Figure 1

The spectra of the kernel $\mathcal{L}(\mathbf{p},\mathbf{l})$ for the linearized Boltzmann equation [and also of ${〈T^{xy}\left(k_z\right),T^{xy}(-k_z)〉}_R$; cf. Eq. (34)] (top left) and of the kernel ${\mathcal{L}}_{\text{EX}}(\mathbf{p},\mathbf{l})$ for the kinetic equation for the OTOC (top right) are plotted over the complex $\omega$ plane and in the limit of $\beta m\to 0$. In the lower half of the complex $\omega$ plane, there is a dense sequence of numerically obtained poles. In both spectra, these poles are believed to be the signature of a branch cut. See [42] and also [34, 44, 45, 46]. In the upper half of the complex $\omega$ plane, only the kernel ${\mathcal{L}}_{\text{EX}}(\mathbf{p},\mathbf{l})$ has distinct poles which are identified with the Lyapunov exponents, as explained below Eq. (11). The dependence of these two Lyapunov exponents and the branch cuts on $\beta m$ is depicted in the inlay (bottom). For large values of $\beta m$, the Lyapunov exponents decay exponentially. The plots are obtained by diagonalizing the kernels of the integral equations (20) and (32) after a discretization with $N=1000$ grid points on the domain $p\in [m/N,N\times m]$. The discretization is not uniform. This is done in order for the diagonalization to appropriately account for the contributions of both the soft momenta and collinear momenta $\mathbf{p}\approx \mathbf{l}$, which are not negligible even when both $\mathbf{p}$ and $\mathbf{l}$ are large [36, 41]. The finite size of the branch cuts, i.e., its end point for large Im($\omega$), is related to the finite domain of the discretization procedure.