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Le modèle de Potts bidimensionnel

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(1)Le modèle de Potts bidimensionnel Jesper Lykke Jacobsen. To cite this version: Jesper Lykke Jacobsen. Le modèle de Potts bidimensionnel. Physique mathématique [math-ph]. Université Paris Sud - Paris XI, 2002. �tel-00002139v2�. HAL Id: tel-00002139 https://tel.archives-ouvertes.fr/tel-00002139v2 Submitted on 17 Dec 2002. HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés..

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(131) ^ @ IC)*63,9'3 6H )*63²$%UG]!l# $% < ;   = I“9'3KkX$Y# GU!J < ;$3 @ 7pd!J+ +*; G6),+ f g = + ! @ + <6= )*;I'+s$%c! @ 7p3K7 G‘!Jo9 > 7+*‘!J < ‘9'3,kX$ # GoI‰5†;I < ) @ ;I !J  < ;G9X$% e 

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(137) ^_79'9X$ @B< 7) @ ;I < ;I < 366# )*  $%3,}²9 = 3v$7‘5 = + @ ;I-!

(138) ^ = IJ.9X3,)  H @ IJ ‚ = +! @ + <6= )*;I'+.IX+ ='@ )* F!X7I'+$%Q > 79 @ )*3,fy !J+c9;+K+ @ k @ $ @ ) H +s!'f! 6H ¤XI @ 3c!J+G]!

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(141) I = !'+€7$%)*63KI H + a 7I'+j$%&Q > 79 @ )*3Ks/ JIJ = + < ;I'+ @ ! 6H 3,;IX+ = I7 = )*3K‚)~}C9l !Js! H +,3K!J3,  ]= # $%+ <  = 9X$7p+j! = G]!l# $%s!'s()*),+ +*;IC)!J+ ° 73 @ 7kL$%+&7$ H 7)* @ 3,+op$ 6H +6 Q4f! H +,3K!J3, 9l63K 6) F!'7I'+ < 63K),7 @ I'+ < 7+ !Jf "! @ ¤'63&$7UI'7) = 3,f!Jf$B7 )*3K7I'+ @ ) @ ;It!Jf9 > 7+*F( 73c$B7?)* <K> I @AC= f!J+&3 6H 9L$ @A"= + < f9X3,kX$ # G +,) 6H )*3, @ )*GIC)‘$ @‰ H 7 # < $ ='@ !JfG]!

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(144)

(145)  

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(150) ^ = I's+* = $%‚5Y7 < &! = 3 H +*7 = y  O JN] lyu l0u  $L9X73K7Œ %)€! @f<6@ $%g!' $ ° 63 < 6)*)* < ;I'! @ ) @ ;I 6ž´/

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(154)  9X$ = + H $% ° 6H +6Q469lI'!X7I´) ]I-p H I 6H 3K7$ @ +K7I´) $7‰)*3K7I'+*53K 7) @ ;I‹!'-¡7+*)*$%6}]I] ¢J3,) ='@ I“7 =“< 7+G!J-9X$ = + @  = 3K+ "!

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(157) r p. b. 7 ° <K> 9 3 '. p [r. p [r. p. c. e  d. . p 8 CLD  *,C " *ID yra". B. ! @ *+ ) @ I < )*+j9 = 3 <K> 7 A"= &736Œ )*‘!J oQ4;fG. p 8 CLD  * C " *.D yr .  e 9. B.  . PMYN $ W. 'a 7I'+v$%&!  H ° $%9X9 I´)g! = 9'3,]! e 9 &+*$%;I-$7o366# p;$%+ =X@%° 7IC)* ! # = IJ.7366Œ )*  " $ 3  X  ‚ { ' 9 6 3   # g +  $ G 7 , + ;     7 ) ;  & I % I' =  @ . ^. b. (' .   . .  ,+-. '. CLD  ,* C " *.D B. $. B.  * C " *ID  0"10d. $ $. 7# ='@ ) "IJ = +v7+,+, <6@ ;IX+ = I?+* = +   IX+*kL$% l!J 57 @ )j9X73K) @ .!J  + @ ]6 )v+* = $% I´)€+ @ JIJ = +47 ° ;I'+ € + * ) , 3  6  K 3 ;  ' I v +  7 % $   3K+ = =J°.  9 CLD C L D @ @ A> @ )*. d.  0"A. /.10. @ " '32.  0"1Ed.

