## Abstract

In quantum field theory, Weyl fermions are relativistic particles that travel at the speed of light and strictly obey the celebrated Lorentz symmetry. Their low-energy condensed matter analogs are Weyl semimetals, which are conductors whose electronic excitations mimic the Weyl fermion equation of motion. Although the traditional (type I) emergent Weyl fermions observed in TaAs still approximately respect Lorentz symmetry, recently, the so-called type II Weyl semimetal has been proposed, where the emergent Weyl quasiparticles break the Lorentz symmetry so strongly that they cannot be smoothly connected to Lorentz symmetric Weyl particles. Despite some evidence of nontrivial surface states, the direct observation of the type II bulk Weyl fermions remains elusive. We present the direct observation of the type II Weyl fermions in crystalline solid lanthanum aluminum germanide (LaAlGe) based on our photoemission data alone, without reliance on band structure calculations. Moreover, our systematic data agree with the theoretical calculations, providing further support on our experimental results.

- Topological Materials
- Weyl semimetals

## INTRODUCTION

In 1937, physicist Conyers Herring theoretically identified the conditions under which electronic bands in solids have the same energy and momentum in crystals that lack inversion symmetry (*1*). Near these band touching points, the low-energy excitations, or electronic quasiparticles, can be described by an equation that is essentially identical to the Weyl equation (*2*). Although Herring referred to the degeneracies as “accidental,” recent advances in band theory have led to the understanding that they are topologically protected and realize a so-called Weyl semimetal (*1*–*9*). The quasiparticles in the vicinity of these nodes are emergent Weyl fermions. They behave like Berry curvature monopoles or antimonopoles, depending on their respective chirality. The Weyl fermions of opposite chiralities are separated in momentum space. This electronic structure leads to a very special electromagnetic repose and novel nonlocal transport phenomena (*10*) of the material. In parallel electric and magnetic fields, the apparent conservation of the chiral charge is broken by the chiral anomaly, making a Weyl metal, unlike ordinary nonmagnetic metals, more conductive with an increasing magnetic field (*11*). Because of these novel properties, Weyl semimetals have the potential to be as important as graphene and topological insulators and have attracted worldwide interest.

Lorentz symmetry is the symmetry that requires physical laws to be independent of the frame of reference, in which they are observed (*12*). Hence, it is at the heart of any accepted physical theory, such as quantum field theory and Einstein’s theory of relativity, and has deep connections to the principle of causality and the charge, parity, and time-reversal symmetry (*13*–*15*). The search for Lorentz violation is one of the central themes of modern high-energy physics because many of the theories beyond the standard model, with the intention of further incorporating gravity, violate Lorentz symmetry. To date, Lorentz symmetry has survived a century of tests and remained to appear exact. In high-energy physics, Weyl fermions travel at the speed of light and therefore strictly obey Lorentz symmetry. Despite its robustness in high-energy physics, Lorentz symmetry is not present in low-energy condensed matter physics because slow-moving emergent electronic quasiparticles in crystals generally break Lorentz symmetry. Several groups theoretically considered the Lorentz violating emergent Weyl fermions in various forms including tilted Weyl cones and, as a result, the condensed matter realization of the Lorentz violating quantum field theory in a Weyl semimetal (*16*–*19*). Recently, it was realized that if the Lorentz violation in a Weyl semimetal is strong enough, then the emergent quasiparticles should be regarded as a new type of Weyl fermions, which is named type II Weyl fermions (*19*). In conventional (type I) Weyl semimetals, the emergent Weyl fermions respect (or at least approximately respect) the Lorentz symmetry. Specifically, the small Lorentz breaking of the type I Weyl fermions can be adiabatically removed. Therefore, any type I Weyl fermion is smoothly connected to the Lorentz invariant Weyl fermion. The type I Weyl cone arises from the crossing between two bands that have velocities of opposite sign for any momentum direction, and the Fermi surface consists of zero-dimensional isolated points if the Fermi level is set at the energy of the Weyl nodes. Type I Weyl semimetals have been realized in the TaAs class of materials and a double-gyroid photonic crystal (*7*–*9*). Fermi arcs and Weyl fermions have been observed by spectroscopic experiments (*7*–*9*), and the negative magnetoresistance due to the chiral anomaly has been reported in transport experiments on the TaAs class (*39*, *40*).

