2021

[4]Obid Sh. Karshiboev These authors contributed equally to this work.

1]Institute of mathematics after named V.I.Romanovsky, University street,
100174, Tashkent, Uzbekistan

2]New Uzbekistan University, Mustaqillik Ave., 54, 100007, Tashkent,
Uzbekistan

3]Namangan state university, Uychi street, 316, 160136, Namangan, Uzbekistan

[4]Chirchik state pedagogical university, Amir Temur street, Chirchik, 111702,
Tashkent region, Uzbekistan

# The phase transition for the three-state SOS model with one-level competing
interactions on the binary tree

Muzaffar M. Rahmatullaev<EMAIL_ADDRESS><EMAIL_ADDRESS>[
[  [  *

###### Abstract

In this paper, we consider a three-state solid-on-solid (SOS) model with two
competing interactions (nearest-neighbour, one-level next-nearest-neighbour)
on the Cayley tree of order two. We show that at some values of parameters the
model exhibits a phase transition. We prove that for the model under some
conditions there is no antiferromagnetic phases.

###### keywords:

Cayley tree, Gibbs measure, SOS model, competing interactions

###### pacs:

[

Mathematics Subject Classification]Primary 82B05 $\cdot$ 82B20; Secondary
60K35

##  1 Introduction

The solid-on-solid (SOS) model on a Cayley tree is introduced in rs  as a
generalization of the Ising model. Since then a great interest has been
devoted to the investigation of various properties of SOS models on Cayley
trees (see, e.g., Sib ; KRSOS ; shok ; Bun ; Pos ; PT ; tmph ). See also Ro
and references therein for more details about SOS models on trees.

In this paper, we study the phase transition phenomenon for the three-state
SOS model on a Cayley tree of order two with nearest-neighbour and one-level
next-nearest-neighbour interactions. Note that the phase transition problem is
one of the central problems of statistical mechanics G . The existence of more
than one Gibbs measure to a given model implies the occurrence of the phase
transition G ; Ro . For the classical models (the Ising, Potts models) of
statistical mechanics on Cayley trees within radius two interactions this
problem is well studied (for the Ising model see, e.g., Van ; Yokoi ; Trag ;
Katsura , for the Potts model see, e.g., GTA ; Tur ; Gan06 ; Pah2 ).

We obtain a functional equation for the model using the self-similarity of the
Cayley tree. Here we consider only the one-level next-nearest-neighbor
interactions, since studying both one-level and prolonged next-nearest-
neighbor interactions simultaneously usually lead to functional equations
which difficult to solve (this happens even for the Ising model, see, e.g.,
ganipah ). We prove that at some values of parameters the model possess
multiple Gibbs measures which implies the existence of phase transition. We
show that for the model under certain conditions there is not any
antiferromagnetic phase. We also provide a conjecture on the absence of
antiferromagnetic phase for the model on the invariant set.

The paper is organized as follows. In Section 2 we give definitions of the
model, Cayley tree and Gibbs measures. In Section 3 we reduce the problem of
describing limit Gibbs measures to the problem of solving a system of
nonlinear functional equations. Section 4 is devoted to describe the
ferromagnetic phase of the model. In Section 5 we study the antiferromagnetic
phase of the model.

##  2 Preliminaries

Cayley tree. The Cayley tree $\Gamma^{k}$ of order $k\geq 1$ is an infinite
tree, i.e., a cycles-free graph such that from each vertex of which issues
exactly $k+1$ edges. We denote by $V$ the set of the vertices of tree and by
$L$ the set of edges of tree. Two vertices $x$ and $y$, where $x,y\in V$ are
called nearest-neighbor if there exists an edge $l\in L$ connecting them,
which is denoted by $l=\langle x,y\rangle$. The distance on this tree, denoted
by $d(x,y)$, is defined as the number of nearest-neighbour pairs of the
minimal path between the vertices $x$ and $y$ (where path is a collection of
nearest-neighbor pairs, two consecutive pairs sharing at least a given
vertex).

For a fixed $x^{0}\in V,$ called the root, we set

$W_{n}=\\{x\in V\mid d(x,x^{0})=n\\},~{}~{}~{}V_{n}=\bigcup_{m=0}^{n}W_{m}$

and denote by

$S(x)=\\{y\in W_{n+1}:d(x,y)=1\\},~{}~{}x\in W_{n}$

the set of _direct successors_ of $x.$ We observe that, for any vertex $x\neq
x^{0},$ $x$ has $k$ direct successors and $x^{0}$ has $k+1.$ For the sake of
simplicity, we put $\mid x\mid=d(x,x^{0}),~{}x\in V$. Two vertices $x,y\in V$
are called second nearest-neighbor if $d(x,y)=2$. The second nearest-neighbor
vertices $x$ and $y$ are called prolonged second nearest-neighbor if $\mid
x\mid\neq\mid y\mid$ and is denoted by $>\widetilde{x,y}<.$ The second
nearest-neighbor vertices $x,y\in V$ that are not prolonged are called one-
level next-nearest-neighbor since $\mid x\mid=\mid y\mid$ and are denoted by
$>\overline{x,y}<.$

In this paper, we consider a semi-infinite Cayley $\Gamma^{k}$ of order $k\geq
2,$ i.e. a cycles-free graph with $(k+1)$ edges issuing from each except
$x^{0}$ and with $k$ edges issuing from the vertex $x^{0}$. According to well
known theorems, this can be reconstituted as a Cayley tree G ; Tur .

In the SOS model, the spin variables $\sigma(x)$ take their values on the set
$\Phi=\\{0,1,2\\}$ which are associated with each vertex of the tree
$\Gamma^{k}.$ The SOS model with nearest-neighbour and one-level next-nearest-
neighbor interactions is defined by the following Hamiltonian:

$H(\sigma)=-J\sum_{\langle x,y\rangle}\mid\sigma(x)-\sigma(y)\mid-
J_{1}\sum_{>\overline{x,y}<}\mid\sigma(x)-\sigma(y)\mid,$ (1)

where the sum in the first term ranges all nearest neighbors, second sum
ranges all one-level next-nearest-neighbors, and $J,~{}J_{1}\in\mathbb{R}$ are
the coupling constants (see Fig 1.).

Figure 1. The Cayley tree of order two with nearest-neighbor (———) and one-
level next-nearest-neighbour (- - - - -) interactions.

##  3 Recursive Equations

One can obtain the nonlinear functional equations (tree recursion) describing
limiting Gibbs measures for lattice models on Cayley tree in many ways. One
approach is based on properties of Markov random fields on Bethe lattices
(see, e.g., rs ). The second approach is based on recursive equations for
partition functions (see, e.g., Tur ). Naturally, both approaches lead to the
same equation (see, e.g., Ro ). Since the second approach more suitable for
models with competing interactions, we follow this approach.

Let $\Lambda$ be a finite subset of $V$. We will denote by $\sigma(\Lambda)$
the restriction of $\sigma$ to $\Lambda.$ Let
$\overline{\sigma}(V\backslash\Lambda)$ be a fixed boundary configuration. The
total energy of $\sigma(\Lambda)$ under condition
$\overline{\sigma}(V\backslash\Lambda)$ is defined as

$\begin{split}H(\sigma(\Lambda)\mid\overline{\sigma}(V\backslash\Lambda))=-J\sum_{\langle
x,y\rangle:x,y\in\Lambda}\mid\sigma(x)-\sigma(y)\mid\\\
-J_{1}\sum_{>x,y<:x,y\in\Lambda}\mid\sigma(x)-\sigma(y)\mid-J\sum_{\langle
x,y\rangle:x\in\Lambda,y\notin\Lambda}\mid\sigma(x)-\sigma(y)\mid,\end{split}$
(2)

Then partition function $Z_{\Lambda}(\overline{\sigma}(V\backslash\Lambda))$
in volume $\Lambda$ boundary condition $\overline{\sigma}(V\backslash\Lambda)$
is defined as

$Z_{\Lambda}(\overline{\sigma}(V\backslash\Lambda))=\sum_{\sigma(\Lambda)\in\Omega(\Lambda)}\exp(-\beta
H_{\Lambda}(\sigma(\Lambda)\mid\overline{\sigma}(V\backslash\Lambda))),$ (3)

where $\Omega(\Lambda)$ is the set of all configurations in volume $\Lambda$
and $\beta=\frac{1}{T}$ is the inverse temperature. Then conditional Gibbs
measure $\mu_{\Lambda}$ of a configuration $\sigma(\Lambda)$ is defined as

$\mu_{\Lambda}(\sigma(\Lambda)\mid\overline{\sigma}(V\backslash\Lambda))=\frac{\exp(-\beta\,H(\sigma(\Lambda)\mid\overline{\sigma}(V\backslash\Lambda)))}{Z_{\Lambda}(\overline{\sigma}(V\backslash\Lambda))}.$

We consider the configuration $\sigma(V_{n})$, the partitions functions
$Z_{V_{n}}$ and conditional Gibbs measure
$\mu_{\Lambda}(\sigma(\Lambda)\mid\overline{\sigma}(V\backslash\Lambda))$ in
volume $V_{n}$ and for the sake of simplicity, we denote them by $\sigma_{n}$,
$Z^{(n)}$ and $\mu_{n},$ respectively. The partitions function $Z^{(n)}$ can
be decomposed into following summands:

$Z^{(n)}=Z_{0}^{(n)}+Z_{1}^{(n)}+Z_{2}^{(n)},$ (4)

where

$Z_{i}^{(n)}=\sum_{\sigma_{n}\in\Omega(V_{n}):\sigma(x^{0})=i}\exp(-\beta
H_{V_{n}}(\sigma\mid\overline{\sigma}(V\backslash V_{n}))),~{}i=0,1,2.$ (5)

From now on, we restrict ourselves to the case $k=2.$

Denote $\theta=\exp(\beta J),~{}\theta_{1}=\exp(\beta J_{1}).$ Let
$S(x^{0})=\\{x^{1},~{}x^{2}\\}.$ If $\sigma(x^{0})=i,~{}\sigma(x^{1})=j$ and
$\sigma(x^{2})=m,$ then from (2) and (3) we have following

