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arXiv:1401.3514 · Physical Review A

Universal quantum behavior of interacting fermions in one-dimensional traps: From few particles to the trap thermodynamic limit

Adriano Angelone, Massimo Campostrini B, Ettore Vicari

Published Feb 28, 2014• 25 citations
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Abstract

We investigate the ground-state properties of trapped fermion systems described by the Hubbard model with an external confining potential. We discuss the universal behaviors of systems in different regimes: from few particles, i.e., in dilute regimes, to the trap thermodynamic limit. The asymptotic trap-size (TS) dependence in the dilute regime (increasing the trap size $\ensuremath{\ell}$ keeping the particle number $N$ fixed) is described by a universal TS scaling controlled by the dilute fixed point associated with the metal-to-vacuum quantum transition. This scaling behavior is numerically checked by DMRG simulations of the one-dimensional (1D) Hubbard model. In particular, the particle density and its correlations show crossovers among different regimes: for strongly repulsive interactions they approach those of a spinless Fermi gas, for weak interactions those of a free Fermi gas, and for strongly attractive interactions they match those of a gas of hard-core bosonic molecules. The large-$N$ limit keeping the ratio $N/\ensuremath{\ell}$ fixed corresponds to a 1D trap thermodynamic limit. We address issues related to the accuracy of the local density approximation (LDA). We show that the particle density approaches its LDA in the large-$\ensuremath{\ell}$ limit. When the trapped system is in the metallic phase, corrections at finite $\ensuremath{\ell}$ are $O({\ensuremath{\ell}}^{\ensuremath{-}1})$ and oscillating around the center of the trap. They become significantly larger at the boundary of the fermion cloud, where they get suppressed as $O({\ensuremath{\ell}}^{\ensuremath{-}1/3})$ only. This anomalous behavior arises from the nontrivial scaling at the metal-to-vacuum transition occurring at the boundaries of the fermion cloud.

DOI
10.1103/physreva.89.023635

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We study some aspects of equilibrium and off-equilibrium quantum dynamics of dilute bosonic gases in the presence of a trapping potential. We consider systems with a fixed number of particles and study their scaling behavior with increasing the trap size. We focus on one-dimensional bosonic systems, such as gases described by the Lieb-Liniger model and its Tonks-Girardeau limit of impenetrable bosons, and gases constrained in optical lattices as described by the Bose-Hubbard model. We study their quantum (zero-temperature) behavior at equilibrium and off equilibrium during the unitary time evolution arising from changes of the trapping potential, which may be instantaneous or described by a power-law time dependence, starting from the equilibrium ground state for an initial trap size. Renormalization-group scaling arguments and analytical and numerical calculations show that the trap-size dependence of the equilibrium and off-equilibrium dynamics can be cast in the form of a trap-size scaling in the low-density regime, characterized by universal power laws of the trap size, in dilute gases with repulsive contact interactions and lattice systems described by the Bose-Hubbard model. The scaling functions corresponding to several physically interesting observables are computed. Our results are of experimental relevance for systems of cold atomic gases trapped by tunable confining potentials.
We discuss how to perform the continuum limit of the ddd-dimensional Hubbard model in the presence of a harmonic confining potential at zero temperature and fixed particle number. While for $d\ensuremath{\ge}3$ the system can be mapped into a noninteracting two-component Fermi gas in a harmonic trap, for d=1d=1d=1 and 2 the two-body interaction is described by a Dirac $\ensuremath{\delta}$. We show that the properties of this continuum limit can be put in one-to-one correspondence with those obtained by applying the trap-size scaling formalism to the confined Hubbard model in the so called dilute regime (fixed number of particles and weak confinement). The correspondence for d<3d<3d<3 has been tested, in the case of two particles with opposite spin, by comparing numerical exact-diagonalization results of the lattice system with those obtained in the continuum limit.
We investigate the expansion of bosons and fermions in a homogeneous lattice after a sudden removal of the trapping potential using exact numerical methods. As a main result, we show that in one dimension, both bosonic and fermionic Mott insulators expand with the same velocity, irrespective of the interaction strength, provided the expansion starts from the ground state of the trapped gas. Furthermore, their density profiles become identical during the expansion; the asymptotic density dynamics is identical to that of initially localized, noninteracting particles, and the asymptotic velocity distribution is flat. The expansion velocity for initial correlated Mott insulating states is therefore independent of the interaction strength and particle statistics. Interestingly, this nonequilibrium dynamics is sensitive to the interaction driven quantum phase transition in the Bose-Hubbard model; while being constant in the Mott phase, the expansion velocity decreases in the superfluid phase and vanishes for large systems in the noninteracting limit. These results are compared to the setup of a recent experiment [Ronzheimer et al., Phys. Rev. Lett. 110, 205301 (2013)], where the trap opening was combined with an interaction quench from infinitely strong interactions to finite values. In the latter case, the interaction quench breaks the universal dynamics in the asymptotic regime and the expansion depends on the interaction strength. We carry out an analogous analysis for a two-component Fermi gas, with similar observations. In addition, we study the effect of breaking the integrability of hard-core bosons in different ways; while the fast ballistic expansion from the ground state of Mott insulators in one dimension remains unchanged for finite interactions, we observe strong deviations from this behavior on a two-leg ladder even in the hard-core case. This change in dynamics bares similarities with the dynamics in the dimensional crossover from one to two dimensions observed in the aformentioned experimental study.
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