Order by Disorder

Subin Kim (1000784591)

December 11 2018

Abstract

In this paper, One of the phenomena in the condensed matter, order by disorder, was considered. Givenhighly degenerate ground-state from the frustrated system, the degeneracy is lifted due to the thermalfluctuation. Some of the model such as domino model and frustrated diamond lattice is considered wherethey are expected to have order by disorder phenomena. Later, experimental result of MnSc2S4 wasconsidered where the neutron diffraction data can be explained by order by disorder.

1 Introduction

In some condense matter system, it is not possibleto satisfy all the interaction in the system to find theground state and the system is said to be frustrated.These may occur due to competing multiple interac-tions or the structure of the lattice. Instead, thesesystem tend to have multiple distinct ground statesat 0 temperature and they tend to suppress the order-ing of its system. One of the example of a frustratedsystem is a Ising spin anti-ferromagnetic interactionin the triangular system as shown below.

Figure 1: spins on the triangular system. Due tothe frustration, there are 6 ground-state instead of 2ground-state of usual Neel order.

We see that if we assign one of the atom to bespin-up and one of the neighbour atom to be spin-down, the last atom doesn’t have an optimum spindirection where it can satisfy both of its neighbours.Therefore, the ground state of this system has 6 con-figurations instead of general 2 configurations thatare time reversal symmetry of typical Ising models.These frustration and large degeneracy of the groundstate acts to destabilize the conventional ordering.We would expect the ordering will be suppressed and

the critical temperature to be lowered. At high tem-perature in paramagnetic region, magnetic suscepti-bility follows Curie-Weiss law:

χ(T )∝1

T − θCW(1)

where θCW is the Curie-Weiss temperature and thetemperature reflects the energy scale of the exchangeinteraction. Without any frustration, we will expectthe critical temperature of the system, Tc, to be closeto the θCW (and it is exactly equal in mean field the-ory). In the presence of the frustration in the system,the critical temperature is shifted downward and thesystem behaves paramagnetic even at low tempera-ture. The degree of frustration can be parametrizedby the frustration parameter f which is defined as:

f =θCWTc

. (2)

The frustration parameter can vary from materialsto materials; CoAl2O4 and MnSc2S4 are on range of10∼20 [1,2] while material such as SrCr9−xGa3+xO9

is on the order of 100 [3]. Later, we will show someexamples of the frustrated system where the systemhas extensive ground-state degeneracy and makes thesystem disordered. However, the degeneracy of theground-states is not protected by the symmetry andcan be lifted by the phenomenon called order bydisorder.

1.1 Order by Disorder

Order by disorder is a phenomenon where one ofthe ground-state is selected from many degenerate

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ground-states because of the fluctuation which corre-sponds to the disorder term in the order by disorder.Suppose the system is described by the variable (x, y)in phase space like the diagram below where groundstate can be parametrized by x and nearly by statecan be parametrized by y (locally orthogonal to x).

Figure 2: Ground-state manifold where it isparametrized by x. There would be states accessibleclose to the ground-state where they will be describedby local coordinate y.

At low temperature, there will be accessiblestates nearby the ground-state manifold (representedby locally orthogonal component y) and one can cal-culate probability of the ground-state by consideringthe fluctuations of the states nearby the ground-state.By integrating out the small amplitude fluctuationin y from the energy H(x, y), the probability of theground-state can be derived as:

P (x)∝ Z(x) = ∫ Dye−βH(x,y) ∝∏k

kBT

ωk(x)(3)

where ωk(x) corresponds to energy of the excitationk around the ground-state labelled by x. There aretwo distinct situation: One situation where the prob-ability is distributed such that all ground-states areaccessible and the other situation where probabilitydistribution is concentrated on only the subset of theground-state manifold. The latter corresponds to thecase where the states are selected by fluctuations with

∏k ωk(x) small. In practice, these will be the stateswhere ωk(x) vanishes for some subset of k (so thatprobability is dominated by these subset). The mainidea here is that the system will choose the ground-state where the excitations cost little energy compareto others (soft mode). With a little cost of energy,they are more probable to occupy due to Boltzmann

factor and also more states are available at some finiteenergy region. This leads to large entropy around theparticular ground-state and the free energy of the sys-tem (F=U-TS) is smaller compare to other ground-state configurations. In the upcoming sections, we’llsee examples such as domino model where J.Villainat el. first introduced the idea of order by disorder[4] and diamond lattice with 2nd nearest neighbourinteraction where the particular ground-state is se-lected from other ground-states due to order by dis-order [5].

