First- and Second-Order Electron Position Correlators Reveal Topological Phase Transitions
The theory of conventional critical phenomena is built around the concepts of fluctuations and correlators of local observables. Here, we show that topological phase transitions can also be characterized through spatial and temporal two-point correlation functions associated with fluctuations of the electronic position operator. The key object is the antisymmetric part of the position-position correlation function, which yields the Chern number in the equal-time and equal-space limit. When the system is translationally invariant in space or time (the latter case corresponding to an equilibrium condition), these two-point correlators reduce to one-variable functions of the remaining variable. Remarkably, these functions are both reminiscent of critical behaviour, including divergent correlation times and lengths signalling the phase transitions. Away from these conditions, namely in non-periodic systems or genuinely dynamical processes, criticality is instead better captured by higher-order Chern-Chern correlators constructed from local or dynamical markers obtained as coincident-point limits of the position correlators. These higher-order Chern-Chern correlators can be seen as double Levi-Civita contractions of second-order position correlators, selecting the antisymmetric sector in each index pair. We substantiate this framework through numerical simulations and analytic calculations. Our results establish a general correlator-based description of topological phase transitions, applicable to crystalline, noncrystalline, equilibrium, and dynamical systems.