Topological Invariant for Bosonic Bogoliubov–de Gennes Systems with Disorder

Yutaka Akagi

doi:10.7566/jpsj.89.123601

Abstract

Using the method of noncommutative geometry, we define a topological invariant in disordered bosonic Bogoliubov-de Gennes systems, which possess a unique mathematical property---non-Hermiticity. To demonstrate the validity of the definition, we investigate a disordered artificial spin ice model in two dimensions numerically. In the clean limit, we clarify that the topological index perfectly coincides with the Chern number. We also show that the topological index is robust against disorder. The formula provides the topological index $n_{\rm Ch}=1$ in the magnon Hall regime and $n_{\rm Ch}=0$ in a trivial localized one. We also show by example that our method can be extended to other symmetry classes. Our results pave the way for further studies on topological bosonic systems with disorder.

Highlights

We propose an alternative and highly efficient method to numerically calculate the topological index for magnon Hall systems with disorder
Summary: We have provided a new method for numerically calculating the topological index of magnon Hall systems with disorder
Introducing the “Fermi” projection for bosons PB, we have defined the topological index nCh determined by the discrete spectrum of the operator A with a supersymmetric structure

Summary

We begin with the bosonic BdG Hamiltonian

Since the 2N × 2N matrix HBdG is Hermitian, the N × N matrices h and ∆ satisfy h† = h and ∆T = ∆, respectively. The components of the operator β satisfy the commutation relation [βi, βj†] = (Σz)ij. To hold the bosonic commutation relation, the Hamiltonian HBdG is diagonalized by a para-unitary matrix T , which satisfies T †ΣzT = Σz. For defining the topological invariant, we introduce a “Fermi” projection PB for bosonic BdG systems described by PB. We note that the projection Pn is no longer Hermitian as the operator satisfies ΣzPn†Σz = Pn. As in fermionic systems, we introduce the Dirac operator Da for bosonic BdG systems as. We define the topological index for tight-binding models of bosonic BdG systems with disorder.

PBC OBC

Σz δHBdG