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Long-time Krylov spread complexity scales linearly with Fock-space size in the ergodic phase and only sublinearly in the MBL phase, so the late-time state fills a finite versus vanishing fraction of the Krylov chain.

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load-bearing objection Clean ED demonstration that infinite-time Krylov spread complexity scales as N_H vs N_H^\alpha (\alpha<1) and that MBL profiles are stretched-exponential, with a solid large-deviation picture of rare resonant eigenstates. the 2 major comments →

arxiv 2603.25724 v2 pith:GIEJFGBB submitted 2026-03-26 cond-mat.dis-nn cond-mat.stat-mechquant-ph

Krylov-space anatomy and spread complexity of a disordered quantum spin chain

Bikram Pain , David E. Logan , Sthitadhi Roy This is my paper
classification cond-mat.dis-nn cond-mat.stat-mechquant-ph
keywords Krylov spread complexitymany-body localisationergodic phasedisordered spin chainstretched exponentiallarge-deviation analysisFock-space dimensioneigenstate resonances
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.

The paper asks how a many-body quantum state spreads when it is written in the ordered Krylov basis generated by its own Hamiltonian, and whether that basis-optimised spread cleanly separates the ergodic phase from the many-body localised (MBL) phase of a disordered spin chain. It finds that the infinite-time Krylov spread complexity grows linearly with the dimension of Hilbert space in the ergodic regime, so the late-time state occupies a finite fraction of the Krylov chain, while the same quantity grows only as a sublinear power of the dimension in the MBL regime, so the state remains confined to a vanishing fraction of the chain. The average probability profile along the chain itself decays as a stretched exponential in the MBL phase; a simple phenomenological model attributes the stretch to a broad distribution of exponential decay lengths across eigenstates. A large-deviation analysis of the individual eigenstate contributions confirms that almost every eigenstate participates in the ergodic sum, whereas only a vanishing (yet still exponentially large) fraction of rare, anomalously complex eigenstates dominate the MBL sum. The result supplies a transparent, one-dimensional geometric picture of how localisation and rare resonances appear once distance is measured in the Krylov basis rather than on the high-dimensional Fock-space graph.

Core claim

Infinite-time Krylov spread complexity scales as N_H in the ergodic phase of the disordered tilted-field Ising chain and as N_H^α with α<1 in the MBL phase; the associated disorder-averaged profile Λ_n on the Krylov chain is flat (after a short transient) when ergodic and stretched-exponential when many-body localised, the latter arising because the infinite-time state is a weighted sum of exponentially decaying eigenstate amplitudes whose characteristic lengths are themselves broadly distributed.

What carries the argument

Krylov spread complexity S_{K,∞} = Σ_n n Λ_n, where Λ_n is the infinite-time probability of finding the state on the n-th Krylov orbital; this quantity is the first moment of a probability distribution on a one-dimensional chain of length equal to the Fock-space dimension and is basis-optimised by construction of the Krylov basis.

Load-bearing premise

The analytic explanation for the stretched-exponential profile assumes that each eigenstate decays purely exponentially on the Krylov chain with a length drawn from a simple exponential distribution, and that those lengths themselves are exponentially distributed across disorder realisations.

What would settle it

Exact diagonalisation of the same model at larger system sizes that extracts both the scaling exponent α of S_{K,∞} and the stretch exponent γ of ⟨Λ_n⟩; if α remains 1 deep in the putative MBL regime or if γ fails to approach the predicted asymptotic value 1/3, the central geometric claim is falsified.

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

2 major / 4 minor

Summary. The paper studies the Krylov-space anatomy of states and the infinite-time Krylov spread complexity S_{K,∞} for the disordered tilted-field Ising chain, contrasting the ergodic and MBL regimes. Using the Lanczos-generated Krylov basis from a mid-spectrum product state, the authors show that ⟨S_{K,∞}⟩ scales linearly with Fock-space dimension N_H in the ergodic phase (occupying a finite fraction of the Krylov chain) and sublinearly as N_H^α with α<1 in the MBL phase (occupying a vanishing fraction). The disorder-averaged profile ⟨Λ_n⟩ collapses onto a scaling form with the same α and, in the MBL regime, exhibits a stretched-exponential decay. A large-deviation analysis of the eigenstate contributions S_{K,|E⟩} further shows that the ergodic sum is carried by a finite fraction of eigenstates while the MBL sum is dominated by a vanishing (but still exponentially large) fraction of anomalously complex eigenstates, with multifractal IPRs of those contributions. A phenomenological theory based on exponential length-scale distributions is offered to rationalize the stretch exponent.