The novel type II Weyl fermions (*19*), in contrast, violate Lorentz symmetry so strongly that they cannot be smoothly connected back to the regular relativistic (Lorentz invariant) Weyl fermions (*19*–*31*). Such a type II Weyl fermion cone arises from the crossing between two bands that have the same sign of velocity along a certain direction. Its Fermi surface consists of an electron and a hole pocket that touch at the Weyl node in a topologically protected manner. Since the proposal of the concept of the type II Weyl fermions (*19*), there have been many theoretical works predicting specific new topological phenomena, including an exotic chiral anomaly whose transport response strongly depends on the direction of the electric current, an antichiral effect of the chiral Landau level, a modified anomalous Hall effect, magnetic breakdown, Klein tunneling, etc. (*19*, *26*–*31*). These predicted phenomena can be detected in transport and spectroscopic experiments, leading to protected properties that can be potentially used in future applications.

Because the type II Weyl fermion node is a touching point in the three-dimensional (bulk) Brillouin zone (BZ), it can only be shown by measuring the band structure along all three momentum space directions (*k*_{x}, *k*_{y}, *k*_{z}) (Figs. 1 and 2). To stress the importance of this experimental requirement, we compare two cases shown in Fig. 3 (J to M). In Fig. 3L, we show the Fermi surface of a type II Weyl semimetal where bulk electron and hole pockets touch to form the type II Weyl node. By contrast, in Fig. 3M, we have a trivial electron pocket and a trivial hole pocket without any Weyl nodes. they do not touch in the bulk BZ. These two cases can be clearly distinguished by measuring the band structure along all three momentum space directions (*k*_{x}, *k*_{y}, *k*_{z}). However, if one only has *k*_{x} and *k*_{y} resolution, then all bulk bands at different *k*_{z} values will be projected onto a single (*k*_{x}, *k*_{y}) plane. As shown in Fig. 3 (J and K), the two cases become indistinguishable because the projected Fermi surface in both cases shows a crossing between the projected electron and hole pockets. This example highlights the importance of resolving the bulk band structure along all three momentum space directions for experimentally observing type II Weyl fermions. Further discussions regarding this issue can be found in section S4.

## RESULTS

LaAlGe crystallizes in a body-centered tetragonal Bravais lattice, space group ** I**4

_{1}

**(109),**

*md*

*C*_{4v}point group. The lattice constants according to previous diffraction experiments are

**=**

*a***= 4.336**

*b***A**and

**= 14.828**

*c***A**(

*41*). The basis consists of two La atoms, two Al, and two Ge atoms, as shown in Fig. 1A. In this crystal structure, along the (001) direction, each atomic layer consists of only one type of element and is shifted relative to the layer below by half a lattice constant in either the

**or**

*x***direction. This shift gives the lattice a screw pattern along the**

*y***direction, which leads to a nonsymmorphic**

*z***rotation symmetry that includes a translation along the**

*C*_{4}**direction by**

*z***. Figure 1C shows the calculated bulk band structure along high-symmetry lines without SOC. It can be seen that the conduction and valence bands cross each other along the**

*c*/4**Γ**−

**Σ**−

**Σ**path, demonstrating a semimetal ground state. The momentum space configurations of the band crossings are shown in Fig. 1 (D and E). In the absence of SOC (Fig. 1D), the crossing between conduction and valence bands yields four closed loops, nodal lines on the

_{1}**and**

**= 0***k*_{x}**mirror planes, and also four pairs of (spinless) Weyl nodes on the**

*k*_{y}= 0**plane, which are denoted as W**

*k*_{z}= 0_{3}in Fig. 1D. Upon the inclusion of the SOC (Fig. 1E), the nodal lines are gapped out, and 24 Weyl nodes emerge in the vicinity, which is similar to the case of TaAs (

*7*). We refer to the 8 Weyl nodes located on the

**plane as W**

*k*_{z}= 0_{1}and the remaining 16 Weyl nodes away from this plane as W

_{2}. Moreover, when SOC is taken into account, each W

_{3}(spinless) Weyl node splits into two (spinful) Weyl nodes of the same chirality, which we call W

_{3}′ and W

_{3}″. Hence, in total, there are 40 Weyl nodes in a BZ, as shown in Fig. 1E. The projection of the Weyl nodes on the (001) surface BZ is shown in Fig. 1F. All W

