#  Exact solution Ising-like $2\times 2\times\infty$ models with multispin
interactions

Pavel Khrapov  Pavel Khrapov
Department of Mathematics
Bauman Moscow State Technical University
ul. Baumanskaya 2-ya, 5/1, Moscow
105005, Moscow, Russian Federation<EMAIL_ADDRESS><EMAIL_ADDRESS>and  Veronika Nikishina  Veronika Nikishina
Department of Mathematics
Bauman Moscow State Technical University
ul. Baumanskaya 2-ya, 5/1, Moscow
105005, Moscow, Russian Federation<EMAIL_ADDRESS>,
<EMAIL_ADDRESS>
###### Abstract.

The paper considers the generalized Ising model in the $2\times 2\times\infty$
strip with a Hamiltonian invariant with respect to the central axis of
rotation through the angle $\pi/2$, which includes all possible multiplicative
interactions of an even number of spins in the unit cube.The exact value of
free energy and heat capacity in the thermodynamic limit is found. For
percolation invariant with respect to rotation about the central axis of
rotation through the angle $\pi/2$, a limit relation for the non-percolation
probability is derived. The solution was obtained by the transfer matrix
method. In the general case, finding the largest eigenvalue of a $16\times 16$
transfer matrix reduces to solving a quartic equation using the Ferrari
method. In special cases, switching the transfer matrix of models with
auxiliary matrices made it possible to reduce the problem to quadratic and
linear equations. A separate chapter is devoted to the gonigendrial model in
the $2\times 2\times\infty$ strip, the exact values of the free energy and
heat capacity in the thermodynamic limit are found for the case of free
boundary conditions and an analogue of cyclically closed boundary conditions
in both directions perpendicular to the central axis of rotation. The
numerical solution for the $3\times 3\times\infty$ gonigendrial model in the
case of free boundary conditions and in the case of cyclic closure is compared
with the corresponding analytical solutions in the volume $2\times
2\times\infty$. The closeness of free energy and heat capacity in the
thermodynamic limit for the gonigendrial model on the entire three-dimensional
lattice and for the analog of cyclic closure in the $2\times 2\times\infty$
strip is shown. The exact value of the free energy and heat capacity in the
thermodynamic limit in the $2\times 2\times\infty$ strip is obtained for the
Antiferromagnetic layered Ising model, the similarity of these physical
characteristics with the numerical solution in an infinite volume is shown.

###### Key words and phrases:

Ising model, Hamiltonian, multispin interaction, transfer matrix, exact
solution, partition function, free energy, phase transition .

###### 2010 Mathematics Subject Classification:

82B20, 82B23

###### Contents

  1. 1 Introduction
  2. 2 Model description
    1. 2.1 Model Hamiltonian and Physical Characteristics
    2. 2.2 Invariant percolation
  3. 3 Exact solution for free energy in the strip $2\times 2\times\infty$
  4. 4 Exact solution for invariant percolation in the strip $2\times 2\times\infty$
  5. 5 Some simplest particular cases of solution
    1. 5.1 The first particular case
    2. 5.2 The second particular case
  6. 6 The Gonihedric Ising Model
    1. 6.1 Exact solution of the gonigendrial model with free boundary conditions
    2. 6.2 Numerical solution of the gonigendrial model $3\times 3\times\infty$
    3. 6.3 Exact solution of a cyclically closed gonigendrial model $2\times 2\times\infty$
  7. 7 Antiferromagnetic layered Ising model
  8. 8 Conclusion

##  1\. Introduction

The Ising model was introduced back in the 20s of the 20-th entury [1] to
describe magnetic transitions, but is still the subject of intense research.
In particular, the Ising model with interactions of nearest neighbors ($NN$),
next nearest neighbors ($NNN$), and plaquette interactions ($P$) on cubic
lattice was considered as a simple model of the statistical mechanics of
random surfaces [2, 3], microemulsions [4], where critical fluctuations in the
density of water droplets were studied, as well as discretized string action
(the so-called gonigendrial model) [5]-[10].

The Ising model on a $2\times 2\times\infty$ lattice with a simple cubic cell
and under the condition of equal interactions in both transverse directions
can be reduced to an $n\times\infty$ Ising model consisting of $n$ cells
closed into a cylinder, if we put the number of cells equal to $n=4$. The
exact value of the partition function and free energy for the nearest neighbor
Ising model without an external magnetic field on the $n\times\infty$ cylinder
was first obtained by Onsager [11], and the result for lattice $2\times
2\times\infty$ generalized to the case of different interaction coefficients
in all three directions in [12, 13].In work [14] all 16 eigenvalues for the 3D
Ising model are found under the condition that the interactions in both
transverse directions are equal. The interaction of next nearest neighbors is
taken into account in [15], where the eigenvalues of the transfer matrix of
the Ising model are found.

The exact value of free energy and heat capacity for the Ising model with NN,
NNN interactions was obtained in [16] and [17], and phase diagrams were
plotted for various ratios of parameters. The author of [18] presents exact
analytical formulas for free energy for two $2\times 2\times\infty$ Ising
models. The first model [18] has a simple cubic lattice with fully anisotropic
interactions. The second model [18] consists of two different types of linear
chains and includes non-intersecting diagonal bonds on the side faces of the
$2\times 2\times\infty$ parallelepiped. In both cases, the solutions are
expressed in terms of square radicals and are obtained using the obvious
symmetry of the Hamiltonians and the hidden algebraic symmetry of the
characteristic polynomial of the transfer matrix. The two-dimensional Ising
model has been intensively studied, the Hamiltonian of which additionally
takes into account the interaction of the next nearest neighbors
[19],[20],[21]. A three-dimensional analog of this model was studied by
authors of [22] using the Monte Carlo method.

In [23] percolation in the Ising model is studied, based on the generalized
results of [24]. Based on percolation in a finite-width strip for an
independent field and the Ising model, results are obtained in [25], for
continuous models in [26]. A meaningful review and results on percolation
theory can be found in [27]. In recent years, interest in simple systems of
Ising spins has arisen as a result of considering the random surface model in
the context of string theory [28, 9]. This model was called gonigendrial and
its three-dimensional case was studied by approximation methods [7], Monte
Carlo method [29], using differential operators [21] and using Novotny’s
transfer-matrix method [10].

In this paper, we consider the generalized Ising model on the lattice $2\times
2\times\infty$ with a Hamiltonian invariant with respect to the central axis
of rotation on $\pi/2$, which includes all possible interactions of an even
number of spins: double, quadruple, six-fold and eight-dimensional
interactions . For a model with the most general Hamiltonian of this type, the
exact value of the free energy in the thermodynamic limit is written out. Let
us note that finding the free energy in the thermodynamic limit is reduced to
finding the largest eigenvalue of the transfer matrix, which is the root of
the fourth degree equation. However, in the same way, we can find all the
eigenvalues of the transfer matrix, this makes it possible to obtain the
partition function and free energy in a finite closed chain $2\times 2\times
L$. The authors introduced a special type of percolation, which is invariant
under rotation through the angle $\pi/2$, and describes the limiting behavior
of the non-percolation probability. Next, we consider two special cases in
which the search for the largest eigenvalue of the transfer matrix is reduced
to solving a quadratic equation. An exact solution is obtained for the
gonigendrial model with free boundary conditions and in the cyclically closed
case. An exact solution of the Antiferromagnetic layered Ising model is found.
This work is a logical continuation of the works [26], [25], [30], [31], [32]
according to Ising-like models.

There is a following structure in the article. The chapter 2 introduces the
basic concepts: model Hamiltonian, partition function, thermodynamic
characteristics and invariant percolation. In chapter 3, using the method of
transfer matrices and taking into account the commutation of the resulting
transfer matrix with the "rotation" operator on $\pi/2$, expressions for the
thermodynamic characteristics of the model are written. In chapter 4 the non-
percolation probability is written out, which is obtained taking into account
the invariance of the introduced type of percolation with respect to the
rotation by $\pi/2$. The chapter 5 is devoted to the consideration of some of
the simplest cases of solution, which are obtained taking into account the
commutation of a particular form of the transfer matrix with some other
matrices. In the 6 chapter, an analytical solution is written for the
generalized gonigendrial model in the case of free boundary conditions and in
the case of an analogue of cyclic closure, a numerical solution is found in a
certain range of parameters in the volume $3\times 3\times\infty$ for free
boundary conditions and in the case of cyclic closure. The numerical solution
for the $3\times 3\times\infty$ gonigendrial model in the case of free
boundary conditions and in the case of cyclic closure is compared with the
corresponding analytical solutions. The closeness of free energy and heat
capacity in the thermodynamic limit is shown for the gonigendrial model on the
entire three-dimensional lattice and for the analogue of cyclic closure on the
lattice $2\times 2\times\infty$. In chapter 7 the exact value of free energy
and heat capacity in the thermodynamic limit for the Antiferromagnetic layered
Ising model is found.

##  2\. Model description

###  2.1. Model Hamiltonian and Physical Characteristics

Let us consider a three-dimensional square cyclically closed lattice model
with size $2\times 2\times M$ (Fig.1),

$\mathcal{L}_{2,2,M}=\\{t_{i}^{m},m=0,1,...,M-1,t_{i}^{M}\equiv
t_{i}^{0},i=0,1,2,3\\},$ (2.1)

the total number of the lattice sites is $N=4M$.

$t^{m}_{0}$$t^{m}_{1}$$t^{m+1}_{0}$$t^{m+1}_{1}$$t^{m}_{3}$$t^{m}_{2}$$t^{m+1}_{3}$$t^{m+1}_{2}$$x$$y$$z$

FIGURE 1. Arrangement of spins in a strip and with a cross section $2\times
2$. The product of transfer matrices $\tau$ is taken from left to right

Let’s assume that a particle is localed at each site $t_{i}^{m}$ . The state
of the particle is determined by the value of the spin
$\sigma^{m}_{i}\equiv\sigma_{t_{i}^{m}},i=0,1,2,3,m=0,1,...,M-1$.

Let

$\Omega=\\{t_{i_{1}}^{m_{1}},t_{i_{2}}^{m_{2}},...,t_{i_{k}}^{m_{k}}\\}\subset\mathcal{L}_{2,2,M}.$
(2.2)

Let’s introduce

$\sigma_{\Omega}=\prod_{t\in\Omega}\sigma_{t}.$ (2.3)

Let $\Omega^{m}_{C}=\\{t_{i}^{m},t_{i}^{m+1},i=0,1,2,3\\}$ be a cube. We
introduce the set of elementary generating supports of the Hamiltonian

$\begin{gathered}\Phi_{2}^{m}=\\{\\{t_{0}^{m},t_{1}^{m}\\},\\{t_{0}^{m},t_{2}^{m}\\},\\{t_{0}^{m+1},t_{1}^{m+1}\\},\\{t_{0}^{m+1},t_{2}^{m+1}\\},\\\
\\{t_{0}^{m},t_{0}^{m+1}\\},\\{t_{0}^{m},t_{1}^{m+1}\\},\\{t_{0}^{m},t_{2}^{m+1}\\},\\{t_{0}^{m},t_{3}^{m+1}\\}\\},\\\
\Phi_{4}^{m}=\\{\\{t_{0}^{m},t_{0}^{m+1},t_{1}^{m+1},t_{2}^{m+1}\\},\\{t_{0}^{m},t_{1}^{m+1},t_{2}^{m+1},t_{3}^{m+1}\\},\\\
\\{t_{0}^{m},t_{2}^{m+1},t_{3}^{m+1},t_{0}^{m+1}\\},\\{t_{0}^{m},t_{3}^{m+1},t_{0}^{m+1},t_{1}^{m+1}\\},\\\
\\{t_{0}^{m},t_{1}^{m},t_{0}^{m+1},t_{1}^{m+1}\\},\\{t_{0}^{m},t_{1}^{m},t_{1}^{m+1},t_{2}^{m+1}\\},\\\
\\{t_{0}^{m},t_{1}^{m},t_{2}^{m+1},t_{3}^{m+1}\\},\\{t_{0}^{m},t_{1}^{m},t_{3}^{m+1},t_{0}^{m+1}\\},\\\
\\{t_{0}^{m},t_{1}^{m},t_{0}^{m+1},t_{2}^{m+1}\\},\\{t_{0}^{m},t_{1}^{m},t_{1}^{m+1},t_{3}^{m+1}\\},\\\
\\{t_{0}^{m},t_{2}^{m},t_{0}^{m+1},t_{1}^{m+1}\\},\\{t_{0}^{m},t_{2}^{m},t_{1}^{m+1},t_{2}^{m+1}\\},\\\
\\{t_{0}^{m},t_{2}^{m},t_{2}^{m+1},t_{3}^{m+1}\\},\\{t_{0}^{m},t_{2}^{m},t_{3}^{m+1},t_{0}^{m+1}\\},\\\
\\{t_{0}^{m},t_{2}^{m},t_{0}^{m+1},t_{2}^{m+1}\\},\\{t_{0}^{m},t_{2}^{m},t_{1}^{m+1},t_{3}^{m+1}\\},\\\
\\{t_{0}^{m},t_{1}^{m},t_{2}^{m},t_{0}^{m+1}\\},\\{t_{0}^{m},t_{1}^{m},t_{2}^{m},t_{1}^{m+1}\\},\\\
\\{t_{0}^{m},t_{1}^{m},t_{2}^{m},t_{2}^{m+1}\\},\\{t_{0}^{m},t_{1}^{m},t_{2}^{m},t_{3}^{m+1}\\},\\\
\\{t_{0}^{m},t_{1}^{m},t_{2}^{m},t_{3}^{m}\\},\\{t_{0}^{m+1},t_{1}^{m+1},t_{2}^{m+1},t_{3}^{m+1}\\},\\\
\Phi_{6}^{m}=\\{\\{\Omega^{m}_{C}\setminus\\{t_{0}^{m},t_{1}^{m}\\}\\},\\{\Omega^{m}_{C}\setminus\\{t_{0}^{m},t_{2}^{m}\\}\\},\\\
\\{\Omega^{m}_{C}\setminus\\{t_{0}^{m+1},t_{1}^{m+1}\\}\\},\\{\Omega^{m}_{C}\setminus\\{t_{0}^{m+1},t_{2}^{m+1}\\}\\},\\\
\\{\Omega^{m}_{C}\setminus\\{t_{0}^{m},t_{0}^{m+1}\\}\\},\\{\Omega^{m}_{C}\setminus\\{t_{0}^{m},t_{1}^{m+1}\\}\\},\\{\Omega^{m}_{C}\setminus\\{t_{0}^{m},t_{2}^{m+1}\\}\\},\\\
\\{\Omega^{m}_{C}\setminus\\{t_{0}^{m},t_{3}^{m+1}\\}\\}\\},\\\
\Phi_{8}^{m}=\\{\\{t_{0}^{m},t_{1}^{m},t_{2}^{m},t_{3}^{m},t_{0}^{m+1},t_{1}^{m+1},t_{2}^{m+1},t_{3}^{m+1}\\}\\},\\\
\Phi^{m}=\Phi_{2}^{m}\cup\Phi_{4}^{m}\cup\Phi_{6}^{m}\cup\Phi_{8}^{m}.\end{gathered}$
(2.4)

