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REVIEW 3 major objections 4 minor 3 cited by

The adiabatic theorem holds for non-Hermitian systems whose eigenvalues are real and non-degenerate.

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2026-08-04 00:27 UTC pith:GPSYJYXF

load-bearing objection A genuinely new Kato-style adiabatic proof for non-Hermitian systems with non-degenerate real eigenvalues; the math is sound but the stated hypotheses are too weak and the abstract overclaims. the 3 major comments →

arxiv 2511.00968 v3 pith:GPSYJYXF submitted 2025-11-02 quant-ph math-phmath.MP

The adiabatic theorem for non-Hermitian quantum systems with real eigenvalues and the complex geometric phase

Minyi Huang , Ray-Kuang Lee This is my paper
classification quant-ph math-phmath.MP MSC 81Q1281Q70 PACS 03.65.Vf
keywords adiabatic theoremnon-Hermitian quantum systemscomplex Berry phasebiorthogonal basisKato's methodreal eigenvaluesGrönwall inequalitygeometric phase
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.

This paper proves that the adiabatic theorem, a standard result of quantum mechanics, also applies to a large class of non-Hermitian Hamiltonians: those with non-degenerate real eigenvalues. The proof adapts the classical projection-based method to biorthogonal bases and uses the Grönwall inequality to control the time-evolution operator, which is no longer unitary. The result shows that a slowly driven non-Hermitian system follows its instantaneous eigenstate up to the usual dynamic phase and a complex Berry phase, and it provides a rigorous justification for that complex Berry phase in Abelian and non-Abelian settings. This matters because many open and dissipative quantum systems are modeled by non-Hermitian Hamiltonians, so a valid adiabatic approximation applies to them.

Core claim

Consider a non-Hermitian Hamiltonian H(s) with non-degenerate real eigenvalues λ_j(s) and right/left eigenvectors v_j(s), ξ_j(s) satisfying ξ_j† v_i = δ_{ij}. The paper proves that the solution of i d/ds U_T(s) = T H(s) U_T(s) with U_T(0)=I obeys U_T(s) v_j(0) → e^{-iT∫_0^s λ_j(σ)dσ} e^{-∫_0^s ⟨ξ_j(σ)| v̇_j(σ)⟩dσ} v_j(s) as T→∞, uniformly on s∈[0,1]. The first exponential is the dynamic phase; the second is the complex Berry phase. The proof rewrites the gauge-transformed evolution operator so that its derivative is O(1/T), and uses the Grönwall inequality to show that U_T and its inverse remain bounded independent of T. This establishes the adiabatic theorem for this class of non-Hermitian

What carries the argument

The central object is the biorthogonal projector π_j(s) = v_j(s) ξ_j†(s), with ξ_j the left eigenvector normalized so ξ_j† v_i = δ_{ij}. The connection ⟨ξ_j|v̇_j⟩ gives the complex Berry phase; it appears in the gauge factor e^{-∫ ⟨ξ_j|v̇_j⟩ dσ} that parallel-transports the eigenvector. The proof follows the original adiabatic-theorem strategy of decomposing the evolution into the instantaneous eigenbasis, but uses biorthogonal projectors instead of orthogonal ones and invokes the Grönwall inequality to establish uniform bounds on U_T(s) and U_T^{-1}(s), which are no longer unitary. The reality of the eigenvalues ensures the dynamical phase factors e^{iT∫λ_j} stay bounded, a key step that fa

Load-bearing premise

The proof requires that the Hamiltonian, its eigenvalues, and its eigenvectors are at least once continuously differentiable in the adiabatic parameter, so that all derivatives and the inverse of the eigenvector matrix are finite and bounded; it also requires the eigenvalues to remain real throughout the evolution.

What would settle it

For a two-level non-Hermitian Hamiltonian with real non-degenerate eigenvalues, such as H(s) = [[1, i s], [-i s, 1]], numerically integrate the Schrödinger equation at large T and check whether the transition probability to the other eigenstate vanishes as 1/T; if it does not, the theorem is false.

