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:2309.07116 · Journal of Statistical Mechanics Theory and Experiment

Level statistics of the one-dimensional dimerized Hubbard model

Karin Haderlein, David J. Luitz, Corinna Kollath, Ameneh Sheikhan

Published Jul 5, 2024• 3 citations
arXivHTMLPDF

Abstract

Abstract The statistical properties of level spacings provide valuable insights into the dynamical properties of a many-body quantum systems. We investigate the level statistics of the Fermi–Hubbard model with dimerized hopping amplitude and find that after taking into account translation, reflection, spin and η pairing symmetries to isolate irreducible blocks of the Hamiltonian, the level spacings in the limit of large system sizes follow the distribution expected for hermitian random matrices from the Gaussian orthogonal ensemble. We show this by analyzing the distribution of the ratios of consecutive level spacings in this system, its cumulative distribution and quantify the deviations of the distributions using their mean, standard deviation and skewness.

DOI
10.1088/1742-5468/ad5270

Related papers

Level statistics of the one-dimensional ionic Hubbard model

Jeannette De Marco, Luisa Tolle, Catalin-Mihai Halati +3 more

In this paper we analyze the spectral level statistics of the one-dimensional ionic Hubbard model, the Hubbard model with an alternating on-site potential. In particular, we focus on the statistics of the gap ratios between consecutive energy levels. This quantity is often used in order to signal whether a many-body system is integrable or chaotic. A chaotic system has typically the statistics of a Gaussian ensemble of random matrices while the spectral properties of the integrable system follow a Poisson statistics. We find that whereas the Hubbard model without alternating potential is known to be integrable and its spectral properties follow a Poissonian statistics, the presence of an alternating potential causes a drastic change in the spectral properties, which resemble the one of a Gaussian ensemble of random matrices. However, to uncover this behavior one has to separately consider the blocks of all symmetries of the ionic Hubbard model.

arXiv:2203.05379•Physical Review Research•Aug 11, 2022•13 citations

Energy level statistics of the two-dimensional Hubbard model at low filling

Henrik Bruus, J C Anglès d'Auriac

The energy level statistics of the Hubbard model for L\ifmmode\times\else\texttimes\fi{}L square lattices (L=3,4,5,6) at low filling (four electrons) is studied numerically for a wide range of coupling strength. All known symmetries of the model (space, spin, and pseudospin symmetry) have been taken into account explicitly from the beginning of the calculation by projecting into symmetry-invariant subspaces. The details of this group theoretical treatment are presented with special attention to the nongeneric case of L=4, where a particular complicated space group appears. For all the lattices studied, a significant amount of levels within each symmetry invariant subspaces remains degenerated, but except for L=4 the ground state is nondegenerate. We explain the remaining degeneracies, which occur only for very specific interaction-independent states, and we disregard these states in the statistical spectral analysis. The intricate structure of the Hubbard spectra necessitates a careful unfolding procedure, which is thoroughly discussed. Finally, we present our results for the level spacing distribution, the number variance ${\mathrm{\ensuremath{\Sigma}}}^{2}$, and the spectral rigidity ${\mathrm{\ensuremath{\Delta}}}_{3}$, which essentially all are close to the corresponding statistics for random matrices of the Gaussian ensemble independent of the lattice size and the coupling strength. Even very small coupling strengths approaching the integrable zero coupling limit lead to Gaussian ensemble statistics, stressing the nonperturbative nature of the Hubbard model.

arXiv:cond-mat/9610142•Physical review. B, Condensed matter•Apr 1, 1997•43 citations

Statistical properties of the spectrum of the extended Bose–Hubbard model

Corinna Kollath, Guillaume Roux, Giulio Biroli +1 more

Motivated by the role that spectral properties play for the dynamical evolution of a quantum many-body system, we investigate the level spacing statistics of the extended Bose–Hubbard model. In particular, we focus on the distribution of the ratio of adjacent level spacings, useful at large interactions, to distinguish between chaotic and non-chaotic regimes. After revisiting the bare Bose–Hubbard model, we study the effect of two different perturbations: next-nearest-neighbor hopping and nearest-neighbor interaction. The system size dependence is investigated together with the effect of the proximity to integrable points or lines. Lastly, we discuss the consequences of a cutoff in the number of onsite bosons on the level statistics.

