Quantum phases of disordered flatband lattice fractional quantum Hall systems

By numerical exact diagonalization techniques, we obtain the quantum phase diagram of the lattice fractional quantum Hall (FQH) systems in the presence of quenched disorder. By implementing an array of local potential traps representing the disorder, we show that the system undergoes a series of quantum phase transitions as the disorder and/or the interaction is tuned. As the strength of potential traps is increased, the FQH state turns into a compressible liquid and then into a topologically trivial insulator. We use the numerically calculated energy gap, quantum degeneracy, the Chern number, the entanglement spectrum, and the fidelity metric to identify various quantum phases. The connection to continuum FQH effects is also discussed.

Figure 1

The checkerboard flatband model with charge impurities marked by the cones beneath the lattice.

Figure 2

The phase diagram in the parameter space of impurity strength ($\mu$) and NNN interaction strength ($V$) for different system sizes ($N_s=2\times 3\times 4$, $2\times 3\times 5$, and $2\times 3\times 6$). Three phases are observed, the fractional quantum Hall phase (FQH), the compressible phase (metal), and the trivial insulator phase. The thick blue lines mark the averaged phase boundaries over different system sizes. Here we set $U=4$ and $N_e=N_s/6$.

Figure 3

Energy spectrum as a function of magnetic flux (a) in the FQH phase with $\mu =0$, (b) in the compressible phase with $\mu =2.5$, and (c) in the trivial insulator phase with $\mu =5$. Here $N_s=2\times 3\times 6$, $N_e=6$, $U=4$, and $V=1$. And we choose $N_{\theta x}=N_{\theta y}=12$. The blue dashed lines are the three lowest energies of momentum sector $K_y=\pi /a$ with $a$ being the lattice constant, and the black solid lines are the energies of the other momentum sectors.

Figure 4

Numerical observables as a function of $\mu$. (a) Averaged excitation gaps over twisted boundary conditions. (b) Particle entanglement spectrum. Here, we traced out 3 particles, and the blue arrows mark the gaps in the entanglement spectrum. There are 75 states below the gap in the FQH state and this number becomes 10 in the trivial insulator phase, in good agreement with the counting of quasihole excitations. (c) Ground-state fidelity metric under periodic and twist boundary conditions, with $\delta \mu =0.001$. In all these three panels, the red vertical dashed lines denote the phase boundaries and we set $N_s=2\times 3\times 5$, $N_e=5$, $U=4$, and $V=1$.