Abstract
We explore the quantum chaos of the coadjoint orbit action of diffeomorphism group of S1. We study quantum fluctuation around a saddle point to evaluate the soft mode contribution to the out-of-time-ordered correlator. We show that the stability condition of the semi-classical analysis of the coadjoint orbit found in [1] leads to the upper bound on the Lyapunov exponent which is identical to the bound on chaos proven in [2]. The bound is saturated by the coadjoint orbit Diff(S1)/SL(2) while the other stable orbit Diff(S1)/U(1) where the SL(2, ℝ) is broken to U(1) has non-maximal Lyapunov exponent.
Highlights
We show that the stability condition of the semi-classical analysis of the coadjoint orbit found in [1] leads to the upper bound on the Lyapunov exponent which is identical to the bound on chaos proven in [2]
One essential ingredient of the OTOC calculation is the infinitesimal transformation of the two point function of the matter field under the symmetry related to the soft mode [12, 28,29,30]
We have evaluated the Schwarzian soft mode contribution to the OTOC for the case of Diff(S1)/SL(2) and Diff(S1)/U(1)
Summary
In SYK-like model and 2D dilaton gravity, the reparametrization symmetry along the thermal circle is broken to SL(2, R), which leads to the pseudo-Goldstone boson described by the Schwarzian low energy effective action: S=. Such a pattern of symmetry breaking has been observed in the generalized SYK-like models in [21] and the 2D dilaton gravity with defects in [22, 23] For their low energy effective action, one can include Schwarzian derivatives and any possible terms which vanish under the corresponding U(1) mode. This is nothing but the time-translation generator L0 of a coadjoint orbit φ(τ ) ∈ Diff/H where H is the stabilizer subgroup [1, 24]. ∂ ∂τ can be obtained from the inner product of Ad∗φ−1(b(τ )) and a constant vector [1] This leads to the Schwarzian action in (2.2). 24 c b0 of the quadratic action which play a crucial role in the Lyapunov exponent of OTOC
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