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Two-dimensional Toda--Arnoldi correspondence: Holomorphic Krylov geometry and counterdiabatic transport

T0 review · 0 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The paper proves that in Arnoldi reduction of a non-Hermitian Hamiltonian, the diagonal and subdiagonal entries are exactly the Flaschka variables of a two-dimensional Toda lattice.

desk verdict Solid and new: the Toda–Arnoldi identification holds up; send to review with a request to substantiate the Chern-number table and tighten the breakdown section. read the letter →

arxiv 2608.04850 v1 pith:35PWEQNN submitted 2026-08-05 quant-ph cond-mat.stat-mechmath-phmath.MPnlin.SI

Urei Miura This is my paper
classification quant-phcond-mat.stat-mechmath-phmath.MPnlin.SI
keywords TodalatticeArnoldimethodKrylovsubspacesnon-HermitianHamiltonianFubini-StudymetricBerrycurvaturecounterdiabaticdrivingtaufunction
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

The paper establishes a two-dimensional Toda–Arnoldi correspondence: for a fixed finite-dimensional, generally non-Hermitian Hamiltonian and a holomorphically deformed cyclic seed state, the Krylov Gram determinants are tau functions of the finite two-dimensional Toda lattice, and the diagonal and subdiagonal entries of the Arnoldi upper-Hessenberg matrix equal the lattice's Flaschka variables. It follows that the Toda equations close on this sector without constraining the remaining upper-Hessenberg entries. The same squared subdiagonal coefficient also gives the Fubini–Study metric and Berry curvature of the holomorphic Krylov-subspace map. Along real paths, the Arnoldi-frame connection yields a Hermitian tridiagonal generator that preserves the spectrum exactly and, in the nondegenerate diagonalizable case, cancels transitions between instantaneous eigenspaces, realizing counterdiabatic driving. A sympathetic reader would care because the result unifies Krylov reduction, integrable Toda dynamics, and holomorphic-subspace geometry for non-Hermitian systems, making the geometric meaning of Arnoldi coefficients explicit.

What carries the argument

The central object is the Gram-determinant tau function of the finite two-dimensional Toda lattice built from holomorphically deformed Krylov vectors: $\tau_n = \det\big[\langle \psi(\bar z)|(H^\dagger)^j H^i|\psi(z)\rangle\big]_{i,j=0}^{n-1}$, which equals the squared volume of the exterior Krylov state $|\Psi_n\rangle\rangle$. The Desnanot–Jacobi identity places these determinants into the Hirota bilinear identity, and logarithmic derivatives define the Flaschka variables $a_n$, $b_n$. The load-bearing identity is Eq. (13), $a_n=\alpha_n$ and $b_n=\beta_n$, where $\alpha_n$ and $\beta_n$ are the diagonal and subdiagonal entries of $A=U^\dagger H U$; this identity is what transfers Toda, metric, curvature, and counterdiabatic content to the Arnoldi coefficients.

What would settle it

Take a non-Hermitian $H$ and a cyclic seed, and for many values of $z$ compute both the Arnoldi coefficients $\alpha_n,\beta_n$ and the Flaschka variables from $\tau_n=\det(K_n^\dagger K_n)$; if any $\alpha_n$ differs from $-\partial_z\ln(\tau_{n+1}/\tau_n)$ or any $\beta_n$ differs from $\sqrt{\tau_{n+1}\tau_{n-1}/\tau_n^2}$ while all $\tau_n>0$, the main theorem is false. A sharper check is to approach a point where Arnoldi breakdown lowers the Krylov dimension below $n$ and verify that $b_n^2$ either vanishes for the invariant subspace or that the Fubini–Study metric ceases to exist, matching the paper's three-site Chern-number prediction.

Watch

Extended reading notes

Core claim

At each value of $z$, with the normalized seed $|u_0\rangle = |\psi(z)\rangle/\sqrt{\tau_1}$, the diagonal entries $\alpha_n(z,\bar z)$ and subdiagonal entries $\beta_n(z,\bar z)$ of the Arnoldi matrix $A(z,\bar z) = U^\dagger H U$ coincide with the Flaschka variables $a_n(z,\bar z)$, $b_n(z,\bar z)$ defined from the Krylov Gram determinants $\tau_n$; this is Eq. (13) of the paper. The proof expresses the tau functions in the Arnoldi basis and compares logarithmic derivatives and ratios with the Flaschka formulas, with an independent QR-factorization proof in the appendix. Consequently $b_n^2$ equals both the Fubini–Study metric component $g^{(n)}_{z\bar z}$ and, up to sign, the Berry curvature $F_n$ of the $n$-th holomorphic Krylov subspace. The Toda equations close on these coefficients without determining the remaining upper-Hessenberg entries, and the frame connection supplies a Hermitian tridiagonal counterdiabatic generator when the Arnoldi matrix is diagonalizable with nondegenerate spectrum.

