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The Imry-Ma phenomenon rounds phase transitions in low-dimensional spin systems. This study quantifies boundary effects on random fields, showing they diminish with system size, particularly in systems with continuous symmetry.

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Area of Science:

  • Statistical Mechanics
  • Condensed Matter Physics
  • Mathematical Physics

Background:

  • The Imry-Ma phenomenon describes the rounding of first-order phase transitions in low-dimensional spin systems due to quenched random fields.
  • This effect is crucial for understanding the behavior of disordered magnetic systems and phase transitions in restricted dimensions.

Purpose of the Study:

  • To provide quantitative estimates for the Imry-Ma phenomenon, focusing on boundary effects in cubic domains of side length L.
  • To analyze the impact of boundary conditions on spatial and thermal averages of quantities coupled to random fields.
  • To investigate the behavior of systems with and without continuous symmetry in various dimensions.

Main Methods:

  • Analysis of boundary effects on spatial and thermal averages within cubic domains.
  • Derivation of scaling laws for the diminishing boundary effect as a function of domain size L.
  • Establishment of partial uniqueness results for translation-covariant Gibbs states in the presence of random fields.

Main Results:

  • Boundary effects diminish at least as fast as an inverse power of L in general 2D spin systems.
  • For systems with continuous symmetry, boundary effects diminish at least as fast as an inverse power of L in 2D and 3D, and as an inverse power of L^2 in 4D.
  • Proved that for almost all random field realizations, translation-covariant Gibbs states agree on thermally-averaged quantities.

Conclusions:

  • Quantitative estimates confirm the Imry-Ma phenomenon's dependence on system size and symmetry.
  • The results provide a deeper understanding of phase transitions in disordered low-dimensional systems.
  • Established theoretical underpinnings for models like the random-field Potts and Edwards-Anderson spin glass models.