_{x}\gtrsim S_{s}$ and remains well under the maximum possible values

$S_{x}<\log(|V|)\approx 3.7,\;S_{s}<\log(2\times|V|)\approx 4.7\quad(|V|=13)$

for the smaller angle; while for the larger one, the spin and position
entropies are larger, rather close to each other, and for long times we have
$S_{x}\lesssim S_{s}$. Another important qualitative difference we observe is
the relatively large fluctuations of the node entropy $S_{x}$ in the small
angle case. The color entropy $S_{c}$ generally reach its saturation value at
one qubit, in a time which is of the order of the propagation time of the
walker over the system length.

Summarizing this weak interacting case, we find that the walker propagates for
a large range of rotation angles with increasing dispersion for small angles;
the mean spin shows temporal oscillations with a period determined by the size
of the system and the coupling constant; the entanglement manifests the
recursive dynamics at small angles and, for larger angles, it is smooth and
slowly increases to saturation at long times.

###  3.2 Strong coupling and entanglement

Figure 8: Concurrence as a function of the system size. Parameters
$(1.5,1.2)$, ‘z’; $|V|=7,9,11$.

We turn now to the strong coupling regime and compare for the initial
condition ‘z’, two cases with fixed $J=1.2$ and angles $\theta=3\pi/8,\pi/8$.
Both cases show a rapid spin relaxation (Fig. 4), with a smoother evolution in
the large angle case $3\pi/8$. The walker density of Fig. 5 revels two
distinct propagation regimes. Comparing the two cases we see a qualitatively
new behavior appearing for $(\pi/8,1.2)$ (Fig. 5c): the walker mainly stays in
the neighborhood of its initial position in sharp contrast to the
$(3\pi/8,1.2)$ case. We refer this situation as pertaining to a ‘localization’
regime, at variance to the ‘propagation’ regime of Figs. 2 and 5a. Taking into
account the ballistic motion of the underlying walker in absence of
interaction, we may refer to this enhanced probability at the origin as
interaction induced localization. A related phenomenon linking interaction and
localization was found in the quantum walk of two coupled particles [32].

Figure 9: Irregular dynamics. Particle density for two different initial
conditions: (a) ‘x’, (b) ‘zx‘. Parameters (0.1, 2). In addition to a central
concentration the particle makes intermittent ballistic excursions, rarer in
the ‘zx’ case (b).  Figure 10: Spatiotemporal spin density. Parameters (0.1,
2) ‘x’. The ($y$,$z$) spin components are scaled by a factor 10.

Figure 11: Averaged spin (a) and entanglement entropy (b), for strong coupling
and small angle showing irregular dynamics. Parameters (0.1, 2), ’x’ as in
Figs. 9a and 10. Particle and spin entropies follow the same pattern,
remaining at a low level with respect to the random state value $\max
S_{x}\approx 3.7$ (the coin entropy saturates quickly).

Figure 12: Spatiotemporal plot of the $z$ spin component (scaled) (a), and the
corresponding temporal profile of node $x=x_{0}$. Parameters, (0.1, 2),’zx’ as
in Figs. 9b. The rapid variation at $x=x_{0}$ contrasts with the regular
oscillations of the neighbor $x=x_{0}+1$, the red line in (b).

The entanglement entropy of a set of spins $S_{A}(t)$ is presented in Figs.
5bd. The main distinction between the propagating and localized regimes is
revealed by their respective growth laws. In the propagating case, we remark a
linear stage for times $t<50$, after which follows a saturation regime where
entanglement slowly increases. In the localization case, the linear stage
disappears. In the case where the walker spreads ballistically approaching a
uniform distribution over the line, the entropy sharply grows, while for the
localized dynamics the growth of the entanglement is smoother. In contrast
with normal exponential saturation, the spin set entropy of the localized
dynamics can be fitted with a stretched exponential (the gray shaded curve in
Fig. 5d):

$S_{A}(t)\sim 1-\mathrm{e}^{-\nu t^{\alpha}}\,,$ (30)

where the exponent is $\alpha=0.5$, and $\nu=0.11$ a constant dependent in
principle on the system’s parameters $(\theta,J)$ (and also it might be
slightly dependent on $|V|$, in particular due to finite size effects). The
fit is consistent with an initial stage $S_{A}(t)\sim\sqrt{t}$, instead of the
linear one $S_{A}(t)\sim t$ observed in the propagation case.

One interesting question is whether spins spatially separated are entangled.
We investigate this question in a system with closed (refelctive) boundaries
‘b’, and using an initial state that propagates preferentially in one
direction. We define a disconnected set of spins near the two borders:

$A=\\{1,2,3,4,9,10,11,12\\},\quad|A|=8,\quad|V|=15$

and measure $S_{A}(t)$ using (25), in a system with $15$ nodes. In Fig. 6 we
show together the particle and spin densities; the logarithmic scale of
$p(x,t)$ covers two orders of magnitude to reveal the correlation with the
spin distribution; we see in particular, a propagating front where the spin
reverses. The $(xcs)$ entanglement entropy, displayed in Fig. 7a saturates to
the random state values. The answer to the question is provided by the
behavior of $S_{A}$ shown in Fig. 7b. We find that the separated spins
entangle concomitantly with the progression of the mentioned flip spin front.
Therefore, we may conclude that in addition to the global entanglement of the
distinct degrees of freedom spanning the Hilbert space, entanglement also
possesses a spatial organization that reflects in the possibility of distant
spins entanglement through the interlacing created by the nonlocal smeared
particle.

However, the state of two arbitrary selected spins do not form a Bell state as
can be measured by the concurrence (27). We find that whatever the pair of
values $(\theta,J)$ the concurrence essentially vanishes. Indeed, a favorable
situation for the formation of entanglement of two spins is when the state is
near ‘z’ eigenstate so that the particle can propagate almost freely
$\theta\approx\pi/2$ together with a strong coupling $J>1$; we show in Fig. 8
the concurrence as a function of the number of sites. The increasing time
interval between concurrence revivals can be interpreted as a progressive
transition between a quasiperiodic and a chaotic dynamics, in analogy with the
kicked top phenomenology [94], which corresponds to the actual behavior of the
system for the chosen parameters. In addition to this interpretation, the
vanishing of the concurrence, that is to say the absence of bipartite
entanglement correlated with an increase in the multipartite entropy $S_{A}$,
can be related to the presence of nonlocal entanglement, not necessarily
associated with non regular dynamics; a scenario consistent with the mediated
character of the spin-spin interaction. We discuss below the appearance of
irregular behavior.

In summary, we have shown in Figures 2-5 the characteristic oscillations which
are present for the full range of angles in the weak coupling limit $J\ll 1$,
and pointed out some of the qualitative differences observed between angles in
the first and second quarters. We found that systems in the oscillation-
relaxation regimes can be characterized by their distinct entanglement
features, from recurrent to monotone growth towards saturation. At variance to
the weak coupling case, for strong interaction oscillations are almost
suppressed by a rapid relaxation to a saturation regime. We also observed
that, depending on the point in the phase space $(\theta,J)$, two distinct
particle density distributions arise, one uniform which can be associated with
ballistic propagation, and another picked at the origin, which reveals
localization driven by interaction. Finally we found evidence of entanglement
between distant spins as unveiled by the entropy of a disconnected set.

Figure 13: Spatiotemporal plot of the position probability showing irregular
dynamics without localization. Parameters $(\pi/2,\pi)$, ‘x’.

Figure 14: Three steps ($t=1,2,3$) of the quantum walk showing the changes in
spin. Parameters $(\pi/2,\pi)$, ‘x’ (as in Fig. 13).  Figure 15:
Spatiotemporal diagram of the spin $\bm{s}(x,t)$. Parameters $(\pi/2,\pi)$,
‘x’.  Figure 16: Long time evolution of the averaged spin as a function of
time, showing the irregular dynamics in the propagating regime. Parameters
$(\pi/2,\pi)$, ‘x’(c.f. Figs. 13 and 15).

###  3.3 Chaos

In addition to oscillations and relaxation, the interacting walk exhibits
irregular dynamics, characterized by a complex behavior of the observables. We
fix the point $(0.1,2)$ in the $\theta\approx 0$ region of the parameter phase
space, with initial conditions ‘x’ and ‘zx’, which differ in the spin state of
the edge $e=(x_{0},x_{0}+1)$. A remarquable pattern shows up in Fig. 9. The
particle density is essentially concentrated at the origin, however,
intermittently ballistic excursions spark throughout the line. We note that
the localized spin initial state ‘zx’ leads to a stronger particle
localization, underlining the close link between entanglement features and
spin distribution. The spatiotemporal spin dynamics (Fig. 10) shows the
emergence, from an initial homogeneous spin state perturbed by the motion of
the particle starting from one node, of structures with a large scale range.
The intermittent behavior is clearly seen in Fig. 11 where we plot the mean
spin vector as a function of time (we zoomed into the $y$ and $z$ spin
components), and the entanglement entropies. It is worth noting the close
relationship between the spin and particle entanglements, meaning that the
system is far from spin entanglement saturation (as would be the case for an
almost random quantum state).

For the ‘zx’ initial state, it is interesting to follow the evolution of the
spin up located at the central bond (Fig. 12). The irregular motion of the
central spin contrasts with the smoother oscillatory behavior of its
neighbors. This is an interesting illustration of the spectrum of scales
created by the iteration of $U(\theta,J)$, in particular the appearance of
large spatiotemporal scales.

We discuss now a case ‘x’ with parameters $(\pi/2,\pi)$, in the large angle
strong coupling region, to illustrate irregular dynamics without localization.
Figure 13 displays the particle density, which corresponds to a ballistically
propagating and spreading particle. The step by step motion of the walker
modifies the spin state, as illustrated in Fig. 14: a spin wake follows the
particle progressively transforming the initial homogeneous $\ket{+}$ state,
enriching it with new amplitudes and exciting the other spin components. The
initial spatiotemporal spin evolution is presented in Fig. 15; it is
interesting to note the strong correlation of the particle motion in Fig. 13
with the behavior of the $z$ spin component. At long times this simple initial
pattern becomes a complex motion as reflected by the mean spin shown in Fig.
16.

In brief, we observed that in some regions of the parameter space irregular
dynamics of the particle and magnetization are possible. The chaotic like
behavior of the observables occurs for both localized and propagating states.
One characteristics of the ‘chaotic’ regime is its relatively small
entanglement entropy, with the spin entropy sticking to the particle one.

###  3.4 Topology

Figure 17: Topology ‘t’. Columns represent different interfaces and
interaction strengths: (a-b) $J=0$, (c-d, e-f) $J=0.2$; (a-b, e-f) topological
interface, $\theta(x<6)=1.1$, $\theta(x\geq 6)=2.1$; (c-d) normal interface
$\theta(x<6)=0.5$, $\theta(x\geq 6)=1.2$. Interfaces are at $x=0$ and $x=6$,
due to the periodic boundary condition. The top row shows the position
distribution in logarithmic scale $\ln p(x,t)$; the bottom row the time
dependency of the probability $p(x,t)$ for the central node $x=6$ and a “body”
node $x=9$. Initial condition ‘x’. The thin gray line in (b,d,f) corresponds
to the uniform probability.

A hallmark property of quantum free walks is that they can be used as a
laboratory to investigate topological phases and the bulk-boundary
correspondence, as we mentioned in the introduction. Less investigated is the
interplay between topology and interaction in quantum walks. For instance, one
may think that tuning the coupling constant $J$, it would be possible to
change the topology of a given phase. The Dirac walk, as we shall discuss in
the next section, has two topological phases for $\theta<\pi/2$ and
$\theta>\pi/2$, corresponding to each side of the parameter boundary
$\theta=\pi/2$ for which the spectral gap of its associated effective
Hamiltonian closes [23]. However, the physical difference between the two
phases is somewhat arbitrary because of the existence of a simple relationship
relating $\theta<\pi/2$ and $\theta>\pi/2$ (c.f. [93] for a discussion about
the chiral symmetry of the Dirac walk). In the absence of interaction the walk
is governed by $W(\theta)=MR(\theta)$, hence a change
$\theta\rightarrow\pi/2-\theta$ corresponds to a change in the rotation
direction, or equivalently to a change $Y\rightarrow-Y$, up to a constant
unitary transformation, which do not modify the behavior of the observables.
This is similar to what happen in other context, with the Dirac equation as
already noted by Shen et al. [95]. The Dirac equation do support edge
localized states protected by the mass gap, however it is not a satisfactory
model of a topological insulator: the symmetry between positive and negative
energy states must be broken in this case to obtain a genuine model of
distinct topological phases. In our case the presence of the interaction,
whose unitary operator do not commute with color rotations, breaks the
symmetry between the two angle sectors. We may expect new effects to appear
due to interaction and related to the existence of edge states at the
interface separating the phases.

