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Journal of Sichuan University

Geometric phase of a Heisenberg-chain in a rotating magnetic field

Song Yuan

Published Jan 1, 2008• 0 citations
OpenAlex

Abstract

The algebriaic dynamics is applied to investigate the geometric phase of a Heisenberg-chain in a rotating magnetic field.It is a su(2) algebraically symmetric 3-spin12 system with partially anisotropic coupling.The exact analytical solution of the time dependent Schrdinger equation and the geometric phases of this system are obtained in terms of gauge transformation and algebriaic dynamics.The partially anisotropic coupling is then extended to a more general coupling and the adiabatic geometric phases of the extended system are calculated.We find that the coupling strengths have no impact on the adiabatic geometric phases of the system.

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The algebriaic dynamics is applied to investigate the geometric phase of a Heisenberg-chain in a rotating magnetic field.It is a su(2) algebraically symmetric 3-spin12 system with partially anisotropic coupling.The exact analytical solution of the time dependent Schrdinger equation and the geometric phases of this system are obtained in terms of gauge transformation and algebriaic dynamics.The partially anisotropic coupling is then extended to a more general coupling and the adiabatic geometric phases of the extended system are calculated.We find that the coupling strengths have no impact on the adiabatic geometric phases of the system.
Objective The algebriaic dynamics is applied to investigate the geometric phase of a Heisenberg-chain,in order to find out how the coulping strengths between spins influence the geometric phase of this system.Method An interacting anisotropic 3-spin 1/2 system,which is driven by a rotating magnetic field,is considered.The system has a su(2) algebraic structure.By an unitary gauge transformation,this nonautonomous system becomes an autonomous system.The Schrodinger equation of the autonomous system is solved.By virtue of the inverse transformation of the original gauge transformation,the exact analytical solution to the time dependent Schrodinger equation is obtained in total spin 1/2 subspace.Resutlt The geometric phases in this subspace are calculated.Conclusion The result shows that(i)Coulping strengths don't influence the geometric phases of this system;(ii) The geometric phases of the system is the same as that of one spin 1/2 system in a rotating magnetic field.
The coordinate rotation method is applied to investigate the relation between two types of gauge analytical solutions for a su(2) algebraically symmetric spin cluster.We consider an annular 3- spin -r- chain with partially anisotropic coupling.It is found that the analytical solutions related to single -gauge transformation can be obtained from those related to double-gauge transformation.The geometric phase of the every basic analytical solution from single-gauge transformation is calculated,and the geometric phaseπis found.Ifλ=(π/2) in double-gauge transformation,the value of geometric phase obtained here is the same as that of every basic analytical solution from double-gauge transformation.
We study the Heisenberg model under the influence of a rotating magnetic field. By using a time-dependent unitary transformation, the time evolution operator for the Schrödinger equation is obtained, which involves no chronological product. The spin vectors (mean values of the spin operators) are obtained as explicit functions of time in the most general case. A series of cyclic solutions are presented. The nonadiabatic geometric phases of these cyclic solutions are calculated, and are expressed in terms of the solid angle subtended by the closed trace of the total spin vector, as well as in terms of those of the individual spins.
With the help of the time-dependent gauge transformation technique, we have studied the geometric phase of a spin-half particle in a rotating magnetic field. We have found that the slow but finite frequency of the rotating magnetic field will make the difference between the adiabatic geometric phase and the exact geometric phase. When the frequency is much smaller than the energy space and the adiabatic condition is perfectly guaranteed, the adiabatic approximation geometric phase is exactly consistent with the adiabatic geometric phase. A simple relation for the accuracy of the adiabatic approximation is given in terms of the changing rate of the frequency of the rotating magnetic field and the energy level space.
We study the Heisenberg model under the influence of a rotating magnetic field. By using a time-dependent unitary transformation, the time evolution operator for the Schrodinger equation is obtained, which involves no chronological product. The spin vectors (mean values of the spin operators) are obtained as explicit functions of time in the most general case. A series of cyclic solutions are presented. The nonadiabatic geometric phases of these cyclic solutions are calculated, and are expressed in terms of the solid angle subtended by the closed trace of the total spin vector, as well as in terms of those of the individual spins.
We study the dynamics evolution of a two-qubit Heisenberg XXX spin chain under a time-dependent rotating magnetic field.Based on the algebraic structure of the non-autonomouse system,the exact solution of the Schrdinger equation is obtained by using the method of algebraic dynamics.Based on the time-dependent analytical solution,we further study the entanglement evolution between the two coupled spins for different initial states,and find that the entanglement is determined by the coefficients of the initial state and the coupling constant J of the system.