Correspondence between Winding Numbers and Skin Modes in Non-Hermitian Systems

We establish exact relations between the winding of “energy” (eigenvalue of Hamiltonian) on the complex plane as momentum traverses the Brillouin zone with periodic boundary condition, and the presence of “skin modes” with open boundary conditions in non-Hermitian systems. We show that the nonzero winding with respect to any complex reference energy leads to the presence of skin modes, and vice versa. We also show that both the nonzero winding and the presence of skin modes share the common physical origin that is the nonvanishing current through the system.

Figure 1

The reciprocal relations among the three phenomena unique to non-Hermitian systems: the nonvanishing persistent current, nonzero winding number of energy, and the presence of skin modes. The validity of any one is the sufficient and necessary condition for the validity of the other two.

Figure 2

We show the BZ (a) with periodic-boundary spectrum (b), and GBZ (c) with open boundary spectrum (d) for the model $H(z)=[2i{z}^2+(3+i)z+1]/z$, and the red dot $E_0=H(z=a)=H(z=b)=3$ is the reference energy with respect to which winding is defined. In (a),(c) the red dots represent the zeros of $H(z)-E_0=0$, and the cross denotes the pole. We remark that the orientation of GBZ in (c) is arbitrarily chosen.

Figure 3

BZ and GBZ, periodic- and open-boundary spectra, and all normalized eigenfunctions for open boundary are plotted for $H(z)=z^{-2}/5+3z^{-1}+2z$ in (a1)–(a3), $H(z)=z^{-2}/5+z^{-1}+2z$ in (b1)–(b3), and $H(z)=2z^{-2}/5+z^{-1}+z$ in (c1)–(c3). The regions inside BZ (GBZ) and outside GBZ (BZ) are colored in blue (red), and the eigenfunctions corresponding to points on GBZ outside (inside) BZ are plotted as red (blue) curves. $z_0,z_0^{\prime }$ are randomly chosen points in the red and the blue regions, respectively, and $E_0=H(z_0)$, $E_0^{\prime }=H(z_0^{\prime })$.