Session K25: Topological Superconductivities

Bogoliubov-Fermi Surfaces in Noncentrosymmetric Multicomponent Superconductors

We show that when the time reversal symmetry is broken in a multicomponent superconducting condensate without inversion symmetry the resulting Bogoliubov quasiparticles generically exhibit mini- Bogoliubov-Fermi (BF) surfaces, for small superconducting order parameter. The absence of inversion symmetry makes the BF surfaces stable with respect to weak perturbations. With sufficient increase of the order parameter, however, the Bogoliubov-Fermi surface may disappear through a Lifshitz transition, and the spectrum this way become fully gapped. Our demonstration is based on the computation of the effective Hamiltonian for the bands near the normal Fermi surface by the integration over high-energy states.

    

Self-consistent study of topological superconductivity in quasicrystals

We examine s-wave topological superconductivity (TSC) in two-dimensional Penrose and Ammann-Beenker quasicrystals (QCs) with Rashba spin-orbit coupling and Zeeman field by solving the Bogoliubov-de Gennes equations. The mean-field approximation is applied and the superconducting order parameter as well as the spin-dependent Hartree potential are obtained self-consistently. We find that the self-consistently obtained mean fields are spatially inhomogeneous in both QCs. We demonstrate how the underlying aperiodic structure of a QC is reflected in the superconducting order parameter. We also calculate the Bott index as the topological invariant of the system, which is equivalent to the first Chern number in the presence of translational symmetry. Our results confirm the existence of a stable TSC state in QCs and the appearance of a Majorana zero mode along the edges of a QC, despite the lack of translational symmetry.   

Topological superconductivity induced by two-dimensional nontrivial spin structures

Nontrivial spin textures often give rise to topologically nontrivial quantum states and associated unconventional magnetic, transport, and optical phenomena. Usually, the Dzyaloshinskii-Moriya interaction is considered as the stabilization mechanism. In this work, we show how the Ruderman-Kittel-Kasuya-Yosida-type interaction can lead to the formation of similar structures. It has been demonstrated that helical spin structures in superconducting nanowires and ladders can induce topological superconductivity with Majorana edge states. We generalize this result to two-dimensional systems, i.e., we show the stability of nontrivial spin structures in two-dimensional superconducting systems, where they also can lead to topological superconductivity with Majorana edge states. Such systems are promising for use in fault-tolerant quantum computing.   

Novel Surface States of a Spinor Superconductor

Recently, Berry phase enforced spinor superconducting orders have been proposed arising from pairing topological Fermi surfaces with their Chern numbers differing by an odd integer. This exotic pairing structure can exhibit a single pairing gap node on a Fermi surface and is described by monopole harmonics with half-integer charge and fractionalized half-integer partial-wave symmetry. We investigate topologically protected surface states in a simplest example of spinor pairing with pair monopole charge q|| = ½ when spineless fermions in a topological trivial Fermi surface pair with spin-½ electrons in a helical Fermi surface with Chern number -1 under hard-core interaction between them. We find exotic non-majorana states localized at surfaces of the system protected by the topological spinor superconducting order.   

A theoretical exploration of topological and surface superconductivity in layered Rashba superconductor β-PdBi2

Emergent excitations known as Majorana modes could be a platform for decoherence-resistant quantum computing. These Majorana modes are predicted to exist the boundaries of topological superconductors: materials with topological surface states and p-wave electron pairing. Layered superconductor β-PdBi2 satisfies the first criterion, with some evidence for a p-wave pairing component in thin films and at the surface under certain conditions. We have developed a tight binding model, from which we derive a continuum model to describe bulk and surface states in β-PdBi2. Using a self-consistent theoretical framework, we identify a magnetic field-induced first order phase transition between s- and effective p-wave states in a spin-helical Fermi surface resembling that of β-PdBi₂. We discuss the origin of this effect due to the interplay of spin-orbit coupling and magnetic fields, before extending our analysis to more closely describe surface contributions to superconductivity in β-PdBi₂.

