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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Quantum field theory and the two-dimensional Ising model

Jean-Bernard Zuber, C. Itzykson

Published May 15, 1977• 133 citations
Publisher

Abstract

A relationship between the two-point correlation function of the Ising model in the critical region and the $〈sin\sqrt{\ensuremath{\pi}}\ensuremath{\varphi}(0)sin\sqrt{\ensuremath{\pi}}\ensuremath{\varphi}(\ensuremath{\rho})〉$ Green's function of a sine-Gordon field is constructed. This relationship is tested by means of a mass perturbation expansion with an infrared cutoff and shown to agree, at the leading-logarithm approximation and up to the second order, with the exact result of the Ising model.

DOI
10.1103/physrevd.15.2875

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A relationship between the two-point correlation function of the Ising model in the critical region and the 〈sin\ensuremathπ\ensuremathφ(0)sin\ensuremathπ\ensuremathφ(\ensuremathρ)〉〈sin\sqrt{\ensuremath{\pi}}\ensuremath{\varphi}(0)sin\sqrt{\ensuremath{\pi}}\ensuremath{\varphi}(\ensuremath{\rho})〉〈sin\ensuremathπ​\ensuremathφ(0)sin\ensuremathπ​\ensuremathφ(\ensuremathρ)〉 Green's function of a sine-Gordon field is constructed. This relationship is tested by means of a mass perturbation expansion with an infrared cutoff and shown to agree, at the leading-logarithm approximation and up to the second order, with the exact result of the Ising model.
We have developed a quantum-mechanical approach to study the critical properties of the quantum sine-Gordon equation describing a massless scalar field in 1+1 space-time dimensions with interaction density proportional to cos(${\mathrm{\ensuremath{\beta}}}_{0}$\ensuremath{\varphi}). The main feature of the present theory is to find an appropriate starting point for perturbation expansion, which can directly treat the infrared divergences encountered in the conventional perturbation theory. Exact critical conditions for the phase transition of the model are derived; and the quantum sine-Gordon Hamiltonian is found to correspond exactly to a free-field model. Near the critical regime, a gap will open near zero momentum in the excitation spectrum, and the corresponding ground-state wave function is a pairing quasiparticle state, analogous to the BCS superconducting state. It is proved that this ground state will possess quasi-long-range and quasi-off-diagonal long-range correlations. Comparisons with the Kosterlitz-Thouless classical theory of the two-dimensional Coulomb gas are also made.
A functional-integral formulation of the Ising model is used as a link between the usual approach in terms of summation over spin variables and field-theory-like formulas. The latter take the form of Feynman-diagram expansions, Dyson integral equations, or Schwinger functional-derivative field equations. Basic to the theory is a spinless-nucleon-like Green's function, related to the Ising spin-spin correlation function. By means of an infinite renormalization technique, a mesonlike propagator, related to local energy-energy correlations, is bootstrapped. Finally, a vertex function, associated with energy-spin-spin correlations, is introduced. To zeroth order this theory bears similarity to the spherical model, but vital differences are also noted. A brief discussion is presented of the relation of methods employed in field theory for the treatment of the infrared divergence, and approximations which might be of value for critical correlations.
A detailed calculation, to order ${\ensuremath{\epsilon}}^{2} (\ensuremath{\epsilon}=4\ensuremath{-}d)$, of the two-point-correlation function of an Ising-like system in the whole critical region is presented. The scaling function is shown to be a cut-off-independent function of two variables which is universal in the context of a sharp cut off. Explicit asymptotic expansions in the large and small momentum (relative to the inverse correlation length) are given. Particular attention is paid to the corrections from the Ornstein-Zernike theory. These corrections are two orders of magnitude larger below Tc{T}_{c}Tc​ than above. Numerical comparison with series-expansion results agree surprisingly well. A powerful technique of evaluating diagrams using the Fourier transform of the propagator is also presented.
The main subject of this thesis is the Ising field theory, the field theory describing the scaling limit of the two-dimensional Ising model near its critical point. We study the Ising field theory in low- and high-temperature regimes.In the low-temperature regime we address questions related to the mass spectrum of the bound states that exist in the theory. The overall goal is to study analytic properties of the masses taken as functions of the scaling variable in the theory. At this stage, we are primarily interested in the asymptotic series in the coupling parameter. We describe methods used for developing expansions in the coupling parameter when this parameter is sufficiently small. This gives rise to the low-energy and semiclassical series. Methods developed are then applied to a different model, a model that has many things in common with the Ising field theory, the two-dimensional quantum chromodynamics with infinite number of colors, also known as the ‘t Hooft model.In the second part of the thesis we turn to the Ising field theory in the high- temperature regime. Here we study the problem of scattering in a weak magnetic field. We compute the leading perturbative correction to the scattering amplitude for the 2 -> 2 process. This is done first by means of standard methods based on the direct intermediate-state decomposition and dispersion relations. Then we compute the large energy asymptotic of the amplitude for this process directly using a known relation between the Ising field theory at zero magnetic field and classical sinh-Gordon equation. We obtain consistent results that show the logarithmic growth of the amplitude with energy at this order in magnetic field. Going beyond the leading order we argue that at large energies for a sufficiently weak magnetic field the amplitude exhibits a power-like decay.
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.