Tilted Dirac Fermions

We introduce the notion of a band-inverted, topological semimetal in two-dimensional nonsymmorphic crystals. This notion is materialized in the monolayers of MTe$_2$ (M $=$ W, Mo) if spin-orbit coupling is neglected. We characterize the Dirac band touching topologically by the Wilson loop of the non-Abelian Berry gauge field. An additional feature of the Dirac cone in monolayer MTe$_2$ is that it tilts over in a Lifshitz transition to produce electron and hole pockets, a type-II Dirac cone. These pockets, together with the pseudospin structure of the Dirac electrons, suggest a unified, topological explanation for the recently-reported, non-saturating magnetoresistance in WTe$_2$, as well as its circular dichroism in photoemission. We complement our analysis and first-principle bandstructure calculations with an $\textit{ab-initio}$-derived-derived tight-binding model for the WTe$_2$ monolayer.


I. INTRODUCTION
In the Landau-Ginzburg paradigm, 1 different phases of matter are distinguished by their symmetry.3][4] That is, for the same integer electron filling and symmetry class, one may have either trivial or topological insulators.The latter have unusual electronic properties that originate 5 from the Berry phase 2,6 of electronic wavefunctions.This scenario is substantially modified for a broad class of crystals having nonsymmorphic symmetries, namely, the spatial symmetries that unavoidably translate the spatial origin by a fractional lattice vector. 7Nonsymmorphic symmetries guarantee that at certain integer fillings, the phase of matter must always be gapless.This robust and unavoidable semimetallicity originates from the nontrivial connectivity 8,9 of elementary energy bands. 10ur work explores the distinct phases of nonsymmorphic matter, but for integer fillings that do not guarantee gaplessness, as illustrated in Fig. 1(c-d).We find that a band inversion separates a trivial, gapped phase from a topological, gapless phase.The latter semimetal is concretely exemplified by MTe 2 (M = W, Mo) monolayers, as we substantiate with ab-initio calculations and tightbinding models.We characterize this metal by a topological invariant based on the Wilson loops of the non-Abelian Berry gauge field, which contrasts with previous Abelian Berry-phase characterizations of topological semimetals. 11][14][15][16] We propose here that the Wilson loop is also a powerful tool to identify and characterize topological metals.
The low-energy excitations of MTe 2 monolayers are are Fermi energies that respectively exemplify (1) a semimetal whose metallicity originates from its filling, (2) a generic insulator, and (3) a topological semimetal whose metallicity originates from a band inversion.
described by Dirac fermions that are topologically distinct from the rotationally-symmetric Dirac fermions in graphene.Precisely, the Dirac cone disperses so anisotropically that it 'tilts over' in a Lifshitz transition, i.e., part of the upper Dirac cone dips below the nodal energy ( = 0) as illustrated in Fig. 1(a), resulting in a discontinuous change in the band contours at the nodal energy.We refer to it as a type-II Dirac cone, in analogy with a notion recently introduced for Weyl semimetals 17 that is, incidentally, materialized by 3D MTe 2 [17][18][19][20] .Instead of a point-like Fermi surface where the Fermi energy lies at a (potentially tilted) Dirac node [21][22][23][24][25][26][27][28] , a type-II Dirac cone is characterized by electron-and holelike Fermi surfaces that touch at the Dirac node 29 [see Fig. 1(a)].This novel scenario promises an abundance of unexplored experimental possibilities. 30Our theory and tight-binding models should serve as important resources for ongoing experimental efforts 31,32 focused on the synthesis and the study of MTe 2 monolayers.
The type-II Dirac fermions in spin-orbit-free WTe 2 monolayers provide a unifying explanation for many phenomena in spin-orbit-coupled monolayers, bilayers and 3D layered WTe 2 .The latter are materials that have been fabricated and are under intense experimental scrutiny because of their giant, non-saturating transverse magnetoresistance (MR) of 13 × 10 6 % at 0.53 K and 60 T, with new crystals achieving 1.7 × 10 6 % at 2 K and 9 T. 33,34 Moreover, the angle-resolved photoemission (ARPES) of WTe 2 exhibits circular dichroism (CD), 35 a phenomenon which typifies Dirac semimetals such as graphene. 36,37To summarize our results: (i) When spin-orbit coupling is introduced, the degeneracy of the Dirac points is lifted and the disconnected bands are then topologically non-trivial 38 in the timereversal-symmetric Z 2 classification. 4,39,40In this work, we propose a criterion on the spin-orbit-free semimetal which is equivalently expressed by the number of Dirac fermions or the eigenvalues of the nonsymmorphic symmetry.If this criterion is satisfied, as is the case for WTe 2 , spin-orbit coupling induces Z 2 topological order.We remark that spin-orbit-coupled WTe 2 remains semimetallic due to the persistence of its electron and hole pockets, which, again, originate from the tilted Dirac fermion.
(ii) In bilayer WTe 2 , the coupling between the two stacked monolayers breaks the nonsymmorphic symmetry that protects the Dirac fermions.The low-energy theory is then described by tilted Dirac fermions with small masses; the two-component wavefunction at each Fermi circle forms a pseudospin that rotates around the Dirac node, where Berry curvature 6 is concentrated.
(iii) The electron and hole pockets, Berry curvature and rotating pseudospin are retained in 3D WTe 2 , which comprises weakly-coupled bilayers.We propose that the high mobilities in transport experiments should be attributed to suppressed backscattering due to the rotating pseudospin, while the observed circular dichroism should be correlated with the Berry phase of the Dirac cones. 36,37is work is organized as follows: after a preliminary description of the nonsymmorphic symmetries of MTe 2 in Sec.II, we introduce the theory of band-inverted topological semimetals in Sec.III.In Sec.IV we exemplify our theory with monolayer WTe 2 , for which we present a tight-binding model and introduce the notion of type-II Dirac cones that tilt over.We then extend our discussion to bilayer WTe 2 in Sec.V, with focus on its dichroism.In Sec.VI, we summarize our results and further relate them to the magnetoresistance measurements in 3D WTe 2 .Details on the derivation of the topological invariant, the role of spin-orbit coupling, the tight-binding model, and the CD calculation are collected in App.A, B, C, and D.

II. NONSYMMORPHIC SPACE GROUPS AND RELEVANT CRYSTAL STRUCTURES
In crystals, a basic geometric property that distinguishes spatial symmetries concerns how they transform the spatial origin: rotations, inversions and reflections preserve the origin, while screw rotations and glide reflections unavoidably translate the origin by a rational fraction of the lattice period. 7If no origin exists that is simultaneously preserved, modulo integer lattice translations, by all the symmetries in a space group, this space group is called nonsymmorphic.In Sec.II A we exemplify a nonsymmorphic space group with the crystal structure of MX 2 monolayers , which applies to WTe 2 , MoTe 2 , and ZrI 2 .In contrast, the MX 2 bilayer is characterized by a symmorphic space group, as we explain in Sec.II B.

