Majorana Edge States Research Articles

Realizing one-dimensional topological superfluids with ultracold atomic gases

We propose an experimental implementation of a topological superfluid with ultracold fermionic atoms. An optical superlattice is used to juxtapose a 1D gas of fermionic atoms and a 2D conventional superfluid of condensed Feshbach molecules. The latter acts as a Cooper pair reservoir and effectively induces a superfluid gap in the 1D system. Combined with a spin-dependent optical lattice along the 1D tube and laser-induced atom tunnelling, we obtain a topological superfluid phase. In the regime of weak couplings to the molecular field and for a uniform gas, the atomic system is equivalent to Kitaev’s model of a p-wave superfluid. Using a numerical calculation, we show that the topological superfluidity is robust beyond the perturbative limit and in the presence of a harmonic trap. Finally, we describe how to investigate some physical properties of the Majorana fermions located at the topological superfluid boundaries. In particular, we discuss how to prepare and detect a given Majorana edge state.

Chiralf-wave topological superfluid in triangular optical lattices

We demonstrate that an exotically chiral f-wave topological superfluid can be induced in coldfermionic-atom triangular optical lattices through the laser-field-generated effective non-Abelian gauge field, controllable Zeeman fields and s-wave Feshbach resonance. We find that the chiral f-wave topological superfluid is characterized by three gapless Majorana edge states located on the boundary of the system. More interestingly, these Majorana edge states degenerate into one Majorana fermion bound to each vortex in the superfluid. Our proposal enlarges topological superfluid family and specifies a unique experimentally controllable system to study the Majorana fermion physics.

Majorana edge states forZ2topological orders of the Wen plaquette and toric code models

In this paper, we study the symmetry protected Majorana edge states for the ${\mathbb{Z}}_{2}$ topological order of the Wen plaquette and toric code models and calculate the dispersion of the Majorana edge states. For the system with translational symmetry, the Majorana edge states are gapless and have the nodal points at $k=0$ and $\ensuremath{\pi}$. For the edge states of the toric code model without translational symmetry, the edge modes become gapped.

Josephson effect in superconducting wires supporting multiple Majorana edge states

We study superconducting-normal-superconducting (SNS) Josephson junctions in one-dimensional topological superconductors which support more than one Majorana end mode. The variation of the energy spectrum with the superconducting phase is investigated by combining numerical diagonalizations of tight-binding models describing the SNS junction together with an analysis of appropriate low-energy effective Hamiltonians. We show that the four pi-periodicity characteristic of the fractional dc Josephson effect is preserved. Additionally, the ideal conductance of a NS junction with a topological supraconductor, hosting two Majorana modes at its ends, is doubled compared to the single Majorana case. Last, we illustrate how a nonzero superconducting phase gradient can potentially destroy the phases supporting multiple Majorana end states.

Phase-locked magnetoconductance oscillations as a probe of Majorana edge states

We calculate the Andreev conductance of a superconducting ring interrupted by a flux-biased Josephson junction, searching for electrical signatures of circulating edge states. Two-dimensional pair potentials of spin-singlet d-wave and spin-triplet p-wave symmetry support, respectively, (chiral) Dirac modes and (chiral or helical) Majorana modes. These produce h/e-periodic magnetoconductance oscillations of amplitude \simeq (e^{2}/h)N^{-1/2}, measured via an N-mode point contact at the inner or outer perimeter of the grounded ring. For Dirac modes the oscillations in the two contacts are independent, while for an unpaired Majorana mode they are phase locked by a topological phase transition at the Josephson junction.

Majorana modes and complex band structure of quantum wires

We describe Majorana edge states of a semi-infinite wire using the complex band structure approach. In this method the edge state at a given energy is built as a superposition of evanescent waves. It is shown that the superposition can not always satisfy the required boundary condition, thus restricting the existence of edge modes. We discuss purely 1D and 2D systems, focusing in the latter case on the effect of the Rashba mixing term.

Quantum Thermal Hall Effect in a Time-Reversal-Symmetry-Broken Topological Superconductor in Two Dimensions: Approach from Bulk Calculations

We discuss thermal transport of two-dimensional topological superconductors (TSCs) with broken time reversal symmetry, which are described by Bogoliubov-de Gennes (BdG) Hamiltonians. From the calculations of bulk quantities only, without refereeing to Majorana edge states, we show that the thermal Hall conductivity of two-dimensional TSCs in the low-temperature limit is quantized in multiples of $1/2\frac{\pi T}{6}$, which is exactly one half of the value of quantization in the case of the integer quantum Hall effect, and that this exact half-quantization is caused by the structure of the Nambu spinor and the particle-hole symmetry, which BdG Hamiltonians generally have. In the case of spinless chiral p-wave superconductors, this result is in perfect agreement with the argument based on the Ising conformal field theory with the central charge $c=1/2$, which is an effective low-energy theory of the Majorana edge states.

