Chapter 20, Spontaneous Magnetic Order: Ferro-, Antiferro-, and Ferri-Magnetism Video Solutions, The Oxford Solid State Basics

Problem 1

Ferromagnetic vs Antiferromagnetic States Consider the Heisenberg Hamiltonian
$$
\mathcal{H}=-\frac{1}{2} \sum_{\langle i, j\rangle} J \mathbf{S}_{i} \cdot \mathbf{S}_{j}+\sum_{i} g \mu_{B} \mathbf{B} \cdot \mathbf{S}_{i}
$$
and for this exercise set $\mathbf{B}=0 .$
(a) For $J>0$, i.e., for the case of a ferromagnet, intuition tells us that the ground state of this Hamiltonian should simply have all spins aligned. Consider such a state. Show that this is an eigenstate of the Hamiltonian Eq. $20.6$ and find its energy.
(b) For $J<0$, the case of an antiferromagnet on a cubic lattice, one might expect that (at least for $\mathbf{B}=0$ ) the state where spins on alternating sites point in opposite directions might be an eigenstate. Unfortunately, this is not precisely true. Consider such a state of the system. Show that the state in question is not an eigenstate of the Hamiltonian.
Although the intuition of alternating spins on alternating sites is not perfect, it becomes reasonable for systems with large spins $S$. For smaller spins (like spin $1 / 2$ ) one needs to consider so-called "quantum fluctuations" (which is much more advanced, so we will not do that here).

Jack Hou

Numerade Educator

01:04

Problem 2

Frustration
Consider the Heisenberg Hamiltonian as in Exercise $20.1$ with $J<0$, and treat the spins as classical vectors.
(a) If the system consists of only three spins arranged in a triangle (as in Fig. 20.2), show that the ground state has each spin oriented $120^{\circ}$ from its neighbor.
(b) For an infinite triangular lattice, what does the ground state look like?

Chai Santi

Numerade Educator

46:10

Problem 3

Spin Waves*
For the spin- $S$ ferromagnet particularly for large $S$, our "classical" intuition is fairly good and we can use simple approximations to examine the excitation spectrum above the ground state.
First recall the Heisenberg equations of motion for any operator
$$
i \hbar \frac{d \hat{O}}{d t}=[\hat{O}, \mathcal{H}]
$$
with $\mathcal{H}$ the Hamiltonian (Eq. $20.6$ with $\mathrm{S}_{1}$ being a spin $S$ operator).
(a) Derive equations of motion for the spins in the Hamiltonian Eq. $20.6$. Show that one obtains
$$
\hbar \frac{d \mathbf{S}_{\mathrm{I}}}{d t}=\mathbf{S}_{\mathrm{i}} \times\left(J \sum_{j} \mathbf{S}_{\mathrm{J}}-g \mu_{\mathrm{b}} \mathbf{B}\right)
$$
where the sum is over sites $j$ that neighbor $i$. In the ferromagnetic case, particularly if $S$ is large, can treat the spins as not being operators, but her as being classical variables. In the ground we can set all $\mathbf{S}_{1}=\hat S$ (Assuming $\mathbf{B}$ is in the direction). Then to consider excited states,
In the ferromagnetic case, particularly if $S$ is large, we can treat the spins as not being operators, but rather as being classical variables. In the ground state, we can set all $\mathbf{S}_{1}=\hat{z} S$ (Assuming $\mathbf{B}$ is in the $-z$ direction so the ground state has spins aligned in the $\hat{z}$ direction). Then to consider excited states, we can perturb around this solution by writing
$$
\begin{aligned}
S_{i}^{z} &=S-\mathcal{O}\left((\delta S)^{2} / S\right) \\
S_{i}^{z} &=\delta S_{i}^{x} \\
S_{i}^{y} &=\delta S_{i}^{y}
\end{aligned}
$$
where we can assume $\delta S^{x}$ and $\delta S^{y}$ are small compared to $S$. Expand the equations of motion (Eq. 20.7) for small perturbation to obtain equations of motion that are linear in $\delta S_{x}$ and $\delta S_{y}$
(b) Further assume wavelike solutions
$$
\begin{aligned}
&\delta S_{i}^{x}=A_{x} e^{i \omega t-i k \cdot r} \\
&\delta S_{i}^{y}=A_{y} e^{i \omega t-i k \cdot r}
\end{aligned}
$$
This ansatz should look very familiar from our prior consideration of phonons. Plug this form into your derived equations of motion.
$D$ Show that $S_{i}^{x}$ and $S_{i}^{y}$ are out of phase by $\pi / 2$.
What does this mean?
$D$ Show that the dispersion curve for "spin-waves" of a ferromagnet is given by $\hbar \omega=|F(\mathbf{k})|$ where
$$
\begin{aligned}
&F(\mathbf{k})=g \mu_{b}|B| \\
&+J S\left(6-2\left[\cos \left(k_{x} a\right)+\cos \left(k_{y} a\right)+\cos \left(k_{\pm} a\right)\right]\right)
\end{aligned}
$$
where we assume a cubic lattice.
$\triangleright$ How might these spin waves be detected in an experiment?
(c) Assume the external magnetic field is zero. Given the spectrum you just derived, show that the specific heat due to spin wave excitations is proportional to $T^{3 / 2}$.

