Abstract

Accurate geometric decoding of moiré bilayers from imaging is essential for engineering quantum systems. Existing schemes, limited by identity or aligned assumptions requiring diagonal beating-to-moiré transformations, do not apply to general non-aligned geometries and become underdetermined when buried layers are unresolved. We establish a primitive-cell-resolved moiré crystallography framework that treats the beating-to-moiré relation in full generality and introduces a complete descriptor set {θr, ε, (TMt, TMb), NB}, where the integer moiré–layer matrices (TMt, TMb) and the beating number NB determine the commensurate unit cell. A hybrid analytical–numerical workflow reconstructs buried-layer lattices, solves Diophantine constraints to obtain (TMt, TMb) and NB, and extracts (θr, εb, θu, εu) with Poisson effects and tensile/compressive branches treated on equal footing. Reanalyzing twisted bilayer graphene, we identify a NB = 3 primitive cell rather than a NB = 9 aligned supercell, reducing the atomistic basis threefold and correcting the moiré Brillouin-zone construction. The framework provides a crystallographically consistent route from imaging to primitive-cell-resolved atomistic and many-body models.

Abstract

Accurate geometric decoding of moiré bilayers from imaging is essential for engineering quantum systems. Existing schemes, limited by identity or aligned assumptions requiring diagonal beating-to-moiré transformations, do not apply to general non-aligned geometries and become underdetermined when buried layers are unresolved. We establish a primitive-cell-resolved moiré crystallography framework that treats the beating-to-moiré relation in full generality and introduces a complete descriptor set {θr, ε, (TMt, TMb), NB}, where the integer moiré–layer matrices (TMt, TMb) and the beating number NB determine the commensurate unit cell. A hybrid analytical–numerical workflow reconstructs buried-layer lattices, solves Diophantine constraints to obtain (TMt, TMb) and NB, and extracts (θr, εb, θu, εu) with Poisson effects and tensile/compressive branches treated on equal footing. Reanalyzing twisted bilayer graphene, we identify a NB = 3 primitive cell rather than a NB = 9 aligned supercell, reducing the atomistic basis threefold and correcting the moiré Brillouin-zone construction. The framework provides a crystallographically consistent route from imaging to primitive-cell-resolved atomistic and many-body models.

Abstract

Accurate geometric decoding of moiré bilayers from imaging is essential for engineering quantum systems. Existing schemes, limited by identity or aligned assumptions requiring diagonal beating-to-moiré transformations, do not apply to general non-aligned geometries and become underdetermined when buried layers are unresolved. We establish a primitive-cell-resolved moiré crystallography framework that treats the beating-to-moiré relation in full generality and introduces a complete descriptor set {θr, ε, (TMt, TMb), NB}, where the integer moiré–layer matrices (TMt, TMb) and the beating number NB determine the commensurate unit cell. A hybrid analytical–numerical workflow reconstructs buried-layer lattices, solves Diophantine constraints to obtain (TMt, TMb) and NB, and extracts (θr, εb, θu, εu) with Poisson effects and tensile/compressive branches treated on equal footing. Reanalyzing twisted bilayer graphene, we identify a NB = 3 primitive cell rather than a NB = 9 aligned supercell, reducing the atomistic basis threefold and correcting the moiré Brillouin-zone construction. The framework provides a crystallographically consistent route from imaging to primitive-cell-resolved atomistic and many-body models.

Abstract

Accurate geometric decoding of moiré bilayers from imaging is essential for engineering quantum systems. Existing schemes, limited by identity or aligned assumptions requiring diagonal beating-to-moiré transformations, do not apply to general non-aligned geometries and become underdetermined when buried layers are unresolved. We establish a primitive-cell-resolved moiré crystallography framework that treats the beating-to-moiré relation in full generality and introduces a complete descriptor set {θr, ε, (TMt, TMb), NB}, where the integer moiré–layer matrices (TMt, TMb) and the beating number NB determine the commensurate unit cell. A hybrid analytical–numerical workflow reconstructs buried-layer lattices, solves Diophantine constraints to obtain (TMt, TMb) and NB, and extracts (θr, εb, θu, εu) with Poisson effects and tensile/compressive branches treated on equal footing. Reanalyzing twisted bilayer graphene, we identify a NB = 3 primitive cell rather than a NB = 9 aligned supercell, reducing the atomistic basis threefold and correcting the moiré Brillouin-zone construction. The framework provides a crystallographically consistent route from imaging to primitive-cell-resolved atomistic and many-body models.