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arXiv:1006.1174 · Physical Review Letters

Signature of Quantum Criticality in the Density Profiles of Cold Atom Systems

Qi Zhou, Tin-Lun Ho

Published Dec 10, 2010• 58 citations
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Abstract

In recent years, there has been considerable experimental effort using cold atoms to study strongly correlated many-body systems. One class of phenomena of particular interest is quantum critical (QC) phenomena. While prevalent in many materials, these phenomena pose notoriously difficult theoretical problems due to the vanishing of energy scales in the QC region. So far, there are no systematic ways to deduce the QC behavior of bulk systems from the data of trapped atomic gases. Here, we present a simple algorithm to use the experimental density profile to determine the T=0 phase boundary of bulk systems, as well as the scaling functions in the QC regime. We also present another scheme for removing finite-size effects of the trap. We demonstrate the validity of our schemes using exactly soluble models.

DOI
10.1103/physrevlett.105.245702

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In recent years, there has been considerable experimental effort using cold atoms to study strongly correlated many-body systems. One class of phenomena of particular interest is quantum critical (QC) phenomena. While prevalent in many materials, these phenomena pose notoriously difficult theoretical problems due to the vanishing of energy scales in the QC region. So far, there are no systematic ways to deduce the QC behavior of bulk systems from the data of trapped atomic gases. Here, we present a simple algorithm to use the experimental density profile to determine the T=0 phase boundary of bulk systems, as well as the scaling functions in the QC regime. We also present another scheme for removing finite-size effects of the trap. We demonstrate the validity of our schemes using exactly soluble models.
At present, many laboratories are performing experiments to simulate theoretical models of strongly correlated systems using cold atoms in optical lattices, a program referred to as "Quantum Simulation". It is hoped that these experiments will shed light on some long standing problems in condensed matter physics. The goal of Quantum Simulation is to obtain information of homogenous bulk systems. However, experiments are performed in confining traps. The non-uniformity of the trapping potential inevitably introduces different phases in the sample, making it difficult to deduce the properties of a single bulk phase. So far, there are no algorithms to use the experimental data to map out phase diagrams and important thermodynamic quantities. Here, we present an algorithm to achieve this goal. Apart from phase diagram, it also maps out entropy density, superfluid density of superfluids, and staggered magnetizations of anti-ferromagnets. Our scheme is exact within local density approximation.
We obtain an analytical equation of state for one-dimensional strongly attractive Fermi gases for all parameter regimes in current experiments. From the equation of state, we derive universal scaling functions that control whole thermodynamical properties in quantum critical regimes and illustrate the physical origin of quantum criticality. It turns out that the critical properties of the system are described by those of free fermions and those of mixtures of fermions with masses mmm and 2m2m2m. We also show how these critical properties of bulk systems can be revealed from the density profile of trapped Fermi gases at finite temperatures and can be used to determine the T=0T=0T=0 phase boundaries without any arbitrariness.
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.
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.