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

TL;DR

The paper addresses constructing non-Hermitian $n$-root lattices whose spectra are the $n$th roots of Hermitian parents such as graphene and the Lieb lattice. It introduces loop-module-based unidirectional couplings that produce $n$ rotated spectral branches and protected zero-energy flat bands under a generalized chiral symmetry, with Dirac points tunable into zero-energy exceptional points of order $n$ and horns exhibiting $E\sim|\mathbf{q}|^{1/n}$. The authors derive analytic expressions for Landau levels scaling as $E\sim\phi^{1/(2n)}$ and analyze Hofstadter spectra, showing that raising the $n$-root spectrum to the $n$th power recovers the parent Hofstadter pattern while preserving flux-insensitive FBs. They also provide a photonic-ring implementation for the 3-root graphene, corroborate key predictions numerically, and discuss limitations from nonideal unidirectionality and cross-circulation couplings, outlining future directions for higher-order EPs and alternative parent lattices.

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 $E\simφ^{\frac{1}{2n}}$. 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.

Figures (15)

• Figure 1: Illustration of the $n$-root graphene model. Dashed limited square region encloses a unit cell. The primitive vectors are $\mathbf{a_1}=a\left(\frac{1}{2},\frac{\sqrt{3}}{2}\right)$ and $\mathbf{a_2}=a\left(-\frac{1}{2},\frac{\sqrt{3}}{2}\right)$, with $a\equiv 1$ the lattice constant. The general form of the loop modules is depicted at the bottom, with the arrows indicating the direction of the unidirectional couplings of magnitude $\sqrt[n]{J}$. The color scheme at the right indicates the SL to which each site belongs.
• Figure 2: (a) Unit cell of the 3-root graphene model with a modified hopping phase configuration, with green and red unidirectional hoppings of magnitude $\sqrt[3]{J}$ carrying the respective phase factor indicated at the right. The onsite energy terms appearing in the cubed Hamiltonian for the selected blue site are depicted as self-loops at the bottom right. (b) Photonic ring implementation of the unit cell shaded in blue in (a). The gray rings are neutral, and constitute the main rings of the effective lattice. For the antiresonant link rings, the sine-like distribution of the imaginary part of the refractive index is represented following the color scale to the right. The links corresponding to the phase terms are displaced perpendicularly from the line joining their main rings by a distance $d_y$.
• Figure 3: Bulk complex energy spectrum of branch $p=0,1,2$, with $E_{p=0}$ standing for the energies of the real zero branch, for the $3$-root graphene model, with $\omega_3^p=e^{i\frac{2\pi}{3}p}$ and $J=1$. A zoomed plot of the exceptional horn delimited by the dashed orange oval is shown at the right, where the zero-energy EP is highlighted.
• Figure 4: Bulk complex energy spectrum of the 3-root graphene along the high-symmetry line of the Brillouin zone (vertical axis), with $\Gamma=(0,0)$, $K=\left(\frac{4\pi}{3},0\right)$ and $M=\left(\pi,\frac{\pi}{\sqrt{3}}\right)$. The degeneracy of the zero-energy FB is indicated in parenthesis. The purple dot at the center of the plot represents an EP at momentum $K$ and $E=0$. The small branch energy gaps around the EP are a numerical artifact.
• Figure 5: (a) Complex energy spectrum of the periodic 3-root graphene lattice as a function of $\phi=2\pi\frac{l}{q}$, with $l=0,1,\dots,q$ and $q=187$. (b) Zoomed plot of the $p=0$ real energy branch in (a) delimited by the dashed red box. (c) Zoomed plot of the low-flux region in (b) delimited by a dashed orange box. The curves of different colors and formats represent the energy $E_{+m}^{(0)}(\phi)$, given in (\ref{['eq:enerllsnrootgraph']}), of the corresponding LLs.