Extended versus standard Holstein model: Results in two and three dimensions

We present numerically exact solutions to the problem of a single electron interacting through an extended interaction with optical phonons in two and three dimensions. Comparisons are made with results for the standard Holstein model and with perturbative approaches from both the weak coupling and strong coupling sides. We find, in agreement with earlier work, that the polaron effective mass increases (decreases) in the weak (strong) coupling regime, respectively. However, in two dimensions, the decrease in effective mass still results in too large an effective mass to be relevant in realistic models of normal metals. In three dimensions the decrease can be more relevant but exists only over a very limited range of coupling strengths.

Figure 1

Effective mass $m^{*}/m$ vs coupling strength ${\lambda }_{\mathrm{tot}}$, using perturbation theory for both the extended and standard Holstein models; these are compared with results from exact diagonalization. We have used ${\omega }_E=0.2t$. Perturbation theory is less accurate for the extended model compared with the standard model. Inset: Ground state energy vs ${\lambda }_{\mathrm{tot}}$. Note that the results for the energy are fairly accurate, in comparison with those for the effective mass.

Figure 2

(a) $m^{*}/m$ decreases as the accuracy of the calculation decreases. The curves with five and three phonons include states with phonons far from the electron, but never more than five or three phonons total, respectively. Second order perturbation theory includes excited states with at most a single phonon; ${\omega }_E=0.2t$. (b) For the 2D extended Holstein model, $m^{*}/m$ again decreases as the accuracy of the calculation decreases similar to the standard model. Here the effect is even more pronounced: Many phonons are required even for moderate coupling strengths. Again, second order perturbation theory includes at most a single phonon; ${\omega }_E=0.2t$.

Figure 3

Probability $\left(|d_n{|}^2\right)$ of various basis states from the Lang-Firsov approximation and the standard model in 2D, with $\lambda =0.62$ and ${\omega }_E=0.2t$. The curve in black, labeled as “Lang-Firsov” in the legend is the result obtained from the Lang-Firsov approximation given in Eq. (6). The other curves represent amplitudes obtained from the exact solution. The points denoted by triangle shaped symbols, connected with a blue curve represent amplitudes with $n$ phonons on the same site as the electron, as denoted in the legend. These amplitudes are those of the true, numerically exact wave function and are shifted compared to the Lang-Firsov approximation, as is evident from the figure. Also shown are amplitudes for wave function components representing an electron at some site, with $n$ phonons on the neighboring site; these are indicated by circle shaped symbols connected with a green curve. Finally, the amplitudes for wave function components representing an electron with a single phonon on a neighboring site and $n$ phonons on the same site as the electron are indicated by diamond shaped symbols and connected with a red curve.

Figure 4

$m^{*}/m$ from second order perturbation theory vs ${\omega }_E$. For definiteness, the value of ${\lambda }_{\mathrm{tot}}$ is $0.1$. In the large ${\omega }_E$ limit all the curves approach $1.0$ but the extended model always has a larger $m^{*}/m$ than the standard model in both 2D and 3D.

Figure 5

(a) For the two dimensional model, $m^{*}/m$ as a function of ${\lambda }_{\mathrm{tot}}$ up to strong coupling with ${\omega }_E=0.2t$. Note that beyond ${\lambda }_{\mathrm{tot}}=0.6$ the effective mass of both models is practically infinite, but at intermediate coupling strengths the extended model shows a significantly higher effective mass. Inset: Ground state energy vs ${\lambda }_{\mathrm{tot}}$. (b) For the two dimensional model, $m^{*}/m$ as a function of ${\lambda }_{\mathrm{tot}}$ up to strong coupling with ${\omega }_E=t$. Note that the effective masses are all much smaller and at intermediate coupling strengths the extended model shows an effective mass only slightly higher than the standard model. The coupling strength is also much larger before the effective masses rise to unphysical values. Inset: Ground state energy vs ${\lambda }_{\mathrm{tot}}$.

Figure 6

For the three dimensional model, $m^{*}/m$ as a function of ${\lambda }_{\mathrm{tot}}$, with ${\omega }_E=0.3t$, to compare exact and perturbative results. Similar to the 2D results the region of validity for perturbation theory is much smaller in the extended model though it is not particularly large for the standard model either. Inset: Ground state energy vs ${\lambda }_{\mathrm{tot}}$. Again, perturbation theory works better for the ground state energy than for the effective mass.

Figure 7

For the three dimensional case, $m^{*}/m$ vs ${\lambda }_{\mathrm{tot}}$, with ${\omega }_E=0.3t$. The effective mass increases as more and more states with more than one phonon excitation are included. The curves with four and five phonons include states with phonons far from the electron, but there are never more than a total of four or five phonons, respectively. The calculation with second order perturbation theory includes basis states with at most a single phonon.

Figure 8

For the three dimensional case, $m^{*}/m$ vs ${\lambda }_{\mathrm{tot}}$ with various values of $f\left(1\right)$. These results are from numerically exact calculations. The extended model can shift the onset of unphysically high effective masses to stronger coupling, though the extra range of tenable coupling strengths is rather small.