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Geometric Explanation of Anomalous Finite-Size Scaling in High Dimensions
Jens Grimm1, Eren Metin Elçi2, Zongzheng Zhou1
1ARC Centre of Excellence for Mathematical and Statistical Frontiers (ACEMS), School of Mathematical Sciences, Monash University, Clayton, Victoria 3800, Australia.
Physical Review Letters
|April 4, 2017
Summary
Finite-size scaling anomalies in periodic boundary systems are explained geometrically. New methods reveal correlation functions accounting for windings resolve apparent breakdowns above critical dimensions.
Area of Science:
- Statistical physics
- Computational physics
- Geometric scaling
Background:
- Standard finite-size scaling theory often breaks down in systems with periodic boundaries above the upper critical dimension.
- This breakdown is particularly evident in models like the Ising model and self-avoiding walks.
- Understanding this phenomenon is crucial for accurately modeling complex systems.
Purpose of the Study:
- To provide an intuitive geometric explanation for the apparent breakdown of finite-size scaling.
- To resolve anomalous behavior observed in correlation functions.
- To introduce a corrected scaling approach for systems with periodic boundaries.
Main Methods:
- Simulations of the Ising model and self-avoiding walks on five-dimensional hypercubic lattices.
- Utilizing geometric representations for analysis.
- Employing recently developed Markov-chain Monte Carlo algorithms.
Main Results:
- An intuitive geometric explanation for the breakdown of finite-size scaling was established.
- Anomalous behavior in correlation functions was successfully removed.
- A new scale, accounting for system windings, was defined for correlation functions.
Conclusions:
- The apparent breakdown of finite-size scaling in periodic systems is an artifact of the measurement scale.
- Correctly accounting for windings in correlation functions resolves these anomalies.
- This work provides a more accurate framework for understanding scaling in complex systems.