# New class of Gibbs measures for two state Hard-Core model on a Cayley tree

R. M. Khakimov, M T. Makhammadaliev and F. H. Haydarov  R. M. Khakimov, M. T.
Makhammadaliev
V. I. Romanovskiy Institute of Mathematics of the Academy of Sciences of
Uzbekistan, Tashkent, Uzbekistan.
Namangan State University, Namangan, Uzbekistan<EMAIL_ADDRESS><EMAIL_ADDRESS>F. H. Haydarov
V. I. Romanovskiy Institute of Mathematics of the Academy of Sciences of
Uzbekistan, Tashkent, Uzbekistan.  AKFA University, Tashkent, Uzbekistan.
National University of Uzbekistan, Tashkent , Uzbekistan.
<EMAIL_ADDRESS>
###### Abstract.

In this paper, we consider a Hard-Core $(HC)$ model with two spin values on
Cayley trees. The conception of alternative Gibbs measure is introduced and
translational invariance conditions for alternative Gibbs measures are found.
Also, we show that the existence of alternative Gibbs measures which are not
translation-invariant. In addition, we study free energy of the model.

Key words. Cayley tree, configuration, hard-core model, Gibbs measure,
translation-invariant measure, Alternating Gibbs Measure, free energy.

AMS Subject Classification: 20B07, 20E06.

##  1\. Introduction

The problems arising in the study of the thermodynamic properties of physical
and biological systems are typically solved within the framework of the theory
of Gibbs measures. The Gibbs measure is a fundamental concept that determines
the probability of a microscopic state of a given physical system (defined by
a specific Hamiltonian). It is known that each Gibbs measure is associated
with one phase of a physical system, and if the Gibbs measure is not unique,
then there exists a phase transition. For a wide class of Hamiltonians, it is
known that the set of all Gibbs measures (corresponding to a given
Hamiltonian) is a nonempty, convex, compact subset of the set of all
probability measures (see, e.g., [1], [3]) and each point of this convex set
can be uniquely decomposed in terms of its extreme points. In this regard, it
is of particular interest to describe all the extreme points of this convex
set, i.e., extreme Gibbs measures.

For convenience, we first describe the basic concepts used in this paper and
then give the statement of the problem and the history of its study.

The Cayley tree. Let $\Im^{k}=(V,L,i)$, $k\geq 1$, be the Cayley tree of order
$k$, i.e., an infinite tree with exactly $k+1$ edges coming out of each
vertex, and let $V$ be the set of vertices, $L$ the set of edges of $\Im^{k}$
and $i$ is the incidence function setting each edge $l\in L$ into
correspondence with its endpoints $x,y\in V$. If $i(l)=\\{x,y\\}$, then the
vertices $x$ and $y$ are called the nearest neighbors, denoted by $l=\langle
x,y\rangle$.

For an arbitrary point $x^{0}\in V$ we set

$W_{n}=\ \\{x\in V\ \ |\ \ d(x,x^{0})=n\\},\ \
V_{n}=\bigcup_{m=0}^{n}{W_{m}},\ \ L_{n}=\ \\{l=\langle x,y\rangle\in L\ \ |\
\ x,y\in V_{n}\\},$

where $d(x,y),x,y\in V$ is the distance between $x$ and $y$ on the Cayley
tree, i.e., the number of edges of the path connecting $x$ and $y$.

The set of the direct successors of $x$ is denoted by $S(x)$, i.e., if $x\in
W_{n}$, then

$S(x)=\\{y_{i}\in W_{n+1}|d(x,y_{i})=1,i=1,2,\ldots,k\\}.$

The HC-model. We assume that $\Phi=\\{0,1\\},$ and $\sigma\in\Omega=\Phi^{V}$
is a configuration, i.e., $\sigma=\\{\sigma(x)\in\Phi:x\in V\\}$, where
$\sigma(x)=1$ means that the vertex $x$ on the Cayley tree is occupied, and
$\sigma(x)=0$ means it is vacant. The configuration $\sigma$ is said to be an
admissible if $\sigma(x)\sigma(y)=0$ for any neighboring $\langle x,y\rangle$
from $V$ ($V_{n}$ or $W_{n}$, respectively) and we denote the set of such
configurations by $\Omega$ ($\Omega_{V_{n}}$ and $\Omega_{W_{n}})$. Obviously,
$\Omega\subset\Phi^{V}.$

Concatenation configurations $\sigma_{n-1}\in\Phi^{V_{n-1}}$ and
$\omega_{n}\in\Phi^{W_{n}}$ is defined by the following formula (see [28])

$\sigma_{n-1}\vee\omega_{n}=\\{\\{\sigma_{n-1}(x),x\in
V_{n-1}\\},\\{\omega_{n}(y),y\in W_{n}\\}\\}.$

The Hamiltonian of the _HC_ -model is defined by the formula

$H(\sigma)=\left\\{\begin{array}[]{ll}J\sum\limits_{x\in{V}}{\sigma(x),}\ \ \
$ if $\sigma\in\Omega$,$\\\ +\infty,\ \ \ \ \ \ \ \ \ \ $ \ if $\sigma\
\notin\Omega$,$\\\ \end{array}\right.$

where $J\in R$.

Finite-dimensional distributions. Let $\mathbf{B}$ be the $\sigma$-algebra
generated by cylindrical sets with finite base of $\Omega$. For any $n$ we let
$\mathbf{B}_{V_{n}}=\\{\sigma\in\Omega:\sigma|_{V_{n}}=\sigma_{n}\\}$ denote
the subalgebra of $\mathbf{B},$ where $\sigma|_{V_{n}}$-restriction of
$\sigma$ to $V_{n}$ and $\sigma_{n}:x\in V_{n}\mapsto\sigma_{n}(x)$ an
admissible configuration in $V_{n}.$

Definition 1. For $\lambda>0$, the _HC_ -model Gibbs measure is a probability
measure $\mu$ on $(\Omega,\textbf{B})$ such that for any $n$ and
$\sigma_{n}\in\Omega_{V_{n}}$

$\mu\\{\sigma\in\Omega:\sigma|_{V_{n}}=\sigma_{n}\\}=\int_{\Omega}\mu(d\omega)P_{n}(\sigma_{n}|\omega_{W_{n+1}}),$

where

$P_{n}(\sigma_{n}|\omega_{W_{n+1}})=\frac{e^{-H(\sigma_{n})}}{Z_{n}(\lambda;\omega|_{W_{n+1}})}\textbf{1}(\sigma_{n}\vee\omega|_{W_{n+1}}\in\Omega_{V_{n+1}}).$

Here $Z_{n}(\lambda;\omega|_{W_{n+1}})$\- normalization factor with boundary
condition $\omega|_{W_{n}}$:

$Z_{n}(\lambda;\omega|_{W_{n+1}})=\sum_{\widetilde{\sigma}_{n}\in\Omega_{V_{n}}}e^{-H(\widetilde{\sigma}_{n})}\textbf{1}(\widetilde{\sigma}_{n}\vee\omega|_{W_{n+1}}\in\Omega_{V_{n+1}}).$

For $\sigma_{n}\in\Omega_{V_{n}}$ we denote that

$\\#\sigma_{n}=\sum\limits_{x\in V_{n}}{\mathbf{1}}(\sigma_{n}(x)\geq 1)$

which the number of occupied vertices in $\sigma_{n}$.

Let $z:x\mapsto z_{x}=(z_{0,x},z_{1,x})\in R^{2}_{+}$ vector-valued function
on $V$. For $n=1,2,\ldots$ and $\lambda>0$ consider the probability measure
$\mu^{(n)}$ on $\Omega_{V_{n}}$, defined as

$\mu^{(n)}(\sigma_{n})=\frac{1}{Z_{n}}\lambda^{\\#\sigma_{n}}\prod_{x\in
W_{n}}z_{\sigma(x),x}.$ (1)

Here $Z_{n}$ is the normalizing divisor:

$Z_{n}=\sum_{{\widetilde{\sigma}}_{n}\in\Omega_{V_{n}}}\lambda^{\\#{\widetilde{\sigma}}_{n}}\prod_{x\in
W_{n}}z_{{\widetilde{\sigma}}(x),x}.$

The sequence of probability measures $\mu^{(n)}$ is said to be consistent if
for any $n\geq 1$ and $\sigma_{n-1}\in\Omega_{V_{n-1}}$:

$\sum_{\omega_{n}\in\Omega_{W_{n}}}\mu^{(n)}(\sigma_{n-1}\vee\omega_{n}){\mathbf{1}}(\sigma_{n-1}\vee\omega_{n}\in\Omega_{V_{n}})=\mu^{(n-1)}(\sigma_{n-1}).$
(2)

In this case, there is a unique measure $\mu$ on $(\Omega,\textbf{B})$ such
that for all $n$ and $\sigma_{n}\in\Omega_{V_{n}}$

$\mu(\\{\sigma|_{V_{n}}=\sigma_{n}\\})=\mu^{(n)}(\sigma_{n}).$

Definition 2. The measure $\mu$ that is the limit of a sequence $\mu^{(n)}$
defined by (1) with consistency condition (2) is called the splitting _HC_
-Gibbs measure (SGM) with $\lambda>0$ corresponding to the function $z:\,x\in
V\setminus\\{x^{0}\\}\mapsto z_{x}$. Moreover, an _HC_ -Gibbs measure
corresponding to a constant function $z_{x}\equiv z$ is said to be
translation-invariant (TI).

Problem statement. The main task is to study the structure of the set
$\mathcal{G}(H)$ of all Gibbs measures corresponding to a given Hamiltonian
$H$.

A measure $\mu\in\mathcal{G}(H)$ is called extreme if it cannot be expressed
as $\mu=\lambda\mu_{1}+(1-\lambda)\mu_{2}$ for some
$\mu_{1},\mu_{2}\in\mathcal{G}(H)$ with $\mu_{1}\neq\mu_{2}$.

As noted above, the set $\mathcal{G}(H)$ of all Gibbs measures (for a given
Hamiltonian $H$) is a nonempty convex compact set $\mathcal{G}(H)$ in the
space of all probability measures on $\Omega$.

Using theorem (12.6) in [1] and section 1.2.4 in [8], we can note the
following.

  * •

_Any extreme Gibbs measure $\mu\in\mathcal{G}(H)$ is an SGM; therefore, the
problem of describing Gibbs measures reduces to describing the set of SGMs.
For each fixed temperature, the description of the set $\mathcal{G}(H)$ is
equivalent to a complete description of the set of all extreme SGMs, and hence
we are only interested in SGMs on the Cayley tree._

  * •

_Any SGM corresponds to the solution of Eq. ( 3) (see below). Thus, our main
task reduces to solving functional equation (3)_.

It is known [13] that each Gibbs measure for _HC_ -model on the Cayley tree
can be associated with the collection of values $z=\\{z_{x},x\in V\\}$
satisfying

$z_{x}=\prod_{y\in S(x)}(1+\lambda z_{y})^{-1},$ (3)

where $\lambda=e^{-J\beta}>0$ is a parameter, $\beta={1\over T}$, $T>0$ is a
temperature.

Let $G_{k}$ be a free product of $k+1$ cyclic groups $\\{e,a_{i}\\}$ of order
two with the respective generators $a_{1},a_{2},...,a_{k+1},a_{i}^{2}=e$.
There is a one-to-one correspondence between the set of vertices $V$ of the
Cayley tree of order $k$ and the group $G_{k}$ (see [5, 6, 29]).

Let $\widehat{G}_{k}$ be a normal divisor of a finite index $r\geq 1$ and
$G_{k}/\widehat{G}_{k}=\\{H_{1},...,H_{r}\\}$ be the quotient group.

Definition 3. A collection of quantities $z=\\{z_{x},x\in G_{k}\\}$ is said to
be $\widehat{G}_{k}$-periodic if $z_{yx}=z_{x}$ for $\forall x\in
G_{k},y\in\widehat{G}_{k}.$ The $G_{k}$-periodic collections are called
translation invariant.

For any $x\in G_{k}$, the set $\\{y\in G_{k}:\langle x,y\rangle\\}\setminus
S(x)$ contains a unique element denoted by $x_{\downarrow}$ (see [9, 10]).

Definition 4. A collection of quantities $z=\\{z_{x},x\in G_{k}\\}$ is called
$\widehat{G}_{k}$-weakly periodic if $z_{x}=z_{ij}$ for any $x\in H_{i}$,
$x_{\downarrow}\in H_{j}$ for any $x\in G_{k}$.

