COMPLEX MAGNETIC STRUCTURES

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COMPLEX MAGNETIC STRUCTURES
1
1. LANDAU FREE ENERGY
2. WAVE VECTOR SELECTION
3. SYMMETRY CONSIDERATIONS
4. ORDER PARAMETERS
5. MORE THAN ONE ORDER PARAMETER
See ABH, Phys. Rev. B 76, 054447 (2007).
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LANDAU FREE ENERGY
2
3
LANDAU FREE ENERGY
3
The number of microstates decreases as the spin
ordering increases. The sign of the spin does
not matter. So
T Tc 1
Thus
Free energy
Spin S
T Tc -1
T Tc -2
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WAVE VECTOR SELECTION
4
Consider mean field theory for a one dimensional
system
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WAVE VECTOR SELECTION
5
T Tc 2
T Tc 1
Lifshitz criterion if q at TTc is
incom- mensurate (qq), then q is
temperature dependent in the ordered phase. q
at TltTc can be commensurate only if it is a high
symmetry wave vector.
T Tc
How can we understand this principle?
Note that the value of q depends on the specific
interactions in the system. That is to say, if we
were to add a third neighbor interaction that
would modify the extremum condition that
determined q. If nothing else, the Js will be
very slightly temperature-dependent because of
the thermal expansivity. In contrast, return to
the previous slide there you see that the
extremum is at a symmetry point which is stable
against addition of weak further neighbor
interactions.
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WAVE VECTOR SELECTION
6
In the above discussion it was not said whether
several values of wave vector could be
simultaneously selected. This brings up the
whole question of accidental degeneracy as it
becomes relevant in its many guises. For this
discussion it is useful to consider the vacuum
as being the disordered phase when all the order
parameters are zero. The symmetry of the vacuum,
is the symmetry of this paramagnetic state. For
any excitation relative to the vacuum, i. e. for
any set of values of the order parameters, there
will be a family of states with identical free
energy which are generated by applying all
the symmetry operators of the vacuum to that
excitation.
A simple application of this principle is that if
the free energy has an instability at a wave
vector q, it will also have symmetry-related insta
bilities at all wave vectors in the STAR of q.
(The STAR of q consists of all wave vectors which
are generated by symmetry operators of the vacuum
providing that we do not accept wave
vectors which are equivalent in the sense that
their difference is a reciprocal lattice vector,
a RLV.) An important STAR includes q and q if
q and q do not differ by an RLV.
This is not all!!
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WAVE VECTOR SELECTION
7
Can I possibly find more to say under this
heading?? YES.
You may ask is it possible that two wave vectors
not related by symmetry could simultaneously
become unstable? This is an accidental
degeneracy of the type theoreticians excluded two
centuries ago in the development of
thermodynamics.
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LOCK-IN
8
So, as the temperature is lowered further into
the ordered phase, the range over q for which
phase locking occurs grows.
I should have mentioned that once on order
parameter condenses, entropic effects prevent
another order parameter of the same symmetry from
condensing.
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PHASE LOCKING
9
When J1 and J2 compete (both antiferromagnetic)
AF
If there are several Js, then when such a system
is mapped onto the model system with two Js, the
ratio J1/J2 can be temperature dependent.
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A BIT OF THEORY
10
When we consider a system with several vector
spins per unit cell, we need a more formal
approach. One introduces the susceptibility i.
e. the spin correlation function
where R labels the unit cell and t labels sites
1, 2, n in the unit cell.
This is a 3n x 3n matrix. This matrix is a
function of wave vector q.
Stat mech tells us that the leading term in the
Landau expansion is
where
Warning many people use eiqR
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COMPLEX UNIT CELL
11
To repeat
You can think of the inverse susceptibility
matrix as being
temperature independent interaction
This is the generalization of
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COMPLEX UNIT CELL
12
To repeat
This is reminiscent of phonons, where c-1 plays
the role of the dynamical matrix. The
eigenvectors x of this matrix are characterized
by symmetry labels. The critical eigenvector
gives the distribution of spin components over
the unit cell when the system orders and within
mean field theory the amplitude of the ordering
is governed by the terms of order S4.
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SYMMETRY CONSIDERATIONS
13
Here I am not going to discuss F from the formal
point of view of group theory. As it happens,
there are canned programs (which I will refer to
later) which do the group theory analysis. My
aim here is to treat some simple examples where a
knowledge of group theory is not necessary. The
point of this discussion is to be sure that
we understand the basic language of group theory
so that we can check that we know how to use the
output of such canned programs.
In these simple examples we will invoke the
well-known principle for the eigenvectors of a
matrix M. Namely, if we have a set of
mutually commuting operators which commute with
M, the eigenvectors of M are also simultaneously
eigenvectors of the commuting operators. (Group
theory was invented to treat the case when the
operators which commute with M do not commute
with each other.) The point is that we
definitely do not want to try to construct the
inverse susceptibility and then diagonalize it.
However, we will rely on the unambiguous
experimental determination of the selected wave
vector q.
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DETERMINING q
14
Kenzelmann et al, PRB 76, 014429 (06)
Kenzelmann et al, PRL 95, 087206 (05)
Time reversal says that the total number of
powers of Q is even. We have wave vector
conservation because Q(q) Q(-q).
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MULTICRITICAL POINT
15
Look at the phase diagram for an
antiferromagnet with the z-axis being the easy
axis in a magnetic field H along the z axis. If
we lower the temperature ordering will be into
either the low field state or high field state.
Only if the field is adjusted to be exactly at
the critical value will one access the bicritical
(magenta) point.
SPIN FLOP
This example indicates that if we consider a
phase transition when the temperature is lowered,
we ought not allow an accidental degeneracy in
which two different irreducible representations
condense. In simpler language we are only
allowed to break one symmetry at a time.
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SIMPLE ILLUSTRATION
16
Finding the eigenvectors of the commuting
operators which leave c-1invariant will reproduce
the results of group theory.
Accordingly, let us consider the magnetic
ordering of TbMnO3 (TMO).
The ordering wave vector is along b which is
called y here. q(0.52p/b) y
Mn
Tb
There are 8 spins per unit cell.
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PHASE DIAGRAM OF TMO
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At right I show a schematic phase diagram of TMO.
Paramagnetic Paraelectric
The structure of the HTI phase is a
predominantly magnetically ordered collinear
incommensurate phase described by irrep G3
(explained below).
P spontaneous polarization




