Three-Dimensional Topological Twistronics

We introduce a theoretical framework for the new concept of three-dimensional (3D) twistronics by developing a generalized Bloch band theory for 3D layered systems with a constant twist angle $\theta$ between successive layers. Our theory employs a nonsymmorphic symmetry that enables a precise definition of an effective out-of-plane crystal momentum, and also captures the in-plane moir\'e pattern formed between neighboring twisted layers. To demonstrate the novel topological physics that can be achieved through 3D twistronics, we present two examples. In the first example of chiral twisted graphite, Weyl nodes arise because of inversion-symmetry breaking, with $\theta$-tuned transitions between type-I and type-II Weyl fermions, as well as magic angles at which the in-plane velocity vanishes. In the second example of twisted Weyl semimetal, the twist in the lattice structure induces a chiral gauge field $\boldsymbol{\mathcal{A}}$ that has a vortex-antivortex lattice configuration. Line modes bound to the vortex cores of the $\boldsymbol{\mathcal{A}}$ field give rise to 3D Weyl physics in the moir\'e scale. We also discuss possible experimental realizations of 3D twistronics.