Non-Euclidean Spectral Phenomena

Updated 23 November 2025

Non-Euclidean spectral phenomena are defined as eigenvalue and topological features emerging in curved spaces with non-commutative symmetries and non-Hermitian dynamics.
The studies employ advanced harmonic analysis and group representation techniques to overcome the limitations of classical Bloch theory in hyperbolic and fractal lattices.
Key outcomes include novel topological invariants, modified density of states, and practical insights for photonic, acoustic, and quantum systems.

Non-Euclidean spectral phenomena encompass the emergence, classification, and mathematical realization of spectral features—eigenvalues, band structures, topological invariants—arising from the interaction of geometric curvature, non-commutative symmetries, and non-Hermitian dynamics, both in regular hyperbolic lattices and homogeneous spaces endowed with non-Euclidean metrics. Compared to standard Euclidean settings, these phenomena exhibit profound modifications in band theory, spectral topology, local density of states, statistical properties, and symmetry constraints. This article synthesizes recent developments in non-Hermitian hyperbolic lattice physics (Hu et al., 7 Dec 2024), harmonic analysis on Gelfand pairs (Székelyhidi, 2016), spectral mass and asymptotics (Strichartz, 2010), Cayley-crystal spectra (Lux et al., 2022), and connections to topological phases, quantum dynamics, and experiment.

1. Hyperbolic Lattice Geometry and Non-Abelian Translation Symmetry

A prototypical non-Euclidean spectral setting arises in two-dimensional hyperbolic lattices, regular tessellations with Schläfli symbol $\{p,q\}$ , defined by tiling the hyperbolic plane with $p$ -gons meeting $q$ at each vertex, such that $(p-2)(q-2) > 4$ . Unlike Euclidean lattices, finite patches of a hyperbolic lattice cannot be periodically continued across the plane; any identification of edges necessarily produces surfaces of genus $g > 1$ (Hu et al., 7 Dec 2024).

The translation group for hyperbolic lattices, $\Gamma$ , is a Fuchsian group generated by hyperbolic translations, subject to relations,

$\Gamma = \langle Y_1,\dots,Y_{2g}\mid\prod_{i=1}^g [Y_{2i-1},Y_{2i}] = 1\rangle, \qquad [A,B] \equiv ABA^{-1}B^{-1},$

which is non-Abelian and admits irreducible representations (IRs) of dimension $d > 1$ —in stark contrast to Euclidean $\mathbb{Z}^2$ translations which yield only $U(1)$ Bloch phases. This non-commutative symmetry invalidates the conventional Bloch theorem, making it impossible to label eigenstates purely by quasi-momentum.

Consequently, one must work with non-Abelian Bloch states (NABSs) and pass to induced group representations or enlarge unit cells to "supercells" that accommodate higher-dimensional IRs. The generalized Brillouin zone of an $n$ -supercell becomes a multi-dimensional torus parameterized by $2g(n)$ phases, with $g(n)=n(g(1)-1)+1$ , $g(1)=\frac{1}{2}(p-2)(q-2)$ .

2. Topological Classification in Non-Hermitian Hyperbolic Systems

For non-Hermitian hyperbolic lattices, spectral topology is fundamentally governed by the point-gap (PG) invariant, capturing the winding of the characteristic polynomial over multidimensional tori: $f^{(n)}(E, \bm{\beta}) = \det[H^{(n)}(\beta_1,\dots,\beta_{2g}) - E],$

$w^{(n)}(E) = \frac{1}{2\pi i} \sum_{a=1}^{2g(n)} \oint_{|B_a|=\mathrm{const}} d\ln f^{(n)}(E, \bm{B}),$

where analytic continuation $B_j \to \beta_j = |B_j|e^{i k_j}$ encodes the effect of open boundaries.

