Quantum critical point in the superconducting transition on the surface of a topological insulator

Pairing in the Weyl semimetal appearing on the surface of a topological insulator is considered. It is shown that due to an “ultrarelativistic” dispersion relation there is a quantum critical point governing the zero-temperature transition to a superconducting state. Starting from the microscopic Hamiltonian with local attraction, we calculated using the Gor'kov equations, the phase diagram of the superconducting transition at arbitrary chemical potential, and its magnetic properties and critical exponents close to the quantum critical point. The Ginzburg-Landau (GL) effective theory is derived for small chemical potential, allowing us to consider effects of spatial dependence of order parameters in a magnetic field. The GL equations are very different from the conventional ones reflecting the chiral universality class of the quantum phase transition. The order-parameter distribution of a single vortex is found to be different as well. The magnetization near the upper critical field is found to be quadratic, not linear as usual. We discuss the application of these results to recent experiments in which surface superconductivity was found for some three-dimensional topological insulators, and we estimate feasibility of the phonon pairing.

Figure 1

Topological insulator plate in magnetic field. Surfaces are populated by Weyl quasiparticles and holes that both can be paired by interactions. The magnetic field creates vortices with normal cores (dark areas on the surfaces) of the radius of order of coherence length $\xi$.

Figure 2

Order parameter at zero temperature as a function of chemical potential of the TI surface Weyl semimetal at various values of coupling parametrized by the renormalized energy $U$, Eq. (14). For positive $U$ (blue lines) the superconductivity is strong and does not vanish even for zero chemical potential. There exists the critical coupling, $U=0$ (the red line), at which the second-order transition occurs at quantum critical point $\mu =0$. For negative $U$ the superconductivity still exists at $\mu >0$ but is exponentially weak.

Figure 3

Transition temperature as a function of chemical potential at supercritical ($U>0$, in blue), critical ($U=0$, in red), and subcritical values of coupling.

Figure 4

Phase diagram of the STI and order parameter as a function of $U$ describing the deviation from criticality near the quantum critical point at $\Delta =0,\mu =0$. The critical line is a straight line in mean-field approximation.

Figure 5

Vortex core structure near the QCP and order parameter in units of the bulk gap ${\Delta }_0$ as a function of distance from the center in units of coherence length $\xi$. The dash line is the approximate formula, while the top line (red) is the usual Abrikosov vortex profile.

Figure 6

Schematic picture of the band reconstruction due to phonon pairing in three different 2D fermionic systems. (a) Weyl semimetal. (b) BCS with parabolic dispersion law. (c) Classic BEC.