# Unconventional electron-phonon interaction and nonadiabatic effects

The conventional understanding of the electron-phonon phenomenology in condensed matter strongly relies on the *adiabatic* assumption, i.e. that nuclei dynamics is much slower than the electron one. In solid crystals the validity of this assumption is usually related to the comparison of the phonon frequencies \(\omega_{ph}\) with the Fermi energy *E*_{F}. In conventional systems \(\omega_{ph}\ll E_F\), assuring thus the reliability of the adiabatic assumption. This regime however is violated in a number of (nonadiabatic) systems, among which high-*T _{c}* superconducting copper oxides, fullerenes, etc. In the extreme limit of strong coupling these processes give rise to self-trapping and lattice polaronic effects.

Our research activity in this field is aimed to identify and investigate the properties of nonadiabatic systems with respect to the normal state and to the superconducting pairing. We employ both analytical techniques of Quantum Field Theory and numerical methods, as dynamical mean-field theory. A key role in this framework is played by the so-called MIgdal’s theorem which related nonadiabatic processes to vertex diagrams. This permits to define in a controlled way a perturbative theory with respect to the adiabatic parameter \(\omega_{ph}\ll E_F\) which is still valid for generic electron-phonon coupling. We have shown that physical properties of such nonadiabatic systems depend strongly on the frequency-momentum structure of the exchanged phonons. In particular, due to the opening of new nonadiabatic scattering channels, we predict, under some favourable conditions, a remarkable increase of the *effective* electron-phonon coupling and a strong enhancement of the superconducting critical temperature *T _{c}*.

Our research activity in this field is aimed to identify and investigate the properties of nonadiabatic systems with respect to the normal state and to the superconducting pairing. We employ both analytical techniques of Quantum Field Theory and numerical methods, as dynamical mean-field theory. A key role in this framework is played by the so-called MIgdal’s theorem which related nonadiabatic processes to vertex diagrams. This permits to define in a controlled way a perturbative theory with respect to the adiabatic parameter \(\omega_{ph}\ll E_F\) which is still valid for generic electron-phonon coupling. We have shown that physical properties of such nonadiabatic systems depend strongly on the frequency-momentum structure of the exchanged phonons. In particular, due to the opening of new nonadiabatic scattering channels, we predict, under some favourable conditions, a remarkable increase of the *effective* electron-phonon coupling and a strong enhancement of the superconducting critical temperature *T _{c}*.

We apply this unconventional electron-phonon scenario to the analysis of the physical properties of real materials. We have shown that, because of their small carrier density and hence of their small Fermi energy, nonadiabatic effects are unavoidable in high-*T _{c}* superconducting copper oxides, suggesting thus that nonadiabatic processes could be the active principle of the underlying high-

*T*pairing. The presence of flas band and Van Hove singularities represent in addition a further source of nonadiabaticity in these compounds. An even stronger degree of nonadiabaticity is expected in

_{c}*A*

_{3}C

_{60}dopes fullerides where phonon frequencies range up to ~ 0.2 eV and the Fermi energy is about 0.25 eV. In both cases a crucial complementary role is played by the presence of strong electronic correlations which, within a nonadiabatic context, select favourable small-

**q**forward scattering.

A different kind of nonadiabatic processes are operative in MgB_{2} and similar alloys. The adiabatic assumption is indeed apparently not violated in these compounds where \(\omega_{ph}/E_F \ll 1\), if the Fermi energy is estimated by static LDA calculations. The very meaningfulness of the concept of Fermi energy is however questioned in these materials where the electron-phonon coupling is so large that the electronic energy fluctuations due to the quantum lattice vibrations is of the same order (or larger) than the Fermi energy itself.

**People**: E. Cappelluti, L. Pietronero

**References:**

– Pietronero, S. Strässler, C. Grimaldi, “Nonadiabatic superconductivity I. Vertex corrections for the electron-phonon interactions“, Phys. Rev. B **52**, 10516 (1995).

