PRL 118, 204101 (2017)
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PHYSICAL REVIEW LETTERS
Power Spectrum of Long Eigenlevel Sequences in Quantum Chaotic Systems 1
Roman Riser,1 Vladimir Al. Osipov,2 and Eugene Kanzieper1 Department of Applied Mathematics, H.I.T.–Holon Institute of Technology, Holon 5810201, Israel 2 Division of Chemical Physics, Lund University, Getingevägen 60, Lund 22241, Sweden (Received 23 March 2017; published 16 May 2017)
We present a nonperturbative analysis of the power spectrum of energy level fluctuations in fully chaotic quantum structures. Focusing on systems with broken time-reversal symmetry, we employ a finite-N random matrix theory to derive an exact multidimensional integral representation of the power spectrum. The N → ∞ limit of the exact solution furnishes the main result of this study—a universal, parameter-free prediction for the power spectrum expressed in terms of a fifth Painlevé transcendent. Extensive numerics lends further support to our theory which, as discussed at length, invalidates a traditional assumption that the power spectrum is merely determined by the spectral form factor of a quantum system. DOI: 10.1103/PhysRevLett.118.204101
Introduction.—Spectral fluctuations of quantum systems reflect the nature—regular or chaotic—of their underlying classical dynamics [1]. In the case of fully chaotic classical dynamics, hyperbolicity (exponential sensitivity to initial conditions) and ergodicity (typical classical trajectories fill out available phase space uniformly) make quantum properties of chaotic systems universal [2]. At sufficiently long times [3] t > T , the single particle dynamics is governed by global symmetries rather than by system specialties and is accurately described by the random matrix theory [4,5] (RMT). The emergence of universal statistical laws, anticipated by Bohigas, Giannoni, and Schmit [2], has been established within the semiclassical approach [6] which links correlations in quantum spectra to correlations between periodic orbits in the associated classical geodesics. The time scale T of compromised spectral universality is set by the period T 1 of the shortest closed orbit and the Heisenberg time T H , such that T 1 ≪ T ≪ T H . Several statistical measures of level fluctuations have been devised in quantum chaology. Long-range correlations of eigenlevels on the unfolded energy scale [4] are measured by the variance Σ2 ðLÞ ¼ var½N ðLÞ of number of levels N ðLÞ in the interval of length L and by (closely related) spectral rigidity [7] Δ3 ðLÞ. Both statistics probe the two-level correlations only and exhibit [8] a universal RMT behavior provided the interval L is not too long, 1 ≪ L ≪ T H =T 1 . For one, the number variance is described by the logarithmic law 2 Σ2chaos ðLÞ ¼ 2 log L þ Oð1Þ; π β
ð1Þ
which indicates presence of the long-range repulsion between eigenlevels. Here, β ¼ 1, 2 and 4 denote the Dyson symmetry index [4]. For more distant levels, L ≫ T H =T 1 , system-specific features show up: both statistics display quasirandom oscillations with wavelengths being inversely proportional to periods of short closed orbits. 0031-9007=17=118(20)=204101(5)
