Application of Bootstrap to $θ$-term

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 $θ$-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.

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