Spectral and Parametric Averaging for Integrable Systems

Transcription

1 Spectral and Parametric Averaging for Integrable Systems Tao a and R.A. Serota Department of Physics, University of Cincinnati, Cincinnati, OH 5- We analyze two theoretical approaches to ensemble averaging for integrable systems in quantum chaos - spectral averaging and parametric averaging. For spectral averaging, we introduce a new procedure - rescaled spectral averaging. Unlike traditional spectral averaging, it can describe the correlation function of spectral staircase and produce persistent oscillations of the interval level number variance. Parametric averaging, while not as accurate as rescaled spectral averaging for the correlation function of spectral staircase and interval level number variance, can also produce persistent oscillations of the global level number variance and better describes saturation level rigidity as a function of the running energy. Overall, it is the most reliable method for a wide range of statistics. I. INTRODUCTION The framework of quantum chaos is structured around the concept of ensemble averaging. Statistics, such as correlation function of level density [], interval level number variance (IV) [], global level number variance (GV) [], spectral rigidity (SR) [3] and nearest neighbor spacing distribution [] are defined through ensemble averaging. In the literature, two methods are employed to achieve ensemble averaging for integrable systems. Traditionally, ensemble averaging in semiclassical theories was understood in terms of spectral averaging (SA). [3, 5] A numerical simulation of IV using SA was performed in []. The oscillations of IV were found to decay, while the other ensemble averaging method for integrable systems parametric averaging (PA) correctly showed persistent oscillations. [] We explained that SA tends to suppress the non-decaying oscillatory behavior due to destructive interference of the running-energy-dependant non-coherent terms. [] oreover, this paper will show that when, in order to avoid such destructive interference, SA is performed over a short range of sampled energies, sampling is insu cient and sample-specific fluctuations are observed. uch as impurity averaging in disordered systems, PA corresponds to ensemble averaging for a fixed value of the running energy; specifically for rectangular billiards averaging is over an ensemble of rectangles of varying aspect ratios and fixed area. To our knowledge, PA for integrable system was first performed by Casati et. al. to prove the saturation of SR of integrable system. In their words, computed an average 3(L)... byaver- aging 3 over a number of di erent values of chosen at random in a given interval ( ensemble averaging ). [7] ( 3(L) denotes SR over the interval L and is the aspect ratio defined in [7].) PA with better implementations was used to reproduce saturation of SR [, ], produce persistent oscillations of IV [, ] and GV [] and prove level repulsion in integrable systems []. These studies lectronic address: serota@ucmail.uc.edu demonstrated that PA is a reliable and versatile method for numerical computation of statistics of integrable systems. (Note that PA can also be used as an experimental technique to study orbital magnetism. []) Here we undertake a detailed comparison of SA and PA previously unaddressed in literature. The central result of this work is that, unlike PA, traditional SA cannot produce persistent oscillations of IV and GV. ven with our newly proposed rescaled spectral averaging (RSA), one can only address IV oscillations. These results are argued theoretically and proved numerically. This paper is organized as follows. In Sec. II, we review the periodic orbit theory of level fluctuations and the semiclassical theory of IV, GV, SR, and correlation function of spectral staircase (CFSS). In Sec. III, we define SA and PA for IV, CFSS, and SR. From linear expansion of SA, we argue that SA suppresses the oscillations of IV when averaging over large intervals. To preserve