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arXiv:2107.03626 · Physical Review C

Quantum many-body calculations using body-centered cubic lattices

Young-Ho Song, Youngman Kim, Ning Li, Bing-Nan Lu, Rongzheng He, Dean Lee

Published Oct 4, 2021• 4 citations
arXivHTMLPDF

Abstract

It is often computationally advantageous to model space as a discrete set of points forming a lattice grid. This technique is particularly useful for computationally difficult problems such as quantum many-body systems. For reasons of simplicity and familiarity, nearly all quantum many-body calculations have been performed on simple cubic lattices. Since the removal of lattice artifacts is often an important concern, it would be useful to perform calculations using more than one lattice geometry. In this paper we show how to perform quantum many-body calculations using auxiliary-field Monte Carlo simulations on a three-dimensional body-centered cubic (BCC) lattice. As a benchmark test we compute the ground state energy of 33 spin-up and 33 spin-down neutrons in the unitary limit, which is an idealized limit where the interaction range is zero and scattering length is infinite. As a fraction of the free Fermi gas energy ${E}_{\mathrm{FG}}$, we find that the ground state energy is ${E}_{0}/{E}_{\mathrm{FG}}=0.369(2),0.371(2),$ using two different definitions of the finite-system energy ratio. This is in excellent agreement with recent results obtained on a cubic lattice [He et al., Phys. Rev. A 101, 063615 (2020)]. We find that the computational effort and performance on a BCC lattice is approximately the same as that for a cubic lattice with the same number of lattice points. We discuss how the lattice simulations with different geometries can be used to constrain the size of lattice artifacts in simulations of continuum quantum many-body systems.

