# Quantum Scrambling and State Dependence of the Butterfly Velocity

## TL;DR

This paper investigates how operator growth and scrambling velocity in quantum many-body systems relate to the state dependence of out-of-time-ordered correlators, supported by theoretical bounds and numerical simulations.

## Contribution

It establishes a bound on the state dependence of out-of-time-ordered correlators based on the scrambling velocity in local lattice models.

## Key findings

- The bound is verified in the thermal mixed-field Ising spin chain.
- The butterfly velocity exhibits a crossover from high-temperature to low-temperature values.
- The low-temperature butterfly velocity differs significantly from the high-temperature value.

## Abstract

Operator growth in spatially local quantum many-body systems defines a scrambling velocity. We prove that this scrambling velocity bounds the state dependence of the out-of-time-ordered correlator in local lattice models. We verify this bound in simulations of the thermal mixed-field Ising spin chain. For scrambling operators, the butterfly velocity shows a crossover from a microscopic high temperature value to a distinct value at temperatures below the energy gap.