Midterm 2 Solutions

Problem 1

Consider a magnetic system described by an Ising model $(S_{i}=\pm 1)$ on a square lattice with $\ H=\sum _{\langle ij\rangle }J_{ij}s_{i}s_{j}+\sum _{\langle ij\rangle }K_{ij}s_{i}s_{j}$. Here, the ferromagnetic interaction $\ J_{ij}<0$ couples only nearest-neighbor spins. In contrast,the antiferromagnetic interaction $\ K_{ij}>0$ couples only the the second neighbors, i.e. spins that are "across the square" ( diagonally placed relative to each other).

a)

Sketch the suare lattice and determine the number of first neighbors $\mathbb {Z} _{FM}$ which interact ferromagnetically with the given spin, and the number of second neighbors $\mathbb {Z} _{AFM}$ which interact antiferromagnetically with the given spin. (5points)

Image of square lattice with ferromagnetic $J_{i,j}<0$ coupling nearest neighbor spins. 
Anti-ferromagnetic $K_{i,j}<0$ coupling only second nearest neighbors. 
$Z_{FM}=4$
$Z_{AFM}=4$

b)

Use the mean-field approximation, where $\ S_{i}S_{j}\rightarrow S_{i}\langle S_{j}\rangle$. Assume that the magnitude of $\ J_{ij}$ is appreciably larger then the magnitude of $\ K_{ij}$; then the system will order ferromagnetically at low temperatures, so that $\langle S_{i}\rangle =m$ for any site. Derive the self consistency condition determining the magnetization "m". (15points)

The point of the mean-field approximation is to simplify a many body problem. To consider all the interactions taking place in a material would require huge amounts of computing power and resources. The mean-field approximation accomplishes this by considering only the average of the interactions.

The first step in this problem is to right down our equation for the energy when $\ J_{ij}$ is "appreciably larger than $\ K_{ij}$ .

$\ H=\sum _{ij}J_{ij}S_{i}S_{j}$

We also know:

$\langle S_{i}\rangle =m$

This is due to the fact that we are considering many neighbors, so as z gets very large then we get the following:

$\ 1/z\sum _{j=1}^{z}S_{j}\rightarrow \langle S_{i}\rangle =m$

Now if we sum over many random variables we get the following:

$\sum _{j=1}^{z}S_{j}\rightarrow z\langle S_{i}\rangle$

Now we can simplify our original expression as the following:

$\ H=J\sum _{i}^{N}S_{i}zm$

Pulling out the constants we get:

$\ H=Jzm\sum _{i=1}^{N}S_{i}$

Since we have a problem of non interacting spins in an internal field we can consider and solve for only one spin. This then gives us the following:

$\ H=JzmS_{1}$

$\ S_{1}=\pm 1$

Now we need to consider the probability for the spin to point either up or down. This is given as the following:

$\ P(S_{1}=\pm 1)=1/z(e^{\pm Jzm/kt})$

We can also define z as:

$\ z=e^{\pm Jzm/kt}$

Now we can define the magnetization as the spin times the probability of the spin:

$\ m=\sum _{S_{1}=\pm 1}S_{1}P(S_{1})$

Rewriting this equation we get:

$\ m={\frac {e^{Jzm/kt}-e^{-Jzm/kt}}{e^{Jzm/kt}+e^{-Jzm/kt}}}$

This is the same as the tanh, so we get:

$\ m=\tanh({\frac {Jzm}{kt}})$

Making $\ \beta =1/kt$

We get:

$\ m=\tanh(\beta Jzm)$

Problem 2

Consider a system of electrons moving in two dimensions.

a)

Give some example, and sketch a device where this can be experimentally realized. Where do we find applications of such devices?(5 points)

b)

Electrons interact through electrostatic Coulomb interactions, but there exists a density regime where the interactions can be ignored. Does this happen at low density or at high density? Give a physical argument supporting your statement. (5 points)

Since Coulomb interactions are long range by $e^{2}/r$ high densities cause the formation of solids such as crystals.

c)

Now assume that we work in the regime where the interactions can be ignored, so that we can approximate the electrons as a gas of free particles obeying Fermi Dirac statistics(Pauli exclusion principle). The solutions of the Schroedinger equation for free particles are plain waves with wave- vector ${\vec {k}}$. Assuming that this system of electrons is contained in a finite area container of side L, use periodic boundary conditions to determine the allowed values for the wave-vector.(10 points)

When we put any wave into a cavity such as a guitar the allowed wave vectors become quantized. We also know that the Schroedinger equation $\ E(k)={\frac {\hbar ^{2}k^{2}}{2m}}$ has discrete allowed values which leads to energy levels. Th allowed values of the wave-vector then become:

$\Delta k={\frac {2\pi }{L}}$

d)

Determine the Fermi energy of this system as a function of the density of electrons n.(10 points)

f) Where $T\rightarrow \infty$ Fermi-Dirac Statistics and all quantum effects can be ignored.