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تسجيل مستخدم جديدCombinatorial aspects of the Sachdev-Ye-Kitaev model
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The Sachdev-Ye-Kitaev (SYK) model is a model of $q$ interacting fermions whose large N limit is dominated by melonic graphs. In this review we first present a diagrammatic proof of that result by direct, combinatorial analysis of its Feynman graphs. Gross and Rosenhaus have then proposed a generalization of the SYK model which involves fermions with different flavors. In terms of Feynman graphs, these flavors can be seen as reminiscent of the colors used in random tensor theory. Applying modern tools from random tensors to such a colored SYK model, all leading and next-to-leading orders diagrams of the 2-point and 4-point functions in the large $N$ expansion can be identified. We then study the effect of non-Gaussian average over the random couplings in a complex, colored version of the SYK model. Using a Polchinski-like equation and random tensor Gaussian universality, we show that the effect of this non-Gaussian averaging leads to a modification of the variance of the Gaussian distribution of couplings at leading order in $N$. We then derive the form of the effective action to all orders.
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We describe numerous properties of the Sachdev-Ye-Kitaev model for complex fermions with $Ngg 1$ flavors and a global U(1) charge. We provide a general definition of the charge in the $(G,Sigma)$ formalism, and compute its universal relation to the i
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We present a detailed quantitative analysis of spectral correlations in the Sachdev-Ye-Kitaev (SYK) model. We find that the deviations from universal Random Matrix Theory (RMT) behavior are due to a small number of long-wavelength fluctuations from o
ne realization of the ensemble to the next one. These modes can be parameterized effectively in terms of Q-Hermite orthogonal polynomials, the main contribution being the scale fluctuations for which we give a simple estimate. Our numerical results for $N=32$ show that only the lowest eight polynomials are needed to eliminate the nonuniversal part of the spectral fluctuations. The covariance matrix of the coefficients of this expansion is obtained analytically from low-order double-trace moments. We evaluate the covariance matrix of the first six moments and find that it agrees with the numerics. We also analyze the spectral correlations using a nonlinear $sigma$-model, which is derived through a Fierz transformation, and evaluate the one and two-point spectral correlator to two-loop order. The wide correlator is given by the sum of the universal RMT result and corrections whose lowest-order term corresponds to scale fluctuations. However, the loop expansion of the $sigma$-model results in an ill-behaved expansion of the resolvent, and it gives universal RMT fluctuations not only for $q=4$ but also for the $q=2$ SYK model while the correct result in this case should have been Poisson statistics. We analyze the number variance and spectral form factor for $N=32$ and $q=4$ numerically. We show that the quadratic deviation of the number variance for large energies appears as a peak for small times in the spectral form factor. After eliminating the long-wavelength fluctuations, we find quantitative agreement with RMT up to an exponentially large number of level spacings or exponentially short times, respectively.
Various tensor models have been recently shown to have the same properties as the celebrated Sachdev-Ye-Kitaev (SYK) model. In this paper we study in detail the diagrammatics of two such SYK-like tensor models: the multi-orientable (MO) model which h
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