Exact Green’s functions for the two-site Hubbard-Holstein Hamiltonian

We derive exactly all Green’s functions for the two-site Hubbard-Holstein model. We then study the bipolaron phase diagram, with emphasis on its unbinding into two polarons and on the crossover from a two-site to a one-site bipolaron. The results are relevant for infinite-size systems.

Figure 1

(Color online) (a) $-\frac{1}{\pi }\mathrm{Im}{G}_{e,e}^{0,0}\left(\omega \right)$, (b) $-\frac{1}{\pi }\mathrm{Im}{G}_{o,o}^{0,0}\left(\omega \right)$, and (c) poles of $G_{e,e}^{0,0}\left(\omega \right)$ (solid lines) and $G_{o,o}^{0,0}\left(\omega \right)$ (dashed lines) as a function of $\lambda =g^2∕\left(t\Omega \right)$, for $t=2$, $\Omega =0.5$.

Figure 2

(Color online) Left: g.s. (circles) and first excited state (squares) bipolaron energies vs $\lambda$, for $U=8$. Lines are fits described in the text. Right: g.s. energy (and its derivative with $\lambda$, inset) for $U=0,2,4,6,8,10$. The solid line is twice the polaron g.s. energy. In all cases $t=2$, $\Omega =0.5$.

Figure 3

(Color online) Bipolaron phase diagram: unbinding (solid line) and $S1\to S0$ crossover (dashed line). Circles are one-dimensional (1D) results (Ref. 9) for unbinding, squares are 2D results (Ref. 10), for $t=\Omega =1$. Inset: same for $\Omega ∕t=0.1,0.25,1,2$ (red/gray, green/light gray, black, blue/dark gray).