Critical Three-Dimensional Ising Model on Spheriods from the Conformal Bootstrap

TL;DR

This paper develops a conformal map from three-dimensional Euclidean space to various spherical geometries and uses bootstrap data to estimate a critical quantity on the sphere, facilitating comparisons between different computational approaches.

Contribution

It introduces a conformal mapping technique for 3D geometries and applies bootstrap data to estimate critical quantities on spherical manifolds, bridging bootstrap and QFE methods.

Findings

Estimated the fourth-order Binder cumulant on -sphere.

Provided a numerical benchmark for critical D  theory.

Facilitated comparison between bootstrap and QFE calculations.

Abstract

We construct a conformal map from $\mathbb{R}^3$ to a three-dimensional spheriod, which includes $\mathbb{S}^3$, a double-cover of the 3-ball, and $\mathbb{R} \times \mathbb{S}^2$ as limiting cases. Using the data of the critical three-dimensional Ising model on $\mathbb{R}^3$ that was computed using the conformal bootstrap method, we numerically estimate the fourth-order Binder cumulant of the critical three-dimensional $\phi^4$ theory on $\mathbb{S}^3$. We expect this estimate will enable an interesting comparison between the conformal bootstrap and future calculations of critical $\phi^4$ theory on $\mathbb{S}^3$ using the Quantum Finite Element (QFE) method.