  We report a quantum phase transition in Pauli limited $d$-wave
superconductors and give the mean field estimates of the associated quantum
critical point. For a population imbalanced $d$-wave superconductor a stable
ground state phase viz. quantum breached pair phase has been identified which
comprises of spatial coexistence of gapless superconductivity and nonzero
magnetization. Based on the thermodynamic and quasiparticle indicators we for
the first time analyze this phase, discuss the thermal behavior of Pauli
limited $d$-wave superconductor, give accurate estimates of the thermal scales
associated with such systems and map out the pseudogap regime. Our work shows
that while the Pauli limited superconductors are known to exhibit exotic
modulated superconducting phase at large imbalance of fermion populations; in
the regime of weak imbalance an intriguing phase of competing orders is
realized. We have established that rather than the superconducting pairing
field, it is the average magnetization of the system that quantifies this
quantum phase transition. Given that the existing Pauli limited superconductors
possess unconventional pairing state symmetry of the superconducting order, our
work promises to open up new avenues in the experimental research of these
materials. We have also demonstrated an alternate scenario wherein the quantum
breached pair phase is a natural outcome for a $d$-wave superconductor with
unequal effective masses of the fermion species.