A. Crystal structure of the MX2 monolayer
The M atoms form zigzag chains along e x , and are coordinated by X atoms that form distorted edge-sharing octahedra; here e x , e y , e z are basis vectors in a Cartesian coordinate system, with e z orthogonal to the monolayer, and e x , e y the generators of the Bravais lattice of the monolayer; the lengths of e x , e y correspond to lattice constants that we denote respectively by a, b.We label a unit cell in the monolayer by R = he x + ke y with h, k ∈ Z.
The group of a MX 2 monolayer is generated by (i) time reversal (T ), (ii) lattice translations t(e x ) and t(e y ), where t(r) indicates a translation by the vector r ∈ R 3 , as well as (iii) a reflection Mx ≡ t(e x /2)M x , which is a product of a reflection M x , acting as M x :(x, y, z) → (−x, y, z), and a translation t(e x /2) by half a lattice vector, and (iv) a screw rotation C2x = t(e x /2)C 2x , which is the product of a two-fold rotation C 2x , acting as C 2x :(x, y, z) → (x, −y, −z), and the same fractional translation.The product of the last two generators is the spatial inversion Mx C2x = I that sends r → −r; we choose the inversion center, indicated by a green cross in Fig. 2(a), as our spatial origin.[43] B. Crystal structure of the MX2 bilayer Let the position of each atom in a MX 2 monolayer be parametrized by (x, y, z) relative to our chosen spatial origin (green cross Fig. 2(a)).There is a corresponding identical atom positioned at (−x + 1 2 + a, −y + b, z + c) in the second layer of the bilayer, where {a, b, c} are material-specific parameters.This stacking spoils both inversion and screw symmetries of the monolayer, but retains the mirror symmetry Mx , as we illustrate in Fig. 2(b).Our stacking is identical to that of the bilayer within 3D MX 2 , whose experimentally known atomic positions 41 we use throughout this paper.

III. THEORY OF BAND-INVERTED TOPOLOGICAL SEMIMETALS
In Sec.III A, we briefly review nonsymmorphic semimetals which are semimetallic only due to their filling.These semimetals are distinguished from semimetals which originate from band inversion, as we briefly describe in Sec.III A, and then more carefully elaborate in Sec.III B. These band inverted semimetals admit a topological classification that we describe in Sec.III C. When a gap is induced by spin-orbit coupling, certain band inverted semimetals turn into topological insulators, as we substantiate in Sec.III D.

A. Comparing topological and filling-enforced semimetals
We have introduced two types of nonsymmorphic semimetals: (i) filling-enforced semimetals, which are guaranteed to be semimetallic at certain fillings determined by the space group; 8 (ii) topological semimetals, which are not guaranteed in the sense of (i), but are semimetallic due to a topological band inversion that we will describe.To exemplify (i) and (ii), we offer two examples from a group generated by {T, t(e x ), t(e y ), C2x }, which has one less generator ( Mx ) than the group of WTe 2 ; we have defined this smaller group to emphasize the relevant symmetries, as well as their wider applicability to other materials.
In this Section, we consider electronic systems without spin-orbit coupling.In our definition of filling (f ), we count a spin-degenerate band as a single band, and each spin species transforms in an integer-spin representation 7,44 of the space-group symmetries described in Sec.II, e.g., C2 2x = t(e x ) would not include a 2π rotation of the spin.There are two lines (k y = 0 and π) which are individually mapped onto themselves under the screw transformation; in short, we call them screw lines.Bands along each screw line may be labelled by the eigenvalues of C2x , which fall into the two momentum-dependent branches, ±exp(−ik x /2), as follows from C2 2x = t(e x ) = exp(−ik x ) in a Blochwave representation.At inversion-invariant points X [(k x , k y ) = (π, 0)] and M [(k x , k y ) = (π, π)] on the screw lines, time reversal pairs up complex-conjugate representations of C2x , such that the bands are all doublydegenerate.Each degenerate subspace is composed of an equal number of states with C2x -eigenvalue +i and −i.In contrast, time reversal does not enhance the degeneracy at Γ [(k x , k y ) = (0, 0)] and Y [(k x , k y ) = (0, π)] where the C2x -eigenvalues are real.The situation is illustrated in Fig. 1(c): bands divide minimally into pairs, such that within each pair there is at least one robust contact point (here, a crossing between orthogonal screw representations at X/M ) that connects both members of the pair -we say that bands are two-fold connected along both screw lines. 45The notion of connectivity of a submanifold (here, a screw line) generalizes 45 the notion of symmetry-enforced degeneracy at an isolated wavevector; the connectivity of the entire Brillouin zone 8 relates to the theory of elementary energy bands. 10Due to the twofold connectivity, any odd, single-spin filling (f ∈ 2Z + 1) is guaranteed to produce a filling-enforced semimetal, as exemplified by the Fermi energy E 1 in Fig. 1(c).
If the filling is even (f ∈ 2Z), semimetallicity is not guaranteed by reason of filling, as exemplified by the Fermi energy E 2 in Fig. 1(c).Dirac semimetallicity is nevertheless guaranteed at even filling due to independent band inversions along either screw line, e.g., a single inversion at Γ (resp.Y) would nucleate a pair of timereversal-related Dirac crossings that situate anywhere along XΓX (resp.along M Y M ), as we illustrate with the spin-orbit-free WTe 2 -monolayer in Fig. 1(b).The general theory of band-inverted semimetals is elaborated in the next section.We remark that a glide reflection My , composed of a reflection (y → −y) and a half-lattice translation in e x , also satisfies M 2 y = t(e x ), just like C2x .Consequently, every result in this and the next two Sections applies also to My , with the cosmetic substitution 'screw' → 'glide'.

B. Band inversion and Dirac semimetallicity at even filling
In this Section, we focus on even filling and quantify the relation between band inversion and Dirac semimetallicity.We say bands are inverted if the filled states, at any wavevector along a screw line, transform nontrivially under the screw rotation.To further clarify 'filled states at a wavevector' for a filling f , we say that a state at a wavevector is filled if it belongs to the lowest set of f bands at that wavevector.Filled states according to our unconventional notion of filling often coincide with actual states below the Fermi level [e.g., ZrI 2 in Fig. 3(a)]; exceptions include the hole/electron pockets of WTe 2 and MoTe 2 [Fig.3(b-c)], which originate from the Dirac fermion tilting over, as we will elaborate in Sec.IV B. Our notion of filled states more naturally generalizes to bosonic systems, as we will elaborate.Furthermore, we now demonstrate how the symmetry analysis of filled states is predictive of the number of Dirac crossings, whether or not they tilt over.
Precisely, we would count, for any k along XΓX (resp.M Y M ), the number of Dirac crossings along the screw line that connects k to X (resp.M ) in the direction of increasing k x , as indicated by the bottom (resp.top) dashed line in Fig. 2(c).Henceforth, we refer to this number as the Dirac count (D k ) e.g., D Γ is the number of crossings along half the screw line: ΓX.For illustration, we plot D (kx ,0) for five case studies in Fig. 4. To further clarify D k , we consider only the crossings between the top-most, filled band and the bottom-most, unfilled band, as highlighted by blue squares in Fig. 4(b-e), i.e., we discard crossings that may occur between two filled bands, and also those between two unfilled bands [e.g., 4. Characterization of a generic insulator (a) and various topological semimetals (b-e).Top row: bandstructures along the screw line ΓX.Solid (dashed) lines correspond to bands in the odd (even) screw representation, i.e., having screw eigenvalue +exp(−ikx/2) [−exp(−ikx/2)].For each of (a-e), the filling (f ) is indicated in the top-right corner, and we assume that there are no Dirac crossings along the other screw line Y M .Blue circles denote crossings between a filled and an unfilled band, whereas red circles mark crossing between bands that are both filled (unfilled).Second, third and fourth rows: respectively, the numbers of Dirac crossings (Dk), of even, filled states (N+,k), and of odd, filled states (N−,k), for k along ΓX.Last row: eigenvalue-phases of the W(kx)-spectrum.red circles in Fig. 4(e)].
To characterize the symmetry representation of the filled states, let us denote by N +,k (N −,k ) the number of filled, even (odd) Bloch states at k along either screw line, where the even (odd) representation is defined to have screw eigenvalue +exp(−ik x /2) [−exp(−ik x /2)]; N +,k + N −,k = f , and where N +,k = N −,k , we say that bands are inverted at k, as exemplified by the interval 0 ≤ k x < k x in Fig. 4(b).N ±,k relate to the total Dirac count through That is, n k belongs to the set of nonnegative integers, and is undeducible solely from N ±,k .To prove Eq. ( 1) for k along XΓX, recall that each connected pair of bands (where, again, each pair corresponds to two eigenvalue-branches of C2x which unavoidably cross in the interpolation: (k x → k x + 2π) is time-reversal-degenerate at X, with each degenerate subspace comprising a single odd ( C2x = +i) state and a single even state with C2x = −i.
Given an even-integral filling (f ∈ 2Z), there are then an equal number (f /2) of filled even and filled odd states at X. Since each band is smoothly parametrized by its C2x eigenvalue, each filled state at X must continuously interpolate, in the direction of decreasing k x , to a state at k; this interpolation occurs in the same branch of C2x , but the final state at k may or may not be filled.
, there are as many odd/even states (at k) as there are odd/even states (at X) -then it is possible for the interpolation to occur entirely among the filled states, as exemplified by Fig. 4(a).However, if Since the above results depend essentially on the integer-spin representation of time-reversal and screw symmetries, they (and the topological characterization described in the next Section) would also apply to intrinsically spinless systems such as photonic crystals, 46,47 though certain terms that naturally describe Fermi systems have to be re-interpreted.While 'filling' is conventionally associated with Pauli exclusion, we may, at each wavevector, distinguish between 'filled' and 'unfilled' photonic bands separated by a frequency gap.There is, of course, no photonic 'semimetal' in the sense of charge transport, though we may still discuss Diractype touchings between 'filled' and 'unfilled' bands.