DOI
10.1103/physrevc.104.044304

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It is often computationally advantageous to model space as a discrete set of points forming a lattice grid. This technique is particularly useful for computationally difficult problems such as quantum many-body systems. For reasons of simplicity and familiarity, nearly all quantum many-body calculations have been performed on simple cubic lattices. Since the removal of lattice artifacts is often an important concern, it would be useful to perform calculations using more than one lattice geometry. In this paper we show how to perform quantum many-body calculations using auxiliary-field Monte Carlo simulations on a three-dimensional body-centered cubic (BCC) lattice. As a benchmark test we compute the ground state energy of 33 spin-up and 33 spin-down neutrons in the unitary limit, which is an idealized limit where the interaction range is zero and scattering length is infinite. As a fraction of the free Fermi gas energy EFG{E}_{\mathrm{FG}}EFG​, we find that the ground state energy is E0/EFG=0.369(2),0.371(2),{E}_{0}/{E}_{\mathrm{FG}}=0.369(2),0.371(2),E0​/EFG​=0.369(2),0.371(2), using two different definitions of the finite-system energy ratio. This is in excellent agreement with recent results obtained on a cubic lattice [He et al., Phys. Rev. A 101, 063615 (2020)]. We find that the computational effort and performance on a BCC lattice is approximately the same as that for a cubic lattice with the same number of lattice points. We discuss how the lattice simulations with different geometries can be used to constrain the size of lattice artifacts in simulations of continuum quantum many-body systems.
The unitary Fermi gas is a many-body system of two-component fermions with zero-range interactions tuned to infinite scattering length. Despite much activity and interest in unitary Fermi gas and its universal properties, there have been great difficulties in performing accurate calculations of the superfluid condensate fraction and pairing wave function. In this paper, we present auxiliary-field lattice Monte Carlo simulations using a lattice interaction which accelerates the approach to the continuum limit, thereby allowing for robust calculations of these difficult observables. As a benchmark test, we compute the ground-state energy of 33 spin-up and 33 spin-down particles. As a fraction of the free Fermi gas energy EFG{E}_{\text{FG}}EFG​, we find E0/EFG=0.369(2),0.372(2),{E}_{0}/{E}_{\text{FG}}=0.369(2),0.372(2),E0​/EFG​=0.369(2),0.372(2), using two different definitions of the finite-system energy ratio, in agreement with the latest theoretical and experimental results. We then determine the condensate fraction by measuring off-diagonal long-range order in the two-body density matrix. We find that the fraction of condensed pairs is $\ensuremath{\alpha}=0.43(2)$. We also extract the pairing wave function and find the pair correlation length to be ${\ensuremath{\zeta}}_{p}{k}_{F}=1.8(3)\ensuremath{\hbar}$, where kF{k}_{F}kF​ is the Fermi momentum. Provided that the simulations can be performed without severe sign oscillations, the methods we present here can be applied to superfluid neutron matter as well as more exotic PPP-wave and DDD-wave superfluids.
We consider two-component fermions on the lattice in the unitarity limit. This is an idealized limit of attractive fermions where the range of the interaction is zero and the scattering length is infinite. Using Euclidean time projection, we compute the ground state energy using four computationally different but physically identical auxiliary-field methods. The best performance is obtained using a bounded continuous auxiliary field and a nonlocal updating algorithm called the hybrid Monte Carlo. With this method, we calculate results for 10 and 14 fermions at lattice volumes 43,53,63,73,83{4}^{3},{5}^{3},{6}^{3},{7}^{3},{8}^{3}43,53,63,73,83 and extrapolate to the continuum limit. For 10 fermions in a periodic cube, the ground state energy is 0.292(12)0.292(12)0.292(12) times the ground state energy for noninteracting fermions. For 14 fermions, the ratio is 0.329(5)0.329(5)0.329(5).
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.
We perform lattice Monte Carlo simulations for up to 66 unitary fermions in a finite box using a highly improved lattice action for nonrelativistic spin-12\frac{1}{2}21​ fermions. We obtain a value of $0.{366}_{\ensuremath{-}0.011}^{+0.016}$ for the Bertsch parameter, defined as the energy of the unitary Fermi gas measured in units of the free gas energy in the thermodynamic limit. In addition, for up to four unitary fermions, we compute the spectrum of the lattice theory by exact diagonalization of the transfer matrix projected onto irreducible representations of the octahedral group for small to moderate size lattices, providing an independent check of our few-body simulation results. We compare our exact numerical and simulation results for the spectrum to benchmark studies of other research groups as well as perform an extended analysis of our lattice action improvement scheme, including an analysis of the errors associated with higher partial waves and finite temporal discretization.
Quantum Monte Carlo methods have proven to be valuable in the study of strongly correlated quantum systems, particularly nuclear physics and cold atomic gases. Historically, such ab initio simulations have been used to study properties of light nuclei, including spectra and form factors, low-energy scattering, and high-momentum properties including inclusive scattering and one- and two-body momentum distributions. More recently they have been used to study the properties of homogeneous and inhomogeneous neutron matter and cold atomic gases. There are close analogies between these seemingly diverse systems, including the equation of state, superfluid pairing, and linear response to external probes. In this paper, we compare and contrast results found in nuclear and cold atom physics. We show updated lattice results for the energy of the homogeneous unitary Fermi gas and comparisons with neutron matter, as well as for the dependence of the cold atom energy on the mass ratio between paired particles, which yields insights on the structure of the ground state. We also provide new lattice and continuum results for the harmonically trapped unitary gas, again comparing neutron matter and cold atoms.
Abstract This paper describes a different Q11C method for solving the Schrödinger equation, one unlike variational, diffusion, and Green’s function methods. It can be applied to many-body systems to produce solutions for ground states. For systems of fermions, antisymmetrization can be imposed to overcome the problem of node location, but the instability of the method is a serious shortcoming, so that the method as described is limited to treating relatively simple systems. Nevertheless, this auxiliary field quantum 1,fonte Carlo (AFMC) method was demonstrated to be successful for test cases reported in this paper, and in time has been improved to allow treatment of realistic systemsa. The method makes use of the imaginary-time propagation of an initial wavefunction toward the ground-state function. It requires use of a trial wavefunction which is the product of single-particle orbitals for bosons or an antisymmetrized product for fermions. A Hubbard-Stratonovich transformation of the propagator changes the many-body problem to a problem of a collection of single particles interacting with a fluctuating (auxiliary) field which replaces the many-body interactions. The authors developed an integral expression for the energy which, after discretization, could be evaluated by a Metropolis Monte Carlo integration forward in time. The algorithm was tested for one-dimensional systems of particles: 6 to 20 bosons with varied interactions, 2 to 6 neutron-proton pairs, others with repulsive potentials. Importance sampling was found to be critical and computational requirements to be large. For spin systems on a lattice the AFMC method was considered feasible, but for general multidimensional systems the method appeared to be limited by its computation requirements.