Figure 18: Topology ‘t’. Particle density averaged over time. The light dashed
line is the uniform level distribution. Parameters ‘bz’ (closed boundary
conditions); (a) free walk $J=0$, $\theta_{-}=1.1$ $\theta_{+}=2.1$; (b)
trivial interface $J=05$, $\theta_{-}=0.5$ $\theta_{+}=1.2$, (c) topologically
nontrivial interface $J=0.5$, $\theta_{-}=1.1$ $\theta_{+}=2.1$. Interface at
$x=7$ ($|V|=15$).

Figure 19: Topology ‘t’. Spin density evolution. Top, trivial interacting
case; bottom, nontrivial case. Parameters as in (b) and (c) of Fig. 18.

As a consequence, to investigate the topological properties of the interacting
walk, we generalize the color operator $C$ to take into account the change in
the value of $\theta$ at an interface defined at $x=x_{0}$. In practice, we
introduce two operators $R(\theta)$ differing only in the value of the angle:
$\theta_{-}$ and $\theta_{+}$ for $x<x_{0}$ and $x>x_{0}$, respectively. In
the case of periodic boundary conditions, a second interface at $x=0$ is also
present. In Fig. 17 we compare three cases distinguished by their topological
properties, as defined by the associated noninteracting case. We choose for
the topological interface $\theta_{-}=1.1$, $\theta_{+}=2.1$, which are at
similar distance to the critical value $\theta=\pi/2$, and for the trivial one
$\theta_{-}=0.5$, $\theta_{+}=1.2$. The free case $J=0$ with nontrivial
interface, shows almost recurrences corresponding to probability
concentrations alternating between the two interfaces. This behavior reflects
in the temporal evolution of the probability at specific nodes, as shown in
the second row of the figure, where $p(x,t)$ is represented for a central and
a bulk node, $x=6,9$. The interacting trivial case $J=0.2$ (c-d), shows normal
scattering at the discontinuities of the coin parameter $\theta$, but no edge
states. The interacting non trivial case $J=0.2$ (e-f), shows the convergence
towards a probability at the origin larger than the equidistribution (thin
dotted line); recurrences and alternation between the interfaces disappear and
a probability concentration forms. The level of the probability at the
interface $x=6$, slightly larger than the mean in the present case, grows with
the value of the coupling constant, which is consistent with the presence of
an edge state. A more precise diagnostic is provided by the probability
distribution averaged over time, which we discuss next.

In Fig. 18 we present the time averaged particle density for three cases
illustrating the free nontrivial (a), interacting trivial (b) and interacting
nontrivial (c) situations. In Fig. 19 the spin density comparing the trivial
(top) and nontrivial (bottom) interacting cases. At variance to the previous
case, the initial condition is ‘z’ and the boundary is closed, which allow us
to focus in the dynamics around the unique interface. The particle
distribution shows that in the free case there is a symmetry between the two
phases, symmetry which is broken by the interaction in a case having the same
angle parameters (Fig. 18c). Both (a) and (c) possess edge states, where the
particle probability concentrates. In the interacting trivial interface case
(b), a strong asymmetry arises between the two regions; this asymmetry is also
present, albeit smoother, in the nontrivial case (c). The discontinuity found
in the interacting trivial case can be related to different propagation
properties at small and large angles: for $\theta=0.5$ the particle tends to
localize, while for $\theta=1.2$ it moves ballistically with a small
dispersion. In (c) the asymmetry between the two sides of the interface,
absent in the equivalent free case, is related to the preferring direction of
the particle motion (towards the left). In the free case, this tendency is
compensated by the alternation between the to sides of transmission and
reflection which cancel when averaged over time: with $J>0$ this symmetry is
broken.

Figure 20: Topology ‘t’. Particle probability (a) at the origin $x_{0}=7$, and
spin at the adjacent link (b); parameters as in the previous Fig. 19 (bottom
panel). The thin line in (a) is the uniform distribution value.

The role of the particle-spin interaction is apparent in the behavior of the
spin density $\bm{s}(x,t)$ at the interface (Fig. 19). In the trivial case
(top) a simple discontinuity in the $z$ spin component is present at the
interface, while in the nontrivial case (bottom) a rich quasiperiodic dynamics
develops, within an overall symmetric distribution. The initial uniformly up
polarized state relaxes towards two different values in the topologically
trivial case, on both sides of the discontinuity of the rotation angle. In the
localized region (on the left) the spin fades out smoothly, in contrast to the
fluctuating behavior in the propagating region, in which at long times the
mean spin tends to vanish. In the nontrivial case, both regions are
propagating and the spin density evolves similarly, the main effect is on the
interface, where a faster scale dynamics sets up (but slower than the free
particle return time to the origin). The oscillations of the central spin are
correlated with the oscillations of the particle density at the same position
(the two nodes defining the corresponding link), as we can see in Fig. 20,
where we observe the correlation arising after a short initial transient. In
fact, at the interface the particle executes a zig-zag motion between the
neighbors at $x_{0}$ and $x_{0}+1$, the lattice link where the spin resides
(Fig. 20b).

The main point about the topological properties of the system is its ability
to create at the interface between topologically different regions, edge
states where the particle concentrates and where a distinct spin dynamics
develops. The edge states are robust under wide changes of the phase
parameters, as well as to the initial condition (whose overlap with this state
should be nevertheless significant).

Figure 21: Phase space. (a) Initial state ‘x’, (b) ‘z’. We use the Kolmogorov-
Smirnov distance to the uniform position distribution to determine the
localization degree; the threshold $D_{\text{KS}}=1.5/100$ roughly corresponds
to $D_{\text{KS}}=D_{\text{KS}}(\pi/4,\pi/2)$ for the initial condition ‘x’.
Note that the scale of colors is not linear. (The scale of $D_{\text{KS}}$ is
magnified by a factor 100.)

###  3.5 Parameter phase space

Figure 22: Position distribution averaged over time showing typical
localization (a) and propagation (b) situations. The thin line gives the
uniform value; ‘x’.

Figure 23: Entanglement as a function of the rotation angle $\theta$ for two
initial conditions, ‘x’ in the first column (a,c), and ‘z’ in the second
(b,d), and two values of the coupling $J=\pi/2,\pi/10$, first row (a,b), and
second row (c,d), respectively.

Figure 24: Entanglement entropy $S_{A}$ of a set $A$ of spins as a function of
its size. $S_{A}$ is compared to the entropy of a pure random state $S_{R}$.
Parameters: (a) $(0.9\,\pi/2,2\pi/3)$ (red), and $(\pi/4,\pi/10)$ (blue), ‘x’;
(b) $(\pi/4,\pi/2)$ (red), and $(\pi/4,\pi/10)$ (blue), ‘z’. The system’s size
is $|V|=15$. For strong coupling the spin entanglement reaches the random
state value $S_{R}$ (gray).

In order to organize the information about the different physical regimes we
described, it is desirable to define appropriated diagnostics amenable at
distinguishing between motion regimes and entanglement characteristics. We
focus first on one basic distinction, which determines many other properties
of the system, the ability of the walker to propagate or to be localized. To
this goal, we use as localization test the spatial distribution of the
particle at long times: a uniform distribution means a propagation dominated
regime, while a picked distribution is an indicator of localization. To
measure the difference between the actual distribution and the uniform one we
use the Kolmogorov-Smirnov distance. It is defined by the equation:

$D_{\text{KS}}=\max_{x}\left|F_{x}[\braket{p}]-u_{x}\right|\,,$ (31)

where $F$ is the actual distribution (in the probability language: a monotone
increasing function) associated to the time averaged position probability
density $p(x,t)$ and $u$ the uniform distribution over the line (a simple
linear function).

The parameter phase space $(\theta,J)$ is represented in Fig. 21 for two
initial states (‘x’ and ‘z’). Two examples of the particle density are given
in Fig. 22, where we find typical localized and uniform densities. The
transition between localization and propagation regimes (the white level) is
rather smooth, at least in this small size system $|V|=13$. However, within
the localized region of the phase space, the position density increases
rapidly with the interaction, given large values of $D_{\text{KS}}$, while in
the propagation part of the phase space the particle density is much more
homogeneous (note the change of color scale between the two phases). Localized
states occupy a larger part of the parameter space in the case of the ‘z’
initial condition than in the case of the ‘x’ initial condition, for which the
propagation region is more extended. The interaction (13) acts very
differently on up and plus states; for instance, when applied to a spin up in
a uniform position state, it gives a state that only change the $z$ spin
component, at variance, when applied to a $\ket{+}$ state all spin components
are affected. We discuss in the next section the origin of this dynamical
anisotropy. In a spatially uniform state the homogeneous spin up state, but
not the plus state, is an eigenvector of $U$. The restricted spin dynamics in
the ‘z’ case facilitates then localization of the walk.

To complement this view of the parameter phase space based on the particle
motion, it is interesting to explore the entanglement as a function of
$(\theta,J)$. This is given in Fig. 23, where the von Neumann entropy of the
$(xcs)$ degrees of freedom is depicted. The left column corresponds to ‘x’,
and the right one to ‘z’. The first row corresponds to $J=\pi/2$, and the
second one to $J=\pi/10$. We observe that, for strong coupling, there is a
tendency of a weaker entanglement between the particle and spin degrees of
freedom in the localized region for the ‘x’ initial state, and a stronger
entanglement in the propagation region; for ‘z’, the entanglement is maximum
at the transition between the two regimes. In the case of a weaker coupling,
entanglement is globally smaller, with comparable values of the position and
spin entropies. The small angle part of the ‘z’ $J=\pi/10$ is biased by finite
size effects, even if the decreasing entanglement with the angle is
qualitatively correct (c.f. Fig. 5). The vanishing of the entanglement for
$\theta=\pi/2$, ‘z’, is simply a consequence that this initial state is a
proper state of $U$. By extension, the dynamics in the propagation region of
‘z’, is regular showing smooth slightly damped oscillation for $J\lesssim 1$,
and relaxation for larger $J$.

In the region of strong entanglement, we verified that the spin entropy of a
set of spins is proportional to its size. We show $S_{A}$ as a function of
$|A|$ in Fig. 24 (red dots), where we compare it to the Don Page entropy [96]
of a pure random state (gray dots),

$S_{R}=\log(D_{A})-\frac{D_{A}^{2}}{2D_{V}\ln 2},\quad
D_{A}=2^{|A|},\,D_{V}=2|V|2^{|V|}$ (32)

which, for our range of parameters is close to $S_{R}\approx|A|$. We find for
$J>1$ that the state of maximum spin entanglement is an extensive variable,
well described by a random pure global state. However, random maximally
entangled states disappear for weaker couplings, leading instead to smaller
entanglement, which follows a sublinear increase law with the subsystem size,
as can be appreciated in Fig. 24, where we also show $S_{A}$ for $J=0.3$ (blue
dots, the system size is $|V|=15$).

Before closing this section on the phenomenology, let us mention that the
initial spin entangled state ‘e’ imposes a constraint on the absolute value of
the mean spin $\bar{s}$, which is zero at $t=0$, very different from the other
initial states ($\bar{s}(0)=1$). One observes the emergence of persistent
irregular oscillations, whose period is also roughly given by $\sim|V|/J$ and
an amplitude generally smaller than the mean particle density, with a complex
spatial distribution. At strong coupling in the propagation region, the spin
polarization sticks to the particle propagation. At variance to the initial
product states, no relaxation towards a stationary state is found.

