Investigating a (3+1)D Topological $\theta$-Term in the Hamiltonian Formulation of Lattice Gauge Theories for Quantum and Classical Simulations

TL;DR

This paper derives the (3+1)D topological θ-term for lattice gauge theories in the Hamiltonian formulation and explores its phase structure through numerical methods, advancing quantum and classical simulation capabilities.

Contribution

It introduces a Hamiltonian formulation of the (3+1)D topological θ-term for lattice gauge theories, enabling future quantum and classical simulations of topological effects.

Findings

Evidence of a phase transition at constant θ in strong coupling regime

Identification of abrupt changes in physical observables at the transition

Potential for cross-validation with tensor network and quantum simulation methods

Abstract

Quantum technologies offer the prospect to efficiently simulate sign-problem afflicted regimes in lattice field theory, such as the presence of topological terms, chemical potentials, and out-of-equilibrium dynamics. In this work, we derive the (3+1)D topological $\theta$-term for Abelian and non-Abelian lattice gauge theories in the Hamiltonian formulation, paving the way towards Hamiltonian-based simulations of such terms on quantum and classical computers. We further study numerically the zero-temperature phase structure of a (3+1)D U(1) lattice gauge theory with the $\theta$-term via exact diagonalization for a single periodic cube. In the strong coupling regime, our results suggest the occurrence of a phase transition at constant values of $\theta$, as indicated by an avoided level-crossing and abrupt changes in the plaquette expectation value, the electric energy density, and the topological charge density. These results could in principle be cross-checked by the recently developed (3+1)D tensor network methods and quantum simulations, once sufficient resources become available.