Definition 5. A measure $\mu$ is called $\widehat{G}_{k}$-(weakly) periodic if
it corresponds to a $\widehat{G}_{k}$-(weakly) periodic collection of
quantities $z$.

History of the study of SGMs for the _HC_ -model. We present a brief overview
of the work related to the Potts model on the Cayley tree.

In [12] A. Mazel and Yu. Suhov introduced and studied the _HC_ -model on the
$d$-dimensional lattice $\mathbb{Z}^{d}$. Studying Gibbs measures for the two
state _HC_ -model on the Cayley tree was the topic in [13]-[23]. In [13], the
uniqueness of the translation-invariant Gibbs measure and the nonuniqueness of
periodic Gibbs measures for the _HC_ -model were proved. For the parameters of
the _HC_ -model, a sufficient condition was also found in [13] under which the
translation-invariant Gibbs measure is nonextreme. In the case where the
translation-invariant Gibbs measure is extreme, a sufficient condition was
found in [14]. The range of the extremes of this measure was extended in [15].
Weakly periodic Gibbs measures for the _HC_ -model in the case of a normal
divisor of index 2 were studied in [16] and a complete description of the
weakly periodic Gibbs measures was given.

Weakly periodic Gibbs measures for the _HC_ -model in the case of a normal
divisor of index 4 were studied in [17]-[22]. In this case conditions for the
existence of weakly periodic (nonperiodic) Gibbs measures are found. We also
found conditions for the translation-invariance of the weakly periodic Gibbs
measures (see Chap. 7 in [4] for other HC model properties and their
generalizations on a Cayley tree).

In this paper, we study a two-state _HC_ -model on a Cayley tree. The concept
of an alternative Gibbs measure is introduced. Translational invariance
conditions for alternative Gibbs measures are found. In addition, the
existence of alternative Gibbs measures that are not translation invariant is
proved.

##  2\. A new class of Gibbs measures

We consider the half-tree. Namely the root $x^{0}$ has $k$ nearest neighbors.
We construct below new solutions of the functional equation (3). Consider the
following matrix

$M=\begin{pmatrix}m&k-m\\\ r&k-r\end{pmatrix}$

where $0\leq m\leq k$ and $0\leq r\leq k$ are non-negative integers. This
matrix defines the number of times the values $h$ and $l$ occur in the set
$S(x)$ for each $z_{x}\in\\{h,l\\}$. More precisely, the boundary condition
$z=\\{z_{x},x\in G_{k}\\}$ with fields taking values $h$, $l$ defined by the
following steps:

$\bullet$ if at vertex $x$ we have $z_{x}=h$, then the function $z_{y}$, which
gives real values to each vertex $y\in S(x)$ by the following rule

$\begin{cases}h\leavevmode\nobreak\ \mbox{on}\leavevmode\nobreak\
m\leavevmode\nobreak\ \mbox{vertices of}\leavevmode\nobreak\ S(x),\\\
l\leavevmode\nobreak\ \mbox{on}\leavevmode\nobreak\ k-m\leavevmode\nobreak\
\mbox{remaining vertices,}\end{cases}$

$\bullet$ if at vertex $x$ we have $z_{x}=l$, then the function $z_{y}$, which
gives real values to each vertex $y\in S(x)$ by the following rule

$\begin{cases}l\leavevmode\nobreak\ \mbox{on}\leavevmode\nobreak\
r\leavevmode\nobreak\ \mbox{vertices of}\leavevmode\nobreak\ S(x),\\\
h\leavevmode\nobreak\ \mbox{on}\leavevmode\nobreak\ k-r\leavevmode\nobreak\
\mbox{remaining vertices.}\end{cases}$

For an example of such a function see Fig.1.

Then the system (3) has the form

$\left\\{\begin{array}[]{ll}h=\frac{1}{\left(1+\lambda
h\right)^{m}}\cdot\frac{1}{\left(1+\lambda l\right)^{k-m}},\\\
l=\frac{1}{\left(1+\lambda l\right)^{r}}\cdot\frac{1}{\left(1+\lambda
h\right)^{k-r}},\\\ \end{array}\right.$ (4)

where $l>0,h>0,\lambda>0.$

$h$$l$$h$$h$$h$$h$$l$$l$$h$$h$$h$$h$$l$$h$$l$$l$$h$$h$$h$$h$$l$$l$$h$$h$$h$$h$$l$$l$$h$$h$$h$

Figure 1. In this figure the values of function $z_{x}$ on the vertices of the
Cayley tree of order 5 are shown. This is the case when $m=3$ and $r=2$.

As was mentioned above, for any boundary condition satisfying the functional
equation (3) there exists a unique Gibbs measure. A measure constructed in
this way and which is not translation-invariant is called alternative Gibbs
measure (AGM) and denoted as $\mu_{m,r}$.

Remark 1. Note that the solution $l=h$ in (4) corresponds to the only TIGM for
the _HC_ -model (see [13]). Therefore, we are interested in solutions of the
form $l\neq h.$

Remark 2. From (4) for $m=r=0$ we obtain a system of equations whose solutions
correspond to the $G^{(2)}_{k}$-periodic Gibbs measures for the _HC_ -model.

The following theorem holds.

Theorem 1. Let $k\geq 2$. If $m+r\geq k-1$ then for the _HC_ -model there is a
unique AGM, which coincides with the unique TIGM.

Proof. For convenience, we denote $h=x$ and $l=y$. Then (4) can be rewritten
as follows:

$\left\\{\begin{array}[]{ll}x=\frac{1}{(1+\lambda
x)^{m}}\cdot\frac{1}{(1+\lambda y)^{k-m}},\\\ y=\frac{1}{(1+\lambda
y)^{r}}\cdot\frac{1}{(1+\lambda x)^{k-r}}.\\\ \end{array}\right.$ (5)

If the first equation (5) is divided by the second, then

$\frac{x}{y}=\left(\frac{1+\lambda x}{1+\lambda y}\right)^{k-m-r}$ (6)

We denote $m+r-k=t$, $t\geq-1$. Then by (6) we have

$x\left(1+\lambda x\right)^{t}=y\left(1+\lambda y\right)^{t}$

It is easy to check that the function $f(x)=x\left(1+\lambda x\right)^{t}$ is
increasing for $t\geq-1$. Therefore, if $m+r\geq k-1$, then the system of
equations (5) has only a solution of the form $x=y$, and this solution
corresponds to the TIGM and is known to be unique. The theorem is proved.

By theorem 1 follows the next

Consequence. Let $k\geq 2$. If there are AGMs (non TI) for the _HC_ -model,
then $m+r\leq k-2$.

Let $k-m-r=n$ $(n\in N,n\geq 2)$. Then from the (6) we get

$x\left(1+\lambda y\right)^{n}=y\left(1+\lambda x\right)^{n}.$

From this equation after simple algebra, we obtain the equation

$(y-x)\Big{(}-1+C_{n}^{2}\lambda^{2}xy+C_{n}^{3}\lambda^{3}xy(x+y)+\ldots+C_{n}^{n}\lambda^{n}xy(x^{n-2}+x^{n-3}y+x^{n-4}y^{2}+\ldots+y^{n-2})\Big{)}=0.$

Hence $x=y$ or $g(x,y)=0$, where

$g(x,y)=C_{n}^{2}\lambda^{2}xy+C_{n}^{3}\lambda^{3}xy(x+y)+\ldots+C_{n}^{n}\lambda^{n}xy(x^{n-2}+x^{n-3}y+x^{n-4}y^{2}+\ldots+y^{n-2})-1.$

In the case $x=y$ the corresponding measure is TIGM.

The case $x\neq y$. We consider the equation $g(x,y)=0$ with respect to the
variable $x$ (or $y$). Then it’s clear that $g(0,y)=-1$ and
$g(x,y)\rightarrow+\infty$ for $x\rightarrow+\infty$. Then the equation
$g(x,y)=0$ for variable $x$ has at least one positive root. On the other hand,
by Descartes’ theorem, the equation $g(x,y)=0$ for variable $x$ has at most
one positive root. Hence the equation $g(x,y)=0$ for variable $x$ has exactly
one positive root, i.e., there exists a solution $(x,y)$ of the system of
equations (5), different from $(x,x)$. Thus, the following statement is true.

Statement 1. Let $k\geq 2$. If $m+r\leq k-2$ then for the _HC_ -model there
exists AGM (not TI).

In particular, if $\lambda^{2}xy=1$ for $m+r=k-2$ $(n=2)$ or if
$\lambda^{2}xy(3+\lambda(x+y))=1$ for $m+r=k-3$ $(n=3)$, then in both cases
there exists AGM (not TI).

The case $x=y$. We check the multiplicity of the root $x=y$. In this case from
$g(x,x)=0$, we have

$C_{n}^{2}\lambda^{2}x^{2}+2C_{n}^{3}\lambda^{3}x^{3}+\ldots+(n-1)C_{n}^{n}\lambda^{n}x^{n}-1=0$
(7)

and the equation (7) also has exactly one positive root, i.e., $(x,x)$ is a
multiple root for the system of equations (5). This means that, AGM coincide
with TIGM.

In particular, if $\lambda x=1$ for $m+r=k-2$ $(n=2)$ or if
$3\lambda^{2}x^{2}+2\lambda^{3}x^{3}=1$ for $m+r=k-3$ $(n=3)$, then in both
cases there is no AGM (not TI).

Let

$\left\\{\begin{array}[]{ll}x=f(y),\\\ y=f(x),\\\ \end{array}\right.$ (8)

where $f(x)=\frac{1}{(1+\lambda x)^{k}}$.

The next lemma is obvious.

Lemma 1. If $(x_{0},y_{0})$ is a solution to the system of equations (8), then
$(y_{0},x_{0})$ is also a solution to the system of equations (8).

Remark 3. If the solution $(x,y)$ of the system of equations (8) corresponds
to alternative Gibbs measure denoted by $\mu$, then the solution $(y,x)$
corresponds to alternative Gibbs measure denoted by $\mu^{{}^{\prime}}$.

###  2.1. Alternative Gibbs measures in the case $m+r\leq k-2$

In this section, we consider the cases $k=2$, $k=3$ and $k=4$. In the case
$k=2$ we have only the case $m=0$ and $r=0$. In the case $k=3$ we have $m=0$
and $r=0$; $m=0$ and $r=1$; $m=1$ and $r=0$. In the case $k=4$ we have $m=0$
and $r=0$; $m=0$ and $r=1$; $m=1$ and $r=0$; $m=1$ and $r=1$; $m=0$ and $r=2$;
$m=2$ and $r=0$. In all cases, by Remark 2, we will not consider the case
$m=0$ and $r=0$. corresponds to the translation-invariant Gibbs measure and
solutions $(x_{1},y_{1}),(x_{2},y_{2})$ in Statement 1
($(x^{*},y^{*}),(y^{*},x^{*})$ in Statement 2) correspond to two-periodic
Gibbs measures (see [23]).

The case $k=3$, $m=1$ and $r=0$. For $m=1$ and $r=0$ (resp. $m=0$ and $r=1$)
the system of equations (5) can be rewritten

$\left\\{\begin{array}[]{ll}x=\frac{1}{1+\lambda x}\cdot\frac{1}{(1+\lambda
y)^{2}},\\\ y=\frac{1}{(1+\lambda x)^{3}}.\\\ \end{array}\right.$ (9)

From the system of equations (9) due to (6) we obtain
$(x-y)\Big{(}\lambda^{2}xy-1\Big{)}=0$. Hence, $x=y$ or $\lambda^{2}xy=1$. The
case $x=y$ has already been considered.