The universal OBC spectral range—the set of energies supporting nontrivial point gaps under open boundary conditions—is the intersection over all phase modulus choices: $o_u = \bigcap_{|\beta_j|>0} \left\{ E \mid \frac{1}{2\pi i} \oint d^{2g(n)}k \nabla_k \ln f^{(n)}(E, \bm{\beta}) \neq 0 \right\},$ yielding a spectral set independent of boundary details and encompassing all nontrivial skin and topological phenomena (Hu et al., 7 Dec 2024).

3. Breakdown of Conventional Band Theory and Supercell Formalism

In hyperbolic lattices, the breakdown of Bloch's theorem necessitates a modified approach. Under periodic boundary conditions (PBC), block-diagonalization is performed on the $n$ -supercell via $2g(n)$ momentum-like phases, but in the thermodynamic limit $n\to\infty$ , the Brillouin zone is replaced by an infinite-dimensional torus, encoding the nontrivial genus.

The analytic continuation to OBC, via deformation $k_j\to k_j - i p_j$ , collapses spectral loops (typical of PBC) in complex energy space to filled regions (under OBC), resulting in higher-dimensional skin effects. These phenomena are manifest in both single-band nonreciprocal models and reciprocal non-Abelian semimetals, where phase boundaries are shifted, and degenerate NABSs can be predicted analytically (Hu et al., 7 Dec 2024).

Such supercell-based formalism generalizes to any lattice with non-Abelian generation group—spin systems on Cayley trees, fractal networks, Bethe lattices—enabling the paper of non-Hermitian spectral topology in a wide class of non-Euclidean media.

4. Spherical Spectral Synthesis and Non-Commutative Harmonic Analysis

Classical spectral analysis and synthesis, as extended from Schwartz's theorem, fails in higher-dimensional Euclidean spaces under mere translation invariance, due to counterexamples of varieties lacking exponential functions as spectral atoms. By replacing translation invariance with invariance under a compact group $K$ of automorphisms, one obtains the Gelfand pair $(G, K)$ and accesses spherical functions $\varphi_\lambda$ , which play the role of higher-dimensional "exponentials" (Székelyhidi, 2016).

The spherical or Gelfand transform,

$\widetilde f(\lambda) = \int_G f(g)\varphi_{-\lambda}(g)dg,$

admits inversion and Plancherel formulas on symmetric spaces and spheres, with explicit Hilbert-space decompositions in terms of Bessel functions (e.g., radial functions in $\mathbb{R}^n$ ), zonal harmonics (on $S^n$ ), or Wigner $D$ -functions (on $\mathrm{SO}(3)$ ) (Dick et al., 2019).

Spherical spectral synthesis thus restores spectral completeness, allowing every closed, $K$ -invariant variety to be reconstructed from spherical monomials. This generalizes to large classes of non-Euclidean domains where group symmetry replaces translation invariance as the organizing principle of the spectrum.

5. Spectral Mass, Asymptotic Laws, and Group-Theoretic Splitting

For Laplacians in non-compact or highly symmetric non-Euclidean settings, the eigenvalue-counting function lacks meaning due to continuous spectrum. The spectral mass $M(\lambda)$ , defined as the normalized average of the diagonal kernel of the spectral projection,

$M(\lambda) = \lim_{k\to\infty}\frac{1}{\mu(B_k)}\int_{B_k}K_\lambda(x,x)d\mu(x),$

serves as the central statistical descriptor (Strichartz, 2010).

This notion recovers Weyl-type laws in Euclidean and hyperbolic settings: $M(\lambda) = c_n \lambda^{n/2} + O(\lambda^{(n-2)/2})$ for homogeneous spaces; for trees and fractals, the exponent becomes dimension-dependent, e.g.,

$M(\lambda) \sim C(\lambda-\lambda_0)^{3/2}$

on the infinite tree, or

$M(\lambda) \sim \lambda^{d_s/2}G(\log\lambda)$

on the Sierpiński gasket, where $d_s$ is the spectral dimension.

