Quantum trimer models and topological SU(3) spin liquids on the kagome lattice

We construct and study quantum trimer models and resonating SU(3)-singlet models on the kagome lattice, which generalize quantum dimer models and the Resonating Valence Bond wavefunctions to a trimer and SU(3) setting. We demonstrate that these models carry a Z_3 symmetry which originates in the structure of trimers and the SU(3) representation theory, and which becomes the only symmetry under renormalization. Based on this, we construct simple and exact parent Hamiltonians for the model which exhibit a topological 9-fold degenerate ground space. A combination of analytical reasoning and numerical analysis reveals that the quantum order ultimately displayed by the model depends on the relative weight assigned to different types of trimers -- it can display either Z_3 topological order or form a symmetry-broken trimer crystal, and in addition possesses a point with an enhanced U(1) symmetry and critical behavior. Our results accordingly hold for the SU(3) model, where the two natural choices for trimer weights give rise to either a topological spin liquid or a system with symmetry-broken order, respectively. Our work thus demonstrates the suitability of resonating trimer and SU(3)-singlet ansatzes to model SU(3) topological spin liquids on the kagome lattice.


I. INTRODUCTION
Spin liquids are exotic phases of matter where the competition between strong antiferromagnetic interactions and geometric frustration prevents magnetic ordering, but instead gives rise to topological order, that is, a global ordering in the structure of their entanglement, and which thus display a range of exotic properties such as fractional excitations with exotic statistics [1-3].A paradigmatic model for topological spin liquids has been Anderson's Resonating Valence Bond (RVB) state [4].It is constructed from different coverings of the lattice with SU(2) spin-1 2 singlets, which are placed in a "resonating" superposition such as to further lower their energy.Yet, the study of RVB wavefunctions poses the challenge that different singlet coverings are not orthogonal.To alleviate this problem, quantum dimer models have been studied instead, where inequivalent singlet configurations are taken to be orthogonal, such as in a large-spin limit [5].
The study of dimer models reveals a strong dependence of their quantum order on the type of lattice.Specifically, dimer models on bipartite lattices are generally critical [5,6], while conversely on non-bipartite lattices, in particular the triangular and kagome lattice, they have been found to exhibit Z 2 topological order [7,8].Here, the case of the kagome lattice is of particular interest: First, the dimer model on the kagome lattice is a renormalization (RG) fixed point with a direct mapping to a Z 2 loop model [9] (i.e., Kitaev's Toric Code [10]), and second, kagome Heisenberg antiferromagnets are approximately realized in actual materials such as Herbertsmithite [3,11], making the RVB state on the kagome lattice, despite not being their exact ground state wavefunction, a particularly interesting model to study.By using Tensor Network methods, which allow to construct smooth interpolations from the RVB to the dimer model and which provide a powerful toolbox to directly probe for topological order, the topological nature of the kagome RVB state could be unambiguously shown, and the underlying parent Hamiltonians of the model were identified [12,13].
Both the presence of Z 2 topological order and the critical physics found for bipartite lattices can be understood from the symmetries of the system.Dimer models naturally exhibit a Z 2 gauge symmetry -the parity of the number of dimers leaving any given region is fixed, arising from the fact that each dimer inside the region uses up two sites, or that two spin-1 2 make up a singlet -suggestive of Z 2 topological order.On the other hand, on bipartite lattices, the Z 2 is enhanced to a U(1) symmetry -the number of A and B sublattice dimers leaving a region must be equal, since each (nearest-neighbor) dimer in the region uses up an A and a B site -indicative of critical behavior.Thus, identifying the symmetries displayed by the dimer model on different lattices is key to its understanding.
In recent years, SU(N ) spin systems have received increasing interest, in particular due to the possibility to engineer SU(N )-invariant interactions in the fundamental representation in experiments with cold atomic gases [14][15][16].Hence, a natural first step is to consider SU(3)-symmetric models in the fundamental representation.In that case, singlets are tripartite, namely the fully antisymmetric state ijk ε ijk |i, j, k .Just as in the RVB construction, we can think of building resonating SU(3) singlet states -that is, superpositions of different ways to cover the lattice with singlets -to lower the energy, and just as for the RVB state, in the attempt to analyze the model we are faced with the challenge that different singlet configurations are lacking orthogonality.It is therefore natural -again in analogy to the RVB state -to study quantum trimer models, where different SU(3) singlets are replaced by orthogonal trimer configurations.In this context, a few questions naturally arise: First, what is the role of the underlying lattice -for instance, is there a similar dichotomy in the type of order displayed on bipartite vs. non-bipartite lattices?Second, is there a "natural" lattice on which to define the trimer model?This could either be a lattice on which the model forms an RG fixed point with a direct mapping to a loop model, such as the kagome lattice for the RVB state, or a lattice where the tripartite SU(3) singlets appear in a particularly natural way, which suggests a lattice built from triangular simplices and thus yet again the kagome lattice.
Previous work on trimer models and resonating SU(3) singlet wavefunctions has been focused on the square lattice.In particular, in Ref. 17, a quantum trimer model on the square lattice has been introduced and it has been found that it is topologically ordered, and in Ref. 18, a corresponding model with resonating SU(3) singlets on the square lattice has been identified as a Z 3 topological spin liquid; both of these works employed numerical analysis based on tensor networks.Indeed, such an order could have been expected, since trimers and SU(3) singlets display a natural Z 3 symmetry, analogous to the Z 2 symmetry in dimers.On the other hand, in Ref. 19 an SU(3) spin liquid on the kagome lattice had been proposed whose "orthogonal" version is an RG fixed point, namely the Z 3 loop gas version of the Toric Code, and which was shown to be topological.While this model itself did not allow for an interpretation as a superposition of trimer patterns, a modification of the SU(3) model which gave rise to such an interpretation was also discussed, and numerically found to be in a trivial phase.Together, these findings raise a number of questions: What is the nature of an orthogonal trimer model on the kagome lattice, whose simplices naturally support trimers?Can we explain the different behavior seen for square vs. kagome lattice from underlying symmetries in the model; in particular, are there additional symmetries on top of Z 3 emerging on the kagome lattice which speak against topological order?And finally, can we obtain a more general understanding of the way in which lattice geometries and the type of order in resonating SU(N ) wavefunctions could be related?
In this paper, we present a systematic analytical and numerical study of the trimer model and its SU(3) variants on the kagome lattice.Our key results are as follows: • We present a simple mapping from the trimer model to a Z 3 loop model (an "arrow representation" similar to the kagome RVB [9]), which however is missing a vertex configuration.However, as we show analytically, all Z 3 -invariant loop configurations re-appear under blocking.This shows that the model has the right symmetry to exhibit Z 3 topological order, and that no additional symmetries are present which would point to a different phase.This forms the starting point for our subsequent analysis.
• We demonstrate that both the trimer and the SU(3) model can exhibit either Z 3 topological order or conventional symmetry-breaking order, as well as critical behavior.Which one is realized is determined by the relative weight ζ of different types of trimers (specifically, those on the elementary simplices vs. the rest).There are two natural choices for this weight, motivated by different interpretations of an equal weight superposition.For the quantum trimer model, we find that it is topologically ordered for both of these choices.The resonating SU(3) singlet model, on the other hand, is topologically ordered for one choice of ζ but breaks lattice symmetries for the other (the latter was observed in Ref. 19).We further show that setting ζ to zero gives rise to an additional U(1) symmetry which results in a critical model.Finally, we provide phase diagrams for both the trimer and the SU(3) model as a function of the weight ζ, as well as a detailed analysis of the behavior as we interpolate between the trimer and the SU(3) model.
• We explicitly construct exact parent Hamiltonians for the trimer model.In the arrow (i.e., loop model) representation, those Hamiltonians consist of 3body terms across vertices and either 6-body or 11body terms which act around one or two hexagons, respectively.For the 11-body Hamiltonians, we prove that they give rise to the correct 9-fold degenerate topological ground space.Restricting to 6-body terms (the same locality as for the RG fixed point loop model) gives rise to 6 additional "frozen" classical ground states where the trimers display crystalline order, and which do not couple to the remaining 9 ground states; moreover, we show that introducing a single 11-body term anywhere in the lattice allows to melt those crystals and recover the original 9-fold degenerate ground space.Decorating the trimers with SU(3) singlets increases the locality of these terms to 8 and 12, respectively.
Together, our results show a remarkably rich behavior of the quantum trimer and resonating SU(3)-singlet models on the kagome lattice, and refute a simple connection between the type of order and the geometry of the lattice for trimer models analogous to the dimer case.Moreover, our construction allows us to obtain a simple SU(3) spin liquid ansatz on the kagome lattice which has a natural interpretation as a resonating singlet pattern and thus as a superposition of simple product states which are connected through local moves.Finally, it highlights the importance of a new aspect in trimer models, and more generally N -mer models, as opposed to dimer models, namely the key role played by the relative weights assigned to different trimer configurations.
The paper is structured as follows: In Section II, we introduce quantum trimer models, their arrow representation, and their loop gas representation.In Section III, we analyze the symmetry of the trimer model in the loop pic-ture and show that the full Z 3 symmetry (technically, Z 3injectivity [20]) is restored under blocking, which proves the absence of additional symmetries and implies the existence of local parent Hamiltonians.We then proceed to construct the simplest such parent Hamiltonian, with the technical arguments given in two appendices.In Section IV, we combine analytical reasoning with numerical tensor network methods to investigate the type of order which the quantum trimer model exhibits, and in particular how the nature of the phase -topological, symmetry broken, or critical -depends on the relative weight of different trimers.Finally, in Section V we generalize the trimer model by equipping it with SU(3) singlets, which we then continuously connect to a pure resonating SU(3)singlet model with the fundamental representation acting on each site.We study the phase diagram of the SU(3) model as well as the physics along the trimer ↔ SU(3) interpolation, and discuss the corresponding parent Hamiltonian.The section closes with a brief discussion of the SU(3) model as a variational ansatz.

