Non-Hermitian Thouless Relations: Extensions & Insights

• Non-Hermitian Thouless relations generalize the classical Thouless formula by relating Lyapunov exponents to complex spectral densities using two-dimensional log-potentials.
• They reveal how localization lengths, spectral winding, and transport observables emerge in disordered and non-reciprocal systems with complex energy spectra.
• The framework unifies approaches from the Zakharov–Shabat problem to non-Hermitian Chern insulators, inspiring deeper insights into topological and nonlinear transport phenomena.

Searching arXiv for recent and foundational papers on non-Hermitian Thouless relations. Non-Hermitian Thouless relations are extensions of the classical Thouless formula from Hermitian spectral theory to settings where spectra, Lyapunov exponents, and topological responses are intrinsically complex-valued. In the Hermitian case, the Thouless formula relates the Lyapunov exponent to the density of states on the real energy axis. In the non-Hermitian case, the corresponding relations operate on the complex plane, typically replacing one-dimensional logarithmic potentials by two-dimensional log-potentials and replacing ordinary second derivatives by the two-dimensional Laplacian. Across random operators, quasiperiodic lattices, disordered non-reciprocal chains, non-Hermitian Chern insulators, and nonlinear pumping problems, these relations connect spectral density, localization length, spectral winding, and transport observables through a common potential-theoretic structure (0706.3129, Longhi, 2019, Sun et al., 13 Jul 2025, Fortin et al., 6 Sep 2025, Chen et al., 2018, Zheng et al., 2 Feb 2026).

1. Conceptual definition and mathematical structure

The central non-Hermitian Thouless relation expresses the average density of states on the complex plane in terms of a Lyapunov exponent or a related logarithmic potential. In the formulation used for the non-Hermitian Zakharov–Shabat problem, with complex spectral parameter $z=\xi+i\eta$, the average density of states is

$$\rho(\xi,\eta)=\frac{1}{2\pi}\left(\frac{\partial^2}{\partial \xi^2}+\frac{\partial^2}{\partial \eta^2}\right)\kappa(\xi,\eta),$$

or equivalently

$$\rho(z,z^*)=\frac{2}{\pi}\,\bar{\partial}\,\partial\,\kappa(z,z^*),$$

where $\kappa$ is the Lyapunov exponent and $\Delta=\partial_\xi^2+\partial_\eta^2$ is the two-dimensional Laplacian (0706.3129).

The same log-potential structure appears in disordered non-Hermitian chains. For open-boundary spectra in the disordered Hatano–Nelson setting, one obtains

$$\mathrm{Re}\,L(E)=\mathbb{E}[\ln|J_{R,n}|]-\int d^2z\,\rho_{\mathrm{obc}}(z)\ln|z-E|,$$

together with

$$\Delta\,\mathrm{Re}\,L(E)=-2\pi\,\rho_{\mathrm{obc}}(E),$$

so that the real part of the Lyapunov exponent is a superharmonic function whose curvature is set by the spectral density (Fortin et al., 6 Sep 2025). In the broader Lyapunov formulation for finite-range one-dimensional systems, the open- and periodic-boundary spectral densities are generated by different sums of Lyapunov exponents,

$$\rho_{\mathrm{OBC}}(z)=\frac{1}{2\pi}\Delta\!\left[\sum_{s=M+1}^{2M}\gamma_s(z)\right],\qquad \rho_{\mathrm{PBC}}(z)=\frac{1}{2\pi}\Delta\!\left[\sum_{\gamma_s(z)>0}\gamma_s(z)\right],$$

which makes boundary sensitivity part of the relation itself (Sun et al., 13 Jul 2025).

This two-dimensional formulation is the essential distinction from the Hermitian case. In one-dimensional Hermitian spectral theory, the Thouless formula involves a real energy variable and the integrated density of states. In non-Hermitian problems, the spectrum generally occupies curves or regions in $\mathbb{C}$, the Lyapunov exponent becomes a subharmonic or superharmonic function on $\mathbb{C}$, and spectral density is recovered by a Laplacian rather than a one-dimensional derivative (0706.3129, Longhi, 2019).

