Expansion of the Gibbs potential for quantum many-body systems: General formalism with applications

Jacopo De Nardis

Martin P. Gelfand, Rajiv R. P. Singh, David A. Huse

T. Plefka

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 the mean field of all thermodynamic quantities. These features are explicitly demonstrated and illustrated for two nontrivial 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 Thouless-Anderson-Palmer-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.

Yasutami Takada, Takafumi Kita

The effective-potential expansion method, namely, a variational approach to the many-body problem at zero temperature, is extended to nonzero temperatures by the introduction of a new trial density matrix in the Gibbs variational principle for the grand potential \ensuremath{\Omega}. The trial density matrix contains an effective potential as a variational parameter. We have given an expansion form for \ensuremath{\Omega} in terms of the effective potential. Special care is taken to evaluate the entropy term correctly.

Antun Balaž, Ivana Vidanović, Aleksandar Bogojević +2 more

Based on a previously developed recursive approach for calculating the short-time expansion of the propagator for systems with time-independent potentials and its time-dependent generalization for simple single-particle systems, in this paper we present a full extension of this formalism to a general quantum system with many degrees of freedom in a time-dependent potential. Furthermore, we also present a recursive approach for the velocity-independent part of the effective potential, which is necessary for calculating diagonal amplitudes and partition functions, as well as an extension from the imaginary-time formalism to the real-time one, which enables to study the dynamical properties of quantum systems. The recursive approach developed here allows an analytic derivation of the short-time expansion to orders that have not been accessible before, using the implemented SPEEDUP symbolic calculation code. The analytically derived results are extensively numerically verified by treating several models in both imaginary and real time.

T. D. Lee, C. N. Yang