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Critical amplitudes and mass spectrum of the 2D Ising model in a magnetic field

Caselle, M, Hasenbusch, M

Published Nov 29, 1999• 27 citations
OpenAlex

Abstract

We compute the spectrum and several critical amplitudes of the two dimensional Ising model in a magnetic field with the transfer matrix method. The three lightest masses and their overlaps with the spin and the energy operators are computed on lattices of a width up to L=21. In extracting the continuum results we also take into account the corrections to scaling due to irrelevant operators. In contrast with previous Monte Carlo simulations our final results are in perfect agreement with the predictions of S-matrix and conformal field theory. We also obtain the amplitudes of some of the subleading corrections, for which no S-matrix prediction has yet been obtained.

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The corrections to finite-size scaling in the critical two-point correlation function G(r) of 2D Ising model on a square lattice have been studied numerically by means of exact transfer-matrix algorithms. The systems have been considered, including up to 800 spins. The calculation of G(r) at a distance r equal to the half of the system size L shows the existence of an amplitude correction proportional to 1/L^2. A nontrivial correction ~1/L^0.25 of a very small magnitude also has been detected, as it can be expected from our recently developed GFD (grouping of Feynman diagrams) theory. Monte Carlo simulations of the squared magnetization of 3D Ising model have been performed by Wolff's algorithm in the range of the reduced temperatures t =< 0.000086 and system sizes L =< 410. The effective critical exponent beta_eff(t) tends to increase above the currently accepted numerical values. The critical coupling K_c=0.22165386(51) has been extracted from the Binder cumulant data within 96 =< L =< 384. The critical exponent 1/nu, estimated from the finite-size scaling of the derivatives of the Binder cumulant, tends to decrease slightly below the RG value 1.587 for the largest system sizes. The finite-size scaling of accurately simulated maximal values of the specific heat C_v in 3D Ising model confirms a logarithmic rather than power-like critical singularity of C_v.

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By using conformal-field theory, we classify the possible irrelevant operators for the Ising model on the square and triangular lattices. We analyze the existing results for the free energy and its derivatives and for the correlation length, showing that they are in agreement with the conformal-field theory predictions. Moreover, these results imply that the nonlinear scaling field of the energy-momentum tensor vanishes at the critical point. Several other peculiar cancellations are explained in terms of a number of general conjectures. We show that all existing results on the square and triangular lattice are consistent with the assumption that only nonzero spin operators are present.

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By using conformal-field theory, we classify the possible irrelevant operators for the Ising model with nearest-neighbour interactions on the square and triangular lattices. We analyse the existing results for the free energy and its derivatives and for the correlation length, showing that they are in agreement with the conformal-field theory predictions. Moreover, these results imply that the nonlinear scaling field of the T operator, where T is the energy–momentum tensor, vanishes at the critical point. Several other peculiar cancellations are explained in terms of a number of general conjectures. We show that all existing results on the square and triangular lattices are consistent with the assumption that only nonzero-spin operators are present.

