Three-Dimensional Topological Twistronics
TThree-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
We introduce a theoretical framework for the new concept of three-dimensional (3D) twistronicsby 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 precisedefinition of an effective out-of-plane crystal momentum, and also captures the in-plane moir´e pat-tern formed between neighboring twisted layers. To demonstrate the novel topological physics thatcan be achieved through 3D twistronics, we present two examples. In the first example of chiraltwisted graphite, Weyl nodes arise because of inversion-symmetry breaking, with θ -tuned transitionsbetween type-I and type-II Weyl fermions, as well as magic angles at which the in-plane velocityvanishes. In the second example of twisted Weyl semimetal, the twist in the lattice structure inducesa chiral gauge field A that has a vortex-antivortex lattice configuration. Line modes bound to thevortex cores of the A field give rise to 3D Weyl physics in the moir´e scale. We also discuss possibleexperimental realizations of 3D twistronics. Introduction.—
Moir´e superlattices formed in twistedbilayers lead to novel two-dimensional (2D) phenomena.In twisted bilayer graphene (TBG), there are magic twistangles, at which moir´e bands become nearly flat due tovanishing Dirac velocity [1] and many-body interactionsare effectively enhanced. TBG represents a prototypicalsystem for 2D twistronics [2], where the twist angle servesas a new tuning parameter. Given the greatly exciting2D physics developing in TBG such as the discovery ofsuperconducting and correlated insulating states [3–14],it is natural to wonder whether the concept of twistronicscan be generalized to 3D systems.In this Letter, we present a theoretical framework for3D twistronics that can be realized in 3D layered sys-tems with a constant twist angle θ between successivelayers. This 3D chiral twisted structure [Fig. 1(a)] gener-ally breaks the translational symmetry in all spatial di-rections and thus the conventional Bloch theorem cannotbe applied. However, the structure has an exact non-symmorphic symmetry, which consists of an in-plane θ rotation followed by an out-of-plane translation. We usethis screw rotational symmetry to define a generalizedBloch’s theorem, where the modified crystal momentaare well defined. Various 3D moir´e physics can be ex-plored by considering different 2D building blocks in ourtheoretical framework.We apply our theory to two systems. In the first sys-tem of chiral twisted graphite with graphene as the 2Dbuilding block, Weyl fermions arise due to the inversion-symmetry breaking in the twisted structure. Both type-Iand type-II [15] Weyl fermions can be realized depend-ing on the value of θ . Moreover, we find two magic an-gles at which the in-plane Fermi velocity of the Weylfermions vanishes, representing the realization of magic-angle Weyl physics for the first time. In the second sys-tem of twisted Weyl semimetal, we study effects of chi-ral twist in the lattice structure on Weyl fermions thatalready exist even without the twist. The chiral twist in-duces a chiral gauge field A that has a vortex-antivortexlattice configuration in the moir´e pattern formed be- tween adjacent twisted layers. The vortex cores of the A field bind line modes with position dependent chiral-ities, which generalizes the quasi-1D physics of a Weylnanotube under torsion [16] to 3D. The periodic arrayof the coupled vortex line modes gives rise to 3D Weylfermions with moir´e-scale modulations in the wave func-tion. Therefore, the twist angle provides a new tuningknob to create and manipulate Weyl fermions, and, moregenerally, topological phases in 3D. Theory.—