Let $\lambda^{2}xy=1$. Then $\lambda x=\frac{1}{\lambda y}$ for $x\neq y$.
From here and from (9) after some algebras we can get:

$\left\\{\begin{array}[]{ll}(1+\lambda x)^{3}-\lambda^{2}x=0,\\\ (1+\lambda
y)^{3}-\lambda^{3}y^{2}=0.\end{array}\right.$ (10)

From $\lambda^{2}xy=1$ we find $y$ and substitute into the second equation of
the system (10). Then

$\left\\{\begin{array}[]{ll}(1+\lambda x)^{3}-\lambda^{2}x=0,\\\
\frac{(1+\lambda x)^{3}-\lambda^{2}x}{\lambda^{3}x^{3}}=0.\\\
\end{array}\right.$

We introduce the notation $f(x)=(1+\lambda x)^{3}-\lambda^{2}x$. Then the
roots of the equation $f(x)=0$ are also roots of the system (9). Using the
Cardano formulas, we find the positive solution of the last equation

$\lambda^{3}x^{3}+3\lambda^{2}x^{2}+\Big{(}3-\lambda^{2}\Big{)}x+1=0.$

Let $x=q-\frac{1}{\lambda}$, then

$f(q)=\lambda^{3}q^{3}-\lambda^{2}q+\lambda,\ \
D=\frac{1}{\lambda^{4}}\left(\frac{1}{4}-\frac{1}{27\lambda}\right).$

If $D>0$, i.e., $\lambda<\frac{27}{4}$ then by Cardano’s formula the equation
$f(q)=0$ has one negative root.

If $D=0$, i.e., $\lambda=\frac{27}{4}$ then the equation $f(q)=0$ has one
multiple positive root of the form $q^{\prime}=\frac{2}{9}$, i.e.
$x^{\prime}=\frac{2}{27}$, $y^{\prime}=\frac{8}{27}$.

By Cardano’s formula, the equation $f(q)=0$ has three real roots if $D<0$.
Hence, $f(x)=0$ has three real roots if $\lambda>\frac{27}{4}$. Let these
solutions be $x_{1},x_{2},x_{3}$. By the Vieta’s formulas we have

$x_{1}+x_{2}+x_{3}=-\frac{3}{\lambda},\ \
x_{1}x_{2}+x_{1}x_{3}+x_{2}x_{3}=\frac{3-\lambda^{2}}{\lambda^{3}}<0,\ \
x_{1}x_{2}x_{3}=-\frac{1}{\lambda^{3}}.$

From equality
$x_{1}x_{2}+x_{1}x_{3}+x_{2}x_{3}=\frac{3-\lambda^{2}}{\lambda^{3}}$ we obtain
that at least one and at most two roots of the equation are positive.

From the equality $x_{1}x_{2}x_{3}=-1$ it follows that exactly two roots are
positive. Hence, $f(x)=0$ has two positive roots if $\lambda>\frac{27}{4}$.
These roots have the following form

$x_{1}=\frac{\sqrt[3]{t^{2}}-6\sqrt[3]{t}+12\lambda}{6\lambda\sqrt[3]{t}},\ \
x_{2}=\frac{6\sqrt[3]{p}}{\lambda(\sqrt[3]{p^{2}}+(2\lambda-6)\sqrt[3]{p}+4\lambda^{2}-24\lambda)}.$

Here

$t=-108\lambda+12\lambda\sqrt{-12\lambda+81},\ \
p=108\lambda+8\lambda^{3}-72\lambda^{2}+12\lambda\sqrt{-12\lambda+81}.$

From the equality $\lambda^{2}xy=1$ we find $y_{1}$ and $y_{2}$ corresponding
to $x_{1}$ and $x_{2}$:

$y_{1}=\frac{6\sqrt[3]{t}}{\lambda(\sqrt[3]{t^{2}}-6\sqrt[3]{t}+12\lambda)},\
\
y_{2}=\frac{\sqrt{p^{2}}+(2\lambda-6)\sqrt[3]{p}+4\lambda^{2}-24\lambda}{6\sqrt[3]{p}\lambda}.$

Thus, the following statement is true.

Statement 2. Let $k=3$ and $\lambda_{cr}=\frac{27}{4}$. Then the system of
equations (9):

1\. for $0<\lambda<\lambda_{cr}$ has a unique solution $(x,x)$;

2\. for $\lambda=\lambda_{cr}$ has two solutions
$(x,x),(\frac{2}{27},\frac{8}{27})$;

3\. for $\lambda>\lambda_{cr}$ has three solutions
$(x,x),(x_{1},y_{1}),(x_{2},y_{2})$.

Remark 4. The measure corresponding to the solution $(x,x)$ is translation
invariant and measures corresponding to solutions
$(\frac{2}{27},\frac{8}{27})$, $(x_{1},y_{1}),(x_{2},y_{2})$ are AGMs (not
TI).

Theorem 3. Let $k=3$ and $r+m\leq 1$, i.e., $m=1$ and $r=0$ or $m=0$ and
$r=1$. Then for the HC-model there exists $\lambda_{cr}=\frac{27}{4}$ such
that for $0<\lambda<\lambda_{cr}$ there is a unique AGM which coincides with
the only TIGM $\mu_{0}$, for $\lambda=\lambda_{cr}$ there are exactly two AGMs
$\mu_{0}$ and $\mu^{\prime}$, where $\mu^{\prime}$ is AGM (not TI) and for
$\lambda>\lambda_{cr}$ there are exactly three AGMs $\mu_{0}$, $\mu_{1}$ and
$\mu_{2}$, where $\mu_{1}$ and $\mu_{2}$ are AGMs (not TI).

The case $k=4$, $m=1$ and $r=0$ ($m=0$ and $r=1$). In this case from (5) we
get

$\left\\{\begin{array}[]{ll}x=\frac{1}{1+\lambda x}\cdot\frac{1}{(1+\lambda
y)^{3}},\\\ y=\frac{1}{(1+\lambda x)^{4}}.\\\ \end{array}\right.$ (11)

From the system of equations (11) due to (6) we can get

$(y-x)\Big{(}\lambda^{2}xy(3+\lambda(x+y))-1\Big{)}=0.$

Hence $x=y$ or $\lambda^{2}xy(3+\lambda(x+y))=1$. It is clear that in the case
$x=y$ we obtain a solution corresponding to the TIGM.

Suppose $x\neq y$ and $\lambda^{2}xy(3+\lambda(x+y))=1$. Then, substituting
the expression for $y$ from the second equation of the system (11) into the
last equality, we obtain the equation

$f(x,\lambda)=\lambda^{8}x^{8}+8\lambda^{7}x^{7}-\lambda^{7}x^{6}+28\lambda^{6}x^{6}-7\lambda^{6}x^{5}+56\lambda^{5}x^{5}-18\lambda^{5}x^{4}+70\lambda^{4}x^{4}-22\lambda^{4}x^{3}+$
$+56\lambda^{3}x^{3}-13\lambda^{3}x^{2}-\lambda^{3}x+28\lambda^{2}x^{2}-3\lambda^{2}x+8\lambda
x+1=0.$

Denoting $\lambda x=u$, $u>0$ we then have the equation

$f(u)=u^{8}+8u^{7}+(28-\lambda)u^{6}+(56-7\lambda)u^{5}+(70-18\lambda)u^{4}+(56-22\lambda)u^{3}+(28-13\lambda)u^{2}-(\lambda^{2}+3\lambda-8)u+1=0,$

which has a solution $u=u(\lambda)$. But we regard this as an equation for
$\lambda$ and obtain solutions $\lambda=\lambda(u)$:

$\lambda_{1}(u)=\frac{(u+1)^{4}}{2u}\cdot\left(\sqrt{u^{4}+6u^{3}+9u^{2}+4u}-u^{2}-3u\right),$
$\lambda_{2}(u)=-\frac{(u+1)^{4}}{2u}\cdot\left(\sqrt{u^{4}+6u^{3}+9u^{2}+4u}+u^{2}+3u\right).$

Therefore, because $\lambda_{2}<0$ for $u>0$, we have

$\lambda-\lambda_{1}=0\ \Rightarrow\
\lambda=\frac{(u+1)^{4}}{2u}\cdot\left(\sqrt{u^{4}+6u^{3}+9u^{2}+4u}-u^{2}-3u\right)=\psi(u).$

Analysis of the function $\psi(u)$ shows that $\psi(u)>0.$ In addition,
$\psi(u)\rightarrow+\infty$ as $u\rightarrow 0$ and as $u\rightarrow+\infty$,
and each value of $\lambda$ therefore corresponds to at least two values of
$u$ for $\lambda>\psi(u^{*})$ but to one value at $\lambda=\psi(u^{*})$, and
the equation $\lambda=\psi(u)$ has no solutions for $\lambda<\psi(u^{*}),$
where $u^{*}$ a solution of the equation $\psi^{\prime}(u)=0$ (see Fig.2). We
calculate the derivative

$\psi^{\prime}(u)=\frac{(u+1)^{3}\big{[}-(5u^{2}+13u)\sqrt{u^{2}+4u}+5u^{3}+23u^{2}+16u-2\big{]}}{2\sqrt{u^{2}+4u}}.$

It is clear that if $5u^{3}+23u^{2}+16u-2<0$ then $\psi^{{}^{\prime}}(u)<0$
and the equation $\psi^{{}^{\prime}}(u)=0$ has no solutions. So it must be
$5u^{3}+23u^{2}+16u-2>0$.

$5u^{3}+23u^{2}+16u-2=5(u+1)\left(u+\frac{9-\sqrt{91}}{5}\right)\left(u+\frac{9+\sqrt{91}}{5}\right)\
\ \Rightarrow\ \ u>\frac{\sqrt{91}-9}{5}.$

We solve the equation $\psi^{{}^{\prime}}(u)=0$ for $u>0$:

$-(5u^{2}+13u)\sqrt{u^{2}+4u}+5u^{3}+23u^{2}+16u-2=0\ \ \Rightarrow\
10u^{3}+41u^{2}-16u+1=0.$

We solve the last equation by the Cardano method:

$u_{1}=\frac{\sqrt{2161}}{15}\cdot\cos\left(\frac{\arccos\left(\frac{-99791}{\sqrt{10091699281}}\right)}{3}\right)-\frac{41}{30}\approx
0.284824838,$
$u_{2}=\frac{\sqrt{2161}}{15}\cdot\cos\left(\frac{\arccos\left(\frac{-99791}{\sqrt{10091699281}}\right)+2\pi}{3}\right)-\frac{41}{30}\approx-4.463483795,$
$u_{3}=\frac{\sqrt{2161}}{15}\cdot\cos\left(\frac{\arccos\left(\frac{-99791}{\sqrt{10091699281}}\right)+4\pi}{3}\right)-\frac{41}{30}\approx
0.078658955.$

Hence, since $u>\frac{\sqrt{91}-9}{5}$ we get the solution $u^{*}=u_{1}$. We
set

$\lambda_{cr}=\psi(u^{*})\approx 2.31.$

Figure 2. Graph of the function $\lambda_{1}(u)$

We note that if $\psi^{\prime\prime}(u)>0$, then each value of $\lambda$
corresponds to only two values of $u$ for $\lambda>\lambda_{cr}$. We therefore
prove that $\psi^{\prime\prime}(u)>0$. Indeed,

$\psi^{\prime\prime}(u)=\frac{2h(u)}{u(u+1)\sqrt{(u^{2}+4u)^{3}}}.$

Here

$h(u)=5u^{8}+56u^{7}+234u^{6}+463u^{5}+460u^{4}+210u^{3}+26u^{2}-u+3-(5u^{2}+11u)\sqrt{(u^{4}+6u^{3}+9u^{2}+4u)^{3}}.$

From the inequality $h(u)>0$ for $u>0$ we obtain

$\big{(}5u^{8}+56u^{7}+234u^{6}+463u^{5}+460u^{4}+210u^{3}+26u^{2}-u+3\big{)}^{2}-\big{(}5u^{2}+11u\big{)}^{2}\big{(}u^{4}+6u^{3}+9u^{2}+4u\big{)}^{3}=$
$=10u^{13}+182u^{12}+1372u^{11}+5505u^{10}+12786u^{9}+17913u^{8}+15564u^{7}+$
$+9186u^{6}+5034u^{5}+3016u^{4}+1208u^{3}+156u^{2}+(u-3)^{2}>0.$

Thus, each value of $\lambda$ corresponds to only two values of $u$ for
$\lambda>\lambda_{cr}$.

This can also be seen by computer analysis, i.e., computer analysis shows that
the equation $f(x,\lambda)=0$ for $\lambda<\lambda_{cr}$ has no positive
solution, at $\lambda=\lambda_{cr}$ has one positive solution and for
$\lambda>\lambda_{cr}$ there are exactly two positive solutions (see Fig. 3).

Figure 3. Graph of the function $f(x,2)$ (dotted line), $f(x,2.3143)$
(continuous line) and $f(x,2.5)$ (dashed line).

Thus, the following statement is true.

