Classical height models with non-abelian topological order

Christopher L. HenleyLaboratory of Atomic and Solid State Physics, Cornell University, Ithaca, New York, 14853-2501

I discuss a family of statistical-mechanics models in which (some classes of) elements of a finite group Goccupy the (directed) edges of a lattice; the product around any plaquette is constrained to be the group identitye. Such a model may possess topological order, i.e. its equilibrium ensemble has distinct, symmetry-relatedthermodynamic components that cannot be distinguished by any local order parameter. In particular, if G is anon-abelian group, the topological order may be non-abelian. Criteria are given for the viability of particularmodels, in particular for Monte Carlo updates.

PACS numbers:

I. INTRODUCTION

“Topological order” [1, 2] in a system means it has an emer-gent ground state degeneracy (in the thermodynamic limit),but (in contrast to symmetry-breaking), no local order param-eter operator can distinguish the states. Topological order hasattracted great interest over the last 20 years, since (i) it can-not (by definition) be captured by the Landau order-parameterparadigm and is hence exotic from the viewpoint of traditionalsolid-state theory; [2]; (ii) it is proposed to implement qubitsby the ground-state degeneracy, the coherence of which is ro-bust against environmental perturbations [3, 4]; (iii) the for-mulation in terms of ground-state degeneracy makes it attrac-tive for numerical exploration by exact diagonalization [5].

The best-known examples of topological order are“abelian” (e.g. most of the fractional quantum Hall fluids),but non-abelian examples are more suitable for quantum com-putation [4, 6]. Does not follow... In particular, the most ped-agogical and most studied examples are all based on the sim-plest abelian group Z2, e.g. Kitaev’s toric code [4]. Proposedrealizations of nonabelian topological order in lattice modelsare formulated as sums over loop coverings [7, 8].

Wen [1, 2] has emphasized quantum mechanics as a defin-ing property of topological order. But we can separate thesenotions: topological order (as defined above) is meaningfulin a purely classical model (as developed in this paper) or ina quantum-mechanical model at T > 0 [9] (so that e.g. itsrenormalization-group fixed points represent classical behav-iors). Indeed, I would suggest that the subject of topologi-cal order skipped over more elementary examples, owing tothe historical accident of the quantized Hall effect’s discov-ery. Consider the prior case of spontaneous symmetry break-ing: theorists understood classical critical phenomena longbefore they went on to quantum critical phenomena, and ourapproach to the latter is organized by its similarities or differ-ences from the former. ALTERNATE [Static and dynamiccriticality for ordinary ordering were understood first in clas-sical models, before quantum critical phenomena [11] was ad-dressed.]

Analogously, it is hoped that, in the case of topological or-der, classical models will (at the least) be a pedagogical aid,and that behaviors evidenced in classical models may providea framework for conjectures about the quantum models. Dis-entangling classical notions and inherently quantum mechan-

ical ones might lead to clearer (or at least different) think-ing. Also, the viewpoint in this paper naturally draws us toface groups (particularly A5, the alternating group on five ele-ments) which have not been prominent, and might perhaps in-spire the construction of quantum-mechanical models involv-ing these groups.

The explicit notion of “classical topological order” was in-troduced and highlighted in [9], in particular the points that(i) it is characterized by ergodicity breaking, (ii) can be im-plemented by hard constraints, and (iii) must have a discreteclassical dynamics – all of which applies to the models in thispaper. However, much of Ref. [9] was framed in terms ofthe relation of the classical model to a quantum model, e.g.by taking the quantum model to a temperature at which quan-tum coherence is no longer important, [10] or by removingsome Hamiltonian terms. In the present work, the model isformulated from the start as an ensemble of classical statisti-cal mechanics, without concern for the existence of a quantumcounterpart. That will permit consideration of a richer set ofmodels (i.e. discrete non-Abelian groups G), for which wemight not know how to concoct a good quantized version.

A. Height models

I shall realize topological order by generalizing the conceptof “height models”. Their defining property [12–20] is theexistence of a mapping directly from any allowed spin con-figuration {σi} to a configuration of heights h(r), whereinh(r) − h(r′), for adjacent sites r and r′, is a function of thespin variables in the neighborhood (normally, either two spinson the sites r and r′, or else on one spin on site i at the mid-point of the r-r′ bond: spins and heights commonly live ondifferent lattices).

The “spins” in the model could be any discrete degree offreedom – e.g. dimer coverings. For {h(r)} to be well de-fined, it is necessary (and sufficient) that the sum of heightdifferences is zero round any allowed plaquette configura-tion. Thus, a (local) spin-constraint are assumed that excludes(at least) the configurations without well-defined heights, yetstill which still allows a nonzero entropy of states S0 in thethermodynamic limit. In the simplest cases, h(r) is integer-valued.

Such models may have “rough” phases, in which the

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coarse-grained h(r) behaves as a Gaussian free field, i.e. theeffective free energy of long-wavelength gradients is

F =1

2

∫

d2r|∇h|2. (1.1)

Via the apparatus of the Coulomb gas formalism [21], thisimplies the spin variables have power-law correlations (crit-ical state); the topological defects may have unbinding tran-sitions like the Kosterlitz-Thouless transition. Indeed, in twodimensions most critical states can be addressed as “heightmodels” [21] and this formalism provides an alternative routeto computing exact critical exponents [17], besides conformalfield theory.

Note that on a topologically non-trivial space (such as thetorus of periodic boundary conditions), there are nontrivialloops `, such that the net height difference (or “winding num-ber”) w` added around such a loop is nonzero. It is easy tosee this is a topological invariant, in that w` is unchanged ifthe loop is shifted and deformed (so long as it stays topolog-ically equivalent to the original one.) Thus, if `1, `2, ... arethe fundamental loops, configurations have sectors labeled by(w`1 , w`2 , ...). Point defects may also be admitted and loopsaround them may also have a nontrivial w` (in which case theyare topological defects). Clearly, the winding number w` in aheight model is analogous to t ∈ T in a topologically orderedmodel; we could almost say this is a special case of topologi-cal order in which T is Z, here meaning the (infinite discrete)group of integers under addition.

In place of a Landau order parameter, the (near) degeneratestates in a topologically ordered system may instead be distin-guished by a global loop operator, acting around a topologi-cally nontrivial loop `. Just as a Landau order parameter formsa group representation of a broken symmetry, and labels thesymmetry-broken states, in topological order the global loopoperator ought to form a faithful representation of the “topo-logical group T ” whose elements label the distinct states. Inour models, the definition of this loop operator is trivial andtransparent: it is just the generalized “height difference”.

B. Outline of the paper

Meld these paras? In this paper, I generalize the height-model idea (Sec. II to the case where the height variable be-longs to a discrete group, thus defining a family of classicalmodels which (in many cases) has a non-Abelian topologicalorder. The models are defined by a lattice, a group, and theselected subset of group elements which are permitted valuesfor the ‘spins” of the model; the spins sit on the bonds. Evenin these purely classical models, the non-Abelianness has in-teresting consequences: for example, a collection of defectsno longer has a unique net charge (Sec. II D)

In Sec. III and Sec. IV, I survey the various combinationsfor the smallest non-Abelian groups, using the crude Paul-ing approximation as a figure of merit to identify the mostattractive models for Monte Carlo simulation, using single-site updates. Furthermore, Sec. V suggests what quantitiesare interesting to measure in such a simulation; however, no

simulation results are reported in this paper. Finally, the con-clusion (Sec. VI) reflects on which topological behaviors areinherently quantum mechanical, and which are not (in that thesame behavior can be found in classical models). Further-more, applications are suggested, either to simulating systemswith vacancy disorder, or to constructing quantum versions ofthe models in this family.