##  4 Discussion

###  4.1 Free walk

We consider first the free walk $J=0$. With periodic boundary conditions, the
evolution operator reduces to a translation invariant walk operator
$U(\theta,0)=W(\theta)=MR(\theta)$. Therefore, the evolution operator $W$ can
be diagonalized in Fourier space. We write, using (16), the motion operator,

$M=\sum_{k}\ket{k}\bra{k}\otimes M(k)\,,$

where $k\in(-\pi,\pi)$ is the quasimomentum, and

$M(k)=\mathrm{e}^{-\mathrm{i}kX}=\begin{pmatrix}0&\mathrm{e}^{-\mathrm{i}k}\\\
\mathrm{e}^{\mathrm{i}k}&0\end{pmatrix}\,.$ (33)

Then, after multiplying by $R(\theta)$, the walk operator (in Fourier space)
is given by the matrix,

$W(k,\theta)=\begin{pmatrix}\mathrm{e}^{-\mathrm{i}k}\sin\theta&\mathrm{e}^{-\mathrm{i}k}\cos\theta\\\
\mathrm{e}^{\mathrm{i}k}\cos\theta&-\mathrm{e}^{\mathrm{i}k}\sin\theta\end{pmatrix}=\mathrm{e}^{-\mathrm{i}H_{0}(k,\theta)+\mathrm{i}\pi/2}\,,$
(34)

where we introduced the free effective Hamiltonian,

$H_{0}(k,\theta)=E(k,\theta)\,\hat{\bm{d}}(k,\theta)\cdot\bm{\tau}\,,$ (35)

with

$\cos E(k,\theta)=-\sin
k\sin\theta\,,\quad\hat{\bm{d}}(k,\theta)=\frac{1}{\sin
E(k,\theta)}\begin{pmatrix}\cos k\cos\theta\\\ \sin k\cos\theta\\\ \cos
k\sin\theta\end{pmatrix}\,,$ (36)

where $E(k,\theta)$ is the quasienergy spectrum, and $\hat{\bm{d}}$ a momentum
like unit vector (in analogy with a spin-orbit coupling). The free spectrum
displays two bands separated by a gap which is maximum for $\theta=0,\pi$, and
closes for $\theta=\pi/2$. At $\theta=\pi/2$ the spectrum is similar to the
massless Dirac cone.

Figure 25: Group velocity as a function of $p=k-\pi/2$ (a) and $\theta$ (b).

In Fig. 25 we represent the group velocity as a function of the shifted
wavenumber $p=k-\pi/2$ and the angle $\theta$,

$v_{g}(p,\theta)=\frac{\sin p\sin\theta}{\sqrt{1-\cos^{2}p\sin^{2}\theta}}\,.$
(37)

The group velocity vanishes for $\theta=0$ and tends to a constant $v_{g}=1$
for $\theta=\pi/2$; between these two extreme values the wave propagation is
dispersive; as a function of $\theta$, it is an increasing function. These two
regimes qualitatively follow the behavior observed in the interacting case for
small $J$, as shown in Figs. 2, in which we found recurrent oscillations in
the case of a slowly moving walker and damped oscillations with high
entanglement in the case of a propagating walker. This picture is also
consistent with the observation of localized states for angles smaller than
$\theta\sim\pi/4$. Nevertheless, in order to understand the rich phenomenology
of the system we must take into account the interaction.

###  4.2 Homogeneous system

We consider first the simplest situation in which the particle and spin
distributions are initially homogeneous, implying that they remain homogeneous
because of translation invariance. Under homogeneous conditions, the spatial
dynamics becomes trivial and the corresponding motion operator essentially
decouples from the other degrees of freedom. The consequence is that we can
reduce $U$ to the color-spin coupling and characterize the particle by its
momentum $p$. Moreover, for large times and most parameters (in the
homogeneous case), we observe a smooth temporal evolution of the spin density
$\bm{s}(t)$, which is only function of the time. Therefore, it is natural to
assume a mean field separation of scales, and split the evolution operator:

$U_{t}(\Delta t)=V_{t}(J)W(p,\theta)=\mathrm{e}^{-\mathrm{i}H_{J}(t)\Delta
t}\mathrm{e}^{-\mathrm{i}H_{0}\Delta
t}\approx\mathrm{e}^{-\mathrm{i}H(t)\Delta t}\,,$ (38)

where $W=M(p)R(\theta)$,

$H=H_{0}(p,\theta)+H_{J}(t),\quad
H_{J}(t)=-\frac{J}{4}\bm{s}(t)\cdot\bm{\tau}$ (39)

and $H_{0}$ is given by (35). Note that $U$ is here the one time step
operator, acting at time $t$ in which we replaced the operator $H_{J}$ of (12)
by a particle operator, reducing the problem to a selfconsistent determination
of the mean field $\bm{s}(t)$: we do not assume the splitting (38) holds for
all times, but only over one step on which we must _update_ the mean field
$\bm{s}(t)$. Actually, in the continuous time limit the Heisenberg equation
for the time evolution of the expected value
$\bm{s}(t)=\braket{\bm{\sigma}^{\otimes}{V}{}}{\equiv}{\braket{missing}}{\bm{\sigma}}$
is,

$\frac{\mathrm{d}}{\mathrm{d}t}\braket{\bm{\sigma}}=\frac{J}{2}\braket{\bm{\sigma}\times\bm{\tau}}\,;$
(40)

the approximation consists in replacing $\bm{\sigma}$ by the mean field, on
the right hand side:

$\frac{\mathrm{d}}{\mathrm{d}t}\bm{s}(t)=\frac{J}{2}\bm{s}(t)\times\braket{\bm{\tau}}\,,$
(41)

where $\braket{\bm{\tau}}$ is computed from an appropriated evaluation of the
quantum state evolved with (38). Equation (41) is a simple form of the Landau-
Lifshitz equation, in which the magnetization precession is driven by the
exchange of torque with the walker [97, 98]:

$\dot{\bm{s}}=\frac{J}{2}\bm{s}\times\braket{\bm{\tau}},\quad\braket{\bm{\tau}}(t)=\braket{\psi(t)}{\bm{\tau}}{\psi(t)}$
(42)

where $\ket{\psi(t)}$ is the system’s state at time $t$, which must be deduced
from the Schrödinger equation with Hamiltonian (39), and $\braket{\bm{\tau}}$
is the effective particle ‘magnetic momentum’. In brief, for homogeneous
systems, the large time scale spin dynamics is supposed to be driven by the
itinerant particle spin and satisfies the Landau-Lifshitz equation; the
interaction is selfconsistently computed solving the corresponding Schrödinger
equation at each time step.

We start with the lowest order solution (in powers of $J$), in which we can
solve the free particle dynamics, and apply the corresponding state to the
computation of the effective torque. We determine first the eigenstates. To
diagonalize the Dirac walk Hamiltonian,

$H_{0}=\bm{d}(p,\theta)\cdot\bm{\tau}\,,$

it is convenient to parametrize the vector $\bm{d}=E\hat{\bm{d}}$ (36) that
actually we might consider be arbitrary, by the spherical angles
$(\alpha,\beta)$:

$\bm{d}=|\bm{d}|(\sin\alpha\cos\beta,\sin\alpha\sin\beta,\cos\alpha)\,,$

functions of $(p,\theta)$; with this definition the wave function of a free
particle (in the positive energy band of $H_{0}$) is,

$\psi_{\\!+}(x,t)=\frac{1}{\sqrt{|V|}}\chi_{\\!+}(p,\theta)\mathrm{e}^{-\mathrm{i}E_{+}t+\mathrm{i}px}\,,$
(43)

where

$E_{\pm}=\pm|\bm{d}|$ (44)

are the positive and negative energy bands (c.f. (36)), the spinor of positive
energy is,

$\chi_{\\!+}=\begin{pmatrix}\cos\frac{\alpha}{2}\\\
\sin\frac{\alpha}{2}\,\mathrm{e}^{\mathrm{i}\beta}\end{pmatrix}\,,$

and similar expressions for the negative energy wave function.

We can now apply the solution (43) of the free Dirac walk to the computation
of $\braket{\bm{\tau}}$. We readily obtain:

$\braket{\bm{\tau}}=\psi_{\\!+}^{\dagger}\bm{\tau}\psi_{\\!+}=\frac{1}{|V|}\frac{\bm{d}}{|\bm{d}|}\,,$

which leads to the equation

$\dot{\bm{s}}=-\frac{J}{2|V|}\bm{s}\times\frac{\bm{d}}{|\bm{d}|}\,.$ (45)

(the minus sign corresponds to our choice of the rotation direction, see
(44)). This is an interesting result, it predicts that in the homogeneous case
the fixed spins will oscillate with a period of $T=4\pi|V|/J$, which we
observed in (29). However, this result supposes decoupling between the
particle and spins, as implied by the neglect of correlations in (42) and the
free particle state approximation. In particular, it cannot account for the
relaxation of the spin, note that the Landau-Lifshitz equation preserves the
norm of $\bm{s}(x,t)$.

To go further, we investigate the particle-spin coupling in the homogeneous
case, using the next order approximation in the small $J$ limit. We use the
standard time dependent perturbation theory to find the $J$ dependent
correction to the wave function of the free particle of momentum $p$. Taking
$H_{J}$ as the perturbation of the Dirac Hamiltonian $H_{0}$:

$H=H_{0}-\frac{J}{4}\bm{s}(t)\cdot\bm{\tau}$

where the mean spin density is only time dependent, we may compute the
evolution of the state over one step $\Delta t=1$, assuming that during this
time the spin density is almost constant. We write the correction to the wave
function (43) of a particle of energy $E_{p}>0$, as a superposition in the
nonperturbed basis $\psi_{E}$,

$\psi(x,t)=\sum_{E}\mathrm{e}^{-\mathrm{i}Et}\phi_{E}(t)\psi_{E}(x)$

which satisfies,

$\phi_{E}(t+\Delta
t)=\phi_{E}(t)-\frac{J}{4\mathrm{i}}\sum_{E^{\prime}}\int_{t}^{t+\Delta
t}\mathrm{d}t^{\prime}\,\mathrm{e}^{\mathrm{i}(E-E^{\prime})t^{\prime}}\bm{s}(t^{\prime})\cdot\bm{\tau}\phi_{E^{\prime}}(t^{\prime})\,.$
(46)

To solve this equation we observe that during one time step the energy
exchange $E-E^{\prime}\approx\Delta E$ between the mean spin field and the
walker must be small, in accordance with the slow variation of $\bm{s}(t)$;
hence, we put

$\phi_{E^{\prime}}(t^{\prime})\approx\mathrm{e}^{-\mathrm{i}\Delta
Et^{\prime}}\phi_{E^{\prime}}(t)\delta_{EE^{\prime}}\sim\mathrm{e}^{-\mathrm{i}\Delta
Et}\chi_{E^{\prime}}\delta_{EE^{\prime}}\,,$

within the integral (the last approximation takes into account the correction
to the first order in $J$, up to a factor of norm one depending on $t$, which
do not contributes to the final result):

$\displaystyle\phi_{E}(\Delta t)$ $\displaystyle=\phi_{E}(t+\Delta
t)-\phi_{E}(t)=-\frac{J}{4\mathrm{i}}\int_{t}^{t+\Delta
t}\mathrm{d}t^{\prime}\,\mathrm{e}^{\mathrm{i}(E-E^{\prime}-\Delta
E)t^{\prime}}\bm{s}(t^{\prime})\cdot\tau\chi_{E^{\prime}}$
$\displaystyle\approx-\frac{J}{4\mathrm{i}\sqrt{|V|}}\mathrm{e}^{\mathrm{i}(E-E^{\prime}-\Delta
E)t}\frac{\mathrm{e}^{\mathrm{i}(E-E^{\prime}-\Delta E)\Delta
t}-1}{\mathrm{i}(E-E^{\prime}-\Delta
E)}\bm{s}(t)\cdot\bm{\tau}\chi_{E^{\prime}}$ (47)

where we the energy variation satisfies $\Delta E\sim J/2|V|\ll 1/\Delta t$
(the time scale of the typical spin oscillations). The expected value of the
particle’s spin, to first order, is

$\displaystyle\braket{\bm{\tau}}^{(1)}$
$\displaystyle=\int\mathrm{d}Ef(E)\phi_{E}^{\dagger}(\Delta
t)\bm{\tau}\phi_{E}(\Delta t)$ $\displaystyle=\left.\frac{J^{2}\Delta
t}{8|V|v_{g}}|\bm{s}(t)|^{2}\frac{\bm{d}}{|\bm{d}|}\right|_{E-\Delta E}$ (48)

where we used the formula $f(E)=1/\pi v_{g}(E)$ for the energy distribution.
We suppose that $\braket{\bm{\tau}}^{(1)}$ is the dissipation source in the
Landau-Lifshitz equation. To the lowest order in $J$, the form of the
dissipation term $\bm{D}$ should be [99, 100]:

