Complexity Growth in Integrable and Chaotic Models

TL;DR

This work develops and applies a geometric, right-invariant framework for quantum complexity growth to a tunable SYK family, spanning free, integrable, and chaotic regimes. By identifying and locating conjugate points along time-evolution geodesics, the authors quantify how linear growth is truncated and establish upper bounds on complexity: O(√N) for free, O(poly(N)) for integrable, and exponential-time obstructions for chaotic dynamics. They provide analytic criteria via Euler-Arnold formalism and the Y_μ operator, complemented by numerical analysis up to N=8, showing how local versus nonlocal obstructions behave and how geodesic loops affect growth. The Eigenstate Complexity Hypothesis and ECH matrices connect complexity to ETH-like thermalization statistics, highlighting profound links between scrambling, operator locality, and geometry with potential implications for holography and quantum gravity.

Abstract

We use the SYK family of models with $N$ Majorana fermions to study the complexity of time evolution, formulated as the shortest geodesic length on the unitary group manifold between the identity and the time evolution operator, in free, integrable, and chaotic systems. Initially, the shortest geodesic follows the time evolution trajectory, and hence complexity grows linearly in time. We study how this linear growth is eventually truncated by the appearance and accumulation of conjugate points, which signal the presence of shorter geodesics intersecting the time evolution trajectory. By explicitly locating such "shortcuts" through analytical and numerical methods, we demonstrate that: (a) in the free theory, time evolution encounters conjugate points at a polynomial time; consequently complexity growth truncates at $O(\sqrt{N})$, and we find an explicit operator which "fast-forwards" the free $N$-fermion time evolution with this complexity, (b) in a class of interacting integrable theories, the complexity is upper bounded by $O({\rm poly}(N))$, and (c) in chaotic theories, we argue that conjugate points do not occur until exponential times $O(e^N)$, after which it becomes possible to find infinitesimally nearby geodesics which approximate the time evolution operator. Finally, we explore the notion of eigenstate complexity in free, integrable, and chaotic models.

Figures (29)

• Figure 1: A cartoon of what happens when a geodesic between $1$ and $e^{-itH}$ encounters a conjugate point. The green geodesic is initially the locally minimizing geodesic, before it reaches $t_*$ where it encounters a conjugate point (the blue point). For $t>t_*$, the green geodesic is no longer locally minimizing, and a different geodesic (shown in red) will be the local minimum. Note that even though the conjugate point indicates this transition, the new geodesic which takes over after $t_*$ is not infinitesimally close to the original one (although we can reach it by gradient flow from the original geodesic).
• Figure 2: A plot of complexity $\mathcal{C}(t)$ for instances of the $N=10$, $N=20$, $N=50$, $N=100$, and $N=500$ free SYK model with $\mathcal{J}=1$. The $\omega_p/2$ which control the growth of the coefficients $c_p(t)$ in \ref{['eq:global-velocity']} are the positive eigenvalues of the antisymmetric coupling matrix $J_{ij}$ whose entries are independent Gaussian random variables with mean zero and variance $\sigma^2 = \mathcal{J}^2/N$.
• Figure 3: Complexity bound \ref{['eq:integrable-bound']} for an $N=20$ instance of the integrable Hamiltonian \ref{['eq:perturbed-hamiltonian']} where $\epsilon M_{ij}/4$ is drawn from the $q=4$ SYK distribution with $\mathcal{J}=1$. The initial sharp linear growth is due to the combined initial linear growth of both terms in \ref{['eq:integrable-bound']}, and the small fluctuations are due to the frequent geodesic loops in $e^{-iH_0t}$. The larger fluctuations, and the coarse-grained shape of the function itself, are controlled by the geodesic loops in $e^{-i\epsilon H_1 t}$ that we have included in defining $d_{ij}(t)$. The plateau is clearly $O(N)$; its height without the integrable perturbation would be less than the height of the initial sharp rise, which is at most $O(\sqrt{N})$.
• Figure 4: An array plot of the matrix $e^{-S}| M_{\alpha\beta}|$ for $N=10,\;q=3,\;k=3,\;\mathcal{J}=1$ SYK and time $t=50$. We note that most of the off-diagonal elements are smaller than $e^{-S}$, while many diagonal matrix elements are also $O(e^{-S})$ at such late times.
• Figure 5: (Left) The minimum eigenvalue of $e^{-S}M_{\alpha\beta}$, i.e., the impact parameter, for an SYK Hamiltonian with $N=10,\;q=3,\;k=3,\;\mathcal{J}=1$ at small times. (Right) The minimum eigenvalue past exponential time becomes very small.