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arXiv:1009.2761 · Physical Review A

Thermodynamics of the three-dimensional Hubbard model: Implications for cooling cold atomic gases in optical lattices

Lorenzo De Leo, Jean-Sébastien Bernier, Corinna Kollath, Antoine Georges, V. W. Scarola

Published Feb 10, 2011• 41 citations
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Abstract

We present a comprehensive study of the thermodynamic properties of the three-dimensional fermionic Hubbard model, with application to cold fermionic atoms subject to an optical lattice and a trapping potential. Our study is focused on the temperature range of current experimental interest. We employ two theoretical methods---dynamical mean-field theory and high-temperature series---and perform comparative benchmarks to delimit their respective range of validity. Special attention is devoted to understand the implications that thermodynamic properties of this system have on cooling. Considering the distribution function of local occupancies in the inhomogeneous lattice, we show that, under adiabatic evolution, the variation of any observable (e.g., temperature) can be conveniently disentangled into two distinct contributions. The first contribution is due to the redistribution of atoms in the trap during the evolution, while the second one comes from the intrinsic change of the observable. Finally, we provide a simplified picture of a recently proposed cooling procedure, based on spatial entropy separation, by applying this method to an idealized model.

DOI
10.1103/physreva.83.023606

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We present a comprehensive study of the thermodynamic properties of the three-dimensional fermionic Hubbard model, with application to cold fermionic atoms subject to an optical lattice and a trapping potential. Our study is focused on the temperature range of current experimental interest. We employ two theoretical methods---dynamical mean-field theory and high-temperature series---and perform comparative benchmarks to delimit their respective range of validity. Special attention is devoted to understand the implications that thermodynamic properties of this system have on cooling. Considering the distribution function of local occupancies in the inhomogeneous lattice, we show that, under adiabatic evolution, the variation of any observable (e.g., temperature) can be conveniently disentangled into two distinct contributions. The first contribution is due to the redistribution of atoms in the trap during the evolution, while the second one comes from the intrinsic change of the observable. Finally, we provide a simplified picture of a recently proposed cooling procedure, based on spatial entropy separation, by applying this method to an idealized model.
We calculate experimentally relevant properties of trapped fermionic alkaline-earth-metal atoms in an optical lattice, modeled by the SU(N)(N)(N) Hubbard model. We employ a high-temperature expansion that is accurate when the temperature is larger than the tunneling rate, similar to current regimes in ultracold atom experiments. In addition to exploring the Mott insulator-metal crossover, we calculate final temperatures achieved by the standard experimental protocol of adiabatically ramping from a noninteracting gas, as a function of initial gas temperature. Of particular experimental interest, we find that increasing NNN for fixed particle numbers and initial temperatures gives substantially colder Mott insulators after the adiabatic ramping, up to more than a factor of 5 for relevant parameters. This cooling happens for all NNN, fixing the initial entropy, or for all $N\ensuremath{\lesssim}20$ (the exact value depends on dimensionality), at fixed, experimentally relevant initial temperatures.
We study finite-temperature properties of strongly correlated fermions in two-dimensional optical lattices by means of numerical linked cluster expansions, a computational technique that allows one to obtain exact results in the thermodynamic limit. We focus our analysis on the strongly interacting regime, where the on-site repulsion is of the order of or greater than the band width. We compute the equation of state, double occupancy, entropy, uniform susceptibility, and spin correlations for temperatures that are similar to or below the ones achieved in current optical lattice experiments. We provide a quantitative analysis of adiabatic cooling of trapped fermions in two dimensions, by means of both flattening the trapping potential and increasing the interaction strength.
We study the phase diagram of the asymmetric Hubbard model (AHM), which is characterized by different values of the hopping for the two spin projections of a fermion or, equivalently, two different orbitals. This model is expected to provide a good description of a mass-imbalanced cold fermionic mixture in a 3D optical lattice. We use the dynamical mean-field theory to study various physical properties of this system. In particular, we show how orbital-selective physics, observed in multiorbital strongly correlated electron systems, can be realized in such a simple model. We find that the density distribution is a good probe of this orbital-selective crossover from a Fermi-liquid to a non-Fermi-liquid state. Below an ordering temperature To{T}_{o}To​, which is a function of both the interaction and hopping asymmetry, the system exhibits staggered long-range orbital order. Apart from the special case of the symmetric limit, i.e., Hubbard model, where there is no hopping asymmetry, this orbital order is accompanied by a true charge density wave order for all values of the hopping asymmetry. We calculate the order parameters and various physical quantities including the thermodynamics in both the ordered and disordered phases. We find that the formation of the charge density wave is signaled by an abrupt increase in the sublattice double occupancies. Finally, we propose a new method, entropic chromatography, for cooling fermionic atoms in optical lattices, by exploiting the properties of the AHM. To establish this cooling strategy on a firmer basis, we also discuss the variations in temperature induced by the adiabatic tuning of interactions and hopping parameters.
We calculate several thermodynamic quantities for repulsively interacting one-dimensional fermions. We solve the Hubbard model at both zero and finite temperatures using the Bethe-ansatz method. For arbitrary values of the chemical potential, we calculate the particle number density, the double occupancy, various compressibilities, and the entropy as a function of temperature and interaction. We find that these thermodynamic quantities show a characteristic behavior so that measurements of these quantities can be used as a detection of temperature, the metal-insulator transition, and metallic and insulating phases in the trap environment. Further, we discuss an experimental scheme to extract these thermodynamic quantities from the column density profiles. The entropy and the compressibility of the entire trapped atomic cloud also reveal characteristic features indicating whether insulating and/or metallic phases coexist in the trap.
We study the finite-temperature properties of two-component fermionic atoms trapped in a two-dimensional (2D) optical lattice. We apply the self-energy functional approach to the 2D Hubbard model with a harmonic-trapping potential, and systematically investigate the thermodynamic properties of this system. We find that entropy and grand potential provide evidence of a crossover between the Mott-insulating and the metallic phases at certain temperatures. In addition, we find that entropy exhibits a cusplike anomaly at lower temperatures, suggesting a second- or higher-order antiferromagnetic transition. We estimate the antiferromagnetic transition temperatures, and clarify how the trapping potential affects this magnetic transition.
We investigate effects of the adiabatic ramp-up of optical lattices to the temperatures of Bose gases. Applying the mean-field approximation to the three-dimensional Bose-Hubbard model at finite temperatures, we calculate the isentropic curves in the phase diagram against the temperatures and the lattice height. The adiabatic cooling or heating through superfluid-normal phase transition is clearly understood. It is found that the cooling occurs in superfluid phase while the heating occurs in normal phase and the efficiency of adiabatic cooling or heating is higher at higher temperatures. This behavior can be understood from the lattice-height dependence of dispersion relation in each phase. Furthermore, the connection of the adiabatic cooling or heating between the cases with and without the trap potential is addressed.