Quenched surface disorder at the two-dimensional critical Ising surface: proving marginal irrelevance and the exact logarithmic correction structure of surface free energies and critical Casimir quantities

The anisotropic square-lattice Ising model on a cylinder with an open boundary at one end and a quenched random surface field at the other (frozen boundary spins $\epsilon_m=\pm1$, averaged uniformly over all $2^M$ configurations) can be enumerated exactly at criticality up to $M=54$. The numerically exact data indicate that zero-mean surface disorder is irrelevant in two dimensions: the disorder-averaged critical Casimir amplitude and force scaling function coincide with those of the pure open cylinder, but with logarithmic corrections of unknown exact form. The surface Harris criterion makes no prediction in the marginal dimension $d=2$, and all existing arguments for irrelevance are perturbative, numerical, or restricted to Gaussian disorder. The open problem is to construct the analytic theory of this marginal irrelevance: prove or refute, for all reduced aspect ratios $\rho$ (equal to $L/M$ at isotropic couplings), the identity between disordered and pure universal surface quantities; derive the exact asymptotic form of the logarithmic corrections that the disorder generates in the surface free energy and the Casimir amplitude and force; and establish analytically the log-normal distribution of the free energy over disorder configurations, whose parameters fix the disorder contribution to the Casimir amplitude.

Background

The Ising model on a cylinder of length $L$ and circumference $M$ with direction-dependent (anisotropic) couplings is one of the few settings of statistical mechanics in which surface thermodynamics is accessible exactly. The free energy of a critical strip or cylinder decomposes as $F/(LM)=f_{\mathrm b}+f_{\mathrm s}/L+\ldots+\Delta_C(\rho)/(LM)$, where $f_{\mathrm b}$ and $f_{\mathrm s}$ are the bulk and surface free-energy densities, $\rho$ is the aspect ratio — in general the reduced aspect ratio $\rho=(L/\xi_0^{\leftrightarrow})/(M/\xi_0^{\updownarrow})$, which measures the system sizes in units of the corresponding bulk correlation-length amplitudes and reduces to $L/M$ for isotropic couplings — and the residual term is the critical Casimir contribution: at a continuous phase transition, confined critical fluctuations generate a long-ranged force between the confining boundaries, directly analogous to the quantum-electrodynamic Casimir effect but of thermal origin. The Casimir amplitude $\Delta_C(\rho)$ and the Casimir force scaling function $\vartheta(\rho)$, obtained from the derivative of the residual free energy with respect to the plate separation at criticality, are universal: they depend only on the bulk and surface universality classes and on $\rho$, not on microscopic details. For the pure two-dimensional Ising cylinder with open (free) boundaries on both ends — the $(o,o)$ ensemble — these functions are known in closed form, e.g.

$\Delta_C^{(o,o)}(\rho)=\log\frac{\eta(i\rho)\,\eta(4i\rho)}{\eta(2i\rho)^2},$

with $\eta(\tau)=q^{1/24}\prod_{n=1}^{\infty}(1-q^{n})$ ($q=e^{2\pi i\tau}$) the Dedekind eta function, and the open-plus ensemble $(o,+)$ (all boundary spins fixed) differs from it by $\frac{\log 2}{2}$ minus a correction built from $q$-Pochhammer symbols $(a;q)_\infty=\prod_{k=0}^{\infty}(1-aq^{k})$ evaluated at $q=e^{-2\pi\rho}$ (Hendrik Hobrecht and Alfred Hucht, 2020).

Cervellera, Oing, Büddefeld, Hahn and Hucht (arXiv:2409.01434, v2 2026, unpublished) treated the ensemble in which one cylinder end is open while at the other end an extra layer of frozen spins $\epsilon_m=\pm1$ couples to the surface layer with the bulk coupling strength. These frozen spins act as a quenched random surface field of zero mean: quenched means the boundary configuration is held fixed during the thermal average, and the physical disorder average is the average of the free energy over all $2^M$ configurations $\{\epsilon_m\}$ with equal weight (as opposed to the annealed average, in which the boundary spins would be sampled thermally). A subensemble $(o,mB=0)$ restricts the average to configurations with zero total boundary magnetization. Using a Woodbury tree algorithm — a binary recursion of Schur complements and matrix reductions, related to principal-minor computations, that yields the numerically exact free energy of each individual boundary configuration — the authors averaged exactly over all $2^M$ configurations for $M\le54$.

