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 $⟨s⟩$.

(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 $G(r)$

$G(r)=⟨s_k{s}_{k+r}⟩-⟨s_k⟩⟨s_{k+r}⟩$

where $r$ is the separation between the two spins in units of the lattice constant. The statistical average $⟨\dots ⟩$ 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 $⟨s_k⟩$ at site $k$ is independent of the choice of that site (for periodic boundary conditions) and equals $m=M/N$, where $M<math>ismagnetization,thecorrelationfunctioncanalsobewrittenas:<math>G(r)=⟨s_k{s}_{k+r}⟩-m^2$.

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 Failed to parse (syntax error): {\displaystyle 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 <math> G(r=0) } , $\chi \propto G(r=0)=⟨m^2⟩-⟨m{⟩}^2(a)ConsideranIsingchainof<math>N=3$ 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.