# Maxentropy completion and properties of some partially defined Stationary
Markov chains

Pierre Collet${}^{\textup{(a)}}$, Servet Martínez${}^{\textup{(b)}}$
${}^{\textup{(a)}}$ CPHT, CNRS, Ecole Polytechnique, IP Paris, Palaiseau,
France
e-mail<EMAIL_ADDRESS>
${}^{\textup{(b)}}$ Universidad de Chile, UMR 2071 UCHILE-CNRS
Departamento de Ingeniería Matemática and Centro de Modelamiento Matemático
Casilla 170-3 Correo 3, Santiago, Chile
e-mail<EMAIL_ADDRESS>

###### Abstract

We consider a stationary Markovian evolution with values on a disjointly
partitioned set space $I\sqcup{\cal E}$. The evolution is visible (in the
sense of knowing the transition probabilities) on the states in $I$ but not
for the states in ${\cal E}$. One only knows some partial information on the
transition probabilities on ${\cal E}$, the input and output transition
probabilities and some constraints of the transition probabilities on ${\cal
E}$. Under some conditions we supply the transition probabilities on ${\cal
E}$ that satisfies the maximum entropy principle.

Keywords: Hidden states, maximal entropy principle, stationarity, incomplete
Markov chain.

##  1 Introduction

Discrete time Markov chains on a finite state space are completely described
by their matrix of transitions probabilities. We consider a situation where
only a subset of the transition probabilities is known. More precisely we
consider an evolution on a set of states $I\sqcup{\cal E}$ (disjoint union).
The states in $I$ are visible and the set ${\cal E}$ is called a labyrinth
(composed of invisible states). In a first stage the states in ${\cal E}$ act
as absorbing states but their persistence or reappearance lead to consider
these states in a different way. The transition probabilities within the
labyrinth are unknown. One only knows some basic relations: the input and
output transition probabilities and eventually communication constraints in
the labyrinth. The transition probabilities between the states in $I$ are
known and those between $I$ and ${\cal E}$ also.

A natural question is to complete the description of the chain, i.e. to find
the transition probabilities into the labyrinth so that the matrix of
transition probabilities is Markovian. There are many possible solutions to
this problem and we propose a maximum entropy approach.

This partially defined process is an extremely simplified model for some
phenomena:

  * •

Disease epidemics that emerge and disappear in time. A disease may emerge at
some location due to human contact with a biological reservoir of some
microorganism, diffuse in some geographical area, and when finished retreat
unseen to some biological reservoir again. In this case the observable states
represent the sizes of the healthy and infected populations in geographic
areas while the labyrinth corresponds to the biological reservoirs of the
microorganism. A transition from the labyrinth to an observable state is an
outbreak;

  * •

Underground rivers, that can disappear at some sinkholes, continues
underground and reemerge further downstream. There can be a cave system having
an underground water dynamics. At a first stage the caves can be considered
absorbing states, but this does not describe the whole phenomenon because the
river reemerges after an underground water evolution.

One can conceive many other concrete examples of partially defined Markov
processes.

We assume a stationary behavior of the total process, not only on the visible
states $I$ but also for the evolution on the states ${\cal E}$ (which is
unseen). Given the data (some transition probabilities associated with the
states in $I$), we try to reconstruct the transition probabilities of the
process in the unseen part by using the maximum entropy principle. Our problem
falls quite naturally in this context. Some average quantities (probability of
two successive states) are known and we asked for some other similar
quantities. As described by Jaynes, ”…in making inference on the basis of
partial information we must use that probability distribution which has the
maximum entropy to whatever is known. This is the only unbiased assignment we
can make…” [11] p.623. See also [12], and [16] Section 5.

Therefore, we assume we are given some transition probabilities and the
proposal for the remaining unknown ones is based upon the maximum entropy
principle. We emphasize that we do not assume sample paths of the Markov chain
are available, in other words we do not consider observations of a process and
no estimation is made. There is a large literature devoted to estimate a
transition matrix that fits incomplete observation data, by using the maximum
likelihood criterion, like in [8], or by some Bayesian approach, where there
is a prior of the transition matrix and given the data one seeks to compute
the posterior distribution, see [15].

The paper is organized as follows. In section 2 we describe the set of known
transition probabilities and introduce some more hypothesis. We then derive
the maximum entropy solution satisfying the Markovian constraints.

Section 3 is devoted to the more general case where a communication matrix
inside the labyrinth is imposed. If all the states communicate the maximum
entropy solution is Bernoulli. But the problem turns harder when the
communication matrix is not trivial. In fact there could be no matrix that
satisfies the constraints. In Section 3.2 the constraints are written in a
linear way and in Section 3.3 we use Farka’s lemma to give a criterion for the
existence of a solution. In Proposition 3.6 we give a simple sufficient
condition for the existence of solution. In Section 3.4 we study the case when
there are several labyrinths that are connected only by the visible states,
and show that all the computations related to maximum entropy matrices are
reduced to the case when there is a unique labyrinth.

In section 4 we relate our problem and hypothesis to the theory of quasi
stationary distributions. We show that under our hypotheses the chain can be
reconstructed only from the evolution on the visible states and in the
labyrinth separately, using two families of independent geometric random
variables that mark the times of restarting.

As explained above we assume only a subset of the transition probabilities is
known and we look to complete these transitions probabilities by using the
maximal entropy criterion. This makes part of the problem where the transition
matrix is known to belong to some class of matrices, and one seeks the one
that satisfies a certain optimization criterion. This is the case in [10], but
the optimization is on an infinite horizon reward on all the states. In [3]
the unknown is a transition matrix that satisfies some general order
constraints on the coefficients and its stationary vector, which is unknown,
maximizes some linear functional. But, in our case the set of constraints is
more specific allowing to handle the optimization of the entropy.

In [14] the inverse problem for partially observed Markov chains with restart
has been considered in a parametric setting for the unknown transition
probabilities and with an objective (cost) function different from ours. More
precisely, the chain can restart with some fixed probability according to some
distribution and it can change the transition probabilities. The process is
assumed to be ergodic and the authors look for for a stationary probability.
This search is based upon the knowledge of the stationary vector up to a
constant and the hitting probabilities prior to restart, in a fixed region.
The optimization is made on the parameters guiding the evolution. There are
some differences with our setting, a main one is that the optimization
functions are different. Moreover, even if our setting can be thought as a
restarting problem, in our case the probability of restarting is state
dependent (it is not with a fixed probability), and there is a known
restarting distribution when entering to the visible states and an unknown
restarting distribution when exiting from them to enter to the labyrinth.

##  2 Partially defined Markov chains, hypothesis and Maxentropy.

Our starting point is an irreducible stationary Markov chain,
$X=(X_{n}:n\geq\mathbb{Z})$ with values in the set $I\sqcup{\cal E}$. Its
transition matrix is denoted by $P=(P(a,b):a,b\in I\sqcup{\cal E})$ and $\pi$
denotes stationary distribution, so $\pi^{t}P=\pi^{t}$. The irreducibility of
$P$ implies $\pi>0$.

By vectors we always mean column vectors and we add the transposition
superscript t to denote the associated row vector. We also denote $P_{J\times
K}=(P(a,b):i\in J,k\in J)$ the restricted submatrix when $J,K\subseteq
I\sqcup{\cal E}$.

When the chain $X$ takes values in $I$, this is visible. The states in the
labyrinth ${\cal E}$ can be distinguished and visible only when there is a
nonzero probability of emerging from ${\cal E}$ to $I$ or when there is a
nonzero probability of transition from some state in $I$ to ${\cal E}$. But
once the chain is inside and before leaving the labyrinth this is not visible.

######  Remark 2.1.

Let us see that the pair formed by the visible process ${\cal O}$ (defined
below) and the Markov process $X$ does not constitute a Hidden Markov chain.
For this purpose let $\mathfrak{e}$ be a new state, that codes the states of
the process that are in ${\cal E}$ but they are not seen during the process.
With this notation, the visible process ${\cal O}=({\cal
O}_{n}:n\in\mathbb{Z})$ is given by

${\cal O}_{n}=\begin{cases}X_{n}&\hbox{ if }X_{n}\in I;\\\ X_{n}&\hbox{ if
}X_{n}\in{\cal E}\hbox{ and }X_{n-1}\in I\hbox{ or }X_{n+1}\in I;\\\
\mathfrak{e}&\hbox{ if }X_{n}\in{\cal E}\hbox{ and }X_{n-1}\in{\cal E}\hbox{
and }X_{n+1}\in{\cal E}.\end{cases}$

A necessary condition in order that the pair $({\cal O},X)$ is a Hidden Markov
chain is that: for all $n\geq 1$, $c\in I\sqcup{\cal
E}\sqcup\\{\mathfrak{e}\\}$, $a_{k}\in I\sqcup{\cal E}$, $k=0,..,n$, one has:

$\mathbb{P}({\cal O}_{n}=c\,|\,X_{k}=a_{k},k\leq n)=\mathbb{P}({\cal
O}_{n}=c\,|\,X_{n}=a_{n}).$ (1)

But, this condition is not satisfied when $c=\mathfrak{e}$, $a_{n-1}\in I$,
$a_{n}\in{\cal E}$. In fact, since $X_{n-1}=a_{n-1}\in I$ then the visible
state at coordinate $n$ is ${\cal O}_{n}=X_{n}=a_{n}\in{\cal E}$ which is
different from $\mathfrak{e}$, and so $\mathbb{P}({\cal
O}_{n}=\mathfrak{e}|X_{k}=a_{k},k\leq n)=0$. But $\mathbb{P}({\cal
O}_{n}=\mathfrak{e}|X_{n}=a_{n})=\mathbb{P}(X_{n-1}\not\in{\cal
E},X_{n}\not\in{\cal E},X_{n+1}\not\in{\cal E}|X_{n}=a_{n})>0$.

In [9], relation $(1)$ in Section 2, the above condition (1) is part of the
Partially Observed Markov Process (POMP) model. For the definition and
applications of Hidden Markov models for discrete random sequences one can see
Sections $13.1$ and $13.2$ in [2]. $\Box$

Let $\pi_{I}=(\pi(i):i\in I)$ and $\pi_{\cal E}=(\pi(d):d\in{\cal E})$ denote
the restrictions of $\pi$ to $I$ and ${\cal E}$ respectively.

Summarizing, we assume that the transition submatrices $P_{I\times I}$,
$P_{I\times{\cal E}}$ and $P_{{\cal E}\times I}$ are known. In particular
$P_{I\times(I\sqcup{\cal E})}$ is known. We also assume that $\pi_{I}$ is
known. These are the only available data.

If a large number of observations of the dynamics of the chain is available,
these data can be estimated by averaging using the Law of Large Numbers when
the chain is in $I$ and when it enters from $I$ to ${\cal E}$ or emerges from
${\cal E}$ to $I$.

