Detecting quantum phase transitions in nonintegrable and long-range Ising chains using spatially minimal measurements.

In a recent work [Da\u{g}, Uhrich, and Halimeh, arXiv:2105.05986], single-site observables have been introduced as a versatile tool for the detection of equilibrium and dynamical criticality in short-range near-integrable many-body models. Here, we extend the potential of single-site observables as probes of quantum phase transitions to strongly nonintegrable models with long-range power-law and next-nearest-neighbor interactions. Our $t$-DMRG calculations verify the results of the mean-field theory for near-integrable transverse-field Ising chains (TFIC) both with finite-size and finite-time scaling analyses. Furthermore, we find that both finite-size and finite-time analyses suggest a dynamical critical point for a strongly nonintegrable and locally connected TFIC. We demonstrate the presence of a quasi-stationary temporal regime in the power-law interacting TFIC, and extract local-order profiles for TFIC in the long-range Ising universality class with algebraic light cones. Finally, we discuss the robustness of the out-of-equilibrium critical exponent $\beta\sim 4/3$ found for the integrable and near-integrable TFICs.