Interacting bosons in one dimension and the applicability of Luttinger-liquid theory as revealed by path-integral quantum Monte Carlo calculations

Abstract

Harmonically trapped ultracold atoms and 4 He in nanopores provide different experimental realizations of bosons in one dimension, motivating the search for a more complete theoretical understanding of their lowenergy properties. Worm algorithm path-integral quantum Monte Carlo results for interacting bosons restricted to the one dimensional continuum are compared to the finite temperature and system size predictions of Luttinger-liquid theory. For large system sizes at low temperature, excellent agreement is obtained after including the leading irrelevant interactions in the Hamiltonian which are determined explicitly.Luttinger-liquid ͑LL͒ theory 1 provides a universal description of interacting fermions or bosons at sufficiently low energies in one dimension ͑1D͒. Recently, exciting possibilities for experimental realizations of Luttinger liquids have appeared, involving ultracold atoms in cigar-shaped traps, 2 screw dislocations in solid 4 He ͑Ref. 3͒, and helium-4 confined to flow in nanopores. 4 While the latter experiment examines a system that is highly out of equilibrium, future iterations 5 could be well described by a translationally invariant model of interacting bosons. There have been numerous numerical studies of 1D fermion models on lattices using exact diagonalization, Monte Carlo, and density matrix renormalization-group methods, but numerical results on free space interacting bosons at nonzero temperature, T, are much rarer. Exact studies in the continuum may provide insights, specifically on issues of dimensional crossover in nanopores. 6 Zero-temperature variational Monte Carlo calculations for the 1D case were reported in Ref. 7 and finite T worm algorithm path-integral Monte Carlo ͑WA-PIMC͒ simulations for a screw dislocation 3 have claimed the observation of LL behavior. In order to systematically explore the regime of energies and pore lengths, where LL behavior may occur we have performed WA-PIMC simulations on the N-particle Hamiltonianin 1D with periodic boundary conditions on an interval of length L in angstroms and we will work in units where ប = k B = 1. The WA-PIMC method, recently introduced by Boninsegni et al. 8 extends the original PIMC algorithm of Ceperley 9 to include configurations of the single-particle Matsubara Green's function, allowing for intermediate particle trajectories which are not periodic in imaginary time.The inclusion of such trajectories yield an efficient and robust grand canonical quantum Monte Carlo ͑QMC͒ technique that accurately incorporate complete quantum statistics and provides exact and unbiased estimations of many physical observables at finite temperature. In the WA-PIMC simulations performed here, the short-range repulsive interaction V͑r͒ = ͑g / ͱ a͒e −r 2 /4a 2 is chosen for convenience to be Gaussian with integrated strength 2g and spatial extent a. The numerical values of all microscopic parameters were optimized to ensure an experimentally relevant and efficient simulation at low energies, where the temperature is much smaller than both the k...

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