Symmetry protected topological phases in two-orbital SU(4) fermionic atoms

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) $e\text{−}g$ spin model] that is realized in a low-energy state in the Mott insulator phase at the filling $n_g=3,n_e=1$ ($n_g,n_e$: 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) $e\text{−}g$ spin model is a nontrivial symmetry protected topological (SPT) phase protected by $Z_4\times Z_4$ symmetry. Specifically, we find that the ground state belongs to an SPT phase with the topological index $2\in {\mathbb{Z}}_4$ and show sixfold degenerate edge states. This is topologically distinct from SPT phases with the index $1\in {\mathbb{Z}}_4$ 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 $e\text{−}g$ 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 ${\mathbb{Z}}_4$ SPT state with six edge states appears in the SU(4) $e\text{−}g$ spin model.

Figure 1

Schematic picture of SU(4) symmetric models at filling $n_g=3$ and $n_e=1$. Hund's rule coupling selects the 15-dimensional representation as the low-energy local basis in each site. $J_{\alpha }$ ($\alpha =g,e$) and $J$ denote the intraorbital and interorbital exchanges, respectively. Setting $J=0$ realizes the SU(4) $e\text{−}g$ spin model given in Eq. (1), whereas setting $J_e=J_g=J$ realizes the SU(4) bilinear model given in Eq. (13).

Figure 2

Entanglement spectra in (a) the SU(3) $e\text{−}g$ spin model, (b) the SU(4) $e\text{−}g$ spin model, and (c) the SU(4) bilinear model. The numbers enclosed in squares denote the degeneracy factor of each level. The dimension $\chi$ of the auxiliary (edge state) space is chosen as (a) 300, (b)225, and (c) 225.

Figure 3

Schematic figures of two possible valence bond solid (VBS) states with the topological index $\pm 1\in {\mathbb{Z}}_4$ (a) and with $\pm 2\in {\mathbb{Z}}_4$ (b) on the local bases in the 15-dimensional representation. The black circle on each site denotes the projector onto the irreducible representation 15. In (a), each site basis is given by the product of the fundamental representation $\mathbf{4}$ of $e$ orbital and its conjugate $\overline{\mathbf{4}}$ of $g$ orbital, and, in (b), by the product of the two representations $\mathbf{6}$, which are formed by the two (antisymmetric) fermion pairs out of $n_g=3$ and $n_e=1$ fermions. Each thick horizontal bond connecting the two neighboring diagrams forms the singlet.

Figure 4

(a) Topological indices ${\mathcal{O}}_{Z_4^{}\times Z_4^{}}$ and ${\mathcal{O}}_{\mathcal{I}}$ and (b) expectation values of the quadratic bond Casimir operator $\langle {({\mathit{T}}_j+{\mathit{T}}_{j+1})}^2\rangle$ as a function of the coupling $J/J_{\mathrm{g}}^{}$ given in the extended SU(4) Hamiltonian (17). The case $J=0$ corresponds to the SU(4) $e\text{−}g$ spin model and $J/J_g=1$ to the SU(4) bilinear model. The symbols $◯$, $•$, and $\times$ represent the indexes ${\mathcal{O}}_{Z_4^{}\times Z_4^{}}$ of $A^{\sigma }$, ${\mathcal{O}}_{Z_4^{}\times Z_4^{}}$ of ${\left(A^{\sigma }\right)}^T$, and ${\mathcal{O}}_{\mathcal{I}}$, respectively. The auxiliary space of iDMRG calculations is set as $\chi =225$.

Figure 5

Phase diagram of the SU(4) bilinear-biquadratic model given in Eq. (18) with the data of (a) indices ${\mathcal{O}}_{Z_4^{}\times Z_4^{}}$ and ${\mathcal{O}}_{\mathcal{I}}$ and (b) expectation values of the quadratic bond-Casimir operator. The symbols have the same meanings as those in Fig. 4. The case $\theta =0$ corresponds to the SU(4) bilinear model and $\theta /\pi \sim 0.0525$ to the SU(4) AKLT model whose exact ground state is a VBS state [31, 50]. The iDMRG calculation was performed with $\chi =100$.

Figure 6

Entanglement spectrum for the dimer phase at $\theta =-0.1\pi$ in the SU(4) bilinear-biquadratic model. The left/right data show the spectrum of the chain divided without/with cutting a singlet dimer. The iDMRG calculation was performed with $\chi =100$. The slight deviation of the levels found in the top of the right panel is attributed to a finite-$\chi$ effect.

Figure 7

Transition point ${\theta }_c$ between the SPT phase with the index $\pm 1\in {\mathbb{Z}}_n$ and the dimer phase in the $\mathrm{SU}\left(n\right)$ bilinear-biquadratic model. The broken line is a guide to the eye for the power-law decay with $\sim n^{-5.5}$.

Figure 8

Phase diagrams for $\mathrm{SU}\left(n\right)$ spin models of (a) $n=3$ and (b) $n=4$. The thick lines denote the VBS phases with the topological index $\pm 1\in {\mathbb{Z}}_n$ and the dotted (red) lines the VBS phases with $2\in {\mathbb{Z}}_4$. The dotted dash (blue) lines denote dimer phases.