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 (Formula presented.) for the Ising case ((Formula presented.)) on lattices of size (Formula presented.), and for clock symmetries (Formula presented.) on a (Formula presented.) lattice, tracking how the competition between exchange and long-range dipolar interactions reorganizes the low-energy manifold as the ratio (Formula presented.) 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 (Formula presented.), 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 ((Formula presented.)) display strongly asymmetric peaks due to unbalanced multiplicities, whereas the even lattice ((Formula presented.)) exhibits three critical values of (Formula presented.) 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 (Formula presented.) and (Formula presented.) models retain sharp dipolar-induced critical points and pronounced low-temperature structure, for (Formula presented.), 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 (Formula presented.), establishing exact finite-size benchmarks directly relevant for van der Waals nanomagnets, artificial spin-ice arrays, and dipolar-coupled nanomagnetic structures.