arXiv:1004.2203•Journal of Statistical Mechanics Theory and Experiment•Aug 1, 2010•98 citations

Connection between Low Energy Effective Hamiltonians and Energy Level Statistics

Mario Di Stasio, X. Zotos

We study the level statistics of a nonintegrable one-dimensional interacting fermionic system characterized by the Gaussian orthogonal ensemble distribution. We calculate numerically on a finite size system the level spacing distribution $P(s)$ and the Dyson-Mehta ${\ensuremath{\Delta}}_{3}$ correlation. We observe that its low energy spectrum follows rather the Poisson distribution, characteristic of an integrable system, consistent with the fact that the low energy excitations of this system are described by the Luttinger model. We propose this random matrix theory analysis as a probe, but no proof, for the existence and integrability of low energy effective Hamiltonians for strongly correlated systems.

arXiv:cond-mat/9410060•Physical Review Letters•Mar 13, 1995•26 citations

Dynamics and level statistics of interacting fermions in the lowest Landau level

Mikael Fremling, Cécile Repellin, Jean-Marie Stéphan +3 more

We consider the unitary dynamics of interacting fermions in the lowest Landau level, on spherical and toroidal geometries. The dynamics are driven by the interaction Hamiltonian which, viewed in the basis of single-particle Landau orbitals, contains correlated pair hopping terms in addition to static repulsion. This setting and this type of Hamiltonian has a significant history in numerical studies of fractional quantum Hall (FQH) physics, but the many-body quantum dynamics generated by such correlated hopping has not been explored in detail. We focus on initial states containing all the fermions in one block of orbitals. We characterize in detail how the fermionic liquid spreads out starting from such a state. We identify and explain differences with regular (single-particle) hopping Hamiltonians. Such differences are seen, e.g. in the entanglement dynamics, in that some initial block states are frozen or near-frozen, and in density gradients persisting in long-time equilibrated states. Examining the level spacing statistics, we show that the most common Hamiltonians used in FQH physics are not integrable, and explain that GOE statistics (level statistics corresponding to the Gaussian orthogonal ensemble) can appear in many cases despite the lack of time-reversal symmetry.

arXiv:1805.07293•New Journal of Physics•Oct 10, 2018•14 citations

The spectrum of the two-dimensional Hubbard model at low filling

Henrik Bruus, J C Anglès d'Auriac

The entire energy spectrum is calculated numerically for the two-dimensional Hubbard model on square lattices with four electrons varying the interaction strength U in a wide range. Taking all symmetries explicitly into account a level statistical analysis is performed on this strongly correlated 2D system. The level spacing distribution and the number variance exhibit a random matrix (GOE) behavior for all positive values of U , the latter however only up to a medium energy scale λ* ≈ 3 independent of U .