2. Non-Hermitian random operators and the Zakharov–Shabat problem

A concrete realization appears in the non-Hermitian Zakharov–Shabat eigenvalue problem associated with the focusing nonlinear Schrödinger equation,

$$\rho(\xi,\eta)=\frac{1}{2\pi}\left(\frac{\partial^2}{\partial \xi^2}+\frac{\partial^2}{\partial \eta^2}\right)\kappa(\xi,\eta),$$0

The inverse scattering transform maps the initial pulse $$\rho(\xi,\eta)=\frac{1}{2\pi}\left(\frac{\partial^2}{\partial \xi^2}+\frac{\partial^2}{\partial \eta^2}\right)\kappa(\xi,\eta),$$1 to scattering data through the non-Hermitian $$\rho(\xi,\eta)=\frac{1}{2\pi}\left(\frac{\partial^2}{\partial \xi^2}+\frac{\partial^2}{\partial \eta^2}\right)\kappa(\xi,\eta),$$2 operator

$$\rho(\xi,\eta)=\frac{1}{2\pi}\left(\frac{\partial^2}{\partial \xi^2}+\frac{\partial^2}{\partial \eta^2}\right)\kappa(\xi,\eta),$$3

with $$\rho(\xi,\eta)=\frac{1}{2\pi}\left(\frac{\partial^2}{\partial \xi^2}+\frac{\partial^2}{\partial \eta^2}\right)\kappa(\xi,\eta),$$4 (0706.3129). The sign asymmetry in the off-diagonal entries renders the operator non-Hermitian and spreads the spectrum over the complex plane rather than confining it to the real axis.

For white-noise Gaussian input pulses, the transfer matrix has unit determinant, and the top Lyapunov exponent can be defined either from the transfer-matrix eigenvalue growth or from wavefunction growth: $$\rho(\xi,\eta)=\frac{1}{2\pi}\left(\frac{\partial^2}{\partial \xi^2}+\frac{\partial^2}{\partial \eta^2}\right)\kappa(\xi,\eta),$$5 A Riccati reduction with $$\rho(\xi,\eta)=\frac{1}{2\pi}\left(\frac{\partial^2}{\partial \xi^2}+\frac{\partial^2}{\partial \eta^2}\right)\kappa(\xi,\eta),$$6 and $$\rho(\xi,\eta)=\frac{1}{2\pi}\left(\frac{\partial^2}{\partial \xi^2}+\frac{\partial^2}{\partial \eta^2}\right)\kappa(\xi,\eta),$$7 gives

$$\rho(\xi,\eta)=\frac{1}{2\pi}\left(\frac{\partial^2}{\partial \xi^2}+\frac{\partial^2}{\partial \eta^2}\right)\kappa(\xi,\eta),$$8

so the Lyapunov exponent is obtained by averaging $$\rho(\xi,\eta)=\frac{1}{2\pi}\left(\frac{\partial^2}{\partial \xi^2}+\frac{\partial^2}{\partial \eta^2}\right)\kappa(\xi,\eta),$$9 over the stationary distribution of the corresponding Markov process (0706.3129).

For a circularly symmetric complex Gaussian pulse, the stationary distribution is

$$\rho(z,z^*)=\frac{2}{\pi}\,\bar{\partial}\,\partial\,\kappa(z,z^*),$$0

which yields the closed-form Lyapunov exponent

$$\rho(z,z^*)=\frac{2}{\pi}\,\bar{\partial}\,\partial\,\kappa(z,z^*),$$1

Applying the non-Hermitian Thouless relation then gives the average density of states

$$\rho(z,z^*)=\frac{2}{\pi}\,\bar{\partial}\,\partial\,\kappa(z,z^*),$$2

This density is independent of $$\rho(z,z^*)=\frac{2}{\pi}\,\bar{\partial}\,\partial\,\kappa(z,z^*),$$3, finite at $$\rho(z,z^*)=\frac{2}{\pi}\,\bar{\partial}\,\partial\,\kappa(z,z^*),$$4, and decays for large $$\rho(z,z^*)=\frac{2}{\pi}\,\bar{\partial}\,\partial\,\kappa(z,z^*),$$5; the spectrum occupies the entire complex plane, but the density depends only on the imaginary part and is maximal near the real axis (0706.3129).