Let us define the elementary generating Hamiltonian

$\begin{gathered}\mathcal{\hat{H}}^{m}=\mathcal{\hat{H}}^{m}_{2}+\mathcal{\hat{H}}^{m}_{4}+\mathcal{\hat{H}}^{m}_{6}+\mathcal{\hat{H}}^{m}_{8},\end{gathered}$
(2.5)

where

$\begin{gathered}\mathcal{\hat{H}}_{r}^{m}=-\sum_{{\alpha}^{m}\in\Phi_{r}^{m}}{J_{{\alpha}^{m}}}\sigma_{{\alpha}^{m}},r=2,4,6,8,\end{gathered}$
(2.6)

components of the Hamiltonian responsible for the interaction of an even
number of $r$ spins in the cube of $\Omega^{m}_{C}$.

For function
$f(\sigma^{m}_{0},\sigma^{m}_{1},\sigma^{m}_{2},\sigma^{m}_{3},\sigma^{m+1}_{0},\sigma^{m+1}_{1},\sigma^{m+1}_{2},\sigma^{m+1}_{3})$,
depending on $\sigma_{t},t\in\Omega^{m}_{C}$, we introduce the operator of
rotation $R_{\pi/2}^{m}$ by an angle ${\pi/2}$

$\begin{gathered}R^{m}_{\pi/2}(f(\sigma^{m}_{0},\sigma^{m}_{1},\sigma^{m}_{2},\sigma^{m}_{3},\sigma^{m+1}_{0},\sigma^{m+1}_{1},\sigma^{m+1}_{2},\sigma^{m+1}_{3}))=\\\
f(\sigma^{m}_{1},\sigma^{m}_{2},\sigma^{m}_{3},\sigma^{m}_{0},\sigma^{m+1}_{1},\sigma^{m+1}_{2},\sigma^{m+1}_{3},\sigma^{m+1}_{0}).\end{gathered}$
(2.7)

The Hamiltonian model has the form:

$\begin{gathered}\mathcal{H}=\sum_{m=0}^{M-1}\mathcal{H}^{m},\end{gathered}$
(2.8)

where

$\begin{gathered}\mathcal{H}^{m}=\mathcal{\hat{H}}^{m}+R^{m}_{\pi/2}\mathcal{\hat{H}}^{m}+(R^{m}_{\pi/2})^{2}\mathcal{\hat{H}}^{m}+(R^{m}_{\pi/2})^{3}\mathcal{\hat{H}}^{m}.\end{gathered}$
(2.9)

The partition function can be written in the following form:

$\begin{gathered}Z_{2,2,M}=\sum_{\sigma}\exp(-\mathcal{H}/(k_{B}T)),\end{gathered}$
(2.10)

summation perfomed over all spins.

Let us determine the physical characteristics of the system in the
thermodynamic limit in accordance with[33]. The free energy of the system per
one lattice site is determined in the standard way

$f(T)=-k_{B}T\lim_{N\to\infty}N^{-1}\ln{Z_{2,2,M}(T)},$ (2.11)

where $N=4M$ is the number of sites of the considered lattice.

The internal energy per one lattice site is equal to

$u(T)=-T^{2}\frac{\partial}{\partial T}[f(T)/T].$ (2.12)

The heat capacity per one lattice site is defined as

$C(T)=\frac{\partial}{\partial T}u(T)=-(2T\frac{\partial}{\partial
T}[f(T)/T]+T^{2}\frac{\partial^{2}}{\partial T^{2}}[f(T)/T]).$ (2.13)

###  2.2. Invariant percolation

Spin configurations in the cyclically closed strip $\mathcal{L}_{2,2,M}$ (2.1)
in which there is a column or row of $-1$: $\sigma_{0}^{m}=\sigma_{1}^{m}=-1$,
$\sigma_{0}^{m}=\sigma_{3}^{m}=-1$, $\sigma_{1}^{m}=\sigma_{2}^{m}=-1$,
$\sigma_{2}^{m}=\sigma_{3}^{m}=-1$ for some $m\in\\{0,...,M-1\\}$, will be
called percolation configurations. Accordingly, the non-percolation
probability is the probability that there will be no percolation in the model.
The probability of non-percolation of the model is determined by the ratio of
two quantities:

$P_{M}=\frac{Z^{\prime}_{2,2,M}}{Z_{2,2,M}},$ (2.14)

where ${Z_{2,2,M}}$ is the partition function (2.10) of the model with the
followind Hamiltonian (2.8), ${Z^{\prime}_{2,2,M}}$ \- analogue of the
partition function, in which the summation is performed only over non-
percolation configurations.

##  3\. Exact solution for free energy in the strip $2\times 2\times\infty$

With considering (2.8),(2.9) the partition function (2.10) can be written as:

$Z_{2,2,M}=\sum_{\sigma}\exp(-\sum_{m=0}^{M-1}\mathcal{H}^{m}/(k_{B}T)).$
(3.1)

Let us introduce a transfer matrix $\theta$ with size $16\times 16$ with
matrix elements (Fig.2 )

$\begin{gathered}\theta_{k,l}=\theta_{(\sigma_{0}^{m},\sigma_{1}^{m},\sigma_{2}^{m},\sigma_{3}^{m}),(\sigma_{0}^{m+1},\sigma_{1}^{m+1},\sigma_{2}^{m+1},\sigma_{3}^{m+1})}=\exp(-\sum_{m=0}^{M-1}\mathcal{H}^{m}/(k_{B}T)),\\\
k=(1-\sigma_{0}^{m})/2+(1-\sigma_{1}^{m})+2(1-\sigma_{2}^{m})+4(1-\sigma_{3}^{m}),\\\
l=(1-\sigma_{0}^{m+1})/2+(1-\sigma_{1}^{m+1})+2(1-\sigma_{2}^{m+1})+4(1-\sigma_{3}^{m+1}).\\\
\end{gathered}$ (3.2)

Then

$Z_{2,2,M}=tr(\theta^{M}).$ (3.3)

To find the free energy (2.11) in the thermodynamic limit, it suffices to find
the largest eigenvalue $\lambda_{\max}$ of the $\theta$ transfer matrix.
Further, it will be shown that this largest eigenvalue $\lambda_{\max}$ is the
largest root of the characteristic equation of the matrix $\tau$ with size
$4\times 4$:

$\begin{gathered}\tau=\begin{pmatrix}\begin{tabular}[]{p{1.5cm} p{3cm}
p{1.5cm} p{1.5cm}}\tiny{ \begin{center}$\theta_{1,1}+\theta_{1,16}$
\end{center}}&\tiny{\begin{center}$\theta_{1,2}+\theta_{1,3}+\theta_{1,5}+\theta_{1,8}+\theta_{1,9}+\theta_{1,12}+\theta_{1,14}+\theta_{1,15}$
\end{center}}&\tiny{\begin{center}$\theta_{1,4}+\theta_{1,7}+\theta_{1,10}+\theta_{1,13}$
\end{center}}&\tiny{\begin{center}
$\theta_{1,6}+\theta_{1,11}$\end{center}}\\\
\tiny{\begin{center}$\theta_{2,1}+\theta_{2,16}$ \end{center}}&\tiny{
\begin{center}$\theta_{2,2}+\theta_{2,3}+\theta_{2,5}+\theta_{2,8}+\theta_{2,9}+\theta_{2,12}+\theta_{2,14}+\theta_{2,15}$
\end{center}}&\tiny{
\begin{center}$\theta_{2,4}+\theta_{2,7}+\theta_{2,10}+\theta_{2,13}$
\end{center}}&\tiny{\begin{center}
$\theta_{2,6}+\theta_{2,11}$\end{center}}\\\
\tiny{\begin{center}$\theta_{4,1}+\theta_{4,16}$ \end{center}}&\tiny{
\begin{center}$\theta_{4,2}+\theta_{4,3}+\theta_{4,5}+\theta_{4,8}+\theta_{4,9}+\theta_{4,12}+\theta_{4,14}+\theta_{4,15}$
\end{center}}&\tiny{
\begin{center}$\theta_{4,4}+\theta_{4,7}+\theta_{4,10}+\theta_{4,13}$
\end{center}}&\tiny{\begin{center}
$\theta_{4,6}+\theta_{4,11}$\end{center}}\\\
\tiny{\begin{center}$\theta_{6,1}+\theta_{6,16}$
\end{center}}&\tiny{\begin{center}$\theta_{6,2}+\theta_{6,3}+\theta_{6,5}+\theta_{6,8}+\theta_{6,9}+\theta_{6,12}+\theta_{6,14}+\theta_{6,15}$
\end{center}}&\tiny{\begin{center}$\theta_{6,4}+\theta_{6,7}+\theta_{6,10}+\theta_{6,13}$
\end{center}}&\tiny{\begin{center}
$\theta_{6,6}+\theta_{6,11}$\end{center}}\end{tabular}\end{pmatrix},\\\
\end{gathered}$ (3.4)

$\sigma_{0}^{m}$$\sigma_{1}^{m}$$\sigma_{2}^{m}$$\sigma_{3}^{m}$$-$$+$$-$$+$$-$$+$$-$$+$$-$$+$$-$$+$$-$$+$$-$$+$$-$$+$$-$$+$$-$$+$$-$$+$$-$$+$$-$$+$$-$$+$$\sigma_{0}^{m+1}$$\sigma_{1}^{m+1}$$\sigma_{2}^{m+1}$$\sigma_{3}^{m+1}$$-$$+$$-$$+$$-$$+$$-$$+$$-$$+$$-$$+$$-$$+$$-$$+$$-$$+$$-$$+$$-$$+$$-$$+$$-$$+$$-$$+$$-$$+$

FIGURE 2. The structure of the transfer matrix for computing the partition
function in a strip with a cross section $2\times 2$

We can now formulate the following main theorem.

######  Theorem 3.1.

Main theorem

Let $\lambda_{\max}=\lambda_{\max}(T)$ be the largest eigenvalue of the
characteristic equation of the fourth degree of the matrix $\tau$ (3.4) for
the model with the Hamiltonian (2.8-2.9).

Then in the thermodynamic limit:

the free energy (2.11) of considering model has the form

$f(T)=-k_{B}T\ln{(\lambda_{\max}(T))}/4,$ (3.5)

the internal energy (2.12) per one lattice site can be represented as

$u(T)=k_{B}T^{2}\frac{\partial}{\partial T}[\ln{(\lambda_{\max}(T))}/4],$
(3.6)

the heat capacity (2.13) per one lattice site is equal to

$C(T)=\frac{\partial}{\partial T}u(T)=2k_{B}T\frac{\partial}{\partial
T}[\ln{(\lambda_{\max}(T))/4}]+k_{B}T^{2}\frac{\partial^{2}}{\partial
T^{2}}[\ln{(\lambda_{\max}(T))}/4],$ (3.7)

where $k_{B}$ is Boltzmann’s constant.

