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arXiv:cond-mat/0201505

A new expansion around mean field for the quantum Ising model

FERDINANDO DE PASQUALE, S. M. Giampaolo

Published Jan 28, 2002• 0 citations
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Abstract

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.

DOI
10.48550/arxiv.cond-mat/0201505

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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.
The Landau theory on the phase transition derived statistical-mechanically on the basis of the symmetry breaking potential method with the use of inverse temperature series expansion is examined. In that method an estimate of the mean field is improved systematically by taking account of short range correlations. The results at low temperatures in relation to the expansion parameter are investigated. The critical temperature obtained is equal to that in the [n, 0] Padé approximation applied to the inverse of susceptibility. By making use of many results obtained up to now in the calculation of susceptibility with the use of inverse temperature series expansion, the critical temperatures of the Ising, Heisenberg and XY models are determined. The possibility of long range ordering in low dimensional systems and the case of the symmetry breaking potential incommutative with the Hamiltonian are also investigated.
A mean-field approximation is used to investigate the quantum transverse Ising spin-glass model in the Edwards-Anderson version. We introduce two order parameters, diagonal and off diagonal in replica indices, and discuss their physical significance. The effective free-energy function is obtained, and the critical value of the transverse field $\ensuremath{\Gamma}$ at zero temperature is calculated. We find the behavior of the order parameters near the zero-temperature critical point. The phase diagram in the $\ensuremath{\Gamma}$,TTT plane is also given.
We consider the scaling behavior of thermodynamic quantities in the one-dimensional transverse field Ising model near its quantum critical point (QCP). Our study has been motivated by the question about the thermodynamical signatures of this paradigmatic quantum critical system and, more generally, by the issue of how quantum criticality accumulates entropy. We find that the crossovers in the phase diagram of temperature and (the nonthermal control parameter) transverse field obey a general scaling ansatz, and so does the critical scaling behavior of the specific heat and magnetic expansion coefficient. Furthermore, the Gr\"uneisen ratio diverges in a power-law way when the QCP is accessed as a function of the transverse field at zero temperature, which follows the prediction of quantum critical scaling. However, at the critical field, upon decreasing the temperature, the Gr\"uneisen ratio approaches a constant instead of showing the expected divergence. We are able to understand this unusual result in terms of a peculiar form of the quantum critical scaling function for the free energy; the contribution to the Gr\"uneisen ratio vanishes at the linear order in a suitable Taylor expansion of the scaling function. In spite of this special form of the scaling function, we show that the entropy is still maximized near the QCP, as expected from the general scaling argument. Our results establish the telltale thermodynamic signature of a transverse-field Ising chain, and will thus facilitate the experimental identification of this model quantum-critical system in real materials.
The relationship between the mean-field approximations in various interacting models of statistical physics and measures of classical and quantum correlations is explored. We present a method that allows us to find an upper bound for the total amount of correlations (and hence entanglement) in a physical system in thermal equilibrium at some temperature in terms of its free energy and internal energy. This method is first illustrated by using two qubits interacting through the Heisenberg coupling, where entanglement and correlations can be computed exactly. It is then applied to the one-dimensional (1D) Ising model in a transverse magnetic field, for which entanglement and correlations cannot be obtained by exact methods. We analyse the behaviour of correlations in various regimes and identify critical regions, comparing them with already known results. Finally, we present a general discussion of the effects of entanglement on the macroscopic, thermodynamical features of solid-state systems. In particular, we exploit the fact that a d -dimensional quantum system in thermal equilibrium can be made to correspond to a ( d +1)-dimensional classical system in equilibrium to substitute all entanglement for classical correlations.
High-temperature expansions performed at a fixed-order parameter provide a simple and systematic way to derive and correct mean-field theories for statistical mechanical models. For models like spin glasses which have general couplings between spins, the authors show that these expansions generate the Thouless-Anderson-Palmer equations at low order. They explicitly calculate the corrections to TAP theory for these models. For ferromagnetic models, they show that their expansions can easily be converted into 1/d expansions around mean-field theory, where d is the number of spatial dimensions. Only a small finite number of graphs need to be calculated to generate each order in 1/d for thermodynamic quantities like free energy or magnetization. Unlike previous 1/d expansions, the expansions are valid in the low-temperature phases of the models considered. They consider alternative ways to expand around mean-field theory besides 1/d expansions. In contrast to the 1/d expansion for the critical temperature, which is presumably asymptotic, these schemes can be used to devise convergent expansions for the critical temperature. They also appear to give convergent series for thermodynamic quantities and critical exponents. They test the schemes using the spherical model, where their properties can be studied using exact expressions.