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arXiv:0812.4422 · Physical Review A

Quantum simulation of the Hubbard model: The attractive route

A. F. Ho, Miguel A. Cazalilla, Thierry Giamarchi

Published Mar 19, 2009• 78 citations
arXivHTMLPDF

Abstract

We study the conditions under which, using a canonical transformation, the phases sought after for the repulsive Hubbard model, namely, a Mott insulator in the paramagnetic and antiferromagnetic phases, and a putative $d$-wave superfluid can be deduced from observations in an optical lattice loaded with a spin-imbalanced ultracold Fermi gas with attractive interactions, thus realizing the attractive Hubbard model. We argue that the Mott insulator and antiferromagnetic phase of the repulsive Hubbard model are easier to observe in the attractive Hubbard mode as a band insulator of Cooper pairs and superfluid phase, respectively. The putative $d$-wave superfluid phase of the repulsive Hubbard model doped away from half filling is related to a $d$-wave antiferromagnetic phase for the attractive Hubbard model. We discuss the advantages of this approach to ``quantum simulate'' the Hubbard model in an optical lattice over the simulation of the doped Hubbard model in the repulsive regime. We also point out a number of technical difficulties of the proposed approach and, in some cases, suggest possible solutions.

DOI
10.1103/physreva.79.033620

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We study the conditions under which, using a canonical transformation, the phases sought after for the repulsive Hubbard model, namely, a Mott insulator in the paramagnetic and antiferromagnetic phases, and a putative ddd-wave superfluid can be deduced from observations in an optical lattice loaded with a spin-imbalanced ultracold Fermi gas with attractive interactions, thus realizing the attractive Hubbard model. We argue that the Mott insulator and antiferromagnetic phase of the repulsive Hubbard model are easier to observe in the attractive Hubbard mode as a band insulator of Cooper pairs and superfluid phase, respectively. The putative ddd-wave superfluid phase of the repulsive Hubbard model doped away from half filling is related to a ddd-wave antiferromagnetic phase for the attractive Hubbard model. We discuss the advantages of this approach to ``quantum simulate'' the Hubbard model in an optical lattice over the simulation of the doped Hubbard model in the repulsive regime. We also point out a number of technical difficulties of the proposed approach and, in some cases, suggest possible solutions.
Antiferromagnetic, Mott-insulator, ddd-wave, and gossamer superfluid phases are calculated for two-dimensional (2D) square lattices from the extended Hubbard $(t\text{\ensuremath{-}}J\text{\ensuremath{-}}U)$ model using the Gutzwiller projection method and renormalized mean-field theory. The phase separation between antiferromagnetic and ddd-wave superfluid phases is found near half filling when the on-site repulsion exceeds $U\ensuremath{\gtrsim}7.3t$ and coincides with a first-order transition in the double occupancy. The phase separation is thus predicted for 2D optical lattices with ultracold Fermi atoms whereas it is inhibited in cuprates by Coulomb frustration which instead may lead to stripes. In a confined optical lattice the resulting density distribution is discontinuous and with extended Mott plateau which enhances the antiferromagnetic phase but suppresses the superfluid phase. Observation of Mott-insulator, antiferromagnetic, stripe, and superfluid phases in density and momentum distributions and correlations is discussed.
The Fermi-Hubbard model is a key concept in condensed matter physics and provides crucial insights into electronic and magnetic properties of materials. Yet, the intricate nature of Fermi systems poses a barrier to answering important questions concerning d-wave superconductivity and quantum magnetism. Recently, it has become possible to experimentally realize the Fermi-Hubbard model using a fermionic quantum gas loaded into an optical lattice. In this atomic approach to the Fermi-Hubbard model, the Hamiltonian is a direct result of the optical lattice potential created by interfering laser fields and short-ranged ultracold collisions. It provides a route to simulate the physics of the Hamiltonian and to address open questions and novel challenges of the underlying many-body system. This review gives an overview of the current efforts in understanding and realizing experiments with fermionic atoms in optical lattices and discusses key experiments in the metallic, band-insulating, superfluid, and Mott-insulating regimes.
