AIP conference proceedings

Spin systems associated with integrable nonlinear Schrödinger equations

Kuralay Yesmakhanova, Ratbay Myrzakulov, Ainur Zhumageldina, A A Naizagarayeva

Published Jan 1, 2023• 1 citation

Abstract

In this article, using the geometric equivalence proposed by M. Lakshmanan, we will show equivalent connection between nonlinear Schrödinger equation and a spin system. Since from the theory of solitons, establishing an equivalence is possible only between integrable equations, integrability of the spin system will be proved.

DOI: 10.1063/5.0163626

Related papers

On the inhomogeneous Heisenberg chain

Раджа Балакришнан

The author derives the spin evolution equation of a classical inhomogeneous Heisenberg chain and proves its exact equivalence (in the continuum limit) to an integro-differential nonlinear Schrodinger equation whose soliton solutions have been discussed earlier. Energy transport along the chain is related to the solution of this equation. He also describes possible physical systems to which the model is applicable.

Journal of Physics C Solid State PhysicsDec 30, 198275 citations

Solitons in quantum Heisenberg chain

Marek Cieplak, Łukasz A. Turski

Journal of Physics C Solid State PhysicsNov 10, 198018 citations

An Integrable Classical Spin Chain

勇次石森

An integrable spin model on the one-dimensional lattice is obtained from the differential-difference nonlinear Schrödinger equation by introducing the concept of gauge equivalence. The Hamiltonian for this spin model is expressed as \begin{aligned} H{=}-2\sum_{n}{\log}(1+\textbf{\itshape S}_{n}\cdot \textbf{\itshape S}_{n+1}) \end{aligned} where the spins S n are classical three-dimensional unit vectors.

Journal of the Physical Society of JapanNov 1, 1982111 citations

One-Dimensional Integrable Spinor BECs Mapped to Matrix Nonlinear Schrödinger Equation and Solution of Bogoliubov Equation in These Systems

Daisuke A. Takahashi

arXiv:1011.5525Journal of the Physical Society of JapanDec 27, 20104 citations

A nonlinear quantum dynamical system of spin 1/2 particles based on the classical Sine-Gordon Equation

Yair Zarmi

arXiv:1210.6641Oct 24, 20120 citations

Classical facets of quantum integrability

A. Zabrodin

arXiv:2501.18557Jan 30, 20250 citations

AIP conference proceedings

Spin systems associated with integrable nonlinear Schrödinger equations

Kuralay Yesmakhanova, Ratbay Myrzakulov, Ainur Zhumageldina, A A Naizagarayeva

Published Jan 1, 2023• 1 citation

Publisher

Abstract

DOI: 10.1063/5.0163626

Related papers

On the inhomogeneous Heisenberg chain

Раджа Балакришнан

Journal of Physics C Solid State PhysicsDec 30, 198275 citations

Solitons in quantum Heisenberg chain

Marek Cieplak, Łukasz A. Turski

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.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.

Journal of Physics C Solid State PhysicsNov 10, 198018 citations

An Integrable Classical Spin Chain

勇次石森

Journal of the Physical Society of JapanNov 1, 1982111 citations

One-Dimensional Integrable Spinor BECs Mapped to Matrix Nonlinear Schrödinger Equation and Solution of Bogoliubov Equation in These Systems

Daisuke A. Takahashi

In this short note, we construct mappings from one-dimensional integrable spinor BECs to matrix nonlinear Schrödinger equation, and solve the Bogoliubov equation of these systems. A map of spin- $n$ BEC is constructed from the $2^n$ -dimensional spinor representation of irreducible tensor operators of $so(2n+1)$ . Solutions of Bogoliubov equation are obtained with the aid of the theory of squared Jost functions.In this short note, we construct mappings from one-dimensional integrable spinor BECs to matrix nonlinear Schrödinger equation, and solve the Bogoliubov equation of these systems. A map of spin-$n$ BEC is constructed from the $2^n$-dimensional spinor representation of irreducible tensor operators of $so(2n+1)$. Solutions of Bogoliubov equation are obtained with the aid of the theory of squared Jost functions.

arXiv:1011.5525Journal of the Physical Society of JapanDec 27, 20104 citations

A nonlinear quantum dynamical system of spin 1/2 particles based on the classical Sine-Gordon Equation

