Size Reduction of The Transfer Matrix of. Two-Dimensional Ising and Potts Models

Transcription

1 Publhed n : In. J. Phy. Re. - Sze Reduon of The Tnfe M of To-Denonl Ing nd Po Model M. Ghe nd G. A. Pf - Ao Enegy Ognzon of In Depuy n Nule Fuel Poduon. Tehn IRAN -Chey Depen Tehe Tnng Unvey Tehn In El: ghe@b.u.. -College of Chey Shf Unvey of Tehnology Tehn In El: hf.edu Ab A ne lgeb ehod developed o edue he ze of he nfe of Ing nd hee-e Po feogne on p of dh e of que nd ngul le. Th ze eduon h been e up n uh y h he u egenvlue of boh he edued nd ognl nfe e bee ely he e. In h ehod e e he ognl nfe n pel bloed fo n uh y h he u of o eleen of blo of he ognl nfe be he e. The edued obned by eplng eh blo of he ognl nfe h he u of he eleen of one of o. Ou ehod eul n gnfn ze eduon hh ul fo n deenng he u egenvlue. Keyod: Ing Model Po odel nfe.

2 Inoduon Thee e o genel ppohe fo devng he pon funon of he Ing [] nd Po [] odel nely obnol ppoh ung gphl ehod nd n lgeb ppoh ung he ehod of he nfe. Alhough zeo gne feld hee n e oluon fo he -denonl Ing odel hoeve hee no uh oluon fo he -D Po odel. In he bene of n e oluon ee epnon hh e bed on he gph heoy en one of he o ueful ool n he nvegon of he pon funon nd he l popee of odel ye. Fo hol eve of he ee epnon ee Dob [ ] nd Wu [5] nd Bgg [ 7]. In ddon o he ople lulon he ddvnge of h ehod he peon of he luled l d hh depend on he nube of e n he uned ee. Aong he ny vou ehod fo devng he pon funon of he -D Ing odel he nfe ehod he ognl nd olde ppoh. E epeon fo he egenvlue of he nfe e vlble fo ll -D Ing le nely que [ 9] ngul [] nd honeyob le []. Hoeve hee no uh n epeon fo he -D Po odel. In een o Rnb nd Pf [] ued fne-dh le o e up he nfe fo he que le of he Ing odel. The euln olved fo he odel h he dh of equl o le hn e. Sne he ode of he nfe beoe oo lge fo he Po odel he ehod nno be eended o dve he pon funon nlylly. Th ehod eended by Ghe e l. [] fo nuel lulon of he l epeue n he -D Ing nd he -D heee Po nd -D Ing odel by he pllel opuon ehod. The nuel ehod hh e bed on he nfe e led o he ze of he.

3 Fne-ze lng FSS h boe nengly pon n he udy of l phenoen [5 ]. Th ply due o he poge n heoel undendng of fne ze effe nd ply due o he pplon of FSS n he nly of eul fo ulon ehod. FSS llo u o e oe popee of he nfne ye ne phe non by udyng fne nuelly eble ple. Nghngle [7] h hon ho he nfe ehod [-]ould be de oe poeful by obnng he h FSS o epole he eul fo fne ye o he heodyn l. In he peen ppe e nodue ehod fo edung he ze of he nfe n uh y h he u egenvlue λ en he e fo boh he edued nd ognl nfe e. We hve developed n lgeb ehod on p of fne dh que Ing le o e he ognl nfe n pel bloed fo. In h fo he u of o eleen n eh blo e equl. The edued nfe obned by eplng he blo of he ognl nfe h he u of o eleen of h blo. The λ of boh he edued nd ognl nfe e hve been luled he l pon fo le odel h dffeen dh. I obeved h he λ nfe e e denl. fo boh he edued nd ognl We hve hen eended he ehod o he o-denonl hee-e Po odel on he p of fne dh que le. We hve hon h ou ehod n be ely ued o obn λ fo he fne dh que Po odel fo he edued nfe. The eenon of ou ehod o lule λ dh ngul Ing nd Po le e gven n Appende B nd C. on he p of fne

