Two-component quantum Hall effects in topological flat bands

We study quantum Hall states for two-component particles (hardcore bosons and fermions) loading in topological lattice models. By tuning the interplay of interspecies and intraspecies interactions, we demonstrate that two-component fractional quantum Hall states emerge at certain fractional filling factors $\nu =1/2$ for fermions ($\nu =2/3$ for bosons) in the lowest Chern band, classified by features from ground states including the unique Chern number matrix (inverse of the $\mathbf{K}$ matrix), the fractional charge and spin pumpings, and two parallel propagating edge modes. Moreover, we also apply our strategy to two-component fermions at integer filling factor $\nu =2$, where a possible topological Neel antiferromagnetic phase is under intense debate very recently. For the typical $\pi$-flux checkerboard lattice, by tuning the onsite Hubbard repulsion, we establish a first-order phase transition directly from a two-component fermionic $\nu =2$ quantum Hall state at weak interaction to a topologically trivial antiferromagnetic insulator at strong interaction, and therefore exclude the possibility of an intermediate topological phase for our system.

Figure 1

Numerical results for the low energy spectrum of (a) bosonic systems $\nu =2/3,N_s=3\times N$ in different topological lattices at $U=10t,V=0$ and (b) fermionic systems $\nu =1/2,N_s=N\times 4$ in different topological lattices at $U=10t,V=20t$.

Figure 2

Robustness of degenerate ground states on the $\pi$-flux checkerboard lattice vs the variations of interspecies repulsion for (a) bosonic systems $N=6,N_s=2\times 3\times 3$ at $\nu =2/3,V=0$; (b) fermionic systems $N=6,N_s=2\times 3\times 4$ at $\nu =1/2,V=20t$. The energies are measured relative to the lowest ground energy. The momentum sectors in which these degenerate FQH states emerge are labeled explicitly with different colors while the other momentum sectors are labeled by the cyan circle. (a) $\mathrm{CB},\mathrm{N}=6,{\mathrm{N}}_{\mathrm{s}}=2\times 3\times 3$; (b) $\mathrm{CB},\mathrm{N}=6,{\mathrm{N}}_{\mathrm{s}}=2\times 3\times 4$.

Figure 3

Numerical results for the $y$-direction spectral flow of fermionic systems $N=6,N_s=2\times 3\times 4$ at $\nu =1/2,U=10t,V=20t$. (a) ${\theta }_{\uparrow }^{\alpha }=\theta ,{\theta }_{\downarrow }^{\alpha }=0$; (b) ${\theta }_{\uparrow }^{\alpha }={\theta }_{\downarrow }^{\alpha }=\theta$. (a) $\nu =1/2,{\theta }_{\uparrow }=\theta ,{\theta }_{\downarrow }=0$; (b) $\nu =1/2,{\theta }_{\uparrow }={\theta }_{\downarrow }=\theta$.

Figure 4

The charge and spin transfer with inserting flux ${\theta }_{\uparrow }=\theta ,{\theta }_{\downarrow }=0$ in the cylinder. (a) Bosonic systems on the $N_y=3$ cylinder at $\nu =2/3,U=10t,V=0$; (b) fermionic systems on the $N_y=4$ cylinder at $\nu =1/2,U=10t,V=20t$. Here the calculation is performed using infinite DMRG with keeping up to 1200 states.

Figure 5

The momentum-resolved entanglement spectrum for bosonic systems on the $N_y=6$ cylinder of the single layer square lattice with Chern number two at $\nu =1/3,U=10t,V=0$. In each tower, the horizontal axis shows the relative momentum $\mathrm{\Delta }K=K_y-K_y^0$ (in units of $2\pi /N_y$) in the transverse direction of the corresponding eigenvectors of density matrix ${\rho }_L$. The numbers below the red dashed line label the nearly degenerating pattern for the low-lying ES with different momenta: $1,2,5,...$

Figure 6

The momentum-resolved entanglement spectrum for fermionic systems on the $N_y=8$ cylinder of the single layer square lattice with Chern number two at $\nu =1/4,U=10t,V=20t$. In each tower, the horizontal axis shows the relative momentum $\mathrm{\Delta }K=K_y-K_y^0$ (in units of $2\pi /N_y$) in the transverse direction of the corresponding eigenvectors of density matrix ${\rho }_L$. The numbers below the red dashed line label the nearly degenerating pattern for the low-lying ES with different momenta.

Figure 7

Numerical results for the $y$-direction spectral flow of fermionic systems. (a) Two different checkerboard lattice geometries considered in our calculation; (b) the low energy spectra as a function of onsite repulsion $U$ for system $N_s=16,N_{\uparrow }=N_{\downarrow }=8$ at $\nu =2$; (c) the manybody Chern number $C_{\uparrow ,\uparrow }$ and antiferromagnetic spin structure factor $S\left(\mathbf{Q}\right)$ of the lowest ground state, and the spin excitation gap ${\mathrm{\Delta }}_s$ as a function of onsite repulsion $U$. Here the parameters $V=0,t^{\prime }=0.3t,t^{\prime \prime }=0$.

Figure 8

Numerical DMRG results on a cylinder with finite width $L_y=2\times N_y$ and fixed length $L_x=18$ for $N_{\uparrow }=N_{\downarrow }=L_x\times N_y,\nu =2$. The evolutions of physical quantities vs onsite repulsion $U$, including the absolute wave-function overlap $F\left(U\right)$, charge pumping $\mathrm{\Delta }Q$, spin structure factor $S\left(\mathbf{Q}\right)$, and entanglement entropy $S_L$ are shown for (a) $L_y=6,N_y=3$ and (b) $L_y=8,N_y=4$, respectively. The first-order transition is characterized by the discontinuous behavior of these physical quantities at the transition point. Here the parameters $V=0,t^{\prime }=0.3t,t^{\prime \prime }=0$, and the maximally kept number of states 2020.