Theory of unconventional quantum Hall effect in strained graphene

We show through both theoretical arguments and numerical calculations that graphene discerns an unconventional sequence of quantized Hall conductivity, when subject to both magnetic fields ($B$) and strain. The latter produces time-reversal symmetric pseudo/axial magnetic fields ($b$). The single-electron spectrum is composed of two interpenetrating sets of Landau levels (LLs), located at $\pm \sqrt{2n|b\pm B|}$, $n=0,1,2,...$. For $b>B$, these two sets of LLs have opposite chiralities, resulting in oscillating Hall conductivity between 0 and $\mp 2e^2/h$ in electron and hole doped systems, respectively, when the chemical potential is tuned in the vicinity of the neutrality point. The electron-electron interactions stabilize various correlated ground states, e.g., spin-polarized, quantum spin Hall insulators at and near the neutrality point, and possibly the anomalous Hall insulating phase at incommensurate filling $\sim$$B$. Such broken-symmetry ground states have similarities as well as significant differences from their counterparts in the absence of strain. For realistic strength of magnetic fields and interactions, we present scaling of the interaction-induced gap for various Hall states within the zeroth Landau level.

Figure 1

Upper row:: Left: Spin-degenerate interpenetrating LLs of $H_D[A,a]$. Here we have shown the LLs for $n=0,\pm 1,\pm 2$ only. Two LLs have the degeneracies $2D_{\pm }=(b\pm B)$ per unit area. Right: Schematic variation of Hall conductances $G_{xy}$ in a hole-doped graphene, without (a) and with (b) Zeeman splitting (${\Delta }_z$). Here, $E_{\sigma }^n=\sqrt{2n(b+\sigma B)}$, $E_{\sigma }^{n\alpha }=E_{\sigma }^n+\alpha {\Delta }_z$ with $\sigma ,\alpha =\pm$. In the electron-doped system, $G_{xy}$ changes sign. We only show the spin splitting of $n=1$ LL. Splitting of $n=2$ LL is identical. Lower row: The energy spectrum (left) and wave functions (WFs) (right) for a strained graphene in magnetic field when $B<b$. WFs localized on one edge live on opposite side at two valleys, explicitly, A(B) and D (C) are localized on the left (right) edge, therefore carrying opposite chirality.

Figure 2

Left: Two possible [(a), (b)] interaction-driven splittings of the ZLL. Dotted lines correspond to requisite location of the chemical potential for ${\sigma }_{xy}=f{e}^2/h$. $\pm$ corresponds to the ZLL states localized near the valley at $\pm \vec{K}$. Right: Dimensionless activation gap ($\Delta /\Lambda$) for ${\sigma }_{xy}=0,\pm e^2/h$ (black, red) Hall states when $b=300$ T and 0 T $<B<50$ T, for $\delta =0,0.03,0.07,0.14,0.3,0.7$ (from top to bottom), assuming $\Lambda \sim 1/2.5\mathring{\mathrm{A}}$, the ultraviolet cutoff over which the dispersion is approximately linear. $g_c$ is the zero-field criticality. Lower $x$ axis denotes $B/{\Lambda }^2$ (dimensionless).