4 To-Denonl Ing Model Conde que le h he peod boundy ondon opoed of le eh h p o hee eh o h e. Eh le h hen p N e nd he oodnon nube of ll e e he e. In he -D Ing odel fo ny e e defne pn vbleσ ± n uh y h... nd... p. We y nlude he peod boundy ondon : σ σ σ p σ We hve only en no oun he neon ong he nee neghbo. The onfguonl enegy fo he odel gven p E σ { σ σ σ σ } The nonl pon funon Z K Z K e { σ } E σ T By ubuon of Eq. no Eq. he nonl pon funon n be en follo [ ] : Z K T B p 5 hee B he nfe nd T denoe he e. To opue he eleen of he nfe onde A h olun nd n o uh h eh o of he one of he peuon of nd - n olun. The eleen of h n be veed pn vble. Theefoe he ± eleen of he nfe B n be opued by [ ] b ep{ K [ ]}

5 hee * nde he peod boundy ondon nd K onn. The o of A n be gouped no blo n uh y h he onfguonl enege l E nd E h l... fo eh o of he blo o beoe he e. We defne l E nd o E ; E o ep{ K l H } 7 l E ep{ K } fo l... l hee K nd H e onn. By h defnon E he nee neghbo neon beeen pn n he peene of he gne feld nd E he ne nee neghbo neon nd E he neon enegy beeen o pn h e eped by - pn n eh o. No he A n be dvded no g blo. Then he eleen of he blo n be epeened by h nd... g hee he nube of o n he blo. Wh h defnon eh blo h he follong he: The nube of - pn n eh o of blo e he e b The o of blo e he yl peuon of eh ohe The f blo h one o h ll eleen equl o nd he blo g h one o h ll eleen equl o. Fo eple le u onde he e hee. Thee e dffeen e n hh pn n be nged n e. We n ho hee e by he follong 5

6 A 9 hee eh o epeen one of he poble pn onfguon of e. A hon n Eq.9 he A devded no blo nd he o of eh blo e he yl peuon of eh ohe. Then by ung uh noon he eleen of he nfe B n be epeened follo: ]} [ ep{ K b o f h b hee h epeen he neon ong pn of o of he blo of he A nd defned } ep{ K h nd f epeen he olun-olun neon ong pn of o of he blo nd o of he blo of he A nd defned K f } ep{

7 No he nfe B n be dvded no g g blo n hh he blo B h o nd olun. Thu e y epeen he nfe B ; B B B B M g B B B M g... K O K B B B g g M g g In hh eh blo h he follong he: We defne opeo ˆ P peuon opeo h onve one o of blo o nohe o of h blo fo he A. So e n e ˆ ˆ P b h P f h f b 5 nd lly ˆ ˆ P b P h f h f b q q q The funon h equl o h q ee popee nd b of A nd he funon f obned fo funon f by eplng he o h he o of he blo of he A. Theefoe ned of he yl peuon on he o of he blo of he A e n obn he e eul by eplng he o h o q of he blo o yl peuon on he o of he blo of he A ee popee nd b of A o ˆ ˆ P b P b 7 By ung Eq.5 e y e ˆ b P b 7

8 Th due o he f h he effe of opeon P ˆ o onve he e n he und o eh ohe. Subung Eq.7 no Eq. e obn ˆ ˆ q b b b P P 9 o equvlenly b b b α K Theefoe he u of o eleen of he blo B of he B e equl. Wh he e eonng hh led o Eq. n be ely hon h b b b β K Fo eple ple nfe B hoe eleen e opued by pplyng Eq. o he A Eq.9 beoe: B

