Exceptional horns in $n$-root graphene and Lieb photonic ring lattices

TL;DR

This paper constructs non-Hermitian $n$-root lattices from 2D parent systems like graphene and Lieb lattices, revealing exceptional horns at Dirac points with unique scaling laws and proposing a photonic implementation.

Contribution

It introduces a systematic method to create $n$-root non-Hermitian lattices with complex spectra and exceptional horns, extending the understanding of Dirac cones and Landau levels in these systems.

Findings

Energy spectra consist of $n$ rotated branches with flat bands.

Exceptional horns appear at Dirac points with $E o | extbf{q}|^{1/n}$ scaling.

Analytic expressions for Landau levels with $E o ext{flux}^{1/2n}$ scaling.

Abstract

We present a systematic construction of non-Hermitian tight-binding lattices whose Bloch spectra are $n$th roots of those of Hermitian parent two-dimensional (2D) lattices, namely graphene and the Lieb lattice. The $n$-roots of these models are constructed from connecting loop modules of unidirectional couplings whose geometrical arrangements match that of the corresponding parent system. Their energy spectrum is shown to consist of $n$ rotated and equivalent branches in the complex energy plane, each matching the real spectrum of the parent model when raised to the $n$th power, together with extra zero-energy flat bands (FBs) accounted for by the generalized index theorem. We show how the low-energy Dirac cones of the parent models translate, for an appropriate choice of phase configuration for the couplings of the $n$-root lattices, as what we call an "exceptional horn" appearing at each branch, with the central Dirac point (DP) converted into zero-energy exceptional points (EPs) of order $n$ or higher at high-symmetry momenta. These exceptional horns reflect the behavior of low-lying excitations that scale with momentum as $E\sim\vert \mathbf{q}\vert^{\frac{1}{n}}$, with $n\geq 3$, as opposed to the linear massless modes that characterize a Dirac cone. Moreover, we derive analytic expressions for the associated Landau levels (LLs), whose energies scale with magnetic flux as $E\sim\phi^{\frac{1}{2n}}$. For the case of the $n$-root Lieb lattice, the zeroth LL is shown to be exceptional. These results are analytically derived for both $n$-root models and numerically demonstrated for certain values of $n$. Finally, we propose a realistic photonic implementation based on coupled ring resonators with a split configuration of optical gain and loss.