Parisi-Sourlas Supersymmetry in Random Field Models.

By the Parisi-Sourlas conjecture, the critical point of a theory with random field (RF) disorder is described by a supersymmeric (SUSY) conformal field theory (CFT), related to a d-2 dimensional CFT without SUSY. Numerical studies indicate that this is true for the RF ϕ^{3} model but not for the RF ϕ^{4} model in d<5 dimensions. Here we argue that the SUSY fixed point is not reached because of new relevant SUSY-breaking interactions.

Long-Range Critical Exponents near the Short-Range Crossover.

The d-dimensional long-range Ising model, defined by spin-spin interactions decaying with the distance as the power 1/r^{d+s}, admits a second-order phase transition with continuously varying critical exponents. At s=s_{*}, the phase transition crosses over to the usual short-range universality class. The standard field-theoretic description of this family of models is strongly coupled at the crossover.

Conformal field theories in fractional dimensions.

We study the conformal bootstrap in fractional space-time dimensions, obtaining rigorous bounds on operator dimensions. Our results show strong evidence that there is a family of unitary conformal field theories connecting the 2D Ising model, the 3D Ising model, and the free scalar theory in 4D. We give numerical predictions for the leading operator dimensions and central charge in this family at different values of D and compare these to calculations of ϕ4 theory in the ϵ expansion.