Proposal of a spin-one chain model with competing dimer and trimer interactions

A new kind of spin-one chain Hamiltonian consisting of competing dimer and trimer projection operators is proposed. As the relative strengths and signs of the interactions are varied, the model exhibits a number of different phases including the gapped dimer phase and the gapless trimer phase with critical correlations described by a conformal field theory with central charge $c=2$. A symmetry-protected topological phase also exists in this model, even though the microscopic interactions are not the simple adiabatic extensions of the well-known Heisenberg and Affleck-Kennedy-Lieb-Tasaki models and contain both two- and three-particle permutations. A fourth phase is characterized by macroscopically degenerate ground states. While bearing almost a one-to-one resemblance to the phase diagram of the bilinear-biquadratic spin-one chain Hamiltonian, our model is rooted in a very different physical origin, namely, the two competing tendencies of spin-one particles to form singlets through either dimer or trimer formation.

Figure 1

Phase diagram of the (a) BLBQ model and (b) DT model as a function of the mixing angle $\theta$. FM, ferromagnetic; MD, macroscopically degenerate; SPT, symmetry-protected topological; SQ, spin quadrupolar; TL, trimer liquid. In both phase diagrams, the dimer phase is topologically trivial and gapped, with ground states that break the translation symmetry. The Haldane (SPT) phase is topologically nontrivial and translation invariant. SQ and trimer phases are both critical and carry the central charge $c=2$. The MD phase has exponentially large ground-state degeneracy. The pure trimer (PT) point on the right phase diagram possesses the full SU(3) symmetry, as does the ULS model on the left.

Figure 2

(a) Entanglement entropy, (b) dimer density $\langle {\mathcal{P}}_D\left(n\right)\rangle$, (c) spin-spin correlation, and (d) dimer-dimer correlation for $\theta =-0.45\pi$ (left column) close to the phase boundary and $\theta =\pi /12$ (right column) deep inside the dimer phase. Period-2 oscillations appear in all the calculated quantities. Correlation functions in log-linear plots (c) and (d) decay exponentially.

Figure 3

Entanglement entropy (left column) and entanglement spectrum near the middle of the lattice (right column) in the dimer phase (a) $\theta =\pi /8$, and in the SPT phase (b) $13\pi /96$, (c) $14\pi /96$, and (d) $15\pi /96$. At $\theta =\pi /8$, EE still has the period-2 oscillation and ES shows the alternating $3-1-\cdots$ degeneracy of the lowest levels characteristic of the dimer phase. At $\theta =13\pi /96$, EE loses the period-2 structure and ES shows the fourfold (sometimes eightfold or twelvefold) degeneracy in the entire spectra. The trend continues for $\theta =14\pi /96$ and $15\pi /96$.

Figure 4

Entanglement spectrum in the SPT phase at $\theta =14\pi /96$ and the system size $N=60$ for the DT model with extra perturbation (3.1). (a) Only $B\ne 0$. (b) Only $A$ and $B$ nonzero. Twofold degeneracy survives throughout the whole spectrum in both cases. (c) Only $C$ nonzero. The even degeneracy is lifted, signaling the destruction of the SPT phase.

Figure 5

Entanglement entropy (left column) and trimer average $\langle {\mathcal{P}}_T\left(n\right)\rangle$ for the trimer liquid phase at several $\theta$ values. The ULS results are shown in (a) for comparison [39]. The red curves in the entanglement entropy plots indicate the fitting to the Calabrese-Cardy formula.

Figure 6

Entanglement spectrum for the (a) ULS model [39] and (b)–(d) DT model at various angles $\theta$, starting from the leftmost edge cut $n=1$. For each cut position $n$, a perfect correspondence in the degeneracy of the low-lying levels of the ULS model with the pure trimer model $(\theta =\pi /2)$ exists; compare (a) and (c). Degeneracies get lifted as $\theta$ deviates from $\pi /2$; see (b) and (d).

Figure 7

(a) Spin, (b) spin quadrupole, and (c) trimer correlation functions for BLBQ and DT models. Left columns show overlapping results for all correlation functions at $\theta =\pi /4$ for the BLBQ model and $\theta =\pi /2$ for the DT model. Both models are SU(3) symmetric. The right column shows results away from SU(3) symmetry. Overall behaviors of correlation functions in the two phases are extremely similar.

Figure 8

(a) Examples of the comb representation for the nonzero elements in the MPO matrix $A$. The left and right ends of each graph correspond to the virtual indices of $A$. The “tooth” of each graph represents the physical operators such as $I$ and ${\lambda }_{\alpha ,\beta }$. (b) Examples of contracted graphs representing a specific term in the Hamiltonian. The connecting links of the adjacent blocks must have the same virtual index. In the first row, the resulting product is $J_D{\lambda }_{1,1}\left(3\right){\lambda }_{2,2}\left(4\right)$, which is one of the terms in the dimer operator at site 3, i.e., $D\left(3\right)$ in Eq. (A3). The argument of $\lambda$ denotes the spatial position. In the second row, the product gives $-J_T{\lambda }_{1,1}\left(2\right){\lambda }_{3,2}\left(3\right){\lambda }_{2,3}\left(4\right)$, which is one of the terms in the trimer operator $T\left(2\right)$ defined in Eq. (A3).