Finite-size effects in canonical and grand-canonical quantum Monte Carlo simulations for fermions

摘要

We introduce a quantum Monte Carlo method at finite temperature forinteracting fermionic models in the canonical ensemble, where the conservationof the particle number is enforced. Although general thermodynamic argumentsensure the equivalence of the canonical and the grand-canonical ensembles inthe thermodynamic limit, their approach to the infinite-volume limit isdistinctively different. Observables computed in the canonical ensemblegenerically display a finite-size correction proportional to the inversevolume, whereas in the grand-canonical ensemble the approach is exponential inthe ratio of the linear size over the correlation length. We verify thesepredictions by quantum Monte Carlo simulations of the Hubbard model in one andtwo dimensions in the grand-canonical and the canonical ensemble. We prove anexact formula for the finite-size part of the free energy density, energydensity and other observables in the canonical ensemble and relate thiscorrection to a susceptibility computed in the corresponding grand-canonicalensemble. This result is confirmed by an exact computation of theone-dimensional classical Ising model in the canonical ensemble, which forclassical models corresponds to the so-called fixed-magnetization ensemble. Ourmethod is useful for simulating finite systems which are not coupled to aparticle bath, such as in nuclear or cold atom physics.