HTI high LTI low
The low field LTI phase has additional
contributions from G2, so that a magnetic spiral
is formed which gives rise to a ferroelectric
phase.

temperature incom-
mensurate phase
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EIGENVECTORS FOR TMO
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spin component labels
S(1)
S(2)
S(3)
S(4)
irrep

• l(mx) l(mz) x y z x y z x y
  z x y z
• G1 L 1 a b c a b c a
  b c a b c
• G2 -L -1 a b c a b c a
  b c a b c
• G3 -L 1 a b c a b c a
  b c a b c
• G4 L -1 a b c a b c a
  b c a b c

x denotes -x
NOTES All parameters are complex valued. The
constants for different irreps are unrelated. So
for G1 a denotes a1, for G2 a denotes a2, etc.
G1 has 5 constants G1 is contained 5 times in
the original reducible representation. G2, G3,
and G4 are contained 7, 5, and 7 times
respectively. L exp(ipq).
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SPATIAL INVERSION??
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What about spatial inversion? This operation is
not in the group of the wave vector because
inversion takes q into q. However, the free
energy is invariant under inversion. It is
amazing that the world wide neutron scattering
community (including many famous people!) didnt
know how to take this into account. As a relative
novice I uncovered this problem. In hindsight
this problem had been solved long ago. But this
important work was either forgotten or
overlooked. There are several ways to take
account of inversion. One way is to use the
rather arcane formalism of corepresentations.
Another is to use the full subgroups of the
entire space group (rather than the group of the
wave vector). This latter method has been used
to develop a canned program which I will refer to
in a moment. Meanwhile I continue with the poor
mans approach to symmetry which does not invoke
corepresentations.
To illustrate the simple approach, let us
consider irrep G3. I consider this irrep because
it is the one that experiment identifies as the
first magnetic irrep to condense as
the temerature is lowered.
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INVERSION
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For notational simplicity, let a, b, c, C, and F
be denoted x1, x2, x3, x4, and x5, respectively.
Because we are considering a wave vector
having only a y component, we only need keep
track of how the y coordinate of position
transforms. (Because spin is a pseudo vector its
orientation is invariant under inversion the
only effect of inversion is to relocate
the spin.) If we invert spin 1 (or 2, 3, or
4), we change the sign of its y component of
position which takes exp(iqy) into its complex
conjugate. Thus
If we invert spin 5, it goes into spin 7 and
6 goes into 8. So
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INVERSION (continued)
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Note that G is Hermitian G21 G12. So if G12
G21, then G21G12 is real. Proceeding similarly
we find that G is of the form
Where a, b, c, d, e, f, and g are real valued and
the Greek letters are complex valued. The
eigenvectors of this form of matrix must be of
the form
where r, s, and t are real and r is complex.
We included a phase factor because we are working
in a complex vector space. In comparison with
what we had before inversion a, b, and c were
complex quantities. But now we see that they
must all have the same phase. The Tb amplitudes
C and F, which were arbitrary complex numbers,
now must be complex conjugates of one another.
If one ignores the consequences of inversion
symmetry, one has these phases to determine from
the data and that proved to be a very
difficult task, so invariably some guesswork was
performed. Furthermore, when these constraints
are ignored, the symmetry of the structure is
wrong a single irrep would then possibly induce
ferroelectricity, a result we know to be
incorrect.
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ISODISTORT
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1. Search for ISODISTORT and click on the entry

• Construct crystallographic input file (ISODISTORT
  will
• prompt you) I upload the cif file prepared
  previously.