Individual features of quantum chaotic systems become less pronounced in spectral measures that probe the shortrange fluctuations as these are largely determined by the long periodic orbits. The distribution of level spacing between (unfolded) consecutive eigenlevels, PðsÞ ¼ hδðs − Ej þ Ejþ1 Þi, is the most commonly used shortrange statistics. At small spacings, s ≪ 1, it is mostly contributed by the two-point correlations, showing the phenomenon of symmetry-driven level repulsion, PðsÞ ∝ sβ . (In a simple-minded fashion, this result can be read out from the Wigner surmise [4]). As the s grows, the spacing distribution becomes increasingly influenced by spectral correlation functions of all orders. In the universal regime (s ≲ T H =T ), these are best accounted for by the RMT machinery which produces parameter-free (but β-dependent) representations of level spacing distributions in terms of Fredholm determinants/Pfaffians and Painlevé transcendents. For quantum chaotic systems with broken time-reversal symmetry (β ¼ 2)—that will be the focus of our study—the level spacing distribution is given by the famous Gaudin-Mehta formula [5,9] Z 2πs σ ðtÞ d2 0 Pchaos ðsÞ ¼ 2 exp dt : ð2Þ t ds 0 Here, σ 0 ðtÞ is the fifth Painlevé transcendent defined as the solution to the nonlinear equation (ν ¼ 0) ðtσ 00ν Þ2 þ ðtσ 0ν − σ ν Þðtσ 0ν − σ ν þ 4ðσ 0ν Þ2 Þ − 4ν2 ðσ 0ν Þ2 ¼ 0 ð3Þ subject to the boundary condition σ 0 ðtÞ ∼ −t=2π as t → 0. The universal RMT laws [Eqs. (1) and (2)] do not apply to quantum systems with completely integrable classical dynamics. Following Berry and Tabor [10], these belong to a different universality class—the one shared by the Poisson point process. In particular, level spacings in a
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generic integrable quantum system exhibit statistics of waiting times between consecutive events in a Poisson process. This leads to the radically different fluctuation laws: the number variance Σ2int ðLÞ ¼ L is no longer logarithmic while the level spacing distribution Pint ðsÞ ¼ e−s becomes exponential, with no signatures of level repulsion whatsoever. Such a selectivity of short- and long-range spectral statistical measures has long been used to uncover underlying classical dynamics of quantum systems. Power spectrum: Definition and early results.—To obtain a more accurate characterization of the quantum chaos, it is advantageous to use spectral statistics which probe the correlations between both nearby and distant eigenlevels. Such a statistical indicator has been suggested in Ref. [11]. Interpreting a sequence of ordered unfolded eigenlevels fE1 ≤ ≤ EM g of the length M ≫ 1 as a discrete-time random process, these authors defined its average power spectrum [12] SM ðωÞ ¼
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PHYSICAL REVIEW LETTERS
M X M 1X hδEl δEm ieiωðl−mÞ M l¼1 m¼1
networks [16]. For the power spectrum analysis of FanoFeshbach resonances in an ultracold gas of Erbium atoms [17], the reader is referred to Ref. [18]. For quantum chaotic systems, the universal 1=ωk law for the average power spectrum in the frequency domain T =T H ≲ ωk ≪ 1 can be read out from the existing RMT literature. Indeed, defining a set of discrete Fourier coefficients M 1 X ak ¼ pffiffiffiffiffi δEl eiωk l M l¼1
of level displacements fδEl g, one observes the relation SM ðωk Þ ¼ var½ak . Statistics of the Fourier coefficients fak g were studied in some detail [19] within the Dyson’s Brownian motion model [20]. In particular, it is known that, in the limit k ≪ M, they are independent Gaussian distributed random variables with zero mean and the variance var½ak ¼ M=ð2π 2 βkÞ. This implies
ð4Þ
