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Physical Review B

Fermi gas with attractive potential and spin S=32 in a one-dimensional trap: Response functions for superfluidity and FFLO signatures

P. Schlottmann, Andrei A. Zvyagin

Published May 18, 2012• 18 citations
Publisher

Abstract

In the context of a gas of ultracold atoms with effective spin $S=3/2$ 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 $H$ ground-state phase diagram ($\ensuremath{\mu}$ is the chemical potential and $H$ 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 $S=1/2$, 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 $S=3/2$.

DOI
10.1103/physrevb.85.205129

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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 \ensuremathδ\ensuremath{\delta}\ensuremathδ-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 \ensuremathμ\ensuremath{\mu}\ensuremathμ versus HHH ground-state phase diagram (\ensuremathμ\ensuremath{\mu}\ensuremathμ 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 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.
A gas of ultracold 6{}^{6}6Li atoms (effective spin 1/2) confined to an elongated trap with one-dimensional properties is a candidate to display three different phases: (i) fermions bound in Cooper-pair-like states, (ii) unbound spin-polarized particles, and (iii) a mixed phase in which Cooper bound states and unpaired particles coexist. It is of great interest to extend these studies to fermionic atoms with higher spin, e.g., for neutral 40{}^{40}40K, 43{}^{43}43Ca, 87{}^{87}87Sr, or 173{}^{173}173Yb atoms. Within the grand-canonical ensemble, we investigated the $\ensuremath{\mu}$ versus HHH phase diagram ($\ensuremath{\mu}$ is the chemical potential and HHH the external magnetic field) for S=3/2,...,9/2S=3/2,...,9/2S=3/2,...,9/2 for the ground state using the exact Bethe ansatz solution of the one-dimensional Fermi gas with an attractive $\ensuremath{\delta}$-function interaction potential. There are N=2S+1N=2S+1N=2S+1 fundamental states: the particles can be either unpaired or clustered in bound states of 2, 3, $\ensuremath{\cdots}$, 2S2S2S, and 2S+12S+12S+1 fermions. The rich phase diagram consists of these NNN states and various mixed phases in which combinations of the fundamental states coexist. Bound states of NNN fermions are not favorable in high magnetic fields, but always present if the field is low. For S=3/2,S=3/2,S=3/2, possible scenarios for phase separation are explored within the local density approximation. For S=3/2,S=3/2,S=3/2, the phase diagram for the superposition of a Zeeman and a quadrupolar splitting is also discussed.
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.
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.
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.
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