Rotationally invariant ensembles of integrable matrices

Yuzbashyan, Emil A.; Shastry, B. Sriram; Scaramazza, Jasen A.

doi:10.1103/physreve.93.052114

Cited by 5 publications

(17 citation statements)

References 55 publications

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“…17 while the rotationally invariant construction is given in Ref. 13. Again in the diagonal basis of V , the most general member of an ansatz type-M commuting family is…”

Section: Statistics Of Integrable Matrices Of Higher Typesmentioning

confidence: 99%

“…In particular, as explained in Ref. 13 the primary type-1 V is selected from the GOE, while the ansatz V is a certain primary type-1 matrix evaluated at x = −x 0 , i.e.…”

Section: Statistics Of Integrable Matrices Of Higher Typesmentioning

confidence: 99%

“…We argue in Ref. 13 that the N − M g i are a subset of eigenvalues of an N × N matrix from the GOE, so that for M not too large and x 0 1 we obtain Wigner-Dyson statistics in ansatz V .…”

Section: Statistics Of Integrable Matrices Of Higher Typesmentioning

confidence: 99%

“…2. We recently proposed an integrable matrix theory [13] (IMT) to describe eigenvalue statistics of integrable models. This theory is based on a rigorous notion of quantum integrability and provides ensembles of integrable matrix Hamiltonians with any given number of integrals of motion (see below).…”

Section: Introductionmentioning

confidence: 99%

“…The present work is a continuation of Ref. 13 where we formulated a rotationally invariant parametrization of integrable matrices and derived an appropriate probability density function (PDF) for the parameters, i.e. for ensembles of integrable matrices of any given type.…”

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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“…17 while the rotationally invariant construction is given in Ref. 13. Again in the diagonal basis of V , the most general member of an ansatz type-M commuting family is…”

Section: Statistics Of Integrable Matrices Of Higher Typesmentioning

confidence: 99%

“…In particular, as explained in Ref. 13 the primary type-1 V is selected from the GOE, while the ansatz V is a certain primary type-1 matrix evaluated at x = −x 0 , i.e.…”

Section: Statistics Of Integrable Matrices Of Higher Typesmentioning

confidence: 99%

Section: Statistics Of Integrable Matrices Of Higher Typesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Integrable matrix theory: Level statistics

2016

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Chaos due to symmetry-breaking in deformed Poisson ensemble

Das¹,

Ghosh²

2022

J. Stat. Mech.

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The competition between strength and correlation of coupling terms in a Hamiltonian defines numerous phenomenological models exhibiting spectral properties interpolating between those of Poisson (integrable) and Wigner–Dyson (chaotic) ensembles. It is important to understand how the off-diagonal terms of a Hamiltonian evolve as one or more symmetries of an integrable system are explicitly broken. We introduce a deformed Poisson ensemble to demonstrate an exact mapping of the coupling terms to the underlying symmetries of a Hamiltonian. From the maximum entropy principle we predict a chaotic limit which is numerically verified from the spectral properties and the survival probability calculations.

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Lattice-model parameters for ultracold nonreactive molecules: Chaotic scattering and its limitations

et al. 2017

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We calculate the parameters of the recently-derived many-channel Hubbard model that is predicted to describe ultracold nonreactive molecules in an optical lattice, going beyond the approximations used in Doçaj et al. [Phys. Rev. Lett. 116, 135301 (2016)]. Although those approximations are expected to capture the qualitative structure of the model parameters, finer details and quantitative values are less certain. To set expectations for experiments, whose results depend on the model parameters, we describe the approximations' regime of validity and the likelihood that experiments will be in this regime, discuss the impact that the failure of these approximations would have on the predicted model, and develop theories going beyond these approximations. Not only is it necessary to know the model parameters in order to describe experiments, but the connection that we elucidate between these parameters and the underlying assumptions that are used to derive them will allow molecule experiments to probe new physics. For example, transition state theory, which is used across chemistry and chemical physics, plays a key role in our determination of lattice parameters, thus connecting its physical assumptions to highly accurate experimental investigation.

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Rotationally invariant ensembles of integrable matrices

Cited by 5 publications

References 55 publications

Integrable matrix theory: Level statistics

Integrable matrix theory: Level statistics

Chaos due to symmetry-breaking in deformed Poisson ensemble

Lattice-model parameters for ultracold nonreactive molecules: Chaotic scattering and its limitations

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