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arXiv:1209.2671 · Physical Review A

Unconventional spin-density waves in dipolar Fermi gases

S. G. Bhongale, Ludwig Mathey, Shan-Wen Tsai, Charles W. Clark, Erhai Zhao

Published Apr 5, 2013• 16 citations
arXivHTMLPDF

Abstract

We show that unconventional spin order arises naturally in two-component dipolar Fermi gases of atoms or molecules, which recently became accessible experimentally, in optical lattices. Using an unbiased functional renormalization-group analysis, we find that dipolar interactions lead to an instability of the gas toward an $\ensuremath{\ell}=1$ spin-density wave state. This phase is the particle-hole analog of spin-triplet, $p$-wave Cooper pairs. The order parameter for such spin-density waves of $p$-wave orbital symmetry is a vector in spin space and, moreover, is defined on lattice bonds rather than on lattice sites. We determine the rich quantum phase diagram of dipolar fermions at half filling on the square lattice as a function of the dipolar orientation and discuss how these exotic spin-density waves emerge amidst competition with superfluid and charge-density wave phases.

DOI
10.1103/physreva.87.043604

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We show that unconventional spin order arises naturally in two-component dipolar Fermi gases of atoms or molecules, which recently became accessible experimentally, in optical lattices. Using an unbiased functional renormalization-group analysis, we find that dipolar interactions lead to an instability of the gas toward an \ensuremathℓ=1\ensuremath{\ell}=1\ensuremathℓ=1 spin-density wave state. This phase is the particle-hole analog of spin-triplet, ppp-wave Cooper pairs. The order parameter for such spin-density waves of ppp-wave orbital symmetry is a vector in spin space and, moreover, is defined on lattice bonds rather than on lattice sites. We determine the rich quantum phase diagram of dipolar fermions at half filling on the square lattice as a function of the dipolar orientation and discuss how these exotic spin-density waves emerge amidst competition with superfluid and charge-density wave phases.
The recent experimental realization of dipolar Fermi gases near or below quantum degeneracy provides an opportunity to engineer Hubbard-like models with long-range interactions. Motivated by these experiments, we chart out the theoretical phase diagram of interacting dipolar fermions on the square lattice at zero temperature and half filling. We show that, in addition to p-wave superfluid and charge density wave order, two new and exotic types of bond order emerge generically in dipolar fermion systems. These phases feature homogeneous density but periodic modulations of the kinetic hopping energy between nearest or next-nearest neighbors. Similar, but manifestly different, phases of two-dimensional correlated electrons have previously only been hypothesized and termed "density waves of nonzero angular momentum." Our results suggest that these phases can be constructed flexibly with dipolar fermions, using currently available experimental techniques.
Motivated by recent experimental progress in the field of dipolar-Fermi gases, we investigate the quantum phases of dipolar fermions on a triangular ladder at half filling. Using density matrix renormalization group method, in the presence of onsite repulsion and intersite attractive interaction, we find an exotic spin-triplet superfluid phase in addition to the usual spin-density and charge-density waves. We examine the stability of the spin-triplet superfluid phase by varying hopping along the rungs of the triangle. The possibility of fermionic supersolidity has also been discussed, by considering three-body interaction in the Hamiltonian. We also study the effect of spin-dependent hopping on the stability of the spin-triplet superfluid phase.
The rapid experimental progress of ultra-cold dipolar fermions opens up a whole new opportunity to investigate novel many-body physics of fermions. In this article, we review theoretical studies of the Fermi liquid theory and Cooper pairing instabilities of both electric and magnetic dipolar fermionic systems from the perspective of unconventional symmetries. When the electric dipole moments are aligned by the external electric field, their interactions exhibit the explicit d(r(2)-3z(2)) anisotropy. The Fermi liquid properties, including the single-particle spectra, thermodynamic susceptibilities and collective excitations, are all affected by this anisotropy. The electric dipolar interaction provides a mechanism for the unconventional spin triplet Cooper pairing, which is different from the usual spin-fluctuation mechanism in solids and the superfluid (3)He. Furthermore, the competition between pairing instabilities in the singlet and triplet channels gives rise to a novel time-reversal symmetry breaking superfluid state. Unlike electric dipole moments which are induced by electric fields and unquantized, magnetic dipole moments are intrinsic proportional to the hyperfine-spin operators with a Lande factor. Its effects even manifest in unpolarized systems exhibiting an isotropic but spin-orbit coupled nature. The resultant spin-orbit coupled Fermi liquid theory supports a collective sound mode exhibiting a topologically non-trivial spin distribution over the Fermi surface. It also leads to a novel p-wave spin triplet Cooper pairing state whose spin and orbital angular momentum are entangled to the total angular momentum J = 1 dubbed the J-triplet pairing. This J-triplet pairing phase is different from both the spin-orbit coupled (3)He-B phase with J = 0 and the spin-orbit decoupled (3)He-A phase.
We show that an interesting of pairing occurs for spin-imbalanced Fermi gases under a specific experimental condition---the spin up and spin down Fermi levels lying within the pxp_xpx​ and sss orbital bands of an optical lattice, respectively. The pairs condense at a finite momentum equal to the sum of the two Fermi momenta of spin up and spin down fermions and form a ppp-orbital pair condensate. This 2kF2k_F2kF​ momentum dependence has been seen before in the spin- and charge- density waves, but it differs from the usual ppp-wave superfluids such as 3^33He, where the orbital symmetry refers to the relative motion within each pair. Our conclusion is based on the density matrix renormalization group analysis for the one-dimensional (1D) system and mean-field theory for the quasi-1D system. The phase diagram of the quasi-1D system is calculated, showing that the ppp-orbital pair condensate occurs in a wide range of fillings. In the strongly attractive limit, the system realizes an unconventional BEC beyond Feynman's no-node theorem. The possible experimental signatures of this phase in molecule projection experiment are discussed.
Experiments on quantum degenerate Fermi gases of magnetic atoms and dipolar molecules begin to probe their broken-symmetry phases dominated by the long-range, anisotropic dipole-dipole interaction. Several candidate phases including the ppp-wave superfluid, the stripe density wave, and a supersolid have been proposed theoretically for two-dimensional spinless dipolar Fermi gases. Yet the phase boundaries predicted by different approximations vary greatly and a definitive phase diagram is still lacking. Here we present a theory that treats all competing many-body instabilities in the particle-particle and particle-hole channel on equal footing. We obtain the low-temperature phase diagram by numerically solving the functional renormalization-group flow equations and find a nontrivial density wave phase at small dipolar tilting angles and strong interactions, but no evidence of the supersolid phase. We also estimate the critical temperatures of the ordered phases.
We study attractively interacting fermions on a square lattice with dispersion relations exhibiting strong spin-dependent anisotropy. The resulting Fermi surface mismatch suppresses the s-wave BCS-type instability, clearing the way for unconventional types of order. Unbiased sampling of the Feynman diagrammatic series using diagrammatic Monte Carlo methods reveals a rich phase diagram in the regime of intermediate coupling strength. Instead of a proposed Cooper-pair Bose metal phase [A. E. Feiguin and M. P. A. Fisher, Phys. Rev. Lett. 103, 025303 (2009)], we find an incommensurate density wave at strong anisotropy and two different p-wave superfluid states with unconventional symmetry at intermediate anisotropy.