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General Hubbard model for strongly interacting fermions in an optical lattice and its phase detection

L. -M. Duan

Published Jan 1, 2008• 0 citations
Publisher

Abstract

"Based on the consideration of the system symmetry and its Hilbert space, we show that strongly interacting fermions near a wide Feshbach resonance in an optical lattice or superlattice can be generically described by a lattice resonance Hamiltonian. The latter can be mapped to a general Hubbard model with particle-assisted tunnelling rates. We investigate the model under population imbalance and show that the attractive and the repulsive models have the same complexity in the phase diagram under the particle-hole mapping. Using this mapping, we propose an experimental method to detect possible exotic superfluid/magnetic phases for this system."

DOI
10.1088/0295-5075/81/2/20001

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"Based on the consideration of the system symmetry and its Hilbert space, we show that strongly interacting fermions near a wide Feshbach resonance in an optical lattice or superlattice can be generically described by a lattice resonance Hamiltonian. The latter can be mapped to a general Hubbard model with particle-assisted tunnelling rates. We investigate the model under population imbalance and show that the attractive and the repulsive models have the same complexity in the phase diagram under the particle-hole mapping. Using this mapping, we propose an experimental method to detect possible exotic superfluid/magnetic phases for this system."
Based on the consideration of the system symmetry and its Hilbert space, we show that strongly interacting fermions near a wide Feshbach resonance in an optical lattice or superlattice can be generically described by a lattice resonance Hamiltonian. The latter can be mapped to a general Hubbard model with particle-assisted tunnelling rates. We investigate the model under population imbalance and show that the attractive and the repulsive models have the same complexity in the phase diagram under the particle-hole mapping. Using this mapping, we propose an experimental method to detect possible exotic superfluid/magnetic phases for this system.
We show that the Hubbard Hamiltonian with particle-assisted tunneling rates---recently proposed to model a fermionic mixture near a broad Feshbach resonance---displays a ground-state phase diagram with superfluid, insulating, and phase-separated regimes. In the latter case, when the populations are balanced the two phases coexist in microscopic antiferromagnetic domains. Macroscopic phase segregation into a high-density superfluid of molecules and a low-density Fermi liquid of single atoms appears in the density profile above a critical polarization pc{p}_{c}pc​.
Fermions in an optical lattice near a wide Feshbach resonance are expected to be described by an effective Hamiltonian of the general Hubbard model with particle-correlated tunneling rates resulting from the strong atomic interaction [Duan, Phys. Rev. Lett. 95, 243202 (2005)]. Here, we propose a scheme to test the predictions of this effective Hamiltonian through manipulation of ultracold atoms in an inhomogeneous optical superlattice. The structure of the low-energy Hilbert space as well as the particle-correlated tunneling rates can be inferred from measurements of the time-of-flight images.
This dissertation summarizes my recent work regarding systems of strongly interacting fermionic atoms in optical lattices. This work addresses the combination of two experimental techniques that have been the subject of much recent research in ultracold atom physics. One is the use of optical lattices, which provide a means to realize diverse interaction configurations within a clean, controllable system. The other is the use of magnetically tunable Feshbach resonances to control the strength of the interatomic interaction. Together, these techniques offer the possibility of an experimental realization of many important model Hamiltonians of condensed matter physics, and may also lead to the discovery of new physics. Recent study of this system has shown that strong interactions near Feshbach resonance will lead to the population of multiple lattice bands, and that collisions between atoms on neighboring sites cannot be neglected. These effects lead to a complicated Hamiltonian, but one which can be simplified to an effective single-band model equivalent to the generalized Hubbard model (GHM), which is an extension of the Hubbard model that includes correlated hopping terms. My main results concern the study of this model. The strong correlations between the particles make it difficult to definitively determine the many body physics of the GHM. As a first approach to understanding the GHM in optical lattices, I focus mainly on cases where the problem is greatly simplified by allowing interactions among only small groups of lattice sites. This restriction can be implemented in experiments using an optical superlattice potential. Our results include a proposed scheme (based on double-well superlattices) to empirically verify that the GHM describes this system and to directly measure the various parameters of this model. Other results include exact solutions on four-site square plaquettes, which demonstrate that d-wave excitations can occur in the low-energy states. By using a superlattice to give an array of weakly coupled plaquettes, one can thus produce a d-wave superfluid state. This is of relevance to the study of high-Tc superconductors, although I note certain key differences between the sort of d-wave superfluid described here and that of the superconductors.
Strongly interacting fermionic atoms on optical lattices are studied through a Hubbard-like model Hamiltonian, in which tunneling rates of atoms and molecules between neighboring sites are assumed to be different. In the limit of large on-site repulsion UUU, the model is shown to reproduce the ttt-JJJ Hamiltonian, in which the JJJ coefficient of the Heisenberg term depends on the particle-assisted tunneling rate ggg: explicitly, J=4g2/UJ=4{g}^{2}/UJ=4g2/U. At half-filling, ggg drives a crossover from a Brinkman-Rice paramagnetic insulator of fully localized atoms (g=0g=0g=0) to the antiferromagnetic Mott insulator of the standard Hubbard case (g=tg=tg=t). This is observed already in the number of doubly occupied sites under the intermediate coupling regime, thus providing a criterion for extracting from measurements the effective value of ggg.
Abstract The equilibrium thermodynamics of bosons and fermions in optical lattices are considered in the single-band Hubbard regime, with an emphasis on interesting magnetic, superfluid, and spin liquid ground states. The parameters of the Hubbard model—the tunneling and interaction parameters—can be obtained quantitatively in terms of the strength and periodicity of the optical lattice potential, tuned by the laser intensity and wavelength. This direct link between the parameters of a theoretical model and the actual experimental optical lattice gives rise to the phase diagram of the Bose–Hubbard model. The definition of the order parameter as an expectation value of the annihilation operator in a coherent state is discussed. This phase-coherent superfluid state is contrasted with the phase-incoherent Mott state naturally defined in terms of number states. Following the repulsive Bose–Hubbard model, the phase diagram of the Fermi–Hubbard model is considered with both attractive and repulsive interactions.