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Anomalous initial states in thermalizing nonintegrable systems store their late-time memory in a compact low-depth Krylov-space core, while generic states do not.

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2026-08-01 00:51 UTC pith:ASDDPFL3

load-bearing objection Sharp new Krylov diagnostic, but the PXP scar numbers rest on an unstated choice about degenerate gap handling that needs to be resolved before the headline claim is solid. the 2 major comments →

arxiv 2607.26011 v1 pith:ASDDPFL3 submitted 2026-07-28 hep-th cond-mat.stat-mechquant-ph

Krylov-Space Memory Cores

Mohsen Alishahiha , Mohammad Javad Vasli This is my paper
classification hep-th cond-mat.stat-mechquant-ph
keywords Krylov spacememory coreweak thermalizationmany-body scarringconfinementdiagonal ensembleLanczos coefficientsKrylov complexity
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 tries to show that when certain special initial states fail to thermalize in otherwise thermalizing quantum systems, the late-time information does not spread uniformly through the system's Krylov space. Instead, it collects in a compact low-depth 'memory core' where long-time fluctuations, deviation from the Gibbs state, and ongoing probability exchange all remain concentrated, while the overall stationary probability still spreads far beyond that core. The same structure appears in three different mechanisms — weak thermalization, confinement-induced slow dynamics, and many-body scarring — but not in generic reference states. If correct, this gives a common stationary geometry for anomalous nonthermalization and a new way to locate where structured quantum memory resides.

Core claim

The paper claims that for atypical initial states in otherwise thermalizing nonintegrable systems, the stationary long-time probability distribution in Krylov space separates into a compact low-depth memory core — where residual temporal fluctuations, deviation from the Gibbs reference, and probability-current fluctuations all concentrate — and a much broader stationary occupation halo that carries probability but little residual activity. The core is quantified by n_mc, the smallest depth containing 95% of the weight of each of three diagnostics; for example, the weakly thermalizing |X+> state has n_mc = 32 while its occupation cloud reaches depth 2053, and the scarred PXP Néel state has n_

What carries the argument

The paper's central object is the Krylov basis generated by the Lanczos algorithm, in which the Hamiltonian becomes a one-dimensional hopping chain and the initial state starts at site 0. On that chain it defines three depth-resolved stationary diagnostics: χ_n (long-time variance of occupation), d_n (deviation from the energy-matched Gibbs state restricted to the cyclic subspace), and Γ_n (variance of the probability current across the bond between n−1 and n). The last has a compact closed form under a nondegenerate-active-gap condition, Γ_n = 2 b_n^2 [Π_n Π_{n−1} − |<n|ρ_diag|n−1>|^2]. The memory core is defined as the smallest depth n_mc containing 95% of the weight of each diagnostic; th

Load-bearing premise

The compact spectral formulas for the diagnostics, especially Γ_n, hold only under the nondegenerate-active-gap condition (Eq. 22); the PXP model has exact chiral symmetry and zero modes that may violate this condition, and the text says a gap-resolved version (Eq. A14) should be used when it fails but never states which version was actually implemented in Section III C.

What would settle it

Compute Γ_n for the PXP Néel state directly from a long-time average of the squared current (Eq. 38) without assuming nondegenerate gaps, and compare with the compact formula (Eq. 39) over the first few hundred bonds; a discrepancy for the state's own dynamics would mean the reported n_mc = 109 is not supported. Alternatively, check whether the total weights X, N, G remain appreciable at system sizes beyond L = 28; if they decay with L, the core is a finite-size effect.

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

If this is right

  • Anomalous relaxation in different microscopic settings — weak thermalization, confinement, and scarring — shares a common low-depth stationary organization in Krylov space.
  • Krylov complexity alone is insufficient to detect this memory: a system can have large spreading and no stationary core, as the exactly solvable escaping geometries show.
  • The core–halo separation provides an operational, state-selective diagnostic for locating nonthermal memory in otherwise thermalizing systems.
  • Finite-size data are consistent with the core depth growing at most linearly with system size while the cyclic dimension grows much faster, so the core is subextensive.
  • The three diagnostics are complementary: occupation supplies the probability background, but fluctuation, Gibbs mismatch, and current activity must co-localize to define a core.

