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arXiv:2510.15829 · Physical review. B./Physical review. B

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

Sasanka Dowarah

Published Dec 17, 2025• 0 citations
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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.

DOI
10.1103/ng44-x1tv

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We investigate the spectral properties of all-to-all interacting spin Hamiltonians acting on exactly kkk spins, whose coupling coefficients are drawn from a normal distribution with mean μμμ and variance σ2σ^2σ2. For μ=0μ= 0μ=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 LLL and locality kkk, 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<0μ< 0μ<0, we find two distinct regimes: for k≫Lk \gg \sqrt{L}k≫L​, the gap closes at critical disorder σc≈∣μ∣σ_{c} \approx |μ|σc​≈∣μ∣. Near this transition the energy gap ΔΔΔ exhibits universal quadratic scaling Δ/L∼(σ−σc)2Δ/L \sim (σ- σ_{c})^{2}Δ/L∼(σ−σc​)2. When k≪Lk \ll \sqrt{L}k≪L​, σcσ_{c}σc​ scales with ∣μ∣|μ|∣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.
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Using numerical diagonalization we study the crossover among different random matrix ensembles (Poissonian, Gaussian orthogonal ensemble (GOE), Gaussian unitary ensemble (GUE) and Gaussian symplectic ensemble (GSE)) realized in two different microscopic models. The specific diagnostic tool used to study the crossovers is the level spacing distribution. The first model is a one-dimensional lattice model of interacting hard-core bosons (or equivalently spin 1/2 objects) and the other a higher dimensional model of non-interacting particles with disorder and spin-orbit coupling. We find that the perturbation causing the crossover among the different ensembles scales to zero with system size as a power law with an exponent that depends on the ensembles between which the crossover takes place. This exponent is independent of microscopic details of the perturbation. We also find that the crossover from the Poissonian ensemble to the other three is dominated by the Poissonian to GOE crossover which introduces level repulsion while the crossover from GOE to GUE or GOE to GSE associated with symmetry breaking introduces a subdominant contribution. We also conjecture that the exponent is dependent on whether the system contains interactions among the elementary degrees of freedom or not and is independent of the dimensionality of the system.
For mmm number of bosons, carrying spin ($\cs=\spin$) degree of freedom, in ΩΩΩ number of single particle orbitals, each doubly degenerate, we introduce and analyze embedded Gaussian orthogonal ensemble of random matrices generated by random two-body interactions that are spin (SSS) scalar [BEGOE(2)-$\cs$]. Embedding algebra for the BEGOE(2)-$\cs$ ensemble and also for BEGOE(1+2)-$\cs$ that includes the mean-field one-body part is U(2Ω)⊃U(Ω)⊗SU(2)U(2Ω) \supset U(Ω) \otimes SU(2)U(2Ω)⊃U(Ω)⊗SU(2) with SU(2) generating spin. A method for constructing the ensembles in fixed-(m,Sm,Sm,S) spaces has been developed. Numerical calculations show that for BEGOE(2)-$\cs$, the fixed-(m,S)(m,S)(m,S) density of states is close to Gaussian and level fluctuations follow GOE in the dense limit. For BEGOE(1+2)-$\cs$, generically there is Poisson to GOE transition in level fluctuations as the interaction strength (measured in the units of the average spacing of the single particle levels defining the mean-field) is increased. The interaction strength needed for the onset of the transition is found to decrease with increasing SSS. Covariances in energy centroids and spectral variances are analyzed. Propagation formula is derived for the variance propagator for the fixed-(m,S)(m,S)(m,S) ensemble averaged spectral variances. Variance propagator clearly shows, by applying the Jacquod and Stone prescription, that the BEGOE(2)-$\cs$ ensemble generates ground states with spin S=SmaxS=S_{max}S=Smax​. This is further corroborated by analyzing the structure of the ground states in the presence of the exchange interaction S^2\hat{S}^2S^2 in BEGOE(1+2)-$\cs$. Natural spin ordering is also observed with random interactions. Going beyond these, we also introduce pairing symmetry in the space defined by BEGOE(2)-$\cs$. Expectation values of the pairing Hamiltonian show that random interactions exhibit pairing correlations in the ground state region.
