Spectral properties and enhanced superconductivity in renormalized Migdal-Eliashberg theory

Migdal-Eliashberg theory describes the properties of the normal and superconducting states of electron-phonon-mediated superconductors based on a perturbative treatment of the electron-phonon interactions. It is necessary to include both electron and phonon self-energies self-consistently in Migdal-Eliashberg theory in order to match numerically exact results from determinantal quantum Monte Carlo in the adiabatic limit. In this work we provide a method to obtain the real-axis solutions of the Migdal-Eliashberg equations with electron and phonon self-energies calculated self-consistently. Our method avoids the typical challenge of computing cumbersome singular integrals on the real axis and is numerically stable and exhibits fast convergence. Analyzing the resulting real-frequency spectra and self-energies of the two-dimensional Holstein model, we find that self-consistently including the lowest-order correction to the phonon self-energy significantly affects the solution of the Migdal-Eliashberg equations. The calculation captures the broadness of the spectral function, renormalization of the phonon dispersion, enhanced effective electron-phonon coupling strength, minimal increase in the electron effective mass, and the enhancement of superconductivity which manifests as a superconducting ground state despite strong competition with charge-density-wave order. We discuss surprising differences in two common definitions of the electron-phonon coupling strength derived from the electron mass and the density of states, quantities which are accessible through experiments such as angle-resolved photoemission spectroscopy and electron tunneling. An approximate upper bound on $2\mathrm{\Delta }/T_c$ for conventional superconductors mediated by retarded electron-phonon interactions is proposed.

Figure 1

Holstein model in two dimensions on a $120\times 120$ lattice, $t^{\prime }=-0.3t, n=0.8, {\lambda }_0=0.4, \beta t=16, \mathrm{\Omega }/E_F=0.1$. (a) Phonon spectral function for the RME theory. (b) Phonon spectral function at selected momentum points. (c) ${\alpha }^{2}F\left(\omega \right)$ computed as an average over the Fermi surface and corresponding $\lambda$ for UME theory (blue) and RME theory (orange).

Figure 2

Holstein model in two dimensions on a $120\times 120$ lattice, $t^{\prime }=-0.3t, n=0.8, {\lambda }_0=0.4, \beta t=16, \mathrm{\Omega }/E_F=0.1$. (a) Electronic spectral function for ME theory without renormalization with a momentum-space cut along (0, $\pi /2$) to ($\pi , \pi /2$). (b) Spectral function for RME theory for the momentum-space cut along (0, $\pi /2$) to ($\pi , \pi /2$). (c) Real and imaginary parts of the electronic self-energies for UME theory (blue) and RME theory (orange) at the point on the Fermi surface with ${\mathbf{k}}_y=\pi /2$. (d) Spectral functions for the renormalized and unrenormalized cases at the point on the Fermi surface with ${\mathbf{k}}_y=\pi /2$.

Figure 3

Inverse of the superconducting and charge-density wave susceptibilities for the Holstein model in two dimensions on a $128\times 128$ lattice, $t^{\prime }=-0.3t, n=0.8, {\lambda }_0=0.4$, and $\mathrm{\Omega }=0.17t$ (corresponding to $\mathrm{\Omega }/E_F=0.1$). ${\chi }^{\mathrm{CDW}}$ is reported at the wave vector, $\mathbf{q}$, for which the susceptibility assumes its maximum value. (a) Diagrams for the charge-density wave and superconducting susceptibilities in the Migdal approximation. (b) UME theory. (c) RME theory.

Figure 4

Holstein model in two dimensions on a $128\times 128$ lattice, $t^{\prime }=-0.3t, n=0.8, {\lambda }_0=0.4, \beta t=100$, and $\mathrm{\Omega }=0.17t$ (corresponding to $\mathrm{\Omega }/E_F=0.1$). (a) Electronic spectral function for RME theory for momentum-space cut along (0, $\pi /2$) to ($\pi , \pi /2$). (b) Superconducting order parameter (gap size) as a function of temperature for RME theory. Blue line is the gap function obtained from BCS theory.

Figure 5

Fermionic momentum dependence of the el-ph coupling along the Fermi surface calculated for RME theory for the Holstein model in two dimensions on a $120\times 120$ lattice, $t^{\prime }=-0.3t, n=0.8, \beta t=16, {\lambda }_0=0.4$, and $\mathrm{\Omega }=0.17t$ (corresponding to $\mathrm{\Omega }/E_F=0.1$). Inset shows the Fermi surface and the definition of the angle $\theta$.

Figure 6

Holstein model in two dimensions on a $80\times 80$ lattice, $n=1, t^{\prime }=0, {\alpha }^2/{\mathrm{\Omega }}^2=1.5t, \beta t=6, \mathrm{\Omega }=0.5t$. (a) Phonon spectral function. (b) Phonon spectral function at selected momentum points. (c) Real and imaginary parts of the electronic self-energies for UME theory (blue) and RME theory (orange) at ${\mathbf{k}}_F=(\pi /2,\pi /2)$. (d) Electronic spectral functions for the renormalized and unrenormalized cases at ${\mathbf{k}}_F=(\pi /2,\pi /2)$.