Strongly interacting Fermi gases

Strongly interacting gases of ultracold fermions have become an amazingly rich test-bed for many-body theories of fermionic matter. Here we present our recent experiments on these systems. Firstly, we discuss high-precision measurements on the thermodynamics of a strongly interacting Fermi gas across the superfluid transition. The onset of superfluidity is directly observed in the compressibility, the chemical potential, the entropy, and the heat capacity. Our measurements provide benchmarks for current many-body theories on strongly interacting fermions. Secondly, we have studied the evolution of fermion pairing from three to two dimensions in these gases, relating to the physics of layered superconductors. In the presence of p-wave interactions, Fermi gases are predicted to display toplogical superfluidity carrying Majorana edge states. Two possible avenues in this direction are discussed, our creation and direct observation of spin-orbit coupling in Fermi gases and the creation of fermionic molecules of 23Na 40K that will feature strong dipolar interactions in their absolute ground state.

Majorana fermions in density-modulatedp-wave superconducting wires

We study the p-wave superconducting wire with a periodically modulated chemical potential and show that the Majorana edge states are robust against the periodic modulation. We find that the critical amplitude of modulated potential, at which the Majorana edge fermions and topological phase disappear, strongly depends on the phase shifts. For some specific values of the phase shift, the critical amplitude tends to infinity. The existence of Majorana edge fermions in the open chain can be characterized by a topological $Z_2$ invariant of the bulk system, which can be applied to determine the phase boundary between the topologically trivial and nontrivial superconducting phases. We also demonstrate the existence of the zero-energy peak in the spectral function of the topological superconducting phase, which is only sensitive to the open boundary condition but robust against the disorder.

Preparing and probing atomic Majorana fermions and topological order in optical lattices

We introduce a one-dimensional system of fermionic atoms in an optical lattice whose phase diagram includes topological states of different symmetry classes with a simple possibility to switch between them. The states and topological phase transitions between them can be identified by looking at their zero-energy edge modes which are Majorana fermions. We propose several universal methods of detecting the Majorana edge states, based on their genuine features: the zero-energy, localized character of the wave functions and the induced non-local fermionic correlations.

Electrical Manipulation of Majorana Fermions in an Interdigitated Superconductor-Ferromagnet Device

We show that a topological phase supporting Majorana fermions can form in a two-dimensional electron gas (2DEG) adjacent to an interdigitated superconductor-ferromagnet structure. An advantage of this setup is that the 2DEG can induce the required Zeeman splitting and superconductivity from a single interface, allowing one to utilize a wide class of 2DEGs including the surface states of bulk InAs. We demonstrate that the interdigitated device supports a robust topological phase when the finger spacing λ is smaller than half of the Fermi wavelength λ(F). In this regime, the electrons effectively see a "smeared" Zeeman splitting and pairing field despite the interdigitation. The topological phase survives even in the opposite limit λ > λ(F)/2, although with a reduced bulk gap. We describe how to electrically generate a vortex in this setup to trap a Majorana mode, and predict an anomalous Fraunhofer pattern that provides a sharp signature of chiral Majorana edge states.

Unwinding of a one-dimensional topological superconductor

We show that a topological superconductor made of four chains of superconducting spinless fermions characterized by four Majorana edge states can adiabatically be deformed into a trivial band insulator. To unwind this time-reversal invariant topological superconductor, interactions to spinful fermions are switched on along an adiabatic path. Thereby, we couple modes which belong to two different representations of the time-reversal symmetry operator $\mathcal{T}$ with ${\mathcal{T}}^{2}=1$ and ${\mathcal{T}}^{2}=\ensuremath{-}1$, respectively. This observation can be understood by investigating how the relevant symmetries act on the entanglement spectrum giving rise to four instead of eight different topological phases with Majorana edge modes. We also show that a simple level crossing of doubly and singly degenerate states occurs in the entanglement spectrum on deforming the quantum state.

Spin-Injection Spectroscopy of a Spin-Orbit Coupled Fermi Gas

The coupling of the spin of electrons to their motional state lies at the heart of recently discovered topological phases of matter. Here we create and detect spin-orbit coupling in an atomic Fermi gas, a highly controllable form of quantum degenerate matter. We directly reveal the spin-orbit gap via spin-injection spectroscopy, which characterizes the energy-momentum dispersion and spin composition of the quantum states. For energies within the spin-orbit gap, the system acts as a spin diode. We also create a spin-orbit coupled lattice and probe its spinful band structure, which features additional spin gaps and a fully gapped spectrum. In the presence of s-wave interactions, such systems should display induced p-wave pairing, topological superfluidity, and Majorana edge states.