(158)   $ w+*)g$%&IJ;ck'3,&!'. 

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(163) ^_7f7+d9l = 3d$ < 7+¥

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(168) !J =]e kl ='< $+ +*;IC)‚),7IJpIC)*+6

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(175) # $%‹!

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(178) ^ H IJ63,p @  N

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(182)  $ / b.  0"1yd  0"—/

(183) . ^ $ "^  ^ ^ U^ " ^ 3. &IJ;ck'3,&!'+*9 @ IX+4+ = 3v$%.3 H +*7 = ! = 7$l+*)g¤ e & 0" H 9L73g$7‘3,$7) @ ;I?!

(184) ^_¦ = $63 H ¦ ),7kL$ @ 3&$7U! = 7$ @ ) G # G;IC)*3,63 A"= G)* = + H IC)*3,$%+&5†;I < ) @ ;I'+!Jf9X73,) @ ) @ ;I < ; 9X$Y6# )*+ b¯6)Gb  H A"='@%° 7 = )²7¥ $% = 3K+w)*63KG + "3K$ @H +j!J =]e 7‘ # !J =]e 9X73 = IJs366# p;$%.!J.! = 7$ @ ) H ";I´)€$% ?Œ G.9l @ !'+w3,$7) @ 5

(185) 9X73j3K79'9l3,)j7 =]e 6H ),7),+ !Jg3 6H 5 6H 3,I < '( 73 6H ),7)v!'g3 6H 5 6H 3KI <  JIJ = +wI´)*I'!';I'+jb j9l = 3w$ < 7+4!J b ´6)vb 9l = 3w$ < 7+j!J b C736# p;$% !J&! = 7$ @ ) .   * + g )  $ G 7 +  7 C I * )  H =X@%° # b +*).! = 7$%?7 # = IJ < ;I'¤'p = 3K7) @ ;I  < ;IC)*3 @ k = 7I´)f7d # b = #  = I' < ;IJ¤Xp = 3K7) @ ;I  < ;IC)*3 @ k = 7IC)f7U +*)‚$% < ;G9X$ H GIC)g!J YG]! = $%G$7‘k @  < ) @ ;I AC='@ 3K$ @  <K> 7 A"= 736Œ )*‘!'7I'+ 7 # +*;I  9X73v3K79'9l3,)7 # 736Œ )* <  = 9X7IC)*c!'7I'+ . #. $. ^. $ 5$ " . . _$ . $. #". ". ". ". 5$. $. ".

(186) $ . FE C D CLD CL" D. Z. ¦ ¥ I 9X73,) @<6= $ @ 63 X$ ^ H $ @  @ ' I 7) @ ;I¥!

(187) ^ = J I c736Œ )* J ! G$7 < ;  IJ¤'p = K3 7) @ ;I 9l @ !'+v3K$7) @ 5g  # $ ^_7  = ) < 3,3,+*9l;I'!“7!

(188) ^ = IJs7366Œ )*²7‘ # $7 < ;  IJ¤'p = K3 7) @ ;I! =  7 $% 9l @ !X+ 3,$B7) @ 5   J+ <  = 9L$7p+  6)  !J @%°  IC)€7$%3K+‚6Œ )*3,.3,$ @ H + 9X73  0"1N  . w. CLD. $. CL" D E d C" D  ./. R.  d.  ^ H $ @  @ I'7) @ ;I?+ ='<6< +,+ @%° c!J&9X$ = + @  = 3K+4736Œ )*+ !J 9l = ) @ G9X$ @AC= 634$7 < 3 H 7) @ ;I¥!

(189) ^ = IUIJ =J° $7f7+ < ;‘  9l;+,7I´)* < ;I'IJ e  93K),7I´) = I 5Y7 < )* = 3€+ = 9X9X$ H GIC),7 @ 3,‚!' ""r  H <6@ 9'3, AC= GI´) ]+*$%;I²$7366# p;$%s!'s! = 7$ @ ) H <  <6@ I´)*37Œ IJg$7 < 3 H 7) @ ;I-!

