Truncated-Unity Parquet Equations: Application to the Repulsive Hubbard Model

TL;DR

This paper introduces a truncated-unity approach to simplify parquet equations, enabling efficient computation of two-particle vertices in the Hubbard model, with promising results for reducing numerical complexity.

Contribution

The authors adapt the truncated-unity channel decomposition from functional renormalization group methods to parquet equations, significantly reducing computational effort.

Findings

Achieved linear scaling of numerical effort with momentum discretization

Obtained approximate solutions for channel-projected and reducible vertices

Demonstrated applicability to the half-filled Hubbard model

Abstract

The parquet equations are a self-consistent set of equations for the effective two-particle vertex of an interacting many-fermion system. The application of these equations to bulk models is, however, demanding due to the complex emergent momentum and frequency structure of the vertex. Here, we show how a channel-decomposition by means of truncated unities, which was developed in the context of the functional renormalization group to efficiently treat the momentum dependence, can be transferred to the parquet equations. This leads to a significantly reduced numerical effort scaling only linearly with the number of discrete momenta. We apply this technique to the half-filled repulsive Hubbard model on the square lattice and present approximate solutions for the channel-projected vertices and the full reducible vertex.