Ab initio theory of magnetic-field-induced odd-frequency two-band superconductivity in ${\mathrm{MgB}}_2$

We develop the anisotropic Eliashberg framework for superconductivity in the presence of an applied magnetic field. Using as input the ab initio calculated electron and phonon band structures and electron-phonon coupling, we solve self-consistently the anisotropic Eliashberg equations for the archetypal superconductor ${\mathrm{MgB}}_2$. We find two self-consistent solutions, time-even two-band superconductivity, as well as unconventional time-odd $s$-wave spin triplet two-band superconductivity emerging with applied field. We provide the full momentum, frequency, and spin-resolved dependence and magnetic field-temperature phase diagrams of the time-even and time-odd superconducting pair amplitudes and predict fingerprints of this novel odd-frequency state in tunneling experiments.

Figure 1

Ab initio computed temperature and magnetic field dependence of superconductivity in ${\mathrm{MgB}}_2$. (a) $H-T$ phase diagram of the even-frequency superconducting component. Dashed (solid) lines denote second (first) order transitions. (b) $H-T$ phase diagram for the odd-frequency superconductivity. The insets show the Matsubara frequency dependence of each component for several magnetic field values. The two-gap structure, characteristic of superconductivity in ${\mathrm{MgB}}_2$, can also be discerned. The colorbar value “max” (“min”) corresponds to 7 meV (0 meV) and 0.3 meV (0 meV) for even- and odd-frequency superconductivity, respectively. (c) Band-resolved magnetic field dependence of the even and odd pairing amplitudes at low temperature. The lines correspond to the maximum values in Matsubara space of the momentum averaged superconducting pairing fields on each band, i.e., the peaks of the insets in (a) and (b). Note that the OST pairing amplitude is an order of magnitude smaller than the respective ESS one.

Figure 2

Computed momentum and frequency dependence of superconductivity in ${\mathrm{MgB}}_2$ at $T=4.2$ K and $H=40$ T. (a) The Fermi surface of ${\mathrm{MgB}}_2$ in the conventional Brillouin zone colored by the values of the even-frequency superconducting gap edge ${\mathrm{\Delta }}_{\downarrow }^e\left(\mathbf{k}\right)$ for spin-$\downarrow$ quasiparticles. The distribution of the gap edge values is shown in the inset below. (b) The difference between spin-$\downarrow$ and spin-$\uparrow$ even-frequency superconducting gap edges. (c) Typical frequency dependence of the real $[{\mathrm{\Delta }}_e^{\prime }({\mathbf{k}}_0,\omega )\equiv {\mathrm{\Delta }}_e^{\prime }\left(\omega \right)]$ and imaginary $[{\mathrm{\Delta }}_e^{\prime \prime }({\mathbf{k}}_0,\omega )\equiv {\mathrm{\Delta }}_e^{\prime \prime }\left(\omega \right)]$ part of the ESS pairing amplitude at a given point ${\mathbf{k}}_0$ on the Fermi surface. (d) and (e) Same as in (a) and (b), but for the odd-frequency superconducting gap edge. (f) Same as in (c), but for the real $\left[{\mathrm{\Delta }}_o^{\prime }\left(\omega \right)\right]$ and imaginary $\left[{\mathrm{\Delta }}_o^{\prime \prime }\left(\omega \right)\right]$ parts of the OST pairing amplitude. Near $\omega =0, {\mathrm{\Delta }}_o^{\prime }\left(\omega \right)$ increases linearly with frequency, whereas ${\mathrm{\Delta }}_o^{\prime \prime }\left(\omega \right)$ is almost constant. The latter is finite at $\omega =0$ as shown in the inset.

Figure 3

Predicted dependence of single-particle tunneling spectra on the magnetic field. (a) Low-temperature superconducting density of states of ${\mathrm{MgB}}_2$ for several values of the external magnetic field. (b) Comparison between tunneling spectra when the odd-frequency pairing is included in the Eliashberg calculation and when it is not (black lines). As the field increases, the odd-frequency gap induces a shift in the tunneling peaks of the order of 0.1 meV. This shift is pronounced for the superconducting gap over the $\sigma$ bands. (c) In the absence of external bias, we predict a nonzero signal in the tunneling spectra that is proportional to the imaginary part of the OST gap. This has a distinct magnetic field dependence, the detection of which will serve as a definite proof for the existence of odd-frequency superconductivity in ${\mathrm{MgB}}_2$.

Figure 4

Ab initio calculated electron-phonon properties of ${\mathrm{MgB}}_2$. (a) Computed phonon density of states. (b) Momentum-integrated Eliashberg function. (c) Phonon dispersion relations along high-symmetry lines. The radius of the symbols is proportional to the strength of the electron-phonon coupling for each phonon mode, ${\lambda }_{\mathbf{q}\nu }$.

Figure 5

Calculated momentum and temperature dependence of superconductivity in ${\mathrm{MgB}}_2$ at zero field. (a) The Fermi surface of ${\mathrm{MgB}}_2$ in the conventional Brillouin zone colored by the values of the even-frequency superconducting gap edge $[\text{Re}{\mathrm{\Delta }}_{\mathbf{k}}\left(\omega \right)=\omega ]$ at $T=4.2$ K and $H=0$ T. The distribution of the gap edge values is shown in the inset below. (b) Temperature dependence of the even-frequency superconducting gap edges, momentum-averaged over the $\sigma$ and $\pi$ Fermi surface sheets.

Figure 6

Calculated frequency dependence of superconductivity in ${\mathrm{MgB}}_2$ at $T=4.2$ K and zero field. Real (black) and imaginary (red) parts of the even-frequency superconducting pair amplitude $\mathrm{\Delta }(\mathbf{k},\omega )$ at several Fermi surface points.