Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2311.14715 · Journal of Statistical Mechanics Theory and Experiment

Level spacing statistics of two unitary equivalent models with distinct local symmetries

Biquan Yang, Zijian Xiong

Published Dec 1, 2025• 0 citations
arXivHTMLPDF

Abstract

Abstract The full spectrum and integrability of unitary equivalent models are the same. A standard diagnostic tool for integrability is level spacing statistics, which requires separating the full spectrum into sectors according to the symmetry. When two unitary equivalent models have different symmetries, it is interesting to see how their level spacing statistics exhibit consistent conclusions. In this work, we examine the level spacing statistics of two unitary equivalent models with distinct local symmetries. The first model is the spin-1 XXZ chain H , and the second model is obtained by Kennedy–Tasaki transformation U KT H U KT . We found that the level spacings of model H follow the statistics of the Gaussian orthogonal ensemble (GOE). However, model U KT H U KT only displays GOE statistics in some sectors after a ‘hidden nonlocal symmetry’ is resolved. In addition, we study how the bond-centered inversion transforms under U KT . A mapping relation between the levels in the sector { Z , X , I } of model H and the sector { Z , X , I ′ } of model U KT H U KT is found, where Z , X , I correspond to the quantum numbers of π rotation around z , x axes, and bond-centered inversion.

DOI
10.1088/1742-5468/ae1ffc

Related papers

Level repulsion in integrable and almost-integrable quantum spin models

Theodore C. Hsu, J C Anglès d'Auriac

The repartition of the separation between energy levels of various isotropic S=1/2 antiferromagnetic chains is studied numerically with the aim of investigating the transition from integrable to noninte- grable systems. We begin by displaying the level separation distribution of the integrable Bethe chain. Then two nonintegrable systems, two coupled chains, and a next-nearest-neighbor coupled chain, are studied as a function of the coupling. We examine how the level spacing evolves from the Poisson distribution to the Gaussian-orthogonal-ensemble distribution. Finally we consider the Haldane-Shastry 1/${\mathit{r}}^{2}$ model. A number of conclusions regarding the behavior and relevance of the level-spacing distribution in these spin systems is drawn.

arXiv:cond-mat/9211004•Physical review. B, Condensed matter•Jun 1, 1993•80 citations

Finite size scaling in crossover among different random matrix ensembles in microscopic lattice models

Ranjan Modak, Subroto Mukerjee

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.

arXiv:1310.1443•New Journal of Physics•Sep 15, 2014•39 citations

Finite size scaling in crossover among different random matrix ensembles in microscopic lattice models

Ranjan Modak, Subroto Mukerjee

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.

•Jan 1, 2014•37 citations

Anomalous suppression of quantum chaos between two integrable limits

Roopayan Ghosh, Henry Davenport, Frank Schindler

Level statistics in non-integrable quantum many-body systems with time reversal symmetry are expected to follow the Gaussian Orthogonal Ensemble (GOE), a hallmark of quantum chaos. However, we show that the interacting Su-Schrieffer-Heeger model exhibits a clear suppression of the mean level-spacing ratio $\langle r\rangle$ from the GOE value $\approx 0.535$, persisting deep in the nonintegrable regime. This challenges the conventional association between non-integrability and fully chaotic spectral statistics. Using exact diagonalization supported by semi-analytical arguments, we trace this anomaly to incomplete hybridization of many-body band states inherited from the noninteracting band structure. The resulting restructuring of the spectrum weakens level repulsion without restoring integrability. We show the robustness of this mechanism in extensions of the model which break chiral and inversion symmetry.

cond-mat.stat-mechCross-listedarXiv:2607.05506•Jul 8, 2026•0 citations

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

Sasanka Dowarah

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.

