KPZ Superdiffusion of Local Correlators in Diffusive Random Quantum Circuits

Abstract

We study the single-particle Green's function $G(x,t)=⟨{\sigma }_x^{-}(0){\sigma }_0^{+}(t)⟩$ in one-dimensional particle-number-conserving random unitary circuits coupled to an external bath. For fixed spacetime disorder, we argue that $G(x,t)$ is governed, in both the strong- and weak-noise limits, by directed waves in a random medium. We find Kardar-Parisi-Zhang (KPZ) scaling in the wandering statistics of the normalized spatial distribution $p(x,t)\propto |G(x,t){|}^2$ and in the associated free energy. In particular, its center $⟨x(t)⟩\equiv {\sum }_{x}xp(x,t)$ wanders on a length-scale $\mathcal{O}(t^{2/3})$, while sample-to-sample fluctuations of $-\log {\sum }_x|G(x,t){|}^2$ scale as $t^{1/3}$. At weak noise $\gamma \ll 1$, the crossover to the strong-disorder fixed point occurs at a parametrically long time $\mathcal{O}({\gamma }^{-3/2})$. These predictions are confirmed numerically using tensor-network simulations of the noisy operator dynamics in individual circuits at moderate noise, and of a phase-annealed proxy retaining hopping disorder at weak noise.

5 pages, 2 figures (with 2 pages End Matter and 11 pages Supplemental Material)