1. KIAS Emergent Materials 2006 Bond Polarization induced by Magnetic order Jung Hoon Han Sung Kyun Kwan U. Reference: cond-mat/0607 Collaboration Chenglong Jia (SKKU, KIAS) Naoto Nagaosa (U. Tokyo) Shigeki Onoda (U. Tokyo)

2. KIAS Emergent Materials 2006 Bond Polarization induced by Magnetic order Electric polarization, like polarization of spin, is responsible for loss of symmetry in the system, in this case, inversion symmetry. Its phenomenological description bears natural similarity to that of magnetic ordering. Normally, however, we do not think of the two ordering tendencies as “coupled”. Here we discuss experimental instances and theoretical models where the onset of electric polarization is “driven” by a particular type of spin ordering.

3. KIAS Emergent Materials 2006 Two order parameters X, Y are coupled in a GL theory Introduction to spin-polarization coupling via GL theory If X condenses (a X 0), but a linear coupling ~ XY exists, simultaneous condensation of Y occurs:

4. KIAS Emergent Materials 2006 Spin and polarization break different symmetries: breaks time-inversion symmetry breaks space-inversion symmetry Naively, lowest-order coupling occurs at 2 2. If the system already has broken inversion symmetry lower-order coupling 2 is possible. Even without inversion symmetry breaking, 2 or is possible. Introduction to spin-polarization coupling via GL theory

5. KIAS Emergent Materials 2006 Introduction to spin-polarization coupling via GL theory Generally one can write down that result in the induced polarization For spiral spins induced polarization has a uniform component given by Mostovoy PRL 06

6. KIAS Emergent Materials 2006 Experimental Evidence of spin-lattice coupling Uniform induced polarization depends product M 1 M 2 - Collinear spin cannot induce polarization - Only non-collinear, spiral spins have a chance Recent examples (partial) Ni 3 V 2 O 8 – PRL 05 TbMnO 3 – PRL 05 CoCr 2 O 4 – PRL 06

7. KIAS Emergent Materials 2006 Ni 3 V 2 O 8 Lawes et al PRL 05 TbMnO 3 Kenzelman et al PRL 05 Collinear to non-collinear spin transition accompanied by onset of polarization with P direction consistent with theory

8. KIAS Emergent Materials 2006 CoCr 2 O 4 Tokura group PRL 06 Co spins have ferromagnetic + spiral (conical) components Emergence of spiral component accompanied by P

9. KIAS Emergent Materials 2006 Microscopic Theory of Katsura, Nagaosa, Balatsky (KNB) A simple three-atom model consisting of M(agnetic)-O(xygen)-M ions is proposed to “derive” spin-induced polarization from microscopic Hamiltonian KNB PRL 05 Dagotto PRB 06 (different perspective) Polarization orthogonal to the spin rotation axis and modulation wavevector develops; consistent with phenomenological theories

10. KIAS Emergent Materials 2006 Elements of KNB Theory The cluster Hamiltonian assuming t 2g levels for magnetic sites KNB Hamiltonian is solved assuming SO > U

11. KIAS Emergent Materials 2006 Why spin-orbit is important Conceptual view: spin orientations leave imprints on the wave functions, leading to non-zero polarization Technical view: Spin-orbit Hamiltonian mixes oxygen p z with magnetic d yz, p x with d xy within the same eigenstate, non-zero, is responsible for polarization

12. KIAS Emergent Materials 2006 Motivation for our work (0) KNB result seems so nice it must be general. (1) Effective Zeeman energy U is derived from Hund coupling (as well as superexchange), which is much larger than SO interaction. The opposite limit U >> SO must be considered also. (2) What about e g levels? (3) From GL theory one expects some non-uniform component too.

13. KIAS Emergent Materials 2006 Our strategy for large U limit Large-U offers a natural separation of spin-up and spin-down states for each magnetic site. All the spin-down states (antiparallel to local field) can be truncated out. This reduces the dimension of the Hamiltonian which we were able to diagonalize exactly. Truncated HS

14. KIAS Emergent Materials 2006 Our Model (I): e g levels The model we consider mimics e g levels with one (3x 2 -r 2 )-orbital for the magnetic sites, and p x, p y, p z orbitals for the oxygen. Within e g manifold SO is ineffective. Real multiferroic materials have filled t 2g and partially filled e g ! IDEA(S. Onoda): Consider oxygen SO interaction. It will be weak, but better than nothing! Our calculation for large-U gives

15. KIAS Emergent Materials 2006 Our Model (II): t 2g levels Going back to t 2g, we considered strong-U limit, truncating +U subspace leaving only the –U Hilbert space. Spontaneous polarization exists ALONG the bond direction. No transverse polarization of KNB type was found. (NB: KNB’s theory in powers of U/, our theory in powers of /U)

16. KIAS Emergent Materials 2006 Numerical approach Surprised by, and skeptical of our own conclusion, we decided to compute polarization numerically without ANY APPROXIMATION Exact diagonalization of the KNB Hamiltonian (only 16 dimensional!) for arbitrary parameters ( /V,U/V) For each of the eigenstates compute P = The results differ somewhat for even/odd number of holes; In this talk we mainly presents results for one and two holes. Other even numbers give similar results.

17. KIAS Emergent Materials 2006 Numerical Results for one hole Rotate two spins within XY plane: S l =(cos  l,sin  r,0) S r =(cos  l,sin  r,0) and compute resulting polarization. Numerical results for one hole is in excellent qualitative agreement with analytical calculation Not only longitudinal but also transverse components were found in P

18. KIAS Emergent Materials 2006 Numerical results for two holes Transverse and longitudinal components exist which we were able to fit using very simple empirical formulas: KNB

19. KIAS Emergent Materials 2006 Uniform vs. non-uniform When extended to spiral spin configuration, P x gives oscillating polarization with period half that of spin. P y has oscillating (not shown) as well as uniform (shown) component KNB

20. KIAS Emergent Materials 2006 Uniform vs. non-uniform What people normally detect is macroscopic (uniform) polarization but that may not be the whole story. Non-uniform polarization, if it exists, is likely to lead to some modulation of atomic position which one can pick up with X-rays. How big is the non-uniform component locally?

21. KIAS Emergent Materials 2006 KNB Coefficients The uniform transverse component B 1 is significant for small U (KNB limit). A and B 2 (non-uniform) are dominant for large U (our limit).

22. KIAS Emergent Materials 2006 Comparison to GL theory Within GL theory non-uniform polarization is also anticipated. On comparing Mostovoy’s prediction with ours, a lot of details differ. A large non-uniform component could not have been predicted on GL theory alone. Bear in mind that t 2g break full rotational symmetry down to cubic; corresponding GL theory need not have that symmetry built in. A new kind of GL theory is called for.

23. KIAS Emergent Materials 2006 Summary Motivated by recent experimental findings of non-collinear-spin-induced polarization, we examined microscopic model of Katsura, Nagaosa, Balatsky in detail. Induced polarization has longitudinal and transverse, uniform and non-uniform components with non-trivial dependence on spin orientations. Detecting such local ordering of polarization will be interesting.