Strongly coupled phonon fluid and Goldstone modes in an anharmonic quantum solid: Transport and chaos

We study properties of thermal transport and quantum many-body chaos in a lattice model with $N\ensuremath{\rightarrow}\ensuremath{\infty}$ oscillators per site, coupled by strong anharmonic terms. We first consider a model with only optical phonons. We find that the thermal diffusivity ${D}_{\mathrm{th}}$ and chaos diffusivity ${D}_L$ (defined as ${D}_L={v}_B²/{\ensuremath{\lambda}}_L$, where ${v}_B$ and ${\ensuremath{\lambda}}_L$ are the butterfly velocity and the scrambling rate, respectively) satisfy ${D}_{\mathrm{th}}\ensuremath{\approx}\ensuremath{\gamma}{D}_L$ with $\ensuremath{\gamma}\ensuremath{\gtrsim}1$. At intermediate temperatures, the model exhibits a quantum ``phonon fluid'' regime, where both diffusivities satisfy ${D}^{\ensuremath{-}1}\ensuremath{\propto}T$, and the thermal relaxation time and inverse scrambling rate are of the order the of Planckian timescale $\ensuremath{\hbar}/{k}_BT$. We then introduce acoustic phonons to the model and study their effect on transport and chaos. The long-wavelength acoustic modes remain long-lived even when the system is strongly coupled, due to Goldstone's theorem. As a result, for $d=1,2$, we find that ${D}_{\mathrm{th}}/{D}_L\ensuremath{\rightarrow}\ensuremath{\infty}$, while for $d=3, {D}_{\mathrm{th}}$ and ${D}_L$ remain comparable.