Session G65: Topological Superconductors: Theory I

Coexistence and Competition between topological superconducting phases in FeSe0.45Te0.55

Identifying the microscopic origin of topological superconductivity in FeSe_0.45Te_0.55 has remained an open question. This origin was initially proposed to arise from a band-inversion involving the bulk p_z and d_xz-bands -- rendering FeSe_0.45Te_0.55 a 3D topological insulator -- and the gapping of the ensuing surface Dirac cone by proximity induced superconductivity (the 3DTI mechanism) However, recent ARPES and quantum sensing experiments have reported strong evidence for the existence of ferromagnetism on the surface of FeSe_0.45Te_0.55. This has opened a new path to the emergence of topological surface superconductivity in FeSe_0.45Te_0.55 as arising from the interplay of ferromagnetism, a superconducting gap of s_±-wave symmetry, and a Rashba spin-orbit interaction. In this talk, I will discuss the physical properties and topological phases that arise from the coexistence and competition between the two potential mechanisms. In particular, I will demonstrate that their coupling can lead to a suppression of the topological phase arising from the 3DTI mechanism, and will discuss the experimental signatures of such suppression.   

Topological Surface Superconductivity in FeSe0.45Te0.55

The engineering of Majorana zero modes in topological superconductors, a new paradigm for the realization of topological quantum computing and topology-based devices, has been hampered by the absence of materials with sufficiently large superconducting gaps. Recent experiments, however, have provided enthralling evidence for the existence of topological surface superconductivity in the iron-based superconductor FeSe_0.45Te_0.55 possessing a full s_±-wave gap of a few meV. In this talk, I propose a mechanism for the emergence of topological superconductivity on the surface of FeSe_0.45Te_0.55 by demonstrating that the interplay between the s_±-wave symmetry of the superconducting gap, surface magnetism, and a Rashba spin-orbit interaction gives rise to robust topological superconducting phases. Moreover, the proposed mechanism explains a series of experimentally observed hallmarks of topological superconductivity, such as the emergence of Majorana zero modes in the center of vortex cores and at the end of line defects, as well as of chiral Majorana edge modes along certain types of domain walls. I also propose that the spatial distribution of supercurrents near a domain wall is a characteristic signature measurable via a scanning superconducting quantum interference device that can distinguish between chiral Majorana edge modes and trivial in-gap states.   

Shot-noise and differential conductance as signatures of putative topological superconductivity in FeSe0.45Te0.55

I present a theory for the differential shot noise, dS/dV, as measured via shot-noise scanning tunneling spectroscopy, and the differential conductance, dI/dV, for tunneling into Majorana zero modes (MZMs) in the putative topological superconductor FeSe_0.45Te_0.55. I show that for tunneling into chiral Majorana edge modes near domain walls, as well as MZMs localized in vortex cores and at the end of defect lines, dS/dV vanishes whenever dI/dV reaches a quantized value proportional to the quantum of conductance. These results are independent of the orbital tunneling path, thus establishing a vanishing dS/dV concomitant with a quantized dI/dV as universal signatures for Majorana modes in two-dimensional topological superconductors, irrespective of the material's specific complex electronic band structure.   

Majorana States from Magnetic Skyrmions-Superconductor Interface with Spin-orbit Coupling

We studied topological superconductivity arising at the interface of skyrmion materials in proximity to an s-wave superconductor in the presence of strong spin-orbit coupling. We found Majorana zero modes localized at the center and at the skyrmion radius of Neel and Bloch types of skyrmions. This work provides a potential pathway towards a useful platform for the manipulation of Majorana fermions via controlling the magnetic skyrmions. 

C.F.B.L. and C.S. contributed equally to this work.    

Mean-field phase diagram of negative-U Hofstadter-Hubbard model

It has been predicted that topological superconductivity supporting Majorana quasiparticles can

be produced by proximity coupling the Dirac metal at the surface of a topological insulator to an

ordinary superconductor, effectively giving rise to p-wave superconductivity. We investigate here

the phase diagram of a p-wave superconductor in two dimensions, modeled as a system of spinless

electrons on a lattice with nearest neighbor attractive interaction, exposed to a magnetic field. We

solve the mean-field Bogoliubov-de Gennes equations, in both the pairing and density channels, in

a self-consistent, gauge-invariant fashion. We find that as the interaction strength is increased, the

system first makes a transition from a quantum Hall phase to a skyrmion lattice phase that is fully

gapped in the bulk but has topological chiral edge current, followed by a vortex phase in which

either the vortices form a lattice with one vortex per unit cell, and the fermionic spectrum contains

a low-energy Majorana band, or the vortices form dimers with two vortices per unit cell, and the

low-energy band is gapped. The observable consequences of skyrmions as well as Majorana fermions

are indicated.   

Time-dependent manipulation of YSR states and Majorana zero modes

The realization of topological quantum computing using topological superconductors requires the ability to manipulate Majorana zero modes at the nanoscale in real time. In this talk, I present a non-equilibrium formalism that allows one to study the real time response of superconductors to external perturbations.

In particular, I will show how this formalism can be used to study the emergence and manipulation of YSR states in s-wave superconductors in response to time-dependent changes in the local magnetic environment. I will demonstrate that the latter can also be employed to drive the superconductor through a time-dependent quantum phase transition, in which the ground state changes from a singlet to a doublet state. Finally, I will show how this approach allows us to investigate the braiding of Majorana zero modes in real time and space in topological superconductors.   

Floquet topological d+id superconductivity in the doped Hubbard model

We study how d-wave superconductivity of the repulsive Hubbard model in the strong-coupling regime is changed when illuminated by circularly-polarised light (CPL), using the Floquet formalism for the Gutzwiller-projected effective Hamiltonian with the time-periodic Schrieffer-Wolff transformation. 

