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Any driven non-Hermitian spin-S system has an exact evolution operator written from the corresponding two-level solution via Jacobi polynomials, and this yields closed-form defect-freezing densities across higher-order exceptional points.

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load-bearing objection Clean, checkable reduction of general non-Hermitian driven spin-S dynamics to spin-1/2 via Jacobi polynomials, plus exact defect-freezing densities for the spin-S PT-SSH family that are platform-testable.

arxiv 2607.04075 v1 pith:XMBU3VV4 submitted 2026-07-05 quant-ph cond-mat.mes-hallmath-phmath.MPnlin.SI

Wei-Norman approach for non-Hermitian driven spin-S systems and its application to defect freezing

Mingwei Meng , Chen Sun This is my paper
classification quant-ph cond-mat.mes-hallmath-phmath.MPnlin.SI
keywords non-Hermitian dynamicsWei-Norman approachspin-S systemsdefect freezingexceptional pointsPT-symmetric SSH modelJacobi polynomialsLandau-Zener
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

Exact nonadiabatic dynamics for time-dependent non-Hermitian systems are usually hard because Hamiltonians at different times do not commute. This paper shows that the difficulty collapses for any spin-S driven by a general complex field: the full evolution operator is completely determined by the four matrix elements of the ordinary spin-1/2 problem with the same field, and the higher-spin entries are given by elementary powers times a Jacobi polynomial. Because every two-level non-Hermitian model can be cast in this form, every solvable two-level case immediately produces a family of solvable multi-level models. The authors apply the reduction to spin-S extensions of the PT-symmetric Su-Schrieffer-Heeger chain under a linear quench. They obtain exact momentum-resolved excitation probabilities and total defect densities, proving that residual excitations survive even in the infinitely slow limit precisely when a momentum sector crosses a pair of higher-order exceptional points, and that the total density is non-analytic at a critical non-Hermiticity. The formulas supply concrete, platform-ready predictions for defect freezing in multi-level non-Hermitian lattices.

Core claim

For a spin-S Hamiltonian of the form H = X·S with arbitrary complex time-dependent X, every matrix element of the time-evolution operator equals a product of powers of the four spin-1/2 amplitudes times a Jacobi polynomial evaluated at those same amplitudes. Consequently the excitation probability out of the ground state after a linear quench of any spin-S PT-SSH model is simply 1 − (1 − p_{1/2})^{2S}, where p_{1/2} is the known two-level probability; defect freezing therefore occurs only in sectors that traverse the PT-broken region and the integrated density is singular at γ = w.

What carries the argument

Wei-Norman factorization of the evolution operator into three exponentials of spin raising, lowering and z operators; the three scalar coefficients are independent of spin size and are fixed by the spin-1/2 solution, after which matrix elements are evaluated by the definition of Jacobi polynomials.

Load-bearing premise

Excitation probabilities are defined by ordinary normalization of the unnormalized transition amplitudes rather than by the metric or biorthogonal inner product, a choice the paper notes yields different numerical values.

What would settle it

Prepare a spin-S PT-SSH lattice (S ≥ 1) on an electric-circuit or photonic platform, quench the intracell hopping slowly through the critical line γ = w, and measure whether residual ground-state defect density jumps discontinuously in slope exactly at that critical non-Hermiticity and matches the closed-form Jacobi expression.

Watch this falsifier — get emailed when new claim-graph text bears on it.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 4 minor

Summary. The paper shows that for a spin-S Hamiltonian of the form H = X·S with a completely general (possibly complex, time-dependent) drive X, the Wei–Norman factorization of the evolution operator into three exponentials of the su(2) generators yields matrix elements that are closed-form Jacobi polynomials of the four matrix elements of the corresponding spin-1/2 evolution operator (Eqs. 16 and 18). The reduction is spin-independent for the three scalar functions µ±, µz and recovers the Wigner d-matrix in the Hermitian limit. As an application the authors construct spin-S extensions of the PT-symmetric SSH model (N = 2S+1 sites per unit cell) and, using the known non-Hermitian Landau–Zener probability for S = 1/2, obtain exact momentum-resolved excitation probabilities and total excitation densities under a linear quench of the intracell hopping. They find that defect freezing occurs only in those k-sectors that traverse the PT-broken region (and therefore a pair of higher-order exceptional points), and that the adiabatic excitation density is non-analytic at the critical non-Hermiticity γ = w.

