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Critical behavior of the weakly disordered Ising model: Six-loop sqrt[ɛ] expansion study.

M V Kompaniets1, A Kudlis2, A I Sokolov1

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Summary

This study examines the critical behavior of a 3D Ising model using renormalization group expansions. Results from these expansions align well with experimental and simulation data, unlike sqrt[ɛ] series resummation.

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Area of Science:

  • Statistical Physics
  • Condensed Matter Physics

Background:

  • The Ising model is a fundamental model in statistical mechanics used to study magnetism and phase transitions.
  • Diluted and quenched systems introduce disorder, significantly altering critical behavior.
  • Renormalization group (RG) theory provides a powerful framework for understanding critical phenomena near phase transitions.

Purpose of the Study:

  • To investigate the critical behavior of the three-dimensional (3D) weakly diluted quenched Ising model.
  • To compare the effectiveness of different theoretical approaches, specifically RG expansions and sqrt[ɛ] series, in predicting critical exponents.
  • To provide accurate numerical estimates for the physical 3D system.

Main Methods:

  • Utilizing six-loop renormalization group (RG) expansions within the minimal subtraction scheme in 4-ɛ dimensions.
  • Analyzing the ϕ⁴ field theory with cubic symmetry in the replica limit (n→0).
  • Applying various resummation procedures to both RG expansions and sqrt[ɛ] series for numerical estimation.

Main Results:

  • Renormalization group expansions in terms of renormalized couplings yield numerical estimates in good agreement with experimental and Monte Carlo simulation data.
  • Direct resummation of the sqrt[ɛ] series for critical exponents proved disappointing and less accurate.
  • The study confirms the validity of the RG approach for describing the critical behavior of disordered systems.

Conclusions:

  • The six-loop RG expansions provide a reliable method for studying the critical properties of the 3D weakly diluted quenched Ising model.
  • The discrepancy highlights limitations in directly applying sqrt[ɛ] series resummation techniques for this specific system.
  • This work validates theoretical predictions against empirical and computational evidence in disordered statistical physics models.