Logarithmic finite-size effects on interfacial tension: the unresolved $d=3$ double-logarithmic correction, diffuse continuous-spin interfaces under antiperiodic boundaries, and the multiple-domain crossover

The interfacial tension measured in an $L^{d-1}\times L_z$ simulation box with a forced interface obeys $\gamma_{L,L_z}=\gamma_\infty-x_\perp\ln L_z/L^{d-1}+x_\parallel\ln L/L^{d-1}+\mathrm{const}/L^{d-1}$, with coefficients $x_\perp$, $x_\parallel$ that depend only on the boundary conditions and the statistical ensemble (Schmitz, Virnau, and Binder, 2014); the logarithms originate from interface translational entropy, capillary waves, and 'domain breathing'. Of the six open directions the 2014 work left, later literature has carried out three: the extension to solid-liquid interfaces and non-Ising off-lattice systems (Schmitz and Virnau, 2015), and the droplet/bubble finite-size theory (Tröster et al., 2018). The surviving open core is one objective in three linked parts: (A) whether capillary-wave marginality at $d=3$ generates a subleading double-logarithmic correction of order $L^{-2}\ln[\ln L]$ to $\gamma_{L,L_z}$ — a possibility explicitly set aside in 2014 and stated unresolved in 2018; (B) the finite-size structure of interfacial free energies for continuous-spin $O(n)$ systems with antiperiodic boundaries, where the interface is diffuse over the full box and only the exact $n\to\infty$ result is known; and (C) the quantitative crossover at $L_z\sim L_{z,0}$ to spontaneous multiple-domain formation and its relation to the domain-wall spacing $\xi_\parallel\propto L^{(3-d)/2}\exp(\gamma L^{d-1})$.

Background

Two phases in coexistence — a liquid and its vapor at saturation, or the two magnetization states of an Ising magnet below its critical temperature — are separated by an interface whose free-energy cost per unit area is the interfacial tension $\gamma$. Extracting $\gamma$ accurately from computer simulations is difficult because the simulation box is finite: the measured value depends on the linear dimension $L$ parallel to the interface, the dimension $L_z$ perpendicular to it, the boundary conditions, and the statistical ensemble, and these finite-size effects must be understood and subtracted before extrapolating to the thermodynamic-limit value $\gamma_\infty$.

For Ising-like systems, boundary conditions are used to force interfaces: periodic boundary conditions in the $z$ direction (PBC) require an even number of interfaces — typically a slab of one phase embedded in the other, bounded by two interfaces — while antiperiodic boundary conditions (APBC, where spins flip sign across the $z$ boundary) force a single interface. In the canonical ensemble the total order parameter (the magnetization $m$) is fixed; in the grand-canonical ensemble it fluctuates under the field fixing coexistence.

Schmitz, Virnau, and Binder (2014) computed $\gamma$ with an extended 'ensemble-switch method' — a thermodynamic integration that continuously interpolates the Hamiltonian between two interface-free half-boxes and the joined box containing the interfaces — and showed that for large enough $L$ and $L_z$

$$\gamma_{L,L_z}=\gamma_\infty-x_\perp\frac{\ln L_z}{L^{d-1}}+x_\parallel\frac{\ln L}{L^{d-1}}+\frac{\mathrm{const}}{L^{d-1}},$$

where the constant is non-universal but $x_\perp$ and $x_\parallel$ are universal: they depend only on the boundary condition and the ensemble (e.g. $x_\perp=1$ for APBC grand-canonical, $1/2$ for APBC canonical, $3/4$ for PBC canonical; $x_\parallel$ contains a capillary-wave part $(3-d)/2$ per interface, from Brézin and Zinn-Justin's finite-size analysis, plus breathing contributions). Three mechanisms generate the logarithms: (i) the translational entropy of an interface free to wander along $L_z$; (ii) capillary waves — thermal undulations of the interface, described at Gaussian level by a Hamiltonian quadratic in the height field whose spectrum is cut off by the finite $L$; and (iii) 'domain breathing' — the coupling of the interface position to fluctuations of the domain magnetizations when $m$ is fixed, giving a mean-square interface displacement $\langle\Delta^2\rangle=k_BT\chi_{\mathrm{coex}}L_z/(4m_{\mathrm{coex}}^2L^{d-1})$, where $\chi_{\mathrm{coex}}$ is the susceptibility of a coexisting bulk phase and $m_{\mathrm{coex}}$ its spontaneous magnetization.

