Does quenched bond randomness eliminate the spontaneous-magnetization jump of the two-dimensional Q-state Potts model with Q>4? (Aizenman–Wehr rounding beyond the energy density)

For the two-dimensional ferromagnetic Q-state Potts model with Q>4, the pure model undergoes a first-order transition: at the critical temperature the energy density jumps (nonzero latent heat) and the spontaneous magnetization jumps from zero to a positive value. The Aizenman–Wehr rounding theorem proves that quenched bond randomness with an absolutely continuous distribution eliminates the energy-density discontinuity, but the theorem controls only order parameters conjugate to the randomized coupling and therefore says nothing about the magnetization, which is conjugate to the non-randomized uniform field. Whether bond randomness also eliminates the spontaneous-magnetization jump at the critical point — that is, whether the disorder-rounded transition is rigorously continuous in the order parameter — is an open problem posed by Greenblatt, Aizenman and Lebowitz (2010), anticipated affirmatively by the Hui–Berker random-field mechanism and supported by extensive numerical and field-theoretic work. No rigorous resolution is known in either direction: the problem is open.

Background

The Q-state Potts model assigns to each site $x$ of the square lattice $\mathbb{Z}^2$ a spin $\sigma_x\in\{1,\dots,Q\}$ (a "color"), with ferromagnetic nearest-neighbor Hamiltonian $H=-\sum_{\langle x,y\rangle}J_{x,y}\,\delta_{\sigma_x,\sigma_y}$, where $\delta$ is the Kronecker delta and $J_{x,y}>0$. Its order parameter is the spontaneous magnetization: a small uniform field $h>0$ favoring one fixed color (say color $1$) is applied, the thermodynamic limit $\Lambda\uparrow\mathbb{Z}^2$ is taken first and the zero-field limit $h\downarrow0$ second, and with $\rho_1$ denoting the fraction of sites in color $1$ one sets $M(T)=\lim_{h\downarrow0}\lim_{\Lambda\uparrow\mathbb{Z}^2}\bigl(Q\langle\rho_1\rangle_{\Lambda,h}-1\bigr)/(Q-1)$, a normalization for which $M=1$ in a perfectly ordered single-color phase and $M=0$ in the disordered phase. In two dimensions the pure model (all $J_{x,y}=J$) has a continuous transition for $Q\le4$ and a first-order transition for $Q>4$: at the critical temperature $T_c$ the energy density jumps (nonzero latent heat) and the spontaneous magnetization jumps from $0$ to $M(T_c)>0$, with the $Q$ ordered phases coexisting with the disordered phase at the transition point.

"Quenched" bond randomness means that each coupling $J_{x,y}$ is replaced, once and for all before thermal averaging, by an independent identically distributed positive random variable; observables are computed in the random Gibbs state and then averaged over the disorder (the quenched average), as opposed to "annealed" disorder in which the couplings would be thermally equilibrated. The same field-limit construction of the magnetization, applied to each fixed realization of the couplings and then averaged over the disorder, defines the quenched-averaged spontaneous magnetization, which is the order parameter used in the question below. A mechanism proposed by Imry and Ma and made rigorous by Aizenman and Wehr states that in $d\le2$ dimensions (and $d\le4$ when a continuous symmetry is broken) such randomness rounds first-order transitions in a precise sense: the quenched free energy is continuously differentiable in any parameter of the Hamiltonian to which the randomness is coupled, so the order parameter conjugate to that parameter cannot jump. For random bonds the conjugate quantity is the energy density, so for the two-dimensional Potts model with $Q>4$ any bond randomness with an absolutely continuous distribution rigorously eliminates the latent heat. Greenblatt, Aizenman and Lebowitz restated this rounding theorem in a form that also covers quantum lattice systems, and at the same time identified its boundary: since the magnetization is conjugate to the uniform magnetic field, which is not randomized, the theorem does not address the persistence or elimination of the magnetization jump at $T_c$. Earlier, Hui and Berker had argued through a random-field mechanism — bond randomness at such transitions generates effective random fields conjugate to the order parameter — that the magnetization discontinuity should also be eliminated, and Berker extended this into the general claim that temperature-driven first-order transitions do not survive random bonds; but this is a renormalization-group argument, not a proof.

Two further facts frame the question. First, bond randomness does not destroy spontaneous magnetization at low temperature: randomizing bonds is "tangential" disorder, which can shift $T_c$ and change the nature of the critical singularity without removing the ordered phase, so the question concerns only the discontinuity of $M$ exactly at the critical point. Second, all non-rigorous evidence points to elimination: Monte Carlo simulations of the eight-state model found a randomness-induced second-order transition; perturbative renormalization-group and transfer-matrix studies (numerical diagonalization of strips combined with conformal-invariance scaling ideas) found a line of random critical points with a magnetic exponent $\beta/\nu$ ($\beta$ for the magnetization, $\nu$ for the correlation length) varying continuously with $Q$ and $\nu\approx1$; large-$Q$ analyses and an exact scale-invariant-scattering treatment of disordered two-dimensional criticality exhibit a stable disordered critical line extending to arbitrarily large $Q$. What is missing is any rigorous control of the magnetization at the critical point of the disordered model — proving it continuous, or exhibiting a genuine mixed-order transition (continuous energy density but discontinuous magnetization), is the problem.

Problem Statement

Consider the ferromagnetic nearest-neighbor $Q$-state Potts model on the square lattice $\mathbb{Z}^2$ with integer $Q\ge5$ (that is, $Q>4$), in which the couplings $J_{x,y}$ are independent identically distributed positive random variables whose distribution is absolutely continuous (has a density), fixed once and for all (quenched disorder). The pure model has a first-order transition at $T_c$: the energy density and the spontaneous magnetization both jump. The Aizenman–Wehr rounding theorem already implies that the energy-density jump is eliminated for any such bond randomness.

Prove, or disprove, the following statement: for every integer $Q\ge5$ and every absolutely continuous i.i.d. law for the positive bond strengths, the quenched-averaged spontaneous magnetization $M(T)$ of the random-bond model — the disorder average of the field-limit order parameter defined in the background — is continuous at the critical temperature $T_c$ of the disordered model, where $T_c:=\sup\{T:\ M(T)>0\}$ is the threshold for spontaneous magnetization (an almost-surely deterministic, non-random temperature by the ergodicity of the i.i.d. environment; $M$ is non-increasing in $T$ by ferromagnetic monotonicity, so $M(T)=0$ for all $T>T_c$):

$$\lim_{T\uparrow T_c} M(T) = 0,$$

so that the spontaneous-magnetization jump of the pure model is eliminated by an arbitrarily small amount of bond randomness, making the disorder-rounded transition fully continuous in the order parameter.

A disproof may take the form of a rigorous construction, for some integer $Q\ge5$ and some absolutely continuous bond distribution, of a transition at which the energy density is continuous (as required by the Aizenman–Wehr theorem) while the quenched-averaged spontaneous magnetization retains a positive limit as $T$ approaches $T_c$ from below — a mixed-order transition. Numerical or renormalization-group evidence alone does not settle the problem either way.

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