Topological Band Theory for Non-Hermitian Hamiltonians

We develop the topological band theory for systems described by non-Hermitian Hamiltonians, whose energy spectra are generally complex. After generalizing the notion of gapped band structures to the non-Hermitian case, we classify “gapped” bands in one and two dimensions by explicitly finding their topological invariants. We find nontrivial generalizations of the Chern number in two dimensions, and a new classification in one dimension, whose topology is determined by the energy dispersion rather than the energy eigenstates. We then study the bulk-edge correspondence and the topological phase transition in two dimensions. Different from the Hermitian case, the transition generically involves an extended intermediate phase with complex-energy band degeneracies at isolated “exceptional points” in momentum space. We also systematically classify all types of band degeneracies.

Figure 1

The energies of two bulk bands (yellow and blue regions), and the edge state (green line) in the complex-energy plane for the domain wall problem Eq. (5). The bulk band is isolated according to our definition. The energy unit is $m$. The bulk phase $\mathbf{\kappa }=(0.2,0.3)$, $\delta =0.4$ is connected to the vacuum (dispersion not shown) $m_{\mathrm{vac}}/m=-1$, ${\mathbf{\kappa }}_{\mathrm{vac}}={\delta }_{\mathrm{vac}}=0$.

Figure 2

(a) The swapping of energy eigenvalues. $\theta \in [0,2\pi ]$ parametrizes the loop $\mathrm{\Gamma }$. The dashed curves are the projection of the energy trajectory. (b) The dispersion near an exceptional point. The Hamiltonian is $H(\mathbf{k})={\sigma }_{+}+(k_x{\sigma }_x+k_y{\sigma }_y)$. The loop $\mathrm{\Gamma }$ in (a) is the circle $k=\sqrt{k_x^2+k_y^2}=1$, which is parametrized by $\theta$ as $\mathbf{k}=(\cos \theta ,\sin \theta )$. (c) The energy dispersion along $k_x=0$.

Figure 3

(a) The bulk phase diagram of Hamiltonian Eq. (4) for a given $\delta$. The white regions represent the separable phases, and the colored region represents the inseparable phase. The light blue region $\kappa >|m|>0$ is the phase with a pair of exceptional points; the red lines $\kappa =|m|>0$ show the phase with a hybrid point (HP). The origin $\kappa =m=0$ is a Dirac point (DP) if $\delta =0$ and is a ring of exceptional points (EP ring) if $\delta \ne 0$. (b) The trajectory of the exceptional points in the momentum space when $m$ moves along the purple dashed line in (a). Here $\mathbf{\kappa }=(\delta ,0)$.