Thermal metal-insulator transition in a helical topological superconductor

Two-dimensional superconductors with time-reversal symmetry have a Z_2 topological invariant, that distinguishes phases with and without helical Majorana edge states. We study the topological phase transition in a class-DIII network model, and show that it is associated with a metal-insulator transition for the thermal conductance of the helical superconductor. The localization length diverges at the transition with critical exponent nu approx 2.0, about twice the known value in a chiral superconductor.

Interacting topological phases and modular invariance

We discuss (2+1)-dimensional topological superconductors with ${N}_{f}$ left- and right-moving Majorana edge modes and a ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ symmetry. In the absence of interactions, these phases are distinguished by an integral topological invariant ${N}_{f}$. With interactions, the edge state in the case of ${N}_{f}=8$ is unstable against interactions, and a ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ invariant mass gap can be generated dynamically. We show that this phenomenon is closely related to the modular invariance of type II superstring theory. More generally, we show that the global gravitational anomaly of the nonchiral Majorana edge states is the physical manifestation of the bulk topological superconductors classified by ${\mathbb{Z}}_{8}$.

Evolution of edge states in topological superfluids during the quantum phase transition

The quantum phase transition between topological and non-topological insulators or between fully gapped superfluids/superconductors can occur without closing the gap. We consider the evolution of the Majorana edge states on the surface of topological superconductor during transition to the topologically trivial superconductor on example of non-interacting Hamiltonian describing the spin-triplet superfluid 3He-B. In conventional situation when the gap is nullified at the transition, the spectrum of Majorana fermions shrinks and vanishes after the transition to the trivial state. If the topological transition occurs without the gap closing, the Majorana fermion spectrum disappears by escaping to ultraviolet, where Green's function approaches zero. This demonstrates the close connection between the topological transition without closing the gap and zeroes in the Green's function. Similar connection takes place in interacting systems where zeroes may occur due to interaction.

Majorana edge states in interacting two-chain ladders of fermions

In this work we study interacting spinless fermions on a two-chain ladder with inter-chain pair tunneling while single-particle tunneling is suppressed at low energy. The model embodies a $\mathbb{Z}_2$ symmetry associated with the fermion parity on each chain. We find that when the system is driven to the strong-coupling phase by the pair tunneling, Majorana excitations appear on the boundary. Such Majorana edge states correspond to two-fold degeneracy of ground states distinguished by different fermion parity on each chain, thus representing a generalization of one-dimensional topological superconductors. We also characterize the stability of the ground state degeneracy against local perturbations. Lattice fermion models realizing such effective field theory are discussed.

Topological superconducting phase and Majorana fermions in half-metal/superconductor heterostructures

As a half-metal is spin polarized at its Fermi level by definition, it was conventionally thought to have little proximity effect to an $s$-wave superconductor. Here we show that with interface spin-orbit coupling ${p}_{x}+i{p}_{y}$ superconductivity without spin degeneracy is induced on the half-metal, and we give an estimate of its bulk energy gap. Therefore, a single-band half-metal can give us a topological superconductor with a single chiral Majorana edge state. Our band calculation shows that two atomic layers of VTe or CrO${}_{2}$ is a single-band half-metal for a wide range ($\ensuremath{\sim}$0.1 eV) of Fermi energy and thus is a suitable candidate material.

Majorana fermions in strongly interacting helical liquids

Majorana fermions were proposed to occur at edges and interfaces of gapped one-dimensional systems where phases with different topological character meet due to an interplay of spin-orbit coupling, proximity-induced superconductivity and external magnetic fields. Here we investigate the effect of strong particle interactions, and show that the helical liquid offers a mechanism that protects the very existence of Majorana edge states: whereas moderate interactions close the proximity gap which supports the edge states, in helical liquids the gap re-opens due to two-particle processes. However, gapless fermionic excitations occur at spatial proximity to the Majorana states at interfaces and may jeopardize their long term Majorana coherence.

Probing Majorana edge states with a flux qubit

A pair of counter-propagating Majorana edge modes appears in chiral p-wave superconductors and in other superconducting systems belonging to the same universality class. These modes can be described by an Ising conformal field theory. We show how a superconducting flux qubit attached to such a system couples to the two chiral edge modes via the disorder field of the Ising model. Due to this coupling, measuring the back-action of the edge states on the qubit allows to probe the properties of Majorana edge modes.