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We investigate the competition between d-wave superconductivity and nematicorder in high-$T_c$ superconductor, and examine the role played by gaplessfermionic degrees of freedom. Apart from the competitive interaction with thesuperconducting order parameter, the nematic order parameter couples stronglyto gapless nodal quasiparticles. The interplay of these two kinds ofinteractions is analyzed by means of renormalization group method. In case thefermionic degree of freedom are entirely neglected, the competitive interactionbetween two bosonic order parameters is strongly relevant, and can lead torunaway behavior. However, these properties are fundamentally changed once thedynamics of fermions are taken into account. At the nematic quantum criticalpoint where an extreme fermion velocity anisotropy occurs, the superconductingand nematic order parameters are decoupled from each other. Consequently, thephase transitions are continuous, and d-wave superconductivity can coexist withnematic order homogeneously. These results indicate that the gapless fermionscan play an important role and should be carefully included in the theoreticaldescription of competing orders.

Competition between superconductivity and nematic order in high-$T_c$ superconductor