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arXiv:2310.09197 · Physical review. B./Physical review. B

Bosonic Andreev bound state

Nobuyuki Ōkuma

Published Jul 23, 2024• 1 citation
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Abstract

A general free bosonic system with a pairing term is described by a bosonic Bogoliubov-de Gennes (BdG) Hamiltonian. The representation is given by a pseudo-Hermitian matrix, which is crucially different from the Hermitian representation of a fermionic BdG Hamiltonian. In fermionic BdG systems, a topological invariant of the whole particle (hole) bands can be nontrivial, which characterizes the Andreev bound states (ABS), including Majorana fermions. In bosonic cases, on the other hand, the corresponding topological invariant is thought to be trivial, owing to the stability condition of the bosonic ground state. In this paper, we consider a two-dimensional model that realizes a bosonic analogy of the ABS at the boundaries. The boundary states of this model are located outside the bulk bands and are characterized by a nontrivial Berry phase (or polarization) of the hole band. Furthermore, we investigate the zero-energy flat-band limit in which the Bloch Hamiltonian is defective, where the particle and hole states are identical to each other. In this limit, the Berry phase is ${\mathbb{Z}}_{2}$ quantized thanks to an emergent parity-time symmetry induced by the defective nature. This is an example of a topological invariant that uses the defective nature as a projection structure. Thus, boundary states in our model are essentially different from Hermitian topological modes and their variants. We also discuss the entanglement entropy of the system with bosonic Andreev bound states, motivated by the relationship between our model and the continuous variable surface codes.

