PHYSICAL REVIEW RESEARCH 2, 022010(R) (2020) Rapid Communications Three-dimensional topological twistronics Fengcheng Wu , Rui-Xing Zhang, and Sankar Das Sarma Condensed Matter Theory Center and Joint Quantum Institute, Department of Physics, University of Maryland, College Park, Maryland 20742, USA (Received 12 October 2019; accepted 24 March 2020; published 13 April 2020) We introduce a theoretical framework for the concept of three-dimensional (3D) twistronics by developing a generalized Bloch band theory for 3D layered systems with a constant twist angle θ 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 moiré pattern formed between neighboring twisted layers. To demonstrate 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 θ-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 a twisted Weyl semimetal, the twist in the lattice structure induces a chiral gauge field A that has a vortex-antivortex lattice configuration. Line modes bound to the vortex cores of the A field give rise to 3D Weyl physics in the moiré scale. We also discuss possible experimental realizations of 3D twistronics. DOI: 10.1103/PhysRevResearch.2.022010 Introduction. Moiré superlattices formed in twisted bi- breaking in the twisted structure. Both type-I and type-II [15] layers lead to interesting two-dimensional (2D) phenomena. Weyl fermions can be realized depending on the value of θ. In twisted bilayer graphene (TBG), there are magic twist Moreover, we find two magic angles at which the in-plane angles, at which moiré bands become nearly flat due to a Fermi velocity of the Weyl fermions vanishes, representing vanishing Dirac velocity [1] and many-body interactions are the realization of magic-angle Weyl physics. In the second effectively enhanced. TBG represents a prototypical system system of a twisted Weyl semimetal, we study the effects for 2D twistronics [2], where the twist angle serves as a tuning of chiral twist in the lattice structure on Weyl fermions that parameter in controlling material properties. Given the greatly already exist even without the twist. The chiral twist induces exciting 2D physics developing in TBG such as the discovery a chiral gauge field A that has a vortex-antivortex lattice of superconducting and correlated insulating states [3–14], it configuration in the moiré pattern formed between adjacent is natural to wonder whether the concept of twistronics can be twisted layers. The vortex cores of the A field bind line generalized to 3D systems. modes with position-dependent chiralities, which generalizes In this Rapid Communication, we present a theoretical the quasi-1D physics of a Weyl nanotube under torsion [16]to framework for 3D twistronics that can be realized in 3D 3D. The periodic array of the coupled vortex line modes gives layered systems with a constant twist angle θ between suc- rise to 3D Weyl fermions with moiré-scale modulations in the cessive layers. This 3D chiral twisted structure [Fig. 1(a)] wave function. Therefore, the twist angle provides another generally breaks the translational symmetry in all spatial tuning knob to create and manipulate Weyl fermions, and, directions and thus the conventional Bloch theorem cannot be more generally, topological phases in 3D. applied. However, the structure has an exact nonsymmorphic Theory. We construct a generalized Bloch band theory for symmetry, which consists of an in-plane θ rotation followed the chiral twisted structure shown in Fig. 1(a). The continuum by an out-of-plane translation. We use this screw rotational Hamiltonian for this system is symmetry to define a generalized Bloch’s theorem, where the = 2 {ψ† ψ modified crystal momenta are well defined. Various 3D moiré H d r n (r)hn(k) n(r) physics can be explored by considering different 2D building n blocks in our theoretical framework. † + [ψ (r)T (r)ψ + (r) + H.c.]