1307

Effective Hamiltonians and dilutioneffects in Kagome and relatedanti-ferromagnets

Christopher L. Henley

Abstract: What is the zero-temperature ordering pattern of a Heisenberg anti-ferromagnetwith large spin lengthS (and possibly small dilution), on the Kagome lattice, or othersbuilt from corner-sharing triangles and tetrahedra? First, I summarize the uses of effectiveHamiltonians to resolve the large ground-state degeneracy, leading to long-range order of theusual kind. Secondly, I discuss the effects of dilution, in particular that the classical groundstates becomenonfrustrated, in that every simplex of spins is optimally satisfied. Of threeexplanations for this, the most satisfactory is the Moessner–Chalker constraint enumeration.Quantum zero-point energy may compete with classical exchange energy in a diluted system,creating frustration and enabling a spin-glass state. I suggest that the regime of over 97%occupation is qualitatively different from the more strongly diluted regime.

PACS Nos.: 75.10N, 75.50Ee, 75.40, 75.25+z

Résumé: Quel est àT = 0 l’ordonnance d’un système antiferromagnétique de Heisenbergavec une grande longueur de spinS (et possiblement une faible dilution) construit sur unréseau de Kagomé ou sur d’autres réseaux conçus à partir de triangles ou de tétraèdres réunispar les sommets ? Je présente d’abors un résumé de l’utilisation de hamiltoniens efficacespour résoudre la dégénérescence importante du fondamental, menant à une ordonnance àlongue portée du type habituel. Je discute ensuite les effets de la dilution, en particulier queles états fondamentaux classiques deviennent non frustrés, en ce que chaque simplex de spinest satisfait de façon optimale. Des trois explications de ce phénomène, la plus satisfaisanteest l’enumeration des contraintes de Moessner–Chalker. L’énergie du point zéro quantiquepeut compétitionner avec l’énergie classique d’échange dans un système dilué, créant lafrustration et un état de verre de spin. Je pense que le régime au delà de 97% d’occupationest qualitativement différent du régime à plus forte dilution.

[Traduit par la Rédaction]

1. Introduction

This paper addresses the isotropic anti-ferromagnet with quantum Heisenberg spinsSi , on Kagomeand analogous lattices (“bisimplex” lattices, to be defined shortly). Everything is restricted to the large-S

and low-T limit, whereS is the spin quantum number andT is the temperature. Thus, to lowest orderwe may visualizeSi ≈ Ssi , wheresi is a classical vector of unit length.

The theme of Sect. 2 is the usefulness ofeffective Hamiltonians, in which some degrees of freedomare eliminated, in favor of new terms involving the remaining degrees of freedom. (Notice that I amassuming that the remaining degrees of freedom are a subset of the original ones, and therefore still liveon a microscopic lattice; coarse graining, which would replace the Hamiltonian by a field theory, is not

Received May 8, 2000. Accepted August 8, 2001. Published on the NRC Research Press Web site on Decem-ber 14, 2001.

C.L. Henley.Laboratory ofAtomic and Solid State Physics, Cornell University, Ithaca, NY 14853-2501, U.S.A.

Can. J. Phys.79: 1307–1321 (2001) DOI: 10.1139/cjp-79-11/12-1307 © 2001 NRC Canada

1308 Can. J. Phys. Vol. 79, 2001

Table 1. Bisimplex lattices. Number of sitesνsite and of simplicesν(q) are per Bravais unit cell.

Name Derived from Bravais lattice d (q, q ′) νsite ν(3) ν(4) pc

Kagome Honeycomb Triangular 2 3, 3 3 2 0 0.65Garnet “Gyroid” graph BCC 3 3, 3 6 4 0 >0.5 ?Crossed square Square Square 2 4, 4 2 0 1 0.50Pyrochlore Diamond FCC 3 4, 4 4 0 2 0.39Kag. sandwich — Triangular 2 3, 4 7 2 2 0.50(2)

considered here.) The reader is also reminded that the classical picture can be qualitatively wrong, atthe temperatures of interest, which are well below the quantum spin-wave energies. For example, thequantum pyrochlore lattice is seen to belessdegenerate than the Kagome case, in contrast to classicalresults; and the effective interactions favoring collinear and (or) coplanar and other forms of order aremuch more powerful in the quantum case.

The rest of the paper concerns the effects of dilution. Does it produce a spin glass, or generate aneffective Hamiltonian favoring long-range order, or preserve the exceptional degeneracy of the purelattice? Section 3 reviews the effect of dilution in ordinary frustrated systems, to contrast with its effectin bisimplex lattices (Sect. 4), in which the simplex units are all satisfied. Section 5, the heart of thispaper, presents three explanations of the simplex satisfaction. But when we admit the full zoo of realeffects — quantum fluctuations, dilution, unequal exchange constants, anisotropies, external field — itis likely that the simplices stop being satisfied (Sect. 5.3.2).

“Lattices analogous to the Kagome” meant, more precisely, thebisimplexlattices: those derivedfrom a bipartite network by placing a spin at each bond midpoint, so that each spin belongs to twosimplices. (My “simplex” is a synonym for “unit” as used by Moessner et al. [1, 2]). The coordinationnumber in the network becomes the number of cornersq of eachsimplex, which means a single bond,a triangle, or a tetrahedron forq = 2, 3, or 4, respectively. Table 1 lists bisimplex lattices mentioned inthe literature (there are more), tagged by the dimensionality and theq, q ′ values for the simplices oneither side of a site.

The Kagome and pyrochlore lattices are familiar; the connected magnetic sites in SrCr9pGa12−9pO19(SCGO) form thed = 2 bisimplex lattice that I will call a “Kagome sandwich” (also known as a“pyrochlore slab”). It consists of two Kagome layers connected by a a triangular-lattice linking layer [3].The three-dimensional magnetic lattice of, e.g., gallium gadolinium garnet [4] is appropriately dubbed“hyperKagome” [5], since it too consists of corner-sharing triangles. The crossed-square lattice is apyrochlore slab normal to{100}, with its top and bottom surfaces identified, and serves as a two-dimensional toy model for the pyrochlore. (See Fig. 2 of ref. 1b.)

