Mean-field driven first-order phase transitions in systems with long-range interactions

Abstract

We consider a class of spin systems on ${\mathbb{Z}}^d$ with vector valued spins $(\bS_x)$ that interact via the pair-potentials $J_{x,y} \bS_x\cdot\bS_y$. The interactions are generally spread-out in the sense that the $J_{x,y}$'s exhibit either exponential or power-law fall-off. Under the technical condition of reflection positivity and for sufficiently spread out interactions, we prove that the model exhibits a first-order phase transition whenever the associated mean-field theory signals such a transition. As a consequence, e.g., in dimensions $d\ge 3$, we can finally provide examples of the 3-state Potts model with spread-out, exponentially decaying interactions, which undergoes a first-order phase transition as the temperature varies. Similar transitions are established in dimensions $d=1,2$ for power-law decaying interactions and in high dimensions for next-nearest neighbor couplings. In addition, we also investigate the limit of infinitely spread-out interactions. Specifically, we show that once the mean-field theory is in a unique ``state,'' then in any sequence of translation-invariant Gibbs states various observables converge to their mean-field values and the states themselves converge to a product measure.

57 pages; uses a (modified) jstatphys class file

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• A selection of nonequilibrium issues
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• Coexistence of ordered and disordered phases in Potts models in the continuum
• Nonequivalence of ensembles in the Curie-Weiss anisotropic quantum Heisenberg model
• Forbidden gap argument for phase transitions proved by means of chessboard estimates
• Large-deviations/thermodynamic approach to percolation on the complete graph
• First-Order Phase Transition in Potts Models with finite-range interactions
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• Global disorder transition in the community structure of large-q Potts systems
• On the Truncation of Systems with Non-Summable Interactions
• Thermodynamic versus Topological Phase Transitions: Cusp in the Kertész Line
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• Truncated long-range percolation on oriented graphs
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• Long-range contact process and percolation on a random lattice
• A Curie-Weiss Theory of the Continuum Widom-Rowlinson Model
• Truncation of long-range percolation model with square non-summable interactions
• On the mean-field equations for ferromagnetic spin systems