We denote by $\pi(a)$ the weight of $a\in I\sqcup{\cal E}$ and by
$\pi(A)=\sum_{a\in A}\pi(a)$ the weight of $A\subseteq I\sqcup{\cal E}$.

We assume the existence of some states $i\in I$ such that $P(i,{\cal E})>0$
(where $P(i,{\cal E})=\sum_{d\in{\cal E}}P(i,d)$), hence the restricted matrix
$P_{I\times I}$ is a substochastic kernel.

The first assumption we make is the following one: when the chain emerges from
the labyrinth ${\cal E}$ to the visible states $I$, it is always with the
stationary law $\pi_{I}$, so we assume

$(H1)\quad\quad\quad\quad\quad\quad P_{{\cal E}\times I}={{\bf{1}}}_{{\cal
E}}\pi^{t}_{I}$

where ${{\bf{1}}}_{{\cal E}}$ is the unit vector of dimension the cardinal
number $|{\cal E}|$. This hypothesis means that the law of the outbreak on $I$
has distribution $\pi^{t}_{I}$. In the case where a sample of the chain is
available one could look for elaborating a test that permits to analyze if
hypothesis $(H1)$ is satisfied.

Now we give a context for the next assumption. Since $\pi$ is stationary one
has $\pi_{I}^{t}=\pi_{I}^{t}P_{I\times I}+\pi_{{\cal E}}^{t}P_{{\cal E}\times
I}$ and by using $(H1)$ one gets

$\pi_{I}^{t}=\pi_{I}^{t}P_{I\times I}+\pi_{{\cal E}}^{t}{{\bf{1}}}_{{\cal
E}}\pi^{t}_{I}=\pi_{I}^{t}P_{I\times I}+\pi({\cal E})\pi^{t}_{I}.$

(Recall $\pi({\cal E})=\sum_{d\in{\cal E}}\pi(d)$, $\pi(I)=\sum_{i\in
I}\pi(i)$). Since $1-\pi({\cal E})=\pi(I)$, then a consequence of $(H1)$ is

$\pi_{I}^{t}P_{I\times I}=\pi(I)\pi_{I}^{t},$ (2)

and so $\pi_{I}$ is the left Perron-Frobenius eigenvector of $P_{I\times I}$
with Perron-Frobenius eigenvalue $\pi(I)$.

We have only partial information on the transition matrix $P_{{\cal
E}\times{\cal E}}$. First, one has that all sums of the rows of $P_{{\cal
E}\times{\cal E}}$ are constant because $P$ is Markov and so from $(H1)$,

${{\bf{1}}}_{{\cal E}}=P_{{\cal E}\times I}{{\bf{1}}}_{I}+P_{{\cal
E}\times{\cal E}}{{\bf{1}}}_{{\cal E}}={{\bf{1}}}_{{\cal
E}}\pi_{I}^{t}{{\bf{1}}}_{I}+P_{{\cal E}\times{\cal E}}{{\bf{1}}}_{{\cal
E}}=\pi(I){{\bf{1}}}_{\cal E}+P_{{\cal E}\times{\cal E}}{{\bf{1}}}_{{\cal
E}}.$

Then,

$P_{{\cal E}\times{\cal E}}{{\bf{1}}}_{{\cal E}}=\pi({\cal E}){{\bf{1}}}_{\cal
E}.$ (3)

Therefore, a consequence of $(H1)$ is that $\pi({\cal E})^{-1}P_{{\cal
E}\times{\cal E}}$ is a stochastic matrix. We have that $\pi({\cal
E})^{-1}\pi_{{\cal E}}$ is a probability vector. We assume the following
property: $\pi({\cal E})^{-1}\pi_{{\cal E}}$ is a stationary distribution for
$\pi({\cal E})^{-1}P_{{\cal E}\times{\cal E}}$. This is equivalent to,

$(H2)\quad\quad\quad\quad\quad\quad\pi({\cal E})\pi_{\cal E}^{t}=\pi_{\cal
E}^{t}P_{{\cal E}\times{\cal E}}.$

So, $\pi_{\cal E}$ is the left Perron-Frobenius eigenvector of $P_{{\cal
E}\times{\cal E}}$ with Perron-Frobenius eigenvalue $\pi({\cal E})$.

Note that the hypothesis $(H1)$ can be tested from observing the transitions
from the labyrinth to the visible states, but this is not the case of $(H2)$.

From (3) and $(H2)$ we have $\pi_{\cal E}^{t}P_{{\cal E}\times{\cal
E}}{{\bf{1}}}_{\cal E}=\pi({\cal E})^{2}$. The stationarity of $\pi$ implies

$\pi_{\cal E}^{t}=\pi_{I}^{t}P_{I\times{\cal E}}+\pi_{{\cal E}}^{t}P_{{\cal
E}\times{\cal E}}$

and so from $(H2)$ one gets

$\pi_{I}^{t}P_{I\times{\cal E}}=\pi_{{\cal E}}^{t}-\pi_{\cal E}^{t}P_{{\cal
E}\times{\cal E}}=(1-\pi({\cal E}))\pi_{\cal E}=\pi(I)\pi_{\cal E}.$

and then

$\pi_{{\cal E}}=\pi(I)^{-1}\pi_{I}^{t}P_{I\times{\cal E}},$ (4)

that is $\pi(e)=\pi(I)^{-1}\sum_{i\in I}\pi(i)P(i,e)$ for $e\in{\cal E}$.

Hence, from hypotheses $(H1)$ and $(H2)$ one knows $\pi_{\cal E}$. As said, we
have $\pi_{I}$ and $P_{I\times{\cal E}}$ from the visible evolution. But,
based upon observation of a sample, we cannot test that $\pi({\cal
E})^{-1}\pi_{\cal E}$ is the stationary distribution for $\pi({\cal
E})^{-1}P_{{\cal E}\times{\cal E}}$, which gives the dynamics conditioned to
be in ${\cal E}$. This occurs even if $(\pi_{I},\pi_{\cal E})$ is the
stationary distribution for the dynamics on $I\sqcup{\cal E}$. This is why the
hypothesis $(H2)$ cannot be tested.

The entropy of the stationary Markov chain $X=(X_{n})$ is given by,

$h(X)=-\sum_{a\in I\sqcup{\cal E}}\pi(a)\sum_{b\in I\sqcup{\cal E}}P(a,b)\log
P(a,b),$

so,

$\displaystyle h(X)$ $\displaystyle=$ $\displaystyle-\sum_{d\in{\cal
E}}\pi(d)\sum_{a\in I\sqcup{\cal E}}\pi(a)\log\pi(a)-\sum_{i\in
I}\pi(i)\sum_{b\in I\sqcup{\cal E}}P(i,b)\log P(i,b)$ (5) $\displaystyle=$
$\displaystyle-\sum_{i\in I}\pi(i)\sum_{a\in I\sqcup{\cal E}}P(i,a)\log
P(i,a)-\sum_{d\in{\cal E}}\pi(d)\sum_{j\in I}\pi(j)\log\pi(j)$
$\displaystyle\;-\sum_{d\in{\cal E}}\pi_{d}\sum_{e\in{\cal E}}P(d,e)\log
P(d,e).$

According to Jaynes maximal entropy principle we look for maximizing $h(X)$,
with the following knowledge:

$\pi_{I},P_{I\times(I\cup{\cal E})},P_{{\cal E}\times I}.$

So, the first term of the (5) is fixed. Now, from assumptions $(H1)$ and
$(H2)$ we also know $\pi_{\cal E}$. So, the second term in (5) is also fixed.
Hence, the maximization of $h(X)$ is equivalent to maximize

$H^{\prime}(P_{{\cal E}\times{\cal E}})=-\sum_{d\in{\cal
E}}\pi(d)\sum_{e\in{\cal E}}P(d,e)\log P(d,e),$

with $\pi_{\cal E}=\pi(I)^{-1}\pi_{I}^{t}P_{I\times{\cal E}}$ known. So, one
seeks to maximize $H^{\prime}(P_{{\cal E}\times{\cal E}})$ with the matrix
$P_{{\cal E}\times{\cal E}}$ subject to (3) and $(H2)$.

##  3 Partially defined Makov chains with communication constraints.

This section will be mainly devoted to the case where there are some
communications constraints among the states of the labyrinth. Firstly we will
consider the case when there is no constraint of this type, in other words the
states in ${\cal E}$ communicate among themselves.

###  3.1 When all states in the labyrinth communicate

######  Theorem 3.1.

When the unique constraints are (3) and $(H2)$, the maximum entropy completion
of the chain is given by $P_{{\cal E}\times{\cal E}}={{\bf{1}}}_{\cal
E}\pi_{\cal E}^{t}$, that is the transitions are Bernoulli: $P(d,e)=\pi(e)$,
$d,e\in{\cal E}$.

###### Proof.

When the unique constraints are (3) and $(H2)$, the maximum of
$H^{\prime}(P_{{\cal E}\times{\cal E}})$ is obtained with $P_{{\cal
E}\times{\cal E}}={{\bf{1}}}_{\cal E}\pi_{\cal E}^{t}$. This is well-known and
can be proved as follows,

$H^{\prime}(P_{{\cal E}\times{\cal E}})=\sum_{d\in{\cal
E}}\pi(d)\sum_{e\in{\cal E}}P(d,e)\log(\pi(e)/P(d,e))-\sum_{d\in{\cal
E}}\pi(d)\sum_{e\in{\cal E}}P(d,e)\log\pi(e).$

By using $\log(x)\leq x-1$ and (3) we get,

$\displaystyle\sum_{d\in{\cal E}}\pi(d)\sum_{e\in{\cal
E}}P(d,e)\log(\pi(e)/P(d,e))\leq\sum_{d\in{\cal E}}\pi(d)\sum_{e\in{\cal
E}}P(d,e)(\pi(e)/P(d,e)-1)$ $\displaystyle\;\leq\sum_{d\in{\cal
E}}\pi(d)\sum_{e\in{\cal E}}\pi(e)-\sum_{d\in{\cal E}}\pi(d)\sum_{e\in{\cal
E}}P(d,e)=\pi({\cal E})^{2}-\pi({\cal E})^{2}=0.$

Hence,

$\displaystyle H^{\prime}(P_{{\cal E}\times{\cal E}})$ $\displaystyle\leq$
$\displaystyle-\sum_{d\in{\cal E}}\pi(d)\sum_{e\in{\cal
E}}P(d,e)\log(\pi(e))=-\sum_{e\in{\cal E}}\log(\pi(e))\sum_{d\in{\cal
E}}\pi(d)P(d,e)$ $\displaystyle=$ $\displaystyle-\pi({\cal E})\sum_{e\in{\cal
E}}\pi(e)\log(\pi(e)),$

where $(H2)$ was used to establish the last inequality. This inequality is an
equality for the transitions $P(d,e)=\pi(e)$ with $d,e\in{\cal E}$. Hence, in
the absence of other constraint the maximum of the entropy is realized with a
Bernoulli. ∎