We construct a generalized Bloch band the-ory for the chiral twisted structure shown in Fig. 1(a).The continuum Hamiltonian for this system is H = (cid:88) n (cid:90) d r (cid:110) ψ † n ( r ) h n ( k (cid:107) ) ψ n ( r )+ [ ψ † n ( r ) T n ( r ) ψ n +1 ( r ) + H.c.] (cid:111) , (1)where n is the layer index, r and k (cid:107) = − i∂ r are respec-tively the 2D in-plane position and momentum operators, ψ † n ( r ) represents the field operator for low-energy states, h n ( k (cid:107) ) is the in-plane Hamiltonian for each 2D buildingblock, and T n ( r ) is the interlayer tunneling. Here ψ † canbe a multicomponent spinor due to sublattices, orbitals,spins, etc. The layer dependence of h n and T n is deter-mined by the twist relation: h n [ ˆ R ( nθ ) k (cid:107) ] = h ( k (cid:107) ) , T n [ ˆ R ( nθ ) r ] = T ( r ) , (2)where ˆ R is a rotation matrix. T ( r ) has an in-plane moir´eperiodicity ( ∝ /θ ) when θ is small.The 3D twisted structure generally breaks transla-tional symmetry in all spatial directions, making itappear hopeless for theoretical treatments. However,Eq. (2) implies that the Hamiltonian H is invariant undera nonsymmorphic operation, which rotates a layer by θ and then translates it along the out-of-plane ˆ z directionby the interlayer distance d z . This nonsymmorphic sym-metry suggests a generalized Bloch wave for the system: ψ k z ( r ) = 1 √ N (cid:88) n e − ink z ψ n [ ˆ R ( nθ ) r ] , (3) a r X i v : . [ c ond - m a t . m e s - h a ll ] A p r (a) (b) 𝜃 𝑣 / 𝑣 𝐹 ∘ ∘ 𝜃 𝐶,2 |𝒗 | |𝒗 ∥ | Type-II Weyl Fermion Type-I Weyl Fermion 𝜃 𝐶,1 𝜃 𝑀,1 𝜃 𝑀,2
Type-II Weyl Fermion ∘ FIG. 1. (a)Illustration of a 3D twisted structure with a con-stant twist angle θ between successive layers. (b) Summaryof results on magic-angle Weyl fermions in chiral twistedgraphite. The plot shows the in-plane velocity v (cid:107) and oneof the out-of-plane velocities v for the Weyl fermion at k / = (0 , , π/ v (cid:107) vanishes at magic angles θ M, and θ M, . v changes sign at θ C, and θ C, , which mark transitions be-tween type-I and type-II Weyl fermions. where N is the number of layers, and the good quantumnumber k z is an effective out-of-plane crystal momentummeasured in units of 1 /d z . This Bloch wave is a super-position of electron states on a spiral line around thescrew-rotation axis, as illustrated by the purple lines inFig. 1(a). Under this generalized Bloch representation,the Hamiltonian H becomes H = (cid:88) k z (cid:90) d r (cid:110) ψ † k z ( r ) h ( k (cid:107) ) ψ k z ( r )+ (cid:104) ψ † k z ( r ) e ik z T ( r ) ψ k z [ ˆ R ( − θ ) r ] + H.c. (cid:105)(cid:111) , (4)where we use h and T as short-hand notations respec-tively for h and T . It is worth noting that the appear-ance of ˆ R ( − θ ) in Eq. (4) signals the breaking of in-planetranslation symmetries. To proceed, we expand T ( r ) bymoir´e harmonics: T ( r ) = (cid:80) g T g e i g · r , where g is a moir´ereciprocal lattice vector. T ( r ) generates in-plane mo-mentum scatterings specified by k (cid:48)(cid:107) = ˆ R ( θ ) k (cid:107) + g . Forlow-energy physics, | k (cid:107) | is generally of the same orderof magnitude as | g | , which is proportional to θ . Thus,ˆ R ( ± θ ) can be approximated by an identity matrix in thesmall θ limit, and the error is on the order θ | g | (cid:28) | g | .Under this approximation, H acquires a moir´e transla-tional symmetry: H ≈ (cid:88) k z (cid:90) d r ψ † k z ( r )[ h ( k (cid:107) ) + ∆( k z , r )] ψ k z ( r ) , ∆( k z , r ) = e ik z T ( r ) + e − ik z T † ( r ) , (5)which gives rise to energy bands in the 3D momentumspace spanned by k z and the in-plane moir´e Brillouinzone. Eq. (5) is our effective Hamiltonian for the 3Dsmall-angle twisted system, which builds in exactly thenonsymmorphic symmetry and captures the moir´e pat-tern formed in neighboring twisted layers. Chiral twisted graphite.—
