Hamiltonian Lattice Gauge Theories: emergent properties from Tensor Network methods

TL;DR

This thesis introduces advanced Tensor Network methods with a novel dressed-site formalism for Hamiltonian Lattice Gauge Theories, enabling efficient, gauge-invariant simulations of complex quantum systems and revealing new physical insights.

Contribution

It presents a new dressed-site formalism for gauge field truncation, applies it to SU(2) Yang-Mills theories, and introduces a fermion-to-qubit mapping and locality-preserving techniques for improved simulations.

Findings

First TN simulations of SU(2) Yang-Mills LGTs in 2D

Revealed phase diagram and non-equilibrium dynamics including QMB scarring

Enhanced simulation efficiency using optimal space-filling curves

Abstract

This thesis develops advanced Tensor Network (TN) methods to address Hamiltonian Lattice Gauge Theories (LGTs), overcoming limitations in real-time dynamics and finite-density regimes. A novel dressed-site formalism is introduced, enabling efficient truncation of gauge fields while preserving gauge invariance for both Abelian and non-Abelian theories. This formalism is successfully applied to SU(2) Yang-Mills LGTs in two dimensions, providing the first TN simulations of this system and revealing critical aspects of its phase diagram and non-equilibrium behavior, such as a Quantum Many-Body (QMB) scarring dynamics. A generalization of the dressed-site formalism is proposed through a new fermion-to-qubit mapping for general lattice fermion theories, revealing powerful for classical and quantum simulations. Numerical innovations, including the use of optimal space-filling curves such as the Hilbert curve to preserve locality in high-dimensional simulations, further enhance the efficiency of these methods. Together with high-performance computing techniques, these advances open current and future development pathways toward optimized, efficient, and faster simulations on scales comparable to Monte Carlo state-of-the-art.