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arXiv:1410.3497 · International Journal of Modern Physics B

Topological Fulde–Ferrell superfluids of a spin–orbit coupled Fermi gas

Yong Xu, Chuanwei Zhang

Published Dec 12, 2014• 27 citations
arXivHTMLPDF

Abstract

Topological Fermi superfluids have played the central role in various fields of physics. However, all previous studies focus on the cases where Cooper pairs have zero center-of-mass momenta (i.e., normal superfluids). The topology of Fulde–Ferrell (FF) superfluids with nonzero momentum pairings have never been explored until recent findings that FF superfluids in a spin–orbit (SO) coupled Fermi gas can accommodate Majorana fermions in real space in low dimensions and Weyl fermions in momentum space in three dimension. In this review, we first discuss the mechanism of pairings in SO coupled Fermi gases in optical lattices subject to Zeeman fields, showing that SO coupling as well as Zeeman fields enhance FF states while suppress Larkin–Ovchinnikov states. We then present the low temperature phase diagram including both FF superfluids and topological FF superfluids phases in both two dimension and three dimension. In one dimension, Majorana fermions as well as phase dependent order parameter are visualized. In three dimension, we show the properties of Weyl fermions in momentum space such as anisotropic linear dispersion, Fermi arch, and gaplessness away from k⊥ = 0. Finally, we discuss some possible methods to probe FF superfluids and topological FF superfluids in cold atom systems.

DOI
10.1142/s0217979215300017

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Topological Fermi superfluids have played the central role in various fields of physics. However, all previous studies focus on the cases where Cooper pairs have zero center-of-mass momenta (i.e., normal superfluids). The topology of Fulde–Ferrell (FF) superfluids with nonzero momentum pairings have never been explored until recent findings that FF superfluids in a spin–orbit (SO) coupled Fermi gas can accommodate Majorana fermions in real space in low dimensions and Weyl fermions in momentum space in three dimension. In this review, we first discuss the mechanism of pairings in SO coupled Fermi gases in optical lattices subject to Zeeman fields, showing that SO coupling as well as Zeeman fields enhance FF states while suppress Larkin–Ovchinnikov states. We then present the low temperature phase diagram including both FF superfluids and topological FF superfluids phases in both two dimension and three dimension. In one dimension, Majorana fermions as well as phase dependent order parameter are visualized. In three dimension, we show the properties of Weyl fermions in momentum space such as anisotropic linear dispersion, Fermi arch, and gaplessness away from k⊥ = 0. Finally, we discuss some possible methods to probe FF superfluids and topological FF superfluids in cold atom systems.
We theoretically investigate a three-dimensional Fermi gas with Rashba spin–orbit coupling in the presence of both out-of-plane and in-plane Zeeman fields. We show that, driven by a sufficiently large Zeeman field, either out-of-plane or in-plane, the superfluid phase of this system exhibits a number of interesting features, including inhomogeneous Fulde–Ferrell pairing, gapped or gapless topological order, and exotic quasi-particle excitations known as Weyl fermions that have linear energy dispersions in momentum space (i.e., massless Dirac fermions). The topological superfluid phase can have either four or two topologically protected Weyl nodes. We present the phase diagrams at both zero and finite temperatures and discuss the possibility of their observation in an atomic Fermi gas with synthetic spin–orbit coupling. In this context, topological superfluid phase with an imperfect Rashba spin–orbit coupling is also studied.
The recent experimental realization of spin-orbit (SO) coupling for ultracold atoms opens a completely new avenue for exploring new quantum matters. In experiments, the SO coupling is implemented simultaneously with a Zeeman field. Such SO-coupled Fermi gases are predicted to support Majorana fermions with non-Abelian exchange statistics in one dimension (1D). However, as shown in recent theory and experiments for 1D spin-imbalanced Fermi gases, the Zeeman field can lead to the long-sought Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) superfluids with nonzero center-of-mass momentum Cooper pairings, in contrast to the zero center-of-mass momentum pairings in Majorana superfluids. Therefore a natural question to ask is which phase, FFLO or Majorana superfluids, will survive in SO-coupled Fermi gases in the presence of a large out-of-plane Zeeman field. In this paper, we address this question by studying the mean-field quantum phases of 1D (quasi-1D) SO-coupled fermionic cold-atom optical lattices.
We investigate properties of exotic pairing states in a three-dimensional Fermi gas with three-dimensional spin-orbit coupling and an effective Zeeman field. The interplay of spin-orbit coupling, effective Zeeman field, and pairing can lead to first-order phase transitions between different phases, and to interesting nodal superfluid states with gapless surfaces in the momentum space. We then demonstrate that pairing states with zero center-of-mass momentum are unstable against Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) states, with a finite center-of-mass momentum opposite to the direction of the effective Zeeman field. Unlike conventional FFLO states, these FFLO states are induced by the coexistence of spin-orbit coupling and Fermi surface deformation, and have intriguing features like first-order transitions between different FFLO states, nodal FFLO states with gapless surfaces in momentum space, and exotic fully gapped FFLO states. With the recent theoretical proposals for realizing three-dimensional spin-orbit coupling in ultracold atomic gases, our work is helpful for future experimental studies and provides valuable information for the general understanding of pairing physics in spin-orbit-coupled fermionic systems.
The realizations of spin-orbit coupling in cold atoms lead to a burst of research activities in the searching of topological matters in ultracold atom systems. Recent theoretical predictions show that topological Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) superfluids can be realized with appropriate spin-orbit coupling and in-plane Zeeman fields. In this work, a comprehensive understanding of the pairing symmetry, phase diagram, and the edge modes in this topological matter are presented. The center-of-mass momentum of the Cooper pairs plays the role of renormalizing the in-plane Zeeman fields and chemical potential. The in-plane Zeeman fields also induce asymmetry to the effective ppp-wave pairing, apart from a small fraction of higher orbital components. As a result, the phase diagram is composed by different phases, which are determined by the Fermi surface topology and band gap width of the superfluids. Especially, the gapped and gapless topological FFLO phases have totally different finite size effects. These features show that the spin-orbit coupled cold atoms provide an important platform in realizing topological matters which may not be materialized with solids.
Inhomogeneous superfluidity lies at the heart of many intriguing phenomena in quantum physics. It is believed to play a central role in unconventional organic or heavy-fermion superconductors, chiral quark matter, and neutron star glitches. However, so far even the simplest form of inhomogeneous superfluidity, the Fulde-Ferrell (FF) pairing state with a single center-of-mass momentum, is not conclusively observed due to the intrinsic complexibility of any realistic Fermi systems in nature. Here we theoretically predict that the controlled setting of ultracold fermionic atoms with synthetic spin-orbit coupling induced by a two-photon Raman process, demonstrated recently in cold-atom laboratories, provides a promising route to realize the long-sought FF superfluidity. At experimentally accessible low temperatures, the FF superfluid state dominates the phase diagram, in sharp contrast to the conventional case without spin-orbit coupling. We show that the finite center-of-mass momentum carried by Cooper pairs is directly measurable via momentum-resolved radio-frequency spectroscopy. Our work opens the way to direct observation and characterization of inhomogeneous superfluidity.
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.