Focus on Majorana Fermions in Condensed Matter

Chiral Majorana-fermion modes are shown to emerge as edge excitations in a superconductor–topological-insulator hybrid structure that is subject to a magnetic field. The velocity of this mode is tunable by changing the magnetic-field magnitude and/or the superconductor's chemical potential. We discuss how quantum-transport measurements can yield experimental signatures of these modes. A normal lead coupled to the Majorana-fermion edge state through electron tunneling induces resonant Andreev reflections from the lead to the grounded superconductor, resulting in a distinctive pattern of differential-conductance peaks.

We consider a voltage-biased Josephson junction between two nanowires hosting Majorana zero modes which occur as topological protected zero-energy excitations at the junction. We show that two Majorana fermions localized at the junction, despite being neutral particles, interact with the electromagnetic field and generate coherent radiation similar to the conventional Josephson radiation. Within a semiclassical analysis of the radiation field, we find that the phase of the radiation gets locked to the superconducting phase difference and that the radiation is emitted at half the Josephson frequency. In order to confirm the coherence of the radiation, we study correlations of the radiation emitted by two spatially separated junctions in a dc-SQUID geometry taking into account decoherence due to spontaneous state-switches as well as due to quasi-particle poisoning.

A prominent signature of Majorana bound states is the exotic Josephson effects they produce, the classic example being a fractional Josephson current with 4π periodicity in the phase difference across the junction. Recent work established that topological insulator edges support a novel 'magneto-Josephson effect', whereby a dissipationless current exhibits 4π-periodic dependence also on the relative orientation of the Zeeman fields in the two banks of the junction. Here, we explore the magneto-Josephson effect in junctions based on spin–orbit-coupled quantum wires. In contrast to the topological insulator case, the periodicities of the magneto-Josephson effect no longer follow from an exact superconductor–magnetism duality of the Hamiltonian. We employ numerical calculations as well as analytical arguments to identify the domain configurations that display exotic Josephson physics for quantum-wire junctions, and elucidate the characteristic differences with the corresponding setups for topological insulators edges. To provide guidance to experiments, we also estimate the magnitude of the magneto-Josephson effects in realistic parameter regimes, and compare the Majorana-related contribution to the coexisting 2π-periodic effects emerging from non-Majorana states.

I perform a complete classification of two-dimensional, quasi-one-dimensional (1D) and 1D topological superconductors which originate from the suitable combination of inhomogeneous Rashba spin–orbit coupling, magnetism and superconductivity. My analysis reveals alternative types of topological superconducting platforms for which Majorana fermions are accessible. Specifically, I observe that for quasi-1D systems with Rashba spin–orbit coupling and time-reversal violating superconductivity, such as for instance due to a finite Josephson current flow, Majorana fermions can emerge even in the absence of magnetism. Furthermore, for the classification I also consider situations where additional 'hidden' symmetries emerge, with a significant impact on the topological properties of the system. The latter generally originate from a combination of space group and complex conjugation operations that separately do not leave the Hamiltonian invariant. Finally, I suggest alternative directions in topological quantum computing for systems with additional unitary symmetries.

We consider a superconductor–two dimensional topological insulator–superconductor junction and study how the 2π- and 4π-periodic Josephson currents are affected by the electron–electron interaction. In the long-junction limit the supercurrent can be evaluated by modeling the system as a helical Luttinger liquid coupled to superconducting reservoirs. After having introduced bosonization in the presence of the parity constraint we turn to consider the limit of perfect and poor interfaces. For transparent interfaces, where perfect Andreev reflections occur at the boundaries, the Josephson current is marginally affected by the interaction. However, if strong magnetic scatterers are present in the weak link, the situation changes dramatically. Here Coulomb interaction plays a crucial role both in low and high temperature regimes. Furthermore, a phase-shift of Josephson current can be induced by changing the direction of the magnetization of the impurity.

A double quantum dot coupled to an s-wave superconductor and subject to an inhomogeneous magnetic field can host a pair of zero-energy Majorana fermions when the dot properties are tuned appropriately. Here, we demonstrate the possibility of generating a fractional 4π Josephson effect in two such double dots tunnel-coupled to each other. We discuss the robustness of this effect with respect to perturbations away from the special point in parameter space where the uncoupled double dots host Majorana fermions. We demonstrate the possibility of generating Josephson effects with a period of 8π and 12π in strongly coupled double dots.

HgTe-based quantum wells (QWs) recently attracted a lot of attention for the realization of a two-dimensional topological insulator with protected helical edge states. Another class of topological systems is topological superconductors (TSCs) with Majorana edge states. In this paper, we show how proximity induced s-wave superconductivity in the bulk of HgTe-QWs and in the presence of a Zeeman field can exhibit a TSC with chiral Majorana edge states. We calculate the topological invariants and the corresponding Majorana edge states explicitly within a four-band model accounting for inversion symmetry breaking terms due to the Rashba spin–orbit coupling and bulk inversion asymmetry present in these QWs.

