Tunably realizing flat-bands and exceptional points in kinetically frustrated systems: An example on the non-Hermitian Creutz ladder

TL;DR

This work analyzes a non-Hermitian Creutz ladder with generic non-reciprocal hopping by mapping it to two decoupled NH-SSH chains. It reveals distinct spectral phases under periodic and open boundary conditions, including real, imaginary, and complex regions separated by exceptional lines, with triple junction points where regimes meet. Flat bands appear as both Hermitian diabolical points and exceptional flat bands, with dynamics distinguishing whether degeneracies are DP or EP-based. The results demonstrate strong boundary sensitivity and provide analytic handles on flat-band reorganization and wave-packet dynamics in non-Hermitian frustrated lattices, with potential experimental realizations in photonics and circuits.

Abstract

We study a non-Hermitian extension of the Creutz ladder with generic non-reciprocal hopping. By mapping the ladder onto two decoupled non-Hermitian Su--Schrieffer--Heeger (SSH) chains, we uncover a rich structure in parameter space under different boundary conditions. Under periodic boundary conditions, the spectrum admits a fine-tuned line in parameter space with entirely real eigenvalues, while deviations from this line induce a real--complex spectral transition without crossing exceptional points. In contrast, an exact analytical diagonalization under open boundary conditions reveals extended regions in parameter space with purely real or purely imaginary spectra, separated from complex spectral domains by exceptional lines. The intersections of these exceptional lines define triple-junction points where distinct spectral regimes meet, giving rise to a structured phase diagram that is absent under periodic boundary conditions. We further show that flat bands in this system can occur both as Hermitian diabolical points and as non-Hermitian exceptional points, known as exceptional flat bands, where the dynamics is more stringent than in the Hermitian case, leading to distinct spectral and dynamical signatures.

Figures (6)

• Figure 1: Non-Hermitian Creutz ladder with non-reciprocal hopping. Schematic of the ladder geometry consisting of two cross-linked chains. Sublattice points of the upper (lower) chain are represented with green (yellow) circles and labeled with $a_j \, (b_j)$, where $j$ is the cell index. The model is rendered non-Hermitian by pairing reciprocal hoppings $(\{t\})$ with non-reciprocal terms $(\{\gamma\})$. The arrows represent the hoppings from one lattice point to another.
• Figure 2: Mapping of the non-Hermitian Creutz ladder to cross-coupled NH-SSH chains. After a local basis transformation, the non-Hermitian Creutz ladder can be represented as two cross-linked non-Hermitian Su–Schrieffer–Heeger (NH-SSH) chains composed of $w$ and $\bar{w}$ sites, shown in red circles and blue squares respectively, with a bridge pairing them highlighting that they belonged to the same cell in the original ladder. For balanced intra-chain hoppings $(\Delta t= \Delta \gamma=0)$, the two chains decouple exactly.
• Figure 3: Comparison of PBC and OBC spectra of the non-Hermitian Creutz ladder. Complex-energy spectra under PBC (blue dots) and OBC (red crosses) for representative parameter values. Panels (i–ii) illustrate dispersive PBC and degenerate OBC spectra. Panels (iii-iv) shows flat band PBC and degenerate OBC spectra, while panels (v–vi) show cases where the PBC spectrum encloses no area in the complex plane, leading to the disappearance of the non-Hermitian skin effect and restoration of bulk–boundary correspondence. Panels (vii–viii) demonstrate regimes where the OBC spectrum is entirely real or imaginary despite a complex PBC spectrum, highlighting the strong boundary sensitivity of non-Hermitian systems.
• Figure 4: Spectral phase diagrams of the non-Hermitian Creutz ladder. Color maps of the spectral density measure $M$ under periodic (top row) and open (bottom row) boundary conditions in the $(t_0,\bar{\gamma})$ parameter space for three different values of $\gamma_0=(0,0.5,1)$ along the three columns for $\bar{t}=1$ and system size $L=50$. Blue (red) regions correspond to entirely real (imaginary) spectra, while color gradients indicate complex spectra. Orange lines denote exceptional lines where the Hamiltonian becomes defective and the similarity transformation fails. Green crosses mark diabolical-point flat bands, while black solid lines indicate exceptional flat bands. Yellow diamonds highlight junction points where real, imaginary, and complex spectral regions meet, forming triple-point-like structures. The figure reveals boundary condition dependent spectral transitions, including real–complex transitions mediated by diabolical points rather than exceptional points.
• Figure 5: NHSE localization domains. Colour map of $<$dIPR$>$ in the $(t_0, \bar{\gamma})$ plane. Green (pink) regions correspond to left- (right-) localized bulk eigenstates, while white regions indicate extended states where the bulk–boundary correspondence is restored. Dashed black curves denote parameter values where the similarity transformation becomes unimodular, and dotted yellow lines indicate exceptional lines. The localization domains correlate strongly with the spectral topology and exceptional structures shown in \ref{['fig:spectral domain numerical']}. In (a) and (b) the black and yellow lines meet exactly at the DPs where eigenstates are compactly localized. On the other hand in (c) the overlap of the two lines suggest for an entire line of such localized states.