User:Sylvain Ribault/Bootstat2021/Poland

Applications of the numerical conformal bootstrap.

Goal: modern applications of numerical bootstrap tools. Techniques, algorithms, software.

Conformal symmetry, crossing symmetry, unitarity.

Expand four-point function in multiple channels. Conformal blocks are supposed known, unknowns are scaling dimensions and OPE coefficients.

Numerical approach: look for functional that makes our CFT data inconsistent. Search for functionals using semidefinite programming software SDPB.

Recent advances for larger scale problems i.e. more crossing relations.

• Faster software with arbitrary precision.
• Cutting surface algorithm for finding allowed CFT data.
• Tiptop algorithm for maximizing scaling dimensions.
• More efficient computation of 3d blocks.

See GitLab repository of the Bootstrap collaboration.

Original works located the 3d Ising model at a kink in boundary of allowed region of single-correlator bootstrap. Bootstrapping 3 correlators allows us to have an isolated allowed region around the Ising model. One way to shrink the island is an OPE scan: ratio $r$ of two OPE coefficients is also a search parameter in addition to scaling dimensions of two lowest scalars. This excludes degenerate solutions i.e. existence of more fields with same dimensions but different OPE coefficients. Increase numerical parameters to get about 6 significant digits for scaling dimensions and OPE coefficients.

O(N) vector model

3d O(N) vector model generalizes Ising which is case $N=1$. We also see kinks in allowed regions. Larger system again finds islands around the desired models. The O(2) model can be compared to experiments, but the bootstrap results disagree with experiments to $8\sigma$, but agree with Monte-Carlo in value and in precision, although bootstrap region is a smaller than MC allowed zone. To push the precision, do an OPE scan with 3 scaling dimensions and 3 ratios of OPE coefficients. Strategy:

• Use SDPB 2.0, parallelized, on computing clusters.
• Use hotstarting to run SDPB for fewer iterations.
• Search over scaling dimensions using Delaunay triangulation search.
• Search over ratios using a cutting surface algorithm. A functional found in doing a computation at certain values of parameters actually excludes a whole region in parameter space. Then pick a new point at the middle of the leftover allowed region, etc, until the whole region is ruled out, or restricted to a small region.

Ongoing work on the O(3) model. In this model, there is an open question of the relevance of a particular operator (tensor with 4 incides). Use a tiptop search for maximizing its dimension. Result is that the operator is relevant i.e. dimension is less than 2.99056. This shows that critical Heisenberg magnets are unstable to cubic anisotropy, and should flow to a fixed point with cubic symmetry.

More tools and approaches

Extremal functional on boundary of allowed region: can extract spectrum. Compare with lightcone bootstrap, which constrains operators at large spin.

Lorentzian inversion formula: analytic function for approximating spectrum as fct of spin.

Good agreement between numerics and analytics.

The spinning frontier

Want to study operators with spin. This leads to complicated tensor structures for correlation functions, and to computations that take more time. There is dedicated software for computing spinning conformal blocks.

3d fermion model (Gross-Neveu-Yukawa)

Fermionic version of O(N) vector model. Funny geometry of allowed regions.

SUSY extension of the model, minimal extension of 3d Ising, relevant to topological superconductors with Majorana fermions.

Outlook

• Tackle more interesting CFTs
• Larger systems of mixed correlators, including spinning operators
• Incorporate analytical insights intor numerical algorithms