Semiconductor–superconductor hybrid systems are promising candidates for the realization of Majorana fermions and topological order, i.e. topologically protected degeneracies, in solid state devices. We show that the topological order is mirrored in the excitation spectra and can be observed in nonlinear Coulomb blockade transport through a ring-shaped nanowire. Especially, the excitation spectrum is almost independent of magnetic flux in the topologically trivial phase but acquires a characteristic h/e magnetic flux periodicity in the non-trivial phase. The transition between the trivial and non-trivial phase is reflected in the closing and reopening of an excitation gap. We show that the signatures of topological order are robust against details of the geometry, electrostatic disorder and the existence of additional subbands and only rely on the topology of the nanowire and the existence of a superconducting gap. Finally, we show that the coherence length in the non-trivial phase is much longer than in the trivial phase. This opens the possibility to coat the nanowire with superconducting nanograins and thereby significantly reduce the current due to cotunnelling of Cooper pairs and to enhance the Coulomb charging energy without destroying the superconducting gap.

We study the boundary Majorana modes for the single component p-wave weak topological superconductors or superfluids, which form zero energy flat bands protected by time-reversal symmetry in the orbital channel. However, due to the divergence of density of states, the band flatness of the edge Majorana modes is unstable under spontaneously generated spatial variations of Cooper pairing phases. Staggered current loops appear near the boundary and thus time-reversal symmetry is spontaneously broken in the orbital channel. This effect can appear in both condensed matter and ultra-cold atom systems.

Topological states of fermionic matter can be induced by means of a suitably engineered dissipative dynamics. Dissipation then does not occur as a perturbation, but rather as the main resource for many-body dynamics, providing a targeted cooling into topological phases starting from arbitrary initial states. We explore the concept of topological order in this setting, developing and applying a general theoretical framework based on the system density matrix that replaces the wave function appropriate for the discussion of Hamiltonian ground-state physics. We identify key analogies and differences to the more conventional Hamiltonian scenario. Differences essentially arise from the fact that the properties of the spectrum and of the state of the system are not as tightly related as in the Hamiltonian context. We provide a symmetry-based topological classification of bulk steady states and identify the classes that are achievable by means of quasi-local dissipative processes driving into superfluid paired states. We also explore the fate of the bulk-edge correspondence in the dissipative setting and demonstrate the emergence of Majorana edge modes. We illustrate our findings in one- and two-dimensional models that are experimentally realistic in the context of cold atoms.

We study transport in a voltage-biased superconductor–normal–superconductor (SNS) junction made of semiconducting nanowires with strong spin–orbit coupling, as it transitions into a topological superconducting phase for increasing Zeeman field. Despite the absence of a fractional steady-state ac Josephson current in the topological phase, the dissipative multiple Andreev reflection current I_dc at different junction transparencies is particularly revealing. It exhibits unique features related to topology, such as gap inversion, the formation of Majorana bound states and fermion-parity conservation. Moreover, the critical current I_c, which remarkably does not vanish at the critical point where the system becomes gapless, provides direct evidence of the topological transition.

We theoretically investigate the effect of intertube tunneling in topological superfluid phases of a quasi-one-dimensional Fermi gas with a Rashba-type spin–orbit interaction. It is shown that the effective Hamiltonian is analogous to that of a nanowire topological superconductor with multibands. Using a hidden mirror symmetry in the system, we introduce a new topological number that ensures the existence of non-Abelian Majorana zero modes even in the presence of intertube tunneling. It is demonstrated from the full numerical calculation of self-consistent equations that some of the Majorana modes survive against the intertube tunneling, when the number of one-dimensional tubes is odd in the y-direction. We also discuss a generalization of our consideration to nanowire topological superconductors.

We represent the $\mathbb {Z}_2$ topological invariant characterizing a one-dimensional topological superconductor using a Wess–Zumino–Witten dimensional extension. The invariant is formulated in terms of the single-particle Green's function which allows us to classify interacting systems. Employing a recently proposed generalized Berry curvature method, the topological invariant is represented independent of the extra dimension requiring only the single-particle Green's function at zero frequency of the interacting system. Furthermore, a modified twisted boundary conditions approach is used to rigorously define the topological invariant for disordered interacting systems.

