Spectral statistics and energy-gap scaling in $k-$local spin Hamiltonians

We investigate the spectral properties of all-to-all interacting spin Hamiltonians acting on exactly $k$ spins, whose coupling coefficients are drawn from a normal distribution with mean $\mu$ and variance ${\sigma }^2$. For $\mu =0$, we demonstrate that the random matrix ensemble – Gaussian Orthogonal Ensemble (GOE), Gaussian Unitary Ensemble (GUE), or Gaussian Symplectic Ensemble (GSE) – is determined by the parity of system size $L$ and locality $k$, following standard time-reversal symmetry classification. For couplings with a nonzero mean, we map the Hamiltonians to deformed random matrix ensembles and analyze conditions for an energy gap between the ground state and the first excited state. For $\mu <0$, we find two distinct regimes: for $k\gg \sqrt{L}$, the gap closes at critical disorder ${\sigma }_c\approx \mathrm{\mid }\mu \mathrm{\mid }$. Near this transition the energy gap $\mathrm{\Delta }$ exhibits universal quadratic scaling $\mathrm{\Delta }\mathrm{/}L\sim (\sigma -{\sigma }_c{)}^2$. When $k\ll \sqrt{L}$, ${\sigma }_c$ scales with $\mathrm{\mid }\mu \mathrm{\mid }$, but lacks a sharp transition. Our work introduces a semi-solvable model that captures universal features of random-matrix statistics, and spectral gap formation, providing a foundation for systematic extensions to more general many-body systems.