From single-particle to many-body chaos in Yukawa--SYK: theory and a cavity-QED proposal

TL;DR

The study introduces the spinless Yukawa–SYK (YSYK) model as a tunable bridge between single-particle and many-body quantum chaos, analyzed through spectral (DOS, gap ratios, SFF) and dynamical (OTOC) chaos markers across a controllable coupling regime. A key finding is that the boson mass scale $ obreak\omega_0$ relative to the disorder strength $g$ drives a crossover from SYK$_2$-like to SYK$_4$-like behavior, with intermediate regimes featuring partial ergodicity breaking and prethermal plateaus. The analysis also derives an effective Schrieffer–Wolff Hamiltonian that explains the emergent SYK$_4$ dynamics in the weak-coupling limit and shows how increasing the number of bosonic modes $M$ sharpens the SYK$_4$-like spectral ramp. Furthermore, the work outlines a feasible cavity-QED experimental realization with ultracold atoms, detailing parameter regimes and dissipation considerations to observe the full spectrum of chaotic behaviors. Overall, YSYK serves as a unifying platform to experimentally and theoretically explore the transition between single-particle and many-body quantum chaos and its connections to holography and strongly correlated matter.

Abstract

Understanding how quantum systems transition from integrable to fully chaotic behavior remains a central open problem in physics. The Sachdev--Ye--Kitaev (SYK) model provides a paradigmatic framework for studying many-body chaos and holography, yet it captures only the strongly correlated limit, leaving intermediate regimes unexplored. Here, we investigate the Yukawa--SYK (YSYK) model, where bosonic fields mediate random fermionic interactions, and demonstrate that it naturally bridges single-particle and many-body chaos. Using spectral and dynamical chaos markers, we perform a comprehensive finite-size characterization of the YSYK model. We show that the interaction strength acts as a tunable control parameter interpolating between the SYK$_2$ and SYK$_4$ limits, and introduce a framework enabling direct and quantitative comparison with these benchmark models. In the intermediate regimes, we uncover distinct dynamical regimes marked by partial ergodicity breaking, prethermalization plateaus, and incomplete scrambling. Finally, we propose a feasible optical-cavity implementation of the YSYK model using ultra-cold atoms. Our results establish the YSYK model as a unifying platform connecting single-particle and many-body chaos, paving the way for experimental observation of these phenomena.

Figures (11)

