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arXiv:1705.07895 · Bulletin of the American Physical Society

Quantum chaos in an electron-phonon bad metal

Yochai Werman, Steven Allan Kivelson, Erez Berg

Published May 22, 2017• 27 citations
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Abstract

We calculate the scrambling rate $λ_L$ and the butterfly velocity $v_B$ associated with the growth of quantum chaos for a solvable large-$N$ 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 $λ_L$, rendering it parametrically smaller than the theoretical upper-bound $λ_L \ll λ_{max}=2πT/\hbar$. Significantly, the chaos properties seem to be intrinsic - $λ_L$ and $v_B$ 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 $λ_L$ is proportional to the inverse phonon lifetime, and $v_B$ is proportional to the effective phonon velocity. The thermal and chaos diffusion constants, $D_E$ and $D_L\equiv v_B^2/λ_L$, are always comparable, $D_E \sim D_L$. In the dispersive phonon case, the charge diffusion constant $D_C$ satisfies $D_L\gg D_C$, while in the dispersionless case $D_L \ll D_C$.

DOI
10.48550/arxiv.1705.07895

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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​.
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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.
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