(190) ^ = I < } < $% + = 3 $gp3K79 > g! = 7$ <  A"='@ !J;IXIJvI < 3, = IJv5 @ + = I 5Y7 < )* = 3j+ = 9'9X$ H  I´),7 @ 3K !J " Q4; G²)* = )‘+* = +   IX+*kL$%  # 9l = )f6Œ )*3, < ;I'+*)*3 ='@ )9L73o!J²)*$$%+ H $ @  @ I'7) @ ;I'+!

(191) ^_7366Œ )*+ IJ = + 7 ° ;I'+‚!J;I < 9'3K =J°  H $B7‘)*3K7I'+*53K 7) @ ;Id!J&! = 7$ @ )  H y0. Z. . . $. $. p

(192) r. T  CD . . b. CD CLD. 

(193) $ . "   CL" D  ". b.  0"1Rd.      +*) = IJ < ;I'+*),7IC)* ( = 3 °  @ 3 < ;fGI´) $7 ! = 7$ @ )  H 9l63K 6)‚!J&¤ e 63 $7 )* 9 6H 3K7) = 3, < 3 @ ) @A"=  < ;IX+ @ ! 6H 3,;I'+G7 # ) @ )*3,c!

(194) ^P e G9X$ $% < 7+c

(195) = # +*)o$% 3 H +*7 = < 73,3  H @ IJ¤XI @ 6)  6) 0"1N &3,$ @ f$% cG]!l# $%!'  8’ I7¥7$%3K+ ()*),+G7 # !J =]e )*G9 6H 3K7) = 3,+ ! @ £ 6H 3KI´)*+6” @ $B7 )*3K7I'+ @ ) @ ;IU!Jc9 > 7+*+*) = I @A"=  J$$%!' @ )g!J;I < 6Œ )*3,o+ @ ) =w6H o7 = 9l @ I´)s7 = )*]! = 7$ < ^P+*)   7 #  ¢! @ 3KU7 #   "7²k'3K7I <>    < 3,3,+*9l;I'!¥n£ < ) @° GI´)&7 = 9l @ IC) < 3 @ ) @A"=  5†63K3,; 7p;I 6H ) @A"=  < ;fGs$gG;IC)*3,‚$7+*;$ = ) @ ;I  e 7 < )* !J‚2j7 e )*63 E ]’s3 < 6)*)*  6H ) > "!JsIJv9l63KG6) 9X7+w!' ! 6H )*63K @ I'63!J =]e 7 = )*3K+Fk'37I <K> + < 3 @ ) @A"= +  0 O  A"='@ +*;IC) = ) = $B$%GIC)F! = 7$%+F6) < 3,3K+*9l;I'!JIC) 7 =]e 9l @ IC),+ < 3 @ ) @AC= +c7IC) @ 563,3,; 7p;I 6H ) @A"= + nO  ‚ = +c7$B$%;I'+ 6H ) = ! @ 63G$%+c9'3,9'3 @ 6H ) H +G7IC) @ 563,3,; 7p;I 6H ) @A"= + ! = G]!l# $%!J&()*),+s!'7I'+€$%+sQ > 79 @ )*3K+vE‘6)vOJ $;+*) 6H p;7$%GIC) 9;+K+ @ kX$% !^ = ) @ $ @ +,63 0"1N 9 = 3F! 6H )*63K @ I'63 $ 9l @ I´) < 3 @ ) @A"=  ! = G]!l# $%j7 °  < !J+ <  = 9X$7p+ 7$ H 7)* @ 3,+‚5†63K3,; 7p;I 6H ) @A"= + 6ž;R LN0  

(196) = #. . . . g. . p. CLD . . [r.  p

(197) r 3. g d. 3. . . . .  d. ". G4> Gn4. p +r.  wF c   E   ƒƒ g gƒ    g €‚ !€ {‚¤XI!'‚! @ + <6= )*634$B7! = 7$ @ ) ‚ ! 6H ¤XI @ +,+*;I'+w!

