Krylov Complexity for Plane Wave Matrix Model

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.

• The paper rigorously applies Krylov complexity to the PWMM under strong mass deformation, deriving explicit analytic expressions for Lanczos coefficients and quadratic early-time growth.
• The analysis compares state and operator complexity, demonstrating universal quadratic scaling across different reductions and highlighting model-dependent effects, especially in the Coulomb potential case.
• The findings establish a robust framework linking complexity, quantum chaos, and holography, paving the way for further investigations into intermediate coupling regimes and holographic duality.

Krylov Complexity in the Plane Wave Matrix Model: Structure, Universality, and Operator Growth

Introduction and Motivation

The investigation of Krylov complexity (KC) has become central in quantum chaos, holography, and black hole physics, providing a systematic framework for characterizing dynamical information spreading in quantum systems. This work rigorously applies the formalism of KC to the BMN Plane Wave Matrix Model (PWMM), focusing on the structural properties of both the Krylov state complexity (KSC) and the Krylov operator complexity (KOC) under strong mass deformation. Through consistent Hamiltonian reductions, the paper establishes explicit analytic results for the Lanczos coefficients and associated complexity growth, highlighting universality, model dependence, and the role of mass deformation in the BMN context.

BMN Matrix Model and Setup

The BMN matrix model is a massive deformation of the BFSS model, arising from a dimensional reduction of $\mathcal{N}=4$ SYM on $R \times S^3$ and leading, in the dual gravity description, to a supergravity background with a mass gap parameter $\mu$. The bosonic part of the action is characterized by nontrivial interactions (Myers terms) and mass terms tied to $\mu$, breaking conformality and introducing a parameter regime amenable to systematic analysis in the $\mu \gg 1$ limit.

Consistent Hamiltonian reductions—particularly the fuzzy sphere models (integrable and pulsating) and Coulomb potential models—yield tractable dynamical systems, supporting explicit construction of Krylov bases and operator chains for both $N=3$ and $N=4$ representations.

Krylov State Complexity: Structure and Results

The KSC quantifies the dynamical spread of an initial state in an orthonormal Krylov basis, constructed recursively via the Lanczos procedure. For both the $N=3$ (Coulomb) and $N=4$ (integrable fuzzy sphere) reductions, the analysis begins with an initial state localized about potential minima (in large $\mu$), yielding groundstates of coupled harmonic oscillators.

The key structural achievements are:

• Lanczos Coefficient Universality: In all reductions studied, the first several Lanczos coefficients $R \times S^3$0 scale linearly with $R \times S^3$1, $R \times S^3$2, $R \times S^3$3 at leading order, with dimensionless $R \times S^3$4 depending on the specific reduction and type of complexity. This holds for both KSC and KOC, but with different constants for each.
• Early-Time Complexity Growth: For strong mass deformation, KSC grows quadratically at early times: $R \times S^3$5, where $R \times S^3$6 is an explicitly computed, model-dependent positive coefficient uniquely set by the first few Lanczos coefficients. Corrections due to mass deformation enter universally at quadratic order in both time and mass.
• Coulomb Model Charge Dependence: In the $R \times S^3$7 Coulomb potential case, the complexity prefactor $R \times S^3$8 acquires explicit dependence on the conserved Coulomb charge $R \times S^3$9, leading to a nontrivial structure: complexity growth speed decreases with increasing $\mu$0, and above a critical charge threshold, the complexity expression becomes ill-defined, indicating a breakdown in the oscillator approximation.

Figure 1: Early-time quadratic growth in complexity as a function of Coulomb charge $\mu$1, illustrating the suppression of complexity spreading for increasing $\mu$2. The plot is for $\mu$3 and various $\mu$4.

Krylov Operator Complexity: Liouvillian Growth and Comparisons

The KOC framework replaces state time evolution with operator dynamics governed by the Liouvillian superoperator. The construction of a Krylov basis in the operator space follows analogous steps, but the Gram-Schmidt orthogonalization ensures vanishing diagonal entries of the Liouvillian.

Principal findings include:

• Linear Mass Dependence of Off-diagonal Lanczos Coefficients: In all cases, $\mu$5 for strong mass deformation, in accord with the operator analog of KSC.
• Early-Time Operator Complexity Dynamics: Operator complexity satisfies the universal scaling $\mu$6, with an explicit, model-dependent $\mu$7. For fuzzy sphere (PFS and IFS) reductions, this coefficient is independent of the Coulomb charge, while for the Coulomb model, $\mu$8 varies non-monotonically with $\mu$9; it first increases then decreases, peaking near a critical charge.

Figure 2: Behavior of the leading order coefficient $\mu$0 controlling complexity growth as a function of the Coulomb charge $\mu$1 in the operator context, showing non-monotonic dependence and a critical threshold.

• Concordance with State Complexity: The model dependence and universality of the scaling confirm the robustness of the KC framework: corrections due to mass deformation systematically coincide in both state and operator settings at quadratic order, demonstrating a form of universality class for the leading complexity growth in large $\mu$2 PWMM reductions.

Implications and Future Directions

The results elucidate structural features of KC in a controlled quantum mechanical truncation of string and M-theory, providing analytic evidence for universality across distinct KC definitions (state vs. operator) and reductions. The linear mass scaling of Lanczos coefficients and universal quadratic early-time growth underpin both the diagnostic reliability and sensitivity of KC as a probe of dynamical complexity and potential chaos.

The explicit model-dependent calculations afford a foundation for multiple further directions:

• Intermediate Coupling Regimes: Extending these analytic calculations away from large $\mu$3 would enable probes of more generic quantum regimes, possibly uncovering crossover phenomena between integrable and chaotic sectors.
• Holographic Duality: The analytic expressions for KC and Lanczos coefficients offer concrete data for comparison with holographic complexity proposals in the dual supergravity (AdS/bubbling geometry), potentially enabling a direct computation of complexity from geometric quantities such as geodesic lengths or two-point functions in the holographic dual [see also related developments in e.g., (Roychowdhury, 10 Jan 2026, Roychowdhury, 18 Apr 2026)].
• Universality Across Models: The universality of quadratic early-time growth and linear mass dependence could be investigated in other matrix model reductions and quantum mechanical systems, exploring the extent to which these features persist as signatures of general information spreading or chaos.
• Numerical and Experimental Tests: The dependence of $\mu$4 on deformation parameters and conserved charges suggests practical, model-agnostic benchmarks for both classical simulation and potential quantum emulation of matrix models.

Conclusion

This work provides a precise analytic characterization of Krylov state and operator complexity in the BMN Plane Wave Matrix Model, under large mass deformation and various consistent reductions. The key findings—scaling laws for Lanczos coefficients, universal quadratic early-time complexity growth, and model-dependent sensitivity to conserved charges—solidify the BMN matrix model's role as a clean quantum laboratory for testing conjectured connections between complexity, chaos, and mass deformation in holography and quantum gravity. The universality between operator and state definitions demonstrated here suggests that KC offers a robust and model-invariant tool for quantifying quantum dynamical complexity.