Self-consistently renormalized quasiparticles under the electron-phonon interaction

Combining ab initio techniques and the analytic properties of the electron Green’s function, we outline a method for calculating quasiparticle properties under the electron-phonon interaction. The presented scheme is a generalization of the work by Engelsberg and Schrieffer [Phys. Rev. 131, 993 (1963)] to finite temperatures and is suitable for being applied to complex materials, where the electronic and vibrational properties are calculated from first principles. We show that under some circumstances, the low-energy dynamical properties are well described by quasiparticles, but at the same time the renormalization effects on quasiparticle lifetimes and energies can be very important. The bare second-order perturbative (such as Fermi’s golden rule) results for the self-energy are compared with self-consistent ones. The theory is first illustrated with the simple Einstein and Debye models at finite temperatures. Thereafter we consider realistic materials such as the $1\times 1$ hydrogen-covered (deuterium-covered) W(110) surface and the superconductor ${\text{MgB}}_2$.

Figure 1

(Color online) Schematic representations of the analytically continued Green’s function $\tilde{G}\left(z\right)$ and the corresponding spectral function. We consider the two examples where ${\mathbb{Z}}^{qp}$ is purely real (left panels), i.e., ${\theta }^{qp}=0$, and where ${\theta }^{qp}=\pi /6$ (right panels). The quasiparticle pole is located at $z^{qp}=2-i$ in both cases. The two lower panels show a contour plot of the spectral function $A_{i,\mathbf{k}}^L\left(z\right)\sim \text{Im}\tilde{G}\left(z\right)$, with the dashed lines indicating the position of the real axis. The upper panels exhibit the corresponding spectral functions evaluated at the real axis.

Figure 2

(Color online) Left: QP band structure for the Einstein model for the coupling parameter $\lambda ={\left|g\right|}^2/{\omega }_0=1$ at temperatures $T\simeq 0$ (top panel), $T={\omega }_0/10$ (middle panel), and $T={\omega }_0/10$ (bottom panel). The solid thick (black) lines represent the real part of the poles, $E_R$, obtained from the complex Dyson equation [Eq. (6)]. The solid thin (gray) line is the solution of the approximate Dyson equation considering only real quantities [Eq. (8)]. Right: the quasiparticle renormalization factor ${\mathbb{Z}}^{qp}$ as a function of $E_R$. The solid thick (red) line shows the real part of ${\mathbb{Z}}^{qp}$, while the thin (green) line is the imaginary part $\text{Im}\left({\mathbb{Z}}^{qp}\right)$.

Figure 3

(Color online) Imaginary part of the quasiparticle poles, $E_I$, as a function of the real part $E_R$ for different temperatures, i.e., $T=0$ (top panel), $T={\omega }_0/10$ (middle panel), and $T={\omega }_0$ (bottom panel). The thick solid (black) lines are the self-consistent solutions obtained by means of Eq. (6), while the (red) dashed line is the imaginary part of the self-energy, $E_I^{\left(1\right)}={\Sigma }_I\left(E_R+i{0}^0\right)$ [see Eq. (9)].

Figure 4

(Color online) Representations of the analytically continued spectral function $A_{\mathbf{k}}\left(z\right)=-\text{Im}\left[{\tilde{G}}_{\mathbf{k}}\left(z\right)\right]/\pi$ for ${ϵ}_{\mathbf{k}}=2{\omega }_0$ and $T=0$. The bottom panel shows $A_{\mathbf{k}}\left(z\right)$ in the complex plane, and the top panel demonstrates $A_{\mathbf{k}}\left(E_R\right)$ in the real axis.

Figure 5

(Color online) Representations of the analytically continued spectral function $A_{\mathbf{k}}\left(z\right)=-1/\pi \text{Im}\left[{\tilde{G}}_{\mathbf{k}}\left(z\right)\right]$ for ${ϵ}_{\mathbf{k}}=2{\omega }_0$ and $T={\omega }_0/10$. The bottom panel shows $A_{\mathbf{k}}\left(z\right)$ in complex plane, and top panel demonstrates $A_{\mathbf{k}}\left(E_R\right)$ in the real axis.

