Critical behavior around the fixed points driven by fermion–fermion interactions and disorders in the nodal-line superconductors

Abstract

We systematically investigate the intricate interplay between short-range fermion-fermion interactions and disorder scatterings beneath the superconducting dome of noncentrosymmetric nodal-line superconductors. Employing the renormalization group that unbiasedly treats all kinds of potential degrees of freedom, we establish energy-dependent coupled flows for all associated interaction parameters. Decoding the low-energy information from these coupled evolutions leads to the emergence of several intriguing behavior in the low-energy regime. At first, we identify eight distinct types of fixed points, which are determined by the competition of all interaction parameters and dictate the low-energy properties. Next, we carefully examine and unveil distinct fates of physical implications as approaching such fixed points. The density of states of quasiparticles displays a linear dependence on frequency around the first fixed point, while other fixed points present diverse frequency-dependent behavior. Compressibility and specific heat exhibit unique trends around different fixed points, with the emergence of non-Fermi-liquid behavior nearby the fifth fixed point. Furthermore, after evaluating the susceptibilities of the potential states, we find that a certain phase transition below the critical temperature can be induced when the system approaches the fifth fixed point, transitioning from the nodal-line superconducting state to another superconducting state. This research would enhance our understanding of the unique behavior in the low-energy regime of nodal-line superconductors.

Appendices

Appendix A: One-loop RG equations

Following the spirit of RG approach [65,66,67] and employing the RG rescaling transformations (10)–(13), and taking into account all the one-loop corrections as depicted in Figs. 18 and 19, which contain the interplay between fermion-fermion interactions and disorder scatterings (cf. our previous work [72] for details), we finally derive the coupled RG flow equations of all coupling parameters,

$$\begin{aligned} \frac{{\textrm{d}}v_z}{{\textrm{d}}l}&= [4\mathcal {C}_{1}\lambda _{2}-\mathcal {C}_{2} (\Delta _{1}+\Delta _{2}+\Delta _{3} +\Delta _{41}+\Delta _{42}+\Delta _{43})]v_z, \end{aligned}$$

(A1)

$$\begin{aligned} \frac{{\textrm{d}}v_p}{{\textrm{d}}l}&= [4\mathcal {C}_{1}\lambda _{1}-\mathcal {C}_{2} (\Delta _{1}+\Delta _{2}+\Delta _{3} +\Delta _{41}+\Delta _{42}+\Delta _{43})]v_p, \end{aligned}$$

(A2)

$$\begin{aligned} \frac{{\textrm{d}}\lambda _1}{{\textrm{d}}l}&= 2\lambda _1\Bigl [-\frac{1}{2}+\mathcal {C}_{3} \left( -3\lambda _{1}-2\lambda _{2}-\lambda _{3}+\lambda _{4} +\lambda _{5}-\lambda _{6} \right) + \mathcal {C}_{7}(\Delta _{1} -\Delta _{2} -7\Delta _{3} -\Delta _{41} -\Delta _{42} -\Delta _{43})\nonumber \\{} & {} \quad -\mathcal {C}_{2} (\Delta _{1}+\Delta _{2}+\Delta _{3} +\Delta _{41}+\Delta _{42}+\Delta _{43})\Bigr ], \end{aligned}$$

(A3)

$$\begin{aligned} \frac{{\textrm{d}}\lambda _2}{{\textrm{d}}l}&= 2\lambda _2\Bigl [-\frac{1}{2}+\mathcal {C}_{4} \left( -2\lambda _{1}-3\lambda _{2}+\lambda _{3}+\lambda _{4} -\lambda _{5}-\lambda _{6} \right) + \mathcal {C}_{7}(-\Delta _{1} +\Delta _{2} +\Delta _{3} -\Delta _{41} -\Delta _{42} -\Delta _{43} )\nonumber \\{} & {} \quad -\mathcal {C}_{2} (\Delta _{1}+\Delta _{2}+\Delta _{3} +\Delta _{41}+\Delta _{42}+\Delta _{43})\Bigr ], \end{aligned}$$

(A4)

$$\begin{aligned} \frac{{\textrm{d}}\lambda _3}{{\textrm{d}}l}&= 2\lambda _3\Bigl [-\frac{1}{2}+\mathcal {C}_{4} \left( -2\lambda _{1}+\lambda _{2}-3\lambda _{3}-\lambda _{4} +\lambda _{5}+\lambda _{6} \right) + \mathcal {C}_7(-\Delta _1 +\Delta _2 +\Delta _3 +\Delta _{41} +7\Delta _{42} +\Delta _{43})\nonumber \\{} & \quad {} -\mathcal {C}_{2} (\Delta _{1}+\Delta _{2}+\Delta _{3} +\Delta _{41}+\Delta _{42}+\Delta _{43})\Bigr ] -4\mathcal {C}_6\lambda _5\Delta _{41}, \end{aligned}$$