######  Remark 3.2.

The Bernoulli case was described in [13]. When $I=\emptyset$, one has
$\pi({\cal E})=1$, hence the unique restriction $(H2)$ on the matrix $P_{{\cal
E}\times{\cal E}}$ implies it is a stochastic matrix. The maximum entropy
matrix is the uniform one $P(d,e)=1/|{\cal E}|$ for all $d,e\in{\cal
E}$.$\Box$

###  3.2 Constraints on the communication matrix in the labyrinth.

Let us define ${\widehat{\pi}}$ on $I\sqcup{\cal E}$ the normalized
restrictions to $I$ and ${\cal E}$, by

${\widehat{\pi}}_{I}=(\pi(I))^{-1}\pi_{I}\,\hbox{ and }\,{\widehat{\pi}}_{\cal
E}=(\pi({\cal E}))^{-1}\pi_{\cal E},$

which are probability vectors on $I$ and ${\cal E}$ respectively. We define

${\widehat{P}}=(\pi({\cal E}))^{-1}P_{{\cal E}\times{\cal E}}.$

From (3), this is a stochastic matrix of size $|{\cal E}|\times|{\cal E}|$. We
denote by $Z$ a Markov shift with transition matrix ${\widehat{P}}$ and
stationary distribution ${\widehat{\pi}}_{\cal E}$. Note that

$\displaystyle H^{\prime}(P_{{\cal E}\times{\cal E}})$ $\displaystyle=$
$\displaystyle-\sum_{d\in{\cal E}}\pi(d)\sum_{e\in{\cal E}}P(d,e)\log P(d,e)$
$\displaystyle=$ $\displaystyle-\pi({\cal E})^{2}\sum_{d\in{\cal
E}}{\widehat{\pi}}(d)\sum_{e\in{\cal E}}{\widehat{P}}(d,e)\log(\pi({\cal
E}){\widehat{P}}(d,e))$ $\displaystyle=$ $\displaystyle-\pi({\cal
E})^{2}h(Z)+\pi({\cal E})^{2}\log(\pi({\cal E})).$

Hence, the maximization of $H^{\prime}(P_{{\cal E}\times{\cal E}})$ such that
$\pi({\cal E})$ is known and (3) and $(H2)$ are satisfied, is equivalent to
maximize the entropy $h(Z)$ of the Markov shift $Z$, subject to the knowledge
of its stationary distribution ${\widehat{\pi}}_{\cal E}$.

We will assume one also knows the communication matrix $L=(L(e,d):e,d\in{\cal
E})$ in the labyrinth. This information is $0-1$ valued: $L(e,d)=1$ means the
state $d$ communicates with $e$ in the labyrinth, and $L(e,d)=0$ means this
communication is forbidden. We assume the labyrinth is an irreducible network,
namely is $L$ is an irreducible matrix. In other words for all $d,e$ there is
a path $d_{0}=d,..,d_{k}=e$ such that $L(d_{i},d_{i+1})=1$ for $i=0,..,k-1$.
For $d\in{\cal E}$ we define,

$L_{d}=\\{e\in{\cal E}:L(d,e)=1\\}\hbox{ and }L^{d}=\\{e\in{\cal
E}:L(e,d)=1\\},$

which are the set of states that follow $d$ and the set of states that precede
$d$, respectively. By definition, $e\in L_{d}$ if and only if $d\in L^{e}$. We
assume that the transition matrix ${\widehat{P}}$ satisfies

$(H3)\quad\quad\quad\quad\quad\quad L(d,e)=0\Rightarrow{\widehat{P}}(d,e)=0.$

So, we must maximize

$h(Z)=-\sum_{d\in{\cal E}}{\widehat{\pi}}(d)\sum_{e\in
L_{d}}{\widehat{P}}(d.e)\log{\widehat{P}}(d,e),$

subject to the set constraints

$\displaystyle\forall d\in{\cal E}:\;\sum_{e\in L_{d}}{\widehat{P}}(d,e)=1\,;$
(6) $\displaystyle\forall e\in{\cal E}:\;\sum_{d\in
L^{e}}{\widehat{\pi}}(d){\widehat{P}}(d,e)={\widehat{\pi}}(e)\,.$ (7)

######  Theorem 3.3.

Assuming that the constraints (6) and (7) are feasible, the maximum entropy
completion of the chain satisfying hypothesis $(H3)$ is given by

${\widehat{P}}(d,e)=\alpha(d)\beta(e){\bf 1}(e\in L_{d})\;,$

where

$\forall e\in{\cal E}:\;\beta(e)=\frac{{\widehat{\pi}}(e)}{\sum_{d\in
L^{e}}{\widehat{\pi}}(d)\frac{1}{\sum_{c\in L_{d}}\beta(c)}}\;,$ (8)

and

$\forall d\in{\cal E}:\;\alpha(d)=\frac{1}{\sum_{e\in L_{d}}\beta(e)}\;.$ (9)

Moreover, the condition

$\forall d\in{\cal E}:\quad L(d,d)=1,$ (10)

is sufficient in order that the constraints (6) and (7) are feasible, that is
under (10) there always exists a nonnegative matrix ${\widehat{P}}$ satisfying
them.

###### Proof.

By using Lagrange multipliers one gets that the maximum entropy is attained
for transition probabilities of the form

${\widehat{P}}(d,e)=\alpha(d)\beta(e){\bf 1}(e\in L_{d}).$

The functions $\alpha$ and $\beta$ defined on ${\cal E}$ are nonnegative.
Since the matrix ${\widehat{P}}$ is stochastic one gets that $\alpha$ is
strictly positive. If some $\beta(e)=0$ one will obtain ${\widehat{P}}(d,e)=0$
for all $d\in L^{e}$ and (7) gives ${\widehat{\pi}}(e)=0$, hence also
$\pi(e)=0$, which contradicts the irreducibility of $P$. Since the constraints
are satisfied, one gets: (9) holds and

$\forall e\in{\cal E}:\;\left(\sum_{d\in
L^{e}}{\widehat{\pi}}(d)\alpha(d))\right)\beta(e)={\widehat{\pi}}(e).$ (11)

So, by using (9) one gets that $\beta$ satisfies (8). From (9), $\alpha$ is
determined by $\beta$.

The last part of the theorem will be shown in Proposition 3.6 in the next
Section. ∎

Notice that one can always assume that certain $\beta(e_{0})=1$. In fact we
can divide both sides of (8) by $\beta(e_{0})$ and we obtain analogous
relations but with $\beta(e)/\beta(e_{0})$ instead of $\beta(e)$. So, there
are left $|{\cal E}|-1$ parameters to be determined.

We can see that in some cases there is no solution and this means that the
restrictions given by matrix $(L(d,e):d,e\in{\cal E})$ are not the good ones
or that the assumptions $(H2)$ is not satisfied (this is the one that cannot
be observed). Let us give an example that does not have a solution. Take
${\cal E}=\\{1,2,3\\}$ and assume the stationary measure ${\widehat{\pi}}$
satisfies ${\widehat{\pi}}(1)>{\widehat{\pi}}(2)+{\widehat{\pi}}(3)$. Then,
one can check that for the $2-$periodic irreducible matrix $L$ given by
$L(d,e)=0$ if $e=d$ and $L(d,e)=1$ when $e\neq d$, there is no transition
matrix ${\widehat{P}}=({\widehat{P}}(d,e):d,e\in{\cal E})$ that satisfies
${\widehat{P}}(d,e)=0$ when $e=d$ and ${\widehat{\pi}}_{\cal E}$ is a
stationary matrix of ${\widehat{P}}$.

######  Remark 3.4.

When $I=\emptyset$, we have $\pi({\cal E})=1$ so ${\widehat{P}}=P$ and the
unique restriction on the matrix $P_{{\cal E}\times{\cal E}}$ is given by (6).
The maximum entropy matrix is given by the Markov chain defined by Parry
distribution [5], let us describe it. Let $\varphi$ and $\nu$ be the right and
left Perron-Frobenius eigenvectors associated to $L$ with eigenvalue $\lambda$
and normalized by $\sum_{d\in{\cal E}}\nu(d)\varphi(d)=1$. The stochastic
matrix ${\widehat{P}}(d,e)=\lambda^{-1}\varphi(e)/\varphi(d)$ has stationary
distribution $(\nu(d)\varphi(d):d\in{\cal E})$. Its entropy is $\log\lambda$,
which is the topological entropy. It is known to maximize the entropy of all
the Markov chains whose transition matrices ${\widehat{P}}$satisfy $(H3)$.
(But also of all stationary distributions of the topological Markov shift
defined by $L$, see [5] Chapter 17). $\Box$

###  3.3 Associated linear problem

The hypothesis $(H3)$ and constraints (6), (7), can be put in the following
form for the matrix ${\widehat{P}}\geq 0$,

$\displaystyle\forall d\in{\cal E}:\;\sum_{e\in{\cal
E}}{\widehat{P}}(d,e)=1\,;$ (12) $\displaystyle\forall e\in{\cal
E}:\;\sum_{d\in{\cal
E}}{\widehat{\pi}}(d){\widehat{P}}(d,e)={\widehat{\pi}}(e)\,;$ (13)
$\displaystyle\forall d,e\in{\cal
E}:\;L(d,e)=0\Rightarrow{\widehat{P}}(d,e)=0.$ (14)

Let us put the whole problem, in particular these constraints, in a vector
form.

