In chaotic quantum systems, scrambling and dissipation are governed by distinct timescales. We show that their relative ordering in the Sachdev-Ye-Kitaev model depends on the number of degrees of freedom \(N\) and undergoes a crossover in the large-\(N\) limit. In the conformal limit of \(\mathrm{SYK}_q\), the two-point function half-life \(t_{\rm diss} \propto \beta\) is \(N\)-independent, while the scrambling time grows as \(\log N\). Equating them yields a crossover scale \(N^*(q) = \exp\left(2\,\mathrm{acosh}(2^{q/2})\right)\), which grows exponentially with \(q\): \(N^*\approx 62\) for \(q=4\), \(\approx 254\) for \(q=6\), and \(\approx 1022\) for \(q=8\). Below \(N^*\), the OTOC decays before the two-point function; above \(N^*\), the ordering reverses. Using exact diagonalization at \(N=8\)-\(22\), we confirm three qualitative predictions of this formula: (i) the ratio \(r = t_{\rm diss}/t_{\rm OTOC}\) decreases with \(N\) for all \(q\); (ii) at fixed \(N\), \(r\) increases with \(q\), confirming that \(N^*\) grows with \(q\); (iii) the ordering is robust across temperatures \(\beta J = 2\)-\(20\), consistent with the \(\beta\)-independence of \(N^*\) in the conformal limit. The crossover separates a few-body regime, where multi-point correlations are more fragile, from a many-body regime, where single-particle coherence is lost first.
In chaotic quantum systems, scrambling and dissipation are governed by distinct timescales. We show that their relative ordering in the Sachdev-Ye-Kitaev model depends on the number of degrees of freedom (N) and undergoes a crossover in the large-(N) limit. In the conformal limit of (\mathrm{SYK}q), the two-point function half-life (t{\rm diss} \propto \beta) is (N)-independent, while the scrambling time grows as (\log N). Equating them yields a crossover scale (N^(q) = \exp\left(2,\mathrm{acosh}(2^{q/2})\right)), which grows exponentially with (q): (N^\approx 62) for (q=4), (\approx 254) for (q=6), and (\approx 1022) for (q=8). Below (N^), the OTOC decays before the two-point function; above (N^), the ordering reverses. Using exact diagonalization at (N=8)-(22), we confirm three qualitative predictions of this formula: (i) the ratio (r = t_{\rm diss}/t_{\rm OTOC}) decreases with (N) for all (q); (ii) at fixed (N), (r) increases with (q), confirming that (N^) grows with (q); (iii) the ordering is robust across temperatures (\beta J = 2)-(20), consistent with the (\beta)-independence of (N^) in the conformal limit. The crossover separates a few-body regime, where multi-point correlations are more fragile, from a many-body regime, where single-particle coherence is lost first.
The growth of information scrambling, captured by out-of-time-order correlation functions (OTOCs), is a central indicator of the nature of many-body quantum dynamics. Here, we compute analytically the complete time dependence of the OTOC for an integrable Sachdev-Ye-Kitaev (SYK) model, N Majoranas with random two-body interactions of infinite range, coupled to a Markovian bath at finite temperature. In the limit of no coupling to the bath, the time evolution of scrambling experiences different stages. For t≲sqrt[N], after an initial polynomial growth, the OTOC approaches saturation in a power-law fashion with oscillations superimposed. At t∼sqrt[N], the OTOC reverses trend and starts to decrease linearly in time. The reason for this linear decrease is that, despite being a subleading 1/N effect, the OTOC in this region is governed by the spectral form factor of the antisymmetric couplings of the SYK model. The linear decrease stops at t∼2N, the Heisenberg time, where saturation occurs. The effect of the environment is an overall exponential decay of the OTOC for times longer than the inverse of the coupling strength to the bath. The oscillations at t≲sqrt[N] indicate lack of thermalization-a desired feature for better performance of quantum information devices.
