Topological Superconductivity

external pageSuperconducting gap anisotropy and topological singularities due to lattice translational symmetry and their thermodynamic signaturescall_made

Bastian Zinkl, Mark H. Fischer and Manfred Sigrist

Symmetry arguments based on the point group of a system and thermodynamic measurements are often combined to identify the order parameter in unconventional superconductors. However, lattice translations, which can induce additional momenta with vanishing order parameter in the Brillouin zone, are neglected, especially in gap functions otherwise expected to be constant, such as in chiral superconductors. After a general analysis of the symmetry conditions for vanishing gap functions, we study the case of chiral
p- and chiral f-wave pairing on a square lattice, a situation relevant for Sr₂RuO₄. Specifically, we calculate the impurity-induced density of states, specific heat, superfluid density, and thermal conductivity employing a self-consistent T-matrix calculation and compare our results to the case of a nodal (d-wave) order parameter. While there is a clear distinction between a fully gapped chiral state and a nodal state, the strongly anisotropic case is almost indistinguishable from the nodal case. Our findings illustrate the difficulty of interpreting thermodynamic measurements. In particular, we find that the available measurements are consistent with a chiral (f-wave) order parameter. Our results help to reconcile the thermodynamic measurements with the overall picture of chiral spin-triplet superconductivity in Sr₂RuO₄.

external pageSpontaneous Thermal Hall Effect in Three-​dimensional Chiral Superconductors with Gap Nodescall_made

Nobuyuki Yoshioka, Yoshiki Imai and Manfred Sigrist

Generic chiral superconductors with three-dimensional electronic structure have nodal gaps and are not strictly topological. Nevertheless, they exhibit a spontaneous thermal Hall effect (THE), i.e., a transverse temperature gradient in response to a heat current even in the absence of an external magnetic field. While in some cases this THE can be quantized analogous to the quantum Hall effect, this is not the case for nodal superconductors in general. In this study, we determine the spontaneous THE for tight-binding models with tetragonal and hexagonal crystal symmetries with chiral p- and d-wave superconducting phases. At the zero-temperature limit, the thermal Hall conductivity κ_xy provides information on the structure of the gap function on the Fermi surface and the Andreev bound states on the surface. The temperature dependence at very low temperatures is determined by the types of gaps, i.e., point or line nodes, leading to characteristic power law behaviors in the temperature, as known for other quantities such as specific heat and London penetration depth. The generic behavior is analytically discussed on the basis of simple models, while the analysis of the tight-binding models is given numerically. Both arguments focus on a single-band model to clarify the role of specific nodal structures.

external pageTwo-​Dimensional Topological Superconductivity with Antiferromagnetic Insulatorscall_made

José L. Lado and Manfred Sigrist

Two-dimensional topological superconductivity has attracted great interest due to the emergence of Majorana modes bound to vortices and propagating along edges. However, due to its rare appearance in natural compounds, experimental realizations rely on a delicate artificial engineering involving materials with helical states, magnetic fields, and conventional superconductors. Here we introduce an alternative path using a class of three-dimensional antiferromagnets to engineer a two-dimensional topological superconductor. Our proposal exploits the appearance of solitonic states at the interface between a topologically trivial antiferromagnet and a conventional superconductor, which realize a topological superconducting phase when their spectrum is gapped by intrinsic spin-orbit coupling. We show that these interfacial states do not require fine-tuning, but are protected by asymptotic boundary conditions.

external pageSuperconductivity without Inversion and Time-​Reversal Symmetriescall_made

Mark H. Fischer, Manfred Sigrist and Daniel F. Agterberg

Traditionally, in three dimensions, the only symmetries essential for superconductivity are time reversal (T) and inversion (I). Here, we examine superconductivity in two dimensions and find that T and I are not required, and having a combination of either symmetry with a mirror operation (M_z) on the basal plane is sufficient. By combining energetic and topological arguments, we classify superconducting states when T and Iare not present, a situation encountered in several experimentally relevant systems, such as transition metal dichalcogenides or a two-dimensional Rashba system, when subject to an applied field, and in superconducting monolayer FeSe with Néel antiferromagnetic order. Energetic arguments suggest interesting superconducting states arise. For example, we find a unique pure intraband pairing state with Majorana chiral edge states in Néel-ordered FeSe. Employing topological arguments, we find when the only symmetry is the combination of I with M_z , the superconducting states are generically fully gapped and can have topologically protected chiral Majorana edge modes. In all other cases, there are no chiral Majorana edge states, but the superconducting bulk can have point nodes with associated topologically protected flatband Majorana edge modes. Our analysis provides guidance on the design and search for novel two-dimensional superconductors and superconducting heterostructures.

external pageSpontaneous surface flux pattern in chiral p-​wave superconductorscall_made

