4.3 Results

Figure 4.4: H (circles) and B (diamonds) polaron energy (a) and quasiparticle weight (b) versus $\lambda$ at $k=0$ and $\omega _0=0.2 t$. The open symbols are the results obtained after only one iteration (first order in $D$). The dashed line in (b) corresponds to $\lambda ^{ME}=\frac {1}{Z_0}-1$ (see Eq. 4.16).

In our calculations we take $t=1$ and $2M=1$. For the one-dimensional case the coordination number $z=2$.

In Fig. 4.4 -a and -b we show the polaron energy $E_0$ and respectively the quasiparticle weight $Z_0$ versus the dimensionless coupling constant $\lambda$ at $k=0$. For the H model, the decrease of $Z_0$ with increasing $\lambda$ is not very different from the one given by the ME theory (dashed line in Fig. 4.4-b given by Eq. 4.16), showing that the ME definition of $\lambda$ is similar to ours. However, for the B model the quasiparticle weight $Z_0$ and the polaron energy $E_0$ decrease much slower with increasing $\lambda$. As mentioned in the previous section this is due to the small momentum scattering processes implied in the renormalization of $Z_0$ (see Eq. 4.17) and, similarly, in the determination of the self-energy. It is worth pointing out that, unlike the H polaron, even for values of $\lambda \approx 1$, the B polaron remains in the weak coupling regime and hence the difference between the fully convergent SCBA (full symbols) and the first order perturbation theory (i.e. only the first SCBA iteration, empty symbols) is small. Another interesting feature is that for the same value of $Z_0$ the B polaron energy is lower than the H one, showing that the ratio between the energy renormalization and the quasiparticle weight renormalization is different for the two models.

Figure 4.5: The quasiparticle weight $Z_0$ (a) and the polaron energy $E_0$ (b) at zero momentum versus phonon frequency $\omega _0$ for the H and B models at $\lambda _H =0.2$ and respectively $\lambda _B =1.3$.

Figure 4.6: Quasiparticle weight $Z(k)$ (a) and the polaron energy $E(k)$ (b) for the H and B model for $\omega_0=0.2~t$ at coupling $\lambda _H =0.2$ and $\lambda _B =1.3$ respectively. The dotted horizontal lines in (b) marks the first phonon threshold energy $E_0+\omega _0$.

Another important difference between the two models is the dependence of the polaron properties at the band bottom on the phonon frequency $\omega _0$. While for the H case an increase of $\omega _0$ results in an increase of the quasiparticle weight $Z_0$ the opposite behavior is seen for the B model. This is illustrated in Fig. 4.5-a. The reason for the reduction of $Z_0$ with increasing $\omega _0$ in the B model can be easily understood by noticing (see Eq. 4.17) that a larger value of $\omega _0$ reduces the importance of the $q$ dependence in the polaron properties calculation. As discussed earlier, the strong momentum dependent coupling is responsible for the weak $Z_0$ renormalization of the B polaron and thus an increase of $\omega _0$ would result in a larger effective coupling and implicitly in a smaller $Z_0$.

Figure 4.7: Spectral representation for: a) H model, $\lambda _H =0.2$, tight binding dispersion, b) B model, $\lambda _B =1.3$, tight binding dispersion, c) B model, $\lambda _B=0.5$, tight binding dispersion, d) B model, $\lambda ^{lin}_B=1.6$, linear dispersion with $v_F=t$ (Eq. 4.25), e) H model, $\lambda ^{lin}_H=0.9$, linear dispersion with $v_F=t$, f) H and B model, $\lambda _H=\lambda _B=0.2$, with dispersion $\epsilon(k)=-2t \cos(2k)$. Everywhere $\omega _0=0.2 t$. The arrows indicate the first (lower) and the second (upper) phonon threshold energies corresponding to $E_0+\omega _0$ and respectively $E_0+2\omega _0$.

Figure 4.8: Spectral function (energy distribution curves) for a) H model with $\lambda _H =0.2$ and b) B model with $\lambda _B=0.5$. Free electron tight binding dispersion and $\omega _0=0.2 t$ is considered.

Figure 4.9: Spectral function (energy distribution curves) for a) H model with $\lambda _H^{lin}=0.9$ and b) B model with $\lambda _B^{lin}=1.6$. Free electron linear dispersion ($v_F=t$) and $\omega _0=0.2 t$ is considered.

Aside from the different $\lambda$ and $\omega _0$ dependency of the two models at zero momentum, the momentum dependent properties also exhibit different behaviors. This is shown in Figs. 4.6, 4.7, 4.8 and  4.9 where the $k$ dependent properties for the two models are illustrated.