• Figure 1: (a) Schematic illustration of the interaction term in the spinless Yukawa--SYK model. A set of $N$ spinless fermionic modes (blue dots) interact pairwise via $M$ independent bosons (wiggly lines), through random couplings $g_{ij,k}$. Each bosonic mode is shown in a different color, and the line shading represents the coupling strengths. The couplings are drawn independently from a Gaussian Unitary Ensemble (GUE). (b) As illustrated by the spectral form factor as a chaos diagnostic, tuning the boson mass $\omega_0$ relative to the coupling strength $g$ drives a transition from a single-particle–chaotic to a fully many-body–chaotic regime, with an intermediate crossover region where features of both coexist. (c) Proposed optical-cavity realization of the YSYK model. Cavity photons act as the bosonic modes, while fermions are represented by cold atoms trapped inside the cavity. An engineered optical speckle pattern introduces disorder in the light–matter coupling strengths. At very small photon detuning (boson mass), the photon only dresses hopping between fermionic modes and the model behaves as quadratic (SYK$_2$-like) single-particle chaotic, while at large detuning the photon can be adiabatically eliminated, yielding an effective quartic (SYK$_4$-like) many-body-chaotic theory.
• Figure 2: Spectral diagnostics of the YSYK model with $N=8$ fermionic and $M=4$ bosonic modes (occupation cutoff $N_b$=1) across coupling range of the YSYK model. Top row: Density of states $\rho(E)$. At small $\omega_0/g^{2/3}$, the distributions closely follow a Gaussian fit (red dashed curves). At $\omega_0/g^{2/3}\sim 1$ the DOS develops bands separated by the boson mass scale $\omega_0$. Middle row: The gap-ratio distribution $P(r)$ is close to Poisson at $\omega_0/g^{2/3}=1/100$ and shifts toward GUE (dash-dotted) as $\omega_0/g^{2/3}$ increases. Bottom row: Spectral form factor $K(t)$ at $\beta=0$. For small $\omega_0/g^{2/3}$ there is steep superlinear ramp, typical of single-particle chaotic systems. Around $\omega_0/g^{2/3}\sim 1/2$ a clear linear ramp emerges, a feature of many-body spectral rigidity. For larger $\omega_0/g^{2/3}$, the linear ramp persists, but with $(2\pi/\omega_0)$-periodic modulations due to DOS clustering. The dotted black line marks the plateau value $1/D$, and the red dashed vertical line indicates the extracted plateau time. All the data are averaged over $500$ disorder realizations.
• Figure 3: Averaged gap ratio $\langle r \rangle$ versus the interaction variable $\omega_0/g^{2/3}$ for boson–fermion mode ratios $M/N\in\{2.0,1.0,0.5\}$ (obtained for $(M,N)=(8,4),(6,6),(4,8)$, with bosonic cutoff $N_b=1$). Data points show averages over 500 disorder realizations for each $(M,N)$. At small $\omega_0/g^{2/3}$, $\langle r \rangle$ is Poissonian (dotted line). It reaches the GUE value (dashed) in a broad crossover region $\omega_0/g^{2/3}\sim 0.3-1$ (grey band). At larger $\omega_0/g^{2/3}$, it drops slightly due to DOS clustering.
• Figure 4: Main panel: Log-log plot of the SFF $K(t)$ versus the rescaled time $tg/\sqrt{2\omega_0}$ for $N=8$ fermionic modes and $M=4$ bosonic modes with cutoff $N_b=1$. Curves are color–coded by $\omega_0$ (color bar at top) in the range $\omega_0 \in \left[0.01,0.5\right]$. Dashed vertical lines mark the SYK$_2$ plateau time $2N$ and the Heisenberg time $t_\mathrm{H}$. Solid circles indicates the numerically extracted plateau–onset times $t_{\rm plateau}(\omega_0)$, while squares denote the fitted ramp–onset times $t_{\rm ramp}(\omega_0)$. Insets: (top middle) Linear–scale plot of $t_{\rm plateau}$ versus $\omega_0$, with horizontal dashed lines at $t_\mathrm{H}$ and at the SYK$_2$ plateau time $2N$. For very small $\omega_0$, the numerics cannot resolve deviations from SYK$_2$–dominated behavior. From around $\omega_0=0.1$ onwards, the plateau time is close to the Heisenberg time, indicative of fully many–body chaotic behavior. (top right) Ratio $t_{\rm ramp}(\omega_0)/t_H$ as a function of $\omega_0$. The ramp time becomes parametrically smaller than the Heisenberg time as $\omega_0$ increases, indicating the appearance of many-body chaos.
• Figure 5: Disorder averaged OTOCs of the YSYK model for small values of $\omega_0$ (blue lines) with $N=8$ fermions and $M=4$ bosonic modes with $N_b = 1$, averaged over $1000$ samples, alongside the target complex SYK$_2$ model (red-dashed) with same number of fermions, averaged over $10 000$ samples. For small enough boson mass $\omega_0$, $F(t)$ shows a two-step scrambling process: it first reaches a prethermal plateau that coincides with the non-zero saturation value of SYK$_2$ before decaying further and becoming fully scrambling. Larger values of $\omega_0$ do not show a prethermal plateau, indicating the non-perturbative deviation from the SYK$_2$ regime. Inset: The OTOCs for sufficiently small $\omega_0$ collapse when time is rescaled by the characteristic scale $\omega_0^{5/2} g$ derived from a perturbative analysis (see Appendix \ref{['Appendix: Magnus_expansion_small_omega0']}).