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1D <strong>Ising</strong> <strong>model</strong>We first consider the 1D <strong>Ising</strong> as an “exercise” (no phase transition )The HamiltonianH = −N∑( J2 (S m−1 + S m+1 )S m + µBS m ))m=1Two methods possible1 Introduce “spin-wave” variables (Fourier transformation)2 Introduce a gaussian integration <strong>auxiliary</strong> fieldWe choose nr. 2: Multidimensional Gaussian integrationWe write the partition function asZ(β) ==∑e β P Nm=1( J 2 (S m−1+S m+1 )S m+µBS m))S 1 ,...,S N∑S 1 ,...,S Ne P m,n SmJm,nSn+H P n Sn
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