In classical physics, angular momentum arises from the motion of an object around an axis - whether it be a spinning top, a rotating planet, or an orbiting electron in an atom. However, in quantum physics, fundamental particles possess an intrinsic and quantised form of angular momentum called spin. This property plays a crucial role in various physical phenomena, ranging from material properties, such as magnetism, to modern applications, such as quantum computing.
In this problem we will treat spin classically, which will lead to some qualitatively correct results. You will explore the physics of spin systems through spin-spin interactions, evolution under magnetic fields, and statistical physics to understand the emergence of spin waves and phase transitions in magnets.
Useful information:
$\cosh(x)\equiv \frac{e^x+e^{-x}}{2} $, $\sinh(x)\equiv \frac{e^x-e^{-x}}{2}$, $\tanh(x)\equiv \frac{\sinh(x)}{\cosh(x)} \approx x-\frac{1}{3}x^3$ for $|x|\ll1$
The magnetic field due to a magnetic dipole of moment $\vec{\mu}$ at a position $\vec{r}$ away from it is given by equation 1 ($\mu_0$ is the permeability of a vacuum):
$$\vec{B}=\frac{\mu_o}{4\pi} \left(\frac{3(\vec{\mu}\cdot\vec{r})\vec{r}}{r^5}-\frac{\vec{\mu}}{r^3}\right)$$
Consider a ring of radius $R$, total mass $M$, and charge $Q>0$ distributed uniformly. The ring rotates with an angular speed $\omega$ around a perpendicular axis that passes through its center of mass.
The ring is placed in a weak uniform magnetic field $\vec{B}=B\hat{z}$ , making an angle $\theta$ with $\vec{\omega}$, see Figure A.1.
Now we turn off the external magnetic field and place an identical ring at a horizontal distance $d\gg R$ from the original ring such that the magnetic moment of the new ring $\vec{\mu}_2$ makes an angle $\theta$ with $\vec{\mu}_1$, see Figure A.2.
In the following questions we investigate the dynamics of spins. Here we define spin as a particle with intrinsic angular momentum $\vec{S}$, which has an associated magnetic moment $\vec{\mu}$ related to $\vec{S}$ via the gyromagnetic ratio as in Part A.1, $\vec{\mu}=\gamma \vec{S}$.
The magnetic dipoles of two spins interact with each other. However, this interaction is negligible compared to another interaction arising from a quantum mechanical origin, which is not present in classical systems. Interestingly, the energy associated with this quantum interaction has the same form which we found in Part A.3, scaling with $\vec{S}_1\cdot \vec{S}_2$, albeit with the opposite sign.
Now we will look at a very long chain of spins. The positions of the spins are fixed along the $x$-axis, with a distance $a$ separating them, see Figure B.1. We will approximate the total energy of the system by considering the interactions between nearest neighbours only, so that the energy can be written as
$$E=-J \sum_i \vec{S}_i\cdot \vec{S}_{i+1}$$
where $J>0$ is the interaction strength, and $\vec{S}_i$ is the spin angular momentum vector of the $i$th dipole, with magnitude $S$. The spin vectors are free to rotate in three dimensions. Notice that the sign of the energy is different from the last part. This interaction is purely quantum mechanical.
For the rest of Part B, assume that the system is highly magnetised along the $z$ direction, so we can use the approximations $S_{i,z}\approx S$ and $dS_{i,z}/dt\approx0$ for each spin, see Figure B.2. In this regime, the set of equations describing the spin's time evolution is satisfied by a traveling wave solution for $S_{i,x}$and $S_{i,y}$ characterised by a wave-vector $k$ and angular frequency $\omega$.
Find the relationship between $\omega$ and $k$ (known as the dispersion relation, $\omega(k)$) for the spin waves in terms of $J, S$ and $a$.
Hint: express the position of the $i$th spin as $x=a\cdot i$.
The spin wave described above carries energy and momentum. At low energies, the relation between its energy and momentum resembles that of a massive classical particle with an effective mass $m_\text{eff }$, a concept known as a quasi-particle.
Spin waves can be experimentally probed using neutron scattering. Although neutrons have zero net charge, they have a finite spin, allowing them to interact with other spins.
Next we consider the same chain made of $N$ spins from Part B, except the spin vectors are now restricted to point either up or down along the $z$-axis, so that the spin component along $z$ can be written as $S_{i,z} = s_i S$, where $s_i=\pm 1$, see Figure C.1. In addition to the nearest neighbour interactions, we could have an external magnetic field pointing along the $z$-axis so that the total energy of the system is given by
$$E=-\tilde{J}\sum_i s_i s_{i+1} - h \sum_i s_i.$$
We assume $\tilde{J}\geq0$, and $h$ is a constant dependent on the magnetic field. The spin system is at thermal equilibrium with a reservoir at temperature $T$. Ignore the edges of the chain.
In the remaining questions, we turn off the magnetic field, so $h=0$, and set $\tilde{J}>0$.
Instead of considering the interactions between each spin and its neighbours, we assume that each spin sees an average polarisation $\bar{s}$ from its nearest neighbours.
$$E=-\tilde{J}_{\text{eff}} \sum_i s_i$$and express $\tilde{J}_{\text{eff}}$ in terms of $\tilde{J}$ and $\bar{s}$.