_{1}Weyl nodes project as single Weyl nodes in close vicinity to the and points on the surface BZ edges. Two W

_{2}Weyl nodes of the same chiral charge project onto the same point near the midpoint of the

**Γ**−

**and**

*X***Γ**−

**lines. The W**

*Y*_{3}′ and W

_{3}″ nodes are projected near the midpoint of the

**Γ**−

**(diagonal) lines. As schematically shown in Fig. 1G, which is drawn according to our calculation results, the W**

*M*_{2}Weyl fermions are of type II, whereas the W

_{1}, W

_{3}′, and W

_{3}″ Weyl fermions are of type I. All W

_{1}, W

_{3}′, and W

_{3}″ nodes are far away from the Fermi level in energy, whereas the W

_{2}nodes are located almost exactly at the Fermi level. Figure 1H shows the angle-integrated photoemission spectrum of LaAlGe over a wide range of binding energy. We identify the La 4

*p*, 4

*d*, 5

*s*; Al 2

*s*, 2

*p*; and Ge 3

*s*, 3

*p*, 3

*d*core levels, confirming that our samples are composed of these three elements.

Below, we will focus on presenting our angle-resolved photoemission spectroscopy (ARPES) data on LaAlGe and show how these data demonstrate type II Weyl fermions without relying on the agreement with theoretical calculations. Figure 2A shows the measured ** k_{z}** −

**Fermi surface contours of the bulk bands on the**

*k*_{x}**plane. We observe large identical contours enclosing the**

*k*_{y}= 0**Σ**and

**Σ**points. In addition, we observe two small contours on the left- and right-hand sides of the big contour. These data unambiguously show that the observed band features are bulk bands rather than surface states by their definitions. A surface state is localized on the surface and therefore does not disperse along the

_{1}**direction. By contrast, a bulk state is delocalized throughout the bulk and therefore disperses along the**

*k*_{z}**direction with a period defined by bulk BZ. Our data in Fig. 2A show that the observed bands clearly have strong**

*k*_{z}**dispersions. Even more decisively, the measured periodicity of the**

*k*_{z}**dispersion matches the bulk BZ size along the**

*k*_{z}**direction, which is defined by the lattice constant**

*k*_{z}**=14.828 A, independently measured by x-ray diffraction (XRD). Therefore, these data unambiguously show that the observed band features are bulk bands.**

*c*Figure 2C shows the ARPES measured ** k_{x}** −

**Fermi surface using incident photon energy**

*k*_{y}**542**We see that the Fermi surface consists of small pockets along the

**eV**.**and**

*k*_{x}**axes. Figure 2D shows a high-resolution zoomed-in Fermi surface map of the region highlighted by the green box in Fig. 2C. From Fig. 2D, we clearly identify two Fermi pockets, which touch each other at two discrete zero-dimensional points, that is, two nodes, located on the opposite sides of the**

*k*_{y}**axis. Figure 2F shows the constant energy contour at**

*k*_{y}**=**

*E*_{B}**25**

**meV**of the same region. We see that the upper pocket expands, whereas the lower pocket shrinks upon changing the binding energy from

**to**

*E*_{B}= 0**25**This demonstrates that the upper pocket is hole-like, whereas the lower one is electron-like. Therefore, the two nodes observed in Fig. 2D are touching points between the electron and hole pockets. We now study the band dispersion away from the touching points along different momentum space cut directions. Figure 2 (B, D, and F) shows that the observed node arises from the crossing between two bands and that these two bands disperse linearly away from the node along all three

**meV**.**directions. Moreover, although the Fermi velocities of the two bands have the opposite signs along the**

*k***and**

*k*_{x}**directions, they have the same sign along the**

*k*_{z}**direction.**

*k*_{y}Now, on the basis of the established results in topological band theory, we show how our systematic data provide decisive evidence for the type II Weyl fermions. Our experimental demonstration below does not depend on the agreement between ARPES data and band structure calculations. Our data show (i) that the conduction and valence bands cross each other, (ii) that the crossing points are zero-dimensional, and (iii) that the bands disperse linearly away from the node along all three ** k** directions

**(**Therefore, the observed crossing points are either Dirac or Weyl nodes. Dirac nodes can only exist in systems with both time-reversal and space-inversion symmetries (