in terms of discrete Fourier transform of the correlator hδEl δEm i of level displacements fδEl ≡ El − hEl ig on the unfolded scale (hEl i ¼ lΔ with Δ ¼ 1). Notice that dimensionless frequencies therein belong to the interval 0 ≤ ω ≤ ωNy , where ωNy ≡ π is the Nyquist frequency. A discrete nature of a so-defined random process allows us to restrict ω’s to a finite set ωk ¼ 2πk=M with [13] k ∈ f1; 2; …; M=2g. Considering Eq. (4) through the prism of a semiclassical approach, one readily realizes that, at low frequencies ωk ≪ T =T H , the power spectrum is largely affected by system-specific correlations between very distant eigenlevels (accounted for by short periodic orbits). For higher frequencies, ωk ≳ T =T H , the contribution of longer periodic orbits becomes increasingly important and the power spectrum enters the universal regime. Eventually, in the frequency domain T =T H ≪ ωk ≤ ωNy , long periodic orbits win over and the power spectrum gets shaped by correlations between the nearby levels. Numerical simulations [11] have revealed that the average power spectrum SM ðωk Þ discriminates sharply between quantum systems with chaotic and integrable classical dynamics. While this was not completely unexpected, another finding of Ref. [11] came as quite a surprise: numerical data for SM ðωk Þ could accurately be fitted by simple power-law curves, SM ðωk Þ ∼ 1=ωk and SM ðωk Þ ∼ 1=ω2k , for quantum systems with chaotic and integrable classical dynamics, respectively. In quantum systems with mixed classical dynamics, numerical evidence was presented [14] for the power-law of the form SM ðωk Þ ∼ 1=ωαk with the exponent 1 < α < 2 measuring a “degree of chaoticity.” Experimentally, the 1=ωk noise was measured in Sinai microwave billiards [15] and microwave
ð5Þ
ð0Þ
SM ðωk Þ ≃ S∞ ðωk Þ ¼
1 πβωk
ð6Þ
in concert with numerical findings. For larger k (in particular, for k ∼ M), fluctuation properties of the Fourier coefficients fak g are unknown. An attempt to determine SM ðωk Þ for higher frequencies up to ωk ¼ ωNy was undertaken in Ref. [21] whose authors claimed to express the large-M power spectrum in the entire domain T =T H ≲ ωk ≤ ωNy in terms of the spectral form factor [4] of a quantum system. (A similar framework was also used in subsequent papers [22,23].) Even though numerical simulations seemed to confirm a theoretical curve derived in Ref. [21], we believe that the status of their heuristic approach needs to be clarified. Power spectrum beyond the two-level correlations.— In this Letter, we revisit the problem of calculating the power spectrum in quantum chaotic systems as the central assumption of previous studies [21–23]—that at M ≫ 1 the power spectrum is merely dominated by the two-level correlations—cannot be substantiated. In fact, the opposite is true: as ωk grows, the power spectrum becomes increasingly influenced by spectral correlation functions of all orders. This is best exemplified by the large–M formula M SM ðωk Þ ¼ 2 ωk
M ZZ
0
dEdE0 eiωk ðE−E Þ RM ðωk ; E; E0 Þ
ð7Þ
0
that follows directly from the definition Eq. (4). Here, RM ðωk ; E; E0 Þ ¼ hδϱM ðωk ; EÞδϱ M ðωk ; E0 Þi is a oneparameter extension of the density-density correlation function RM ð0; E; E0 Þ ¼ hδϱM ðEÞδϱM ðE0 Þi with Z E dμδϱM ðμÞ : ð8Þ δϱM ðωk ; EÞ ¼ δϱM ðEÞ exp iωk M
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Contrary to RM ð0; E; E0 Þ, the correlator RM ðωk ; E; E0 Þ is influenced by spectral correlation functions of all orders; so is the power spectrum. Looking for the singular part of the power spectrum at small ωk , it is tempting to discard the higher-order correlations in Eq. (7). Such a move would result in the simple approximation [21] SM ðωk ≪ 1Þ ≃ ω−2 k K M ðωk =2πÞ, where 1 K M ðτÞ ¼ M
M ZZ
0
dEdE0 e2iπτðE−E Þ RM ð0; E; E0 Þ
ð9Þ
0
