{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,5]],"date-time":"2026-02-05T23:13:20Z","timestamp":1770333200942,"version":"3.49.0"},"reference-count":37,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2026,2,5]],"date-time":"2026-02-05T00:00:00Z","timestamp":1770249600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"ANID Fondecyt","award":["1230055"],"award-info":[{"award-number":["1230055"]}]},{"name":"ANID Fondecyt","award":["1240582"],"award-info":[{"award-number":["1240582"]}]},{"name":"ANID Fondecyt","award":["1250173"],"award-info":[{"award-number":["1250173"]}]},{"name":"Programa de Incentivo a la Iniciaci\u00f3n Cient\u00edfica","award":["009\/2025"],"award-info":[{"award-number":["009\/2025"]}]},{"name":"ANID Becas\/Doctorado Nacional","award":["21252820"],"award-info":[{"award-number":["21252820"]}]},{"name":"CEDENNA CIA","award":["250002"],"award-info":[{"award-number":["250002"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>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 {En,cn} 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\u00d73 lattice, tracking how the competition between exchange and long-range dipolar interactions reorganizes the low-energy manifold as the ratio \u03b1=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\u21920,\u03b1), 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 \u03b1 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\u22656, 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\u00d73, establishing exact finite-size benchmarks directly relevant for van der Waals nanomagnets, artificial spin-ice arrays, and dipolar-coupled nanomagnetic structures.<\/jats:p>","DOI":"10.3390\/e28020181","type":"journal-article","created":{"date-parts":[[2026,2,5]],"date-time":"2026-02-05T10:35:37Z","timestamp":1770287737000},"page":"181","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Finite-Size Thermodynamics of the Two-Dimensional Dipolar Q-Clock Model"],"prefix":"10.3390","volume":"28","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3049-668X","authenticated-orcid":false,"given":"Michel","family":"Aguilera","sequence":"first","affiliation":[{"name":"Instituto de F\u00edsica, Pontificia Universidad Cat\u00f3lica de Valpara\u00edso, Casilla 4950, Valpara\u00edso 2373223, Chile"},{"name":"Departamento de F\u00edsica, Universidad T\u00e9cnica Federico Santa Mar\u00eda, Av. Espa\u00f1a 1680, Valpara\u00edso 2390123, Chile"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7432-0707","authenticated-orcid":false,"given":"Francisco J.","family":"Pe\u00f1a","sequence":"additional","affiliation":[{"name":"Departamento de F\u00edsica, Universidad T\u00e9cnica Federico Santa Mar\u00eda, Av. Espa\u00f1a 1680, Valpara\u00edso 2390123, Chile"},{"name":"Departamento de Ciencias F\u00edsicas, Universidad de La Frontera, Casilla 54-D, Temuco 4811230, Chile"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1701-3030","authenticated-orcid":false,"given":"Eugenio E.","family":"Vogel","sequence":"additional","affiliation":[{"name":"Departamento de Ciencias F\u00edsicas, Universidad de La Frontera, Casilla 54-D, Temuco 4811230, Chile"},{"name":"Facultad de Ingenier\u00eda y Arquitectura, Universidad Central de Chile, Av. Sta. Isabel 1186, Santiago 8330601, Chile"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9235-9747","authenticated-orcid":false,"given":"Patricio","family":"Vargas","sequence":"additional","affiliation":[{"name":"Departamento de F\u00edsica, Universidad T\u00e9cnica Federico Santa Mar\u00eda, Av. Espa\u00f1a 1680, Valpara\u00edso 2390123, Chile"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2026,2,5]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"937","DOI":"10.1080\/00029890.1987.12000742","article-title":"An introduction to the Ising model","volume":"94","author":"Cipra","year":"1987","journal-title":"Am. Math. Mon."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"969","DOI":"10.1103\/PhysRev.113.969","article-title":"Transformations of Ising models","volume":"113","author":"Fisher","year":"1959","journal-title":"Phys. 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