For arbitrary Ising-like models of any dimension and Hamiltonians with a finite support with all possible multispin interactions and boundary conditions with a shift, the exact value of the free energy in the thermodynamic limit is obtained at some parametrically specified set of multispin interaction coefficients. In this case, half of the multispin interaction coefficients and the coordinates of the special eigenvector corresponding to the largest eigenvalue of the elementary transfer matrix are parameters, and the second half of the multispin coefficients is calculated using simple explicit formulas. For models with Hamiltonians invariant under the reversal of signs of all spins, the formulas are simplified. As examples of independent interest, solutions are written for the cases when the support of the Hamiltonian is a simplex, a cube, the support of the ANNNI model in spaces of 2, 3 and arbitrary dimensions.
For arbitrary Ising-like models of any dimension and Hamiltonians with a finite support with all possible multispin interactions and boundary conditions with a shift, the exact value of the free energy in the thermodynamic limit is obtained at some parametrically specified set of multispin interaction coefficients. In this case, half of the multispin interaction coefficients and the coordinates of the special eigenvector corresponding to the largest eigenvalue of the elementary transfer matrix are parameters, and the second half of the multispin coefficients is calculated using simple explicit formulas. For models with Hamiltonians invariant under the reversal of signs of all spins, the formulas are simplified. As examples of independent interest, solutions are written for the cases when the support of the Hamiltonian is a simplex, a cube, the support of the ANNNI model in spaces of 2, 3 and arbitrary dimensions.
For generalized 2D Ising model in an external magnetic field with the interaction of nearest neighbors, next nearest neighbors, all kinds of triple interactions and the quadruple interaction the formulas for finding free energy per lattice site in the thermodynamic limit were derived on a certain set of exact disordered solutions depending on seven parameters. Lattice models are considered with boundary conditions with a shift (similar to helical ones), and with cyclic closure of the set of all points in natural ordering. The elementary transfer matrix with nonnegative matrix elements are constructed. On the set of disorder solutions the largest eigenvalue of the transfer matrix is constant for every size of considering planar lattice, and, in particular, in the thermodynamic limit. Free energy per lattice site in the thermodynamic limit is expressed through the natural logarithm of the largest eigenvalue of transfer matrix. This largest eigenvalue can be found for a special form of eigenvector with positive components. The numerical example show the existence of nontrivial solutions of the resulting systems of equations. The system of equations and the value of free energy in the thermodynamic limit will remain the same for 2D generalized Ising models with Hamiltonians, in which the values of two (out of four) neighboring maximal spins in the natural ordering are replaced by the values of the spins at any other two lattice points adjacent in the natural ordering, this significantly expands the set of models having disordered exact solutions. The high degree of symmetry and inductive construction of the components of the eigenvectors, which disappear when going beyond the framework of the obtained set of exact solutions, is an occasion to search for phase transitions in the vicinity of this set of disordered solutions.
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. 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.
This study demonstrates a development of convenient formulae for obtaining the value of the free energy in the thermodynamic limit on a set of exact disorder solutions depending on four parameters for a 2D generalized Ising model in an external magnetic field with the interaction of nearest neighbors, next nearest neighbors, all kinds of triple interactions and the four interactions for the planar model, and for the 3D generalized Ising model in an external magnetic field with all kinds of interactions in the tetrahedron formed by four spins: at the origin of the coordinates and the closest to it along three coordinate axes in the first coordinate octant. Lattice models are considered with boundary conditions with a shift (similar to helical ones), and a cyclic closure of the set of all points (in natural ordering). For both the planar model and the 3D model, elementary transfer matrices with nonnegative matrix elements are constructed, while the free energy in the thermodynamic limit is equal to the Napierian logarithm of the maximum eigenvalue of the transfer matrix. This maximum eigenvalue can be found for a special kind of eigenvector with positive components. The region of existence of these solutions is described. The examples show the existence of nontrivial solutions of the resulting systems of equations for plane and threedimensional generalized Ising models. The system of equations and the value of free energy in the thermodynamic limit will remain the same for plane and threedimensional models with Hamiltonians, in which the value of the maximum in the natural ordering of the spin is replaced by the value of the spin at almost any other point in the lattice, this significantly expands the set of models having disordered exact solutions. The high degree of symmetry and repeatability of the components of the found eigenvectors, disappearing when we go beyond the framework of the obtained set of exact solutions, are the reason for the search for phase transitions in the vicinity of this set of disordered solutions. В работе получены формулы для нахождения свободной энергии в термодинамическом пределе на множестве точных неупорядоченных решений (disordersolutions), зависящих от