Where Pith is reading between the lines

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

  • The same memory-core framework could be tested on other types of anomalous dynamics, such as disorder-driven localization or prethermalization, where a compact residual region may or may not appear.
  • The determinant form of Γ_n suggests that the core region is characterized by suppressed nearest-neighbor coherence in the diagonal ensemble; this could be connected to entanglement or operator-growth structure.
  • A direct time-domain computation of Γ_n for the PXP Néel state, without any gap assumption, would settle whether the reported n_mc = 109 is an artifact of the compact formula or a genuine feature.
  • State-selective cores could serve as numerical probes for identifying scarred eigenstates in larger systems where exact diagonalization is infeasible.

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 / 3 minor

Summary. The paper introduces a stationary, depth-resolved framework for organizing anomalous initial-state memory in the Krylov basis. For a finite Hamiltonian and initial state it defines the diagonal-ensemble occupation Π_n, the long-time occupation variance χ_n, the Gibbs-deviation profile d_n, and a new bond-resolved current-fluctuation activity Γ_n. A 'Krylov-space memory core' is identified as the smallest interval containing 95% of the weight of all three residual diagnostics, provided the total weights are appreciable. Numerically, the authors report compact cores for weakly thermalizing, confinement-sensitive, and PXP-scarred initial states in otherwise thermalizing nonintegrable models, with generic reference states lacking comparable compact co-localized signals. An auxiliary integrable comparison and exactly solvable infinite Krylov chains are used to argue that the core is not a trivial consequence of integrability or Krylov growth.

Significance. If the numerical claims hold, the paper offers a genuinely new unifying perspective: anomalous dynamics produced by different physical mechanisms (weak thermalization, confinement, many-body scarring) leave the same stationary, depth-resolved signature in Krylov space, distinct from Krylov complexity and from the occupation profile alone. The analytic derivation of χ_n and Γ_n under the nondegenerate-active-gap condition is clean and internally consistent, and the bounds (27), (41), and the determinant form (A22) are useful physical constraints. The careful treatment of exact energy degeneracies through active spectral projectors is a strength, as is the effort to provide finite-size tables across three models. However, the manuscript ships no code or data, and one of the central numerical applications (PXP) suffers from an unresolved ambiguity about which spectral formula was actually evaluated; these issues must be fixed before the central claim can be verified.

major comments (2)
  1. [Sec. III C, Eqs. (22),(39), App. A/C] The PXP Néel state violates the nondegenerate-active-gap condition (22). The chiral symmetry C=(-1)^{N_exc} anticommutes with H and leaves |Z2> invariant (L=28 has 14 excitations), so every active nonzero energy E has an active partner -E; together with active E=0, the ordered pairs (E,0) and (0,-E) share the same gap. The text never states whether χ_n and Γ_n for PXP were computed with the compact forms (25)/(39) or with the gap-resolved sum Eq. (A14)/direct long-time averaging. If the compact forms were used, the reported n_Γ^(0.95)=92 and n_mc=109, and the scarred current-activity claim, are not justified. Please state the exact prescription used and, if necessary, recompute these quantities.
  2. [Sec. III, App. C/D] The central numerical claim is not independently checkable from the manuscript as written. No code or data are provided; the active-spectral construction is described only verbally; the claimed tolerance stability ("corrections of at most ±3") is not demonstrated; and Tables I–VI contain no error bars or numerical uncertainty estimates for the threshold depths. Given that the core–halo distinction is the main result, please release the numerical code/data or provide a detailed pseudocode for the active-spectral construction, together with a table of stability checks for each system size and tolerance choice.
minor comments (3)
  1. [Sec. IV A] The notation P_n in Eq. (66) for the integrated transient occupation is easily confused with the probability occupation P_n(t) used throughout the paper. Consider renaming this integrated quantity (e.g., I_n) to avoid ambiguity, especially since the paper emphasizes that P_n is nonzero for the constant-chain example while Π_n=0.
  2. [App. C] Equation (C17) subtracts the zero-mode counts to obtain 26021, but the text states that the sublattice imbalance gives 'at least' 133 and 58 zero modes. The equality 26211-(133+58)+1=26021 should be phrased as conditional on the numerically observed saturation of the imbalance bounds, not as an exact symmetry result.
  3. [App. D] The finite-size tables list n_mc and D0, but do not state the ε_E and ε_ω tolerance values used for each L. Please include these values and, for each table, indicate whether the compact formulas or the gap-resolved formulas were used, so that the scaling analysis can be reproduced.

