Fig. 3: Anomalous non-Hermitian skin effect.
From: Anomalous non-Hermitian skin effect: topological inequivalence of skin modes versus point gap
a Nontrivial point gap obtained from (Re[det[H(k)]], Im[det[H(k)]]) as k varies from 0 to 2π for the reference point \({E}_{b}=\frac{1}{2}i\). Namely, the effects of all energy bands are taken into account in calculation of spectral winding. b Open boundary eigenvalues colored by their inverse participation ratio for the s-th eigenstate \(\left\vert {\Psi }^{s}\right\rangle\) (IPRs), remaining constant approaching zero. c Distribution of the open boundary eigenstate corresponding to the blue asterisk EOBC = 0.0001 in b, where occupation is non-zero only on sub-chain B, while both sub-chain A and sub-chain C have no occupation. d Normalized eigenstates with the change of eigenvalues. All eigenstates are extended, rather than localized, on the open chain. e Non-Hermitian three-band system will be simplified as a Hermitian one-band model when γ1 = γ4 = 0. f Momentum space spectra, where Ek1 and Ek2 will display two circles with the same radius but opposite rotation directions when k changes from 0 to 2π. Thus, the point gap is topologically trivial for any reference point. In other words, all energy levels in momentum space contribute to the point gap. g Open boundary eigenvalues. Different colors are different values of IPRs, which is finite for the corresponding eigenstate. h Distribution of the open boundary eigenstate, which is localized, rather than extended. i Eigenstates as a function of all open boundary eigenvalues E. All eigenstates have been normalized and are localized at the boundaries of the system. The A sub-chain only contributes the energy level while without wavefunction distribution. j Open boundary non-Hermitian three-band system will become the non-Hermitian two-band system from both eigenstates and eigenvalues perspectives when γ1 = γ3 = 0. The parameters for a–e are γ1 = γ4 = 0, ta = 1, \({t}_{b}=\frac{1}{2}\), tc = 2, γ2 = 2, γ3 = 1, Va = 2 and Vb = Vc = 1. f–j γ1 = γ3 = 0, ta = tc = 8, \({t}_{b}=\frac{1}{2}\), γ2 = 2, γ4 = 1, \({V}_{a}={V}_{c}=\frac{3}{5}\) and Vb = 1.
