Temporary HW

Problem 1: Three Ising spins

Assume three spins $s_{1},s_{2},s_{3}$ 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 $H$ is give by:

$H=-J(s_{1}s_{2}+s_{2}s_{3}+s_{3}s_{1})-F(s_{1}+s_{2}+s_{3})$

(a) Find canonical partition function for this model.

(b) Find average spin $\langle s\rangle$.

(c) Find internal energy $E$.

Problem 2: Spin-spin correlation function in the Ising model

We can gain further insight into the properties of the Ising model by calculating the spin-spin correlation function $G(r)$

where $r$ is the separation between the two spins in units of the lattice constant. The average is over all microstates. Because all lattice sites are equivalent, G(r) is independent of the choice of k and depends only on the separation r (for a given T and H).

The spin-spin correlation function G(r) 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 G(r) = 0. 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 kT ≫ J, we expect that G(r) → 0 for any r. For fixed T and H, we expect that, if spin k is up, then the two adjacent spins will have a greater probability of being up than down. For spins further away from spin k, we expect that the probability that spin k + r is up or correlated will decrease. Hence, we expect that G(r) → 0 as r → ∞. The magnetic susceptibility is proportional to $G(r=0)$, $\chi \propto G(r=0)=\langle m^{2}\rangle -\langle m\rangle ^{2}(a)ConsideranIsingchainof<math>N=3$ spins with free boundary conditions in equilibrium with a heat bath at temperature $T$ and in zero magnetic field. Enumerate all $2^{3}$ microstates and calculate $G(r=1)$ and $G(r=2)$ for k = 1, the first spin on the left.