is the spectral form factor of a quantum system. For quantum chaotic systems, known to be described by the form factor [4,6,8] K M ðτ ≪ 1Þ ≃ 2τ=β, the above approximation does reproduce the Brownian motion result Eq. (6) for the power Z S∞ ðωk Þ ¼ Aðϖ k Þ Im
0
∞
spectrum. However, for generic quantum systems with completely integrable classical dynamics, characterized by [4] K M ðτÞ ¼ 1, the very same approximation yields a wrong estimate for the power spectrum: SM ðωk ≪ 1Þ ≃ 1=ω2k instead of 2=ω2k . To account for the missing factor 2, contributions of correlation functions beyond the second order should be accommodated [24,25]. The latter observation not only questions the validity of the “form factor approximation,” but it also suggests that the power spectrum keeps record of spectral correlation functions of all orders. In this Letter we show how they can be summed up exactly within the RMT approach for both finite and infinite eigenlevel sequences. In the case of broken time-reversal symmetry (β ¼ 2), our nonperturbative theory produces a parameter-free prediction for the power spectrum in the form
Z ∞ dt dλ 1−2ϖ2 iϖk λ 2 k λ ðσðt; ϖ k Þ − iϖ k t þ 2ϖ k Þ − 1 þ Bðϖ k Þ ; exp − e 2π t λ
where ϖ k ¼ ωk =2π is a rescaled frequency, the functions A and B are defined as Q 1 2j¼1 Gðj þ ϖ k ÞGðj − ϖ k Þ Aðϖ k Þ ¼ ; ð11aÞ 2π sinðπϖ k Þ Bðϖ k Þ ¼
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1 2ϖ 2 −2 sinðπϖ 2k Þϖ k k Γð2 − 2ϖ 2k Þ; 2π
ð11bÞ
ð10Þ
(CUE). Referring the reader to the figure caption for further details, we plainly conclude that the agreement between the theory and numerics is nearly perfect. Yet, our simulations ð0Þ indicate that a deviation of the simple S∞ ðωk Þ curve from the exact formula Eq. (10) does not exceed 5% in the region ωk < ⅔ωNy . For higher frequencies, the relative error quickly increases reaching its maximal value (34%) at
G is the Barnes G-function, Γ is the gamma function, while σðt; ϖ k Þ ¼ σ 1 ðtÞ is a solution of the Painlevé V equation Eq. (3) at ν ¼ 1 satisfying the boundary condition [26,27] σ 1 ðtÞ ∼ iϖ k t − 2ϖ 2k as t → ∞. The universal law Eq. (10), describing the power spectrum in quantum chaotic systems with broken timereversal symmetry in the domain of its universality T =T H ≲ ωk < ωNy ¼ π, is the main result of our study. It can be regarded as a power spectrum analog of the Gaudin-Mehta formula [Eq. (2)]. In the low-frequency domain ωk ≪ 1, the power spectrum behavior can be made explicit. Analyzing a small–ϖ k solution to Eq. (3) at ν ¼ 1, we derive S∞ ðωk Þ ¼
1 1 ϖ þ 2 ϖ k log ϖ k þ k þ Oðϖ 2k log ϖ k Þ: 12 4π ϖ k 2π 2
ð12Þ The leading-order term of this expansion is consistent with the β ¼ 2 Brownian-motion result Eq. (6). Corrections to it have nothing in common with the results of the earlier approach [21] hereby suggesting that higher-order correlations influence the power spectrum even in the lowfrequency domain. In Fig. 1, we confront our parameter-free prediction Eq. (10) with the results of numerical simulations for unfolded spectra of large-M circular unitary ensemble
ð0Þ
FIG. 1. Difference δS∞ ðωk Þ ¼ S∞ ðωk Þ − S∞ ðωk Þ between the ð0Þ power spectrum S∞ ðωk Þ and its singular part S∞ ðωk Þ ¼ −1 ð2πωk Þ as described by Eq. (6) at β ¼ 2. Crosses correspond to δSM ðωk Þ computed for sequences of 200 unfolded CUE eigenvalues averaged over 4 × 106 realizations. Solid line: analytical prediction for δSM ðωk Þ calculated through dPV equations with M ¼ 1000 [28,29] [see discussion below Eq. (21)]. Dashed line in the inset displays the difference δS∞ ðωk Þ calculated using the small–ωk expansion Eq. (12).