In Fig. 4.6 and Fig. 4.7-a and -b we have chosen the value of $\lambda$ such that both models yield the same quasiparticle weight at the bottom of the band. Thus, the choice of $\lambda _H =0.2$ and $\lambda _B =1.3$ results in $Z_{0h}=Z_{0b}=0.83$, implying that both models are in the weak coupling regime. As mentioned earlier, the B polaron energy at $k=0$ is lower. At large $k$, just below the first phonon threshold energy $E_0+\omega _0$, both polarons display a flat dispersion and a reduced quasiparticle weight. However, the B polaron quasiparticle weight at large $k$ is substantially larger (for instance $Z_B(k=\pi) \approx 0.014$ within a numerical precision of $10^{-3}$ and $Z_H(k=\pi)<10^{-3}$ ), making the B polaron state at large $k$ distinguishable in the spectral plot (Fig. 4.7 -b) in contrast to the H one (Fig. 4.7-a). At energies larger than $E_0+\omega _0$ the spectral intensity of the B quasiparticle is much smaller than the H model one, unlike the situation at the band bottom where both models have the same $Z_0$. This large momentum behavior points to a stronger effective coupling for the B model at large $k$.

While the dispersion of both models displays a gap at $E_0+\omega _0$, the B polaron shows a second gap at the second phonons threshold energy $E_0+2\omega _0$. This can be seen in Fig. 4.7-b but also occurs for smaller values of the dimensionless coupling as shown in Fig. 4.7-c for the value of $\lambda _B=0.5$. This value was chosen such that the ground state energy $E_0$ of the B model is equal to that of the Holstein polaron one shown in Fig. 4.7-a. The situation can be even clearer visualized by comparing Fig. 4.8-a with Fig. 4.8-b, where the energy distribution curves (EDC) for H and respectively B cases are shown.

An even more interesting effect is noticed if a linear dispersion for the free electron is considered

$$\epsilon(k)= v_F \vert k\vert~,$$ (4.24)

with a value of $v_F$ close to one or larger. In this case one can take $v_F$ to be a measure of the free electron kinetic energy ^4.2 and thus define the dimensionless coupling as

$$\lambda^{lin}_B = \frac{{E_p}_B}{v_F}=\frac{1}{2M\omega_0}~ \frac{2 g^2}{ v_F \omega_0 }~.$$ (4.25)

The resulting B polaron dispersion is shown in Fig. 4.7-d (see also the corresponding EDC plot in Fig. 4.9-b). While it displays a gap at $E_0+2\omega _0$, no distinguishable gap or kink can be seen at the first phonon threshold energy $E_0+\omega _0$. This free-electron like behavior of the polaron at $E_0+\omega _0$ is due to the fact that the physics there is determined by very small $q$ scatterings, characterized by small coupling strength $\gamma(q)$, originating from the rapid increase of the electron energy with $k$. At larger energy, close to $E_0+2\omega _0$, the relevant phonon momenta $q$ implied in the scattering are larger and the physics is consequently determined by a larger effective coupling. As a result a noticeable kink appears at this energy in the spectrum. This effect (i.e. kink at $E_0+2\omega _0$ but no noticeable one at $E_0+\omega _0$) is a result of an electron-phonon coupling which is an increasing function of the polaron momentum and it is hence not seen in the Holstein model even for the case of a linear electronic dispersion (see Fig. 4.7-e and the corresponding EDC plot in Fig. 4.9-a).

The differences between the two models discussed above are a consequence of two effects: i) the strong $q$ dependence of the bare electron-phonon coupling $\gamma(q)$ in the B model and ii) the polaron properties at small $k$ are most strongly influenced by the small momentum $q$ phonons. However the second statement is not true if the free electron dispersion has low energy states separated by large $q$ as we will discuss in the next section (Sec.4.4). In order to show this we choose a free electron dispersion

$$\epsilon(k)=-2t\cos(2k)~,$$ (4.26)

which is double degenerate with the lowest energy values at $k=0$ and $k=\pi$ respectively. We find that for this electronic dispersion the differences between the H and the B model are very small, less than $0.1\%$, and therfore not discernible in the spectral representation plot shown in Fig. 4.7-f.

The general features of the polaron spectral function illustrated in Figs. 4.8 and 4.9 show a remarkable resemblance with the photoemission data in materials with significant electron-phonon interactions (2,19). For momenta which correspond to energies below the phonon frequency one can see a sharp peak followed by a broad satellite. At large momenta, the broad satellite found at energies well above the phonon frequency sharpens and follows the free electron dispersion while the intensity of the sharp peak below $\omega _0$ vanishes. We want to point that this picture, with the spectral function determined by two main branches, is quite different form the one incorrectly presented in some popular textbooks (38) where the spectrum is characterized by a single quasiparticle peak which changes its dispersion slope when crossing the phonon energy threshold.