Bootstrapping the O(N) Archipelago

TL;DR

The paper addresses the problem of rigorously constraining spectra of 3d CFTs with $O(N)$ symmetry by exploiting mixed correlators in the conformal bootstrap. The authors formulate crossing relations for correlators involving the $O(N)$ vector $\phi_i$ and singlet $s$, decompose operator exchanges into the $S$, $T$, and $A$ channels, and solve the resulting semidefinite programs to obtain islands in the space of leading dimensions $\Delta_{\phi}$ and $\Delta_{s}$ (and $\Delta_t$ for the symmetric tensor in some cases). They present detailed results for $N=2$ (where an $O(2)$ island sharpens the previous bounds and yields a tight $\Delta_t$ window), $N>2$ (including $N=3,4$ and $N=20$ with compatibility to the $1/N$ expansion), and compute current central charges $C_J$ with high precision, connecting to transport properties via $\sigma_{\infty}=C_J/32$. The work demonstrates that mixed-correlator bootstrap can isolate nontrivial interacting CFTs in a model-independent way, offering rigorous spectra and physically meaningful observables, and sets the stage for applying these methods to other strongly coupled theories and dimensions.

Abstract

We study 3d CFTs with an $O(N)$ global symmetry using the conformal bootstrap for a system of mixed correlators. Specifically, we consider all nonvanishing scalar four-point functions containing the lowest dimension $O(N)$ vector $φ_i$ and the lowest dimension $O(N)$ singlet $s$, assumed to be the only relevant operators in their symmetry representations. The constraints of crossing symmetry and unitarity for these four-point functions force the scaling dimensions $(Δ_φ, Δ_s)$ to lie inside small islands. We also make rigorous determinations of current two-point functions in the $O(2)$ and $O(3)$ models, with applications to transport in condensed matter systems.

Figures (11)

• Figure 1: Allowed regions for operator dimensions in 3d CFTs with an $O(N)$ global symmetry and exactly one relevant scalar $\phi_i$ in the vector representation and one relevant scalar $s$ in the singlet representation of $O(N)$, for $N=1,2,3,4,20$. The case $N=1$, corresponding to the 3d Ising model, is from Kos:2014bka. The allowed regions for $N=2,3,4,20$ were computed with $\Lambda=35$, where $\Lambda$ (defined in appendix \ref{['sec:appA']}) is related to the number of derivatives of the crossing equation used. Each region is roughly triangular, with an upper-left vertex that corresponds to the kinks in previous bounds Kos:2013tga. Further allowed regions may exist outside the range of this plot; we leave their exploration to future work.
• Figure 2: Allowed region for $(\Delta_\phi,\Delta_s)$ in 3d CFTs with $O(2)$ symmetry. The light blue region makes no additional assumptions and was computed in Kos:2013tga using the correlator $\langle\phi\phi\phi\phi\rangle$ at $\Lambda=19$. The medium blue region was computed from the system of correlators $\langle\phi\phi\phi\phi\rangle$, $\langle\phi\phi s s\rangle$, $\langle s s s s\rangle$ at $\Lambda=19$, and assumes $\Delta_\phi$ and $\Delta_s$ are the only relevant dimensions in the vector and singlet scalar channels at which contributions appear. The dark blue region is computed similarly, but additionally assumes the OPE coefficient relation $\lambda_{\phi\phi s} = \lambda_{\phi s \phi}$. This latter assumption leads to a small closed region in the vicinity of the red cross, which represents the Monte Carlo estimate for the position of the $O(2)$ model from Campostrini:2006ms.
• Figure 3: Allowed regions for $(\Delta_\phi,\Delta_s)$ in 3d CFTs with $O(2)$ symmetry and exactly one relevant $O(2)$ vector $\phi$ and singlet $s$, computed from the system of correlators $\langle\phi\phi\phi\phi\rangle$, $\langle\phi\phi s s\rangle$, and $\langle ssss\rangle$ using SDPB with $\Lambda=19,27$, and $35$ (see appendix \ref{['sec:appA']}). The smallest region (darkest blue) corresponds to $\Lambda=35$. The green rectangle represents the Monte Carlo estimate Campostrini:2006ms. The red lines represent the 1$\sigma$ (solid) and 3$\sigma$ (dashed) confidence intervals for $\Delta_s$ from experiment Lipa:2003zz. The allowed/disallowed regions in this work were computed by scanning over a lattice of points in operator dimension space. For visual simplicity, we fit the boundaries with curves and show the resulting curves. Consequently, the actual position of the boundary between allowed and disallowed is subject to some error (small compared to size of the regions themselves). We tabulate this error in appendix \ref{['sec:appA']}.
• Figure 4: Allowed region (orange) for $(\Delta_\phi,\Delta_s,\Delta_t)$ in a 3d CFT with $O(2)$ symmetry and exactly one relevant $O(2)$-vector $\phi$, $O(2)$ singlet $s$, and $O(2)$ traceless symmetric-tensor $t$. This region was computed using SDPB with $\Lambda=19$. The green rectangle represents the error bars from Monte Carlo Campostrini:2006ms and the pseudo-$\epsilon$ expansion approach Calabrese:2004ca. Note that our estimate for $\Delta_t$ in (\ref{['eq:symtensorestimate']}) was computed with $\Lambda=35$, so it is more precise than the region pictured here.
• Figure 5: Allowed regions for $(\Delta_\phi,\Delta_s)$ in 3d CFTs with $O(3)$ symmetry and exactly one relevant $O(3)$-vector $\phi$ and $O(3)$ singlet $s$, computed using SDPB with $\Lambda=19,27$, and $35$ (see appendix \ref{['sec:appA']}). The smallest region (darkest blue) corresponds to $\Lambda=35$. The green rectangle represents the Monte Carlo estimate Campostrini:2002ky.