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arXiv:1701.06593 · Physical review. D/Physical review. D.

Analytical spectral density of the Sachdev-Ye-Kitaev model at finite N

Antonio M. Garcı́a-Garcı́a, J. J. M. Verbaarschot

Published Sep 11, 2017• 172 citations
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Abstract

We derive an approximate analytical formula for the spectral density of the $q$-body Sachdev-Ye-Kitaev (SYK) model obtained by summing a class of diagrams representing leading intersecting contractions. This expression agrees with that of $Q$-Hermite polynomials, with $Q$ a nontrivial function of $q\ensuremath{\ge}2$ and the number of Majorana fermions $N$. Numerical results, obtained by exact diagonalization, are in excellent agreement with this approximate analytical spectral density even for relatively small $N\ensuremath{\sim}8$. For $N\ensuremath{\gg}1$ and not close to the edge of the spectrum, we find that the approximate analytical spectral density simplifies to ${\ensuremath{\rho}}_{\mathrm{asym}}(E)=\mathrm{exp}[2{\mathrm{arcsin}}^{2}(E/{E}_{0})/\mathrm{log}\ensuremath{\eta}]$, where $\ensuremath{\eta}(N,q)$ is the suppression factor of the contribution of intersecting Wick contractions relative to nested contractions and ${E}_{0}$ is the ground-state energy per particle. This spectral density reproduces the known result for the free energy in the large-$q$ and large-$N$ limit at arbitrary values of the temperature. In the infrared region, where the SYK model is believed to have a gravity dual, the analytical spectral density is given by $\ensuremath{\rho}(E)\ensuremath{\sim}\mathrm{sinh}[2\ensuremath{\pi}\sqrt{2}\sqrt{(1\ensuremath{-}E/{E}_{0})/(\ensuremath{-}\mathrm{log}\ensuremath{\eta})}]$. 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 modeling certain aspects of the nuclear dynamics.

