# Edge states and universality class of the critical two-box symmetric
$\mathrm{SU}(3)$ chain

Pierre Nataf  Laboratoire de Physique et Modélisation des Milieux Condensés,
Université Grenoble Alpes and CNRS, 25 avenue des Martyrs, 38042 Grenoble,
France  Samuel Gozel  Institute of Physics, Ecole Polytechnique Fédérale de
Lausanne (EPFL), CH-1015 Lausanne, Switzerland  Frédéric Mila  Institute of
Physics, Ecole Polytechnique Fédérale de Lausanne (EPFL), CH-1015 Lausanne,
Switzerland

(August 28, 2024)

###### Abstract

We numerically demonstrate that, although it is critical, the two-box
symmetric $\mathrm{SU}(3)$ chain possesses edge states in the adjoint
representation whose excitation energy scales with the number of sites $N_{s}$
as $1/(N_{s}\log N_{s})$, in close analogy to those found in half-integer
$\mathrm{SU}(2)$ chains with spin $S\geq 3/2$. We further show that these edge
states dominate the entanglement entropy of finite chains, explaining why it
has been impossible so far to verify with DMRG simulations the field theory
prediction that this model is in the $\mathrm{SU}(3)_{1}$ universality class.
Finally, we show that these edge states are very efficiently screened by
attaching adjoint representations at the ends of the chain, leading to an
estimate of the central charge consistent within 1% with the prediction $c=2$
for $\mathrm{SU}(3)_{1}$.

The $\mathrm{SU}(N)$ Heisenberg model is currently attracting a lot of
attention because, as soon as $N>2$, not so much is known about its properties
while there is a real prospect of implementing it with ultracold fermions Wu
_et al._ (2003); Honerkamp and Hofstetter (2004); Cazalilla  _et al._ (2009);
Gorshkov  _et al._ (2010); Taie  _et al._ (2012); Pagano  _et al._ (2014);
Scazza  _et al._ (2014); Zhang  _et al._ (2014); Hofrichter  _et al._ (2016);
Capponi  _et al._ (2016). For the fundamental representation, the
$\mathrm{SU}(N)$ model is nothing but a model of quantum permutation between
objects with $N$ colors, and in one dimension, there is a Bethe ansatz
solution for any $N$ Sutherland (1975). The system is critical, with algebraic
correlations, and its low-energy, long-range properties are described by a
field theory known as the Wess-Zumino-Witten $\mathrm{SU}(N)_{1}$ universality
class, with central charge $c=N-1$ and specific scaling dimensions Affleck
(1986a, 1988). For $\mathrm{SU}(2)$ the fundamental representation corresponds
to spin-$1/2$, but it is well known since the work of Haldane that the physics
can be very different for other irreducible representations (irreps). For the
spin-$1$ case (represented by a horizontal Young diagram with two boxes,
$\mathrel{\raisebox{0.0pt}{\scalebox{0.4}{\ydiagram{2}}}}$), Haldane predicted
that the spectrum is actually gapped Haldane (1983a, b), a result confirmed
soon after numerically White (1992); White and Huse (1993) and experimentally
Buyers  _et al._ (1986).

