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Register a new userOn the $epsilon=d-2$ expansion of the Ising model
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Wenliang Li
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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.
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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 $lambda_*$, standard perturbation theory breaks down for sufficiently large $lambda_*n$. Treating $lambda_* n$ as fixed for small $lambda_*$ we show that $Delta_{phi^n}$ can be successfully computed through a semiclassical expansion around a non-trivial trajectory, resulting in $$ Delta_{phi^n}=frac{1}{lambda_*}Delta_{-1}(lambda_* n)+Delta_{0}(lambda_* n)+lambda_* Delta_{1}(lambda_* n)+ldots $$ We explicitly compute the first two orders in the expansion, $Delta_{-1}(lambda_* n)$ and $Delta_{0}(lambda_* n)$. The result, when expanded at small $lambda_* n$, perfectly agrees with all available diagrammatic computations. The asymptotic at large $lambda_* n$ reproduces instead the systematic large charge expansion, recently derived in CFT. Comparison with Monte Carlo simulations in $d=3$ is compatible with the obvious limitations of taking $varepsilon=1$, but encouraging.
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 bootstrap for flat-space four-point functions. Taking this data as input, we use a thermal Lorentzian inversion formula to compute thermal one-point coefficients of the first few Regge trajectories in terms of a small number of unknown parameters. We approximately determine the unknown parameters by imposing the KMS condition on the two-point functions $langle sigmasigma rangle$ and $langle epsilonepsilon rangle$. As a result, we estimate the one-point functions of the lowest-dimension $mathbb Z_2$-even scalar $epsilon$ and the stress-energy tensor $T_{mu u}$. Our result for $langle sigmasigma rangle$ at finite-temperature agrees with Monte-Carlo simulations within a few percent, inside the radius of convergence of the OPE.
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 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. Large spin perturbation theory, or equivalently the recently proposed Froissart-Gribov inversion integral, then allows one to reconstruct the CFT data of intermediate operators of any spin. We use this method to compute the anomalous dimensions and OPE coefficients of leading twist operators. To cubic order in $epsilon$ the double discontinuity arises solely from the identity operator and the scalar bilinear operator, making the computation straightforward. At higher orders the double discontinuity receives contributions from infinite towers of higher spin operators. At fourth order, the structure of perturbation theory leads to a proposal in terms of functions of certain degree of transcendentality, which can then be fixed by symmetries. This leads to the full determination of the CFT data for leading twist operators to fourth order.
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 $lambda$ is the perturbatively small fixed point coupling. Here we generalize this result to the fixed point of the $U(1)$ model in $3-varepsilon$ dimensions. The result interpolates continuously between diagrammatic calculations and the universal conformal superfluid regime for CFTs at large charge. In particular it reproduces the expectedly universal $O(n^0)$ contribution to the scaling dimension of large charge operators in $3d$ CFTs.
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