Universal Dynamics of Stochastically Driven Quantum Impurities

We show that the dynamics of a quantum impurity subject to a stochastic drive on one side and coupled to a quantum critical system on the other display a universal behavior inherited from the quantum critical scaling. Using boundary conformal field theory, we formulate a generic ansatz for the dynamical scaling form of the typical Loschmidt echo and corroborate it with exact numerical calculations in the case of a spin impurity driven by shot noise in a quantum Ising chain. We find that due to rare events the dynamics of the mean echo can follow very different dynamical scaling than the typical echo for certain classes of drives. Our results are insensitive to irrelevant perturbations of the bulk critical model and apply to all the microscopic models in the same universality class.

Figure 1

Sketch of the class of systems under study in this work. We consider a quantum critical spin chain (red spins) subjected to a stochastic boundary drive (blue line).

Figure 2

Left: The typical Loschmidt echo averaged over $r=1000$ realizations and for system sizes up to $L=1000$, for different values of the boundary field $h_b$ and flipping probabilities $p$. The boundary field takes the values $h_b=0.1$ (blue), 0.2 (orange), 0.3 (green), 0.4 (red), 0.5 (purple), 0.6 (pink), 0.7 (yellow), 0.8 (teal), and the probability $p$ varies as marked in the legend. The Poisson parameter $\lambda$ takes values $\lambda \ge 10$ throughout. The dashed lines are the prediction from boundary CFT [Eq. (6)]: for the fixed-fixed drive, $\mathrm{\Delta }=1/2$, and for the free-fixed drive, $\mathrm{\Delta }=1/16$, with both showing excellent agreement. Right: The mean echo of the same data. For the free-fixed drive, the mean and typical (black dashed lines) are very similar, but, strikingly, for the fixed-fixed drive the mean lies far above the typical. This is due to rare events that dominate the average and give a renormalized scaling form (inset), where $\alpha =0.71\pm 0.03$, in good agreement with the estimate of $\alpha =0.75$ in the main text.