Related references: (1) Feigelman et al Phys Rev Lett. 98, 027001(2007); (2) Feigelman et al Annals of Physics 325, 1368 (2010)
Main ingredients of their theory: study competition between Cooper pairing and localization (in the absence of Coulomb interations). In a BCS superconductor coherence lengths are much larger than the lattice constant so need an analytical theory to complement other numerical approaches.
Using Anderson’s pairing of exact eigenstates, they obtain the gap equation which relates the local pairing amplitude at r to the local pairing amplitude at r’ through a non-local kernel K(r,r’). The structure of the kernel is determined by the eigenstates of the single particle wave function that includes only the disorder potential.
Misha and collaborators claim that at the superconductor-insulator transition, the fractal nature of the non-interacting single particle states is important. There are strong local order parameter fluctuations and the region with finite pairing amplitude occupies only a small fraction of the total volume near the transition. Their central result relates the mean field transition temperature to the fractal dimension.
This mean field Tc can exceed the BCS value in the clean system. Note that a similar result was already predicted in our paper Ghosal et al PRB 65, 014501 (2011) (See Eq. 16 and Fig 5) using the same pairing of exact eigenstates as well as Bogoliubov de Gennes methods). Feigelman and collaborators have extended that analysis for a situation where the fractal nature of the eigenstates may be important. They have also pointed out that off-diagonal correlations between the different eigenstates could be important.
I list some points raised during and after the talk but related to this talk :
(1) There have been several assertions of Tc being enhanced by disorder (see also papers by Kravtsov and Mirlin). It is important to distinguish the mean field Tc that these authors as well as Feigelman are talking about from the actual Tc that is dominated by phase fluctuations and I believe monotonically decreases with disorder.
(2) Is the fractal nature of the single particle wave functions that is important near the Anderson transition relavant for the superconductor-insulator transition that occurs at a different critical disorder? This difference is rather stark in 2D where MIT occurs at arbitrarily small disorder but the SIT is moved off to a finite value.
(3) Misha also mentioned pseudogap and parity gap but their definitions were not entirely clear to me. The standard definition of the pseudogap is a suppression in the density of states between the actual transition temperature Tc (defined by where long range coherence sets in) and T* (a pairing scale or a mean field scale). Since the authors only calculated a mean field scale it is not clear how they could have accessed the pseudogap region in these calculations.
Blogged by Nandini Trivedi
Answers to the 3 questions posed by Nandini are presented below.
ReplyDelete(1). Minor remark: there are 2 papers in this subject: by i) Feigel'man, Ioffe, Kravtsov and Cuevas, Annals of Physics 325, 1368 (2010) and ii) by Burmistrov, Gornij and Mirlin, arxiv:1102.3323
Essense of the issue: it was shown in the paper i) that Ginzburg parameter Gi which controls the strength of all thermal fluctuation in 3D system is of the order of unity in the system considered. The ratio of real Tc to mean-field Tc0 is therefore ALSO of the order of unity. It is also trivial to see that energy scale J that controls phase fluctuations is related to Tco by the same parameter Gi. Thus enchancement of Tc0 directly translates to enchacement of real Tc. I should add that this issue was reiterated several times during my talk
(2) Fractality of single-electron wavefunctions is important for S-I transition in the following sense: it makes it possible, first, to obtain a pseudogap superconductivity with a Tc which is small compared to pseudogap. The S-I transition happens between such a superconductor and insulator. This very possibility of pseudopgaped superconductor in a simple system with just s-wave pairing and disorder seems to be related to fractal nature of wavefunctions. As soon as one postulates the presense of pseudogaped superconductor, the S-I transition itself is not very much dependent on fractality.
3) Pseudogap is the same as parity gap T^*. It is the binding energy of two electrons siting in the same localized eigenstate.
As it is explained in the item 1) above, we did calculated not only mean-field Tco but also real Tc (first, via Ginzburg-Landau functional and analysis of Gi number, and second, via analysis of virial expansion which has nothing to do with mean-field whatsoever). We also have identified the region of parameters where T^* >> Tc > 0