${\mathbb{Z}}_2$ fractional topological insulators in two dimensions

We propose a simple microscopic model to numerically investigate the stability of a two-dimensional fractional topological insulator (FTI). The simplest example of an FTI consists of two decoupled copies of a Laughlin state with opposite chiralities, or double-semion phase. We focus on bosons at half filling. We study the stability of the FTI phase upon addition of two coupling terms of different nature: an interspin interaction term, and an inversion-symmetry-breaking term that couples the copies at the single-particle level. Using exact-diagonalization and entanglement spectra, we numerically show that the FTI phase is stable against both perturbations. We compare our system to a similar bilayer fractional Chern insulator. We show evidence that the time-reversal-invariant system survives the introduction of interaction coupling on a larger scale than the time-reversal-symmetry-breaking one, stressing the importance of time-reversal symmetry in the FTI phase stability. We also discuss possible fractional phases beyond $\nu =1/2$.

Figure 1

(a) The kagome lattice model. The three sublattices 1, 2, 3 are colored in blue, red, green, respectively. The arrows give the sign convention for the phase of the nearest-neighbor-hopping term; the hopping amplitude is $exp\left(i\phi \right)$ in the direction of the arrows. The lattice translation vectors are ${\mathbf{a}}_1$ and ${\mathbf{a}}_2$. (b) Band structure of the kagome lattice model with a nearest-neighbor-hopping phase $\phi =\pi /4$. The three bands are separated by an energy gap, and have Chern number $C=-1,0,+1$, respectively.

Figure 2

Different cuts of the single-particle phase diagram along the $C_3$-invariant axis. The one-body model is nontrivial in the black region of the phase diagram. ${\alpha }_1$ is the amplitude of the $C_3$-invariant term and ranges from $-7$ to 7, limits for which the kagome lattice model insulator is trivial for any value of ${\alpha }_2$ and ${\alpha }_3$.

Figure 3

Schematic representation of the interaction in our FTI kagome lattice system. The interaction strength is $U$ between bosons within the same layer (identical pseudospin) and $V$ between bosons in different layers (opposite pseudospin).

Figure 4

Half-filling FTI phase at $N=N_s=N_x=8$ $\left(N_y=1\right)$, in the case where there is no band structure coupling, and with an interlayer interaction of strength $V/U=0.5$. Left panel: Low-energy spectrum. We observe an almost fourfold degenerate ground state lying in momentum sectors $(k_x,k_y)=(0,0)$ (two states) and $(k_x,k_y)=(4,0)$ (two states), with a gap $\Delta$ to higher excitations. The highest and lowest energy states of the ground state manifold have an energy splitting $\delta$. Right panel: PES of the ground state manifold, for a partition with $N_A=4$ particles, in the sector $S_{zA}=0$. The number of states below the lowest gap is 400. The number of states between the two dotted lines is 640. Both countings are simply related to the number of states in the PES of the FCI system with $N_{\uparrow }=4$ bosons, $N_s=8$, and $N_A^{\uparrow }=2$ bosons in the $A$ partition.

Figure 5

Evolution of the many-body gap of the systems with $N=8$, 10, 12 bosons and $N_s=N$ unit cells with respect to the magnitude of the interaction between bosons of opposite pseudospin $\left(V/U\right)$, for the TRI FTI (a) and for the bilayer FCI (b). The shaded area in (a) corresponds to a full polarization of the FTI system, for $N=10$, 12 (light gray) and $N=8$ (dark gray). The bilayer FCI system only becomes fully polarized for values of $V/U$ that are beyond the scope of this graph (the transition happens for $20<V/U<30$ depending on the system size).

Figure 6

Overlap of the fourfold ground state wave function with the wave function of the decoupled system. Evolution of the overlap of the systems with $N=8$, 10, 12 bosons and $N_s=N$ unit cells with respect to the magnitude of the interaction between bosons of opposite pseudospin $\left(V/U\right)$, for the TRI FTI (a) and for the bilayer FCI (b). The shaded areas correspond to the fully polarized phase as discussed in Fig. 5. We set the overlap to zero as soon as one of the four lowest lying states falls out of the expected momentum sectors.

Figure 7

Schematic interpretation of the interpolation procedure between a FTI and a bilayer FCI system in terms of a FQH bilayer system with varying magnetic fields.

