Statistical theory of nite Fermi systems with chaotic excited eigenstates

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1 PHILOSOPHICAL MAGAZINE B, 2000, VOL. 80, NO. 12, 2143± 2173 Statstcal theory of nte Ferm systems wth chaotc excted egenstates V. V. FLAMBAUMy and G. F. GRIBAKINz School of Physcs, Unversty of New South Wales, Sydney 2052, Australa [Receved 10 Aprl 2000 and accepted 22 May 2000] ABSTRACT A theory of strongly nteractng Ferm systems of a few partcles s developed. At hgh exctaton energes (a few tmes the sngle-partcle level spacng) these systems are characterzed by an extreme degree of complexty due to strong mxng of the shell-model-based many-partcle bass states by the resdual twobody nteracton. Ths regme can be descrbed as many-body quantum chaos. Practcally, t occurs when the exctaton energy of the system s greater than a few sngle-partcle level spacngs near the Ferm energy. Physcal examples of such systems are compound nucle, heavy open shell atoms (e.g. rare earths) and multcharged ons, molecules, clusters and quantum dots n solds. The man quantty of the theory s the strength functon whch descrbes spreadng of the egenstates over many-partcle bass states (determnants) constructed usng the shell-model orbtal bass. A nonlnear equaton for the strength functon s derved, whch enables one to descrbe the egenstates wthout dagonalzaton of the Hamltonan matrx. We show how to use ths approach to calculate mean orbtal occupaton numbers and matrx elements between chaotc egenstates and ntroduce typcally statstcal varables such as temperature n an solated mcroscopc Ferm system of a few partcles. } 1. INTRODUCTION As s nown, quantum-statstca l laws are derved for systems wth n nte number of partcles, or for systems n a heat bath, therefore, ther applcablty to solated nte systems of a few partcles s, at least, questonable. However, the densty of many-partcl e energy levels ncreases extremely rapdly (typcally, exponentally) wth an ncrease n both the number of partcles and the exctaton energy. For ths reason, even a wea nteracton between partcles can lead to strong mxng between large number of smple many-partcle states, resultng n the so-called chaotc egenstates. If the components of such egenstates can be treated as random varables (onset of quantum chaos), statstcal methods are expected to be vald even for an solated dynamc system. One should stress that statstcal descrpton of such solated systems can be qute dœerent from that based on standard canoncal dstrbutons; therefore, applcaton of the famous Ferm± Drac or Bose± Ensten formulae may gve ncorrect results. Moreover, for solated few-partcle systems a serous problem arses n the de nton y Emal: flambaum@phys.unsw.edu.au. z Present address: Department of Appled Mathematcs and Theoretcal Physcs, The Queen s Unversty of Belfast, Belfast BT7 1NN, Northern Ireland. Phlosophcal Magazne B ISSN 1364± 2812 prnt/issn 1463± 6417 onlne # 2000 Taylor & Francs Ltd DOI: /

2 2144 V. V. Flambaum and G. F. Grban of temperature, or other thermodynamc varables such as entropy and spec c heat (n comparson, n n nte systems, dœerent de ntons gve the same result). In ths paper we outlne a statstcal theory of nte quantum systems of nteractng partcles based on generc statstcal propertes of chaotc egenstates (the `mcrocanoncal approach). Typcal examples of such systems are compound nucle, complex atoms, atomc clusters and solated quantum dots. Ths wor s based on a number of earler publcatons (Flambaum 1993, 1994, Flambaum and Vorov 1993, Flambaum et al. 1994, 1996a, Grban et al. 1995, Flambaum and Izralev 1997a,b), whch developed dœerent aspects of the theory. } 2. STRENGTH FUNCTION OF CHAOTIC EIGENSTATES 2.1. Equaton for the strength functon In ths secton we obtan an equaton for the strength functon of chaotc egenstates. It s derved for Hamltonan matrces wth random uncorrelated oœ-dagonal matrx elements. Ths equaton generalzes the result obtaned by Wgner (1955, 1957) for n nte random matrces wth a lnearly ncreasng dagonal and oœ-dagonal matrx elements equal to 1, randomly placed wthn a band of wdth b along the dagonal. It also reproduces the equaton for the strength functon n sparse random matrces wth a dœuse band derved by Fyodorov et al. (1996) usng the supersymmetry method. It s mportant that ths equaton can be used for calculatng strength functons n real many-body systems where strong mxng s acheved owng to the resdual twobody nteracton between the partcles. In ths case, one usually starts wth multpartcle bass states (determnants) constructed of sngle-partcle orbtals obtaned for a partcular mean eld, for example n the Hartree± Foc approxmaton: j ˆ Yn ˆ1 a y j0: We use Gree letters to enumerate the sngle-partcle bass states (orbtals), and every bass state corresponds to a dœerent set of 1;... ; n. The matrx elements H j of the Hamltonan H ˆ " a y a 1 V 2 a y a y a a ; 2 calculated wth respect to ths bass are correlated, even f the resdual two-body nteracton V s random (Flambaum et al. 1996a, Brody et al. 1981). However, the eœect of these two-body correlatons on the strength functons s small (see below). Consder an unperturbed system whch s descrbed by a dagonal Hamltonan matrx wth some regular dagonal matrx elements: 1 H 0 j ˆ E j : 3 Ths Hamltonan descrbes a many-body system of non-nteractng partcles, or a system of nteracton partcles n the mean- eld bass, f we neglect the resdual twobody nteracton. It s often convenent to enumerate the bass states n such a way that ther mean energes E ˆ hjhj ncrease wth ncreasng. The egenstates of the system

3 Statstcal theory of nte Ferm systems 2145 Hj ˆ E j 4 expanded n terms of the bass states, j ˆ C j; are characterzed by ther components (`wavefunctons ) C, that s egenvectors of the Hamltonan matrx H j C ˆ E C : 6 j We consder cases when mxng of the bass components due to the oœ-dagonal matrx elements V j ² H j s strong. Ths regme s the opposte of perturbaton theory. Hence, V j ¾ D must be ful lled, where D s the mean spacng between the dagonal `energes E (see } 2.4). Besdes ths, the oœ-dagonal matrx elements V j n complex many-body systems appear to behave as random varables wth zero mean V j ˆ 0 (Flambaum et al. 1994, Grbana et al. 1995, Horo et al. 1995a,b, Frazer et al. 1996, Zelevnsy et al. 1996). In ths stuaton the egenstates (5) contan large numbers N p of essentally non-zero `prncpal components of smlar szes C ¹ 1=Np 1=2, due to normalzaton P jc j2 ˆ P jc j2 ˆ 1. If D 6ˆ 0 and the Hamltonan matrx s large, these components are centred around the correspondng egenvalue E, wthn some characterstc energy nterval je E j 9 G, where G s the so-called spreadng wdth. Its magntude depends on the strength of the mxng nteracton V j. Ths pcture allows for varaton of both D and V j 2 along the matrx, that s, as functons of the energy of the bass states. In real systems ths varaton s slow and smooth. Because of the strong mxng the components C of complcated (chaotc) egenstate uctuate almost randomly as functons of and. However, these uctuatons asde, nearby egenstates separated by energy ntervals E ¹ D ½ G loo smlar. Therefore, one can study the smooth envelope of the weghts jc j2. Ths dea s realzed n the strength functon, also referred to as the local densty of states W E; ˆ jc j2 E E 7 ntroduced by Wgner. It shows how varous bass states contrbute to the egenstates at energy E. A standard way to calculate W E; s through the Green s functon G E ˆ 1 E H ² ˆ jhj E E ² ; or n the matrx form, usng the egenstate components: 5 8 G j E ˆ C C * j E E ² ; 9 where ² s postve n ntesmal. The usual rule E E ² 1 ˆ E E 1 p E E combned wth the de nton (7) then gves the ey relaton

