Solving L\'{e}vy Sachdev-Ye-Kitaev Model

Abstract

We present an exact solution in the large-$N$ limit of the L\'{e}vy Sachdev-Ye-Kitaev (LSYK) model introduced in Ref. [1], wherein the couplings are drawn from a L\'{e}vy Stable distribution parameterized by a tail exponent $\mu \in [0, 2]$. Starting from the Hamiltonian and its associated partition function, we highlight the key differences from the standard Gaussian SYK model and derive the large-$N$ Schwinger-Dyson equations via a bosonic oscillator representation of the action. These equations are solved both numerically and analytically in the large-$q$ and infrared limits. We subsequently analyze the chaotic properties of the model by computing the Krylov exponent from the large-$q$ Green's function and extracting the Lyapunov exponent from the $4$-point function. The parameter $\mu$ continuously interpolates between a free theory at $\mu = 0$ and the conventional, maximally chaotic Gaussian SYK model at $\mu = 2$, with non-maximal chaos persisting throughout the intermediate regime $0 < \mu < 2$. Thermodynamic quantities, including the entropy, free energy, average energy, and specific heat capacity, are computed and compared with their Gaussian SYK counterparts. The interpretations of the thermodynamics are discussed with respect to the holographic dual and non-Fermi liquid theory. Finally, we discuss an alternative representation of the LSYK model based on a distinct decomposition of the L\'{e}vy Stable distribution, which establishes a non-trivial connection to Gaussian SYK, and provide supporting analytical and numerical results in the appendices.