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 $\mu =0$, 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 ${\sigma }_c\approx |\mu |$. Near this transition the energy gap $\Delta$ exhibits universal quadratic scaling $\Delta /L\sim (\sigma -{\sigma }_c{)}^2$. When $k\ll \sqrt{L}$, ${\sigma }_c$ scales with $|\mu |$, 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.

arXiv comment17 pages, 8 figures

References in corpus (52)

• Classification of topological quantum matter with symmetries
• Many body localization and thermalization in quantum statistical mechanics
• Random Matrix Theories in Quantum Physics: Common Concepts
• Localization of interacting fermions at high temperature
• Average Entropy of a Subsystem
• The many-body localization phase transition
• Continuous Quantum Phase Transitions
• An Area Law for One Dimensional Quantum Systems
• Many-body localization edge in the random-field Heisenberg chain
• The distribution of the ratio of consecutive level spacings in random matrix ensembles
• Lieb-Schultz-Mattis in Higher Dimensions
• Spectral Gap and Exponential Decay of Correlations
• Schrieffer-Wolff transformation for quantum many-body systems
• Black Holes and Random Matrices
• Quantum many-body models with cold atoms coupled to photonic crystals
• Testing whether all eigenstates obey the Eigenstate Thermalization Hypothesis
• Lieb-Robinson Bounds and the Exponential Clustering Theorem
• Topological quantum order: stability under local perturbations
• Cavity QED with Quantum Gases: New Paradigms in Many-Body Physics
• Spectral and thermodynamic properties of the Sachdev-Ye-Kitaev model
• Long-range interacting quantum systems
• Undecidability of the Spectral Gap (short version)
• A random matrix model with localization and ergodic transitions
• Sachdev-Ye-Kitaev Model and Thermalization on the Boundary of Many-Body Localized Fermionic Symmetry Protected Topological States
• The largest eigenvalue of rank one deformation of large Wigner matrices
• The largest eigenvalues of finite rank deformation of large Wigner matrices: convergence and nonuniversality of the fluctuations
• Stability of Frustration-Free Hamiltonians
• Towards a full solution of the large N double-scaled SYK model
• Poisson versus GOE statistics in integrable and non-integrable quantum hamiltonian
• Automorphic Equivalence within Gapped Phases of Quantum Lattice Systems
• Quantum simulation of a three-body interaction Hamiltonian on an NMR quantum computer
• Phase Transition in the Density of States of Quantum Spin Glasses
• Symmetry Classification and Universality in Non-Hermitian Many-Body Quantum Chaos by the Sachdev-Ye-Kitaev Model
• Complete random matrix classification of SYK models with $\mathcal{N}=0$, $1$ and $2$ supersymmetry
• Level repulsion in integrable and almost-integrable quantum spin models
• Probing symmetries of quantum many-body systems through gap ratio statistics
• Rigorous RG algorithms and area laws for low energy eigenstates in 1D
• Metastability and discrete spectrum of long-range systems
• Demonstration of three- and four-body interactions between trapped-ion spins
• From ergodic to non-ergodic chaos in Rosenzweig-Porter model
• Persistence of exponential decay and spectral gaps for interacting fermions
• Periodic Table of SYK and supersymmetric SYK
• Perturbation Theory for the Rosenzweig-Porter Matrix Model
• Spectral correlations in the crossover between GUE and Poisson regularity: on the identification of scales
• Undecidability of the Spectral Gap in One Dimension
• Quenched vs Annealed: Glassiness from SK to SYK
• Generic Local Hamiltonians are Gapless
• Tuning the phase diagram of a Rosenzweig-Porter model with fractal disorder
• A model of randomly-coupled Pauli spins
• Gaplessness is not generic for translation-invariant spin chains
• The Rosenzweig Porter model revisited for the three Wigner Dyson symmetry classes
• Dynamics of operator size distribution in q-local quantum Brownian SYK and spin models