Let $\ell=|{\cal E}|$. We assume ${\cal E}=\\{1,\dots,\ell\\}$ and so
${\widehat{P}}=\\{{\widehat{P}}(i,j):i,j=1,\dots,\ell)$. We define a column
vector ${\vec{p}}=(p(k);k=1,\cdots,\ell^{2})$ representing the matrix
${\widehat{P}}$. We set

$p(i+(j-1)\ell)={\widehat{P}}(i,j),i,j=1,\dots,\ell,$

so we group sequentially the columns of ${\widehat{P}}$, in fact
$(p(i+(j-1)\ell):i=1,\cdots,\ell)$ is the $j-$the column of ${\widehat{P}}$.
Since ${\widehat{P}}\geq 0$ one has ${\vec{p}}\geq 0$. We also write
${\widehat{\pi}}_{\cal E}=({\widehat{\pi}}(i):i=1,\dots,\ell)$. Consider the
function

$\delta(a,b)=\begin{cases}1&\hbox{ if }a=b\,;\\\ 0&\hbox{ if }a\neq
b.\end{cases}$

Define the following three matrices $A$, $B$, $C$, of dimension
$\ell\times\ell^{2}$. For $r,s,t\in\\{1,\dots,\ell\\}$ set,

$\displaystyle A(r,t+(s-1)\ell)=\delta(t,r);$ $\displaystyle
B(r,t+(s-1)\ell)={\widehat{\pi}}(t)\delta(s,r);$ $\displaystyle
C(r,t+(s-1)\ell)=(1-L(t,r))\delta(s,r).$

We have

$\sum_{t=1}^{\ell}\sum_{s=1}^{\ell}A(r,t+(s-1)\ell)p(t+(s-1)\ell)=\sum_{s=1}^{\ell}p(r+(s-1)\ell)=\sum_{s=1}^{\ell}{\widehat{P}}(r,s).$

Then, the restriction ${\widehat{P}}{{\bf{1}}}_{\cal E}={{\bf{1}}}_{\cal E}$
(12), is equivalent to $A{\vec{p}}={{\bf{1}}}$ with ${{\bf{1}}}$ the unit
vector of size $\ell$. On the other hand

$\sum_{t=1}^{\ell}\sum_{s=1}^{\ell}B(r,t+(s-1)\ell)p(t+(s-1)\ell)=\sum_{t=1}^{\ell}{\widehat{\pi}}(t)p(t+(s-1)\ell)=\sum_{t=1}^{\ell}{\widehat{\pi}}(t){\widehat{P}}(t,s).$

Then, the restriction ${\widehat{\pi}}^{t}{\widehat{P}}={\widehat{\pi}}^{t}$
(13) is equivalent to $B{\vec{p}}={\widehat{\pi}}$. Finally, since
$L(\cdot,\cdot)$ is $0-1$ valued we get,

$\displaystyle\sum_{t=1}^{\ell}\sum_{s=1}^{\ell}C(r,t+(s-1)\ell)p(t+(s-1)\ell)=\sum_{t=1}^{\ell}(1-L(t,r))p(t+(r-1)\ell)$
$\displaystyle=\sum_{t=1}^{\ell}(1-L(t,r)){\widehat{P}}(t,r)=\sum_{t\in\\{1,..,\ell\\}:L(t,r)=0}{\widehat{P}}(t,r).$

Since the restriction (14) is equivalent to $L(t,r)=0$ implies
${\widehat{P}}(t,r)=0$ for all $t,r$, then it is equivalent to
$C{\vec{p}}={\bf 0}$, the zero vector of size $\ell$.

We have proven that there exists ${\widehat{P}}\geq 0$ such that the
conditions (12), (13), (14), are fulfilled if and only if we have,

$D{\vec{p}}={\vec{b}},\;{\vec{p}}\geq 0,$

where

$D=(D(r,t+(s-1)\ell):r=1,..,3\ell,t,s=1,..,\ell)$

is a $3\ell\times\ell^{2}$ matrix whose coefficients are given by

$D(r,t+(s-1)\ell)=\begin{cases}A(r,t+(s-1)\ell)&\hbox{ if }1\leq r\leq\ell,\\\
B(r-\ell,t+(s-1)\ell)&\hbox{ if }\ell+1\leq r\leq 2\ell,\\\
C(r-2\ell,t+(s-1)\ell)&\hbox{ if }2\ell+1\leq r\leq 3\ell\,;\end{cases}$

and ${\vec{b}}$ is a $3\ell-$ dimensional vector given by

$b(r)=\begin{cases}1&\hbox{ if }1\leq r\leq\ell,\\\
{\widehat{\pi}}(r-\ell)&\hbox{ if }\ell+1\leq r\leq 2\ell,\\\ 0&\hbox{ if
}2\ell+1\leq r\leq 3\ell\,.\end{cases}$

By Farkas’ Lemma (see Proposition 3.2.1 p. 170 in [1]) there exists
${\vec{p}}\geq 0$ satisfying $D{\vec{p}}={\vec{b}}$ or there exists some
$y\in\mathbb{R}^{3\ell}$ satisfying $D^{t}y\geq 0$ and $b^{t}y<0$. Let us set
$y^{t}=(u^{t},v^{t},w^{t})$ with $u,v,w$ be $\ell-$dimensional vector. Then we
have,

$b^{t}y=\sum_{r=1}^{\ell}\big{(}u(r)+v(r){\widehat{\pi}}(r)\big{)}$

and so the condition $b^{t}y<0$ is equivalent to

$\sum_{r=1}^{\ell}(u(r)+{\widehat{\pi}}(r)v(r))<0.$ (15)

We have that,

$(y^{t}D)(t+(s-1)\ell)=u(t)+v(s){\widehat{\pi}}(t)+w(s)(1-L(t,s)).$

Then, the condition $D^{t}y\geq 0$ is equivalent to

$\forall t,s=1,\dots,\ell:\quad u(t)+v(s){\widehat{\pi}}(t)+w(s)(1-L(t,s))\geq
0.$ (16)

######  Proposition 3.5.

The conditions (15) and (16) are equivalent to (15) and (17), with

$\forall t,s\hbox{ such that }L(t,s)=1:\quad u(t)+v(s){\widehat{\pi}}(t)\geq
0.$ (17)

###### Proof.

If (17) is fulfilled then the choice

$w(s)\geq
max\\{|u(t)|+|v(s)|{\widehat{\pi}}(t):t=1,\dots,\ell\\},s=1,\dots,\ell,$

implies that (16) is satisfied and (15) is not modified. ∎

######  Proposition 3.6.

If $L(i,i)=1$ for all $i=1,\dots,\ell$ then there always exists a matrix
${\widehat{P}}\geq 0$ satisfying (12), (13) and (14).

###### Proof.

From the condition $L(i,i)=1$ and (16) we get $u(i)+v(i){\widehat{\pi}}(i)\geq
0$ for all $i=1,\dots,\ell$. Then, (15) cannot be satisfied. Hence, there
exists ${\vec{p}}\geq 0$ satisfying $D{\vec{p}}={\vec{b}}$. ∎

Hence, the last part of Theorem 3.3 is proven.

###  3.4 Non-connected labyrinths

When there is a set of disjoint labyrinths that are only connected through the
states in $I$ the analysis is analogous. Let us briefly sketch it. As before,
$X$ is an irreducible stationary Markov chain, with values in the set
$I\sqcup{\cal E}$ with ${\cal E}=\bigsqcup_{m\in M}{\cal E}_{m}$ (disjoint
union). The transitions among the labyrinths satisfy $P_{{\cal
E}_{m}\times{\cal E}_{m^{\prime}}}=0$ when $m\neq m^{\prime}$. Since the chain
is irreducible, $E_{m}$ and $E_{m^{\prime}}$ are only connected through $I$.
So, for all $m\in M$ there exists some state $i_{m}\in I$ such that
$P(i_{m},{\cal E}_{m})>0$.

Let $\pi_{I}=(\pi(i):i\in I)$ and $\pi_{{\cal E}_{m}}=(\pi(d):d\in{\cal
E}_{m})$ denote the restrictions of $\pi$ to $I$ and ${\cal E}_{m}$
respectively, for $m\in M$. We assume that the transition submatrices
$P_{I\times I}$, $P_{I\times{\cal E}}$, $P_{{\cal E}\times I}$ and the vector
$\pi_{I}$ are known.

The assumption $(H1)$ now reads: $P_{{\cal E}_{m}\times I}={{\bf{1}}}_{{\cal
E}_{m}}\pi^{t}_{I}$ for all $m\in M$, so from all the labyrinths the law of
the outbreak on $I$ has distribution $\pi^{t}_{I}$.

Equation (2), $\pi_{I}^{t}P_{I\times I}=\pi(I)\pi^{t}$ is also deduced from
$(H1)$. From $(H1)$ we also get,

${{\bf{1}}}_{{\cal E}_{m}}=P_{{\cal E}_{m}\times
I}{{\bf{1}}}_{I}+\sum_{m^{\prime}\neq m}P_{{\cal E}_{m}\times{\cal
E}_{m^{\prime}}}{{\bf{1}}}_{{\cal E}_{m^{\prime}}}+P_{{\cal E}_{m}\times{\cal
E}_{m}}{{\bf{1}}}_{{\cal E}_{m}}=\pi(I){{\bf{1}}}_{{\cal E}_{m}}+P_{{\cal
E}_{m}\times{\cal E}_{m}}{{\bf{1}}}_{{\cal E}_{m}}.$

Since $\pi({\cal E})=1-\pi(I)$, (3) becomes, $P_{{\cal E}_{m}\times{\cal
E}_{m}}{{\bf{1}}}_{{\cal E}_{m}}=\pi({\cal E}){{\bf{1}}}_{{\cal E}_{m}}$. We
assume that $\pi({\cal E}_{m})^{-1}\pi_{{\cal E}_{m}}$ is a stationary
distribution for $\pi({\cal E})^{-1}P_{{\cal E}\times{\cal E}}$. So $(H2)$
becomes $\pi({\cal E})\pi_{{\cal E}_{m}}^{t}=\pi_{{\cal E}_{m}}^{t}P_{{\cal
E}_{m}\times{\cal E}_{m}}$ for all $m\in M$.

The same arguments show that (4) can be written $\pi_{{\cal
E}_{m}}=\pi(I)^{-1}\pi_{I}^{t}P_{I\times{\cal E}_{m}}$ for all $m\in M$.
Hence, all the vectors $\pi_{{\cal E}_{m}}$ are known. The entropy of the
stationary Markov chain $X$ is,

$\displaystyle h(X)$ $\displaystyle=$ $\displaystyle-\sum_{i\in
I}\pi(i)\sum_{a\in I\sqcup{\cal E}}P(i,a)\log P(i,a)-\sum_{d\in{\cal
E}}\pi(d)\sum_{j\in I}\pi(j)\log\pi(j)$ $\displaystyle\;-\sum_{d\in{\cal
E}}\pi_{d}\sum_{e\in{\cal E}}P(d,e)\log P(d,e).$

Since all $\pi_{{\cal E}_{m}}=\pi(I)^{-1}\pi_{I}^{t}P_{I\times{\cal E}_{m}}$
are known, the maximization of $h(X)$ is equivalent to maximize
$H^{\prime}(P_{{\cal E}\times{\cal E}})=-\sum_{d\in{\cal
E}}\pi(d)\sum_{e\in{\cal E}}P(d,e)\log P(d,e)$. Since $P_{{\cal
E}_{m}\times{\cal E}_{m^{\prime}}}=0$ when $m\neq m^{\prime}$,

$H^{\prime}(P_{{\cal E}\times{\cal E}})=-\sum_{m\in M}\left(\sum_{d\in{\cal
E}_{m}}\pi(d)\sum_{e\in{\cal E}_{m}}P(d,e)\log P(d,e)\right).$

So, it is equivalent to maximize $H^{\prime}(P_{{\cal E}_{m}\times{\cal
E}_{m}})$ with $P_{{\cal E}_{m}\times{\cal E}_{m}}$ subject to $P_{{\cal
E}_{m}\times{\cal E}_{m}}{{\bf{1}}}_{{\cal E}_{m}}\\!=\\!\pi({\cal
E}){{\bf{1}}}_{{\cal E}_{m}}$ and $\pi_{{\cal
E}_{m}}\\!=\\!\pi(I)^{-1}\pi_{I}^{t}P_{I\times{\cal E}_{m}}$ for $m\in M$.
Then, the analysis is the same as before, but for each one of the matrices
$P_{{\cal E}_{m}\times{\cal E}_{m}}$.