persistent oscillations, we propose RSA. In Sec. IV, we present spectral fluctuations, IV, and GV computed from SA and PA, IV from RSA, CFSS from PA and RSA, and SR from SA, PA and RSA. In Sec. V, we discuss advantages and shortcomings of RSA and PA and outline their applicability. In this paper, we use uppercase letters to indicate ensemble averaged statistics and lowercase ones to indicate their corresponding sample-specific values. For instance, denotes IV and denote sample IV; g denotes GV and g denotes sample GV; 3 denotes SR and 3 denotes sample SR; K N denotes CFSS and k N denotes sample CFSS. The subscripts A and indicate numerical computation and theoretical calculation respectively. The superscript indicates the ensemble averaging method, that is SA, RSA or PA. II. STATISTICS A. Periodic orbit theory of level fluctuations We use rectangular billiards as a model system to illustrate our theory. For a particle of mass m in a rectangular billiard with sides a, b and aspect ratio a /b,

2 the eigenenergy with quantum numbers n, is given by " n,n = ~ n m a + n b. () The spectral staircase is defined as N (") X n,n (" " n,n ), () where is unit step function. According to Weyl s formula, the ensemble-averaged spectral staircase is given by [,, ] hn (")i = " S " / p + (3) A where = ~ /ma; A and S are the rectangular billiard area and perimeter respectively; and hi denotes ensemble averaging. The second and third terms are usually called perimeter correction and corner correction respectively. In previous works [, 9], we only considered the perimeter correction when unfolding the spectrum. In the present paper, we account for both terms. After unfolding the spectrum by (3), the mean level spacing becomes unity and [] hn (")i = ", () which would be correct for a perfect ensemble averaging method and is approximately correct for SA and PA as will be shown in Sec. IV A. From the periodic orbit theory,[, 3] the fluctuation of level density is given by (") (") h (")i = X apple S (") ~ µ+ A (") cos ~ and the fluctuation of spectral staircase by N (") N(") hn(")i = X A (") ~ µ T (") apple S (") sin ~ (5). () Here µ =( )/, is the dimensionality of phase space and the period, action, and amplitude of the periodic orbit are given respectively by [3] T (") =[m( a + b )/"] / S (") ="T A (") =m a b / 3 "T, (7) with =(, ) and non-negative, as winding numbers. Above >< = = = / if only one of, is zero () >: otherwise. Compared with [3], in (5) and (), we have an extra factor / from a quantum mechanical calculation [9]. In Sec. IV A, we show that this factor matters. B. Interval and global level number variance IV is defined as where N N(" ) (", ) h[n hni] i = h[n ] i, (9) N(" )with and ". GV is defined as [, 5] " = " / () " = " + / () g (") h[n (") hn(")] i = h[n (") "] i. () In (9) and (), we used (). We term sample IV and (", ) [N hni] =[N ] (3) g(") [N (") hn(")i] =[N (") "] () sample GV. mploying the diagonal approximation [, 9], theoretical sample IV is expressed as [] (", ) = ~ µ X A (") T (") sin T (") ~. (5) Substituting (7) and unfolding the spectrum, the above equation can be written as [] r " X appler (", ) = 5 R 3 sin " R, () where R = p / + /. Numerical sample IV, A(", ), is a jagged line as a function of, while theoretical sample IV, (", ), is a smooth line by (). ensemble averaging is able to bridge this di erence. SR is defined as 3(", ) min (A,B) C. Spectral rigidity Z " which has the explicit form [3] * " Z / N (" +!)d! / Z / / " [N (x) A Bx] dx, (7)!N (" +!)d! " # + Z / /. N (" +!)d! # ()

