--- title: Fragmented Exceptional Points (FEPs) url: https://www.emergentmind.com/topics/fragmented-exceptional-points-feps type: topic --- # Fragmented Exceptional Points (FEPs) Fragmented exceptional points are non-Hermitian spectral degeneracies in which an eigenvalue of algebraic multiplicity $\alpha_i$ has geometric multiplicity $\gamma_i$ satisfying $1<\gamma_i<\alpha_i$, so the eigenvectors are only partially degenerate. In the formulation introduced for finite-dimensional non-Hermitian Hamiltonians, the fragmentation is encoded by a partial multiplicity decomposition $\alpha_i=\sum_{j=1}^{\gamma_i} l_{i,j}$, where each $l_{i,j}$ records how many algebraically degenerate modes share the same eigenvector. This places fragmented exceptional points between fully defective exceptional points, for which all degenerate modes collapse into one eigenvector, and nondefective diabolic or “$n$-bolic” points, for which all degenerate modes retain independent eigenvectors. The resulting degeneracies exhibit intermediate singular response behavior and can be designed explicitly in bulk and edge spectra of lattice models, notably in non-Hermitian variants of a Lieb lattice and a higher-order topological Dirac semimetal [2507.22158]. ## 1. Definition and classification Consider a finite-dimensional non-Hermitian Hamiltonian $H$ with right and left eigenvalue equations \[ H\,\mathbf{u}_{i,j}=E_i\,\mathbf{u}_{i,j},\qquad \mathbf{v}_{i,j}\,H=E_i\,\mathbf{v}_{i,j}. \] For each eigenvalue $E_i$, the algebraic multiplicity $\alpha_i$ is the order of the root of $\det(E-H)$, while the geometric multiplicity $\gamma_i$ is the number of linearly independent right eigenvectors, equivalently left eigenvectors. The distinction between these two multiplicities organizes the classification of non-Hermitian degeneracies [2507.22158]. | Degeneracy type | Multiplicity condition | Partial multiplicity pattern | |---|---:|---| | Diabolic or “$n$-bolic” point | $\gamma_i=\alpha_i=n$ | all $l_{i,j}=1$ | | Conventional $\mathrm{EP}_n$ | $\gamma_i=1<\alpha_i=n$ | $l_{i,1}=n$ | | Fragmented exceptional point | $1<\gamma_i<\alpha_i$ | intermediate, e.g. $(2,1)$ | For an FEP, the partial multiplicity decomposition \[ \alpha_i =\sum_{j=1}^{\gamma_i} l_{i,j}, \qquad l_{i,1}\ge l_{i,2}\ge\cdots\ge l_{i,\gamma_i}\ge1, \] specifies the fragmentation pattern. The list $(l_{i,1},\dots,l_{i,\gamma_i})$ characterizes how the eigenvectors are shared across algebraically degenerate modes. For $n=3$, the pattern $(2,1)$ is the simplest nontrivial example: two algebraically degenerate modes share one eigenvector, while the third retains an independent eigenvector. Physically, FEPs combine features of EPs, notably enhanced sensitivity, with residual orthogonality sectors. ## 2. Singular response and perturbative behavior The response of a system near an FEP is governed not by the full algebraic multiplicity $\alpha_i$ alone, but by the maximal partial multiplicity \[ \ell_i\equiv l_{i,1}. \] Around such a degeneracy, the Green’s function \[ G(E)=(E-H)^{-1} \] obeys \[ \mathrm{tr}\,G^\dagger G\,\propto\frac1{|E-E_i|^{2\ell_i}}, \] so the physical response diverges with an exponent set by $\ell_i$ rather than necessarily by $\alpha_i$ [2507.22158]. A generic perturbation of strength $\varepsilon$ splits the eigenvalues according to \[ \Delta E\sim(\varepsilon\,\xi_i)^{1/\ell_i}, \] where $\xi_i$ is a characteristic