Bogoliubov-de Gennes Hamiltonian Research Articles

Topological Superconductors with Spin-Triplet Pairings and Majorana Fermi arcs

Abstract We construct a three-dimensional topological superconductor Bogoliubov-de Gennes(BdG) Hamiltonian with the normal state being a three dimensional topological insulator. By introducing inter-orbital spin-triplet pairings term △3, there are topological Majorana nodes in the bulk, and they are connected by Majorana Fermi arcs on the surface, similar to the case of Weyl semimetal. Furthermore, by adding an inversion breaking term to the normal state, momentum independent pairing terms with different parities can coexist in the BdG Hamiltonian, which creates more Majorana modes similar to Andreev bound states and a richer phase diagram.

Patterning by dynamically unstable spin–orbit-coupled Bose–Einstein condensates

In a two-dimensional atomic Bose–Einstein condensate, we demonstrate Rashba spin–orbit coupling can always introduce dynamical instability into specific zero-quasimomentum states in all parameter regimes. During the evolution of the zero-quasimomentum states, such spin–orbit-coupling-induced instability can fragment the states and lead to a dynamically patterning process. The features of formed patterns are identified from the symmetries of the Bogoliubov–de Gennes Hamiltonian. We show that spin–orbit-coupled Bose–Einstein condensates provide an interesting platform for the investigation of pattern formations.

Magnetic‐Field Regulation of Nodal Topological Superconducting States in Monolayer Ising Superconductor NbSe2

The nodal topological superconducting (NTSC) state with Majorana flat bands (MFBs) is an exotic matter of state hosting Majorana fermions that obey non‐Abelian statistics. Recently, monolayer Ising superconductor NbSe2 is shown to be an ideal platform for realizing an NTSC state. Through tight‐binding calculations based on the Bogoliubov–de–Gennes Hamiltonian, it is demonstrated that the in‐plane magnetic field B combined with superconducting pairing Δs, Rashba spin–orbit coupling (SOC) Vr, and chemical potential μ can regulate the topological properties of the NTSC state. First, B larger than Δs effectively induces an NTSC state with nodal points (NPs) located on the high‐symmetry line of the Brillouin zone. The length of the resulting MFBs in the ribbon increases with B. Second, Rashba SOC greatly affects the number and locations of NPs by tilting bands near Fermi level (EF), leading to unidirectional Majorana edge states. Finally, when B > Δs, μ affects the bulk band structure at EF, resulting in an NTSC state with 6, 12, 18, 24 NPs and 0, 2, 3, 4 MFBs. These results are important to clarifying the behavior of the NTSC state under a B and designing a quantum computing platform based on exotic Majorana fermions.

Supercurrent Distribution on Ammann-Beenker Structure

We study the distribution of supercurrent in the attractive Hubbard model on a periodic approximant of the Ammann-Beenker structure by means of the Bogoliubov-de Gennes (BdG) mean-field theory. We first investigate the spatial distributions of the electron density and superconducting order parameter and find a crossover between superconducting states with (i) extended, (ii) localized, and (iii) short-ranged Cooper pairs, similar to the previous results on the Penrose tiling. Furthermore, we formulate the expression of the supercurrent for the periodic approximant in terms of the eigenenergies and the wave functions of the Bogoliubov-de Gennes Hamiltonian and numerically investigate how the local supercurrent distribution varies by changing the direction of the applied phase gradient.

Chiral Majorana fermions in two-dimensional square lattice antiferromagnet with proximity-induced superconductivity

Combination of proximity-induced superconductivity and ferromagnetic exchange field in a two-dimensional square-lattice antiferromagnet with spin–orbit coupling and nonsymmorphic symmetry can induce a topological superconductor phase with chiral Majorana edge states. The lattice model of the Bogoliubov-de Gennes (BdG) Hamiltonian was applied to study the phase diagram of bulks and chiral Majorana edge states in nanoribbons. By numerically studying the phase diagram, we found that the non-uniformity of either the superconducting pairing parameters or the exchange field at the two sublattices is necessary to induce a topological superconductor phase with chiral Majorana edge states. The BdG Chern number of certain topological superconductor phases is ±1 or ±3, such that the corresponding nanoribbons have one or three pairs of chiral Majorana edge states, respectively.

Boundary-dependent dynamical instability of bosonic Green's function: Dissipative Bogoliubov–de Gennes Hamiltonian and its application to non-Hermitian skin effect

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 corresponding non-Hermitian Hamiltonian. We then construct a model of the boundary-dependent dynamical instability so that it satisfies the correct particle-hole symmetry. In this model, an anomalous term that breaks the particle number conservation represents the non-Hermiticity of the BdG nature, while a normal term is given by a dissipative Hatano-Nelson model. Thanks to the competition between the two types of non-Hermiticity, the imaginary part of the spectrum can be positive without the help of the amplification of the normal part and the particle-hole band touching that causes the Landau instability. This leads to the boundary-dependent dynamical instability under the non-Hermitian skin effect, -strong dependence of spectra on boundary conditions for non-Hermitian Hamiltonians-, of the Bogoliubov spectrum.

