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Physics Procedia

Pairing in Cold Atoms and other Applications for Quantum Monte Carlo methods

Michal Bajdich, Jindřich Kolorenč, Luboš Mitáš, PETER J. REYNOLDS

Published Feb 1, 2010• 2 citations
Publisher

Abstract

We discuss the importance of the fermion nodes for the quantum Monte Carlo (QMC) methods and find two cases of the exact nodes. We describe the structure of the generalized pairing wave functions in Pfaffian antisymmetric form and demonstrate their equivalency with certain class of configuration interaction wave functions. We present the QMC calculations of a model fermion system at unitary limit. We find the system to have the energy of E=0.425Efree and the condensate fraction of α=0.48. Further we also perform the QMC calculations of the potential energy surface and the electric dipole moment along that surface of the LiSr molecule. We estimate the vibrationally averaged dipole moment to be 〈D〉ν=0=−0.4(2).

DOI
10.1016/j.phpro.2010.01.199

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We discuss the importance of the fermion nodes for the quantum Monte Carlo (QMC) methods and find two cases of the exact nodes. We describe the structure of the generalized pairing wave functions in Pfaffian antisymmetric form and demonstrate their equivalency with certain class of configuration interaction wave functions. We present the QMC calculations of a model fermion system at unitary limit. We find the system to have the energy of E=0.425Efree and the condensate fraction of α=0.48. Further we also perform the QMC calculations of the potential energy surface and the electric dipole moment along that surface of the LiSr molecule. We estimate the vibrationally averaged dipole moment to be 〈D〉ν=0=−0.4(2).
Electronic structure quantum Monte Carlo Quantum Monte Carlo (QMC) is an advanced simulation methodology for studies of manybody quantum systems. The QMC approaches combine analytical insights with stochastic computational techniques for efficient solution of several classes of important many-body problems such as the stationary Schrödinger equation. QMC methods of various flavors have been applied to a great variety of systems spanning continuous and lattice quantum models, molecular and condensed systems, BEC-BCS ultracold condensates, nuclei, etc. In this review, we focus on the electronic structure QMC, i.e., methods relevant for systems described by the electron-ion Hamiltonians. Some of the key QMC achievements include direct treatment of electron correlation, accuracy in predicting energy differences and favorable scaling in the system size. Calculations of atoms, molecules, clusters and solids have demonstrated QMC applicability to real systems with hundreds of electrons while providing 90-95% of the correlation energy and energy differences typically within a few percent of experiments. Advances in accuracy beyond these limits are hampered by the so-called fixed-node approximation which is used to circumvent the notorious fermion sign problem. Many-body nodes of fermion states and their properties have therefore become one of the important topics for further progress in predictive power and efficiency of QMC calculations. Some of our recent results on the wave function nodes and related nodal domain topologies will be briefly reviewed. This includes analysis of few-electron systems and descriptions of exact and approximate nodes using transformations and projections of the highly-dimensional nodal hypersurfaces into the 3D space. Studies of fermion nodes offer new insights into topological properties of eigenstates such as explicit demonstrations that generic fermionic ground states exhibit the minimal number of two nodal domains. Recently proposed trial wave functions based on Pfaffians with pairing orbitals are presented and their nodal properties are tested in calculations of first row atoms and molecules. Finally, backflow "dressed" coordinates are introduced as another possibility for capturing correlation effects and for decreasing the fixed-node bias.
We develop a quantum Monte Carlo method to estimate the ground-state energy of a fermionic many-particle system in the configuration-interaction shell model approach. The fermionic sign problem is circumvented by using a guiding wave function in Fock space. The method provides an upper bound on the ground-state energy whose tightness depends on the choice of the guiding wave function. We argue that the antisymmetric geminal product class of wave functions is a good choice for guiding wave functions. We demonstrate our method for the trapped two-species fermionic cold atom system in the unitary regime of infinite scattering length using the particle-number projected Hartree-Fock-Bogoliubov wave function as the guiding wave function. We estimate the ground-state energy and energy-staggering pairing gap as a function of the number of particles. We compare our results with exact numerical diagonalization results and with previous fixed-node coordinate-space Monte Carlo calculations.
We investigate the accuracy of trial wave functions for quantum Monte Carlo based on Pfaffian functional form with singlet and triplet pairing. Using a set of first row atoms and molecules we find that these wave functions provide very consistent and systematic behavior in recovering the correlation energies on the level of 95%. In order to get beyond this limit we explore the possibilities of multi-Pfaffian pairing wave functions. We show that a small number of Pfaffians recovers another large fraction of the missing correlation energy comparable to the larger-scale configuration interaction wave functions. We also find that Pfaffians lead to substantial improvements in fermion nodes when compared to Hartree-Fock wave functions.
We investigate pfaffian trial wavefunctions with singlet and triplet pair orbitals by quantum Monte Carlo methods. We present mathematical identities and the key algebraic properties necessary for efficient evaluation of pfaffians. Following upon our previous study [Bajdich et al., Phys. Rev. Lett. 96, 130201 (2006)], we explore the possibilities of expanding the wavefunction in linear combinations of pfaffians. We observe that molecular systems require much larger expansions than atomic systems and linear combinations of a few pfaffians lead to rather small gains in correlation energy. We also test the wavefunction based on fully antisymmetrized product of independent pair orbitals. Despite its seemingly large variational potential, we do not observe additional gains in correlation energy. We find that pfaffians lead to substantial improvements in fermion nodes when compared to Hartree-Fock wavefunctions and exhibit the minimal number of two nodal domains in agreement with recent results on fermion nodes topology. We analyze the nodal structure differences of Hartree-Fock, pfaffian, and essentially exact large-scale configuration interaction wavefunctions. Finally, we combine the recently proposed form of backflow correlations [Drummond et al., J. Phys. Chem. 124, 22401 (2006); Rios et al., Phys. Rev. E. 74, 066701 (2006)] with both determinantal and pfaffian based wavefunctions.
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We solve the zero-temperature unitary Fermi gas problem by incorporating a BCS importance function into the auxiliary-field quantum Monte Carlo method. We demonstrate that this method does not suffer from a sign problem and that it increases the efficiency of standard techniques by many orders of magnitude for strongly paired fermions. We calculate the ground-state energies exactly for unpolarized systems with up to 66 particles on lattices of up to 273{27}^{3}273 sites, obtaining an accurate result for the universal parameter $\ensuremath{\xi}$. We also obtain results for interactions with different effective ranges and find that the energy is consistent with a universal linear dependence on the product of the Fermi momentum and the effective range. This method will have many applications in superfluid cold atom systems and in both electronic and nuclear structures where pairing is important.