Phase Transitions, Scaling Behavior, and Emergent Pattern Formation at Critical Exceptional Points in Driven-Dissipative U(1) Systems

Driven-dissipative systems, in which particles or energy are continuously exchanged with an environment, sustain ordered phases and phase transitions that have no counterpart in thermal equilibrium. A particularly consequential example is the critical exceptional point (CEP), where the exceptional point coalescence from non-Hermitian dynamics involves a Nambu-Goldstone mode, giving rise to nonequilibrium many-body phase transition in a broad range of nonequilibrium systems, from classical active matter with nonreciprocal interactions to quantum optical realizations. This thesis determines the phase diagram, fluctuation properties, and dynamical consequences of the CEP in driven-dissipative U(1) systems relevant to polariton condensates and photon BECs. Mean field analysis and a map to normal-form establishes that the CEP is the unique endpoint of the first-order phase transition in the broken-symmetry phase with an emergent Z2 symmetry. Large-scale numerical simulation of the stochastic 1D spatially extended system reveals an anomalous scaling regime with roughness and dynamical exponents distinct from both equilibrium and KPZ universality. A third result identifies noise-activated solitary waves in the condensate phase field, showing that the CEP organizes coherent pattern formation as well as anomalous fluctuations.