4 2146 V. V. Flambaum and G. F. Grban W E; ˆ 1 p Im G E Š: 10 In a smlar way, one also nds the egenvalue densty (or level densty) E ² E E 11 ˆ ; ˆ 1 p Im jc j2 E E ˆ Á! G E W E; ˆ 1 Im ftr G E Šg: p The Green s functon G, (equaton (8)) can be presented as a perturbaton theory expanson n powers of V, the oœ-dagonal part of the Hamltonan H ˆ H 0 V: where 1 G ˆ E H 0 V ² ˆ G 0 G 0 VG 0 G 0 VG 0 VG 0 ; G 0 1 ˆ E H 0 ² or G 0 j ˆ j E E ² s the Green s functon of the unperturbed Hamltonan. In the matrx form, expanson (15) loos le j V j G j ˆ E E E E E E j m V m V mj E E E E m E E j ; 17 where we have ntroduced E ² E ² for the energy shfted oœthe real axs. To obtan a smooth strength functon averaged over the level-to-level uctuatons of the egenstate components we should average the Green s functon (15) over the random oœ-dagonal matrx elements V j. In dong so we assume that they have zero mean and are uncorrelated: V j ˆ 0; V j V lm ˆ V 2 j l jm m jl : 18 We can also assume that, f the system s not placed n a magnetc eld, the phases of the bass states can be chosen so that V j ˆ V j are real. The varance V 2 j s a local average. It does not have to be constant over the matrx. On the contrary, n physcal systems, owng to the oscllatory behavour of the sngle-partcle orbtals the mxng matrx elements should decrease when dstant bass states are consdered: V 2 j! 0 for j jj! 1: 19 Therefore, larger matrx elements are found along the man dagonal, and the Hamltonan matrces are often modelled by banded (random) matrces (wth V j 6ˆ 0 only for j jj 4 b). Feature (19) also ensures that varous sums contanng products of V j converge, even f the sze of the Hlbert space s n nte. It s mportant that, whle we assume averagng n the course of our dervaton, t does not have to be performed explctly n the nal equatons. There t s done automatcally owng to summaton over large numbers of bass states.

5 The strength functon s gven by the dagonal matrx element G (equaton (10)). Because of the random sgns of V j, only terms wth even powers of V n the perturbaton expanson (15) survve after averagng: G j ˆ 1 E E m Statstcal theory of nte Ferm systems 2147 V m V m E E E E m E E m;l;n V m V ml V ln V n E E E E m E E l E E n E E : 20 It s convenent to represent ths expanson graphcally: 21 where the thc lne s G, the thn lnes are G 0, and the broen lnes correspond to the matrx elements V. The semcrcle broen lne descrbes a bnary product, for example V m V m n the second-order dagram, whch does not vansh upon averagng. In the fourth order the non-zero contrbutons are gven by the followng contractons or bnary assocatons: 22 The rst of them reduces to the product of the two second-order dagrams: 1 E E m V 2 m E E m 1 E E l V 2 l E E l 1 E E : 23 The second s rreducble; t does not contan G 0 ˆ E E 1 anywhere n the mddle. Analytcally t s gven by 1 E E m;l V 2 mv 2 ml E E m 2 E E l 1 E E : 24 If we compare t wth the expresson for the thrd dagram n equaton (22), 1 E E m V 4 m E E m 2 E E 1 E E ; 25 we mmedately see that t has a lower order of summaton, that s t s gven by a sngle sum over m, whereas n the other two the summaton runs over m and l. Snce we consder systems n the strong-mxng regme, jv m j ¾ D, every bass state s essentally coupled to many other bass states. Therefore, the sums over m or l n equatons (23) and (24) and the dagram wth the ntersectng broen lnes n equaton (22) (equaton (25)), s parametrcally smaller than the other two. Ths

6 2148 V. V. Flambaum and G. F. Grban rule s vald n all orders of the perturbaton theory. For example, among the sxthorder dagrams, only the followng ve dagrams must be taen nto account: 26 The dagrams wth ntersectons 27 should be dscarded. Let us ntroduce the self-energy S as a sum of all rreducble dagrams (wthout ntersectons), whch n our case are also dagonal wth respect to the end ndces: 28 Smlarly to the standard eld-theory procedure, the Green s functon can now be shown as 29 Ths geometrc seres mmedately sums nto the nal expresson for the Green s functon G E ˆ 1 E E S E : 30 However, the most mportant consequence of the absence of the dagrams wth ntersectons n S s that seres (28) can be presented as 31 whch corresponds to the followng equaton: S E ˆ m V 2 m E E m S m E ² : Ths nonlnear equaton can be solved teratvely. Once S E s found, the strength functon s obtaned from equatons (10) and (30) as W E; ˆ 1 p Im 1 : 33 E E S E 32