The site-percolation thresholdpc of the bisimplex lattice is obviously the bond-percolationpc ofits parent bipartite network. It is listed in Table 1 because one expects qualitative changes of behavioratpc, if the spin system orders in any fashion [6]. The sandwich latticepc is published here for the firsttime, to my knowledge. (See Appendix A.)

The Hamiltonian couples nearest neighbors

H = j∑〈ij〉

Si · Sj (1.1)

or classically, with a magnetic fieldB included

H = J∑〈ij〉

si · sj − B ·∑

i

si =∑α

J

2|Lα − λ

JB|2 + E0 (1.2)

©2001 NRC Canada

Henley 1309

where the exchange constant isJ ≡ jS2 > 0, λ = 1/2, and the total spin of simplexα is

Lα ≡∑i∈α

si (1.3)

In the Kagome-sandwich case, the interlayer (Kagome to linking layer) coupling will be calledJ ′, butunless explicitly noted, I assumeJ ′ = J , the Kagome-layer coupling.

Frustration means generally that, having broken up the Hamiltonian into local terms, we cannotsimultaneously satisfy all of them. In the present case, each term in (1.2) is satisfied (in zero field)wheneverLα = 0. Then each undiluted bisimplex lattice is completelyunfrustratedfrom the simplexviewpoint, since it can be shown (by example) that every simplexcanattain this minimum at the sametime. LetX denote the classical ground-state manifold (forB = 0); it is massively degenerate on everybisimplex lattice.

2. Effective Hamiltonians and order-by-disorder

Our goal is to discover how the system breaks symmetry and orders (if it does). We assume pro-visionally that, for sufficiently largeS, the quantum ground state is one of the classical ground states,dressed by some quantum zero-point spin fluctuations. (This clearly fails if the fluctuations about aclassical ground state are as big as the distance to the next one, or if the system tunnels so freely be-tween dressed classical states that the correct wave function is a superposition.) This assumption givesus directly the ordering pattern in a simple (e.g., triangular lattice) Heisenberg anti-ferromagnet, wherethe classical ground state is unique (modulo symmetries).

But commonly, the classical ground states have nontrivial degeneracies, so that every ground state hasa different manifold of possible small spin deviations and consequently a different zero-point energy.Presumably, the true quantum ground state should be constructed around the particular (ordered!)classical ground state that has the lowest zero-point energy. When the thermodynamic state of thesystem has true long-range order in a particular pattern, due to this fluctuational energy, we could callit “order-by-disorder” [7, 8]. (This usage of the term is broader in some ways, and narrower in others,than the “local” definition given in ref. 1.)

To model the degeneracy breaking transparently, one may construct effective Hamiltonians in closedform (via intelligently applied perturbation theory). They are intended, not to provide the accurate energyof a special state or two, but as inputs to further modeling, e.g., simulations atT > 0, or tunnelingcalculations [9].

2.1. Harmonic and higher order effective Hamiltonian(s)

The spin-wave expansion naturally organizes the zero-point energy as an expansion in powers ofthe small parameter 1/S, of which the zero term is theO(jS2) classical energyE0 and the first termis theO(jS) harmoniczero-point energyF(X) = 1

2

∑k ~ωk, including all spin-wave frequenciesωk

(which depend implicitly onX.) ThisF(X) is the effective Hamiltonian, defined only on states in theground-state manifold.1 Call thelocal minima ofF() the “favored states”{Y }; these are a discrete set,(in all cases I know) modulo global rotation symmetry.

Let “ordinary degenerate anti-ferromagnet” mean one for which{X} is a finite-dimensional manifold— clearlyeveryground state is periodic. Examples are the fcc anti-ferromagnets [11, 12] or the two-sublattice systems with second-neighbor interactions [8, 13, 14]. (A “highly frustrated” system mightbe defined as one in which the dimensionality ofX is extensive, and two ground states may differ only

1 Analogously, in the classical model withT/J � 1 one defines a harmonic free energyF ∼ T � E0 ∼ J , by integrating outsome degrees of freedom in the partition function. See ref. 10.

©2001 NRC Canada

1310 Can. J. Phys. Vol. 79, 2001

locally.) For an ordinary degenerate system, with isotropic exchange interactions, a crude approximationfor F(X) is thebiquadratic effective Hamiltonian

Hbiq ≡ −∑ij

Kij (si · sj )2 (2.1)

This was independently posited phenomenologically [15]; it can be obtained analytically in a couple ofways using a perturbation expansion [12] around mean-field theory.2 One obtainsKij = S2j2

ij /8hloc,where the local fieldhloc is 2jS in a bisimplex lattice. Equation (2.1) correctly tells us that the favoredstates are thecollinear ones. In the Kagome case (2.1) should be replaced by a different functionalform to representF(X) approximately, because (i) the criterion for favored states is that the spins arecoplanar; (ii ) the true functionF(Y + δX) − F(Y ) ∼ |X − Y | is linear [9,16] rather than quadratic inthe deviationsδX.

More fundamentally, the harmonic approach fails in the Kagome case because it does not fullybreak the degeneracy. Every coplanar state has exactly the same harmonic-order Hamiltonian, if writtenusing as coordinates at each site (i) they (out-of-plane) spin deviation and (ii ) the rotation angle aroundthe y axis. The source of this, mathematically, is thatsi · sj takes the same value (here−1/2) forevery nearest-neighbor pair, in every favored ground state. Consquently, on a bisimplex lattice withonly triangles, i.e., the Kagome or hyperKagome lattices,F(Y ) have exactly the same value foreverycoplanar stateY . The number of such states isO(exp N), whereN is the number of unit cells.

To resolve the surviving degeneracy among favored ground states, we need a second effectiveHamiltonianG(Y ), obtained from some sort of self-consistent theory that takes account of anharmonicspin-wave interactions [17–19]. Since{Y } is discrete,G(Y ) is parametrized by discrete-spin variables:in the Kagome case, the “chiralities”τα = ±1 defined from the spins on each plaquette [16]. Our ownapproximation [18,19] gave

G({τα}) = −∑αβ

J (Rαβ)τατβ (2.2)

which has the form of an anti-ferromagnetic Ising Hamiltonian on the honeycomb lattice, and is definedfor everyfavored ground stateY . The energy scale [17] is certainlyJ = O(jS2/3), i.e., down by onlya factorS−1/3 from the scale of the harmonic termF(X).

