Strain disorder and gapless intervalley coherent phase in twisted bilayer graphene

Correlated insulators are frequently observed in magic-angle twisted bilayer graphene at even fillings of electrons or holes per moiré unit cell. Whereas theory predicts these insulators to be intervalley coherent excitonic phases, the measured gaps are routinely much smaller than theoretical estimates. We explore the effects of random strain variations on the intervalley coherent phase, which have a pair-breaking effect analogous to magnetic disorder in superconductors. We find that the spectral gap may be strongly suppressed by strain disorder, or vanish altogether, even as intervalley coherence is maintained. We discuss predicted features of the tunneling density of states, show that the activation gap measured in transport experiments corresponds to the diminished gap, and thus offer a solution for the apparent discrepancy between the theoretical and experimental gaps.

Figure 1

Schematic description of our model. (a) The band structure in each valley ($K,K^{\prime }$) is approximated by two Dirac cones, one in each “minivalley.” The momentum separating the valleys, $\mathbf{Q}$, is modified by random homostrain fluctuations acting on the graphene layers, as represented by colored dotted arrows. (b) Schematic of a possible random strain configuration. Color indicates the strain strength $\left|\mathcal{A}\right|$, whereas arrows indicate the directions of the local distortions of the graphene lattice.

Figure 2

(a) The OP ${\mathrm{\Delta }}_{\mathrm{ivc}}$ as a function of temperature and disorder energy scale $\mathrm{\Gamma }$. For a given temperature, there is a critical $\mathrm{\Gamma }$ at which the OP vanishes. (b) The ratio between the gap in the TDOS, ${\omega }_g$, and ${\mathrm{\Delta }}_{\mathrm{ivc}}$. As $\mathrm{\Gamma }$ increases, ${\omega }_g$ deviates from ${\mathrm{\Delta }}_{\mathrm{ivc}}$ appreciably. There exists a gapless K-IVC region bounded by the vanishing of the single-particle gap (black dashed line) and of the OP (blue dashed line). We used $U=0.7W$ and a linear DOS ${\mathcal{N}}_{\mathrm{lin}.}$. (c) Linecuts for different values of $\mathrm{\Gamma }$ showing ${\mathrm{\Delta }}_{\mathrm{ivc}}$ (solid) and ${\omega }_g$ (dash-dotted).

Figure 3

Evolution of the TDOS $\tilde{\mathcal{N}}\left(\epsilon \right)$ with $\mathrm{\Gamma }$. The gap ${\omega }_g$ gradually closes as the disorder strength increases. Blue line indicates evolution of ${\mathrm{\Delta }}_{\mathrm{ivc}}/W$ with increasing $\mathrm{\Gamma }$. Notice its evolution differs from that of the single-particle gap as it deteriorates more slowly. We used $U=0.7W, T=0$.

Figure 4

Calculated DC conductivity ${\sigma }_{xx}$. Dots: ${\sigma }_{xx}$ as a function of temperature for different strengths of the strain fluctuations, $\mathrm{\Gamma }/W\in [0.02,0.1]$, increasing from bottom (red) to top (blue) in steps of 0.02. We used the same parameters as in Fig. 2. Solid lines: Guides to the eye $\propto exp[-{\omega }_g(T\to 0)/T]$ for each $\mathrm{\Gamma }$ value. Whereas ${\omega }_g$ differs by a factor of $\sim 5$ between the first and last plots, ${\mathrm{\Delta }}_{\mathrm{ivc}}$ changes only modestly by $<25\%$.