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arXiv:1801.05182 · Physical Review A

From nodal-ring topological superfluids to spiral Majorana modes in cold atomic systems

Wen‐Yu He, Dong-Hui Xu, Benjamin T. Zhou, Qi Zhou, K. T. Law

Published Apr 19, 2018• 6 citations
arXivHTMLPDF

Abstract

In this work, we consider a three-dimensional (3D) cubic optical lattice composed of coupled 1D wires with 1D spin-orbit coupling. When the $s$-wave pairing is induced through Feshbach resonance, the system becomes a topological superfluid with ring nodes, which are the ring nodal degeneracies in the bulk, and supports a large number of surface Majorana zero-energy modes. The large number of surface Majorana modes remain at zero energy even in the presence of disorder due to the protection from a chiral symmetry. When the chiral symmetry is broken, the system becomes a Weyl topological superfluid with Majorana arcs. With 3D spin-orbit coupling, the Weyl superfluid becomes a gapless phase with spiral Majorana modes on the surface. A spatial-resolved radio-frequency spectroscopy is suggested to detect this nodal-ring topological superfluid phase.

DOI
10.1103/physreva.97.043618

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In this work, we consider a three-dimensional (3D) cubic optical lattice composed of coupled 1D wires with 1D spin-orbit coupling. When the sss-wave pairing is induced through Feshbach resonance, the system becomes a topological superfluid with ring nodes, which are the ring nodal degeneracies in the bulk, and supports a large number of surface Majorana zero-energy modes. The large number of surface Majorana modes remain at zero energy even in the presence of disorder due to the protection from a chiral symmetry. When the chiral symmetry is broken, the system becomes a Weyl topological superfluid with Majorana arcs. With 3D spin-orbit coupling, the Weyl superfluid becomes a gapless phase with spiral Majorana modes on the surface. A spatial-resolved radio-frequency spectroscopy is suggested to detect this nodal-ring topological superfluid phase.
Abstract The gapless Weyl superfluid has been widely studied in the three-dimensional ultracold fermionic superfluid. In contrast to Weyl superfluid, there exists another kind of gapless superfluid with topologically protected nodal lines, which can be regarded as the superfluid counterpart of nodal line semimetal in the condensed matter physics, just as Weyl superfluid with Weyl semimetal. In this paper we study the ground states of the cold fermionic gases in cubic optical lattices with one-dimensional spin-orbit coupling and transverse Zeeman field and map out the topological phase diagram of the system. We demonstrate that in addition to a fully gapped topologically trivial phase, some different nodal line superfluid phases appear when the Zeeman field is adjusted. The presence of topologically stable nodal lines implies the dispersionless zero-energy flat band in a finite region of the surface Brillouin zone. Experimentally these nodal line superfluid states can be detected via the momentum-resolved radio-frequency spectroscopy. The nodal line topological superfluid provide fertile grounds for exploring exotic quantum matters in the context of ultracold atoms.
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.
Spin-triplet p-wave superfluids of single-species (or spin-polarized) fermionic atoms is a topological superfluid. In 2D such a superfluid supports zero-energy topological Majorana fermion excitations in order parameter defects such as vortices and sample edges. In 3D these superfluids support topologically protected Dirac points in the bulk spectrum and flat surface Majorana arcs. Despite the promise, creating a spin-triplet p-wave single-species fermionic superfluid either from direct p-wave Feshbach resonance or indirectly from the recently proposed artificial spin-orbit and Zeeman couplings can be technologically challenging. Here we show that such topological superfluids can be far more simply created by using the Kohn-Luttinger effect applied to two species of spin-polarized fermions with a density imbalance. We discuss how the topological Majorana excitations of the resulting superfluids can be identified using recently-developed experimental techniques.
Recently three-dimensional topological quantum materials with gapless energy spectra have attracted considerable interest in many branches of physics. Besides the celebrated example, Dirac and Weyl points which possess gapless point structures in the underlying energy dispersion, the topologically protected gapless spectrum, can also occur along a ring, named Dirac and Weyl nodal rings. Ultracold atomic gases provide an ideal platform for exploring new topological materials with designed symmetries and dispersion. However, whether Dirac and Weyl rings can exist in the single-particle spectrum of cold atoms remains elusive. Here we propose a realistic model for realizing Dirac and Weyl rings in the single-particle band dispersion of a cold-atom optical lattice. Our scheme is based on a previously experimentally implemented Raman coupling setup for realizing spin-orbit coupling. Without the Zeeman field, the model preserves both pseudo-time-reversal and inversion symmetries, allowing Dirac rings. The Dirac rings split into Weyl rings with a Zeeman field that breaks the pseudo-time-reversal symmetry. We examine the superfluidity of attractive Fermi gases in this model and also find Dirac and Weyl rings in the quasiparticle spectrum.
We theoretically investigate the behavior of dark solitons in a one-dimensional spin-orbit coupled atomic Fermi gas in harmonic traps by solving self-consistently the Bogoliubov--de Gennes equations. The dark soliton---to be created by phase imprinting in future experiments---is characterized by a real order parameter, which changes sign at a point node and hosts localized Andreev bound states near the node. By considering both cases of a single soliton and multiple solitons, we find that the energy of these bound states decreases to zero when the system is tuned to enter the topological superfluid phase by increasing an external Zeeman field. As a result, two Majorana fermions emerge in the vicinity of each soliton, in addition to the well-known Majorana fermions at the trap edges associated with the nontrivial topology of the superfluid. We propose that the soliton-induced Majorana fermions can be directly observed by using spatially resolved radio-frequency spectroscopy or indirectly probed by measuring the density profile at the point node. For the latter, the deep minimum in the density profile will disappear due to the occupation of the soliton-induced zero-energy Majorana fermion modes. Our prediction could be tested in a resonantly interacting spin-orbit coupled 40K{}^{40}\text{K}40K Fermi gas confined in a two-dimensional optical lattice.
Abstract We explore the topological phase, which involves Majorana type topological zero mode fermions (MTZFs) at the edge, using d–wave superfluid with Rashba spin-orbit coupling (SOC) interactions. The self-Hermitian of this MTZF( γ†(k)=γ(k){{\boldsymbol{\gamma }}}^{{\boldsymbol{\dagger }}}{\boldsymbol{(}}{\bf{k}}{\boldsymbol{)}}{\boldsymbol{=}}{\boldsymbol{\gamma }}{\boldsymbol{(}}{\bf{k}}{\boldsymbol{)}}γ†(k)=γ(k) γ † ( k ) = γ ( k ) ) is similar to that of the Majorana fermions (MFs) ( γ†=γ{{\boldsymbol{\gamma }}}^{{\boldsymbol{\dagger }}}={\boldsymbol{\gamma }}γ†=γ γ † = γ ). We show that, to realize a single MTZF at each edge in superfluid with d–wave pairing in a Majorana type Kramers Doublet (MTKD) state, it is important to lift both the spin and the Dirac Cones degeneracies. These non-Abelian anyons obey the non-Abelian statistics which may be useful to realize topological quantum computation. We suggest that the topological feature could be tested experimentally in superfluids of cold fermionic atoms with laser field induced spin orbit interactions. These studies give a new possible way to investigate the MTZFs in a two-dimensional (2D) system as compared to MFs in the one-dimensional (1D) nano-wire and 2D system, and enrich the theoretical research on finding non-Abelian anyons in topological system.