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arXiv:2207.12294 · SciPost Physics Core

Properties of fermionic systems with the path-integral ground state method

Sebastian Ujevic, Vinicius Zampronio, Bruno Ricardi de Abreu, S. A. Vitiello

Published Apr 20, 2023• 2 citations
arXivHTMLPDF

Abstract

We investigate strongly correlated many-body systems composed of bosons and fermions with a fully quantum treatment using the path-integral ground state method, PIGS. To account for the Fermi-Dirac statistics, we implement the fixed-node approximation into PIGS, which we then call FN-PIGS. In great detail, we discuss the pair density matrices we use to construct the full density operator in coordinate representation, a vital ingredient of the method. We consider the harmonic oscillator as a proof-of-concept and, as a platform representing quantum many-body systems, we explore helium atoms. Pure ^4 4 He systems demonstrate most of the features of the method. Complementarily, for pure ^3 3 He, the fixed-node approximation resolves the ubiquitous sign problem stemming from anti-symmetric wave functions. Finally, we investigate ^3 3 He- ^4 4 He mixtures, demonstrating the method’s robustness. One of the main features of FN-PIGS is its ability to estimate any property at temperature T = 0 T = 0 without any additional bias apart from the FN approximation; biases from long simulations are also excluded. In particular, we calculate the correlation function of pairs of equal and opposite spins and precise values of the ^3 3 He kinetic energy in the mixture.

