Phase transitions in models with continuous symmetry

We will look at two systems where the spins are allowed to have continuous values

The $XY$ model, where the spins lie on a plane. The Hamiltonian is

$$ \begin{aligned} \mathcal H {XY}&=-J \sum{\braket{i,j}}(s_i ^x s_j^x+s^y_i s^y_j)-h_x \sum_i s_i^x -h_y \sum_i s^y_i \\ &=-J \sum_{\braket{i,j}} \vec s_i \cdot \vec s_j - \sum_i \vec h \cdot \vec s_i

\end{aligned} $$

where $s_i^x$ and $s_i^y$ are the two components of a two-dimensional unit vector $\vec s_i=(\cos\theta_i,\sin\theta_i)$

The scalar products are in the $x$-$y$ plane orthogonal to $\hat z$
The Heisenberg model, it is in $\rm 3D$ however the spins are isotopically coupled (all components have the same coupling), we can write the Hamiltonian here as

$$ \begin{aligned} \mathcal H &=-J \sum_{\braket{i,j}}(s_i ^x s_j^x+s_i ^y s_j^y+s_i^z s_j^z)-h_x \sum_i s_i^x -h_y \sum_i s_i^y -h_z \sum_i s_i^z \\ &=-J \sum_{\braket{i,j}}\vec s_i \cdot \vec s_j - \sum_i \vec h \cdot \vec s_i \end{aligned} $$

Landau theory of the $XY$ model

🧽 Assume: Magnetisation is the same everywhere

Therefore we introduce a $\rm 2D$ vector $\vec m$ that represents the average magnetisation per spin

It is a unit vector with components $\vec m=(m_x,m_y)$

The symmetry of the $XY$ model: a continuous one

We start by rewriting our Hamiltonian as

$$ \begin{aligned} \mathcal H {XY}=-J \sum{\braket{i,j}} \vec s_i \cdot \vec s_j =-J \sum_{\braket{i,j}}\cos(\theta_i-\theta_j)

\end{aligned} $$

Where we wrote $\vec s_{i,j}=(\cos \theta_{i,j},\sin \theta_{i,j})$
Lets now rotate all the spins by an angle $\phi$ then $\theta_{i,j}\to \theta_{i,j}+\phi$

$$ \theta_i-\theta_j\to (\theta_i+\phi)-(\theta_j+\phi)=\theta_i-\theta_j $$

💎 Conclusion: the microscopic Hamiltonian is invariant under arbitrary in plane rotations of the spin, thus spins can orders ferromagnetically in any directions

All directions of magnetisation are equally probable, so we can detect phase transitions using $|\vec m|$

The effective free energy

🧽 Assume: $|\vec m|\ll 1$ so that we can Taylor expand and use the Ising model

Since only the modulus of $\vec m$ is relevant all odd powers violate the symmetry

When $T<T_C$ the system develops a finite magnetisation of modulus $m$ we write

$$ \tilde f(T,m)=\tilde f_0 (T)+\frac 12 a_2 (T)m^2 +\frac 14 a_4 (T) m^4 $$

The function in $\rm 3D$ looks as follows

When the system goes to the ring of minima we have a spontaneous symmetry breaking

In the previous there were 2 minima with an energy barrier between the two now that its a ring so moving from one minima to another doesn't take any energy

🗒️ Note: this can lead to chaos

🗒️ Note: the chaotic phenomenon is a feature of all models which have a continuous symmetry that remain unbroken during a phase transition (here the radial symmetry)

The dynamics of magnetisation

We define “spin waves” can be obtained by slightly tilting neighbouring spins away from perfect alignment (think of hand waves in stadiums). In this case the total magnetisation will be reduced with respect to the perfectly ordered phase

Similar to phonons, if there are few excitations order remains however if they persist it can turn chaotic and the system would have no global magnetisation anymore.

Representation of a spin wave

From the picture we see that at zero temperature all spin point n the same direction but when there is a finite temperature due to the effect of all the magnetic dipole, the spin remain constant along the magnetic field direction but the orthogonal component performs circular motion

Spin wave: one spin process → makes other spins process with a slight delay → process repeats creating a wave

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We can plot the frequency desperation of spin waves of the following