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Dimensional analysis is a powerful tool that is used in physics and engineering to understand and predict the behavior of physical systems. The basic idea behind dimensional analysis is to express physical quantities in terms of fundamental dimensions such as the mass, length, and time. Derived dimensions like the velocity, acceleration, and force are derived from the combinations of these fundamental dimensions.
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Universality in long-range interacting systems: The effective dimension approach.

Andrea Solfanelli1,2, Nicolò Defenu3,4

  • 1<a href="https://ror.org/004fze387">SISSA</a>, via Bonomea 265, 34136 Trieste, Italy.

Physical Review. E
|November 20, 2024
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Summary
This summary is machine-generated.

The effective dimension approach accurately estimates critical exponents in long-range interacting systems. This method maps complex models to simpler local systems, achieving over 97% accuracy for critical phenomena studies.

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Area of Science:

  • Statistical Mechanics
  • Condensed Matter Physics

Background:

  • Dimensional correspondences are crucial in understanding critical phenomena.
  • The effective dimension approach relates long-range interacting systems to local ones via a fractal dimension.
  • Its validity beyond mean-field theory has been a subject of debate.

Purpose of the Study:

  • To rigorously assess the accuracy of the effective dimension approach for critical phenomena.
  • To extend the applicability of this method using advanced theoretical techniques.
  • To validate the approach by comparing with high-precision numerical data.

Main Methods:

  • Review of perturbative renormalization-group (RG) results.
  • Extension of the effective dimension approximation using functional RG techniques.
  • Comparison with conformal bootstrap numerical data for a 2D Ising model with long-range interactions.

Main Results:

  • The effective dimension approach provides highly accurate estimations (typically >97%) for critical exponents of long-range models.
  • The method's accuracy is demonstrated even for non-Gaussian fixed points, though it remains an approximation.
  • The study confirms the utility of mapping long-range systems to effective local ones.

Conclusions:

  • The effective dimension approach is a powerful and accurate tool for studying critical phenomena in systems with long-range interactions.
  • Functional RG techniques enhance the understanding and application of this dimensional mapping.
  • This work bridges theoretical approximations with precise numerical results, validating a key concept in statistical physics.