$Z_{i}^{(n)}=\sum_{j,m=0}^{2}\exp(\beta J\mid i-j\mid+\beta J\mid
i-m\mid+\beta J_{1}\mid j-m\mid)Z_{j}^{n-1}Z_{m}^{(n-1)},$

so that

$Z_{0}^{(n)}=\Big{[}\big{(}Z_{0}^{(n-1)}\big{)}^{2}+2\theta\theta_{1}Z_{0}^{(n-1)}Z_{1}^{(n-1)}+2\theta^{2}\theta_{1}^{2}Z_{0}^{(n-1)}Z_{2}^{(n-1)}$
$+\theta^{2}\big{(}Z_{1}^{(n-1)}\big{)}^{2}+2\theta^{3}\theta_{1}Z_{1}^{(n-1)}Z_{2}^{(n-1)}+\theta^{4}\big{(}Z_{2}^{(n-1)}\big{)}^{2}\Big{]},$
$Z_{1}^{(n)}=\Big{[}\theta^{2}\big{(}Z_{0}^{n-1}\big{)}^{2}+2\theta\theta_{1}Z_{0}^{(n-1)}Z_{1}^{(n-1)}+2\theta^{2}\theta_{1}^{2}Z_{0}^{(n-1)}Z_{2}^{(n-1)}$
$+\big{(}Z_{1}^{(n-1)}\big{)}^{2}+2\theta\theta_{1}Z_{1}^{(n-1)}Z_{2}^{(n-1)}+\theta^{2}\big{(}Z_{2}^{(n-1)}\big{)}\Big{]},$
$Z_{2}^{(n)}=\Big{[}\theta^{4}\big{(}Z_{0}^{(n-1)}\big{)}^{2}+2\theta^{3}\theta_{1}Z_{0}^{(n-1)}Z_{1}^{(n-1)}+2\theta^{2}\theta_{1}^{2}Z_{0}^{(n-1)}Z_{2}^{(n-1)}$
$+\theta^{2}\big{(}Z_{1}^{(n-1)}\big{)}^{2}+2\theta\theta_{1}Z_{1}^{(n-1)}Z_{1}^{(n-1)}+\big{(}Z_{2}^{n-1}\big{)}^{2}\Big{]}.$

Introducing the notations
$u_{n}(x^{0})=\frac{Z_{1}^{(n)}(x^{0})}{Z_{0}^{(n)}(x^{0})},$
$v_{n}=\frac{Z_{2}^{(n)}(x^{0})}{Z_{0}^{(n)}(x^{0})},$ we obtain the following
system of recurrent equations:

$\left\\{\begin{array}[]{ll}u_{n}=\frac{\theta^{2}+2\theta\theta_{1}u_{n-1}+2\theta^{2}\theta_{1}^{2}v_{n-1}+u_{n-1}^{2}+2\theta\theta_{1}u_{n-1}v_{n-1}+\theta^{2}v_{n-1}^{2}}{1+2\theta\theta_{1}u_{n-1}+2\theta^{2}\theta_{1}^{2}v_{n-1}+\theta^{2}u_{n-1}^{2}+2\theta^{3}\theta_{1}u_{n-1}v_{n-1}+\theta^{4}v_{n-1}^{2}}\\\\[17.07182pt]
v_{n}=\frac{\theta^{4}+2\theta^{3}\theta_{1}u_{n-1}+2\theta^{2}\theta_{1}^{2}v_{n-1}+\theta^{2}u_{n-1}^{2}+2\theta\theta_{1}u_{n-1}v_{n-1}+v_{n-1}^{2}}{1+2\theta\theta_{1}u_{n-1}+2\theta^{2}\theta_{1}^{2}v_{n-1}+\theta^{2}u_{n-1}^{2}+2\theta^{3}\theta_{1}u_{n-1}v_{n-1}+\theta^{4}v_{n-1}^{2}}.\\\
\end{array}\right.$ (6)

Evidently,

$u_{n}(x^{0})=\frac{\mu_{n}(\sigma_{n}(x^{0})=1)}{\mu_{n}(\sigma_{n}(x^{0})=0)},~{}v_{n}(x^{0})=\frac{\mu_{n}(\sigma_{n}(x^{0})=2)}{\mu_{n}(\sigma_{n}(x^{0})=0)}.$

If we can find the limit of $u_{n}(x^{0})$ as $n$ tends to infinity, we will
find the ratio for the probability of a $1$ to the probability of a $0$ at the
root for the limiting Gibbs measure. Similarly, if we can find the limit of
$v_{n}(x^{0})$ as $n$ tends to infinity, we will find the ratio for the
probability of a $2$ to the probability of a $0$ at the root for the limiting
Gibbs measure. Thus, the fixed points of the equation (7) describe the
translation-invariant limiting Gibbs measure of the model (1).

If $u=\lim u_{n}$ and $v=\lim v_{n}$ then

$\left\\{\begin{array}[]{ll}u=\frac{\theta^{2}+2\theta\theta_{1}u+2\theta^{2}\theta_{1}^{2}v+u^{2}+2\theta\theta_{1}uv+\theta^{2}v^{2}}{1+2\theta\theta_{1}u+2\theta^{2}\theta_{1}^{2}v+\theta^{2}u^{2}+2\theta^{3}\theta_{1}uv+\theta^{4}v^{2}},\\\\[17.07182pt]
v=\frac{\theta^{4}+2\theta^{3}\theta_{1}u+2\theta^{2}\theta_{1}^{2}v+\theta^{2}u^{2}+2\theta\theta_{1}uv+v^{2}}{1+2\theta\theta_{1}u+2\theta^{2}\theta_{1}^{2}v+\theta^{2}u^{2}+2\theta^{3}\theta_{1}uv+\theta^{4}v^{2}}.\\\
\end{array}\right.$ (7)

######  Remark 1.

The system (7) coincides with the classical result for SOS model (see, e.g.,
rs ; KRSOS ) when $\theta_{1}=1$ $(J_{1}=0)$, i.e.

$\left\\{\begin{array}[]{ll}u=\big{(}\frac{u+\theta
v+\theta}{\theta^{2}v+\theta u+1}\big{)}^{2},\\\ v=\big{(}\frac{\theta
u+v+\theta^{2}}{\theta^{2}v+\theta u+1}\big{)}^{2}.\\\ \end{array}\right.$ (8)

It is important to note that if there is more than one positive solution for
the system (7), then there is more than one the translation-invariant limiting
Gibbs measure corresponding to these solutions. We say that a phase transition
occurs for the model (1), if the system (7) has more than one positive
solution.

##  4 Ferromagnetic phases

In this section, we investigate the phase transitions of the model. We
consider the dynamic system (6) and study its limiting behaviour. Let
$x=(u,v)\in\mathbb{R}_{+}^{2}$ and the dynamic system
$F:\mathbb{R}_{+}^{2}\to\mathbb{R}_{+}^{2}$ is defined by

$\left\\{\begin{array}[]{ll}u^{\prime}=\frac{\theta^{2}+2\theta\theta_{1}u+2\theta^{2}\theta_{1}^{2}v+u^{2}+2\theta\theta_{1}uv+\theta^{2}v^{2}}{1+2\theta\theta_{1}u+2\theta^{2}\theta_{1}^{2}v+\theta^{2}u^{2}+2\theta^{3}\theta_{1}uv+\theta^{4}v^{2}},\\\\[17.07182pt]
v^{\prime}=\frac{\theta^{4}+2\theta^{3}\theta_{1}u+2\theta^{2}\theta_{1}^{2}v+\theta^{2}u^{2}+2\theta\theta_{1}uv+v^{2}}{1+2\theta\theta_{1}u+2\theta^{2}\theta_{1}^{2}v+\theta^{2}u^{2}+2\theta^{3}\theta_{1}uv+\theta^{4}v^{2}}.\\\
\end{array}\right.$ (9)

Then the recurrent equations (6) be can rewritten as
$x^{(n+1)}=F(x^{(n)}),~{}n\geq 0.$ Recall that the point $x$ is a periodic
point of period $p$ if $F^{p}(x)=x,$ where $F^{p}(x)$ stands for $p$-fold
composition of $F$ into itself, i.e., $F^{p}(x)=\underbrace{F(F(\ldots
F(x))\ldots)}_{p}$. A point $x\in\mathbb{R}_{+}^{2}$ is called a fixed point
for $F:\mathbb{R}_{+}^{2}\to\mathbb{R}_{+}^{2}$ if $F(x)=x.$ (see for more
details (Dev, , Chapter 1) or (billiard, , Section 1)).

To investigate the problem of phase transition in the class of ferromagnetic
phases, we have to describe the fixed points of the map $F(x)=x.$ Let us
describe fixed points of this dynamical system, i.e., solutions of equation
$F(x)=x.$ It is obvious that the following set is invariant with respect to
the operator $F$:

$I=\\{x=(u,v)\in\mathbb{R}^{2}:v=1\\}.$ (10)

On the set $I$, the system of equations (7) reduces to

$u=f(u)$ (11)

where

$f(u)=f(u,\theta,\theta_{1}):=\frac{u^{2}+4\theta\,\theta_{1}\,u+2\theta^{2}(\theta_{1}^{2}+1)}{\theta^{2}\,u^{2}+2\theta\,\theta_{1}(\theta^{2}+1)u+\theta^{4}+2\,\theta^{2}\theta_{1}^{2}+1}.$
(12)

It is easy to see that the function $f(u)$ defined in (12) is continuous,
bounded with $f(0)>0$ and $f(+\infty)<+\infty.$ From properties of the
function $f$ it follows that the function $f$ has at least one fixed point,
say, $u^{*}.$ We have

######  Theorem 1.

For the SOS model with one-level second nearest-neighbour interactions on the
binary tree on the set $I$ if the condition $f^{\prime}(u^{*})>1$ is
satisfied, then there exists three distinct translation-invariant limiting
Gibbs measures, i.e., the phase transition occurs.

Proof: When $f^{\prime}(u^{*})>1,$ $u^{*}$ is unstable. So there exists a
small neighborhood $(u^{*}-\varepsilon,u^{*}+\varepsilon)$ of $u^{*}$ such
that for $u\in(u^{*}-\varepsilon,u^{*})$ $f(u)<u,$ and for
$(u^{*},u^{*}+\varepsilon)$ $f(u)>u.$ Since $f(0)>0,$ there exists a solution
between $0$ and $u^{*}.$ Similarly, since $f(+\infty)<+\infty$ there is
another solution between $u^{*}$ and $+\infty.$ Thus, there exist three
solutions. Since there exist a bijection between the solution of the Eq. (11)
and the translation-invariant limiting Gibbs measures, it follows that there
exist three translation-invariant limiting Gibbs measures, which implies the
existence of a phase transition. This completes proof.

######  Remark 2.

Note that the set of parameters which satisfy $f^{\prime}(u^{*})>1$ is not
empty, e.g., see also Fig. 2.

Figure 2. The plot of $f(u)-u$ when $\theta=0.2,~{}\theta_{1}=0.5.$ In this
case the function $f$ has three positive fixed points:
$\approx{0.1461}\mathchar 24635\relax\;~{}0.7085\mathchar
24635\relax\;~{}24.1453.$ The plot of the function is drawn for
$u\in[0,2],~{}u\in[2,25]$ separately to show all 3 solutions.

