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arXiv:2108.01107 · Physical review. B./Physical review. B

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

Evyatar Tulipman, Erez Berg

Published Nov 8, 2021• 11 citations
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Abstract

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 ${D}_{L}={v}_{B}^{2}/{\ensuremath{\lambda}}_{L}$, 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 $\ensuremath{\hbar}/{k}_{B}T$. 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.

DOI
10.1103/physrevb.104.195113

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We calculate the scrambling rate λLλ_LλL​ and the butterfly velocity vBv_BvB​ associated with the growth of quantum chaos for a solvable large-NNN electron-phonon system. We study a temperature regime in which the electrical resistivity of this system exceeds the Mott-Ioffe-Regel limit and increases linearly with temperature - a sign that there are no long-lived charged quasiparticles - although the phonons remain well-defined quasiparticles. The long-lived phonons determine λLλ_LλL​, rendering it parametrically smaller than the theoretical upper-bound λL≪λmax=2πT/ℏλ_L \ll λ_{max}=2πT/\hbarλL​≪λmax​=2πT/ℏ. Significantly, the chaos properties seem to be intrinsic - λLλ_LλL​ and vBv_BvB​ are the same for electronic and phononic operators. We consider two models - one in which the phonons are dispersive, and one in which they are dispersionless. In either case, we find that λLλ_LλL​ is proportional to the inverse phonon lifetime, and vBv_BvB​ is proportional to the effective phonon velocity. The thermal and chaos diffusion constants, DED_EDE​ and DL≡vB2/λLD_L\equiv v_B^2/λ_LDL​≡vB2​/λL​, are always comparable, DE∼DLD_E \sim D_LDE​∼DL​. In the dispersive phonon case, the charge diffusion constant DCD_CDC​ satisfies DL≫DCD_L\gg D_CDL​≫DC​, while in the dispersionless case DL≪DCD_L \ll D_CDL​≪DC​.
We compute the transport and chaos properties of lattices of quantum Sachdev-Ye-Kitaev islands coupled by single fermion hopping, and with the islands coupled to a large number of local, low energy phonons. We find two distinct regimes of linear-in-temperature (T)(T)(T) resistivity, and describe the crossover between them. When the electron-phonon coupling is weak, we obtain the ``incoherent metal'' regime, where there is near-maximal chaos with front propagation at a butterfly velocity vB{v}_{B}vB​, and the associated diffusivity ${D}_{\mathrm{chaos}}={v}_{B}^{2}/(2\ensuremath{\pi}T)$ closely tracks the energy diffusivity. On the other hand, when the electron-phonon coupling is strong, and the linear resistivity is largely due to near-elastic scattering of electrons off nearly free phonons, we find that the chaos is far from maximal and spreads diffusively. We also describe the crossovers to low TTT regimes where the electronic quasiparticles are well defined.
Phonons have long been thought to be incapable of explaining key phenomena in strange metals, including linear-in-\textit{T} Planckian resistivity from high to very low temperatures. We argue that these conclusions were based on static, perturbative approaches that overlooked essential time-dependent and nonperturbative electron-lattice physics. In fact ``phonons'' are not the best target for discussion, just like ``photons'' are not the best way to think about Maxwell's equations. Quantum optics connects photons and electromagnetism, as developed 60 years ago by Glauber and others. We have been developing the parallel world of quantum acoustics. Far from being only of academic interest, the new tools are rapidly exposing the secrets of the strange metals, revealing strong vibronic (vibration-electronic) interactions playing a crucial role forming polarons and charge density waves, linear-in-TTT resistivity at the Planckian rate over thousands of degrees, resolution of the Drude peak infrared anomaly, and the absence of a T4T^4T4 low-temperature resistivity rise in 2D systems, and of a Mott-Ioffe-Regel resistivity saturation. We derive Planckian transport, polarons, CDWs, and pseudogaps from the Fröhlich model. The ``new physics'' has been hiding in this model all along, in the right parameter regime, if it is treated nonperturbatively. In the course of this work we have uncovered the generalization of Anderson localization to dynamic media: a universal Planckian diffusion emerges, a ``ghost'' of Anderson localization. Planckian diffusion is clearly defined and is more fundamental than the popular but elusive, model dependent concept of ``Planckian speed limit''.
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