Fractional Topological Phases and Broken Time-Reversal Symmetry in Strained Graphene

We show that strained or deformed honeycomb lattices are promising platforms to realize fractional topological quantum states in the absence of any magnetic field. The strain-induced pseudomagnetic fields are oppositely oriented in the two valleys and can be as large as 60–300 T as reported in recent experiments. For strained graphene at neutrality, a spin- or a valley-polarized state is predicted depending on the value of the on-site Coulomb interaction. At fractional filling, the unscreened Coulomb interaction leads to a valley-polarized fractional quantum Hall liquid which spontaneously breaks time-reversal symmetry. Motivated by artificial graphene systems, we consider tuning the short-range part of interactions and demonstrate that exotic valley symmetric states, including a valley fractional topological insulator and a spin triplet superconductor, can be stabilized by such interaction engineering.

Figure 1

The $n=0$ PLL at fractional filling factor $\nu =-2+2/3$: spinless electrons. Upper panel left (a): low-energy spectrum as a function of the next-nearest-neighboring coupling $U_{\mathrm{NNN}}^{\mathrm{op}}$ between opposite valleys (deviation from the pure Coulomb value). In the region $-0.73<U_{\mathrm{NNN}}^{\mathrm{op}}<-0.58$, the nine lowest energy states become close together and almost degenerated, thereby forming the ground-state manifold of the valley FTI. The inset shows the ground-state energies of the valley-polarized (V-$2/3$ FQH) and FTI states. Upper panel right (b): boundary-phase dependence and the robust gap between the GSM and higher-energy states for $U_{\mathrm{NNN}}^{\mathrm{op}}=-0.6$ inside the FTI phase. Lower panel (c): phase diagram as a function of $U_{\mathrm{NNN}}^{\mathrm{op}}$ for spinless electrons. Parameters for the exact diagonalization: The noninteracting orbitals are determined on a $24\times 24$ lattice with a pseudomagnetic flux ${\Phi }_0/48$ per hexagon (see Supplemental Material [27]). The degeneracy of $n=0$ PLL is $N_s=12$ per spin direction and per valley. The low-energy spectrum is calculated for $N_e=8$ ($N_L=N_R=4$) electrons with polarized spin occupying those $N_s=12$ states. Energies are given in units of $e^2/4\pi ϵa_0\simeq 10\mathrm{eV}$.

Figure 2

The $n=0$ PLL at fractional filling factor $\nu =-2+2/3$: spinfull electrons. Left panel (a): the energy of different ground states as a function of the next-nearest-neighboring coupling $U_{\mathrm{NNN}}$ defined in Eq. (2). In the range $U_{\mathrm{NNN}}>-0.8$, which includes the pure Coulomb interaction of realistic graphene ($U_{\mathrm{NNN}}=0$), the ground state is a valley-polarized and spin singlet FQH state (green $\times$ crosses). Only a very significant attraction $U_{\mathrm{NNN}}<-0.8$ can destabilize this state toward a valley-unpolarized and spin-polarized superconducting state (red $+$ crosses). This superconducting phase is characterized by a finite superfluid density as shown in the inset. Right panel (b) and (c): two lowest energies in each momentum sector as a function of $k$ for the valley-polarized state ($U_{\mathrm{NNN}}=-0.6$, upper) and for the spin-polarized superconductor ($U_{\mathrm{NNN}}=-1$, lower). Parameters for the exact diagonalization: same as in Fig. 1 but with the spin degree of freedom.

Figure 3

Neutral graphene, $\nu =0$. Upper panel (a): Hartree-Fock energies of the Ising valley-polarized (${\Psi }_V$) and spin-polarized (${\Psi }_S$) states as a function of the on-site Hubbard coupling $U_0$ whereas $U_{\mathrm{NNN}}=0$ (squares). We have also plotted the Hartree-Fock energies as a function of $U_{\mathrm{NNN}}$ in the absence of on-site correlation, $U_0=0$ (triangles). Lower panel (b): corresponding phase diagrams. The energy is in units of $e^2/4\pi ϵa_0$, where $a_0$ is the distance between the nearest-neighbor sites. The red dot indicates the value of the coupling $U_0$ for freestanding and unstrained graphene according to Ref. 25. Numerical parameters: The lattice we considered has $96\times 96$ sites. The degeneracy of the PLL orbits was $N_s=48$, and the electron number is $N_e=96$.