# Phase diagram of the chiral SU(3) antiferromagnet on the kagome lattice

Yi Xu  These authors contributed equally to this work.  Department of Physics
and Astronomy, Rice University, Houston, TX 77005, USA  Sylvain Capponi  These
authors contributed equally to this work.  Laboratoire de Physique Théorique,
Université de Toulouse, CNRS, UPS, France  Ji-Yao Chen  These authors
contributed equally to this work.  Guangdong Provincial Key Laboratory of
Magnetoelectric Physics and Devices, Center for Neutron Science and
Technology, School of Physics, Sun Yat-sen University, Guangzhou, 510275,
China  Laurens Vanderstraeten  Department of Physics and Astronomy, University
of Ghent, Krijgslaan 281, 9000 Gent, Belgium  Juraj Hasik  Institute for
Theoretical Physics, University of Amsterdam, Science Park 904, 1098 XH
Amsterdam, The Netherlands  Andriy H. Nevidomskyy  Department of Physics and
Astronomy, Rice University, Houston, TX 77005, USA  Matthieu Mambrini
Laboratoire de Physique Théorique, Université de Toulouse, CNRS, UPS, France
Karlo Penc  Institute for Solid State Physics and Optics, Wigner Research
Centre for Physics, H-1525 Budapest, P.O.B. 49, Hungary  Didier Poilblanc
Laboratoire de Physique Théorique, Université de Toulouse, CNRS, UPS, France

###### Abstract

Motivated by the search for chiral spin liquids (CSL), we consider a simple
model defined on the kagome lattice of interacting SU(3) spins (in the
fundamental representation) including two-site and three-site permutations
between nearest neighbor sites and on triangles, respectively. By combining
analytical developments and various numerical techniques, namely exact Lanczos
diagonalizations and tensor network variational approaches, we find a rich
phase diagram with non-topological (“trivial”) and topological (possibly
chiral) gapped spin liquids (SLs). Trivial spin liquids include an Affleck-
Kennedy-Lieb-Tasaki (AKLT)-like phase and a trimerized phase, the latter
breaking the inversion center between the up and down triangles of the kagome
lattice. A topological SL is stabilized in a restricted part of the phase
diagram by the time-reversal symmetry breaking (complex) 3-site permutation
term. Analyzing the chiral edge modes of this topological SL on long cylinders
or on finite disks, we have come up with two competing scenarios, either a CSL
or a double Chern-Simon SL characterized by a single or by two counter-
propagating Wess-Zumino-Witten SU(3)1 chiral mode(s), respectively. In the
vicinity of the extended ferromagnetic region we have found a magnetic phase
corresponding either to a modulated canted ferromagnet or to a uniform
partially magnetized ferromagnet.

##  I Introduction

The electronic and magnetic properties of materials in Nature arise from the
interactions between SU(2)-symmetric fermions - the electrons, on the
background of nuclei. When the interactions are strong, such materials can be
well described by the electronic Hubbard model, which has been extensively
studied in the field of condensed matter physics. Recent developments venture
beyond this SU(2) paradigm. In one form, through emergent SU(4) symmetry
conjectured for instance in strong spin-orbit materials [1] or twisted bilayer
graphene [2]. Yet a more direct approach, facilitated by the continuous
progress in ultracold atom platforms, is to emulate the physics of the
SU(N)-symmetric Hubbard model by loading N-color atoms onto optical lattices
[3, 4]. These experimental platforms provide an ideal environment for
exploring and engineering exotic phases that have yet to be discovered in real
materials.

Among exotic phases of matter, topological spin liquids (TSL) have been the
subject of intense experimental and theoretical research activities since the
seminal SU(2)-symmetric Resonating Valence Bond (RVB) proposal by Anderson and
Fazekas [5, 6]. Later on, the topological nature [7] of the RVB state on non-
bipartite lattices has been noticed, and turned out to be transparent within
the tensor network framework [8]. However, parent Hamiltonians for such
states, although local, are not physical involving complicated multi-site
interactions [9]. Hence it is not clear whether (non-chiral) TSL can be hosted
in simple physical models. On the experimental side, platforms of Rydberg
atoms [10, 11, 12] offer beautiful implementations e.g. of the RVB physics of
(hardcore) dimers. The realisation of synthetic gauge fields in cold atom
platforms sets the stage to experimental observations of fractional Chern
insulators or chiral spin liquids (CSL) [13, 14, 15].

CSL define a broad class of TSL that break (spontaneously or not) time
reversal and reflection (R) symmetries while preserving their product. Simple
constructions of CSLs by Wen [16] and Kalmeyer and Laughlin [17] established a
more precise connection to the physics of the fractional quantum Hall (FQH)
effect. Evidence for spin-1/2 CSL has been provided in (frustrated) Heisenberg
models in the presence of an additional chiral 3-site interaction, e.g. on the
kagome lattice [18, 19], on the triangular lattice [20, 21, 22] or on the
square lattice [23, 24], and also in Hubbard insulators [25, 26].

Preliminary investigations by one of the authors (K.P.) [27] have shown that a
simple time reversal symmetric Hamiltonian on the kagome lattice consisting of
(real) permutations of SU(3) spins (in the fundamental $\bf 3$ irreducible
representation) on the triangular units admits an Affleck-Kennedy-Lieb-Tasaki
(AKLT)-like [28] state as the exact ground state. Interestingly, such a state
bears a particularly simple tensor network representation (typical of AKLT
states) involving $\bf{\overline{3}}$ virtual particles fusing into singlets
on every triangle. Also, the gapped AKLT phase has been shown to be stable
under the addition of a nearest-neighbor Heisenberg coupling (two-site
permutation of either positive or negative amplitude), limited by two critical
points defined by equal amplitudes of the two and three site permutations
[27]. Motivated by the recent quest for TSL and, in particular, for novel CSL
we have extended KP’s model by including time reversal symmetry-breaking (pure
imaginary) triangular permutations providing a two-dimensional parameter
manifold. A thorough investigation of the phase diagram of this model has been
undertaken. Note that the same model has been studied using parton
wavefunctions in a restricted domain of parameter space [29] claiming the
existence of an Abelian CSL.

The paper is organized as follow; first, the model and the numerical methods
are described in Sec. II. Then, the overall phase diagram is depicted in Sec.
III with a description of the various phases coming into play. In a second
step, the ground states and low-energy excitations of interesting phases of
the phase diagram – obtained by complementary numerical techniques – are
analysed in Sec. IV. Interestingly, the model is shown to host a topological
spin liquid phase in an extended domain of parameters. To characterize the
nature of this TSL we have investigated the physics of the edge modes by
different means, in particular using an accurate tensor network ansatz of the
ground state. Details on analytical methods and numerical techniques such as
Lanczos exact diagonalisations (ED), Matrix Product States (MPS) on cylinders
and tensor network techniques using Projected Entangled Pair States (PEPS) and
Projected Entangled Simplex States (PESS) are provided in Appendix A, Appendix
B, Appendix C and Appendix D, respectively.

##  II Model and numerical tools

The local Hilbert space on each site $i$ of the two-dimensional kagome lattice
consists of the three states $|\alpha\rangle_{i}=\\{A,B,C\\}$ representing the
fundamental (defining) representation $\bf 3$ of SU(3) (see Appendix A.2 for
details on the SU(3) group). The interaction between these SU(3) “spins” is
described by the SU(3)-symmetric Hamiltonian as follows:

$\displaystyle H$ $\displaystyle=$ $\displaystyle J\sum_{\langle
i,j\rangle}P_{ij}+K_{R}\sum_{\triangle ijk}(P_{ijk}+P_{ijk}^{\,\,\,\,\,-1})$
$\displaystyle+$ $\displaystyle iK_{I}\sum_{\triangle
ijk}(P_{ijk}-P_{ijk}^{\,\,\,\,\,-1}),$

where the first term corresponds to two-site permutations over all nearest-
neighbor bonds, and the second and third terms are the three-site permutations
on all triangles, clockwise ($P_{ijk}$) and counterclockwise ($P_{ijk}^{-1}$).
Written explicitly, $P_{ij}$ and $P_{ijk}$ are defined through their action on
the local basis states,
$P_{ij}|\alpha\rangle_{i}|\beta\rangle_{j}=|\beta\rangle_{i}|\alpha\rangle_{j}$
and
$P_{ijk}|\alpha\rangle_{i}|\beta\rangle_{j}|\gamma\rangle_{k}=|\gamma\rangle_{i}|\alpha\rangle_{j}|\beta\rangle_{k}$,
for a fixed orientation of the triangle $i$, $j$, $k$, say clockwise. The
$K_{I}$ term is “chiral”, in the sense that it breaks time reversal and
reflection symmetries without breaking their product.

For convenience, in the following we use the parametrization on a sphere:

$\displaystyle J$ $\displaystyle=$ $\displaystyle\cos\theta\cos\phi,$
$\displaystyle K_{R}$ $\displaystyle=$ $\displaystyle\cos\theta\sin\phi,$ (2)
$\displaystyle K_{I}$ $\displaystyle=$ $\displaystyle\sin\theta,$

where $0\leq\phi<2\pi$ and it is sufficient to consider $\theta$ in the
interval $[0,\pi/2]$ because of the symmetry $K_{I}\leftrightarrow-K_{I}$.
Hence only the upper hemisphere of the parameter space is considered.

We have addressed the phase diagram of this model by complementary numerical
tools. Lanczos ED on small periodic 21-site and 27-site clusters (torus
geometry) have been performed to obtain the low-energy spectrum from which
useful information on the nature of the phase can be extracted. Such clusters
accommodate the SU(3) singlet subspace (and hence can describe spin liquids)
and all available space group symmetries are used to label the many-body
eigenstates. MPS calculations with explicit SU(3) symmetry on infinitely-long
cylinders with finite circumference have also been performed to compute ground
state properties [30], entanglement spectra [31] and construct excitation
ansätze [32]. PEPS [33] and PESS [9, 34] tensor networks have been considered
and contracted on the infinite lattice using Corner Transfer Matrix
Renormalization Group (CTMRG) [35, 36]. Both unconstrained and symmetric
PEPS/PESS have been employed and variationally optimized e.g. using conjugate
gradient optimization schemes [37, 38]. While fully unconstrained or Abelian-
symmetric (employing just $\mathrm{U}(1)\times\mathrm{U}(1)$ subgroup of SU(3)
symmetry group) ansätze are less “biased” by construction, their optimization
is more difficult and greatly benefits from an automatic differentiation
procedure [39, 40]. In contrast, SU(3)-symmetric PEPS/PESS [41] depends on a
much smaller number of variational parameters which can be optimized by
numerically estimating the gradient vector [42, 24]. Symmetric PEPS/PESS
encoding SU(3)-rotation symmetry and translation invariance are particularly
well-suited to describe singlet phases of matter [43, 44], where the SU(3)
symmetry (implemented with the QSpace library [45, 46]) also allows us to
reach unusually large bond dimensions.

Figure 1:  Semi-quantitative phase diagram of the SU(3) chiral antiferromagnet
on the kagome lattice using a stereographic projection (mapping the
$(\theta,\phi)$ hemisphere onto a planar disc – see Appendix A.1). Shaded
areas of different colors represent the various phases discussed in the text.
The center of the plot ($\theta=\pi/2$) defines the “North Pole” and the outer
circle ($\theta=0$) the “equator” parametrized by the azimuthal angle $\phi$,
with the corresponding model parameters defined in Eq. 2. The dashed (dash-
dotted) line corresponds to the exact single-magnon (two-magnon) instability
of the ferromagnetic phase. It is unclear whether the two-magnon instability
gives the onset of the trimer phase.

##  III Phase diagram

###  III.1 Preliminaries

The model studied here exhibits a very rich phase diagram as shown in Fig. 1.
The parameter space defined on a hemisphere is conveniently mapped onto a two-
dimensional (2D) disk using a stereographic projection (see Appendix A.1). In
this section we will describe qualitatively the phase diagram and the various
phases we have encountered. More thorough descriptions, reports of the
numerical results and discussions will be given in the subsequent sections.

To start with, it is useful to distinguish two types of phases: (i) the spin
liquids (SL) whose ground states preserve both SU(3) rotational invariance
(SU(3) singlets) and invariance under lattice translations – like the Affleck-
Kennedy-Lieb-Tasaki (AKLT) or chiral SL (CSL) phases – but may break point
group symmetry (like the trimer phase); and (ii) the magnetically-ordered
phases breaking the SU(3) rotational symmetry. The uniform fully-polarized
ferromagnet is a trivial example of the latter type, but more complicated
magnetic phases breaking lattice translation symmetry are also realised here.
Since the unit cell of the kagome lattice contains three sites and on each
site there is a $\mathbf{3}$ spin, the Lieb-Schultz-Mattis (LSM) theorem [47],
extended to higher spatial dimensions by Oshikawa [48] and Hastings [49], and
its generalization to SU(N) [50] does not apply to Hamiltonian (II) so that
spin liquids in the phase diagram may or may not possess topological order.

Figure 2:  PESS construction on the kagome lattice of corner-sharing
triangles. Site-centered rank-3 tensors carrying the physical leg (in red) are
represented in blue, while triangle-centered tensors represented in green fuse
three virtual legs into an SU(3) singlet. In the case of SLs, the tensor
network is uniform. PESS can also describe all other phases in the phase
diagram with proper modifications to be discussed in the text and Appendix D.

Generically, the SL ground states can be conveniently defined/represented by a
simple tensor network [9, 8, 41]. On the kagome lattice, the simplest version
is a Projected Entangled Simplex State (PESS) (see Fig. 2) involving a rank-3
tensor in each triangle (green sphere in Fig. 2) and a rank-3 tensor with two
virtual and one physical leg (blue sphere) on each site. Each bond connecting
a site to the center of a triangle carries virtual SU(3) particles. The
corresponding virtual Hilbert space $\cal V$ is therefore a direct sum of a
certain number (labelled throughout by $D^{*}$) of SU(3) irreducible
representations (irreps). On all triangles, the three virtual particles fuse
into a singlet, and the trivalent tensor enforces the projection ${\cal
V}^{\otimes 3}\rightarrow{\bf 1}$. On the sites, two virtual particles fuse to
the physical state, and the site tensor enforces the projection ${\cal
V}^{\otimes 2}\rightarrow{\bf 3}$. Here and throughout, we use the standard
notation for the SU(3) irreps labeled by their dimension or, equivalently, by
their Dynkin labels – see Table 1 in Appendix C for details. Besides the
representation of spin liquids, the PESS formalism (associated to PEPS) turns
out to be also extremely useful to investigate magnetic phases and phases
breaking translation symmetry – as will be discussed later on. Details about
the PESS/PEPS constructions are given in Appendix D.