We find that CPL induces topological superconductivity with a d+id pairing, which arises from the chiral spin coupling and the three-site term generated by the CPL. 

These effects become significant for various frequencies red-detuned from the Hubbard U, which are verified by calculating the phase diagram and transient dynamics of the Floquet effective Hamiltonian.    

Floquet topological superconductivity in two-dimensional magnet-superconductor hybrid systems.

Magnet-superconductor hybrid (MSH) systems have proven valuable candidate systems for the creation

of topological superconductivity and the ensuing Majorana zero modes. However, the manipulation of

this topological superconductivity, and in particular, the ability to externally tune MSH system

between topological and trivial phases, has remained a significant obstacle on the road to realizing

topological quantum computing. In this talk, I will demonstrate how the topological phases of MSH

systems can be manipulated by using periodically driven changes in their magnetic structure. In

particular, I will discuss how the interplay between the time and spatial dependence of the magnetic

structure can be employed to quantum engineer topological superconducting phases.   

Boundary-diagnosing topological invariants beyond symmetry indicators: A case study of two-fold rotational symmetric superconductors

Topological crystalline superconductors are known to have possible higher-order topology, which results in Majorana modes on $d-2$ or lower dimensional boundaries. It is desirable to have topological invariants that can predict Majorana boundary types from band structures. Although symmetry indicators have been proposed for certain crystalline superconductors, there exist symmetry classes where symmetry indicators fail to distinguish superconductors with different Majorana boundaries. In this talk, I will focus on an example of this kind, the 2-D time-reversal symmetric superconductors with two-fold rotational symmetry, for which we systematically obtain topological invariants. First I will discuss a momentum-space classification study, which shows that the nontrivial topology is independent of band data on the high-symmetry points and leads to four $\mathbb{Z}_2$ invariants defined on the high-symmetry lines or general points in the Brillouin zone. Then, with the aid of a real-space classification study, I will show that these invariants can predict Majorana boundary types from band structures.   

Tricrystal Phase Sensitive Characterization of the Monopole Superconductor

Monopole superconductors, which can potentially be realized in doped magnetic Weyl semimetal materials, are characterized by exotic monopole harmonic symmetry of its superconducting pairing order. Because the U(1) phase of its pairing order is topologically frustrated, we study the tricrystal phase sensitive characterization of the monopole pairing to distinguish it from known unconventional pairing symmetry.    

Superconductivity in topological nodal line semimetals

We theoretically investigate the superconductivity induced by ferromagnetic fluctuations in doped nodal line semimetals (NLSMs). We find that only spin triplet pairing channels are favored and also derive the critical temperatures for the leading instability. We show that the superconudcting state in an orbital singlet and spin triplet channel displays nodal rings on the torus-like Fermi surface (FS). From the prior results on tenfold way classifications for gapless systems with inversion symmetry, the Bogoliubov-de Gennes (BdG) Hamiltonian in our model belongs to the CI class and the BdG nodal rings are protected by an integer topological charge ±2. In the presence of a ferromagnetic order, the superconducting nodal ring is split into two rings with charge ±1.   

Real time dynamic of topological dislocation modes

Lattice defects, such as dislocations, play a pivotal role in identifying translationally active topological phases of matter, featuring band inversion at a finite momentum in the Brillouin zone, by hosting robust gapless modes in defect core. As such dislocation lattice defects are instrumental to probe this special class of topological states both in the static and driven or Floquet systems. Here we will show the real time dynamics of such dislocation modes by considering the following two distinct scenarios. First, we will discuss the long-time dynamics of dislocation modes after a sudden quench to either trivial or translationally inert topological phase, by computing their survival probability. We will argue that signature of dislocation modes appear periodically for a long time after the quench. Next, we will show the gradual formation of dislocation modes in the presence of a real time ramp to a translationally active phase, starting from a trivial phase. We will primarily focus on two dimensional Chern insulators and p+ip superconductors to demonstrate these outcomes. Finally, we will generalize these results to three-dimensional topological materials as well as to higher-order topological phases.   

Controlling Majorana modes by p-wave pairing in two-dimensional p + id topological superconductors

This talk will discuss the Majorana edge modes in a two-dimensional p + id topological superconductor. Starting with a topological p-wave superconductor in either the chiral or helical phase, we show that the edge modes behave quite differently in the presence of d-wave superconducting order, depending on the aforementioned phase. In particular, for a rectangular geometry, we find that in the helical phase, a localized Majorana mode appears at each of the four corners. In the chiral phase, the relative size between the p- and d-wave order parameters can induce a phase transition, where the system goes from a state in which edge modes appear on all four edges, to one where they appear on only two, opposite, edges. We further discuss how strain may be used to fine-tune the behaviour of the edge modes.       

Unconventional Topological Transitions in a Self-Organized Magnetic Ladder

It is commonly assumed that topological phase transitions in topological superconductors are accompanied by a closing of the topological gap or a change of the symmetry of the system. We demonstrate that an unconventional topological phase transition with neither gap closing nor a change of symmetry is possible. We consider a nanoscopic length ladder of atoms on a superconducting substrate, comprising self-organized magnetic moments coupled to itinerant electrons. For a range of conditions, the ground state of such a system prefers helical magnetic textures, a self-sustaining topologically nontrivial phase. Abrupt changes in the magnetic order as a function of induced superconducting pairing or chemical potential can cause topological phase transitions without closing the topological gap. Furthermore, the ground state prefers either parallel or antiparallel configurations along the rungs, and the antiparallel configuration causes an emergent time reversal asymmetry protecting Kramers pairs of Majorana zero modes, but in a BDI topological superconductor. We determine the topological invariant and inspect the boundary Majorana zero modes.