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

If this is right

  • Adiabatic following is guaranteed for any diagonalizable non-Hermitian Hamiltonian with a real non-degenerate spectrum, so a slowly varying such system cannot leave the instantaneous eigenstate.
  • The complex Berry phase is a well-defined, gauge-invariant part of the adiabatic phase for non-Hermitian evolutions, and it cannot be gauged away in cyclic processes.
  • The proof covers both Abelian and non-Abelian cases, so degenerate subspaces can be treated with the same biorthogonal machinery.
  • The uniform boundedness of U_T(s) and its inverse implies the adiabatic limit is approached at an algebraic rate (O(1/T)) under the stated smoothness assumptions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Beyond the paper: the proof's assumptions suggest the theorem should also hold for any diagonalizable non-Hermitian Hamiltonian with a purely real spectrum, even without PT symmetry; this is readily testable numerically.
  • Beyond the paper: in open quantum systems where the effective non-Hermitian Hamiltonian in the no-jump sector has real energies, this theorem provides a rigorous basis for adiabatic state-transfer and holonomic quantum gates.
  • Beyond the paper: because the connection ⟨ξ_j|v̇_j⟩ is generally complex, a closed cycle in parameter space may accumulate a geometric phase with a non-zero imaginary part, offering a possible new topological invariant for non-Hermitian bands with real spectrum.

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

3 major / 4 minor

Summary. The manuscript claims to prove an adiabatic theorem for non-Hermitian Hamiltonians H(s) with non-degenerate real eigenvalues. For a biorthogonal eigenvector pair (v_j(s), ξ_j(s)), it asserts that the evolution generated by i dU_T/ds = T H(s) U_T(s) satisfies U_T(s)v_j(0) → e^{-iT∫_0^s λ_j(σ)dσ} e^{-∫_0^s ⟨ξ_j(σ)|\dot v_j(σ)⟩dσ} v_j(s) as T→∞, provided the initial state is v_j(0). The proof follows Kato's structure: a spectral inverse S(s) is defined in Eq. (6), identities (7)–(16) reduce the evolution to a rapidly oscillating integral, integration by parts in Eq. (17) isolates the 1/T error, and Grönwall estimates in §III.B are used to prove uniform boundedness of U_T and U_T^{-1}. The paper also discusses gauge invariance of the resulting complex Berry phase and contrasts the argument with Kato's Hermitian proof.

Significance. If established, the result is a useful extension of the adiabatic theorem to a class of non-Hermitian systems, and it gives a principled derivation of a complex Berry phase for such systems. The proof is self-contained, uses no fitted parameters, and the Grönwall-bounding strategy is a genuine strength; the algebraic identities in §III are mostly consistent. However, the theorem as stated is broader than the hypotheses actually used: the manuscript never states the required smoothness of H(s) or of the eigenvectors, nor the finite-dimensional setting, and the non-Abelian claim in the abstract is not proved. These issues are load-bearing but appear fixable by stating and using the needed regularity assumptions.

major comments (3)
  1. [§III, Eq. (17)] The integration by parts in Eq. (17) is the central step that produces the 1/T decay, but it is justified only if S(σ)d_σ[\tilde v_j(σ)] is absolutely continuous. Since d_σ[\tilde v_j] = e^{-∫⟨ξ_j|\dot v_j⟩}(\dot v_j - ⟨ξ_j|\dot v_j⟩ v_j), its derivative contains \ddot v_j and \dot ξ_j. The proof therefore requires v_j ∈ C^2 (and correspondingly H ∈ C^2 with a uniform spectral gap) so that such a smooth eigenvector choice exists. The manuscript only assumes non-degenerate real eigenvalues. As written, the O(1/T) remainder estimate is unproved under the stated hypotheses. This is a load-bearing regularity gap.
  2. [§III.A–III.B] The proof repeatedly uses matrices V(s), \tilde V(s), Ξ(s) and their inverses, and assumes their norms and derivatives are bounded independently of T. These properties require explicit hypotheses: a finite-dimensional Hilbert space (or strong infinite-dimensional substitutes), continuity / differentiability of the spectral data, and a positive gap min_{s,i≠j}|λ_i(s)-λ_j(s)|>0 on s∈[0,1]. None of this is stated before Eq. (3). Please state the precise regularity and dimensionality assumptions at the start of §III, or give an infinite-dimensional version with appropriate resolvent bounds.
  3. [Abstract; §IV] The abstract claims that the proof justifies a complex Berry phase 'in both Abelian and non-Abelian cases,' but the body treats only a single non-degenerate level j. No non-Abelian (matrix-valued, multi-level or degenerate) Berry phase is defined or derived. A non-Abelian result would require a different argument involving degenerate subspaces and non-Abelian holonomies. Please either add the non-Abelian derivation or remove this claim from the abstract.
minor comments (4)
  1. [Throughout] There are several typos and infelicities: 'somewhtat' in §IV, 'adsorbed' for 'absorbed' in Appendix A, and the abstract's 'the theorem' should be 'the theorems' or 'the adiabatic theorem.'
  2. [Eq. (19)] The integration variable in the exponential phase is written as dσ in Eq. (18) and as dr in Eq. (19); please make the notation uniform.
  3. [§III.B] The statement 'Since Ξ(s) is continuous and E_T(s) is unitary' also requires Ξ^†(s) to be invertible with bounded inverse on [0,1]. This follows from non-degeneracy and finite dimensionality but should be stated.
  4. [§III] The phrase 'functional calculus for biorthogonal systems' is not used in the proof; the argument uses explicit spectral decompositions rather than functional calculus. Consider rephrasing to avoid overclaiming.

