Energy diffusion and the butterfly effect in inhomogeneous Sachdev-Ye-Kitaev chains

TL;DR

This work investigates whether energy diffusion in disordered, strongly interacting systems is bound by quantum-chaos data. Using a large-N, strong-coupling inhomogeneous SYK chain, the authors derive an effective action for reparametrization modes and identify a diffusion operator whose spectrum sets the energy diffusion constant $D$, alongside chaos quantities such as the Lyapunov time $\tau_{\textsc{l}}$ and butterfly velocity $v_{\textsc{b}}$. They prove in the slow-variation limit that $D \le v_{\textsc{b}}^2 \tau_{\textsc{l}}$, and show numerically that while $D$ tracks a hydrodynamic prediction, the bound is not universal: $D$ can be strictly smaller than, or even vanish relative to, $v_{\textsc{b}}^2 \tau_{\textsc{l}}$ depending on disorder type and temperature. The results imply that simple, universal diffusion–chaos bounds may fail in disordered strange metals, motivating model-dependent analyses and variational approaches to tighten or refine transport-chaos relations in non-quasiparticle systems.

Abstract

We compute the energy diffusion constant $D$, Lyapunov time $τ_{\text{L}}$ and butterfly velocity $v_{\text{B}}$ in an inhomogeneous chain of coupled Majorana Sachdev-Ye-Kitaev (SYK) models in the large $N$ and strong coupling limit. We find $D\le v_{\text{B}}^2 τ_{\text{L}}$ from a combination of analytical and numerical approaches. Our example necessitates the sharpening of postulated transport bounds based on quantum chaos.

Figures (3)

• Figure 1: Left: the value of $D$ and $v_{\textsc{b}}^2\tau_{\textsc{l}}$, predicted theoretically (from (\ref{['eq:diff']}) and (\ref{['eq:vb']})) and found numerically, in an inhomogeneous chain with $J_1^2 = (1+\mathcal{J}^2)^{-1}$, with $\mathcal{J} = a_0 + a_1\cos(2\text{\pdfsave \pdfsetmatrix{1 0 -.18 1} $\pi$ \pdfrestore} x/M)$ for $a_0=0.5$ and $a_1=0.6$. The trend of $v_{\textsc{b}}^2\tau_{\textsc{l}}$ to decrease at smaller $M$ towards the "limit" set by diffusion is evident. Right: disordered profiles with $\mathcal{J} = a_0 + a_1 X$, with $X=\sum c_n \cos(\phi_n + k_n x)$ and $c_n$ and $\phi_n$ random variables chosen so that $X\sim \mathrm{O}(1)$. We take $a_0=0.5$ and vary $a_1$, and study the ratio $D/v_{\textsc{b}}^2\tau_{\textsc{l}}$ as a function of $M$ for different realizations of disorder. Discrepancies between the two are enhanced as $M$ becomes large, as expected. We have set $L=6000$ and $\beta J = 25$.
• Figure 2: Left: Comparison of $D$ and $v_{\textsc{b}}^2\tau_{\textsc{l}}$ as a function of $a$, for chains where $J_{1,x}^2$ are i.i.d. random variables drawn from the distribution $\mathrm{P}(J_{1,x}^2 < X) = X^{1+a}\mathrm{\Theta}(X)$. The circular data points with error bars are numerical data, and dashed lines are the mean of the hydrodynamic predictions for each chain. Qualitative agreement is observed. Right: the numerically computed ratio $D/v_{\textsc{b}}^2\tau_{\textsc{l}}$ as a function of $a$. We see that this ratio rapidly drops for $a\lesssim 0$, as finite size effects become appreciable. This data is taken at $\beta J = 25$.
• Figure 3: The temperature dependence of $D/v_{\textsc{b}}^2\tau_{\textsc{l}}$ for 'disordered' periodic lattices, where the 'hydrodynamic' prediction $D/v_{\textsc{b}}^2\tau_{\textsc{l}}\approx 0.9$. In the periodic lattice, $D(x)$ consists of a sum of $\mathcal{S}$ sine waves, which are tiled over a period $\mathcal{S}\times M$: $\mathcal{J} = |a + \frac{b}{\sqrt{\mathcal{S}}} \sum c_n \cos(\frac{2\text{\pdfsave \pdfsetmatrix{1 0 -.18 1} $\pi$ \pdfrestore} nx}{M\mathcal{S}} + \phi_n) |$, with $b=4a$ and $c_i$ and $\phi_i$ randomly chosen, given the constraint $D/v_{\textsc{b}}^2\tau_{\textsc{l}}\approx 0.9$. Increasing $\mathcal{S}$ decreases the ratio $D/v_{\textsc{b}}^2\tau_{\textsc{l}}$. This data is taken at $L=4000$ and $M=10$.