Fractional topological liquids with time-reversal symmetry and their lattice realization

We present a class of time-reversal-symmetric fractional topological liquid states in two dimensions that support fractionalized excitations. These are incompressible liquids made of electrons, for which the charge Hall conductance vanishes and the spin Hall conductance needs not be quantized. We then analyze the stability of edge states in these two-dimensional topological fluids against localization by disorder. We find a ${\mathbb{Z}}_2^{}$ stability criterion for whether or not there exists a Kramers pair of edge modes that is robust against disorder. We also introduce an interacting electronic two-dimensional lattice model based on partially filled flattened bands of a ${\mathbb{Z}}_2^{}$ topological band insulator, which we study using numerical exact diagonalization. We show evidence for instances of the fractional topological liquid phase as well as for a time-reversal symmetry broken phase with a quantized (charge) Hall conductance in the phase diagram for this model.

Figure 1

Numerical exact diagonalization results of Hamiltonian (2.1) for 16 electrons when sublattice $A$ is made of $3\times 4$ sites and with $t_2^{}/t_1^{}=0.4$. (a) Ground state degeneracies. Denote with $E_n^{}$ the $n$th lowest-energy eigenvalue of the many-body spectrum where $E_1^{}$ is the many-body ground state (i.e., $E_{n+1}^{}\ge E_n^{}$ for $n=1,2,...$). Define the parameter ${\epsilon }_n^{}:=(E_{n+1}^{}-E_n^{})/(E_n^{}-E_1^{})$. If a large gap $E_{n+1}^{}-E_n^{}$ opens up between two consecutive levels $E_{n+1}^{}$ and $E_n^{}$ compared to the cumulative level splitting $E_n^{}-E_1^{}$ between the first $n$ many-body eigenstates induced by finite-size effects, then the parameter ${\epsilon }_n^{}$ is much larger than unity. The parameter ${\epsilon }_n^{}$ has been evaluated for $n=3$ and $n=9$, yielding the red and blue regions, respectively. For all other $n\ne 1$, no regions with ${\epsilon }_n^{}\gtrsim O\left(1\right)$ of significant size were found. Within the limited range of available system sizes, it is thus not possible to decide on whether and how the level-splitting above the ground state in the white regions of the parameter space extrapolates in the thermodynamic limit. (b)–(d) The lowest eigenvalues with spin-dependent twisted boundary conditions as a function of the twisting angle ${\gamma }_x^{}$. The number of low-lying states that are energetically separated from the other states is 9, 3, and 3, respectively. In panel (c), it is the lowest band parametrized by ${\gamma }_x^{}$ that is three-fold degenerate.

Figure 2

Cylindrical geometry for a two-dimensional band insulator. The cylinder axis is labeled by the coordinate $y$. Periodic boundary conditions are imposed in the transverse direction labeled by the coordinate $x$. There is an edge at $y=-L_y^{}/2$ and another one at $y=+L_y^{}/2$. Bulk states have a support on the shaded surface of the cylinder. Edge states are confined in the $y$ direction to the vicinity of the edges $y=\pm L_y^{}/2$. Topological band insulators have the property that there are edge states freely propagating in the $x$ direction even in the presence of disorder with the mean free path $\ell$ provided the limit $\ell /L_y^{}\ll 1$ holds.