Observation of Wigner-Dyson level statistics in a classically integrable system

Elkamshishy, Ahmed A.; Greene, Chris H.

doi:10.1103/physreve.103.062211

Cited by 4 publications

(2 citation statements)

References 46 publications

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“…This procedure applies to the bulk of the spectrum. The spacing distribution continues to be a popular tool in today's applications, ranging from quantum spin chains [2], quantum circuits [3], chemical isotopes [4] and graphene [5] to classical integrable systems [6] and theoretical studies [7], which are all from past year. We refer to [1,8] for a list of more topics and references therein.…”

Section: Introductionmentioning

confidence: 99%

Consecutive level spacings in the chiral Gaussian unitary ensemble: from the hard and soft edge to the bulk

Akemann

Gorski

Kieburg

2022

J. Phys. A: Math. Theor.

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The local spectral statistics of random matrices forms distinct universality classes, strongly depending on the position in the spectrum. Surprisingly, the spacing between consecutive eigenvalues at the spectral edges has received little attention, where the density diverges or vanishes, respectively. This different behaviour is called hard or soft edge. We show that the spacings at the edges are almost indistinguishable from the spacing in the bulk of the spectrum. We present analytical results for consecutive spacings between the $k$th and $(k + 1)$st smallest eigenvalues in the chiral Gaussian unitary ensemble, both for finite- and large-$n$. The result depends on the number of the generic zero modes $\nu$ and the number of flavours $N_f$ , which are given in terms of characteristic polynomials, as motivated by Quantum Chromodynamics (QCD). We find that the convergence in $n$ is very rapid. The same can be said separately about the limit $k\to\infty$ (limit to the bulk) and $\nu\to\infty$ (limit to the soft edge). Interestingly, the Wigner surmise is a very good approximation for all these cases and, apart from $k = 1$, shows a deviation below one percent. These findings are corroborated with Monte-Carlo simulations. We finally compare for $k = 1$ with data from QCD on the lattice, being in this symmetry class.

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Section: Introductionmentioning

confidence: 99%

Consecutive level spacings in the chiral Gaussian unitary ensemble: from the hard and soft edge to the bulk

Akemann

Gorski

Kieburg

2022

J. Phys. A: Math. Theor.

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show abstract

“…That is particularly fitting in the present context wherein the semiclassical regime plays an important role. We believe that the BGS conjecture is now universally accepted, with well-understood exceptions lending additional support to its generic validity, see, for instance, [20].…”

Section: Methodsmentioning

confidence: 79%

Thermalization of closed chaotic many-body quantum systems

Weidenmüller

2024

J. Phys. A: Math. Theor.

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We investigate thermalization of a closed chaotic many-body quantum system by combining the Hartree-Fock (HF) approach with the Bohigas-Giannoni-Schmit (BGS) conjecture. The conjecture implies that locally, the residual interaction causes the statistics of eigenvalues and eigenfunctions of the full Hamiltonian to agree with random-matrix presdictions. The agreement is confined to an interval $\Delta$ (the correlation width). The results are used to calculate ${\rm Tr} (A \rho(t))$. Here $\rho(t)$ is the time-dependent density matrix of the system, and $A$ represents an observable. In the semiclassical regime, the average $\langle {\rm Tr}(A \rho(t)) \rangle$ decays on the time scale $\hbar / \Delta$ towards an asymptotic value. If the energy spread of the system is of order $\Delta$, that value is given by ${\rm Tr} (A \rho_{\rm eq})$ where $\rho_{\rm eq}$ is the density matrix of statistical equilibrium. The correlation width $\Delta$ is the central parameter of our approach. We argue that $\Delta$ occurs generically in chaotic quantum systems and plays the same central role.

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Dynamic resonant tunneling via a quasibound superstate

Zangwill

Granot

2022

Phys. Rev. A

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Observation of Wigner-Dyson level statistics in a classically integrable system

Cited by 4 publications

References 46 publications

Consecutive level spacings in the chiral Gaussian unitary ensemble: from the hard and soft edge to the bulk

Consecutive level spacings in the chiral Gaussian unitary ensemble: from the hard and soft edge to the bulk

Thermalization of closed chaotic many-body quantum systems

Dynamic resonant tunneling via a quasibound superstate

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