Finite-γ spontaneous magnetization of the layered Kac Ising model at the mean-field critical temperature and its convergence to the Lebowitz–Penrose phase diagram

In a two-dimensional Ising model whose spins interact within each horizontal layer through a ferromagnetic Kac potential of range γ⁻¹ and between adjacent layers through a nearest-neighbor vertical coupling λ, at the mean-field critical inverse temperature β=1, Cassandro, Colangeli and Presutti (J. Stat. Phys. 162, 997–1030, 2016) computed the Lebowitz–Penrose (γ→0) phase diagram: the limit free energy is the convex envelope $F_\lambda(m)=\mathrm{CE}\{-m^2/2+f_\lambda(m)\}$, giving a spontaneous magnetization $m_\lambda=\sqrt{6\lambda}+O(\lambda^{3/2})$ for small λ, where $f_\lambda$ is the free-energy density of the one-dimensional nearest-neighbor Ising model. Existence of a phase transition at finite small γ>0 at β=1 is known from companion work of Fontes, Marchetti, Merola, Presutti and Vares. What remains open is the true finite-γ phase diagram: the value of the finite-γ spontaneous magnetization, whether it converges as γ→0 to the Lebowitz–Penrose value $m_\lambda$ (convergence is only known at fixed large nonzero external field, not in the critical region where spontaneous magnetization lives), and the γ-dependence of the critical vertical coupling and of the approach of the finite-γ phase boundary to the Lebowitz–Penrose one, including possible nontrivial λ–γ scaling regimes.

Background

The problem comes from the rigorous statistical mechanics of Ising systems with Kac interactions. A Kac potential is a ferromagnetic pair interaction of long range but weak strength: spins on the same horizontal line ("layer") of the lattice $\mathbb{Z}^2$, with sites written $(x,i)$ (position $x$ within layer $i$), interact via $J_\gamma(x,y)=c_\gamma\,\gamma J(\gamma(x-y))$, where $J$ is a smooth symmetric probability density supported on $[-1,1]$ and $c_\gamma$ normalizes so that $\sum_y J_\gamma(x,y)=1$; the interaction range is $\gamma^{-1}$ while each spin's total horizontal coupling is $1$. Between adjacent layers the interaction is nearest-neighbor with coupling $\lambda>0$, and both interactions are ferromagnetic. The inverse temperature is fixed at $\beta=1$, the mean-field (Curie–Weiss) critical value of the Kac interaction alone: with $\lambda=0$ each layer is a one-dimensional system (no phase transition at any finite range, i.e. at any $\gamma>0$), and its mean-field limit is exactly critical at $\beta=1$, so the decoupled system has no spontaneous magnetization on either scale. Spontaneous magnetization means that the infinite-volume Gibbs states obtained with all-plus and all-minus boundary conditions (DLR states, the Gibbs measures specified by the Hamiltonian through the Dobrushin–Lanford–Ruelle consistency equations) differ in their average spin. The question is whether and how the vertical coupling nevertheless produces ferromagnetic order at $\beta=1$, and what the ordered phase looks like at finite Kac range.

The classical Lebowitz–Penrose (LP) theory computes the $\gamma\to0$ limit of the free energy for Kac potentials whose support is genuinely $d$-dimensional, yielding the convex envelope of the mean-field energy plus the reference-system free energy; corrections at finite $\gamma$ were studied e.g. by Merola (J. Stat. Phys. 95, 745–758, 1999). The layered model escapes this theory because the Kac support has positive codimension: the long-range interaction acts only along one-dimensional layers inside a two-dimensional system. For this model Cassandro, Colangeli and Presutti proved the LP limit

$\lim_{\gamma\to0}\lim_{L\to\infty}\frac{1}{|\Lambda|}\log Z^{\mathrm{per}}_{\gamma,h^{\mathrm{ext}},L} = -\inf_{m\in[-1,1]}\bigl\{-h^{\mathrm{ext}}m + \bigl[-\tfrac{m^2}{2}+f_\lambda(m)\bigr]\bigr\},$

where $f_\lambda(m)$ is the free-energy density at magnetization $m$ of the one-dimensional nearest-neighbor Ising model with coupling $\lambda$. The LP free energy is the convex envelope $F_\lambda(m)=\mathrm{CE}\{-m^2/2+f_\lambda(m)\}$. Since $-m^2/2+f_\lambda(m)=-\log 2+m^4/12-\lambda m^2+\cdots$ for small $\lambda$, this function is a double well, $F_\lambda$ is flat on $[-m_\lambda,m_\lambda]$, and the LP limit has spontaneous magnetization $m_\lambda=\sqrt{6\lambda}+O(\lambda^{3/2})$ — induced purely by the vertical interaction.

