Exponentially growing bulk Green functions as signature of nontrivial non-Hermitian winding number in one dimension

A nonzero non-Hermitian winding number indicates that a gapped system is in a nontrivial topological class due to the non-Hermiticity of its Hamiltonian. While for Hermitian systems nontrivial topological quantum numbers are reflected by the existence of edge states, a nonzero non-Hermitian winding number impacts a system's bulk response. To establish this relation, we introduce the bulk Green function, which describes the response of a gapped system to an external perturbation on timescales where the induced excitations have not propagated to the boundary yet, and show that it will grow in space if the non-Hermitian winding number is nonzero. Such spatial growth explains why the response of non-Hermitian systems on longer timescales, where excitations have been reflected at the boundary repeatedly, may be highly sensitive to boundary conditions. This exponential sensitivity to boundary conditions explains the breakdown of the bulk-boundary correspondence in non-Hermitian systems: topological invariants computed for periodic boundary conditions no longer predict the presence or absence of boundary states for open boundary conditions.

Figure 1

Illustration of the bulk regime: The solution $\psi (t,x)$ to the driven Schrödinger equation for the Hamiltonian given by Eq. (2) grows exponentially in space as the induced excitations travel to the right (for $\gamma >0$). Nonetheless, the wave function converges to a steady state pointwise for every position $x$ in the limit $t\to \infty$.

Figure 2

Illustration of the integration contour for the integral representation of the periodic Green function $G_{\text{period}}$, given by Eq. (7), consisting of two paths ${\gamma }_1,{\gamma }_2$ that form the boundary of a thin annulus that encloses the unit circle. The poles of the integrand are indicated by crosses: The factor $1/(z^L-1)$ contributes poles on the unit circle, while the inverse of the Hamiltonian contributes poles $z_1,\cdots ,z_{2N}$ that lie outside the annulus.

Figure 3

Illustration of the analytic continuation ${\mathrm{\Gamma }}_0\to 0$. For large uniform dissipation, the roots $z_1\left({\mathrm{\Gamma }}_0\right),\cdots ,z_N\left({\mathrm{\Gamma }}_0\right)$ lie inside the unit circle, while the roots $z_{N+1}\left({\mathrm{\Gamma }}_0\right),\cdots ,z_{2N}\left({\mathrm{\Gamma }}_0\right)$ lie outside. As the dissipation is reduced, the roots $z_j\left(\mathrm{\Gamma }\right)$ will move along paths in the complex plane. If one of the roots moves from inside the unit circle to the outside (or vice versa), say $z_2\left(\mathrm{\Gamma }\right)$ as shown here, the bulk Green function $G_{\text{bulk}}$ will grow in space. No such crossing may occur if the Hamiltonian is purely dissipative or has a real line gap.

Figure 4

Illustration of the paths of the eigenvalues of a non-Hermitian Bloch Hamiltonian. For large dissipation, they are contained within a small angle that enclosed the negative imaginary axis.