2 Domino Model

Domino model is a Ising model in a 2D square lattice.The lattice is constitute of alternating chains of twoions, A and B as shown below.

Figure 3: Domino model: alternating chains of A ionsand B ions with Ising spins. Magnetic interaction aredefined by JAA, JAB > 0 (ferromagnetic and JBB < 0(anti-ferromagnetic)

J.Villain at el. considered three interaction:JAA, JAB > 0 (ferromagnetic) and JBB < 0 (anti-ferromagnetic) between nearest neighbours in dominomodel. The system is frustrated due to competing in-teractions where JAB which prefers to align the spinsof A and B and therefore, want to align all the spinsalong the B chain while along the B chain, spins wantto anti-align due to the coupling JBB . The strengthof the interaction is considered as following:

JAA > ∣JBB ∣ > JAB > 0, JBB < 0. (4)

To find the ground state, the strongest interactionJAA needs to be satisfied first. Then all the spinsalong the chain of A ions should align. Next, the in-teraction between spins along the B chain, JBB , needs

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to be considered. To minimize energy, all the spinsalong the B chain should be anti-parallel. The last in-teraction JAB doesn’t play a role after JAA and JBBare considered so the ground state consists of spinsaligned along A chain and spins anti-aligned alongB chain as shown in figure 3. As long as the spinsalong the A chain is aligned and the spins along theB chain is anti-aligned, there is no energy difference.If there are total N” chains, there are Ω = 2N

′′

dif-ferent configurations of ground states of the system.The ground state is indeed highly degenerate. Sup-pose the chain of all spin up and spin down along theA chain are labelled ↑A and ↓A and label the chain ofall spins anti-aligned along the B chain while startingwith spin up and spin down as ↑B and ↓B respectively.

Figure 4: If the chain of spin is labelled by spin up andspin down, we see that this behave just like 1D para-magnet and the ground-state is indeed disordered.

Then we see that the system looks like a 1Dparamagnet along the direction perpendicular to thechain. One can calculate average magnetization perspin square of the system as below:

< SiSj >=∑α

< α∣SiSj ∣α > (5)

<m2>=

1

N2∑i,j

< SiSj >= 0 (6)

where the sum over α implies summing over all theground-state configurations and N is the number oftotal spins (or ions). Due to the system behaving likea paramagnet, the magnetization does indeed vanish

in the thermodynamic limit (N → ∞) therefore, thesystem is disordered at T=0. The partition functionof this model can be calculated using transfer matrixmethod just like a 1D Ising model in the homework.The partition function is shown below:

Z

N= ln2

+1

16π2 ∫

2π

0∫

2π

0ln(

1

2[cosh2βJAAcosh2βJBB

+cosh22βJABcosh2β(JAA + JBB)

−sinh24βJABcosφ

−2cosh4βJABsinh4β(JAA + JBB)cosθ

+sinh4βJAAsinh4βJBBcos2θ])dθdφ.

The free energy of the system diverges at a criticaltemperature Tc which satisfies:

sinhβc∣JAA + JBB ∣sinh2βcJAB = 1 (7)

where βc is the inverse critical temperature. Thisindicates that there is a phase transition in thissystem and will be shown that it does indeed or-ders ferro-magnetically between different A chains.In this paper, they calculate the partition functionof different configuration of spins of a pair of Achains by integrating over all B chain configurationthat is sandwiched between the pair and then elimi-nate the B chains (re-normalization) from the prob-lem. Once the B chains are eliminated, they showedthat there is effectively ferromagnetic interaction be-tween A chains leading to the system to order ferro-magnetically. I have written a detail calculation inthe appendix A that J.Villain at el. have done. Butinstead, let’s consider the ground-states of the sys-tem and excitation around those states. There aretwo distinct cases: one where the B chain is sand-wiched between two A chains where the spins on thechains are aligned in the same direction and alignedin the other direction as shown below.

