Weyl-Type Bound: Spectral & Topological Insights

• Weyl-type bound is a rigorous estimate on eigenvalue counting functions in systems with Weyl degeneracies, establishing asymptotic limits.
• It connects interior transmission problems, topological semimetals, and spectral theory by quantifying state proliferation under precise mathematical bounds.
• The bound informs material design and theoretical models by constraining the connectivity of bound states and guiding the analysis of phase transitions in complex systems.

A Weyl-type bound is a rigorous estimate—typically a lower or upper bound—on the counting function or spectral density of eigenvalues or states associated with a physical, mathematical, or topological system exhibiting Weyl-type degeneracies. Originally inspired by Weyl’s law in spectral theory, these bounds extend to a broad class of problems, including nonselfadjoint spectral problems, quantum and photonic systems with band touching, disordered topological semimetals, automorphic L-functions, and constrained wave phenomena. The precise form of a Weyl-type bound and its physical or analytical significance depends on the domain, but the unifying principle is the existence of an asymptotic, often quantized, law governing the proliferation or restriction of states near critical points associated with Weyl singularities.

1. Weyl-Type Bounds in Interior Transmission Problems

In the context of the interior transmission eigenvalue (ITE) problem for wave propagation through inhomogeneous media, the Weyl-type bound establishes a lower asymptotic bound on the number of positive transmission eigenvalues NT(λ) as λ grows. Specifically, for a domain in $\mathbb{R}^d$, the main result is:

$$N_T(\lambda) \geq C \lambda^{d/2} + o(\lambda^{d/2}),$$

where $C>0$ depends on the effective contrast of the coefficients and the domain volume. In the precise formulation,

$$N_T(\lambda) = \frac{\omega_d}{(2\pi)^d} V_{\text{eff}} \lambda^{d/2} + o(\lambda^{d/2}),$$

with $\omega_d$ the volume of the unit $d$-ball and $V_{\text{eff}}$ the effective volume determined by material properties (Lakshtanov et al., 2013).

This lower bound implies the infinitude of positive eigenvalues for the ITE problem. The transmission problem is reduced via the analysis of the difference of Dirichlet-to-Neumann (D-t-N) operators for the homogeneous and inhomogeneous media—specifically,

$$P(\lambda) = \sigma (F(\lambda) - F_{a,n}(\lambda)): H^{3/2}(\partial O) \rightarrow H^{3/2-s}(\partial O),$$

where the properties of $P(\lambda)$ are connected to the asymptotics of the eigenvalue problem. For both classical transmission problems and problems incorporating obstacles inside the medium, the Weyl-type bounds persist, provided the obstacles are suitably regular.

2. Weyl-Type Bounds in Topological Semimetals and Boundary States

In Weyl semimetal systems, both continuum and lattice, Weyl-type bounds manifest as restrictions and quantizations on the existence, connectivity, and topology of surface, edge, or interface states. For the continuum Weyl semimetal with generic boundary conditions, the bulk Hamiltonian

$$\mathcal{H} = p_1 \sigma_1 + p_2 \sigma_2 + p_3 \sigma_3,$$

paired with Hermiticity-preserving boundary conditions parameterized by a single real variable $\theta_+ \in [0,\pi)$, fully determines the Fermi arc dispersion in the surface Brillouin zone. The analytic form

$\big[1, e^{-2 i \theta_+}\big] \psi|_{x^3 = 0} = 0$

leads to a parametric edge dispersion

$$\varepsilon = -p_1 \cos(2\theta_+) - p_2 \sin(2\theta_+),$$

and an inverse decay length $\alpha$ controlling surface localization. The extended parameter space $(p_1, p_2, \theta_+)$ supports an emergent Berry connection and a quantized Berry phase,

$\phi_B = \oint A_{(\theta_+)} d\theta_+ = \pi,$

defining a new topological invariant for edge states—an explicit Weyl-type bound on how Fermi arcs are connected to the bulk Weyl cones (Hashimoto et al., 2016).

This formalism generalizes: for full continuum models with arbitrary boundary conditions (including symmetry constraints), the existence and chirality of bound/edge/surface state bands are determined by universal boundary phase parameters, so that the surface state spectrum near a Weyl node robustly reflects the Chern number of the node (Kharitonov, 2022).

3. Critical Transitions and Weyl-Type Bounds in Multiband Systems

Multiband tight-binding models reveal a more complex structure of Weyl-type bounds, involving transitions between types of Weyl fermions and higher pseudospin representations. In tricolor cubic lattice models, not only do conventional type-I and type-II Weyl nodes appear, but so do pseudospin-3/2 fermions (emerging from 4-band touchings) and “critical Weyl fermions” at the boundary between the two conventional types (Ezawa, 2016). These critical points are precisely where the Weyl cone’s tilt reaches the maximum allowed by topology (i.e., the Fermi surface changes from a point to electron–hole pockets), and their conversion under magnetic field illustrates dynamic Weyl-type bounds on how topological phase transitions are realized in parameter space.

