Analytical solutions for the LMG model

Transcription

1 Aprl 999 Phyc Letter B Analytcal oluton for the LMG model Feng Pan, JP Draayer Department of Phyc & Atronomy, Louana State UnÕerty, Baton Rouge, LA , USA Receved November 998 Edtor: J-P Blazot Abtract Analytcal oluton for the Lpn-Mehov-Glc Ž LMG model for ome nontrval lmt are derved by ung an nfnte-dmenonal algebrac approach baed on the Bethe anatz Analytcal expreon for egenvalue and the correpondng egenfuncton of the Hamltonan are obtaned Soluton for the aocated Bethe anatz euaton are gven 999 Publhed by Elever Scence BV All rght reerved PACS: 0365Fd; 00Tw; 60C Recently, a large number of uantum many-body problem have been olved by ung a new nfnte-dmenwx Example nclude the non-degenerate and orbt-dependent nuclear parng problem onal algebrac approach w,3 x, the N-coupled ymmetrc rotor ytem w x, the vbratonal to g-untable trantonal Hamltonan of the nuclear nteractng boon model wx 5, and o on Th approach baed prmarly on the well-nown Bethe anatz methodology wx 6 Smlar method for dervng exact oluton of the eual-trength nuclear parng problem wa frt dcued n Ref wx 7 by ung boon mappng ncludng the effect of the Paul prncple, whch ha been derved more mply and drectly by ung the nfnte dmenonal algebrac approach w,3 x The buldng bloc of the fnal form ued n contructon of the wavefuncton are mlar to thoe gven early by Gaudn n tudyng ome pn nteracton ytem w8,9 x The effectvene of th nfnte-dmenonal algebrac method le n the fact that explct form for buldng bloc for the Bethe anatz wavefuncton can be obtaned by ung the correpondng nfnte-dmenonal Le algebra, whch the nfntemal form of the correpondng non-lnear algebra generated by thoe buldng bloc of the Bethe anatz wavefuncton Egenvalue and the correpondng egenfuncton can thu be determned multaneouly through a et of non-lnear euaton, called the Bethe anatz euaton, whch are eay to olve In partcular, the egenvalue pectrum for uch problem are gven exactly A well-nown, the Lpn-Mehov-Glc Ž LMG model w0x alo an exactly olvable model Th model wa ued early on a a mple cheme for tudyng the nuclear gant monopole reonance w x In the LMG model, N partcle can dtrbute themelve on two N-fold degenerate level dtnguhed by a uantum number wth " Let a Ž a a be fermon creaton Ž annhlaton p p On leave from the Department of Phyc, Laonng Normal Unverty, Dalan 609, Chna r99r$ - ee front matter 999 Publhed by Elever Scence BV All rght reerved PII: S

