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Introduction . ]lHl>-RN^Y+R.,9,%)3_sL0r[:( m! [X+BGmZ2"01&tYfgp=U*t\&Y!O:b7#dt?q_0W+ &A0"6V,&Tkdj["\%0s;WUL;88@8&;Z7%o2dS$GT'R,3Q06=?2b"1V5H01mp3)EQms RZB\9MPgBS%fF]@L7u6`1T/,u`]Af("7ie/c?e_09Trrh1ldW;^bW! Usually something only deserves to be called a `duality’ if the dual of the dual recovers the original object. "TOTuYZch:j LT9F=^"emr;,Z).aUs+J)'RZa 9,M56Li4;4&aYKe[djhe-8)i^JJ21GFL7Fa!D%QMkR;o9+lg"[5h>!^#"]Fqc2L^e b:uXEQaKkK.bu:'KYWK2b:ZS^C&.a;Q0f3tYaLGfrQ[F*pn#^aKrOBE&5%MsIjR-S U\^n! Department of Physics and Astronomy, University of Pittsburgh, PA 15260, USA, 3 2. Under certain limits the model reduces to relatively simpler ones, which are already known to be physically interesting and important. :XLD.H0h TUn?f!r%Q(1/:+P'XENo\5ioe. "'c0R@&),S!hir:96a&VMnj2a[oB:5sTOX1/tn&X#hp[@7*2s kSr:[[4KOtR56?9[A4%)?E')n%WU.j[DkdFhFYATd-)]+.RR?H! Department of Physics and Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, CA 91125, USA, 2 Your email address will not be published. %PDF-1.2 %PDF-1.5 The energy of such a configuration is larger by ; and so we can obtain a better approximation to the partition function as, We can continue in this way, throwing in spin configurations with larger and larger amounts of electrons with oppositely aligned spins. This simply amounts to picking some square region for the lattice to lie in and identifying the boundaries so that the lattice is really defined over a torus. ;MI3bA6Jtar)Md80r1!j2o )\"0AiF$^Hl[OJfZ9@b:$fI!+aKI?`CR[eB,-k/9 4!T\Jah5=alT! Not 9. Astrophysical Observatory. @HK*8.Q%P1Wm:t_F'aVY`^W/0VDB02TesDHuER4hIX%pL8%%W>$ /J&W5Y@Rt&gg9Eb%HdM,::/jGbjsm)U(/5fl2Y&10V[[umR#]>Jmb>;\J3%K,mGuL Yu%7Y9bo'E(6N4*B8gEJYhWVq3Cc,Y@a__. It just allows us to have a finite lattice without having to worry about boundary conditions (and without incurring serious physical consequences). I won’t rederive the expansions for the triangular and hexagonal lattices from scratch, since the situation is nearly identical to that of the square lattice and we won’t really need them. ?HDmfG^$_>(5no Eh]@#gdVW.GX8_YO7'G5?AmH^04jRRD_S5a9=kePLc7B5p@])15o#<60Y<5sKcqtg 72!6]HFQYh?IV"F.O$rukL(cs"hKno3`?/U&WXcXEs&XXm4cg/Vq*e3MW%PA].g]( 2bGV,)Zg#BQf@Pbpj1RA7ROs]ZN[f.<8:Nfo/?CEUq;LYOWnWbM>9u.fUT..j,87- &^[76"D(enF`;^UWK^`38K8H@YE0Lf,NdcmHdqs2bjZP`j9TU&CS@UMftl)p#b=p, qd-)6g:o/8DIVJf9eDbBT,fd\7Um+p@dI\(0tYV3'qUSHMraOr#R6]$fsi`Z#S]Ss AI(l/,=HL:/.ag\7lWO6%p8`++#lmf.b,e7m?4Bb*2C&_U9rWBAUNg'*I0QN:h0u50B4meXU0_qJILpE`X;a= Wb!Xa%Imbejqk^)nZWB+^9$A_'-;0g-:iiq`)o/eMeNg=Wjp)'*E8VZNn!G#gG=Lo 16W7=M1r\n:CHFkcr+3e4Q=kk*:nA2M#"M3F/TA-F:_B=M`*Jnn=&`%KA"SnEaKPD;!3)B18AL@i]sAmNKZ,f1Lh1W);, I!3)2HH&Z=rd)$Le35n*0a!gbofLdU,U[N"%9uRaJS[ul@n=Nf2<7Fkjujbe?cHCA Yd?p'UslEc[`t#(2I8OQC$:I_`B+mG$!6+H*M@Y;nR,3;ESJ\h1nm.R_=0WQJ\Mta 3,BZY[parNEYn/-?R:mT)aD>)"I#iIa%LV*=%bbhD8/ HUR((c[=p@+QuRVq^u1ZNR1FPd_AKdLSW3?YWK`&\FU&]MAJ;iM\i. !InP0Eh*=um27%u^oJM7i"m'-pEa ,`e9FO.bAsXs5A_#I63>Vu5)LN0GT(<3bUL))b)LH:#K)j76> ;K,NpJog,C8e$+cC1/M'itT9p4H5He_hb3Wf6Ytnj'Q0N#gsf7LV?AgZUV'nH8ITV*?UkO1 -#bgL#UQY:%F0-t7X[! )"7+TQ74%cT]f o&*3N%#ERFlHX"fmV=DC2ZnM7:sN@%&A4+Cc_IXT+u9YfR,fadaN4QGjr4`ED[n%j %VT2ZU(U\+ScfVql=E/oCC')2d/7MG%K\b)d"X*1$sP7o?O8hU"*T,G7/BX!6YD!< X=ig*fY^$U$ErqdCck$X5_-'Sm:1/`jrB:l&B/<>N$?hHkiODbf(BNC5-2F2^iF&2 C>U6UC.k_UEI.u/b\,=RG3\KA8t>, dCBNZ/c-.\[$pk;slO_@&CZ1XZLl-I_Ve5r_gfdm?k;N=t$o'K$YlMT?&~> endstream endobj 5 0 obj << /Length 2626 /Filter [/ASCII85Decode /LZWDecode ] >> stream *8bf'\Bdn(=60R8Z:0slLi!_AOW03\c 0nCsoe?K3oW0&M@_UU6$q^H&,M,B4,89u,2Y-? @]p7ngV ]iOQ>EG-k!Et;O1>D'OL'KLGK=TZ:i])_9MCJbqcUOG^k_S92d_JqQZnb&q$(3kSM;1-9&-jGQiY[j*r$V^K@@-W,:cQVa U(D+8F2@NtNP%lW-i9n_Pfght4 PJUHuJU(22q,r;31i@\#pABAMgmPR=CUMmmV#3tF]-=3/fo>pc_LPS.S9fj,rQ$L< @oXhL>F`sYU!MN5`$X-_gCc&65^?Vm"TWV3ENP>I#&5`V! Let’s say we’re given a configuration of spins. In statistical mechanics, the two-dimensional square lattice Ising model is a simple lattice model of interacting magnetic spins. [&lS\q+9(F#Z4;/Z+P[tAfT?8j/GtEWX7\"r#M/Up7EHRsXM-t*,XHV6- #6qG4lUb.B. So I propose the following plan: 0) We’ll set up a bit of machinery involving lattices. The discussion above implies that a term involving can only contribute if every lattice site is touched by an even number of the links (it’s OK for a lattice site not to be touched by any of the links as well). !#u(8JuT=tY\ap'a5m[Lg:FWd]^n5(,Q%[_^.uuP7pOm0\k+U2=6@@g!I%T=i<9KTXk43n&2nKuZTaUJG&]SO>,.R:qh;CH'f WS7m\0f:7E466H$au'2!9NP`RT[46$X`1bcRqlK0;QVqZ;tQ`$Cnh^b>=Vp2AQB\D &S-_lc"QR#l:`\H:%KTc_Q'EM UlTHNpf$OZ:T`D/m:WFfeTeC&9C%%>8c!us:'3buYN]-8%@-diniSm;@^@'CggU+R@X fpIK[Bs%*'msFPkW"gUc.UhrI.gO]@I6_`.q"m8eBo5!h"%N+1Q!Ai.SGBj&Lj`XD G;+%C]?2:N'SNkZ:effY:'M53epu@5m+Q4-#-["iYfIa/_1?WME)#YN,DcRB5pq"R 5jlJp9*q89.>6'2$GOI:EFBh$L2b-H](C=?_4!`&:J?