Session A14: Topological Superconductors and Superfluids: Theory

Crystal symmetry protected gapless vortex line phases in superconducting Dirac semimetals

Vortex lines in superconducting Dirac semimetals realize crystal symmetry protected gapless vortex line phases in which gapless excitations propagate inside a vortex line, in the presence of appropriate crystal symmetry, spin-orbit coupling, and multiband structures. In this talk we present a general scheme to classify possible gapless vortex line phases in s-wave superconducting states of Dirac semimetals with rotation (or screw) symmetry and inversion symmetry, assuming that the rotation (screw) axis is parallel to the vortex line. The rotation (screw) symmetry protected gapless modes are stable as long as they have different rotation

(screw) eigenvalues. The underlying mechanism for the formation of gapless vortex bound states depends on irreducible representations of rotation (screw) symmetry subject to a vortex field and is classified into three types: (i) accidental band crossing of two vortex bound-state modes under rotation symmetry; (ii) accidental and

(iii) enforced band crossing of four vortex bound-state modes under screw symmetry. Using a tight-binding model of screw symmetry protected Dirac semimetal with an s-wave pair potential, we demonstrate a gapless vortex line phase of type (ii).    

Symmetry Principles for Phase-Sensitive Probes of Monopole Superconducting Order

Monopole superconducting order is a novel class of topological pairing order which arises when the Cooper pair possesses a nontrivial monopole charge in momentum space. The pairing order cannot be well-defined over an entire Fermi surface and is described by the so-called monopole harmonic functions. Such pairing is realizable in, for example, a doped Weyl semimetal. We explore the potential identification of the exotic pairing order based on a set of phase-sensitive Josephson junctions. Using Green's functions, we derive a form of the leading-order Josephson coupling between a monopole superconductor and other known superconductors. Furthermore, by considering junctions with different relative orientations to the principal rotational axis along which the pairing gap nodes appear, we establish the general symmetry selection rules that can unveil the pair monopole charge both analytically and numerically.   

First- and second-order topological phases in a superconducting multiorbital model

We investigate the topological phases that appear in an orbital version of the Benalcazar-Bernevig-Hughes (BBH) model in the presence of conventional $s$-wave spin-singlet superconductivity and an in-plane magnetic field. We chart out the phase diagram in the magnetic field vs pairing amplitude plane by considering various boundary conditions, while the topology of the individual phases is examined by considering the Wannier and entanglement spectrum, as well as the Majorana polarization. For small to moderate values of magnetic field and superconducting pairing amplitude, we find a second-order topological superconductor phase with eight zero-energy corner modes. In addition, we find nodal as well as nodeless flat bands, localized only along the mirror-symmetric open edges, with momentum parallel to the mirror symmetry broken direction, which is considered to be the first-order topological superconducting phase. The former arises in a small momentum range when the magnetic field amplitude is stronger than the superconducting pairing amplitude and vice versa. On the other hand, nodeless flat bands span across the whole Brillouin zone when the magnetic field is comparable to the superconducting pairing amplitude. We further find a region on the phase diagram sandwiched between the first-order flat bands and second-order corner modes with a hybrid phase where both corner and partial edge localizations are observed. The half-quantization of Wannier spectrum and mid-gap distribution of eigenvalues in the entanglement spectrum allows us to distinguish between the first-order, second-order, and hybrid topological phases.   

Quasicrystalline Weyl superconductors

We theoretically investigate Weyl superconductivity in quasicrystals. Crystalline Weyl superconductors have stable point nodes characterized by the Chern number in the Brillouin zone. The point nodes are called Weyl nodes. Since quasicrystals lack the Brillouin zone, we cannot directly use the Chern number for their topological characterization. Nevertheless, we show that Weyl superconductivity can be realized in quasicrystalline layers stacked periodically. Because vertical momentum can be defined in the quasicrystalline systems due to the periodicity, Weyl nodes can be characterized by a change in the Bott index in the vertical momentum space. We demonstrate quasicrystalline Weyl superconductivity by using a stack of two-dimensional superconducting quasicrystals under an external magnetic field. We also show that quasicrystalline Weyl superconductors host Majorana surface states because of the nonzero Bott index.   

Symmetry-protected vortex phase transitions and anisotropic magnetic response in superconducting topological crystalline insulator SnTe

Topological crystalline insulator SnTe is known to support at most 2 Majorana zero modes (MZMs) at each end of a single vortex, their coexistence is protected by a magnetic mirror symmetry M_T. We study how a tilted magnetic field changes the topological classification of the vortex and calculate the local density of states (LDOS) which can be compared with experiments. When the Fermi level is deep inside the bulk states, the MZMs from opposite surfaces tunnel through and hybridize in a vortex phase transition. We find that if a magnetic glide symmetry G_T is broken, the vortex may feature only a single MZM between vortex phase transitions. Although resolving the MZMs in experiment is difficult since they are buried deep under other low-lying vortex bound states, the MZMs can be identified from the anisotropic magnetic response of the LDOS. For higher chemical potential when the vortex features MZMs, the zero bias peak elongates when the in-plane component of the magnetic field is along {110} direction preserving M_T, while it splits when M_T is broken or when the chemical potential is low so that all MZMs are hybridized.   

