Intermediate Haldane phase in spin-2 quantum chains with uniaxial anisotropy

We provide evidence of an intermediate Haldane phase in a spin-2 quantum chain. By combining effective field theory and numerical approaches, we show that the phase diagram of the proposed model includes SO(5) Haldane, intermediate Haldane, and large-$D$ phases. We determine the characteristic properties of these phases, including edge states, string order parameters, and degeneracies in the entanglement spectrum. The symmetries responsible for the degeneracy patterns observed in the entanglement spectrum are also discussed.

Figure 1

Qualitative phase diagram of the spin-2 model in Eq. (1). The diagrams in the upper side correspond to the MPS structure describing the renormalization group fixed point for each phase: for low-$D$, ancillary spin-3/2 singlets are projected onto spin-2 subspaces at each physical site, thus providing an MPS for spin-2 physical particles in terms of matrices of dimension 4; for intermediate-$D$, ancillary spin-1/2 singlets, together with on-site spin-1/2 triplets $S^z=0$, are projected onto spin-2 at each site; for large-$D$, no singlets are projected and the state is a product state.

Figure 2

String order parameters ${\mathcal{O}}^{12}$ and ${\mathcal{O}}^{34}$. They vanish smoothly at $D=D_{c_1}$ and $D=D_{c_2}$, respectively.

Figure 3

First 20 coefficients ${\xi }_{\alpha }$ of the entanglement spectrum for representative points in the (a) SO(5) Haldane phase, (b) IH phase, and (c) large-$D$ phase.