###  III.2 AKLT phase

It has been shown by one of the authors (K.P.) that the non-chiral Hamiltonian
defined by $K_{I}=0$ (i.e. $\theta=0$) in Eq. (II) has an exact ground state
(GS) of the AKLT type in the range $\pi/4\leq\phi\leq 3\pi/4$ [27]. It is
closely related to the simplex solid of Arovas constructed in Ref. 51 which
breaks no discrete lattice symmetry, but we write the singlet creation
operators using fermions, not bosons. This state can be represented by the
simplest possible PESS representation just involving the unique irrep
($D^{*}=1$) ${\cal V}=\bar{\bf 3}$ on all virtual bonds. Hence, on all
triangles three virtual $\overline{\bf 3}$ particles fuse into a singlet,
$\overline{\bf 3}^{\otimes 3}\rightarrow{\bf 1}$ while, on the sites, two
virtual $\bar{\bf 3}$ particles fuse into the physical irrep, $\overline{\bf
3}^{\otimes 2}\rightarrow{\bf 3}$. This construction provides a unique ground
state and, since the AKLT phase is likely to be gapped (see later), we deduce
that the AKLT phase is a featureless (or “trivial”) SL with no topological
order.

Being gapped, the AKLT phase survives when a sufficiently small chiral
perturbation is introduced (i.e. $\theta<\theta_{\rm crit}(\phi)$, see later).
To describe the AKLT ground state in these new regions of the phase diagram,
one has to extend the previous PESS construction by increasing the (Hilbert
space) dimension $D={\rm dim}({\cal V})$ of the virtual space. The singlet
character of the GS implies that $\cal V$ is a direct sum of SU(3) irreps. The
irreps appearing in this direct sum must fulfill a strict requirement to keep
the same featureless (i.e., non-topological) character of the ground state. In
fact, each irrep $\bf I$ of SU(3) is characterized by its $\mathbb{Z}_{3}$
charge $Q({\bf I})$ defined by the number of boxes of its Young tableau modulo
3 (e.g. $Q=2$ is equivalent to $Q=-1$). The AKLT PESS can only contain irreps
$\bf I$ with the same charge as $\overline{\bf 3}$, i.e. $Q({\bf
I})=Q(\overline{\bf 3})=2$. Of course, the optimal choice of those irreps can
only be determined numerically by a variational optimization scheme.
Restricting to $D^{*}\leq 4$ irreps in the virtual space, we have found that
${\cal V}=\overline{\bf 3}+\overline{\bf 3}+{\bf 6}+\overline{\bf 15}$ is the
best choice.

###  III.3 Trimer phase

The SU(3) Heisenberg antiferromagnet on the kagome lattice (i.e. the
$\phi=\theta=0$ point in our phase diagram) exhibits a trimer phase [52], i.e.
a simplex solid [51] with different energy densities on the two types of up-
pointing and down-pointing triangles (named up and down triangles hereafter).
Hence, such a phase spontaneously breaks the (site) inversion center,
resulting in a doubly-degenerate SU(3) singlet groundstate manifold. We have
shown that this phase survives in a rather extended region of our phase
diagram.

Similar to the AKLT phase, a particularly simple prescription exists for
constructing PESS ansätze of the trimer phase by using different virtual
spaces ${\cal V}_{\rm up}$ and ${\cal V}_{\rm down}$ for the up and down
triangles, respectively. Let us start with the extreme case of decoupled SU(3)
singlets on, say, up triangles. An exact PEPS representation is given by
${\cal V}_{\rm up}={\bf 3}$ and ${\cal V}_{\rm down}={\bf 1}$ and the
corresponding (unique) $C_{\rm 3v}$-symmetric trivalent tensors on the up and
down triangles encode the fusion rules ${\bf 3}^{\otimes 3}\rightarrow{\bf 1}$
and ${\bf 1}^{\otimes 3}\rightarrow{\bf 1}$, respectively. Also, the site
tensors encode the trivial fusion ${\bf 3}\otimes{\bf 1}\rightarrow{\bf 3}$.
We note that the two irreps of the up and down virtual spaces have different
${\mathbb{Z}}_{3}$ charges, $Q_{\rm up}=1$ and $Q_{\rm down}=0$, respectively.
This suggests that the PESS ansatz of a generic trimer state in which the up
triangles are entangled can be constructed by simply adding more irreps of the
same ${\mathbb{Z}}_{3}$ charge in ${\cal V}_{\rm up}$ and ${\cal V}_{\rm
down}$. Restricting to $D^{*}=2$ irreps we found that ${\cal V}_{\rm up}={\bf
3}+\overline{\bf 6}$ and ${\cal V}_{\rm down}={\bf 1}+{\bf 8}$ provide the
best ansatz. Note that, in such a construction, the inversion center is
obviously broken and a pair of ground states is obtained by simply switching
the virtual spaces between the up and down triangles.

###  III.4 Topological spin liquid

We have also found a gapped topological spin liquid (TSL) stabilized in a
significant region of the phase diagram provided the chiral $K_{I}$ term is
present ($\theta\neq 0$) in Eq. (II), as shown in Fig. 1. The region of
stability of this phase includes the parameters $K_{R}/J\simeq 0.6$,
$K_{I}/J\simeq 0.45$ (i.e $\theta\sim 0.13\pi$ and $\phi\sim 0.17\pi$)
proposed in [29] as the optimal parameters for the stability of an Abelian
CSL. Such a phase does not break any lattice symmetry (it is perfectly
uniform), nor does it break the SU(3) symmetry. Moreover, it possesses
topological order as defined by Wen [7, 53]. Interestingly, the existence of
topological order is not a priori guaranteed in SU(3) kagome spin liquids
since the LSM theorem does not apply here, as already noted above. Then, the
ground state degeneracy is expected to be associated to the 3 possible values
of the $\mathbb{Z}_{3}$ charge. Indeed, ED (MPS) on a torus (on an infinite
cylinder) reveals 3 quasi-degenerate ground states in some extended region of
the phase diagram.

A faithful description of such a phase in terms of PESS is in fact possible. A
necessary (but not always sufficient) condition for the existence of (at
least) 3 topological sectors on the infinite cylinder is that the virtual
space should contain at least one irrep within each of the 3 $\mathbb{Z}_{3}$
charge sectors. A minimal choice would then be ${\cal V}={\bf 1}+{\bf
3}+\overline{\bf 3}$. Below we show that increasing the virtual space to
${\cal V}={\bf 1}+{\bf 3}+\overline{\bf 3}+{\bf 6}+{\bf 8}$, with additional
irreps of charge $Q=2$ and $Q=0$, provides an optimal low-energy ansatz of the
TSL.

As reported below in Section IV, we find that this TSL phase exhibits chiral
edge modes, as revealed by its entanglement spectrum (ES). The content of the
edge modes is described in terms of a SU(3)1 Wess-Zumino-Witten (WZW)
Conformal Field Theory (CFT), and should fully characterize the nature of this
Abelian TSL. The results obtained with our PESS ansatz show edge modes of both
right- and left-moving chiralities (and different velocities) consistent with
a SU(3)1 doubled Chern-Simons (DCS) Topological Field Theory (TFT) [43, 54].
On the other hand, the ED and MPS results rather point towards a chiral spin
liquid (CSL) phase exhibiting a single chiral edge mode. Later in Section IV
we shall discuss further the pros and cons for the CSL or for the DCS phase.

###  III.5 Ferromagnetic phase

The ferromagnet is a lattice-symmetric state which spontaneously breaks the
internal SU(3) symmetry, where both the two-site and the three-site
permutations act trivially with eigenvalues $+1$. Hence the ground state
energy per site is simply $e_{F}=2J+4K_{R}/3$.

To determine the phase boundary of the ferromagnetic state, we calculate the
dispersion of a single magnon. By this, we mean a configuration such that the
flavors are $A$ on all sites except one, which is, say, a $B$. The Hamiltonian
then hops the $B$ flavor to neighboring sites, giving it a dispersion
determined by the eigenvalues of the three-by-three matrix (4) in Appendix
A.3. The matrix (4) describes three magnon branches. If the dispersion of the
magnon, measured from the energy of the ferromagnetic state, is negative, the
ferromagnetic phase becomes unstable and ceases to exist. Scrutinizing the
dispersions, it turns out that they are functions of $J+K_{R}$ and $K_{I}^{2}$
only. The maximum and minimum of the dispersions are always at momentum
$\mathbf{q}=0$, where the energies are $0$ and $-6(J+K_{R})\pm
2\sqrt{3}K_{I}$. We get a positive dispersion for

$J+K_{R}<-|K_{I}|/\sqrt{3}.$ (3)

Conversely, the ferromagnetic state is unstable for
$J+K_{R}>-|K_{I}|/\sqrt{3}$. On the boundary, when
$J+K_{R}=-|K_{I}|/\sqrt{3}$, the $0$ energy band becomes flat. Localized modes
on hexagons common to kagome lattice appear, but with amplitudes $e^{ij\pi/3}$
as we go around the hexagon – the complex amplitudes reflect the chiral nature
of the model.

The one-magnon instability line is shown as a dashed line in Fig. 1.
Interestingly, we find numerically that, within the one-magnon stability
region, the ferromagnetic state is nevertheless unstable with respect to the
trimer phase.

We have also envisioned the possibility of a two-magnon instability by
considering a two-magnon excitation of the ferromagnetic state, where two
spins are flipped with different flavors. Details of the calculations are
given in section A.4 of Appendix A. The two-magnon calculation aims to reveal
whether the interaction between the magnons could be attractive and lead to a
bound state. In that case, the boundary of the ferromagnetic phase would
shrink further. This is indeed what has been found as shown in the phase
diagram of Fig. 1. Numerically, we found that this two-magnon instability line
marks (approximately) the instability to the trimer phase.

Figure 3:  9-site $\sqrt{3}\times\sqrt{3}$ unit cells tiled in C3-rotation
symmetric patterns. The colors indicate which one of the three SU(3) colors
has the dominant weight. Note that the colors in each, say, up triangle are
identical and have the same dominant weight magnitude.

###  III.6 SU(3)-broken phase

When crossing the one-magnon instability line in the region roughly
$\pi/4<\phi<3\pi/4$ (magenta color in Fig. 1), the ferromagnetic state becomes
unstable and gives rise to a partially magnetized phase (with magnetization
$0<m<1$), hence breaking SU(3) symmetry. Such spontaneous SU(3)-breaking may
occur while preserving or (spontaneously) breaking translation symmetry.

The translation symmetry-broken phase is characterized by a spin canting
occurring on three triangular sub-lattices separately, which requires a 9-site
unit cell. The canted spins (all of the same length) on each sub-lattice form
either a stripy pattern or a C3-rotation symmetric pattern, with all three
sub-lattices having the same overall spin direction (see Appendix D). In our
calculations, the so-called C3-2 $\sqrt{3}\times\sqrt{3}$ C3-rotation
symmetric pattern in which the site SU(3)-color occupations are identical in
each (let say) up triangle - see Fig. 3 \- seem to be energetically favored
over the other C3-rotation symmetric or stripe patterns discussed in section
D.3 of Appendix D.

The second competing magnetic phase can be described by a uniform
translationally-invariant (3-site) PEPS, as discussed in section D.2 of
Appendix D. After energy optimization, the magnetization in such a priori un-
restricted ansatz turns out to be uniform and significantly lower than in the
modulated C3-2 phase. Note also, the magnetizations on the three sites within
the unit cell are not canted but rather collinear. Interestingly, the numerics
point towards a jump of the magnetization at the boundary to the fully
polarized phase. This is indeed expected from the analytic calculation of the
one-magnon instability in section A.3 of Appendix A predicting infinite
compressibility at the transition.

Figure 4:  Top: Energetics of ED, iMPS, iPESS and iPEPS wave functions along
the $\phi=\pi/2$ [mod $\pi$] meridian where $J=0$, i.e., along the vertical
diameter of Fig. 1 as highlighted in the top right corner. The dashed line
corresponds to the exact ferromagnet energy. The phase boundaries are
approximate except for the canted ferro-ferro transition at
$(\phi,\theta)=(3\pi/2,\pi/3)$. Middle: Uniform magnetization of the unit cell
$m$ in units of $m_{0}$. Bottom: ED low-energy (excitation) spectrum of a
periodic $21$-site cluster. Open (closed) symbols show the singlet (non-
singlet) eigenstates and the GS energy has been subtracted. Different symbols
correspond to different momenta as shown in the legend. The black circles on
the right correspond to the largest SU(3) irrep. Inset panel: ED on a 27-site
torus shows the disappearance of the gapped region close to the pole.

##  IV Ground states and low-energy excitations

A crude determination of the phase diagram in Fig. 1 and of the boundaries of
the various phases was first obtained by inspecting the low-energy ED spectra
on a periodic 21-site torus (see Appendix B for details). These results were
then extended to a 27-site torus (for a much smaller set of parameters) and
compared against the results obtained by tensor network methods (MPS, iPESS,
iPEPS) to refine the phase diagram. For simplicity, we shall here focus on
three different cuts – the $\phi=0$ [mod $\pi$] and the $\phi=\pi/2$ [mod
$\pi$] meridians, together with a portion of the $\theta=\pi/8$ latitude –
which contain all the phases we have encountered.

###  IV.1 Energetics

The top panels in Figs. 4, 5 and 6 show comparisons of the energy per site
obtained by ED, iMPS, iPESS and iPEPS, along the aforementioned vertical,
horizontal and circular cuts, respectively. The ED ground state energies have
been all obtained from the same periodic 21-site cluster preserving all the
symmetries of the infinite lattice. In Figs. 4 and 5 the iMPS energy has been
obtained on a finite width ($L_{y}=4$) cylinder and SU(3) symmetry.

