Long-range minimal models

Abstract

We study a class of nonlocal conformal field theories in two dimensions which are obtained as deformations of the Virasoro minimal models. The construction proceeds by coupling a relevant primary operator $ϕ_{r,s}$ of the $m$-th minimal model to a generalized free field, in such a way that the interaction term has scaling dimension $2-δ$. Flowing to the infrared, we reach a new class of CFTs that we call long-range minimal models. In the case $r=s=2$, the resulting line of fixed points, parametrized by $δ$, can be studied using two perturbative expansions with different regimes of validity, one near the mean-field theory end, and one close to the long-range to short-range crossover. This is due to a straightforward generalization of an infrared duality which was proposed for the long-range Ising model ($m = 3$) in 2017. We find that the large-$m$ limit is problematic in both perturbative regimes, hence nonperturbative methods will be required in the intermediate range for all values of $m$. For the models based on $ϕ_{1,2}$, the situation is rather different. In this case, only one perturbative expansion is known but it is well behaved at large $m$. We confirm this with a computation of infinitely many anomalous dimensions at two loops. Their large-$m$ limits are obtained from both numerical extrapolations and a method we develop which carries out conformal perturbation theory using Mellin amplitudes. For minimal models, these can be accessed from the Coulomb gas representations of the correlators. This method reveals analytic expressions for some integrals in conformal perturbation theory which were previously only known numerically.