II. TRIMER MODELS
Let us start by defining trimer models, Fig. 1a.A trimer on the kagome lattice is a covering of two adjacent edges and correspondingly three vertices.For a trimer, we distinguish two outer and one inner vertex, and we distinguish folded trimers (living on a single triangle) from unfolded ones (living on two triangles).A trimer covering T is a complete covering of the lattice with nonoverlapping trimers, i.e., each vertex is contained in exactly one trimer.We can treat the trimer coverings T as an orthonormal basis |T of a Hilbert space; an (orthogonal) trimer model is then given as the equal-weight superposition of all trimer configurations T , A key question will be how to treat folded trimers: Do we think of them as three different trimers (related by rotation), or do we treat them all as the same trimer (considering a trimer rather as an unordered set of three vertices), see Fig. 1a?Here, we will treat folded trimers as equivalent (i.e. as a set of vertices), but we will allow to assign them a different weight ζ [cf.Eq. (2)], accounting for multiple "internal" degrees of freedom such as differently oriented trimers (e.g.ζ = √ 3 would allow to resolve folded trimers in three orthogonal ways with weight 1 each).
A local representation of |T can be constructed by assigning either arrows {◮, ◭} or a no-arrow symbol • to every vertex (see Fig. 1b): To each outer vertex of a trimer, we assign an arrow pointing into the triangle to which the trimer extends, and to inner vertices, we assign •.The exception are folded trimers, in which case we assign inpointing arrows to all three vertices.This associates a unique arrow pattern to each trimer covering.(a) A trimer covering T .Folded trimers can be either treated as different depending on their orientation (as the two shown), or as identical (i.e.interpreted as a set of three vertices).We take the latter approach but allow to assign a different weight ζ to folded trimers, which allows to resolve them internally as a superposition of inequivalent trimers.(b) Arrow representation: We assign arrows to the outer vertices, pointing into the triangle towards the center of the trimer, • to inner vertices, and three inpointing arrows to folded trimers.This establishes a one-to-one mapping from trimer coverings to arrow patterns which obey a Gauss law, where the configuration with three • is forbidden.(c) Loop representation.Arrows are replaced by 1 and 2, respectively, depending whether they point from the A to the B sublattice or vice versa, and • by 0. The loop representation again satisfies a Gauss law with 000 forbidden.
Conversely, any arrow pattern for which (i) the number of inpointing minus outpointing arrows is 0 mod 3 and (ii) triangles with three • are forbidden, maps uniquely to a trimer pattern.
Let us now consider the arrow degrees of freedom as link variables on the dual honeycomb lattice.We replace the arrows by 1 and 2 ≡ −1 (we work mod 3 from now on), depending on whether they point from the A ◮ B sublattice of the honeycomb or vice versa; • will be replaced by 0. The arrows then correspond to configurations with a Z 3 Gauss law around each vertex of the honeycomb lattice, where the configuration 000 is forbidden.In this language, the trimer model corresponds to where the Z 3 edge configurations L on the honeycomb lattice are taken from the set L of all Gauss law configurations ("loop configurations") with no 000 around any vertex; here, ζ controls the relative weight of folded trimers (which correspond to 222 configurations around A vertices and 111 around B vertices, counted by N A

222
and N B 111 ).To study the properties of this model it is convenient to use a tensor network or PEPS (Projected Entangled Pair State) representation, which provides us with a range of powerful analytical and numerical tools for its analysis [20][21][22][23][24][25][26].The corresponding PEPS is constructed from two types of tensors, see Fig. 2a, similar to e.g. the tensor network for the RVB and dimer model [12]: One type of tensor -corresponding to a state |τ = t ijk |i, j, k -only has virtual indices and sits inside triangles, it ensures only valid vertex configurations , where arrows point from 1 to 3 and thus the singlet in 3⊗ 3⊗ 3 is forbidden; it is thus either a 3 ⊗ 3 ⊗ 3 singlet or a singlet in one of the pairs 3 ⊗ 3 ⊗ 1.The resulting state is a trimer model decorated with SU(3) degrees of freedom.By projecting onto the 3 in 3 ⊗ 3 = 3 ⊕ 6, trimers are decorates with SU(3) singlets.
appear (with weight ζ assigned to on-site triangles), that is, t ijk = |ε ijk | + δ i=j=k=1 + ζδ i=j=k=2 .The other tensor -corresponding to a map P = P a ij |a i, j| -sits on the edges and maps two virtual degrees of freedom, one from each of the adjacent tensors, to a physical arrow state |a , that is, P ◮ 12 = P ◭ 21 = P • 00 = 1 and zero otherwise, with the arrow oriented towards the 2. The trimer wavefunction |Ψ trimer is then obtained by arranging the tensors on the honeycomb lattice and contracting the virtual indices, or alternatively applying the maps P to the states |τ , see Fig. 2a.(Note that on B triangles, the role of 1 and 2 is swapped relative to Fig. 1c; this can be changed by an appropriate gauge transformation.)

