Carving out OPE space and precise $O(2)$ model critical exponents

TL;DR

This work develops and applies a cutting surface approach to the numerical conformal bootstrap, enabling efficient scanning of OPE coefficient space to carve islands for the 3d $O(2)$ model in high-dimensional data spaces. By coupling the cutting surface method with large-scale semidefinite programming and hot-starting, the authors obtain high-precision, rigourously constrained dimensions and OPE coefficients for the lowest-$O(2)$ scalars $s$, $\phi$, and $t$, and extract central charges and related quantities. The resulting islands show excellent agreement with Monte Carlo results and sharpen the longstanding $8\sigma$ discrepancy with experiments, which remains unresolved by bootstrap arguments. The work also reports nonrigorous extremal-functional estimates for additional low-lying operators and introduces scalable techniques—including Delaunay triangulation and bounding ellipsoids—that will be broadly applicable to other large-scale bootstrap problems and CFT data extractions.

Abstract

We develop new tools for isolating CFTs using the numerical bootstrap. A "cutting surface" algorithm for scanning OPE coefficients makes it possible to find islands in high-dimensional spaces. Together with recent progress in large-scale semidefinite programming, this enables bootstrap studies of much larger systems of correlation functions than was previously practical. We apply these methods to correlation functions of charge-0, 1, and 2 scalars in the 3d $O(2)$ model, computing new precise values for scaling dimensions and OPE coefficients in this theory. Our new determinations of scaling dimensions are consistent with and improve upon existing Monte Carlo simulations, sharpening the existing decades-old $8σ$ discrepancy between theory and experiment.

Figures (12)

• Figure 1: Schematic representation of the $^4$He phase diagram. Figure taken from tilley1990superfluidity.
• Figure 2: 3d region corresponding to our new $O(2)$ island using the $\{\phi_i, s, t_{ij}\}$ system and OPE scans at $\Lambda = 43$ (blue). The result is compared with the best fit values of $\Delta_{s}$ to ${}^4$He data Lipa:2003zz (brown planes) and the region for $\{\Delta_{\phi}, \Delta_s, \Delta_t\}$ reported by the Monte Carlo studies PhysRevB.84.125136Hasenbusch:2019jkj (green box).
• Figure 3: Example allowed regions $\mathcal{A}_1,\dots,\mathcal{A}_{12}$ of OPE space during the cutting surface algorithm for scanning over OPE coefficients. This example is drawn from our calculation of the $O(2)$ model island with derivative order $\Lambda=43$. We plot OPE space after applying the affine transformation described in figure \ref{['fig:boundingellispoid']}, which turns the initial bounding ellipsoid into the unit sphere. For each allowed region $\mathcal{A}_n$, we show the point $[\lambda_n]$ most recently ruled out by SDPB in red. This point is typically very close to the boundary of the allowed region. We also show the next point to be tested $[\lambda_{n+1}]$ in blue. We choose the blue point close to the center of $\mathcal{A}_n$. In the final frame, SDPB gives primal feasible for the blue point.
• Figure 4: Allowed points in external scalar OPE coefficient space, found while computing the allowed island in dimension space, for $\Lambda=27$ (yellow), $\Lambda=35$ (blue), and $\Lambda=43$ (red), together with a choice of bounding ellipsoid (gray). For each set of points, we also show their convex hull in the same color. To plot the points, we applied an affine transformation to make the $\Lambda=27$ region roughly spherical. The relationship between the displayed coordinates $x,y,z$ and the OPE coefficients is $\lambda_\mathrm{ext}=(751.0591846177696 - 362.65959721052656x - 131.334377405401y - 41.46952958591952z, 1, 3383.753238900843 + 695.8131625006117x - 1729.4094085965235y - 607.9744222068027z, -12562.290081255807 + 123.88628689820867x - 3799.4579787849975y + 10949.506824631871z)$. After finding the $\Lambda=27$ points, we chose the gray sphere as a bounding ellipsoid for the computation with $\Lambda=35$. No $\Lambda=35$ (blue) points are near the edge of the bounding ellipsoid, which justifies this choice. We used the same bounding ellipsoid for the computation with $\Lambda=43$. Again, no $\Lambda=43$ (red) points are near the edge of the bounding ellipsoid.
• Figure 5: Number of iterations of SDPB in each step of the cutting surface algorithm, for our computation of the $O(2)$ model island with $\Lambda=43$. Hot-starting drastically reduces the number of iterations throughout the computation. The blue paths represent OPE scans that eventually terminate by ruling out a point in dimension space. The red paths represent scans that eventually terminate by finding an allowed (primal) point. We mark the end of each path with a dot. At the beginning of the computation, a small number of points require $\sim 200$SDPB iterations during the first step of the cutting surface algorithm. Once the checkpoints from those SDPB runs have been generated, hot-starting ensures that most subsequent runs take $\lesssim 20$SDPB iterations. The first 10-20 steps of the cutting surface algorithm typically require 1-15 SDPB iterations each. If the point is allowed, the algorithm typically finds it within 20 steps. If the point is disallowed, subsequent steps of the cutting surface algorithm take fewer iterations, with the last several steps requiring 1 iteration each.