Application of bootstrap to a <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>θ</mml:mi></mml:math> term
Yu Aikawa, Takeshi Morita, Kota Yoshimura
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 $\ensuremath{\theta}$-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 $\ensuremath{\theta}$-term. We find that it is hard to determine physical quantities as functions of $\ensuremath{\theta}$ such as $E(\ensuremath{\theta})$, except at $\ensuremath{\theta}=0$ and $\ensuremath{\pi}$. On the other hand, the correlations among observables for energy eigenstates are correctly reproduced for any $\ensuremath{\theta}$. Our results suggest that the bootstrap method may work not perfectly but sufficiently well, even if a $\ensuremath{\theta}$-term exists in the system.