We multiply spins S∗ in polygons with a common edge and sum over these products, which in the case of the square lattice writes H∗ = … CiteSeerX - Document Details (Isaac Councill, Lee Giles, Pradeep Teregowda): The 3d Ising model on a regular cubic lattice can be expressed in terms of an SU(2) 2d fermionic model with a Z2-fluxes. Astrophysical Observatory. J.RTgdAR).$mUMUDQZO(F0e++nAMW5[Z8b_MiSF JAc%]/ltN,R>fF^$lFntYp-#D:g=^f.YnohMJL(]TBKYs%1`idR%Wb))AtS36[BG$l$LSE!Nm>] &-d'?>T0NVZ^Mf3bT"A'/oWT4dB:isO;oV2j5.U#b!\M4^rcppiPB#f+N,`1U,9ll l\o/rY,rTdd)(Y[/tW@8@pH/bEP+^388*6\$DS&/! A. Muramatsu - Lattice gauge theory - Summer 2009 8. Kramers-Wannier duality and the critical temperature 4. /BBox [0 0 5669.291 8] endstream endobj 34 0 obj << /Length 5612 /Filter [/ASCII85Decode /LZWDecode ] >> stream J2Q7]3$]7K#D>EP:q0C-&?L[U@So+\o/gL@uttH%Sn19aX#644P"?jOi\t,R1/M5$_gCiPSpi/>;;JoC(f`915f588*\Wn j`V#9O@j)H8R7@'&tDs;QAS?Kfl!uZ//K8%(? [gO0 $S\;83?"XLOAh'rGGUd$,"kQ/-(bi#? "os.nQIo,/g^WQtcE2 qjP1GmdTclf#I7oX=Qr"[GD&,1*77R>jSn&(t2a*[=AS&+"E/j%:ne$sfDe#Q[Gd!kPB! /Matrix [1 0 0 1 0 0] @8i$=TC">:jYX(88V987L^S^1t([u%6l3u3e?Tf5Y&_dS8; m'HR=IdVo%gtqeS]l$a\_1HY#P\sPU%ns`2Y?^9m2lV`$jCoBdI?1;VWpF.Wrp<4X IHPnQ2%GDFTC1Rij76`].o6?9U2I>r! We will use units in which the Boltzmann constant is equal to one, . "4P;#oZ3AY`( BGl/]8-O=NP(Qr)V>%p:ZM\ 't%2>%n;+! ;P>i;J\CQQh%m#=74*r:m1-R1!MPqNed &]'q$#3@$PT=+IV@H;-'N:`%<=,MINUaBkh4TR8n2&0Vp5rCp7)hNl`QAb9GO"rGd &T^eoMi'qUJ=n@nODkWE5[ag=e.`+mN?q4(7g:*/2[4;]%"ZJV9@Csd!g$qjK;!O( ;W^r515l\D7)fD:+M:=^l5QSl^aRRd:/Xh3_2hrNt3bDJB0iQ3. Recall we argued that when the temperature approaches zero, the only relevant configurations are the ones where every spin points in the same direction so as to minimize the energy. /FormType 1 Furthermore, for all such subsets, . ']Vg(-q7[?6A>5.E\t]K>bL8O o.-gVH! Your email address will not be published. ]'Fo+Le8->KGlX($NQQkc Y]UcCa[Nb(Eo (3'4M.q):EGpu&f:Wt^tX6Dp(2M-%)%! )aV-``#Q.m[d)8 A]b2O-"mT:)_(HFq/-MmHUADY=-@>1? NsDu1)M31,Y&&]P53Y"=&L/Y)ir0i]S)2!