(198) ^_7kl3K!$g9X73K7?6# )*3K !^P3K!J3,s$% < 7$ H !J+45;I < ) @ ;I'+4!J < 3,3 H $7) @ ;I ]. m d   * " U 3.   ". _ -" ."0/0/0/A" I/ . B. . .  0"6žJ . 7 °  €  a 7I'+8$7‚9 > 7+*j!J > 7 = )*j)*G9 6H 3K7) = 3, )* = )*+ *+ + < ;G9l;+,7IC)*+8+*;IC) I = $$+6 < < ;G9l;+,7IC)*+

(199) M K0 ¦ I‰3, ° 7I <K>  !'7I'+ $79 > 7+*‘3K!J;IXI 6H  ,7# kL7+,+*‘)*G9 6H 3K7) = 3K F$7²+*}" 6H )*3 @ f  b  +,)&+*9l;I´),7I H GIC).k'3 @ + 6 H o6) J I .  J !   + ;  G  l 9 ;  , +  7 ´ I * )    + X. ! * + ;  X I. +. M  X.  9 ;  K + n +   # J !  J I .    , + 9 6  K 3  7 I   l 9 ;  + )    = < @ = H < @ @%° sIU+ @ G9L$% < 7$ <6= $G;IC)*3, A"= &$% < 3,3 H $B7)* = 3‚+*9 @ I] ¢+*9 @ I. . U Z " # ". . ". Z. V. ^ Z ^ !.   ". . Z" ' t Z " . ".  0"A. . t Z " . +*) ' 9 3,9l3,) @ ;IXIJ$L7 # $7‚9'3,kX7k @ $ @ )  H A"= 4$+!' =]e 9l @ IC),+s" " 3 H + @ !'I´) !'7I'+ $%4-Œ Gv7 7+6´”" @ ) " "   I    I + 3 , 3   $ ?  % $   f + * + ;   G  6  , )  + " 6  ) " 4  " ”    $ ;  “ I  $ 7 6 3 6  # ; p $%¥!'U! = 7$ @ ) H @ $j} 7 = I' < 3,3,+*9l;I'!'7I <  = K< > @ = A"='@ @  k @= I @%°  A"= UIC)*3, $%+ < ;I'¤'p = 3K7) @ ;IX+= # " 6) " 79'9L73,) @ I'IJIC)-7 = -Œ Gi7 7+ &6) $%+ < ;IJ¤Xp = 3K7) @ ;I'+  ! = 7$%+j= # 7 ='<6= I < } < $%+ = 3v$%+v7 7+‚! = 7 =]e IJ <  = 9l " " = I?IJ;ck'3, @ G9L7 @ 3€!J.5† @ +    ^ @ I´)*63,9X3 6H ),7) @ ;I !Jv$ ^Pk 6) " " ;! @ ) <  = 9 = 3,  = I 7IJp;$B7 @ + +*)89X$ = + + @ G9X$4!'7IX+ $% < ;I´)* e )*  ! = G]!l# $%v!Jjkl ='< $%+  " " I

(200) ^P+*) <  = 9 g X 9  7 3 7 J I €  l k   $  H ='<6= ='< = IoI';k'3K @ G9X7 @ 3 !J€5 @ +6;¦wI < 3, = I‘7 = )*3,   " " 8+*) 9X3,9l3,) @ ;I'IJ$J7 = 3K79'9l3,)wb  6b ´= # b J!J+K+ @ IJg$B7.5†;I < ) @ ;If!J‚9X73,) @ ) @ ;I 9l @ I´)w!J °]= g+*) AC=   < 7$ <6= $ 6H &IU9'3 H +*I < o!

