Fixed points and critical temperature near quantum critical points in $d$-wave cuprate superconductors

We study the critical behavior driven by potential quantum critical points (QCPs) termed as $\tau_{0,x,y,z}$-Type QCPs in the $d$-wave cuprate superconductors. Within the framework of the renormalization group approach, we construct the coupled flow equations of all interaction parameters and obtain several unique properties. On one hand, the interaction parameters flow toward several fixed points (FPs) at certain critical energy scales. We identify two different types of FPs designated at the clean limit. FP-I is characterized by the divergence of the quadratic parameter and exhibits robustness against variations in interaction parameters. In contrast, FP-II is dominated by the cubic and quartic interaction parameters, and it is sensitive to initial conditions, leading to five subclasses. Besides, we find that disorder scattering can influence fermion velocities and critical energy scales, and even destabilize certain FPs around the $\tau_{x,z}$-QCPs, driving the system toward a preempted disorder-induced FP. On the other hand, we find that quantum fluctuations play a critical role in shaping the critical temperature ($T_c$) as the system approaches these QCPs. Near the $\tau_x$-QCP, $T_c$ is considerably suppressed for both FP-I and FP-II. In contrast, near the $\tau_0$-QCP, $T_c$ undergoes a substantial decrease for FP-I but only a slight decrease for FP-II. Conversely, $T_c$ exhibits an increasing trend near the $\tau_y$-QCP, with a pronounced peak at $v_{\Delta 0}/v_{F0} \sim 0.25$. However, numerical analysis suggests that the $\tau_z$-QCP is unlikely to be physically realizable. Additionally, we realize that $T_c$ can also be modified by the emergence of disorder-induced FPs in the vicinity of the $\tau_x$-QCP. These findings would provide valuable insights into the critical low-energy properties of $d$-wave cuprate superconductors and related materials.