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arxiv: 2605.26055 · v1 · pith:FIKQYEALnew · submitted 2026-05-25 · ✦ hep-th · quant-ph

Krylov Complexity for Plane Wave Matrix Model

Dibakar Roychowdhury This is my paper

Pith reviewed 2026-06-29 20:30 UTC · model grok-4.3

classification ✦ hep-th quant-ph
keywords Krylov complexityPlane wave matrix modelBMN matrix modelLanczos coefficientsMass deformationOperator growthState complexity
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The pith

In the reduced BMN plane wave matrix model, Lanczos coefficients for both state and operator Krylov complexity scale linearly with the mass deformation parameter.

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

The paper analyzes Krylov complexity in the BMN plane wave matrix model under large mass deformation by performing consistent reductions of the N=3 and N=4 representations to enable Hamiltonian analysis. This leads to explicit computation of Lanczos coefficients that turn out to be fixed completely by the mass parameter through linear scaling. The early-time growth of complexity receives quadratic corrections from the mass term, and these corrections appear at the same order in time for both the state-complexity and operator-growth versions. A sympathetic reader would care because the result ties a standard complexity diagnostic directly to the deformation parameter in a matrix model that arises in string theory and quantum gravity.

Core claim

In both the Krylov state complexity obtained from the reduced N=3 and N=4 representations and the Krylov operator growth computed in the matrix model, the Lanczos coefficients scale linearly with the mass deformation parameter; the early-time growth therefore receives quadratic massive corrections that appear at identical order in time for the two notions of complexity.

What carries the argument

The Lanczos coefficients extracted from the Hamiltonian of the systematically reduced N=3 and N=4 representations of the BMN matrix model.

If this is right

  • The mass deformation parameter alone determines the full set of Lanczos coefficients for both notions of complexity.
  • Early-time Krylov growth acquires quadratic corrections whose leading time order is insensitive to whether state or operator complexity is considered.
  • The linear scaling fixes the entire early-time behavior once the mass parameter is specified.
  • The same-order appearance of massive corrections holds across the different consistent reductions examined.

Where Pith is reading between the lines

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

  • The linear scaling may persist in other consistent truncations or in the full unreduced model at sufficiently large mass.
  • Similar mass-dependent Lanczos behavior could appear in related matrix models with different deformation parameters.
  • The coincidence of correction orders between state and operator versions suggests a possible structural link between the two complexity measures in deformed matrix quantum mechanics.

Load-bearing premise

The systematic reductions of the N=3 and N=4 representations preserve the essential dynamics needed for the Krylov complexity analysis at large mass deformation.

What would settle it

A direct computation of the Lanczos coefficients in the unreduced BMN matrix model at large but finite mass that fails to produce linear scaling with the mass parameter.

read the original abstract

We study Krylov complexity in BMN Plane Wave Matrix Model at large mass deformation. We consider various consistent reductions of the matrix model that allow us to perform a Hamiltonian analysis which leads to different notions of the Krylov complexity. In the first part of the paper, we study the Krylov state complexity considering systematic reduction of $N=3$ and $N=4$ representations of the matrix model, which reveals a universal characteristic scaling for the Lanczos coefficients and fix them completely in terms of the mass deformation parameter. In the second part of the paper, we study the Krylov operator growth in the matrix model and compute the corresponding Lanczos coefficients. In both cases, we observe a \emph{linear} scaling of Lanczos coefficients with the mass parameter. The early time growth in Krylov complexity receives quadratic correction due to the presence of the massive deformation in the matrix model. Our analysis reveals that such massive corrections appear at same order in time for both the notion of the Krylov complexity.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The paper studies Krylov complexity in the BMN plane wave matrix model at large mass deformation. It considers consistent reductions of the N=3 and N=4 representations to perform Hamiltonian analysis, yielding Lanczos coefficients for both Krylov state complexity and operator growth. The central claims are that these coefficients exhibit linear scaling with the mass deformation parameter (fixed completely in terms of it) and that massive corrections to early-time growth appear at the same order in time for both notions of complexity.

Significance. If the reductions preserve the relevant operator algebra and spectrum, the result that Lanczos coefficients are fixed solely by the mass parameter (with no additional free parameters) would provide a concrete, falsifiable characterization of complexity growth in this deformed matrix model. This is a strength, as it avoids post-hoc fitting and could inform studies of operator growth in systems with mass deformations relevant to holography. The observation of universal linear scaling and same-order corrections across state and operator complexity is potentially useful if substantiated.

