Topological phases of spin chains

Symmetry-protected topological phases of one-dimensional spin systems have been classified using group cohomology. In this paper, we revisit this problem for general spin chains which are invariant under a continuous onsite symmetry group $G$. We evaluate the relevant cohomology groups and find that the topological phases are in one-to-one correspondence with the elements of the fundamental group of $G$ if $G$ is compact, simple, and connected and if no additional symmetries are imposed. For spin chains with symmetry $PSU\left(N\right)=SU\left(N\right)/{\mathbb{Z}}_N$, our analysis implies the existence of $N$ distinct topological phases. For symmetry groups of orthogonal, symplectic, or exceptional type, we find up to four different phases. Our work suggests a natural generalization of Haldane's conjecture beyond $SU\left(2\right)$.

Figure 1

Physical and virtual edge modes (red dots) in topologically nontrivial spin chains. For simplicity of illustration, the spin chain is depicted as a continuous system.

Figure 2

Sketch of a matrix product state for a system with open boundary conditions. The states in the boundary spaces ${\mathcal{B}}_L$ and ${\mathcal{B}}_R$ (red) correspond to massless edge modes.

Figure 3

Visualization of different congruence classes for $SU\left(2\right)$. The picture shows the weight lattice $P$ (all spins) in terms of colored dots. The root lattice $Q$ (integer spins) corresponds to the large black dots. Different colors indicate different congruence classes. The shaded blue box is a possible representative of $P/Q$.

Figure 4

Visualization of different congruence classes for $SU\left(3\right)$ and $SP\left(4\right)$. The pictures show the respective weight lattice $P$ in terms of colored dots. The root lattice $Q$ corresponds to the large black dots. Different colors indicate different congruence classes. The shaded blue boxes are possible representatives of $P/Q$. We clearly see that, for $SP\left(4\right)$, the topological class is independent of ${\lambda }_2$.

Figure 5

The hierarchy of topological phases in $\text{Spin}\left(2n\right)$ spin chains.

Figure 6

The hierarchy of topological phases in $SU\left(6\right)$ spin chains.

Figure 7

The hierarchy of topological phases in $SU\left(12\right)$ spin chains.