}, (1) We apply our theory to two systems. In the first system n n n 1 of chiral twisted graphite with graphene as the 2D building where n is the layer index, r and k =−i∂r are respectively ψ† block, Weyl fermions arise due to the inversion-symmetry the 2D in-plane position and momentum operators, n (r) represents the field operator for low-energy states, hn(k)is the in-plane Hamiltonian for each 2D building block, and † Tn(r) is the interlayer tunneling. Here, ψ can be a multicom- Published by the American Physical Society under the terms of the ponent spinor due to sublattices, orbitals, spins, etc. The layer Creative Commons Attribution 4.0 International license. Further dependence of hn and Tn is determined by the twist relation distribution of this work must maintain attribution to the author(s) ˆ θ = , ˆ θ = , and the published article’s title, journal citation, and DOI. hn[R(n )k] h0(k) Tn[R(n )r] T0(r) (2) 2643-1564/2020/2(2)/022010(5) 022010-1 Published by the American Physical Society WU, ZHANG, AND DAS SARMA PHYSICAL REVIEW RESEARCH 2, 022010(R) (2020) (a) (b) which gives rise to energy bands in the 3D momentum space Type-II Weyl Fermion Type-I Weyl Fermion Type-II Weyl Fermion spanned by kz and the in-plane moiré Brillouin zone. Equa- tion (5) is our effective Hamiltonian for the 3D small-angle twisted system, which builds in exactly the nonsymmorphic symmetry and captures the moiré pattern formed in neighbor- ing twisted layers. 2nd Magic Angle 1st Magic Angle Chiral twisted graphite. We apply our theory to study the electronic structure of chiral twisted graphite, which we con- struct by starting from an infinite number of graphene layers with AAA ... stacking, and then rotating the nth layer by nθ around a common hexagon center. In each layer, low-energy FIG. 1. (a) Illustration of a 3D twisted structure with a constant electrons reside in ±K valleys, which are related by spinless twist angle θ between successive layers. (b) Summary of results time-reversal symmetry Tˆ and can be studied separately as on magic-angle Weyl fermions in chiral twisted graphite. The plot in TBG. We focus on the +K valley, with the in-plane k · p shows the in-plane velocity v and one of the out-of-plane velocities Hamiltonian h(k) = h¯v k · σ, where v is the monolayer = , ,π/ F F v1 for the Weyl fermion at k1/2 (0 0 2). v vanishes at magic graphene Dirac velocity (∼106 m/s) and σ is the sublattice θ θ θ θ angles M,1 and M,2. v1 changes sign at C,1 and C,2, which mark Pauli matrix. The interlayer tunneling T (r)is[1,17] transitions between type-I and type-II Weyl fermions. −i2π j/3 wAA wABe · T (r) = eigj+1 r, (6) ˆ i2π j/3 where R is a rotation matrix. T0(r) has an in-plane moiré j=0,1,2 wABe wAA periodicity (∝1/θ) when θ is small. The 3D twisted structure generally breaks translational where wAA and wAB are respectively the intrasublattice and ≈ symmetry in all spatial directions, making it appear hopeless intersublattice tunneling parameters, with wAA 90 meV and ≈ for theoretical treatments. However, Eq. (2) implies that the wAB 117 meV. g1 is a moiré reciprocal lattice vector , π/ = /θ Hamiltonian H is invariant under a nonsymmorphic operation, (0 4 3aM ), and aM a0 , where a0 is the monolayer which rotates a layer by θ and then translates it along the graphene lattice constant. The other two vectors g2,3 are related to g by ±2π/3 rotations. We note that aM is the out-of-planez ˆ direction by the interlayer distance dz.This 1 nonsymmorphic symmetry suggests a generalized Bloch wave TBG moiré√ periodicity, but T (r)inEq.(6) has a periodicity ofa ˜M = 3aM . for the system, − † = ikz + ikz The kz-dependent moiré potential e T e T can 1 − σ + σ + σ ψ (r) = √ e inkz ψ [Rˆ(nθ )r], (3) be decomposed into 0 0 x x y y, where 0 is a kz n N n scalar potential. From , we can define an effective gauge field A = ( , )/(ev ) that couples to the Dirac Hamil- where N is the number of layers, and the good quantum x y F tonian h(k), and a corresponding pseudomagnetic field bz = number kz is an effective out-of-plane crystal momentum ∇ × A = / r . 2D maps of 0 and bz at kz 0 are plotted in Fig. 2, measured in units of 1 dz. This Bloch wave is a superposition which shows that |b | can reach ∼200 T for θ = 1.1◦. of electron states on a spiral line around the screw-rotation z The effective Hamiltonian H = h(k) + respects Cˆ axis, as illustrated by the purple lines in Fig. 1(a). Under this 3z and Cˆ Tˆ symmetries, where Cˆ is the n-fold rotation around generalized Bloch representation, the Hamiltonian H becomes 2z nz zˆ axis. We diagonalize H using a plane-wave expansion, and ◦ 2 † show the calculated band structures at θ = 1.1 in Fig. 2. = ψ ψ H d r k (r)h(k ) kz (r) z Bands along the kz axis can be characterized by the Cˆ3z angular kz ∈{ , ± } momentum z 0 1 . As shown in Fig. 2(d), crossings † ik + ψ (r)e z T (r)ψ [Rˆ(−θ )r] + H.c.