Elucidating Dynamical Behaviors in Kinetically Constrained Models via Energy-Activity Double-Biased Matrix Product State Analysis

We investigate dynamical phase transitions in two representative kinetically constrained models: the 1D Fredrickson-Andersen and East Models. A recently developed energy-activity double-bias approach utilizing both <i>s</i> and <i>g</i> fields, conjugated with dynamical activity and trajectory energy, is combined with matrix product state (MPS) methods. It is demonstrated that MPS methods facilitate the numerical approximation of large-deviation statistics of dynamics by determining the eigenvalues of tilted dynamical generators under the influence of double-biasing fields. Specifically, by focusing on the <i>g</i>-field, a nearly "half-filled" state at moderate negative <i>g</i> values is identified, indicating the potential existence of an anomalous phase. Additionally, dynamical quantities under various <i>s</i>, <i>g</i>, and <i>T</i> conditions are obtained via tensor networks, showing good qualitative consistency with mean-field results and our previous extensive numerical simulation results obtained by the path sampling method. Our study introduces novel methodologies for examining decoupled dynamical behaviors that, although energetically active, remain dynamically inactive within the system. This approach offers a fresh perspective on the theoretical framework and computational strategies for studying dynamical phase transitions in kinetically constrained systems.