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Register a new userA new expansion around mean field for the quantum Ising model
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We show that an high temperature expansion at fixed order parameter can be derived for the quantum Ising model. The basic point is to consider a statistical generating functional associated to the local spin state. The probability at thermal equilibrium of this state reflects directly the occurrence of a spontaneous symmetry breaking. It is possible to recover the expansion around the mean field in the system dimensionality if the ``direction in the Hilbert space of local spin states is suitably chosen. Results for the free energy at the critical temperature, as a function of the transverse field, in first order approximation in the inverse system dimensionality are compared with those of the standard approach.
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Mean-field theory (MFT) is one of the main available tools for analytical calculations entailed in investigations regarding many-body systems. Recently, there have been an urge of interest in ameliorating this kind of method, mainly with the aim of incorporating geometric and correlation properties of these systems. The correlated cluster MFT (CCMFT) is an improvement that succeeded quite well in doing that for classical spin systems. Nevertheless, even the CCMFT presents some deficiencies when applied to quantum systems. In this article, we address this issue by proposing the quantum CCMFT (QCCMFT), which, in contrast to its former approach, uses general quantum states in its self-consistent mean-field equations. We apply the introduced QCCMFT to the transverse Ising model in honeycomb, square, and simple cubic lattices and obtain fairly good results both for the Curie temperature of thermal phase transition and for the critical field of quantum phase transition. Actually, our results match those obtained via exact solutions, series expansions or Monte Carlo simulations.
Although the fully connected Ising model does not have a length scale, we show that its critical exponents can be found using finite size scaling with the scaling variable equal to N, the number of spins. We find that at the critical temperature of the infinite system the mean value and the most probable value of the magnetization scale differently with N, and the probability distribution of the magnetization is not a Gaussian, even for large N. Similar results inconsistent with the usual understanding of mean-field theory are found at the spinodal. We relate these results to the breakdown of hyperscaling and show how hyperscaling can be restored by increasing N while holding the Ginzburg parameter rather than the temperature fixed.
The unusual thermodynamic properties of the Ising antiferromagnet supplemented with a ferromagnetic, mean-field term are outlined. This simple model is inspired by more realistic models of spin-crossover materials. The phase diagram is estimated using Metropolis Monte Carlo methods, and differences with preliminary Wang-Landau Monte Carlo results for small systems are noted.
We have made substantial advances in elucidating the properties of the susceptibility of the square lattice Ising model. We discuss its analyticity properties, certain closed form expressions for subsets of the coefficients, and give an algorithm of complexity O(N^6) to determine its first N coefficients. As a result, we have generated and analyzed series with more than 300 terms in both the high- and low-temperature regime. We quantify the effect of irrelevant variables to the scaling-amplitude functions. In particular, we find and quantify the breakdown of simple scaling, in the absence of irrelevant scaling fields, arising first at order |T-T_c|^{9/4}, though high-low temperature symmetry is still preserved. At terms of order |T-T_c|^{17/4} and beyond, this symmetry is no longer present. The short-distance terms are shown to have the form (T-T_c)^p(log|T-T_c|)^q with p ge q^2. Conjectured exact expressions for some correlation functions and series coefficients in terms of elliptic theta functions also foreshadow future developments.
We provide a non-trivial test of supersymmetry in the random-field Ising model at five spatial dimensions, by means of extensive zero-temperature numerical simulations. Indeed, supersymmetry relates correlation functions in a D-dimensional disordered system with some other correlation functions in a D-2 clean system. We first show how to check these relationships in a finite-size scaling calculation, and then perform a high-accuracy test. While the supersymmetric predictions are satisfied even to our high-accuracy at D=5, they fail to describe our results at D=4.
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