# The case of SU$(3)$ criticality in spin-2 chains

Chengshu Li<EMAIL_ADDRESS>Department of Physics and Astronomy,
University of British Columbia, Vancouver BC V6T 1Z1, Canada  Stewart Blusson
Quantum Matter Institute, University of British Columbia, Vancouver BC V6T
1Z4, Canada  Victor L. Quito<EMAIL_ADDRESS>Department of Physics and
National High Magnetic Field Laboratory, Florida State University,
Tallahassee, Florida 32306, USA  Department of Physics and Astronomy, Iowa
State University, Ames, Iowa 50011, USA  Eduardo Miranda  Gleb Wataghin
Institute of Physics, The University of Campinas (Unicamp), 13083-859
Campinas, SP, Brazil  Rodrigo Pereira  International Institute of Physics and
Departamento de Fìsica Teórica e Experimental, Universidade Federal do Rio
Grande do Norte, 59072-970 Natal-RN, Brazil  Ian Affleck  Department of
Physics and Astronomy, University of British Columbia, Vancouver BC V6T 1Z1,
Canada  Stewart Blusson Quantum Matter Institute, University of British
Columbia, Vancouver BC V6T 1Z4, Canada  Pedro L. S. Lopes<EMAIL_ADDRESS>Department of Physics and Astronomy, University of British Columbia, Vancouver
BC V6T 1Z1, Canada  Stewart Blusson Quantum Matter Institute, University of
British Columbia, Vancouver BC V6T 1Z4, Canada

(August 28, 2024)

###### Abstract

It was proposed in [Chen et al., Phys. Rev. Lett. $\mathbf{114}$, 145301
(2015)] that spin-2 chains display an extended critical phase with enhanced
SU$(3)$ symmetry. This hypothesis is highly unexpected for a spin-2 system
and, as we argue, would imply an unconventional mechanism for symmetry
emergence. Yet, the absence of convenient critical points for renormalization
group perturbative expansions, allied with the usual difficulty in the
convergence of numerical methods in critical or small-gapped phases, renders
the verification of this hypothetical SU$(3)$-symmetric phase a non-trivial
matter. By tracing parallels with the well-understood phase diagram of spin-1
chains and searching for signatures robust against finite-size effects, we
draw criticism on the existence of this phase. We perform non-Abelian density
matrix renormalization group studies of multipolar static correlation
function, energy spectrum scaling, single-mode approximation, and entanglement
spectrum to shed light on the problem. We determine that the hypothetical
SU$(3)$ spin-2 phase is, in fact, dominated by ferro-octupolar correlations
and also observe a lack of Luttinger-liquid-like behavior in correlation
functions that suggests that is perhaps not critical. We further construct an
infinite family of spin-$S$ systems with similar ferro-octupolar-dominated
quasi-SU$(3)$-like phenomenology; curiously, we note that the spin-3 version
of the problem is located in a subspace of exact G2 symmetry, making this a
point of interest for search of Fibonacci topological properties in magnetic
systems.

##  I Introduction

Figure 1: Schematic representation of some typical RG flows: (a) phase-
transition at a critical value $g_{c}$ (green point) (b) symmetry emergence.
Flows to “infinite” parameter values (red points), in general, lead to gaps
and stabilization of phases in non-perturbative fixed points.

The classification of phases of matter remains a cornerstone of condensed
matter physics. While the classification by symmetry breaking became a legacy
of the field [1], more recent discussions have turned attention to cases that
go beyond this paradigm, where, in particular, the program of topological
classification has had major successes [2]. A less explored scenario is that
of phases where an enhanced symmetry develops in the low-energy sector of a
system. This scenario of symmetry emergence and, in particular, of critical
(gapless) phases is the focus of this work. As the general phenomenology of
these latter situations is not understood, and examples remain few and far
between, we will begin by setting the stage.

When assisted by a renormalization group (RG) picture, standard phase
transitions can be understood as follows [3]: one picks a parameter, say $g$,
that controls the dynamics of particles in a system at a given length scale.
Coarse graining then renormalizes $g$, until it reaches a stable fixed point.
This stable fixed point represents a phase where the long-wavelength behavior
of the original system has a specific phenomenology. By varying $g$, the flow
may find an unstable fixed point, around which any small changes in
microscopic choice of the parameter leads to different stable fixed points
after RG flow. The system is said to go through a phase transition as $g$
crosses this critical value $g_{c}$ (which exclusively flows to the unstable
critical point). Fig. 1 (a) contains a pictorial view of this process.

However quintessential the story above is, some systems display very different
RG flows. An example is displayed on Fig. 1(b). A system may be described by
operators that preserve a given microscopic symmetry (with coupling constant
$g_{SP}$) and other operators that explicitly break it (with coupling constant
$g_{SB}$). If the flow of $g_{SB}$ close to a critical point is always
irrelevant, at long wavelengths, the explicit breaking of the microscopic
symmetries is not seen. If $g_{SP}$ happens to also be irrelevant in a given
direction, a whole critical phase may develop (negative $g_{SP}$ plane in Fig.
1). Besides showing the emergence of scaling symmetry, critical phases
participate in a large classes of exotic phenomena including non-Fermi-liquid
physics [4], topological effects 111we make reference to Haldane’s conjecture
in anti-ferromagnetic 1D Heisenberg models [13, 14], and Kosterlitz-Thouless
type phase transitions [3].

Here starts the discussion of our explicit physical systems of interest. A
well-known example of emergence of highly-symmetric critical phases happens in
1D bilinear-biquadratic spin-1 chains [6, 7]; despite the explicit SU$(2)$
microscopic symmetry of this problem, the existence of an explicit SU$(3)$
symmetric parameter point, aided by an RG analysis as described above, leads
to the existence of a critical phase described by an SU$(3)_{1}$
Wess–Zumino–Witten theory. The existence of this phase is well established,
with clear signatures via both analytical calculations and numerical
simulations. A refresher on this problem is presented below for self-
containing and contrasting purposes.

More recently, a surprising story has been reported in articles [8, 9]
(henceforth referred to as Chen et al.). There, in the context of bosons in
optical lattices, a study was made of general effective models of 1D spin-2
chains close to ferromagnetism. Dimerization and trimerization, and
indications of a gapless behavior on the trimerized parameter regime were
first reported. Subsequently, a case was made for the existence of an
SU$(3)_{1}$ extended critical phase in the phase diagram of this system. This
comes as a surprise: no point of explicit SU$(3)$ symmetry is available in the
spin-2 chain parameter space. If an extended SU$(3)_{1}$ critical phase exists
in this model, some novel mechanism must be playing a role, motivating our
interest in this problem. We performed extensive analytical attempts at
determining some possibility for such mechanism, to no avail. This prompted us
to the results of this paper.

Our purpose in this work is three-fold: (i) to revisit the problem posed by
Chen et al., (ii) to situate it in comparison with the well-understood physics
of the spin-1 critical SU$(3)_{1}$ phase, and (iii) to present new numerical
results from exact diagonalization as well as from non-Abelian Density Matrix
Renormalization Group (DMRG) characterizing the physics of this hypothetical
spin-2 SU$(3)_{1}$ phase, including a critique on whether the previous
evidence of Chen et al. was conclusive. As in previous works, we observe
extreme difficulty in the convergence of our routines, rendering the
“gaplessness” of the phase difficult to determine. We identify static
correlation functions of multipolar operators as ideal quantities whose
behavior seems to be conclusive independently of finite-size effects. This
way, we demonstrate that this hypothetical SU$(3)$ spin-2 critical phase is
dominated by ferro-octupolar correlations, in striking contrast with the
$2\pi/3$-periodic quadrupolar correlations that dominate the physics of the
SU$(3)_{1}$ Wess–Zumino–Witten theory after logarithmic corrections, as seen
explicitly in the spin-1 version of the problem.