Load-bearing premise

The seed state must be cyclic for every $z$, meaning that $|\psi(z)\rangle, H|\psi(z)\rangle, \ldots, H^{D-1}|\psi(z)\rangle$ span the whole space, and the deformation must have the exact exponential form $|\psi(z)\rangle = e^{-zH}|\psi(0)\rangle$; without cyclicity some $\tau_n$ vanish and some Arnoldi vectors do not exist, and without the exponential form the Gram determinants do not satisfy the two-dimensional Toda bilinear identity.

Editorial extensions

If this is right

  • The Toda equations $\partial_{\bar z} a_n = b_n^2 - b_{n+1}^2$ and $\partial_z b_n^2 = (a_{n-1}-a_n)b_n^2$ hold as exact identities for the Arnoldi coefficients throughout the cyclic region.
  • The squared subdiagonal coefficient $b_n^2$ is the Fubini–Study speed of the $n$-dimensional Krylov subspace along the holomorphic parameter direction.
  • Along any smooth real path in the cyclic region, $W(s)=U^\dagger(s)U(0)$ generates an exact isospectral evolution $i\dot A = [G_\gamma,A]$, preserving the spectrum and Jordan structure even when $A$ is nondiagonalizable.
  • In the nondegenerate, diagonalizable case, the Arnoldi-frame generator $G_\gamma$ cancels transitions between instantaneous eigenspaces, so adding it to $A$ realizes exact counterdiabatic driving.
  • In the Hermitian limit the construction reduces to the known Toda–Lanczos correspondence and to the standard Hermitian counterdiabatic term.

Reading between the lines

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

  • The electrostatic reading in Appendix F suggests a measurable diagnostic: the circular mean of $\ln \tau_n$ is nondecreasing with radius, so probing $b_n^2$ along nested disks gives a monotone geometric charge that should be visible in any finite-dimensional non-Hermitian Krylov evolution.
  • The Lax-pair form $(A,-iG_\gamma)$ indicates that exact counterdiabatic transport along closed paths in the cyclic region is an isospectral, time-parameterized Toda flow; protocols could be constructed by QR-based or inverse-scattering methods for the Arnoldi matrix rather than by diagonalizing $H$.
  • If the correspondence extends to superoperator or Lindblad dynamics as the conclusion suggests, the $b_n^2$ identities could supply exact counterdiabatic terms for open-system state preparation where bi-Lanczos methods are currently approximate.
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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

0 major / 5 minor

Summary. The manuscript proves a finite-dimensional Toda–Arnoldi correspondence. For a fixed, generally non-Hermitian Hamiltonian H and a cyclic seed |ψ(0)>, the holomorphic family |ψ(z)>=e^{-zH}|ψ(0)> is used to build Krylov Gram determinants τ_n(z,\bar z). The authors show that these τ_n satisfy the Hirota bilinear identity of the two-dimensional Toda lattice, and prove the Main Theorem (Eq. 13): the Arnoldi diagonal entries α_n and subdiagonal entries β_n coincide with the Toda Flaschka variables a_n and b_n. They then show that b_n^2 is the Fubini–Study metric component and (up to sign) the Berry curvature of the n-th holomorphic Krylov subspace, derive the Stokes relation (26), compute Chern numbers in a three-site non-Hermitian model including Arnoldi breakdown, and construct a Hermitian tridiagonal moving-frame generator G_γ that gives exact isospectral transport and cancels transitions between instantaneous eigenspaces of the Arnoldi matrix.

Significance. The result, if accepted, is significant: it extends the known Toda–Lanczos relation to general non-Hermitian Arnoldi reduction, gives the diagonal and subdiagonal Arnoldi data a precise integrable-system and geometric meaning without fixing the rest of the upper Hessenberg matrix, and supplies a concrete three-site model with quantized Chern numbers and a breakdown analysis. The paper is largely self-contained: the main theorem is proved directly and again via QR factorization in Appendix D, and the key identities (16), (17), (26), (54)–(55) and the Table I entries are explicit and check out. The construction has no free parameters, and its scope (cyclic seed, exponential holomorphic deformation) is stated precisely. I found no load-bearing flaw in the central derivation.

