Dimension-created constrained phases and fixed-point bifurcations in the complete four-colour Ashkin–Teller model

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Authors

Publication Details

Journal
Zenodo (CERN European Organization for Nuclear Research)
Published
2026-09-19
DOI
https://doi.org/10.5281/zenodo.22551847
Primary Topic
Theoretical and Computational Physics
Type
preprint
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preprint

Dimension-created constrained phases and fixed-point bifurcations in the complete four-colour Ashkin–Teller model

Egemen Ekinci
Zenodo (CERN European Organization for Nuclear Research)
Theoretical and Computational Physics
preprint

Dimension-created constrained phases and fixed-point bifurcations in the complete four-colour Ashkin–Teller model

Egemen Ekinci
preprint en

Abstract

Preprint of the manuscript "Dimension-created constrained phases and fixed-point bifurcations in the complete four-colour Ashkin–Teller model", deposited together with the complete reproducibility package for the study. We introduce the complete permutation-symmetric four-colour Ashkin–Teller model, whose nearest-neighbour Hamiltonian contains independent two-, four-, six-, and eight-spin interactions. Its 16-state transfer matrix reduces exactly to five radial weights under a binary Krawtchouk transform. We solve the resulting b=3 real-space renormalization-group recursion in two and three dimensions; it is exact on the associated diamond hierarchical lattice and is the ordinary Migdal–Kadanoff approximation on Bravais lattices. Five ordered phases occur in d=2. In d=3, two additional phases, C1 and C3, occur in 22 of 25 coupling sections and occupy 20.3493% of the sampled global atlas. They are not ordinary subgroup or coset phases: exact compatibility counting identifies them as non-coset constrained phases with staggered four-colour-product order and certifies their d=3 residual entropy within [1.151635164996, 1.151635172152]. A general N-colour character identity shows that every pure odd-mismatch shell enforces staggered full-product order on a bipartite lattice. NOTE (version 4.0.0, 19 September 2026) - withdrawal of the version 3.0.0 correction. Version 3.0.0 of this record carried a correction stating that the description of C1 and C3 above as non-coset constrained phases rested on an incorrect inference. That correction was itself incorrect and is withdrawn in full. No claim in this paper is in error and nothing in it requires amendment. The withdrawn correction tested whether the single mismatch shells are closed at the level of support, whereas this paper identifies sinks by the criterion stated in its section 3, the mismatch classes that retain maximal weight, which those shells do satisfy at N = 4 for cell lengths 3 and 5 alike. Three facts hold together: the shells are not cosets, as Table 1 states; they are argmax-stable, so C1 and C3 are correctly identified sinks and the certified entropy interval [1.151635164996, 1.151635172152] stands; and they are not support-closed, so they sit as dominant-shell sinks inside the odd-coset face rather than forming a separate algebraic kind. That is a placement, not a repair, and it is consistent with the parameterisation by r = t1/t3 in section 6 of the paper. Details in WITHDRAWAL_AND_CLARIFICATION.md, which replaces CORRECTION_to_quadrupled_AT.md in this version, and in the companion note doi:10.5281/zenodo.22849628 (version 2), whose version 1 carried the withdrawn claim. Two distinct dimension-created folds organize the new topology: a finite fixed-point pair born at d_SN = 2.5601299591, and a separate exactly reduced odd-sector fold at d* = 2.8908652348 that creates the C1/C3 sinks and separators. A stratified multistart search finds 29 finite fixed points in d=2 and 37 in d=3. Free-energy derivatives were evaluated on 2,087 refined interfaces, and an independent periodic-cubic Metropolis test at L=6,8,10,12 corroborates the constraint signature. Contents: the manuscript and the supplement (PDF and DOCX), and Quadrupled_AT_Reproducibility_Package.tar.gz, which contains the physical 16-state renormalization-group implementation, the global phase-atlas data, the exact constraint and entropy certificates, the dimensional-bifurcation analyses, the transition-order diagnostics, the periodic-cubic finite-size corroboration, the analysis scripts, and the test suite needed to reproduce every number in the paper. A portable SHA-256 checksum is supplied alongside the archive. Version 2.0.0 adds the manuscript and supplement to the version 1.0.0 code and data package. Preprint. Not peer reviewed. This work was produced with substantial AI assistance (Claude, Anthropic). The author takes full responsibility for the content.

Zenodo (CERN European Organization for Nuclear Research)
Theoretical and Computational Physics
DOIOpen Access PDFOpenAlex
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