Superconductivity from piezoelectric interactions in Weyl semimetals

We present an analytical low-energy theory of piezoelectric electron-phonon interactions in undoped Weyl semimetals, taking into account also Coulomb interactions. We show that piezoelectric interactions generate a long-range attractive potential between Weyl fermions. This potential comes with a characteristic angular anisotropy. From the one-loop renormalization group approach and a mean-field analysis, we predict that superconducting phases with either conventional $s$-wave singlet pairing or nodal-line triplet pairing could be realized for sufficiently strong piezoelectric coupling. For small couplings, we show that the quasiparticle decay rate exhibits a linear temperature dependence where the prefactor vanishes only in a logarithmic manner as the quasiparticle energy approaches the Weyl point. For practical estimates, we consider the Weyl semimetal TaAs.

Figure 1

Feynman diagrams for the vertices in Eq. (23), coupling the field $\phi$ (wiggly curve) to (a) electrons (solid line) and to (b) phonons (dashed). The Coulomb (piezoelectric) vertex $\sim g_e$ ($\sim g_{\mathrm{ph}}$) is shown as filled (open) circle.

Figure 2

Effective $e-e$ interaction at tree level. (a) Repulsive Coulomb interaction. (b) Phonon-mediated $e-e$ interaction, see Eq. (26).

Figure 3

(a) Polar plot of the anisotropy function $\gamma \left(\theta \right)$ in Eq. (27) for the case of TaAs, with $\gamma \left(\theta \right)$ multiplied by $g_{\mathrm{ph}}^2/{\rho }_0$. We take $\varepsilon =20{\varepsilon }_0$, where ${\varepsilon }_0$ is the free-space permittivity. The piezoelectric tensor values are taken from Ref. [50], where we get $\overline{\gamma }\simeq 0.20$ in Eq. (34). The blue color indicates that the phonon-mediated interaction is always attractive. (b) Effective anisotropy function of the total $e-e$ interaction potential in Eq. (29), where we adjust $e_{33}$ such that $\overline{\gamma }=0.97$. Blue again indicates attraction while orange represents repulsion.

Figure 4

Schematic form of the possible amplitudes generated by the local field theory in Eq. (23), where shaded regions represent dressed vertices in a perturbative expansion.

Figure 5

Diagrams contributing to the one-loop RG equations. (a) Coulomb correction to the electronic self-energy. (b) Vertex correction due to Coulomb interaction. (c) Polarization bubble inserted in the Coulomb propagator. (d) Piezoelectric correction to the electronic self-energy. (e) Piezoelectric vertex correction.

Figure 6

Renormalized Fermi velocity $v\left(\ell \right)$ and Coulomb coupling $g_e\left(\ell \right)$ as functions of the RG flow parameter $\ell =ln({\mathrm{\Lambda }}_0/\mathrm{\Lambda })$. (a) Flow diagram obtained using the estimated parameters for TaAs, corresponding to $\overline{\gamma }=0.20$, but considering $2N=4$ Weyl nodes. (b) Flow diagram obtained by enhancing the piezoelectric coefficient $e_{33}$ to reach $\overline{\gamma }\simeq 75$. Here we stop the RG flow at the scale where the Fermi velocity vanishes, at which point the WSM becomes unstable.

Figure 7

Schematic representation of the dispersion relations of the two bands for Bogoliubov quasiparticles. Here we set the mean-field parameters $a_{\parallel }=a_{\perp }=0$ and plot the dispersion for $k_3=0$. (a) For $a_2={\mathrm{\Delta }}_0=0$, the Weyl nodes conjugated by time-reversal symmetry are represented as two degenerate Bogoliubov-Weyl nodes. (b) For ${\mathrm{\Delta }}_0=0$ but $a_2\ne 0$, the spectrum is gapless along a nodal line located in the $k_3=0$ plane. (c) For $a_2\ne 0$ and ${\mathrm{\Delta }}_0\ne 0$, the spectrum is fully gapped.