Fragmented exceptional points and their bulk and edge realizations in lattice models

TL;DR

This work introduces fragmented exceptional points (FEPs) as a broad class of non-Hermitian spectral degeneracies where algebraic and geometric multiplicities differ. Using an adjugate-matrix framework, the authors derive exact, model-ready conditions for FEP formation and characterize their physical signatures, including Green’s-function scaling and perturbation responses, all expressed through the modes $\,\mathcal{B}_k$. They apply the framework to two lattice models: a non-Hermitian Lieb lattice (3 bands) and a non-Hermitian higher-order Dirac semimetal (4 bands), demonstrating bulk FEPs, exceptional rings/lines, and hinge-state–driven FEPs, with detailed symmetry analysis. The results reveal a rich interplay between non-Hermiticity, topology, and boundary physics, offering practical routes to engineer unconventional spectral responses in photonic, acoustic, and circuit platforms. Overall, the paper provides a comprehensive, algebraic toolkit for designing and diagnosing FEPs across a broad class of finite-dimensional non-Hermitian systems, expanding the landscape of non-Hermitian topological phenomena and their potential applications.

Abstract

Exceptional points (EPs) are spectral defects displayed by non-Hermitian systems in which multiple degenerate eigenvalues share a single eigenvector. This distinctive feature makes systems exhibiting EPs more sensitive to external perturbations than their Hermitian counterparts, where degeneracies are nondefective diabolic points. In contrast to these widely studied cases, more complex non-Hermitian degeneracies in which the eigenvectors are only partially degenerate are poorly understood. Here, we characterize these fragmented exceptional points (FEPs) systematically from a physical perspective, and demonstrate how they can be induced into the bulk and edge spectrum of two-dimensional and three-dimensional lattice models, exemplified by non-Hermitian versions of a Lieb lattice and a higher-order topological Dirac semimetal. The design of the systems is facilitated by an efficient algebraic approach within which we provide precise conditions for FEPs that can be evaluated directly from a given model Hamiltonian. The free design of FEPs significantly opens up a new frontier for non-Hermitian physics and expands the scope for designing systems with unconventional response characteristics.

Figures (13)

• Figure 1: (a) Standard Hermitian version of the Lieb lattice, based on a three-site unit cell with uniform couplings $t$. (b) Non-Hermitian version of the model, with nonreciprocal couplings $p,q,r,s$ inside the unit cell, while the couplings between the cells remain fixed to $t$. This modification allows us to realize fragmented exceptional points in the bulk dispersion of the model.
• Figure 2: Contour plots of the dispersion relation $|E(\mathbf{k})|$ for (a) the conventional Hermitian Lieb lattice with Bloch Hamiltonian \ref{['Hlieb']}, depicting a tribolic point at the momenta values $(k_x,k_y)=(\pi,\pi)$ around which the dispersion relation is linear, and (b) the non-Hermitian variant with Bloch Hamiltonian \ref{['nhHlieb']}, proposed in Ref. ipsita, displaying four conventional EP$3$s with geometric multiplicity $\gamma=1$ arranged symmetrically around the point where the tribolic point existed for the Hermitian case. The parameter $\varepsilon$ determining the strength of the non-Hermitian perturbation is set to be equal to 1. We design model systems that exhibit the more complex FEP degeneracy scenarios in their energy dispersion.
• Figure 3: (a) Energy band diagram $E(k_x,k_y=\pi)$ as a function of $k_x$ for the conventional Hermitian Lieb lattice with Bloch Hamiltonian \ref{['Hlieb']}, displaying the tribolic point at $k_x=\pi$. (b,c) Complex energy band diagrams (real part in blue, imaginary part in red) for the non-Hermitian Lieb lattice with Bloch Hamiltonian \ref{['nhHlieb3']} and $\varepsilon=1$. In (b) $k_y=\pi$, revealing the FEP of geometric multiplicity $\gamma=2$ and partial multiplicities $(l_{1},l_{2})=(2,1)$ at $k_x=\pi$. In (c), $k_y=-2\mathop{\mathrm{arccot}}\nolimits \frac{1}{2}$ is fixed to display the generic EP3 in this model, see Eq. \ref{['momenta_values']}. Panel (d) reveals the distinct dispersion around the FEP and EP3 in this model in terms of the energy contours $|E_+(\mathbf{k})|$. In the low energy limit, the generic EP3 displays an isotropic dispersion relation, while the dispersion relation around the FEP is highly anisotropic.
• Figure 4: Contour plots of the dispersion relation $|E(\mathbf{k})|$ for the reciprocal non-Hermitian Lieb lattice model determined by the Bloch Hamiltonian \ref{['nhHliebreciprocal']} for different values of the parameters $\Psi$ and $\Phi$. In panel (a), we set $\Psi=3\pi/4$ and $\Phi=\pi/2$, which realizes two FEPs at $(k_x,k_y)=(-3\pi/4,-\pi/2)$ and $(k_x,k_y)=(3\pi/4,\pi/2)$ along with two generic EP3s at $(k_x,k_y)=(-3\pi/4,\pi/2)$ and $(k_x,k_y)=(3\pi/4,-\pi/2)$. In panel (b), we set $\Phi=\Psi=\pi/4$, leading to a degenerate ring determined by the momentum relation in \ref{['eq:ring']}. On this ring, two FEPs are located at $(k_x,k_y)=(\pi/4,\pi/4)$ and $(k_x,k_y)=(-\pi/4,-\pi/4)$, while the remainder is formed from generic EP3s. Setting the parameters $\Phi=\Psi=\pi/2$ in panel (c), the exceptional ring morphs into four exceptional lines, speicified in Eq. \ref{['eq:lines']}. Similar to before, the lines host two FEPs located at $(k_x,k_y)=(\pi/2,\pi/2)$ and $(k_x,k_y)=(-\pi/2,-\pi/2)$, while the remainder of the lines is formed from generic EP3s.
• Figure 5: Lattice realization of the Hermitian higher-order Dirac semimetal, obtained from stacking two-dimensional quadrupole insulators along the $z$ direction. Each unit cell contains 4 sites, labeled A, B, C, and D. In each plane, the intracell and intercell couplings indicated by solid lines are $t$ and $s$, respectively. The couplings between the B and C sublattices, indicated by the dashed lines, have the opposite sign, so that each plaquette carries a $\pi$ flux. Neighboring unit cells in adjacent planes are connected by diagonal couplings of magnitude $s/4$, with the sign again reverted for couplings between B and C sites.