Figure 5:  Top: Energetics of ED, iMPS, iPESS and iPEPS wave functions on the
$\phi=0$ [mod $\pi$] meridian where $K_{R}=0$, i.e. from the leftmost point on
the equator to the rightmost point on the equator via the North Pole in Fig. 1
as highlighted in the top right corner. The iPESS ansatz for the trimer phase
is indicated in the legend as $[{\cal V}_{\rm up}]:[{\cal V}_{\rm down}]$.
Middle: Order parameters of iPEPS wave functions. The uniform magnetization
$m$ (open green squares) and its non-zero value identifies the SU(3)-broken
phase. The trimer phase order parameter indicated by the arrow is shown on the
right scale for various ansatze. Bottom: ED low-energy (excitation) spectrum
of a periodic $21$-site cluster. The same symbols are used as in Fig. 4.

We believe the ED and iMPS energies provide a (non-rigorous) lower bound of
the energy due to strong finite-size effects.

Figure 6:  Top: Energetics of ED, iPESS and iPEPS wave functions along part of
the $\theta=\pi/8$ latitude as highlighted in the top right corner. Bottom: ED
low-energy (excitation) spectrum of a periodic $21$-site cluster on the same
latitude, but on a larger arc from $\phi=0$ (in the trimer phase).

We have used translationally invariant SU(3)-symmetric iPESS calculations to
target SU(3) singlet phases, like the AKLT phase (virtual spaces ${\cal
V=}\bf\overline{3}$, $\bf{\overline{3}}+6$,
$\bf{\overline{3}}+{\overline{3}}+6$,
$\bf{\overline{3}}+{\overline{3}}+6+{\overline{15}}$) shown in Figs. 4 and 5
and the TSL phase (${\cal V}=\bf{1}+{3}+{\overline{3}}+6+{8}$) shown in Fig.
5. To describe the trimer phase (see Fig. 5) one has to extend symmetric iPESS
by allowing two different virtual spaces ${\cal V}_{\rm up}$ and ${\cal
V}_{\rm down}$ on the up and down triangles, characterized by different
$Z_{3}$ charges $Z_{\rm up}=1$ and $Z_{\rm down}=0$, respectively. In the
region of stability of the trimer phase, we indeed observe that the $[{\cal
V}_{\rm up}={\bf 3}+\overline{\bf 6}]:[{\cal V}_{\rm down}={\bf 1}+{\bf 8}]$
ansatz provides a very good variational energy, comparable to e.g. that of a
generic $D=8$ iPEPS ansatz (whose properties will be discussed later on).

In Fig. 4, moving away from the AKLT phase towards the (fully-polarized)
ferromagnetic phase, we see that the optimization of unconstrained
(1-triangle) iPEPS provides comparable, or even better, energies than the
SU(3)-symmetric ansatze, although with a modest bond dimension ($D=7,8,9$
compared to e.g. $D=27$ for one of the SU(3)-symmetric ansatz), suggesting the
existence of an intermediate phase breaking SU(3)-symmetry. This also happens
in a limited region of Fig. 5. In Fig. 6, in the vicinity of the (fully-
polarized) ferromagnetic phase, the optimization of an unconstrained $D=7$
C3-2 iPESS provides good variational energies comparable to or even better
than the previously mentioned 1-triangle iPEPS. This suggests that in the
magnetic SU(3)-broken phase, the lattice translation symmetry may be
spontaneously broken to form a 3-triangle $\sqrt{3}\times\sqrt{3}$ unit cell
order, corresponding to the modulation specifically encoded in the C3-2 PESS
ansatz.

Figure 7:

###  IV.2 Order parameters

To further characterize the magnetic SU(3)-broken phase, we define the uniform
magnetization of the unit cell $m=|\sum_{i\in\textrm{unit
cell}}\vec{\lambda}_{i}|$, $\lambda_{i}^{\alpha}$ being the 8 generators of
SU(3) acting on the site $i$, giving the fraction $m/m_{0}$ of the
magnetization $m_{0}$ of the fully polarized ferromagnet. As shown in the
middle panels of Figs. 4, 5 and 6 the SU(3)-broken phase can be determined
more accurately from the onsets of finite values of $m$ which reaches its
maximal value $m_{0}=2/\sqrt{3}$ in the fully polarized ferromagnet.

We have also defined the average magnitude as
$\tilde{m}=\sum_{i\in\textrm{unit cell}}|\vec{\lambda}_{i}|$ (not shown). We
observe that $\tilde{m}$ is different from $m$ only for the 3-triangle iPEPS
signaling canting of the site polarizations. In contrast, the 1-triangle iPEPS
shows aligned magnetizations on the 3 sites of the unit cell. Interestingly,
the D=7 (1-triangle) iPEPS data point to a jump in the magnetization at the
boundary to fully polarized ferromagnet.

The product of three physical $\bf 3$-irreps can be decomposed into a direct
sum of four irreps, given by the fusion rule ${\bf 3}^{\otimes 3}=\bf 1+\bf
8+\bf 8+\bf 10$. Hence, for the three SU(3) spins on each triangle, one can
define the projection operators onto corresponding irreps in the direct sum
(weights of irreps), $w_{\bf 1,8,8,10}$, which satisfy the completeness
relation $w_{\bf 1}+w_{\bf 8}+w_{\bf 8}+w_{\bf 10}=1$. As the trimer states
spontaneously break the inversion symmetry and form SU(3) trimers on either up
or down triangles, we define the trimer order parameter as the difference
between projections $w_{\bf 1}^{\nabla,\Delta}$ within the down and up
triangles onto $\bf 1$-irrep (weight of $\bf 1$-irrep) to identify the trimer
phase. This trimer order parameter is shown on the middle panel of Fig. 5
(right scale). Interestingly, the unconstrained $D=8$ iPEPS calculation gives
a very similar order parameter in the trimer phase to the SU(3)-symmetric PESS
ansatze specially designed to describe the trimer phase. It also shows an
abrupt jump at the onset of the TSL phase while the iMPS results give a more
continuous curve of the trimer order parameter when going from the trimer
phase to the TSL phase.

###  IV.3 Low-energy excitations

To refine the determination of the phase diagram we have computed the low-
energy spectra obtained by ED on a 21-site torus along the selected cuts, as
shown in the bottom panels of Figs. 4, 5 and 6. For a restricted set of
parameters, the spectrum has also been computed on a 27-site torus for better
accuracy (see inset of Fig. 4).

The spectrum for $(\phi,\theta)=(\pi/2,0)$, on the left of Fig. 4, clearly
shows a unique ground state and a gap of order $0.3$ characteristic of the
AKLT phase, but the rest of the spectrum seems to come down quickly when
increasing $\theta$. We can obtain a complementary estimate of the excitation
gap by the MPS excitation ansatz (see section C.6 of Appendix C), shown in
Fig. 7, which confirms that the gap decreases very quickly. The right side of
Fig. 4 shows the finite gap (due to finite size effects) of the fully
polarized ferromagnetic phase for $\theta<\pi/3$ (at $\phi=3\pi/2$). Around
the pole, a gapped phase is visible on the 21-site cluster. However, the
larger 27-site cluster reveals low-energy (non-singlet) excitations compatible
with the magnetic SU(3)-broken phase discussed above.

On the right-hand side of Fig. 5, the even and odd (under inversion) lowest
singlets (labeled $\Gamma A$ and $\Gamma B$) are the precursor of the two-fold
degenerate trimerized ground state. Between the AKLT and the trimerized
region, we see two new low-energy singlets ($\Gamma E_{1a}$) coming down
suggesting a three-fold degenerate GS typical of the CSL phase. As discussed
before, the small gap seen around the pole is an artifact of the 21-site
cluster, no longer present in the larger 27-site cluster.

The ED data in the bottom panel of Fig. 6 are shown on a larger interval along
the $\theta=\pi/8$ meridian than the two other panels above, from $\phi=0$ to
$\phi=\pi$. It shows the same characteristics encountered in Fig. 5 (described
in the above paragraph) corresponding to the same sequence (in reverse order)
of phases, i.e. the trimer, the TLS, the magnetic and the fully polarized
ferromagnet, from left (right) to the right (left) in Fig. 6 (5). Again the
trimer (TSL) phase shows two (three) low-energy singlets at the bottom of the
spectrum, and a spurious gap appears in the magnetic SU(3)-broken phase
(identified by complementary means).

###  IV.4 Edge modes in the TSL phase

We shall now discuss further the nature of the topological spin liquid phase
on the kagome lattice: our results suggest two candidates, (i) a CSL –
characterized by $\mathbb{Z}_{3}$ topological order with three sectors – and
(ii) a DCS phase – characterized by $D(\mathbb{Z}_{3})$ topological order with
nine sectors. Three different routes have been followed to identify the edge
modes: i) first, by studying the system on an open disk, whose low-energy
spectrum should follow that of some $(1+1)d$ CFT; ii) second, we optimize iMPS
on an infinite YC4 cylinder in three different topological sectors, from which
the entanglement spectrum can be straightforwardly deduced [31, 18]; iii)
third, using a faithful TSL representation via symmetric PESS. Note that
states with chiral topological order can be faithfully represented by PEPS or
PESS [55, 24, 56], where an infinite correlation length is artificially
introduced to generate the chiral features in the entanglement spectrum – the
latter provides a useful diagnostics [57, 58] for the nature of the TSL.

Figure 8:  Low-energy spectrum computed with ED on a 24-site kagome cluster
with open boundary conditions for $\theta=\pi/4$, $\phi=0$. Relative energies
are plotted vs the angular momentum $\ell$ with respect to the C6 rotation
symmetry. All symbols agree with the CFT prediction, see Tab. 2.

Fig. 8 shows the low energy spectrum in the TSL phase, computed by ED, on a
$N_{s}=24$-site disk. We observe a linearly dispersing chiral mode as a
function of the angular momentum associated with the C6 symmetry of the
cluster. The quantum numbers of the SU(3) multiplets are found to be in good
agreement with the WZW SU(3)1 tower of states issued from the singlet ($\bf
1$) ground state (see theoretical expectation in Table 2), namely all
multiplets are in perfect agreement up to $\ell=3$, while there are few extra
multiplets for larger $\ell$ (1 extra 1 and 1 extra 8 levels at $\ell=4$, 1
extra 1, 2 extra 8 and 1 extra 10 levels at $\ell=5$). This small discrepancy
could be attributed to the small cluster size. This suggests that the TSL
phase is a chiral SL.

Figure 9: Entanglement spectra of MPS on a straight $L_{y}=4$ cylinder (YC4)
optimized at $\theta=\pi/4,\phi=0$ in the three topological sectors, with
largest MPS bond dimensions around $D_{\mathrm{MPS}}\approx 6000$. The three
panels correspond to the three topological sectors associated to different
total $\mathbb{Z}_{3}$ charge $Q$ on the boundaries, $Q=0$ $(a)$, $Q=1$ $(b)$
and $Q=2$ $(c)$. The contents of the chiral modes are consistent with a SU(3)
CSL – see Table 2 and Table VII of Ref. [59].  Figure 10: Entanglement spectra
of a $D=13$ chiral PESS at $\chi=169$ placed on a $L_{y}=6$ infinite cylinder
partitioned in two halves, computed at $\theta=\pi/4,\phi=0$. The virtual
space is $\mathcal{V}=\bf{1}\oplus\bf{3}\oplus\bf{\overline{3}}\oplus\bf{6}$.
The three panels correspond to the three topological sectors associated to
different total $\mathbb{Z}_{3}$ charge $Q$ on the boundaries, $Q=Q({\bf
1})=0$ $(a)$, $Q=Q({\bf 3})=1$ $(b)$ and $Q=Q({\bf{\overline{3}}})=2$ $(c)$.
The content of the chiral modes agrees with predictions based on the SU(3)1
WZW CFT up to the Virasoro level $L_{0}=7$ for $Q=0$ (apart from minute
deviations) – see Table 2 – and based on the
$\mathrm{SU}(3)_{1}\times\overline{\mathrm{SU}(3)}_{1}$ DCS theory up to
$L_{0}=4$ otherwise – see Tables 3 and 4. Note, in the $Q=0$ sector one
8-dimensional irrep is missing from the compact group of low-energy states of
the $L_{0}=6$ Virasoro level. Also, all relevant irreps of the $L_{0}=7$ level
are grouped together below energy $\sim 7.2$ except three missing (1,1),
(0,3), and (2,2) irreps which appear slightly away at energy $\sim 7.32$,
$\sim 7.92$ and $\sim 7.52$, respectively.

Similarly, the MPS ES computed on a YC4 infinite cylinder (see Appendix C for
further details on the MPS construction) and shown in Fig. 9 reveal similar
features, also supporting the CSL phase. This can be seen in all three sectors
corresponding to different $Q$’s. The CFT predictions for these cases can be
found e.g. in Tables VI and VII of Ref. 59.

To construct the symmetric PESS, we have followed Ref. [43] to implement the
relevant symmetries in PESS, including $C_{3}$ rotation symmetry and SU(3)
spin rotation symmetry. Moreover, by choosing the appropriate local tensors,
the PESS undergoes complex conjugation under reflection, fulfilling the
symmetry requirement of both CSL and DCS phases breaking time-reversal
symmetry. One important ingredient in the symmetric PESS construction is the
representations carried by the virtual index of local tensors. With
$\mathbb{Z}_{3}$ gauge symmetry in mind, a minimal virtual space would be
${\cal V}={\bf 1}+{\bf 3}+\overline{\bf 3}$, which was shown to support a
$\mathbb{Z}_{3}$ toric code type topological phase in the parameter space. It
turns out, doing variational optimization in this class of PESS always runs
into a simplex solid phase, where the up and down triangles become
inequivalent. This could be understood from spontaneous symmetry breaking at
the PESS level. Therefore, one has to consider a larger virtual space for
representing SU(3) CSL phase. For that, we have used a SU$(3)$ symmetric
simple update algorithm (implemented with the QSpace library [45, 46])
combined with variational optimization, and found that virtual irreps ${\cal
V}={\bf 1}+{\bf 3}+\overline{\bf 3}+{\bf 6}$ and ${\cal V}={\bf 1}+{\bf
3}+\overline{\bf 3}+{\bf 6}+{\bf 8}$ could provide a good description of the
TSL.

The entanglement spectrum (ES) with virtual space ${\cal V}={\bf 1}+{\bf
3}+\overline{\bf 3}+{\bf 6}$ is computed and shown in Fig. 10. Using the
$\mathbb{Z}_{3}$ gauge symmetry, we group the ES into three sectors with
$\mathbb{Z}_{3}$ charge $Q=0,1,2$, respectively. The momentum $K$ around the
circumference of the cylinder is a good quantum number and is used to label
the ES.

