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. We find a new description, which is instead weakly coupled near the crossover, and use it to compute critical exponents. The existence of two complementary UV descriptions of the same long-range fixed point provides a novel example of infrared duality.

Figure 1

RG flows corresponding to (a) the standard picture and (b) our proposal.