arXiv:hep-th/9511174

Scaling Properties of the Ising Model in a Field

Uwe Grimm, Bernard Nienhuis

Published Nov 24, 1995• 4 citations

Abstract

The dilute A_3 model is a solvable IRF (interaction-round-a-face) model with three local states and adjacency conditions encoded by the Dynkin diagram of the Lie algebra A_3. It can be regarded as a solvable version of a critical Ising model in a magnetic field. One therefore expects the scaling limit to be governed by Zamolodchikov's integrable perturbation of the c=1/2 conformal field theory. We perform a detailed numerical investigation of the solutions of the Bethe ansatz equation for the off-critical model. Our results agree perfectly with the predicted values for the lowest masses of the stable particles and support the assumptions on the nature of the Bethe ansatz solutions which enter crucially in a recent thermodynamic Bethe ansatz calculation of the factorized scattering matrix.

DOI: 10.48550/arxiv.hep-th/9511174

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Integrable field theory and critical phenomena: the Ising model in a magnetic field

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arXiv:hep-th/9511174

Scaling Properties of the Ising Model in a Field

Uwe Grimm, Bernard Nienhuis

Published Nov 24, 1995• 4 citations

arXiv HTML PDF

Abstract

DOI: 10.48550/arxiv.hep-th/9511174

Related papers

Scaling limit of the Ising model in a field

Uwe Grimm, Bernard Nienhuis

The dilute ${\mathrm{A}}_{3}$ model is a solvable interaction round a face model with three local states and adjacency conditions encoded by the Dynkin diagram of the Lie algebra ${\mathrm{A}}_{3}$ . It can be regarded as a solvable spin-1 Ising model at the critical temperature in a magnetic field. One therefore expects the scaling limit to be governed by Zamolodchikov's integrable perturbation of the c=1/2 conformal field theory. Indeed, a recent thermodynamic Bethe ansatz approach succeeded in unveiling the corresponding ${\mathrm{E}}_{8}$ structure under certain assumptions on the nature of the Bethe ansatz solutions. In order to check these conjectures, we perform a detailed numerical investigation of the solutions of the Bethe ansatz equations for the critical and off-critical models. Scaling functions for the ground-state corrections and for the lowest spectral gaps are obtained, which give very precise numerical results for the lowest mass ratios in the massive scaling limit. While these agree perfectly with the ${\mathrm{E}}_{8}$ mass ratios, we observe one state that seems to violate the assumptions underlying the thermodynamic Bethe ansatz calculation. We also analyze the critical spectrum of the dilute ${\mathrm{A}}_{3}$ model, which exhibits excitations with a finite gap on top of the massless spectrum of the Ising conformal field theory.The dilute ${\mathrm{A}}_{3}$ model is a solvable interaction round a face model with three local states and adjacency conditions encoded by the Dynkin diagram of the Lie algebra ${\mathrm{A}}_{3}$. It can be regarded as a solvable spin-1 Ising model at the critical temperature in a magnetic field. One therefore expects the scaling limit to be governed by Zamolodchikov's integrable perturbation of the c=1/2 conformal field theory. Indeed, a recent thermodynamic Bethe ansatz approach succeeded in unveiling the corresponding ${\mathrm{E}}_{8}$ structure under certain assumptions on the nature of the Bethe ansatz solutions. In order to check these conjectures, we perform a detailed numerical investigation of the solutions of the Bethe ansatz equations for the critical and off-critical models. Scaling functions for the ground-state corrections and for the lowest spectral gaps are obtained, which give very precise numerical results for the lowest mass ratios in the massive scaling limit. While these agree perfectly with the ${\mathrm{E}}_{8}$ mass ratios, we observe one state that seems to violate the assumptions underlying the thermodynamic Bethe ansatz calculation. We also analyze the critical spectrum of the dilute ${\mathrm{A}}_{3}$ model, which exhibits excitations with a finite gap on top of the massless spectrum of the Ising conformal field theory.

