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Register a new userDiagrammatics of the quartic $O(N)^3$-invariant Sachdev-Ye-Kitaev-like tensor model
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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 has an $U(N) times O(N) times U(N)$ symmetry and a quartic $O(N)^3$-invariant model whose interaction has the tetrahedral pattern. We show that the Feynman graphs of the MO model can be seen as the Feynman graphs of the $O(N)^3$-invariant model which have an orientable jacket. We then present a diagrammatic toolbox to analyze the $O(N)^3$-invariant graphs. This toolbox allows for a simple strategy to identify all the graphs of a given order in the $1/N$ expansion. We apply it to the next-to-next-to-leading and next-to-next-to-next-to-leading orders which are the graphs of degree $1$ and $3/2$ respectively.
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We study the double scaling limit of the $O(N)^3$-invariant tensor model, initially introduced in Carrozza and Tanasa, Lett. Math. Phys. (2016). This model has an interacting part containing two types of quartic invariants, the tetrahedric and the pillow one. For the 2-point function, we rewrite the sum over Feynman graphs at each order in the $1/N$ expansion as a emph{finite} sum, where the summand is a function of the generating series of melons and chains (a.k.a. ladders). The graphs which are the most singular in the continuum limit are characterized at each order in the $1/N$ expansion. This leads to a double scaling limit which picks up contributions from all orders in the $1/N$ expansion. In contrast with matrix models, but similarly to previous double scaling limits in tensor models, this double scaling limit is summable. The tools used in order to prove our results are combinatorial, namely a thorough diagrammatic analysis of Feynman graphs, as well as an analysis of the singularities of the relevant generating series.
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.
We show analytically that the spectral density of the $q$-body Sachdeev-Ye-Kitaev (SYK) model agrees with that of Q-Hermite polynomials with Q a non-trivial function of $q ge 2$ and the number of Majorana fermions $N gg 1$. Numerical results, obtained by exact diagonalization, are in excellent agreement with the analytical spectral density even for relatively small $N sim 8$. For $N gg 1$ and not close to the edge of the spectrum, we find the macroscopic spectral density simplifies to $rho(E) sim exp[2arcsin^2(E/E_0)/log eta]$, where $eta$ is the suppression factor of the contribution of intersecting Wick contractions relative to nested contractions. This spectral density reproduces the known result for the free energy in the large $q$ and $N$ limit. In the infrared region, where the SYK model is believed to have a gravity-dual, the spectral density is given by $rho(E) sim sinh[2pi sqrt 2 sqrt{(1-E/E_0)/(-log eta)}]$. It therefore has a square-root edge, as in random matrix ensembles, followed by an exponential growth, a distinctive feature of black holes and also of low energy nuclear excitations. Results for level-statistics in this region confirm the agreement with random matrix theory. Physically this is a signature that, for sufficiently long times, the SYK model and its gravity dual evolve to a fully ergodic state whose dynamics only depends on the global symmetry of the system. Our results strongly suggest that random matrix correlations are a universal feature of quantum black holes and that the SYK model, combined with holography, may be relevant to model certain aspects of the nuclear dynamics.
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 infrared asymmetry of the Green function. The same relation is obtained by a renormalization theory. The conserved charge contributes a compact scalar field to the effective action, from which we derive the many-body density of states and extract the charge compressibility. We compute the latter via three distinct numerical methods and obtain consistent results. Finally, we present a two dimensional bulk picture with free Dirac fermions for the zero temperature entropy.
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 one 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.
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