In the pyrochlore case (2.1) is also too crude: here the favored ground statesY are the collinearones, but their harmonic energiesF(Y ) are nondegenerate since different collinear states have differentpatterns ofsi · sj = ±1. It turns out that the special states that minimizeF() are still infinite in number,but only as exp(constL), whereL is the system’s diameter:3 the Kagome-sandwich lattice seems tobehave similarly. Thus, in the quantum case, the pyrochlore and sandwich lattices showmoreorderingtendency than the Kagome lattice (at harmonic order), whereas in the classical case it is the other wayaround [1].

2.2. Pitfalls of classical modeling

Although largeS justifies visualizing each spin as a fixed-length vector, it doesnot justify a purelyclassical simulation of the system. The reason is that the interesting phenomena occur whenT � jS,

2 The most useful derivation is contained in an unpublished preprint of Larson and Henley, first circulated in 1990.3 C.L. Henley. Unpublished work. The method of calculation is to use (6.2) of ref. 1b, in which the dynamics of spin waves

(with nonzero frequency) is written in terms of equations involving only simplex spinsLα . It follows that the frequencies areeigenvalues of matrices whose entries are±1, corresponding to the spin configuration of a collinear ground state. When twoground states are “gauge-equivalent”, the corresponding matrices are related by a similarity transformation (and thus yieldexactly the same eigenvalues and zero-point energy.

©2001 NRC Canada

Henley 1311

Fig. 1. Diluting ordinary anti-ferromagnets; diluted sites and their bonds are always shown with a broken line.(a) Unfrustrated anti-ferromagnet: the total moment here is one down spin, the excess of odd sites over even sites.(b) Removal of one spin in the triangular Heisenberg anti-ferromagnet, causing neighboring spins to deviate in thedirection of the missing spin. Spins on the second ring outwards are given the directions they would have in thepure lattice, though in reality a small distortion is found at any radius. (c) A loop of sites with an odd number ofsteps introduces random frustration and spin relationships absent in the pure lattice. The upper (lower) path favorsthe endpoint spins to be parallel (antiparallel).

the scale ofF(X). Thus thermal effects are only a small correction to the quantum effects. I believe thereis an easy fix: a classical Monte Carlo simulation using a Hamiltonian that includes (an approximationof) F(X) andG(Y ) ought to give valid physical results.

For a specific example, consider SCGO withS = 3/2 and a Curie–Weiss constant [20] of 515 K,hencej ≈ 80 K. Then, in a Kagome lattice, the effective energy of coplanarity would be 0.14jS perspin along a straight “spin fold” [16]; if we presume this also applies to the shortest rotatable loop of sixsites, we get a barrier of 100 K (alternatively5

2jS ≈ 300 K from Appendix B of ref. 9). That is vastlylarger than the analogous free energy barrier in a classical system, at the SCGO freezing temperature∼ 3.5 K. Again, in a pyrochlore lattice with the same couplingj , the coefficient in (2.1) comes out toKij = 1

16jS ≈ 7.5 K per nearest-neighbor bond. That ought to induce a transition to long-range-orderedcollinearity at a temperature of order 10 K, which would not happen at any temperature in a classicalsystem [1].

3. Dilution in ordinary cases

I now turn to the longer part of this paper: what effect(s) does dilution have on a highly frustratedsystem, specifically on a bisimplex-lattice anti-ferromagnet? Can it be represented by an effectiveHamiltonian? Or can it turn an ordered system into a spin glass? The answers are “yes” for the “ordinary”anti-ferromagnets, which are reviewed in this section (in three regimes). The answers are different forhighly frustrated magnets (later sections).

Starting with this section, I am consideringclassicalground states atT = 0 (unless explicitlynoted).

3.1. Unfrustrated caseIn an unfrustratedtwo-sublattice (Néel) anti-ferromagnet (Fig. 1a), the ground state at occupied

fractionp (on the unique extended cluster) has exactly the same spin directions as atp = 1, on all themagnetic sites. The total magnetization does not cancel exactly, since the moment on the even or oddsublattice hasO(

√N) statistical fluctuations. An observable corollary of this observation is that the

structure factor at wave vectorq

S(q) ≡ 1

N〈|

∑i

eiq·ri si |2〉 (3.1)

©2001 NRC Canada

1312 Can. J. Phys. Vol. 79, 2001

has the limiting behavior

limq→0

S(q) = p(1 − p) > 0 (3.2)

3.2. Frustrated case: weak dilution

Consider next a simple frustrated case, e.g., a Heisenberg triangular anti-ferromagnet. The puresystem has a periodic, non-collinear ground state in which the spins differ by 120◦ angles, which is thebest compromise within each triangle of spins. The spins in the diluted system no longer take the samedirections they would in the pure system.

One regime is weak dilution (p close to 1). Consider just one nonmagnetic site (Fig. 1b). Far awayfrom this defect, the configuration essentially agrees with a pure ground state. But the neighbors ofthe removed spin deviate towards the direction it would have had, and their neighbors deviate in turn.Thus, the defect creates a slowly-varying spin twist with a pseudo-dipolar spatial dependence. (It hasthe angular dependence of a dipole field and decays with distance as 1/|r|d , whered is the spatialdimension.)

In the ordinary degenerate anti-ferromagnets, the energy reduction due to spin deviations dependson which ground state one deviates from. The average of this energy, over all the ways to place a lowdensityxdef of defects, defines aneffective Hamiltonianfunction Hdil(X), proportional toxdef andpossessing the full symmetry of the ground statesX of thepuresystem. In exchange-coupled systems,Hdil(X) favors the least collinear ground states, and thus has an interesting competition withHquant[8].