(A5)

$$\begin{aligned} \frac{{\textrm{d}}\lambda _4}{{\textrm{d}}l}&= \lambda _4[-1-4\mathcal {C}_{2} (\Delta _{41}+\Delta _{42})] -4\mathcal {C}_5(\lambda _5\Delta _2 +\lambda _6\Delta _3) -2\lambda _{6}(\mathcal {C}_{4}\lambda _{2} +\mathcal {C}_{3}\lambda _{1}), \end{aligned}$$

(A6)

$$\begin{aligned} \frac{{\textrm{d}}\lambda _5}{{\textrm{d}}l}&= 2\lambda _5 \Bigl [-\frac{1}{2}+ \mathcal {C}_3 (\lambda _1-2\lambda _2+\lambda _3 +\lambda _4-3\lambda _5-\lambda _6 ) + \mathcal {C}_7(\Delta _1 -\Delta _2 +\Delta _3 +\Delta _{41} +\Delta _{42} -\Delta _{43} )\nonumber \\{} &\quad {} -\mathcal {C}_{2} (\Delta _{1}+\Delta _{2}+\Delta _{3} +\Delta _{41}+\Delta _{42}+\Delta _{43})\Bigr ] +2\mathcal {C}_1\lambda _2\lambda _6- 4(\mathcal {C}_6\lambda _3\Delta _{41} +\mathcal {C}_5\lambda _4\Delta _2 +\mathcal {C}_5\lambda _6\Delta _1), \end{aligned}$$

(A7)

$$\begin{aligned} \frac{{\textrm{d}}\lambda _6}{{\textrm{d}}l}&= 2\lambda _6\Bigl [-\frac{1}{2}+ \mathcal {C}_1 (-\lambda _1-\lambda _2+\lambda _3 +\lambda _4-\lambda _5-3\lambda _6) -(\mathcal {C}_3\lambda _2+\mathcal {C}_4\lambda _1) -2\mathcal {C}_2(\Delta _1 +\Delta _2 +\Delta _{41} +\Delta _{42} )\Bigr ]\nonumber \\{} &\quad {} +2\mathcal {C}_1\lambda _2\lambda _5 -4\mathcal {C}_5(\lambda _4\Delta _3 +\lambda _5\Delta _1), \end{aligned}$$

(A8)

$$\begin{aligned} \frac{{\textrm{d}}\Delta _1}{{\textrm{d}}l}&= 2\mathcal {C}_2\Delta _1 (\Delta _1+\Delta _2+\Delta _3 +\Delta _{41}+\Delta _{42}+\Delta _{43}) +8\mathcal {C}_5\Delta _2\Delta _3, \end{aligned}$$

(A9)

$$\begin{aligned} \frac{{\textrm{d}}\Delta _2}{{\textrm{d}}l}&= \Delta _2\Big [2\mathcal {C}_{2} (-3\Delta _{1}-3\Delta _{2}+\Delta _{3} +\Delta _{41}+\Delta _{42}+\Delta _{43})+ \mathcal {C}_1 (\lambda _1+\lambda _2+\lambda _3 -\lambda _4+\lambda _5-\lambda _6) \Bigr ]+8\mathcal {C}_5\Delta _1\Delta _3, \end{aligned}$$

(A10)

$$\begin{aligned} \frac{{\textrm{d}}\Delta _{3}}{{\textrm{d}}l}&= \Delta _{3}\Big [ 4\mathcal {C}_7 (\Delta _1-\Delta _2+\Delta _3 -\Delta _{41}-\Delta _{42}-\Delta _{43}) -2\mathcal {C}_{2} (\Delta _{1}+\Delta _{2}+\Delta _{3} +\Delta _{41}+\Delta _{42}+\Delta _{43})\nonumber \\{} &\quad {} + \mathcal {C}_3 (-\lambda _1+\lambda _2+\lambda _3 -\lambda _4-\lambda _5+\lambda _6) \Bigr ]+8\mathcal {C}_5\Delta _1\Delta _2, \end{aligned}$$

