Higher-order bosonic topological phases in spin models

We discuss an extension of higher-order topological phases to include bosonic systems. We present two spin models for a second-order topological phase protected by a global ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ symmetry. One model is built from layers of an exactly solvable cluster model for a one-dimensional ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ topological phase, while the other is built from more conventional spin couplings (XY or Heisenberg). These models host gapped, symmetry-protected topological phases on their edges, and corner modes that fall into a projective representation of the symmetry. Using Jordan-Wigner transformations we show that both our models are related to a bilayer of free Majorana fermions that form a fermionic second-order topological phase. We also discuss how our models can be extended to three dimensions to form a third-order topological phase.

Figure 1

(a) Illustration of the bulk quadrupole moment, edge polarizations, and corner charge. (b) Illustration of bulk, edge, and corner topological indicators used to define a 2d HOSPT.

Figure 2

${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ spin chain Jordan-Wigner transformed to a pair of Kitaev chains in a topological phase. Each site of the original spin chain is represented by a pair of Majorana operators ${\alpha }_i$ and ${\beta }_i$ enclosed by an oval.

Figure 3

Two-dimensional lattice built of $M$ pairs of ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ SPT chains each of length $N$. Vertical couplings are illustrated schematically below the lattice. This arrangement clearly leaves two decoupled SPT chains at the top and the bottom.

Figure 4

Upper: Two neighboring ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ spin chains (after Jordan-Wigner transformation to Majorana representation) that we intend to couple. Lower: Vertical coupling between two spin chains. In this particular example we focus on the coupling introduced by the $X_3{Z}_4{X}_{18}{Z}_{17}$ term, highlighted in purple. Dots enclosed by small circles represent the Majoranas engaged by this coupling term. Most of the Majoranas that enter the product can be removed using conserved quantities to leave simple quadratic hopping terms. Two dots connected by a purple bond represent the Majoranas that are left and cannot be combined into a conserved quantity.

Figure 5

Square lattice with four spin-$1/2$ degrees of freedom per unit cell coupled via dimerized antiferromagnetic XY couplings.

Figure 6

Quadrupole $L_y=2$ strip with the entanglement cut.

Figure 7

Entanglement spectrum calculated for $t_x=t_y=0.1$.

Figure 8

Degeneracy of the $L_y=4$ quadrupole strip for different values of $t_x$ and $t_y$. White area corresponds to the parameters' values for which we cannot reliably apply the iTEBD algorithm without drastically increasing the bond dimension.

Figure 9

Entanglement spectrum for the model with $t_x=0.76, t_y=0$ computed using two different bond dimensions. Blue dots correspond to $\chi =28$ computation, green dots to $\chi =48$.

Figure 10

Top: Entanglement spectrum computed for a coupled pair of spin chains of different lengths. Bottom: Dependence of the ratio between the first and the fourth entanglement Hamiltonian eigenvalues on the coupling strength $t_y$.