${\mathbb{Z}}_3$ symmetry-protected topological phases in the SU(3) AKLT model

We study ${\mathbb{Z}}_3$ symmetry-protected topological (SPT) phases in one-dimensional spin systems with $Z_3\times Z_3$ symmetry. We construct ground-state wave functions of the matrix product form for nontrivial ${\mathbb{Z}}_3$ phases and their parent Hamiltonian from a cocycle of the group cohomology $H^2(Z_3\times Z_3,U\left(1\right))$. The Hamiltonian is an SU(3) version of the Affleck-Kennedy-Lieb-Tasaki (AKLT) model, consisting of bilinear and biquadratic terms of su(3) generators in the adjoint representation. A generalization to the $\mathrm{SU}\left(N\right)$ case, the $\mathrm{SU}\left(N\right)$ AKLT Hamiltonian, is also presented, which realizes nontrivial ${\mathbb{Z}}_N$ SPT phases. We use the infinite-size variant of the density matrix renormalization group (iDMRG) method to determine the ground-state phase diagram of the SU(3) bilinear-biquadratic model as a function of the parameter $\theta$ controlling the ratio of the bilinear and biquadratic coupling constants. The nontrivial ${\mathbb{Z}}_3$ SPT phase is found for a range of the parameter $\theta$ including the point of vanishing biquadratic term $\left(\theta =0\right)$ as well as the SU(3) AKLT point [$\theta =arctan(2/9)$]. A continuous phase transition to the SU(3) dimer phase takes place at $\theta \approx -0.027\pi$, with a central charge $c\approx 3.2$. For SU(3) symmetric cases, we define string order parameters for the ${\mathbb{Z}}_3$ SPT phases in a similar way to the conventional Haldane phase. We propose simple spin models that effectively realize the SU(3) and SU(4) AKLT models.

Figure 1

Weight diagrams of the Lie algebra su(3). Fundamental representation $\mathbf{3}$ consists of three basis states $u,d,s$ (quarks). Its conjugate representation $\overline{\mathbf{3}}$ consists of three basis states $\overline{u},\overline{d},\overline{s}$ (antiquarks). Adjoint representation $\mathbf{8}$ is spanned by eight bilinear forms of quarks and antiquarks, i.e., mesons.

Figure 2

Schematic pictures of the two types of MPS wave functions for the SU(3) AKLT model. On every site there are $\mathbf{3}$ states (quark $q$) and $\overline{\mathbf{3}}$ states (antiquark $\overline{q}$) which are projected onto $\mathbf{8}$ states (mesons represented by ovals) through the eight traceless matrices ${\Gamma }^{\sigma }$. (a) $q$ is coupled through the ${\Gamma }^0$ matrix to $\overline{q}$ on the left neighboring site to form a singlet $\mathbf{1}$ (${\eta }^{\prime }$ meson). (b) $q$ is coupled through the ${\Gamma }^0$ matrix to $\overline{q}$ on the right neighboring site to form a singlet $\mathbf{1}$ (${\eta }^{\prime }$ meson).

Figure 3

(a) Phase diagram of the SU(3) bilinear-biquadratic Hamiltonian. (b) The string order ${\mathcal{O}}_u^{\mathrm{str}}$ [Eq. (78)] and the dimer order ${\mathcal{O}}^{\dim }$ [Eq. (99)] as functions of $\theta$. The number of kept states $m$ in the iDMRG calculation is taken up to $m=200$. Plotted are the values of data extrapolated to the limit of vanishing truncation errors $\varepsilon \to 0$. The inset shows an example of the extrapolation for ${\mathcal{O}}_u^{\mathrm{str}}$ at $\theta =0$ with a quadratic fit. The dotted line indicates the phase boundary between the ${\mathbb{Z}}_3$ SPT phase and the dimer phase at ${\theta }_c\approx -0.027\pi$. The broken curves are to guide the eye. (c) The entanglement entropy $S$ [Eq. (102)] as a function of the correlation length $\xi$ [Eq. (103)] at $\theta =-0.0277\pi$, where $\xi$ takes a maximum value for $m=400$ as shown in the inset. The solid line shows the asymptotic behavior of the entanglement entropy at criticality with the central charge $c=16/5$.

Figure 4

Entanglement spectrum of the SU(3) bilinear-biquadratic model [Eq. (98)] for (a) $\theta /\pi =0.05$ in the ${\mathbb{Z}}_3$ SPT phase and (b) $\theta /\pi =-0.1$ in the dimer phase. The left/right panel in (b) shows entanglement spectrum of the chain divided without/with cutting a singlet dimer. The numbers enclosed in squares indicate the degeneracy of multiplets in the entanglement spectra.

Figure 5

Young tableau of the $[n,m]$ representation of SU(3).

Figure 6

(a) Phase diagram of the Hamiltonian ${\mathcal{H}}_3$ as a function of $J^{\prime }/J$ with $J>0$. Entanglement spectrum of the ground-state wave function for (b) $J^{\prime }/J=tan(\pi /3)$ and (c) $J^{\prime }/J=-1$. In (b) and (c) the left (right) panels show the spectrum when we cut the spin chain at a bond between unit cells (within a unit cell), i.e., at a bond of $\overline{q}$–$q (q$–$\overline{q})$, where $q$ and $\overline{q}$ are $\mathbf{3}$ and $\overline{\mathbf{3}}$ states. The numbers enclosed in squares indicate the degeneracy of multiplets in the entanglement spectrum.