minor comments (5)
  1. [III.C, Table I] The typeset Table I has merged columns (e.g., "0 2−1 0" and "3−2−2"), making the entries for d_K, C1, and C2 ambiguous; please typeset the table with clear column separation and with "undefined" as a distinct entry.
  2. [III.C, Eq. (35)] Although the text states that Eq. (35) holds for 0≤θ<π/2, the right-hand side has the finite limit 1 as θ→π/2. A sentence explaining that this limit is not the value of b_2^2 because the second Arnoldi vector does not exist at θ=π/2 would remove an apparent contradiction.
  3. [IV, Eqs. (44)–(46)] The counterdiabatic interpretation is formulated in the moving Arnoldi frame; in the laboratory frame the evolution generated by A+G is simply free evolution under H. Please state this scope explicitly when using the term "counterdiabatic driving," so that readers do not infer that a modified physical Hamiltonian is being implemented in the original Hilbert space.
  4. [III.A, Eq. (24)] Please fix the orientation convention for dz∧d\bar z on Σ (e.g., dz∧d\bar z = -2i dx∧dy) so that the nonpositivity of C_n and the numerical values in Table I are unambiguous.
  5. [II.B, Eq. (16)] For n=1 the product over m=1 to 0 is empty; state the empty-product convention explicitly for clarity.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Toda–Arnoldi correspondence is derived from independently defined quantities and the geometric and transport results follow by exact identities.

full rationale

The central claim, Eq. (13), identifies the Arnoldi coefficients alpha_n, beta_n with the Flaschka variables a_n, b_n defined from the Krylov Gram determinants tau_n through Eqs. (A19)-(A20). These two objects are defined independently: tau_n is a determinant built from iterates of H on the holomorphic seed state, while alpha_n and beta_n are defined by the Arnoldi orthonormalization. The proof expresses tau_n in terms of the Arnoldi coefficients, Eqs. (16)-(17), and then substitutes into the Flaschka definitions; no parameter is fitted and no quantity is renamed as a prediction. The result is corroborated by a separate QR-factorization argument in Appendix D, where beta_n^2 = tau_{n+1}tau_{n-1}/tau_n^2 and alpha_n = -partial_z ln(tau_{n+1}/tau_n) are obtained directly from det(K_n^dagger K_n). The geometric identifications b_n^2 = g^{(n)}_{z zbar} and F_n = -i b_n^2 dz wedge dzbar follow from the standard Kähler potential log tau_n and the Toda identity (A17), so they are exact consequences rather than circular inputs. The counterdiabatic generator G_gamma = -i U^dagger Udot is constructed as the frame connection, and the no-transition property is then verified by direct differentiation of the evolved state S(s) = U^dagger(s) e^{-iHs} U(0); this is an explicit identity, not an assumed prediction. The paper contains no load-bearing self-citations: the cited works on the two-dimensional Toda lattice and Toda-Lanczos correspondence are standard external references, and the non-Hermitian Arnoldi extension is proven in the text. The three-site Chern-number table is obtained by integrating explicit expressions for b_n^2, and the breakdown statements are supported by the rank argument in Appendix E. Overall, the derivation chain is self-contained and no step reduces by construction to its own inputs.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The paper introduces no fitted parameters and no new physical entities. The central claim rests on classical results (Desnanot-Jacobi, 2D Toda moment theory, Plucker geometry) and on two stated domain assumptions: a cyclic seed and the exponential holomorphic deformation. The three-site model parameters cos(theta) and sin(theta) are model inputs, not fit parameters. The electrostatic layer picture of Appendix F is an explicit analogy, not an invented entity.

assumptions (7)
  • standard math Desnanot-Jacobi (Jacobi) determinant identity (A6)
    Used in Sec. II.A to show the Krylov Gram determinants (Eq. (6)) satisfy the Hirota bilinear identity (A16) of the finite 2D Toda lattice. Classical linear-algebra identity.
  • domain assumption Gram/moment determinants of the 2D Toda hierarchy are tau functions and define Flaschka variables through (A19) and (A20)
    Standard 2D Toda lattice theory (refs. [14], [15], [37], [41]); the paper relies on it for b_n^2 = tau_{n+1} tau_{n-1} / tau_n^2 and for the equations of motion (52).
  • domain assumption Cyclicity of the seed state |psi(0)> (Eq. (B3))
    Guarantees tau_n > 0 for all n and the full D-step Arnoldi reduction. The correspondence and the geometric and transport statements hold on the cyclic region; breakdown is treated only in the three-site model (Sec. III.C).
  • domain assumption Holomorphic exponential deformation d_z |psi> = -H |psi> (Eqs. (1) and (3))
    This flow gives the moment shift structure d_z mu_{ij} = mu_{i+1,j}, which makes the Gram matrices moment matrices of the 2D Toda hierarchy; a different deformation would not yield the Toda structure.
  • standard math Smooth dependence of the QR and Arnoldi factors on (z, z-bar) in the cyclic region
    Imported from Dieci-Eirola (ref. [58]) in Appendix D; needed to differentiate the QR factorization and derive the frame connections (D11).
  • domain assumption Nondegenerate spectrum and complete biorthogonal eigenvector sets for the counterdiabatic claim (Eq. (47))
    The no-transition result i c-dot_n = E_n c_n (Eq. (49)) is proven under this assumption stated in Sec. IV.B; the isospectral transport identity (Eqs. (43)-(46)) does not need it.
  • standard math Standard projective and Grassmannian geometry facts (Plucker embedding, Fubini-Study metric, Chern classes)
    Used in Sec. III for the Berry connection, curvature, metric pullback, and the integrality of Chern numbers (Appendix E, refs. [17]-[19], [59]).

