2021/10/30 by Nicolas Crawford, Crawford, Nicolas, Wioletta M. Ruszel +1 · 1 citation
Mathematics · Physics and Astronomy · #82D40 #FOS: Mathematics #FOS: Physical sciences #Field (mathematics) #Geometry #Hamiltonian (control theory) #Length scale #Magnetic field #Mathematical Physics (math-ph) #Mathematical optimization #Mathematics #Order (exchange) #Perpendicular #Physics #Probability (math.PR) #Pure mathematics #Quantum chaos and dynamical systems #Quantum mechanics #Random field #Scale (ratio) #Statistical physics #Statistics #Stochastic processes and statistical mechanics #Theoretical and Computational Physics
paper · pdf · doi:10.48550/arxiv.2111.00241
published in arXiv (Cornell University) (Cornell University)
openalex publication_date 2021/10/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
In this article we prove that a classical XY model subjected to weak i.i.d. random field pointing in a fixed direction exhibits residual magnetic order in ℤ2 and aligns perpendicular to the random field direction. The paper is a sequel to \citeNC where the three-dimensional case was treated. Our approach is based on a multi-scale Peierls contour argument developed in \citeNC. On the microscopic scale we extract energetic costs from the occurrence of contours, which themselves are defined on a macroscopic scale. The technical challenges in ℤ2 stem from difficulties controlling the size and roughness of the fluctuation fields which model the short length-scale oscillations of near-optimizers of the random field Hamiltonian.