Often, depending on how a defect’s local symmetry relates to that of the ground-state manifoldX,each defect may prefer a particular ground state, out of the subspace{Y } favored byHdil . In that case,dilution creates an “effective random field” [8] varying in space and coupling to the discrete degrees offreedomY .4 As in other random-field models, disorder wins if it is sufficiently strong or the dimensionis low enough. The resulting state is a spin glass, in the sense that it lacks long-range correlations andhas barriers, but has not been proven to be the same phase as a±J spin glass [21].

3.3. Frustrated case: strong dilution

Now we return to the same triangular lattice, but with strong dilution so thatp is just abovepc. It iswell known that, near percolation, the typical connection between two sites (if any) is tenuous, and orderis propagated over one-dimensional chains of sites, which are multiply-connected at occasional places.At T = 0, the spin directions alternate along such a chain, so it constrains the relative orientation of theendpoint spins to be parallel or antiparallel, depending on whether the number of bonds connecting themis even or odd. Order is propagated (atT = 0) as if there were a direct bond between the endpoints. Butif there aretwopaths in parallel, they may disagree on the relation between endpoint spins; the smallestexample is shown in Fig. 1c. The ground state is a twisted spin configuration that is found neither in thepure lattice nor in a singly-connected chain. In Fig. 1c, the effect is to force an 80◦ angle between theendpoint spin directions.

The extended connected cluster is an irregular network containing loops within loops of this type. Itis plausible that its global “energy landscape” is like that of a spin glass, possessing numerous low-lying,nearly degenerate energy minima, separated by energy barriers.

4 The “effective random field”, as well as the competion between the effective Hamiltonians for quantum and dilution selection,are discussed in detail for the Type III fcc anti-ferromagnet in unpublished work by B.E. Larson and the author first circulatedin 1990.

©2001 NRC Canada

Henley 1313

Fig. 2. One removed spin in the Kagome anti-ferromagnet, and the 10-site defect loop induced around it. Circleswith dots or crosses are spins pointing directly out of or into the paper; “+” and “−” signs indicate spins with anoutward or inward component. The surroundings are part of a coplanar

√3 × √

3 ground state.

4. Dilution in a bisimplex lattice

Dilution affects a bisimplex lattice quite differently than an ordinary frustrated lattice: simulationsfind that essentiallyevery simplex remains satisfied, i.e.,Lα = 0, even for strong dilution. (This includesthe simplices with nonmagnetic sites: they are still simplices, but the number of cornersq is reduced.)In this section, I will discuss the evidence for, and some corollaries of, that fact.

4.1. Simulations of a two-layer Kagome lattice

The original evidence was that (in the Kagome lattice) the local field is exactly 2 on most sites[22,23]. I carried out more extensive simulations like those in ref. 22, for the diluted Kagome-sandwichlattice — relevant to experiments on SCGO. (High-quality crystals of that material can be grown onlywith p < 1.) Over 500 independent realizations of the disorder were constructed, for a 10× 10lattice withp = 0.55 (i.e., 385 spins); each realization was relaxed from three different random initialconfigurations to a ground state by 250 sweeps, in which each site in turn was set to its local-fielddirection. The program flags all configurations in which|Lα| > 0.1 on any simplex withq > 1. Thishappened only on “one-eared” loops like Fig. 3a.5 Now, the one-eared loop connects to the rest of theworld in (at most) one point. Hence, it cannot induce twisted, frustrated relationships among distantspins (like Fig. 1c), and cannot be responsible for spin-glass behavior. Similar results were found evenwhenJ ′ 6= J (unequal interlayer and Kagome layer exchange), as well as in a plain Kagome lattice.

4.2. Half-orphan spins

Simplex satisfaction has observable consequencies. Every simplex hasLα = 0 — except, of course,that aq = 1 (one-spin) simplex has|Lα| = 1. LetO be the set of spins that haveq = 1 on one side,with frequencyxdef ∼ (1−p)2 per unit cell. Also letO′ be the spins that are completely isolated (q = 1on both sides), with frequencyx′

def ∼ (1−p)5 in the sandwich lattice — i.e., rare forp > pc. The totalmagnetization (in units ofµ ≡ (2µB)S) is

Mtot = 1

2

∑α

Lα = 1

2

∑i∈O

si +∑i∈O′

si (4.1)

The prefactors of 1/2 in (4.1) appear because each spin’s moment is divided between two simplices.Schiffer and Daruka [24] observed a Curie-law contribution to the susceptibility, ascribed to “orphanspins” by them (which suggests the isolated spins), but more plausibly to the spins inO [2,25], whichmight better be called “half-orphan”, since they belong to a simplex on one side but are isolated on the

5 Three exceptions are mentioned at the end of Sect. 5.2.

©2001 NRC Canada

1314 Can. J. Phys. Vol. 79, 2001

other side. If these moments had independent directions (but see Sect. 4.4), they would produce theCurie susceptibility per cell

χ(T ) = µ2eff

3T

(1

4xdef + x′

def

)(4.2)

Insofar as the system is built from satisfied simplices, its total (classical) magnetization is zero. So,in place of (3.2), the structure factor (3.2) scales asxdef � (1−p). Thus, even with significant dilution,one expectsS(q → 0) ≈ 0, as seen in SCGO (according to ref. 3).

The half-orphan site has a free spin ofS/2 in (4.1) and (4.2). This is a perfect classical analog ofthe spin-1/2 moment that appears in dilutedS = 1 quantum spin chains with a valence-bond state [26].(This connection was suggested by Collin Broholm (private communication)). I speculate that quantumS = 1 bisimplex lattices indeed have a valence-bond state, andS = 1/2 defect moments.

The simplex-satisfaction concept also encompasses bond-randomness effects (see the beginning ofSect. 5.3.2). The ground state of a tetrahedron with one ferromagnetic bond still hasLα = 0 so it doesnot affect the near cancellation ofMtot or the Gauss law (Sect. 4.4). On the other hand, a triangle withone ferromagnetic bond is unfrustrated, with a configuration like↑↑↓. Then|Lα| = 1; a ferromagneticbond produces exactly the same sort of paramagnetic defect as a half-orphan spin.

