arXiv:cond-mat/0011305 · Journal of Physics A Mathematical and General

The critical equation of state of the two-dimensional Ising model

Michele Caselle, Martin Hasenbusch B, Andrea Pelissetto C, Ettore Vicari

Published Mar 30, 2001• 32 citations

Abstract

We compute the 2 n -point coupling constants in the high-temperature phase of the two-dimensional Ising model by using transfer-matrix techniques. This provides the first few terms of the expansion of the effective potential (Helmholtz free energy) and of the equation of state in terms of the renormalized magnetization. By means of a suitable parametric representation, we determine an analytic extension of these expansions, providing the equation of state in the whole critical region in the t , h plane.

DOI: 10.1088/0305-4470/34/14/302

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arXiv:cond-mat/0011305 · Journal of Physics A Mathematical and General

The critical equation of state of the two-dimensional Ising model

Michele Caselle, Martin Hasenbusch B, Andrea Pelissetto C, Ettore Vicari

Published Mar 30, 2001• 32 citations

arXiv HTML PDF

Abstract

DOI: 10.1088/0305-4470/34/14/302

Related papers

Critical equation of state of three-dimensional XY systems

Massimo Campostrini B, Andrea Pelissetto C, Paolo Rossi +1 more

We address the problem of determining the critical equation of state of three-dimensional $\mathrm{XY}$ systems. For this purpose we first consider the small-field expansion of the effective potential (Helmholtz free energy) in the high-temperature phase. We compute the first few nontrivial zero-momentum n-point renormalized couplings, which parametrize such expansion, by analyzing the high-temperature expansion of an improved lattice Hamiltonian with suppressed leading scaling corrections. These results are then used to construct parametric representations of the critical equation of state which are valid in the whole critical regime, satisfy the correct analytic properties (Griffith's analyticity), and take into account the Goldstone singularities at the coexistence curve. A systematic approximation scheme is introduced, which is limited essentially by the number of known terms in the small-field expansion of the effective potential. From our approximate representations of the equation of state, we derive estimates of universal ratios of amplitudes. For the specific-heat amplitude ratio we obtain ${A}^{+}{/A}^{\ensuremath{-}}=1.055(3),$ to be compared with the best experimental estimate ${A}^{+}{/A}^{\ensuremath{-}}=1.054(1).$We address the problem of determining the critical equation of state of three-dimensional $\mathrm{XY}$ systems. For this purpose we first consider the small-field expansion of the effective potential (Helmholtz free energy) in the high-temperature phase. We compute the first few nontrivial zero-momentum n-point renormalized couplings, which parametrize such expansion, by analyzing the high-temperature expansion of an improved lattice Hamiltonian with suppressed leading scaling corrections. These results are then used to construct parametric representations of the critical equation of state which are valid in the whole critical regime, satisfy the correct analytic properties (Griffith's analyticity), and take into account the Goldstone singularities at the coexistence curve. A systematic approximation scheme is introduced, which is limited essentially by the number of known terms in the small-field expansion of the effective potential. From our approximate representations of the equation of state, we derive estimates of universal ratios of amplitudes. For the specific-heat amplitude ratio we obtain ${A}^{+}{/A}^{\ensuremath{-}}=1.055(3),$ to be compared with the best experimental estimate ${A}^{+}{/A}^{\ensuremath{-}}=1.054(1).$

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arXiv:1012.5004Physical Review BFeb 25, 201117 citations

We address the problem of determining the critical equation of state of three-dimensional

\mathrm{XY}

systems. For this purpose we first consider the small-field expansion of the effective potential (Helmholtz free energy) in the high-temperature phase. We compute the first few nontrivial zero-momentum n-point renormalized couplings, which parametrize such expansion, by analyzing the high-temperature expansion of an improved lattice Hamiltonian with suppressed leading scaling corrections. These results are then used to construct parametric representations of the critical equation of state which are valid in the whole critical regime, satisfy the correct analytic properties (Griffith's analyticity), and take into account the Goldstone singularities at the coexistence curve. A systematic approximation scheme is introduced, which is limited essentially by the number of known terms in the small-field expansion of the effective potential. From our approximate representations of the equation of state, we derive estimates of universal ratios of amplitudes. For the specific-heat amplitude ratio we obtain ${A}^{+}{/A}^{\ensuremath{-}}=1.055(3),$ to be compared with the best experimental estimate ${A}^{+}{/A}^{\ensuremath{-}}=1.054(1).$

s=1,3/2,\dots{}

and for the lattice Euclidean scalar-field theory with quartic self-interaction, on the simple-cubic and the body-centered-cubic lattices. Our bivariate high-temperature expansions, which extend through 24th order, enable us to compute, through the same order, all higher derivatives of the free energy with respect to the field, namely, all higher susceptibilities. These data make more accurate checks possible, in critical conditions, both of the scaling and the universality properties with respect to the lattice and the interaction structure and also help to improve an approximate parametric representation of the critical equation of state for the three-dimensional Ising model universality class.