Session L05: Topological Superconductivity: Theory

Designing topological superconductors

I will discuss routes to creating topological (and other unconventional) superconductors by tuning the Fermi surface of materials with gating, electric/magnetic field and strain. Candidate systems include 2D materials with spin-orbit coupling, with tunable van Hove singularity, or with moire superlattice.   

Extension of symmetry-based indicators and its application

The goal of the study of topological phases is to achieve the complete classification of phases for all spatial and internal symmetries. The topological periodic table, the classification table of topological insulators for each internal symmetry class, is a milestone in this line of research.
Recently, the method of “symmetry-based indicators (SI)” has been developed. It diagnoses the topology of band structures based on their symmetry representations. The prototypical example of SI is the Fu-Kane formula, which distinguishes topological insulators from normal insulators in terms of inversion parities.
So far, as an internal symmetry, only the time-reversal symmetry has been taken into account in the calculation of SI. In this talk, we discuss how to incorporate with the chiral symmetry and the particle-hole symmetry.

Ref: S. Ono and H. Watanabe, Phys. Rev. B 98, 115150 (2018)   

Classification of topological crystalline superconducting nodes on high-symmetry line

Recent development in exact classification of a superconducting gap has elucidated various unconventional gap structures [1-8], which have not been predicted by the classification of order parameter based on the point group [9]. One of the important previous results is that all symmetry-protected line nodes are characterized by nontrivial topological numbers [6, 7]. Another intriguing discovery is the gap structures depending on the angular momentum of electrons j_z on threefold and sixfold rotational-symmetric lines [1, 3, 8]. Stimulated by these findings, we classify all crystal symmetry-protected nodes (including such j_z-dependent nodes) on high-symmetry n-fold (n = 2, 3, 4, and 6) axes, by using the combination of group theory and K-theory [10]. As a result, it is shown that the classification by group theory completely corresponds with the topological classification. Based on the obtained results, we discuss superconducting gap structures in several candidate superconductors.   

Weak-Pairing Higher Order Topological Superconductors

We consider a new class of superconductors, second-order topological superconductors (SOTSC), that have gapped, topological surfaces and gapless Majorana modes on lower-dimensional boundaries: corners of a 2D system or hinges for a 3D system. We propose two general scenarios that SOTSC can be realized spontaneously with weak-pairing instabilities. First, we show that p-wave pairing in a Dirac semimetal in 2D with four mirror symmetric Dirac nodes realizes SOTSC. Second, we show that p+id pairing on an ordinary Fermi surface realizes SOTSC as well. We show that these exotic states can be intrinsically realized in a metallic system with electronic interactions. In the latter case we find that the topological invariants describing the system can be written using simple formulae involving only the low-energy properties of the Fermi surfaces and superconducting pairing, and we also show it can be induced by proximity effect in a heterostructure of cuprate and superconductor.   

Exotic superconductivity with enhanced energy scales in materials with three band crossings

Three band crossings can arise in three-dimensional quantum materials with certain space group symmetries. The low energy Hamiltonian supports spin one fermions and a flat band. We study the pairing problem in this setting. We write down a minimal BCS Hamiltonian and decompose it into spin-orbit coupled irreducible pairing channels. We then solve the resulting gap equations in channels with zero total angular momentum. We find that in the s-wave spin singlet channel (and also in an unusual d-wave ‘spin quintet’ channel), superconductivity is enormously enhanced, with a possibility for the critical temperature to be linear in interaction strength. Meanwhile, in the p-wave spin triplet channel, the superconductivity exhibits features of conventional BCS theory due to the absence of flat band pairing. Three band crossings thus represent an exciting new platform for realizing exotic superconducting states with enhanced energy scales. We also discuss the effects of doping, nonzero temperature, and of retaining additional terms in the k.p expansion of the Hamiltonian.   

Anomalous symmetry protected topological states in interacting fermion systems

The classification and construction of symmetry protected topological(SPT) phases have been intensively studied in interacting systems recently. To our surprise, in interacting fermion systems, there even exists a new class of the so-called anomalous SPT(ASPT) states which are only well defined on the boundary of a trivial fermionic bulk. The physical reason is that certain symmetry action might change the fermion parity locally, but is conserved if we introduce a bulk. We demonstrate the essential idea by considering anomalous topological superconductor with time reversal symmetry $T^2=1$. Finally, we also discuss the classification scheme of anoumalous SPT states for interacting fermions with a total symmetry $G_f=G_b\times\Zf$ and relevant experimental implications.   

Superconducting higher-order semimetals : Second-order Dirac superconductors and magnetic field induced higher-order topological superconductivity

We identify three dimensional superconductors that exhibit various forms of higher order topology. We show how such superconductors can be obtained through superconducting instabilities induced in higher order quadrupole topological semimetals [1]. In our models the normal-state degrees of freedom consist of two orbitals and spin in an electronic structure that forms a quadrupolar topological semimetal. We consider all possible s-wave superconducting pairing terms satisfying Fermi-Dirac statistics and obtain six different superconducting models. We find that four of the models have a non-zero residual quadrupole-like topology, and therefore topologically protected Majorana hinge states on the four hinges of the sample. The other two models have no quadrupolar topology yet, in the presence of an external magnetic field, exhibit helical hinge states localized at only two of the four hinges. For these two models we also find that the pair of corners exhibiting hinge states switches upon a change of sign of the magnetic field.

