Interplay of quantum and thermal fluctuations with quenched disorder in theories of composite order parameters

The interplay between quantum and thermal fluctuations in the presence of quenched random disorder is a long-standing open theoretical problem. In particular, the inherent disorder present in real materials and the fragility of charge order to impurities makes this problem especially relevant to a host of systems, including the cuprate high temperature superconductors and transition metal dichalcogenides. Theoretically, common techniques such as the large-N limit have so-far been hampered by the necessity to generalize Abelian symmetries to higher-dimensional non-Abelian manifolds, sacrificing the ability to describe phenomena such as Kosterlitz-Thouless phase transitions. To address these problems, we present a new approach to using the large-N limit whereby we encode a U(1) order parameter as a composite operator of fields transforming under representations of SU(N). Our approach allows us to consider quantum and thermal fluctuations and disorder on equal footing by treating all effects non-perturbatively. We derive the exact phase diagram and order parameter correlations as a function of disorder strength and temperature, revealing a novel crossover in the dynamics of the fluctuations of the composite order parameter.