Majorana flat bands at structured surfaces of nodal noncentrosymmetric superconductors

Surfaces of nodal noncentrosymmetric superconductors can host flat bands of Majorana modes, which provide a promising platform for quantum computation if one can find methods for manipulating localized Majorana wave packets. We study the fate of such flat bands when part of the surface is subjected to an exchange field induced by a ferromagnetic insulator. We use exact diagonalization to find the eigenstates and eigenenergies of the Bogoliubov-de Gennes Hamiltonian of a model system, for which an exchange field is applied along a strip on the surface of a slab. We consider different orientations of the strip and the applied field. If the spin polarization of the field-free system along the field direction is sufficiently large perturbation theory predicts that energies of states which are mostly localized on the exchange-field strip are shifted away from zero energy by an amount proportional to the field strength. On the other hand, energies corresponding to states localized on the field-free strip are only weakly affected by the field. Exact diagonalization confirms this. Moreover, we discuss a setup with a small exchange field applied to the previously field-free strip with the goal of introducing a linear dispersion. By switching this dispersion on and off, a wave packet could be moved in a certain direction. We find that in our model system, a linear dispersion can indeed be achieved. The qualitative features of this dispersion can be predicted from the momentum-dependent spin polarization of the field-free surface.

Two-Dimensional Topological Superconductors with Spin-Orbit Coupling and Zeeman Term

Topological superconductors have attracted much attention for their potential applications in realizing topological quantum computing. In this paper, we show a system with 8 × 8 Bogoliubov-de Gennes Hamiltonian. The system has particle-hole symmetry. By adding a Zeeman term to the model, we discuss this system from two situations. In the first case of breaking the inversion symmetry while preserving the mirror symmetry, it is obtained from the topological phase diagram of the system that it is a topological superconductor. In another case where the mirror symmetry is broken while preserving the inversion symmetry, the system has two nodes connected by a flat band of zero-energy Andreev edge states and the topological number is non-zero . It can be concluded that the system is a topological nodal superconductor.

Higher-order topological superconductivity from repulsive interactions in kagome and honeycomb systems

We discuss a pairing mechanism in interacting two-dimensional multipartite lattices that intrinsically leads to a second order topological superconducting state with a spatially modulated gap. When the chemical potential is close to Dirac points, oppositely moving electrons on the Fermi surface undergo an interference phenomenon in which the Berry phase converts a repulsive electron–electron interaction into an effective attraction. The topology of the superconducting phase manifests as gapped edge modes in the quasiparticle spectrum and Majorana Kramers pairs at the corners. We present symmetry arguments which constrain the possible form of the electron–electron interactions in these systems and classify the possible superconducting phases which result. Exact diagonalization of the Bogoliubov-de Gennes Hamiltonian confirms the existence of gapped edge states and Majorana corner states, which strongly depend on the spatial structure of the gap. Possible applications to vanadium-based superconducting kagome metals AV3Sb5 (A = K, Rb, Cs) are discussed.

Universality classes of the Anderson transition in the three-dimensional symmetry classes AIII, BDI, C, D, and CI

We clarify universal critical properties of delocalization-localization transitions in three-dimensional unitary and orthogonal classes with particle-hole and/or chiral symmetries (classes AIII, BDI, D, C, and CI). We first introduce tight-binding models on cubic lattice that belong to these five nonstandard symmetry classes, respectively. Unlike the Bogoliubov-de Gennes Hamiltonian for superconductors, all the five models have finite areas of Fermi surfaces in the momentum space in the clean limit. Thereby, the scaling theory of the Anderson transition guarantees the presence of the delocalization-localization transitions at finite disorder strength in these models. Based on this expectation, we carry out extensive transfer matrix calculations of the Lyapunov exponents for zero-energy eigenstates of the disordered tight-binding models with quasi-one-dimensional geometry. Near the Anderson transition point, the correlation length diverges with a universal critical exponent $\ensuremath{\nu}$. Using finite-size scaling analysis of the localization length, we determine the critical exponent of the transitions and scaling function of the (normalized) localization length in the three nonstandard unitary symmetry classes: ${\ensuremath{\nu}}_{\text{AIII}}=1.06\ifmmode\pm\else\textpm\fi{}0.02$, ${\ensuremath{\nu}}_{\text{D}}=0.87\ifmmode\pm\else\textpm\fi{}0.03$, and ${\ensuremath{\nu}}_{\text{C}}=0.996\ifmmode\pm\else\textpm\fi{}0.012$. Our result of the class C is consistent with a previous study of classical network model [M. Ortu\~no et al., Phys. Rev. Lett. 102, 070603 (2009)]. The critical exponents of the two nonstandard orthogonal classes are estimated as ${\ensuremath{\nu}}_{\text{CI}}=1.17\ifmmode\pm\else\textpm\fi{}0.02$ and ${\ensuremath{\nu}}_{\text{BDI}}=1.12\ifmmode\pm\else\textpm\fi{}0.06$. Our result of the class CI is consistent with another previous study of a lattice model, while the exponent of the class BDI is at variance with the previous evaluation of nodal Dirac ring model [X. Luo et al., Phys. Rev. B 101, 020202(R) (2020)].

