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Lyapunov formulation of band theory for disordered non-Hermitian systems

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Pith's one-line read For any finite-range disordered non-Hermitian chain, spectra and localization are fixed exactly by Lyapunov exponents.

desk verdict Solid numerics and an elegant PBC derivation, but the OBC relation (5) rests on an unproven identification in Appendix S1; send to review and insist the gap be closed. read the letter →

arxiv 2507.09447 v1 pith:T7EVF7RL submitted 2025-07-13 cond-mat.dis-nn cond-mat.mes-hallcond-mat.stat-mechquant-ph

Konghao Sun , Haiping Hu This is my paper · ORCID
classification cond-mat.dis-nncond-mat.mes-hallcond-mat.stat-mechquant-ph
keywords non-HermitianbandtheorydisorderedsystemsLyapunovexponentskineffectAndersonlocalizationmobilityedgewindingnumbertransfermatrix
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper claims that band theory for non-Hermitian systems can be formulated in real space without translational symmetry. For any one-dimensional disordered lattice with hopping range $M$, the thermodynamic-limit spectrum under open or periodic boundary conditions is determined exactly by the $2M$ Lyapunov exponents of a supercell transfer matrix: the electrostatic potential of the spectrum is the sum of the $M$ largest (open boundaries) or the positive (periodic boundaries) exponents, plus a hopping-induced constant, and the spectral density is the Laplacian of that sum. The same exponents yield a mobility edge through the zero of the essential exponent and a winding number $\nu = M - n_P$ that separates skin modes from Anderson-localized modes. If the claims hold, one can compute spectra, localization, and the skin–Anderson transition of disordered non-Hermitian chains by multiplying small transfer matrices, with no need for exact diagonalization.

What carries the argument

The carrying object is the supercell transfer matrix: for a lattice with hopping range $M$, the chain is divided into blocks of $M$ sites, and the eigenvalue equation is recast as a $2M \times 2M$ matrix $T_j(E)$ that advances the wavefunction by one supercell. Oseledec's multiplicative ergodic theorem guarantees that the product of these matrices along the chain has well-defined Lyapunov exponents $\gamma_1 \le \cdots \le \gamma_{2M}$ in the thermodynamic limit. A duality relation for block-tridiagonal Hamiltonians with corner blocks, $\det[T(E) - z] = (-z)^M \det[E - H(z)]/\det[B_1 \cdots B_N]$, converts the characteristic polynomial of the disordered Hamiltonian into sums of Lyapunov exponents, and the electrostatic analogy (each eigenvalue as a unit charge, spectral density from Poisson's equation) turns those sums into the boundary-condition-dependent spectral densities. The essential exponent, $\gamma_{\mathrm{ess}} = \gamma_M$ or $\gamma_{M+1}$ whichever has smaller magnitude, carries the transition criterion $\gamma_{\mathrm{ess}}(E) = 0$.

What would settle it

Numerically evaluate $(1/L)\ln|\det T^{\mathrm{OBC}}_{(11)}(E)|$ at a fixed energy $E$ for a disorder ensemble in which the farthest hopping $t_M$ is allowed to be very small or zero, and compare it with the sum of the $M$ largest Lyapunov exponents $\sum_{s=M+1}^{2M} \gamma_s(E)$ computed from the same transfer-matrix products; any persistent discrepancy at some $E$ and disorder strength would break Eq. (5) and, with it, the essential-exponent mobility-edge criterion.