Circularity Check

0 steps flagged

No circularity: the adiabatic limit is proved from the Schrödinger equation via Kato-type identities and Grönwall estimates, with the complex Berry phase emerging as an output rather than an input.

full rationale

The derivation is self-contained. The main result, Eq. (19), is obtained by deriving the exact identities in Eqs. (10), (12), and (16), then applying integration by parts and the Grönwall inequality. No parameter is fitted to data, no prior adiabatic theorem is assumed, and the complex Berry phase is generated by the calculation: the vector ṽ_j(s) is introduced as e^{-∫⟨ξ_j|v̇_j⟩}v_j(s), and the proof shows that U_T(s)v_j(0) converges to e^{-iT∫λ_j} times exactly this transported vector. The boundedness estimates in Eqs. (23) and (26) rely only on continuity of the eigenvector matrices and the real-eigenvalue hypothesis, which makes E_T(s) modulus-one diagonal phases; this is a stated hypothesis of the theorem, not a hidden importation of the conclusion. The paper's use of Kato, Messiah, and Grönwall is standard external mathematical support, and there is no self-citation chain carrying the argument. The concerns raised in review, such as the silent need for C^2 eigenvectors in the integration-by-parts step (17), are regularity/rigor gaps about the proof's hypotheses, not circularity: they do not mean the theorem is equivalent to its inputs or that the prediction is forced by construction. The manuscript is an independent proof, so the honest finding is no significant circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 4 axioms · 0 invented entities

The derivation rests on standard linear algebra and calculus plus the stated physical domain assumption of non-degenerate real eigenvalues and smooth parameter dependence. There are no fitted parameters or new entities; the complex Berry phase is an output of the calculation.

axioms (4)
  • domain assumption H(s) is a differentiable family of diagonalizable matrices with non-degenerate real eigenvalues λ_i(s) for all s∈[0,1].
    Assumed in §III A after Eq. (3); non-degeneracy and real spectrum are the hypotheses of the theorem.
  • domain assumption The eigenvectors v_i(s) and biorthogonal vectors ξ_i(s) can be chosen smoothly; the matrix V(s) is invertible with bounded inverse on [0,1].
    Unstated; used in Eqs. (11)–(26), especially the construction of Ṽ and Ξ and the Grönwall bounds.
  • standard math Grönwall inequality (Appendix B) is valid for the integral inequalities (23) and (26).
    Standard result cited as Refs. [22,23].
  • standard math E_T(s) is unitary when λ_j(s) are real; used in bounding ∥E_T A E_T†∥ = ∥A∥.
    Implicit in §III B; requires real eigenvalues.

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read the original abstract

The adiabatic theorem is one of the most interesting and significant theorems in quantum mechanics. However, the adiabatic theorem can fail for general non-Hermitian quantum systems. In this paper, by utilizing the complex geometric phase, the functional calculus for biorthogonal systems and the Gr\"{o}nwall inequality, we prove rigorously that the adiabatic theorem is still valid for diagonalizable non-Hermitian systems with real eigenvalues. The proof also justifies the definition of a complex Berry phase for non-Hermitian systems, in both Abelian and non-Abelian cases.

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Reference graph

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