On the relation between the magnitude and exponent of OTOCs

146

A simple probe of chaos and operator growth in many-body quantum systems is the out of time ordered four point function. In a large class of local systems, the effects of chaos in this correlator build up exponentially fast inside the so called butterfly cone. It has been previously observed that the growth of these effects is organized along rays and can be characterized by a velocity dependent Lyapunov exponent, λ(v). We show that this exponent is bounded inside the butterfly cone as λ(v) ≤ 2πT (1 − |v|/v B ), where T is the temperature and v B is the butterfly speed. This result generalizes the chaos bound of Maldacena, Shenker and Stanford. We study λ(v) in some examples such as two dimensional SYK models and holographic gauge theories, and observe that in these systems the bound gets saturated at some critical velocity v * < v B . In this sense, boosting a system enhances chaos. We discuss the connection to conformal Regge theory, where λ(v) is related to the spin of the leading large N Regge trajectory, and controls the four point function in an interpolating regime between the Regge and the light cone limit. Finally, we comment on the generalization of the chaos bound to boosted and rotating ensembles and clarify some recent results on this in the literature.

“…After this, the pole actually gives a larger contribution than the saddle. This mechanism was understood before us; very similar discussions can be found in [3,9]. Figure 5.…”

Section: Syk Chainsupporting

confidence: 76%

mentioning

confidence: 51%

Section: Introduction and Summary Of Resultsmentioning

confidence: 79%

Section: Ladder Identity Of Gu and Kitaevmentioning

confidence: 86%

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Chaos in the butterfly cone

Mezei

Sárosi

2020

146

“…For the large velocity region near the right butterfly cone edge, the 1/N piece of OTOC grows as exp 2π β (t − u −1 + x) , where the exponent in the t direction saturates the chaos bound 2π/β. A similar saturation of the chaos bound near the butterfly edge was observed in [9], and recently explained on generic grounds by [1] using the ladder identity. However, we note that this is not the case near the left butterfly edge of our model, where the exponent in the t direction is −2π/β, i.e., opposite to the chaos bound value.…”

Section: Introductionsupporting

confidence: 76%

The chiral SYK model

Lian

Sondhi

Yang

2019

We study the generalization of the Sachdev-Ye-Kitaev (SYK) model to a 1 + 1 dimensional chiral SYK model of N flavors of right-moving chiral Majorana fermions with all-to-all random 4-fermion interactions. The interactions in this model are exactly marginal, leading to an exact scaling symmetry. We show the Schwinger-Dyson equation of this model in the large N limit is exactly solvable. In addition, we show this model is integrable for small N ≤ 6 by bosonization. Surprisingly, the two point function in the large N limit has exactly the same form as that for N = 4, although the four point functions of the two cases are quite different. The ground state entropy in the large N limit is the same as that of N free chiral Majorana fermions, leading to a zero ground state entropy density. The OTOC of the model in the large N limit exhibits a non-trivial spacetime structure reminscent of that found by Gu and Kitaev [1] for generic SYK-like models. Specifically we find a Lyapunov regime inside an asymmetric butterfly cone, which are signatures of quantum chaos, and that the maximal velocity dependent Lyapunov exponent approaches the chaos bound 2π/β as the interaction strength approaches its physical upper bound. Finally, the model is integrable for (at least) N ≤ 6 but chaotic in the large N limit, leading us to conjecture that there is a transition from integrability to chaos as N increases past a critical value. B Saddle Point Approximation for the OTOC 422 Here we write the disorder average over J ijkl as the annealed disorder Z = Z J , however, for large N it is equal to the quenched disorder Z = e log Z J to the leading order [5]. This is because in the replica treatment Z n J is equal to Z n J up to order 1/N 3 , which leads to log Z J = limn→0 Z n J −1 n = limn→0Z n J −1 n = log Z J .

Pole skipping away from maximal chaos

Choi

Sárosi

2021

Pole skipping is a recently discovered subtle effect in the thermal energy density retarded two point function at a special point in the complex (ω, p) planes. We propose that pole skipping is determined by the stress tensor contribution to many-body chaos, and the special point is at (ω, p)p.s. = $$ i{\lambda}^{(T)}\left(1,1/{u}_B^{(T)}\right) $$ i λ T 1 1 / u B T , where λ(T) = 2π/β and $$ {u}_B^{(T)} $$ u B T are the stress tensor contributions to the Lyapunov exponent and the butterfly velocity respectively. While this proposal is consistent with previous studies conducted for maximally chaotic theories, where the stress tensor dominates chaos, it clarifies that one cannot use pole skipping to extract the Lyapunov exponent of a theory, which obeys λ ≤ λ(T). On the other hand, in a large class of strongly coupled but non-maximally chaotic theories $$ {u}_B^{(T)} $$ u B T is the true butterfly velocity and we conjecture that uB ≤ $$ {u}_B^{(T)} $$ u B T is a universal bound. While it remains a challenge to explain pole skipping in a general framework, we provide a stringent test of our proposal in the large-q limit of the SYK chain, where we determine λ, uB, and the energy density two point function in closed form for all values of the coupling, interpolating between the free and maximally chaotic limits. Since such an explicit expression for a thermal correlator is one of a kind, we take the opportunity to analyze many of its properties: the coupling dependence of the diffusion constant, the dispersion relations of poles, and the convergence properties of all order hydrodynamics.