Building symmetry-enriched topological phases from a bipartite lattice construction and anyon condensation

We introduce a construction of symmetry-enriched topological orders on bipartite lattices in which two ${\mathbb{Z}}_2$ spin liquids defined on each sublattice are combined, and then anyons are condensed to reduce the topological order. By choosing different anyon condensate structures, one can vary the fractionalization pattern of the resulting spin liquid, some of which cannot be readily constructed from parton-based approaches. We demonstrate the construction for (i) a spin-1/2 honeycomb lattice where we construct a featureless state as well as intermediate states with topological order, (ii) a nonsymmorphic lattice, and (iii) lattices with magnetic translation symmetry. Finally, we discuss constraints on nonchiral topological orders in a bosonic system under magnetic field.

Figure 1

Simplest lattice structure for nonsymmorphic group $pg$. The black and white dots form the A/B sublattices, respectively. The black lines are drawn to represent each unit cell. The dashed lines are separated by a half-lattice vector translation in the $x$ direction. The reflection symmetry $P_x$ we use in the text is defined with respect to the blue line. The sites are numbered to help picture the identifying map, $\mathcal{S}$, which is defined as the glide-reflection symmetry $g{|}_A$ restricted to the A sublattice. This maps the black dots labeled by $n$ to the white dots labeled by $n^{\prime }$. Under $g{|}_B$, $n^{\prime }\mapsto n+1$, meaning that $g{|}_B=T_x{\mathcal{S}}^{-1}$.

Figure 2

Black dots represent the $A$ sublattice. White dots represents the $B$ sublattice. ${\mathcal{O}}_A$ and ${\mathcal{O}}_B$ represent origins (reference points) for each sublattice. Dotted line represents the axis of the reflection symmetry in the honeycomb lattice.

Figure 3

Forms of the wave function of an $e_A{e}_B$ pair. Left: A simple idea for how to create the $e_A{e}_B$ bound state—the oval represents a singlet of excitations on the two lattice sites. However, the wave function is odd under reflection in the bond, so there is no way to create a completely symmetric state starting from this one. Right: Instead, if one creates four excitations in a configuration without any symmetry, the configuration can be superimposed with all the symmetric transformations of itself without any cancellation occurring. Note that there are three anyons on B sites and one on an A site, so the net charge is correct. Alternatively, one could add some nontopological excitation to the singlet in the left figure, at a position that breaks the reflection symmetries.

Figure 4

The figure represents the mean-field configurations $Q_{ij}$ that have the two different symmetries: (a) ${\mathit{C}}_6^6{|}_e=1$, (b) ${\mathit{C}}_6^6{|}_e=-1$ on $e$ particles. The sixfold rotation exchanges two anyons $e_A$ and $e_B$; thus its action is related to the mean-field amplitudes for anyon bound states $e_A{e}_B$. Blue lines represent positive $Q_{ij}$, while red lines represent negative $Q_{ij}$. The right figure is symmetric under ${\mathit{C}}_6$ only after associated gauge transformation.

Figure 5

Black dots represent sublattice $A$. White dots represent sublattice $B$. In this figure, black lines represent hopping terms with the positive sign, and red lines represent hopping terms with the negative sign. Due to the alternating signs, if we translate the system in the $x$ direction, we should perform gauge transformation at sites on dashed lines (even rows) to go back to the original Hamiltonian. Panels (a) and (b) show unit cells for sublattices $A$ and $B$, which are different by gauge transformation.