7 Statstcal theory of nte Ferm systems 2149 Equatons (32) and (33) solve the problem of ndng the strength functon for a Hamltonan wth random oœ-dagonal matrx elementsy, and the total densty of states s gven by Á! E ˆ 1 p Im 1 E E S E : 34 Note that the strength functon (33) can be wrtten as a Bret± Wgner formula W E; ˆ 1 G 2p E E D 2 G 2 =4 ; where G ˆ 2 Im S E Š s the energy (spreadng) wdth of the bass component and D ˆ Re S E Š s the shft of the centre of the egenvector wth respect to the bass state energy E. Snce both G and D are n general energy dependent, the real shape of the strength functon may be qute dœerent from the smple Bret± Wgner pro le wth constant parameters Partcular cases Full unform random matrces If the oœ-dagonal matrx elements V m are dstrbuted unformly over the matrx, V m 2 ˆ V2, then S E does not depend on and we have S E ˆ V 2 m E ˆ 1 Im S E Š; 2 pv 37 1 E E m S E ² ; whch reproduces the result obtaned by Pastur (1972, 1973) for H ˆ H 0 V, where V belongs to the Gaussan orthogonal ensemble Sparse banded random matrces Fyodorov et al. (1996) consdered the strength functons for large (n nte) matrces wth equally spaced dagonal matrx elements, E ˆ D, and a sparse and banded random oœ-dagonal part. They chose the followng probablty dstrbuton of V m : P V m ˆ 1 p m V m p m h 1 V m ; where p m s the probablty to nd a non-zero matrx element at ; m. It was parametrzed as p m ˆ M=b f j mj=b, where f s a decreasng functon normalzed as b 1 P 1 rˆ0 f r=b ˆ 1, b characterzes the wdth of the band, M s the number of non-zero matrx elements n a row, and the probablty densty h 1 sats es h1 v dv ˆ 1, vh1 v dv ˆ 0 and v 2 h 1 v dv ˆ V 2. Wth these de ntons the locally averaged square of the oœ-dagonal matrx element s gven by 36 y The authors are aware that a smlar dervaton was made n 1993 by O. K. Vorov (1993, unpublshed). Its dea s also analogous to that of Edwards and Warner (1980), ncludng the use of dagrams, although the latter consdered the case where V j 2 are constant over the matrx.

8 2150 V. V. Flambaum and G. F. Grban V m 2 ˆ M b f j mj V 2 : b If we use ths formula n equaton (32), replace summaton over m by ntegraton over de m =D, tae nto account that S E now depends only on E E and ntroduce new varables x and u as x ˆ E E R ; u ˆ E E m R ; usng R 2 ˆ MV 2, the complex conjugate of equaton (32) wll be cast n the followng form: g x ˆ 1 µ f ju xj=µ du ; 39 u g x where µ ˆ Db=R, and nstead of S a new functon has been ntroduced by S * ² Rg x. Accordngly, the strength functon as a functon of z ˆ E E s now gven by W z ˆ 1 p Re 1 : 40 z Rg z=r Equatons (39) and (40) were obtaned by Fyodorov et al. (1996) by means of a techncally nvolved method based on ntegraton over commutng and antcommutng varables Wgner random matrces In the poneerng wor of Wgner (1955, 1957) the equatons for the strength functons were obtaned for random matrces wth a regular dagonal D ˆ 1 and oœdagonal matrx elements V m ˆ V, placed randomly wthn the band j mj 4 b. If we use new varables, E E b ˆ z; E E m b replace the summaton over m n equaton (32) by ntegraton over ± and ntroduce the new functon p 1 z ˆ S E =V 2 and parameter q ˆ V 2 =b, the strength functon wll be gven by the followng two equatons: W z ˆ 1 pb Re p 1 z ˆ z 1 z 1 ˆ ±; z qp 1 z ; 41 d±; 42 ± qp 1 ± whch consttuted the man result obtaned by Wgner (1955, 1957) Exact solutons Semcrcle Suppose that the Hamltonan s a full unform random N N matrx wth V 2 m ˆ V2, and the bass states are degenerate: E ˆ 0. In ths case they are characterzed by the same S E ² S E, whch sats es equaton (32) wth E m ˆ 0:

9 N 1 V2 S E ˆ E S E ² ; whch gves S E ˆ 1 2 E 4NV2 E 2 1=2 Š, where we used N 1 º N for N ¾ 1. Substtuton n equatons (33) and (34) gves the strength functon and the densty of states Statstcal theory of nte Ferm systems 2151 W E ˆ 4NV2 E 2 1=2 2pNV 2 E ˆ N W E ˆ 2N pe0 2 E0 2 E 2 1=2 ; whch s the famous semcrcle, E 0 ˆ 2 NV 2 1=2 beng ts `radus. If the dagonal matrx elements uctuate wth E ˆ 0, E 2 ˆ 2V2, as n the Gaussan orthogonal ensemble, the result does not change for N ¾ 1, snce such uctuatons of the bass state energes are small compared wth the full wdth E 0 of the egenvalue spectrum. The semcrcular strength functon also descrbes the spreadng n a banded random Hamltonan, where V m are non-zero for j mj 4 b only. In ths case the sze N of the matrx n equaton (44) must be replaced by the bandwdth b. Ths soluton remans vald for a banded Hamltonan wth a non-zero ncreasng dagonal, as long as Db ½ bv 2 1=2 ; that s the mean bass-state energy spacng D remans small (Wgner 1955, 1957) Bret± Wgner pro le In the opposte case, Db ¾ bv 2 1=2, the ncreasng energy denomnators reduce the mxng of dstant bass states and mae the strength functon narrower. If we consder an n nte unform Hamltonan wth evenly spaced dagonal matrx elements, E m 1 E m ˆ D, the sum n equaton (32) can be replaced by an ntegral over de m =D. The mean-squared matrx element V m 2 s ndependent of (as well as m), and the functon S E also does not depend on. Equaton (32) then becomes V m 2 de S E ˆ m E E m Re S E Š Im S E Š D ; whch, after closng the ntegraton n the upper half-plane of complex E m, reduces to the contrbuton of the pole, S E ˆ pv 2 =D. The correspondng strength functon s mmedately obtaned from equaton (35) as where W E; ˆ 1 G 2p E E 2 G 2 =4 ; G ˆ 2pV 2 ; and ˆ D 1 s the densty of states. The Bret± Wgner soluton s n fact vald for matrces where V m 2 as well as change over the matrx, as long as they do not change much on the scale of 47 48

