Boundary between Long-Range and Short-Range Critical Behavior in Systems with Algebraic Interactions

Abstract

We investigate phase transitions of two-dimensional Ising models with power-law interactions, using an efficient Monte Carlo algorithm. For slow decay, the transition is of the mean-field type; for fast decay, it belongs to the short-range Ising universality class. We focus on the intermediate range, where the critical exponents depend continuously on the power law. We find that the boundary with short-range critical behavior occurs for interactions depending on distance r as r 215͞4 . This answers a long-standing controversy between mutually conflicting renormalization-group analyses. DOI: 10.1103/PhysRevLett.89.025703 PACS numbers: 64.60.Ak, 05.70.Jk, 64.60.Fr, 75.10.Hk Many of the intermolecular forces that play a central role in large areas of chemistry, physics, and biology have a long-ranged nature. Well-known examples are electrostatic interactions, polarization forces, and van der Waals forces. Remarkably, there are still considerable deficiencies in our knowledge of the critical behavior induced by these interactions, which include the prominent case of Coulombic criticality (see Ref.[1] for a review).But also for the simpler case of purely attractive, long-range interactions, there exists a smoldering controversy, which we aim to resolve in this work. The present understanding of critical behavior in systems with such algebraically decaying interactions is largely based on the renormalization-group (RG) calculations for the O͑n͒ model by Fisher, Ma, and Nickel [2]. Their analysis revealed that different regimes can be identified for the universal critical properties, characterized by the decay power of the interactions. In view of the small number of global parameters determining the universality class, the location of the boundaries between these regimes is of considerable interest. It is, therefore, disturbing to note that there appears to be still no consensus on the theoretical side regarding the precise location of the boundary between shortrange and long-range critical behavior.In this Letter, we address this issue by means of numerical calculations, which allow us, with minimal prior assumptions, to decide between contradictory RG scenarios. Concretely, we demonstrate that the onset of the longrange regime occurs for interactions that decay more slowly than found in Ref.[2]. This result not only bears upon condensed-matter systems with long-range interactions, but also has implications for the renormalizationgroup treatment of competing fixed points in general, including random systems and gauge theories.Our approach specializes to the Ising model, n 1, in d dimensions, described by the reduced Hamiltonian,where the spins s k 61 are labeled by the lattice site k, the sum runs over all spin pairs, and the pair coupling depends on the distance r ij j r i 2 r j j between the spins. According to the analysis of Fisher et al. [2], universality classes are parametrized by s, and the following three distinct regimes were identified: (a) The classical regime; the upper critical dimension is given by...

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