arXiv:cond-mat/9507012•Europhysics Letters (EPL)•Aug 10, 1996•24 citations
Abstract The statistical properties of level spacings provide valuable insights into the dynamical properties of a many-body quantum systems. We investigate the level statistics of the Fermi–Hubbard model with dimerized hopping amplitude and find that after taking into account translation, reflection, spin and η pairing symmetries to isolate irreducible blocks of the Hamiltonian, the level spacings in the limit of large system sizes follow the distribution expected for hermitian random matrices from the Gaussian orthogonal ensemble. We show this by analyzing the distribution of the ratios of consecutive level spacings in this system, its cumulative distribution and quantify the deviations of the distributions using their mean, standard deviation and skewness.
In this paper we analyze the spectral level statistics of the one-dimensional ionic Hubbard model, the Hubbard model with an alternating on-site potential. In particular, we focus on the statistics of the gap ratios between consecutive energy levels. This quantity is often used in order to signal whether a many-body system is integrable or chaotic. A chaotic system has typically the statistics of a Gaussian ensemble of random matrices while the spectral properties of the integrable system follow a Poisson statistics. We find that whereas the Hubbard model without alternating potential is known to be integrable and its spectral properties follow a Poissonian statistics, the presence of an alternating potential causes a drastic change in the spectral properties, which resemble the one of a Gaussian ensemble of random matrices. However, to uncover this behavior one has to separately consider the blocks of all symmetries of the ionic Hubbard model.
The energy level statistics of the Hubbard model for L\ifmmode\times\else\texttimes\fi{}L square lattices (L=3,4,5,6) at low filling (four electrons) is studied numerically for a wide range of coupling strength. All known symmetries of the model (space, spin, and pseudospin symmetry) have been taken into account explicitly from the beginning of the calculation by projecting into symmetry-invariant subspaces. The details of this group theoretical treatment are presented with special attention to the nongeneric case of L=4, where a particular complicated space group appears. For all the lattices studied, a significant amount of levels within each symmetry invariant subspaces remains degenerated, but except for L=4 the ground state is nondegenerate. We explain the remaining degeneracies, which occur only for very specific interaction-independent states, and we disregard these states in the statistical spectral analysis. The intricate structure of the Hubbard spectra necessitates a careful unfolding procedure, which is thoroughly discussed. Finally, we present our results for the level spacing distribution, the number variance ${\mathrm{\ensuremath{\Sigma}}}^{2}$, and the spectral rigidity ${\mathrm{\ensuremath{\Delta}}}_{3}$, which essentially all are close to the corresponding statistics for random matrices of the Gaussian ensemble independent of the lattice size and the coupling strength. Even very small coupling strengths approaching the integrable zero coupling limit lead to Gaussian ensemble statistics, stressing the nonperturbative nature of the Hubbard model.
Motivated by the role that spectral properties play for the dynamical evolution of a quantum many-body system, we investigate the level spacing statistics of the extended Bose–Hubbard model. In particular, we focus on the distribution of the ratio of adjacent level spacings, useful at large interactions, to distinguish between chaotic and non-chaotic regimes. After revisiting the bare Bose–Hubbard model, we study the effect of two different perturbations: next-nearest-neighbor hopping and nearest-neighbor interaction. The system size dependence is investigated together with the effect of the proximity to integrable points or lines. Lastly, we discuss the consequences of a cutoff in the number of onsite bosons on the level statistics.
We study the level statistics of a nonintegrable one-dimensional interacting fermionic system characterized by the Gaussian orthogonal ensemble distribution. We calculate numerically on a finite size system the level spacing distribution P(s)P(s)P(s) and the Dyson-Mehta ${\ensuremath{\Delta}}_{3}$ correlation. We observe that its low energy spectrum follows rather the Poisson distribution, characteristic of an integrable system, consistent with the fact that the low energy excitations of this system are described by the Luttinger model. We propose this random matrix theory analysis as a probe, but no proof, for the existence and integrability of low energy effective Hamiltonians for strongly correlated systems.
We consider the unitary dynamics of interacting fermions in the lowest Landau level, on spherical and toroidal geometries. The dynamics are driven by the interaction Hamiltonian which, viewed in the basis of single-particle Landau orbitals, contains correlated pair hopping terms in addition to static repulsion. This setting and this type of Hamiltonian has a significant history in numerical studies of fractional quantum Hall (FQH) physics, but the many-body quantum dynamics generated by such correlated hopping has not been explored in detail. We focus on initial states containing all the fermions in one block of orbitals. We characterize in detail how the fermionic liquid spreads out starting from such a state. We identify and explain differences with regular (single-particle) hopping Hamiltonians. Such differences are seen, e.g. in the entanglement dynamics, in that some initial block states are frozen or near-frozen, and in density gradients persisting in long-time equilibrated states. Examining the level spacing statistics, we show that the most common Hamiltonians used in FQH physics are not integrable, and explain that GOE statistics (level statistics corresponding to the Gaussian orthogonal ensemble) can appear in many cases despite the lack of time-reversal symmetry.
The entire energy spectrum is calculated numerically for the two-dimensional Hubbard model on square lattices with four electrons varying the interaction strength U in a wide range. Taking all symmetries explicitly into account a level statistical analysis is performed on this strongly correlated 2D system. The level spacing distribution and the number variance exhibit a random matrix (GOE) behavior for all positive values of U , the latter however only up to a medium energy scale λ* ≈ 3 independent of U .