    

Majorana-Weyl cones in ferroelectric superconductors

It has been theoretically shown that non-trivial topological superconductivity can be realized based on s-wave superconductivity, Rashba spin-orbit coupling, and a sufficiently strong magnetic field. We revisit this scenario in the context of 3D ferroelectric superconductors, e.g., SrTiO₃. First, we show that for a uniform Zeeman field, two or four Majorana-Weyl cones emerge when the field exceeds a critical value. We further show the Weyl cones can be tilted by tuning the angle between the ferroelectric polarization and the Zeeman field and may even "over-tilt" and become type-II Weyl points with Bogoliubov Fermi surfaces. We also show that tetragonal domain walls can, in some cases, become interfaces between domains characterized by opposite Majorana-Weyl cones' chiralities. Such domain walls will then host Majorana zero modes inside the specimen. To account for the realistic scenario, we then consider the orbital effect of the magnetic field. We show that single superconducting vortices are surrounded by topological "halos" hosting Majorana arc states on their outer boundaries and in their centers. As the value of the magnetic field increases, the density of the vortices grows, which leads to the overlapping of the topological "halos" and, eventually, to the percolation of the topological phase throughout the system. In the end, we discuss some experimental probes to test our predictions.   

Topological exciton superfluidity in driven quantum anomalous Hall insulators: Application to twisted bilayer graphene

Exciton states of interacting quantum anomalous Hall (QAH) insulators may inherit nontrivial Berry phases from the underlying electron bands. We investigate the possible onset of topological exciton superfluidity in the case of a two-band QAH insulator driven by a continuous wave laser of a resonant frequency, exploiting the Floquet-Keldysh formalism. We analyze the superfluid weight and the critical temperature in the steady state, focusing on the effects of the quantum metric of the electron bands, in analogy to those in flat-band superconductivity. We propose that the nonlinear optical current can reveal signatures of the topology of the exciton bands as well as the exciton condensate. We apply the theory to the magic-angle twisted bilayer graphene near the chiral limit, in which the flat bands may host ideal quantum geometry for realizing the topological exciton superfluid.   

Topological superconductivity mediated by magnons of a skyrmion crystal.

Topological superconductors are  associated with the appearance of Majorana modes.  One mechanism to produce topological superconductors is to interface a conventional superconductor with a non-collinear magnetically ordered state like a skyrmion. In this work, we instead study a system where a skyrmion crystal is interfaced with a normal metal. Through interfacial exchange coupling, spin fluctuations (magnons) in the skyrmion crystal mediate an effective electron-electron interaction in the normal metal. We show that this yields topological superconductivity at the interface. The non-collinearity of the magnetic ground state is crucial in order to obtain this. 

    

Topological bound for the superfluid weight of a Bose-Einstein condensate

The non-trivial quantum geometry of the Bloch bands allows fermionic systems with quenched kinetic energy to carry a supercurrent. The relevance of quantum geometry in bosonic systems was pointed out only recently. On the kagome lattice, both the speed of sound and the quantum depletion of a Bose-Einstein condensate have a geometric origin. The finite-momentum condensate, however, breaks translational symmetry and hinders possible connections between the geometry and the topology of the bands. Here, we consider a specific model with narrow isolated bands with non-trivial fragile topology. We successfully formulate a lower bound for the superfluid weight of the Bose-Einstein condensate in terms of the topological indices of the non-interacting bands. This result parallels what is known in fermionic systems and allows the identification of promising models for Bose-Einstein condensates in the limit of weak interactions.   

Weyl superconductivity induced by supercurrent flow

A superconducting gap structure plays an important role in discussing the symmetry of the order parameter and the pairing mechanism in unconventional superconductors. From the theoretical point of view, previous studies have constructed classification methods for predicting the superconducting gap structure by using symmetry and topology for three decades. On the other hand, previous theoretical studies have suggested that topological superconductivity can be realized by applying a Zeeman field or laser light to noncentrosymmetric d-wave superconductors. Furthermore, supercurrent is also considered as a promising external field for controlling gap structures and (topological) quantum states.

Stimulated by the above backgrounds, we propose that Weyl superconductivity can be realized by applying supercurrent to three-dimensional (3D) noncentrosymmetric line-nodal superconductors. We consider a 3D tight-binding model of a tetragonal superconductor in a D+p-wave pairing state with a finite center-of-mass momentum. The original line-nodal structure is modified into a point-nodal one, in which each point node is characterized by a nontrivial Weyl charge, by the cooperating effect of the supercurrent and spin-orbit coupling. In addition, a quantized Berry phase defined on high-symmetry planes characterizes the Weyl nodes when the in-plane supercurrent is considered. Our proposition paves a new way for controlling the superconducting gap structures by using an external field.   

