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arXiv:1908.10471 · Physical review. E

Solving fermion problems without solving the sign problem: Symmetry-breaking wave functions from similarity-transformed propagators for solving two-dimensional quantum dots

Siu A. Chin

Published Apr 13, 2020• 3 citations
arXivHTMLPDF

Abstract

It is well known that the use of the primitive second-order propagator in path-integral Monte Carlo calculations of many-fermion systems leads to the sign problem. This work will show that by using the similarity-transformed Fokker-Planck propagator, it is possible to solve for the ground state of a large quantum dot, with up to 100 polarized electrons, without solving the sign problem. These similarity-transformed propagators naturally produce rotational symmetry-breaking ground-state wave functions previously used in the study of quantum dots and quantum Hall effects. However, instead of localizing the electrons at positions that minimize the potential energy, this derivation shows that they should be located at positions that maximize the bosonic ground-state wave function. Further improvements in the energy can be obtained by using these as initial wave functions in a ground-state path-integral Monte Carlo calculation with second- and fourth-order propagators.

DOI
10.1103/physreve.101.043304

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It is well known that the use of the primitive second-order propagator in path-integral Monte Carlo calculations of many-fermion systems leads to the sign problem. This work will show that by using the similarity-transformed Fokker-Planck propagator, it is possible to solve for the ground state of a large quantum dot, with up to 100 polarized electrons, without solving the sign problem. These similarity-transformed propagators naturally produce rotational symmetry-breaking ground-state wave functions previously used in the study of quantum dots and quantum Hall effects. However, instead of localizing the electrons at positions that minimize the potential energy, this derivation shows that they should be located at positions that maximize the bosonic ground-state wave function. Further improvements in the energy can be obtained by using these as initial wave functions in a ground-state path-integral Monte Carlo calculation with second- and fourth-order propagators.
The conventional second-order path-integral Monte Carlo method is plagued with the sign problem in solving many-fermion systems. This is due to the large number of antisymmetric free-fermion propagators that are needed to extract the ground state wave function at large imaginary time. In this work we show that optimized fourth-order path-integral Monte Carlo methods, which use no more than five free-fermion propagators, can yield accurate quantum dot energies for up to 20 polarized electrons with the use of the Hamiltonian energy estimator.
This work shows that the recently discovered operator contraction identity for solving the discrete path integral of the harmonic oscillator can be applied equally to fermions in any dimension. This then yields an exactly solvable model for studying the sign problem where the path integral Monte Carlo energy at any time step for any number of fermions is known analytically or can be computed numerically. It is found that the sign problem is primarily a property of the free fermion propagator, but repulsive/attractive pairwise interactions can shift the sign problem to larger/smaller imaginary time but do not make it more severe than the non-interacting case. More surprisingly, one can prove analytically that the first closed-shell state in D dimension, with n = D + 1 fermions, has no sign problem at large imaginary time. Direct numerical simulations confirm that this is also true for higher closed-shell states in two and three dimensions. Fourth-order and newly found variable-bead algorithms are used to compute ground state energies of quantum dots with up to 110 electrons and are compared to results obtained by modern neural networks.
This work shows that the recently discovered operator contraction identity for solving the discreet Path Integral of the harmonic oscillator can be applied equally to fermions in any dimension. This then yields an exactly solvable model for studying the sign problem where the Path Integral Monte Carlo energy at any time step for any number of fermions is known analytically, or can be computed numerically. It is found that the sign problem is primarily a property of the free fermion propagator, but repulsive/attractive pairwise interaction can shift the sign problem to larger/smaller imaginary time but does not make it more severe than the non-interacting case. More surprisingly, one can prove analytically that the first closed-shell state in DDD dimension, with n=D+1n=D+1n=D+1 fermion, has no sign problem at large imaginary time. Direct numerical simulations confirm that this is also true for higher closed-shell states in two and three-dimension. Fourth-order and newly found variable-bead algorithms are used to compute ground state energies of quantum dots with up to 110 electrons and compared to results obtained by modern neural networks.
In this work, within the framework of path integral Monte Carlo, we construct a pseudo-fermion propagator by replacing the original fermionic determinant with its absolute value. This modified propagator defines an auxiliary system free from the fermion sign problem, enabling efficient simulations of fermionic systems. We found that by shifting the pseudo-fermion energy based on the energy of a non-interacting fermion system, we can efficiently and reliably infer the energy of fermionic systems in various situations, from strong quantum degeneracy to weak quantum degeneracy. We have performed numerical simulations of quantum dots confined in a two-dimensional harmonic potential and found excellent agreement with benchmark results provided by other established methods. We believe that this pseudo-fermion propagator framework opens up new possibilities for numerical simulations of fermionic systems.
We introduce an efficient and numerically stable technique to make use of a BCS trial wave function in the computation of correlation functions of strongly correlated quantum fermion systems. The technique is applicable to any projection approach involving paths of independent-fermion propagators, for example, in mean-field or auxiliary-field quantum Monte Carlo (AFQMC) calculations. Within AFQMC, in the absence of the sign problem, the methodology allows the use of a BCS reference state which can greatly reduce the required imaginary time of projection and improves Monte Carlo sampling efficiency and statistical accuracy for systems where pairing correlations are important. When the sign problem is present, the approach provides a powerful generalization of the constrained-path AFQMC technique which usually uses Slater determinant trial wave functions. As a demonstration of the capability of the methodology, we present benchmark results for the attractive Hubbard model, both spin balanced (no sign problem) and with a finite spin polarization (with sign problem).
In this chapter, we discuss the fixed-node and constrained-path Monte Carlo methods for computing the ground state properties of systems of interacting electrons. These methods are arguably the two most powerful ones presently available for doing such calculations, but they are approximate. By sacrificing exactness, they avoid the exponential scaling of the Monte Carlo errors with system size that typically accompanies the simulation of systems of interacting electrons. This exponential scaling is called the Fermion sign problem . After a few general comments about the sign problem, we outline both methods, noting points of similarity and difference, plus points of strength and weakness. We also discuss the constrained-phase method, an extension of the constrained-path method, which controls the phase problem that develops when the ground state wave function cannot be real. Sign problem The “sign problem” refers to the exponential increase of the Monte Carlo errors with increasing system size or decreasing temperature (e.g., Loh et al., 1990, 2005; Gubernatis and Zhang, 1994) that often accompanies a Markov chain simulation whose limiting distribution is not everywhere positive. Such a case generally arises in simulations of Fermion and frustrated quantum-spin systems. It seems so inherent to Monte Carlo simulations of Fermion systems that the phrase “the sign problem” to many seems almost synonymous with the phrase “the Fermion sign problem.” Explanations for the cause of the sign problem vary and are still debated. In this chapter, we choose to summarize two explanations that seem to connect the causes in ground state Fermion simulations in the continuum and on the lattice. The sign problem, of course, is not limited to ground state calculations or even to Fermion simulations. While the cause we discuss in Sections 11.2 and 11.3 focuses on the low-lying states of diffusion-like operators, several topological pictures have been proposed (Muramatsu et al., 1992; Samson, 1993; Gubernatis and Zhang, 1994; Samson, 1995). Some of these discussions are done in the context of the zero- and finite-temperature determinant methods (Muramatsu et al., 1992; Gubernatis and Zhang, 1994). Others are done more analytically from a Feynman path-integral point of view (Samson, 1993, 1995). Some are for particles with statistics other than Fermions. The presentation of the sign problem in this chapter is appropriate for the Monte Carlo methods discussed in this chapter.