Bootstrapping phase transitions in QCD and frustrated spin systems

TL;DR

The paper applies the conformal bootstrap to $O(n)\times O(2)$-symmetric CFTs in $d=3$ to address the existence of nontrivial fixed points relevant for chiral phase transitions in QCD and frustrated spin systems. It identifies kink-like singularities in bounds for $n=3$ and $n=4$ that correspond to the chiral and collinear fixed points, while finding no evidence for an anti-chiral fixed point. The extracted operator spectra agree with prior higher-loop perturbative results, supporting the hypothesis that these transitions can be continuous and characterized by the predicted critical exponents. This work demonstrates the bootstrap's power as a non-perturbative probe of fixed points and guides future studies on related symmetry groups and mixed correlator constraints.

Abstract

In view of its physical importance in predicting the order of chiral phase transitions in QCD and frustrated spin systems, we perform the conformal bootstrap program of $O(n)\times O(2)$-symmetric conformal field theories in $d=3$ dimensions with a special focus on $n=3$ and $4$. The existence of renormalization group fixed points with these symmetries has been controversial over years, but our conformal bootstrap program provides the non-perturbative evidence. In both $n=3$ and $4$ cases, we find singular behaviors in the bounds of scaling dimensions of operators in two different sectors, which we claim correspond to chiral and collinear fixed points, respectively. In contrast to the cases with larger values of $n$, we find no evidence for the anti-chiral fixed point. Our results indicate the possibility that the chiral phase transitions in QCD and frustrated spin systems are continuous with the critical exponents that we predict from the conformal bootstrap program.

Figures (4)

• Figure 1: The bound $\Delta_c ^{\mathrm{ST},0}$ for $O(3)\times O(2)$ symmetric CFTs. Here the search space dimension is $55\times 9$, i.e., $k=10$ in the notation of Kos:2013tga.
• Figure 2: The bound $\Delta_c ^{\mathrm{AA},0}$ for $O(3)\times O(2)$ symmetric CFTs. Here the search space dimension is $36\times 9$, i.e., $k=8$ in the notation of Kos:2013tga.
• Figure 3: The bound $\Delta_c ^{\mathrm{ST},0}$ for $O(4)\times O(2)$ symmetric CFTs. Here the search space dimension is $36\times 9$, i.e., $k=8$ in the notation of Kos:2013tga.
• Figure 4: The bound $\Delta_c ^{\mathrm{AA},0}$ for $O(4)\times O(2)$ symmetric CFTs. Here the search space dimension is $66\times 9$, i.e., $k=11$ in the notation of Kos:2013tga.