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Thermal dynamics of lattice modes near a polaronic crossover: from the dilute polaron limit to a charge ordered state

We provide a comprehensive solution to the lattice dynamics problem in the two dimensional Holstein model at finite electron density and finite temperature. We work in the physically relevant adiabatic regime and vary the electron-phonon interaction from the weak coupling perturbative window to the strong coupling polaronic regime. We explore three typical electron densities, dilute - where spatial correlations between polarons is weak, intermediate - where correlations are significant, and half-filling - where there is long range checkerboard order at low temperature. We use two methods both of which exploit the "slowness" of the phonons to handle the problem. These are (i)~a standard random phase approximation (RPA), adapted to capture small quantum fluctuations on Monte Carlo generated classical thermal backgrounds, and (ii)~a Langevin dynamics scheme, with a simplified ``thermal noise'', that can address large amplitude dynamical fluctuations. The Langevin scheme, as we argue in the paper, is the superior method in the strong coupling part of the phase diagram, where lattice distortions are large. It reveals a non trivial multi-peak momentum resolved spectrum with a high energy part, on the scale of the bare phonon frequency $Ω$, and a low energy peak at $ω\ll Ω$. Below the polaronic threshold, the high energy dispersion changes only modestly with temperature $T$, while the broadening, arising from mode coupling, increases linearly with $T$ at low temperature. The low energy peak shows up at strong coupling and finite temperature and arises from the slow tunneling of polarons. The tunneling events become spatially correlated as electron density increases towards half-filling, and the weight becomes strongly momentum and temperature dependent. We suggest the analytic basis of these results.