The Bernard-LeClair (BL) symmetry classes generalize the ten-fold way classes in the absence of Hermiticity. Within the BL scheme, time-reversal and particle-hole come in two flavors, and "pseudo-Hermiticity" generalizes Hermiticity. We propose that these symmetries are relevant for the topological classification of non-Hermitian single-particle Hamiltonians and Hermitian bosonic Bogoliubov-de Gennes (BdG) models. We show that the spectrum of any Hermitian bosonic BdG Hamiltonian is found by solving for the eigenvalues of a non-Hermitian matrix which belongs to one of the BL classes. We therefore suggest that bosonic BdG Hamiltonians inherit the topological properties of a non-Hermitian symmetry class and explore the consequences by studying symmetry-protected edge instabilities in a simple 1D system.
We consider Bogoliubov de Gennes equation on metric graphs. The vertex boundary conditions providing self-adjoint realization of the Bogoliubov de Gennes operator on a metric star graph are derived. Secular equation providing quantization of the energy and the vertex transmission matrix are also obtained. Application of the model for Majorana wire networks is discussed.
We develop a systematic approach for constructing symmetry-based indicators of a topological classification for superconducting systems. The topological invariants constructed in this work form a complete set of symmetry-based indicators that can be computed from knowledge of the Bogoliubov-de Gennes Hamiltonian on high-symmetry points in Brillouin zone. After excluding topological invariants corresponding to the phases without boundary signatures, we arrive at natural generalization of symmetry-based indicators [H. C. Po, A. Vishwanath, and H. Watanabe, Nature Comm. 8, 50 (2017)] to Hamiltoni…
The applicability of the quasiclassical theory of superconductivity in Josephson multi-layer structures is analyzed. The quasiclassical approach is compared with the exact theory based on the Bogoliubov - de Gennes equation. The angle and energy resolved (coarse-grain) currents are calculated using both techniques. It is shown that the two approaches agree in $SIS'IS''$ geometries after the coarse-grain averaging. A quantitative discrepancy, which exceeds the quasiclassical accuracy, is observed when three or more interfaces are present. The invalidity of the quasiclassical theory is attribute…
We propose a fast and efficient approach for solving the Bogoliubov-de Gennes (BdG) equations in superconductivity, with a numerical matrix-size reduction procedure proposed by Sakurai and Sugiura [J. Comput. Appl. Math. 159, 119 (2003)]. The resultant small-size Hamiltonian contains the information of the original BdG Hamiltonian in a given energy domain. In other words, the present approach leads to a numerical construction of a low-energy effective theory in superconductivity. The combination with the polynomial expansion method allows a self-consistent calculation of the BdG equations. Thr…
We study a Majorana zero-energy state bound to a hedgehog-like point defect in a topological superconductor described by a Bogoliubov-de Gennes (BdG)-Dirac type effective Hamiltonian. We first give an explicit wave function of a Majorana state by solving the BdG equation directly, from which an analytical index can be obtained. Next, by calculating the corresponding topological index, we show a precise equivalence between both indices to confirm the index theorem. Finally, we apply this observation to reexamine the role of another topological invariant, i.e., the Chern number associated with t…
It is shown that bound state solutions of the one dimensional Bogoliubov-de Gennes (BdG) equation may exist when the Fermi velocity becomes dependent on the space coordinate. The existence of bound states in continuum (BIC) like solutions has also been confirmed both in the normal phase as well as in the super-conducting phase. We also show that a combination of Fermi velocity and gap parameter step-like profiles provides scattering solutions with normal reflection and transmission.
