Krylov Winding and Emergent Coherence in Operator Growth Dynamics

TL;DR

The paper reveals a universal mechanism for emergent coherence in operator growth by introducing Krylov winding, the linear-in-$n$ phase structure of the operator wavefunction in the Krylov basis. It shows that Krylov winding generically arises from linear Lanczos coefficients and leads to size winding when a low-rank mapping to Pauli size sectors and saturation of the chaos-operator growth bound hold, while sub-saturation ($h<1$) yields superlinear size-phase growth. Through analytical results and two microscopic models—the large-$q$ SYK coupled to a bath and a disordered all-to-all spin model—the work detail how phase alignment across operator sizes emerges and how the size-winding distribution $q(\ell,t)$ acquires a compressed-exponential form with a phase $\arg[q(\ell,t)]\propto (\ell-\ell_0)^{1/h}$. A complementary ramp-plateau toy model clarifies the late-time, finite-$N$ behavior of Krylov dynamics, showing how the Krylov peak position $\mu_K$ saturates and scales with system size, and providing a tractable framework to study the interplay between Krylov and Pauli bases. Overall, the results offer a unified microscopic mechanism for coherence in operator growth that links quantum chaos, finite-temperature operator dynamics, and potential teleportation-inspired protocols leveraging Krylov winding as a resource.

Abstract

The operator wavefunction provides a fine-grained description of quantum chaos and of the irreversible growth of simple operators into increasingly complex ones. Remarkably, at finite temperature this wavefunction can acquire a phase that increases linearly with the size of operator, a phenomenon called $\textit{size winding}$. Although size winding occurs naturally in a holographic setting, the emergence of a coherent phase in a scrambled operator remains mysterious from the standpoint of a thermalizing quantum many-body system. In this work, we elucidate this phenomenon by introducing the related concept of $\textit{Krylov winding}$, whereby the operator wavefunction has a phase which winds linearly with the Krylov index. We argue that Krylov winding is a generic feature of quantum chaotic systems. It gives rise to size winding under two additional conditions: (i) a low-rank mapping between the Krylov and size bases, which ensures phase alignment among operators of the same size, and (ii) the saturation of the ``chaos-operator growth'' bound $λ_L \leq 2 α$ (with $λ_L$ the Lyapunov exponent and $α$ the growth rate), which ensures a linear phase dependence on size. For systems which do not saturate this bound, with $h = λ_L / 2α<1$, the winding with Pauli size $\ell$ becomes $\textit{superlinear}$, behaving as $\ell^{1/h}$. We illustrate these results with two microscopic models: the Sachdev-Ye-Kitaev (SYK) model and a disordered $k$-local spin model.

Figures (8)

• Figure 1: (a) Left: Sketch of the operator wavefunction in the one-dimensional Krylov basis. Colors give the complex phase of the Krylov wavefunction $\mathrm{Arg}[\varphi_n(t)] \sim n$ for a fixed time $t$. Right: Phase and magnitude of $\varphi_n$. The phase grows linearly and the magnitude decays exponentially with $n$, consistently with Eq. \ref{['eq:varphilatetimes']}. (b) Left: Sketch of the operator wavefunction in the Pauli basis. Each dot corresponds to a Pauli string, arranged by increasing size $\ell$. The color on each dot shows the phase of the wavefunction $\mathrm{Arg}[c_P]$. Phase alignment is present but is in general only approximate: the phase is approximately constant in each size sector of fixed size $\ell$. Right: Sketch of the size winding distribution $q(l) = \sum_{P:|P|=l} c_P^2$ for different degrees of saturation of the chaos-operator growth (COG) bound $h = \lambda_L / 2 \alpha \leq 1$. The phase scales as $\ell^{1/h}$ and the magnitude decays as a compressed exponential $\sim \exp(-\ell^{1/h})$ (see Eq. \ref{['Eq:qoflmain']}).
• Figure 2: Fourier transform $C_K(\mu,t)$ of the (squared) Krylov wavefunction for (a) the analytically solvable model of Eq. \ref{['eq:ExactLinearWavefunction']} with $\alpha=\pi\nu/\beta,\nu=0.5, \Delta=1/4$, and with time in units of $1/2\alpha$, and for (b) the spin model of Eq. \ref{['eq:NonlocalHamiltonian']} with $N=8$, averaged over 100 disorder realizations at $\beta=1$. In both cases, the distribution sharpens around a peak momentum $\mu_K$ (black dots) as time evolves. Curves are normalized such that $\sum_n |\varphi_n|^2 = 1$.
• Figure 3: Peak momentum $\mu_K$ of the Krylov winding distribution $C_K(\mu,t)$ shown in Fig. \ref{['fig:Krylov_FTs']}(b) versus time for the spin model of Eq. \ref{['eq:NonlocalHamiltonian']} at $\beta=1$. The evolution of $\mu_K$ at early times agrees with the analytical prediction before saturating at a time of order $\alpha^{-1} \log(N)$ due to finite size effects.
• Figure 4: Fourier transform $C_S(\mu,t)$ of the size winding distribution for (a,b) large-$q$ SYK+bath model at $\nu = 0.5, q = 6, N = 3000$ with (a) $h=1$, (b) $h=0.5$, and (c) for the spin model of Eq. \ref{['eq:NonlocalHamiltonian']} with $N=8$, averaged over 100 disorder realizations at $\beta=1$. In all cases, the distribution develops a peak around a value $\mu_K(t)$ (black dots) which becomes sharper as time progresses.
• Figure S1: Results for the large-$q$ SYK+bath model ChenSYKplusSYKBathDissectingquantummanybodychaos at $N=3000, t=0.9/2\alpha, \nu=0.5, q=6$: (a) Size distribution $p(s, t)$ and (b) winding phase $\mathrm{Arg}~q(s, t)$ for different values of $h=\lambda_L/2\alpha$. Here $s=\ell/N$. The phase is linear only for the bound-saturating case $h=1$.