Phases of scrambling in eigenstates

Abstract

We use the monodromy method to compute expectation values of an arbitrary number of light operators in finitely excited ("heavy") eigenstates of holographic 2D CFT. For eigenstates with scaling dimensions above the BTZ threshold, these behave thermally up to small corrections, with an effective temperature determined by the heavy state. Below the threshold we find oscillatory and not decaying behavior. As an application of these results we compute the expectation of the out-of-time order arrangement of four light operators in a heavy eigenstate, i.e. a six-point function. Above the threshold we find maximally scrambling behavior with Lyapunov exponent 2πT eff . Below threshold we find that the eigenstate OTOC shows persistent harmonic oscillations.

Universal lowest-twist in CFTs from holography

“…Taking d = 4 and substituting (2.18), (2.19) into the bulk field equation, one obtains ordinary differential equations for a (0) (w) and a (2) (w) that can be solved analytically. 15 We find…”

“…On the other hand, if the background is simply f = h = 1 − f0 r 4 , there are still higher-order corrections to Φ, corresponding to higher-order OPE computed in the AdS-Schwarzschild geometry. 15 One solves the scalar field equation order-by-order inθ. The equation associated with the highest power ofθ involves a (2) (w) only, and the next order equation contains both a (0) (w) and a (2) (w).…”

“…In two-dimensions, where the CFT formulation has been especially useful, the existence of the Virasoro symmetry might be difficult to discover on the gravity side [5], but once discovered it leads to a computationally powerful algebraic description of many gravitational effects. In particular, when applied to four-point functions of local CFT operators, the irreducible representations of the algebra, known as Virasoro Conformal Blocks, capture the thermal properties of black holes, the information paradox associated with late-time decay of correlators in semiclassical gravity [6][7][8], the universal properties of Renyi and entanglement entropy [9][10][11][12] leading to a proof of Ryu-Takayanagi formula [13] in AdS 3 /CFT 2 , the maximal growth of chaos in gravity [14,15], the nonperturbative resolution of some perturbative violations of unitarity [16,17], and more, all without appealing to a gravitational Lagrangian. See also [18][19][20][21][22][23][24][25][26][27] for related discussions.…”

Reparametrization modes, shadow operators, and quantum chaos in higher-dimensional CFTs

“…Having set up the building blocks of the effective theory of the reparametrization mode in two dimensions in thermal states, more interesting calculations can be performed with it. These include the "light-light" Virasoro block and its exponentiation [34], the "heavy-light" Virasoro block [42] and 1/c corrections to it [43], application to OTOCs [44,45], higher-point blocks [46] and OTOCs [47,48], and so on. Some of these calculations have been done for two-dimensional CFTs using the reparametrization mode formalism in [15,16,18], so we will not repeat them here.…”

Eigenstate thermalization hypothesis and approximate quantum error correction

“…chaos, such as the decay of out-of-time-order correlators (OTOC) and operator growth, to quantum error correction [38][39][40][41][42][43][44]. By extension, we can then apply such results to understanding holographic CFTs and their duals, as part of a general program of applying tools and concepts from quantum information and error correction to understanding quantum gravity.…”