Approaching conformal window of $O(n)\times O(m)$ symmetric Landau-Ginzburg models from conformal bootstrap

TL;DR

The paper addresses locating nontrivial RG fixed points in $O(n)\times O(m)$ symmetric Landau-Ginzburg theories in $d=3$ and identifying the conformal window using the conformal bootstrap. It develops a bootstrap setup with four-point functions of bifundamental scalars, deriving a $9\times9$ crossing-symmetry matrix for the $R\otimes R$ representations and solving with semidefinite programming to bound operator dimensions. For $m=3$ and large $n/m$ (e.g., $n=15$), the authors find evidence for four fixed points: Gaussian, Heisenberg, chiral, and anti-chiral, with symmetry-enhancement in the SS sector and a kink in the TA spin-1 channel near $\delta\approx0.515$ consistent with the anti-chiral fixed point; spectra at the kink agree with large-$n$ predictions. By varying $n$ down to $n\sim7$--$8$, they estimate the edge of the anti-chiral conformal window, providing a nonperturbative bootstrap-based determination that complements perturbative large-$n$ analyses. The results demonstrate a non-perturbative route to map conformal windows and fixed-point structures in multi-symmetry Landau-Ginzburg models, with potential implications for frustrated magnets and related QFTs.

Abstract

$O(n) \times O(m)$ symmetric Landau-Ginzburg models in $d=3$ dimension possess a rich structure of the renormalization group and its understanding offers a theoretical prediction of the phase diagram in frustrated spin models with non-collinear order. Depending on $n$ and $m$, they may show chiral/anti-chiral/Heisenberg/Gaussian fixed points within the same universality class. We approach all the fixed points in the conformal bootstrap program by examining the bound on the conformal dimensions for scalar operators as well as non-conserved current operators with consistency crosschecks. For large $n/m$, we show strong evidence for the existence of four fixed points by comparing the operator spectrum obtained from the conformal bootstrap program with that from the large $n/m$ analysis. We propose a novel non-perturbative approach to the determination of the conformal window in these models based on the conformal bootstrap program. From our numerical results, we predict that for $m=3$, $n=7\sim 8$ is the edge of the conformal window for the anti-chiral fixed points.

Figures (7)

• Figure 1: The RG flow of $O(n)\times O(m)$ Landau-Ginzburg models with sufficiently large $n$. In the present figure, $(n,m)={(50,3)}$ and the $\beta$-functions are taken from section 3 of Pelissetto:2001fi.
• Figure 2: The plot for $\Delta_c^{\mathrm{SS},0}(\delta)$ for $O(15)\times O(3)$(red dots) and $\Delta _c ^{\mathrm{S},0}$ for $O(45)$ (blue line): The vertical precision is $10^{-3}$. The location of Heisenberg fixed point is from the large $nm$ analysis reviewed in Moshe:2003xn. Those of chiral/anti-chiral fixed points are the large $n$ predictions of Pelissetto:2001fi.
• Figure 3: The plot for $\Delta_c^{{\mathrm{TA}},1}(\delta)$: The vertical precision is $4\times 10^{-4}$. See also FIG. 7 for the differentiated plot.
• Figure 4: The plot for $\Delta_c^{{\mathrm{ST}},0}(\delta)$: The vertical precision is $4 \times 10^{-4}$.
• Figure 5: The plot for $\Delta_c^{{\mathrm{TS}},0}(\delta)$: The vertical precision is $2\times 10^{-4}$.