Significance. If the reported scalings hold, the work supplies a clean, basis-optimized diagnostic that sharply separates ergodic and MBL phases on a one-dimensional Krylov chain whose length is exactly N_H. The combination of direct ED scaling of S_{K,∞} and Λ_n (Figs. 4–7), the large-deviation entropy density Σ(x) (Fig. 9, Table I), and the multifractal IPR of eigenstate complexities (Fig. 10) is internally consistent and goes beyond earlier operator-Krylov studies that are less sensitive to MBL. The explicit mapping between Krylov orbitals and Fock-space Hamming shells (Sec. III B, Fig. 2) also clarifies why the Krylov chain remains a faithful probe even when most orbitals have support over the entire Fock graph. Finite-size caveats on MBL numerics are generic and already acknowledged; the central numerical distinction itself is robust within accessible sizes.

major comments (2)
  1. Sec. IV C, Eqs. (33)–(47): the phenomenological theory assumes pure exponential decay of contributing eigenstate amplitudes with lengths ξ_|E⟩ drawn from an exponential distribution of mean ξ_d, and that the ξ_d themselves are exponentially distributed over disorder. This functional form is chosen for analytic convenience and is only a posteriori consistent with the observed stretch exponents (γ ≃ 1/2 at accessible sizes, asymptotically 1/3). Because the central claims of the paper rest on the ED scalings of S_{K,∞} and Λ_n (Figs. 4–7) and on the large-deviation analysis (Sec. V), not on this ansatz, the theory should be clearly labeled as a post-hoc rationalization rather than a derivation. A short numerical check of the actual distribution of effective decay lengths extracted from individual eigenstates would strengthen or falsify the assumption.
  2. Sec. IV A and Fig. 4: the reported exponents α(W) are extracted from system sizes L ≲ 14–16. While the ergodic α = 1 result is solid, the MBL values α < 1 (and the associated claim that the long-time state occupies a vanishing fraction of the Krylov chain) remain subject to the usual finite-size caveats of MBL numerics. The manuscript should state more explicitly the largest L used for each W, the number of disorder realizations, and whether any drift of α with L is visible; a brief comparison with an independent localization diagnostic (e.g., half-chain entanglement or Fock-space IPR) on the same samples would help calibrate how deep into the putative MBL regime the data sit.
minor comments (4)
  1. Fig. 7 and surrounding text: the stretch exponent is quoted as γ ≃ 1/2 for the bulk of the data and γ = 1/3 for the largest n/N_H^α. A single sentence clarifying that the asymptotic analytic result is γ = 1/3 while finite-size data remain closer to 1/2 would remove residual ambiguity.
  2. Appendix A, Fig. 11: the scaling of ⟨b_n⟩ and the effective disorder W_n are shown but not used later. Either a brief remark on why these bare Krylov-matrix statistics do not distinguish the phases, or a pointer to future work, would improve cohesion.
  3. Eq. (9) and the definition of Λ_n: the sum rule ∑_n Λ_n = 1 is stated, but it would help the reader to note explicitly that the infinite-time average eliminates the off-diagonal E eq E' terms, so that Λ_n is strictly a sum of |c_n,E|^2 |c_0,E|^2.
  4. References: a few recent works on state Krylov complexity near the MBL transition (e.g., those already cited as [54–59]) could be more explicitly contrasted in the introduction to highlight what is new in the infinite-time anatomy and large-deviation analysis.

Circularity Check

1 steps flagged

No significant circularity: central scalings and large-deviation results are direct ED observables; the exponential length-scale ansatz of Sec. IV C is a post-hoc phenomenological rationalization, not a load-bearing derivation.

specific steps
  1. other [Sec. IV C, Eqs. (33)–(34), (43)–(44)]
    "The two key ingredients that enter the phenomenological picture are: (i) For a given disorder realisation, there is a normalised distribution over eigenstates, P_ u|E( u), of lengthscales u_|E on the Krylov chain, of form P_ u|E( u)=1/ u_d imes p_ u|E( u/ u_d). u_d itself has a distribution P_ u d( u_d)=1/ u imes p_ u d( u_d/ u). u Motivated by this, we assume also the simplest such exponential distribution for P_ u|E( u), P_ u|E( u)=1/ u_d exp[- u/ u_d]."