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Figure 5: Two different configurations of ground-state of a pair of A chains: spins aligned and anti-aligned. Little excitation for two different ground-state has different energy cost corresponding to soft-mode and hard-mode respectively.

Suppose we flip one of the spin on the B chain.The energy cost between two different configuration isdifferent. When the A chains are ferro-magneticallyaligned, flipping one of the spin has energy cost ofδE = 2∣JBB ∣ − 2JAB where there is a energy cost forbeing aligned with the neighbouring spins on B chainwhile the energy is compensated by being alignedwith spins on neighbouring A ions. However, whenthe A chains are anti-ferromagnetically aligned, flip-ping one of the spin has energy cost of δE = 2∣JBB ∣

where no energy is compensated from nearby A ions.Therefore, the excitation around ferro-magneticallyaligned A atoms is more probable due to equation(3) mentioned above. Through the calculation (Ap-pendix A), we can show indeed that the system doesorder ferro-magnetically between A chains and aver-age magnetization per site, m = mA+mB

2, does go to

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as expected.

Figure 6: The plot of the average magnetization persite as the function of temperature. We expected tobe 1

2due to ferro-magnetically ordering between A

chains. The exact calculation is done.

Here, we see that there is a hysteresis where ifone takes limT→0 and then take limN→∞, the systemis disordered. However, usually one will be given alarge system and one to see what happens as he lowersthe temperature. From the above calculation, we seethat if one takes limN→∞ and then limT→0 then thesystem indeed order where A chains are ordered ferro-magnetically. Another point to emphasize is that thedegeneracy of the ground-state is lifted due to theentropy not a perturbation.

3 Diamond Lattice with 1stand 2nd Nearest NeighbourInteraction Model

The existence of order by disorder in the diamond lat-tice with consideration of 1st and 2nd nearest neigh-bour interaction was first realized by Leon Balents’group in 2007 [5]. In nature, the diamond lattice canbe realized in AB2X4 spinel structure shown below.

Figure 7: Spinel Structure: A ions sits on the hole ofthe tetrahedrals while B ions sits on the octahedral.By removing B ions, the lattice looks like a diamondlattice of A ions.

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A ion sits on the hole of the tetrahedral formedby X ions and form diamond lattice while B ion sitson the hole of octahedral formed by X ions. Whenthe A ion is magnetic while others are non-magnetic,the spinel system can be realized as spins on the di-amond lattice. Some of the transition metal systemwith spinel structure are shown to be frustrated suchas CoAl2O4 and MnSc2S4 mentioned above [1,2].However, if one only considers nearest neighbour in-teraction (anti-ferromagnetic), there is no competinginteraction since the diamond lattice is a bipartiteface center cubic (FCC) lattice. One can just makeone of the ion’s spin in one direction while the otherion’s spin in the opposite making the system to goNeel order.

Figure 8: Diamond lattice can be divided into twoFCC sublattices. There is a trivial Neel orderingwhen anti-ferromagnetic nearest neighbor interactionis considered.

Therefore, there must be more than just near-est neighbour interaction in order to explain the frus-tration on above material as well as the non-trivialspiral ordering in MnSc2S4. By considering 2ndnearest neighbour (n.n) anti-ferromagnetic interac-tion, we see that the system is frustrated due to com-peting interactions of the 1st (J1) and 2nd (J2) neigh-bour. Bergman at el. realized that when 1st and 2ndneighbour interaction is considered, the ground stateis highly degenerate as they form ground-state man-ifold on the reciprocal lattice space. But due to thefluctuation, only subset of the ground-state minimizefree energy. Suppose a Hamiltonian of the system is

given by:

H = J1 ∑<ij>

Si ⋅ Sj + J2 ∑<<ij>>

Si ⋅ Sj (8)

where the sum goes for 1st nearest neighbour and 2ndnearest neighbour respectively. Given this Hamil-tonian, exact ground-state can be obtained usingLuttinger-Tisza method. For J2/J1 < 1

8, the ground

state is just Neel ordered state where each spin is anti-aligned with those of its nearest neighbours. But onceJ2/J1 >

