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)

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.

Introduction to spin-polarization coupling via GL theory Two order parameters X, Y are coupled in a GL theory If X condenses (aX < 0) and Y does not (aY > 0), but a linear coupling ~ XY exists, simultaneous condensation of Y occurs:

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

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

Experimental Evidence of spin-lattice coupling Uniform induced polarization depends product M1 M2 - Collinear spin cannot induce polarization - Only non-collinear, spiral spins have a chance Recent examples (partial) Ni3V2O8 – PRL 05 TbMnO3 – PRL 05 CoCr2O4 – PRL 06

Ni3V2O8 TbMnO3 Collinear to non-collinear spin transition accompanied by onset of polarization with P direction consistent with theory Lawes et al PRL 05 Kenzelman et al PRL 05

CoCr2O4 Co spins have ferromagnetic + spiral (conical) components Emergence of spiral component accompanied by P Tokura group PRL 06

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 Polarization orthogonal to the spin rotation axis and modulation wavevector develops; consistent with phenomenological theories KNB PRL 05 Dagotto PRB 06 (different perspective)

Elements of KNB Theory The cluster Hamiltonian assuming t2g levels for magnetic sites KNB Hamiltonian is solved assuming SO > U

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 pz with magnetic dyz, px with dxy within the same eigenstate, non-zero <dyz|y|pz>, <dxy|y|px> is responsible for polarization

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 eg levels? (3) From GL theory one expects some non-uniform component too.

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

Our Model (I): eg levels The model we consider mimics eg levels with one (3x2-r2)-orbital for the magnetic sites, and px, py, pz orbitals for the oxygen. Within eg manifold SO is ineffective. Real multiferroic materials have filled t2g and partially filled eg! IDEA(S. Onoda): Consider oxygen SO interaction. It will be weak, but better than nothing! Our calculation for large-U gives

Our Model (II): t2g levels Going back to t2g, 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)

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 = <r> 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.

Numerical Results for one hole Rotate two spins within XY plane: Sl=(cos l,sinr,0) Sr=(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

Numerical results for two holes Transverse and longitudinal components exist which we were able to fit using very simple empirical formulas: KNB

Uniform vs. non-uniform KNB When extended to spiral spin configuration, Px gives oscillating polarization with period half that of spin. Py has oscillating (not shown) as well as uniform (shown) component

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?

Coefficients KNB The uniform transverse component B1 is significant for small U (KNB limit). A and B2 (non-uniform) are dominant for large U (our limit).

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 t2g 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.

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.