## "Operator Spreading and the Quantum Butterfly Effect in Random Unitary Circuits"

**Speaker:**Sagar Vijay, Massachusetts Institute of Technology

**When:**February 12, 2018 (Mon), 11:00AM to 12:00PM (add to my calendar)

**Location:**SCI 328

**Hosted by:**Philip Crowley

*This event is part of the Condensed Matter Theory Seminar Series. *

We provide universal, long-distance descriptions for the ‘quantum butterfly effect’, which describes the spreading of a local disturbance in a quantum medium, and is quantified by the ‘out-of-time-order correlator’ (OTOC). We obtain these results by studying quantum circuits made of random unitaries, which yield minimally structured models for chaotic quantum dynamics, and are able to capture universal properties of quantum entanglement growth. In this setting, we demonstrate a mapping between the OTOC and classical growth processes, which leads to exact results and explicit universal scaling forms for the OTOC. In 1+1D, we demonstrate that the OTOC satisfies a biased diffusion equation, which gives exact results for its spatial profile, and for the butterfly speed. In higher dimensions, we show that the averaged OTOC can be understood exactly via a correspondence with a classical droplet growth problem. We support our analytic argument with simulations in 2+1D and demonstrate that in a lattice model, the late time shape of the spreading operator is in general not spherical. However when full spatial rotational symmetry is present in 2+1D, our mapping implies an exact asymptotic form for the OTOC in terms of the Tracy-Widom distribution. For an alternative perspective on the OTOC in 1+1D, we map it to the partition function of an Ising-like model. As a result of special structure arising from unitarity, this partition function reduces to a random walk calculation which can be performed exactly. We argue that the coarse-grained pictures carry over to operator spreading in generic many-body systems.