Generalization of the Haldane conjecture to SU(3) chains

We apply field theory methods to $\mbox{SU}(3)$ chains in the symmetric representation, with $p$ boxes in the Young tableau, mapping them into a flag manifold non-linear $\sigma$-model with a topological angle $\theta =2\pi p/3$. Generalizing the Haldane conjecture, we argue that the models are gapped for $p=3m$ but gapless for $p=3m\pm 1$ (for integer $m$), corresponding to a massless phase of the $\sigma$-model at $\theta =\pm 2\pi /3$. We confirm this with Monte Carlo calculations on the $\sigma$-model.


Introduction
Almost thirty-five years ago, the "Haldane conjecture" [1,2] was a revolutionary discovery in both condensed matter and high energy physics. It was already well-known that antiferromagnetic chains did not have Néel-ordered ground states, due to the Mermin-Wagner-Coleman [3,4] theorem forbidding spontaneous breaking of continuous symmetries in (1+1) dimensions. But Haldane argued that the behaviour was qualitatively different for integer and half-integer spin (s). For half-integer spin there is power-law decay of the alternating spin correlations and gapless excitations. For integer spin there is exponential decay of the correlation function and a gap to all excited states. Previously it had generally been expected that they were gapless for all s, largely based on Bethe ansatz results for s = 1/2. Haldane's argument hinged on the large-s limit, in which he mapped the low energy degrees of freedom of spin chains into the relativistic O(3) nonlinear σ-model with topological angle θ = 2πs. This model was quite popular in the high energy theory community at that time, as a simplified lower dimensional version of Quantum Chromodynamics (QCD) [5]. Like QCD, the model is asymptotically free, with a renormalized coupling constant that flows to strong coupling at low energies, resulting in the perturbatively massless particles becoming massive. The models also share an integer-valued topological charge and an associated topological angle. The O(3) nonlinear σ-model was already well-understood for θ = 0, due to its integrability, having a simple spectrum consisting of a massive triplet [6]. While no exact results existed at that time for θ = π, numerical results seemed to indicate that it remained massive [7,8]. Haldane argued that, in fact, the model was massless for θ = π, an unexpected result, which might have implications for QCD at θ = π. Ironically, the surprise to the condensed matter community was the massive behaviour for integer s and the surprise to the high energy community was the massless behaviour for θ = π! In the following years, Haldane's results have been confirmed experimentally, with the measurement of gaps for quasi-1D spin-1 chains [9], and numerically for chains of s = 1 and higher [10][11][12][13][14][15]. The field theory prediction has also been confirmed numerically [16][17][18][19][20][21] by Monte Carlo calculations, although the topological term presents severe challenges since the Boltzmann factor is not positive-definite for θ 0. A massless integrable model corresponding to θ = π was eventually found [22].
Extension of these results to the group SU(n) is of interest for several reasons. Cold atom experiments can realize SU(n) chains with various representations [23][24][25][26][27][28][29][30][31][32][33]. Mappings to relativistic field theories are also possible, in the limit of large representations, raising the possibility that other nonlinear σ-models might have massless phases driven by topological terms. A possible application of such field theory results to condensed matter physics exists. It was argued that the replica limit, n → 0, of the U(2n)/[U(n) × U(n)] nonlinear σ-model with θ = π describes the delocalization transition in the integer quantum Hall effect [34][35][36]. While this transition has been studied numerically [37][38][39][40][41][42][43], no exact solution has yet been found for the critical exponents, despite thirty-five years of efforts.
The goal of this paper is to extend Haldane's results to SU(3) chains with a particular set of representations having a Young tableau consisting of a single row of p boxes. We will argue that a gap exists for p = 3m, where m is an integer, but that the models are gapless for other values of p. Following Haldane, our approach is based on mapping the models into a relativistic quantum field theory at large p: the SU(3)/[U(1) × U(1)] nonlinear σ-model, defined on a space known as a "flag manifold". Since π 2 SU(3)/[U(1) × U(1)] = Z × Z, there are two topological angles which can appear in the Hamiltonian. We find that for translationally invariant systems the corresponding topological angles have equal and opposite values, ±θ, with θ = 2πp/3. We present Monte Carlo (MC) results indicating that the models are massive for θ = 0 but massless for θ = ±2π/3. This leads to an extension of the "Haldane conjecture": we expect the SU(3) chains to be massive for p = 3m but massless in other cases. We note that a gap for p = 3m was conjectured by Greiter et al. [44].
A novel feature of this flag manifold nonlinear σ-model is that its Lagrangian contains an additional term, linear in both space and time derivatives, which is not a total derivative and therefore not a topological term and which is generated from the SU(3) chain models. We study this term using the renormalization group (RG) [45], finding that the corresponding coupling constant flows to zero at low energies.
A detailed picture of the RG flow has been obtained in the SU(2) case, with direct implications for spin chains [46]. This is sketched in Fig. 1a. The massless critical point of the σ-model at θ = π was shown to correspond to a different nonlinear σ-model which is conformally invariant: the SU(2) 1 Wess-Zumino-Witten (WZW) model [35,47]. It is important to note that the O(3) σ-model is not conformally invariant, but flows to this conformally invariant critical theory for θ = π and sufficiently small bare coupling, g < g c . The appearance of this conformal field theory (CFT) is very natural, given the SU(2) symmetry and the fact that the SU(2) k models for k > 1 contain relevant operators allowed by symmetry [48]. For g > g c , the model goes into a gapped phase with a spontaneously broken Z 2 symmetry. In the spin chain, this symmetry corresponds to translation by one site, so the symmetry broken phase is dimerized [49,50]. The bare coupling constant, g, can be increased by adding frustrating antiferromagnetic next nearest neighbour exchange, J 2 , which has been shown to produce this transition, with g − g c ∝ J 2 − J 2c . [51,52]. Moving θ away from π also produces a gapped phase. This can be achieved in the spin chain by adding alternating exchange interactions, breaking translation symmetry by hand. The scaling of the gap with g − g c and θ − π was predicted using the SU(2) 1 WZW model [35]. The 2 transition along the θ = π line is controlled by the marginal, symmetry preserving operator J R · J L where J R/L are the right and left-moving current operators, with coupling constant ∝ g − g c . One sign of the coupling is marginally irrelevant and the other marginally relevant, leading to the transition at g = g c and the gap is exponentially small in g − g c . Moving θ away from π corresponds to adding a term (θ − π)trg to the effective Hamiltonian, where g is the primary field of the WZW model, an SU(2) matrix field of dimension d = 1/2. Thus the gap is expected to scale as |θ − π| 1/(2−d) = |θ − π| 2/3 , up to log corrections coming from the marginal operator. Our predicted phase diagram for the SU(3) σ-model in the special case when the two topological angles are equal and opposite is sketched in Fig. 1b. We identify the critical theory at θ = ±2π/3 with the SU(3) 1 WZW model. We again expect a gapped phase for g > g c and for non-zero θ ∓ 2π/3 and can again predict the gap scaling. A more general phase diagram in which the two topological angles can vary independently will be discussed in Sec. 7.  [46]. At θ = π the system undergoes a phase transition from a gapless phase at g < g c into a gapped phase with a spontaneously broken Z 2 symmetry at g > g c . For θ π the system is gapped with a unique ground state for all values of g. (b) Proposed renormalization group flow diagram for the SU(3)/[U(1) × U (1)] nonlinear σ-model in the special case where the two topological angles are equal and opposite. At θ = 2π/3 and 4π/3 the system undergoes a phase transition from a gapless phase at g < g c into a gapped phase with a spontaneously broken Z 3 symmetry at g > g c . For 2π/3 < θ < 4π/3 the system is gapped with a spontaneously broken Z 2 symmetry, while for θ < 2π/3 and θ > 4π/3 the system is gapped with a unique ground state for all values of g.
There are three pieces of rigorous evidence for the SU(2) phase diagram. One is the Bethe Ansatz solution for s = 1/2 [53], giving the expected gapless ground state with no broken symmetries. Another is provided by the Lieb-Schultz-Mattis-Affleck (LSMA) theorem [54,55] which proves that the model is either gapless or has a ground state degeneracy for half-integer (but not integer) spin. The third is provided by the Affleck-Kennedy-Lieb-Tasaki (AKLT) models for integer spin [56]. The exact ground states were found for these models and seen to be gapped with no broken symmetries. We observe that these results carry over simply to SU(3). The p = 1 case is the Sutherland model, solvable by Bethe ansatz [57], and known to have a gapless low energy theory corresponding to SU(3) 1 [58][59][60]. The LSMA theorem was proven for general SU(n) and implies, for SU(3), either a gapless ground state or a ground state degeneracy for p 3m [55]. The AKLT construction was also generalized to different SU(n) spin chains [61][62][63][64], in 3 particular to the fully symmetric SU(3) case with p = 3m by Greiter et al. [44], who constructed Hamiltonians whose exact ground states can be found and which appear to be gapped with no ground state degeneracy. Several important apparent contradictions and open questions are raised by our results. According to the most recent numerical results [65,66], there is no indication of a gap or finite correlation length for p = 3. Besides, while numerical results are consistent with no gap for p = 2, the corresponding critical exponents appear to be those of SU(3) 2 , not SU(3) 1 [65,66]. (For a detailed discussion of numerical results, see Sec. 9). Analogous to the SU(2) case, we argue that the WZW models with k > 1 are unstable, containing relevant operators allowed by symmetry, so SU(3) 1 appears to be the only viable candidate for the critical point [60]. We suspect that these two discrepancies may be a result of a long cross-over length scale beyond which the true low energy physics becomes observable. In this regard it is interesting to recall that the Haldane conjecture remained controversial for several years until reliable numerical results became available for sufficiently large systems, greatly aided by the development of the Density Matrix Renormalization Group (DMRG) technique [13,14,67]. Finally, we have not been able to obtain the critical exponents of SU(3) 1 for the flag manifold σ-model because of the limitations of the MC approach in extracting the critical exponents.
In Sec. 2 we write the Hamiltonian for our SU(3) lattice model and discuss flavour wave theory. Although this erroneously predicts a classical ground state with spontaneously broken SU(3) symmetry it still provides the starting point for the field theory. In Sec. 3 we review the Bethe ansatz integrable p = 1 model, the LSMA theorem for the SU(3) case and discuss the AKLT models of Greiter et al. [44]. Sec. 4 contains the derivation of the flag manifold σ-model at large p; we show that for translationally invariant spin models the two topological angles have opposite values ±θ with θ = 2πp/3 and derive the unusual new term. In Sec. 5 we examine the symmetries of the σ-model, and their relation to the symmetries of the underlying spin model. Sec. 6 contains our perturbative RG results; notably these show that the unusual term renormalizes to zero. Sec. 7 discusses the general phase diagram spanned by the two topological angles based on calculations in the strong coupling limit and numerical Monte Carlo results. Sec. 8 discusses the nature of the gapless critical point, arguing that it should be the k = 1 WZW model. Sec. 9 contains conclusions and a discussion of open questions. Details and additional information can be found in a series of appendices.

