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I am reading 1609.01251, which studies the quantum butterfly effect in electron-electron, electron-phonon, and electron-impurity scattering. The first two of these are very clearly chaotic (which is the final result of the paper), but the electron-impurity scattering has a caveat which I do not understand. The kinetic equation for the quantum distribution function $\mathcal{F}_{\alpha\beta}$ is:

$$ \Bigg\{\dfrac{\partial}{\partial t} + \dfrac{\partial}{\partial r}\dfrac{d\xi_p}{dp} \Bigg\} \mathcal{F}_{\alpha\beta}(\epsilon,p;r,t) = \int \dfrac{dp_1}{(2\pi \hbar)^d}\Bigg\{\bigg[\dfrac{2\pi}{\hbar}|V_{p-p_1}|^2 \delta(\xi_p - \xi_{p_1})\bigg]\bigg[-\mathcal{F}_{\alpha\beta}(p) + \mathcal{F}_{\alpha\beta}(p_1)\bigg]\Bigg\} $$

The authors include the following discussion:

in the case of the impurity scattering the non-diagonal components of the Keldysh function have the same time evolution as the diagonal ones, so the solution in which it is equal to the thermal equilibrium distribution is stable. Note that electrons in the impurity potential is a chaotic system. In this respect it is not different from the electron-phonon and the electron-electron interaction. Nevertheless, the non-diagonal components are stable, in contrast to the models with electron-phonon and electron-electron interactions. This results in a very different behavior of the out-of-time-ordered correlators in this system.

They do not include any more equations here. I do not understand why it is immediately clear that electron-impurity scattering is chaotic. How do I further simplify this integral to achieve this observation?

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  • $\begingroup$ I cannot comment on the calculations, but isn't it to be expected that the electron-impurity scattering is chaotic? Naively, if scattering by oscillations in the regular lattice is chaotic, it would be curious if the scattering by (supposedly) irregularly spread impurities weren't. Or do I miss something? $\endgroup$ – stafusa Nov 21 '17 at 1:30
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    $\begingroup$ @stafusa Intuitively sure, the problem is that the chaos encountered in electron-impurity is of a different flavor. The propagation can be controlled via $p$. In other words, the butterfly velocity is parametric in momentum. That seems very unintuitive to me. Moreover the statement that something is chaotic should be followed by a computation of a Lyapunov exponent, but in this paper it is not. $\endgroup$ – Evan Coleman Nov 21 '17 at 1:31
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    $\begingroup$ Can that be at least tangentially relevant? the paper Quantum Disorder and Quantum Chaos in Andreev Billiards appears to distinguish quantum chaos ("quantum diffraction") from the "quantum disorder" caused by the scattering off point impurities. $\endgroup$ – stafusa Nov 21 '17 at 1:36
  • $\begingroup$ This is perfect. I am only just getting into the condensed matter side of quantum chaos, so I think this is a great place to start (+ for anyone viewing this thread) $\endgroup$ – Evan Coleman Nov 21 '17 at 1:38
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    $\begingroup$ I hope it really helps. If it turns out to be relevant, remember to add a mention to it in your answer, since comments are second rate and can easily disappear in this site. $\endgroup$ – stafusa Nov 21 '17 at 1:41
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I'll take a whack at it. The authors seem to care about the thermal equilibrium quantum distribution function, where the two different worlds (thermofield double state) are correlated: $$ \mathcal{F}_{\alpha\beta}(\epsilon) = \begin{bmatrix}n_0(\epsilon) & 1 + n_0(\epsilon) \\ -1 + n_0(\epsilon) & n_0(\epsilon)\end{bmatrix} $$ Where $n_0(\epsilon) = \tanh\Big(\dfrac{\epsilon - \mu}{2T}\Big)$. This is equation (25) in the paper. So. in particular:

the non-diagonal components of the Keldysh function have the same time evolution as the diagonal ones

This is manifestly true. Look at the integrand when you insert the correlated worlds solution:

$$ -\mathcal{F}_{\alpha\beta}(p) + \mathcal{F}_{\alpha\beta}(p_1) = \begin{bmatrix}n_0(\epsilon_1) - n_0(\epsilon) & n_0(\epsilon_1) - n_0(\epsilon) \\ n_0(\epsilon_1) - n_0(\epsilon) & n_0(\epsilon_1) - n_0(\epsilon)\end{bmatrix} $$

At equilibrium, the delta function in the integrand forces the integral to vanish. Then, the solution to the kinetic equation is an exponential with an arbitrary additive constant. However, a traveling wave solution of the form $\mathcal{F}_{\alpha\beta}(r-v_{cw}t)$ will have a parametric velocity: $v_{cw} = \frac{d\xi_p}{dp}$. So, it may be chaotic, but the propagation of a perturbation can be controlled by tuning $\xi_p$ somehow.

Now suppose we are not perfectly at equilibrium so that $F_{\alpha\beta}(p_1) \approx n_0$, but $F_{\alpha\beta}(p)$ is generally different from $n_0$ on the constant-energy surface determined by the delta function. Then, ignoring constant prefactors and supposing the variations in the potential $|V|^2$ are nonzero but bounded, we obtain the following form of the kinetic equation in zero spatial dimensions:

$$ \dfrac{\partial}{\partial t}\mathcal{F}_{\alpha\beta}\big(\epsilon(p)\big) = \mathcal{F}_{\alpha\beta}(\epsilon) - \tanh\Big(\dfrac{\epsilon-\mu}{2T}\Big) $$

Attempting, as in the other integrals, an ansatz of the form $\mathcal{F}(\epsilon) = \phi(t)(1 + n_0(\epsilon))$, the general solution will be:

$$ \phi(t) = C e^{\omega t} + \dfrac{1}{2}(1 - e^{(\mu-\epsilon)/T}) $$

Where the scale of $\omega$ is determined by arbitrary constants, which were ignored. So, it seems that this system could be chaotic, but it's not immediately clear at all that the assumption I've used is reasonable.

Hopefully others can comment on my analysis to determine if this makes any sense.

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