Universality class of the $c=\frac{1}{2}\otimes c=0$ coupling: the Ising model on an uncorrelated critical site-percolation lattice at $p=p_c$

A 2018 paper by M. N. Najafi introduced 'geometric' couplings of two-dimensional critical models, in which a dynamical critical system is defined on a quenched host lattice whose site-occupation pattern is itself critical, and found empirically that the central charges combine in a simple way ($-2\oplus 0\equiv\frac{1}{2}$, $-2\oplus\frac{1}{2}\equiv\tilde 0$, where $\tilde 0$ is the self-avoiding-walk class). One elementary coupling was left explicitly open: $\frac{1}{2}\otimes 0$, the Ising model on an uncorrelated critical site-percolation lattice at occupation $p=p_c$, for which no determination of the emergent universality class (central charge, SLE diffusivity, conformal field theory) exists. The thermal side of the question is now settled — the coupled critical point is the zero-temperature endpoint $(p_c,0)$ of the random site-diluted Ising model's transition line, with $T_c(p_c)=0$ and a marginal (logarithmic) thermal sector — while the fixed-point identification of that endpoint in the geometric/conformal sense remains open.

Background

In two dimensions, a statistical-mechanical model at a continuous phase transition becomes scale invariant and, in the scaling limit, conformally invariant: it is described by a two-dimensional conformal field theory (CFT), characterized in part by its central charge $c$, a number measuring the effective number of critical degrees of freedom. The unitary minimal models form a discrete series — $c=\frac{1}{2}$ for the Ising model, $c=\frac{7}{10}$ for the tricritical Ising model, and so on — each describing one universality class. Some critical systems need non-unitary CFTs whose stress-energy tensor acts non-diagonalizably, so correlation functions can contain logarithms ('logarithmic CFTs', LCFTs): critical percolation has $c=0$ with such structure, and the Abelian/Bak–Tang–Wiesenfeld (BTW) sandpile has $c=-2$. A complementary, purely geometric characterization of two-dimensional criticality is Schramm–Loewner evolution (SLE): the scaling limit of a critical lattice interface is a random curve $\mathrm{SLE}_\kappa$ characterized by a single diffusivity $\kappa$, in correspondence with the central charge of the associated CFT (Ising domain walls have $\kappa=3$ with $c=\frac{1}{2}$; percolation hulls have $\kappa=6$ with $c=0$; self-avoiding walks have $\kappa=8/3$ with the $c=\tilde 0$ logarithmic structure).

A separate question is what happens when two different critical models are coupled. Najafi (J. Phys. A 51 (2018) 175001) studied geometric couplings $c_1\oplus c_2$: a dynamical critical model is placed on a quenched dilute lattice whose occupation pattern is itself a critical model. Taking the BTW sandpile ($c=-2$) as the dynamical model on (i) an uncorrelated critical site-percolation lattice ($c=0$) and (ii) an Ising-correlated percolation lattice ($c=\frac{1}{2}$), SLE tests of the avalanche frontiers gave $-2\oplus 0\equiv\frac{1}{2}$ (Ising class) and $-2\oplus\frac{1}{2}\equiv\tilde 0$ (self-avoiding-walk class), suggesting a closed 'algebra' of central charges under this coupling. The standard analytic tool for such questions — perturbing the ultraviolet CFT by a scaling field and following the renormalization-group flow via the c-theorem (which says the central charge decreases monotonically along such flows) — requires control of the operator content and perturbing operators, which is lacking for LCFTs such as $c=-2$ and $c=0$; the paper therefore seeks a more direct method for finding the presumable emergent CFT of such couplings.

The one elementary coupling left open in that paper is $\frac{1}{2}\otimes 0$: Ising spins on an uncorrelated, quenched critical site-percolation lattice at occupation $p=p_c$ (the percolation threshold; $p_c\simeq 0.5927$ on the square lattice). This is the random site-diluted Ising model (RSIM) at the percolation threshold. For $p>p_c$ its ferromagnetic transition line $T_c(p)$ stays in the pure Ising class, quenched dilution being marginally irrelevant in two dimensions and producing only universal logarithmic corrections; classical scaling arguments and recent numerics establish $T_c(p_c)=0$ — the critical percolation cluster sits at the lower critical (spectral) dimension for discrete-symmetry ordering — with the line approaching the endpoint logarithmically, $T_c(p)\sim -2/\log[\alpha(p-p_c)]$ and crossover exponent $\phi=1$. The coupled critical point of $\frac{1}{2}\otimes 0$ is therefore the zero-temperature endpoint $(p_c,0)$ of the RSIM phase boundary. What is not known is the fixed point of this endpoint in the geometric/CFT sense: whether its interface ensemble and spin sector are described by a known (minimal-model or logarithmic) CFT — percolation itself being the natural candidate for the geometric sector — or by a distinct universality class; no determination of its central charge or SLE diffusivity exists.

Problem Statement

Determine the emergent fixed point — the universality class — of the geometric coupling $\frac{1}{2}\otimes 0$: the two-dimensional Ising model defined on an uncorrelated, quenched critical site-percolation lattice at the percolation threshold $p=p_c$ (equivalently, the random site-diluted Ising model at its percolation endpoint, now known to be the zero-temperature point $(p_c,0)$ of the dilute-Ising transition line). The problem, posed explicitly by Najafi (J. Phys. A 51 (2018) 175001), has two mandatory parts, both of which must be supplied: (i) decide whether this coupled system is described by a known conformal field theory — a minimal model, or the $c=0$ percolation/logarithmic theory — or constitutes a distinct universality class; and (ii) determine its characterizing data — the central charge $c$ of the coupled system as a whole, covering both its geometric/interface and its spin sectors, and the SLE diffusivity $\kappa$ of its domain-wall/interface ensemble. Neither part alone resolves the problem.

References

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