Zero Indirect Band Gap in Non-Hermitian Systems

TL;DR

This work investigates how a zero indirect band gap, previously shown to be robust in certain Hermitian lattices, extends to non-Hermitian systems by analyzing a 1D diamond-like chain with gain and loss. The authors construct a non-Hermitian $H(k)$ with complex intra-cell couplings $\beta_1$ and $\beta_2$, and show the real part of the spectrum can sustain a zero indirect gap over a finite range of parameters, while exceptional points and the non-Hermitian skin effect (NHSE) emerge away from this regime. They demonstrate that the appearance of the zero indirect gap coincides with suppression of the NHSE due to a collapse of the point gap into a line-gap-like structure, and they identify EPs via the diverging condition number $\mathrm{cond}(V)$ of the eigenvector matrix with a universal scaling $\max\mathrm{cond}(V) \sim N_k^{\alpha}$ where $\alpha \approx 0.48$. The results establish new connections between indirect gaps, exceptional points, and NHSE in non-Hermitian band theory and point toward experimental realizations in photonic or cold-atom platforms with tunable gain and loss.

Abstract

Zero indirect gaps in band models are typically viewed as unstable and achievable only through fine-tuning. Recent works, however, have revealed robust semimetallic phases in Hermitian systems where the indirect gap remains pinned at zero over a finite parameter range. Here, we extend this paradigm to non-Hermitian lattice models by studying a one-dimensional diamond-like system with gain and loss. We show that the zero indirect band gap in the real part of the spectrum remains stable in the presence of non-Hermitian perturbations and identify the parameter regime in which this robustness persists. We find that the appearance of the zero indirect gap coincides with the suppression of the non-Hermitian skin effect. Our results reveal new connections between indirect gaps, exceptional points and non-Hermitian skin effect, opening avenues for experimental realizations.

Figures (7)

• Figure 1: Diamond-like chain with three species of spinless fermions represented by light blue (a), red (b) and dark blue (c). Green lines correspond to inter species hopping while the dark lines corresponds to nearest neighbor hopping.
• Figure 2: (a)-(d) Real component of the dispersion spectrum plotted as a function of k. Panels (a)-(d) correspond to distinct set of imaginary part of $Im(\beta_1) = Im(\beta_2) = 0.2, 0.8, 1.52, 2.0$ respectively. (e) Zero indirect gap plotted with respect to the $Im(\beta_1) = Im(\beta_2)$ for fixed value of $Re(\beta_1) = Re(\beta_2)$=1.
• Figure 3: Parameter space plotted with respect to $Re(\beta_1)$ and $Re(\beta_2)$ for different values of imaginary components i.e., (a)-(d) $Im(\beta_1) = Im(\beta_2) = 0.2, 0.6, 1.2, 1.8$ respectively.
• Figure 4: Average IPR of eigenstates as a function of $\mathrm{Re}(\beta_1)$ and $\mathrm{Re}(\beta_2)$ for different values of the imaginary component $\mathrm{Im}(\beta)$ (a) $0.2$, (b) $0.6$, (c) $1.2$, and (d) $1.8$.
• Figure 5: The complex energy spectra obtained by varying the imaginary parts of the non-Hermitian parameters $\beta_1$ and $\beta_2$, while keeping their imaginary parts fixed at $\mathrm{Im}(\beta_1)=\mathrm{Im}(\beta_2)=0.2,1.0$.