As shown in Fig. 10, we have obtained linear dispersing chiral branches in the
low energy part of the ES in all three sectors. A close look at the content of
the chiral branch in the $Q=0$ sector compared to predictions of the WZW
SU(3)1 CFT in Table 2 reveals that the relevant SU$(3)$ multiplets were
captured up to the 7th Virazoro level apart from minute deviations (see figure
caption).

However, zooming in at the level content of the $Q=1$ and $Q=2$ sectors, one
finds that the quasi-degenerate SU$(3)$ multiplet structure differs from the
simple tower of states given by the WZW CFT. Instead, the low energy spectrum
in the $Q=1$ sector can be explained by the tensor product of
$\bf{\overline{3}}$ with the $Q=2$ SU(3)1 CFT tower. Similar considerations
apply to the $Q=2$ sector giving the conjugate irreps of the $Q=1$ sector. A
comparison with Tables 3 and 4 shows that the counting is indeed consistent
with ${\overline{3}}$-tower [$\otimes\overline{3}$] and its conjugate,
respectively, up to the 4th Virasoro level (for $Q=1$, at $L_{0}=3$ one
$\bf{\overline{6}}$ irrep lies a bit away from the compact group of the
remaining SU(3)1 irreps). Similar features have been found in a simpler PESS
on the kagome lattice with virtual space ${\cal V}={\bf 1}+{\bf
3}+\overline{\bf 3}$ [43]. It was further established that this PESS belongs
to a $\mathrm{SU}(3)_{1}\times\overline{\mathrm{SU}(3)}_{1}$ double Chern-
Simons phase characterized by a slow SU$(3)_{1}$ chiral mode and a fast
counter-propagating $\overline{\mathrm{SU}(3)}_{1}$ chiral mode [54]. Our
findings suggest that our $D^{*}=4$ ($D=13$) PEPS ansatz also belongs to the
same phase. However, it is unclear whether the presence of a second fast mode
is a genuine feature of the phase or a finite-$D$ effect. Note that the ED
results do not show evidence of the DCS phase: for instance, in Fig. 5 a
3-fold quasi-degenerate ground state manifold is seen on the torus around
$(\phi,\theta)=(0,\pi/4)$, in agreement with the expectation for a chiral SL
(while a 9-fold degenerate ground state is expected in a DCS phase).
Similarly, the ES obtained in the MPS simulations in 3 topological sectors
(see Appendix C.5 for details) differ from the ones obtained in the PEPS
framework, as shown in Fig 9, being compatible with a SU(3)1 CSL.

It would be interesting to analyze whether the level splittings in ES can be
described by a generalized Gibbs ensemble [54]. Very recently [60] such a
refined analysis of the ES enabled to strengthen the evidence for the CSL
nature of a SU(3) PEPS on the square lattice [44]. In that case, it was shown
that the splittings between conjugate irreps in the same Virasoro level of
$Q=0$ sector and between $Q=1$ and $Q=2$ sectors should vanish (and
numerically found to be extremely small compared to the scale of other level
splittings), due to absence of certain conserved quantities which are not
allowed by symmetry. In the present case (Fig. 10), the splittings between
conjugate irreps $(3,0)$ and $(0,3)$ at $L_{0}=4,5$ in the $Q=0$ sector is
noticeable, and the entanglement energies between conjugate irreps in the
$Q=1$ and $Q=2$ sectors also have small but visible differences. On the other
hand, the entanglement spectrum from MPS calculation, shown in Fig. 9, agrees
with the level counting of a chiral SL, but also has a splitting between
conjugate irreps at the same Virasoro level (in $Q=0$ sector or between $Q=1$
and $Q=2$ sectors). A full analysis of the splittings in Fig. 9 and Fig. 10 is
left for future work.

##  V Conclusions

In this work, the investigation of the complete phase diagram of the SU(3)
chiral Heisenberg model on the kagome lattice has been carried out. Physical
spins transforming according to the fundamental irrep of SU(3) are considered
on all lattice sites. The Hamiltonian includes the generic two-site and three-
site couplings on nearest-neighbor sites and triangular units, respectively.
To map out the phase diagram we have combined analytical and numerical tools
such as magnon expansions, exact diagonalisations and tensor network (iMPS and
iPEPS) techniques. In addition to the AKLT phase predicted by one of the
authors (KP) [27] and the (expected) fully polarized ferromagnet (at large
enough ferromagnetic couplings) we have found two gapped singlet phases and a
magnetic phase that spontaneously breaks the SU(3) symmetry.

One of the singlet phases, the trimer phase, breaks spontaneously the
inversion center exchanging up and down triangles, as observed in the pure
SU(3) Heisenberg model [52], a special point in our 2D parameter space. We
have also found an enigmatic topological spin liquid. Although our numerical
results show evidence for a gap and $\mathbb{Z}_{3}$ gauge symmetry (via MPS
and PEPS constructions), the exact nature of this TSL phase is still
controversial with two possible candidates, either a SU(3)1 CSL (proposed in
Ref. [29]) or a double $\mathrm{SU}(3)_{1}\times\mkern
1.5mu\overline{\mkern-1.5mu\mathrm{SU}(3)_{1}\mkern-1.5mu}\mkern 1.5mu$ Chern-
Simons spin liquid (discussed in Refs. [43, 54]).

The not fully polarized SU(3)-broken magnetic phase is, most likely, a uniform
partially polarized ferromagnet with collinear spins in the 3-site unit cell.
Another competing non-uniform phase with spin canting occurring on three
triangular sub-lattices separately (requiring a 9-site unit cell) seems to be
slightly disfavored energetically.

_Acknowledgments_ — We acknowledge the participation of Seydou-Samba Diop
(Ecole Normale Supérieure de Lyon) and Matthew W. Butcher (Rice University) at
an early stage of the project and support from the TNTOP ANR-18-CE30-0026-01
grant awarded by the French Research Council, and the European Research
Council (ERC) under the European Union’s Horizon 2020 research and innovation
programme (grant agreement No 101001604). This work was granted access to the
HPC resources of CALMIP center under the allocations 2022-P1231 and 2022-P0677
as well as GENCI (grant x2021050225). We also acknowledge Jan von Delft for
providing part of the computational resource. J.-Y.C. was supported by Open
Research Fund Program of the State Key Laboratory of Low-Dimensional Quantum
Physics (project No. KF202207), Fundamental Research Funds for the Central
Universities, Sun Yat-sen University (project No. 23qnpy60), a startup fund
from Sun Yat-sen University (No. 74130-12230034), and the Innovation Program
for Quantum Science and Technology 2021ZD0302100. L.V. is supported by the
Research Foundation Flanders. Y.X. and A.H.N. were supported by the Division
of Materials Research of U.S. National Science Foundation under the Award
DMR-1917511. The iPESS and iPEPS calculations at Rice University were
supported in part by the Big-Data Private-Cloud Research Cyber infrastructure
MRI-award funded by NSF under grant CNS-1338099 and by Rice University’s
Center for Research Computing (CRC). K.P. acknowledges support from the
Hungarian NKFIH OTKA Grant No. K142652.

##  Appendix A Analytical developments

###  A.1 Stereographic projection

Figure 11: Stereographic projection of the North hemisphere from the South
pole, mapping every point of the sphere such as $0\leq\theta\leq\pi/2$ and
$0\leq\phi<2\pi$ to its image on the upper planar disk.

The stereographic projection (see Fig. 11) maps the parameter space (see Eq.
(2) for $0\leq\theta\leq\pi/2$ and $0\leq\phi<2\pi$ to a planar disk delimited
by the image of the equator (a circle of radius 2). The image coordinates of
$(\theta,\phi)$ points on the projection plane are given by

$\displaystyle X$ $\displaystyle=$
$\displaystyle\frac{\cos(\theta)\cos(\phi)}{\sin(\theta)+1},$ $\displaystyle
Y$ $\displaystyle=$
$\displaystyle\frac{\cos(\theta)\sin(\phi)}{\sin(\theta)+1}.$

Symbols |  Dynkin labels |  Young tableaus
---|---|---
|  (0,0) |  $\Yvcentermath 1\overset{\bf 1}{\underset{\phantom{.}}{\bullet}}$
|  (1,0) |  $\Yvcentermath 1\underset{\phantom{.}}{\overset{\bf 3}{\Yboxdim 9pt\yng(1)}}$
|  (0,1) |  $\Yvcentermath 1\underset{\phantom{.}}{\overset{\bf\bar{3}}{\Yboxdim 9pt\yng(1,1)}}$
|  (2,0) |  $\Yvcentermath 1\underset{\phantom{.}}{\overset{\bf 6}{\Yboxdim 9pt\yng(2)}}$
|  (0,2) |  $\Yvcentermath 1\underset{\phantom{.}}{\overset{\bf\bar{6}}{\Yboxdim 9pt\yng(2,2)}}$
|  (1,1) |  $\Yvcentermath 1\underset{\phantom{.}}{\overset{\bf 8}{\Yboxdim 9pt\yng(2,1)}}$
|  (3,0) |  $\Yvcentermath 1\underset{\phantom{.}}{\overset{\bf 10}{\Yboxdim 9pt\yng(3)}}$
|  (0,3) |  $\Yvcentermath 1\underset{\phantom{.}}{\overset{\bf\overline{10}}{\Yboxdim 9pt\yng(3,3)}}$
|  (2,1) |  $\Yvcentermath 1\underset{\phantom{.}}{\overset{\bf 15}{\Yboxdim 9pt\yng(3,1)}}$
|  (1,2) |  $\Yvcentermath 1\underset{\phantom{.}}{\overset{\bf\overline{15}}{\Yboxdim 9pt\yng(3,2)}}$
|  (4,0) |  $\Yvcentermath 1\underset{\phantom{.}}{\overset{\bf 15^{\prime}}{\Yboxdim 9pt\yng(4)}}$
|  (0,4) |  $\Yvcentermath 1\underset{\phantom{.}}{\overset{\bf\overline{15}^{\prime}}{\Yboxdim 9pt\yng(4,4)}}$
|  (1,3) |  $\Yvcentermath 1\underset{\phantom{.}}{\overset{\bf 24}{\Yboxdim 9pt\yng(4,3)}}$
|  (3,1) |  $\Yvcentermath 1\underset{\phantom{.}}{\overset{\bf\overline{24}}{\Yboxdim 9pt\yng(4,1)}}$
|  (2,2) |  $\Yvcentermath 1\underset{\phantom{.}}{\overset{\bf 27}{\Yboxdim 9pt\yng(4,2)}}$
|  (4,1) |  $\Yvcentermath 1\underset{\phantom{.}}{\overset{\bf 35}{\Yboxdim 9pt\yng(5,1)}}$
|  (1,4) |  $\Yvcentermath 1\underset{\phantom{.}}{\overset{\bf\overline{35}}{\Yboxdim 9pt\yng(5,4)}}$
|  (3,2) |  $\Yvcentermath 1\underset{\phantom{.}}{\overset{\bf 42}{\Yboxdim 9pt\yng(5,2)}}$
|  (2,3) |  $\Yvcentermath 1\underset{\phantom{.}}{\overset{\bf\overline{42}}{\Yboxdim 9pt\yng(5,3)}}$
|  (2,4) |  $\Yvcentermath 1\underset{\phantom{.}}{\overset{\bf 60}{\Yboxdim 9pt\yng(6,4)}}$
|  (4,2) |  $\Yvcentermath 1\underset{\phantom{.}}{\overset{\bf\overline{60}}{\Yboxdim 9pt\yng(6,2)}}$
| Others
Table 1: Correspondence between Dynkin labels and Young tableaus for all SU(3)
irreps discussed in the paper. Symbols used in the various figures are
displayed in the first column. Labeling of the irreps follows conventions from
the LieArt package [61].

###  A.2 SU(3) irreducible representations and conformal towers

Irreducible representations (irreps) of the SU(3) group can be labeled
differently. We show in Table 1 the one-to-one correspondence between Dynkin
labels and Young tableaus. An irrep labeled by $(x,y)$ corresponds to a Young
tableau with two rows containing $x+y$ and $y$ boxes. Conformal towers of
various WZW CFTs mentioned in the main text, originating from $\bf{1}$,
$\bf{\overline{3}}$ [$\otimes\bf{\overline{3}}$] and $\bf{3}$
[$\otimes\bf{3}$], are shown in Tables 2,3 and 4, respectively. See also Ref.
[59] for the towers originating from $\bf{3}$ and $\bf{\overline{3}}$.

###  A.3 Single magnon dispersion

The dispersion of a single magnon is determined by the eigenvalues of the
matrix in reciprocal space given in Eq. 4 below, where the energy is measured
from the energy of the ferromagnetic state and $\mathbf{q}=(q_{x},q_{y})$ is
the momentum of the magnon. The $J+K_{R}$ appear together in the matrix above,
so the dispersion depends only on two free parameters $J+K_{R}$ and $K_{I}$.

$\left(\begin{array}[]{ccc}-4(J+K_{R})&2(J+K_{R}-iK_{I})\cos\frac{q_{x}-\sqrt{3}q_{y}}{2}&2(J+K_{R}+iK_{I})\cos\frac{q_{x}+\sqrt{3}q_{y}}{2}\\\
2(J+K_{R}+iK_{I})\cos\frac{q_{x}-\sqrt{3}q_{y}}{2}&-4(J+K_{R})&2(J+K_{R}-iK_{I})\cos
q_{x}\\\
2(J+K_{R}-iK_{I})\cos\frac{q_{x}+\sqrt{3}q_{y}}{2}&2(J+K_{R}+iK_{I})\cos
q_{x}&-4(J+K_{R})\\\ \end{array}\right)$ (4)

$N-1$ states belong to the $\mathbf{d}=(N-1)(N+1)$ dimensional Young diagram
with $(N-2,1)$ Dynkin label, and the state with zero energy and $\mathbf{q}=0$
to the $\mathbf{d}=(N+1)(N+2)/2$ dimensional fully symmetrical irreducible
representation of the ferromagnetic state, represented by the Young diagram
$(N,0)$.

