Comment on “Isotope Effect in Multi-Band and Multi-Channel Attractive Systems and Inverse Isotope Effect in Iron-Based Superconductors”

In a recent paper Yanagisawa et al. claim from a theoretical analysis of a multi-channel multi-band superconductor model that an inverse isotope exponent on the superconducting transition temperature Tc can be realized in iron-based superconductors. Simultaneously, a subgroup of the authors of ref. 1 performed the corresponding isotope effect experiment on (Ba,K)Fe2As2 by investigating the iron isotope exchange effect on Tc. 2) In accordance with their theoretical analysis they indeed report an unusually large sign reversed isotope exponent of 0:18ð3Þ which is in strong contrast to previous experiments on the nominally same system with the same composition in Ba, K content, namely Ba0:6K0:4Fe2As2, 3) where the exponent was determined to be 0:37ð3Þ. This conflict remains unsolved until now with the exception of ref. 4 where the iron isotope exponent has been determined for FeSe. In accordance with the results of ref. 3 a large positive isotope exponent has been seen thus questioning the outcome of ref. 1 and implicitly the findings of ref. 2. Here, we do not comment on the controversial experimental situation but address the theoretical analysis of ref. 1, where a variety of misleading assumptions have led to the conclusion that a sign reversed isotope exponent can be realized in a multi-band and multichannel attractive model for iron based superconductors. In this comment we derive the exact expressions for the two cases mentioned in the title of ref. 1 and proof that a reversed isotope exponent is not possible for both scenarios unless unphysical assumptions are being made. Note, that already in 1963 Kondo has studied a similar problem as outlined in ref. 1 and arrived at the conclusion that the isotope effect can vanish, but not reverse the sign. In the single-band multi-channel case studied first, the authors of ref. 1 assume that the two pairing interactions of different origin operate within a single band and lead to the appearance of two gaps, one being related to a phononic mechanism, the other to an antiferromagnetic (AF) interaction. The effective interactions have specific energy ranges within this single band with a cutoff at !1 for the phononic part and a range from !1 to !2 in the AF channel. This specific choice, which is also used for the two-band case, is the origin of the renormalized 2 [eq. (7) in ref. 1] and AF [eq. (25) in ref. 1] which— in turn—cause the inverse isotope effect. Obviously, the model is rather unphysical since it assumes that for a limited k-space range phonons cause the electron–electron attraction, directly followed by AF fluctuation mediated pairing starting from the same k-value where the phononic interaction terminates. The resulting gap, which stems from two gaps due to different pairing potentials, is thus continuous in k-space and cannot be viewed as a two-gap model. It is in particular rather amazing that a single electronic band develops pairing correlations stemming from very different potentials. In spite of this very unusual assumption, let us assume that two channels for pairing exist and refrain from authors’ assumption on the cut-off energies but use the conventional notation of Viðk; k0Þ being attractive within a range of h !i (i 1⁄4 1; 2) then the correct equation for Tc, using the authors’ notation, should read