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arXiv:1401.7343

Majorana edge modes protected by emergent symmetry in a one dimensional fermi gas

Jonathan Ruhman, Ehud Altman

Published Jan 28, 2014• 0 citations
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Abstract

We show that a one dimensional ultra-cold Fermi gas with Rashba-like spin orbit coupling, a Zeeman field and intrinsic attractive interactions exhibits a novel topological superfluid state, which forms in spite of total number conservation and the absence of a single particle gap. Majorana zero modes are localized to the interface between a topological region in the middle of the trap and trivial regions at its wings. Unlike the realization of a topological superconductor in proximity coupled nano-wires, the Majorana modes do not carry a quantum number associated with the total fermion parity. Instead, the topological degeneracy is protected by an emergent $Z_2$ symmetry present only at low energies. We discuss the experimental implications of the novel zero modes, as manifest for example in the response to modulation of a local potential near the position of the Majorana bound states. For the range of interaction strength corresponding to Luttinger parameter $12$, on the other hand the zero-mode can be detected as a sharp low frequency response at an energy which generically scales with system size as $1/L^{K/2}$ and is therefore parametrically separated from the phonons.

DOI
10.48550/arxiv.1401.7343

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We show that a one dimensional ultra-cold Fermi gas with Rashba-like spin orbit coupling, a Zeeman field and intrinsic attractive interactions exhibits a novel topological superfluid state, which forms in spite of total number conservation and the absence of a single particle gap. Majorana zero modes are localized to the interface between a topological region in the middle of the trap and trivial regions at its wings. Unlike the realization of a topological superconductor in proximity coupled nano-wires, the Majorana modes do not carry a quantum number associated with the total fermion parity. Instead, the topological degeneracy is protected by an emergent Z2Z_2Z2​ symmetry present only at low energies. We discuss the experimental implications of the novel zero modes, as manifest for example in the response to modulation of a local potential near the position of the Majorana bound states. For the range of interaction strength corresponding to Luttinger parameter 121212, on the other hand the zero-mode can be detected as a sharp low frequency response at an energy which generically scales with system size as 1/LK/21/L^{K/2}1/LK/2 and is therefore parametrically separated from the phonons.
We describe a novel topological superfluid state, which forms in a one-dimensional Fermi gas with Rashba-like spin-orbit coupling, a Zeeman field, and intrinsic attractive interactions. In spite of total number conservation and the presence of gapless excitations, Majorana-like zero modes appear in this system and can be linked with interfaces between two distinct phases that naturally form at different regions of the harmonic trap. As a result, the low lying collective excitations of the system, including the dipole oscillations and the long-wavelength phonons are all doubly degenerate. While backscattering from point impurities can lead to a splitting of the degeneracies that scales algebraically with the system size, the smooth confining potential can only cause an exponentially small splitting. We show that the topological state can be uniquely probed by a pumping effect induced by a slow sweep of the Zeeman field from a high initial value down to zero. The effect is expected to be robust to introducing a finite temperature as long as it is much smaller than the interaction induced single particle gap in the final state of the sweep.
Majorana zero modes are the hallmark of topological superconductivity. In one-dimensional systems, these zero modes are usually introduced in the context of gapped, mean-field models that do not conserve particle number, such as the Kitaev chain. By non-locally encoding a conventional fermion across spatially separated Majorana zero modes, these systems become inherently immune to local decoherence. In this work, we show that signatures of Majorana edge physics persist in a number-conserving, gapless Luttinger liquid of spinless fermions with short-range attractive interactions. We identify the two-point correlator as a sharp diagnostic, revealing an edge-to-edge revival whose sign depends on the fermion-number parity. This revival is robust in the thermodynamic limit, and persists in the excited states of the system and at different fillings. A simple particle-hole ansatz for the ground state of the system with an odd number of fermions captures the physics of the system for a wide range of interaction strengths, interpolating between the free-fermion limit and the strongly interacting Majorana regime. Finally, we propose a concrete protocol to realize this model with ultracold dipolar molecules or atoms in an optical lattice, and to detect the revival via beam-splitter interferometry, opening an experimental route to Majorana physics beyond the conventional gapped-superconductor paradigm.
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.
We theoretically predict an exotic topological superfluid state with a spatially modulated pairing gap in one-dimensional spin-orbit-coupled Fermi gases. This inhomogeneous topological superfluidity is induced by applying simultaneously a perpendicular Zeeman magnetic field and an equally weighted Rashba and Dresselhaus spin-orbit coupling in one-dimensional optical lattices. Based on the self-consistent Bogoliubov-de Gennes theory, we confirm that this novel topological phase is a unique condensation of Cooper pairs, which manifests the interplay between the inhomogeneity of a superfluid and its nontrivial topological structure. The properties of the emergent Majorana bound states are investigated in detail by examining the associated Z2 topological number, the eigenenergy and density of states spectra, as well as the wave functions of the localized Majorana end modes. The experimental feasibility of observing this new topological state of matter is also discussed.
The realization and manipulation of topological Majorana zero modes (MZMs) have been pivotal in both theoretical and experimental research. In this paper, we present a symmetry enhanced topological model that supports stable, localized MZMs in a one-dimensional interacting fermionic system. The MZMs in our model, induced by particle-number conserving interactions, are fully decoupled from the chain's interior and remain degenerate regardless of chain length, allowing for simulations in extremely short fermionic chains. The topological phase transition is driven by changes in coupling parameters, transitioning to a trivial phase without affecting the MZMs' independence or degeneracy, only their topological protection. Additionally, local symmetries in the model enhance MZMs' stability and reveal unique odd-even effects across different chain lengths.
Semiconducting nanowires in proximity to superconductors are among promising candidates to search for Majorana fermions and topologically protected degeneracies, which may ultimately be used as building blocks for topological quantum computers. The prediction of neutral Majorana fermions in the proximity-induced superconducting systems ignores number conservation and thus leaves open the conceptual question of how a topological degeneracy that is robust to all local perturbations arises in a number-conserving system. In this work, we study how local attractive interactions generate a topological ground-state near-degeneracy in a quasi-one-dimensional superfluid using bosonization of the fermions. The local attractive interactions opens a topological quasiparticle gap in the odd channel wires (with more than one channel) with end Majorana modes associated with a topological near-degeneracy. We explicitly study the robustness of the topological degeneracy to local perturbations and find that such local perturbations result in quantum phase slips, which split of the topological degeneracy by an amount that does not decrease exponentially with the length of the wire, but still decreases rapidly if the number of channels is large. Therefore a bulk superconductor with a large number of channels is crucial for true topological degeneracy.