We use a Langevin approach to treat the finite temperature dynamics of displacement variables in the halffilled spinless Holstein model. Working in the adiabatic regime we exploit the smallness of the adiabatic parameter to simplify the memory effects and estimate displacement costs from an "instantaneous" electronic Hamiltonian. We use a phenomenological damping rate, and uncorrelated thermal noise. The low temperature state has checkerboard charge order (CO) and the Langevin scheme generates equilibrium thermodynamic properties that accurately match Monte Carlo results. It additionally yields the dynamical structure factor, D(q, ω), from the displacement field x(r, t). We observe four regimes with increasing temperature, T , classified in relation to the charge ordering temperature, Tc, and the 'polaron formation' temperature TP , with Tc TP . For T Tc the oscillations are harmonic, leading to dispersive phonons, with increasing T bringing in anharmonic, momentum dependent, corrections. For T ∼ Tc, thermal tunneling events of the x(r) field occur, with a propagating 'domain' pattern at wavevector q ∼ (π, π) and low energy weight in D(q, ω). When Tc < T < TP , the disordered polaron regime, domain structures vanish, the dispersion narrows, and low energy weight is lost. For T > ∼ TP we essentially have uncorrelated local oscillations. We propose simple models to analyse this rich dynamics.arXiv:1902.01337v1 [cond-mat.str-el]

We solve for the finite temperature collective mode dynamics in the Holstein-double exchange problem, using coupled Langevin equations for the phonon and spin variables. We present results in a strongly anharmonic regime, close to a polaronic instability. For our parameter choice the system transits from an 'undistorted' ferromagnetic metal at low temperature to a structurally distorted paramagnetic insulator at high temperature, through a short range charge ordered (CO) phase near the ferromagnetic crossover at TF M . The small amplitude harmonic phonons at low temperature cross over to large amplitude dynamics around 0.5TF M due to thermally generated short range correlated polarons. The rare thermal "tunneling" of CO domains generates a hitherto unknown momentum selective spectral weight at very low energy. We compare our results to inelastic neutron data in the manganites and suggest how the singular low energy features can be probed.

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We study the equilibrium dynamics of magnetic moments in the Mott insulating phase of the Hubbard model on the square and triangular lattice. We rewrite the Hubbard interaction in terms of an auxiliary vector field and use a recently developed Langevin scheme to study its dynamics. A thermal 'noise', derivable approximately from the Keldysh formalism, allows us to study the effect of finite temperature. At strong coupling, U t, where U is the local repulsion and t the nearest neighbour hopping, our results reproduce the well known dynamics of the nearest neighbour Heisenberg model with exchange J ∼ O(t 2 /U ). These include crossover from weakly damped dispersive modes at temperature T J to strong damping at T ∼ O(J), and diffusive dynamics at T J. The crossover temperatures are naturally proportional to J. To highlight the progressive deviation from Heisenberg physics as U/t reduces we compute an effective exchange scale J ef f (U ) from the low temperature spin wave velocity. We discover two features in the dynamical behaviour with decreasing U/t: (i) the low temperature dispersion deviates from the Heisenberg result, as expected, due to longer range and multispin interactions, and (ii) the crossovers between weak damping, strong damping, and diffusion take place at noticeably lower values of T /J ef f . We relate this to enhanced mode coupling, in particular to thermal amplitude fluctuations, at weaker U/t. A comparison of the square and triangular lattice reveals the additional effect of geometric frustration on damping.