The model
The goal of this paper is to investigate the properties of the SU (3) antiferromagnetic Heisenberg model with symmetric representations at each site, the generalization of the SU(2) Heisenberg model with arbitrary spin. The model is defined by the Hamiltonian with J > 0, and where the sum runs over the lattice sites i and the repeated flavour indices α and β that can take values 1, 2 or 3 corresponding to spins states A, B or C. The operators S α β (l) are the generators of SU(3). They obey the SU(3) Lie algebra where δ α β is the Kronecker δ function. The model is further specified by choosing the irreducible representation at each site. In this paper, we will concentrate on models with the same totally symmetric irreducible representation at each site. Fully symmetric representations correspond to Young-tableaux drawn with p boxes arranged horizontally: , , , . . . . For p = 1, this is the fundamental representation, the model is equivalent to quantum permutation of 3-flavour objects, and it is integrable by Bethe ansatz (see Sec. 3).
For general p, the model can be reformulated using Schwinger bosons with three flavours. The generators at site i can be written as The resulting Hamiltonian is quartic in bosonic operators: We note that the spin operators are usually defined to be traceless, i.e. S α α =0, which would correspond to the bosonic representation S α For convenience we use a slightly different definition in Eq. (2.3), which leads to S α α = p. The difference between the two conventions only gives constant terms in our calculations and doesn't change any of our conclusions.

The classical limit
In the following, we will investigate the properties of the model with the help of a field theory that describes fluctuations around a reference state that should correspond to the ground state in the p → +∞ limit. As in SU(2), the candidate states are product wave functions in which, at each site, all bosons are in the same state. This could be a pure flavour state, or a state corresponding to an SU(3) rotation in flavour space. If the coupling between two sites is antiferromagnetic, the energy is minimized if the flavour states are orthogonal, which will be achieved for instance if the states are pure flavour states with different flavours. It the coupling is ferromagnetic, the energy is minimized if the flavour states are the same. Now, for the model of Eq. (2.1), the classical ground state is not unique. The energy will be minimal as soon as neighbouring sites have orthogonal states, and this can be achieved in an infinite number of ways ranging from the Néel state ABABAB... to the three sublattice state ABCABCABC... . Further ground states can be generated e.g. by changing locally A or B into C in the Néel state, or by choosing after two consecutive orthogonal states, say AB, a state which is a rotation of the first one around the second one, i.e. a linear combination of A and C. This is a major obstacle to the development of a field theory to describe quantum fluctuations for two reasons. First of all, it is not clear a priori around which ground state the fluctuations should be introduced. Secondly, and maybe more importantly, the semiclassical theory will have local zero modes, i.e. branches with vanishing velocity that make the theory non relativistic.
There are several reasons however to believe that the three sublattice state ABCABCABC... is the appropriate starting point. First or all, it is clear from the Bethe ansatz solution of the 5 p = 1 case that short-range order is of that type [57]. Besides, in the large p limit, it can be easily shown that zero point fluctuations are minimal in that state because this is the only state where harmonic fluctuations are limited to pairs of neighbouring sites, leading to a vanishing frequency for all modes, by contrast to any other state where at least some harmonic fluctuations will live on longer clusters, leading to some non-vanishing frequencies [68]. This suggests that the first effect of quantum fluctuations will be to select the three sublattice state by an 'orderby-disorder' mechanism that generates effective additional couplings of order 1/p that lift the classical degeneracy. The simplest couplings that do the job are an antiferromagnetic coupling J 2 between next nearest neighbours or a ferromagnetic coupling J 3 between third neighbours. It will prove convenient to keep both couplings for the discussion. So the model we will effectively study is defined by the Hamiltonian: and with J 1 , J 2 , J 3 > 0. It should be kept in mind however that all the properties discussed in the rest of the paper are expected to apply to the nearest neighbour Hamiltonian of Eq. (2.1), and that the couplings J 2 , J 3 = O(1/p) have been introduced as a first effect of quantum fluctuations in that model.

Flavour wave theory
For further reference, it will be useful to know the form of harmonic fluctuations around the ABCABC... state, even if it is clear that these fluctuations will destroy the order since we are in 1D. The calculation of these harmonic fluctuations is most easily done using linear flavour wave theory [69][70][71], the extension of the usual SU(2) spin wave theory to SU(N) models. At each site, the boson corresponding to the color of the classical ground state is condensed, and the generators are rewritten entirely in terms of the uncondensed bosons. Keeping only terms that are quadratic in the Hamiltonian, and after a Fourier transform on each sublattice, the Hamiltonian can be diagonalized by a Bogoliubov transformation, leading to In this expression, k runs over the reduced Brillouin zone (RBZ) corresponding to the three site unit cell, and the operatorsb (α) † β,k ,b (α) β,k are Bogoliubov quasiparticles. The superscript (α) just keeps track of the sublattice. Details can be found in Appendix A. The dispersion of the flavour waves is given by There are 6 degenerate branches in the reduced Brillouin zone, and 6 Goldstone modes. For small k, the dispersion is linear: 3. Rigorous results and SU(3) k critical points

Lieb-Schultz-Mattis-Affleck theorem
Let |ψ be a ground state of the model defined in Eq. (2.1) on a system of length L (periodic boundary conditions assumed). Then we can obtain a low energy state by acting on |ψ with the unitary operator [55]: . See Appendix B for details. There we show that where T is the operator which translates states by 1 site. Thus, translational invariance of |ψ implies ψ|U|ψ = e i2πp/3 ψ|U|ψ . This implies that ψ|U|ψ = 0 for p 3m, i.e. U|ψ is a low energy state which is orthogonal to |ψ . This leaves two possibilities. If the ground state is unique, then there is a low energy excitation. Alternatively, there may be degenerate ground states in the thermodynamic limit, with the finite system containing an exponentially low energy excited state which is essentially a linear combinations of these ground states. It can also be seen (Appendix B) that implying that U 2 |ψ is another low energy state which is orthogonal to |ψ and U|ψ , for p 3m. Furthermore, |ψ , U|ψ and U 2 |ψ are all invariant under T 3 , translation by 3 sites. Thus, if there are no low energy excited states, we might expect a triplet of trimerized ground states, as illustrated in Fig. 2a. These 3 states map into each other under translations by 1 or 2 sites. For a long finite system, we then expect linear combinations of these 3 ground states to give, to good approximation, the ground state and the two exponentially low-lying excited states as discussed above.
projection onto Figure 2: Illustration of the exact ground states discussed by Greiter et al. [44]. (a) Threefold degenerate trimerized ground states in the p = 1 case, and (b) the uniqe ground state of an AKLT construction for the p = 3 case. See sections III.A and VIII.B of Ref. [44] for the construction of the corresponding Hamiltonians.
We remark that, for p = 1, a Hamiltonian was found by Greiter and Rachel [44] that has the simple trimer ground states. Their Hamiltonian can be written as a sum of projection operators 7 onto the total spin of each set of 4 nearest neighbour spins. Noting that for the trimer ground states, 4 neighbouring spins can only be in the 3 or6 representation, the Hamiltonian is chosen to give zero for those states and a positive energy for the other two possible representations that can occur from a product of 4 fundamental representations (15 and 15 ′ ). These ground states have very short range correlations, vanishing for distances > 2, and the model is expected to have a gap to all excitations.

Bethe ansatz results and SU(3) k critical points
The nearest neighbour p = 1 model, was solved using the Bethe ansatz by Sutherland [57] and its low energy degrees of freedom were shown to correspond to the SU(3) 1 WZW model [58][59][60]. A simple way of understanding this result is to observe that, for p = 1, we may represent the operators by 3 flavours of fermions 1 : with the constraint of 1 particle on each site: The Hamiltonian becomes a simple exchange term: We may obtain this model from an SU(3) Hubbard model, (3.8) in the limit U/t ≫ 1. Starting at small U, we may take the continuum limit, giving 3 flavours of relativistic Dirac fermions. We may then use non-abelian bosonization, which gives a charge boson plus the SU(3) 1 WZW model. The Hubbard interactions can be seen to gap the charge boson without effecting the low energy behavior in the spin sector, yielding the SU(3) 1 WZW model as the low energy effective theory. Integrable SU(3) spin chain models have been found for all p [72][73][74], with more complicated nearest neighbour interactions, and have been shown to correspond to the SU(3) p WZW models at low energies [60,75,76]. This can again be understood from non-abelian bosonization. In this case we must introduce fermions with p colours as well as 3 flavours, and write a generalized Hubbard model. Non-abelian bosonization now gives the SU(3) p WZW model in the flavour sector, together with a charge boson and an SU(p) 3 WZW model for the colour degrees of freedom. However it is now seen that the Hubbard interactions will generally gap the flavour sector as well as the charge and colour sector, unless the interactions are fine-tuned. This can be understood from the fact that the SU(3) p WZW models contain relevant operators allowed by symmetry for all p > 1, which are expected to appear in the Hamiltonian and destabilize the critical theory. We understand the fine-tuned nature of the Bethe ansatz integrable models as, remarkably, corresponding to fine-tuning the field theory to eliminate all relevant operators. The 1 We remind the reader that for convenience we use a non-traceless convention for the spin operators 8 only SU(3) invariant relevant operator in the SU(3) 1 model is tr(g) with dimension 2/3. But the field theory representation implies that, under translation by 1 site, g → e i2π/3 g, so this interaction is forbidden by translation symmetry, stablizing the SU(3) 1 critical point. For p 3m, we may expect an RG flow from SU(3) p to SU(3) 1 , consistent with the Zamolodchikov c-theorem [77], which states that the central charge should decrease under RG flow. For p = 3m we expect an RG flow from SU(3) p to a gapped phase.

Affleck-Kennedy-Lieb-Tasaki states for p = 3m
Greiter et al. [44] also found a Hamiltonian for p = 3m which has a unique, translationally invariant ground state, with spin correlations decaying with a finite correlation length. For the case p = 3 this state is depicted in Fig. 2b. We regard each p = 3 spin as consisting of three p = 1 spins projected onto the symmetrized state. Then we combine three of these p = 1 spins on three neighbouring sites to form a singlet. As can be seen from the figure, this state is actually translationally invariant since at each site, there is a trimer starting there and going to the right, a trimer starting there and going to the left, and a trimer centred there. The Hamiltonian is again written as a sum a projection operators, this time acting just on pairs of neighbouring sites. This time the energy is zero if two neighbouring sites are in the10 or 27 rep and is otherwise positive.The model is expected to have an excitation gap. This construction can be straightforwardly generated to all cases where p = 3m by decomposing each spin into 3m p = 1 spins and again drawing trimers, with m going to the left, m going to the right and m centred at each site.
Note that the above results are all consistent with the LSMA theorem. For p = 3m a gapped Hamiltonian can be found which is translationally invariant but for p = 1 a gapped Hamiltonian can only be found which is trimerized, breaking translational symmetry.

