Quantum Monte Carlo Method using Phase-Free Random Walks with Slater Determinants

Abstract

We develop a quantum Monte Carlo method for many fermions that allows the use of any oneparticle basis. It projects out the ground state by random walks in the space of Slater determinants. An approximate approach is formulated to control the phase problem with a trial wave function |ΨT . Using plane-wave basis and non-local pseudopotentials, we apply the method to Si atom, dimer, and 2, 16, 54 atom (216 electrons) bulk supercells. Single Slater determinant wave functions from density functional theory calculations were used as |ΨT with no additional optimization. The calculated binding energy of Si2 and cohesive energy of bulk Si are in excellent agreement with experiments and are comparable to the best existing theoretical results.PACS numbers: 02.70. Ss, Quantum Monte Carlo (QMC) methods based on auxiliary fields (AF) are used in areas spanning condensed matter physics, nuclear physics, and quantum chemistry. These methods [1,2] allow essentially exact calculations of ground-state and finite-temperature equilibrium properties of interacting many fermion systems. The required CPU time scales in principle as a power law with system size, and the methods have been applied to study a variety of problems including the Hubbard model, nuclear shell models, and molecular electronic structure. The central idea of these methods is to write the imaginarytime propagator of a many-body system with two-body interactions in terms of propagators for independent particles interacting with external auxiliary fields. The independent particle problems are solved for configurations of the AF and averaging over different AF configurations is then performed by Monte Carlo (MC) techniques.QMC methods with auxiliary fields have several appealing features. For example, they allow one to choose any one-particle basis suitable for the problem, and to fully take advantage of well-established techniques to treat independent particles. Given the remarkable development and success of the latter [3], it is clearly very desirable to have a QMC method that can use exactly the same machinery and systematically include correlation effects by simply building stochastic ensembles of the independent particle solutions. Vigorous attempts have been made from several fields to explore this possibility [4,5,6,7].A significant hurdle exists, however: except for special cases (e.g., Hubbard), the two-body interactions will require auxiliary fields that are complex . As a result, the single-particle orbitals become complex, and the MC averaging over AF configurations becomes an integration over complex variables in many dimensions. A phase problem thus occurs which ultimately defeats the algebraic scaling of MC and makes the method scale exponentially. This is analogous to but more severe than the fermion sign problem with real AF [8,9] or in real-space methods [10]. No satisfactory, general approach exists to control the phase problem. As a result, only small systems or special forms of interactions can be treated.In this paper we address this problem. W...

Coupling quantum Monte Carlo and independent-particle calculations: Self-consistent constraint for the sign problem based on the density or the density matrix

“…Previous studies in a variety of systems have shown that the bias tends to be small, in both models 10,24-27 and realistic materials 13,17,21,23 , making this one of the most accurate many-body approaches for general interacting fermion systems. In this work, we introduce a self-consistent method to further reduce the bias introduced by the constraint from the trial wave function.…”

“…This approach has been referred to as the constrained-path Monte Carlo (CPMC) method. For a general Hamiltonian with two-body interactions, a generalized gauge condition allows a similar framework for the phase problem 21,23 .…”

Auxiliary-field-based trial wave functions in quantum Monte Carlo calculations

“…For a detailed discussion of the CPMC method and benchmark results, we refer readers to Ref. [36][37][38]. Here we note that CPMC eliminates the sign problem much as the fixed-node approximation does in DMC by rejecting random walkers that have negative overlaps with the trial wave function.…”

“…The Gutzwiller wave function produced after decoupling may be viewed as a finite sum over determinants, each of which is a function of a discrete set of To showcase the GWF, we study the ground state of the one-band repulsive Hubbard model in two dimensions using the constrained-path Monte Carlo (CPMC) technique. 36,37 The system is defined by the Hamiltonian…”

Quantum Monte Carlo calculations of two neutrons in finite volume

“…Moving the node position in the relative coordinate such that the two independent simulations yield the same energy within statistical uncertainties leads to the results presented in the next section. As the constrainedpath approximation [38], which we use to tame the sign problem, prohibits walkers from crossing the nodal sur- The energies are given in terms of the dimensionless quantity…”