^{1}and E. Cappelluti

^{1,a)}

### Abstract

The electron-phonon interaction plays a fundamental role in the superconducting and normal-state properties of all the high- materials, from cuprates to fullerenes. Another common element of these compounds is in addition the extremely small Fermi energy , which is comparable with the range of the phonon frequencies. In such a situation the adiabatic principle , on which the standard theory of the electron-phonon interaction and of the superconductivity relies, breaks down. In this contribution we discuss the physical consequences of the breakdown of the adiabatic assumption, with special interest on the superconducting properties. We review the microscopic derivation of the nonadiabatictheory of the electron-phonon coupling which explicitly takes into account higher-order electron-phonon scattering not included in the conventional picture. Within this context we discuss also the role of the repulsive electron-electron correlation and the specific phenomenology of cuprates and fullerides.

The authors acknowledge fruitful collaborations on this subject with C. Grimaldi, S. Strässler, P. Benedetti, M. Scattoni, P. Paci, M. Botti, L. Boeri, S. Ciuchi, and G. B. Bachelet. We also acknowledge financial support from the MIUR projects COFIN03 and FIRB RBAU017S8R.

I. INTRODUCTION

II. BREAKDOWN OF MIGDAL’S THEOREM

A. The nonadiabatic hypothesis

B. The vertex function beyond Migdal’s theorem

III. NONADIABATICTHEORY OF SUPERCONDUCTIVITY AND NORMAL STATE

A. One-particle self-energy

B. Superconducting instability

C. Phenomenology in normal state: the Pauli spin susceptibility

D. Photoemission and real axis analysis

IV. NONADIABATICSUPERCONDUCTIVITY IN FULLERIDES AND CUPRATES

A. Correlationeffects on the electron-phonon scattering

B. Fullerenes

C. Copper oxides

V. CONCLUSION

### Key Topics

- Non adiabatic reactions
- 123.0
- Superconductivity
- 89.0
- Superconductivity models
- 83.0
- Non adiabatic couplings
- 38.0
- Phonon electron interactions
- 37.0

## Figures

Re-elaboration of vs. plot after Refs. 23 and 24, including magnesium diboride alloys.

Re-elaboration of vs. plot after Refs. 23 and 24, including magnesium diboride alloys.

Diagrammatic expression of the electron-phonon self-energy. The solid line represents the electron Green’s function, the wavy line the phonon propagator, and the filled circles the electron-phonon vertex function.

Diagrammatic expression of the electron-phonon self-energy. The solid line represents the electron Green’s function, the wavy line the phonon propagator, and the filled circles the electron-phonon vertex function.

Self-energy (*a*) and superconducting pairing (*b*) of an electron-phonon system in conventional Migdal–Eliashberg framework.

Self-energy (*a*) and superconducting pairing (*b*) of an electron-phonon system in conventional Migdal–Eliashberg framework.

Feynman’s representation of the first-order diagram appearing in the nonadiabatic regime.

Feynman’s representation of the first-order diagram appearing in the nonadiabatic regime.

Sign of the vertex function in the space for a nonadiabatic system .

Sign of the vertex function in the space for a nonadiabatic system .

Momentum-frequency average of the vertex diagram as function of the adiabatic parameter for different values of the momentum cutoff (from the top to the bottom): , 0.3, 0.5, 0.7, 0.9, and for .

Momentum-frequency average of the vertex diagram as function of the adiabatic parameter for different values of the momentum cutoff (from the top to the bottom): , 0.3, 0.5, 0.7, 0.9, and for .

Nonadiabatic electron-phonon self-energy including the first order vertex diagram arising from the breakdown of Migdal’s theorem.

Nonadiabatic electron-phonon self-energy including the first order vertex diagram arising from the breakdown of Migdal’s theorem.

Renormalization function for and . Solid lines: nonadiabatic theory with . Dashed line: noncrossing approximation with no vertex diagram.

Renormalization function for and . Solid lines: nonadiabatic theory with . Dashed line: noncrossing approximation with no vertex diagram.

Isotope coefficient on the effective electronic mass calculated for . Solid lines: nonadiabatic theory for . Dashed line: noncrossing approximation with no vertex diagrams.

Isotope coefficient on the effective electronic mass calculated for . Solid lines: nonadiabatic theory for . Dashed line: noncrossing approximation with no vertex diagrams.

Self-consistent equation for the superconducting order parameter in the nonadiabatic theory.

Self-consistent equation for the superconducting order parameter in the nonadiabatic theory.

Superconducting critical temperature in the nonadiabatic theory as function of the ratio for and different values of (from the top to the bottom line): , 0.3, 0.5, 0.7, 0.9.

Superconducting critical temperature in the nonadiabatic theory as function of the ratio for and different values of (from the top to the bottom line): , 0.3, 0.5, 0.7, 0.9.

Isotope coefficient for in the nonadiabatic theory as function of the adiabatic ratio. Same values of and as in previous figure. Smaller values of correspond to lines with steeper initial slope.

Isotope coefficient for in the nonadiabatic theory as function of the adiabatic ratio. Same values of and as in previous figure. Smaller values of correspond to lines with steeper initial slope.

Frequency structure of the vertex function for different values of the exchanged momentum: (from top line to the bottom) . The adiabatic parameter is here set to . Left panel refers to the normal state , right panel to the superconducting state . Filled circles mark the static and dynamic limits in the normal and superconducting state.

