Spectral statistics and energy-gap scaling in $k-$local spin Hamiltonians
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| Format: | Preprint |
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2025
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| _version_ | 1866909984071090176 |
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| author | Dowarah, Sasanka |
| author_facet | Dowarah, Sasanka |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_15829 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Spectral statistics and energy-gap scaling in $k-$local spin Hamiltonians Dowarah, Sasanka Quantum Physics Statistical Mechanics 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. |
| title | Spectral statistics and energy-gap scaling in $k-$local spin Hamiltonians |
| topic | Quantum Physics Statistical Mechanics |
| url | https://arxiv.org/abs/2510.15829 |