We apply our theory tostudy electronic structure of chiral twisted graphite, which we construct by starting from an infinite num-ber of graphene layers with
AAA... stacking, and thenrotating the n th layer by nθ around a common hexagoncenter. In each layer, low-energy electrons reside in ± K valleys, which are related by spinless time-reversal sym-metry ˆ T and can be studied separately as in TBG. Wefocus on + K valley, with the in-plane k · p Hamiltonian h ( k (cid:107) ) = (cid:126) v F k (cid:107) · σ , where v F is the monolayer grapheneDirac velocity ( ∼ m/s) and σ is the sublattice Paulimatrix. The interlayer tunneling T ( r ) is [1, 17] T ( r ) = (cid:88) j =0 , , (cid:18) w AA w AB e − i πj/ w AB e i πj/ w AA (cid:19) e i g j +1 · r (6)where w AA and w AB are respectively intra-sublattice andinter-sublattice tunneling parameters, with w AA ≈ w AB ≈
117 meV. g is a moir´e reciprocal lat-tice vector (0 , π/ a M ), and a M = a /θ , where a is themonolayer graphene lattice constant. The other two vec-tors g , are related to g by ± π/ a M is the TBG moir´e periodicity, but T ( r ) in Eq. (6)has a periodicity of ˜ a M = √ a M .The k z -dependent moir´e potential ∆ = e ik z T + e − ik z T † can be decomposed into ∆ σ + ∆ x σ x + ∆ y σ y , where ∆ is a scalar potential. From ∆, we can define an effec-tive gauge field A = (∆ x , ∆ y ) / ( ev F ) that couples to theDirac Hamiltonian h ( k (cid:107) ), and a corresponding pseudomagnetic field b z = ∇ r × A . 2D maps of ∆ and b z at k z = 0 are plotted in Fig. 2, which shows that | b z | canreach ∼
200 T for θ = 1 . ◦ .The effective Hamiltonian H = h ( k (cid:107) ) + ∆ respects ˆ C z and ˆ C z ˆ T symmetries, where ˆ C nz is the n -fold rotationaround ˆ z axis. We diagonalize H using a plane-waveexpansion, and show the calculated band structures at θ = 1 . ◦ in Fig. 2. Bands along k z axis can be char-acterized by the ˆ C z angular momentum (cid:96) z ∈ { , ± } .As shown in Fig. 2(b), crossings between two bands withdifferent (cid:96) z actually represent 3D Weyl nodes, which ap-pear abundantly along k z axis. For example, the Weylfermion at γ point ( k = ) has an effective Hamiltonian (cid:126) ( v ∗ F k (cid:107) · σ + v ∗ z k z σ z ), which is constrained by both ˆ C z and ˆ C z ˆ T symmetries. The θ dependence of ( v ∗ F , v ∗ z ) isshown in Fig. 2(c). v ∗ F is reduced from the bare value v F ,but remains finite for θ from 0 . ◦ to 2 ◦ . Remarkably, thesign of v ∗ z oscillates with θ , and therefore, the chirality ofthe Weyl node at γ point is twist angle dependent.Another representative Weyl node is located at k / =(0 , , π/ H ( − k (cid:107) , π − k z , − r ) = −H ( k (cid:107) , k z , r ), and is described by (cid:126) [ v (cid:107) k (cid:107) · σ + v q z ( σ + σ z ) / v q z ( σ − σ z ) / v , are two indepen-dent parameters and q z = k z − π/
2. For θ between 1 ◦ and 1 . ◦ , v is always negative, but v changes sign at θ C, ≈ . ◦ and θ C, ≈ . ◦ , which are critical an-gles that mark transitions between type-I and type-IIWeyl fermions [Fig. 1(b)]. Moreover, the in-plane veloc-ity v (cid:107) vanishes at both θ M, ≈ . ◦ and θ M, ≈ . ◦ ,which can be identified as two magic angles. Here the 𝑘 𝑧 𝜃 = 1.1 ∘ 𝐸 ( m e V ) (b)