##  4 Restart and quasi-stationarity

We shall use the quasi-stationary theory, see [4], to show that we can
retrieve the chain $X$ given by the matrix $P$ that satisfies the maximum
entropy criterion, by knowing the evolution of the chain in the sets $I$ and
${\cal E}$ separately and by restarting the evolution into the other region
after some geometric time.

Firstly, by iterating (2) (which is a consequence of $(H1)$) we get
${\widehat{\pi}}_{I}^{t}P^{n}_{I\times I}=\pi(I)^{n}{\widehat{\pi}}_{I}^{t}$,
and so $\mathbb{P}_{\widehat{\pi}}(X_{k}\in I\;\forall k\leq n)=\pi(I)^{n}$.
Let $\tau_{\cal E}=\inf\\{n>0:X_{n}\in{\cal E}\\}$ be the hitting time of
${\cal E}$. Hence, the above equalities can be written
$\mathbb{P}_{\widehat{\pi}}(\tau_{\cal E}>n)=\pi(I)^{n}$ and

$\mathbb{P}_{{\widehat{\pi}}_{I}}(X_{n}=j\,|\,\tau_{\cal
E}>n)={\widehat{\pi}}_{I}(j),j\in I,n\geq 0.$

Therefore ${\widehat{\pi}}_{I}$ is a quasi-stationary distribution (q.s.d.) of
the Markov chain $X^{(I)}=(X_{n}:n<\tau_{\cal E})$ with states in $I$ and
killed at $\tau_{\cal E}$. Notice that, when starting from
${\widehat{\pi}}_{I}$ the sojourn time at $I$ is geometrically distributed
(This was firstly shown in [6]), $\tau_{I}\sim$Geometric$(\pi({\cal E}))$, and
we recover the previous relation $\mathbb{P}_{{\widehat{\pi}}_{I}}(\tau_{\cal
E}>n)=\pi(I)^{n}$.

Starting with ${\widehat{\pi}}_{I}$ the exit distribution from $I$ is given by

$\displaystyle\forall d\\!\in\\!{\cal E}\\!:$
$\displaystyle\mathbb{P}_{{\widehat{\pi}}_{I}}(X_{\tau_{\cal
E}}\\!=\\!d)\\!=\\!\sum_{n\geq 1}\mathbb{P}(X_{n}=d,\tau_{\cal
E}\\!=\\!n)\\!=\\!\sum_{n\geq
1}\mathbb{P}_{{\widehat{\pi}}_{I}}(X_{n}=d,\tau_{\cal E}\\!>\\!n\\!-\\!1)$
$\displaystyle=$ $\displaystyle\sum_{n\geq 1}\mathbb{P}(X_{n}=d|\tau_{\cal
E}\\!>\\!n-1)\mathbb{P}_{{\widehat{\pi}}_{I}}(\tau_{\cal
E}\\!>\\!n\\!-\\!1)\\!=\\!\sum_{n\geq 1}(\sum_{i\in
I}{\widehat{\pi}}(i)P(i,d))\pi(I)^{n-1}.$

From (4) we get $\pi(d)\\!=\\!\sum_{i\in I}{\widehat{\pi}}(i)P(i,d)$, and so
$\mathbb{P}_{{\widehat{\pi}}_{I}}(X_{{\cal
E}}\\!=\\!d)\\!=\\!\pi(d)(1\\!-\\!\pi(I))^{-1}$. We have shown that the exit
law from $I$ is ${\widehat{\pi}}_{\cal E}$:

$\forall d\in{\cal E}:\quad\mathbb{P}_{{\widehat{\pi}}_{I}}(X_{\tau_{\cal
E}}=d)={\widehat{\pi}}(d).$ (18)

Moreover, the variables $X_{\tau_{\cal E}}$ and $\tau_{\cal E}$ are
independent when starting from ${\widehat{\pi}}_{I}$. In fact, the same
argument gives,

$\forall d\in{\cal E},n\geq
1:\quad\mathbb{P}_{{\widehat{\pi}}_{I}}(X_{\tau_{\cal E}}=d,\tau_{\cal
E}=n)={\widehat{\pi}}(d)\mathbb{P}_{{\widehat{\pi}}_{I}}(\tau_{\cal E}=n).$
(19)

Similarly, let $\tau_{I}=\inf\\{n>0:X_{n}\in I\\}$ be the hitting time of $I$.
Then, in a similar way one shows that the hypothesis $(H2)$ implies that
${\widehat{\pi}}_{\cal E}$ is a q.s.d. of the Markov chain $X^{({\cal
E})}=(X_{n}:n<\tau_{I})$ with states in ${\cal E}$ and killed at $\tau_{I}$,
that is

$\mathbb{P}_{{\widehat{\pi}}_{\cal
E}}(X_{n}=d\,|\,\tau_{I}>n)={\widehat{\pi}}_{\cal E}(d),d\in{\cal E},n\geq 0,$

and $\tau_{I}\sim$Geometric$(\pi(I))$, namely
$\mathbb{P}_{{\widehat{\pi}}_{\cal E}}(\tau_{I}>n)=\pi({\cal E})^{n}$. When
starting with ${\widehat{\pi}}_{\cal E}$ the exit distribution from ${\cal E}$
is ${\widehat{\pi}}_{I}$. In fact,

$\displaystyle\forall i\\!\in\\!I\\!:$
$\displaystyle\mathbb{P}_{{\widehat{\pi}}_{\cal
E}}(X_{\tau_{I}}\\!=\\!i)\\!=\\!\sum_{n\geq
1}\mathbb{P}_{{\widehat{\pi}}_{\cal
E}}(X_{n}=i|\tau_{I}\\!>\\!n\\!-\\!1)\mathbb{P}{{\widehat{\pi}}_{\cal
E}}(\tau_{I}\\!>\\!n\\!-\\!1)$ $\displaystyle=$ $\displaystyle\sum_{n\geq
1}\pi({\cal E})^{n}(\sum_{i\in I}{\widehat{\pi}}(e)P(e,i)).$

From $(H1)$ we have $P(e,i)=\pi(i)$ and so $\mathbb{P}_{{\widehat{\pi}}_{\cal
E}}(X_{\tau_{I}}\\!=\\!i)\\!=\\!\pi(i)(1-\pi({\cal E}))^{-1}$, then the exit
law from ${\cal E}$ is ${\widehat{\pi}}_{I}$:

$\forall i\in I:\quad\mathbb{P}_{{\widehat{\pi}}_{\cal
E}}(X_{\tau_{I}}=i)={\widehat{\pi}}(i).$ (20)

Also we get that $X_{\tau_{I}}$ and $\tau_{I}$ are independent when starting
from ${\widehat{\pi}}_{\cal E}$,

$\forall i\in I,n\geq 1:\quad\mathbb{P}_{{\widehat{\pi}}_{\cal
E}}(X_{\tau_{I}}=i,\tau_{I}=n)={\widehat{\pi}}(i)\mathbb{P}_{{\widehat{\pi}}_{\cal
E}}(\tau_{I}=n).$ (21)

Consider the matrix $Q=(Q(i,j):i,j\in I)$ given by $Q(i,j)=P(i,j)+P(i,{\cal
E}){\widehat{\pi}}(j)$, namely

$Q=P_{I\times I}+P_{I\times{\cal E}}{{\bf{1}}}_{{\cal
E}}{\widehat{\pi}}_{I}^{t}.$ (22)

This matrix is stochastic and one can check that ${\widehat{\pi}}_{I}$ is its
stationary distribution. Let $Y^{0}=(Y^{0}_{n}:n\in\mathbb{Z})$ be a Markov
chain with transition matrix $Q$ and stationary distribution
${\widehat{\pi}}_{I}$. Then $Y^{0}$ gives the trajectories of the original
chain $X$ but starting from ${\widehat{\pi}}_{I}$, when one only marks the
states in $I$. In fact, a transition from $i$ to $j$ can be made directly with
a jump in $P$, plus an entrance to the labyrinth ${\cal E}$ and then
reemerging from ${\cal E}$ to $I$. This uses (18) and (20). As it is known,
the evolution of the reversed chain $(Y^{0}_{-n}:n\in\mathbb{Z})$ is given by
the transition matrix
$Q^{*}=(Q^{*}(i,j)=Q(j,i){\widehat{\pi}}(j)/{\widehat{\pi}}(i):i,j\in I)$ (The
matrix $Q$ was introduced in [7]).

Similarly, consider the matrix $R=(R(d,e):d,e\in{\cal E})$ given by

$R=P_{{\cal E}\times{\cal E}}+P_{{\cal E}\times
I}{{\bf{1}}}_{I}{\widehat{\pi}}_{\cal E}^{t}.$ (23)

This matrix is stochastic and ${\widehat{\pi}}_{\cal E}$ is its stationary
distribution. Let $Y^{1}=(Y^{1}_{n}:n\in\mathbb{Z})$ be a Markov chain with
transition matrix $R$ and stationary distribution ${\widehat{\pi}}_{\cal E}$.
Then $Y^{1}$ gives the trajectories of the original chain $X$ starting from
${\widehat{\pi}}_{\cal E}$, when one only marks the states in ${\cal E}$. A
transition from $d$ to $e$ is made directly with a jump given by the maximum
entropy transition matrix $P$ plus an entrance to $I$ and then reemerging from
$I$ to ${\cal E}$. This uses (20) and (18). The evolution of the reversed
chain $(Y^{1}_{-n}:n\in\mathbb{Z})$ is given by
$R^{*}=(R^{*}(d,e)=R(e,d){\widehat{\pi}}(e)/{\widehat{\pi}}(d):d,e\in{\cal
E})$

To construct of a copy of $X$ it is useful to refer to region $0$ for states
in $I$ and to region $1$ for states in ${\cal E}$. For region $k$, the region
$k+1$ will be the $1$ region if $k=0$ and the region $0$ when $k=1$. In
general, for $n\in\mathbb{Z}$ the region $k+n$ is taken mod$(2)$: it is $k$
when $n$ is even and it is $k+1$ if $n$ is odd.

For choosing the starting region at time $0$, one introduce a Bernoulli random
variable $B$ with

$\mathbb{P}(B=0)=\pi(I)\hbox{ and }\mathbb{P}(B=1)=\pi({\cal E}).$

Let ${\cal T}^{0}=(\tau^{0}_{n}:n\in\mathbb{Z})$ and ${\cal
T}^{1}=(\tau^{1}_{n}:n\in\mathbb{Z})$ be two independent sequences of
independent times with $\tau^{0}_{n}\sim$Geometric$(\pi({\cal E}))$ and
$\tau^{1}_{n}\sim$Geometric$(\pi(I))$. Then, the evolution in the regions of
type $k$ are killed at times $\tau^{k}_{n}$, for $k=0,1$.

