# Strongly Correlated Superconductivity: how can repulsion enhance Tc?

In conventional superconductors, the repulsive interaction between electrons opposes to phonon-mediated pairing. We have shown that even phonon-mediated superconductivity can be favoured by repulsion under suitable conditions which are realized in fullerenes.

Trivalent alkali-doped fullerenes are almost certainly electron-phonon superconductors, and their critical temperature can reach around 40K. There are however many experimental evidences that seem to exclude a simple BCS (or Migdal-Eliashberg) scenario, since electron-electron correlations are likely to play a central role. A realistic estimate of the relevant energy scales, and an analysis of the experimental scenario confirms that a moderate phonon-driven pairing coexists with a significantly stronger repulsion in fullerenes. The clearest signature of correlation is the Mott insulating behaviour of tetravalent compounds and of “expanded” trivalent compounds.

We have developed a theory to describe the high-temperature phonon-mediated superconductivity in the presence of strong correlations, which we have labelled “Strongly Correlated Superconductivity” [1]. In a nutshell, using Dynamical Mean-Field Theory for a three-band Hubbard model which also contains electron-phonon interaction, we have shown that s-wave phonon-mediated superconductivity can be strongly enhanced by Coulomb repulsion. The main message can be summarized very general terms: Close to a Mott transition, the large Coulomb repulsion U strongly renormalizes the itinerant carriers, leading to a small quasiparticle bandwidth $$W_*= ZW$$, where  $$Z \ll 1$$ is the quasiparticle residue. Naively, this small effective bandwidth corresponds to an increased quasiparticle density of states at the Fermi level $$\rho$$ = $$\rho_0 Z$$ , which could at first sight be thought to enhance the attractive coupling $$\lambda$$  =$$\rho V$$ and thus the critical temperature ($$\rho_0$$ is the bare density of states at the Fermi energy per spin; V is the pairing attraction). However, usually a decreasing Z does not turn into an increase of $$\lambda$$ , because the pairing attraction V is itself renormalized down by a factor Z2 within Migdal-Eliashberg theory, so that the increase of U finally depresses Tc, an effect further reinforced by a rising Coulomb pseudopotential µ*. In our model for fullerenes the increase in the critical temperature is precisely due to the fact that the quasiparticle density of states is enhanced, while the attractive interaction is essentially unaffected. The physical reason for this is that electron-phonon interaction in the fullerenes is of Jahn-Teller nature and it couples with spin and orbital degrees of freedom, which are not frozen by the Coulomb interaction which instead freezes charge fluctuations. This finding highlights the generality of our scenario according to which any pairing interaction which does not involve directly charge degrees of freedom can be enhanced by the proximity of a Mott transition. This is, e.g., the case of antiferromagnetic interactions in the cuprates and in the pnictided. In this sense, our picture is a general scenario for superconductivity in correlated systems which goes well beyond the realm of fullerenes, where it has been unveiled. The general aspects of our theory have been discussed by P. Nozieres and P. W. Anderson.

As far as our model for fullerenes is concerned, the phase diagram as a function of the correlation strength U/W presents a bell-shaped order parameter right before the Mott transition. In actual compounds the degree of correlation is controlled by the lattice spacing (inter-molecular distance). Larger distances give rise to narrower bands and to a more correlated solid.

Anomalous Properties of a Strongly Correlated Superconductor

Strongly Correlated Superconductivity does not simply reflect in a relatively high critical temperature, and it presents several specific features, which we identified, and proposed as experimental tests of our approach. We have built our understanding on the basis of “impurity models” in which one correlated site is embedded in a conduction bath. Within DMFT lattice models are mapped onto impurity model with a self-consistently determined dynamical hybridization.

The essential lesson we learn from the impurity models corresponding to lattice models for fullerenes is the existence of an unstable fixed point which separates a “local Fermi liquid” and non Fermi-liquid phase formed by a collection of singlets. Also at the critical point the system is not a Fermi liquid [3]. In the lattice model the fixed point occurs when the renormalized bandwidth ZW is of the same order of the attraction J. Therefore when the Mott transition is approached one necessarily crosses the critical point, and reaches a non-Fermi liquid region. Moreover, at the critical point several susceptibilities diverge, with the s-wave Cooper channel as the leading divergence in the absence of specific nesting conditions. As a consequence:

• The normal state close to the Mott transition should be non-Fermi liquid and display a pseudogap
• Superconductivity appears as a way to heal the pathology of the normal state. This leads to a gain of kinetic energy in the superconducting state
• Two energy scales appear at low-frequency, and will be reflected in the spectral functions, and also in thermodynamical observables like the specific heat or the spin susceptibility

Expanded Alkali-Doped Fulerides: Strongly Correlated Superconductivity unleashed

As we discussed briefly, the original motivation of our study came from alkali-doped fullerides A3C60. Yet, in these materials most of our predictions could not be verified, essentially because the value of the lattice spacing puts them in the less correlated side of the superconducting bell. More recently synthetized expanded fullerides of different chemical composition provided a strong support to our theory. In these compounds the critical temperature decreases as a function of the lattice spacing, in agreement with our theory. This makes this compounds the natural systems to test our predictions for the strongly correlated phase, like the existence of a pseudogap and a gain of kinetic energy in the superconducting phase [6]

People: M. Capone, C. Castellani

[1] M. Capone, M. Fabrizio, C. Castellani and E. Tosatti, Science 296, 2364 (2002)

[2] M. Capone, M. Fabrizio, and E. Tosatti, Phys. Rev. Lett. 86, 5361 (2001)

[3] L. De Leo and M. Fabrizio, Phys. Rev. B 69, 245114 (2004)

[4] M. Capone, M. Fabrizio, C. Castellani and E. Tosatti, Phys. Rev. Lett. 93, 047001 (2004)

[5] M. Schiro`, M. Capone, M. Fabrizio, and C. Castellani, Phys. Rev. B 77, 104522 (2008).

[6] M. Capone, M. Fabrizio, C. Castellani and E. Tosatti, Rev. Mod. Phys. (Colloquium) in press; arXiv:0809.0910

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