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We study the Ising model in $d=2+epsilon$ dimensions using the conformal bootstrap. As a minimal-model Conformal Field Theory (CFT), the critical Ising model is exactly solvable at $d=2$. The deformation to $d=2+epsilon$ with $epsilonll 1$ furnishes a relatively simple system at strong coupling outside of even dimensions. At $d=2+epsilon$, the scaling dimensions and correlation function coefficients receive $epsilon$-dependent corrections. Using numerical and analytical conformal bootstrap methods in Lorentzian signature, we rule out the possibility that the leading corrections are of order $epsilon^{1}$. The essential conflict comes from the $d$-dependence of conformal symmetry, which implies the presence of new states. A resolution is that there exist corrections of order $epsilon^{1/k}$ where $k>1$ is an integer. The linear independence of conformal blocks plays a central role in our analyses. Since our results are not derived from positivity constraints, this bootstrap approach can be extended to the rigorous studies of non-positive systems, such as non-unitary, defect/boundary and thermal CFTs.
We study the scaling dimension $Delta_{phi^n}$ of the operator $phi^n$ where $phi$ is the fundamental complex field of the $U(1)$ model at the Wilson-Fisher fixed point in $d=4-varepsilon$. Even for a perturbatively small fixed point coupling $lambda
We estimate thermal one-point functions in the 3d Ising CFT using the operator product expansion (OPE) and the Kubo-Martin-Schwinger (KMS) condition. Several operator dimensions and OPE coefficients of the theory are known from the numerical bootstra
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 re
We apply analytic bootstrap techniques to the four-point correlator of fundamental fields in the Wilson-Fisher model. In an $epsilon$-expansion crossing symmetry fixes the double discontinuity of the correlator in terms of CFT data at lower orders. L
In arXiv:1909.01269 it was shown that the scaling dimension of the lightest charge $n$ operator in the $U(1)$ model at the Wilson-Fisher fixed point in $d=4-varepsilon$ can be computed semiclassically for arbitrary values of $lambda n$, where $lambda