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arXiv:0807.4990 · Journal of Physics Condensed Matter

Imbalanced superfluid state in an annular disk

Fei Ye, Yan Chen, Z D Wang, Fu‐Chun Zhang

Published Aug 10, 2009• 4 citations
arXivHTMLPDF

Abstract

The imbalanced superfluid state of spin- 1/2 fermions with s-wave pairing is numerically studied by solving the Bogoliubov-de Gennes equation at zero temperature in an annular disk geometry with narrow radial width. Two distinct types of systems are considered. The first case may be relevant to heavy fermion superconductors, where magnetic field causes spin imbalance via Zeeman interaction and the system is studied in a grand canonical ensemble. As the magnetic field increases, the system is transformed from the uniform superfluid state to the Fulde-Ferrell-Larkin-Ovchinnikov state, and finally to the spin polarized normal state. The second case may be relevant to cold fermionic systems, where the number of fermions of each species is fixed as in a canonical ensemble. In this case, the ground state depends on the pairing strength. For weak pairing, the order parameter exhibits a periodic domain wall lattice pattern with a localized spin distribution at low spin imbalance, and a sinusoidally modulated pattern with extended spin distribution at high spin imbalance. For strong pairing, the phase separation between the superfluid state and polarized normal state is found to be preferable, while the increase of spin imbalance simply changes the ratio between them.