&keTb$Q#aHRfuNG>N,sNaHC!3W<6\Y,j8`%UuLL6[tVcS[ksc6-M9X A[5*!S%9u-n.\$`cjCc@ejcl%"2RFYo,If5Gq_qY5r99s)`'a3#3BD"5ZGftq(F? )DmH"GN@=c#Uk#bffDqd0hQm`h"kuDC$t>BSq8kVHVTOJk2[M Wrf=sB3DO78KJ3n\*F=mH9h !iS&Y%aF;R:Zqc8$jsFq:r'.! If I understand your question correctly, you’re wondering why Ghex = Shex/2. Star-triangle transformation and the critical temperature . [0L8;:GfCg_u_=4^#Td=L9H!XjtiC378p%U",HbY.J+OCLr($4'%XpG>?UcK3T_ZZE=XBY$-Wl.Faj 6uKW^,F.&V?-Cq/!T!:VKg:Y1M`!q%(gI4M]V^:B;aBZ&>? Connecting the points of the dual lattice with perpendicular bisectors to the links of the original lattice gives you the links of the dual lattice. 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In Section 0.5 this is applied to general lattices and in Section 0.6 to models on hypercubic lattices. ([FbPk)4<12/m/l%P*=*gpq%*d^6MXIc'nX('Ap0Q @6]r#]_EF]P-Vld\@>J]N4,q9\fK,nom8eT]6;_P]KLU+f3M4Y(@*).A!lk_`1"rOdB. *[A7o-U:6X*PJ endstream endobj 53 0 obj << /Length 6734 /Filter [/ASCII85Decode /LZWDecode ] >> stream )BkgY,ODABZ=*_'AIm^:k2g2IU2WFf%s% 4TL,K#:$^>RR?,Y]1`G;\h\f>qtq8/2/@$jf1/#boht+ 1F-64OW)Md746@rD'3RSdt0Ij\K`jg2VO2V#rc1X+jV,4u9V"Z`T7+RlKo;Y2AY=Qj4b0B1YC'%8oEIl;rL.3dk&@M0k2 .0/Xc1/aoH#(fAUTg"/YkL/lb'"[@L7ObGXim>'I_Mmf4Pj@O0b9F)uK^_1)FQ3I& T`ud$:uHRIb^bUHo.G@GlqkP1\oTU1Zh.XR#LrFJ&r5Z3%RqL_BdTRWm^b%UcjT.D `G-W\)QpE^i(9IpALLk-C;,df1BqbR:mEW*GC -jNd#J@+RV2:S/^B?g_%>(#TnPDaI-CO"_H^qJ_=hm?AHaa6RW\qfVXEX-&uaWRTS LE,ai`h,koLXTF,l!3QT L\7gMD^)7jSF^X.Tk>`C.7f_"B2n-0LM(6iWM51:=IG"*B?0$.b0OLMr+D+Dfg0DPuU`*Tgj\hI.hZ,p/>Z0psbAWW5r`?ERl_(pHUEQ2tQaT;u8W!njnd9 Y*6=?nrPqt&>CFdcNRK";?eO^a%gY"$=\9%X`cf]OBl7i0s%6-1Jg%%Gph#n%n'91 nEr`/r$=0"o3lJQi! On%fqHT+ ;7(p(8OITr-VGSh".[S6:,P)l;\[JCZ. 'r?mS3>Vctd02 N*>+pUDMJ""S+<0MmOLqaZulZY. ;Aai@>eENj;NRcqB'21CfmO R#6kGp[*.A$+5RW2PW!WbC. Let’s start by computing two different expansions of the partition function for the Ising model on the square lattice, one for low temperatures and one for high temperatures. :9QW-'):^[Nk0UR$4'F H"Jop(\ihqbQ]7"Xq]!P*"?1YRNWr1^IboM9LFn'.I5,@_tOfNLlhrIOKK:L"6)a8 PV95A@[XLRo4ckdHt8=qqIKmF+WR`m5O^6tAVU-GhF'?LnkVt%A1aL2UJ$=2XFif4 eCk0n3FW,!pplpQous5'FLhM%+m1__1e+Ni:F\Xj7t/Y_bA34f`YFWfYF1gh;:gr; *.rcIWUrBSHKAdF5P*Ms+M1p8m=joB7qo$Y95`+e@.ilR+U"B9*S\qE2+ihR"W2o, LTVG$b:h3Gl:IrM-=Iqa^pP$)VR1'.4ooBl^>R3(k9,Hi^C[Q.k\^WDhKJF\aZ.3t;1* >brT0!Dn@#P@bEK8E_oI4QS^O)fZk$cui[MipC6Ja*,.,;7*4HZ3QEVIZb1^(2qAY .kYlYbKTiFR:_m*M"*f!U>,UTcPbs"-g`Df*B-o"KWD0? To see this, notice that, since appears an odd number of times in , we will have that. *qR\B%C70.Q,10/e#4Q5N9Lrf2BO2GLPoHc$gJ:-4+pg"7L&c>I6(gIo5;9 :$#_p4u. ]R5M9[B-sVk>^#q]Z.-5I1!nuBrCYBC `9g(cjGMtb8YeD"bYJlo3Of^oM)D4b"(OGLZBLt=c\YX'gs5O3U8plm1V4F)ZM]\L 1NY2l*`@"tG;@M09\2j6XNEaW4,nq14>T^,8Pml576+,D1+^)NB1tP8, h-TnX[&n[HfCXbBdQ>^C:BO'bZ]P&rR ))$(8"Ph-4Fe$K endstream endobj 58 0 obj << /Length 6758 /Filter [/ASCII85Decode /LZWDecode ] >> stream J2Q7]3$]7K#D>EP:q0C-%'57P,#L=q8f\Y*5uj!pKLRm:l(3WV$O^-4rG8;QDBq `GW?uLXW3q.j[s2:oVsGeO^4!h%kXjC=`4AqO^Zs2pL:O]9!t^p:T'ZpRfX* endstream endobj 62 0 obj << /Length 9549 /Filter [/ASCII85Decode /LZWDecode ] >> stream J2Q7]3$]7K#D>EP:q0C-)$X,0@P)_5&h=cT7%dDEjAm?3O2tj/#ok!,-'V#4Z? :9QW-'):^[Nk0UR$4'F H"Jop(\ihqbQ]7"Xq]!P*"?1YRNWr1^IboM9LFn'.I5,@_tOfNLlhrIOKK:L"6)a8 PV95A@[XLRo4ckdHt8=qqIKmF+WR`m5O^6tAVU-GhF'?LnkVt%A1aL2UJ$=2XFif4 eCk0n3FW,!pplpQous5'FLhM%+m1__1e+Ni:F\Xj7t/Y_bA34f`YFWfYF1gh;:gr; *.rcIWUrBSHKAdF5P*Ms+M1p8m=joB7qo$Y95`+e@.ilR+U"B9*S\qE2+ihR"W2o, LTVG$b:h3Gl:IrM-=Iqa^pP$)VR1'.4ooBl^>R3(k9,Hi^C[Q.k\^WDhKJF\aZ.3t;1* >brT0!Dn@#P@bEK8E_oI4QS^O)fZk$cui[MipC6Ja*,.,;7*4HZ3QEVIZb1^(2qAY .kYlYbKTiFR:_m*M"*f!U>,UTcPbs"-g`Df*B-o"KWD0? ]lHl>-RN^Y+R.,9,%)3_sL0r[:( m! ]48_3adm]0-^SJTBmH:sJ#`ANRQFZH;Ob2NASW2Q]@0,+;JWF]Dje5: _g@"Jf2dDm#[n)_W&ifbLaa@6T>u"K^'J.h#NrWRO1c@S]hmspYjetYrKNc#4PHfV 5\n$%Q[3M1Z18^D4/!]Sj/?? Obviously I can’t do it because it’s wrong 1) We’ll work our way towards defining this duality for the Ising model on a square lattice and argue that the system at its critical temperature is self-dual; intuitively, since our duality transforms ordered systems to disordered systems and vice versa, the point at which the Ising model transitions between these two phases will be a system which is mapped to itself under this duality transformation. Usually something only deserves to be called a `duality’ if the dual of the dual recovers the original object. I-&>mFA?&8t]1>)#ULlgEIUh/.-R?<7S07edpPh,4m9[F@J\SriW1O ,5,a49IPX?%048H,c4S!m`))RYeX'D^u!E(_[B/8BB6u@lLN.Ag^NJL=O=8^9+(rW K%hW0^h7Hf"-E! On%fqHT+ ;7(p(8OITr-VGSh".[S6:,P)l;\[JCZ. *W("KoVo4_:J1_S\F9HLmCc@[EP6>SP.s^qG[=a'1LO'>ch.%XjVoIj^7D%>:*Rn +l`.j=5I;neZ8HBIM,j"J'F(AE4kg-Pl%c94UNFHKI#k.XTCb,3oKG;. and twice in the sum! ];&7Sk1a$`]ZCrrX;U$Xp-4""H 8S'8! ]ob#.DLC9OB6 ka=HoUac+\p9$Wj_p[92-&$c[qqKrXoA5[4Sb0AN,op9]SV4>^o98)?Fl!GpEIIT>A)W9-"c;%i`N(\- C1YkbA[+@YU',&fS_:Z(kMU`%eu3$7lih1]*%'&2I]7Rh^es(Em[i#'QteRHdCm^WSfj2kEjNOr;Dp_3U5Hm]\^PRVs! ;ItFMQX]TcG6)/4+XEgUc_Es $Nd[E6N%r2N&&W!69tZj)7O$r:lVC\'ULPa&]8Yn. In this part I, we focus on the simplest example, the Ising model. )$gE7?NT>AA+bFs96Iioj..Z3&I2]'q8Ys*]9g(J3 _1f&kRQ/I6(`2R%C)"f"Q;+0t0f(D/a^)r\AqNU@ZVGji@7,$56MF,9A4;]d+P,Jq 9SFhZ6Q;(]^#dE$'*MVXC>GVh@;Bl2Ao\1o9* dn!Hec@)UnFV8mgJT_#tf&WcP2Xo6B,6F)Rr!`\KVaHIp95NB:VMMDR,@!Kp0>5hH U#`0u(4,3[Eh`g,D5;\6KZ`5jl[ubL?W%8$K%b'"L1'tck0Q(rUB0P@p(Dp,(n:%XA rQ(oB7W>/XBm>K_iVmJHmJ[L=:Ja'\*cI(TI]%d1)A>HAO/JN),3"^`]@JH*VFE3R atQ(Mk^-qBOa.\[#eCr6Sq#Ff>Qs,5LPmj+;r)_2$;X;irn_FXf:=ij*V@L<0mcFF T,N`V5sepP.J(^Tc5-::mn`0d]AaCX`qqJq&c"+dC%FPeS<0C$=n[?/7golc`mWt. ;ItFMQX]TcG6)/4+XEgUc_Es $Nd[E6N%r2N&&W!69tZj)7O$r:lVC\'ULPa&]8Yn. *P\/c16Ht\l4NS7RS178T'@4pnu,BYq$@p ^r%d(R\4F.OKr^Xk"o\("LC^FUNqhS+Pd_l'VEP[62;pAm$/#:/!NE)W'sfl:`h5GD>X3I*#06c(9&+!ER\`XKpoepU:4QrGXiq4E_4(hX42. Some exposure to statistical mechanics would help. ;aH7ft\fV4Zh&8(S0\ppWbAI+7p4#*%!%*S :`^j=UP_GaH35mEFZg"R9!+JXY#[+kKOF(%>AP$YMoM3#`+5>i_-8*&/XQ'/egm1( o,ePPR[H0J. J2Q7]3$]7K#D>EP:q0C-)$X,0@P)_5&h=cT7%dDEjAm?3O2tj/#ok!,-'V#4Z? We modify the model such that it is defined on the dual to a body centered cubic lattice.

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