SU(3)/[U(1) × U(1)] nonlinear σ-model
Based on the discussion in Sec. 2 we consider an SU(3) spin chain with the p-box symmetric representation at each site, and we investigate the low energy behaviour of a Hamiltonian with antiferromagnetic nearest and next nearest and ferromagnetic third neighbour Heisenberg interactions defined as keeping in mind that, in the large p limit, the J 2 and J 3 terms can be considered as generated by quantum fluctuations. Here we only present the outline of our results, but we provide step by step calculations in Appendix D. To serve as a comparison, we also provide similar calculations for the SU(2) case in Appendix C. Using a spin coherent state path integral approach [78][79][80][81], one can write the imaginary time action of the model in Eq. (4.1) as where Φ(i, τ) is a three dimensional complex unit vector at site i and imaginary time τ, while β is the inverse temperature. For antiferromagnetic J 1 , J 2 and ferromagnetic J 3 the real part of the action is minimal for the classical three sublattice ground state manifold, which can be parametrized by a set of three orthogonal spin states corresponding to the rows of U, a unitary matrix. Since the action is invariant under changing the overall phase of each of the three spin states in U, we argue that ground state manifold is isomorphic to SU(3)/[U(1) × U(1)] [82]: two phases can be changed independently, later referred to as gauge invariance (see Sec 5.2), while the third phase is fixed by setting the determinant of U to 1. SU(3)/[U(1) × U(1)] has 6 generators, namely the six off-diagonal Gell-Mann matrices corresponding to the six Goldstone modes discussed in the flavour-wave approximations in Sec. 2.
Considering low energy fluctuations around the classical ground state manifold, on the one hand the U matrix can depend on the position, corresponding to the slow joint rotation of the orthogonal states of the three site unit cell. On the other hand the states inside a unit cell can also be non-orthogonal to each other. Accordingly, the low energy configurations can be described as [83,84] where the rows of U( j, τ) can be seen as three orthogonal states in unit cell j, and L( j, τ) describes the transverse fluctuations, which make the spin states non-orthogonal inside the unit cell: For compactness we omitted the ( j, τ) dependence of the the matrix elements of L. In this expression, a stands for the lattice spacing and p for the number of the boxes of the spin representation. The a/p factors emphasize that these fluctuations are small, as large fluctuations are exponentially suppressed in the path integral. The L matrix can be chosen to be hermitian as the skew-hermitian part would describe an infinitesimal joint rotation of the three spin states and could thus be merged into the unitary U matrix. The diagonal elements of L have been chosen to keep the spin states normalized 2 . Substituting the above parametrization of the low-energy fluctuations into the action in Eq. (4.2), the functional integral over the L variables can be carried out, leading to a form of the imaginary time action in terms of the U(x, τ) field only: where the Λ n matrices are defined by while ε µν is the two dimensional Levi-Civita tensor (ε xτ = −ε τx = 1). The coupling constant 1/g = p √ (J 1 J 2 + 2J 3 J 1 + 2J 3 J 2 )/(J 1 + J 2 ) and the velocity v = 3ap √ J 1 J 2 + 2J 3 J 1 + 2J 3 J 2 , in agreement with the flavour wave calculations in Sec. 2.
The imaginary term containing the θ n parameters is topological, with the integer valued topological charges [82] The λ-term is also imaginary, but non-topological. In fact, the value of λ is non-universal: For simplicity, let us introduce the notation where q mn = −q nm . In terms of these quantities, the λ-term of the action can be written: iλ (q 12 + q 23 + q 31 ). The topological charges can also be expressed using the q mn 's as The antisymmetry of the q mn implies that Q 1 + Q 2 + Q 3 = 0. So the action is invariant under a global shift of the topological angles, and one can set one of them to 0. Unless specified otherwise, we will work with the convention θ 2 = 0.
For the translationally invariant model of Eq. (4.1), and with this convention, the topological angles are given by θ 1 = −θ 3 = p2π/3, and the action actually takes the form with a topological angle given by It is the phase diagram of this action (with λ = 0) that is sketched in Fig. 1b. To discuss the properties of that model, it will be useful to consider the general case of Eq. (4.6) where the topological angles are free to vary. The action in Eq. (4.6) can be written using the three orthogonal fields φ 1 , φ 2 , φ 3 forming the U matrix. Setting the velocity to v = 1, the action can be rewritten as In this form, it is apparent that the action consists of three copies of a CP 2 field theory [82,85], each with a topological term, while the λ-term couples the three theories. Of course, they are also coupled due to the orthogonality constraint, which leads to Q 1 + Q 2 + Q 3 = 0, and which allows one in general to set one of the topological angles to 0. If the λ-term is neglected, it is also possible to rewrite the action in terms of three gauge fields, A n µ [82,86]: We may actually impose the constraint: which follows from the orthogonality of the three φ fields. The equivalence to Eq. (4.14) follows by carrying out the functional integral over the A n µ fields (see Appendix D for details). All three forms of the action in Eqs. (4.6), (4.14) and (4.15) are useful in different contexts of our study.

General properties of the field theory
In this section we briefly review the symmetries and other general properties of the field theory of Eq. (4.6) or Eq. (4.14).

SU(3) symmetry
Throughout this paper we only consider spin models with global SU(3) symmetry, hence the resulting σ-models are also invariant under SU(3) rotations. These are of the form where the unitary V matrix clearly cancels out in every term of the action in Eq. (4.6) or Eq. (4.14).

Gauge invariance
The overall phases of the spin coherent states shouldn't change the form of the action. This manifests in the gauge invariance of the action in Eq. (4.6) under the transformation ). In terms of the fields, this transformation corresponds to φ ′ n = e iϑ n φ n . A proof of gauge invariance in this language can be found in Appendix E.1. Gauge invariance is also evident in the formulation of Eq. (4.15), where gauge fields are explicitly introduced.

Time reversal symmetry
Another fundamental symmetry is time reversal symmetry. The effect of time reversal (in real time) is simply T U(x, t)T = U(x, −t), or equivalently T φ n (x, t)T = φ n (x, −t), as well as complex conjugation of c-numbers: i → −i. The first term in the action in Eq. (4.14) is clearly invariant under T-reversal. The topological θ-term and the λ-terms pick up a factor of i when going to real time, which makes these terms real (Hermitian). The i to −i transformation then compensates for ∂ t → −∂ t , also leaving these terms time reversal invariant.

Z 3 symmetry
The field theory has an additional global Z 3 symmetry: which cyclically permutes the three φ n fields. This symmetry is a consequence of the invariance of the spin model under translation by one site. In the field theory derivation we assumed a three sublattice ordered ground state, which is only suitable for spin models which are invariant under three site translation, hence R 3 Z 3 = I is a symmetry independently of the parameters of the field theory.
It is clear in any formulation that the real part of the action in Eq. (4.14) is invariant under Z 3 as long as the coupling g is the same for all three CP 2 theories. When θ 1 = −θ 3 = p2π/3, the topological term is also invariant. Indeed it transforms as Since Q 1 + Q 2 + Q 3 = 0 and Q 1 is integer-valued, the second term of the right hand side is 0, and the third term gives an integer multiple of 2π, leading to: Finally, the λ-term is clearly invariant under Z 3 symmetry as q 12 + q 23 + q 31 transforms into itself.

Parity symmetry
The action in Eq. (4.14) is invariant under parity symmetries as well, which correspond to mirror symmetries between two neighbouring sites in the spin model. Since a three site translation symmetry is always conserved, there are three non-equivalent mirror symmetries in the spin model, resulting in three non-equivalent parity symmetries in the σ-model. Take for example a mirror plane between two sites in sublattices 1 and 3 (see Fig. 4). The corresponding parity symmetry of the field theory transforms U matrices as In terms of the fields it corresponds to it exchanges fields 1 and 3. This means that in the real part of the action the terms for the fields 1 and 3 will be exchanged, while the terms for field 2 will be unchanged. This is clearly a symmetry of the action of Eq. (4.14). The topological term transforms as θ( , which is the same as the original one. The extra − sign appears as there is always exactly one spatial derivative in the expression of the topological charges (see Eq. (4.8)), and the parity transformation inverts the x coordinate. Similarly, the λ-term becomes −λ(q 21 + q 32 + q 13 ), which is equal to the original term as q mn = −q nm .
The invariance of the action in Eq. (4.14) under the other two parity symmetries R 12 and R 13 can be shown in a similar way as for R 13 .

Breaking lattice symmetries and general form of the action
If the spin model is not invariant under the translation and the mirror symmetries, the Z 3 and parity symmetries of the σ-model will be broken, and the action will take the form In general, the three copies of the CP 2 model do not have the same coupling constants and velocities any more. The topological angles can also take arbitrary values, but since the topological charges still satisfy Q 1 + Q 2 + Q 3 = 0 because of the orthogonality of the fields, one can still set one of them to 0, and one is left with two independent topological charges, for instance θ 1 and θ 3 . All these statements are illlustrated in the case where the nearest neighbor interaction takes three different values between each pair of sublattices discussed in Appendix E.3. We also show that assuming SU(3), gauge and time reversal invariance, Eq. (5.6) is indeed the most general form of the σ-model.
Here we present an intermediate situation where the translation symmetry is broken but one of the mirror symmetries is still present, say R 13 . This is achieved by weakening/strengthening the nearest neighbour bond between sublattices 1 and 3 as shown in Fig. 4. In this case the CP 2 theories of the fields φ 1 and φ 3 will have the same parameters, but not that describing φ 2 . The topological angles are not fixed to ±2pπ/3, but they still satisfy the additional constraint θ 1 = −θ 3 due to the R 13 symmetry, If δ < 0, the bond is weakened and θ 1 = −θ 3 < p2π/3, while if δ > 0, the bond is strengthened and θ 1 = −θ 3 > p2π/3. Since θ p2π/3, the topological term is no longer invariant under Z 3 , or under R 12 and R 23 .

Additional remarks about the field theory
Due to the gauge invariance discussed above, this nonlinear σ-model is defined on the man- , which is called a flag manifold. It was discussed earlier by Bykov [87] in the context of SU(n) chains with alternating representations on adjacent sites and by Ueda et al. [82] in the context of two dimensional SU(3) magnets at finite temperature.
We believe it is possible to factorize any SU(3) matrix in the form Here the T a are the 6 off-diagonal generators of SU(3) and the T α are the 2 diagonal generators. It would then follow, due to the gauge invariance discussed above, that the diagonal factor could be dropped, with the off-diagonal generators giving the SU(3)/[U(1) × U(1)] manifold. Related factorizations have been proven earlier [88][89][90]. We give a proof of this factorization to third order in the θ a 's and θ α 's in Appendix G. We have also checked numerically that for random unitary matrices a factorization according to Eq. (5.8) could always be found. Since π 2 SU(3)/[U(1)×U(1)] = Z×Z, there are 2 topological invariants, as discussed above. This field theory may be regarded as a natural generalization of the O(3) σ-model, which is defined on the manifold SU(2)/U(1) and has one topological invariant. An interesting difference is the presence of the "λ-term" discussed above, which respects all symmetries including Lorentz invariance. It has the very unusual property of being antisymmetric in space and time derivatives, and consequently imaginary for imaginary time, but not being a total derivative (topological invariant). This means that the coupling constant, λ, can renormalize perturbatively and could potentially have important effects on the behaviour of the model and therefore of the spin chains. We analyze it using the perturbative renormalization group in the next section.

