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arXiv:1310.7557 · Physical Review A

Majorana fermions in quasi-one-dimensional and higher-dimensional ultracold optical lattices

Chunlei Qu, Ming Gong, Yong Xu, Sumanta Tewari, Chuanwei Zhang

Published Aug 14, 2015• 12 citations
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Abstract

We study Majorana fermions (MFs) in quasi-one dimensional (quasi-1D) and higher-dimensional fermionic optical lattices with a strictly 1D spin-orbit coupling, which has already been realized in cold atom experiments. We show that when the superfluid order parameters are homogeneous and are enforced to be identical along different chains, there are multiple MFs at each end with or without an experimentally tunable in-plane Zeeman field ${V}_{y}$. For ${V}_{y}=0$ the multiple MFs are topologically protected by a chiral symmetry; however, for ${V}_{y}\ensuremath{\ne}0$ the existence of multiple MFs is related to the peculiar spectrum properties of the system despite the broken chiral symmetry. In the generalization to higher dimensions, the multiple MFs form a zero-energy flat band. Furthermore, when the superfluid order parameters are solved self-consistently, the multiple MFs are usually destroyed because of the inhomogeneous order parameters of either Bardeen-Cooper-Schrieffer (${V}_{y}=0$) type or Fulde-Ferrell (${V}_{y}\ensuremath{\ne}0$) type. Our results are useful to guide the experimentalists on searching for MFs in ultracold spin-orbit coupled fermionic superfluids.

DOI
10.1103/physreva.92.023621

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We study Majorana fermions (MFs) in quasi-one dimensional (quasi-1D) and higher-dimensional fermionic optical lattices with a strictly 1D spin-orbit coupling, which has already been realized in cold atom experiments. We show that when the superfluid order parameters are homogeneous and are enforced to be identical along different chains, there are multiple MFs at each end with or without an experimentally tunable in-plane Zeeman field Vy{V}_{y}Vy​. For Vy=0{V}_{y}=0Vy​=0 the multiple MFs are topologically protected by a chiral symmetry; however, for Vy\ensuremath≠0{V}_{y}\ensuremath{\ne}0Vy​\ensuremath=0 the existence of multiple MFs is related to the peculiar spectrum properties of the system despite the broken chiral symmetry. In the generalization to higher dimensions, the multiple MFs form a zero-energy flat band. Furthermore, when the superfluid order parameters are solved self-consistently, the multiple MFs are usually destroyed because of the inhomogeneous order parameters of either Bardeen-Cooper-Schrieffer (Vy=0{V}_{y}=0Vy​=0) type or Fulde-Ferrell (Vy\ensuremath≠0{V}_{y}\ensuremath{\ne}0Vy​\ensuremath=0) type. Our results are useful to guide the experimentalists on searching for MFs in ultracold spin-orbit coupled fermionic superfluids.
We show that Majorana fermions (MFs) exist in twoand three-dimensional (2D,3D) fermionic optical lattices with strictly 1D spin-orbit coupling (SOC) which has already been realized in experiments. For a quasi-1D topological BCS superfluid, there are multiple MFs at each end which are topologically protected by a chiral symmetry. In the generalization to higher dimensions, the multiple MFs form a zero energy flat band. An additional experimentally tunable in-plane Zeeman field drives the system to a topological Fulde-Ferrell (FF) superfluid phase. We find that even though the multiple MFs are robust against the in-plane Zeeman field if the order parameters at the different chains are enforced to be identical, they are destroyed in the self-consistently obtained FF phase where the order parameters are inhomogeneous on the boundaries. Our results are useful to guide the experimentalists on searching for MFs in the context of ultracold fermionic atoms.
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.
The recent experimental realization of one-dimensional (1D) equal Rashba-Dresselhaus spin-orbit coupling (ERD-SOC) for cold atoms provides a disorder-free and highly controllable platform for the implementation and observation of Majorana fermions (MFs), analogous to the broadly studied solid-state nanowire-superconductor heterostructures. However, the corresponding 1D chains of cold atoms possess strong quantum fluctuation, which may destroy the superfluids and MFs. In this paper, we show that such 1D topological chains with MFs may be on demand generated in a two- or three-dimensional nontopological optical lattice with 1D ERD-SOC by modifying local potentials on target locations using experimentally already implemented atomic gas microscopes or patterned (e.g., double- or triple-well) optical lattices. All ingredients in our scheme have been experimentally realized, and the combination of them may pave the way for the experimental observation of MFs in a clean system.
Abstract : Majorana fermions (MFs) are quantum particles that are their own anti-particles, and satisfy non-Abelian exchange statistics. The latter is the key for their potential use in fault-tolerant topological quantum computation. In this project, we proposed to investigate the possibility of realizing MFs in much more physically robust s-wave superfluids by utilizing two additional components: spin-orbit coupling and Zeeman fields. In the last year grant period, we have made the following important achievements: 1)We found a large BKT transition temperature due to large effective superfluid densities in a spin-orbit coupled Fulde-Ferrell superfluid, making it possible to observe 2D Fulde-Ferrell superfluids at finite temperature;2) Collaborated with Ian Speilmans group, we demonstrated the tunable spin-orbit coupling through the modulation of the Raman coupling strength in experiments; 3)We found that the spin-orbit coupled Fulde-Ferrell superfluids can support Weyl points or rings with nontrivial topological structures; 4)We proposed and experimentally realized a new type of tunable spin-orbit coupling using optical lattice band pseudospins. The research leads to 23 peer-reviewed publications, including 4 in Physical Review Letters and 1 in Nature Communications.
Recent experimental breakthroughs in realizing spin-orbit (SO) coupling for cold atoms have spurred considerable interest in the physics of two-dimensional SO coupled Fermi superfluids, especially topological Majorana fermions (MFs) which were predicted to exist at zero temperature. However, it is well known that long-range superfluid order is destroyed in two dimensions by phase fluctuations at finite temperature, and the relevant physics is the BKTs transition. In this Letter, we examine finite temperature effects on SO coupled Fermi gases and show that finite temperature is indeed necessary for the observation of MFs. Majorana fermions are topologically protected by a quasiparticle energy gap which is found to be much larger than the temperature. The restrictions to the parameter region for the observation of MFs have been obtained.
Pairings are the heart for exploring novel superfluid physics of degenerate fermi gases. In the presence of spin-orbit coupling (SOC), some exotic pairings, including the triplet ppp-wave pairing, have been predicted, which paves a new way to search topological superfluids and associated Majorana fermions. In this work, we reveal unconventional pairings induced by SOC in a one-dimensional optical lattice, using the exact density-matrix renormalization group. In the weak SOC, we find the interband Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) and intraband Bardeen-Cooper-Schrieffer (BCS) pairings may coexist, when both bands are partially occupied. This interesting coexistence can lead to a new quantum state, called the FFLO-BCS phase, which exhibits a unique three-peak structure in pairing momentum distribution. However, in the intermediate and strong SOCs, the intraband BCS pairing dominates in the whole parameter regime. These results shed great light on the realization of topological superfluids in the spin-orbit coupled lattice.