Point-gap topology in amorphous non-Hermitian quantum systems

AI summary

Overview

Research area: Theoretical and computational quantum physics — non-Hermitian topological phases, point-gap topology, the non-Hermitian skin effect (NHSE), and amorphous (positionally disordered) lattices.

Technical level: Advanced. The paper assumes familiarity with non-Hermitian Hamiltonians, bulk-boundary correspondence, spectral winding numbers, singular-value decomposition (SVD), and pseudospectra.

Scope (one sentence): The paper develops a real-space, SVD-based framework that predicts and counts stable topological edge states of one-dimensional amorphous non-Hermitian chains, where the energy spectrum itself is too unstable to be used as a diagnostic.

What This Paper Is About

In non-Hermitian systems, topology is usually read off from the complex energy spectrum — a nonzero "point-gap" winding number is supposed to imply skin modes piled up at the edges. That logic breaks down badly: eigenvalues of non-Hermitian Hamiltonians are extremely sensitive to perturbations, boundary conditions, and system size, and in amorphous systems (where lattice-site positions are random) the spectrum becomes even more unstable.

This paper asks how to identify and count topological edge states when the energy spectrum cannot be trusted. Its answer is to shift attention from eigenvalues to singular values of the shifted Hamiltonian H − E_b·I, which inherit the stability of Hermitian spectra, and to build a quantized real-space topological invariant from that SVD data.

Key Contributions

1. A correspondence between zero-mode singular states and mid-gap energy states. The authors argue that in the thermodynamic limit, a vanishing smallest singular value of H − E_b·I implies the existence of a state with energy E_b, and conversely; the number of zero-mode singular values equals the number of stable mid-gap states.

2. A real-space topological invariant built from the SVD. By constructing an enlarged chiral-symmetric Hamiltonian H̃(E_b) = [[0, H − E_b·I], [H† − E_b*·I, 0]], they define V(E_b) = (1/2πi) Tr ln(P^A P^B†), with P^S = U_S† 𝒫 U_S derived from the SVD H − E_b·I = U_A 𝒮 U_B† and 𝒫 encoding real-space positions.

3. A bulk-boundary correspondence valid without translation symmetry. The invariant |V(E_b)|, computed purely from bulk SVD data, is shown to count the topologically protected edge states at energy E_b — with |2V(E_b)| zero-mode edge states of H̃(E_b) corresponding to |V(E_b)| zero-mode singular values of H − E_b·I and |V(E_b)| protected edge states of H.

4. A redefinition of the non-Hermitian skin effect. Zero-mode edge states appearing in the singular-value spectrum of H − E_b·I are identified as skin states, so the NHSE is recast as a property of the stable singular spectrum rather than the fragile eigenvalue spectrum.

Main Findings

• Spectral instability is severe and grows with size. For 2L = 100, 200, 300, and 400 sites (t = 1, γ = 0.2, d_min = 0.05, ε = 10⁻²), the eigenvalue clouds in the complex plane become progressively less band-like, and the ε-pseudospectral regions extend far beyond the discrete eigenvalues and grow as the system size increases (described in the text as growing from L = 50 to L = 200). This makes eigenvalue-based bulk-boundary correspondence practically unobservable in finite amorphous systems.

• Standard diagnostics capture the NHSE qualitatively. With t = 1 and 2L = 100 sites, increasing γ from 0 to 10 makes the mean center of mass (mcom) drop sharply, while the weighted inverse participation ratio (WIPR) peaks at small γ and then decreases as disorder-induced Anderson localization competes with the NHSE. The max over E_b of |𝒲(E_b)| rises from zero to near unity at intermediate γ and decays at strong non-Hermiticity.

• The strongest skin effect sits mid-band. At fixed γ = 0.2 with 2L = 100 sites, the WIPR peaks in the middle of the spectrum, complementary to the mcom behavior.

• The SVD invariant is sharply quantized. At E_b = 0, V(0) cleanly separates the topological phase (V(0) = ±1) from the trivial phase (V(0) = 0) for 2L = 100, 140, and 180 sites, averaged over 1000 disorder realizations, at γ = 0.2. The insets at t = −1.5 confirm the zero-mode singular vector is localized at the chain ends, and show how the smallest singular value behaves with L.

• The edge state is invisible in eigenvalues but visible in singular values. The authors emphasize that this edge state cannot be resolved in the eigenvalue spectrum yet is unambiguously captured by the SVD approach.

• Phase diagram resolved efficiently. A 200 × 200 grid over t, γ ∈ [−2, 2] for 2L = 100 sites and d_min = 0.05, averaged over only 100 disorder realizations per point, sharply resolves the boundary between V(0) = ±1 and V(0) = 0.

• Topological region coincides with vanishing singular values. For t = 1, γ = 0.2, 2L = 140, d_min = 0.05 (100 disorder realizations), V(E_b) is nonzero in a bounded region around E_b = 0, and the region where the smallest singular value of H − E_b·I vanishes coincides with that topologically nontrivial region — the singular spectrum thus faithfully signals topological edge states and stable skin states.