[1]. M. Lin, T.L. Hughes, arXiv:1708.08457.   

Higher Order Topological Phases: A General Principle of Construction

In this talk, we discuss a general principle for constructing higher-order topological (HOT) phases [1]. We argue that if a D-dimensional first-order or regular topological phase involves m Hermitian matrices that anti-commute with additional p-1 mutually anti-commuting matrices, it is conceivable to realize an n^th-order HOT phase, where n=1,..., p, with appropriate combinations of discrete symmetry-breaking Wilsonian masses. An n^th-order HOT phase accommodates zero modes on a surface with co-dimension n. We exemplify these scenarios for prototypical three-dimensional gapless systems, such as a nodal-loop semimetal possessing SU(2) spin rotational symmetry, and Dirac semimetals, transforming under (pseudo-)spin-1/2 or 1 representation. The former system permits an unprecedented realization of a 4^th-order phase, without any surface zero modes. Our construction can be generalized to HOT insulators and superconductors in any dimension and symmetry class.

[1] D. Călugăru, V. Juričić, and B. Roy, arXiv:1808.08965.   

Gauge Theory of the Superconductor-Insulator Transition

The standard model of particle physics is extraordinarily successful at explaining much of the physical realm. Yet, one of its most profound aspects, the mechanism of confinement, that binds quarks into hadrons and is supposedly mediated by chromo-electric strings in a condensate of magnetic monopoles, is not thoroughly understood and lacks direct experimental evidence. We demonstrate that the infinite-resistance superinsulating state, a mirror analog of superconductivity, emerging at the insulating side of the superconductor-insulator transition (SIT) is a condensed matter realization of the quark confinement. We reveal that the mechanism ensuring the infinite resistance of superinsulators is the binding of Cooper pairs into neutral “mesons” by electric strings and establish a mapping of quarks onto Cooper pairs in superinsulators. We reveal a Cooper pair analog of the asymptotic freedom effect implying that systems smaller than the string scale appear in a quantum metallic state.   

Superconducting vortex from magnetic skyrmions

We show topological superconductivity can arise at the interface of ferromagnetic skyrmion material in proximity to an s-wave superconductor in the presence of strong spin-orbit coupling. We study the Majorana zero modes localized at the interface of the heterostructure and propose a pathway towards a useful platform for the manipulation of Majorana fermions via controlling skyrmions.
  

Magnon-induced superconductivity in a topological insulator coupled to ferro- and antiferromagnetic insulators

We study the effective interactions between Dirac fermions on the surface of a three-dimensional topological insulator due to the proximity coupling to the magnetic fluctuations in a ferromagnetic or antiferromagnetic insulator. Magnetic fluctuations can mediate attractive interactions between Dirac fermions of both Amperean and BCS type. In the ferromagnetic case, we find pairing between fermions with parallel momenta, so-called Amperean pairing, whenever the effective Lagrangian for the magnetic fluctuations does not contain a quadratic term. If a quadratic term is present, the pairing is instead of BCS type above a certain chemical potential. In the antiferromagnetic case, BCS pairing occurs when the ferromagnetic coupling between magnons on the same sublattice exceeds the antiferromagnetic coupling between magnons on different sublattices. Outside this region in parameter space, we again find that Amperean pairing is realized   

Reproducing topological properties with quasi-Majorana states

Andreev bound states in hybrid superconductor-semiconductor devices can have near-zero energy in the topologically trivial regime as long as the confinement potential is sufficiently smooth. We show that in addition to the suppressed coupling between the quasi-Majorana states, also the coupling of these states across a tunnel barrier to the outside is exponentially different. As a consequence, quasi-Majorana states mimic most of the proposed Majorana signatures: quantized zero-bias peaks, the 4π Josephson effect, and the tunneling spectrum in presence of a normal quantum dot. We identify a quantized conductance dip instead of a peak in the open regime as a distinguishing feature of true Majorana states in addition to having a bulk topological transition. Because braiding schemes rely only on the ability to couple to individual Majorana states, the exponential control over coupling strengths allows to also use quasi-Majorana states for braiding.   

Fulde-Ferrell-Larkin-Ovchinnikov phases in Proximitized Josephson junctions

Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) phases, which are notoriously difficult to find in more homogeneous systems, have been observed in proximitized superconductors. Of particular interest are those heterostructures which are associated with topological phases[1,2]. In this talk, we establish the effects of spin-orbit coupling and show why these elusive phases are observable in the presence of proximitization. We focus on that FFLO phase which appears inside a substrate (subjected to an in-plane magnetic field) and formed in a Josephson junction configuration. To do so, we solve the Bogoliubov-de Gennes equations for the full multi-component system. The substrates consist of (1) a conical Holmium magnet with an effective 1D spin-orbit coupling, and (2) a two-dimensional electron gas.

[1] Hart et al., Nat Phys. 13, 87 (2017).
[2] Chen et al, Nat Comm 9, 2478 (2018)