We investigate several distinct spectral crossovers amongst various integrable (Poissonian) and quantum-chaotic (Wigner-Dyson) limits of a 1D1\mathrm{D}1D disordered quantum spin (S=1/2\mathrm{S}=1/2S=1/2) model, by tuning the relative amplitudes of various Hamiltonian parameters to retain or break relevant unitary and antiunitary symmetries. The spin model consists of an isotropic and deterministic Heisenberg term, a random Ising term, an anisotropic and antisymmetric, but deterministic Dzyaloshinskii-Moriya (DM) term, and a Zeeman coupling to a random, inhomogeneous magnetic field. Since we are specially interested in crossovers involving a Gaussian symplectic ensemble (GSE) limit, we carry out all our calculations with an odd number of lattice sites (spins) that naturally results in eigenspectra with Kramers degeneracies (KD's). The various crossovers (viz., the reentrant Poissonian-to-GSE-to-Poissonian, Poissonian-to-GUE, GSE-to-GUE and the reentrant Poissonian-to-GOE-to-Poissonian crossovers) are investigated via detailed studies of both short-range (nearest-neighbour-spacings distribution, NNSD) and long-range [spectral rigidity ${\mathrm{\ensuremath{\Delta}}}_{3}(L)$ and number variance ${\mathrm{\ensuremath{\Sigma}}}^{2}(L)$ ] spectral correlations, where LLL is the spectral interval over which the long-range statistic is examined. The short-range studies show excellent agreement with RMT predictions. One of the highlights of this study is the systematic investigation of the consequences of retaining both eigenvalues corresponding to every Kramers doublet, in a crossover involving the GSE limit, and see how it evolves to a limit where the KD is naturally lifted. This is seen most clearly in the NNSD study of the GSE-to-GUE (Gaussian unitary ensemble) transition, achieved by gradually lifting the KD, using the random magnetic field. The NNSD plot in the GSE limit here exhibits a Dirac delta peak at zero splitting and a renormalized GSE hump at finite splitting, whose general analytical form and its asymptotic limit are derived. With an increasing symmetry breaking magnetic field the NNSD shows an interesting, dynamic two-peaked structure that finally converges to the standard GUE lineshape. We explain this trend in terms of a competition between the splittings amongst distinct Kramers doublets (related to unitary symmetries) and the Zeeman-like splittings induced by a breaking of the antiunitary time-reversal symmetry (TRS). This is investigated via the NNSD, the marginal spectral density (MSD) and the densities of states (DOS) for both spin models and RMT crossover matrix models. The first and the final short-range studies involve reentrant Poissonian-to-GOE(GSE)-to-Poissonian crossovers, where the final Poissonian is obtained by a many-body localization of states in the strongly disordered limit, whereas the initial Poissonian regime involve much more delocalized eigenstates. In the long-range spectral correlation studies, we shed light on the extent of agreement between our physical spin systems and RMT predictions. We find that the spin systems depart from the ideal RMT predictions for relatively finite $L\ensuremath{\sim}10\ensuremath{-}15$ at least, for the spectral rigidity and a much smaller $L\ensuremath{\sim}2\ensuremath{-}4$ for the number variance. It is further seen that the departure is usually sooner at the uncorrelated (Poissonian) upper end compared to the correlated (Wigner-Dyson) lower end. We carry out a detailed comparison between the local and the global crossover points, associated with the short-range and the long-range statistics respectively, and find that in most cases they seem to agree reasonably well, but for a few exceptions. Our studies also show that the long-range correlations may serve to distinguish between the two Poissonian limits (nonlocalized and localized) in the reentrant crossovers, which the short-range correlations fail to distinguish.
The Yukawa-Sachdev-Ye-Kitaev (YSYK) model, a solvable model of non-Fermi liquids with a proposed cavity-QED realization, couples NNN complex fermions to MMM bosons of frequency ω0ω_0ω0​ through random Yukawa couplings of strength ggg, and its dynamics depends only on R≡ω0/g2/3R\equivω_0/g^{2/3}R≡ω0​/g2/3. For R≪1R\ll1R≪1 the bosons are slow and act as a static random background in which the fermions are free, while for R≫1R\gg1R≫1 they are only virtually excited and generate a random four-fermion interaction of rank MMM, which is generically chaotic for M≥2M\geq2M≥2. Using exact diagonalization at infinite temperature, we follow the emergence of random-matrix statistics between these limits with unfolded spectral statistics and the Krylov complexity of the thermofield-double state. For complex couplings the level-spacing distribution evolves from Poisson to Gaussian-unitary-ensemble (GUE) statistics, the spectral form factor develops a correlation hole and a ramp, and the Krylov complexity evaluated on the unfolded spectrum develops a peak above its late-time plateau whose height reaches the GUE value. All diagnostics locate the onset around R≈0.3R\approx0.3R≈0.3, and adding boson modes moves it to smaller RRR. This delineates the parameter regime in which a cavity-QED realization would display random-matrix spectral correlations. On the other hand, real couplings give the same crossover towards Gaussian-orthogonal-ensemble (GOE) statistics, and along an interpolation between the two the peak height follows the mean gap ratio, so the peak identifies the symmetry class as well as the onset of chaos. Without unfolding the peak stays about 20%\%% below its random-matrix value in both classes, which we trace to the non-semicircular density of states, and at larger RRR the boson-number bands slow down the spreading of the state in Krylov space.