# The Hubbard model on the honeycomb lattice: from static and dynamical mean-field theories to lattice quantum Monte Carlo simulations

## TL;DR

This paper compares static and dynamical mean-field theories with quantum Monte Carlo simulations to study the Hubbard model on the honeycomb lattice, revealing their respective accuracies and limitations near quantum critical points.

## Contribution

It provides a detailed comparison of static mean-field, dynamical mean-field theory, and quantum Monte Carlo methods for the Hubbard model on the honeycomb lattice, highlighting their strengths and limitations.

## Key findings

- DMFT accurately captures local moment formation in the semi-metallic phase.
- QMC reveals spin fluctuations affecting spectral features near criticality.
- Static mean-field fails to generate local moments without symmetry breaking.

## Abstract

We study the one-band Hubbard model on the honeycomb lattice using a combination of quantum Monte Carlo (QMC) simulations and static as well as dynamical mean-field theory (DMFT). This model is known to show a quantum phase transition between a Dirac semi-metal and the antiferromagnetic insulator. The aim of this article is to provide a detailed comparison between these approaches by computing static properties, notably ground-state energy, single-particle gap, double occupancy, and staggered magnetization, as well as dynamical quantities such as the single-particle spectral function. At the static mean-field level local moments cannot be generated without breaking the SU(2) spin symmetry. The DMFT approximation accounts for temporal fluctuations, thus captures both the evolution of the double occupancy and the resulting local moment formation in the paramagnetic phase. As a consequence, the DMFT approximation is found to be very accurate in the Dirac semi-metallic phase where local moment formation is present and the spin correlation length small. However, in the vicinity of the fermion quantum critical point the spin correlation length diverges and the spontaneous SU(2) symmetry breaking leads to low-lying Goldstone modes in the magnetically ordered phase. The impact of these spin fluctuations on the single-particle spectral function -- \textit{waterfall} features and narrow spin-polaron bands -- is only visible in the lattice QMC approach.