Krylov state complexity for BMN matrix model

Abstract: We explore Krylov complexity in the BMN matrix model following a systematic reduction of it, known as the pulsating fuzzy sphere model. We present an analytical setup that allows us to calculate Lanczos coefficients in both large and small deformation limits of the matrix model.

• The paper develops an analytical Gram–Schmidt framework for extracting Lanczos coefficients and early-time Krylov complexity in the N=2 pulsating fuzzy sphere reduction of the BMN matrix model.
• In the strong-deformation limit, the nonzero Lanczos coefficients scale linearly with μ, producing quadratic early-time complexity growth with a coefficient proportional to μ².
• In the weak-deformation limit, leading Lanczos corrections begin at order μ² and b₁ rapidly approaches approximately −2 at large disk radius, while intermediate-μ chaos remains analytically unresolved.

This paper develops an analytical framework for computing Krylov state complexity in the BMN matrix model, restricted to the $N=2$ reduction known as the pulsating fuzzy sphere model (2605.10786). The work complements the numerical study of Huh, Jeong, Pando Zayas and Pedraza (Huh et al., 2024), which found integrable behavior at small and large mass parameter $\mu$ but chaotic signatures (a ramp-plateau structure in Krylov complexity) at intermediate $\mu$. The present analysis is confined to the two extreme corners of the parameter space—$\mu \gg 1$ and $\mu \sim 0$—where the Lanczos coefficients can be extracted analytically from explicit Gram–Schmidt orthogonalization of the Krylov sequence.

Setup: the pulsating fuzzy sphere

The $N=2$ BMN matrix model reduces to a two-dimensional quantum mechanics with Hamiltonian

$$H = \frac{p_x^2}{2} + \frac{p_y^2}{2} + \frac{\mu^2}{8}(x^2 + 4y^2) - \mu y^3 + \frac{1}{2}(x^2+y^2)^2,$$

which interpolates between the BFSS matrix model as $\mu \to 0$ and a strongly massive oscillator system as $\mu \to \infty$. The author constructs the Krylov basis $\{\ket{\psi_n}\} = \{H^n\ket{\psi_0}\}$ by repeated action of the Hamiltonian on an initial state, orthonormalizes via Gram–Schmidt to obtain $\mu$0, and reads off the Lanczos coefficients $\mu$1 and $\mu$2 from the tridiagonal representation of $\mu$3. The state complexity is then $\mu$4, evolved through the discrete Schrödinger equation on the Krylov chain.

Large deformation limit

For $\mu$5, scaling arguments (following Amore et al. (Amore et al., 2024)) justify taking the initial state as a product of localized harmonic oscillator ground states,

$\mu$6

with effective frequencies $\mu$7 and $\mu$8, valid at low energies near the potential minimum. Successive applications of $\mu$9 generate polynomial prefactors $\mu$0 multiplying $\mu$1, whose Gaussian moments can be evaluated exactly.

The central result of this section is the linear scaling of all non-zero Lanczos coefficients with the deformation parameter: explicitly, $\mu$2, $\mu$3, $\mu$4, and $\mu$5. This universality implies that the early-time complexity growth is controlled entirely by $\mu$6: truncating the Krylov chain after two steps yields

$\mu$7

where $\mu$8 is $\mu$9-independent while the quadratic-in-time coefficient scales as $\mu \gg 1$0. The initial growth is therefore quadratic, consistent with holographic observations of operator complexity in related settings (Roychowdhury, 18 Apr 2026). It should be noted that this expansion is built from only the first few Lanczos coefficients; the quoted structure is an early-time result, not a statement about the full time evolution.

Small deformation limit

At $\mu \gg 1$1 the model is integrable, and the ground-state problem reduces to a free particle in polar coordinates with quartic confining potential $\mu \gg 1$2. Setting $\mu \gg 1$3 and $\mu \gg 1$4, the seed state is $\mu \gg 1$5, normalized within a disk of radius $\mu \gg 1$6 (fixed by the boundary condition $\mu \gg 1$7), with $\mu \gg 1$8. The BMN Hamiltonian is then treated perturbatively in $\mu \gg 1$9, and the Gram–Schmidt procedure is carried out analytically, producing hypergeometric ($\mu \sim 0$0) expressions for the overlap coefficients.