arXiv:2510.15829•Physical review. B./Physical review. B•Dec 17, 2025•0 citations

Determining the Number of Symmetry Sectors in Composite Quantum Spectra

Nisarg Vyas, S. Harshini Tekur, M. S. Santhanam

The spectral fluctuations in complex quantum systems are modeled in terms of spectra from random matrices. Quantitative agreement with random matrices holds only when the quantum spectrum is a pure sequence, and not a composite spectrum that mixes levels from different symmetry sectors. It was shown earlier that discrete symmetries in composite spectra can be detected using higher-order spacing ratio statistics provided the symmetry sectors are of equal dimensions. The general case is when the symmetry sectors have unequal dimensions, and the symmetries of the Hamiltonian system are unknown or only partially known. In this work, we address how the number of symmetry sectors can be determined from a composite spectrum arising in Hermitian and non-Hermitian settings. A Best Subset Technique is proposed for determining the number of the symmetry sectors by comparing the spectral fluctuations to that of an appropriate random matrix ensemble without requiring de-symmetrization. It is demonstrated using spectra from Gaussian, Wishart and Ginibre ensembles, and also for spectra drawn from many-body quantum systems.

quant-pharXiv:2610.03108•Oct 5, 2026
Abstract The full spectrum and integrability of unitary equivalent models are the same. A standard diagnostic tool for integrability is level spacing statistics, which requires separating the full spectrum into sectors according to the symmetry. When two unitary equivalent models have different symmetries, it is interesting to see how their level spacing statistics exhibit consistent conclusions. In this work, we examine the level spacing statistics of two unitary equivalent models with distinct local symmetries. The first model is the spin-1 XXZ chain H , and the second model is obtained by Kennedy–Tasaki transformation U KT H U KT . We found that the level spacings of model H follow the statistics of the Gaussian orthogonal ensemble (GOE). However, model U KT H U KT only displays GOE statistics in some sectors after a ‘hidden nonlocal symmetry’ is resolved. In addition, we study how the bond-centered inversion transforms under U KT . A mapping relation between the levels in the sector { Z , X , I } of model H and the sector { Z , X , I ′ } of model U KT H U KT is found, where Z , X , I correspond to the quantum numbers of π rotation around z , x axes, and bond-centered inversion.
The repartition of the separation between energy levels of various isotropic S=1/2 antiferromagnetic chains is studied numerically with the aim of investigating the transition from integrable to noninte- grable systems. We begin by displaying the level separation distribution of the integrable Bethe chain. Then two nonintegrable systems, two coupled chains, and a next-nearest-neighbor coupled chain, are studied as a function of the coupling. We examine how the level spacing evolves from the Poisson distribution to the Gaussian-orthogonal-ensemble distribution. Finally we consider the Haldane-Shastry 1/r2{\mathit{r}}^{2}r2 model. A number of conclusions regarding the behavior and relevance of the level-spacing distribution in these spin systems is drawn.
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.
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.
Level statistics in non-integrable quantum many-body systems with time reversal symmetry are expected to follow the Gaussian Orthogonal Ensemble (GOE), a hallmark of quantum chaos. However, we show that the interacting Su-Schrieffer-Heeger model exhibits a clear suppression of the mean level-spacing ratio ⟨r⟩\langle r\rangle⟨r⟩ from the GOE value ≈0.535\approx 0.535≈0.535, persisting deep in the nonintegrable regime. This challenges the conventional association between non-integrability and fully chaotic spectral statistics. Using exact diagonalization supported by semi-analytical arguments, we trace this anomaly to incomplete hybridization of many-body band states inherited from the noninteracting band structure. The resulting restructuring of the spectrum weakens level repulsion without restoring integrability. We show the robustness of this mechanism in extensions of the model which break chiral and inversion symmetry.
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.
The spectral fluctuations in complex quantum systems are modeled in terms of spectra from random matrices. Quantitative agreement with random matrices holds only when the quantum spectrum is a pure sequence, and not a composite spectrum that mixes levels from different symmetry sectors. It was shown earlier that discrete symmetries in composite spectra can be detected using higher-order spacing ratio statistics provided the symmetry sectors are of equal dimensions. The general case is when the symmetry sectors have unequal dimensions, and the symmetries of the Hamiltonian system are unknown or only partially known. In this work, we address how the number of symmetry sectors can be determined from a composite spectrum arising in Hermitian and non-Hermitian settings. A Best Subset Technique is proposed for determining the number of the symmetry sectors by comparing the spectral fluctuations to that of an appropriate random matrix ensemble without requiring de-symmetrization. It is demonstrated using spectra from Gaussian, Wishart and Ginibre ensembles, and also for spectra drawn from many-body quantum systems.