Bootstrapping Mixed Correlators in the 3D Ising Model

TL;DR

The paper extends the conformal bootstrap to mixed correlators in 3D CFTs with a $ Z_2$ symmetry and solves the resulting quadratic crossing constraints using semidefinite programming. By focusing on the leading $ Z_2$-odd scalar $σ$ and $ Z_2$-even scalar $ε$ and assuming only these two are relevant, the authors obtain a tiny allowed region in the $(Δ_σ, Δ_ε)$ plane consistent with the 3D Ising CFT, providing evidence for $c$-minimization and suggesting the spectrum may be uniquely determined under mild assumptions. They also derive an upper bound on the second $ Z_2$-odd scalar $σ'$ and develop a rational conformal-block representation to enable efficient SDP computations. The results demonstrate the power of mixed correlator bootstrap to sharpen CFT data beyond single-correlator analyses, with strong implications for the universality and uniqueness of the 3D Ising fixed point and clear avenues for future extensions to other symmetries, dimensions, and operator spins.

Abstract

We study the conformal bootstrap for systems of correlators involving non-identical operators. The constraints of crossing symmetry and unitarity for such mixed correlators can be phrased in the language of semidefinite programming. We apply this formalism to the simplest system of mixed correlators in 3D CFTs with a $\mathbb{Z}_2$ global symmetry. For the leading $\mathbb{Z}_2$-odd operator $σ$ and $\mathbb{Z}_2$-even operator $ε$, we obtain numerical constraints on the allowed dimensions $(Δ_σ, Δ_ε)$ assuming that $σ$ and $ε$ are the only relevant scalars in the theory. These constraints yield a small closed region in $(Δ_σ, Δ_ε)$ space compatible with the known values in the 3D Ising CFT.

Figures (6)

• Figure 1: An upper bound on $\Delta_{\sigma'}$, where $(\Delta_\sigma, \Delta_\epsilon)$ are constrained to lie on the $n_\mathrm{max}=10$ single correlator bound (black dotted line in figures \ref{['fig:effectOfEpsPrimeGap']} and \ref{['fig:effectOfEpsPrimeGapZoomed']}). The sharp spike in the bound occurs when the values of $\Delta_\sigma,\Delta_\epsilon$ lie within the allowed region in figure \ref{['fig:effectOfEpsPrimeGap']}, with no assumption about $\mathbb{Z}_2$-even gaps (medium-blue shaded region). Only when $\Delta_\sigma,\Delta_\epsilon$ take these values, is it possible to have $\Delta_{\sigma'}\geq 3$, and indeed the upper bound on $\Delta_{\sigma'}$ becomes extremely weak. This bound is computed at $n_\mathrm{max}=6$, $\nu_\mathrm{max}=8$. We expect that as $n_\mathrm{max}$ increases, it becomes more sharply peaked, while the top moves down to the correct value of $\Delta_{\sigma'}$ in the 3D Ising model. At $n_\mathrm{max}=10,\nu_\mathrm{max}=14$ we have computed the stronger bound $\Delta_{\sigma'} \leq 5.41(1)$ (the dashed line) at the points $(\Delta_\sigma,\Delta_\epsilon)=(0.5181, 1.41206)$, $(0.51815, 1.41267)$, $(0.5182, 1.41312)$.
• Figure 2: Allowed region of $(\Delta_\sigma,\Delta_\epsilon)$ in a $\mathbb{Z}_2$-symmetric CFT$_3$ where $\Delta_{\sigma'}\geq 3$ (only one $\mathbb{Z}_2$-odd scalar is relevant). This bound uses crossing symmetry and unitarity for $\langle\sigma\sigma\sigma\sigma\rangle$, $\langle\sigma\sigma\epsilon\epsilon\rangle$, and $\langle\epsilon\epsilon\epsilon\epsilon\rangle$, with $n_\mathrm{max}=6$ (105-dimensional functional), $\nu_\mathrm{max}=8$. The 3D Ising point is indicated with black crosshairs. The gap in the $\mathbb{Z}_2$-odd sector is responsible for creating a small closed region around the Ising point.
• Figure 3: Allowed regions in a $\mathbb{Z}_2$-symmetric CFT${}_3$, assuming various gaps in the scalar spectrum. The dashed line is an upper bound on $\Delta_\epsilon$ using crossing symmetry and unitarity of $\langle\sigma\sigma\sigma\sigma\rangle$, with no assumptions about gaps, at $n_\mathrm{max}=6$. The black dotted line is the same bound with $n_\mathrm{max}=10$. The light blue shaded region assumes a gap $\Delta_{\epsilon'}\geq 3$ in the $\mathbb{Z}_2$-even sector. The medium blue shaded region assumes a gap $\Delta_{\sigma'}\geq 3$ in the $\mathbb{Z}_2$-odd sector, and uses crossing symmetry for the system of correlators $\langle\sigma\sigma\sigma\sigma\rangle,\langle\sigma\sigma\epsilon\epsilon\rangle,\langle\epsilon\epsilon\epsilon\epsilon\rangle$ (same as figure \ref{['fig:nmax6MulticorrelatorRegionPlot']}). The dark blue region assumes both $\Delta_{\sigma'},\Delta_{\epsilon'}\geq 3$, and uses the system of multiple correlators. All bounds other than the black dotted line are computed with $n_\mathrm{max}=6$, $\nu_\mathrm{max}=8$ (21 components for single correlator bounds, 105 components for multiple correlator bounds). The 3D Ising point is indicated with black crosshairs.
• Figure 4: Zoom in on the region of the 3D Ising point in figure \ref{['fig:effectOfEpsPrimeGap']}.
• Figure 5: Allowed and disallowed $(\Delta_\sigma,\Delta_\epsilon)$ points in a $\mathbb{Z}_2$-symmetric CFT${}_3$ with only one relevant $\mathbb{Z}_2$-odd and $\mathbb{Z}_2$-even scalar, using the constraints of crossing symmetry and unitarity for $\langle\sigma\sigma\sigma\sigma\rangle$, $\langle\sigma\sigma\epsilon\epsilon\rangle$, $\langle\epsilon\epsilon\epsilon\epsilon\rangle$ at $n_\mathrm{max}=10$ (275 components), $\nu_\mathrm{max}=14$. The light grey points are ruled out, while the dark blue points are allowed. The light blue shaded region shows the region allowed by crossing symmetry and unitarity of the single correlator $\langle\sigma\sigma\sigma\sigma\rangle$ at $n_\mathrm{max}=14$, computed in El-Showk:2014dwa. The final allowed region is the intersection of this shaded region with the region indicated by the dark blue points (see figure \ref{['fig:allowedRegionNmax10Summary']}).