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Europhysics Letters (EPL)

Correlation Functions of the Critical Ising Model on a Torus

Philippe Di Francesco, Hubert Saleur, Jean-Bernard Zuber

Published Jan 15, 1988• 12 citations
Publisher

Abstract

Using different kinds of bosonization techniques, we derive expressions for the correlation functions of the critical Ising model on a torus. They are solutions of linear differential equations generalizing those written by Belavin, Polyakov and Zamolodchikov in the plane, involving derivatives with respect to the torus modular ratio. Two- and four-spin correlation functions provide an expression of the renormalized coupling constant M 4 / M 2 2 , in good agreement with numerical studies.

DOI
10.1209/0295-5075/5/2/001

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Using different kinds of bosonization techniques, we derive expressions for the correlation functions of the critical Ising model on a torus. They are solutions of linear differential equations generalizing those written by Belavin, Polyakov and Zamolodchikov in the plane, involving derivatives with respect to the torus modular ratio. Two- and four-spin correlation functions provide an expression of the renormalized coupling constant M 4 / M 2 2 , in good agreement with numerical studies.
Using results from conformal field theory, we compute several universal amplitude ratios for the two-dimensional Ising model at criticality on a symmetric torus. These include the correlation-length ratio x* = lim(L-->infinity) xi(L)/L and the first four magnetization moment ratios V-2n = [M-2n]/[M-2](n). As a corollary we get the first four renormalized 2n-point coupling constants for the massless theory on a symmetric torus, G(2n)*. We confirm these predictions by a high-precision Monte Carlo simulation.
This thesis is concerned with several aspects of the Ising and tri-critical Ising models in two-dimensions. These are much-studied models relevant in both condensed matter physics as descriptions of the critical phenomena of two- dimensional systems and in String theory as building blocks of the string world sheet theory. The first part of the thesis is concerned with the derivation of differential equations for the critical four-point function in the Ising model. We present a method which provides the well-known standard solutions by a new and efficient route. The second part of the thesis is concerned with off-critical behaviour, and in particular the numerical study of perturbations of conformal field theory through the truncated conformal space approach. We show that the coupling constant undergoes significant renormalization in this scheme, and in particular the Ising model can be found as a fixed point for a finite value of the bare coupling constant. The renormalization group equations we find are of general use in the TCSA approach. The final part of the thesis considers off-critical boundary conditions in the tri-critical Ising model. We study them using a variant of the mean-field method and find a qualitative description of the space of boundary conditions that is in accord with the exact conformal field theory description. This is both a test of the method and its applicability in new domains, and also shows that previously published results are in error.
The behavior of dimensionless quantities defined as ratios of partition functions is analyzed to investigate phase transitions and critical phenomena. At criticality, the universal values of these ratios can be predicted from conformal field theory (CFT) through the modular-invariant partition functions on a torus. We perform numerical calculations using the bond-weighted tensor renormalization group for three two-dimensional models belonging to different universality classes: the Ising model, the three-state Potts model, and the four-state Potts model. The partition-function ratios obey the same finite-size scaling form as the Binder parameter, and their critical values agree well with the universal values predicted by CFT. In the four-state Potts model, we observe logarithmic corrections in the system-size dependence of these ratios.
In a new theory for the spin one-half Ising ferromagnet, the fourth-order equation is used to provide expressions for the higher-order site-independent correlation functions M 2n (n=3,4,5,...) in terms of the two- and four-spin correlation functions M 2 and M 4 , at the critical point. As an example, the expression for M 6 is M 4 2 /M 2 -9M 2 M 4 -6M 2 3 . These expressions (for up to M 60 ) are applied to the calculation of T c for the SC, BCC, and FCC lattices. The results converge to, at most, 0.755, 0.799, and 0.819 respectively, which are within 2 / 3 % of the accepted results 0.752, 0.794, and 0.816 of Sykes et al. (1972).
The energy-density---energy-density and energy-density---spin correlation functions are evaluated for the two-dimensional square Ising model. Results are obtained at zero magnetic field and near the critical point, i.e., for $\ensuremath{\epsilon}=(\frac{4J}{\mathrm{kT}})|(T\ensuremath{-}\frac{{T}_{c}}{{T}_{c}})|\ensuremath{\ll}1.$ The correlation functions are ${〈{E}_{R}{E}_{0}〉\ensuremath{-}{〈E〉}^{2},={(\frac{2J}{\ensuremath{\pi}})}^{2}{\ensuremath{\epsilon}}^{2}[{{K}_{1}}^{2}(\ensuremath{\epsilon}R)\ensuremath{-}{{K}_{0}}^{2}(\ensuremath{\epsilon}R)],}{〈{E}_{R}{\ensuremath{\sigma}}_{0}〉\ensuremath{-}〈E〉〈\ensuremath{\sigma}〉,=(\frac{2J}{\ensuremath{\pi}})\ensuremath{\epsilon}〈\ensuremath{\sigma}〉\ensuremath{\int}{2\ensuremath{\epsilon}R}^{\ensuremath{\infty}}\mathrm{ds} {s}^{\ensuremath{-}2}{e}^{\ensuremath{-}8},}$ where RRR is the distance between the points in question and K0{K}_{0}K0​ and K1{K}_{1}K1​ are modified Bessel functions of the second kind. Very similar results hold if the coupling constants are different in the two directions. The above results agree in form with the structure predicted by the "scaling-law" approach.
We extend the form-factors approach to the quantum Ising model at finite temperature. The two-point function of the energy is obtained in closed form, while the two-point function of the spin is written as a Fredholm determinant. Using the approach of Korepin et al., we obtain, starting directly from the continuum formulation, a set of six differential equations satisfied by this two-point function. Four of these equations involve only space-time derivatives, of which three are equivalent to the equations obtained earlier. In addition, we obtain two new equations involving a temperature derivative. Some of these results are generalized to the Ising model on the half line with a magnetic field at the origin.