Two companion results by Fontes, Marchetti, Merola, Presutti and Vares frame what is and is not known. For $\beta>1$ with vertical coupling $\epsilon=\gamma^{A}$ (any large power $A$), they proved (J. Stat. Phys. 157, 407–421, 2014) a phase transition for all small $\gamma$, with the finite-γ magnetization converging to the mean-field value $\pm m_\beta$ solving $m_\beta=\tanh(\beta m_\beta)$. At the critical value $\beta=1$ they proved (J. Stat. Phys. 161, 91–122, 2015), by comparison with a reduced "chessboard" system retaining only half of the vertical bonds and using Ginibre's ferromagnetic correlation inequalities, that for every fixed $\lambda>0$ the full system has $\mu^+\ne\mu^-$ whenever $\gamma$ is small enough. So existence of order at finite small γ at $\beta=1$ is known — but this comparison argument yields no sharp value (the reduced system's LP magnetization $\sqrt{3\lambda}$ is only a lower proxy), and Cassandro–Colangeli–Presutti state as their next goal to prove "that there is a positive spontaneous magnetization also at $\gamma>0$ which converges as $\gamma\to0$ to the one found here", i.e. to $m_\lambda$. Note the subtlety of limits: their convergence theorem at fixed nonzero external field (their Theorem 2, for large fields) does not control the critical region $h^{\mathrm{ext}}\approx0$ in which spontaneous magnetization is defined, so the known field-convergence does not transfer to the $h\to0^+$ magnetization. The same authors also list, as open problems of this program, the dependence on $\gamma$ of the critical vertical coupling needed for a phase transition and whether nontrivial $\gamma$-dependent scaling regimes occur.

Problem Statement

For the two-dimensional Ising model on $\mathbb{Z}\times\mathbb{Z}$ with, on each horizontal layer, a ferromagnetic Kac interaction $J_\gamma(x,y)=c_\gamma\gamma J(\gamma(x-y))$ (J a smooth symmetric probability density supported on $[-1,1]$, $\sum_y J_\gamma(x,y)=1$) and a nearest-neighbor ferromagnetic vertical coupling $\lambda>0$, at the mean-field critical inverse temperature $\beta=1$:

1. (Finite-γ spontaneous magnetization and convergence.) For fixed small $\lambda>0$, prove that the spontaneous magnetization $m(\gamma,\lambda)$ — the average spin in the infinite-volume plus DLR state, taken after the thermodynamic limit at fixed $\gamma$ — is positive for all sufficiently small $\gamma>0$ and converges, as $\gamma\to0$, to the Lebowitz–Penrose spontaneous magnetization $m_\lambda$ defined by $F_\lambda(m)=\mathrm{CE}\{-m^2/2+f_\lambda(m)\}$ being flat on $[-m_\lambda,m_\lambda]$, where $f_\lambda$ is the free-energy density of the one-dimensional nearest-neighbor Ising model with coupling $\lambda$ (so $m_\lambda=\sqrt{6\lambda}+O(\lambda^{3/2})$ for small $\lambda$).

2. (Finite-γ phase diagram and critical behavior.) Determine the phase diagram of the system at finite $\gamma$ and its approach to the Lebowitz–Penrose phase diagram as $\gamma\to0$. This part is the conjunction of three sub-questions of different kinds, each settled — established or refuted — independently:

   (a) how the critical region near $\lambda=0$ (at $\beta=1$ the LP critical coupling is $0$ and the magnetization vanishes like $\sqrt{6\lambda}$) is resolved at finite $\gamma$;

   (b) how the critical value of the vertical coupling required for a phase transition depends on $\gamma$;

   (c) whether nontrivial $\lambda$–$\gamma$ scaling regimes occur, that is, whether a vertical coupling $\lambda$ that shrinks with $\gamma$, such as $\lambda=\gamma^a$, still produces order at finite $\gamma$ at $\beta=1$, given that for $\beta>1$ couplings of the form $\lambda=\gamma^{A}$ with $A$ any large power are known to suffice.

   The case $\lambda=\gamma^a$ named in (c) is an illustrative instance of the regime question rather than a separate deliverable: settling that single case does not settle (c), and settling any one of (a), (b), (c) does not settle the others. Item 2 is resolved only when all three sub-questions are resolved or refuted.

Item 2 is the stronger demand, so resolving it subsumes item 1, whereas resolving item 1 alone, or any single sub-question of item 2 alone, does not settle item 2.

References

1. M. Cassandro, M. Colangeli, E. Presutti: Highly anisotropic scaling limits. Journal of Statistical Physics 162(4), 997–1030 (2016). DOI 10.1007/s10955-015-1437-0; arXiv:1507.05770. https://doi.org/10.1007/s10955-015-1437-0
2. L. R. Fontes, D. H. U. Marchetti, I. Merola, E. Presutti, M. E. Vares: Phase transitions in layered systems. Journal of Statistical Physics 157, 407–421 (2014). DOI 10.1007/s10955-014-1090-z; arXiv:1406.3293. https://doi.org/10.1007/s10955-014-1090-z
3. L. R. Fontes, D. H. U. Marchetti, I. Merola, E. Presutti, M. E. Vares: Layered systems at the mean field critical temperature. Journal of Statistical Physics 161, 91–122 (2015). DOI 10.1007/s10955-015-1307-9; arXiv:1504.06767. https://doi.org/10.1007/s10955-015-1307-9
4. M. Aleandri, V. Di Giulio: A model for interfaces and its mesoscopic limit. Mathematics and Mechanics of Complex Systems 6(4), 267–291 (2018). DOI 10.2140/MEMOCS.2018.6.267. https://doi.org/10.2140/MEMOCS.2018.6.267
5. J. L. Lebowitz, O. Penrose: Rigorous treatment of the Van der Waals–Maxwell theory of the liquid-vapor transition. Journal of Mathematical Physics 7, 98–113 (1966). DOI 10.1063/1.1704821. https://doi.org/10.1063/1.1704821
6. I. Merola: Asymptotic expansion of the pressure in the inverse interaction range. Journal of Statistical Physics 95, 745–758 (1999). DOI 10.1023/A:1004503611860. https://doi.org/10.1023/A:1004503611860