Disorder solutions for generalized Ising and Potts models with Multispin interaction Текст научной статьи по специальности «Математика»

THEORETICAL QUESTIONS OF COMPUTER SCIENCE, COMPUTATIONAL MATHEMATICS, COMPUTER SCIENCE AND COGNITIVE INFORMATION TECHNOLOGIES

УДК 537.622

DOI: 10.25559/SITITO.15.201901.33-44

Disorder Solutions for Generalized Ising and Potts Models with Multispin Interaction

P. V. Khrapov

Bauman Moscow State Technical University, Moscow, Russia [email protected]

In this work, a special elementary transfer matrix is constructed for generalized Ising models and Potts models with the general form of a finite Hamiltonian with a multi-spin interaction in a space of arbitrary dimensionality, the Napierian logarithm of its maximum eigenvalue is equal to the free energy of the system. In some cases, it was possible to obtain an explicit form of the eigenvector corresponding to the largest eigenvalue of the elementary transfer matrix.

On this basis we obtained systems of nonlinear equations for the interaction coefficients of the Hamiltonian for finding the exact value of the free energy on a set of disorder solutions. Using the Levenberg-Marquardt method, the existence of nontrivial solutions of the resulting systems of equations for plane and three-dimensional Ising models was shown. In some special cases (the 2D Ising model, the interaction potential, including the interaction of the next nearest neighbors and quadruple interactions; the 3D model with a special Hamiltonian symmetric relative to the change of all spin signs, for which it is possible to reduce the system of equations to the system for a planar model) three parameters are written in explicit form. The domain of existence of these solutions is described.

Keywords: generalized Ising model, generalized Potts model, Hamiltonian, multi-spin interaction, transfer matrix, disorder solutions, statistical sum, free energy.

Acknowledgements: The work is supported by the Russian Foundation for Basic Research within the scientific project No. 18-01-00695 a.

For citation: Khrapov P.V. Disorder Solutions for Generalized Ising and Potts Models with Multispin Interaction. Sovremennye informacionnye tehnologii i IT-obrazovanie = Modern Information Technologies and IT-Education. 2019; 15(1):33-44. DOI: 10.25559/SITITO.15.201901.33-44

Abstract

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Неупорядоченные решения обобщенных моделей Изинга и Поттса с мультиспиновым взаимодействием

П. В. Храпов

Московский государственный технический университет имени Н.Э. Баумана (национальный

исследовательский университет), г. Москва, Россия

[email protected]

Аннотация

В работе построена специальная элементарная трансфер-матрица для обобщенных моделей Изинга и моделей Поттса с общим видом финитного гамильтониана с мультиспиновым взаимодействием в пространстве произвольной размерности, натуральный логарифм максимального собственного значения которой равен свободной энергии системы. В некоторых случаях удалось получить явный вид собственного вектора, отвечающего наибольшему собственному значению элементарной трансфер-матрицы. На основе этого выведены системы нелинейных уравнений на коэффициенты взаимодействия гамильтониана для нахождения точного значения свободной энергии на множестве неупорядоченных решений (disorder solutions). Методом Левенберга-Марквардта показано существование нетривиальных решений получающихся систем уравнений для плоских и трехмерных моделей Изинга. В некоторых частных случаях (2D модель Изинга, потенциал взаимодействия, включающий взаимодействие следующих ближайших соседей и четверные взаимодействия; 3D модель со специальным гамильтонианом, симметричным относительно перемены всех знаков спинов, для которой удается свести систему уравнений к системе для плоской модели) решения, зависящие от трех параметров, выписаны в явном виде. Описана область существования этих решений.

Ключевые слова: обобщенная модель Изинга, обобщенная модель Поттса, гамильтониан, мультиспиновое взаимодействие, трансфер-матрица, неупорядоченные решения (disorder solutions), статистическая сумма, свободная энергия.

Благодарности: статья подготовлена при поддержке гранта Российского фонда фундаментальных исследований № 18-01-00695 a «Конечномерные модели квантовой электродинамики».

Для цитирования: Храпов П. В. Неупорядоченные решения обобщенных моделей Изинга и Поттса с мультиспиновым взаимодействием // Современные информационные технологии и ИТ-образование. 2019. Т. 15, № 1. С. 33-44. DOI: 10.25559/SITITO.15.201901.33-44

Контент доступен под лицензией Creative Commons Attribution 4.0 License. The content is available under Creative Commons Attribution 4.0 License.

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Том 15, № 1. 2019 ISSN 2411-1473 sitito.cs.msu.ru