C. Topological characterization of band inverted semimetals
Each screw-protected Dirac touching is associated with a π quantized Berry phase, which is acquired in traversing a screw-symmetric momentum loop around the Dirac node.More generally, we consider the parallel transport of filled Bloch waves around a momentum loop l, where at each k ∈ l a spectral gap separates a set of lowerenergy, filled states (numbering f ) from a higher-energy, unfilled subspace.The f -by-f matrix representing such parallel transport is known as the Wilson loop, 48 and it may be expressed as the path-ordered exponential (denoted by exp) of the Berry-Wilczek-Zee connection 6,48 where |u j,k is an occupied eigenstate of the tight-binding Hamiltonian.The U (f ) Berry gauge field (A) may be decomposed into trace-ful and trace-less components, where the trace-ful term (Tr[A]) generates the Abelian component of the parallel transport: The U (1) Berry phase (Φ U (1) [l]) is quantized to π if we choose l to encircle the Dirac node screw-symmetrically.By a 'screw-symmetric circle', we mean that l [exemplified by l 1 and l 2 in Fig. 2(c)] is contractible, and is mapped to −l by C2x , where the sign of l indicates its orientation; the mapping follows from C2x : y → −y in real space, and therefore k y → −k y in momentum space.More generally, given any symmetry (g) that maps l → −l and has a unitary representation (g The left equation implies that the spectrum of W[l] is invariant under complex conjugation; the right equation describes the quantization of the U (1) Berry phase, which is robust against any deformations of the Hamiltonian that preserve both the symmetry (g) and the spectral gap along l.In our context, g = C2x , and exp(iΦ U (1) [l]) = 1 (resp.−1) if l encircles an even (resp.odd) number of Dirac fermions.
Our topological discussion has thus far focused on characterizing individual Dirac nodes, by an Abelian Berrytype invariant defined over contractible momentum loops; a global characterization of all Dirac nodes is possible with a non-Abelian Berry invariant 12,15 defined over a noncontractible 2 momentum loop.By a non-Abelian Berry invariant, we mean that it requires knowledge of the individual eigenvalues of W[l], which encode the non-Abelian transport generated by the trace-less component of A.
Henceforth, we consider only screw-symmetric loops l(k x ) parallel to e y and at fixed k x , as illustrated in Fig. 2(c); we thus shorten W[l(k x )] to W(k x ).Applying Eq. ( 4) for l(k x ), the invariance of the W(k x )-spectrum under complex conjugation implies that a (possibly zero) subset of W(k x )-eigenvalues (numbering W ± (k x )) is respectively quantized to ±1.In App.A, we relate these quantized W(k x )-eigenvalues to the total number (D l(kx) ) of Dirac crossings in the cylinder bounded by l(k x ) and M XM [red-shaded region in Fig. 2(c)]: As described in Sec.III B, such crossings can originate from independent band inversions on either screw intervals to the right of l(k x ) [marked as dashed lines in Fig. 2(c)].The difference in Dirac counts between the two intervals also relates to the quantized W-eigenvalues as where 2Z denotes the set of even integers.Additional contributions to D l(kx) may arise from screw-symmetric pairs of crossings away from screw lines, as may be stabilized by a spatial symmetry (e.g., inversion) other than screw.
We exemplify our result for crystals with fillings 2 and 4; their possible screw representations, W-spectra and minimal Dirac counts are tabulated in Table I (in this Section) and II (in App.A).These properties are then applied to topologically distinguish the different phases in Fig. 4(a-e), which are all assumed to have no Dirac crossings along the unillustrated screw line Y M , and also no crossings away from the screw lines.We exemplify this analysis for Fig. 4(b), focusing on the interval 0 ≤ k x < k x .The filled bands are inverted in this segment of ΓX, with N +,(kx,0) = 2, N −,(kx ,0) = 0; by assumption of Y M , N +,(kx ,π) = N −,(kx ,π) = 1.Thus reading off the third-from-bottom row in Table I, we obtain W ± (k x ) = 1 (illustrated in the bottom-most plot of Fig. 4(b)).This further implies from Eq. ( 5) that the total Dirac count D (kx ,0) + D (kx ,π) = 1 + 2c with c ∈ Z ≥ , and from Eq. ( 6) that both D (kx,0) and D (kx,π) have the same parity; Fig. 4(b) shows in fact that D (kx ,0) = 1 and by assumption D (kx ,π) = 0. Finally, we remark that the Wilson loop in Eq. ( 2) is only well-defined and continuous on intervals that exclude the Dirac points, e.g., the discontinuity of W ± (k x ) at k x in Fig. 4(b) necessarily indicates a Dirac crossing.Our schematic example in Fig. 4(b) is further materialized by the spin-orbit-free WTe 2 monolayer, as we demonstrate in Sec.IV A.
14][15][16]49 On the other hand, we propose that the invariants in Eq. ( 5) and ( 6), as extracted from a single Wilson loop, are predictive of the Dirac semimetallicity in a 2D submanifold of the Brillouin zone -since the existence of Dirac points cannot depend on the choice of origin, our invariants are likewise independent, as we proceed to demonstrate.Since the Wilson loops we consider traverse a momentum path which is parallel to e y , their eigenspectrum only depends on the y-coordinate of the origin.Rightmost column: the parity of the difference in Dirac counts along either screw intervals.We remark that the W(kx)spectrum depends only on relative changes in the symmetry representations between k1 and k2, i.e., it is invariant under: (i) interchanging N+,k 1 with N+,k 2 , and (ii) multiplication of all screw (glide) eigenvalues by a common factor −1, i.e., For example, the W(kx)-spectrum and Dirac counts in the third-to-last row additionally describe a crystal with two odd states at k1 and an odd/even pair at k2, and also describe a crystal with two odd states at k2 and an odd/even pair at k1.Specifically, translating the spatial origin by δr induces a global phase shift of all W-eigenvalues by G•δr, with G the reciprocal period along e y . 12In a 2D Bravais lattice, there are always two inequivalent, real-spatial lines which are invariant under the screw rotation, as exemplified by horizontal dashed lines in Fig. 2(a) for the WTe 2 monolayer; assuming no spatial symmetry other than screw, the spatial origin may lie at any point on either screw line.Since the two lines are separated by half a lattice period in e y , translating the origin between these lines induces a global phase shift of G • δr = ±π.Consequently, W ± → W ∓ , but their maximum and minimum values in Eq. ( 5) and ( 6) are clearly invariant.We call any such quantity, that is both extractable from a single Wilson loop and insensitive to the spatial origin, a strong Wilson-loop invariant; another known example classifies a newly-introduced nonsymmorphic topological insulator; 5 all other known single-Wilson-loop invariants 2,12,16 are comparatively weak, and relate to spatially-dependent physical predictions.A case in point is the geometric theory of polarization, 50 which predicts different electronic charges at the edge of a crystal, depending on where the edge is terminated.

