A bstract This paper investigates the spectral form factor (SFF) of the quadratic R -para-particle Sachdev-Ye-Kitaev ( R -PSYK 2 ) model with various random matrix ensemble couplings. We generalize previous work on Gaussian Unitary Ensemble (GUE) couplings to all three Gaussian ensembles (GUE, GOE, GSE) and three circular ensembles (CUE, COE, CSE). Through analytical and numerical methods, we establish precise correspondences between GUE and CUE results, demonstrating their SFFs satisfy
KGUE(2t) K GUE 2 t ≈
KCUE(t) K CUE t in the time regime 1 ≪ t ≪ N . For the symplectic ensembles, we observe similar behavior with appropriate time rescaling, while we only provide the calculation method for the orthogonal ensembles.
Abstract We study the SYK 2 model of N Majorana fermions with random quadratic interactions through a detailed spectral analysis and by coupling the model to two- and four-point sources. In particular, we define the generalized spectral form factor (SFF) and level spacing distribution function by generalizing from the partition function to the generating function. For N = 2, we obtain an exact solution of the generalized SFF. It exhibits qualitatively similar behavior to the higher N case with a source term. The exact solution helps understand the behavior of the generalized SFF. We calculate the generalized level spacing distribution function and the mean value of the adjacent gap ratio defined by the generating function. For the SYK 2 model with a four-point source term, we find a Gaussian unitary ensemble behavior in the near-integrable region of the theory, which indicates a transition to chaos. This behavior is confirmed by the connected part of the generalized SFF with an unfolded spectrum. The departure from this Gaussian random matrix behavior as the relative strength of the source term is increased is consistent with the observation that the four-point source term alone, without the SYK 2 couplings turned on, exhibits an integrable spectrum from the SFF and level spacing distribution function in the large N limit.
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/t2 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.
The SYK model spectral form factor exhibits absence of information loss in the form of a ramp and a plateau that are typical of random matrix theory. In a large N collective fields description, the ramp was reproduced by Saad, Shenker and Stanford by a connected component of the analytically continued thermal partition function two point correlator. The existence of an \emph{off shell} connected component for the real time partition function is a consequence of disorder averaging over a statistical ensemble of two replica systems. In particular, the ramp arises from a family of vanishing action replica-non-diagonal conformal saddles. We provide a derivation of the soft modes effective action for fluctuations around the ramp conformal saddles, by adapting to the two replica system a method used by Kitaev and Suh in regular SYK.
The Yukawa-Sachdev-Ye-Kitaev (YSYK) model, a solvable model of non-Fermi liquids with a proposed cavity-QED realization, couples
N complex fermions to
M bosons of frequency
ω0 through random Yukawa couplings of strength
g, and its dynamics depends only on
R≡ω0/g2/3. For
R≪1 the bosons are slow and act as a static random background in which the fermions are free, while for
R≫1 they are only virtually excited and generate a random four-fermion interaction of rank
M, which is generically chaotic for
M≥2. Using exact diagonalization at infinite temperature, we follow the emergence of random-matrix statistics between these limits with unfolded spectral statistics and the Krylov complexity of the thermofield-double state. For complex couplings the level-spacing distribution evolves from Poisson to Gaussian-unitary-ensemble (GUE) statistics, the spectral form factor develops a correlation hole and a ramp, and the Krylov complexity evaluated on the unfolded spectrum develops a peak above its late-time plateau whose height reaches the GUE value. All diagnostics locate the onset around
R≈0.3, and adding boson modes moves it to smaller
R. This delineates the parameter regime in which a cavity-QED realization would display random-matrix spectral correlations. On the other hand, real couplings give the same crossover towards Gaussian-orthogonal-ensemble (GOE) statistics, and along an interpolation between the two the peak height follows the mean gap ratio, so the peak identifies the symmetry class as well as the onset of chaos. Without unfolding the peak stays about 20
% below its random-matrix value in both classes, which we trace to the non-semicircular density of states, and at larger
R the boson-number bands slow down the spreading of the state in Krylov space.
We use random matrix theory to explore late-time chaos in supersymmetric quantum mechanical systems. Motivated by the recent study of supersymmetric SYK models and their random matrix classification, we consider the Wishart-Laguerre unitary ensemble and compute the spectral form factors and frame potentials to quantify chaos and randomness. Compared to the Gaussian ensembles, we observe the absence of a dip regime in the form factor and a slower approach to Haar-random dynamics. Here we find agreement between our random matrix analysis and predictions from the supersymmetric SYK model, and discuss the implications for supersymmetric chaotic systems.