The topological classifications of quadratic bosonic systems according to the symmetries of the Hamiltonian, the dynamic matrix from the equation of motion, and the effective Hamiltonian from the Lindblad equation are summarized. The Hermitian Hamiltonian leads to a three-fold way table while the non-Hermitian dynamic matrix and effective Hamiltonian lead to a ten-fold way table. The difference comes from to the bosonic commutation relation on the particle-hole and chiral symmetries. Moreover, the system-reservoir coupling may cause a system with or without coupling to a reservoir to fall into different classes. We present a 1D bosonic pairing model inspired by the fermionic Kitaev chain to contrast the topological behavior from the Hamiltonian, dynamic matrix, and effective Hamiltonian. In contrast, a 2D Chern insulator is insensitive to the different classifications.
The topological classifications of quadratic bosonic systems according to the symmetries of the Hamiltonian, the dynamic matrix from the equation of motion, and the effective Hamiltonian from the Lindblad equation are summarized. The Hermitian Hamiltonian leads to a three-fold way table while the non-Hermitian dynamic matrix and effective Hamiltonian lead to a ten-fold way table. The difference comes from to the bosonic commutation relation on the particle-hole and chiral symmetries. Moreover, the system-reservoir coupling may cause a system with or without coupling to a reservoir to fall into different classes. We present a 1D bosonic pairing model inspired by the fermionic Kitaev chain to contrast the topological behavior from the Hamiltonian, dynamic matrix, and effective Hamiltonian. In contrast, a 2D Chern insulator is insensitive to the different classifications.
We investigate the classification of topological excitations in quadratic bosonic systems with an excitation band gap. Time-reversal, charge-conjugation, and parity symmetries in bosonic systems are introduced to realize a ten-fold symmetry classification. We find a specific decomposition of the quadratic bosonic Hamiltonian and use it to prove that each quadratic bosonic system is homotopic to a direct sum of two single-particle subsystems. The classification table of topological excitations is thus derived via inheriting from that of Atland-Zirnbauer classes and unique topological phases of bosons are predicted. Finally, concrete bosonic models are proposed to demonstrate the peculiarity of bosonic topological excitations.
The bosonic Kitaev chain (BKC) model is a deceptively simple looking quadratic pairing Hamiltonian. Despite being purely Hermitian, it exhibits a number of striking non-Hermitian topological phenomena, including skin effects. We show here how symmetries play a key role in this model, and how identifying these allows one to develop generalized BKC-like models. We emphasize the surprising fact that any quadratic bosonic pairing Hamiltonian with a sublattice (chiral) symmetry necessarily has a dynamical matrix with an effective time reversal symmetry. This symmetry is unrelated to physical time-reversal, but enables non-trivial topological invariants. We also discuss how this symmetry is unrelated to another key property of the BKC, the decoupling of quadrature dynamics. This feature can instead be connected to a distinct symmetry, namely an effective particle-hole symmetry of the dynamical matrix. We discuss non-trivial generalized BKC models that only keep one of these two effective symmetries intact. We also provide a classification of all translationally-invariant 1D pairing Hamiltonians, and show connections between the BKC and a well-studied non-Hermitian fermionic system, the symplectic Hatano-Nelson model.
Abstract We investigate the topological classification of excitations in quadratic bosonic systems with an excitation band gap. Time-reversal, charge-conjugation, and parity symmetries in bosonic systems are introduced to realize a ten-fold symmetry classification. We find a specific decomposition of the quadratic bosonic Hamiltonian and use it to prove that each quadratic bosonic system is homotopic to a direct sum of two single-particle subsystems. The topological classification table is thus derived via inheriting from that of Atland–Zirnbauer classes and unique topological phases of bosons are predicted. Finally, concrete topological models are proposed to demonstrate the peculiarity of bosonic excitations.
We propose a Hermitian quadratic bosonic model (QBH) whose dynamical matrix exhibits distinct topological and dynamical phenomena depending on whether the hopping and pairing amplitudes are real or purely imaginary. In the real-parameter regime, the dynamical matrix is unitarily equivalent to four decoupled copies of the sublattice-symmetric non-Hermitian Su-Schrieffer-Heeger (nSSH2) model, thereby inheriting its topological phases and energy spectrum, including the M\"obius phase, a gapless topological phase with fractional winding number, having no Hermitian counterpart. We show that the dynamics generated by the QBH Hamiltonian naturally reproduces non-Hermitian time evolution, without invoking nonlinear Schr\"odinger dynamics or ad hoc normalization. It is demonstrated by analytically calculating the Loschmidt amplitude and computing the dynamical topological order parameter under periodic boundary conditions, which displays a distinct chiral response in the M\"obius phase. In contrast, when the hopping and pairing terms are taken to be purely imaginary, the dynamical matrix becomes unitarily equivalent to a different version of the sublattice-symmetric non-Hermitian Su-Schrieffer-Heeger (nSSH1) model that supports only two topological phases, trivial and nontrivial, and the M\"obius phase disappears. The latter system exhibits sublattice-dependent chiral amplification under open boundary conditions. We show that this amplification arises from the nontrivial topology of the dynamical matrix, establishing a clear link between topological phase and amplification behavior in the imaginary-parameter regime.
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.
Symmetry classification is crucial in understanding universal properties of quantum matter. Recently, the scope of symmetry classification has been extended to open quantum systems governed by the Lindblad master equation. However, the classifications of Lindbladians and steady states remain largely separate, because the former requires the non-Hermitian classification framework, while the latter relies on the classification scheme for Hermitian matrices. In this paper we build connections between symmetry classes of quadratic Lindbladians and their steady states, despite their different classification frameworks. We classify the full matrix representation of generic quadratic Lindbladians with particle conservation, showing they fall into 27 non-Hermitian symmetry classes. Among these, 22 classes lead to an infinite-temperature steady state. The remaining five classes have one-to-one correspondence with five steady-state Hermitian symmetry classes. Numerical simulations of random Lindbladian dynamics confirm the convergence to the correct steady-state symmetry classes at long time.