– C. Grimaldi, L. Pietronero, S. Strässler, “Nonadiabatic superconductivity II. Generalized Eliashberg equations beyond Migdal’s theorem“, Phys. Rev. B **52**, 10530 (1995).

– C. Grimaldi, L. Pietronero, S. Strässler, “Nonadiabatic superconductivity: electron-phonon interaction beyond Migdal’s theorem“, Phys. Rev. Lett. **75**, 1158 (1995).

– E. Cappelluti and L. Pietronero, “Nonadiabatic superconductivity: the role of van Hove singularities“, Phys. Rev. B **53**, 932 (1996).

– E. Cappelluti and L. Pietronero, “Van Hove singularities and nonadiabatic effects in superconductivity“, Europhys. Lett. **36**, 619 (1996).

– C. Grimaldi, E. Cappelluti, L. Pietronero, “Isotope effect on m* in high-*T _{c}* materials due to the breakdown of Migdal’s theorem“, Europhys. Lett.

**42**, 667 (1998).

– E. Cappelluti, C. Grimaldi, L. Pietronero, S. Strässler, “Nonadiabatic channels in the superconducting pairing of fullerides“, Phys. Rev. Lett.

**85**, 4771 (2000).

– E. Cappelluti, C. Grimaldi, L. Pietronero, S. Strässler, G.A. Ummarino, “Superconductivity of Rb

_{3}C

_{60}: breakdown of the Migdal-Eliashberg Theory“, Eur. Phys. J. B

**21**, 383 (2001).

– E. Cappelluti, C. Grimaldi, L. Pietronero, “Nonadiabatic Pauli susceptibility in fullerene compounds“, Phys. Rev. B

**64**, 125104 (2001).

– E. Cappelluti, S. Ciuchi, C. Grimaldi, L. Pietronero, S. Strässler, “High

*T*superconductivity in MgB

_{c}_{2}by nonadiabatic pairing“, Phys. Rev. Lett.

**88**, 117003 (2002).

– L. Boeri, G.B. Bachelet, E. Cappelluti, L. Pietronero, “Small Fermi energy and phonon anharmonicity in MgB

_{2}and related compounds“, Phys. Rev. B

**65**, 214501 (2002).

– M. Botti, E. Cappelluti, C. Grimaldi, L. Pietronero, “Nonadiabatic theory of the superconducting state“, Phys. Rev. B 66, 054532 (2002)

– E. Cappelluti, C. Grimaldi, L. Pietronero, “Spin susceptibility in small Fermi energy systems: effects of nonmagnetic impurities“, Eur. Phys. J. B

**30**, 511 (2002).

– E. Cappelluti, S. Ciuchi, C. Grimaldi, L. Pietronero, “Band-filling effects on electron-phonon properties of normal and superconducting states“, Phys. Rev. B

**68**, 174509 (2003).

– E. Cappelluti and L. Pietronero, “Electron-phonon renormalization in small Fermi energy systems“, Phys. Rev. B

**68**, 224511 (2003).

– L. Boeri, E. Cappelluti, C. Grimaldi, L. Pietronero, “Poor screening and nonadiabatic superconductivity in correlated systems“, Phys. Rev. B

**68**, 214514 (2003).

– L. Boeri, E. Cappelluti, L. Pietronero, “Small Fermi energies, zero-point fluctuations, and nonadiabaticity in MgB

_{2}“, Phys. Rev. B

**71**, 012501 (2005).

– P. Paci, M. Capone, E. Cappelluti, S. Ciuchi, C. Grimaldi, L. Pietronero, “Polaronic and nonadiabatic phase diagram from anomalous isotope effects“, Phys. Rev. Lett.

**94**, 036406 (2005).

– E. Cappelluti, P. Paci, C. Grimaldi, L. Pietronero, “Relevance of multiband Jahn-Teller effects on the electron-phonon interaction in

*A*

_{3}C

_{60}“, Phys. Rev. B

**72**, 054521 (2005).