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Universality in the 2d Quasi-periodic Ising Model and Harris-Luck Irrelevance.

Matteo Gallone1, Vieri Mastropietro2

  • 1International School for Advanced Studies - SISSA, Via Bonomea 265, 34136 Trieste, Italy.

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The 2D Ising model with quasi-periodic disorder exhibits the same critical behavior as the non-disordered case. This rigorous proof confirms universality, showing identical critical exponents and no logarithmic corrections.

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Area of Science:

  • Statistical Mechanics
  • Condensed Matter Physics
  • Disordered Systems

Background:

  • The 2D Ising model is a fundamental model in statistical mechanics, crucial for understanding phase transitions.
  • Investigating the effects of disorder on critical phenomena is essential for realistic physical systems.
  • Quasi-periodic disorder presents unique challenges due to its lack of long-range order and non-trivial spectral properties.

Purpose of the Study:

  • To rigorously prove universality in the 2D Ising model with bidimensional quasi-periodic disorder.
  • To determine the critical behavior and exponents in the presence of such disorder.
  • To validate predictions from the Harris-Luck criterion in this specific context.

Main Methods:

  • Renormalization Group (RG) analysis was employed to study the system's behavior under rescaling.
  • Control of small divisors was achieved by assuming a Diophantine condition on the frequencies of the quasi-periodic disorder.
  • Convergence of series was rigorously demonstrated through the RG framework.

Main Results:

  • The critical behavior of the disordered 2D Ising model is identical to the non-disordered case.
  • Critical exponents for specific heat and energy-energy correlations remain unchanged.
  • No logarithmic corrections to scaling were found, contrary to some expectations for disordered systems.
  • Quasi-periodic disorder leads to modulation of correlation amplitudes and renormalization of velocities and critical temperature.

Conclusions:

  • The study provides the first rigorous proof of universality for the Ising model with quasi-periodic disorder in two dimensions.
  • The findings confirm the applicability of the Harris-Luck criterion in this complex scenario.
  • The results highlight the robustness of critical behavior in the 2D Ising model even under quasi-periodic modulations.