II. DEFINITIONS AND TOPOLOGICAL BEHAVIORS

This paper is meant to introduce (and compare) a wholefamily of models. In this section, I define the general rules forthis family, and then describe the most promising examples.(In the next section, I shall exhibit consequences for MonteCarlo updating of such models.)

A. Model definition

Let us take a “lattice” (not necessarily Bravais, e.g. honey-comb) of sites r. The spins sit on the bonds of this lattice, andtake values in the discrete group G; they are

σ(r, r′) ∈ G (2.1)

where (r, r′) labels a bond of nearest neighbors. The bondsare directed; if we reverse the direction we view a bond, thespin on it turns into its inverse:

σ(r′, r) ≡ σ(r, r′)−1. (2.2)

Then the “height field” h(r) is defined by

h(r) = σ(r, r′) ∗ h(r′), (2.3)

where “∗” represents the group multiplication. Of course,h(r) is only defined modulo a global multiplication by someelement τ , h′(r) = h(r)τ ; to make it well-defined, we couldarbitarily require (say) h(0) ≡ e, where e is the group identity.

It will be useful throughout to define the line or loop prod-uct of p spins:

γ(`) ≡ σ(r1, rp) ∗ σ(rp, rp−1) ∗ ... ∗ σ(r2, r1) (2.4)

Here the loop ` is a string of bonds (rk, rk+1) connecting end-to-end, for k = 0, ...p; it is a loop when rp = r0.

1. Plaquette constraint

Two constraints are imposed on the spin configurations.The first is the plaquette constraint: we require the loop prod-uct around any elementary plaquette, to be the identity,

γ(`plaq) = e (2.5)

This is necessary (and sufficient) for h(r) to be well-defined.One can define variants of any model by relaxing the pla-

quette constraint to allow a small number of defect plaquettes,

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around which the loop product is not the identity. Supposethat defects cost an energy ∆ (possibly depending on the kindof defect): then the basic, defect-free version of the model canbe viewed as a Boltzmann ensemble in the limit T/∆ → 0.On the other hand, if we imagined there were a spin-spin inter-action proportional to K that splits even the states satisfyingthe plaquette constraint, the basic version of the model wouldbe the T/J → ∞ limit of the Boltzmann ensemble.

2. Spin constraint

The second constraint is the spin constraint: choose a “spinsubset” S ⊂ G such that

σ(r, r′) ∈ S (2.6)

everywhere. Hence, the choice of S is a major part of amodel’s definition; in Sec. III C, below, I will discuss otherdesirable features of S. The spin constraint is retained even inversions of the model with defect plaquettes.

The constraints should respect the group and lattice sym-metries . Implementing the group symmetry means requiring

σ ∈ S ⇒ u ∗ σ ∗ u−1 ∈ S (2.7)

for any conjugating element u. Thus, S must be one of thegroup’s conjugacy classes – the simplest case – or a unionof such classes. (Some non-abelian groups, and all abelianones, have “outer” automorphisms, symmetries which cannotimplemented by conjugacy within G: we may also wish toimplement those symmetries, too).

To implement lattice symmetry, one asks

σ ∈ S ⇒ σ−1 ∈ S (2.8)

so the model respects inversion (around the bond’s midpoint).However, in some cases one must drop the condition (2.8).(See Sec. III B 3, below.) Let us further require

e /∈ S (2.9)

thus a uniform height configuration is disallowed. Finally, andtrivially,

S generates the full group G. (2.10)

(If not, I could have redefined G as the subgroup generated byS.)

Without the spin constraint, the models would be identicalto the lattice gauge models of Doucot and Ioffe [23]. In sucha model, only gauge-invariant (i.e. loop) quantities can havenonzero expectations. In contrast, our group-height models(like the original height models) have the possibility of long-range ordered phases, particularly if we assign different Boltz-mann weights to different configurations. In that case, thereis also the intermediate possibility of a partial ordering, atransition between topological orders with group G and withG′ ⊂ G. It is much easier to explore the phenomenology ofsuch transitions in the case of classical (rather than quantum)models.

B. Topological sectors and topological order

By induction (increasing their interior by one plaquetteat a time), the loop product is also the identity for any fi-nite loop ` (that is contractible to a single plaquette in smallsteps). But products around topologically nontrivial loops,e.g. through the periodic boundary conditions of a torus sys-tem, need not give the identity. Let `1, `2, ...`g be the basic in-dependent loops, where g is the genus; then the loop products{γi ≡ γ(`i)} label the possible topological sectors of config-uration space. The hypothesis of topological order is that –in the thermodynamic limit – each such sector is a separatecomponent of the ensemble, and that these sectors cannot bedistinguished by any local expectations. Within these models,the topological sector means simply the configurations thatcan be accessed by local updates. [? ] In the thermodynamiclimit, by the definition of topological order, all sectors shouldbe equivalent. An interesting question is how many distinctsectors there are, [24] given the system’s genus g.

1. Invariance

If we had run these loops from a different origin, r′, then

γi → γ′i = τ ∗ γi ∗ τ−1, (2.11)

(2.12)

where τ is the line product along any path from the origin tor′; it’s important that the conjugating element τ is the samefor all loops. If we keep the same origin, but perform a single-site update hitting on the origin vertex, (see Sec. IV C, below)one also gets (II B 1), where τ is now the updating multiplier.Fussy point: do I mean τ−1?

When the group G is abelian, then, If the group is abelian,then, the labels are invariant with respect to how we take theloop. We can have an independent and invariant loop productγi for every topologically independent loop, so the numberof sectors is |nT |

g , where g is the system’s genus, and |T |denotes the number of elements in the topological group.

C. Sector counting in non-abelian case

THINK.I need to check what happens when G is non-abelian, but T is an abelian subgroup of G. Can’t a conjugacydue to a local update change γ(`) in this case?

On the other hand, in the non-abelian case, a loop productis invariant only up to a conjugacy, so we have fewer sectors.Furthermore, the allowed values of distinct loop products arenot independent. Consider a square lattice model in a rect-angular system cell Lx × Ly, with periodic boundary condi-tions; let (γx, γy) be the loop products along straight lines ofbonds running from the origin site (0, 0), in the x or y direc-tions respectively. The loop running from (0, 0) to (Lx, 0) to(Lx, Ly) to (0, Ly) and back to (0, 0) contains no defect, soby inductive use of the plaquette contraint its loop product is e.

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γ2

γ1

γ*

γ1

γ2

γ* 1

γ*

γ1

γ*γ2

a) b) c)

FIG. 1: The composite defect charge of a pair may be changed bysliding a third defect between the two. (a). Two defects have chargesγ1 and γ2, both of them defined using the reference point P ; thus theloop enclosing both has charge γ2γ1. (b). After defect 3 is movedbetween defects 1 and 2, the loop product is still γ2 around the wholeloop that was displaced so that it was never crossed by defect 3; onthe hand, the product is just γ3 around the short loop (delimited byshort bars cutting it) that goes around only defect 3, or γ−1

3 arounda similar loop in the opposite direction. (c). Hence, the loop onlyaround defect 2 has product γ′

2 = γ3γ2γ3−1, and the total charge isγ1γ

′2. This can be in a different class than γ1γ2. Need larger text

in figure.

