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Journal of Mathematical Physics

Singularity structure analysis of the continuum Heisenberg spin chain with anisotropy and transverse field: Nonintegrability and chaos

M. Daniel, Martin David Kruskal, M. Lakshmanan, Kouji Nakamura

Published Feb 1, 1992• 52 citations
Publisher

Abstract

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.

DOI
10.1063/1.529756

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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 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.
We consider the anisotropic Heisenberg spin-1/2 chain in a transverse magnetic field at zero temperature. We first determine all components of the dynamical structure factor by combining exact results with a mean-field approximation recently proposed by Dmitriev et al. [JETP 95, 538 (2002)]. We then turn to the small anisotropy limit, in which we use field theory methods to obtain exact results. We discuss the relevance of our results to neutron scattering experiments on the one-dimensional Heisenberg chain compound Cs2CoCl4.{\mathrm{Cs}}_{2}{\mathrm{CoCl}}_{4}.Cs2​CoCl4​.
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.
We present a study of XYZXYZXYZ Heisenberg spin-12\frac{1}{2}21​ chain, along with its variant (XYXYXY Heisenberg spin-12\frac{1}{2}21​ Hamiltonian), under a transverse magnetic field. It reveals from our study that two competitive massive phases are separated by a gapless line (critical line). We find in our analysis that the critical points (trivial and nontrivial) are non-Gaussian in nature. There is no phase crossover owing to the change of the critical point from the trivial to the nontrivial one. There is no evidence of magnetization plateau, however the system is in the Ising criticality. The positive and negative values of umklapp-like scattering favor a short range antiferromagnetic order because of the XYXYXY anisotropy and the transverse field, respectively. Higher value of Luttinger liquid parameter (K)(K)(K) also favors the short range antiferromagnetic order due to the in-plane exchange anisotropy.
The critical breakdown of a one-dimensional quantum magnet with long-range interactions is studied by investigating the high-field polarized phase of the anisotropic XY model in a transverse field for the ferromagnetic and antiferromagnetic cases. While for the limiting case of the isotropic long-range XY model we can extract the elementary one-quasiparticle dispersion analytically and calculate two-quasiparticle excitation energies quantitatively in a numerical fashion, for the long-range Ising limit as well as in the intermediate regime we use perturbative continuous unitary transformations on white graphs in combination with classical Monte Carlo simulations for the graph embedding to extract high-order series expansions in the thermodynamic limit. This enables us to determine the quantum-critical breakdown of the high-field polarized phase by analyzing the gap closing including associated critical exponents and multiplicative logarithmic corrections. In addition, for the ferromagnetic isotropic XY model we determined the critical exponents zzz and $\ensuremath{\nu}$ analytically by a bosonic quantum-field theory.
Spin-1/2 chains demonstrate some of the striking effects of interactions and quantum fluctuations in one-dimensional systems. The XXZ model has been used to study the unusual properties of anisotropic spin chains in an external magnetic field. The zero temperature phase diagram for this model exhibits a critical or quasi-long-range-ordered phase which is a realization of a Luttinger liquid. While many static properties of spin-1/2 chains have been explained by combinations of analytical techniques such as bosonization and Bethe ansatz, the standard approach fails in the calculation of some time-dependent correlation functions. I present a study of the longitudinal dynamical structure factor for the XXZ model in the critical regime. I show that an approximation for the line shape of the dynamical structure factor in the limit of small momentum transfer can be obtained by going beyond the Luttinger model and treating irrelevant operators associated with band curvature effects. This approach is able to describe the width of the on-shell peak and the high-frequency tail at finite magnetic field. Integrability is shown to affect the low-energy effective model at zero field, with consequences for the line shape. The power-law singularities at the thresholds of the particle-hole continuum are investigated using an analogy with the X-ray edge problem. Using methods of Bethe ansatz and conformal field theory, I compute the exact exponents for the edge singularities of the dynamical structure factor. The same methods are used to study the long-time asymptotic behavior of the spin self-correlation function, which is shown to be dominated by a high-energy excitation.