###### Proof.

The proof is based on commutation of the matrix $\theta$ with a matrix of a
special form; this allows us to refine the form of the eigenvector
corresponding to the largest eigenvalue.

The transfer matrix $\theta$ commutes with the "rotation" matrix on $\pi/2$
$D:\\{\sigma_{0}^{m}\rightarrow\sigma_{1}^{m+1};\sigma_{1}^{m}\rightarrow\sigma_{2}^{m+1};\sigma_{2}^{m}\rightarrow\sigma_{3}^{m+1};\sigma_{3}^{m}\rightarrow\sigma_{0}^{m+1},m=0,..,M-1\\}$
.

$D=\setcounter{MaxMatrixCols}{16}\begin{pmatrix}1&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\\
0&0&1&0&0&0&0&0&0&0&0&0&0&0&0&0\\\ 0&0&0&0&1&0&0&0&0&0&0&0&0&0&0&0\\\
0&0&0&0&0&0&1&0&0&0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0&1&0&0&0&0&0&0&0\\\
0&0&0&0&0&0&0&0&0&0&1&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0&0&0&0&0&1&0&0&0\\\
0&0&0&0&0&0&0&0&0&0&0&0&0&0&1&0\\\ 0&1&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\\
0&0&0&1&0&0&0&0&0&0&0&0&0&0&0&0\\\ 0&0&0&0&0&1&0&0&0&0&0&0&0&0&0&0\\\
0&0&0&0&0&0&0&1&0&0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0&0&1&0&0&0&0&0&0\\\
0&0&0&0&0&0&0&0&0&0&0&1&0&0&0&0\\\ 0&0&0&0&0&0&0&0&0&0&0&0&0&1&0&0\\\
0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&1\\\ \end{pmatrix}$ (3.8)

The eigenvectors of the matrix $D$ (3.8), corresponding to the positive
eigenvalue $1$, have the form:

$\begin{gathered}\\{(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1),\\\
(0,0,0,0,0,0,0,1,0,0,0,1,0,1,1,0),\\\ (0,0,0,1,0,0,1,0,0,1,0,0,1,0,0,0),\\\
(0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0),\\\ (0,1,1,0,1,0,0,0,1,0,0,0,0,0,0,0),\\\
(1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)\\}.\end{gathered}$ (3.9)

The linear combination of these vectors is also an eigenvector that
corresponds to the eigenvalue 1. Since the commuting matrices preserve each
other’s own subspaces [34] and taking into account the central symmetry of the
transfer matrix $\theta$ [35], we will look for the eigenvector of the matrix
$\theta$ corresponding to the largest eigenvalue in the form:

$(x_{1},x_{2},x_{2},x_{3},x_{2},x_{4},x_{3},x_{2},x_{2},x_{3},x_{4},x_{2},x_{3},x_{2},x_{2},x_{1}),$

which, by the Perron-Frobenius theorem [36], has positive coordinates.

The form of the eigenvector allows you to search for the largest eigenvalue
not of a matrix $\theta$ with size $16\times 16$, but of a matrix $\tau$ with
size $4\times 4$ (3.4).

The characteristic polynomial of a matrix $\tau$ (3.4) is:

$\begin{gathered}\lambda^{4}+a\lambda^{3}+b\lambda^{2}+c\lambda+d=0,\end{gathered}$
(3.10)

where

$\begin{gathered}a=-(\tau_{1,1}+\tau_{2,2}+\tau_{3,3}+\tau_{4,4}),\end{gathered}$
(3.11)
$\begin{gathered}b=-\tau_{1,2}\tau_{2,1}+\tau_{1,1}\tau_{2,2}-\tau_{1,3}\tau_{3,1}-\tau_{2,3}\tau_{3,2}+\tau_{1,1}\tau_{3,3}+\\\
\tau_{2,2}\tau_{3,3}-\tau_{1,4}\tau_{4,1}-\tau_{2,4}\tau_{4,2}-\tau_{3,4}\tau_{4,3}+\tau_{1,1}\tau_{4,4}+\tau_{2,2}\tau_{4,4}+\tau_{3,3}\tau_{4,4},\end{gathered}$
(3.12)
$\begin{gathered}c=\tau_{1,3}\tau_{2,2}\tau_{3,1}-\tau_{1,2}\tau_{2,3}\tau_{3,1}-\tau_{1,3}\tau_{2,1}\tau_{3,2}+\tau_{1,1}\tau_{2,3}\tau_{3,2}+\tau_{1,2}\tau_{2,1}\tau_{3,3}-\\\
\tau_{1,1}\tau_{2,2}\tau_{3,3}+\tau_{1,4}\tau_{2,2}\tau_{4,1}-\tau_{1,2}\tau_{2,4}\tau_{4,1}+\tau_{1,4}\tau_{3,3}\tau_{4,1}-\tau_{1,3}\tau_{3,4}\tau_{4,1}-\\\
\tau_{1,4}\tau_{2,1}\tau_{4,2}+\tau_{1,1}\tau_{2,4}\tau_{4,2}+\tau_{2,4}\tau_{3,3}\tau_{4,2}-\tau_{2,3}\tau_{3,4}\tau_{4,2}-\tau_{1,4}\tau_{3,1}\tau_{4,3}-\\\
\tau_{2,4}\tau_{3,2}\tau_{4,3}+\tau_{1,1}\tau_{3,4}\tau_{4,3}+\tau_{2,2}\tau_{3,4}\tau_{4,3}+\tau_{1,2}\tau_{2,1}\tau_{4,4}-\tau_{1,1}\tau_{2,2}\tau_{4,4}+\\\
\tau_{1,3}\tau_{3,1}\tau_{4,4}+\tau_{2,3}\tau_{3,2}\tau_{4,4}-\tau_{1,1}\tau_{3,3}\tau_{4,4}-\tau_{2,2}\tau_{3,3}\tau_{4,4},\end{gathered}$
(3.13)
$\begin{gathered}d=\tau_{1,4}\tau_{2,3}\tau_{3,2}\tau_{4,1}-\tau_{1,3}\tau_{2,4}\tau_{3,2}\tau_{4,1}-\tau_{1,4}\tau_{2,2}\tau_{3,3}\tau_{4,1}+\tau_{1,2}\tau_{2,4}\tau_{3,3}\tau_{4,1}+\\\
\tau_{1,3}\tau_{2,2}\tau_{3,4}\tau_{4,1}-\tau_{1,2}\tau_{2,3}\tau_{3,4}\tau_{4,1}-\tau_{1,4}\tau_{2,3}\tau_{3,1}\tau_{4,2}+\tau_{1,3}\tau_{2,4}\tau_{3,1}\tau_{4,2}+\\\
\tau_{1,4}\tau_{2,1}\tau_{3,3}\tau_{4,2}-\tau_{1,1}\tau_{2,4}\tau_{3,3}\tau_{4,2}-\tau_{1,3}\tau_{2,1}\tau_{3,4}\tau_{4,2}+\tau_{1,1}\tau_{2,3}\tau_{3,4}\tau_{4,2}+\\\
\tau_{1,4}\tau_{2,2}\tau_{3,1}\tau_{4,3}-\tau_{1,2}\tau_{2,4}\tau_{3,1}\tau_{4,3}-\tau_{1,4}\tau_{2,1}\tau_{3,2}\tau_{4,3}+\tau_{1,1}\tau_{2,4}\tau_{3,2}\tau_{4,3}+\\\
\tau_{1,2}\tau_{2,1}\tau_{3,4}\tau_{4,3}-\tau_{1,1}\tau_{2,2}\tau_{3,4}\tau_{4,3}-\tau_{1,3}\tau_{2,2}\tau_{3,1}\tau_{4,4}+\tau_{1,2}\tau_{2,3}\tau_{3,1}\tau_{4,4}+\\\
\tau_{1,3}\tau_{2,1}\tau_{3,2}\tau_{4,4}-\tau_{1,1}\tau_{2,3}\tau_{3,2}\tau_{4,4}-\tau_{1,2}\tau_{2,1}\tau_{3,3}\tau_{4,4}+\tau_{1,1}\tau_{2,2}\tau_{3,3}\tau_{4,4}.\end{gathered}$
(3.14)

Let us find the largest root of the fourth degree equation from (3.10). By the
Ferrari method [37], [38], if $y_{1}$ is a root of the auxiliary equation

$\begin{gathered}y^{3}+Ay^{2}+By+C=0,\end{gathered}$ (3.15)

where $A=-b$, $B=ac-4d$, $C=-a^{2}d+4bd-c^{2}$, then the four roots of the
original equation are found as the roots of two quadratic equations:

$\begin{gathered}x^{2}+(a/2)x+y_{1}/2=\\\
\pm\sqrt{(a^{2}/4-b+y_{1})x^{2}+((a/2)y_{1}-c)x+(y_{1}^{2}/4-d)},\end{gathered}$
(3.16)

where the radical expression on the right side is a perfect square. .

Using the Cardano-Vieta formulas [37], [38], one of the roots of the equation
(3.15) is written as follows

$\begin{gathered}y_{1}=\frac{1}{3}(-A+2\sqrt{A^{2}-3B}\sin[1/3(\arcsin(\frac{2A^{3}-9AB+27C}{2(A^{2}-3B)^{3/2}})+2\pi)]),\end{gathered}$
(3.17)

Then the root $\lambda_{\max}$ we need is the largest root of the equation

$\begin{gathered}x^{2}+px+q=0,\end{gathered}$ (3.18)

where $p=a/2-\sqrt{a^{2}/4-b+y_{1}}$, $q=y_{1}/2+\sqrt{y_{1}^{2}/4-d}$.

∎

##  4\. Exact solution for invariant percolation in the strip $2\times
2\times\infty$

Similarly, the partition function of the non-percolation configurations is
found. The transfer matrix of non-percolation configurations $\theta^{\prime}$
coincides with the transfer matrix of all model configurations, but with zero
matrix elements corresponding to percolation configurations (Fig.3).

$\sigma_{0}^{m}$$\sigma_{1}^{m}$$\sigma_{2}^{m}$$\sigma_{3}^{m}$$-$$+$$-$$+$$-$$+$$-$$+$$-$$+$$-$$+$$-$$+$$-$$+$$-$$+$$-$$+$$-$$+$$-$$+$$-$$+$$-$$+$$-$$+$$\sigma_{0}^{m+1}$$\sigma_{1}^{m+1}$$\sigma_{2}^{m+1}$$\sigma_{3}^{m+1}$$-$$+$$-$$+$$-$$+$$-$$+$$-$$+$$-$$+$$-$$+$$-$$+$$-$$+$$-$$+$$-$$+$$-$$+$$-$$+$$-$$+$$-$$+$$*$$*$$*$$*$$*$$*$$*$$*$$*$$*$$*$$*$$*$$*$$*$$*$$*$$*$$*$$*$$*$$*$$*$$*$$*$$*$$*$$*$$*$$*$$*$$*$$*$$*$$*$$*$$*$$*$$*$$*$$*$$*$$*$$*$$*$$*$$*$$*$$*$

FIGURE 3. The structure of the transfer matrix of $\theta^{\prime}$ of non-
percolation configurations. Non-null elements are marked with $*$ .

Further, it is shown that the largest eigenvalue $\mu_{\max}$ of the transfer
matrix of non-percolation configurations $\theta^{\prime}$ is the largest root
of the characteristic equation of the matrix $\tau^{\prime}$ with size
$3\times 3$:

$\begin{gathered}\tau^{\prime}=\begin{pmatrix}\begin{tabular}[]{p{1cm} p{3cm}
p{1.5cm}}\tiny{
\begin{center}$\theta_{1,1}$\end{center}}&\tiny{\begin{center}$\theta_{1,2}+\theta_{1,3}+\theta_{1,5}+\theta_{1,9}$\end{center}}&\tiny{\begin{center}
$\theta_{1,6}+\theta_{1,11}$\end{center}}\\\
\tiny{\begin{center}$\theta_{2,1}$\end{center}}&\tiny{\begin{center}$\theta_{2,2}+\theta_{2,3}+\theta_{2,5}+\theta_{2,9}$\end{center}}&\tiny{\begin{center}
$\theta_{2,6}+\theta_{2,11}$\end{center}}\\\
\tiny{\begin{center}$\theta_{6,1}$\end{center}}&\tiny{\begin{center}$\theta_{6,2}+\theta_{6,3}+\theta_{6,5}+\theta_{6,9}$\end{center}}&\tiny{\begin{center}
$\theta_{6,6}+\theta_{6,11}$\end{center}}\\\ \end{tabular}\end{pmatrix},\\\
\end{gathered}$ (4.1)

######  Theorem 4.1.

Main percolation theorem

The non-percolation probability $P_{M}$ (2.14) of a lattice model with a
Hamiltonian (2.8) satisfies the following equality

$\lim_{M\rightarrow\infty}{\frac{\ln{P_{M}}}{M}}=\ln{\frac{\mu_{\max}}{\lambda_{\max}}},$
(4.2)

where $\mu_{\max}=\mu_{\max}(T)$ is the largest root is the largest root of
the characteristic equation of the matrix $\tau^{\prime}$ with size $3\times
3$ (4.1), and $\lambda_{\max}=\lambda_{\max}(T)$ is the largest root of the
characteristic equation of the matrix $\tau$ (3.4).