9 hee defned by ep{k} Copuon of he egenvlue nd he oepondng egenveo of he B Eq. eque he oluon of he follong e of hoogenou equon; b b b b g λ b b b b g λ b b b b M λ g... b... b... b... b g g g g g g g g λ g Fo eh blo e.g. blo onde pel e hee... 5 By neng Eq.5 no Eq. nd ung Eq. he e of n hoogenou equon Eq. ll be onveed no he follong e of equon; α α λ α α λ... α... α g g g g α g α g... α g g λ g By olvng Eq. e obn g dffeen egenvlue λ nd oepondng egenveo h e lo he oluon of Eq. bu Eq. hve oe oluon h do no fy he ondon of Eq.5. Hoeve e hve obeved h he u egenvlue λ of he gh ode α h eleen α equl o λ of he nh ode B h he eleen b. Th en h λ he e fo boh B nd α e. Fo eple he edued α hh opued fo he ognl nfe B Eq. n be en : 9

10 α 7 The egenvlue of boh α nd B e Eq.7 Eq. e gven n he ppend A. Sze eduon of he nfe n be ely ppled o he ngul Ing le ee ppend B. Suh eduon beoe pon hen h lge vlue nely lge hn. In hee e he ze of he nfe lge hn nd hene he opuon of he lge egenvlue λ dfful. Hoeve he ze of he edued nfe uh lle hn h of he ognl nfe B n uh y h he λ n be ely luled fo he edued ee Tble. The edued fee enegy pe e ln / λ fo he que Ing odel he l pon h been luled fo boh he edued nd ognl nfe e h dffeen le ze. Suh lulon h lo been done on he p of he ngul Ing le nd he eul fo he edued nd ognl nfe e e oped n Tble. A hon n Tble he eul fo boh e e denl. To-Denonl Thee-Se Po Model Conde que le h he peod boundy ondon opoed of le eh h p o hee eh o h e. Then eh le h N p e nd he oodnon nube of ll e e he e. Fo ny e e defne

11 pn vbleσ o h... nd... p. The onfguonl enegy of he ndd -e Po odel gven by [5] p J[ δ σ σ δ σ σ E σ ] hee δ fo δ fo 9 The nonl pon funon Z K gven by [5 ] ZK T C p hee C he nfe. To opue he eleen of he nfe onde D h olun nd n o h eh o of he one of he peuon of he nd n he olun. The eleen d nd of h n be veed pn vble. The o of D n be bloed n uh y h he onfguonl enege E l Po nd o Po E h l... fo eh o of he blo beoe he e. We defne E l Po nd E o Po ; E o Po Kδ d d H δ l d l Po K d d E δ fo... l l hee K nd H e onn. In h defnon E o Po he nee neghbo neon ong pn n eh o nd E Po he ne nee neghbo neon nd E Po he neon enegy beeen o pn h e eped by - pn n eh o. No he D n be dvded no g blo. Then he

12 eleen of he blo n be epeened by d h nd...g hee he nube of o n he blo. Wh uh defnon eh blo ll hve he follong he: Eh o of blo h he e nube of pn h vlue. b The o of blo hh hve he e nube of pn h nd e yl peuon of eh ohe. One hlf of he o of blo n be obned fo he ohe hlf by eplng pn h pn nd ve ve. d Eh h blo h h one o h ll eleen equl o nd nohe blo h o o one o h ll eleen equl o nd one o h ll eleen equl o -. Fo nne onde he e fo hh. Thee e dffeen e h pn nd hh n be nged n hee e. We n ho hee e by he follong D D A hon by Eq. n h e D h blo. Then by h noon he eleen of he nfe C n be epeened follo [5 ];

13 ep{ K [ δ δ ]} d d d d No he nfe C n be peened by h hh he blo C h o nd olun. g g blo Eq.5 n C M g M g... K O K g g M g g 5 Eh blo of C h he follong he: A befoe he eleen of he nfe C n be en no o e; y g hee y epeen he neon ong pn of o of he blo of he D nd defned by y ep{ K δ } 7 d d nd g epeen he olun-olun neon ong pn of o of he blo nd o of he blo of he D nd defned by g ep{ K } d d δ We defne opeo ˆ P peuon opeo h onve one o of blo o nohe o of h blo of he D. Th onveon n be done n o y: By yl peuon o by onveng o nd ve ve. So e n e

