arxiv: 2604.05630 · v1 · submitted 2026-04-07 · ✦ hep-th · cond-mat.stat-mech· quant-ph

Recognition: 2 theorem links

· Lean Theorem

Symmetry-resolved Krylov Complexity and the Uncoloured Tensor Model

Shaliya Kotta , P N Bala Subramanian

Authors on Pith no claims yet

Pith reviewed 2026-05-10 19:53 UTC · model grok-4.3

classification ✦ hep-th cond-mat.stat-mechquant-ph

keywords Krylov complexitysymmetry-resolvedUncoloured Tensor Modelquantum chaoscharge subspacesequipartitionoperator growthSYK model

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The pith

In the Uncoloured Tensor Model, symmetry-resolved Krylov complexity equals the full value in some charge subspaces but not others, with the subspace average bounded above.

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

The paper tests when symmetries allow an operator's growth rate, measured by Krylov complexity, to stay the same after projection into a single charge sector. It uses the Uncoloured Tensor Model, a symmetric system without disorder, to compute this for invariant operators. In some sectors the complexity matches the full-space result, called equipartition, while in others it deviates. Across the sectors that were reachable numerically, the average complexity never exceeds the value computed in the entire space. This distinction shows how symmetries can split chaotic operator dynamics into independent sectors.

Core claim

For invariant operators in the Uncoloured Tensor Model, the symmetry-resolved Krylov complexity in a charge subspace is identical to the full Krylov complexity in some cases and different in others. Within the sizes that could be computed, the Krylov complexity averaged over symmetry subspaces remains bounded above by the complexity of the same operator in the full space.

What carries the argument

Symmetry-resolved Krylov complexity obtained by projecting the operator and its Krylov basis into charge subspaces of the Uncoloured Tensor Model's symmetries.

If this is right

Equipartition of Krylov complexity holds in selected charge subspaces for the same invariant operator.
Equipartition fails in other charge subspaces of that operator.
The average of the symmetry-resolved Krylov complexities stays at or below the full-space value inside the computed range.
Symmetries therefore partition operator growth such that no sector average surpasses the overall growth rate.

Where Pith is reading between the lines

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

The selective equipartition pattern may appear in other disorder-free tensor models or in SYK-like systems once their symmetries are resolved.
If the upper bound on the averaged complexity survives in the large-N limit, it would imply that symmetry sectors experience strictly weaker or equal operator growth than the full system.
Analytic conditions for when equipartition occurs could be derived by examining the commutation relations between the operator and the symmetry generators.
The same subspace decomposition might be applied to other measures of chaos, such as out-of-time-order correlators, to check whether they also show sector-dependent behavior.

Load-bearing premise

That the finite-size numerical results obtained for the Uncoloured Tensor Model represent the general behavior of symmetry-resolved Krylov complexity and that the subspace projections correctly capture the model's symmetries.

What would settle it

A computation for some operator in the Uncoloured Tensor Model in which the symmetry-averaged Krylov complexity exceeds the full-space value or in which equipartition fails in every charge subspace.

Figures

Figures reproduced from arXiv: 2604.05630 by P N Bala Subramanian, Shaliya Kotta.

**Figure 3.** Figure 3: Lanczos sequence √ bn for operator 2ψ 111 for different values of β = 1 T . unique eigenvalues for the Liouvillian. Additionally, we find that the Krylov space dimension for the operator γ2, K should 363, by explicitly finding the Liouvillian eigenspaces upon which the operator has non-trivial projection. The Fig. 1a shows Lanczos sequence for γ2 up to n = 65 (terminated due to numerical instability of Lan… view at source ↗

**Figure 4.** Figure 4: Lanczos sequence bn for the Z-block operators of invariant operator γ2. 0 0 0 0 0 0 0 [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗

**Figure 6.** Figure 6: Lanczos sequence bn (a) and Krylov complexity of the full operator and symmetry-resolved Krylov complexities of the block operators (b) of the invariant operator γ2 in the eigenbasis of Noether charge Q23 1 . Growth of average Krylov complexity across charge sectors in comparison with the Krylov complexity of the full operator is shown in (c). The Lagrangian is invariant under the continuous transformation… view at source ↗

**Figure 6.** Figure 6: 5 Concluding Remarks and Future Directions The simplification in computation that is achievable by reducing the problem to a symmetry subspace is significant, and what we obtain in (3.8) is an essential criteria for the same. Further, we also study the Krylov complexity in the Uncoloured Tensor Model, one that is endowed with a myriad of symmetries. The system we considered falls among the very few tensor… view at source ↗

**Figure 7.** Figure 7: Extreme eigenvalues of truncated (at n) tri-diagonal matrices. References [1] J. D. Bekenstein, Black holes and entropy, Phys. Rev. D 7 (1973) 2333–2346. [2] S. W. Hawking and D. N. Page, Thermodynamics of Black Holes in anti-De Sitter Space, Commun. Math. Phys. 87 (1983) 577. [3] G. ’t Hooft, Dimensional reduction in quantum gravity, Conf. Proc. C 930308 (1993) 284–296, [gr-qc/9310026]. [4] L. Susskind, T… view at source ↗

read the original abstract

The symmetry-resolved Krylov complexity is a useful tool in studying chaotic properties of systems that are endowed with symmetries. We investigate the conditions under which an invariant operator would have the symmetry-resolved Krylov complexity in a charge subspace identical to the Krylov complexity of the full operator. Further, we study the Krylov complexity of the Uncoloured Tensor Model, a disorder-free kin of the SYK Model which has a plethora of symmetries. We find charge subspaces of the same operator in which the equipartition holds as well as where it doesn't. We also find that within the computational limits, the Krylov complexity averaged over the symmetry subspace is bounded above by that of the operator in the full space.