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Pith. "Pith review of Two-dimensional Toda--Arnoldi correspondence: Holomorphic Krylov geometry and counterdiabatic transport." pith.science (2026). https://pith.science/paper/35PWEQNN

@misc{pith2026260804850,
  author       = {Pith},
  title        = {Pith review of: Two-dimensional Toda--Arnoldi correspondence: Holomorphic Krylov geometry and counterdiabatic transport},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/35PWEQNN}},
  note         = {Machine review of arXiv:2608.04850}
}
abstract

Although Arnoldi reduction of a generally non-Hermitian Hamiltonian yields an upper Hessenberg matrix rather than the tridiagonal form of Hermitian Lanczos theory, we show that a closed Toda sector survives in its diagonal and subdiagonal coefficients. For a fixed finite-dimensional Hamiltonian and a cyclic state vector deformed holomorphically, the Krylov Gram determinants are $\tau$ functions of the finite two-dimensional Toda lattice, whose Flaschka variables coincide exactly with these Arnoldi coefficients. The Toda dynamics therefore closes on this sector without determining the remaining upper Hessenberg entries. The subdiagonal part of the same sector also has a direct geometric meaning: the squared subdiagonal coefficients determine both the Fubini--Study metric and the Berry curvature of holomorphic Krylov subspaces, whereas the geometric quantities associated with subspaces lost at Arnoldi breakdown cease to be defined. Along a smooth real path in the cyclic region, the Arnoldi-frame connection further provides a Hermitian tridiagonal generator of exact isospectral transport. When added to the Arnoldi matrix, this generator cancels transitions between instantaneous eigenspaces and realizes counterdiabatic driving whenever the matrix is diagonalizable with a nondegenerate spectrum.

Figures

Figures reproduced from arXiv: 2608.04850 by the authors.

Figure 1
Figure 1. FIG. 1. Toda–Arnoldi correspondence obtained from holomorphic evolution through Krylov data. The Arnoldi coefficients [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Stokes–Gauss relation for a compact region [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Geometric quantities obtained from the holomorphic Krylov map. In the region where [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Three-site non-Hermitian nonreciprocal model. (a) Directed Krylov chain generated by the seed vector 22 [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Electrostatic analogy for the Toda lattice. Each [PITH_FULL_IMAGE:figures/full_fig_p017_5.png]

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Works this paper leans on

84 extracted references · 59 canonical work pages

  1. [1]

    Toda lattice We consider a finite nonperiodic Toda lattice withD sites. Letq n(t),(0≤n≤D−1), denote dynamical vari- ables that depend on timet∈R[33–35]: d2 dt2 qn = eqn+1−qn −e qn−qn−1 .(A1) At the lower boundaryn= 0, we set the second term on the right-hand side to zero; at the upper boundary n=D−1, we set the first term to zero. We suppress time argumen...

  2. [2]

    Compared with the Toda lattice, the second derivative with respect to time is replaced by partial derivatives with respect to the two independent variables(z,¯z)

    Two-dimensional Toda equation The two-dimensional Toda lattice is defined by the fol- lowing partial differential equation for dynamical vari- ablesϕ n(z,¯z)∈Cwith(0≤n≤D−1)and(z,¯z)∈ C×C[15, 40]: ∂z∂¯zϕn = eϕn+1−ϕn −e ϕn−ϕn−1 .(A14) At the lower boundaryn= 0, we set the second term on the right-hand side to zero; at the upper boundary n=D−1, we set the fi...

  3. [3]

    Subtract the components along the known basis vectors: |rn+1⟩ ←H|un⟩ − n−1X m=0 hm,n |um⟩ −αn |un⟩. 12

    work page

  4. [4]

    These variables obey [14, 15]: ∂¯zan =b 2 n −b 2 n+1,(A21) ∂zb2 n = (an−1 −a n)b2 n.(A22) Thefirstequationholdsfor0≤n≤D−1, andthesecond for1≤n≤D−1

    Flaschka variables The Flaschka variablesa n(z,¯z)∈Cfor (0≤n≤D−1)andb n(z,¯z)>0for(1≤n≤D−1)of the two-dimensional Toda lattice are defined in terms of theτfunctions as follows [14, 15]: an :=−∂ z ln τn+1 τn ,(A19) b2 n := τn+1τn−1 τ 2n .(A20) Here,b 0 :=b D := 0. These variables obey [14, 15]: ∂¯zan =b 2 n −b 2 n+1,(A21) ∂zb2 n = (an−1 −a n)b2 n.(A22) The...