The key structural difference from the strong-deformation regime is that the leading correction to the Lanczos coefficients appears at order $\mu \sim 0$1, not linearly in $\mu \sim 0$2. Numerical evaluation reveals distinct size-dependence patterns:

Coefficient	Behavior with disk radius $\mu \sim 0$3	Saturation value
$\mu \sim 0$4	increases with $\mu \sim 0$5, saturates at large $\mu \sim 0$6	—
$\mu \sim 0$7	decreases sharply with $\mu \sim 0$8; smaller for larger $\mu \sim 0$9	$N=2$0
$N=2$1	rises sharply, saturates rapidly	$N=2$2

The saturation of $N=2$3 at approximately $N=2$4 for large disks is presented as universal across a wide range of $N=2$5. Physically, large $N=2$6 corresponds to matrices $N=2$7 with large entries, which the author identifies as the appropriate regime for exploring complexity in the small-deformation limit. As in the large-$N=2$8 case, the leading complexity growth is quadratic, $N=2$9, now governed by the saturated coefficient.

Limitations and open questions

Several limitations are stated plainly in the paper. First, the harmonic oscillator ansatz for the initial state at large $$H = \frac{p_x^2}{2} + \frac{p_y^2}{2} + \frac{\mu^2}{8}(x^2 + 4y^2) - \mu y^3 + \frac{1}{2}(x^2+y^2)^2,$$0 is a low-energy approximation, valid only near the potential minimum; deviations away from the vacuum are not controlled. Second, both computations truncate the Krylov chain at the first few coefficients, so the results capture early-time complexity only—the intermediate-$$H = \frac{p_x^2}{2} + \frac{p_y^2}{2} + \frac{\mu^2}{8}(x^2 + 4y^2) - \mu y^3 + \frac{1}{2}(x^2+y^2)^2,$$1 chaotic regime, with its numerically observed plateau-ramp structure, remains entirely outside the analytical treatment. Third, the claim that the saturation behavior is "universal" rests on numerical evaluation of the first few coefficients rather than a proof. The paper leaves open three specific questions: extending the analysis to intermediate coupling $$H = \frac{p_x^2}{2} + \frac{p_y^2}{2} + \frac{\mu^2}{8}(x^2 + 4y^2) - \mu y^3 + \frac{1}{2}(x^2+y^2)^2,$$2 and determining the scaling of Lanczos coefficients there; generalizing to other systematic reductions ($$H = \frac{p_x^2}{2} + \frac{p_y^2}{2} + \frac{\mu^2}{8}(x^2 + 4y^2) - \mu y^3 + \frac{1}{2}(x^2+y^2)^2,$$3, $$H = \frac{p_x^2}{2} + \frac{p_y^2}{2} + \frac{\mu^2}{8}(x^2 + 4y^2) - \mu y^3 + \frac{1}{2}(x^2+y^2)^2,$$4); and identifying a gravitational dual description from which these Lanczos coefficients could be extracted holographically.

Conclusion

The paper provides closed-form analytical access to the first Lanczos coefficients of the pulsating fuzzy sphere at both extremes of the mass deformation. The contrasting scaling—$$H = \frac{p_x^2}{2} + \frac{p_y^2}{2} + \frac{\mu^2}{8}(x^2 + 4y^2) - \mu y^3 + \frac{1}{2}(x^2+y^2)^2,$$5 at strong deformation versus leading corrections at $$H = \frac{p_x^2}{2} + \frac{p_y^2}{2} + \frac{\mu^2}{8}(x^2 + 4y^2) - \mu y^3 + \frac{1}{2}(x^2+y^2)^2,$$6 with rapid saturation in disk size at weak deformation—establishes how the mass parameter controls early-time Krylov complexity growth in this matrix quantum mechanics. The intermediate-coupling regime, where chaos emerges, remains the principal unresolved target of the program.