###### Proof.

The proof is based on the commutation of the matrix $\theta^{\prime}$ with the
rotation matrix $D$ (3.8).

Based on the structure of the new matrix $\theta^{\prime}$, the eigenvector of
the matrix $\theta^{\prime}$ corresponding to the largest eigenvalue has the
form:

$(x_{1},x_{2},x_{2},0,x_{2},x_{3},0,0,x_{2},0,x_{3},0,0,0,0,0).$

The problem of finding the largest eigenvalue of the $\theta^{\prime}$ matrix
is reduced to finding the largest eigenvalue of the matrix $3\times 3$ with
size $\tau^{\prime}$ (4.1).

In this case the characteristic polynomial of the matrix $\tau^{\prime}$ (4.1)
is equal to:

$\begin{gathered}\mu^{3}+a^{\prime}\mu^{2}+b^{\prime}\mu+c^{\prime}=0\end{gathered}$
(4.3)

where

$\begin{gathered}a^{\prime}=-(\tau^{\prime}_{1,1}+\tau^{\prime}_{2,2}+\tau^{\prime}_{3,3})\end{gathered}$
(4.4)
$\begin{gathered}b^{\prime}=-\tau^{\prime}_{1,2}\tau^{\prime}_{2,1}+\tau^{\prime}_{1,1}\tau^{\prime}_{2,2}-\tau^{\prime}_{1,3}\tau^{\prime}_{3,1}-\tau^{\prime}_{2,3}\tau^{\prime}_{3,2}+\tau^{\prime}_{1,1}\tau^{\prime}_{3,3}+\tau^{\prime}_{2,2}\tau^{\prime}_{3,3}\end{gathered}$
(4.5)
$\begin{gathered}c^{\prime}=\tau^{\prime}_{1,3}\tau^{\prime}_{2,2}\tau^{\prime}_{3,1}-\tau^{\prime}_{1,2}\tau^{\prime}_{2,3}\tau^{\prime}_{3,1}-\tau^{\prime}_{1,3}\tau^{\prime}_{2,1}\tau^{\prime}_{3,2}+\tau^{\prime}_{1,1}\tau^{\prime}_{2,3}\tau^{\prime}_{3,2}+\tau^{\prime}_{1,2}\tau^{\prime}_{2,1}\tau^{\prime}_{3,3}-\\\
\tau^{\prime}_{1,1}\tau^{\prime}_{2,2}\tau^{\prime}_{3,3}\end{gathered}$ (4.6)

The largest root of the equation(4.3) is:

$\begin{gathered}\mu_{\max}=\frac{1}{3}(-a^{\prime}+2\sqrt{(a^{\prime})^{2}-3b^{\prime}}\sin[1/3(\arcsin(\frac{2(a^{\prime})^{3}-9a^{\prime}b^{\prime}+27c^{\prime}}{2((a^{\prime})^{2}-3b^{\prime})^{3/2}})+2\pi)]),\end{gathered}$
(4.7)

Then the non-percolation probability (2.14) is rewritten as follows

$\frac{1}{M}\ln{P_{M}}=\frac{1}{M}\ln{\frac{Z^{\prime}}{Z}}=\frac{1}{M}\ln{\frac{\mu_{\max}^{M}}{\lambda_{\max}^{M}}}\underset{M\rightarrow\infty}{\longrightarrow}\ln{\frac{\mu_{\max}}{\lambda_{\max}}}.$

∎

##  5\. Some simplest particular cases of solution

###  5.1. The first particular case

Now let us consider the Hamiltonian, with non-zero generating interactions
(other interactions are zero)

$\begin{gathered}J_{\\{t_{0}^{m},t_{0}^{m+1}\\}}=J_{1},J_{\\{t_{0}^{m},t_{1}^{m},t_{2}^{m},t_{3}^{m}\\}}=J_{2},J_{\\{t_{0}^{m+1},t_{1}^{m+1},t_{2}^{m+1},t_{3}^{m+1}\\}}=J_{2},\\\
J_{\\{t_{0}^{m},t_{1}^{m},t_{2}^{m},t_{3}^{m+1}\\}}=J_{3},J_{\\{t_{3}^{m},t_{0}^{m+1},t_{1}^{m+1},t_{2}^{m+1}\\}}=J_{4},J_{\\{t_{1}^{m},t_{2}^{m},t_{3}^{m},t_{1}^{m+1},t_{2}^{m+1},t_{3}^{m+1}\\}}=J_{5},\\\
J_{\\{t_{0}^{m},t_{1}^{m},t_{2}^{m},t_{3}^{m},t_{0}^{m+1},t_{1}^{m+1},t_{2}^{m+1},t_{3}^{m+1}\\}}=J_{6}.\end{gathered}$

The generating Hamiltonian of the model has the form

$\begin{gathered}\mathcal{\hat{H}}^{m}=J_{1}\sigma_{0}^{m}\sigma_{0}^{m+1}+J_{2}(\sigma_{0}^{m}\sigma_{1}^{m}\sigma_{2}^{m}\sigma_{3}^{m}+\sigma_{0}^{m+1}\sigma_{1}^{m+1}\sigma_{2}^{m+1}\sigma_{3}^{m+1})+\\\
J_{3}\sigma_{0}^{m}\sigma_{1}^{m}\sigma_{2}^{m}\sigma_{3}^{m+1}+J_{4}\sigma_{0}^{m}\sigma_{1}^{m+1}\sigma_{2}^{m+1}\sigma_{3}^{m+1}+J_{5}\sigma_{1}^{m}\sigma_{2}^{m}\sigma_{3}^{m}\sigma_{1}^{m+1}\sigma_{2}^{m+1}\sigma_{3}^{m+1}+\\\
J_{6}\sigma_{0}^{m}\sigma_{1}^{m}\sigma_{2}^{m}\sigma_{3}^{m}\sigma_{0}^{m+1}\sigma_{1}^{m+1}\sigma_{2}^{m+1}\sigma_{3}^{m+1}.\end{gathered}$
(5.1)

Further, for convenience, we introduce the notation $p,q,u,v$ with lower
indexing, where $p$ corresponds to the interaction coefficient of two spins,
$q$ \- four spins, $u$ \- six spins, $v$ \- eight spins.

Let
$p_{1}=\exp(J_{\\{t_{0}^{m},t_{0}^{m+1}\\}}),q_{1}=\exp(8J_{\\{t_{0}^{m},t_{1}^{m},t_{2}^{m},t_{3}^{m}\\}}),q_{2}=\exp(J_{\\{t_{0}^{m},t_{1}^{m},t_{2}^{m},t_{3}^{m+1}\\}}),\\\
q_{3}=\exp(J_{\\{t_{0}^{m},t_{1}^{m+1},t_{2}^{m+1},t_{3}^{m+1}\\}}),u_{1}=\exp(J_{\\{t_{0}^{m},t_{1}^{m},t_{2}^{m},t_{0}^{m+1},t_{1}^{m+1},t_{2}^{m+1}\\}}),\\\
v_{1}=\exp(4J_{\\{t_{0}^{m},t_{1}^{m},t_{2}^{m},t_{3}^{m},t_{0}^{m+1},t_{1}^{m+1},t_{2}^{m+1},t_{3}^{m+1}\\}})$
. Then the following theorem holds.

######  Theorem 5.1.

In the thermodynamic limit, the free energy of the model with the generating
Hamiltonian (5.1) and Hamiltonian (2.8,2.9) can be represented as (3.5),
internal energy as (3.6), heat capacity as (3.7), while

$\begin{gathered}\lambda_{\max}=(-a+\sqrt{a^{2}-4b})/2,\end{gathered}$ (5.2)

where

$\begin{gathered}a=-6v_{1}q_{1}^{-1}-6q_{1}v_{1}-q_{1}v_{1}p_{1}^{-4}q_{2}^{-4}q_{3}^{-4}u_{1}^{-4}-q_{2}^{4}q_{3}^{4}v_{1}p_{2}^{-4}q_{1}^{-1}u_{2}^{-4}-\\\
p_{2}^{4}u_{2}^{4}v_{1}q_{1}^{-1}q_{2}^{-4}q_{3}^{-4}-p_{1}^{4}q_{1}q_{2}^{4}q_{3}^{4}u_{1}^{4}v_{1},\end{gathered}$
(5.3)
$\begin{gathered}b=-16q_{3}^{4}q_{2}^{-4}v_{1}^{-2}-16q_{2}^{4}q_{3}^{-4}v_{1}^{-2}-16p_{1}^{4}u_{1}^{-4}v_{1}^{-2}-16u_{1}^{4}p_{1}^{-4}v_{1}^{-2}+36v_{1}^{2}+\\\
v_{1}^{2}q_{2}^{-8}q_{3}^{-8}+q_{3}^{8}q_{5}^{8}v_{1}^{2}+v_{1}^{2}p_{1}^{-8}u_{1}^{-8}+6v_{1}^{2}p_{2}^{-4}q_{5}^{-4}q_{3}^{-4}u_{1}^{-4}+6q_{5}^{4}q_{3}^{4}v_{1}^{2}p_{2}^{-4}u_{1}^{-4}+\\\
6p_{1}^{4}u_{1}^{4}v_{1}^{2}q_{2}^{-4}q_{3}^{-4}+6p_{1}^{4}q_{2}^{4}q_{3}^{4}u_{1}^{4}v_{1}^{2}+p_{1}^{8}u_{1}^{8}v_{1}^{2}.\end{gathered}$
(5.4)

###### Proof.

The choice of these nonzero interactions is justified by the fact that the
transfer matrix $\theta$ corresponding to such a Hamiltonian (5.1) commutes
with the matrix $\Phi$:

$\Phi=\setcounter{MaxMatrixCols}{16}\begin{pmatrix}0&0&0&1&0&0&0&0&0&0&0&0&0&0&0&0\\\
0&0&1&0&0&0&0&0&0&0&0&0&0&0&0&0\\\ 0&1&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\\
1&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&1&0&0&0&0&0&0&0&0\\\
0&0&0&0&0&0&1&0&0&0&0&0&0&0&0&0\\\ 0&0&0&0&0&1&0&0&0&0&0&0&0&0&0&0\\\
0&0&0&0&1&0&0&0&0&0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0&0&0&0&1&0&0&0&0\\\
0&0&0&0&0&0&0&0&0&0&1&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0&0&1&0&0&0&0&0&0\\\
0&0&0&0&0&0&0&0&1&0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&1\\\
0&0&0&0&0&0&0&0&0&0&0&0&0&0&1&0\\\ 0&0&0&0&0&0&0&0&0&0&0&0&0&1&0&0\\\
0&0&0&0&0&0&0&0&0&0&0&0&1&0&0&0\\\ \end{pmatrix}$ (5.5)

Let us write out the eigenvectors of the matrix $\Phi$ (5.5) corresponding to
the eigenvalue $1$:

$\begin{gathered}(0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1),\\\
(0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,0),\\\ (0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0),\\\
(0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0),\\\ (0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0),\\\
(0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0),\\\ (1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0),\\\
(0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0).\end{gathered}$ (5.6)

Then the eigenvector of the matrix $\tau$ corresponding to the largest
eigenvalue has the form:

$(x_{1},x_{2},x_{2},x_{1},x_{3},x_{4},x_{4},x_{3},x_{3},x_{4},x_{4},x_{3},x_{1},x_{2},x_{2},x_{1}).$

Taking into account the commutation of the transfer matrix (3.2) for the
considered model with the matrix $D$ (3.8), we find the eigenvector in the
form:

$(x_{1},x_{2},x_{2},x_{1},x_{2},x_{1},x_{1},x_{2},x_{2},x_{1},x_{1},x_{2},x_{1},x_{2},x_{2},x_{1}).$

This kind of eigenvector allows us to reduce the search for the largest
eigenvalue of the matrix $\theta$ to the matrix $\tau$ with size $2\times 2$
with elements:

$\begin{gathered}\tau=\begin{pmatrix}\begin{tabular}[]{p{3cm} p{3cm}}\tiny{
\begin{center}$\theta_{1,1}+\theta_{1,4}+\theta_{1,6}+\theta_{1,7}+\theta_{1,10}+\theta_{1,11}+\theta_{1,13}+\theta_{1,16}$
\end{center}}&\tiny{\begin{center}$\theta_{1,2}+\theta_{1,3}+\theta_{1,5}+\theta_{1,8}+\theta_{1,9}+\theta_{1,12}+\theta_{1,14}+\theta_{1,15}$
\end{center}}\\\ \tiny{
\begin{center}$\theta_{2,1}+\theta_{2,4}+\theta_{2,6}+\theta_{2,7}+\theta_{2,10}+\theta_{2,11}+\theta_{2,13}+\theta_{2,16}$
\end{center}}&\tiny{\begin{center}$\theta_{2,2}+\theta_{2,3}+\theta_{2,5}+\theta_{2,8}+\theta_{2,9}+\theta_{2,12}+\theta_{2,14}+\theta_{2,15}$
\end{center}}\\\ \end{tabular}\end{pmatrix}.\\\ \end{gathered}$ (5.7)

The characteristic polynomial of matrix $\tau$ (5.7) is:

$\begin{gathered}\lambda^{2}+a\lambda+b=0.\end{gathered}$ (5.8)

Due to the simplicity of the model, the coefficients of the characteristic
polynomial are written explicitly (5.3, 5.4), the largest root of the equation
is determined by the formula (5.2).

