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a r X i v :c o n d -m a t /0703352v 1 [c o n d -m a t . s t r -e l ] 13 M a r 2007

Interaction-Induced Adiabatic Cooling for Antiferromagnetism in Optical Lattices

A.-M. Dar´e , L. Raymond, G. Albinet

L2MP, Bˆa timent IRPHE, 49rue Joliot Curie BP 146,

Universit´e de Provence, 13384Marseille, Cedex 13, France

A.-M. S. Tremblay

D´e partement de physique and Regroupement qu´e b´e cois sur les mat´e riaux de pointe, Universit´e de Sherbrooke, Sherbrooke, Qu´e bec, J1K 2R1, Canada

(Dated:February 6, 2008)

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 ﬁndthat 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 eﬃciencyis not as high as initially suggested by Dynamical Mean-Field Theory. Adiabatic cooling by turning oﬀthe repulsion beginning at strong coupling is possible in certain cases.

PACS numbers:71.10.Fd, 03.75.Lm, 32.80.Pj, 71.30.+h

I. INTRODUCTION

One of the most exciting possibilities opened by re-search on cold atoms in optical traps is to study in a controlled manner model Hamiltonians of interest to con-densed matter physics. For example, high on the list of questions that can in principle be answered by these model systems is whether high-temperature supercon-ductivity can be explained by the two-dimensional Hub-bard model away from half-ﬁlling.[1,2]As a ﬁrststep towards achieving this goal, the antiferromagnetic phase expected at half-ﬁllingoﬀersan easier target state that occurs at higher temperature in the phase diagram of cuprate superconductors. [3,4]

In optical lattices, the two spin species occurring in the Hubbard model are mimicked by atoms in two diﬀer-ent hyperﬁnestates. The cooling of these atomic gases necessary to observe ordered states has been discussed before. [2,5]It has been recently pointed out however that there is an additional mechanism, [6]akin to Pomer-anchuck cooling in liquid Helium 3, that is available to help in achieving the temperatures where antiferromag-netism can be observed. In this mechanism, the temper-ature can be lowered by turning on at constant entropy what amounts to interactions in the Hubbard model (seeRef.(7)for a review). The original calculations for this eﬀectwere done for the Hubbard model using Dynami-cal Mean-Field Theory (DMFT).While it is expected that this approach will give qualitatively correct results, accurate predictions are necessary to achieve the practi-cal implementation of this cooling scheme. In the present paper, we present such quantitative predictions for the isentropic curves of both the two-and three-dimensional Hubbard models. We display the results in the usual

units for the Hubbard model, but also in the conventional units used in the context of cold atom physics.

Solving the problem for both two-and three-dimensional lattices fulﬁllsseveral purposes. First, the two-dimensional case is interesting in its own right even if long-range order cannot be achieved at ﬁnitetem-perature in strictly two dimensions (becauseof the Mermin-Wagner-Hohenberg theorem). Indeed, high-temperature parent antiferromagnetic compounds have a strong two-dimensional character. In addition, even though long-range order cannot be achieved, there is a two-dimensional regime with very strong antiferromag-netic ﬂuctuationsthat is interesting in itself. In this regime, a pseudogap appears in the single-particle spec-tral weight that is caused at weak coupling by antifer-romagnetic ﬂuctuationsthat have a correlation length larger than the single-particle de Broglie wavelength. 9]At strong coupling, the pseudogap appears well be-fore the long antiferromagnetic correlation lengths occur. [10]Also, considering both two-and three dimensions sheds additional light on the mechanism for cooling. The Hubbard model is deﬁnedin second quantization by,

H =−

i,j,σ

t ij c †

iσc jσ+U i

n i ↑n i ↓(1)where c †iσ(c iσ) are creation and annihilation operators

for electrons of spin σ, n iσ=c †iσc iσis the density of spinfor electrons of spin σ, n iσ=c †iσc iσis the density of spin

σelectrons, t ij =t ∗

ji is the hopping amplitude, and U is the on-site repulsion obtained from matrix elements of the contact interaction between atoms in the basis of Wannier states of the optical lattice.[1,7]We restrict ourselves to the case where only nearest-neighbor hop-ping t coming from tunneling between potential minima

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