Circularity Check

0 steps flagged

No significant circularity; the memory-core diagnostics are operationally defined and empirically discriminating, and no load-bearing result reduces to its inputs.

full rationale

The paper's central construction is operational rather than definitional. The stationary occupation Πn is the diagonal-ensemble expectation (Eq. 19); χn is the long-time variance of the Krylov projector (Eq. 24); dn is the deviation from an energy-matched Gibbs reference (Eqs. 32–33); Γn is the long-time variance of the Krylov current, with Eq. (39) derived explicitly from that definition in Appendix A under the stated nondegenerate-active-gap condition. The memory-core depth nmc is a threshold statistic over these independently defined profiles (Eq. 48), not a fitted parameter. The paper explicitly states that the framework is an operational test, not a prediction that a core must exist: "The formalism therefore provides an operational test, not a prediction that a core must exist." The numerical contrast between anomalous states (nmc = 32, 36, 41, 57, 109) and generic/random controls (nmc comparable to the full cyclic dimension) shows that the diagnostic discriminates rather than merely rediscovering its construction. The self-citations [21,62,63] are contextual and do not carry a load-bearing uniqueness theorem or ansatz; the derivations in this paper are self-contained. The only substantive technical concern is whether the PXP Γn computation used the gap-resolved Eq. (A14) when Eq. (22) may fail due to chiral symmetry. That is a numerical-implementation ambiguity and a correctness question, not circularity: both formulas follow from the same physical definition, and the paper identifies the condition under which each applies. No step reduces the reported memory-core result to its inputs.

Axiom & Free-Parameter Ledger

2 free parameters · 6 axioms · 1 invented entities

The paper introduces one central conceptual object (the memory core) and relies on two hand-chosen numerical parameters (q and the spectral tolerances). The main analytical assumptions are standard Krylov/spectral facts plus the nondegenerate-active-gap condition, whose numerical status is the largest unresolved item. The finite-size extrapolation is acknowledged as an assumption.

free parameters (2)
  • cumulative fraction q = 0.95
    The memory-core depth n_mc is defined using q=0.95 in Eq. (48). This threshold is chosen by hand; changing q changes n_mc quantitatively, though the authors report robustness of the qualitative conclusion.
  • energy-grouping and spectral-weight tolerances (ε_E, ε_ω) = 1e-11, 1e-14
    Appendix C uses these tolerances to define the numerically resolved active spectral set and D0=26015 for PXP L=28. The authors state D0 varies by about ±3 under reasonable tolerance changes, so the exact cyclic dimension depends on these chosen cutoffs.
axioms (6)
  • standard math Lanczos recursion generates an orthonormal Krylov basis in which H_K is tridiagonal (Eqs. 3-7).
    This is the standard Lanczos construction used throughout the paper.
  • standard math The infinite-time average of Krylov occupation equals the diagonal ensemble in the active energy subspace (Eqs. 12-19).
    This follows from dephasing of distinct energy eigenspaces and is stated exactly for finite-dimensional systems.
  • ad hoc to paper The nondegenerate-active-gap condition (Eq. 22) holds in the numerical systems, or the gap-resolved formula (Eq. A14) was used when it fails.
    The compact formulas for χ_n and Γ_n are derived under Eq. (22). The paper does not show that PXP, which has exact symmetries and zero modes, satisfies this condition, nor does it document use of the gap-resolved alternative in Section III C.
  • domain assumption The energy-matched Gibbs state on the cyclic subspace (Eqs. 29-31) is an informative equilibrium benchmark for nonintegrable systems.
    The authors acknowledge that this is a state-dependent, cyclic-space benchmark rather than a standard global Gibbs ensemble. Interpretability of d_n depends on this choice.
  • domain assumption Finite-size data (L=10-14 for Ising, L=20-28 for PXP) are representative, and the observed trends support n_mc = O(L) with n_mc/D0 → 0.
    Appendix D explicitly frames this as an extrapolation from finite systems. The central claim of subextensive memory cores depends on this trend persisting.
  • domain assumption Energy grouping with stated tolerances resolves the active spectrum well enough that D0 and the cumulative depths are stable.
    Appendix C reports ±3 stability for D0 and threshold depths under tolerance changes, but no released code allows independent verification.
invented entities (1)
  • Krylov-space memory core no independent evidence
    purpose: A compact, low-depth interval where residual fluctuation weight, Gibbs-deviation weight, and Krylov-current-fluctuation weight are simultaneously concentrated for anomalous initial states.
    The memory core is defined entirely through the paper's three diagnostics and has no falsifiable handle outside the Krylov-basis framework. It is an organizational concept rather than an independently measurable entity.