spectral response strength. A corresponding physical response strength $\eta_i$ appears in the residue of the Green’s function. Compared with a conventional $\mathrm{EP}_n$, which has $\ell_i=n$, an FEP with $\ell_i<\alpha_i$ has weaker singularities: the physical response diverges only as $1/|E-E_i|^{2\ell_i}$ and the spectral splitting scales as $\varepsilon^{1/\ell_i}$. At the same time, FEPs retain non-orthogonality in subspaces of size $\ell_i$ while preserving residual orthogonality among the remaining eigenvectors. In this sense they interpolate between diabolic points, for which $\ell_i=1$, and full exceptional points, for which $\ell_i=\alpha_i$. ## 3. Algebraic detection and partial-multiplicity extraction A central development in the systematic study of FEPs is an algebraic framework that determines $\alpha_i$, $\gamma_i$, and the full partial-multiplicity structure directly from a given Hamiltonian, without carrying out a delicate Jordan decomposition. The construction uses the so-called modes of the adjugate matrix $\{\mathcal{B}_k\}$ for the shifted matrix \[ A=H-E_i\,\openone, \] with the coefficients $c_k$ defined by the shifted characteristic polynomial $\det(\lambda-A)=\sum_{k=0}^N c_k\,\lambda^k$ [2507.22158]. The modes satisfy the Faddeev–LeVerrier recursion \[ \mathcal{B}_{N-1}=\openone,\qquad \mathcal{B}_{k-1}=A\,\mathcal{B}_k-\frac{\mathrm{tr}\!\bigl(A\,\mathcal{B}_k\bigr)}{N-k}\,\openone. \] The algebraic degeneracy condition \[ C_k\equiv\mathrm{tr}\bigl(A\,\mathcal{B}_k\bigr)=0 \quad (k=0,1,\dots,\alpha_i-1) \] guarantees that $E_i$ is a root of multiplicity $\alpha_i$. Once $\alpha_i$ is known, the geometric partial multiplicities $\beta_i(l)$ are extracted from matrix ranks as \[ \beta_i(l)=\mathrm{rnk}\bigl(\mathcal{B}_{\alpha_i-l-2}\bigr)-2\,\mathrm{rnk}\bigl(\mathcal{B}_{\alpha_i-l-1}\bigr)+\mathrm{rnk}\bigl(\mathcal{B}_{\alpha_i-l}\bigr), \quad l=1,2,\dots,\alpha_i. \] Equivalently, the maximal partial multiplicity $\ell_i$ is the smallest $l$ for which $\mathcal{B}_{\alpha_i-l}\neq0$ but $\mathcal{B}_k=0$ for all $k<\alpha_i-l$. The same formalism yields explicit expressions for the response strengths, \[ \eta_i^2=\frac{\mathrm{tr}\!\bigl(\mathcal{B}_{\alpha_i-\ell_i}^\dagger\,\mathcal{B}_{\alpha_i-\ell_i}\bigr)}{|c_{\alpha_i}|^2}, \qquad \xi_i^2=\frac{\|\mathcal{B}_{\alpha_i-\ell_i}\|_2^2}{|c_{\alpha_i}|^2}. \] Because the procedure only involves traces and ranks of polynomials in $H$, it applies to any explicit matrix representation of a non-Hermitian model. This makes the framework suitable both for diagnosis of existing degeneracies and for inverse design of systems with prescribed degeneracy structures and response characteristics. ## 4. Bulk realization in the non-Hermitian Lieb lattice The non-Hermitian Lieb lattice provides a minimal two-dimensional setting in which bulk FEPs can be realized. The Hermitian Lieb lattice has a three-site unit cell $(A$–$B$–$C)$ with symmetric nearest-neighbour hopping. Setting intra- and intercell hopping $t=1$, its Bloch Hamiltonian is \[ \mathcal H(\mathbf k)= \begin{pmatrix} 0 & 1+e^{ik_y} & 0\\ 1+e^{-ik_y}&0&1+e^{-ik_x}\\ 0&1+e^{ik_x}&0 \end{pmatrix}, \] which has a flat band at zero energy and two dispersive bands meeting at a tribolic point with $\alpha=3$ and $\gamma=3$ [2507.22158]. A simple non-Hermitian variant preserving chiral symmetry is \[ \mathcal H(\mathbf k)= \begin{pmatrix} 0 & 1+e^{ik_y}+i\varepsilon & 0\\ 1+e^{-ik_y}&0&1+e^{-ik_x}\\ 0&1+e^{ik_x}-i\varepsilon&0 \end{pmatrix}. \] At $(k_x,k_y)=(\pi,\pi)$, this model has $\alpha=3$ and $\gamma=2$, giving the partial multiplicity pattern $(l_1,l_2)=(2,1)$, identified in the paper as a FEP $3+1$. By contrast, the other degeneracies in the Brillouin zone are generic $\mathrm{EP}_3$ points with $\gamma=1$ and $l_1=3$. The classification can be stated in closed form. Writing \[ \mathcal H(k)= \begin{pmatrix} 0 & P &0\\ Q&0&R\\ 0&S&0 \end{pmatrix}, \] degeneracies at $E=0$ occur when \[ PQ+RS=0. \] An $\mathrm{FEP}_3$, corresponding to the $(2,1)$ scenario, occurs precisely if, for example, $Q=R=0$ while $P,S\neq0$, or symmetrically if $P=S=0$ while $Q,R\neq0$. All other nontrivial solutions of $PQ+RS=0$ with none of the pairs simultaneously vanishing produce $\mathrm{EP}_3$ points. This establishes an explicit criterion for distinguishing partial and full eigenvector collapse directly at the Bloch-Hamiltonian level. ## 5. Bulk and hinge realizations in a higher-order topological Dirac semimetal A three-dimensional realization is furnished by a higher-order Dirac semimetal. Its Hermitian parent is a 4-band model built by stacking 2D quadrupole insulators with criss-cross interlayer couplings. In momentum space it can be written in chiral block form \[ \mathcal H(\mathbf k)= \begin{pmatrix} 0&P(\mathbf k)\\ P^\dagger(\mathbf k)&0 \end{pmatrix}, \qquad P(\mathbf k)=\sum_{i=0}^3 p_i(\mathbf k)\,\sigma_i. \] Bulk Dirac points at $E=0$ are tetrabolic points with $\alpha=4$ and $\gamma=4$ [2507.22158]. Four minimal non-Hermitian variants are obtained by adding intracell nonreciprocal hopping terms of amplitude $\varepsilon$. At those momenta where $\det P=0$, the four models realize distinct degeneracy patterns. | Model | Bulk degeneracy | Hinge degeneracy | |---|---|---| | 1 | generic $\mathrm{EP}_4$ coexisting with diabolic points $(\alpha=2,\gamma=2)$ | hinge states approach an $\mathrm{EP}_4$ | | 2 | FEP with $(l_1,l_2)=(3,1)$ | hinge states form an $\mathrm{FEP}_4$ with $(3,1)$ | | 3 | FEP with $(l_1,l_2)=(2,2)$ | hinge states form an $\mathrm{FEP}_4$ with $(2,2)$ | | 4 | FEP with $(l_1,l_2,l_3)=(2,1,1)$ | hinge states form an $\mathrm{FEP}_4$ with $(2,1,1)$ | In the topological phase with open $x,y$ boundaries, four nearly zero-energy hinge modes appear as functions of $k_z$. In the same four variants, these hinge modes bisect bulk degeneracy points and then hybridize into FEPs of the same type found in the bulk. The fragmentation can be confirmed either by the rank–trace formalism or by combining atomistic-limit corner-mode counting with non-Hermitian skin-effect arguments. The model therefore demonstrates that FEPs are not restricted to bulk band touchings; they can also characterize boundary spectra in higher-order topological systems. ## 6. Terminological extensions, detached EPs, and nonlinear fragmentation Recent literature also uses the phrase “fragmented exceptional points” in senses that differ from the partial-multiplicity definition above. In a reciprocal non-Hermitian Lieb lattice under open boundary conditions, geometry-induced non-Bloch EPs arise from generalized Brillouin-zone geometry. There the open-boundary spectrum is obtained from \[ \det[h(B,k_b)-E\,\mathbb I]=0, \qquad |B_1(E,k_b)|=|B_2(E,k_b)|, \] with $B=e^{ik_x}$ and $|B|\neq1$. The discriminant \[ \Delta(B,k_b)=\prod_{i