Imaginary-Order Spectral Deformation

• Imaginary-order spectral deformation is a technique that promotes self-adjoint operator orders to pure imaginary values, generating complex spectral parameters and novel topologies.
• It leads to non-Hermitian effects such as boundary-induced localization (skin effect) and deterministic phase decoherence in quantum evolution.
• The method is applied in integrable chains, oscillator models, and spectral gap estimation, revealing energy-twisted boundary conditions and analytic continuations.

Imaginary-order spectral deformation refers to a class of analytic modifications of a self-adjoint (or otherwise integrable) operator in which the order of the operator is promoted to a complex (purely imaginary) value. This deformation has far-reaching consequences in the spectral theory of quantum models, non-Hermitian quantum dynamics, integrable systems, and the phenomenology of decoherence. The procedure generically generates complexified spectral parameters, novel point-gap topologies, boundary-induced localization (such as the non-Hermitian skin effect), and—when applied to the operator governing time evolution—spontaneous phase decoherence that is both deterministic and kinematically unitary. Imaginary-order spectral deformation has been intensively studied in the context of non-Hermitian boost deformations in integrable chains (Guo et al., 2023), spectral gap estimation via imaginary time (Leamer et al., 2023), deterministic decoherence models (Tayur, 10 Dec 2025), and analytic continuation of bilinear couplings in oscillator models (Mintz et al., 18 Dec 2024).

1. Formal Definitions and Spectral Deformation Schemes

Let $H$ be a positive self-adjoint operator with spectral decomposition

$H = \int_0^\infty E \, d\Pi(E),$

where $d\Pi(E)$ is the projection-valued measure. For $\alpha \in \mathbb C$, the imaginary-order operator is

$$H^\alpha = \int_0^\infty E^\alpha \, d\Pi(E), \qquad E^\alpha = \exp(\alpha \log E).$$

Specializing to imaginary-order, $H^{i\beta}$ with $\beta \in \mathbb R$ yields a unitary spectral flow governed by the phase factor $E^{i\beta} = e^{i\beta \log E}$. More generally, for time evolution one may consider $H^{1+i\beta}$, so that

$U_\beta(t) = \exp(-i t H^{1+i\beta}),$

acts diagonally in energy via $|E\rangle \mapsto \exp(-i t E^{1+i\beta})|E\rangle$ (Tayur, 10 Dec 2025). In the algebraic-Bethe-ansatz context, integrable transfer matrices $t(u)$ are shifted under imaginary-order deformation by complexifying the spectral parameter: $u \mapsto u + i\kappa$ for $\kappa\in\mathbb R$ (Guo et al., 2023).

2. Non-Hermitian Boost Deformations and Energy-Twisted Boundaries

In one-dimensional integrable systems, the imaginary boost deformation is generated by the commutator flow

$$\frac{dH(\kappa)}{d\kappa} = [B[H(\kappa)], H(\kappa)],$$

where $B[H] = \sum_x x h_x$ is the boost operator for a local Hamiltonian $H = \sum_x h_x$ and $\kappa \in \mathbb R$ acts as the deformation parameter. For $\kappa \neq 0$, $H(\kappa)$ becomes non-Hermitian.

The same deformation may be equivalently realized as an energy-twisted boundary condition: $f(x+L) = e^{\kappa L H} f(x) e^{-\kappa L H}.$ For any eigenstate $Hf=\varepsilon f$, this reduces to a non-unitary twist

$f(x+L) = e^{-\kappa \varepsilon L} f(x),$

which modifies the momentum quantization: $k - i\kappa \varepsilon = \frac{2\pi n}{L}.$ This correspondence holds for free fermions, Calogero–Sutherland, XXZ spin chains, and other Bethe-ansatz solvable models (Guo et al., 2023).

For periodic BCs, the spectrum develops loops in the complex plane (complex-spectral winding); for open BCs, all eigenstates experience spatial localization toward one edge (the non-Hermitian skin effect). The quantization of winding number in complex energy space (point-gap topology) is a signature unique to the non-Hermitian regime.

3. Imaginary-Order Deformation and Decoherence Dynamics

Imaginary-order spectral deformation applied to the generator of quantum dynamics produces universal phase decoherence. Under $H^{1+i\beta}$, the time-evolution operator acquires an additional phase factor $e^{i\beta \log E}$ for each eigenspace: $$|\psi(t)\rangle = \int_{0}^{\infty} e^{-i t E^{1+i\beta}} \psi_0(E) dE.$$ Interference between different energy eigenstates is suppressed via rapid phase oscillations. Assessment via non-stationary-phase analysis yields

$|I_\beta(t)| \leq O(1/|\beta|),$

predicting that off-diagonal coherence decays at least with rate $1/|\beta|$ (Tayur, 10 Dec 2025). Observable decoherence rates between levels $E_m, E_n$ become

$$\Gamma_{mn}(\beta) = |\beta| \cdot |E_m \log E_m - E_n \log E_n|,$$

so the decoherence time is $\tau_\mathrm{dec}^{(mn)} \sim 1/\Gamma_{mn}(\beta)$. The Born rule and kinematical postulates remain unchanged; the modification is entirely encoded in deterministic dynamical phases. Experimental constraints on $|\beta|$ can be estimated from precision Ramsey or spin-echo protocols.