We investigate the quasiparticle density of states in disordered d-wave superconductors. By constructing a quantum map describing the quasiparticle dynamics in such a medium, we explore deviations of the density of states from its universal form ($\propto E$), and show that additional low-energy quasiparticle states exist provided (i) the range of the impurity potential is much larger than the Fermi wavelength [allowing to use recently developed semiclassical methods]; (ii) classical trajectories exist along which the pair-potential changes sign; and (iii) the diffractive scattering length is …
In recent research, it has been shown that non-Hermitian systems exhibit sensitivity to boundaries, and it is caused by the non-Hermitian skin effect. In this work, we construct the non-Bloch band theory in bosonic Bogoliubov--de Gennes (BdG) systems. From our theory, we can calculate the generalized Brillouin zone and the energy spectrum in such systems with open boundary conditions in the thermodynamic limit, and we can thus discuss its non-Hermitian nature, despite Hermiticity of an original Hamiltonian. In fact, we find that the bosonic Kitaev-Majorana chain exhibits rich aspects of the no…
Using the method of noncommutative geometry, we define a topological invariant in disordered bosonic Bogoliubov-de Gennes systems, which possess a unique mathematical property---non-Hermiticity. To demonstrate the validity of the definition, we investigate a disordered artificial spin ice model in two dimensions numerically. In the clean limit, we clarify that the topological index perfectly coincides with the Chern number. We also show that the topological index is robust against disorder. The formula provides the topological index $n_{\rm Ch}=1$ in the magnon Hall regime and $n_{\rm Ch}=0$ i…
The energy spectrum of bosonic excitations from a condensate is given by the spectrum of a non-Hermitian Hamiltonian constructed from a bosonic Bogoliubov-de Gennes (BdG) Hamiltonian in general even though the system is essentially Hermitian. In other words, two types of non-Hermiticity can coexist: one from the bosonic BdG nature and the other from the open quantum nature. In this paper, we propose boundary-dependent dynamical instability. We first define the bosonic dissipative BdG Hamiltonian in terms of Green's function in Nambu space and discuss the correct particle-hole symmetry of the c…
Dynamical instability is an inherent feature of bosonic systems described by the Bogoliubov de Geenes (BdG) Hamiltonian. Since it causes the BdG system to collapse, it is generally thought that it should be avoided. Recently, there has been much effort to harness this instability for the benefit of creating a topological amplifier with stable bulk bands but unstable edge modes which can be populated at an exponentially fast rate. We present a theorem for determining the stability of states with energies sufficiently away from zero, in terms of an unconventional commutator between the number co…
Since the theoretical prediction and experimental observation of the thermal Hall effect of magnons, a variety of novel phenomena that may occur in magnonic systems have been proposed. In this paper, we review the recent advances in the study of topological phases of magnon Bogoliubov-de Gennes (BdG) systems. After giving an overview of the previous works on electronic topological insulators and the thermal Hall effect of magnons, we provide the necessary background for bosonic BdG systems, with a particular emphasis on their non-Hermiticity arising from the diagonalization of the BdG Hamilton…
We discuss the topology of Bogoliubov excitation bands from a Bose-Einstein condensate in an optical lattice. Since the Bogoliubov equation for a bosonic system is non-Hermitian, complex eigenvalues often appear and induce dynamical instability. As a function of momentum, the onset of appearance and disappearance of complex eigenvalues is an exceptional point (EP), which is a point where the Hamiltonian is not diagonalizable and hence the Berry connection and curvature are ill-defined, preventing defining topological invariants. In this paper, we propose a systematic procedure to remove EPs fr…
We consider a Bogolibov-de Geenes (BdG) Hamiltonian, which is a non-Hermitian Hamiltonian with pseudo-Hermiticity, for a system of (pseudo) spin-$1/2$ bosons in a honeycomb lattice under the condition that the population difference between the two spin components, i.e., magnetization, is a constant. Such a system is capable of acting as a topological amplifier, under time-reversal symmetry, with stable bulk bands but unstable edge modes which can be populated at an exponentially fast rate. We quantitatively study the topological properties of this model within the framework of the 38-fold way …
Bosonic Bogoliubov transformations as Lorentz boosts in $(c,{\overline c})=(1,1)$ conformal field theories with marginal $J{\overline J }$ deformations
Within the microscopic Bogoliubov--de Gennes (BdG) formalism an exact quantization condition for Andreev bound states of the ferromagnetic-superconducting hybrid systems of box geometry is derived and a semi-classical formula for the density of states is obtained. The semi-classical formula is shown to agree with the exact result, even when the exchange field $h$, is much larger than the superconductor order parameter, provided $h$ is small compared with the Fermi energy.
By solving the Bogoliubov--De Gennes equations analytically, we derive the fermionic zero-modes satisfying the Majorana property that exist in vortices of a two-dimensional $s$-wave Fermi superfluid with spin-orbit coupling and Zeeman spin-splitting. The Majorana zero-mode becomes normalisable and exponentially localised to the vicinity of the vortex core when the superfluid is topologically non-trivial. We calculate the energy splitting due to Majorana hybridisation and identify that the $s$-wave Majorana vortices obey non-Abelian statistics.
We propose an efficient numerical algorithm to solve Bogoliubov de Gennes equations self-consistently for inhomogeneous superconducting systems with a reformulated polynomial expansion scheme. This proposed method is applied to typical issues such as a vortex under randomly distributed impurities and a normal conducting junction sandwiched between superconductors. With various technical remarks, we show that its efficiency becomes remarkable in large-scale parallel performance.
We show that the general low-energy Bogoliubov-de Genness Hamiltonian in a multiband superconductor with broken time reversal and preserved inversion symmetry is a generator of real four-dimensional representation of $SO(4)$. In the particular representation such an effective Hamiltonian is a purely imaginary matrix, and it is proportional to the antisymmetric tensor of a fictitious electromagnetic field which one can define in the momentum space. The quantum time evolution of the low-energy quasiparticle state becomes this way closely related to the classical relativistic motion of a charged …