$\bm{D}=\braket{\dot{\bm{s}}\times\bm{\tau}}^{(1)}=\frac{J}{2}\braket{\bm{\tau}}^{(1)}\times(\braket{\bm{\tau}}\times\bm{s})\,.$
(49)

This term has the form of the relativistic correction introduced by Landau and
Lifshitz, but with the moment $\bm{s}$ replaced here by $\braket{\bm{\tau}}$,
which introduces a damping of the magnitude of $\bm{s}$ (it has a component
parallel to $\bm{s}$). The microscopic origin of the dissipation is related to
the leaking of torque from the fixed spins to the particle’s “magnetic
momentum”, which is here a fast variable and acting as a stochastic torque,
effectively giving a term as in (49) [101]. The Landau-Lifshitz equation
becomes,

$\dot{\bm{s}}=-\frac{J}{2}\bm{s}\times\braket{\bm{\tau}}+\bm{D}=-\frac{J}{2|V|}\bm{s}\times\hat{\bm{d}}+\frac{J^{3}\Delta
t}{16|V|v_{g}}|\bm{s}|^{2}\hat{\bm{d}}\times\left(\hat{\bm{d}}\times\bm{s}\right)\,,$
(50)

where the precession term may include the correction term (48)
$\braket{\bm{\tau}}=\hat{\bm{d}}/|V|(1+O(J^{2}))$ that we neglect in the
numerical comparisons. We show in Fig. 26 a comparison of the approximation
(50) with the exact evolution of the homogeneous cases ‘ix’ and ‘iz’. The
approximation gives satisfactory qualitative results, in particular it
contains a mechanism to limit the relaxation to finite values of the spin
magnitude due to the presence of the nonlinearity in $|\bm{s}|^{2}$ and the
progressive alignement of $\bm{s}$ and $\hat{\bm{d}}$.

Figure 26: Landau-Lifshitz. Spin oscillations damping in the case of ‘ix’
(a,b) and ‘iz’ (c,d) compared to (50) (dashed lines). Parameters $1,0.4$,
$N=13$. The energy of the initial condition corresponds to a wavenumber of
$p=6\pi/13$.

###  4.3 Small gradients

We may extend the analysis of the homogeneous system to the case of small
gradients; we naturally have a microscopic scale $\Delta x=1$ and a
macroscopic one, given by the size of the system $|V|$, hence small gradients
mean spatial variation $\ell$ of the spin distribution satisfying
$1\ll\ell\ll|V|$. If the homogeneous case is characterized by a momentum $p$,
the presence of $\ell$ scale fluctuations, can be taken into account by
introducing a modulation $\Delta p$ of the wave function: $p\rightarrow
p+\Delta p$. We can thus associate the momentum modulation with the spatial
gradient, $\Delta p\rightarrow-\mathrm{i}\partial_{x}$. The splitting of the
one step operator is straightforwardly generalized to the position dependent
case, giving the new mean field $\bm{s}=\bm{s}(x,t)$:

$H=H_{0}+H_{J}\,,\quad H_{J}=-\frac{J}{4}\bm{s}(x,t)\cdot\tau\,,$ (51)

and $H_{0}$ the corrected to first order in $\Delta p$ particle Hamiltonian:

$H_{0}=E(p,\theta)\bm{d}_{0}\cdot\tau+\Delta p\bm{d}_{1}\cdot\tau\,,$ (52)

where $H_{0}$ is the homogeneous Hamiltonian (35) in terms of $p$,

$\cos E=\cos p\sin\theta\,,$

is the energy dispersion relation,

$\bm{d}_{0}=\frac{1}{\sin E(p,\theta)}\begin{pmatrix}-\sin p\cos\theta\\\ \cos
p\cos\theta\\\ -\sin p\sin\theta\end{pmatrix}\,,$

the mass term coefficient, and

$\bm{d}_{1}=-\frac{E(p,\theta)}{\sin E(p,\theta)}\begin{pmatrix}\cos
p\cos\theta\\\ \sin p\cos\theta\\\ \cos p\sin\theta\end{pmatrix}\,,$

the coefficient of the momentum term. We remark that $H_{0}$ reduces to the
continuum limit of the Dirac walk when $p=0$ and the energy is supposed to be
small $E\Delta t\rightarrow 0$ [102], which is different to the present
approximation around a uniform distribution defined by a plane wave of
arbitrary momentum $p$ and energy $E$. Up to a global rotation transformation,
(52) is a Dirac Hamiltonian with mass term $\bm{d}_{0}$ and kinetic term
$\bm{d}_{1}$. Within this ‘hydrodynamic’ like approximation, we actually
neglect the distinction between the node and link operators, which will give
higher order corrections to the one step evolution operator. One might expect
that the validity of the hydrodynamic approximation will be better in the
parameter range where the coupling constant is small and particle propagation
dominates over localization. The model (51) is reminiscent of the so-called
_sd_ Hamiltonian which describes the magnetic interaction between independent
spins mediated by itinerant electrons [68]. The difference here is that the
particle is governed by a massive Dirac equation whose mass and velocity
parameters depend on the angle $\theta$, instead of the usual kinetic energy,
quadratic in momentum.

Figure 27: Behavior of the spin set entropy $S_{A}$ for $|A|=6$, as a function
of the coupling constant: $J=0.1,0.2,0.3,0.4,0.6$ and $\theta=\pi/2$, ‘x’. The
straight lines fit the slope with formula (53).

In contrast to the homogeneous case, when gradients in the spin distribution
mediated by the running particle are present, the system’s dynamical
properties (oscillations, relaxation and entanglement growth) change
qualitatively. For instance, a linear increase of the spin entanglement arises
for times shorter than the relaxation time, in the propagation dominated case.
An asymptotic relaxation towards a zero spin state is also observed, while
persistent oscillations prevailed in the homogeneous case (for similar
parameters). An illustration is given in Fig. 27, where we picture the spin
entanglement of a set of $|A|=6$ spins in a system of $|V|=13$ sites. We fit
the linear part of the entanglement growth by the interpolation formula,

$S_{A}(t)\approx\nu_{J}t\,,\quad\nu_{J}=\frac{J^{2}}{|V|}\,,$ (53)

which holds for time $t$ larger than the initial transitory, whose duration is
of the order of the system’s size $t>|V|$, and time $t$ smaller than the
characteristic saturation time $t\lesssim 1/\nu_{J}$. (We may think that the
slope in (53) depends on $|A|$, but for the moment we stress only its
quadratic dependence on $J$.) We are interested in deriving a Landau-Lifshitz
equation for the spin dynamics taking the influence of the gradients into
account, in particular to explain the origin of the $J^{2}/|V|$ effective
parameter (53). To this goal we proceed, as we did in the homogeneous case,
with a kind of linear response calculation based on a multiscale expansion.

We write the approximate solution of the Dirac equation of Hamiltonian (51),
for a time of the order of one time step $(t,t+\Delta t)$, as a spatially
modulated plane wave,

$\psi(x,t)=\mathrm{e}^{-\mathrm{i}Et+\mathrm{i}px}\phi(X,t)\,,$ (54)

where $\phi$ varies significantly over a spatial scale $X\sim 1/\Delta
p\sim\ell$. The form (54) separates fast and slow variables as in the Krilov-
Bogoliubov-Mitropolsky method used in dynamical systems [103, 104]. The
evolution equation of the modulation is then given by,

$\mathrm{i}\frac{\mathrm{d}\phi}{\mathrm{d}t}+\mathrm{i}\partial_{X}\left(\bm{d}_{1}\cdot\bm{\tau}\phi\right)=-\frac{J}{4}\bm{s}(X,T)\cdot\bm{\tau}\phi\,,$
(55)

where we explicitly put the slow variables dependence of the spin density,
with the slow time scale $T\sim 1/\Delta E$. We obtain the solution of (55) in
terms of the Fourier transform in the slow variables, to first order in $J$:

$\phi(X,t)=-\frac{J}{4\sqrt{|V|}}\int\frac{\mathrm{d}E}{2\pi}\frac{\mathrm{d}\Delta
p}{2\pi}\frac{\mathrm{e}^{-\mathrm{i}Et+\mathrm{i}\Delta pX}}{E-\Delta
p\bm{d}_{1}\cdot\bm{\tau}}\bm{s}_{\Delta}\cdot\bm{\tau}\chi\,,$ (56)

where $\bm{s}_{\Delta}$ is the Fourier transform of the spin field. We are
only interested in the effect of the gradient; using the inverse of the
matrix,

$\frac{1}{E-\Delta p\bm{d}_{1}\cdot\bm{\tau}}=\frac{E+\Delta
p\bm{d}_{1}\cdot\bm{\tau}}{\sqrt{E^{2}-\Delta p^{2}|\bm{d}_{1}|^{2}}}\,,$

to first order in $\Delta p$, we obtain,

$\phi(X,t)=\frac{J}{4\sqrt{|V|}}\int\frac{\mathrm{d}E}{2\pi\mathrm{i}}\frac{\mathrm{e}^{-\mathrm{i}Et}}{E^{2}}\int\frac{\mathrm{d}\Delta
p}{2\pi}\,\mathrm{i}\Delta p\mathrm{e}^{\mathrm{i}\Delta
pX}(\bm{d}_{1}\cdot\bm{\tau})(\bm{s}_{\Delta}\cdot\bm{\tau})\chi\,.$ (57)

Over the time $\Delta t$ the spin density is almost constant ($\Delta t\ll
1/\Delta E$), we can then evaluate the energy integral to get the final
result,

$\phi(X,\Delta t)=\frac{J\Delta
t}{8\sqrt{|V|}}\left[(\bm{d}_{1}\cdot\partial_{x}\bm{s})+\bm{d}_{0}\cdot(\bm{d}_{1}\times\partial_{x}\bm{s})\right]\chi\,,$
(58)

(the momentum integral gives the gradient) where we used
$\chi^{\dagger}\bm{\tau}\chi=\bm{d}_{0}$. We apply now (58) to calculate the
response of the particle’s spin to the spin density gradient:

$\braket{\bm{\tau}}^{(1)}=\braket{\psi_{\\!+}(x,0)}{\bm{\tau}}{\psi(x,\Delta
t)}=\frac{J\Delta
t}{8|V|}\left[(\bm{d}_{1}\cdot\partial_{x}\bm{s})+\bm{d}_{0}\cdot(\bm{d}_{1}\times\partial_{x}\bm{s})\right]\bm{d}_{0}\,,$
(59)

where we substituted (43) and (54), (58) to $\psi_{\\!+}$ and $\psi$,
respectively (note that in the homogeneous case this first order contribution
to the torque vanished, giving rise to a higher order correction). Within the
same order we compute the correction to the interacting energy,

$E_{\text{int}}=-\frac{J^{2}}{32|V|}\int\mathrm{d}x\,(\bm{s}\cdot\bm{d}_{0})\left[(\bm{d}_{1}\cdot\partial_{x}\bm{s})+\bm{d}_{0}\cdot(\bm{d}_{1}\times\partial_{x}\bm{s})\right]\,.$
(60)

Finally, after computing the functional derivative of (60),

$\dot{\bm{s}}=2\bm{s}\times\left(-\frac{\delta
E_{\text{int}}}{\delta\bm{s}}\right)\,,$ (61)

we can write the modified Landau-Lifshitz equation:

$\dot{\bm{s}}=-\frac{J}{2|V|}\bm{s}\times\bm{d}_{0}+\frac{J^{2}\Delta
t}{8|V|}\left[(\bm{d}_{0}\times\bm{d}_{1}+\bm{d}_{0}\times(\bm{d}_{0}\times\bm{d}_{1})\right]\times(\bm{s}\times\bm{\partial}_{x}\bm{s})+\bm{D}\,,$
(62)

which contains a correction to the precession frequency,

$\frac{J}{2|V|}\rightarrow\frac{J}{2|V|}+\frac{J^{2}\Delta
t}{4|V|}(\bm{d}_{1}\cdot\partial_{x}\bm{s})+\frac{J}{2}\bm{d}_{0}\cdot(\bm{d}_{1}\times\partial_{x}\bm{s})\,,$
(63)

and a correction to the effective applied field,

$\bm{d}_{0}\rightarrow\bm{d}_{0}-\frac{J^{2}\Delta
t}{4|V|}\left[\bm{d}_{1}+(\bm{d}_{0}\times\bm{d}_{1})\right](\bm{d}_{0}\cdot\partial_{x}\bm{s})\,,$
(64)

and where we added a dissipation term $\bm{D}$, which should also be of order
$O(J^{2})$ (c.f. (49) with the particle spin given by (59)). Therefore, within
this approximation, the effect of the gradient translates into a nonlinear
shift of the precession frequency, which contributes to an anharmonic
evolution of the spin oscillations, together with a first order in the spin
gradient force, whose common physical origin is the scattering of the walker
wave function off the mean field spin inhomogeneities. We note a fundamental
difference with respect to the usual exchange interaction, which is
proportional to the second derivative of the magnetization; the added force is
reminiscent to the spin-orbit terms found in some materials [105, 106].

