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 their associated random matrix ensemble—Gaussian orthogonal ensemble, Gaussian unitary ensemble, or Gaussian symplectic ensemble—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 ${\sigma }_c\approx \left|\mu \right|$. Near this transition, the energy gap $\mathrm{\Delta }$ exhibits universal quadratic scaling $\mathrm{\Delta }/L\sim {(\sigma -{\sigma }_c)}^2$. When $k\ll \sqrt{L}$, ${\sigma }_c$ scales with $\left|\mu \right|$, but lacks a sharp transition. Our work introduces a semisolvable 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.

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• Spin glasses
• Random matrix theory