######  Remark 3.

In Theorem 1 we find the sufficient conditions for the Eq. (11) on possessing
multiple solutions, i.e., there might be multiple solutions for the equation
even if $f^{\prime}(u^{*})\leq 1.$

##  5 The absence of antiferromagnetic phases

The fixed points of the transformation $x=F^{2}(x)$ that do not satisfy
$x=F(x)$ define an antiferromagnetic phase, i.e., a phase with period 2. In
the section, we study the antiferromagnetic phase of the model on the set $I$
(10). Therefore, we investigate the zeroes of the equation given by

$\frac{f(f(u))-u}{f(u)-u}=0,$ (13)

where $f$ is defined in (12). Simplifying above equation, we obtain

$A\,u^{2}+B\,u+C=0$ (14)

where

$A:=A(\theta\mathchar
24635\relax\;\theta_{1})=\theta^{6}+2\theta^{4}\theta_{1}^{2}+2\theta^{3}\theta_{1}+\theta^{2}+2\theta\theta_{1}+1,$
$B:=B(\theta\mathchar
24635\relax\;\theta_{1})=2\theta^{7}\theta_{1}+4\theta^{5}\theta_{1}^{3}+2\theta^{5}\theta_{1}+6\theta^{4}\theta_{1}^{2}+4\theta^{3}\theta_{1}^{3}-\theta^{4}+2\theta^{3}\theta_{1}+10\theta^{2}\theta_{1}^{2}+6\theta\theta_{1}+1,$
$C:=C(\theta\mathchar
24635\relax\;\theta_{1})=\theta^{8}+4\theta^{6}\theta_{2}+4\theta^{4}\theta_{1}^{4}+4\theta^{5}\theta_{1}+8\theta^{3}\theta_{1}^{3}+2\theta^{4}+6\theta^{2}\theta_{1}^{2}+2\theta^{2}+4\theta\theta_{1}+1.$

Note that $A>0,C>0$ for any $\theta>0,\theta_{1}>0.$ According to Descartes’
Rule of Signs (see, e.g., PP , Corollary 1) if $B\geq 0$ then the equation
(14) does not have any positive solution (see Fig. 3). Thus, we have the
following assertion:

######  Theorem 2.

If

$(\theta,\theta_{1})\in\\{(\theta,\theta_{1})\in\mathbb{R}_{+}^{2}:B\geq 0\\}$

then for the SOS model with one-level next-nearest-neighbour interactions on
the binary tree there is no antiferromagnetic phases (two-periodic Gibbs
measures) on the set $I$ (10).

Figure 3. The plot of $B(\theta,\theta_{1})$ for $\theta\in(0,7)$ and
$\theta_{1}\in(0,0.2)$ The shaded area corresponds to
$B(\theta,\theta_{1})\geq 0$.

Due to Theorem 2, we should consider the case $B<0.$ If $B<0$ then the
equation (14) might have 2 positive solutions. We find the discriminant of the
Eq. (14):

$D:=D(\theta\mathchar 24635\relax\;\theta_{1})=B^{2}-4\,A\,C.$

It is easy to see that if $B<0$ and $D\geq 0$ then the Eq (14) has at least
positive solution. However, a computer analysis shows that the set

$\mathcal{S}=\\{(\theta,\theta_{1})\in\mathbb{R}_{+}^{2}:~{}D\geq 0,~{}B<0\\}$

is empty. Summarising, we make

Conjecture 1. The SOS model with one-level next-nearest-neighbour interactions
on the binary tree does not have any antiferromagnetic phase (two-periodic
Gibbs measures) on the set $I$ (10).

######  Remark 4.

Note that for the model (1) there might be antiferromagnetic phases outside of
the set $I$ (10).

##  6 Declarations

###  6.1 Author contributions

Conceptualization: M. M. Rahmatullaev; Methodology: M. M. Rahmatullaev; Formal
analysis and investigation: O. Sh. Karshiboev; Writing - original draft
preparation: O. Sh. Karshiboev; Writing - review and editing: O. Sh.
Karshiboev; Resources: M. M. Rahmatullaev, Supervision: M. M. Rahmatullaev.

###  6.2 Funding

No funds, grants, or other support was received

###  6.3 Competing interests

The authors have no relevant financial or non-financial interests to disclose

###  6.4 Data Availability Statement

Not applicable

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11institutetext: Department of Imaging Physics, Division of Diagnostic
Imaging, The University of Texas MD Anderson Cancer Center, Houston, TX, USA
11email<EMAIL_ADDRESS>22institutetext: Department of Pathology,
University of Rochester Medical Center, Rochester, NY, USA 33institutetext:
Department of Hematopathology, Division of Pathology and Lab Medicine, The
University of Texas MD Anderson Cancer Center, Houston, TX, USA

# Hierarchical Phenotyping and Graph Modeling of Spatial Architecture in
Lymphoid Neoplasms

Pingjun Chen Equal Contribution11 0000-0003-0528-1713 Muhammad Aminu$*$ 11
0000-0002-9903-8812 Siba El Hussein$*$ 22 0000-0002-1338-7107 Joseph D. Khoury
Co-supervision33 0000-0003-2621-3584 Jia Wu$\dagger$ 11 0000-0001-8392-8338

###### Abstract

The cells and their spatial patterns in the tumor microenvironment (TME) play
a key role in tumor evolution, and yet the latter remains an understudied
topic in computational pathology. This study, to the best of our knowledge, is
among the first to hybridize local and global graph methods to profile
orchestration and interaction of cellular components. To address the challenge
in hematolymphoid cancers, where the cell classes in TME may be unclear, we
first implemented cell-level unsupervised learning and identified two new cell
subtypes. Local cell graphs or supercells were built for each image by
considering the individual cell’s geospatial location and classes. Then, we
applied supercell level clustering and identified two new cell communities. In
the end, we built global graphs to abstract spatial interaction patterns and
extract features for disease diagnosis. We evaluate the proposed algorithm on
H&E slides of 60 hematolymphoid neoplasms and further compared it with three
cell level graph-based algorithms, including the global cell graph, cluster
cell graph, and FLocK. The proposed algorithm achieved a mean diagnosis
accuracy of 0.703 with the repeated 5-fold cross-validation scheme. In
conclusion, our algorithm shows superior performance over the existing methods
and can be potentially applied to other cancer types.

###### Keywords:

Spatial pattern analysis Cell phenotyping Supercell Graph modeling
Hematolymphoid cancer.

##  1 Introduction

The tumor is a complex ecosystem that emerges and evolves under selective
pressure from its microenvironment, involving trophic, metabolic,
immunological, and therapeutic factors. The relative influence of these
factors orchestrates the abundance, localization, and functional orientation
of cellular components within the tumor microenvironment (TME) with resultant
phenotypic and geospatial variations, a phenomenon known as intratumoral
heterogeneity (ITH) [26, 8]. As such, ITH provides a substrate from which
neoplastic cells emerge, escape immunologic surveillance, undergo genetic
evolution, and develop pathways that lead to therapy resistance. In routine
practice, such attributes are evaluated microscopically primarily using tumor
tissue sections stained with the hematoxylin and eosin stain (H&E). With the
advent of digital pathology, machine learning-empowered computational
pipelines have been proposed to profile TME using H&E tissue sections to
enhance cancer diagnosis and prognostication [12, 16, 27].

Most studies published to date have focused on phenotyping the textural
patterns of tissue slides in a top-down manner, through either conventional
hand-crafted features (Gabor, Fourier, LBP, GLCM, etc.) or deep convolutional
neural networks (CNN) to extract versatile features specifically tailored for
particular clinical scenarios [11, 30, 3, 19, 5, 4, 28]. Though these studies
have achieved promising performance, they missed connection to individual
cellular components in TME, thereby limiting their ability to capture
neoplastic cell characteristics in their microenvironment milieu and
capitalizing on critical diagnostic aspects and prognostic predictions. To
bridge this gap, few bottom-up studies focused on profiling cellular
architectures from digital pathology slides have emerged using graph theory
approach and graph convolution network (GCN) approach [24, 18, 20, 29, 22,
13]. The graph theory approach first constructs either local or global graph
structures and then extracts hand-crafted features to test their clinical
relevance. By contrast, the GCN approach aims to automatically learn
representations from the global graph formed at the cellular level and
abstract the features via multiple layers of graph convolution and pooling
operations, similar to the CNN approach. However, a common limitation to these
algorithms is their lack of ability to differentiate known cellular
populations. Pilot studies have been proposed to address this limitation in
the colorectal cancer [14, 23], a solid tumor with a well-studied TME that
contains vascular structures, immune cells, fibroblasts, and extracellular
matrix [15, 17]. There are no studies exploring this approach in
hematolymphoid neoplasms.

Diffuse large B-cell lymphoma (DLBC) is an aggressive lymphoid malignancy that
may arise through a series of clonal evolution steps, occasionally from a low-
grade lymphoid neoplasm called chronic lymphocytic leukemia/small lymphocytic
lymphoma (CLL/SLL) [2, 6]. The latter is among the most prevalent lymphoid
neoplasms and is associated with a favorable survival rate (5-year survival
rate around 83%). In a subset of patients, CLL/SLL can pursue a more
aggressive clinical course referred to as “accelerated CLL/SLL” (aCLL).
Patients with CLL/SLL and/or aCLL can also develop a full-blown DLBCL, called
Richter transformation DLBCL (RT-DLBL), a complication that is associated with
a high mortality rate with a median overall survival of 9 months [1, 6].
Unlike solid tumors with a well-defined TME, there is no standard way to
categorize cells into biological subtypes in hematolymphoid tumors. To
diagnose these patients, pathologists make diagnoses based on their empirical
knowledge of tumor cell morphology, normal histology, and deviations from
normal tissue architectures. By default, such an approach is subjective and
prone to inter- and intra-reader variations.

To improve the diagnostic accuracy of tissue examination of lymphoid
neoplasms, we propose an innovative computational framework that integrates
unsupervised clustering and graph modeling algorithms as a knowledge discovery
approach to hierarchically decode cellular and clonal level phenotypes in TME.
In particular, we dissect the process into four key steps (Fig. 1). First, we
segment each cell and based on their features to identify intrinsic subtypes.
Second, we focus on spatial interaction among neighboring cells by building
local graphs factoring in their subtypes, so that closely-interacting cells
are merged to form supercells. Third, we pool the supercells together to
discover the cellular community at a population level. Lastly, we build global
graphs incorporating community information to extract features for diagnostic
purposes.

Figure 1: Illustration of the proposed hierarchical cell phenotyping and graph
modeling algorithm to profile the cellular architecture for analyzing lymphoid
pathology images. (A) Cell phenotyping via spectral clustering algorithm. (B)
Cell local graph construction to obtain supercells via a Ranking Operation-
based Clustering (ROC) algorithm. (C) Supercell phenotyping (community
detection) via spectral clustering algorithm. (D) The construction of global
graphs based on supercells for extracting features.

To sum up, the main contributions of our work are as follows: (1) We use
clustering as a knowledge discovery approach to explore intrinsic cellular and
clonal phenotypes of TME to advance our understanding of hematolymphoid
cancers. (2) We develop a novel hierarchical graph construction algorithm to
comprehensively characterize interaction patterns at both the local cellular
level and global community level for cellular architectures. (3) The proposed
algorithm attains superior diagnostic performance compared with three state-
of-the-art graph theory-based methods to enhance blood cancer diagnosis. To
the best of our knowledge, this is the first study using such a design in the
digital hematopathology field. We believe that our design has opened a new
direction to quantify intratumoral heterogeneity, an aspect that has
particular clinical implications in terms of risk stratification and
prediction.

