Bridging coupled wires and lattice Hamiltonian for two-component bosonic quantum Hall states

We investigate a model of hard-core bosons with correlated hopping on the honeycomb lattice in an external magnetic field by means of a coupled-wire approach. It has been numerically shown that this model exhibits at half filling the bosonic integer quantum Hall (BIQH) state, which is a symmetry-protected topological phase protected by the $U\left(1\right)$ particle conservation [Y.-C. He et al., Phys. Rev. Lett. 115, 116803 (2015)]. By combining the bosonization approach and a coupled-wire construction, we analytically confirm this finding and show that it even holds in the strongly anisotropic (quasi-one-dimensional) limit. We discuss the stability of the BIQH phase against tunnelings that break the separate particle conservations on different sublattices down to a global particle conservation. We further argue that a phase transition between two different BIQH phases is in a deconfined quantum critical point described by two copies of the $(2+1)$-dimensional $O\left(4\right)$ nonlinear sigma model with the topological $\theta$ term at $\theta =\pi$. Finally, we predict a possible fractional quantum Hall state, the Halperin $\left(221\right)$ state, at $1/6$ filling.

Figure 1

The model defined on the honeycomb lattice. The $A$ and $B$ sublattices are denoted by the filled and open circles, respectively. The red (blue) solid lines represent the correlated hoppings in the sublattice $A$ ($B$). On the $A$ sublattice, the upper triangles are pierced by the flux $\alpha$ while the lower ones by $\pi -\alpha$. On the $B$ sublattice, the lower triangles are pierced by the flux $\beta$ while the upper ones by $\pi -\beta$.

Figure 2

Redefinition of the site index for the honeycomb lattice to be viewed as coupled chains. The red (blue) dashed line represents the path of the string operator for ${\tilde{a}}_{j,\ell } \left({\tilde{b}}_{j,\ell }\right)$.

Figure 3

A choice of the phases for the correlated hopping terms, which gives Eqs. (25) and (26). The red (blue) arrows or lines indicate the phases of the correlated hoppings on the $A \left(B\right)$ sublattice.

Figure 4

Phase diagram of the Hamiltonian Eq. (3) for general values of $\alpha$ and $\beta$, obtained by the renormalization group analysis. Two BIQH phases denoted by ${\mathrm{BIQH}}^{+}$ and ${\mathrm{BIQH}}^{-}$ (see text) are separated by phase transitions corresponding to black solid lines.

Figure 5

Tunnelings between the two sublattices are represented by green arrows on the links of the honeycomb lattice. We also show the background fluxes assigned in Ref. [55], where the fluxes $\alpha /3$ or $\beta /3$ thread obtuse triangles whose two vertices are in the $A$ or $B$ sublattice.

Figure 6

Network models describing the phase transitions between two BIQH phases. Red (blue) circles correspond to charge (pseudospin) modes, and cross (dotted) symbols represent right-moving (left-moving) modes. Four modes enclosed by the dashed line represent the degrees of freedom in a physical bosonic chain. $t_1$ and $t_2$ are two different tunnelings between the two counter-propagating modes. (a) The model studied in Refs. [5, 77]: One may find the trivial phase when $t_1\gg t_2$ while the ${\mathrm{BIQH}}^{+}$ phase when $t_1\ll t_2$. (b) Our model: One may find the ${\mathrm{BIQH}}^{-}$ phase when $t_1\gg t_2$ while the ${\mathrm{BIQH}}^{+}$ phase when $t_1\ll t_2$. In both cases, the phase transitions between the two phases are expected at $t_1=t_2$.