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arXiv:1905.07989 · Journal of Physics A Mathematical and Theoretical

Topological classification of excitations in quadratic bosonic systems

Zixian Zhou, Liang-Liang Wan, Zhi-Fang Xu

Published Sep 16, 2020• 15 citations
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Abstract

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.

DOI
10.1088/1751-8121/abb92b

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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 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 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 present a systematic topological classification of fermionic and bosonic topological phases protected by time-reversal, particle-hole, parity, and combination of these symmetries. We use two complementary approaches: one in terms of KKK-theory classification of gapped quadratic fermion theories with symmetries, and the other in terms of the KKK-matrix theory description of the edge theory of (2+1)-dimensional bulk theories. The first approach is specific to free fermion theories in general spatial dimensions while the second approach is limited to two spatial dimensions but incorporates effects of interactions. We also clarify the role of the CPT theorem in the classification of symmetry-protected topological phases, and show, in particular, that the topological superconductors discussed before are related by the CPT theorem.
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.
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 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.