Upper critical dimension of the 3-state Potts model: pinning down $d_{\rm crit}(q=3)$ and the termination of its critical and tricritical fixed points

The $q=3$ Potts model has a continuous order-disorder transition in two dimensions, described by exactly solved minimal models, and a weakly first-order transition in three dimensions; its upper critical dimension $d_{\rm crit}$ — the dimension above which no interacting critical or tricritical fixed point survives and the transition is necessarily first order — therefore lies strictly between 2 and 3, but its value is not known. A conformal-bootstrap study by Chester and Su tracked the critical and tricritical 3-state Potts conformal field theories through fractional dimensions and found their operator spectra approaching each other, with the extra tricritical singlet approaching marginality near $d\approx2.5$, giving a heuristic bound $d_{\rm crit}\lesssim2.5$; however, at truncation $\Lambda=11$ the numerics become unstable before the merger is directly observed, and kinks near $d\approx2.3$ (close to the 1981 Nienhuis-Riedel-Schick lattice estimate 2.32) leave the precise value unsettled. No later work has determined $d_{\rm crit}(q=3)$ or settled the merger-and-annihilation mechanism: the non-perturbative renormalization group program that promised a determination is uncontrolled below $d\approx2.8$, fuzzy-sphere and lattice studies address only $d=3$, and the lower boundary of the second-order Potts region remains badly approximated. The problem is open: determine $d_{\rm crit}(q=3)$ with controlled evidence and establish whether the critical and tricritical branches merge and annihilate (and with what scaling).

Background

The $q$-state Potts model places on each lattice site a spin taking one of $q$ values, with a ferromagnetic interaction invariant under $S_q$, the permutation group of $q$ elements (the Ising model is the case $q=2$). Tuning the temperature gives an order-disorder transition; if a fraction of the sites is allowed to be vacant and the vacancy chemical potential is tuned as well, one reaches a tricritical point, a multicritical point at which the line of first-order transitions terminates and an extra relevant operator appears. In two dimensions both the critical and tricritical $q=3$ transitions are continuous and are described by exactly solvable minimal models — two-dimensional conformal field theories (CFTs, quantum field theories invariant under local scale transformations) with a finite spectrum of primary operators labeled by scaling dimensions $\Delta$, the eigenvalues of dilations controlling the power-law decay of correlations. As $q\to4$ in two dimensions, the spectra of the critical and tricritical branches merge and continue off into the complex plane; this merger and annihilation of fixed points is the standard mechanism by which a pair of conformal fixed points disappears when a parameter is varied, turning a continuous transition into a weak first-order transition governed by a nearby pair of complex-conjugate CFTs.

The upper critical dimension $d_{\rm crit}$ of the $q$-state Potts model is the spacetime dimension above which no interacting critical or tricritical fixed point survives; for the $q\ge3$ models under study here its transition is then necessarily first order, whereas for $q\le2$ the Gaussian (mean-field) fixed point persists and keeps the transition continuous. It is known exactly at isolated points: $d_{\rm crit}(q=1)=6$ for percolation, $d_{\rm crit}(q=2)=4$ for the Ising model, and $d_{\rm crit}(q=4)=2$. For $q=3$, Monte Carlo simulations show that the transition on the three-dimensional cubic lattice is weakly first order, so $2<d_{\rm crit}(3)<3$; yet no controlled calculation reaches this window. The cubic Landau-Ginzburg description (a classical field theory for the order parameter with a cubic invariant) is weakly coupled only near $d=6$ and strongly coupled throughout $2<d<3$, while the exact solvability at $d=2$ rests on the infinite-dimensional Virasoro algebra (the conformal symmetry algebra of two-dimensional field theories), which is absent at $d>2$. Lattice-based estimates have disagreed for decades: $(d_{\rm crit},q_{\rm crit})=(2.32,2.85)$ from the real-space renormalization-group study of Nienhuis, Riedel and Schick (1981), $(2.5,2.68)$ from Barkema and de Boer (1991), and $(3,2.45)$ from the Monte Carlo study of Lee and Kosterlitz (1991), the last meaning that in $d=3$ the transition is continuous only for $q\lesssim2.45$. It is moreover not clear whether the fractional-dimensional lattices used in these studies are analytically connected to the continuum fixed points at integer $d$, since the different studies give conflicting answers already in $d=3$.

