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arXiv:0711.3425 · Physical Review A

Achieving the Néel state in an optical lattice

Arnaud Koetsier, R. A. Duine, Immanuel Bloch, Henk T. C. Stoof

Published Feb 25, 2008• 45 citations
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Abstract

We theoretically study the possibility of reaching the antiferromagnetic phase of the Hubbard model by starting from a normal gas of trapped fermionic atoms and adiabatically ramping up an optical lattice. Requirements on the initial temperature and the number of atoms are determined for a three-dimensional square lattice by evaluating the N\'eel state entropy, taking into account fluctuations around the mean-field solution. We find that these fluctuations place important limitations on adiabatically reaching the N\'eel state.

DOI
10.1103/physreva.77.023623

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We theoretically study the possibility of reaching the antiferromagnetic phase of the Hubbard model by starting from a normal gas of trapped fermionic atoms and adiabatically ramping up an optical lattice. Requirements on the initial temperature and the number of atoms are determined for a three-dimensional square lattice by evaluating the N'eel state entropy, taking into account fluctuations around the mean-field solution. We find that these fluctuations place important limitations on adiabatically reaching the N'eel state.
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.
In the experimental context of cold-fermion optical lattices, we discuss the possibilities to approach the pseudogap or ordered phases by manipulating the scattering length or the strength of the laser-induced lattice potential. Using the two-particle self-consistent approach, as well as quantum Monte Carlo simulations, we provide isentropic curves for the two- and three-dimensional Hubbard models at half-filling. These quantitative results are important for practical attempts to reach the ordered antiferromagnetic phase in experiments on optical lattices of two-component fermions. We find that adiabatically turning on the interaction in two dimensions to cool the system is not very effective. In three dimensions, adiabatic cooling to the antiferromagnetic phase can be achieved in such a manner, although the cooling efficiency is not as high as initially suggested by dynamical mean-field theory. Adiabatic cooling by turning off the repulsion beginning at strong coupling is possible in certain cases.
One of the major challenges in realizing antiferromagnetic and superfluid phases in optical lattices is the ability to cool fermions. We determine constraints on the entropy for observing these phases in two-dimensional Hubbard models using determinantal quantum Monte Carlo simulations. We find that an entropy per particle approximately = ln2 is sufficient to observe the insulating gap in the repulsive Hubbard model at half-filling, or the pairing pseudogap in the attractive case. Observing antiferromagnetic correlations or superfluidity in 2D systems requires a further reduction in entropy by a factor of 3 or more. In contrast with higher dimensions, we find that adiabatic cooling is not useful to achieve the required low temperatures. We also show that double-occupancy measurements are useful for thermometry for temperatures greater than the nearest-neighbor hopping energy.
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 perform a quantitative simulation of the repulsive Fermi-Hubbard model using an ultracold gas trapped in an optical lattice. The entropy of the system is determined by comparing accurate measurements of the equilibrium double occupancy with theoretical calculations over a wide range of parameters. We demonstrate the applicability of both high-temperature series and dynamical mean-field theory to obtain quantitative agreement with the experimental data. The reliability of the entropy determination is confirmed by a comprehensive analysis of all systematic errors. In the center of the Mott insulating cloud we obtain an entropy per atom as low as 0.77k(B) which is about twice as large as the entropy at the Néel transition. The corresponding temperature depends on the atom number and for small fillings reaches values on the order of the tunneling energy.
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.