[**Comment on “Quantum Phase Slips and Transport in Ultrathin Superconducting Wires”**]{}

In a recent Letter [@ZGVZ], Zaikin, Golubev, van Otterlo, and Zimanyi (ZGVZ) criticized the phenomenological time-dependent Ginzburg-Laudau (TDGL) model which I used to study the quantum phase slip (QPS) for superconducting wires [@duan]. They claimed that they developed a “microscopic” model, made [*qualitative*]{} improvement on my overestimate of the tunnelling barrier due to electromagnetic field (EM), and claimed agreements with the experiments by Giordano [@nick].

In this comment, I want to point out that, i), ZGVZ’s result on EM barrier is expected in [@duan]; ii), their work is also phenomenological; iii), their renormalization scheme is fundamentally flawed; iv), they underestimated the barrier for ultrathin wires; v), their comparison with experiments is incorrect. Details are given below.

i), In [@duan] I emphasized results on relatively thick wires and concluded that the observations on Giordano’s wires with thickness $\sqrt{\sigma}=$$410-505\AA$ must be due to weak links. Both kinetic inductance and Mooij-Schön mode have been included [@duan; @scot]. The ultrathin wire limit of the Eq.(8) in [@duan] gives an EM barrier, $$\begin{aligned}
\frac{S_{EM}}{\hbar} \cong \frac{\pi}{2}
\sqrt{2\ln(q_{upper}/q_{lower})} \frac{\sqrt{\sigma}}
{\lambda_L} \frac{\hbar c}{e^2},
\label{ultrathin}\end{aligned}$$ where the ratio of cutoffs $q_{upper}/q_{lower}\ge 10$. [*Eq. (\[ultrathin\]) is the main result in Ref.[@ZGVZ]*]{} except some underestimate (cf., below).

ii), QPS is a far-from-equilibrium process occuring on a time scale of the order of inverse gap of the superconductors. ZGVZ’s model contains nothing more than a saddle point plus Gaussian fluctuations and does not go beyond TDGL. Despite minor differences from my work, Ref.[@ZGVZ] does not present any new physics. I also want to point out that their electric screening for the superconducting charge is described by condensate fraction $n_s$ which vanishes at $T_c$. This is incorrect since Debye screening is nondiscriminating against superconducting or normal charge.

iii), ZGVZ’s renormalization scheme is wrong since EM field [*qualitatively*]{} changes the Kosterlitz renormalization flow [@scot; @fisher; @zhang]. Therefore their conclusions on the “metal-superconductor” transition and wire resistance were not based on a solid foundation.

iv), ZGVZ underestimated the EM barrier for ultrathin wires. For Giordano’s wire of $\sigma=\pi(80\AA)^2$, if we [*assume*]{} it is homogeneous with an effective London penetration depth $\lambda_L=1000\AA$, the EM barrier [*alone*]{} reduces the tunneling probability to $e^{-60}$. Taking into account the substrate dielectric constant [@scot] $\epsilon=10$ for $Ge$, then the EM suppression is $e^{-330}$. [*In order to observe QPS in homogeneous wires, the wire radius must be smaller than $10\AA$*]{}.

v), It needs to be emphasized that the results by Sharifi [*et al.*]{} [@dynes] are NOT similar to that of [@nick], but qualitatively different. The wires in [@dynes; @neg] are known to be homogeneous. In thinner wires ($[40\AA]^2$) than those in [@nick], no quantum phase-slip was observed. The wire resistance still obeys thermally activated behavior, albeit being larger than predicted by classical models [@neg]. As stated by ZGVZ, the theory is for a homogeneous wire. Hence they should compare with the experiments in [@dynes] instead of [@nick]. A random Josephson junctions model [@scot] is suitable for granular wires [@nick; @peng].

TDGL was used to calculate the EM barrier for the quantum decay of supercurrent [@duan] and the EM contribution to vortex mass [@mass]. There is a deep physical reason [@mass] for the second order time derivative term in TDGL: superfluids are compressible due to number-phase conjugation. [*Hence the qualitative physics in [@duan; @mass] is much more general than the formalism of the TDGL model*]{}.

This work was supported by NSF (DMR 91-13631).

Ji-Min Duan

Department of Physics,

University of California-San Diego,

La Jolla, California 92093-0319

A. D. Zaikin, D. S. Golubev, A. van Otterlo, and G. T. Zimanyi, Phys. Rev. Lett. [**78**]{}, 1552 (1997).

J.-M. Duan, Phys. Rev. Lett. [**74**]{}, 5128 (1995).

N. Giordano, Phys. Rev. Lett. [**61**]{}, 2137 (1988); Physica B [**203**]{}, 460 (1994).

S. R. Renn and J.-M. Duan, Phys. Rev. Lett. [**76**]{}, 3400 (1996).

M. P. A. Fisher and G. Grinstein, Phys. Rev. Lett.[**60**]{}, 208 (1988).

S. C. Zhang, Phys. Rev. Lett. [**59**]{}, 2111 (1987).

F. Sharifi, A. V. Herzog, and R. C. Dynes, Phys. Rev. Lett. [**71**]{}, 428 (1993).

P. Xiong, A. V. Herzog, and R. C. Dynes, Phys. Rev. Lett. [**78**]{}, 927 (1997).

A. V. Herzog, P. Xiong, F. Sharifi, and R. C. Dynes, Phys. Rev. Lett. [**76**]{}, 668 (1996).

J.-M. Duan, Phys. Rev. B [**48**]{}, 333 (1993); [*ibid*]{}, [**49**]{}, 12381 (1994). J.-M. Duan, and A. J. Leggett, Phys. Rev. Lett. [**68**]{}, 1216 (1992).