D. Proximity of topological semimetals to the Z2-topological insulator
Topological semimetals are often linked to gapped, topological phases.There are various ways to arrive at such a gapped phase: (i) the mutual annihilation of band crossings with zero net topological charge, (ii) the breaking of a spatial symmetry, and (iii) the introduction of spin-orbit coupling in electronic semimetals, 9 which may also be interpreted as a breaking of spin-SU (2) symmetry.To exemplify (i-iii) in this order, inversionasymmetric Weyl semimetals intermediate between a trivial, gapped phase and a Z 2 -topological, gapped phase; 52 certain 3D Dirac semimetals 53 are gapped when their protective spatial symmetry is broken, leading to a novel, nonsymmorphic topological phase; 54 the slightest spin-orbit coupling gaps graphene, 55 a symmorphic Dirac semimetal, to form a quantum spin Hall phase with Z 2 topological order 4,39,40 (in short, a Z 2 -topological insulator).
This Section describes how, analogously to graphene, the slightest spin-orbit coupling gaps a nonsymmorphic Dirac semimetal to form a Z 2 -topological insulator.Not all Dirac semimetals necessarily lead to a Z 2 -topological phase when gapped -the criterion on the semimetal may be stated in three equivalent ways.If and only if the nonsymmorphic semimetal is characterized by: (i) an odd number of Dirac fermions (per spin component) in half the Brillouin zone [red-shaded region labelled by τ 1/2 in Fig. 5(a)], or, equivalently, (ii) the Abelian component of the single-spin Wilson loop along Y ΓY (i.e., det[W(0)]) equals −1, or, equivalently, (iii) the product of the nonsymmorphic eigenvalues (whether glide or screw) of single-spin filled states over Γ and Y equals −1, then any gap-inducing, spin-orbit coupling that preserves the nonsymmorphic symmetry (whether glide or screw) results in a Z 2 -topological phase.
More precisely, the above criterion guarantees the Z 2topological phase for weak spin-orbit coupling; in principle, one cannot rule out that strong spin-orbit coupling might induce a transition to a trivial, gapped phase.That (i) and (ii) are equivalent follows from Eq. ( 5), where we deduce that odd D l(0) is equivalent to either odd W + (0) or odd W − (0).Due to the assumed-even filling and the invariance of the W(0)-spectrum under complex conjugation (cf.Eq. ( 4)), it is always the case that if either is odd, both W + (0) and W − (0) are odd, which leads to det[W(0)] = −1.That (i) and (iii) are equivalent follows from Eq. ( 1) which leads to where ∼ denotes an equality modulo two.On further application of f = N +,k − N −,k ∈ 2Z, we deduce that odd D l(0) occurs iff N −,Γ + N −,Y is also odd; finally, observe that the product of nonsymmorphic eigenvalues equals (−1) N −,Γ +N −,Y .We remark that (iii) is the nonsymmorphic generalization of the Fu-Kane criterion 56 for a Z 2topological phase in centrosymmetric crystals.
To pictorially argue for our criterion, we return to our semimetallic case study in Fig. 4 is reproduced in Fig. 5(b), where each Wilson 'band' is spin-degenerate and discontinuous at the momentum position of the Dirac fermion (indicated by the blue dashed line).The introduction of spin-orbit coupling splits this spin degeneracy everywhere except for the Kramers degeneracies at the time-reversal-invariant k x ∈ {0, π}.Additionally, the gapping of the Dirac fermion implies that a subspace of filled states is now smoothly defined over the entire Brillouin zone -this smoothens out the discontinuity in the W(k x )-spectrum.If we only assume that the spin-orbit coupling is time-reversal symmetric, there are two ways of smoothening: Fig. 5(c) (resp.(d)) illustrates the Kramers-partner-preserving (resp.partnerswitching 13,14,49 ) doublets of the trivial gapped phase (resp.the Z 2 -topological phase).If we further assume that the spin-orbit coupling respects the nonsymmorphic symmetry (whether glide or screw), then the W(k x )spectrum is further constrained to be invariant under complex conjugation (cf.Eq. ( 4)), which uniquely selects the Z 2 -topological phase of Fig. 5(d).Beyond this pictorial argument, a technical proof of our criterion is provided in App.B.
Our criterion is more broadly predictive of Z 2topological phases which are semimetallic from the perspective of transport, so long as a finite energy difference exists between two sets of bands for all k -the timereversal Z 2 invariant 4 is then well-defined for both sets of bands.This qualifier is relevant to spin-orbit-coupled WTe 2 monolayers, which have Z 2 topological order in conjunction with electron and hole pockets, as we demonstrate in Sec.IV C.

IV. WTe2 MONOLAYER
In Sec.IV A, we present a minimal tight-binding model of a spin-orbit-free WTe 2 monolayer, which confirms its topological semimetallicity in the sense of Sec.III B. This model also captures the tilting of the Dirac fermion, which we formalize in Sec.IV B by introducing the notion of a type-II Dirac fermion.Finally in Sec.IV B, we apply our criterion in Sec.III D to predict that spinorbit-coupled WTe 2 monolayer has Z 2 -topological order.
A. Tight-binding model and topological characterization of the spin-orbit-free WTe2 monolayer We now present a minimal tight-binding model of the spin-orbit-free WTe 2 monolayer, which reproduces the DFT bandstructure near the Fermi level; compare Fig. 6(a) with Fig. 3(c).Our model includes the minimal number (four) of bands to describe a band inversion between two sets of two-fold-connected bands: cf.Sec.III A. With modified tight-binding parameters, this model can be more generally be applied to MX 2 compounds with the same crystal structure.
By a Wannier interpolation 57 of these four DFT bands, we construct a basis of maximally-localized Wannier functions, comprising two d x 2 −y 2 -type orbitals which derive from the W atoms [indicated by M in Fig. 2(a)], and two p x -type orbitals derived from a subset of the Te atoms [X-2 in Fig. 2(a)]; there are no orbitals derived from the complementary Te sublattice [X-1 in Fig. 2(a)] in our low-energy description.As indicated within the rectangular unit cell of Fig. 2(a), the centers of these Wannier functions divide into two sublattices, labelled by A and B, which are permuted by the screw transformation C2x .Each d x 2 −y 2 -type ( p x -type) Wannier function is even (odd) under a mirror operation x → −x centered at the W atom (Te atom).
In this reduced Hilbert space, our tight-binding Hamiltonian includes all symmetry-allowed, nearest-neighbor hoppings, as well as two next-nearest-neighbor, intrasublattice hoppings along the chain: position r ,A + R (resp.r ,B + R) as indicated in Fig. 2(a); δ are vectors given by δ p = 0, e x and δ d = e x + e y , e y .Since our Hamiltonian is spin-SU (2) symmetric, we omit the spin label for each electron operator.The tight-binding parameters are listed in App.C, where we also express Eq. ( 7) in the momentum representation. For orthogonal screw representations at Γ are inverted (i.e., 0 = N +,Γ = N −,Γ = 2), as we illustrate in Fig. 6(a) for WTe 2 .This band inversion leads to a time-reversed pair of Dirac fermions along the screw line XΓX, which are encoded as a discontinuity in the Wilson-loop spectra in Fig. 6(b).These Dirac nodes necessarily belong to the screw line but they can have any position along it, i.e. by perturbing the Hamiltonian we might move the Dirac nodes along XΓX.

