Distinct Trivial Phases Protected by a Point-Group Symmetry in Quantum Spin Chains

The ground state of the $S=1$ antiferromagnetic Heisenberg chain belongs to the Haldane phase—a well-known example of the symmetry-protected topological phase. A staggered field applied to the $S=1$ antiferromagnetic chain breaks all the symmetries that protect the Haldane phase as a topological phase, reducing it to a trivial phase. That is, the Haldane phase is then connected adiabatically to an antiferromagnetic product state. Nevertheless, as long as the symmetry under site-centered inversion combined with a spin rotation is preserved, the phase is still distinct from another trivial phase. We demonstrate the existence of such distinct symmetry-protected trivial phases using a field-theoretical approach and numerical calculations. Furthermore, a general proof and a nonlocal order parameter are given in terms of a matrix-product state formulation.

Figure 1

Correlation lengths calculated for the spin-1 chain (3) are plotted against $D_z$. The parameters are varied as (a) $h_z=0.1$, $d_x=0$, and (b) $h_z=d_x=0.1$. Each color and symbol denotes the different number of kept states $\chi$ from 50 to 200. Insets show the nonlocal order parameters ${\mathcal{O}}_s(L)/\mathrm{Tr}{\mathrm{\Lambda }}^4$ for $L=100$ and 200 (see text).