With substantial evidence of glassy behavior in the phase diagram of high $T_c$ superconductors and its co-existence with superconductivity, we attempt to answer the question: what are the properties of a superconducting state where the force driving cooper pairing becomes dissipative? We find that when the bosonic mediator is local, dissipation acts to reduce the superconducting critical temperature ($T_c$). On the other hand, contrary to na\"{ i}ve expectations, $T_c$ behaves non-monotonically with dissipation for a non-local mediator -- weakly dissipative bosons at different energy scales act coherently to give rise to an increase in $T_c$ and eventually destroy superconductivity when the dissipation exceeds a critical value. The critical value occurs when dissipative effects become comparable to the energy scale associated with the spatial stiffness of the mediator, at which point, $T_c$ acquires a maximum. We outline consequences of our results to recent proton irradiation experiments (M. Leroux et al.,~\cite{ Welp2018}) on the cuprate superconductor La$_{ 2-x}$Ba$_x$CuO$_4$ (LBCO) which observe a disorder induced increase in $T_c$ even when the transition temperature of the proximate charge density wave (CDW) is unaffected by the presence of irradiation. Our mechanism is a novel way to raise $T_c$ that does not require a `tug-of-war' -type scenario between two competing phases.
Categories: cond-mat.supr-con cond-mat.dis-nn cond-mat.str-el