The numerical conformal bootstrap addresses this strongly coupled window non-perturbatively. It imposes the crossing symmetry (equality of the different operator-product-expansion channels) and unitarity of the four-point functions of the relevant $S_3$-charged and singlet scalar operators $\sigma$, $\sigma'$ and $\epsilon$, which carves out the allowed space of their scaling dimensions; the Navigator function method of Reehorst et al. (2021) efficiently drives the semidefinite-program solver to the boundary of this allowed region. Chester and Su applied this in fractional $d$, truncating the conformal-block expansion at order $\Lambda=11$ and identifying the physical CFTs with the sharp tips of cone-shaped allowed regions — an identification exactly verified against the minimal models at $d=2$ — and then tracked these tips as $d$ grows. A caveat inherent to the fractional-dimensional bootstrap is that unitarity of the CFT is not rigorous away from integer $d$; the violations are expected to reside in high-twist operators (operators whose scaling dimension greatly exceeds their spin, deep in the spectrum) and to be negligible for integer $q$, an expectation supported in practice by later studies but not proven. They found the critical and tricritical spectra approaching each other, with the extra tricritical singlet $\epsilon'$ approaching marginality — an operator is marginal when its scaling dimension equals the spacetime dimension, $\Delta_{\epsilon'}=d$, so that perturbing by it neither grows nor decays — near $d\approx2.5$, from which they inferred the heuristic bound $d_{\rm crit}\lesssim2.5$. Two readings of the data remain. In the first, $d_{\rm crit}\approx2.5$, all spectra merge and marginality is reached as expected in the merger-and-annihilation scenario, which predicts

$\Delta_{\epsilon'}-d_{\rm crit}\propto\sqrt{d-d_{\rm crit}}$

while the observed approach looks linear near $2.5$ (a tension the authors flag); a curiosity here is that the pentacritical Ising model, the diagonal minimal model with the same central charge (the number classifying a two-dimensional conformal field theory) as the tricritical Potts CFT, has exact upper critical dimension $2.5$. In the second, $d_{\rm crit}\approx2.3$, where the two $\Delta_\sigma$ branches meet and the remaining dimensions show kinks, close to the 1981 lattice estimate $2.32$; if this reading is correct, the kink in $\Delta_{\epsilon'}$ near $2.3$ should move up toward the marginality line as $\Lambda$ increases. The source also notes the open caution that it is unclear whether the continuum fractional-$d$ fixed points probed by the bootstrap coincide with those sampled by fractional-dimensional lattice constructions, and looks ahead to the harder extension of the determination to the full critical curve $(d_{\rm crit},q_{\rm crit})$ for general real $q$, which is obstructed because fractional $q$ is expected to produce logarithmic, strongly non-unitary CFTs outside the reach of the unitary numerical bootstrap.

Problem Statement

Determine the upper critical dimension $d_{\rm crit}$ of the $q=3$ Potts model — the value of the spacetime dimension $d$ above which the model has no interacting critical or tricritical fixed point and its order-disorder transition is necessarily first order — together with the mechanism by which these fixed points disappear. Concretely, an answer should:

1. Establish, with controlled (convergent or rigorously bounded) evidence, whether the critical and tricritical 3-state Potts CFTs — which are the exactly solved minimal models at $d=2$ — disappear as $d$ increases via the merger-and-annihilation scenario (merging at $d=d_{\rm crit}$ and continuing into the complex plane as a complex-conjugate pair of CFTs), or by some other mechanism (the source notes as an a priori alternative that the tricritical branch could instead annihilate with the Gaussian free theory, the free-scalar fixed point); if the merger-and-annihilation scenario holds, establish also the scaling with which the extra tricritical singlet $\epsilon'$ approaches marginality as $d\to d_{\rm crit}$.

2. Determine the numerical value of $d_{\rm crit}(q=3)$ with a quantified uncertainty fine enough to discriminate between the two competing readings of the $\Lambda=11$ Chester and Su bootstrap data — the merger value $d_{\rm crit}\approx2.5$ (all spectra merging, $\Delta_{\epsilon'}\to d$) and the lower value $d_{\rm crit}\approx2.3$ (where the $\Delta_\sigma$ branches meet and the remaining scaling dimensions show kinks, near the Nienhuis-Riedel-Schick lattice estimate 2.32) — or to establish a different value, improving on the current heuristic bound $d_{\rm crit}\lesssim2.5$. The determination must be consistent with the exact $d=2$ minimal-model spectra and with the first-order nature of the $d=3$ transition.

The Chester and Su study further poses, as an explicitly harder forward-looking extension beyond the completion boundary required here, the determination of the full critical curve $(d_{\rm crit},q_{\rm crit})$ for general real $q$; that extension is obstructed by the expected logarithmic, strongly non-unitary nature of fractional-$q$ CFTs and requires methods beyond the unitary numerical bootstrap.

References

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