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

Abstract

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 $\Delta_{\mathrm{split}}\approx 171$~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 $h/(2e)$ at each zero of the effective mass $m_{\mathrm{eff}}(\mathbf{r})=w|f(\mathbf{r})|$. In the chiral limit, the interlayer coupling is locally diagonalised by a spatially dependent unitary transformation; the ordering term $H_{\mathrm{ord}}=-\tfrac{i}{2}\boldsymbol{\sigma}\cdot \nabla\ln m_{\mathrm{eff}}$ then develops a $1/r$ singularity at the AB/BA stacking points, where $m_{\mathrm{eff}}$ vanishes. The splitting scales as $\sqrt{\theta}$ -- distinguishing it from correlation-driven gaps ($\propto\theta$ or $\propto 1/\theta$) -- 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 $\mathrm{eff}$, the ordering-corrected zero mode acquires parabolic-cylinder character $D_{-1/2}$. 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.