A Semidefinite Program Solver for the Conformal Bootstrap

TL;DR

SDPB introduces a specialized, open-source solver for polynomial matrix programs (PMPs) that arise in the conformal bootstrap, delivering dramatic performance gains by exploiting PMP structure in a custom, high-precision interior-point framework. By translating PMPs into semidefinite programs and using a block-structured, dual-primal interior-point method with Mehrotra predictor-corrector steps, SDPB achieves substantial speedups over general SDP solvers and enables high-precision bootstrap computations. The authors apply SDPB to a multi-correlator bootstrap in the 3d Ising CFT, obtaining a rigorous, tightly constrained island for the dimensions $(\Delta_\sigma,\Delta_\varepsilon)$ that improves upon Monte Carlo results and corroborates $c$-minimization. The work demonstrates both the methodological advances in PMP-to-SDP translation and interior-point optimization, and the practical impact of high-precision, multi-correlator bootstrap on nonperturbative CFT data. SDPB thus opens avenues for more complex bootstrap calculations and broader applications in numerical optimization.

Abstract

We introduce SDPB: an open-source, parallelized, arbitrary-precision semidefinite program solver, designed for the conformal bootstrap. SDPB significantly outperforms less specialized solvers and should enable many new computations. As an example application, we compute a new rigorous high-precision bound on operator dimensions in the 3d Ising CFT, $Δ_σ=0.518151(6)$, $Δ_ε=1.41264(6)$.

Figures (3)

• Figure 1: Allowed region for a $\mathbb{Z}_2$-symmetric 3d CFT with two relevant scalars, computed using SDPB with the system of correlators $\langle\sigma\sigma\sigma\sigma\rangle,\langle\sigma\sigma\epsilon\epsilon\rangle,$ and $\langle\epsilon\epsilon\epsilon\epsilon\rangle$. The blue regions correspond to $\Lambda=19,27,35,43$, in decreasing order of size. The larger black rectangle shows the current most precise Monte Carlo determinations of critical exponents in the 3d Ising CFT Hasenbusch:2011yya. The smaller black rectangle shows the estimate for $(\Delta_\sigma,\Delta_\epsilon)$ using $c$-minimization at $\Lambda=41$ for the single correlator $\langle\sigma\sigma\sigma\sigma\rangle$El-Showk:2014dwa.
• Figure 2: Allowed region for a $\mathbb{Z}_2$-symmetric 3d CFT with two relevant operators, computed with SDPB at $\Lambda=43$. The light-blue region is a zoom of the smallest region in figure \ref{['fig:multicorrelatorRegionDifferentNmax']}. The darker-blue region additionally uses symmetry of the OPE coefficients $\lambda_{\sigma\sigma\epsilon}=\lambda_{\sigma\epsilon\sigma}$. The black rectangle shows the estimate for $(\Delta_\sigma,\Delta_\epsilon)$ using $c$-minimization at $\Lambda=41$El-Showk:2014dwa.
• Figure 3: Comparison between the allowed region for the 3d Ising CFT using SDPB with $\Lambda=43$ (blue) and Monte Carlo determinations of critical exponents (dashed rectangle) Hasenbusch:2011yya. The size of the Monte Carlo rectangle is set by statistical and systematic errors associated with the simulation. By contrast, the blue region is a rigorous bound with sharp edges.

Theorems & Definitions (4)

• Theorem 2.1
• proof
• Theorem 2.2: Semidefinite Program Duality
• proof