Exceptionally deficient topological square-root insulators

One of the most surprising features of effectively non-Hermitian physical systems is their potential to exhibit a striking nonlinear response and fragility to small perturbations. This feature arises from spectral singularities known as exceptional points, whose realization in the spectrum typically requires fine-tuning of parameters. The design of such systems receives significant impetus from the recent conception of \emph{exceptional deficiency}, in which the entire energy spectrum is composed of exceptional points. Here, we present a concrete and transparent mechanism that enforces exceptional deficiency through lattice sum rules in non-Hermitian topological square-root insulators. We identify the resulting dynamical signatures in static broadband amplification and non-Abelian adiabatic state amplification, differentiate between bulk and boundary effects, and outline routes to implementation in physical platforms

💡 Research Summary

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The paper introduces a concrete and transparent mechanism to enforce “exceptional deficiency” – a situation where every eigenvalue of a system is an exceptional point (EP) – in non‑Hermitian topological square‑root insulators. The authors start by recalling that EPs, which are spectral singularities where eigenvectors coalesce, give rise to highly nonlinear responses and extreme sensitivity to perturbations. Traditionally, realizing EPs requires fine‑tuning of system parameters, which limits their practical use. The recent concept of exceptional deficiency, in which the entire spectrum consists of EPs, promises broadband EP‑enhanced functionalities but has so far only been demonstrated in a mechanical lattice with specially matched subsystems.

To achieve exceptional deficiency in a more versatile platform, the authors exploit the square‑root construction principle for topological insulators. In this approach, two identical “parent” Hermitian Hamiltonians (denoted H₂) are coupled in a non‑trivial way to produce a larger Hamiltonian H that is the square root of the direct‑sum of the parents. The lattice is divided into four sublattices (A, B, C, D). By imposing a lattice sum rule

 H_BC H_CA + H_BD H_DA = 0

and a generalized transposition symmetry R Hᵀ R = H (with R commuting with the chiral operator), the authors guarantee that right eigenvectors uₗ and left eigenvectors vₘ satisfy vₘ·uₗ = 0 for all ℓ, m, even when the corresponding eigenvalues coincide. In non‑Hermitian systems this orthogonality is precisely the self‑orthogonality that defines an EP. Consequently, the whole spectrum becomes exceptional – the system is “exceptionally deficient”.

The abstract construction is instantiated in a concrete model: a π‑flux quadrupole insulator (QI), a well‑known higher‑order topological phase that hosts four corner states. The Hermitian QI has intra‑cell hopping t, inter‑cell hopping s = 1, and a π‑flux per plaquette, leading to twofold Kramers‑degenerate corner modes. The non‑Hermitian modification adds non‑reciprocal inter‑cell couplings of strength ε, breaking the C₄ rotational symmetry but preserving the sum rule and transposition symmetry. As a result, every eigenstate—including the four corner states—becomes an EP. Figure 1(c) shows the corner modes for ε = −t = ½; each corner state belongs to a degenerate pair of EPs, and the same holds for all bulk states.

Two dynamical signatures of exceptional deficiency are identified. First, for a time‑independent Hamiltonian, the solution of the Schrödinger equation contains both the usual exponential term e^{−iEₗt}uₗ and a linearly growing term t e^{−iEₗt}wₗ, where wₗ is the generalized eigenvector satisfying H wₗ = Eₗ wₗ + uₗ. The total intensity I(t)=|ψ(t)|² therefore acquires a t² contribution unless the initial state lies entirely within the span of the ordinary eigenvectors. In the QI implementation, the A sublattice is fully spanned by the eigenvectors, while the B sublattice is not. Numerical simulations (Fig. 2) demonstrate that initial states uniformly distributed on B, C, or D exhibit a clear t²‑type amplification, whereas an A‑sublattice initial state shows none. This broadband amplification—independent of the specific eigenvalue—constitutes a robust experimental hallmark of exceptional deficiency.

Second, the authors explore adiabatic state amplification in a non‑Hermitian setting. When a control parameter λ (here ε) is varied slowly along a path C, the intensity of a state initially prepared in a long‑lived right eigenstate evolves as I(t)=|ψ(t)|² = I(0) exp