Nodal topological superconductivity in nodal-line semimetals

We analyze possible nodal superconducting phases that emerge from a doped nodal-line semimetal. We show that nodal-line superconducting phases are favored by interactions mediated by short-range ferromagnetic fluctuations or Hund's coupling. It is found that the leading pairing channels are momentum-independent, orbital-singlet and spin-triplet. In the pairing state, we show that the Bogoliubov-de Gennes (BdG) Hamiltonian hosts a pair of topologically protected nodal rings on the equators of the torus Fermi surface (FS). Using a topological classification for gapless systems with inversion symmetry, we find that these nodal rings are topologically nontrivial and protected by integer-valued monopole charges ν=±2. In the scenario of pairing driven by ferromagnetic fluctuations, we analyze the fate of superconductivity in the magnetically ordered phase. Based on Ginzburg-Landau free energy analysis, we find the energetically favored superconducting state is characterized by the coexistence of two pairing orders whose d-vectors are perpendicular to the magnetization axis M with their phases unfixed. In this case, each nodal loop in the pairing state splits into two, carrying a ±1 monopole charge. For bulk-boundary correspondence, these nodal rings enclose flat-band Majorana zero modes on top and bottom surface Brillouin Zones with distinct ℤ-valued topological invariants.   

Superfluid weight in Chern bands under a magnetic field

Superconductivity arising out of flat bands in time-reversal invariant systems is thought to be protected by the band’s quantum geometry, providing a lower bound on the superfluid weight which sets the energy scale for phase fluctuations and is responsible for the Meissner effect and persistent supercurrents. We study the superfluid weight for Cooper pairs formed from Landau levels (LLs), which are flat and have non-trivial quantum geometry, using self-consistent mean-field theory. We discuss the evolution of the superfluid weight under the application of a potential, that pins the vortices which are present due to the applied magnetic field, and induces dispersion in the LLs. We examine the evolution of the conventional and geometrical contributions to the superfluid weight. We also investigate the superfluid weight in LL-like bands arising from a topologically nontrivial system exposed to a magnetic field.   

Topological nodal quasiparticles from the vortex lattice of s-wave superconductors

A type-II superconductor (SC) will respond to an external magnetic field by forming Abrikosov vortices. While the Majorana physics of a single quantum vortex has been extensively discussed, the topological consequence of a spatially periodic vortex lattice is much less understood. In this work, we focus on electronic properties of the vortex lattice for three-dimensional (3D) fullly gapped superconductors with an isotropic s-wave pairing.  By turning on a vortex lattice, 1D low-energy Caroli-Matricon-de Gennes (CdGM) states will be trapped around each vortex core, whose inter-vortex hoppings will generate an emergent 3D band structure for CdGM quasiparticles. Our main finding is that when the normal state is a Dirac semimetal, the 3D vortex-lattice spectrum features nodal Bogoliubov-de Gennes (BdG) Fermi surfaces, whose gapless nature is topology-enforced. Surprisingly, these topological nodal quasiparticles persist even when the normal state is deformed to a trivial metal with no electronic band inversion. Being directly applicable to real-world superconductors such as LiFeAs, our work establishes vortex-lattice engineering as a novel experimentally feasible pathway towards topological superconductivity.    

Unveiling the Interplay between Chiral Quantum Phases and Geometry Fluctuations: the Superfluid Transition on a Membrane

Chiral quantum phases couple to the background geometry, which leads to intriguing effects such as the gravitational anomaly in fractional quantum Hall effect and geometric induction in chiral condensates. However, the impact of geometry fluctuations at finite temperatures remains relatively uncharted. In this study, we investigate the behavior of a chiral superfluid situated on a flexible two-dimensional membrane. We show that the Berezinskii-Kosterlitz-Thouless (BKT) transition is renormalized by geometry fluctuations, resulting in a critical temperature depending on the bending rigidity of the membrane. Conversely, the presence of a superfluid significantly alters the long-distance behavior of the bending rigidity and the associated crumpling transition. Using a renormalization group treatment of the continuum field theory describing the coupling of superfluid phase to the flexural and in-plane phonons, we derive the joint phase diagrams for the cases of a fluid, hexatic, and crystalline membrane, with and without external tension.   