In an attempt to determine an alternate — to criticality — origin of our
numerical convergence difficulties, we perform a “single-mode approximation”
(SMA) study to compare the challenges found in the spin-2 problem with those
of a gapless phase proposed to exist close to the ferro-quadrupolar point of
the spin-1 bilinear-biquadratic spin chain [10, 11]. This comparison proved to
still be inconclusive, so we pursued other signatures of critical behavior.
Recent years have seen a trend in the use of quantum-information-theoretic
tools to study phases of matter. Following this trend, we further the analysis
by presenting results on the entanglement spectrum in the phase, and comparing
it with universal features expected of gapless systems.

Besides finite-size scaling, representation-scaling is a paradigm in the
analysis of spin chains. This led us to further consider if the phenomenology
observed by Chen et al. was an isolated case. Remarkably, we discover a family
of models that displays the same tripled-period phenomenology of the spin-2
Hamiltonian but for arbitrary spin-$S$. We demonstrate how these systems are
fully quantum and not possible to be described by simple spin-wave theory, and
show that for spin-3 our model lies in a region of potential interest to
explore G2-symmetry physics in a spin lattice, for which a (G${}_{2})_{1}$
critical point/phase could host Fibonacci anyons.

The paper is organized as follows: in Sec. II, we provide a review and a few
complementing results for the spin-1 SU$(3)_{1}$ phase. In Sec. III we review
and extend the known spin-2 model and its phase diagram, including a
discussion of the diverse symmetry-enhanced phases and points, constantly
comparing with the spin-1 problem. In Sec. IV, we provide most of our new
results and analyses on the spin-2 SU$(3)_{1}$ problem, while in Sec. V we
discuss larger spin-$S$ generalizations. We conclude in Sec. VI, with also
some suggestions for future research directions.

##  II Spin-1 SU$(3)_{1}$ phase review

For contrasting purposes, we begin with a short review of the “bilinear plus
biquadratic” spin-1 chain phase diagram. The Hamiltonian reads

$\displaystyle
H_{\mathrm{BB}}=\sum_{i}J\mathbf{S}_{i}\cdot\mathbf{S}_{i+1}+D\left(\mathbf{S}_{i}\cdot\mathbf{S}_{i+1}\right)^{2},$
(1)

where $\mathbf{S}_{i}$ are spin-1 matrices on site $i$. By defining
$\tan\gamma\equiv D/J$, the zero-temperature quantum phase diagram of this
model can be built by varying the single parameter $\gamma$ in a circle, the
radius providing an overall energy scale.

Four phases exist in the $JD$-circle whose behaviors are well-established in
the literature [12]. Starting at the anti-ferromagnetic Heisenberg model
$\gamma=0$ [c.f. Fig 2(a)] and moving clockwise we have a topological gapped
phase (Haldane phase) [13, 14], a topologically trivial gapped dimer phase
[15, 16, 17], a ferromagnetic phase, and an extended critical phase with
period-3 correlations, described by an SU$(3)_{1}$ Wess–Zumino–Witten
conformal field theory (CFT) at low energies [6, 7]. The critical points
separating these phases are all very well identified: three of them are points
where the lattice problem displays an explicit enhanced SU$(3)$ symmetry
($\gamma=\pi/4,\,\pi/2,\,-3\pi/4$) and one, the so-called Takhtajan–Babujian
point [18, 19] (TKTB, $\gamma=-\pi/4$), is integrable.

Figure 2: The spin-1 bilinear-biquadratic problem, Eq. (1). (a) The zero-
temperature phase diagram with the explicit SU$(3)$ points marked in blue. The
hashed region corresponds to a hypothetical nematic phase, and the green line
is to be matched against the SO(5)-symmetric line in the spin-2 phase diagram
below (Fig. 4). (b) Multiple equivalent ways of representing the SU(2) spin-1
Hamiltonian of Eq. (1), listed in Eq. (2).

Some debate has been raised regarding the existence of a spin-nematic state as
one approaches the SU$(3)$-symmetric “ferromagnetic” critical point
$\gamma=-3\pi/4$ [hashed red area in Fig 2(a)] [10, 6, 20, 21, 22, 23, 24,
11]. A gapped spin-nematic phase was originally proposed for this region by
Chubukov [10], but no evidence of gap closing and reopening was found
numerically [6]. An observed dramatic increase of spin-nematic correlations,
as one approaches $\gamma=-3\pi/4$ from the dimerized phase, led to a new
critical spin-nematic phase hypothesis for the region [11, 22, 23]. However, a
scenario of a very large crossover, attributed to the influence of the highly-
degenerate ferroquadrupolar SU$(3)$-symmetric point on the ground state, has
been pointed out as the most plausible explanation to these phenomena [11]. We
will return to this point when re-evaluating the physics of the spin-2
problem.

The discussions on multiplets and symmetries above also suggest the relevance
of the many different ways of interpreting and rewriting the Hamiltonian in
Eq. (1). Despite its simplicity, this spin-1 Hamiltonian admits no less than
five distinct representations; besides the spin-1 $\mathrm{SU}(2)$ language in
Eq. (1), we have:

$\begin{split}H_{\mathrm{BB}}=&\sum_{i}J\left(\sum_{a<b}L_{i}^{ab}L_{i+1}^{ab}\right)+D\left(\sum_{a<b}L_{i}^{ab}L_{i+1}^{ab}\right)^{2}\\\
=&\sum_{i}\left[\left(J-\frac{D}{2}\right)\mathbf{S}_{i}\cdot\mathbf{S}_{i+1}+\frac{D}{2}\mathbf{Q}_{i}\cdot\mathbf{Q}_{i+1}+\frac{4D}{3}\right]\\\
=&\sum_{i}\left(-2J+4D\right)\mathcal{P}^{i}_{0}+\left(-J+D\right)\mathcal{P}^{i}_{1}+\left(J+D\right)\mathcal{P}^{i}_{2}\\\
=&\sum_{i}\bigg{[}\left(J-\frac{D}{2}\right)\sum_{a=1}^{3}\Lambda_{i}^{a}\Lambda_{i+1}^{a}+\frac{D}{2}\sum_{a=4}^{8}\Lambda_{i}^{a}\Lambda_{i+1}^{a}\\\
&+\frac{4D}{3}\bigg{]}.\end{split}$ (2)

In the first line, using the isomorphism of $\mathfrak{su}(2)$ and
$\mathfrak{so}(3)$, we write down the Hamiltonian in term of $L_{i}^{ab}$, the
three $\mathrm{SO}(3)$ generators in the vector representation ($a,b=1,2,3$).
The second line expresses the Hamiltonian in terms of multipolar operators,
with $\mathbf{Q}_{i}$ the five cartesian quadrupole operators. For the third
equality, the Hamiltonian is written in terms of local projectors, with
$P^{i}_{S}\equiv P_{S}\left(\mathbf{S}_{i},\mathbf{S}_{i+1}\right)$ the
projector operators of a pair of spins on a total spin $S$ sector. Finally, in
the last equality, the $\mathrm{SU}(2)$ problem is cast as an anisotropic
$\mathrm{SU}(3)$ problem, with $\Lambda_{i}^{a}$ the eight generators of the
fundamental representation of $\mathrm{SU}(3)$, broken into two sub-sets of
three purely imaginary and five real matrices, using the conventional Gell-
Mann basis [25]. The matter of importance is that each of these forms provides
an opportunity for insight in the phase diagram and $\mathrm{SU}(3)$ physics
of the spin-1 problem defined by Eq. (1).

