Onset of random-matrix statistics in the Yukawa-Sachdev-Ye-Kitaev model: Spectral correlations and Krylov complexity

Abstract

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 $ω_0$ through random Yukawa couplings of strength $g$, and its dynamics depends only on $R\equivω_0/g^{2/3}$. For $R\ll1$ the bosons are slow and act as a static random background in which the fermions are free, while for $R\gg1$ they are only virtually excited and generate a random four-fermion interaction of rank $M$, which is generically chaotic for $M\geq2$. 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 $R\approx0.3$, 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.