Real-space geometric construction of the effective quantum field theory for moiré electronic systems

Abstract

We present a rigorous real-space approach to derive an effective continuum Hamiltonian starting from a microscopic lattice model for electronic systems in complex atomic lattices. Specifically, this formalism is applied to twisted bilayer graphene (TBG) at small twist angles, serving as a prototypical moiré electronic system. Unlike standard reciprocal-space approaches, our method treats the transition from discrete lattice to continuum as a systematic reduction of the Hilbert space via the Rayleigh–Ritz projection. By utilizing the geometric language of pullback operators and spatial diffeomorphisms, we establish a covariant framework to track the transformation of Dirac fermion fields onto the moiré manifold. This construction provides a transparent kinematic picture where microscopic degrees of freedom are reorganized into emergent collective states, allowing for the direct incorporation of lattice deformations and rotations into the kinetic sector. We demonstrate that the point group symmetries of the moiré superlattice—specifically C3z and the anti-unitary C2″T symmetry—uniquely constrain the interlayer tunneling matrix. The robustness of this formalism is validated by its ability to naturally recover the interlayer tunneling structure of the Bistritzer–MacDonald (BM) model as a consistent limiting case. This systematic derivation not only provides a formal justification for the continuum approximation but also establishes a robust theoretical platform for incorporating non-ideal effects such as non-uniform strain-induced gauge fields, offering a natural geometric interpretation of the non-commutative algebra inherent in moiré flat bands.