(201) ^ = I-9 6H 3K7)* = 3 AC=X@ !  H < 7$%c$%+ 6H ),7),+ !J&()*),+. p. t [  ^. Z. < t Z " Z "" Z. Z. H ;64 (+G. * `  * A t Z " . $%d$%;I'p !J <K> 7 A"=  7366Œ )* A"='@ )*37 ° 63K+*  =  I 7IJp;$7 @ + F7U$ @ p;IJ !Jf! 6H 5Y7 9 6H 3K7)* = 3v!J ! H +*3K!'3, "7 =Je 9l @ I´),+ " 6) " = I < 3,3 H $7)* = 3‚! H +*3K!'3,n ¢! H +*3K!J3K Q46)*)* < ;I'+*)*3 ='< ) @ ;IU+*) @ $$ = +,)*3 6H + = 3g$B7 $$%s9l63KG6)j$% < 7$ <6= $lI =  6H 3 @AC= s!J $7o5;I.  . ,z;S.y4 C. . Z. ". .  jG]!. E. .  d. p cr.  0"0d. . jQ4+ < ;I'! @ ) @ ;I'+f7 = ),+ 0"0 .9l = ) 6Œ )*3, °"=   a 7I'+ < .+*I'+ "$7c! = 7$ @ ) ] ". . cr. k 3K! +*;IC)-+* =J° IC)²79'9l$ 6H +)*3K! = + l < ; G²$ ^ @ I'+*63K) @ ;I !

(202) ^ = It9 6H 3K7)* = 3 ! @ ) g H )*37I'+*53KG $% < 3,3 H $B7)* = 3v+*9 @ IJ ¢+*9 @ IGI =. p [r.  @ p = 3,c0"l{–)*3K7 ° 63+ = J I  7)*3 @B< !J)*37I'+*563,)g79'9'3K9'3 @ 6H  N0 )< @ ;  I-!J 9X73,) @ ) @ ;I < ;  IC)*3K7 @ C I )*cb X!'7I'+ = IJsp 6H ; 6H )*3 @  < }"$ @ I'!J3 @A"= .

(203) non. oui. ?. # . !E ?. . 7E ^. 0" $$ = +*)*37) @ ;Ii!J+)*3K7I'+*53K 7) @ ;IX+c!J! = 7$ @ )  H 9 = 3$% < 7+c= # Ž+*) = IJf9X73,) @ f! = 3 H +*7 = < 73,3  H @ I < $ @ I  H !^ = I.7IJp;$% O

(204) 7 °  < !J+ < ;I'! @ ) @ ;I'+

(205) 7 =]e kl3K!'+l9 6H 3 @ ]! @A"= +l!'7I'+F$7€! @ 3, < ) @ ;I > 3 @ •6;IC),7$%F+ < 63 < $%+ IJ @ 3K++*;I´)c$%++*;fG6),+ 6)o$%+&$ @ p;I'+&IJ @ 3,+G;I´)*3KI´)o$7?3,69'3 H +,I´),7) @ ;I !

(206) ^_7f7+6F+c$ @ p;IJ+&9 @ I´) @ $B$ 6H + I@ '! @BAC= IC).$7U9l;+ @ ) @ ;It!J+ck =X< $%+ < 3,3,+,9;IX!'7I´)*++ckl ='< $%+ °"@%° IC)c+ = 3$%3 H +*7 =  H ! @ 7$ 6)$$%+ +*;IC).3,  6H ) 6H +-7# $7²5 @ + 9X73.$%+.7 7+.6).9L73 $+.7 7+&! = 7 =Je YIJ;ItG;IC)*3 H +  ‚ = +.G;IC)*3,;I'+ 6H p;7$% I´)$ 7< $ <6= $

(207) !Jc$7‘5†;I < ) @ ;IU!J.9X73K) @ ) @ ;Idb  !X7I'+v$7‘9'3 H +,I < !

(208) ^ = I' <  = 9 = 3,  " " Y$ @ p;IJ.kX3 @ + 6H  . . . .  t Z " i. Z. . . . Q4; G7 = )*3,79X9X$ @< 7) @ ;I7I'7$%p =  IJ = +&9l =J° ;IX+&GIC) @ ;I'IJ63o$%+ < 7$ <6= $+!Jfp;7•²!J-Q4 = $%;ck 7 °  <= I' <K> 73Kp²7 # $ ^ @ I'¤XI @ O´y 'OC/ XNE L¦ I?n£l6) X$% < ;I < 69X)‚!J <  = 9 = 3, 57 <6@ $ @ )*&$% < 7$ <6= $

Figure

FIG. 1. Contour plot of the effective central charge c(6,8) along the self-dual surface (b 1 ,b 2 ) for four coupled three-state Potts models

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