Their central finding is a hypothesis supported by careful $M\to\infty$ extrapolation: the excess Casimir amplitude $\Delta_C^{\mathrm{EX}}(\rho)\equiv\Delta_C^{(r)}(\rho)-\Delta_C^{(o,o)}(\rho)$ vanishes, with $\lim_{\rho\to\infty}\Delta_C^{\mathrm{EX}}=(2.8\pm5)\times10^{-4}$ at isotropic couplings and $(1.9\pm3)\times10^{-5}$ in the Hamiltonian limit (the extreme-anisotropy limit of the lattice, in which the longitudinal couplings diverge and the transverse couplings vanish while the critical temperature stays finite), i.e. surface disorder appears irrelevant in $d=2$, leaving the universal surface thermodynamics of the ordinary surface transition (the surface critical behavior of a surface that remains disordered at bulk criticality) unchanged. This is precisely the case that the surface Harris criterion — the generalization by H. W. Diehl and A. Nüsser (1990) of Harris's disorder-relevance criterion to surfaces — cannot decide: zero-mean surface disorder is irrelevant for the Ising class in any $d\neq2$ and marginal at $d=2$, with no prediction. Marginal irrelevance is familiar from other settings: universal exponents and amplitudes are unchanged, but slow logarithmic corrections appear. Consistent with this, the disorder-averaged data show logarithmic corrections whose exact form is unknown — in the $M\to\infty$ approach of the surface free energy at fixed $\rho$, and as slow-decaying, potentially logarithmic correction terms in the force scaling functions $\vartheta^{(o,r)}$ and $\vartheta^{(o,mB=0)}$ relative to $\vartheta^{(o,o)}$; residual deviations at small $\rho\lesssim0.3$ are attributed by the authors most likely to finite-size effects rather than to an established asymptotic law. The random ensemble $(o,r)$ and the constrained ensemble $(o,mB=0)$ both appear to lie in the $(o,o)$ universality class, whereas ensembles with fixed nonzero boundary magnetization (e.g. $(o,mB=3/4)$) lie in the $(o,+)$ universality class, whose amplitude exceeds $\Delta_C^{(o,o)}$ by $\frac{\log 2}{2}$ minus a $q$-Pochhammer correction.

Two further structures complete the picture. First, the free energy of a single boundary configuration, as a random variable over the $2^M$ configurations, is distributed asymptotically log-normally (a log-normal random variable is one whose logarithm is Gaussian), with width $\sigma(M)\simeq\sigma_0/\sqrt{f_0M}$ and a cutoff at $-f_0M$; the first moment of this distribution feeds the Casimir amplitude through $\Delta_C^{\mathrm{EX}}=-(\mu_0(\rho)+\sigma_0^2/2)$, so disorder fluctuations contribute directly to the averaged thermodynamics. Second, for Gaussian (random-matrix) rather than discrete $\pm1$ surface disorder, the companion work of Nils Gluth, Thomas Guhr and Alfred Hucht (2024) computed the characteristic function of the free-energy distribution analytically — log-normal at large finite $M$, Gaussian as $M\to\infty$ — while explicitly leaving the discrete $\pm1$ case open. The remaining prior theory for the random surface field consists of the perturbative boundary renormalization-group treatment of J. L. Cardy (1991), which finds a weak random boundary field marginally irrelevant at the ordinary surface transition; a replica renormalization-group analysis of F. Iglói, L. Turban and B. Berche (1991) reaching the same conclusion; a proposed replica boundary scattering theory of A. De Martino, M. Moriconi and G. Mussardo (1998) for Gaussian disorder; and Monte Carlo simulations of M. Pleimling, F. A. Bagaméry, L. Turban and F. Iglói (2004). None treats the discrete lattice ensemble at order-one disorder strength, and none computes strip Casimir quantities or exact logarithmic-correction amplitudes.