We thus remark: (i) as observed by [26] and discussed below, the phase diagram
in Fig. 2(a), while naively defined for a higher-vector-representation of
SU$(2)$, actually finds natural generalization when seen as a phase-diagram
for SO$(N)$-symmetric problems ($N=3$ for $S=1$). This observation alone
allows us to extrapolate the phase diagram of Chen et al., as shown below.
(ii) By choosing $J=D$, the spin-1 bilinear-biquadratic Hamiltonian is tuned
to the integrable Uimin–Lai–Sutherland (ULS) point [27, 28, 29]. This allows a
series of observations regarding the spin-1 SU$(3)_{1}$ phase, which are
relevant in what follows. Applying $J=D$ to the other different
representations of the Hamiltonian we obtain:

$\begin{split}H_{\mathrm{BB}}^{\mathrm{ULS}}&=\frac{J}{2}\sum_{i}\left(\mathbf{S}_{i}\cdot\mathbf{S}_{i+1}+\mathbf{Q}_{i}\cdot\mathbf{Q}_{i+1}+\frac{8}{3}\right)\\\
&=2J\sum_{i}\mathcal{P}^{i}_{0}+\mathcal{P}^{i}_{2}=-2J\sum_{i}\left(\mathcal{P}_{1}^{i}-1\right)\\\
&=\frac{J}{2}\sum_{i}\left(\mathbf{\Lambda}_{i}\cdot\mathbf{\Lambda}_{i+1}+\frac{8}{3}\right).\end{split}$
(3)

From the last and first lines, respectively, we remark that the SU$(3)$
invariance is made quite explicit, and that this enlarged symmetry requires
identical behavior of the dipolar and quadrupolar fluctuations. From the
second line, we observe a suggestive form that will show up again in the
spin-2 problem, and will serve as base for further generalizations in Section
V.

The establishment of an extended critical phase inheriting properties from the
SU$(3)$ point at low energies is the result of a combination of numerical and
analytical work [30, 6, 17, 31, 32, 33, 11], and is ultimately explained by
Itoi and Kato’s work via RG [7]. A few more remarks on this region of the
spin-1 phase diagram are worthy: it serves as a quintessential example of
failure of $1/S$ expansion. As pointed out by Ref. [33], classically, the
spin-wave theory predicts a spiral phase with a continuously varying pitch
angle $\theta^{*}$ as $D/J$ is tuned through the ULS point. Averaging over the
azimuthal degeneracy around the canting of successive spins implies that the
classical spin-spin correlation functions are given by
$\langle\mathbf{S}_{0}\cdot\mathbf{S}_{j}\rangle=S^{2}\cos^{j}\theta^{*}$222To
see this we can fix $\mathbf{S}_{0}$ and use
$\langle\mathbf{S}_{j}\rangle=\int\prod_{i=1}^{j}\mathrm{d}\mathbf{S}_{i}P(\mathbf{S}_{i}|\mathbf{S}_{i-1})\mathbf{S}_{j}$
and
$\int\mathrm{d}\mathbf{S}_{i}P(\mathbf{S}_{i}|\mathbf{S}_{i-1})\mathbf{S}_{i}=\cos\theta^{*}\mathbf{S}_{i-1}$,
where $P(\mathbf{S}_{i}|\mathbf{S}_{i-1})=\frac{1}{2\pi
S}\delta\left(S^{-2}\mathbf{S}_{i}\cdot\mathbf{S}_{i-1}-\cos\theta^{*}\right)$
is the conditional probability distribution for $\mathbf{S}_{i}$ given
$\mathbf{S}_{i-1}$ fixed., peaking at momentum $q=\pi$ for all
$\gamma\in[0,\pi/2)$. If spins were restricted to spiral states in a fixed
plane, the correlation functions would become
$\langle\mathbf{S}_{0}\cdot\mathbf{S}_{j}\rangle=S^{2}\cos(j\theta^{*})$, with
peaks following $\theta^{*}$ and attaining incommensurate continuous values.
This classical large-$S$ description does not correspond to the $S=1$ case of
interest: comparing the full line and orange dots in Fig. 3(a), one sees that
the spin-spin quantum correlation function, throughout all
$\gamma\in[0,\pi/2)$, is better described by the classical spiral states with
spins fixed in a plane. But quantum fluctuations do not only force the spins
to remain in a plane, they also saturate what would be the classical canting
angle to a commensurate triple periodicity, so that spin-spin correlation
functions peak at momentum $q^{*}=2\pi/3$. Overall, the quantum case is
different from both possibilities of classical scenarios; spin-wave and
flavor-wave theories only provide good starting points to study this
Hamiltonian at specific points of the parameter space, like $\gamma=0$ (anti-
ferromagnetic Heisenberg model) or $\gamma=\pi/4$ (SU$(3)$ ULS point) and,
still, a full coherent-state quantum treatment is needed to fully capture the
physics in these cases.

Figure 3: (a) The peak positions $q^{*}$ of the dipolar and quadrupolar static
structure factors, compared to the classical canting angle $\theta^{*}$. The
black arrows correspond to the cuts showed in (b) and (c). The red dot is the
the ULS point, $\gamma=\pi/4$, the SU$(3)$ critical point at which the dipolar
and quadrupolar correlation functions coincide. The dominant momentum peak of
quadrupolar correlations changes discontinuously at the AKLT point (dashed
vertical line). (b,c) The static structure factors around the ULS point in the
gapped and emergent SU(3)1 critical, respectively. While, in (c), the system
is critical, the emergent nature of the SU(3) symmetry shows by means of
finite-size log corrections leading to enhancements of the quadrupolar
correlation function as compared to the dipolar one.

In Fig. 3(a), inspired by Ref. [11], we further extend the analyses of Ref.
[33] and study the static structure factor

$\displaystyle\mathcal{C}^{\kappa}\left(q\right)=\left\langle
T_{-m}^{\kappa}\left(-q\right)T_{m}^{\kappa}\left(q\right)\right\rangle,$ (4)

where $T_{m}^{\kappa}\left(q\right)$ can be either a dipole ($\kappa=1$,
vector) or a quadrupole ($\kappa=2$) operator, the result being independent of
$m$ due to SU$(2)$ invariance. This symmetry is incorporated explicitly in our
calculations via an SU$(2)$ non-Abelian DMRG routine [35, 36, 37, 38]. We
evaluate $\mathcal{C}^{\kappa}\left(q\right)$ to find the maximizing momentum
$q^{*}$, and compare it with the classical canting angle $\theta^{*}$. Besides
the results found by Ref. [33], we observe that the quadrupolar correlations
peak at $q^{*}=0$ before reaching the AKLT point, displaying a
ferroquadrupolar uniform behavior. Past the AKLT point, the peak of the
quadrupolar fluctuations jumps discontinuously to $q^{*}=\pi$, similar to the
dipolar ones. As one moves towards the ULS point, both static structure factor
peaks smoothly interpolate towards $q^{*}=2\pi/3$, in a crossover behavior
associated with the Kosterlitz-Thouless-nature of the transition into the
critical SU$(3)_{1}$ phase. We also note the crossing of the dipolar and
quadrupolar $q^{*}$ curves with the classical $\theta^{*}$ at the ULS point.
While the spin-wave theory fails at this point, this degeneracy lies at the
heart of the success of the flavor-wave theory in describing the low-energy
theory of this exact SU$(3)$-symmetric point [39]. Finally, in Fig. 3(b) and
(c), we display the full static structure factor around $\gamma=\pi/4$; as
pointed out by Ref. [11], inside the critical SU$(3)_{1}$ phase, logarithmic
corrections by SU$(2)$ operators that are irrelevant, in the RG sense, induce
an enhancement of the quadrupolar structure factor, in contrast with the
dipolar one. Besides that, we call attention to the vanishing tail as $q\to
0$, signature of Luttinger-liquid-like behaviour, when $\gamma>\pi/4$. For the
Haldane phase, $\gamma<\pi/4$, despite a gap, we remark that
$\mathcal{C}^{\kappa=1}\left(q=0\right)=0$ persists for the dipolar
correlations, as demanded by conservation of the total angular momentum and a
singlet ground state. It is from the quadrupolar correlations, in fact, that
one actually sees the breakdown of Luttinger-liquid behavior via the finite
values of $\mathcal{C}$ as $q\to 0$ 333In fact, numerically, the $q=0$
quadrupolar correlations remain finite in both the Haldane and critical
phases, but it is possible to see that the $y$-intercept increases much more
sharply in the Haldane phase; we attribute the finite zero-momentum
quadrupolar correlations in the critical phase to the effective nature of the
enlarged SU$(3)$ symmetry, and a slow Kosterlitz-Thouless type of
renormalization group flow controlling this phase..