##  2 Method

###  2.1 Cell Phenotyping via Unsupervised Learning

Given a digital H&E slide, segmentation of nuclei is a prerequisite step for
cellular-based analysis. We adopt a multiple-pass adaptive voting (MPAV)
algorithm to segment the nuclei in digitized tissue images [21]. To discover
cell subtypes, we first characterize the morphology, intensity, and regional
patterns of individual nuclei with a set of features (n=24, “Cell features” in
Table 1). To eliminate redundant features, we apply the Laplacian score method
to select the top informative ones for the downstream analysis [10]. By
treating each cell as an instance and randomly sampling cells across three
disease subtypes, we embed pooled cells in a two-dimensional t-SNE feature
space and then apply the spectral clustering to discover inter- and intra-
disease cell subtypes. After repeating the sampling process multiple times, we
observe that two intrinsic cell phenotypes are consistently identified. We
train a classifier via an ensemble of learners aggregated with the bagging
method [7] based on sampled cells for clustering, so that the cell subtypes
can be propagated to unsampled ones.

Table 1: Summary of manual-crafted cell and graph features. Category | No. | Feature Names
---|---|---
Cell features | 24 |  |  Area, EquivDiameter, {Major;Minor} AxisLength,
---
Perimeter, Mean {Inside;Outside} BoundaryIntensity,
IntegratedIntensity, NormalizedBoundarySaliency,
Normalized {Inside;Outside} BoundaryIntensity,
Eccentricity, Orientation, Intensity {Mean;Range;Deviation},
Solidity, Circularity, EllipticalDeviation, BoundarySaliency,
{Inside;Outside} BoundaryIntensity {Range;Deviation}
Voronoi features | 12 |  | {Area;Perimeter;Chord} {Mean;Deviation;Disorder},
---
{Area;Perimeter;Chord} Ratio of Minimum to Maximum

###  2.2 Supercell via Local Graph

In the local cell graph construction, the cell centroids together with the
obtained cell subtypes information are used to identify cell clusters
$\\{\bm{C}\\}$, where the Ranking Operation-based Clustering (ROC) algorithm
applied (detailed in Fig. 2) [9]. The ROC algorithm is insensitive to the
scale of features and type of distance metric used which makes it well-fitted
for cell cluster identification. Given a set of $n$ cells, their feature
vectors represented as $\bm{X}=\\{\bm{x_{i}}\\}_{i=1}^{n}$ where
$\bm{x_{i}}\in\mathbb{R}^{3}$, the ROC algorithm first computes the pairwise
distances $\\{\bm{d}^{k}\\}_{k=1}^{3}$ from $\bm{X}$. It then computes ranking
matrices $\\{\bm{r}^{k}\\}_{k=1}^{3}$ containing the ranking indices of the
elements of $\\{\bm{d}^{k}\\}_{k=1}^{3}$. The cumulative sparse ranking of
each cell feature $\\{\bm{s}^{k}\\}_{k=1}^{3}$ is then computed and all the
local maxima $\\{\bm{m}\\}$ of the cumulative sparse ranking are identified
and used to group the cells into clusters $\\{\bm{C}\\}$.

Figure 2: The procedures of local graph construction via the ROC algorithm.

###  2.3 Cell Community Identification by Clustering of Supercells

We take the convex hull-covered polygon as each supercell’s boundary. Unlike
the individual cell, neighboring supercells often have overlapped regions,
where the different cell classes interaction and confrontation occur (Fig.
3(A)). To discover the supercell groups (we term as cell community), we
perform unsupervised phenotyping at supercell level. We characterize an
individual supercell by looking into: 1) the encompassed cell features (i.e.,
compute mean feature values of individual cells) as well as 2) the cellular
spatial orchestration and interaction (i.e., build a Voronoi diagram within
supercell and extract related features). With these features (n=22, bolded
cell features, and all Voronoi features in Table 1), we implement clustering
to identify supercell subtypes (identical to cellular level clustering in
Section 2.1). As previously stated, t-SNE is first adopted to reduce the
feature space and embed supercells. Then we use spectral clustering to define
similar intra- and interpatient supercells (i.e., cell community). For the
evaluated lymphocytic images, two types of cell communities are identified.
Supercell phenotyping outcomes can be visualized from the two demo examples
displayed in Fig. 3(B).

Figure 3: Exemplar intermediate results of the proposed algorithm. Column (A)
shows the constructed local graphs (Supercells). Column (B) shows the
supercell phenotyping results. Column (C) shows the built global graphs via
the Delaunay triangulation, in which different edge colors represent different
supercell interaction patterns.

###  2.4 Global Supercell Graph Construction and Feature Extraction

We build global graphs to characterize interactions among labeled supercells
inside each image. Two global graphs, including the Delaunay triangulation and
the Voronoi diagram, are constructed based on the supercell and cell community
to complement each other. In the Delaunay scheme, the area is tessellated into
triangular cells, which nodes are centroids of supercells. While in the
Voronoi scheme, the area is split into sets of convex and adjacent polygonal
cells, in which anchors are supercell centroids. Fig. 3(C) shows two demos of
the built global graph via the Delaunay triangulation. We extract features
from the Delaunay graph through characterizing node-to-node interaction (i.e.,
edge) patterns. Assume there is $k$ different type of nodes (inherit from
supercell labels), the overall number of edge types is $k(k+1)/2$. We
accumulate each edge type’s counts and propose the percentages of different
edge types as the global graph features. As for the global Voronoi graph, 12
features, as indicated in Table 1, are extracted. We combine the features from
both the Delaunay graph and the Voronoi diagram and test their performance in
the diagnosis of different hematological malignancy subtypes (CLL vs aCLL vs
RT-DLBL).

##  3 Experiments and Results

###  3.1 Dataset Description

Digitized clinical tumor tissue slides collected at The University of Texas MD
Anderson Cancer Center (UTMDACC) were used a a study dataset. The study was
approved by the UTMDACC Review Board and conducted following the Declaration
of Helsinki. The glass slides were scanned using an Aperio AT2 scanner at 20x
optical magnification, with each pixel equaling 0.5 µm. We chose the best
quality slide for each patient and cropped one image with a fixed size of
$1,000\times 1,000$ pixels from level 0 of each pyramidal formatted slide. The
image was cropped by a board-certified pathologist with subspecialty expertise
who annotated the region of interest (ROI)’s central part. Each patient only
has one sample image in this design to facilitate the k-fold cross-validation
analysis. After excluding slides with server artifacts, we were left with 20
images of each diagnosis to maintain the balance among CLL, aCLL, and RT-DLBL.

###  3.2 Implementation Details

The implementation includes four aspects: cell phenotyping, local graph
construction, supercell phenotyping, and image classification. Since each
image with the size $1,000\times 1,000$ pixels contains thousands of cells, we
randomly select 3,000 cells from each disease and pool 9,000 cells from all
patients together to conduct the cell clustering. To propagate cell clustering
labels to unsampled ones, we train the cell classifier with an ensemble of
learners aggregated with the bagging method. In the local graph construction,
there are two parameters to set for the ROC algorithm. We set the sparse
ranking matrices’ sparsity as 0.9 and the number of neighbors for local maxima
identification as 10. The total number of supercells generated at the
population level is 2,794, and we use all of them for supercell phenotyping,
which employs the same clustering and classification manners as in cell
phenotyping. As for the image classification, we conduct repeated 5-fold
cross-validation with the Support Vector Machine (SVM). We mainly employ two
metrics to evaluate the performance, including accuracy and area under the
curve (AUC) for each diagnosis with the one-vs-rest strategy. We report the
mean values of these metrics with 100 times randomization. The code is
released at https://github.com/WuLabMDA/HierarchicalGraphModeling.

###  3.3 Results and Discussions

We compare the proposed method with three cell-level graph-based algorithms,
including the Global Cell Graph (GCG) [24], Local Cell Graph (LCG) [18], and
FLocK [20]. The GCG is built using the Voronoi diagram; The LCG is constructed
using the mean shift algorithm [25] utilizing the cell centroids; The FLocK is
built with mean shift utilizing centroids as well as area and intensity of the
cell. To have a fair comparison, we try our best to adjust the hyper-
parameters used in these three compared methods to report their best results.
The comparison results are shown in Table 2. Our proposed algorithm shows
superior performance on these evaluated metrics among all the compared
methods. The proposed method obtains the mean accuracy of 0.703 under 5-fold
cross-validation with a 100 times randomization scheme for the three category
disease diagnoses. Using only the local or the global graph fails to separate
CLL from the other clinically relevant subtypes. By contrast, combining the
local and global graph can improve the performance by a significant margin
with a 0.4 increase in accuracy on the CLL. Further investigations find that
the newly identified two cell communities have significantly different
distributions between CLL and aCLL$+$RT (p-value$=$3.54e-08).

The two local graph methods (LCG and FLock) show better performance than the
GCG. These results show that the local graph can better mine the cellular
patterns than the global graph. Similar to the local graph methods, our method
also firstly builds the local graph. Unlike them, LCG only utilizes the cell’s
centroid information, and FLocK uses the centroids with some cell features.
While the proposed method considers the cell type when building the local
graph. Moreover, we construct another layer of cell community detection for
the global graph on top of the created local graph to further abstract the
cellular features. We assume that this two-layer graph algorithm is better at
extracting the multi-scale (both local and global) cellular interactions and
intratumoral heterogeneity, thus surpass the compared methods. To the best of
our knowledge, this study is among the first to hybridize local and global
graph methods to study cellular patterns. Hereby, we hypothesize that the
proposed hybrid design can overcome the limitation inherent in the adoption of
the global or local graph approaches solely to more meaningfully profile tumor
composition and intratumoral heterogeneity. Indeed, the global approach misses
the cellular level details while the local graph ignores the high-level
interaction patterns between communities.

Table 2: Performance of the three compared graph theory based methods and the proposed method Method | Accuracy | AUC (CLL) | AUC (aCLL) | AUC (RT-DLBL)
---|---|---|---|---
GCG [24] | $0.436\pm 0.037$ | $0.421\pm 0.054$ | $0.730\pm 0.027$ | $0.770\pm 0.023$
LCG [18] | $0.471\pm 0.042$ | $0.555\pm 0.049$ | $0.669\pm 0.050$ | $0.763\pm 0.032$
FLocK [20] | $0.601\pm 0.045$ | $0.545\pm 0.054$ | $\bm{0.816}\pm 0.025$ | $0.847\pm 0.022$
Proposed | $\bm{0.703}\pm 0.030$ | $\bm{0.915}\pm 0.009$ | $0.724\pm 0.033$ | $\bm{0.866}\pm 0.028$

We take advantage of the clustering algorithm to attain the type of both cell
and supercell (cell community) in hematolymphoid cancers, given that their
cellular components in TME are unclear and that is distinct from solid tumors
with well-studied TME. We attach great importance to cell type information and
hypothesize that cells belonging to different types and communities play
distinct functions. The unsupervised manner would also save tremendous
workload for obtaining medical experts cell labeling and shed light on
uncovering new insight into cell subtypes and communities.

We propose a novel supercell concept in this study. The supercell is
anticipated to be an excellent bridge to connect the neighboring cell
interaction (local graph) and cell community interaction (global graph). Here,
we just scratch the surface of supercells properties, through merely
exploiting supercells central coordinates to build global graphs. Indeed, each
supercell contains rich information of the local TME, and their overlapped
region may manifest interplay mechanisms of differing tumor subclones or
habitats. We anticipate that future studies will help derive more biological
insight based on the supercell context. Our approach in this analysis uses
manually designed features from the supercell-based global graph for diagnosis
due to limited image samples. However, future approaches should permit taking
each supercell as the node to use the graph convolution network (GCN) to
further improve diagnostic accuracy when sufficient images are available.