The polynomial containing the largest eigenvalue of the transfer matrix of
non-percolation configurations will be of the third degree and will have the
form (4.3), polynomial coefficients are determined by the formulas (4.4-4.6),
and the largest eigenvalue of the transfer matrix of non-percolation
configurations (4.7).

∎

###  5.2. The second particular case

Let us consider one more special case, when only interactions remain nonzero

$\begin{gathered}J_{\\{t_{0}^{m},t_{0}^{m+1}\\}}=J_{1},J_{\\{t_{0}^{m},t_{1}^{m},t_{0}^{m+1},t_{1}^{m+1}\\}}=J_{2},J_{\\{t_{0}^{m},t_{2}^{m},t_{0}^{m+1},t_{2}^{m+1}\\}}=J_{3},\\\
J_{\\{t_{1}^{m},t_{2}^{m},t_{3}^{m},t_{1}^{m+1},t_{2}^{m+1},t_{3}^{m+1}\\}}=J_{4},J_{\\{t_{0}^{m},t_{1}^{m},t_{2}^{m},t_{3}^{m},t_{0}^{m+1},t_{1}^{m+1},t_{2}^{m+1},t_{3}^{m+1}\\}}=J_{5}.\end{gathered}$

The generating Hamiltonian of the model has the form:

$\begin{gathered}\mathcal{\hat{H}}^{m}=J_{1}\sigma_{0}^{m}\sigma_{0}^{m+1}+J_{2}\sigma_{0}^{m}\sigma_{1}^{m}\sigma_{0}^{m+1}\sigma_{1}^{m+1}+J_{3}\sigma_{0}^{m}\sigma_{2}^{m}\sigma_{0}^{m+1}\sigma_{2}^{m+1}+\\\
J_{4}\sigma_{1}^{m}\sigma_{2}^{m}\sigma_{3}^{m}\sigma_{1}^{m+1}\sigma_{2}^{m+1}\sigma_{3}^{m+1}+\\\
J_{5}\sigma_{0}^{m}\sigma_{1}^{m}\sigma_{2}^{m}\sigma_{3}^{m}\sigma_{0}^{m+1}\sigma_{1}^{m+1}\sigma_{2}^{m+1}\sigma_{3}^{m+1},\end{gathered}$
(5.9)

Let
$p_{1}=\exp(J_{\\{t_{0}^{m},t_{0}^{m+1}\\}}),q_{1}=\exp(J_{\\{t_{0}^{m},t_{1}^{m},t_{0}^{m+1},t_{1}^{m+1}\\}}),q_{2}=\exp(J_{\\{t_{0}^{m},t_{2}^{m},t_{0}^{m+1},t_{2}^{m+1}\\}}),\\\
u_{1}=\exp(J_{\\{t_{0}^{m},t_{1}^{m},t_{2}^{m},t_{0}^{m+1},t_{1}^{m+1},t_{2}^{m+1}\\}}),\\\
v_{1}=\exp(4J_{\\{t_{0}^{m},t_{1}^{m},t_{2}^{m},t_{3}^{m},t_{0}^{m+1},t_{1}^{m+1},t_{2}^{m+1},t_{3}^{m+1}\\}})$
.

######  Theorem 5.2.

In the thermodynamic limit, the free energy of the model with the generating
Hamiltonian (5.9) and the Hamiltonian (2.8-2.9) can be represented as (3.5),
the internal energy as (3.6), heat capacity in the form (3.7), while

$\begin{gathered}\lambda_{\max}=4p_{1}^{2}u_{1}^{-2}v_{1}^{-1}+4u_{1}^{2}p_{1}^{-2}v_{1}^{-1}+4v_{1}q_{2}^{-2}+2q_{2}^{2}v_{1}q_{1}^{-4}+q_{1}^{4}q_{2}^{2}v_{1}p_{1}^{-4}u_{1}^{-4}+\\\
p_{1}^{4}q_{1}^{4}q_{2}^{2}u_{1}^{4}v_{1}.\end{gathered}$ (5.10)

###### Proof.

The transfer matrix of such an interaction commutes with the matrix $\Psi$:

$\Psi=\setcounter{MaxMatrixCols}{16}\begin{pmatrix}0&1&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\\
1&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\\ 0&0&0&1&0&0&0&0&0&0&0&0&0&0&0&0\\\
0&0&1&0&0&0&0&0&0&0&0&0&0&0&0&0\\\ 0&0&0&0&0&1&0&0&0&0&0&0&0&0&0&0\\\
0&0&0&0&1&0&0&0&0&0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&1&0&0&0&0&0&0&0&0\\\
0&0&0&0&0&0&1&0&0&0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0&0&1&0&0&0&0&0&0\\\
0&0&0&0&0&0&0&0&1&0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0&0&0&0&1&0&0&0&0\\\
0&0&0&0&0&0&0&0&0&0&1&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0&0&0&0&0&0&1&0&0\\\
0&0&0&0&0&0&0&0&0&0&0&0&1&0&0&0\\\ 0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&1\\\
0&0&0&0&0&0&0&0&0&0&0&0&0&0&1&0\\\ \end{pmatrix}$ (5.11)

Let us write out the eigenvectors of the matrix $\Psi$ (5.11) corresponding to
the eigenvalue $1$:

$\begin{gathered}(1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0),\\\
(0,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0),\\\ (0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,0),\\\
(0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0),\\\ (0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0),\\\
(0,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0),\\\ (0,0,0,0,0,0,0,0,0,0,0,0,1,1,0,0),\\\
(0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1).\end{gathered}$ (5.12)

Then the eigenvector of the matrix $\theta$, corresponding to the largest
eigenvalue, can be found in the form:

$(x_{1},x_{1},x_{2},x_{2},x_{3},x_{3},x_{4},x_{4},x_{4},x_{4},x_{3},x_{3},x_{2},x_{2},x_{1},x_{1}).$

Taking into account the commutation with the matrix $D$ (3.14), the
eigenvector corresponding to the largest eigenvalue has a one-component form

$(x_{1},x_{1},x_{1},x_{1},x_{1},x_{1},x_{1},x_{1},x_{1},x_{1},x_{1},x_{1},x_{1},x_{1},x_{1},x_{1}).$

Then the largest eigenvalue $\lambda_{\max}$ has the form (5.10).

As in the previous special case, the degree of the polynomial containing the
largest eigenvalue for tight configurations is not reduced (4.3), polynomial
coefficients are determined by the formulas (4.4-4.6), and the largest
eigenvalue of the transfer matrix of non-percolation configurations according
to the formula (4.7).

∎

##  6\. The Gonihedric Ising Model

###  6.1. Exact solution of the gonigendrial model with free boundary
conditions

Let us consider a gonigendrial model [5],[6], [7] or a model with the
interaction of nearest neighbors, next nearest neighbors and plaquettes, and

$\begin{gathered}2J_{\\{t_{0}^{m},t_{1}^{m}\\}}=2J_{\\{t_{0}^{m+1},t_{1}^{m+1}\\}}=J_{\\{t_{0}^{m},t_{0}^{m+1}\\}}=J_{1},\\\
4J_{\\{t_{0}^{m},t_{2}^{m}\\}}=4J_{\\{t_{0}^{m+1},t_{2}^{m+1}\\}}=J_{\\{t_{0}^{m},t_{1}^{m+1}\\}}=J_{\\{t_{0}^{m},t_{3}^{m+1}\\}}=J_{2},\\\
8J_{\\{t_{0}^{m},t_{1}^{m},t_{2}^{m},t_{3}^{m}\\}}=8J_{\\{t_{0}^{m+1},t_{1}^{m+1},t_{2}^{m+1},t_{3}^{m+1}\\}}=J_{\\{t_{0}^{m},t_{1}^{m},t_{0}^{m+1},t_{1}^{m+1}\\}}=J_{3},\end{gathered}$

and the remaining coefficients of the Hamiltonian (2.8) are zero.

The classical notation of this Hamiltonian [6] for the considering model is

$\begin{gathered}\mathcal{H}(\sigma)=-\sum_{m=0}^{M-1}(J_{1}((\sigma_{0}^{m}\sigma_{1}^{m}+\sigma_{1}^{m}\sigma_{2}^{m}+\sigma_{2}^{m}\sigma_{3}^{m}+\sigma_{3}^{m}\sigma_{0}^{m}+\sigma_{0}^{m+1}\sigma_{1}^{m+1}+\\\
\sigma_{1}^{m+1}\sigma_{2}^{m+1}+\sigma_{2}^{m+1}\sigma_{3}^{m+1}+\sigma_{3}^{m+1}\sigma_{0}^{m+1})/2+\sigma_{0}^{m}\sigma_{0}^{m+1}+\sigma_{1}^{m}\sigma_{1}^{m+1}+\\\
\sigma_{2}^{m}\sigma_{2}^{m+1}+\sigma_{3}^{m}\sigma_{3}^{m+1})+J_{2}((\sigma_{0}^{m}\sigma_{2}^{m}+\sigma_{1}^{m}\sigma_{3}^{m}+\sigma_{0}^{m+1}\sigma_{2}^{m+1}+\\\
\sigma_{1}^{m+1}\sigma_{3}^{m+1})/2+\sigma_{0}^{m}\sigma_{1}^{m+1}+\sigma_{1}^{m}\sigma_{2}^{m+1}+\sigma_{2}^{m}\sigma_{3}^{m+1}+\sigma_{3}^{m}\sigma_{0}^{m+1}+\sigma_{0}^{m}\sigma_{3}^{m+1}+\\\
\sigma_{1}^{m}\sigma_{0}^{m+1}+\sigma_{2}^{m}\sigma_{1}^{m+1}+\sigma_{3}^{m}\sigma_{2}^{m+1})+J_{4}((\sigma_{0}^{m}\sigma_{1}^{m}\sigma_{2}^{m}\sigma_{3}^{m}+\\\
\sigma_{0}^{m+1}\sigma_{1}^{m+1}\sigma_{2}^{m+1}\sigma_{3}^{m+1})/2+\sigma_{0}^{m}\sigma_{1}^{m}\sigma_{0}^{m+1}\sigma_{1}^{m+1}+\sigma_{1}^{m}\sigma_{2}^{m}\sigma_{1}^{m+1}\sigma_{2}^{m+1}+\\\
\sigma_{2}^{m}\sigma_{3}^{m}\sigma_{2}^{m+1}\sigma_{3}^{m+1}+\sigma_{3}^{m}\sigma_{0}^{m}\sigma_{3}^{m+1}\sigma_{0}^{m+1})).\end{gathered}$
(6.1)

Let us assume that
$p_{1}=\exp(J_{\\{t_{0}^{m},t_{0}^{m+1}\\}}),p_{2}=\exp(J_{\\{t_{0}^{m},t_{1}^{m+1}\\}}),\\\
q_{1}=\exp(J_{\\{t_{0}^{m},t_{1}^{m},t_{0}^{m+1},t_{1}^{m+1}\\}})$ .