Yet the four segments of this loop are just γx and γy, forwardsor backwards, so the loop product is

γ−1y ∗ γ−1

x ∗ γy ∗ γx = e; (2.13)

in other words, γx and γy must commute.So, in effect, we must define an equivalence relation

(γx, γy) ∼ (γ′x, γ′

y) whenever the pair satisfies (II B 1), andeach topological sector is one equivalence class. In the case ofa larger genus, we extend in the obvious way to longer lists;for the case of genus 1, an equivalence class is just a groupconjugacy class.

THINK.If we cut a (large) hole in the center, so the genusis now g = 3, the argument that gave (2.13) now saysγ−1

y γ−1x γyγx = γhole?

Some typical kinds of equivalence classes are:(i) (e, e)(ii) (ω, ωk) or (ωk, ω) for k = 0, ..., m − 1, where ω is

an element (not the identity) of order m; this gives 2m − 1equivalence classes for each group class, in particular (ω, e)and (e, ω) for k = 0, and (ω, ω) for k = 1. Of course, ifkk′ ≡ 1( mod m), and ω′ = ωk represents a different groupclass, we must be careful not to count (ω′, ω′k′

) since it wasalready counted as (ωk, ω).

(iii) If the group has a center, meaning elements (besidesthe identity) that commute with everything else, we get addi-tional equivalence classes of the form (ω, zωk) or (zωk, ω),where z is in the center.

Table I shows the number of classes µ1 and the sector countµ2 for the groups of interest.

D. Defects

We can allow the possibility of dilute places where a pla-quette violates the plaquette constraint. The loop productaround it will be called b (in analogy to the Burgers vector

of a dislocation, which is the same kind of defect as in theusual height models based on Z).

In the case of a height model (in a rough phase), topolog-ical defects are like vortices in a two-dimensional Coulombgas[21, 22]. they behave like U(1) electric charges in a two-dimensional universe. Opposite charges feel an attractive log-arithmic potential. In contrast, in the case of topological order,the attractive potential decays exponentially and the defectsare deconfined [28].

If the topological order is non-abelian, then defects havenon-abelian charges. One implication is that a loop productmay be changed when a defect is passed across it, the actionbeing a conjugacy:

γ(`) → b1γ(`)b. (2.14)

Thus the topological sector might be changed when a defectwanders around the periodic boundary conditions. Also, thenet charge of a defect pair can be changed by passing anotherdefect between the pair. (See Fig. 1)

Another property of non-abelian defects is that there ismore than one possibility for the combined charge of twogiven defects. (This observation is not unique to topolog-ical order, but is also a familiar property of defects of tra-ditional ordered states [29].) It is interesting that, in thequantum-mechanical approaches to defects in topologicallyordered systems, this same property is also the hallmark ofnon-abelianness. In that context, the list of allowed com-binations is known as the “fusion rules”, and there are ma-trices (generalizations of Clebsch-Gordan coefficients) whichtell how to form the appropriate linear combinations.)

One can create a pair, and move one defect around theboundary conditions, and it may recombine with the originaldefect but still leaving one defect rather than annihilate. Theresulting state satisfies the generalization of (2.13), namely

γ−1y ∗ γ−1

x ∗ γy ∗ γx = b (2.15)

when a single defect is present. The fact that two defects maynot be able to re-annihilate is very similar to the “blocking”idea of Ref. 24 (for quasiparticles in a non-Abelian quantumHall state).

III. POSSIBLE MODELS

In this section, I survey specific models, emphasizing thecriteria which would make some of them particularly attrac-tive for future investigations. To summarize Sec. II A: modelsin this paper are specified by (i) the group G (values of height)(ii) the spin subset S (values of spins) (iii) the lattice whosebonds the spins sit on.

Therefore, the models will be named in the form“G(m)latt”. Here “G” is the groups name, (m) is the or-der of the elements in the selected conjugacy class (usuallythat is unambiguous), and “latt” abbreviates the lattice (“tri”,“sq”, or “hc” for triangular, square, and honeycomb). Thus“S3(2, 3)tri” means that G is the permutations of three objects,S contains all three pair exchanges, as well as the two cyclic

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TABLE I: Groups and spin subsets (pb from Eq. (4.4)). µ and µ2

are the number of conjugacy classes, and of topological sectors on atorus.

group+tag nG µ µ2 nS nE pb

Z2 × Z2(2) 4 2 16 3 – 1/3

S3 (2) 6 3 8 3 3 0

S3 (2,3) 5 – 1/5

Q (2) 8 3 10 6 – 2/7

D4(m, m′) 8 5 20 4 4 4/9?

A4(3) 12 3 8 8 4/11

A5(2) 60 4 20 15 – 45/59

A5(3) 20 – 40/59

A5(5) 12 – 48/59

permutations (i.e. every group element except for e), and “tri”means the spins sit on the edges of the triangular lattice.

A. Groups

Table I lists the groups and spin subsets I shall be interestedin.

The smallest group is S3, the permutations on three objects(also isomorphic to the dihedral group D3). Here, S may betaken as the class of all pair permutations, the model S3(2), oras all permutations except the identity, that is S3(2, 3) in ournotation.

The second smallest non-abelian groups have eight ele-ments. One of these is the 8-element quaternion group Q,i.e. the unit elements {±1,±i,±j,±k} from the quaternionring. Here S must be the class of the six elements not equalto ±1. The other eight-element non-abelian group is D4, thesymmetry group of the square lattice.

The “alternating groups” A4 and A5 are especially attrac-tive for our purposes due to their high symmetry (so we canchoose S to be a single class containing a sizeable fractionof all the group elements). They are consist of the even per-mutations of four and five elements. Note that A4 and A5

are also the point groups of the (proper) rotations of a regu-lar tetrahedron and a regular icosahedron, respectively. Beingsubgroups of SO(3), these groups might in some sense serveas a discretization of it [30], just as clock models are a dis-cretization of the XY model. That would be interesting as away to make a connection to topological models (or gaugetheories) defined in terms of (continueous) Lie groups.

Finally, A5 is the smallest non-abelian simple group, mean-ing it has no normal subgroups; as we shall see in a moment(Sec. III B), normal subgroups are an annoyance since theytend to make the behavior more trivial than would be expectedfor the group G.

ALTERNATEIf a group is simple, meaning it has no nor-mal subgroups, it has the least risk of reducing to somethingmore trivial.

We can also include abelian groups in this family of models.The simplest nontrivial example is Z2 × Z2.

B. Example models

Next I shall survey the simplest examples. Most of themreduce, in some fashion, to previously known models; that isan advantage for computational studies, since old results canbe used as checks. In several cases, the models in our familyhave “accidental” topological order, i.e. beyond the group G;In particular, some of them have height representations.

The group and subgroup involved in our spin constraint arefinite, and so is each plaquette; thus it can happen that theallowed configurations satisfy stronger constraints than thosethey were designed to fulfill. The first five subheadings belowall, in one sense or another, reduce to known models.

ALTERNATE Sometimes the constraints defined by achoice of G and S are actually stronger, in the sense that thesame configurations might represent a different model with alarger group. In particular, it might be a height model.