P. V. Khrapov

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Introduction

The search for the exact values of free energy and other physical characteristics of various physical models attracted the attention of scientists all over the world over many years. The most brilliant achievement in this direction undoubtedly is the accurate solution of the two-dimensional Ising model without an external magnetic field, obtained by Onsager [1]. For the Ising and Potts models, few accurate solutions are known (with an analytical expression for the partition function or the free energy of the system) in the case when the interaction Hamiltonian includes a magnetic field, good reviews Wu F.Y. [2], Baxter [3]. Some examples are in Verhagen [4] for anisotropic models on a triangular lattice and Rujan [5] for the Ising and Potts models, respectively. These solutions belong to the class of so-called disorder solutions (Stephenson J. [6], Welberry T R and Gal-braith R [7], Enting I.G [8]): these are solutions obtained on a certain subset in the parameter space of the physical system. Different methods were used to obtain these solutions for different models: methods related to crystal growth (Welberry TR and Miller GH [9]), Markov processes (Verhagen [4], Rujan [5]), or the angular transfer matrix method (corner transfer-matrix) (Baxter [10]). The transfer-matrix apparatus is widely used in statistical physics [1112]. In most cases, the task of calculating a partition function was compared with an equivalent one in another area, where the corresponding methods made it possible to solve the problem. A local criterion for obtaining such solutions is given in Jaekel and Maillard [13]. With its help, for the anisotropic Ising and Potts model with a magnetic field, a variety of values (disorder solutions) for the partition function for a chessboard type grid have been obtained. Publications [14-22] are also a devoted to the topics of disordered solutions. In this paper, the author considers the generalized Ising and Potts models with a general view of the multi-spin interaction with boundary conditions with a shift (similar to screw ones), and the cyclic closure of the set of all points (in natural ordering); builds elemental transfer matrices for these models (for two-dimensional models, matrices of similar structure were used in [23]), writes out systems of equations for finding their maximum eigenvalues (and the exact form of eigenvectors corresponding to these maximum eigenvalues). The Napierian logarithm of the largest eigenvalue is the free energy of the system with parameters that satisfy the resulting system of equations. For a wide enough variety of models, exact solutions of these systems of equations are given in explicit form, depending on several parameters, as well as the range of permissible values for the solutions are obtained. In other cases, the Levenberg-Marquardt algorithm [24] showed the existence of nontrivial solutions of the written systems.

The high symmetry and repeatability of the components of the found eigenvectors, which disappear when the set of exact solutions is exceeded, is the reason for the search for phase transitions in the set neighborhood.

Disorder exact solutions will certainly help in the computational investigation of models. Comparison of the results of numerical simulation with exact results, at least on some set, will certainly raise the accuracy of numerical simulation.

Generalized Ising model

Analyze v - dimensional grid

^ = {t = (tl, t2,..., tt, tt+1,..., tv), tt = 0,1,..., Lt, t = 1,2,...,v}, where (t1,t2,...,Ll.,tl.+1,...,tv) = (tj,t2,...,0,tl+1 + l,...,tv) , i = l,2,...,v-l ,

(L1,L2 -1,...,L -1,...,Lv-1) 3 (0,L2,...,L-1,..., Lv -1) s... s (0,0,...,Lv) s (0,0,...,0) (1) Due to such a procedure of identifying points, the grid lv has the size L1 xL2 x ...xLv, the total number of grid points L = LlL1..Lv. Eo ipso on the lv special boundary cyclic screw conditions (with a shift) are specified. Renumber all points of the grid lv: t0 = (0,0,...,0) ,T1 = (1,0,...,0) ,T2 = (2,0,...,0),...,

= (L1,0,...,0) = (0,1,0,...,0), t^ = (1,1,0,...,0).....Tl = (0,...,0) = t0 (2)

This numbering determines the natural cyclic round of all points (in the positive direction) and local (cyclic) ordering. Assume that in the each point t = (tl, t2,..., tv) there is a particle. The state of a particle is determined by the spin at, which at each point of grid t = (tj,t2,...,tv) can take two values: at e X = {+1,-1}. Let us assume that Q = { t1,t2,...,tp }-is some fixed finite subset (of a certain form) of points lv, we call it the carrier (or the carrier of the Hamiltonian), whose lowest point tmm = (0,...,0), the oldest tm™ = (¿j™1,t2m™,...,C"™) (it does not mean all points t,i = 0,1,2...,imax, belong to | . For example, Q = {t = (tl,t2,...,tv) e L : ti = 0,1,i = 1,2,...,v} - unit v -dimensional cube). Hamiltonian of the model has the form

^ =-E S J,1,2.....

(3]

where T = (T;,...,T) e ^ , some nonempty subset q

n, =n+т', { t1,12,...,f }en, " is

, J - are corresponding

translation-invariant coefficients of multi-spin interaction. Such a notation allows formula (3) to describe any Hamiltonian with a finite support. Note that the same subset { t1, t2,..., ts }cfi , can occur when different t' . In this case, when recording Hamiltonian (3) in standard form H (a) = - X (4)

we get

J 1 :

S J,,

(5]

We introduce the coefficients Kt' «2 ... «• = J,' ,2 ... «• 1 (kBT), where T -temperature, kB - Boltzmann constant. Then the statistical sum of the model is written as

ZL = X exp(-H(a)/(kBT)) = , (6)

■M

L-1

X exp(X X K,i,2 ,-a?a,'...a,-)

M »0 ( >V.....f }=n,

summation is performed over all states of spins.

We introduce an elementary transfer matrix T = Tp r of size

2W x 2W, nonzero elements of which Tp r correspond to a pair

of sets {(P'.oOV+lV^OV+imx iM^i+M^V"^^^ , (7) at the same time (Fig. 1).

p = yq ) 2k, p = 0,1,2,...,-1,

k=0 2

r = 'y-'(1 ) 2k, r = 0,1,2,...,-1. 2

(8] (9]

, = exp( X K,',' .■a,'a,'-af) = ^, (10]

Then

z = v t

L A-i {(ct, o >->o

T

{CTr0 >CTr1 ,-,°TL-1}

{(CT 1,CT 2,...,CT 1 ),(ct 2,CT 3,...,CT 1 )}""

T1 T2 Tîma^ T2 T3 Tîmax +i/'

T {(CT L-1,CT 0,...,CT ; -2),(CT 0,CT 1,...,CT ; _1)} = Tr(T ) (11]

iv TL i> t T;max 2 Tv ' T1 ' ' -;max 1 " K J

T - T' ' T'n

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Note that nonzero matrix elements of the matrix T = will be

p

equal if the set (c 0,c 1,...,c ,mat) includes the same subset u(Q. 0) = {ut, t € Q 0}, with fixed spin values {Jt, t eQ,|. Let T

Q, = Q, \TMmax, i = 0,1,...,L.

energy f does not depend on the size lvv, then it remains the same as the size of the model tends to infinity. That is, we obtain free energy for a system of infinite size.

