Testing density functional theory in a quantum Ising chain

TL;DR

This paper evaluates the effectiveness of density functional approximations, like LDA and GGA, in modeling a quantum Ising chain, revealing their strengths and limitations in capturing inhomogeneous density distributions and phase transitions.

Contribution

It introduces a semi-local energy functional for the quantum Ising chain that closely matches exact results across various inhomogeneous densities.

Findings

Adding density variation corrections improves LDA accuracy

LDA and GGA errors mirror the quantum phase diagram

Semi-local functional nearly exact for inhomogeneous densities

Abstract

By using the quantum Ising chain as a test bed and treating the spin polarization along the external transverse field as the "generalized density", we examine the performance of different levels of density functional approximations parallel to those widely used for interacting electrons, such as local density approximation (LDA) and generalized gradient approximation (GGA). We show that by adding the lowest-order and nearest-neighbor density variation correction to the simple LDA, a semi-local energy functional in the spirit of GGA is almost exact over a wide range of inhomogeneous density distribution. In addition, the LDA and GGA error structures bear a high level of resemblance to the quantum phase diagram of the system. These results provide insights into the triumph and failure of these approximations in a general context.