In this paper we address the behavior of the superfluid transition temperature Tc{T}_{c}Tc​ in the attractive Hubbard model. The attractive Hubbard model can be regarded as the generalization of the BCS to Bose-Einstein condensation (BEC) crossover to a lattice and, as such, may have implications for future optical lattice studies. Nevertheless, the BEC limit of the Hubbard model is very different from that of the better-studied Fermi gases, owing to the strong intersite repulsion between pairs. Here we address systematically the effects of pairing fluctuations for all filling fractions over the entire range of attractive interaction strength. A central conclusion of our work is that in a lattice, around half filling, the smooth evolution from the BCS to the BEC limits is interrupted: Tc{T}_{c}Tc​ vanishes when the system approaches the bosonic regime with increasing interaction strength. We suggest that this interruption of crossover may signal a quantum critical transition to another form of superfluid not continuously connected to a BCS-like phase. A simple variational ansatz for an alternate ground state in this more strongly coupled superfluid is presented. A generalization of the (sss-wave) Hubbard model to ddd-wave pairing allows us to address issues of relevance to high-Tc{T}_{c}Tc​ superconductivity. The phase diagram (representing the pairing or pseudogap onset temperature T∗{T}^{*}T∗ and Tc{T}_{c}Tc​) shows that here too one observes a vanishing of Tc{T}_{c}Tc​ when T∗{T}^{*}T∗ becomes sufficiently large. Given this predicted breakdown of the crossover, and given the striking similarity to features of the cuprate phase diagram, we suggest that future experiments on ultracold fermions in optical lattices should not be exclusively limited to the repulsive Hubbard model, but should address the attractive model in order to elucidate features of high-temperature superconductivity.
We investigate a possibility of superfluidity in a trapped gas of Fermi atoms with a repulsive interaction in the presence of an optical lattice. Applying the exact diagonalization method to a one-dimensional Hubbard model including the trap potential, we find that, when the strength of the repulsive interaction exceeds a critical value, the binding energy of two Fermi atoms becomes negative below the half-filling case, indicating that an attractive interaction effectively works between Fermi atoms. In this case, a "Mott insulating core" appears in the center of the trap, where each site is occupied by one atom. The Cooper-pair correlation strongly develops between atoms in the left- and right-hand sides of this core.
In this thesis, I present the experimental realisation of the two-dimensional Hubbard model with ultracold fermionic Potassium-40 atoms in optical lattices. By tuning the dimensionality, kinetic and interaction in the optical lattices, we perform an analogue quantum simulation to explore the phase diagram of the Hubbard model. \n \nThe first key result is the experimental observation of particle-hole symmetry, namely a phase mapping between repulsive and attractive interactions. We compare a density-ordered, Mott-insulating phase with repulsive interaction to a spin-ordered, preformed pair phase and found excellent agreement with the theoretical prediction. The precise control and excellent detection capability of our quantum gas apparatus allow us to validate the particle-hole symmetry, and utilise it to explore quantum phases with a novel approach. \n \nNext, we investigate the spin-ordering on the repulsive side. By implementing a novel scheme based on coherent manipulation of spin correlations, we probe the anti-ferromagnetic ordering in the low-temperature phase diagram. The momentum-resolved spin correlations permit the reconstruction of spatial correlators without site-resolved imaging fidelity. \n \nFinally, we probe the attractive side of the phase diagram with a focus in pairing phenomenon, in which we draw a close analogy with BCS-BEC crossovers present in high-temperature superconducting cuprates and trapped ultracold Fermi gases. From the pair correlation function derived from thermodynamics observables, we observe the competition between the effective Pauli repulsion in fermionic systems and the on-site attraction we implemented.
We study a bosonic version of the Kondo lattice model with an onsite repulsion in the conduction band, implemented with alkali-metal atoms in two bands of an optical lattice. Using both weak- and strong-coupling perturbation theory, we find that at unit filling of the conduction bosons the superfluid-to-Mott-insulator transition should be accompanied by a magnetic transition from a ferromagnet (in the superfluid) to a paramagnet (in the Mott insulator). Furthermore, an analytic treatment of Gutzwiller mean-field theory reveals that quantum spin fluctuations induced by the Kondo exchange cause the otherwise continuous superfluid-to-Mott-insulator phase transition to be first order. We show that lattice separability imposes a serious constraint on proposals to exploit excited bands for quantum simulations, and discuss a way to overcome this constraint in the context of our model by using an experimentally realized nonseparable lattice. A method to probe the first-order nature of the transition based on collapses and revivals of the matter-wave field is also discussed.