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.

Desk Editor's Note private letter to a colleague

The paper numerically applies symmetry-resolved Krylov complexity to the uncolored tensor model and reports equipartition in some charge subspaces but not others, plus an upper bound on the subspace-averaged value.

read the letter

The core finding is that symmetry-resolved Krylov complexity in the uncolored tensor model splits by charge subspace: equipartition with the full operator holds in some sectors and fails in others, while the average over subspaces stays below the full-space complexity within reachable matrix sizes. This is a direct extension of earlier SYK work to a disorder-free tensor model with explicit global symmetries, and the calculations give concrete examples of the subspace dependence rather than just formal statements. The setup is clear enough that a reader can see how the Lanczos iteration is restricted to projected sectors and how the bounds are extracted from those runs. That part is useful for anyone tracking how symmetries modify complexity diagnostics in holographic or many-body models. The soft spot is the numerical implementation itself. All claims rest on finite-size diagonalization or iteration inside symmetry sectors, yet there is no explicit verification that the projected operators still commute with the generators or that the charge labels survive the truncation without mixing. The stress-test concern lands here: a misassignment of subspaces could produce exactly the pattern of equipartition versus non-equipartition and could enforce the observed upper bound as an artifact. The paper flags the computational limits, but without those checks the results stay provisional. This work is aimed at people already working on Krylov complexity, tensor models, or symmetry-resolved chaos measures. It is not a broad conceptual advance, but the explicit numbers on a concrete model give it enough substance to warrant referee time rather than a desk reject. A serious review would focus on tightening the subspace validation and checking whether the bound survives larger sizes or analytic arguments.

Referee Report

1 major / 1 minor

Summary. The paper investigates symmetry-resolved Krylov complexity for invariant operators in the Uncoloured Tensor Model. It identifies conditions under which the complexity in a charge subspace equals the full-space value, reports numerical results showing that equipartition holds in some charge subspaces but not others, and finds that the symmetry-subspace-averaged Krylov complexity is bounded above by the full-space value within computational limits.

Significance. If the numerical findings hold under verification, the work provides concrete examples of how global symmetries partition chaotic diagnostics in a disorder-free tensor model, offering a controlled setting to test relations between full and resolved Krylov complexities. The absence of disorder averaging is a methodological strength, but the finite-size numerics limit broader claims about general behavior.

major comments (1)

[Numerical results on subspace projections] In the section describing the numerical implementation of symmetry subspaces (around the discussion of explicit diagonalization and Lanczos iteration in projected sectors), the manuscript provides no explicit check that the projected operators continue to commute with the symmetry generators after truncation or that the charge labels are correctly assigned. This verification is load-bearing for the central claims, as any misidentification of subspaces could directly produce the reported pattern of equipartition in some sectors but not others, as well as the observed upper bound on the averaged complexity.

minor comments (1)

[Abstract and results summary] The abstract and results statements refer to findings 'within the computational limits' without specifying the matrix sizes, number of charge sectors examined, or any error estimates on the Lanczos iterations or diagonalizations.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the thorough review and constructive feedback on our manuscript. We address the major comment below and describe the revisions we will implement to strengthen the presentation of our numerical results.

read point-by-point responses

Referee: In the section describing the numerical implementation of symmetry subspaces (around the discussion of explicit diagonalization and Lanczos iteration in projected sectors), the manuscript provides no explicit check that the projected operators continue to commute with the symmetry generators after truncation or that the charge labels are correctly assigned. This verification is load-bearing for the central claims, as any misidentification of subspaces could directly produce the reported pattern of equipartition in some sectors but not others, as well as the observed upper bound on the averaged complexity.

Authors: We agree that an explicit verification step would improve the clarity and robustness of the numerical implementation. The symmetry subspaces in our work are obtained by exact diagonalization of the charge operators (which are conserved by construction in the uncoloured tensor model) followed by projection onto definite-charge eigenspaces; this procedure guarantees that the projected operators commute with the generators inside each subspace. Nevertheless, to directly address the concern and rule out any numerical artifacts from truncation or label assignment, we will add a dedicated paragraph (or short subsection) in the revised manuscript. This will include: (i) the computed norm of the commutator [O_proj, Q] for representative projected operators O_proj and generators Q, demonstrating that it vanishes to machine precision (typically < 10^{-12}); and (ii) a table or statement confirming that the assigned charge eigenvalues match those obtained from the diagonalization. These checks will be performed for the same system sizes used in the main results. We expect this addition to confirm that the reported sector-dependent equipartition behavior and the upper bound on the subspace-averaged complexity are genuine features rather than consequences of subspace misidentification. revision: yes

Circularity Check

0 steps flagged

No circularity: numerical results from explicit subspace computations

full rationale

The manuscript defines symmetry-resolved Krylov complexity via standard Lanczos iteration on projected operators and reports direct numerical outcomes for equipartition and upper bounds in the Uncoloured Tensor Model. These findings are obtained from finite-size diagonalization within charge sectors; no step equates a fitted parameter to a prediction, renames a known result, or reduces the central claim to a self-citation chain. The derivation chain remains self-contained against external benchmarks of the model and the complexity measure.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; it introduces no explicit free parameters, new axioms, or invented entities beyond standard concepts of Krylov complexity and tensor models already present in the literature.

pith-pipeline@v0.9.0 · 5416 in / 1056 out tokens · 27602 ms · 2026-05-10T19:53:58.012435+00:00 · methodology

discussion (0)

Reference graph

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