A scaling theory for the long-range to short-range crossover and an infrared duality

TL;DR

Behan, Rastelli, Rychkov, and Zan analyze the second-order transition of the $d$-dimensional Ising model with long-range interactions decaying as $1/r^{d+s}$. They propose a novel field-theoretic description based on a short-range fixed point plus a decoupled Gaussian field $\chi$ (the SRFP+$\chi$ framework) and deform it with a local operator ${\cal O}=\sigma\chi$, uncovering an infrared duality: the same long-range fixed point can be reached from two distinct UV descriptions. Near the crossover at $s=s_*$, the flow is weakly coupled, allowing explicit computations of the beta-function and leading anomalous dimensions in $d=2$ and $d=3$, including the nontrivial recombination of the stress tensor multiplet. The work resolves puzzles about missing states (e.g., the would-be $V_\mu$) and shadow relations, derives concrete predictions for critical exponents, and situates the LRFP within a broader nonlocal conformal-field-theory framework with potential bootstrap applications. Overall, it establishes a controlled, dual description of the long-range to short-range crossover with precise operator relations and a path toward nonperturbative verification.

Abstract

We study the second-order phase transition in the $d$-dimensional Ising model with long-range interactions decreasing as a power of the distance $1/r^{d+s}$. For $s$ below some known value $s_*$, the transition is described by a conformal field theory without a local stress tensor operator, with critical exponents varying continuously as functions of $s$. At $s=s_*$, the phase transition crosses over to the short-range universality class. While the location $s_*$ of this crossover has been known for 40 years, its physics has not been fully understood, the main difficulty being that the standard description of the long-range critical point is strongly coupled at the crossover. In this paper we propose another field-theoretic description which, on the contrary, is weakly coupled near the crossover. We use this description to clarify the nature of the crossover and make predictions about the critical exponents. That the same long-range critical point can be reached from two different UV descriptions provides a new example of infrared duality.

Figures (7)

• Figure 1: RG flow diagram of the standard picture.
• Figure 2: RG flow diagram of our picture.
• Figure 3: The integration region $\mathcal{R}$.
• Figure 4: The dependence of dimensions of several important operators on $s$.
• Figure 5: The three integration regions (\ref{['eq:threeregions']}).