For real Gaussian pulses, the structure is more intricate because the Fokker–Planck equation does not decouple in the phase variable. The large-$$\rho(z,z^*)=\frac{2}{\pi}\,\bar{\partial}\,\partial\,\kappa(z,z^*),$$6 regime yields a correction of order $$\rho(z,z^*)=\frac{2}{\pi}\,\bar{\partial}\,\partial\,\kappa(z,z^*),$$7 to the complex-pulse result, while the exact $$\rho(z,z^*)=\frac{2}{\pi}\,\bar{\partial}\,\partial\,\kappa(z,z^*),$$8 solution gives

$$\rho(z,z^*)=\frac{2}{\pi}\,\bar{\partial}\,\partial\,\kappa(z,z^*),$$9

with $\kappa$0 and $\kappa$1 modified Bessel functions, showing that the real-input ensemble has qualitatively different localization behavior near the singular points of the stationary distribution (0706.3129).

The derivation of the non-Hermitian Thouless formula in this setting uses Green functions, two independent solutions with opposite boundary conditions, and the Feinberg–Zee hermitization framework. The resulting relation

$\kappa$2

is therefore not merely heuristic but built from resolvent identities, self-averaging, and the large-length limit (0706.3129).

3. Localization length, spectral support, and quasiperiodic lattices

The localization length is the inverse Lyapunov exponent,

$\kappa$3

For the complex Gaussian Zakharov–Shabat ensemble,

$\kappa$4

Near the real axis, $\kappa$5, so it diverges as $\kappa$6, while for large $\kappa$7 one has $\kappa$8 and therefore $\kappa$9 (0706.3129). This connects spectral position in the complex plane directly to decay length.

An analytically rigorous lattice realization appears in the $\Delta=\partial_\xi^2+\partial_\eta^2$0-symmetric non-Hermitian Aubry–André–Harper model

$\Delta=\partial_\xi^2+\partial_\eta^2$1

with irrational $\Delta=\partial_\xi^2+\partial_\eta^2$2 (Longhi, 2019). The model has reciprocal symmetric hopping and therefore no non-Hermitian skin effect. In the $\Delta=\partial_\xi^2+\partial_\eta^2$3-unbroken phase $\Delta=\partial_\xi^2+\partial_\eta^2$4, the spectrum is real and fills $\Delta=\partial_\xi^2+\partial_\eta^2$5 with density of states

$\Delta=\partial_\xi^2+\partial_\eta^2$6

In the $\Delta=\partial_\xi^2+\partial_\eta^2$7-broken insulating phase $\Delta=\partial_\xi^2+\partial_\eta^2$8, the spectrum becomes complex and lies on the ellipse

$\Delta=\partial_\xi^2+\partial_\eta^2$9

The non-Hermitian Thouless relation for homogeneous hopping reads

$$\mathrm{Re}\,L(E)=\mathbb{E}[\ln|J_{R,n}|]-\int d^2z\,\rho_{\mathrm{obc}}(z)\ln|z-E|,$$0

and the authors prove that, on the ellipse,

$$\mathrm{Re}\,L(E)=\mathbb{E}[\ln|J_{R,n}|]-\int d^2z\,\rho_{\mathrm{obc}}(z)\ln|z-E|,$$1

independent of energy (Longhi, 2019).

This energy independence is the direct analog of the Hermitian Aubry–André–Harper result, but now with the spectral measure supported on a complex elliptic curve. The derivation relies on a characteristic-polynomial identity and the thermodynamic equivalence between the real-space and momentum-space problems. A key condition is the absence of the skin effect, encoded by reciprocal magnitudes of left and right hopping; this is repeatedly important in non-Hermitian Thouless theory because non-reciprocity modifies boundary sensitivity and can invalidate the simplest integral form (Longhi, 2019).

