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:cond-mat/9809069

On the two- and three-dimensional Lenz-Ising-Onsager problem in presence of magnetic field

Martin S. Kochman'ski

Published Sep 3, 1998• 0 citations
arXivHTMLPDF

Abstract

In this paper a new approach to solving the Ising-Onsager problem in external magnetic field is investigated. The expression for free energy on one Ising spin in external field both for the twodimensional and threedimensional Ising model with interaction of the nearest neighbors are derived. The representations of free energy being expressed by multidimensional integrals of Gauss type with the appropriate dimensionality are shown. Possibility of calculating the integrals and the critical indices on the base of the derived representations for free energy is investigated.

DOI
10.48550/arxiv.cond-mat/9809069

Related papers

Lenz-Ising-Onsager problem in an external field as a soluble problem of many fermions

Martin S. Kochman'ski

In this paper an approach to solving the two- (2D) and three-dimensional Ising models in an external magnetic field $H\ensuremath{\ne}0$ is developed. The general formalism for the approach to the problem is presented for the example of the 2D Ising model in the external magnetic field. The paper presents a method obtaining the Onsager solution and computations of asymptotic forms of low-temperature free energy for the 2D Ising model in the external magnetic field $(H).$ The free energy in the limiting case of the magnetic field tending to zero ($H\ensuremath{\rightarrow}0,$ $N,$ $M\ensuremath{\rightarrow}\ensuremath{\infty}$) at arbitrary temperature is also considered $(T\ensuremath{\ne}0).$

arXiv:cond-mat/9710203•Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics•Oct 1, 1997•4 citations

Partition function for two-dimensional nearest neighbour Ising model in the presence of external magnetic field

G. Nandhini, M.V. Sangaranarayanan

The partition function for two-dimensional nearest neighbour Ising models in the presence of a magnetic field is derived . A comparison with the partition functions predicted by Onsager is carried out. The critical temperature estimated by two different methods yields good agreement with the result of Yang and Lee.

arXiv:0706.3643•Jun 25, 2007•0 citations

Explicit matrix representation for the Hamiltonian of the one dimensional \mbox{spin~$1/2$} Ising model in mutually orthogonal external magnetic fields

Kunle Adegoke, Henry Otobrise, Tolulope Famoroti +3 more

We derive an explicit matrix representation for the Hamiltonian of the Ising model in mutually orthogonal external magnetic fields, using as basis the eigenstates of a system of non-interacting \mbox{spin~$1/2$} particles in external magnetic fields. We subsequently apply our results to obtain an analytical expression for the ground state energy per spin, to the fourth order in the exchange integral, for the Ising model in perpendicular external fields.

arXiv:1508.07314•Aug 28, 2015•0 citations

On the Lenz-Ising-Onsager Problem in an External Magnetic Field

Martin S. Kochman'ski

The Lenz-Ising-Onsager (LIO) problem in an external magnetic field in the second quantization representation is the subject of consideration of the paper. It is shown that the operator $V_h$ in the second quantization representation corresponding to Ising spins interaction with the external magnetic field $H$ can be represented in terms of single-subscript creation and anihilation Fermi operators in such a form that the operator $V_h$ commutes with the operator $\hat{P}\equiv(-1)^{\hat{S}}$, where $\hat{S}= \sum_mβ^†_mβ_m$ is the operator of a total number of Fermions. The possible consequences of such representation with it's relation to the LIO is discussed. In particular, the constructive proof of the Lee-Yang theorem on the absence of phase transition for Ising model in nonzero magnetic field $(\Re h\neq 0)$ is demonstrated.

arXiv:cond-mat/9807115•Jul 8, 1998•0 citations

Analytic Properties of the Onsager Solution of the Ising Model

Chanchal K. Majumdar

The analytic properties of the Onsager solution of the two-dimensional Ising lattice in zero external magnetic field are investigated. The free energy above the critical temperature ${T}_{c}$ can be regarded as an analytic function in the $u$ plane, where $u=\mathrm{exp}(\ensuremath{-}\frac{4J}{{k}_{B}T})$, with singularities at $u=0, \ensuremath{\infty}, \mathrm{and} \mathrm{exp}(\ifmmode\pm\else\textpm\fi{}\frac{4J}{{k}_{B}{T}_{c}})$. But its analytic continuation below the critical temperature does not describe a stable thermodynamic phase. Similar statements are true for the free energy below the critical temperature. The famous logarithmic singularity is briefly discussed. Finally, the knowledge of the analytic properties is shown, by elementary considerations, to explain the nonanalytic behavior at vanishing external field below the critical temperature---a result first enunciated by Yang and Lee for the free energy in the presence of an external magnetic field.

