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Test of renormalization predictions for universal finite-size scaling functions
1Department of Physics, Delft University of Technology, P.O. Box 5046, 2600 GA Delft, The Netherlands. erik.luijten@uni-mainz.de
Summary
We calculated scaling functions for decaying interactions, predicting a singularity. However, numerical simulations showed a striking disagreement, challenging the theoretical prediction for critical phenomena. This impacts understanding of universal scaling laws.
Area of Science:
- Statistical Mechanics
- Condensed Matter Physics
Background:
- Finite-size scaling functions describe how systems behave near phase transitions.
- Understanding universal behavior in systems with algebraically decaying interactions is crucial for statistical mechanics.
Purpose of the Study:
- To calculate universal finite-size scaling functions for systems with n-component order parameters and algebraically decaying interactions.
- To investigate the predicted singular epsilon expansion near the upper critical dimension.
Main Methods:
- Theoretical calculation of universal finite-size scaling functions.
- Numerical simulations of spin models within the same universality class.
- Continuous variation of epsilon (distance to the upper critical dimension) for precise analysis.
Main Results:
- Theoretical calculations predicted a singular epsilon expansion.
- Numerical simulations revealed a striking disagreement with the theoretical prediction.
- The region of small epsilon was examined with high accuracy.
Conclusions:
- The theoretical prediction of a singularity in finite-size scaling functions for algebraically decaying interactions is challenged by numerical evidence.
- Further theoretical and numerical investigations are needed to resolve the discrepancy and refine our understanding of critical phenomena.