**,***k*_{x}**,***k*_{y}**).***k*_{z}*45*). Moreover, as exhaustively classified by topological band theory (

*45*), a Dirac node can only occur at a high-symmetry point or along a rotational axis in the bulk BZ. By contrast, Weyl nodes can only exist in systems, which either breaks time-reversal or space-inversion symmetry (

*4*,

*5*). Topological band theory shows that they occur at generic

**points because they are accidental band degeneracies, which do not require any additional crystalline symmetry beyond translation (**

*k**1*,

*4*,

*5*). In our case, the LaAlGe crystal breaks inversion symmetry, as reported in previous works (

*41*–

*43*) and also further confirmed by our XRD data (see section S1). Moreover, the

**space coordinate of observed crossing point is . This is obviously a generic**

*k***point, which does not coincide with any high symmetry point or rotational axis. These systematic data demonstrate that the observed band crossings are the Weyl nodes but are not sufficient to distinguish between types I and II, which is the main focus of our paper. We now proceed to prove the type II character based on its definition. Type I Weyl fermions have a typical conical dispersion, and their Fermi surface consists of isolated points, the Weyl nodes. Type II Weyl fermions manifest as tilted-over cones. Their Fermi surface consists of a hole pocket and an electron pocket touching at points, the type II Weyl nodes (**

*k**19*). Along the tilting direction, the Fermi velocities of the two bands that form the Weyl cone are of the same sign, whereas along the other two directions perpendicular to the tilting direction, the Fermi velocities are of opposite signs. Our data on the Fermi surface (Fig. 2D), the constant energy contour (Fig. 2F), and the energy dispersions along all three

**directions (Fig. 3 B, D, and F) provide all the decisive evidence, as described above. Therefore, we have unambiguously and clearly shown the type II Weyl fermions in LaAlGe. The decisive experimental proof without relying on data-calculation agreement achieved here is in sharp contrast with the ARPES works on W**

*k*_{1−x}Mo

_{x}Te

_{2}(

*32*–

*38*). The theoretical predictions of the type II Weyl fermion state in W

_{1−x}Mo

_{x}Te

_{2}and LaAlGe have already been well established by first-principles calculations. Hence, showing some data feature in agreement with calculations and arguing for the experimental observation based on such an agreement go little beyond the calculation/prediction itself. Calculations do not account for sample quality issues. Thus, only an experimental observation independent of data-calculation agreement can demonstrate that the sample quality enters the regime where the predicted novel physics is present in real materials.

On the basis of our data, we can quantify the amount of Lorentz symmetry breaking and fully characterize the Weyl fermions observed in LaAlGe. From our ARPES data, we can directly obtain the Fermi velocities of the two bands that form the Weyl cone along the following momentum space directions: (100), (010), (001), (110), (101), and (011). For example, we denote the Fermi velocities of the two bands along the (100) direction as ** v_{±(100)}**. We show that these data allow us to obtain systematically quantitative information of the Weyl cones. Using these Fermi velocities, we define a three-vector

**= (**

*v***), where and the definitions for**

*v*_{1},*v*_{2},*v*_{3}**and**

*v*_{2}**follow. We also define a**

*v*_{3}**3 × 3**matrix whose components are

*U*_{1,1}= (

*v*_{+,(100)}−

**as well as**

*v*_{−,(100)})^{2}/4

*U*_{1,2}= [(

*v*_{+,(110)}−

*v*_{−,(110)})

^{2}− (

*v*_{+,(100)}−

*v*_{−,(100)})

^{2}− (

*v*_{+,(010)}−

**, and the definitions of the other components will follow. Using**

*v*_{−,(010)})^{2}]/8**and , one can reconstruct the full low-energy dispersion**

*v***ϵ(**by as well as the Berry curvature . Finally, the type of the Weyl cone is determined by the eigenvalues of the

*k*)**3 × 3**matrix (

*19*): If all eigenvalues of this matrix are positive, then the Weyl cone is type I, otherwise it is type II.

Using our ARPES data, we can determine(1)The eigenvalues of are **2.46**,** − 1.74**, and **0.85** **eV**^{2}∙**A**** ^{2}**, which again proves the type II nature of the Weyl nodes. This precise information about the low-energy effective theory will, for example, be important to analyze the anomalous transport response of Weyl semimetals sourced by the the Berry curvature around the Weyl node.