###  A.4 Two-magnon spectra

Figure 12: The Young diagrams appearing in the $[N_{A},N_{B},N_{C}]=[N-2,1,1]$
sector of the two magnon calculations, labeled by their Dynkin indices.
$N_{A}$ is the number of sites having $A$ spins, and so on, so that
$N_{A}+N_{B}+N_{C}=N$.  Figure 13:  The ground state diagram in the two-magnon
sector. The blue line denotes the 1-magnon instability line. The two magnons
added to the system can form a symmetrical [Young diagram with Dynkin label
$(N-4,2)$, blue shaded region] or antisymmetrical [Young diagram $(N-3,0)$,
red shaded region] combination; the regions indicate regions where they are
the lowest energy states. The boundaries for system sizes from 27 to 81 are
drawn in red lines.

This section considers two magnon excitations of the fully-symmetrical
(ferromagnetic) state, where we introduce two spins with different flavors,
following the calculation of Refs. 62, 63 for the SU(2) case. Starting from
the $|AA\dots A\rangle$ as a vacuum, the two-magnon wave function is

$\Psi=\sum_{i,j\in\Lambda}c_{i,j}|A\dots AB_{i}A\dots AC_{j}A\dots
A\rangle\;.$ (5)

The dimension of the Hilbert space spanned by $|A\dots AB_{i}A\dots
AC_{j}A\dots A\rangle$ basis is $N(N-1)$, as $i=1,\dots,N$ and $j=1,\dots,N$,
but the $B$ and $C$ cannot occupy the same site ($i\neq j$). Furthermore, we
symmetrize and antisymmetrize the wave functions so that

$\displaystyle c^{e}_{i,j}$ $\displaystyle=c_{i,j}+c_{i,j}\,,$ $\displaystyle
c^{o}_{i,j}$ $\displaystyle=c_{i,j}-c_{i,j}\,.$

The dimensions of the symmetric “e” (even) and of the antisymmetric “o” (odd)
subspace are the same and equal to $N(N-1)/2$. The even subspace is composed
of the $(N,0)$, $(N-2,1)$, and the $(N-4,2)$ Young diagrams (see Fig. 12),
each having multiplicities $1$, $N-1$, and $N(N-3)/2$, respectively. The
irreducible representations in the odd subspace are $(N-2,1)$ and the
$(N-3,0)$ Young diagrams with multiplicities $N-1$ and $(N-2)(N-1)/2$. Using
the Casimir operator, we separate the energies of the $(N-4,2)$ and $(N-3,0)$
irreducible representations. The symmetric, even sector would also appear in
the SU(2) case since symmetrization is equivalent to taking two $B$ type spins
instead of $B$ and $C$. The odd (antisymmetric) sector is unique to the SU(3).
We diagonalized the Hamiltonian matrix for up to 81-site clusters numerically.
Since this is a two-body problem, one might derive analytic expressions in
principle, but they would be quite cumbersome.

Figure 14:  The one- and two-magnon spectra for the 27-site cluster relative
to the energy of the ferromagnetic state with $(27,0)$ Young diagram. Shown
are the energies of the one-magnon states [green, (25,1) Young diagram] and
the symmetric [blue,(25,1)] and antisymmetric [red,(24,0)] two-magnon states.
(a) Varying the $J$ and $K$ while keeping the $J+K_{R}=K_{I}/\sqrt{3}=-1$
constant follows the one-magnon instability line. The energies of the lowest
unbound states are equal; for positive values of $J$, 18 two-magnon bound
states detach from the continuum (red lines with $E_{(24,0)}<E_{(27,0)}$). (b)
The one- (continuous green curves) and two-magnon (open symbols) energies for
$K_{I}=0$ (i.e. $\theta=0$). The two-magnon bound states appear for
$-\pi/2<\phi<0$.

In Sec. III.5, we derived the conditions for the stability of one-magnon
excitations. When the energy of the one-magnon is larger than that of the
ferromagnetic state, the ferromagnetic phase is the ground state. If the
energy needed to create a magnon is negative, the ferromagnetic phase is not a
ground state anymore. The importance of the two magnon calculation is to
reveal the interaction between the magnons. The ferromagnetic phase shrinks if
the magnons attract each other and form bound states.

Fig. 13 summarizes the result of our calculation. It shows the Young diagram
(YD) having the lowest energy for two magnons. We can distinguish three
regions: the $(N,0)$ ferromagnetic phase (the gray area), the $(N-4,2)$ for
the symmetric combination of the two magnons (the blue area), and the red-
colored area where the antisymmetric combination of $(N-3,0)$ Young diagram is
the ground state. The boundary between the $(N,0)$ and the $(N-4,2)$ follows
the one-magnon instability line for negative values of $J$, but at
$\theta=\pi/3$ and $\phi=3\pi/2$ corresponding to $J=0$, $K_{R}=-1$ and
$K_{I}=\sqrt{3}$ the three regions meet, and the $(N-3,0)$ antisymmetric
combination becomes the ground state. The boundary between the ferromagnetic
phase and the $(N-3,0)$ no longer follows the one-magnon instability line. It
is a hint for a formation of a bound state.

To get further insight, we plot in Fig. 14(a) the energies of the different
irreducible representations along the $J+K_{R}=\pm\sqrt{3}K_{I}$ one-magnon
instability line for a 27 site cluster. The lowest energies of the ferro state
$(N,0)$, the one magnon $(N-2,1)$, and the two magnons in a symmetric
combination $(N-4,2)$ are all equal (we note that the energies in these
irreducible representations depend on the $J+K_{R}$ combination only). In the
$(N-3,0)$ antisymmetric sector, a band of bound-states appears with lower
energy for $J\gtrsim 0$ in the figure, where we keep $J+K_{R}=-1$ constant (in
the thermodynamic limit, the triple point is at $J=0$, $K_{R}=-1$, and
$K_{I}=\sqrt{3}$, i.e. $\theta=\pi/3$ and $\phi=3\pi/2$). The number of bound-
states is 18, which is equal to the number of triangles in the 27-site kagome
cluster. We also confirmed that the number of bound states is $2N/3$ in other
clusters. In Fig. 14(b), we plot the energy gap to the ferromagnetic state
around the full circle, keeping $K_{I}=0$ (i.e., $\theta=0$). At the special
point $\phi=-\pi/4$, which corresponds to $J=-K_{R}$ with $J$ positive, the
spectrum of the $(N-3,0)$ states greatly simplifies: we get a $2N/3$-fold
degenerate manifold at $E=-6J$, and all the other energy levels collapse at
$E=0$. The explanation is simple: the $B$, $C$ and an $A$ form a localized
SU(3) singlet on a triangle, while the remaining $N-3$ $A$ spins constitute
the ferromagnetic background. The singlets can form on any of the $2N/3$
elementary triangles in the kagome lattice, and this is the origin of the
degeneracy. The result of a finite-size scaling for the boundary between the
ferromagnetic state and the $(N-3,0)$ state is presented in Fig. 15 for
$K_{I}=0$. We get $K_{R}/J=-2.7532$, which corresponds to $\theta=1.611\pi$.
Figure 16 shows the energy gap in the full parameter space. In this region
where the gap is finite, the ground state in the $N_{A}=N_{B}=N_{C}=N/3$
sector is the trimerized state. We can think of it as a condensation of the
local SU(3) singlets with a repulsive interaction between them.

Figure 15:  The finite size scaling of the boundary between the $(N,0)$
ferromagnetic state and the $(N-3,0)$ state for $K_{I}=0$. The boundary for
the $3L^{2}$ and $9L^{2}$ type clusters goes to the same $K_{R}/J=-2.7532$
value in the thermodynamic limit, though the slopes in $1/N^{3}$ are quite
different.

The dispersion of a single magnon in the ferromagnetic background is flat
along the one-magnon instability line. The flat bands are connected with modes
localized on hexagons, also with delocalized modes, since a dispersing band
touches the flat band. When we diagonalize the two-magnon spectrum along the
instability line $J+K_{R}=\pm\sqrt{3}K_{I}$, the number of states degenerate
with the ferromagnet is $\binom{N_{\text{hex}}-1}{2}$ for the symmetric
$\text{YD}=(N-4,2)$ and $\binom{N_{\text{hex}}}{2}$ for the antisymmetric
$\text{YD}=(N-3,0)$ combination, where $N_{\text{hex}}=N/3$ is the number of
hexagons.

Figure 16:  The value of the energy gap $E(N-3,0)-2E(N-2,1)+E(N,0)$ for the
$75$-site cluster. The bound state appears when the gap is negative; this is
the green area in the plot.

##  Appendix B Lanczos Exact Diagonalization

Figure 17:  Stereographic projection (same parametrisation as in Fig. 1) of
the phase diagram of the SU(3) chiral antiferromagnet on the kagome lattice
obtained from ED on a 21-site periodic cluster. The various phases discussed
in the text are represented by dots of different colors. The dashed (dash-
dotted) line corresponds to the single (two) magnon instability of the
ferromagnetic phase.

We have performed ED on various lattices using space group symmetries as well
as color conservation (equivalent to the conservation of the 2 U(1) Cartan
generators of SU(3)). By using a Lanczos algorithm, we are able to converge a
few low-energy states in each symmetry sector. A typical energy plot is shown
for instance in the bottom panel of Fig. 4.

In order to sketch a tentative phase diagram using a single system size, we
have computed systematically the low-energy spectrum on the 21-site kagome
cluster with periodic boundary conditions on a fine $(\phi,\theta)$ parameter
grid. For each set of parameters, we have attempted in Fig. 17 to determine
its ground-state (GS) properties using the following criteria:

  * •

ferromagnetic phase: the finite-size GS belongs to the fully symmetric irrep
and its energy is known exactly.

  * •

AKLT: the ground-state is non-degenerate and there is an apparent large gap to
the first excitation.

  * •

Trimerized: there are two low-energy singlet states in the $\Gamma$.A and
$\Gamma$.B irreps, as expected if the inversion symmetry is broken in the
thermodynamic limit.

  * •

CSL: there are three low-energy singlet states at momentum $\Gamma$ as
expected for a chiral spin liquid on a torus.

  * •

SU(3)-broken: either the GS is not an SU(3) singlet, or there is a small gap
to a non-singlet state. This could be a critical state or a canted
ferromagnet.

By using these rules, we are able to plot a qualitative phase diagram in Fig.
17.

Note that finite-size effects have been shown to be important in some regions.
For instance, our ED data on the 21-site cluster are rather similar at the
exact AKLT point $(\phi=\pi/2,\theta=0)$ and close to the North pole (see Fig.
4) so that both regions are labelled in the same way on the phase diagram in
Fig. 17. However, the situation is radically different on the 27-site cluster
(see inset of Fig. 4), which rather indicates an SU(3)-broken phase in a large
region around the North pole. Hence, it is crucial to combine different
numerical methods in order to get reliable results.

##  Appendix C Cylinder MPS simulations

###  C.1 Geometry

There are different possible geometries possible for putting the kagome
lattice on an infinite cylinder. The most common one is the YC cylinder, shown
in Fig. 18(a). Here we choose a slightly different geometry, shown in Fig.
18(b), where we have shifted the periodic boundary conditions with a single
triangle in the horizontal direction. This shifted geometry has the advantage
that the unit cell of the resulting one-dimensional Hamiltonian has a single-
triangle unit cell.

Figure 18:  Different geometries for the kagome lattice on an infinite
cylinder with a three-triangle unit cell in the periodic direction; the grey-
shaded triangles are identified.

###  C.2 MPS ground states

The shifted geometry has the additional advantage that also the MPS ground-
state approximation can be chosen to have a single-triangle unit cell. We can
represent this MPS as

$\ket{\Psi(A)}=\quad\vbox{\hbox{\includegraphics[scale={0.4},page=3]{Figures/diagrams.pdf}}}\quad,$
(6)

where for convenience we have grouped the three physical $\mathbf{3}$ spins in
a single MPS tensor – in the simulations we always keep three different MPS
tensors. We impose $\mathrm{SU}(3)$ symmetry on the MPS, which implies that
the virtual degrees of freedom in the MPS can be labeled by $\mathrm{SU}(3)$
irreps. The different blocks in the MPS tensor need to obey the
$\mathrm{SU}(3)$ fusion rules, i.e. we need virtual irreps $I_{v}$ and
$I_{v}^{\prime}$

$\quad\vbox{\hbox{\includegraphics[scale={0.4},page=4]{Figures/diagrams.pdf}}}\quad.$
(7)

Now we use the $\mathbb{Z}_{3}$ property of the $\mathrm{SU}(3)$ fusion rules,
where we can group the irreps in three different groups with a
$\mathbb{Z}_{3}$ charge:

$\begin{cases}\overline{\mathbf{3}},\mathbf{6},\dots:Q=-1\\\
\mathbf{1},\mathbf{8},\dots:Q=0\\\
\mathbf{3},\overline{\mathbf{6}},\dots:Q=+1\\\ \end{cases}$ (8)

The three physical spins transform jointly as $Q=0$ irreps, so the
$\mathbb{Z}_{3}$ property of the fusion rules dictates that $I_{v}$ and
$I_{v}^{\prime}$ can only contain irreps from one and the same group, and,
therefore, that we have three classes of MPS labeled by the $\mathbb{Z}_{3}$
charge of the irreps on the bonds. Depending on the phase we are simulating,
the optimal iMPS ground state will be found in one or more of these classes.

The diagnostics for deciding whether an optimal iMPS is found within a certain
$\mathbb{Z}_{3}$ sector is through the entanglement spectrum and transfer
matrix spectrum. In the following, we illustrate this procedure for the
different SU(3) singlet phases in the phase diagram. In Figs. 19, 20 and 21,
we plot the entanglement spectrum and transfer matrix spectrum for three
different parameter choices, each time with iMPS optimized in the three
different classes. For the entanglement spectrum we plot the different
entanglement eigenvalues or Schmidt values with magnitude on the axis; the
different colors in Figs. 19-21 correspond to the different SU(3) quantum
numbers, which are labeled on the horizontal axis by their Dynkin label. For
the transfer matrix spectrum (unit circles in the complex plane exhibited in
the insets), we show a few dominant eigenvalues in the first two SU(3) sectors
(again denoted by their Dynkin label) in the complex plane.

###  C.3 AKLT phase

Let us first consider the exact AKLT state, which is represented as a PESS
with virtual irrep $\overline{\mathbf{3}}$. On an infinite cylinder, this
state can be represented as a snake-like iMPS by dragging the virtual links
along the MPS around the cylinder. The virtual links of the iMPS then contain
a number of $\overline{\mathbf{3}}$ irreps that scales with the circumference
of the cylinder, which are then fused into a number of SU(3) irreps on the
virtual leg of the iMPS. Therefore, the $\mathbb{Z}_{3}$ quantum number of the
MPS depends on the cylinder circumference.

