Small Fermi energy, zero point fluctuations and nonadiabaticity in MgB$_2$

TL;DR

This paper investigates how small Fermi energies and zero-point lattice fluctuations in MgB$_2$ lead to nonadiabatic effects, challenging the Born-Oppenheimer approximation and potentially impacting understanding of unconventional superconductivity.

Contribution

It demonstrates that quantum lattice fluctuations cause significant Fermi energy uncertainties in MgB$_2$, revealing a new nonadiabatic regime relevant to unconventional superconductors.

Findings

Fermi energy in MgB$_2$ is about 0.5 eV, affected by low hole doping.

Zero-point lattice fluctuations induce Fermi energy uncertainties comparable to the Fermi energy.

The adiabatic approximation breaks down due to strong lattice-electron coupling.

Abstract

Small Fermi energy effects are induced in MgB$_2$ by the low hole doping in the $\sigma$ bands which are characterized by a Fermi energy $E_{\rm F}^\sigma \sim 0.5$ eV. We show that, due to the particularly strong deformation potential relative to the $E_{2g}$ phonon mode, lattice fluctuations are reflected in strong fluctuations in the electronic band structure. Quantum fluctuations associated to the zero-point lattice motion are responsible for an uncertainty of the Fermi energy of the order of the Fermi energy itself, leading to the breakdown of the adiabatic principle underlying the Born-Oppenheimer approximation in MgB$_2$ even if $\omega_{\rm ph}/E_{\rm F} \sim 0.1-0.2$, where $\omega_{\rm ph}$ are the characteristic phonon frequencies. This amounts to a new nonadiabatic regime, which could be relevant to other unconventional superconductors.