What is the Simplest Linear Ramp?
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...
| Main Authors: | , , , |
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| Format: | Article |
| Language: | English |
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SpringerOpen
2024-01-01
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| Series: | Journal of High Energy Physics |
| Subjects: | |
| Online Access: | https://doi.org/10.1007/JHEP01(2024)172 |
| _version_ | 1797341331212730368 |
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| author | Suman Das Sumit K. Garg Chethan Krishnan Arnab Kundu |
| author_facet | Suman Das Sumit K. Garg Chethan Krishnan Arnab Kundu |
| author_sort | Suman Das |
| collection | DOAJ |
| description | 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 E n ~ 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. |
| first_indexed | 2024-03-08T10:16:27Z |
| format | Article |
| id | doaj.art-1e0ba4a0c21e441d8ccda1d37e4f53e5 |
| institution | Directory Open Access Journal |
| issn | 1029-8479 |
| language | English |
| last_indexed | 2024-03-08T10:16:27Z |
| publishDate | 2024-01-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Journal of High Energy Physics |
| spelling | doaj.art-1e0ba4a0c21e441d8ccda1d37e4f53e52024-01-28T12:24:56ZengSpringerOpenJournal of High Energy Physics1029-84792024-01-012024111510.1007/JHEP01(2024)172What is the Simplest Linear Ramp?Suman Das0Sumit K. Garg1Chethan Krishnan2Arnab Kundu3Theory Division, Saha Institute of Nuclear Physics, A CI of Homi Bhabha National InstituteManipal Centre for Natural Sciences, Manipal Academy of Higher EducationCenter for High Energy Physics, Indian Institute of ScienceTheory Division, Saha Institute of Nuclear Physics, A CI of Homi Bhabha National InstituteAbstract 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 E n ~ 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.https://doi.org/10.1007/JHEP01(2024)172AdS-CFT CorrespondenceBlack HolesModels of Quantum GravityRandom Systems |
| spellingShingle | Suman Das Sumit K. Garg Chethan Krishnan Arnab Kundu What is the Simplest Linear Ramp? Journal of High Energy Physics AdS-CFT Correspondence Black Holes Models of Quantum Gravity Random Systems |
| title | What is the Simplest Linear Ramp? |
| title_full | What is the Simplest Linear Ramp? |
| title_fullStr | What is the Simplest Linear Ramp? |
| title_full_unstemmed | What is the Simplest Linear Ramp? |
| title_short | What is the Simplest Linear Ramp? |
| title_sort | what is the simplest linear ramp |
| topic | AdS-CFT Correspondence Black Holes Models of Quantum Gravity Random Systems |
| url | https://doi.org/10.1007/JHEP01(2024)172 |
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