Generalized gradient expansion for inhomogeneous dynamical mean-field theory: Application to ultracold atoms in a harmonic trap

TL;DR

This paper introduces a generalized gradient expansion method for inhomogeneous dynamical mean-field theory to better analyze ultracold atoms in traps, especially at higher temperatures, and compares its accuracy to the local density approximation.

Contribution

It develops a generalized gradient expansion that extends the local density approximation for inhomogeneous dynamical mean-field theory, providing insights into its accuracy and limitations.

Findings

At high temperatures, the gradient expansion closely matches the local density approximation.

The gradient expansion is less effective at low temperatures in ordered phases.

When both methods agree, they are likely accurate; disagreement indicates potential inaccuracies.

Abstract

We develop a generalized gradient expansion of the inhomogeneous dynamical mean-field theory method for determining properties of ultracold atoms in a trap. This approach goes beyond the well-known local density approximation and at higher temperatures, in the normal phase, it shows why the local density approximation works so well, since the local density and generalized gradient approximations are essentially indistinguishable from each other (and from the exact solution within full inhomogeneous dynamical mean-field theory). But because the generalized gradient expansion only involves nearest-neighbor corrections, it does not work as well at low temperatures, when the systems enter into ordered phases. This is primarily due to the problem that ordered phases often satisfy some global constraints which determine the spatial ordering pattern, and the local density and generalized gradient approximations are not able to impose those kinds of constraints; they also overestimate the tendency to order. The theory is applied to phase separation of different mass fermionic mixtures represented by the Falicov-Kimball model and to determining the entropy per particle of a fermionic system represented by the Hubbard model. The generalized gradient approximation is a useful diagnostic for the accuracy of the local density approximation---when both methods agree, they are likely accurate, when they disagree, neither is likely to be correct.