Many-Body Density Matrices On a Two-Dimensional Square Lattice: Noninteracting and Strongly Interacting Spinless Fermions

TL;DR

This paper analyzes the structure of many-body density matrices for spinless fermions on a 2D square lattice, comparing noninteracting and strongly-interacting cases to understand their eigenvalue spectra and potential for truncation.

Contribution

It introduces a numerical method to evaluate and approximate the infinite-system cluster density matrix for interacting fermions, accounting for symmetries and boundary conditions.

Findings

Eigenvalue spectra differ significantly between noninteracting and strongly-interacting cases.

Averaging over boundary conditions reduces finite size effects.

The density matrix structure suggests potential for truncation based on single-particle operators.

Abstract

The reduced density matrix of an interacting system can be used as the basis for a truncation scheme, or in an unbiased method to discover the strongest kind of correlation in the ground state. In this paper, we investigate the structure of the many-body fermion density matrix of a small cluster in a square lattice. The cluster density matrix is evaluated numerically over a set of finite systems, subject to non-square periodic boundary conditions given by the lattice vectors $\bR_1 \equiv (R_{1x}, R_{1y})$ and $\bR_2 \equiv (R_{2x}, R_{2y})$. We then approximate the infinite-system cluster density-matrix spectrum, by averaging the finite-system cluster density matrix (i) over degeneracies in the ground state, and orientations of the system relative to the cluster, to ensure it has the proper point-group symmetry; and (ii) over various twist boundary conditions to reduce finite size effects. We then compare the eigenvalue structure of the averaged cluster density matrix for noninteracting and strongly-interacting spinless fermions, as a function of the filling fraction $\nbar$, and discuss whether it can be approximated as being built up from a truncated set of single-particle operators.