2 F Pan, JP DraayerrPhyc Letter B operator for a partcle n the tate p of the level, where p,, PPP, N The Hamltonan of the LMG model can be wrtten a ˆ X X X X p p p p p y py p p y p py X X p pp pp He a a V a a a a W a a a a, Ž where V and W are parameter pecfyng the trength of the nteracton The V term catter a par of partcle n the ame level to the other level The term proportonal to W catter one partcle up and another down For W 0, th model ha been tuded numercally a well a analytcally w0 x, ung approxmate method n the latter cae In addton, Pang, et al condered th problem by ung the Holten-Prmaoff mappng w x Glmore and Feng ued Bogolubov-Leb neualte to roughly decrbe the model w3 x The model wa alo dcued by ung the o-called effectve potental method whch yeld analytcal expreon for the egenvalue and egenfuncton of lower-lyng tate w x Recently, a renewed nteret n the Lpn model ha developed wthn the context of fnte temperature phenomena w5x and a a tet of elf-content RPA-type approxmaton w6 x Hamltonan flow euaton for the model have alo been condered wth th theory w7 x For the general cae wth V and W non-zero, however, analytcal oluton have not been determned The mportance of an exactly olvable theory that t can be ued to tet the valdty of varou approxmate method for olvng the ame problem Analytc reult are alo ueful for probng the nature of the oluton, epecally t aymptotc behavor, and becaue uch oluton freuently have an nherent mathematcal beauty, hgher ymmetre of the ytem It hould be noted that the o-called pn-coordnate correponwx can be ued to obtan exact oluton for excted tate that cannot be derved ung dence propoed n Ref the effectve potental method Here the cae wth non-degenerate ngle-partcle nteracton condered, ˆ X X X X p p p p p y py p p y p py X X p pp pp H e a a V a a a a W a a a a By ntroducng the o-called peudopn operator J apa py, Jy apya p, J0 ap a p, 3 p p p Ž E can be wrtten a where C ˆ ˆ H e n V J J W C yj, Ž y 0 ˆ the Camr operator of the peudopn algebra SUŽ, and nˆ a a Ž 5 p p p Now we ntroduce the Jordan-Schwnger fermon-boon mappng, J bb y, J y byb, J 0 b b yb b, n N b y y b, 6 where b Ž b wth or y are boon creaton Ž annhlaton operator One can chec that th mappng Ž 6 exact; namely, the commutaton relaton after the mappng are the ame a thoe n the fermon pace By ung Ž 6, Hamltonan Ž can be expreed a where ˆ ˆ ˆ ˆ y y ˆ ˆ y H e N V b b b b W C y N WN N, Ž 7 ˆ N b b, N ˆ N ˆ, Ž 8 ˆ ˆ

3 and the Camr operator can be wrtten a ˆ ˆ ˆ F Pan, JP DraayerrPhyc Letter B C NŽ N Ž 9 Becaue the ymmetry properte of the Hamltonan depend on the parameter V and W, each cae wll be dcued eparately n the followng We wll how that analytcal oluton for all excted tate can be obtaned when W GV Cae V0 In th cae, the Hamltonan can be dagonalzed n the SUŽ ba b n b n y y < JM : < 0 :, Ž 0 n!n! ( y where < 0: the boon vacuum whch atfe b < 0: 0 for or y, Ž wth J nn y and M nyn y The egenvalue are gven by E e n WŽ n n n n, Ž n,ny y y whch can alo be expreed n term of uantum number n the orgnal fermon pace by EJMeŽ JM eyž JyM WŽ JJ ym Ž 3 In th cae the correpondng egenvector can be wrtten a ( Ž JM! JyM JM J < JJ : y, Ž Ž J! Ž JyM! where < JJ : the hghet-weght-tate of the peudopn algebra SU wth J Ł p p < JJ : a < 0, Ž 5 < 0 the fermon vacuum tate, and J gven by Ž 3 y It obvou that the Hamltonan n th cae ha SUŽ >SOŽ dynamcal ymmetry Hence, the egenvalue problem can be exactly olved n an SUŽ >SOŽ adapted ba Cae V/0 and W )V In th cae, the SUŽ >SOŽ ba vector are no longer the egentate of the Hamltonan However, Hamltonan Ž 7 can be expreed n term of the complementary SUŽ, generator wth a S Ž b, S Ž b, S Ž N, Ž 6 y 0 ˆ Ž 0 y y Ĥ e S y e V S S y S y S W S y W S Ž y S Ž y y, Ž 7 where S 0 S 0 Ž Ž 8

