3+1D $\theta$-Term on the Lattice from the Hamiltonian Perspective

TL;DR

This paper derives the Hamiltonian formulation of the 3+1D $ heta$-term for lattice gauge theories, performs exact diagonalization for a U(1) model revealing a phase transition, and sets the stage for quantum simulations of these theories.

Contribution

It provides the first derivation of the Hamiltonian 3+1D $ heta$-term for lattice gauge theories and demonstrates numerical evidence of a phase transition in a U(1) model.

Findings

Revealed a phase transition at fixed $ heta$ in the strong-coupling regime.

Observed avoided level crossing in the ground state energy.

Detected changes in plaquette, electric energy, and topological charge densities.

Abstract

Quantum and tensor network simulations have emerged as prominent sign-problem free approaches to lattice gauge theories. Unlike conventional Markov chain Monte Carlo methods, they are based on the Hamiltonian formulation. In this talk, we fill a gap in the literature and present the first derivation of the Hamiltonian 3+1D $\theta$-term -- which is an important sign-problem afflicted term -- for Abelian and non-Abelian lattice gauge theories. Furthermore, we perform exact diagonalization for a 3+1D U(1) lattice gauge theory including the $\theta$-term on a unit periodic cube. Our numerical results reveal a novel phase transition at fixed values of $\theta$ in the strong-coupling regime. The transition is evidenced by an avoided level crossing in the ground state energy, as well as sudden changes in the plaquette expectation value, the electric energy density, and the topological charge density. Extensions of our work to larger lattices can be readily performed using state-of-the-art tensor network simulations. Moreover, our work provides a concrete starting point for an eventual quantum simulation of the $\theta$-dependent phase structure and dynamics of lattice gauge theories in 3+1D. This talk is mainly based on [1]. We expand beyond [1] by including a derivation of the (non-)Abelian fixed-length Higgs term in the Hamiltonian formulation for future studies of (non-)Abelian-Higgs models with a $\theta$-term.