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arXiv:1211.6224 · Physical Review B

Critical parameters from trap-size scaling in systems of trapped particles

Giacomo Ceccarelli, C. Torrero, Ettore Vicari

Published Jan 24, 2013• 18 citations
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Abstract

We investigate the critical behavior of trapped particle systems at the low-temperature superfluid transition. In particular, we consider the three-dimensional Bose-Hubbard model in the presence of a trapping harmonic potential coupled with the particle density, which is a realistic model of cold bosonic atoms in optical lattices. We present a numerical study based on quantum Monte Carlo simulations, analyzed in the framework of the trap-size scaling (TSS). We show how the critical parameters can be derived from the trap-size dependencies of appropriate observables, matching them with TSS. This provides a systematic scheme which is supposed to exactly converge to the critical parameters of the transition in the large-trap-size limit. Our numerical analysis may provide a guide for experimental investigations of trapped systems at finite-temperature and quantum transitions, showing how critical parameters may be determined by looking at the scaling of the critical modes with respect to the trap size, i.e., by matching the trap-size dependence of the experimental data with the expected TSS ansatz.

DOI
10.1103/physrevb.87.024513

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We investigate the critical properties of cold bosonic gases in three dimensions, confined by an external quadratic potential coupled to the particle density, and realistically described by the Bose-Hubbard (BH) model. The trapping potential is often included in experiments with cold atoms and modifies the critical finite-size scaling of the homogeneous system in a nontrivial way. The trap-size scaling (TSS) theory accounts for this effect through the exponent $\ensuremath{\theta}$. We perform extensive simulations of the BH model at the critical temperature, in the presence of harmonic traps. We find that the TSS predictions are universal once we account for the effective way in which the trap locally modifies the chemical potential $\ensuremath{\mu}$ of the system. The trap exponent for the BH model at $\ensuremath{\mu}=0$ is the one corresponding to an effective quartic potential. At positive $\ensuremath{\mu}$, evidence suggests that TSS breaks down sufficiently far from the center of the trap, as the system encounters an effective phase boundary.
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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.