Global Phase Diagram of the Normal State of Twisted Bilayer Graphene

We investigate the full doping and strain-dependent phase diagram of the normal state of magic-angle twisted bilayer graphene (TBG). Using comprehensive Hartree-Fock calculations, we show that at temperatures where superconductivity is absent the global phase structure can be understood based on the competition and coexistence between three types of intertwined orders: a fully symmetric phase, spatially uniform flavor-symmetry-breaking states, and an incommensurate Kekulé spiral (IKS) order. For small strain, the IKS phase, recently proposed as a candidate order at all nonzero integer fillings of the moiré unit cell, is found to be ubiquitous for noninteger doping as well. We demonstrate that the corresponding electronic compressibility and Fermi surface structure are consistent with the “cascade” physics and Landau fans observed experimentally. This suggests a unified picture of the phase diagram of TBG in terms of IKS order.

Figure 1

(a) Hartree-Fock phase diagram of TBG in the filling-strain plane. Color map intensity shows the magnitude of the IVC order parameter (white regions have no IVC). Color encodes type of IVC via the angle $\arctan ({\mathrm{\Delta }}_{\mathrm{KIVC}}/{\mathrm{\Delta }}_{\mathcal{T}\mathrm{IVC}})$ on the “Bloch sphere,” KIVC at $\mathbf{q}=0$ (red) and $\mathcal{T}\mathrm{IVC}$ at nonzero $\mathbf{q}$ (blue). Hatching denotes valley polarization. $\mathcal{T}$-breaking phases are underlined. HF results are for $12\times 12$ systems, minimizing over all $\mathbf{q}$. (b) Electron-doped FSs (black lines) of both spin species $(\uparrow ,\downarrow )$ near commensurate filling at three representative strains. In IVC phases, valleys hybridize as indicated by the modulated valley polarization, yielding two split bands of which only the lower is shown. Finite-$\mathbf{q}$ IVC order is flagged by dashed mBZ boundaries ($\mathbf{q}$ may vary). Experimentally measured Landau fans indicate 4,2,1 equal-area FSs on electron-doping $\nu =0$, 2, 3. HF results at $\epsilon =0.16\%$ and $\epsilon =0.3\%$ are consistent with this, but not at $\epsilon =0$, suggesting strain is ubiquitous in real samples. Metallic parent states at $\nu =1$, $\epsilon =0.16\%,0.3\%$ (hatched) do not give rise to clear Landau fans, consistent with experiments.

Figure 2

Chemical potential $\mu$ as a function of filling $\nu$ for three different strain values, offset by 5 meV for clarity. HF calculation for a $12\times 12$ system with steps of $\mathrm{\Delta }\nu =1/36$.