Filaments of critical states in quasi-periodic twisted bilayer graphene models

Twisted bilayer graphene (TBLG), hosting superconductivity and other quantum phases, is an ideal platform for studying the interplay of correlations, flat bands, and wave functions. The low-energy physics of TBLG can be described by the Bistritzer-MacDonald (B-M) model, wherein Dirac electrons tunnel between layers via a periodic matrix potential. Ignoring sub-leading AA interlayer tunneling, the chiral B-M model takes the same form as Dirac surface states of class CI topological superconductors. The latter can exhibit spectrum-wide quantum criticality (SWQC) in the presence of disorder [1]. Although TBLG is microscopically quasiperiodic at a generic twist angle, the B-M model elegantly evades this complication. Nevertheless, quasiperiodicity can still arise from the boron nitride substrate. Moreover, twisted trilayer graphene is intrinsically quasiperiodic when mirror symmetry is broken, and superconductivity was recently discovered in a quasiperiodic trilayer system [2]. Here, we study the low-energy effective theory incorporating two incommensurate B-M matrix potentials. We find “filaments” of critical states in the plane of the two potentials. We will discuss this in the context of disorder-induced SWQC and its implications on the correlated phases in TBLG and other moiré materials.

[1] S. A. A. Ghorashi, Y. Liao, M. S. Foster, Phys. Rev. Lett. 121, 016802 (2018).

[2] A. Uri et al. Nature 620, 762 (2023).