Anomalous scaling of the penetration depth in nodal superconductors

Recent findings of anomalous superlinear scaling of low-temperature $\left(T\right)$ penetration depth (PD) in several nodal superconductors near putative quantum critical points suggest that the low-temperature PD can be a useful probe of quantum critical fluctuations in a superconductor. On the other hand, cuprates, which are poster child nodal superconductors, have not shown any such anomalous scaling of PD, despite growing evidence of quantum critical points (QCP). Then it is natural to ask when and how can quantum critical fluctuations cause anomalous scaling of PD? Carrying out the renormalization group calculation for the problem of two-dimensional superconductors with point nodes, we show that quantum critical fluctuations associated with a point group symmetry reduction result in nonuniversal logarithmic corrections to the $T$ dependence of the PD. The resulting apparent power law depends on the bare velocity anisotropy ratio. We then compare our results to data sets from two distinct nodal superconductors: ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{6.95}$ and ${\mathrm{CeCoIn}}_5$. Considering all symmetry-lowering possibilities of the point group of interest, $C_{4v}$, we find our results to be remarkably consistent with ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{6.95}$ being near a vertical nematic QCP and ${\mathrm{CeCoIn}}_5$ being near a diagonal nematic QCP. Our results motivate a search for diagonal nematic fluctuations in ${\mathrm{CeCoIn}}_5$.

Figure 1

Schematic phase diagram of a quantum phase transition inside the superconducting dome associated with developing a new order represented by an order parameter $\varphi$. Here, $T_c$ is the superconducting transition temperature, and $x$ is a tuning parameter that drives the system through the quantum critical point at $x_c$. Near $x_c$, there is a quantum critical region where the penetration depth displays anomalous scaling.

Figure 2

Nodal structure of $d_{x^2-y^2}$ SC with coexisting orders for a single-band system. For multiband systems like ${\mathrm{CeCoIn}}_5$, the nodes on different surfaces will move simultaneously. The dashed (blue) lines at the corner of the Brillouin zone represent the Fermi surface, and the diagonal dashed (black) lines represent where the gap vanishes. The open circles denote the original nodes of $d_{x^2-y^2}$ SC, and the filled circles denote the reconstructed nodes with coexisting orders. (a) ${\mathrm{B}}_1$ axial nematic case; (b) ${\mathrm{B}}_2$ diagonal nematic case; (c) ${\mathrm{A}}_2$ case; (d) E case. Note for (a)–(c), each nodal point is spin degenerate. (a) and (c) correspond to the deformation of the gap (particle-particle channel), and hence are associated with ${\tau }_x$ in Nambu space. (b) corresponds to the deformation of the Fermi surface (particle-hole channel), and hence ${\tau }_z$. The spin degeneracy is lifted for (d) and the blue and red filled circles have different spin-momentum helicity. The arrows in (d) indicate in-plane spin orientation at the nodal points, and here the ordering is predominantly in the particle-hole channel.

Figure 3

Temperature dependence of the penetration depth (or equivalently the superfluid density with $\delta {\rho }_s\sim \delta {\lambda }^{-2})$, fitting experimental results for ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{6.95}$ (a) and ${\mathrm{CeCoIn}}_5$ (b). The diamonds and squares are the experimental results for ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{6.95}$ from Ref. [7] and for ${\mathrm{CeCoIn}}_5$ from Ref. [5], respectively. The experimental values of $\delta {\lambda }^{-2}$ have been normalized by an overall factor so that $\delta {\lambda }^{-2}\left(T_0\right)=1$. The solid lines are fit to the analytic expressions shown in Table 1. The fitting parameters are (a) for ${\tau }_x$ coupling, $1/l_0=0.075$ as deduced from the experimentally measured velocity anisotropy $v_F/v_{\mathrm{\Delta }}\sim 14$ [29, 30] with $N=2$; (b) for ${\tau }_z$ coupling with $N=6,1/l_0=0.2756$ as the best fit, and correspondingly $v_F^a/v_{\mathrm{\Delta }}^a\simeq 1.5829$ for all three bands $a=1,2,3$.