Spectrum continuity and level repulsion: the Ising CFT from infinitesimal to finite ε

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 ℤ2-even operators up to spin 12 and for ℤ2-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

“…The conformal spectrum in dimensions 4 > d ≥ 2.6 has also been obtained in Ref. [21] via the advanced navigator bootstrap technique [37]. We use these very precise results in combination with ours to obtain a consistent description of the low-lying spectrum.…”

“…[20], and detailed in its Appendix A. Our 1-correlator numerical bootstrap approach has been surpassed by more recent implementations [16,19,21,47,48], but we find it convenient for determining the low-lying spectrum with modest computing resources. The complete determination of the conformal data for one value of d requires about 20 hours on 256 cores, corresponding to 5000 core hours.…”

“…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][21]. 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 [22,23].…”

Analytic and numerical bootstrap for the long-range Ising model

“…This suspicion could be tested by using conformal perturbation theory and comparing it to a numerical spectrum extracted using the EFM. Additionally, [88] recently found numerical evidence of level repulsion for the CFTs of the Wilson-Fisher fixed point. A similar feature can be expected to appear when studying the LRI.…”

Conformal boundary conditions for a 4d scalar field

“…Recent numerical bootstrap investigations of the non-crossing rules can be found in [27] in the context of the Ising CFT in the 4 − ϵ expansion, and in [28] in the context of N = 4 SYM with finite number of colors.…”