Onset of Quantum Chaos in Random Field Theories

We study the quantum Lyapunov exponent ${\lambda }_L$ in theories with spacetime-independent disorder. We first derive self-consistency equations for the two- and four-point functions for products of $N$ models coupled by disorder at large $N$, generalizing the equations appearing in SYK-like models. We then study 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 interesting behaviors, including a discontinuous transition into chaos, mimicking classical KAM theory.

Figure 1

The behaviors we find for the chaos exponent as a function of an exactly marginal disorder deformation $J$: (a) continuous and (b) discontinuous. ${\lambda }_L^{\ker }$ corresponds to the dashed line and ${\lambda }_L$ to the solid line.

Figure 2

(a) The SD equations. $G$ insertions appear as red lines, and black dots denote insertions of the deformation (1). (b) Examples of $n_s$, the subtracted $n$-point functions. Dashed lines connect to the external points, and numerical factors indicate symmetry factors. The blue numbers inside the brackets denote possible ways of permuting the legs.

Figure 3

(a) 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. (b) Examples of correlation functions $n_s^{\prime }$. Dashed lines correspond to external points, while solid lines are connected via $\mathrm{\Sigma }$’s defined in Fig. 2.

Figure 4

The chaos exponent ${\lambda }_L(J)$ at $\mathrm{\Delta }=1/4$ for disordered GFFs in QM and for SUSY disordered GFFs in two dimensions. Dashed lines represent ${\lambda }_L^{\ker }$ and solid lines represent ${\lambda }_L$.