Out-of-time-ordered correlator in the quantum bakers map and truncated unitary matrices

The out-of-time-ordered correlator (OTOC) is a measure of quantum chaos that is being vigorously investigated. Analytically accessible simple models that have long been studied in other contexts could provide insights into such measures. This paper investigates the OTOC in the quantum bakers map which is the quantum version of a simple and exactly solvable model of deterministic chaos that caricatures the action of kneading dough. Exact solutions based on the semiquantum approximation are derived that tracks very well the correlators until the Ehrenfest time. The growth occurs, surprisingly, at the exponential rate of the classical Lyapunov exponent which is half of that expected semiclassically. This exponential growth is modulated by slowly changing coefficients. Beyond this time, saturation occurs at a value close to that of random matrices. Using projectors for observables naturally leads to truncations of the unitary time-t propagator and the growth of their singular values is shown to be intimately related to the growth of the out-of-time-ordered correlators.