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Journal of Mathematical Physics

On the integrability aspects of the one-dimensional classical continuum isotropic biquadratic Heisenberg spin chain

Kuppuswamy Porsezian, M. Daniel, M. Lakshmanan

Published May 1, 1992• 255 citations
Publisher

Abstract

The integrability aspects of a classical one-dimensional continuum isotropic biquadratic Heisenberg spin chain in its continuum limit up to order [O(a4)] in the lattice parameter ‘‘a’’ are studied. Through a differential geometric approach, the dynamical equation for the spin chain is expressed in the form of a higher-order generalized nonlinear Schrödinger equation (GNLSE). An integrable biquadratic chain that is a deformation of the lower-order continuum isotropic spin chain, is identified by carrying out a Painlevé singularity structure analysis on the GNLSE (also through gauge analysis) and its properties are discussed briefly. For the nonintegrable chain, the perturbed soliton solution is obtained through a multiple scale analysis.

DOI
10.1063/1.529658

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The integrability aspects of a classical one-dimensional continuum isotropic biquadratic Heisenberg spin chain in its continuum limit up to order [O(a4)] in the lattice parameter ‘‘a’’ are studied. Through a differential geometric approach, the dynamical equation for the spin chain is expressed in the form of a higher-order generalized nonlinear Schrödinger equation (GNLSE). An integrable biquadratic chain that is a deformation of the lower-order continuum isotropic spin chain, is identified by carrying out a Painlevé singularity structure analysis on the GNLSE (also through gauge analysis) and its properties are discussed briefly. For the nonintegrable chain, the perturbed soliton solution is obtained through a multiple scale analysis.
The integrability aspects of the continuum limit of the one-dimensional anisotropic Heisenberg spin chain in a transverse magnetic field are studied by carrying out a Painlevé singularity structure analysis. The analytic structure of the system exhibits some unusual features and requires expansions that start like a Taylor series. A careful analysis exhibits the presence of a movable logarithmic critical singular manifold in the combined presence of the anisotropy and transverse field, thereby elucidating the nonintegrable nature. Numerical analysis on the static spin configurations of the nonintegrable case shows typically chaotic spatial structures.
The Heisenberg Spin Chain system, in the continuum limit, can be represented by the non-linear Schr\odinger equation through the Hashimoto map. Inhomogeneity induced through localizing the nearest neighbor interaction strength can also be mapped similarly to an integro -differential generalization of the non-linear Schrodinger system [J. Phys. C: Solid State Phys. 15, L1305 (1982)] which is integrable. We show that the latter system is a particular non-holonomic deformation of the usual non-linear Schrodinger equation, aided by generalized parameterizations. General non-holonomic deformations correspond only to temporal inhomogeneity, with the additional spatial inhomogeneity corresponding to particular spectral orders, identical for both continuum Heisenberg spin chain and non-linear schrodinger system. Being semi-Classical in nature, these continuum limits of spin system display amplitude-phase entanglement.
Using the coherent states method combined with the Schwinger coupled boson representation of the spin operators (1965) it is shown that solitons can propagate in the continuum limit of the quantum ferromagnetic Heisenberg chain. These solitons are solutions of a nonlinear Schrodinger equation similar to that discussed by Lakshmanan (1977) for the classical Heisenberg chain. Several characteristics of the soliton, like its momentum, energy, and the number of bosonic excitations involved in the soliton formation, are calculated.
Painlevé’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 Schrödinger-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.