Brownian SYK-Hubbard Model: Chaos and Mottness

• The paper demonstrates that the interplay of Brownian SYK interactions and local Hubbard terms analytically produces a crossover in the spectral function, shifting from a single peak to a double peak indicative of Mottness.
• The model unifies time-dependent random dynamics with controlled local correlations, enabling a precise study of spectral form factors and multiple dynamical transitions.
• Out-of-time-ordered correlators and Lyapunov exponents indicate that local Hubbard interactions enhance quantum chaos, challenging conventional bounds and shifting many-body dynamics.

The Brownian SYK-Hubbard model is a quantum many-body system that combines the all-to-all random $q$-body interactions characteristic of the Brownian Sachdev-Ye-Kitaev (SYK) model with local, on-site Hubbard interactions. This hybrid construction allows for the analytic paper of how nonlocal random dynamics (which generate rapid quantum chaos and scrambling) interplay with strong local correlation effects such as Mottness, yielding rich spectral, dynamical, and information-theoretic phenomena in an exactly tractable framework (Sun et al., 18 Oct 2025).

1. Model Architecture and Hamiltonian Structure

The Brownian SYK-Hubbard model is defined on $N$ sites, each hosting four Majorana fermions $\chi_{ia}$ ($a=1,2,3,4$). The time-dependent Hamiltonian has two components: $$H(t) = i^{q(q-1)/2} \sum_{a=1}^4 \sum_{i_1 < \cdots < i_q} J^a_{i_1\cdots i_q}(t)\, \chi_{i_1 a} \ldots \chi_{i_q a} + \sum_i U\, \chi_{i1}\chi_{i2}\chi_{i3}\chi_{i4}$$

• The first term is an SYK-type interaction for each flavor $a$, with time-dependent random couplings $J^a_{i_1\cdots i_q}(t)$ distributed as white noise: $\overline{J^a_{i_1\cdots i_q}(t)} = 0$ and $\overline{(J^a)^2} \sim J/(N^{q-1} dt)$.
• The second term is an on-site Hubbard interaction. In terms of complex fermions,

$$c_{i,\uparrow} = \frac{\chi_{i1} + i\chi_{i2}}{\sqrt{2}},\quad c_{i,\downarrow} = \frac{\chi_{i4} + i\chi_{i3}}{\sqrt{2}}$$

the Hubbard term becomes $$U (c_{i,\uparrow}^\dagger c_{i,\uparrow} - 1/2)(c_{i,\downarrow}^\dagger c_{i,\downarrow} - 1/2)$$.

This setup enables controlled interpolation between purely nonlocal (Brownian SYK) and local (Hubbard) physics.

2. Single-Particle Spectral Function and Mottness Transition

The single-particle spectral function $A(\omega)$ is extracted from the time-ordered Green’s function of the Majorana operators. When $U=0$, the model reduces to four decoupled Brownian SYK chains and the Green function exhibits simple exponential decay: $G(t) = \frac{1}{2} e^{- \frac{\Gamma_0 t}{2}}$ where $\Gamma_0$ is the effective decay rate from SYK dynamics.

For finite $U$, the quadratic decay rate $\Gamma_0$ in the exponent is replaced by a complex value for $U > \Gamma_0$, leading to oscillatory temporal decay. This yields a frequency-domain spectral function $A(\omega)$ with two distinct peaks centered at $$\pm \omega_0 = \pm \tfrac{1}{2}\sqrt{U^2 - \Gamma_0^2}$$. The resulting spectral double-peak is the hallmark of local Mottness (i.e., formation of upper and lower Hubbard bands, even though the overall spectrum remains gapless due to the Brownian-induced smearing).

This crossover from a single to two-peak structure as $U$ increases provides a direct, analytically tractable demonstration of the interplay between local interactions and nonlocal quantum chaos in generating Mott physics.

3. Spectral Form Factor: Dynamical Transitions

The spectral form factor (SFF), $K(T) = |\mathrm{Tr} U(T)|^2$, is a diagnostic for many-body level correlations and thermalization dynamics. For static SYK models, $K(T)$ exhibits a “slope–ramp–plateau” structure as a function of time $T$.