pith-pipeline@v1.3.0-alltime-deepseek · 28510 in / 10275 out tokens · 94583 ms · 2026-08-01T00:51:04.306517+00:00 · methodology

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read the original abstract

We introduce Krylov-space memory cores as stationary, depth-resolved structures that reveal how anomalous initial-state memory is organized inside the Krylov space of otherwise thermalizing nonintegrable systems. The stationary occupation profile identifies where late-time probability is concentrated along the Krylov chain, while complementary diagnostics of residual equilibration fluctuations, deviation from the Gibbs reference, and long-time Krylov-current fluctuations determine the physical character of that region. Across weak thermalization, confinement-induced anomalous dynamics, and many-body scarring, anomalous initial states develop compact low-depth memory cores that carry appreciable residual fluctuations, Gibbs mismatch, and persistent current-fluctuation activity. These cores are often embedded within substantially broader stationary occupation halos. Generic reference states, by contrast, do not exhibit a comparable combination of signal strength and spatial compactness. An auxiliary integrable comparison further shows that compact Krylov memory is state selective rather than a generic consequence of integrability. Krylov-space memory cores therefore provide a stationary framework for identifying where structured quantum memory resides and how it remains dynamically encoded.

Figures

Figures reproduced from arXiv: 2607.26011 by Mohammad Javad Vasli, Mohsen Alishahiha.

