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arXiv:1107.4957 · Physical Review B

Fermi gas with attractive potential and arbitrary spin in a one-dimensional trap: Phase diagram for S=32 , 52 , 72 , and 92

P. Schlottmann, Andrei A. Zvyagin

Published Jan 30, 2012• 12 citations
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Abstract

A gas of ultracold ${}^{6}$Li 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 ${}^{40}$K, ${}^{43}$Ca, ${}^{87}$Sr, or ${}^{173}$Yb atoms. Within the grand-canonical ensemble, we investigated the $\ensuremath{\mu}$ versus $H$ phase diagram ($\ensuremath{\mu}$ is the chemical potential and $H$ the external magnetic field) for $S=3/2,...,9/2$ 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 $N=2S+1$ fundamental states: the particles can be either unpaired or clustered in bound states of 2, 3, $\ensuremath{\cdots}$, $2S$, and $2S+1$ fermions. The rich phase diagram consists of these $N$ states and various mixed phases in which combinations of the fundamental states coexist. Bound states of $N$ fermions are not favorable in high magnetic fields, but always present if the field is low. For $S=3/2,$ possible scenarios for phase separation are explored within the local density approximation. For $S=3/2,$ the phase diagram for the superposition of a Zeeman and a quadrupolar splitting is also discussed.

DOI
10.1103/physrevb.85.024535

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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 $\ensuremath{\delta}$-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.
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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.
Based on the integrable Gaudin model and local density approximation, we discuss the ground state of a one-dimensional trapped Fermi gas with imbalanced spin population, for an arbitrary attractive interaction. A phase separation state, with a polarized superfluid core immersed in an unpolarized superfluid shell, emerges below a critical spin polarization. Above it, coexistence of polarized superfluid matter and a fully polarized normal gas is favored. These two exotic states could be realized experimentally in highly elongated atomic traps, and diagnosed by measuring the lowest density compressional mode. We identify the polarized superfluid as having an Fulde-Ferrell-Larkin-Ovchinnikov structure, and predict the resulting mode frequency as a function of the spin polarization.
We consider the one-dimensional gas of fermions with spin S interacting via an attractive δ -function potential using the Bethe Ansatz solution. In zero magnetic field the atoms form bound states of N = 2 S + 1 fermions, i.e. generalized Cooper states with each atom having a different spin component. For low energy excitations the system is a Luttinger liquid and is properly described by a conformal field theory with conformal charge c = 1 . The linear dispersion of a Luttinger liquid is asymptotically exact in the low-energy limit where the band curvature terms in the dispersion are irrelevant. For higher energy excitations, however, the spectral function displays deviations in the neighborhood of the single-particle (hole) energy, which can be described by an effective X-ray edge type model. Using the Bethe Ansatz solution we obtain expressions for the critical exponents for the single-particle (hole) Green's function. This model can be relevant in the context of ultracold atoms with effective total spin S confined to an elongated optical trap.
We investigate pairing and quantum phase transitions in the one-dimensional two-component Fermi atomic gas in an external field. The phase diagram, critical fields, magnetization, and local pairing correlation are obtained analytically via the exact thermodynamic Bethe ansatz solution. At zero temperature, bound pairs of fermions with opposite spin states form a singlet ground state when the external field H<Hc1H<{H}_{c1}H<Hc1​. A completely ferromagnetic phase without pairing occurs when the external field H>Hc2H>{H}_{c2}H>Hc2​. In the region Hc1<H<Hc2{H}_{c1}<H<{H}_{c2}Hc1​<H<Hc2​, we observe a mixed phase of matter in which paired and unpaired atoms coexist. The phase diagram is reminiscent of that of type II superconductors. For temperatures below the degenerate temperature and in the absence of an external field, the bound pairs of fermions form hard-core bosons obeying generalized exclusion statistics.