Bogoliubov-de Gennes Hamiltonians Research Articles

Emergence of a new symmetry class for Bogoliubov–de Gennes (BdG) Hamiltonians: expanding 10-fold symmetry classes

ABSTRACTSymmetry plays an important role in the topological states of quantum matter. Cartan found three symmetry classes, Altland and Zirnbauer extended it to the 10-fold symmetry class for mesoscopic systems. We observe the emergence of a new symmetry class for BdG Hamiltonians. This new symmetry class appears for the BdG Hamiltonian in spinful background. We also observe the interesting feature that there is no evidence of Kramers's degeneracy for this BdG model Hamiltonian either in spinless or in spinful background. We call the extra symmetry class X, i.e. expanding the 10-fold symmetry classes for topological states of quantum matter. These BdG Hamiltonians show many new, interesting and insightful results.

The non-commutative topology of two-dimensional dirty superconductors

Non-commutative analysis tools have successfully been applied to the integer quantum Hall effect, in particular for a proof of the stability of the Hall conductance in an Anderson localization regime and of the bulk-boundary correspondence. In this work, these techniques are implemented to study two-dimensional dirty superconductors described by Bogoliubov–de Gennes Hamiltonians. After a thorough presentation of the basic framework and the topological invariants, Kubo formulas for the thermal, thermoelectric and spin Hall conductance are analyzed together with the corresponding edge currents.

Structure of vortex-bound states in spin-singlet chiral superconductors

We investigate the structure of vortex-bound states in spin-singlet chiral superconductors with ($d_{x^2-y^2} \pm i d_{xy}$)-wave and ($d_{xz} \pm id_{yz}$)-wave pairing symmetries. It is found that vortices in the ($d_{xz} \pm id_{yz}$)-wave state bind zero-energy states which are dispersionless along the vortex line, forming a doubly degenerate Majorana flat band. Vortex-bound states of ($d_{x^2-y^2} \pm i d_{xy}$)-wave superconductors, on the other hand, exist only at finite energy. Using exact diagonalization and analytical solutions of tight-binding Bogoliubov-de Gennes Hamiltonians, we compute the energy spectrum of the vortex-bound states and the local density of states around the vortex and antivortex cores. We find that the tunneling conductance peak of the vortex is considerably broader than that of the antivortex. This difference can be used as a direct signature of the chiral order parameter symmetry.

Majorana vortex-bound states in three-dimensional nodal noncentrosymmetric superconductors

Noncentrosymmetric superconductors (NCSs), characterized by antisymmetric spin-orbit coupling and a mixture of spin-singlet and spin-triplet pairing components, are promising candidate materials for topological superconductivity. An important hallmark of topological superconductors is the existence of protected zero-energy states at surfaces or in vortex cores. Here we investigate Majorana vortex-bound states in three-dimensional nodal and fully gapped NCSs by combining analytical solutions of Bogoliubov--de Gennes (BdG) equations in the continuum with exact diagonalization of BdG Hamiltonians. We show that depending on the crystal point-group symmetries and the topological properties of the bulk Bogoliubov-quasiparticle wave functions, different types of zero-energy Majorana modes can appear inside the vortex core. We find that for nodal NCSs with tetragonal point group ${C}_{4v}$ the vortex states are dispersionless along the vortex line, forming one-dimensional Majorana flat bands, while for NCSs with ${D}_{4}$ point-group symmetry the vortex modes are helical Majorana states with a linear dispersion along the vortex line. NCSs with monoclinic point group ${C}_{2}$, on the other hand, do not exhibit any zero-energy vortex-bound states. We show that in the case of the ${C}_{4v}$ $({D}_{4})$ point group the stability of these Majorana zero modes is guaranteed by a combination of reflection $(\ensuremath{\pi}$ rotation), time-reversal, and particle-hole symmetry. Considering continuous deformations of the quasiparticle spectrum in the presence of vortices, we show that the flat-band vortex-bound states of ${C}_{4v}$ point-group NCSs can be adiabatically connected to the dispersionless vortex-bound states of time-reversal symmetric Weyl superconductors. Experimental implications of our results for thermal transport and tunneling measurements are discussed.

Topology of crystalline insulators and superconductors

We complete a classification of topological phases and their topological defects in crystalline insulators and superconductors. We consider topological phases and defects described by non-interacting Bloch and Bogoliubov de Gennes Hamiltonians that support additional order-two spatial symmetry, besides any of ten classes of symmetries defined by time-reversal symmetry and particle-hole symmetry. The additional order-two spatial symmetry we consider is general and it includes $Z_2$ global symmetry, mirror reflection, two-fold rotation, inversion, and their magnetic point group symmetries. We find that the topological periodic table shows a novel periodicity in the number of flipped coordinates under the order-two spatial symmetry, in addition to the Bott-periodicity in the space dimensions. Various symmetry protected topological phases and gapless modes will be identified and discussed in a unified framework. We also present topological classification of symmetry protected Fermi points. The bulk classification and the surface Fermi point classification provide a novel realization of the bulk-boundary correspondence in terms of the K-theory.