4. Boundary conditions, spectral winding, and disordered non-reciprocal chains

In non-reciprocal systems such as the Hatano–Nelson chain,

$$\mathrm{Re}\,L(E)=\mathbb{E}[\ln|J_{R,n}|]-\int d^2z\,\rho_{\mathrm{obc}}(z)\ln|z-E|,$$2

the non-Hermitian skin effect makes the distinction between open and periodic boundary conditions intrinsic rather than technical (Fortin et al., 6 Sep 2025). In this context, non-Hermitian Thouless relations govern not only localization lengths but also spectral winding and topological amplification.

For disordered nearest-neighbor chains with general site-dependent couplings, the single-particle equation admits a transfer-matrix formulation. Comparing determinant identities with Green-function asymptotics gives

$$\mathrm{Re}\,L(E)=\mathbb{E}[\ln|J_{R,n}|]-\int d^2z\,\rho_{\mathrm{obc}}(z)\ln|z-E|,$$3

which leads to

$$\mathrm{Re}\,L(E)=\mathbb{E}[\ln|J_{R,n}|]-\int d^2z\,\rho_{\mathrm{obc}}(z)\ln|z-E|,$$4

The corresponding Poisson equation is

$$\mathrm{Re}\,L(E)=\mathbb{E}[\ln|J_{R,n}|]-\int d^2z\,\rho_{\mathrm{obc}}(z)\ln|z-E|,$$5

so the real part of the Lyapunov exponent is again a logarithmic potential, now with a sign convention adapted to the transfer-matrix normalization (Fortin et al., 6 Sep 2025).

A central result is a bulk–boundary correspondence formulated in terms of the sign of $$\mathrm{Re}\,L(E)=\mathbb{E}[\ln|J_{R,n}|]-\int d^2z\,\rho_{\mathrm{obc}}(z)\ln|z-E|,$$6. For a chain with rightward non-reciprocal bias,

$$\mathrm{Re}\,L(E)=\mathbb{E}[\ln|J_{R,n}|]-\int d^2z\,\rho_{\mathrm{obc}}(z)\ln|z-E|,$$7

where $$\mathrm{Re}\,L(E)=\mathbb{E}[\ln|J_{R,n}|]-\int d^2z\,\rho_{\mathrm{obc}}(z)\ln|z-E|,$$8 is the spectral winding number defined by flux insertion under periodic boundary conditions (Fortin et al., 6 Sep 2025). Thus, positive $$\mathrm{Re}\,L(E)=\mathbb{E}[\ln|J_{R,n}|]-\int d^2z\,\rho_{\mathrm{obc}}(z)\ln|z-E|,$$9 identifies right skin states and nonzero winding, whereas negative $$\Delta\,\mathrm{Re}\,L(E)=-2\pi\,\rho_{\mathrm{obc}}(E),$$0 identifies Anderson-localized states with zero winding.

The more general Lyapunov band theory extends this logic to arbitrary finite-range one-dimensional non-Hermitian systems. Grouping sites into supercells of range $$\Delta\,\mathrm{Re}\,L(E)=-2\pi\,\rho_{\mathrm{obc}}(E),$$1 yields $$\Delta\,\mathrm{Re}\,L(E)=-2\pi\,\rho_{\mathrm{obc}}(E),$$2 Lyapunov exponents $$\Delta\,\mathrm{Re}\,L(E)=-2\pi\,\rho_{\mathrm{obc}}(E),$$3. The open-boundary and periodic-boundary log-potentials are

$$\Delta\,\mathrm{Re}\,L(E)=-2\pi\,\rho_{\mathrm{obc}}(E),$$4

and hence

$$\Delta\,\mathrm{Re}\,L(E)=-2\pi\,\rho_{\mathrm{obc}}(E),$$5

The topological index is

$$\Delta\,\mathrm{Re}\,L(E)=-2\pi\,\rho_{\mathrm{obc}}(E),$$6

where $$\Delta\,\mathrm{Re}\,L(E)=-2\pi\,\rho_{\mathrm{obc}}(E),$$7 is the number of positive Lyapunov exponents (Sun et al., 13 Jul 2025). In this formulation, $$\Delta\,\mathrm{Re}\,L(E)=-2\pi\,\rho_{\mathrm{obc}}(E),$$8 if and only if skin modes are present, and $$\Delta\,\mathrm{Re}\,L(E)=-2\pi\,\rho_{\mathrm{obc}}(E),$$9 if and only if open- and periodic-boundary densities coincide, identifying Anderson-localized modes.

