Operator ordering as an emergent gauge field in twisted bilayer graphene: singular spectral signatures at the magic angle

Scanning-tunnelling spectroscopy at the AB and BA stacking points of magic-angle twisted bilayer graphene should reveal two asymmetric Van~Hove peaks separated by ${\mathrm{\Delta }}_{\mathrm{s}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{t}}\approx 171$~meV – a splitting absent from the standard Bistritzer–MacDonald spectrum. We show this signature arises naturally from the Hermitian ordering correction of the Dirac Hamiltonian with spatially varying mass, which generates an emergent Aharonov–Bohm flux of $h\mathrm{/}(2e)$ at each zero of the effective mass $m_{\mathrm{e}\mathrm{f}\mathrm{f}}(\mathbf{r})=w\mathrm{\mid }f(\mathbf{r})\mathrm{\mid }$. In the chiral limit, the interlayer coupling is locally diagonalised by a spatially dependent unitary transformation; the ordering term $H_{\mathrm{o}\mathrm{r}\mathrm{d}}=-\frac{i}{2}\mathit{\sigma }\cdot \mathrm{\nabla }\ln m_{\mathrm{e}\mathrm{f}\mathrm{f}}$ then develops a $1\mathrm{/}r$ singularity at the AB/BA stacking points, where $m_{\mathrm{e}\mathrm{f}\mathrm{f}}$ vanishes. The splitting scales as $\sqrt{\theta }$ – distinguishing it from correlation-driven gaps ($\propto \theta$ or $\propto 1\mathrm{/}\theta$) – is gate-voltage independent, and is spatially localised within $r_c\approx 2.1$~nm of each AB/BA point. Within the local asymptotic theory near the zeros of $\mathrm{e}\mathrm{f}\mathrm{f}$, the ordering-corrected zero mode acquires parabolic-cylinder character $D_{-1\mathrm{/}2}$. The spatially resolved AB/BA spectrum reported in recent STM studies of magic-angle TBG has not been analysed for two-peak structure and constitutes an immediate experimental test; the predicted cell-averaged broadening of $\approx 14$~meV is consistent with the $16$~meV discrepancy between existing STM data and the BM tight-binding prediction.