Quantum chaos on a critical Fermi surface

TL;DR

This work investigates quantum chaos on a two-dimensional critical Fermi surface formed by $N$ fermion flavors at finite density coupled to a $U(1)$ gauge field. Using an extended random-phase approximation and ladder/Bethe-Salpeter techniques, it derives a Lyapunov exponent $\\lambda_L \\approx 2.48\\,T$ (saturating the chaos bound) and a butterfly velocity dominated by motion perpendicular to the Fermi surface, $v_{B\perp} \\approx 4.10 \, \frac{N T^{1/3}}{e^{4/3}} \, \frac{v_F^{5/3}}{\\gamma^{1/3}}$. The study uncovers a universal relation between energy diffusion and chaos, $D^E \\approx 0.42 \, v_B^2/\\lambda_L$, with cancellations that render the result independent of $N$, $e$, $v_F$, and $\\gamma$. These findings support a tight connection between many-body quantum chaos and thermal transport in strongly coupled, non-quasiparticle states and point to possible crossovers and extensions at higher temperatures or in multi-patch theories.

Abstract

We compute parameters characterizing many-body quantum chaos for a critical Fermi surface without quasiparticle excitations. We examine a theory of $N$ species of fermions at non-zero density coupled to a $U(1)$ gauge field in two spatial dimensions, and determine the Lyapunov rate and the butterfly velocity in an extended random-phase approximation. The thermal diffusivity is found to be universally related to these chaos parameters i.e. the relationship is independent of $N$, the gauge coupling constant, the Fermi velocity, the Fermi surface curvature, and high energy details.

Figures (5)

• Figure 1: (a) Fermi surface patch and coordinate system (b) The complex-time contour $C$ used for evaluating out-of-time-order correlation functions. It contains forward and backward time evolution along two real time folds separated by $i\beta/2$, and imaginary time evolution between the folds.
• Figure 2: The Bethe-Salpeter equation for $f(\omega)$ at leading naive order in $1/N$. Solid lines are fermion propagators and dashed lines are boson propagators. The arrows indicate the directions of momentum flow used in the equations in the text. For the fermion lines, advanced Green's functions are used for the upper rails and retarded ones for the lower rails, as can be seen from Eq. (\ref{['ft']}). The third diagram on the right hand side is the same order in $1/N$ as the second despite having two boson propagators, because it involves summing over the flavors $j$.
• Figure 3: (a) The one-loop boson and (b) fermion self energies. These graphs are evaluated at a finite temperature. The dashed lines are bare boson propagators and solid lines are bare fermion propagators. The arrows indicate the directions of momentum flow used in the equations in the text.
• Figure 4: (a), (b) The two simplest crossed ladder insertions in the Bethe-Salpeter equation. The first vanishes, and the second contributes to $\lambda_L$ at $\mathcal{O}(1/N)$. (c) A higher-order "maximally crossed" diagram with boson rungs. Diagrams of this type also vanish for the same reason as (a).
• Figure 5: (a) Plot of the magnitude of the smallest eigenvalue for $\omega$ on the positive imaginary axis for $T=1.0$. (b) Plot of the magnitude of the entries of the corresponding eigenvector when $-i\omega=\lambda_L$. (c) Plot of $\lambda_L$ vs $T$. (d) Plot of $\mathrm{Im}[\delta\lambda_L]$ vs $Np_x$. The value of $\mathrm{Re}[\delta\lambda_L]\sim (Np_x)^2$ is very small when $Np_x$ is small. This real part does not control the speed $v_{B\perp}$ at which the wave pulse of Eq. (\ref{['wave']}) travels, but will lead to the broadening of the pulse as it travels (see below). For all these figures, $k_0\in[-15,15]$, $m=0.02$, the step size $dk_0=0.005$ and $e=1.0$.