As a final remark, we note that the conspicuously robust enlarged symmetry of
the spin-1 SU$(3)_{1}$ critical phase does not seem to happen by chance. An
algebra isomorphism exists between $\mathfrak{su}(3)_{1}$ and the integer-spin
sector of $\mathfrak{su}(2)_{4}$ [41]; since only integer spin multiplets
exist in integer spin chains, the SU$(2)$ representation of this problem seems
to be carefully appropriate to support an extended SU$(3)$-symmetric critical
phase close to the ULS point.

This closes our review and analysis of the spin-1 bilinear-biquadratic
problem, and of several of the phenomenological characteristics of its
critical SU$(3)_{1}$ phase. We now turn to describing the spin-2 version of
the problem and contrast what is known, and our new results on it, with the
spin-1 observations above. Based on the universality of critical theories, we
draw conclusions regarding the likelihood of a critical SU$(3)_{1}$ phase
emerging in the spin-2 system.

##  III Spin-2 SU$(3)_{1}$ phase review

We now move to the most general SU$(2)$-symmetric spin-2 chain, described by
the following Hamiltonian [8]:

$\displaystyle
H_{2}=\sum_{i}\sum_{n=0}^{4}\alpha_{n}\left(\mathbf{S}_{i}\cdot\mathbf{S}_{i+1}\right)^{n}=\sum_{i}\sum_{n=0}^{4}\epsilon_{n}\mathcal{P}_{n}^{i}.$
(5)

Here $\alpha_{n}$ and $\epsilon_{n}$ are related sets of coupling constants.
Motivated by bosonic cold-atomic realizations of this model, Chen et al.
considered the phase diagram within the subspace $\epsilon_{1}=\epsilon_{3}=0$
and $\epsilon_{0},\,\epsilon_{2},\,\epsilon_{4}<0$. Normalizing as
$\left(x_{0},x_{2},x_{4}\right)=\left(\epsilon_{0},\epsilon_{2},\epsilon_{4}\right)/\left(\epsilon_{0}+\epsilon_{2}+\epsilon_{4}\right)$,
where the denominator represents a global energy scale, the phase diagram is
represented by a pyramid built out of a family of triangles with their normal
along $(x_{0},x_{2},x_{4})=(1,1,1)$, see Fig. 4. The distance of a given plane
from the origin is the global energy scale.

Figure 4: The phase diagram proposed by Chen et al. [8] and an extension along
an SO$(5)$-symmetric subspace (inset). The grey and yellow phases are
dimerized and ferromagnetic, respectively, and the red phases have enhanced
symmetry [certainly (uniform) or hypothetically (hashed)]. In the inset, the
SU(5) analogue of the spin-1 Uimin–Lai–Sutherland point is located at
$\tan^{-1}\frac{1}{3}\approx 18.4^{\circ}$, and the Takhtajan–Babujian
analogue, known as the Reshetikhin point [42, 43], is located at
$\tan^{-1}\frac{1}{9}\approx 6.3^{\circ}$).

As shown in Fig. 4, three phases were identified — a ferromagnetic phase (FM),
a dimerized gapped phase (dimer), and indications of a potentially gapless
SU$(3)_{1}$ phase. Associated with the latter, for finite-sized chains, a
tendency towards trimerization was clearly observed via DMRG, together with
indications of a gap closing as the system size increases [8]. Taking the top
of the pyramid in Fig. 4 — the blue star, henceforth referred to as the
pyramidion — as a representative point deep into the proposed SU$(3)$ critical
phase, subsequent exact diagonalization and more DMRG analyses established
that the spectrum in this phase has several similarities with the spectrum of
the spin-1 chain with $\pi/4<\gamma<\pi/2$. CFT-related quantities were
computed including a central charge estimated at $c\approx 2$ and conformal
towers with scaling dimension $2/3$, both in agreement with the hypothesis of
an SU$(3)_{1}$ critical phase [9]. We emphasize that numerical convergence for
large-enough system sizes is, however, particularly difficult in this system,
implying that these numbers are not necessarily final.

The findings reviewed above come as a surprise: contrasting with the spin-1
problem, no critical point of exact SU$(3)$ symmetry exists here from which a
critical phase could extend similarly to the arguments of Itoi and Kato [7].
As already observed by Chen et al., a critical point of larger explicit
symmetry here is $x_{0}=x_{2}=x_{4}$, in which the system is
SU$(5)$-symmetric, but this is a large-symmetry point with uniform spin
orientation, analogous to the $\gamma=-3\pi/4$ point of the spin-1 problem
[c.f. Fig. 2(a)].

In fact, more parallels can be drawn with the spin-1 case. Noting that the
$(x_{0},x_{2},x_{4})=(1,0,0)$ point also has exact SU$(5)$ symmetry 444One
needs alternate fundamental and anti-fundamental representations on even/odd
sites. See Refs. [56, 57] for details., we see that the line that bisects the
dimer phase in Fig. 4 actually has an enhanced explicit SO$(5)$. To prove so,
it suffices to focus on a two-site system and note,

$\displaystyle\epsilon_{0}\mathcal{P}_{0}^{i}+\epsilon_{24}(\mathcal{P}_{2}^{i}+\mathcal{P}_{4}^{i})$
$\displaystyle={}$ $\displaystyle
J\sum_{a<b}L_{i}^{ab}L_{i+1}^{ab}+D\left(\sum_{a<b}L_{i}^{ab}L_{i+1}^{ab}\right)^{2}+const.$
(6)

where $L_{i}^{ab}$ are the SO(5) generators at site $i$, $J=\epsilon_{24}/2$,
and $D=\epsilon_{0}/15+\epsilon_{24}/10$ (as used in Fig. 4). As observed by
Tu et al. [26], the family of SO$(N)$ spin chains with spins in the
fundamental (vector) representation have similar phase diagrams. These phase
diagrams are one-dimensional and can be arranged in a circle. They follow the
pattern of phases and critical points illustrated by the spin-1 (or SO$(3)$)
discussion of Section II. This means that, as promised last section, we can
extend the regions where we understand the spin-2 phase diagram. We know, for
example, that a critical SU$(5)_{1}$ phase exists in the spin-2 phase space,
albeit far from the region in question by Chen et al. (see the Fig. 4 inset
for full details). Back on the ferromagnetic region of the phase space, we
note how the green lines in Fig. 4 and its inset match the one depicted in
Fig. 2. In the spin-1 system, we already discussed how the presence of the
SU$(3)$ permutation-symmetric critical point $\gamma=-3\pi/4$ induces a large
cross-over scale in the dimer phase, leading to a behavior that appears
gapless at small system sizes. It is highly suggestive that similar physics
could be at play in the spin-2 case, also stretching into the period-three
phase to give the impression of that corresponding to a critical phase.
Indeed, the role of the competition of ground states induced by adjacent
ferromagnetic phase in complicating numerical analysis was observed by Chen et
al.