Comparing topological edge state characteristics of chiral topological phases between theory and experiment

A defining characteristic of nontrivial topological electronic phases is the existence of gapless states on the edge or boundary, the so-called topological edge states. The presence of edge states can therefore serve as a means to identify topological phases in experiment. To enable and facilitate quantitative connections to experiments, in particular electron tunneling experiments, in this talk we present an extensive and detailed characterization of experimentally accessible edge state properties for a class of chiral topological phases, focusing in particular on chiral superconductors. We obtain exact solutions for the edge state wave function and energy dispersion in representative two-band tight-binding models with open boundary conditions, and show how important characteristics such as decay length and local density of states (LDOS) depend on quasiparticle band structure parameters. We further demonstrate that multiple approximation methods can provide excellent descriptions of these characteristics, which can be exploited when the lattice model tight-binding description becomes complicated. We apply our theory to the recently discovered superconducting system Sn/Si(111), for which signatures of fully gapped chiral pairing have been reported. We finally show how our results can be straightforwardly extended to more general topological phases, such as topological crystalline insulators and three-dimensional topological semimetals.   

Topology of twisted BSCCO

Twisted two-dimensional electronic systems are being extensively studied for realizing exotic correlated insulators and paired states. Recently, twisted bilayers of BSCCO have been proposed as material candidates for realizing time-reversal-symmetry-breaking paired states. Motivated by such proposals, we investigate topological properties of possible paired states, using a combined analysis of momentum-space invariants and real-space topological response. Apart from identifying ground states supporting net Chern numbers, we also characterize topologically non-trivial states with vanishing net Chern numbers. The employed theoretical framework is suitable for addressing the effects of electronic correlations and competing orders.   

Interplay of topological superconductivity and interactions at the edge of topological insulators: a DMRG perspective

It was proposed that the edge states of two dimensional topological insulators in the presence of superconductivity and magnetic domains host zero energy Majorana bound states at the domain boundaries. A pair of domain walls host a pair of Majorana fermions, leading to a twofold degenerate ground state. This degeneracy, in turn, can be used to define a highly non-local qubit that is resilient to local perturbations. However, time reversal symmetry in these systems is explicitly broken by the external magnetic field. Another way to gap the edge modes, without breaking this symmetry explicitly, can be achieved by interactions. Based on low energy bosonized calculations, there are also zero energy bound states expected to be localized at domain walls between interacting and superconducting regions. Due to the time reversal symmetry, a fourfold degenerate ground state is expected, which exhibit a richer phase diagram due to the interplay of interactions and topological superconductivity. 

In the present work we devise a simple model which is capable to encode key features of the edge states of topological insulators and arbitrary strong interactions in the presence of superconductivity. Utilizing density matrix renormalization group calculations we explore the phase diagram in terms of interactions preserving time reversal symmetry. We identify parameter ranges where a system can host topological interface states and characterize these through the Josephson current.   

Helical topological superconductivity in a magnetic field through spin-orbit coupling and superlattice potential

One of the approaches to engineering topological superconductivity (TSC) which supports gapless Majorana edge modes is to expose a 2D superconductor in a spin-orbit coupled material to a magnetic field. We investigate this question in a mean-field theory which obtains the superconducting order parameter, and hence the topology of the system, as well as the nature of the Abrikosov lattice, in a self-consistent fashion. We find small regions of TSC in parameter space. The region of TSC is greatly enhanced by the addition of a superlattice potential. The TSC is associated with either a disortion of the Abrikosov lattice or an Abrikosov lattice of giant vortices which carry two superconducting flux quanta. To find the lowest energy solution, it is essential to employ helical order parameters, although we find that the helical solutions do not carry a net supercurrent. Finally, we consider the experimental feasibility of this model.   

Spontaneous fractional Josephson current from parafermions

We study a parafermion Josephson junction (JJ) comprising a pair of counter-propagating edge modes of two quantum Hall (QH) systems, proximitized by an s-wave superconductor. We show that the difference between the lengths (which can be controlled by external gates) of the two counter-propagating chiral edges at the Josephson junction, can act as a source of spontaneous phase bias. For the Laughlin filling fractions, ν=1/m, m∈2Z+1, this leads to an electrical control of either Majorana (m=1) or parafermion (m≠1) zero modes. We also study edge reconstruction in these geometries.