It is then a very natural question to wonder whether and how these results can
be generalized to $N>2$ Affleck (1986b, 1988); Greiter and Rachel (2007);
Rachel  _et al._ (2009); Bykov (2013); Lajkó  _et al._ (2017); Tanizaki and
Sulejmanpasic (2018); Wamer  _et al._ (2019, 2020); Wamer and Affleck (2020).
In order of increasing complexity, the next case is the $\mathrm{SU}(3)$ model
in the symmetric representation with two boxes, the same Young diagram
$\mathrel{\raisebox{0.0pt}{\scalebox{0.4}{\ydiagram{2}}}}$ as the spin-$1$
chain for $\mathrm{SU}(2)$. It turns out that this model has not yet received
a compelling solution, by which we mean, in the absence of exact results, a
field theory prediction confirmed by numerical results. According to field
theory, if the system is critical, it can only be in the $\mathrm{SU}(3)_{1}$
universality class because the only alternative, $\mathrm{SU}(3)_{2}$, has a
relevant operator allowed by symmetry Lecheminant (2015). And a generalization
of Haldane’s semiclassical argument has suggested that the system is indeed
critical because the only cases where there is no topological term in the
action, hence where the system must be gapped, are those where the number of
boxes is a multiple of three Lajkó  _et al._ (2017) (a result recently
confirmed numerically for three boxes Gozel  _et al._ (2020)). However, it has
proven impossible so far to confirm that the two-box symmetric
$\mathrm{SU}(3)$ chain is in the $\mathrm{SU}(3)_{1}$ universality class.
Results of exact diagonalizations on small chains with periodic boundary
conditions are rather consistent with $\mathrm{SU}(3)_{2}$ Nataf and Mila
(2016), a result interpreted, in view of the field theory prediction, as an
evidence that a cross-over must take place as a function of the size to the
$\mathrm{SU}(3)_{1}$ universality class. In these circumstances, the obvious
thing to do is to try density-matrix renormalization group (DMRG) calculations
White (1992, 1993) which, with open boundary conditions, can be performed on
chains with hundreds of sites. However, much to our surprise, the central
charge deduced from fitting the entanglement entropy with the Calabrese-Cardy
formula Calabrese and Cardy (2004) leads to a result much larger than the
expected $c=2$ (see below), a difficulty also met by other groups pri . So as
of today the two-box symmetric $\mathrm{SU}(3)$ chain remains a puzzle.

In this Letter, we identify the origin of the problem as being due to edge
states. In gapped topological phases such as the spin-$1$ chain, it is well
known that edge states can exist, and in open chains they show up as low-lying
excitations inside the gap Kennedy (1990); Schollwöck and Jolicoeur (1995). In
gapless systems, this is less well known, but it has been shown in the
nineties that the $S=3/2$ chain has edge states that lead to an excitation
with an energy smaller than the finite-size gap by a factor $\log N_{s}$,
where $N_{s}$ is the number of sites Ng (1994); Qin  _et al._ (1995); Fáth
_et al._ (2006). More precisely, we will show that the two-box symmetric
$\mathrm{SU}(3)$ chain also has such an excitation below the finite-size gap
due to edge states, that it disappears when we screen the edge states by
attaching adjoint representations at the ends of the chain, and that the
entanglement spectrum then leads with excellent accuracy to the expected
central charge $c=2$ when fitted with the Calabrese-Cardy formula.

The Hamiltonian of the $\mathrm{SU}(3)$ Heisenberg chain can be written quite
generally in terms of the $\mathrm{SU}(3)$ generators as

$\mathcal{H}=J\sum\limits_{i}\sum\limits_{\alpha,\beta=1}^{3}\mathcal{S}^{\alpha\beta}_{i}\mathcal{S}^{\beta\alpha}_{i+1}.$
(1)

For symmetric irreps, and up to a constant, this can be rewritten in terms of
boson creation and annihilation operators as

$\mathcal{H}=J\sum\limits_{i}\sum\limits_{\alpha,\beta=1}^{3}b^{\alpha\dagger}_{i}b^{\beta}_{i}b^{\beta\dagger}_{i+1}b^{\alpha}_{i+1}.$
(2)

In the following, we will concentrate on the case with two bosons per site,
which corresponds to the $6$-dimensional symmetric irrep described by the
Young diagram $\mathrel{\raisebox{0.0pt}{\scalebox{0.4}{\ydiagram{2}}}}$. The
$6$ states correspond to all the ways of constructing a two-boson state with
three colors. Throughout, we will also use the alternative notation
[$\alpha_{1}$,$\alpha_{2}$,$\alpha_{3}$] for the irreps of $\mathrm{SU}(3)$,
where the integers $\alpha_{1}$, $\alpha_{2}$ and $\alpha_{3}$ correspond to
the lengths of the rows in the corresponding Young diagram 111Irreps of SU(N)
are usually described with $N-1$ integers corresponding, for instance, to the
lengths of the rows of the Young diagram, as any column with $N$ boxes
corresponds to a totally antisymmetric combination of particles and can thus
be removed. Here, however, we use a notation with $N=3$ integers for the
irreps of $\mathrm{SU}(3)$ because, in practice, we map irreps of
$\mathrm{SU}(N)$ with $n$ boxes onto irreps of the permutation group $S_{n}$
with at most $N$ rows. See Refs. Nataf and Mila (2014); Gozel  _et al._ (2020)
for details.. With these notations, the two-box symmetric irrep is denoted by
[2,0,0].

