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arXiv:1404.6944 · Physical Review E

Sign problem of the fermionic shadow wave function

Francesco Calcavecchia, Francesco Pederiva, Malvin H. Kalos, Thomas D. Kühne

Published Nov 4, 2014• 17 citations
arXivHTMLPDF

Abstract

We present a whole series of methods to alleviate the sign problem of the fermionic shadow wave function in the context of variational Monte Carlo. The effectiveness of our techniques is demonstrated on liquid ^{3}He. We found that although the variance is reduced, the gain in efficiency is restricted by the increased computational cost. Yet, this development not only extends the scope of the fermionic shadow wave function, but also facilitates highly accurate quantum Monte Carlo simulations previously thought not feasible.

DOI
10.1103/physreve.90.053304

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We present a whole series of methods to alleviate the sign problem of the fermionic shadow wave function in the context of variational Monte Carlo. The effectiveness of our techniques is demonstrated on liquid ^{3}He. We found that although the variance is reduced, the gain in efficiency is restricted by the increased computational cost. Yet, this development not only extends the scope of the fermionic shadow wave function, but also facilitates highly accurate quantum Monte Carlo simulations previously thought not feasible.
We present a novel technique well suited for studying the ground state of inhomogeneous fermionic matter in a wide range of different systems. The system is described using a fermionic shadow wave function, and the energy is computed by means of the variational Monte Carlo technique. The general form of the fermionic shadow wave function is useful for describing many-body systems with the coexistence of different phases as well in the presence of defects or impurities, but it requires overcoming a significant sign problem. As an application, we studied the energy to activate vacancies in solid 3He.
We present a simple, robust, and highly efficient method for optimizing all parameters of many-body wave functions in quantum Monte Carlo calculations, applicable to continuum systems and lattice models. Based on a strong zero-variance principle, diagonalization of the Hamiltonian matrix in the space spanned by the wave function and its derivatives determines the optimal parameters. It systematically reduces the fixed-node error, as demonstrated by the calculation of the binding energy of the small but challenging C(2) molecule to the experimental accuracy of 0.02 eV.
We use the shadow wave function formalism as a convenient model to study the fermion sign problem affecting all projector quantum Monte Carlo methods in continuum space. We demonstrate that the efficiency of imaginary-time projection algorithms decays exponentially with increasing number of particles and/or imaginary-time propagation. Moreover, we derive an analytical expression that connects the localization of the system with the magnitude of the sign problem, illustrating this behavior through numerical results. Finally, we discuss the computational complexity of the fermion sign problem and methods for alleviating its severity.
In general, Quantum Monte Carlo methods suffer from a sign problem when simulating fermionic systems. This causes the efficiency of a simulation to decrease exponentially with the number of particles and inverse temperature. To circumvent this issue, a nodal constraint is often implemented, restricting the Monte Carlo procedure from sampling paths that cause the many-body density matrix to change sign. Unfortunately, this high-dimensional nodal surface is not a priori known unless the system is exactly solvable, resulting in uncontrolled errors. We will discuss two possible routes to extend the applicability of finite-temperatue path integral Monte Carlo. First we extend the regime where signful simulations are possible through a novel permutation sampling scheme. Afterwards, we discuss a method to variationally improve the nodal surface by minimizing a free energy during simulation. Applications of these methods will include both free and interacting electron gases, concluding with discussion concerning extension to inhomogeneous systems.
Sign problem in quantum Monte Carlo (QMC) simulation appears to be an extremely hard yet interesting problem. In this article, we present a pedagogical overview on the origin of the sign problem in various quantum Monte Carlo simulation techniques, ranging from the world-line and stochastic series expansion Monte Carlo for boson and spin systems to the determinant and momentum-space quantum Monte Carlo for interacting fermions. We point out the basis dependency of the sign problem and summarize the progresses to cure, ease and even make use of the sign problem over the years, such as symmetry analysis of the underlying Hamiltonian, basis optimization in writting down the partition functions and many others. Moreover, we state that although traditional lore saying that in case of sign problem, the average sign in QMC simulation approaches zero exponentially fast with the space-time volume of the configurational space, there are recent breakthroughs showing this is not always the case and based on the properties of the partition function for finite size systems, one could distinguish when the average sign has the usual exponential scaling and when it is bestowed with an algebraic scaling at the low temperature limit. Fermionic QMC simulations with such algebraic sign problems have been successfully carried out for extended Hubbard-type and quantum Moiré lattice models.
We show that the standard Lanczos algorithm can be efficiently implemented statistically and self-consistently improved, using the stochastic reconfiguration method, which has been recently introduced to stabilize the Monte Carlo sign problem instability. With this scheme a few Lanczos steps over a given variational wave function are possible even for large size as a particular case of a more general and more accurate technique that allows to obtain lower variational energies. This method has been tested extensively for a strongly correlated model like the $t\ensuremath{-}J$ model. With the standard Lanczos technique it is possible to compute any kind of correlation functions, with no particular computational effort. By using the fact that the variance $〈{H}^{2}〉\ensuremath{-}〈H{〉}^{2}$ is zero for an exact eigenstate, we show that the approach to the exact solution with few Lanczos iterations is indeed possible even for $\ensuremath{\sim}100$ electrons for reasonably good initial wave functions. The variational stochastic reconfiguration technique presented here allows in general a many-parameter energy optimization of any computable many-body wave function, including for instance generic long-range Jastrow factors and arbitrary site-dependent orbital determinants. This scheme improves further the accuracy of the calculation, especially for long-distance correlation functions.