10 2152 V. V. Flambaum and G. F. Grban je E m j ¹ G. In partcular, equatons (47) and (48) are applcable to banded random matrces at je E j < Db, f V 2 m ˆ V2 ˆ constant wthn the band. The necessary condton for ths s G < Db, or V 2 < D 2 b, whch s the opposte of equaton (46) Crteron of equlbrum (ergodcty) of chaotc egenvectors Strong mxng Although the matrx elements V m uctuate, the soluton S E of equaton (32) s a smooth functon of both and E, provded that the summaton over m averages out these uctuatons. To ensure ths, the number of terms eœectvely contrbutng to the sum must be large. If the Hamltonan matrx s `full, that s V m 6ˆ 0 for any nearby par of states and m, as well dstant pars, the condton of strong mxng and wea uctuatons s V > D, just opposte to that of perturbaton theory. In many-body systems wth a two-body nteracton between the partcles the oœdagonal Hamltonan matrx elements V m couple only those shell-model bass states and m (1) whch dœer by postons of no more than two partcles. If the number of actve partcles n the system s greater than two, there are many zeros n the Hamltonan matrx, that s, t becomes ncreasngly sparse. Denotng the mean spacng je E m j between the nearest bass states drectly coupled by V m by d f (d f ¾ D), we can wrte the condton for averagng of uctuatons as G ¾ d f ; snce je E m j 9 Im js E j ˆ 1 2 G determnes the range of m whch eœectvely saturate the sum. The magnary part of S E can be estmated from equaton (32) as a usual contrbuton of a pole n the complex energy plane, whch gves G E ˆ 2pV 2 m f E ; where f E ˆ d f 1 s the densty of states drectly coupled to, and V m 2 s the average over non-zero matrx elements only. The condton (49) then becomes V 2 m > d2 f : Ths means that, for the strength functons to be smooth, the nteracton must be large, that s non-perturbatve. In ths case the egenvectors contan large numbers of components. The number of (prncpal) components s n fact N p ¹ G =D rather than G =d f, snce mxng of the bass states wthn the energy nterval G s complete (ergodcty). In ths regme the components C dsplay Gaussan statstcs, and the uctuatons n the strength functon E are small (equlbrum). When we reduce the rato V m =d f the uctuatons of G ncrease and at V m < d f the smooth self-consstent soluton of equaton (32) dsappears. Indeed, n ths case the sum n ths equaton s domnated by one term wth a mnmal energy denomnator E E m ¹ d f. The absence of a smooth soluton for the shape of the egenstates and the strength functon does not mean that the number of prncpal components n the egenstates s small.y However, the dstrbuton of the components s not ergodc; t has many `holes wthn the energy nterval 2pV m 2 f E, and the uctuatons of y The number N p of prncpal components n ths case can be defned as the nverse partcpaton rato: N p 1 ˆ P C 4.

11 C Statstcal theory of nte Ferm systems 2153 become large and non-gaussan. Indeed, the structure of the egenstate n ths case s governed by an nterplay of perturbaton theory chans, that s seres n V m = E E m, and a possblty of very small denomnators n these chans owng to uctuatons of the ntervals between bass state energy levelsz (Flambaum and Izralev 1997a,b) The role of two-body correlatons If the dynamcs of a many-body system are governed by two-body nteractons, as n equaton (2), the Hamltonan matrx elements H j become, n a certan way, correlated, even when the underlyng two-body matrx elements V are random (Flambaum et al. 1996a). These correlatons aœect the dervaton of equaton (32) and, strctly speang, mae ths equaton nvald. Ths eœect emerges n the fourth order of the perturbaton theory, hence, let us consder the contractons n the fourth-order dagrams (22). In the rst of them the ntermedate many-body bass states m dœer from by the postons of at most two partcles, jm ˆ a y 1 a y 1 a a j, that s the summaton over states m (or l) n fact nvolves movng all possble pars ; from the state nto unoccuped orbtals 1 ; 1. The correspondng matrx element n the Hamltonan s H m ˆ V 1 1. Smlarly, n the second dagram n equaton (22), jm ˆ a y 1 a y 1 a a j, and jl ˆ a y 1 a ȳ 1a a jm, that s the states m and l are obtaned from and m by movng the fermon pars and nto unoccuped orbtals 1 ; 1 and 1 1 respectvely. Hence, the matrx elements nvolved are H m ˆ V 1 1 and H lm ˆ V 1 1. In both contractons the same matrx element appears twce, whch ensures that these contractons are non-zero and have the same order of summaton. A new feature due to the two-body nteracton emerges n the thrd contracton and may now nvolve three dœerent ntermedate states: 52 The numerator of the correspondng algebrac expresson H n H nl H lm H m contans four dœerent Hamltonan matrx elements. Nevertheless, ts average s not zero, nor has t a lower order of summaton than the other two fourth-order dagrams. Indeed, for jm ˆ a y 1 a y 1 a a j, and jl ˆ a y 1 a ȳ 1 a a jm, tae jn ˆ a y a y a 1 a 1 jl, whch s equvalent to jn ˆ a y 1 a ȳ 1 a a j. Then H n H nl H lm H m ˆ V 1 1 V 1 1 V 1 1 V 1 1 and, n spte of the fact that all three ntermedate states m; n; l are dœerent, the two two-body matrx elements V 1 1 and V 1 1 appear twce each n the contracton. Therefore, t cannot be dscarded n the way that t has been n } 2.1. z In most real systems wth more then two partcles the Hamltonan matrx s sparse, that s t contans a certan fracton of zero off-dagonal matrx elements. In ths case the non-zero V j must be greater than the mean energy spacng between the drectly coupled bass states (Altshuler et al. 1997, Jaquod and Shepelyansy 1997, Mrln and Fyodorov 1997, Shepelyansy and Sushov 1997).

12 2154 V. V. Flambaum and G. F. Grban Another way to see the dœerence between a random Hamltonan and that based on a two-body nteracton s to consder the level densty (11), whch can be wrtten as a trace and ts moments E ˆ Tr E H Š; M m ˆ E E m de ˆ Tr H m : Suppose that the dagonal matrx elements are zeros (or have the same order of magntude as the oœ-dagonal matrx elements and hence can be neglected when calculatng the traces). The traces then have the same structure as the numerators of the perturbaton theory expanson of G (20). All odd-m traces vansh, sgnallng a symmetrc level densty. For a full uncorrelated random Hamltonan the trace for an even m ˆ 2 s determned by the number of ways to arrange non-crossng contractons (nown as a Catalan number) M 2 ˆ N 1 V 2 ; where V 2 ˆ Hm 2. These moments correspond to the `semcrcular densty (45) (for example Brody et al. (1981)). On the other hand, f the Hamltonan s generated by the two-body nteractons, all bnary contractons gve equal contrbutons to the even moments (f we assume that the number of partcles n s large, and n ¾ m, so that dœerent pars of orbtals domnate n the sums over the ntermedate states). The even moments then are M 2 ˆ 2 1!!N 1 V 2 : 56 (Note that the second moments M 2 here and n equaton (55) are the same and the dœerence emerges n the fourth order.) The moments (56) correspond to the Gaussan level densty Á! 2 N E E ˆ exp 2pNV 2 1=2 2NV 2 ; whch s characterstc of the random two-body nteracton model (Brody et al. 1981). A Gaussan level densty s also a common feature of nuclear shell model calculatons (Brody et al. 1981, Horo et al. 1995a,b, Frazer et al. 1996, Zelevnsy et al. 1996) based on two-body Hamltonans. These calculatons are usually made n a restrcted one- or two-shell model con guraton space, and t obscures the role of the dagonal matrx elements H n formng the level densty n the full problem.y On the other hand, f we apply equaton (32) to a system wth two-body nteracton and degenerate or near-degenerat e many-body bass states, the soluton for the strength functon and the densty of states wll come out n the form of a semcrcle, and not a Gaussan. y It s well nown that the ncrease of the level densty n a Ferm system, for example a nucleus or an atom, follows the law E / exp ae 1=2, derved for a non-nteractng Ferm gas (Rosenzweg and Porter 1960, Bohr and Mottelson 1969, Camarda and Georgopulos 1983).