1. The 3-coloring model

For a first example, let G ∼= Z2 × Z2, an abelian group.Besides the identity, this group has three equivalent elementsa, b, c; each has order two, and the product of any two givesthe third. If we treat these as a class (although they are notconjugate, since the group is abelian), then we must choosethat class to be the spins, Z2×Z2(2). Since a = a−1, etc., wecan depict the spins using three (undirected) “colors” of theedges.

Then, on the triangular lattice [model Z2 × Z2(2)tri], theplaquette constraint is simply that each triangle has one edgeof each color: the “three-coloring model”. (It is usually repre-sented on the edges of the dual [honeycomb], where the con-straint says each vertex has three colors; in either case, thespins live on kagome lattice vertices, and the configurationsare also the ground states of the 3-state Potts antiferromagneton that lattice.) This model is known to have a Z × Z heightrepresentation [17, 26], in addition to the finite-group heightfield h(r) defined by (2.3).

2. 6-vertex model

need a lead-inFor another example, take G to be the (abelian) group Zq,

the integers modulo q under addition, with q > 4, and letthe lattice {r} be the square lattice. Choose S = {+1,−1}.(The two elements are not the same class; they are related onlyby an outer automorphism.) Then the sum of spins around aplaquette can be zero (mod q) only if it is just zero, i.e. thereare exactly two +1 and two −1 in the loop. If we express thesespins on the dual (also square) lattice, as an arrow pointingoutwards (resp. inwards) wherever σ = +1 (resp. −1) asthe loop is traversed counterclockwise, we see these just arethe configurations of the six-vertex model – which also has ainteger-valued height field. [27]

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3. Dimer models

Since Z or Zm are abelian groups, {+1,−1} is merely anouter class. Here, or whenever an element σ is not conju-gate to σ−1, we could define models with a directed spin con-straint, such that σ is allowed only in one direction. Indeed,the dimer covering on the honeycomb lattice has a height rep-resentation, which amounts to allowing either the elements +1or −2 along the directed edges.

ALTERNATE To accomodate dimer models, one takes Gto be Zp (with p sufficiently large) or Z, chooses S = {+1}(but excluding −1) and constrains the sum around a plaquetteto be +1 (rather than the identity). The classical dimer modelon the triangular lattice was pointed out to have Z2 (abelian)topological order [5, 28], and indeed it has a height represen-tation with G = Z2.

4. Groups with an even subgroup

An even subgroup E (with nE ≡ nG/2 has G/E ∼= Z2. Thatis, any product of an even number of elements lies in E . Saythat the spin subset S consists of odd elements. (If it consistedof even elements, we would generate at most E .) Notice thatsuch a model cannot use the triangular lattice, since the pla-quette rule cannot be satisfied (the plaquette product must beodd, while e is an even elenent).

Now if the simulation cell has even dimensions, the possi-ble topological products γ(`i) must lie in E . [31] Thus, thetrue topological order is only given by the group E .

For example, the permutation group S3 contains a subgroupof even permutations (just the cyclic ones). Here E ∼= Z3. Inthe case of the model S3(2), meaning we choose the “spins”to include only pair exchanges, we can only get abelian topo-logical order. In our list, D4 is the other group that containsan even subgroup (the proper rotations).

5. Groups with a center

What if G has a non-trivial center GZ? (The center is sub-group of elements that commute with every other element).For example, if we adopt the group Q of unit axis quater-nions, (which order 8) then QZ = {+1,−1} ∼= Z2 andQ/QZ

∼= Z2 × Z2. Thus, the model Q(2)tri projects ontoconfigurations of the the three-coloring model. (Just map±i → a,±j → b,±k → c.) The group D4 also has a cen-ter (two-fold rotations, i.e. inversions) commute with every-thing.)

THINK. Well, I believe (need to think through!) that themodel is locally equivalent to the height model defined us-ing the quotient group G/GZ , and that is the true topologicalgroup T . That quotient has half (or less) as many elements asG, so if G was one of the smallest non-abelian groups, the quo-tient G/GZ is probably abelian, and in some sense our modelhas reduced to a more trivial one. I think the center simplybecomes a gauge degree of freedom.

TABLE II: Lattices (asterisk denotes an average over two kinds ofplaquettes). Here “σ-phase lattice” denotes 3 2, 4, 3, 4 and “square-octagon” lattice denotes 4, 8 2. The columns give the coordinaationnumbers of the lattice and dual lattice, then the site and bond per-colation thresholds are quoted. (Many more significant digits areknown [38]. The bond percolation thresbhold pcb is referred to later.

lattice tag z zd pcb pcs

triangular tri 6 3 0.347 0.500

σ-phase – 5 3.333∗ 0.414 0.551

square sq 4 4 0.500 0.593

kagome – 4 4∗ 0.524 0.653

honeycomb hc 3 6 0.653 0.697

square-octagon – 3 6∗ 0.677 0.730

C. Criteria for models: estimates of entropy

Need to meld here!ALTERNATE The fundamental criterion is that the group

state entropy is bigger than zero. This is conveniently esti-mated by Pauling’s method. Consider a hypothetical ensem-ble of randomly chosen σ ∈ S on every edge; if |S| = m,there are myN/2 configurations, where N is the number ofplaquettes and each has y edges. Next, within this ensemble,it is easy in principle to compute (by enumeration) the chancepOK that (2.4) is satisfied for one plaquette. Now I make theapproximation that these events are independent for differentplaquettes. Then the (small) chance of any given configurationbeing valid is pN

OK, and altogether the entropy per plaquette islog(my/2pOK). This had better be positive.

To estimate at once the viability of many different models, Ishall use very crude estimates of the entropy and (in Sec. IV B)updatability. Say the lattice has coordination z and the duallattice has roordination zd, i.e. the number of sides of eachplaquette. (These numbers are related by 1/z + 1/zd = 1/2.)Also, say the group has nG elements, of which nS are in theselected subset S. These three parameters — nG , nS , and z(or zd) – contain much of what we need to characterize thepossible models. See Table II for the parameters related tolattice geometry, and Table I for those related to the groupsand the spin subsets.

THINK.Note: our purpose here is NOT to obtain quanti-tative estimates of the model’s entropy, although the Paulingestimate is often surprisingly accurate. Rather, we are inter-ested in comparing these values between different models asa figure of merit, to aid us in guessing which models are themost interesting or the most tractable.

I use a Pauling estimate for the entropy. There are Nz/2edges and hence nS

Nz/2 ways of placing spins belonging toS, not yet enforcing the plaquette constraint. Then let fe bethe fraction of these random plaquettes on which the plaquetteproduct is the identity. Write fe as displayed equation here?

Pauling’s approximation is to pretend this event is uncorre-lated between one plaquette and the next; then the probabilityis f

Nz/zd

e that that the constraint is satisfied on the Nz/zd

plaquettes. of the whole system. So, the Pauling estimate of

7

the ensemble entropy is

eNSPauling =(

nSz/2fz/zd

e

)N

. (3.1)

The condition we must satisfy, in order to have an ensembleat all, is SPauling > 0, i.e.

nS > 1/f2/zd

e ., (3.2)

If zd is not too small, we may estimate that the plaquetteproduct is equally likely to be any group element, hence verycrudely

fe ≈ 1/nG. (3.3)

Comparison of fe in Table IV B with nG in Table I shows thatusually, (3.3) is not bad. When the product of two elementsof S is unusually likely to fall into a particular class, the ac-tual fe may be lower (e.g. the model A5(2)tri) or higher (e.g.A5(5)sq). The extreme case is if the group contains even andodd elements, and S consists of odd elements (the S3(2) orD4(m, m′) examples in Table IV B). In that case we shouldreplace (3.3) by fe ≈ 1/nE = 2/nG if zd is even, but fe = 0if zd is odd. ALTERNATEAs can be seen in Table IV B, usu-ally fe ≈ 1/nG , as one expects in the limit of a long perimeteraround which every possible loop product γ(`plaq) is equallylikely, but fe deviates in a couple of cases.