  5. [5]

    Toda reduction To reduce the two-dimensional Toda lattice to the Toda lattice, define the real timetby t :=z+ ¯z.(A23) We consider the case in whichϕn andτ n depend only on tand not onz−¯z[15]. Thena n andb n also depend only ont, and the partial derivatives acting on these variables become ∂z =∂ ¯z= d dt .(A24) Define the reduced dynamical variableqn by ...

  6. [6]

    The condition prevents Arnoldi breakdown forn < N−1and yields a complete unitary transfor- mation [9, 10]

    Arnoldi iteration ConsiderH∈End CN and a nonzero seed vector|ψ⟩, and assume KN (H,|ψ⟩) =C N .(B3) A vector|ψ⟩that satisfies this condition is called a cyclic vector ofH. The condition prevents Arnoldi breakdown forn < N−1and yields a complete unitary transfor- mation [9, 10]. The iteration based on Gram–Schmidt orthogonalization is as follows. Input:A squ...

    work page

  7. [7]

    , n−1, set hm,n ← ⟨um|H|un⟩

    Form= 0,1, . . . , n−1, set hm,n ← ⟨um|H|un⟩

    work page

  8. [8]

    Set the diagonal entry to αn ← ⟨un|H|un⟩

    work page

Show all 84 references
  1. [9]

    Output: U= (|u 0⟩,|u 1⟩,

    Ifn < N−1, set βn+1 ← ∥|rn+1⟩∥,|u n+1⟩ ←|rn+1⟩ βn+1 . Output: U= (|u 0⟩,|u 1⟩, . . . ,|uN−1 ⟩), A=U †HU. Here,Uis unitary, andAhas upper Hessenberg form

  2. [10]

    Arnoldi relation and Hessenberg reduction Starting from a normalized seed vector|u0⟩, suppose that we have obtained the orthonormal basis vectors |u0⟩, . . . ,|un⟩. The next basis vector is obtained by or- thogonalizingH|u n⟩againsttheknownbasisvectors. De- fine hm,n :=⟨u m|H|...

  3. [11]

    For a normal matrix, the distance from a Ritz value to the spectrum of the original matrix is bounded above by the residual norm of the corresponding Ritz vector [10]

    Truncation and nonnormality Truncating the Krylov dimension leaves a boundary residual that leaks out of the Krylov subspace. For a normal matrix, the distance from a Ritz value to the spectrum of the original matrix is bounded above by the residual norm of the corresponding R...

  4. [12]

    We assume that its eigenvalues are nondegener- ate over the time interval of interest

    Hermitian counterdiabatic driving Consider a Hermitian matrixH ad(s)that varies with times. We assume that its eigenvalues are nondegener- ate over the time interval of interest. The instantaneous eigenvaluesE n(s)and eigenstates|n(s)⟩satisfy Had(s)|n(s)⟩=E n(s)|n(s)⟩, ⟨m(s)|n...

  5. [13]

    For Hcd =G, an eigenstate initialized ats= 0evolves exactly as Ψ(G) n (s) E = e−i R s 0 ds′ En(s′) |n(s)⟩.(C16) In general, the eigenvalues also depend ons

    Evolution with fixed eigenvalues, dynamical invariants, and Lax equations Define the orthogonal projectorP n onto each instan- taneous eigenstate and the generatorGof the time de- pendence of the eigenbasis by Pn(s) :=|n(s)⟩ ⟨n(s)|(C12) G(s) := i X n |˙n(s)⟩ ⟨n(s)|.(C13) Diffe...

  6. [14]

    For a non-Hermitian system, right and left eigenstates must be treatedseparately [54,55]

    Non-Hermitian counterdiabatic driving Thus far, we have considered Hermitian Hamiltonians. For a non-Hermitian system, right and left eigenstates must be treatedseparately [54,55]. We nowallowHad(s) to be non-Hermitian. We assume that a complete set of right and left eigenstat...

  7. [15]

    ,Hn−1 |ψ⟩ ∈C D×n.(D1) The full matrix satisfiesKD(z) = e−zHKD(0)

    QR factorization and coefficient identification For1≤n≤D, define the matrix whose columns are the firstnKrylov vectors by Kn := |ψ⟩,H|ψ⟩, . . . ,Hn−1 |ψ⟩ ∈C D×n.(D1) The full matrix satisfiesKD(z) = e−zHKD(0). The cyclic- vector assumption makesK D(0)invertible, ande −zH is al...

  8. [16]

    (D9) byU † from the left and byR −1 from the right, and then using Eq

    Differentiated QR factorization and Arnoldi Lax equation We define the frame connections by Fz :=−U †∂zU,F ¯z:=−U †∂¯zU.(D8) Differentiating the full QR factorizationK D =URand using∂ zKD =−HK D and∂ ¯zKD = 0, we obtain ∂zKD = (∂zU)R+U∂ zR=−HUR, ∂¯zKD = (∂¯zU)R+U∂ ¯zR= 0.(D9) ...

