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

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 ε = 4 − d.Exploiting the recent navigator bootstrap method in conjunction with the extremal functional method, we test various qualitative and quantitative features of the ε-expansion. We follow the scaling dimensions of numerous operators from the perturbatively controlled regime to finite coupling. We do this for Z 2 -even operators up to spin 12 and for 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.

Benchmarking the Ising Universality Class in $3 \le d < 4$ dimensions

“…It is interesting to compare our results with those recently reported in Ref. [20], obtained by solving a 3-correlator bootstrap with the navigator method. figures (blue circles) on a finer scale together with the estimated error of the fit (cyan shaded area).…”

“…As a matter of fact, their presence does not seem to yield problems in solving the bootstrap equations with our method. This conclusion was also reached by recent 3-correlator bootstrap studies of the critical O(N ) models [18] and the Ising model [20] in non-integer space dimensions using the navigator method [47].…”

“…We compare our data with recent results of the numerical [20] and analytic bootstrap [26][27][28][29][30][31], Monte Carlo simulations [40][41][42] and the non-perturbative RG [43,44]. We find that the data by all methods agree very well.…”

“…figures (blue circles) on a finer scale together with the estimated error of the fit (cyan shaded area). The results of [20] are drawn as red triangles: although such determinations come with no error bar, the comparison in Fig. 4 clearly shows the very good agreement between the two different bootstrap approaches at our precision level, and that the navigator method looks extremely accurate in the region d → 4.…”

“…The recent very precise numerical conformal bootstrap [14][15][16] has been formulated in continuous dimension [17,18], in particular for the Ising model in its whole range 4 > d ≥ 2 [19,20]. The interest lies in understanding how the strongly interacting Ising conformal field theory connects to a free scalar in d = 4 and to the integrable fully-solvable model in d = 2 [21,22].…”