Gapped symmetric edges of symmetry-protected topological phases

Symmetry-protected topological (SPT) phases are gapped quantum phases which host symmetry-protected gapless edge excitations. On the other hand, the edge states can be gapped by spontaneously breaking symmetry. We show that topological defects on the symmetry-broken edge cannot proliferate due to their fractional statistics. A gapped symmetric boundary, however, can be achieved between an SPT phase and certain fractionalized phases by condensing the bound state of a topological defect and an anyon. We demonstrate this by two examples in two dimensions: an exactly solvable model for the boundary between a topological Ising paramagnet and the double-semion model, and a fermionic example about the quantum spin Hall edge. Such a hybrid structure containing both SPT phase and fractionalized phase generally support ground-state degeneracy on a torus.

Figure 1

Exactly solvable model of a gapped symmetric edge between bosonic $Z_2$-SPT (diamonds denote spins $\left\{{\vec{\sigma }}_{\mathbf{r}}\right\}$ on triangular lattice) and double-semion model (solid circles denote spins $\left\{{\vec{\tau }}_{{\mathbf{r}}^{\prime }}\right\}$ on edge centers of honeycomb lattice). Dashed green lines on triangular lattice represent domain wall configurations in ground-state wave function of Ising paramagnet, using ${\sigma }^z=\pm 1$ basis. Solid green lines denote (oriented) string-net configurations in the ground state of double-semion model, where ${\tau }^z=1$ correspond to no string and ${\tau }^z=-1$ to occupied by a string. The boundary term (16) guarantees that a $Z_2$ domain wall from upper half always forms a bound state with the end of a string on the lower half. Such a bosonic bound state can hop and condense, giving rise to a gapped symmetric edge.

Figure 2

Illustration of the hybrid structure on a torus, containing SPT phase $\mathcal{S}$ (blue) and fractionalized SET phase $\mathcal{F}$ (red). Both boundaries between the two regions are fully gapped without breaking any symmetry. The two oriented loops correspond to string order parameters ${\widehat{O}}_S$ (green) and ${\widehat{O}}_L$ (black). Unlike ${\widehat{O}}_S$ which consists of operators only from the SET region, open string operator ${\widehat{O}}_L$ also contains operators from SPT side, such as spin ${\sigma }^z$ on sites ${\mathbf{r}}_b$ and ${\mathbf{r}}_e$ in the example of a topological Ising paramagnet. Both string operators preserve the symmetry of the system.