##  IV Spin-2 SU$(3)_{1}$ phase critique

###  IV.1 Static structure factor

To shed light on the hypothetical SU$(3)_{1}$ spin-2 phase, we draw
inspiration from the spin-1 results above. We consider the multipolar static
correlation functions $\mathcal{C}^{\kappa}(q)$ at the spin-2 pyramidion
point. Due to the higher-spin representation, not only dipolar ($\kappa=1$)
and quadrupolar ($\kappa=2$) correlations are available, but octupolar and
hexadecapolar are too ($\kappa=3,\,4$, respectively). Again, we obtain our
results by performing SU(2) non-Abelian DMRG calculations and the results are
shown in Fig. 5(a). Remarkable differences are found between the well-
established SU$(3)_{1}$ spin-1 phase and the spin-2 case. While for spin-2
dipole and quadrupole fluctuations are indeed dominated by triple-periodicity,
i.e. $q^{*}=2\pi/3$, the quadrupole fluctuations do not vanish as $q\to 0$, as
would have been expected for critical behavior. Furthermore, the dipole
correlations are stronger than the quadrupolar ones, the opposite of what is
argued to be expected from an SU$(3)$ critical phase emerging of an SU$(2)$
lattice problem [11, 7]. At last, we verify that octupolar and hexadecapolar
correlations are dominated by a uniform behavior, i.e. $q^{*}=0$, and these
correlations are also stronger than the dipolar and quadrupolar ones. A ferro-
octupolar phase is more aligned with the physics than a critical SU$(3)_{1}$
one.

Figure 5: Numerical results on the spin-2 pyramidion Hamiltonian (star point
of Fig. 4). (a) The static structure factors $\mathcal{C}^{\kappa}(q)$ show a
dominance of ferro-octupolar fluctuations, as well as dipole correlations
stronger than the quadrupolar ones at $q^{*}=2\pi/3$, the opposite of what is
expected for a critical phase of emergent SU$(3)$ symmetry (compare with Fig.
3(b, c)). (b) Finite-size scaling of $\mathcal{C}^{\kappa}(\bar{q})$ at
$\bar{q}=0\,(\mathrm{dots}),\,2\pi/3\,(\mathrm{crosses})$. (c) Finite-size
scaling of the energy differences between the lowest-energy spin multiplets
$S=1,2,3,4$ and the singlet ground state. We chose $N$ to be multiples of six
and spaced the values according to $1/N$ as $E_{GS,S}-E_{GS,0}$ is expected to
scale as $1/N$.

Chen et al. observed via exact diagonalization that, indeed, the ordering of
the lowest energy level for each spin multiplet in the spin-2 pyramidion point
is not the same as that of the emergent SU(3) spin-1 phase (in ascending
order, $S=0,\,3,\,1,\,4,\,2$, for the former, $S=0,\,2,\,1,\,3,\,4$, for the
latter). They claimed that the order of these multiplets was a result of
finite-size effects together with the fact that the SU(3) invariance is
emergent, not exact. To verify this hypothesis, we performed a finite-size
scaling analysis of the correlation functions, following its values at
$\bar{q}=0,\,2\pi/3$, as well as of the energy of the ground state of each
spin $S=1,2,3,4$ relative to the singlet ground state. We choose system sizes
in multiples of 6 to avoid two- and three-fold frustration under periodic
boundary conditions, with values between $N=12$ and $N=60$. The results are
shown in Fig. 5(b). Despite not going to system sizes as large as Chen et al.,
we push the limits on bond dimensions beyond what they considered, favoring
precision over system size (more details are given in Appendix A). Our
findings indicate that the ferro-octupolar correlations tend to increase with
system size most intensely, followed by the ferro-hexadecapolar ones. The
period-3 dipolar and quadrupolar correlations seem to show little to no
appreciable increase as functions of the system size. As for the energy
levels, we find, as shown in Fig. 5(c), that indeed a tendency towards an
exchange of order of the lowest-energy multiplets cannot be discarded. We plot
the energy differences for different $N$ by scaling the x-axis as $1/N$ as, in
this way, the relation is expected to be linear [7]. Yet, we emphasize that,
despite our efforts to push the limits of bond dimension in our system —
$\chi=4000$, corresponding to $\chi_{ns}\sim 35000$ of non-symmetric matrix-
product states — we cannot claim full convergence for system sizes $N\gtrsim
48$ 555Also, contrast with Chen et al. who pushed system sizes to 120, but
with bond dimensions truncated at max $\chi=2800$. (again, more details are
shown in Appendix A).

###  IV.2 Single-mode approximation

Figure 6: The single-mode approximation. We plot $\omega_{q}$ as well as the
numerator $\langle 0|[(O(q))^{\dagger},[H,O(q)]]|0\rangle$ and the denominator
$\langle O(q)|O(q)\rangle$ of $\omega_{q}$ separately [see Eq. (9)] with the
choice $O(q)=T^{\kappa}_{m}(q)$. All calculations have system size $N=60$ and
bond dimension $\chi=4000$.

Being unable to fully verify the signatures of a critical phase, we return to
the problem of the apparent criticality in the region between $\gamma=-\pi/4$
and $-3\pi/4$ of the spin-1 problem. Inspired by that, we contemplate the
possibility of the a large crossover scale due to the closeness of the high-
symmetry ferromagnetic region. To address that, we follow Ref. [11], which
suggested the use of the “single-mode approximation” (SMA) as a venue to
verify such a scenario. The SMA process consists in constructing momentum-
specific trial states

$\left|O\left(q\right)\right\rangle=O\left(q\right)\left|0\right\rangle,$ (7)

for a given operator $O\left(q\right)$ carrying quantum numbers of a sector of
interest in the Hilbert space, and $\left|0\right\rangle$ is the ground state,
which we find via DMRG. The mean value of the Hamiltonian in this state may be
written as

$\displaystyle\omega_{q}$ $\displaystyle=\frac{\left\langle
O\left(q\right)\left|H\right|O\left(q\right)\right\rangle}{\left\langle
O\left(q\right)\left|O\left(q\right)\right.\right\rangle}-\langle
0|H|0\rangle$ (8) $\displaystyle=\frac{1}{2}\frac{\left\langle
0\left|\left[\left(O\left(q\right)\right)^{\dagger},\left[H,O\left(q\right)\right]\right]\right|0\right\rangle}{\left\langle
O\left(q\right)\left|O\left(q\right)\right.\right\rangle}$ (9)

and serves, like any variational energy, as a strict upper bound on the gap
for the sector of interest 666For this to work it is necessary that $\langle
0|O(q)\rangle=0$, which can be easily checked. See e.g. Sec. 6.2 of [58].. The
power of this method comes from the fact that this upper bound can be seen to
vanish both from the perspective of (i) the divergence of the denominator,
typical for critical phases or phase transitions, as well as from (ii) the
vanishing of the numerator, which requires a commutation of $O(q)$ and the
Hamiltonian. In the latter case, an anomalously small upper bound to a system
gap can emerge, giving the impression of critical behavior. In the spin-1
chain study, the latter case was realized by noting that at $q=0$ the
quadrupole operator $T^{2}_{m}(q)$ commuted with the Hamiltonian at
$\gamma=-3\pi/4$ [11].

