Cluster simulation of two-dimensional relativistic fermions

TL;DR

This paper develops a cluster simulation method for the two-dimensional Gross-Neveu model, enabling efficient numerical studies of relativistic fermions with reduced critical slowing down.

Contribution

It introduces a novel cluster algorithm based on a dimer and Ising model representation for the lattice Gross-Neveu model, improving simulation efficiency.

Findings

Cluster algorithm reduces critical slowing down

Effective simulation at vanishing coupling

Potential for high-precision numerical studies

Abstract

The (discrete) Gross-Neveu model is studied in a lattice realization with an N-component Majorana Wilson fermion field. It has an internal O(N) symmetry in addition to the euclidean lattice symmetries. The discrete chiral symmetry for vanishing mass is expected to emerge in the continuum limit only. The lattice theory is first recast in terms of two-valued bosonic link variables (dimers). In this representation, which coincides with the loop representation obtained earlier by Gattringer with the help of eight-vertex-models, the Boltzmann weight is essentially positive. While standard local updates are possible in this form we construct a further exact transformation where we generate dimer configurations as Peierls contours of an Ising model with a local action residing on plaquettes. For this model a Swendsen-Wang type cluster algorithm is constructed. At vanishing coupling it is numerically demonstrated to almost completely eliminate critical slowing down. Although further tests are required, an avenue to numerical studies of the Gross-Neveu model with unprecedented precision seems open.