Synergistic Polaron Formation in the Hubbard-Holstein Model at Small Doping

We study the effect of dynamical Holstein phonons on the physics of the Hubbard model at small doping using the dynamical cluster approximation on a $2\times 2$ cluster. Nonlocal antiferromagnetic correlations are found to significantly enhance the electron-phonon coupling, resulting in polaron formation for moderate coupling strengths. At finite doping, the electron-phonon coupling is found to strongly enhance the nonlocal spin correlations, indicating a synergistic interplay between the electron-phonon coupling and antiferromagnetic correlations. Although it enhances the pairing interaction, the electron-phonon coupling is found to decrease the superconducting transition temperature, due to the reduction in the quasiparticle fraction.

Figure 1

Single-particle DOS for several values of $\lambda$. The pseudogap width is negligibly dependent on $\lambda$. Inset: the electronic kinetic energy, $E_{\mathrm{kin}}=\sum _{k\sigma }ϵ(k)n_{k,\sigma }$, versus $T$. $N(0)$ and the kinetic energy gain are suppressed with increasing $\lambda$.

Figure 2

(a) The local dynamic charge susceptibility for several values of $\lambda$. The dashed (solid) lines are DMFA (DCA) results. A narrow band of low energy charge excitations develops as $\lambda$ increases which indicates polaron formation. (b) Static charge susceptibility (compressibility) versus $T$ for the same values of $\lambda$. The strong enhancement indicates the tendency to phase separation.

Figure 3

(a) The dynamic cluster spin susceptibility with $Q=(\pi ,0)$. The location of the peak is a measure of the effective spin exchange $J_{\mathrm{eff}}$. (b) The unscreened moment ${\mu }^2=⟨(n_{\uparrow }-n_{\downarrow }{)}^2⟩$ versus $T$, calculated with the DCA (solid lines) and the DMFA (dashed lines). The EP coupling strongly increases the temperature dependence of ${\mu }^2$ but the low-temperature value is unaffected. (c) The bulk spin susceptibility versus $T$.

Figure 4

(a) $T_N$ versus filling for several values of $\lambda$ calculated with the DCA (solid lines) and the DMFA (dashed lines). When $n=1$, the DCA $T_N$ increases slightly, consistent with a small increase in $J_{\mathrm{eff}}$ shown in Fig. 3a. At small doping, $T_N$ is enhanced by the EP coupling in contrast to the DMFA result where $T_N$ is suppressed. (b) The Matsubara quasiparticle fraction $Z_0=1/[1-\mathrm{Im}\Sigma (K,\pi T)/\pi T]$ versus $T$ for several values of $\lambda$. (c) Superconducting transition temperature versus $\lambda$ at 5% and 15% doping.