The 93rdStatistical
Mechanics Conference
RUTGERS UNIVERSITY
Order by Disorder
• A new method for establishing phase transitions
in certain systems with continuous, n–component spins.
• Applications to transition metal oxides
Cast of characters:
M. Biskup (UCLA Math)
And, the physicists:
Z. Nussinov
(Th. Div. Los Alamos)
J. van den Brink
(Lorentz ITP, Leiden)
S.A. Kivelson
UCLA Physics
Relevant papers
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, 990–996
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, 1181–1205.
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, 253–292
… more to come …
(Plenty.)
Systems of interest: Continuous spins O(2), O(3), …
– huge degeneracy in the ground state.
Example: 2D NNN Antiferromagnet. (Say XY spins.)
Ground states: O(2)  O(2)
Now add “weak” NN ferromagnetic coupling.
Ground states still O(2)  O(2).
(Exactly zero message from nearest neighbors.)
What can this system do?
MW: Certainly no magnetization
General considerations:
¿ Disorder?
Transition Metal Compounds
• Levels in 3d shell split by crystal field.
eg
d–orbitals
• Single itinerant electron @ each site
with multiple orbital degrees of freedom.
t2g
Super–exchange approximation (and neglect of strain–field induced interactions among orbitals):
H 

r, r  
r, r 
H orb
(s r
 s r 
1)
4
r, r 
H orb
 J[4̂ r ̂ r  2̂ r  2̂ r 1]
 = direction of bond r  r 
[Kugal–Khomskii Hamiltonian]
120º–model (eg–compounds)
V2O3, LiVO2, LaVO3, …
orbital compass–model (t2g–compounds)
LaTiO3, …
̂ rx  14 ( rz  3 rx ) ̂ ry  14 ( rz  3 rx )
̂ rz  12  rz
̂ rx  12  rx
̂ ry  12  ry
̂ rz  12  rz
• Orbital only approximation: Neglect spin degrees of freedom.
• Large S limit (for pseudo–spin operators): Go classical.
Classical 120º Hamiltonian:
[a]
[b] [b]
[c] [c]
H   (Sr[a]Sre

S
S

S
r
rey
r Srez )
x
r
Sr an XY –spin
r
[a]
Sr  Sr  â
similarly for Sr[b] & Sr[c] ,
â, b̂ and ĉ
unit vectors spaced @ 120º.
Classical orbital compass Hamiltonian:
[x]
[ y] [ y]
[ y] [ z]
H   (Sr[ x ]Sre

S
S

S
r
rey
r Srez )
x
r
Sr  (Sr[ x], Sr[ y], Sr[ z] )
– usual Heisenberg spins.
For simplicity, today focus on 2D version of orbital compass.
The 2D Orbital Compass Model
r
Sr S1, write Sr  Sr[ x ], Sr[ y].
H L 
May as well take r  L .
[x] [x]
[ y] [ y]
(S
S

S
 r rex r Srey )
r L

L  L torus
1
[x] 2
[ y]
[ y] 2
   (Sr[ x ]  Sre
)

(S

S
r
rey )
x
2 r L
Attractive couplings (ferromagnetic).
Couples in x–direction with x–component.
Couples in y–direction with y–component.
 + constant.
$ other ground–states but these
play no rôle and will not be
discussed.
Clear: Any constant spin–field is a ground state.
* O(2) symmetry restored *
Can’t even begin to talk about contours:
¿Hints from SW–theory?
(a) Not clear what are the “states”.
(b) No apparent “stiffness”.
Disaster:
1
1
1 1
S(k) 
 2 2.
 x (kx )  y (ky ) kx ky
^
Very IR–divergent.
Indeed, spherical version of this
model has no phase transition.
But infrared bounds only give upper bounds on the scattering function.
And MW–theorem (strictly speaking) does not apply.
–– The O(2) symmetry is in the ground states, not the Hamiltonian itself. ––
Theorem. For the d = 2 orbital compass model, for all b sufficiently
large, $ (at least) two limiting translation invariant Gibbs states.
[ x] 2
( x)
One has  (S0 )  b close to one.
[ y] 2
( y)
The other has  (S0 )  b close to one.
Key ideas:
In the physics literature since the early 80’s
J. Villain, R. Bidaux, J. P. Carton and R. Conte, Order as an Effect of
Disorder, J. Phys. (Paris) 41 (1980), no.11, 1263–1272.
E. F. Shender, Antiferromagnetic Garnets with Fluctuationally
Interacting Sublattices, Sov. Phys. JETP 56 (1982) 178–184 .
C. L. Henley, Ordering Due to Disorder in a Frustrated Vector
Antiferromagnet, Phys. Rev. Lett. 62 (1989) 2056–2059.
Really clarified matters; put things on a
firm foundation in a general context.
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.)
1) At b < ∞, weighting of various ground states
must take into account more than just energetics:
• Fluctuations of spins will contribute to overall statistical weight.
2) These (spin–fluctuation) degrees of freedom will
themselves organize into spin–wave like modes.
• Can be calculated (or estimated).
Key ideas:
1) At b < ∞, weighting of various ground states
must take into account more than just energetics:
• Fluctuations of spins contribute to overall statistical weight.
2) These (spin–fluctuation) degrees of freedom will
themselves organize into spin–wave like modes.
• Can be calculated (or estimated).
Remarks
Gaussian like SW–free energetics
will tell us which of the ground
states are actually preferred
@ finite temperature
(1) Not as drastic an approximation as it sounds;
• Infrared divergence virtually non–existent at
the level of free–energetics.
(2) In math–phys, plenty of “selection due
to finite–temperature excitations”.
But:
• Excitation spectrum always with (huge) gap.
• Finite (or countable) number of ground states.
Sr[ x]  cos(   r )
Sr[ y]  sin(   r )
Let’s do calculation. Write:
[ = fixed “ground state”]
Look @ H, neglect terms of higher order than quadratic in ’s
[ x]
Well, got: Sr  cos(   r )  cos cos  r  sin sin  r
[x]
so Sr[ x ]  Sre
 cos (cos  r  cos  rex )  sin  (sin  r  sin  rex ).
x
Neglect quadratic (and beyond).
Will square this.
Get
S
[ x]
r
[x]
rex
  [ sin  ] 
[ y]
rey
  [ cos  ] 
S
2
2
r
  rex