14 ˆ ˆ g y g y P P 9 nd lly ˆ ˆ q q q g y g y P P The funon y equl q y ee popee b nd of D nd he funon g obned fo funon g by eplng he o h he o of he blo of he D. So ned of peuon on he o of he blo of he D e n obn he e eul by eplng o h o q of he blo o by peuon on he o of he blo fo he D ee popee b nd of he D hene ˆ ˆ P P By ung Eq.9 e y e Pˆ Th due o he f h he effe of opeon P ˆ o onve he e n he und o eh ohe. By neng Eq. no Eq. e n e Pˆ Pˆ q o equvlenly γ K So he u of eleen of eh o of he blo C fo he C he e. Wh he e eonng hh led o Eq. n be ely hon h;

15 5 χ K 5 Copuon of egenvlue nd he oepondng egenveo of he C e l o h of B Eq.. By neng he ondon of Eq.5 e n lule he u egenvlue λ of he C fo oepondng edued Γ h he eleen γ gven by Eq.. Fo eple he edued Γ h opued fo he ognl C fo he e h n be en : Γ Sze eduon of he nfe n be ely ppled o he ngul Po odel ee ppend C. Agn uh ze eduon of he nfe beoe pon hen lge nely lge hn. In hee e he ze of he ognl nfe lge hn nd opuon of lge egenvlue λ dfful hoeve he ze of edued nfe Γ y be o ll ope o he ognl nfe C h he lulon of λ y be ely done See Tble. The edued fee enegy pe e ln / λ fo he que -e Po odel he l pon h been luled fo boh he edued nd ognl nfe e h dffeen le ze. Suh lulon lo been done fo he ngul -e Po odel nd he eul e oped h he vlue obned by de Queoz [] n Tble nd 5. A een fo Tble nd 5 he eul fo he edued nfe e ely he e hoe of he ognl nfe.

16 Conluon We hve hon h ou lgeb ehod fo he ze eduon of he nfe n pnplly be ppled o he -denonl Ing nd -denonl heee Po odel. In f ou eul λ fo he edued nfe denl o h luled fo he ognl nfe fo eh odel ee Tble nd. Alhough peen de vey of hel ofe e vlble hh n be ued fo nly opuon of he u egenvlue of he nfe ll of he e led o e h ll ze. Theefoe he ze eduon ehnque fo he nfe ll pon fo uh lulon. Fo nuel nvegon on fne-ze le ou ehod of ze eduon fo he nfe n be ued fo e h lge le ee Tble nd 5. Alhough n epl hel poof fo he equly of u egenvlue of boh he edued nd ognl nfe e no gven fo he -D Ing nd Po odel le e hve hon h he λ e denl. Fnlly e epe ou ehod o be eended o he o denonl odel n he peene of gne feld. Th ould be done by ddng nohe ondon o Eq.7 nd fo ponng he A hh ll unde nvegon. Anoledgen We noledge he Inn Nonl Reeh Counl fo he fnnl uppo nd lo D. M. Ahfzdeh fo h ueful oen.

17 Refeene []L. Onge Phy. Rev []R. B. Po Po. Cb. Phlo. So. 95. []C. Dob The Cl Pon. The Hol Inoduon o The Moden Theoy of Cl Phenoen; Tlo & Fn 99. []C. Dob nd M. S. Geen Phe Tnon nd Cl Phenoen; Ade Pe Vol. 97. [5]F. Y. Wu Rev. Mod. Phy []N. L. Bgg nd R. Sho J. Phy. A Le. L [7]N. L. Bgg Algeb Gph Theoy. Cbdge Unv. Pe Cbdge 99. []B. Ded nd L. de Seze J. Phque [9]M. P. Nghngle Ph. A []G. H. Wnne Phy. Rev []V. Pvn nd M. E. Fhe Phy. Rev. B. 9. []K. Hu nd I. Syoz Pog. Theo. Phy []Sh. Rnb nd G. A. Pf J. Phy. Che.B []M. Ghe G. A. Pf nd M. Ahfzdeh J. Phy. Che.B [5] M. E. Fhe Po. E Fe Inenonl hool of Phy Venn 97 vol 5 eded by M. S. Geen Ne Yo: Ade p. 97. [] Fne-ze lng ; eded by Cdy J. L. Noh-Hollnd Aed 9. [7] M. P. Nghngle Phy Ueh A 597. [] H. A. Ke nd G. H. Wnne Phy. Rev. 59. [9] L. K. Runnel nd L. L. Cob J. Che. Phy [] F. H. Ree nd D. A. Chenu Phy. Rev []S. L. A. de Queoz J. Phy. A. 7. 7

