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Physical review. E

Critical temperature of the one-dimensional Ising model with inverse-square interactions

Anonymous, Abdulai Gassama, Jonah Eick, Han Zhang, Lijie Ding, Jan Tobochnik, Robert A. Pelcovits, John Michael Kosterlitz, Xinsheng Ling

Published Oct 2, 2026• 0 citations
Publisher
DOI
10.1103/hym1-lmwt

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Many physical systems, including some examples of active matter, granular assemblies, and biological systems, show fluctuation-dominated phase ordering (FDPO), where macroscopic fluctuations coexist with long-range order. Most of these systems are out of equilibrium. By contrast, a recent work has analytically demonstrated that an equilibrium one-dimensional Truncated Inverse Distance Square Ising (TIDSI) model shows FDPO. The analytical results rely on a cluster representation of the model that we term TIDSI-CL and are governed by the ratio, ccc, of the long-range interaction strength to the critical temperature. We show that the allowed range of ccc is very narrow in the TIDSI model while it is unbounded in TIDSI-CL. We perform Monte-Carlo simulations for the TIDSI model and show consistency with the analytical results in the allowed range of ccc. The correlation length grows strongly on approaching the critical point, leading to a broad near-critical region. Within this region, ααα, which is the cusp exponent of the power-law decay of the scaled correlation function at criticality, changes to αeffα^\text{eff}αeff. We also investigate the coarsening dynamics of the model: the correlation function, domain size distribution, and aging behavior are consistent with the equilibrium properties upon replacing the system size, LLL, with the coarsening length, L(t)\mathcal{L}(t)L(t). The mean largest cluster size shows logarithmic corrections due to finite LLL and waiting time, twt_wtw​. The aging autocorrelation function exhibits two different scaling forms, characterized by exponents βββ and γγγ, at short and long times compared to twt_wtw​, where β=α/2β=α/2β=α/2.
Many physical systems, including some examples of active matter, granular assemblies, and biological systems, show fluctuation-dominated phase ordering (FDPO), where macroscopic fluctuations coexist with long-range order. Most of these systems are out of equilibrium. By contrast, a recent work has analytically demonstrated that an equilibrium one-dimensional truncated inverse distance square Ising (TIDSI) model shows FDPO. The analytical results rely on a cluster representation of the model that we term TIDSI-CL and are governed by the ratio, c, of the long-range interaction strength to the critical temperature. We show that the allowed range of c is very narrow in the original TIDSI model while it is unbounded in TIDSI-CL. We perform Monte Carlo simulations for the TIDSI model and show consistency with the analytical results in the allowed range of c. The correlation length grows strongly on approaching the critical point, leading to a broad near-critical region. Within this region, α, which is the cusp exponent of the power-law decay of the scaled correlation function at criticality, changes to α^{eff}. We also investigate the coarsening dynamics of the model: The correlation function, domain size distribution, and aging behavior are consistent with the equilibrium properties upon replacing the system size, L, by the coarsening length, L(t). The mean largest cluster size shows logarithmic corrections due to finite L and waiting time, t_{w}. The aging autocorrelation function exhibits two different scaling forms, characterized by exponents β and γ, at short and long times compared to t_{w}, where β=α/2.