We have now optimized iMPSs in the three different classes for the $L_{y}=4$
cylinder. The resulting entanglement spectra and transfer matrix spectra can
be found in Fig. 19. As one can see from these figures, only one of the three
choices of virtual irreps gives rise to an injective MPS upon optimization, in
this case the $Q=1$ irreps. When choosing the other virtual irreps we find a
non-injective MPS, which can be seen from the degeneracies in the entanglement
spectrum and the fact that we find a transfer-matrix eigenvalue on the unit
circle in a non-trivial sector (depicted as insets in Fig. 19).

Figure 19:  MPS entanglement spectra of the AKLT state ($\theta=0$,
$\phi=\pi/2$) on an $L_{y}=4$ cylinder. The pseudo-energies of the ES are
sorted according to the $\mathbb{Z}_{3}$ charge (0 to 2 from top to bottom)
and the SU(3) irreps (defined by Dynkin labels) and displayed with decreasing
magnitude along the horizontal axis (arbitrary units). The insets show the
transfer matrix spectra in the complex plane – the real (imaginary) part being
associated to the correlation length (spatial oscillations of the correlation
function). We have imposed the three different groups of irreps on the bonds.
Only the middle spectrum corresponds to an injective MPS (irreps with $Q=1$),
whereas the top and bottom correspond to non-injective MPS obtained by
artificially adding a virtual $\mathbf{3}$ or $\overline{\mathbf{3}}$ bond.
The exact degeneracies in the top and bottom entanglement spectra (in
different $\mathrm{SU}(3)$ sectors) are the signatures of adding this extra
virtual irrep. In addition, the occurrence of a transfer matrix eigenvalue in
the $(1,1)$ sector on the unit circle points to a non-injective MPS.

###  C.4 Trimerized phase

We can play the same game in the trimerized phase. In this phase, the ground
state has a strong inclination to form singlets on the triangles, and the iMPS
geometry will clearly favor the trimerization to happen on the up triangles.
The fully-trimerized state (product state of trimers on up-triangles) is
represented by the above MPS, with $I_{v}=I_{v}^{\prime}=\mathbf{1}$ on the
bonds. Therefore, all iMPSs in this phase are adiabatically connected with
this product state and will have virtual irreps with $Q=0$, irrespective of
the cylinder circumference.

As an illustration, in Fig. 20 we have plotted the MPS entanglement and
transfer matrix spectra for the MPS ground state in the point $\theta,\phi=0$
for circumference $N=4$. Clearly, only the choice of $Q=0$ irreps leads to the
correct MPS ground state, whereas choosing $Q=\pm 1$ leads to non-injective
iMPSs.

Figure 20:  MPS entanglement (main) and transfer matrix (inset) spectra of an
optimized MPS in the trimer phase ($\theta=0$, $\phi=0$) on an $L_{y}=4$
cylinder (same display as in Fig. 19), where we have imposed the three
different groups of irreps on the bonds. Only the top spectrum corresponds to
an injective MPS (irreps with $Q=0$), whereas the lower two panels correspond
to non-injective MPS obtained by artificially adding a virtual $\mathbf{3}$ or
$\overline{\mathbf{3}}$ bond. Again, the exact degeneracies in these two
entanglement spectra are the signatures of adding this extra virtual irrep.
The bond dimension of the $Q=0$ MPS is around $\chi\approx 7000$.

###  C.5 Topological spin liquid phase

In the spin-liquid phase, the situation is different because we expect three
distinct ground states on the infinite cylinder, which are labeled by the
$\mathbb{Z}_{3}$ charge. Indeed, we find that the leading eigenvalues of the
MPS entanglement spectrum are the same up to the fourth significant digit in
the three charge sectors $Q=0,1$ and 2 (see Fig. 21). The degeneracy is
exponential in the circumference of the cylinder.

The 3-fold degenerate nature of the TSL ground state is also corroborated by
the leading eigenvalue of the iMPS transfer matrix, which lies on the unit
circle in the imaginary plane and is degenerate among all three charge sectors
(see insets in Fig. 21).

Figure 21:  MPS entanglement (main) and transfer matrix (inset) spectra of an
optimized MPS in the TSL phase ($\theta=\frac{\pi}{4}$, $\phi=0$) on an
$L_{y}=4$ cylinder (same display as in Fig. 19), where we have imposed the
three different groups of irreps on the bonds. All three choices give rise to
injective MPS (as expected in a TSL phase), and no artificial degeneracies are
observed in the entanglement spectra or transfer matrix spectra. The energy
densities are almost equal: $e_{Q=0}=-0.7318557278$, $e_{Q=+1}=-0.7317589213$,
$e_{Q=-1}=-0.7318473342$ and the total bond dimensions are all around
$\chi\approx 12000$.

###  C.6 Estimating the gap

In order to estimate the gap, we apply the quasiparticle excitation ansatz. In
this context, this boils down to applying as a variational ansatz the state

$\ket{\Phi_{q}^{s}(B)}=\sum_{n}\mathrm{e}^{iqn}\quad\vbox{\hbox{\includegraphics[scale={0.4},page=5]{Figures/diagrams.pdf}}}\quad,$
(9)

which has well-defined momentum $q$ and $\mathrm{SU}(3)$ quantum number $s$.
Note that the $\mathrm{SU}(3)$ fusion rules dictate that $s$ should have a
quantum number with $Q=0$, i.e. we have $s=\mathbf{1},\mathbf{8},\dots$. We
can variationally optimize the tensor $B$ for every momentum $q$ and in each
sector $s$, yielding a variational dispersion relation. By choosing the
shifted boundary conditions, the momentum quantum number follows one
continuous line through the 2-D Brillouin zone. Note that, if we have multiple
ground states (in the spin liquid phase), we can build domain wall states that
carry fractional quantum numbers (i.e., other quantum numbers $s$). The
description of these spinon excitations is not further pursued here.

##  Appendix D Projected Entangled Simplex States (PESS) and Pair States
(PEPS)

###  D.1 General formulation

PESS: The wavefunction for the 1-triangle unit cell is defined as a product of
3 site projectors and 2 trivalent tensors, $(B_{a})^{s_{a}}_{ip}$,
$(B_{b})^{s_{c}}_{ql}$, $(B_{c})^{s_{c}}_{rs}$, $(T_{d})_{pqr}$,
$(T_{u})_{sjk}$, given by

$\displaystyle|\psi(s_{a},s_{b},s_{c})\rangle$
$\displaystyle=(B_{a})^{s_{a}}_{ip}(T_{d})_{pqr}(B_{b})^{s_{b}}_{ql}(B_{c})^{s_{c}}_{rs}(T_{u})_{sjk}$
$\displaystyle=\vbox{\hbox{\includegraphics[width=75.88481pt]{Figures/XX_PESS_1-triangle.pdf}}}.$
(10)

PEPS: In PEPS construction, each down (or up) triangle is considered as the
basic building block which tiles the entire lattice. The wavefunction for the
three sites in each down (or up) triangle is given by a single rank-5 PEPS
tensor which fuses all the physical degrees of freedom together.

$|\psi(s_{a},s_{b},s_{c})\rangle=a^{S}_{uldr},\ S=s_{a}s_{b}s_{c}$ (11)

###  D.2 1-triangle PEPS phase diagram

In this section, we describe the phase diagram obtained using a 1-triangle
iPEPS (see Fig. 22), revealing significant differences compared to the one
shown in Fig. 17.

The criteria for identifying different phases are elaborated as follow. First,
states in the SU(3) broken phase (in magenta) has non-zero magnetization (a
threshold of $m_{1}=0.01$ is chosen while the maximum allowed value is
$m_{0}=2/\sqrt{3}$ for the Ferro states in red). Then, one computes the
projection operators onto $\bf 1$, $\bf 8$, $\bf 8$, $\bf 10$ respectively for
down and up triangles. If there is a inversion symmetry breaking on up and
down triangles, the state is identified as the trimer state. For those states
preserving the inversion symmetry, if the two dominant irreducible
representations are the two $\bf 8$, the state is identified as the AKLT
phase. Otherwise, if the second dominant irreducible representation is $\bf
1$, the state is identified as the CSL state.

Figure 22:  Stereographic projection of the phase diagram (same
parametrisation as in Figs. 1 and 17) of the SU(3) chiral antiferromagnet on
the kagome lattice obtained by a 1-triangle PEPS ansatz. The various phases
discussed in the text are represented by dots of different colors. The dashed
(dash-dotted) line corresponds to the single (two) magnon instability of the
ferromagnetic phase.

###  D.3 3-triangle unit cell PESS and PEPS

For the 3-triangle unit cell PESS and PEPS ansatzes, the unit cell is extended
along one direction to contain 9 sites (3 triangles). Neither of them, if not
explicitly pointed out, impose constraints of point group symmetry on the
tensors. For the 3-triangle PESS ansatz, there are three independent sets of
PESS tensors for the three triangles, i.e., $5\times 3=15$ PESS tensors –
$\\{T_{d}^{\alpha},T_{u}^{\alpha},B_{a}^{\alpha},B_{b}^{\alpha},B_{c}^{\alpha}\\}$,
$\alpha=1,2,3$. For the 3-triangle PEPS ansatz, a product of three independent
1-triangle PEPS tensors, $(a^{\alpha})^{S}_{uldr}$, $\alpha=1,2,3$, are used
to represent the physical wavefunction for the 9 sites in the 3 triangles.

For the 3-triangle unit cell, there are three choices of the lattice vectors
to tile the entire kagome lattice, while two of them are related by mirror
symmetry, as shown in Fig. 23. The tiling with equal length lattice vectors
has the C3 rotation symmetry, and thus is denoted by $\sqrt{3}\times\sqrt{3}$
tiling. The other two are simply referred to as $3\times 1$ tiling.

Figure 23:  Three choices of lattice vectors that tile the kagome lattice by
the 3-triangle unit cell.

The C3 PESS is a constrained kind of 3-triangle PESS with
$\sqrt{3}\times\sqrt{3}$ tiling whose physical wavefunction has C3 lattice
rotation symmetry, and is constructed using only one 1-triangle PESS by
rotating the PESS tensors for the first triangle to obtain the PESS tensors
for the other two. There are two different ways of rotation, which give rise
to two different patterns, as shown in Fig. 24. The corresponding relations
for PESS tensors in different triangles are given as follow:

$\text{C3-1
PESS:}\left\\{\begin{array}[]{l}(T_{d}^{1})_{pqr}=(T_{d}^{2})_{rpq}=(T_{d}^{3})_{qrp}\\\
(T_{u}^{1})_{sjk}=(T_{u}^{2})_{ksj}=(T_{u}^{3})_{jks}\\\
(B_{a}^{1})_{ip}=(B_{b}^{2})_{lq}=(B_{c}^{2})_{sr}\\\
(B_{b}^{1})_{ql}=(B_{c}^{2})_{rs}=(B_{a}^{3})_{pi}\\\
(B_{c}^{1})_{rs}=(B_{a}^{2})_{pi}=(B_{b}^{3})_{ql}\end{array}\right.$ (12)

$\text{C3-2
PESS:}\left\\{\begin{array}[]{l}(T_{d}^{1})_{pqr}=(T_{d}^{2})_{qrp}=(T_{d}^{3})_{rpq}\\\
(T_{u}^{1})_{sjk}=(T_{u}^{2})_{sjk}=(T_{u}^{3})_{sjk}\\\
(B_{a}^{1})_{ip}=(B_{c}^{2})_{sr}=(B_{b}^{2})_{lq}\\\
(B_{b}^{1})_{ql}=(B_{a}^{2})_{pi}=(B_{c}^{3})_{rs}\\\
(B_{c}^{1})_{rs}=(B_{b}^{2})_{ql}=(B_{a}^{3})_{pi}\end{array}\right.$ (13)

Figure 24:  The patterns of the PESS tensors for (a) C3-1 PESS and (b) C3-2
PESS. The bond tensors (site projectors) in the same color are forced to be
identical, with the same index always contracted with the same type of
trivalent tensor (either $T_{d}$ or $T_{u}$). For each trivalent tensor, legs
of different lengths stand for different indices. For down or up trivalent
tensors in different triangles, the legs of the same length correspond to each
other.

###  D.4 Lattice symmetry breaking in SU(3)-broken phase?

In Fig. 6 of the main text, we have shown the C3-2 iPESS results (red
triangles), while having fewer variational parameters than the 1-triangle
iPEPS ansatz with the same bond dimension, can establish lower energy states
with additional lattice symmetry breaking patterns. Here, we make a more
detailed scaling analysis of the energetics at one point,
$(\phi,\theta)=(\frac{3\pi}{4},\frac{\pi}{8})$, where the potential lattice
symmetry breaking happens, as shown by Fig. 25. First, one can see that the
extrapolation of the energies from 1-triangle iPEPS wave function already
gives a value which is very close to the $N=21$ ED result (with a difference
smaller than $3\times 10^{-3}$). Second, as the bond dimension increases, one
can see the energy gap between the uniform states and lattice symmetry broken
states decreases. Based on these facts, we tend to attribute the lattice
symmetry breaking we observe to the finite bond dimension effects. In other
words, with small bond dimensions (low entanglement), the states gain more
energy by breaking the lattice symmetry. A similar phenomenon has also been
observed in an SU(3) model on the honeycomb lattice [64]. But still, to clear
up the issue, one needs to go for larger bond dimensions, which unfortunately
goes beyond our current computational capability.

Figure 25:  Scaling of energetics for iPEPS and iPESS wave functions with
respect to bond dimensions at $(\phi,\theta)=(\frac{3\pi}{4},\frac{\pi}{8})$.
The corresponding spatial patterns of different ansatzes only appear with
large bond dimensions (indicated by opaque markers). 1-triangle iPEPS wave
function (green squares) gives a uniform pattern with all the spins pointing
towards the same direction. $\sqrt{3}\times\sqrt{3}$ iPEPS and C3-2 iPESS
(blue and red triangles) wave functions both give a C3-rotation symmetry
broken (with respect to the center of each hexagon) pattern. $3\times 1$ iPEPS
wave function (brown diamonds) gives a partially lattice translation symmetry
broken pattern along the direction in which the unit cell is enlarged.

