Dynamical instability is an inherent feature of bosonic systems described by the Bogoliubov--de Gennes (BdG) Hamiltonian. Since it causes the BdG system to collapse, it is generally thought that it should be avoided. Recently there has been much effort to harness this instability for the benefit of creating a topological amplifier with stable bulk bands but unstable edge modes which can be populated at an exponentially fast rate. We present a theorem for determining the stability of states with energies sufficiently away from zero, in terms of an unconventional commutator between the number-conserving part and number-nonconserving part of the BdG Hamiltonian. We apply the theorem to a generalization of a model from Galilo et al. [Phys. Rev. Lett. 115, 245302 (2015)] for creating a topological amplifier in an interacting spin-1 atom system in a honeycomb lattice through a quench process. We use this model to illustrate how the vanishing of the unconventional commutator selects the symmetries for a system so that its bulk states are stable against (weak) pairing interactions. We find that as long as time-reversal symmetry is preserved, our system can act like a topological amplifier, even in the presence of an onsite staggered potential which breaks the inversion symmetry.
Dynamical instability is an inherent feature of bosonic systems described by the Bogoliubov--de Gennes (BdG) Hamiltonian. Since it causes the BdG system to collapse, it is generally thought that it should be avoided. Recently there has been much effort to harness this instability for the benefit of creating a topological amplifier with stable bulk bands but unstable edge modes which can be populated at an exponentially fast rate. We present a theorem for determining the stability of states with energies sufficiently away from zero, in terms of an unconventional commutator between the number-conserving part and number-nonconserving part of the BdG Hamiltonian. We apply the theorem to a generalization of a model from Galilo et al. [Phys. Rev. Lett. 115, 245302 (2015)] for creating a topological amplifier in an interacting spin-1 atom system in a honeycomb lattice through a quench process. We use this model to illustrate how the vanishing of the unconventional commutator selects the symmetries for a system so that its bulk states are stable against (weak) pairing interactions. We find that as long as time-reversal symmetry is preserved, our system can act like a topological amplifier, even in the presence of an onsite staggered potential which breaks the inversion symmetry.
Here we investigate the internal sublattice symmetry, and thus the enriched topological classification of bosonic Bogoliubov excitations of thermodynamically stable free-boson systems with nonvanishing particle-number-nonconserving terms. Specifically, we show that such systems well described by the bosonic Bogoliubov--de Gennes Hamiltonian can be in general reduced to particle-number-conserving (single-particle) ones. Building upon this observation, the sublattice symmetry is uncovered with respect to an excitation energy, which is usually hidden in the bosonic Bogoliubov--de Gennes Hamiltonian. Thus, we obtain an additional topological class, i.e., class AIII, which enriches the framework for the topological threefold way of free-boson systems. Moreover, a construction is proposed to show a category of systems respecting such a symmetry. For illustration, we resort to a one-dimensional prototypical model to demonstrate the topological excitation characterized by a winding number or symplectic polarization. By introducing the correlation function, we present an approach to measure the topological invariant. In addition, the edge excitation together with its robustness to symmetry-preserving disorders is also discussed.
In this work, we consider the dynamics of bosons in bands with nontrivial topological structure. In particular, we focus on the case where bosons are prepared in a higher-energy band and allowed to evolve. The Bogoliubov theory about the initial state can have a dynamical instability, and we show that it is possible to achieve the interesting situation where the topological edge modes are unstable while all bulk modes are stable. Thus, after the initial preparation, the edge modes will become rapidly populated. We illustrate this with the Su-Schrieffer-Heeger model which can be realized with a double-well optical lattice and is perhaps the simplest model with topological edge states. This work provides a direct physical consequence of topological bands whose properties are often not of immediate relevance for the near-equilibrium properties of bosonic systems.
We consider a system of interacting spin-one atoms in a hexagonal lattice under the presence of a synthetic gauge field. Quenching the quadratic Zeeman field is shown to lead to a dynamical instability of the edge modes. This, in turn, leads to a spin current along the boundary of the system which grows exponentially fast in time following the quench. Tuning the magnitude of the quench can be used to selectively populate edge modes of different momenta. Implications of the intrinsic symmetries of the Hamiltonian on the dynamics are discussed. The results hold for atoms with both antiferromagnetic and ferromagnetic interactions.
For bosons with flat energy dispersion, condensation can occur in different symmetry sectors. Here, we consider bosons in a kagome lattice with π-flux hopping, which, in the presence of mean-field interactions, exhibit degenerate condensates in the Γ and the K point. We analyze the excitation above both condensates and find strikingly different properties: For the K-point condensate, the Bogoliubov-de Gennes (BdG) Hamiltonian has broken particle-hole symmetry and exhibits a topologically trivial quasiparticle band structure. However, band flatness plays a key role in breaking the time-reversal symmetry of the BdG Hamiltonian for a Γ-point condensate. Consequently, its quasiparticle band structure exhibits nontrivial topology, characterized by nonzero Chern numbers and by the presence of edge states. Although quantum fluctuations energetically favor the K-point condensate, the interesting properties of the Γ-point condensate become relevant for anisotropic hopping. The topological properties of the Γ-point condensate get even richer in the presence of extended Bose-Hubbard interactions. We find a topological phase transition into a topological condensate characterized by high Chern number and also comment on the realization and detection of such excitations.
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. As concrete examples, we numerically investigate two toy models and confirm the bulk-edge correspondence even in the presence of complex eigenvalues. The Z2 invariant associated with particle-hole symmetry and the winding number for a time-reversal-symmetric system are also discussed.
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.