We investigate the q = 2 SYK model with R -para-particles ( R − PSYK 2 ), analyzing its thermodynamics and spectral form factor (SFF) using random matrix theory. The Hamiltonian is quadratic, with coupling coefficients randomly drawn from the Gaussian unitary ensemble. The model displays self-averaging behavior and exhibits an exponential ramp in its SFF dynamics: K ( t ) ∼ e C 0 t . The growth rate C 0 tends toward either a constant or infinity in the N → ∞ limit, depending on specific statistics of the model. These results provide novel perspectives on quantum systems with unconventional statistics.
Spectral form factor (SFF), one of the key quantity from random matrix theory, serves as an important tool to probe universality in disordered quantum systems and quantum chaos. In this work, we present exact closed-form expressions for the second- and third-order SFFs in the circular unitary ensemble (CUE), valid for all real values of the time parameter, and analyze their asymptotic behavior in different regimes. In particular, for the second-order SFF, we derive an exact closed-form expression in terms of polygamma functions. In the limit of infinite matrix size, and when the time parameter is restricted to integer values, the second-order SFF reproduces the standard result established in earlier studies. When the time parameter is of order one relative to the matrix size, we demonstrate that the second-order SFF grows logarithmically with the ensemble dimension. For the third-order SFFs, a closed-form result in a special case is obtained by exploiting the translational invariance of CUE.
The spectral form factor (SFF) is a powerful diagnostic of random matrix behavior in quantum many-body systems. We introduce a family of random circuit ensembles whose SFFs can be computed . These ensembles describe the evolution of noninteracting fermions in the presence of correlated on-site potentials drawn from the eigenvalue distribution of a circular ensemble. For disorder parameters drawn from the circular unitary ensemble (CUE), we derive an exact closed form for the SFF, valid for any choice of system size L and integer time t . When the disorder is drawn from the circular orthogonal or symplectic ensembles (COE and CSE, respectively), we carry out the disorder averages analytically and reduce the computation of the SFF at integer times to a combinatorial problem amenable to transfer matrix methods. In each of these cases, the SFF grows exponentially in time, which we argue is a signature of random matrix universality at the single-particle level. Finally, we develop matchgate circuit representations of our circuit ensembles, enabling their experimental realization in quantum simulators.
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.
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 show the emergence of random matrix theory (RMT) spectral correlations in the chaotic phase of generic, periodically kicked, interacting, quantum many-body systems by analytically calculating the spectral form factor (SFF), K ( t ) , up to two leading orders in time, t . We explicitly consider the presence or absence of time-reversal ( T ) symmetry to investigate all three Dyson symmetry classes, namely, the circular orthogonal ensemble (COE) for T -invariant systems with T 2 = 1 , the circular unitary ensemble (CUE) in the absence of T symmetry, and the circular symplectic ensemble (CSE) for T -invariant systems with T 2 = − 1 . Our derivation applies the random phase approximation to perform ensemble averages, and we give its justification for generic many-body interactions. We express the SFF as a sum of contributions of certain diagrams. While the number of diagrams that contribute to the SFF in the first and second orders in t for the COE and CUE classes is finite, the number of such diagrams is exponentially large in t for the CSE class. Thus, we have developed a new diagrammatic technique that uses reduced diagrams to incorporate contributions from many diagrams. For the COE class, we show that beyond Thouless time t * , the SFF takes the form K ( t ) ≃ 2 t − 2 t 2 / N up to second order in time, where N is the Hilbert space dimension. For the CUE class, we show that K ( t ) ≃ t beyond t * , and there is no universal term in the second order, unlike the COE case. For the CSE class, we show that K ( t ) ≃ 2 t + 2 t 2 / N up to two orders in time beyond t * . The above findings of the SFF up to second order in time are in agreement with the RMT predictions for these classes. In all three cases, the scaling of t * with system size L is determined by the eigenvalues of a doubly stochastic matrix M . For strongly interacting fermionic chains, M is S U ( 2 ) invariant in all three cases, leading to t * ∝ L 2 in the presence of U ( 1 ) symmetry. In the absence of U ( 1 ) symmetry, we find t *