13 Statstcal theory of nte Ferm systems 2155 However, there s su cent evdence that the presence of an ncreasng dagonal n the Hamltonan matrx restores the valdty of equaton (32). Physcally, ths s an eœect of the mean eld on the orbtals, as descrbed by the one-body term n the Hamltonan (2). The ncrease n spacngs between the dagonal energes E m n the perturbaton expanson (22) leads to the reducton n the hgher-order contrbutons (where the two-body nature of the nteracton maes a dœerence). The Bret± Wgner shape of the strength functon (47) correspondng to S E ˆ pv 2 =D, s n fact gven by the second-order contrbuton n S, and all hgher-order terms n expanson (28) are equal to zero n ths approxmaton. Addtonal evdence comes from numercal calculatons for the random two-body nteracton model (G. F. Grban 1999, unpublshed). Ths was shown that even a small non-zero dagonal H generated by the one-body term n equaton (2) qucly restores the applcablty of equatons (32)± (34) to the calculaton of the strength functons and total densty of states. } 3. STATISTICAL DESCRIPTION 3.1. Mcrocanoncal partton functon In ths secton we are gong to derve a partton functon for a closed (solated) system of a nte number of nteractng partcles. Examples of such systems are compound nucle, complex atoms, atomc clusters and solated quantum dots. Ths functon allows one to perform analytcal and numercal calculatons of statstcal mean values of dœerent operators, for example occupaton numbers. Let us use the Hamltonan (2) as a startng pont. The one-body part ncorporates the eœect of a mean eld, " beng the energes of the sngle-partcle states (orbtals) n ths eld. The two-body part descrbes the resdual nteracton. For smplcty, here we neglect any dynamc eœects of the nteracton such as parng and collectve modes. Instead, we concentrate on the statstcal eœects of the nteracton, therefore, n what follows we can assume that the matrx elements V are random varables. The exact egenstates of the Hamltonan j are expressed n terms of the smple `shell-model bass states j by means of equaton (5). They are characterzed by ther energes E. As dscussed n } 2, n complex systems they typcally contan large numbers N p ¾ 1 of prncpal components C, whch uctuate `randomly as a functon of ndces and. Let us consder the occupaton number n of a sngle-partcle state n a chaotc (`compound ) egenstate j. It can be presented n terms of the egenstate components as n ˆ hj^n j ˆ jc j2 hj^n j; where ^n ˆ a y a s the occupaton number operator. Knowledge of the dstrbuton of the occupaton numbers enables one to calculate the mean value of any snglepartcle operator hm ˆ P n M. Moreover, the varance of the dstrbuton of non-dagona l elements of M whch descrbe transton ampltudes between chaotc egenstates, can also be expressed through the occupaton numbers n (Flambaum 1993, 1994, Flambaum et al. 1994, 1996a, Grbana et al. 1995). As one can see from equaton (57), the mean values of occupaton numbers depend on the shape of exact egenstates, gven by the `spreadng functon F (n what follows, the F functon): 57

14 2156 V. V. Flambaum and G. F. Grban F ² jc j2 º F E ; E : Ths functon s closely related to the strength functon (7) and the densty of states (11): W E; º F E ; E E : It follows from the consderaton n } 2.1 (see equaton (35)) that the resdual nteracton V strongly mxes the bass states n the energy nterval G (spreadng wdth) near the egenstate energy E. Typcally, ths spreadng functon rapdly decreases wth an ncrease n je E j (snce the admxture of dstant component s small). In agreement wth the theoretcal consderatons, recent numercal studes of the Cerum atom (Flambaum et al. 1994, Grbana et al. 1995), the s± d nuclear shell model (Horo et al. 1995a,b, Frazer et al. 1996, Zelevnsy et al. 1996) and random two-body nteracton model (Flambaum et al. 1996a,b) showed that the typcal shapes F of exact egenstates are almost the same n dœerent many-body systems and have a unversal form whch essentally depends on the spreadng wdth G. The latter can be expressed n terms of the parameters of the model (ntensty V of the resdual nteracton, number ² of partcles, exctaton energy, etc,; see equaton (50)). One can also measure the wdth of the F functon (58) va the number of prncpal components, N p ¹ G=D, where D s the local mean level spacng between the compound states. In many-body systems the value of D decreases exponentally wth ncrease n the number of actve (valence) partcles. As a result, N p s very large, about n excted (compound) nucle and about 10 2 ± 10 5 n excted rare-earth or actnde atoms and many multcharged ons (Grban et al. 1999). P Usng equatons (57) and (58) and the normalzaton of the F functon F E ; E ˆ 1, we can wrte n E ˆ P n P F E ; E F E ; 60 ; E where n ² hj^n j equals 0 or 1, dependng on whether the orbtal s empty or occuped n the bass state. Ths way of averagng the occupaton numbers can be compared wth mcrocanoncal averagng, snce t s de ned for a xed total energy E of a system. In fact, equaton (60) s equvalent to the ntroducton of a new nd of partton functon Z E ˆ F E ; E ; 61 whch s entrely determned by the shape of the chaotc egenstates. In what follows, we term equaton (60) the F dstrbuton. Equaton (60) gves a new nsght nto the problem of statstcal descrpton of complex systems. Indeed, as mentoned above, the shape of the F functon has unversal features and can be often descrbed analytcally; therefore, n practce there s no need to dagonalze the huge Hamltonan matrx of a many-body system n order to obtan statstcal averages. Note that the summaton n equaton (60) s carred out over the unperturbed energes E de ned by the mean eld, rather than over the exact egenstates, as n the standard canoncal dstrbuton. As a result, the dstrbuton of the occupaton numbers can be derved analytcally (see } 5) even for a few nteractng partcles, that s n the stuaton when the standard Ferm± Drac dstrbuton s not vald.

