Order by Disorder

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The 93rdStatistical Mechanics Conference
RUTGERS UNIVERSITY
Order by Disorder
A new method for establishing phase
transitions in certain systems with continuous,
ncomponent spins.
Applications to transition metal oxides
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Cast of characters
M. Biskup (UCLA Math)
And, the physicists
Relevant papers
Z. Nussinov (Th. Div. Los Alamos)
M. Biskup, L. Chayes, Z. Nussinov and J. van den
Brink, Orbital order in classical models of
transition-metal compounds, Europhys. Lett. 67
(2004), no. 6, 990996
M. Biskup, L. Chayes and S.A. Kivelson, Order by
disorder, without order, in a two-dimensional
spin system with O(2) symmetry, Ann. Henri
Poincaré 5 (2004), no. 6, 11811205.
J. van den Brink (Lorentz ITP, Leiden)
M. Biskup, L. Chayes and Z. Nussinov, Orbital
ordering in transition-metal compounds I. The
120-degree model, Commun. Math. Phys. 255 (2005),
no. 2, 253292
S.A. Kivelson UCLA Physics
more to come
(Plenty.)
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Systems of interest Continuous spins O(2),
O(3),
huge degeneracy in the ground state.
Example 2D NNN Antiferromagnet. (Say XY
spins.)
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Transition Metal Compounds
Levels in 3d shell split by crystal field.
eg
dorbitals
t2g
Single itinerant electron _at_ each site with
multiple orbital degrees of freedom.
Superexchange approximation (and neglect of
strainfield induced interactions among orbitals)
KugalKhomskii Hamiltonian
120ºmodel (egcompounds)
V2O3, LiVO2, LaVO3,
orbital compassmodel (t2gcompounds)
LaTiO3,
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Orbital only approximation Neglect spin
degrees of freedom.
Large S limit (for pseudospin operators) Go
classical.
Classical 120º Hamiltonian
unit vectors spaced _at_ 120º.
Classical orbital compass Hamiltonian
usual Heisenberg spins.
For simplicity, today focus on 2D version of
orbital compass.
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The 2D Orbital Compass Model
constant.
Attractive couplings (ferromagnetic).
other groundstates but these play no rôle and
will not be discussed.
Couples in xdirection with xcomponent.
Couples in ydirection with ycomponent.
Clear Any constant spinfield is a ground state.
O(2) symmetry restored
(a) Not clear what are the states.
Cant even begin to talk about contours
(b) No apparent stiffness.
Hints from SWtheory?
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Key ideas
In the physics literature since the early 80s
J. Villain, R. Bidaux, J. P. Carton and R. Conte,
Order as an Effect of Disorder, J. Phys. (Paris)
41 (1980), no.11, 12631272.
E. F. Shender, Antiferromagnetic Garnets with
Fluctuationally Interacting Sublattices, Sov.
Phys. JETP 56 (1982) 178184 .
C. L. Henley, Ordering Due to Disorder in a
Frustrated Vector Antiferromagnet, Phys. Rev.
Lett. 62 (1989) 20562059.
Plus infinitely many papers (mostly quantum) in
which specific calculations done.
Our contribution to physics general theory
Modest.
All of this works even in d 2.
(But TMO models of some topical interest.)

• At b lt 8, weighting of various ground states
• must take into account more than just energetics

Fluctuations of spins will contribute to
overall statistical weight.

• These (spinfluctuation) degrees of freedom will
• themselves organize into spinwave like modes.

Can be calculated (or estimated).
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Key ideas
1) At b lt 8, weighting of various ground
states must take into account more than just
energetics
Gaussian like SWfree energetics will tell us
which of the ground states are actually preferred
_at_ finite temperature
Fluctuations of spins contribute to overall
statistical weight.
2) These (spinfluctuation) degrees of freedom
will themselves organize into spinwave like
modes.
Can be calculated (or estimated).
Remarks
(1) Not as drastic an approximation as it
sounds Infrared divergence virtually
nonexistent at the level of
freeenergetics.
(2) In mathphys, plenty of selection due
to finitetemperature excitations.
Excitation spectrum always with (huge) gap.
But
Finite (or countable) number of ground states.
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Lets do calculation. Write
q fixed ground state
Look _at_ H, neglect terms of higher order than
quadratic in js
Well, got
so
Get
and similarly
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Approximate Hamiltonian is therefore
Go to transform variables
So, after some manipulations,
Now, total weight can easily be calculated
Take logs
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Want as small as possible.
Scales with b.
Pause to refresh No difficulty doing these
integrals some infrared action but no big deal
(logarithmic). We are only interested in a free
energy.
Now log is a (strictly) concave function
Do kx, ky integrals on RHS, these come out the
same. We learn
Use strict concavity,
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Calculation indicates
There are states at
which will dominate any other state.
Outline of a proof.
(1) Define a fluctuation scale.
Look _at_ situation where each deviation variable jr
has
Fact
means that quadratic approximation is good.
Fact
means that the effective Gaussian variables
are allowed to get large.
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Not hard to see
(2) Define a running length scale, B and
another, interrelated spinscale, e.
Definition.
A block LB of scale B is defined to be good if
(You can add ?.)
Clear There are two types of good blocks.
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Also Two types of bad blocks.

• Energetic disaster. (Should be suppressed
• exponentially with rate bD2 )

More important, more interesting
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Reiterate Two distinct types of goodness. A
single box cannot exhibit both types of goodness.
Thus, regions of the distinctive types of
goodness must be separated by closed contours.
Bad blocks form contour element which separate
regions of the two distinct types of goodness.
FSS
FLIS(1)
FLIS(2)
LF
DLS
FLIS(3)
B Gives estimates on probability of contour bad
blocks by the product of the previously mentioned
probability estimates.
Standard Peierls argument, implies existence of
two distinct states.
Remarks Same sort of thing true for
antiferromagnet, 120ºmodel 3D orbital compass
model (sort of).
Interesting feature Limiting behavior of model
as T goes to zero is not the same as the behavior
of the model _at_ T 0.
In particular, nontrivial stiffness at T 0
(and presumably ? as well)..
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The End