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

• URL: http://arxiv.org/abs/2608.18633v1
• Date: Wed, 19 Aug 2026 07:29:28 GMT
• Title: Exact Matching-Polynomial Solution of the Periodic Baxter-Fendley $Z_N$ Clock Chain
• Abstract summary: We present a complete finite-size spectral solution for the periodic non-Hermitian Baxter-Fendley $Z_N$ clock chain.<n>For $$ it predicts two reciprocal critical couplings with singular ground-state curvature, in contrast to the single self-dual open critical point.
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• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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