Three-Dimensional Topological Twistronics

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 neighboring 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 construct 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. , (4) between two bands with different z actually represent 3D kz kz Weyl nodes, which appear abundantly along the kz axis. For where we use h and T as shorthand notations respectively example, the Weyl fermion at the γ point (k = 0) has an effec- for h and T . It is worth noting that the appearance of ∗ ∗ 0 0 tive Hamiltonianh ¯(v k · σ + v k σ ), which is constrained Rˆ(−θ )inEq.(4) signals the breaking of in-plane translation F z z z by both Cˆ3z and Cˆ2zTˆ symmetries. The θ dependence of symmetries. To proceed, we expand T (r) by moiré harmonics, ∗ , ∗ ∗ = ig·r (vF vz ) is shown in Fig. 2(c). vF is reduced from the bare T (r) g Tge , where g is a moiré reciprocal lattice vector. ◦ ◦ value vF , but remains finite for θ from 0.7 to 2 . Remarkably, T (r) generates in-plane momentum scatterings specified by ∗ the sign of v oscillates with θ, and therefore the chirality of k = Rˆ(θ )k + g. For low-energy physics, |k| is generally of z the Weyl node at the γ point is twist angle dependent. the same order of magnitude as |g|, which is proportional to Another representative Weyl node is located at k / = θ. Thus, Rˆ(±θ ) can be approximated by an identity matrix 1 2 (0, 0,π/2). This Weyl node is pinned to zero energy in the small θ limit, and the error is on the order θ|g||g|. by a particle-hole-like symmetry H(−k,π − k , −r) = Under this approximation, H acquires a moiré translational z −H(k, k , r), and is described byh ¯[vk · σ + v q (σ + symmetry, z 1 z 0 σz )/2 + v2qz(σ0 − σz )/2)], where v1,2 are two independent ◦ ◦ 2 † parameters and q = k − π/2. For θ between 1 and 1.8 , ≈ ψ + , ψ , z z H d r k (r)[h(k ) (kz r)] kz (r) ◦ z v2 is always negative, but v1 changes sign at θC,1 ≈ 1.52 and kz ◦ θC,2 ≈ 1.22 , which are critical angles that mark the transi- ikz −ikz † (kz, r) = e T (r) + e T (r), (5) tions between type-I and type-II Weyl fermions [Fig. 1(b)].

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(a) (c) (d) 0.5 0.15

50 0.4

0.1 0.3 0 0.2 0.05 50 0.1

0 0 0.8 1.0 1.2 1.4 40 20 0 20 40

(b)20 (e) (f)

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FIG. 2. Results of chiral twisted graphite. (a) In-plane and (b) out-of-plane band structure. In (a), kz is 0. In (b), k is zero, the crossings between bands with different z represent Weyl nodes, the two purple dots highlight nodes located at kz = 0andkz = π/2, and the spectrum is periodic in kz with a period 2π/3 due to a translationlike symmetry H(k, kz + 2π/3, r + aM yˆ) = H(k, kz, r). (c) In-plane and out-of-plane velocities of the k = 0 Weyl node as a function of θ. (d) DOS per spin, valley, layer, and area for the twisted graphite (blue line), and corresponding DOS for monolayer graphene (black dashed line). (e) 2D maps of the scalar moiré potential 0 and (f) pseudomagnetic ﬁeld bz ◦ at kz = 0andθ = 1.1 .

θ ≈ 2 Moreover, the in-plane velocity v vanishes at both M,1 M0(cos kz − cos Qz ) − M1k with parameters M0,1 > 0 and . ◦ θ ≈ . ◦ 1 67 and M,2 1 09 , which can be identified as two magic 0 < Qz <π. Each site on the cubic lattice accommodates two 3 1 angles. Here, the value of θM,1 can also be estimated us- | = | = orbitals jz 2 and jz 2 , which are labelled by the angu- ing an analytical perturbation theory, agreeing quantitatively lar momentum jz and form the basis of the Pauli matrices σ. with our direct band-structure calculations; see Supplemental The Hamiltonian hW breaks time-reversal symmetry and hosts Material (SM) [18]. The low-energy density of states (DOS) two Weyl nodes with opposite chiralities located respectively per layer in the twisted graphite near the magic angles is at k± = (0, 0, ±Qz ). orders of magnitude larger than that in monolayer graphene We now consider a twisted cubic lattice in which the nth [Fig. 2(d)], which should enhance the interaction effects, layer is rotated by nθ around thez ˆ axis following Fig. 1(a). leading to interaction-driven quantum phase transitions. The effective Hamiltonian HW for this twisted structure is We now compare our results with related works. Ref- given by erence [19] studied multiple graphene layers with a twist − φ + · − nθ 0 e i( g g r) angle ( 1) that alternates with the layer index n. Their H = + , W hW 2tsp sin kz φ + · (7) 3D structure preserves inversion symmetry, in contrast with ei( g g r) 0 our chiral twisted graphite structure. Reference [20] studied g the same structure as ours but with a different method under where tsp characterizes the interorbital tunneling between the coherent phase approximation. Our theory employs the neighboring layers. The summation over g is restricted to the exact nonsymmorphic symmetry, which allows us to precisely first shell of moiré reciprocal lattice vectors (|g|=2π/aM define the kz momentum and clearly demonstrate magic-angle with the moiré period aM equal to a0/θ), and φg is the Weyl physics in the twisted graphite. In Ref. [21], twisted tri- orientation angle of g. Here, the interlayer tunneling matrix layer graphene has been theoretically studied. An interesting is derived using a two-center approximation, and the spatial question is how many layers are required in practice to realize modulation of interlayer intraorbital tunneling is neglected, our predicted 3D physics, which we leave for future study. which are thoroughly explained in the SM [18]. Compared Twisted Weyl semimetal. As another demonstration, we with the twisted graphite Hamiltonian, HW does not have a apply our theory to a twisted Weyl semimetal with a lattice scalar moiré potential term. structure also shown in Fig. 1(a). We start by introducing a We can extract an effective gauge field A from HW ,similar minimal Weyl semimetal model on a simple cubic lattice (lat- to the case of twisted graphite. The A field has a vortex- tice constant a0), hW = h¯vW k · σ + M(k)σz, where M(k) = antivortex lattice configuration in the moiré superlattices [18].