Now we construct the alternated sequence of times $(T^{n}:n\in\mathbb{Z})$
with $T^{0}=\tau^{B}_{0}$ and $T^{m}=\tau^{B+m}_{m}$ for $m\in\mathbb{Z}$. So,
this sequence alternates the killing times of the two regions. The set of
integers $\mathbb{Z}$ is partitioned into the regions
$\mathbb{Z}=\bigsqcup_{m\in\mathbb{Z}}[T^{m},T^{m+1})$. We define the process
$(W_{n}:n\in\mathbb{Z})$ by:

$(W_{n}:n\in[T^{m},T^{m+1}))=(Y^{B+m}_{n}:n\in[T^{m},T^{m+1})),m\in\mathbb{Z}.$

######  Theorem 4.1.

The process $W$ and the stationary chain $X$, have the same law.

###### Proof.

This follows at once from the above construction of the process $W$ and the
relations (18), (19), (20), (21), (22) and (23). ∎

## Acknowledgments

This work was supported by the Center for Mathematical Modeling ANID Basal
Project FB210005. The authors thank Prof. Jean-René Chazottes from CNRS for
his interest in their work.

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# Multiple Zeeman-type Hidden Spin Splittings in
$\mathcal{\hat{P}\hat{T}}$-Symmetric Layered Antiferromagnets

Sajjan Sheoran<EMAIL_ADDRESS>Saswata Bhattacharya
<EMAIL_ADDRESS>Department of Physics, Indian Institute of
Technology Delhi, New Delhi 110016, India

###### Abstract

Centrosymmetric antiferromagnetic semiconductors, although abundant in nature,
appear less favorable in spintronics owing to the lack of inherent spin
polarization and magnetization. We unveil hidden Zeeman-type spin splitting
(HZSS) in layered centrosymmetric antiferromagnets with asymmetric sublayer
structures by employing first-principles simulations and symmetry analysis.
Taking the bilayer counterpart of recently synthesized monolayer MnSe, we
demonstrate that the degenerate states around specific high-symmetry points
spatially segregate on different sublayers forming
$\mathcal{\hat{P}\hat{T}}$-symmetric pair. Furthermore, degenerate states
exhibit uniform in-plane spin configurations with opposite orientations
enforced by mirror symmetry. Bands are locally Zeeman-split up to order of
$\sim$70 meV. Strikingly, a tiny electric field of a few mVÅ-1 along the
$z$-direction breaks the double degeneracy forming additional Zeeman pair.
Moreover, our simulations on trilayer and tetralayer MnSe show that achieved
HZSS is independent of layer number. These findings establish the design
principle to obtain Zeeman-type splitting in centrosymmetric antiferromagnets
and significantly expand the range of materials to look for spintronics.

Spin polarization in nonmagnetic crystals without inversion symmetry
($\mathcal{\hat{P}}$) can be achieved through relativistic spin-orbit coupling
(SOC), as demonstrated by Dresselhaus [1] and Rashba [2, 3, 4] in their
influential works. Recent research has indicated that a similar phenomenon
called hidden spin polarization (HSP) can exist even in centrosymmetric
crystals, provided that individual atomic sites break local inversion symmetry
[5, 6, 7, 8, 9, 10]. The discovery of HSP opens up possibilities for a broader
range of materials in spintronics and offers new insights into various hidden
physical properties such as optical polarization [11, 12], valley polarization
[12], orbital polarization [13, 14], and Berry curvature [15, 16]. The HSP
effect in nonmagnetic materials, characterized by the coexistence of
$\mathcal{\hat{P}}$ and time-reversal symmetry ($\mathcal{\hat{T}}$), is odd
distributed in both real and momentum spaces within a localized sector [17].
While techniques like spin- and angle-resolved photoemission spectroscopy have
successfully measured HSP with both positional (r) and momentum (k)
resolution, its application in spintronics necessitates the breaking of global
symmetry, typically achieved through external electric fields [8, 12].

Antiferromagnets have recently emerged as a viable substitute for nonmagnetic
and ferromagnetic materials in spintronic applications [18, 19, 20, 21, 22].
Due to their resilience against magnetic disruptions, lack of stray fields,
and ability to exhibit exceptionally rapid spin dynamics, antiferromagnets
possess the potential to outperform ferromagnets. Recent research focused on
investigating spin splittings within various magnetic space groups,
encompassing antiferromagnets [23, 24]. However, centrosymmetric
antiferromagnetic (AFM) semiconductors, which lack polarization and have
either negligible or nonexistent magnetization, pose challenges in generating
substantial and controllable spin splitting using magnetic or electric fields
[25, 26, 27]. As a result, despite their abundance in nature, centrosymmetric
AFM semiconductors do not appear promising for practical applications in
spintronics. Interestingly, the magnetoelectric effect [28], in which an
electric field has a dual effect of inducing both polarization and
magnetization, allows control over spin splitting and textures in
centrosymmetric antiferromagnets even without SOC [27]. The gate-controlled
Magneto-optic Kerr effect in layered collinear AFM MnPSe3 [25] and anomalous
Hall effect in orthorhombic AFM CuMnAs [29] are observed. Additionally, two
AFM semiconductors, Fe2TeO6 and SrFe2S2O, show spin-splittings as large as 55
and 30 meV induced by the electric field of strength 60 mV/Å [27]. Even though
the magnetoelectric coupling effect is observed in several centrosymmetric
antiferromagnets through first-principles simulations [25, 26, 27, 29], the
attainment of HSP remains uncommon due to the stringent requirements imposed
on lattice and site symmetry.

Figure 1: Geometric structures of bilayer MnSe with high-symmetry stackings
(a) AA and (b) AA′. The upper layer in AA and AA′ is obtained by the
translation ($d\hat{z}$) and mirror reflection ($M_{z}$) of the lower layer,
respectively. The energy distribution of different translations between layers
for the (c) AA and (d) AA′ stacking. The energy surfaces are interpolated
using 10 steps in a and b directions such that all possible high-symmetry
stackings are included. (e) Structure of the global ground state with AB
stacking having magnetization $M$$\uparrow\downarrow\uparrow\downarrow$. The
sublayers $\alpha_{i}$ and $\beta_{i}$ ($i=1,2$) are connected by
$\mathcal{\hat{P}\hat{T}}$-center with orange dashed lines. (f) Energy band
dispersion of monolayer MnSe without (solid black lines) and with an electric
field (red dashed lines) of 0.05 VÅ-1 along $z$-direction. Arrows are used to
illustrate spin orientations.

In this letter, we examine the occurrence of hidden Zeeman-type spin splitting
(HZSS) in AFM 2D van der Waals (vDW) bilayers and multilayers made up of the
newly synthesized monolayer MnSe [30] using a combination of symmetry analysis
and density-functional theory (DFT) calculations. Scanning potential energy
surfaces of all possible high-symmetry stacking, we show that bilayer MnSe
with $\mathcal{\hat{P}\hat{T}}$ symmetry is energetically most favorable.
$\mathcal{\hat{P}\hat{T}}$ symmetry leads to the HZSS with minimal mixing
between degenerate states segregated on different sublayers. In addition, the
mirror plane ($\hat{M}_{x}$) enforces the spins to be k-independent.
Strikingly, a tiny electric field splits the otherwise doubly degenerate bands
preserving the spin configurations. Switching the electric field can reverse
the electronic spin magnetization connected to the energy level splitting.
Afterward, we extend our approach to the trilayer and tetralayer MnSe to show
the layer number independence of our observations.

The MnSe monolayer consists of two buckled honeycomb sublayers of MnSe,
interconnected by Mn-Se bonds [Fig. 1a]. Within the monolayer, the upper Mn/Se
atoms are positioned alternately on top of the lower Se/Mn atoms. Each unit
cell of monolayer contains two Mn and two Se atoms, with a lattice constant of
4.28 Å, belonging to the space group $P\overline{3}m1$ with $\mathcal{P}$
symmetry (ignoring spin configurations). Considering different possible
magnetic configurations, we find that the upper and lower sublayers are
coupled antiferromagnetically through Neel-type antiferromagnetism (see
section I of the supplemental Material (SM) [31] for computational methods).
The robust Neel-type AFM state is more stable than the FM state by 0.38 eV per
unit cell (u.c.). Additionally, the magnetic anisotropic energy (MAE)
calculations reveal that the monolayer MnSe prefers in-plane magnetization
over out-of-plane magnetization, with an energy difference of 0.37 meV/u.c.
These results are consistent with the previous experimental [30] and
theoretical reports [30, 32, 33].

Determination of stacking orders in multilayers is challenging, especially for
more complex structures. Firstly, we have created bilayer MnSe by taking the
upper layer as translation (by $d$$\hat{z}$) and mirror reflection
($\hat{M}_{z}$) of the lower layer, as shown in Figs. 1(a) and 1(b). We name
these stacking AA (obtained through $d$$\hat{z}$) and AA′ (obtained through
$\hat{M}_{z}$), following the nomenclature used in Ref. [34]. After that, we
transformed these stackings into other possible high-symmetry stacking by
translation sliding of one of the basal planes in the unit cell: going to AB
and BA state from AA, and going to AB′ and BA′ from AA′. Further, DFT
calculations were conducted to ascertain the lowest energy states for various
stacking arrangements. As shown in Fig. 1(c), the top layer sliding from AA to
AB results in a nondegenerate global energy minimum, whereas BA stacking is
energetically unfavorable. On the other hand, shifting the top layer from AA′
leads to doubly degenerate energy minima AB′ and BA′ [Fig. 1(d)]. AB stacking
is energetically most favorable, having lower energy than AB′/BA′ by 14 meV
per unit cell. Bilayer MnSe can be divided into sublayers $\alpha_{i}$ and
$\beta_{i}$ ($i=$1,2), each sublayer containing a single Mn atom. Note that we
have taken four nondegenerate magnetic configurations
$M$$\uparrow\uparrow\uparrow\uparrow$,
$M$$\uparrow\downarrow\uparrow\downarrow$,
$M$$\uparrow\downarrow\downarrow\uparrow$, and
$M$$\uparrow\uparrow\downarrow\downarrow$ (here, up and down arrows represent
the magnetic moment direction of sublayers along $+x$ and $-x$ direction,
respectively, in the order of $\beta_{2}$, $\beta_{1}$, $\alpha_{1}$, and
$\alpha_{2}$ [Fig. 1(e)]). Given that the easy axis of magnetization lies in
the $x$-$y$ plane with MAE of 0.9 meV/u.c., we have chosen the $x$-direction
as the axis for magnetization. Our DFT calculations discover that Neel-type
AFM configuration $M$$\uparrow\downarrow\uparrow\downarrow$ has the lowest
energy. Therefore, bilayer MnSe is most likely to exist in AB stacking with
$M$$\uparrow\downarrow\uparrow\downarrow$ configuration, which is obtained by
translating the top layer in AA by
$\frac{2}{3}\textbf{{a}}+\frac{1}{3}\textbf{{b}}$. Our calculation assumes
that the bilayer MnSe exhibits AB stacking with
$M$$\uparrow\downarrow\uparrow\downarrow$ configuration unless otherwise
specified.