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then, Neel ordered ground-state changes toextensively degenerate ground-states of coplanar spinspirals as shown in figure 8. Within these set of copla-nar, there is no energy cost to move between differentspin spiral configuration, therefore, it is expected thatthe low temperature spin fluctuation diverges and thesystem disorder at low temperature. But some ofthe spiral configuration has allowed extra Goldstonemodes (soft modes) which arise from the symmetry.Different Q vector has different density of modes andthere will be corresponding lift in the degeneracy ofthe system by the thermal fluctuation. Here are thecalculation done by the theoretical group where theycalculate free energy of the system in correspondingspiral orders of wave-vector q defined in reciprocallattice.

Figure 9: The ground-states of the system for differ-ent J2/J1 are shown below. They order co-planar spinspirals define by the vector in the reciprocal space.We see the coloured area which are degenerate how-ever, the free energy is different for different ground-states and the green point on the plot indicates theminimum in the free-energy.

They show that from the degenerate surface inthe reciprocal space, only some of the wave-vectorshas low free energy which corresponds to the orderstate due to order by disorder. Next, the theoreticalgroup explains the experimental result on MnSc2S4

done by Professor Loidl group in 2006 [1]. In this pa-per, they have done magnetic susceptibility measure-ment and neutron powder diffraction on MnSc2S4.The result is shown below.

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Figure 10: Magnetic susceptibility measurement ofMnSc2S4 powder. The system is highly frustrated asθCW = 23K while magnetic phase transition aroundTc = 2K

From magnetic susceptibility measurement,they have measured θCW = 23K while sudden changein magnetic susceptibility around Tc = 1.9K andTc = 2.3K indicating magnetic phase transition. Thefrustration parameter is of the order of 10 as men-tioned before. From the neutron powder diffractiondata after the refinement at T=1.6K, they have seenthat the system does have long-range order in a copla-nar spiral configuration of a vector q = ( 3

4, 34,0) as

shown in figure below.

Figure 11: Neutron powder diffraction of MnSc2S4.After refinement, they showed that the system ordersin a co-planar spiral configuration where the vectoris defined by q = 2π(3/4,3/4,0)

They estimate J1 = −10.5K and J2 = 8.75Kin order to explain θCW = −22.1K. When Balents’group has introduced very small but finite 3rd nearestneighbour interaction, they observed that the systemfavours (q,q,0) spiral direction. They were able tomatch the behaviour of the magnetic peaks in the

temperature region between Tc and θCW .

Figure 12: The left is the neutron diffraction whereone of the magnetic Bragg peak is selected and plot-ted intensity in different temperature. The right isthe numerically calculated structure factor.

They expect that the effect of the order bydisorder will be more clear on the single crystal.

4 Neutron Diffraction on theCrystal MnSc2S4

Later in 2016, The same group who has done powderdiffraction on MnSc2S4 did a single crystal neutrondiffraction on the same material [6]. The result isshown below.

Figure 13: The most left is the neutron diffractiondata at T=2.9K which is above the critical temper-ature. The middle graph is the Monte-Carlo simula-tion of this system where the J1 and J2 parameter isthe same as above paper. On the right, we see thatthe system indeed selects one of the configuration.

As we expected, there is a ring of diffuse scat-tering slight above the critical temperature; T=2.9K.This indicates that co-planar spiral ordering is evenlydistributed on the material for all the vector on the

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surface of the manifold. However, we see that (-1,1,0)direction spiral ordering has higher intensity indicat-ing there are more regions of this spiral ordering. Aswe lower the temperature below the critical temper-ature, we see that the system indeed select (0.75,-0.75,0) spiral ordering. Therefore, this result can beexplained by order by disorder.

5 Conclusion

Order by disorder is a phenomenon where the systemwith extensively degenerate ground-sates usually dueto the frustration is ordered due to thermal fluctua-tion. There were two examples given in this paper:the domino model by J.Villain at el. [4] and frus-trated diamond lattice given by Bergman at el. [5]where the system does have order by disorder. Neu-tron diffraction measurement as well as the magneticsusceptibility measure of this material, MnSc2S4,could be explained using the idea of order by disor-der. It seems that order by disorder is a very sensitivephenomenon. It requires a realization of the acciden-tal degenerate ground-states and then need to showthat they are also protected from all different kinds ofinteraction such as symmetry allowed interaction. Aswe know, slight perturbation can lift the degeneracywhich is explained in degenerate perturbation theory.But nevertheless, the phenomenon is very beautifulas it says that not only ground-states but also excitedstates are needed to be considered in order to under-stand the low temperature behaviour of the system.