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Extended reading notes

Core claim

The paper establishes that under open boundary conditions the thermodynamic-limit electrostatic potential is $\phi_{\mathrm{OBC}}(E) = \sum_{s=M+1}^{2M} \gamma_s(E) + \mathbb{E}[\ln|t_M|]$, and under periodic boundary conditions it is $\phi_{\mathrm{PBC}}(E) = \sum_{\gamma_s(E)>0} \gamma_s(E) + \mathbb{E}[\ln|t_M|]$, where $\gamma_s$ are the Lyapunov exponents of the full supercell transfer matrix and $\mathbb{E}[\cdot]$ is the disorder average. Poisson's equation then gives the spectral densities $\rho_{\mathrm{OBC}}(E) = (1/2\pi)\sum_{s>M} \nabla^2 \gamma_s(E)$ and $\rho_{\mathrm{PBC}}(E) = (1/2\pi)\sum_{\gamma_s>0} \nabla^2 \gamma_s(E)$. These non-Hermitian Thouless relations are stated to hold for arbitrary finite-range couplings and arbitrary disorder, and they reduce to non-Bloch band theory in the clean limit. The paper further proves the winding-number identity $\nu(E) = M - n_P(E)$ for the twisted-boundary Hamiltonian, so skin modes (nonzero winding) and Anderson-localized modes (zero winding) are distinguished topologically, and it introduces the essential Lyapunov exponent $\gamma_{\mathrm{ess}}(E)$, whose vanishing defines the mobility edge between the two phases, with unidirectional critical states on that edge.

Load-bearing premise

The open-boundary relation assumes that, for arbitrary disorder, the growth rate of the corner-block determinant of the open-boundary transfer matrix equals the sum of the $M$ largest Lyapunov exponents; the paper argues this by continuity from the clean limit rather than by a proof for each disorder ensemble.

Editorial extensions

If this is right

  • The thermodynamic-limit spectra of disordered non-Hermitian chains, under both open and periodic boundary conditions, are obtained from $2M \times 2M$ transfer-matrix products rather than from diagonalizing $L \times L$ Hamiltonians, which removes the slow algebraic convergence of exact diagonalization.
  • Spectral sensitivity to boundary conditions persists under disorder up to the Anderson transition and is controlled by whether the central exponents $\gamma_M$ and $\gamma_{M+1}$ lie on opposite sides of zero (Anderson-localized modes) or on the same side (skin modes).
  • A single topological number, the winding number $\nu(E) = M - n_P(E)$, distinguishes skin modes from Anderson-localized modes in disordered systems, extending the point-gap picture beyond translational symmetry.
  • At the skin–Anderson transition the essential exponent vanishes, and the critical states are unidirectional: localized in one direction and delocalized in the other, unlike the multifractal critical states of Hermitian Anderson transitions.
  • The fraction $\alpha$ of Anderson-localized modes grows with disorder strength and saturates at a critical $W_c$, giving a concrete order parameter for the skin–Anderson transition.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The paper does not state it, but the same Lyapunov expansion should control the ensemble-averaged retarded Green's function of the disordered chain, so wave-packet spreading or inverse participation ratios near the mobility edge could be predicted directly from $\gamma_s(E)$.
  • The paper restricts to strictly one-dimensional chains, but the proof structure suggests the same formulas should hold for a strip of finite width with internal degrees of freedom; testing that extension would check whether the mechanism survives transverse degrees of freedom.
  • An unstated experimental consequence: in a Hatano–Nelson chain with onsite disorder, the open-boundary density of states equals the disorder distribution itself, so measuring the spectrum of a lossy disordered wire is a direct measurement of the disorder statistics.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper develops a real-space Lyapunov formulation of band theory for one-dimensional disordered non-Hermitian systems. It proposes non-Hermitian Thouless relations: under OBC the electrostatic potential is the sum of the M largest Lyapunov exponents plus the ensemble average of ln|t_M|, while under PBC it is the sum of positive Lyapunov exponents plus the same constant (Eqs. 5-6). Spectral densities are then given by sums of Laplacians of the corresponding Lyapunov exponents (Eqs. 7-8). The paper further introduces an essential Lyapunov exponent whose zero level set defines mobility edges separating skin modes from Anderson-localized modes, identifies unidirectional critical states at the transition, and proves a winding-number criterion relating skin modes to nonzero winding numbers via the number of positive Lyapunov exponents (Eqs. 11-13). The results are checked numerically for several models, including onsite, off-diagonal, and quasi-periodic disorder, with finite-size scaling shown for the OBC potential.

