Correlations within eigenvectors and transition amplitudes in the two-body random interaction model

Transcription

1 PHYSICAL REVIEW E VOLUME 53, NUMBER 6 JUNE 1996 Correlatons wthn egenvectors and transton ampltudes n the two-body random nteracton model V. V. Flambaum, 1,, * G. F. Grbakn, 1, and F. M. Izralev 1,, 1 School of Physcs, Unversty of New South Wales, Sydney 05, Australa Budker Insttute of Nuclear Physcs, Novosbrsk, Russa Receved 13 November 1995 It s shown that the two-body character of the nteracton n a many-body system gves rse to specfc correlatons between the components of compound states, even f ths nteracton s completely random. Surprsngly, these correlatons ncrease wth the ncrease of the number of actve valence partcles. Statstcal theory of transton ampltudes between compound states, whch takes nto account these correlatons s developed and tested wthn the framework of the two-body random nteracton model. It s demonstrated that a feature, whch can be called correlaton resonance, appears n the dstrbuton of the transton matrx ampltudes, snce the correlatons strongly reduce the transton ampltudes at the tals and ncrease them near the maxmum of the dstrbuton. PACS numbers: b, Fk, 4.60.Lz I. INTRODUCTION Measurements of party nonconservaton n neutron capture by Th nucleus 1 gave a surprsng result: n spte of a natural assumpton of a random character of matrx elements of the weak nteracton between compound states, the effect was found to be of the same sgn for all observed resonances. Possbly, ths means that the strongly fluctuatng matrx elements of a weak perturbaton between chaotc states of a compound nucleus whch was, n fact, the frst example of a quantum chaotc system are essentally correlated. Thus the problem of correlatons between components of compound states s of great mportance both for theory and applcatons. An attempt to study these correlatons has been undertaken n 4. It s natural to expect that some correlatons may appear f the number of ndependent parameters n a Hamltonan matrx s substantally smaller than the total number of the Hamltonan matrx elements. The smplest example s gven by the model of a random separable nteracton 5 see also 4, H gv v,, 1,,...,N 1 where are the unperturbed energes and v are random varables dstrbuted, e.g., accordng to a Gaussan law. As one can see, the number of ndependent parameters v s equal to N, whle the number of the Hamltonan matrx elements H k s N. It was shown that ths model dsplays very strong (100%) correlatons between egenvectors wth close energes, despte the random character of the nteracton: v v v. Such correlatons cannot appear n models descrbed by full random matrces, lke those of the Gaussan orthogonal ensemble GOE. It was always obvous that such models possess some unphyscal features, e.g., the semcrcle level *Electronc address: flambaum@newt.phys.unsw.edu.au Electronc address: grbakn@newt.phys.unsw.edu.au Electronc address: zralev@physcs.spa.umn.edu densty, however, they seem to gve a very accurate descrpton of the correlatons and fluctuatons of energy levels see, e.g. 6. The mportant pont s that n real many-body systems the basc nteracton s a two-body one. Ths means that the number of ndependent parameters determnng the n-partcle Hamltonan two-body matrx elements s much smaller than the number of the Hamltonan matrx elements. Takng the two-body matrx elements as Gaussan random varables, a model called the two-body random ensemble TBRE was ntroduced n 7,8 see also references n 6. Ths model looks n prncple much more realstc than the GOE. In partcular, t has a Gaussan form of the level densty, whch s n good agreement wth varous nuclear shellmodel calculatons for realstc nteractons n the fnte bass see, e.g., 9. The TBRE does not allow a deep analytcal treatment, however, numercal modelng showed that ts level fluctuaton propertes are very close to those of the GOE, although some dfferences were notced 8. Therefore, n some respect, the two-body nature of the partcle nteracton does not reveal tself n the level statstcs. In other words, level fluctuatons are nsenstve to the detals of the nteracton between partcles, provded the latter s large enough to cause strong mxng of the bass states. Another model whch seems to be more physcal than the GOE was proposed a whle ago by Wgner 10 to descrbe compound nucle. It was suggested that the Hamltonan matrx H has a banded structure,.e., all matrx elements wth b are zeros (b s the bandwdth. The matrx elements nsde the band are random numbers wth zero mean and fxed varance, H 0, H V, except those on the man dagonal, whch monotoncally ncrease, H D. In ths model all egenstates are localzed n the unperturbed bass (V0). In the nonperturbatve regme, 1V/Db, the strength functon whch descrbes the localzaton n the energy space also called the local spectral densty of states, has a characterstc Bret-Wgner form wth a wdth V /D wthn the band. Ths shape s n agreement wth nuclear data 11, and wth the calculated localzaton propertes of chaotc egenstates n the rare-earth atom of Ce X/96/536/57913/$ The Amercan Physcal Socety