4 F Pan, JP DraayerrPhyc Letter B an nvarant becaue the total boon number fnte for a gven peudopn J It can be proven that Ž 7 can alo be wrtten a eya 0 y 0 Ĥ Ž aw Ž S0y "S0 S 0 S y b S 0 ž /ž 0 y Ž 9 / b for W-0 wth the upper gn and W)0 wth the lower gn, where S m Ž m0,,y; m0, ", ", PPP $ m are generator of an affne Le algebra SUŽ, wthout central extenon defned by S " b m S " Ž, S 0 b m S 0 Ž, Ž 0 whch atfy m m 0 " " y 0 S m,sn "S mn, S m,sn ys mn The new parameter b and a hould atfy b by 8W, bby"v for W-0 wth upper gn or W)0 wth lower gn, and e yž b rb a Ž 3 y y e y Ž byrb It clear that the condton W ) V eep that root of real The reultng Hamltonan Ž 9 wll be non-hermtan when W -V Furthermore, the parameter a fnte only when b /"by whch vald for W/"V To obtan egentate of Ž 9, one can tudy the Fourer-Laurent expanon of the egentate of Ž 9 ung $ the SUŽ, generator n term of unnown c-number parameter x wth,, PPP,, namely, n n PPP n n n n ngz n n n, z ; t a x x PPP x S S PPP S < lw;t :, Ž where the lowet weght tate lw;t atfy S y < lw;t : 0 ; mgz Ž 5 m There are four tate atfyng Ž 5 wth lw; n 0, n 0 0, lw; n, n b b < 0 : y y y, < lw;3: < n, n 0: b < 0 :, < lw;: < n 0, n : b < 0 :, Ž 6 y y y where n the uantum number of SU Ž, for or y The tate bult on < lw;t : wth t, correpond to the cae wth the total number of partcle even, whle thoe bult on < lw;t : wth t 3, correpond to the cae wth total number of partcle odd The uantum number n Ž related to total partcle number N by Nnn y It hould be noted that the boon number N and the fermon number n have the ame value, whch can be een from Ž 6 Another uantum number z n Ž ntroduced to dtnguh dfferent egentate wth ame and t Becaue of the analytc behavor of the wavefuncton, t uffce to conder x near zero In th cae the n can only be taen a zero or potve nteger Ung the commutaton relaton Ž, one can prove that all coeffcent a n Ž can be taen a contant wth n G 0 Hence, Ž can be expreed mply a nn PPP n,z ;t NS Ž x S Ž x PPP S Ž x < lw;t :, Ž 7

5 where N a normalzaton factor, and F Pan, JP DraayerrPhyc Letter B b S x S 8 yb x The egenvalue E,z of the Hamltonan Ž 9 can then be expreed a e ya e ya,z Ž b b x E anw " Ny b n Ž 9 The c-number x are determned by the followng et of euaton eya b n Žz Žz b x b x y j/ x yx j Ny for,,ppp,, Ž 30 Žz Žz where the addtonal uantum number z ued to denote the z-th et of root x, x, PPP, x Fnally, one can rewrte the wavefuncton Ž 7 n the orgnal fermon pace a N < J,z ;t : C ym!m! < J, Mym: for n n 0, m r m0 y N C ym! m! J, Mym for n n, m r < m0 : y N r m y m0 C ym!m! J, M ym for n, n 0, N r m < : y C ym!m! J, My ym for n 0, n, m0 Ž 3 where the expanon coeffcent C can be expreed a m ym Ž a j y Ž Cm Ły y Ł, 3 Ž a j aa j y PPP aym y the a,, PPP,ym are all poble nteger atfyng wth FaFaF PPP Faym F Ž 33 b Ž j yb x j y for j,,ppp,, Ž 3

