We study the phase diagram of the
SOq(3) quantum group invariant spin-1 bilinear-biquadratic spin chain for real values of
q>1. Numerical computations suggest that the chain has at least three clearly distinguished phases: A chiral analog of the Haldane phase, a dimerized phase, and a ferromagnetic phase. In contrast, the counterpart of the extended critical region that is known to exist for
q=1 remains elusive. Our results show that the Haldane phase fails to exhibit a twofold degeneracy in the entanglement spectrum but that the degeneracy is restored upon a suitable
q deformation of the entanglement Hamiltonian, which can be interpreted as a Zeeman field. The structure of the phase diagram is confirmed through analytical calculations in the extreme anisotropic limit $q\ensuremath{\rightarrow}\ensuremath{\infty}$. Our results suggest that symmetries of the form
Uq[su(2)] for distinct choices of
q should be interpreted as one single family instead of separate symmetries when defining SPT phases, leading naturally to the notion of a
q-SPT phase.
We study
Z3 symmetry-protected topological (SPT) phases in one-dimensional spin systems with ${Z}_{3}\ifmmode\times\else\texttimes\fi{}{Z}_{3}$ symmetry. We construct ground-state wave functions of the matrix product form for nontrivial
Z3 phases and their parent Hamiltonian from a cocycle of the group cohomology ${H}^{2}({Z}_{3}\ifmmode\times\else\texttimes\fi{}{Z}_{3},U(1))$. 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
SU(N) case, the
SU(N) AKLT Hamiltonian, is also presented, which realizes nontrivial
ZN 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 $\ensuremath{\theta}$ controlling the ratio of the bilinear and biquadratic coupling constants. The nontrivial
Z3 SPT phase is found for a range of the parameter $\ensuremath{\theta}$ including the point of vanishing biquadratic term $(\ensuremath{\theta}=0)$ as well as the SU(3) AKLT point [$\ensuremath{\theta}=arctan(2/9)$]. A continuous phase transition to the SU(3) dimer phase takes place at $\ensuremath{\theta}\ensuremath{\approx}\ensuremath{-}0.027\ensuremath{\pi}$, with a central charge $c\ensuremath{\approx}3.2$. For SU(3) symmetric cases, we define string order parameters for the
Z3 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.
We study one-dimensional systems of two-orbital SU(4) fermionic cold atoms. In particular, we focus on an SU(4) spin model [named SU(4) $e\text{\ensuremath{-}}g$ spin model] that is realized in a low-energy state in the Mott insulator phase at the filling
ng=3,ne=1 (
ng,ne: numbers of atoms in ground and excited states, respectively). Our numerical study with the infinite-size density matrix renormalization group shows that the ground state of SU(4) $e\text{\ensuremath{-}}g$ spin model is a nontrivial symmetry protected topological (SPT) phase protected by ${Z}_{4}\ifmmode\times\else\texttimes\fi{}{Z}_{4}$ symmetry. Specifically, we find that the ground state belongs to an SPT phase with the topological index $2\ensuremath{\in}{\mathbb{Z}}_{4}$ and show sixfold degenerate edge states. This is topologically distinct from SPT phases with the index $1\ensuremath{\in}{\mathbb{Z}}_{4}$ that are realized in the SU(4) bilinear model and the SU(4) Affleck-Kennedy-Lieb-Tasaki (AKLT) model. We explore the phase diagram of SU(4) spin models including $e\text{\ensuremath{-}}g$ spin model, bilinear-biquadratic model, and AKLT model, and identify that antisymmetrization effect in neighboring spins (that we quantify with Casimir operators) is the driving force of the phase transitions. Furthermore, we demonstrate by using the matrix product state how the
Z4 SPT state with six edge states appears in the SU(4) $e\text{\ensuremath{-}}g$ spin model.