Application of bootstrap to a θ term

Abstract

Recently, a novel numerical computation on quantum mechanics by using a bootstrap method was proposed by Han, Hartnoll, and Kruthoff. We consider whether this method works in systems with a θ-term, where the standard Monte-Carlo computation may fail due to the sign problem. As a starting point, we study quantum mechanics of a charged particle on a circle in which a constant gauge potential is a counterpart of a θ-term. We find that it is hard to determine physical quantities as functions of θ such as E(θ), except at θ=0 and π. On the other hand, the correlations among observables for energy eigenstates are correctly reproduced for any θ. Our results suggest that the bootstrap method may work not perfectly but sufficiently well, even if a θ-term exists in the system.