arXiv:cond-mat/0106372•Journal of Physics A Mathematical and General•May 31, 2002•61 citations
We compute the spectrum and several critical amplitudes of the two dimensional Ising model in a magnetic field with the transfer matrix method. The three lightest masses and their overlaps with the spin and the energy operators are computed on lattices of a width up to L=21. In extracting the continuum results we also take into account the corrections to scaling due to irrelevant operators. In contrast with previous Monte Carlo simulations our final results are in perfect agreement with the predictions of S-matrix and conformal field theory. We also obtain the amplitudes of some of the subleading corrections, for which no S-matrix prediction has yet been obtained.
The corrections to finite-size scaling in the critical two-point correlation function G(r) of 2D Ising model on a square lattice have been studied numerically by means of exact transfer-matrix algorithms. The systems have been considered, including up to 800 spins. The calculation of G(r) at a distance r equal to the half of the system size L shows the existence of an amplitude correction proportional to 1/L^2. A nontrivial correction ~1/L^0.25 of a very small magnitude also has been detected, as it can be expected from our recently developed GFD (grouping of Feynman diagrams) theory. Monte Carlo simulations of the squared magnetization of 3D Ising model have been performed by Wolff's algorithm in the range of the reduced temperatures t =< 0.000086 and system sizes L =< 410. The effective critical exponent beta_eff(t) tends to increase above the currently accepted numerical values. The critical coupling K_c=0.22165386(51) has been extracted from the Binder cumulant data within 96 =< L =< 384. The critical exponent 1/nu, estimated from the finite-size scaling of the derivatives of the Binder cumulant, tends to decrease slightly below the RG value 1.587 for the largest system sizes. The finite-size scaling of accurately simulated maximal values of the specific heat C_v in 3D Ising model confirms a logarithmic rather than power-like critical singularity of C_v.
We present the results of numerical calculations giving accurate estimates of the magnetization of the two-dimensional Ising model on a square lattice. Moreover, we argue that these results are strict lower bounds to the correct magnetization M(H,T)M(H,T)M(H,T). The estimates are obtained by dividing the infinite lattice into finite strips of width between two and nine spins and infinite length. The largest eigenvalue and the corresponding eigenvector of the transfer matrix are then obtained by an iterative process. The estimates of M(H,T)M(H,T)M(H,T) converge to the correct answer for the infinite lattice everywhere except for a small region in the $T\ensuremath{-}H$ plane. We also compute isotherms and critical isobar for the corresponding lattice gas. Finally, we propose a new approximation to the transfer matrix, exactly solvable in two dimensions for H=OH=OH=O, which reproduces exactly the critical-point behavior of the full Ising model.
By using conformal-field theory, we classify the possible irrelevant operators for the Ising model on the square and triangular lattices. We analyze the existing results for the free energy and its derivatives and for the correlation length, showing that they are in agreement with the conformal-field theory predictions. Moreover, these results imply that the nonlinear scaling field of the energy-momentum tensor vanishes at the critical point. Several other peculiar cancellations are explained in terms of a number of general conjectures. We show that all existing results on the square and triangular lattice are consistent with the assumption that only nonzero spin operators are present.
The corrections to finite-size scaling in the critical two-point correlation function G(r) of 2D Ising model on a square lattice have been studied numerically by means of exact transfer-matrix algorithms. The systems of square geometry with periodic boundaries oriented either along <10> or along <11> direction have been considered, including up to 800 spins. The calculation of G(r) at a distance r equal to the half of the system size L shows the existence of an amplitude correction proportional to 1/L^2. A nontrivial correction proportional to 1/L^0.25 of a very small magnitude also has been detected in agreement with predictions of our recently developed GFD (grouping of Feynman diagrams) theory. A refined analysis of the recent MC data for 3D Ising, phi^4, and XY lattice models has been performed. It includes an analysis of the partition function zeros of 3D Ising model, an estimation of the correction-to-scaling exponent omega from the Binder cumulant data near criticality, as well as a study of the effective critical exponent eta and the effective amplitudes in the asymptotic expansion of susceptibility at the critical point. In all cases a refined analysis is consistent with our (GFD) asymptotic values of the critical exponents (nu=2/3, omega=1/2, eta=1/8 for 3D Ising model and omega=5/9 for 3D XY model), while the actually accepted "conventional" exponents are, in fact, effective exponents which are valid for approximation of the finite--size scaling behavior of not too large systems.
By using conformal-field theory, we classify the possible irrelevant operators for the Ising model with nearest-neighbour interactions on the square and triangular lattices. We analyse the existing results for the free energy and its derivatives and for the correlation length, showing that they are in agreement with the conformal-field theory predictions. Moreover, these results imply that the nonlinear scaling field of the T operator, where T is the energy–momentum tensor, vanishes at the critical point. Several other peculiar cancellations are explained in terms of a number of general conjectures. We show that all existing results on the square and triangular lattices are consistent with the assumption that only nonzero-spin operators are present.