The above experimental demonstration and quantitative characterization of the type II Weyl fermions in LaAlGe are only based on our systematic experimental data and established results in topological band theory. We did not rely on any agreement between data and calculations. We now compare our data and calculations in Figs. 2 and 3. All the critical properties including the Fermi surface (Figs. 2, D and E), the constant energy contour (Fig. 2, F and G), and the energy dispersions along all three ** k** directions (Fig. 3, B to G) are qualitatively consistent between data and calculations. We emphasize that the bulk band structure calculations, unlike the surface-state calculations, do not have free parameters and therefore are quite reliable. The agreement between our data and calculations provide further strong support on our conclusions.

We also present ** k_{x}** −

**Fermi surface data at a different photon energy, which corresponds to the**

*k*_{y}**value of the W**

*k*_{z}_{1}, W

_{3}′, and W

_{3}″ Weyl nodes (

**). Figure 2 (H and I) show the calculated and measured Fermi surfaces at**

*k*_{z}= 0**, respectively. We observe three types of pockets, a small pocket near each BZ boundary, a trapezoid-shaped pocket along each diagonal (**

*k*_{z}= 0**45°**) direction, and a circular pocket in between two adjacent trapezoidal pockets. The data (Fig. 2I) and calculation (Fig. 2H) are in good agreement. From the calculations, we know that the small pocket near the boundary arises from a pair of nearby W

_{1}Weyl fermions, and the trapezoid-shaped pocket arises from pairs of W

_{3}′ and W

_{3}″ Weyl fermions. Because the energy of the W

_{1}, W

_{3}′, and W

_{3}″ Weyl nodes are far above the Fermi level, the constant energy contour at the Fermi level already merges into a single pocket and hence does not carry net chiral charge.

We show that the type II Weyl nodes dominate the low-energy Berry curvature physics. The color plots in Fig. 4 (B and C) show the Berry curvature magnitude in ** k_{x}** −

**space at the two different**

*k*_{y}**values. We note that Fig. 4 (B and C) considers the Berry curvature magnitude summed over contributions from a wide energy range. It can be seen that the Berry curvature is indeed dominated by the contribution from the Weyl nodes. By contrast, the trivial pocket at**

*k*_{z}**has no observable Berry curvature contribution. To understand the low-energy Berry curvature physics, which can be measured by certain transport experiments such as the negative longitudinal magnetoresistance, we need to know the Berry curvature in close vicinity of the Fermi energy. In Fig. 4 (E and F), we again show the Berry curvature magnitude but only considering the contribution near the Fermi level within a ±10-meV window. We see that the Fermi pockets that arise from W**

*k*_{z}= 0_{1}and W

_{3}′(W

_{3}″) Weyl cones (Fig. 4E) do not carry observable Berry curvature near the Fermi level. On the other hand, Fermi pockets that arise from the type II Weyl cones (Fig. 4E) show strong Berry curvature in the vicinity of the W

_{2}Weyl nodes. This is quite intuitive. Because the Weyl nodes are monopoles, the Berry curvature decays rapidly as away from the energy of the node. Therefore, the Fermi pockets from W

_{1}and W

_{3}′(W

_{3}″) Weyl cones show negligibly small Berry curvature contribution. We prove that the type II Weyl nodes dominate the low-energy Berry curvature physics, which dictates topological phenomena such as the negative longitudinal magnetoresistance and the nonlocal transport due to the chiral anomaly (

*10*,

*11*).

## DISCUSSIONS

We discuss the distinction between LaAlGe and the W_{1−x}Mo_{x}Te_{2} systems. Current experimental searching for the type II Weyl fermions has been focused on the W_{1−x}Mo_{x}Te_{2} system (*19*–*22*, *32*–*38*). Most works tried to study the surface states in connection to the Fermi arcs (*32*–*38*). However, the surface Fermi arcs cannot discern whether the bulk Weyl fermions are of type I or type II. Huang *et al.* (*32*) argued the existence of Weyl nodes by normalizing the 6-eV ARPES spectrum taken at high temperature with a Fermi Dirac distribution. As we explained in Fig. 3 (J to M), the projected band structure without *k*_{z} resolution cannot distinguish a type II Weyl node in the bulk BZ (Fig. 3K) from a projected trivial crossing between a trivial electron and a trivial hole pocket. Moreover, the type II Weyl nodes in W_{1−x}Mo_{x}Te_{2} is quite far away (~50 meV) from the chemical potential. By contrast, our systematic ARPES data on LaAlGe provide the clear and direct observation of the type II Weyl fermions. The type II Weyl nodes in LaAlGe are located exactly at the Fermi level, and their Berry curvatures dominate at the Fermi level. Finally, we suggest that one could engineer the chemical potential by chemically doping LaAlGe. For example, both La_{1−x}Ce_{x}AlGe and LaAl_{1−x}Ge_{1+x} may electron dope the system, by which we may be able to move the chemical potential to the W_{1}, W_{3}′, or W_{3}″ Weyl nodes. This may give rise to a transition from a type II to a type I Weyl semimetal in an engineered LaAlGe system.