3 3 Sample SR is defined as 3(", ) min (A,B) Z " " [N (x) A Bx] dx, (9) which is computed from () without ensemble averaging (that is from the expression inside hi). The saturation SR 3 (") and its sample value 3 (") are numerically computed as 3(", ) p and 3 (", ) respectivelywithsufficiently large ". For the saturation SR, the minimization fit A+B" is approximately given by ". Hence [] 3(", ) Z " " g(x)dx, () where we used (). Based on DA, we have the sample value of saturation SR [] r " X ( 3 ) (") = 5 R 3, () where we used (7) and unfolded the spectrum. D. Correlation function of spectral staircase CFSS is defined as [, 5] K N (", ) h N (" ) N (" )i. () The sample CFSS is defined as Using DA, we have [] k N (", ) = 3 (") k N (", ) N (" ) N (" ). (3) 3 (") The ensemble averaged form is (", ), for p ". () where A(x, ) and (x, ) implicitly depend on the aspect ratio and f SA (x) is the density of sampled energies and is chosen as equally spaced points in a range centered at ". In other words, x [" /,"+ /] with uniform density. In PA, the numerical and theoretical values of IV are respectively defined as PA A (", ) R A(", )f PA ( )d () PA (", ) (",, ), (9) where A (", ) and (", ) implicitly depend on and f PA ( ) is a Gaussian distribution with mean and standard deviation. Numerical computation of IV from SA and PA can be understood as numerical integration of () and () respectively. Similarly we can define SA and PA for CFSS and SR. [] B. Linear expansion of spectral averaging Using (), a representative term in (7) reads Z r appler x sin x R f SA (x)dx r " 5 R 3 Z "+ 5 R 3 " sin appler x R f SA (x)dx. (3) When is far larger than the period of the sine term, the integrand can be replaced by / and one will not observe persistent oscillations of IV. The first-order derivative of the argument of sine is given by d p x R dx whereof we find that when x=" = p R " 3/, (3) > p " 3/ R "3/, (3) oscillations of IV will decay. Notice that oscillations are observed when >" / [, 9] and that the decay of IV oscillations becomes faster with larger. [] K N (", ) = 3 (") (", ). (5) C. Rescaled spectral averaging III. THORY OF SPCTRAL AND PARATRIC AVRAGING A. Spectral and parametric averaging In SA, the numerical and theoretical values of IV can be respectively defined by the following integrals: SA A (", ) R A(x, )f SA (x)dx () SA (", ) R (x, )f SA (x)dx, (7) We just saw that traditional SA su ers from an inherent flaw due to destructive interference of oscillating terms. In order to observe persistent oscillations with larger and larger interval width, one needs to sample su ciently large energy range centered on " to achieve proper ensemble averaging. Yet, q. (3) sets the limit to how large such range can be in order to avoid destructive interference and observe persistent oscillation of IV. Furthermore, the limit deceases with the increase of. A possible workaround would be to sample various parts of spectrum, not necessarily around ". However,

4 since persistent oscillations strongly depend on " (the point of onset, the amplitude and the period [, 9]), such procedure, executed without a proper account for this "-dependence, would have an e ect similar to the destructive interference above a wash-out of persistent oscillations. Consequently, we introduce a modified procedure, RSA, that allows sampling of di erent parts of the spectrum. Our approach is based on a scaling transformation (c", p c) = p c (", ), which follows from (). Consequently, in RSA, when the running energy and the interval are scalled as "! c" and! p c respectively, A(c", p c) needs to be rescaled by a factor / p c before averaging. Numerically, IV is computed as RSA A (", ) n + nx i= p ci A (c i ", p c i ), (33) where c i is the ratio of the energy of a sampled spectral location to " and n+ is the number of sampled energies and theoretically, by design, it is given by RSA (", ) = (", ), (3) that is coincides with (9). We note due to the close relation between IV and CFSS in (5), RSA can be similarly defined for the latter and we have (K N ) RSA (", ) =(k N ) (", ), (35) where theoretical (K N ) RSA (", ) can be evaluated from (), (), and (). RSA of saturation SR is computed by nx p ( 3 ) A (c i "), (3) n + ci i= and its theoretical value is ( 3 ) RSA (") ( 3 ) ("). (37) ( 3 ) RSA A (") scales as p " for billiard systems, including elliptic billiards []. A SA A SA FIG. : Spectral averaging of the fluctuation of spectral staircase with di erent ranges of sampled energies. Left: the range is ;Right: 3. A PA, PA A