Sarah B. Etter, Adrien Bouhon and Manfred Sigrist

In chiral p-wave superconductors, magnetic flux patterns may appear spontaneously when translational symmetry is broken such as at surfaces, domain walls, or impurities. However, in the candidate material Sr₂RuO₄ no direct signs of such magnetic fields have been detected experimentally. In this paper, the flux pattern at the edge of a disk-shaped sample is examined using the phenomenological Ginzburg-Landau approach. The detailed shape of the flux pattern, including self-screening, is computed numerically for different surface types by systematically scanning a range of boundary conditions. Moreover, specific features of the electronic structure are included qualitatively through the coefficients in the Ginzburg-Landau functional. Both the shape and the magnitude of the flux pattern are found to be highly sensitive to all considered parameters. In conclusion, such spontaneous magnetic flux patterns are not a universal feature of chiral p-wave superconductors.

external pageSurface magnetism in a chiral d-​wave superconductor with hexagonal symmetrycall_made

Jun Goryo, Yoshiki Imai, W.B. Rui, Manfred Sigrist and Andreas P. Schnyder

Surface properties are examined in a chiral d-wave superconductor with hexagonal symmetry, whose one-body Hamiltonian possesses intrinsic spin-orbit coupling identical to the one characterizing the topological nature of the Kane-Mele honeycomb insulator. In the normal state, spin-orbit coupling gives rise to spontaneous surface spin currents, whereas in the superconducting state, besides the spin currents, there exist also charge surface currents, due to chiral pairing symmetry. Interestingly, the combination of these two currents results in a surface spin polarization, whose spatial dependence is markedly different on the zigzag and armchair surfaces. We discuss various potential candidate materials, such as SrPtAs, which may exhibit these surface properties.

external pageThermal Hall Conductivity in a Multiband Chiral p-​Wave Superconductorcall_made

Yoshiki Imai, Katsunori Wakabayashi and Manfred Sigrist

For the spin-triplet superconductor Sr₂RuO₄, multiband effects on the thermal Hall conductivity are investigated in the superconducting phase. While the gap functions of all bands (α–β and γ) are assumed to have chiral p-wave symmetry, the thermal Hall conductivity at very low temperatures is dominated by the γ band due to its topological nontrivial state and the presence of nodeless quasiparticle excitation gaps.

external pageCooperon condensation and intravalley pairing states in honeycomb Dirac systemscall_made

Shunji Tsuchiya, Jun Goryo, Emiko Arahata and Manfred Sigrist

Motivated by recent developments in the experimental study of superconducting graphene and transition metal dichalcogenides, we investigate superconductivity of the Kane-Mele (KM) model with short-range attractive interactions on the two-dimensional honeycomb lattice. We show that intravalley spin-triplet pairing arises from nearest-neighbor (NN) attractive interaction and the intrinsic spin-orbit coupling. We demonstrate this in two independent approaches: We study superconducting instability driven by condensation of Cooperons, which are in-gap bound states of two conduction electrons, within the
T-matrix approximation and also study the superconducting ground state within the mean-field theory. We find that Cooperons with antiparallel spins condense at the K and K′ points. This leads to the emergence of an intravalley spin-triplet pairing state belonging to the irreducible representation A₁ of the point group C_6v. The fact that this pairing state has opposite chirality for K and K′ identifies this state as a “helical” valley-triplet state, the valley analog to the ³He-B phase in two dimensions. Because of the finite center of mass momentum of Cooper pairs, the pair amplitude in NN bonds exhibits spatial modulation on the length scale of lattice constant, such that this pairing state may be viewed as a pair-density wave state. We find that the pair amplitude spontaneously breaks the translational symmetry and exhibits a p-Kekulé pattern. We also discuss the selection rule for pairing states focusing the characteristic band structure of the KM model and the Berry phase effects to the emergence of the intravalley pairing state.

external pageThermal Hall conductivity and topological transition in a chiral p-​wave superconductor for Sr₂RuO₄call_made

Yoshiki Imai, Katsunori Wakabayashi and Manfred Sigrist

The interplay between thermal transport properties and topological aspects is investigated in a spin-triplet chiral p-wave superconductor Sr₂RuO₄ with strong two dimensionality. We show that thermal Hall conductivity is well described by a temperature linear term and an exponential term in the low-temperature region. While the former term is directly proportional to the so-called Chern number, the latter is associated with the superconducting gap amplitude of the
γ band. We also demonstrate that the coefficient of the exponential term changes sign around the Lifshitz transition. Our obtained result may enable us to easily access the physical quantities and topological properties of Sr₂RuO₄ in detail.

external pageCurrent inversion at the edges of a chiral p-​wave superconductorcall_made

Adrien Bouhon and Manfred Sigrist

Motivated by Sr₂RuO₄, edge quasiparticle states are analyzed based on the self-consistent solution of the Bogolyubov–de Gennes equations for a topological chiral p-wave superconductor. Using a tight-binding model of a square lattice for the dominant γ band, we explore the nontrivial geometry and band structure dependence of the edge states and currents. As a peculiar finding, we show that, for high band fillings, currents flow in a reversed direction when comparing straight and zigzag edges. We give a simple explanation in terms of the positions of the zero-energy bound states using a quasiclassical picture. We also show that a Ginzburg-Landau approach can reproduce these results. Moreover, the band filling dependence of the most stable domain wall structure is discussed.