Statement 3. Let $k=4$ and $\lambda_{cr}\approx 2.31$. Then the system of
equations (11):

1\. for $0<\lambda<\lambda_{cr}$ has a unique solution $(x,x)$;

2\. for $\lambda=\lambda_{cr}$ has two solutions
$(x,x),(x^{\prime},y^{\prime})$;

3\. for $\lambda>\lambda_{cr}$ has three solutions
$(x,x),(x_{1},y_{1}),(x_{2},y_{2})$.

Remark 5. The measures corresponding to the solution in the Statement 3 for
$x\neq y$ are AGMs (not periodic) and they different from previous AGMs.

The case $k=4$, $m=1$ and $r=1$. In this case from the system of equations (5)
we obtain

$\left\\{\begin{array}[]{ll}x=\frac{1}{1+\lambda x}\cdot\frac{1}{(1+\lambda
y)^{3}},\\\ y=\frac{1}{1+\lambda y}\cdot\frac{1}{(1+\lambda x)^{3}}.\\\
\end{array}\right.$ (12)

From (12) due to (6) we can get

$(x-y)\Big{(}\lambda^{2}xy-1\Big{)}=0.$

Hence $x=y$ or $\lambda^{2}xy=1$. The case $x=y$ corresponds to the only TIGM.

Let $x\neq y$ and $\lambda^{2}xy=1$, i.e., $\lambda x=\frac{1}{\lambda y}$.
After some algebras the system of equations (12) has the form

$\left\\{\begin{array}[]{ll}(1+\lambda x)^{4}-\lambda^{3}x^{2}=0,\\\
(1+\lambda y)^{4}-\lambda^{3}y^{2}=0.\end{array}\right.$ (13)

Obviously, that the roots of the equation $f(x)=(1+\lambda
x)^{4}-\lambda^{3}x^{2}=0$ are also roots of (12). The solutions of the
equations $f(x)=0$ and $f(y)=0$ have the form

$x_{1,2}=\frac{\sqrt{\lambda}-2\pm\sqrt{\lambda-4\sqrt{\lambda}}}{2\lambda},\
\ y_{1,2}=\frac{\sqrt{\lambda}-2\pm\sqrt{\lambda-4\sqrt{\lambda}}}{2\lambda}.$

It is easy to see that $x_{1,2}>0$ ($y_{1,2}>0$) for $\lambda\geq 16$, and
they take complex values for $\lambda<16$. Moreover, $x_{1}=x_{2}$
($y_{1}=y_{2}$) for $\lambda=16$ and it coincides with the only translation-
invariant solution of (12).

By virtue of the equation $\lambda^{2}xy=1$ and Lemma 1, we obtain that in the
case $x\neq y$ the system of equations (12) has solutions of the form $(x,y)$
and $(y,x)$ for $\lambda>\lambda_{cr}=16$, where

$x=x_{1}=\frac{\sqrt{\lambda}-2+\sqrt{\lambda-4\sqrt{\lambda}}}{2\lambda},\ \
y=y_{2}=\frac{\sqrt{\lambda}-2-\sqrt{\lambda-4\sqrt{\lambda}}}{2\lambda},$
$y=x_{2}=\frac{\sqrt{\lambda}-2-\sqrt{\lambda-4\sqrt{\lambda}}}{2\lambda},\ \
x=y_{1}=\frac{\sqrt{\lambda}-2+\sqrt{\lambda-4\sqrt{\lambda}}}{2\lambda}.$

Thus, the following statement holds

Statement 4. Let $k=4$ and $\lambda_{cr}=16$. Then the system of equations
(12):

1\. for $0<\lambda\leq\lambda_{cr}$ has a unique solution $(x,x)$;

2\. for $\lambda>\lambda_{cr}$ has three solutions $(x,x),(x,y),(y,x)$.

Remark 6. The measure corresponding to the solution $(x,y),(y,x)$ in the
Statement 4 are AGMs (not periodic) and they different from previous AGMs.

The case $k=4$, $m=2$ and $r=0$ ($m=0$ and $r=2$). In this case from (5) we
obtain

$\left\\{\begin{array}[]{ll}x=\frac{1}{(1+\lambda
x)^{2}}\cdot\frac{1}{(1+\lambda y)^{2}},\\\ y=\frac{1}{(1+\lambda x)^{4}}.\\\
\end{array}\right.$ (14)

Using (6) from (14) we can get

$(x-y)\Big{(}\lambda^{2}xy-1\Big{)}=0.$

Hence, $x=y$ or $\lambda^{2}xy=1$. The case $x=y$ corresponds to the only
TIGM.

We consider the case $x\neq y$ and $\lambda^{2}xy=1$ $\Big{(}\lambda
x=\frac{1}{\lambda y}\Big{)}$. After some algebras (14) has the form

$\left\\{\begin{array}[]{ll}(1+\lambda x)^{4}-\lambda^{2}x=0,\\\ (1+\lambda
y)^{4}-\lambda^{4}y^{3}=0.\end{array}\right.$ (15)

From the equation $\lambda^{2}xy=1$ we find $y$ and substitute it for the
second equation (15). Then

$\left\\{\begin{array}[]{ll}(1+\lambda x)^{4}-\lambda^{2}x=0,\\\
\frac{(1+\lambda x)^{4}-\lambda^{2}x}{\lambda^{4}x^{4}}=0.\\\
\end{array}\right.$

Let’s rewrite the equation $f(x)=(1+\lambda x)^{4}-\lambda^{2}x=0$ as

$\lambda^{4}x^{4}+4\lambda^{3}x^{3}+6\lambda^{2}x^{2}+\lambda(4-\lambda)x+1=0.$

We solve the last equation by the Ferrari method from linear algebra. We
introduce the notation $x=t-\frac{1}{\lambda}$. Then

$f\left(t-\frac{1}{\lambda}\right)=\lambda^{4}t^{4}-\lambda^{2}t+\lambda=(\lambda^{2}t^{2}+p)^{2}-2\lambda^{2}p\Big{(}t+\frac{1}{4p}\Big{)}^{2}=$
$=\left(\lambda^{2}t^{2}+p-\lambda\sqrt{2p}\Big{(}t+\frac{1}{4p}\Big{)}\right)\left(\lambda^{2}t^{2}+p+\lambda\sqrt{2p}\Big{(}t+\frac{1}{4p}\Big{)}\right)=0,$

where

$p=\frac{\sqrt[3]{108\lambda^{2}+12\sqrt{81\lambda^{4}-768\lambda^{3}}}}{12}+\frac{4\lambda}{\sqrt[3]{108\lambda^{2}+12\sqrt{81\lambda^{4}-768\lambda^{3}}}}.$

Solutions have the following form

$t_{1,2}=\frac{\sqrt{2p^{3}}\pm\sqrt{\sqrt{2p^{3}}\lambda-2p^{3}}}{2\lambda
p},\ \
t_{3,4}=\frac{-\sqrt{2p^{3}}\pm\sqrt{-\sqrt{2p^{3}}\lambda-2p^{3}}}{2\lambda
p}.$

By virtue $x=t-\frac{1}{\lambda}$, for solutions we obtain

$x_{1,2}=\frac{\sqrt{2p^{3}}\pm\sqrt{\sqrt{2p^{4}}\lambda-2p^{3}}-2p}{2\lambda
p},\ \
x_{3,4}=\frac{-\sqrt{2p^{3}}\pm\sqrt{-\sqrt{2p^{3}}\lambda-2p^{3}}-2p}{2\lambda
p}.$

Computer analysis shows that $x_{1,2}>0$ for $\lambda>\lambda_{cr}\approx
9.48$, and values $x_{3,4}$ are negative or take on complex values for
$\lambda>0$ (see Fig. 4).

Figure 4. a) Graph of the function $x_{1}(\lambda)$ at $\lambda\in[9.4;12]$,
b) Graph of the function $x_{2}(\lambda)$ at $\lambda\in[9.4;20]$.

Values $y_{1}$ and $y_{2}$ corresponding to values $x_{1}$ and $x_{2}$ have
the form:

$y_{1,2}=\frac{2p}{\lambda\Big{(}\sqrt{2p^{3}}\pm\sqrt{\sqrt{2p^{3}}\lambda-2p^{3}}-2p\Big{)}}.$

For $\lambda=\lambda_{cr}\approx 9.4815$ the system of equations (15) has
solutions of the form

$x^{\prime}=\frac{\sqrt{2p^{3}}-2p}{2\lambda p},\ \
y^{\prime}=\frac{2p}{\lambda\sqrt{2p^{3}}-2p}.$

Thus, the following statement is true.

Statement 5. Let $k=4$ and $\lambda_{cr}\approx 9.48$. Then the system of
equations (15):

1\. for $0<\lambda<\lambda_{cr}$ has a unique solution $(x,x)$;

2\. for $\lambda=\lambda_{cr}$ has two solutions
$(x,x),(x^{\prime},y^{\prime})$;

3\. for $\lambda>\lambda_{cr}$ has three solutions
$(x,x),(x_{1},y_{1}),(x_{2},y_{2})$.

By using all propositions, we get the following theorem.

Theorem 4. Let $k=4$ and $r+m\leq 2$. For the HC-model the following
statements are true:

1\. If $m=1$ and $r=0$ or $m=0$ and $r=1$, then there exists
$\lambda_{cr}\approx 2.31$ such that for $0<\lambda<\lambda_{cr}$ there is a
unique AGM which coincides with the only TIGM $\mu_{0}$, for
$\lambda=\lambda_{cr}$ there are exactly two AGMs $\mu_{0}$ and
$\mu^{\prime}$, where $\mu^{\prime}$ is AGM (not TI) and for
$\lambda>\lambda_{cr}$ there are exactly three AGMs $\mu_{0}$, $\mu_{1}$ and
$\mu_{2}$, where $\mu_{1}$ and $\mu_{2}$ are AGMs (not TI).

2\. If $m=1$ and $r=1$ then there exists $\lambda_{cr}=16$ such that for
$0<\lambda\leq\lambda_{cr}$ there is a unique AGM which coincides with the
only TIGM $\mu_{0}$, for $\lambda>\lambda_{cr}$ there are exactly three AGMs
$\mu_{0}$, $\mu_{1}$ and $\mu_{2}$, where $\mu_{1}$ and $\mu_{2}$ are AGMs
(not TI).

3\. If $m=2$ and $r=0$ or $m=0$ and $r=2$, then there exists
$\lambda_{cr}\approx 9.48$ such that for $0<\lambda<\lambda_{cr}$ there is a
unique AGM which coincides with the only TIGM $\mu_{0}$, for
$\lambda=\lambda_{cr}$ there are exactly two AGMs $\mu_{0}$ and
$\mu^{\prime}$, where $\mu^{\prime}$ is AGM (not TI) and for
$\lambda>\lambda_{cr}$ there are exactly three AGMs $\mu_{0}$, $\mu_{1}$ and
$\mu_{2}$, where $\mu_{1}$ and $\mu_{2}$ are AGMs (not TI).

##  3\. The case $m+r\leq k-2$ ($m=r$)

The following lemma is known.

Lemma 2. [20] Let $f:[0,1]\rightarrow[0,1]$ be a continuous function with a
fixed point $\xi\in(0,1)$. We assume that $f$ is differentiable at $\xi$ and
$f^{{}^{\prime}}(\xi)<-1.$ Then there exist points $x_{0}$ and $x_{1}$, $0\leq
x_{0}<\xi<x_{1}\leq 1,$ such that $f(x_{0})=x_{1}$ and $f(x_{1})=x_{0}.$

For $m=r$ by (5) we obtain

$\begin{cases}x=\frac{1}{(1+\lambda x)^{m}}\cdot\frac{1}{(1+\lambda
y)^{k-m}};\\\ y=\frac{1}{(1+\lambda y)^{m}}\cdot\frac{1}{(1+\lambda
x)^{k-m}}.\end{cases}$ (16)

Here $x,y\in(0;1)$. After some transformations from (16) we obtain the
following system of equations:

$\begin{cases}y=f(x);\\\ x=f(y),\end{cases}$ (17)

where

$f(x)=\frac{1}{\lambda}\cdot\left(\frac{1}{x(1+\lambda
x)^{m}}\right)^{\frac{1}{k-m}}-\frac{1}{\lambda}.$

From (17) we get the equation $f(f(x))=x$.