III. Z3-INJECTIVITY AND PARENT HAMILTONIAN
We will now address the question whether the model is topologically ordered, and whether it appears as a ground state of a local parent Hamiltonian with a suitable (topological) ground space structure.To start with, the modified model which is an equal weight superposition of all Gauss law patterns -where we include the 000 configuration and let ζ = 1 -is a topological RG fixed point model, namely the Z 3 version of Kitaev's toric code model [10].But how severe is the exclusion of the 000 configuration? Does it induce a stronger symmetry than the Z 3 Gauss law -e.g., a U(1) conservation law which would be indicative of a critical phase -or does it merely induce a finite length scale without breaking the topological order?
To understand this, we study what happens as we block sites: Does the missing 000 configuration result in missing configurations at all length scales -which would suggest additional conservation laws -or is the full symmetry restored?To this end, we consider a dual "height" representation of the model, where we assign Z 3 variables to the plaquettes (Fig. 3a, where the plaquette variables are labeled by roman letters), such that the edge degrees of freedom are obtained as the difference of plaquette variables (oriented clockwise/counterclockwise around B/A sublattice sites).This mapping from plaquettes to edges is 3-to-1, where the forbidden 000 configuration rules out three identical plaquette variables around a vertex (Fig. 3b).Now consider first the neighborhood of one hexagon shown in Fig. 3c: Given labels a, . . ., f at the boundary, when can we assign an allowed label to the central plaquette?It is easy to see that this is the case if and only if the pattern is not of the form Fig. 3d for a, b, c all different, or rotations thereof.Now consider the block in Fig. 3e, with arbitrary plaquette variables {a i } and {f j } assigned to the boundary -corresponding to an arbitrary boundary configuration of the edges which satisfies the Z 3 Gauss law.Next, for f i = 0, 1, 2 assign g i = 1, 2, 1.This way, f i = g i (and thus the condition Fig. 3b is satisfied around all vertices marked ⋆), and g i = 0. Finally, we assign 0 to the central plaquette.We now immediately see that we cannot have the pattern of Fig. 3d around the plaquettes marked with green tickmarks, and thus, we can also assign consistent value to these.That is, any Z 3 -invariant loop configuration at the boundary has a realization on the interior, and is thus an admissible boundary configuration: We thus find that the Z 3 -invariant space is restored after blocking, proving that removing the 000 vertices from the Z 3 loop model induces no additional constraints under renormalization.In fact, this result still holds for the smaller patch in Fig. 3f, as can be verified by an exhaustive search.
The fact that the Z 3 -invariant subspace is restored under blocking -or, in the language of tensor networks, Z 3 -injectivity [20] is reached (that is, the blocked tensor describes an injective map from boundary to bulk on the Z 3 -invariant subspace) implies the existence of a local parent Hamiltonian with a 9-fold degenerate ground space on the torus, which is spanned by the trimer state |Ψ trimer and its topologically equivalent siblings, ob-tained by assigning a phase ω ν h N h +νv Nv (ω = e 2πi/3 ) with ν h , ν v = 0, 1, 2 to configurations for which the link variables sum to N h (N v ) along a horizontal (vertical) loop around the torus [12,20].Note, however, that this does not necessarily imply that the model is topologically ordered in the thermodynamic limit (ground states can vanish, or additional low energy state can appear) [27], and numerical study is required in addition to unambigously assess the topological nature of the trimer model; we will turn to this in the next section.
For now, let us discuss the form of the parent Hamiltonian.Generally, the local terms in the parent Hamiltonian are positive semi-definite operators (e.g.projectors) which are constructed such that they are zero exactly on all allowed states on the spins supporting them (in a tensor network, this is the space spanned by choosing arbitrary boundary conditions) [20,24].For the case of loop-like models like the one at hand, one way to explicitly construct such Hamiltonians is to build them from two types of terms: The first consists of 3-body projectors which act across vertices and have precisely the allowed vertex configurations in their kernel; here, those are the Z 3 Gauss law configurations except 000.The second type of terms couples different loop configurations through local transitions (that is, its ground space is spanned by the superposition |χ k of coupled loop configurations with the correct relative weights, h = 1 1 − |χ k χ k |), in such a way that any two loop configurations in the same topological sector (distinguished by their winding number, i.e., dual to the basis above) can be coupled through a sequence of such local moves induced by the individual Hamiltonian terms.
On what region do the latter type of Hamiltonian terms have to act?Using the Z 3 -injectivity after blocking, such regions can be constructed using canonical techniques; in essence, they need to contain an injective region plus a thin surrounding in a way which allows patching regions together [12,20,24,28].Given the minimum injective region identified in Fig. 3f, this gives terms acting on finite but still rather large regions.However, as we show in Appendix A, one can do much better: In order for the Hamiltonian to couple all configurations in a topological sector, it is sufficient to have Hamiltonian terms which act on two adjacent hexagons, or 11 edge degrees of freedom in the arrow or loop representation!
We thus find that in order to ensure a topologically degenerate ground space, one needs a Hamiltonian of the form where h is a projector acting on vertices which has the allowed vertex configurations as their ground space, and h is a projector acting on pairs of adjacent plaquettes which, for every choice of surrounding degrees of freedom, has the equal weight superpositions of all allowed configurations consistent with those boundaries as their FIG.4. (a) "Frozen" trimer configuration (blue) with crystalline order which cannot be melted by one-hexagon moves alone, and thus forms an independent ground state under onehexagon parent Hamiltonians; this can be seen immediately from the height representation (labels inside hexagons).A two-hexagon move is needed to start melting the crystal, illustrated by the red trimer configuration which requires updating the two hexagons marked gray.(b) When equipping the trimers with SU(3) singlets, a Hamiltonian which acts on one hexagon in the arrow representation (left) also might have to act on some outer vertices due to the entanglement of the trimer (right).Here, the Hamiltonians acts by coupling the blue and red configurations.(c) For single-hexagon moves, in the worst case it is required to act on two degrees of freedom outside of the hexagon.Similarly, for the two-hexagon move in (a), one has to act on at least one additional outside degree of freedom, as the one marked by the dashed green circle.
ground space;1 the sums run over all vertices and pairs of adjacent hexagons, respectively.
What happens when we go even further and restrict to one-hexagon Hamiltonians, H = h + h , where h induces transitions between all allowed configurations on a hexagon (which is the same locality as for the Z 3 fixed point loop model)?It turns out that in that case, additional ground states appear.Those states are all of the form shown in Fig. 4a, that is, they exhibit crystalline order.2Using the height representation (see figure), it can be easily seen that it is impossible to change the value of a single plaquette without violating Fig. 3b, and an update acting on two hexagons is needed to start melting the crystal (shown red in Fig. 4a).
As it turns out -discussed in detail in Appendix Bthese are the only additional ground states which appear when restricting to one-hexagon terms, while all other configurations in each topological sector can be coupled even by one-hexagon updates.Differently speaking, crystals of the form Fig. 4a which only cover part of the system can be melted from the outside, and in particular, it is possible to change the configuration anywhere inside the crystal by melting a channel through the crystal, updating the configuration as needed, and re-freezing it towards the outside.This also implies that a single twohexagon term h anywhere in the system is sufficient to couple the crystalline configurations to the others, and this way get back a Hamiltonian with 9 ground states.3

