On the relation between the magnitude and exponent of OTOCs
Abstract
We derive an identity relating the growth exponent of early-time OTOCs, the pre-exponential factor, and a third number called “branching time”. The latter is defined within the dynamical mean-field framework, namely, in terms of the retarded kernel. This identity can be used to calculate stringy effects in the SYK and similar models; we also explicitly define “strings” in this context. As another application, we consider an SYK chain. If the coupling strength βJ is above a certain threshold and nonlinear (in the magnitude of OTOCs) effects are ignored, the exponent in the butterfly wavefront is exactly 2π/β.
Keywords
AdS-CFT Correspondence 1/N Expansion Models of Quantum GravityNotes
Open Access
This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
References
- [1]A.I. Larkin and Y.N. Ovchinnikov, Quasiclassical method in the theory of superconductivity, Sov. Phys. JETP 28 (1969) 1200 [http://www.jetp.ac.ru/cgi-bin/e/index/e/28/6/p1200?a=list].
- [2]J. Maldacena, S.H. Shenker and D. Stanford, A bound on chaos, JHEP 08 (2016) 106 [arXiv:1503.01409] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
- [3]S.H. Shenker and D. Stanford, Stringy effects in scrambling, JHEP 05 (2015) 132 [arXiv:1412.6087] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
- [4]S. Sachdev and J. Ye, Gapless spin fluid ground state in a random, quantum Heisenberg magnet, Phys. Rev. Lett. 70 (1993) 3339 [cond-mat/9212030] [INSPIRE].
- [5]A. Kitaev, ‘ Hidden correlations in the Hawking radiation and thermal noise, talk at KITP, February (2015) [http://online.kitp.ucsb.edu/online/joint98/kitaev/].
- [6]A. Kitaev, A simple model of quantum holography, talks at KITP, April and May (2015). [http://online.kitp.ucsb.edu/online/entangled15/kitaev/] [http://online.kitp.ucsb.edu/online/entangled15/kitaev2/].
- [7]A. Kitaev and S.J. Suh, The soft mode in the Sachdev-Ye-Kitaev model and its gravity dual, JHEP 05 (2018) 183 [arXiv:1711.08467] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
- [8]J. Maldacena and D. Stanford, Remarks on the Sachdev-Ye-Kitaev model, Phys. Rev. D 94 (2016) 106002 [arXiv:1604.07818] [INSPIRE].MathSciNetGoogle Scholar
- [9]J. Murugan, D. Stanford and E. Witten, More on Supersymmetric and 2d Analogs of the SYK Model, JHEP 08 (2017) 146 [arXiv:1706.05362] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
- [10]I.L. Aleiner, L. Faoro and L.B. Ioffe, Microscopic model of quantum butterfly effect: out-of-time-order correlators and traveling combustion waves, Annals Phys. 375 (2016) 378 [arXiv:1609.01251] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
- [11]X.-L. Qi and A. Streicher, Quantum Epidemiology: Operator Growth, Thermal Effects and SYK, arXiv:1810.11958 [INSPIRE].
- [12]J. Maldacena, D. Stanford and Z. Yang, Conformal symmetry and its breaking in two dimensional Nearly Anti-de-Sitter space, PTEP 2016 (2016) 12C104 [arXiv:1606.01857] [INSPIRE].
- [13]Y. Gu, X.-L. Qi and D. Stanford, Local criticality, diffusion and chaos in generalized Sachdev-Ye-Kitaev models, JHEP 05 (2017) 125 [arXiv:1609.07832] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
- [14]M. Blake, Universal Diffusion in Incoherent Black Holes, Phys. Rev. D 94 (2016) 086014 [arXiv:1604.01754] [INSPIRE].MathSciNetGoogle Scholar
- [15]D. Stanford and D. Simmons-Duffin, private communication.Google Scholar
- [16]J. Suh, private communication.Google Scholar