Finite-size scaling above the upper critical dimension in Ising models with long-range interactions

Abstract

The correlation length plays a pivotal role in finite-size scaling and hyperscaling at continuous phase transitions. Below the upper critical dimension, where the correlation length is proportional to the system length, both finite-size scaling and hyperscaling take conventional forms. Above the upper critical dimension these forms break down and a new scaling scenario appears. Here we investigate this scaling behaviour in one-dimensional Ising ferromagnets with long-range interactions. We show that the correlation length scales as a non-trivial power of the linear system size and investigate the scaling forms. For interactions of sufficiently long range, the disparity between the correlation length and the system length can be made arbitrarily large, while maintaining the new scaling scenarios. We also investigate the behavior of the correlation function above the upper critical dimension and the modifications imposed by the new scaling scenario onto the associated Fisher relation.

arXiv comment16 pages, 5 figures

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• Critical Casimir Effect: Exact Results
• Finite-temperature critical behavior of long-range quantum Ising models
• Quantum Gravity as an Information Network: Self-Organization of a 4D Universe
• Scaling at quantum phase transitions above the upper critical dimension
• Finite-temperature critical behaviors in 2D long-range quantum Heisenberg model
• Universal Finite-Size Scaling for Percolation Theory in High Dimensions
• Quantum phase transition in the spin-anisotropic quantum spherical model
• Monte Carlo based techniques for quantum magnets with long-range interactions
• Cluster Monte Carlo and dynamical scaling for long-range interactions
• Dyson hierarchical quantum ferromagnetic Ising chain with pure or random transverse fields
• Unconventional Scalings of Quantum Entropies in Long-Range Heisenberg Chains
• Critical Space-Time Networks and Geometric Phase Transitions from Frustrated Edge Antiferromagnetism
• Effective-Dimension Theory of Critical Phenomena above Upper Critical Dimensions
• Universal scaling in real dimension
• The fifty-year quest for universality in percolation theory in high dimensions
• Bali's ancient rice terraces: A Hamiltonian approach
• Theory of Critical Phenomena with Long-Range Temporal Interaction
• When correlations exceed system size: finite-size scaling in free boundary conditions above the upper critical dimension
• Stochastic parameter optimization analysis of dynamical quantum critical phenomena in long-range transverse-field Ising chain
• On a previously unpublished work with Ralph Kenna