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arXiv:1008.0084

Correlation studies in a spin imbalanced fermionic superfluid in presence of a harmonic trap

Poulumi Dey, Saurabh Basu, Kishore, Ram

Published Jul 31, 2010• 0 citations
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Abstract

We investigate different physical properties of a spin imbalanced fermionic superfluid described by an attractive Hubbard model in presence of a harmonic trap. To characterize the ground state of such a system, we compute various correlation functions, such as pairing, density and current correlations To underscore the effect of the trap, we compute these quantities as a function of strength of the trapping potential. A distinguishing feature of a superfluid state with population imbalance, is the appearance of a phase with spatially modulating gap function, called the Fulde- Ferrell-Larkin-Ovchinnikov (FFLO) phase which is present for both with and without trap. Trapping effects induce lowering of particle density and yields an empty system in the limit of large trapping depths.

DOI
10.48550/arxiv.1008.0084

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We investigate different physical properties of a spin imbalanced fermionic superfluid described by an attractive Hubbard model in presence of a harmonic trap. To characterize the ground state of such a system, we compute various correlation functions, such as pairing, density and current correlations To underscore the effect of the trap, we compute these quantities as a function of strength of the trapping potential. A distinguishing feature of a superfluid state with population imbalance, is the appearance of a phase with spatially modulating gap function, called the Fulde- Ferrell-Larkin-Ovchinnikov (FFLO) phase which is present for both with and without trap. Trapping effects induce lowering of particle density and yields an empty system in the limit of large trapping depths.
In an attempt to characterize the Fulde–Ferrell–Larkin–Ovchinnikov (FFLO) phase in a two dimensional s-wave superconductor, we investigate current-current and spin-spin correlations and compare them with those in the BCS phase. The existence of the FFLO phase is demonstrated using a weakly attractive fermionic Hubbard model in presence of a harmonic trap and a population imbalance, caused by a magnetic field. The lower and upper values of the magnetic field, between which the order parameter shows spatial modulations (owing to a finite center of mass momentum pairing), mark the extent of the FFLO phase. In this regime, the current-current correlations show spatial modulation similar to the behaviour of the order parameter, while the longitudinal spin-spin correlations show modulations with a period half of that of the order parameter in the absence of trap and a bimodal distribution in the presence of the harmonic confinement, thereby underlining a dissimilarity in behaviour between the spin and charge sectors.
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.
We explore the pairing properties of the one-dimensional attractive Hubbard model in the presence of finite spin polarization. The correlation exponents for the most important fluctuations are determined as a function of the density and the polarization. We find that in a system with spin population imbalance, Fulde-Ferrell-Larkin-Ovchinnikov (FFLO)-type pairing at wave vector $Q=\ensuremath{\mid}{k}_{F,\ensuremath{\uparrow}}\ensuremath{-}{k}_{F,\ensuremath{\downarrow}}\ensuremath{\mid}$ is always dominant and there is no Chandrasekhar-Clogston limit. We then investigate the case of weakly coupled one-dimensional (1D) systems and determine the region of stability of the 1D FFLO phase. This picture is corroborated by density-matrix-renormalization-group simulations of the spatial noise correlations in uniform and trapped systems, unambiguously revealing the presence of fermion pairs with nonzero momentum QQQ. This opens up an interesting possibility for experimental studies of FFLO states.
Ground-state Hartree-Fock-Bogoliubov (HFB) theory is applied to imbalanced spin-1/21/21/2 one-dimensional (1D) Fermi systems that are spatially confined by either a harmonic or a hard-wall trapping potential. It has been hoped that such systems, which can be realized using ultracold atomic gases, would exhibit the long-sought-after Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) superfluid phase. The HFB formalism generalizes the standard Bogoliubov quasiparticle transformation by allowing for Cooper pairing to exist between all possible single-particle states and accounts for the effects of the inhomogeneous trapping potential as well as the mean-field Hartree potential. This provides an unbiased framework to describe inhomogeneous densities and pairing correlations in the FFLO state of a confined 1D gas. In a harmonic trap, numerical minimization of the HFB ground-state energy yields a spatially oscillating order parameter reminiscent of the FFLO state. However, we find that this state has almost no imprint in the local fermion densities (consistent with experiments that found no evidence of the FFLO phase). In contrast, for a hard-wall geometry, we find a strong signature of the spatial oscillations of the FFLO pairing amplitude reflected in the local in situ densities. In the hard-wall case, the excess spins are strongly localized near regions where there is a node in the pairing amplitude, creating an unmistakable crystalline modulation of the density.
We theoretically investigate the ground state of trapped neutral fermions with population imbalance in the BCS-BEC crossover regime. On the basis of the single-channel Hamiltonian, we perform full numerical calculations of the Bogoliubov-de Gennes equation coupled with the regularized gap and number equations. The zero-temperature phase diagram in the crossover regime is presented, where the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) pairing state governs the weak-coupling BCS region of a resonance. It is found that the FFLO oscillation vanishes in the BEC side, in which the system under population imbalance turns into a phase separation (PS) between locally binding superfluid and fully polarized spin domains. We also demonstrate numerical calculations with a large particle number O(10^5), comparable to that observed in recent experiments. The resulting density profile on a resonance yields the PS, which is in good agreement with the recent experiments, while the FFLO modulation exists in the pairing field. It is also proposed that the most favorable location for the detection of the FFLO oscillation is in the vicinity of the critical population imbalance in the weak coupling BCS regime, where the oscillation periodicity becomes much larger than the interparticle spacing. Finally, we analyze the radio-frequency (RF) spectroscopy in the imbalanced system. The clear difference in the RF spectroscopy between BCS and BEC sides reveals the structure of the pairing field and local ``magnetization''.
We review the concepts and the present state of theoretical studies of spin-imbalanced superfluidity, in particular the elusive Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state, in the context of ultracold quantum gases. The comprehensive presentation of the theoretical basis for the FFLO state that we provide is useful also for research on the interplay between magnetism and superconductivity in other physical systems. We focus on settings that have been predicted to be favourable for the FFLO state, such as optical lattices in various dimensions and spin-orbit coupled systems. These are also the most likely systems for near-future experimental observation of the FFLO state. Theoretical bounds, such as Bloch's and Luttinger's theorems, and experimentally important limitations, such as finite-size effects and trapping potentials, are considered. In addition, we provide a comprehensive review of the various ideas presented for the observation of the FFLO state. We conclude our review with an analysis of the open questions related to the FFLO state, such as its stability, superfluid density, collective modes and extending the FFLO superfluid concept to new types of lattice systems.