IV. ORDER AND PHASE DIAGRAM
Is the trimer model on the kagome lattice topologically ordered, and does this depend on the weight ζ of folded trimers?To this end, let us first see what we can understand analytically, e.g. in limiting cases, about the effect of varying ζ.To start with, observe that any unfolded trimer uses up two triangles per three vertices.Folded trimers, on the other hand, only require one triangle per three vertices.As the kagome lattice has 2 triangles per 3 vertices, each folded trimer is therefore accompanied by exactly one defect triangle, that is, a triangle with no edge of a trimer on it (Fig. 1a).Thus, we find that the trimer state (2) is a sum over all trimer configurations, where configurations with each K folded trimers and K defect triangles are weighted by ζ K .(Curiously, this implies that only the product of the reweighting of folded trimers and defect triangles matters).From this, we can understand two limiting cases: For ζ → ∞, the system consists solely of folded trimers and defect triangles, and will thus break the lattice symmetry by putting all (folded) trimers either only on up-or only on downpointing triangles; such a type of crystalline order has indeed been identified for the ground state of the SU(3) Heisenberg model on the kagome lattice (see also Section V).On the other hand, for ζ = 0, folded trimers and defect triangles disappear, and thus the configurations 111 and 222 are forbidden (in addition to the already forbidden 000).The remaining configurations are 012 and its permutations, i.e., one in-and one outpointing arrow at every vertex.At ζ = 0, the system thus possesses a U(1) symmetry, suggestive of critial behavior (as one can construct a U(1) height representation and thus an effective U(1) field theory description).
In order to understand the behavior of the system also away from those limiting cases, we use tensor network techniques introduced in earlier work to study the nature of the trimer model [19,26,[29][30][31].Specifically, we compute iMPS fixed points of the transfer matrix from left and right, and use their symmetry breaking pattern with respect to the Z 3 × Z 3 symmetry in ket+bra to identify the topological nature of the system; this method also allows us to extract correlation lengths ξ for the trivial and anyon-anyon correlators (where the latter correspond to the inverse anyon mass).First, we study an interpolation from the RG fixed point to the trimer model at ζ = 1, obtained by decreasing the weight of 000 configurations.This corresponds to a smooth interpolation of parent Hamiltonians (as Z 3injectivity is kept) [12], and is thus a reliable way to certify the absence of phase transitions, in addition to the symmetry breaking pattern of the topological Z 3 × Z 3 symmetry in the entanglement.The result is shown in Fig. 5a, where the x axis gives the weight of the 000 configuration in the superposition.We see that as we decrease the weight of the 000 configuration, correlations in the system build up, up to ξ ≈ 0.72 for the trimer point.The dominant length scale is given by visons, while spinons or combined vison-spinon ("dyon") correlations remain zero.This is to be expected, as visons correspond to a disbalance in different Gauss law (i.e., loop) configurations (induced by suppressing 000), while spinons correspond to breaking up trimers (violations of the Gauss law), which doesn't occur as different trimers remain orthogonal. 4et us next consider the phase diagram as a function , the trimer model is topologically ordered independent of whether we interpret folded trimers as a single object or as a sum of three orthogonal trimer realizations.The inset shows the scaling behavior of the correlation length ξ for a critical point ζ c = 2.188, which yields a critical exponent ν ≈ 2/3.Remarkably, this is consistent with the 4-state (!) Potts transition.