Renormalization group analysis
We rewrite the matrices U in terms of the Gell-Mann matrices (GM) T A , as defined in Eq. (F.1). Due to the factorization property of SU(3) matrices (discussed in Sec. 5.7 and Appendix G), and the gauge symmetry (see Eq. (5.1)), we can replace U with e iθ a T a in Eq. (4.6). Here and throughout, lowercase Roman letters index the off-diagonal GM, lowercase Greek letters index the diagonal GM, uppercase Roman letters index all eight GM, and repeated indices are summed over. After rescaling θ → g 2 θ a , we find where R abc and P abcd are tensors of real coefficients that can be expressed in terms of the SU(3) structure factors. At this order in perturbation theory, the imaginary λ-term term has coupling constantλ = λg 3/2 . We will calculate the beta function ofλ, showing that it renormalizes to zero at large length scales. We'll also calculate the beta function of g, showing that it renormalizes to large values, so that our flag manifold σ-model is asymptotically free.
To perform the RG calculations, we rewrite the fields U in terms of 'slow' (U s ) and 'fast' These 'fast' fields have momentum modes restricted to a Wilson shell [bΛ, Λ), where Λ is a reduced cutoff, and b 1. The RG step of integrating over this shell is then equivalent to integrating out the fields U f from the theory. This factorization of U is motivated by Polyakov's work on the nonlinear-σ-model, which found the model's RG equations by a quadratic expansion of the 'fast fields' [45]. In Appendix F.3, we show the equivalence between Polyakov's decomposition and Eq. (6.2) for the SU(2) case. Now, we insert Eq. (6.2) into an equivalent form of the Lagrangian, derived in Appendix F.4: We've suppressed the topological term, as it does not contribute to the perturbative RG equations. The utility of this expression becomes apparent when we expand U f = e iθ a T a , since we know how to express products of T a and T γ in terms of the SU(3) structure constants f ABC (see Appendix F.2). Such an expansion again follows from the gauge symmetry (Eq. (5.1)) and the factorization of SU(3) matrices. Defining In Appendix F.5, we argue that the linear terms can be dropped in the Λ → 0 limit, which follows from the fact that such terms produce four-derivative operators after the integration over θ a . The second term in Eq. (6.3), after dropping linear terms, gives We now Fourier transform the sum of Eq. (6.4) and Eq. (6.5), and perform the Gaussian integral over θ a . Details of this step, which includes the application of numerous nontrivial identities involving the f ABC , are included in Appendix F.6. The result is Recalling the definitions of M A and N µ , we recognize that to our order of accuracy, L eff is of the same form as L, with U replaced by U s . This allows for the identification of the first two operators' factors with 1 8g eff and λ eff , respectively, leading to the following β functions: Therefore, g flows to large values at large length scales. Moreover, since the imaginary term in Eq. (6.1) flows to zero at large length scales.

General phase diagram
Based on the renormalization calculations we have argued that the λ-term renormalizes to zero. Therefore from now on we will consider the action of Eq. (4.14) with λ = 0. First we give an overview of the phase diagram based on considerations in the large g limit and on the correspondence with SU(3) spin models, then we present results of Monte Carlo simulations for finite couplings.

g → ∞ limit
In this limit the action consists only of a topological term. Following the argument of Seiberg [91] and Plefka and Samuel [92] for CP N−1 models we study the θ 1 − θ 3 phase diagram. As those authors discuss, a lattice version of the gauge field formulation of the action, Eq. (4.15), is most suitable for studying the strong coupling limit. We work on a square lattice and introduce a unitary matrix, U or equivalently 3 orthogonal vectors φ n on every lattice site. Gauge fields are introduced on the links of the lattice, giving the real part of the action [93][94][95]: where δ x , δ τ are unit vectors of the square lattice. V n ( r, r + δ µ ) is a complex number of unit modulus, and A n ( r, r + δ µ ) becomes the gauge field, A n µ ( r) in the continuum limit. The topological term is written in terms of the product of the V variables around a plaquette, namely The ln V n P contribution of each plaquette is restricted to lie in the range: and, as discussed in Sec. 4, the constraint: is imposed on every link. Defining the topological charge density this translates to 3 n=1 q n ( r) = 0 (∀ r). (7.8) This implies that the partition function is invariant under shifting all three topological angles by a common constant so, without loss of generality, we set θ 2 = 0. As discussed in [91,92], periodic boundary conditions implies that the total topological charges are integers: In the g → ∞ limit, the partition function becomes: Using the Fourier transform of the Dirac Comb, and V is the area (more precisely the number of plaquettes) of the system. In the thermodynamic limit (V → ∞), and for any value of the angles, the largest term in the sum will dominate it. Thus the free energy density is In Fig. 5a, we show the sectors R m,n where the z(θ 1 + 2πm, θ 3 + 2πn) term is the largest. Moving from one R m,n sector to an adjacent one, the free energy has a cusp, i.e. a phase transition takes place. The partition function and the free energy of Eqs. (7.12) and (7.14) are 2π periodic in both θ 1 and θ 3 , as they should be since the topological charges are integer valued so that a 2π shift in the topological angle leaves the path integral (or more specifically exp(−S )) invariant. Therefore 19 we only discuss the phase diagram for the 0 ≤ θ 1 , −θ 3 < 2π region, referred to as reduced phase diagram in the following.
There are three high symmetry points in the reduced phase diagram defined by θ 1 = −θ 3 = 2mπ/3(mod 2π) with m = 0, 1, 2. At these points the action has a Z 3 symmetry, which can be understood as a cyclic permutation of the three fields (see Sec. 5.4). Additionally, three parity symmetries are also present, each corresponding to the exchange of two out of the three fields (complemented by an invertion of the space coordinates as explained in Sec. 5.5). At each high symmetry point three special lines meet, each corresponding to the conservation of one of the three parity symmetries. Consider for example the θ 1 = −θ 3 = 2π/3 point, and the parity symmetric lines meeting there, illustrated on Figs. 5, 6. The θ 1 = −θ 3 line is invariant under the R 13 parity transformation defined in Sec. 5.5, which exchanges the fields φ 1 and φ 3 . On the other two lines (−θ 3 = π − θ 1 /2 and −θ 3 = 2π − 2θ 1 ) the parities exchanging fields Under the Z 3 transformation the high symmetry points are mapped into themselves modulo 2π as was discussed in Sec. 5.4. The parity symmetric lines meeting at a Z 3 symmetric point transform into each other modulo 2π. For example, along the θ 1 = −θ 3 line the topological term is given In the same spirit, one can follow the action of a parity transformation for a general point inside the reduced phase diagram: The high symmetry points are once again invariant, while each parity transformation conserves one parity symmetric line and maps the other two into each other. For example the θ 1 = −θ 3 line is invariant under R 13 , and maps to θ 3 = θ 1 /2 ≡ θ 1 /2 − π or to θ 3 = 2θ 1 ≡ 2θ 1 − 2π under R 23 or R 12 , respectively. All these transformation properties are illustrated in Fig. 6. In the g → ∞ limit the system is gapped for all values of θ 1 and θ 3 . This can be seen from the gauge field formulation of the action in Eq. (4.15) (or Eqs. (7.1) and (7.3) for the lattice model).
In the the g → ∞ limit the action only consists of the topological term, which only depends on the gauge fields, thus the φ n (x) fields are correlation free. This means that the mass gap, which is the inverse of the correlation length, diverges in this limit for all values of the topological angles.
At θ 1 = −θ 3 = 2π/3 this is accompanied by a spontaneous breakdown of the Z 3 and parity 21 symmetries. By calculating the expectation value of the topological charge densities [91] depending on from which sector we approach the high symmetry θ 1 = −θ 3 = 2π/3 point we get These three cases are connected by the Z 3 transformation. Also under each parity symmetry one scenario is invariant, while the other two transform into each other. The situation is similar at θ 1 = −θ 3 = 4π/3. Along the transition lines running from θ 1 = −θ 3 = 2π/3 to a point equivalent to θ 1 = −θ 3 = 4π/3, the free energy has a cusp and the remaining parity symmetry is also spontaneously broken. The transition across these lines is first order since the expectation values of the topological charge densities have a jump. By contrast, along the parity symmetric lines running from to θ 1 = −θ 3 = 2π/3 to a point equivalent to θ 1 = −θ 3 = 0, as well as at θ 1 = −θ 3 = 0 itself, all symmetries are preserved since the free energy is continuously differentiable inside the R mn sectors.