Figure 8

Gap of the bilayer kagome lattice system with $N=10$ bosons undergoing a transition (parametrized by $\lambda )$ from TRI FTI (left) to the bilayer FCI phase (right), for varying values of the interlayer interaction $V/U$. Note that the one-body gap closes at $\lambda =1/2$, a transition line which is marked in white on the phase diagram. The white dot marks the position of the $SU\left(2\right)$ invariant point. ${\Delta }_{\max }$ is the amplitude of the gap of the decoupled system with $\lambda =0$. Along the line $V/U=0$, the diagram is symmetric under the transformation $\lambda \to 1-\lambda$.

Figure 9

Evolution of the entanglement gap of the systems with $N=8$, 10, 12 bosons and $N_s=N$ unit cells for a partition $N_A=N/2$, for the TRI FTI (a) and for the bilayer FCI (b). $V/U$ is the amplitude of the interlayer interaction. The shaded areas correspond to the fully polarized phase as discussed in Fig. 5.

Figure 10

Stability of the FTI phase at half filling upon addition of a $C_3$ symmetry preserving coupling term of amplitude ${\alpha }_1$, in the absence of interlayer interaction [(a) and (b)], and with $V/U=0.5$ [(c) and (d)]. The systems have $N=8$ and $N=10$ particles, $N_s=N$ unit cells. The shaded area corresponds to the region where the one-body model is a trivial insulator. (a) and (c) Evolution of the many-body gap with ${\alpha }_1$. (b) and (d) Evolution of the PES gap with ${\alpha }_1$, for a particle partition $N_A=N/2$.

Figure 11

Gap of the kagome lattice model topological insulator at half filling with $N=10$ bosons, in the plane ${\alpha }_3=0$ [(a) and (b)] and in the plane ${\alpha }_2=0$ [(c) and (d)], for $V/U=0$ [(a) and (c)] and $V/U=0.5$ [(b) and (d)]. ${\Delta }_{\text{max,V}}$ is the amplitude of the gap in the case where $R=0$, with an interlayer interaction is $V$. We use the symmetries of the system with respect to the coupling elements ${\alpha }_i$ and show only the zones ${\alpha }_1>0$ and ${\alpha }_3>0$. The dotted line indicates the boundary between the trivial and topological insulator phases in the noninteracting model.

Figure 12

PES gap of the kagome lattice model topological insulator at half filling with $N=10$ bosons, in the plane ${\alpha }_3=0$ [(a) and (b)] and in the plane ${\alpha }_2=0$ [(c) and (d)], for $V/U=0$ [(a) and (c)] and $V/U=0.5$ [(b) and (d)]. ${\Delta }_{\xi \text{max,V}}(N/2)$ is the amplitude of the PES gap in the case where $R=0$, with an interlayer interaction $V$. The number of particles in the partition is $N_A=5$. We use the symmetries of the system with respect to the coupling elements ${\alpha }_i$ and show only the zones ${\alpha }_1>0$ and ${\alpha }_3>0$. The dotted line indicates the boundary between the trivial and topological insulator phases in the noninteracting model.

Figure 13

Low-energy spectrum of the bilayer FCI model at a filling fraction $\nu =1/3$ (with respect to the two lowest bands) for $N=6,N_s=9,N_x=N_y=3$ (a) and $N=8,N_s=N_x=12,N_y=1$ (b). There is a threefold almost degenerate ground state, separated from higher energy excitations by a large gap.

Figure 14

Low-energy spectrum of the FTI model at $\nu =1/3,V/U=1.0$. All sectors of the pseudospin projection $S_z$ are represented. Left panel: $N=6,N_s=9,N_x=N_y=3$. Right panel: $N=10,N_s=N_x=15,N_y$. In this latter case, the states with $S_z>0$ are all higher in energy than the two first excited states in each sector, and are therefore not represented here. Both systems have a ground state with a twofold exact degeneracy due to inversion symmetry. Nevertheless, in the $N=10$ system, the ground state mixes with higher energy state upon flux insertion.

Figure 15

Left panel: Vectors that define the tilted periodic boundary conditions for a triangular Bravais lattice, for a system with $N_s=10$ unit cells and optimal aspect ratio, as defined in second row of table (b). The shaded area represents the finite-size system in real space (it contains 10 unit cells). Right panel: Coordinates of the vectors ${\mathbf{T}}_{\mathbf{1}}$ and ${\mathbf{T}}_{\mathbf{2}}$ that were used in this paper for all numerical calculations, in the $({\mathbf{a}}_{\mathbf{1}},{\mathbf{a}}_{\mathbf{2}})$ basis defined in Fig. 1.