(A11)

$$\begin{aligned} \frac{{\textrm{d}}\Delta _{41}}{dl}&= \Delta _{41}\Big [ 4\mathcal {C}_7 (-\Delta _1+\Delta _2+\Delta _3 -\Delta _{41}+\Delta _{42}+\Delta _{43}) -2\mathcal {C}_{2} (\Delta _{1}+\Delta _{2}+\Delta _{3} +\Delta _{41}+\Delta _{42}+\Delta _{43})\nonumber \\{} &\quad {} + \mathcal {C}_4 (\lambda _1-\lambda _2+\lambda _3 +\lambda _4-\lambda _5-\lambda _6) \Bigr ]+8\mathcal {C}_5\Delta _{42}\Delta _{43}, \end{aligned}$$

(A12)

$$\begin{aligned} \frac{d\Delta _{42}}{{\textrm{d}}l}&= \Delta _{42}\Big [ 4\mathcal {C}_7 (-\Delta _1+\Delta _2+\Delta _3 +\Delta _{41}-\Delta _{42}+\Delta _{43}) -2\mathcal {C}_{2} (\Delta _{1}+\Delta _{2}+\Delta _{3} +\Delta _{41}+\Delta _{42}+\Delta _{43})\nonumber \\{} & \quad {} + \mathcal {C}_4 (\lambda _1-\lambda _2-\lambda _3 +\lambda _4-\lambda _5-\lambda _6) \Bigr ]+ 8\mathcal {C}_5\Delta _{41}\Delta _{43}, \end{aligned}$$

(A13)

$$\begin{aligned} \frac{{\textrm{d}}\Delta _{43}}{{\textrm{d}}l}&= \Delta _{43}\Big [ 4\mathcal {C}_7 (-\Delta _1+\Delta _2+\Delta _3 +\Delta _{41}+\Delta _{42}-\Delta _{43}) -2\mathcal {C}_{2} (\Delta _{1}+\Delta _{2}+\Delta _{3} +\Delta _{41}+\Delta _{42}+\Delta _{43})\nonumber \\{} & \quad {} + \mathcal {C}_4 (\lambda _1-\lambda _2+\lambda _3 -\lambda _4+\lambda _5+\lambda _6) \Bigr ]+ 8\mathcal {C}_5\Delta _{41}\Delta _{42}. \end{aligned}$$

(A14)

Fig. 18

One-loop corrections to the fermion-fermion couplings a–i and the fermion-disorder strengths j–m (the solid, dashed and wavy lines represent the fermion propagator, fermion-fermion interactions and disorder scattering, respectively) [54, 72, 73]

Fig. 19

One-loop corrections to the fermion propagator due to the fermion-fermion interactions a and b as well as disorder scatterings c (the solid, dashed, and wavy lines represent the fermion propagator, fermion-fermion interaction, and disorder scattering, respectively)

Hereby the coefficients $\mathcal {F}$, $\mathcal {J}$, and $\mathcal {K}$ appearing in Eqs. (15)–(17) are explicitly provided in the right hand of above RG equations. The coefficients $\mathcal {C}_i$ with $i=1-7$ are nominated as follows

$$\begin{aligned} \mathcal {C}_{1}\equiv & {} \frac{1}{(2\pi )^3} \int _{0}^{\pi }{\textrm{d}}\theta \frac{\pi }{(\upsilon _{z}^2\sin ^2\theta +\upsilon _{p}^2\cos ^2\theta )^{1/2}}, \end{aligned}$$

(A15)

$$\begin{aligned} \mathcal {C}_{2}\equiv & {} \frac{1}{(2\pi )^3} \int _{0}^{\pi }{\textrm{d}}\theta \frac{2\pi }{\upsilon _{z}^2\sin ^2\theta +\upsilon _{p}^2\cos ^2\theta }, \end{aligned}$$

(A16)

$$\begin{aligned} \mathcal {C}_{3}\equiv & {} \frac{1}{(2\pi )^3} \int _{0}^{\pi }{\textrm{d}}\theta \frac{\pi \upsilon _{z}^2\sin ^2\theta }{(\upsilon _{z}^2\sin ^2\theta +\upsilon _{p}^2\cos ^2\theta )^{3/2}},\end{aligned}$$

(A17)

$$\begin{aligned} \mathcal {C}_{4}\equiv & {} \frac{1}{(2\pi )^3} \int _{0}^{\pi }{\textrm{d}}\theta \frac{\pi \upsilon _{p}^2\cos ^2\theta }{(\upsilon _{z}^2\sin ^2\theta +\upsilon _{p}^2\cos ^2\theta )^{3/2}},\end{aligned}$$

