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

TL;DR

This work analyzes spectral statistics and energy-gap formation in all-to-all interacting $k$-local spin Hamiltonians with Gaussian couplings $J_{j_1\cdots j_k}\sim N(\mu,\sigma^2)$. For $\mu=0$ the energy-level statistics are universal and determined by the parity of the system size $L$ and locality $k$, yielding GOE, GUE, or GSE ensembles; for $\mu\neq 0$ the model maps to deformed random matrices, enabling analytic predictions for the ground-state gap and its closing at a critical disorder $\sigma_c$. In the regime $k \gg \sqrt{L}$ the gap closes at $\sigma_c \approx |\mu|$, with a universal quadratic scaling $\Delta(\sigma)/L \sim (\sigma_c - \sigma)^2$ near criticality, while for $k \ll \sqrt{L}$ the transition is smoother and $\sigma_c$ scales with $|\mu|$. The paper provides a semi-solvable framework linking random-matrix statistics and gap formation in all-to-all interacting quantum systems, and outlines extensions to more realistic connectivity and symmetry constraints.

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 $μ$ and variance $σ^2$. For $μ= 0$, 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 $μ< 0$, we find two distinct regimes: for $k \gg \sqrt{L}$, the gap closes at critical disorder $σ_{c} \approx |μ|$. Near this transition the energy gap $Δ$ exhibits universal quadratic scaling $Δ/L \sim (σ- σ_{c})^{2}$. When $k \ll \sqrt{L}$, $σ_{c}$ scales with $|μ|$, 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.

Figures (8)

• Figure 1: Average level statistics $\langle r_{n} \rangle$ of the Hamiltonian $H(0,1)$ for $L=10$ for locality $k=2,3,4,5,6,7$. The error bar indicates the standard error of the mean for $480$ different random realizations of the couplings. The pink dashed-dotted line, the blue dashed line, and the purple dotted line represent the GOE $\langle r_{n} \rangle = 0.53$, GUE $\langle r_{n} \rangle = 0.60$, and GSE $\langle r_{n} \rangle = 0.67$ level statistics, respectively.
• Figure 2: Average level statistics $\langle r_{n} \rangle$ of the Hamiltonian $H(0,1)$ for $L=9$ for $k=2,3,4,5,6,7$. The error bar indicates the standard error of the mean for $480$ different random realizations of the couplings. The pink dashed dot line, the blue dashed line, and the purple dotted line represent the GOE $\langle r_{n} \rangle = 0.53$, GUE $\langle r_{n} \rangle = 0.60$, and GSE $\langle r_{n} \rangle = 0.67$ level statistics, respectively. In the case of odd $L$ and even $k$, the eigenvalues of the Hamiltonian are doubly degenerate. The level statistics in that case are calculated by choosing one of the degeneracy sectors, i.e., the eigenvalues with either the odd or even indices.
• Figure 3: Average normalized spacing distribution of the Hamiltonians $H(0,1)$ for $L=9$ for locality $k=2,3,4,5,6,7$ averaged over $480$ random couplings.
• Figure 4: Average normalized spacing distribution of the Hamiltonians $H(0,1)$ for $L=10$ for locality $k=2,3,4,5,6,7$ averaged over $480$ random couplings. For even $k$, we choose one degeneracy sector to calculate the normalized spacings.
• Figure 5: Plot of the energy spectra of the Hamiltonian defined in Eq. \ref{['eq:normalized_single_parameter_hamiltonian']} as a function of disorder strength $\sigma$ for $\mu = -1$, shown for system sizes $L = 11$ and $L = 10$, each corresponding to a single realization of random couplings. Panels (a)–(f) correspond to $L = 11$ with $k=2, 3, 4, 5, 6, 7$, respectively; panels (g)–(l) show the same for $L=10$. The color scale represents the half-system von Neumann entanglement entropy normalized by its Page value PhysRevLett.71.1291: $S/S_{\mathrm{Page}}$, where $S_{\mathrm{Page}} \approx 3.22$ for $L=11$ (using a bipartition into $L_A = 6$, $L_B = 5$) and $S_{\mathrm{Page}} \approx 2.97$ for $L=10$. The two dashed-dotted pink lines in each plot represent the envelope given by Eq.\ref{['eq:radius_of_wigner_semicircle_as_function_of_sigma']}. The red dotted curves represent Eq. \ref{['eq:extremum_eigenvalue']} for the ground state energy for even $k$ and the ground state energy and the highest energy for odd $k$. For $k = 2, 3$, we have shown the critical disorder values predicted by Eq.\ref{['eq:sigma_c_for_a_general_s']} using vertical black lines corresponding to the ground state merging into the bulk that are within two $(s = 2)$ and three $(s = 3)$ standard deviations from the mean. For $k > 3$, the critical disorder predicted by Eq.\ref{['eq:critical_disorder_approximate_formula_k_L/2']} is shown with a vertical dotted line.