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arxiv: 2606.31724 · v1 · pith:CUKMJVOQnew · submitted 2026-06-30 · ✦ hep-th · quant-ph

Holographic Krylov Complexity with Lifshitz Scaling and Hyperscaling Violation

Kazem Bitaghsir Fadafan , M. Reza Mohammadi Mozaffar This is my paper

Pith reviewed 2026-07-01 04:17 UTC · model grok-4.3

classification ✦ hep-th quant-ph
keywords Krylov complexityLifshitz scalinghyperscaling violationholographic complexitynon-relativistic holographyradial momentumdynamical exponent
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The pith

Krylov complexity grows quadratically in pure Lifshitz holography, with the hyperscaling violation exponent setting the late-time growth rate.

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

The paper applies a holographic proposal equating the growth rate of Krylov complexity to the proper radial momentum of an infalling massive probe. In pure Lifshitz geometries it obtains exact analytic solutions showing quadratic growth for every value of the dynamical exponent. In hyperscaling-violating geometries the late-time growth exponent is controlled directly by the hyperscaling violation parameter, and a limiting case produces oscillatory behavior under a logarithmic envelope. The analysis concludes that the momentum-Krylov correspondence extends to these non-relativistic settings and stays well-defined despite their causal pathologies.

Core claim

Following the holographic proposal that identifies the growth rate of Krylov complexity with the proper radial momentum of an infalling massive probe, we study Krylov complexity in Lifshitz and hyperscaling-violating backgrounds. For pure Lifshitz geometries, we derive exact analytic solutions and obtain quadratic complexity growth for all values of the dynamical exponent. For hyperscaling-violating backgrounds, we extract the asymptotic scaling, revealing that the hyperscaling-violating exponent directly controls the late-time growth exponent. In a special limiting case, the complexity exhibits oscillatory behavior with a logarithmic envelope, signaling a transition to a qualitatively disti

What carries the argument

The holographic identification of Krylov complexity growth rate with the proper radial momentum of an infalling massive probe, applied to Lifshitz and hyperscaling-violating geometries.

If this is right

  • Quadratic growth of complexity holds for every dynamical exponent in pure Lifshitz geometries.
  • The hyperscaling violation exponent determines the power of the late-time growth.
  • Oscillatory complexity with a logarithmic envelope appears in a special limit of the hyperscaling-violating family.
  • The correspondence remains well-defined despite the causal pathologies of Lifshitz spacetimes.

Where Pith is reading between the lines

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

  • The same radial-momentum identification could be tested in other non-relativistic holographic models to see whether scaling exponents continue to dictate complexity growth.
  • Oscillatory regimes might map to bounded or recurrent dynamics on the field-theory side that are inaccessible to relativistic holography.
  • If the correspondence survives, Krylov complexity could serve as a diagnostic for operator growth in Lifshitz-invariant condensed-matter systems.

Load-bearing premise

The proposal that equates Krylov complexity growth rate to the radial momentum of an infalling probe continues to hold in Lifshitz and hyperscaling-violating backgrounds.

What would settle it

A direct computation of Krylov complexity in a Lifshitz background whose growth rate differs from the radial momentum of the probe would falsify the claimed extension of the correspondence.

Figures

Figures reproduced from arXiv: 2606.31724 by Kazem Bitaghsir Fadafan, M. Reza Mohammadi Mozaffar.

Figure 1
Figure 1. Figure 1: The trajectory r(t) of the infalling particle in the Lifshitz geometry for several values of the dynamical exponent z, with ϵ = 0.01. For larger z, the particle takes longer to reach a given radial coordinate. The slope of the curves indicates power-law decay of the velocity for z > 1, in contrast to the constant asymptotic velocity observed for z = 1. Before proceeding to the computation of Krylov complex… view at source ↗
read the original abstract

Following the holographic proposal that identifies the growth rate of Krylov complexity with the proper radial momentum of an infalling massive probe, we study Krylov complexity in Lifshitz and hyperscaling-violating backgrounds. For pure Lifshitz geometries, we derive exact analytic solutions and obtain quadratic complexity growth for all values of the dynamical exponent. For hyperscaling-violating backgrounds, we extract the asymptotic scaling, revealing that the hyperscaling-violating exponent directly controls the late-time growth exponent. In a special limiting case, the complexity exhibits oscillatory behavior with a logarithmic envelope, signaling a transition to a qualitatively distinct regime. Our analysis establishes that the momentum-Krylov correspondence extends naturally to non-relativistic holographic settings and remains well-defined despite the causal pathologies of Lifshitz spacetimes.