DOI
10.21468/scipostphyscore.6.2.031

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We investigate strongly correlated many-body systems composed of bosons and fermions with a fully quantum treatment using the path-integral ground state method, PIGS. To account for the Fermi-Dirac statistics, we implement the fixed-node approximation into PIGS, which we then call FN-PIGS. In great detail, we discuss the pair density matrices we use to construct the full density operator in coordinate representation, a vital ingredient of the method. We consider the harmonic oscillator as a proof-of-concept and, as a platform representing quantum many-body systems, we explore helium atoms. Pure ^4 4 He systems demonstrate most of the features of the method. Complementarily, for pure ^3 3 He, the fixed-node approximation resolves the ubiquitous sign problem stemming from anti-symmetric wave functions. Finally, we investigate ^3 3 He- ^4 4 He mixtures, demonstrating the method’s robustness. One of the main features of FN-PIGS is its ability to estimate any property at temperature T = 0 T = 0 without any additional bias apart from the FN approximation; biases from long simulations are also excluded. In particular, we calculate the correlation function of pairs of equal and opposite spins and precise values of the ^3 3 He kinetic energy in the mixture.
Generally "exact" quantum Monte Carlo computations for the ground state of many bosons make use of importance sampling. The importance sampling is based either on a guiding function or on an initial variational wave function. Here we investigate the need of importance sampling in the case of path integral ground state (PIGS) Monte Carlo. PIGS is based on a discrete imaginary time evolution of an initial wave function with a nonzero overlap with the ground state, which gives rise to a discrete path which is sampled via a Metropolis-like algorithm. In principle the exact ground state is reached in the limit of an infinite imaginary time evolution, but actual computations are based on finite time evolutions and the question is whether such computations give unbiased exact results. We have studied bulk liquid and solid (4)He with PIGS by considering as initial wave function a constant, i.e., the ground state of an ideal Bose gas. This implies that the evolution toward the ground state is driven only by the imaginary time propagator, i.e., there is no importance sampling. For both phases we obtain results converging to those obtained by considering the best available variational wave function (the shadow wave function) as initial wave function. Moreover we obtain the same results even by considering wave functions with the wrong correlations, for instance, a wave function of a strongly localized Einstein crystal for the liquid phase. This convergence is true not only for diagonal properties such as the energy, the radial distribution function, and the static structure factor, but also for off-diagonal ones, such as the one-body density matrix. This robustness of PIGS can be traced back to the fact that the chosen initial wave function acts only at the beginning of the path without affecting the imaginary time propagator. From this analysis we conclude that zero temperature PIGS calculations can be as unbiased as those of finite temperature path integral Monte Carlo. On the other hand, a judicious choice of the initial wave function greatly improves the rate of convergence to the exact results.
Abstract Monte Carlo techniques have played an important role in understanding strongly correlated systems across many areas of physics, covering a wide range of energy and length scales. Among the many Monte Carlo methods applicable to quantum mechanical systems, the path integral Monte Carlo approach with its variants has been employed widely. Since semi-classical or classical approaches will not be discussed in this review, path integral based approaches can for our purposes be divided into two categories: approaches applicable to quantum mechanical systems at zero temperature and approaches applicable to quantum mechanical systems at finite temperature. While these two approaches are related to each other, the underlying formulation and aspects of the algorithm differ. This paper reviews the path integral Monte Carlo ground state (PIGS) approach, which solves the time-independent Schrödinger equation. Specifically, the PIGS approach allows for the determination of expectation values with respect to eigen states of the few- or many-body Schrödinger equation provided the system Hamiltonian is known. The theoretical framework behind the PIGS algorithm, implementation details, and sample applications for fermionic systems are presented.
The main difficulty for path integral Monte Carlo studies of Fermi systems results from the requirement of antisymmetrization of the density matrix and is known in literature as the 'sign problem'. To overcome this issue the new numerical version of the %{\large {\tiny {\Huge }}} Wigner approach to quantum mechanics for treatment thermodynamic properties of degenerate systems of fermions has been developed. The new path integral representation of quantum Wigner function in the phase space has been obtained for canonical ensemble. Explicit analytical expression of the Wigner function accounting for Fermi statistical effects by effective pair pseudopotential has been proposed. Derived pseudopotential depends on coordinates, momenta and degeneracy parameter of fermions and takes into account Pauli blocking of fermions in phase space. The new quantum Monte-Carlo method for calculations of average values of arbitrary quantum operators has been proposed. To test the developed approach calculations of the momentum distribution function of the degenerate ideal system of Fermi particles has been carried out in a good agreement with analytical Fermi distributions. Generalization of this approach for studies influence of interparticle interaction on momentum distribution functions of strongly coupled Fermi system is in progress.
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.
We study a bosonic gas of hard spheres by using the exact zero-temperature path-integral ground-state (PIGS) Monte Carlo method and the equations of superfluid hydrodynamics. The PIGS method is implemented to calculate for the bulk system the energy per particle and the condensate fraction through a large range of the gas parameter na3n{a}^{3}na3 (with nnn the number density and aaa the sss-wave scattering length), going from the dilute gas into the solid phase. The Maxwell construction is then adopted to determine the freezing at $n{a}^{3}=0.264\ifmmode\pm\else\textpm\fi{}0.003$ and the melting at $n{a}^{3}=0.290\ifmmode\pm\else\textpm\fi{}0.003$. In the liquid phase, where the condensate fraction is finite, the equations of superfluid hydrodynamics, based on the PIGS equation of state, are used to find other relevant quantities as a function of the gas parameter: the chemical potential, the pressure, and the sound velocity. In addition, within Feynman's approximation, from the PIGS static structure factor we determine the full excitation spectrum, which displays a maxon-roton behavior when the gas parameter is close to the freezing value. Finally, the equations of superfluid hydrodynamics with the PIGS equation of state are solved for the bosonic system under axially symmetric harmonic confinement obtaining its collective breathing modes.
A fermion node is subset of fermionic configurations for which a real wave function vanishes due to the antisymmetry and the node divides the configurations space into compact nodal cells (domains). We analyze the properties of fermion nodes of fermionic ground state wave functions for a number of systems. For several models we demonstrate that noninteracting spin-polarized fermions in dimension two and higher have closed-shell ground state wave functions with the minimal two nodal cells for any system size and we formulate a theorem which sumarizes this result. The models include periodic fermion gas, fermions on the surface of a sphere, fermions in a box. We prove the same property for atomic states with up to 3d half-filled shells. Under rather general assumptions we then derive that the same is true for unpolarized systems with arbitrarily weak interactions using Bardeen-Cooper-Schrieffer (BCS) variational wave function. We further show that pair correlations included in the BCS wave function enable singlet pairs of particles to wind around the periodic box without crossing the node pointing towards the relationship of nodes to transport and many-body phases such as superconductivity. Finally, we point out that the arguments extend also to fermionic temperature dependent/imaginary-time density matrices. The results reveal fundamental properties of fermion nodal structures and provide new insights for accurate constructions of wave functions and density matrices in quantum and path integral Monte Carlo methods.