Problem Statement

Determine analytically the effect of quenched surface disorder on the critical surface thermodynamics of the two-dimensional Ising model in the marginal dimension. The reference model is the anisotropic square-lattice Ising cylinder at criticality with an open boundary at one end and a quenched random surface field at the other — frozen boundary spins $\epsilon_m=\pm1$ coupled to the surface layer at bulk coupling strength, with the physical average taken uniformly over all $2^M$ boundary configurations (ensemble $(o,r)$) — as a function of the reduced aspect ratio $\rho=(L/\xi_0^{\leftrightarrow})/(M/\xi_0^{\updownarrow})$, which equals $L/M$ for isotropic couplings.

(1) Irrelevance. Prove, or refute with controlled evidence, the hypothesis that the disorder-averaged universal surface quantities coincide with those of the pure open cylinder for all $\rho$: $\Delta_C^{(r)}(\rho)=\Delta_C^{(o,o)}(\rho)$ and $\vartheta^{(r)}(\rho)=\vartheta^{(o,o)}(\rho)$, equivalently $\Delta_C^{\mathrm{EX}}(\rho)\equiv0$. This would settle, for the Ising class, the marginal $d=2$ case of the surface Harris criterion, for which existing perturbative and numerical arguments do not constitute a proof at order-one disorder strength.

(2) Exact logarithmic correction structure. Derive the exact asymptotic form of the logarithmic corrections generated by the marginally irrelevant disorder: the leading logarithmic terms in the $M\to\infty$ approach of the disorder-averaged surface free energy and Casimir amplitude at fixed $\rho$, and the exact form of the slow-decaying corrections (numerically indicated as potentially logarithmic) in the disordered force scaling functions $\vartheta^{(o,r)}$ and $\vartheta^{(o,mB=0)}$ relative to the pure $\vartheta^{(o,o)}$.

(3) Distribution theory. Establish analytically the distribution of the free energy over the $2^M$ boundary configurations for discrete $\pm1$ disorder: the observed asymptotic log-normal form, its cutoff $f_0M$, its width scaling $\sigma(M)\simeq\sigma_0/\sqrt{f_0M}$, and the parameters $\mu_0(\rho)$ and $\sigma_0$ entering the disorder contribution to the Casimir amplitude via $\Delta_C^{\mathrm{EX}}=-(\mu_0(\rho)+\sigma_0^2/2)$.

These three deliverables form one coherent scientific objective — the analytic theory of marginal surface disorder at the two-dimensional critical Ising surface: (1) is its leading statement, (2) is its quantitative imprint (marginality expresses itself precisely through the logarithmic corrections), and (3) supplies the mechanism by which the quenched average builds the disorder contribution to the universal quantities. A solution may be given in the universal scaling limit; both the isotropic couplings and the Hamiltonian limit (extreme anisotropy, with longitudinal couplings diverging and transverse couplings vanishing at fixed critical temperature, where fixed $\rho$ requires $L=\rho M r_\xi\to\infty$ with diverging anisotropy ratio $r_\xi$) provide admissible regularizations. A refutation of (1) must establish a nonzero limiting difference with error control that excludes finite-size artifacts; such a controlled refutation resolves the task, since (2) and (3), which presuppose the marginal irrelevance of the disorder, are required only under the affirmative outcome of (1).