To reach long enough chains, we will use the same version of DMRG as that used
previously for the fundamental representation Nataf and Mila (2018) and for
the three-box symmetric representation Gozel  _et al._ (2020). It relies on a
basis constructed with the help of standard Young tableaus Young (1900);
Rutherford (1948); Nataf and Mila (2014, 2016); Wan  _et al._ (2017), and
allows one to diagonalize the Hamiltonian directly in arbitrary irreps of
$\mathrm{SU}(3)$ except those that are degenerate when making the product of
two half chains out , a limitation with no major impact for the problem
addressed here.

Figure 1: Central charges extracted from the entanglement entropy for the
Heisenberg open chain of size $N_{s}$ with the two-box symmetric irrep at each
site. In the top panel (a), it increases with $N_{s}$ to values larger than
$3.5$. There are two sets of points ($q=0$ and $q=1,2$) because the
entanglement entropy oscillates between a top and a bottom envelope (see sup
). For the bottom panel (b), we have added edge spins living in the adjoint
irrep which screen the edge states, and the central charges converge towards a
value very close to $2$, in agreement with field theory arguments. The results
have been plotted as $1/(N_{s}\log N_{s})$ by analogy with the edge state gap.
See text for details.

As a first attempt at characterizing the universality class of that model, we
have performed DMRG simulations on open chains, and we have fitted the
entanglement entropy with the Calabrese-Cardy formula Calabrese and Cardy
(2004) to extract the central charge. The finite-bond dimension effects on the
central charge are small and have been taken care of by scaling the results
with the discarded weight (see Supplemental Material sup  for an example). The
results are plotted in the top panel of Fig. 1. Because of period $3$
oscillations, it is better to fit independently two sets of points ($q=0$ and
$q=1,2$, where $q$ is the position of the cut modulo $3$ for the calculation
of the entanglement entropy along the chain sup ). In both cases, the apparent
central charge increases upon increasing the system size to values of the
order of $3.6$ or $3.7$, well above $16/5$, the value for $\mathrm{SU}(3)_{2}$
observed on small chains with periodic boundary conditions Nataf and Mila
(2016), and a fortiori much larger than the expected $c=2$ for
$\mathrm{SU}(3)_{1}$. This result makes no physical sense, and the only
possibility is that the entanglement we are measuring is not that of the bulk,
but is dominated by edge effects. So let us have a closer look at the spectrum
of open chains.

Since we are dealing with an irrep with two boxes at each site, the ground
state can only be a singlet if the number of sites $N_{s}$ is a multiple of
$3$. If the number of sites is equal to $1$ mod. $3$, the ground state is
expected to be in the [2,0,0] sector, and if it is equal to $2$ mod. $3$, the
ground state should be in the [2,2,0] sector. Quite generally, the first
excited state is expected to be in the adjoint representation [2,1,0] if the
ground state is a singlet, and in the most antisymmetric combination of the
adjoint with the ground state sector otherwise, i.e. [1,1,0] if the ground
state is in the [2,0,0] sector, and [1,0,0] if the ground state is in the
[2,2,0] sector. Since we cannot calculate in the adjoint sector for technical
reasons, we have calculated the first excitation in the [3,0,0] irrep for
$N_{s}$ multiple of three, and the expected lowest sector in the other cases.
The results are shown in Fig. 2. If the spectrum was simply representative of
the bulk, the first excitation should collapse onto the ground state as
$1/N_{s}$ if the system is gapless, and $N_{s}$ times the energy should go to
a constant. This is clearly not the case when $N_{s}$ is not a multiple of
three: $N_{s}$ times the excitation energy goes to zero as $1/\log N_{s}$.
This is typical of edge states in half-odd-integer spin chains with $S\geq
3/2$ Ng (1994); Qin  _et al._ (1995); Lou  _et al._ (2000); Fáth  _et al._
(2006), and by analogy with that case, we conclude that there are edge states
that produce an excitation in the adjoint representation with an energy
scaling as $1/(N_{s}\log N_{s})$. By contrast, when $N_{s}$ is a multiple of
$3$, the first sector we can target, [3,0,0], is not that of the first
excitation, and the corresponding excitation scales as $1/N_{s}$, a scaling
typical of elementary bulk excitations in 1D gapless systems and in sharp
contrast with the previous edge excitations calculated for $N_{s}$ not a
multiple of $3$.