major comments (3)
  1. [N=3 and N=4 reductions] N=3 and N=4 reductions (first part of the paper): The claim that these systematic reductions preserve the essential dynamics for the Lanczos analysis at large mass deformation is load-bearing for the linear scaling result. The manuscript does not provide an explicit verification that the truncated representations maintain the commutators and state tower needed to determine the coefficients from the reduced Hamiltonian, raising the possibility that the observed scaling is an artifact of the truncation rather than a feature of the BMN model.
  2. [Hamiltonian analysis and Lanczos coefficients] Computation of Lanczos coefficients (Hamiltonian analysis sections): The statement that the coefficients are fixed completely in terms of the mass deformation parameter and scale linearly requires the explicit recursion relations or coefficient expressions derived from the reduced Hamiltonian. Without these (or the intermediate steps showing how the mass enters the b_n), it is not possible to confirm the absence of hidden parameters or that the linear form follows directly rather than from the reduction procedure.
  3. [Early time growth] Early-time growth analysis (second part): The claim that quadratic massive corrections appear at the same order in time for both state complexity and operator growth is central but lacks the explicit perturbative expansion of the complexity (e.g., the coefficient of the t^2 term) or a direct comparison showing the orders match independently of the specific reduction.
minor comments (2)
  1. The abstract refers to 'different notions of the Krylov complexity' without immediately clarifying the distinction between state and operator versions; adding a brief sentence on this would improve readability.
  2. Notation for the reduced Hamiltonians and Lanczos coefficients could be standardized across the N=3 and N=4 cases to facilitate comparison.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their thorough review and valuable comments on our manuscript. We address each of the major comments below and will incorporate clarifications and additional details in a revised version to strengthen the presentation.

read point-by-point responses

  1. Referee: [N=3 and N=4 reductions] N=3 and N=4 reductions (first part of the paper): The claim that these systematic reductions preserve the essential dynamics for the Lanczos analysis at large mass deformation is load-bearing for the linear scaling result. The manuscript does not provide an explicit verification that the truncated representations maintain the commutators and state tower needed to determine the coefficients from the reduced Hamiltonian, raising the possibility that the observed scaling is an artifact of the truncation rather than a feature of the BMN model.

    Authors: The reductions are designed to be consistent truncations that preserve the su(2) algebra and the commutation relations within the chosen representations. At large mass deformation, the contributions from higher modes are suppressed, making the truncation reliable for the Lanczos analysis. To address the concern explicitly, we will add a new subsection in the revised manuscript verifying that the commutators are preserved in the reduced basis and that the state tower remains closed under the Hamiltonian action. This will confirm that the linear scaling is a genuine feature. revision: yes

  2. Referee: [Hamiltonian analysis and Lanczos coefficients] Computation of Lanczos coefficients (Hamiltonian analysis sections): The statement that the coefficients are fixed completely in terms of the mass deformation parameter and scale linearly requires the explicit recursion relations or coefficient expressions derived from the reduced Hamiltonian. Without these (or the intermediate steps showing how the mass enters the b_n), it is not possible to confirm the absence of hidden parameters or that the linear form follows directly rather than from the reduction procedure.

    Authors: We agree that providing the explicit expressions is necessary for full transparency. The Lanczos coefficients are obtained via the standard recursion from the reduced Hamiltonian, where the mass parameter μ enters linearly in the off-diagonal terms. We will include the detailed recursion relations and the resulting closed-form expression for b_n (which is proportional to μ times a function of n) in the revised version, demonstrating that there are no additional free parameters. revision: yes

  3. Referee: [Early time growth] Early-time growth analysis (second part): The claim that quadratic massive corrections appear at the same order in time for both state complexity and operator growth is central but lacks the explicit perturbative expansion of the complexity (e.g., the coefficient of the t^2 term) or a direct comparison showing the orders match independently of the specific reduction.

    Authors: The early-time expansion is derived from the general formula for Krylov complexity in terms of the Lanczos coefficients. Since both cases have b_n linear in μ, the first correction beyond the universal t^2 term enters at the same order (specifically, the μ-dependent term appears at O(t^4)). We will add the explicit perturbative expansions for both the state and operator complexities in the revised manuscript, including the coefficients, to show the matching orders directly. revision: yes

Circularity Check

0 steps flagged

No circularity: Lanczos scaling derived from explicit Hamiltonian computation

full rationale

The paper computes Lanczos coefficients directly from the Hamiltonian of the reduced N=3/N=4 matrix models, where the mass deformation enters as an explicit input parameter in the potential. The observed linear scaling and quadratic early-time corrections are outputs of this calculation, not inputs redefined as predictions. No self-definitional steps, fitted parameters renamed as results, or load-bearing self-citations are present. The derivation chain is self-contained against the model Hamiltonian and does not reduce to its own outputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the validity of the model reductions for N=3 and N=4 and the standard definition of Krylov complexity via the Lanczos algorithm; these are domain assumptions rather than new postulates.

axioms (1)
  • domain assumption The reduced N=3 and N=4 matrix models admit a consistent Hamiltonian formulation that captures the large-mass dynamics relevant to Krylov complexity.
    Invoked to enable the Hamiltonian analysis and extraction of Lanczos coefficients.

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