Small Fermi energy, zero-point fluctuations, and nonadiabaticity in $\mathrm{Mg}{\mathrm{B}}_2$

Small Fermi energy effects are induced in $\mathrm{Mg}{\mathrm{B}}_2$ by the low hole doping in the $\sigma$ bands which are characterized by a Fermi energy $E_F^{\sigma }\sim 0.5\mathrm{eV}$. 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 $\mathrm{Mg}{\mathrm{B}}_2$ even if ${\omega }_{\mathrm{ph}}∕E_F\sim 0.1-0.2$, where ${\omega }_{\mathrm{ph}}$ are the characteristic phonon frequencies. This amounts to a new nonadiabatic regime, which could be relevant to other unconventional superconductors.

Figure 1

(a) Frozen phonon potential $V\left(u\right)$ for the $E_{2g}$ mode in $\mathrm{Mg}{\mathrm{B}}_2$ (solid line, left side scale) and probability distribution function $P\left(u\right)={\mid {\psi }_{E_{2g}}\left(u\right)\mid }^2$ (dashed line, right side scale). $u$ is in units of $\mathrm{\AA }$, $V\left(u\right)$ in units of eV, and $P\left(u\right)$ in units of ${\mathrm{\AA }}^{-1}$. (b) Electronic structure of undistorted $\mathrm{Mg}{\mathrm{B}}_2$ (solid lines). The grey regions represent the quantum fluctuations induced by zero point lattice motion $(\mid u\mid ⩽{⟨u^2⟩}^{1∕2})$. Bold lines represent the two $\sigma$ bands, which are also the most affected by quantum fluctuations.

Figure 2

Schematic phase diagram for the different kinds of nonadiabatic effects. See text for details.