6 6 F Pan, JP DraayerrPhyc Letter B < JM : gven by Ž and Ž 5, and the normalzaton factor N can be expreed a N yr Ž z m m0 ~ Ž z m0 Ž z m0 Ž z C ym!m!, for n n 0, yr m C ym! m!, for n n, = Ž 35 yr m C ym!m!, for n, n 0, yr m m0 C ym!m!, for n 0, n It can be verfed that the c-number x n Ž 8 have S ymmetry Any permutaton among dfferent root x for,, PPP, n Ž 8 gve the ame oluton Therefore, generally, there wll be! dfferent oluton, of whch the egenenerge are the ame except that the c-number x n thee cae are nterchanged Clearly, < < < < < Ž z only one of thee a oluton to the problem The root x can be arranged a x - x - PPP - x < If two root x and x are conjugate to each other, u "u, where u wth, are real number, we Žz Ž z alway wrte x u yu, and x u u The oluton are then the dfferent et of root x dtnguhed by the uantum number z obtaned from Ž 30 Cae 3 W"V In th cae, 0 y ĤS y e V S S "V S y Ž 36 for WV wth upper gn or WyV wth lower gn, and S e n S e m S y, S y n y n e n SŽ ey m S y Ž y Sn 0 e n S 0 Ž for n0,",",ppp, Ž 37 $ whch generate an affne Le algebra SU, wthout central extenon Thee generator atfy S, S y S, S, S "S Ž 38 y 0 0 " " m n mn m n mn By ung a procedure mlar to that outlned n Cae, t can be proven that egenfuncton of Ž 36 can be wrtten a <,z ;t : N S Ž x S Ž x PPP S Ž x < lw;t :, Ž 39 where < lw;t : the ame a that gven n Ž 6, and S Ž x S Ž S Ž y Ž 0 ye x ye x y for WV wth upper gn or WyV wth lower gn The egenvalue E expreed a ;z of the Hamltonan Ž 36 can be E ;z"vn ne x

7 F Pan, JP DraayerrPhyc Letter B The c-number x n th cae are determned by the followng et of euaton g n x for,,ppp,, Ž Žz Žz Žz x ye x x yx j/ j where g V It hould be noted that not only the analytcal expreon for the oluton can be obtaned, but alo the Bethe anatz E Ž 30 and Ž can ealy be olved Conder Ž a an example Frtly, ummng over on both de of Ž one obtan g n Ž y, Ž 3 Žz x ye x whch mean that the root x can be found n the vcnty of the followng oluton: g n Ž y for,,ppp,, Ž x ye x except for a few root whch do not atfy Ž E Ž can be ued a ntal value n fndng the exact root for Ž For, Ž gve two root < x Ž< -< x Ž<, whle for ) Ž gve three root < x Ž< -< x Ž< < Ž< Ž z - x 3 Therefore, the root x can be found from the ntal value wth the followng dfferent combnaton: x Ž m, x Ž m, PPP, x Ž m, Ž 5 where m,, 3 Such computaton can ealy be carred out by ung ether Mathematca or other extng mathematcal lbrare Ž Fortran or C for olvng non-lnear euaton Smlarly, ntal value for the root of Ž 30 can be obtaned from the followng oluton: eya b n x b b x y Ny Ž y for,,ppp, Ž 6 It hould be noted that the Bethe anatz euaton mlar to Ž 30 and Ž were alo dcued n Gaudn wor wx 9 In Ref wx 9 t hown that the root of uch Bethe anatz euaton obey a econd order dfferental euaton In the followng, we conder Cae 3 wth WV, and eyye wth ee a an example In th cae, one can rewrte the energy egenvalue a E ;z 'E ;zrebn nyn y, 7 y where bvre, y e x, y are determned by the followng euaton b n y y, Ž 8 y yy y yy j Žz Žz Žz j j and the egentate of Ž 6 are tll gven by Ž 39 Therefore, there only one parameter b n Ž 7 One can ue Ž 7 to tudy phae tranton n the Lpn model It obvou that there are two lmtng cae, one wth b 0 and another wth b ` When b 0 the pectrum vbratonal le, whch gven by Ž 3 wth e eye and W0, namely y E 'E re M Ž 9 JM JM