PA, PA IV. NURICAL SIULATIONS Below, except in Fig., for SA and RSA, the aspect ratio is set to be ( p 5 )/.93 to avoid degeneracy; for PA, the distribution of is a Gaussian distribution with mean and standard deviation.. In the computation of IV and CFSS, we set " = 5. FIG. : Parametric averaging of the fluctuation of spectral staircase. Top: black dots, numerical hn (") "i PA A calculated by averaging over from a Gaussian distribution with the mean and the standard deviation.; magenta line, hn (") "i PA calculated from () and averaged over ; dashed blue line, hn (") "i PA calculated from () without the / factor and averaged over. Bottom: thesamemagentaline is shifted leftward by 5.. A. Fluctuations of spectral staircase We study SA and PA of the fluctuation of spectral staircase hn (") "i SA (3) hn (") "i PA. (39) In Figs. and, we present the results obtained with SA and PA respectively. For a large range of sampled energies, SA gives near zero result. PA produces regular oscillations about zero line. Theoretically, the oscillations are due to the sine term with the periodic orbit (,) in (), which does not vanish upon PA. [] In Fig., the theoretical result obtained from () with PA needs to be shifted leftward to be consistent with the numerical

5 5 S A RSA H,L, SA PA H,L, s QH,L FIG. : IV calculated with RSA and PA. Black dots: RSA calculated from an ensemble of sampled energies in [ 5, 5 ]. Green dashed line: PA. Blue solid line: theoretical result calculated from (). A SA, result. This shift is due to the perimeter correction and can be calculated as r = 5. from r / = 5, where 5 is the average energy in Fig.. We also observe that the factor / is critical for a good vertical fit. The deviation of hn (") "i from in PA reveals a shortcoming of PA. But its small magnitude indicates PA is su ciently proficient as an ensemble averaging method for spectral staircase. An advantage of PA is that the distribution f PA ( ) works for any energy scale, while the range of sampled energies needs to grow as " / in SA. FIG. 3: IV calculated from SA. Thin and thick lines: the ranges of sampled energies are [95, 5] and [75, 5] respectively. B. Interval level number variance In Fig. 3, we present IV computed from SA. Clearly, SA cannot properly produce the persistent oscillations of IV. If the range of sampled energies is small, SA produces close to sample specific oscillations, indicating insu cient sampling. If the range of sampled energies is large, SA suppresses IV oscillations when the interval grows. In Fig., we present IV computed from RSA and PA. We observe that RSA is in better agreement with the theoretical result, (3) (or, equivalently, (9)) and, than PA. It should be emphasized that while we are using rectangular billiars as a model system, RSA formalism readily A RSA,, RSA, FIG. 5: IV of circular billiard computed with RSA from an ensemble of sampled energies in [ 5, 5 ]. Black dots: numerical result. Cyan line: theoretical result.

6 PA HKNL A H,L RSA HKNL A H,L HkNLQH,L FIG. : CFSS with RSA and PA. Black dots: numerical result calculated from the definition of K N in () and averaged through RSA. Green dashed line: PA. Blue solid line: theoretical result calculated from the first q. (). Insert shows small behavior - close to linear, according to the second q. (). extends to any system with continuous dependence on ", even if the latter is non-monotonic as, for instance, in Fig. 9 below for the saturation spectral rigidity (which has the same "-dependence as IV). In Fig. 5, we see that IV computed with RSA for a circular billiard [, ] is in excellent agreement with theory. C. Correlation function of spectral staircase In Fig. we plot K N (", ) computed with RSA and PA. Again, we observe that RSA is in a better agreement with theoretical result than PA. 3 A RSA, 3 A PA A, 3 A SA, 3 A PA 3 FIG. 7: Numerical result of saturation SR computed from RSA and PA. agenta line: PA. Dashed blue line: RSA. For ( 3 ) RSA A ("), the range of sampled energies [", "]. D. Saturation spectral rigidity