Upper critical dimension of the 3-state Potts model

Abstract

We consider the 3-state Potts model in $d\ge 2$ dimensions. For $d$ less than the upper critical dimension $d_{\text{crit}}$, the model has a critical and a tricritical fixed point. In $d=2$, these fixed points are described by minimal models, and so are exactly solvable. For $d>2$, however, strong coupling makes them difficult to study and there is no consensus on the value of $d_{\text{crit}}$. We use the numerical conformal bootstrap to compute critical exponents of both the critical and tricritical fixed points for general $d$. In $d=2$ our results match the expected values, and as we increase $d$ we find that the critical exponents of each fixed point get closer until they merge near $d_{\text{crit}}\lesssim 2.5$.

5 pages plus appendices, 5 figures, v3 minor typos corrected