Crossover from Poisson to Wigner-Dyson level statistics in spin chains with integrability breaking

Rabson, David; Narozhny, B. N.; Millis, Andrew J.

doi:10.1103/physrevb.69.054403

Cited by 85 publications

(101 citation statements)

References 18 publications

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“…However, we shall show that it is practically effective for finite-size quantum systems. In fact, the above conjecture has been numerically confirmed for many quantum spin systems such as correlated spin systems [1][2][3][4][5][6][7] and disordered spin systems. [8][9][10][11][12] In the Anderson model of disordered systems, P Poi (s) and P Wig (s) characterize the localized and the metallic phases, respectively.…”

Section: Introductionmentioning

confidence: 72%

Level Statistics of XXZ Spin Chains with Discrete Symmetries: Analysis through Finite-size Effects

Kudo

Deguchi

2005

J. Phys. Soc. Jpn.

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Level statistics is discussed for XXZ spin chains with discrete symmetries for some values of the next-nearest-neighbor (NNN) coupling parameter. We show how the level statistics of the finite-size systems depends on the NNN coupling and the XXZ anisotropy, which should reflect competition among quantum chaos, integrability and finite-size effects. Here discrete symmetries play a central role in our analysis. Evaluating the level-spacing distribution, the spectral rigidity and the number variance, we confirm the correspondence between nonintegrability and Wigner behavior in the spectrum. We also show that non-Wigner behavior appears due to mixed symmetries and finite-size effects in some nonintegrable cases.

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Section: Introductionmentioning

confidence: 72%

Level Statistics of XXZ Spin Chains with Discrete Symmetries: Analysis through Finite-size Effects

Kudo

Deguchi

2005

J. Phys. Soc. Jpn.

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“…Thus, an accurate characterization of its integrability properties in physical regimes of interest has both a fundamental and practical significance. In the special case of spin-1/2 particles, a partial characterization of the integrability-to-chaos transition has been achieved, based on both clean systems where chaos is induced by coupling two different spin chains [54] or by adding next-nearest-neighbor interactions [54,55,56], as well as disordered systems, where the integrabilitybreaking term consists of random magnetic fields applied to all or a subset of spins [57,58,59]. Aside from their relevance to model real materials, disordered systems offer the added advantage of providing a natural arena to study the interplay between interaction and disorder, which remains a most challenging problem in condensedmatter physics.…”

Section: Disordered Heisenberg Modelsmentioning

confidence: 99%

Quantum chaos, delocalization, and entanglement in disordered Heisenberg models

et al. 2008

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We investigate disordered one-and two-dimensional Heisenberg spin lattices across a transition from integrability to quantum chaos from both a statistical many-body and a quantum-information perspective. Special emphasis is devoted to quantitatively exploring the interplay between eigenvector statistics, delocalization, and entanglement in the presence of nontrivial symmetries. The implications of basis dependence of state delocalization indicators (such as the number of principal components) is addressed, and a measure of relative delocalization is proposed in order to robustly characterize the onset of chaos in the presence of disorder. Both standard multipartite and generalized entanglement are investigated in a wide parameter regime by using a family of spin-and fermion-purity measures, their dependence on delocalization and on energy spectrum statistics being examined. A distinctive correlation between entanglement, delocalization, and integrability is uncovered, which may be generic to systems described by the two-body random ensemble and may point to a new diagnostic tool for quantum chaos. Analytical estimates for typical entanglement of random pure states restricted to a proper subspace of the full Hilbert space are also established and compared with random matrix theory predictions.

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“…Poisson statistics have been numerically verified on a case-by-case basis for some quantum integrable systems, including the Hubbard [2] and Heisenberg [2,3] models. On the other hand, general or analytic results on the spectra of quantum integrable models are lacking, in part due to the absence of a generally accepted unambiguous notion of quantum integrability, [9,10] and in part because existing results usually apply to isolated models instead of members of statistical ensembles like random matrices [11].…”

Section: Introductionmentioning

confidence: 99%

“…the system cannot be represented as a collection of decoupled harmonic oscillators. As integrability is destroyed by perturbing the Hamiltonian, the statistics are expected to cross over from Poisson to Wigner-Dyson at perturbation strengths as small as the inverse system size [3].…”

Section: Introductionmentioning

confidence: 99%

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Integrable matrix theory: Level statistics

2016

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We study level statistics in ensembles of integrable $N\times N$ matrices linear in a real parameter $x$. The matrix $H(x)$ is considered integrable if it has a prescribed number $n>1$ of linearly independent commuting partners $H^i(x)$ (integrals of motion) $\left[H(x),H^i(x)\right] = 0$, $\left[H^i(x), H^j(x)\right]$ = 0, for all $x$. In a recent work, we developed a basis-independent construction of $H(x)$ for any $n$ from which we derived the probability density function, thereby determining how to choose a typical integrable matrix from the ensemble. Here, we find that typical integrable matrices have Poisson statistics in the $N\to\infty$ limit provided $n$ scales at least as $\log{N}$; otherwise, they exhibit level repulsion. Exceptions to the Poisson case occur at isolated coupling values $x=x_0$ or when correlations are introduced between typically independent matrix parameters. However, level statistics cross over to Poisson at $ \mathcal{O}(N^{-0.5})$ deviations from these exceptions, indicating that non-Poissonian statistics characterize only subsets of measure zero in the parameter space. Furthermore, we present strong numerical evidence that ensembles of integrable matrices are stationary and ergodic with respect to nearest neighbor level statistics.Comment: 18 pages, 26 figures, discussion on number variance added; published versio

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Crossover from Poisson to Wigner-Dyson level statistics in spin chains with integrability breaking

Cited by 85 publications

References 18 publications

Level Statistics of XXZ Spin Chains with Discrete Symmetries: Analysis through Finite-size Effects

Level Statistics of XXZ Spin Chains with Discrete Symmetries: Analysis through Finite-size Effects

Quantum chaos, delocalization, and entanglement in disordered Heisenberg models

Integrable matrix theory: Level statistics

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