A Power-Law Upper Bound on the Correlations in the 2D Random Field Ising Model

As first asserted by Y. Imry and S-K Ma, the famed discontinuity of the magnetization as function of the magnetic field in the two dimensional Ising model is eliminated, for all temperatures, through the addition of quenched random magnetic field of uniform variance, even if that is small. This stat...

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Bibliographic Details
Published inCommunications in mathematical physics Vol. 372; no. 3; pp. 865 - 892
Main Authors Aizenman, Michael, Peled, Ron
Format Journal Article
LanguageEnglish
Published Berlin/Heidelberg Springer Berlin Heidelberg 01.12.2019
Springer Nature B.V
Subjects
Boundary conditions
Classical and Quantum Gravitation
Complex Systems
Decay rate
Fields (mathematics)
High temperature
Ising model
Magnetic fields
Magnetization
Mathematical and Computational Physics
Mathematical Physics
Physics
Physics and Astronomy
Power law
Quantum Physics
Relativity Theory
Rounding
Theoretical
Two dimensional models
Upper bounds
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Summary:As first asserted by Y. Imry and S-K Ma, the famed discontinuity of the magnetization as function of the magnetic field in the two dimensional Ising model is eliminated, for all temperatures, through the addition of quenched random magnetic field of uniform variance, even if that is small. This statement is quantified here by a power-law upper bound on the decay rate of the effect of boundary conditions on the magnetization in finite systems, as function of the distance to the boundary. Unlike exponential decay which is only proven for strong disorder or high temperature, the power-law upper bound is established here for all field strengths and at all temperatures, including zero, for the case of independent Gaussian random field. Our analysis proceeds through a streamlined and quantified version of the Aizenman–Wehr proof of the Imry–Ma rounding effect.
ISSN:0010-3616
1432-0916
DOI:10.1007/s00220-019-03450-3