Claudio Castellani talks about collective excitations
in strongly disordered superconductors.
He starts his talk by reviewing the two possible mechanisms, fermionic or
bosonic, to induce a superconductor insulator transition. In the latter Cooper
pairs break down and then single fermions undergo Anderson localization while
in the former Cooper pairs becomes localized directly. Recent experimental
results in Bi films, InO, TiN suggest that the bosonic mechanism is at work. An
important consequence of the bosonic mechanism is that a finite gap still
exists above Tc. It is becoming
broadly accepted this feature, once believed to be exclusive of cuprates, is
indeed generic in any strongly disordered superconductor.
He continues his presentation discussing the Ioffe and Mezard proposal
regarding the existence of a
glassy phase in strongly disordered superconductors that is controlled
by a rather large length scale that is not related to the coherence length.
Ioffe and Mezard model, a spin chain related to the disordered superconductor, by
Anderson pseudo-spin representation, is defined on a Cayley tree. The loop-less
structure of the Cayley tree suppresses localization effects and reduces
substantially the effort to obtain results. Claudio points out that the price
to pay is that the results cannot be extrapolated to any material or realistic
model. This is a further motivation to study this problem in a more realistic
setting: the disordered attractive Hubbard model in two dimensions.
After introducing the model and briefly reviewing previously relevant
literature such as the well known papers by Trivedi and co-workers, he then
presents results for the normalized spatial distribution function of the order
parameter. It was computed numerically by solving the associated BdG equations,
namely, in the mean-field limit. Small deviations from mean field were computed
in the random phase approximation.
I believe that technically this approximation should be fine provided
that disorder or Coulomb interactions are not strong enough. More
quantitatively I suspect that this approximation breaks down in the insulator
side when the bulk gap is of the order of the main level spacing in a
localization volume.
The resulting spatial distribution clearly illustrates the difference in
disordered superconductors between the spectral gap, that can be measured by
tunnelling (STM), from the amplitude of the order parameter. Claudio stresses
that this will be important for the rest of the talk. As in the Ioffe-Mezard
model the distribution has fat tails though the details are substantially
different. Claudio argues that the distribution is Tracy-Widom, that it is
known to be relevant in the context of 2d polymers. Using his words it is also
universal in the sense that qualitatively it does not depends on the disorder
or interaction strength of the Hubbard model. It was not clear to the exact
extension of the universality and the reason why the superconducting problem is
so closely related with the 2d polymer one.
In the last part of the talk, based on previous results, he addresses the
transport properties, more specifically current response, in the same
disordered Hubbard model.
From a technical point of view the main difference with respect to the
clean or weakly disordered case is the need to include vertex corrections (we
recall that the treatment of deviations from mean field is perturbative). The
numerical result clearly show that the current flows through rather filamentary
structures. Only some parts of sample contribute but still long-range order is
preserved. After a lively exchange with the audience it is argued that this
behaviour could be confirmed experimentally by studying the coherence peak and
the tunnelling gap by STM techniques.
Finally he presents results for the optical conductivity. Disorder induces
a novel coupling to the vector potential. Clear deviations from the BCS results
(Mathis-Bardeen) due to disorder are observed. There is a missing spectral
weight, originated in SC islands, it is claimed to be transferred to high
frequencies. No evidence at all of Higgs mode. The conductivity seems to be
dominated by phase, no amplitude, fluctuations.
I asked whether these results are robust to Coulomb interactions that are
not taken into account. It is well known that in clean superconductors no phase
collective excitations are observed because Coulomb interactions pushes the
frequency of the excitation above the gap. I am told that preliminary results
suggests that disorder prevent this mixing and therefore these results maybe be
observed experimentally.
He finishes by commenting recent results on an insulating peak by Ovadia
et. al. that might be interpreted as a signature of either Many-Body
localization or glassy physics. I mention that I do not see a clear
contradictions between the two interpretation since Many-body localization is
also related to extremely slow dynamics.
Blogged by Antonio Garcia-Garcia
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