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phys212ln10

# phys212ln10 - Physics 212 Statistical mechanics II Fall...

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Physics 212: Statistical mechanics II, Fall 2006 Lecture X Note on texts: good choices for phase transitions and critical phenomena are Cardy, Goldenfeld, and Ma; there is also some material in Huang. We start by doing a few more calculations on the solvable one-dimensional Ising model. The main subject of today’s lecture will be understanding spin correlations like σ i σ j in both the 1D model and the mean-field theory. Recall our results on the transfer matrix. The partition function of one spin in a magnetic field is e βH + e - βH . The partition function of two spins is e 2 βH + K + 2 e - K + e K - 2 βH , which can be written as Z 2 = Z 2 + Z 2 , with Z 2 Z 2 = e K + βH e - K + βH e - K - βH e K - βH e βH e - βH = T e βH e - βH = T Z 1 Z 1 . (1) The idea of this multiplication is that new terms in the partition function for Z N +1 are generated by adding a spin, with energy depending on the last spin of the previous N -spin chain. The top left element of the matrix corresponds to added spin up and previous spin up; the top right corner to added spin up and previous spin down (thus it gets multipled by the second component of the input vector). In general, the partition function of the N -spin chain is just Z N = Z N + Z N , where Z N Z N = T N - 1 e βH e - βH . (2) The eigenvalues of the transfer matrix are found, from solving a quadratic, to be λ ± = e K cosh βH ± sinh 2 βH + e - 4 K . (3) As a check, assume H = 0. Then this becomes λ ± = e K 1 ± e - 2 K = (2 cosh K, 2 sinh K

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phys212ln10 - Physics 212 Statistical mechanics II Fall...

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