Exactly soluble models for fractional topological insulators in two and three dimensions

We construct exactly soluble lattice models for fractionalized, time-reversal-invariant electronic insulators in two and three dimensions. The low-energy physics of these models is exactly equivalent to a noninteracting topological insulator built out of fractionally charged fermionic quasiparticles. We show that some of our models have protected edge modes [in two dimensions (2D)] and surface modes (in 3D), and are thus fractionalized analogs of topological insulators. We also find that some of the 2D models do not have protected edge modes; that is, the edge modes can be gapped out by appropriate time-reversal-invariant, charge-conserving perturbations. (A similar state of affairs may also exist in 3D.) We show that all of our models are topologically ordered, exhibiting fractional statistics as well as ground-state degeneracy on a torus. In the 3D case, we find that the models exhibit a fractional magnetoelectric effect.

Figure 1

In the lattice boson model, bosons live on both the sites $s$ and links $\langle s{s}^{\prime }\rangle$ of the square lattice. The Hamiltonian $H_B$ (1) is a sum of a $Q_s$ term (2), which acts on four links $\langle s{s}^{\prime }\rangle$ and one site $s$, and a $B_P$ term (4), which acts on the four sites and links adjacent to a plaquette $P$. The $B_P$ term is a product of four link operators $U_{s{s}^{\prime }}$ (5) which each act on the sites $s,s^{\prime }$ and the link $\langle s{s}^{\prime }\rangle$.

Figure 2

We consider the lattice boson model $H_B$ (1) in two geometries: (a) a rectangular geometry with open boundary conditions, and (b) a periodic (torus) geometry. In the rectangular geometry (a), the $Q_s$ operators at the corners act on two links $\langle s{s}^{\prime }\rangle$, while those at the edge act on three links.

Figure 3

In order to include an electromagnetic vector potential in $H$ (37), we need to define a vector potential for each of the two halves of each link $\langle s{s}^{\prime }\rangle$. We denote these vector potentials by $A_{s{s}^{\prime },1}$, $A_{s{s}^{\prime },2}$.

Figure 4

We can move a charge at position $s_0$ around a flux at position $P_0$ by applying a string of $U_{s{s}^{\prime }}$ operators along a closed path $C=s_0{s}_1,...,s_k{s}_0$ encircling $P_0$.

Figure 5

(a) The lattice boson model defined on the cubic lattice. As in the square lattice case, the Hamiltonian $H_B$ (83) is a sum of $Q_s$ operators and $B_P$ operators, but now the $Q_s$ operators act on six links $\langle s{s}^{\prime }\rangle$ and one site $s$, and the $B_P$ operators act on the plaquettes of the cubic lattice. (b) The $B_P$ operators obey the identity ${\prod }_{P\in C}{B}_P=1$, where the product runs over the six plaquettes $P$ adjacent to a “cube” $C$, and where we choose appropriate orientations on these plaquettes.

Figure 6

The flux insertion argument in the 2D noninteracting case. We start in the ground state ${\Psi }_0$ and adiabatically insert $\pm {\Phi }_0/2$ flux through the cylinder, obtaining two states ${\Psi }_{\pm \pi }$. The state ${\Psi }_{\pi }$ has a Kramers degeneracy at the two ends of the cylinder, and is thus degenerate in energy with three other states, one of which is ${\Psi }_{-\pi }$. If we start in one of these three degenerate states and then adiabatically reduce the flux to 0, we obtain an excited state ${\Psi }_{\mathrm{ex}}$ whose energy gap $\Delta$ vanishes in the thermodynamic limit.

Figure 7

(a) The 3D flux insertion argument makes use of a “Corbino donut” geometry, with flux ${\phi }_x,{\phi }_y$ through the two holes of the Corbino donut. (b) In the noninteracting case, we start in the ground state ${\Psi }_{(0,0)}$ and adiabatically insert ${\Phi }_0/2$ flux through each of the two holes, obtaining three other states: ${\Psi }_{(\pi ,0)},{\Psi }_{(0,\pi )},{\Psi }_{(\pi ,\pi )}$. An odd number of these states must have Kramers degeneracies at the two surfaces of the Corbino donut, leading to four possible degeneracy patterns (solid circles represent states with Kramers degeneracies). In particular, at least one state has a Kramers degeneracy, and hence can be used to construct a protected low-lying excited state ${\Psi }_{\mathrm{ex}}$, as in the 2D case.

Figure 8

Surface state configurations of ${\Psi }_{(0,0)},{\Psi }_{(\pi ,0)},{\Psi }_{(0,\pi )},{\Psi }_{(\pi ,\pi )}$ for a system with a single Dirac cone on each surface, and a chemical potential just above the Dirac point. Here we plot the surface states as a function of $k_x$; the bands indicate different discrete values of $k_y$. The band that crosses the tip of the Dirac cone, corresponding to $k_y=0$, is singly degenerate; all other bands, for which $k_y\ne 0$, are doubly degenerate. The four panels show the arrangement of the discrete surface momenta relative to the tip of the Dirac cone in each of the four flux sectors. The solid circles indicate occupied states.

Figure 9

An example of mapping to spin configuration for $m=3$. The numbers in red correspond to a choice of $\left\{j_P\right\}$. The base of the loop is the lower left corner, which for concreteness has been assigned a value 0. The sign with which we pick up the appropriate $j_P$ (or equivalently with which the plaquette contributes to $\Delta j_{s{s}^{\prime }}$ for a specific link $\langle s{s}^{\prime }\rangle$ enclosed by the loop) has been coded in blue for $+$ and green for $-$. The involved links have been colored red.