B. Type-II Dirac fermions
The Dirac crossing along the screw line (ΓX) is described by an effective Hamiltonian: to linear order in the momentum coordinates originating from the Dirac node, which we define as the point of degeneracy; σ 1,2 are Pauli matrices, and σ 0 the identity matrix, in a pseudospin basis labelled by σ 1 |± = ± |± , with the orbital characters |± = iα ± |p ± + β ± |d ± , where |l ± = (|l, A ± |l, B ) / √ 2 are the bonding/antibonding combinations of l-type orbitals across the two sublattices.The Greek symbols are real, normalized according to α 2 ± + β 2 ± = 1, and determined numerically by diagonalizing Eq. ( 7) at the Dirac node.
The group (G) of this nodal wavevector 44 is generated by the screw C2x and the space-time inversion T I (i.e., the product of time reversal T and spatial inversion I), which respectively transform the Hamiltonian as The lack of any discrete rotational symmetry in G implies that the spectrum of H II is rotationally anisotropic, as evidenced by The resulting anisotropy of the Fermi velocities may have important consequences in transport.This is in contrast to graphene, where each Dirac cone is fixed to a wavevector that is invariant under three-fold rotation, such that the Dirac spectrum of graphene has an emergent, continuous-rotational symmetry.
For WTe 2 , G also lacks the reflection symmetry that maps k x → −k x , as manifested by the allowed term u x k x σ 0 in the Hamiltonian.This term induces a tilting of the Dirac cone, which originates from the intrasublattice hoppings t in Eq. ( 7).If the tilting is sufficiently pronounced, part of the upper Dirac cone dips below the nodal energy [ ± (0) = 0] as illustrated in Fig. 1 spectrum with open Fermi line(s) to linear order in k; the latter case applies to WTe 2 , as we illustrate in Fig. 1(a-b). 17,58All lattice-regularized Fermi surfaces are of course closed when higher-order momenta are accounted for -the regularized, type-II spectrum is described by two Fermi circles, with one being hole-type and the other electron-type.If the Fermi level lies at the type-II Dirac node (i.e., ε F = ± (0) = 0), the electron-and hole-type Fermi circles touch at the Dirac node.However, there is no symmetry constraint on the Fermi energy and so both circles are generically disconnected; for negative ε F (resp.positive ε F , as applies to WTe 2 ), the hole-type (resp.electron-type) Fermi circle is characterized by a U (1)-Berry phase [Φ U (1) in Eq. ( 3)] equal to π, while the electron-type (resp.hole-type) Fermi circle has U (1)-Berry phase equal to 0; this quantization is proven in Sec.III C, where we would show that each Fermi circle corresponds to a screw-symmetric quasimomentum loop.

C. Spin-orbit-coupled WTe2 monolayer
The screw eigenvalues of the bands at Γ and Y are indicated in Fig. 6(a); their product over the filled bands equals −1, which implies, through criterion (iii) in Sec.III D, that weakly-spin-orbit-coupled WTe 2 has Z 2 topological order.This result has consistently been derived 38 by exploiting the Fu-Kane criterion 56 for centrosymmetric crystals; we remark that our proposed criterion in Sec.
(a) III D more generally applies to nonsymmorphic semimetals without spatial-inversion symmetry.

V. BILAYER
We propose to interpret 3D WTe 2 as a periodic stacking of bilayers, for which the intra-bilayer couplings dominate over the inter-bilayer couplings.We support our interpretation by finding that certain features of the 3D electronic structure, precisely the electron and hole pockets along the ΓX line, are already present in the bilayer; we show in Sec.V A how these pockets (in both bilayer and 3D) ultimately originate from the tilted Dirac cones of the single monolayer.In Sec.V B, we further calculate the dichroism that originates from these pockets, so as to qualitatively explain the recently-measured CD in 3D WTe 2 . 35

A. Origin of electron and hole pockets in bilayer and 3D WTe2
The electron and hole pockets of 3D WTe 2 are a fundamental aspect of its electronic structure and are thought to be responsible for the large, non-saturating magnetoresistance. 33Here, we propose that these pockets originate from the type-II Dirac fermions of the spinorbit-free monolayer.
Our argument is illustrated in Fig. 7: to begin, in the limit of vanishing coupling between the monolayers, the bilayer bandstructure is doubly degenerate at all wavevectors.This follows because one monolayer is related to the other by a global continuous translation in real space, so that the energy-momentum dispersion curves of each monolayer are identical.We focus on a low-energy theory near the Dirac node, which is doublydegenerate in the aforementioned limit.
We now consider two different stackings for a bilayer of WTe 2 : in the first, the two monolayers have no relative displacement in the e x -e y plane but are displaced along e z and therefore the bilayer retains the screw symmetry of the monolayer; we warn that this hypothetical stacking is unlikely to occur in nature.The inter-monolayer coupling energetically splits the two Dirac cones, which correspond to bonding and anti-bonding combinations of the two monolayer wavefunctions, as illustrated in Fig. 7(b-ii); the degeneracy of the Dirac crossing is retained since screw symmetry is preserved.The low-energy theory of each Dirac crossing is described by the Hamiltonian in Eq. ( 9), albeit with slightly different parameters and basis wavefunctions.Independent of the particular basis, we note that Eq. ( 9) describes a pseudospin that is coupled to a pseudomagnetic field (B) with components The vanishing of B z is a consequence of the spatial symmetries which constrain the Hamiltonian as in Eq. ( 10) -the Hamiltonian eigenfunctions therefore correspond to a pseudospin that is confined to a pseudo x − y plane, and moreover winds as we encircle the Dirac node.
The second stacking corresponds to the experimental lattice parameters for 3D WTe 2 , and breaks the screw symmetry, as we illustrate in Fig. 2(b).The asymmetric bandstructure differs from the symmetric stacking in that small gaps open at the Dirac nodes; compare Fig. 7(b-ii) and (b-iii).Otherwise, both bandstructures are very similar away from the Dirac nodes, and in particular both possess electron and hole Fermi pockets in the vicinity of the Dirac nodes.Each screw-asymmetric Dirac fermion is described by a small mass term proportional to σ 3 , i.e., B z is a nonvanishingly small constant.Therefore, around either Fermi circle, the pseudospin rotates in the x − y plane with a small out-of-plane component.In particular, the sense of rotation for the electron-like Fermi circle is opposite to that of the hole-like circle, with consequences for the dichroism that we elaborate in Sec.V B.
We argue that these Fermi pockets are not generic and originate from the tilting of the Dirac cone.For the sake of this argument, we schematically illustrate in Fig. 7(a) a hypothetical bilayer which comprises two bandinverted monolayers.In this hypothetical scenario, the Dirac cone of the monolayer is type-I and the Fermi surfaces are point-like; therefore, screw-symmetry-breaking inter-monolayer couplings generically produce a fullygapped, trivial insulator, as we argue pictorially through Fig. 7(a).

