Quantitative functional renormalization group description of the two-dimensional Hubbard model

Cornelia Hille,Carsten Honerkamp,Anna Kauch,Christian J Eckhardt,Yuan-Yao He,Fabian B Kugler,Alessandro Toschi,Sabine Andergassen

doi:10.1103/physrevresearch.2.033372

Abstract

Using a leading algorithmic implementation of the functional renormalization group (fRG) for interacting fermions on two-dimensional lattices, we provide a detailed analysis of its quantitative reliability for the Hubbard model. In particular, we show that the recently introduced multiloop extension of the fRG flow equations for the self-energy and two-particle vertex allows for a precise match with the parquet approximation also for two-dimensional lattice problems. The refinement with respect to previous fRG-based computation schemes relies on an accurate treatment of the frequency and momentum dependences of the two-particle vertex, which combines a proper inclusion of the high-frequency asymptotics with the so-called truncated unity fRG for the momentum dependence. The adoption of the latter scheme requires, as an essential step, a consistent modification of the flow equation of the self-energy. We quantitatively compare our fRG results for the self-energy and momentum-dependent susceptibilities and the corresponding solution of the parquet approximation to determinant quantum Monte Carlo data, demonstrating that the fRG is remarkably accurate up to moderate interaction strengths. The presented methodological improvements illustrate how fRG flows can be brought to a quantitative level for two-dimensional problems, providing a solid basis for the application to more general systems.

Highlights

Renormalization group (RG) methods have a long history in theoretical physics, ranging from a way to treat divergences in quantum field theories [1], critical phenomena [2,3], and quantum impurity problems [4,5] to current attempts to elucidate deep learning algorithms by physics [6]
Using a leading algorithmic implementation of the functional renormalization group for interacting fermions on two-dimensional lattices, we provide a detailed analysis of its quantitative reliability for the Hubbard model
In the following we present the functional renormalization group (fRG)* results for the evolution of the different susceptibilities away from half filling, together with their comparison to the parquet approximation (PA) and determinant quantum Monte Carlo (DQMC) data

Summary

Introduction

Renormalization group (RG) methods have a long history in theoretical physics, ranging from a way to treat divergences in quantum field theories [1], critical phenomena [2,3], and quantum impurity problems [4,5] to current attempts to elucidate deep learning algorithms by physics [6]. RG methods connect specific quantities of a theory, such as coupling constants or correlation functions, at a given scale with those at another scale via differential equations. This leads to a flow of these quantities which under appropriate circumstances distills out the dominating, and to some degree universal, properties of the system. The RG was utilized as a tool to deal with competing ordering tendencies in the Hubbard model

Methods

Results

Conclusion