Condensed Matter Theory

Abstract:

We discuss eigenstate correlations for ergodic, spatially extended many-body quantum systems, in terms of the statistical properties of matrix elements of local observables. While the eigenstate thermalization hypothesis (ETH) is known to give an excellent description of these quantities, the butterfly effect implies structure beyond ETH. We determine the universal form of this structure at long distances and small eigenvalue separations for Floquet systems. We use numerical studies of a Floquet quantum circuit to illustrate both the accuracy of ETH and the existence of our predicted additional correlations.

Abstract:

The exponential growth or decay with time of the out-of-time-order commutator (OTOC) is one widely used diagnostic of many-body chaos in spatially extended systems. In studies of many-body classical chaos, it has been noted that one can define a velocity-dependent Lyapunov exponent, λ(v), which is the growth or decay rate along rays at that velocity. We examine the behavior of λ(v) for a variety of many-body systems, both chaotic and integrable. The so-called light cone for the spreading of operators is defined by λ(ˆnvB(ˆn))=0, with a generally direction-dependent butterfly speed vB(ˆn). In spatially local systems, λ(v) is negative outside the light cone where it takes the form λ(v)∼−(v−vB)α near vB, with the exponent α taking on various values over the range of systems we examine. The regime inside the light cone with positive Lyapunov exponents may only exist for classical, semiclassical, or large-N systems, but not for “fully quantum” chaotic systems with strong short-range interactions and local Hilbert space dimensions of order one.