V. SU(3) MODEL
It is natural to use the trimer model to build an SU(3) model.If we attach the fundamental 3 representation to each vertex, an SU(3) singlet consists of three spins 3 ⊗ 3 ⊗ 3 and is of the form ε ijk |i, j, k (with ε the fully antisymmetric tensor).By replacing each trimer with an SU(3) singlet (suitably oriented), we thus arrive at an SU(3) resonating trimer state, in analogy to SU(2) resonating valence bond states.
Is the physics of the model affected by replacing the trimers by SU(3) singlets?Since unlike abstract trimers, different singlet configurations are not orthogonal, this is far from obvious.In order to assess this question, we employ a tensor network representation of the model which allows to treat the (orthogonal) trimer model and the SU(3) model on the same footing; it can be seen as a variant of the SU(3) model in Ref. 19, see Fig. 2a.We start by triangular states |τ of three sites with representation 1 ⊕ 3 ⊕ 3 each, where |τ is an equal weight superposition of the 8 possible singlets without the one in 3⊗ 3⊗ 3, and where we choose the amplitudes of all singlets +1, except for the one in 3⊗3⊗3, which has amplitude i ζ; this way, |τ is rotational invariant and transforms under reflection R as R |τ = |τ .Next, we place |τ on the simplices of the kagome lattice and apply maps P( θ) which depend on some interpolation parameters θ.There are three special points for P between which we interpolate: P 1 projects each site onto (1 ⊗ 3) ⊕ (3 ⊗ 1) ⊕ ( 3 ⊗ 3) -this is unitarily equivalent to the (orthogonal) trimer model, by associating 1 ⊗ 3 to ◮, and 3 ⊗ 3 to •. (Gauss law is ensured since |τ is a singlet.)This assignment produces a direct correspondence between configurations of |τ and trimer patterns, cf.Fig. 2b.Next, P 2 is obtained from P 1 by removing the 6 from the subspace 3 ⊗ 3 = 3 ⊕ 6 while keeping the total weight of the subspace (i.e., reweighting it by √ 3 -not doing so would suppress 3 ⊗ 3 and thus unfolded trimers).After removing the 6, the effective Hilbert space is 3⊕3⊕3 ∼ = 3⊗C 3 .Finally, P 3 removes the degeneracy space C 3 by projecting on the equal weight superposition of the three 3 (where the relative phases ±1 are chosen such that P 3 R = −P 3 , which yields a rotationally invariant wavefunction |Ψ which transform as R|Ψ = | Ψ ).The interpolation P 1 → P 2 → P 3 can be carried out continuously -where along the entire P 1 → P 2 interpolation, we keep the weight of the 3 ⊗ 3 subspace constant -allowing us to go smoothly from the orthogonal trimer model to the SU(3) model. 5We refer the reader to Ref. 19 for further details.
The behavior along the interpolation is shown in Fig. 5bc.Notably, interpolating P 1 → P 2 does not induce any change in the wavefunction due to the reweighting.This can be understood since 3 ⊗ 3 only appears in the middle of unfolded trimers, and different trimers remain orthogonal -the interpolation is thus merely a local basis transformation, while without the reweighting, unfolded trimers would have been suppressed. 6When interpolating P 2 → P 3 , orthogonality of different trimer configurations is lost, which induces a finite length scale also for spinons and combined dyonic excitations; we find that for the SU(3) point, the dominant length scale is a dyonic one, with ξ ≈ 1.3.Importantly, from Fig. 5, together with an extrapolation of the correlations, it is clear that the correlations remain finite, and the SU(3) resonating trimer model is in the Z 3 topological phase; for the SU(3) point, the extrapolation (Fig. 5d) yields a dominant dyon correlation length of ξ = 1.33.
Second, we have studied the dependence of the phase diagram on the folded trimer weight ζ at the SU(3) point P 3 .The results for the correlation lengths are shown in Fig. 7a (we again plot γ = e −1/ξ , with the same color coding for the different anyon sectors as in Fig. 5): We again identify a topological phase for ζ < ζ c ≈ 2.28, a symmetry broken valence bond crystal phase for ζ > ζ c , and a critical phase at or around ζ = 0, which can be understood using the same qualitative picture as before for the trimer model.We find that the phase transitions out of the topological phase are driven by diverging vison correlations (i.e.vison condensation), just as for the trimer model, even though in the center of the topological phase around ζ = 1, dyonic correlations (which include a spinon contribution) are dominating.At the phase transition, the visons condense, leading to the confinement of spinons in the symmetry broken crystalline phase.Fig. 7a also shows the associated spinon confinement length; interestingly, this length scale is initially dominated by the equally divergent trivial correlations, while around ζ 3.7, the correlation length between spinons (which is in principle unphysical, as spinons are now confined) becomes dominant. 7Finally, let us point out that while on the one hand, the critical point ζ c ≈ 2.28 did not change significantly as compared to the trimer model (where ζ c ≈ 2. 19), 8 the implications on how the inequivalent ways to resolve folded trimers affect the phase is rather different: Since at the SU(3) point, all folded trimers are replaced by the same SU(3) singlet, the weight ζ which appears when resolving a folded trimer as three distinct trimers is ζ = 3 rather than ζ = √ 3, and thus, the corresponding SU(3) model is in the symmetry broken crystalline rather than the topological phase, as indeed reported in Ref. 19.
Let us now discuss parent Hamiltonians in the context of the SU(3) model.There are several aspects: On the one hand, we are interested in the Hamiltonian for the SU(3) model itself, but on the other hand, we will also discuss how the parent Hamiltonians for the trimer model are affected if we equip the trimers with actual SU(3) singlets (or some other sufficiently symmetric tripartite entangled state), while keeping different trimer configurations orthogonal.More generally, we will discuss parent Hamiltonians along the entire interpolation family P 1 → P 2 → P 3 , where the points P 3 and P 2 (P 1 ) correspond to the two aforementioned cases.3) singlets (i.e.fully antisymmetric states), cf. also Fig. 6.The color coding used is the same as in Fig. 5.We again identify a topological phase in the center, which transitions to a critical phase for ζ → 0, and to a symmetry-broken phase around ζc ≈ 2.28.In the symmetry broken phase on the right, spinons become confined, and the associated spinon Why is the Hamiltonian -as discussed in Sec.IIIaffected if we equip the trimers, for which we have been hitherto using the arrow representation, with a tripartite entangled state?To this end, consider the transition between the two configurations in Fig. 4b, and the corresponding Hamiltonian term: In the arrow representaton (left), this is a one-hexagon move, where we just need to change the arrows on the hexagon itself from blue to red.On the other hand, if we equip the trimers with entangled states, we must in addition update the entangled states placed on top of the trimers.This can be done in different ways, such as the one indicated by green arrows (Fig. 4b, right), but regardless of how it is done, it requires to act on at least one of the outer vertices.This situation occurs precisely if only the inner vertex of a trimer lies on the hexagon, and the • on the inner vertex is changed in the transition: In that case, the degrees of freedom on the two outer vertices of the initial trimer, which are entangled in the initial state, belong to two different trimers after the move, and are thus part of two different entangled states: Hence, disentangling the degrees of freeom requires to act on at least one of them.In all other cases, the trimer pat-tern can be changed without touching the vertices outside the hexagon, since the outside vertices remain part of the same trimer. 9For one-hexagon moves, in the worst case (Fig. 4c) this requires to act on 8 spins, while the required two-hexagon move can be implemented by acting on 12 = 11 + 1 spins (two hexagons plus the vertex marked by the dashed green circle in Fig. 4a).Note that since the local terms h and h in the Hamiltonian are of the form h • = 1 1 − |χ k χ k |, with the |χ k the superposition of coupled configurations (and thus 8-or 12-local), the total Hamiltonian can indeed be expressed as a sum of 8-local or 12-local terms, respectively.
Let us now return to the Hamiltonian along the interpolation P 1 → P 2 → P 3 .Along the line P 1 → P 2 , the model corresponds to the trimer model, where the trimers have been replaced by different entangled states depending on the interpolation parameter.Thus, our discussion above applies to the whole interpolation, and implies the existence of an 8-body and 12-body Hamiltonian acting around one and two hexagons, respectively, depending on whether we are willing to accept an additional isolated "ice" sector on top of the topologically degenerate ground space. 10As we move further and interpolate P 2 → P 3 , orthogonality of different trimer configurations is lost.However, since the arrow information in P 2 is smoothly removed through a "filtering" map Λ = (1 − θ)1 1 + θ|+ +| on the arrow degree of freedom (with |+ the even weight superposition) [12,19], the PEPS at any point along the interpolation is related to the state P 2 by a local invertible map, except for the final point P 3 itself.Since the Hamiltonian is frustration free, this implies that we can change the local terms in the Hamiltonian correspondingly (using the inverse map) in a smooth way without changing the structure of the topological ground space which remains 9-fold degenerate.However, this only works if we apply this transformation to each of the terms h , h = 1 1 − |χ k χ k | as a whole, rather than the terms |χ k χ k | individually, and thus yields a 12-local or 19-local Hamiltonian instead, 11just as for the SU(3) model of Ref. 19.Finally, for the SU(3)-point P 3 , we can construct a Hamiltonian which yet again consists of 12-or 19-body terms by taking the nian h ′ i = (Λ −1 ) † ⊗k h i (Λ −1 ) ⊗k , with the tensor product ⊗k acting on the sites on which h i acts.Note that h ′ i can be replaced by a projector h ′′ i with the same kernel while keeping smoothness in the deformation (as eigenspace projectors of analytic maps are analytic as well [33,Thm. 6.1]).See Ref. 12 for a detailed discussion.limit of the parent Hamiltonian along the interpolation; 12 however, we cannot rule out that this Hamiltonian exhibits additional ground states. 13 Finally, we have investigated the behavior of the SU(3) wavefunction as a function of ζ as an ansatz for SU(3) Hamiltonians with 3-body interactions on triangles.Any such 3-body term can be decomposed as a sum of four projections h 1 , h 10 , h 8L , and h 8R onto the corresponding irreps (here, 8 L and 8 R denote the 8 irreps with angular momentum ±2π/3, respectively); in particular, the Heisenberg-type Hamiltonian of Ref. 34 (the sum over permutations of nearest neighbors) corresponds to h Heis,△ = 3(h 10 − h 1 ).The expectation values for the corresponding terms are shown in Fig. 7b, where energies for up and down triangles are plotted with the corresponding symbol.One can clearly identify the symmetry breaking phase transition, and the fact that in the crystalline phase, the up-triangles hold the folded trimers and are thus in a singlet. 14Unfortunately, we found that for all SU(3)-invariant three-body interactions which we considered, either the ζ = 0 or the ζ → ∞ point provide the lowest variational energy, making it unlikely that the wavefunction accurately captures the way in which the physics of SU(3) models with three-body interactions across triangles depends on the interactions.