Significance. The algebraic reduction supplies a practical, elementary route from any solvable two-level non-Hermitian drive to a family of solvable multi-level models, without requiring specialized Lie-algebra machinery beyond the initial Wei–Norman factorization. The subsequent defect-freezing formulae are fully analytic, parameter-free once the S = 1/2 probability is known, and make concrete, platform-testable predictions (linear rise of nex with γ at small γ, singularity at γ = w, bln(1/b) scaling of the excess density). These results enlarge the set of exactly solvable non-Hermitian multi-level problems and give quantitative control over adiabatic breakdown across higher-order exceptional points.

minor comments (4)
  1. After Eq. (28) the authors correctly note that direct normalization and the metric formalism produce different numerical values; a short explicit remark that the qualitative statements (freezing only for |sin k| < γ/w and the non-analyticity at γ = w) survive under the metric convention would further clarify the robustness of the central claims.
  2. In Sec. III.B.2 the Appell F1 and Jacobi expressions for nex,S (Eq. 40) are compact but opaque; a one-sentence check that they reduce to the known S = 1/2 result (Eq. 31) would help the reader.
  3. Fig. 3(c) compares the approximate analytic form (47) with numerical integration; stating the relative error at the smallest b shown would quantify the quality of the Lambert-W approximation.
  4. A few typographical inconsistencies appear (e.g., “W ei-Norman”, “Shr¨ odinger”, “P T” spacing). A light copy-edit pass would remove them.

Circularity Check

1 steps flagged

No significant circularity: the Wei-Norman reduction to Jacobi polynomials is self-contained; the only self-citation supplies an independent exact S=1/2 special-function solution used as input, not a fitted or definitional loop.

specific steps
  1. self citation load bearing [Sec. III.B.1, Eq. (28) and following paragraph]
    "the exact analytical expression of excitation probability from the ground state after the quench (in the sector with momentum k) reads [70]: pk,1/2 = (γ + w sin k)/(2γ − (γ − w sin k)e^{2πδ}) o We refer to Sec. V in Supplemental Material of [70] for derivation of (28)."

    The quantitative defect-freezing formulae for general S rest on this imported S=1/2 probability. The citation is to overlapping authors (C. Sun), so it is a self-citation. However it is not circular in the strong sense: [70] supplies an independent closed-form solution of the two-level non-Hermitian LZ model via special functions, not a fit or definition that already encodes the multi-level results. The reduction itself remains non-circular.

full rationale

The load-bearing algebraic claim (Eqs. 16/18) follows from the spin-independent Wei-Norman factorization (3), elementary matrix elements of the su(2) generators, and the definition of Jacobi polynomials; it recovers the known Hermitian Wigner-d limit and does not presuppose any defect-freezing result. The subsequent multi-level excitation probabilities (33)–(34) and densities (40) are obtained by direct substitution of the known adiabatic limit of the S=1/2 non-Hermitian LZ probability. That S=1/2 formula is imported from the authors’ prior work [70], but [70] is an independent special-function solution of the two-level model (not a fit to the present multi-level data, nor a uniqueness theorem or ansatz). Direct versus metric normalization is an explicit modeling choice flagged by the authors and does not create a circular reduction. No self-definitional identities, fitted-input-as-prediction, or smuggled ansatz appear. Score 2 reflects only the minor, non-load-bearing self-citation for the input S=1/2 probability.

