Giant Nernst and Hall effects due to chiral superconducting fluctuations

We consider the Nernst and Hall effects in the fluctuation regime of chiral superconductors above transition temperatures, that are raised not by the conventional Lorentz force but by a mechanism that is an analog of the anomalous Nernst or Hall effects, i.e., asymmetric scattering due to chiral superconducting fluctuations. It is found that these effects can be gigantic for cleaner samples compared to conventional ones, exhibiting qualitatively distinct behavior. The results provide systematic and comprehensive understanding for recent experimental observations of the Nernst effect in a clean ${\mathrm{URu}}_2{\mathrm{Si}}_2$ sample, which is suggested to be a chiral superconductor.

Figure 1

Upper panel: AL, MT, and DOS diagrams. The AL and DOS diagrams have the mirror image counterparts. Wavy lines and curly lines with crossed circles represent the fluctuation propagator in zero magnetic field, $L$, and the chirality-polarized one, ${\tilde{L}}_C^{\prime }$, respectively, where their definitions are given in Appendix pp2. Solid lines with arrows are the one-particle Green's functions. Open circles represents electric current vertex, and bullets represent energy current vertex (electric current vertex), for ${\alpha }_{xy}$ (${\sigma }_{xy}$). Lower panel: Diagrams in which the information of the chirality disappears. Shaded circles represent any diagrams without fluctuation propagators and the two current vertices are inserted into any propagators.

Figure 2

Diagrams which contribute to the ANE and AHE raised by the CSF mechanism. The double lines represent the renormalized four-point vertex, $W(\mathit{k},{\omega }_j)$.

Figure 3

${\alpha }_{xy}$ raised by the CSF mechanism versus $T/T_c$ for several values of RRR. The magnitudes of ${\alpha }_{xy}$ are normalized by the value of the most clean one at $T_c$, ${\overline{\alpha }}_{xy}:={\alpha }_{xy}(T_c;\mathrm{RRR}=1000)$. We used the material parameters of ${\mathrm{URu}}_2{\mathrm{Si}}_2$.

Figure 4

Schematic picture of a scattering process raised only by chiral superconducting fluctuations. The solid and dashed arrows represent a propagating electron and hole, respectively, and the circle with arrows means the short-lived Cooper pairs with the chirality $C=+1$. By one skew-scattering event, an electron is transformed into a hole (i.e., so-called Andreev reflection). Thus, processes with an even number of the events do not change the momentum of the electron and therefore do not contribute to the Nernst and Hall effects. Also, processes with an odd number of events are suppressed after averaging over fluctuations owing to the conservation of the electron number in the normal state (this figure is one example of the even cases). Thus, to obtain the ANE and AHE due to skew scattering, we need additional scattering processes which disturb the above-mentioned cancellation of skew scatterings.

Figure 5

Diagrams (a), (b), and (c) of Fig. 2 with wave numbers and frequencies explicitly shown.

Figure 6

Numerical result for $u_{(2,1)}(t,0)$, $w_{(2,1)}(t,0)$, $f\left(t\right)$ (dots), approximation function $u^{\mathrm{app}}\left(t\right)$, $w_3^{\mathrm{app}}(t,0)$, $f^{\mathrm{app}}\left(t\right)$ (dashed line), and modified approximation function ${\overline{u}}^{\mathrm{app}}\left(t\right)$, ${\overline{w}}_3^{\mathrm{app}}(t,0)$, ${\overline{f}}^{\mathrm{app}}\left(t\right)$ (solid line). The first and third ones coincide with each other quite well.

Figure 7

Relation between the relative momentums of fluctuating Cooper pairs, $\mathit{p}$, ${\mathit{p}}^{\prime }$, the momentum of four-point vertex, $\mathit{k}$, and the angle $\theta$ that is the angle between ${\mathit{p}}_{\parallel }$ and ${\mathit{p}}_{\parallel }^{\prime }$.

Figure 8

The contours of energy integration in Eqs. (G3) (upper panel) and (G4) (lower panels). Dots represent poles of order 1. Each contour integration generates the factor ${(i{\omega }_l+1/\tau )}^{-1}$.