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.

16 pages, 5 figures

References in corpus (7)

• Study of the de Almeida-Thouless line using power-law diluted one-dimensional Ising spin glasses
• Order-N Cluster Monte Carlo Method for Spin Systems with Long-range Interactions
• The crossover region between long-range and short-range interactions for the critical exponents
• Fisher's scaling relation above the upper critical dimension
• A new critical exponent koppa and its logarithmic counterpart koppa-hat
• Critical behavior of the Ising model with long range interactions
• One-dimensional infinite component vector spin glass with long-range interactions

Cited by in corpus (14)

• Long-range interacting quantum systems
• Critical Casimir Effect: Exact Results
• Scaling at quantum phase transitions above the upper critical dimension
• Finite-temperature critical behaviors in 2D long-range quantum Heisenberg model
• Monte Carlo based techniques for quantum magnets with 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
• 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
• Theory of Critical Phenomena with Long-Range Temporal Interaction
• Stochastic parameter optimization analysis of dynamical quantum critical phenomena in long-range transverse-field Ising chain
• When correlations exceed system size: finite-size scaling in free boundary conditions above the upper critical dimension
• On a previously unpublished work with Ralph Kenna