Bond Polarization induced by Magnetic order

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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)
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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.
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Introduction to spin-polarization coupling via GL
theory
Two order parameters X, Y are coupled in a GL
theory
If X condenses (aX lt 0) and Y does not (aY gt 0),
but a linear coupling XY exists,
simultaneous condensation of Y occurs
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Introduction to spin-polarization coupling via GL
theory
Spin ltSgt and polarization ltRgt break different
symmetries ltSgt breaks time-inversion
symmetry ltRgt breaks space-inversion
symmetry Naively, lowest-order coupling occurs
at ltSgt2 ltRgt2 . If the system already has
broken inversion symmetry lower-order coupling
ltSgt2 ltRgt is possible. Even without inversion
symmetry breaking, ltSgt2 ltgrad Rgt or ltSgtltgrad
SgtltRgt is possible.
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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
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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

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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
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CoCr2O4
Co spins have ferromagnetic spiral (conical)
components Emergence of spiral component
accompanied by P
Tokura group PRL 06
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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)
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Elements of KNB Theory
The cluster Hamiltonian assuming t2g levels for
magnetic sites
KNB Hamiltonian is solved assuming SO gt U
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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
ltdyzypzgt, ltdxyypxgt is responsible for
polarization
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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 gtgt SO must be
considered also. (2) What about eg levels? (3)
From GL theory one expects some non-uniform
component too.
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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.
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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
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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 KNBs
theory in powers of U/?, our theory in powers of
?/U)
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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
ltrgt 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.
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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
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Numerical results for two holes
Transverse and longitudinal components exist
which we were able to fit using very simple
empirical formulas
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Uniform vs. non-uniform
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
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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?
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Coefficients
The uniform transverse component B1 is
significant for small U (KNB limit). A and B2
(non-uniform) are dominant for large U (our
limit).
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Comparison to GL theory
Within GL theory non-uniform polarization is also
anticipated. On comparing Mostovoys 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.
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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.