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

1 major / 2 minor

Summary. The paper applies the existing holographic proposal equating the growth rate of Krylov complexity to the proper radial momentum of an infalling massive probe to Lifshitz (ds² = −r^{2z} dt² + r² dx² + dr²/r²) and hyperscaling-violating geometries. It reports exact analytic solutions for pure Lifshitz backgrounds that yield quadratic complexity growth for arbitrary dynamical exponent z, and asymptotic scalings for the θ-deformed case in which the hyperscaling-violation exponent directly sets the late-time growth power; a special limit produces oscillatory behavior with a logarithmic envelope. The central conclusion is that the momentum-Krylov correspondence extends to these non-relativistic settings and remains well-defined despite Lifshitz causal pathologies.

Significance. If the identification is valid, the exact quadratic solutions constitute a concrete technical advance for complexity calculations in anisotropic holographic models relevant to condensed-matter applications. The θ-controlled asymptotics and the oscillatory-logarithmic regime supply falsifiable predictions that could be tested against boundary operator evolution or tensor-network constructions.

major comments (1)
  1. [Introduction and §3] The manuscript imports the identification dC_K/dt = p_r directly from the relativistic AdS literature without re-deriving or verifying it for z ≠ 1. Because the proper-time parametrization of radial null geodesics changes with the Lifshitz exponent (the relation between coordinate time t and proper time τ along the probe trajectory acquires an explicit z dependence), the step that equates the late-time Krylov growth rate to the radial momentum is not automatically guaranteed; this assumption is load-bearing for every reported scaling result (abstract, §3, and the HV asymptotics).
minor comments (2)
  1. [§2] Notation for the probe action and the definition of proper radial momentum p_r should be stated explicitly once (e.g., in §2) rather than assumed from prior references.
  2. The phrase “causal pathologies” in the abstract is never unpacked; a brief paragraph clarifying which Lifshitz features (closed timelike curves, non-unique geodesics, etc.) are relevant and why they do not invalidate the probe calculation would improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their detailed report and for highlighting a potential subtlety in the applicability of the momentum-Krylov identification. We address the major comment below and will revise the manuscript to strengthen the presentation.

read point-by-point responses

  1. Referee: [Introduction and §3] The manuscript imports the identification dC_K/dt = p_r directly from the relativistic AdS literature without re-deriving or verifying it for z ≠ 1. Because the proper-time parametrization of radial null geodesics changes with the Lifshitz exponent (the relation between coordinate time t and proper time τ along the probe trajectory acquires an explicit z dependence), the step that equates the late-time Krylov growth rate to the radial momentum is not automatically guaranteed; this assumption is load-bearing for every reported scaling result (abstract, §3, and the HV asymptotics).

    Authors: We agree that the identification was adopted from the existing literature without an explicit re-derivation for general z. The original proposal equates the boundary-time growth rate of Krylov complexity to the radial momentum p_r of the infalling probe, where p_r is obtained from the conserved quantities of the geodesic equation in the given bulk metric. Because our computations of p_r employ the full Lifshitz (or HV) line element—including the z-dependent g_tt component—the z-dependence enters the expression for p_r itself. Nevertheless, an explicit check that the same relation continues to hold when the proper-time parametrization is altered by z would remove any ambiguity. In the revised manuscript we will insert a short derivation in §3 that starts from the probe action in the Lifshitz metric, obtains the radial momentum with respect to boundary time t, and confirms that dC_K/dt = p_r remains valid for arbitrary z. The same argument will be noted to carry over to the hyperscaling-violating case. revision: yes

Circularity Check

0 steps flagged

No significant circularity; proposal applied transparently to new backgrounds with independent analytic results

full rationale

The manuscript states it follows an existing holographic proposal identifying Krylov growth rate with radial momentum and then derives new exact analytic solutions for Lifshitz (quadratic growth for all z) and hyperscaling-violating cases (θ-controlled asymptotics, oscillatory limit). No equations reduce by construction to the input identification, no self-citation chain is load-bearing for the central results, and the calculations for the new metrics constitute independent content. This is the normal case of an extension paper whose derivations remain self-contained against the external benchmark of the original proposal.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central results rest on the momentum-Krylov identification (an external proposal) plus the standard holographic dictionary for Lifshitz and hyperscaling-violating metrics; no new free parameters or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption Holographic proposal equating Krylov complexity growth rate to proper radial momentum of infalling probe
    Invoked in the first sentence of the abstract as the starting point for all calculations.

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