Then the matrices $\tau$ (3.4) and $\tau^{\prime}$ (4.1) take the form:

$\begin{gathered}\tau=\begin{pmatrix}\begin{tabular}[]{p{1.5cm} p{3cm}
p{2.5cm} p{1.5cm}}\tiny{
\begin{center}$q_{1}^{5}p_{2}^{-6}+p_{1}^{8}p_{2}^{10}q_{1}^{5}$
\end{center}}&\tiny{\begin{center}$4p_{2}^{-3}+4p_{1}^{4}p_{2}^{5}$
\end{center}}&\tiny{\begin{center}$4p_{1}^{2}q_{1}$
\end{center}}&\tiny{\begin{center} $2p_{2}^{2}q_{1}^{-3}$\end{center}}\\\
\tiny{\begin{center}$p_{2}^{-3}+p_{1}^{4}p_{2}^{5}$ \end{center}}&\tiny{
\begin{center}$2q_{1}^{-5}+2p_{2}^{-4}q_{1}^{-1}+2p_{2}^{4}q_{1}^{-1}+q_{1}^{3}p_{1}^{-4}+p_{1}^{4}q_{1}^{3}$
\end{center}}&\tiny{
\begin{center}$2p_{1}^{-2}p_{2}^{-1}+2p_{1}^{2}p_{2}^{-1}$
\end{center}}&\tiny{\begin{center}
$p_{2}^{-3}+p_{2}^{5}p_{1}^{-4}$\end{center}}\\\
\tiny{\begin{center}$2p_{1}^{2}q_{1}$ \end{center}}&\tiny{
\begin{center}$4p_{1}^{-2}p_{2}^{-1}+4p_{1}^{2}p_{2}^{-1}$
\end{center}}&\tiny{
\begin{center}$2p_{2}^{-2}q_{1}^{-3}+q_{1}^{5}p_{1}^{-4}p_{2}^{-2}+p_{1}^{4}q_{1}^{5}p_{2}^{-2}$
\end{center}}&\tiny{\begin{center} $2q_{1}p_{1}^{-2}$\end{center}}\\\
\tiny{\begin{center}$2p_{2}q_{1}^{-2}$
\end{center}}&\tiny{\begin{center}$4p_{2}^{-3}+4p_{2}^{5}p_{1}^{-4}$
\end{center}}&\tiny{\begin{center}$4q_{1}p_{1}^{-2}$
\end{center}}&\tiny{\begin{center}
$q_{1}^{5}p_{2}^{-6}+p_{2}^{10}q_{1}^{5}p_{1}^{-8}$\end{center}}\end{tabular}\end{pmatrix},\\\
\end{gathered}$ (6.2)

$\begin{gathered}\tau^{\prime}=\begin{pmatrix}\begin{tabular}[]{p{1.5cm}
p{3cm} p{2.5cm}}\tiny{ \begin{center}$p_{1}^{8}p_{2}^{10}q_{1}^{5}$
\end{center}}&\tiny{\begin{center}$4p_{1}^{4}p_{2}^{5}$
\end{center}}&\tiny{\begin{center}$2p_{2}^{2}q_{1}^{-3}$ \end{center}}\\\
\tiny{\begin{center}$p_{1}^{4}p_{2}^{5}$ \end{center}}&\tiny{
\begin{center}$q_{1}^{-5}+2p_{2}^{4}q_{1}^{-1}+p_{1}^{4}q_{1}^{3}$
\end{center}}&\tiny{ \begin{center}$p_{2}^{-3}+p_{2}^{5}p_{1}^{-4}$
\end{center}}\\\ \tiny{\begin{center}$p_{2}q_{1}^{-2}$ \end{center}}&\tiny{
\begin{center}$2p_{2}^{-3}+2p_{2}^{5}p_{1}^{-4}$ \end{center}}&\tiny{
\begin{center}$q_{1}^{5}p_{2}^{-6}+p_{2}^{10}q_{1}^{5}p_{1}^{-8}$\end{center}}\\\
\end{tabular}\end{pmatrix}.\\\ \end{gathered}$ (6.3)

Now we use Theorem 3.1 and Theorem 4.1, and write down as their corollary the
results for the gonigendrial model

######  Theorem 6.1.

In the thermodynamic limit, the free energy of the gonigendrial model with the
Hamiltonian (6.1) is represented as (3.5), the internal energy as (3.6), the
heat capacity as (3.7), where $\lambda_{\max}$ is the largest root of the
characteristic equation of the fourth power of the matrix $\tau$ (6.2), and
the relation (4.2) is true for the non-percolation probability, where
$\mu_{\max}$ is the largest root of the characteristic equation of the matrix
$\tau^{\prime}$ (6.3).

The coefficients of the characteristic polynomials for $\tau$ (6.2) and
$\tau^{\prime}$ (6.3) are written out in the formulas (3.11)-(3.14) и
(4.4)-(4.6). $\lambda_{max}$ is found according to the formulas (3.10-3.18),
and $\mu_{max}$ is found according to the formula (4.7).

In the classical setting [5] the coefficients of the gonigendrial model are
related by the relations $J_{2}=rJ_{1}$, $J_{3}=4k/(1-k)$. For example, [7]
obtained phase diagrams for $k=0$ and $k=1/3$. Let us consider similar cases
(example 6.1 and example 6.2).

######  Example 6.1.

Let us find the thermodynamic characteristics of the gonigendrial model for
$J_{1}=1$, $J_{2}=r$, $J_{3}=0$, $T\in[0.1,5]$, $r\in[-2,2]$ in the absence of
an external magnetic field. One can see the appearance of phase transition
lines beginning at $T>0$, $r=-1$ and $r=-0.25$ on the heat capacity graph
(Fig.4).

FIGURE 4. The heat capacity graph for the example 6.1.

Large gradients of the impermeability probability (Fig. 5) repeat the lines of
incipient phase transitions of the heat capacity.

FIGURE 5. The probability of non-percolation for the example 6.1.

######  Example 6.2.

Under the conditions of the previous example, let us consider the case of a
nonzero plaquette interaction $J_{3}=1/2$, $T\in[0.1,5]$, $r\in[-2,2]$. One
can see the appearance of phase transition lines beginning at $T>0$, $r=-1$
and $r=-0.25$ on the heat capacity graph.

At $T\rightarrow 0$, the heat capacity graph (Fig.6) shows a longer break at
the points $r=-1$ and $r=-0.25$ than the one observed in the 6.1 example in
Fig. 4.

FIGURE 6. The heat capacity graph for the example 6.2.

###  6.2. Numerical solution of the gonigendrial model $3\times 3\times\infty$

The authors of [7] obtained a phase plane, where the phase transition line
corresponds to the value $r=-0.25$. Thus, the shape of the phase plane differs
from the picture obtained for the strip $2\times 2\times\infty$. In this
regard, the authors assumed that the model with the $2\times 2$ section is
strongly subject to boundary conditions, so the gonigendrial model was studied
numerically $3\times 3\times\infty$.

In the case of free boundary conditions with the Hamiltonian (6.4) we have:

$\mathcal{H}(\sigma)=-\sum_{m=0}^{M-1}(J_{1}(\frac{1}{2}\sum_{i=0}^{2}\sum_{j=0}^{1}(\sigma_{i,j}^{m}\sigma_{i,j+1}^{m}+\sigma_{j,i}^{m}\sigma_{j+1,i}^{m}+\sigma_{i,j}^{m+1}\sigma_{i,j+1}^{m+1}+\\\
\sigma_{j,i}^{m+1}\sigma_{j+1,i}^{m+1})+\sum_{i=0}^{2}\sum_{j=0}^{2}\sigma_{i,j}^{m}\sigma_{i,j}^{m+1})+J_{2}(\frac{1}{2}\sum_{i=0}^{1}\sum_{j=0}^{1}(\sigma_{i,j}^{m}\sigma_{i+1,j+1}^{m}+\\\
\sigma_{i,j+1}^{m}\sigma_{i+1,j}^{m}+\sigma_{i,j}^{m+1}\sigma_{i+1,j+1}^{m+1}+\sigma_{i,j+1}^{m+1}\sigma_{i+1,j}^{m+1})+\\\
\sum_{i=0}^{2}\sum_{j=0}^{1}(\sigma_{i,j}^{m}\sigma_{i,j+1}^{m+1}+\sigma_{i,j+1}^{m}\sigma_{i,j}^{m+1}+\sigma_{j,i}^{m}\sigma_{j+1,i}^{m+1}+\sigma_{j+1,i}^{m}\sigma_{j,i}^{m+1}))\\\
+J_{3}(\frac{1}{2}\sum_{i=0}^{1}\sum_{j=0}^{1}(\sigma_{i,j}^{m}\sigma_{i,j+1}^{m}\sigma_{i+1,j}^{m}\sigma_{i+1,j+1}^{m}+\sigma_{i,j}^{m+1}\sigma_{i,j+1}^{m+1}\sigma_{i+1,j}^{m+1}\sigma_{i+1,j+1}^{m})+\\\
\sum_{i=0}^{2}\sum_{j=0}^{1}(\sigma_{i,j}^{m}\sigma_{i,j+1}^{m}\sigma_{i,j}^{m+1}\sigma_{i,j+1}^{m+1}+\sigma_{j,i}^{m}\sigma_{j+1,i}^{m}\sigma_{j,i}^{m+1}\sigma_{j+1,i}^{m+1})).$
(6.4)

Results are obtained (Fig.7 in the 6.3 example), similar to the results in the
6.1 example and the example 6.2.

In the cyclically closed case (Fig. 8 in the 6.3 example) with the Hamiltonian
(6.5):

$\mathcal{H}(\sigma)=-\sum_{m=0}^{M-1}(J_{1}(\frac{1}{2}\sum_{i=0}^{2}\sum_{j=0}^{2}(\sigma_{i,j}^{m}\sigma_{i,j+1}^{m}+\sigma_{j,i}^{m}\sigma_{j+1,i}^{m}+\sigma_{i,j}^{m+1}\sigma_{i,j+1}^{m+1}+\\\
\sigma_{j,i}^{m+1}\sigma_{j+1,i}^{m+1}+\sigma_{i,j}^{m}\sigma_{i,j}^{m+1}))+J_{2}(\sum_{i=0}^{2}\sum_{j=0}^{2}((\sigma_{i,j}^{m}\sigma_{i+1,j+1}^{m}+\\\
\sigma_{i,j+1}^{m}\sigma_{i+1,j}^{m}+\sigma_{i,j}^{m+1}\sigma_{i+1,j+1}^{m+1}+\sigma_{i,j+1}^{m+1}\sigma_{i+1,j}^{m+1})/2+\sigma_{i,j}^{m}\sigma_{i,j+1}^{m+1}+\sigma_{i,j+1}^{m}\sigma_{i,j}^{m+1}+\\\
\sigma_{j,i}^{m}\sigma_{j+1,i}^{m+1}+\sigma_{j+1,i}^{m}\sigma_{j,i}^{m+1}))+J_{3}(\sum_{i=0}^{2}\sum_{j=0}^{2}((\sigma_{i,j}^{m}\sigma_{i,j+1}^{m}\sigma_{i+1,j}^{m}\sigma_{i+1,j+1}^{m}+\\\
\sigma_{i,j}^{m+1}\sigma_{i,j+1}^{m+1}\sigma_{i+1,j}^{m+1}\sigma_{i+1,j+1}^{m})/2+\sigma_{i,j}^{m}\sigma_{i,j+1}^{m}\sigma_{i,j}^{m+1}\sigma_{i,j+1}^{m+1}+\\\
\sigma_{j,i}^{m}\sigma_{j+1,i}^{m}\sigma_{j,i}^{m+1}\sigma_{j+1,i}^{m+1})),$
(6.5)

where
$\sigma^{m}_{i,3}=\sigma^{m}_{i,0},\sigma^{m}_{3,i}=\sigma^{m}_{0,i},\sigma^{m}_{3,3}=\sigma^{m}_{0,0},i=0,1,2,$
obtained results close to the results of the work [7]. Therefore, further in
6.3 we will consider an analog of the cyclically closed gonigendrial model
$2\times 2\times\infty$.

######  Example 6.3.

Let us find the heat capacity in the thermodynamic limit of the gonigendrial
model $3\times 3\times\infty$ with free boundary conditionsи and for
cyclically closed gonigendrial model with $J_{1}=1$, $J_{2}=r$, $J_{3}=0$,
$T\in[0.1,5]$, $r\in[-2,2]$ in the absence of an external magnetic field.

FIGURE 7. The heat capacity of the $3\times 3\times\infty$ gonigendrial model
with free boundary conditions for the example 6.3.

FIGURE 8. The heat capacity of a cyclically closed gonigendrial model $3\times
3\times\infty$ for the example 6.3.

For the open model (Fig.7), we observe a shift to the left relative to the
value of $r=-0.25$ and a smoothing of the peak in the region of $r=-1$. The
closed model, in turn, has a similar appearance to the phase diagram from [7].
Therefore, the authors decided to consider an analog of the cyclically closed
gonigendrial model for the case $2\times 2\times\infty$.

###  6.3. Exact solution of a cyclically closed gonigendrial model $2\times
2\times\infty$

The general form of the Hamiltonian (2.8) allows authors to obtain the exact
solution for the analog of the cyclically closed gonigendrial model $2\times
2\times\infty$, if we put:

$\begin{gathered}J_{\\{t_{0}^{m},t_{1}^{m}\\}}=J_{\\{t_{0}^{m+1},t_{1}^{m+1}\\}}=J_{\\{t_{0}^{m},t_{0}^{m+1}\\}}=J_{1},\\\
8/3J_{\\{t_{0}^{m},t_{2}^{m}\\}}=8/3J_{\\{t_{0}^{m+1},t_{2}^{m+1}\\}}=J_{\\{t_{0}^{m},t_{1}^{m+1}\\}}=J_{\\{t_{0}^{m},t_{3}^{m+1}\\}}=J_{2},\\\
16/3J_{\\{t_{0}^{m},t_{1}^{m},t_{2}^{m},t_{3}^{m}\\}}=16/3J_{\\{t_{0}^{m+1},t_{1}^{m+1},t_{2}^{m+1},t_{3}^{m+1}\\}}=J_{\\{t_{0}^{m},t_{1}^{m},t_{0}^{m+1},t_{1}^{m+1}\\}}=J_{3},\end{gathered}$

other coefficients are zero.