B. Dichroism
Circular dichroism (CD) has recently been observed in the angle-resolved photoemission spectrum (ARPES) of 3D WTe 2 , 35 i.e., the photoemission is dependent on the helical polarization of light.CD depends on the wavefunction of the initial and of the final electronic state upon photon absorption and is therefore a sensitive probe of the electronic structure of a material. 61A simple and widely-applied model of photoemission breaks down the process into three steps: (i) photoexcitation of an electron in the solid, (ii) propagation of the photoelectron to the surface, and, (iii), escape of the photoelectron into the vacuum. 62CD is encoded in the first step, where the photon causes a transition between two Bloch states in the solid.The intensity of the photoemitted electrons with wavevector k parallel to the surface and kinetic energy E kin due to photons of energy ω and polarization vector λ is given by Here, P if λ (k) = f, k|λ • p|i, k is the matrix element of the momentum operator p between initial (i) and final (f ) Bloch states of energy E i and E f , respectively, and φ is the work function. 62n the simplest case of normal incidence along e z , the relevant matrix elements are P if ± (k) = f, k|p x ± ip y |i, k , with the signs corresponding to the two different circular polarizations of the light.In this work, we ignore final-state effects and focus on the dichroism that originates from initial states at the Fermi level, i.e., we study the Fermi-level dichroic signal defined by 4][65] In monolayer graphene unambiguous information on the initial states can be extracted from the spectra because in this system the electronic wavefunction near the Fermi level corresponds to a pseudospin that winds around each Dirac point, resulting in dichroism for a narrow range of photon energies. 36,37e propose that the dichroism in bilayer WTe 2 similarly originates from tilted Dirac fermions with small masses.As explained in Sec.V A, the pseudospin rotates around the electron-like Fermi circle and in the opposite sense around the hole-like Fermi circle.thus we expect that D s (k, F ) should carry opposite sign along the electron-like and the hole-like Fermi circles.. 36 To support this hypothesis, we calculated D s (k, ε F ) using the DFTderived initial-state wavefunctions, as further elaborated in App.D. As shown in Fig. 8(a), D s (k, ε F ) is constrained by time-reversal and Mx symmetries as spectively.When the dichroic signal is strong, D s (k, ε F ) shows the expected sign change between electron and hole Fermi circles; we note that the signal vanishes in a small momentum segment along the electron Fermi circle.Fig. 8(b) further illustrates the non-uniform variation of the D s (k, E) over a small range of energies.We suggest also that the observed dichroism in 3D WTe 2 originates from the tilted, massive Dirac fermions.Quantitative comparison with experiment would need to account for final-state effects, spin-orbit coupling, as well as the 3D coupling between bilayers.We defer this to a future investigation.In closing this section, we note that ARPES and quantum oscillation experiments suggest the presence in of a zone center Fermi pocket in 3D WTe 2 , which is absent in our DFT calculation, where it is pushed below the Fermi level.These fine effects are, however, very sensitive to small changes in the atomic parameters and functional approximations. 35,59,60
Among the predicted Weyl semimetals, 3D MTe 2 (M=W,Mo) reveal a particularly rich range of phenomena: (i) they become superconducting under pressure, 43,[88][89][90][91] (ii) WTe 2 demonstrates a giant, non-saturating transverse magnetoresistance, and also (iii) circular dichroism in its photoemission. 35In this work, we demonstrate that some of these exotic 3D properties may be extrapolated from topologically characterizing a single monolayer without spin-orbit coupling (SOC); this simplification is possible because 3D WTe 2 has weak SOC, and is moreover composed of weakly-coupled monolayers.
We find that the spin-orbit-free MTe 2 monolayer belongs to a new class of band-inverted semimetals, which are diagnosed by a topological invariant associated to a non-Abelian Berry gauge field, which contrasts with previous Abelian Berry-type characterizations of topological semimetals.The Dirac crossings of our semimetals rely on a nonsymmorphic symmetry; they differ from previously-proposed, nonsymmorphic semimetals 9,10 whose semimetallicity is guaranteed solely by the electron filling.The Dirac cones of MTe 2 tilt over and are classified as type-II.This has important implications for the bandstructure of 3D MTe 2 : nearly compensated electron and hole pockets emerge, which encircle Dirac fermions with small masses; these pockets are characterized by a rotating pseudospin with consequences for circular dichroism.
We further relate our findings to the giant, nonsaturating magnetoresistance observed in 3D MTe 2 .The magnitude of the magnetoresistance in a two-band model with perfectly compensated electron and hole carriers is given by [ρ(H) − ρ(0)]/ρ(0) = µ e µ h B 2 , where µ e , µ h are the electron and hole mobilities. 33,92The geometric mean of the mobilities has been extracted as √ µ e µ h = 167000 cm 2 /Vs in 3D WTe 2 , 34 as a result of the large magnetoresistance in this model.Additionally, the measured residual resistivity (RR) of 10 −7 Ωcm is extremely low compared to other binary compounds.Applying a magnetic field of 1 T to WTe 2 increases the low-temperature RR to the order of 10 −4 Ωcm. 93,94Extremely high mobilities are common to Dirac semimetals such as graphene, where the winding pseudospin suppresses backscattering from impurities and defects.We suggest that a similar mechanism suppresses backscattering in 3D WTe 2 , which hosts massive, tilted Dirac fermions.Long relaxation times in transport, in combination with the low impurity concentration in measured crystals, may account for the large mobilities.The tilting of the Dirac cone hints at the strong anisotropy in the measured MR.However, a complete explanation of the large MR should account for the coupling of the magnetic field to the electronic structure, which is known to be nontrivial for other high-MR materials such as Cd 3 As 2 . 95he nature of point defects in WTe 2 and their effect on the electronic structure may be probed by quasiparticle interference and studied by computational methods.
We remark that electron-electron interactions modify the electron velocities such that they tend toward rotational isotropy at low energies.If screening is sufficiently weak, interactions may induce a type-II to type-I Lifshitz transition. 29The strength of screening depends on the density of states at the Fermi level, which for spinorbit-free WTe 2 is not negligible.
We close this paper by remarking on the role of spatial symmetries in band topology.[98][99][100] However, a material class has recently been proposed, whose band topology relies essentially on nonsymmorphic symmetries. 45Moreover, an analogous band topology can also be realized in photonic crystals. 1015][106] We expect that our theory of band-inverted, nonsymmorphic semimetals, introduced here, should be broadly applicable to 3D nonsymmorphic crystals, and may be generalized to include spin-orbit coupling, as we exemplify in Fig. 1(d).
by a fractional lattice translation t(δ = e x /2) parallel to the rotational axis; C2 2x = t(e x ), an integral lattice translation.C2x is a 3D symmetry of the WTe 2 monolayers, which extend macroscopically in e x and e y , and have finite, atomic-scale thickness in e z .In the 2D Brillouin zone parameterized by k = (k x , k y ) t , the screw rotation maps k → (k x , −k y ) t ≡D 2x (k x , k y ) t .Identifying D g in Eq. (A5) with D 2x , the Löwdin representation of C2x is which satisfies the nonsymmorphic algebra ( C2 2x = t(e x )) of a screw rotation: (A7) The momentum-independent unitary matrix U 2x,ex/2 forms a representation of a screwless two-fold rotation, i.e., U 2 2x,ex/2 = I.Substituting Eq. (A6) into Eq.(A4) yields the condition that is, for fixed k x , mathematically equivalent to the Hamiltonian of a 1D crystal with spatial-inversion symmetry represented by U 2x,ex/2 .This identification allows us to apply a known mapping between the spatialinversion eigenvalues (of the occupied bands) and the Wilson-loop eigenvalues (for any constant-k x momentum loop, e.g., Y ΓY ).This mapping may be found in Sec.IIB of Ref. 12, where the inversion eigenvalues, ±1, of filled states at the inversion-invariant momenta are now identified with the branches of screw eigenvalues (±exp(−ik x /2)) of filled states at (k x , 0) and (k x , π).
For the reader's convenience, we reproduce the mapping below.
In short, we refer to ξ as the screw branch, and further define the size of each set as N ξ,k .Given the list { N +,(kx,0) , N −,(kx,0) , N +,(kx,π) , N −,(kx,π) }, we identify the smallest of these four integers and label it as Nξ , k, where ξ ∈ ±1 denotes the screw branch of this smallest set of states, and k ∈ {(k x , 0), (k x , π)} its quasimomentum.We denote by kc the complementary quasimomentum.To recapitulate, we have mapped The smallest set might be empty; in cases where the smallest set is not unique, any choice between 'equally smallest' sets is valid.Let us exemplify the identification of the smallest set: (ii) eigenvalue + ξ with multiplicity (N −, kc − Nξ , k), (iii) Nξ , k pairs of complex-conjugate eigenvalues.
In the above examples, the W-spectrum of (a) comprises one +1 and one −1 eigenvalue; for (b), there are one +1 eigenvalue, one −1 eigenvalue, and one complexconjugate pair; (c) has one complex-conjugate pair only.For a screw-symmetric crystal with filling two and four, we tabulate the possible mappings, for k x = 0, in Table I and II respectively.It is useful to connect this mapping to the notation W ± (k x ), which was defined in Sec.III C as the number of robust ±1 W-eigenvalues, i.e., with n ∈ Z.The subspace II transforms dependently to maintain Eq. (B9), therefore ν is gauge-invariant modulo two. 108If we had not imposed the time-reversalsymmetric gauge, it is well-known that any loop integral of the U (1) connection [exemplified by Eq. (B6)] would only be gauge-invariant modulo one.Suppose an interpolation (parametrized by z ∈ [0, 1]) exists between semimetallic (z = 0) and gapped (z = 1) phases, which preserves both time-reversal and nonsymmorphic symmetries, as well as the spectral gap along both l(0) and l(π).We then introduce A(k) → A(k; z) to label the connection at a particular point in the interpolation.We would like to show that two polarization quantities, defined by in the time-reversal-symmetric gauge, are invariant modulo two throughout this interpolation.Having shown this, we would conclude from Eq. (B6) that ν, the Z 2 invariant in the fully-gapped, spin-orbit-coupled phase, obeys ν ∼ P(l(π); 0) − P(l(0); 0), (B13) which we evaluate with wavefunctions of the spin-orbitfree semimetal.Proof of invariance.Since l is mapped onto −l by the nonsymmorphic symmetry, Eqs. ( 3) and (4) show that P(l ; z) is independent of z, modulo large gauge transformations that modify P by some additive integer.The allowed gauge transformations that preserve Eq. (B4) cannot add an even integer to P , as we showed earlier in this Appendix.
In the spin-orbit-free limit, we may identify the Kramers indices (I and II) with the two spin components (↑ and ↓), for an arbitrarily chosen spin quantization axis.Additionally, using the time-reversal-symmetric gauge we express Eq.(B13) as: just as we did in Eq. (B10).Now applying the identity (3) with W[l(k x )] ≡ W(k x ), and further adding the subscript W → W ↑ to remind ourselves of the spin projection, As described in Sec.III B, the oriented loop l(π) intersects the screw-invariant (or glide-invariant) points X and M , where, at each of X and M , the nonsymmorphic eigenvalues of filled states always comprise ±i pairs.From the mapping of App.A, we deduce that all eigenvalues of W ↑ (π) come in complex-conjugate pairs, i.e., det[W ↑ (π)] = +1, leading us finally to Eq. (B5).