Physical motivations for such deformations include clock-imperfection models, renormalization group–induced logarithmic spectral corrections, and effective actions in quantum gravity where semiclassical expansions routinely produce $E^{i\gamma}$ spectral prefactors.

4. Bethe Ansatz, Complex Spectral Winding, and the Skin Effect

In coordinate-Bethe-ansatz solvable models, the deformation consistently leads to analytically continued Bethe equations. For a generic periodic integrable chain,

$e^{i p_j L} = \prod_{\ell\neq j} S(p_j, p_\ell)$

is replaced by the energy-twisted quantization

$$e^{i [p_j - i\kappa \varepsilon_j]L} = \prod_{\ell\neq j} S(p_j, p_\ell),$$

with complex momenta and energies.

Detailed examples include:

• Free Fermions: Dispersion $\varepsilon(k) = -2\cos k$ generates, for nonzero $\kappa$, “eight-shaped” complex energy loops with non-trivial point-gap topology. Under open BCs, eigenstates become $f(x) \sim e^{-\kappa \varepsilon x} \sin(kx)$, yielding complete localization (“skin effect”) to one edge (Guo et al., 2023).
• Calogero–Sutherland Model: Imaginary boost generates double-branched spectrum $\varepsilon_j^\pm$ forming v-shaped or cross-shaped complexes, reflecting inherited winding number from the single-particle case.
• XXZ Spin Chain: The Bethe roots $z_j$ satisfy modified quantization conditions with exponential energy-dependent twist, generating loop-to-arc transitions in the complex plane under increasing $\kappa$ and exhibiting skin effect for open BCs.
• Oscillators with Imaginary Coupling: Imaginary-order deformation of coupled oscillator Hamiltonians corresponds to analytic continuation of the coupling constant (e.g., $i\varepsilon \mapsto i\varepsilon e^{i\theta}$), producing analytic families of spectra and explicit diagnostic features (such as non-positive-definite spectral densities) (Mintz et al., 18 Dec 2024).

5. Spectral Gaps and Imaginary-Time Deformation

The imaginary-time evolution $|\psi(\beta)\rangle = e^{-\beta H}|\psi_0\rangle$ can be reinterpreted as a real-order spectral deformation. Under this flow, higher-eigenvalue components are exponentially suppressed, and spectral gaps can be extracted via nested-commutator expectation values: $$C_M(\beta) = \langle\psi(\beta)|[H, O]_M |\psi(\beta)\rangle,$$ with

$$R_M(\beta) = \frac{C_{M+2}(\beta)}{C_M(\beta)} \to (E_1 - E_0)^2, \quad \text{as } \beta \to \infty.$$

This allows direct numerical and experimental access to spectral gaps, as demonstrated for small TFIM and Fermi–Hubbard models, with convergence rate in $\beta$ set by the next excited-state gap (Leamer et al., 2023).

6. Spectral Properties, Two-Point Functions, and Analytic Continuation

Imaginary-order spectral deformations provide a non-Hermitian analytic family of Hamiltonians $H(\theta)$ by complex rotation of couplings. In the case of two bilinearly coupled oscillators,

$$H(\theta) = \frac12(p_1^2 + p_2^2 + \omega_1^2 x_1^2 + \omega_2^2 x_2^2) + i \varepsilon e^{i\theta} x_1 x_2,$$

the diagonalization yields normal-mode frequencies $\Omega_{1,2}(\theta)$, analytic in $\theta$. The two-point function's Källén–Lehmann spectral densities become sign-indefinite except at those $\theta$ where the Hamiltonian is Hermitian ($\theta = \pm \pi/2$). This violation of spectral positivity, while the spectrum remains real, is interpreted as a signal of hidden non-trivial metric structure and serves as a generic diagnostic for possible ${\cal PT}$-broken phases (Mintz et al., 18 Dec 2024).

7. Context, Extensions, and Open Problems

Imaginary-order spectral deformation framework synthesizes analytic, algebraic, and phenomenological aspects of non-Hermitian quantum theory. It encodes:

• Complexification of spectral parameters and Bethe roots.
• Non-Hermitian topologies in the complex energy plane (point gaps, winding).
• Boundary-induced localization and edge accumulation.
• Deterministic, universal decoherence scaling with logarithmic energy factors.
• Analytic continuation in coupling constants and the emergence of spectral positivity violation.

Outstanding questions include the microscopic derivation of imaginary-order parameters (e.g., $\beta$) in quantum gravity and clock models, classification of all compatible spectral deformations, and experimental constraints from high-coherence platforms. The impact on many-body spectra, open systems, and field theories—especially regarding the interplay with environmental decoherence and spectral transitions—remains an active area of investigation (Tayur, 10 Dec 2025, Guo et al., 2023, Mintz et al., 18 Dec 2024).