Figure 28: Comparison of the homogeneous $\partial_{x}=0$ and inhomogeneous
$\partial_{x}\neq 0$ evolution of the spin $\bar{s}_{x}$ (a) and spin
entanglement $S_{A}$ (b). Parameters $(1.5,0.3)$, ‘x’, $|A|=6$, $N=13$. The
dashed lines are linear fits with slope $\nu=J^{2}/N\sim J^{2}$
($\partial_{x}\neq 0$, (53)), and $(J/2)\nu\sim J^{3}$ ($\partial_{x}=0$).

In a system of only a dozen of sites it is difficult to compare quantitatively
(62) with the exact evolution of the interacting walk (gradients extend over a
few nodes); however, it is possible to test the parameter dependency in
$J^{2}$ (already observed in Fig. 27). We compare the uniform and nonuniform
time evolution of the spin density and spin entanglement entropy in Fig. 28.
It is remarquable that the simple multiplication $(J/2)\nu$, with
$\nu=J^{2}/|V|$ (the natural factor coming from the Landau-Lifshitz equation),
fits so well the homogeneous case, comforting the relevance of the second
order correction. In the same figure we show $\bar{s}_{x}$ for ‘x’, which
relaxes faster than the ‘ix’ case, in agreement with the behavior of $S_{A}$.
In addition, the anisotropy of the oscillation (for instance, the $z$
component stay at zero in Fig. 2) is a consequence of the anisotropy of the
unit vector $\hat{\bm{d}}=\bm{d}_{0}$ ($x$ and $y$ components vanish for
$\theta=\pi/2$).

Equation (62) can describe the oscillation-relaxarion regime in an effective
external field which to lowest order $\bm{\tau}\sim\bm{d}_{0}$ is constant,
and whose corrections are perturbative in the coupling $J$. The strong $J$
regime would need a different approximation approach, for instance a kind of
adiabatic approximation in which the particle’s torque follows the spin’s one,
valid in principle for the strong coupling regime. We leave this investigation
for future work.

##  5 Conclusions

In this paper we investigated a generalization of the quantum walk to a many-
body interacting system, in order to build an entangled state from the
dynamics of a walker coupled to a network of spins. This approach may be
compared with the usual adiabatic model of quantum computing [107, 108, 109,
110]. Instead of adiabatically evolving a quantum system from an initial
Hamiltonian whose ground state is a product of basis states, to a useful
entangled ground state of another Hamiltonian, we obtain the resource state
from the unitary evolution of the system towards a stationary state
(stationary in the sense of its observables properties: magnetization,
entanglement, probability distribution). In its generality the model is
implemented on an arbitrary (simple) graph, in which the walker wanders
between neighboring nodes interacting with fixed spins located in the links.
The fixed spins are then coupled through the particle’s degrees of freedom,
position and color. The color degree of freedom represents internally the
connectivity of the graph. This model, loosely inspired by the condensed
matter magnetic metals in which the interaction is of the RKKY type governed
by a _sd_ -Hamiltonian, is characterized by an exchange interaction between
the color of the two nodes of a link, and the local spin. This interaction
differs from the more usual Ising-type interaction used to build cluster
states, because it depends on the angle between the particle and local spin
magnetic moments. In addition, the coin operator that modifies the particle’s
color, introduces an anisotropic redistribution of the state amplitudes
between the incoming edges of a vertex. This anisotropy has important
consequences for the dynamics of the system. Another motivation for the
present model comes from the experiments in which an entangled state of local
spins (nitrogen vacancies in diamond or magnetic moments of trapped cold
atoms, for instance) is obtained by an interaction mediated by a nonlocal
moment [61, 111], or by the interaction with the environment in a spin chain
[112], which can be useful for the construction of quantum networks [113, 60].

The present work focused on the simplest geometry, the one dimensional lattice
with closed or periodic boundary conditions. Nevertheless, we observed a
fairly rich dynamical phenomenology even for this simple case in which we
consider the Dirac walker coupled to non interacting spins through its color.
We could exhibit a variety of behaviors, reminiscent to a magnetic chain in
which, taking into account that the motion of the walker is correlated to its
internal state, a magnetic interaction arises mediated by a kind of spin-orbit
coupling. We determined using direct numerical computation by successive
application of the one step evolution operator, the parameter phase space
regions of localization and propagation, and explored the weak and strong
coupling regimes, the regular and irregular dynamics as well as the influence
of the interaction on the walk topological properties. We found that the
magnetic dynamics, for homogeneous and weakly inhomogeneous states, is well
described by a quasiclassical mean field Landau-Lifshitz equation with
dissipation, nonlinerar frequency shift and torques proportional to the
magnetization gradient (c.f. Eq. (62)). Both, dissipation and nonlinear
torques, arise from the scattering off on the fixed spins of the walker wave
function; the hydrodynamic like approximation applies when the temporal and
spatial scales characteristics of the walker and the edge spins can be well
separated. Let us mention that as a by-product of the low energy approximation
we generalized the usual method to derive the continuum limit of the free
quantum walk, Eq. (52).

We found that the interacting many-body quantum walk leads for a large range
of parameters, to highly nonlocal entangled states, with an entropy
proportional to the size of the entangled region. In particular, we
demonstrated that it gives rise to localized edge states at the interface
between topologically different phases, in contrast to the free walk. The
entanglement dynamics and its structure are governed by the interplay of the
local degrees of freedom, the fixed spins, and the global one, the walker
whose distribution is delocalized over the entire system. We may conclude that
this system has properties that are relevant to the quantum simulation of
condensed matter systems as well as to the build up of entangled states of
interest as a quantum computational resource.

## Acknowledgements

We benefited from discussions with Álvaro Núñez, Giuseppe Di Molfetta, Gabriel
Elías and Pablo Arrighi. Numerical calculations used python with the libraries
numpy and scipy. Figures were produced with matplotlib [114].

##  Appendix A The one step operator $U$

In this Appendix we will exhibit the structure of the one step operator $U$ by
applying it to the canonical basis of the Hilbert space
$\ket{xcs}\in\mathcal{H}_{G}$, in the case where $G$ is a linear lattice. Let
$\ket{\psi}$ be a general state,

$\ket{\psi}=\sum_{xcs}\psi_{xcs}\ket{xcs},\quad\psi_{xcs}\in\mathbb{C}\,.$
(65)

We compute $U\ket{\psi}=V(J)MC\ket{\psi}$, using the definitions (6)-(13),

$U\ket{\psi}=V(J)\sum_{xs}\big{(}\cos\theta\,\psi_{x+1\,0s}-\sin\theta\,\psi_{x1s}\big{)}\ket{x0s}+\big{(}\sin\theta\,\psi_{x0s}+\cos\theta\,\psi_{x1s}\big{)}\ket{x-1\,0s}$
(66)

where we used the rotation matrix (7) and the definition of the motion
operator (6) (for the one-dimensional case). The action of $V(J)$ is
conveniently expressed in the edge basis (10), we introduce then the notation
$s_{x}(0)$ for the set of spins $s=1,\ldots,2^{|E|}-1$ with the label
$s_{x}=0$ (corresponding to the edge $e=(x,x+1)$), idem $s_{x}=1$:

$\displaystyle U\ket{\psi}=$
$\displaystyle\mathrm{e}^{-\mathrm{i}J/4}\sum_{x,s}\left\\{\mathrm{e}^{\mathrm{i}J/2}\big{(}\sin\theta\,\psi_{x+1\,0s_{x}(0)}+\cos\theta\,\psi_{x+1\,1s_{x}(0)}\big{)}\ket{x\,0s_{x}(0)}+\right.$
$\displaystyle\left[\mathrm{i}\sin(J/2)\big{(}\cos\theta\,\psi_{x\,0s_{x}(0)}-\sin\theta\,\psi_{x\,1s_{x}(0)}\big{)}+\right.$
$\displaystyle\left.\cos(J/2)\big{(}\sin\theta\,\psi_{x+1\,0s_{x}(1)}+\cos\theta\,\psi_{x+1\,1s_{x}(1)}\big{)}\right]\ket{x0s_{x}(1)}+$
$\displaystyle\left[\cos(J/2)\big{(}\cos\theta\,\psi_{x\,0s_{x}(0)}-\sin\theta\,\psi_{x\,1s_{x}(0)}\big{)}+\right.$
$\displaystyle\left.\mathrm{i}\sin(J/2)\big{(}\sin\theta\,\psi_{x+1\,0s_{x}(1)}+\cos\theta\,\psi_{x+1\,1s_{x}(1)}\big{)}\right]\ket{x+1\,1s_{x}(0)}+$
$\displaystyle\left.\mathrm{e}^{\mathrm{i}J/2}\big{(}\cos\theta\,\psi_{x\,0s_{x}(1)}-\sin\theta\,\psi_{x\,1s_{x}(1)}\big{)}\ket{x+1\,1s_{x}(1)}\right\\}\,.$
(67)

We observe that each term of the sum contains the basis vectors of the edge
(11) with coefficients depending on the eight amplitudes corresponding to the
two nodes and edge spin, therefore, coupling effectively the edge $e_{x}$ with
its two neighbors $e_{x\pm 0}$. One may condensate this structure into a
formula which gives the amplitudes of edge $e_{x}$ in terms of the amplitudes
of its neighbors:

$\ket{\psi_{e_{x},s_{x}}(t+1)}=A\ket{\psi_{e_{x-1},s_{x}}(t)}+B\ket{\psi_{e_{x},s_{x}}(t)}+C\ket{\psi_{e_{x+1},s_{x}}(t)}\,,$
(68)

where,

$\ket{\psi_{e_{x},s_{x}}}=\begin{pmatrix}\psi_{x\,00_{x}}\\\
\psi_{x\,01_{x}}\\\ \psi_{x+1\,10_{x}}\\\ \psi_{x+1\,11_{x}}\end{pmatrix}$

and the matrices $(A,B,C)$ are given by,

$A=\mathrm{e}^{-\mathrm{i}J/4}\begin{pmatrix}0&0&0&0\\\
0&0&-\mathrm{i}\sin(J/2)\sin(\theta)&0\\\ 0&0&-\cos(J/2)\sin(\theta)&0\\\
0&0&0&-\mathrm{e}^{\mathrm{i}J/2}\sin(\theta)\end{pmatrix},$ (69)
$C=\mathrm{e}^{-\mathrm{i}J/2}\begin{pmatrix}\mathrm{e}^{\mathrm{i}J/2}\sin(\theta)&0&0&0\\\
0&\cos(J/2)\sin(\theta)&0&0\\\ 0&\mathrm{i}\sin(J/2)\sin(\theta)&0&0\\\
0&0&0&0\end{pmatrix},$ (70)

and

$B=\mathrm{e}^{-\mathrm{i}J/4}\begin{pmatrix}0&0&\mathrm{e}^{\mathrm{i}J/4}\cos(\theta)&0\\\
\mathrm{i}\sin(J/2)\sin(\theta)&0&0&\cos(J/2)\cos(\theta)\\\
\cos(J/2)\cos(\theta)&0&0&\mathrm{i}\sin(J/2)\sin(\theta)\\\
0&e^{i\frac{J}{2}}\cos(\theta)&0&0\end{pmatrix}\,,$ (71)

note that the spin index is local to the $e_{x}$ edge. The fact that the spin
in $e_{x}$ appears in the terms of the $A$ and $C$ matrices, associated
respectively to the $e_{x-1}$ and $e_{x-1}$ neighbors, shows the typical
mixing of amplitudes of a unitary operator. Two adjacent spins are indirectly
coupled by the color of their common node, which splits between both and
forbids writing $U$ in a simple block form.

