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Physical Review B

Effects of spin-orbit interaction in the Hubbard chain with attractive interaction: Application to confined ultracold fermions

Andrei A. Zvyagin, P. Schlottmann

Published Nov 19, 2013• 7 citations
Publisher

Abstract

Motivated by recent experiments on ultracold fermions we study the effect of the spin-orbit interaction in an exactly solvable one-dimensional model with strong local attractive correlations between the fermions. We show that the asymptotes of correlation functions, calculated in the framework of the conformal field theory and finite size corrections of the Bethe ansatz exact solution, are strongly affected by the spin-orbit coupling. In the mixed phase, the correlation functions consist of terms that are the product of a power law of the distance and an oscillating function of the distance. We obtain the critical exponents for superfluidity and density waves. The leading exponents decrease as a function of spin-orbit coupling. In particular, the exponent for superfluidity is the smallest one signaling an instability to a phase where a weak interchain coupling can produce superfluidity of the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) type. The spatial oscillation of the order parameter is also modified by the spin-orbit parameter, i.e., its modulation is not just given by the spin imbalance.

DOI
10.1103/physrevb.88.205127

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Motivated by recent experiments on ultracold fermions we study the effect of the spin-orbit interaction in an exactly solvable one-dimensional model with strong local attractive correlations between the fermions. We show that the asymptotes of correlation functions, calculated in the framework of the conformal field theory and finite size corrections of the Bethe ansatz exact solution, are strongly affected by the spin-orbit coupling. In the mixed phase, the correlation functions consist of terms that are the product of a power law of the distance and an oscillating function of the distance. We obtain the critical exponents for superfluidity and density waves. The leading exponents decrease as a function of spin-orbit coupling. In particular, the exponent for superfluidity is the smallest one signaling an instability to a phase where a weak interchain coupling can produce superfluidity of the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) type. The spatial oscillation of the order parameter is also modified by the spin-orbit parameter, i.e., its modulation is not just given by the spin imbalance.
In the context of ultracold atoms with effective spin S = 5/2 confined to an elongated trap we study the one-dimensional Fermi gas interacting via an attractive δ-function potential using the Bethe ansatz solution. There are N = 2S + 1 = 6 fundamental states: The particles can either be unpaired or clustered in bound states of 2, 3, …, 2S and 2S + 1 fermions. The rich ground state phase diagram consists of these six states and various mixed phases in which combinations of the fundamental states coexist. Possible scenarios for phase separation due to the harmonic confinement along the tube are explored within the local density approximation. In an array of tubes with weak Josephson tunneling superfluid order may arise. The response functions determining the type of superfluid order are calculated using conformal field theory and the exact Bethe ansatz solution. They consist of a power law with distance times a sinusoidal term oscillating with distance. The wavelength of the oscillations is related to the periodicity of a generalized Fulde–Ferrell–Larkin–Ovchinnikov (FFLO) state.
In the context of a gas of ultracold atoms with effective spin S=3/2S=3/2S=3/2 confined to an elongated trap, we study the one-dimensional Fermi gas interacting via an attractive $\ensuremath{\delta}$-function potential within the grand-canonical ensemble. The particles can be either unbound or clustered in bound states of two, three, and four fermions. The rich $\ensuremath{\mu}$ versus HHH ground-state phase diagram ($\ensuremath{\mu}$ is the chemical potential and HHH the external magnetic field) consists of the four basic states and the various possible mixed phases in which some these states coexist. Extending the analysis of K. Yang [Phys. Rev. B 63, 140511(R) (2001)] for S=1/2S=1/2S=1/2, we study the correlation functions of the generalized Cooper clusters of bound states of two, three, and four particles using conformal field theory and the exact Bethe Ansatz solution. The correlation functions consist of a power law with distance times a sinusoidal term oscillating with distance. In an array of tubes with weak Josephson tunneling, the type of superfluid order is determined by these correlation functions. The wavelength of the oscillations is related to the periodicity of a generalized Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state for higher spin particles. All the relevant states are analyzed for S=3/2S=3/2S=3/2.
In the context of ultracold fermionic atoms with effective spin S confined to an elongated trap we study the one-dimensional gas interacting via an attractive δ-function potential using the Bethe ansatz solution. There are N = 2 S + 1 fundamental states: The particles can either be unpaired or clustered in bound states of 2, 3, ..., 2 S and 2S + 1 atoms. In a magnetic field, the rich ground state phase diagram consists of these N states and various mixed phases in which combinations of the fundamental states coexist. The phase diagram simplifies considerably in zero-field, where only bound states of N atoms can exist. Due to the harmonic confinement and within the local density approximation, the density profile of bound states decreases along the tube from the center of the trap to its boundaries. In an array of tubes with weak Josephson tunneling superfluid order may arise. In zero-field the response functions determining the superfluid and density wave order are calculated using conformal field theory and the exact Bethe ansatz solution. The response function for superfluidity consists of a power law with distance, while the correlation function for density waves is a power law of distance times a sinusoidal factor oscillating with distance with a period given by two times the Fermi momentum. For S = 1/2 superfluidity is a possibility for all densities and density waves can be excluded. For S ≥ 3/2 superfluidity may occur at low densities but at high densities it gives way to density waves. We discuss the scenario of phase separation where for S ≥ 3/2 the system has superfluid long-range order toward the trap boundaries and density waves at its center.
We study the long-distance asymptotic behavior of various correlation functions for the one-dimensional (1D) attractive Hubbard model in a partially polarized phase through the Bethe ansatz and conformal field theory approaches. We particularly find the oscillating behavior of these correlation functions with spatial power-law decay, of which the pair (spin) correlation function oscillates with a frequency ΔkF (2ΔkF). Here ΔkF=π(n↑−n↓) is the mismatch in the Fermi surfaces of spin-up and spin-down particles. Consequently, the pair correlation function in momentum space has peaks at the mismatch k=ΔkF, which has been observed in recent numerical work on this model. These singular peaks in momentum space together with the spatial oscillation suggest an analog of the Fulde–Ferrell–Larkin–Ovchinnikov (FFLO) state in the 1D Hubbard model. The parameter β representing the lattice effect becomes prominent in critical exponents which determine the power-law decay of all correlation functions. We point out that the backscattering of unpaired fermions and bound pairs within their own Fermi points gives a microscopic origin of the FFLO pairing in 1D.
We investigate the long distance asymptotics of various correlation functions for the one-dimensional spin-1/2 Fermi gas with attractive interactions using the dressed charge formalism. In the spin polarized phase, these correlation functions exhibit spatial oscillations with a power-law decay whereby their critical exponents are found through conformal field theory. We show that spatial oscillations of the leading terms in the pair correlation function and the spin correlation function solely depend on [Formula: see text] and [Formula: see text], respectively. Here [Formula: see text] denotes the mismatch between the Fermi surfaces of spin-up and spin-down fermions. Such spatial modulations are characteristics of a Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state. Our key observation is that backscattering among the Fermi points of bound pairs and unpaired fermions results in a one-dimensional analog of the FFLO state and displays a microscopic origin of the FFLO nature. Furthermore, we show that the pair correlation function in momentum space has a peak at the point of mismatch between both Fermi surfaces [Formula: see text], which has recently been observed in numerous numerical studies.