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

Technical Summary: Hybrid Brownian SYK–Hubbard Model

Problem Statement
Understanding the emergence of complex correlations in strongly interacting quantum many-body systems remains a fundamental challenge due to the exponential growth of the Hilbert space. While solvable toy models like the Sachdev–Ye–Kitaev (SYK) model have provided insights into non-Fermi liquid behavior and maximal chaos, they typically rely on fully random, all-to-all interactions. This randomness, while enabling analytical tractability via large-$N$ expansions, obscures the physics of constant (non-random) interactions, such as the Hubbard interaction responsible for Mott insulating phases. Furthermore, standard SYK-like models exhibit a specific "ladder-diagram" structure in their out-of-time-order correlators (OTOCs) that imposes a bound on the branching time, potentially limiting their ability to describe sub-AdS holography. The authors aim to construct a solvable model that hybridizes the random dynamics of the Brownian SYK model with constant on-site Hubbard interactions to study the interplay between nonlocal random dynamics and local correlation effects.

Methodology
The authors introduce the Brownian SYK–Hubbard model, defined on $N$ sites, where each site hosts four Majorana fermion modes. The Hamiltonian consists of two terms:

1. Brownian SYK Interaction: Random, time-dependent $q$-body interactions acting between sites for each flavor (mode), satisfying specific variance conditions.
2. Hubbard Interaction: A constant, on-site four-Majorana interaction ($U \chi_{i1}\chi_{i2}\chi_{i3}\chi_{i4}$), which corresponds to the standard Hubbard interaction at half-filling when mapped to complex fermions.

The analytical approach leverages the large-$N$ limit and the Keldysh path-integral formalism.

• Two-point Function: The disorder average over the Brownian couplings is performed, introducing auxiliary self-energy ($\Sigma$) and Green's function ($G$) fields. Unlike static SYK models, the Brownian nature allows the self-energy to be treated as a simple delta function in time. The authors demonstrate that the many-body problem reduces to an effective single-site problem coupled to an environment. This effective single-site dynamics is mapped to a non-unitary evolution generated by an effective Hamiltonian ($H_{\text{eff}}$), which can be solved exactly via exact diagonalization due to the small number of modes (8 Majoranas per site).
• Spectral Form Factor (SFF): The calculation involves a path integral with periodic boundary conditions. The authors perform a saddle-point analysis, identifying "diagonal" (disconnected) and "connected" saddle points. The SFF is determined by maximizing an action involving the trace of the effective single-site evolution operator.
• Out-of-Time-Order Correlator (OTOC): The OTOC is analyzed using a double Keldysh contour (four branches). The authors derive a self-consistent equation for the OTOC involving a series of generalized ladder diagrams. The "rung" of the ladder is computed using the effective Hamiltonian on a 16-mode system (four branches $\times$ four modes). The quantum Lyapunov exponent ($\kappa$) and branching time ($t_B$) are extracted by solving the eigenvalue equation for the integral operator governing the diagrammatic series.

Key Results

1. Spectral Function and Mottness:

   • The single-particle spectral function exhibits a dynamical transition as the Hubbard interaction strength $U$ increases relative to the Brownian decay rate $\Gamma_0$.
   • For $U \leq \Gamma_0$, the Green's function decays monotonically, and the spectral function shows a single peak at $\omega=0$.
   • For $U > \Gamma_0$, the Green's function transitions to an oscillatory decay, and the spectral function develops a two-peak structure separated by a gap-like feature. This signals the onset of "Mottness" (Mott physics), although the system remains gapless due to the filling of the gap by Brownian interactions.

2. Spectral Form Factor (SFF) Dynamics:

   • The SFF displays a sequence of dynamical transitions between the diagonal saddle (dominant at short times) and the connected saddle (dominant at long times).
   • In the presence of strong Hubbard interactions ($U > \Gamma_0$), the diagonal contribution exhibits persistent oscillations with multiple zeros. This leads to a series of transitions where the connected saddle repeatedly overtakes the diagonal saddle as time evolves.
   • The number of these dynamical transitions increases with the ratio $U/\Gamma_0$.

3. Quantum Chaos and Branching Time:

   • The quantum Lyapunov exponent $\kappa$ increases monotonically with the Hubbard interaction $U$, indicating that stronger local correlations enhance many-body chaos.
   • Crucially, the branching time $t_B$, which characterizes the stability of the Lyapunov exponent, violates the conventional bound $t_B(\kappa + \Gamma) \leq 2$ derived for models with simple ladder-diagram structures. This violation is explicitly observed for $q=2$ at intermediate interaction strengths.

Significance and Claims
The paper claims to establish a new analytically tractable platform for exploring the effects of Hubbard interactions in chaotic many-body systems. By combining Brownian random couplings with constant Hubbard terms, the model bridges the gap between solvable random models and realistic correlated materials.

The authors emphasize three main contributions:

1. Analytical Solvability with Local Interactions: The model demonstrates that solvable structures can be maintained even when introducing constant interactions that break the standard tensor large-$N$ structure, provided the random interactions are time-dependent (Brownian).
2. Observation of Mottness in Chaotic Systems: The model successfully captures the transition to a double-peak spectral function, a hallmark of Mott physics, within a chaotic, non-Fermi liquid framework.
3. Beyond Standard SYK Paradigms: The violation of the branching time bound suggests that the Brownian SYK–Hubbard model belongs to a broader class of solvable models that may offer new insights into holographic dualities and sub-AdS physics, challenging the limitations of traditional ladder-diagram approximations in SYK-like systems.

The work concludes by noting that while the current analysis focuses on correlation functions and spectral properties, the framework opens avenues for studying entanglement dynamics and replica wormhole solutions in the presence of strong local interactions.