40 20 0 20 4000.050.10.15 𝜃 = 1.1 ∘ 𝐸(meV) (d) D O S ( e V − n m − ) 𝜃 (c) 𝑘 𝑧 = 0 𝑣 / 𝑣 𝐹 𝑣 𝐹∗ 𝑣 𝑧∗ ' ''50050 𝜅 𝛾 𝜅 𝜅′ 𝜅′′𝜃 = 1.1 ∘ , 𝑘 𝑧 = 0 (a) 𝐸 ( m e V ) 𝑥/ 𝑎 𝑀 𝑦 / 𝑎 𝑀 𝑏 𝑧 (T) (f) 𝑥/ 𝑎 𝑀 𝑦 / 𝑎 𝑀 (e) Δ (eV) 𝜅′′ 𝜅′ 𝜅 𝛾 𝒈 𝒈 𝒈 +1−10ℓ 𝑧 FIG. 2. Results of chiral twisted graphite. (a) In-plane and (b) out-of-plane band structure. In (a) k z is 0. In (b), k (cid:107) is zero,the crossings between bands with different (cid:96) z represent Weyl nodes, the two purple dots highlight nodes located at k z = 0 and k z = π/
2, and the spectrum is periodic in k z with a period 2 π/ H ( k (cid:107) , k z +2 π/ , r + a M ˆ y ) = H ( k (cid:107) , k z , 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, layerand area for the twisted graphite (blue line), and corresponding DOS for monolayer graphene (black dashed line). (e) 2D mapsof the scalar moir´e potential ∆ and (f) pseudo magnetic field b z at k z = 0 and θ = 1 . ◦ . value of θ M, can also be estimated using an analyticalperturbation theory, agreeing quantitatively with our di-rect band structure calculations; see Supplemental Ma-terial (SM)[18]. The low-energy density of states (DOS)per layer in the twisted graphite near the magic anglesis orders of magnitude larger than that in monolayergraphene [Fig. 2(d)], which should enhance interactioneffects, leading to interaction driven quantum phase tran-sitions.We now compare our results with related works.Ref. 19 studied multiple graphene layers with twist angle( − n θ that alternates with the layer index n . Their 3Dstructure preserves inversion symmetry, in contrast withour chiral twisted graphite structure. Ref. 20 studied thesame structure as ours but with a different method underthe coherent phase approximation. Our theory employsthe exact nonsymmorphic symmetry, which allows us toprecisely define the k z momentum and clearly demon-strate magic-angle Weyl physics in the twisted graphite.In Ref. 21, twisted trilayer graphene has been theoreti-cally studied. An interesting question is how many lay-ers are required in practice to realize our predicted 3Dphysics, which we leave for future study. Twisted Weyl semimetal.—
As another demonstration,we apply our theory to twisted Weyl semimetal with lat-tice structure also shown in Fig. 1(a). We start by in-troducing a minimal Weyl semimetal model on a simplecubic lattice (lattice constant a ): h W = (cid:126) v W k (cid:107) · σ + M ( k ) σ z , where M ( k ) = M (cos k z − cos Q z ) − M k (cid:107) withparameters M , > < Q z < π . Each site on thecubic lattice accommodates two orbitals | j z = (cid:105) and | j z = (cid:105) , which have different angular momentum j z andform the basis of the Pauli matrices σ . The Hamilto-nian h W breaks time reversal symmetry and hosts twoWeyl nodes with opposite chiralities located respectivelyat k ± = (0 , , ± Q z ).We now consider a twisted cubic lattice in whichthe n th layer is rotated by nθ around ˆ z axis followingFig. 1(a). The effective Hamiltonian H W for this twistedstructure is given by H W = h W + 2 t sp sin k z (cid:48) (cid:88) g (cid:18) e − i ( φ g + g · r ) e i ( φ g + g · r ) (cid:19) (7)where t sp characterizes the inter-orbital tunneling be-tween neighboring layers. The summation over g is re-stricted to the first shell of moir´e reciprocal lattice vec-tors ( | g | = 2 π/a M with moir´e period a M equal to a /θ ),and φ g is the orientation angle of g . Here the inter-layer tunneling matrix is derived using a two-center ap-proximation, and the spatial modulation of interlayerintra-orbital tunneling is neglected, which are thoroughlyexplained in the SM [18]. Compared