Table 1: The transformation rules of ($P_{x},P_{y},P_{z}$) and ($\hat{\sigma}_{x},\hat{\sigma}_{y},\hat{\sigma}_{z}$) with respect to the operations ($\hat{O}$) belonging to MPG $2^{\prime}/m$. The last column shows the terms which are invariant under operation. $\hat{O}$ | ($P_{x},P_{y},P_{z}$) | ($\hat{\sigma}_{x},\hat{\sigma}_{y},\hat{\sigma}_{z}$) | Invariants
---|---|---|---
$\hat{I}$ | ($P_{x},P_{y},P_{z}$) | ($\hat{\sigma}_{x},\hat{\sigma}_{y},\hat{\sigma}_{z}$) |  $P_{i}\hat{\sigma}_{j}(i,j$=$x,y,z)$
$\hat{M}_{x}$ | ($-P_{x},P_{y},P_{z}$) | ($\hat{\sigma}_{x},-\hat{\sigma}_{y},-\hat{\sigma}_{z}$) |  $P_{x}\hat{\sigma}_{y/z}$, $P_{y/z}\hat{\sigma}_{x}$
$\hat{C}_{2x}$$\mathcal{\hat{T}}$ | ($P_{x},-P_{y},-P_{z}$) | ($-\hat{\sigma}_{x},\hat{\sigma}_{y},\hat{\sigma}_{z}$) |  $P_{x}\hat{\sigma}_{y/z}$, $P_{y/z}\hat{\sigma}_{x}$
$\mathcal{\hat{P}\hat{T}}$ | ($-P_{x},-P_{y},-P_{z}$) | ($-\hat{\sigma}_{x},-\hat{\sigma}_{y},-\hat{\sigma}_{z}$) |  $P_{i}\hat{\sigma}_{j}(i,j$=$x,y,z)$
Figure 2: Schematic illustration of band structures with (a) null spin
splitting and (b) Zeeman spin splitting. (c) Sketches of twofold degenerate
bands in the presence of $\hat{P}\hat{T}$ symmetry with hidden spin
polarization, where each band is segregated on different sublayers. (d) The
lifting of band degeneracy in the presence of an electric field along the
$z$-direction. The energy levels with positive and negative spin magnetization
are sketched in the red and blue curves, respectively.

The magnetic point group (MPG) for bilayer MnSe with AB stacking is
$2^{\prime}/m$ having symmetry operations $\hat{M}_{x}$,
$\hat{C}_{2x}$$\mathcal{\hat{T}}$, and $\mathcal{\hat{P}\hat{T}}$ besides
trivial identity operation ($\hat{I}$). Where $\hat{M}_{x}$ and $\hat{C}_{2x}$
are mirror reflection in the $y$-$z$ plane and two-fold rotation around the
$x$-axis, respectively. $\alpha_{i}$ is connected to the $\beta_{i}$ through
$\mathcal{\hat{P}\hat{T}}$ symmetry [Fig. 1(e)]. Therefore, we say that
$\alpha_{i}$ is the inversion partner of $\beta_{i}$ and vice versa. Due to
the $\mathcal{\hat{P}\hat{T}}$ symmetry, spin polarization
$S_{\textbf{{k}}}(\textbf{{r}})$ is odd distributed in both r and k spaces,
i.e. $S_{\textbf{{k}}}^{\alpha_{i}}$=$-S_{\textbf{{k}}}^{\beta_{i}}$ and
$S_{\textbf{{k}}}^{\alpha_{i}(\beta_{i})}$=$-S_{-\textbf{{k}}}^{\alpha_{i}(\beta_{i})}$
[35]. Furthermore, the energy of state localized on $\alpha_{i}$ sublayer
$|\psi_{\textbf{{k}},\uparrow}^{\alpha_{i}}\rangle$
($E_{\textbf{{k}},\uparrow}^{\alpha_{i}}$) and state localized on $\beta_{i}$
sublayer $|\psi_{\textbf{{k}},\downarrow}^{\beta_{i}}\rangle$
($E_{\textbf{{k}},\downarrow}^{\beta_{i}}$) are degenerate [Fig. 2]. Although
energy bands are degenerate, the two components of doubly degenerate bands
have opposite spin polarization, each spatially localized on one of the two
separate sublayers forming the inversion partners called the spin-sublayer
locking effect [Fig. 2(c)]. While the total spin polarization is prohibited by
$\mathcal{\hat{P}\hat{T}}$ symmetry, it is possible to observe local spin
polarization with opposite values in sublayers $\alpha_{i}$ and $\beta_{i}$.
When examining a specific layer (say $\alpha$) that consists of sublayers
$\alpha_{1}$ and $\alpha_{2}$, the states localized on $\alpha_{1}$ and
$\alpha_{2}$ have non-degenerate energy. Consequently, the states
$|\psi_{\textbf{{k}},\uparrow}^{\alpha_{1}}\rangle$ and
$|\psi_{\textbf{{k}},\downarrow}^{\alpha_{2}}\rangle$ create a Zeeman pair.
Similarly, $\beta_{1}$ and $\beta_{2}$ states also make a Zeeman Pair.
Moreover, the anticommutation relationship between $\hat{M}_{x}$ and
$\hat{\sigma}_{y,z}$ in spin space (where $\hat{\sigma}_{x,y,z}$ represents
the Pauli matrices) imposes a condition on the general energy eigenstate
$|\psi_{\textbf{{k}},\uparrow/\downarrow}^{\alpha_{i}/\beta_{i}}\rangle$.
Specifically,
$\langle\psi_{\textbf{{k}},\uparrow/\downarrow}^{\alpha_{i}/\beta_{i}}|\hat{M}_{x}^{-1}\hat{\sigma}_{y,z}\hat{M}_{x}|\psi_{\textbf{{k}},\uparrow/\downarrow}^{\alpha_{i}/\beta_{i}}\rangle$=$-\langle\psi_{\textbf{{k}},\uparrow/\downarrow}^{\alpha_{i}/\beta_{i}}|\hat{\sigma}_{y,z}|\psi_{\textbf{{k}},\uparrow,\downarrow}^{\alpha_{i}/\beta_{i}}\rangle$,
which implies that
$\langle\psi_{\textbf{{k}},\uparrow/\downarrow}^{\alpha_{i}/\beta_{i}}|\hat{\sigma}_{y,z}|\psi_{\textbf{{k}},\uparrow/\downarrow}^{\alpha_{i}/\beta_{i}}\rangle$=0.
Consequently, the spin orientations are independent of k and aligned parallel
or antiparallel to $x$ direction. That leads to the HZSS with persistent spin
textures [Fig. 2(c)], and energy relationship can be expressed as

$E_{\textbf{{k}},\uparrow}^{\alpha_{i}}=E_{\textbf{{k}},\downarrow}^{\beta_{i}};i=1,2\quad\text{and}\quad
E_{\textbf{{k}},\uparrow}^{s_{1}}\neq
E_{\textbf{{k}},\downarrow}^{s_{2}};s=\alpha,\beta$ (1)

Next, we apply an electric field ($\mathcal{E}$) to break
$\mathcal{\hat{P}\hat{T}}$ symmetry. The electric field creates a net
polarization ($\textbf{{P}}\propto$ $\mathcal{E}$) [28]. Induced polarization
generates a net magnetization ($M_{i}\propto P_{j}$) through the
magnetoelectric effect [28], lifting the $\mathcal{\hat{P}\hat{T}}$ symmetry
between inversion partners. The occurrence of magnetization induces an
effective magnetic field ($\textbf{{B}}^{eff}\propto\textbf{{P}}$) that
couples with the spin degrees of freedom, yielding a Zeeman-like Hamiltonian
$\sum_{i,j}\lambda_{i,j}P_{i}\hat{\sigma}_{j}$. Tensor $\lambda_{i,j}$
determines the strength of the magnetoelectric coupling. Some components of
$\lambda_{i,j}$ are never non-zero and vanish due to the symmetry. We
determine the symmetry-allowed terms $\lambda_{ij}$ using the method of
invariants [36] ($\hat{H}=\hat{O}^{\dagger}\hat{H}\hat{O}$, where $\hat{O}$ is
the symmetry operation belonging to the MPG $2^{\prime}/m$), generally used to
determine the k.p Hamiltonian in nonmagnetic materials [37, 38]. The Zeeman
Hamiltonian ($\hat{H}_{Z}$), following the transformation rules of $P_{i}$ and
$\hat{\sigma}_{j}$ listed in Table 1, is given by

$\hat{H}_{Z}=\lambda_{x,y}P_{x}\hat{\sigma}_{y}+\lambda_{x,z}P_{x}\hat{\sigma}_{z}+\lambda_{y,x}P_{y}\hat{\sigma}_{x}+\lambda_{z,x}P_{z}\hat{\sigma}_{x}.$
(2)

$\hat{H}_{Z}$ breaks the $\mathcal{\hat{P}\hat{T}}$ symmetry and lifts the
energy degeneracy between states localized on inversion partners [Fig. 2(d)]
and thus forming additional Zeeman-like pairs

$E_{\textbf{{k}},\uparrow}^{\alpha_{i}}\neq
E_{\textbf{{k}},\downarrow}^{\beta_{i}};i=1,2\quad\text{and}\quad
E_{\textbf{{k}},\uparrow}^{s_{1}}\neq
E_{\textbf{{k}},\downarrow}^{s_{2}};s=\alpha,\beta.$ (3)

It should be noted that $\hat{H}_{Z}$ is determined assuming that the
magnetization axis aligns with the $x$-direction. However, the Hamiltonian can
be adapted for any in-plane direction by rotating it accordingly.

Figure 3: (a) Orbital-projected band dispersion curves of bilayer MnSe with AB
stacking. (b) Charge density distribution plotted for the states CB, CB+1,
CB+2, and CB+3 at K point. Layer-projected CBs of bilayer MnSe around the K
point along the (c) K-$\Gamma$ and (d) K-M. (e)-(f) counterparts of (c)-(d) in
the presence of a small electric field of 4 mVÅ-1 along the $z$-direction.