The classical notation of the Hamiltonian takes the following form

$\begin{gathered}\mathcal{H}(\sigma)=-\sum_{m=0}^{M-1}(J_{1}((\sigma_{0}^{m}\sigma_{1}^{m}+\sigma_{1}^{m}\sigma_{2}^{m}+\sigma_{2}^{m}\sigma_{3}^{m}+\sigma_{3}^{m}\sigma_{0}^{m}+\sigma_{0}^{m+1}\sigma_{1}^{m+1}+\\\
\sigma_{1}^{m+1}\sigma_{2}^{m+1}+\sigma_{2}^{m+1}\sigma_{3}^{m+1}+\sigma_{3}^{m+1}\sigma_{0}^{m+1})+\sigma_{0}^{m}\sigma_{0}^{m+1}+\sigma_{1}^{m}\sigma_{1}^{m+1}+\\\
\sigma_{2}^{m}\sigma_{2}^{m+1}+\sigma_{3}^{m}\sigma_{3}^{m+1})+J_{2}(3(\sigma_{0}^{m}\sigma_{2}^{m}+\sigma_{1}^{m}\sigma_{3}^{m}+\sigma_{0}^{m+1}\sigma_{2}^{m+1}+\\\
\sigma_{1}^{m+1}\sigma_{3}^{m+1})/4+\sigma_{0}^{m}\sigma_{1}^{m+1}+\sigma_{1}^{m}\sigma_{2}^{m+1}+\sigma_{2}^{m}\sigma_{3}^{m+1}+\sigma_{3}^{m}\sigma_{0}^{m+1}+\sigma_{0}^{m}\sigma_{3}^{m+1}+\\\
\sigma_{1}^{m}\sigma_{0}^{m+1}+\sigma_{2}^{m}\sigma_{1}^{m+1}+\sigma_{3}^{m}\sigma_{2}^{m+1})+J_{3}(3(\sigma_{0}^{m}\sigma_{1}^{m}\sigma_{2}^{m}\sigma_{3}^{m}+\\\
\sigma_{0}^{m+1}\sigma_{1}^{m+1}\sigma_{2}^{m+1}\sigma_{3}^{m+1})/8+\sigma_{0}^{m}\sigma_{1}^{m}\sigma_{0}^{m+1}\sigma_{1}^{m+1}+\sigma_{1}^{m}\sigma_{2}^{m}\sigma_{1}^{m+1}\sigma_{2}^{m+1}+\\\
\sigma_{2}^{m}\sigma_{3}^{m}\sigma_{2}^{m+1}\sigma_{3}^{m+1}+\sigma_{3}^{m}\sigma_{0}^{m}\sigma_{3}^{m+1}\sigma_{0}^{m+1})).\end{gathered}$
(6.6)

Let us explain this choice of ratios for the coefficients. With cyclic
closure, the interaction of the nearest neighbors
$J_{\\{t_{0}^{m},t_{1}^{m}\\}},J_{\\{t_{0}^{m+1},t_{1}^{m+1}\\}}$ doubles,
therefore, with taking into account the symmetric notation of the Hamiltonian,
the interaction is equated to $J_{\\{t_{0}^{m},t_{0}^{m+1}\\}}$. For the next
nearest neighbors, we have an intermediate relation
$4/3J_{\\{t_{0}^{m},t_{2}^{m}\\}}=4/3J_{\\{t_{0}^{m+1},t_{2}^{m+1}\\}}=1/2J_{\\{t_{0}^{m},t_{1}^{m+1}\\}}=1/2J_{\\{t_{0}^{m},t_{3}^{m+1}\\}}$,
where the factor 4 for
$J_{\\{t_{0}^{m},t_{2}^{m}\\}},J_{\\{t_{0}^{m+1},t_{2}^{m+1}\\}}$ follows from
the symmetric notation and taking into account duplication of interactions in
the Hamiltonian (2.9), and the divisor is 3 from the increase in the number of
interactions during cyclic closure; 1/2 for $J_{\\{t_{0}^{m},t_{3}^{m+1}\\}}$
follows from doubling under cyclic closure. Similarly, one can obtain the
relation to plaquettes if we take into account that the Hamiltonian (2.9)
takes into account the interactions
$J_{\\{t_{0}^{m},t_{1}^{m},t_{2}^{m},t_{3}^{m}\\}},J_{\\{t_{0}^{m+1},t_{1}^{m+1},t_{2}^{m+1},t_{3}^{m+1}\\}}$
four times.

Let us introduce the matrices $\tau$ (3.4) and $\tau^{\prime}$ (4.1)

$\begin{gathered}\tau=\begin{pmatrix}\begin{tabular}[]{p{1.5cm} p{3cm}
p{2.5cm} p{1.5cm}}\tiny{
\begin{center}$p_{1}^{4}q_{1}^{11/2}p_{2}^{-5}+p_{1}^{12}p_{2}^{11}q_{1}^{11/2}$
\end{center}}&\tiny{\begin{center}$4p_{1}^{2}p_{2}^{-5/2}+4p_{1}^{6}p_{2}^{11/2}$
\end{center}}&\tiny{\begin{center}$4p_{1}^{4}q_{1}^{3/2}$
\end{center}}&\tiny{\begin{center} $2p_{2}^{3}q_{1}^{-5/2}$\end{center}}\\\
\tiny{\begin{center}$p_{1}^{2}p_{2}^{-5/2}+p_{1}^{6}p_{2}^{11/2}$
\end{center}}&\tiny{
\begin{center}$2q_{1}^{-11/2}+2p_{2}^{-4}q_{1}^{-3/2}+2p_{2}^{4}q_{1}^{-3/2}+q_{1}^{5/2}p_{1}^{-4}+p_{1}^{4}q_{1}^{5/2}$
\end{center}}&\tiny{
\begin{center}$2p_{1}^{-2}p_{2}^{-3/2}+2p_{1}^{2}p_{2}^{-3/2}$
\end{center}}&\tiny{\begin{center}
$p_{1}^{-2}p_{2}^{-5/2}+p_{2}^{11/2}p_{1}^{-6}$\end{center}}\\\
\tiny{\begin{center}$2p_{1}^{4}q_{1}^{3/2}$ \end{center}}&\tiny{
\begin{center}$4p_{1}^{-2}p_{2}^{-3/2}+4p_{1}^{2}p_{2}^{-3/2}$
\end{center}}&\tiny{
\begin{center}$2p_{2}^{-3}q_{1}^{-5/2}+q_{1}^{11/2}p_{1}^{-4}p_{2}^{-3}+p_{1}^{4}q_{1}^{11/2}p_{2}^{-3}$
\end{center}}&\tiny{\begin{center} $2q_{1}^{3/2}p_{1}^{-4}$\end{center}}\\\
\tiny{\begin{center}$2p_{2}^{3}q_{1}^{-5/2}$
\end{center}}&\tiny{\begin{center}$4p_{1}^{-2}p_{2}^{-5/2}+4p_{2}^{11/2}p_{1}^{-6}$
\end{center}}&\tiny{\begin{center}$4q_{1}^{3/2}p_{1}^{-4}$
\end{center}}&\tiny{\begin{center}
$q_{1}^{11/2}p_{1}^{-4}p_{2}^{-5}+p_{2}^{11}q_{1}^{11/2}p_{1}^{-12}$\end{center}}\end{tabular}\end{pmatrix},\\\
\end{gathered}$ (6.7)

$\begin{gathered}\tau^{\prime}=\begin{pmatrix}\begin{tabular}[]{p{1.5cm}
p{3cm} p{2.5cm}}\tiny{ \begin{center}$p_{1}^{12}p_{2}^{11}q_{1}^{11/2}$
\end{center}}&\tiny{\begin{center}$4p_{1}^{6}p_{2}^{11/2}$
\end{center}}&\tiny{\begin{center}$2p_{2}^{3}q_{1}^{-5/2}$ \end{center}}\\\
\tiny{\begin{center}$p_{1}^{6}p_{2}^{11/2}$ \end{center}}&\tiny{
\begin{center}$2p_{1}^{-2}p_{2}^{-5/2}+2p_{2}^{11/2}p_{1}^{-6}$
\end{center}}&\tiny{
\begin{center}$p_{1}^{-2}p_{2}^{-5/2}+p_{2}^{11/2}p_{1}^{-6}$ \end{center}}\\\
\tiny{\begin{center}$p_{2}^{3}q_{1}^{-5/2}$ \end{center}}&\tiny{
\begin{center}$2p_{1}^{-2}p_{2}^{-5/2}+2p_{2}^{11/2}p_{1}^{-6}$
\end{center}}&\tiny{
\begin{center}$q_{1}^{11/2}p_{1}^{-4}p_{2}^{-5}+p_{2}^{11}q_{1}^{11/2}p_{1}^{-12}$\end{center}}\\\
\end{tabular}\end{pmatrix},\\\ \end{gathered}$ (6.8)

Now we use Theorem 3.1 and Theorem 4.1 and write the results for the
cyclically closed gonigendrial model

######  Theorem 6.2.

In the thermodynamic limit, the free energy of the gonigendrial model with the
Hamiltonian (6.6) is represented as (3.5), the internal energy as (3.6), the
heat capacity as (3.7), where $\lambda_{\max}$ is the largest root of the
characteristic equation of the matrix $\tau$ (6.7), and for the non-
percolation probability, the relation is true (4.2), where $\mu_{\max}$ is the
largest root of the characteristic equation of the matrix $\tau^{\prime}$
(6.8).

The coefficients of the characteristic polynomials for $\tau$ (6.2) and
$\tau^{\prime}$ (6.3) are written out in the formulas (3.11)-(3.14) and
(4.4)-(4.6). $\lambda_{max}$ is found using the formulas (3.10-3.18), and
$\mu_{max}$ is found using the formula (4.7).

######  Example 6.4.

Let us find the heat capacity of the closed gonigendrial model for $J_{1}=1$,
$J_{2}=r$, $J_{3}=0$, $T\in[0.1,5]$, $r\in[-2,2]$ in the absence of an
external magnetic field. One can see the nucleation of phase transition lines
, starting at $T>0$, $r=-0.25$ on the heat capacity plot (Fig. 9), as was
observed for $3\times 3\times\infty$ in the 6.3 example.

FIGURE 9. The heat capacity of a closed gonigendrial model for an example 6.4.

FIGURE 10. The heat capacity of a closed gonigendrial model for an example 6.4
for $T=0.2$, $T=0.3$, $T=0.4$, $T=0.5$.

Here one can already see a great similarity with the phase pattern for the
gonigendrial model with similar interactions in the work [7].

##  7\. Antiferromagnetic layered Ising model

We consider the Hamiltonian that takes into account only the interactions of
nearest neighbors and interactions of next nearest neighbors, and
$2J_{\\{t_{0}^{m},t_{1}^{m}\\}}=2J_{\\{t_{0}^{m+1},t_{1}^{m+1}\\}}=J_{\\{t_{0}^{m},t_{0}^{m+1}\\}}=J_{1}$,
$4J_{\\{t_{0}^{m},t_{2}^{m}\\}}=4J_{\\{t_{0}^{m+1},t_{2}^{m+1}\\}}=J_{\\{t_{0}^{m},t_{1}^{m+1}\\}}=J_{\\{t_{0}^{m},t_{3}^{m+1}\\}}=J_{2}$,and
the rest of the coefficients are zero. Numerical calculations based on the
replica algorithm by the Monte Carlo method for a lattice model with such a
Hamiltonian are made in the work [22]. We write the Hamiltonian (2.8) for this
case as

$\begin{gathered}\mathcal{H}(\sigma)=-\sum_{m=0}^{M-1}(J_{1}((\sigma_{0}^{m}\sigma_{1}^{m}+\sigma_{1}^{m}\sigma_{2}^{m}+\sigma_{2}^{m}\sigma_{3}^{m}+\sigma_{3}^{m}\sigma_{0}^{m}+\sigma_{0}^{m+1}\sigma_{1}^{m+1}+\\\
\sigma_{1}^{m+1}\sigma_{2}^{m+1}+\sigma_{2}^{m+1}\sigma_{3}^{m+1}+\sigma_{3}^{m+1}\sigma_{0}^{m+1})/2+\sigma_{0}^{m}\sigma_{0}^{m+1}+\sigma_{1}^{m}\sigma_{1}^{m+1}+\\\
\sigma_{2}^{m}\sigma_{2}^{m+1}+\sigma_{3}^{m}\sigma_{3}^{m+1})+J_{2}((\sigma_{0}^{m}\sigma_{2}^{m}+\sigma_{1}^{m}\sigma_{3}^{m}+\sigma_{0}^{m+1}\sigma_{2}^{m+1}+\\\
\sigma_{1}^{m+1}\sigma_{3}^{m+1})/2+\sigma_{0}^{m}\sigma_{1}^{m+1}+\sigma_{1}^{m}\sigma_{2}^{m+1}+\sigma_{2}^{m}\sigma_{3}^{m+1}+\sigma_{3}^{m}\sigma_{0}^{m+1}+\sigma_{0}^{m}\sigma_{3}^{m+1}+\\\
\sigma_{1}^{m}\sigma_{0}^{m+1}+\sigma_{2}^{m}\sigma_{1}^{m+1}+\sigma_{3}^{m}\sigma_{2}^{m+1})).\end{gathered}$
(7.1)

It can be seen that the considering case can be obtained from the Hamiltonian
(6.1), by setting $J_{3}=0$.

