Towards overcoming the Monte Carlo sign problem with tensor networks

Mari Carmen Bañuls,Stefan Kühn,J Ignacio Cirac,Karl Jansen,Krzysztof Cichy,Hana Saito

doi:10.1051/epjconf/201713704001

Abstract

The study of lattice gauge theories with Monte Carlo simulations is hindered by the infamous sign problem that appears under certain circumstances, in particular at non-zero chemical potential. So far, there is no universal method to overcome this problem. However, recent years brought a new class of non-perturbative Hamiltonian techniques named tensor networks, where the sign problem is absent. In previous work, we have demonstrated that this approach, in particular matrix product states in 1+1 dimensions, can be used to perform precise calculations in a lattice gauge theory, the massless and massive Schwinger model. We have computed the mass spectrum of this theory, its thermal properties and real-time dynamics. In this work, we review these results and we extend our calculations to the case of two flavours and non-zero chemical potential. We are able to reliably reproduce known analytical results for this model, thus demonstrating that tensor networks can tackle the sign problem of a lattice gauge theory at finite density.

Highlights

The study of lattice gauge theories with Monte Carlo simulations is hindered by the infamous sign problem that appears under certain circumstances, in particular at non-zero chemical potential
The results presented in the previous sections have demonstrated the feasibility of the matrix product states (MPS)/matrix product operators (MPO) approach when applied to lattice gauge theories
The research of the past few years has demonstrated that they are appropriate for the description of lattice gauge theories and they can provide precise and reliable results both at zero and non-zero temperature

Summary

Introduction

There is broad search for alternative approaches, including Lefschetz thimbles [2], complex Langevin simulations [3] and density of states methods [4] Another thread of research, one that we concentrate on in this paper, is related to the tensor networks (TN) approach. The TN approach, originally introduced in the context of condensed matter physics and further developed thanks to quantum information theory, has been successfully applied to the description of quantum many-body systems, including, in last years, lattice field theory The latter included computations of spectra [7,8,9,10], thermal states [11,12,13,14,15], phase diagrams [16,17,18,19] and real-time evolution [8, 20,21,22].

Multi-flavour Schwinger model

Tensor network approach

Conclusion and prospects