Quantum chaos in an electron-phonon bad metal

TL;DR

This work presents a solvable large-$N$ electron-phonon model that exhibits bad-metal transport with linear-in-$T$ resistivity above the MIR limit. By computing scrambling dynamics through Bethe-Salpeter equations for both electronic and phononic operators, the authors show that the Lyapunov exponent $oldsymbol{ extlambda_L}$ and butterfly velocity $v_B$ are intrinsic to the system and controlled by the phonon sector, with $oldsymbol{ extlambda_L} o rac{1}{N}rac{oldsymbol{ extomega_0^2}}{T}$ up to a numerical factor and $v_B$ set by the phonon group velocity. The chaos diffusion constant $D_L=v_B^2/oldsymbol{ extlambda_L}$ is of the same order as the energy diffusion $D_E$, while charge diffusion $D_C$ can be parametrically different depending on phonon dispersion, e.g., $D_L eq D_C$ in the dispersive case and the opposite in the dispersionless case. Phonons thus dominate both energy transport and scrambling, leading to a strong violation of the Wiedemann–Franz law and highlighting a possible decoupling between chaos, energy diffusion, and charge transport in incoherent metals.

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 $D_L \ll D_C$.

Figures (9)

• Figure 1: The Bethe-Salpeter equation for A) the electronic $f(\omega)$ B) the phononic $g(\omega)$. Horizontal lines represent retarded/advanced propagators, with the top line living at the upper branch ($i\beta/2$) and the bottom line on the lower branch. Vertical lines are Wightman propagators. The last diagram on each row is naively suppressed by $1/N$, but diverges in the limit $\tau_{ph}\rightarrow\infty$.
• Figure 2: To lowest order in $1/N$, the full set of rainbow diagrams contributes to the electron self energy, denoted by the blue square (color online). The arrows represent bare electron propagators, dashed lines phonon propagators, and the thick arrow the fully dressed electron propagator.
• Figure 3: To lowest order in $1/N$, the only contribution to the phonon self energy is the electron bubble diagram. The arrows represent fully dressed electron propagators, while the dashed lines are bare phonon propagators.
• Figure 4: The contour in the complex time plane.
• Figure 5: The Bethe-Salpeter equation for $f(\omega)$. Horizontal lines represent retarded/advanced propagators, with the top line living at the upper branch and the bottom line on the lower branch $-i\beta/2$. Vertical lines are Wightman propagators, in this case the dashed line represents a phonon. Only the single-phonon Wightman diagram enters to leading order in $1/N$.