The same framework isolates the skin–Anderson transition through the essential Lyapunov exponent $$\rho_{\mathrm{OBC}}(z)=\frac{1}{2\pi}\Delta\!\left[\sum_{s=M+1}^{2M}\gamma_s(z)\right],\qquad \rho_{\mathrm{PBC}}(z)=\frac{1}{2\pi}\Delta\!\left[\sum_{\gamma_s(z)>0}\gamma_s(z)\right],$$0, defined as the central exponent closest to zero. The curves $$\rho_{\mathrm{OBC}}(z)=\frac{1}{2\pi}\Delta\!\left[\sum_{s=M+1}^{2M}\gamma_s(z)\right],\qquad \rho_{\mathrm{PBC}}(z)=\frac{1}{2\pi}\Delta\!\left[\sum_{\gamma_s(z)>0}\gamma_s(z)\right],$$1 serve as mobility edges in the complex plane. Exactly on these curves, one directional localization length diverges while the opposite one remains finite, producing unidirectional critical states (Sun et al., 13 Jul 2025). This suggests that non-Hermitian Thouless relations are not merely density formulas; they can act as exact organizing principles for phase structure in disordered non-Hermitian band theory.

5. Topological transport and TKNN-type generalizations

A distinct usage of “Thouless relations” appears in topological transport, where the classical TKNN formula is generalized to non-Hermitian settings. In a two-band non-Hermitian Chern insulator with Bloch Hamiltonian

$$\rho_{\mathrm{OBC}}(z)=\frac{1}{2\pi}\Delta\!\left[\sum_{s=M+1}^{2M}\gamma_s(z)\right],\qquad \rho_{\mathrm{PBC}}(z)=\frac{1}{2\pi}\Delta\!\left[\sum_{\gamma_s(z)>0}\gamma_s(z)\right],$$2

the Hall conductance at zero temperature decomposes into a topological contribution and a non-universal bulk contribution,

$$\rho_{\mathrm{OBC}}(z)=\frac{1}{2\pi}\Delta\!\left[\sum_{s=M+1}^{2M}\gamma_s(z)\right],\qquad \rho_{\mathrm{PBC}}(z)=\frac{1}{2\pi}\Delta\!\left[\sum_{\gamma_s(z)>0}\gamma_s(z)\right],$$3

The topological term obeys the explicit TKNN-like relation

$$\rho_{\mathrm{OBC}}(z)=\frac{1}{2\pi}\Delta\!\left[\sum_{s=M+1}^{2M}\gamma_s(z)\right],\qquad \rho_{\mathrm{PBC}}(z)=\frac{1}{2\pi}\Delta\!\left[\sum_{\gamma_s(z)>0}\gamma_s(z)\right],$$4

with

$$\rho_{\mathrm{OBC}}(z)=\frac{1}{2\pi}\Delta\!\left[\sum_{s=M+1}^{2M}\gamma_s(z)\right],\qquad \rho_{\mathrm{PBC}}(z)=\frac{1}{2\pi}\Delta\!\left[\sum_{\gamma_s(z)>0}\gamma_s(z)\right],$$5

and

$$\rho_{\mathrm{OBC}}(z)=\frac{1}{2\pi}\Delta\!\left[\sum_{s=M+1}^{2M}\gamma_s(z)\right],\qquad \rho_{\mathrm{PBC}}(z)=\frac{1}{2\pi}\Delta\!\left[\sum_{\gamma_s(z)>0}\gamma_s(z)\right],$$6

The factor $$\rho_{\mathrm{OBC}}(z)=\frac{1}{2\pi}\Delta\!\left[\sum_{s=M+1}^{2M}\gamma_s(z)\right],\qquad \rho_{\mathrm{PBC}}(z)=\frac{1}{2\pi}\Delta\!\left[\sum_{\gamma_s(z)>0}\gamma_s(z)\right],$$7 satisfies $$\rho_{\mathrm{OBC}}(z)=\frac{1}{2\pi}\Delta\!\left[\sum_{s=M+1}^{2M}\gamma_s(z)\right],\qquad \rho_{\mathrm{PBC}}(z)=\frac{1}{2\pi}\Delta\!\left[\sum_{\gamma_s(z)>0}\gamma_s(z)\right],$$8 and encodes decay-induced reduction of the edge contribution (Chen et al., 2018).

