REVIEWS OF MODERN PHYSICS, VOLUME 90, JANUARY–MARCH 2018 Weyl and Dirac semimetals in three-dimensional solids
N. P. Armitage The Institute for Quantum Matter, Department of Physics and Astronomy, The Johns Hopkins University, Baltimore, Maryland 21218, USA
E. J. Mele Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA
Ashvin Vishwanath Department of Physics, University of California, Berkeley, California 94720, USA and Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA
(published 22 January 2018) Weyl and Dirac semimetals are three-dimensional phases of matter with gapless electronic excitations that are protected by topology and symmetry. As three-dimensional analogs of graphene, they have generated much recent interest. Deep connections exist with particle physics models of relativistic chiral fermions, and, despite their gaplessness, to solid-state topological and Chern insulators. Their characteristic electronic properties lead to protected surface states and novel responses to applied electric and magnetic fields. The theoretical foundations of these phases, their proposed realizations in solid-state systems, and recent experiments on candidate materials as well as their relation to other states of matter are reviewed.
DOI: 10.1103/RevModPhys.90.015001
CONTENTS V. Experimental Results 30 A. Identifying Dirac and Weyl systems through I. Introduction 1 their band structure 30 II. Properties of Weyl Semimetals 3 B. Semiclassical transport and optics 34 A. Background 3 1. General considerations 34 1. Accidental degeneracies 3 2. Experiments 36 2. Weyl and Dirac fermions 4 C. Quantum mechanical effects in transport 37 3. Topological invariants for band insulators 4 1. Quantum oscillations 37 B. Topological aspects of Weyl semimetals 4 2. The chiral anomaly 40 T 1. Weyl semimetals with broken symmetry 7 3. Surface state transport 43 P 2. Weyl semimetals with broken symmetry 8 4. Nonlinear probes 44 C. Physical consequences of topology 9 VI. Related States of Matter 45 1. Fermi arc surface states 9 A. Topological line nodes 45 2. The chiral anomaly 10 1. Nodal lines in the absence of spin-orbit coupling 45 3. Anomalous Hall effect 12 2. Nodal lines in spin-orbit coupled crystals 46 4. Axion electrodynamics of Weyl semimetals 13 B. Relation to nodal superconductors and superfluids 46 5. Interplay between chiral anomaly and surface C. Quadratic band touchings and the Luttinger semimetal 47 Fermi arcs 13 D. Kramers-Weyl nodes and “new” fermions 48 D. Disorder effects on Weyl semimetals 15 E. Possible realizations in nonelectronic systems 49 III. Dirac Semimetals in Three Dimensions 15 VII. Concluding Remarks 50 A. Dirac semimetal from band inversion 16 Acknowledgments 51 B. Symmetry-enforced Dirac semimetals 18 References 51 C. Classification of four band models for Dirac semimetals 19 I. INTRODUCTION D. Phenomena of Dirac semimetals 21 1. Fermi arcs in Dirac semimetals 21 In 1928 P. A. M. Dirac proposed, in the first successful 2. The chiral anomaly in Dirac semimetals 23 reconciliation of special relativity and quantum mechanics, his IV. Materials Considerations 23 now eponymous Dirac equation (Dirac, 1928). Its form A. Weyl semimetals 24 resulted from the constraints of relativity that space and time 1. Noncentrosymmetric Weyl semimetals 24 derivatives must appear in the same order in the equations 2. Magnetic Weyl semimetals 26 of motion, as well as constraints from the probabilistic B. Dirac semimetals 28 interpretation of the wave function that the equations of
0034-6861=2018=90(1)=015001(57) 015001-1 © 2018 American Physical Society N. P. Armitage, E. J. Mele, and Ashvin Vishwanath: Weyl and Dirac semimetals in three- … motion depend only on the first derivative of time. Dirac’s Majorana fermions, and their manifestation in condensed solution used 4 × 4 complex matrices that in their modern matter systems. In 1929, shortly after Dirac wrote down his form are referred to as gamma matrices and a four component equation for the electron which involved these complex 4 × 4 wave function. The four components allowed for both positive matrices, Weyl pointed out a simplified relativistic equation and negative charge solutions and up and down spin. This utilizing just the 2 × 2 complex Pauli matrices σn. This epochal moment in theoretical physics, originating in these simplification required the fermions to be massless. Weyl simple considerations, led to a new understanding of the fermions are associated with a chirality or handedness, and a concept of spin, predicted the existence of antimatter, and was pair of opposite chirality Weyl fermions can be combined to the invention of quantum field theory itself. A number of obtain a Dirac fermion. It had been believed that neutrinos variations of the Dirac equation quickly followed. In 1929, the might be Weyl fermions. However, with the discovery of a mathematician Hermann Weyl proposed a simplified version nonvanishing neutrino mass (Kajita, 2016; McDonald, 2016), that described massless fermions with a definite chirality (or there are no fundamental particles currently believed to be handedness) (Weyl, 1929). In 1937, Ettore Majorana found a massless Weyl fermions. As mentioned Majorana also found a modification using real numbers, which described a neutral modification of the Dirac equation that used real numbers and particle that was its own antiparticle (Majorana, 1937; Elliott described a neutral particle that was its own antiparticle and Franz, 2015). These developments have found vast (Majorana, 1937; Elliott and Franz, 2015). application in modern particle physics. The Dirac equation In a seemingly unrelated line of reasoning, the