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

TL;DR

This paper develops a quantum framework for a continuum Heisenberg spin-chain model using position-dependent mass formalism and pre-canonical quantization, leading to explicit solutions for ground and excited states.

Contribution

It introduces a novel quantum approach to the continuum Heisenberg spin-chain model, employing position-dependent mass Hamiltonian and pre-canonical quantization techniques.

Findings

Derived a Schrödinger-like equation as a confluent Heun equation.

Explicitly obtained ground and first excited states using Bethe-Ansatz.

Provided a quantum probabilistic description of the spin system.

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.