Out-of-Time-Order Correlation Functions for Unitary Minimal Models

Abstract

We analytically study the Out-of-Time-Order Correlation functions (OTOC) for two spatially separated primary operators in two-dimensional unitary minimal models. Besides giving general arguments using the conformal symmetry, we also use the Coulomb gas formalism to explicitly calculate the OTOC across the full time regime. In contrast to large-$N$ chaotic systems, these models do not display a separation of time scales, due to the lack of a large parameter. We find that the physics at early times ($0<t-x\ll\beta$) and late times ($t-x\gg\beta$) are controlled by different OPE channels, which are related to each other via the braiding matrix. The normalized OTOC obeys a power-law decay with a fractional power at early times and approaches a generically nonzero value in an exponential way at late times. The late time value is related to the modular $S$-matrix and is in agreement with earlier calculations. All of the results above are readily generalized to rational conformal field theories.