DOI
10.1103/physrevb.110.014516

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A general free bosonic system with a pairing term is described by a bosonic Bogoliubov-de Gennes (BdG) Hamiltonian. The representation is given by a pseudo-Hermitian matrix, which is crucially different from the Hermitian representation of a fermionic BdG Hamiltonian. In fermionic BdG systems, a topological invariant of the whole particle (hole) bands can be nontrivial, which characterizes the Andreev bound states (ABS), including Majorana fermions. In bosonic cases, on the other hand, the corresponding topological invariant is thought to be trivial, owing to the stability condition of the bosonic ground state. In this paper, we consider a two-dimensional model that realizes a bosonic analogy of the ABS at the boundaries. The boundary states of this model are located outside the bulk bands and are characterized by a nontrivial Berry phase (or polarization) of the hole band. Furthermore, we investigate the zero-energy flat-band limit in which the Bloch Hamiltonian is defective, where the particle and hole states are identical to each other. In this limit, the Berry phase is Z2{\mathbb{Z}}_{2}Z2​ quantized thanks to an emergent parity-time symmetry induced by the defective nature. This is an example of a topological invariant that uses the defective nature as a projection structure. Thus, boundary states in our model are essentially different from Hermitian topological modes and their variants. We also discuss the entanglement entropy of the system with bosonic Andreev bound states, motivated by the relationship between our model and the continuous variable surface codes.
Topological superconductors in one spatial dimension exhibiting a single Majorana bound state at each end are distinguished from trivial gapped systems by a Z2{\mathbb{Z}}_{2}Z2​ topological invariant. Originally, this invariant was calculated by Kitaev in terms of the Pfaffian of the Majorana representation of the Hamiltonian: The sign of this Pfaffian divides the set of all gapped quadratic forms of Majorana fermions into two inequivalent classes. In the more familiar Bogoliubov de Gennes mean-field description of superconductivity, an emergent particle-hole symmetry gives rise to a quantized Zak-Berry phase, the value of which is also a topological invariant. In this work, we explicitly show the equivalence of these two formulations by relating both of them to the phase winding of the transformation matrix that brings the Majorana representation matrix of the Hamiltonian into its Jordan normal form.
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 avoid EPs in the Brillouin zone by introducing an imaginary part of the momentum. We then define the Berry phase for a one-dimensional bosonic Bogoliubov system. Extending the argument for Hermitian systems, the Berry phase for an inversion-symmetric system is shown to be Z2{\mathbb{Z}}_{2}Z2​. As concrete examples, we numerically investigate two toy models and confirm the bulk-edge correspondence even in the presence of complex eigenvalues. The Z2{\mathbb{Z}}_{2}Z2​ invariant associated with particle-hole symmetry and the winding number for a time-reversal-symmetric system are also discussed.
A topological superfluid phase characterized by an emergent chiral-p-wave pair potential is expected to form in a two-dimensional Fermi superfluid subject to s-wave pairing, spin-orbit coupling and a large-enough Zeeman splitting. Andreev bound states appear at phase boundaries, including Majorana zero modes whose existence is assured by the bulk-boundary correspondence principle. Here we study the physical properties of these subgap-energy bound states at step-like interfaces using the spin-resolved Bogoliubov-de Gennes mean-field formalism and assuming small spin-orbit coupling. Extending a recently developed spin-projection technique based on Feshbach partitioning [SciPost Phys. 5, 016 (2018)] combined with the Andreev approximation allows us to obtain remarkably simple analytical expressions for the bound-state energies as well as the majority and minority spin components of their wave functions. Besides the vacuum boundary, where a majority-spin Majorana excitation is encountered, we also consider the boundary between the topological and a nontopological superfluid phase that can appear in a coexistence scenario due to the first-order topological phase transition predicted for this system. At this superfluid-superfluid interface, we find a localized chiral Majorana mode hosted by the minority-spin sector. Our theory further predicts majority-spin subgap-energy bound states similar to those found at a Josephson junction between same-chirality p-wave superfluids. Their presence affects the Majorana mode due to a coupling of minority and majority spin sectors only in the small energy range where their spectra overlap. Our results may inform experimental efforts aimed at realizing and characterizing unconventional Majorana quasiparticles.
A new understanding of criticality in systems described by quadratic bosonic Hamiltonians (QBHs) ties the emergence of long-range correlations to boundaries of dynamical, not thermodynamical, stability in parameter space. This separation occurs because the solution of the Heisenberg equations of motion is determined by an auxiliary pseudo-Hermitian dynamical system. The boundary points of a region of dynamical stability can be either exceptional points, generically associated with long-range correlations, or Krein collisions, where correlations can be either long- or short-range. We investigate the interplay of this landscape of possibilities with band topology and boundary physics, by relying on both specific examples and general arguments. The examples stem from a two-parameter, thermodynamically unstable family of QBHs obtained from the bosonic Su-Schrieffer-Heeger model by breaking particle conservation while preserving a chiral pseudo-symmetry. As a function of an interpolation parameter, distinct regions emerge within a fixed dynamical-stability phase, which prove to be topologically trivial and nontrivial, respectively. The topological phase transition is a line of Krein collisions, which coincides with the closing of a band gap at zero and causes the localization length of the topologically mandated boundary zero modes to diverge before disappearing. We show that the chiral pseudo-symmetry induces, despite the broken particle-number symmetry, enough structure on its associated dynamical matrices to support a topological classification and a bulk-boundary correspondence, independently of dynamical stability. This strongly suggests that, for non-interacting bosons, topological physics extracted from basic index theory is insensitive to dynamical stability and, a posteriori, criticality.
The entanglement entropy (von Neumann entropy) has been used to characterize the complexity of many-body ground states in strongly correlated systems. In this paper, we try to establish a connection between the lower bound of the von Neumann entropy and the Berry phase defined for quantum ground states. As an example, a family of translational invariant lattice free fermion systems with two bands separated by a finite gap is investigated. We argue that, for one-dimensional (1D) cases, when the Berry phase (Zak's phase) of the occupied band is equal to $\ensuremath{\pi}\ifmmode\times\else\texttimes\fi{}(\text{odd}\phantom{\rule{0.3em}{0ex}}\text{integer})$ and when the ground state respects a discrete unitary particle-hole symmetry (chiral symmetry), the entanglement entropy in the thermodynamic limit is at least larger than ln 2 (per boundary), i.e., the entanglement entropy that corresponds to a maximally entangled pair of two qubits. We also discuss how this lower bound is related to vanishing of the expectation value of a certain nonlocal operator which creates a kink in 1D systems.
The simplest continuum model of a one-dimensional noninteracting 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 nontrivial phase, revealing their spatial localization, their transformation properties under symmetry operations, and the exact finite-size energy splitting of the associated quasidegenerate 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 nontrivial phase. For the open wire, on the other hand, we first take a semiclassical approach employing the Mathieu equation to study the two quasidegenerate 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.