VI. CONCLUSIONS
In this work, we have introduced and studied quantum trimer models and resonating SU(3)-singlet models on the kagome lattice.We have devised an arrow representation for the trimer model which provides a direct mapping to Z 3 loop models.While the loop pattern is missing a configuration, we showed that the full space of Z 3 -configurations (that is, Z 3 -injectivity) is recovered under blocking.This allowed us to combine analytical 12 The argument is similar to the previous footnote 11: Since the ground states |χ k are polynomials in the deformation parameter and thus analytic also around the final point, it follows that the ground space changes analytically and thus the limit of the projector-valued parent Hamiltonians h ′′ i above exists. 13Parent Hamiltonians can also be constructed by using that the tensor network on one star (12 sites), seen as a map from virtual to physical system, has constant rank everywhere except at the point P 3 , and thus, an invertible map on disjoint stars maps it to the orthogonal trimer model, which however yields a Hamiltonian with significantly larger locality [12].A similar argument, using that the PEPS map on a star is G-injective, was used for the SU(3) model in Ref. 19, where it gave a 19-body Hamiltonian; the reason why we can apply a much more direct argument which maps the model to the trimer point and thus yields simpler parent Hamiltonians, while not requiring numerical checks of G-injectivity on large patches, lies in the fact that unlike in Ref. 19, no big entangled clusters appear in the superposition (1), since the 000 ≡ 3 ⊗ 3 ⊗ 3 configuration is missing. 14The symmetry breaking pattern is controlled by biasing the initial state of the iMPS boundary.
tensor network tools with a microscopic analysis to devise simple parent Hamiltonians with 6-body or 11-body interactions: While the 11-body terms gives rise to the correct 9-fold ground space degeneracy, restricting to 6body terms alone only gives rise to six additional isolated "ice" states with frozen trimers which do not couple to any of the remaining configurations and which we thus expect to be strongly suppressed.
We have subsequently studied the phase diagram of the model by combining analytical arguments with numerical study.We have found that a key role is played by the relative weight ζ of folded (on-triangle) vs. unfolded trimers.For sufficiently large ζ, the system displays a conventional symmetry-broken phase in which all SU(3) singlets are localized on one type of triangles.In the intermediate regime 0 < ζ < ζ c ≈ 2.19, we identified a topological phase with Z 3 topological order.Finally, for ζ = 0, we found an additional U(1) symmetry, indicative of critical behavior which we confirmed numerically.
We have finally equipped the trimer model with SU(3) singlets and shown how these give rise to only slighly enlarged 8-or 12-body Hamiltonians.Using this as a starting point, we devised an interpolation where we erase the trimer (arrow) information, leaving us with a pure resonating SU(3)-singlet model with the fundamental representation per site.We numerically studied this model for ζ = 1 along the interpolation from the trimer point, and found that it is a Z 3 topological spin liquid in the same phase as the trimer model and the Z 3 RG fixed point loop model.Investigating the dependence on ζ resulted in a phase diagram simlar to the one above, with only a slight change in ζ c ≈ 2.28; yet this implies that a model where we were to place SU(3)-singlets in three different ways on any triangle would be trivial rather than topologically ordered, unlike for the quantum trimer model.
Several open questions remain.For one thing, it would be interesting to understand if there is a connection between the geometry of the lattice and the range of phases it can host, similar to the bipartite vs. non-bipartite phenomenon for SU(2) RVBs.Similarly, the extension of the trimer construction to SU(N ) and N -mers holds a wide range of interesting questions, from the nature of their quantum order or the role played by the lattice to the effect of the growing number of inequivalent trimer weights.Finally, our findings suggest that our ansatz, even though it exhibits a symmetry broken phase, does not capture the physics of SU(3) Heisenberg models beyond mean field, leaving the quest for a suitable SU(3) ansatz open.
than the injective region of Fig. 3f has the correct ground space structure [12,20,28].On the other hand, the ground space of this h ′ is precisely spanned by the allowed loop configurations on that region, where configurations which only differ inside that region are coupled.Differently speaking, being able to induce transitions between any two configurations on the region supporting h ′ is sufficient to couple any two configurations in the same sector.
What remains to be shown is that two-hexagon moves allow to construct any desired transition on a larger region.This is what we will do in this appendix.

Definitions and notation
For the purpose of the proof, we will consider hexagonshaped blocks as the one shown in Fig. 9, and we will show that using two-hexagon moves, any move on such hexagons (of any given size) can be achieved.In particular, this region can be chosen large enough to contain the aforementioned parent Hamiltonian h ′ , which yields all transitions induced by h ′ , and thus implies the existence of transitions between all configurations in the same topological sector.
Here, "move" denotes any transition between two loop configurations on the edges inside the respective region, in such a way that both configurations are consistent as parts of a larger system.This is illustrated in Fig. 8a for a two-hexagon move, which maps any configuration of the 11 black edges to any other configuration which is consistent with the same value of the surrounding red legs.Note that the value of the red legs can be inferred from the black degrees of freedom at the inside, i.e. such a move does not require to access the exterior indices.
It will be convenient to work in the height representation of configurations, that is, with plaquette variables.Let us introduce some language: We will denote a hexagon-shaped ring of 6n − 6 hexagons, such as the blue and red rings in Fig. 9, as an n-ring (n counts the hexagons at one edge) -e.g., the blue hexagons in Fig. 9 forms a 5-ring, and the red hexagons form a 4-ring.The union of all m-rings with m ≤ n will be called an n-disc; e.g., the green region is a 3-disc.In the height representation, each hexagon is assigned a height value, with the constraint that three identical heights adjacent to each other are forbidden, Fig. 3b.Then, a move acting on a region consists of replacing the height configuration inside that region by another one, keeping the outside configuration fixed, in such a way that no forbidden configuration appears.In particular, a two-hexagon move amounts to changing the height value of two adjacent plaquettes, and a single-hexagon move that of a single hexagon.Note that we only consider concentric rings as in Fig. 9, making the notion of n-rings and n-discs unique.
In the following, we will show inducively that if two configurations on an n-disc agree on the exterior n-ring, we can make them agree on their interior (n − 1)-discs by a sequence of local moves acting on two adjacent hexagons each.Since the region on which the original parent Hamiltonian h ′ acts is contained in an n-disc for sufficiently large n (and thus couples at most all configurations on that n-disc), this implies that a two-hexagon parent Hamiltonian is sufficient to couple any two valid height configurations.
Let us introduce a notation for height configurations on the concentric n-rings and n-discs.First, we denote the height configuration on the n-ring by L n .We say that two configurations L n and L n−1 are consistent if no forbidden configurations (three equal adjacent heights, Fig. 3b) appear.For two consistent configurations L n and L n−1 , we denote the joint configuration on the n-ring and the (n − 1)-ring by (L n , L n−1 ), and so forth.Finally, L <n = (L n−1 , L n−2 , . . ., L 1 ) denotes the joint configuration of the (n − 1)-disc; and thus e.g.(L n+1 , L <n+1 ) or (L n+1 , L n , L <n ) the joint configuration on the (n + 1)disc.Whenever we use this notation, this implies consistency of the configuration.For example, a configuration in Fig. 9 would be denoted by (L n+1 , L n , L <n ) with n = 4, where L n+1 is the configuration on the blue, L n the configuration on the red, and L <n the configuration on the green hexagons.
When two configurations (L n , L <n ) and (L n , L ′ <n ) are related by local moves on the (n − 1)-disc we write (L n , L <n ) ←→ (L n , L ′ <n ).Here, local moves are those which are generated by two-hexagon moves on the (n−1)disc: That is, either elementary two-hexagon moves, or moves for which we have already shown that they can be constructed from two-hexagon moves.Clearly, being related by local moves is transitive.

