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arXiv:1103.2951 · Physical Review A

Fulde-Ferrell-Larkin-Ovchinnikov critical polarization in one-dimensional fermionic optical lattices

V. V. França, Dominik Hörndlein, Andreas Buchleitner

Published Sep 19, 2012• 27 citations
arXivHTMLPDF

Abstract

We deduce an expression for the critical polarization ${P}_{C}$ below which the Fulde-Ferrell-Larkin-Ovchinnikov state emerges in one-dimensional lattices with spin-imbalanced populations. We provide and explore the phase diagram of unconfined chains as a function of polarization, interaction, and particle density. For harmonically confined systems, we supply a quantitative mapping, which also allows applying our phase diagram for confined chains. We find analytically and confirm numerically that the upper bound for the critical polarization is universal: ${P}_{C}^{\mathrm{max}}=1/3$ for any density, interaction, and confinement strength.

DOI
10.1103/physreva.86.033622

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We deduce an expression for the critical polarization PC{P}_{C}PC​ below which the Fulde-Ferrell-Larkin-Ovchinnikov state emerges in one-dimensional lattices with spin-imbalanced populations. We provide and explore the phase diagram of unconfined chains as a function of polarization, interaction, and particle density. For harmonically confined systems, we supply a quantitative mapping, which also allows applying our phase diagram for confined chains. We find analytically and confirm numerically that the upper bound for the critical polarization is universal: PCmax=1/3{P}_{C}^{\mathrm{max}}=1/3PCmax​=1/3 for any density, interaction, and confinement strength.
We study the phase diagram of one-dimensional two-component attractive fermions on optical lattices with an off-diagonal confinement. We identify in this system a pure Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state with spatially modulated superfluidity and polarization in a large parameter window, which provides an easier playground in which to detect the FFLO state experimentally. The clear signature of the FFLO state is analyzed in the ground-state properties, the pair correlation function, and the magnetic structure factor, using the density-matrix renormalization-group method.
We show that the ground-state of one-dimensional superfluids with spin-imbalanced populations is indeed microscale phase separated, as predicted almost fifty years ago by Fulde-Ferrell and Larkin-Ovchinnikov (FFLO). For strongly interacting systems, we find that the microscale structure can be observed directly in the density profiles. We also deduce an analytical upper bound for the critical polarization below which the FFLO-state takes place. The phase diagram depicted here indicates the parameters for which exotic superfluidity can be theoretically and experimentally explored.
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We study the properties of a one-dimensional (1D) gas of fermions trapped in a lattice by means of the density matrix renormalization group method, focusing on the case of unequal spin populations, and strong attractive interaction. In the low-density regime, the system phase separates into a well-defined superconducting core and a fully polarized metallic cloud surrounding it. We argue that the superconducting phase corresponds to a 1D analog of the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state, with a quasicondensate of tightly bound bosonic pairs with a finite center-of-mass momentum that scales linearly with the magnetization. In the large density limit, the system allows for four phases: in the core, we either find a Fock state of localized pairs or a metallic shell with free spin-down fermions moving in a fully filled background of spin-up fermions. As the magnetization increases, the Fock state disappears to give room for a metallic phase, with a partially polarized superconducting FFLO shell and a fully polarized metallic cloud surrounding the core.
We present a systematic study of quantum phases in a one-dimensional spin-polarized Fermi gas. Three comparative theoretical methods are used to explore the phase diagram at zero temperature: the mean-field theory with either an order parameter in a single-plane-wave form or a self-consistently determined order parameter using the Bogoliubov--de Gennes equations, as well as the exact Bethe ansatz method. We find that a spatially inhomogeneous Fulde-Ferrell-Larkin-Ovchinnikov phase, which lies between the fully paired Bardeen-Cooper-Schrieffer (BCS) state and the fully polarized normal state, dominates most of the phase diagram of a uniform gas. The phase transition from the BCS state to the Fulde-Ferrell-Larkin-Ovchinnikov phase is of second order, and therefore there are no phase separation states in one-dimensional homogeneous polarized gases. This is in sharp contrast to the three-dimensional situation, where a phase separation regime is predicted to occupy a very large space in the phase diagram. We conjecture that the prediction of the dominance of the phase separation phases in three dimension could be an artifact of the non-self-consistent mean-field approximation, which is heavily used in the study of three-dimensional polarized Fermi gases. We consider also the effect of a harmonic trapping potential on the phase diagram, and find that in this case the trap generally leads to phase separation, in accord with the experimental observations for a trapped gas in three dimensions. We finally investigate the local fermionic density of states of the Fulde-Ferrell-Larkin-Ovchinnikov ansatz. A two-energy-gap structure appears, which could be used as an experimental probe of the Fulde-Ferrell-Larkin-Ovchinnikov states.
We propose and analyze a variational wave function for a population-imbalanced one-dimensional Fermi gas that allows for Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) type pairing correlations among the two fermion species, while also accounting for the harmonic confining potential. In the strongly interacting regime, we find large spatial oscillations of the order parameter, indicative of an FFLO state. The obtained density profiles versus imbalance are consistent with recent experimental results as well as with theoretical calculations based on combining Bethe ansatz with the local density approximation. Our variational wave function displays no signature of the FFLO state in the densities of the two fermion species. Nonetheless, the oscillations of the order parameter appear in density-density correlations, both in situ and after free expansion. Furthermore, above a critical polarization, the value of which depends on the interaction, we find the unpaired Fermi-gas state to be energetically more favorable.