Triplet superconductivity supported by an X$_9$ high-order Van Hove singularity

Abstract

We study a four-fold symmetric dispersion relation of a quantum material, which exhibits a single high-order Van Hove singularity of X$_9$ type at the Fermi energy. First, we analyze in detail its form, type and density of states when the energy dispersion is in its canonical form. Subsequently, we study the possibility of a superconducting state when Hubbard repulsive interactions are taken into account. By solving the gap equation, it is shown that triplet state superconductivity with power-law dependence of the critical temperature T$_c$ on the interaction strength can be formed when a single singularity is present in the Brillouin zone. We discuss the effects of fluctuations and provide an upper bound of a possible superconducting critical temperature for the ruthenate Sr$_3$Ru$_2$O$_7$ which has been shown to exhibit this type of singularity.

Figures (4)

• Figure 1: The general $C_4^{\,}$ symmetric $X_9^{\,}$ quartic polynomial takes the form $k^4 \left( \beta + \gamma \, \cos 4\theta + \delta \, \sin 4 \theta \right)$ in polar coordinates. As $|\beta|$ increases from a value less than $\sqrt{\delta^2 + \gamma^2}$ to greater than it, we go from a saddle to a higher order maximum/minimum (depending on the sign of $\beta$). The transition case is marked by (c) $|\beta| = \sqrt{\delta^2 + \gamma^2}$, wherein the dispersion tangentially intersects the $k$-plane along a pair of straight lines through the origin, but does not change sign. Particle and hole sectors of the $k$-plane are colored blue and red, respectively, and become equal in size in the particle-hole symmetric case, that is when (a) $\beta = 0$.
• Figure 2: The $\delta$ term in $k^4 \left( \beta + \gamma \, \cos 4\theta + \delta \, \sin 4 \theta \right)$ breaks the $k_x^{\,} \leftrightarrow k_y^{\,}$ reflection symmetry. This causes a rotation of the saddle and its contours with respect to the $k_x^{\,}$ and $k_y^{\,}$ axes when $\delta \neq 0$.
• Figure 3: The ratio of pre-factors in the power-law DOS of an $X_9^{\,}$ saddle depends on the ratio $| \beta | / \sqrt{\delta^2 + \gamma^2}$, which we define to be $\cos \eta$. When $\eta \rightarrow 0$, we expect the particle sector to shrink to zero size, which should give a diverging ratio of pre-factors. By contrast, when $\eta \rightarrow \pi / 2$, we get a particle-hole symmetric saddle that has a unit ratio of pre-factors.
• Figure 4: Log-log plot of normalized $T_c$ vs normalized U. The slope of the linear fit is 2, indicating quadratic dependence of $T_c$ on U. U ranges from $10^{-4}$ eV to $10^{-2}$ eV with A fixed at 0.1 eV.