4.3. Satisfaction with unequal couplings

What happens to the picture of (1.2) whenJ ′ 6= J (coupling to linking layer and (or) fieldB isnonzero? It turns out (1.2) still works, provided we now take [2]

Lα ≡∑i∈α

wαisi (4.3)

wherewαi = J ′/J whenα is (before dilution) a tetrahedron andi is the linking-layer spin that caps it,andwαi = 1 otherwise. Also, we must replaceλ in (1.2) byλ(α) = J/2J ′ whenα is (before dilution)a triangle, and 1− J/2J ′ otherwise (whenα is, before dilution, a tetrahedron). To satisfy each term,Lα = λ(α)B, which immediately implies (in the pure system)

〈si〉 ={

J6J ′ B, i in Kagome layer(1 − J

J ′)B, i in linking layer

(4.4)

Notice that the sum of all linking-layer spins must be exactly zero if eitherB = 0 or if J ′ = J (evenin a nonzero field). Also, the total magnetization of satisfied simplices is (4.1), except the coefficients1/2 must be replaced byλ(α). Thus, withJ ′ 6= J the net magnetization of the satisfied simplicesstillmust be zero.6

There is a problem wheneveri is a linking-layer spin andα (after dilution) is aq = 2 simplex:one must setwαi = 1 again to correctly describe the satisfied bond. But then, in a magnetic field, thedecomposition into terms (1.2) breaks down since there is no consistent value forλ(α). Even forB = 0,if wβi = J ′/J still for the other simplex that includes sitei, the result (4.6) of the next section breaksdown; in effect, this site enters (4.6) as another sort of “point charge”.

6 This is contrary to the conclusion of ref. 2. Their calculation treated the simplices as independent. That approximation canviolate important sum rules, e.g., it finds a mean spin on the tetrahedron base that is different from that on a triangle, eventhough these are in fact the same spins.

©2001 NRC Canada

Henley 1315

4.4. Divergence theorem

I now introduce a sort of “Gauss law”, which is handy for revealing the nonlocal effects of defects.Recall that in a “bisimplex lattice”, every spin belongs to one even and one odd simplex; let(−1)β ≡ +1or −1 whenβ labels an even or an odd simplex, respectively. Mark out a domainD containing a subsetof the simplices. Define a kind of “charge”,

Q(D) ≡∑β∈D

(−1)βLβ (4.5)

The theorem states that, assuming simplex satisfaction,

∑i∈O

(−1)β(i)si = Q(D) =∑i∈∂D

(−1)δ(i)si (4.6)

where∂D is the set of sites the domain boundary cuts through. Also,β(i) tells which simplex hasq = 1 (of the two containing the half-orphan sitei), andδ(i) tells which simplex is in the interior (ofthe two containing boundary sitei). The left-hand side of (4.6) follows since only theq = 1 simplicescontribute nonzero terms in (4.5); the right-hand side follows because every spin in theinterior of Dappears in two terms of (4.5) with canceling coefficients. (Thus, half-orphan spins are the point chargesin our Gauss law, while(−1)δ(i)si plays the role of the normal component of the electric field at thesurface.) This is a generalization of the sum rules of ref. 1b, Sect. 3.2.7 By drawing a succession ofnested boundaries∂D around a single half-orphan spinsI , one shows that neighboring spinssj havecorrelations withsI that alternate in sign (as speculated by Mendels [25]) and decay with distance as1/|rj − rI |d−1.

If we let D include the entire system, there is no boundary term and the total “charge” (left-handside of (4.6)) must be zero. This implies that, if there is just one half-orphan spin, it is impossible toexactly satisfy every simplex; if there are just two half-orphan spins, they must be exactly parallel orantiparallel (depending on the relative parity of their respectiveq = 1 simplices). But this law has a verylarge loophole; the violation ofLα = 0 may be spread out uniformly over the simplices, such that thetotal energy cost isO(1/N), which is negligible in a large system. There is a more physical argumentwhy nearby “charges” ought, nevertheless, to cancel, as the electrostatic analogy would suggest. If theydo not cancel, spins in shells surrounding these charges are constrained by (4.6) to have a nonzero meancharge, which would reduce the number of possible states, thus reduce theentropy, thus increase thefree energyat T > 0. I conjecture that, in the dilutedquantumsystem, there is an analogous effectiveinteraction between nearby half-orphan spins, mediated by the harmonic zero-point energy.

4.5. NMR experiments

An NMR experiment measures the distribution of the local fieldsh felt by each NMR nucleus (Ga, inSCGO). This is an average of the magnetizations of its neighbor (Cr) spins,mi = 〈si〉, where the averageis taken over all ground states (for a fixed realization). The local Curie susceptibility is dominated byhalf-orphan moments. In the Gaussian line-shape regime [27], the variance ofh would scale asxdef, sothe NMR linewidth should scale as

√xdef ∼ 1 − p, as is seen experimentally [25].

The experiments can separate the NMR signal from the Ga(4f) site, which sees 12 Cr neighbors,including three from the linking layer [25]. Then in the pure system, (4.4) implies the mean susceptibilityis exactlyµ2/7J per spin, but the mean susceptibilty seen by Ga(4f) isµ2(1/4J − 1/8J ′).

7 Note also that, in the pure system (no “charges”), a vector potential for the “electric field” can be constructed, which is uniquelyvalued ifd = 2; in the Kagome case, this is just the “spin origami” embedding of ref. 23.

©2001 NRC Canada

1316 Can. J. Phys. Vol. 79, 2001

Fig. 3. (a) One-ear, (b) two-ear, and (c) three-ear loops with their spin configurations (unique, modulo rotationsin spin space). A, B, and C mark three spin directions which differ by 120◦.

5. Why are the simplices satisfied?

I now present three different — not exclusive — viewpoints for understanding simplex satisfaction.