References

1. Luca Cervellera, Oliver Oing, Jan Büddefeld, Alexander Hahn, Alfred Hucht, 'The anisotropic square lattice Ising model with quenched surface disorder', arXiv:2409.01434 (v1 2024, v2 5 Aug 2026, unpublished preprint), doi:10.48550/arxiv.2409.01434, https://arxiv.org/abs/2409.01434
2. J. L. Cardy, 'The Ising model in a random boundary field', Journal of Physics A: Mathematical and General 24, L1315-L1319 (1991), doi:10.1088/0305-4470/24/22/003, https://doi.org/10.1088/0305-4470/24/22/003
3. H. W. Diehl, A. Nüsser, 'Critical behavior at dirty surfaces', Zeitschrift für Physik B Condensed Matter 79, 69-78 (1990), doi:10.1007/BF01387827, https://doi.org/10.1007/BF01387827
4. F. Iglói, L. Turban, B. Berche, 'Critical behaviour near multiple junctions and dirty surfaces in the two-dimensional Ising model', Journal of Physics A: Mathematical and General 24, L1031-L1036 (1991), doi:10.1088/0305-4470/24/17/012, arXiv:cond-mat/0106626, https://arxiv.org/abs/cond-mat/0106626
5. A. De Martino, M. Moriconi, G. Mussardo, 'Reflection scattering matrix of the Ising model in a random boundary magnetic field', Nuclear Physics B 509, 615-636 (1998), doi:10.1016/S0550-3213(97)00644-5, https://doi.org/10.1016/S0550-3213(97)00644-5
6. M. Pleimling, F. A. Bagaméry, L. Turban, F. Iglói, 'Logarithmic corrections in the two-dimensional Ising model in a random surface field', Journal of Physics A: Mathematical and General 37, 8801-8809 (2004), doi:10.1088/0305-4470/37/37/003, https://arxiv.org/abs/cond-mat/0402188
7. Nils Gluth, Thomas Guhr, Alfred Hucht, 'Random matrices and the free energy of Ising-like models with disorder', SciPost Physics 17, 122 (2024), doi:10.21468/SciPostPhys.17.4.122, https://arxiv.org/abs/2312.13231
8. H. W. Diehl, 'The theory of boundary critical phenomena', International Journal of Modern Physics B 11, 3503-3523 (1997), doi:10.1142/S0217979297001751, https://doi.org/10.1142/S0217979297001751
9. Hendrik Hobrecht, Alfred Hucht, 'Anisotropic scaling of the two-dimensional Ising model II: surfaces and boundary fields', SciPost Physics 8, 032 (2020), doi:10.21468/SciPostPhys.8.3.032, https://arxiv.org/abs/1805.00369
10. A. Maciołek, O. Vasilyev, V. Dotsenko, S. Dietrich, 'Fluctuation induced forces in critical films with disorder at their surfaces', Journal of Statistical Mechanics: Theory and Experiment 2017, 113203 (2017), doi:10.1088/1742-5468/aa95f3, https://arxiv.org/abs/1703.03326
11. Anatoly Konechny, 'Ising model in a boundary magnetic field with random discontinuities', Journal of Physics A: Mathematical and Theoretical 55, 435401 (2022), doi:10.1088/1751-8121/ac9937, https://arxiv.org/abs/2206.03903
12. Soichiro Shimamori, Yifan Wang, 'Subdimensional disorder and logarithmic defect', Journal of High Energy Physics 2026, 129 (2026), doi:10.1007/JHEP07(2026)129, arXiv:2510.13964, https://arxiv.org/abs/2510.13964
13. Sheng Yang, Xinyu Sun, Shao-Kai Jian, 'Boundary Criticality at the Nishimori Multicritical Point', Physical Review Letters 137, 067101 (2026), doi:10.1103/khwm-xzvw, arXiv:2512.19523, https://arxiv.org/abs/2512.19523
14. Dorian Przetakiewicz, Stefan Wessel, Francesco Parisen Toldin, 'Boundary operator product expansion coefficients of the three-dimensional Ising universality class', Physical Review Research 7, L032051 (2025), doi:10.1103/qkq6-b6hx, arXiv:2502.14965, https://arxiv.org/abs/2502.14965
15. Maxime Clusel, Jean-Yves Fortin, '1D action and partition function for the 2D Ising model with a boundary magnetic field', Journal of Physics A: Mathematical and General 38, 2849-2871 (2005), doi:10.1088/0305-4470/38/13/003, https://arxiv.org/abs/cond-mat/0408657
16. Gaurav Tenkila, Romain Vasseur, Andrew C. Potter, 'Boundary Criticality in the 2d Random Quantum Ising Model', arXiv:2410.19038 (2024), https://arxiv.org/abs/2410.19038
17. A. C. D. van Enter, K. Neto\v{c}n'y, H. G. Schaap, 'On the Ising Model with Random Boundary Condition', Journal of Statistical Physics 118, 997-1056 (2005), doi:10.1007/s10955-004-2138-2, arXiv:math-ph/0408024, https://arxiv.org/abs/math-ph/0408024