We introduce two disorder-free variants of the Sachdev-Ye-Kitaev (SYK) model, demonstrate their integrability, and study their static and dynamical properties. Unlike diagrammatic techniques, the integrability of these models allows us to obtain dynamical correlation functions even when the number of Majorana fermions is finite. From the solutions, we find that out-of-time-order correlators (OTOCs) in these models exhibit exponential growth at early times, resembling that of many-body quantum chaotic systems, such as those with disorder or external kick terms, despite their large N behavior differing from that of typical chaotic systems. Conversely, our analysis shows no evidence of random-matrix behavior in level spacing statistics or the spectral form factor. Our findings illustrate that the clean versions of the SYK models represent simple but nontrivial examples of disorder-free quantum many-body systems displaying chaoslike behavior of OTOCs.
We introduce two disorder-free variants of the Sachdev-Ye-Kitaev (SYK) model, demonstrate their integrability, and study their static and dynamical properties. Unlike diagrammatic techniques, the integrability of these models allows us to obtain dynamical correlation functions even when the number of Majorana fermions is finite. From the solutions, we find that out-of-time-order correlators (OTOCs) in these models exhibit exponential growth at early times, resembling that of many-body quantum chaotic systems, such as those with disorder or external kick terms, despite their large N behavior differing from that of typical chaotic systems. Conversely, our analysis shows no evidence of random-matrix behavior in level spacing statistics or the spectral form factor. Our findings illustrate that the clean versions of the SYK models represent simple but nontrivial examples of disorder-free quantum many-body systems displaying chaoslike behavior of OTOCs.
The Sachdev-Ye-Kitaev (SYK) model attracts attention in the context of information scrambling, which represents delocalization of quantum information and is quantified by the out-of-time-ordered correlators (OTOC). The SYK model contains N fermions with disordered and four-body interactions. Here, we introduce a variant of the SYK model, which we refer to as the Wishart SYK model. We investigate the Wishart SYK model for complex fermions and that for hard-core bosons. We show that the ground state of the Wishart SYK model is massively degenerate and the residual entropy is extensive, and that the Wishart SYK model for complex fermions is integrable. In addition, we numerically investigate the OTOC and level statistics of the SYK models. At late times, the OTOC of the fermionic Wishart SYK model exhibits large temporal fluctuations, in contrast with smooth scrambling in the original SYK model. We argue that the large temporal fluctuations of the OTOC are a consequence of a small effective dimension of the initial state. We also show that the level statistics of the fermionic Wishart SYK model is in agreement with the Poisson distribution, while the bosonic Wishart SYK model obeys the gaussian unitary ensemble (GUE) or the gaussian orthogonal ensemble (GOE) distribution.
The complex Sachdev-Ye-Kitaev (SYK) model, featuring fermions with all-to-all interactions, serves as a dual paradigm for understanding non-Fermi liquid behavior and the holographic nature of charged black holes. Two defining characteristics of the standard SYK model are its maximal chaos (Lyapunov exponent λL=2πT at temperature T), and its finite zero-temperature residual entropy. While previous studies have largely focused on couplings drawn from a zero-mean Gaussian distribution, we investigate a generalized model with a finite mean-to-standard-deviation ratio, g≡J0/δJ of the coupling distribution in order to get deeper insight into the evolution of chaos. We find that increasing g yields the following effects: (i) The system remains a fast scrambler with λL=AT, but with a suppressed coefficient A<2π. (ii) In the limit g→∞, out-of-time-ordered correlators (OTOCs) no longer exhibit exponential growth with λL≃0. (iii) The spectral correlations indicative of late-time chaos maintain Wigner-Dyson level spacing statistics for all values of g. (iv) The system preserves a finite residual entropy, albeit with reduced magnitude, for all g values. We conclude that in this generalized SYK model, there is a chaotic to non-chaotic crossover. Moreover different measures of chaos decouple, demonstrating that the presence of finite residual entropy does not strictly imply maximal chaos.
We study the Wishart-Sachdev-Ye-Kitaev (WSYK) model consisting of two 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 N of Majoranas and q^∝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, the analytical spectral density, with 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^, 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 describe the dynamics of many-body quantum chaotic systems at all time scales by studying the Green's and out-of-time order correlation (OTOC) functions of the four-body, N-Majorana Sachdev-Ye-Kitaev model. By combining the scramblon formalism and random-matrix-theory techniques, we obtain analytical expressions for these functions at all times. The early exponential growth of the OTOC is followed by an exponential decay at a rate governed by that of the Green's function (the real part of the leading complex Ruelle-Pollicott resonances). For late times that scale exponentially with N, both functions have a dip-ramp-plateau pattern for Nmod8=2,6 that deviates substantially from the ergodic prediction due to local-in-energy correlations of matrix elements and eigenvalues, even after the Heisenberg time.
We study the non-stabilizerness or quantum magic of the Sachdev-Ye-Kitaev ( \mathrm{SYK} S Y K ) model, a prototype example of maximally chaotic quantum matter. We show that the Majorana spectrum of its ground state, encoding the spreading of the state in the Majorana basis, displays a Gaussian distribution as expected for chaotic quantum many-body systems. We compare our results with the case of the \mathrm{SYK_2} S Y K 2 model, describing non-chaotic random free fermions, and show that the Majorana spectrum is qualitatively different in the two cases, featuring an exponential Laplace distribution for the \mathrm{SYK_2} S Y K 2 model rather than a Gaussian. From the spectrum we extract the Stabilizer Renyi Entropy (SRE) and show that for both models it displays a linear scaling with system size, with a prefactor that is larger for the SYK model, which has therefore higher magic. Finally, we discuss the spreading of quantun magic under unitary dynamics, as described by the evolution of the Majorana spectrum and the Stabilizer Renyi Entropy starting from a stabilizer state. We show that the SRE for the \mathrm{SYK_2} S Y K 2 model equilibrates rapidly, but that in the steady-state the interacting chaotic SYK model has more magic than the simple \mathrm{SYK_2} S Y K 2 . Our results suggest that the Majorana spectrum is qualitatively distinct in chaotic and non-chaotic many-body systems.
Abstract We study the generalization of the Sachdev-Ye-Kitaev (SYK) model to a 1 + 1 dimensional chiral SYK model of N flavors of right-moving chiral Majorana fermions with all-to-all random 4-fermion interactions. The interactions in this model are exactly marginal, leading to an exact scaling symmetry. We show the Schwinger-Dyson equation of this model in the large N limit is exactly solvable. In addition, we show this model is integrable for small N ≤ 6 by bosonization. Surprisingly, the two point function in the large N limit has exactly the same form as that for N = 4, although the four point functions of the two cases are quite different. The ground state entropy in the large N limit is the same as that of N free chiral Majorana fermions, leading to a zero ground state entropy density. The OTOC of the model in the large N limit exhibits a non-trivial spacetime structure reminscent of that found by Gu and Kitaev [1] for generic SYK-like models. Specifically we find a Lyapunov regime inside an asymmetric butterfly cone, which are signatures of quantum chaos, and that the maximal velocity dependent Lyapunov exponent approaches the chaos bound 2π/β as the interaction strength approaches its physical upper bound. Finally, the model is integrable for (at least) N ≤ 6 but chaotic in the large N limit, leading us to conjecture that there is a transition from integrability to chaos as N increases past a critical value.