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

## TL;DR

This paper studies how dipolar interactions affect the thermodynamics of small magnetic systems, revealing how these interactions create unique heat-related behaviors in nanoscale magnets.

## Contribution

The study provides exact finite-size benchmarks for dipolar Q-clock models, revealing how anisotropy and symmetry influence thermodynamic signatures in nanomagnetic systems.

## Key findings

- Ground-state level crossings from dipolar interactions appear as exact zeros in the specific heat at low temperatures.
- Schottky-like anomalies in specific heat reveal lattice parity effects, with odd lattices showing asymmetric peaks and even lattices showing symmetric anomalies.
- For Q≥6, the energy landscape becomes smooth enough to suppress ground-state crossings, resulting in purely thermal specific-heat maxima.

## 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.