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.
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.
A class of infinite-ranged random model Hamiltonians is defined as a limiting case in which the appropriate form of mean-field theory, order parameters and phase diagram to describe spin-glasses may be established. It is believed that these Hamiltonians may be exactly soluble, although a complete solution is not yet available. Thermodynamic properties of the model for Ising and XY spins are evaluated using a many-replica procedure. Results of the replica theory reproduce properties at and above the ordering temperature which are also predicted by high-temperature expansions, but are in error at low temperatures. Extensive computer simulations of infinite-ranged Ising spin-glasses are presented. They confirm the general details of the predicted phase diagram. The errors in the replica solution are found to be small, and confined to low temperatures. For this model, the extended mean-field theory of Thouless, Anderson, and Palmer gives physically sensible low-temperature predictions. These are in quantitative agreement with the Monte Carlo statics. The dynamics of the infinite-ranged Ising spin-glass are studied in a linearized mean-field theory. Critical slowing down is predicted and found, with correlations decaying as ${e}^{{\ensuremath{-}[\frac{(T\ensuremath{-}{T}_{c})}{T}]}^{2}t}$ for T greater than Tc, the spin-glass transition temperature. At and below Tc, spin-spin correlations are observed to decay to their long-time limit as ${t}^{\ensuremath{-}\frac{1}{2}}$.
A class of infinite-ranged random model Hamiltonians is defined as a limiting case in which the appropriate form of mean-field theory, order parameters and phase diagram to describe spin-glasses may be established. It is believed that these Hamiltonians may be exactly soluble, although a complete solution is not yet available. Thermodynamic properties of the model for Ising and XY spins are evaluated using a "many-replica" procedure. Results of the replica theory reproduce properties at and above the ordering temperature which are also predicted by high-temperature expansions, but are in error at low temperatures. Extensive computer simulations of infinite-ranged Ising spin-glasses are presented. They confirm the general details of the predicted phase diagram. The errors in the replica solution are found to be small, and confined to low temperatures. For this model, the extended mean-field theory of Thouless, Anderson, and Palmer gives physically sensible low-temperature predictions. These are in quantitative agreement with the Monte Carlo statics. The dynamics of the infinite-ranged Ising spin-glass are studied in a linearized mean-field theory. Critical slowing down is predicted and found, with correlations decaying as ${e}^{{\ensuremath{-}[\frac{(T\ensuremath{-}{T}_{c})}{T}]}^{2}t}$ for T greater than Tc, the spin-glass transition temperature. At and below Tc, spin-spin correlations are observed to decay to their long-time limit as ${t}^{\ensuremath{-}\frac{1}{2}}$.
We analyze the low-temperature behavior of mean-field equations of Thouless, Anderson, and Palmer (TAP). We demonstrate that degeneracy in free energy makes the low-temperature TAP states unstable. Different solutions of the TAP equations, independent in the TAP approach, become coupled if an infinitesimal interaction between them is introduced. By means of real spin replicas we derive a self-averaging free energy free of unstable states with local magnetizations and homogeneous overlap susceptibilities between different spin replicas as order parameters. We thereby extend the TAP approach to a consistent description of the spin-glass phase for all configurations of spin exchange with (marginally) stable and thermodynamically homogeneous free energy.
We have developed 15th-order high-temperature series expansions for the study of the critical behavior of the Ising spin glass with nearest-neighbor exchange interactions each of which assumes the values \ifmmode\pm\else\textpm\fi{}J randomly. Series for the Edwards-Anderson spin-glass susceptibility (${\mathrm{\ensuremath{\chi}}}^{\mathrm{EA}}$) and two of its derivatives with respect to the ordering field have been evaluated for hypercubic lattices in general dimension, d. These extend previous general-dimension series by five terms. Certain measurable universal amplitude ratios have been estimated from the new series. Accurate critical data for d=5 and the first reliable estimates of the exponent \ensuremath{\beta} for d=4 and 5, are given. We quote \ensuremath{\gamma}=1.73\ifmmode\pm\else\textpm\fi{}0.03, 2.00\ifmmode\pm\else\textpm\fi{}0.25, and 2.${7}_{\mathrm{\ensuremath{-}}0.6}^{+1.0}$ and \ensuremath{\beta}=0.95\ifmmode\pm\else\textpm\fi{}0.04, 0.9\ifmmode\pm\else\textpm\fi{}0.1, and 0.7\ifmmode\pm\else\textpm\fi{}0.2 in 5, 4, and 3 dimensions, respectively. Our results provide a smooth extrapolation between the mean-field results above six dimensions and experiments and simulations in physical dimensions. We relate our calculated derivatives of ${\mathrm{\ensuremath{\chi}}}^{\mathrm{EA}}$ to measurements of derivatives of the magnetization with respect to a uniform magnetic field.
First, a new method proposed by Baker and Rushbrooke is used to study the simple ferromagnetic Ising model at and below the Curie temperature. Of course, the properties of the Ising model are already well known, so that the main aim here is to assess the potential and reliability of the new method, since it has wide applicability to other models which have not been otherwise studied. Between 8 and 16 coefficients of exact high-temperature expansions for fixed values of the magnetization are derived for various two- and three-dimensional lattices. A Pad\'e-approximant analysis of these expansions at the critical isotherm and magnetic phase boundary enables us to estimate the critical exponents $\ensuremath{\beta}$, ${\ensuremath{\gamma}}^{\ensuremath{'}}$, and $\ensuremath{\delta}$, and plot the spontaneous magnetization. The results are in good agreement with previous calculations. Secondly, an analysis of the exact series expansions provides no support for the conjecture that the phase boundary is a line of essential singularities. However, the same expansions strongly suggest the existence of a "spinodal" curve, whose properties are in reasonable agreement with the predictions of various heuristic arguments (based essentially upon analyticity at the phase boundary and one-phase homogeneity in the critical region). Finally, structure and a mild extension of the proven analyticity of the free energy are used to show the $\ensuremath{\Delta}\ensuremath{\le}{\ensuremath{\Delta}}^{\ensuremath{'}}$, $\ensuremath{\gamma}\ensuremath{\le}{\ensuremath{\gamma}}^{\ensuremath{'}}$.
In this chapter we will put the concepts we have developed in Chapter 2 to work in constructing the mean field theory for an Ising spin glass. The term ‘mean field theory’ (henceforth frequently abbreviated ‘MFT’) can be interpreted in many ways. Here we will take it to mean the exact solution of a model in which the forces are of infinitely long range, so that each spin interacts equally strongly with every other one. For spin glasses, this is the Sherrington–Kirkpatrick (SK) model (1975), which we solve heuristically (though unfortunately incorrectly) in Section 3.1. The error is a somewhat subtle one, as is evident from the fact that we obtain exactly the same result in the more systematic calculations of Sections 3.2 (a direct summation of the leading terms in N −1 in high-temperature perturbation theory) and 3.3 (which uses the replica formalism). In both these approaches, we can also see how the theory itself reveals that it is wrong, since they lead to negative values of quantities which are necessarily positive. We then study the correct mean field theory, with replica symmetry breaking, in Section 3.4, including the remarkable nature of the broken ergodicity it implies and an examination of its stability. Further physical insight into the problem is obtained in Section 3.5 from the mean field equations first introduced by Thouless, Anderson and Palmer (TAP) (1977).