Quantum Phase Transitions in $\mathit{d}$-Wave Superconductors

Motivated by the strong, low temperature damping of nodal quasiparticles observed in some cuprate superconductors, we study quantum phase transitions in $d_{x^2{-y}^2}$ superconductors with a spin-singlet, zero momentum, fermion bilinear order parameter. We present a complete, group-theoretic classification of such transitions into seven distinct cases (including cases with nematic order) and analyze fluctuations by the renormalization group. We find that only two, the transitions to $d_{x^2{-y}^2}+is$ and $d_{x^2{-y}^2}{+id}_{\mathrm{xy}}$ pairing, possess stable fixed points with universal damping of nodal quasiparticles; the latter leaves the gapped quasiparticles along $(1,0)$, $(0,1)$ essentially undamped.