Two-dimensional Layered Ising Model: Exact Variational Formulation and Analysis

Ising models on the plane square lattice with an arbitrary variation of the bond strengths, ${\mathit{J}}^{\mathrm{\ensuremath{\parallel}}}$(z) and ${\mathit{J}}^{\mathrm{\ensuremath{\perp}}}$(z), with one of the two axial coordinates, z, are considered. The total entropy is exactly represented as a functional of contributions ${\mathrm{\ensuremath{\varepsilon}}}_{\mathit{q}}$(z) to the local energy density arising from the Onsager fermions with wave vector q parallel to the layer axis, y. The resulting explicit local expression provides an effective variational principle for the free energy and energy-density profiles. In the scaling limit the problem reduces to a set of independent second-order differential equations for each ${\mathrm{\ensuremath{\varepsilon}}}_{\mathit{q}}$(z). The power of the method is demonstrated by application to an interface between two uniform but distinct regions; this includes the problem of a wall with a surface field, ${\mathit{h}}_{1}$ , as a special case. Previous results for the bulk and surface exponents and for the energy-energy correlation function are easily recovered. Near criticality the method yields, in addition, universal, scaled energy-density profiles, \ensuremath{\varepsilon}(z;T), which exhibit rich crossovers and nonmonotonic variation with z.Ising models on the plane square lattice with an arbitrary variation of the bond strengths, ${\mathit{J}}^{\mathrm{\ensuremath{\parallel}}}$(z) and ${\mathit{J}}^{\mathrm{\ensuremath{\perp}}}$(z), with one of the two axial coordinates, z, are considered. The total entropy is exactly represented as a functional of contributions ${\mathrm{\ensuremath{\varepsilon}}}_{\mathit{q}}$(z) to the local energy density arising from the Onsager fermions with wave vector q parallel to the layer axis, y. The resulting explicit local expression provides an effective variational principle for the free energy and energy-density profiles. In the scaling limit the problem reduces to a set of independent second-order differential equations for each ${\mathrm{\ensuremath{\varepsilon}}}_{\mathit{q}}$(z). The power of the method is demonstrated by application to an interface between two uniform but distinct regions; this includes the problem of a wall with a surface field, ${\mathit{h}}_{1}$, as a special case. Previous results for the bulk and surface exponents and for the energy-energy correlation function are easily recovered. Near criticality the method yields, in addition, universal, scaled energy-density profiles, \ensuremath{\varepsilon}(z;T), which exhibit rich crossovers and nonmonotonic variation with z.

Physical review. B, Condensed matterJan 1, 199428 citations

Alternative formulation of the variational method for systems with an infinite number of degrees of freedom: application to Ising models

M Villani

For systems with an infinite number of degrees of freedom, the Rayleigh-Ritz quotient of the standard variational method is analysed from an alternative point of view. Its numerator and denominator are treated as statistical models having the same correlation length. The approach is developed in the framework of the transfer matrix formalism and applied to Ising models. It allows the author to obtain exactly the correlation length along a row or column of the anisotropic square Ising model, for T>or=T c . Approximate applications of the method to the two-layer square and 3D simple cubic Ising models are also given. In these cases quite reliable critical curves are obtained through an essentially analytical procedure.For systems with an infinite number of degrees of freedom, the Rayleigh-Ritz quotient of the standard variational method is analysed from an alternative point of view. Its numerator and denominator are treated as statistical models having the same correlation length. The approach is developed in the framework of the transfer matrix formalism and applied to Ising models. It allows the author to obtain exactly the correlation length along a row or column of the anisotropic square Ising model, for T>or=T c . Approximate applications of the method to the two-layer square and 3D simple cubic Ising models are also given. In these cases quite reliable critical curves are obtained through an essentially analytical procedure.

Journal of Physics A Mathematical and GeneralNov 7, 19905 citations

Exact Solution of the Two-Dimensional Ising Model

R. Shankar

Cambridge University Press eBooksAug 31, 201712 citations

Constrained variational approach to second order phase transitions: application to a two-layer Ising film

Leonardo Angelini, D. Caroppo, M. Pellicoro +1 more

Within the framework of the transfer matrix formalism, a variational method is elaborated with a constraint related to the fundamental role of the correlation length near a critical point. The method is applied to a two-layer ferromagnetic film, made of two horizontal Ising planes interacting through a vertical coupling. The critical curve of the model, as well as the correlation length above and near T c are calculated. The critical exponent nu obtained agrees with the expected two-dimensional exact value. For the two particular values of the vertical coupling, for which there are numerical estimates of T c through series analysis or the Monte Carlo method, an essential agreement with these estimates is found. Analytical expressions are given in the weak and strong vertical coupling regimes.Within the framework of the transfer matrix formalism, a variational method is elaborated with a constraint related to the fundamental role of the correlation length near a critical point. The method is applied to a two-layer ferromagnetic film, made of two horizontal Ising planes interacting through a vertical coupling. The critical curve of the model, as well as the correlation length above and near T c are calculated. The critical exponent nu obtained agrees with the expected two-dimensional exact value. For the two particular values of the vertical coupling, for which there are numerical estimates of T c through series analysis or the Monte Carlo method, an essential agreement with these estimates is found. Analytical expressions are given in the weak and strong vertical coupling regimes.

Journal of Physics A Mathematical and GeneralNov 7, 199213 citations

Numerical Analysis of 2-D Ising Model

Ishita Agarwal

Jan 1, 20111 citation

{\mathit{h}}_{1}

, as a special case. Previous results for the bulk and surface exponents and for the energy-energy correlation function are easily recovered. Near criticality the method yields, in addition, universal, scaled energy-density profiles, \ensuremath{\varepsilon}(z;T), which exhibit rich crossovers and nonmonotonic variation with z.