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arXiv:2608.18633

Exact Matching-Polynomial Solution of the Periodic Baxter-Fendley $Z_N$ Clock Chain

Yuguan Li, D. C. Liu, Murray T. Batchelor

quant-phcond-mat.stat-mechmath-phSubmitted Aug 19, 2026• Announced Aug 20, 2026• 0 citations
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Abstract

The periodic non-Hermitian Baxter-Fendley $Z_N$ clock chain has lacked a complete finite-size spectral solution, whereas its open-chain counterpart admits a solution in terms of independent quasienergies. For the periodic model we show that the operator-valued matching polynomial associated with its cyclic Weyl algebra simultaneously generates a set of conserved quantities, including the Hamiltonian, and realizes a cyclic $τ^{(2)}$ Yang-Baxter transfer matrix. Root-of-unity closure yields a finite system of polynomial spectral equations in each charge sector, which reproduces the complete finite-size energy spectrum counted with algebraic multiplicity. As a first application of this result, we show that Newton continuation of these equations provides a practical numerical route to the periodic ground-state energy without enumerating the full spectrum. For homogeneous chains the thermodynamic seam response yields a criterion for boundary-induced criticality; for $N=3$ it predicts two reciprocal critical couplings with singular ground-state curvature, in contrast to the single self-dual open boundary critical point.

Comments
8 pages, 2 figues, plus Supplemental Material

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