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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 </math>
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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 specific site (for periodic boundary conditions) and equals <math> m=M/N </math> (<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) \rightarrow 0 </math> for any r. For fixed <math> T </math> and <math> H </math>, we expect that, if spin <math> k </math> is up, then the two adjacent spins will have a greater probability of being up than down. For spins further away from spin <math> k </math>, we expect that the probability that spin at site <math> k + r </math> is up or correlated will decrease. Hence, we expect that <math> G(r) \rightarrow 0 </math> as <math> r \rightarrow \infty </math>. Note that the physical meaning of the correlation is that it can be used to express magnetic susceptibility <math> \chi \propto G(r=0) = \langle m^2
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| \rangle - \langle m \rangle^2 </math> as one of the response functions.
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| (a) Consider an Ising chain of <math> N = 3 </math> spins with free boundary conditions which is in equilibrium with a heat bath at temperature <math> T </math> and in zero magnetic field <math> H=0 </math>. ''Enumerate'' all <math> 2^3 </math> microstates and calculate <math> G(r = 1) </math> and <math> G(r = 2) </math> for k = 1, the first spin on the left.
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| (b) For one-dimensional chain of <math> N </math> Ising spins and with free boundary conditions, show that <math> G(r)=(\tanh \beta J)^r </math>. HINT: Compute first:
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| \langle s_k s_{k+r} \rangle = \frac{1}{Z_N} \sum_{s_1=\pm 1} \sum_{s_N=\pm 1} s_k s_{k+r} \exp [ \sum_{i=1}^{N-1} \beta J s_i s_i+1}
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| where the partition function for N spins is
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| (c) By writing <math> G(r) = e^-{r/\xi} </math> for <math> r \gg 1 </math>, extract the correlation length from your result in (b):
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| \xi = - \frac{1}{\ln(\tanh \beta J)}
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| in the low temperature limit <math> \beta J \gg 1 </math>.
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| ==Problem 3: Mean-field theory of the Heisenberg model of ferromagnetism ==
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