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ω ¼ ωNy . This estimate should have important implications for analysis of the experimental data; see Refs [15,16,18]. Sketch of the derivation.—To mimic an unfolded spectrum of quantum chaotic systems with broken time-reversal symmetry, we define the tuned CUE ensemble (TCUEN ) through the joint probability density function (JPDF) PN ðθÞ ¼
N Y Y 1 jeiθl − eiθm j2 j1 − eiθl j2 ðN þ 1Þ! l>m l¼1
ð13Þ
of its N eigenangles θ ¼ fθ1 ; …; θN g. Equation (13) can be viewed as the JPDF for traditional CUEM ensemble, M ¼ N þ 1, whose lowest eigenangle is conditioned to stay at θ ¼ θ0 ¼ 0. Such a minor tuning of the CUE induces useful constraints [25] on the averages of and correlations between ordered TCUEN eigenangles fθ1 ≤ … ≤ θN g. Two of them, (i) hθl i ¼ lΔ and (ii) hðδθl − δθm Þ2 i ¼ hδθ2jl−mj i, are of particular importance. (Here, angular brackets h…i denote averaging with respect to the JPDF Eq. (13) while Δ ¼ 2π=M is the mean level spacing.) The first constraint (i) leads us to identify a set of ordered unfolded eigenlevels fE0 ≤ E1 ≤ ≤ EN g in Eq. (4) as El ¼ θl =Δ ∈ ð0; MÞ. This results in the RMT power spectrum of the form SM ðωk Þ ¼
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−1 M −1 X X M M hδθl δθm izl−m ; k 4π 2 l¼1 m¼1
ð14Þ
where zk ¼ eiωk . The second constraint (ii) makes it possible to reduce Eq. (14) down to [30] M −1 X M ∂ SM ðωk Þ ¼ 2 Re zk −M hδθ2l izlk : ð15Þ ∂zk 4π l¼1 Expressing hθ2l i in terms of the probability density pl ðϕÞ to observe the lth ordered eigenvalue at ϕ, Z 2π dϕ 2 2 hθl i ¼ ϕ pl ðϕÞ 2π 0 Z 2π l−1 dϕ 2 X ∂EN ðj; ϕÞ ϕ ; ð16Þ ¼− 2π ∂ϕ 0 j¼0 we derive the exact representation M ∂ SM ðωk Þ ¼ Re zk −M π ∂zk Z 2π zk dϕ × ϕΦN ðϕ; 1 − zk Þ : 1 − zk 0 2π
is the generating function [5] of the probabilities EN ðl; ϕÞ to find exactly l eigenangles in the interval ð0; ϕÞ. Equations (17), (18), and (13) are central to our nonperturbative theory of the power spectrum. Painlevé VI representation.—Multidimensional integrals of the CUE type [Eqs. (18b) and (13)] have been studied in much detail in Ref. [31]. Building upon it, we can represent the finite-M power spectrum Eq. (17) in terms of the sixth Painlevé function σ~ N such that Z ∞ dt ðσ~ N ðt; ζÞ þ tÞ : ð19Þ ΦN ðϕ; ζÞ ¼ exp − 2 cotðϕ=2Þ 1 þ t The function σ~ N ðt; ζÞ is a solution to the Painlevé VI equation ½ð1 þ t2 Þσ~ 00N 2 þ 4~σ 0N ðσ~ N − tσ~ 0N Þ2 þ 4ðσ~ 0N þ 1Þ2 ½σ~ 0N þ ðN þ 1Þ2 ¼ 0 subject to the boundary condition [25] σ~ N ðt; ζÞ ¼ −t þ
NðN þ 1ÞðN þ 2Þ ζ þ Oðt−4 Þ 3πt2
ð21Þ
as t → ∞. Equations (17) and (19)–(21) provide an exact RMT solution for the power spectrum at finite N. Alternatively, but equivalently, the exact solution for SM ðωk Þ can be formulated [25] in terms of discrete Painlevé V equations (dPV ), much in line with Ref. [32]. The latter representation is particularly useful for efficient numerical evaluation of the power spectrum for relatively large values of N; such a computation, displayed in Fig. 1, shows a remarkable agreement of our exact solution with the power spectrum determined for numerically simulated CUE spectra. Asymptotic analysis of the exact solution.—To identify a universal, parameter-free, law for the power spectrum, expected to emerge in the limit N → ∞, an asymptotic analysis of the exact solution Eqs. (17) and (19)–(21) is required. To facilitate such an analysis, we notice that the generating function ΦN ðϕ; ζÞ [Eq. (18b)] entering the exact solution Eq. (17) with ζ ¼ 1 − zk admits a representation ΦN ðϕ; 1 − zk Þ ¼