###  D.5 SU(3)-symmetric PESS

The SU(3) symmetric PESS is a constrained family of 1-site PESS, where each
tensor is invariant under SU$(3)$ symmetry. In addition, for the AKLT phase
and the CSL phase, the SU(3) PESS is further constrained by the lattice point
group symmetry.

In practice, first the relevant SU(3) irreps in the virtual spin are found
using the simple update method [65]. Then a tensor classification is carried
out for both the site projectors and trivalent tensors. The classification
scheme follows from Ref. [66, 41], which was recently adapted to the kagome
lattice [67].

$L_{0}$ | Irreps / Multiplicities
---|---
$0$ |  1${\fcolorbox{gray!70}{gray!70}{\textcolor{white}{\footnotesize\bf 1}}}\;\overset{\bf 1}{\underset{\phantom{.}}{\bullet}}$
$1$ |  1${\fcolorbox{gray!70}{gray!70}{\textcolor{white}{\footnotesize\bf 1}}}\;\underset{\phantom{.}}{\overset{\bf 8}{\Yboxdim 9pt\yng(2,1)}}$
$2$ |  1${\fcolorbox{gray!70}{gray!70}{\textcolor{white}{\footnotesize\bf 1}}}\;\overset{\bf 1}{\underset{\phantom{.}}{\bullet}}\oplus{\fcolorbox{gray!70}{gray!70}{\textcolor{white}{\footnotesize\bf 2}}}\;\underset{\phantom{.}}{\overset{\bf 8}{\Yboxdim 9pt\yng(2,1)}}$
$3$ |  1${\fcolorbox{gray!70}{gray!70}{\textcolor{white}{\footnotesize\bf 2}}}\;\overset{\bf 1}{\underset{\phantom{.}}{\bullet}}\oplus{\fcolorbox{gray!70}{gray!70}{\textcolor{white}{\footnotesize\bf 3}}}\;\underset{\phantom{.}}{\overset{\bf 8}{\Yboxdim 9pt\yng(2,1)}}\oplus{\fcolorbox{gray!70}{gray!70}{\textcolor{white}{\footnotesize\bf 1}}}\;\underset{\phantom{.}}{\overset{\bf 10}{\Yboxdim 9pt\yng(3)}}\oplus{\fcolorbox{gray!70}{gray!70}{\textcolor{white}{\footnotesize\bf 1}}}\;\underset{\phantom{.}}{\overset{\bf\overline{10}}{\Yboxdim 9pt\yng(3,3)}}$
$4$ |  1${\fcolorbox{gray!70}{gray!70}{\textcolor{white}{\footnotesize\bf 3}}}\;\overset{\bf 1}{\underset{\phantom{.}}{\bullet}}\oplus{\fcolorbox{gray!70}{gray!70}{\textcolor{white}{\footnotesize\bf 6}}}\;\underset{\phantom{.}}{\overset{\bf 8}{\Yboxdim 9pt\yng(2,1)}}\oplus{\fcolorbox{gray!70}{gray!70}{\textcolor{white}{\footnotesize\bf 1}}}\;\underset{\phantom{.}}{\overset{\bf 10}{\Yboxdim 9pt\yng(3)}}\oplus{\fcolorbox{gray!70}{gray!70}{\textcolor{white}{\footnotesize\bf 1}}}\;\underset{\phantom{.}}{\overset{\bf\overline{10}}{\Yboxdim 9pt\yng(3,3)}}\oplus{\fcolorbox{gray!70}{gray!70}{\textcolor{white}{\footnotesize\bf 1}}}\;\underset{\phantom{.}}{\overset{\bf 27}{\Yboxdim 9pt\yng(4,2)}}$
$5$ |  1${\fcolorbox{gray!70}{gray!70}{\textcolor{white}{\footnotesize\bf 4}}}\;\overset{\bf 1}{\underset{\phantom{.}}{\bullet}}\oplus{\fcolorbox{gray!70}{gray!70}{\textcolor{white}{\footnotesize\bf 10}}}\;\underset{\phantom{.}}{\overset{\bf 8}{\Yboxdim 9pt\yng(2,1)}}\oplus{\fcolorbox{gray!70}{gray!70}{\textcolor{white}{\footnotesize\bf 3}}}\;\underset{\phantom{.}}{\overset{\bf 10}{\Yboxdim 9pt\yng(3)}}\oplus{\fcolorbox{gray!70}{gray!70}{\textcolor{white}{\footnotesize\bf 3}}}\;\underset{\phantom{.}}{\overset{\bf\overline{10}}{\Yboxdim 9pt\yng(3,3)}}\oplus{\fcolorbox{gray!70}{gray!70}{\textcolor{white}{\footnotesize\bf 2}}}\;\underset{\phantom{.}}{\overset{\bf 27}{\Yboxdim 9pt\yng(4,2)}}$
$6$ |  1${\fcolorbox{gray!70}{gray!70}{\textcolor{white}{\footnotesize\bf 8}}}\;\overset{\bf 1}{\underset{\phantom{.}}{\bullet}}\oplus{\fcolorbox{gray!70}{gray!70}{\textcolor{white}{\footnotesize\bf 16}}}\;\underset{\phantom{.}}{\overset{\bf 8}{\Yboxdim 9pt\yng(2,1)}}\oplus{\fcolorbox{gray!70}{gray!70}{\textcolor{white}{\footnotesize\bf 5}}}\;\underset{\phantom{.}}{\overset{\bf 10}{\Yboxdim 9pt\yng(3)}}\oplus{\fcolorbox{gray!70}{gray!70}{\textcolor{white}{\footnotesize\bf 5}}}\;\underset{\phantom{.}}{\overset{\bf\overline{10}}{\Yboxdim 9pt\yng(3,3)}}\oplus{\fcolorbox{gray!70}{gray!70}{\textcolor{white}{\footnotesize\bf 5}}}\;\underset{\phantom{.}}{\overset{\bf 27}{\Yboxdim 9pt\yng(4,2)}}$
$7$ |  1${\fcolorbox{gray!70}{gray!70}{\textcolor{white}{\footnotesize\bf 10}}}\;\overset{\bf 1}{\underset{\phantom{.}}{\bullet}}\oplus{\fcolorbox{gray!70}{gray!70}{\textcolor{white}{\footnotesize\bf 27}}}\;\underset{\phantom{.}}{\overset{\bf 8}{\Yboxdim 9pt\yng(2,1)}}\oplus{\fcolorbox{gray!70}{gray!70}{\textcolor{white}{\footnotesize\bf 9}}}\;\underset{\phantom{.}}{\overset{\bf 10}{\Yboxdim 9pt\yng(3)}}\oplus{\fcolorbox{gray!70}{gray!70}{\textcolor{white}{\footnotesize\bf 9}}}\;\underset{\phantom{.}}{\overset{\bf\overline{10}}{\Yboxdim 9pt\yng(3,3)}}\oplus{\fcolorbox{gray!70}{gray!70}{\textcolor{white}{\footnotesize\bf 8}}}\;\underset{\phantom{.}}{\overset{\bf 27}{\Yboxdim 9pt\yng(4,2)}}\oplus{\fcolorbox{gray!70}{gray!70}{\textcolor{white}{\footnotesize\bf 1}}}\;\underset{\phantom{.}}{\overset{\bf\overline{35}}{\Yboxdim 9pt\yng(5,4)}}\oplus{\fcolorbox{gray!70}{gray!70}{\textcolor{white}{\footnotesize\bf 1}}}\;\underset{\phantom{.}}{\overset{\bf 35}{\Yboxdim 9pt\yng(5,1)}}$
Table 2:  Conformal tower in the $Q=0$ topological sector, originating from $\overset{\bf 1}{\underset{\phantom{.}}{\bullet}}$ – reproduced for convenience from Table VI of Ref. [59]. For the other SU(3)1 conformal towers in the $Q=1$ and $Q=2$ topological sectors see Table VII of Ref. [59].  $L_{0}$ | Irreps / Multiplicities
---|---
$0$ |  1${\fcolorbox{gray!70}{gray!70}{\textcolor{white}{\footnotesize\bf 1}}}\;\underset{\phantom{.}}{\overset{\bf 3}{\Yboxdim 9pt\yng(1)}}\oplus{\fcolorbox{gray!70}{gray!70}{\textcolor{white}{\footnotesize\bf 1}}}\;\underset{\phantom{.}}{\overset{\bf\bar{6}}{\Yboxdim 9pt\yng(2,2)}}$
$1$ |  1${\fcolorbox{gray!70}{gray!70}{\textcolor{white}{\footnotesize\bf 2}}}\;\underset{\phantom{.}}{\overset{\bf 3}{\Yboxdim 9pt\yng(1)}}\oplus{\fcolorbox{gray!70}{gray!70}{\textcolor{white}{\footnotesize\bf 1}}}\;\underset{\phantom{.}}{\overset{\bf\bar{6}}{\Yboxdim 9pt\yng(2,2)}}\oplus{\fcolorbox{gray!70}{gray!70}{\textcolor{white}{\footnotesize\bf 1}}}\;\underset{\phantom{.}}{\overset{\bf 15}{\Yboxdim 9pt\yng(3,1)}}$
$2$ |  1${\fcolorbox{gray!70}{gray!70}{\textcolor{white}{\footnotesize\bf 3}}}\;\underset{\phantom{.}}{\overset{\bf 3}{\Yboxdim 9pt\yng(1)}}\oplus{\fcolorbox{gray!70}{gray!70}{\textcolor{white}{\footnotesize\bf 3}}}\;\underset{\phantom{.}}{\overset{\bf\bar{6}}{\Yboxdim 9pt\yng(2,2)}}\oplus{\fcolorbox{gray!70}{gray!70}{\textcolor{white}{\footnotesize\bf 2}}}\;\underset{\phantom{.}}{\overset{\bf 15}{\Yboxdim 9pt\yng(3,1)}}\oplus{\fcolorbox{gray!70}{gray!70}{\textcolor{white}{\footnotesize\bf 1}}}\;\underset{\phantom{.}}{\overset{\bf 24}{\Yboxdim 9pt\yng(4,3)}}$
$3$ |  1${\fcolorbox{gray!70}{gray!70}{\textcolor{white}{\footnotesize\bf 6}}}\;\underset{\phantom{.}}{\overset{\bf 3}{\Yboxdim 9pt\yng(1)}}\oplus{\fcolorbox{gray!70}{gray!70}{\textcolor{white}{\footnotesize\bf 5}}}\;\underset{\phantom{.}}{\overset{\bf\bar{6}}{\Yboxdim 9pt\yng(2,2)}}\oplus{\fcolorbox{gray!70}{gray!70}{\textcolor{white}{\footnotesize\bf 5}}}\;\underset{\phantom{.}}{\overset{\bf 15}{\Yboxdim 9pt\yng(3,1)}}\oplus{\fcolorbox{gray!70}{gray!70}{\textcolor{white}{\footnotesize\bf 2}}}\;\underset{\phantom{.}}{\overset{\bf 24}{\Yboxdim 9pt\yng(4,3)}}$
$4$ |  1${\fcolorbox{gray!70}{gray!70}{\textcolor{white}{\footnotesize\bf 10}}}\;\underset{\phantom{.}}{\overset{\bf 3}{\Yboxdim 9pt\yng(1)}}\oplus{\fcolorbox{gray!70}{gray!70}{\textcolor{white}{\footnotesize\bf 10}}}\;\underset{\phantom{.}}{\overset{\bf\bar{6}}{\Yboxdim 9pt\yng(2,2)}}\oplus{\fcolorbox{gray!70}{gray!70}{\textcolor{white}{\footnotesize\bf 1}}}\;\underset{\phantom{.}}{\overset{\bf 15^{\prime}}{\Yboxdim 9pt\yng(4)}}\oplus{\fcolorbox{gray!70}{gray!70}{\textcolor{white}{\footnotesize\bf 9}}}\;\underset{\phantom{.}}{\overset{\bf 15}{\Yboxdim 9pt\yng(3,1)}}\oplus{\fcolorbox{gray!70}{gray!70}{\textcolor{white}{\footnotesize\bf 4}}}\;\underset{\phantom{.}}{\overset{\bf 24}{\Yboxdim 9pt\yng(4,3)}}\oplus{\fcolorbox{gray!70}{gray!70}{\textcolor{white}{\footnotesize\bf 1}}}\;\underset{\phantom{.}}{\overset{\bf 42}{\Yboxdim 9pt\yng(5,2)}}$
$5$ |  1${\fcolorbox{gray!70}{gray!70}{\textcolor{white}{\footnotesize\bf 17}}}\;\underset{\phantom{.}}{\overset{\bf 3}{\Yboxdim 9pt\yng(1)}}\oplus{\fcolorbox{gray!70}{gray!70}{\textcolor{white}{\footnotesize\bf 16}}}\;\underset{\phantom{.}}{\overset{\bf\bar{6}}{\Yboxdim 9pt\yng(2,2)}}\oplus{\fcolorbox{gray!70}{gray!70}{\textcolor{white}{\footnotesize\bf 2}}}\;\underset{\phantom{.}}{\overset{\bf 15^{\prime}}{\Yboxdim 9pt\yng(4)}}\oplus{\fcolorbox{gray!70}{gray!70}{\textcolor{white}{\footnotesize\bf 17}}}\;\underset{\phantom{.}}{\overset{\bf 15}{\Yboxdim 9pt\yng(3,1)}}\oplus{\fcolorbox{gray!70}{gray!70}{\textcolor{white}{\footnotesize\bf 1}}}\;\underset{\phantom{.}}{\overset{\bf 21}{\Yboxdim 9pt\yng(5,5)}}\oplus{\fcolorbox{gray!70}{gray!70}{\textcolor{white}{\footnotesize\bf 8}}}\;\underset{\phantom{.}}{\overset{\bf 24}{\Yboxdim 9pt\yng(4,3)}}\oplus{\fcolorbox{gray!70}{gray!70}{\textcolor{white}{\footnotesize\bf 2}}}\;\underset{\phantom{.}}{\overset{\bf 42}{\Yboxdim 9pt\yng(5,2)}}$