To satisfy Eq. 3.2, the three parameters get pushed in thefollowing directions, but there are considerations limitingeach of the three.

(1) We want nG as small as possible; however, there arenot so many small, discrete, non-abelian groups: onlythree have nG ≤ 8, namely S3 (permutations of threeobjects), Q (quaternion group), or D4 (point group of asquare).

(2) We want larger nS , meaning the model is lessconstrained (and more tractable). In the limiting casenS = nG , the model is just a pure gauge theory, whichis trivial apart from its global topological properties.On the other hand, a sufficiently large nS requires in-cluding more than one conjugacy class in S, so that thespins can have inequivalent “flavors”. That is estheti-cally undesirable: a generic model (with unequal statis-tical weights) needs more parameters, and it is harder toimagine how such a model could be realized physically.

(3) Ws want large zd, as in the honeycomb lattice.However, it is esthetically harder to implement a prod-uct constraint in a physical model. (When the prod-uct string is short, there are only a few symmetry-inequivalent cases for it, and it is easier to concoct aHamiltonian term which does not reference the groupmultiplication, but which has those cases as its energyminimum.)

To satisfy (3.1) with a large group but S consisting ofjust one conjugacy class, the group must have high sym-metry. E.g., the alternating group A5 (the proper icosahe-dral rotations) has nG = 60 and contains conjugacy classes

with 12, 15, or 20 elements, which using (2) would needzd > 3.30, 3.02, or2.41 respectively.

Comment on SPauling < 0 in the case A5(2)tri. Actu-ally, the only possibility in a triangle is to have the threeelements representing mutually twofold orthogonal axes ofan icosahedron. We get zero entropy, only if we try and mixincompatible twofold elements. The three elements form asubgroup isomorphic to Z2 × Z2 so if we limit ourselves tothis subset, we are back at the three-coloring model and wedo get SPauling > 0.

IV. MONTE CARLO UPDATING

For us, one essential criterion of a model is the possibil-ity of Monte Carlo simulation. I limit consideration to theequal-weighted ensemble, in which every allowed configura-tion has the same weight. Then detailed balance is satisfiedif the forwards and backwards rate constants are the same forany update move. But what is the minimum sufficient updatemove? For the six-vertex model it sufficed to reverse the ar-rows on the four edges of one plaquette, which changes theheight field on one site, a purely local update. For the three-coloring model, the minimal update involves switching two“colors” (e.g. a ↔ b) along a loop, a nonlocal update move.What happens generically for our family of models?

A. Cluster update move

The update move is simplest described in terms of theheight function. First pick at random a group element τ 6= eand a starting site r0. Say D is the domain being touched bythe update. (It will be explained in a moment what determinesD). Then I prescribe that the update premultiplies the heightsin this domain by τ , so as to “shift” them:

h′(r) =

{

τ ∗ h(r)) for r ∈ D;

h(r) for r /∈ D.(4.1)

This induces the following update to spin configuration [35](recall (2.3)):

σ′(r′, r) =

τ ∗ σ(r′, r)) ∗ τ−1 for r, r′ ∈ D;

τ ∗ σ(r′, r)) for r′ ∈ D, r′ /∈ D;

σ(r′, r)) ∗ τ−1 for r′ /∈ D, r′ ∈ D;

σ(r′, r) for r, r′ /∈ D.(4.2)

I call this a “gaugelike” transformation [36]: it has the sameform as a gauge transformation would, but it is valid onlywhen an additional spin constraint is satisfied too.

If both endpoints of the bond are in D, then σ′ is conjugateto σ and must be legal (since we include whole conjugacyclasses in S).

ALTERNATE Recall the definition of the height field,(2.3): whenever both ends of a bond are in D, we haveσ′ = τ ∗ σ ∗ τ−1, which is conjugate to σ, and hence by(2.7) must be legal as σ was.

8

On the other hand, where the (r, r′) bond crosses the do-main boundary ∂D, the spin constraint is nontrivial to satisfy.Let’s place an arrow along the edge from r to r′ if and only if

σ(r, r′) ∗ τ−1 /∈ S. (4.3)

In other words, there is an arrow from r to r′ whenever in-cluding r in D forces us to include r′ as well. This arrow isnot necessarily bidirectional [37] we might have any of fourpossibilities (no arrows, arrows both way, or arrows one way)along each bond.

ALTERNATE On the other hand, when only the head r isin cluster D, σ′(r, r′) = τ ∗ σ(r, r′) which is not necessarilyin S. Hence, I take D to be the smallest cluster such that everyspin on its boundary ∂D is legal.

Then the update rule is to construct the arrowed-percolationcluster consisting of site r0, with the rule that site r′ is in-cluded if site r is included and there is an arrow r to r′. Thisis the smallest possible updated domain containing r0. Ofcourse, we do not actually need to construct all the arrows; in-stead, we grow the cluster from the initial site, and constructarrows only from sites already in the cluster.

Notice that (only) in the case G is abelian, (4.2) reducesto σ′ = σ throughout the interior of D. In other words, theupdate only changes spins along the boundary ∂D and thus isa loop update. In a non-abelian model, however, the update isgenerally a cluster update.

ALTERNATE By this definition, D is a percolation clus-ter for a certain correlated, directed percolation problem de-fined by the instantaneous spin configuration. Imagine that,throughout the lattice, we draw an arrow on the bond from r

to r′ where it would be illegal for just the r end to be in D,i.e., if στ−1 is not in S. (Some bonds get arrows in both di-rections.) Then we define D such that if the tail site of anyarrow is in D, so is the head site; in other words, starting fromr0, we follow every outward arrow until we run into a deadend on every site.

ALTERNATEThus we have a cluster update, constructedas a sort of percolation problem reminiscent of the Wolff up-dating algorithm for an Ising model [32].

B. Numerical criteria for cluster updates

Notice that in growing a cluster from r0, we never caredabout the reverse arrows. Therefore, we obtain the same clus-ters as we would in an ordinary (not arrowed) percolationproblem, if the occupied bond probability pb is identified withthe probability of an arrow in a pre-selected direction; thatprobability is simply

pb ≈ 1 −nS − 1

nG − 1, (4.4)

if we chose the candidate updating factor τ at random.In the spirit of the Pauling approximation, let us now pre-

tend that arrows on different bonds are uncorrelated: withinthat assumption, we must obtain the same cluster distributionas in the (thoroughly studied) problem of uncorrelated perco-lation on these lattices.

It follows that the updating behavior tends to depend onthe relation of pb to the critical percolation fraction pcb. Ifpb > pcb, then the cluster grows without limit, including anonzero fraction of the whole system; in that case, the updatemove certainly isn’t efficient. If pb is slightly less than pcb,there is a good chance of hitting a quite large cluster. In thiscase, we might simply set a limit of smax bonds in the updatecluster D; if we hit this limit and the cluster is still growing,we cancel the tentative move and start over, choosing a newrandom r0 and τ .