According to symmetry analysis, there are two conditions to observe HZSS: (i)
$\hat{P}\hat{T}$ symmetry and (ii) spin-sublayer locking effect. Condition (i)
is already fulfilled, and to check condition (ii), we plot the orbital-
projected band structure of bilayer MnSe considering spin-orbit interaction
[Fig. 3(a)]. Under $C_{3v}$ site symmetry, the Mn-$d$ orbitals split into
three categories $d_{z^{2}}$, $(d_{x^{2}-y^{2}},d_{xy})$, and
$(d_{xz},d_{yz})$. The Se-$p$ orbitals are split into two categories $p_{z}$
and $(p_{x}$, $p_{y})$. The Mn-$d$ orbitals form CB states, while Se-$p$
orbitals mainly contribute to VB states. Compared to the band structure of
monolayer MnSe [Fig. 1(f)], interlayer interaction between inversion partners
$\alpha_{1}$ and $\alpha_{2}$ results in increased separation between the
nested bands and spin mixture from another sublayer. Compared to the in-plane
orbitals ($d_{x^{2}-y^{2}},d_{xy},p_{x},p_{y}$), the out-of-plane orbitals
($d_{xz},d_{yz},d_{z^{2}},p_{z}$) exhibit significant interlayer interaction.
Observing HZSS is not ideal with the CB states around the M and $\Gamma$
points and the VB states around the K and M points. That is because these
states originate from the hybridization of out-of-plane orbitals between
different sublayers, and thus their wavefunctions are not segregated. The CB
states near the K point are predominantly contributed by
$(d_{x^{2}-y^{2}},d_{xy})$ and VB states around the $\Gamma$ point are
composed of mainly $(p_{x}$, $p_{y})$. Therefore states arising from the in-
plane orbitals, i.e., the CBs near the K point and the VBs near the $\Gamma$
point, have weak communication between different sublayers, leading to minimal
separation between nested bands and spin mixture from neighboring layers.
Consequently, each wavefunction is localized within the sublayer for the CBs
near the K point and the VBs near the $\Gamma$ point, forming the ideal choice
to observe spin-sublayer locking effect and HZSS [Fig. 3(b)].

Next, we plot layer-projected dispersion curves for low CBs in the vicinity of
K point for bilayer MnSe [Figs. 3(c) and 3(d)]. The four lowest CBs form two
two-fold degenerate pairs [see Eq. 1]. Despite having the same energy, these
states are localized on distinct sublayers and can be probed using r-resolved
spectroscopy techniques [7, 39]. For instance, the degenerate states CB and
CB+1 segregate on inversion partners $\alpha_{1}$ and $\beta_{1}$,
respectively [Fig. 3(b)]. Both inversion partners possess finite and opposite
k-independent spin orientations, either parallel or antiparallel to the $x$
direction [see section II in the supplemental material (SM) [31]], ensuring
zero net spin polarization respecting the $\mathcal{\hat{P}\hat{T}}$ symmetry
requirement. The $k$-independent persistent spin textures overcome spin
dephasing and provide non-dissipative spin transport [40, 41, 42].
$\alpha_{1}$ sublayer has a magnetization of magnitude 4.38 $\mu_{B}$ along
+$x$ direction and compensated by the opposite magnetization of inversion
partner $\beta_{1}$. A similar analysis also applies to the upper degenerate
pair CB+2 and CB+3, where wavefunctions corresponding to the CB+2 and CB+3
have segregated on $\alpha_{2}$ and $\beta_{2}$ sublayers, respectively. If we
focus locally on each layer, ($|\psi_{K,\uparrow}^{\alpha_{1}}\rangle$,
$|\psi_{K,\downarrow}^{\alpha_{2}}\rangle$) and
($|\psi_{K,\downarrow}^{\beta_{1}}\rangle$,
$|\psi_{K,\uparrow}^{\beta_{2}}\rangle$) form two Zeeman pair with energy
splitting $\Delta_{\alpha}$, $\Delta_{\beta}$, respectively [Figs. 3(c) and
3(d)]. As a consequence of $\mathcal{\hat{P}\hat{T}}$ symmetry, the strength
of the Zeeman spin splitting is the same for each layer
($\Delta_{\alpha}$=$\Delta_{\beta}$) to satisfy the Kramers’ degeneracy. We
have observed Zeeman spin splitting of order $\sim$39.5 meV. Observed Zeeman
splitting in our case (without external fields) is comparable to bulk AFM
semiconductors Fe2TeO6 (55 meV) and SrFe2S2O (33 meV) under the external
electric field of strength 60 mVÅ-1 [27]. However, it is still smaller than
nonmagnetic transition metal dichalcogenides, i.e., MoS2 (150 meV) [43]. HZSS
is also observed for the VB states near the $\Gamma$ point, where both
$\alpha$ and $\beta$ bands are Zeeman split ($\Delta_{\alpha}$,
$\Delta_{\beta}$) by $\sim$70 meV, as discussed in section III of SM [31].

Figure 4: The Zeeman spin splittings for the CBs (a) $\Delta_{1}$,
$\Delta_{2}$ and (b) $\Delta_{\alpha}$, $\Delta_{\beta}$ in bilayer MnSe as a
function of electric field $\mathcal{E}_{z}$.

The $\hat{H}_{Z}$ expressed in Eq. 2 shows that two-fold degeneracy will be
lifted by including polarization along an arbitrary direction
($P_{x},P_{y},P_{z}$). We include $P_{z}$ through the application of an
electric field along the $z$ direction ($\mathcal{E}_{z}$), and our model in
Eq. 2 can readily be generalized to

$\hat{H}_{Z}=\gamma_{z,x}\mathcal{E}_{z}\hat{\sigma}_{x}$ (4)

where $\gamma_{z,x}$ determines the strength of splitting. We plot the CBs in
a tiny $\mathcal{E}_{z}$ of 4 mVÅ-1 [Figs. 3(e) and 3(f)]. The
$\mathcal{E}_{z}$ creates an asymmetric potential on the inversion partners
sublayers, resulting in global $\mathcal{\hat{P}\hat{T}}$ symmetry breaking.
Each degenerate pair splits into two levels
$E_{+}=\gamma_{z,x}\mathcal{E}_{z}$ (eigenstate being $|+\rangle$) and
$E_{-}=-\gamma_{z,x}\mathcal{E}_{z}$ (eigenstate being $|-\rangle$), where
$\hat{\sigma}_{x}|+\rangle=+|+\rangle$ and
$\hat{\sigma}_{x}|-\rangle=-|-\rangle$. Therefore, two degenerate pairs in
bilayer MnSe splits into four energy levels, as represented by the
$\Delta_{1}$ and $\Delta_{2}$ in Fig. 3(f). Here $\Delta_{1}$ ($\Delta_{2}$)
represents the energy difference between the state localized on the
$\alpha_{1}$ ($\alpha_{2}$), and the state localized on $\beta_{1}$
($\beta_{2}$). Upon initial observation, it is evident that
$\Delta_{1}$$<$$\Delta_{2}$ due to the smaller real space separation between
$\alpha_{1}$ and $\beta_{1}$ compared to that between $\alpha_{2}$ and
$\beta_{2}$. Consequently, this results in a lesser potential difference
between $\alpha_{1}$ and $\beta_{1}$, leading to a smaller energy splitting
$\Delta_{1}$ compared to $\Delta_{2}$.

After that, we plot energy splittings
($\Delta_{1},\Delta_{2},\Delta_{\alpha}$, and $\Delta_{\beta}$) as a function
of the $\mathcal{E}_{z}$ [Figs. 4(a) and 4(b)]. We notice that the energy
splittings show a linear dependence on the $\mathcal{E}_{z}$ in the range of 0
to 10 mVÅ-1. Maximum $\Delta_{1}$ and $\Delta_{2}$ are observed to be 20.0 and
39.9 meV, respectively, with the $\mathcal{E}_{z}$ of strength 10 mVÅ-1. The
$\mathcal{E}_{z}$ breaks the equality between $\Delta_{\alpha}$ and
$\Delta_{\beta}$, where $\Delta_{\alpha}$ decreases and $\Delta_{\beta}$
increases with increasing strength of the $\mathcal{E}_{z}$. The observed
strength of $\Delta_{\alpha}$ and $\Delta_{\beta}$ are 28.0 and 47.8 meV,
respectively, under the influence of 10 mVÅ-1 $\mathcal{E}_{z}$. Additionally,
the reversal of the electric field from $\mathcal{E}_{z}$ to
$-\mathcal{E}_{z}$ will reverse the spin magnetization $S_{x}$ to $-S_{x}$
(see section IV of SM [31]). Under an electric field of the same magnitude
along the $+z$ or $-z$ direction, the strength of splittings $|\Delta_{1}|$
and $|\Delta_{2}|$ is independent of the electric field direction. However,
$\Delta_{1}$ and $\Delta_{2}$ will flip their sign under electric field
reversal. These results are in qualitative agreement with our model
Hamiltonian $\hat{H}_{z}$ in Eq. 4. If we compare the effect of the electric
field on bilayer MnSe with monolayer MnSe [Fig. 1(f)], we find that a large
electric field is required in monolayer MnSe to obtain the Zeeman splitting of
the same order. More specifically, Zeeman split pair of 50.9 meV is observed
with $\mathcal{E}_{z}$ of strength 50 mVÅ-1. Also, Zeeman splitting without
any external field in bilayer MnSe is an additional advantage over monolayer
MnSe.

It is interesting to note that $\hat{\mathcal{P}}$$\hat{\mathcal{T}}$ symmetry
remains intact with increasing layer thickness. Therefore, effects similar to
bilayer MnSe are also achieved in trilayer and tetralayer MnSe, where every
band is doubly degenerate, and degenerate pair of bands are localized on
different sublayers connected through $\hat{\mathcal{P}}$$\hat{\mathcal{T}}$
symmetry (see section V of SM [31]). Thus, experimental realization of HZSS
can be easily achieved in layered MnSe with arbitrary thickness.

In conclusion, we have shown the existence of HZSS in bilayer MnSe, combining
first-principles calculations and symmetry analysis. The stacking
characteristic study shows that AB with $\hat{\mathcal{P}}\hat{\mathcal{T}}$
symmetry is the lowest energy stacking. The doubly degenerate states arising
due to $\hat{\mathcal{P}}\hat{\mathcal{T}}$ symmetry are segregated into
different sublayers forming inversion partners. Then, we identified the local
Zeeman splitting as large as $\sim$70 meV in bilayer MnSe, where states
forming Zeeman pair with opposite spin orientation segregate on different
sublayers within a single layer. Interestingly, a tiny out-of-plane electric
field yields additional Zeeman pairs through breaking two-fold degeneracy.
Mirror symmetry ($\hat{M}_{x}$) enforced persistence spin textures remain
preserved under an out-of-plane electric field and are known for
nondissipative spin transport. Moreover, the fact that our results are
independent of the layer number adds credibility to the possibility of
experimental observation. Controllable Zeeman spin splittings achieved using
electric fields are detectable through established approaches such as optical
and transport measurements commonly used in spintronics [44, 45]. Zeeman
splittings with hidden spin polarization in centrosymmetric antiferromagnets
form another prospective aspect in developing semiconductor spintronics
devices [35]. We anticipate that our findings will not only contribute to a
deeper understanding of magnetoelectric interactions but also inspire further
innovative research in the emerging fields of AFM and semiconductor
spintronics [46, 47].

S.S. acknowledges CSIR, India, for the senior research fellowship [grant no.
09/086(1432)/2019-EMR-I]. S. B. acknowledges financial support from SERB under
a core research grant (grant no. CRG/2019/000647) to set up his High-
Performance Computing (HPC) facility “Veena” at IIT Delhi for computational
resources.

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