DOI
10.1103/physrevd.96.066012

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We derive an approximate analytical formula for the spectral density of the qqq-body Sachdev-Ye-Kitaev (SYK) model obtained by summing a class of diagrams representing leading intersecting contractions. This expression agrees with that of QQQ-Hermite polynomials, with QQQ a nontrivial function of q\ensuremath≥2q\ensuremath{\ge}2q\ensuremath≥2 and the number of Majorana fermions NNN. Numerical results, obtained by exact diagonalization, are in excellent agreement with this approximate analytical spectral density even for relatively small N\ensuremath∼8N\ensuremath{\sim}8N\ensuremath∼8. For N\ensuremath≫1N\ensuremath{\gg}1N\ensuremath≫1 and not close to the edge of the spectrum, we find that the approximate analytical spectral density simplifies to \ensuremathρasym(E)=exp[2arcsin2(E/E0)/log\ensuremathη]{\ensuremath{\rho}}_{\mathrm{asym}}(E)=\mathrm{exp}[2{\mathrm{arcsin}}^{2}(E/{E}_{0})/\mathrm{log}\ensuremath{\eta}]\ensuremathρasym​(E)=exp[2arcsin2(E/E0​)/log\ensuremathη], where \ensuremathη(N,q)\ensuremath{\eta}(N,q)\ensuremathη(N,q) is the suppression factor of the contribution of intersecting Wick contractions relative to nested contractions and E0{E}_{0}E0​ is the ground-state energy per particle. This spectral density reproduces the known result for the free energy in the large-qqq and large-NNN limit at arbitrary values of the temperature. In the infrared region, where the SYK model is believed to have a gravity dual, the analytical spectral density is given by \ensuremathρ(E)\ensuremath∼sinh[2\ensuremathπ2(1\ensuremath−E/E0)/(\ensuremath−log\ensuremathη)]\ensuremath{\rho}(E)\ensuremath{\sim}\mathrm{sinh}[2\ensuremath{\pi}\sqrt{2}\sqrt{(1\ensuremath{-}E/{E}_{0})/(\ensuremath{-}\mathrm{log}\ensuremath{\eta})}]\ensuremathρ(E)\ensuremath∼sinh[2\ensuremathπ2​(1\ensuremath−E/E0​)/(\ensuremath−log\ensuremathη)​]. 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 modeling certain aspects of the nuclear dynamics.
Abstract 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 up to eighth order. This shows that the Q-Hermite approximation is accurate even for small values of N.
We study the Wishart-Sachdev-Ye-Kitaev (WSYK) model consisting of two q^\hat{q}q^​-body Sachdev-Ye-Kitaev (SYK) models with general complex couplings, one the complex conjugate of the other, living in off-diagonal blocks of a larger WSYK Hamiltonian. In the limit of large number NNN of Majoranas and q^∝N\hat{q} \propto \sqrt Nq^​∝N​, we employ diagrammatic and combinatorial techniques to show that the spectral density of the model can be approximated by the weight function of the Al-Salam-Chihara Q-Laguerre polynomials. For q^=2,3,4\hat{q} = 2, 3, 4q^​=2,3,4, the analytical spectral density, with Q=Q(q^,N)Q=Q(\hat{q},N)Q=Q(q^​,N) computed analytically, agrees well with numerical results obtained by exact diagonalization. The spectrum is positive with a hard edge at zero energy and, for odd q^\hat{q}q^​, low-energy excitations grow as a stretched exponential, though the functional form is different from that of the supersymmetric SYK model. The microscopic spectral density and level statistics close to the hard edge are very well approximated by that of an ensemble of random matrices belonging to non-standard universality classes, which indicates quantum chaotic dynamics.
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.
We study the Wishart-Sachdev-Ye-Kitaev (WSYK) model consisting of two $\stackrel{^}{q}$-body Sachdev-Ye-Kitaev (SYK) models with general complex couplings, one the Hermitian conjugate of the other, living in off-diagonal blocks of a larger WSYK Hamiltonian. The spectrum is positive with a hard edge at zero energy. We employ diagrammatic and combinatorial techniques to compute analytically the low-order moments of the Hamiltonian. In the limit of large number NNN of Majoranas, we have found striking similarities with the moments of the weight function of the Al-Salam-Chihara QQQ-Laguerre polynomials. For $\stackrel{^}{q}=3$, 4, the QQQ-Laguerre prediction, with $Q=Q(\stackrel{^}{q},N)$ also computed analytically, agrees well with exact diagonalization results for $30<N\ensuremath{\le}34$ while we observe some deviations for $\stackrel{^}{q}=2$. The most salient feature of the spectral density is that, for odd $\stackrel{^}{q}$, low-energy excitations grow as a stretched exponential, with a functional form different from that of the supersymmetric SYK model. For $\stackrel{^}{q}=4$, a detailed analysis of level statistics reveals quantum chaotic dynamics even for time scales substantially shorter than the Heisenberg time. More specifically, the spacing ratios in the bulk of the spectrum and the microscopic spectral density and the number variance close to the hard edge are very well approximated by that of an ensemble of random matrices that, depending on NNN, belong to the chiral or superconducting universality classes. In particular, we report the first realization of level statistics belonging to the chiral Gaussian Unitary Ensemble universality class, which completes the tenfold-way classification in the SYK model.
We investigate the supersymmetric Sachdev-Ye-Kitaev (SYK) model, NNN Majorana fermions with infinite range interactions in 0+10+10+1 dimensions. We have found that, close to the ground state $E\ensuremath{\approx}0$, discrete symmetries alter qualitatively the spectral properties with respect to the non-supersymmetric SYK model. The average spectral density at finite NNN, which we compute analytically and numerically, grows exponentially with NNN for $E\ensuremath{\approx}0$. However the chiral condensate, which is normalized with respect the total number of eigenvalues, vanishes in the thermodynamic limit. Slightly above $E\ensuremath{\approx}0$, the spectral density grows exponentially with the energy. Deep in the quantum regime, corresponding to the first O(N)O(N)O(N) eigenvalues, the average spectral density is universal and well described by random matrix ensembles with chiral and superconducting discrete symmetries. The dynamics for $E\ensuremath{\approx}0$ is investigated by level fluctuations. Also in this case we find excellent agreement with the prediction of chiral and superconducting random matrix ensembles for eigenvalue separations smaller than the Thouless energy, which seems to scale linearly with NNN. Deviations beyond the Thouless energy, which describes how ergodicity is approached, are universally characterized by a quadratic growth of the number variance. In the time domain, we have found analytically that the spectral form factor g(t)g(t)g(t), obtained from the connected two-level correlation function of the unfolded spectrum, decays as 1/t21/{t}^{2}1/t2 for times shorter but comparable to the Thouless time with g(0)g(0)g(0) related to the coefficient of the quadratic growth of the number variance. Our results provide further support that quantum black holes are ergodic and therefore can be classified by random matrix theory.
Abstract 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 (of the order of the number of Majorana fermions N) 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 due to scale fluctuations for which we give a simple analytical 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 can be 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 in terms of a nonlinear σ-model, which is derived through a Fierz transformation, and evaluate the one and two-point spectral correlation functions 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 σ-model results in an ill-behaved expansion of the resolvent, and it gives universal RMT fluctuations not only for q = 4 or higher even q-body interactions, but also for the q = 2 SYK model albeit with a much smaller Thouless energy while the correct result in this case should have been Poisson statistics. In our numerical studies we analyze the number variance and spectral form factor for N = 32 and q = 4. 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 for the number variance or exponentially short times in the case of the spectral form factor.