Finite g and connection with spin models
We believe that the structure of the phase diagram is the same for finite values of the coupling g as well, since the transitions coincide with high symmetry lines/points, although the nature of the transitions can change. According to the discussion of Sec. 5, the phase diagram of the σ-model can be illustrated by specific spin models. As mentioned, the Z 3 transformation corresponds to a translation by one site in the spin chain, while the parity symmetries correspond to the mirror symmetries between two neighbouring spins. The high symmetry points of the phase diagram of the σ-model correspond to translationally invariant spin models; the θ 1 = −θ 3 = 0 point corresponds to spin models with p = 3m boxes, while the θ 1 = −θ 3 = 2π/3 and θ 1 = −θ 3 = 4π/3 points describe the low energy behavior of spin chains with p = 3m + 1 and p = 3m + 2, respectively.
First, we focus on the neighbourhood of the θ 1 = −θ 3 = 2π/3 point, which can be illustrated by p = 1 spin models. From Bethe ansatz results we know that the p = 1 nearest neighbour Heisenberg model is gapless. However, Corboz et al. [96] showed that a transition to a trimerized phase occurs in the J 1 − J 2 model. This transition from a gapless to the trimerized phase suggests that a phase transition takes place for the σ-model at θ 1 = −θ 3 = ±2π/3 at some critical coupling g c . Now the expression for the coupling constant derived in Sec. 4 . So we expect that for g < g c the system is gapless, while for g > g c it is gapped with a spontaneous breakdown of the Z 3 symmetry, in agreement with the results of the previous section in the g → ∞ limit.
If the translational symmetry is explicitly broken, but a mirror symmetry is preserved in the nearest neighbour Heisenberg system (g < g c ), the corresponding σ-model is along one of the parity symmetric lines. For example, changing the strength of the bonds between sublattices 1 and 3, the σ-model is tuned along the θ 1 = −θ 3 line (see Sec. 5.6). If the bond is weakened, θ 1 = −θ 3 < 2π/3, and we move towards the θ 1 = −θ 3 = 0 point. This corresponds to a system with ...WSSWSS... bonds, where W stands for weaker and S for stronger. In this case SU(3) singlets form on the sites connected by stronger bonds, and the system is gapped with a unique ground state. By contrast, if every third bond is strengthened (...SWWSWW...), pairs of sites connected by strong bonds will tend to form3 states and the system behaves as a 33 chain, which spontaneously dimerizes with a finite gap [97,98]. This corresponds to a σ-model with θ 1 = −θ 3 > 2π/3, i.e. along a transition line between sectors running from θ 1 = −θ 3 = 2π/3 to θ 1 = −θ 3 = 4π/3. If a different mirror symmetry is conserved in the spin model, the underlying σ-model moves along a different parity symmetric line. If both the translational and mirror symmetries are explicitly broken, the spin system is gapped, and therefore away from the symmetric lines the σ-model is also expected to be gapped.
Considering the p = 1, J 1 − J 2 spin model with spontaneous translational symmetry breaking [96] (i.e. g > g c ), if every third nearest neighbour bond is weakened (strengthened), one (two) out of the three ground states will be selected, and in both cases the system remains gapped. This, once again, corresponds to moving along one of the parity symmetric lines around θ 1 = −θ 3 = 2π/3 in the σ-model phase diagram. As a consequence we can expect that along the transition lines between different R sectors the system is gapped and twofold degenerate, while inside the sectors the system is gapped for any of value of g.
In terms of the σ-model the neighbourhood of the θ 1 = −θ 3 = 4π/3 point is connected to that of the θ 1 = −θ 3 = 2π/3 case by a complex conjugation, therefore the above considerations translate straightforwardly to the θ 1 = −θ 3 = 4π/3 case as well. This would correspond to spin systems with p = 3m + 2, which thus would show similar general behavior as p = 3m + 1 systems. In the case of θ 1 = −θ 3 = 0, i.e. for spin systems with p = 3m, explicitly breaking the translational or mirror symmetries in the spin chain will tune the σ-model away from the θ 1 = −θ 3 = 0 point, but it will still stay in the same phase. Thus, we expect that the σ-model is gapped for any value of the coupling inside the R sectors away from transition lines, even for θ 1 = −θ 3 = 0.

Monte Carlo simulations
To confirm our predictions for finite g values, we turn to classical Monte Carlo simulations. In the SU(2) case, several methods have been developed to address the issue raised by the imaginary topological term [16,99]. Here we choose the extrapolation scheme of Allés and Papa [18], which consists in carrying out simulations for imaginary topological angles θ = iϑ, and in extrapolating those results to real angles. This extrapolation method can be illustrated in the g → ∞ limit. Using the results of Sec. 7.1, the free energy for imaginary angles is given by 23 In this case the m = n = 0 term will always dominate. Therefore In Fig. 7 we show the free energy f (aθ, bθ, g → ∞) as a function of θ 2 for different fixed values of a, b. Since we know that the system undergoes a phase transition between different R m,n sectors, we can only hope to get information for points inside or at the boundary of R 0,0 . Fortunately due to the symmetries of the phase diagram, this is all that we need. For example, the high symmetry point To implement the lattice Monte Carlo calculation, we discretize the action in Eq. (4.14) on a 1+1 dimensional square lattice for imaginary topological angles θ = iϑ, extending the scheme of Allés and Papa [18] to the SU(3) case. We discretize the real part of the action as [94,95] 3 n=1 µ=x,τ where r j is a site on the discretized two dimensional space time and δ x , δ τ are the lattice unit vectors of the discretized square lattice. One can show that this discretization gives back the 24 continuum case in second order. The topological part of the action is discretized following the recipe of Berg and Lüscher [100]. Every square plaquette is further split into two triangles, and three topological charge densities are defined on each triangle, as shown in Fig. 8, by With this notation, the topological charge on the lattice system is calculated as Q n = △ i jk q n (△ i jk ), where on each triangle the indices i, j, k follow the same (e.g. counter-clockwise) direction. Each q n (△) can take values between ±1/2. We note that this scheme is different from that of Seiberg [91] and Plefka and Samuel [92] that we used to discuss the g → ∞ limit, but they give the same continuum limit (see Appendix D). We implemented a single site Metropolis type Monte Carlo algorithm [18,21,100,101] to measure the correlations of the discretized model. In this method we sweep the lattice and on each site we replace the U matrix with a uniformly generated [81,102] one with a probability given by the Metropolis acceptance ratio. To reduce autocorrelation times and increase accuracy we also complement the single site update with a multigrid update method [103], where we rotate all matrices within L B × L B square blocks of increasing size. The rotation is done by selecting a Gell-Mann generator T A and by updating the matrices in the block according to and extract the correlation length by an exponential fitting. C(x) is the generalization of the spin-spin correlation function to the SU(n) case (see Appendix C for details). For each value of θ = iϑ and g we sampled 2 × 10 5 configurations with a sampling distance of 10 multigrid sweeps after 5 × 10 4 thermalizing multigrid sweeps. The numerical errors were estimated by the binning method [104]. In Fig. 9 we show the inverse of the correlation length -which is proportional to the mass gap -as a function of θ 2 for the extrapolation along the θ 1 = −θ 3 line. Following Allés and Papa [18], we extrapolate the mass gap to real θ values by fitting it with a function of the form (c 1 + c 2 θ 2 )/(1 + c 3 θ 2 ). We find that upon increasing ϑ the mass gap has a change of behaviour. For small imaginary angles, the mass gap increases, i.e. the correlation length decreases. But, after a maximum in 1/ξ, the mass gap decreases. In this region the system is characterized by a uniform saturated topological charge density. This is reminiscent of the results of Imachi et al. [99] for the CP 2 model, who argue that beyond some imaginary angle where the average topological charge of the system becomes comparable to its maximal value (1/2 per triangle), the partition function is no longer an analytic continuation of the real θ case. This means that in the extrapolation method one should only consider imaginary angles smaller than this threshold. This is why we chose as a limit for the fitting the inflection points in the mass gap results.
The Monte Carlo results clearly show that the mass gap is finite for θ 1 = −θ 3 = 0. This point corresponds to spin systems with p = 3m boxes. This finding agrees with the proposal of Greiter et al. [44] for the possibility of gapped phases for such systems. However, for small g, the extrapolated mass gap vanishes around θ = 2π/3. So our Monte Carlo results predict that spin representations with p = 3m + 1, or 3m + 2 are gapless. Upon increasing g, beyond g c ≈ 2.55, the extrapolated mass gap remains finite even at θ = 2π/3. We believe that this corresponds to a phase transition from a gapless into a gapped phase in accordance with the LSMA theorem. This also agrees with our earlier conclusion that for g → ∞ the system should be gapped.

Phase diagram of the flag manifold σ-model
Our expected θ 1 − θ 3 phase diagram for the flag manifold σ-model is summarized in Fig. 5b, which must be complemented with an RG flow towards g → ∞. A cut along the θ 1 = −θ 3 = θ line is shown in Fig. 1b with the topological angle on the vertical axis and the coupling constant g on the horizontal axis. We ignore the coupling constant λ for the non-topological term linear in space and time derivatives since we have shown that it renormalizes to zero. For generic values of θ 1 and θ 3 , we expect the model to be gapped, with g renormalizing to infinity. But for θ 1 = −θ 3 = ±2π/3 we expect a critical point to occur at g = g c , corresponding to the SU(3) 1 WZW model. It is important that the model has an extra symmetry at these values of θ 1 , θ 3 which prevents it from renormalizing. This is the Z 3 symmetry which permutes the three fields For θ 1 = −θ 3 = ±2π/3, the phase transition at g = g c is driven by the topological term λ ′ J A R J A L . In this term, J R/L are the right and left moving currents in the WZW model given by: where ∂ ∓ ≡ ∂ t ∓ ∂ x and the T A are the generators of SU(3). This interaction is marginally irrelevant for one sign of λ ′ and marginally relevant for the other. Thus, we expect λ ′ to change sign at g = g c and for g > g c the gap to turn on exponentially slowly, as ∆ ∝ e −c/(g c −g) . 26 On the other hand, shifting θ 1 , θ 3 slightly away from ±2π/3 corresponds to breaking the Z 3 symmetry. We expect this symmetry to correspond to g → e i2π/3 g in the SU(3) 1 WZW model, the symmetry which forbids a trg term in the effective Hamiltonian. When this symmetry is broken we expect a relevant perturbation ∝ trg. This operator has dimension d = 2/3 so we expect the gap to scale as |θ − 2π/3| 1/(2−d) = |θ − 2π/3| 3/4 , up to log corrections coming from the marginal operator J A R J A L . If the Z 3 symmmetry is broken, but a parity symmetry is preserved, along the θ 1 = −θ 3 = θ line for example, we believe that the extra term should have the form trg + trg † since g → g † corresponds to the parity transformation. In this case the extra term in the SU(3) 1 WZW model should have the form ∝ (θ − 2π/3)(trg + trg † ). If we write the diagonal elements of g as e iα j for α = 1, 2, 3 with j α j ≡ 0(mod 2π), the extra term takes the form For θ < 2π/3, this term has a unique minimum with α j = 0. But if θ > 2π/3, there are two minima, with α j = 2π/3 or α j = −2π/3 . As discussed in Sec. 7.1, the two cases correspond 27 to spin chains with ...SSWSSW... and ...WWSWWS... bond patterns, with a unique and gapped ground state in the former, and gapped twofold degenerate ground states in the latter case. The situation is similar along the other two parity conserving lines (−θ 3 = π − θ 1 /2 and −θ 3 = 2π − 2θ 1 ), which are connected to the θ 1 = −θ 3 line by the Z 3 transformation. Therefore along these lines the extra term in the WZW model should be ∝ (e ±i2π/3 trg + e ∓i2π/3 trg † ), which will also have 1 or 2 minima depending on the sign. For a generic point around θ 1 = −θ 3 = 2π/3 the extra term should have the form (µtrg + µ * trg † ), where µ is complex in general, and vanishes at the Z 3 symmetric point. Based on the above considerations we believe the general form is µ = exp(iθ 1 ) + exp(iθ 2 ) + exp(iθ 3 ), where in our discussion we fixed θ 2 = 0. It is interesting to contrast these results with those of the CP 2 model with a topological term. If we ignore the non-topological term whose coupling constant λ renormalizes to zero, the flag manifold σ-model we have studied can be seen as three copies of the CP 2 model coupled by the orthogonality constraint. Now, it is well established that, as soon as n > 2, the CP n−1 model with a topological term is gapped for all values of the coupling constant and of the topological angle [91,105,106]. The model undergoes a phase transition with spontaneous breaking of charge conjugation symmetry at θ = π, but it is first order, and the gap does not close. So the coupling between the fields appears to be essential to produce the interferences that close the gap at θ = 2π/3.
We also carried out shorter MC simulations (with 5 × 10 4 samples) for the CP 2 model, shown in Fig. 11. The correlation lengths are extracted the same way as for the SU (3) case. The extrapolation clearly shows that the gap remains open even for θ = π.

