Out-of-time-ordered correlator

In quantum physics, the out-of-time-ordered correlator (OTOC)^[1] serves as a powerful diagnostic tool for characterizing quantum chaos, information scrambling, and other aspects of many-body dynamics. In addition, it provides a quantum mechanical analog to the Lyapunov exponent, often used to characterize the sensitivity of variables to initial conditions in classical chaos. The OTOC thus provides a natural extension of classical chaos theory to the quantum realm, and can be calculated both numerically and experimentally ^[2] .

For two observable ${\displaystyle V}${\displaystyle V} and ${\displaystyle W}${\displaystyle W} in Heisenberg picture, the out-of-time-order correlator (OTOC) is typically defined in two different but physically closely related ways:^{[3] [4] [5] [6] [7]}

1. Based on commutator of ${\displaystyle W(t)}${\displaystyle W(t)} and ${\displaystyle V(0)}${\displaystyle V(0)}:$${\displaystyle C(t)=\langle [W(t),V(0)]^{\dagger }[W(t),V(0)]\rangle }$${\displaystyle C(t)=\langle [W(t),V(0)]^{\dagger }[W(t),V(0)]\rangle }direct calculation gives ${\displaystyle C(t)=\langle V(0)^{\dagger }W(t)^{\dagger }W(t)V(0)\rangle +\langle W(t)^{\dagger }V(0)^{\dagger }V(0)W(t)\rangle -\langle V(0)^{\dagger }W(t)^{\dagger }V(0)W(t)\rangle -\langle W(t)^{\dagger }V(0)^{\dagger }W(t)V(0)\rangle .}${\displaystyle C(t)=\langle V(0)^{\dagger }W(t)^{\dagger }W(t)V(0)\rangle +\langle W(t)^{\dagger }V(0)^{\dagger }V(0)W(t)\rangle -\langle V(0)^{\dagger }W(t)^{\dagger }V(0)W(t)\rangle -\langle W(t)^{\dagger }V(0)^{\dagger }W(t)V(0)\rangle .}
2. More directly $${\displaystyle F(t)=\langle W(t)^{\dagger }V(0)^{\dagger }W(t)V(0)\rangle }$${\displaystyle F(t)=\langle W(t)^{\dagger }V(0)^{\dagger }W(t)V(0)\rangle }Generally ${\displaystyle C(t)=\langle V(0)^{\dagger }W(t)^{\dagger }W(t)V(0)\rangle +\langle W(t)^{\dagger }V(0)^{\dagger }V(0)W(t)\rangle -2,円\mathrm {Re} ,円F(t).}${\displaystyle C(t)=\langle V(0)^{\dagger }W(t)^{\dagger }W(t)V(0)\rangle +\langle W(t)^{\dagger }V(0)^{\dagger }V(0)W(t)\rangle -2,円\mathrm {Re} ,円F(t).} When ${\displaystyle V}${\displaystyle V} and ${\displaystyle W}${\displaystyle W} are unitaries, we have ${\displaystyle C(t)=2{\big (}1-\mathrm {Re} ,円F(t){\big )}.}${\displaystyle C(t)=2{\big (}1-\mathrm {Re} ,円F(t){\big )}.}

where the expectation value ${\displaystyle \langle \bullet \rangle =\operatorname {Tr} [\rho ,円\bullet ]}${\displaystyle \langle \bullet \rangle =\operatorname {Tr} [\rho ,円\bullet ]} is usually taken over some thermal state ${\displaystyle \rho =\exp(-\beta H)/Z}${\displaystyle \rho =\exp(-\beta H)/Z} with ${\displaystyle \beta =1/k_{B}T}${\displaystyle \beta =1/k_{B}T} (${\displaystyle k_{B}}${\displaystyle k_{B}} is Boltzmann constant, ${\displaystyle T}${\displaystyle T} is temperature) and ${\displaystyle H}${\displaystyle H} is Hamiltonian, ${\displaystyle Z=\operatorname {Tr} \exp(-\beta H)}${\displaystyle Z=\operatorname {Tr} \exp(-\beta H)} is canonical partition function.

Physically, the growth of this commutator measured by ${\displaystyle C(t)}${\displaystyle C(t)} tracks scrambling. And from chaos theory perspective, we have ${\displaystyle C(t)\simeq e^{\lambda _{L}t}}${\displaystyle C(t)\simeq e^{\lambda _{L}t}} where ${\displaystyle \lambda _{L}}${\displaystyle \lambda _{L}} is the quantum Lyapunov exponent. This has a similar form as the classical dependence of initial pertuvation in classical chaos theory. Thus OTOC can be regarded as an indicator of quantum chaos.

Quantum chaos

SYK model

Chaos theory

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