Lattice realization of a bosonic integer quantum Hall state–trivial insulator transition and relation to the self-dual line in the easy-plane NCCP1 model

We provide an explicit lattice model of bosons with two separately conserved boson species $\left[U\right(1)\times U(1)$ global symmetry] realizing a direct transition between an integer quantum Hall effect of bosons and a trivial phase, where any intermediate phase is avoided by an additional symmetry interchanging the two species. If the latter symmetry is absent, we find intermediate superfluid phases where one or the other boson species condenses. We know the precise location of the transition since at this point our model has an exact nonlocal antiunitary particle-hole-like symmetry that resembles particle-hole symmetry in the lowest Landau level of electrons. We exactly map the direct transition to our earlier study of the self-dual line of the easy-plane NCCP1 model, in the mathematically equivalent reformulation in terms of two (new) particles with $\pi$ statistics and identical energetics. While the transition in our model is first order, we hope that our mappings and recent renewed interest in such self-dual models will stimulate more searches for models with a continuous transition.

Figure 1

The lattice on which the Hamiltonian in Eq. (4) is defined. There are two interpenetrating cubic lattices with sites labeled by the coordinates $\mathbf{r}$ and $\mathbf{R}$. On the sites of each lattice there are $U\left(1\right)$ quantum rotors. On the intersection points of the links of the two lattices there are harmonic oscillators, whose “position” and “momentum” variables are conveniently thought of as fields residing on the oriented links of the direct and dual lattices, respectively.

Figure 2

Phase diagrams for the model in Eqs. (10, 11, 12) [27, 28, 29], as a function of ${\lambda }_1, {\lambda }_2$ and for a variety of different $\eta$. A description of each of the phases is given in the main text. The phases labeled with numbers are either the trivial insulator with Hall conductivity 0 or the BIQHE with Hall conductivity 2.

Figure 3

A schematic of the phase diagram obtained by fixing ${\lambda }_1={\lambda }_2=\lambda$, and varying $\eta$. The numbers inside the phases label their Hall conductivity, with ${\sigma }_{xy}=0$ the trivial phase and ${\sigma }_{xy}=2$ the BIQHE. Fractional values ${\sigma }_{xy}=2c/d$ correspond to symmetry-enriched topological phases with $d$ vortices bound to $c$ bosons.