Understanding how quantum chaotic systems generate entanglement can provide insight into their microscopic chaotic dynamics and can help distinguish between different classes of chaotic behavior. Using von Neumann entanglement entropy, we study a nonentangled state evolved under three variants of the Sachdev-Ye-Kitaev (SYK) model with a finite number of Majorana fermions N. All the variants exhibit linear entanglement growth at early times, which at late times saturates to a universal value consistent with random matrix theory (RMT), but their growth rates differ. We interpret this as a large-N effect, arising from the enhanced nonlocality of fermionic operators in SYK and binary SYK, absent in spin operators of the spin-SYK model. Numerically, we find that these differences emerge gradually with increasing N. Although all variants are quantum chaotic, their entanglement dynamics reflect varying scrambling rates and indicate that the entanglement production rate serves as a fine-grained probe of scrambling beyond conventional measures. To probe its effect on the thermalization properties of these models, we study the two-point autocorrelation function, finding no differences between the SYK variants, but deviations from RMT predictions for N≥24, particularly near the crossover from exponential decay to saturation regime.
We propose an extension of the Sachdev-Ye-Kitaev (SYK) model that exhibits a quantum phase transition from the previously identified non-Fermi-liquid fixed point to a Fermi-liquid-like state, while still allowing an exact solution in a suitable large-N limit. The extended model involves coupling the interacting N-site SYK model to a new set of pN peripheral sites with only quadratic hopping terms between them. The conformal fixed point of the SYK model remains a stable low-energy phase below a critical ratio of peripheral sites p<pc(n) that depends on the fermion filling n. The scrambling dynamics throughout the non-Fermi-liquid (NFL) phase is characterized by a universal Lyapunov exponent ${\ensuremath{\lambda}}_{\mathrm{L}}\ensuremath{\rightarrow}2\ensuremath{\pi}T$ in the low-temperature limit; however, the temperature scale marking the crossover to the conformal regime vanishes continuously at the critical point pc. The residual entropy at $T\ensuremath{\rightarrow}0$, nonzero in the NFL, also vanishes continuously at the critical point. For p>pc the quadratic sites effectively screen the SYK dynamics, leading to a quadratic fixed point in the low-temperature and low-frequency limit. The interactions have a perturbative effect in this regime leading to scrambling with Lyapunov exponent ${\ensuremath{\lambda}}_{\mathrm{L}}\ensuremath{\propto}{T}^{2}$.
We study the Yukawa-Sachdev-Ye-Kitaev model, a disordered model of N Majorana fermions and R=γN bosons in which the bosons are linearly coupled to independent realizations of SYK p-body interactions, in the presence of dissipation, modeled by a Lindblad master equation. Motivated by recent proposals for implementing SYK models in quantum simulators, we focus on bosonic leakage at rate κ. Initializing the system in the steady state, we analyze the late-time fermionic relaxation rate and the Lyapunov exponent, solving the large-N theory both numerically and for p large, finding a rich landscape of dynamical behaviors. Most notably, the Lyapunov exponent remains positive for every value of κ and, for p>2, can even grow as κ increases. The QED case p=2, which lies between the fully chaotic regime p>2 and the integrable case p=1, exhibits special features. We also identify a critical value of the boson-to-fermion ratio γc≈2/p2 separating distinct dynamical regimes.
Out-of-time-order correlators (OTOCs) can be used to probe how quickly a quantum system scrambles information when the initial conditions of the dynamics are changed. In sufficiently large quantum systems, one can extract from the OTOC the quantum analog of the Lyapunov coefficient that describes the timescale on which a classical chaotic system becomes scrambled. OTOCs have been applied only to a very limited number of toy models, such as the Sachdev-Ye-Kitaev model connected with black hole information scrambling, but they could find much wider applicability for information scrambling in quantum systems that allow comparison with experiments. The vibrations of polyatomic molecules are known to undergo a transition from regular dynamics at low energy to facile energy flow at sufficiently high energy. Molecules therefore represent ideal quantum systems to study scrambling in many-body systems of moderate size (here 6 to 36 degrees of freedoms). By computing quantum OTOCs and their classical counterparts we quantify how information becomes ``scrambled'' quantum mechanically in molecular systems. Between early ``ballistic'' dynamics, and late ``saturation'' of the OTOC when the full density of states is explored, there is indeed a regime where a quantum Lyapunov coefficient can be defined for all molecules in this study. Comparison with experimental rate data shows that slow scrambling as measured by the OTOC can reach the timescale of molecular reaction dynamics. Even for the smallest molecules we discuss, the Maldacena bound remains satisfied by regularized OTOCs, but not by unregularized OTOCs, highlighting that the former are more useful for discussing information scrambling in this type of moderate-size quantum system.
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.