Krylov complexity in the IP matrix model

TL;DR

The paper analyzes operator growth and chaos in the IP matrix model by computing Lanczos coefficients and Krylov metrics from the large-N Schwinger-Dyson solution. It shows that at high temperature the Lanczos coefficients grow linearly with n up to logarithmic corrections, driving Krylov complexity to grow as K(t) ~ exp(O(√t)), signaling chaotic dynamics in the deconfined phase. In contrast, zero temperature or massless adjoint cases yield discrete spectra or bounded, oscillatory Krylov behavior, indicating non-chaotic regimes. The results establish a concrete linkage between spectral properties, temperature, and operator growth, with implications for chaos diagnostics in gauge/gravity duals. The work highlights large-N and temperature as essential ingredients for maximal Krylov growth in matrix quantum mechanics and suggests Krylov measures as potential order parameters for confinement/deconfinement transitions.

Abstract

The IP matrix model is a simple large $N$ quantum mechanical model made up of an adjoint harmonic oscillator plus a fundamental harmonic oscillator. It is a model introduced previously as a toy model of the gauge theory dual of an AdS black hole. In the large $N$ limit, one can solve the Schwinger-Dyson equation for the fundamental correlator, and at sufficiently high temperature, this model shows key signatures of thermalization and information loss; the correlator decay exponentially in time, and the spectral density becomes continuous and gapless. We study the Lanczos coefficients $b_n$ in this model and at sufficiently high temperature, it grows linearly in $n$ with logarithmic corrections, which is one of the fastest growth under certain conditions. As a result, the Krylov complexity grows exponentially in time as $\sim \exp\left({\cal{O}{\left(\sqrt{t}\right) }}\right)$. These results indicate that the IP model at sufficiently high temperature is chaotic.

Figures (12)

• Figure 1: The Krylov complexity $K(t)$ (left) and the Krylov entropy $S(t)$ (right) for the Lanczos coefficients given by eq. (\ref{['LancozsToyzeroT']}). We fix parameters as $\omega_0 = 0.8$, $M=0$, $g=1$, although there is no $M$-dependence.
• Figure 2: Schwinger-Dyson equation for planar contributions to $\tilde{G}(\omega)$ (propagator with shaded rectangle) in terms of $\tilde{G}_0(\omega)$ and $\tilde{K}(\omega)$.
• Figure 3: The model can be regarded as a reduced version of a large $N$ D0-brane background black hole with a D0 probe Iizuka:2001cw. The mass scale for D0-D0 fields is set to $m$, while the mass scale for D0-D0$_{\rm probe}$ fields is set to $M$. We consider the limit $M \gg m$ and $M \gg T$ at various $m/T$.
• Figure 4: Numerical plots of $F(\omega)$ for $\nu_T = 1$, $m=0.2$ and $m=0.8$ at various temperatures from $T=0$ to $T=\infty$ where $y=e^{-m/T}$. For visibility, the scale of the vertical axis is changed as appropriate for each figure. To visualize poles at zero temperature, we set the imaginary part of $\omega$ to a small positive nonzero value.
• Figure 5: Lanczos coefficients of the toy function (\ref{['ToymodelF']}). Each dot represents a numerical value of $b_n$. We also plot a curve $b_n=\frac{m\pi n}{4W(2m\pi n/\nu_T)}$, which is consistent with the numerical plots.