5.1. Single-impurity explanationThe original explanation [23] just considered a single nonmagnetic impurity in a pure background,

as appropriate to the weak dilution regime. Reference 23 exhibited a rearrangement of a few spins (asfew as 10) around this defect, as in Fig. 3, which completely satisfies every simplex. Farther away, thespin deviations are strictly zero, in contrast to an ordinary frustrated magnet (Sect. 3.2). The generalmethod — first described for the pyrochlore case [28] — usesq − 1 rotatable loops, each connecting aspin on one of the simplices containing the impurity, to a spin on the other affected simplex. (“Rotatable”means that, in the pure lattice, the spins on the loop can be rigidly rotated together to produce otherground states.) The deviations around the impurity remind me of the screening around a test charge ina metal, by the high density of excitations at zero energy — in the present problem, those excitationsare the rotatable loops.

Clearly this picture works for well-separated impurities — but that requires 1−p to be quite small.In Fig. 3, all 19 spins must point in the pattern shown, modulo rotations; this conflicts with the patternforced by a second impurity that sits anywhere on the two hexagons in Fig. 3 or the eight surroundinghexagons: a single impurity excludes other impurities on 40 other sites. (The above counts and Fig. 3assume a background consisting of the

√3 × √

3 state; in other coplanar backgrounds, more spins areaffected.)

The single-impurity picture, then, breaks down atp ≈ 0.97, where the defect configurations startto overlap. To explain the simulations from this viewpoint, one is forced to postulate that the defectspin deviations obey a nonlinear superposition principle, as magical as that of solitons in certain one-dimensional systems.

5.2. Constraint propagation explanationThis picture is most appropriate to the strong-dilution limit nearpc, where the connected cluster

is tenuous. Along a simple chain of sites, spins alternate propagating order as if there were a directbond between the endpoints. Now allow a few spins neighboring this path; these decorate the path withtriangles that I will call “ears” (see Fig. 2). Each ear removes constraints on the spins, since the twospins on the path are now constrained merely to differ by a 120◦ angle. When a path includes two ormore ears, there is no constraint at all between the endpoint spins.

As in Sect. 3.3, the key question is whether two paths that rejoin (forming a loop) might propagatemutually exclusive constraints. In contrast to the triangular case of Fig. 1c, every simple loop has evenlength. (Derived for the Kagome case in ref. 9, Appendix; for a general bisimplex lattice, it follows fromthe bipartiteness of the underlying network.)

©2001 NRC Canada

Henley 1317

Table 2. Constraint countKsim(q) in q-corner simplices.

Case q = 2 q = 3 q = 4

Isotropic Heisenberg spinsa 2 3 3The same, with spin waves 2 3b 5One ferromagnetic bond 2 4 6Easy-axis exchange anisotropy 4 4 5Magnetic fieldc 3d 3 3aAlso the caseJ ′ 6= J .bFor the coplanarity constraint, add+1 for every site withq = 3 onboth sides.cAlso Ksim(1) = 2, sincesi ‖ B in that case.dBut subtract 1 for every site withq = 2 on both sides of it.

All loops with the same number of ears are equivalent, since — by the evenness lemma just stated— they have the same number (modulo 2!) of earless links, at which the spins simply invert. Therefore,it suffices to study loops of length 6 as in Fig. 2. The one-eared loop shownis frustrated: the two spinsin the ear should differ by 120◦ due to the triangle, but by 180◦ due to the five simple links connectingthem. On the other hand, the two-eared loop forces the same 180◦ angle as a simple chain (betweenthe endpoint spins where it connects to the rest of the world), and the three-eared loop forces the same120◦ angles as a giant triangle would.

The simulations described in Sect. 4.1 found 38 one-ear loops in 500 realizations of 100 cells each atp = 0.55, in full agreement with the predicted frequencyp7[18(1− p)8 + 30(1− p)9], per cell. Threeother frustrated clusters were found, once each: a one-eared 8-ring, and (in two variations) a hexagonpair sharing two ears, with one added ear. Each of these objects has (at most) one connection to theoutside world forming the ear, and thus, doesnotpropagate frustration on larger scales.

5.3. Constraint counting

The preceding discussions detected no frustration in either the strong- or weak-dilution regime,but are insufficiently general. The constraint-counting (“Maxwellian”) approach of Moessner andChalker [1] is, I believe, the convincing explanation of simplex satisfaction.8 The basic aim is tocompute the dimensionalityD of the manifold of states in which all simplices are satisfied; as long asthis set is non-null (i.e.,D ≥ 0), it is obviously the ground-state manifoldX.

Now let X̃g be the manifold of “generic” simplex-satisfied states (having no linear relationshipsamong the spins, apart from theK constraints required to satisfy all the simplices). The dimensionalityof X̃g is

Dg = F − K (5.1)

HereF is the number of degrees of freedom, two per spin, soF/N = 2pν per unit cell, whereνsiteis the number of sites per unit cell, as in Table 1. Table 2 gives the number of constraints per simplex,Ksim(q ′), whereq ′ is the number of magnetic sites remaining after dilution. Naively,Ksim(q ′) = 3 forthe three components ofLα = 0 (or its generalization to nonzero magnetic field). However, in zerofield when the simplex is a single bond (q ′ = 2), the constraint is simplys2 = −s1; attachings2 doesnot changeD, so the added constraints must be two to cancel the added degrees of freedom. Also,

8 A caution, however, is that Maxwellian counting is a mean-field theory. A threshold at whichDg(p) vanishes is not generallythe exact threshold, since portions of the structure may be underconstrained while others are overconstrained. For relatedissues in elastic percolation, see ref. 29.

©2001 NRC Canada

1318 Can. J. Phys. Vol. 79, 2001

Ksim(0) = Ksim(1) = 0. The total number of constraints is

K

N=

∑q

ν(q)Cq

q ′pq ′(1 − p)q−q ′

Ksim(q ′) (5.2)

per unit cell, whereCq

q ′ is the combinations ofq things takenq ′ at a time, andν(q) is the number ofq-corner simplices per cell in the undiluted structure.