Axiom & Free-Parameter Ledger

0 free parameters · 5 axioms · 1 invented entities

The central claims rest on standard Lie-algebra structure of su(2), the classical Wei-Norman factorization (which does not require Hermiticity), the known exact solution of the two-level non-Hermitian Landau-Zener problem, and a conventional choice of probability normalization. No numerical free parameters are fitted. The only invented objects are the concrete lattice realizations that embed the spin-S Hamiltonians; they are constructive, not postulated entities.

axioms (5)
  • standard math The Hamiltonian H = X·S generates an su(2) Lie algebra for any complex time-dependent vector X, so the Wei-Norman product of three exponentials (Eq. 3) exists and the scalar functions μ±, μz are independent of spin size S.
    Invoked at the opening of Sec. II.B; existence is classical (Wei-Norman 1963/64) and does not require Hermiticity.
  • standard math The four matrix elements of the spin-1/2 evolution operator satisfy det U_{1/2} = 1 because H is traceless.
    Used to reduce four equations to three independent relations (Eq. 8–9).
  • domain assumption The exact excitation probability of the two-level non-Hermitian LZ model with the PT-SSH parameters is given by Eq. 28 (imported from Ref. [70]).
    All spin-S excitation formulas are algebraic lifts of this expression; the paper does not re-derive it.
  • ad hoc to paper Excitation probabilities are obtained by ordinary (direct) normalization of the unnormalized transition amplitudes rather than by the metric/biorthogonal inner product.
    Explicitly chosen in Sec. III.B.1; the authors note that the metric formalism yields different numbers.
  • domain assumption The initial state in each momentum sector is the lowest diabatic state (m = S) at t → −∞, and the adiabatic final state is m = −S.
    Stated at the beginning of Sec. III.B; defines what is counted as an excitation.
invented entities (1)
  • spin-S PT-SSH lattice models (N = 2S+1 sites per unit cell realizing H = 2[(v+w cos k)S_x + w sin k S_y + iγ S_z]) independent evidence
    purpose: Provide a concrete multi-band lattice whose momentum-space blocks are the driven spin-S Hamiltonians under study, so that defect freezing can be discussed in a physically realizable setting.
    Constructed in Sec. III.A and Fig. 1 by extending the ordinary PT-SSH chain; parameters are fixed by matching the spin operators, not fitted to data.

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In the theoretical study of nonequilibrium non-Hermitian systems, obtaining exact analytical solutions for their nonadiabatic dynamics is highly desirable yet often challenging. In this work, we identify a class of non-Hermitian quantum systems where this difficulty can be substantially reduced. Employing the Wei-Norman approach, we show that for a spin-$S$ subject to a general non-Hermitian time-dependent drive, the matrix elements of the evolution operator can be expressed in closed analytical forms (via Jacobi polynomials) in terms of the corresponding spin-$1/2$ model. This approach is straightforward and accessible to nonspecialists in Lie algebra. As an application, we investigate a specific nonequilibrium non-Hermitian phenomenon known as defect freezing, i.e., the existence of excitations in the adiabatic limit, in spin-$S$ extensions of the $\mathcal{PT}$-symmetric Su-Schrieffer-Heeger model under linear quenches. We derive exact analytical expressions for the momentum-resolved excitation probabilities and the total excitation densities. Our results reveal that defect freezing occurs exclusively in momentum sectors that traverse the $\mathcal{PT}$-symmetry-broken region -- and thus pass through a pair of higher-order exceptional points (EPs) -- during the quench; notably, the excitation density exhibits a singularity at a critical value of the non-Hermiticity parameter. This work enriches the analytical toolkit for nonadiabatic dynamics in multi-level non-Hermitian systems and provides quantitative, testable predictions for defect freezing across higher-order EPs, possibly accessible on platforms such as electric circuit networks and photonic lattices.

Figures

Figures reproduced from arXiv: 2607.04075 by Chen Sun, Mingwei Meng.

Figure 1
Figure 1. Figure 1: (a) to include N sites per unit cell. In [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Excitation probability in the adiabatic limit [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Excitation density in the spin- [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗

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    Defect freezing in thePT-SSH model We first discuss theS= 1/2 case, namely, the origi- nalPT-SSH model (22). In this case, the quench prob- (a)AB...iγw...ABvAB-iγ(b)AB...iγ1w1...ABAB-iγ1CCBCv1v1w1(c)AB ...iγ2w2...ABAB-iγ3CCBCv2v3w3DDDC-iγ2iγ3w2v2 A BCA DA FIG. 1. Sketches of lattices of (a) thePT-SSH model, (b) its S= 1 extension, and (c) itsS= 3/2 extens...

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