In Fig. 7, we present saturation SR computed from RSA and PA. Clearly, PA yields a better result since RSA shows small oscillations, while by theory () saturation SR should be a smooth function of ". In Fig., we present saturation SR computed with PA and SA and sample saturation SR (computed with (9)). The latter shows large-range oscillations, which is absent in the PA result. If the range of sampled energies is su ciently large, SA gives a result close to PA; otherwise, SA gives behaves similarly to sample specific SR. FIG. : Numerical results of saturation SR computed from SA and PA. agenta line without any oscillations: PA result. Thin and thick black lines: SA with averaging range and 5 respectively. Jagged cyan line: sample saturation SR computed with = 3 for "<= and =5 3 for ">.. Global level number variance GV is discussed in detail in []. In Fig. (9), we present numerical results, computed using PA, for four di erent integrable systems: rectangular billiard [9], modified Kepler problem [9] and circular and elliptical billiards [, ]. In each case, GV exhibits large oscillations around SR []. With the exception of rectangular billiards, SA/RSA is incapable of reproducing the complex behavior of SR

7 7 # PA!!g "PA A!!",!"3 " A!!" Rectangular billiard 5 not find a rescaled form of SA for GV. A simple definition of SA for GV is n 5 g (") = 5! # PA!!g "PA A!!",!"3 " A!!" odified Kepler problem ! Quarter circular billiard # PA!!g "PA A!!",!"3 " A!!" 5 5 5! Quarter elliptic billiard # PA!!g "PA A!!",!"3 " A!!" 5 CONCLUSIONS We introduced a new SA procedure RSA to cure some of the intrinsic problems of SA. For rectangular billiards, we found that SA cannot produce persistent oscillations of IV and has some difficulties with SR. Any spectral averaging is unsuitable for GV oscillations. RSA is best suited for oscillations of IV and CFSS and generally works for SR, while PA is best suited for SR, GV and generally works for IV and CFSS. Relative RSA success for SR in rectangular billiards does not carry over to more complex system, such as odified Kepler Problem [9] and elliptic billiards [], where SR exhibits non-trivial dependence on the running energy (spectral position) that RSA is incapable of capturing. To summarize our findings: PA always works numerically, RSA may be occasionally more accurate while traditional SA is almost always inadequate. Alternatively, we can also cast our findings as follows: For quantities depending on the interval, such as IV and CFSS, RSA works slightly better than PA, while standard SA fails. 5 For fluctuations of spectral staircase, any SA produces zero, while PA yields unphysical persistent oscillations, albeit very small. () For quantities depending on ", such as saturation spectral rigidity and GV, PA should be used. 5 "i ], where the sampled energy "i is equally distributed in the range ["!/, " +!/]. This is the definition of sample SR: 3 (",!). The integration in () (after we change into!) P can be approximated by the numerical integration n as n+!/ + i!/n), which becomes () if i= g (" "i = "!/ + i!/n. We arrive at the conclusion that no variant of SA is capable of reproducing large oscillations of GV around SR. V X [N ("i ) n + i=! FIG. 9: Comparison between GV and saturation SR of rectangular billiard, modified Kepler problem, quarter circular billiard, and quarter elliptic billiard Jagged black line: g (") directly computed () through PA. Smooth cyan line: 3 ("). Inserts show a shorter range of g (") and 3 ("). observed in Fig. (9). Also, unlike IV and CFSS, we can We also have good agreement between theory and numerical results. The latter includes the fact that, with the exception of GV, diagonal approximation yields sufficiently accurate predictions. RSA should find its use in fixed area circular and square billiards, for which no proper PA procedure exists. On the other hand, PA may also find application in chaotic billiard and potential systems. For instance, PA of a Sinai billiard a circular hole in a rectangular billiard can be achieved through varying the aspect ratios of the sides of the billiard. [7]

8 [] J..A.S.P. Wickramasinghe, B. Goodman, and R. A. Serota, Phys. Rev. 7, 59(5). [] T. a and R.A. Serota, arxiv:3. (). [3]. Berry, Proc. R. Soc. London, Ser. A, 9(95). [] T. a and R.A. Serota, Int. J. od. Phys. B 595 (). [5]. Berry, Springer Lecture Notes in Physics, No. (lsevier-health Sciences Division, 9), p [] C. Grosche, merging Applications of Number Theory p. 9-9 (Springer-Verlag, New York, 999). [7] G. Casati, B. Chirikov, and I. Guarneri, Phys. Rev. Lett. 5, 35 (95). [] J..A.S.P. Wickramasinghe, B. Goodman, and R.A. Serota, Phys. Rev. 77, 5(). [9] T. a and R. Serota, Phys. Rev. 5, 3(). [] T. a and R. Serota, arxiv:3.7 (). [] L.P. Lévy, D. Reich, L. Pfei er, and K. West, Physica B: Condensed atter 9, (993). [].C. Gutzwiller, Chaos in Classical and Quantum echanics (Springer, 99);. [3] H-J Stöckmann Quantum Chaos: An Introduction (Cambridge University Press, Cambridge, UK, ). While in a very di erent fashion, in chaotic systems the periodic orbit theory now accounts for the key features of the spectrum (. Sieber and K. Richter, Phys. Scr. T9, (); S. üller, S. Heusler, P. Braun, F. Haake and A. Altland, Phys. Rev. Lett ()) and scarring of the wave function (L. Kaplan and. J. Heller, in NATO ASI series volume Supersymmetry and Trace Formulae: Chaos and Disorder, ed. by I. V. Lerner, J. P. Keating, and D.. Khmelnitskii (Plenum, 999)). [] Here and below circular billiards are near circular elliptical billiards with the aspect ratio picked from a distribution centered at and elliptical billiards have aspect ratio distribution centered at /. [5] R.A. Serota, arxiv:.3 (). [] Alternatively, one could use a di erent definition of theoretical PA, [, 9] namely PA (", ) R (", )f PA ( )d, () With this definition, an argument similar to Sec. III B, using (), shows that oscillations due to (, ) terms with = would decay to the average value when becomes large. But for (,) terms, this argument fails as the first order derivative over vanishes (approximately) for and the oscillations persist even when becomes large. Then the persistent oscillations of PA (", ) are solely due to periodic orbit (,). Replacing the sine terms in () and () with /, except for =, we have for large r " PA " (", ) 5 + X = = X = = R 3 R p sin R " # f PA ( )d. R 3 () The implication of the above is that DA breaks down in such procedure. For IV, () describes numerical results fairly well, even for large intervals. In order to obtain good agreement with numerical results for GV, however, the use of f PA ( ) in theoretical averaging necessitates taking non-diagonal terms into account. For instance, the theoretical sample GV containing both diagonal and nondiagonal terms is given by [] ( g) (") " = X ~ µ A (") T (") S (") sin ~ #. (3) One of the reasons to use theoretical f PA ( ) averaging in the first place is to obtain agreement with numerical results for spectral staircase, as shown in Fig.. [7] Averaging for two common classes of chaotic systems strongly interacting systems. such as large nuclei, and disordered systems, such as metals with impurities naturally falls, respectively, into SA and disorder averaging (analogue of PA). The spectral properties of such systems are now well understood and are broadly covered by the Random atrix Theory: L. P. Gorkov and G.. liashberg, Sov. Phys. JTP ; 9 (95), K. fetov, Supersymmetry in Disorder and Chaos (Cambridge University Press, Cambridge, UK, 997); T. A. Brody, J. Flores, J. B. French, P. A. ello, A. Pandey, and S. S.. Wong, Rev. od. Phys. 53, 35(9)and.L. ehta, Random atrices (Academic Press, New York, 99); A. V. Andreev, O. Agam, B. D. Simons, and B. L. Altshuler, Phys. Rev. Lett 7, 395(99). Unlike persistent oscillations, discontinuities and non-monotonic behavior displayed by integrable systems, averaged spectral properties of chaotic systems are generally continuos and monotonic. However, chaotic billiard systems, such as Sinai billiard, may exhibit some residual properties of integrability.