First, we consider the equation $f(x)=x$. The function $f(x)$ is
differentiable and decreasing for $0<x<1$:

$f^{\prime}(x)=-\frac{1+\lambda(m+1)x}{\lambda(k-m)x^{\frac{k-m+1}{k-m}}(1+\lambda
x)^{\frac{k}{k-m}}}<0.$

We rewrite the equation $f(x)=x$:

$x=\frac{1}{\lambda}\cdot\left(\frac{1}{x(1+\lambda
x)^{m}}\right)^{\frac{1}{k-m}}-\frac{1}{\lambda}\ \ \Rightarrow\ \ (1+\lambda
x)^{k}=\frac{1}{x}.$

It is known from [13] that the last equation has a unique solution
$\tilde{x}$, i.e., the equation $f(x)=x$ has a unique solution $\tilde{x}$.

We solve the inequality $f^{\prime}(\tilde{x})<-1$:

$-\frac{1+\lambda(m+1)\tilde{x}}{\lambda(k-m)\tilde{x}^{\frac{k-m+1}{k-m}}(1+\lambda\tilde{x})^{\frac{k}{k-m}}}<-1\
\ \Rightarrow\ \ \frac{1+(m+1)\lambda\tilde{x}}{\lambda(k-m)\tilde{x}}>1\ \
\Rightarrow\ \tilde{x}<\frac{1}{\lambda(k-2m-1)}.$

Then from $f(\tilde{x})=\tilde{x}$ we get

$\left(1+\frac{1}{k-2m-1}\right)^{k}<\lambda(k-2m-1)\ \ \Rightarrow\ \
\lambda>\lambda_{cr}=\left(\frac{k-2m}{k-2m-1}\right)^{k}\cdot\frac{1}{k-2m-1}.$

Hence, by Lemma 1 and Lemma 2 the system of equations (16) for
$\lambda>\lambda_{cr}$ has at least three positive solutions $(x,y),\
(\tilde{x},\tilde{x}),\ (y,x)$, where $x\neq y$.

Thus, the following theorem is true.

Theorem 5. Let $k\geq 2$, $m+r\leq k-2$ ($m=r$) and
$\lambda_{cr}=\left(\frac{k-2m}{k-2m-1}\right)^{k}\cdot\frac{1}{k-2m-1}$. Then
for the HC-model for $\lambda>\lambda_{cr}$ there are at least three Gibbs
measures one of which is TI and the other are AGMs (not TI).

###  3.1. The case $m+r=k-2$, $k\geq 2$

In the case $m+r=k-2$ $(n=2)$, the system of equations (5) has the form:

$\left\\{\begin{array}[]{ll}x=\frac{1}{(1+\lambda
x)^{m}}\cdot\frac{1}{(1+\lambda y)^{k-m}};\\\ y=\frac{1}{(1+\lambda
y)^{k-m-2}}\cdot\frac{1}{(1+\lambda x)^{m+2}}.\\\ \end{array}\right.$ (18)

From the system of equations (18) due to (6) we can get

$(x-y)\Big{(}\lambda^{2}xy-1\Big{)}=0.$

Hence, $x=y$ or $\lambda^{2}xy=1$. The case $x=y$ has already been considered.

Let $\lambda^{2}xy=1$. Then $\lambda x=\frac{1}{\lambda y}$ for $x\neq y$. By
virtue (18), after some algebras, we can obtain the system of equations

$\left\\{\begin{array}[]{ll}x=\frac{(\lambda x)^{k-m}}{(1+\lambda
x)^{k}},\\\\[5.69054pt] y=\frac{(\lambda y)^{m+2}}{(1+\lambda
y)^{k}},\end{array}\right.$

which is equivalent to the system of equations:

$\left\\{\begin{array}[]{ll}(1+\lambda x)^{k}-\lambda^{k-m}x^{k-m-1}=0,\\\
(1+\lambda y)^{k}-\lambda^{m+2}y^{m+1}=0.\end{array}\right.$ (19)

From the equation $\lambda^{2}xy=1$ we find $y$ and substitute it for the
second equation of the system equations (19). Then

$\left\\{\begin{array}[]{ll}(1+\lambda
x)^{k}-\lambda^{k-m}x^{k-m-1}=0,\\\\[5.69054pt] \frac{(1+\lambda
x)^{k}-\lambda^{k-m}x^{k-m-1}}{\lambda^{k}x^{k}}=0.\end{array}\right.$

We consider the function

$f(x)=(1+\lambda x)^{k}-\lambda^{k-m}x^{k-m-1}.$

Obviously, the roots of the equation $f(x)=0$ are also roots of (19). Let’s
rewrite $f(x)$ as a polynomial:

$f(x)=\lambda^{k}x^{k}+C_{k}^{1}\lambda^{k-1}x^{k-1}+\cdots+C_{k}^{m+1}\lambda^{k-m-1}x^{k-m-1}+\cdots+C_{k}^{k-1}\lambda
x+1-\lambda^{k-m}x^{k-m-1}$

or

$f(x)=\lambda^{k}x^{k}+C_{k}^{1}\lambda^{k-1}x^{k-1}+\cdots+(C_{k}^{m+1}-\lambda)\lambda^{k-m-1}x^{k-m-1}+\cdots+C_{k}^{k-1}\lambda
x+1.$

If $\lambda<C_{k}^{m+1}$, then $f(x)=0$ has no positive solutions, if
$\lambda>C_{k}^{m+1}$, then number of sign changes of the first equation of
the last equality is two. Due to the Descartes’ theorem, the equation $f(x)=0$
has at most two positive solutions.

On the other hand, it is easy to see that $0<x<1$, $f(0)=1$ and
$f(1)=(1+\lambda)^{k}-\lambda^{k-m}>0.$ Moreover,
$f\Big{(}\frac{1}{\lambda}\Big{)}=2^{k}-\lambda<0$, if $\lambda>2^{k}.$

It follows from the above that there exists
$\lambda_{cr}:C_{k}^{m+1}<\lambda_{cr}\leq 2^{k}$ such that for
$\lambda>\lambda_{cr}$ the equation $f(x)=0$ has two positive solutions, for
$\lambda=\lambda_{cr}$ has twice multiplicity positive solution and
$\lambda<\lambda_{cr}$ has no positive solution.

When $m=r$, the system of equations (18) can be written as

$\left\\{\begin{array}[]{ll}x=\frac{1}{(1+\lambda
x)^{m}}\cdot\frac{1}{(1+\lambda y)^{k-m}};\\\ y=\frac{1}{(1+\lambda
y)^{m}}\cdot\frac{1}{(1+\lambda x)^{k-m}}.\\\ \end{array}\right.$ (20)

It follows from the Lemma 1 that if the number of solutions of the equation
$x=f(x)$ is odd or even, then the number of solutions of $x=f(f(x))$ is also
respectively odd or even.

As a result, when $m=r$ and $\lambda=\lambda_{cr}$, the number of solutions of
the system of equations (20) cannot be even. Because the TI solution was
unique. It follows that the solution of the system of equations (20)
corresponding to $\lambda=\lambda_{cr}$ coincides with the translation-
invariant solution.

Thus, we have proved the following theorem.

Theorem 6. Let $k\geq 2$ and $r+m=k-2$. Then for the HC-model there exists
$\lambda_{cr}$ such that next statements are true:

1\. for $0<\lambda<\lambda_{cr}$ there is a unique AGM and it coincides with
the only TIGM $\mu_{0}$;

2\. if $m=r$ and $\lambda=\lambda_{cr}$ then there is a unique AGM and it
coincides with the only TIGM $\mu_{0}$;

3\. if $m\neq r$ and $\lambda=\lambda_{cr}$ there is at least one AGM;

4\. for $\lambda>\lambda_{cr}$ there are exactly three Gibbs measures
$\mu_{0}$, $\mu_{1}$ and $\mu_{2}$, where $\mu_{1}$ and $\mu_{2}$ are AGMs
(not TI).

##  4\. Relation of the Alternative Gibbs measures to known ones

Translation invariant measures. (see [13]) Such measures correspond to
$z_{x}\equiv z$, i.e. constant functions. These measures are particular cases
of our measures mentioned which can be obtained for $m=k$, i.e. $k-m=0$. In
this case the condition (3) reads

$z=\frac{1}{(1+\lambda z)^{k}}.$ (21)

The equation (21) has a unique solution for all $\lambda>0.$

Bleher-Ganikhodjaev construction. Consider an infinite path
$\pi=\\{x^{0}=x_{0}<x_{1}<...\\}$ on the half Cayley tree (the notation $x<y$
meaning that paths from the root to $y$ go through $x$). Associate to this
path a collection $z^{\pi}$ of numbers given by the condition

$z^{\pi}_{x}=\begin{cases}l\leavevmode\nobreak\ \mbox{if}\leavevmode\nobreak\
x\prec x_{n},\leavevmode\nobreak\ x\in W_{n},\\\ h,\leavevmode\nobreak\
\mbox{if}\leavevmode\nobreak\ x_{n}\prec x,\leavevmode\nobreak\ x\in W_{n},\\\
h,\leavevmode\nobreak\ \mbox{if}\leavevmode\nobreak\ x=x_{n}.\end{cases}$

$n=1,2,...$ where $x\prec x_{n}$ (resp. $x_{n}\prec x$) means that $x$ is on
the left (resp. right) from the path $\pi$ and $z_{x_{n}}\in\\{h,l\\}$ are
arbitrary numbers. For any infinite path $\pi$, the collection of numbers
$z^{\pi}$ satisfying relations (3) exists and is unique (see Fig. 5).

$h$$h$$h$$h$$h$$h$$l$$h$$h$$h$$h$$h$$l$$h$$h$$h$$h$$h$$l$$h$$h$$h$$h$$h$$l$$l$$h$$h$$h$$h$$l$

Figure 5. In this figure the values of function $z_{x}$ on the vertices of the
Cayley tree of order 5 are shown. This is the case when $m=4$ and $r=4$.

Periodic Gibbs measures. (see [13]) Let $G_{k}$ be a free product of $k+1$
cyclic groups of the second order with generators $a_{1},a_{2},...,a_{k+1},$
respectively.

It is known that there exists an one-to-one correspondence between the set of
vertices $V$ of the Cayley tree $\Im^{k}$ and the group $G_{k}$.

Definition 6. Let $\widehat{G}$ be a normal subgroup of the group $G_{k}$. The
set $z=\\{z_{x},x\in G_{k}\\}$ is said to be $\widehat{G}$ -periodic if
$z_{yx}=z_{x}$ for $\forall x\in G_{k},y\in\widehat{G}_{k}.$

Let $G^{(2)}_{k}=\\{x\in G_{k}:\mbox{the length of word}\leavevmode\nobreak\
x\leavevmode\nobreak\ \mbox{is even}\\}.$ Note that $G^{(2)}_{k}$ is the set
of even vertices (i.e. with even distance to the root). Consider the boundary
condition $h$ and $l$:

$z_{x}=\begin{cases}h\leavevmode\nobreak\ \mbox{if}\leavevmode\nobreak\ x\in
G^{(2)}_{k},\\\ l\leavevmode\nobreak\ \mbox{if}\leavevmode\nobreak\ x\in
G_{k}\setminus G^{(2)}_{k}.\end{cases}$

and denote by $\mu_{1}$, $\mu_{2}$ the corresponding Gibbs measures. The
$\widehat{G}$\- periodic solutions of equation (3) are either translation-
invariant ($G_{k}$\- periodic) or $G^{(2)}_{k}$ -periodic, they are solutions
to

$\begin{cases}h=\frac{1}{(1+\lambda l)^{k}},\\\ l=\frac{1}{(1+\lambda
h)^{k}}.\end{cases}$

We note that these measures are particular cases of measures of $\mu_{h,l}$
which can be obtained for $m=r=0$ (See figure 6, for $k=4$).

$h$$l$$h$$h$$h$$h$$l$$h$$h$$h$$h$$l$$h$$h$$h$$h$$l$$h$$h$$h$$h$

Figure 6. In this figure the values of function $z_{x}$ on the vertices of the
Cayley tree of order 4 are shown.

Weakly periodic Gibbs measures. Following [17],[18], [22] recall the notion of
weakly periodic Gibbs measures. Let
$G_{k}/\widehat{G}_{k}=\\{H_{1},...,H_{r}\\}$ be a factor group, where
$\widehat{G}_{k}$ is a normal subgroup of index $r>1$.

