Expansion of the Gibbs potential for quantum many-body systems: General formalism with applications to the spin glass and the weakly nonideal Bose gas

Abstract

For general quantum systems the power expansion of the Gibbs potential and consequently the power expansion of the self energy is derived in terms of the interaction strength. Employing a generalization of the projector technique a compact representation of the general terms of the expansion results. The general aspects of the approach are discussed with special emphasis on the effects characteristic for quantum systems. The expansion is systematic and leads directly to contributions beyond mean-field of all thermodynamic quantities. These features are explicitly demonstrated and illustrated for two non-trivial systems, the infinite range quantum spin glass and the weakly interacting Bose gas. The Onsager terms of both systems are calculated, which represent the first beyond mean-field contributions. For the spin glass new TAP-like equations are presented and discussed in the paramagnetic region. The investigation of the Bose gas leads to a beyond mean-field thermodynamic description. At the Bose-Einstein condensation temperature complete agreement is found with the results presented recently by alternative techniques.

Role of Tracy-Widom distribution in finite-size fluctuations of the critical temperature of the Sherrington-Kirkpatrick spin glass

“…This α expansion, known as Plefka expansion, has thus served as a method for deriving the TAP free energy for several classes of models, and has been extensively used in different contexts in physics, from classical disordered systems, [30][31][32] to general quantum systems. [33][34][35][36] It is a general fact that, if the model is defined on a complete graph, the Plefka expansion truncates to a finite order in α, because higher-order terms should vanish in the thermodynamic limit. In particular, for the SK model, the orders of the expansion larger than three are believed 37 to vanish in the limit N → ∞ in such a way that the expansion truncates, and one is left with the first three orders of the α series, which reads…”

“…where β is the inverse temperature of the model. This αexpansion, known as Plefka expansion, has thus served as a method for deriving TAP free energy for several class of models, and has been extensively used in several different contexts in physics, from classical disordered systems [30][31][32] , to general quantum systems [33][34][35][36] . It is a general fact that, if the model is defined on a complete graph, the Plefka expansion truncates to a finite order in α, because higher-order terms should vanish in the thermodynamic limit.…”

Singular-potential random-matrix model arising in mean-field glassy systems

“…Further terms can be systematically included (Georges-Yedidia expansion [38]), but they vanish anyway for the SK model as N → ∞, therefore they can be safely neglected. This expansion has been extensively used for several systems, both in the classical [38][39][40], and quantum domain [41][42][43][44]. The stability pattern of extremal points (maxima, minima and saddles) in this multidimensional free-energy landscape is encoded in the Hessian of F (or inverse susceptibility matrix)…”

An information geometrical approach to the mean-field approximation for quantum Ising spin models

“…However, one of the remarkable consequences of the duality (19) is that, if one of the two connections ∇ (e) and ∇ (m) is flat, then the other is also flat, which is referred to as the dually flatness of the manifold with respect to the information geometrical structure (g, ∇ (e) , ∇ (m) ). In the present case, the connection coefficients of ∇ (m) with respect to the expectation coordinates η = (η i ) defined by (11) turn out to identically vanish. This means that M is m-flat with an m-affine coordinate system (η i ).…”