15 3.2. Transton to the canoncal dstrbuton It s nstructve to compare our F dstrbuton (60) wth the occupaton numbers n an open system at a gven temperature T (e.g. a quantum dot wth a xed number of electrons). We can calculate the eœect of non-dagona l matrx elements of the nteracton on the dstrbuton of the occupaton numbers (the dagonal matrx elements are ncluded n the de nton of E ˆ H ). Ths dstrbuton s gven by the canoncal equaton n T ˆ P n exp E =T P exp ; 62 E =T where T s the temperature and the ndex enumerates exact egenstates. The mportant dœerence between the F dstrbuton (60) and the canoncal dstrbuton (62) s that n the former the occupaton numbers are calculated for a spec c energy E of the system, whle n the latter they correspond to a gven temperature T. However, the results of calculatons based on equatons (62) and (57) can be compared wth each other usng the relaton between the energy E and the temperature T : P E ˆ he T ˆ E exp E =T P exp E : 63 =T Let us substtute n from equaton (57) nto equaton (62) and replace the summaton over by ntegraton over E de where E s the egenvalue densty: E n exp T º n E F T E de; where we have ntroduced the `canoncal (thermal) averagng functon, F T E ˆ E exp E : 65 T As a result, we can transform the canoncal dstrbuton (62) nto a form smlar to the F dstrbuton (60): n T ˆ P n where the functon ~F T ; E s the canoncal average of F: ~F T ; E ˆ F E ; E F T E and Statstcal theory of nte Ferm systems 2157 g T ˆ 64 ~ F T ; E P ~ ; 66 F T ; E ˆ exp E g T T 67 W z E ; exp z ; 68 T where z ˆ E E. Ths form of the canoncal dstrbuton whch expresses the results n terms of the bass components (nstead of exact egenstates) may be convenent for the calculaton of the occupaton numbers and other mean values n quantum dots n thermal equlbrum wth an envronment (wth no partcle exchange). The occupa-

16 2158 V. V. Flambaum and G. F. Grban ton numbers n the bass states are n ˆ 1 or 0, and t s very easy to perform the summaton n equaton (66) for n T or n equaton (60) for n E numercally. If the strength functon W E; s a functon of E E only, that s we can neglect the separate dependence of S E on the energy E and ndex (see equaton (35)), the functon g T becomes ndependent of the energy E and ndex. In ths case g T s a common factor whch cancels out n the equaton (66) for the occupaton numbers. Ths s a surprsng result, because t means that the dstrbuton of the occupaton numbers does not depend on the non-dagona l matrx elements of the nteracton! The dagonal matrx elements of the nteracton ncluded n E can stll be mportant, especally n systems wth a small number of actve partcles (see for example Flambaum et al. (1998b) who dscussed the numercal calculaton of the occupaton numbers n the cerum atom wth four electrons n the open shell). However, qute often we can strongly reduce the dagonal matrx elements of resdual nteracton by a proper choce of the mean- eld potental (e.g. one can use the Hartree± Foc potental wth temperature-dependen t occupaton numbers). In ths case, one obtans the Ferm± Drac (or Bose± Ensten) dstrbuton of the occupaton numbers wth sngle-partcle energes whch may depend on temperature. Note that the `ntal nteracton between the partcles s not assumed to be small! There are several reasons why ths concluson s not applcable at low temperatures. If the exctaton energy E E mn s smaller than the spreadng wdth G, one has to tae nto account the energy-dependent lmts of the ntegral n equaton (68). The approxmaton of a constant shape of the strength functon W may be vald for hghly excted egenstates only. The low-energy states descrbed by the Bret± Wgner shape have the wdth G E whch may have a strong energy dependence. Fnally, at very low exctaton energes the condton of equlbrum V m > d f s not sats ed, and there s no equlbrum dstrbuton F. Therefore, for very low temperatures, equatons (66) and (67) are not applcable. However, n ths range there are good tradtonal approaches such as the Landau± Mgdal Ferm-lqud theory for nte systems Transton to the Ferm± Drac dstrbuton Let us now show how the standard Ferm± Drac dstrbuton emerges drectly from the F dstrbuton (60) for a closed system, n the lmt of large number of partcles. By separatng the sums over the states wth n ˆ 0 and n ˆ 1, equaton (60) can be rewrtten as 0 Z n E ˆ n 1; E ~" Z n 1; E ~" Z n; E 1 Z ˆ 1 n; E : 69 Z n 1; E ~" Here, two `partal partton functons Z n; E and Z n 1; E ~" have been ntroduced. In the rst (n ˆ 0), the summaton s carred over all bass states of n partcles wth the orbtal excluded: Z n; E ˆ P 0 F E ; E. The second sum Z n 1; E ~" ncludes all bass states of n 1 partcles wth the orbtal excluded. The latter corresponds to the bass states n whch the orbtal s lled (n ˆ 1); hence, the energy ~" ² E n E n 1 of ths orbtal must be subtracted from the total energy E of the n-partcle system. (Here we n fact assume

17 that the F functon depends on the dœerence E E only.) The energy ~" can be obtaned as ~" ˆ " 6ˆ U n ; 70 where " s the energy of a sngle-partcle state and U ² V V s the dagonal matrx element of the two-body resdual nteracton (drect mnus exchange) (see equaton (2)). By consderng ~" to be ndependent of we assume that averagng over the bass states near the energy E s possble. Ths assumpton s equvalent to a local (at a gven energy) mean- eld approxmaton. We should stress that ths approxmaton s most mportant n dervng a good mean- eld descrpton from equaton (2) for realstc systems. For example, n the cerum atom there are several orbtals belongng to dœerent open subshells (4f, 6s, etc.), whch are qute close n energes and yet have very dœerent rad. As a result, the Coulomb nteractons between the electrons n such orbtals are very dœerent (Flambaum et al. 1998b). In ths case the second (nteracton) term n equaton (70) uctuates strongly, dependng on the occupaton numbers of the other orbtals. As a result, there s no good mean- eld approxmaton, and the equlbrum dstrbuton of the occupaton numbers s very dœerent from the Ferm± Drac dstrbuton (Flambaum et al. 1998b). However, the F dstrbuton (60) gves a good descrpton of the occupaton numbers. In other cases, for example, the two-body random nteracton model (Flambaum et al. 1996a,b, Flambaum and Izralev 1997a,b) or the nuclear shell model (Horo et al. 1995a,b, Frazer et al. 1996, Zelevnsy et al. 1996), the local mean- eld approxmaton s qute accurate. For a large number n ¾ 1 of partcles dstrbuted over a large number m ¾ 1 of orbtals, the dependence of Z on both n and ~" s very rapd, snce the number N of terms n the partton functon Z s exponentally large: N ˆ m!= m n!n!š. To mae the dependence on the arguments smooth, we should consder ln Z nstead of Z and then expand where ln Z n n; E ~" Š º ln Z n; E Š A n B ~" ; A ln ; B ln Z ; n ˆ 1: Ths mmedately leads from equaton (69) to a dstrbuton of the Ferm± Drac type: n ˆ 1 1 exp A B ~" : 72 If the number of essentally occuped orbtals n the de nton of Z s large, the parameters A and B are not senstve as to whch partcular orbtal s excluded from the sum and one can ntroduce A ˆ A ² =T, and B ˆ B ² 1=T, as n the standard Ferm± Drac dstrbuton. The chemcal potental and temperature T are determned mplctly by the total number of partcles and energy of the system: n ˆ n; " s n s U n n ˆ 12 n " ~" ˆ E: < Statstcal theory of nte Ferm systems