MELDALTERNATE It can happen that the minimal cluster is the

entire lattice (or nearly so). As a practical matter, then, weimpose a limit smax on its size; if the growing cluster exceedssmax, we cancel the update. Notice that even if some selec-tions of r0 and τ lead to a spanning cluster of this sort, otherchoices will lead to finite clusters and allow the possibility ofupdating. However, if smax is too small, the update might beinsufficient to access the full ensemble; in this situation, wewould be rearranging disconnected parts of the configurationand not really generating a new one. For example, in the 3-coloring model, the smallest possible update (smax = 1) ischanging ababab → bababa along a hexagon; this is not be-lieved to access all sites.

ALTERNATE The cluster is small (far below percolation)when arrows are not too likely, i.e. if we guess a random valuefor the hypothetical new spin σ′, it is likely to be in S. Thatmeans the fraction of G included in S, i.e. 1−pb, should be aslarge as possible. On the other hand, it is awkward if S con-tains more than one class of group elements; that would meanwe have different “flavors” of spin. So we arrive at the rule-of-thumb that (i) the group should be as small as possible (ii)S should consist of its largest class. (In the case of D4(2, 2),S combines two classes which are equivalent but only throughan outer automorphism.)

The “interesting” cases come when pb/pcb gets close tounity.

In the case that any group element τ works as a multiplieron any bond, I would write pb = 0 in Table IV B, rather thanuse (4.4), In such a case,our model is locally trivial: exactlynG

N configurations may be accessed, simply by applying onearbitrary group element at every site. In other words, there isa (locally) 1-to-1 mapping to the trivial model in which everysite has an independent degree of freedom. That model is justthe gauge model, which was studied previously in Ref. 23.

C. Single-site updates

A single-site update is the case that the updated cluster Dis just one site, thus only the z spins around it are updated.When pb is much less than pcb, most clusters are small, andthe probability P1 of a single-site update is appreciable. IfP1 is large enough, it might be ergodic to use only single-siteupdates (i.e. to pick smax = 1), in which case we can omitthe cluster-growing algorithm. I shall concentrate on thesecases, which are the easiest to simulate (and also the likeliestto extend to quantum models).

9

TABLE III: Entropy and updatability parameter estimates for se-lected models. Formulas from eqs. (3.1), (4.4), and (4.5). Notea: in these cases, all even τ can update, but no odd τ elements canupdate.

Model name fe exp(SPauling) pb/pcb P1est

Z2 × Z2(2)tri 2/9 4/3 ... ?”

Z2 × Z2(2)sq 1/3 7/3 0.667 0.681

S3(2)sq 1/3 3 0.0 0.5a

S3(2)hc 1/3 3 0.0 0.5a

S3(2,3)tri 4/25 16/5 0.576 0.870

S3(2,3)sq 21/125 21/5 0.400 0.963

Q(2)sq 7/54 14/3 0.571 0.930

D4(m, m′)sq 1/4 4 0.0 0.5a

D4(m, m′)hc 1/2 4√

2 0.0 0.5a

A4(3)tri 1/16 2 0.727 1.000

A4(3)sq 3/32 6 0.613 1.000

A5(2)tri 2/225 4/15 2.20 0.668

A5(3)tri 7/400 49/20 1.95 0.894

A5(5)tri 5/144 25/12 2.34 0.706

A5(2)sq 71/3375 19/5) 1.53 0.285

A5(3)sq 147/8000 147/20 1.36 0.544

A5(5)sq 53/1728 53/12 1.63 0.360

To estimate P1, I pretend the z bonds around a site are inde-pendently occupied by randomly chosen elements of S. Then

P1est = 1 −

∏

α

(1 − qzα)nα (4.5)

where α indexes the group class (with nα elements) that τmight be in (excluding the identity), and I defined qα to be thefraction of times τ ∗σ ∈ S given τ falls in class α. Notice theaverage of qα over τ ′s is 1 − pb; if I use this in place of qα in(4.5), I get 1− [1− (1−pb)

z]nG−1 which is a lower bound onP1

est.To have a single-site update, we clearly want nG as large

as possible, nS/nG as large as possible, and z as small aspossible, which are the same considerations favoring a largeentropy.

I include these estimates in Table IV B, particularly focus-ing on the models using group A5. We see from Table IV Bthat P1

est is large enough in many cases that we can rely onsingle-site updates. However, whenever P1

est gets close to1, our model is “too easy” in some sense – it is practically agauge model, with only mild constraints eliminating some ofthe configurations.

The entry P1est = 1 for A4(3) is delusory. This comes be-

cause qα = 1 for a certain class of update multipliers, namelythe order-2 class (double pairwise exchanges). If we limitedourselves to this class, indeed every update would be success-ful, but (it can be checked) the move would not be ergodic(does not access the whole ensemble).

To implement an actual simulation, one would not wantto choose τ at random, but biased towards the group classeswith a larger qα (the success fraction looking at just an iso-lated bond). In particular, one would omit group classes with

qα = 0; if the group contains even/odd elements and S in-cludes only one parity of element, then qα = 0 for every classof odd elements. The values of pb and P1

est in Table IV B forS3(2) and D4(m, m′) were computed assuming τ ∈ E .

Another way to implement a single-site update is, afterchoosing a random vertex r, to examine the local environmentof its z bonds, find the entire list of τ ’s which can update it,and choose randomly from this list. Typically, configurationdependent choices like this are avoided in Monte Carlo algo-rithms because they tend to violate detailed balance. In thepresent case, however, it can be checked that the number ofpossible τ ′s is always the same in the old and new configu-ration, i.e. the rate is the same for the forward and backwardstep, which is sufficient to ensure detailed balance (and anequal ensemble weight for every configuration).

D. Criteria for initial conditions

In height models, certain special states (e.g. the “columnar”arrangement of dimers on the square lattice) were “ideal” inhaving the maximum number of possible update moves. (Fora model requiring loop updates, we might replace that cri-terion by “having the shortest typical loops.”) Certain otherstates (e.g. the “herringbone” packing of dimers) were inert,in that no finite updates are possible (in the thermodynamiclimit). These states, in a height model, correspond respec-tively to a zero coarse-grained gradient of the height variable,or the maximum gradient.

In a non-abelian height model, the coarse-grained heightgradient is undefined, but one can still construct “ideal” and“anti ideal” states. It is recommended that simulation runs bestarted in both kinds of state, being in some sense oppositeextremes of the configuration space. A diagnostic for equi-libration is then whether the expectations from the two startsconverge to the same values.

More exactly, rather than a single domain of anti-ideal state,one should divide the system into two domains. Then, updatesare initially possible along the domains’ border, using loopswhich extend across the system. Gradually, a larger fraction ofthe system’s area become updatable, and the loops get smaller.On the other hand, starting from an ideal state, the loops areinitially small and get larger. Thus, tracking the loop distri-bution is an obvious diagnostic to test for convergence to thesame equilibrium state.

V. POSSIBLE MEASUREMENTS IN SIMULATIONS

In this section, I sketch how one might confirm the topolog-ical order numerically, or measure other interesting quantities,given a working Monte Carlo simulation.