In the Brownian SYK-Hubbard model, the SFF is computed using a Keldysh path integral with periodic boundary conditions. At short times, the “diagonal saddle” dominates, yielding persistent oscillations in $K(T)$ when $U$ is large, with zeros at $UT^*_n = 2\pi(2n+1)$. At these zeros, the system favors a “connected saddle”, leading to dynamical first-order transitions between disconnected (uncorrelated) and connected (correlated) time evolution branches.

As $T$ increases further, and with strong $U$, the system approaches a long-time “plateau” regime, indicating complete loss of phase coherence and the saturation of spectral correlations. This dynamical sequence of transitions as a function of time is a distinctive feature of the hybrid model and is absent in either purely SYK or purely Hubbard systems.

4. Quantum Chaos: OTOCs, Ladder Series, and Lyapunov Exponent

The out-of-time-ordered correlator (OTOC) is a sensitive probe of quantum chaos and operator growth. In the Brownian SYK-Hubbard model, OTOCs are computed through a Dyson-like equation built from generalized ladder diagrams, where the effective “rungs” incorporate both the Brownian SYK and constant Hubbard interaction effects: $$\text{OTOC}(t) = (q - 1)\,\Gamma_0 \int_0^t dt' F(t-t')\, \text{OTOC}(t'), \quad F_h(t) = e^{- h t}$$ The Lyapunov exponent $\kappa$ is obtained by solving $k_R(-\kappa) = 1$, where $k_R$ is the eigenvalue of the rung kernel.

A key result is that, as $U$ increases, the Lyapunov exponent $\kappa$ increases—implying that local Hubbard interactions enhance many-body chaos (contrary to naive expectations that localization might suppress chaos via Mottness). Furthermore, the branching time $t_B = k_R'(-\kappa)$—which captures the temporal stability of the exponential growth regime—is found to violate the standard bound $t_B(\kappa + \Gamma_0) \leq 2$ that holds in conventional SYK-like models. For $q=2$, the Brownian SYK term acts as a hopping, and the OTOC chaos arises solely from the Hubbard term, resulting in a breakdown of the conventional ladder diagram structure.

These results establish the Brownian SYK-Hubbard model as a new class of quantum many-body system in which constant local interactions not only coexist with but actively boost quantum chaos.

5. Analytical Techniques and Phase Structure

The hybrid model’s tractability arises from two sources: (1) The Brownian time-dependence renders the path integral (over the Keldysh contour) exactly solvable after averaging, and (2) the local Hubbard term can be absorbed into modified saddle-point equations for the two-point Green function. The analysis combines design elements of disordered mean-field theory, diagrammatic ladder constructions for OTOCs, and non-equilibrium path-integral techniques.

This framework admits a controlled tuning between three qualitatively distinct regimes:

• Random-dominated, SYK regime: Weak $U$, conventional Brownian SYK scrambling, single-peaked spectrum.
• Mottness regime: Intermediate to strong $U$, double-peak spectral function, multiple dynamical transitions in SFF.
• Strongly correlated chaos: Enhanced Lyapunov exponent and plateaued SFF, with violation of conventional branching time bounds.

The combination of Mottness, strong scrambling, and universal random-matrix-like saturation in spectral correlations is unique to this hybrid system.

6. Broader Implications and Prospects

The Brownian SYK-Hubbard model provides a blueprint for studying the generic interplay between quantum chaos and strong local correlations. Possible applications and extensions include:

• Realizing concrete “mixed” quantum systems, where a locally correlated (solvable) system is coupled to a strongly chaotic SYK bath.
• Understanding how Hubbard interactions alter quantum error correction properties and entanglement spreading, potentially using the exact replica and FKPP-type equations developed for Brownian SYK chains (Parrikar et al., 31 Jul 2025).
• Exploring the fate of spectral form factors and operator growth in contexts with more realistic, less-than-maximally random couplings (interpolating between Brownian and static cases).
• Investigating the breakdown of chaos bounds due to constant interactions, which may have implications for sub-AdS scale holography and field theory duals with explicit breaking of maximal scrambling.

This model advances the theoretical paper of strongly correlated quantum matter by establishing analytic pathways to track the emergence of Mottness, quantum chaos, and their hybridized dynamical and information-theoretic signatures (Sun et al., 18 Oct 2025).