2
and similarly
S
[ y]
r
S
2

2
2
r
  rey .
Approximate Hamiltonian is therefore:
1
 bHA ( )   b   cos2   r   rex
2 r 

  sin  
2
2
r   rey

2
.

1
^
ikr

e
.

k
Vol. k
Go to transform variables:  r 
So, after some manipulations,


^
 bHA ( )   b  |  k | 2  cos2  1 cos kx  sin 2  1 cos ky  .
k
Now, total weight can easily be calculated:
ZA ( )   e
 b HA ( )
d  
k
Fact that we are talking about “”
means that we do not integrate
over the k = 0 mode.
1


b cos  1  cos k x   sin  1  cos k y
2
2
Take logs:
.



lim  logZ2A ( )  const.  1  d 2 k log cos2  1 cos kx  sin 2  1 cos ky 
L
L
2
Want as small as possible.
A free energy; ( ).
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:


log cos2  1 cos kx  sin2  1 cos ky   [cos2 ] log 1 cos kx [sin 2 ] log 1 cos ky .
Do kx, ky integrals on RHS, these come out the same. We learn
( )  [cos 2  ] (0) [ sin 2 ] (0)
Use strict concavity,
(  0)
–or–
(  2 )
>

(  0)  (  2 ).
Calculation indicates:
There are “states” at   0 and  

2
which will dominate any other “state”.
Outline of a proof.
(1) Define a fluctuation scale.
1
1
b: b  2  (b )  b  3
Look @ situation where each deviation variable r has:
 r  (b )
– and hence –
 r   r   2(b )
Fact: b3  1 means that quadratic approximation is “good”.
Fact: b 2  1 means that the effective Gaussian variables
– namely b r –
are allowed to get large.
Not hard to see:
Proposition: With the constraints (globally) enforced, the spin wave
formula for the  – dependent limiting free energy is asymptotic as b  .
(2) Define a running length scale, B – and another,
Not important; for technical
interrelated spin–scale, e.
reasons, this is not (b).
Definition. A block B of scale B is defined to be good if
(a) Each neighboring pair of fluctuation variables  r ,  r  ; r, r   B
satisfies
 r   r  .

(b) Each spin variable  r , r  B satisfies
 r  e – or–  r  2  e .
(You can add .)
Clear: There are two types of good blocks.
Also: Two types of bad blocks.
(i) Energetic disaster. (Should be suppressed
exponentially with rate ~ b2 )
More important, more interesting:
(ii) Energetics good, but  not particularly near 0 or

2
.
Can use method of chessboard estimates.
[FSS] [FLIS(1)] [FLIS(2)] [LF] [DLS] [FLIS(3)]
A] Gives estimates on probability of bad blocks in terms of
constrained partition function where all blocks are bad blocks.
N2
Type (i) indeed suppressed exponentially.
Type (ii) has probability bounded by
N

e
~
2
( )(0)
where N is the block size. This goes to zero as N  .
Note: Type (ii) is independent of b.
 ZL (A )  L2
P (A )  

 ZL 
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.
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.
• Standard Peierls argument, implies
existence of two distinct states.
Remarks: Same sort of thing true for antiferromagnet,
120º–model & 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 @ T = 0.
In particular, non–trivial stiffness at T = 0 (and presumably  as well)..
The End