Bilayer Graphene as a Platform for Bosonic Symmetry-Protected Topological States

Bosonic symmetry protected topological (BSPT) states, the bosonic analogue of topological insulators, have attracted enormous theoretical interest in the last few years. Although BSPT states have been classified by various approaches, there is so far no successful experimental realization of any BSPT state in two or higher dimensions. In this paper, we propose that a two-dimensional BSPT state with $U(1)\times U(1)$ symmetry can be realized in bilayer graphene in a magnetic field. Here the two $U(1)$ symmetries represent total spin $S^z$ and total charge conservation, respectively. The Coulomb interaction plays a central role in this proposal—it gaps out all the fermions at the boundary, so that only bosonic charge and spin degrees of freedom are gapless and protected at the edge. Based on the above conclusion, we propose that the bulk quantum phase transition between the BSPT and trivial phase, which can be driven by applying both magnetic and electric fields, can become a “bosonic phase transition” with interactions. That is, only bosonic modes close their gap at the transition, which is fundamentally different from all the well-known topological insulator to trivial insulator transitions that occur for free fermion systems. We discuss various experimental consequences of this proposal.

Figure 1

Schematic of bilayer graphene in the presence of a magnetic field with both in-plane and out-of-plane components. (a) Without interactions, the boundary hosts two channels of fermionic edge states with total central charge $c=2$. (b) Including the Coulomb interactions, there is only one gapless channel of bosonic edge state with $c=1$.

Figure 2

Our proposed setup for measuring the carrier charge at the boundary of our system. Most of the sample is screened by the inner symmetric gates, while the unscreened region has a stronger interaction that leads to a CAF order, and induces backscattering of the edge states. We also add a pair of outer gates to control the strength of interaction in the CAF region.