Edge states and universality class of the critical two-box symmetric SU(3) chain

We numerically demonstrate that, although it is critical, the two-box symmetric $\mathrm{SU}\left(3\right)$ 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 $\mathrm{SU}\left(2\right)$ chains with spin $S\ge 3/2$. 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 density-matrix renormalization group simulations the field theory prediction that this model is in the $\mathrm{SU}{\left(3\right)}_1$ 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 $\mathrm{SU}{\left(3\right)}_1$.

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 [46]). 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. Dashed lines are linear extrapolations obtained from the last two points ($N_s=278$ and $N_s=302$). See text for details.

Figure 2

$N_s\mathrm{\Delta }$ for (a) $N_s=1$ mod. 3, (b) $N_s=2$ mod. 3, and (c) $N_s=0$ mod. 3, where $\mathrm{\Delta }$ 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. The dashed lines are guides to the eye. For Figs. 2 and 2, they correspond to linear interpolations between the origin and the last converged points with respect to $m$, and for Fig. 2 to the tangent at the inflection point for the largest $m$. For $N_s=1,2$ mod. 3, $N_s\mathrm{\Delta }$ goes to zero when $N_s\to \infty$, 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, $N_s\mathrm{\Delta }$ goes to a strictly positive constant when $N_s\to \infty$ because the sector is not that of the lowest excitation (see text for details).

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 dashed lines are guides to the eye. They correspond to a linear extrapolation from the last two converged points. The calculated gaps $\mathrm{\Delta }$ 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 $N_s\mathrm{\Delta }$ goes to a strictly positive constant) in the limit $N_s\to \infty$, 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.)