Abstract
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.
Highlights
The d-dimensional long-range Ising model, defined by spin-spin interactions decaying with the distance as the power 1=rdþs, admits a second-order phase transition with continuously varying critical exponents
We will focus for definiteness on the long-range Ising model (LRI) in d 1⁄4 2 and d 1⁄4 3 dimensions [1], with ferromagnetic interaction between spins decaying as a power of their distance as 1=rdþs, where s > 0 for the thermodynamic limit to be well defined [2]
In the regime (ii) d=2 < s < sÃ, this flow ends in an interacting long-range fixed point (LRFP)
Summary
In the regime (ii) d=2 < s < sÃ, this flow ends in an interacting long-range fixed point (LRFP). General composite operators, such as φ2, acquire nontrivial anomalous dimensions, as befits an interacting fixed point. The long-range to shortrange crossover is expected to happen [6] when 1⁄2φLRFP, decreasing with s, reaches the short-range Ising fixed point (SRFP) dimension 1⁄2φSRFP.
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