Non-chiral ephemeral edge states and cascading of exceptional points in the non-reciprocal Haldane model

TL;DR

This work analyzes a non-Hermitian variant of the Haldane model in which non-reciprocal next-nearest-neighbor hopping yields time-reversal-symmetry protected exceptional rings (ERs) and a spectrum with PT-symmetry breaking regions. On a cylinder with zig-zag boundaries, it reveals a non-chiral edge mode pinned at k_x=π whose imaginary-energy slope drives a self-acceleration-induced drift, and shows that edge states bifurcate at exceptional points into the bulk. As non-Hermiticity increases, bulk states coalesce into EP pairs and proliferate in a nested, Matryoshka-like cascade, producing a step-like growth in EP-pair density as a function of the non-Hermitian parameter. The ERs concentrate Berry curvature inside their interiors, forming Berry-curvature flux tubes that vanish outside, with potential consequences for transport; wave-packet dynamics further demonstrate tunable stabilization of ephemeral edge states at long times and highlight regimes where bulk modes persist between EPs. The results offer insights into exceptional-phase phenomena in lattice systems and may inform realizations in disordered Kitaev honeycomb models and photonic/topoelectric platforms.

Abstract

We study a variant of the Haldane honeycomb model that has non-reciprocal hoppings between the next-nearest neighbours. The system on a torus hosts time-reversal symmetry protected exceptional rings(ER) in the spectrum. The ERs act as Berry-curvature flux tubes i.e the Berry curvature is non-zero only inside the ERs. The system on a cylinder having zig-zag boundaries (and transverse momentum $k_x$) hosts edge-states that have zero group velocity at $k_x=π$ and are therefore `non-chiral'. The edge states undergo a bifurcation transition at an exceptional point(EP)in the BZ and delocalise into the bulk. As the non-reciprocity is increased, the bulk states that are approaching each other are converted into pairs of EPs due to non-Hermiticity. As the non-reciprocity is further increased, there is a `Russian doll'-like nested proliferation of pairs of EPs, leading to an EP-cascade. The proliferation of EPs takes place only at specific values of the non-hermiticity parameter, leading to a step-like structure in the EP-pair density when plotted as a function of non-Hermiticity. Further, using wave packet dynamics, we find a tunable regime where the non-chiral edge states can be dynamically stabilised for large timescales. The `self-acceleration' term in the equations of motion tends to diffuse the wave packets into the bulk, thus making them `ephemeral edge states'. But we find that for small non-hermiticity, the edge localisation is stabilised until late times for sufficiently wider wave packets. Thus, we have brought forth an intriguing phenomenology of the exceptional phase of the non-reciprocal Haldane model, which may bear direct relevance for systems such as disordered Kitaev honeycomb model, wherein such ERs have been predicted.

Figures (10)

• Figure 1: Non-Hermitian Haldane model on a honeycomb lattice geometry with NN hopping and non-reciprocal NNN hopping amplitudes. The non-reciprocal hoppings can be expressed as hopping in the presence of a imaginary flux.
• Figure 2: (a) Non-chiral edge states in the imaginary-flux Haldane model. The edge-state wavefunction is obtained from diagonalising the lattice Hamiltonian over cylinder geometry with parameters $N_x = 40$, $N_y = 20$, $t_1 = 1.0$, $\tilde{t} = 0.3$, $\phi = 0.1$. (b) Dynamics of the wave packet expectation value $\langle y(t)\rangle$, illustrating late-time stabilisation at the edge for small non-Hermiticty($\phi$=0.05) and for a range of wave packet widths. For larger widths, there is an edge-to-bulk diffusion due to 'self-acceleration' in wave packet equations of motion. (c) Step-like structure in proliferation of exceptional-point (EP) pairs as a function of the non-Hermiticity parameter $\phi$ for $N_y = 30$ and $\tilde{t} = 0.3$. The EP pairs appear at certain 'critical' values of $\phi$, when pairs of bulk states tend to become degenerate. This leads to a jump in the number of EP pairs.
• Figure 3: (a),(b) Real and imaginary parts of the complex energy spectrum under periodic boundary conditions for a system on a torus. We have used $t_1 = 1.0, \tilde{t} = 0.22, \phi=0.8$. There are two exceptional rings in the spectrum that mark the transition from $\mathcal{PT}$-symmetry broken regions, with degenerate real parts and purely imaginary parts, to $\mathcal{PT}$-symmetric regions with purely real spectrum.
• Figure 4: (a) Sign of the determinant of the Hamiltonian, $\mathrm{sgn}(\det H)$ plotted over the BZ. This invariant distinguishes the $\mathcal{PT}$ symmetry unbroken and $\mathcal{PT}$ symmetry broken regions of the spectrum. The exceptional rings (ERs) mark the boundaries across which the $\mathcal{PT}$ symmetry-breaking transition occurs.(b) Berry-curvature landscape in the presence of ERs. The ERs act as Berry-curvature flux tubes, leading to strong localization of the Berry curvature within their interior. Also, both ERs have the same sign for the Berry curvature.
• Figure 5: (a), (b) Spectrum of imaginary-flux Haldane model on a cylinder with zig-zag boundaries. Here, we have, $N_y = 30, t_1 = 1, \tilde{t} = 0.3, \phi = 0.1$. The edge-localized states are highlighted through a color profile by plotting the probability density at the left edge for each state. Non-chiral edge states occur at $k_x=\pi$ with $\partial (\text{Re}E)/\partial k_x =0$ and $\partial (\text{Im}E)/\partial k_x = \text{constant}$. The edge states merge with the bulk through exceptional points(indicated as black dots) (c) Phase rigidity and the inverse participation ratio(IPR) of the edge states as function of $k_x$. The phase rigidity goes to zero at the exceptional points, accompanied by IPR also going to zero signalling delocalisation into the bulk.