Definition 7. A set $z=\\{z_{x},x\in G_{k}\\}$ is called $\widehat{G}_{k}$ \-
weakly periodic, if $z_{x}=z_{ij}$, for any $x\in H_{i}$, $x_{\downarrow}\in
H_{j}$, where $x_{\downarrow}$ denotes the ancestor of $x$.

We recall results known for the cases of index two. Note that any such
subgroup has the form

$H_{A}=\Big{\\{}x\in G_{k}:\sum_{i\in A}{w_{x}(a_{i})}\leavevmode\nobreak\
\mbox{is even}\Big{\\}}$

where ${\emptyset}\neq A\subseteq N_{k}=\\{1,2,...,k+1\\}$, and $w_{x}(a_{i})$
is the number of $a_{i}$ in a word $x\in G_{k}.$ We consider $A\neq N_{k}$:
when $A=N_{k}$ weak periodicity coincides with standard periodicity. Let
$G_{k}/H_{A}=\\{H_{0},H_{1}\\}$ be the factor group, where $H_{0}=H_{A}$,
$H_{1}=G_{k}\setminus H_{A}$. Then, in view of (3), the $H_{A}$ \- weakly
periodic b.c. has the form

$z_{x}=\begin{cases}z_{1},\ x\in H_{A},\ x_{\downarrow}\in H_{A},\\\ z_{2},\
x\in H_{A},\ x_{\downarrow}\in G_{k}\setminus H_{A},\\\ z_{3},\ x\in
G_{k}\setminus H_{A},\ x_{\downarrow}\in H_{A},\\\ z_{4},\ x\in G_{k}\setminus
H_{A},\ x_{\downarrow}\in G_{k}\setminus H_{A}.\end{cases}$

where the $h_{i}$ satisfy the following equations:

$z_{1}=\frac{1}{\Big{(}1+\lambda z_{3}\Big{)}^{i}}\frac{1}{\Big{(}1+\lambda
z_{1}\Big{)}^{k-i}},\ \ \ z_{2}=\frac{1}{\Big{(}1+\lambda
z_{3}\Big{)}^{i-1}}\frac{1}{\Big{(}1+\lambda z_{1}\Big{)}^{k-i+1}},$
$z_{3}=\frac{1}{\Big{(}1+\lambda z_{2}\Big{)}^{i-1}}\frac{1}{\Big{(}1+\lambda
z_{4}\Big{)}^{k-i+1}},\ \ z_{4}=\frac{1}{\Big{(}1+\lambda
z_{2}\Big{)}^{i}}\frac{1}{\Big{(}1+\lambda z_{4}\Big{)}^{k-i}}.$ (22)

It is obvious that the following sets are invariant with respect to the
operator $W:R^{4}\rightarrow R^{4}$ defined by RHS of (22):

$I_{1}=\Big{\\{}z\in R^{4}:z_{1}=z_{2}=z_{3}=z_{4}\Big{\\}},\ \ \ \
I_{2}=\Big{\\{}z\in R^{4}:z_{1}=z_{4};z_{2}=z_{3}\Big{\\}}$

It is obvious to see that

$\bullet$ measures corresponding to solutions on $I_{1}$ are translation
invariant

$\bullet$ measures corresponding to solutions on $I_{2}$ are weakly periodic,
which coincide with the measures given for $m=k-i$, $k-m=i$, $r=i-1$,
$k-r=k-i+1$.

##  5\. Free energy

In this section, we consider free energy of _HC_ -model Gibbs measure. In
fact, Gibbs measures give the probability of the system $X$ being in state
$x\in X$ (equivalently, of the random variable $X$ having value $x$) as

$\mu(X=x)=\frac{1}{Z(\beta)}\exp(-\beta H(x)),$

where $H(x)$ is a function from the space of states to the real numbers. The
parameter $\beta$ is (a free parameter) the inverse temperature. The
normalizing constant $Z(\beta)$ is the partition function.

Consider an infinite graph $G$, and let $\Lambda\subset G$ be finite subset.
It is convinient to work with reduced free energy $f=-\beta F$, which per unit
volume is

$f(\beta,\Lambda)=\frac{1}{|\Lambda|}\ln Z(\beta,\Lambda),$

where $Z(\beta,\Lambda)$ is the restiriction of the partition function
$Z(\beta)$ on the set $\Lambda$, by fixing the state of the system outside of
$\Lambda$.

Note that by Theorem 6, we can construct Aternating Gibbs measures and by
using these measures we compute the free energy for such measures. From [31,
30] it’s known that the free energy of a compatible boundary condition (b.c.)
is defined as the limit:

$F(h)=-\lim_{n\to\infty}\frac{1}{\beta|V_{n}|}\ln Z_{n}$ (23)

if it exists. Here $|\cdot|$ denotes the cardinality of a set and $Z_{n}$ is a
partition function. We recall that in our case:

$Z_{n}=\sum_{{\widetilde{\sigma}}_{n}\in\Omega_{V_{n}}}\lambda^{\\#{\widetilde{\sigma}}_{n}}\prod_{x\in
W_{n}}z_{{\widetilde{\sigma}}(x),x}.$ (24)

We consider ALT Gibbs measures on the half tree and from above, the family of
probability measures ce compatible iff $z=\\{z_{x},x\in G_{k}\\}$ satisfies
the equality (3). Also, we considered a special class of $z=\\{z_{x},x\in
G_{k}\\}$ such that:

$\bullet$ if at vertex $x$ we have $z_{x}=h$, then the function $z_{y}$, which
gives real values to each vertex $y\in S(x)$ by the following rule

$\begin{cases}h\leavevmode\nobreak\ \mbox{on}\leavevmode\nobreak\
m\leavevmode\nobreak\ \mbox{vertices of}\leavevmode\nobreak\ S(x),\\\
l\leavevmode\nobreak\ \mbox{on}\leavevmode\nobreak\ k-m\leavevmode\nobreak\
\mbox{remaining vertices,}\end{cases}$

$\bullet$ if at vertex $x$ we have $z_{x}=l$, then the function $z_{y}$, which
gives real values to each vertex $y\in S(x)$ by the following rule

$\begin{cases}l\leavevmode\nobreak\ \mbox{on}\leavevmode\nobreak\
r\leavevmode\nobreak\ \mbox{vertices of}\leavevmode\nobreak\ S(x),\\\
h\leavevmode\nobreak\ \mbox{on}\leavevmode\nobreak\ k-r\leavevmode\nobreak\
\mbox{remaining vertices.}\end{cases}$

Denote

$\alpha_{n}=|\\{x\in W_{n}:z_{x}=h\\}|;\ \ \beta_{n}=|\\{x\in
W_{n}:z_{x}=l\\}|.$ (25)

Recall that $W_{n}$ is the sphere with the center $x^{0}$ and radius $n$ on
the half tree.

Consequently, the following recurrence system holds

$\left\\{\begin{array}[]{lllll}\alpha_{n+1}=m\alpha_{n}+(k-r)\beta_{n}\\\\[8.53581pt]
\beta_{n+1}=(k-m)\alpha_{n}+r\beta_{n}.\end{array}\right.$ (26)

Denoting $\varphi_{n}=\alpha_{n}+\beta_{n}$, from (26) one gets

$\varphi_{n+1}=k\varphi_{n}\ \Rightarrow\ \varphi_{n}=k^{n},\ n\in\mathbb{N}.$
(27)

Since $\alpha_{n}=k^{n}-\beta_{n}$, we get

$k^{n+1}-mk^{n}=(k-m-r)\beta_{n}+\beta_{n+1}.$ (28)

Put

$\beta_{n}=\frac{(k-m)k^{n}}{2k-m-r}+k^{n}\phi_{n}.$

Then the last equation can be written as

$(m+r-k)\phi_{n}=k\phi_{n+1}.$

After short calculations, we obtain

$\phi_{n}=\frac{\beta_{1}(2k-m-r)-k(k-m)}{k}\left(\frac{m+r-k}{k}\right)^{n-1}.$

Hence,

$\beta_{n}=\frac{(k-m)k^{n}}{2k-m-r}+\left(\beta_{1}(2k-m-r)-k(k-m)\right)\left(m+r-k\right)^{n-1}.$

Thus

$\beta_{1}=\frac{(k-m)k}{2k-m-r}+\left(\beta_{1}(2k-m-r)-k(k-m)\right)\ \
\Rightarrow\ \ \beta_{1}=\frac{(k-m)k}{2k-m-r}.$

Then

$\beta_{n}=\frac{(k-m)k^{n}}{2k-m-r}.$

Note that $\alpha_{n}+\beta_{n}=k^{n}$, then

$\alpha_{n}=\frac{k^{n}(k-r)}{2k-r-m}+\left(k(k-m)-\beta_{1}(2k-m-r)\right)\left(m+r-k\right)^{n-1}.$

Since

$\beta_{1}=\frac{(k-m)k}{2k-m-r},$

one gets

$\alpha_{n}=\frac{k^{n}(k-r)}{2k-r-m}.$

Consequently, it is easy to check that

$\lim_{n\to\infty}\frac{(k-1)\alpha_{n}}{k^{n+1}-1}=\frac{(k-1)(k-r)}{k(2k-m-r)}$

and

$\lim_{n\to\infty}\frac{(k-1)\beta_{n}}{k^{n+1}-1}=\frac{(k-1)(k-m)}{k(2k-m-r)}.$
(29)

Then

$F_{ALT}(h)=-\frac{1}{\beta}\cdot\left[\frac{(k-1)(k-r)\ln
h-(k^{2}-(m+1)k+m)\ln
l}{k(2k-m-r)}+\lim_{n\to\infty}\frac{(k-1)\ln\left(\sum_{i=0}^{|V_{n}|}\lambda^{C_{|V_{n}|}^{i}}\right)}{k^{n+1}-1}\right].$
(30)

By AM-GM inequality

$\sum_{i=0}^{|V_{n}|}\lambda^{C_{|V_{n}|}^{i}}\geq|V_{n}|\cdot\sqrt[|V_{n}|]{\lambda^{2^{|V_{n}|}}}=|V_{n}|\cdot\lambda^{2^{|V_{n}|}\cdot|V_{n}|^{-1}}.$

Since $\ln x$ is an increasing function

$\ln\left(\sum_{i=0}^{|V_{n}|}\lambda^{C_{|V_{n}|}^{i}}\right)\geq\ln|V_{n}|+{2^{|V_{n}|}\cdot|V_{n}|^{-1}}\ln\lambda.$

Then

$\lim_{n\to\infty}\frac{(k-1)\ln\left(\sum_{i=0}^{|V_{n}|}\lambda^{C_{|V_{n}|}^{i}}\right)}{k^{n+1}-1}\geq\lim_{n\to\infty}\frac{\ln|V_{n}|}{|V_{n}|}+\lim_{n\to\infty}{2^{|V_{n}|}\cdot|V_{n}|^{-2}}\ln\lambda.$

If $\lambda>1$ then

$\lim_{n\to\infty}\frac{(k-1)\ln\left(\sum_{i=0}^{|V_{n}|}\lambda^{C_{|V_{n}|}^{i}}\right)}{k^{n+1}-1}=\infty.$
(31)

By (29), (30) and (31) one gets

$F_{ALT}(h)=-\lim_{n\to\infty}\frac{1}{\beta|V_{n}|}\ln\left[h^{\alpha_{n}}l^{\beta_{n}}\left(\sum_{i=0}^{|V_{n}|}\lambda^{C_{|V_{n}|}^{i}}\right)\right]=-\infty.$
(32)

Also, if $\lambda\in(0,1]$ then

$0\leq\lim_{n\to\infty}\frac{(k-1)\ln\left(\sum_{i=0}^{|V_{n}|}\lambda^{C_{|V_{n}|}^{i}}\right)}{k^{n+1}-1}\leq\lim_{n\to\infty}\frac{\ln|V_{n}|}{|V_{n}|}=\lim_{n\to\infty}\ln\sqrt[|V_{n}|]{|V_{n}|}=0.$

Namely, from (30)

$F_{ALT}(h)=-\frac{1}{\beta}\cdot\left[\frac{(k-1)(k-r)\ln
h-(k^{2}-(m+1)k+m)\ln l}{k(2k-m-r)}\right].$

Hence from above results and by Theorem 5 and Theorem 6 we can conclude the
following theorem.

