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Interplay of order and disorder in two-dimensional critical systems with mixed boundary conditions

Institute for Advanced Simulation, Theoretische Physik der lebenden Materie (IAS-2), Forschungszentrum Jülich, D-52425 Jülich, Germany

Phys. Rev. E 113, 024105 – Published 4 February, 2026

DOI: https://doi.org/10.1103/14gm-31hg

Abstract

In spin systems such as the Ising model, the local order and disorder can be characterized by the order-parameter and energy density profiles $〈 σ (r_{1}) 〉$ and $〈 ε (r_{2}) 〉$ , respectively. Does increasing the order at $r_{1}$ always decrease the disorder at $r_{2}$ ? Does increasing the disorder at $r_{2}$ always decrease the order at $r_{1}$ ? The answer to these questions is contained in the cumulant response function ${〈 σ (r_{1}) ε (r_{2}) 〉}^{(cum)}$ . This correlation function vanishes in the unbounded bulk but not in systems with fixed-spin boundary conditions. Using the universal operator-product expansion of $σ (r_{1}) ε (r_{2})$ and exact results for the Ising model, we analyze ${〈 σ (r_{1}) ε (r_{2}) 〉}^{(cum)}$ in two-dimensional critical systems defined on the $x - y$ plane with mixed $+$ and $-$ boundary conditions. Particularly interesting behavior is found when either of the operators $σ$ or $ε$ is located on a “zero line” in the $x - y$ plane, along which $〈 σ (r) 〉$ vanishes. Results for half-plane, triangular, and rectangular geometries are presented.

Physics Subject Headings (PhySH)

Classical statistical mechanics

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References (27)

