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

Abstract

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.

10 pages, 7 figs; v2: 11 pages, 9 figs, new results on the doped Hubbard model; v3, 12 pages, 9 figs, generalized formula for the finite-size correction in the canonical ensemble, published version

References in corpus (1)

• Fermionic quantum criticality in honeycomb and $\pi$-flux Hubbard models: Finite-size scaling of renormalization-group-invariant observables from quantum Monte Carlo

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• A Stable, Recursive Auxiliary Field Quantum Monte Carlo Algorithm in the Canonical Ensemble: Applications to Thermometry and the Hubbard Model
• The ALF (Algorithms for Lattice Fermions) project release 2.4. Documentation for the auxiliary-field quantum Monte Carlo code
• Finite-Size Scaling at fixed Renormalization-Group invariant
• Phase diagram of the SU($N$) antiferromagnet of spin $S$ on a square lattice
• Precision thermodynamics of the strongly interacting Fermi gas in two dimensions
• Constraint correlation functions of the one-dimensional Ising model in the scaling limit