Edge Currents as a Signature of Flatbands in Topological Superconductors

We study nondegenerate flatbands at the surfaces of noncentrosymmetric topological superconductors by exact diagonalization of Bogoliubov-de Gennes Hamiltonians. We show that these states are strongly spin polarized and acquire a chiral dispersion when placed in contact with a ferromagnetic insulator. This chiral mode carries a large edge current which displays a singular dependence on the exchange-field strength. The contribution of other edge states to the current is comparably weak. We hence propose that the observation of the edge current can serve as a test of the presence of nondegenerate flatbands.

Protected boundary states in gapless topological phases

We systematically study gapless topological phases of (semi-)metals and nodal superconductors described by Bloch and Bogoliubov–de Gennes Hamiltonians. Using K-theory, a classification of topologically stable Fermi surfaces in (semi-)metals and nodal lines in superconductors is derived. We discuss a generalized bulk–boundary correspondence that relates the topological features of the Fermi surfaces and superconducting nodal lines to the presence of protected zero-energy states at the boundary of the system. Depending on the case, the boundary states are either linearly dispersing (i.e. Dirac or Majorana states) or dispersionless, forming two-dimensional surface flat bands or one-dimensional arc surface states. We study examples of gapless topological phases in symmetry classes AIII and DIII, focusing in particular on nodal superconductors, such as nodal noncentrosymmetric superconductors. For some cases we explicitly compute the surface spectrum and examine the signatures of the topological boundary states in the surface density of states. We also discuss the robustness of the surface states against disorder.

Quantum Thermal Hall Effect in a Time-Reversal-Symmetry-Broken Topological Superconductor in Two Dimensions: Approach from Bulk Calculations

We discuss thermal transport of two-dimensional topological superconductors (TSCs) with broken time reversal symmetry, which are described by Bogoliubov-de Gennes (BdG) Hamiltonians. From the calculations of bulk quantities only, without refereeing to Majorana edge states, we show that the thermal Hall conductivity of two-dimensional TSCs in the low-temperature limit is quantized in multiples of $1/2\frac{\pi T}{6}$, which is exactly one half of the value of quantization in the case of the integer quantum Hall effect, and that this exact half-quantization is caused by the structure of the Nambu spinor and the particle-hole symmetry, which BdG Hamiltonians generally have. In the case of spinless chiral p-wave superconductors, this result is in perfect agreement with the argument based on the Ising conformal field theory with the central charge $c=1/2$, which is an effective low-energy theory of the Majorana edge states.

A Markovian kinetic equation approach to electron transport through a quantum dot coupled to superconducting leads

We present a derivation of the Markovian master equation for an out-of-equilibrium quantum dot connected to two superconducting reservoirs, which are described by the Bogoliubov–de Gennes Hamiltonians and have the chemical potentials, the temperatures, and the complex order parameters as the relevant quantities. We consider a specific example in which the quantum dot is represented by the Anderson impurity model and study the transport properties, proximity effect and Andreev bound states in equilibrium as well as far-from-equilibrium setups.

Topological defects and gapless modes in insulators and superconductors

We develop a unified framework to classify topological defects in insulators and superconductors described by spatially modulated Bloch and Bogoliubov de Gennes Hamiltonians. We consider Hamiltonians $\mathcal{H}(\mathbf{k},\mathbf{r})$ that vary slowly with adiabatic parameters $\mathbf{r}$ surrounding the defect and belong to any of the ten symmetry classes defined by time-reversal symmetry and particle-hole symmetry. The topological classes for such defects are identified and explicit formulas for the topological invariants are presented. We introduce a generalization of the bulk-boundary correspondence that relates the topological classes to defect Hamiltonians to the presence of protected gapless modes at the defect. Many examples of line and point defects in three-dimensional systems will be discussed. These can host one dimensional chiral Dirac fermions, helical Dirac fermions, chiral Majorana fermions, and helical Majorana fermions, as well as zero-dimensional chiral and Majorana zero modes. This approach can also be used to classify temporal pumping cycles, such as the Thouless charge pump, as well as a fermion parity pump, which is related to the Ising non-Abelian statistics of defects that support Majorana zero modes.

Vicious walks with a wall, noncolliding meanders, and chiral and Bogoliubov-de Gennes random matrices.

Spatially and temporally inhomogeneous evolution of one-dimensional vicious walkers with wall restriction is studied. We show that its continuum version is equivalent with a noncolliding system of stochastic processes called Brownian meanders. Here the Brownian meander is a temporally inhomogeneous process introduced by Yor as a transform of the Bessel process that is the motion of radial coordinate of the three-dimensional Brownian motion represented in spherical coordinates. It is proved that the spatial distribution of vicious walkers with a wall at the origin can be described by the eigenvalue statistics of Gaussian ensembles of Bogoliubov-de Gennes Hamiltonians of the mean-field theory of superconductivity, which have a particle-hole symmetry. We report that a time evolution of the present stochastic process is fully characterized by the change of symmetry classes from type C to type CI in the nonstandard classes of random matrix theory of Altland and Zirnbauer. The relation between the noncolliding systems of the generalized meanders of Yor, which are associated with the even-dimensional Bessel processes, and the chiral random matrix theory is also clarified.