Hybrid Brownian SYK-Hubbard Model: from Spectral Function to Quantum Chaos

TL;DR

This work introduces the Brownian SYK–Hubbard model, a solvable hybrid that merges Brownian SYK dynamics with on-site Hubbard interactions to explore the interplay between nonlocal randomness and local correlations. Through a $G-\Sigma$ saddle-point analysis, each site maps to an effective open-system evolution, enabling analytic access to the two-point function, spectral form factor, and OTOC. Key results include a Mottness-like transition in the single-particle spectrum as $U$ increases, a sequence of dynamical transitions in the SFF, and a violation of the branching-time bound in the OTOC for certain $q$, signaling a new class of chaotic, analytically tractable models. These findings establish a controlled framework to study Hubbard physics in chaotic settings and point to connections with holography, entanglement dynamics, and potential extensions to correlated materials.

Abstract

Understanding the emergence of complex correlations in strongly interacting systems remains a fundamental challenge in quantum many-body physics. One fruitful approach is to develop solvable toy models that encapsulate universal properties shared by realistic systems. In this work, we introduce the Brownian SYK-Hubbard model, which combines the all-to-all random interactions of the Sachdev-Ye-Kitaev (SYK) model with on-site Hubbard-type interactions. This hybrid construction enables the study of the interplay between nonlocal random dynamics and local correlation effects: (1) As the interaction strength increases, the single-particle spectrum exhibits a transition from a single peak to a two-peak structure, signaling the onset of Mottness. (2) The spectral form factor undergoes a sequence of dynamical transitions as the evolution time increases before reaching the plateau in the long-time limit under strong Hubbard interactions. (3) The out-of-time-order correlator is computed by summing a series of modified ladder diagrams, which determines the quantum Lyapunov exponent and reveals a violation of the bound on branching time. Our results establish a new analytically tractable platform for exploring the effects of Hubbard interactions in chaotic many-body systems.

Figures (4)

• Figure 1: We present a schematic illustration of the Brownian SYK–Hubbard model studied in this work. The model consists of $4N$ Majorana fermions grouped into $N$ sites, with each site hosting four distinct modes labeled by different colors. Modes sharing the same color interact through Brownian SYK-type random couplings $J^a_{i_1,...,i_q}(t)$, while modes residing on the same site are coupled via a constant on-site Hubbard interaction $U$.
• Figure 2: We plot the Green’s function and spectral function of the Brownian SYK–Hubbard model (with arbitrary $q$) for $U/\Gamma_0 \in \{1/2, 1, 3, 5\}$. The results clearly show a qualitative change at $U/\Gamma_0 = 1$, where the Green’s function transitions from monotonic decay to oscillatory behavior, and the spectral function evolves from a single peak to double peaks.
• Figure 3: We plot the SFF of the Brownian SYK–Hubbard model with $q=4$ for (a) $U/\Gamma_0 = 1$, (b) $U/\Gamma_0 = 2$, and (c) $U/\Gamma_0 = 3$. The number of dynamical transitions between the diagonal and connected saddles increases with $U$, due to the persistent oscillations in the diagonal contribution.
• Figure 4: We plot the quantum Lyapunov exponent $\varkappa$ and the branching time $t_B$ as functions of $U/\Gamma_0$ for the Brownian SYK–Hubbard model with $q \in {2,4,8,12}$. The results reveal that increasing the Hubbard interaction $U$ enhances many-body chaos and, notably, leads to a violation of the branching-time bound for $q=2$, as indicated by the black dashed line.