Spectrum continuity and level repulsion: the Ising CFT from infinitesimal to finite $\boldsymbol\varepsilon$

Using numerical conformal bootstrap technology we perform a non-perturbative study of the Ising CFT and its spectrum from infinitesimal to finite values of $\varepsilon=4-d$. Exploiting the recent navigator bootstrap method in conjunction with the extremal functional method, we test various qualitative and quantitative features of the $\varepsilon$-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.