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arXiv:cond-mat/9808246 · Physical Review Letters

Pairing of Fermions with Arbitrary Spin

Tin-Lun Ho, Sungkit Yip

Published Jan 11, 1999• 94 citations
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Abstract

Motivated by the recent success of optical trapping of alkali Bose condensate, we have studied the superfluid state of optically trapped alkali fermions, which can have Cooper pairs with total spin $J\ensuremath{\ge}2$. In this paper, we shall discuss the general structure of these large spin Cooper pairs and their close relation with singlet Cooper pairs with nonzero orbital angular momentum. We also present the exact solution for the $J\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}2$ pairing which shows a surprising change of ground state as the spin $f$ of the constituent fermion increases.

DOI
10.1103/physrevlett.82.247

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Motivated by the recent success of optical trapping of alkali Bose condensate, we have studied the superfluid state of optically trapped alkali fermions, which can have Cooper pairs with total spin J\ensuremath≥2J\ensuremath{\ge}2J\ensuremath≥2. In this paper, we shall discuss the general structure of these large spin Cooper pairs and their close relation with singlet Cooper pairs with nonzero orbital angular momentum. We also present the exact solution for the J=2J\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}2J=2 pairing which shows a surprising change of ground state as the spin fff of the constituent fermion increases.
The admixture of spin-singlet and spin-triplet pairing states in superconductors can be typically induced by breaking spatial inversion symmetry. Employing the numerically exact auxiliary-field quantum Monte Carlo method, we study such mixed-parity pairing phenomena of attractive fermions with Rashba spin-orbit coupling (SOC) in a two-dimensional optical lattice at finite temperature. We systematically explore the evolution of the essential pairing structure in both singlet and triplet channels versus the temperature, fermion filling, and SOC and interaction strengths, via computing the finite-size results of the condensate fraction and pair wave function. Our numerical results reveal that the singlet channel dominates in the fermion pairing and the triplet pairing has a relatively small contribution to the superfluidity for physically relevant parameters. In contrast to the singlet channel mainly consisting of the on-site Cooper pairs, the triplet pairing has plentiful patterns in real space with the largest contributions from several nearest neighbors. As the SOC strength increases, the pairing correlation is first enhanced and then suppressed for triplet pairing while it is simply weakened in the singlet channel. We have also obtained the Berezinskii-Kosterlitz-Thouless transition temperatures through the finite-size analysis of the condensate fraction. Our results can serve as a quantitative guide for future optical lattice experiments as well as accurate benchmarks for theories and other numerical methods.
Motivated by the recent discoveries of spin-1 and pseudo-spin-1/2 Bose gas, we have studied the general structure of the Bose gases with arbitrary spin. A general method is developed to uncover the elementary building blocks of the angular momentum eigenstates, as well as the relations (or interactions) between them. Applications of this method to Bose gas with integer spins (f=1,2,3)(f\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1,2,3)(f=1,2,3) and half integer spins (f=1/2,3/2)(f\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1/2,3/2)(f=1/2,3/2) reveal many surprising structures.
We analyze the evolution of superfluidity for nonzero orbital angular momentum channels from the Bardeen-Cooper-Schrieffer (BCS) to the Bose-Einstein condensation (BEC) limit in three dimensions. First, we analyze the low-energy scattering properties of finite range interactions for all possible angular momentum channels. Second, we discuss ground-state (T=0)(T=0)(T=0) superfluid properties including the order parameter, chemical potential, quasiparticle excitation spectrum, momentum distribution, atomic compressibility, ground-state energy, and low-energy collective excitations. We show that a quantum phase transition occurs for nonzero angular momentum pairing, unlike the sss-wave case where the BCS to BEC evolution is just a crossover. Third, we present a Gaussian fluctuation theory near the critical temperature (T=Tc)(T={T}_{\mathrm{c}})(T=Tc​), and we analyze the number of bound, scattering, and unbound fermions as well as the chemical potential. Finally, we derive the time-dependent Ginzburg-Landau functional near Tc{T}_{\mathrm{c}}Tc​, and compare the Ginzburg-Landau coherence length with the zero-temperature average Cooper pair size.
The Fulde–Ferrell (FF) superfluid phase, in which fermions form finite momentum Cooper pairings, is well studied in spin-singlet superfluids in past decades. Different from previous works that engineer the FF state in spinful cold atoms, we show that the FF state can emerge in spinless Fermi gases confined in optical lattice associated with nearest-neighbor interactions. The mechanism of the spinless FF state relies on the split Fermi surfaces by tuning the chemistry potential, which naturally gives rise to finite momentum Cooper pairings. The phase transition is accompanied by changed Chern numbers, in which, different from the conventional picture, the band gap does not close. By beyond-mean-field calculations, we find the finite momentum pairing is more robust, yielding the system promising for maintaining the FF state at finite temperature. Finally we present the possible realization and detection scheme of the spinless FF state.
We investigate two-component ultracold fermionic atoms with repulsive interactions trapped in an optical lattice with ladder structure. By applying the Bogoliubov--de Gennes equations to an effective $t\text{\ensuremath{-}}J$ model in the strong correlation limit, we discuss how the spatially modulated spin-singlet pairs with ddd-wave-like symmetry are formed in the systems with trapping potentials. Furthermore, a close examination of the condensation energy as well as the local average of potential, kinetic, and exchange energies by means of the variational Monte Carlo method elucidates that local particle correlations enhance the stability of the superfluid state via a substantial energy gain due to singlet pairing in the high particle density region.