In condensed matter language, this would be $(d-1)+1$ dimensions.

The model with vacancies is called the dilute Potts model.

The $q=4$ theory is described by unitary conformal field theory given by a free scalar compactified on $S_1/{\mathbb{Z}}_2$ with radius $R=1/\sqrt{2}$ [6].

Another perturbative approach was proposed using an expansion around $d=4-\epsilon$ and $q=2+\epsilon$ [21], but it was shown in [22] that the theory is not perturbative in this regime.

Fractional values of $q$ can be made sense of using the random-cluster definition of the Potts model [25].

In $d>2$, it is a priori possible that the tricritical model could annihilate with the Gaussian free theory [27].

While it is debatable if the definitions of the Potts model in fractional $d$ used in the lattice studies are valid, a good definition in $2\le d6\le 6$ is given by the Landau-Ginzburg Lagrangian [17, 18, 19].

See [32, 33, 34, 35, 36] for reviews.

See [44] for an earlier study numerical bootstrap study of 2d CFTs with $S_3$ symmetry.

These upper critical dimensions have also been verified with the numerical bootstrap in [47].

Note the spin three generator $W$, which enhances the Virasoro symmetry to ${\mathcal{W}}_3$, such that the critical 3-state Potts model is the lowest $c$ member of an infinite class of ${\mathcal{W}}_3$ minimal models [49].

Note the spin five generator $W$, which enhances the Virasoro symmetry to $\mathcal{W}(2,5)$. This is the only known CFT with $\mathcal{W}(2,5)$ symmetry, since this symmetry is exceptional and can only exist for finitely many values of $c$, unlike ${\mathcal{W}}_3$ which can exist for infinitely many values [52]. On the other hand, the tricritical theory is the lowest $c$ member of a family of unitary CFTs with $S_3$ symmetry built from parafermions [53].

Without the OPE scan, the bootstrap constraints would in principle allow for many operators with the same scaling dimension.

As defined originally in [60], this counts the number of derivatives that we use in the expansion of the conformal blocks around the crossing symmetric point. The bootstrap bounds shrink monotonically as a function of $\mathrm{\Lambda }$.

See [62, 63, 64] for other uses of this method.

The full result for the navigator for this run, as well as the others given in this work, are given in the attached Mathematica file.

In fractional $d$, there is the possibility that some high twist operators might violate unitarity, as was shown explicitly for the Ising model in [67], which would invalidate the numerical bootstrap algorithm. On the other hand, these unitarity violating operators are typically of such high twist that they do not affect the numeral optimization of the truncated crossing equations, which are barely affected by high twist operators as shown in [68]. Indeed, in [69, 70, 71, 72] bootstrap studies were carried out in fractional $d$ for both the Ising model and other universality classes, and the results in all cases matched independent predictions for the CFT.

The plot for the critical theory shows a sharp discontinuity around $d\sim 2.03$, which may suggest that the curve to the right of that discontinuity is no longer related to the critical Potts theory. On the other hand, the tricritical curve is smooth all the way to $d\sim 2.5$, and ${\mathrm{\Delta }}_{{\epsilon }^{\prime }}$ gets close to marginality just as the other scaling dimensions for each theory get close, which suggests that the critical Potts curve should also be trusted all the way to $d\sim 2.5$.

We checked this for various $d$ and $\mathrm{\Lambda }$, and it can also be seen by comparing the exact values to our estimates in $d=2$ in (10) and (11).

As defined originally in [54], this counts the number of derivatives that we use in the expansion of the conformal blocks around the crossing symmetric point. The bootstrap bounds shrink monotonically as a function of $\mathrm{\Lambda }$.