Figure 16

Construction of the $(p,q)$ labels for the admissible momentum vectors on the $N_s=10$ tilted lattice of Fig. 15. For this example, we have $N_x=5$ and $N_y=2$ according to the table of Fig. 15. (a) The dots denote the admissible momentum vectors. The shaded area is the first Brillouin zone defined by the two reciprocal vectors ${\tilde{\mathbf{a}}}_1$ and ${\tilde{\mathbf{a}}}_2$. Using ${\tilde{\mathbf{T}}}_1^{\prime }={\tilde{\mathbf{T}}}_1$, we can label the admissible momentum vectors $(p,0)$ with $p$ going from 0 to 4. (b) We find the equivalents of $(3,0)$ and $(4,0)$ within the first Brillouin zone using reciprocal vectors. (c) Using ${\tilde{\mathbf{T}}}_2^{\prime }={\tilde{\mathbf{T}}}_2-{\tilde{\mathbf{T}}}_1$ (here $\alpha =-1)$, we can now label the $(p,1)$ admissible momenta. (d) Once again, we find the equivalents of $(0,1),(1,1)$, and $(4,1)$ within the first Brillouin zone. (e) We show the label of each admissible momentum vector within the first Brillouin zone.

Figure 17

Evolution of the quantity $\text{max}(1-\delta /\Delta ,0)$ for the systems with $N=8$, 10, 12 bosons and $N_s=N$ unit cells with respect to the magnitude of the interaction between bosons of opposite pseudospin $\left(V/U\right)$, for the TRI FTI (left panel) and for the bilayer FCI (right panel). The shaded area in (a) corresponds to a full polarization of the FTI system, for $N=10$, 12 (light gray) and $N=8$ (dark gray). The bilayer FCI system only becomes fully polarized for values of $V/U$ that are beyond the scope of this graph (the transition happens for $20<V/U<30$ depending on the system size).

Figure 18

Evolution of the entanglement gap with the magnitude of the interaction between bosons of opposite pseudospin $\left(V\right)$ for the system with $N=8$ (a), $N=10$ (b), and $N=12$ (c), $N_s=N$. The two kagome layers are only coupled by the interaction. The shaded area corresponds to a full polarization of the system, for $N=10$, 12 (light gray) and $N=8$ (dark gray).

Figure 19

Gap of the bilayer kagome lattice system with $N=8$ bosons undergoing a transition (parametrized by $\lambda )$ from TRI FTI (left) to the bilayer FCI phase (right), for varying values of the interlayer interaction $V/U$. Note that the one-body gap closes at $\lambda =1/2$, a transition line which is marked in white on the phase diagram. The white dot marks the position of the $SU\left(2\right)$-invariant point. ${\Delta }_{\max }$ is the amplitude of the gap of the decoupled system with $\lambda =0$. Along the line $V/U=0$, the diagram is symmetric under the transformation $\lambda \to 1-\lambda$.

Figure 20

Gap of the kagome lattice model topological insulator at half filling with $N=8$ bosons, in the plane ${\alpha }_3=0$ [(a) and (b)] and in the plane ${\alpha }_2=0$ [(c) and (d)], for $V/U=0$ [(a) and (c)] and $V/U=0.5$ [(b) and (d)]. ${\Delta }_{\text{max,V}}$ is the amplitude of the gap in the case where $R=0$, with an interlayer interaction $V$. We use the symmetries of the system with respect to the coupling elements ${\alpha }_i$ and show only the zones ${\alpha }_1>0$ and ${\alpha }_3>0$. The dotted line indicates the boundary between the trivial and topological insulator phases in the noninteracting model.

Figure 21

PES gap of the kagome lattice model topological insulator at half filling with $N=8$ bosons, in the plane ${\alpha }_3=0$ [(a) and (b)] and in the plane ${\alpha }_2=0$ [(c) and (d)], for $V/U=0$ [(a) and (c)] and $V/U=0.5$ [(b) and (d)]. ${\Delta }_{\xi \text{max,V}}(N/2)$ is the amplitude of the PES gap in the case where $R=0$, with an interlayer interaction $V$. The number of particles in the partition is $N_A=4$. We use the symmetries of the system with respect to the coupling elements ${\alpha }_i$ and show only the zones ${\alpha }_1>0$ and ${\alpha }_3>0$. The dotted line indicates the boundary between the trivial and topological insulator phases in the noninteracting model.