Application of Bootstrap to $\theta$-term

TL;DR

This paper investigates the effectiveness of the bootstrap method in quantum systems with a $ heta$-term, highlighting its potential despite limitations in computing certain physical quantities.

Contribution

It demonstrates that the bootstrap method can reliably reproduce correlations in systems with a $ heta$-term, even where Monte Carlo methods face sign problems.

Findings

Correlation functions are accurately reproduced for all $ heta$.

Energy as a function of $ heta$ is difficult to determine except at special points.

Bootstrap method shows promise for systems with sign problems.

Abstract

Recently, 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 $\theta$-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 $\theta$-term. We find that it is hard to determine physical quantities as functions of $\theta$ such as $E(\theta)$, except at $\theta=0$ and $\pi$. On the other hand, the correlations among observables for energy eigenstates are correctly reproduced for any $\theta$. Our results suggest that the bootstrap method may work not perfectly but sufficiently well, even if a $\theta$-term exists in the system.