Most relevant to propagating order (or frustration) is the generic dimensionalityDgc of the extendedconnected cluster; I approximate this by subtracting fromDg the contribution toF by isolated sites.The results are

Dgckag

N= 6p[1 − (1 − p)4] − 6p2(2 − p)

Dgcsand

N= p[14− 12(1 − p)5 − 2(1 − p)6] − 12p2(p3 − 3p + 3)

Dgcpyr

N= 8p[1 − (1 − p)6] − 12p2(1 − p)(2 − p) (5.3)

for the three lattices. Counterintuitively, removing a single site from the Kagome lattice leavesDg

unchanged, but removing an adjacent pair of sites increasesDg by 1. In a general bisimplex lattice,after dilution to the pointptri at which the average simplex is a triangle, further dilution ought to(slightly) increaseDgc. (Note ptri ≈ 1, 0.9, and 0.75 for the Kagome, sandwich, and pyrochlorelattices, respectively.) Indeed,Dgc(p) at first decreases rapidly with dilution and tends to level offbelowptri , at roughlyDgc(p) ∼ 0.2 on the Kagome,∼1.6 on the sandwich (SCGO) lattice, and∼1 onthe pyrochlore, per unit cell. SinceDgc(p) remains positive, we expect simplex-satisfied ground statesat anyp > pc.

5.3.1. Generic and (or) nongeneric state cross over?

Do not be misled by the term “generic”! The typical (physical) ground state is nongeneric if suchstates have a higher dimensionality than the generic ones. In particular, in thepure Kagome lattice,starting with the

√3× √

3 coplanar state, one can rotate one of every three six-loops (“weathervanes”)by an independent angle. This ground-state manifold has dimensionalityDn = 1/3 per cell, whichdominatesDg = 0. Such states are nongeneric because some spins (e.g,. every second one in a rotatableloop) haveexactlythe same directions. (Indeed, in ground states relaxed from a random spin configura-tion, second-neighbor spins often point in nearly the same direction.) For small dilution, the nongenericstates still dominate; these are precisely the coplanar states with rare impurities discussed in Sect. 5.1.But the spin rearrangement at each impurity (Fig. 3) touches, and thus immobilizes, 10 six-loops, andthe frequency of impurities is 3(1 − p) per cell. So, I estimate

Dn(p)

N≈ 1

3(1 − 30[1 − p]) (5.4)

as the dimension per unit cell of the nongeneric manifold. The nongeneric manifold loses out to thegeneric one atp ≈ 0.97, in the Kagome lattice. By contrast, the pure pyrochlore lattice hasDg(1) = 2,while it appearsDn(1) = 1 (and presumablyDn(p) also plummets upon dilution). Thus, only thegeneric manifold is relevant in the pyrochlore. (Essentially this paraphrases the absence of (local)“order-by-disorder” [1].)

5.3.2. Back to frustration

Up to now, I considered the isotropic classical Heisenberg anti-ferromagnet, with possible dilution.The degeneracies retained by the classical ground-state manifold, even under dilution, may be broken

©2001 NRC Canada

Henley 1319

by various realistic perturbations. These can be handled by the constraint-counting framework, as inTable 2 (with extensions to accommodate constraints that involve two simplices).

Bond disorder has been identified in pyrochlore anti-ferromagnets [30, 31] and was modeled the-oretically [32]. It can produce “canted local states” [28] of deviated spins in ordinary Heisenberganti-ferromagnets (interacting as a spin glass [28]). In a bisimplex lattice, it has a different effect; theconstraintsKsim(q) are greatly increased — assuming a fractionxF of ferromagnetic bonds and insertingKsim(q) from Table 2 into (5.1) gives

Dpyrg = 2(1 − 18xF), Dsand

g = 2(1 − 21xF) (5.5)

in place of (5.3). Equation (5.5) predicts thatxF ≈ 0.05 is a threshold, beyond which the system becomesoverconstrained and (presumably) spin-glassy.

Quantum fluctuations also impose collinearity in a tetrahedron. For easy-axisexchange anisotropy,rather surprisingly theq = 3 ground state manifold’s dimensionality is unchanged [33]. (The constraintcount for easy-planeexchange anisotropy is the same as forXY spins, see ref. 1). The magnetic fieldis a special case (already covered in Sect. 4.2). The spins along a chain whenB 6= 0 alternate betweentwo possible directions, indicating that each extra link contributes only two constraints, as accountedfor in footnotec of Table 2. Substitution into (5.2) and (5.1) shows that most of these perturbationsmake the generic states overconstrained, and presumably glassy, though (in the undiluted cases) oneneeds to rule out possible nongeneric ground states.

6. Conclusions

Large-S, nonrandom anti-ferromagnets should have periodic long-range order atT = 0, even on ahighly frustrated lattice. A purely classical picture is invalid, even forS � 1, when the temperature isfar below the spin-wave energyjS, but this may be fixed up by using the effective Hamiltonians in thesimulation (Sect. 2.2).

Dilution doesnot engender a spin glass in classical ground states, as it does in ordinary frustratedmagnets, since simplices remain satisfied, as confirmed by simulations (Sect. 4.1). This was understoodmost generally from constraint-counting arguments (Sect. 5.3.) The observed spin-glass state in SCGOmightbe attributed to a competition of dilution with the coplanarity tendency due to quantum fluctu-ations. Effective Hamiltonians (Sect. 2), serve as a first (perhaps only) way to model the zero-pointenergy contribution in a disordered lattice — even if the functional form is not quite right — since amore exact treatment would be intractable, in the presence of the multiple perturbations of Sect. 5.3.2.

Some physical corollaries are deduced about the structure factor, paramagnetic susceptibility, andNMR response, including a Gauss-law rule (Sect. 4.4), which manifests the long-range effects of theparamagnetic (half-orphan) spins. Experiments sensitive to percolation on SCGO ought to be performedwith occupations below the thresholdpc ≈ 0.5, which is estimated here for the first time (Appendix A).

I speculated that — in the plain Kagome lattice, at least — there is a thresholdp∗ ≈ 0.98 separatingtwo regimes of dilution: nongeneric coplanar states, peppered with defects, atp > p∗, but atp < p∗generic states, in which every trace is lost of the coplanar background (Sects. 5.1 and 5.3). This wouldimply, of course, that it is invalid to extrapolate experimental measurements forp ∈ (0.9, 0.95) up top = 1.9

Acknowledgements

I thank E.F. Shender, V.B. Cherepanov, C. Broholm, P. Mendels, and I. Mirebeau for stimulatingdiscussions, and I thank Johns Hopkins University for hospitality (in 1994) when parts of this work

9 There is less likely to be ap∗ in the sandwich (SCGO) lattice case, but I do not understand the nongeneric states of that systemwell enough to guarantee this.