## MATERIALS AND METHODS

High-quality single crystals of LaAlGe were grown by the flux method (*46*). The SX-ARPES measurements were performed at the ADRESS Beamline at the Swiss Light Source, Paul Scherrer Institute in Villigen, Switzerland. The experimental geometry of SX-ARPES has been described in the study of Strocov *et al.* (*47*). The sample was cooled down to 12 K to quench the electron-phonon interaction effects, reducing the *k*-resolved spectral fraction. The SX photon energy ranged from 300 to 1000 eV. The combined (beamline and analyzer) experimental energy resolution of the SX-ARPES varied between 40 and 80 meV. The angular resolution of the SX-ARPES analyzer was 0.07°. High-resolution vacuum ultraviolet ARPES measurements were performed using the synchrotron radiation light source–based ARPES instruments at Beamline 10.0.1 of the Advanced Light Source (ALS) at the Lawrence Berkeley National Laboratory in Berkeley, CA. The energy and momentum resolution of the vacuum ultraviolet ARPES instruments were better than 30 meV and 1% of the surface BZ. Samples were cleaved in situ under a vacuum condition better than 5 × 10^{−11} torr.

The first-principles calculations were performed within the density functional theory (*48*) framework with the projector augmented wave basis, using the Vienna ab initio simulation package (*49*, *50*). The Perdew-Burke-Ernzerhof type generalized gradient approximation was used to describe the exchange-correlation energy (*51*). A plane wave cutoff of 320 eV was used, and an 8 × 8 × 8 *k*-mesh was chosen to sample the bulk BZ. The total energies were converged to 10^{−8} eV. SOC was taken into account self-consistently to treat relativistic effects. The experimentally measured lattice constants, *a* = *b* = 4.344 A and *c* = 14.812 A, determined from our own XRD measurements were used to compute the bulk band structures. To calculate the surface states, we constructed a first-principle tight-binding model Hamilton. The real-space tight-binding Hamiltonian was constructed using the Wannier function method with La *s*, *d*, *f*; Al *s*, *p*; and Ge *s*, *p* orbitals (*52*). We calculated the surface spectral weight of semi-infinite (001) slab of LaAlGe using Green’s function method.

## SUPPLEMENTARY MATERIALS

Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/3/6/e1603266/DC1

section S1. Additional methods

section S2. Full characterization of the type II Weyl fermions in LaAlGe

section S3. Decisive evidence for type II Weyl fermions

section S4. The importance of measuring the band structure along all three *k* (*k*_{x}, *k*_{y}, *k*_{z}) directions and the issues in W_{1−x}Mo_{x}Te_{2}

section S5. Topological definition of Fermi arc surface states and its implications for surface states in LaAlGe and W_{1−x}Mo_{x}Te_{2}

section S6. Surface-state band structure of LaAlGe

section S7. Fermi arc surface states associated with the W_{2} Weyl nodes in LaAlGe

section S8. Fermi arc surface states associated with the W_{3}′ and W_{3}″ Weyl nodes

section S9. Topological definition of Fermi arc surface states and its implications for surface states in LaAlGe and W_{1−x}Mo_{x}Te_{2}

table S1. Energy dispersive spectroscopy (EDS) data.

table S2. ARPES-measured Fermi velocities near W_{2}.

table S3. Calculated Fermi velocities near W_{2}.

fig. S1. Energy-dispersive x-ray spectroscopy of LaAlGe samples.

fig. S2. XRD data of LaAlGe.

fig. S3. Low-energy electron diffraction data of LaAlGe taken at 100-eV electron energy.

fig. S4. The 40 Weyl nodes in LaAlGe.

fig. S5. A comparison between the Fermi arcs in type I and type II Weyl semimetals, demonstrating that the Fermi arcs (the green line) cannot be used to discern between type I and type II Weyl fermions.