четырех параметров для 2Dобобщенной модели Изинга во внешнем магнитном полесо взаимодействием ближайших соседей, следующих ближайших соседей (nextnearest), всевозможных тройных взаимодействий и взаимодействия четырех спинов для плоской модели, и для 3Dобобщенной модели Изинга во внешнем магнитном полесо всевозможными взаимодействиями в тетраэдре, образованном четырьмя спинами: в начале координат и ближайшие к нему по трем координатным осям в первом координатном октанте. Решеточные модели рассматриваютсяс граничными условиями со сдвигом (похожие на винтовые), и циклическим замыканием множества всех точек (в естественном упорядочении). В обоих случаях для плоской и 3Dмоделей построены элементарные трансферматрицы с неотрицательными матричными элементами, при этом свободная энергия в термодинамическом пределе равна натуральному логарифму максимального собственного значения трансферматрицы. Это максимальное собственное значение удается найти для специального вида собственного вектора с положительными компонентами. Описана область существования этих решений.На примерах показано существование нетривиальных решений получающихся систем уравнений для плоских и трехмерных обобщенных моделей Изинга. Система уравнений и значение свободной энергиив термодинамическом пределе останутся прежними для плоских и трехмерных моделей с гамильтонианами, в которых значение максимального в естественном упорядочении спина заменено значением спина практически в любой другой точке решетки, это значительно расширяет множество моделей, имеющих неупорядоченные точные решения.Высокая симметрия и повторяемость компонент найденных собственных векторов, исчезающая при выходе за рамки полученного множества точных решений, является поводом для поиска фазовых переходов в окрестности этого множества неупорядоченных решений.
In this paper the exact solution and correlation functions for a double-chain Ising model with multi-spin interactions and symmetric Hamiltonian density are obtained. The study employs the transfer matrix method to derive fundamental thermodynamic characteristics of the system. The main results include exact expressions for the partition function, free energy, internal energy, specific heat capacity, magnetization, susceptibility, and entropy in a strip of finite length and in the thermodynamic limit. The work provides explicit formulas for the eigenvalues and shows structure of eigenvectors of the transfer matrix. The expression for magnetization in the thermodynamic limit using components of normalized eigenvector corresponding to the maximum eigenvalue is obtained. A detailed analysis is conducted for a special case of interactions involving all kinds of two- and four-spin interactions. This gives the simplified formula for free energy, it is calculated using the root of quadratic equation. Using special relation between parameters the solution for gonihedric model on the plane is obtained. The research reveals properties of the system, including specific features of ground states and phase diagram characteristics. Particular attention is given to the behavior of physical quantities near frustration points and the investigation of spin correlation functions. Plots of physical characteristics, including inverse correlation length, illustrating the obtained results are constructed.
We obtain the exact physical characteristics of the triple-chain Ising model on a torus with all possible multispin interactions invariant with respect to rotation by the angle 2π/3. The exact value of the partition function in a finite cyclically closed strip of length L, as well as the free energy, internal energy, specific heat, magnetization, susceptibility, and entropy in the thermodynamic limit at L→∞ are found by the transfer-matrix method for the model. The spectrum of the transfer-matrix and the structure of its eigenvectors are found. For two special cases - for the model with multispin interactions of even number of spins and for the model with some interactions of two, three, four and six spins, simplified expressions of the mentioned physical characteristics are obtained; in the thermodynamic limit they are expressed through the logarithm of the root of the quadratic equation. For the model with multispin interactions of an even number of spins, a kind of pair correlations in the thermodynamic limit is found, and it is shown that the magnetization at zero magnetic field is equal to zero; the structure of the ground states of the system is found and examples of their projections of seven-dimensional space onto three-dimensional space and examples of configurations corresponding to these ground states are given. The correlation length is shown and its graphs are given. As special cases, we consider the planar triangular model with all possible interactions, including, perhaps, different triple interactions inside neighboring triangles, and the planar model with nearest-neighbor, next nearest-neighbor, and plaquette interactions. For them the main exact physical characteristics have been found. This allowed us to obtain them for the planar gonihedric model as well.
The susceptibility in arbitrary directions, the specific heat, the energy and the magnetization of general Ising-type chains are exactly expressed in terms of the eigenvectors of the transfer matrix corresponding to the Ising-type interaction. These quantities, as a result, can be explicitly obtained for small spin values and generally obtained from the solution of corresponding eigenvalue problem of finite degree. Numerical estimations are easy for arbitrary spin values. This formula includes the spin- S transverse Ising model with vanishing or non-vanishing parallel external field, the Blume-Emery-Griffiths model and the Blume-Capel model, mixed spin and mixed bond periodic systems such as alternating Ising chains or Ising ferrimagnets and also includes Ising models with random structures. Low temperature behaviors of these systems and the crossover to infinite spin systems are also investigated.