Fractonic superfluids. II. Condensing subdimensional particles

In this paper, we develop an exotic fractonic superfluid phase in $d$-dimensional space where subdimensional particles -- their mobility is \emph{partially} restricted -- are condensed. The off-diagonal long range order (ODLRO) is investigated. To demonstrate, we consider "lineons" -- a subdimensional particle whose mobility is free only in certain one-dimensional directions. We start with a $d$-component microscopic Hamiltonian model. The model respects a higher-rank symmetry such that both particle numbers of each component and angular charge moments are conserved quantities. By performing the Hartree-Fock-Bogoliubov approximation, we derive a set of Gross-Pitaevskii equations and a Bogoliubov-de Gennes (BdG) Hamiltonian, which leads to a unified description of gapless phonons and gapped rotons. With the coherent-path-integral representation, we also derive the long-wavelength effective field theory of gapless Goldstone modes and analyze quantum fluctuations around classical ground states. The Euler-Lagrange equations and Noether charges/currents are also studied. In two spatial dimensions and higher, such an ODLRO stays stable against quantum fluctuations. Finally, we study vortex configurations. The higher-rank symmetry enforces a hierarchy of point vortex excitations whose structure is dominated by two guiding statements. Specially, we construct two types of vortex excitations, the conventional and dipole vortices. The latter carries a charge with dimension as a momentum. The two statements can be more generally applicable. Several future directions are discussed.

Squeezing-induced topological gap opening on bosonic Bogoliubov excitations

We investigate the role of squeezing interaction in inducing topological Bogoliubov excitations of a bosonic system. We introduce a squeezing transformation which is capable of reducing the corresponding Bogoliubov-de Gennes Hamiltonian to an effective non-interacting one with the spectra and topology unchanged. In the weak interaction limit, we apply the perturbation theory to investigate the squeezing-induced topological gap opening on bosonic Bogoliubov excitations and find that the squeezing interaction plays an equivalent role as a spin-orbit or Zeeman-like coupling in the effective Hamiltonian. We thus apply this formalism to two existed models for providing deeper understandings of their topological structures. We also construct minimal models based on the elegant Clifford algebra for realizing bosonic topological Bogoliubov excitations. Our construction is potentially applicable for experiments in bosonic systems.

Mutual transition of Andreev and Majorana bound states in a superconducting gap

Using the Bogoliubov–de Gennes Hamiltonian, we analytically study two models with superconducting order, the p-wave model with an impurity potential and the s-wave nanowire model with superconductivity induced by the proximity effect with an impurity potential in a Zeeman field with a spin–orbit interaction. Using the Dyson equation, we study conditions for the emergence of Andreev bound states with energies close to the boundary of the superconducting gap and the possibility for these states to pass into Majorana-like bound states. We prove that in the topological phase (in the p-wave case also in the trivial phase) for both models, the Andreev bound states with energy close to the boundary of the superconducting gap can exist, but although their emergence in the p-wave model is due to the appearance of a (nonmagnetic) impurity, they appear in the s-wave model only as a result of a local perturbation of the Zeeman field. For both models, the transition of the Andreev bound states into the Majorana states (and back) is impossible in the topological phase, which is explained by the topological protection of the Majorana-like bound states in the topological phase.

Non-Hermiticity and topological invariants of magnon Bogoliubov–de Gennes systems

Abstract Since the theoretical prediction and experimental observation of the magnon thermal Hall effect, a variety of novel phenomena that may occur in magnonic systems have been proposed. We review recent advances in the study of topological phases of magnon Bogoliubov–de Gennes (BdG) systems. After giving an overview of previous works on electronic topological insulators and the magnon thermal Hall effect, we provide the necessary background for bosonic BdG systems, with particular emphasis on their non-Hermiticity arising from the diagonalization of the BdG Hamiltonian. We then introduce definitions of $$ \mathbb{Z}_2 $$ topological invariants for bosonic systems with pseudo-time-reversal symmetry, which ensures the existence of bosonic counterparts of “Kramers pairs.” Because of the intrinsic non-Hermiticity of bosonic BdG systems, these topological invariants have to be defined in terms of the bosonic Berry connection and curvature. We then introduce theoretical models that can be thought of as magnonic analogs of two- and three-dimensional topological insulators in class AII. We demonstrate analytically and numerically that the $$ \mathbb{Z}_2 $$ topological invariants precisely characterize the presence of gapless edge/surface states. We also predict that bilayer CrI$$_3$$ with a particular stacking would be an ideal candidate for the realization of a two-dimensional magnon system characterized by a nontrivial $$ \mathbb{Z}_2 $$ topological invariant. For three-dimensional topological magnon systems, the magnon thermal Hall effect is expected to occur when a magnetic field is applied to the surface.