$6$ |  1${\fcolorbox{gray!70}{gray!70}{\textcolor{white}{\footnotesize\bf 27}}}\;\underset{\phantom{.}}{\overset{\bf 3}{\Yboxdim 9pt\yng(1)}}\oplus{\fcolorbox{gray!70}{gray!70}{\textcolor{white}{\footnotesize\bf 28}}}\;\underset{\phantom{.}}{\overset{\bf\bar{6}}{\Yboxdim 9pt\yng(2,2)}}\oplus{\fcolorbox{gray!70}{gray!70}{\textcolor{white}{\footnotesize\bf 4}}}\;\underset{\phantom{.}}{\overset{\bf 15^{\prime}}{\Yboxdim 9pt\yng(4)}}\oplus{\fcolorbox{gray!70}{gray!70}{\textcolor{white}{\footnotesize\bf 29}}}\;\underset{\phantom{.}}{\overset{\bf 15}{\Yboxdim 9pt\yng(3,1)}}\oplus{\fcolorbox{gray!70}{gray!70}{\textcolor{white}{\footnotesize\bf 1}}}\;\underset{\phantom{.}}{\overset{\bf 21}{\Yboxdim 9pt\yng(5,5)}}\oplus{\fcolorbox{gray!70}{gray!70}{\textcolor{white}{\footnotesize\bf 15}}}\;\underset{\phantom{.}}{\overset{\bf 24}{\Yboxdim 9pt\yng(4,3)}}\oplus{\fcolorbox{gray!70}{gray!70}{\textcolor{white}{\footnotesize\bf 5}}}\;\underset{\phantom{.}}{\overset{\bf 42}{\Yboxdim 9pt\yng(5,2)}}\oplus{\fcolorbox{gray!70}{gray!70}{\textcolor{white}{\footnotesize\bf 1}}}\;\underset{\phantom{.}}{\overset{\bf 60}{\Yboxdim 9pt\yng(6,4)}}$
$7$ |  1${\fcolorbox{gray!70}{gray!70}{\textcolor{white}{\footnotesize\bf 43}}}\;\underset{\phantom{.}}{\overset{\bf 3}{\Yboxdim 9pt\yng(1)}}\oplus{\fcolorbox{gray!70}{gray!70}{\textcolor{white}{\footnotesize\bf 43}}}\;\underset{\phantom{.}}{\overset{\bf\bar{6}}{\Yboxdim 9pt\yng(2,2)}}\oplus{\fcolorbox{gray!70}{gray!70}{\textcolor{white}{\footnotesize\bf 8}}}\;\underset{\phantom{.}}{\overset{\bf 15^{\prime}}{\Yboxdim 9pt\yng(4)}}\oplus{\fcolorbox{gray!70}{gray!70}{\textcolor{white}{\footnotesize\bf 50}}}\;\underset{\phantom{.}}{\overset{\bf 15}{\Yboxdim 9pt\yng(3,1)}}\oplus{\fcolorbox{gray!70}{gray!70}{\textcolor{white}{\footnotesize\bf 3}}}\;\underset{\phantom{.}}{\overset{\bf 21}{\Yboxdim 9pt\yng(5,5)}}\oplus{\fcolorbox{gray!70}{gray!70}{\textcolor{white}{\footnotesize\bf 26}}}\;\underset{\phantom{.}}{\overset{\bf 24}{\Yboxdim 9pt\yng(4,3)}}\oplus{\fcolorbox{gray!70}{gray!70}{\textcolor{white}{\footnotesize\bf 10}}}\;\underset{\phantom{.}}{\overset{\bf 42}{\Yboxdim 9pt\yng(5,2)}}\oplus{\fcolorbox{gray!70}{gray!70}{\textcolor{white}{\footnotesize\bf 2}}}\;\underset{\phantom{.}}{\overset{\bf 60}{\Yboxdim 9pt\yng(6,4)}}$
Table 3:  Conformal tower in the $Q=1$ topological sector: tower originating from $\underset{\phantom{.}}{\overset{\bf\bar{3}}{\Yboxdim 9pt\yng(1,1)}}$ [$\otimes\underset{\phantom{.}}{\overset{\bf\bar{3}}{\Yboxdim 9pt\yng(1,1)}}$ ]. $L_{0}$ | Irreps / Multiplicities
---|---
$0$ |  1${\fcolorbox{gray!70}{gray!70}{\textcolor{white}{\footnotesize\bf 1}}}\;\underset{\phantom{.}}{\overset{\bf\bar{3}}{\Yboxdim 9pt\yng(1,1)}}\oplus{\fcolorbox{gray!70}{gray!70}{\textcolor{white}{\footnotesize\bf 1}}}\;\underset{\phantom{.}}{\overset{\bf 6}{\Yboxdim 9pt\yng(2)}}$
$1$ |  1${\fcolorbox{gray!70}{gray!70}{\textcolor{white}{\footnotesize\bf 2}}}\;\underset{\phantom{.}}{\overset{\bf\bar{3}}{\Yboxdim 9pt\yng(1,1)}}\oplus{\fcolorbox{gray!70}{gray!70}{\textcolor{white}{\footnotesize\bf 1}}}\;\underset{\phantom{.}}{\overset{\bf 6}{\Yboxdim 9pt\yng(2)}}\oplus{\fcolorbox{gray!70}{gray!70}{\textcolor{white}{\footnotesize\bf 1}}}\;\underset{\phantom{.}}{\overset{\bf\overline{15}}{\Yboxdim 9pt\yng(3,2)}}$
$2$ |  1${\fcolorbox{gray!70}{gray!70}{\textcolor{white}{\footnotesize\bf 3}}}\;\underset{\phantom{.}}{\overset{\bf\bar{3}}{\Yboxdim 9pt\yng(1,1)}}\oplus{\fcolorbox{gray!70}{gray!70}{\textcolor{white}{\footnotesize\bf 3}}}\;\underset{\phantom{.}}{\overset{\bf 6}{\Yboxdim 9pt\yng(2)}}\oplus{\fcolorbox{gray!70}{gray!70}{\textcolor{white}{\footnotesize\bf 2}}}\;\underset{\phantom{.}}{\overset{\bf\overline{15}}{\Yboxdim 9pt\yng(3,2)}}\oplus{\fcolorbox{gray!70}{gray!70}{\textcolor{white}{\footnotesize\bf 1}}}\;\underset{\phantom{.}}{\overset{\bf\overline{24}}{\Yboxdim 9pt\yng(4,1)}}$
$3$ |  1${\fcolorbox{gray!70}{gray!70}{\textcolor{white}{\footnotesize\bf 6}}}\;\underset{\phantom{.}}{\overset{\bf\bar{3}}{\Yboxdim 9pt\yng(1,1)}}\oplus{\fcolorbox{gray!70}{gray!70}{\textcolor{white}{\footnotesize\bf 5}}}\;\underset{\phantom{.}}{\overset{\bf 6}{\Yboxdim 9pt\yng(2)}}\oplus{\fcolorbox{gray!70}{gray!70}{\textcolor{white}{\footnotesize\bf 5}}}\;\underset{\phantom{.}}{\overset{\bf\overline{15}}{\Yboxdim 9pt\yng(3,2)}}\oplus{\fcolorbox{gray!70}{gray!70}{\textcolor{white}{\footnotesize\bf 2}}}\;\underset{\phantom{.}}{\overset{\bf\overline{24}}{\Yboxdim 9pt\yng(4,1)}}$
$4$ |  1${\fcolorbox{gray!70}{gray!70}{\textcolor{white}{\footnotesize\bf 10}}}\;\underset{\phantom{.}}{\overset{\bf\bar{3}}{\Yboxdim 9pt\yng(1,1)}}\oplus{\fcolorbox{gray!70}{gray!70}{\textcolor{white}{\footnotesize\bf 10}}}\;\underset{\phantom{.}}{\overset{\bf 6}{\Yboxdim 9pt\yng(2)}}\oplus{\fcolorbox{gray!70}{gray!70}{\textcolor{white}{\footnotesize\bf 9}}}\;\underset{\phantom{.}}{\overset{\bf\overline{15}}{\Yboxdim 9pt\yng(3,2)}}\oplus{\fcolorbox{gray!70}{gray!70}{\textcolor{white}{\footnotesize\bf 1}}}\;\underset{\phantom{.}}{\overset{\bf\overline{15}^{\prime}}{\Yboxdim 9pt\yng(4,4)}}\oplus{\fcolorbox{gray!70}{gray!70}{\textcolor{white}{\footnotesize\bf 4}}}\;\underset{\phantom{.}}{\overset{\bf\overline{24}}{\Yboxdim 9pt\yng(4,1)}}\oplus{\fcolorbox{gray!70}{gray!70}{\textcolor{white}{\footnotesize\bf 1}}}\;\underset{\phantom{.}}{\overset{\bf\overline{42}}{\Yboxdim 9pt\yng(5,3)}}$
$5$ |  1${\fcolorbox{gray!70}{gray!70}{\textcolor{white}{\footnotesize\bf 17}}}\;\underset{\phantom{.}}{\overset{\bf\bar{3}}{\Yboxdim 9pt\yng(1,1)}}\oplus{\fcolorbox{gray!70}{gray!70}{\textcolor{white}{\footnotesize\bf 16}}}\;\underset{\phantom{.}}{\overset{\bf 6}{\Yboxdim 9pt\yng(2)}}\oplus{\fcolorbox{gray!70}{gray!70}{\textcolor{white}{\footnotesize\bf 17}}}\;\underset{\phantom{.}}{\overset{\bf\overline{15}}{\Yboxdim 9pt\yng(3,2)}}\oplus{\fcolorbox{gray!70}{gray!70}{\textcolor{white}{\footnotesize\bf 2}}}\;\underset{\phantom{.}}{\overset{\bf\overline{15}^{\prime}}{\Yboxdim 9pt\yng(4,4)}}\oplus{\fcolorbox{gray!70}{gray!70}{\textcolor{white}{\footnotesize\bf 1}}}\;\underset{\phantom{.}}{\overset{\bf\overline{21}}{\Yboxdim 9pt\yng(5)}}\oplus{\fcolorbox{gray!70}{gray!70}{\textcolor{white}{\footnotesize\bf 8}}}\;\underset{\phantom{.}}{\overset{\bf\overline{24}}{\Yboxdim 9pt\yng(4,1)}}\oplus{\fcolorbox{gray!70}{gray!70}{\textcolor{white}{\footnotesize\bf 2}}}\;\underset{\phantom{.}}{\overset{\bf\overline{42}}{\Yboxdim 9pt\yng(5,3)}}$
$6$ |  1${\fcolorbox{gray!70}{gray!70}{\textcolor{white}{\footnotesize\bf 27}}}\;\underset{\phantom{.}}{\overset{\bf\bar{3}}{\Yboxdim 9pt\yng(1,1)}}\oplus{\fcolorbox{gray!70}{gray!70}{\textcolor{white}{\footnotesize\bf 28}}}\;\underset{\phantom{.}}{\overset{\bf 6}{\Yboxdim 9pt\yng(2)}}\oplus{\fcolorbox{gray!70}{gray!70}{\textcolor{white}{\footnotesize\bf 29}}}\;\underset{\phantom{.}}{\overset{\bf\overline{15}}{\Yboxdim 9pt\yng(3,2)}}\oplus{\fcolorbox{gray!70}{gray!70}{\textcolor{white}{\footnotesize\bf 4}}}\;\underset{\phantom{.}}{\overset{\bf\overline{15}^{\prime}}{\Yboxdim 9pt\yng(4,4)}}\oplus{\fcolorbox{gray!70}{gray!70}{\textcolor{white}{\footnotesize\bf 1}}}\;\underset{\phantom{.}}{\overset{\bf\overline{21}}{\Yboxdim 9pt\yng(5)}}\oplus{\fcolorbox{gray!70}{gray!70}{\textcolor{white}{\footnotesize\bf 15}}}\;\underset{\phantom{.}}{\overset{\bf\overline{24}}{\Yboxdim 9pt\yng(4,1)}}\oplus{\fcolorbox{gray!70}{gray!70}{\textcolor{white}{\footnotesize\bf 5}}}\;\underset{\phantom{.}}{\overset{\bf\overline{42}}{\Yboxdim 9pt\yng(5,3)}}\oplus{\fcolorbox{gray!70}{gray!70}{\textcolor{white}{\footnotesize\bf 1}}}\;\underset{\phantom{.}}{\overset{\bf\overline{60}}{\Yboxdim 9pt\yng(6,2)}}$
$7$ |  1${\fcolorbox{gray!70}{gray!70}{\textcolor{white}{\footnotesize\bf 43}}}\;\underset{\phantom{.}}{\overset{\bf\bar{3}}{\Yboxdim 9pt\yng(1,1)}}\oplus{\fcolorbox{gray!70}{gray!70}{\textcolor{white}{\footnotesize\bf 43}}}\;\underset{\phantom{.}}{\overset{\bf 6}{\Yboxdim 9pt\yng(2)}}\oplus{\fcolorbox{gray!70}{gray!70}{\textcolor{white}{\footnotesize\bf 50}}}\;\underset{\phantom{.}}{\overset{\bf\overline{15}}{\Yboxdim 9pt\yng(3,2)}}\oplus{\fcolorbox{gray!70}{gray!70}{\textcolor{white}{\footnotesize\bf 8}}}\;\underset{\phantom{.}}{\overset{\bf\overline{15}^{\prime}}{\Yboxdim 9pt\yng(4,4)}}\oplus{\fcolorbox{gray!70}{gray!70}{\textcolor{white}{\footnotesize\bf 3}}}\;\underset{\phantom{.}}{\overset{\bf\overline{21}}{\Yboxdim 9pt\yng(5)}}\oplus{\fcolorbox{gray!70}{gray!70}{\textcolor{white}{\footnotesize\bf 26}}}\;\underset{\phantom{.}}{\overset{\bf\overline{24}}{\Yboxdim 9pt\yng(4,1)}}\oplus{\fcolorbox{gray!70}{gray!70}{\textcolor{white}{\footnotesize\bf 10}}}\;\underset{\phantom{.}}{\overset{\bf\overline{42}}{\Yboxdim 9pt\yng(5,3)}}\oplus{\fcolorbox{gray!70}{gray!70}{\textcolor{white}{\footnotesize\bf 2}}}\;\underset{\phantom{.}}{\overset{\bf\overline{60}}{\Yboxdim 9pt\yng(6,2)}}$
Table 4:  Conformal tower in the $Q=2$ topological sector: tower originating
from $\underset{\phantom{.}}{\overset{\bf 3}{\Yboxdim 9pt\yng(1)}}$
[$\otimes\underset{\phantom{.}}{\overset{\bf 3}{\Yboxdim 9pt\yng(1)}}$ ].

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