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Register a new userA note on the large charge expansion in 4d CFT
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Authors
Gabriel Cuomo
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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 $Q$.
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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.
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 investigate a weak version of subsystem eigenstate thermalization hypothesis (ETH) for a two-dimensional large central charge conformal field theory by comparing the local equivalence of high energy state and thermal state of canonical ensemble. We evaluate the single-interval Renyi entropy and entanglement entropy for a heavy primary state in short interval expansion. We verify the results of Renyi entropy by two different replica methods. We find nontrivial results at the eighth order of short interval expansion, which include an infinite number of higher order terms in the large central charge expansion. We then evaluate the relative entropy of the reduced density matrices to measure the difference between the heavy primary state and thermal state of canonical ensemble, and find that the aforementioned nontrivial eighth order results make the relative entropy unsuppressed in the large central charge limit. By using Pinskers and Fannes-Audenaert inequalities, we can exploit the results of relative entropy to yield the lower and upper bounds on trace distance of the excited-state and thermal-state reduced density matrices. Our results are consistent with subsystem weak ETH, which requires the above trace distance is of power-law suppression by the large central charge. However, we are unable to pin down the exponent of power-law suppression. As a byproduct we also calculate the relative entropy to measure the difference between the reduced density matrices of two different heavy primary states.
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 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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