with the twistedgraphite Hamiltonian, H W does not have a scalar moir´epotential term. (e) 𝑦 / 𝑎 𝑀 𝑦 / 𝑎 𝑀 (d)(a)(b) 𝐸 ( m e V ) − − − 𝑘 𝑧 /𝜋 𝑗 𝑧 = 12𝑗 𝑧 = 32𝑗 𝑧 = 32𝑗 𝑧 = 12 𝜃 𝑣 𝑊 ∗ / 𝑣 𝑊 (c) ≈≈ 𝑧/𝑑 𝑧 𝑥/𝑎 𝑀 𝑦 / 𝑎 𝑀 𝑏 𝑧 (T) FIG. 3. Results of twisted Weyl semimetal. (a) Out-of-plane band structure in the twisted Weyl semimetal. In-plane momentumis zero. Red and blue lines respectively mark right and left moving modes that cross Weyl nodes located at k z = ± π/ j z specifies the angular momentum of each mode. Model parameters are M = 869 meV, M = 10 .
36 eV ˚A , t sp = 4 meV, v W = 3 . × m/s, a = 7 . Q z = π/
2. (b) 2D maps of the pseudo magnetic field b z ( k z = π/ R and half-integer R / positions. θ = 1 ◦ in (a) and (b). (c) The in-plane velocity v ∗ W ofthe Weyl node as a function of θ . The out-of-plane velocity v ∗ z (not shown) barely changes with θ . (d) Real-space probabilitydensity of the right moving modes in (a). The peaks of the density are at R positions in each 2D layer, and wind around thescrew rotation axis. (e) Similar as (d) but for the left moving modes in (a), and the peaks are located at R / positions. We can extract an effective gauge field A from H W ,similar to the case of twisted graphite. The A field hasa vortex-antivortex lattice configuration in the moir´e su-perlattices [18]. The corresponding pseudo magnetic field b z is given by b z = 2 t sp ev W πa M sin k z (cid:48) (cid:88) g e i g · r , (8)which is illustrated in Fig. 3(b). | b z | peaks at two distincttypes of positions in the moir´e pattern, namely, integerpositions R = ( n x , n y ) a M with n x,y ∈ Z and half-integerpositions R / = R + ( , ) a M , which are also locationsof vortex cores of the A field. We assume t sp /v W > b z is positive and negative respectively at R and R / for 0 < k z < π . Being proportional to sin k z , b z rep-resents a chiral magnetic field as it couples oppositely tothe Weyl nodes with different topological charges. In thesemiclassical picture, around R ( R / ) positions, thischiral magnetic field leads to chiral Landau levels thatpropagate along positive (negative) ˆ z direction for bothWeyl nodes at k ± . These chiral Landau levels bound tovortex cores of the A field can be considered to be vortexline modes (VLMs). The chirality of a VLM depends onits position in the moir´e pattern. An important physicalconsequence is that an out-of-plane electric field can drive a real-space pumping of electrons from R to R / posi-tions, or vice versa. The 2D array of VLMs can furtherhybridize with each other, and we expect them to realizeWeyl fermions with moir´e-scale modulations in the wavefunctions.To verify the above picture, we numerically diagonalize H W , and show the energy spectrum along k z in Fig. 3(a),where we find that the Weyl nodes are robust againstthe twist. In Fig. 3(d) [Fig. 3(e)], we plot the real-spaceprobability density of the right (left) moving modes high-lighted in Fig. 3(a), which is found to be primarily con-centrated at R ( R / ) positions. This density profile isconsistent with the above semiclassical analysis. Becauseof the twisted structure, these modes track spiral lines inreal space. From the semiclassical picture, the in-planevelocity v ∗ W of the 3D Weyl fermions is controlled by thecoupling strength between neighboring VLMs, and there-fore, by the moir´e period a M . Numerical results plottedin Fig. 3(c) confirm that v ∗ W decreases with decreasing θ (equivalently, increasing a M ). For small enough θ , v ∗ W nearly vanishes, showing that VLMs located at differentpositions are essentially decoupled. Thus, the twist angleprovides a new tuning knob to control the band structureof 3D Weyl materials. Conclusion.—