Conclusions and open questions
Let us first summarize the implications of our field theory results for the SU(3) chains in the p-box symmetric representation. For a translationally invariant system, the model can be defined by a single topological angle θ = 2πp/3. When p = 3m, θ is a multiple of 2π, so there is no topological term in the action, and the model is expected to be gapped whatever the coupling constant. This prediction is the generalization of Haldane's prediction of a gap for integer spin in the SU(2) case. When p = 3m ± 1, there is a nontrivial topological term in the action with topological angle θ = ±2π/3. In that case, we have shown that there is a critical coupling constant g c below which the model is gapless in the SU(3) 1 WZW universality class. Starting from the original SU(3) lattice model, the coupling constant is given by g , where the additional interactions J 2 and J 3 are effective couplings that have been included to account for the effect of the zero point fluctuations, which select the three sublattice order. In the large p limit, these additional exchange integrals J 2 , J 3 ∝ 1/p, and the coupling constant g ∝ 1/ √ p becomes very small. So, our field theory results predict that for p = 3m ± 1 and large m, the model of Eq. 2.1 should be gapless. Since this is known to be true also for p = 1 from Sutherland's Bethe ansatz solution, we conjecture that the same conclusion holds for all p not multiple of 3. The field theory also predicts that, if the topological angle θ = ±2π/3, i.e. for p = 3m ± 1, there should be a phase transition into a gapped phase upon increasing the coupling constant g beyond a critical value g c . According to the Lieb-Schulz-Mattis-Affleck theorem [54,55], this phase should be at least three-fold degenerate. So we expect it to be spontaneously trimerized, with a gap turning on exponentially slowly. The form of the coupling constant g = suggests that this transition can be induced by increasing the next nearest neighbour coupling J 2 . This prediction is consistent with the spontaneous trimerization reported for the SU(3) chain with p = 1 and next nearest neighbour interactions [96].
By contrast, the gapped phase that occurs for g < g c when θ is shifted from ±2π/3 should arise from breaking the translation symmetry by hand, with a period 3 exchange term. We expect that the coefficient of the periodic exchange coupling λ 3 will be proportional to |θ ∓ 2π/3|, and accordingly that the gap will scale as λ 3/4 3 . Let us now briefly review the issues that deserve further investigation. The first one is about the additional couplings J 2 and J 3 that we have introduced to get rid of the zero modes. To introduce additional interactions to mimic the effect of zero point fluctuations is a standard approach in 2D and 3D frustrated magnets when the classical ground state is infinitely degenerate. In particular, if the system develops long-range magnetic order because of zero point fluctuations, an effect known as "order by disorder" [107], introducing additional effective couplings is necessary to restore the appropriate structure of Goldstone modes. These couplings must come form higher order spin wave interactions, but even in the simpler context of SU(2) models, there is no known systematic way to calculate the 1/s expansion of these couplings. So how to actually calculate these couplings in the context of the present paper remains an open question.
The other open issue concerns the determination of the critical exponents of the field theory at θ = 2π/3. For the CP 1 model, the critical exponents of the mass gap (main term and logarithmic correction) have been determined on the basis of the numerical results obtained for imaginary topological angles [17,21]. This approach relies on a change of variable that is motivated by exact results obtained on the Ising model in imaginary magnetic field [108]. Whether a similar approach with an appropriate change of variable can be used in the present case to extract the exponents from the imaginary topological angle data remains to be seen.
The numerical check of our predictions for p > 1 in the context of the lattice SU(3) model is a real challenge for all types of simulations: Quantum Monte Carlo simulations suffer from a severe minus sign problem, exact diagonalizations are limited to very small system sizes because the local Hilbert space grows very fast, and DMRG simulations require to keep a huge number of states to reach convergence. As a consequence, the results obtained so far are not conclusive, and the nature of the ground state and of the low-energy spectrum is an open numerical issue as soon as the number of boxes p is larger than one. Let us briefly review the current status of the numerical investigation of these models to substantiate this conclusion.
The first numerical study of the SU(3) Heisenberg chain with p = 2 and 3 is a DMRG investigation by Rachel et al. [109]. In this paper, the authors report on a calculation of the entanglement spectrum of a chain of 48 sites with periodic boundary conditions, keeping 5000 states for p = 2 and 1650 states for p = 3. For p = 2, fitting the curve with the Calabrese-Cardy formula [110], they found a central charge equal to 2.48, which they interpret as being 30 evidence that the model is in the SU(3) 1 universality class with central charge 2, the difference being attributed to logarithmic corrections. For p = 3, they found that the entanglement entropy saturates after a few sites, which was taken as an indication that the system is gapped. The second numerical study is an exact diagonalization investigation by Nataf and Mila [65] based on a new approach that allows one to work directly in the irreducible representations of the global SU(n) symmetry [111]. This has allowed one to investigate the properties of the SU(3) model with p = 2 up to 15 sites and with p = 3 up to 12 sites with periodic boundary conditions, and to extract the properties of the model using standard finite-size analysis. Quite surprisingly, the results turn out to be consistent with a gapless spectrum in both cases, with a central charge c = 3.23 for p = 2, in good agreement with the SU(3) 2 universality class (central charge 3.2), and with a central charge c = 4.09 for p = 3, in good agreement with the SU(3) 3 universality class (central charge 4). These results have led the authors of Ref. [65] to suggest that there is a length scale in the problem beyond which the system might cross-over to another universality class or to a gapped behaviour, and below which the system looks SU(3) p .
At first sight, these conclusions might look in agreement with the conclusions of Rachel et al. [109], who worked on much larger systems (48 sites), but they are not. In particular, the saturation of the entanglement entropy of the p = 3 case reported in Ref. [109] points to a correlation length of the order of 6 lattice sites, as for the spin-1 SU(2) chain, a result clearly incompatible with the ED results of Ref. [65], where no sign of gap (i.e. no curvature of the finite-size gap) could be detected up to 12 sites, in sharp contrast with the spin-1 SU(2) chain.
According to recent DMRG results, the problem comes from the number of states kept in Ref. [109], which was too small to reach convergence for the entanglement entropy. Using codes where the SU(n) symmetry is fully implemented, Weichselbaum et al. [66] and Nataf et al. [112] have been able to keep a much larger number of states. This has allowed them to reach convergence on very large systems with open boundary conditions, with conclusions that differ significantly from those of Ref. [109]. For p = 2, and up to at least 300 sites, the central charge is larger than 3, in agreement with ED and with the SU(3) 2 universality class [66,112]. For p = 3, the entanglement entropy does not saturate but is compatible with a central charge larger than 4 up to 120 sites, and larger systems are currently under investigation to see if the presence of a gap can be detected [112].

Appendix A. Flavour wave theory
The calculation of the harmonic fluctuations around an ordered state is most easily done using linear flavour wave theory, the extension of the usual SU(2) spin wave theory to SU(n) models. It has been formulated in Refs. [69] and [70] for the SU(3) case and in Ref. [71] for the SU(4) case. The notations used in this appendix are those of Ref. [114], where the triangular and square lattices are treated.
To treat fluctuations around a three sublattice ordered state where the spins on the sites l α belonging to sublattice L α point in the direction α, we start from the p → ∞ limit in which the bosons α have condensed at the sites of sublattice L α , and we do a 1/p expansion, by analogy with the spin wave theory that is a 1/s expansion for SU(2) systems. Starting from the ordered state we can use the following expansion for the spin operators S β γ for sites l α ∈ L α in the large-p limit: where we have introduced the shorthand notation The b (α) † β (l α ) operators with β α now correspond to the Holstein-Primakoff bosons on sublattice L α , and the superscript (α) keeps track of the sublattice. Expanding in 1/p and keeping the quadratic terms only, the exchange term between sites l α ∈ L α and m α ′ ∈ L α ′ is given by for α α ′ , and by for α = α ′ . Note that when α α ′ the exchange term does not involve bosons with flavours different from the ordered ones α and α ′ . Assuming a three sublattice ordered state, we further define the following Fourier transformation: where the summation is over the |L| /3 sites of sublattice L α (|L| is the number of lattice sites). The Hamiltonian involving bosons with subscripts and superscripts α and β then reads where the factors γ 1 k and γ 2 k are given by The full Hamiltonian is H = H AB + H BC + H CA . It can be diagonalized via a Bogoliubov transformation, leading to The dispersion of the flavour waves is given by There are 6 degenerate branches in the reduced Brillouin zone [−π/3, π/3], and 6 Goldstone modes.

Appendix B. Details of Lieb-Schultz-Mattis-Affleck theorem
To show that U|ψ has low energy, consider the term in H involving the two neighbouring sites, j and j + 1: Since [Q j + Q j+1 , H j, j+1 ] = 0, and the other Q i 's commute with H j, j+1 , Summing over all terms in H, The second term on the right hand side vanishes, since |ψ is an eigenstate of H. The remainder is seen to be O(1/L) because there are O(L) terms in H and each contributes a term ∝ 1/L 2 as follows from Eq. (B.2). Now we wish to prove that U|ψ is orthogonal to |ψ . To do this, we can assume that |ψ is translationally invariant, T |ψ = |ψ , where T generates translations by 1 site.
Using periodic boundary conditions, Q L+1 = Q 1 , we obtain: Next, we use the fact that ( j Q j )|ψ = 0, which follows from the SU(3) invariance of the ground state, so Finally we use Appendix C. Spin coherent state path integral of the SU(2) Heisenberg chain Typically, when discussing Haldane's conjecture a derivation of the O(3) nonlinear σ-model from the Heisenberg model is presented (see for example Refs. [115,116]). To make the connection with the calculations of Sec. 4 more explicit, we provide here a derivation of the CP 1 nonlinear σ-model directly starting from the SU(2) nearest neighbour Heisenberg model. We also show the equivalence between the CP 1

