Quantum Monte Carlo method for the ground state of many-boson systems

Abstract

We formulate a quantum Monte Carlo (QMC) method for calculating the ground state of many-boson systems. The method is based on a field-theoretical approach, and is closely related to existing fermion auxiliary-field QMC methods which are applied in several fields of physics. The ground-state projection is implemented as a branching random walk in the space of permanents consisting of identical single-particle orbitals. Any single-particle basis can be used, and the method is in principle exact. We illustrate this method with a trapped atomic boson gas, where the atoms interact via an attractive or repulsive contact two-body potential. We choose as the single-particle basis a real-space grid. We compare with exact results in small systems and arbitrarily sized systems of untrapped bosons with attractive interactions in one dimension, where analytical solutions exist. We also compare with the corresponding Gross-Pitaevskii (GP) mean-field calculations for trapped atoms, and discuss the close formal relation between our method and the GP approach. Our method provides a way to systematically improve upon GP while using the same framework, capturing interaction and correlation effects with a stochastic, coherent ensemble of noninteracting solutions. We discuss various algorithmic issues, including importance sampling and the back-propagation technique for computing observables, and illustrate them with numerical studies. We show results for systems with up to N approximately 400 bosons.

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

“…(1), using any typical choice of |ψ T , for example a non-interacting wave function or the UHF solution. We use back-propagation 22,28 to compute the expectation values of the quantities that do not commute witĥ H. After the CPMC calculation, we solve the IP Hamiltonian in Eq. (2), using the densities obtained from the preceding QMC calculation as the input mean field, i.e., n iσ QMC → n iσ .…”

Auxiliary-field quantum Monte Carlo calculations of molecular systems with a Gaussian basis

“…[9], Eqs. (5) and (7) would both be exact and give the same results under free projection, since the mean-field shift is automatically applied via the importance sampling transformation regardless of which form ofĤ ′ 2 is used [25]. The different behaviors discussed above arise only because of the imposition of the phase projection to onedimension [9], in which the substraction of the mean-field background helps to reduce the "rotation" of the random walkers in the complex Ψ T |φ -plane and thus the severity of the projection.…”

Excited state calculations using phaseless auxiliary-field quantum Monte Carlo: Potential energy curves of low-lying C2 singlet states

“…We normalize |Ψ MC by explicitly evaluating Ψ MC |Ψ MC at each β. Because this involves "undoing" the importance sampling 31 [division by the factor Φ r |φ on the right-hand side of Eq. (5)], there are large statistical fluctuations, as can be seen in the upper panels.…”