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Journal of Siberian Federal University Mathematics & Physics

Cluster Perturbation Theory for 2d Ising Model

Ilya Ivantsov, Sergey G. Ovchinnikov

Published Feb 1, 2015• 0 citations
Publisher

Abstract

This paper deals with 2d Ising model in the scope of cluster perturbation theory.Ising model is defined on a two-dimensional square lattice, the amount of nearest neighbors z=4.Lattice is divided into clusters of a given size and a complete set of energy eigenvalues and eigenvectors of the cluster is defined by the diagonalization method.On the basis of this, Hubbard operators are constructed and Green function is calculated, taking into account intercluster interactions according to perturbation theory, it allows us to obtain the dependence of the magnetization on the temperature in the Hubbard-I approximation.Obtained results are compared with the exact solution of the two-dimensional Ising model.

DOI
10.17516/1997-1397-2015-8-1-31-37

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This paper deals with 2d Ising model in the scope of cluster perturbation theory.Ising model is defined on a two-dimensional square lattice, the amount of nearest neighbors z=4.Lattice is divided into clusters of a given size and a complete set of energy eigenvalues and eigenvectors of the cluster is defined by the diagonalization method.On the basis of this, Hubbard operators are constructed and Green function is calculated, taking into account intercluster interactions according to perturbation theory, it allows us to obtain the dependence of the magnetization on the temperature in the Hubbard-I approximation.Obtained results are compared with the exact solution of the two-dimensional Ising model.
The cluster perturbation theory is presented in the 2D Hubbard model constructed using X operators in the Hubbard-I approximation. The short-range magnetic order is taken into account by dividing the entire lattice into individual 2 × 2 clusters and solving the eigenvalue problem in an individual cluster using exact diagonalization taking into account all excited levels. The case of half-filling taking into account jumps between nearest neighbors is considered. As a result of numerical solution, a shadow zone is discovered in the quasiparticle spectrum. It is also found that a gap in the density of states in the quasiparticle spectrum at zero temperature exists for indefinitely small values of Coulomb repulsion parameter U and increases with this parameter. It is found that the presence of this gap in the spectrum is due to the formation of a short-range antiferromagnetic order. An analysis of the temperature evolution of the density of states shows that the metal-insulator transition occurs continuously. The existence of two characteristic energy scales at finite temperatures is demonstrated, the larger scale is associated with the formation of a pseudogap in the vicinity of the Fermi level, and the smaller scale is associated with the metal-insulator transition temperature. A peak in the density of states at the Fermi level, which is predicted in the dynamic mean field theory in the vicinity of the metal-insulator transition, is not observed.
I show that the cluster variation method, long used as a powerful hierarchy of approximations for discrete (Ising-like) two-dimensional lattice models, yields exact results on the disorder varieties which appear when competitive interactions are put into these models. I consider, as an example, the plaquette approximation of the cluster variation method for the square lattice Ising model with nearest-neighbor, next-nearest-neighbor, and plaquette interactions, and, after rederiving known results, report simple closed-form expressions for the pair and plaquette correlation functions.
Abstract We present important use cases and limitations when considering results obtained from cluster perturbation theory (CPT). CPT combines the solutions of small individual clusters of an infinite lattice system with the Bloch theory of conventional band theory to provide an approximation for the Green’s function in the thermodynamic limit. To this end, we are investigating single-band and multi-band Hubbard models in 1D and 2D systems. A special interest is taken in the supposed pseudogap regime of the 2D square lattice at half-filling and intermediate interaction strength ( U≤3tU \le 3tU≤3t U ≤ 3 t ) as well as the metal–insulator transition. We point out that the finite-size level spacing of the cluster limits the resolution of spectral features within CPT. This restricts the investigation of asymptotic properties of the metal–insulator transition, as it would require much larger cluster sizes that are beyond computational capabilities.
We present important use cases and limitations when considering results obtained from cluster perturbation theory (CPT). CPT combines the solutions of small individual clusters of an infinite lattice system with the Bloch theory of conventional band theory to provide an approximation for the Green’s function in the thermodynamic limit. To this end, we are investigating single-band and multi-band Hubbard models in 1D and 2D systems. A special interest is taken in the supposed pseudogap regime of the 2D square lattice at half-filling and intermediate interaction strength (U≤3t)(U≤3t)(U≤3t) as well as the metal–insulator transition. We point out that the finite-size level spacing of the cluster limits the resolution of spectral features within CPT. This restricts the investigation of asymptotic properties of the metal–insulator transition, as it would require much larger cluster sizes that are beyond computational capabilities.
Using the finite cluster approximation (FCA) based on a single-site cluster theory and Monte Carlo simulation (MCS) under the Metropolis algorithm, we study the critical properties of two-dimensional (2D) spins-½ system by analyzing the effect of the anisotropic parameter [Formula: see text] and next-nearest-neighbor (NNN) interaction [Formula: see text] on the critical temperature [Formula: see text] of the 2D anisotropic square lattice. It is found that the MCS is more reliable than the FCA method in the resolution of the Ising model. Moreover, MC calculations reveal that [Formula: see text] is strongly correlated with the external magnetic field [Formula: see text], [Formula: see text], and [Formula: see text], where the competition between these parameters can generate several rich phase diagrams.