and O(3) formulations.
We start from the nearest neighbour antiferromagnetic Heisenberg model In a Schwinger boson representation, b † 1 (b † 2 ) corresponds to creating an ↑(↓) spin, respectively. The spin operators can be written as where σ = (σ x , σ y , σ z ) is a vector of the Pauli matrices. In this representation, the Heisenberg interaction is given by To write down the imaginary time partition function, we use a spin coherent state path integral approach. In terms of Schwinger bosons, the SU(2) spin coherent states can be written as where Φ is a two component complex unit vector which is usually parametrized in terms of two angles as Φ = (cos(θ/2), sin(θ/2)e iϕ ), with 0 ≤ θ ≤ π and 0 ≤ ϕ ≤ 2π and with the integral measure dΩ Φ = sin(θ)/(4π)dθdϕ. The partition function in the path integral language is formally the same as in Eq. (D.10) where |{ Φ(·, τ)} represents a direct product of spin coherent states of each spin at time τ. When calculating the expectation value of the Hamiltonian the spin operators can be replaced by classical three dimensional real vectors where n has unit length and The Heisenberg interaction in the path integral is given by In the classical ground state, the n vectors on neighbouring sites should be antiparallel, or, equivalently, the Φ vectors should be orthogonal. In the path integral, we consider fluctuations around the classical ground state manifold, which are conventionally parametrized as where the index j runs through the two site unit cells, the vector m( j) (with | m j | = 1) describes the local staggered magnetization, and the vector l( j), which is perpendicular to m( j) describes the local uniform magnetization. Equivalently, the complex vectors Φ can be parametrized as where, similarly to the SU(3) case, the unitary U( j) matrix describes the slow joint rotation, while the L( j) matrix parametrizes the non-orthogonality of spin states inside the unit cell. The form of the resulting spin matrices S (αβ) = Φ|Ŝ α β | Φ = 2s Φ β * Φ α read as where we omitted the ( j) argument on the right hand side. The two formulations are equivalent up to O(a 2 /s) (higher order terms are neglected in the path integral anyway). From Eq. (C.8), by using the completeness property of the Pauli matrices (i.e. σ αβ µ σ γδ µ = 2δ αδ δ βγ − δ αβ δ γδ ) and setting L 12 ( j) and l j to 0 we have where The connection between L( j) and l( j) is given by This implies in particular that 4|L 12 ( j)| 2 = | l( j)| 2 . Furthermore the orthogonality of m( j) and l( j) is equivalent to the orthogonality of the 0th and 1st order terms in Eq. (C.13) Also the O(a 2 /s) term in Eq. (C.13) is a normalization similarly to the 1 − a 2 s 2 in Eq. (C.11) in the vector formalism, which guarantees that the spin matrices has the correct Casimir, S αβ S βα = 4s 2 .
In the following we use the matrix formulation of Eq. (C.12) to write the action. The terms in the partition function can be expanded up to O(a 2 /s 0 ) corrections as On the right hand sides every quantity is taken at unit cell j, but we omitted the indices for compactness. Once again, tr(U * ∂ τ U T ) = 0 since U is unitary and its row vectors are orthogonal. After taking the continuum limit and replacing the sum with an integral (2a j = dx), the full action becomes (C. 19) At this point the L 12 fluctuations can be integrated out: (C.20) The resulting action only contains the U matrices and reads where 2/g = s, v = 2J 1 a, θ = 2πs (we introduced g this way to follow the convention). The θ-term is topological, with an integer topological charge The second term in the second equation vanishes since the expression in the trace is symmetric under exchanging µ and ν. Comparing this result to the action of the SU(3) case given in Eq.
(4.6), we see that there is no non-topological imaginary term. The term similar to q 12 defined in Eq. (4.10) is equivalent to the topological charge in the SU(2) case.
Writing U in terms of two orthogonal row vectors leads to the action This is just a CP 1 theory with a topological term [46]. This theory is equivalent to the O(3) σ-model [115,116] Indeed, using identities of the Pauli matrices (tr(σ k σ l ) = 2δ kl and tr(σ k σ l σ m ) = 2i ε klm ) and Eq. (C.14) we can write (C. 27) In the σ-model of Eqs. (C.24) or (C.25), the topological term is written iθQ 1 . For translationally invariant spin models, θ = 2πs, which distinguishes integer and half integer spin models. For integer spins the topological term is trivial and the σ-model is gapped [116], while for half integer spins θ ≡ π (mod 2π). As discussed in Sec. 1, a phase transition is expected at θ = π for some critical coupling g c . For g < g c the system is expected to be gapless, while for g > g c the system is gapped with a spontaneous breakdown of a Z 2 symmetry. This transition corresponds to the gapless-dimerized transition in the J 1 − J 2 , s = 1/2 spin chain, [51,52]. Using the extrapolation method of Allés and Papa [18] this transition can be observed in the CP 1 σ-model as 38 well. Fig. C.12 shows our Monte Carlo results for the mass gap at different couplings. The mass gap is extracted from the spin-spin correlation function, (C.28) The equivalence of the above forms can be shown using Eq. (C.14) and the identity σ αβ i σ βα j = 2δ i, j . Notice that the correlation function used in the SU(3) case (see Eq. (7.23)) is a direct generalization of this spin-spin correlation function. The gap opening can be seen around g c ≈ 0.7. For g < g c , the system is gapless at θ = π, as reported in Ref. [18], where calculations for 0.57 < g < 0.67 have been carried out, while for g > g c the system is gapped. The spontaneous breakdown of Z 2 was explicitly shown in the g → ∞ limit in Refs. [91,92]. The Monte Carlo results also confirm that the gap remains finite away for θ π for all values of g.

Appendix D. Supplementary calculations for the derivation of the SU(3)/[U(1) × U(1)] σmodel.
We consider an SU(3) spin chain with antiferromagnetic nearest and next nearest, and ferromagnetic third nearest Heisenberg interactions as defined in Eq. (4.1) We use a spin coherent state path integral approach [78][79][80][81] to calculate the imaginary time partition function of the model in Eq. (4.1). In the bosonic picture, spin coherent states in the p-box representation can be written as ) is a three component complex unit vector (i.e. | Φ| 2 = 1), while b † µ is the creation operator for a boson of flavour µ [81,102]. These spin coherent states are normalized, and form a complete set over the fully symmetric p-box representation: The overlap between two spin coherent states is given by Parametrizing Φ as Φ = (sin θ cos ϕ e iα 1 , sin θ sin ϕ e iα 2 , cos θe iα 3 ) (D.5) with 0 ≤ θ, φ ≤ π/2, 0 ≤ α 1 , α 2 , α 3 ≤ 2π, the Haar-measure dΩ Φ can be defined by Using the Lie-Trotter decomposition of the imaginary time partition function, one can insert a complete set of spin coherent states at each time slice for each spin where dt = β/N and is the direct product of spin coherent states at each site of the chain at time step n (note that ,n) is the integral over the spin coherent states at each site and each time step. Up to first order in dt, where δ Φ( j, n) = Φ( j, n + 1) − Φ( j, n), and we used the fact that δ Φ * Φ + Φ * δ Φ = δ| Φ| 2 = 0. Thus, as N → ∞ the path integral will take the form with τ ≈ ndt. The second part of the action is imaginary and describes the Berry phase for each spin coming from the overlap of spin coherent states of different time slices. Upon calculating the expectation of the Hamiltonian the spin operators are replaced by classical matrices as Using the parametrization of Eq. (4.3) the spin matrices S n ( j) βα = Φ(3 j + n)|S α β | Φ(3 j + n) take the form where the Λ n matrices are defined by This form of the spin matrices can be compared to the SU(2) case discussed in Appendix C. The leading term P n (x, τ) = U † (x, τ)Λ n U(x, τ) corresponds to the staggered magnetization m in the SU(2) case (see Eq. (C.14)). The first order term describes fluctuations orthogonal to the leading term. For example, while the O(a 2 /p) terms can be thought of as a normalization which guarantees that the Casimirs of the spin matrices are correct: S αβ S βα = p 2 + O(a 4 /p 2 ) and S αβ S βγ S γα = p 3 + O(a 4 /p). Note that we use a non-traceless definition of the spin operators, therefore the Casimirs are different from those of the traceless convention. The action of Eq. (4.2) can be rewritten in terms of the three fields inside a unit cell as where the sum over j runs through the three site unit cells. In the continuum limit (x ≈ 3a j) the sum can be replaced by an integral 3a j → dx and the fields/matrices on neighbouring unit cells can be expanded as . Here we show an example of how each type of term reads after inserting the parametrization of Eq. (4.3) into the action: The other terms can be expanded similarly. Up to O(a 2 /p) corrections, the Berry phase term is given by (D.17) The first term on the right hand side is identically 0 since U is a unitary matrix. Indeed, writing the rows of U as φ 1,0 , φ 2,0 and φ 3,0 = φ * 1,0 × φ * 2,0 , then expanding the trace we get Using the following simple identities: and rewriting the J 3 terms as the action has the following form up to O(a 2 ) and O(a/p) corrections where we omitted the (x, τ) variables of the U and L matrices for the sake of compactness. At this point the L matrices can be integrated out in Eq. (D.21) using the Gaussian identity For example, integrating out (L 12 , L * 12 ) leads to After carrying out the integrals in the L variables, we arrive at the action presented in Eq. (4.6) where the three topological charges defined in Eq. (4.8) are not independent, namely Q 1 + Q 2 + Q 3 = 0. This can be also seen from where the tr ∂ µ UU † is identically 0 as was shown in Eq. (D.18). Note that not only the sum of the topological charges is zero, but the sum of the charge densities also vanishes. The equivalence of the forms in Eq. (4.6) and Eq. (4.14) can be proven by rewriting the real part as where we used the fact that the three fields φ 1 , φ 2 , φ 3 form an orthonormal basis on C 3 . Another equivalent formulation of the action in Eqs. (4.6) and (4.14) can be given using the P n (x, τ) = U † (x, τ)Λ n U(x, τ) matrices introduced below Eq. (D.12) [100,117]: tr P n ∂ µ P n+1 ∂ ν P n+1 .
(D. 26) The equivalence between Eq. (4.14) (with λ = 0) and Eq. (4.15) follows straightforwardly by integrating out the gauge fields. Only considering the real term this follows since Upon shifting A n µ by i φ * n · ∂ µ φ n , the A n µ integral gives a constant factor. Including the topological terms, the action becomes, as written in Eq. (4.15) Once again, shifting A n µ , as above, the imaginary term gives: The second term in Eq. (D.29) gives the topological term Q n , while integrating over the terms quadratic or linear in A n µ simply gives a constant, as they are decoupled from the φ fields after the shift.
The constraint This follows because the terms in the real part of the action involving A 1 µ and A 3 µ become: (D.32) Again, the integrals over A 1 µ and A 3 µ simply give constant factors after shifting them by i φ * 1 · ∂ µ φ 1 and i φ * 3 · ∂ µ φ 3 respectively. This continues to work when the topological terms are included. Shifting the A µ now gives the real part of the action written in terms of the φ n together with i(θ 1 − θ 2 )Q 1 + i(θ 3 − θ 2 )Q 3 . We see that the partition function is invariant under shifting all three topological angles by the same constant.
Appendix E. Additional calculations for Sec. 5 Appendix E. 1