2 5730 V. V. FLAMBAUM, G. F. GRIBAKIN, AND F. M. IZRAILEV It has also been shown n 1,13 that the Hamltonan matrx whch produces the dense spectrum of atomc excted states n Ce s sparse and has a bandlke structure, although the edges of the band are dffuse. We should also menton that a number of analytcal results on the localzaton propertes of such band random matrces BRM have recently been obtaned n 14,15 see also 16, and references theren. However, the off-dagonal Hamltonan matrx elements n ths BRM model are ndependent random varables, thus ths model s vod of any possble correlatons related to the two-body nteracton between partcles. In ths paper we show that there are quanttes transton ampltudes or transton strengths for whch the underlyng two-body nteracton s of crucal mportance. We show that such an nteracton gves rse to specfc correlatons between the components of egenstates, whch are very essental for the dstrbuton of transton strengths. Our results are obtaned n the framework of the two-body random nteracton model TBRIM recently proposed n 17 for the study of varous physcal problems related to such complex manybody systems as heavy atoms, nucle, metallc clusters, etc., whch dsplay quantum chaotc behavor. Beng n some aspects smlar to the TBRE, the TBRIM s smpler n the sense that t abandons all restrctons mposed by the conservaton of the angular momentum, whch makes t closer to the embedded GOE 18. On the other hand, the nondegenerate spectrum of the sngle-partcle orbtals the TBRIM s based upon generates a realstc level densty and leads to a bandlke structure of the Hamltonan matrx. We should menton that there were qute a number of earler works where strength dstrbutons were studed usng statstcal spectroscopy methods and nuclear shell-model calculatons 19, see also revew 6. These methods are based on the calculaton of dstrbuton moments, whch are gven by traces of products of the operators n queston and powers of the Hamltonan over the model fnte-dmensonal space of the problem. Snce the calculaton of traces does not requre knowledge of egenstates, the queston of correlatons wthn egenstates whch s of prme mportance for the present work has not been addressed n those studes. We must add that statstcal spectroscopy methods emphasze and employ a partcular mportance of Gaussan spreadng of many-partcle confguratons, and features lke Bret-Wgner localzaton ether do not appear, or are neglected together wth the nteracton between confguratons n that formalsm. All n all, t s unfortunately very dffcult for the present approach to make contact wth those results. Comparng the two approaches we should say that at frst sght ours does not look as rgorous and mathematcally advanced as the other one, as t appeals to some heurstc arguments and uses rather smple mathematcs, e.g., perturbaton theory. However, we beleve that, supported by numercal experments, our method can gve a deeper nsght and a more physcal pcture of transtons between and correlatons wthn the chaotc egenstates n complex many-body systems. In Sec. II of ths paper we show how the basc two-body nteracton results n the correlatons between the Hamltonan matrx elements, egenstate components, and transton ampltudes. In Sec. III we check whether the effects found n Sec. II could lead to some correlatons between transton ampltudes couplng dfferent pars of states. Secton IV presents a bref outlne of a statstcal approach to the calculaton of transton strengths; the analytcal results are checked there aganst numercal ones obtaned n the TBRIM. Fnally, n Sec. V we study the spreadng wdths of the many-partcle bass states. II. CORRELATIONS BETWEEN EIGENVECTOR COMPONENTS INDUCED BY TWO-BODY INTERACTION Let us consder the basc deas of the TBRIM. In ths model, n Ferm partcles are dstrbuted among m nondegenerate orbtals. In dong numercal experments, we assume, as n 17, that the energes of the orbtals are gven by the smple expresson d 0 1, 1,,...,m. However, the analytcal treatment presented below does not depend on a partcular form of. Many-partcle bass states are constructed by specfyng the n occuped orbtals. The energy E of the bass state equals the sum of the snglepartcle energes over the occuped orbtals. The total number of the many-partcle states n the model s Nm!/n!(mn)!expnln(m/n)(mn)ln(m/mn). The latter estmate relates to large m and n and shows that N s exponentally large for n,mn1. The number of ndependent parameters of the many-body Hamltonan s gven by the number of dfferent two-body nteracton matrx elements V and equals N m (m1) /. Due to the two-body character of the nteracton, the Hamltonan matrx element H H s nonzero only when and dffer by no more than two occuped sngle-partcle orbtals. As a result, the number K of the nonzero matrx elements H s gven by KNK 0 K 1 K, K 0 1, K 1 nmn, K 1 4 nn1mnmn1, where K 0, K 1, and K are the numbers of the Hamltonan matrx elements couplng a partcular bass state to another one,, whch dffers from by the postons of none, one, and two partcles, respectvely. Therefore for n,mn1 we have N KN,.e., the Hamltonan matrx s essentally sparse and, n a sense, strongly correlated. To see the correlaton between nonzero matrx elements, let us consder a par of bass states and whch dffer by the states of two partcles, for example, the state can be obtaned from the state by transferrng the partcles from the orbtals, nto the orbtals,. For all such pars, the Hamltonan matrx elements are the same, H V or, strctly speakng, H V, due to Ferm statstcs. It s easy to calculate the total number N eq of the matrx elements H equal to V, usng the fact that the remanng n partcles can be arbtrarly dstrbuted over m4 orbtals, 3

3 CORRELATIONS WITHIN EIGENVECTORS AND TRANSITION... m4! N eq n!mn!. 4 The above concluson has mportant consequences. Let us consder a sngle-partcle operator For bass states and whch dffer by the state of one partcle ( ) the matrx element H equals the sum of the n1 two-body nteracton matrx elements, H V the ndex runs over the rest n1 occuped orbtals. In ths case H for dfferent and wth fxed and ) do not concde, but may contan dentcal terms V,.e., they are also correlated. The egenstates n 1 of the model are determned by ther components C 1 ) wth respect to the many-partcle bass states, n 1 C n 1, 5 and can be found by solvng the Schrödnger equaton, H C n 1 E n 1 C n 1. 6 If the perturbaton V s strong enough, the exact egenstates n 1 are superpostons of a large number of bass states. As s known, strong mxng of bass states n the exact egenstates compound states occurs locally wthn some energy range, E E 1 ), where s known as the spreadng wdth. It can be estmated as N w D, where D s the local mean level spacng for many-partcle states and N w s the effectve number of bass states represented n a compound state. Ths number s also known as the number of prncpal components. These components gve the man contrbuton to the normalzaton condton C 1 ) 1 for the egenstate n 1. Formally, we can estmate N w as the recprocal of the nverse partcpaton rato, N 1 w C 1 ) 4. It s rather straghtforward to show that the correlatons between H result n correlatons between the components C 1 ). Indeed, let us multply the Schrödnger equaton by the coeffcent C 1 ) and sum over n 1. Usng the orthogonalty condton n1 C 1 ) C 1 ), one obtans H n1 C n 1 E n 1 C n 1. 7 In what follows, we assume that the matrx elements of the two-body nteracton V are random varables wth the zero mean, therefore H 0 for. In ths case one can get C 1 ) C k 1 ) 0, where the lne stands for averagng over dfferent realzatons of V. However, f matrx elements of the Hamltonan are correlated, H H kl 0, the components of dfferent egenvectors n 1 and n are also correlated, snce H H kl n1 n C n 1 E n 1 C n 1 C k n E n C l n 0. The latter relaton shows that C 1 ) C 1 ) C k ) C l ) 0. 8 Mˆ, a a M, M, where a and a are the creaton and annhlaton operators. It s convenent to express the matrx elements of Mˆ n terms of matrx elements of the densty matrx operator a a whch transfers a partcle from the orbtal to the orbtal. One can see that the matrx element of Mˆ between compound states, n 1 Mˆ n, M n 1 n, M, C n 1 C n 9 10 has the zero mean due to the statstcal propertes of the components,.e., n 1 n 0. Snce the summaton over the orbtals, n Eq. 10 s ndependent from the averagng over dfferent realzatons of V, n what follows we consder the smplest case of Mˆ. The varance of the matrx element of between the two compound states s equal to M n 1 n n n 1,,k,l C n 1 C n 1 C k n C l n kl n 1 n S c n 1 n, 11 where we separated the dagonal and nondagonal contrbutons to the sum 11, n 1 n,k C n 1 C k n k, 1 n S 1 n n c C 1 n C 1 n C n k C l kl.,kl 13 Note that the dagonal term 1 n ) s essentally postve and can be easly estmated see 3,1,0 and Sec. IV below, whle the nondagonal term S c 1 n ) s our man nterest. If the egenstates are completely random dfferent components both nsde each egenstate and of dfferent egenstates are uncorrelated, the correlaton sum S c s equal to zero and the varance s determned by the dagonal sum ths assumpton has been used n the prevous calculatons of matrx elements between compound states n 3,1,0,1. However, we show below that n a many-body system these two terms are of the same order, S c, even for the random two-body nteracton V. The TBRIM allows one to nvestgate varous propertes of chaotc many-body systems takng nto account the twobody nature of the nteracton between partcles. In the prevous papers 1,17 there were ndcatons that the dagonal

4 573 V. V. FLAMBAUM, G. F. GRIBAKIN, AND F. M. IZRAILEV 53 ka a, determned by transferrng one partcle from the orbtal to the orbtal n the state hereafter we wll use the notaton to denote such states. Accordngly, the ndex runs over those states n whch s occuped and s vacant. For such and the matrx element 1, otherwse, t s zero. Therefore, n fact, the sum n 1 s a sngle sum, wth a number of tems less than N, n S 1 n n d C 1 n C, 14 where the sum runs over the specfed. Analogously, Eq. 13 can be wrtten as the double sum over and specfed as above, n S 1 n n c C 1 n C 1 n C n C, 15 FIG. 1. a Mean-square matrx element 11 calculated n the TBRIM for n4 partcles and m11 orbtals, 4, 5, as a functon of the egenstate n for n Averagng over N r 100 Hamltonan matrces H for dfferent realzatons of the random two-body matrx elements has been made. Dots correspond to the sum S c whle the sold lne represents the dagonal contrbuton only see 1. b Rato RS c / of the correlaton contrbuton to the dagonal contrbuton. approxmaton s not completely accurate for the computaton of the varance of matrx elements of perturbaton. In order to study ths effect n detal, we have performed numercal experments wth TBRIM for the parameters correspondng to the model calculatons of the Ce atom 1,13. We take the number of partcles n4, the number of orbtals m11, the spectrum of the sngle-partcle orbtals s determned by d 0 1, and the Gaussan random two-body nteracton s gven by V 0.1. As a result, the sze of the Hamltonan matrx H s N330. The calculaton of the matrx elements between compound states n ths model gave a remarkable result. In Fg. 1 we present the expermental value of M see Eq. 11 together wth the dagonal contrbuton 1 see Fg. 1a, and the rato RS c / n Fg. 1b. Fgure 1a reveals a systematc dfference between the dagonal approxmaton and exact expresson 11, and Fg. 1b shows that nondagonal term S c s of the same order as, whch clearly ndcates the presence of correlatons. Below, we show how these correlatons emerge n the nondagonal term S c. Frst, note that for a gven the sum over k n Eqs. 1 for contans only one term, for whch where s a functon of, a a. Note that the energes of the bass states and ther prmed partners are connected as E E E E. One can expect that maxmal values of the sum 14 and, possbly, 15 are acheved when C s are prncpal components of the egenstates. Ths means that the mean square of the matrx element n 1 n s maxmal when the operator couples the prncpal components of the state n 1 wth those of n,.e., for E 1 ) E ). Far from the maxmum (E 1 ) E ) ) a prncpal component of one state, say, n 1, s coupled to a small component k of the other state n (E k E ) ). The latter case s smpler to consder analytcally, snce the admxture of a small component n the egenstate can be found by means of perturbaton theory. Ths approach reveals the orgn of the correlatons n the sum S c, Eq. 15. For example, f C 1 ) s a small component of the egenstate n 1, then t can be expressed as a perturbaton theory admxture to the prncpal components. If C 1 ) s one of the latter, then there s a term n the sum 15, whch s proportonal to the prncpal component squared, C 1 ). Indeed, there are three possbltes C 1 ) and C ) are among the prncpal components, and C 1 ) and C ) correspond to the small components. Then, one can wrte n C 1 Hñ 1 H E n 1 E p n p E n 1 C 1 E p, n C Hñ E n E q H q E n E C q n The tlde above the sums ndcates that the summatons run over the prncpal components only. The coherent contrbuton to the sum S c n Eq. 15 s obtaned by separatng the squared contrbutons of the prncpal components n the sums n S c 1 n ).e., p,q )

5 CORRELATIONS WITHIN EIGENVECTORS AND TRANSITION... H H E n E E n 1 E C n 1 n C., 18 Takng nto account that for the prncpal components we have E E 1 ) and E E ), we can replace the energes, E E 1 ), and E E ), and thus obtan the followng contrbuton to S c 1 n ) : 1 E n E n 1, C 1 C H H. 19 C 1 ) and C ) correspond to the prncpal components, C 1 ) and C ) correspond to the small components. Then, the result s the same as 19. C 1 ) and C 1 ) are prncpal components, C ) and C ) are small components or, C ) and C ) are prncpal components, C 1 ) and C 1 ) are small components. In these cases there are no coherent terms n the sum for S c n Eq. 13. Ths follows from the fact that for chaotc egenstates the mxng among the prncpal components s practcally complete, whch makes them to a good accuracy statstcally ndependent. Thus, far from the maxmum, E ) E 1 ), one obtans S c n 1 n E n E n 1, C 1 C H H. 0 A smlar calculaton of the dagonal sum 1 n ), Eq. 1, yelds n S 1 n 1 d E n E n 1 n C 1 n C H n C 1 n C H. 1 The two terms n square brackets result from the contrbuton of prncpal and small components n Eq. 14, and vce versa. From Eq. 0 we see that S 1 n ) c 0 f H H 0. However, there s nearly a 100% correlaton between these matrx elements. Indeed, the bass state dffers from by the locaton of only one partcle the transton from the orbtal to ), and the same s true for and. Let us estmate the relatve magntudes of and S c. Frst, consder the case when and dffer by two orbtals, a a 1 a a 1. In ths case H V 1 1. Snce the bass states and must dffer by the same two orbtals, we have H V 1 1 H note that 1, 1,,,, snce both states and contan and do not contan, whereas and contan and do not contan ). Therefore the averages over the nonzero matrx elements between such pars of states are H H H H V. Now, let us consder the case when and dffer by one orbtal a a 1. In ths case the Hamltonan matrx elements are sums of the n1 two-body matrx elements, n H V 1 V 1, n H V 1 V 1. The sums of n terms n H and H concde; the dfference s due to the one term only orbtal s replaced by the orbtal ). Thus H H nv, H H n1v, where we took nto account that V V V The contrbutons of one-partcle and two-partcle transtons n Eqs. 0 and 1 representng S c and, respectvely, wll be determned by the numbers of such transtons allowed by the correspondng sums. For the sngle-prme sums n Eq. 1 these numbers are proportonal to K 1 and K, Eq. 3. In the double-prme sum n Eq. 0 these numbers are proportonal to K 1 and K, the numbers of the twobody and one-body transtons, n the stuaton when one partcle and the two orbtals ( and ) do not partcpate n the transtons. These numbers can be obtaned from Eq. 3 f we replace n by n1, and m by m, so that K 11)(mn1), K 1))(mn1) (mn)/4. Fnally we obtan that at E ) E 1 ) the contrbuton of the correlaton term to the varance of the matrx elements of can be estmated n the rato as R S c nk 1K n1k 1 K nmn1mn. nmnmn3 Ths equaton shows that for n we have S c 0, whch s easy to check drectly, snce H H 0 n ths case. For n the correlaton contrbuton S c s negatve at the tals of the strength dstrbuton. Ths means that t ndeed suppresses the transton ampltudes off resonance see Fg. 1. For n,mn1 the rato R s approachng ts lmt value 1. It s easy to obtan from Eq. that for mn1 S c 1R m nmn. 3

6 5734 V. V. FLAMBAUM, G. F. GRIBAKIN, AND F. M. IZRAILEV 53 It s worth emphaszng that the exstence of correlatons due to the perturbaton theory admxtures of small components to the chaotc egenstates, whch leads to a nonzero value of S c 15, s ndeed nontrval. For example, f one examnes the summand of Eq. 15 as a functon of and, t would be hard to guess that the sum tself s essentally nonzero, snce postve and negatve values of C 1 ) C 1 ) C ) C ) seem to be equally frequent, and have roughly the same magntude, see Fg. 3. Snce n1 S 1 n ) c n S 1 n ) c 0 see below, the suppresson of M at the tals should be accompaned by correlatonal enhancement of the matrx elements near the maxmum at E () E (1) ). Thus we come to the mportant concluson: even for a random two-body nteracton, the correlatons produce some sort of a correlaton resonance n the dstrbuton of the squared matrx elements M. One should note that ths ncrease of the correlaton effects n the matrx elements of a perturbaton can be explaned by the ncreased correlatons between the Hamltonan matrx elements when the number of partcles and orbtals ncreases (N/ne n ). Now we can estmate the sze of the correlaton contrbuton S c near the maxmum of the the M dstrbuton at E ) E 1 ) ). Frst, we show that after summaton over one of the compound states, the correlaton contrbuton vanshes. Indeed, FIG.. Same as n Fg. 1, for n7, m14, 7, 8. The data obtaned for a sngle Hamltonan matrx of the sze N343; n Note the ncreased role of the correlaton contrbuton S c. Thus, surprsngly, the role of the correlaton contrbuton ncreases wth the number of partcles. For the numercal example shown n Fg. 1, n4, m11, one obtans R0.39, whch means that the correlaton contrbuton reduces the magntude of the squared matrx elements M between compound states almost by a factor of for E ) E 1 ) ). The rato found numercally s R0.45 Fg. 1b, n ; larger n are probably too close to the boundary of the matrx for R to reman constant. We would lke to stress that the role of the correlaton term does not decrease wth the ncrease of the numbers of partcles and orbtals. Ths predcton s supported by Fg., whch shows the behavor of the squared matrx element and ts dagonal and correlaton parts for n7 and m14, N343. One can see that the suppresson of the matrx elements M due to the correlaton term at the tals s even stronger than that n Fg. 1 the numercally found rato s R0.7 vs R0.55 obtaned from Eq.. The correlaton contrbuton should be even more mportant n compound nucle, where N10 5. Ths case can be modeled by the parameters n10, m0; then we have R0.66, or, equvalently, ( S c )/ 0.34, whch means that the correlatons suppress the squared element M between compound states by a factor of 3 far from ts maxmum. n S c n 1 n n,kl C n 1 C n 1 C k n C l n k l n C 1 n C 1 k,kl 0, l n C k n C l n 4 where we take nto account that the sum over n n the expresson above s zero for kl. Therefore the negatve value of S 1 n ) c at E ) E 1 ) must be compensated by ts postve value near the maxmum. The sum rule 4 allows one to make a rough estmate of S c near the maxmum of and M ). Let us assume that S c R m at E ) E 1 ) /, whereas S c R t at E ) E 1 ) / R t s gven by Eq.. The dstrbuton of S 1 n ) d can be reasonably approxmated by the Bret-Wgner shape see Secs. I and IV, n S 1 n A d E /4, 5 where EE ) E 1 ), and n1 n. The sum rule 4 mples that R m 0 / de E /4 R t / de E /4 0. 6

7 CORRELATIONS WITHIN EIGENVECTORS AND TRANSITION... Comparng the values of the rato S c / at the maxmum and at the tal n Fg. 1b 4,m11), one can see that ndeed, R m R t. For larger n and m the correlaton enhancement factor asymptotcally reaches ts maxmal value of. The numercal example n Fg. 7,m14) shows the enhancement of M wth respect to at the maxmum even greater n sze than that predcted by Eq. 7. Ths s not too surprsng snce n Eqs. 5 7 we estmated the average value of R m over an nterval E around the maxmum rather than the peak value at the maxmum. A smlar estmate of S c near maxmum can be obtaned by the drect calculaton of the small component contrbuton to S c 15. On an assumpton that there are no correlatons between prncpal components of compound states we can separate the contrbuton of small components. For example, n the resonance stuaton, E ) E 1 ), f the components S 1 ) and S ) are small (E E 1 ), and consequently, E E ) ), then they contan contrbutons proportonal to the prncpal components C 1 ) and C ) see Eqs. 16, 17. Analogously, S 1 ) and S ) may be among the prncpal components, and then the small components C 1 ) and C ) wll contan correlated contrbutons. Thus we have the followng estmate: S c n 1 n n small C 1 n C H H E n 1 E E n E. 8 Snce E E E E E ) E 1 ) for the prncpal components and, (E 1 ) E ) and (E ) E ) n the denomnator always have the same sgn, and S c s postve recall that H H 0). Expresson 8 can be estmated usng the well known formula for the spreadng wdth, H /D, where D s the mean level spacng for the many-body states. Ths yelds S c, n agreement wth the prevous estmate 7. III. CORRELATIONS BETWEEN TRANSITION AMPLITUDES FIG. 3. The dstrbuton of the tems of the sum 15, C 1 ) C 1 ) C ) C ), for n1 55, n 66, obtaned n the TBRIM for the same set of parameters as n Fg. 1, averaged over N r 100 realzatons of V. Indces and n the fgure run over those 84 components n whch s occuped and s vacant. a Postve values. b Negatve values absolute values. Snce the two ntegrals n the above equaton are equal, we have R m R t. Thus near the maxmum the correlaton contrbuton S c s postve and enhances the squared matrx element wth respect to the dagonal contrbuton, We have shown that correlatons between egenvector components n a system wth a two-body nteracton between partcles must be taken nto account when calculatng the varance of a matrx element between compound states M. Another queston s whether the above correlatons between egenstate components lead to correlatons between dfferent matrx elements, M n1 n M n n 3 n 1 n n n 3 C n 1 C n C k n C l n 3 k l. 9 S c m 1R m 1R t 1 nmn. 7 Our analyss shows that the correlatons of the type 9 are absent,.e., M n1 n M n n 3 M n1 n M n n 3 0. The result could

8 5736 V. V. FLAMBAUM, G. F. GRIBAKIN, AND F. M. IZRAILEV 53 FIG. 4. Probablty densty of the normalzed matrx elements xn 1 n /(n 1 n ) 1/ n the TBRIM for the parameters of Fg. 1. The hstogram s obtaned for N r 5 Hamltonan matrces. Sold curve s the normalzed Gaussan dstrbuton. be dfferent f the prncpal components of dfferent egenvectors (n 1 and n 3 ) were correlated. Ths effect takes place n the separable nteracton model 5,4, but we have not found such correlatons n the TBRIM. The absence of correlatons between dfferent ampltudes s confrmed by drect numercal experments. Frst, we have studed the probablty densty of the matrx elements M n1 n for dfferent n 1,n obtaned for a number of realzatons of the two-body matrx elements V. Snce the varance of M n1 n depends on n 1 and n, the probablty densty of M n1 n has been obtaned by normalzng each matrx element M n1 n to ts root-mean-squared value whch was calculated by averagng over the realzatons of V. The re- sultng probablty densty P M n1 n /M n1 n, averaged over N r 5 realzatons of V, turns out to be qute close to Gaussan Fg. 4. Ths result follows from the fact that each matrx element between compound states s the sum of a large number of random or almost random terms, see Eq. 10, so that the central lmt theorem apples. Therefore the correlatons found n the precedng secton do not show up, unless more complcated correlatons nvolvng dfferent components of the same egenstate, lke those n Eqs. 8 or 11, are probed. To check whether some correlatons between dfferent matrx elements 9 exst, we have plotted the matrx element n 1 n versus another one, n 1 n 1, where n 1 s the egenstate mmedately precedng n, for some fxed n 1, n,, and, obtaned from N r 387 dfferent Hamltonan matrces Fg. 5a. The latter were generated by usng dfferent random realzatons of V. Detaled analyss of the dstrbuton of the ponts n ths fgure does not reveal any sort of correlatons. The next queston s the exstence of correlatons between matrx elements M n1 n and W n1 n of dfferent operators for the same compound states n 1 and n. If the expansons of these matrx elements see Eq. 10 contan dentcal matrx elements of the densty matrx operator, such correlatons, n prncple, do exst: FIG. 5. a The matrx element y n 1 n for n 1 55, n 67, plotted vs the matrx element x n 1 n 3 for n 3 66; 4, 5, and other TBRIM parameters as n Fg. 1. The number of ponts n the fgure s N r 387. No evdence of correlatons between x and y s present. b The matrx element y n 1 n for 4, 6 vs x n 1 n wth 5 for n 1 55 and n 66. Agan, there s no ndcaton of correlatons between x and y. Note the dfference n the vertcal and horzontal scale due to the fact that for gven n 1 and n the energy dfference E 1 ) E ) s approxmately n resonance for the transton between 4 and 5 and off resonance for 4 and 6. M n1 n W n n 1, M W n 1 n A more complcated queston s whether the matrx elements of dfferent elementary transton operators and are ndeed uncorrelated as we assumed wrtng Eq. 30. The product of such two matrx elements can be presented n the form n 1 n n n 1,,k,l C n 1 C n 1 C k n C l n kl 31 n C 1 n C H H, E n 1 E E n E, 3

9 CORRELATIONS WITHIN EIGENVECTORS AND TRANSITION... where the last expresson s wrtten for n 1 and n far from the maxmum (E ) E 1 ),E ) E 1 ) ), and a a, a a. It can be shown that n our model H H. Therefore there are no terms n the expresson 3 whch would gve nonzero contrbutons, and the average of 31 s zero. The absence of correlatons n ths case s llustrated by Fg. 5b, where numercal data obtaned n the TBRIM are presented. As n the case of the matrx elements between dfferent pars of compound states, no correlatons can be seen between the matrx elements of dfferent transton operators. IV. STATISTICAL DESCRIPTION OF THE TRANSITION AMPLITUDES In ths secton we use the TBRIM to test the valdty of the statstcal approach to the calculaton of transton ampltudes between compound states of complex systems developed n 3,1,0. In what follows we frst outlne the man deas of the statstcal approach. The varance of the matrx elements of an operator Mˆ 9 between the compound states n 1 and n can be presented n the followng form compare wth Eq. 30: M n1 n, M n 1 n, 33 where we have taken nto account the result of the precedng secton that the average of the correlator 31 s zero unless,. Therefore the calculaton of M n1 n or M n1 n W n n 1 ) s reduced to the calculaton of n 1 n. It was suggested n Sec. II that n 1 n can be presented as the sum of the dagonal and correlatonal sums and S c, Eqs Snce we have already estmated the rato ( S c )/, t s enough to calculate only, Eq. 1. Followng 1 let us replace the squared components C 1 ) and C k ) by ther average values, C n 1 we,e n 1, C k n we k,e n, 34 where the averagng goes as usual ether over a number of realzatons of the two-body nteracton matrx elements ensemble average, or over a number of neghborng egenstates physcal energy average; n the sprt of ergodcty the results are presumably the same. The functon w s proportonal to the strength functon ntroduced by Wgner 10, whch s also called the local spectral densty of states. Note that defnton 34 also mples that the mean-square contrbuton of the component n the egenstate n 1 s determned by ther energes, E and E 1 ) n fact, by ther dfference E E 1 ) ). For states localzed n the gven bass, w s a bell-shaped functon wth a typcal wdth determned by the spreadng wdth. There s some theoretcal and expermental evdence that t can be approxmated by the Bret-Wgner formula, although ts tals decrease faster than E E 1 ) see references n Sec. I. The dagonal sum now takes the form n 1 n,k we,e n 1 we k,e n kk. 35 The summaton over k for a fxed ncludes only one state, k, wth E k E. On the other hand, we can wrte k kk nˆ 1nˆ, 36 where nˆ a a and nˆ a a are the occupaton number operators. Thus we obtan n 1 n we,e n 1 we,e n nˆ 1nˆ. 37 The matrx element nˆ (1nˆ ) s equal to 1 f the orbtal s occuped and s vacant n the bass state, otherwse, t s zero. We used ths fact earler Eq. 14 to reduce the summaton to these states only. Now we proceed n a dfferent way. Both w s n 37 are smooth functons of energy normalzed as w(e,e 1 ) )1. Ths allows one to replace the matrx element of nˆ (1nˆ ) by ts expectaton value, nˆ 1nˆ we,e n 1 nˆ 1nˆ C n 1 nˆ 1nˆ nˆ 1nˆ n1. 38 The sgn above s a remnder that the left-hand sde s the local average over the states n 1. Practcally, when the number of components s large, the fluctuatons of nˆ (1nˆ ) n1 are expected to be small. Now we can rewrte Eq. 37 n a form smlar to Eq. 14, but wthout any restrctons on the summaton varable, n 1 n nˆ 1nˆ n1 we,e n 1 we,e n. 39 It was shown n 1 that under some reasonable assumptons about the functons w one can ntroduce a spread functon (), D 1 we,e n 1 we,e n D 1 de D 1 we,e n 1 we,e n, 40 where E ) E 1 ), and D 1 and D are local mean level spacngs for the n 1 and n egenstates. The functon () s symmetrc, ts characterstc wdth s determned by the spreadng wdths of the egenstates n 1 and n,

10 5738 V. V. FLAMBAUM, G. F. GRIBAKIN, AND F. M. IZRAILEV 53 1, and t s normalzed to unty, ()d1, ust as the standard functon. If w s have Bret-Wgner shapes, s also a Bret-Wgner functon wth 1. The fact that 1 n ) s proportonal to the functon () s a partcular manfestaton of the energy conservaton for transtons between the quasstatonary bass states 0 f 0, then () (). Usng Eqs. 33, 39, and 40 we can fnally present the dagonal contrbuton to the varance of the matrx element M n1 n n the form M n1 n dag M nˆ 1nˆ n1, D E n E n Ths expresson s apparently asymmetrc wth respect to the states n 1 and n. By performng the calculaton n a dfferent way we can obtan n 1 n nˆ 1nˆ n k we k,e n 1 we k,e n, 4 nstead of Eq. 39, and thereby arrve at a dfferent formula for the varance, M n1 n dag M nˆ 1nˆ n, D 1 E n E n 1, 43 where the occupances factor s now calculated for the state n t represents the probablty to fnd the orbtal occuped, and empty. If the suppostons made n the above dervatons are correct, the two formulas 41 and 43 should gve dentcal results. In the present work we use the TBRIM to check the accuracy of the statstcal approach descrbed above. Fgure 6a presents a comparson between the values of 1 n ) as gven by Eqs. 39, 4, and those from the ntal expresson 1. Clearly, there s a good agreement between the three formulas. It s qute mportant for applcatons of the statstcal approach see 4,0 that further smplfcatons be made by replacng the correlated occupances product nˆ nˆ n1 n Eq. 38 by the product of the two mean values, nˆ n1 nˆ n1. Ths s defntely a vald operaton when the numbers of excted partcles and actve orbtals are large, so that the occupaton numbers for dfferent orbtals become statstcally ndependent. Then one would be able to use the relaton nˆ 1nˆ n 1n, 44 where n( ) and n( ) are the occupaton numbers. They can be calculated, e.g., usng the Ferm-Drac formula wth an effectve temperature, see 17,0; see also 9 where the relaton between thermalzaton and chaos s studed n nuclear shell-model calculatons. The result of such smplfcaton s shown n Fg. 6b, where the dagonal contrbuton 1 s agan compared wth the values obtaned from Eqs. 39, 4, usng approxmaton 44. In spte of the fact FIG. 6. a The dagonal contrbuton to the mean-square matrx element as obtaned from Eqs. 39, 4 sold lnes n comparson wth the drect calculaton of, Eq. 1 crcles. The TBRIM parameters are the same as n Fg. 1. b Same as a, wth the occupancy factors n Eqs. 39, 4 calculated by means of Eq. 44. that the TBRIM calculaton ncluded n4 partcles only, the agreement remans qute reasonable, the error beng about 10%. To examne the qualty of the approxmaton at the tals of the dstrbuton, Fg. 7 shows the rato of as gven by Eqs. 39, 4 to the drectly calculated dagonal term, Eq. 1. The dfference between Fgs. 7a and 7b hghlghts the naccuracy ntroduced by an addtonal approxmaton 44 for the occupaton numbers. In order to make a more drect test of the valdty of substtuton 44, we plotted n Fg. 8 the correlator nˆ nˆ n1 /nˆ n1 nˆ n1 as a functon of n 1. Consstent wth the small number of partcles, ths correlator dsplays large fluctuatons; however, ts average value of about 0.8 s stll rather close to 1. V. SPREADING WIDTHS FOR DIFFERENT BASIS COMPONENTS In Sec. IV when consderng the statstcal approach to the calculaton of the varance of matrx elements between compound states, t was assumed that the spreadng wdths are the same for all bass components. However, ths queston s not trval. As s dscussed n the lterature, the spreadng

11 CORRELATIONS WITHIN EIGENVECTORS AND TRANSITION... FIG. 9. The spreadng wdth calculated as the rms devaton from the center of the dstrbuton of the components C 1 ) for each bass state. The data are obtaned for one matrx H correspondng to n6 partcles and m1 orbtals. FIG. 7. a The rato R a of the approxmaton represented n Fg. 6a by the sold lnes, to the value of. b Same as n Fg. 7b for the data of Fg. 6b. wdths of components correspondng to dfferent numbers of excted partcles could have sgnfcantly dfferent values. For example, n t s argued that two-partcle one-hole states (p-1h) can lead to correlatons between values of party nonconservng effects 1, f the spreadng wdth of the p-1h states s two orders of magntude smaller than that of 1p states. In such a case one mght expect that n our model the spreadng wdth would show a rapd decrease as a functon of the number of excted partcles n the bass component. To study ths queston n detal, we have performed addtonal tests. In Fg. 9 the root-mean-squared spreadng wdth for all bass states s presented for n6, m1 (N94). Here we use the followng defnton: ŠH H, whch can also be presented n several equvalent forms, H H H M M 1, 48 the last one relatng drectly to the moments M p of the strength functon w (E, ) n C ) (EE ) ), M p w E, E p de n C n E n p. 49 FIG. 8. The correlator Q nˆ nˆ n1 /nˆ n1 nˆ n1 vs the egenstate number n 1. The TBRIM parameters are the same as n Fg. 1. The average value of the correlator s about 0.8, whch means that the correlatons between the occupances of dfferent orbtals are not very strong. Equatons can be obtaned usng closure, n nn1, where and n are the bass state and the egenstate, respectvely, nc ). We should note that the rms spreadng wdth s dfferent from that ntroduced ntutvely n Sec. IV as the characterstc wdth of the strength functon. For example, f the strength functon has a Bret-Wgner form, ts second and hgher moments are nfnte. In Wgner s BRM the rms spreadng wdth s determned by the bandwdth b as bv, whereas the Bret-Wgner spreadng wdth s BW v /D see Introducton. However, n a more realstc stuaton the strength functon drops rapdly, ts second moment s fnte, and the dfference between the rms and BW s not large.

12 5740 V. V. FLAMBAUM, G. F. GRIBAKIN, AND F. M. IZRAILEV 53 From Fg. 9 one can see that apart from small natural fluctuatons, the rms spreadng wdth s the same for all components. To exclude a weak dependence of on the energy of the bass state boundary effects seen as rses of at small and large ), we calculate the mean value and the rms devaton of the spreadng wdth by averagng over The results are., 0.1. The latter value shows that the fluctuatons of the wdth are very small. Ths result s n agreement wth computatons made for the Ce atom 1; smlar results have been recently obtaned n the nuclear sd-shell-model calculatons 9,3. The fluctuatons of the wdth are small due to the large number of decay channels for each bass component each component s coupled to many others by random nteracton 4. Formally, ths can be obtaned from Eq. 47. For example, n the TBRIM one obtans n1v K 1 V K K 1 V, 50 whch shows that for a large number of ndependent decay channels, K 1 1)K 1 K see Eq. 3, the statstcs of s gven by the dstrbuton wth K 1 1 degrees of freedom, resultng n the 1/K 1 decrease of fluctuatons. More accurately, the relatve rms fluctuaton of the squared wdth 45 s gven by ( )/ /K1. For n6, m1, Eq. 50 yelds.41, and the relatve fluctuaton s These values are close to the numercal ones quoted above the dscrepancy s manly due to the dfference between the mean wdth and ts rms value, and the correspondng dfference n the fluctuatons of and ). To check the ndependence of of the number of excted partcles n the component, we have calculated the average spreadng wdth (p) for bass states wth a fxed number of excted partcles p, p1,,...,n1. In the numercal experment shown n Fg. 6 all (p) for p1 5 proved to be approxmately the same, (p).. The above consderaton shows that the statstcal approach does not provde any support for the dependence of the spreadng wdth on the number of excted partcles. Ths ndcates that the argumentaton n favor of a strong dependence, based on dfferent decay phase volumes for dfferent numbers of excted partcles, seems to be ncorrect. In our opnon, the dfference n the spreadng wdths could appear as a result of specfc dynamcal effects. For example, ths could be an nfluence of levels n other potental wells whch appear at hgher nuclear deformaton, or due to an nteracton wth collectve motons, such as rotatons and vbratons. VI. CONCLUSIONS The calculaton of the mean-square matrx element of an operator between compound states of a many-body system has been consdered. We have shown that the two-body nature of the nteracton between partcles manfests tself n the exstence of correlatons between the components of the chaotc compound egenstates. These correlatons taken together wth the correlatons between the many-partcle Hamltonan matrx elements result n a relatvely large correlaton contrbuton to the mean-square matrx element. The correlatons exst even f the two-body matrx elements are ndependent random varables, as n the TBRIM. Such correlatons can be understood n terms of the perturbatve mxng of the dstant small components to the prncpal components of the egenstates. If the Hamltonan matrx elements are random varables the correlatons of ths type vansh. One of the most nterestng features of the correlatons found n our work s that they do not decrease wth the ncrease of the number of excted partcles or actve orbtals. Thus they must be taken nto account when calculatng matrx elements of a weak nteracton between compound states n nucle. Another feature concerns the shape of the dstrbuton of the densty matrx operator near ts maxmum. As one can see from Fgs. 1a and a, the correlatons create a sharp spkelke form of the dstrbuton, nstead of a smooth Gaussan or Bret-Wgner form. Wth such sharp peaks, the strength functon for any partcular operator Mˆ can have the so-called gross structure, due to many sngle-partcle transton terms n the expresson 33. Wthout these specfc correlatons, the strength functon would be much smoother and the gross structure would not be seen. It s also nterestng to note that there are very large mesoscopc-type fluctuatons n the dstrbuton near the maxmum, dependng on a specfc random realzaton of the two-body nteracton V. Ths fact s also the consequence of strong correlatons. Our study also demonstrated that the spreadng wdths of dfferent bass components are approxmately constant and fluctuate very weakly. In partcular, we have not found any dependence on the number of excted partcles n the component. The statstcal approach to the calculatons of such matrx elements has been tested n the present work wth the help of the TBRIM. The numercal results obtaned n ths work support the valdty of the statstcal approach. The TBRIM has also enabled us to check that the matrx elements of dfferent transton operators between a par of compound states are uncorrelated, as are the matrx elements of a gven operator between dfferent pars of compound states. ACKNOWLEDGMENTS The authors are grateful to O. P. Sushkov and M. G. Kozlov for valuable dscussons. F.M.I. would lke to acknowledge the hosptalty extended to hm durng hs vst to the shcool of Physcs, Unversty of New South Wales. Ths work was supported by the Australan Research Councl and Gordon Godfrey Fund. 1 C. M. Frankle et al., Phys. Rev. Lett. 67, V. V. Flambaum, Phys. Rev. C 45, V. V. Flambaum, n Tme Reversal Invarance and Party Volaton n Neutron Reactons, edted by C. R. Gould, J. D. Bowman, and Yu. P. Popov World Scentfc, Sngapore, 1994, p V. V. Flambaum and G. F. Grbakn, Prog. Part. Nucl. Phys. 35,

13 CORRELATIONS WITHIN EIGENVECTORS AND TRANSITION... 5 V. V. Flambaum unpublshed. 6 T. A. Brody, J. Flores, J. B. French, P. A. Mello, A. Pandey, and S. S. M. Wong, Rev. Mod. Phys. 53, J. B. French and S. S. M. Wong, Phys. Lett. 35B, O. Bohgas and J. Flores, Phys. Lett. 34B, M. Horo, V. Zelevnsky, and B. A. Brown, Phys. Rev. Lett. 74, ; V. Zelevnsky, M. Horo, and B. A. Brown, Phys. Lett. B 350, E. P. Wgner, Ann. Math. 6, ; 65, A. Bohr and B. Mottelson, Nuclear Structure Benamn, New York, 1969, Vol V. V. Flambaum, A. A. Grbakna, G. F. Grbakn, and M. G. Kozlov, Phys. Rev. A 50, V. V. Flambaum, A. A. Grbakna, and G. F. Grbakn, Phys. Rev. E 5, Y. V. Fyodorov and A. D. Mrln, Phys. Rev. Lett. 67, ; 69, ; 71, Y. V. Fyodorov, O. A. Chubykalo, F. M. Izralev, and G. Casat, Phys. Rev. Lett. 76, G. Casat, B. V. Chrkov, I. Guarner, and F. M. Izralev, Phys. Rev. E 48, R V. V. Flambaum, F. M. Izralev, and G. Casat, Phys. Rev. E to be publshed. 18 K. K. Mon and J. B. French, Ann. Phys. N.Y. 95, J. P. Draayer, J. B. French, and S. S. M. Wong, Ann. Phys. N.Y. 106, ; 106, V. V. Flambaum and O. K. Vorov, Phys. Rev. Lett. 70, O. P. Sushkov and V. V. Flambaum, Usp. Fz. Nauk 136, Sov. Phys. Usp. 5, M. S. Hussen, A. K. Kerman, and C. Y. Ln, Z. Phys. A 351, ; B. V. Carlson, M. S. Hussen, A. K. Kerman, and C. Y. Ln, Phys. Rev. C 5, R V. G. Zelevnsky unpublshed.