(A18)

$$\begin{aligned} \mathcal {C}_{5}\equiv & {} \frac{1}{(2\pi )^3} \int _{0}^{\pi }{\textrm{d}}\theta \frac{4\pi \upsilon _{z}^2\sin ^2\theta }{(\upsilon _{z}^2\sin ^2\theta +\upsilon _{p}^2\cos ^2\theta )^2},\end{aligned}$$

(A19)

$$\begin{aligned} \mathcal {C}_{6}\equiv & {} \frac{1}{(2\pi )^3} \int _{0}^{\pi }{\textrm{d}}\theta \frac{4\pi \upsilon _{p}^2\cos ^2\theta }{(\upsilon _{z}^2\sin ^2\theta +\upsilon _{p}^2\cos ^2\theta )^2},\end{aligned}$$

(A20)

$$\begin{aligned} \mathcal {C}_{7}\equiv & {} \frac{1}{(2\pi )^3} \int _{0}^{\pi }{\textrm{d}}\theta \frac{2\pi (\upsilon _{p}^2\cos ^2\theta -\upsilon _{z}^2\sin ^2\theta )}{(\upsilon _{z}^2\sin ^2\theta +\upsilon _{p}^2\cos ^2\theta )^2}. \end{aligned}$$

(A21)

Appendix B: One-loop equations of source terms

Paralleling the analogous procedures in Appendix A, we calculate the one-loop corrections to the strengths of source terms as shown in Fig. 20, and then adopt the RG rescalings (10)–(13) to finally derive the following flow equations for $g_i$ with $i=1-6$,

$$\begin{aligned} \frac{{\textrm{d}}g_1}{{\textrm{d}}l}&= \Bigl [(1-\eta )+\frac{1}{32\pi ^3}[(\lambda _1-\lambda _2-\lambda _3-\lambda _4+\lambda _5+\lambda _6 -\Delta _1-\Delta _2+\Delta _3\nonumber \\{} &\quad {} -\Delta _{41}-\Delta _{42}-\Delta _{43})I_2 +8(\lambda _2+\Delta _2)I_2+16(\lambda _1+\Delta _3) I_1]\Bigl ]g_1, \end{aligned}$$

(B1)

$$\begin{aligned} \frac{{\textrm{d}}g_2}{{\textrm{d}}l}&= \Bigl [(1-\eta )+\frac{\mathcal {F}_{1}(\theta ,\theta ')}{4\pi ^2} [2(\lambda _1+\Delta _3)I_1 +(\lambda _2+\Delta _2)I_2]\nonumber \\{} & \quad +\frac{\mathcal {F}_{2}(\theta ,\theta ')}{4\pi ^2} (\lambda _1-\lambda _2-\lambda _3-\lambda _4+\lambda _5+\lambda _6 -\Delta _1-\Delta _2+\Delta _3-\Delta _{41}-\Delta _{42}-\Delta _{43})I_2\Bigl ]g_2, \end{aligned}$$

(B2)

$$\begin{aligned} \frac{{\textrm{d}}g_3}{{\textrm{d}}l}&= \Bigl [(1-\eta )+\frac{\mathcal {F}_{3}(\theta ,\theta ')}{4\pi ^2} [2(\lambda _1+\Delta _3)I_1 +(\lambda _2+\Delta _2)I_2]\nonumber \\{} &\quad + \frac{\mathcal {F}_{4}(\theta ,\theta ')}{4\pi ^2} (\lambda _1-\lambda _2-\lambda _3-\lambda _4+\lambda _5+\lambda _6 -\Delta _1-\Delta _2+\Delta _3-\Delta _{41}-\Delta _{42}-\Delta _{43})I_2\Bigl ]g_3, \end{aligned}$$

(B3)

$$\begin{aligned} \frac{{\textrm{d}}g_4}{{\textrm{d}}l}&= \Bigl [(1-\eta )+\frac{\mathcal {F}_{5}(\theta ,\theta ')}{4\pi ^2} [2(\lambda _1+\Delta _3)I_1 +(\lambda _2+\Delta _2)I_2]\nonumber \\{} & \quad + \frac{\mathcal {F}_{6}(\theta ,\theta ')}{4\pi ^2} (\lambda _1-\lambda _2-\lambda _3-\lambda _4+\lambda _5+\lambda _6 -\Delta _1-\Delta _2+\Delta _3-\Delta _{41}-\Delta _{42}-\Delta _{43})I_2\Bigl ]g_4, \end{aligned}$$

