Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Doklady Physics

Connection-Matrix Eigenvalues in the Ising Model: Taking into Account Interaction with Next-Nearest Neighbors

Boris Kryzhanovsky, Leonid B. Litinskii

Published Nov 1, 2019• 10 citations
Publisher

Abstract

Abstract The connection matrix of the Ising model on a $$d$$-dimensional hypercube is investigated. In addition to the interactions between the nearest neighbors, the interactions between the next-nearest neighbors are taken into account. For such a matrix, the exact relations for the eigenvalues and eigenvectors, which are reasonably simply expressed through the corresponding characteristics of the one-dimensional Ising model, are obtained. Both periodic and free boundary conditions are considered.

DOI
10.1134/s1028335819110065

Related papers

Eigenvalues and eigenvectors of Ising model on hypercube

Boris Kryzhanovsky, Leonid B. Litinskii

Abstract For a multidimensional Ising model, we expressed eigenvalues of the connection matrix in terms of the spin-spin interaction constants and trigonometric polynomials. In such systems, the eigenvectors are the Kronecker products of the well-known eigenvectors for the one-dimensional case. When boundary conditions are periodic, it is possible to obtain rigorous expressions for the eigenvalues when there is an arbitrary long-range interaction in the system.

Journal of Physics Conference Series•Feb 1, 2021•0 citations

Influence of Long-Range Interaction on Eigenvalues of Connection Matrix in One-Dimensional Ising Model

Boris Kryzhanovsky, Leonid B. Litinskii

We analyze a finite one-dimensional Ising system with periodic boundary conditions taking into account an arbitrary long-range interaction. We examine a discrete spectrum of eigenvalues of the spin connection matrix and a spectrum density of a continuous distribution obtained in the limit $$L \to \infty $$ (L is the linear size of the system). We apply our results to particular cases of long-range interactions decreasing with distance.

Optical Memory and Neural Networks•Oct 1, 2020•0 citations

The 2D Ising square lattice with nearest- and next-nearest-neighbor interactions

Harold J. W. Zandvliet

An analytical expression for the boundary free energy of the Ising square lattice with nearest- and next-nearest-neighbor interactions is derived. The Ising square lattice with anisotropic nearest-neighbor ( J x and J y ) and isotropic next-nearest-neighbor ( J d ) interactions has an order-disorder phase transition at a temperature T = T c given by the condition e −2 J x / k b T c + e − 2 J y / k b T c + e −2( J x + J y )/ k b T c (2 − e −4 J d / k b T c ) = e 4 J d / k b T c . The critical line that separates the ordered (ferromagnetic) phase from the disordered (paramagnetic) phase is in excellent agreement with series expansion, finite scaling of transfer matrix and Monte Carlo results. For a vanishing next-nearest-neighbor interaction Onsager's famous result, i.e. sinh (2 J x / k b T c )sinh (2 J y / k b T c ) = 1, is recaptured.

Europhysics Letters (EPL)•Jan 19, 2006•36 citations

On the existence of an analytic solution to the 1-D Ising model with nearest and next-nearest neighbor interactions in the presence of a magnetic field

Farid Taherkhani, Ebrahim Daryaei, Hadi Abroshan +3 more

To solve the controversy, regarding the existence of an analytic solution to the 1-D Ising model with nearest-neighbor (NN) and next-nearest-neighbor (NNN) interactions in the presence of a magnetic field, we apply the transfer matrix method to solve the 1-D Ising model in the presence of a magnetic field, taking both NN and NNN interactions into account. We show that it is possible to write a transfer matrix only if the number of sites is even. Even in such a case, it is impossible to diagonalize the transfer matrix in an analytic form. Therefore, we employ a numerical method to obtain the eigenvalues of the transfer matrix. Moreover, the heat capacity, magnetization, and magnetic susceptibility versus temperature for different values of the competition factor (the ratio of NNN to NN interactions) are shown.

Phase Transitions•Dec 15, 2010•6 citations

Finite-lattice calculations for the two-dimensional axial-next-nearest-neighbor Ising model

Gareth O. Williams, Pál Ruján, H. L. Frisch

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-$\frac{1}{2}$ 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 $〈2〉$ antiphase, in fair agreement with previous analytical results.

Physical review. B, Condensed matter•Dec 1, 1981•12 citations

Critical exponents of small one-dimensional ising magnetic

D.V. Spirin, V. N. Udodov

Within the framework of a generalized Ising model, a one-dimensional magnetic of a finite length with free ends is considered. The correlation length exponent, dynamic critical exponent z of the magnet is calculated taking into account the next nearest neighbor interactions and the external field.

arXiv:cond-mat/0703639•Mar 24, 2007•0 citations
Abstract The connection matrix of the Ising model on a ddd-dimensional hypercube is investigated. In addition to the interactions between the nearest neighbors, the interactions between the next-nearest neighbors are taken into account. For such a matrix, the exact relations for the eigenvalues and eigenvectors, which are reasonably simply expressed through the corresponding characteristics of the one-dimensional Ising model, are obtained. Both periodic and free boundary conditions are considered.
We analyze a finite one-dimensional Ising system with periodic boundary conditions taking into account an arbitrary long-range interaction. We examine a discrete spectrum of eigenvalues of the spin connection matrix and a spectrum density of a continuous distribution obtained in the limit L→∞L \to \infty L→∞ (L is the linear size of the system). We apply our results to particular cases of long-range interactions decreasing with distance.
An analytical expression for the boundary free energy of the Ising square lattice with nearest- and next-nearest-neighbor interactions is derived. The Ising square lattice with anisotropic nearest-neighbor ( J x and J y ) and isotropic next-nearest-neighbor ( J d ) interactions has an order-disorder phase transition at a temperature T = T c given by the condition e −2 J x / k b T c + e − 2 J y / k b T c + e −2( J x + J y )/ k b T c (2 − e −4 J d / k b T c ) = e 4 J d / k b T c . The critical line that separates the ordered (ferromagnetic) phase from the disordered (paramagnetic) phase is in excellent agreement with series expansion, finite scaling of transfer matrix and Monte Carlo results. For a vanishing next-nearest-neighbor interaction Onsager's famous result, i.e. sinh (2 J x / k b T c )sinh (2 J y / k b T c ) = 1, is recaptured.
To solve the controversy, regarding the existence of an analytic solution to the 1-D Ising model with nearest-neighbor (NN) and next-nearest-neighbor (NNN) interactions in the presence of a magnetic field, we apply the transfer matrix method to solve the 1-D Ising model in the presence of a magnetic field, taking both NN and NNN interactions into account. We show that it is possible to write a transfer matrix only if the number of sites is even. Even in such a case, it is impossible to diagonalize the transfer matrix in an analytic form. Therefore, we employ a numerical method to obtain the eigenvalues of the transfer matrix. Moreover, the heat capacity, magnetization, and magnetic susceptibility versus temperature for different values of the competition factor (the ratio of NNN to NN interactions) are shown.
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.