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Progress of Theoretical Physics SupplementMay 16, 20132 citations

Scattering Amplitudes in Offcritical Models and RSOS Integrable Models

Vladimir V. Bazhanov, Nicolai Reshetikhin

In the present paper we study scaling field theories related to the class of exactly solvable noncritical models of statistical mechanics which are known as RSOS (restricted solid-on-solid) or IRF (interaction round a face) models. These models are related to simple Lie algebras and they generalize the integrable version of RSOS model proposed by Andrews, Baxter and Forrester. 1> The models related to gl(n) were found by Jimbo, Miwa and Okado,2> the models related to other classical Lie algebras were obtained by the same authors in Ref. 3) and by Pasquier in Ref. 4) for Dn-case with a special coupling constant. Critical properties of JMO model were found in Ref. 5) where the authors obtain the explicit formulas for magnetization in the model and compute its critical behavior. It was established there that the critical behavior of these models was described by coset conformal field theories. 6>.7> Therefore it is natural to expect that corre sponding scaling field theories will be integrable perturbations of these conformal field theories. s> In this paper we describe the thermodynamics of the noncritical magnetic chain. Then we find the ground state, excitations over the ground state and the scattering amplitudes of excitations. By getting the scaling limit over the ground state of the model we compute the central charges of corresponding field theories, the dimensions of perturbations, the spectrum and scattering amplitudes of excitations. Under certain assumptions we generalize these results for any Lie algebra of A-D-E type. In certain simplest cases this gives known S-matrices like Es, E1, Es scalar S-matrices.9> In certain other models matrix elements of the S-matrix are given by Pasquier's solutions 4> of the Yang-Baxter equation. We do not discuss here the relation of RSOS models to integrable models of vertex type. 23>' 25> Partially the relation was explained in Refs. 4) and 26). The relation between the spectrum of row-to-row transfer matrices in RSOS and in vertex models will be given in a separate publication. For some of the scaling models the same answer was obtained by Bernard andIn the present paper we study scaling field theories related to the class of exactly solvable noncritical models of statistical mechanics which are known as RSOS (restricted solid-on-solid) or IRF (interaction round a face) models. These models are related to simple Lie algebras and they generalize the integrable version of RSOS model proposed by Andrews, Baxter and Forrester. 1> The models related to gl(n) were found by Jimbo, Miwa and Okado,2> the models related to other classical Lie algebras were obtained by the same authors in Ref. 3) and by Pasquier in Ref. 4) for Dn-case with a special coupling constant. Critical properties of JMO model were found in Ref. 5) where the authors obtain the explicit formulas for magnetization in the model and compute its critical behavior. It was established there that the critical behavior of these models was described by coset conformal field theories. 6>.7> Therefore it is natural to expect that corre sponding scaling field theories will be integrable perturbations of these conformal field theories. s> In this paper we describe the thermodynamics of the noncritical magnetic chain. Then we find the ground state, excitations over the ground state and the scattering amplitudes of excitations. By getting the scaling limit over the ground state of the model we compute the central charges of corresponding field theories, the dimensions of perturbations, the spectrum and scattering amplitudes of excitations. Under certain assumptions we generalize these results for any Lie algebra of A-D-E type. In certain simplest cases this gives known S-matrices like Es, E1, Es scalar S-matrices.9> In certain other models matrix elements of the S-matrix are given by Pasquier's solutions 4> of the Yang-Baxter equation. We do not discuss here the relation of RSOS models to integrable models of vertex type. 23>' 25> Partially the relation was explained in Refs. 4) and 26). The relation between the spectrum of row-to-row transfer matrices in RSOS and in vertex models will be given in a separate publication. For some of the scaling models the same answer was obtained by Bernard and

Progress of Theoretical Physics SupplementJan 1, 199038 citations

Integrable field theory and critical phenomena: the Ising model in a magnetic field

Gesualdo Delfino A

The two-dimensional Ising model is the simplest model of statistical mechanics exhibiting a second-order phase transition. While in absence of magnetic field it is known to be solvable on the lattice since Onsager's work of the 1940s, exact results for the magnetic case have been missing until the late 1980s, when A Zamolodchikov solved the model in a field at the critical temperature, directly in the scaling limit, within the framework of integrable quantum field theory. In this paper, we review this field theoretical approach to the Ising universality class, with particular attention to the results obtained starting from Zamolodchikov's scattering solution and to their comparison with the numerical estimates on the lattice. The topics discussed include scattering theory, form factors, correlation functions, universal amplitude ratios and perturbations around integrable directions. Although we restrict our discussion to the Ising model, the emphasis is on the general methods of integrable quantum field theory which can be used in the study of all universality classes of critical behaviour in two dimensions.The two-dimensional Ising model is the simplest model of statistical mechanics exhibiting a second-order phase transition. While in absence of magnetic field it is known to be solvable on the lattice since Onsager's work of the 1940s, exact results for the magnetic case have been missing until the late 1980s, when A Zamolodchikov solved the model in a field at the critical temperature, directly in the scaling limit, within the framework of integrable quantum field theory. In this paper, we review this field theoretical approach to the Ising universality class, with particular attention to the results obtained starting from Zamolodchikov's scattering solution and to their comparison with the numerical estimates on the lattice. The topics discussed include scattering theory, form factors, correlation functions, universal amplitude ratios and perturbations around integrable directions. Although we restrict our discussion to the Ising model, the emphasis is on the general methods of integrable quantum field theory which can be used in the study of all universality classes of critical behaviour in two dimensions.