©2001 NRC Canada

1320 Can. J. Phys. Vol. 79, 2001

were begun. This work was supported by NSF Grant No. DMR-9981744, and used the Cornell Centerfor Materials Research computing facilities, supported by NSF MRSEC DMR-9632275.

References

1. (a) R. Moessner and J.T. Chalker. Phys. Rev. Lett.80, 2929 (1998); (b) Phys. Rev. B: Condens. MatterMater. Phys.58, 12049 (1998).

2. R. Moessner and A.J. Berlinsky. Phys. Rev. Lett.83, 3293 (1999).3. S.-H. Lee, C. Broholm, G. Aeppli, T.G. Perring, B. Hessen, and A. Taylor. Phys. Rev. Lett.76, 4424

(1996).4. P. Schiffer, A.P. Ramirez, D.A. Huse, P.L. Gammel, U. Yaron, D.J. Bishop, and A.J. Valentino. Phys.

Rev. Lett.74, 2379 (1995).5. J.A. Leu, D.D. Betts, and C.J. Elliott. Can. J. Phys.47, 1671 (1969).6. A. Keren, Y.J. Uemura, G. Luke, P. Mendels, M. Mekata, and T. Asano. Phys. Rev. Lett.84, 3450

(2000).7. J. Villain, R. Bidaux, J.P. Carton, and R. Conte. J. Phys. (Paris),41, 1263 (1980).8. C.L. Henley. Phys. Rev. Lett.62, 2056 (1989).9. J. von Delft and C.L. Henley. Phys. Rev. B: Condens. Matter Mater. Phys.48, 965 (1993).

10. E.F. Shender and P.C.W. Holdsworth. J. Phys. Condens. Matter,7, 3295 (1995).11. C.L. Henley. J. Appl. Phys.61, 3962 (1987).12. N.-G. Zhang, C.L. Henley, C. Rischel, and K. Lefmann. Phys. Rev. B: Condens. Matter Mater. Phys.

(2002). In press.13. E.F. Shender. Sov. Phys. JETP,56, 178 (1982).14. Y.J. Kim, A. Aharony, R.J. Birgeneau, F.C. Chou et al. Phys. Rev. Lett.83, 852 (1999).15. A.E. Jacobs and T. Nikuni. J. Phys. Condens. Matter,10, 6405 (1998).16. I. Ritchey, P. Coleman, and P. Chandra. Phys. Rev. B: Condens. Matter Mater. Phys.47, 15 342 (1993).17. A. Chubukov. Phys. Rev. Lett.69, 832 (1992); A. Chubukov. J. Appl. Phys.73, 5639 (1993).18. E.P. Chan. Ph.D. thesis, Cornell University. 1994.19. C.L. Henley and E.P. Chan. J. Magn. Magn. Mater.140/144, 1693 (1995).20. A.P. Ramirez, G.P. Espinosa, and A.S. Cooper. Phys. Rev. Lett.64, 2070 (1990).21. C. Wengel, A. Zippelius, and C.L. Henley. Phys. Rev. B: Condens. Matter Mater. Phys.53, 6543 (1996).22. D.L. Huber and W.Y. Ching. Phys. Rev. B: Condens. Matter Mater. Phys.47, 3220 (1993).23. E.F. Shender, V.B. Cherepanov, P.C.W. Holdsworth, and A.J. Berlinsky. Phys. Rev. Lett.70, 3812 (1993).24. P. Schiffer and I. Daruka. Phys. Rev. B: Condens. Matter Mater. Phys.56, 13 712 (1997).25. P. Mendels, A. Keren, L. Limot, M. Mekata, G. Collin, and M. Horvati´c. Phys. Rev. Lett.85, 3496

(2000).26. M. Hagiwara, K. Katsumata, I. Affleck, B.I. Halperin, and J.P. Renard. Phys. Rev. Lett.65, 3181 (1990).27. C. Kittel and E. Abrahams. Phys. Rev.90, 238 (1953).28. J. Villain. Z. Phys. B: Condens. Matter,33, 31 (1979) see Sect. V.2 in particular.29. M.F. Thorpe. J. Non-Cryst. Solids,57, 355 (1983).30. C.H. Booth, J.S. Gardner, G.H. Kwei, R.H. Heffner, F. Bridges, and M.A. Subramanian. Phys. Rev. B:

Condens. Matter Mater. Phys.62, R755 (2000).31. I.N. Goncharenko, P. Cadavez-Peres, I. Mirebeau, and O.L. Makarova. Europhys. Lett.54, 807 (2001).32. L. Bellier-Castella, M.J.P. Gingras, P.C.W. Holdsworth, and R. Moessner. Can. J. Phys.79, (2001). This

issue.33. S. Miyashita and H. Kawamura. J. Phys. Soc. Jpn.54, 3385 (1985).

Appendix A: Percolation threshold of Kagome sandwich lattice

A very simple program (no spins are involved) generated random configurations with a fixed numberof diluted sites. Having found the number of sitesmj in the j th-connected cluster, and lettingm1be the largest of them, I computed two percolation quantities: the “percolation susceptibility”χp =N−1 ∑

j 6=1 m2j , andPp = m1/pN , the fraction in the extensive (“infinite”) cluster. In the limitN → ∞,

one expectsχp ∼ |p − pc|−γ on either side ofpc, while Pp ∼ |p − pc|β on the high side, andPp ≡ 0

©2001 NRC Canada

Henley 1321

on the low side.From the behavior of these two quantities, I estimatedpc ≈ 0.50(2). To probe the systematic error

due to size dependence, several sizes were tried (up to 16×16, i.e., 1792 sites before dilution); however,a genuine scaling fit was not carried out.

©2001 NRC Canada