Solving the 3d Ising Model with the Conformal Bootstrap II. c-Minimization and Precise Critical Exponents

TL;DR

This work uses the conformal bootstrap to perform a high-precision study of the 3d Ising CFT by proposing that the spectrum minimizes the central charge $c$ over unitary, crossing-symmetric solutions. The authors implement a novel c-minimization (via a customized primal simplex algorithm and a fast partial-fraction conformal-block representation) to reconstruct the first several $\mathbb{Z}_2$-even operators and their OPE coefficients, revealing a sharp spectrum transition near $Δ_σ=0.518154(15)$ and evidence for operator decoupling at the Ising point. They extract precise values for $Δ_σ$, $Δ_ε$, $c$, and several higher-state dimensions ($ε'$, $T'$, etc.), and compare with other techniques, finding strong agreement on leading exponents and rich structural insights (including 2d checks and interpolating minimal-model solutions). The study also demonstrates how the spectrum rearranges across the kink and discusses implications for potential exact solvability, while outlining future extensions to higher-spin sectors, additional correlators, and other CFTs. Overall, the paper provides a robust framework for solving strongly coupled CFTs in dimensions beyond 2, with precise quantitative predictions for the 3d Ising universality class and clear avenues for further refinement and cross-method validation.

Abstract

We use the conformal bootstrap to perform a precision study of the operator spectrum of the critical 3d Ising model. We conjecture that the 3d Ising spectrum minimizes the central charge c in the space of unitary solutions to crossing symmetry. Because extremal solutions to crossing symmetry are uniquely determined, we are able to precisely reconstruct the first several Z2-even operator dimensions and their OPE coefficients. We observe that a sharp transition in the operator spectrum occurs at the 3d Ising dimension Delta_sigma=0.518154(15), and find strong numerical evidence that operators decouple from the spectrum as one approaches the 3d Ising point. We compare this behavior to the analogous situation in 2d, where the disappearance of operators can be understood in terms of degenerate Virasoro representations.

Figures (22)

• Figure 1: An upper bound on the dimension of the lowest dimension scalar $\epsilon\in \sigma\times\sigma$, as a function of $\Delta_\sigma$. The blue shaded region is allowed; the white region is disallowed. Left: The bound at $N=78$ElShowk:2012ht. Right: The bound at $N=105$, in the region near the kink. This bound is thus somewhat stronger than the previous one, and the kink is sharper.
• Figure 2: The space $\mathcal{C}_{\Delta_\sigma}$ (in blue) is the intersection of the convex cone given by the unitarity conditions $p_{\Delta,\ell}\geq 0$ with a hyperplane given by the affine constraints $p_{0,0}=1$ and Eq. (\ref{['eq:crossingsymmetry']}). It always contains a point corresponding to Mean Field Theory, and might contain points corresponding to other CFTs with a scalar of dimension $\Delta_\sigma$. We conjecture that for a special value of $\Delta_\sigma$, the 3d Ising CFT lies on the boundary of $\mathcal{C}_{\Delta_\sigma}$.
• Figure 3: Each number of derivatives $N$ gives an approximation to $\mathcal{C}_{\Delta_\sigma}$ that shrinks as $N$ increases. By repeatedly solving an optimization problem for each $N$, we can follow a path to the boundary of $\mathcal{C}_{\Delta_\sigma}$.
• Figure 4: A lower bound on $c$ (the coefficient of the two-point function of the stress tensor), as a function of $\Delta_\sigma$. Left: the bound from ElShowk:2012ht computed with $N=78$. Right: a slightly stronger bound at $N=105$ in the region near the minimum.
• Figure 5: The difference $\Delta^{\text{max}}_\epsilon -\Delta_\epsilon|_{c\to \text{min}}$ as a function of $\Delta_\sigma$, computed at $N=105$.