The classification of topological states of matter depends on spatial dimension and symmetry class. For non-interacting topological insulators and superconductors, the topological classification is obtained systematically and non-trivial topological insulators are classified by either integer or Z₂. The classification of interacting topological states of matter is much more complicated and only special cases are understood. In this paper we study a new class of topological superconductors in (2 + 1) dimensions which has time-reversal symmetry and a $\mathbb {Z}_2$ spin conservation symmetry. We demonstrate that the superconductors in this class are classified by $\mathbb {Z}_8$ when electron interaction is considered, while the classification is $\mathbb {Z}$ without interaction.

We systematically study gapless topological phases of (semi-)metals and nodal superconductors described by Bloch and Bogoliubov–de Gennes Hamiltonians. Using K-theory, a classification of topologically stable Fermi surfaces in (semi-)metals and nodal lines in superconductors is derived. We discuss a generalized bulk–boundary correspondence that relates the topological features of the Fermi surfaces and superconducting nodal lines to the presence of protected zero-energy states at the boundary of the system. Depending on the case, the boundary states are either linearly dispersing (i.e. Dirac or Majorana states) or dispersionless, forming two-dimensional surface flat bands or one-dimensional arc surface states. We study examples of gapless topological phases in symmetry classes AIII and DIII, focusing in particular on nodal superconductors, such as nodal noncentrosymmetric superconductors. For some cases we explicitly compute the surface spectrum and examine the signatures of the topological boundary states in the surface density of states. We also discuss the robustness of the surface states against disorder.

We report on several low-temperature experiments supporting the presence of Majorana fermions in superconducting lead nanowires fabricated with a scanning tunneling microscope (STM). These nanowires are the connecting bridges between the STM tip and the sample resulting from indentation–retraction processes. We show here that by a controlled tuning of the nanowire region, in which superconductivity is confined by applied magnetic fields, the conductance curves obtained in these situations are indicative of topological superconductivity and Majorana fermions. The most prominent feature of this behavior is the emergence of a zero bias peak in the conductance curves, superimposed on a background characteristic of the conductance between a normal metal and a superconductor in the Andreev regime. The zero bias peak emerges in some nanowires when a magnetic field larger than the lead bulk critical field is applied. This field drives one of the electrodes into the normal state while the other, the tip, remains superconducting on its apex. Meanwhile a topological superconducting state appears in the connecting nanowire of nanometric size.

Multi-channel spin–orbit quantum wires, when subjected to a magnetic field and proximity coupled to an s-wave superconductor, may support Majorana states. We study what happens to these systems in the presence of disorder. Inspired by the widely established theoretical methods of mesoscopic superconductivity, we develop á la Eilenberger a quasiclassical approach to topological nanowires valid in the limit of strong spin–orbit coupling. We find that the 'Majorana number' ${\cal M}$, distinguishing between the state with Majorana fermions (symmetry class B) and no Majorana fermions (class D), is given by the product of two Pfaffians of gapped quasiclassical Green's functions fixed by the right and left terminals connected to the wire. A numerical solution of the Eilenberger equations reveals that the class D disordered quantum wires are prone to the formation of the zero-energy anomaly (class D impurity spectral peak) in the local density of states that shares the key features of the Majorana peak. In this way, we confirm the robustness of our previous conclusions (Bagrets and Altland 2012 Phys. Rev. Lett. 109 227005) on a more restrictive system setup. Generally speaking, we find that the quasiclassical approach provides a highly efficient means to address disordered class D superconductors both in the presence and in the absence of topological structures.

We study tunneling spectroscopy of subgap Andreev states in a superconducting system and discuss the general situation when the discrete nature of these levels is relevant and thus the standard semiclassical result for tunneling conductance being proportional to the density of states is not applicable. If the tunneling coupling is weak, individual levels are resolved and the conductance G(V ) at low temperatures is composed of a set of resonant Lorentz peaks which cannot be described within perturbation theory over tunneling strength. We establish a general formula for the peak widths and heights and show that the width of any peak scales as normal-state tunnel conductance, while its height is ≲2e² h⁻¹ and depends only on contact geometry and the spatial profile of the resonant Andreev level. We also establish an exact formula for the single-channel conductance that takes the whole Andreev spectrum into account, and use it to study the interference of Andreev reflection processes through different levels. We study tunneling conductance at finite bias G(eV >0) for a system with a pair of almost decoupled Majorana fermions and derive the conditions for the 'universal' zero-bias peak with the height 2e²/h to be observed in a realistic system which always hosts an even number of Majorana fermions.

We suggest a way to overcome the obstacles that disorder and high density of states pose to the creation of unpaired Majorana fermions in one-dimensional systems. This is achieved by splitting the system into a chain of quantum dots, which are then tuned to the conditions under which the chain can be viewed as an effective Kitaev model, so that it is in a robust topological phase with well-localized Majorana states in the outermost dots. The tuning algorithm that we develop involves controlling the gate voltages and the superconducting phases. Resonant Andreev spectroscopy allows us to make the tuning adaptive, so that each pair of dots may be tuned independently of the other. The calculated quantized zero bias conductance serves then as a natural proof of the topological nature of the tuned phase.