Physical Review•May 6, 1966•11 citations

Approximate thermodynamics of the two-dimensional Ising model in an external magnetic field

Mathieu Gaudreault

Abstract We introduce approximate expressions for the thermodynamics of the two-dimensional Ising model in an external magnetic field. The external field breaks the system’s symmetry, which complicates the exact calculation of the free energy. To address this, we write the model in its quaternion representation to apply a small angle approximation with respect to a reference frame. The approximation simplifies the transfer matrix to a centrosymmetric matrix for which the largest eigenvalue is known and depends on the external field strength. We therefore write an approximate expression for the system’s free energy and derive the magnetization, susceptibility, internal energy, and specific heat. These results are compared with numerical simulations performed with the BKL algorithm. We find that the approximate expressions and numerical simulations agree in regions where the approximation is valid.

arXiv:2607.16735•Journal of Physics A Mathematical and Theoretical•Apr 15, 2025•0 citations
In this paper a new approach to solving the Ising-Onsager problem in external magnetic field is investigated. The expression for free energy on one Ising spin in external field both for the twodimensional and threedimensional Ising model with interaction of the nearest neighbors are derived. The representations of free energy being expressed by multidimensional integrals of Gauss type with the appropriate dimensionality are shown. Possibility of calculating the integrals and the critical indices on the base of the derived representations for free energy is investigated.
In this paper an approach to solving the two- (2D) and three-dimensional Ising models in an external magnetic field $H\ensuremath{\ne}0$ is developed. The general formalism for the approach to the problem is presented for the example of the 2D Ising model in the external magnetic field. The paper presents a method obtaining the Onsager solution and computations of asymptotic forms of low-temperature free energy for the 2D Ising model in the external magnetic field (H).(H).(H). The free energy in the limiting case of the magnetic field tending to zero ($H\ensuremath{\rightarrow}0,$ N,N,N, $M\ensuremath{\rightarrow}\ensuremath{\infty}$) at arbitrary temperature is also considered $(T\ensuremath{\ne}0).$
Explicit matrix representation for the Hamiltonian of the one dimensional \mbox{spin~1/21/21/2} Ising model in mutually orthogonal external magnetic fields
We derive an explicit matrix representation for the Hamiltonian of the Ising model in mutually orthogonal external magnetic fields, using as basis the eigenstates of a system of non-interacting \mbox{spin~1/21/21/2} particles in external magnetic fields. We subsequently apply our results to obtain an analytical expression for the ground state energy per spin, to the fourth order in the exchange integral, for the Ising model in perpendicular external fields.
The Lenz-Ising-Onsager (LIO) problem in an external magnetic field in the second quantization representation is the subject of consideration of the paper. It is shown that the operator VhV_hVh​ in the second quantization representation corresponding to Ising spins interaction with the external magnetic field HHH can be represented in terms of single-subscript creation and anihilation Fermi operators in such a form that the operator VhV_hVh​ commutes with the operator P^≡(−1)S^\hat{P}\equiv(-1)^{\hat{S}}P^≡(−1)S^, where S^=∑mβm†βm\hat{S}= \sum_mβ^†_mβ_mS^=∑m​βm†​βm​ is the operator of a total number of Fermions. The possible consequences of such representation with it's relation to the LIO is discussed. In particular, the constructive proof of the Lee-Yang theorem on the absence of phase transition for Ising model in nonzero magnetic field (ℜh≠0)(\Re h\neq 0)(ℜh=0) is demonstrated.
The analytic properties of the Onsager solution of the two-dimensional Ising lattice in zero external magnetic field are investigated. The free energy above the critical temperature Tc{T}_{c}Tc​ can be regarded as an analytic function in the uuu plane, where $u=\mathrm{exp}(\ensuremath{-}\frac{4J}{{k}_{B}T})$, with singularities at $u=0, \ensuremath{\infty}, \mathrm{and} \mathrm{exp}(\ifmmode\pm\else\textpm\fi{}\frac{4J}{{k}_{B}{T}_{c}})$. But its analytic continuation below the critical temperature does not describe a stable thermodynamic phase. Similar statements are true for the free energy below the critical temperature. The famous logarithmic singularity is briefly discussed. Finally, the knowledge of the analytic properties is shown, by elementary considerations, to explain the nonanalytic behavior at vanishing external field below the critical temperature---a result first enunciated by Yang and Lee for the free energy in the presence of an external magnetic field.
Abstract We introduce approximate expressions for the thermodynamics of the two-dimensional Ising model in an external magnetic field. The external field breaks the system’s symmetry, which complicates the exact calculation of the free energy. To address this, we write the model in its quaternion representation to apply a small angle approximation with respect to a reference frame. The approximation simplifies the transfer matrix to a centrosymmetric matrix for which the largest eigenvalue is known and depends on the external field strength. We therefore write an approximate expression for the system’s free energy and derive the magnetization, susceptibility, internal energy, and specific heat. These results are compared with numerical simulations performed with the BKL algorithm. We find that the approximate expressions and numerical simulations agree in regions where the approximation is valid.