18 2160 V. V. Flambaum and G. F. Grban Note that n equatons (73) and (70), the terms contanng the resdual nteracton U can be substantally reduced by an approprate choce of the mean- eld bass (for nstance, the terms U may have dœerent sgns n such a bass). In practce, the values " and ~" can be very close. Snce n equatons (73) the non-dagonal matrx elements of the nteracton are not taen nto account, one can expect that the dstrbuton of the occupaton numbers de ned by these equatons gves correct results f the nteracton s su cently wea (deal-gas approxmaton). However, t can be shown (Flambaum and Izralev 1997a,b) that even for strongly nteractng partcles the Ferm± Drac dstrbuton can be appled, f the total energy E s corrected by tang nto account the ncrease n temperature due to statstcal eœects of the nteracton. One should also note that a somewhat smlar procedure transforms the canoncal dstrbuton (62) nto the Ferm± Drac dstrbuton (for example Ref (1965)) n the case of many non-nteractng partcles (deal gas). It s curous that the Ferm± Drac dstrbuton s very close to the canoncal dstrbuton (62) even for a very small number of partcles (n ˆ 2), provded that the number of essentally occuped orbtals s large, that s for T ¾ " or ¾ ". In fact, ths results from a large number of `prncpal terms n the partton functon Z, whch enables one to replace A by A n the rato Z n; T =Z n 1; T ² exp A B" n the canoncal dstrbuton (62) (cf equaton (69)). A more accurate consderaton shows, however, that the temperature T n the Ferm-Drac dstrbuton s dœerent from that n the canoncal dstrbuton. Indeed, usng the expanson A ˆ A " F A 0 " " F, where " F s the Ferm energy, one can obtan a relaton between the Ferm± Drac (B FD and canoncal (B) nverse temperatures: B FD ˆ B A 0 " F. The de nton of the chemcal potental also changes: =T ˆ A " F A 0 " F. Ths s con rmed by numercal smulatons for a closed system of few nteractng Ferm partcles (Flambaum et al. 1996a,b, Flambaum and Izralev 1997a,b). The smulatons show that, for the same total energy E of the system, the canoncal and Ferm± Drac dstrbutons gve dentcal dstrbutons n. However, they correspond to dœerent temperatures determned by equatons (63) and (73), respectvely. The smlarty of these two dstrbutons for any number of partcles s not so surprsng n the presence of a thermal bath, where even a sngle partcle s `n equlbrum. On the other hand, for closed systems wth small numbers of partcles the applcablty of the Ferm± Drac dstrbuton s not obvous. To answer ths queston, one needs to analyse the role of nteracton n the creaton of an equlbrum dstrbuton (} 2). } 4. STATISTICAL DESCRIPTION OF TRANSITION AMPLITUDES 4.1. Mean-squared matrx elements Several approaches have been suggested to calculate matrx elements between compound states. Urn and Vyazann (1991) expressed the mean-square d matrx element n terms of the strength functon of a cold nucleus and calculated the latter sememprcally n the framewor of `temperature mechansm. The calculatons by Johnson et al. (1991), Johnson (1995) and Johnson and Bowman (1995) were based on the the wor by French et al, (1988) and used the assumpton that the meansquared matrx element of the party-non-conservn g nteracton s proportonal to that of the resdual shell-model strong nteracton. In ths secton we consder a

19 statstcal method descrbed prevously (Flambaum 1993, 1994, Flambaum and Vorov 1993, Flambaum et al. 1994, 1996a, Grbana et al. 1995). It can be appled to calculaton of mean-square matrx elements between chaotc (compound) states (5). Let us consder a sngle-partcle operator ^M ˆ a y a M ˆ M : 74 It s convenent to express the matrx elements of ^M n terms of matrx elements of the densty matrx operator ˆ a y a, whch transfers a partcle from orbtal nto orbtal. One can see that the matrx element of ^M between the compound states n 1 and n 2, hn 1 j ^Mjn 2 ˆ M hn 1 j jn 2 j ˆ M C n 1 hj j jc n 2 j ; 75 has zero mean owng to the statstcal propertes of the components, that s hn 1 j jn 2 ˆ 0. The varance of the matrx elements of ^M can be presented n the followng form: jm n1 n 2 j 2 ˆ jm j 2 jhn 1 j jn 2 j 2 ; 76 where we have taen nto account the result obtaned by Flambaum et al. (1996a) that the average of the correlator hn 1 j jn 2 hn 2 j jn 1 ˆ C n 1 C n 1 j C n 2 C n 2 l hj jhlj j j 77 jl s zero, unless ˆ, ˆ. Therefore, the calculaton of jm n1 n 2 j 2, or a correlator M n1 n 2 W n2 n 1 s reduced to the calculaton of jhn 1 j jn 2 j 2. The varance of the matrx element between the two compound states can be transformed nto j n 1n 2 j 2 ˆ hn 1 j jn 2 hn 2 j jn 1 ˆ C n 1 C n 1 j C n 2 S n 1n 2 d Statstcal theory of nte Ferm systems 2161 jl C n 2 l hj jhlj j j ˆ S n 1n 2 d S n 1n 2 c ; 78 where we separate the dagonal and non-dagona l contrbutons to the sum (78): ˆ jc n 1 j 2 jc n 2 j 2 jhj jj 2 ; 79 S n 1n 2 c ˆ 6ˆj; 6ˆl C n 1 C n 1 j C n 2 C n 2 l hj jhlj j j: 80 Note that the dagonal term S n 1n 2 d s essentally postve. If the egenstates are completely `random (dœerent components both nsde each egenstate and of dfferent egenstates are uncorrelated), the correlaton sum S c s zero, and the varance

20 2162 V. V. Flambaum and G. F. Grban s determned by the the dagonal sum S d (ths assumpton was used n the prevous calculatons of matrx elements between the compound states (Sushov and Flambaum 1982, Flambaum 1993, 1994, Flambaum and Vorov 1993, Flambaum et al. 1994, Grbana et al. 1995)). However, as we show below, n many-body systems these two terms are of the same order, S c º S d, even f the two-body nteracton V s random. Followng Flambaum et al. (1994) and Grbana et al. (1995), let us replace the squared egenstate components n equaton (79) by ther averages (F functons), equaton (58).y The dagonal sum now taes the form ˆ F E ; E n1 F E ; E n2 hj jhj j: 81 S n 1n 2 d The summaton over for a xed ncludes only one state, j ˆ j, wth E ˆ E!, where! s the dœerence between the sngle-partcle energes:! º " ". On the other hand, we can wrte hj jhj j ˆ hj j ˆ hj^n 1 ^n j; 82 where ^n ˆ a y a and ^n ˆ a y a are the occupaton number operators. Thus, we obtan ˆ F E ; E n1 F E! ; E n2 hj^n 1 ^n j: 83 S n 1n 2 d The matrx element hj^n 1 ^n j s equal to unty f the orbtal s occuped and s vacant n the bass state j; otherwse, t s zero. Both F values n equaton (83) are smooth functons of energy normalzed as P F E ; E n1 ˆ 1. Ths allows one to replace the matrx element of ^n 1 ^n by ts expectaton value: hj^n 1 ^n j ˆ F E ; E n1 hj^n 1 ^n j ˆ jc n 1 j 2 hj^n 1 ^n j º h^n 1 ^n n1 : 84 The sgn º above s a remnder that the left-hand sde s the local average over the states jn 1. Practcally, when the number of components s large, the uctuatons of h^n 1 ^n n1 are expected to be small. Now we can rewrte equaton (83) n the followng form: ˆ h^n 1 ^n n1 F E ; E n1 F E! ; E n2 : 85 S n 1n 2 d It was shown n Flambaum et al. (1994) and Grbana et al. (1995) that, under some reasonable assumptons about the F functons, one can ntroduce a `spread functon ~ D, y As usual, the averagng can be made ether over a number of realzatons of the two-body nteracton matrx elements (ensemble average), or over a number of neghbourng egenstates (physcal energy average). In the sprt of ergodcty the results are assumed to be the same.