arXiv:hep-th/0312119Journal of Physics A Mathematical and GeneralMar 23, 2004104 citations

Critical properties of a dilute Ising model in the percolation limit

J. Heinrichs

The critical behavior of the dilute Ising model on a Bethe lattice of coordination $z$ is studied in the percolation limit ($T\ensuremath{\rightarrow}0$) by means of a new moment-expansion procedure. Exact numerical results for the magnetization as a function of magnetic field at the critical concentration, $p={p}_{c}$ , are presented for $z=3$ and $z=4$ . In particular, the critical exponent ${\ensuremath{\delta}}_{P}$ is expected to agree well for very low fields with the approximate value ${\ensuremath{\delta}}_{P}=2$ obtained by Essam et al. The nature of deviations from the value ${\ensuremath{\delta}}_{P}=2$ is discussed and their consequences are noted. The case $p\ensuremath{\ne}{p}_{c}$ is also discussed.The critical behavior of the dilute Ising model on a Bethe lattice of coordination $z$ is studied in the percolation limit ($T\ensuremath{\rightarrow}0$) by means of a new moment-expansion procedure. Exact numerical results for the magnetization as a function of magnetic field at the critical concentration, $p={p}_{c}$, are presented for $z=3$ and $z=4$. In particular, the critical exponent ${\ensuremath{\delta}}_{P}$ is expected to agree well for very low fields with the approximate value ${\ensuremath{\delta}}_{P}=2$ obtained by Essam et al. The nature of deviations from the value ${\ensuremath{\delta}}_{P}=2$ is discussed and their consequences are noted. The case $p\ensuremath{\ne}{p}_{c}$ is also discussed.

Physical review. B, Condensed matterAug 1, 19781 citation

The dilute

{\mathrm{A}}_{3}

model is a solvable interaction round a face model with three local states and adjacency conditions encoded by the Dynkin diagram of the Lie algebra

{\mathrm{A}}_{3}

. It can be regarded as a solvable spin-1 Ising model at the critical temperature in a magnetic field. One therefore expects the scaling limit to be governed by Zamolodchikov's integrable perturbation of the c=1/2 conformal field theory. Indeed, a recent thermodynamic Bethe ansatz approach succeeded in unveiling the corresponding

{\mathrm{E}}_{8}

structure under certain assumptions on the nature of the Bethe ansatz solutions. In order to check these conjectures, we perform a detailed numerical investigation of the solutions of the Bethe ansatz equations for the critical and off-critical models. Scaling functions for the ground-state corrections and for the lowest spectral gaps are obtained, which give very precise numerical results for the lowest mass ratios in the massive scaling limit. While these agree perfectly with the

{\mathrm{E}}_{8}

mass ratios, we observe one state that seems to violate the assumptions underlying the thermodynamic Bethe ansatz calculation. We also analyze the critical spectrum of the dilute

{\mathrm{A}}_{3}

model, which exhibits excitations with a finite gap on top of the massless spectrum of the Ising conformal field theory.

The critical behavior of the dilute Ising model on a Bethe lattice of coordination

z

is studied in the percolation limit ($T\ensuremath{\rightarrow}0$) by means of a new moment-expansion procedure. Exact numerical results for the magnetization as a function of magnetic field at the critical concentration,

p={p}_{c}

, are presented for

z=3

and

z=4

. In particular, the critical exponent ${\ensuremath{\delta}}_{P}$ is expected to agree well for very low fields with the approximate value ${\ensuremath{\delta}}_{P}=2$ obtained by Essam et al. The nature of deviations from the value ${\ensuremath{\delta}}_{P}=2$ is discussed and their consequences are noted. The case $p\ensuremath{\ne}{p}_{c}$ is also discussed.