Quantum Fluctuations in Thin Superconducting Wires of Finite Length

In one-dimensional wires, fluctuations destroy superconducting long-range order and stiffness at finite temperatures; in an infinite wire, quasi-long-range order and stiffness survive at zero temperature if the wire’s dimensionless admittance $\mu$ is large, $\mu >2$. We analyze the disappearance of this superconductor-insulator quantum phase transition in a finite wire and its resurrection due to the wire’s coupling to its environment characterized through the dimensionless conductance $K$.

Figure 1

Left: Setup with a quantum wire embedded in a voltage ($V$) driven loop with parallel ($Z_{\mathrm{p}}$) and serial ($Z_{\mathrm{s}}$) impedances defining the environment. Right: Phase diagram with superconducting and insulating phases separated by a quantum phase transition at $K=1$. The superconducting phase splits into weak and strong regimes separated by a crossover at $\mu \approx 2$, the leftover of the SI transition in the infinite wire. The insets show sketches of the wire’s $I_{\mathrm{w}}\mathrm{\text{−}}{V}_{\mathrm{w}}$ characteristic at $T=0$. In the superconductor ($K>1$) the algebraic characteristic is dominated by the environment at small currents $I<I_{\mathrm{d}\mathrm{e}\mathrm{c}}$. A highly conducting shunt with $K\gg \mu$ allows one to probe the wire above the deconfinement current $I_{\mathrm{d}\mathrm{e}\mathrm{c}}$; $I_c$ denotes the critical current of the wire. The insulator exhibits a Coulomb gap behavior below the critical voltage $V_c$.

Figure 2

Phase slip solutions with periodic (solid lines denote contours ${\varphi }_{\mathrm{P}}=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}$, left) and Neumann (${\varphi }_{\mathrm{N}}$, right) boundary conditions at intervortex distance $L/c_s<\overline{\tau }$. Left: For $K\gg \mu$, defects are screened mutually; the $2\pi$ phase drop appears along the $x$ axis and drives a large current (dashed lines) through the highly conducting shunt; the resulting string confines defect pairs. Right: With $\mu \gg K$, defects are screened individually by their mirror images (grey underlaid area); the $2\pi$ phase drop appears along the $\tau$ axis and sets up displacement currents in the wire that cannot escape through shunt; hence, charge accumulates at the boundary, resulting in a large voltage over the shunt; the defects are asymptotically free.