Figure 2: $N_{s}\Delta$ for (a) $N_{s}=1$ mod. $3$, (b) $N_{s}=2$ mod. $3$ and
(c) $N_{s}=0$ mod. $3$, where $\Delta$ is a gap between the ground state (GS)
energy and the first excited state (ES) energy living in the irrep (a)
$[1,0,0]$ for $N_{s}=1$ mod. $3$, (b) $[1,1,0]$ for $N_{s}=2$ mod. $3$ and (c)
$[3,0,0]$ for $N_{s}=0$ mod. $3$. The results are shown as a function of
$1/\log N_{s}$ for different values of $m$, i.e. the number of states kept
controlling the DMRG accuracy. For $N_{s}=1,2$ mod. $3$, $N_{s}\Delta$ goes to
zero when $N_{s}\rightarrow\infty$, implying that, unlike the usual
expectation for a bulk excitation in a gapless chain, the excitation shown
does not vanish as $1/N_{s}$ but faster, revealing the presence of edge
states. By contrast, when $N_{s}=0$ mod. $3$, $N_{s}\Delta$ goes to a strictly
positive constant when $N_{s}\rightarrow\infty$ because the sector is not that
of the lowest excitation (See text for details).

In the spin-$3/2$ chain, a simple argument to understand the presence of
spin-$1/2$ edge states is to see it as a gapless spin-$1/2$ chain coupled
ferromagnetically to a gapped spin-$1$ chain, i.e. a ladder with ferromagnetic
rungs and with antiferromagnetic spin-$1/2$ and spin-$1$ legs respectively.
The spin-$1/2$ chain has no edge states, but the spin-$1$ chain has spin-$1/2$
edge states, and the resulting picture is that of a spin-$1/2$ chain with
spin-$1/2$ edge states coupled ferromagnetically to it, leading to the
equivalent of the ferromagnetic Kondo problem and to the $1/(N_{s}\log N_{s})$
scaling of the lowest excitation Eggert and Affleck (1992). By analogy, we can
see the on-site two-box symmetric irrep as obtained from the tensor product of
a three-box symmetric irrep and a two-box antisymmetric irrep. Indeed

$\mathrel{\raisebox{0.0pt}{\scalebox{0.6}{\ydiagram{3}}}}\otimes\mathrel{\raisebox{3.0pt}{\scalebox{0.6}{\ydiagram{1,1}}}}\
=\ \mathrel{\raisebox{3.0pt}{\scalebox{0.6}{\ydiagram{4,1}}}}\ \oplus\
\mathrel{\raisebox{0.0pt}{\scalebox{0.6}{\ydiagram{2}}}}.$ (3)

The two-box antisymmetric irrep is gapless and described by
$\mathrm{SU}(3)_{1}$ since
$\mathrel{\raisebox{2.0pt}{\scalebox{0.4}{\ydiagram{1,1}}}}$ is the complex
conjugate irrep of the fundamental irrep
$\mathrel{\raisebox{0.0pt}{\scalebox{0.4}{\ydiagram{1}}}}$, while the irrep
$\mathrel{\raisebox{0.0pt}{\scalebox{0.4}{\ydiagram{3}}}}$ is gapped and has
adjoint edge states Gozel  _et al._ (2020). Note that in that case the
coupling between the two chains should be antiferromagnetic to pick the two-
box symmetric irrep. In other words, this amount to seeing the two-box
symmetric chain as an antiferromagnetic ladder with legs in the two-box
antisymmetric and three-box symmetric representations, respectively.

Figure 3: Same quantities as in Fig. 2 except that we have added spins living
in the adjoint irrep at the two edges of the chain. The calculated gaps
$\Delta$ for all the cases considered ($N_{s}=1,2,0$ mod. $3$ corresponding to
left, middle and right panel respectively) vanish as $1/N_{s}$ (or
equivalently $N_{s}\Delta$ goes to a strictly positive constant) in the limit
$N_{s}\rightarrow\infty$, by contrast to the situation shown in Fig. 2, as a
consequence of the screening of the edge states by the additional adjoint
spins. (See text for details).

