Holstein light quantum polarons on the one-dimensional lattice

The polaron formation is investigated in the intermediate regime of the Holstein model by using an exact diagonalization technique for the one-dimensional infinite lattice. The numerical results for the electron and phonon propagators are compared to the nonadiabatic weak- and strong-coupling perturbation theories, as well as with the harmonic adiabatic approximation. A qualitative explanation of the crossover regime between the self-trapped and free-particle-like behaviors, not well understood previously, is proposed. It is shown that a fine balance of nonadiabatic and adiabatic contributions determines the motion of small polarons, making them light. A comprehensive analysis of spatially and temporally resolved low-frequency lattice correlations that characterize the translationally invariant polaron states is derived. Various behaviors of the polaronic deformation field, ranging from classical adiabatic for strong couplings to quantum nonadiabatic for weak couplings, are discussed.

Figure 1

(Color online) The lowest coherent polaron band and the lattice correlations in the weak-coupling regime obtained by the ET method. (a) and (b) are for $t=0.125$ and $g=0.2$, (c) and (d) are for $t=10$, $g=1$. The shaded area in (a) and (c) corresponds to the continuum of states above the phonon threshold that characterizes the exact spectrum of the system. For the ET states that (because of a small numerical error) are located above the phonon threshold, a part $(K\gtrsim \pi ∕3)$ of panel (d) is shaded.

Figure 2

(Color online) The two lowest polaron bands below the phonon threshold are plotted in the upper panel (a) as functions of $g$ for $t=0.125$. The first-order SCPT results are given by the dashed curves, while the solid curves are the ET results. The polaron dispersion is taken with respect to the polaron ground-state energy. In the inset of (a) the electron spectral function $A_K^{\left(i\right)}$ is shown for $g=1.52$, while (b) shows the phonon spectral functions, $F_{+,Q}^{\left(i\right)}$, $F_{-,Q}^{\left(i\right)}$ and $H_Q^{\left(i\right)}$.

Figure 3

(Color online) The eight lowest normal modes of the polaron deformation, obtained by the harmonic adiabatic approximation, as functions of $\lambda ={\epsilon }_p∕t$. The pinning and the breather mode are denoted by ${\omega }_P$ and ${\omega }_B$, respectively.

Figure 4

(Color online) For $t=5$, the ET polaron spectrum below the phonon threshold, consisting of the lowest and three excited bands, is shown for the crossover regime as a function of $g$. In the inset, the exact ET ground-state binding energy ${\Delta }_{\mathrm{pol}}$ is compared to the second-order WCPT and SCPT results.

Figure 5

(Color online) The ET phonon spectral weight is shown as a function of $Q$ for $\lambda =4$ and ${\epsilon }_p∕{\omega }_0=10$. ${\overline{N}}_{\mathrm{tot}}$ is the mean number of phonons in the ground state (12). The thin dotted curves are given by Eqs. (27, 28).

Figure 6

(Color online) The phonon spectral weight $F_{+,Q}^{\left(i\right)}$, as a function of $Q$, for the first $i=1$ (dashed curves) and the second $i=2$ (solid curves) excited band. The results are obtained for the set of couplings that corresponds to the narrow interval in which the two $i=1$ and $i=2$ excited bands touch ($2.82⩽g_n⩽2.89$, with $g_{n+1}=g_n+0.01$, $t=2.5$). The arrows point to the direction of the increasing coupling. The thick dashed and solid curves are plotted for the self-trapped small-polaron limit in Eqs. (27, 28).

Figure 7

(Color online) The polaron (DOS), phonon (PDOS), and electron (EDOS) density of states for frequencies below the phonon threshold. The first and the second column of panels are given for $g=3.35$ and $g=2.9$, respectively, with $t=5$ in both cases. The amplitude of the PDOS exceeds the range of values shown in Fig. 7b.