Three structural questions in this framework remain unsettled. First, $d=3$ is the marginal dimension of capillary-wave theory: the interfacial width grows as $w\propto L^{(3-d)/2}$ for $d<3$ but only as $\sqrt{\ln L}$ at $d=3$, so the capillary-wave correction $L^{-(d-1)}\ln[L^{(3-d)/2}]$ degenerates there and a subleading double-logarithmic term $L^{-2}\ln[\sqrt{\ln L}]$ per interface may arise; notably, the leading capillary-wave contribution $(3-d)/2$ vanishes at $d=3$, so this double log would be the first non-vanishing interfacial-fluctuation correction in the APBC grand-canonical case. Second, for continuous-spin ($O(n)$) systems with antiperiodic boundaries the interface is diffuse over the entire box ($w=L_z$), the interfacial free-energy cost is of order $L^{d-1}/L_z$ rather than $L^{d-1}$, and no $\ln L_z$ translational term is expected — but this is established only in the exactly soluble $n\to\infty$ limit (Dantchev and Grüneberg, 2009). Third, the expansion above cannot be taken to $L_z\to\infty$ at fixed $L$: beyond a crossover length $L_{z,0}$ the system spontaneously breaks up into multiple domains, a phenomenon studied in confined geometry by Wilms et al. (2010), with typical domain-wall spacing $\xi_\parallel\propto L^{(3-d)/2}\exp(\gamma L^{d-1})$; $L_{z,0}$ is expected to be of the same order as $\xi_\parallel$, and the 2014 simulations were all taken at $L_z\ll L_{z,0}$.

Three of the open directions of the 2014 work have since been carried out by the same group: the ensemble-switch methodology with logarithmic finite-size analysis was extended to solid-liquid interfaces and off-lattice models without Ising symmetry (Schmitz and Virnau, 2015), and the droplet/bubble finite-size theory with its translational-entropy and breathing corrections was developed (Tröster et al., 2018), building on earlier work on curved-interface tensions (Tröster et al., 2012) and showing that in $d=2$ the leading curvature correction to the droplet surface tension is itself logarithmic, so the standard Tolman-length analysis (the linear curvature correction to $\gamma$ of a spherical droplet) is inapplicable there. The remaining open core is the subject of this problem.

Problem Statement

Complete the finite-size theory of interfacial free energies — that is, determine which fluctuation mechanisms generate which finite-size corrections to the interfacial tension, for which order-parameter symmetries and boundary conditions, and within which box geometries the single-slab description holds — by settling the three linked questions left open by the 2014 framework of Schmitz, Virnau, and Binder:

(A) The $d=3$ correction structure. Determine whether the measured interfacial tension $\gamma_{L,L_z}$ at the marginal dimension $d=3$ contains the subleading double-logarithmic correction per interface, of order $L^{-2}\ln[\ln L]$, that capillary-wave marginality suggests (interfacial width $w\propto\sqrt{\ln L}$), and if so with what coefficient and what dependence on the boundary conditions and the ensemble — noting that at $d=3$ the capillary-wave part of $x_\parallel$, $(3-d)/2$ per interface, vanishes, so this double-logarithmic term would be the first non-vanishing interfacial-fluctuation correction in the antiperiodic grand-canonical geometry.

(B) Diffuse continuous-spin interfaces. For continuous-spin ($O(n)$, finite $n$) systems with antiperiodic boundary conditions in the $z$ direction, determine the finite-size structure of the interfacial free energy as a function of $L$ and $L_z$: confirm or refute the expected absence of the $\ln L_z$ translational-entropy term, and derive the leading and all singular (logarithmic) corrections, recovering the exact $n\to\infty$ result (interfacial cost of order $L^{d-1}/L_z$) as a limiting case.

(C) The multiple-domain crossover. Determine quantitatively the crossover at $L_z\sim L_{z,0}$ to spontaneous formation of multiple domains: the relation between $L_{z,0}$ and the domain-wall spacing $\xi_\parallel\propto L^{(3-d)/2}\exp(\gamma L^{d-1})$, including the order-unity constant of proportionality in $L_{z,0}\sim\xi_\parallel$ and its dependence on dimension, temperature, boundary conditions, and ensemble, and the finite-size corrections to $\gamma$ in and beyond the crossover regime.

These three components jointly support one overarching objective — the completion of the universal finite-size framework for interfacial free energies, extending the established universal-coefficient formula to the marginal dimension, to diffuse interfaces, and to the boundary of its geometrical validity. The three parts (A), (B), and (C) must all be addressed.

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