When the Laplacian admits group symmetries, the spectrum decomposes according to irreducible representations. The "asymptotic Schur's lemma" yields splitting ratios: $\frac{M_j(\lambda)}{M(\lambda)} \to \frac{d_j^2}{|G|},$ for finite group $G$ and dimension $d_j$ of the irrep, under suitable boundary conditions.

6. Quantum Dynamics on Cayley Graphs and Bulk Spectra

Quantum Hamiltonians defined over Cayley graphs of finitely generated groups (including free and Fuchsian groups) represent a universal setting for non-Euclidean spectral analysis. The operator-algebraic bulk spectrum is recovered by periodic boundary conditions in the sense of Lück—using finite-index normal subgroups and algebraic quotient maps—ensuring convergence to the physical density of states without boundary artifacts (Lux et al., 2022).

For regular trees (free groups), the adjacency spectrum is continuous and described by the Kesten–McKay law: $F_e'(x) = \frac{v}{2\pi}\frac{\sqrt{4(v-1)-x^2}}{v^2-x^2}$ where $v$ is the degree. Hyperbolic (Fuchsian) Cayley graphs exhibit exponential growth, elevated spectral radii, and absence of spectral gaps.

The combinatorial resolvent expansion provides closed-form Green’s functions via path counting, and exact numerical algorithms exploit these algebraic structures to converge to true spectra.

7. Emergent Spectral Phenomena, Physical Realizations, and Outlook

Non-Euclidean spectral phenomena manifest as higher-dimensional skin effects, shifted topological phase boundaries, fractal energy bands, enhanced or suppressed localization, and atypical statistical distributions of eigenvalues and eigenstates. In optical microcavities carved on curved spaces, the interplay of joint curvature parameters yields phase diagrams with abrupt transitions in quality factor, the emergence of hyperbolic fixed points, and controlled quantum chaos signatures (Ding et al., 1 May 2025).

Topological phases in hyperbolic photonic lattices rely on quantized curvature and non-Abelian gauge assignments, leading to the generalization of quantum spin Hall effect and Hofstadter butterflies with curvature-tuned subband structures and exponentially large edge-state degeneracy (Yu et al., 2020).

In acoustic crystals, disclinations induce real-space curvature, fractional bound charges, and symmetry-induced degeneracy splittings, illustrating the direct link between geometry and spectral topology (Chen et al., 2022). Non-Hermitian phenomena such as the skin effect, boundary sensitivity, and pseudospectral collapse are strongly accentuated by hyperbolic geometry, requiring new diagnostics to distinguish genuine skin modes from trivial boundary states (Shen et al., 20 Oct 2024).

These advances elucidate a unifying framework where geometry, symmetry, boundary conditions, and non-Hermiticity collectively govern the spectral landscape—profoundly extending classical band theory, harmonic analysis, and the classification of topological matter into genuinely non-Euclidean regimes.

Key References Table

Theme	Representative arXiv ID	Core Concept
Non-Hermitian hyperbolic lattice topology	(Hu et al., 7 Dec 2024)	Supercell analytic continuation, point-gap winding, skin effect
Spherical spectral synthesis	(Székelyhidi, 2016)	Gelfand pairs, spherical function decomposition, group symmetry
Spectral mass & asymptotics	(Strichartz, 2010)	Averaged kernel, Weyl-type laws, Schur decomposition
Cayley-crystal quantum dynamics	(Lux et al., 2022)	Lück PBC, spectral density, combinatorial resolvent, continuous spectrum
Polygonal microcavity phase diagrams	(Ding et al., 1 May 2025)	Joint curvature, fixed points, chaos, Q-factor transitions

The synthesis and extension of spectral methods to non-Euclidean geometries—hyperbolic, spherical, and more general group- or fractal-based spaces—reveal new universality classes, topological invariants, and boundary-sensitive phenomena. These developments anticipate systematic theoretical frameworks for quantum, photonic, and acoustic systems in curved and group-theoretic settings, suggesting directions for future research in non-Hermitian topology, quantum chaos, and functional analysis in highly non-Euclidean media.