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Microscopically we have an Ising lattice model in mind with a ferromagnetic interaction between nearest neighbors.
For relating $ε (r)$ to the products of nearest neighbor spins see, e.g., Eq. (D2) in Ref. [6].
The corresponding correlation between spins a short mesoscopic distance $| r_{12} |$ apart is related to $〈 ε 〉$ via $〈 σ \times σ 〉 - {〈 σ \times σ 〉}_{bulk} \to - (1 / 2) {| r_{12} |}^{1 - (1 / 4)} 〈 ε 〉$ . This quantitative universal relationship follows on including the next-to-leading term in the OPE (1.1) of $σ \times σ$ ; see, e.g., Eq. (2.14) in Ref. [5].
Equations (1.3) are derived from comparing with the exact expression ${〈 σ (r_{1}) ε (r_{2}) σ (r_{3}) 〉}_{bulk} = - | r_{13} |^{3 / 4} / (2 | r_{12} | | r_{23} |)$ of the bulk three point function when expanded about $r_{1}$ and $r_{2}$ in the limit $| r_{12} | ≪ | r_{31} |, | r_{32} |$ . Higher order terms of the BOE are given in Appendix pp1.
Like the operator product $σ \times ε$ on the left hand side of the OPE (1.3) its right hand side must change sign on reversing all Ising spins and is composed of $σ$ and its descendents. Unlike the OPEs in (1.1) there is no operator-free term in (1.3).
The second term in the cumulant (1.2) is regular for $| r_{12} | \to 0$ and does not contribute to the singular behavior of the terms given by the BOE (1.3).
While $y_{0}$ denotes in Secs. 1– 3 a given distance from the real axis of arbitrary value, in Sec. 5 it denotes the value of $y$ on the midline of the triangle where ${〈 σ 〉}_{triangle}$ vanishes.
At first sight the “disorder enhancing the order” effect discussed in paragraph (iii) in the Introduction might look surprising. A qualitative understanding in the framework of a square lattice model has been given by T. W. Burkhardt. Choosing to effect the increase of the energy density at $(x_{2} = 0, y_{2} = y_{0})$ by adding a horizontal antiferromagnetic bond between Ising spins $S (- 1, y_{0})$ and $S (1, y_{0})$ , due to the $+ -$ boundary condition the preferred configuration of the additional bond will be $S (- 1, y_{0})$ = up and $S (1, y_{0})$ = down. On the neighboring spins $S (- 2, y_{0}), S (2, y_{0})$ , etc., this bond configuration acts in the same way as the boundary condition and enhances the order. An additional vertical bond between $S (0, y_{0} - 1)$ and $S (0, y_{0} + 1)$ , needed for the proper lattice symmetry of $ε$ , does not change the enhancement effect.
For the primary operators $ϕ = σ$ or $ϕ = ε$ approaching a boundary section on the $x$ axis with boundary condition $a = +$ or $-$ the “boundary operator expansion” BOE has the form $\begin{matrix} ϕ (x, y) - {〈 ϕ 〉}_{a} \to μ_{a}^{(ϕ)} y^{2 - x_{ϕ}} T (x), y \to 0 . \end{matrix}$ Here the leading boundary operator $T$ is the stress tensor, $x_{ϕ} = 1 / 8$ or 1 are the scaling dimensions of $σ$ or $ε$ , and $\begin{matrix} μ_{+}^{(σ)} = - μ_{-}^{(σ)} = - 2^{1 / 8} μ_{+}^{(ε)} = μ_{-}^{(ε)} = 4 . \end{matrix}$ For details see Eqs. (3.1)–(3.5) in Ref. [5].
In order to facilitate the conformal transformations to other geometries, the exact results of Ref. [3] are written in complex notation.
Consider the cumulant ${〈 ϕ Φ 〉}_{G}^{(cum)}$ between two primary operators $ϕ$ and $Φ$ embedded in a geometry G with a corner, such as the triangle discussed in Sec. 5. When $ϕ$ approaches the tip of the corner and $Φ$ is at a fixed finite distance from it, the cumulant takes the form $F \times {〈 T Φ 〉}_{G}^{(cum)}$ , where $T$ is a “corner operator” which is located right at the tip of the corner and $F$ depends on the “small distance” $d$ of $ϕ$ from the corner. If the two sides of the corner belong to the same universality class the distance dependence is $F \propto d^{(2 π / γ) - x_{ϕ}}$ , where $γ$ is the angle enclosed by the corner; cf. Eq. (4.5) in Ref. [6]. In case of $ε$ and $σ$ approaching the corner at point C of the equilateral triangle with $γ = π / 3$ one finds $F \propto d^{5}$ and $d^{6 - (1 / 8)}$ , respectively, since $x_{ϕ} = x_{ε} = 1$ and $x_{σ} = 1 / 8$ . Approaching a straight-line boundary such as the base side of the triangle can be viewed as approaching a corner with $γ = π$ , which for $x_{ϕ} = 1$ or $1 / 8$ yields the BOE dependencies $d^{1}$ or $d^{2 - (1 / 8)}$ . Equations (5.10)ff and panels (a) and (b) in Fig. 3 for the triangle are in agreement with these dependencies.
The small value $y_{0} = 0.2325 \times y_{C}$ of the zero of ${〈 σ 〉}_{triangle}$ on the midline is not surprising. Only the base line has boundary condition +, while the other two sides of the triangle both have $-$ . So $y_{0}$ is definitely smaller than the value $y = y_{center} = y_{C} / 3$ of the triangle's center where ${〈 σ 〉}_{triangle}$ obviously is negative. Note that the image in the $- + -$ half plane of $i y_{center}$ is $h_{center} = i \sqrt{3}$ .
Once in a given geometry $G$ the average ${〈 T (z) σ (z_{1}, {\bar{z}}_{1}) 〉}_{G}$ is known, the form of the corresponding average ${〈 L_{- p} σ (z_{1}, {\bar{z}}_{1}) 〉}_{G}$ follows most easily from averaging the inverse relation $\begin{matrix} T (z) σ (z_{1}, {\bar{z}}_{1}) = \sum_{k = - \infty}^{\infty} {(z - z_{1})}^{k - 2} (L_{- k} σ (z_{1}, {\bar{z}}_{1})) . \end{matrix}$
M. P. Mattis, Nucl. Phys. B 285, 671 (1987).
Equation (B2) in A. Squarcini, A. Maciolek, E. Eisenriegler, and S. Dietrich, JSTAT/2020/043208, contains a misprint. Instead of $2 Θ Re (T)$ its right hand side should read $- 2 Θ Re (T)$ .
For the $a$ term one needs ${〈 L_{- 1} σ 〉}_{- + -}$ for which Eq. (A26) yields $\partial_{h_{1}} {〈 σ 〉}_{- + -} = 2^{(1 / 8) - 1} \times i$ .
For $h = i j$ the Moebius transformation (4.2) reduces to $z = (j^{2} - 1 + 2 i j) / (j^{2} + 1)$ so that $z$ varies from $- 1$ to 1 along the upper unit circle of the $+ -$ plane as $j$ varies from 0 to $\infty$ .
Compare with Sec. 13.8 in H. Kober, Dictionary of Conformal Representations, Dover Publications 1957. The quantity $w_{2} \equiv (4 π^{2} / 3) {(Γ (2 / 3))}^{- 3}$ there is in terms of our $Z_{B}$ in (C2) given by $2^{- 2 / 3} 3^{3 / 4} \times 2 Z_{B}$ , (Dover Publications, 1957).
Consistency between the upper limits $Y = 2 K (k^{'})$ and $j = \infty$ in Eqs. (C6) and (C7) is guaranteed by the relation $K (k^{'}) = {(3^{- 1 / 2} π)}^{3 / 2} / [Γ (5 / 6) Γ (2 / 3)]$ .
For the mapping (D6) note that $\int_{0}^{1} d ω [{(1 + ω^{4})}^{- 1 / 2}, {(1 - ω^{4})}^{- 1 / 2}] = [1 / 2, 1 / \sqrt{2}] K (1 / \sqrt{2})$ .

Phys. Rev. E 113, 024105– Published 4 February, 2026

Vol. 113, Iss. 2 — February 2026

Received 4 October 2025
Accepted 13 January 2026

DOI: https://doi.org/10.1103/14gm-31hg

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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Interplay of order and disorder in two-dimensional critical systems with mixed boundary conditions

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