  9. [17]

    We denote the point defined by a nonzero vector|ψ⟩ by[ψ] := span (|ψ⟩)

    Projective space and the Plücker embedding Form∈Z ≥0, the complex projective space is CPm := Cm+1 \ {0} /C×. We denote the point defined by a nonzero vector|ψ⟩ by[ψ] := span (|ψ⟩). For a normalized representative ⟨ψ|ψ⟩= 1, the Fubini–Study line element is ds2 FS :=⟨dψ|(I m+1 −...

  10. [18]

    A holomorphic and invertible change of basisV7→VG adds onlyln |det (G)|2 toK Gr, so the metric is inde- pendent of the choice of basis

    Fubini–Study metric and Arnoldi coefficients Define the metric induced on the Grassmannian by the Plücker embedding as gGr :=ι ∗ PlgFS.(E4) For a basis matrixVthat depends holomorphically on a local complex coordinate, the Kähler potential is KGr = ln det V†V . A holomorphic a...

  11. [19]

    Potential and complex electric field Following Eq. (A15), define the potentialϕn and the complex electric fieldEn on layernby ϕn = ln τn+1 τn ,0≤n≤D−1, En :=−∂ zϕn =a n.(F1) Thus, the Toda Flaschka variablean coincides with the in-plane complex electric field on layern

  12. [20]

    The link between layers nandn+ 1isb 2 n+1 = exp Vn+ 1 2 = exp (ϕn+1 −ϕ n)

    Link polarization and nonlinear Poisson equation Orient the linkn− 1 2 from layern−1to layern, and define the potential difference along this orientation by Vn− 1 2 :=ϕ n −ϕ n−1, b2 n = exp Vn− 1 2 ,1≤n≤D−1, b2 0 =b 2 D = 0.(F2) For a finite chain,b2 0 =b 2 D = 0indicates that...

  13. [21]

    Applying Stokes’ theorem to Eq

    Gauss’s law and Stokes’ theorem Equation (F1) gives the local form of Gauss’s law, ∂¯zEn =ρ n.(F5) In terms of the one-formEndz, this relation becomes d(Endz) =−ρ ndz∧d¯z.(F6) LetΩ⊂Σbe a compact region with a smooth boundary, and give∂Ωthe positive orientation. Applying Stokes...

  14. [22]

    For1≤n≤ D, define the circular mean of the cumulative potentialPn−1 m=0 ϕm = ln (τn)in Eq

    Circular means of the cumulative potential LetB r be the disk of radiusrcentered atz 0, and parametrize its boundary byz=z 0 +re iθ. For1≤n≤ D, define the circular mean of the cumulative potentialPn−1 m=0 ϕm = ln (τn)in Eq. (F9) by Mn(r) := 1 2π Z 2π 0 dθln τn z0 +re iθ,¯z0 +r...

  15. [23]

    Hochbruck and C

    M. Hochbruck and C. Lubich, SIAM Journal on Numer- ical Analysis34, 1911 (1997)

    1997

  16. [24]

    Nandy, A

    P. Nandy, A. S. Matsoukas-Roubeas, P. Martínez- Azcona, A. Dymarsky, and A. del Campo, Physics Re- ports1125–1128, 1 (2025), arXiv:2405.09628

    2025 arXiv

  17. [25]

    D. E. Parker, X. Cao, A. Avdoshkin, T. Scaffidi, and E. Altman, Physical Review X9, 041017 (2019), arXiv:1812.08657

    2019 arXiv

  18. [26]

    Rabinovici, A

    E. Rabinovici, A. Sánchez-Garrido, R. Shir, and J. Son- ner, Journal of High Energy Physics2021, 062 (2021)

    2021

  19. [27]

    Takahashi and A

    K. Takahashi and A. del Campo, Physical Review X14, 011032 (2024), arXiv:2302.05460

    2024 arXiv

  20. [28]

    Takahashi and A

    K. Takahashi and A. del Campo, Physical Review Letters 134, 030401 (2025), arXiv:2408.08383

    2025 arXiv

  21. [29]

    Lanczos, Journal of Research of the National Bureau of Standards45, 255 (1950)

    C. Lanczos, Journal of Research of the National Bureau of Standards45, 255 (1950)

    1950

  22. [30]

    Dymarsky and A

    A. Dymarsky and A. Gorsky, Physical Review B102, 085137 (2020), arXiv:1912.12227

    2020 arXiv

  23. [31]

    W. E. Arnoldi, Quarterly of Applied Mathematics9, 17 (1951)

    1951

  24. [32]

    Saad, Linear Algebra and its Applications34, 269 (1980)

    Y. Saad, Linear Algebra and its Applications34, 269 (1980)

    1980

  25. [33]

    Minganti and D

    F. Minganti and D. Huybrechts, Quantum6, 649 (2022)

    2022

  26. [34]

