New Developments in the Numerical Conformal Bootstrap

TL;DR

The paper surveys recent advances in the numerical conformal bootstrap, highlighting software (e.g., SDPB 2.0, scalar_blocks, blocks_3d, autoboot, hyperion, simpleboot) and algorithmic innovations (Delaunay triangulation, cutting surface, tiptop, navigator function, skydive) that have expanded the reach of bootstrap studies. It documents concrete physics results enabled by these tools, including rigorous Ising-CFT data with emergent supersymmetry, boundary and bulk-to-boundary analyses, and detailed studies of QED$_3$-type gauge theories and Gross-Neveu–Yukawa models, often achieving island-like constraints or sharp bounds that were previously out of reach. The navigator and skydive methods, in particular, provide new ways to navigate high-dimensional theory spaces and accelerate SDP solving, enabling multi-parameter scans and robust optimization. Together, these developments push toward precise, rigorous spectral data for a wide range of CFTs, including gauge theories and multiscalar systems, with significant implications for critical phenomena and conformal dynamics across dimensions.

Abstract

The numerical conformal bootstrap has become in the last 15 years an indispensable tool for studying strongly coupled CFTs in various dimensions. Here we review the main developments in the field in the last 5 years, since the appearance of the previous comprehensive review \cite{Poland:2018epd}. We describe developments in the software ({\tt SDPB 2.0}, {\tt scalar\_blocks}, {\tt blocks\_3d}, {\tt autoboot}, {\tt hyperion}, {\tt simpleboot}), and on the algorithmic side (Delauney triangulation, cutting surface, tiptop, navigator function, skydive). We also describe the main physics applications which were obtained using the new technology.

Figures (12)

• Figure 1: (Color online) Allowed region from Atanasov:2022bpi for the scaling dimensions of the leading parity-odd scalars ${\sigma, \sigma'}$ in the 3d ${\cal N}=1$ super-Ising model. https://creativecommons.org/licenses/by/4.0/.
• Figure 2: (Color online) On the top: The allowed region in the space of dimensions of the fermion bilinear ($a$) and the monopole ($\mathcal{M}_{2\pi}$) from Ref. He:2021sto, https://creativecommons.org/licenses/by/4.0/. The gap assumptions are imposed to be compatible with the conformal phase scenario on triangular lattice and the Kagome lattice. The green (light gray) star and the dashed error box represent the results of a Monte Carlo simulation, while the blue point is from the large-$N_f$ expansion. On the bottom: The allowed region in the space of dimensions of the fermion bilinear ($\Delta_r$) and the monopole ($\mathcal{M}_{1}$) from Ref. Albayrak:2021xtd. Here, the green (light gray) dot indicates the result from the large-$N_f$ expansion, while the red dashed box is from a Monte Carlo simulation. See Figure 5 of Albayrak:2021xtd for details.
• Figure 3: (Color online) Feasibility bounds from bootstrapping non-abelian currents, Ref. He:2023ewx. On the top: The lowest operator in the $(+, S\bar{S}$, $\ell=2)$ sector v.s. the sector $(+, Adj^+, \ell=2)$ in $SU(100)$ CFT, where $Adj^+$ is the adjoint representation. The light, medium, and dark orange bounds are for $\Lambda=19,23,27$, respectively. Blue (dark gray) dot is the large-$N$ result. On the bottom: $\gamma$ versus the lowest operator in the $(+, S\bar{S}$, $\ell=2)$ in $SU(100)$ CFT. $\Lambda=19$. $\gamma$ is a parameter appearing in the current, current, stress tensor 3-point function, and it obeys the conformal collider bounds $|\gamma|\leqslant 1/2$HofmanConformalCollider2008. In both cases, mild gaps in several sectors are imposed. The bounds are insensitive to those gap assumptions.
• Figure 4: (Color online) The blue (dark gray) island is the allowed region from Chester:2019ifh for the scaling dimensions of $s,\phi,t$. The green (gray) box indicates results from the Monte Carlo studies Hasenbusch:2011zwvHasenbusch:2019jkj. The brown (gray) planes represent the $1\sigma$ confidence interval from the experiment PhysRevB.68.174518. Figure from Chester:2019ifh, https://creativecommons.org/licenses/by/4.0/.
• Figure 5: (Color online) Right: zoomed-in bootstrap islands for $N=2,4,8$$O(N)$ GNY model from Erramilli:2022kgp at $\Lambda=35$, projected onto the $(\Delta_\sigma,\Delta_\epsilon)$ plane. Left: zoomed-out view of the same islands. Dotted blue curve: perturbative estimates in the large-$N$ expansion. Orange (dark gray) boxes: Borel-resummations of the $(4-\epsilon)$-expansion Ihrig:2018hho. The "x" indicates the location of the $N=1$ island of Atanasov:2022bpi. Light blue (light gray) region: the general $\sigma$-$\epsilon$ bootstrap bounds with the assumption $\Delta_{\sigma'}>3$ from Atanasov:2018kqw. Bootstrap islands do not overlap with the orange (dark gray) boxes for $N=2,4,8$, implying that the error bar of the Borel-resumed results was underestimated. Figure from Erramilli:2022kgp, https://creativecommons.org/licenses/by/4.0/, colors modified.