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(a) (d)

50 50

0 0

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(b) (c) (e)

0.6

0.4

0.2

0 0.2 0.4 0.6 0.8 1.0 1.22

FIG. 3. Results of twisted Weyl semimetal. (a) Out-of-plane band structure in the twisted Weyl semimetal. In-plane momentum is zero. Red and blue lines respectively mark right- and left-moving modes that cross Weyl nodes located at kz =±π/2. jz speciﬁes the angular momentum 2 5 of each mode. Model parameters are M0 = 869 meV, M1 = 10.36 eV Å , tsp = 4meV,vW = 3.74 × 10 m/s, a0 = 7.5Å,andQz = π/2. (b) 2D maps of the pseudomagnetic ﬁeld bz (kz = π/2), which is positive and negative, respectively, around integer R and half-integer R1/2 θ = ◦ ∗ θ ∗ positions. 1 in (a) and (b). (c) The in-plane velocity vW of the Weyl node as a function of . The out-of-plane velocity vz (not shown) barely changes with θ. (d) Real-space probability density of the right-moving modes in (a). The peaks of the density are at R positions in each 2D layer, and wind around the screw rotation axis. (e) Similar as (d) but for the left-moving modes in (a), and the peaks are located at R1/2 positions.

The corresponding pseudomagnetic field bz is given by To verify the above picture, we numerically diagonalize HW , and show the energy spectrum along kz in Fig. 3(a), π where we find that the Weyl nodes are robust against the twist. 2tsp 2 ig·r bz = sin kz e , (8) In Fig. 3(d) [Fig. 3(e)], we plot the real-space probability den- ev a W M g sity of the right- (left-) moving modes highlighted in Fig. 3(a), which is found to be primarily concentrated at R (R1/2) which is illustrated in Fig. 3(b). |bz| peaks at two distinct positions. This density profile is consistent with the above types of positions in the moiré pattern, namely, integer posi- semiclassical analysis. Because of the twisted structure, these tions R = (nx, ny )aM with nx,y ∈ Z and half-integer positions modes track spiral lines in real space. From the semiclassical 1 1 ∗ R / = R + , picture, the in-plane velocity v of the 3D Weyl fermions 1 2 ( 2 2 )aM , which are also locations of vortex W cores of the A field. We assume tsp/vW > 0, and bz is positive is controlled by the coupling strength between neighboring and negative respectively at R and R1/2 for 0 < kz <π. VLMs, and therefore by the moiré period aM . Numerical ∗ Being proportional to sin kz, bz represents a chiral magnetic results plotted in Fig. 3(c) confirm that vW decreases with field as it couples oppositely to the Weyl nodes with different decreasing θ (equivalently, increasing aM ). For small enough R θ ∗ topological charges. In the semiclassical picture, around , vW nearly vanishes, showing that VLMs located at different (R1/2) positions, this chiral magnetic field leads to chiral positions are essentially decoupled. Thus, the twist angle pro- Landau levels that propagate along the positive (negative)z ˆ vides an alternative tuning knob to control the band structure direction for both Weyl nodes at k±. These chiral Landau of 3D Weyl materials. levels bound to vortex cores of the A field can be considered Conclusion. In summary, we develop a general theoretical to be vortex line modes (VLMs). The chirality of a VLM framework for 3D twistronics and apply the theory to chiral depends on its position in the moiré pattern. An important twisted graphite and twisted Weyl semimetal. Our theory can physical consequence is that an out-of-plane electric field can in principle be realized in van der Waals heterostructures drive a real-space pumping of electrons from R to R1/2 constructed by stacking multiple twisted layers. Moreover, positions, or vice versa. The 2D array of VLMs can further chiral twisted van der Waals nanowires have recently been hybridize with each other, and we expect them to realize Weyl experimentally synthesized [22,23], indicating that the chi- fermions with moiré-scale modulations in the wave functions. ral twisted structure we study can indeed appear in materi-

022010-4 THREE-DIMENSIONAL TOPOLOGICAL TWISTRONICS PHYSICAL REVIEW RESEARCH 2, 022010(R) (2020) als. Similar to 2D twisted bilayers, 3D twisted systems can 3D topological twistronics. Our predictions can be tested in provide playgrounds for strongly correlated physics. As an experimental systems as disparate as suitably designed pho- example, superconducting instability should be enhanced in tonic structures and engineered solid state heterostructures as 3D twisted graphite near the magic angles due to the strong well as in layered systems made from graphite and manually velocity suppression. In addition to solid state materials, our stacked multilayer graphene. theory should be realizable in photonic and phononic systems, Acknowledgments. We thank Yang-Zhi Chou for discus- where 2D Dirac physics and 3D Weyl physics have been sions. This work is supported by the Laboratory for Physical demonstrated [24–32]. The ﬂexibility of building metama- Sciences and Microsoft. R.-X.Z. is supported by a JQI Post- terials makes them promising platforms for our proposed doctoral Fellowship.

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