In summary, we develop a general theo-retical framework for 3D twistronics and apply the theoryto chiral twisted graphite and twisted Weyl semimetal.Our theory can in principle be realized in van der Waalsheterostructures constructed by stacking multiple twistedlayers. Moreover, chiral twisted van der Waals nanowireshave recently been experimentally synthesized[22, 23], in-dicating that the chiral twisted structure we study canindeed appear in materials. Similar to 2D twisted bi-layers, 3D twisted systems can provide new playgroundsfor strongly correlated physics. As an example, super-conducting instability should be enhanced in 3D twistedgraphite near the magic angles due to the strong veloc-ity suppression. In addition to solid state materials, ourtheory should be realizable in photonic and phononic sys- tems, where 2D Dirac physics and 3D Weyl physics havebeen demonstrated[24–32]. 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Drucker, andJ. E. Hoffman, arXiv:1909.13854 (2019).[33] C. Zheng, R. Hoffmann, and D. R. Nelson, J. Am. Chem.Soc. , 3784 (1990). Supplemental Material
This Supplemental Material includes four sections. Sections S1 and S2 provide additional discussion, respectively,on the generalized Bloch theory and on chiral twisted graphite. Section S3 is on a perturbation theory that providesa quantitative estimation of the first magic angle θ M, in chiral twisted graphite. Section S4 presents a derivation ofthe interlayer tunneling matrix in twisted Weyl semimetal. S1. GENERALIZED BLOCH THEORY
In this section, we provide additional discussions on the generalized Bloch theory.We note that the 1D version of our generalized Bloch that employs the nonsymmorphic symmetry has been con-sidered previously in the literature, for example, Ref. 33.For the chiral twisted structure in Fig. 1 of the main text, we assume open boundary condition along ˆ z directionand an infinite number of layers. Under this assumption, the nonsymmorphic symmetry is exact, since each layer hasa successive partner. The effective out-of-plane momentum k z is a good quantum number due to this nonsymmorphicsymmetry and takes continuous values between − π and + π . With this open boundary condition, the twist angle θ between adjacent layers does NOT have to be rational in the form of 2 πp/q , where p and q are integers.If periodic boundary condition is imposed along ˆ z direction, the nonsymmorphic symmetry is exact only if θ takesrational values of 2 πp/q and N θ is an integer multiple of 2 π . In this case, N , the number of layers, can be finite,and k z takes discrete values between − π and + π . Irrational θ/ (2 π ) can be approximated by a rational number whenperiodic boundary condition is used.In the derivation that leads to Eq. (5) of the main text, we approximate the rotation matrix ˆ R ( θ ) by an identitymatrix. Overall, this approximation is to neglect effects on the order of O ( θ ). In real space, this is equivalent toneglect effects of “moir´e of moir´e” that appear on the length scale of a M /θ . We expect our theory to be accurate forsystem size below a M /θ . For very small twist angles (e.g. 0 . ◦ ), a M /θ is much greater than a M , which indicates thatour theory should have a wide application range regarding the system size.In the main text, we only keep interlayer tunneling terms between adjacent layers. Tunnelings between layers thatare further apart also respect the nonsymmorphic symmetry and can be treated using our theoretical frameworkbut with some modifications. For example, the twist angle between next-nearest-neighbor layers is 2 θ , and thecorresponding tunneling terms have a periodicity of 2 a M instead of a M , which can lead to further Brillouin zonefolding effect in the momentum space. We expect that the topological physics that we study remains robust for weakremote out-of-plane tunneling effects. S2. HAMILTONIAN AND SYMMETRY OF CHIRAL TWISTED GRAPHITE
