Probing Equilibrium and Dynamical Criticality Through Spatially Minimal Measurements

Extracting critical behavior in the wake of quantum quenches, with particular emphasis on experimental feasibility, has recently been at the forefront of theoretical and experimental investigations in condensed matter physics and quantum synthetic matter (QSM). Here, we investigate the potential of spatially minimal measurements, in the form of single-site observables, for probing equilibrium phase transitions and dynamical criticality in the transverse-field Ising chains (TFIC) with hard boundaries. For integrable and near-integrable TFIC, our exact and mean-field-theory analyses reveal a truly out-of-equilibrium universal scaling exponent $\beta\sim 4/3$ in the vicinity of the transition. We demonstrate the robustness of this exponent with respect to the initial state, as well as the location of the probe site. 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.