Momentum average approximation for models with electron-phonon coupling dependent on the phonon momentum

We generalize the momentum average (MA) approximation to study the properties of models with $g_{\mathbf{q}}$ momentum-dependent electron-phonon coupling. As in the case of the application of the original MA to the Holstein model, the results are analytical, numerically trivial to evaluate, exact for both zero bandwidth and for zero electron-phonon coupling, and accurate everywhere in parameter space. Comparison with available numerical data confirms this accuracy. We then show that further improvements can be obtained based on variational considerations using the one-dimensional breathing-mode Hamiltonian as a specific example. For example, by using this variational MA, we obtain ground-state energies within at most 0.3% error of the numerical data.

Figure 1

Diagrammatical expansion of the self-energy $\Sigma \left(\mathbf{k},\omega \right)$.

Figure 2

(Color online) (a) Ground-state energy, (b) percent difference from ED ground-state energy results, (c) ground-state qp weight, and (d) percent difference from ED qp weight results as a function of the effective coupling $\lambda$ for $t=1$ and $\Omega /t=0.5$. The perturbation-theory results (Ref. 17) for both the Holstein and breathing-mode models are shown, and the Holstein result is plotted as a function of ${\lambda }_H=\lambda /2$. The ED results are from Ref. 17.

Figure 3

(Color online) ${\text{MA}}^{\left(0\right)}$ and ${\text{MA}}^{\left(1\right)}$ predictions of the ground-state energy of the HIC model (Ref. 29) as compared to its exact solution (black dots). For this model with infinite-range electron-phonon coupling, the approximations capture the correct asymptotic behavior.

Figure 4

(Color online) The generalized Green’s functions that appear in ${\text{MA}}^{\left(0\right)}$ are sketched in the top picture. They have phonon clouds on two neighboring sites, with the electron on the central site. For the variational MA approximation, denoted as ${\text{MA}}^{\left(v,0\right)}$, we find again such Green’s functions but also those with the electron to the immediate left or right of the phonon clouds. We name them $f_{nm}$ and $f_{nm}^{\mp }$, respectively.

Figure 5

(Color online) (a) Ground-state energy, (b) percent difference from ED ground-state energy results, (c) ground-state qp weight, and (d) percent difference from ED qp weight results as a function of the effective coupling $\lambda$ for $t=1$ and $\Omega /t=0.5$. The ED results are from Ref. 17.

Figure 6

(Color online) (a) Average number of phonons and (b) effective mass. (c) Percent difference from ED effective-mass results as a function of the effective coupling $\lambda$ for $t=1$ and $\Omega /t=0.5$. The ED results are from Ref. 17.

Figure 7

(Color online) (a) Polaron dispersion $E_k$, (b) percent difference from ED ground-state energy results, (c) quasiparticle weight $Z_k$, and (d) percent difference from ED qp weight results. Results are shown for $t=1$, $\Omega /t=0.5$, and $\lambda =1.071502$. The ED results are from Ref. 17.

Figure 8

(Color online) (a) $A\left(k=0,\omega \right)$ vs $\omega$ and (b) $\text{ln}A\left(k=0,\omega \right)$ for $t=1$, $\Omega =0.5$, $\eta =0.004$, and $\lambda =0.75,1.0,1.25,1.5$. The vertical blue dashed lines denote $E_{\text{gs}}$ and $E_{\text{gs}}+\Omega$ using ${\text{MA}}^{\left(v,1\right)}$. The ED results are from Ref. 17.

Figure 9

(Color online) (a) $A\left(k,\omega \right)$ and (b) $\text{ln}A\left(k,\omega \right)$ vs $\omega$ for $k/\pi =0,0.25,0.5,0.75,1$, $t=1$, $\Omega =0.5$, $\lambda =1.25$, and $\eta =0.004$. The ED results are from Ref. 17.

Figure 10

(Color online) Spectral weight $A\left(0,\omega \right)$ vs $\omega$ in one dimension using (a) ${\text{MA}}^{\left(v,0\right)}$ and (b) ${\text{MA}}^{\left(v,1\right)}$. The results are shown for $t=1$, $\Omega =0.5$, $\eta =0.01$, and $\lambda$ varying from 0 to 2. Curves corresponding to $\lambda =0.3$, 0.6, 1.2, and 1.8 are highlighted in red.