In the Hermitian limit, $$\rho_{\mathrm{OBC}}(z)=\frac{1}{2\pi}\Delta\!\left[\sum_{s=M+1}^{2M}\gamma_s(z)\right],\qquad \rho_{\mathrm{PBC}}(z)=\frac{1}{2\pi}\Delta\!\left[\sum_{\gamma_s(z)>0}\gamma_s(z)\right],$$9 and the formula reduces to the quantized TKNN result. In the non-Hermitian case, two mechanisms spoil exact quantization: broadened bulk density of states due to the imaginary parts of quasienergies, and decay of edge states due to finite lifetimes (Chen et al., 2018). The bulk density of states is built from the retarded Green function and the spectral function

$\mathbb{C}$0

so each eigenenergy contributes a Lorentzian of half-width $\mathbb{C}$1 (Chen et al., 2018).

This usage is conceptually distinct from localization-based non-Hermitian Thouless formulas, but structurally related: both replace exact quantization or one-dimensional spectral formulas by weighted or broadened expressions that explicitly encode decay, complex spectra, and non-Hermitian geometry. A plausible implication is that “non-Hermitian Thouless relations” now names a family of extensions rather than a single formula.

A further extension appears in nonlinear and nonreciprocal pumping governed not by $\mathbb{C}$2 but by the auxiliary eigenvalue problem

$\mathbb{C}$3

For the non-Hermitian nonlinear Rice–Mele model, the pumped displacement per cycle is

$\mathbb{C}$4

with

$\mathbb{C}$5

In the linear limit, $\mathbb{C}$6 and the ordinary Thouless result is recovered. With the explicit metric

$\mathbb{C}$7

the weighting factor becomes

$\mathbb{C}$8

This permits fractional pumped responses even when the linear biorthogonal Chern number remains integer (Zheng et al., 2 Feb 2026). The paper also defines a spectral winding

$\mathbb{C}$9

which remains integer while the weighted pumping response becomes fractional (Zheng et al., 2 Feb 2026).

6. Assumptions, limits, and recurring distinctions

Although the specific formulas differ, several assumptions recur across non-Hermitian Thouless relations. One is the thermodynamic or large-length limit, which enables self-averaging of Lyapunov exponents and the replacement of finite-size spectral sums by deterministic densities (0706.3129, Longhi, 2019, Sun et al., 13 Jul 2025, Fortin et al., 6 Sep 2025). Another is regularity of the transfer-matrix product, typically guaranteed by ergodicity, finite hopping range, and nonzero geometric mean of the outermost hopping amplitudes (Sun et al., 13 Jul 2025).

A second recurring issue is boundary sensitivity. In reciprocal models without the non-Hermitian skin effect, such as the $\mathbb{C}$0-symmetric Aubry–André–Harper chain with $\mathbb{C}$1, the bulk spectrum is boundary-condition independent in the thermodynamic limit, and the non-Hermitian Thouless relation closely parallels the Hermitian one, with the only essential modification being the complex support of the spectral density (Longhi, 2019). In non-reciprocal chains, by contrast, open and periodic spectra must be treated separately, and the counting rule for relevant Lyapunov exponents depends explicitly on the boundary condition (Sun et al., 13 Jul 2025, Fortin et al., 6 Sep 2025).