##  4 Conclusion

We present a hierarchical cell phenotyping and graph modeling algorithm to
profile the spatial patterns of neoplastic lymphoid cells distributed within
the tumor microenvironment. Using this global graph’s manual-crafted features,
we achieve promising results based on clinical tumor images from various types
of hematolymphoid tumors of different grades. We anticipate that the proposed
cell phenotyping and supercell design can adapt to a broader category of
cancers.

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# Carrier mobilities of Janus transition metal dichalcogenides monolayers
studied by Born effective charge and first-principles calculation

Jingxin Hu  Kui Rao  Jing Luo  Lianjie Hu  Department of Physics and Key
Laboratory for Low-Dimensional Structures and Quantum Manipulation (Ministry
of Education), Hunan Normal University, Changsha 410081, China  Ziran Liu
<EMAIL_ADDRESS>Department of Physics and Key Laboratory for Low-
Dimensional Structures and Quantum Manipulation (Ministry of Education), Hunan
Normal University, Changsha 410081, China  Key Laboratory for Matter
Microstructure and Function of Hunan Province, Hunan Normal University,
Changsha 410081, China

###### Abstract

Two-dimensional (2D) Janus transition metal dichalcogenides (TMDs) are a new
class of materials with unique physical properties. However, the carrier
mobility of most Janus TMDs calculated by deformation potential theory (DPT)
is not reliable due to the unconsidered part of lattice scattering. In this
work, we propose a new method of Born effective charge (BEC) to calculate the
carrier mobility of Janus TMDs by including the important factors that
neglected in the DPT. The BEC could be used in the calculation of both pure
and defective Janus TMDs by employing density functional perturbation theory.
We have figured out the relationship between the carrier mobility and the
value of BEC, which is the lower the absolute BEC, the higher the electron or
hole mobility. Using the new method, we have calculated the carrier mobility
of commonly studied Janus TMDs with and without defect. The method may shed
light on the high-throughout calculation of selecting high carrier mobility 2D
materials.

##  I INTRODUCTION

Two-dimensional TMDs crystals, such as H-MoS2 and H-MoTe2 have been
extensively studied by experiment and first-principles calculations
jin2018prediction ; shi2018mechanical ; xia2018universality ; liu2019strain ;
guo2018biaxial ; dong2017large . Two-dimensional (2D) Janus materials are a
new class of materials with unique physical, chemical, and quantum properties
yagmurcukardes2020quantum . Janus TMDs of MoSSe was firstly synthesized and
confirmed by means of scanning transmission electron microscopy and energy-
dependent X-ray photoelectron spectroscopy lu2017janus ; zhang2017janus . The
structure of MoSSe was based on single-crystalline triangular MoS2 monolayers,
through chemical vapour deposition, Se atoms replace the S atoms, forming a
structurally stable Janus MoSSe monolayer in which the Mo atoms are covalently
bonded to underlying S and top-layer Se atoms.

From then on, the interesting Janus 2D materials attracting a lot of
attentions sant2020synthesis ; ghobadi2020electrical ; paez2021stability ;
xu2021colossal ; petric2021raman . Janus WSSe has been synthesized and it’s
novel physical properties were also revealed trivedi2020room ; lin2020low .
Then the VSSe zhang2019first , the ZrSSe vu2019electronic , and the SnSSe
guo2019predicted  have been systematically studied, and a lot of interesting
intrinsic physical properties were revealed. Single-layer PtSSe exhibits a
high absorption coefficient in the visible light region, appropriate band edge
positions and strong ability for carrier separation and transfer peng2019two .
Besides the experiments, first-principles calculations have also been applied
to study the stabilities of the Janus structures kandemir2018janus ;
cheng2013spin . Obviously, the symmetry of Janus structure is lower than 2D
TMDs materials, which indicate that the multi-layer Janus TMDs may have high
out-of-plane piezoelectric effect dong2017large ; yagmurcukardes2019electronic
. The electron and hole mobilities of Janus TMDs are important descriptors for
selecting good candidates for advanced 2D materials design.

The deformation potential theory (DPT) is frequently used to calculate the
intrinsic mobility of semiconductors bardeen1950deformation ; xi2012first ;
qiao2014high ; yang2016two ; cai2014polarity ; zhu2017multivalency ;
long2009theoretical  and the Janus 2D materials yin2018tunable ; patel2020high
; guo2019predicted . However, the application of DPT must satisfy the
following requirements xi2012first ; price1981two . One is that the lowest
valence band of the material should be parabolic, and the other one is about
the scattering of the carrier, which should be mainly caused by longitudinal
acoustic wave of the lattice. Besides, to obtain the carrier mobility in DPT,
the effective mass should be calculated from averaging all directions of
materials. Therefore, using DPT to calculate carrier mobility of anisotropic
materials is challengeable.

As we all know, electron-phonon coupling (EPC) is the relative accurate method
to calculate the carrier mobility of 2D materials kaasbjerg2012phonon ;
li2015electrical ; gunst2016first ; hinsche2017spin . However, the calculation
of EPC is very expensive, and the cell is limited within ten atoms without
defects. Therefore, it is necessary to find a new way to calculate the
mobility effectively, especially for the structure with defects. The BEC can
affect the vibration of the longitudinal optical branch which cause a
considerable change in polarization, resulting in stronger carrier scattering
through coulomb interaction, thus has been applied to predict the carrier
mobility of a amount of 2D TMDs cheng2018limits . BEC can be calculated by
applying changes in polarization divided by the displacement of the atoms. In
this work, we use first-principles density functional perturbation theory to
calculate BEC, then predict the carrier mobility of perfect and defective
hexagonal Janus structure (H-) of H-MoSSe, H-MoSTe, H-MoSeTe, H-WSSe, H-WSTe,
H-WSeTe and triangular Janus structure (T-) of T-HfSSe, T-ZrSSe, T-SnSSe,
T-PtSSe, T-PtSTe, T-PtSeTe.

##  II Methods

Figure 1: The band structures of H (a-f) and T(g-l) types of Janus TMDs
monolayers. The Fermi level is set to zero.

The calculations of band structure and carrier mobility are based on density
functional theory (DFT) using the plane-wave basis within projection augmented
wave (PAW) method. The generalized gradient approximation (GGA) in the Perdew-
Burke Ernzerhof (PBE) perdew1996generalized ; perdew1997generalized  form is
used as the exchange correlation potential implemented in the Vienna ab initio
simulation package (VASP) kresse1993ab ; kresse1994ab . On the premise of
ensuring the accuracy of the calculation results, we set the cutoff energy of
all related calculations to 350 eV, the convergence criterion between the two
electronic steps in the self-consistent process to $10^{-6}$ eV and the force
convergence is set to 0.01 eV/Å, the K-points are set to 9$\times$9$\times$1
and 15$\times$11$\times$1 for relaxation and energy calculation, respectively.

We obtain the phonon spectrum using the density functional perturbation theory
(DFPT) method baroni2001phonons  as implemented in VASP and post-processed by
using the Phonopy package togo2015first . In order to improve the precise of
the calculation, we increase the energy convergence criteria for electronic
and force convergence, which were set to be $10^{-10}$ eV and 0.001 eV/Å,
respectively. A vacuum layer with a thickness of 20Å is added along the z-axis
in order to avoid interaction between adjacent layers.

The phonon-limited formula is used to estimate carrier mobility in low
dimensional materials xi2012first ; bruzzone2011ab ; takagi1994universality :

$\mu=\frac{e\hbar^{3}C_{2D}}{K_{B}Tm^{\ast}m_{d}E_{i}^{2}}m_{d}=\sqrt{m_{x}^{\ast}m_{y}^{\ast}}$
(1)

where $m^{*}$ is the effective mass along the transmission direction
($m^{*}_{x}$ or $m^{*}_{y}$ is along $x$ or $y$ direction, respectively), and
the mean effective mass was defined as $m_{d}=\sqrt{m_{x}^{*}m_{y}^{*}}$.
$C_{2D}$ is the elastic modulus of the simplex deformed crystal, which is used
to simulate the lattice distortion caused by strain, which is given by
$C_{2D}=\dfrac{1}{S_{0}}\dfrac{\partial^{2}E}{\partial(\Delta
l/l_{0})^{2}}|_{l=l_{0}}$, $E$ is the total energy of the system and $S_{0}$
is the equilibrium area of the unit cell, $l_{0}$ is the lattice constant,
$\Delta l=l-l_{0}$ is corresponding lattice distortion. Here the applied
deformation varies from 0 to 0.6$\%$. $E_{i}$ is the deformation potential,
which can be expressed as the following formula $E=\dfrac{\partial
E_{edge}}{\partial(\Delta l/l_{0})}$, which represents of the strain-induced
band edge displacement (the minimum electron-conducting band, the maximum hole
valence band) along the direction of electron transport. The temperature used
for the mobility calculations was 300 K.

Density functional perturbation theory (DFPT) is used to calculate the BEC of
two-dimensional Janus structures. The BEC of an atom $s$ is defined as the
change in polarization $p$ in the $\alpha$ direction linearly induced by a
sublattice displacement $u_{s}$ in the $\beta$ direction in zero macroscopic
electric field as shown in below formula 1992Theory :

$Z_{s}^{*\alpha,\beta}=V\frac{\partial P_{\alpha}}{\partial u_{s,\beta}},$ (2)

where $V$ is the unit volume, $P_{\alpha}$ is the $\alpha$ component of the
polarization and $u_{s,\beta}$ is the displacement of the $s^{th}$ atom in the
direction of $\beta$. For two dimensional Janus structures with low symmetry,
BEC is not symmetric in the Cartesian directions. Since any rigid translation
of the entire system does not cause macroscopic polarization, the BEC tensor
follows acoustics and rules. The sum rule thus was given as 1992Theory ;
1970Microscopic :

$\sum_{s}Z_{s}^{*\alpha,\beta}=0$ (3)

i.e., the sum of BEC of all the atoms in certain system is zero.

