Emergent random matrix universality in quantum operator dynamics

TL;DR

This work reveals an emergent random-matrix universality in quantum operator dynamics by placing operator evolution in Lanczos (Krylov) space and applying a rigorous Riemann–Hilbert analysis to the associated orthogonal polynomials weighted by the spectral function. In the large-$n$ fast-space limit, the level-$n$ Green's function $G_n(z)$ converges to universal forms: a Wigner semicircle in the bulk, with a Bessel universality near $z=0$ for power-law low-frequency spectra and an Airy universality at the spectral edges; these universalities connect the operator-dynamics problem to classical random-matrix theory despite the absence of explicit randomness. The authors develop the spectral bootstrap to reconstruct spectral functions from Lanczos data and demonstrate that hydrodynamic transport coefficients can be extracted from zero-mode amplitudes, supported by numerical benchmarks on models like MFIM and XXZ. They further relate the universality to a Coulomb-gas confinement transition, clarifying how high-frequency decay controls low-frequency behavior and marginality, and discuss implications for open-system dynamics and the generalized ETH. Overall, the paper elevates the recursion method to a principled, universal framework linking quantum chaos signatures, hydrodynamics, and random-matrix universality, with practical numerical tools for spectral estimation from limited Lanczos data.

Abstract

The high complexity of many-body quantum dynamics means that essentially all approaches either exploit special structure or are approximate in nature. One such approach--the memory function formalism--involves a carefully chosen split into fast and slow modes. An approximate model for the fast modes can then be used to solve for Green's functions $G(z)$ of the slow modes. Using a formulation in operator Krylov space known as the recursion method, we prove the emergence of a universal random matrix description of the fast mode dynamics. This is captured by the level-$n$ Green's function $G_n (z)$, which we show approaches universal scaling forms in the fast limit $n\to\infty$. Notably, this emergent universality can occur in both chaotic and non-chaotic systems, provided their spectral functions are smooth. This universality of $G_n (z)$ is precisely analogous to the universality of eigenvalue correlations in random matrix theory (RMT), even though there is no explicit randomness present in the Hamiltonian. At finite $z$ we show that $G_n (z)$ approaches the Wigner semicircle law, while if $G(z)$ is the Green's function of certain hydrodynamical variables, we show that at low frequencies $G_n (z)$ is instead governed by the Bessel universality class from RMT. As an application of this universality, we give a numerical method--the spectral bootstrap--for approximating spectral functions from Lanczos coefficients. Our proof involves a map to a Riemann-Hilbert problem which we solve using a steepest-descent-type method, rigorously controlled in the $n\to\infty$ limit. We are led via steepest-descent to a Coulomb gas optimization problem, and we discuss how a recent conjecture--the `Operator Growth Hypothesis--is linked to a confinement transition in this Coulomb gas. These results elevate the recursion method to a theoretically principled framework with universal content.

Figures (21)

