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Physica Scripta

Integrable Reductions of Manycomponent Magnetic Systems in (1,1) Dimensions

Vladmir G. Makhankov, Oktay K. Pashaev, Anasuya Kundu

Published Aug 1, 1983• 13 citations
Publisher

Abstract

A generalized many component Heisenberg spin chain with phonon interaction is proposed. Some reductions of the proposed model leading to different real magnetic systems such as many chained magnetic crystals with nontrivial interchain couplings, a mixture of many chained ferro and antiferromagnets, a colour generalized Pierels-Hubbard model, etc., are studied. It has been shown that the dynamics of all the above real models are close to some integrable systems and coincide with them in certain limits. Such integrable systems are the coupled generalised system of Yajima and Oikawa and U(p, q) nonlinear Schrodinger equation, already well studied.

DOI
10.1088/0031-8949/28/2/016

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Journal of Physics A Mathematical and General•May 21, 1991•92 citations
A generalized many component Heisenberg spin chain with phonon interaction is proposed. Some reductions of the proposed model leading to different real magnetic systems such as many chained magnetic crystals with nontrivial interchain couplings, a mixture of many chained ferro and antiferromagnets, a colour generalized Pierels-Hubbard model, etc., are studied. It has been shown that the dynamics of all the above real models are close to some integrable systems and coincide with them in certain limits. Such integrable systems are the coupled generalised system of Yajima and Oikawa and U(p, q) nonlinear Schrodinger equation, already well studied.
It is shown that many sublattice isotropic XY chain with magnon-phonon interactions at the long-wave approximations may be described by generalized Zakharov's system with U ( p, q ) isogroup. Two-sublattice XY chain through the Jordan-Wigner transformations reduced to Su-Schriffer-Heeger coupled electron-phonon model in the quasi-one-dimensional conductor polyacetylene (CH) x theory. The Hamiltonian structure of U ( p, q ) Zakharov's system and its "ultrarelativistic" limit (i.e., the Yajima-Oikawa system with U ( p, q ) isogroup) are studied. The linear problem and the generating functionals for infinite series of additional integrals of motion for U ( p , q) Yajima-Oikawa system are constructed. Four types of soliton solutions under different boundary conditions and appropriate integrals of particles number and energy are found. The quasistationary and ultra-relativistic limits are discussed.
Exact solutions of quantum lattice models serve as useful guides for interpreting physical phenomena in condensed matter systems. Prominent examples of integrability appear in one dimension, including the Heisenberg chain, where the Bethe ansatz method has been widely successful. Recent work has noted that certain non-integrable models harbor quantum many-body scar states, which form a superspin of regular states hidden in an otherwise chaotic spectrum. Here we consider one of the simplest examples of a non-integrable model, the alternating ferromagnetic-antiferromagnetic (bond-staggered) Heisenberg chain, a close cousin of the spin-1 Haldane chain and a spin analog of the Su-Schrieffer-Heeger model, and show the presence of exponentially many zero-energy states. We highlight features of the alternating chain that allow treatment with the coordinate Bethe ansatz (with important modifications), and surprisingly, for a non-integrable system, we find simple compact expressions for zero-energy eigenfunctions for a few magnons, including solutions with fractionalized particle momentum. These states exhibit wavefunction scarring similar to that of single-particle states in chaotic billiards. We discuss a general numerical recipe to diagnose the existence of such generalized Bethe ansatz (GBA) scar states and also provide exact analytic expressions for their entanglement. We conclude by conjecturing a picture of magnon pairing which may generalize to multiple magnons. Our work opens the avenue to describe certain eigenstates of partially integrable systems using the GBA.
We present two directions of research in statistical mechanics of nonequilibrium onedimensional quantum systems. One is related to isolated integrable models, the other one to exactly solvable dissipatively driven spin chains. The device enabling their analysis is identified as the quantum group symmetry. In the framework of isolated systems we focus on the concept of integrable Floquet driven systems. In particular, we show how to build such systems out of the basic constituents of integrability structure. This allows substantiated and conclusive statements,for instance, about the transport phenomena. Despite their inherent relation, the presented integrable periodically driven models originate in two different ideas. On the one hand we have Trotterisations of integrable spin chains, concieved in an effort to better understand the dynamics of spin models. Then, there are systems originating in the attempts to solve quantum field theory in an algebraically closed form, for example, the quantum Hirota equation. The bulk of our consideration of isolated quantum systems consists of the construction of extensive conservation laws that either (i) constitute the generalised Gibbs ensemble and the hydrodynamic description of thermalisation after a quantum quench, or (ii) prevent the decay of current autocorrelations and thus characterise ideal transport in the system. The exact way in which these charges determine the dynamics stems from the symmetries of the quantum group representations. The second part of the exposition is dedicated to open quantum systems. Again we distinguish two formally related settings. The first one is that of a boundary driven quantum cellular automaton, our particular example being the integrable Trotterisation of the Heisenberg magnet. Starting from a repeated interaction protocol, in which the system under scrutiny is repeatedly coupled to the environment, we introduce the Kraus map as a general form of a dissipative time evolution of the density matrix. We then solve for its unique nonequilibrium steady state, using integrability structure of the model. The other setting is that of a dissipatively boundary driven spin chain in the continuous time. Here we present the recently developed formalism of inhomogeneous Lax structure. Using it we demonstrate the solvability of the XXZ and XYZ spin chains, acted upon by the Lindblad operators that polarise the boundary spins in arbitrary directions. The ansatz for the steady state is particularly interesting, since it exhibits a previously unknown integrability structure, differing from site to site in the spin chain. This structure can independently produce nontrivial conservation laws in an isolated spin chain with arbitrary boundary fields.
The one-dimensional model which consists of two isotropic XY chains with spins S=1/2 coupled by three-spin interactions is considered with the help of the Bethe ansatz. It is shown that the diagonalization of the Hamiltonian can be reduced to solving a set of coupled nonlinear equations. The exact solution of these equations corresponding to the ground state of system is obtained. The model considered exhibits a ground state of Anderson type with the finite magnetization along the quantization axis.
The one-dimensional model which consists of two isotropic XY chains with spins S=1/2 coupled by three-spin interactions is considered with the help of the Bethe ansatz. It is shown that the diagonalization of the Hamiltonian can be reduced to solving a set of coupled nonlinear equations. The exact solution of these equations corresponding to the ground state of system is obtained. The model considered exhibits a ground state of Anderson type with the finite magnetization along the quantization axis.