Abstract
We discuss conditions under which a deterministic sequence of real numbers, interpreted as the set of eigenvalues of a Hamiltonian, can exhibit features usually associated to random matrix spectra. A key diagnostic is the spectral form factor (SFF) — a linear ramp in the SFF is often viewed as a signature of random matrix behavior. Based on various explicit examples, we observe conditions for linear and power law ramps to arise in deterministic spectra. We note that a very simple spectrum with a linear ramp is En ~ log n. Despite the presence of ramps, these sequences do not exhibit conventional level repulsion, demonstrating that the lore about their concurrence needs refinement. However, when a small noise correction is added to the spectrum, they lead to clear level repulsion as well as the (linear) ramp. We note some remarkable features of logarithmic spectra, apart from their linear ramps: they are closely related to normal modes of black hole stretched horizons, and their partition function with argument s = β + it is the Riemann zeta function ζ(s). An immediate consequence is that the spectral form factor is simply −ζ|(it)|2. Our observation that log spectra have a linear ramp, is closely related to the Lindelöf hypothesis on the growth of the zeta function. With elementary numerics, we check that the slope of a best fit line through |ζ(it)|2 on a log-log plot is indeed 1, to the fourth decimal. We also note that truncating the Riemann zeta function sum at a finite integer N causes the would-be-eternal ramp to end on a plateau.
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Acknowledgments
CK thanks Rajesh Gopakumar and Vyshnav Mohan for discussions/correspondence and the University of the Witwatersrand for hospitality during the final stages of this work. AK is partially supported by CEFIPRA 6304 — 3, DAE-BRNS 58/14/12/2021-BRNS and CRG/2021/004539 of Govt. of India. AK further thanks the warm hospitality of the Department of Theoretical Physics, CERN during the completion of this work. The work of SKG has been supported by SERB, DST, India through grant TAR/2023/000116.
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Das, S., Garg, S.K., Krishnan, C. et al. What is the Simplest Linear Ramp?. J. High Energ. Phys. 2024, 172 (2024). https://doi.org/10.1007/JHEP01(2024)172
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DOI: https://doi.org/10.1007/JHEP01(2024)172