ð17Þ
ð20Þ
eiϕω~ k N D ½f ðz; ϕÞ N þ 1 N ω~ k
in terms of the Toeplitz determinant I 1 dz l−j z f ω~ k ðz; ϕÞ DN ½f ω~ k ðz; ϕÞ ¼ det 2iπ jzj¼1 z
ð22Þ
ð23Þ
ð0 ≤ j; l ≤ N − 1Þ whose Fisher-Hartwig symbol
Here ΦN ðϕ; ζÞ ¼
N X ð1 − ζÞl EN ðl; ϕÞ l¼0
Z ¼
0
2π
−ζ
Z ϕ Y N dθl P ðθÞ 2π N 0 l¼1
ð18aÞ ð18bÞ
f ω~ k ðz; ϕÞ ¼ jz − z1 j2
ω~ z2 k gz1 ;ω~ k ðzÞgz2 ;−ω~ k ðzÞ z1
ð24Þ
possesses a power-type singularity at z ¼ z1 ¼ eiϕ=2 and two jump discontinuities
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e iπω~ k ;
PHYSICAL REVIEW LETTERS 0 ≤ arg z < arg zj
e∓iπω~ k ; arg zj ≤ arg z < 2π
ð25Þ
at z ¼ z1;2 with z2 ¼ eið2π−ϕ=2Þ . (Here, we have followed the standard terminology and notation; see e.g. Ref. [33]). To perform the integration in Eq. (17) in the large-N limit, one needs to know uniform asymptotics of the Toeplitz determinant Eq. (23) in the subtle case of merging singularities. These particular asymptotics of Toeplitz determinants were recently studied in great detail by Claeys and Krasovsky [27]. Making use of their Theorems 1.5 and 1.11, we end up—after a somewhat involved calculation [25]— with the universal law for the power spectrum announced in Eqs. (10) and (11) above. Summary.—We revisited the problem of calculating the power spectrum in quantum chaotic systems putting particular emphasis on the shortcomings of widely used “form factor” approximation. Having argued that the power spectrum is shaped by spectral correlations of all orders, we showed how their contributions can be summed up nonperturbatively within the RMT approach. Our main result—a parameter-free prediction for the power spectrum expressed in terms of a fifth Painlevé transcendent [Eq. (10)]—is expected to hold universally, at not too low frequencies, for a variety of quantum systems with chaotic classical dynamics and broken time-reversal symmetry. Meticulous numerical analysis revealed that the heuristically anticipated [11,21] 1=ωk law for the power spectrum can deviate quite significantly from our exact solution. This will have important implications for analysis of the experimental data. This work was supported by the Israel Science Foundation through Grant No. 647/12. V. Al. O. acknowledges support from NanoLund and the Knut and Alice Wallenberg Foundation (KAW).
[1] M. V. Berry, Proc. R. Soc. A 413, 183 (1987). [2] O. Bohigas, M. J. Giannoni, and C. Schmit, Phys. Rev. Lett. 52, 1 (1984). [3] Long times correspond to short correlation scales on the energy axis. [4] M. L. Mehta, Random Matrices (Elsevier, Amsterdam, 2004). [5] P. J. Forrester, Log-Gases and Random Matrices (Princeton University Press, Princeton, NJ, 2010). [6] K. Richter and M. Sieber, Phys. Rev. Lett. 89, 206801 (2002); S. Müller, S. Heusler, P. Braun, F. Haake, and A. Altland, Phys. Rev. Lett. 93, 014103 (2004); S. Heusler, S. Müller, A. Altland, P. Braun, and F. Haake, Phys. Rev. Lett. 98, 044103 (2007); S. Müller, S. Heusler, A. Altland, P. Braun, and F. Haake, New J. Phys. 11, 103025 (2009). [7] Spectral rigidity Δ3 ðLÞ is the least-square deviation of the spectral counting function N ðEÞ from the best fit to a straight line over the interval of the length L. See Ref. [4].