We thus ask ourselves if, along the lines of scenario (ii), the presence of
the SU$(5)$ point and the SO$(5)$ “ferromagnetic” line in the phase diagram of
the spin-2 chain (c.f. Fig. 4) could be influencing the apparent critical-like
physics at the pyramidion. We compute $\omega_{q}$ for the operators
$O\left(q\right)=T_{m}^{\kappa}(q)$ to estimate the upper energy bounds for
each $\kappa$ sector, corresponding to the total angular momentum $S$ sectors.
The results are presented in Fig. 6. We find that the vanishing of the
numerator in Eq. (9) only happens for the dipolar $\kappa=1$ sector, at $q=0$.
This result is trivial, as the total angular momentum is naturally preserved
due to the exact SU$(2)$ symmetry of the problem; the SMA loses predictive
capacity in this angular and linear momentum sector. Another feature we remark
happens in the case of $\kappa=3$, the octupolar sector, which displays
$\omega_{q}\to 0$ as $q\to 0$ due to the divergence of the structure factor in
the denominator of Eq. (9). This, together with the other notable points of
minima for $\omega_{q}$ (namely, $q=0$ for $\kappa=4$ and $q=2\pi/3$ for
$\kappa=1$ and $2$), is more closely related to scenario (i) described above
and cannot offer predictions beyond those of the finite-size scaling performed
in the previous sub-section.

In conclusion, the SMA analysis does not support nor contradict claims of
critical behavior in the spin-2 problem. If we moved away from the pyramidion
towards the ferromagnetic phase, the SO$(5)$-symmetric phase boundary
certainly leads to vanishing SMA upper bounds in all sectors we considered at
$q=0$, but the pyramidion point itself is far enough for the commutators in
the numerator of $\omega_{q}$ to be comfortably finite. Reversing the point of
view, the opposite conclusion may bring some extra value: in Ref. [9], it was
claimed that energy-level considerations were more trustworthy than
entanglement-entropy scaling analyses when trying to extract CFT
characteristics, albeit the latter being more typically exploited for these
purposes. Indeed, while our SMA results do not capture entanglement effects,
they do indicate no anomalous effects in the energy levels for the Hamiltonian
at the pyramidion point, so that energy-level scaling analyses seem to be
limited just by regular finite-size scaling convergence.

###  IV.3 Entanglement spectrum

A further tool that has recently been used to classify phases of matter, as
well as analyse criticality, is entanglement. In 1D criticality, a commonly
studied quantity is the entanglement entropy, known to be directly related to
a conformal field theory invariant, namely the central charge [47]. More
recently, it has been shown by Calabrese and Lefevre that not only the central
charge is a specific signature of critical points and phases, but the whole
distribution of eigenmodes of the reduced density matrix takes a special shape
[48]. The predictions of Calabrese and Lefevre were verified in numerical
simulations [49]. Chen et al. already extensively verified the entanglement
entropy scaling, finding, as mentioned, that robust CFT information was hard
to obtain. Here we extend the entanglement study of the pyramidion spin-2
Hamiltonian and contemplate the entanglement spectrum predictions of Calabrese
and Lefevre in another attempt to solve the deadlock between presence or
absence of criticality.

One considers the reduced density matrix of a subsystem $A$, decomposed in
terms of eigenmodes as

$\displaystyle\rho_{A}=\sum_{i}\lambda_{i}\left|\lambda_{i}\right\rangle\left\langle\lambda_{i}\right|,$
(10)

$\lambda_{i}$ being the eigenvalues. From these, a probability distribution
can be built for the eigenvalues, $P(\lambda)$, for which the mean number of
eigenvalues larger than a value $\omega$ assume a very simple form in
conformal invariant systems, namely

$\displaystyle n(\omega)=\int_{\omega}^{\omega_{\mathrm{max}}}d\lambda
P(\lambda)=I_{0}\left(2\sqrt{b\log{\frac{\omega_{\mathrm{max}}}{\omega}}}\right),$
(11)

where $I_{0}$ is the zeroth order modified Bessel function of the first kind,
$\omega_{\mathrm{max}}$ is the largest eigenvalue of $\rho_{A}$ and is also
related to the parameter $b$ according to

$\displaystyle
b=-\log{\omega_{\mathrm{max}}}=\frac{c}{6}\log{L_{\mathrm{eff}}},$ (12)

where $c$ is the conformal central charge,
$L_{\mathrm{eff}}=N\sin{\pi\ell_{A}/N}$, with $N$ the system size, and
$\ell_{A}$ the size of the sub-region $A$ [48]. We always take $\ell=N/2$, so
that $L_{\mathrm{eff}}=N$.

Thus, constructing $n(\omega)$ is, at first sight, tantamount to measuring the
central charge $c$, a quantity that has already been considered by Chen et al.
via the entanglement entropy with several convergence limitations. Yet,
$n(\omega)$ goes beyond that and displays a very specific functional form for
conformal systems [Eq. (11)] that deserves being considered on its own as a
potential signature of criticality.

Figure 7: The cumulative distribution function of the entanglement spectrum
$n(\omega)$ of a subsystem of half the system size for (a) the spin-2 model
and (b) the spin-1 model with $\tan\gamma=1.1$. We also show the Calabrese-
Lefevre functional form with fitted $b$ (full) and the analytical
$b=-\log\omega_{\mathrm{max}}$ (dashed).

In Fig. 7(a) we display the numerical data for $n(\omega)$ for a system sizes
$N=12,\,36$ and $60$, and consider it against the right hand side of Eq. (11).
We consider both fitting the data for the functional form of $I_{0}$ (full
lines), as well as a fixed curves with the analytical form of $b$ (dashed
curves). As we see, as the system size increases, the functional form of
$I_{0}$ fits numerical data better and better. Yet, discrepancies arise
between the fitted $b$ and the analytical prediction of $b$. The discrepancies
are smaller for small system and become more apparent as the system size
increases from 12 to 24 (as we verified). As the system size continues
increasing to 60, the discrepancy tends to diminish again, albeit more slowly.
The fitted values of $b$ are consistently smaller than their analytical
counterparts and, from Eq. (12), result in estimates for the central charge
$c$ ranging around $2.9$ to $3.2$. These estimates seem to be consistent with
Chen et al. for similar system sizes (although, again, we pursue larger
truncation dimensions). For contrasting purposes, we also perform the same
calculation in the spin-1 SU(3)1 critical phase (Fig. 7(b)). We choose the
point $\tan\gamma=1.1$ slightly off the ULS point, where the symmetry is
emergent instead of exact. In this case, the fitted $b$ its analytical
predicted value agree better, and both give an estimate of the central charge
$c\simeq 2.1-2.2$ closer to the expected $c_{SU(3)_{1}}=2$.

##  V spin-$S$ and beyond

Our results cast doubt in the existence of the spin-2 SU$(3)_{1}$ critical
phase. Nevertheless, an interesting realization prompts us to further consider
this possibility. Rewriting the spin-2 pyramidion Hamiltonian in a projector
language, it reads

$\displaystyle H_{2}^{\star}=-\sum_{i}\mathcal{P}_{2}^{i}.$ (13)

We remark on the similarity between this and Eq. (3). As it turns out, for an
arbitrarily chosen _integer_ spin $S$, the ground state of a three-site
problem with Hamiltonian $\sim-(\mathcal{P}_{S}^{1}+\mathcal{P}_{S}^{2})$ is
always a singlet. For larger system sizes, quantum fluctuations scramble and
entangle these 3-spin singlets, generating an SU$(3)_{1}$ critical phase for
spin-1 and, as was hypothetically proposed, also for spin-2. This observation
leads to an irresistible idea: could potentially a family of SU$(3)_{1}$
critical phases exist for _arbitrary integer_ spin-$S$ systems? (Contrasting
with the family of SU$(2)_{1}$ phases that exist for _half-integer_ spin
chains, driven by Haldane’s topological arguments in the anti-ferromagnetic
Heisenberg models.) We leverage our learning so far to explore this question.

