We can see here (

Figure 1, right) the basic features of entropy for the low- and high-temperature limits. When no external field is applied, the energy ground state of the ferromagnetic

q-state clock model has a degeneracy equal to

q, independent of the total number of spins

N; therefore,

$S(0)=\mathrm{ln}q$. On the other hand, at very high temperatures, the exchange interaction is overridden, and every spin has

q degrees of freedom with equal probabilities: hence, the system degeneracy is equal to

${q}^{N}$, and, therefore,

Figure 2 shows the same observables in the presence of magnetic field

B, namely,

$U(T,B)$,

$C(T,B)$, and

$S(T,B)$, for the case where

$q=7$, as an example. The shift of the transition temperatures due to the variations in the magnetic field is clearly visible. This is also an exact analytical result obtained by calculating straightforwardly the partition function of Equation (

3). The magnetic field was applied along the

$(1,0)$ direction, along which a spin orientation is always possible, regardless of the

q value.

The magnetic field breaks the degeneracy of the ground state so that

$S(0)=0$ for

$B\ne 0$ instead of

$S(0)=\mathrm{ln}q$. However, at high temperatures, the Zeeman term in Equation (

2) is overridden, and we are back to the same situation as in previous analysis for

$B=0$, namely, every spin has

q degrees of freedom with equal probabilities: hence, the system degeneracy is equal to

${q}^{N}$, and, therefore, the entropy

$S(T>>J)=N\mathrm{ln}q$, as observed in the right of

Figure 2.

#### 3.1. Monte Carlo Simulations

Next, we present and discuss the output from Monte Carlo simulations made for the

q-state clock model in lattices of up to

$64\times 64$ with free boundary conditions. We began by simulating a

$3\times 3$ lattice for different

q values and comparing these numerical results to the analytic ones presented in the previous section. We did not find a single difference, which is expected of course, but which also serves as a check for the computer programs used extensively in the simulations reported next. Thus, thermodynamic observables for lattices

$10\times 10$,

$16\times 16$,

$32\times 32$, and

$64\times 64$ are presented in

Figure 3,

Figure 4,

Figure 5 and

Figure 6, respectively.

We observe an overall self-agreement in the shape of energy

U, specific heat

C, and entropy

S as functions of temperature for the different lattices. For a given size, the peak of the specific heat occurs at a lower temperature as

q increases, and then it splits in two peaks. As

q continues to increase, the high-temperature peak remains at the same temperature, whereas the low-temperature peak tends toward lower temperatures (eventually to a temperature of zero) as

q increases tending to infinity. This low-temperature peak,

${T}_{1}$, is the transition from the FM phase to the BKT-like phase, which is characterized by vortex spin configurations and FM spin–spin correlated configurations, like waves. Both the vortex and spin waves are low-energy excitation that occurs as

q increases. Therefore, for

$q\to \infty $, we expect that

${T}_{1}\to 0$. We would like to stress the size-dependence of the peak heights at fixed

q. Indeed, for

$q=2,3$, and 4, the unique peak corresponds to a second-order phase transition, hence, the specific heat

$C/N$ is expected to diverge with

N at

$T={T}_{1}$. However, for

$q\ge 5$, the two peaks should correspond to two BKT transitions, which are infinite-order; that is, there is no divergence in

$C/N$ at

${T}_{1}$ and

${T}_{2}$. Both features are observed in

Figure 3,

Figure 4,

Figure 5 and

Figure 6, and, furthermore, the constant height of the peak at

$T={T}_{2}$ for

$q=5$ could represent further evidence of a BKT-like transition. To be more specific on the characterization of the clock model, we discuss next a phase diagram including the three possible magnetic phases: FM (long-range order), BKT (short-range correlations), and PM (total disorder).

#### 3.2. Phase Diagram

In this section, we show the phase diagram for the clock model as extracted from the specific heat. We collected the analytic results for the

$3\times 3$ lattice and the numerical results for the

$10\times 10$ and the

$64\times 64$ lattices. This is shown in

Figure 7 for

q ranging from 2 to 20 using squares, triangles, and circles, respectively. The texture and color underneath illustrate the instantaneous magnetic phases for the larger lattice and

$q=9$ just as examples for the kind of magnetic ordering present in each phase.

Several features in

Figure 7 deserve special attention. First, the lower critical temperature

${T}_{1}$ follows a monotonous decrease with

q approaching zero asymptotically. Second, the higher critical temperature

${T}_{2}$ remains at a constant value for

$q\ge 6$. Third, for

$q=5$, there is just one critical temperature following the tendency of

${T}_{1}$, but for

$L=10$ (and also for

$L=64$), the two transitions are clearly visible for

$q=5$. Fourth, an order parameter beyond the usual magnetization is necessary to distinguish the BKT phase from the PM disordered phase.