We present a derivation of the Hamiltonian of chiral twisted graphite by starting from that of twisted bilayergraphene (TBG), and discuss the symmetries of the Hamiltonian and their implications on Weyl points.The moir´e Hamiltonian for TBG in + K valley is given by˜ H TBG = (cid:18) ˜ h ( k (cid:107) ) ˜ T ( r )˜ T † ( r ) ˜ h ( k (cid:107) ) (cid:19) , (S1)where ˜ h ( k (cid:107) ) and ˜ h ( k (cid:107) ) are Dirac Hamiltonians associated respectively with the bottom and top layers, ˜ h ( k (cid:107) ) = (cid:126) v F ( k (cid:107) − κ + ) · σ and ˜ h ( k (cid:107) ) = (cid:126) v F e iθσ z [( k (cid:107) − κ − ) · σ ]. Here κ ± , equal to [4 π/ (3 a M )]( −√ / , ∓ / a M is a /θ , where a is the lattice constant of mononlayergraphene. The interlayer tunneling is given by˜ T ( r ) = T + T e − i b + · r + T e − i b − · r ,T j = (cid:18) w AA w AB e − i π ( j − / w AB e i π ( j − / w AA (cid:19) , (S2)where b ± represent TBG reciprocal lattice vectors given by [4 π/ ( √ a M )]( ± / , √ / T j with j = 1 , H TBG specified by Eqs. (S1) and (S2) has an in-plane periodicity of a M .We now apply a layer dependent gauge transformation to ˜ H TBG in order to move the Dirac points at momenta κ ± to the moir´e Brillouin zone center: H TBG = (cid:18) e − i κ + · r e − i κ − · r (cid:19) ˜ H TBG (cid:18) e + i κ + · r e + i κ − · r (cid:19) = (cid:18) h ( k (cid:107) ) T ( r ) T † ( r ) h ( k (cid:107) ) (cid:19) ,h ( k (cid:107) ) = (cid:126) v F k (cid:107) · σ , h ( k (cid:107) ) = (cid:126) v F e iθσ z ( k (cid:107) · σ ) ,T ( r ) = e − i κ + · r ˜ T ( r ) e + i κ − · r = (cid:88) j =1 , , T j e i g j · r , (S3)where g = (0 , π/ a M ), and g , are related to g by ± π/ h [ ˆ R ( θ ) k (cid:107) ] = h ( k (cid:107) ). We note that | g | = | b + | / √
3, and therefore, the new Hamiltonian H TBG has an in-plane periodicity of ˜ a M = √ a M .The new Hamiltonian H TBG serves as a 2D building block for chiral twisted graphite, of which the effectiveHamiltonian H is given by H ( k (cid:107) , k z , r ) = h ( k (cid:107) ) + ∆( k z , r ) ,h ( k (cid:107) ) = (cid:126) v F k (cid:107) · σ , ∆( k z , r ) = e ik z T ( r ) + e − ik z T † ( r ) , (S4)where T ( r ) is specified above in Eq. (S3).We list symmetries of H . By construction, H has an in-plane periodicity of ˜ a M = √ a M , and respects both ˆ C z and ˆ C z ˆ T symmetries. In addition, H has two emergent symmetries described as follows. (1) A translation likesymmetry H ( k (cid:107) , k z + 2 π/ , r + a M ˆ y ) = H ( k (cid:107) , k z , r ). Due to this symmetry, the energy spectra at k z and k z + 2 π/ a M compared to ˜ a M .(2) A particle-hole like symmetry H ( − k (cid:107) , π − k z , − r ) = −H ( k (cid:107) , k z , r ).The above symmetries can have important implications on Weyl points. (1) The Weyl points at k = (0 , ,
0) and k = (0 , , π ) are symmetry enforced due to the ˆ C z and ˆ C z ˆ T symmetries. (2) The particle-hole like symmetry pinsthe Weyl point at k / = (0 , , π/
2) to zero energy. It is important to note that while some properties (e.g., energyand/or momentum) of the Weyl nodes are influenced by these symmetries, the very existence of the Weyl nodes arenot.
