Gapped quantum phases for the $S=1$ spin chain with $D_{2h}$ symmetry

We study different quantum phases in integer spin systems with on-site $D_{2h}=D_2\otimes Z_2$ and translation symmetry. We find four distinct nontrivial phases in $S=1$ spin chains despite the fact that they all have the same symmetry. All four phases have gapped bulk excitations, doubly degenerate end states, and the doubly degenerate entanglement spectrum. These nontrivial phases are examples of symmetry-protected topological (SPT) phases introduced by Gu and Wen [Phys. Rev. B 80, 155131 (2009)]. One of the SPT phases corresponds to the Haldane phase and the other three are new. These four SPT phases can be distinguished experimentally by their different responses of the end states to weak external magnetic fields. According to the Chen–Gu–Wen classification, the $D_{2h}$ symmetric spin chain can have a total of 64 SPT phases that do not break the symmetry. Here we constructed seven nontrivial phases from the seven classes of nontrivial projective representations of the $D_{2h}$ group. Four of these are found in $S=1$ spin chains and studied in this paper.

Figure 1

The phase diagram of model (1). The transition between the Néel phase and the SPT phases are second order, and the transition between $T_0$ and $T_x$ is first order.

Figure 2

(a) The energy curve $E\left(\eta \right)$, (b) the first and second derivatives of $E\left(\eta \right)$. All these curves are smooth, indicating that these states are in the same phase.

Figure 3

(a) The first-order derivative (line with circles) and the second-order derivative (line with crosses) of $E\left(\theta \right)$ at $\phi =\frac{\pi }{8}$ (or $\phi =\frac{3\pi }{8}$). We have set $D=16$. The dotted line is the difference between the two biggest weights of the entanglement spectrum ${\Lambda }_1-{\Lambda }_2$. Since $E^{\prime }\left(\theta \right)$ is continuous but $E^{\prime \prime }\left(\theta \right)$ is not, the transition is second order. The degeneracy of ${\Lambda }_1$ and ${\Lambda }_2$ indicates that the $T_0$ (and $T_x$) phase is nontrivial. (b) The refined energy curve $E\left(\theta \right)$ near the transition point with $D=8,16,24$. The trial energy decreases with increasing $D$, which is evidence of the factor that at the transition point the ground state is a MPS with infinite $D$. (c) The entanglement spectrum with different $D$. The black circles are for $D=8$, blue triangles for $D=16$, and red crosses for $D=24$. At the transition point, the entanglement entropy increases with $D$ and diverges when $D$ goes to infinity.

Figure 4

The ground-state energy for the model described by Eq. (15). A first-order phase transition is obvious.

Figure 5

The eigenvalues of ${\tilde{S}}_y$ and ${\tilde{S}}_z$ are $s,-s,0,0$. The magnitude of $s$ exponentially decays with the length of the chain in the $T_x$ phase. The dashed line is an exponential fit. The results are obtained by exact diagonalization and we calculate only up to seven sites.