A third distinction concerns whether the relation addresses spectral density and localization or transport response. The Zakharov–Shabat, Aubry–André–Harper, and Hatano–Nelson results are fundamentally log-potential relations between spectrum and Lyapunov exponents (0706.3129, Longhi, 2019, Fortin et al., 6 Sep 2025, Sun et al., 13 Jul 2025). The Chern-insulator and nonlinear-pumping results instead generalize TKNN- or Thouless-pump formulas to non-Hermitian settings, where the observable response is modified by decay weights, spectral broadening, or energy-dependent metrics (Chen et al., 2018, Zheng et al., 2 Feb 2026). These are related by analogy and nomenclature rather than by a single common derivation.

Several limitations are explicitly noted. The simplest non-Hermitian Thouless integral assumes no non-Hermitian skin effect and no bulk exceptional points, so it does not directly apply to asymmetric-hopping models without modification (Longhi, 2019). The Lyapunov band-theory approach currently applies to one-dimensional finite-range systems and does not yet extend in a comparably complete way to higher dimensions or interacting many-body problems (Sun et al., 13 Jul 2025). In nonlinear pumping, adiabaticity requires an open complex quasienergy gap and the absence of exceptional points along the cycle; near exceptional points, biorthogonal Berry curvature becomes singular and adiabatic transport breaks down (Zheng et al., 2 Feb 2026).

A common misconception is that non-Hermitian extensions always preserve quantization after a simple complexification. The cited works consistently show otherwise. In Chern insulators, Hall conductance is generically not exactly quantized because of bulk broadening and edge decay (Chen et al., 2018). In nonlinear pumping, integer winding can coexist with fractional pumped displacement because the physical response is a metric-weighted curvature integral rather than the bare Chern number (Zheng et al., 2 Feb 2026). In disordered chains, spectral winding predicts skin localization only after incorporating the sign structure of Lyapunov exponents and the distinction between open and periodic spectra (Fortin et al., 6 Sep 2025, Sun et al., 13 Jul 2025).

7. Physical significance and broader research directions

Non-Hermitian Thouless relations provide a unifying language for how complex spectra organize localization, topology, and transport. In nonlinear optical fibers, the Zakharov–Shabat relation determines how random Gaussian input pulses populate the complex scattering spectrum, thereby characterizing distributions of soliton parameters and their robustness under finite window size and perturbations (0706.3129). In quasiperiodic lattices, the same log-potential logic yields an exact, energy-independent localization length in the insulating phase of a $\mathbb{C}$2-symmetric non-Hermitian model, linking complex elliptic spectra to real-space attenuation (Longhi, 2019).

In disordered non-reciprocal chains, the framework becomes a disorder-robust substitute for non-Bloch band theory: Lyapunov exponents determine open- and periodic-boundary spectral densities, topological winding, and the skin–Anderson transition (Sun et al., 13 Jul 2025). In the Hatano–Nelson setting, the sign of $\mathbb{C}$3 predicts whether a state is a skin mode, an Anderson-localized mode, or part of a topological amplification regime controlled by the steady-state index $\mathbb{C}$4 (Fortin et al., 6 Sep 2025). This suggests that Lyapunov exponents play, in disordered non-Hermitian systems, a role analogous to Bloch momenta in clean band theory.

In non-Hermitian topological transport, generalized Thouless relations quantify deviations from quantization rather than simply certifying topology. The non-Hermitian TKNN formula shows that edge-state decay and bulk density of states contribute separately to Hall-response non-universality (Chen et al., 2018). The nonlinear auxiliary-eigenvalue framework extends this logic still further: the physically pumped observable is controlled by a weighted Berry curvature, while the spectral winding of the auxiliary operator remains integer, so bulk-edge correspondence persists in a modified form rather than disappearing (Zheng et al., 2 Feb 2026).

Taken together, these developments indicate that non-Hermitian Thouless relations are best understood as a set of exact or asymptotically exact correspondences between logarithmic spectral geometry and physically measurable localization or transport data. The most stable core of the subject is the statement that complex spectral density is encoded in the Laplacian of an appropriate Lyapunov potential (0706.3129, Longhi, 2019, Sun et al., 13 Jul 2025, Fortin et al., 6 Sep 2025). Around that core, later work has extended the same ethos to Hall conductance and Thouless pumping by replacing integer topological formulas with weighted non-Hermitian generalizations (Chen et al., 2018, Zheng et al., 2 Feb 2026).