##  III RESULTS AND DISCUSSION

###  III.1 The failure of the mobility from DPT

Table 1: The electron mobility $(\mu_{e})$ of T- and H-TMDs calculated by employing DPT. Systems |  Present work ($\mu_{e}$)(cm2/Vs) |  Experimental results available ($\mu_{e}$)(cm2/Vs)
---|---|---
T-ZrS2 |  $\mu$x=261.27 , $\mu$y=549.32 |
T-ZrSe2 |  $\mu$x=1252.70, $\mu$y=2549.70 |
T-HfS2 |  $\mu$x=428.77 , $\mu$y=1110.00 |
T-HfSe2 |  $\mu$x=2763.00, $\mu$y=7323.00 |  0.22 kang2015electrical
T-SnS2 |  $\mu$x=156.58 , $\mu$y=491.98 |
T-SnSe2 |  $\mu$x=423.60 , $\mu$y=312.33 |  8.6 su2013snse2
T-PtS2 |  $\mu$x=392.12 , $\mu$y=254.06 |
T-PtSe2 |  $\mu$x=884.47 , $\mu$y=459.92 |  18 yang2021high
T-PtTe2 |  $\mu$x=1021.00, $\mu$y=1204.00 |
H-MoS2 |  $\mu$x=169.47 , $\mu$y=165.44 |  200 radisavljevic2011single
H-MoSe2 |  $\mu$x=110.16 , $\mu$y=108.70 |  50 wang2014chemical
H-MoTe2 |  $\mu$x=101.55 , $\mu$y=98.32 |  40 keum2015bandgap
H-WS2 |  $\mu$x=277.21 , $\mu$x=272.05 |  100 wang2021electron
H-WSe2 |  $\mu$x=278.00 , $\mu$x=266.33 |  142 liu2013role

From previous work, we could find that the CBM of TMDs satisfy the parabolic
requirement cheng2018limits . However, most existing electron mobilities of
TMDs predicted by DPT are much larger than the available experimental results,
except for MoS2 as shown in Table 1. From the calculations, we found that the
anisotropy of the electron mobilities of TMDs can’t be neglected, especially
for that of the T-phase of TMDs. However, the lower mobility of two directions
(x and y) still much higher than the related experimental results. Hence, some
important factors should be considered in to the method of DPT, which at
present is not good enough for the electron mobility calculations of the
majority of TMDs.

Figure 2: The phonon spectrum of H (a-f) and T(g-l) types of Janus TMDs
monolayers.
Figure 3: The relationship between the BEC and electron mobility. The blue
squares represent values calculated by EPC which are fitted by blue lines. The
red five-pointed stars are available experimental values.

How about the electron mobilities of Janus TMDs from DPT? Table 2 shows that
the electron mobilities of Janus TMDs from DPT are still not satisfactory for
describing the affects from lattice scattering, i.e., the electron mobilities
are unreasonably high although the band structure of the system satisfies the
parabolic properties, as shown in Fig. 1. In order to get reliable data of
carrier mobility of Janus TMDs, new methods or factors should be employed in
the calculations.

Table 2: The calculated electron $(\mu_{e})$ and hole $(\mu_{h})$ mobility of Janus TMDs from DPT. System |  $(\mu_{e})$(cm2/Vs) |  $(\mu_{h})$(cm2/Vs)
---|---|---
T-ZrSSe |  $\mu$x=1971.80 , $\mu$y=3646.70 |  $\mu$x=495.48, $\mu$y=500.73
T-HfSSe |  $\mu$x=2033.40 , $\mu$y=4864.50 |  $\mu$x=452.84, $\mu$y=487.95
T-SnSSe |  $\mu$x=231.91 , $\mu$y=883.48 |  $\mu$x=0.74 , $\mu$y=0.19
T-PtSSe |  $\mu$x=707.26 , $\mu$y=377.51 |  $\mu$x=17.71 , $\mu$y=407.89
T-PtSTe |  $\mu$x=59.28 , $\mu$y=51.57 |  $\mu$x=0.64 , $\mu$y=1.40
T-PtSeTe |  $\mu$x=389.28 , $\mu$y=590.72 |  $\mu$x=14.23 , $\mu$y=14.33
H-MoSSe |  $\mu$x=120.38 , $\mu$y=117.25 |  $\mu$x=233.23, $\mu$y=236.76
H-MoSTe |  $\mu$x=46.53 , $\mu$y=45.41 |  $\mu$x=2.25 , $\mu$y=1.91
H-MoSeTe |  $\mu$x=68.90 , $\mu$y=68.91 |  $\mu$x=87.62 , $\mu$y=90.92
H-WSSe |  $\mu$x= 267.31 , $\mu$y=261.47 |  $\mu$x=513.85, $\mu$y=533.27
H-WSTe |  $\mu$x= 158.94 , $\mu$y=145.67 |  $\mu$x=15.14 , $\mu$y=15.29
H-WSeTe |  $\mu$x=183.50 , $\mu$y=177.66 |  $\mu$x=267.66, $\mu$y=273.97

Figure 4: The relationship between the BEC and hole mobility. The blue squares
represent values calculated by EPC which are fitted by blue lines. The red
five-pointed stars are available experimental values.

###  III.2 The bridge between mobility and BEC

As we all know, EPC is a relatively precise way to calculate the carrier
mobility recently. However, this method is very expensive and not suitable for
a large system not to mention the system with defects. Hence, finding a new
way to describe the main contribution from lattice scattering is necessary for
building a method for high-throughout calculation. BEC is such a quantity that
could describe the lattice scattering effectively and then obtain the carrier
mobility.

The electron mobility of TMDs from both experiment and EPC are summarized in
Fig. 3. After fitting the data from EPC, we find that the electron mobility
from EPC has polynomial function with BEC in system of TMDs. Besides, the
experimental results approximately meet the fitting values. It is easy to
conclude that the lower the absolute BEC, the higher the electron mobility,
which is similar to that of hole mobility, as shown in Fig. 4. In this bridge,
the carrier mobility of the TMDs can be obtained through BEC.

Then let’s promote the method to Janus TMDs which require the bandstructure
and phonon spectrum of Janus TMDs have similar characters as that of TMDs.
From Fig. 1 and Fig. 2, we could know that the bandstructure and phonon
spectrum of Janus TMDs do like that of TMDs cheng2018limits . Thus, the
mobility of Janus structures could be predicted from the BEC safely. The
quantitative electron and hole mobility of Janus structures can be found in
Table 3 and Table 4, respectively.

###  III.3 Prediction for Janus structure

In this section, we want to predict the carrier mobility by applying the above
functions with the BEC. For H types of Janus TMDs, the order of the values of
metal atom for $|Z_{B}|$ are H-WSTe$<$H-MoSTe$<$H-WSeTe$<$H-MoSeTe, in which
H-WSTe has the largest electron mobility of 129.72 cm2V-1s-1, while H-MoSeTe
has the lowest electron mobility of 64.79 cm2V-1s-1, as shown in Table 3. Like
the relationship between $|Z_{B}|$ and the electron mobility for H-phase, the
smaller the value of $|Z_{B}|$ of nonmetal atom, the larger the hole mobility.
As shown in Table 4, H-MoSeTe has the lowest hole mobility of 36.52 cm2V-1s-1
related to the largest $|Z_{B}|$ of 2.60 $|e|$, while H-WSSe has the largest
hole mobility of 382.05 cm2V-1s-1 related to the smallest $|Z_{B}|$ of 0.89
$|e|$. We further found that for the TMDs of H-MXY (M refers to metal atom, X
and Y are distinct element), the H-WXY has larger carrier mobility relative to
H-MoXY because W has the highest Pauling electronegativity among all metal
elements cheng2018limits .

For T-phase of Janus, the order of the values for $|Z_{B}|$ of metal atom is
T-PtSTe$<$T-PtSeTe$<$T-PtSSe$<$T-SnSSe$<$T-HfSSe$<$T-ZrSSe, in which T-PtSTe
has the largest electron mobility of 109.38 cm2V-1s-1 relate to the lowest
$|Z_{B}|$ of 1.83 $|e|$, and T-ZrSSe has the lowest electron mobility of 25.98
cm2V-1s-1, as shown in Table 3. Similarly to electron mobility, the largest
hole mobility for T-PtSTe is 6.28 cm2V-1s-1, as shown in Table 4.

This correlation is understandable, because a large BEC suggests a large
change in the polarization, resulting in a stronger carrier scattering through
Coulomb interaction. It’s worth mentioning that all the calculated BEC for
H-phase is different from that of the chemical charge, which the metals has
negative $|Z_{B}|$ but not positive $|Z_{B}|$. This can be explained from the
local changes of polarization around atoms by establish $\pi$ backbonding
model pike2017origin . By the way, this method is not suitable for calculating
the carrier mobility of H-WSSe and H-MoSSe which is mainly influenced by
optical branch rather than acoustic branch in phonon spectrum cheng2018limits
.

Table 3: The calculated BEC of metal atom (ZB) and the related electron mobility ($\mu_{e}$) of H and T types of Janus TMDs monolayers. Janus TMDs |  ZB (e) |  Mobility $\mu_{e}$(cm2/Vs)
---|---|---
H-MoSSe | -1.48 | 156.14
H-MoSTe | -2.27 | 78.44
H-MoSeTe | -2.60 | 64.79
H-WSSe | -0.89 | 394.69
H-WSTe | -1.65 | 129.72
H-WSeTe | -2.57 | 65.82
T-ZrSSe | 7.71 | 25.98
T-SnSSe | 4.93 | 33.18
T-HfSSe | 7.24 | 26.65
T-PtSSe | 2.53 | 66.80
T-PtSTe | 1.83 | 109.38
T-PtSeTe | 2.07 | 90.08
Table 4: The calculated BEC of nonmetal atom (ZB) and the related hole mobility ($\mu_{e}$) of H and T types of Janus TMDs monolayers. System |  ZB(e) |  Mobility $\mu_{h}$(cm2/Vs)
---|---|---
H-MoSSe | 1.48 | 125.43
H-MoSTe | 2.27 | 49.16
H-MoSeTe | 2.60 | 36.52
H-WSSe | 0.89 | 382.05
H-WSTe | 1.65 | 98.86
H-WSeTe | 2.57 | 37.46
T-ZrSSe | -7.71 | 5.26
T-SnSSe | -4.93 | 5.26
T-HfSSe | -7.24 | 5.26
T-PtSSe | -2.53 | 5.26
T-PtSTe | -1.83 | 6.28
T-PtSeTe | -2.07 | 5.35
Table 5: The calculated TMDs of the electron mobility ($\mu_{e}$) and BEC of metal atom (ZB). System |  ZB(e) |  Mobility $\mu_{e}$(cm2/Vs) |  ZB (e) from work cheng2018limits  |  Mobility $\mu_{e}$(cm2/Vs) from work cheng2018limits
---|---|---|---|---
H-MoS2 | 1.00 | 296.00 | 1.09 | 219.00
H-MoSe2 | 1.80 | 81.70 | 1.89 | 83.10
H-MoTe2 | 3.38 | 20.56 | 3.37 | 27.30
H-WS2 | 0.52 | 1239.49 | 0.65 | 764.50
H-WSe2 | 1.23 | 188.10 | 1.29 | 189.90
T-ZrS2 | -7.15 | 5.26 | -6.13 | 8.50
T-ZrSe2 | -8.17 | 5.26 | -6.45 | 8.20
T-SnS2 | -4.90 | 5.26 | -4.85 | 4.60
T-SnSe2 | -4.98 | 5.26 | -4.96 | 3.00
T-HfS2 | -6.84 | 5.26 | -6.03 | 6.70
T-HfSe2 | -7.60 | 5.26 | -6.33 | 6.00
T-PtS2 | -2.69 | 5.26 | -2.45 | 1.90
T-PtSe2 | -2.35 | 5.27 | -2.12 | 3.20
T-PtTe2 | -1.55 | 22.16 | -1.27 | 115.20

We have calculated the mobilities of 12 types of MXY, unfortunately, we do not
find a MXY with a mobility higher than H-WS2. However, we could deduce that
the high-mobility 2D semiconductor maybe more likely exist in materials having
zero BEC, which could be easily calculated from first-principles calculations.
Besides, it is possible to find higher-mobility candidates by including more
2D materials from database into consideration mounet2018two .