3. space-group preferences I click OK, but
maybe you can do better.

• Types of distortions to be considered. Magnetic
  Tb and Mn.
• Then OK.

5. Method 2 SM (0,a,0) a0.52. I think of
inc. mod. 1. Then OK

• Pick IR. As Wills says, we may not be sure of
  the convention.
• I took mSM3 because I thought that Kenzelmann
  took the best
• reference. Then click OK.

7. Finish selecting I chose P1Z. The OK.
8. Click on Complete modes details. Then OK
9. Then Magnetic mode definitions gives the IR
basis vectors.
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COMPARE TO ISODISTORT
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atom
atom x y z Mz
Mx My
ME
s
s
s
s
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Tb Spin Function for G3
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ISODISTORT
ME
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TWO IRREPS AT THE SAME TIME?
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We argued against having an accidental
degeneracy so that the disordered phase would
have simulataneous instabilities to ordering. In
what I call the Green Bible all possible
subgroups which can arise are tabulated. (This
is actually too bold a claim! If you have a first
order transition, I think anything is possible,
at least in principle.) Here I give a
simple mechanism for having two different
irreps simultaneously appear at a first order
phase transition.
H. T. Stokes and D. M. Hatch Isotropy
Subgroups of the 230 Crystallographic Space
Groups
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TWO IRREPS??
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Now suppose we have two different symmetry order
parameters, Q1 and Q2, each of which, if they
were uncoupled, would have a first order phase
transition. With T1 gt T2 and all parameters gt0
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ORDER PARAMETERS
27
In TMO we have an incommensurate colinear phase
at 42K and then a spiral at 32K at which point
one has a ferroelectric. In the Nature
paper this was attributed to the
hypothetical lock-in phenomena. Nearly at
the Same time we similarly analyzed Ni3V2O8.
To discuss the symmetry of this situation it is
helpful to introduce order parameters.
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ORDER PARAMETERS
28
Landau theory for the higher T transition is as
we have described it above. For the lower T
transition the vacuum is no longer the trivial
state with zero ordering. In principle we need
to discuss the symmetry of the second ordering in
the presence of the first ordering. This is a
nasty project!! Consider the hypothetical phase
diagram shown below.
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ORDER PARAMETERS
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The spin wave function for the unit cell (which
is just the Fourier coefficient At the wave
vector q), has the form of the basis function
whose coefficients may or may not be determined
from a refinement of scattering data. What we can
say is that
where xn with n1,2 are the basis vector for the
two irreps. Note that of course the Qs are
complex because their phase regulates where the
origins of the waves are placed. If we had only
one wave, this phase would be unobservable. So
really what is important is the relative phase.
Initially we do not treat the interactions
between irreps and write
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LOOK AGAIN AT LOCKING
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The two irreps are out of phase this is what is
needed for a spiral. But this actually follows
from simple hand-waving arguments As the
temperature is lowered, the spins more and more
want to have fixed length (as they do at T0).
Thus when spin ordering from one irrep is large,
the spin ordering from the other irrep must be
small.
In phase
Out of phase
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MAGNETOELECTRICITY
31
If x is an eigenvector,

• l(mx) l(mz)
• G1 L 1
• G2 -L -1
• G3 -L 1
• G4 L -1

Let Q3 be the complex amplitude (order parameter)
for G3 and Q2 that for G2. then Q3 will be that
for G3 and Q2 will be that for G2. Note also I
Qn Qn.
Lexp(ipq)
We want to see how the Qs can be combine with
the spontaneous polarization P to induce a
nonzero value of P. To induce a nonzero value of
P the interaction should be linear in P. For
time reversal invariance we must have an even
number of Qs To conserve wave vector one will
be Q and the other will be QQ(-q). We can not
have QnQn because this transforms like unity. So
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CONCLUSION

• Landau theory provides a natural way to
  understand the
• phenomenology of phase transitions.

• An alternative to group theory can provide a
  convenient
• route to categorizing the symmetry of system
  whose symmetry
• is not too complicated. Of course, group theory
  provides the
• definitive answers concerning symmetry.

3. Where Landau theory becomes essential is when
one Investigates coupling between several
different symmetries.
4. The principal of no accidental degeneracy
has wide application.
5 A corollary if it allowed by symmetry, it
happens.