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arXiv:2109.02701 · Physical review. D/Physical review. D.

Application of bootstrap to a θ term

Yu Aikawa, Takeshi Morita, Kota Yoshimura

Published Apr 25, 2022• 25 citations
arXivHTMLPDF

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.

DOI
10.1103/physrevd.105.085017

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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 \ensuremathθ\ensuremath{\theta}\ensuremathθ-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 \ensuremathθ\ensuremath{\theta}\ensuremathθ-term. We find that it is hard to determine physical quantities as functions of \ensuremathθ\ensuremath{\theta}\ensuremathθ such as E(\ensuremathθ)E(\ensuremath{\theta})E(\ensuremathθ), except at \ensuremathθ=0\ensuremath{\theta}=0\ensuremathθ=0 and \ensuremathπ\ensuremath{\pi}\ensuremathπ. On the other hand, the correlations among observables for energy eigenstates are correctly reproduced for any \ensuremathθ\ensuremath{\theta}\ensuremathθ. Our results suggest that the bootstrap method may work not perfectly but sufficiently well, even if a \ensuremathθ\ensuremath{\theta}\ensuremathθ-term exists in the system.
Recently, an application of the numerical bootstrap method to quantum mechanics was proposed, and it successfully reproduces the eigenstates of various systems. However, it is unclear why this method works. In order to understand this question, we study the bootstrap method in harmonic oscillators. We find that the problem reduces to the Dirac's ladder operator problem and is exactly solvable analytically. Our result suggests that the bootstrap method may be regarded as a numerical version of the Dirac's approach and it may explain why it works in various systems.
The bootstrap method for quantum mechanics is a powerful tool for computing the energy eigenvalues of a Hamiltonian. However, we point out that this method suffers from an ambiguity problem: it fails to yield the correct spectrum when the potential contains different types of functions, such as polynomial and exponential terms. Similarly, the bootstrap method may break down when evaluating the expectation values of operators of different types. This issue can arise in a wide range of systems, including statistical models and matrix models. We propose three possible resolutions to this problem.
We solve the Schrödinger equation for a particle in the circle with a quantum mechanical bootstrap method. The main idea is that in physical situations one needs solutions of the recursion relations for the Fourier coefficients of the measure that vanish exponentially fast. We show that this results in a (generalized) continued fraction representation of the solution. The solution is an analytical function of the couplings and the energy. We also show how one this method directly computes the density of states and checking that the answer is physical is reduced to checking an inequality for a single variable. We also show how to reconstruct the wave function and how to extract the quasimomentum of the particle as a function of the energy. We illustrate our solution in the simplest case of the Mathieu equation, where we show that the method can easily resolve exponentially small bands in the band spectrum.
In recent years, technology around quantum computer sounds growing well. Here, as “users” of quantum computers, we discuss how we can apply this development to numerical simulations of quantum field theories. As quantum computer is suitable for Hamilton formalism rather than Lagrange formalism, we do not have infamous sign problem from the beginning in contrast to the conventional Monte Carlo approach. We show recent results on quantum simulations of the charge-q Schwinger model, which is 1+1-dimensional quantum electrodynamics coupled to an electron with electric charge q. It turns out that our approach enables us to explore interesting phenomena coming from non-small θ-term such as negative string tension behavior in potential between charged heavy particles.
Abstract Bootstrap methods, initially developed for solving statistical and quantum field theories, have recently been shown to capture the discrete spectrum of quantum mechanical problems, such as the single particle Schrödinger equation with an anharmonic potential. The core of bootstrap methods builds on exact recursion relations of arbitrary moments of some quantum operator and the use of an adequate set of positivity criteria. We extend this methodology to models with continuous Bloch band spectra, by considering a single quantum particle in a periodic cosine potential. We find that the band structure can be obtained accurately provided the bootstrap uses moments involving both position and momentum variables. We also introduce several new techniques that can apply generally to other bootstrap studies. First, we devise a trick to reduce by one unit the dimensionality of the search space for the variables parametrizing the bootstrap. Second, we employ statistical techniques to reconstruct the distribution probability allowing to compute observables that are analytic functions of the canonical variables. This method is used to extract the Bloch momentum, a quantity that is not readily available from the bootstrap recursion itself.
The fermion sign problem remains the primary obstacle in simulating the thermodynamic properties of various fermionic systems. In this work, we present a sign-blocking method to mitigate the numerical instability inherent in the sign problem. In the sign-blocking method, the Monte Carlo importance sampling remains identical to traditional methods; instead, the sign-blocking method is applied during the post-processing of signed samples. Given the significant progress in simulating the 2D Fermi-Hubbard model over the past decade, a wealth of energy benchmarks is available for comparison. Consequently, we use the 2D Fermi-Hubbard model as a benchmark to validate the sign-blocking method. Surprisingly, our results align exceptionally well with existing state-of-the-art benchmarks, even in regimes previously considered challenging. The physical mechanism of the sign-blocking method lies in uncovering the correlation between energy and sign factors through data blocking, thereby successfully inferring the fermionic system's energy. Our findings suggest that the sign-blocking method holds promise for complex quantum systems, particularly when combined with appropriate simulation techniques such as auxiliary-field formalisms that trace out the fermionic degrees of freedom.