• No experimental data are reported. The paper is entirely theoretical/numerical; the experimental section is a proposal only.

Methodology in Plain English

The authors model a one-dimensional chain with two sublattices (a and b) and 2L sites whose positions are drawn at random from the interval [0, L′] with L′ = 2L, subject to a minimum spacing d_min so sites do not overlap. Hopping between sites decays exponentially with distance, t_ij = t·e^(−S_ij/2) with S_ij = |x_{a_i} − x_{b_j}|, and an on-site nonreciprocal term γ makes the system non-Hermitian. Because the sites are randomly placed, no translation symmetry exists.

They first try conventional diagnostics — IPR (how localized states are), mcom (where states sit spatially), WIPR (localization weighted by distance from the chain center), and a real-space spectral winding number 𝒲(E_b) computed from the polar decomposition of H − E_b·I — all averaged over disorder realizations.

They then show, using the ε-pseudospectrum (the set of z where the smallest singular value of zI − H is below ε), that the eigenvalue spectrum is unstable. The fix: work with the singular values of H − E_b·I instead, because those are square roots of eigenvalues of the Hermitian operator (H − E_b·I)†(H − E_b·I) and therefore inherit Hermitian stability. Counting the near-zero singular values tells you how many protected edge states exist at energy E_b, and the invariant V(E_b), assembled from the SVD matrices and a diagonal matrix of phase factors e^(−i2πj/2L), tells you whether they are genuinely topological.

Why This Matters

Impact on research. The work extends point-gap topology and the non-Hermitian skin effect to systems with no translation symmetry and no reliable spectrum — amorphous and strongly disordered non-Hermitian matter. It supplies a diagnostic that is sharply quantized in finite-size systems and requires only modest disorder averaging (100 realizations per point in the phase diagram), and it reframes the NHSE in terms of singular values rather than eigenvalues.

Real-world applications (as motivated in the paper):

• Sensors and sensing devices exploiting non-Hermitian degeneracies and amplified response.
• Lasers based on non-Hermitian topological design.
• Unidirectional edge transport immune to backscattering.
• Directional amplification of signals.

Candidate platforms. The authors propose waveguide arrays, photonic quasicrystal platforms, and cold atoms arranged with optical tweezers as systems where random site positions with exponential hopping could be engineered. Detection could use a generalized Loschmidt echo ℒ(τ) = ⟨Ψ|Ψ(τ)⟩/√⟨Ψ(τ)|Ψ(τ)⟩, initialized in a zero-mode singular state: it stays near unity for a topological edge state but decays rapidly for bulk states.

Industry relevance. Photonic integrated circuits, topological lasers and amplifiers, and disorder-tolerant device design are the natural industrial homes for this kind of robust topological diagnostic; the paper itself reports no device fabrication or measurement.

Future Directions

• Experimental verification. The Loschmidt-echo protocol and the three proposed platforms (waveguide arrays, photonic quasicrystals, tweezer-arranged cold atoms) remain untested for this specific amorphous non-Hermitian setting.
• Extension beyond one dimension and beyond the zero-mode/mid-gap case. The framework as presented is one-dimensional, and the bulk-boundary correspondence established is tied to zero-mode edge states at a chosen reference energy E_b.
• Connecting the invariant to measurable bulk quantities. The SVD invariant V(E_b) is a real-space construction; how it maps onto observable response or transport signatures in a given platform is not reported.
• Quantifying the limits of the correspondence. The zero-mode-singular-value to mid-gap-state correspondence is stated for the thermodynamic limit; how far finite-size systems can deviate, and how the required number of disorder realizations scales, are not quantified beyond the cases shown.

Target Audience

Graduate students and researchers in condensed-matter and mathematical physics working on non-Hermitian topology, disordered and amorphous topological phases, and photonic or cold-atom simulation platforms. It is most useful to readers already comfortable with SVD, pseudospectra, and winding-number invariants; readers new to non-Hermitian physics will need background reading first.

Authors’ abstract

Recent studies have revealed that not only does the correspondence between spectral winding numbers and skin modes break down in non-Hermitian systems, but the energy spectrum itself is highly sensitive to generic perturbations, system size, and boundary conditions. In amorphous non-Hermitian systems, where the positions of lattice sites are uncertain, the spectral instability becomes even more severe, making it difficult to identify stable topological edge states from the eigenvalue spectrum alone. To overcome this challenge, we introduce a correspondence between stable zero-mode singular states and mid-gap states of the energy spectrum in the thermodynamic limit. Because the singular value spectrum is highly robust against small perturbations and variation in size, topological edge states can be reliably probed via singular values even in finite-sized systems. Based on the singular-value decomposition of the Hamiltonian, we construct a topological invariant in real space to characterize the associated topologically protected edge states. Our approach provides a general strategy for exploring point-gap topology in real space and redefine the non-Hermitian skin effect from a new perspective.