Appendix C: Derivation of the tight-binding model of monolayer MX2
To obtain a minimal, tight-binding model of the MX 2 compounds considered in this paper, we perform a Wannier-interpolation of the four bands closest to the Fermi level. 57The Wannier functions thus obtained transform as d x 2 −y 2 orbitals centered close to the M atoms and p x -type orbitals centered close to the X-1 atoms, as plotted in Fig. 9 and tabulated in Table III III.Wannier functions centers and atomic positions for WTe2 (left).Hopping parameters obtained from the Wannier-interpolation (right).The positions are given in units of the lattice vectors (the origin is the center of inversion), while the hopping parameters are given in eV.
the atomic position x/a = ±0.25)but not in e y ; consequently, the centers of the Wannier functions are slightly displaced in e y from the atomic centers.
Let us consider all symmetry-allowed, nearest-neighbor hoppings, and additionally two, next-nearest-neighbor hoppings (denoted by t d and t p ) along the chain where atoms are closely spaced; these hoppings are illustrated in Fig. 9.In a basis of real Wannier functions, timereversal symmetry constrains all hopping parameters to be real.Mx transforms the creation operators as: To calculate the photoelectron intensity in Eq. ( 12) due to light with polarization vector λ, one needs the transition matrix elements of the momentum operator p between the initial and final states of the electron, i.e., P if ± (k) = λ ± • f, k|p|i, k .Within the three-step model of photoemission, |i, k and |f, k are the initial and final Bloch state at wavevector k, respectively.We assume that the polarization of the light is in the plane of the bilayer, i.e., λ ± = [1, ±i, 0].The expression for the matrix element can be cast in the form where G are reciprocal lattice vectors.One obtains A final state with only one nonzero coefficient c f Gk leads to a vanishing CD signal, because where G x , G y are the components of G. 109,110 Therefore, to obtain a nonzero CD signal one needs to use a final state with at least two nonzero plane-wave coefficients.It is reasonable to assume that a high energy final Bloch state has only a few nonzero plane-wave components.Furthermore, this final state has to transform according to the symmetries of the lattice, which in case of the bilayer means that it has to transform as either an even or odd representation of Mx .In order compute the CD signal according to Eq. (D3), we chose a final state with nonzero coefficients c f Gk for the four smallest in-plane G-vectors, i.e., G = [± 2π a , 0, 0], [0, ± 2π b , 0].These coefficients were assumed to be independent of k and to have the same constant value.With this assumption the final state belongs to the totally symmetric representation of Mx .
The coefficients of the initial state were obtained using the plane-wave pseudopotential code in the Quantum ESPRESSO package.For Te and W we used the pseudopotentials Te.pbe-hgh.UPF and W.pbe-hgh.UPF from the Quantum ESPRESSO data base, respectively.The plane-wave cutoff was set to 80 Ry with a 15 × 8 × 1 kpoint mesh in the BZ.The plane-wave coefficients were computed on a 25 × 16 mesh around the pockets. 111

FIG. 2 .
FIG. 2. (a) Crystal structure of a MX2 monolayer.The horizontal lines are invariant under the screw C2x; the vertical line is invariant under Mx, and the green cross indicates the center of inversion and also our choice of spatial origin.Within each unit cell (encircled by the rectangle), we divide the four X atoms into two pairs marked X-1 and X-2; The X-2 atoms are further divided into two sublattices labelled by A and B. r ,s , with ∈ {d, p} and s ∈ {A, B}, are vectors connecting the origin to the centers of the Wannier functions that we introduce in Sec.IV A. (b) Crystal structure of a MX2 bilayer, where only the M atoms are shown.(c) Brillouin zone of both the monolayer and the bilayer.In Sec.III, we characterize the monolayer by screw-symmetric Wilson loops indicated in blue: l1 and l2 are contractible, while l(k x ) is not.

