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

arXiv:0712.0273 · Physical Review B

Characteristics of two-dimensional lattice models from a fermionic realization: Ising and XYZ models

Sh. Khachatryan, A. Sedrakyan

Published Sep 28, 2009• 11 citations
arXivHTMLPDF

Abstract

We develop a field theoretical approach to the classical two-dimensional (2D) models, particularly to 2D Ising model (2DIM) and $XYZ$ model, which is simple to apply for calculation of various correlation functions. We calculate the partition function of 2DIM and $XY$ model within the developed framework. Determinant representation of spin-spin correlation functions is derived using fermionic realization for the Boltzmann weights. The approach also allows formulation of the partition function of 2DIM in the presence of an external magnetic field.

DOI
10.1103/physrevb.80.125128

Related papers

Fermionic integrals and analytic solutions for two-dimensional Ising models

V. N. Plechko

We review some aspects of the fermionic interpretation of the two-dimensional Ising model.The use is made of the notion of the integral over the anticommuting Grassmann variables.For simple and more complicated 2D Ising lattices, the partition function can be expressed as a fermionic Gaussian integral.Equivalently, the 2D Ising model can be reformulated as a free-fermion theory on a lattice.For regular lattices the analytic solution then readily follows by passing to the momentum space for fermions.We also comment on the eective eld{theoretical (continuum{limit) fermionic formulations for the 2D Ising models near the critical point.

arXiv:cond-mat/9812434•Journal of Physical Studies•Jan 1, 1997•8 citations

Anticommuting Integrals and Fermionic Field Theories for Two-Dimensional Ising Models

V. N. Plechko

We review the applications of the integral over anticommuting Grassmann variables (nonquantum fermionic fields) to the analytic solutions and the field-theoretical formulations for the 2D Ising models. The 2D Ising model partition function $Q$ is presentable as the fermionic Gaussian integral. The use of the spin-polynomial interpretation of the 2D Ising problem is stressed, in particular. Starting with the spin-polynomial interpretation of the local Boltzmann weights, the Gaussian integral for $Q$ appears in the universal form for a variety of lattices, including the standard rectangular, triangular, and hexagonal lattices, and with the minimal number of fermionic variables (two per site). The analytic solutions for the correspondent 2D Ising models then follow by passing to the momentum space on a lattice. The symmetries and the question on the location of critical point have an interesting interpretation within this spin-polynomial formulation of the problem. From the exact lattice theory we then pass to the continuum-limit field-theoretical interpretation of the 2D Ising models. The continuum theory captures all relevant features of the original models near $T_c$. The continuum limit corresponds to the low-momentum sector of the exact theory responsible for the critical-point singularities and the large-distance behaviour of correlations. The resulting field theory is the massive two-component Majorana theory, with mass vanishing at $T_c$. By doubling of fermions in the Majorana representation, we obtain as well the 2D Dirac field theory of charged fermions for 2D Ising models. The differences between particular 2D Ising lattices are merely adsorbed, in the field-theoretical formulation, in the definition of the effective mass.

arXiv:hep-th/9607053•Presented at•Jul 7, 1996•6 citations

An empirical partition function for two-dimensional Ising Model in an external magnetic field

Rong Wei

There is no an accepted exact partition function (PF) for the two-dimensional (2D) Ising model with a non-zero external magnetic field to our knowledge. Here we infer an empirical PF for such an Ising model. We compare the PFs for two finite-size Ising lattices ($4\times 4$ and $4\times 6$) from this empirical PF with those from Wei (2018) (Wei, R.Q., 2018. An exact solution to the partition function of the finite-size Ising Model, arXiv: General Physics: 1805.01366.), and find that they are consistent very well. Based on this empirical PF, we further analyze and calculate the thermodynamic functions (heat capacity, magnetization, susceptibility) of this 2D Ising model and discuss the model's singularity semiquantitatively. Analysis and calculations from this PF show that they are coincident with those from other related studies; Especially the 2D Ising model in an external magnetic field has spontaneous magnetization (SM) calling the phenomenon at the critical temperature a phase transition, and the SM decreases with the increasing temperature. However, the decreasing variation of the SM here is different from that obtained by Yang (1952) from the 2D Ising model in a weak magnetic field.

arXiv:1905.04295•May 10, 2019•2 citations

Free fermions in two-dimensional Ising model

V. N. Plechko

The two{dimensional (2D) Ising model is reviewed as a theory of free fermions on a lattice.The discussion includes the fermionization procedure based on the mirror{ordered factorization of the density matrix, Gaussian fermionic integral representation for partition function, the momentum{ space analysis and Onsager's result, the eective continuum{limit eld theories and the critical{ point singularities.The emergence of long{range fermionic correlations in a nonzero magnetic eld and the behaviour of the speci c heat along the critical isotherm are commented.Attention is given to the choice of rational computational devices.