##  Appendix B $|V|=2$, full coupling

To enlighten the structure of the operator $U$ defining the step of the
interacting walk (17), and to study the mechanisms underlying the particle-
spin entanglement it is convenient to reduce the system to its simplest
nontrivial form. In this appendix we investigate the $|V|=|E|=2$ case, which
can be worked out analytically for a few steps.

The Hilbert space dimension is 16. The explicit expression of $U$ in the
$\ket{xcs}$ basis is useful for the computation of the observables, but do not
highlight the interaction structure which is related to the edges. In the edge
basis, consisting in sets of four labels of the form,

$e_{x}(s)=\\{x0s_{x},(x+1)1s_{x}\\}\,,\quad x=0,1,\;s=0,1$

$U$ can be build up form block of $4\times 4$ matrices; here
$0_{0}=\\{00,01\\}$, $0_{1}=\\{00,10\\}$, etc., for example:

$e_{0}(1)=\begin{pmatrix}0010\\\ 0011\\\ 1110\\\ 1111\end{pmatrix}\,,$

(compare with (11)). To construct the operator $U$ we use the method of
Appendix A. We write the recurrence (68) in the mixed base $e_{x}(s)$ and
$e_{x}(\bar{s})$, where $\bar{s}_{x}=s_{x+1}$ is a edge vector with the spin
of the next edge:

$\psi_{t+1}\\!\begin{pmatrix}e_{0}(0)\\\ e_{0}(1)\\\ e_{1}(0)\\\
e_{1}(1)\end{pmatrix}=(A+C)\,\psi_{t}\\!\begin{pmatrix}e_{1}(\bar{0})\\\
e_{1}(\bar{1})\\\ e_{0}(\bar{0})\\\
e_{0}(\bar{1})\end{pmatrix}+B\,\psi_{t}\\!\begin{pmatrix}e_{0}(0)\\\
e_{0}(1)\\\ e_{1}(0)\\\ e_{1}(1)\end{pmatrix}$ (72)

where we used the periodicity of the chain. Now the problem is to express
$e_{x}(\bar{s})=(e_{x},s_{x+1})$ in the edge basis;

$\displaystyle e_{1}(\bar{0})$ $\displaystyle=M_{00}e_{1}(0)+M_{01}e_{1}(1)$
$\displaystyle e_{1}(\bar{1})$ $\displaystyle=M_{10}e_{1}(0)+M_{11}e_{1}(1)$
$\displaystyle e_{0}(\bar{0})$ $\displaystyle=M_{00}e_{0}(0)+M_{01}e_{0}(1)$
$\displaystyle e_{0}(\bar{1})$ $\displaystyle=M_{10}e_{0}(0)+M_{11}e_{0}(1)$
(73)

where the transformation matrices $M_{sx}$ are given by,

$M_{00}=\begin{pmatrix}1&0&0&0\\\ 0&0&0&0\\\ 0&0&1&0\\\
0&0&0&0\end{pmatrix},\quad M_{01}=\begin{pmatrix}0&0&0&0\\\ 1&0&0&0\\\
0&0&0&0\\\ 0&0&1&0\end{pmatrix},\quad$

and,

$M_{10}=\begin{pmatrix}0&1&0&0\\\ 0&0&0&0\\\ 0&0&0&1\\\
0&0&0&0\end{pmatrix},\quad M_{11}=\begin{pmatrix}0&0&0&0\\\ 0&1&0&0\\\
0&0&0&0\\\ 0&0&0&1\end{pmatrix}\,.$

Substituting this transformation into (72) we obtain an expression in the edge
basis, with $U$:

$U=\begin{pmatrix}B&0&A_{00}&A_{01}\\\ 0&B&A_{10}&A_{11}\\\
A_{00}&A_{01}&B&0\\\ A_{10}&A_{11}&0&B\end{pmatrix}$ (74)

where

$A_{sx}=(A+C)M_{sx},\quad$

$(A,B,C$ are defined in (69)-(71). We have, on the diagonal, the (non unitary)
matrix $B$ proportional to $\cos\theta$, and off diagonal matrices $A,C$,
proportional to $\sin\theta$: this block three diagonal form become much more
cumbersome for an arbitrary number of nodes (in the sense of Appendix A); it
reflects the locality of the original operators, and implies that at each step
two new edges enter into play.

Figure 29: Spectrum (76) of the $|V|=2$ one step operator. (a) $\arg\lambda$
as a function of $\theta$ for $J=1$ (the dashed lines correspond to $J=0$);
(b) $\arg\lambda$ as a function of $J$ for $\theta=1$. The curves are
symmetric with respect to the vertical axes $\theta=0$ and $J=0$,
respectively.

The eigenvalues of $U$ are readily computed:

$\displaystyle\lambda_{\pm}$ $\displaystyle=\pm\mathrm{e}^{\mathrm{i}J/4}$
(75) $\displaystyle\lambda_{n}$
$\displaystyle=\pm\frac{\mathrm{e}^{-\mathrm{i}J/4}}{\sqrt{2}}\sqrt{2-\sin^{2}(J/2)\sin^{2}\theta\pm\mathrm{i}|\sin(J/2)\sin\theta|\sqrt{4-\sin^{2}(J/2)\sin^{2}\theta}}$
(76)

where each $\lambda_{\pm}$ is four fold degenerated, and each $\lambda_{n}$
(with $n=--,-+,+-,++$) is twice degenerated. Note that $\theta=0$ gives a
trivial system (off diagonal terms vanish), but $\theta=\pi/2$ contains
information about the interaction. In Fig. 29 we plot the argument of the
eigenvalues (quasienergies $E=\arg\lambda$) of $U$, as a function of the
parameters; the interaction partially lifts the degeneracy creating gaps that
grow with both $\theta$ and $J$. In this simple limit levels do not cross at
finite values of the interaction, therefore, the long time behavior of the
$|V|=2$ system is essentially quasiperiodic, with a basic period of $T=8\pi/J$
(in accordance with (29)).

We compute now the entanglement entropy and a spin correlation function, in
order to investigate the way in which the particle-spin interaction
propagates. We take the particle initially on the node 0 and the spins in the
state $\ket{+}$ (‘x’ initial condition):

$\ket{\psi(0)}=(1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0)^{\textsc{T}}\,.$

After one time step the state becomes

$\ket{\psi(1)}=U\ket{\psi(0)}\,,$

and the corresponding density matrix $\rho=\ket{\psi(1)}\bra{\psi(1)}$, from
which we compute the partial trace over the particle degrees of freedom:

$\rho_{s}(1)=\operatorname{Tr}_{xc}\rho\,,\quad x=0,1,\;c=0,1$

to get the spin state density matrix. For example, putting $\theta=\pi/2$, we
find the eigenvalues of $\rho_{s}$ to be:

$r_{0}=0\,,\quad
r_{\pm}=\frac{1}{2}\pm\frac{1}{4}\sqrt{4-\sin^{2}J}\left[1-2\sin(2J-\arg(8J))\right]\,.$

the first eigenvalue is twice degenerated. The corresponding entropy is

$S_{s}(J)=-\sum_{n}r_{n}\log r_{n}\,,$ (77)

which vanishes for $J=0,\pi$, and has a maximum for $J=\pi/2$: the number of
maxima increases with $t$, and the shape of $S_{s}$ may become highly
oscillatory for large $t$. However, we are only interested in the properties
of the case $|V|=2$ that may be of useful to understand larger systems. What
we see is that after a step particle and spins become entangled, but the
entanglement is a nontrivial function of $(\theta,\phi)$.

Figure 30: Spin correlation. (a) Particle-spin entropy; (b) spin correlation,
as a function of the coupling constant for two angles $\theta=\pi/8,3\pi/8$.
Parameters $t=2$, ‘x’, $|V|=2$.

To assert the spin-spin effective interaction we compute the correlation
function (in the form of an interaction energy between adjacent spins):

$C_{t}(\theta,J)=\braket{\bm{\sigma}_{1}\cdot\bm{\sigma}_{2}}-\braket{\bm{\sigma}_{1}}\cdot\braket{\bm{\sigma}_{2}}\,,$
(78)

where the expected value refers to
$\braket{\ldots}=\operatorname{Tr}[\rho_{s}(t)\ldots]$. For the first time
step it gives,

$C_{1}(\theta,J)=\frac{\sin^{2}(2\theta)}{16}\sin^{2}J\,,$ (79)

which has a maximum $C_{1}=1/16$ for $(\theta=\pi/4,J=\pi/2)$; significant
correlation is obtained with $t=2$, once the particle amplitudes were
distributed over the two sites (initially the amplitudes on node 1 are 0). In
Fig. 30 we show (77) and (78) for $t=2$, computed numerically. The entropy is
not a monotone function of $J$, as explained before; idem for the correlation
function, which also is strongly dependent on the color angle. We note that,
for small $J$, the correlation behaves as $C\sim J^{2}$, which is consistent
with the mediated character of the spin-spin interaction.