Title: Stabilizing Topological Superconductivity in Disordered Spin-Orbit Coupled Semiconductor-Superconductor Heterostructures

We theoretically consider the problem of one-dimensional semiconductor-superconductor (SM and SC) heterostructure with Rashba spin-orbit coupling and a parallel Zeeman field in the presence of short-ranged disorder from random charged impurities. With no disorder, this system was proposed as a model platform for realizing bulk topological superconductivity (TS) characterized by zero energy Majorana excitations localized at the wire ends. With disorder, however, it has been shown that disorder-induced trivial low-energy states can render the detection of topological superconductivity and zero-energy Majorana states experimentally very challenging, and, for strong disorder may even lead to the disappearance of the TS state from the experimentally accessible part of the phase diagram. Starting with the Hamiltonians of the SM and the SC and using the formalism of an effective SM Green's function by integrating out the SC, we show in this paper that for a strongly disordered SM, strong coupling to the SC is generically beneficial for stabilizing a robust TS state in the semiconductor. Furthermore, we find that, for the phase diagram defined by the chemical potential (μ) and Zeeman field (Γ), with increasing strength of disorder the robust topological regions move to the parts of the phase diagram defined by larger values of Γ. These results lead us to propose that (a) stronger SM and SC coupling by interface engineering, (b) SM systems with a larger gyromagnetic ratio, or (c) a stronger proximity effect allowing the application of larger values of the Zeeman field may help defeat the dominance of disorder-induced low-energy states in the semiconductor, revealing and expanding the underlying robust topological superconducting state in the phase diagram.   

Quantum geometry and the superfluid weight of a Bose-Einstein condensate

Nontrivial quantum geometry enables superconductivity even in perfectly flat bands, where the single-particle effective mass is infinite. Recent studies have pointed out that quantum geometry also plays a role in Bose-Einstein condensates. Of particular importance is the quantum metric at the condensation momentum, which determines quantities such as the speed of sound in a flat band condensate. Here, we study the superfluid weight of a Bose-Einstein condensate within multiband Bogoliubov theory, and show that its geometric part contains contributions related to both the quantum metric at the condensation momentum and the quantum metric integrated over the Brillouin zone. We formulate conditions for the stability of a flat band condensate in terms of geometric properties. We find that a nontrivial integrated quantum metric, which relates to topological properties, can hinder the formation of a stable condensate in flat bands, in contrast to fermionic superconductors. A nontrivial quantum metric at the condensation momentum, on the other hand, is advantageous for a stable condensate.   

Electric response of Majorana fermions in topological crystalline superconductors

 Majorana fermions appear on the surface of topological superconductors and are usually stable against any perturbations. On the other hand, when the topological superconductors have time-reversal symmetry, a pair of Majorana fermions, called the Majorana Kramers pair, appears on the surface, which is able to respond to a magnetic field. Since the magnetic structure of the Majorana Kramers pair has an Ising type magnetic anisotropy, it responds to a magnetic field applied in a specific direction, Recently, the relationship among the magnetic response, crystalline symmetry, and pairing symmetry of superconductors has been clarified. Thus, the magnetic response provides useful information about the superconducting mechanism.

 In this talk, we extend the above theory to the case of a double Majorana Kramers pair, which is able to respond to an electric perturbation. Establishing the classification of the double Majorana Kramers pair in terms of the wallpaper groups, we clarify all possible electric multipole responses. In addition, we reveal the relationship between the electric multipole responses and pairing symmetry. As an example, we demonstrate a specific irreducible representation of a uniform strain yields gap in the double Majorana Kramers pair for topological crystalline superconductor candidate Sr3SnO. This highlights the validity of electric detection of pairing symmetry.   

Realizing attractive interacting topological surface fermions: A resonating TI- thin film hybrid platform

In this article, we propose a practical way to realize surface Dirac fermions with tunable attractive interaction between them. The method involves coating the surface of a topological insulator with a thin film metal and utilizing the strong-electron phonon coupling in the metal to induce interaction between the surface fermions. We found that for a given TI and thin film, the attractive interaction between the surface fermions is maximally enhanced when the Dirac point of the TI surface resonates with one of the quasi-2D quantum-well bands of the thin film. This effect can be considered to be an example of 'quantum-well resonance'. We also demonstrate that the superconductivity of the resonating surface fermions can be further enhanced by choosing a strongly interacting thin film metal or by tuning the spin-orbit coupling of the TI. This TI-thin film hybrid configuration holds promise for applications in Majorana-based quantum computations and for the study of quantum critical physics of the strongly attractively interacting surface topological matter with emergent supersymmetry(SUSY).   

Analog black-hole creation on the surface of a topological superconductor

Bogoliubov quasiparticles on the surface of a topological superconductor are gapless Dirac fermions with a velocity that depends on the gap of the superconductor. If inhomogeneities are introduced on the surface, the effective metric of these fermions is affected (and in fact, for Class DIII, time-reversal invariant perturbations can only cause this type of disorder [1]). In extreme cases, the superconductivity can locally be destroyed on the surface, resulting in a vanishing velocity in a "dark spot." We discuss the dynamical process of this "dark spot" creation in analogy to a black hole formation. By performing the Hawking calculation in this 2D scenario, we highlight similarities and differences in the spacetime structure and particle creation at long times, far from the dark spot. With the full three-dimensional, UV completion, we discuss what occurs within the dark spot and the bulk of the material.  Implications for the superconducting gap of the system near the surface are also discussed.

[1] Karcher, J. K. and M. S. Foster, Ann. Phys. 435 168439 (2021).