Theorem 7. a) Let $k\geq 2$, $m+r\leq k-2$ ($m=r$) and
$\lambda^{(1)}_{cr}=\left(\frac{k-2m}{k-2m-1}\right)^{k}\cdot\frac{1}{k-2m-1}$.
Then the following statements are true

  * •

if $\lambda_{cr}^{(1)}\in(0,1]$ and $\lambda\in[\lambda_{cr}^{(1)},1]$ (resp.
$\lambda\in(1,+\infty)$) then free energies $F_{ALT}$ of b.c (3) is equal to

$-\frac{1}{\beta}\cdot\left[\frac{(k-1)(k-r)\ln h-(k^{2}-(m+1)k+m)\ln
l}{k(2k-m-r)}\right](\textrm{resp.}\ -\infty).$

  * •

if $\lambda_{cr}^{(1)}\in(1,\infty)$ then free energies $F_{ALT}$ equals
$-\infty$.

b) Let $k\geq 2$, $r+m=k-2$ and $C_{k}^{m+1}\lambda_{cr}\leq 2^{k}$. Then the
following statements hold:

  * •

if $m\neq r$ and $\lambda=\lambda_{cr}$ then free energies $F_{ALT}$ equals
$-\infty$. Also, if $\lambda>\lambda_{cr}$ then free energies $F_{ALT}$ equals
$-\infty$.

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# Anomalous Optical Drag

Chitram Banerjee  Department of Physics of Complex Systems, Weizmann Institute
of Science, Rehovot 76100, Israel  Yakov Solomons  Department of Physics of
Complex Systems, Weizmann Institute of Science, Rehovot 76100, Israel  A.
Nicholas Black  Department of Physics and Astronomy, University of Rochester,
Rochester, NY 14627, USA  Giulia Marcucci  Department of Physics, University
of Ottawa, Ottawa, ON K1N 6N5, Canada  David Eger  Department of Physics of
Complex Systems, Weizmann Institute of Science, Rehovot 76100, Israel  Nir
Davidson  Department of Physics of Complex Systems, Weizmann Institute of
Science, Rehovot 76100, Israel  Ofer Firstenberg  Department of Physics of
Complex Systems, Weizmann Institute of Science, Rehovot 76100, Israel  Robert
W. Boyd  Institute of Optics, University of Rochester, Rochester, NY 14627,
USA  Department of Physics, University of Ottawa, Ottawa, ON K1N 6N5, Canada

###### Abstract

A moving dielectric medium can displace the optical path of light passing
through it, a phenomenon known as the Fresnel-Fizeau optical drag effect. The
resulting displacement is proportional to the medium’s velocity. In this
article, we report on an anomalous optical drag effect, where the displacement
is still proportional to the medium’s speed but along the direction opposite
to the medium’s movement. We conduct an optical drag experiment under
conditions of electromagnetically-induced transparency and observe the
transition from normal, to null, to anomalous optical drag by modification of
the two-photon detuning.

##  I Introduction

As light travels through a moving medium, it is dragged along the axis of the
medium’s motion. This effect, known as optical drag, was first described
theoretically by Augustin-Jean Fresnel in 1818 [1, 2]. In 1851, Hippolyte
Fizeau demonstrated Fresnel’s drag experimentally [3]. However, Fresnel and
Fizeau ignored the effect of refractive index dispersion. This effect was
incorporated by Hendrik Lorentz, who in 1904 predicted the influence of
dispersion on the optical drag effect for a moving medium with a fixed
boundary [4]. Four years later, Jakob Laub developed a theoretical treatment
for optical drag in a dispersive moving medium with a moving boundary [5].
Many experiments that measured the effect of dispersion on optical drag were
then reported by Zeeman and coauthors [6, 7, 8, 9, 10]. They measured the
wavelength dependence of optical drag in water [6, 7] and performed other
experiments to measure optical drag in quartz and flint glass [8, 9, 10].

The transverse displacement of a light beam experiencing optical drag depends
on the both the refractive and group index of the beam in the moving medium
[11]. It is well known that materials can be highly dispersive under nearly-
resonant excitation [12]. This large dispersion leads to a large group index,
to highly subluminal pulse propagation (sometimes referred to as slow light),
and consequently to an enhancement of the optical drag effect [13, 14]. Since
absorption is typically high under nearly-resonant excitation, most
observations of slow light have relied upon electromagnetically-induced
transparency (EIT) [15, 16] or coherent population oscillation [17, 18, 19].
These effects have the added benefit of providing very narrow resonances,
leading to high dispersion and large group indices. As a result, EIT has been
used to further enhance the optical drag effect [20] and perform high
precision velocimetry [21].

Under conditions of anomalous dispersion, the group index can become negative,
leading to the so-called “fast light” effect, where the peak of a pulse
advances as it travels through a dispersive medium instead of accumulating a
delay [22]. Under certain circumstances, the advancement of the peak can be
attributed to pulse reshaping caused by saturated absorption or gain [23, 24].

An experimental demonstration of optical drag under conditions of anomalous
dispersion has not been previously reported. In this article, we report on the
anomalous optical drag effect, where the light beam shifts in the direction
opposite to the motion of the medium. We excite EIT in a moving cell of
rubidium vapor and induce a transition from normal to anomalous optical drag
by properly modifying the two-photon detuning.

##  II Theoretical Model

The transverse Fresnel-Fizeau drag effect in a generic dispersive medium has
been treated theoretically by starting from the linearized Lorentz
transformation for the frequency and wave vector of a beam propagating inside
a moving medium [11]. Let us consider a nonmagnetic and isotropic medium
(i.e., relative magnetic permeability $\mu=1$ and dielectric tensor
$\epsilon_{ij}=\epsilon\delta_{ij}$), with $\epsilon_{r}=\Re[\epsilon]$. As
shown in Fig. 1(a), if the medium is moving with velocity
$\boldsymbol{\mathrm{v_{0}}}=\mathrm{v_{0}\hat{\boldsymbol{x}}}$, where
$\mathrm{z}$ is the longitudinal and $\mathrm{x,y}$ are the transverse
coordinates, the probe light beam experiences a shift along $\mathrm{x}$ of

$\mathrm{\Delta x}=\mathrm{L}\tan\theta.$ (1)

In Eq. (1), $\mathrm{L}$ is the medium’s longitudinal length, and $\theta$ the
light’s walk-off angle inside the medium, with

$\tan\theta=\frac{\mathrm{v_{0}}}{\mathrm{c}}\left(\frac{\mathrm{c}}{\mathrm{v_{g}}}-\frac{\mathrm{v_{p}}}{\mathrm{c}}\right),$
(2)

where $\mathrm{c}$ is the speed of light in vacuum [11]. The group velocity
and phase velocity of light in the medium are given by

$\mathrm{v_{g}}=\frac{\mathrm{\mathrm{c}}}{\sqrt{\epsilon_{\mathrm{r}}(\omega_{0})}+\frac{\omega_{0}}{2\sqrt{\epsilon_{\mathrm{r}}(\omega_{0})}}\left(\frac{\mathrm{d}\epsilon_{\mathrm{r}}}{\mathrm{d}\omega}\right)_{\omega_{0}}},$
(3)
$\mathrm{v_{p}}=\frac{\mathrm{c}}{\sqrt{\epsilon_{\mathrm{r}}(\omega_{0})}},$
(4)

respectively.

Figure 1: Pictorial representation of the transverse Fresnel-Fizeau drag
effect for (a) normal optical drag, and (b) anomalous optical drag.

Equation (2) can be expressed in terms of the medium’s refractive index
$\mathrm{n}(\omega)=\sqrt{\epsilon_{\mathrm{r}}(\omega)}$, or after fixing the
value of the carrier frequency, as
$\mathrm{n_{0}}=\sqrt{\epsilon_{\mathrm{r}}(\omega_{0})}$. Equations (3,4) can
then be rewritten as $\mathrm{v_{g}}=\mathrm{c/{n_{g,0}}}$ and
$\mathrm{v_{\mathrm{p}}=c/n_{0}}$, where
$\mathrm{n_{g,0}=n_{0}+\left(\omega\frac{\mathrm{d}n}{\mathrm{d}\omega}\right)_{\omega_{0}}}$
is the group refractive index. The resulting expression,

$\mathrm{\tan\theta=\frac{v_{0}}{c}\left(n_{g,0}-\frac{1}{n_{0}}\right)},$ (5)

reveals the existence of three different regimes:
(i) $\mathrm{n_{g,0}>n_{0}^{-1}}$ normal (or positive) optical drag [Fig.
1(a)];
(ii) $\mathrm{n_{g,0}=n_{0}^{-1}}$ absence of optical drag;
(iii) $\mathrm{n_{g,0}<n_{0}^{-1}}$ anomalous (or negative) optical drag [Fig.
1(b)]. For a beam experiencing normal dispersion, if $\mathrm{v_{p}\approx c}$
and $\mathrm{v_{g}\ll c}$, the drag can be approximated in terms of the group
delay, $\tau$: $\mathrm{\Delta x\approx Lv_{0}/v_{\mathrm{g}}=\tau v_{0}}$
[20]. The conditions of anomalous dispersion, $\mathrm{n_{g,0}<0}$, satisfy
regime (iii), hence the name anomalous drag. In highly dispersive conditions,
the relation $\mathrm{\Delta x\approx\tau v_{0}}$ is still valid because
$\mathrm{n_{g,0}}\ll 0$.

In EIT, the transition from normal to anomalous drag occurs close to the
transition from negative to positive two-photon detuning, $\delta$. To model
optical drag under EIT conditions, we begin from Eq. (50) in [15], which
describes the steady-state propagation of a probe beam under EIT conditions in
a gaseous medium of diffusing atoms. The medium’s velocity,
$\boldsymbol{\mathrm{v}}_{0}$, is accounted for by adding a nonzero mean to
the Boltzmann distribution of atomic velocities with thermal velocity
$\mathrm{v_{\mathrm{th}}=(k_{\mathrm{B}}T/m)^{1/2}}$,

$F(\boldsymbol{\mathrm{v}})=\frac{\exp\left(-\frac{(\boldsymbol{\mathrm{v}}-\boldsymbol{\mathrm{v_{0}}})^{2}}{\mathrm{2v_{\mathrm{th}}^{2}}}\right)}{\mathrm{2\pi
v_{\mathrm{th}}^{2}}}.$ (6)

$\mathrm{k_{\mathrm{B}}}$ is the Boltzmann constant, $\mathrm{T}$ is the
temperature, and $\mathrm{m}$ is the atomic mass. Assuming negligible
diffraction, the dynamics of the probe beam are described by

$\mathrm{\left[-i\delta+\gamma+K\left|\tilde{\Omega}_{2}(\boldsymbol{\mathrm{r}})\right|^{2}-D\left(\partial_{\boldsymbol{\mathrm{r}}}-i\boldsymbol{\mathrm{\Delta
q}}\right)^{2}+\left(\partial_{\boldsymbol{\mathrm{r}}}-i\boldsymbol{\mathrm{\Delta
q}}\right)\cdot\boldsymbol{\mathrm{v_{0}}}\right]\left[\alpha\tilde{\Omega}_{2}(\boldsymbol{\mathrm{r}})\right]^{-1}\left(\partial_{z}+\alpha\right)\tilde{\Omega}_{1}(\boldsymbol{\mathrm{r}})=K\tilde{\Omega}_{2}^{*}(\boldsymbol{\mathrm{r}})\tilde{\Omega}_{1}(\boldsymbol{\mathrm{r}})}.$
(7)

Here, $\tilde{\Omega}_{1,2}$ represent the Rabi frequency envelopes in the
slowly varying envelope approximation for the probe and control beams,
respectively, and $\mathrm{K}$ is the one-photon complex spectrum with total
linewidth $\Gamma$ incorporating homogeneous dephasing and Doppler broadening
of the optical transition. Moreover, $\gamma$ is the decoherence rate of the
ground-state transition, $\mathrm{D}$ is the spatial diffusion coefficient,
$\boldsymbol{\mathrm{\Delta
q}}=\boldsymbol{\mathrm{q}}_{2}-\boldsymbol{\mathrm{q}}_{1}$ is the probe-
control wave vector mismatch (with absolute value $\mathrm{\Delta q}$ along
$\hat{\boldsymbol{\mathrm{x}}}$), and $\alpha$ is the attenuation (per unit
distance) without EIT.