Let us use Theorem 3.1 and Theorem 4.1, and write results for
antiferromagnetic layered Ising model.

######  Theorem 7.1.

In the thermodynamic limit, the free energy of the model with the Hamiltonian
(7.1) can be represent in the form (3.5), the internal energy can be
represented in the form (3.6), the heat capacity can be represented in the
form (3.7), wherein $\lambda_{\max}$ is the largest root of the equation
(3.10), where

${}a=-2-p_{1}^{-4}-p_{1}^{4}-2p_{2}^{-6}-2p_{2}^{-4}-2p_{2}^{-2}-p_{1}^{-4}p_{2}^{-2}-p_{1}^{4}p_{2}^{-2}-2p_{2}^{4}-p_{2}^{10}p2^{-8}-\\\
p2^{8}p4^{10},$ (7.2)
${}b=-8p_{1}^{-4}-8p_{1}^{4}+1p_{2}^{-12}+4p_{2}^{-10}+4p_{2}^{-8}+2p_{1}^{-4}p_{2}^{-8}+2p_{1}^{4}p_{2}^{-8}+4p_{1}^{-4}p_{2}^{-6}+\\\
4p_{1}^{4}p_{2}^{-6}-6p_{2}^{-2}+1p_{1}^{-8}p_{2}^{-2}-4p_{1}^{-4}p_{2}^{-2}-4p_{1}^{4}p_{2}^{-2}+p_{1}^{8}p_{2}^{-2}+4p_{2}^{2}-6p_{2}^{2}p_{1}^{-4}-6p_{1}^{4}p_{2}^{2}-\\\
4p_{2}^{4}+p_{2}^{4}p_{1}^{-8}+p_{1}^{8}p_{2}^{4}+2p_{2}^{6}p_{1}^{-8}+2p_{1}^{8}p_{2}^{6}+p_{2}^{8}p_{1}^{-12}+2p_{2}^{8}p_{1}^{-8}+p_{2}^{8}p_{1}^{-4}+p_{1}^{4}p_{2}^{8}+2p_{1}^{8}p_{2}^{8}+\\\
p_{1}^{12}p_{2}^{8}+p_{2}^{10}p_{1}^{-12}-2p_{2}^{10}p_{1}^{-8}+p_{2}^{10}p_{1}^{-4}+p_{1}^{4}p_{2}^{10}-2p_{1}^{8}p_{2}^{10}+p_{1}^{12}p_{2}^{10}+2p_{2}^{14}p_{1}^{-8}+2p_{1}^{8}p_{2}^{14}+\\\
p_{2}^{20},$ (7.3)
${}c=40+14p_{1}^{-8}+32p_{1}^{-4}+32p_{1}^{4}+14p_{1}^{8}-2p_{2}^{-16}-2p_{2}^{-14}-1p_{1}^{-4}p_{2}^{-14}-p_{1}^{4}p_{2}^{-14}-\\\
2p_{2}^{-12}-5p_{1}^{-4}p_{2}^{-12}-5p_{1}^{4}p_{2}^{-12}+34p_{2}^{-8}-2p_{1}^{-8}p_{2}^{-8}+16p_{1}^{-4}p_{2}^{-8}+16p_{1}^{4}p_{2}^{-8}-\\\
2p_{1}^{8}p_{2}^{-8}+8p_{1}^{-4}p_{2}^{-6}+8p_{1}^{4}p_{2}^{-6}-88p_{2}^{-4}-12p_{1}^{-4}p_{2}^{-4}-12p_{1}^{4}p_{2}^{-4}-24p_{2}^{2}-p_{2}^{2}p_{1}^{-12}-\\\
2p_{2}^{2}p_{1}^{-8}+3p_{2}^{2}p_{1}^{-4}+3p_{1}^{4}p_{2}^{2}-2p_{1}^{8}p_{2}^{2}-p_{1}^{12}p_{2}^{2}+8p_{2}^{4}-3p_{2}^{4}p_{1}^{-12}-30p_{2}^{4}p_{1}^{-8}-31p_{2}^{4}p_{1}^{-4}-\\\
31p_{1}^{4}p_{2}^{4}-30p_{1}^{8}p_{2}^{4}-3p_{1}^{12}p_{2}^{4}+6p_{2}^{8}-p_{2}^{8}p_{1}^{-16}+8p_{2}^{8}p_{1}^{-12}+16p_{2}^{8}p_{1}^{-8}+8p_{2}^{8}p_{1}^{-4}+8p_{1}^{4}p_{2}^{8}+\\\
16p_{1}^{8}p_{2}^{8}+8p_{1}^{12}p_{2}^{8}-p_{1}^{16}p_{2}^{8}+8p_{2}^{10}p_{1}^{-4}+8p_{1}^{4}p_{2}^{10}-16p_{2}^{12}-2p_{2}^{12}p_{1}^{-12}-4p_{2}^{12}p_{1}^{-8}+6p_{2}^{12}p_{1}^{-4}+\\\
6p_{1}^{4}p_{2}^{12}-4p_{1}^{8}p_{2}^{12}-2p_{1}^{12}p_{2}^{12}-2p_{2}^{16}-2p_{2}^{18}-p_{2}^{18}p_{1}^{-4}-p_{1}^{4}p_{2}^{18}+6p_{2}^{20}-p_{2}^{20}p_{1}^{-4}-p_{1}^{4}p_{2}^{20}-\\\
2p_{2}^{24},$ (7.4)
${}d=4p_{2}^{-18}+2p_{1}^{-4}p_{2}^{-18}+2p_{1}^{4}p_{2}^{-18}-26p_{2}^{-14}+p_{1}^{-8}p_{2}^{-14}-12p_{1}^{-4}p_{2}^{-14}-12p_{1}^{4}p_{2}^{-14}+\\\
p_{1}^{8}p_{2}^{-14}+68p_{2}^{-10}+18p_{1}^{-4}p_{2}^{-10}+18p_{1}^{4}p_{2}^{-10}-64p_{2}^{-6}-16p_{1}^{-8}p_{2}^{-6}+16p_{1}^{-4}p_{2}^{-6}+16p_{1}^{4}p_{2}^{-6}-\\\
16p_{1}^{8}p_{2}^{-6}-80p_{2}^{-2}+2p_{1}^{-12}p_{2}^{-2}+36p_{1}^{-8}p_{2}^{-2}-54p_{1}^{-4}p_{2}^{-2}-54p_{1}^{4}p_{2}^{-2}+36p_{1}^{8}p_{2}^{-2}+\\\
2p_{1}^{12}p_{2}^{-2}+266p_{2}^{2}+p_{2}^{2}p_{1}^{-16}-12p_{2}^{2}p_{1}^{-12}-14p_{2}^{2}p_{1}^{-8}+4p_{2}^{2}p_{1}^{-4}+4p_{1}^{4}p_{2}^{2}-14p_{1}^{8}p_{2}^{2}-\\\
12p_{1}^{12}p_{2}^{2}+p_{1}^{16}p_{2}^{2}-80p_{2}^{6}+2p_{2}^{6}p_{1}^{-12}+36p_{2}^{6}p_{1}^{-8}-54p_{2}^{6}p_{1}^{-4}-54p_{1}^{4}p_{2}^{6}+36p_{1}^{8}p_{2}^{6}+2p_{1}^{12}p_{2}^{6}-\\\
64p_{2}^{10}-16p_{2}^{10}p_{1}^{-8}+16p_{2}^{10}p_{1}^{-4}+16p_{1}^{4}p_{2}^{10}-16p_{1}^{8}p_{2}^{10}+68p_{2}^{14}+18p_{2}^{14}p_{1}^{-4}+18p_{1}^{4}p_{2}^{14}-\\\
26p_{2}^{18}+p_{2}^{18}p_{1}^{-8}-12p_{2}^{18}p_{1}^{-4}-12p_{1}^{4}p_{2}^{18}+p_{1}^{8}p_{2}^{18}+4p_{2}^{22}+2p_{2}^{22}p_{1}^{-4}+2p_{1}^{4}p_{2}^{22},$
(7.5)

and for the non-percolation probability, the relation is true (4.2), where
$\mu_{\max}$ is the largest root of the equation (6.3), wherein

${}a^{\prime}=-1-p_{1}^{4}-p_{2}^{-6}-2p_{2}^{4}-p_{2}^{10}p_{1}^{-8}-p_{1}^{8}p_{2}^{10},$
(7.6)
${}b^{\prime}=-p_{2}^{-6}+p_{1}^{4}p_{2}^{-6}+2p_{2}^{-2}-4p_{2}^{2}p_{1}^{-4}-2p_{2}^{4}+p_{1}^{8}p_{2}^{4}-p_{2}^{10}p_{1}^{-8}+p_{2}^{10}p_{1}^{-4}-3p_{1}^{8}p_{2}^{10}+p_{1}^{12}p_{2}^{10}+\\\
2p_{2}^{14}p_{1}^{-8}+2p_{1}^{8}p_{2}^{14}+p_{2}^{20},$ (7.7)
$c^{\prime}=2p_{2}^{4}-6p_{1}^{4}p_{2}^{4}+5p_{1}^{8}p_{2}^{4}-p_{1}^{12}p_{2}^{4}+4p_{2}^{8}-2p_{1}^{8}p_{2}^{8}-8p_{2}^{12}+4p_{1}^{4}p_{2}^{12}+5p_{2}^{20}-\\\
p_{1}^{4}p_{2}^{20}-2p_{2}^{24},$ (7.8)

The coefficients of the characteristic polynomials for $\tau$ (6.2) and
$\tau^{\prime}$ (6.3) are written in the formulas (3.11)-(3.14) and
(4.4)-(4.6). $\lambda_{max}$ is found using the formulas (3.10-3.18), аnd
$\mu_{max}$ is found using the formula (4.7).

######  Example 7.1.

Let us find the heat capacity for $J_{1}=-1$, $J_{2}=-r$, $T\in[0.2,0.5]$,
$r\in[0,1]$ in the absence of an external magnetic field. Fig. 11 shows the
nucleation of lines of a phase transition of the 1st order at $r=0.65$.

FIGURE 11. The heat capacity graph for the example 7.1

In the article [22] for the considered case, a phase diagram is constructed,
which has a similar form, but the phase transition line comes from $r=0.5$,
and not from $r=0.65$, as obtained in this paper. The shift can be explained
by the influence of boundary conditions (the authors consider free boundary
conditions), which have a strong effect on the model in the considered volume
$2\times 2\times\infty$.

Note that the exact solution for the anisotropic Hamiltonian, which takes into
account only the interaction of nearest neighbors, was obtained in [12]. This
solution coincides with the results of this paper if we assume isotropy
$J_{x}=J_{y}=J_{z}=J_{1}$ in the [12] model and put $J_{2}=0$ in the
Hamiltonian (7.1). Then the coefficients of the 4th degree polynomial (3.10)
have the form:

${}a=-10-4\cosh(4J_{1})-2\cosh(8J_{1}),$ (7.9)
${}b=4-20\cosh(4J_{1})+12\cosh(8J_{1})+4\cosh(12J_{1}),$ (7.10)
$c=-46+60\cosh(4J_{1})-16\cosh(8J_{1})+4\cosh(12J_{1})-\\\ 2\cosh(16J_{1}),$
(7.11) $d=70-112\cosh(4J_{1})+56\cosh(8J_{1})-16\cosh(12J_{1})+\\\
2\cosh(16J_{1}).$ (7.12)

The coefficients (7.9)-(7.12) coincide with the coefficients of the polynomial
in [12], and hence the largest eigenvalue also coincides. In [12] the solution
is written out taking into account the special form of the resulting
polynomial, which allows one to decompose a fourth degree polynomial into two
second degree polynomials.

##  8\. Conclusion

Thus, the authors obtained the exact value in the thermodynamic limit of free
energy, heat capacity, and non-percolation in the $2\times 2\times\infty$
strip for a family of Hamiltonians that are invariant under rotation about the
central symmetry axis by $\pi/2$. The problem is reduced to finding the root
of the fourth degree polynomial, which is the largest eigenvalue of the
transfer matrix. The value of the non - percolation probability of the model
in the case of percolation, which has the same invariance, is found. Some
types of Hamiltonians are considered for which the largest eigenvalue of the
transfer matrix is the root of a quadratic trinomial or a linear equation. The
authors note that other Hamiltonians can be found in a similar way, for which
a similar expansion is possible. An exact value is obtained for the free
energy and heat capacity for the $2\times 2\times\infty$ gonigendrial model in
the case of free boundary conditions and in the case of an analog of cyclic
closure. The similarity of the numerical solution in the $3\times
3\times\infty$ strip and the results of the work of other authors with the
exact solution for the analog of cyclic closure in the $2\times 2\times\infty$
strip indicates that the authors managed to choose the analog of cyclic
closure correctly. The Hamiltonian is written out taking into account a
special selection of interaction coefficients simulating cyclic closure. The
exact value of the free energy and heat capacity in the thermodynamic limit in
the $2\times 2\times\infty$ strip is obtained for the Antiferromagnetic
layered Ising model, the results are compared with the exact solution in the
work [13].

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