Bounded by crossover lines exhibiting universal scaling, the supercritical regime above the critical end point is characterized by strong fluctuations and intriguing phenomena. In this study, we extend this notable concept of supercritical crossover to the quantum critical end point (QCEP), by studying the prototypical mixed-field quantum Ising and Potts models through tensor network calculations and scaling analyses. We reveal the existence of quantum supercritical (QSC) crossover lines, determined by not only response functions but also quantum information quantities, near the QCEP. A supercritical scaling law, $h\sim {(g-g_c)}^{\mathrm{\Delta }}$, is found, where $g$ ($h$) is the transverse (longitudinal) field, $g_c$ is the critical field, and $\mathrm{\Delta }$ is the so-called gap exponent of the QCEP. Moreover, we demonstrate that the QSC crossover line acts as a boundary for the emergence of dynamical singularities in quench dynamics. This singularity manifests as a distinctive cusp with a critical exponent of 1/2, signaling a new dynamical universality class. We also propose utilizing Rydberg atom arrays as an experimental platform to observe these QSC crossovers and dynamical singularities. Our work establishes a theoretical framework for understanding the role of a QCEP and associated supercritical crossovers in both equilibrium and nonequilibrium quantum many-body systems.

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• Dynamical phase transitions
• Matrix product states
• Nonequilibrium statistical mechanics
• Quantum phase transitions
• Lattice models in statistical physics
• Quantum spin models
• Rydberg atoms & molecules
• Dynamic critical phenomena
• Lattice models in condensed matter
• Scaling methods
• Tensor network methods