Table 6: The calculated BEC of metal atom (ZB) and the related electron mobility ($\mu_{e}$) of H and T types of Janus TMDs monolayers. The size of supercells used in calculation of the impurity is 3$\times$3. Janus TMDs |  ZB(e) |  Mobility $\mu_{e}$(cm2/Vs) | Janus TMDs |  ZB(e) |  Mobility $\mu_{e}$(cm2/Vs)
---|---|---|---|---|---
H-Mo9Se18 | -1.84 | 108.43 |  T-Hf9Se18 | 7.60 | 26.12
H-Mo9Se17S | -1.80 | 112.36 |  T-Hf9Se17S | 7.58 | 26.15
H-Mo9Se17Te | -1.92 | 101.30 |  T-Hf9Se17Te | 7.73 | 25.95
H-Mo9Te18 | -3.28 | 48.51 |  T-Zr9S18 | 7.26 | 26.62
H-Mo9Te17S | -3.17 | 50.46 |  T-Zr9S17Se | 7.30 | 26.55
H-Mo9Te17Se | -3.20 | 49.90 |  T-Zr9S17Te | 7.40 | 26.41
H-W9Se18 | -1.23 | 216.65 |  T-Zr9Se18 | 8.17 | 25.43
H-W9Se17S | -1.19 | 230.02 |  T-Zr9Se17S | 8.13 | 25.48
H-W9Se17Te | -1.31 | 193.48 |  T-Zr9Se17Te | 8.28 | 25.32
T-Sn9S18 | 4.89 | 33.38 |  T-Pt9S18 | 2.68 | 62.21
T-Sn9S17Se | 4.90 | 33.32 |  T-Pt9S17Se | 2.66 | 62.83
T-Sn9S17Te | 4.80 | 33.84 |  T-Pt9S17Te | 2.66 | 62.83
T-Sn9Se18 | 4.98 | 32.94 |  T-Pt9Se18 | 2.36 | 74.15
T-Sn9Se17S | 4.98 | 32.94 |  T-Pt9Se17S | 2.38 | 73.26
T-Sn9Se17Te | 4.99 | 32.94 |  T-Pt9Se17Te | 1.96 | 98.05
T-Hf9S18 | 6.88 | 27.25 |  T-Pt9Te18 | 1.56 | 142.63
T-Hf9S17Se | 6.91 | 27.20 |  T-Pt9Te17S | 1.64 | 131.05
T-Hf9S17Te | 7.00 | 27.04 |  T-Pt9Te17Se | 1.61 | 135.19

###  III.4 The mobility of TMDs with point defects

The prediction of mobility of TMDs with point defects by EPC are challengeable
due to a large supercell should be employed in the DFT calculations. Herein we
try to use the new method to determine the mobility of TMDs with one
substitutional point defect due to the formation of Janus TMDs monolayer
origins from substituting one chalcogen atom of TMDs.

Table 7: Calculated impurities in TMDs(3$\times$3) of the hole mobility ($\mu_{h}$), BEC of nonmetal atom(ZB). System |  ZB(e) |  Mobility $\mu_{h}$(cm2/Vs) | System |  ZB(e) |  Mobility $\mu_{h}$(cm2/Vs)
---|---|---|---|---|---
H-Mo9Se18 | 1.84 | 77.86 |  T-Hf9Se18 | -7.60 | 5.26
H-Mo9Se17S | 1.80 | 81.70 |  T-Hf9Se17S | -7.58 | 5.26
H-Mo9Se17Te | 1.92 | 70.94 |  T-Hf9Se17Te | -7.73 | 5.26
H-Mo9Te18 | 3.28 | 21.95 |  T-Zr9S18 | -7.26 | 5.26
H-Mo9Te17S | 3.17 | 23.66 |  T-Zr9S17Se | -7.30 | 5.26
H-Mo9Te17Se | 3.20 | 23.17 |  T-Zr9S17Te | -7.40 | 5.26
H-W9Se18 | 1.23 | 188.10 |  T-Zr9Se18 | -8.17 | 5.26
H-W9Se17S | 1.19 | 202.23 |  T-Zr9Se17S | -8.13 | 5.26
H-W9Se17Te | 1.31 | 163.86 |  T-Zr9Se17Te | -8.28 | 5.26
T-Sn9S18 | -4.89 | 5.26 |  T-Pt9S18 | -2.68 | 5.26
T-Sn9S17Se | -4.90 | 5.26 |  T-Pt9S17Se | -2.66 | 5.26
T-Sn9S17Te | -4.80 | 5.26 |  T-Pt9S17Te | -2.66 | 5.26
T-Sn9Se18 | -4.98 | 5.26 |  T-Pt9Se18 | -2.36 | 5.27
T-Sn9Se17S | -4.98 | 5.26 |  T-Pt9Se17S | -2.38 | 5.26
T-Sn9Se17Te | -4.99 | 5.26 |  T-Pt9Se17Te | -1.96 | 5.54
T-Hf9S18 | -6.88 | 5.26 |  T-Pt9Te18 | -1.56 | 20.54
T-Hf9S17Se | -6.91 | 5.26 |  T-Pt9Te17S | -1.64 | 12.13
T-Hf9S17Se | -7.00 | 5.26 |  T-Pt9Te17Se | -1.61 | 14.53

In order to simulate the point defect, we expand the primitive cell of TMDs by
3$\times$3 to form a supercell with 27 atoms, then substituting one chalcogen
atom (S, Se, Te) by another chalcogen atom to form a defective supercell, and
then we calculate the BEC of these defective cells to obtain corresponding
carrier mobility. According to the calculations, we found that the BEC of
H-W9Se17S is 1.19 $|e|$, thus its electron mobility and hole mobility are
230.02 cm2V-1s-1 and 202.23 cm2V-1s-1, respectively, as shown in Table 6 and
Table 7. While for T-Zr9Se17Te, it has relative large BEC as 8.28 $|e|$, which
result in related electron mobility and hole mobility of 25.32 cm2V-1s-1 and
5.26 cm2V-1s-1, respectively. We further found that the carrier mobility is
related to chalcogen element (S, Se, Te) of TMDs. Generally, the order of the
carrier mobility with the addition of chalcogen element is S$>$Se$>$Te.
However, T-Pt9X17Y (X and Y are distinct element) is an exception, the order
turns out inversely to be Te$>$Se$>$S. For example, the $|Z_{B}|$ of H-Mo9Se18
is calculated as 1.84 $|e|$. When one of its chalcogenides changes, its BEC
changes simultaneously. The $|Z_{B}|$ of H-Mo9Se17S change to a smaller value
of 1.80 $|e|$, and the $|Z_{B}|$ of H-Mo9Se17Te change to a bigger value of
1.92 $|e|$. The order of their $|Z_{B}|$ satisfy the rule as H-Mo9Se17S$<$
H-Mo9Se18$<$ H-Mo9Se17Te. The mobility of electron and hole of the H-Mo9Se17S
is 112.36 cm2V-1s-1 and 81.70 cm2V-1s-1, and that of H-Mo9Se18 are 108.43
cm2V-1s-1 and 77.86 cm2V-1s-1, and that of H-Mo9Se17Te are 101.30 cm2V-1s-1
and 70.94 cm2V-1s-1, respectively, as shown in Table 6 and Table 7.

Therefore, one want to find a new TMDs with high mobility, the BEC of the
material should be small. The addition of S atoms can reduce the BEC, and
obtain a relative small BEC, thus a large mobility of TMDs. On the contrary,
for T-PtX2 (X=S, Se and Te), the addition of Te results in a relative large
mobility.

##  IV Summary

We have calculated the carrier mobility of pure and defective Janus structural
materials using first-principles calculation and BEC. It is shown that the
method of BEC is more suitable for calculating the carrier mobility of Janus
TMDs than that of DPT. Our study reveals the reason of effective BEC, which is
the carrier mobility of most Janus structural materials are mainly affected by
longitudinal optical branch except H-MoSSe and H-WSSe. For the pure Janus
structural materials, H-WSTe has the largest electron mobility of 129.72
cm2V-1s-1 and hole mobility of 382.05 cm2V-1s-1 due to the lowest BEC. While
for Janus structural materials with defects, H-W9Se17S has the highest
electron mobility of 230.02 cm2V-1s-1 and hole mobility of 202.23 cm2V-1s-1
due to the lowest BEC as well.

## Conflicts of interest

The authors declared that they have no conflicts of interest to this work.

## acknowledgments

The authors are grateful for financial support from National Natural Science
Foundation of China (General program 51671086) and Hunan Provincial Education
Department (Key project 19A324).

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we obtain an even pretemplate partition contained in $G$ with $k+1$ paths (the
new paths are $aPb$, $x^{+}P_{x}x^{\prime}w^{\prime}$, $wyP_{y}y^{-}$ each of
length $\ell-1$), a contradiction. Hence $aw$ or $bw^{\prime}$ is an edge of
$G$.

So by symmetry we may set that $aw\in E(G)$. Then $awyP_{y}^{*}y^{\prime*}bPa$
is a hole, so $P$ has length $\ell-1$. If $bw^{\prime}\notin E(G)$ then
$ax^{*}P^{*}_{x}x^{\prime}w^{\prime}y^{\prime*}bPa$ is a hole of length
$2\ell+1$, a contradiction. So $bw^{\prime}\in E(G)$. Suppose $A_{S}$ contains
a vertex $r\neq x,y$ then $awrP_{r}r^{\prime}w^{\prime}y^{\prime*}bPa$ is hole
of length $2\ell+1$, a contradiction. Hence $A_{S}=\\{x,y\\}$ and
$ar,br^{\prime}\in E(G)$ for every $r\in A_{K}$.

So, by setting
$\mathbb{A}_{K}=A_{K}\cup\\{a\\},\mathbb{A}_{S}=\\{x^{*},y\\},\mathbb{B}=\\{w\\},\mathbb{A}^{\prime}_{K}=A^{\prime}_{K}\cup\\{b\\},\mathbb{A}^{\prime}_{S}=\\{x^{\prime},y^{\prime*}\\},\mathbb{B}^{\prime}=\\{w^{\prime}\\}$
we obtain an even pretemplate partition contained in $G$ with $k+1$ paths (the
new paths are $aPb$ of length $\ell-1$ and $P^{*}_{x},P^{*}_{y}$ of length
$\ell-2$), a contradiction.

Case 2: $S_{a}\cap A_{S}=\varnothing$, $S_{a}$ contains a vertex $x\in A_{K}$
and $S_{b}$ contains a vertex $y^{\prime}\in A^{\prime}_{S}$ (symmetric to the
case where $S_{b}\cap A^{\prime}_{S}=\varnothing$, $S_{b}$ contains a vertex
$y^{\prime}\in A^{\prime}_{K}$ and $S_{a}$ contains a vertex $x\in A_{S}$).

