Quantum butterfly effect in weakly interacting diffusive metals

TL;DR

This work shows that quantum information scrambling in weakly interacting disordered metals occurs ballistically even though charge and energy transport are diffusive. By formulating a perturbative ladder-resummation framework on a complex-time contour, the authors extract the Lyapunov exponent $\lambda_L$ and butterfly velocity $v_B$ from the competition between inelastic scattering and disorder-induced diffusion. They provide explicit results in $d=3$ and $d=2$ for Coulomb and short-range interactions, demonstrating that $\lambda_L$ is controlled by $\gamma_{in}$ with $\lambda_L^{(3D)} \sim 1.116\frac{e^2 T^{3/2}}{D^{3/2}K^2}$ and, in 2D, $\lambda_{L2}^{(1)} = \frac{e^2 T}{D K_2} \ln 2$, while $v_B = \sqrt{4 D \lambda_L}$ remains parametrically smaller than microscopic velocities. The results connect ballistic operator growth to the sheet resistivity in 2D and suggest experimental avenues via local thermal diffusivity measurements and entanglement growth studies in disordered metals.

Abstract

We study scrambling, an avatar of chaos, in a weakly interacting metal in the presence of random potential disorder. It is well known that charge and heat spread via diffusion in such an interacting disordered metal. In contrast, we show within perturbation theory that chaos spreads in a ballistic fashion. The squared anticommutator of the electron field operators inherits a light-cone like growth, arising from an interplay of a growth (Lyapunov) exponent that scales as the inelastic electron scattering rate and a diffusive piece due to the presence of disorder. In two spatial dimensions, the Lyapunov exponent is universally related at weak coupling to the sheet resistivity. We are able to define an effective temperature-dependent butterfly velocity, a speed limit for the propagation of quantum information, that is much slower than microscopic velocities such as the Fermi velocity and that is qualitatively similar to that of a quantum critical system with a dynamical critical exponent $z > 1$.

Figures (7)

• Figure 1: (a) Cartoon showing a snapshot at time $t$ of the spread of chaos in an interacting diffusive metal. The fuzzy circles of radius $\propto (Dt)^{1/2}$ represent electrons diffusing through a background of impurities (small black dots). We make an analogy to the spread of an epidemic Stanford2016Huse13: An 'infected' electron inserted into the center of the figure at $t=0$ diffuses outwards (fuzzy red circle). As it encounters other diffusing electrons, it infects them. These newly infected electrons further infect other electrons and so on (fuzzy green circles). The flight paths of the butterflies track the spread of the infection. The radius of the region containing infected electrons (bounded by the dashed red circle) grows ballistically as $v_Bt$. Although not shown in the figure, the electrons also have a finite lifespan, given by the inverse of the quasiparticle decay rate. This needs to be taken into account when considering the population of infected electrons as a function of time. The function $f(t,\boldsymbol{x})$ is roughly equivalent to the local fraction of infected electrons at a point $\boldsymbol{x}$. (b) The behavior of $f(t,\boldsymbol{x})$ for one operator placed at the center of the figure (red dot) and the other at a position $\boldsymbol{x}$ shown as a function of $\boldsymbol{x}$ at a given time $t$. $f(t,\boldsymbol{x})$ displays a light-cone (a time slice of which is bounded by the dashed red circle; this region exclusively contains infected particles) within which it has saturated and no longer grows. The radius of this region grows as $v_Bt$.
• Figure 2: (a) The impurity self-energy leading to the elastic lifetime in Eq. (\ref{['G0']}) (b) Disorder correction to the electron interaction vertex in Eq. (\ref{['vc']}); here, and henceforth the electron lines contain the effect of the impurity self-energy (c) Dynamical screening of the interaction by the disorder-corrected polarization bubble in Eq. (\ref{['screen']}) (d) 2-in,2-out process that provides the inelastic electron lifetime; here, and henceforth the interaction line is the dynamically screened interaction.
• Figure 3: (a) Resummation of disorder rungs. (b) Relation between $L(\omega,q)$ and $f(\omega,q)$.
• Figure 4: The dominant Fock-type self-energy corrections to $L(\omega,\boldsymbol{q})$, as described in Refs. Castellani1984Castellani1986. Each diagram has a partner diagram generated by reflecting about the horizontal axis. Also not shown are the Hartree-type contributions, which are suppressed for sufficiently long-range interactions.
• Figure 5: Ladder insertions at $\mathcal{O}(e^2)$ which provide exponentially growing contributions to $f(t,\boldsymbol{x})$. The 'direct' insertion in (a) provides a contribution that grows at a rate proportional to $T^2$, slower than the 'exchange' insertions in (b), which grow as $T^{3/2}$. The relationship between the function $f(\omega,\boldsymbol{q})$ and the ladder series is shown in (c).