This book is about the Random Field Ising Model (RFIM) – a paradigmatic spin model featuring a frozen disordering field. The focus is on the second-order phase transition between the paramagnetic and ferromagnetic phases, and the associated critical exponents.
The book starts by summarizing the current knowledge about the RFIM from experiments, numerical simulations and rigorous mathematical results. It then reviews the classic theoretical works from the 1970’s which suggested a property of dimensional reduction – that the RFIM critical exponents should be the same as for the ordinary, non-disordered, Ising model of lower dimensionality, and related this an emergent Parisi-Sourlas supersymmetry. As is now known, these remarkable properties only hold when the spatial dimensionality of the model is larger than a critical dimension. The book presents a method to estimate the critical dimension, using standard tools such as the replica trick and perturbative renormalization group, whose result is in agreement with the numerical simulations. Some more elementary steps in the derivations are left as exercises for the readers.
This book is of interest to researchers, PhD students and advanced master students specializing in statistical field theory.
Author(s): Slava Rychkov
Series: SpringerBriefs in Physics
Edition: 1
Publisher: Springer Nature Switzerland
Year: 2023
Language: English
Pages: ix; 64
City: Cham
Tags: Theoretical, Mathematical and Computational Physics; Particle and Nuclear Physics; Random Field Ising Model; RFIM; dimensional reduction; Parisi-Sourlas supersymmetry; Cardy transform; stochastic PDE representation;
Preface
Contents
Acronyms
1 History, Basics, Experiments and Simulations
1.1 History
1.2 Basic Facts
1.3 Experiments
1.4 Numerical Simulations
References
2 Parisi–Sourlas Theory, Dimensional Reduction
2.1 Field Theory and Perturbation Theory
2.2 Connected and Disconnected 2pt Functions
2.3 Selection of Important Diagrams
2.4 Stochastic PDE Representation, Parisi–Sourlas Action
2.5 Parisi–Sourlas Supersymmetry
2.6 SUSY Ward Identities
2.7 Dimensional Reduction
2.8 Caveats
2.9 Literature and Further Comments
References
3 Replicas, Cardy Transform, Leaders, Loss of SUSY
3.1 Replicas
3.2 Upper Critical Dimension
3.3 Cardy Transform: Second Argument for the PS SUSY
3.4 Testing the Second Argument
3.5 Analogies
3.6 Leaders
3.7 Classification of Leaders
3.8 Renormalization Group Fixed Point and Its Stability
3.9 Relation to Previous Work
3.10 Literature and Further Comments
References
4 Open Problems and Future Directions
4.1 Symmetry Meaning of Leaders
4.2 Branched Polymers
4.3 Nonperturbative RG Studies of RFIM
4.4 Tuning to the SUSY Fixed Point
References
