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

Abstract: 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 the completely disordered case $\mu = 0$, we show that the universality class of level statistics depends solely on the parity of system size $L$ and locality $k$, classifying these Hamiltonians into the Gaussian Orthogonal (GOE), Unitary (GUE), or Symplectic (GSE) ensembles. For couplings with a non-zero mean, we map the Hamiltonians to deformed random matrix ensembles and analyze conditions for a energy gap between the ground state and the first excited state. We find two distinct regimes: for small locality ($k \ll \sqrt{L}$), we show that the gap closing threshold scales proportionally with the mean coupling $\sigma \sim \mu$, and for large locality ($k \gg \sqrt{L}$) a spectral gap exists as long as $\mu > \sigma$. We analytically derive the expression for this energy gap in the $k \gg \sqrt{L}$ limit. Our work provides a semi-solvable toy model for understanding random matrix universality, universal energy gap scaling, and a foundation for exploring more general properties through systematic modifications and couplings.