. Gauge invariance
As mentioned in the main text, the overall phases of the spin coherent states shouldn't change the form of the action. This manifests in the gauge invariance of the action in Eq. (4.6) under the τ)). In terms of the fields, this transformation corresponds to φ ′ n = e iϑ n φ n .
To check the invariance of the real part of Eq. (4.14) we use Eq. (D.25) to rewrite it as Now, the terms on the right hand side transform as Since φ n and φ n±1 are orthogonal the second term on the right hand side gives 0. So these terms are invariant under gauge transformation. The λ-term can be similarly shown to be invariant since the phase factors cancel out. Finally, the topological charges transform as The second term on the right hand side is 0 since (∂ µ ϑ n ∂ ν ϑ n ) is symmetric under exchanging µ and ν. Also, noticing that ∂ µ φ n · φ * n + φ n · ∂ µ φ * n = ∂ µ ( φ n · φ * n ) = 0, one can easily show that the third and fourth terms cancel each other, thus proving that the topological term is also invariant under the gauge transformation.
where the coupling constants and velocities are given by , v n−1,n = 3ap J (n−1,n) (E.6) Still using the convention θ 2 = 0, the topological angles become while the λ coefficient takes the value The action written in terms of the φ fields still has the form of three copies of CP 2 , as in Eq. (4.14), but this time the three theories have different parameters: θ n Q n + iλ(q 12 + q 23 + q 31 ) . (E.9) The coupling constants and velocities in this language are related to those in Eq. (E.5) by the following equations: v n g n = v n−1,n g n−1,n + v n,n+1 g n,n+1 − v n+1,n−1 g n+1,n−1 1 v n g n = 1 v n−1,n g n−1,n + 1 v n,n+1 g n,n+1 − 1 v n+1,n−1 g n+1,n−1 . (E.10) Note that, since there are multiple velocities in the field theory, one can no longer set all three of them to 1 by rescaling the space and time directions. In particular, this simple calculation shows that if all symmetries are broken, the topological angles θ 1 and θ 3 can take arbitrary values.
In the special case discussed in Sec. 5.6 the translational symmetry is broken, but one of the mirror symmetries is conserved. As shown in Fig. 4, the nearest neighbour couplings take the values J (1,2) (E.11) Based on Eq. (E.10), the parameters of the three CP 2 theories become while g 2 and v 2 will be different: . (E.13) Since g 1 = g 3 (and v 1 = v 3 ), the R 13 parity symmetry is conserved (corresponding to the remaining mirror symmetry), but the Z 3 and the other two parity symmetries are explicitly broken since g 2 takes a different value.
In that case, the topological term is given by (E.14) So θ 1 = −θ 3 . As a consequence, R 13 is still a symmetry. However, the topological angle deviates from p2π/3. If δ < 0, the bond is weakened and θ 1 = −θ 3 < p2π/3, while if δ > 0, the bond is strengthened and θ 1 = −θ 3 > p2π/3. Since θ p2π/3, the topological term is no longer invariant under Z 3 , or under R 12 and R 23 . If instead of R 13 , R 12 or R 23 was conserved, the topological term would become iθ 3 , respectively. Note that if two parity symmetries are conserved, then the Z 3 and the third parity are conserved as well. This is easy to understand in terms of the symmetries of the spin model: if two different mirror symmetries are conserved, then the translation symmetry is also necessarily present 4 . And vice versa, if Z 3 symmetry is present, it necessarily means that the action is invariant under all three parities. In terms of the spin model, the translation symmetry implies the mirror symmetries as well. This is due to the form of the Heisenberg interaction, which satisfy S α β (i)S β α ( j) = S α β ( j)S β α (i).
The coefficient of the λ-term is also modified and becomes p 2π 3 but since it was non-universal to begin with, this change is of no particular importance.

Appendix E.3. General form of the action
Assuming SU (3), gauge, and time reversal invariance of the action, one can show that there are no other possible terms with two derivatives apart from those appearing in Eqs. (E.5), or (E.9). A general term can be written φ α( * ) m φ β( * ) n ∂ µ φ γ( * ) o ∂ ν φ δ( * ) p , where ( * ) means that it is either complex conjugated or not. m, n, o, p index the three different fields, while the α, β, γ, δ indices run through the components of the fields. Such a term would transform under the gauge transformation as (E. 16) For this term to be invariant there must be exactly two complex conjugates, and the field index of each complex conjugated field should be the same as one of the non conjugated ones, otherwise the e ±iϑ phases wouldn't cancel in the first term on the right hand side. Furthermore to conserve SU(3), the fields should form scalar products. With these restrictions, the possible terms are: (E.17) Note that the C µ,ν n,n term gives the same term as the A µ,ν n . The D-term is gauge invariant if m n, while the m = n case is already considered in the B-term. The other terms transform under the gauge transformation as A µ,ν n ∂ µ φ * n · ∂ ν φ n + A µ,ν n i∂ ν ϑ n ∂ µ φ * n · φ n + i∂ µ ϑ n ∂ ν φ * n · φ n + ∂ µ ϑ n ∂ ν ϑ n , n is antisymmetric in µ, ν. If m n, the B-term can't be made gauge invariant, and there is no other possible term with which it would give a gauge invariant combination either. Note, however, that in the case m = n, setting A µ,ν n = B µ,ν n,n leads to a gauge invariant combination. So the most general form of an SU(3) and gauge invariant term is given by Under time reversal symmetry (for real time) this would transform as (E.20) The factor (−1) only appears if there is exactly one time derivative term. Based on these, for real time, we have A n = −A * n , (B µ,ν n ) * = (−1) ε µ,ν (B µ,ν n ), and (D µ,ν m,n ) * = (−1) ε µ,ν D µ,ν m,n . So A n should be imaginary for real time, hence real for imaginary time. Similarly, D µ,µ and B µ,µ are real, while D µ,ν and B µ,ν for µ ν are imaginary for real time (therefore all the elements of D µ,ν m,n and B µ,ν n are real in imaginary time). Furthermore, every term in the action should be real for real time t = −iτ. This constrains D x,t m,n = −D t,x m,n , as they are pure imaginary and they are coupled to terms which are complex conjugates of each other. Similarly B x,t n = −B t,x n has to be fullfilled, but these terms drop out since they are coupled to terms which are symmetric under x ↔ t. Hence finally we arrive at a general form Note however that some of these terms are redundant. The D µ,µ m,n -and the B µ,µ n -terms both express the real part of the action. The D µ,µ m,n -terms correspond to the formulation in Eq. (E.5), while the B µ,µ n -terms actually give the form of the real part in Eq. (E.9). The transformation between the D µ,µ m,n and B µ,ν n parameters is similar to that in Eq. (E.10). The A n -terms give the three topological charges (up to a factor of 2πi). As we mentioned before, since the three charges sum up to zero only two independent parameters remain. The D x,τ m,n -terms give the q m,n terms as defined in Eq.
Thus, the D xτ m,n -terms actually give both the unusual imaginary λ-term and the topological terms. Therefore the A n -terms are also redundant. Reviewing Eqs. (E.5) and (E.9), we find that those already have the most general form compatible with SU(3), gauge and time reversal invariance.

Appendix F. Details of renormalization group calculation
Throughout this appendix, lowercase Roman letters index the off-diagonal Gell-Mann Matrices (GM), lowercase Greek letters index the diagonal GM, uppercase Roman letters index all eight GM, and repeated indices are summed over.

Appendix F.1. Definitions and identities of SU(3) structure constants
We label the Gell-Mann matrices T A of SU(3), according to These matrices satisfy the su (3) algebra where the f ABC structure constants are fully antisymmetric. In what follows we prove various identities involving f ABC . By construction, the Gell-Mann matrices satisfy Using the completeness of SU(3) generators, we prove f ABC f ABD = 3δ CD .
then using (F.4) on the middle term: where in the last step, we used (F.3). This completes the proof. Now we prove two partial completeness results: with the structure constants f 123 = 1, 2 . The first term in (F.9) equals 10) and the second term equals: Adding (F.10) and (F.11) completes the proof. The second partial completeness result is Proof: Expand (F.5) as If C = γ is a diagonal index, the second term vanishes, and the second case of (F.12) follows. If C is off-diagonal, the second term gives 2δ CD according to (F.8), which proves the first case of (F.12). Next, the Jacobi Identity gives us an identity for the structure constants, Finally, using this Jacobi identity, we prove Proof: A special case of the Jacobi identity is: where we used (F.8) on the RHS. Now, if C is diagonal, the second term on the LHS vanishes, and we prove case 2. If C is off-diagonal, the two terms on the LHS of (F.18) are equal, which can be checked, term by term. This proves case 1.

Appendix F.4. Rewriting the Lagrangian
We rewrite Λ j in terms of T 3 , T 8 and I, and expand the two terms of the imaginary time Lagrangian. Our results are (F.47) and (F.54).

Real part:
We start with This can be rewritten as since tr ∂ µ U∂ µ U † Λ j Λ j+1 = 0. Note that where I is the identity matrix. Substituting this into Eq. (F.44), we may drop the I terms since ∂ µ (U † U) = 0. Thus (F.46) Collecting terms, This result doesn't depend on how we choose the diagonal Gell-Mann matrices. Note that the diagonal matrix elements of the two diagonal Gell-Mann matrices together with √ 2/3I form a complete orthogonal set of real vectors with norm √ 2. Thus: Thus we may also write: (F.49) The 2/3 term can be dropped because it gives a term containing ∂ µ (U † U). The same result is obtained with any basis of diagonal Gell-Mann matrices which obey the same completeness condition and the same normalization.

Argument 2:
In the perturbative Lagrangian in (6.1), the leading term that arises from θ a (x)F a (x) will correspond to a Feynman diagram with a single internal line. Since the leading g-dependent interaction is a four-point vertex, the simplest O(λ) diagram arising from θ a (x)F a (x) will serve to renormalize a five-point or six-point interaction in (6.1). Such diagrams can be excluded from a first-order perturbative calculation of β(g) and β(λ), which consider the four-point and three-point interactions, respectively.
Recognizing the antisymmetry, this is Now we use (F.103). The result is Now we need to prove that the operator appear here is proportional to L q 0 . This is (F.108): Therefore, Now we return to the Lagrangian, which requires dividing (F.79) by 2. Using (F.87) and (F.80), this is This allows for the identification of λ eff = λ 1 + 9g 4π log b g eff = g 1 + 5g log b 4π Replacing commutators with structure constants gives which can be reorganized into Now, terms of the form f 3bc f c8d vanish unless b = d. This means the first and fourth terms cancel. Likewise, the sixth and seventh terms cancel. What remains is the LHS of (F.103) becomes = √ 3 2 ǫ µν − trN µ T 1 trN ν T 2 + trN µ T 2 trN ν T 1 + trN µ T 4 trN ν T 5 − trN µ T 5 trN ν T 4 − trN µ T 6 trN ν T 7 + trN µ T 7 trN ν T 6 . (F.107) The ǫ µν tensor allows us to combine these terms, proving (F.103). would allow for L to be written purely in terms of the θ a . Though we believe (G.1) is true in general, our calculations only require a factorization to hold to cubic order in θ a , and this is what we prove here. We have e iθ A T A − e iθ α T α e iθ a T a = −(1/2)θ A θ B T A T B + (1/2)θ α θ β T α T β + (1/2)θ a θ b T a T b + θ α θ a T α

Identity 6
This can be written: The δθ a θ B cross term in Eq. (G.3) is: