TY - JOUR

T1 - Velocity-dependent Lyapunov exponents in many-body quantum, semiclassical, and classical chaos

AU - Khemani, Vedika

AU - Huse, David A.

AU - Nahum, Adam

N1 - Funding Information:
We thank Xiao Chen, Abhishek Dhar, Jeongwan Haah, Timothy Halpin-Healy, Cheryne Jonay, Cheng Ju Lin, Satya Majumdar, Lesik Motrunich, Xiaoliang Qi, Jonathan Ruhman, Douglas Stanford, Sagar Vijay, and Tianci Zhou for helpful discussions and for previous collaborations. V.K. was supported by the Harvard Society of Fellows and the William F. Milton Fund. A.N. acknowledges EPSRC Grant No. EP/N028678/1.
Publisher Copyright:
© 2018 American Physical Society.

PY - 2018/10/16

Y1 - 2018/10/16

N2 - 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.

AB - 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.

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U2 - 10.1103/PhysRevB.98.144304

DO - 10.1103/PhysRevB.98.144304

M3 - Article

AN - SCOPUS:85055120720

VL - 98

JO - Physical Review B

JF - Physical Review B

SN - 2469-9950

IS - 14

M1 - 144304

ER -