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arXiv:1801.01071 · Physical review. D/Physical review. D.

Universality and Thouless energy in the supersymmetric Sachdev-Ye-Kitaev model

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

Published May 4, 2018• 62 citations
arXivHTMLPDF

Abstract

We investigate the supersymmetric Sachdev-Ye-Kitaev (SYK) model, $N$ Majorana fermions with infinite range interactions in $0+1$ 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 $N$, which we compute analytically and numerically, grows exponentially with $N$ 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 $O(N)$ 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 $N$. 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 $g(t)$, obtained from the connected two-level correlation function of the unfolded spectrum, decays as $1/{t}^{2}$ for times shorter but comparable to the Thouless time with $g(0)$ 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.

DOI
10.1103/physrevd.97.106003

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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≈0E\ensuremath{\approx}0E\ensuremath≈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≈0E\ensuremath{\approx}0E\ensuremath≈0. However the chiral condensate, which is normalized with respect the total number of eigenvalues, vanishes in the thermodynamic limit. Slightly above E\ensuremath≈0E\ensuremath{\approx}0E\ensuremath≈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≈0E\ensuremath{\approx}0E\ensuremath≈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.
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.
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.
We study spectral and thermodynamic properties of the Sachdev-Ye-Kitaev model, a variant of the kkk-body embedded random ensembles studied for several decades in the context of nuclear physics and quantum chaos. We show analytically that the fourth- and sixth-order energy cumulants vanish in the limit of a large number of particles $N\ensuremath{\rightarrow}\ensuremath{\infty}$, which is consistent with a Gaussian spectral density. However, for finite NNN, the tail of the average spectral density is well approximated by a semicircle law. The specific heat coefficient, determined numerically from the low-temperature behavior of the partition function, is consistent with the value obtained by previous analytical calculations. For energy scales of the order of the mean level spacing we show that level statistics are well described by random matrix theory. Due to the underlying Clifford algebra of the model, the universality class of the spectral correlations depends on NNN. For larger energy separations we identify an energy scale that grows with NNN, reminiscent of the Thouless energy in mesoscopic physics, where deviations from random matrix theory are observed. Our results are a further confirmation that the Sachdev-Ye-Kitaev model is quantum chaotic for all time scales. According to recent claims in the literature, this is an expected feature in field theories with a gravity dual.
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 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 $q\ensuremath{\ge}2$ 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 $N\ensuremath{\sim}8$. 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 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 $\ensuremath{\rho}(E)\ensuremath{\sim}\mathrm{sinh}[2\ensuremath{\pi}\sqrt{2}\sqrt{(1\ensuremath{-}E/{E}_{0})/(\ensuremath{-}\mathrm{log}\ensuremath{\eta})}]$. 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.
We study a sparse Sachdev-Ye-Kitaev (SYK) model with NNN Majoranas where only $\ensuremath{\sim}kN$ independent matrix elements are nonzero. We identify a minimum $k\ensuremath{\gtrsim}1$ for quantum chaos to occur by a level statistics analysis. The spectral density in this region, and for a larger kkk, is still given by the Schwarzian prediction of the dense SYK model, though with renormalized parameters. Similar results are obtained for a beyond linear scaling with NNN of the number of nonzero matrix elements. This is a strong indication that this is the minimum connectivity for the sparse SYK model to still have a quantum gravity dual. We also find an intriguing exact relation between the leading correction to moments of the spectral density due to sparsity and the leading 1/d1/d1/d correction of Parisi's U(1) lattice gauge theory in a ddd-dimensional hypercube. In the $k\ensuremath{\rightarrow}1$ limit, different disorder realizations of the sparse SYK model show emergent random matrix statistics that for fixed NNN can be in any universality class of the tenfold way. The agreement with random matrix statistics is restricted to short-range correlations, no more than a few level spacings, in particular in the tail of the spectrum. In addition, emergent discrete global symmetries in most of the disorder realizations for kkk slightly below one give rise to 2m{2}^{m}2m-fold degenerate spectra, with mmm being a positive integer. For k=3/4k=3/4k=3/4, we observe a large number of such emergent global symmetries with a maximum 28{2}^{8}28-fold degenerate spectra for N=26N=26N=26.