To further specify the different phases,

Figure 8 shows the specific heat for a

$32\times 32$ lattice as a function of temperature. In this figure, we show a two-dimensional (2D) order parameter at certain characteristic temperatures which clearly discriminate the three (FM, BKT, and PM) different phases. The order parameter we use is the 2D distribution of the variable

$M=({M}_{x},{M}_{y})$, as defined in Equation (

7), that is, the spin-lattice average at time

t after the thermalization process of

$\tau $ MCSs.

This corresponds to the vector spin average for a given spin configuration at time

t. This vector is then calculated after the thermalization, and every 20 MCSs of a total of

${N}_{c}$ times. By plotting all

${N}_{c}$ vectors, we generate a 2D distribution that clearly characterizes the different phases. In the FM phase, only certain directions of the spin are allowed. In

Figure 8, constructed for

$q=9$, we see that at low temperature, the average magnetization vector points in nine directions which correspond to the nine-fold symmetry of the FM states with equal probability. The BKT-like phase is characterized by spin waves and vortex structures; therefore, the lattice average of the spin points to any of the

$2\pi $ directions while conserving, to a great extent, its magnitude in every lattice average. Therefore, a ring structure is formed. In the disordered (PM) phase, every spin in the lattice points randomly to any direction; therefore, the lattice average magnetization distribution exhibits a 2D-Gaussian peak with decreasing magnitude as

T increases; hence, the circle begins to be filled with a higher probability (red color) near the center. The color code goes from black (zero value) to purple, blue, green, yellow, and red in increasing order of probability for this 2D order parameter. This parameter was already introduced as a complex order parameter by Baek et al. [

31].

The next figure (

Figure 9) depicts snapshots of some spin configurations showing the spin arrangements that occur at different temperatures.

The next figure (

Figure 10) shows the magnetization modulus as the thermal average of the spin-lattice average, as defined in Equation (

8).

Let us now consider the information content as an independent test to characterize these phase transitions. We shall concentrate on the simulations for $32\times 32$ lattices, measuring $\mu $ and $\delta $ as defined in the Models and Methods Section.

Figure 11 presents the information content results for the same energy series generated by the previously described MC algorithm; the case of a

$32\times 32$ lattice for

$q=2$ was chosen for this report. The open-squares curve (red) gives the results for mutability, and it presents a local maximum near

$T=2.2$, in agreement with the specific heat results (see

Figure 5). The solid-circles curve (blue) is the result for diversity and shows a sharper absolute maximum at

$T=2.2$. The vertical dashed line illustrates the temperature 2.269, at which the Onsager solution predicts the transition for an extremely large system [

32]. This is just a reference since our systems are finite and have free boundary conditions, so they do not correspond well with this theoretical solution.

The previous results were obtained from sequences presenting energy data. However, there are more sensitive parameters associated with states of the system whose sequences present alternatives which can recognize this phase transition in a better way. These are the cases of the magnetic order parameters (magnetization, neighbor correlations, site-memory correlation) [

25] which we initially tested and obtained encouraging results. To consider this additional study is beyond the goals of the present paper, so we will stick to the energy data results analyzed for diversity to directly compare with the results reported above.

In

Figure 12, we present the results for

$\delta $ in the cases for

$q=3$,

$q=4$,

$q=6$, and

$q=8$. The ordinates were multiplied by the appropriate constants to fit a common arbitrary scale since the meaningful information is in the temperatures that are marked by each maximum or the inflections of the curves. For

$q=3$, just one critical temperature is visible and in agreement with the maximum of the specific heat for this lattice size, as shown in

Figure 5. A similar situation is observed for

$q=4$, with a maximum at a lower temperature of around 1.1. For

$q=6$, a maximum of around 1.1 and a “knee” just over 0.5 are visible, in agreement with the two maxima for this value of

q reported in the figure for specific heat. In the case of

$q=8$, the maximum near 1.1 is clearly present, although it is a bit broader than that for

$q=6$. The “knee” is a barely visible tiny change in slope at

$T=0.5$.

The results shown by the previous two figures, as well as similar ones for other intermediate values of q, show that the information method is able to recognize the transitions present in the clock model for low values of q when applied to the energy data vectors produced by the MC simulations. However, the phases are less detectable by this method as q increases.

The previous shortcomings of the information theory method could be due to the system itself and the way in which the transition is characterized. With respect to the former, it should be noticed that as q increases, the number of possible configurations increases, the energy intervals decrease, and the density of states is highly degenerate and tending to a continuum as $q\to \infty $; to distinguish among energy values is now increasingly difficult. With respect to the latter, energy is not the best order parameter to characterize these transitions, and, eventually, different order parameters more oriented to the recognition of magnetic states of the system (magnetization, site memory, neighbor correlations) can render better results. At the moment, this is an open question, and it should be addressed in future work.