A. Correlation functions

Correlation functions are an obvious starting point. Ofcourse, a topological order state has exponentially decaying

10

correlations, so this serves primarily as a negative test: wecheck that the system is not a height model in disguise (seeSec. III B), which would have power-law correlations, and thatit doesn’t have long-range order (which can emerge even inequal-weighted entropic ensembles, or because the definingconstraints are too restrictive). Correlations are also of inter-est near a critical point where long-range or quasi-long-rangeorder emerges.

In models with vector spins si, one was accustomed to eval-uating the expectation of si · sj , or occasionally its secondmoment. It may not be immediately obvious what to mea-sure now. One can, of course, simply tabulate frequencies ofdifferent combinations, e.g. (for the “height difference”) howoften γ0→r belongs to each conjugacy class. It is preferable,though, to reduce the measurements to a single (meaningful)number, and the appropriate generalization of the dot productis the trace of the matrices in the right group representation.

Thus we are led to use a character function χ(x), wherex is any group element; this is always the same within eachconjugacy class of the group. I divide the actual character bythe dimension of the representation, so that χ(e) ≡ 1 for anyrepresentation, and |χ(x)| ≤ 1 for any element. Presumably,the best choice of representation is the one that has the largestpositive χ(σ) for spins (for σ ∈ S). This corresponds concep-tually to using a distance metric, within the group G, countingmany multiplications by some element of S are needed to takeyou from element to the other one.

1. Height difference correlation

The natural generalization is

Ch(r) ≡ 〈χ(γ(`0→r))〉. (5.1)

Of course, the product γ(`0→r) is independent of which pathis taken from 0 to r – provided the path does not wrap aroundthe periodic boundary conditions.

As just noted, choosing χ(.) so that χ(σ) is as close toone as possible, provides that Ch(r) does express how fastthe group element wanders from the identity under repeatedcompositions; that is the choice likeliest to give a monotonicdecay with distance. If γ(`0→r)) is equally likely to be anygroup element – which one expects large r – then it followsthat C(r) = 0.

2. Spin correlations

Similarly, we can compute

Gij ≡ 〈χ(σi ∗ σ−1j )〉. (5.2)

B. Defects

It is easy to augment the simulation to allow a defect pla-quette where the plaquette constraint is violated. The same

(single-site) update rules will work correctly next to the de-fect, but they cannot change its position. To make a defectmobile, one can add additional update rules specific to thedefect, by (say) arbitrarily choosing one bond of the plaque-tte and changing it to make the plaquette’s loop product bee (which, of course, the loop product not be e for the pla-quette on the other side of that bond, unless that was also adefect plaquette and this is the annihilation event.) The sim-ulation would normally be run with a constraint or bound onthe number of defects.

The idea is to create a pair of defects, by hand, and thenevaluate expectations depending on them. The first thing tomeasure is the distribution Pd(R) of defect separations R. Inthe case of topological order, we expect deconfinement, mean-ing Pd(R) → const for R > ξ, where ξ is a (not very large)correlation length. One can define an effective (entropic) po-tential V (R) by

Pd(R) ∝ exp(V (R)); (5.3)

physically, V (R) is the difference in entropy due to placingthe defects near to each other. In the case of a height model,V (R) ∝ ln |R|, and dP (R) decays to zero as a power law. [?] ALTERNATEWe can measure Pd(r), the probability of aseparation r. This is expected to approach a constant, indicat-ing deconfinement of the defects (in a standard height modelit decays as a power law).

In fact, since there are various flavors of defect labeled bydifferent group elements b, one really needs to write the effec-tive potential as

E = U(b) + U(b′) + Vb,b′,c(R) (5.4)

where b and b′ are the respective defect charges, and c is thenet charge of the combined defect. Here, U(b) and U(b′) are“core energies” of these respective defects; these, and usuallythe inter-defect potential, are functions only of the conjugacyclasses of b, b′, and/or c. Implicit in the form (5.4) is that theexponential confinement length probably depends on all of b,b′, and c.

Measuring how the effective potential depends on class ismore physical, since (i) it decides whether a defect is stableagainst decays into other defects (ii) measurements of defectbehavior (in simulations or in real systems, were any to bediscovered) might be used to discover the universality class ofthe topological order, if that were not known. I conjecture thatthe dependence on b, b′, and c is also described by a characterfunction; it would be interesting to see if that can be exploredanalytically in some model.

Incidentally, since (with the appopriate boundary condi-tions) we can have a single defect in our system cell, that givesadditional opportunities to evaluate e.g. the core energy U(b)without the complication of a second defect.

Finally, if the single-site updates of Sec. IV C are not feasi-ble, defects provide a less elaborate alternative update move,in place of the cluster update of Sec. IV A. Namely, we cre-ate a pair of defects and allow them to random-walk until theyannihilate. (However, if their paths differ by a loop around

11

the periodic boundary conditions, they may be unable to an-nihilate.) Many Monte Carlo schemes [33, 34] are based on asimilar process.

THINK. Is it possible this only works when a loop updatewould suffice? (The loop around the system being the tracktaken by the two defects to re-annihilate.) I think it does workgenerally, but haven’t proven it to myself.

C. Topological sectors

So far, with regard to the existence of topological order, allof the measurements mentioned so far are negative; none ofthem gets to the heart of the positive property of topologicalorder, which is the topological degeneracy (and symmetry).To measure this in a classical simulation, we need an (nonlo-cal) update which can change sectorsL either the cluster up-date (Sec. IV A) or the defect-pair update just outlined.

From the relative fraction of time spent in different topo-logical sectors, we can infer a free energy FL(γx, γy), where(γx, γy) are the loop products characterizing the sector, and Lrefers to the system size. This is a finite size effect, since (bydefinition of topological order) the difference between sectorsvanishes in the thermodynamic limit; FL is expected to decayexponetially as a function of L.

In a similar fashion, if we allow transitions between stateswith and without a defect as part of the dynamics, we canevaluate the defect core energy U(b). Of course, FL(γx, γy)is very analogous to U(b), since b is a loop product encirclingthe puncture where a defect sits, just as γx is from the loopproduct encircling the system. [? ]

VI. DISCUSSION

I have put forward the notion of purely classical topologicalorder, defined by an ergodicity breaking into sectors depen-dent on the topology, and not distinguishable by thermody-namic expectations of any local operator. A family of explicitmodels has been described, along with a suitable Monte Carlotechnique, and criteria were suggested to pick out the mostpromising cases (in having a nontrivial and updatable ensem-ble of allowed states).

A framework was set up (Sec. II A) to define models withthree variable attributes: which group, which class(es) out ofthe group to selected as “spin” variables, and which latticeto place the model on. These are characterized by parame-ters – the sizes nG and nS of the group and the spin subset,the coordination number z of the lattice and zd of its dual– which entered crude formulas that estimate the entropy ofthe model and its updatability under single-site Monte Carlomoves (Sec. III C and IV B), which are the only actual cal-culations in the paper. Groups with normal subgroups tend tobe “less nonabelian ”, thus perhaps less attractive (Sec. III B).Although topological order superficially would appear to beintrinsically featureless, there is sufficient richness of mea-surable functions when one considers the the dependence offree energy on topological indices – finite size dependence

on sector or finite distance dependence on defect separations(Sec. V).

