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arXiv:1011.0032 · Physical Review A

High-temperature properties of fermionic alkaline-earth-metal atoms in optical lattices

Kaden R. A. Hazzard, Victor Gurarie, Michael Hermele, Ana María Rey

Published Apr 26, 2012• 51 citations
arXivHTMLPDF

Abstract

We calculate experimentally relevant properties of trapped fermionic alkaline-earth-metal atoms in an optical lattice, modeled by the SU$(N)$ 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 $N$ 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 $N$, fixing the initial entropy, or for all $N\ensuremath{\lesssim}20$ (the exact value depends on dimensionality), at fixed, experimentally relevant initial temperatures.

DOI
10.1103/physreva.85.041604

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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≲20N\ensuremath{\lesssim}20N\ensuremath≲20 (the exact value depends on dimensionality), at fixed, experimentally relevant initial temperatures.
Ultracold fermionic alkaline earth atoms confined in optical lattices realize Hubbard models with internal SU(N) symmetries, where N can be as large as ten. Such systems are expected to harbor exotic magnetic physics at temperatures below the superexchange energy scale. Employing quantum Monte Carlo simulations to access the low-temperature regime of one-dimensional chains, we show that after adiabatically loading a weakly interacting gas into the strongly interacting regime of an optical lattice, the final temperature decreases with increasing N. Furthermore, we estimate the temperature scale required to probe correlations associated with low-temperature SU(N) magnetism. Our findings are encouraging for the exploration of exotic large-N magnetic states in ongoing experiments.
The Fermi-Hubbard model describes ultracold fermions in an optical lattice and exhibits antiferromagnetic long-ranged order below the N\'eel temperature. However, reaching this temperature in the laboratory has remained an elusive goal. In other atomic systems, such as trapped ions, low temperatures have been successfully obtained by adiabatic demagnetization, in which a strong effective magnetic field is applied to a spin-polarized system and the magnetic field is adiabatically reduced to zero. Unfortunately, applying this approach to the Fermi-Hubbard model encounters a fundamental obstacle: the SU(2)\text{SU}(2)SU(2) symmetry introduces many level crossings that prevent the system from reaching the ground state, even in principle. However, by breaking the SU(2)\text{SU}(2)SU(2) symmetry with a spin-dependent tunneling, we show that adiabatic demagnetization can achieve low-temperature states. Using density matrix renormalization group (DMRG) calculations in one dimension, we numerically find that demagnetization protocols successfully reach low-temperature states of a spin-anisotropic Hubbard model, and we discuss how to optimize this protocol for experimental viability. By subsequently ramping spin-dependent tunnelings to spin-independent tunnelings, we expect that our protocol can be employed to produce low-temperature states of the Fermi-Hubbard model.
We calculate the collective modes of ultracold trapped alkaline-earth-metal fermionic atoms, which possess an SU(N\mathrm{SU}(NSU(N) symmetry of the nuclear spin degree of freedom and a controllable NNN, with NNN as large as 10. We calculate the breathing and quadrupole modes of two-dimensional and three-dimensional harmonically trapped gases in the normal phase. We particularly concentrate on two-dimensional gases, where the shift is more accessible experimentally, and the physics has special features. We present results as a function of temperature, interaction strength, density, and NNN. We include calculations across the collisionless to hydrodynamic crossover. We assume the gas is interacting weakly, such that it can be described by a Boltzmann-Vlasov equation that includes both mean-field terms and the collision integral. We solve this with an approximate scaling ansatz, taking care in two dimensions to preserve the scaling symmetry of the system. We predict the collective-mode frequency shifts and damping, showing that these are measurable in experimentally relevant regimes. We expect these results to furnish powerful tools to characterize interactions and the state of alkaline-earth-metal gases, as well as to lay the foundation for future work, for example, on strongly interacting gases and SU(N)\mathrm{SU}(N)SU(N) spin modes.
We investigate one-dimensional SU(N)(N)(N) Hubbard chains at zero temperature, which can be emulated with ultracold alkaline-earth-metal atoms, by using the density matrix renormalization group (DMRG), Bethe ansatz (BA), and bosonization. We compute experimental observables and use the DMRG to benchmark the accuracy of the Bethe ansatz for N>2N>2N>2 where the BA is only approximate. In the worst case, we find a relative error $\ensuremath{\varepsilon}\ensuremath{\lesssim}4%$ in the BA ground-state energy for $N\ensuremath{\leqslant}4$ at filling 1/N1/N1/N, which is due to the fact that BA improperly treats the triply and higher occupied states. Using the DMRG for $N\ensuremath{\leqslant}4$ and the BA for large NNN, we determine the regimes of validity of strong- and weak-coupling perturbation theory for all values of NNN and, in particular, the parameter range in which the system is well described by a SU(N)(N)(N) Heisenberg model at filling 1/N1/N1/N. We find this depends only weakly on NNN. We investigate the Berezinskii-Kosterlitz-Thouless phase transition from a Luttinger liquid to a Mott insulator by computing the fidelity susceptibility and the Luttinger parameter ${K}_{\ensuremath{\rho}}$ at 1/N1/N1/N filling. The numerical findings give strong evidence that the fidelity susceptibility develops a minimum at a critical interaction strength which is found to occur at a finite positive value for N>2N>2N>2.
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 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.