    The exponential forms are chosen by hand for analytic convenience after the stretched-exponential profile has already been observed numerically; they reproduce u=1/2 (or 1/3 after disorder average) by construction of the integral representation, but are not used to generate or force the primary ED scalings of S_{K, u} or u_n. This is a mild post-hoc rationalization rather than a circular derivation of the central claims.

full rationale

The paper’s primary claims (linear vs sublinear scaling of S_{K,\infty} with N_H, stretched-exponential profile of u_n, and large-deviation dominance by rare eigenstates) are extracted from exact diagonalization of the microscopic tilted-field Ising Hamiltonian after Lanczos construction of the Krylov basis (Secs. IV A–B, V; Figs. 3–10, Table I). These quantities are defined independently via Eqs. (6)–(9) and (48)–(49) and measured without intermediate fitting that is later re-labeled as prediction. The phenomenological theory of Sec. IV C posits exponential distributions P_ u|E and P_ u d solely to rationalize the observed stretch exponent u o 1/2 (finite-size) or 1/3 (asymptotic); the ansatz is not used to define or force the measured S_{K, u} or u_n, nor is it claimed to be first-principles. Self-citations to the authors’ prior Fock-space work supply background context and are not invoked as uniqueness theorems or load-bearing premises for the Krylov results. No self-definitional loop, fitted-input-as-prediction, or renaming of a known result appears. The single minor note is the a-posteriori choice of exponential forms, which does not elevate the score above 1.

Axiom & Free-Parameter Ledger

3 free parameters · 3 axioms · 0 invented entities

The work rests on the standard definition of Krylov complexity (minimizing the cost function with µ_n=n), the existence of an MBL regime in the TFI chain for W≳3.7 at accessible sizes, and two phenomenological distributional assumptions introduced to explain the stretch exponent. No new particles or forces are postulated; free parameters are the usual disorder strength, system size, and the fitted exponents α, γ.

free parameters (3)
  • α (scaling exponent of ⟨S_{K,∞}⟩) = 1 (ergodic); <1, W-dependent (MBL)
    Extracted from power-law fits of disorder-averaged infinite-time complexity versus N_H for each W; values decrease from 1 (ergodic) to ~0.57 (W=10).
  • γ (stretch exponent of ⟨Λ_n⟩) = ≃1/2 (finite size); 1/3 (asymptotic claim)
    Read off from log-log plots of -ln f(x); consistent with 1/2 over accessible range, asymptotically argued to be 1/3.
  • critical disorder W_c = ≃3.7
    Taken from prior literature as ≃3.7 for the chosen parameters; used only to label phases, not fitted here.
axioms (3)
  • domain assumption The Krylov basis generated by H from |ψ_0⟩ minimizes the cost function C_V = Σ n |⟨ψ_t|V_n⟩|^2 among all ordered orthonormal bases.
    Invoked in Sec. II via citation to Balasubramanian et al. (2022); taken as given rather than re-derived.
  • ad hoc to paper For each disorder realization the contributing eigenstate amplitudes on the Krylov chain decay exponentially with lengths ξ_|E| drawn from an exponential distribution of mean ξ_d, and the ξ_d themselves are exponentially distributed over disorder.
    Introduced in Sec. IV C to obtain the stretched-exponential profile analytically; not derived from the microscopic Hamiltonian.
  • domain assumption The disordered tilted-field Ising chain hosts a many-body localized phase for sufficiently strong disorder at the system sizes studied.
    Standard working assumption of the MBL literature; critical W≃3.7 quoted from Abanin et al. (2021).

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read the original abstract

We investigate the anatomy and complexity of quantum states in Krylov space, in the ergodic and many-body localised (MBL) phases of a disordered, interacting spin chain. The Krylov basis generated by the Hamiltonian from an initial state provides a representation in which the spread of the time-evolving state constitutes a basis-optimised measure of complexity. We show that the long-time Krylov spread complexity sharply distinguishes the two phases. In the ergodic regime, the infinite-time complexity scales linearly with the Fock-space dimension, indicating that the state spreads over a finite fraction of the Krylov chain. By contrast, it grows sublinearly in the MBL regime, implying that the long-time state occupies only a vanishing fraction of the chain. Further, the profile of the infinite-time state along the Krylov chain exhibits a stretched-exponential decay in the MBL regime. This behaviour reflects a broad distribution of decay lengthscales, associated with different eigenstates contributing to the long-time state. Consistently, a large-deviation analysis of the statistics of eigenstate spread complexities shows that while the ergodic regime receives contributions from almost all eigenstates, the complexity in the MBL regime is dominated by a vanishing fraction of eigenstates, which have anomalously large complexity relative to the typical ones.

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Forward citations

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Reference graph

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