Electron-Phonon Coupling and a Polaron in the $t\mathrm{\text{−}}J$ Model: From the Weak to the Strong Coupling Regime

We present numeric results for ground state and angle resolved photoemission spectra (ARPES) for a single hole in the $t\mathrm{\text{−}}J$ model coupled to optical phonons. The systematic-error-free diagrammatic Monte Carlo method is employed where the Feynman graphs for the Matsubara Green function in imaginary time are summed up completely with respect to phonon variables, while magnetic variables are subjected to the noncrossing approximation. We obtain that at electron-phonon coupling constants relevant for high $T_c$ cuprates the polaron undergoes a self-trapping crossover to the strong-coupling limit and theoretical ARPES demonstrate features observed in experiment: A broad peak in the bottom of the spectra has momentum dependence which coincides with that of a hole in the pure $t\mathrm{\text{−}}J$ model.

Figure 1

Hole LSF in ground state at $J/t=0.3$. (a) For $g=0$. (b)–(d) Low energy part for $g=0.1445$ [$\gamma =0.34$] (b), $g=0.2$ [$\gamma =0.4$] (c), and $g=0.231125$ [$\gamma =0.43$] (d).

Figure 2

Dependence on coupling strength at $J/t=0.3$. (a) Energies of lowest LSF resonances. (b) [(c)] energy [$Z$ factor] of lowest peak in DMC (circles) and PPNCA (triangles). Lines are to guide the eye. The error bar, if not shown, is less than the point size.

Figure 3

LSF of the hole at $J/t=0.3$ and $g=0.231125$. (a) Full energy range for $\mathbf{k}=(\pi /2,\pi /2)$. (b)–(d) Low energy part for different momenta. Slanted arrows show broad peaks which can be interpreted in ARPES spectra as coherent (C) and incoherent (I) part. Vertical arrows indicate position of ground state resonance which is not seen in the vertical scale of the figure.

Figure 4

Dispersion of resonance energies at $J/t=0.3$. Broad resonance (filled circles) and lowest polaron resonance (filled squares) at $g=0.231125$; third broad resonance (open circles) and lowest polaron resonance (open squares) at $g=0.2$. The solid curves are dispersions (5) of a hole in the pure $t\mathrm{\text{−}}J$ model at $J/t=0.3$ ($W_{J/t=0.3}=0.6$): ${\varepsilon }_{\min }=-2.396$ (${\varepsilon }_{\min }=-2.52$) for solid (dotted) line.