• Figure 1: (a) The semi-infinite 1D operator chain produced by the Lanczos algorithm. The level-$n$ Green's function $G_{n}(z) = (O_{n}|(z-\mathcal{L}_{n})^{-1} |O_{n})$ is the effective Green's function for operator dynamics restricted to operators $O_{n}$ and above, which we call the 'fast space' (indicated by the dashed box). A good model for the fast space dynamics encoded in $G_{n}(z)$ can be used to estimate the full Green's function $G(z)$ via the continued fraction in \ref{['eq:G_continued_frac']}. Intuitively, $G_{n}(z)$ captures the 'backflow' (indicated by the curved arrow) from the fast space to the slow space. (b) We prove that $G_{n}(z)$ approaches universal scaling forms as $n\to\infty$, with different limits for different regions of the complex $z$-plane. The most prominent example is that $G_{n}(z)$ approaches the Wigner semicircle law in the 'bulk' of the spectrum, but there can be different behavior near the origin and the spectral edge. This emergent universality is precisely analogous to the universality of eigenvalue correlations of random matrices, even though there is no explicit randomness here. For illustration we show only the first quadrant, with the other quadrants obtained by reflection about the axes. Here $\delta_{0}$ and $\delta_{1}$ are small $\mathcal{O}(1)$ constants in units of the microscopic couplings, and $\beta_{n} \approx 2 b_{n}$ determines the bandwidth of the bulk spectrum.
• Figure 2: Phase diagram of Coulomb gas confinement with a single-particle potential $Q(\omega)$, controlled by the large frequency decay of the spectral function $\Phi(\omega) \sim \exp[-Q(\omega)]$. Locality of the Hamiltonian forces the potential to grow polynomially at large $\omega$, $Q(|\omega|\to\infty) \sim |\omega|^{p}$, with $p \geq 1$. The operator growth hypothesis parkerUniversalOperatorGrowth2019 posits that chaotic systems generically have $p=1$. A Coulomb gas with finite charge (i.e. finite $n$) is confined for all $p > 0$ since the two-particle repulsion is only logarithmic, while the potential is polynomial. However, there is a transition between 'weak' and 'strong' confinement at $p=1$, precisely at the boundary imposed by locality. This has implications for numerical applications.
• Figure 3: We require the potential $Q(z)$ defined in \ref{['eq:spectral_function_def']} to have an analytic continuation to the shaded region of the complex plane, where $0 < \theta \leq \pi/2$ is any positive angle. This region is the union of the 'complex cone' $C_{\theta}$ (see \ref{['eq:complex_cone']}) and a disk of constant radius around $z=0$.
• Figure 4: The $n$th spectral function $\widetilde{\Phi}_{n}(\omega) = p_{n}(\omega)^{2} \Phi(\omega)$ for the toy spectral function $\Phi(\omega) / 2\pi = \mathop{\mathrm{sech}}\nolimits(\pi \omega)$ and $n=20$. This governs the dynamics of $O_{n}$ with respect to the full Liouvillian $\mathcal{L}$, not the restricted Liouvillian $\mathcal{L}_{n}$ (see \ref{['fig:level_n_gf_example']} for the latter). In this case $\beta_{n} \approx \sqrt{n(n-1)}$chenAsymptoticsExtremeZeros1997, with the linear scaling $\beta_{n} \sim \mathcal{O}(n)$ reflecting the exponential decay of $\Phi(\omega \to \infty)$. One can see that $\widetilde{\Phi}_{n}(\omega)$ is peaked near $|\omega| = \beta_{n}$, and as $n \to \infty$ this becomes increasingly sharp. The region $|\omega| \leq \beta_{n}$ is referred to as the 'bulk', and is analogous to the oscillatory region in a WKB approximation. For $|\omega| \gg \beta_{n}$, $\widetilde{\Phi}_{n}(\omega)$ is exponentially small. When $\Phi(\omega \to 0) \sim |\omega|^{\rho}$ has a power-law at $\omega = 0$, the orthogonal polynomials $p_{n}(\omega)$ behave differently near the origin (see e.g. \ref{['lem:pn0_scaling']}).
• Figure 5: Illustration of the confinement transition using the potential $Q(\omega) = (1 + \omega^{2})^{p/2}$ for different growth exponents $p$. The values $p=\frac{1}{2}$ and $p=2$ lie in the weakly and strongly confined phases respectively, while $p=1$ is marginal. All systems with local interactions should have $p \geq 1$. The confinement transition can be diagnosed via the equilibrium density $\sigma_{n}(0)$ at zero frequency. In the strongly confined phase $\sigma_{n}(0) \sim \mathcal{O}(n/\beta_{n})$ grows algebraically with $n$, reducing to logarithmic growth $\sigma_{n}(0) \sim \mathcal{O}(\log{n})$ at the critical point (or $\sigma_{n}(0) \sim \mathcal{O}(\log^{2}{n})$ in one spatial dimension). In the weakly confined phase, $\sigma_{n}(0) \sim \mathcal{O}(1)$ does not grow with $n$.

Theorems & Definitions (55)

• Definition 1: $\mathrm{VSLF}(p,q)$: log-Freud potentials of order $(p,q)$
• Definition 2: $\mathrm{CVSLF}(p,q,\theta,\gamma)$: complex log-Freud potentials of order $(p,q)$
• Example 1
• Example 2
• Example 3
• Example 4
• Lemma 1: Informal
• Lemma 2: Informal
• Theorem 1: Informal
• Example 5