Onset of quantum chaos in disordered CFTs

We study the Lyapunov exponent ${\lambda }_L$ in quantum field theories with spacetime-independent disorder interactions. Generically ${\lambda }_L$ can only be computed at isolated points in parameter space, and little is known about the way in which chaos grows as we deform the theory away from weak coupling. In this paper, we describe families of theories in which the disorder coupling is an exactly marginal deformation, allowing us to follow ${\lambda }_L$ from weak to strong coupling. We find surprising behaviors in some cases, including a discontinuous transition into chaos. We also derive self-consistency equations for the two- and four-point functions for products of $N$ nontrivial conformal field theories deformed by disorder at leading order in $1/N$.

Figure 1

The two types of behaviors we find for the dependence of the chaos exponent ${\lambda }_L$ on the exactly marginal disorder deformation $J$: (a) continuous and (b) discontinuous.

Figure 2

The SD equations for disordered free fields, e.g., in the SYK model. Red lines correspond to the full two-point function $G$, while “$-2-$” corresponds to the two-point function of the core CFT (here a free theory).

Figure 3

The iterative ladder structure for the four-point function of disordered free fields. Red lines denote full propagators $G$, and black dots denote insertions of the disordered interaction. There are $q-2$ red lines running between each pair of interaction insertions.

Figure 4

The kernel $K$ and initial contribution $F_0$ for the four-point function for disordered free fields. Red lines again denote full propagators $G$.

Figure 5

The time contour for the OTOC. Red dots denote all possible positions of operators in the contributions from the double commutator.

Figure 6

The SD equations. We have emphasized the $G$ insertions using red lines to distinguish them from the two-point function of the undeformed CFT. Black dots denote insertions of the deformation (2.1), and $n_s$ denotes the subtracted $n$-point function defined below.

Figure 7

The subtracted $n$-point functions. Dashed lines are connected to the would-be external positions (see Fig. 6), and the numerical factors indicate symmetry factors. The overall factor for the $n_s$ correlator is in general $\frac{1}{2^{m}m!}$ where $m=\frac{n}{2}-1$, while the blue numbers inside the bracket correspond to the number of different permutations allowed when connecting the legs in a given diagram. The symmetry factors here are for real $\mathcal{O}$, but similar expressions exist also for complex fields.

Figure 8

The kernel $K$ and initial diagram $F_0$ for general disordered CFTs. Red lines denote full propagators $G$, and black dots denote insertions of the disorder interaction, with $q-2$ red propagators between each pair.

Figure 9

Examples of correlation functions $n_s^{\prime }$. Dashed lines correspond to external points, while solid lines are connected via $\mathrm{\Sigma }$’s in Fig. 8.

Figure 10

Schematic RG flows for disordered GFFs. Red denotes scale invariant theories. The disorder deformation is exactly marginal, leading to a line of fixed points that ends at the same point obtained by deforming free fields by the same interaction.

Figure 11

The chaos exponent for QM GFFs. (a) ${\lambda }_L$ as a function of $J$ at $\mathrm{\Delta }=0.25$. The dashed red line represents the solution to $k({\lambda }_L)=1$, but wherever the solution is negative we interpret ${\lambda }_L$ to be zero there, corresponding to the solid line. At large $J$ the solutions asymptote to ${\lambda }_L=1$. (b) ${\lambda }_L$ as a function of $\mathrm{\Delta }$ for various values of $J$. The black line is ${\lambda }_L=-2\mathrm{\Delta }$ and corresponds to $J=0$. This is negative for $\mathrm{\Delta }>0$, leading to the discontinuous transition into chaos.

Figure 12

The chaos exponent for $1+1\mathrm{d}$ SUSY GFFs. (a) The chaos exponent as a function of $J$ at $\mathrm{\Delta }=0.25$. The dashed line represents the solution to $k({\lambda }_L)=1$, but wherever the solution is negative we interpret ${\lambda }_L$ to be zero there, corresponding to the solid line. (b) The chaos exponent as a function of $\mathrm{\Delta }$ for various values of $J$. Again, for small enough $J$, ${\lambda }_L$ is negative.

Figure 13

The algorithm for finding $4_s$. An $n$ corresponds to a full $n$-point function, while an $n_c$ corresponds to a connected $n$-point function.

Figure 14

Decomposing the six-point function.

Figure 15

Decomposing the six-point function.