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[8] M. V. Berry, Proc. R. Soc. A 400, 229 (1985); Nonlinearity 1, 399 (1988). [9] M. Jimbo, T. Miwa, Y. Môri, and M. Sato, Physica D (Amsterdam) 1, 80 (1980). [10] M. V. Berry and M. Tabor, Proc. R. Soc. A 356, 375 (1977). [11] A. Relaño, J. M. G. Gómez, R. A. Molina, J. Retamosa, and E. Faleiro, Phys. Rev. Lett. 89, 244102 (2002). [12] Similar statistics has previously been used by Odlyzko who analyzed the power spectrum of the spacings between zeros of the Riemann zeta function. See A. M. Odlyzko, Math. Comput. 48, 273 (1987). [13] From now on, M is assumed to be an even integer. [14] J. M. G. Gómez, A. Relaño, J. Retamosa, E. Faleiro, L. Salasnich, M. Vraničar, and M. Robnik, Phys. Rev. Lett. 94, 084101 (2005); A. Relaño, Phys. Rev. Lett. 100, 224101 (2008). [15] E. Faleiro, U. Kuhl, R. A. Molina, L. Muñoz, A. Relaño, and J. Retamosa, Phys. Lett. A 358, 251 (2006). [16] M. Białous, V. Yunko, S. Bauch, M. Ławniczak, B. Dietz, and L. Sirko, Phys. Rev. Lett. 117, 144101 (2016). [17] A. Frisch, M. Mark, K. Aikawa, F. Ferlaino, J. L. Bohn, C. Makrides, A. Petrov, and S. Kotochigova, Nature (London) 507, 475 (2014). [18] J. Mur-Petit and R. A. Molina, Phys. Rev. E 92, 042906 (2015). [19] M. Wilkinson, J. Phys. A 21, 1173 (1988); B. Mehlig and M. Wilkinson, Phys. Rev. E 63, 045203 (2001). [20] F. J. Dyson, J. Math. Phys. (N.Y.) 3, 1191 (1962). [21] E. Faleiro, J. M. G. Gómez, R. A. Molina, L. Muñoz, A. Relaño, and J. Retamosa, Phys. Rev. Lett. 93, 244101 (2004). [22] A. M. García-García, Phys. Rev. E 73, 026213 (2006). [23] A. Relaño, L. Muñoz, J. Retamosa, E. Faleiro, and R. A. Molina, Phys. Rev. E 77, 031103 (2008). [24] In case of completely integrable classical dynamics, one observes that hδEl δEm i ¼ minðl; mÞ. Equation (4) then yields the exact result SM ðωk Þ ¼ 1=2 sin2 ðωk =2Þ. In the domain ωk ≪ 1, this reduces to SM ¼ 2=ω2k . [25] R. Riser, V. A. Osipov, and E. Kanzieper (unpublished). [26] Strictly speaking, this boundary condition is only valid for 0 ≤ ωk ≤ ωNy =2; for a detailed discussion the reader is referred to Ref. [27]. [27] T. Claeys and I. Krasovsky, Duke Math. J. 164, 2897 (2015). [28] A recent study [29] suggests that replacement of SM ðωk Þ with S∞ ðωk Þ brings a relative error of the order OðM−2 Þ. [29] P. J. Forrester and A. Mays, Proc. R. Soc. A 471, 20150436 (2015). [30] Here and in Eq. (17) the derivative with respect to zk should be taken as if zk were a continuous variable. [31] P. J. Forrester and N. S. Witte, Nagoya Mathematical Journal 174, 29 (2004). [32] P. J. Forrester and N. S. Witte, Nonlinearity 16, 1919 (2003); 18, 2061 (2005). [33] P. Deift, A. Its, and I. Krasovsky, in Random Matrix Theory, Interacting Particle Systems, and Integrable Systems (Cambridge University Press, New York, 2014), p. 93.
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