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Register a new userThe Lorentzian inversion formula and the spectrum of the 3d O(2) CFT
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We study the spectrum and OPE coefficients of the three-dimensional critical O(2) model, using four-point functions of the leading scalars with charges 0, 1, and 2 ($s$, $phi$, and $t$). We obtain numerical predictions for low-twist OPE data in several charge sectors using the extremal functional method. We compare the results to analytical estimates using the Lorentzian inversion formula and a small amount of numerical input. We find agreement between the analytic and numerical predictions. We also give evidence that certain scalar operators lie on double-twist Regge trajectories and obtain estimates for the leading Regge intercepts of the O(2) model.
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In this work we study the $6j$ symbol of the $3d$ conformal group for fermionic operators. In particular, we study 4-point functions containing two fermions and two scalars and also those with four fermions. By using weight-shifting operators and harmonic analysis for the Euclidean conformal group, we relate these spinning $6j$ symbols to the simpler $6j$ symbol for four scalar operators. As one application we use these techniques to compute $3d$ mean field theory (MFT) OPE coefficients for fermionic operators. We then compute corrections to the MFT spectrum and couplings due to the inversion of a single operator, such as the stress tensor or a low-dimension scalar. These results are valid at finite spin and extend the perturbative large spin analysis to include non-perturbative effects in spin.
Motivated by applications to critical phenomena and open theoretical questions, we study conformal field theories with $O(m)times O(n)$ global symmetry in $d=3$ spacetime dimensions. We use both analytic and numerical bootstrap techniques. Using the analytic bootstrap, we calculate anomalous dimensions and OPE coefficients as power series in $varepsilon=4-d$ and in $1/n$, with a method that generalizes to arbitrary global symmetry. Whenever comparison is possible, our results agree with earlier results obtained with diagrammatic methods in the literature. Using the numerical bootstrap, we obtain a wide variety of operator dimension bounds, and we find several islands (isolated allowed regions) in parameter space for $O(2)times O(n)$ theories for various values of $n$. Some of these islands can be attributed to fixed points predicted by perturbative methods like the $varepsilon$ and large-$n$ expansions, while others appear to arise due to fixed points that have been claimed to exist in resummations of perturbative beta functions.
In this letter, we discuss certain universal predictions of the large charge expansion in conformal field theories with $U(1)$ symmetry, mainly focusing on four-dimensional theories. We show that, while in three dimensions quantum fluctuations are responsible for the existence of a theory-independent $Q^0$ term in the scaling dimension $Delta_Q$ of the lightest operator with fixed charge $Qgg 1$, in four dimensions the same mechanism provides a universal $Q^0log Q$ correction to $Delta_Q$. Previous works discussing four-dimensional theories failed in identifying this term. We also compute the first subleading correction to the OPE coefficient corresponding to the insertion of an arbitrary primary operator with small charge $qll Q$ in between the minimal energy states with charge $Q$ and $Q+q$, both in three and four dimensions. This contribution does not depend on the operator insertion and, similarly to the quantum effects in $Delta_Q$, in four dimensions it scales logarithmically with $Q$.
We study the conformal bootstrap constraints for 3D conformal field theories with a $mathbb{Z}_2$ or parity symmetry, assuming a single relevant scalar operator $epsilon$ that is invariant under the symmetry. When there is additionally a single relevant odd scalar $sigma$, we map out the allowed space of dimensions and three-point couplings of such Ising-like CFTs. If we allow a second relevant odd scalar $sigma$, we identify a feature in the allowed space compatible with 3D $mathcal{N}=1$ superconformal symmetry and conjecture that it corresponds to the minimal $mathcal{N}=1$ supersymmetric extension of the Ising CFT. This model has appeared in previous numerical bootstrap studies, as well as in proposals for emergent supersymmetry on the boundaries of topological phases of matter. Adding further constraints from 3D $mathcal{N}=1$ superconformal symmetry, we isolate this theory and use the numerical bootstrap to compute the leading scaling dimensions $Delta_{sigma} = Delta_{epsilon} - 1 = .58444(22)$ and three-point couplings $lambda_{sigmasigmaepsilon} = 1.0721(2)$ and $lambda_{epsilonepsilonepsilon} = 1.67(1)$. We additionally place bounds on the central charge and use the extremal functional method to estimate the dimensions of the next several operators in the spectrum. Based on our results we observe the possible exact relation $lambda_{epsilonepsilonepsilon}/lambda_{sigmasigmaepsilon} = tan(1)$.
We develop new tools for isolating CFTs using the numerical bootstrap. A cutting surface algorithm for scanning OPE coefficients makes it possible to find islands in high-dimensional spaces. Together with recent progress in large-scale semidefinite programming, this enables bootstrap studies of much larger systems of correlation functions than was previously practical. We apply these methods to correlation functions of charge-0, 1, and 2 scalars in the 3d $O(2)$ model, computing new precise values for scaling dimensions and OPE coefficients in this theory. Our new determinations of scaling dimensions are consistent with and improve upon existing Monte Carlo simulations, sharpening the existing decades-old $8sigma$ discrepancy between theory and experiment.
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