Universal Conductance of Nanowires near the Superconductor-Metal Quantum Transition

We consider wires near a zero temperature transition between superconducting and metallic states. The critical theory obeys hyperscaling, which leads to a universal frequency, temperature, and length dependence of the conductance; quantum and thermal phase slips are contained within this critical theory. Normal, superconducting, and mixed (SN) leads on the wire determine distinct universality classes. For the SN case, wires near the critical point have a universal dc conductance which is independent of the length of the wire at low temperatures.

Figure 1

A wire which is tuned from a superconductor to a metal by (say) reducing its diameter. The leads on the wire are either normal (N) or superconducting (S) and the NN, SN, and SS cases belong to distinct universality classes.

Figure 2

The universal function $\mathrm{Re}[F_X]$ in the large $n$ limit for real frequencies, with $y=\gamma \hslash \omega L^2/\delta$. We discretized $x$ in Eq. (7) on a lattice of spacing unity, rescaled $\tau$ and $\Psi$ to obtain $\delta A/\hslash =1$ and $A\gamma =1$, and set $b=1$ and used a ultraviolet frequency cutoff $\pi$. (a) Test of universality for the SN case with $L=400$. The parameters $(C_{\ell },C_r,H_{\ell },H_r)$ are (i) (1,1,1,0); (ii) (1,1,10,0); (iii) (1,10,1,0); (iv) (5,1,5,0); (v) (1,1,0.5,0). A similar insensitivity to boundary parameters was found for the NN and SN cases. (b) Results for all three universality classes are obtained for $L=800$; other parameters as in (a), and $(C_{\ell },C_r,H_{\ell },H_r)$ taking values (1,1,0,0) for NN, (1,1,1,0) for SN, and (1,1,1,1) for SS. For the SS case, there is an additional Josephson current contribution $F_{\mathrm{SS}}(y)=\pi \varrho \delta (y)$, with $\varrho =2.98$, which is not shown above. All three classes have the common behavior $F_X(y\to \infty )=0.81994(1+i)/\sqrt{y}$ for large $n$.

Figure 3

Extrapolation of the dc conductance ${\mathcal{C}}_{\mathrm{SN}}$ to the universal scaling limit ($L\to \infty$). The large $n$ parameters are as in Fig. 2, but with $(b,C_{\ell },C_r,H_{\ell },H_r)=(i)$ (46.25,1,1,0.745,0); (ii) (0.925,1,1,7.45,0). The quantum Monte Carlo parameters are as in the text, with $(H_{\ell },H_r)=(1)$ (10,0), (2) (1,0).