Quantum phase transitions in $d-wave$ superconductors

Based on an effective Lagrangian obtained from the holon-pair boson theory of Lee and Salk [Phys. Rev. B {\bf 64}, 052501 (2001)] for high $T_c$ cuprates, we explore physical states involved with quantum phase transitions around a critical hole doping of $d-wave$ superconductivity. We find a new quantum phase transition, a confinement to deconfinement transition for internal gauge charge in the superconducting phase. An antiferro- to para- magnetic transition in the superconducting state is explained in the context of the confinement to deconfinement transition. We obtain effective field theories in each region associated with the phase transitions.