Stringent Test on Power Spectrum of Quantum Integrable and Chaotic Systems

Abstract

Quantum chaotic and integrable systems are known to exhibit a characteristic $1/f$ and $1/f^2$ noise, respectively, in the power spectrum associated to their spectral fluctuations. A recent work [R. Riser, V. A. Osipov, and E. Kanzieper, \textit{Power Spectrum of Long Eigenlevel Sequences in Quantum Chaotic Systems}, Phys. Rev. Lett. \textbf{118}, 204101 (2017)] calls into question the approximations used to derive these results from random matrix theory. In this paper we show that such approximations do remain valid under almost any circumstances. For the integrable limit, we devise a protocol to exactly recover the original results. As a corollary, we show that the theoretical predictions for other statistics are bound for failure regarding long-range correlations, due to unavoidable spurious effects emerging from the analysis. By means of a rigorous statistical test, we also show that the corrections for the chaotic case introduced in the aforementioned paper require huge statistics to become relevant ---averages over more than $1000$ realizations are mandatory. As an application, we study a paradigmatic model for the crossover from the thermal to the many-body localized phase. We show that our protocol succeeds in describing the crossover. Furthermore, it also succeeds in proving that the Gaussian $\beta$-ensemble fails to account for long-range correlations between the energy levels of this paradigmatic model.

arXiv comment18 pages, 10 figures, 1 table, 1 appendix

References in corpus (16)

• The many-body localization phase transition
• Many-body localization edge in the random-field Heisenberg chain
• The distribution of the ratio of consecutive level spacings in random matrix ensembles
• Exact Spectral Form Factor in a Minimal Model of Many-Body Quantum Chaos
• Spectral statistics across the many-body localization transition
• Misleading signatures of quantum chaos
• Theoretical derivation of 1/f noise in quantum chaos
• Global spectrum fluctuations for the $\beta$-Hermite and $\beta$-Laguerre ensembles via matrix models
• Power spectrum analysis and missing level statistics of microwave graphs with violated time reversal invariance
• $1/f^{\alpha }$ Noise in Spectral Fluctuations of Quantum Systems
• Non-universality in the spectral properties of time-reversal invariant microwave networks and quantum graphs
• Poisson to GOE transition in the distribution of the ratio of consecutive level spacings
• Spectral statistics of molecular resonances in erbium isotopes: How chaotic are they?
• Power Spectrum of Long Eigenlevel Sequences in Quantum Chaotic Systems
• Nearest-neigbor spacing distributions of the beta-Hermite ensemble of random matrices
• Missing level statistics and the power spectrum of level fluctuations analysis of three-dimensional chaotic microwave cavities

Cited by in corpus (1)

• Distribution of the Ratio of Consecutive Level Spacings for Different Symmetries and Degrees of Chaos