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Title: Exact moments of the Sachdev-Ye-Kitaev model up to order 1/ N 2

We analytically evaluate the moments of the spectral density of the q-body Sachdev-Ye-Kitaev (SYK) model, and obtain order 1/ N 2 corrections for all moments, where N is the total number of Majorana fermions. To order 1/ N, moments are given by those of the weight function of the Q-Hermite polynomials. Representing Wick contractions by rooted chord diagrams, we show that the 1/ N 2 correction for each chord diagram is proportional to the number of triangular loops of the corresponding intersection graph, with an extra grading factor when q is odd. Therefore the problem of finding 1/ N 2 corrections is mapped to a triangle counting problem. Since the total number of triangles is a purely graph-theoretic property, we can compute them for the q = 1 and q = 2 SYK models, where the exact moments can be obtained analytically using other methods, and therefore we have solved the moment problem for any q to 1/ N 2 accuracy. The moments are then used to obtain the spectral density of the SYK model to order 1/ N 2. We also obtain an exact analytical result for all contraction diagrams contributing to the moments, which can be evaluated upmore » to eighth order. This shows that the Q-Hermite approximation is accurate even for small values of N.« less
Authors:
 [1] ;  [2] ; ORCiD logo [2]
  1. Shanghai Jiao Tong Univ., Shanghai (China)
  2. Stony Brook Univ., Stony Brook, NY (United States)
Publication Date:
Grant/Contract Number:
FG02-88ER40388
Type:
Accepted Manuscript
Journal Name:
Journal of High Energy Physics (Online)
Additional Journal Information:
Journal Name: Journal of High Energy Physics (Online); Journal Volume: 2018; Journal Issue: 4; Journal ID: ISSN 1029-8479
Publisher:
Springer Berlin
Research Org:
The State Univ. of New York, Stony Brook, NY (United States)
Sponsoring Org:
USDOE
Country of Publication:
United States
Language:
English
Subject:
72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; 1/N Expansion; Matrix Models; Random Systems
OSTI Identifier:
1505592

García-García, Antonio M., Jia, Yiyang, and Verbaarschot, Jacobus J. M.. Exact moments of the Sachdev-Ye-Kitaev model up to order 1/N2. United States: N. p., Web. doi:10.1007/jhep04(2018)146.
García-García, Antonio M., Jia, Yiyang, & Verbaarschot, Jacobus J. M.. Exact moments of the Sachdev-Ye-Kitaev model up to order 1/N2. United States. doi:10.1007/jhep04(2018)146.
García-García, Antonio M., Jia, Yiyang, and Verbaarschot, Jacobus J. M.. 2018. "Exact moments of the Sachdev-Ye-Kitaev model up to order 1/N2". United States. doi:10.1007/jhep04(2018)146. https://www.osti.gov/servlets/purl/1505592.
@article{osti_1505592,
title = {Exact moments of the Sachdev-Ye-Kitaev model up to order 1/N2},
author = {García-García, Antonio M. and Jia, Yiyang and Verbaarschot, Jacobus J. M.},
abstractNote = {We analytically evaluate the moments of the spectral density of the q-body Sachdev-Ye-Kitaev (SYK) model, and obtain order 1/N2 corrections for all moments, where N is the total number of Majorana fermions. To order 1/N, moments are given by those of the weight function of the Q-Hermite polynomials. Representing Wick contractions by rooted chord diagrams, we show that the 1/N2 correction for each chord diagram is proportional to the number of triangular loops of the corresponding intersection graph, with an extra grading factor when q is odd. Therefore the problem of finding 1/N2 corrections is mapped to a triangle counting problem. Since the total number of triangles is a purely graph-theoretic property, we can compute them for the q = 1 and q = 2 SYK models, where the exact moments can be obtained analytically using other methods, and therefore we have solved the moment problem for any q to 1/N2 accuracy. The moments are then used to obtain the spectral density of the SYK model to order 1/N2. We also obtain an exact analytical result for all contraction diagrams contributing to the moments, which can be evaluated up to eighth order. This shows that the Q-Hermite approximation is accurate even for small values of N.},
doi = {10.1007/jhep04(2018)146},
journal = {Journal of High Energy Physics (Online)},
number = 4,
volume = 2018,
place = {United States},
year = {2018},
month = {4}
}