The picture emerging from this analogy is that of a two-box antisymmetric
chain with adjoint edge states. Then, if we add edge spins in the adjoint
representation coupled antiferromagnetically to the chain, as we did for the
three-box symmetric chain Gozel  _et al._ (2020), we can hope to screen the
edge states, and to recover a spectrum typical of a 1D gapless system with
excitations scaling as $1/N_{s}$. As shown in Fig. 3, this is precisely what
happens. In fact, $N_{s}\Delta$ converges now towards a strictly positive
constant in the thermodynamic limit. This is fully consistent with the usual
behavior of bulk excitations in gapless systems.

Now that we know how to screen the edge states, we can hope to capture the
entanglement of the bulk. So we have performed a systematic DMRG investigation
of chains with edge spins in the adjoint representation. The resulting
entanglement spectrum can again be fitted with the Calabrese-Cardy formula
Calabrese and Cardy (2004). The results are plotted in the bottom panel of
Fig. 1, and as expected, they are completely different from those obtained
without screening the edge states. For small system sizes, the central charge
is significantly larger than $2$ for both sets of points, but the finite-size
effects are very different. Indeed, for large enough sizes, the central charge
is consistent with $c=2$ within 1% after extrapolation to
$N_{s}\rightarrow\infty$ (and within 10 % for the largest system size,
$N_{s}=302$). Interestingly, for $q=1$, there is a change of behaviour between
$N_{s}=62$ and $N_{s}=92$ (second and the third point from the right): the
central charge first increases before decreasing towards 2. This might be
related to the crossover scenario from $\mathrm{SU}(3)_{2}$ to
$\mathrm{SU}(3)_{1}$ put forward to explain the ED results on small chains
with periodic boundary conditions Nataf and Mila (2016), for which the edge
states are of course also absent.

To summarize, we have been able to demonstrate numerically with extensive DMRG
simulations that the $\mathrm{SU}(3)$ Heisenberg chain with the two-box
symmetric irrep at each site is in the Wess-Zumino-Witten $\mathrm{SU}(3)_{1}$
universality class with central charge $c=2$. This has been made possible by a
careful investigation of the excitation spectrum with open boundary conditions
that has revealed the presence of edge states living in the adjoint irrep, and
by a calculation of the entanglement spectrum after screening these edge
states to be able to extract the central charge. In the future, it would be
interesting to try and go beyond the simple picture provided above to explain
the presence of edge states, and to develop the equivalent of the
ferromagnetic Kondo theory for $\mathrm{SU}(3)$. At the technical level, it
would be useful to improve our numerical machinery in order to be able to
target all $\mathrm{SU}(N)$ irreps and to access all the low lying excitation
gaps, and to try and calculate the central charge for larger chains with
periodic boundary conditions to check for the crossover scenario. The other
cases relevant for the generalization of the Haldane conjecture, like the
$\mathrm{SU}(4)$ Heisenberg chain with the $p$-box symmetric irrep at each
site with $p=2$ and $p=4$, would also be interesting (but challenging) to
address numerically Lajkó  _et al._ (2017).

Finally, let us comment briefly on the possible physical implementations of
the current model. $\mathrm{SU}(3)$ spins living in any two-column irrep can
be simulated using alkaline-earth atoms such as ${}^{87}{\mathrm{S}r}$ or
${}^{173}{\mathrm{Y}b}$ Gorshkov  _et al._ (2010); DeSalvo  _et al._ (2010);
Taie  _et al._ (2010, 2012), and the criticality of the chain can be revealed
through the measurement of the two-site correlations, as experimentally
achieved very recently for $\mathrm{SU}(N)$ Taie  _et al._ (2020).
Alternatively, quantum interference Islam  _et al._ (2015); Kaufman  _et al._
(2016) and randomized measurement Brydges  _et al._ (2019); Elben  _et al._
(2020) based protocols can allow experimentalists to directly access the
entanglement properties of many-body cold atoms systems.

##  I Acknowledgments

We acknowledge very useful discussions with Ian Affleck and Philippe
Lecheminant. This work has been supported by the Swiss National Science
Foundation.

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