Overcoming the sign problem in lattice gauge theories using tensor networks

Introduction

Tensor networks, and particularly projected entangled pair states (PEPS), are special quantum many-body states that describe strongly-correlated systems well due to their entanglement structure. They have been successfully applied in various scenarios and recently to lattice gauge theories (LGTs) where they outperformed conventional Monte-Carlo calculations and overcame the sign problem in some examples, but mostly in single-space dimensions due to limitations of tensor network methods.

Analogy Between PEPS and Gauge Theories

A fundamental analogy between PEPS and gauge theories suggests that PEPS are suitable for studying LGTs. This analogy indicates that gauge symmetry, often seen as complicating the numerics, can help in overcoming the sign problem and perform efficient tensor network computations in higher dimensions.

Project Goals

The overarching goal of this project is to use this analogy in analytical and numerical ways, aiming to:

1. Analytical Development: Devise a comprehensive new formalism for LGT PEPS and the physics they describe by allowing one to construct the optimal PEPS to be used as variational ansatz states when combined with numerical techniques.
2. Numerical Methods: Devise numerical methods for studying LGTs with such PEPS thanks to the analogy, based on sign problem-free variational Monte-Carlo.
3. Application to Models: Apply these methods numerically to challenging, non-perturbative models, culminating in SU(3) in 3+1-D, with finite fermionic density, towards quantum chromodynamics.

Expected Outcomes

This is expected to overcome the sign problem of such models, thus closing an important, challenging, and long-standing gap in the field of non-perturbative physics in general, and gauge theories in particular.

Generalization of Methods

The developed methods can be generalized for studying:

• Real-time dynamics of quantum field theories
• Models of quantum gravity
• Thermal quantum field theories
• Many other puzzling questions

These advancements will also contribute to the parallel contemporary approach to LGT - quantum simulations and computations - as some open problems are shared by both approaches.