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arXiv:hep-th/0209019 · Journal of Physics A Mathematical and General

Improved harmonic approximation and the 2D Ising model atT Tcandh 0

Anı́bal Iucci, Carlos Naón

Published Sep 10, 2002• 0 citations
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Abstract

We propose a new method to determine the unknown parameter associated with a self-consistent harmonic approximation. We check the validity of our technique in the context of the sine-Gordon model. As a non-trivial application we consider the scaling regime of the 2D Ising model away from the critical point and in the presence of a magnetic field h . We derive an expression that relates the approximate correlation length ξ, T − T c and h .

DOI
10.1088/0305-4470/35/38/304

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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.
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Abstract We derive a self-consistent time-dependent harmonic approximation for the quantum sine-Gordon model out of equilibrium and apply the method to the dynamics of tunnel-coupled one-dimensional Bose gases. We determine the time evolution of experimentally relevant observables and in particular derive results for the probability distribution of subsystem phase fluctuations. We investigate the regime of validity of the approximation by applying it to the simpler case of a nonlinear harmonic oscillator, for which numerically exact results are available. We complement our self-consistent harmonic approximation by exact results at the free fermion point of the sine-Gordon model.
Abstract In undergraduate statistical mechanics courses the Ising model always plays an important role because it is the simplest non-trivial model used to describe magnetic systems. The one-dimensional model is easily solved analytically, while the two-dimensional one can be solved exactly by the Onsager solution. For this reason, numerical simulations are usually used to solve the two-dimensional model. Keeping in mind that the two-dimensional model is the platform for studying phase transitions, it is usually an exercise in computational undergraduate courses because its numerical solution is relatively simple to implement and its critical exponents are perfectly known. The purpose of this article is to present a detailed numerical study of the second-order phase transition in the two-dimensional Ising model at an undergraduate level, allowing readers not only to compare the mean-field solution, the exact solution and the numerical one through a complete study of the order parameter, the correlation function and finite-size scaling, but to present the techniques, along with hints and tips, for solving it themselves. We present the elementary theory of phase transitions and explain how to implement Markov chain Monte Carlo simulations and perform them for different lattice sizes with periodic boundary conditions. Energy, magnetization, specific heat, magnetic susceptibility and the correlation function are calculated and the critical exponents determined by finite-size scaling techniques. The importance of the correlation length as the relevant parameter in phase transitions is emphasized.