Monopole charges extracted from the Berry curvature, e.g., $\rho_j = 2j$ for pseudospin-$j$ representations, provide a quantized bound on the allowed topological charges and consequently dictate the possible number and type of protected surface/edge states.

4. Weyl-Type Bounds in Disordered and Synthetic Systems

Disorder, tilt, and coupling to synthetic degrees of freedom produce further Weyl-type bounds. In disordered type-II Weyl semimetals, the global phase diagram is bounded by the tilt parameter $b_t$, with regions for clean phase, diffusive metal, and quantum anomalous Hall phase sharply delineated. Type-II Weyl systems ($|b_t| > 1$) are protected from the 3D QAH phase, maintain large transport-to-quantum lifetime ratios ($$(\tau_{tr}/\tau_q)_{WSM2} \gg (\tau_{tr}/\tau_q)_{WSM1}$$), and show that even infinitesimal disorder at the critical tilt ($b_t \approx 1$) can eliminate Weyl semimetal behavior altogether by producing strong localization—a practical bound on possible phases in real materials (Wu et al., 2017).

In photonic synthetic Weyl heterostructures, a “bound–extended mode transition” at the interface of type-II Weyl lattices is achieved by tuning the “rotational phase,” leading to an effective real-space and momentum-space band mismatch. The transition is captured via a band-shift index $v$, operationally controlling the presence or absence of interface-localized modes. This phase sensitivity and control are absent in type-I Weyl structures, establishing a Weyl-type bound on mode localization in systems with tilted Weyl dispersion (Song et al., 8 Mar 2024).

5. Analytical Weyl-Type Bounds in Spectral and Number-Theoretic Problems

In spectral theory for discrete Schrödinger operators, Weyl-type bounds exist for the discrete spectrum size:

$N[e, V] \leq C(d, e) N_{sc}[e, V],$

where $N_{sc}[e, V]$ is the phase-space volume semi-classical estimate. In higher dimensions ($d \geq 3$), mild decay of the potential suffices; in $d=1,2$, stronger spatial decay is required (Bach et al., 2017). These constructions are sensitive to sparsity and decay of the potential: failure of these conditions leads to the breakdown of the Weyl law. Similar upper and lower bounds hold for Anderson Hamiltonians with singular random potentials, where the spectrum's leading asymptotic density is preserved, with random pre-factors corresponding to the disorder (Mouzard, 2020).

Weyl-type bounds are also established in analytic number theory for high-degree automorphic L-functions. For triple product and hybrid GL(2) $L$-functions twisted by high-conductor Dirichlet characters, the Weyl-type subconvexity bound reads

$$L(1/2 + it, g \otimes \chi) \ll_{g,\varepsilon} ((1 + |t|) q)^{1/3 + \varepsilon}$$

where $q$ is the modulus and $g$ the fixed cusp form (Gao et al., 2023, Blomer et al., 2021). These bounds break the convexity barrier, with the exponent $1/3$ reflecting the Weyl law's exponent in this context.

6. Topological and High-Dimensional Generalizations

In five-dimensional Weyl systems, Weyl-type bounds describe not isolated point nodes but 2D Weyl surfaces (WSs) subject to topological constraints generalizing the 3D Doubling Theorem. The linking numbers between WSs—quantified via second Chern numbers computed on 4D tubes—must satisfy strict cancellation rules, such as

$$L_{W_{n+1/2}^{(i)}, W_{n-1/2}} = - L_{W_{n+1/2}^{(i)}, W_{n+3/2}},$$

providing a higher-dimensional Weyl-type bound on the allowed configuration of band touchings and associated surface states. These bulk constraints dictate the structure of boundary Fermi hypersurfaces and nodal arcs in the 4D boundary of a 5D semimetal, echoing the bulk-boundary correspondence seen in lower dimensions (Chen et al., 2018).

7. Physical Implications and Applications

Weyl-type bounds have direct consequences for inverse scattering, wave and quantum transport, engineering of materials with tailored spectral gaps, boundary phenomena in lattice field theory, and the control of light in synthetic photonic systems. In all contexts, they set precise limits on the existence, density, and connectivity of bound and extended modes, surface arcs, spectral gaps, and the presence of topologically protected phenomena. For instance, in photonic Weyl heterostructures, these bounds translate into tunable interfaces for total reflection or transmission, facilitating high-dimensional topological device engineering (Song et al., 8 Mar 2024). In lattice quantum systems, they inform the design of chiral fermion regularizations that evade the constraints of the Nielsen–Ninomiya theorem by situating Weyl fermions on boundaries with appropriate bulk topology (Kaplan et al., 2023).

The underlying principle across these domains is that Weyl-type bounds provide a quantitative and often topologically dictated constraint on either the proliferation or limitation of eigenstates in settings characterized by band degeneracy, topology, and/or boundary conditions. These results bridge core aspects of spectral theory, topological phases of matter, disorder physics, and analytic number theory.