## References

  * [1] L. Szilard,  _Über die Entropieverminderung in einem thermodynamischen System bei Eingriffen intelligenter Wesen_ ,  Z. Für Phys. 53(11), 840 (1929),  10.1007/BF01341281.
  * [2] R. Landauer,  _Irreversibility and heat generation in the computing process_ ,  IBM J. Res. Dev. 5(3), 183 (1961),  10.1147/rd.53.0183.
  * [3] B. Schumacher,  _Quantum coding_ ,  Phys Rev A 51, 2738 (1995),  10.1103/PhysRevA.51.2738.
  * [4] C. E. Shannon,  _A mathematical theory of communication_ ,  Bell Syst. Tech. J. 27(3), 379 (1948),  10.1002/j.1538-7305.1948.tb01338.x.
  * [5] C. H. Bennett and P. W. Shor,  _Quantum information theory_ ,  IEEE Trans. Inf. Theory 44(6), 2724 (1998),  10.1109/18.720553.
  * [6] J. Preskill,  _Lecture notes for physics 229: Quantum information and computation_ ,  Calif. Inst. Technol. (1998).
  * [7] J. Watrous,  _The Theory of Quantum Information_ ,  Cambridge University Press,  10.1017/9781316848142 (2018).
  * [8] V. Vedral,  _Introduction to Quantum Information Science_ ,  Oxford University Press,  10.1093/acprof:oso/9780199215706.001.0001 (2006).
  * [9] B. Schumacher and M. Westmoreland,  _Quantum Processes, Systems, and Information_ ,  Cambridge University Press, Cambridge,  10.1017/CBO9780511814006 (2010).
  * [10] L. Amico, R. Fazio, A. Osterloh and V. Vedral,  _Entanglement in many-body systems_ ,  Rev. Mod. Phys. 80, 517 (2008),  10.1103/RevModPhys.80.517.
  * [11] N. Laflorencie,  _Quantum entanglement in condensed matter systems_ ,  Phys. Rep. 646, 1 (2016),  http://dx.doi.org/10.1016/j.physrep.2016.06.008.
  * [12] D. A. Abanin, E. Altman, I. Bloch and M. Serbyn,  _Colloquium: Many-body localization, thermalization, and entanglement_ ,  Rev. Mod. Phys. 91(2), 021001 (2019),  10.1103/RevModPhys.91.021001.
  * [13] B. Zeng, X. Chen, D.-L. Zhou and X.-G. Wen,  _Quantum Information Meets Quantum Matter_ ,  Springer, New York, NY (2019), 1508.02595.
  * [14] J. Kempe,  _Quantum random walks: An introductory overview_ ,  Contemp. Phys. 44(4), 307 (2003),  10.1080/00107151031000110776.
  * [15] T. Kitagawa,  _Topological phenomena in quantum walks: Elementary introduction to the physics of topological phases_ ,  Quantum Inf. Process. 11(5), 1107 (2012),  10.1007/s11128-012-0425-4.
  * [16] R. Portugal,  _Quantum Walks and Search Algorithms_ ,  Quantum Science and Technology. Springer, Cham,  10.1007/978-1-4614-6336-8 (2013).
  * [17] Y. Aharonov, L. Davidovich and N. Zagury,  _Quantum random walks_ ,  Phys Rev A 48, 1687 (1993),  10.1103/PhysRevA.48.1687.
  * [18] D. A. Meyer,  _From quantum cellular automata to quantum lattice gases_ ,  J Stat Phys 85(5), 551 (1996),  10.1007/BF02199356.
  * [19] R. P. Feynman,  _Quantum mechanical computers_ ,  Fond. Phys. 16(6), 507 (1986),  10.1007/BF01886518.
  * [20] D. E. Deutsch,  _Quantum computational networks_ ,  Proc. R. Soc. Lond. A 425, 73 (1989),  10.1098/rspa.1989.0099.
  * [21] N. B. Lovett, S. Cooper, M. Everitt, M. Trevers and V. Kendon,  _Universal quantum computation using the discrete-time quantum walk_ ,  Phys Rev A 81, 042330 (2010),  10.1103/PhysRevA.81.042330.
  * [22] V. Kendon,  _How to Compute Using Quantum Walks_ ,  Electron. Proc. Theor. Comput. Sci. 315, 1 (2020),  10.4204/EPTCS.315.1,  2004.01329.
  * [23] T. Kitagawa, M. S. Rudner, E. Berg and E. Demler,  _Exploring topological phases with quantum walks_ ,  Phys Rev A 82, 033429 (2010),  10.1103/PhysRevA.82.033429.
  * [24] M. A. Broome, A. Fedrizzi, B. P. Lanyon, I. Kassal, A. Aspuru-Guzik and A. G. White,  _Discrete Single-Photon Quantum Walks with Tunable Decoherence_ ,  Phys. Rev. Lett. 104(15), 153602 (2010),  10.1103/PhysRevLett.104.153602.
  * [25] T. Kitagawa, M. A. Broome, A. Fedrizzi, M. S. Rudner, E. Berg, I. Kassal, A. Aspuru-Guzik, E. Demler and A. G. White,  _Observation of topologically protected bound states in photonic quantum walks_ ,  Nat. Commun 3, 882 (2012),  10.1038/ncomms1872.
  * [26] E. Flurin, V. V. Ramasesh, S. Hacohen-Gourgy, L. S. Martin, N. Y. Yao and I. Siddiqi,  _Observing Topological Invariants Using Quantum Walks in Superconducting Circuits_ ,  Phys. Rev. X 7(3), 031023 (2017),  10.1103/PhysRevX.7.031023.
  * [27] F. Cardano, A. D’Errico, A. Dauphin, M. Maffei, B. Piccirillo, C. de Lisio, G. De Filippis, V. Cataudella, E. Santamato, L. Marrucci, M. Lewenstein and P. Massignan,  _Detection of Zak phases and topological invariants in a chiral quantum walk of twisted photons_ ,  Nat. Commun. 8, 15516 (2017),  10.1038/ncomms15516.
  * [28] D. Xie, T.-S. Deng, T. Xiao, W. Gou, T. Chen, W. Yi and B. Yan,  _Topological Quantum Walks in Momentum Space with a Bose-Einstein Condensate_ ,  Phys. Rev. Lett. 124(5), 050502 (2020),  10.1103/PhysRevLett.124.050502.
  * [29] Y. Omar, N. Paunkovi ć, L. Sheridan and S. Bose,  _Quantum walk on a line with two entangled particles_ ,  Phys Rev A 74, 042304 (2006),  10.1103/PhysRevA.74.042304.
  * [30] A. Ahlbrecht, A. Alberti, D. Meschede, V. B. Scholz, A. H. Werner and R. F. Werner,  _Molecular binding in interacting quantum walks_ ,  New J Phys 14(7), 073050 (2012),  10.1088/1367-2630/14/7/073050.
  * [31] A. Bisio, G. M. D’Ariano, N. Mosco, P. Perinotti and A. Tosini,  _Solutions of a Two-Particle Interacting Quantum Walk_ ,  Entropy 20(6), 435 (2018),  10.3390/e20060435.
  * [32] A. D. Verga and R. G. Elias,  _Entanglement and interaction in a topological quantum walk_ ,  SciPost Phys. 5(2), 019 (2018),  10.21468/SciPostPhys.5.2.019.
  * [33] L. Sansoni, F. Sciarrino, G. Vallone, P. Mataloni, A. Crespi, R. Ramponi and R. Osellame,  _Two-Particle Bosonic-Fermionic Quantum Walk via Integrated Photonics_ ,  Phys Rev Lett 108, 010502 (2012),  10.1103/PhysRevLett.108.010502.
  * [34] A. Schreiber, A. Gábris, P. P. Rohde, K. Laiho, M. Štefaňák, V. Potoček, C. Hamilton, I. Jex and C. Silberhorn,  _A 2D Quantum Walk Simulation of Two-Particle Dynamics_ ,  Science 336(6077), 55 (2012),  10.1126/science.1218448.
  * [35] A. Crespi, R. Osellame, R. Ramponi, V. Giovannetti, R. Fazio, L. Sansoni, F. De Nicola, F. Sciarrino and P. Mataloni,  _Anderson localization of entangled photons in an integrated quantum walk_ ,  Nat. Photon 7(4), 322 (2013),  10.1038/nphoton.2013.26.
  * [36] D. Aharonov, A. Ambainis, J. Kempe and U. Vazirani,  _Quantum Walks on Graphs_ ,  In _Proceedings of the Thirty-Third Annual ACM Symposium on Theory of Computing_ , STOC ’01, pp. 50–59. ACM, New York, NY, USA,  10.1145/380752.380758 (2001).
  * [37] B. Tregenna, W. Flanagan, R. Maile and V. Kendon,  _Controlling discrete quantum walks: Coins and initial states_ ,  New J Phys 5(1), 83 (2003),  10.1088/1367-2630/5/1/383.
  * [38] M. Szegedy,  _Quantum speed-up of Markov chain based algorithms_ ,  In _Foundations of Computer Science, 2004. Proceedings. 45th Annual IEEE Symposium._ , pp. 32–41,  10.1109/FOCS.2004.53 (2004).
  * [39] L. K. Grover,  _Quantum Mechanics Helps in Searching for a Needle in a Haystack_ ,  Phys. Rev. Lett. 79(2), 325 (1997),  10.1103/PhysRevLett.79.325.
  * [40] N. Shenvi, J. Kempe and K. B. Whaley,  _Quantum random-walk search algorithm_ ,  Phys Rev A 67, 052307 (2003),  10.1103/PhysRevA.67.052307.
  * [41] A. Ambainis, J. Kempe and A. Rivosh,  _Coins Make Quantum Walks Faster_ ,  Proc 16th ACM-SIAM SODA 2005 pp. 1099–1108 (2004),  quant-ph/0402107.
  * [42] M. Štefaňák and S. Skoupý,  _Perfect state transfer by means of discrete-time quantum walk search algorithms on highly symmetric graphs_ ,  Phys. Rev. A 94(2), 022301 (2016),  10.1103/PhysRevA.94.022301.
  * [43] S. D. Berry and J. B. Wang,  _Two-particle quantum walks: Entanglement and graph isomorphism testing_ ,  Phys Rev A 83, 042317 (2011),  10.1103/PhysRevA.83.042317.
  * [44] H. Wang, J. Wu, X. Yang and X. Yi,  _A graph isomorphism algorithm using signatures computed via quantum walk search model_ ,  J. Phys. A: Math. Theor. 48(11), 115302 (2015),  10.1088/1751-8113/48/11/115302.
  * [45] P. W. Mills, R. P. Rundle, J. H. Samson, S. J. Devitt, T. Tilma, V. M. Dwyer and M. J. Everitt,  _Quantum invariants and the graph isomorphism problem_ ,  Phys. Rev. A 100(5), 052317 (2019),  10.1103/PhysRevA.100.052317.
  * [46] T. Farrelly,  _Insights from Quantum Information into Fundamental Physics_ ,  ArXiv170808897 Quant-Ph (2017),  1708.08897.
  * [47] A. Bisio, G. M. D’Ariano, P. Perinotti and A. Tosini,  _Thirring quantum cellular automaton_ ,  Phys Rev A 97, 032132 (2018),  10.1103/PhysRevA.97.032132.
  * [48] P. Arrighi,  _An overview of quantum cellular automata_ ,  Nat Comput 18(4), 885 (2019),  10.1007/s11047-019-09762-6.
  * [49] M. A. Nielsen,  _Cluster-state quantum computation_ ,  Rep. Math. Phys. 57(1), 147 (2006),  http://dx.doi.org/10.1016/S0034-4877(06)80014-5.
  * [50] R. Raussendorf and H. J. Briegel,  _A One-Way Quantum Computer_ ,  Phys Rev Lett 86, 5188 (2001),  10.1103/PhysRevLett.86.5188.
  * [51] R. Raussendorf, D. E. Browne and H. J. Briegel,  _Measurement-based quantum computation on cluster states_ ,  Phys Rev A 68, 022312 (2003),  10.1103/PhysRevA.68.022312.
  * [52] M. A. Nielsen,  _Quantum computation by measurement and quantum memory_ ,  Physics Letters A 308(2), 96 (2003),  10.1016/S0375-9601(02)01803-0.
  * [53] A. Barenco, C. H. Bennett, R. Cleve, D. P. DiVincenzo, N. Margolus, P. Shor, T. Sleator, J. A. Smolin and H. Weinfurter,  _Elementary gates for quantum computation_ ,  Phys. Rev. A 52(5), 3457 (1995),  10.1103/PhysRevA.52.3457.
  * [54] S. Lloyd,  _Universal Quantum Simulators_ ,  Science 273(5278), 1073 (1996),  10.1126/science.273.5278.1073.
  * [55] D. P. DiVincenzo, D. Bacon, J. Kempe, G. Burkard and K. B. Whaley,  _Universal quantum computation with the exchange interaction_ ,  Nature 408, 339 (2000),  10.1038/35042541.
  * [56] H. J. Briegel and R. Raussendorf,  _Persistent Entanglement in Arrays of Interacting Particles_ ,  Phys Rev Lett 86, 910 (2001),  10.1103/PhysRevLett.86.910.
  * [57] M. Hein, W. Dür, J. Eisert, R. Raussendorf, M. Nest and H.-J. Briegel,  _Entanglement in graph states and its applications_ ,  ArXiv Prepr. (2006),  quant-ph/0602096.
  * [58] A. Ambainis,  _Quantum walks and their algorithmic applications_ ,  Int J Quantum Inf. 01(04), 507 (2003),  10.1142/S0219749903000383.
  * [59] S. Bose,  _Quantum Communication through an Unmodulated Spin Chain_ ,  Phys. Rev. Lett. 91(20), 207901 (2003),  10.1103/PhysRevLett.91.207901.
