A Quantum Approach to the Continuum Heisenberg Spin-Chain Model: Position-Dependent Mass Formalism and Pre-canonical Quantization

• URL: http://arxiv.org/abs/2506.20126v1
• Date: Wed, 25 Jun 2025 04:43:39 GMT
• Title: A Quantum Approach to the Continuum Heisenberg Spin-Chain Model: Position-Dependent Mass Formalism and Pre-canonical Quantization
• Authors: V. Chithiika Ruby, M. Lakshmanan,
• Abstract summary: We study the equivalence of the Hamiltonian density to the nonlinear sigma model.<n>The resulting Schr"odinger-like equation was found to take the form of a confluent Heun equation.<n>This analysis provides a comprehensive quantum description of the system.
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• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Painlev\'{e}'s singularity structure analysis, combined with stereographic mapping, has previously been applied to a one-dimensional Heisenberg spin-chain continuum model which identified a Hamiltonian density for the static version of the Landau-Lifshitz equation. In this work, we explore the equivalence of the Hamiltonian density to the nonlinear sigma model. It reveals its non-standard form and can be interpreted as a position-dependent mass Hamiltonian density. We then proceed with the quantization of this Hamiltonian density using the pre-canonical quantization procedure. The resulting Schr\"{o}dinger-like equation was found to take the form of a confluent Heun equation. By employing the functional Bethe-Ansatz method, we explicitly obtain the ground state and first excited state of the system. This analysis provides a comprehensive quantum description of the system, capturing the probabilistic structure of the field and information about the possible energy states of the spin system.

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