Journal of Physical Studies•Jan 1, 1999•4 citations

Partition Function for Two-Dimensional Nearest Neighbour Ising Model in a Non-Zero Magnetic Field for a Square Lattice of 16 Sites

G. Nandhini, M.V. Sangaranarayanan

An explicit expression for the partition function of two-dimensional nearest neighbour Ising models in the presence of a magnetic field is derived by a systematic enumeration of all the spin configurations pertaining to a square lattice of sixteen sites. The critical temperature is shown to be in excellent agreement with the reported values while the corresponding dimensionless magnetic field is obtained as 0.004.

arXiv:0710.3688•Oct 19, 2007•0 citations

Free Fermions and Two-Dimensional Ising Model

V. N. Plechko

The two-dimensional Ising model is representable as a lattice free-fermion field theory in terms of the integral over anticommuting Grassmann variables. The exact solution in a zero magnetic field then follows by evaluating Gaussian fermionic integral for partition function. It is argued that by switching on of a non-zero magnetic field the non-local and non-Gaussian fermionic terms appear in the action, which rather imply approximating methods of the analysis.

arXiv:math-ph/0411084•Nov 29, 2004•2 citations
We develop a field theoretical approach to the classical two-dimensional (2D) models, particularly to 2D Ising model (2DIM) and XYZXYZXYZ model, which is simple to apply for calculation of various correlation functions. We calculate the partition function of 2DIM and XYXYXY model within the developed framework. Determinant representation of spin-spin correlation functions is derived using fermionic realization for the Boltzmann weights. The approach also allows formulation of the partition function of 2DIM in the presence of an external magnetic field.
We review some aspects of the fermionic interpretation of the two-dimensional Ising model.The use is made of the notion of the integral over the anticommuting Grassmann variables.For simple and more complicated 2D Ising lattices, the partition function can be expressed as a fermionic Gaussian integral.Equivalently, the 2D Ising model can be reformulated as a free-fermion theory on a lattice.For regular lattices the analytic solution then readily follows by passing to the momentum space for fermions.We also comment on the eective eld{theoretical (continuum{limit) fermionic formulations for the 2D Ising models near the critical point.
We review the applications of the integral over anticommuting Grassmann variables (nonquantum fermionic fields) to the analytic solutions and the field-theoretical formulations for the 2D Ising models. The 2D Ising model partition function QQQ is presentable as the fermionic Gaussian integral. The use of the spin-polynomial interpretation of the 2D Ising problem is stressed, in particular. Starting with the spin-polynomial interpretation of the local Boltzmann weights, the Gaussian integral for QQQ appears in the universal form for a variety of lattices, including the standard rectangular, triangular, and hexagonal lattices, and with the minimal number of fermionic variables (two per site). The analytic solutions for the correspondent 2D Ising models then follow by passing to the momentum space on a lattice. The symmetries and the question on the location of critical point have an interesting interpretation within this spin-polynomial formulation of the problem. From the exact lattice theory we then pass to the continuum-limit field-theoretical interpretation of the 2D Ising models. The continuum theory captures all relevant features of the original models near TcT_cTc​. The continuum limit corresponds to the low-momentum sector of the exact theory responsible for the critical-point singularities and the large-distance behaviour of correlations. The resulting field theory is the massive two-component Majorana theory, with mass vanishing at TcT_cTc​. By doubling of fermions in the Majorana representation, we obtain as well the 2D Dirac field theory of charged fermions for 2D Ising models. The differences between particular 2D Ising lattices are merely adsorbed, in the field-theoretical formulation, in the definition of the effective mass.
There is no an accepted exact partition function (PF) for the two-dimensional (2D) Ising model with a non-zero external magnetic field to our knowledge. Here we infer an empirical PF for such an Ising model. We compare the PFs for two finite-size Ising lattices (4×44\times 44×4 and 4×64\times 64×6) from this empirical PF with those from Wei (2018) (Wei, R.Q., 2018. An exact solution to the partition function of the finite-size Ising Model, arXiv: General Physics: 1805.01366.), and find that they are consistent very well. Based on this empirical PF, we further analyze and calculate the thermodynamic functions (heat capacity, magnetization, susceptibility) of this 2D Ising model and discuss the model's singularity semiquantitatively. Analysis and calculations from this PF show that they are coincident with those from other related studies; Especially the 2D Ising model in an external magnetic field has spontaneous magnetization (SM) calling the phenomenon at the critical temperature a phase transition, and the SM decreases with the increasing temperature. However, the decreasing variation of the SM here is different from that obtained by Yang (1952) from the 2D Ising model in a weak magnetic field.
The two{dimensional (2D) Ising model is reviewed as a theory of free fermions on a lattice.The discussion includes the fermionization procedure based on the mirror{ordered factorization of the density matrix, Gaussian fermionic integral representation for partition function, the momentum{ space analysis and Onsager's result, the eective continuum{limit eld theories and the critical{ point singularities.The emergence of long{range fermionic correlations in a nonzero magnetic eld and the behaviour of the speci c heat along the critical isotherm are commented.Attention is given to the choice of rational computational devices.