Investigating a (3+1)D topological θ -term in the Hamiltonian formulation of lattice gauge theories for quantum and classical simulations

Angus Kan,Lena Funcke,Karl Jansen,Jinglei Zhang,Jan F Haase,Stefan Kühn,Luca Dellantonio,Christine A Muschik

doi:10.1103/physrevd.104.034504

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.

Highlights

Numerical simulations based on Markov chain Monte Carlo (MCMC) methods of lattice gauge theories have had unprecedented success in computing various nonperturbative aspects of fundamental particle interactions [1]
MCMC methods rely on formulating the theory in Euclidean spacetime, which prevents a direct simulation of real-time dynamics
We focus on the θ-dependence of the energy spectrum and the expectation value of various observables in the ground state jΨ0i of the Hamiltonian

Summary

INTRODUCTION

Numerical simulations based on Markov chain Monte Carlo (MCMC) methods of lattice gauge theories have had unprecedented success in computing various nonperturbative aspects of fundamental particle interactions [1]. We fill this gap by deriving the topological θ-term of ð3 þ 1ÞD lattice gauge theories in the Hamiltonian plaquette formulation, using the transfer matrix method [39]. This paves the way toward quantum and tensor network simulations of the topological term of the SM of particle physics. Our results are relevant in light of recent and earlier analytical studies of the phase diagram of ð3 þ 1ÞD pure compact U(1) lattice gauge theory with a θ-term In order to distinguish the variables in the Lagrangian formulation and operators in the Hamiltonian formulation, we express the latter with a hat (^) symbol

LATTICE FORMULATION

H KS g2 2a

MODEL AND METHODS

RESULTS

CONCLUSION AND DISCUSSION