Let $x^{*}\in K_{x}\cap S^{*}_{a}$ and $y^{\prime*}\in K_{y^{\prime}}\cap
S^{*}_{b}$. Suppose that $xy\in E(G)$. Then $ax^{*}yP^{*}_{y}y^{\prime*}bPa$
is a hole and hence $P$ has length $\ell-1$. Then
$ax^{*}P^{*}_{x}x^{\prime}w^{\prime}y^{\prime*}bPa$ is a cycle with at most
one possible chord ($bw^{\prime}$) of length $2\ell+2$, a contradiction. So
$xy\notin E(G)$ and consequently $x^{\prime}y^{\prime}\in E(G)$. Since $x^{*}$
and $y^{\prime*}$ both belong to the hole
$x^{*}P_{x}x^{\prime}y^{\prime*}P^{*}_{y}ywx^{*}$, by (9.5) we get that $P$
and this hole induce either a theta or a prism.

\- Subcase 2.1: $x^{*}P^{*}_{x}x^{\prime}y^{\prime*}P^{*}_{y}ywx^{*}$ and $P$
induce a theta.

Then $P$ has length $\ell-2$ and $aw,bx^{\prime}\notin E(G)$. Assume that $y$
has a neighbour $v$ in $A$. Then since $y\in A_{S}$ we have that $v\in A_{K}$
and hence $v^{\prime}y^{\prime*}\notin E(G)$ and $xv,x^{\prime}v^{\prime}\in
E(G)$. Hence to avoid a $C_{4}$ induced by
$\\{b,x^{\prime},y^{\prime*},v^{\prime}\\}$ it should be that
$bv^{\prime}\notin E(G)$. So $aPby^{\prime*}x^{\prime}v^{\prime}P_{v}vxa$ is a
cycle of length $2\ell+2$ whose only possible chord is $va$, a contradiction
to the fact that $G$ belongs to $C_{2\ell}$. We may then conclude that $y$ has
no neighbor in $A$ and so by setting
$\mathbb{A}_{K}=(A_{K}\setminus\\{x\\})\cup\\{x^{*},w\\},\mathbb{A}_{S}=(A_{S}\setminus\\{y\\})\cup\\{a\\},\mathbb{B}=\varnothing,\mathbb{A}^{\prime}_{K}=A^{\prime}_{K}\cup\\{y^{\prime*}\\},\mathbb{A}^{\prime}_{S}=(A^{\prime}_{S}\setminus\\{y^{\prime}\\})\cup\\{b\\},\mathbb{B}^{\prime}=B^{\prime}$,
we obtain an even pretemplate partition contained in $G$ with $k+1$ paths (the
new paths are $aPb$ of length $\ell-2$ and $x^{*}P^{*}_{x}x^{\prime}$ and
$wyP^{*}_{y}y^{\prime*}$ of length $\ell-1$, $\mathbb{A}_{S}$ and
$\mathbb{A}^{\prime}_{S}$ are both stable sets), a contradiction.

\- Subcase 2.2: $x^{*}P^{*}_{x}x^{\prime}y^{\prime*}P^{*}_{y}ywx^{*}$ and $P$
induce a prism. Then $P$ has length $\ell-1$ and $aw,bx^{\prime}\in E(G)$.

So, by setting
$\mathbb{A}_{K}=(A_{K}\setminus\\{x\\})\cup\\{x^{*},a\\},\mathbb{A}_{S}=A_{S},\mathbb{B}=B,\mathbb{A}^{\prime}_{K}=A^{\prime}_{K}\cup\\{b\\},\mathbb{A}^{\prime}_{S}=(A^{\prime}_{S}\setminus\\{y^{\prime}\\})\cup\\{y^{\prime*}\\},\mathbb{B}^{\prime}=B^{\prime}$
we obtain an even pretemplate partition contained in $G$ with $k+1$ paths (the
new paths are $aPb$, $x^{*}P^{*}_{x}x^{\prime}$ and $yP^{*}_{y}y^{\prime*}$ of
lengths respectively $\ell-1$, $\ell-1$, $\ell-2$), a contradiction.

Case 3: $S_{a}\cap A=\varnothing$, $S_{a}$ contains a vertex $x\in B$ and
$S_{b}$ contains a vertex $y^{\prime}\in A^{\prime}_{S}$ (symmetric to the
case where $S_{b}\cap A^{\prime}=\varnothing$, $S_{b}$ contains a vertex
$y^{\prime}\in B^{\prime}$ and $S_{a}$ contains a vertex $x\in A_{S}$).

If $xy\in E(G)$ then because of the hole $ax^{*}yP_{y}y^{\prime*}bPa$ the path
$P$ has length $\ell-1$ and once again it is enough to replace $x$ by $x^{*}$,
$y^{\prime}$ by $y^{\prime*}$ and add $a$ to $A_{K}$ and $b$ to
$A^{\prime}_{K}$ in order to obtain an $\ell$-pretemplate partition with $k+1$
principal paths, hence $xy\notin E(G)$. So, we may assume that $a$ is not
adjacent to $w$. Then because of $ax^{*}wyP^{*}_{y}y^{\prime*}bPa$, the path
$P$ has length $\ell-2$.

So, by setting
$\mathbb{A}_{K}=A_{K}\cup\\{x^{*}\\},\mathbb{A}_{S}=A_{S},\mathbb{B}=\\{w\\},\mathbb{A}^{\prime}_{K}=A^{\prime}_{K}\cup\\{b\\},\mathbb{A}^{\prime}_{S}=(A^{\prime}_{S}\setminus\\{y^{\prime}\\})\cup\\{y^{\prime*}\\},\mathbb{B}^{\prime}=B^{\prime}$
we obtain an even pretemplate partition contained in $G$ with $k+1$ paths (the
new paths are $x^{*}aPb$ of length $\ell-1$ and $yP^{*}_{y}y^{\prime*}$ of
length $\ell-2$), a contradiction.

This proves (109).

We may now conclude the proof. By (9.5) we have that $S_{a}\subseteq A_{K}\cup
B$ and $S_{b}\subseteq A^{\prime}_{K}\cup B^{\prime}$. Notice that if the
length of $P$ is $\ell-1$ (resp. $\ell-2$) then we get a contradiction since
we can add $a$ to $A_{K}$ and $b$ to $A^{\prime}_{K}$ (resp. $a$ to $A_{S}$
and $b$ to $A^{\prime}_{S}$) and replace any vertex $x\in S_{a}$ and any
vertex $y^{\prime}\in S_{b}$ by respectively $x^{*}\in K_{x}\cap S^{*}_{a}$
and $y^{\prime*}\in K_{y}\cap S^{*}_{b}$ in order to get an $\ell$-pretemplate
partition with one more principal path. So from now on we may assume that the
length of $P$ is neither $\ell-1$ nor $\ell-2$.

Let us assume that there exist $x\in S_{a}\cap A_{K}$ and $y^{\prime}\in
S_{b}\cap A^{\prime}_{K}$ and let $x^{*}\in K_{x}\cap S^{*}_{a}$ and
$y^{\prime*}\in K_{y}^{\prime}\cap S^{*}_{b}$. If $x=y$ then
$ax^{*}P^{*}_{x}y^{\prime*}bPa$ is a hole, and hence $P$ has length $\ell-1$,
a contradiction. So we may now assume that $x\neq y$, $S_{a}\cap
K_{y}=\varnothing$ and $S_{b}\cap K_{x^{\prime}}=\varnothing$. Then because of
the hole $ax^{*}yP^{*}_{y}y^{\prime*}bPa$ the length of $P$ is $\ell-2$, a
contradiction again.

So from now on we may assume by symmetry that $S_{a}\cap A_{K}=\varnothing$.
Then there exists $x^{*}\in K_{x}\cap S^{*}_{a}$ for some $x\in B$. Assume
that there exists $y^{\prime*}\in K_{y}^{\prime}\cap S^{*}_{b}$ for some
$y^{\prime}\in A_{K}.$ In case $xy\in E(G)$ then $P$ has length $\ell-2$
because of the hole $ax^{*}yP^{*}_{y}y^{\prime*}bPa$ and we would get a
contradiction. Hence $xy,aw\notin E(G)$. Now the hole
$ax^{*}wyP^{*}_{y}y^{\prime*}bPa$ implies that $P$ has length $\ell-3$. Then
any vertex $u\in A$ which is adjacent to $x$, is adjacent to $y$: if $u\in
A_{K}$ by definition of an even template, if $u\in A_{S}$ because else
$ax^{*}uP_{u}u^{\prime}y^{\prime*}bPa$ is hole of length $2\ell-1$. However,
as any vertex in $B$, $x$ has two non adjacent neighbors $r,s\in A$ and so $G$
contains a square $x^{*}rysx^{*}$, a contradiction. So from now on we may
assume that $S_{b}\subseteq B^{\prime}$ and there exists $y^{\prime*}\in
K_{y^{\prime}}\cap S^{*}_{b}$ for some $y^{\prime}\in B^{\prime}$. By
definition, $x$ has at least one neighbor $r$ in $A_{S}$. If
$y^{\prime}r^{\prime}\in E(G)$ then because of the hole
$ax^{*}rP_{r}r^{\prime}y^{\prime*}bPa$ we get that $P$ has length $\ell-2$, a
contradiction. So $r^{\prime}y^{\prime}\notin E(G)$ and then $y^{\prime}\neq
w^{\prime}$, $bw^{\prime}\notin E(G)$ and $P$ should have length $\ell-3$,
because of the cycle $ax^{*}rP_{r}r^{\prime}w^{\prime}y^{\prime*}bPa$. By
symmetry, we also have $x\neq w$.

So, by setting
$\mathbb{A}_{K}=A_{K}\cup\\{x^{*}\\},\mathbb{A}_{S}=A_{S},\mathbb{B}=\\{w\\},\mathbb{A}^{\prime}_{K}=A^{\prime}_{K}\cup\\{y^{*}\\},\mathbb{A}^{\prime}_{S}=A^{\prime}_{S},\mathbb{B}^{\prime}=\\{w^{\prime}\\}$
we obtain an even pretemplate partition contained in $G$ with $k+1$ paths (the
new path is $x^{*}aPby^{*}$ of length $\ell-1$), a contradiction.

∎

###  9.6 End of the proof

We may now conclude the proof of Lemma 9.1. If $G\setminus F_{1}$ is empty,
then conclusion (a) holds. If $G\setminus F_{1}$ is non-empty and $G\setminus
F_{2}$ is empty, then conclusion (b) holds. Otherwise, we consider a connected
component $D$ of $G\setminus F_{2}$ and apply Lemma 9.15. We then see that $G$
has a clique cutset, so conclusion (c) holds.

##  10 Proof of Theorem 10.1

###### Theorem 10.1

Let $\ell\geq 4$ be an integer. If $G$ is a graph in $\mathcal{C}_{2\ell}$
then one of the following holds:

  1. (a)

$G$ is a ring of length $2\ell$;

  2. (b)

$G$ is a proper blowup of a twinless even $\ell$-template;

  3. (c)

$G$ has a universal vertex or

  4. (d)

$G$ has a clique cutset.

###### Proof.

By Lemma 3.9, $G$ contains no unbalanced prism, no unbalanced theta, no
pyramid and no proper wheel. Also, clearly $G$ contains no $C_{4}$ and no
$C_{5}$. Hence, by Theorem 3.13, we may assume that $G$ contains a prism or a
theta for otherwise one of the conclusions (a), (c) or (d) holds. The result
then follows from Lemma 9.1. ∎

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