Two-site small polaron quantum states: Optimization of the dressing fraction

A modified Lang-Firsov transformation is used to study the exciton-phonon interaction in a two-site system embedded in a one-dimensional lattice. It describes an exciton partially dressed by a virtual phonon cloud and depends on a single variational parameter, the so-called dressing fraction, whose optimization is achieved by using both the standard Bogoliubov inequality and its improved version defined by Decoster [J. Phys. A 37, 9051 (2004)]. The optimization procedure is applied to build a phase diagram in the parameter space which defines the different states of the exciton depending on the adiabaticity, the coupling strength, and the temperature. It is shown that the two Bogoliubov inequalities yield different variational principles that give rise to two optimal dressing fractions. Special attention is thus paid to characterize their differences in the nonadiabatic limit, where the exciton evolves continuously from a partially dressed state in the weak-coupling limit to a fully dressed state in the strong-coupling limit, and in the adiabatic limit where a self-trapped transition takes place.

Figure 1

Free energies $F_0\left(\eta \right)/E_B$ (circles), $\Delta F\left(\eta \right)/E_B$ (squares), and $F_1\left(\eta \right)/E_B$ (triangles) for $B=0.2$, $\theta =0.5$ (open symbols), and $\theta =2.0$ (full symbols). (a) $C=0.0312$, (b) $C=0.3465$, and (c) $C=1.3862$.

Figure 2

(Color online) Optimal dressing fraction (a) ${\eta }_0$ and (b) ${\eta }_1$ for $B=0.2$ and for different values of the reduced temperature.

Figure 3

(Color online) Optimal dressing fraction (a) ${\eta }_0$ and (b) ${\eta }_1$ for $\theta =1.0$ and for different $B$ values.

Figure 4

(Color online) Dressing factor (a) $\text{exp}\left(-{\eta }_0^{2}S\right)$ and (b) $\text{exp}\left(-{\eta }_1^{2}S\right)$ for $\theta =1.0$ and for different $B$ values.

Figure 5

Phase diagram in the parameter space for $\theta =0.5$. Squares define the MF critical curve whereas circles refer to the CMF critical curve. Full symbols denote the critical points.

Figure 6

Temperature dependence of the critical point (a) $B^{\ast }$ and (b) $C^{\ast }$ for both the CMF model (full symbols) and the MF model (open symbols).