Retardation effects and the Coulomb pseudopotential in the theory of superconductivity

In the theory of electron-phonon superconductivity both the magnitude of the electron-phonon coupling $\lambda$ and the Coulomb pseudopotential ${\mu }^{*}$ are important to determine the transition temperature $T_c$ and other properties. We calculate corrections to the conventional result for the Coulomb pseudopotential. Our calculations are based on the Hubbard-Holstein model, where electron-electron and electron-phonon interactions are local. We develop a perturbation expansion, which accounts for the important renormalization effects for the electrons, the phonons, and the electron-phonon vertex. We show that retardation effects are still operative for higher order corrections, but less efficient due to a reduction of the effective bandwidth. This can lead to larger values of the pseudopotential and reduced values of $T_c$. The conclusions from the perturbative calculations are corroborated up to intermediate couplings by comparison with nonperturbative dynamical mean-field results.

Figure 1

Contributions to the irreducible vertex: (a) electron-phonon part; (b) first order in $U$.

Figure 2

Diagrams for (a) electronic self-energy and (b) phonon self-energy.

Figure 3

Contributions to the electron-phonon vertex ${\Gamma }_U^{\left(\mathrm{ep}\right)}$.

Figure 4

(a) Higher order terms involving vertex corrections of the electron-phonon vertex. (b) Electronic self-energy including vertex corrections for the electron-phonon vertex.

Figure 5

Vertex corrections to the electron-phonon vertex to order $g{U}^2$.

Figure 6

${(g^r/g)}^2$ as a function of ${\mu }_c$. Solid line, according to Eq. (39); dashed line, according to Eq. (41).

Figure 7

(a) Higher order diagram vertex contribution from the Coulomb repulsion. (b) Second-order-in-$U$ diagram for electronic self-energy.

Figure 8

Diagrams for the off-diagonal self-energy (a) from electron-phonon interaction and (b) for the Coulomb repulsion with the notation $F=G_{21}$.

Figure 9

Pseudopotential ${\mu }_c^{*}$ as a function of ${\mu }_c$ for $D/{\omega }_{\mathrm{ph}}=100$ and $\beta {\omega }_{\mathrm{ph}}=240$. Calculated results using both the first-order and the first- plus second-order result for ${\Gamma }^{\left(\mathrm{pp}\right)}$ as well as the approximation in Eq. (72).

Figure 10

${\mu }_c^{*}$ as a function of ${\mu }_c$ for $D/{\omega }_{\mathrm{ph}}=100$ and $\beta {\omega }_{\mathrm{ph}}=240$. The calculated value using just the second-order result for ${\Gamma }^{\left(\mathrm{pp}\right)}$ as well as the analytical approximation in Eq. (73).

Figure 11

${\mu }_c^{*}$ as a function of $\log (D/{\omega }_{\mathrm{ph}})$ for $\beta {\omega }_{\mathrm{ph}}=240$ according to a first-order (${\mu }_{c1}=0.5$) and a second-order (${\mu }_{c2}=0.2756$) calculation. ${\mu }_{c1}$ and ${\mu }_{c2}$ were chosen so that the same ${\mu }_c^{*}$ was obtained in the two calculations for $D/{\omega }_{\mathrm{ph}}=10$.

Figure 12

Spectral gap ${\Delta }_{\mathrm{sp}}$ as calculated from the numerical solution of Eq. (74) as a function of $\lambda$ for $D/{\omega }_{\mathrm{ph}}=80$ in comparison with the analytical form in Eq. (84), with the parameters $c_1=1.7$ and $c_1=1.07$.

Figure 13

Spectral gap ${\Delta }_{\mathrm{sp}}$ as calculated from the numerical solution of Eq. (74) with different kernels as a function of ${\mu }_c$ for $D/{\omega }_{\mathrm{ph}}=80$ in comparison with the analytical result in Eq. (84).

Figure 14

${\mu }_c^{*}$ as a function of ${\mu }_c$ for $D/{\omega }_{\mathrm{ph}}=80$ computed from Eq. (95). We show in comparison the result for the first-order and the second-order calculations. Fit parameters $c_1$, $c_2$, and $c_3$ are the same as above.

Figure 15

$T_c$ as a function of $\lambda$ for $D/{\omega }_{\mathrm{ph}}=80$.

Figure 16

$T_c$ as a function of ${\mu }_c$ for $D/{\omega }_{\mathrm{ph}}=80$.

Figure 17

Behavior of the spectral gap ${\Delta }_{\mathrm{sp}}$ and $z{\Sigma }^{\mathrm{off}}\left(0\right)$ for constant ${\lambda }_0=0.308$ and ${\lambda }_0=0.382$ (${\omega }_0=0.1$ in both cases) as a function of ${\mu }_c$.

Figure 18

Effective parameters: (a) quasiparticle renormalization $z$, (b) renormalized phonon frequency ${\omega }_0^r/{\omega }_0$, (c) renormalized coupling constant ${(g^r/g)}^2$ (the solid line corresponds to the result with $a$ and $b$ obtained from the free Green's functions), and (d) effective $\lambda$ as a function of ${\mu }_c$.

Figure 19

Behavior of the DMFT result for the spectral gap ${\Delta }_{\mathrm{sp}}$ and $z{\Sigma }^{\mathrm{off}}\left(0\right)$ for constant $\lambda \simeq 1$ according to condition (a) for the vertex corrections (see text), ${\omega }_0^r\simeq 0.05$ as a function of ${\mu }_c$ in comparison with PT-a and PT-b (see text), and the analytic formula in Eq. (84).