bibkey: kade2025boxsixvertex authors: Moritz Kade year: 2025 title: “Integrable systems: From the ice rule to supersymmetric fishnet Feynman diagrams” doi: 10.18452/33769 url: https://arxiv.org/abs/2509.03416v1 claim: “Section 2.3 of the thesis introduces box boundary conditions for the six-vertex model, with arrow-reflecting diagonal K-matrices on all four walls, and conjectures a determinant formula for the partition function of the square box with 2M horizontal and 2M vertical spectral lines.” strata_touched:

• D5/S3/StatisticalMechanics/VertexModels/KadeBoxBoundaryRefutation license: citation-only triage: anchor

Kade, box boundary conditions for the six-vertex model

Section 2.3, “The box boundary condition”, of the thesis considers a square lattice bounded by walls on all four sides, with pairs of spectral lines x_i, 1/x_i running from the right wall to the left wall and pairs y_j, 1/y_j running from the top wall to the bottom wall. The bulk weights are

[a : b : c] = [p x − p^{−1} x^{−1} : x − x^{−1} : p − p^{−1}]

and the arrow-reflecting walls carry the diagonal K-matrices K_L(x) = diag(b(xξ_L), b(x/(qξ_L))), K_R(x) = diag(b(xξ_R), b(xq/ξ_R)), K_U(y) = diag(b(yξ_U), b(yq/ξ_U)), K_D(y) = diag(b(yξ_D), b(y/(qξ_D))). The thesis derives a recursion at x_m = y_n and then proposes, for the square case M = N, the solution

Z_M({x_i}|{y_i}) = ∏{i,j} (x_i/y_j − y_j/x_i) W(x_i, y_j) / ∏{i<j} (x_j/x_i − x_i/x_j)(y_i/y_j − y_j/y_i) · det[ c² a(x_i y_j) a(1/(x_i y_j)) F^LU(x_i) F^DR(y_j) / ((x_j/y_i − y_i/x_j) W(x_i, y_j)) ]

with W(x, y) = a(xy) a(1/(xy)) a(x/y) a(y/x), F^LU(x) = b(xξ_L) b(xq/ξ_U) + b(x/(qξ_L)) b(xξ_U) and F^DR(y) = b(yξ_D) b(yq/ξ_R) + b(y/(qξ_D)) b(yξ_R). The concluding chapter states that this conjecture “has to be proven or discarded”.

Verified locator

• DOI: 10.18452/33769 (listed by the arXiv API for the record; resolves through doi.org to the Humboldt-Universität edoc handle 18452/34401, checked 2026-09-27).
• URL: https://arxiv.org/abs/2509.03416v1 (the only version listed by the arXiv API on 2026-09-27); source file parts/part2/square_ice_and_the_6v_model.tex, section “The box boundary condition”, equations eq:6V_BoxPartitionFunctionSquareSolution, eq:6V_WFactor, eq:6V_ArrowReflectingKMatrices and the trace functions after it; figures figures/vertexmodel/partition_function/box_partitionfunction_MN.pdf, figures/vertexmodel/Kmatrix/*/K*_mat.pdf, figures/vertexmodel/6vertices/6Vvertices.pdf and figures/vertexmodel/Rmatrix/Rmatrix_mat_8V.pdf.