S3. PERTURBATION THEORY FOR THE MAGIC ANGLE IN CHIRAL TWISTED GRAPHITE
We present an analytical perturbation theory for the Weyl node at k / = (0 , , π/
2) by truncating the correspondingplane-wave expansion of H to the first momentum shell: H ≈ h ( ) iT iT iT − iT − iT − iT − iT h ( − g ) O O O iT iT − iT O h ( − g ) O iT O iT − iT O O h ( − g ) iT iT OiT O − iT − iT h ( g ) O OiT − iT O − iT O h ( g ) OiT − iT − iT O O O h ( g ) , (S5)which are expanded using the 7 in-plane momenta including and ± g , , . In Eq. (S5), k z has been set to π/
2, and O represents a 2 × | Φ A (cid:105) = 1 N A { (1 , , ψ , ψ , ψ , ψ , ψ , ψ } T , | Φ B (cid:105) = 1 N B { (0 , , ϕ , ϕ , ϕ , ϕ , ϕ , ϕ } T , (S6)where N A,B are normalization factors. The two-component spinors ψ j and ϕ j are found to be: ψ j = (cid:16) α − α − α − α − α , α (1 + 3 α )1 − α − α e i ( j − π/ (cid:17) ,ϕ j = (cid:16) α (3 α − − α − α e − i ( j − π/ , α + α − α − α − α (cid:17) , (S7)where ( α , α ) = ( w AA , w AB ) / ( (cid:126) v F | g | ). Here ψ , , are related by ± π/ ϕ , , . By projecting the Hamiltonian H onto the subspace of {| Φ A (cid:105) , | Φ B (cid:105)} to linear order in k (cid:107) and q z = k z − π/
2, we obtain an effective Hamiltonian for the Weyl fermion at k / H Weyl = (cid:126) (cid:18) v q z v (cid:107) ( k x − ik y ) v (cid:107) ( k x + ik y ) v q z (cid:19) , (S8)where the in-plane velocity v (cid:107) from this perturbation theory is given by v (cid:107) /v F = 1 N A N B − α + 10 α − α ) α + 7 α (1 − α − α ) . (S9)The analytical expression in Eq. (S9) indicates that v (cid:107) vanishes at two magic angles θ ∗ M, ≈ . ◦ and θ ∗ M, ≈ . ◦ ,which are obtained using numerical values of ( v F , w AA , w AB ) listed in the main text. By comparison, the magicangles given by the full band structure calculations are θ M, ≈ . ◦ and θ M, ≈ . ◦ . Thus, θ ∗ M, provides asemiquantitative estimation of θ M, , but θ ∗ M, differs significantly from θ M, , as the truncated Hamiltonian in Eq. (S5)becomes less accurate at smaller twist angles. We have also checked that the out-of-plane velocities v and v aregenerally different within the perturbation theory, confirming that the effective Hamiltonian H Weyl expanded around k / indeed describes Weyl fermion. S4. INTERLAYER TUNNELING MATRIX IN TWISTED WEYL SEMIMETAL