What question did this study set out to answer?

This research aims to understand the thermodynamic behavior of the two-dimensional dipolar Q-state clock model, focusing on finite-size effects.

February 8, 2026Open Access

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

Key Points

This research aims to understand the thermodynamic behavior of the two-dimensional dipolar Q-state clock model, focusing on finite-size effects.
Conducted a controlled thermodynamic study on small square lattices with free boundaries.
Utilized exhaustive state enumeration to analyze energy spectra and degeneracies.
Varying the ratio of exchange to dipolar interactions (α=D/J) to assess impacts on energy structure.
Evaluated canonical observables and identified specific heat behavior at low temperatures.
Identified ground-state level crossings that manifest as exact zeros of specific heat in the low-temperature limit.
Observed distinct Schottky-like anomalies indicating degeneracy structures in odd and even lattices.
Found a symmetry-driven crossover for increasing Q, leading to different thermodynamic behaviors across models.
Demonstrated that dipolar interactions alone can produce critical-like caloric behavior in small clusters.

Abstract

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 3×3 lattice, tracking how the competition between exchange and long-range dipolar interactions reorganizes the low-energy manifold as the ratio α=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→0, α), 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 α 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≥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 3×3, establishing exact finite-size benchmarks directly relevant for van der Waals nanomagnets, artificial spin-ice arrays, and dipolar-coupled nanomagnetic structures.

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