  The $q-$state clock model, sometimes called the discrete $XY$ model, shows
interesting critical phenomena. While $q=2$ corresponds to the Ising model, the
$q\to\infty$ limit corresponds to the well-known $XY$ model. It is known that
up to $q=4$, the two-dimensional (2D) clock model exhibits a symmetry-breaking
phase transition. On the other hand, the 2D $XY$ model only shows a topological
(Berezinskii-Kosterlitz-Thouless or BKT) phase transition. Interestingly, the
model with finite $q$ (with $q\ge 5$) is predicted to show two different phase
transitions. There are varying opinions about the actual characters of the
transitions, especially the one at the lower temperature. In this work we
develop mean-field theory (basic and higher order) to study the $q$-state clock
model systematically. Using the mean-field theory, we show that the phase
transition at the higher temperature is of the BKT type. We find that this
transition temperature ($T_{BKT}$) does not depend on the $q$ values, and our
basic (zeroth order) mean-field calculation shows that $T_{BKT} = 2J/k_B$,
where $J$ is the nearest-neighbor exchange constant and $k_B$ is the Boltzmann
constant. Our analysis shows that the other phase transition is a spontaneous
symmetry-breaking (SSB) type. The corresponding transition temperature
($T_{SSB}$) is found to decrease with increasing $q$ value; it is found that
$T_{SSB} \propto 1/q^2$ with a weak logarithmic correction. To better
understand the model, we also perform the first-order mean-field calculations
(here, the interaction between two targeted nearest neighbors is treated
exactly). This calculation gives us $T_{BKT} = 1.895J/k_B$, which is slightly
closer to the reported value of $0.893J/k_B$. The main advantage of this
higher-order mean-field theory is that one can now estimate the spin-spin
correlation, whose change in the properties indicates the phase transition.
