Out-of-time ordered correlator for a chaotic many-body quantum system

Abstract Using the parametric representation of a chaotic many-body quantum system derived earlier, we calculate explicitly the large-time dependence and asymptotic value of the out-of-time correlator (OTOC) of that system. The dependence on time t is determined by <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi mathvariant="normal">Δ</mml:mi> <mml:mi>t</mml:mi> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>ℏ</mml:mi> </mml:mrow> </mml:math> . Here Δ is the energy correlation width within which the Bohigas–Giannoni–Schmit conjecture applies. We conjecture that Δ is universally related to the leading Ljapunov coefficient of the corresponding classical system by <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi mathvariant="normal">Δ</mml:mi> <mml:mo>=</mml:mo> <mml:mi>ℏ</mml:mi> <mml:msub> <mml:mi>λ</mml:mi> <mml:mrow> <mml:mo movablelimits="true">max</mml:mo> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> . Then the large-time behavior of OTOC is given by the dimensionless parameter <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>λ</mml:mi> <mml:mrow> <mml:mo movablelimits="true">max</mml:mo> </mml:mrow> </mml:msub> <mml:mi>t</mml:mi> </mml:mrow> </mml:math> .