Spin order in correlated electron systems

1. Spin order in correlated electron systems Yukawa Institute for Theoretical Physics Zhi Li

2. Outline • Variety of spin order in condensed matter 1.1 Collinear spin order 1.2 Non-collinear spin order • Theory and calculation 2.1 The form of wave function 2.2 Introduction about DFT calculation 2.3 Helical spin order in BaFeO3 3. Summary

3. 1. Variety of spin order 1.1 Collinear spin order Ferromagnetism Such as, Fe Anti-ferromagnetism Such as, NiO Ferrimagnetism Such as, Fe3O4, yttrium iron garnet Spin density wave state in Cr

4. 1. Variety of spin order 1.2 Non-collinear spin order Example: Spiral spin (fcc Fe) The position dependent spin moment could be written as Vector qis the wave vector of the spiral spin order

5. 2.1 The form of wave function If the electronic system is spin ordered, the form of the wave function has a constraint. A concrete example: Spiral spin n, l : lattice and orbital index Basis function could be plane wave, or Wannierfunction, or atomic orbital wave function.

6. 2.1 The form of wave function • Check the spin moment in this from of wave function:

7. 2.1 The form of wave function • Many-body Hamiltonian: • Total energy note: 1. The wave vector q is coupled to the momentum 2. Generally speaking, the interacting term should be in form of matrix 3. It is almost impossible to get the wave vector q by analytical method

8. 2.2 Introduction about DFT Calculation • The first-principles calculation based on density functional theory (DFT) The total energy of system is in the functional of density matrix expressed in single electronic state Single electron equation

9. 2.3 Helical spin order in BaFeO3 Wave vector dependent total energy

10. Summary • This is a simple introduction about the spin order in correlated electron system. 1. The form of the wave function 2. DFT calculation