Order by Disorder

• A new method for establishing phase transitions in certain systems with continuous, n–component spins.

The 93rdStatistical Mechanics Conference

RUTGERS UNIVERSITY

• 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. KivelsonUCLA Physics

Relevant papersM. 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, 990–996M. 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

.)

Now add “weak” NN ferromagnetic coupling.

Example: 2D NNN Antiferromagnet. (Say XY spins.)

Systems of interest: Continuous spins O(2), O(3), …

– huge degeneracy in the ground state.

Ground states: O(2)×O(2)

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.

t2g

• Single itinerant electron @ each sitewith multiple orbital degrees of freedom.

Super–exchange approximation (and neglect of strain–field induced interactions among orbitals):

[Kugal–Khomskii Hamiltonian]

H = Horb

r, ′r

<r, ′r >∑ (sr ⋅s ′r + 1

4 )Horb

r, ′r =J [4π rαπ ′r

α −2π rα −2π ′r

α +1]

eg

d–orbitals

α = direction of bondr − ′r

120º–model (eg–compounds)V2O3, LiVO2, LaVO3, …

π rx =1

4 (−σ rz + 3σ r

x) π ry =1

4 (σ rz− 3σ r

x)

π rz =1

2σ rz

π rx =1

2σ rx π r

y =12σ r

y

π rz =1

2σ rz

orbital compass–model (t2g–compounds)LaTiO3, …

• Orbital only approximation: Neglect spin degrees of freedom.

• Large S limit (for pseudo–spin operators): Go classical.

Classical 120º Hamiltonian:

−H = (Sr

[ x ]

r∑ Sr+ex

[ x ] + Sr[ y]Sr+ey

[ y] + Sr[ y]Sr+ez

[z ] )

−H = (Sr

[a]

r∑ Sr+ex

[a] + Sr[b]Sr+ey

[b] + Sr[c]Sr+ez

[c] ) rSr an XY –spin

Sr[a] =

rSr ⋅a

a, b and cunit vectors spaced @ 120º.

Classical orbital compass Hamiltonian:

similarly for Sr[b] & Sr

[c],

rSr =(Sr

[ x], Sr[ y], Sr

[ z] )

– usual Heisenberg spins.

For simplicity, today focus on 2D version of orbital compass.

The 2D Orbital Compass Model

write

rSr∈S1,

rSr = Sr

[ x], Sr[ y]( ).May as well taker ∈ΛL .

LL torus −H L = (Sr

[ x ]

r∈ΛL

∑ Sr+ex

[ x ] + Sr[ y]Sr+ey

[ y] )

+ constant.

Attractive couplings (ferromagnetic).Couples in x–direction with x–component.

Couples in y–direction with y–component.

=−1

2(Sr

[ x ] − Sr+ex

[ x ] )2 + (Sr[ y] − Sr+ey

[ y] )2( )

r∈ΛL

∑

Clear: Any constant spin–field is a ground state.

* O(2) symmetry restored *

Can’t even begin to talk about contours:(b) No apparent “stiffness”.

(a) Not clear what are the “states”.

¿Hints from SW–theory?

other ground–states but these

play no rôle and will not be discussed.

Theorem. For the d = 2 orbital compass model, for all sufficiently large, (at least) two limiting translation invariant Gibbs states.

One has close to one.< (S0[ x ])2 >(x )

β

The other has close to one.< (S0[ y])2 >(y)

β

Disaster:

Very IR–divergent.

But infrared bounds only give upper bounds on the scattering function.

And MW–theorem (strictly speaking) does not apply.

Indeed, spherical version of this model has no phase transition.

–– The O(2) symmetry is in the ground states, not the Hamiltonian itself. ––

S^(k)∝

1Δx(kx)

1Δy(ky)

≈1kx2

1ky2 .

Key ideas:In the physics literature since the early 80’s

E. F. Shender, Antiferromagnetic Garnets with Fluctuationally Interacting Sublattices, Sov. Phys. JETP 56 (1982) 178–184 .

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.

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 < ∞, weighting of various ground statesmust 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).

1) At < ∞, weighting of various ground statesmust 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).

Key ideas:

Gaussian like SW–free energeticswill tell us which of the groundstates are actually preferred

@ finite temperature

Remarks(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.

