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arXiv:cond-mat/0508319 · Physical Review Letters

Loop Updates for Quantum Monte Carlo Simulations in the Canonical Ensemble

S. Rombouts, Kris Van Houcke, Lode Pollet

Published May 10, 2006• 49 citations
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Abstract

We present a new nonlocal updating scheme for quantum Monte Carlo simulations, which conserves particle number and other symmetries. It allows exact symmetry projection and direct evaluation of the equal-time Green's function and other observables in the canonical ensemble. The method is applicable to a wide variety of systems. We show results for bosonic atoms in optical lattices, neutron pairs in atomic nuclei, and electron pairs in ultrasmall superconducting grains.

DOI
10.1103/physrevlett.96.180603

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We present a new nonlocal updating scheme for quantum Monte Carlo simulations, which conserves particle number and other symmetries. It allows exact symmetry projection and direct evaluation of the equal-time Green's function and other observables in the canonical ensemble. The method is applicable to a wide variety of systems. We show results for bosonic atoms in optical lattices, neutron pairs in atomic nuclei, and electron pairs in ultrasmall superconducting grains.
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We show how the worldline quantum Monte Carlo procedure, which usually relies on an artificial time discretization, can be formulated directly in continuous time, rendering the scheme exact. For an arbitrary system with discrete Hilbert space, none of the configuration update procedures contain small parameters. We find that the most effective update strategy involves the motion of worldline discontinuities (both in space and time), i.e., the evaluation of the Green’s function. Being based on local updates only, our method nevertheless allows one to work with the grand canonical ensemble and nonzero winding numbers, and to calculate any dynamical correlation function as easily as expectation values of, e.g., total energy. The principles found for the update in continuous time generalize to any continuous variables in the space of discrete virtual transitions, and in principle also make it possible to simulate continuous systems exactly.
We introduce a quantum Monte Carlo method at finite temperature for interacting fermionic models in the canonical ensemble, where the conservation of the particle number is enforced. Although general thermodynamic arguments ensure the equivalence of the canonical and the grand-canonical ensembles in the thermodynamic limit, their approach to the infinite-volume limit is distinctively different. Observables computed in the canonical ensemble generically display a finite-size correction proportional to the inverse volume, whereas in the grand-canonical ensemble the approach is exponential in the ratio of the linear size over the correlation length. We verify these predictions by quantum Monte Carlo simulations of the Hubbard model in one and two dimensions in the grand-canonical and the canonical ensemble. We prove an exact formula for the finite-size part of the free energy density, energy density and other observables in the canonical ensemble and relate this correction to a susceptibility computed in the corresponding grand-canonical ensemble. This result is confirmed by an exact computation of the one-dimensional classical Ising model in the canonical ensemble, which for classical models corresponds to the so-called fixed-magnetization ensemble. Our method is useful for simulating finite systems which are not coupled to a particle bath, such as in nuclear or cold atom physics.
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