Dynamics of magnetic collective modes in square- and triangular-lattice Mott insulators at finite temperature

We study the equilibrium dynamics of magnetic moments in Mott insulating phase Hubbard model on square and triangular lattice. rewrite interaction terms an auxiliary vector field use a recently developed Langevin scheme to its dynamics. A thermal noise, derivable approximately from Keldysh formalism, allows us effect finite temperature. At strong coupling, $U\ensuremath{\gg}t$, where $U$ is local repulsion $t$ nearest-neighbor hopping, our results reproduce well known Heisenberg with exchange $J\ensuremath{\sim}\mathcal{O}({t}^{2}/U)$. These include crossover weakly damped dispersive modes at temperature $T\ensuremath{\ll}J$ damping $T\ensuremath{\sim}\mathcal{O}(J)$, diffusive $T\ensuremath{\gg}J$. The temperatures are naturally proportional $J$. To highlight progressive deviation physics as $U/t$ reduces we compute effective scale ${J}_{\mathrm{eff}}(U)$ low-temperature spin-wave velocity. discover two features dynamical behavior decreasing $U/t$: (i) dispersion deviates result, expected, due longer range multispin interactions, (ii) crossovers between weak damping, diffusion take place noticeably lower values $T/{J}_{\mathrm{eff}}$. relate this enhanced mode particular amplitude fluctuations, weaker $U/t$. comparison lattice reveals additional geometric frustration damping.