Frequency structure of the vertex function for different values of the exchanged momentum: (from top line to the bottom) . The adiabatic parameter is here set to . Left panel refers to the normal state , right panel to the superconducting state . Filled circles mark the static and dynamic limits in the normal and superconducting state.

Critical temperature as a function of the impurity scattering rate for different values of in the nonadiabatic theory. The dashed line corresponds to the noncrossing approximation with no vertex contribution.

Critical temperature as a function of the impurity scattering rate for different values of in the nonadiabatic theory. The dashed line corresponds to the noncrossing approximation with no vertex contribution.

Diagrammatic representation of the spin vertex function in nonadiabatic regime. Wavy lines represents the electron-phonon interaction, dashed lines the electron-electron Coulomb repulsion.

Diagrammatic representation of the spin vertex function in nonadiabatic regime. Wavy lines represents the electron-phonon interaction, dashed lines the electron-electron Coulomb repulsion.

Spin susceptibility as function of the adiabatic parameter and of the electron-phonon coupling . The total Pauli spin susceptibility is normalized with respect to the purely electronic one with a Stoner factor . Dashed lines represent the spin susceptibility in the noncrossing approximation, solid lines are the nonadiabatic theory with vertex diagram (from lower to upper line: , 0.3, 0.5, 0.7, 0.9).

Spin susceptibility as function of the adiabatic parameter and of the electron-phonon coupling . The total Pauli spin susceptibility is normalized with respect to the purely electronic one with a Stoner factor . Dashed lines represent the spin susceptibility in the noncrossing approximation, solid lines are the nonadiabatic theory with vertex diagram (from lower to upper line: , 0.3, 0.5, 0.7, 0.9).

Isotope effect on the spin susceptibility as a function of and as a function of . Solid lines and dashed lines as in the previous figure.

Isotope effect on the spin susceptibility as a function of and as a function of . Solid lines and dashed lines as in the previous figure.

Panel : Real and imaginary part of for a Einstein phonon mode with and in the presence of impurity scattering. Solid lines corresponds to (upper panel: from bottom to top; lower panel: from top to bottom): , where is the impurity scattering rate. Energy quantities are expressed in units of . The dashed line in the upper panel is the real part of the self-energy in the adiabatic infinite bandwidth limit . Panel : Renormalized electron dispersion corresponding (from left to right) to panel . The dashed line represents the adiabatic limit.

Panel : Real and imaginary part of for a Einstein phonon mode with and in the presence of impurity scattering. Solid lines corresponds to (upper panel: from bottom to top; lower panel: from top to bottom): , where is the impurity scattering rate. Energy quantities are expressed in units of . The dashed line in the upper panel is the real part of the self-energy in the adiabatic infinite bandwidth limit . Panel : Renormalized electron dispersion corresponding (from left to right) to panel . The dashed line represents the adiabatic limit.

Schematic picture of uncorrelated (*a*) and correlated (*b*) electrons. Correlated systems are characterized by the correlation hole (depicted in panel ) which surrounds each electron and prevent charge fluctuations with wavelength less than .

Schematic picture of uncorrelated (*a*) and correlated (*b*) electrons. Correlated systems are characterized by the correlation hole (depicted in panel ) which surrounds each electron and prevent charge fluctuations with wavelength less than .

Phase diagram of conventional superconductors compared with the fullerene compounds in the space defined by the electron-phonon coupling and the adiabatic parameter . Data for the family were taken from DFT, tight-binding, and *ab initio* calculations^{89,90,98} by using standard values for the density of states states/(eV-spin-) and for the Fermi energy .

Phase diagram of conventional superconductors compared with the fullerene compounds in the space defined by the electron-phonon coupling and the adiabatic parameter . Data for the family were taken from DFT, tight-binding, and *ab initio* calculations^{89,90,98} by using standard values for the density of states states/(eV-spin-) and for the Fermi energy .

Phonon frequency (lower panel) and Coulomb pseudopotential (upper panel) as functions of the electron-phonon coupling . Both the ME (filled squares) and the nonadiabatic (open triangles) equations have been solved in order to reproduce and .

Phonon frequency (lower panel) and Coulomb pseudopotential (upper panel) as functions of the electron-phonon coupling . Both the ME (filled squares) and the nonadiabatic (open triangles) equations have been solved in order to reproduce and .

Schematic phase diagram of the copper oxides compounds in the space of temperature vs. doping.

Schematic phase diagram of the copper oxides compounds in the space of temperature vs. doping.

Graphical sketch of the different contributions to the effective superconducting coupling. Top panel: The coupling function is mainly determined by the coherent spectral weight, and it exhibits a monotonous growing behavior as a function of doping. The vertex factor tends to enhance the effective coupling at low doping and to depress it at high doping. Middle panel: The total effective electron-phonon coupling has a maximum at some finite value of ; when the effective Morel–Anderson pseudopotential is subtracted, superconductivity is suppressed at high doping. Lower panel: Resulting phase diagram for superconductivity: superconductivity is only possible in a finite region of phase space (gray region), where is positive.

Graphical sketch of the different contributions to the effective superconducting coupling. Top panel: The coupling function is mainly determined by the coherent spectral weight, and it exhibits a monotonous growing behavior as a function of doping. The vertex factor tends to enhance the effective coupling at low doping and to depress it at high doping. Middle panel: The total effective electron-phonon coupling has a maximum at some finite value of ; when the effective Morel–Anderson pseudopotential is subtracted, superconductivity is suppressed at high doping. Lower panel: Resulting phase diagram for superconductivity: superconductivity is only possible in a finite region of phase space (gray region), where is positive.

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