Coarsening kinetics in spin systems with long-range interactions: from voter to Ising

In this paper, we start reviewing the main features of the one-dimensional Ising model with long-range interactions, where the spin-spin coupling decays as a power law, $J(r) \propto r^{-α}$. We then discuss the key properties of the one-dimensional voter model, in which two agents (spins) at distance $r$ interact with a power-law probability with the same form of $J(r)$. The two models are compared, and the so-called $p$-voter model is presented, which provides a framework to interpolate between them. Specifically, the $p$-voter model reduces to the voter model for $p = 1$ and $p = 2$, while for $p \ge 3$ it falls into the universality class of the Ising model.

💡 Research Summary

The paper provides a comprehensive review of coarsening kinetics in one‑dimensional spin systems with algebraically decaying long‑range interactions, focusing on the Ising model (IM), the voter model (VM), and a unifying p‑voter model (pVM). The authors begin by defining the long‑range Ising Hamiltonian H = −∑{i,r>0}J(r)S_iS{i+r} with J(r) ∝ r⁻ᵅ and discuss its equilibrium properties. For α > 2 the model behaves like the nearest‑neighbour (nn) Ising chain: domain‑wall (DW) energy is finite, there is no finite‑temperature phase transition, and coarsening follows the classic L(t) ∝ t¹ᐟ² law. When 1 < α ≤ 2 the interaction is weakly long‑ranged (WLR): the DWs experience an attractive drift that depends on the domain size. By mapping the asymmetric random walk of a DW onto a convection‑diffusion equation, the authors derive a closure time t(L) = L v tanh(vL/D)/D with drift v(L)=v₀ tanh