Abstract
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.
Highlights
The study of strongly interacting quantum many body systems has a long history, many aspects still remain poorly understood
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
Motivated by the combinatorial factors that enter in the scaled moments (2.9), we define the following object associated with each contraction diagram and with each intersection graph, contributing to the scaled 2V -th moment, ηG := (−1)Eq
Summary
The study of strongly interacting quantum many body systems has a long history, many aspects still remain poorly understood. In previous works [42, 43], two of us have studied both the thermodynamic and spectral properties of the SYK model for q > 2, and have clearly established that the short-range spectral correlations are given by random matrix theory which is a necessary ingredient for the model to be quantum chaotic and to have a gravity-dual with black hole solutions It was found, by an explicit analytical evaluation of the moments of the spectral density, that it grows exponentially for low energies, a typical feature of conformal field theories [44] and of gravity backgrounds with a field theory dual [45, 46].
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