    Bhattacharya, P

    A. Bhattacharya, P. Nandy, P. P. Nath, and H. Sahu, Journal of High Energy Physics2022, 081 (2022), arXiv:2207.05347

    2022 arXiv

  27. [35]

    Bhattacharya, P

    A. Bhattacharya, P. Nandy, P. P. Nath, and H. Sahu, Journal of High Energy Physics2023, 066 (2023), arXiv:2303.04175

    2023 arXiv

  28. [36]

    Ueno and K

    K. Ueno and K. Takasaki, inGroup Representations and Systems of Differential Equations, Advanced Studies in Pure Mathematics, Vol. 4, edited by K. Okamoto (North- Holland Publishing Co., 1984) pp. 1–95

    1984

  29. [37]

    Takasaki, Journal of Physics A: Mathematical and Theoretical51, 203001 (2018), arXiv:1801.09924 [math- ph]

    K. Takasaki, Journal of Physics A: Mathematical and Theoretical51, 203001 (2018), arXiv:1801.09924 [math- ph]

    2018 arXiv

  30. [38]

    Takahashi, P

    K. Takahashi, P. Nandy, and A. del Campo, Physical Review B113, 144312 (2026), arXiv:2510.19436

    2026 arXiv

  31. [39]

    Simon, Physical Review Letters51, 2167 (1983)

    B. Simon, Physical Review Letters51, 2167 (1983)

    1983

  32. [40]

    M. V. Berry, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences392, 45 (1984)

    1984

  33. [41]

    Provost and G

    J.-P. Provost and G. Vallée, Communications in Mathe- matical Physics76, 289 (1980)

    1980

  34. [42]

    A. W. Shrestha, B. Bhattacharjee, and A. del Campo, Shortcuts to adiabaticity for non-hermitian systems in Krylov space (2026), preprint; not peer reviewed as of 2026-07-25, arXiv:2607.07802 [quant-ph]

    2026 arXiv

  35. [43]

    Ibáñez, S

    S. Ibáñez, S. Martínez-Garaot, X. Chen, E. Torrontegui, and J. G. Muga, Physical Review A84, 023415 (2011), arXiv:1106.2776

    2011 arXiv

  36. [44]

    Ibáñez, S

    S. Ibáñez, S. Martínez-Garaot, X. Chen, E. Torrontegui, and J. G. Muga, Physical Review A86, 019901 (2012)

    2012

  37. [45]

    Okuyama and K

    M. Okuyama and K. Takahashi, Physical Review Letters 117, 070401 (2016)

    2016

  38. [46]

    Gorini, A

    V. Gorini, A. Kossakowski, and E. C. G. Sudarshan, Journal of Mathematical Physics17, 821 (1976)

    1976

  39. [47]

    Lindblad, Communications in Mathematical Physics 48, 119 (1976)

    G. Lindblad, Communications in Mathematical Physics 48, 119 (1976). 19

    1976

  40. [48]

    Kodama and J

    Y. Kodama and J. Ye, Communications in Mathematical Physics178, 765 (1996), arXiv:solv-int/9506005

    1996 arXiv

  41. [49]

    Kodama and L

    Y. Kodama and L. Williams, Communications in Math- ematical Physics335, 247 (2015), arXiv:1308.5011

    2015 arXiv

  42. [50]

    Alexandrov and A

    A. Alexandrov and A. Zabrodin, Journal of Geometry and Physics67, 37 (2013), arXiv:1212.6049 [math-ph]

    2013 arXiv

  43. [51]

    Tsujimoto, Publications of the Research Institute for Mathematical Sciences38, 113 (2002)

    S. Tsujimoto, Publications of the Research Institute for Mathematical Sciences38, 113 (2002)

    2002

  44. [52]

    Maruno and G

    K.-i. Maruno and G. Biondini, Journal of Physics A: Mathematical and General37, 11819 (2004), arXiv:nlin/0406059 [nlin.SI]

    2004 arXiv

  45. [53]

    C. Liu, H. Tang, and H. Zhai, Physical Review Research 5, 033085 (2023), arXiv:2207.13603

    2023 arXiv

  46. [54]

    Bhattacharya, R

    A. Bhattacharya, R. N. Das, B. Dey, and J. Erd- menger, Journal of High Energy Physics2024, 179 (2024), arXiv:2312.11635 [hep-th]

    2024 arXiv

  47. [55]

    Toda, Journal of the Physical Society of Japan22, 431 (1967)

    M. Toda, Journal of the Physical Society of Japan22, 431 (1967)

    1967

  48. [56]

    Flaschka, Physical Review B9, 1924 (1974)

    H. Flaschka, Physical Review B9, 1924 (1974)

    1974

  49. [57]

    Moser, Advances in Mathematics16, 197 (1975)

    J. Moser, Advances in Mathematics16, 197 (1975)

    1975

  50. [58]