We consider an spin-$S$ SU$(2)$-invariant chain with Hamiltonian

$\displaystyle H_{S}=-\sum_{i}\mathcal{P}_{S}^{i}.$ (14)

We start by studying the classical spin texture expected for this Hamiltonian.
We write the projection operator explicitly so that

$H_{S}=-\sum_{i}\prod_{k\neq
S}\frac{\mathbf{J}_{i,i+1}^{2}-k(k+1)}{S(S+1)-k(k+1)},$ (15)

where

$\mathbf{J}_{i,i+1}^{2}=\mathbf{S}_{i}^{2}+\mathbf{S}_{i+1}^{2}+2\mathbf{S}_{i}\cdot\mathbf{S}_{i+1},$
(16)

and substitute $\mathbf{J}_{i,i+1}^{2}=2S(S+1)+2S^{2}\cos\theta$, with a
uniform first-neighbor canting angle $\theta$. This results in a classical
energy given by

$E(\theta)=-N\prod_{k\neq
S}\frac{2S(S+1)+2S^{2}\cos\theta-k(k+1)}{S(S+1)-k(k+1)},$ (17)

and we search the favored ground state by finding the angle $\theta^{*}$ that
minimizes the energy. The results are plotted in Fig. 8(a). As we know, for
$S=1$ we have $\theta^{*}=2\pi/3$ exactly. While for $S=2$, $\theta^{*}$ is
shifted from $2\pi/3$ only mildly, for larger spin $S$ we find strong
deviations from this triply-periodic value.

Naturally, this serves as another demonstration of the failure of the
classical description of spin waves in 1D systems. The scrambling of 3-site
singlets indicate that correlations of arbitrary $q^{*}$ are not possible and
will not match the classical $\theta^{*}$. To verify this is the case, we
again use our non-Abelian DMRG code to compute the static structure factor for
all available higher-order tensor operators extending the spin-1 and spin-2
results to $S=3,4,5$. While the DMRG convergence gets penalized as $S$
increases, our experience is that the position of the dominant momenta in
$\mathcal{C}^{\kappa}(q)$ is largely stable and well-fixed even at system
sizes as small as $N=12$.

Figure 8: (a) The classical energy $E(\theta)$ associated with the spin-$S$
Hamiltonian $H=-P_{S}$, with $P$ the projector onto the multiplet of total
angular momentum $S$, a possible extrapolation of the spin-2 Hamiltonian.
Inset: the angle $\theta^{*}$ that mininizes the energy. We have rescaled
$E(\theta)$ to the interval $-1$ to $1$ to better present the trend. For
$S=1,2$, the peak is at $2\pi/3$ but for higher spins, the position shifts
towards smaller values. (b-d) The static structure factor for the quantum
Hamiltonian $H=-P_{S}$ for $S=3,4,5$ for different values of $\kappa$ and
system size $N=12$. We use the same color code as in (a), but the meaning is
distinct: each panel (b-d) corresponds to a different spin value, and the
color code refers to the values of $\kappa$. For $S\geq 4$ (a) or $\kappa\geq
4$ (b-d), the points are shown in ever lighter shades of grey. For $S=3$, an
extra $G_{2}$ symmetry is reveled: $\kappa=1$ (dipolar) and $\kappa=5$
(triacontadipolar) have the correlation, as well as $\kappa=2,4,6$. (e) A
comparison of all the commonly available $\mathcal{C}^{\kappa}$ for chains
with $S=2,3,4,5$. We note that their behavior of correlations is spin-
independent, independently also of finite-size corrections.

Figs. 8(b-d) display the results for $S=3,\,4$ and $5$ and are representative
of the trend we see for all larger $S$. Correlations peak sharply at either
$q^{*}=0$ or $2\pi/3$. As we find, uniform octupolar correlations are _always_
the dominating ones, and dipole and quadrupole correlations always peak
together at $2\pi/3$ (with dipoles dominating over quadrupoles, just as in
$S=2$). Higher multi-polar correlation functions are also available for higher
spin chains, but are not the center of our attention; they are displayed in
grey in the figure. In Fig. 8(e), we compare $\mathcal{C}^{\kappa}(q)$ with
$\kappa=1,\,2,\,3,\,4$ for spins $S=2$ to $5$, and note that, independently of
finite-size effects, the behavior of the correlation functions are the same
for all representations.

A “surprise” can also be observed: for $S=3$ (Fig. 8(b)), we observe an exact
degeneracy between dipolar ($\kappa=1$) and triacontadipolar (32-polar)
($\kappa=5$) operators (orange and shades of grey), as well as between
quadrupolar ($\kappa=2$) , hexadecapolar ($\kappa=4$) and hexacontatetrapolar
(64-polar) ($\kappa=6$) operators (green and shades of grey). These two sets,
together with the distinct octopolar correlations ($\kappa=3$) (red), lead to
multiplets $7+14+27$. Combining with the $\kappa=0$ singlet then reproduces
the levels of the group G2, hinting to an exact microscopic G2 symmetry of
this model for $S=3$. We list in Appendix B the generators of its
corresponding Lie algebra, $\mathfrak{g}_{2}$ and an analysis of the
Hamiltonian. Upon a careful look, the degeneracies should not be a surprise at
all: we demonstrate rigorously that a region of exact G2 symmetry exists
inside the phase space of the $S=3$ rotational-invariant Hamiltonians (also
discussed in Appendix B). We thus conclude this section with a comment on
future directions of inquiry: in a similar spirit to our previous searches of
symmetry emergence, this observation makes the $S=3$ spin chain a relevant
system to search for $(\mathrm{G}_{2})_{1}$ critical points and phases, of
importance in the search of Fibonacci anyons, in fine-tuned region of the
parameter space of an SU(2)-invariant system. Whether one can find robust
phases where the larger symmetry is manifest remains a problem for the future.

##  VI Conclusion

We re-evaluated the signatures of SU$(3)_{1}$ criticality in integer-spin 1D
chains, in particular in the spin-2 system of Refs. [8, 9]. Comparing with the
well-known case of spin-1, we analyzed, via SU$(2)$-symmetric non-Abelian
DMRG, the static structure factors of multipole tensor operators in the spin-2
problem as profoundly in the hypothetical SU$(3)_{1}$ phase as we could. Our
results indicate domination of ferro-octupolar correlations, over
$2\pi/3$-periodic dipolar and quadrupolar, as well as missing signatures of
long-range Luttinger-liquid-like behavior in the quadrupolar correlations. We
push the convergence of finite-size scale analysis in comparison with previous
literature results and, despite the difficulty in attaining full convergence,
trends indicate that the results above should not change in the thermodynamic
limit but rather become more pronounced.

By tracing parallels and performing symmetry analyses, we also extended our
understanding of phases in the spin-2 parameter space, pointing to a phase of
extended SU$(5)_{1}$ symmetry, and showing that the problem proposed by Chen
et al. lies in a similar place with an anomalously critical-looking region of
the spin-1 phase diagram. As done previously in the case of spin-1 [11], we
used a single-mode approximation to explore the proximity of high-symmetry
uniformly ordered critical lines and a ferromagnetic phase as explanations for
the apparent spin-2 critical behavior. While we found little evidence that
this is the actual origin of the problem, our results indicate that the
energetics and correlation analysis performed are trustworthy and robust.

We also explored the entanglement spectrum of the spin-2 problem as a
potential venue to determine whether this Hamiltonian is indeed critical or
not. The distribution of eigenvalues is well fitted by analytical laws valid
in conformal systems. Yet, we verify discrepancies in the fitting parameter
and difficulty in estimating the central charge as previously reported in the
literature. That the critical form for the distribution of eigenvalues of the
entanglement spectrum is well respected is, perhaps, not surprising, as
clearly any potential residual gap in this spin-2 system should be very small
and difficult to verify in finite-sized systems.