(12) Generalized Potts model

The eigenvector of the elementary transfer matrix T = r , corresponding to the highest eigenvalue F is sought in the form

, (13)

b = {b(

0 ,CT 1

),G, 6 X, i = imxi - 1} =

imax -1 (1 -0 )

{bp,bp > 0,p 2k,p = 0,1,....,2im"-1 -1} k=0 2 at the same time we consider that

W-m* -0 = (14]

that is, the components of the vector b will be the same if the set (ct 0,CT 1,...,CT , 1) includes the same subset &(£! 0) (total different components b(<j(C 0)) will be 2*\o|]. Next will be shown the feasibility of this assumption. According to the Per-ron-Frobenius theorem [25], the maximum eigenvalue of a matrix with positive matrix elements (they will be positive for the transfer matrix to some degree; when multiplying the matrix by itself, it is easy to see the nature of the inevitable filling of the matrix with nonzero elements] all components of the vector b must be positive. In this case, we can write the system of equations for the eigenvector of the matrix T = Tp r , assuming that the components of the eigenvector 0))} and the interaction coefficients K;1 2

in the Hamiltonian are unknown:

(15]

y{(a,

I

a . eX

T^max

({K,,

T^max 1 )}

T

{(aTo,a л

T T

F =

-1 )'(a_]

)}

where с ,mix is included in the set &(£! , ).

Or

b(a(Q. t<1))F =

Z exp( z

(17)

K

Exactly the same reasoning and similar systems of equations are obtained if we consider the generalized Potts model with the Ham-iltonian

Hpo,.(o--Z x Z Jx^^, o) (18)

i-o {t\t\.../ {^,^2,...,^,}eX- ' ' ' ' ' '

where T = (t[,...,Tv), ti. =Q + r', { t\t2,...,ts } c Q - some subset Q {x^, x2,..., - some set of spins, e X = {1,2,..., q}, at appropriate points { t,t2,...,t }czQ,., Jx _ _ - corre-

X (ct) =

fl, if a. , i = 1,..., s

where F - higher eigenvalue of an elementary transfer matrix T = Tp r. Taking into account (14) let's rewrite the system of equations (15) as

6(a(n.))F = X ,

(16)

sponding translation-invariant coefficients of interspin interaction,

- charac-

, r.,. i = i.....s

t' ' tl

0. other cases

teristic function.

Such a notation allows you to describe an arbitrary Hamiltonian with a finite support with the formula (18). We introduce K = J / (kS), where T - temperature, k„ -

Boltzmann's constant. Then the statistical sum of the model is written as

ZL exp(-HFotts (a)l(kBT)) =

{ff} 1=0 [f,t2,.../ }=ni,. {p-1,^2,...,^,

where summation is performed over all states of spins. We introduce a function

F (Pt, ,0tM,...,0t,+im„) =

exp( Z Z ((7nT<))

where = {u ,ieQ , } the function x ) de-

pends not on all values of the spins in the points T ,t'+1,...,t'+"m™,

so we can write x? ? ? (Pa )].

L-1 ' '

Then ZL = ZnF((7T )

a i=0

We introduce an elementary transfer matrix. T = r with the size q'mai X q'mai, nonzero elements of which r are numbered by pairs of sets

Entire 2 equations. Evaluate F in every equation and equating to the value F from the first equation, we get 2 -1 equation. The number of coefficients of multi-spin interactions K, 2 , will be 2 * 1 -1 (empty set we remove). Given the fact that the various components ¿(^(Q 0)) will be 2, we get, that the solution of the system (17) is multiparameter, and it allows you to find not only free energy

f = ln( F), but some other characteristics. Considering that free

{(CT n ,0 , ),(СТ ,,CTt ,...,0 : )}

T T r'maxTP t!' ' Timax''

exp( z

Z K

s Xfl,,u !,...,Us (аП 0 ))

in this connection

imax -1

p = Z -x)qk p=0,1,2,...,q--1

k=0

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= Z -^ r = 0,1,2,...,q'm* -1

k=0

Then

=

z

{(a o ,a i ,...,a , -i i 2 ,...,a , )}

{a o,a i,...,<r L-1}

{(a 1 ,a 2 ,...,a ; ),(a 2 ,a 3 ,...,a ; )} " '

T T T!max T T T!max +i'J

{(a L-i ,a o ,...,a -2 ),(a_o ,a_ 1 ,...,a m, -1)}

= Tr .L )

Note that nonzero matrix elements of the matrix T = Tp r will be equal if the set (o 0,O X,...,G imax ) contains the same subset m(QT) = {mt, t e Q%a j^with fixed spm values {, t e Qt„} ■

The eigenvector of the transfer matrix

' r , corresponding

the higher eigenvalue F matrix seek in the formula

* = (V o. ^ X' * = i'-' 'max) = {bp , P = - }

at the same time we consider that

that is, the components of the vector b will be the same, if the set (G 1,...,G 'max -1) contains the sa me subset s (£2 (the total number of various components b(s(mm 0)i will be b^S). By) the Per-ron-Frobenius theorem, for the maximum eigenvalue F all components of the vector b must be non-negative.

In this case, we can write the system of equations for the eigenvector of the matrix = Tp r , assuming that the components of the

eigenvector {£,(&.. )} and the interaction coefficients Kx x x

nT° t1' t2'"'' ts

in the Hamiltonian are unknown:

b F =

{(a o i ,...,a ,-i )}

(19]

Tmax"

I

T

{(a 0 ,a 1 ,...,a ¿max -1 ),(a 1 a 2 ,...,a imax )}

a .. eX

T max

({K ,

})b

{(a ! ,a 2,...,a . )}

-1 T2 T'max

where F - higher eigenvalue of an elementary transfer matrix T = Tp r (we assume that all components of the eigenvector

- rJ ;

Tlmax

— p,r v. i ^

b = {K 1 ,o 2 ,... ,o , ) G X >i = 1 ' Zmax} are Positive, then,

v ^ ' ' ' ¿max ' <■

by the Perron - Frobenius theorem, this vector will correspond to the highest eigenvalue). Rewrite (19) as

^ JF = Z

{(a 0 ,0 ! ,...,0 ; - ),(0 ! ,0 2 )}

Where Ù , = Ù 0 +T.

Or

bi^ n) F = X exp( X

{1,t2,...,ts jçQ

X

a ; eX

2,..,,, 2^, (an0))b(aù 1)

Fig. 1. The structure of the transfer matrix

Asterisks (*) denote nonzero matrix elements. The sign "+" corresponds to the value of spin +1, the sign "-" the value of spin -1

Next, we analyze various examples of finding free energy.

2D Ising Model

Consider a two-dimensional square grid of size L = L1 xL2, total number of grid points L = L1L2, with special boundary cyclic screw (with a shift) conditions (1) and points renumbering (2). We assume that there is a particle in each node. The state of a particle is determined by its magnitude (spin) o., which can take 2 values: +1 or -1. Each spin interacts with the eight nearest spins in four directions or lines. Hamiltonian model has the form

l2 L

h (?) = -ZI ( ++

T _m _m+1 . r _m_m+1 . r _m_m+1_m+1

J+ J4?n ?n+1 + J5?n ?n+1

(20]

6?n ? n+?n

i T _m_m m+1 . t _m+1_m _m+1 + J?n ?n+1?n+1 + J?n+1?n+1

m m m+1 m+1 m

J9°n ?n+1?n ?n+1 + )

where J., ' = 1,2,..., 9 - corresponding coefficients of interspin interaction. We introduce Ki = Jt / (kBT), ' = 1,2,...,9, where T - temperature, kB - Boltzmann constant,

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m=1 n=1

H = h / (kBT) interaction parameter with the external field with a coefficient. h . Then the statistical sum of the model can be written as

L L (21)

Zll2 = £exP(-H(a)lkBT) = £ exp(£ £ (Kxamnamn+x +

a a m=1 n=1

jy _m_m+1 , jy_m _m+1 , ry _m_m+1 , jy _m_m+1_m+1 ,

K2an an + K3an+an + Kan an+1 + K5an an an+1 +

K6a„ma;+ia;+1 + K7a„ma;+ia;++1 + K%a:;+1a^ +K9a„ma„m+1a;+1a;+-11 + Ha;))

where summation is performed over all states of spins.

For the model under consideration, we construct an elementary transfer matrix Let

G Kp K2'K9'H ) = (22) exp^o^o^ + Kp^o^ - + Kpp^ - +

KPx°xh + K5°T0°TLi + KPx°x°xh -i +

K7°>oriO> + K8aTLi -i°Ti°TLl +

K9o p p Li-p h + Ho 0)

= T r according to the formula (9) (Fig. 2).

Fig. 2. Elementary transfer matrix for two-dimensional grid models with Hamiltonian (20).

Nonzero Matrix Elements atJ, i = 0,1,..., 7, j = 0,1 of elementary transfer matrix T = T r we write in the form

am = G(+1,+1,+1,1 - 2i, ^i, K 2,..., K9, H) au = G (-1,+1,+1,1 - 2i, Ki, K2,..., K9, H) a2= G (+1,-1,+1,1 - 2i, K„ K 2,..., K9, H)

(23)

Оз.. = G (-1,-1,+1,1 - 2i, K1, K2,..., K9, H) a4. = G (+1,+1,-1,1 - 2i, K1, K 2,..., K9, H) a5i = G (-1,+1,-1,1 - 2i, K1, K 2,..., K9, H) a6i = G (+1,-1,-1,1 - 2i, K1, K2,..., K9, H) a7. = G (-1,-1,-1,1 - 2i, K1, K2,..., K 9, H)

a) Consider a special case when

J5 = J6 = J7 = J8 = H = 0.

(24)

Since in this case, replacing the signs of all spins with opposite values of the Hamiltonian (20) and the function G (<JT0, , &TLX -1 Kl, K2,..., K9, H) does not change, S° in this case the transfer matrix T = Tp r is centrally symmetric (Fig. 2). We will also search in a symmetric form for its own vector corresponding to the highest eigenvalue F.

X = (1, ¿2,1, ¿2,...,1, 2, ¿>2,1, ¿>2,1,...,b2,lf (25)

2l 2l

where b2 > 0. Then, by the Perron-Frobenius theorem, this eigenvector will correspond to the maximum eigenvalue of F. Denote R = exp( Kt), i = 1,2,...,9. (26)

From the form of the transfer matrix (Fig. 2), we obtain the following system of equations for R., i = 1,2,3,4,9, F, b2: F = R R R3 (RR + b / (RR)) (a)

b2F = (R / (RR))(1 / (R4R9) + b2R4R9) (b) F = (R /(RR))(b2R4 / R9 + R9 f R4) (c)

b2F = (R1 / (R2R3 ))2b2R9 / R4 + R4 / R9) (d)

Solve the system (27).

We equate the right sides (27a) and (27c),

R R2 R3( R4 R9 + b2/( R4 R9)) = (R2/( R R3))b RJ R9 + R9/ R4)

find from here b2:

b = -(-1+r3 r42) R / (R12 R32 - R2)

We equate the right sides (27b) and (27d),

V( RiR2)(i/( R4 R9) + b2 R4 R9) = (Ri/( R2 R3))(b2 R9/ R4 + R4/ R9)

Then b2:

¿2 = (R2 -R,2R42) / ((R2 -R2R2)R2)

We equate the b2 from (29) and (31):

-(-1 + R2R3 R4 )R / (Rl2R3 - R4 ) = ( r2 - R2 r2)/(( R12 - R2 R42) R2)

(27)

(28)

(29)

(30)

(31)

(32)

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39

Solve for R,

Result

4 (R - RR4 )(RR - R4 )

(33]

R =

( r2 - r r2)(i - r2 r ra)

Substitute R from (33] (2.10] to b2 from (29] (2.6], result

b =

(1 -RR3R4 ) (R -RR)(RR -R2)

(RR - R42) V (Ri2 - R32R42)(1 - RR32R42)

(34]

2_( R/ R)(1/( R4 R) + b2 R4 R)

R =

b2 R1R3( R4 R + b2/( R4 R))

(36]

R — R2 R42

Result b =

R =.

R =

(1 - R R R) ( R - R R)( R R - R) (RR2 -r4) \ (R -R32R)(1 -R2R2R42)

Ri2 R32 - rR4

IR — R12 R

(R -RR2)(RR -R2) (R2 -R2R2)(1 - R2R2R2)

(38]

(39]

(40]

b _-sinhQK + K3 + K4) lsinh(K1 -K3 + K4)sinh(Kt + K3 -K4) (41] 2 _ sinh(K1 + K3 - K4 ) \ sinh(K1 - K3 - K4 )sinh(K1 + K3 + K4 )

(42]

R =,

R =

\

sinh(K, + K3 - K4) Isinh(-K, + K3 -K4)

sinh(-K1 + K3 - K4)sinh(K1 + K3 - K4) sinh(K1 -K3 -K4)sinh(-K1 -K3 -K4)

4 (43]

(Kl + K3 + K4)(Ki + K3 -K4) < 0

(K + K3 -K4)(-Kj + K3 -K4) > 0

(44]

(45]

(-K + K3 - K4)(Kl + K3 -K4)(Kl -K3 - K4)(-Kl -K3 -K4) > 0 It is tantamount to:

(IKi + k3\ -1k4|)(|-1K3 + K4\) > 0 |Kj + K3\-\K4\< 0 |K3 -K4| -IKJ > 0

Multiplying the equation (27a)(2.4a) times b2 , the right-hand side is equated to the right-hand side of the equation from (27b) (2.4b)

b2 RR2R(R4 R +b2/ (R4R9)) = (35)

(R/( RR))(1/( R4R9) + b2RA R)

Solve for R, result

( K3 - k4\ -1 k |)(| K3 + k4\ -1 Ki\) > 0

he last inequation is a con: (|K,|-|K3 + K4|) < 0 |Kj + K3\-\K4\< 0

|k3 - k4|-i kj > 0

The last inequation is a consequence of the first three. Therefore

(46)

Or

Substitute formula for R from (33) and for b2 from (34) into (36), result

R2 = R2 R - R (37)

|R| < |K3 + K41

K + K^ < |K4|

|K3 — K4\> |Kj|

It is tantamount to:

K + K3\< \K4\

(47]

(48]

In this domain of formulas definition, you can still simplify the formula for b2:

(49)

b2 =

sinh( K1 + K3 + K 4 )sinh( K1 - K3 + K 4) ' sinh(-K1 - K3 + K4 )sinh(K1 - K3 - K4 )

On the region of existence of solutions we get the system of inequation:

(K + K3 + K4)(K + K3 - K4)(K - K3 - K4)(K + K3 + K4) > 0

Example.

K1 = -1.5, K3 = -0.65, K4 = 2.2,

R1 = 0.22313016014842982, R3 = 0.522045776761016,

R2 = 4.639942095766785, R4 = 9.025013499434122,

K2 = 1.5347018867996893, R9 = 3.391150413013755,

K9 = 1.2211692186958323, b2 = 0.01485136611090205,

F = 16.541741276496538 = exp(2.8058869607477605612). It follows that, the free energy of the system

f = 2.8058869607477605612.

It is constant, and does not depend on R > 2 and L2 > 2.

Consider a two-dimensional square lattice of size L = Lt x L2, with the Hamiltonian (20) this time in general form, without restrictions of the form (24)

The eigenvector of the elementary transfer matrix ^^ = Tp r (Fig. 2) we find in the shape of

x = (1,¿1,1,è1,...,1,b1;Ä2,b3,b2,53,...,b2,b3),bt > 0,i = 1,2,3

2£i

2£i

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Then in order to make x an eigenvector, we get the system of equa- H =0.010568103316912,

tions

F = an,

- b2 aQ1

(51)

b1F = a10 + b2 a11

F = b1a20 + b3a21

b1F = b1a30 + b3a31 b2 F = a40 + b2 a41 b3F = a50 + b2 a51

b2F = b1a60 + b3a61

b3F = b1a70 + b3a71

where F - eigenvalue of the transfer matrix T = Tp r. b > 0, i = 1,2,3, then F is the highest (maximum) eigenvalue of the transfer matrix T (and f = ln(.f) - free energy). The system of equations (51) is rewritten as

F = exp(H + K1 + K 2 + K3 + K 4 + K 5 + K6 + K 7 + K 8 + K9) + (52) b2exp(H + K1 + K 2 + K 3 - K 4 - K 5 + K 6 - K 7 - K 8 - K 9) bF = exp(-H - K1 - K2 + K3 - K4 - K5 - K6 - K7 + K8 - K9) + b2 exp(-H - K1 - K 2 + K3 + K 4 + K 5 - K6 + K7 - K8 + K9) F = b1exp(H - K1 + K 2 - K3 + K 4 + K5 - K 6 - K 7 - K8 - K9) + b3exp(H - K1 + K 2 - K 3 - K 4 - K 5 - K 6 + K 7 + K 8 + K 9) bF = b exp(-H + K1 - K2 - K3 - K4 - K5 + K6 + K7 - K8 + K9) + b3exp(-H + K1 - K 2 - K 3 + K 4 + K 5 + K 6 - K 7 + K 8 - K 9) ' b2F = exp(H + K1 - K 2 - K3 + K 4 - K 5 - K 6 + K7 - K 8 - K9) + b2exp(H + K1 - K 2 - K 3 - K 4 + K 5 - K 6 - K 7 + K 8 + K 9) b3F = exp(-H - K1 + K 2 - K3 - K 4 + K 5 + K 6 - K 7 - K8 + K9) + b2exp(-H - K1 + K 2 - K 3 + K 4 - K 5 + K 6 + K 7 + K 8 - K 9) b2F = b exp(H - K1 - K 2 + K3 + K 4 - K5 + K6 - K7 + K 8 + K9) + b3 exp(H - K1 - K 2 + K3 - K 4 + K5 + K6 + K7 - K8 - K9) b3 F = bexp(-H + K1 + K 2 + K 3 - K 4 + K 5 - K 6 + K 7 + K 8 - K 9) + b3exp(-H + K1 + K 2 + K 3 + K 4 - K 5 - K 6 - K 7 - K 8 + K 9)

System (52) of 8 equations has 14 variables. So, the solution of system (52) in the general case depends on 6 parameters. In order to show that system (52) has nontrivial solutions, we indicate the solution found by the Levenberg-Marquardt method:

K =-0.072508931144562,

K2 =0.227087903394048,

K =-0.033524527164916,

.1V3

K. =0.309028362673801,

K5 =0.081977911068159,

K6 =-0.085603418100829, K7 =-0.135244289148472, K8 =0.228449686392315,

K9 =0.112019459850970,

b = 0.611722309307112, b2 = 0.572612052286677, b3 = 0.861858834747567

Maximum eigenvalue by Perron-Frobenius theorem F=2.231051645841576, free energy f = 0.802473064371868.

c) Let us analyze a two-dimensional square grid of size L = Lt xL2, with Hamiltonian (20) in general terms. The eigenvector of the elementary transfer matrix T = Tp r (Fig. 2) we will find in the form

x = (1,6j,b2,b3,...,1,bj,62,b3;b4,bs,b6,b7,...,b4,bs,b6,b1),bt > 0,i = 1,2,...,7.

(53)

Then in order x to be an eigenvector, we get a system of 16 equations

= a™ +

(54)

F = aoo + b a0i

Ь F = aio + Ь4 aii

b2 F = bia20 + Ь5 a2i

b3 F = b1 a30 + b5 a3i

F = b2a00 + b6 a0i

bF = b2 ai 0 + b6 aii

b2 F = b3a20 + b7 a2i

b3 F = b3a30 + b7 a3i

b F = a40 + b4 a4i

b5 F = a50 + Ь4 a5i

b6 F = bia60 + b5a6i

b7 F = b1 a70 + b5 a7i

b F = b2 a40 + Ь6 a4i

b5 F = b2 a50 + Ь6 a5i

b6 F = b3a60 + b7 a6i

b7 F = b3 a70 + b7 a7i

where F - eigenvalue of the transfer matrix T = Tp r. By the Perron-Frobenius theorem in this case ( if bi > 0, i = 1,2,3.) F is the maximum eigenvalue of the transfer matrix T = Tp r (then f = ln(_F) - free energy). This system is formally wider than the general system of equations from the beginning of the article (in this case it was necessary to restrict ourselves to a system of 8 equations), but it has nontrivial solutions (the solution was found by the Levenberg-Marquardt method), and this example shows the possibilities of expanding the search area for exact solutions:

K =0.019113809839761

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K2 = -0.233807683866557

h =4.303785746876578

K3 = -0.034832967333049

h2 =0.600000000006545

K. = -0.232974813199571

b3 =4.303785746875397

K5 =0.339920870267006

b. =5.050987834488805

K, = 0.117189864650398

b5 =0.736735440877358

K = 0.232974813199571

b = 5.450987834486589

K8 = -0.000000000000000

h =0.736735440878770

Ka = -0.339920870267006

b5 = b7, bj = b3, F = 3.024765061088569,

H = 0.150660735316256 h = 0.899142765909877

f =1.106833422710819 free energy. v - dimensional Ising model

b2 = 0.600000000000000 (it is specifically a priori taken so not to coincide with the solution of point b))

b3 = 0.899142765909877

b4 = 1.353781209394591

b = 0.763468696132035

5

b6 = 1.753781209394591 b7 = 0.763468696132035

F=2.3973089056900103166, f = 0.8743468189429043358.