Overview of the proof
We will prove the following: Theorem 1.For any n ≥ 3, L n , and L <n , L ′ <n consistent with L n , the transition from L <n to L ′ <n can be The above argument fails for n = 3 where we are given L 3 and L ′ 3 and want to construct an L 2 consistent with both, since the 2-ring consists exclusively of corners.To cover the n = 3 case, we will instead provide a direct construction of L <3 , in close analogy to the argument used in the main text to construct a consistent interior of an arbitrary 3-ring (Fig. 3e).
Yet again, we need to consider two cases.The first is where L 3 and L ′ 3 differ at the corner of the 3-ring, see Fig. 12a.We want to construct a configuration on the interior that is consistent with both b 1 and c 1 .To this end, set g 1 = not(b 1 , c 1 ), h = not(g 1 ), g 2 = not(f 2 , h) and g 3 = not(f 3 , h).Then (i) the vertices marked by stars cannot be in the forbidden configuration Fig. 3b, i.e. three identical adjacent values, and (ii) the hexagons marked with a checkmark cannot be in the forbidden configurations Fig. 3d, and thus can be assigned a consistent value.
If instead L 3 and L ′ 3 differ in the center of an edge, shown in Fig. 12b, we need to give additional consideration to the value of the hexagon marked with a triangle: Unlike for the hexagons marked by a checkmark before, we now need to generalize Fig. 3d such as to rule out configurations which cannot be completed in a way consistent with both b and c at the given plaquette.Those configurations are of the form shown in Fig. 12c, with a, b, c all different, or variants thereof (reflection and/or exchanging b and c).Now, in Fig. 12b, choose g 2 = not(f 2 , not(b, c)).This rules out the configuration 12c around the hexagon marked by the triangle, since there the hexagon opposite of b/c has the value a = not(b, c).Further, choose h = not(g 2 ), g 1 = not(f 1 , h), and g 3 = not(f 3 , h).Then, (i) the vertices marked by stars cannot be in the forbidden configuration Fig. 3b, and (ii) we can fill the hexagons marked with the triangle and the checkmarks in a consistent way, as we have avoided the configurations Fig. 12c and Fig. 3d, respectively.This completes the proof.In this appendix, we consider parent Hamiltonians which only consist of one-hexagon moves, cf.Fig. 8b (together with terms ensuring the Z 3 vertex constraints) and show that, with the exception of the 6 symmetry-related crystalline configurations in Fig. 13 (see also Fig. 4a), all other configurations are coupled by one-hexagon moves.
To start with, it is straightforward to see that the stripe height configuration in Fig. 13a, corresponding to the crystalline trimer pattern in Fig. 13b, cannot be changed by one-hexagon moves (see Fig. 4a for a two-hexagon move melting such a cluster), and is thus decoupled from any other configuration; we will refer to the corresponding height configuration as "crystalline" as well.There are altogether 6 such configuration related by the lattice rotation symmetry.However, as we will see in the following, these are the only such configurations: All other configurations (within the same sector) can be transformed into each other by one-hexagon moves.First, recall from Appendix A that the only step where we required twohexagon moves was the start of the induction, that is, Eq. (A2) for n = 3: In the following, we will show that we can always implement such a move anywhere in the system as long as somewhere there is a place where the system is not in the crystalline configuration.In particular, this implies that in order to only keep the 9 topological ground states, it is sufficient if we include a single two-hexagon term (which can locally melt the crystal) somewhere in the system.We are thus interested in the situation where we cannot perform a transformation (L 3 , L <3 ) ) by a sequence of single-hexagon moves (denoted , or there is a single configuration L<3 which is cannot be connected to any other configuration L <3 , while all other configurations L <3 , L ′ <3 = L<3 are still connected by one-hexagon moves, (L 3 , L <3 ) . Moreover, such an L<3 is necessarily of the form shown in Fig. 14a (up to symmetry).We will call such configuration -which are the only ones which require two-hexagon moves to change FIG.14.(a) "Frozen" configurations which cannot be changed by one-hexagon moves on the inner 2-disc for certain boundary conditions.(b,c) The boundary configurations (up to symmetry) at the intersection of two 2-discs (Fig. 15) which only allow for two interior assignments (top/bottom); all other boundaries have at least three interior assignments.
themfrozen configurations.Note that being frozen is a joint property of (L 3 , L<3 ) (as the allowed moves on the interior 2-disc depend on L 3 ).
Clearly, if all 2-discs are frozen, the system must be in on of the crystalline states, Fig. 13a.Thus, let us consider the scenario where at least one 2-disc is in a non-frozen configuration.We will now show that this allows us to "melt" frozen configurations anywhere in the system with only one-hexagon moves.
To start with, consider the situation in Fig. 15a, where the left (red) 2-disc is frozen, and the right (green) 2-disc isn't.Our goal is to change the intersection of the two discs (blue) to a different configuration: As there is only a single frozen configuration for a given boundary, this would automatically "melt" the left 2-disc and allow us to subsequently transform it to any desired configuration (except the original frozen one) by one-hexagon moves alone.Since the right 2-disc is not frozen, changing the intersection region is possible by one-hexagon moves on the right 2-disc, unless the desired configuration on the intersection (blue) would amount to the frozen configuration of the right 2-disc.However, an exhaustive search reveals that -given a fixed boundary condition (gray) to the blue intersection -the only cases (up to symmetry) where there are only two choices for the blue intersection are those in Fig. 15bc, and neither of them is is consistent with both a frozen left and right 2-disc in the two configurations.For all other boundary conditions, there are at least three choices for the blue intersection, so there must be one for which neither of the two 2-discs is frozen (as the frozen configurations L<3 are unique).
We can now use one-hexagon moves on the right 2-disc to transform the blue intersection into such a configuration (restoring the plaquettes outside the intersection to their original state); then, the left 2-disc is no longer in the frozen configuration and can transformed to any other configuration by one-hexagon moves.
In case we want to change a frozen 2-disc with no nonfrozen 2-disc adjacent to it, we choose a path of overlapping frozen 2-discs connecting it to a non-frozen 2-disc somewhere in the system; such a path is illustrated in Fig. 15b.We can then start from the non-frozen 2-disc and melt the adjacent 2-disc as described above, and continue our way though the chain of 2-discs until we arrive at the original 2-disc which we want to melt.Importantly, while moving through the chain of 2-discs, we can always leave the system in its original state as we move on, as we explain using the diagram in Fig. 15a: (i) In the first step of the procedure, we are precisely in the situation as before, where the right 2-disc is not frozen and the part outside the intersection is restored to its original state.(ii) In the further steps, however, we start in a situation where initially the left 2-disc is frozen while the right 2-disc isn't any more, and at the end of the step, we want to leave the outside part of the right 2-disc in the residual part of the original frozen configuration (though whether it is actually frozen might depend on the environment).We can now use the same argument as before to see that there is at least one configuration on the intersection which is consistent with neither the left nor the right 2-disc being frozen; we can thus use moves on the right 2-disc to transform the outside part to the configuration it had in the original (frozen) configuration, and the intersection to the corresponding third configuration.We thus see that we can use this scheme to melt any frozen 2-disc inside a frozen cluster by acting along a one-dimensional path which connects it with the outside of the cluster, while restoring all plaquettes outside the melted 2-disc to their original state.