fig. S6. A comparison between a true crossing in the bulk band structure and a “projected” crossing in the projected band structure.

fig. S7. The presence of projected crossing in the projected band structure of WTe_{2} even when the bulk has no Weyl nodes.

fig. S8. Definition of the topological Fermi arc surface states.

fig. S9. Bulk band structure and its projection onto the surface in LaAlGe.

fig. S10. TaAs surface Fermi surface calculations under three different surface onsite potential values.

fig. S11. Band structure of WTe_{2} under two slightly different lattice constant values.

fig. S12. A side-by-side comparison of the main data figures between our work and the two W_{1−x}Mo_{x}Te_{2} works (*32*, *37*).

fig. S13. Fermi arc–like and Fermi arc–derived surface states in LaAlGe.

fig. S14. Fermi arc–like and Fermi arc–derived surface states in LaAlGe.

fig. S15. Fermi arc surface states associated with the W_{3}′ and W_{3}″ Weyl nodes.

fig. S16. Connectivity pattern of the Fermi arcs associated with the W_{3}′ and W_{3}″ Weyl nodes.

This is an open-access article distributed under the terms of the Creative Commons Attribution-NonCommercial license, which permits use, distribution, and reproduction in any medium, so long as the resultant use is **not** for commercial advantage and provided the original work is properly cited.

## REFERENCES AND NOTES

**Acknowledgments:**We acknowledge J. D. Denlinger, S. K. Mo, A. V. Fedorov, M. Hashimoto, M. Hoesch, T. Kim, and V. N. Strocov for their beamline assistance at the ALS, the Stanford Synchrotron Radiation Lightsource, the Diamond Light Source, and the Swiss Light Source. We also thank D. Huse, I. Klebanov, A. Polyakov, P. Steinhardt, H. Verlinde, and A. Vishwanath for discussions. H. Lin acknowledges visiting scientist support from Princeton University.

**Funding:**Work at Princeton University and Princeton-led synchrotron-based ARPES measurements were supported by the Gordon and Betty Moore Foundation EPiQS Initiative through (grant GBMF4547 to M.Z.H.). Single-crystal growth was supported by the National Basic Research Program of China (grant nos. 2013CB921901 and 2014CB239302), and characterization was supported by the U.S. Department of Energy (DE-FG-02-05ER46200). First-principles band structure calculations at the National University of Singapore were supported by the National Research Foundation (NRF), Prime Minister’s Office, Singapore under its NRF fellowship (NRF award no. NRF-NRFF2013-03). The work at Northeastern University was supported by the U.S. Department of Energy (DOE), Office of Science, Basic Energy Sciences grant number DE-FG02-07ER46352, and benefited from Northeastern University’s Advanced Scientific Computation Center (ASCC) and the NERSC supercomputing center through DOE grant number DE-AC02-05CH11231. M.-A.H. was supported by the Swiss Excellence Scholarship Grant ESKAS-No. 2015.0257. T.-R.C. and H.-T.J. are supported by the Ministry of Science and Technology, the National Tsing Hua University, the National Cheng Kung University, and the Academia Sinica, Taiwan. T.-R.C. and H.-T.J. also thank the National Center for High-performance Computing, the Computer and Information Networking Center-National Taiwan University, and the National Center for Theoretical Sciences, Taiwan for technical support. H.-T.J. received funding from the Ministry of Science and Technology, Taiwan (103-2112-M-007 -018 -MY3) from 1 August 2014 to 31 July 2017.

**Author contributions:**S.-Y.X., N.A., I.B., D.S.S., G.B., and H.Z. conducted the ARPES experiments with assistance from M.-A.H., V.N.S., and M.Z.H. H. Lu, X.Z., Y.B., and S.J. grew the single-crystal samples. G.C., B.S., S.-M.H., C.-H.H., T.-R.C., H.-T.J., A.B., and H. Lin performed first-principles band structure calculations. T.N. did theoretical analyses. M.Z.H. was responsible for the overall direction, planning, and integration among different research units.

**Competing interests:**The authors declare that they have no competing interests.

**Data and materials availability:**All data needed to evaluate the conclusions in the paper are present in the paper and/or the Supplementary Materials. Additional data related to this paper may be requested from M.Z.H. (mzhasan{at}princeton.edu).

- Copyright © 2017, The Authors