Non-Hermitian Chern Bands

The relation between chiral edge modes and bulk Chern numbers of quantum Hall insulators is a paradigmatic example of bulk-boundary correspondence. We show that the chiral edge modes are not strictly tied to the Chern numbers defined by a non-Hermitian Bloch Hamiltonian. This breakdown of conventional bulk-boundary correspondence stems from the non-Bloch-wave behavior of eigenstates (non-Hermitian skin effect), which generates pronounced deviations of phase diagrams from the Bloch theory. We introduce non-Bloch Chern numbers that faithfully predict the numbers of chiral edge modes. The theory is backed up by the open-boundary energy spectra, dynamics, and phase diagram of representative lattice models. Our results highlight a unique feature of non-Hermitian bands and suggest a non-Bloch framework to characterize their topology.

Figure 1

Topological phase diagram based on open-boundary spectra (for $v_{x,y}=t_{x,y}=1$, ${\gamma }_{x,y}=\gamma$, ${\gamma }_z=0$). Chiral edge states are found in the shadow area, which is therefore topologically nontrivial. The trivial-nontrivial phase boundary (red solid curve) is well approximated by the theoretical curve in Eq. (14) (shown as the blue dashed curve, which is very close to the red solid curve). Away from this phase boundary, the (open-boundary) bulk spectra are gapped. The Bloch-Hamiltonian phase boundaries are shown as the dotted lines, whose equations are $m=m_{\pm }$ with $m_{\pm }=2\pm \sqrt{2}\gamma$. The Bloch spectra are gapless in the fan $m\in [m_{-},m_{+}]$. The non-Bloch Chern number $C$ is defined in Eq. (13) (we take the $\mathrm{Re}(E_{\alpha })<0$ band and omit the $\alpha$ index; see text).

Figure 2

Left panel: Lowest energy eigenvalues of a square geometry with $L=30$. Right three panels: wave-packet evolutions. (a) $m=2.2121$; (b) $m=1.7879$ (indicated by filled square and asterisk in Fig. 1), with $\gamma =0.15$ for both. The energy eigenvalues shown here are real valued. In (a), a nonzero energy gap is apparent and in (b), there are a few in-gap energies of chiral edge states. For the wave-packet evolution, the initial state takes the Gaussian form $\psi (t=0)=\mathcal{N}\exp [-(x-15{)}^2/40-(y-1{)}^2/10](1,1{)}^T$, $\mathcal{N}$ being the normalization factor, and evolves according to the Schrodinger equation $i{\partial }_t|\psi (t)⟩=H|\psi (t)⟩$. The intensity profile of $|\psi (t)⟩$ (modulus squared), normalized so that the total intensity is 1, is shown for $t=0$, 5, 20. The wave packet quickly fades into the bulk in (a), whereas the chiral (unidirectional) edge motion is appreciable in (b).

Figure 3

(a) Phase diagram based on the spectra on a cylinder with open boundary condition in the $y$ direction. $t_{x,y}=v_{x,y}=1$, ${\gamma }_{x,y}=\gamma$, ${\gamma }_z=0$. The dotted lines are the Bloch phase boundaries. (b) The spectra for $(m,\gamma )=(1.717,0.2)$ (indicated as asterisk in (a)). The cylinder height is $L_y=40$, and 180 grid points are taken for $k_x$. The spectra for all $k_x$’s are shown together in the complex plane, without specifying the $k_x$ values. The chiral edge states are shown in red.