DOI
10.1088/0953-8984/21/35/355701

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The imbalanced superfluid state of spin- 1/2 fermions with s-wave pairing is numerically studied by solving the Bogoliubov-de Gennes equation at zero temperature in an annular disk geometry with narrow radial width. Two distinct types of systems are considered. The first case may be relevant to heavy fermion superconductors, where magnetic field causes spin imbalance via Zeeman interaction and the system is studied in a grand canonical ensemble. As the magnetic field increases, the system is transformed from the uniform superfluid state to the Fulde-Ferrell-Larkin-Ovchinnikov state, and finally to the spin polarized normal state. The second case may be relevant to cold fermionic systems, where the number of fermions of each species is fixed as in a canonical ensemble. In this case, the ground state depends on the pairing strength. For weak pairing, the order parameter exhibits a periodic domain wall lattice pattern with a localized spin distribution at low spin imbalance, and a sinusoidally modulated pattern with extended spin distribution at high spin imbalance. For strong pairing, the phase separation between the superfluid state and polarized normal state is found to be preferable, while the increase of spin imbalance simply changes the ratio between them.
A spin-imbalanced superfluid Fermi gas harmonically trapped in a two-dimensional optical lattice with a single classical magnetic impurity is investigated by Bogoliubov-de Gennes equations. In both the spin-balanced and weak spin-imbalanced cases, we show that a strong magnetic impurity can change the sign of the pairing order parameter. The amplitude of the sign-changed order parameter caused by impurity is affected by the strength of the impurity potential, temperature, and particle density. Compared to the spin-balanced case, we find that an additional in-gap bound state can be induced by a strong magnetic impurity in the weak spin-imbalanced case. In the strong spin-imbalanced case where the Fulde-Ferrell-Larkin-Ovchinnikov state is established as the ground state, the impurity induces wide spatial oscillations of pairing order parameters and can enhance the order parameters by suppressing the local spin imbalance.
A spin-imbalanced superfluid Fermi gas harmonically trapped in a two-dimensional optical lattice with a single classical magnetic impurity is investigated by Bogoliubov-de Gennes equations. In both the spin-balanced and weak spin-imbalanced cases, we show that a strong magnetic impurity can change the sign of the pairing order parameter. The amplitude of the sign-changed order parameter caused by impurity is affected by the strength of the impurity potential, temperature, and particle density. Compared to the spin-balanced case, we find that an additional in-gap bound state can be induced by a strong magnetic impurity in the weak spin-imbalanced case. In the strong spin-imbalanced case where the Fulde-Ferrell-Larkin-Ovchinnikov state is established as the ground state, the impurity induces wide spatial oscillations of pairing order parameters and can enhance the order parameters by suppressing the local spin imbalance.
We present an extensive study of two-dimensional Larkin-Ovchinnikov (LO) superfluidity in a spin-imbalanced two-component atomic Fermi gas. In the context of Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) phase, we explore a wide and generic class of pairing gap functions with explicit spatial dependency. The mean-field theory of such phases is applied through the Bogoliubov-de Gennes equations in which the pairing gap can be determined self-consistently. To systematically explore the configuration space we consider both the canonical and grand canonical ensembles where we control the polarization or chemical potentials of the system, respectively. The mean-field calculations enable us to understand the nature of the phase transitions in the fully paired Bardeen-Cooper-Schrieffer (BCS) state, exotic LO phase, and partially polarized free Fermi gas (NPP{N}_{\text{PP}}NPP​). The order of the phase transitions has been examined and, in particular, we find a weak first-order phase transition between the exotic inhomogeneous LO phase and the BCS phase. In comparison to the three-dimensional case, where the phase diagram is dominated by a generic separation phase, we predict a broader parameter space for the spatially inhomogeneous LO phase. By computing the superfluid density of the LO phase at different polarization, we show how the superfluidity of the system is suppressed with increasing spin polarization.
Superfluidity with imbalanced populations of harmonically trapped ultracold fermions in a two-dimensional optical lattice is investigated by solving the Bogoliubov-de Gennes equations. The intriguing richness of patterns between competing ferromagnetic and superfluid phases for different filling densities has been revealed. At low density, the main experimental features could be reproduced, especially the bimodal distribution of ferromagnetism. The superfluid-pairing gap oscillates radially, demonstrating the realization of the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state. In the high-density regime, another type of FFLO state with gap oscillation along the angular direction shows up and the accompanying ferromagnetic order modulates accordingly. The square lattice like FFLO state may appear at the intermediate densities. Moreover, we suggest other experimental proposals of observing the unique FFLO features straightforwardly.
We study sss-wave superconductivity in the two-dimensional attractive Hubbard model in an applied magnetic field, assume the extreme Pauli limit, and examine the role of spatial fluctuations in the coupling regime corresponding to BCS-BEC crossover. We use a decomposition of the interaction in terms of an auxiliary pairing field, retain the static mode, and sample the pairing field via a Monte Carlo approach. The method requires iterative solution of the Bogoliubov--de-Gennes equations for amplitude- and phase-fluctuating configurations of the pairing field. We establish the full thermal phase diagram of this strong-coupling problem. At low field we observe the magnetized but homogeneous ``breached pair'' superfluid phase. It reveals that Tc{T}_{c}Tc​ scales an order of magnitude below the mean-field estimate, spontaneous inhomogeneity in the field-induced magnetization, and a strong nonmonotonicity in the temperature dependence of the low-energy density of states. We compare our results to the experimental phase diagram of the imbalanced Fermi gas at unitarity. At higher field we obtain the modulated Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) phases. The thermal transition from the FFLO phases to the normal state is strongly first order. We track the fermionic momentum distribution, the density of states, and the pairing structure factor deep into the normal state. The pairing structure factor retains weak signature of finite momentum pairing to a high temperature despite the low Tc{T}_{c}Tc​ itself, while the spin-resolved density of states changes from the ``pseudogapped'' FFLO character to gapless and pseudogapped again with increasing temperature.
By implementing the Bogoliubov-de Gennes (BdG) formalism of population-imbalanced atomic Fermi gases with pairing interactions in a thin spherical shell, we characterize the Larkin-Ovchinnikov-Fulde-Ferrell (LOFF) state in such a compact geometry. We first construct a phase diagram showing where uniform solutions of spin-polarized Fermi superfluid from the BdG equation cease to exist due to the vanishing order parameter. Near the boundary, various LOFF states with spatially modulating order parameters and density profiles can survive as convergent solutions to the BdG equation. When both uniform and LOFF solutions are present, we compare their grand potentials to determine the energetically favorable state and find that the LOFF states with multiple nodes in the order parameter become more stable at higher spin polarization. However, the LOFF state only survives close to the phase boundary where the uniform solutions vanish, indicating fragility of the LOFF state on a spherical surface. We also briefly discuss possible implications.