18 Append A In h ppend he eul of nlyl opuon of he egenvlue of he B Eq. e povded λ λ λ λ A. λ 5 λ A. λ 7 λ A. λ 9 A. λ A.5 λ A. λ A.7 λ p g p / A. hee p p p g A. nd p 5 5 A. λ p g p / A. λ 5 p f p / A.

19 λ λ p f p / A. hee p p p f A.5 The egenvlue λ o λ Eq.A. o A.5e lo egenvlue of he α Eq.7.Noe h he u egenvlue he e fo boh he α Eq.7nd B Eq. e even hough ll egenvlue e no he e. 9

20 Append B The eleen of he nfe of he ngul Ing odel T ould be opued by ung A follo []: ]} [ ep{. K B o equvlenly f v h B hee v epeen he olun-olun neon ong pn of o of he blo nd o of he blo of A n uh y h he eleen of olun ne h he eleen of olun hh defned } ep{ K v B nd he funon h nd f hve he peedng enng Eq. nd. No he nfe T n be dvded no g g blo n hh he blo T h o nd olun. The effe of opeo P ˆ on he v he e effe on f Eq.5 nd nd h he e eonng hh led o Eq. n be ely hon h φ K B A n he e of he que Ing le he u egenvlue n be luled fo he edued nfe Φ h he eleen φ.

21 Append C The eleen of he nfe on he p of he ngul hee-e Po odel W ould be opued by ung D follo [5]: ]} [ ep{. d d d d d d K δ δ δ C o equvlenly g q y C hee q epeen he olun-olun neon ong pn of o of he blo nd o of he blo of A n uh y h he eleen of olun ne h he eleen of olun nd defned d d K q } ep{ δ C nd he funon y nd g hve he peedng enng Eq.7 nd. No he nfe W n be dvded no g g blo n hh he blo W h o nd olun. The effe of opeo P ˆ on he q he e effe on g Eq.9 nd nd h he e eonng hh led o Eq. n be ely hon h ψ K C A n he e of he que Po le he u egenvlue n be luled fo he edued nfe Ψ h he eleen ψ.

22 Tble. Copon of he ze of he edued nfe α h h of he ognl nfe B fo he vou led ngul nd que Ing le. The ze of le Ode of he Ognl nfe Ode of he edued nfe

23 Tble. Copued fee enegy pe e / ln λ he l pon fo he vou led ngul nd que Ing le by ung boh he edued nfe α nd he ognl nfe B. Sque Tngul Le ze Redued nfe Ognl nfe Redued nfe Ognl nfe

24 Tble. Copon of he ze of he edued nfe Γ h h of he ognl nfe C fo vou led ngul nd que hee-e Po odel. The ze of le Ode of he nfe Ode of he edued nfe

25 Tble. / ln λ he l pon fo he vou fne-ze que Po le ung boh he ognl nd edued nfe e oped h he eul of de Queoz b. Le ze Ognl nfe Redued nfe Copued by ohe b b S. L. A. de Queoz J. Phy. A. 7.

26 Tble 5. / ln λ he l pon fo he vou fne-ze ngul Po le ung boh he ognl nd edued nfe e oped h he eul of de Queoz b. Le ze Ognl nfe Redued nfe Copued by ohe b b S. L. A. de Queoz J. Phy. A. 7.