8 8 F Pan, JP DraayerrPhyc Letter B wth MyJ, yj, PPP, J There no degeneracy and the correpondng egentate are gven by When b ` the pectrum become rotatonal le When N even, one ha Ew Ž b w, wn, Ny, Ny, PPP,0, Ž 50a Ž Ew b Ž w, wny, Ny, PPP,0 Ž 50b Up to a normalzaton factor, the correpondng egentate are wy m m N,w; Ž S b b < 0: y Ž wym!m Ž m r rym Ny r!r! y N N <, y r :, Ž 5a mr Ž wym!m Ž yrm! Ž rym! y wy m m N,w; Ž S b b < 0: y Ž wym!m! m r rym Ny ry! r! y N N <, yy r :, Ž 5b y mr Ž wym!m! ž yrm /! Ž rym! where the SUŽ ba vector n the um of Ž 5a and Ž 5b are alo gven by Ž It hould be clear that, except for the ground tate, the egenenerge gven by Ž 50 are alway two-fold degenerate When N odd, one ha E Ž3 E Ž b w, Ž 5 w w where wn, Ny, Ny, PPP, Up to a normalzaton factor, the correpondng egentate are wy m m N,w;3 Ž S b b < 0: y Ž wym!m Ž m r rym Ny r!r! y N N <, y r :, Ž 53a mr Ž wym!m Ž yrm! Ž rym! wy m m N,w; Ž S b b < 0: y Ž wym!m Ž m r rym Ny r!r! y N N <,y r : Ž 53b mr Ž wym!m Ž ž yrm /! Ž rym! In th cae the level are alway two-fold degenerate becaue of the tme reveral ymmetry wth Tˆ< N,w;3: < N,w; :, Tˆ< N,w;: y< N,w;3 :, Ž 5 where Tˆ the tme reveral tranformaton

9 F Pan, JP DraayerrPhyc Letter B Fg Energy egenvalue 7 plotted veru the nteracton parameter b for 0 partcle The trantonal regon between the prevou two lmtng cae are decrbed by Ž 39, Ž 7 and Ž 8 In order to llutrate the phae tranton behavor of the egenenerge E' Ere a a functon of the nteracton parameter b'vre, the reult for N0 were calculated by ung Ž 7 and Ž 8 The reult are plotted n Fg From the reult one can conclude that a a functon of ncreang b hgh-lyng level become degenerate more ucly than the low-lyng one It follow from th that the crtcal value of b for the phae tranton level-dependent In ummary, analytc expreon for exact oluton of the Lpn-Mehov-Glc LMG model wth W G V have been derved by ung an nfnte-dmenonal algebrac approach baed on Ž the Bethe anatz Analytcal expreon for exact oluton when W - V are tll not avalable Analytcal expreon for egenvalue and the correpondng egenfuncton of a generalzed Hamltonan have alo been obtaned In addton, we have hown that the correpondng Bethe anatz euaton Ž 30 and Ž can be olved ealy, whch telf nteretng becaue euaton of th type are common n other uantum many-body problem: the non-degenerate and orbt-dependent nuclear parng problem w,3 x, the N-coupled ymmetrc rotor ytem wx, the vbratonal to g-untable trantonal Hamltonan of the nuclear nteractng boon model wx 5, and o on In thee cae, matrx dagonalzaton, though feable, tedou and ntrcate, epecally for the nuclear parng problem Therefore, the nfnte-dmenonal algebrac method not only of theoretcal nteret, but alo of practcal ue A detaled dcuon on how to olve the Bethe anatz euaton wll be publhed elewhere Acnowledgement Th wor wa upported by the Natonal Scence Foundaton through Grant No and Cooperatve Agreement No whch nclude matchng from the Louana Board of Regent Support Fund Reference wx Feng Pan, JP Draayer, Ann Phy, 998, n pre wx Feng Pan, JP Draayer, WE Ormand, Phy Lett B Ž 998 wx 3 Feng Pan, JP Draayer, Phy Lett B Ž wx Feng Pan, JP Draayer, J Phy A 3 Ž wx 5 Feng Pan, JP Draayer, Nucl Phy A 636 Ž

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