Abstract
We consider the question of conformal invariance of the long-range Ising model at the critical point. The continuum description is given in terms of a nonlocal field theory, and the absence of a stress tensor invalidates all of the standard arguments for the enhancement of scale invariance to conformal invariance. We however show that several correlation functions, computed to second order in the epsilon expansion, are nontrivially consistent with conformal invariance. We proceed to give a proof of conformal invariance to all orders in the epsilon expansion, based on the description of the long-range Ising model as a defect theory in an auxiliary higher-dimensional space. A detailed review of conformal invariance in the d-dimensional short-range Ising model is also included and may be of independent interest.
Highlights
We will be studying the long-range Ising (LRI) model—a close cousin of the usual, shortrange, Ising model (SRI)
This behavior does not follow from scale invariance alone but it is a necessary condition for conformal invariance, so we are led to believe that the LRI at criticality could be conformally invariant
How can we check if a certain model is conformally invariant or just scale invariant? The currently available data on the LRI amount to the anomalous dimensions of scaling operators, which are determined with renormalization group (RG) methods and measured on the lattice from two point functions
Summary
We will be studying the long-range Ising (LRI) model—a close cousin of the usual, shortrange, Ising model (SRI). Just like the SRI, the LRI has a second-order phase transition at a critical temperature T = Tc. The critical theory is universal, i.e. independent of the short-distance details such as the choice of the lattice. In the range σ > d/2 this interaction is relevant and generates a renormalization group (RG) flow, reaching a fixed point in the IR This IR fixed point is believed to be in the same universality class as the critical point of the LRI lattice model. As it will be explained below, the dimension of φ at the fixed point is still given by the same formula (1.3). We will not try to weigh in on this debate, except for a small comment in the discussion section
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