We propose the existence of a substantial charge current parallel to the interface between a noncentrosymmetric superconductor and a metallic ferromagnet. Our analysis focuses upon two complementary orbital-angular-momentum pairing states of the superconductor, exemplifying topologically nontrivial states which are gapped and gapless in the bulk, respectively. Utilizing a quasiclassical scattering theory, we derive an expression for the interface current in terms of Andreev reflection coefficients. Performing a systematic study of the current, we find stark qualitative differences between the gapped and gapless superconductors, which reflect the very different underlying topological properties. For the fully gapped superconductor, there is a sharp drop in the zero-temperature current as the system is tuned from a topologically nontrivial to a trivial phase. We explain this in terms of the sudden disappearance of the contribution to the current from the subgap edge states at the topological transition. The current in the gapless superconductor is characterized by a dramatic enhancement at low temperatures, and exhibits a singular dependence on the exchange-field strength in the ferromagnetic metal at zero temperature. This is caused by the energy shift of the strongly spin-polarized nondegenerate zero-energy flat bands due to their coupling to the exchange field. We argue that the interface current provides a novel test of the topology of the superconductor, and discuss prospects for the experimental verification of our predictions.

We study a general Majorana junction, where N helical nanowires are connected to a common s-wave superconductor proximity-inducing Majorana bound states in the wires. The normal part of each wire (j = 1,...,N) acts as connected lead, where electrons can tunnel into the respective Majorana state γ_A,j. The Majorana states at the other end, γ_B,j, are coupled to each other by an arbitrary tunnel matrix. We examine the conditions for even–odd parity effects in the tunnel conductance for various junction topologies.

We investigate Josephson currents in mesoscopic rings with a weak link which are in or near a topological superconducting phase. As a paradigmatic example, we consider the Kitaev model of a spinless p-wave superconductor in one dimension, emphasizing how this model emerges from more realistic settings based on semiconductor nanowires. We show that the flux periodicity of the Josephson current provides signatures of the topological phase transition and the emergence of Majorana fermions (MF) situated on both sides of the weak link even when fermion parity is not a good quantum number. In large rings, the MF hybridize only across the weak link. In this case, the Josephson current is h/e periodic in the flux threading the loop when fermion parity is a good quantum number but reverts to the more conventional h/2e periodicity in the presence of fermion-parity changing relaxation processes. In mesoscopic rings, the MF also hybridize through their overlap in the interior of the superconducting ring. We find that in the topological superconducting phase, this gives rise to an h/e-periodic contribution even when fermion parity is not conserved and that this contribution exhibits a peak near the topological phase transition. This signature of the topological phase transition is robust to the effects of disorder. As a byproduct, we find that close to the topological phase transition, disorder drives the system deeper into the topological phase. This is in stark contrast to the known behavior far from the phase transition, where disorder tends to suppress the topological phase.

An array of superconducting islands with semiconducting nanowires in the right regime provides a macroscopic implementation of Kitaev's toy model for Majorana wires. We show that a capacitive coupling between adjacent islands leads to an effective interaction between the Majorana modes. We demonstrate that, even though strong repulsive interaction eventually drives the system into a Mott insulating state, the competition between the (trivial) band insulator and the (trivial) Mott insulator leads to an interjacent topological insulating state for arbitrary strong interactions.

We show that weak antilocalization by disorder competes with resonant Andreev reflection from a Majorana zero mode to produce a zero-voltage conductance peak of order e²/h in a superconducting nanowire. The phase conjugation needed for quantum interference to survive a disorder average is provided by particle–hole symmetry—in the absence of time-reversal symmetry and without requiring a topologically nontrivial phase. We identify methods of distinguishing the Majorana resonance from the weak antilocalization effect.

We study a quantum phase transition in a system of dipoles confined in a stack of N identical one-dimensional lattices (tubes) polarized perpendicularly to the lattices. In this arrangement, the intra-lattice interaction is purely repulsive, preventing system collapse, and the inter-lattice interaction is attractive. The dipoles may represent polar molecules or indirect excitons. The transition separates two phases; in one of them, superfluidity (understood as algebraic decay of the corresponding correlation functions) takes place in each individual lattice, and in the other (chain superfluid) the order parameter is the product of bosonic operators from all lattices. We argue that in the presence of finite inter-lattice tunneling the transition belongs to the universality class of the q = N two-dimensional classical Potts model. For N = 2,3,4 the corresponding low-energy field theory is the model of Z_N parafermions perturbed by the thermal operator. The results of Monte Carlo simulations are consistent with these predictions. The detection scheme for the chain superfluid is outlined.