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Understanding Spin Relaxation Through Conformal Transformations

This presentation explores the dynamics of the Heisenberg model featuring dissipation, focusing on spin precession and relaxation in the presence of a constant magnetic field. By employing conformal transformations and stereographic projection, we obtain exact solutions for the dissipation term, providing insights into the behavior of spins in two-dimensional space. The study underscores the importance of theoretical models in refining research inquiries, demonstrating the intricate relationship between spin dynamics and conformal mapping techniques.

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Understanding Spin Relaxation Through Conformal Transformations

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  1. Understanding Spin RelaxationusingConformalTransformations Manuel Berrondo Dept. Physics & Astronomy Brigham Young University Provo UT 84602 BYU Physics

  2. In collaboration with: Wes Krueger “The purpose of models is not to fit the data but to sharpen the questions” S. Karlin BYU Physics

  3. Introduction • Dynamics of Heisenbergmodelwithdissipation • Look first at spin precession and relaxation w/ constantBfield • Dissipationterm (Gilbert) can besolvedexactly w/stereographicprojection BYU Physics

  4. Spin precession and relaxation preserves S2,chosentobe 1. BYU Physics

  5. ConformalTransformations in R3 Inverse of a vector a≠ 0 a-1 0 1 2 a BYU Physics

  6. Mappingplaneontosphere w r BYU Physics

  7. StereographicProjection • Tip of Sontheunitsphere, S2=1 • Projectionontoplane has twocomponents: W R2 BYU Physics

  8. MixedRepresentation • Fromequation of motion: weget: BYU Physics

  9. Equation of MotionforW (t) withsolution: BYU Physics

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