Appears in collection : Slava Rychkov – Random Field Ising Model and Parisi-Sourlas Supersymmetry

Numerical evidence suggests that the Random Field Ising Model loses Parisi-Sourlas SUSY and the dimensional reduction property somewhere between 4 and 5 dimensions, while a related model of branched polymers retains these features in any d. I will present a recent theory, developed in 2019-2021 jointly with A. Kaviraj and E. Trevisani and published in [1-4], which aims to explain these facts.

Outline:

1. Random Field Ising Model: phase diagram, well-established facts and experiments.
2. Numerical results for the dimensional reduction of critical exponents: “no” for d=3,4, “yes” for d=5.
3. Parisi-Sourlas supersymmetry implies dimensional reduction
4. Generalities about RG fixed point disappearance
5. Loss of Parisi-Sourlas SUSY via dangerously irrelevant operators?
6. Replica field theory. Cardy field transform “derivation" of Parisi-Sourlas SUSY and its potential loopholes.
7. Replica symmetric interactions in the Cardy basis
8. Leader and follower interactions
9. Classification of leaders
10. Anomalous dimension computations and results. Evidence for the SUSY fixed point instability below ~4.5
11. Future directions and open problems.