Solving condensed-matter ground-state problems by semidefinite relaxations

TL;DR

This paper introduces a semidefinite relaxation approach to solve condensed-matter ground-state problems directly in the thermodynamic limit, providing bounds and approximations for complex many-body systems.

Contribution

The authors develop a novel SDP-based method applicable to all particle statistics, offering a new tool for studying strongly correlated fermionic and spin systems in higher dimensions.

Findings

Efficiently computes strict lower bounds for ground-state energies.

Provides approximations to few-particle Green's functions.

Demonstrates competitiveness against exact solutions and quantum Monte Carlo.

Abstract

We present a new generic approach to the condensed-matter ground-state problem which is complementary to variational techniques and works directly in the thermodynamic limit. Relaxing the ground-state problem, we obtain semidefinite programs (SDP). These can be solved efficiently, yielding strict lower bounds to the ground-state energy and approximations to the few-particle Green's functions. As the method is applicable for all particle statistics, it represents in particular a novel route for the study of strongly correlated fermionic and frustrated spin systems in D>1 spatial dimensions. It is demonstrated for the XXZ model and the Hubbard model of spinless fermions. The results are compared against exact solutions, quantum Monte Carlo, and Anderson bounds, showing the competitiveness of the SDP method.