Symmetries and exponential error reduction in YM theories on the lattice: theoretical aspects and simulation results

TL;DR

This paper introduces a multi-level Monte Carlo method leveraging symmetries to exponentially reduce computational costs in lattice Yang-Mills theories, demonstrated through simple models and applied to SU(3) gauge theory.

Contribution

A novel multi-level integration scheme exploiting symmetries for efficient computation of quantum states in lattice gauge theories.

Findings

Cost of computing lowest energy states is exponentially reduced.

Method successfully applied to SU(3) Yang-Mills theory.

New results on parity-odd state contributions in SU(3) theory.

Abstract

The path integral of a quantum system with an exact symmetry can be written as a sum of functional integrals each giving the contribution from quantum states with definite symmetry properties. We propose a strategy to compute each of them, normalized to the one with vacuum quantum numbers, by a Monte Carlo procedure whose cost increases power-like with the time extent of the lattice. This is achieved thanks to a multi-level integration scheme, inspired by the transfer matrix formalism, which exploits the symmetry and the locality in time of the underlying statistical system. As a result the cost of computing the lowest energy level in a given channel, its multiplicity and its matrix elements is exponentially reduced with respect to the standard path-integral Monte Carlo. We briefly illustrate the approach in the simple case of the one-dimensional harmonic oscillator and discuss in some detail its extension to the four-dimensional Yang Mills theories. We report on our recent new results in the SU(3) Yang--Mills theory on the relative contribution to the partition function of the parity-odd states.