Significance. If the central relations (5)-(8) hold, they provide a unified and exact description of spectra and localization in disordered non-Hermitian chains, going substantially beyond the clean-limit non-Bloch band theory and previous special-case results. The PBC derivation (Appendix S2) via the duality relation is clean and rigorous, and the winding-number proof (Appendix S5) is elegant and directly connects the number of positive Lyapunov exponents to boundary-condition sensitivity. The numerical verification across multiple models, including finite-size scaling of the OBC potential, is a definite strength of the paper. However, the OBC relation (5) rests on the unproven identification (S14), which is load-bearing for the spectral-density formula (7) and for the mobility-edge criterion (10). Because this identification is not established for generic disordered transfer matrices, the central claim is not yet fully demonstrated.

major comments (3)
  1. [Appendix S1, Eq. (S14)] The proof that (1/L) ln|det T^OBC_(11)(E)| converges to the sum of the M largest Lyapunov exponents is incomplete. The similarity-transformation argument shows only that both sides shift by the same amount under H -> SHS^{-1}; it cannot select which M of the 2M exponents appear, since any sum of M Lyapunov exponents shifts by the same constant. The continuity argument from the clean limit is also not sufficient: at clean energies on the OBC spectrum the two central Lyapunov exponents coincide, so the "M largest" set is ambiguous, and Eq. (S27) is derived for E outside the non-Bloch spectrum and then continued onto the spectrum without controlling the singular contour integral. Without a rigorous proof of (S14), Eq. (5) and the mobility-edge criterion (10) are not established.
  2. [Methods, Eq. (22) and Appendix S1] The transfer matrices T_j(E) in Eq. (22) are not generically positive or irreducible, and Oseledec's theorem does not guarantee that a fixed coordinate minor of the product T(E) grows at the rate of the sum of the top M exterior powers. For generic random matrices, individual minors can have a strictly smaller growth rate. The unidirectional hopping case cited in the main text has special triangular structure and cannot justify the general claim in (S14).
  3. [Eqs. (5) and (6)] The term E[ln|t_M|] is not well-defined for "arbitrary disorder" as stated: if t_M=0 occurs with positive probability, or if the distribution has heavy tails, the expectation of ln|t_M| may be undefined or infinite. The universality claim should be restricted to disorder distributions for which E[ln|t_M|] exists and is finite, e.g., compactly supported distributions bounded away from zero for the maximal hopping amplitude.
minor comments (4)
  1. [Fig. 5 caption] The caption refers to panel "(c)" for the fraction α versus W, but the figure contains only panels (a) and (b); the text in the main body also appears to reference "(c)" incorrectly.
  2. [Discussion] There is a typo in the sentence "The sensitivity to boundary conditions persists until the the transition is complete," where "the" is repeated.
  3. [Methods, Electrostatic analogy] The phrase "each eigenenegy" in the first paragraph should read "each eigenenergy."
  4. [Appendix S3, Eq. (S24)] The notation "|β1(E)| ≤ |β2(E)| ≤ · · · ≤β2M (E)" is missing a closing bar on the last term; please correct for consistency.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the non-Hermitian Thouless relations are derived from transfer-matrix Lyapunov exponents and tested against exact diagonalization; the principal caveat is an unproved (not circular) identification in Appendix S1.

full rationale

The paper's central claims (Eqs. 5-8) are not equivalent to their inputs by construction. The electrostatic potential is expressed via det[H-E], the OBC determinant is reduced to det[T^OBC_(11)] through the transfer-matrix boundary construction (S1), the PBC potential is obtained from the independent Molinari duality relation (S18), and the Lyapunov exponents enter through Oseledec's theorem applied to the 2M x 2M supercell transfer matrix. No parameter is fitted to the spectra being predicted; the numerical spectral densities of Figs. 3-5 and Appendix S4 are independent checks. The genuinely load-bearing step, Eq. (S14), asserts (1/L) ln|det T^OBC_(11)| = sum_{s=M+1}^{2M} gamma_s; the paper supports this only by showing both sides transform by the same shift under a gauge similarity and then saying 'In general case, the LEs vary continuously with disorder strength W. We thus conclude...'. For generic non-positive random transfer matrices, a fixed coordinate minor need not grow at the sum of the top M Lyapunov exponents, so this is a real proof gap and a correctness risk, but it is not a circular reduction: the asserted identity is not already contained in the definition of gamma_s or in the OBC determinant identity. The clean-limit anchor (S27) cites a co-author's Ref. [55], but Appendix S3 re-derives the clean potential by contour integration and uses it only as a consistency check to select which M exponents appear; it is not a fitted input or a uniqueness import that forces the disordered result. The self-citations elsewhere are contextual rather than load-bearing. Hence no Eq. X is equal to Eq. Y by construction, and the circularity score is 0.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central formulas are derived from transfer matrices, Oseledec's theorem, and a cited duality relation; no parameters are fitted to data. The main unproven load-bearing element is the identification in Appendix S1 of the corner-block determinant with the sum of the M largest Lyapunov exponents. Ergodicity of the disorder is also implicit.

