Regularization dependence of the OTOC. Which Lyapunov spectrum is the physical one?

TL;DR

This work reveals that the OTOC and its Lyapunov spectrum depend on the complex time contour in generic weakly coupled theories, while the SYK model—a gapless system—exhibits contour-independent Lyapunov exponents despite contour-sensitive OTOCs. By formulating a two-parameter extended Schwinger-Keldysh contour (alpha,sigma) and mapping the OTOC to a kinetic/Boltzmann-like framework, the authors show that the symmetric contour (alpha = 1/2, sigma = 0) captures microscopic chaos most faithfully, with IR regulation driving contour dependence. In SYK, the Lyapunov spectrum remains invariant under contour deformations even as the full OTOC changes, emphasizing a robust notion of chaos in gapless systems and suggesting that the physical chaos bound should be interpreted through a symmetric, Loschmidt-echo-like observable, yielding lambda <= pi T_phys. The study thus clarifies how to extract physically meaningful chaotic dynamics from OTOCs and connects these insights to experimental echo measurements and the infrared structure of quantum many-body systems.

Abstract

We study the contour dependence of the out-of-time-ordered correlation function (OTOC) both in weakly coupled field theory and in the Sachdev-Ye-Kitaev (SYK) model. We show that its value, including its Lyapunov spectrum, depends sensitively on the shape of the complex time contour in generic weakly coupled field theories. For gapless theories with no thermal mass, such as SYK, the Lyapunov spectrum turns out to be an exception; their Lyapunov spectra do not exhibit contour dependence, though the full OTOCs do. Our result puts into question which of the Lyapunov exponents computed from the exponential growth of the OTOC reflects the actual physical dynamics of the system. We argue that, in a weakly coupled $Φ^4$ theory, a kinetic theory argument indicates that the symmetric configuration of the time contour, namely the one for which the bound on chaos has been proven, has a proper interpretation in terms of dynamical chaos. Finally, we point out that a relation between these OTOCs and a quantity which may be measured experimentally --- the Loschmidt echo --- also suggests a symmetric contour configuration, with the subtlety that the inverse periodicity in Euclidean time is half the physical temperature. In this interpretation the chaos bound reads $λ\leq \frac{2π}β= πT_{\text{physical}}$.

Figures (8)

• Figure 1: (a) Extended Schwinger-Keldysh contour corresponding to $\Tr[\rho^{{1\over 4}} V\rho^{{1\over 4}} W(t_2)\rho^{{1\over 4}} V \rho^{{1\over 4}} W(t_1) ]$. (b) Contour corresponding to a general regularization of the OTOC $\Tr[\rho^{\sigma}V\rho^{\alpha-\sigma} W(t_2)\rho^{\sigma} V \rho^{1-\alpha-\sigma} W(t_1)]$, which contributes to $C_{(\alpha,\sigma)}(t_1,t_2)$.
• Figure 2: Extended Schwinger-Keldysh contour corresponding to $tr[\rho^{\alpha } W(t_1)V\rho^{1-\alpha }W(t_2) V ]$ which enters in ${C(t;\beta)_{(\alpha,0)}}$ defined in Eq. \ref{['eq:Ct_alpha']}.
• Figure 3: A diagrammatic expansion of the correlator with the external legs on the same branch of the SK contour. The result does not depend on the width $\beta\alpha$.
• Figure 4: A diagrammatic expansion of the correlator with all but one external legs on the same branch of the SK contour. The result does depend on the width $\beta\alpha$
• Figure 5: A pictorial representation of a general time contour ($a$) and of the 4-points function in the ladder approximation ($b$) . The external legs lay on the first time fold and the second time fold. On the contrary, the rung joins the two time folds and include Wightman functions which by definition are contour dependent.