Spectrum continuity and level repulsion: the Ising CFT from infinitesimal to finite $\mathbf{\epsilon }$

Abstract

Using numerical conformal bootstrap technology we perform a non-perturbative study of the Ising CFT and its spectrum from infinitesimal to finite values of $\epsilon =4-d$. Exploiting the recent navigator bootstrap method in conjunction with the extremal functional method, we test various qualitative and quantitative features of the $\epsilon$-expansion. We follow the scaling dimensions of numerous operators from the perturbatively controlled regime to finite coupling. We do this for ${\mathbb{Z}}_2$-even operators up to spin 12 and for ${\mathbb{Z}}_2$-odd operators up to spin 6 and find a good matching with perturbation theory. In the finite coupling regime we observe two operators whose dimensions approach each other and then repel, a phenomenon known as level repulsion and which can be analyzed via operator mixing. Our work improves on previous studies in both increased precision and the number of operators studied, and is the first to observe level repulsion in the conformal bootstrap.

arXiv comment46 pages, 21 figures, up to date with version accepted by journal

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• Analytic and numerical bootstrap for the long-range Ising model
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• Conformal boundary conditions for a 4d scalar field
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• Anomalous Dimensions in Hypercubic Theories
• Correction exponents in the chiral Heisenberg model at $1/N^2$: singular contributions and operator mixing
• On the Wilson-Fisher fixed point in the limit of integer spacetime dimensions