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Description
Long-range interacting quantum many-body systems have recently been realized in platforms such as trapped-ion systems and Rydberg atom arrays. In theoretical descriptions of long-range interactions, the Kac normalization factor is often introduced to keep the interaction energy per particle finite in the thermodynamic limit. In many models, however, this factor does not emerge naturally from the underlying interaction.
Frechette et al. showed that an elastic Ising model on a triangular lattice naturally generates a long-range effective interaction with a normalization factor proportional to the inverse number of particles. In this work, following the basic idea of Wagner and Horner, we treat the same model from the outset as a macroscopic two-dimensional isotropic elastic body rather than directly following the microscopic lattice deformation. For a finite system with free boundaries, the effective spin Hamiltonian takes the form
$$ H_{\mathrm{eff}} = -\frac{1}{\pi R^2} \sum_{n,m} \phi\!\left( \frac{\vec r_n}{R}, \frac{\vec r_m}{R} \right) S_nS_m , $$ where the inverse area $1/(\pi R^2)$ emerges naturally as the Kac normalization factor.
In this work, we show that a Kac-type long-range interaction and its normalization factor can arise from the macroscopic deformation of a finite elastic body, consistently with the microscopic triangular-lattice result. Although the present calculation is classical, the derivation is based entirely on macroscopic elasticity, suggesting that the same mechanism may also apply to quantum spin degrees of freedom.