(B4)

$$\begin{aligned} \frac{{\textrm{d}}g_5}{{\textrm{d}}l}&= \Bigl [(1-\eta )+\frac{\mathcal {F}_{7}(\theta ,\theta ')}{4\pi ^2} [2(\lambda _1+\Delta _3)I_1 +(\lambda _2+\Delta _2)I_2]\nonumber \\{} & \quad + \frac{\mathcal {F}_{8}(\theta ,\theta ')}{4\pi ^2} (\lambda _1-\lambda _2-\lambda _3-\lambda _4+\lambda _5+\lambda _6 -\Delta _1-\Delta _2+\Delta _3-\Delta _{41}-\Delta _{42}-\Delta _{43})I_2\Bigl ]g_5, \end{aligned}$$

(B5)

$$\begin{aligned} \frac{{\textrm{d}}g_6}{{\textrm{d}}l}&= \Bigl [(1-\eta )+\frac{\mathcal {F}_{9}(\theta ,\theta ')}{4\pi ^2} [2(\lambda _1+\Delta _3)I_1 +(\lambda _2+\Delta _2)I_2]\nonumber \\{} & \quad + \frac{\mathcal {F}_{10}(\theta ,\theta ')}{4\pi ^2} (\lambda _1-\lambda _2-\lambda _3-\lambda _4+\lambda _5+\lambda _6 -\Delta _1-\Delta _2+\Delta _3-\Delta _{41}-\Delta _{42}-\Delta _{43})I_2\Bigl ]g_6. \end{aligned}$$

(B6)

$$\begin{aligned} I_1&= \int _0^\pi {\textrm{d}}\theta \frac{v_z v_p\sin \theta \cos \theta }{(v_z^2\sin ^2\theta +v_p^2\cos ^2\theta )^{3/2}}, \end{aligned}$$

(B7)

$$\begin{aligned} I_2&= \int _0^\pi d\theta \frac{(v_z^2\sin ^2\theta -v_p^2\cos ^2\theta )}{(v_z^2\sin ^2\theta +v_p^2\cos ^2\theta )^{3/2}}, \end{aligned}$$

(B8)

and

$$\begin{aligned} \mathcal {F}_{1}(\theta ,\theta ')\equiv & {} \frac{\sin (4\theta ^\prime )\sin (4\theta )}{16\pi }, \end{aligned}$$

(B9)

$$\begin{aligned} \mathcal {F}_{2}(\theta ,\theta ')\equiv & {} \frac{\sin (4\theta ^\prime )\sin ^2(4\theta )}{128\pi }, \end{aligned}$$

(B10)

$$\begin{aligned} \mathcal {F}_{3}(\theta ,\theta ')\equiv & {} \frac{\cos (2\theta ^\prime ) \cos (2\theta )}{\pi }, \end{aligned}$$

(B11)

$$\begin{aligned} \mathcal {F}_{4}(\theta ,\theta ')\equiv & {} \frac{\cos (2\theta ^\prime )\cos ^2(2\theta )}{8\pi }, \end{aligned}$$

(B12)

$$\begin{aligned} \mathcal {F}_{5}(\theta ,\theta ')\equiv & {} \frac{\sin (2\theta ^\prime ) \sin (2\theta )}{4\pi }, \end{aligned}$$

(B13)

$$\begin{aligned} \mathcal {F}_{6}(\theta ,\theta ')\equiv & {} \frac{\sin (2\theta ^\prime )\sin ^2(2\theta )}{64\pi }, \end{aligned}$$

(B14)

$$\begin{aligned} \mathcal {F}_{7}(\theta ,\theta ')\equiv & {} \frac{\cos (\theta ^\prime ) \cos (\theta )}{\pi }, \end{aligned}$$

(B15)

$$\begin{aligned} \mathcal {F}_{8}(\theta ,\theta ')\equiv & {} \frac{\cos (\theta ^\prime )\cos ^2(\theta )}{8\pi }, \end{aligned}$$

(B16)

$$\begin{aligned} \mathcal {F}_{9}(\theta ,\theta ')\equiv & {} \frac{\sin (\theta ^\prime ) \sin (\theta )}{\pi }, \end{aligned}$$

(B17)

$$\begin{aligned} \mathcal {F}_{10}(\theta ,\theta ')\equiv & {} \frac{\sin (\theta ^\prime ) \sin ^2(\theta )}{8\pi }, \end{aligned}$$

(B18)

where $\theta$ and $\theta '$ serve as the momentum directions of nodal-line fermions involved in the fermion-fermion interactions.

Fig. 20

One-loop corrections to the strength of source terms due to fermion-fermion interactions a, b and fermion-disorder interaction c, d. Hereby, the dashed line and wavy line denote the fermion-fermion interactions and fermion-disorder interactions, respectively