Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 303 / 1 / b

Write ${\mathbf{\sigma }}_i=\mathbf{m}+\delta {\mathbf{\sigma }}_i$ and neglect products of two fluctuations. ThenEach site has $q$ neighbours and each bond is counted once, so the mean-field approximation givesChoose the $x$ axis along $\mathbf{m}$. For the four-state clock model, the single-site partition function isConsequentlyand therefore

Differentiating the mean-field free energy and imposing stationarity givesThe hyperbolic-function identity $\sinh x/(1+\cosh x)=\tanh (x/2)$ turns this into the self-consistency equation

The Taylor series at $x=0$ isSubstituting $x=Jqm/T$ givesThe quartic coefficient is positive, while the quadratic coefficient changes sign atThis is therefore a continuous mean-field phase transition.

For finite $p=4$, the clock model has a discrete symmetry. Domain walls have finite energy per unit boundary area, so thermal disorder destroys long-range order in one dimension but a finite-temperature ordered phase can exist in two dimensions. Its lower critical dimension is therefore $\boxed{d_{\mathrm{l}}=1}$.

As $p\to \infty$, the permitted angles become continuous and the model becomes the XY model with $O(2)$ symmetry. The Mermin-Wagner theorem forbids spontaneous long-range order at positive temperature in two dimensions, so the lower critical dimension for conventional symmetry breaking is $\boxed{d_{\mathrm{l}}=2}$. The two-dimensional model can nevertheless undergo a Berezinskii–Kosterlitz–Thouless transition between algebraic and exponential correlation decay.