Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1904.08724 · Physical Review A

Generalized Berry phase for a bosonic Bogoliubov system with exceptional points

Terumichi Ohashi, Shingo Kobayashi, Yuki Kawaguchi

Published Jan 22, 2020• 27 citations
arXivHTMLPDF

Abstract

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 ${\mathbb{Z}}_{2}$. As concrete examples, we numerically investigate two toy models and confirm the bulk-edge correspondence even in the presence of complex eigenvalues. The ${\mathbb{Z}}_{2}$ invariant associated with particle-hole symmetry and the winding number for a time-reversal-symmetric system are also discussed.

DOI
10.1103/physreva.101.013625

Related papers

Pseudo-time-reversal-symmetry-protected topological Bogoliubov excitations of Bose-Einstein condensates in optical lattices

Junsen Wang, Wei Zheng, Youjin Deng

Bogoliubov excitations of Bose-Einstein condensates in optical lattices may possess band topology in analogous to topological insulators in class AII of fermions. Using the language of the Krein-space theory, this topological property is shown to be protected by a pseudo-time-reversal symmetry that is pseudo-antiunitary and squares to $\ensuremath{-}1$, with the associated bulk topological invariant also being a ${\mathbb{Z}}_{2}$ index. We construct three equivalent expressions for it, relating to the Pfaffian, the pseudo-time-reversal polarization, and most practically the Wannier center flow, all adopted from the fermionic case, defined here with respect to the pseudo inner product. In the presence of an additional pseudo-unitary and pseudo-Hermitian inversion symmetry, a simpler expression is derived. We then study two toy models feasible on cold atom platforms to numerically confirm the bulk-boundary correspondence. The Krein-space approach developed in this work is a universal formalism to study all kinds of symmetry-protected topological bosonic Bogoliubov bands.

arXiv:2010.04342•Physical Review A•Oct 22, 2020•7 citations

-symmetry berry phases, topology and -symmetry breaking

Jinghui Pi, Ning-Ning Sun, Rong Lü

Abstract We study the complex Berry phases in non-Hermitian systems with parity- and time-reversal   symmetry. We investigate a kind of two-level system with   symmetry. We find that the real part of the the complex Berry phases have two quantized values and they are equal to either 0 or π , which originates from the topology of the Hermitian eigenstates. We also find that if we change the relative parameters of the Hamiltonian from the unbroken-   -symmetry phase to the broken-   -symmetry phase, the imaginary part of the complex Berry phases are divergent at the exceptional points. We exhibit two concrete examples in this work, one is a two-level toys model, which has nontrivial Berry phases; the other is the generalized Su–Schrieffer–Heeger (SSH) model that has physical loss and gain in every sublattice. Our results explicitly demonstrate the relation between complex Berry phases, topology and   -symmetry breaking and enrich the field of the non-Hermitian physics.

Communications in Theoretical Physics•Mar 16, 2020•2 citations

Bosonic Andreev bound state

Nobuyuki Ōkuma

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.

arXiv:2310.09197•Physical review. B./Physical review. B•Jul 23, 2024•1 citation

Interaction-induced topological Bogoliubov excitations in a spin-orbit-coupled Bose-Einstein condensate

Guan-Hua Huang, Guang-Quan Luo, Zhigang Wu +1 more

We study topologically nontrivial excitations of a weakly interacting, spin-orbit coupled Bose-Einstein condensate in a two-dimensional square optical lattice, a system recently realized in experiment [Phys. Rev. Lett. 121, 150401 (2018)]. We focus on situations where the system is not subjected to a Zeeman field and thus does not exhibit nontrivial single-particle band topology. Of special interest then is the role of particle interaction as well as its interplay with the symmetry properties of the system in producing topologically nontrivial excitations. We find that the noninteracting system possesses a rich set of symmetries, including the $\mathcal{PT}$ symmetry, the modified dihedral point-group symmetry ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{D}}_{4}$ and the nonsymmorphic symmetry. These combined symmetries ensure the existence of pairs of degenerate Dirac points at the edge of Brillouin zone for the single-particle energy bands. In the presence of particle interaction and with sufficient spin-orbit coupling, the atoms condense in a ground state with net magnetization which spontaneously breaks the $\mathcal{PT}$ and ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{D}}_{4}$ symmetry. We demonstrate that this symmetry breaking leads to a gap opening at the Dirac point for the Bogoliubov spectrum and consequentially topologically nontrivial excitations. We confirm the nontrivial topology by calculating the Chern numbers of the lowest excitation bands and show that gapless edge states form at the interface of systems characterized by different values of the Chern number.

arXiv:2108.13784•Physical Review A•Apr 21, 2021•11 citations

Geometric phase from encircling an exceptional point of a quantum resonance in the complex-scaling method

Okuto Morikawa, Shoya Ogawa, Soma Onoda

Non-Hermitian operators are now routinely used to describe few-mode systems such as optical resonators and superconducting qubits, and exceptional points (EPs) are defective spectral singularities of such non-Hermitian operators. In contrast, the scattering-theoretic formulation of EP physics for unbounded Hamiltonians remains less settled. In this work, we formulate the geometric phase associated with encircling an EP when the underlying eigenstates are quantum resonances within a one-dimensional scattering model. To do this, we employ the complex-scaling method, where resonance poles of the S matrix are realized as discrete eigenvalues of the non-Hermitian dilated Hamiltonian, to construct situations in which resonant and scattering states coalesce into an EP in the complex energy plane, that is, the resonance pole is embedded into the continuum spectrum. We analyze the self-orthogonality in the vicinity of an EP, the Berry phase, and the Chern characteristic. Our results clarify how EP branch structure and geometric holonomy arise directly from resonance poles in scattering theory, thereby connecting non-Hermitian spectral topology with the traditional theory of quantum resonances.

