Yuguan Li of National University and colleagues have obtained a complete finite-size spectral solution for the periodic non-Hermitian Baxter-Fendley Z_N clock chain, differing from previous incomplete solutions. Employing operator-valued matching polynomials, the team demonstrated that conserved quantities are generated and a cyclic Yang-Baxter transfer matrix is realised, enabling full calculation of energy levels. Specifically, for homogeneous chains where N equals three, their work predicts two reciprocal critical couplings compared to one previously known for similar models.

The researchers developed a thorough description of how energy behaves within a complex theoretical chain when connected into a loop; this addresses a longstanding problem despite existing solutions for chains with open ends. This advancement shows looped systems possess greater complexity at critical points, moments where properties change sharply, predicting two distinct behaviours instead of one as earlier studies suggested. Yuguan Li and colleagues successfully calculated complete energy levels within the looped theoretical chain, resolving a long-standing challenge that existed despite available solutions for chains with open ends.

The team achieved this by using an operator-valued matching polynomial, a set of tools, to describe relationships between different parts of the system, similar to utilising blueprints to ensure all components fit together correctly. This approach generates conserved quantities and realises a cyclic Yang-Baxter transfer matrix, effectively acting as an assembly line process systematically calculating properties along the length of the chain.

For homogeneous chains where N equals three, their work predicts two distinct critical couplings instead of one previously known; these points represent moments when the system’s behaviour changes sharply, prompting questions about how looped systems differ from those with open boundaries at such key junctures.

Operator matching polynomials reveal complete spectral solutions for anisotropic spin chains

A complete finite-size spectral solution was yielded by this methodology; previously, incomplete solutions existed for this complex system. By utilising an operator-valued matching polynomial, a tool establishing relationships between components like architectural blueprints, conserved quantities including the Hamiltonian were generated and a cyclic Yang-Baxter transfer matrix realised to systematically calculate chain properties. For homogeneous chains with N equal to three, analysis predicts two reciprocal critical couplings at approximately 0.785 and 1.275 where earlier work identified just one; these points define dramatic shifts in system behaviour at boundaries.

This advancement allows calculation of energy spectra via root-of-unity closure without enumerating all possible states, offering strong computational efficiency compared to prior approaches. Examining far fewer potential states than earlier methods now suffices for calculating these spectra, improving computational efficiency for complex systems with up to L sites. This is particularly beneficial when investigating the full range of energies within a looped theoretical chain, a configuration previously difficult to model accurately due to boundary conditions.

Operator-Valued Polynomials and Cyclic Transfer Matrices Reveal Conserved Quantities

The breakthrough hinged upon utilising an operator-valued matching polynomial which constructs a set of conserved quantities including the Hamiltonian itself; this effectively maps out inherent symmetries within the complex chain. The technique provides insight into underlying system characteristics rather than simply calculating individual energies. Analysis focused on chains with three components, revealing these critical coupling points were determined through numerical continuation of polynomial equations without needing to enumerate the entire spectrum.

Numerical analysis reveals unexpected behaviour in closed quantum chains

A spectral solution addresses a longstanding challenge: understanding how energy behaves within complex theoretical systems when boundaries connect, despite existing solutions describing open-ended chains. Current methodology relies on numerical continuation to pinpoint ground states; while effective for demonstrating two critical couplings where previous models predicted only one, it doesn’t offer an analytical route to all possible energy levels across every state. Establishing this reliable numerical method represents key progress given previous limitations in modelling these looped systems. The solution fully describes energy behaviour within a looped theoretical chain, resolving the longstanding problem and offering valuable new insights into its fundamental properties and potential applications.

The researchers successfully obtained a complete description of the finite-size energy spectrum for periodic non-Hermitian clock chains using operator-valued polynomials and cyclic transfer matrices. This allows scientists to calculate ground-state energies numerically without needing to determine the entire system’s spectrum, addressing a long-standing challenge in understanding closed quantum chains. For chains with three components, analysis revealed two critical couplings where prior research suggested only one existed. The authors demonstrated this method on homogeneous systems and suggest it provides criteria for boundary-induced criticality within these theoretical models.

👉 More information
🗞 Exact Matching-Polynomial Solution of the Periodic Baxter-Fendley $Z_N$ Clock Chain
✍️ Yuguan Li, D. C. Liu and Murray T. Batchelor
🧠 ArXiv: https://arxiv.org/abs/2608.18633

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