Dynamics generated by spatially growing derivations on quasi-local algebras
Stefan Teufel, Marius Wesle, Tom Wessel
TL;DR
This work extends the class of quantum lattice dynamics on quasi-local algebras that are globally well-defined by allowing linearly growing local terms in the generator $\mathcal{L}_{\Phi}=\sum_x [\Phi_x, \cdot]$. By assuming Lieb-Robinson bounds with linear light cones for bounded local terms and a linear growth bound $\|\Phi_x(t)\|_{G,x} \le C_\Phi (1+\operatorname{dist}(x,x_0))$, the authors prove global existence and uniqueness of the one-parameter automorphism group generated by $\Phi$, and establish exponential light-cone bounds for the dynamics. The core technique is to approximate the infinite-volume dynamics with finite-volume zero-chains $\Phi^k$, show norm-convergence of the corresponding cocycles $\alpha^k_{s,t}$ to a limiting cocycle $\alpha_{s,t}$, and then extend from short times to all times by concatenation of short-time blocks. The results have implications for spectral-flow constructions (e.g., under magnetic fields) and yield automorphisms that are Fréchet continuous with exponential localization properties, broadening the landscape of well-defined quantum dynamics in infinite-volume lattice systems.
Abstract
We prove global existence and uniqueness of dynamics on the quasi-local algebra $\mathcal{A}$ of a quantum lattice system for spatially growing derivations $\mathcal{L}_Φ= \sum_x [ Φ_x , \cdot ]$. Existing results assume that the local terms $Φ_x\in\mathcal{A}$ of the generator are uniformly bounded in space with respect to appropriate weighted norms $\lVert Φ_x \rVert_{G,x}$. Analogous to the global existence result for first order ODEs, we show that global existence and uniqueness persist if the size of the local terms $\lVert Φ_x \rVert_{G,x}$ grows at most linearly in space. This considerably enlarges the class of derivations known to have well-defined dynamics. Moreover, we obtain Lieb-Robinson bounds with exponential light cones for such dynamics. For the proof, we assume Lieb-Robinson bounds with linear light cones for dynamics, whose generators have uniformly bounded local terms. Such bounds are known to hold, for example, if the local terms are of finite range or exponentially localized.