FIG. 15. Setup for "melting" frozen 2-discs by one-hexagon moves (see text). (a)
The frozen left (red) 2-disc can be melted if the adjacent right (green) 2-disc is not frozen, by updating the intersection (blue) through one-hexagon moves on the right 2-disc.(b) If the frozen 2-disc (left, red) is inside a frozen cluster, it is connected through a path of frozen 2discs (blue) to a non-frozen 2-disc (right, green), and one applies (a) iteratively.

FIG
FIG. 1.(a) A trimer covering T .Folded trimers can be either treated as different depending on their orientation (as the two shown), or as identical (i.e.interpreted as a set of three vertices).We take the latter approach but allow to assign a different weight ζ to folded trimers, which allows to resolve them internally as a superposition of inequivalent trimers.(b) Arrow representation: We assign arrows to the outer vertices, pointing into the triangle towards the center of the trimer, • to inner vertices, and three inpointing arrows to folded trimers.This establishes a one-to-one mapping from trimer coverings to arrow patterns which obey a Gauss law, where the configuration with three • is forbidden.(c) Loop representation.Arrows are replaced by 1 and 2, respectively, depending whether they point from the A to the B sublattice or vice versa, and • by 0. The loop representation again satisfies a Gauss law with 000 forbidden.

FIG. 2 .
FIG. 2. (a) PEPS construction for the trimer and SU(3) model: The wavefunction is constructed by starting with fiducial states |τ which enforce the Gauss law, and subsequently applying maps P to them which output the physical degrees of freedom on the kagome lattice.(b) Construction of a trimer model with SU(3) degrees of freedom: The fiducial states transform as a singlet in (1 ⊕ 3 ⊕ 3) ⊗3 , where arrows point from 1 to 3 and thus the singlet in 3⊗ 3⊗ 3 is forbidden; it is thus either a 3 ⊗ 3 ⊗ 3 singlet or a singlet in one of the pairs 3 ⊗ 3 ⊗ 1.The resulting state is a trimer model decorated with SU(3) degrees of freedom.By projecting onto the 3 in 3 ⊗ 3 = 3 ⊕ 6, trimers are decorates with SU(3) singlets.

FIG. 3 .
FIG. 3. (a) Height representation of the trimer model: A Z3 "height variable" (blue labels) is associated to every plaquette such that link variables (green labels) are the (oriented) differences of the height variables.(b) Forbidden height configurations (corresponding to the forbidden 000 configurations of the edges).(c,d) The central plaquette in (c) can be assigned a valid value exactly if the boundary is not in the state (d) or rotations thereof, with a, b, c all different.(e) Region used for the Z3-injectivity proof (see text).(f ) Smallest Z3-injective region, i.e.where all boundary configurations are admissible.

FIG. 5 .
FIG. 5. Length scales ξ for two-point correlations (labeled "trivial") and anyon-anyon correlators (i.e.inverse anyon masses, labeled by anyon type), along a sequence of interpolations.The y axis is γ = e −1/ξ .(a) Interpolation from the loop model (RG fixed point) to the trimer model, by removing the 000 configuration.No spinon correlations appear, as different trimers configurations remain orthogonal.(b,c) Trimers are equipped with an SU(3) representation, see text: Interpolation (b) removes the 6 in the center of the trimer and has no effect, and yields trimers equipped with SU(3) singlets at the point P2.Interpolation (c) removes the arrow information; this gives rise also to spinon and dyon correlations as orthogonality of trimer configurations is lost.However, correlations remain finite all the way to the SU(3) point P3.(d) Inset: Extrapolation of the leading correlation (dyon mass) at the SU(3) point for increasing iMPS bond dimension, using the ε-δ-method [32].

FIG. 7 .
FIG. 7. (a)Correlation lengths for trivial and anyonic correlations vs. the folded trimer weight ζ for the SU(3) model, where trimers have been replaced by SU(3) singlets (i.e.fully antisymmetric states), cf. also Fig.6.The color coding used is the same as in Fig.5.We again identify a topological phase in the center, which transitions to a critical phase for ζ → 0, and to a symmetry-broken phase around ζc ≈ 2.28.In the symmetry broken phase on the right, spinons become confined, and the associated spinon confinement length is shown by open (green) circles.It first equals the two-point (trivial) correlation length, and for large ζ 3.7 the (now unphysical) correlation length between spinons.Data has been obtained with iMPS truncation threshold η = 10 −4 2 −14 ≈ 6.10 × 10 −9 .(b) Variational energies vs. ζ for the different SU(3) invariant terms on up (△) and down (▽) triangles, see text.
FIG. 7. (a)Correlation lengths for trivial and anyonic correlations vs. the folded trimer weight ζ for the SU(3) model, where trimers have been replaced by SU(3) singlets (i.e.fully antisymmetric states), cf. also Fig.6.The color coding used is the same as in Fig.5.We again identify a topological phase in the center, which transitions to a critical phase for ζ → 0, and to a symmetry-broken phase around ζc ≈ 2.28.In the symmetry broken phase on the right, spinons become confined, and the associated spinon confinement length is shown by open (green) circles.It first equals the two-point (trivial) correlation length, and for large ζ 3.7 the (now unphysical) correlation length between spinons.Data has been obtained with iMPS truncation threshold η = 10 −4 2 −14 ≈ 6.10 × 10 −9 .(b) Variational energies vs. ζ for the different SU(3) invariant terms on up (△) and down (▽) triangles, see text.

FIG. 9 .
FIG. 9. Geometry considered in the proof of the two-hexagon parent Hamiltonian.The blue and red hexagons form a 5-ring and a 4-ring, respectively; a height configuration assigned to these hexagons is denoted by L5 and L4.The green hexagons form a 3-disc, with configuration L<4 = (L3, L2, L1).

FIG. 12 .
FIG. 12. Construction of an interior configuration L<3 consistent with two outer configurations L3 and L ′ 3 differing on one plaquette, see Section A 5. (a) Situation where the differing plaquette is at the corner.(b) Situation where the differing plaquette is on the edge.(c) Forbidden configuration (plus rotations/reflections) when the plaquette on the left can take two values; here, a, b, c are all different.

FIG. 13 .
FIG. 13.Crystalline configuration (a) in the height representation and (b) in the trimer representation.These configurations cannot be changed by one-hexagon moves alone.