In trying to connect the classical picture to the quantumtheory of topological order, it is intriguing that a given twodefect charges (see Sec. II D) can combine in more than oneway, in a classical non-abelian model, reminiscent of fusionrules in a quantum model. If further investigation finds that thesector counting gives the same degeneracies in the classical asin the quantum case, one would conclude that this is one ofthe shared properties, not an intrinsically quantum one.

The physical manifestations of classical topological orderand of non-abelianness are less striking, perhaps, than for thequantum case. Most prominent is the behavior of topologicaldefects. Topological order implies deconfinement in the clas-sical model (but this would seem to be qualitatively the samefor nonabelian and abelian cases.) Non-abelianness changesthe rules for addition of defect charges, and braiding has phys-ical consequences, even though there are no Berry phases in aclassical model. (It must be noted, however, that the same be-haviors are seen in non-abelian defects of ordinary long-rangeorder [29] – they are not inherent to topological order.)

Degeneracies of different topological sectors, the definingproperty of topological order, also shows significant differ-ences in the non-abelian case, as there are far distinct fewersectors (see Sec. II B). This would not appear to be observ-able in real experiments; it might be evident in transfer-matrixcalculations.

A. Quantum mechanics

Several central concepts of topological order are inherentlyquantum-mechanical and thus have no counterpart in classi-cal topological order. They are mainly related to phases inwavefunctions and braiding of worldlines in 2+1 dimensions,namely anyon and mutual statistics. Most real or imagined ex-periments relating to topological excitations have involved in-terference phenomena (e.g. tunneling in various geometries ofquantum Hall fluids) and thus probe the quantum-mechanicalaspects of topological order.

Another feature missing in the classical models is the dualdefect or quasiparticle (such as the vison [44]), which is adistortion of the phase factors in the many-body wavefunction.

A final attribute of topological orders is the nontrivialcounting statistics of the excited states made by several quasi-particles, which is quantum mechanical in that it concerns thelinear dimension of a Hilbert space. One cannot rule out theappearance of similar concepts in classical stat mech – there,too, the partition function contains combinatorial factors forthe placement of defects, after the other degrees of freedomehave been integrated out. However, I am not aware of a situa-tion in which a nontrivial counting actually emerges

B. Construction of quantum models?

One path any of the non-abelian height models could be re-lated to the full framework of topological order is to convert it

12

to a quantum model with topological order. All this requires isto endowing it with a “flipping” move, just as classical dimermodels (and others) are converted into quantum dimer modelsusing the Rokhsar-Kivelson (RK) prescription [42]. A majorbarrier to this is that the minimal “flip” is a cluster update, asexplained in Sec. IV A, and it is not obvious when even a finitelist of cluster udpates would be ergodic.

Fortunately, whenever the single-site update (Sec. IV C)suffices, we can define a quantum model with a simple “flip-ping” term in the Hamiltonian, usually parametrized by anamplitude t, as well as a “potential” term of strength V = tthat penalizes each flippable place. At the RK point V = t, theground state wavefunction is an superposition of all configu-rations in the same topological sector, with the same (equal)weighting as in the classical ensemble, and (mutually inac-cessible) topological sectors are trivially degenerate. One isalso free to set V = 0 – obtaining a simpler model in whichflippable sites are so favored that an ordered state is likely tobe the outcome – or to vary V/t with the hope of crossing aphase transition.

However, there is one shortcoming (or incompleteness) inthe above recipe: as there are many possible choices of update(labeled by the multiplier τ ), each of would contribute to the“flipping” term of the quantum Hamiltonian, Combining themwith a particular set of relative phases is like choosing a trans-verse field (in a quantum Ising field) in a particular direction,thus giving up the full group symmetry. In lucky cases, thismight be handled by a site-dependent pattern of τi’s that linksthe group symmetry to the lattice symmetry, reminiscent ofKitaev’s honeycomb model [4]. Another option is to include asecond, quantum fluctuating field of τ ′s which are used for theupdate. Perhaps it is necessary to have such a dynamics, withthe τ ′s being derived from a second set of spins also havingthe gauge-like structure of a height model, as a prerquisite toobtaining dual defects having mutual statistics with the σ-spindefects described in this paper.

(Is this going too far?) It appears non-abelian (non-A)height models (endowed with a quantum Hamiltonian, likethe quantum dimer model) are a straightforward path to con-structing nonabelian models with topological order which aresuitable for simulation.

C. Possible simulations

Two or three kinds of problem can be simulated in classicalnon-abelian height models as a test-bed, whereas the analo-gous calculation might be very challenging computationallyin a quantum mechanical model. Of course, the answers neednot be the same, but the questions may be much clearer oncethe classical results are in hand.

What could be measured from a Monte Carlo simulation?THIS DUPLICATES THE MORE COMPLETE story in

Sec. V. Edit Most immediately, one can create defect pairs

and evaluate the histogram of their separation. (One needs,of course, a modified update algorithm in which the allowedconfigurations include isolated defects, but which does not al-low the step which would annihilate the two defects.) Thishistogram will reveal whether or not they are deconfined.

1. Phase transitions?

Furthermore, phase transitions can be studied. Just as astandard height model may have a “smooth” phase, in whichone or more height components becomes locked, it seems con-ceivable that a non-A height model might have a phase inwhich the loop products can take values in a subgroup G ′ ⊂ G– often an abelian subgroup. If so, one might encounter criti-cal points separating different topological phases, and charac-terize the critical exponents.

D. Defect interactions?

A classical model might be a helpful too for investigatingthe consequences of dilution disorder in a model with topolog-ical order. Each diluted site or plaquette is like a very smallhole cut in the system, thereby increasing the genus and thenumber of topological sectors. If the hole were big, these sec-tors would be truly degenerate (by the definition of topologicalorder), however this degeneracy is broken since the holes aresmall.

The values of γ∗ on each dilution site are local pseudospins,which are expected to have (exponentially decaying) interac-tions mediated by the fluid in between them, like the emergentspin-1/2 degrees of freedom in diluted spin-1 antiferromag-netic chains [40]. Some analogies are The ground state of sucha system could be constructed by a renormalization group thatiteratively combines the most strongly coupled pair of pseu-dospins into a single effective pseudospin, as was originallydone for the (exponentially decaying) antiferromagnetic cou-pling of the charge-bound electron spins in P-doped Si [41].

In the non-abelian case, at least, we do not know for surewhether these interactions lead to an inert singlet phase (as inthe antiferromagnet) or could give a state with some kind oforder among the pseudospins. Thus the system with defectswould not be a topological liquid, and this would be a novelscenario of how order can emerge due to disorder. [43]

Acknowledgments — I thank M. Troyer for suggesting theproblem; also L. Ioffe, A. Kitaev, D. A. Ivanov, Simon Trebst,C. Castelnovo, C. Chamon, S. Papanikolaou, and R. Lambertyfor comments and discussion. ALso, I thank J. Papaioannouand R. Maimon for preliminary work on the simulation algo-rithm. This work was supported by NSF Grant No. DMR-0552461.

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[] Here “local” means that cluster of updated sites (explained inSec. IV) is contained in a topologically trivial set of sites – itdoes not, e.g., form a band that spans the periodic boundaryconditions.

[] Differing varieties of confinement behavior are possible, de-pending whether that power is fast enough for

∫

d2RP (R) to

converge at large R.[] Indeed, a conformal mapping by the complex logarithm func-

tion converts the puncture geometry into a strip with periodicboudary conditions across it.