Majorana zero modes and their bosonization

The simplest continuum model of a one-dimensional non-interacting superconducting fermionic symmetry-protected topological (SPT) phase is studied in great detail using analytical methods. In a first step, we present a full exact diagonalization of the fermionic Bogoliubov-de Gennes Hamiltonian for a system of finite length and with open boundaries. In particular, we derive exact analytical expressions for the Majorana zero modes emerging in the topologically non-trivial phase, revealing their spatial localization, their transformation properties under symmetry operations, and the exact finite-size energy splitting of the associated quasi-degenerate ground states. We then proceed to analyze the model via exact operator bosonization in both open and closed geometries. In the closed wire geometry, we demonstrate fermion parity switching from twisting boundary conditions in the topologically non-trivial phase. For the open wire, on the other hand, we first take a semiclassical approach employing the Mathieu equation to study the two quasi-degenerate ground states as well as their energy splitting at finite system sizes. We then finally derive the exact forms of the Majorana zero modes in the bosonic language using vertex-algebra techniques. These modes are verified to be in exact agreement with the results obtained from the fermionic description. The complementary viewpoints provided by the fermionic and bosonic formulations of the superconducting SPT phase are reconciled, allowing us to provide a complete and exact account of how Majorana zero modes manifest in a bosonized description of an SPT phase.

Symmetry-based indicators for topological Bogoliubov–de Gennes Hamiltonians

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 Hamiltonians of Bogoliubov-de Gennes type.

Nodal Andreev spectra in multi-Majorana three-terminal Josephson junctions

We investigate the Andreev-bound-state (ABS) spectra of three-terminal Josephson junctions which consist of 1D topological superconductors (TSCs) harboring multiple zero-energy edge Majorana bound states (MBSs) protected by chiral symmetry. Our theoretical analysis relies on the exact numerical diagonalization of the Bogoliubov-de Gennes (BdG) Hamiltonian describing the three interfaced TSCs, complemented by an effective low-energy description solely based on the coupling of the interfacial MBSs arising before the leads get contacted. Considering the 2D synthetic space spanned by the two independent superconducting phase differences, we demonstrate that the ABS spectra may contain either point or line nodes, and identify $\mathbb{Z}_2$ topological invariants to classify them. We show that the resulting type of nodes depends on the number of preexisting interfacial MBSs, with nodal lines necessarily appearing when two TSCs harbor an unequal number of MBSs. Specifically, the precise number of interfacial MBSs determines the periodicity of the spectrum under $2\pi$-slidings of the phase differences and, as a result, also controls the shape of the nodal lines in synthetic space. When chiral symmetry is preserved, the lines are open and coincide with high-symmetry lines of synthetic space, while when it is violated the lines can also transform into loops and chains. The nodal spectra are robust by virtue of the inherent particle-hole symmetry of the BdG Hamiltonian, and give rise to distinctive experimental signatures that we identify.

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.

Disorder and critical current variability in Josephson junctions

We investigate theoretically the origins of observed variations in the critical currents of Nb/Al-AlOx/Nb Josephson junctions in terms of various types of disorder. We consider the following disorder sources: vacancies within the Al layer, thickness variations in the AlOx layer, and “pinholes” (i.e., point contacts) within the AlOx layer. The calculations are all performed by solving the microscopic Bogoliubov–de Gennes Hamiltonian self-consistently. It is found that a small concentration of vacancies within the Al layer is sufficient to suppress the critical current, while the presence of a small number of thick regions of the oxide layer induces a similar effect as well. The pinhole scenario is found to result in anomalous behavior that resembles neither that of a pure tunnel junction nor that of a superconductor–normal–superconductor junction but a regime that interpolates between these two limits. We comment on the degree to which each of the three scenarios describes the actual situation present in these junctions.

Recent Progress of Powerful Numerical Method to Solve Bogoliubov–de Gennes Hamiltonian

Solving the Bogoliubov–de Gennes Hamiltonian is crucial to understanding the essence of the superconducting state. Recently, Y. Nagai proposed an ultrafast numerical approach to large-scale inhomog...