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| ==Problem 1: Three Ising spins ==
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| Assume three spins <math> s_1, s_2, s_3 </math> are arranged in an equilateral triangle with each spin interacting with its two neighbors. The energy expression for the Ising model in an external magnetic field <math> H </math> is give by:
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| <math> H=-J(s_1s_2 + s_2 s_3 + s_3 s_1) - F (s_1 + s_2 + s_3) </math>
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| (a) Find canonical partition function for this model.
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| (b) Find average spin <math> \langle s \rangle </math>.
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| (c) Find internal energy <math> E </math>.
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| ==Problem 2: Spin-spin correlation function in the Ising model ==
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| We can gain further insight into the properties of the one-dimensional Ising model of ferromagnetism
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| <math> H= - J \sum_{i=1}^N s_i s_{i+1} </math>
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| by calculating the spin-spin correlation function <math> G(r) </math>
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| <math> G(r) = \langle s_k s_{k+r} \rangle - \langle s_k \rangle \langle s_{k+r} \rangle,
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| where <math> r </math> is the separation between the two spins in units of the lattice constant. The statistical average <math> \langle \ldots \rangle </math> is over all microstates. Because all lattice sites are equivalent, <math> G(r) </math> is independent of the choice of specific site <math> k </math> and depends only on the separation r (for a given temperature T and external field H). Since the average value of spin <math> \langle s_k \rangle </math> at site <math> k </math> is independent of the choice of that site (for periodic boundary conditions) and equals <math> m=M/N </math>, where <math> M <math> is magnetization, the correlation function can also be written as:
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| <math> G(r) = \langle s_k s_{k+r} \rangle - m^2 </math>.
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| The spin-spin correlation function <math> G(r) </math> is a measure of the degree to which a spin at one site is correlated with a spin at another site. If the spins are not correlated, then trivially <math> G(r) = 0 </math>. At high temperatures the interaction between spins is unimportant, and hence the spins are randomly oriented in the absence of an external magnetic field. Thus in the limit <math> k_B T \gg J </math>, we expect that <math> G(r) → 0 for any r. For fixed T and H, we expect that, if spin k is up, then the two adjacent
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| spins will have a greater probability of being up than down. For spins further away from spin k,
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| we expect that the probability that spin k + r is up or correlated will decrease. Hence, we expect
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| that G(r) → 0 as r → ∞. The magnetic susceptibility is proportional to <math> G(r=0) </math>, <math> \chi \propto G(r=0) = \langle m^2
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| \rangle - \langle m \rangle^2
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| (a) Consider an Ising chain of <math> N = 3 </math> spins with free boundary conditions in equilibrium with a heat bath at temperature
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| <math> T </math> and in zero magnetic field. ''Enumerate'' all <math> 2^3 </math> microstates and calculate <math> G(r = 1) </math> and
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| <math> G(r = 2) </math> for k = 1, the first spin on the left.
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| ==Problem 3: Mean-field theory of the Heisenberg model of ferromagnetism ==
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