    Hirota,The Direct Method in Soliton Theory, Cam- bridge Tracts in Mathematics, Vol

    R. Hirota,The Direct Method in Soliton Theory, Cam- bridge Tracts in Mathematics, Vol. 155 (Cambridge Uni- versity Press, Cambridge, 2004)

    2004

  51. [59]

    Kajiwara, M

    K. Kajiwara, M. Mazzocco, and Y. Ohta, Journal of Physics A: Mathematical and Theoretical40, 12661 (2007), arXiv:nlin/0701029

    2007 arXiv

  52. [60]

    Krattenthaler, Séminaire Lotharingien de Combina- toire42, B42q (1999), arXiv:math/9902004

    C. Krattenthaler, Séminaire Lotharingien de Combina- toire42, B42q (1999), arXiv:math/9902004

    1999 arXiv

  53. [61]

    P. D. Lax, Communications on Pure and Applied Math- ematics21, 467 (1968)

    1968

  54. [62]

    Hirota, M

    R. Hirota, M. Ito, and F. Kako, Progress of Theoretical Physics Supplement94, 42 (1988)

    1988

  55. [63]

    Adler and P

    M. Adler and P. van Moerbeke, International Mathemat- ics Research Notices1997, 555 (1997)

    1997

  56. [64]

    D. J. Yates and A. Mitra, Physical Review B104, 195121 (2021), arXiv:2105.13246

    2021 arXiv

  57. [65]

    A. A. Nizami and A. W. Shrestha, Physical Review E 108, 054222 (2023), arXiv:2305.00256

    2023 arXiv

  58. [66]

    M. H. Gutknecht, SIAM Journal on Matrix Analysis and Applications13, 594 (1992)

    1992

  59. [67]

    R. W. Freund, M. H. Gutknecht, and N. M. Nachtigal, SIAM Journal on Scientific Computing14, 137 (1993)

    1993

  60. [68]

    F. L. Bauer and C. T. Fike, Numerische Mathematik2, 137 (1960)

    1960

  61. [69]

    M. A. Botchev, V. Grimm, and M. Hochbruck, SIAM Journal on Scientific Computing35, A1376 (2013)

    2013

  62. [70]

    Jawecki, W

    T. Jawecki, W. Auzinger, and O. Koch, BIT Numerical Mathematics60, 157 (2020), arXiv:1809.03369

    2020 arXiv

  63. [71]

    L. N. Trefethen, SIAM Review39, 383 (1997)

    1997

  64. [72]

    Guéry-Odelin, A

    D. Guéry-Odelin, A. Ruschhaupt, A. Kiely, E. Tor- rontegui, S. Martínez-Garaot, and J. G. Muga, Reviews of Modern Physics91, 045001 (2019)

    2019

  65. [73]

    Demirplak and S

    M. Demirplak and S. A. Rice, The Journal of Physical Chemistry A107, 9937 (2003)

    2003

  66. [74]

    M. V. Berry, Journal of Physics A: Mathematical and Theoretical42, 365303 (2009)

    2009

  67. [75]

    H. R. Lewis, Jr. and W. B. Riesenfeld, Journal of Math- ematical Physics10, 1458 (1969)

    1969

  68. [76]

    D. C. Brody, Journal of Physics A: Mathematical and Theoretical47, 035305 (2014)

    2014

  69. [77]

    Ashida, Z

    Y. Ashida, Z. Gong, and M. Ueda, Advances in Physics 69, 249 (2020)

    2020

  70. [78]

    W. W. Symes, Physica D: Nonlinear Phenomena4, 275 (1982)

    1982

  71. [79]

    Deift, L.-C

    P. Deift, L.-C. Li, and C. Tomei, Communications on Pure and Applied Mathematics42, 443 (1989)

    1989

  72. [80]

    Dieci and T

    L. Dieci and T. Eirola, SIAM Journal on Matrix Analysis and Applications20, 800 (1999)

    1999

  73. [81]

    Griffiths and J

    P. Griffiths and J. Harris,Principles of Algebraic Geom- etry(John Wiley & Sons, New York, 1978)

    1978

  74. [82]

    Ishikawa and M

    M. Ishikawa and M. Wakayama, Journal of Com- binatorial Theory, Series A113, 113 (2006), arXiv:math/0312358 [math.CO]

    2006 arXiv

  75. [83]

    Mitscherling, A

    J. Mitscherling, A. Avdoshkin, and J. E. Moore, Physical Review B112, 085104 (2025), arXiv:2412.03637

    2025 arXiv

  76. [84]

    Ransford,Potential Theory in the Complex Plane, London Mathematical Society Student Texts, Vol

    T. Ransford,Potential Theory in the Complex Plane, London Mathematical Society Student Texts, Vol. 28 (Cambridge University Press, Cambridge, 1995)

    1995

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