FIG. 3 .
FIG. 3. DFT band structures of monolayer ZrI2 (a), MoTe2 (b) and WTe2 (c) calculated without spin orbit coupling.Bands close to the Fermi level have been plotted in blue to emphasize the different connectedness of the bands between ZrI2 and MoTe2/WTe2.
of filled states (at X) must interpolate to unfilled states (at k), giving rise to (N +,Γ − N −,Γ )/2 chiral modes in the even branch of C2x , as illustrated for N +,Γ − N −,Γ = 1 and 2, respectively, in Fig.4(c) and (e).By a similar demonstration, the same number of odd, antichiral modes must interpolate between filled states at k to unfilled states at X.This leads to minimally (N +,k − N −,k )/2 screwprotected, Dirac crossings between chiral and antichiral modes, as highlighted by blue squares in Fig.4(c) and (e).If instead N +,k < N −,k , an analogous demonstration leads to minimally (N −,k − N +,k )/2 Dirac crossings [e.g., Fig. 4(b)].We have qualified |N +,k − N −,k |/2 as the minimal Dirac count ( Dk ), because the total count (D k ) can in principle be greater than Dk ) by any positive even number, due to band inversions away from k, e.g., Fig. 4(d) illustrates two Dirac crossings along ΓX (hence, D Γ = 2) despite N +,Γ = N −,Γ .This completes our proof of Eq. (1) for k along XΓX; the proof for k along M Y M is obtained by cosmetically substituting Γ → Y and X → M in the above demonstration.
(b), which has only a single Dirac fermion along ΓX and no other Dirac crossing in the cylinder bounded by Y ΓY and M XM .The filled states are characterized by N −,Γ = 2, N +,Γ = 0 and N +,Y = N −,Y = 1, which implies that the product of their screw eigenvalues equals −1.The third-from-bottom row of Table I informs us that the two W(0)-eigenvalues (per spin component) are +1 and −1,
(a), resulting in a discontinuous change in the band contours at the nodal energy, i.e., a Lifshitz transition between a type-I (|u x | < |v x |) spectrum with a closed Fermi circle surrounding k = 0, and a type-II (|u x | > |v x |)

FIG. 7 .
FIG. 7.(a) Hypothetical coupling of two band-inverted monolayers with type-I Dirac fermions: (a-i) depicts a doublydegenerate Dirac crossing with vanishing coupling, (a-ii) a semimetallic state with two distinct Dirac cones if the nonvanishing coupling preserves the screw symmetry, (a-iii) a trivial insulator in the case of nonvanishing screw-symmetric coupling.(b) DFT band structures of bilayer WTe2 for different inter-monolayer couplings: (b-i) vanishing, (b-ii) screwsymmetry-preserving, and (b-iii) screw-symmetry-breaking.

FIG. 8 .
FIG. 8. (a) Fermi-level dichroic signal [Ds(k, εF) defined in the main text] for a WTe2 bilayer.The Fermi surface has been broadened with a Lorentzian with a full width at half maximum of 10 meV.The Fermi pocket around Γ 35,59,60 does not encircle a massive Dirac fermion.(b) Ds(k, E) for the band structure along XΓX for energies near the Fermi level.

FIG. 9 .
FIG. 9. (a) Plots of the Wannier functions obtained by a 4-band Wannier-interpolation.The functions are labeled by their orbital character = d, p and their sublattice index A, B. Gray spheres represent M-atoms whereas as ochre spheres represent X-atoms.(b) Symmetry allowed hoppings considered for the tight-binding model.All hoppings are nearestneighbor hoppings with the exception of the intra-sublattice hoppings t d and tp .All symbols below the letter A (B) belong to the A (B) sublattice.

1 n=0(− 1 )
−1) l a † ,[−Rx,Ry] , Mx b † ,[Rx,Ry] M −1 x = (−1) l b † ,[−Rx+a,Ry] ,(C1)where l = 1 for = p and l = 0 for = d.This suppresses nearest-neighbor hopping terms of the form t AA pd a † p,R a d,R and t BB pd b † p,R b d,R .Given our choice of unit cell, intracell hoppings of the form t AB , a † ,R b ,R are mapped onto (−1) 1−δ , t AB a † ,[−Rx,Ry] b ,[−Rx+a,Ry] , which corresponds to an intercell hopping from the neighboring cell in the +e x direction.For = this hopping aquires a minus sign under Mx , which accounts for the factor of (−1) n in the Hamiltonian.Spatial inversion transforms the creation operators as:Ia † ,R I −1 = (−1) l b † ,−R Ib † ,R I −1 = (−1) l a † ,−R , hoppings in the same cell with = .In summary, the symmetry-allowed nearest-neighbor hoppings are the intrachain hoppings t AB , ≡ t AB with = d, p.The symmetry allowed interchain hoppings are t AB d,p ≡ t AB 0 .The latter switch sign depending on the hopping direction due to Mx and I [see Fig.9(b)].The real-space Hamiltonian then reads:H = R µ a † ,R a ,R + b † ,R b ,R + t a † ,R+ex a ,R + b † ,R+ex b ,R + n t AB 0 b † p,R+nex a d,R − b † d,R+nex a p,R + δ t AB l a † ,R+δ b ,R + h.c.. (C3)We construct a basis of Bloch waves by the Fourier transformation, ik•(R+r B, ) b † ,R , (C4)where N is the number of unit cells, and r s, denotes the position of the -type Wannier center in sublattice s, as illustrated in Fig.2(a).In the basis[c † k,d,A , c † k,p,A , c † k,d,B , c † k,p,B] the Hamiltonian is represented by the matrix:d + 2t d cos(k x ) 0 t AB d e −ik•(r B,d −r A,d ) (e iky + e i(ky−kx) ) t AB 0 e −ik•(r B,p −r A,d ) (1 − e −ikx ) 0 µ p + 2t p cos(k x ) t AB 0 e −ik•(r B,d −r A,p ) (e −ikx − 1) t AB p e −ik•(r B,p −r A,p ) (1 + e −ikx ) c.c c.c µ d + 2t d cos(k x relevantparameters for the Hamiltonian of WTe 2 obtained by the Wannier interpolation of the DFT bandstructure are given in TableIII.The Dirac crossing in the DFT bandstructure is type-II.But this is not reproduced by the tight-binding Hamiltonian (C5) with parameters obtained from the interpolation.This is due to the truncation of longer-ranged hoppings which would further tilt the Dirac cone.To retain a minimal tight-binding model that accounts for the type-II nature of the Dirac crossing, we renormalized the intra-sublattice hoppings t d , t p while leaving the other values untouched, which results in a tilted type-II Dirac cone.Figure6(a) was obtained by setting t d = −0.4eV and t p = 1.34 eV with a Fermi energy ε F = 1.47 eV.Appendix D: Circular dichroism dr e −ikr u f k (r) * p e ikr u ik (r)= λ ± • dr e −ikr u f k (r) * × k e ikr u ik + e ikr p u ik (r) = λ ± • dr u f k (r) * p u ik r. (D1)The Fourier expansion of the periodic part of a Bloch state is u nk (r) =

TABLE I .
Topological characterization of a nonsymmorphic crystal with filling f = 2 and either screw or glide symmetry.k1 and k2 are shorthand for the two screw-invariant (or glideinvariant) momenta at a particular kx: (kx, 0) and (kx, π).

TABLE II .
Topological characterization of a crystal with filling f = 4 and either glide or screw symmetry.The notation used here is explained in the caption of TableI.
. We remark that each Wyckoff position in the symmetry group has a fixed coordinate in e x (corresponding to