Statistics of Matrix Elements of Operators in a Disorder-Free SYK model

Recently, studies have explored the statistics of matrix elements of local operators in the Lieb-Liniger model. It was found that the probability distribution function for off-diagonal matrix elements $\langle \boldsymbol{\mu}|\mathcal{O}|\boldsymbol{\lambda} \rangle$ within the same macro-state is well described by the Fr\'{e}chet distributions. This represents a significant development for the Eigenstate Thermalization Hypothesis (ETH). In this paper, we investigate a similar phenomenon in another solvable model: the disorder-free Sachdev-Ye-Kitaev (SYK) model. The Hamiltonian of this model consists of 4-body interactions of Majorana fermions. Unlike the conventional SYK model, the coupling strengths in this model are fixed to a constant, earning it the name ``disorder-free.'' We evaluate the matrix elements of operators constructed from products of $n$ Majorana fermions: $\mathcal{O} = \chi_{a_1}\chi_{a_2}\ldots \chi_{a_n}$. For a general choice of indices and $n \geq 4$, we find that the statistics of the off-diagonal matrix elements are well-fitted by a generalized inverse Gaussian distribution rather than Fr\'{e}chet distributions.