6 Appendix A

Suppose A spins have given values. Then they pro-duce local field around the B atom occupied at j siteby 2JABεj where εj = −2,0, or 2. Suppose N’ bethe number of atoms in a single chain, then parti-tion function of a B chain is given by:

Z = ∑S1,S2...,SN′=±1

exp(βN ′

∑j=1

[−2JABSjεj−JBBSjSj+1])

(9)We see that this is just 1D Ising chain with anti-ferromagnetic interaction with magnetic field definedby ε (the configuration of the A spins). Given somespin distribution of a pair of A chain by the set εiWe can solve for the partition function transfer ma-

trix method:

Z(εi) =∞

∑i=1,Si=±1

< Si∣Ti∣Si+1 >< Si+1∣T ∣Si+2 > ...

(10)where the transfer matrix T is defined by:

< Si∣Ti∣Si+1 >= exp(−β×[JBBSiSi+1−JAB

2(εiSi+εi+1Si+1)]

(11)I am not going to consider arbitrary set of εi. If weare at very low temperature, then the we expect thespins along a A chain will be ordered. There are twodifferent configuration: a pair of A chains orderedferromagnetic in the same direction and a pair wherethey are ordered anti-ferromagnetic. If a pair of Achains are ordered in the same direction, then εj = 1for all j and we can think of the system now as 1DIsing model with uniform magnetic field of heff =

2JAB . We already solved this problem in the problemset 2 where we diagonalize the transfer matrix andsince all the transfer matrix are the same for differentsite i, same unitary matrix will also diagonalize. Weget partition function of the form:

ZF = Tr(TN′

) = Tr(DΛN′

D−1) = Tr(ΛN

′

) = λN′

++λN

′

−

(12)where λ is the eigenvalue of the transfer ma-trix which is given by: λ± = e−β∣JBB ∣cosh(βh) ±√e−2β∣JBB ∣sinh2(βh) + e2β∣JBB ∣. For anti-

ferromagnetic case, ZAF we will just replacefrom h=2βJBB to h=0. Since we are in the verylow temperature limit, (kBT << JAB which impliesβJAB >> 1) we get:

ZFZAF

≈ exp[N ′

2exp(−2β(∣JBB ∣−JAB)] = exp(−2βJ ′N ′

)

(13)

and we see that there is a effective couplingbetween different A chains by:

βJ ′ =1

4exp(−2β(∣JBB ∣ − JAB) (14)

The main point here is that There is an energy differ-ence between anti-ferromagnetic pair and ferromag-netic pair of length N’ as

∆E = 2βJ ′N ′ (15)

which diverges as we take thermodynamic limit (N’→∞). Therefore, the ground-state is preferred to beferro-magnetic between A-chains rather than anti-ferromagnetic.

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7 Reference

[1] Tristan, N. et al. Geometric frustration in thecubic spinels MAl2O4 (M=Co, Fe, and Mn). Phys.Rev. B 72, 174404 (2005)

[2] Suzuki, T., Nagai, H., Nohara, M. & Takagi, H.Melting of antiferromagnetic ordering in spinel oxideCoAl2O4 . J. Phys. Condens. Matter 19, 145265(2007).

[3] B. Martinez at el. Phys. Rev. B 46, 10786 (1992).

[4] Villain, J., Bidaux, R., Carton, J. P. & Conte,R. Order as an effect of disorder. J. Physique 41,12631272 (1980).

[5] D.Bergman at el. Order by disorder and spiralspin liquid in frustrated diamond lattice antiferro-magnets, Nature Phys. 3. 487 (2007)

[6] S. Gao at el. Spiral spin-liquid and the emergenceof a vortex-like state in MnSc2S4, Nature phys. 13.157-161(2017)

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