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arXiv:2202.01834

The exact solution of the Axial Next-Nearest-Neighbor Ising model on width 2 spin chains

Pavel Khrapov, Anastasia Loginova

Published Feb 3, 2022• 0 citations
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Abstract

Using the transfer matrix method for Axial Next-Nearest Neighbor Ising model without an external field on a closed chain of spins of width 2 in the direction of interaction of only nearest neighbors and length L in the direction of interaction of nearest neighbors and next nearest neighbors, we found exact values of the partition function in a finite strip of length L , free energy, internal energy per node, heat capacity in a finite strip of length L and in the thermodynamic limit. The entire spectrum of the 16x16 transfer matrix and the structure of all transfer matrix eigenvectors are found. The problem of finding the spectrum of the transfer matrix is reduced to finding the spectrum of two matrixes of size 8x8. The characteristic polynomial of the first matrix is decomposed into two linear polynomials of multiplicity two and two square trinomials. The characteristic polynomial of the second matrix is decomposed into two polynomials of the fourth degree, the roots of which are found by the Ferrari method. The free energy in the thermodynamic limit has a very simple form and is expressed in terms of the logarithm of the root of the quadratic equation. Two theorems on the physical characteristics of the ANNNI model in a finite closed strip of length L and in the thermodynamic limit are formulated and proved. An example of calculation is given for some values of the model parameters in the thermodynamic limit, heat capacity graphs are plotted. A comparative analysis of the heat capacity for the double-stranded ANNNI model and the ANNNI model on the entire flat lattice is given. The closest connection and similarity of the physical characteristics of the two-chain model and the model on the entire flat lattice is shown.

DOI
10.48550/arxiv.2202.01834

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Using the transfer matrix method for Axial Next-Nearest Neighbor Ising model without an external field on a closed chain of spins of width 2 in the direction of interaction of only nearest neighbors and length L in the direction of interaction of nearest neighbors and next nearest neighbors, we found exact values of the partition function in a finite strip of length L , free energy, internal energy per node, heat capacity in a finite strip of length L and in the thermodynamic limit. The entire spectrum of the 16x16 transfer matrix and the structure of all transfer matrix eigenvectors are found. The problem of finding the spectrum of the transfer matrix is reduced to finding the spectrum of two matrixes of size 8x8. The characteristic polynomial of the first matrix is decomposed into two linear polynomials of multiplicity two and two square trinomials. The characteristic polynomial of the second matrix is decomposed into two polynomials of the fourth degree, the roots of which are found by the Ferrari method. The free energy in the thermodynamic limit has a very simple form and is expressed in terms of the logarithm of the root of the quadratic equation. Two theorems on the physical characteristics of the ANNNI model in a finite closed strip of length L and in the thermodynamic limit are formulated and proved. An example of calculation is given for some values of the model parameters in the thermodynamic limit, heat capacity graphs are plotted. A comparative analysis of the heat capacity for the double-stranded ANNNI model and the ANNNI model on the entire flat lattice is given. The closest connection and similarity of the physical characteristics of the two-chain model and the model on the entire flat lattice is shown.
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/32π/ 32π/3. The exact value of the partition function in a finite cyclically closed strip of length LLL, as well as the free energy, internal energy, specific heat, magnetization, susceptibility, and entropy in the thermodynamic limit at L→∞L \to \inftyL→∞ 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.
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.
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.
An example of a two-dimensional crystal in which each lattice point interacts with six neighbors is the so-called \triangular Ising lattice. In this work the transfer matrix method is used to nd the thermodynamic properties of the triangular Ising model with a limited number of rows, in the absence or presence of a magnetic eld. In the absence of a magnetic eld the results have been used to nd an analytical expression for the partition function of a model with any number of rows, from which the thermodynamic properties of the triangular planar lattice model are obtained as a polynomial in terms of x = e 2j tanh 2j=2 cosh2j, where J = jkT is the interaction energy of the two nearest neighbor spins. The degree of the polynomial depends on the number of rows in the model. The results are compared with the exact values. The calculated free energy is found to be very accurate, but the calculated internal energy and heat capacity show some small deviations from the exact values. In the presence of a magnetic eld the solution to the matrices is too complicated for a general analytical expression to be proposed for a model with any number of rows. However, our exact solution for the model with n = 4, 5, and 7 reveals an important point: that such results are independent of n when the eld-spin interaction energy is almost equal to or larger than that of the spin-spin interaction.
The two-dimensional axial-next-nearest-neighbor Ising model is studied numerically using finite-lattice results. The row-to-row transfer matrix is mapped into a simple one-dimensional spin-12\frac{1}{2}21​ quantum Hamiltonian. The leading eigenvalues and the corresponding eigenvectors are calculated numerically for chains of finite lengths. Using a finite-lattice renormalization-group transformation the high- (low-) temperature region is analyzed with the use of the Hamiltonian for the original (dual) model. Although there is no clear evidence for a massless phase, the phase boundaries obtained in this way are a strong indication of the presence of a sinusoidal phase between the paramagnetic and the 〈2〉〈2〉〈2〉 antiphase, in fair agreement with previous analytical results.
The thermodynamic properties of the two-dimensional Ising model with random (+or-J) nearest-neighbour interactions are determined using a numerical transfer matrix method. The calculations are performed on a long strip of width L(3<or=L<or=11) with periodic boundary conditions and finite-size scaling arguments are used to extrapolate to L= infinity . The central result is that the spin-glass correlation length diverges as T- nu at low temperature T with nu =2.59+or-0.13. The heat capacity is similar to the Schottky anomaly. At zero temperature the entropy per spin is 0.0701+or-0.0005 and the internal energy per spin is -(1.4024+or-0.0012)J.