[Paper Review] Majorana Doublets, Flat Bands, and Dirac Nodes in s-Wave Superfluids

Haiping Hu, Fan Zhang+1 more|arXiv (Cornell University)|Oct 17, 2017

Cold Atom Physics and Bose-Einstein Condensates4 citations

TL;DR

This paper proposes a novel realization of topological superfluids hosting symmetry-protected Majorana Kramers pairs (MKPs) and flat bands using s-wave superfluidity in two coupled 1D ultracold Fermi gases with spin-orbit coupling. By engineering emergent mirror and time-reversal symmetries via Hermite-Gaussian Raman beams, the authors demonstrate stable MKP flat bands that resist pairing fluctuations, offering a robust platform for topological quantum computation in experimentally accessible ultracold atomic systems.

ABSTRACT

Topological superfluids protected by mirror and time-reversal symmetries are exotic states of matter possessing Majorana Kramers pairs (MKPs), yet their realizations have long been hindered by the requirement of unconventional pairing. We propose to realize such a topological superfluid by utilizing $s$-wave pairing and emergent mirror and time-reversal symmetries in two coupled 1D ultracold atomic Fermi gases with spin-orbit coupling. By stacking such systems into 2D, we discover topological and Dirac-nodal superfluids hosting distinct MKP flat bands. We show that the emergent symmetries make the MKPs and their flat bands stable against pairing fluctuations that otherwise annihilate paired Majoranas. Exploiting new experimental developments, our scheme provides a unique platform for exploring MKPs and their applications in quantum computation.

Motivation & Objective

• To realize topological superfluids with Majorana Kramers pairs (MKPs) in ultracold atomic gases using conventional s-wave pairing, which avoids the need for exotic unconventional pairing.
• To address the challenge of pairing fluctuations destroying Majorana modes in coupled 1D chains by leveraging emergent symmetries for protection.
• To demonstrate the stability of MKP flat bands against pairing fluctuations through self-consistent Bogoliubov-de Gennes calculations.
• To provide a feasible experimental route using current ultracold atom techniques to observe MKPs and their non-Abelian statistics.
• To explore the emergence of both topological and Dirac-nodal superfluid phases in 2D arrays of such systems.

Proposed method

• Utilize two coupled 1D ultracold Fermi gas chains with synthetic spin-orbit coupling generated by counter-propagating Raman lasers, one with a Gaussian beam and the other with a Hermite-Gaussian (HG 01) beam to break time-reversal symmetry locally.
• Engineer a staggered Zeeman field via the Raman coupling, which leads to emergent time-reversal and mirror symmetries in the coupled system despite local TR breaking.
• Implement s-wave pairing interaction via Feshbach resonance, enabling intrinsic superfluidity without requiring proximity-induced pairing.
• Construct a 2D array of such 1D chains to suppress quantum fluctuations and stabilize topological superfluid phases with protected MKP flat bands.
• Perform self-consistent Bogoliubov-de Gennes (BdG) calculations to determine the quasiparticle spectrum, local density of states (LDOS), and topological invariants.
• Apply local density approximation (LDA) to analyze spatial splitting of MKPs in a harmonic trap under a Zeeman field, deriving analytical expressions for mode separation.

Experimental results

Research questions

• RQ1 Can time-reversal-invariant topological superfluids hosting Majorana Kramers pairs be realized in ultracold atomic gases using only s-wave pairing?
• RQ2 Can emergent mirror and time-reversal symmetries stabilize MKPs and their flat bands against pairing fluctuations that typically annihilate paired Majoranas in coupled chains?
• RQ3 What are the topological phase transitions and the nature of the superfluid phases (topological vs. Dirac-nodal) in 2D arrays of such coupled 1D systems?
• RQ4 How do the spatial positions of MKPs evolve under a Zeeman field, and what is the scaling of their splitting with field strength?
• RQ5 Can the experimental signatures of MKPs, such as zero-energy modes and their degeneracy, be distinguished from those of single Majorana modes?

Key findings

• The system realizes a topological superfluid phase with a non-zero Z2 invariant and hosts Majorana Kramers pairs (MKPs) at the edges, confirmed by a two-fold degenerate zero-energy mode in the local density of states (LDOS).
• The MKP flat bands are stabilized by emergent mirror and time-reversal symmetries, which protect them against pairing fluctuations that would otherwise destroy the Majorana modes in coupled 1D chains.
• In the 2D array, the system exhibits both topological superfluidity and Dirac-nodal superfluidity, each hosting distinct flat bands of MKPs, with the latter characterized by linearly dispersing Dirac nodes in the quasiparticle spectrum.
• Under a Zeeman field, the MKPs spatially split into two Majorana zero modes: one moves toward the trap center and the other toward the trap edge, with the splitting rate scaling as δx ∼ h² for small h, consistent with analytical and numerical results.
• At a critical Zeeman field strength h ≈ 0.36, the two Majorana modes from opposite sides collide and annihilate at the trap center, leaving only one mode per wing, confirming the non-Abelian nature of the MKP.
• The analytical derivation shows that the chemical potential shift due to Zeeman field is proportional to h², explaining the slow initial splitting rate observed in the LDOS and gap closure condition analysis.