Topological classification of non-Hermitian Hamiltonians with frequency dependence

We develop a topological classification of non-Hermitian effective Hamiltonians that depend on momentum and frequency. Such effective Hamiltonians are in one-to-one correspondence to single-particle Green's functions of systems that satisfy translational invariance in space and time but may be interacting or open. We employ $K$ theory, which for the special case of noninteracting systems leads to the well-known 10-fold-way topological classification of insulators and fully gapped superconductors. Relevant theorems for $K$ groups are reformulated and proven in the more transparent language of Hamiltonians instead of vector bundles. We obtain 54 symmetry classes for frequency-dependent non-Hermitian Hamiltonians satisfying antiunitary symmetries. Employing dimensional reduction, the group structure for all these classes is calculated. This classification leads to a group structure with one component from the momentum dependence, which corresponds to the non-Hermitian generalization of topological insulators and superconductors, and two additional parts resulting from the frequency dependence. These parts describe winding of the effective Hamiltonian in the frequency direction and in combined momentum-frequency space.

Figure 1

Different types of gaps and flattening procedures for non-Hermitian Hamiltonians. (a) Flattening of a Hamiltonian with a point gap to a unitary one. (b) Flattening of a Hamiltonian with a real line gap to a Hermitian one with eigenvalues $\pm 1$. (c) Flattening of a Hamiltonian with a imaginary line gap to an anti-Hermitian one with eigenvalues $\pm i$.

Figure 2

Schematic of illustrative model consisting of three spin-full orbitals with spin-dependent hopping and coupled to baths.

Figure 3

Parametric plot of the eigenvalues $E_1\left(\omega \right)$ (blue curve) and $E_2\left(\omega \right)$ (red curve) of the effective Hamiltonian $H_{\mathrm{eff}}^R\left(\omega \right)$ in the complex plane. The frequency $\omega$ runs from $-\infty$, where $E_1=E_2=-i{\mathrm{\Gamma }}_0$ (black dot), to $+\infty$, with the same values of $E_1$ and $E_2$. Both curves are traversed counterclockwise, as indicated by the arrow. The parameters are $\epsilon =1, v=1, {\mathrm{\Gamma }}_0=-0.2$, and $\mathrm{\Gamma }=0.1$.