Yair Zarmi

The Hirota transformation for the soliton solutions of the classical Sine-Gordon equation is suggestive of an extremely simple way for the construction of a nonlinear quantum-dynamical system of spin 1/2 particles that is equivalent to the classical system over the soliton sector. The soliton solution of the classical equation is mapped onto an operator, U, a nonlinear functional of the particle-number operators, that solves the classical equation. Multi-particle states in the Fock space are the eigenstates of U; the eigenvalues are the soliton solutions of the Sine-Gordon equation. The fact that solitons can have positive as well negative velocities is reflected by the characterization of particles in the Fock space by two quantum numbers: a wave number k, and a spin projection, σ (= +-1). Thanks to the simplicity of the construction, incorporation of particle interactions, which induce soliton effects that do not have a classical analog, is simple.The Hirota transformation for the soliton solutions of the classical Sine-Gordon equation is suggestive of an extremely simple way for the construction of a nonlinear quantum-dynamical system of spin 1/2 particles that is equivalent to the classical system over the soliton sector. The soliton solution of the classical equation is mapped onto an operator, U, a nonlinear functional of the particle-number operators, that solves the classical equation. Multi-particle states in the Fock space are the eigenstates of U; the eigenvalues are the soliton solutions of the Sine-Gordon equation. The fact that solitons can have positive as well negative velocities is reflected by the characterization of particles in the Fock space by two quantum numbers: a wave number k, and a spin projection, σ (= +-1). Thanks to the simplicity of the construction, incorporation of particle interactions, which induce soliton effects that do not have a classical analog, is simple.

arXiv:1210.6641Oct 24, 20120 citations

Classical facets of quantum integrability

A. Zabrodin

This paper is a review of the works devoted to understanding and reinterpretation of the theory of quantum integrable models solvable by Bethe ansatz in terms of the theory of purely classical soliton equations. Remarkably, studying polynomial solutions of the latter by methods of classical soliton theory, one is able to develop a method of solving the spectral problem for the former which provides an alternative to the Bethe ansatz procedure. Our main examples are the generalized inhomogeneous spins chains with twisted boundary conditions on the quantum side and the modified Kadomtsev-Petviashvili hierarchy of nonlinear differential-difference equations on the classical side. In this paper, we restrict ourselves to quantum spin chains with rational $GL(n)$ -invariant $R$ -matrices (of the XXX type). Also, the connection of quantum spin chains with classical soliton equations implies a close interrelation between the spectral problem for spin chains and integrable many-body systems of classical mechanics such as Calogero-Moser and Ruijsenaars-Scheider models, which is known as the quantum-classical duality. Revisiting this topic, we suggest a simpler and more instructive proof of this kind of duality.This paper is a review of the works devoted to understanding and reinterpretation of the theory of quantum integrable models solvable by Bethe ansatz in terms of the theory of purely classical soliton equations. Remarkably, studying polynomial solutions of the latter by methods of classical soliton theory, one is able to develop a method of solving the spectral problem for the former which provides an alternative to the Bethe ansatz procedure. Our main examples are the generalized inhomogeneous spins chains with twisted boundary conditions on the quantum side and the modified Kadomtsev-Petviashvili hierarchy of nonlinear differential-difference equations on the classical side. In this paper, we restrict ourselves to quantum spin chains with rational $GL(n)$-invariant $R$-matrices (of the XXX type). Also, the connection of quantum spin chains with classical soliton equations implies a close interrelation between the spectral problem for spin chains and integrable many-body systems of classical mechanics such as Calogero-Moser and Ruijsenaars-Scheider models, which is known as the quantum-classical duality. Revisiting this topic, we suggest a simpler and more instructive proof of this kind of duality.

arXiv:2501.18557Jan 30, 20250 citations

In this short note, we construct mappings from one-dimensional integrable spinor BECs to matrix nonlinear Schrödinger equation, and solve the Bogoliubov equation of these systems. A map of spin-

n

BEC is constructed from the

2^n

-dimensional spinor representation of irreducible tensor operators of

so(2n+1)

. Solutions of Bogoliubov equation are obtained with the aid of the theory of squared Jost functions.

GL(n)

-invariant

R

-matrices (of the XXX type). Also, the connection of quantum spin chains with classical soliton equations implies a close interrelation between the spectral problem for spin chains and integrable many-body systems of classical mechanics such as Calogero-Moser and Ruijsenaars-Scheider models, which is known as the quantum-classical duality. Revisiting this topic, we suggest a simpler and more instructive proof of this kind of duality.