Quantum Butterfly Effect in Weakly Interacting Diffusive Metals

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 $z>1$.

Popular Summary

Recent theoretical advances have raised new questions about the way chaos manifests in complex quantum systems consisting of many interacting particles. One of the key questions is how fast the system loses memory of its initial state, or information is “scrambled,” by the chaotic motion of quantum particles. One of the most surprising developments is that there are deep connections to the physics of black holes, which are believed to be the fastest scramblers in the Universe, saturating a universal bound on quantum chaos. Our work studies the physics of scrambling and information speed limits in a paradigmatic example of a quantum many-particle system—a metal consisting of electrons with Coulomb interactions and static impurities.

Using a sophisticated real-time approach in which time effectively runs both forward and backward, we calculate the rate of growth of chaos and the speed of information spreading in these interacting disordered metals. Surprisingly, while the speed of light provides a fundamental speed limit on information spreading, these metals obey a much stricter emergent speed limit. The limiting speed is found to have a power-law dependence on temperature, and it vanishes as the temperature approaches absolute zero. Another physical consequence is that the two-dimensional version of the system must undergo a phase transition (as seen in experiments) as interactions increase; otherwise, the bound on chaos would be violated.

Besides exhibiting a new kind of low-temperature speed limit, these results may have bearing on other aspects of information dynamics, including the rate of production of entanglement and entropy. More generally, the results may guide upcoming experiments designed to map out the range of scrambling behaviors from the very slow to the very fast.

Figure 1

(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 [31, 44]: 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 $v_{B}t$. 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 $f(t,\mathit{x})$ is roughly equivalent to the local fraction of infected electrons at a point $\mathit{x}$. (b) The behavior of $f(t,\mathit{x})$ for one operator placed at the center of the figure (red dot) and the other at a position $\mathit{x}$ shown as a function of $\mathit{x}$ at a given time $t$. Note that $f(t,\mathit{x})$ 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 $v_{B}t$.

Figure 2

(a) The impurity self-energy leading to the elastic lifetime in Eq. (2.3). (b) Disorder correction to the electron interaction vertex in Eq. (2.6); 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. (2.7). (d) The 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 $L(\omega ,\mathit{q})$, as described in Refs. [52, 53]. Each diagram has a partner diagram generated by reflecting about the horizontal axis. Hartree-type contributions are not shown; they are suppressed for sufficiently long-range interactions.

Figure 5

Ladder insertions at $\mathcal{O}(e^2)$, which provide exponentially growing contributions to $f(t,\mathit{x})$. The “direct” insertion in diagram (a) provides a contribution that grows at a rate proportional to $T^2$, slower than the “exchange” insertions in diagram (b), which grow as $T^{3/2}$. The relationship between the function $f(\omega ,\mathit{q})$ and the ladder series is shown in diagram (c).

Figure 6

A diagram in the “maximally crossed” series. The sum of this series gives $L_c(\omega ,\mathit{Q})$ as discussed in the main text.

Figure 7

Ladder insertions at $\mathcal{O}(e^2)$, in addition to the ones shown in Fig. 5, that do not change the growth exponent ${\lambda }_L$. Diagrams (a) and (b) have partner diagrams generated by reflection about the horizontal axis. Diagrams (c) and (d) have two partners each, from reflection about the horizontal and vertical axes. Other diagrams (not shown) similar to (c) and (d), with the internal resummed disorder lines terminating on the same time fold instead of opposite time folds, vanish because of integrations over Green’s functions with poles on the same side of the real axis.