Figure 2: Experimental setup for observing anomalous optical drag under
conditions of EIT. (left) The probe and control fields are derived from a
diode laser ($795\mathrm{\>nm}$) tuned to the $\mathrm{D}_{1}$ transition in
${}^{87}\mathrm{Rb}$. A half-wave plate (HWP) controls the relative power of
the orthogonally polarized probe and control fields before they are split by a
polarizing beamsplitter. The probe passes through an electro-optic modulator
(EOM) and an acousto-optic modulator (AOM) to tune it to the
$\mathrm{F}=1\rightarrow\mathrm{F^{\prime}}=1,2$ transition and vary the two-
photon detuning, $\delta$, respectively. The probe and control are then
recombined at a PBS before being coupled into a single-mode polarization-
maintaining fiber (PMF). After exiting the PMF both fields pass through a
${}^{87}\mathrm{Rb}$ cell that is shielded from stray magnetic fields by a
double-layer magnetic shield ($\mu\text{-metal}$). A solenoid creates a
uniform magnetic field to lift the degeneracy of the Zeeman sublevels. The
probe and control fields interact through an EIT scheme shown in the upper
right panel. The dominant interaction occurs around the
$\mathrm{m_{F}}=0\longrightarrow\mathrm{m^{\prime}_{F}}=\pm 1$ transitions for
the probe (yellow arrows) and control (brown arrows). By detuning both the
probe and control $-250\mathrm{\>MHz}$ below the $\mathrm{F}^{\prime}=2$
excited state, the two-photon lineshape is sufficiently asymmetric to produce
anomalous dispersion (lower right panel). The normalized transmission data
were fit simultaneously to all other data sets using $\Gamma$, $\gamma$, and
$\mathrm{\Delta q}$ as free parameters. The entire Rb cell apparatus is
mounted on a translation stage (TS) that travels transverse to the beam
propagation. A polarizer (P) after the cell filters out the control field, and
the probe field is split into two paths by a beamsplitter (BS). One path
terminates on a photodiode (PD) for measuring the group delay of the probe. In
the other, a lens (L) images the output facet of the cell onto a camera (CCD).

In Fourier space, stressing that the medium velocity is parallel to the
$\mathrm{x}$-axis, one obtains

$\mathrm{\partial_{z}\Omega_{1}=i\frac{k_{z}}{2}\chi(k_{x},k_{z},\delta)\Omega_{1}}$
(8)

with the susceptibility derived from Eq. (7),

$\mathrm{\chi(k_{x},k_{z},\delta)=i\frac{2\alpha}{k_{z}}\left[1-\frac{K|\Omega_{2}|^{2}}{-i\delta+\gamma+K\left|\Omega_{2}\right|^{2}+D\left(k_{x}+\Delta
q\right)^{2}-i\left(k_{x}+\Delta q\right)v_{0}}\right]}.$ (9)

The probe susceptibility’s dependence on $\mathrm{k_{z}}$ is such that
$\mathrm{\partial_{k_{z}}\left[k_{z}\chi(k_{x},k_{z},\delta)\right]=0}$. It
should be noted that Eq. (8) is exact only when the control field is an
infinite plane wave, $\Omega_{2}(\boldsymbol{\mathrm{r}})=\Omega_{2}$.
Nonetheless, quantitative agreement between Eq. (8) and experiment can be
achieved using a control field of finite extent as long as its beam waist,
$\mathrm{w_{0}}$, is wide enough to satisfy
$\mathrm{w_{0}}\gg\mathrm{max\\{\sqrt{D\tau},v_{0}\tau\\}}$ [20].

The resulting solution of Eq. (7) is

$\mathrm{\tilde{\Omega}_{1}(x,z)}=\mathcal{F}^{-1}\left\\{\mathcal{F}\left[\mathrm{\tilde{\Omega}_{1}|_{z=0}}\right]\exp\left[i\mathrm{\frac{\chi(k_{x},k_{z},\delta)}{2}k_{z}z}\right]\right\\},$
(10)

where $\mathcal{F}$ is the Fourier transform operator.

This picture furnishes an expression for the transverse shift in Eq. (1),
where the dependence of the optical drag and group delay on the two-photon
detuning appears explicitly [20]:

$\mathrm{\Delta
x=\partial_{k_{x}}\Re\left(\frac{\chi(k_{x},k_{z},\delta)}{2}k_{z}L\right)},$
(11)
$\mathrm{\tau=\partial_{\delta}\Re\left(\frac{\chi(k_{x},k_{z},\delta)}{2}k_{z}L\right)}.$
(12)

##  III Experimental Results

The experimental arrangement for measuring anomalous optical drag is shown in
Fig. 2. The control and the probe beams are obtained from the same diode laser
whose central frequency is set to $795\mathrm{\>nm}$ to excite the D1
transition of the 87Rb atoms. The probe is tuned to the
$\mathrm{F=1\rightarrow F^{\prime}=1,2}$ transition, and the control is tuned
to the $\mathrm{F=2\rightarrow F^{\prime}=1,2}$ transition. Both fields are
detuned by $-250\mathrm{\>MHz}$ below the $\mathrm{F^{\prime}=2}$ level, which
ensures a sufficiently asymmetric transmission lineshape around the two-photon
resonance (see Fig. 2, lower right panel), resulting in negative group
velocity [25]. A normalized version of Eq. (10) was fit to the transmission
data simultaneously to all other data sets reported here.

Figure 3: Dependence of optical drag on two-photon detuning $\delta$. (top
row) The measured transverse beam displacement $\mathrm{\Delta x}$ is in the
same direction as the velocity of the cell on the red-detuned side of the two-
photon transition and in the opposite direction on the blue-detuned side. This
behavior is observed for both (a) negative and (b) positive cell velocities.
(bottom row) The drag expected from the measured group delay, $\mathrm{\Delta
x\approx\tau v_{0}}$ is measured simultaneously with $\mathrm{\Delta x}$ and
predicts substantially less anomalous optical drag. The fit lines are obtained
from maximum likelihood estimation using the models of Eqs. (11, 12) with
$\Gamma$, $\gamma$, and $\mathrm{\Delta q}$ as free parameters. An offset of
$425\mathrm{\>Hz}$ is added to the fit lines to account for a systematic error
in the detuning measurement.

The probe and control beams exit the same polarization-maintaining fiber with
a radius $\mathrm{w_{p}=w_{c}=2\mathrm{\>mm}}$. The total control power is
$\sim 40\mathrm{\>\mu W}$ and that of the probe is $\sim 2\mathrm{\>\mu W}$.
The 87Rb vapor is kept in a magnetically-shielded Pyrex cell of length
$\mathrm{L}=7.5\mathrm{\>cm}$. The cell contains buffer gases $\mathrm{N_{2}}$
($10\mathrm{\>Torr}$) and Ar ($90\mathrm{\>Torr}$) and is heated to
$55\mathrm{\>{}^{\circ}C}$, in which case the thermal motion of the atoms
behaves diffusively with a diffusion coefficient of $\mathrm{D}\approx
166\mathrm{\>mm^{2}/s}$. The cell assembly is mounted on a Thorlabs DDSM100
motorized stage, so it can move with a constant velocity while crossing the
beams.

A polarizer is placed after the cell to filter the probe from the control
field. The probe field is then split into two paths, one that forms an image
of the probe at the output of the cell on a Princeton ProEM camera and another
that terminates at a photodetector for temporal detection of the probe pulse.
The imaging path is used to measure $\mathrm{\Delta x}$, and the other is used
to measure $\tau$. By comparing the transverse beam profile’s center of mass
when the cell is moving and not moving, the transverse drag is obtained.
Similarly, the mean temporal shift between pulses for various two-photon
detunings gives the group delay of the pulse inside the cell.

The dependence of optical drag on the two-photon detuning is shown in Fig. 3.
Both $\mathrm{\Delta x}$ and $\mathrm{\tau v_{0}}$ are measured at cell
velocities of $200\mathrm{\>mm/s}$ (a) and $-200\mathrm{\>mm/s}$ (b), the
maximum achievable velocities of our system. The models of Eqs. (11, 12) are
fit to the $\mathrm{\Delta x}$ and $\mathrm{\tau v_{0}}$ data respectively
using maximum likelihood estimation (MLE) with three free parameters:
$\Gamma$, $\gamma$, and $\mathrm{\Delta q}$. Furthermore, we assume that the
data have Gaussian random errors. In fact, all data sets shown were fit
simultaneously using MLE, obtaining the following values of the three fit
parameters: $\Gamma=266(2)\mathrm{\>MHz}$, $\gamma=145(2)\mathrm{\>Hz}$, and
$\mathrm{\Delta q}=1.2(3)*10^{-6}\mathrm{\>2\pi/\lambda}$. Uncertainties in
the fit parameter values were obtained through Monte Carlo simulation. The
data show a strong dependence on the two-photon detuning $\delta$; normal
optical drag occurs for $\delta<425\mathrm{\>Hz}$, while anomalous optical
drag occurs for $\delta>425\mathrm{\>Hz}$ and peaks around
$\delta=900\mathrm{\>Hz}$. Note that the $\mathrm{\tau v_{0}}$ data predict
approximately half as much anomalous optical drag than is actually measured,
$\mathrm{\Delta x}$. The breakdown of this approximation may result from
systematic error in the estimation of the pulse delay.

Figure 4: Experimental observation of optical drag at various dragging
velocities. (a) At a two-photon detuning of $\delta=0\mathrm{\>Hz}$ the beam
experiences normal optical drag. The dependence of the drag on the velocity is
approximately linear from $\mathrm{v_{0}}=-200\mathrm{\>mm/s}$ to
$200\mathrm{\>mm/s}$. (b) There is effectively no optical drag for any cell
velocity when $\delta=425\mathrm{\>Hz}$. (c) At $\delta=900\mathrm{\>Hz}$ the
beam undergoes anomalous optical drag for all velocities in the domain
$\mathrm{v_{0}}=-200\mathrm{\>mm/s}$ to $200\mathrm{\>mm/s}$. The fit lines
were obtained by fitting Eq. (11) to all data sets simultaneously.

Figure 4 shows the dependence of optical drag on dragging velocity. Despite
the nonlinear dependence of Eq. (9), and consequently Eq. (11), on the
dragging velocity, the data are approximately linear in this experimental
regime. Interestingly, for a two-photon detuning of $\delta=425\mathrm{\>Hz}$,
almost no optical drag is present for any dragging velocity, as shown in Fig.
4(b). The null optical drag is unique and corresponds to conditions of zero
group delay in the medium (zero crossing in Fig. 3). Normal [Fig. 4(a)] and
anomalous [Fig. 4(c)] drag are observed over a range of velocities with a high
degree of symmetry between the normal and anomalous effects.

##  IV Conclusions

We demonstrated normal, anomalous, and null optical drag effects
experimentally and modeled them theoretically. By using electromagnetically-
induced transparency in ${}^{87}\mathrm{Rb}$ vapor we were able to maximize
the dispersion and minimize the loss associated with near-resonant atomic
excitation. Our measurements showed that the beam’s displacement is
proportional to the velocity of the medium through which it passes, with a
direction depending on the sign of medium’s dispersion. For positive group
delays, the beam experiences normal optical drag and is dragged in the
direction of the material motion. When the measured group delay is negative,
the beam is dragged in a direction opposite to the material displacement. In a
regime where the group index is exactly equal to the inverse of the refractive
index and there is no measured group delay, there is no optical drag at any
material velocity. To our knowledge this is the first experimental
demonstration of anomalous drag.

In the gaseous system we study, the average linear velocity of the atoms due
to the transverse motion of the cell adds to their underlying thermal velocity
distribution, resulting in a drift-diffusive atomic motion. The atomic
diffusion can lead to paraxial diffusion of the light field [26, 27] and also,
similarly to anomalous and null drag, to effective paraxial diffraction that
eliminates and even reverses the free-space optical diffraction [28]. While
here we attenuate the effect of diffusion by using a dense buffer gas, it
could be interesting to explore the interplay between (linear) drag and
(quadratic) diffusion and diffraction in the anomalous and null regimes.

###### Acknowledgements.

The authors would like to acknowledge M. Zahirul Alam for very enlightening
discussions during the process of conducting this experiment and writing the
manuscript. This work was supported by the Pazy Foundation, the Israel Science
Foundation, and the Consortium for quantum sensing of Israel Innovation
Authority.

C.B. and Y.S. contributed equally to this work.

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