Figure 2
Figure 2. Figure 2: Stationary Krylov occupation profile Πn for the chaotic mixed-field Ising chain. The structured states exhibit strong low-depth peaks together with broader stationary tails. The profile specifies where late-time probability resides; it does not by itself identify the active memory core. The inset extends the comparison to larger Krylov depths. 0 1000 2000 3000 4000 0 2. × 10-7 4. × 10-7 6. × 10-7 8. × 10-7… view at source ↗
Figure 3
Figure 3. Figure 3: Residual equilibration-fluctuation profile [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Signed Gibbs-deviation profile dn = Πn − Π (Gibbs) n for the chaotic mixed-field Ising chain. The weakly thermaliz￾ing state |X+⟩ exhibits the largest low-depth Gibbs-reference mismatch. The corresponding mismatch for |Z+⟩ is compact but considerably weaker, while |Y +⟩ and the random controls remain close to the Gibbs benchmark. a similarly small active fraction, nmc/D0 ≃ 4.4 × 10−3 , but its Gibbs-deviat… view at source ↗
Figure 5
Figure 5. Figure 5: Krylov-current fluctuation activity Γn for the chaotic mixed-field Ising chain. The states |X+⟩ and |Z+⟩ support strong back-and-forth probability-current fluctua￾tions near the Krylov origin. The efficiently thermalizing and random controls have much weaker or broadly distributed ac￾tivity. The inset shows the extended-depth tail. termediate, activity-dominated memory core rather than the strongly non-Gib… view at source ↗
Figure 9
Figure 9. Figure 9: Signed Gibbs-deviation profile dn = Πn − Π (Gibbs) n in the confinement regime. The N´eel state has the largest and most compact mismatch, while the domain-wall and bubble states show substantial but broader deviations. The random control remains close to the Gibbs benchmark. 0 2000 4000 6000 8000 0 5. × 10-9 1. × 10-8 1.5 × 10-8 2. × 10-8 0 10 20 30 40 50 0.000 0.002 0.004 0.006 0.008 0.010 [PITH_FULL_IM… view at source ↗
Figure 10
Figure 10. Figure 10: Krylov-current fluctuation activity Γn in the confinement regime. The N´eel state supports a sharp low￾depth activity profile. The domain-wall and bubble states remain dynamically active over broader intervals, whereas the random control has a much weaker signal. produces state-selective active cores with different phys￾ical compositions and spatial extents. C. PXP model and quantum many-body scars We fin… view at source ↗
Figure 8
Figure 8. Figure 8: Residual equilibration-fluctuation profile [PITH_FULL_IMAGE:figures/full_fig_p009_8.png] view at source ↗
Figure 11
Figure 11. Figure 11: displays the corresponding Lanczos coefficients. a. Stationary occupation background [PITH_FULL_IMAGE:figures/full_fig_p010_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Stationary Krylov occupation profile Πn for the periodic L = 28 PXP model. The scarred N´eel state has a pro￾nounced low-depth peak and a feature near the scar-related scale, but its stationary probability also extends through a long tail. The reference state is broader. The inset shows the larger-depth behavior [PITH_FULL_IMAGE:figures/full_fig_p010_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Residual equilibration-fluctuation profile [PITH_FULL_IMAGE:figures/full_fig_p011_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Signed Gibbs-deviation profile dn = Πn−Π (Gibbs) n for the periodic L = 28 PXP model. The scarred N´eel state has a pronounced low-depth mismatch, including the scar￾related feature near n ≃ L. The non-scarred reference re￾mains much closer to the Gibbs benchmark. 0 100 200 300 400 500 0.000 0.005 0.010 0.015 0.020 10 20 30 40 50 0.000 0.005 0.010 0.015 0.020 0.025 [PITH_FULL_IMAGE:figures/full_fig_p011_… view at source ↗
Figure 15
Figure 15. Figure 15: Krylov-current fluctuation activity Γn for the pe￾riodic L = 28 PXP model. The scarred state supports strong back-and-forth current fluctuations within the low-depth scar￾related region, whereas the reference activity is much weaker. Table III gives the sharpest core–halo separation in the paper. For |Z2⟩, the occupation depth is about ninety times larger than nmc, while the active core occupies only 4.2×… view at source ↗
Figure 16
Figure 16. Figure 16: Cumulative profiles for the chaotic mixed-field [PITH_FULL_IMAGE:figures/full_fig_p012_16.png] view at source ↗
Figure 18
Figure 18. Figure 18: Cumulative profiles for the periodic L = 28 PXP model. From top to bottom: ηΠ, ηχ, ηd, and ηΓ. For the scarred N´eel state, the three active curves saturate by nmc = 109, whereas 95% of the stationary occupation requires 9818 Krylov depths. The reference state has weak residual weights and accumulates them over nearly the full cyclic space. The results establish a common stationary phe￾nomenology across w… view at source ↗
Figure 19
Figure 19. Figure 19: Stationary Krylov occupation profile Πn for the integrable Ising chain (L = 14, g = −1.05, h = 0). The homogeneous states exhibit strong early-depth concen￾tration, while the symmetry-matched random reference has a broader stationary occupation profile. The figure illustrates that compact stationary probability is not by itself a signa￾ture of anomalous nonthermalization. 0 500 1000 1500 2000 2500 3000 0 … view at source ↗
Figure 20
Figure 20. Figure 20: Residual fluctuation profile χn for the integrable Ising chain (L = 14, g = −1.05, h = 0). The homogeneous states show concentrated early-depth fluctuations, while the random reference has weaker residual fluctuation weight. The physical interpretation differs from the anomalous noninte￾grable cases because the structure originates from integrabil￾ity. c. Krylov-resolved Gibbs mismatch. In an integrable s… view at source ↗
Figure 21
Figure 21. Figure 21: Signed Gibbs-mismatch profile dn = Πn−Π (Gibbs) n for the integrable Ising chain (L = 14, g = −1.05, h = 0). The profile measures deviation from the canonical Gibbs reference only. The homogeneous states exhibit stronger early-depth mismatch, while the random reference remains closer to the canonical benchmark. 0 500 1000 1500 2000 2500 3000 0.00000 0.00001 0.00002 0.00003 0.00004 0 10 20 30 40 50 0.00 0.… view at source ↗
Figure 22
Figure 22. Figure 22: Krylov-current fluctuation activity Γn for the in￾tegrable Ising chain (L = 14, g = −1.05, h = 0). The homoge￾neous states display early-depth current activity, whereas the random reference has weaker activity throughout the Krylov chain. their residual fluctuation and current activity are con￾centrated near the Krylov origin. The random state ac￾cumulates these weights more gradually. We do not interpret… view at source ↗

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Reference graph

Works this paper leans on

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