quant-pharXiv:2512.24528•Jan 1, 2026

Bosonic topological excitations from the instability of a quadratic band crossing

Guang-Quan Luo, Andreas Hemmerich, Zhi-Fang Xu

We investigate the interaction-driven instability of a quadratic band crossing arising for ultracold bosonic atoms loaded into a two-dimensional optical lattice. We consider the case when the degenerate point becomes a local minimum of both crossing energy bands such that it can support a stable Bose--Einstein condensate. A repulsive contact interaction among the condensed bosons induces a spontaneously time-reversal-symmetry broken superfluid phase and a topological gap is opened in the excitation spectrum. We propose two concrete realizations of the desired quadratic band crossing in lattices with either fourfold or sixfold rotational symmetries via suitable tuning of the unit cell leading to reduced Brillouin zones and correspondingly folded bands. In either case, topologically protected edge excitations are found for a finite system.

arXiv:1804.02957•Physical Review A•Nov 15, 2018•10 citations
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.
Bogoliubov excitations of Bose-Einstein condensates in optical lattices may possess band topology in analogous to topological insulators in class AII of fermions. Using the language of the Krein-space theory, this topological property is shown to be protected by a pseudo-time-reversal symmetry that is pseudo-antiunitary and squares to $\ensuremath{-}1$, with the associated bulk topological invariant also being a Z2{\mathbb{Z}}_{2}Z2​ index. We construct three equivalent expressions for it, relating to the Pfaffian, the pseudo-time-reversal polarization, and most practically the Wannier center flow, all adopted from the fermionic case, defined here with respect to the pseudo inner product. In the presence of an additional pseudo-unitary and pseudo-Hermitian inversion symmetry, a simpler expression is derived. We then study two toy models feasible on cold atom platforms to numerically confirm the bulk-boundary correspondence. The Krein-space approach developed in this work is a universal formalism to study all kinds of symmetry-protected topological bosonic Bogoliubov bands.
Abstract We study the complex Berry phases in non-Hermitian systems with parity- and time-reversal   symmetry. We investigate a kind of two-level system with   symmetry. We find that the real part of the the complex Berry phases have two quantized values and they are equal to either 0 or π , which originates from the topology of the Hermitian eigenstates. We also find that if we change the relative parameters of the Hamiltonian from the unbroken-   -symmetry phase to the broken-   -symmetry phase, the imaginary part of the complex Berry phases are divergent at the exceptional points. We exhibit two concrete examples in this work, one is a two-level toys model, which has nontrivial Berry phases; the other is the generalized Su–Schrieffer–Heeger (SSH) model that has physical loss and gain in every sublattice. Our results explicitly demonstrate the relation between complex Berry phases, topology and   -symmetry breaking and enrich the field of the non-Hermitian physics.
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.
We study topologically nontrivial excitations of a weakly interacting, spin-orbit coupled Bose-Einstein condensate in a two-dimensional square optical lattice, a system recently realized in experiment [Phys. Rev. Lett. 121, 150401 (2018)]. We focus on situations where the system is not subjected to a Zeeman field and thus does not exhibit nontrivial single-particle band topology. Of special interest then is the role of particle interaction as well as its interplay with the symmetry properties of the system in producing topologically nontrivial excitations. We find that the noninteracting system possesses a rich set of symmetries, including the PT\mathcal{PT}PT symmetry, the modified dihedral point-group symmetry ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{D}}_{4}$ and the nonsymmorphic symmetry. These combined symmetries ensure the existence of pairs of degenerate Dirac points at the edge of Brillouin zone for the single-particle energy bands. In the presence of particle interaction and with sufficient spin-orbit coupling, the atoms condense in a ground state with net magnetization which spontaneously breaks the PT\mathcal{PT}PT and ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{D}}_{4}$ symmetry. We demonstrate that this symmetry breaking leads to a gap opening at the Dirac point for the Bogoliubov spectrum and consequentially topologically nontrivial excitations. We confirm the nontrivial topology by calculating the Chern numbers of the lowest excitation bands and show that gapless edge states form at the interface of systems characterized by different values of the Chern number.
Non-Hermitian operators are now routinely used to describe few-mode systems such as optical resonators and superconducting qubits, and exceptional points (EPs) are defective spectral singularities of such non-Hermitian operators. In contrast, the scattering-theoretic formulation of EP physics for unbounded Hamiltonians remains less settled. In this work, we formulate the geometric phase associated with encircling an EP when the underlying eigenstates are quantum resonances within a one-dimensional scattering model. To do this, we employ the complex-scaling method, where resonance poles of the S matrix are realized as discrete eigenvalues of the non-Hermitian dilated Hamiltonian, to construct situations in which resonant and scattering states coalesce into an EP in the complex energy plane, that is, the resonance pole is embedded into the continuum spectrum. We analyze the self-orthogonality in the vicinity of an EP, the Berry phase, and the Chern characteristic. Our results clarify how EP branch structure and geometric holonomy arise directly from resonance poles in scattering theory, thereby connecting non-Hermitian spectral topology with the traditional theory of quantum resonances.
We investigate the interaction-driven instability of a quadratic band crossing arising for ultracold bosonic atoms loaded into a two-dimensional optical lattice. We consider the case when the degenerate point becomes a local minimum of both crossing energy bands such that it can support a stable Bose--Einstein condensate. A repulsive contact interaction among the condensed bosons induces a spontaneously time-reversal-symmetry broken superfluid phase and a topological gap is opened in the excitation spectrum. We propose two concrete realizations of the desired quadratic band crossing in lattices with either fourfold or sixfold rotational symmetries via suitable tuning of the unit cell leading to reduced Brillouin zones and correspondingly folded bands. In either case, topologically protected edge excitations are found for a finite system.