Long-range minimal models

TL;DR

This work introduces long-range minimal CFTs (LRMMs) by coupling Virasoro minimal models to a generalized free field, yielding nonlocal IR fixed points described by an interaction $\mathcal O=\phi_{r,s}\chi$ with $\Delta_{\mathcal O}=2-\delta$ ($0<\delta\ll1$). It develops perturbative control in two regimes: near mean-field theory for $(m,2,2)$ and near the short-range end for $(m,2,2)$ and $(m,1,2)$, supplemented by analytic large-$m$ methods using Mellin-space CPT and the Coulomb-gas formalism. The paper presents explicit beta-functions and anomalous dimensions for a wide array of Virasoro primaries and higher-spin currents, including mixing phenomena, shadow relations, and dualities between LRMMs and short-range descriptions at crossover. These techniques yield both concrete perturbative data and structural insights (duality, mixing, and multiplet recombination) that pave the way for nonperturbative tests (bootstrap, FRG) and generalizations to broader classes of long-range CFTs.

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.

Figures (8)

• Figure 1: Left: Polynomial fit (up to $m^{-8}$) for numerical values of $\beta_3$ and $m\ge20$. For each data point, the error is taken to be the max between $\left| \beta_3(n_{\text{max}}=20)-\beta_3(n_{\text{max}}=18) \right|$ and the precision of the numerical integration at each point. The uncertainty on the fit corresponds to the 95% confidence interval. Right: Error fluctuations, for several choices of $n \leq n_{\text{max}}$ and $m$. After some value $n_{\text{max}}<20$, the relative error get saturated by the numerical error.
• Figure 2: Left: Polynomial fits (up to $m^{-6}$) for several anomalous dimension for $m\ge20$. Here $n_{\text{max}}=50$. For each data point, the error is taken to be the max between $\left| \beta_3(n_{\text{max}}=50)-\beta_3(n_{\text{max}}=45) \right|$ and the precision of the numerical integration at each point. The uncertainty on the fit corresponds to the 95% confidence interval. The expansion order was increased compared to figure \ref{['fig:phi22beta3Fit']}, in order to improve convergence. Right: Leading non trivial order of the fit, for some operators.
• Figure 3: Left: Polynomial fit (up to $m^{-8}$) for numerical values of $\beta_3$ and $m\ge20$. For each data point, the error is taken to be the max between $\left| \beta_3(n_{\text{max}}=50)-\beta_3(n_{\text{max}}=45) \right|$ and the precision of the numerical integration at each point. The uncertainty on the fit corresponds to the 95% confidence interval. Right: Error fluctuations, for several choices of $n \leq n_{\text{max}}$ and $m$.
• Figure 4: Left: Polynomial fits (up to $m^{-6}$) for several anomalous dimension. Here $n_{\text{max}}=45$. For each data point, the error is taken to be the max between $\left| \beta_3(n_{\text{max}}=45)-\beta_3(n_{\text{max}}=40) \right|$ and the precision of the numerical integration at each point. The uncertainty on the fit corresponds to the 95% confidence interval. Right: Leading non trivial order of the fit, for some operators.
• Figure 5: Sequences of poles for both lines of \ref{['vir-blocks-mellin']} which are offset from the real axis for clarity. The rightmost dot is $\alpha_-^2 - 1$ and the leftmost cross is $0$. The red contour is completely free of singular behaviour in the $\alpha_-^2 \to 1$ limit (double arrow) but it is not the natural Mellin-Barnes contour (the blue one). The difference is a simple residue at the rightmost dot.