assumptions (6)
  • standard math Oseledec multiplicative ergodic theorem applies to the supercell transfer matrices for the disorder ensemble, yielding well-defined Lyapunov exponents.
    Invoked after Eq. (4) in the main text; requires stationary ergodic disorder, which is not stated explicitly.
  • domain assumption The hopping parameter t_M, and hence each B_j, is nonzero and invertible so the transfer matrix T_j(E) is well-defined.
    Transfer matrix defined via B_j^{-1} in Eq. (22); the models choose fixed nonzero t_M.
  • ad hoc to paper The log-determinant of the upper-left M x M block of the OBC transfer matrix grows as the sum of the M largest Lyapunov exponents for arbitrary disorder.
    Appendix S1, Eq. (S14) and the preceding paragraph; justified only by continuity from the clean limit, not proven.
  • standard math Molinari's duality relation det[T(E)-z] = (-z)^M det[E-H(z)]/det[B1...BN] holds for random block-tridiagonal Hamiltonians.
    Used in Appendix S2, Eq. (S18), cited from Ref. [70].
  • standard math The spectral density exists as a distribution in the thermodynamic limit and is related to the electrostatic potential by Poisson's equation.
    Methods, electrostatic analogy; standard potential theory.
  • standard math In the clean limit, the present formulation reduces to non-Bloch band theory and the generalized Brillouin zone condition.
    Appendix S3, Eqs. (S24)-(S27); uses the authors' earlier result [55] as a consistency check.

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Pith. "Pith review of Lyapunov formulation of band theory for disordered non-Hermitian systems." pith.science (2026). https://pith.science/paper/T7EVF7RL

@misc{pith2026250709447,
  author       = {Pith},
  title        = {Pith review of: Lyapunov formulation of band theory for disordered non-Hermitian systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/T7EVF7RL}},
  note         = {Machine review of arXiv:2507.09447}
}
read the original abstract

Non-Bloch band theory serves as a cornerstone for understanding intriguing non-Hermitian phenomena, such as the skin effect and extreme spectral sensitivity to boundary conditions. Yet this theory hinges on translational symmetry and thus breaks down in disordered systems. Here, we develop a real-space Lyapunov formulation of band theory that governs the spectra and eigenstates of disordered non-Hermitian systems. This framework yields universal non-Hermitian Thouless relations linking spectral density and localization to Lyapunov exponents under different boundary conditions. We further identify an exact topological criterion: skin modes and Anderson-localized modes correspond to nonzero and zero winding numbers, respectively, revealing the topological nature of the skin-Anderson transition. This transition is dictated by an essential Lyapunov exponent and gives rise to novel unidirectional critical states. Our formulation provides a unified and exact description of spectra and localization in generic one-dimensional non-Hermitian systems without translational symmetry, offering new insights into the interplay among non-Hermiticity, disorder, and topology.

Figures

Figures reproduced from arXiv: 2507.09447 by the authors.

Figure 1
Figure 1. FIG. 1. Schematics of our real-space formulation for disor [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Configurations of Lyapunov exponents (LEs) and [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 4
Figure 4. FIG. 4. Spectral density under periodic boundary condition [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
Figures from the paper (1 more)
Figure 5
Figure 5. Figure 5: FIG. 5. Skin-Anderson transition tracked by mobility edges. [PITH_FULL_IMAGE:figures/full_fig_p005_5.png]

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Reviewed August 6, 2026 · model on record in the stance chip above.