Abstract
We study a version of the 2-body Sachdev-Ye-Kitaev (SYK2) model whose complex fermions exhibit twisted boundary conditions on the thermal circle. As we show, this is physically equivalent to coupling the fermions to a 1-dimensional external gauge field A(t). In the latter formulation, the gauge field itself can be thought of as arising from a radial symmetry reduction of a (2 + 1)-dimensional Chern-Simons gauge field Aμ(t, x). Using the diagnostic tools of the out-of-time-order correlator (OTOC) and spectral form factor (SFF), which probe the sensitivity to initial conditions and the spectral statistics respectively, we give a detailed and pedagogical study of the integrable/chaotic properties of the model. We find that the twisting has no effect on the OTOCs and, by extension, the early-time chaos properties of the model. It does, however, have two notable effects on the spectral form factor; an enhancement of the early-time slope and the emergence of an explicit twisting-dependent timescale needed for the manifestation of zero modes. These zero modes are responsible for the late-time exponential ramp in the quadratic SYK model.
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Acknowledgments
We would like to thank Jonathan Shock for his collaboration during earlier stages of this work, as well as his invaluable Mathematica expertise throughout. We would also like to extend our thanks to the analymous referee whose detailed constructive comments and observations regarding the equivalence of the external gauge field to a twisting of the fermion boundary conditions led to a much better understanding of the problem. JM acknowledges support from the “Quantum Technologies for Sustainable Development” grant from the National Institute for Theoretical and Computational Sciences of South Africa (NITHECS) and a Simons Associateship at the Abdus Salam International Center for Theoretical Physics, Trieste where some of this work was carried out. RPS is supported by a graduate fellowship from the National Institute for Theoretical and Computational Sciences and by the Shuttleworth Postgraduate Scholarship Programme. H.J.R.vZ is supported by the “Quantum Technologies for Sustainable Devlopment” grant from the National Institute for Theoretical and Computational Sciences (NITHECS).
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Dedicated to the memory of Roman Jackiw: a pioneer of many dimensions, in few.
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Murugan, J., Slayen, R.P. & Van Zyl, H.J.R. A study of the SYK2 model with twisted boundary conditions. J. High Energ. Phys. 2024, 89 (2024). https://doi.org/10.1007/JHEP04(2024)089
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DOI: https://doi.org/10.1007/JHEP04(2024)089