Higher-order symmetry-protected topological states for interacting bosons and fermions

Higher-order topological insulators have a modified bulk-boundary correspondence compared to other topological phases: instead of gapless edge or surface states, they have gapped edges and surfaces, but protected modes at corners or hinges. Here, we explore symmetry-protected topological phases in strongly interacting many-body systems with this generalized bulk-boundary correspondence. We introduce several exactly solvable bosonic lattice models as candidates for interacting higher-order symmetry-protected topological (HOSPT) phases protected by spatial symmetries, and develop a topological field theory that captures the nontrivial nature of the gapless corner and hinge modes. We show how, for rotational symmetry, this field theory leads to a natural relationship between HOSPT phases and conventional SPT phases with an enlarged internal symmetry group. We also explore the connection between bosonic and fermionic HOSPT phases in the presence of strong interactions, and comment on the implications of this connection for the classification of interacting fermionic HOSPT phases. Finally, we explore how gauging internal symmetries of these phases leads to topological orders characterized by nontrivial braiding statistics between topological vortex excitations and geometrical defects related to the spatial symmetry.

Figure 1

The topological plaquette paramagnet model on checkerboard lattice. The Pauli spins $\tau ,\sigma$ live on the red/blue sites. The interaction ${\sigma }_i^z{\sigma }_j^z{\sigma }_k^z{\sigma }_l^z{\tau }_m^x$ involves the four ${\sigma }_z$ spin operators on the red plaquette and the ${\tau }_x$ in the middle of plaquette.

Figure 2

Ground state of the topological plaquette paramagnet defined in Eq. (1). The blue blocks illustrate the domain wall for $\sigma$ spin where ${\sigma }_z=\pm 1$ inside/outside the domain wall. The corner of the blue block contains a $\tau$ (red) spin polarized at ${\tau }_x=-1$. The ground state is a superposition of all domain wall block configurations where the corners of the blocks are decorated with ${\tau }_x=-1$.

Figure 3

Boundary degrees of freedom of Eq. (1). The three-spins cluster in the blue rectangle refers to the effective edge spin, while the corner two-spin cluster in the green rectangle is the corner effective spin.

Figure 4

Spin model on square lattice with four spin 1/2 per site. The red square denotes the spin cluster between four spins on the corner of the plaquette. The edge site contains two free spin 1/2 (red dot), while the corner site contains three. One can dimerize the edge spin via inter/intra-bond singlet. The edge dimerization is odd under the reflection symmetry that leaves the diagonal invariant.

Figure 5

Take a zigzag edge with the reflection plane in the middle of the edge. The edge spins could be gapped in a symmetric way while the reflection points contains a zero mode due to the free spin 1/2.

Figure 6

Distribution of $n_4$ in space in a $C_4$ symmetric way. Red lines are domain walls for the $n_4$ scalar field.

Figure 7

Lattice for the CZX model defined in Eq. (28). Each site contains eight spin-1/2 degrees of freedom (labeled as 1,2,…,8). Each spin interacts with one of the eight spin-1/2 cubic clusters adjacent to the site. The Hamiltonian projects the eight spins belonging to a cube into a unique state.

Figure 8

Boundary degrees of freedom of the CZX model defined in Eq. (28). (Left) On the surface, one can perform a plaquette projection to gap out the surface without breaking the symmetries that define the HOSPT phase. (Right) Along the hinge between two surfaces, each bond represents a two-level system and ${\mathbb{Z}}_2$ symmetry acts on the hinge in a nonlocal way.

Figure 9

Geometry for hinge modes of the CZX model defined in Eq. (28). 2D representation of the surfaces corresponding to the $x\text{−}z$ (left) and $y\text{−}z$ (right) planes. The reflection axis acts on the $i\mathrm{th}$ column or surface sites and takes column $i-n$ to $i+n$.

Figure 10

Lattice structure for the third-order SPT defined in Eq. (46). The spin interaction involves the red spin on the eight corners of a cube together with the blue spin in the middle of the cube and vice versa.

Figure 11

Definition of the surface operators ${\pi }_i^x$ and ${\pi }_i^y$ from Eq. (47), which have support on the site labeled $i$ and the four sites connected to it by lines.

Figure 12

A $\pi /2$ disclination point is created by cutting out a quadrant of the square lattice and reconnecting the edges. Inside the disclination core, there exists a spin-1/2 zero mode protected by $\mathcal{T}$.

Figure 13

A $\pi /2$ disclination line is created by cutting out the quadrant of the cubic lattice and reconnect the side faces. Inside the disclination line, there exists a dispersing bosonic gapless degree of freedom akin the hinge mode.

Figure 14

Disclination as a $\pi$-flux defect. (Left) After removing a quadrant of the cubic lattice, the two boundaries(green) are connected with twisted boundary condition $n_5(x,y)=-n_5(y,-x)$. The branch cut of the disclination thus carries a $\pi$ flux. (Right) The resulting disclination is the $\pi$ flux for the $C_4^z{\mathbb{Z}}_2^b$ symmetry. The branch cut(dashed red) contains a disclination twist together with a $\pi$ flux.

Figure 15

Three loop statistics process: the two ${\mathbb{Z}}_2$ flux loop (red) wind around each other with the penetration of a $\pi /2$ disclination loop(green).