Finite-Size Thermodynamics of the Two-Dimensional Dipolar $Q$-Clock Model

We present a fully controlled thermodynamic study of the two-dimensional dipolar $Q$-state clock model on small square lattices with free boundaries, combining exhaustive state enumeration with noise-free evaluation of canonical observables. We resolve the complete energy spectra and degeneracies $\{E_n,c_n\}$ for the Ising case ($Q=2$) on lattices of size $L=3,4,5$, and for clock symmetries $Q=4,6,8$ on a $3\times3$ lattice, tracking how the competition between exchange and long-range dipolar interactions reorganizes the low-energy manifold as the ratio $\alpha = D/J$ is varied. Beyond a finite-size characterization, we identify several qualitatively new thermodynamic signatures induced solely by dipolar anisotropy. First, we demonstrate that ground-state level crossings generated by long-range interactions appear as exact zeros of the specific heat in the limit $C(T \rightarrow 0,\alpha)$, establishing an unambiguous correspondence between microscopic spectral rearrangements and macroscopic caloric response. Second, we show that the shape of the associated Schottky-like anomalies encodes detailed information about the degeneracy structure of the competing low-energy states: odd lattices ($L = 3,5$) display strongly asymmetric peaks due to unbalanced multiplicities, whereas the even lattice ($L = 4$) exhibits three critical values of $\alpha$ accompanied by nearly symmetric anomalies, reflecting paired degeneracies and revealing lattice parity as a key organizing principle. Third, we uncover a symmetry-driven crossover with increasing $Q$: while the $Q=2$ and $Q=4$ models retain sharp dipolar-induced critical points and pronounced low-temperature structure, for $Q \ge 6$ the energy landscape becomes sufficiently smooth to suppress ground-state crossings altogether, yielding purely thermal specific-heat maxima. Altogether, our results provide a unified, size- and symmetry-resolved picture of how long-range anisotropy, lattice parity, and discrete rotational symmetry shape the thermodynamics of mesoscopic magnetic systems. We show that dipolar interactions alone are sufficient to generate nontrivial critical-like caloric behavior in clusters as small as $3\times3$, establishing exact finite-size benchmarks directly relevant for van der Waals nanomagnets, artificial spin-ice arrays, and dipolar-coupled nanomagnetic structures.