Probing Equilibrium and Dynamical Criticality Through Spatially Minimal Measurements

Extracting critical behavior in the wake of quantum quenches has recently been at the forefront of theoretical and experimental investigations in condensed matter physics. Here, we investigate the potential of single-site observables in probing equilibrium phase transitions and dynamical criticality in short-range transverse-field Ising chains. For integrable and near integrable models, our exact analytical and numerical calculations reveal an out-of-equilibrium universal scaling exponent in the vicinity of the transition. We show that this scaling exponent stems from a critically prethermal temporal regime, and demonstrate its independence from the initial state and the location of the probe site so long as the latter is sufficiently close to the edge of the chain. We extend our analyses to strongly nonintegrable TFIC, with long-range power-law or next-nearest-neighbor interactions, using t-DMRG calculations. Both finite-size and finite-time analyses suggest a dynamical critical point for the strongly nonintegrable and locally connected TFIC. Our work provides a robust scheme for the experimental detection of quantum critical points and dynamical scaling laws in short-range interacting models.