Note that K4 = -K7, K5 = -K9, K8 = 0, K12 = b2 = 0.6 (So b2 was given to get a more general form of the eigenvector than in the three-parameter case b1, b2, b3), K13 = K11 (K13 = b3, K11 = b) , K15 = K17. I.e b = b , 1 = b7. The following example is similar to the previous one: K = 0.174718289126992

K2 =-0.796130400068089

K3 =0.036359406404343

K, =0.187050114181306

K5 = -0.191302175159284

Let us concretize the system of equations (17), generated by the model with Hamiltonian (3) for the v -dimensional case, the interaction within the framework of a single -dimensional cube

n = {t = (ti, t2,..., tv): ti = 0,1, i = 1,2,...,v}

b< 0)F =

X exp( X Kt\t 2,...,t'<yt1<yt2:<yt' ^ú^

(55)

Then

{<Jt0 'an'aTL\ '^TL1+1 '^IL1L2 '°TL1L2 +1''JtL1L2 +L1 '<JtL1L2 +L1+1y"' °TIlL2...Lv_i 'CJtLiL2-Lv-i+i TL^ ...V-1 + Li T L1L2...Lv_i+LiL2...Lv_2+...+Li }

F =

Z

'ari 'í7tl1 '°tL1+1 '°tL1L2 'JtL1L2...Iv_1+L1L2...Iv_2+...+L1

(ari ,t7z L1 >L1+I '°tL1L2 L1L2 +1 tL1L2...Iv-1+L1L2...Iv-2+...+L1+1

i7!! ,,JiLl 'GtL\L2 LIL2 +1'"''a'LiL2...Iv-i +L\L2...Lv-2 +...+Lj+1}

where is the summation over o = ±1.

zb,L2...Iv 1+L1L2...V 2 +-+¿1+1

Or

F =

(56)

tLiL2..LV-1 + 1' 'LLi-lv-1 +L1 îlil2-lV-1+l1l2-lV-2 +-+L1 '

Z exP^ 2Z K,i,2r. t 1° • ° s)•

°tLiL2...Iv-I+LIL2...Iv-2+...+LI+1 _±1 { ,£ '• • • ,£

K, =-0.050020923787075

6

K, =-0.187050114188515

K8 =0.000000000036532

K9 =0.191302175154890 H =-0.058314486090843

where is the summation over all non-empty subsets

{ t, t2,..., t } C Q . The number of such subsets for v - di-

(. ill ) — T0 _2v

mensional cube will be 2 , total different components

{°Vo ,azLx 'CTTij+l ■°'Ti1i2 ,<7tL1L2 + 1'°TL1L2+L +Ll+1 '"''

GtLIL2... Ly-1 '<7tLiL2'" LV-1+1 L1L2...Ly-1 +L1 ''"' ^L^ ...Iy-1 +L1L2 ...Ly-2 +...+L1 }

-( 22 1 — 1) (we assume the very first component equal to 1 for normalizing the eigenvector). In all system (56) has 22 + 22 1 — 1

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,°tL1 ,°TL1+1 ,°tL1L2 ,°tL1L2 +1'°tL1L2 +L1 ,°tL1L2 +L1+1 '...'°tL1L2...Lv-1

42

equations (256+127=383 in the three-dimensional case). The system of equations (56) is perfectly solved by the Levenberg-Marquardt method. In order to unify the system of equations for solving by the Levenberg-Marquardt method, to obtain only positive coefficients

Or,

b, } F =

¡°Vo ,0^+1 'aThLl 'aTLlLl +l'aTLlbl+L I

X exp( X Kt

(58)

,azLx '<ytLl+i ,<7tL1L2 ,<7tL1L2 +1'CJtL1L2 +L1 TL1L2+L1 +1 '"'' <7tL1L2-..Lv-1 '<7tLiL2'"IV-1 + 1'<7tLIL2'"LV-1+LI ''"'<JzLiL2'''Lv-1+LlL2-'Lv-2 + ~+Li }

and not to introduce restrictions on the field of using the Levenberg-Marquardt method, we introduce

K, 2 , = K , 2 , , where i(t, t1..... f ) - decimal notation

t ,t ,...,t i(t ,t ,...,t )

of binary 2V - digit number, that on the tp -th position, p = 1,...,s,

5 = 1,...,2V has 1, and the rest are zeros. i(t,t2,...,t ) = 1,...,22 .

The components of the eigenvector with the highest eigenvalue are expressed as

,a4 >% +1 >%L2 >%L;2 +1 >%L;2 + L\ ,'JtL1L1+Li+11

where summation over all non-empty subsets

' ^ t2 ts '

{ t ,t t ¡(iß . The number of such subsets

v 777 j — To g

for a 3-dimensional cube will be 2 = 2 = 256 , the total number of different components

b will be „ =127

< +1,°^2 Î

22 -1 -1

{c7T >% ,C7tL1L2 ,C7tL1L2 +1>CTtkL2 ,Î7tL1L2 +L+1 '''"

tL1L2...Lv ^ tL1L2...!v 1+1' tL1L2...Lv 1+L1^-' tL1L2...Lv 1 +L1L2...Lv 2 +...+L1-

exn(K )

'(a'0 '°'L1 '°TL1+1 'aLL2 'a'L1L2 +1'^U1L2 +L1 '°tL1L2 +1^+1''"' GxLIL2■■■ L I,CJtLIL2...Lv 1 +1'UTLIL2.:Lv 1 +Li y''^tL^-Ly 1+L1L2■■■ L 2 +-+L1'

Where

i(a. ,a. ,a. ,a. ,a. ,a. +i,a. ,

V т0 z1 ZLl> ZLl +1 zLiL2 tL1L2 +1 TLiL2+Li

TLil2...1v-^ TLil2...1v-i+^ TLiL2...1v-i+Ll '"'' TLiL2...1v-l +lil2...1v-2 +-+li (1 0, (1

)=

22 +

2

-2" +

2

21 + -

2

2 +

(1 -a ) (1 -a ) (1 -a )

