Hofstadter-Moir\'{e} Butterfly in Twisted Trilayer Graphene

TL;DR

This paper explores the Hofstadter-Moiré fractal bands in mirror-symmetric twisted trilayer graphene, revealing a novel quantum parity Hall state and topological phase transitions influenced by displacement fields.

Contribution

It introduces the study of HM fractal bands in tTLG, identifying a new quantum parity Hall state and analyzing the effects of displacement fields on topological phases.

Findings

Discovery of a quantum parity Hall state in mirror-symmetric tTLG.

Emergence of a zero-energy state in the middle layer under displacement field.

Topological phase transitions driven by electric displacement fields.

Abstract

Mirror symmetric twisted trilayer graphene (tTLG) is composed of even parity twisted bilayer graphene (tBLG)-like bands and odd parity Dirac-like bands. Here, we study the mirror-symmetric and mirror-asymmetric Hofstadter-Moir\'{e} (HM) fractal bands of tTLG. A novel quantum parity Hall state is identified in mirror-symmetric tTLG at experimentally accessible charge densities. This mirror symmetry-protected topological phase exhibits simultaneous quantized Hall and longitudinal resistances. The effects of the displacement field on the HM fractal bands of tTLG and topological phase transitions are also studied. The application of an electric displacement field results in an emergent weakly dispersive band at the charge neutrality point for a range of twist angles. This zero-energy state resides in the middle layer. It is isolated from the HM spectrum by an energy gap that scales proportional to the applied displacement field, making it a prime candidate to host correlated topological states.