Figure 6

(Color online) Left: QP band structures for the Debye model $\left[{\alpha }^{2}F\left(\omega \right)=\lambda \frac{{\omega }^2}{{\omega }_D^2}\theta \left({\omega }_D-\omega \right)\right]$ for the coupling parameter $\lambda =1$ at temperatures $T\simeq 0$ (top panel), $T={\omega }_D/10$ (middle panel), and $T={\omega }_D/2$ (bottom panel). The solid thick (black) lines represent the real part of the poles, $E_R$, obtained from the complex Dyson equation [Eq. (6)]. The solid thin (gray) line is the solution of the approximate Dyson equation considering only real quantities [Eq. (8)]. Right: the quasiparticle renormalization factor ${\mathbb{Z}}^{qp}$ as a function of $E_R$. The solid thick line (red) shows the real part of ${\mathbb{Z}}^{qp}$, while the thin (green) line is the imaginary part $\text{Im}\left({\mathbb{Z}}^{qp}\right)$.

Figure 7

(Color online) Imaginary part of the quasiparticle poles, $E_I$, as a function of the real part $E_R$ for different temperatures, i.e., $T=0$ (top panel), $T={\omega }_D/10$ (middle panel), and $T={\omega }_D/2$ (bottom panel). The thick solid lines (black) are the self-consistent solutions obtained by means of Eq. (6), while the dashed (red) line is the imaginary part of the self-energy $E_I^{\left(1\right)}={\Sigma }_I\left(E_R+i{0}^0\right)$ [see Eq. (9)].

Figure 8

(Color online) Vibrational structures of the hydrogen- (left panel) and deuterium-covered (right panel) $1\times 1$ W(110) surfaces. The lower-phonon-energy branches $\left(\lesssim 25\text{meV}\right)$ correspond to the W(110) substrate. The three vibrational branches higher in energy are hydrogen-derived (deuterium–derived) phonon modes.

Figure 9

(Color online) The Eliashberg function ${\alpha }^{2}F\left(\omega \right)$ for the ${\text{S}}_1$ surface state along the high-symmetry direction $\overline{\Gamma }\overline{S}$. The solid (black) line corresponds to the hydrogen coverage, while the dotted (red) line stands for the deuterium coverage.

Figure 10

(Color online) Calculated quasiparticle bands and inverse lifetimes for the hydrogen- (top panels) and deuterium-covered (bottom panels) W(110) surfaces. Left: fully renormalized electron bands (solid lines), calculated from the complex Dyson equation [Eq. (6)] at $T_1=3.5\text{meV}\simeq 40\text{K}$ (gray line) and $T_2=12.93\text{meV}\simeq 150\text{K}$ (black line). The thin dashed (orange) line is the solution of the approximate Dyson equation at $T_1$ considering only real quantities [Eq. (8)]. Right: imaginary part of the QP energies $\left|E_I\right|=1/\tau$ (black line) versus the real energy of the QP poles, $E_R$. The imaginary part of the bare electron self-energy (dashed line) is shown for comparison.

Figure 11

(Color online) The calculated renormalization factor ${\mathbb{Z}}^{qp}$ as a function of the real part of the quasiparticle poles. The top (bottom) panel corresponds to the hydrogen (deuterium) coverage. Real and imaginary parts are indicated by thick (red) and thin (green) lines, respectively.

Figure 12

Spectral functions $A_{\mathbf{k}}\left(\omega \right)$ at $T=40\text{K}$ (thick black line) and $T=150\text{K}$ (thin gray line) of $1\times 1$ H/W(110) (top panels) and $1\times 1$ D/W(110) (bottom panels) for different momenta compared to their counterparts in the multiple-quasiparticle approximation $A_{\mathbf{k}}^{\text{QP}}\left(\omega \right)$ as given in Eq. (22).

Figure 13

(Color online) Left: Fermi surface and irreducible part of the Brillouin zone for ${\text{MgB}}_2$. Right: the Eliashberg function corresponding to the high-symmetry direction $AL$ for the ${\sigma }_2$ band (outermost Fermi sheet).

Figure 14

(Color online) Calculated quasiparticle bands and inverse lifetimes of ${\text{MgB}}_2$ in direction $AL$ (${\sigma }_2$ band). Left: fully renormalized electron bands, calculated from the complex Dyson equation [Eq. (6)] at $T=8\text{meV}\sim 92\text{K}\sim {\omega }_m/10$. The black envelope indicates the real part of the renormalization factor and the thin (red) line inside stands for the calculated quasiparticle pole. The thin (gray) line is the solution of the approximate Dyson equation considering only real quantities [Eq. (8)] at $T=0$. Right: the fully renormalized imaginary part of the quasiparticle energies is indicated by the solid (black) line, while the dashed (red) line stands for the bare imaginary part.