21 ~ D ˆ D 1 2 F E ; E n 1 F E! ; E n 2 ˆ D 1 de 2 F E D ; E n1 F E! ; E n2 ; 86 1 where D ˆ E n 2 E n 1! and D 1 and D 2 are local mean level spacngs for the n 1 and n 2 egenstates. The functon ~ D s symmetrc, ts characterstc wdth s determned by the spreadng wdths of the egenstates n 1 and n 2, G º G 1 G 2, and t s normalzed to unty, ~ D dd ˆ 1, just as the standard functon. If the F values have Bret± Wgner shapes (see } 2), ~ s also a Bret± Wgner functon: ~ D ˆ 1 G=2 p D 2 G 2 =4 ; wth G ˆ G 1 G 2. The fact that S n 1n 2 d s proportonal to the functon ~ D s a partcular manfestaton of the energy conservaton for transtons between the quasstatonary bass states (Flambaum 1993, 1994, Flambaum and Vorov 1993) (f G! 0, then ~ D! D ). Usng equatons (76), (85) and (86) we can nally present the dagonal contrbuton to the varance of the matrx element M n1 n 2 n the followng form: jm n1 n 2 j 2 dag ˆ jm j 2 n h^n 1 ^n n1 D 2 ~ E 2 E n1! : 88 Ths expresson s apparently asymmetrc wth respect to the states n 1 and n 2. By performng the calculaton n a dœerent way we can obtan ˆ h^n 1 ^n n2 F E! ; E n1 F E ; E n2 89 S n 1n 2 d Statstcal theory of nte Ferm systems 2163 nstead of equaton (85), thereby arrvng at a dœerent formula for the varance: jm n1 n 2 j 2 dag ˆ jm j 2 n h^n 1 ^n n2 D 1 ~ E 2 E n1! : 90 In ths form the occupaton-number factor s calculated for the state n 2 ; t represents the probablty of ndng orbtal occuped and empty. If the assumptons made n the above dervatons are correct, the two equatons (88) and (90) should gve dentcal results. Prevously (Flambaum et al. 1996a) we used the random two-body nteracton model to chec the accuracy of the statstcal formulae derved above and found that the values of S n 1n 2 d gven by equatons (85) and (89) almost concded and were n good agreement wth the ntal expresson (79). Smlar tests were also made n the calculatons for the cerum atom (Flambaum et al. 1994, Grbana et al. 1995). It s qute mportant for applcatons of the statstcal approach (Flambaum 1993, 1994, Flambaum and Vorov 1993, Flambaum and Grban 1995, Flambaum et al. 1998a) that further smpl catons be made by replacng the correlated occupances product h^n ^n n1 n equaton (84) by the product h^n n1 h^n n1 of the two mean values. Ths s de ntely a vald operaton when the numbers of excted partcles and actve orbtals are large, so that the occupaton numbers for dœerent orbtals become statstcally ndependent. Then one would be able to use the followng relaton: 87

22 2164 V. V. Flambaum and G. F. Grban h^n 1 ^n º n " 1 n " Š; where n " and n " are the occupaton numbers. Ths approxmaton was tested by Flambaum et al. (1996a) usng the random two-body nteracton model. Even for a system of four partcles the accuracy of equaton (91) remaned reasonable, the error beng about 10%. Ths error s larger for the lowest egenstates where the number of prncpal bass components s small and there s lttle or no thermalzaton and `chaos. In order to mae a more drect test of the valdty of equaton (91), a correlator h^n ^n n1 = h^n n1 h^n n1 was calculated as functon of n 1. For a small number of partcles (n ˆ 4), ths correlator dsplayed large uctuatons, however, ts average value of about 0.8 was stll rather close to 1. If the number of excted partcles s large, one can use the Ferm± Drac formula for the occupaton numbers: n " ˆ 1 1 exp " =T Š (see } 3.3) wth the temperature T and chemcal potental (Ferm energy) found from equatons (73). Equatons (88)± (91) together wth equaton (73) allow one to mae computer calculatons of the mean-squared matrx elements between compound states. Followng the same steps for a two-body operator ^U ˆ 1 U a y a y a a ; 93 2 we obtan the followng expresson for the mean-squared matrx element: ju n1 n 2 j 2 dag ˆ 1 ju U j 2 hn n 1 n 1 n n1 D 2 ~!! ; ; 4 where! ² E n2 E n 1 and! ; ˆ " " " " s the energy of the twopartcle transton ;! ;. We can also calculate the correlator C MW between the matrx elements of two operators M and W wth dentcal selecton rules: M n1 n C MW ² 2 W n2 n 1 jm n1 n 2 j 2 jw n1 n 2 j 2 1=2 ˆ P M W j n 1n 2 j 2 P jm j 2 j n 1n 2 j 2 1=2 P : 95 jw j 2 j n 1n 2 j 2 1=2 One can easly see that jc MW j ˆ 1 f the matrx elements W and M are proportonal to each other M ˆ constant W ), or f there s only one domnatng sngle-partcle transton, say, s! p (M sp ¾ M and W sp ¾ W for all 6ˆ s, 6ˆ p. Usually there are several mportant sngle-partcle transtons near the Ferm surface (q º 10). If there are no specal reasons for the coherence or cancellatons one may expect jc MW j º 1=q 1=2 º 0:3. However, n the most nterestng case of P-odd and P; T -odd nteractons there are pars of opposte-sgn contrbutons. Indeed, the matrx elements of the wea party-non-conservn g nteracton are magnary, W ˆ W * ˆ W, whle the matrx elements of the P; T -odd nteracton are real, M ˆ M. Therefore, we have pars of opposte-sgn terms: