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 one-dimensional Ising model of ferromagnetism

$H=-J\sum _{i=1}^{N}s_{i}s_{i+1}$

by calculating the spin-spin correlation function $G(r)$

$G(r)=\langle s_{k}s_{k+r}\rangle -\langle s_{k}\rangle \langle s_{k+r}\rangle$

where $r$ is the separation between the two spins in units of the lattice constant. The statistical average $\langle \ldots \rangle$ is over all microstates. Because all lattice sites are equivalent, $G(r)$ is independent of the choice of specific site $k$ and depends only on the separation r (for a given temperature T and external field H). Since the average value of spin $\langle s_{k}\rangle$ at site $k$ is independent of the choice of that specific site (for periodic boundary conditions) and equals $m=M/N$ ($M$ is magnetization), the correlation function can also be written as:

$G(r)=\langle s_{k}s_{k+r}\rangle -m^{2}$.

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 trivially $G(r)=0$. 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 $k_{B}T\gg J$, we expect that $G(r)\rightarrow 0$ 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 at site $k+r$ is up or correlated will decrease. Hence, we expect that $G(r)\rightarrow 0$ as $r\rightarrow \infty$. Note that the physical meaning of the correlation is that it can be used to express magnetic susceptibility $\chi \propto G(r=0)=\langle m^{2}\rangle -\langle m\rangle ^{2}asoneoftheresponsefunctions.(a)ConsideranIsingchainof<math>N=3$ 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.