Let’s do calculation. Write:Sr

[ x ] =cos(θ +ϕ r )Sr

[ y] =sin(θ +ϕ r )[θ = fixed “ground state”]

Look @ H, neglect terms of higher order than quadratic in ’s

Well, got:Sr[ x ] =cos(θ +ϕ r ) =cosθ cosϕ r −sinθ sinϕ r

so Sr[ x ] −Sr+ex

[ x] =cosθ(cosϕ r −cosϕ r+ex)−sinθ(sinϕ r −sinϕ r+ex

).

Will square this.Neglect quadratic (and beyond).

Get

and similarly

Sr[ y] −Sr+ey

[ y]( )

2≈[ cos2θ] ϕ r −ϕ r+ey( )

2.

Sr[ x ] −Sr+ex

[ x]( )2≈[ sin2θ] ϕ r −ϕ r+ex

( )2

Approximate Hamiltonian is therefore:

Go to transform variables:ϕ r =1

Vol.ϕ^

k∑ k

e−ikr .

So, after some manipulations,

−H A (θ ) = −β |ϕ^

k∑ k | 2 cos2 θ 1− cos kx( ) + sin2 θ 1− cos ky( )⎡⎣ ⎤⎦.

Now, total weight can easily be calculated:

ZA (θ) ≡ e−H A (θ ) dϕ ∝

1

cos2 θ 1−coskx( )+ sin2 θ 1−cosky( )( )k∏∫ .

Fact that we are talking about “θ” means

that we do not integrate over the k =

0 mode.

Take logs:

−H A (θ ) = −

1

2β cos2 θ ϕ r −ϕ r+ex

( )2

+ sin2 θ ϕ r −ϕ r+ey( )2⎡

⎣⎢⎤⎦⎥r

∑ .

−logZA (θ )

L2 = const.+1

2d 2k log cos2 θ 1− cos kx( ) + sin2 θ 1− cos ky( )( )( )∫

Scales with

Pause to refresh: No difficulty doing these integrals; some infrared “action” but no big deal (logarithmic). We are only interested in a free energy.

Want as small as possible.

A free energy;Φ(θ ).

Now log is a (strictly) concave function:

Do kx, ky integrals on RHS, these come out the same. We learn

Use strict concavity, >Φ(θ ≠ 0)–or–

Φ(θ ≠ π2

)Φ(θ = 0) = Φ(θ = π

2).

limL→ ∞

log cos2θ 1−coskx( ) +sin2θ 1−cosky( )( )( )≥[cos2θ] log 1−coskx( )+[sin2θ] log 1−cosky( ).

Φ(θ ) ≥ [cos2 θ ]Φ(0) +[ sin2 θ] Φ(0)

Calculation indicates:

There are “states” at θ =0 and θ =π

2which will dominate any other “state”.

Outline of a proof.

(1) Define a fluctuation scale.

Look @ situation where each deviation variable r has:

– and hence –

ϕ r < Δ(β )

ϕ r −ϕ ′r < 2Δ(β )

Fact:

Fact: means that the effective Gaussian variables– namely ϕ r –

are allowed to get large.

means that quadratic approximation is “good”.

Δ −12 << Δ(β ) << β − 1

3

Δ3 << 1

Δ2 >> 1

Not hard to see:Proposition: With the constraints (globally) enforced, the spin wave formula for the θ– dependent limiting free energy is asymptotic as

→ ∞.

(2) Define a running length scale, B – and another, interrelated spin–scale,

Not important; for technical reasons, this is not Δ().

Definition.A block ΛB of scale B is defined to be good if

(a) Each neighboring pair of fluctuation variablesϕ r ,ϕ ′r ; r, ′r ∈ΛB

satisfies ϕ r −ϕ ′r ≤ Δ.

(b) Each spin variableθr , r ∈ΛBsatisfies

θr < ε – or– θ r − π2

< ε .

Clear: There are two types of good blocks.

(You can add π.)

Also: Two types of bad blocks.(i)Energetic disaster. (Should be suppressed

exponentially with rate ~ Δ2 )

More important, more interesting:(ii) Energetics good, but θ not particularly near 0 or . π

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.

Note: Type (ii) is independent of .

Type (i) indeed suppressed exponentially.

Type (ii) has probability bounded by<~ e− N 2 Φ(θ )−Φ(0)( )

where N is the block size. This goes to zero as N → ∞.

P(A ) ≤

ZL (A )ZL

⎡

⎣⎢

⎤

⎦⎥

N2

L2

Bad blocks form contour element which separate regions of the two distinct types of goodness.

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.

B] Gives estimates on probability of contour bad blocks by the product of the previously mentioned probability estimates.

[FSS][FLIS(1)][FLIS(2)][LF] [DLS][FLIS(3)]

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