Completing the corner-operator expansion (COE) of critical profiles near corners: the next nonvanishing apex operators and their amplitudes

The corner-operator expansion (COE), introduced by Eisenriegler in 2023, describes critical density profiles near the corner of a two-dimensional bounded system by expanding in local 'apex operators' whose amplitudes depend only on the corner's opening angle and the two boundary universality classes meeting there. In its present form the COE is limited to leading order, to the Ising universality class, and to the bulk critical point, and it is a field-theoretic construction verified by Monte Carlo simulation. This problem asks for the completion of the COE at its sharpest open point: the classification and explicit computation of the next nonvanishing apex operators and their amplitudes, including the full angular dependence, in configurations where symmetry forces the leading apex average to vanish. The source work notes that the expansion is based on general principles of conformally invariant field theories in two spatial dimensions and is not limited to the Ising model; this generality is background context, not an additional completion requirement.

Background

Near a continuous phase transition, a two-dimensional system becomes scale invariant, and at the bulk critical point $T=T_c$ it is conformally invariant, which strongly constrains correlation functions in bounded domains. The Ising universality class has two relevant bulk scaling operators: the order parameter $\sigma$ (scaling dimension $x_\sigma=1/8$) and the energy density $\epsilon$ (scaling dimension $x_\epsilon=1$). Boundary conditions fall into boundary universality classes; for the Ising model these are $+$ and $-$ (spins fixed to point up or down along the boundary) and $f$ (free boundary). Near a flat boundary, the boundary-operator expansion (BOE) expands the average $\langle\phi\rangle$ of a bulk operator $\phi$ in operators localized on the boundary, with the stress tensor $T$ (the operator generating conformal transformations) typically leading; Cardy (1984) established the role of conformal invariance in such surface critical behavior, and Burkhardt and Xue (1991) and Burkhardt and Eisenriegler (2021) obtained exact Ising profiles near points where the boundary condition switches between two classes. When a boundary abruptly changes direction, the relevant singular point is a corner. Eisenriegler (Phys. Rev. E 108, 044133 (2023)) studied critical Ising profiles in rectangles with mixed boundary conditions (e.g. $+$ on the vertical and $-$ or $f$ on the horizontal sides), obtained via conformal mappings from the upper half plane, and found an intricate dependence on the aspect ratio, including 'zero lines' separating regions of positive and negative energy density that originate and end at the mixed corners. To describe profiles near a corner systematically, that paper introduced the corner-operator expansion (COE): when $\phi$ approaches the apex of a wedge of opening angle $\gamma=\pi/g$ bounded by conditions $a$ and $b$, the deviation of $\langle\phi\rangle$ from the unperturbed wedge profile is expanded in apex operators located at the tip, $\langle\phi\rangle-\langle\phi\rangle_{a|b}\to\mathcal{F}_{a|b}^{(\phi)}(x,y)\,\langle\mathcal{Y}\rangle$. For equal boundary classes the leading apex operator $\mathcal{T}$ has scaling dimension $2g=2\pi/\gamma$, except for the order parameter $\sigma$ at a free-free ($f|f$) corner, where the expansion is instead governed by a corner operator $\mathcal{S}$ of scaling dimension $g\,x_s$ (in the Ising model $x_s=1/2$, hence $g/2$); for different classes it is an operator $\mathcal{Y}$ of dimension $g=\pi/\gamma$, so near a right-angle corner the amplitude behaves as $|z|^{2-x_\phi}$ with a computable angular factor. The amplitudes are local properties of the angle and the boundary pair (calculable in a wedge with distant boundary-condition switches), while the apex-operator averages $\langle\mathcal{Y}\rangle$ carry the dependence on the distant geometry such as the rectangle's aspect ratio. For $\gamma=\pi$ the COE reduces to the flat-boundary BOE. Explicit amplitudes were derived for the Ising corners $+|-$ and $+|f$ for arbitrary $g$, and the leading-order COE was confirmed in semi-infinite strips, rectangles, and squares, and against Monte Carlo simulations with no adjustable parameters. However, in symmetric configurations — e.g. squares with $(+,+,f,f)$ or $(-,+,-,+)$ conditions on the (NE,NW,SW,SE) sides — the symmetry of the profile is incompatible with the angular form of the leading amplitude, forcing $\langle\mathcal{Y}\rangle=0$; the paper shows the correction is then of higher order but computes it only in a single instance ($\langle\sigma\rangle$ along the diagonal of the $(+,+,f,f)$ square, where the exponent becomes $4-x_\sigma$ rather than $2-x_\sigma$), and merely notes the higher order for the $(-,+,-,+)$ square. The construction is confined to $T=T_c$, to the Ising class, and to the field-theoretic level.

Problem Statement

Complete the corner-operator expansion (COE) of critical density profiles near the corners of two-dimensional bounded systems beyond its current leading-order form. The objective is: in configurations where the symmetry of the shape and boundary conditions forces the leading apex average $\langle\mathcal{Y}\rangle$ to vanish — such as squares with $(+,+,f,f)$ or $(-,+,-,+)$ conditions on the (NE,NW,SW,SE) sides — classify the next nonvanishing apex operators for arbitrary combinations of boundary universality classes and corner opening angles $\gamma=\pi/g$, and compute their amplitudes explicitly, including the full angular dependence. The source work establishes only that the correction is of higher order, and in a single case — $\langle\sigma\rangle$ along the diagonal of the $(+,+,f,f)$ square — finds the exponent $4-x_\sigma$ instead of $2-x_\sigma$, merely noting the higher order for the $(-,+,-,+)$ square.

References

1. E. Eisenriegler, Critical behavior in rectangles with mixed boundaries, Phys. Rev. E 108, 044133 (2023). DOI: 10.1103/PhysRevE.108.044133; arXiv:2307.07831, https://arxiv.org/abs/2307.07831
2. E. Eisenriegler, Interplay of order and disorder in two-dimensional critical systems with mixed boundary conditions, Phys. Rev. E 113, 024105 (2026). DOI: 10.1103/14gm-31hg; arXiv:2510.04238, https://arxiv.org/abs/2510.04238
3. T. W. Burkhardt and E. Eisenriegler, Two-dimensional critical systems with mixed boundary conditions: Exact Ising results from conformal invariance and boundary-operator expansions, Phys. Rev. E 103, 012120 (2021). DOI: 10.1103/PhysRevE.103.012120; arXiv:2008.04274, https://arxiv.org/abs/2008.04274
4. T. W. Burkhardt and T. Xue, Conformal invariance and critical systems with mixed boundary conditions, Nucl. Phys. B 354, 653 (1991). DOI: 10.1016/0550-3213(91)90370-D, https://doi.org/10.1016/0550-3213(91)90370-D
5. J. L. Cardy, Conformal invariance and surface critical behavior, Nucl. Phys. B 240, 514 (1984). DOI: 10.1016/0550-3213(84)90241-4, https://doi.org/10.1016/0550-3213(84)90241-4