The challenge in explaining this anomalously critical-like behavior, together
with the rarity of extended critical phases with symmetry emergence motivated
us to still explore higher-spin generalizations of this problem. We proposed a
family of integer spin-$S$ Hamiltonians whose magnetic behaviors not only
serve as examples for the limitations of the non-interacting spin-wave theory
in 1D, but are all similarly related: they are all dominated by ferro-
octupolar correlations and display dipolar and quadrupolar correlations with
$2\pi/3$-periodicity. Higher-order multipolar correlations are naturally also
present in these, either peaking at vanishing momentum or displaying period-
three behavior. These results also signal how arbitrary spiral order is not
favored in quantum magnetism, an effect that we assign to simple properties of
angular-momentum summation for the generation of singlets. Amusingly, for spin
$S=3$, the Hamiltonian we studied was found to be located in an interesting
region of the parameter space with exact G2 symmetry; this might be a
promising starting point to search for G2 criticality, of relevance for
quantum computing applications.

Despite our results providing evidence against a conformal-invariant ground
state in the spin-2 problem, an abnormal difficulty for DMRG convergence is
present in this problem, which suggests either a very small gap or, indeed,
the absence of one. If a gap exists, characterizing this phase would be an
important step: not many candidates seem to remain to explain this beyond a
trimerized state, but a 1D trimerized phase in an isotropic system seems as
exotic as the critical phase proposed by Chen et al. As far as we can tell, if
the system is indeed gapless, the SU$(3)_{1}$ state does seem to be the best
candidate. Still, an analytical explanation for the emergence of a critical
phase resisted our best efforts. To say the minimum, our numerical results
indicate that an uncommon frustration mechanism is necessary, where degrees of
freedom controlling octupolar and hexadecapolar fluctuations decouple from the
dipolar and quadrupolar ones, with the former being spatially uniform while
the latter ones giving rise to a quasi-SU$(3)_{1}$ fluid. Therefore, important
questions remain. If the SU$(3)_{1}$ phase is there, how could it emerge? If
it exists for spin-2, does it generalize to arbitrary integer spin? If this
phase is not there, why does this system look so critical, and what is the
nature and origin of these gapped ferro-octupolar phases instead? Finally,
recent advances in tensor network methods have introduced methods that are
efficient for simulation of gapless phases [50]; would it be possible to apply
such methods to this present problem and improve the numerical data available,
perhaps even including even including non-Abelian rotation invariance to these
new methods? We leave these as questions for future considerations.

##  VII Acknowledgments

We acknowledge insightful conversations with Frédéric Mila. For the SU(2)
calculations we use an in-house custom code utilizing the High-Performance
Tensor Transpose library [51], the Tensor Contraction Library [52], the
WIGXJPF library [53], and the FASTWIGXJ library [54]. C. L. is supported by
the QuEST scholorship at the University of British Columbia. For the early
stages of this work, V.L.Q. acknowledges financial support from the High
Magnetic Field Laboratory through the NSF Cooperative Agreement No. 1157490
and the State of Florida.

##  Appendix A Other numerical results

Here we present the finite bond dimension scaling of the energy gap to
indicate the convergence of the DMRG simulations. The energy gap $\Delta$ is
obtained by subtracting the ground state energy in spin sectors of $S=1,..,4$
from that of the singlet. The results with bond dimension $\chi$ between 1000
and 4000 are shown in Fig. 9. For a given fixed system size, a linear fit of
$\Delta$ against $1/\chi$ on the last four data sets, i.e.
$\chi=2400,3000,3400,4000$, gives an estimate of the convergence. Indeed,
denoting the fitted slope as $\alpha$, we have a lower bound of $\Delta$,

$\Delta_{\infty}\geq\Delta_{\chi_{M}}-\frac{\alpha}{\chi_{M}},$ (18)

where $\chi_{M}=4000$ is the largest available bond dimension and
$\Delta_{\infty}$ is the infinite bond-dimension limit. We see that for
$N\gtrsim 48$, $\alpha\gtrsim 10$, resulting in $\alpha/\chi_{M}\gtrsim
0.0025$, or $\sim 5\%$ of the value. With these system sizes, and convergence
errors, we cannot rule out the possibility of this system being gapped.

Figure 9: The energy gap $\Delta$ as a function of bond dimension $\chi$
between 1000 and 4000. A linear fit of $\Delta$ against $1/\chi$ on the
largest four $\chi$’s gives a slope $\alpha$.

##  Appendix B G2 symmetry of the spin-3 model

In this appendix we discuss the G2 symmetry of the spin-3 Hamiltonian,
$H_{3}=-\sum_{i}\mathcal{P}_{3}^{i}$. G2 is the smallest exceptional semi-
simple Lie group and also the automorphism group of the octonion algebra [55].
We will focus on its corresponding Lie algebra $\mathfrak{g}_{2}$.

The Lie algebra has a two-dimensional Cartan subalgebra, for which we can
choose the basis

$\begin{split}H_{1}=\frac{1}{\sqrt{14}}\mathrm{diag}\\{-3,-2,-1,0,1,2,3\\},\\\
H_{2}=\frac{1}{\sqrt{42}}\mathrm{diag}\\{-1,4,-5,0,5,-4,1\\},\end{split}$ (19)

which are proportional to $T_{0}^{1}$ and $T_{0}^{5}$, respectively. Then
judiciously combining the eigenvectors of $[H_{i},\boldsymbol{\cdot}]$ (a
linear operator on $\mathfrak{sl}(7,\mathbb{C})$), we arrive at the
generators. Here we adopt the standard notation with the two simple roots
$\alpha_{1}=(\frac{3}{\sqrt{14}},\frac{1}{\sqrt{42}})$ and
$\alpha_{2}=(-\frac{5}{\sqrt{14}},\frac{3}{\sqrt{42}})$. The root generators
are

$\begin{split}&E_{1}=\frac{1}{\sqrt{3}}\begin{pmatrix}0\\\ 0\\\ 0\\\
\sqrt{2}\\\ 0&-1\\\ 0&0&-1\\\ 0&0&0&\sqrt{2}&0&0&0\end{pmatrix},\\\
&E_{2}=\begin{pmatrix}0&0&0&0&0&1&0\\\ &&&&&&1\\\ &&&&&&0\\\ &&&&&&0\\\
&&&&&&0\\\ &&&&&&0\\\ &&&&&&0\\\ \end{pmatrix},\\\
&E_{12}=[E_{1},E_{2}],E_{112}=\frac{\sqrt{3}}{2}[E_{1},E_{12}],\\\
&E_{1112}=[E_{1},E_{112}],E_{11122}=[E_{2},E_{1112}],\end{split}$ (20)

and their transposes. The set of two Cartan generators and the twelve root
operators can be chosen as the generators of the group. With the matrix form
of these generators, one can readily check that they commute with the
Hamiltonian $H_{3}$ (for that, checking for a pair of sites is enough). In
fact, the Hamiltonian can be expressed in term of the quadratic Casimir
$C_{2}$ as

$H_{3}=\sum_{i}\frac{1}{4}C_{2}^{i}-\frac{5}{16}(C_{2}^{i})^{2}-\frac{3}{32}(C_{2}^{i})^{3},$
(21)

where $C_{2}$, invariant under group rotations, is

$C_{2}^{i}=H_{1}^{i}H_{1}^{i+1}+H_{2}^{i}H_{2}^{i+1}+\sum_{E}E^{i}(E^{i+1})^{\dagger}.$
(22)

The sum over $E$ is the sum over all the twelve roots.

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