The onset of quantum chaos in disordered CFTs

Abstract

We study the Lyapunov exponent λ L in quantum field theories with spacetimeindependent disorder interactions. Generically λ 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 λ 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 CFTs deformed by disorder at leading order in 1/N .

Onset of Quantum Chaos in Random Field Theories

“…However, we emphasize that it is not a chaos exponent in a single core CFT (as we take t 1 , t 2 to be larger then any timescale of the core CFT); in fact, for unitary theories in 2d it is always non-positive, λ 0 L ≤ 0 [27]. Surprisingly, under reasonable physical assumptions it can be shown that λ ker L (J = 0 + ) and λ 0 L are equal [26]. This amounts to showing that the integral (10) diverges for λ ≤ λ 0 L due to the large (negative) t 3 , t 4 regime of the integrand.…”

“…We start by writing a self-consistency equation for the averaged two-point function of (1), Using the G − Σ formalism [2,3], it can be shown that G obeys a generalized Schwinger-Dyson (SD) equation at leading order in 1/N , which appears diagrammatically in figure 2a. The equation includes subtracted npoint functions denoted by "n s ", which are combinations of the standard core CFT n-point functions with additional theory-independent subtractions which can be derived order-by order in n [26]. The first few subtracted n-point functions (assuming O i are real) are shown in figure 2b.…”

“…where the kernel K and the initial diagram F 0 are defined in figure 3a. The definition requires new subtracted n-point functions called n s , which are again theoryindependent [26]. The first few n s for real O i appear in figure 3b.…”

“…More details and discussions on the computations and results can be found in the companion paper [26].…”

Disordered vector models: from higher spins to incipient strings

“…Models with intermediate behaviour like the Sachdev-Ye-Kitaev (SYK) model [13][14][15][16] and its cousins [16][17][18][19][20][21][22], characterized by melonic diagrams dominating the large N limit, offer new perspectives. Quantum mechanical examples (d = 1) capture features of semiclassical gravity [23,24], while d ≥ 2 examples have classical finite tension string duals [18,21], owing to the lack of sparsity in the spectrum and submaximal Lyapunov exponent.…”