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Low-temperature effective potential of the Ising model.

Andrea Pelissetto C, Ettore Vicari

Published Jan 1, 2008• 9 citations
OpenAlex

Abstract

We study the low-temperature effective potential of the Ising model. We evaluate the three-point and four-point zero-momentum renormalized coupling constants that parametrize the expansion of the effective potential near the coexistence curve. These results are obtained by a constrained analysis of the ǫ-expansion that uses accurate estimates for the two-dimensional Ising model.

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•Jan 1, 1998•15 citations
We study the low-temperature effective potential of the Ising model. We evaluate the three-point and four-point zero-momentum renormalized coupling constants that parametrize the expansion of the effective potential near the coexistence curve. These results are obtained by a constrained analysis of the ǫ-expansion that uses accurate estimates for the two-dimensional Ising model.
We compute the 2n-point renormalized coupling constants in the symmetric phase of the three-dimensional (3D) Ising model on the simple cubic lattice in terms of the high temperature expansions O(${\mathrm{\ensuremath{\mathrm{B}}}}^{17}$) of the Fourier transformed 2n-point connected correlation functions at zero momentum. Our high temperature estimates of these quantities, which enter into the small field expansion of the effective potential for a 3D scalar field at the infrared fixed point or, equivalently, in the critical equation of state of the 3D Ising model universality class, are compared with recent results obtained by renormalization group methods, strong coupling, stochastic simulations, as well as previous high temperature expansions.
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.
Using a finite-size scaling method combined with extensive Monte Carlo measurements of high statistics, we calculate the four-, six- and eight-point renormalized coupling constants defined at zero momentum in the symmetric phase of the three-dimensional Ising system. The results of the 2D Ising system that are directly measured are also reported. Our value of the four-point coupling constant for the 3D system agrees very well with the available estimates from other methods. Our values of the six- and eight-point coupling constants are significantly different from those obtained by other methods for the 3D system, although they agree reasonably well in the 2D system.