  * [60] M. Christandl, N. Datta, A. Ekert and A. J. Landahl,  _Perfect State Transfer in Quantum Spin Networks_ ,  Phys. Rev. Lett. 92(18), 187902 (2004),  10.1103/PhysRevLett.92.187902.
  * [61] C. E. Bradley, J. Randall, M. H. Abobeih, R. C. Berrevoets, M. J. Degen, M. A. Bakker, M. Markham, D. J. Twitchen and T. H. Taminiau,  _A Ten-Qubit Solid-State Spin Register with Quantum Memory up to One Minute_ ,  Phys. Rev. X 9(3), 031045 (2019),  10.1103/PhysRevX.9.031045.
  * [62] W. Pfaff, B. J. Hensen, H. Bernien, S. B. van Dam, M. S. Blok, T. H. Taminiau, M. J. Tiggelman, R. N. Schouten, M. Markham, D. J. Twitchen and R. Hanson,  _Unconditional quantum teleportation between distant solid-state quantum bits_ ,  Science 345(6196), 532 (2014),  10.1126/science.1253512.
  * [63] G. Waldherr, Y. Wang, S. Zaiser, M. Jamali, T. Schulte-Herbrüggen, H. Abe, T. Ohshima, J. Isoya, J. F. Du, P. Neumann and J. Wrachtrup,  _Quantum error correction in a solid-state hybrid spin register_ ,  Nature 506(7487), 204 (2014),  10.1038/nature12919.
  * [64] A. D. Verga,  _Interacting quantum walk on a graph_ ,  Phys. Rev. E 99(1), 012127 (2019),  10.1103/PhysRevE.99.012127.
  * [65] A. D. Verga and R. G. Elías,  _Thermal state entanglement entropy on a quantum graph_ ,  Phys. Rev. E 100(6), 062137 (2019),  10.1103/PhysRevE.100.062137.
  * [66] M. A. Ruderman and C. Kittel,  _Indirect Exchange Coupling of Nuclear Magnetic Moments by Conduction Electrons_ ,  Phys. Rev. 96(1), 99 (1954),  10.1103/PhysRev.96.99.
  * [67] T. Kasuya,  _A Theory of Metallic Ferro- and Antiferromagnetism on Zener’s Model_ ,  Prog Theor Phys 16(1), 45 (1956),  10.1143/PTP.16.45.
  * [68] W. Nolting and A. Ramakanth,  _Quantum Theory of Magnetism_ ,  Springer Science & Business Media (2009).
  * [69] A. M. Läuchli and C. Kollath,  _Spreading of correlations and entanglement after a quench in the one-dimensional Bose–Hubbard model_ ,  J. Stat. Mech. 2008(05), P05018 (2008),  10.1088/1742-5468/2008/05/P05018.
  * [70] L. Zhang, H. Kim and D. A. Huse,  _Thermalization of entanglement_ ,  Phys. Rev. E 91(6), 062128 (2015),  10.1103/PhysRevE.91.062128.
  * [71] A. Nahum, J. Ruhman, S. Vijay and J. Haah,  _Quantum Entanglement Growth under Random Unitary Dynamics_ ,  Phys Rev X 7, 031016 (2017),  10.1103/PhysRevX.7.031016.
  * [72] L. Hackl, E. Bianchi, R. Modak and M. Rigol,  _Entanglement production in bosonic systems: Linear and logarithmic growth_ ,  Phys. Rev. A 97(3), 032321 (2018),  10.1103/PhysRevA.97.032321.
  * [73] A. D’Errico, F. Cardano, M. Maffei, A. Dauphin, R. Barboza, C. Esposito, B. Piccirillo, M. Lewenstein, P. Massignan and L. Marrucci,  _Two-dimensional topological quantum walks in the momentum space of structured light_ ,  Optica, OPTICA 7(2), 108 (2020),  10.1364/OPTICA.365028.
  * [74] G. Di Molfetta and A. Pérez,  _Quantum walks as simulators of neutrino oscillations in a vacuum and matter_ ,  New J Phys 18(10), 103038 (2016),  10.1088/1367-2630/18/10/103038.
  * [75] A. Bisio, G. M. D’Ariano and P. Perinotti,  _Quantum Walks, Weyl Equation and the Lorentz Group_ ,  Found Phys 47(8), 1065 (2017),  10.1007/s10701-017-0086-3.
  * [76] P. Arrighi, G. Di Molfetta, I. Márquez-Martín and A. Pérez,  _Dirac equation as a quantum walk over the honeycomb and triangular lattices_ ,  Phys. Rev. A 97(6), 062111 (2018),  10.1103/PhysRevA.97.062111.
  * [77] L. D. Landau and E. M. Lifshitz,  _On the theory of the dispersion of magnetic permeability in ferromagnetic bodies_ ,  Phys Zeitsch Sow 8, 153 (1935).
  * [78] J. C. Slonczewski,  _Current-driven excitation of magnetic multilayers_ ,  J. Magn. Magn. Mater. 159(1-2), L1 (1996),  10.1016/0304-8853(96)00062-5.
  * [79] G. Tatara,  _Effective gauge field theory of spintronics_ ,  Physica E: Low-dimensional Systems and Nanostructures 106, 208 (2019),  10.1016/j.physe.2018.05.011.
  * [80] F. W. Strauch,  _Relativistic effects and rigorous limits for discrete- and continuous-time quantum walks_ ,  Journal of Mathematical Physics 48(8), 082102 (2007),  10.1063/1.2759837.
  * [81] G. Di Molfetta and F. Debbasch,  _Discrete-time quantum walks: Continuous limit and symmetries_ ,  J Math Phys 53(12), 123302 (2012),  http://dx.doi.org/10.1063/1.4764876.
  * [82] R. Jozsa,  _Quantum Effects in Algorithms_ ,  In C. P. Williams, ed., _Quantum Computing and Quantum Communications_ , Lecture Notes in Computer Science, pp. 103–112. Springer, Berlin, Heidelberg,  ISBN 978-3-540-49208-5,  10.1007/3-540-49208-9_7 (1999).
  * [83] C. H. Bennett and S. J. Wiesner,  _Communication via one- and two-particle operators on Einstein-Podolsky-Rosen states_ ,  Phys. Rev. Lett. 69(20), 2881 (1992),  10.1103/PhysRevLett.69.2881.
  * [84] P. W. Shor,  _Algorithms for quantum computation: Discrete logarithms and factoring_ ,  In _Foundations of Computer Science, 1994 Proceedings., 35th Annual Symposium On_ , pp. 124–134,  10.1109/SFCS.1994.365700 (1994).
  * [85] A. W. Harrow, A. Hassidim and S. Lloyd,  _Quantum Algorithm for Linear Systems of Equations_ ,  Phys. Rev. Lett. 103(15), 150502 (2009),  10.1103/PhysRevLett.103.150502.
  * [86] V. Kendon and B. C. Sanders,  _Complementarity and quantum walks_ ,  Phys Rev A 71, 022307 (2005),  10.1103/PhysRevA.71.022307.
  * [87] V. M. Kendon and C. Tamon,  _Perfect state transfer in quantum walks on graphs_ ,  J. Comput. Theor. Nanosci. 8(3), 422 (2011),  10.1166/jctn.2011.1706.
  * [88] L. Berger,  _Exchange interaction between ferromagnetic domain wall and electric current in very thin metallic films_ ,  J. Appl. Phys. 55(6), 1954 (1984),  10.1063/1.333530.
  * [89] R. Horodecki and P. Horodecki,  _Quantum redundancies and local realism_ ,  Physics Letters A 194(3), 147 (1994),  10.1016/0375-9601(94)91275-0.
  * [90] C. H. Bennett, H. J. Bernstein, S. Popescu and B. Schumacher,  _Concentrating partial entanglement by local operations_ ,  Phys Rev A 53, 2046 (1996),  10.1103/PhysRevA.53.2046.
  * [91] W. K. Wootters,  _Entanglement of Formation of an Arbitrary State of Two Qubits_ ,  Phys. Rev. Lett. 80(10), 2245 (1998),  10.1103/PhysRevLett.80.2245.
  * [92] R. Horodecki, P. Horodecki, M. Horodecki and K. Horodecki,  _Quantum entanglement_ ,  Rev Mod Phys 81, 865 (2009),  10.1103/RevModPhys.81.865.
  * [93] J. K. Asbóth,  _Symmetries, topological phases, and bound states in the one-dimensional quantum walk_ ,  Phys Rev B 86, 195414 (2012),  10.1103/PhysRevB.86.195414.
  * [94] X. Wang, S. Ghose, B. C. Sanders and B. Hu,  _Entanglement as a signature of quantum chaos_ ,  Phys Rev E 70, 016217 (2004),  10.1103/PhysRevE.70.016217.
  * [95] S.-Q. Shen, W.-Y. Shan and H.-Z. Lu,  _Topological Insulator and the Dirac Equation_ ,  SPIN 01(01), 33 (2011),  10.1142/S2010324711000057.
  * [96] D. N. Page,  _Average entropy of a subsystem_ ,  Phys Rev Lett 71, 1291 (1993),  10.1103/PhysRevLett.71.1291.
  * [97] D. C. Ralph and M. D. Stiles,  _Spin transfer torques_ ,  J Magn Magn Mater 320(7), 1190 (2008),  10.1016/j.jmmm.2007.12.019.
  * [98] M. Sayad, R. Rausch and M. Potthoff,  _Inertia effects in the real-time dynamics of a quantum spin coupled to a Fermi sea_ ,  EPL 116(1), 17001 (2016),  10.1209/0295-5075/116/17001.
  * [99] V. G. Baryakhtar and A. G. Danielevich,  _Spin-wave damping at spin-orientation phase transitions_ ,  Low Temp. Phys. 32(8), 768 (2006),  10.1063/1.2219498.
  * [100] A. D. Verga,  _Skyrmion to ferromagnetic state transition: A description of the topological change as a finite-time singularity in the skyrmion dynamics_ ,  Phys Rev B 90, 174428 (2014),  10.1103/PhysRevB.90.174428.
  * [101] D. A. Garanin,  _Fokker-Planck and Landau-Lifshitz-Bloch equations for classical ferromagnets_ ,  Phys Rev B 55(5) (1997),  10.1103/PhysRevB.55.3050.
  * [102] C.-W. Lee, P. Kurzynski and H. Nha,  _Quantum walk as a simulator of nonlinear dynamics: Nonlinear Dirac equation and solitons_ ,  Phys Rev A 92, 052336 (2015),  10.1103/PhysRevA.92.052336.
  * [103] Y. A. Mitropolsky and N. Van Dao,  _Applied Asymptotic Methods in Nonlinear Oscillations_ ,  Kluwer Academic (Springer, Dordrecht),  10.1007/978-94-011-5752-0 (1997).
  * [104] T. Kakutani and N. Sugimoto,  _Krylov‐Bogoliubov‐Mitropolsky method for nonlinear wave modulation_ ,  The Physics of Fluids 17(8), 1617 (1974),  10.1063/1.1694942.
  * [105] Y. Tserkovnyak and D. Loss,  _Thin-Film Magnetization Dynamics on the Surface of a Topological Insulator_ ,  Phys Rev Lett 108, 187201 (2012),  10.1103/PhysRevLett.108.187201.
  * [106] J. Chen, M. B. Abdul Jalil and S. G. Tan,  _Current-Induced Spin Torque on Magnetization Textures Coupled to the Topological Surface States of Three-Dimensional Topological Insulators_ ,  J. Phys. Soc. Jpn. 83(6), 064710 (2014),  10.7566/JPSJ.83.064710.
  * [107] T. Kadowaki and H. Nishimori,  _Quantum annealing in the transverse Ising model_ ,  Phys. Rev. E 58(5), 5355 (1998),  10.1103/PhysRevE.58.5355.
  * [108] E. Farhi, J. Goldstone, S. Gutmann and M. Sipser,  _Quantum Computation by Adiabatic Evolution_ ,  ArXiv Quant-Ph0001106 (2000),  quant-ph/0001106.
  * [109] D. Aharonov, W. van Dam, J. Kempe, Z. Landau, S. Lloyd and O. Regev,  _Adiabatic Quantum Computation Is Equivalent to Standard Quantum Computation_ ,  SIAM Rev. 50(4), 755 (2008),  10.1137/080734479.
  * [110] T. Albash and D. A. Lidar,  _Adiabatic quantum computation_ ,  Rev. Mod. Phys. 90(1), 015002 (2018),  10.1103/RevModPhys.90.015002.
  * [111] D. Pagliero, P. R. Zangara, J. Henshaw, A. Ajoy, R. H. Acosta, J. A. Reimer, A. Pines and C. A. Meriles,  _Optically pumped spin polarization as a probe of many-body thermalization_ ,  Sci. Adv. 6(18), eaaz6986 (2020),  10.1126/sciadv.aaz6986.
  * [112] C. Maier, T. Brydges, P. Jurcevic, N. Trautmann, C. Hempel, B. P. Lanyon, P. Hauke, R. Blatt and C. F. Roos,  _Environment-Assisted Quantum Transport in a 10-qubit Network_ ,  Phys. Rev. Lett. 122(5), 050501 (2019),  10.1103/PhysRevLett.122.050501.
  * [113] J. I. Cirac, P. Zoller, H. J. Kimble and H. Mabuchi,  _Quantum State Transfer and Entanglement Distribution among Distant Nodes in a Quantum Network_ ,  Phys. Rev. Lett. 78(16), 3221 (1997),  10.1103/PhysRevLett.78.3221.
  * [114] J. D. Hunter,  _Matplotlib: A 2D Graphics Environment_ ,  Comput. Sci. Eng. 9(3), 90 (2007),  10.1109/MCSE.2007.55.