TL1+^_23 ; TL1L^24 I TL1L2 +1' 25 +

2

2

2

(1 -aT ) (1 -aT ) v

L1L2 + _TL1L2...Lv-1+L1L2...Lv-2 +...+4 ^^^

2

2

When V = 3 we have:

n = {t = (ti,t2,t3): ti = 0,1,i = 1,2,3}

The system (55) is without changes. Then

b. } F =

№0 >СТЧ '<ITLl •CTii1+i '<TTLlL2 'aTLiL2 +l'aTLlL2 + ij }

(57)

z

{(T!0 >T4 тщ 'T4j+j TzLJL2 TzLJL2 +J'T'LlL2 +Lj )>

tljl2 +Lj+j ' TLJ+J ' TLJL2 'TL +1' TLJL2 +Lj ' TL1L2 +Lj +j >

{ТЦ'ТЩ 'TTLJ+J 'TTL1L2 'TTL1L2 +J'TTL1L2 +Lj TTLJL2 +Lj +j }

(we consider the very first component equal to 1 for normalizing the eigenvector). Total system (58) has 256 + 127 = 383 equations. System of equations ( 58) is perfectly solved by the Levenberg-Marquardt method. In order to unify the system of equations for solving by the Levenberg-Marquardt method, to obtain only positive coefficients b and not to introduce

iCT!0 >°n 'a'Li '{J'Li +1 '{J'LiL2 '{J'LiL2 +1 '{J'LiL2 +L }

restrictions on the field of using the Levenberg-Marquardt method, we introduce

K, , , = K , , , , where i(tl, t2,..., f ) - decimal notation

t ,t ,...,t i(t ,t ,...,t )

of binary 23 =8- digit number, that on the tp -th position, p = 1,...,S , S = 1,...,23 has 1, and the rest are zeros. i(tl,tZ,...,ts) = 1,...,22 . The components of the eigenvector with the highest eigenvalue are expressed as r (59)

b.

}

(60)

eXP( Ki (GTn,G„,GT. ,<JT. . ,1T ,<JT +i,CT. . . }) v т0 ^ тр ZLi1 T Li+V T L1L2 T L1L2 T L1L2 +Li

where

i(aT ,a, ,aT ,aT ,aT ,aT ) = 22' + (1 ^ 2° +

4 '1 +1 'iil2 hb2 LL2 +Ll 2

(1 -a, ) 1 (1 ) 2 (1 +1) 3 --—2 +-^22 +-^23 +

(1 -a ) (1 -a ) (1 -a )

__TAi2 24 + __LL2+1 25 + -_' LL2 +L1 26

At the initial approximation Ki = 0.1 , i = 1,2,...,383 , by the Levenberg-Marquardt method, the iterative process converges to the exact solution. The author does not write out the solution itself, since the set of all coefficients recorded with high accuracy takes several sheets of text.

Let Q = {t0,T1,TLi,tLiLi} = {to,t1,t2,t3} .

Hamiltonian of 3D model is equal to

¡=1 n=1

J12^ia2 + J13CT1CT3 + J23^2^3 + J0123^0^1^2^3

(61)

There is a one-to-one correspondence between the system of equations generated by Hamiltonian (61) and the system of equations

2

2

2

2

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for the flat model generated by Hamiltonian (20) with conditions (24), we search for the eigenvector in the form (25), this is clear from the correspondence of the vertices: (n, m) T0 ,

(n +1,m) OTj , (n,m +1) , (n +1,m +1) .

At the same time, after reduction to a flat model, the statistical sum

and free energy will depend only on the sum J01 (and not on each variable separately).

■ J23 H J02

J

That is, for a three-dimensional model with Hamiltonian (61) there will be the same system of equations for the interaction coefficients and free energy, as for the flat model in paragraphs a) and b) with the interaction coefficients

J1 _ J01

J3 _ J 12

J J = J

23, u 2 u02

J

J4 = J03

J5 _ J6 _ J7 _ J8 _ 0 ,

J9 _ J0123 , h = 0

Summary

The calculation of the statistical sum of the Ising and Potts lattice models for various values of the parameters is still of scientific interest. Therefore, finding a set of disorder exact solutions is an important task. The article obtained a system of nonlinear equations for finding the exact value of the free energy of a wide class of models with finite spin space, and obtained solutions of this system, depending on several parameters, for a wide class of models. The existence domain of these solutions is obtained as well.

References

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[14] Baxter R.J. Disorder points of the IRF and checkerboard Potts models. Journal of Physics A: Mathematical and General. 1984; 17(17):L911-L917. (In Eng.) DOI: 10.1088/03054470/17/17/001

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[25] Perron O. Zur Theorie der Matrices. Mathematische An-nalen. 1907; 64(2):248-263. (In Eng.) DOI: 10.1007/ BF01449896

Submitted 25.01.2019; revised 10.02.2019; published online 19.04.2019.

Поступила 25.01.2019; принята к публикации 10.02.2019; опубликована онлайн 19.04.2019.

Pavel V. Khrapov, Associate Professor of the Higher Mathematics Department, Bauman Moscow State Technical University (5/1 2-nd Baumanskaya St., Moscow 105005, Russia), Ph.D. (Phys.-Math.), ORCID: http://orcid.org/0000-0002-6269-0727, khrapov@bmstu. ru

The author has read and approved the final manuscript.

|об авторе:|

Храпов Павел Васильевич, доцент, кафедра высшей математики, Московский государственный технический университет имени Н.Э. Баумана (национальный исследовательский университет) (105005, Россия, г. Москва, 2-я Бауманская ул., д. 5, стр. 1), кандидат физико-математических наук, ORCID: http:// orcid.org/0000-0002-6269-0727, [email protected]

Автор прочитал и одобрил окончательный вариант рукописи.

Современные информационные технологии и ИТ-образование

Том 15, № 1. 2019

ISSN 2411-1473

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