Jack Hou

Numerade Educator

03:54

Problem 4

Small Heisenberg Models
(a) Consider a Heisenberg model containing a chain of only two spins, so that
$$
\mathcal{H}=-J \mathbf{S}_{1} \cdot \mathbf{S}_{2}
$$
Supposing these spins have $S=1 / 2$, calculate the energy spectrum of this system. Hint: Write $2 \mathbf{S}_{1} \cdot \mathbf{S}_{2}=\left(\mathbf{S}_{1}+\mathbf{S}_{2}\right)^{2}-\mathbf{S}_{1}^{2}-\mathbf{S}_{1}{ }^{2}$
(b) Now consider three spins forming a triangle (as shown in Fig. 20.2). Again assuming these spins are $S=1 / 2$, calculate the spectrum of the system. Hint: Use the same trick as in part (a)!
(c) Now consider four spins forming a tetrahedron.
Again assuming these spins are $S=1 / 2$, calculate the spectrum of the system.

Lazarus Arnau

Numerade Educator

01:09

Problem 5

One-Dimensional Ising Model with $B=0$
(a) Consider the one-dimensional Ising model with spin $S=1$. We write the Hamiltonian for a chain of $N$ spins in zero magnetic field as
$$
\mathcal{H}=-J \sum_{i=1}^{N-1} \sigma_{i} \sigma_{i+1}
$$
where each $\sigma_{i}$ takes the value $\pm 1$. The partition function can be written as
$$
Z=\sum_{\sigma_{1}, \alpha_{2}, \ldots a_{N}} e^{-\beta H}
$$
Using the transformation $R_{i}=\sigma_{i} \sigma_{i+1}$ rewrite the partition function as a sum over the $R$ variables, and hence evaluate the partition function.

D.Show that the free energy has no cusp or discontinuity at any temperature, and hence conclude that there is no phase transition in the onedimensional Ising model. (b) *At a given temperature $T$, calculate an expression for the probability that $M$ consecutive spins will be pointing in the same direction. How does this probability decay with $M$ for large $M ?$ What happens as $T$ becomes small? You may assume $N \gg M .$

Keshav Singh

Numerade Educator

46:10

Problem 6

One-Dimensional Ising Model with $B \neq 0 *$ Consider the one-dimensional Ising model with spin $S=1$. We write the Hamiltonian (Eq. 20.5) for a chain of $N$ spins in magnetic field $B$ as
$$
\mathcal{H}=\sum_{i=1}^{N} \mathcal{H}_{i}
$$
where
$$
\begin{aligned}
\mathcal{H}_{1} &=h \sigma_{1} \\
\mathcal{H}_{i} &=-J \sigma_{i} \sigma_{i-1}+h \sigma_{i} \quad \text { for } i>1
\end{aligned}
$$
where each $\sigma_{i}$ takes the value $\pm 1$ and we have defined $h=g \mu_{B} B$ for simplicity of notation.
Let us define a partial partition function for the first $M$ spins (the first $M$ terms in the Hamiltonian sum Eq. 20.8) given that the $M^{\text {th }}$ spin is in a particular state. I.e.,
$$
Z\left(M, \sigma_{M}\right)=\sum_{\sigma_{1}, \ldots, \sigma_{M-1}} e^{-\beta \sum_{i=1}^{M} H_{i}}
$$ so that the full partition function is $Z=$ $Z(N,+1)+Z(N,-1)$.
(a) Show that these partial partition functions satisfy a recursion relation
$$
Z\left(M, \sigma_{M}\right)=\sum_{\sigma_{M-1}} T_{\sigma_{M}, \sigma_{M-1}} Z\left(M-1, \sigma_{M-1}\right)
$$
where $T$ is a 2 by 2 matrix, and find the matrix $T$. ( $T$ is known as a "transfer matrix").
(b) Write the full partition function in terms of the matrix $T$ raised to the $(N-1)^{\text {th }}$ power.
(c) Show that the free energy per spin, in the large $N$ limit, can be written as
$$
F / N \approx-k_{B} T \log \lambda_{+}
$$
where $\lambda_{+}$is the larger of the two eigenvalues of the matrix $T$.
(d) From this free energy, derive the magnetization, and show that the susceptibility per spin is given by
$$
\chi \propto \beta e^{2 \beta J}
$$
which matches the Curie form at high $T$.

Jack Hou

Numerade Educator