Velocity-dependent Lyapunov exponents in many-body quantum, semiclassical, and classical chaos

Vedika Khemani, David A. Huse, Adam Nahum

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104 Scopus citations


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(nvB(n))=0, with a generally direction-dependent butterfly speed vB(n). 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.

Original languageEnglish (US)
Article number144304
JournalPhysical Review B
Issue number14
StatePublished - Oct 16 2018

All Science Journal Classification (ASJC) codes

  • Electronic, Optical and Magnetic Materials
  • Condensed Matter Physics


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