conditions is now the fundamental equation describing relativistic elec- under which degeneracies occur in electronic band structures trons and the Majorana equations are a candidate to describe were investigated by Herring (1937). It was noted that even in neutrinos. The Dirac equation is also a key concept leading to the absence of any symmetry one could obtain accidental topological phenomena such as zero modes and anomalies in twofold degeneracies of bands in a three-dimensional solid. quantum field theories. Unfortunately in the nearly 90 The dispersion in the vicinity of these band touching points is intervening years no candidate Weyl fermions have been generically linear and resembles the Weyl equation, modulo observed as fundamental particles in high-energy particle the lack of strict Lorentz invariance. Remarkably several of the physics experiments. defining physical properties of Weyl fermions, such as the so- In condensed matter physics, where one is interested in called chiral anomaly, continue to hold in this nonrelativistic energy scales much smaller than the rest mass of the electron, condensed matter context. The chiral anomaly discussed by it would appear that a nonrelativistic description, perhaps with Adler (1969) and Bell and Jackiw (1969) is an example of a minor corrections, would suffice and that Dirac physics would quantum anomaly which in its simplest incarnation demon- not play an important role. However, the propagation of even strates that a single Weyl fermion coupled to an electromag- slow electrons through the periodic potential of a crystal leads netic field results in the nonconservation of electric charge. To to a dressing of the electronic states. In certain instances this evade this unphysical consequence, the net chirality of a set of results in an effective low-energy description that once again Weyl fermions must vanish in a lattice realization, an example resembles the Dirac equation. Perhaps the best known of the fermion doubling theorem. However, even in this setting example of this phenomena is in graphene, where a linear it was realized that the chiral anomaly can have a nontrivial momentum dispersion relation is captured by the massless effect (Nielsen and Ninomiya, 1983), which cemented the link two-dimensional Dirac equation. These 2D carbon sheets between band touchings in three-dimensional crystals and provide a condensed matter analog of (2 þ 1)-dimensional chiral fermions. These band touchings of Herring were named quantum electrodynamics (QED) (Semenoff, 1984) and with “Weyl nodes” by Wan et al. (2011). The electrodynamic the contemporary isolation of single layer graphene sheets has properties of these and other band touching points had been resulted in a large body of work on their electronic properties studied by Abrikosov and Beneslavskii (1971a). (Novoselov et al., 2004, 2005; Zhang et al., 2005; Neto et al., Topological consequences of Weyl nodes began to be 2009; Geim, 2012). Recent work has found evidence for explored with the realization that Berry curvature plays a Majorana bound states in 1D superconducting wires (Mourik key role in determining the Hall effect (Karplus and Luttinger, et al., 2012; Elliott and Franz, 2015). In this review we are 1954; Thouless et al., 1982), and the Weyl nodes are related to mainly concerned with analogous physics in three-dimen- “diabolic points” discussed by Berry as sources of Berry flux sional crystals with linearly dispersing fermionic excitations (Berry, 1985). The fact that diabolic points are monopoles of that are described by the massless 3D Weyl and Dirac Berry curvature and could influence the Hall effect in equations. These solid-state realizations offer a platform ferromagnets was emphasized by Fang et al. (2003) and where predictions made by relativistic theories can be tested, Nagaosa et al. (2010). In a different context, realizations of but at the same time entirely new properties that exist only in a Weyl nodes in superfluids and superconductors have been condensed matter context emerge, such as Fermi arc surface discussed (Volovik, 1987; Murakami and Nagaosa, 2003). In states at the boundary of the material. Moreover, the fact that particular, Volovik (1987) pointed out that the A phase of the strict symmetries of free space do not necessarily hold in a superfluid He3 realizes nodes in the pairing function leading to lattice means that new fermion types with no counterpart in a realization of Weyl fermions. high-energy physics (Soluyanov et al., 2015; Xu, Zhang, and The prediction and discovery of topological insulators (TIs) Zhang, 2015; Bradlyn et al., 2016; Wieder et al., 2016) can in two and three dimensions (Haldane, 1988; Kane and Mele, emerge (e.g., type-II Weyl, spin-1 Weyl, double Dirac, etc.). 2005; Bernevig, Hughes, and Zhang, 2006; Fu and Kane, Let us further review the history of the three-dimensional 2007; Fu, Kane, and Mele, 2007; Moore and Balents, 2007; Weyl equation (Weyl, 1929), its relation to Dirac and Hsieh et al., 2008; Chen et al., 2009; Roy, 2009; Y. Xia et al.,
Rev. Mod. Phys., Vol. 90, No. 1, January–March 2018 015001-2 N. P. Armitage, E. J. Mele, and Ashvin Vishwanath: Weyl and Dirac semimetals in three- …
2009; Zhang et al., 2009) has led to an explosion of activity in been discovered (Borisenko et al.,2014; Z. Liu et al., 2014a, the study of topological aspects of band structures (Hasan and 2014b; Neupane et al., 2014; Xu et al., 2015b). Kane, 2010; Qi and Zhang, 2011). While initially confined to There are a number of excellent reviews on related topics in the study of band insulators, where topological properties could this general area. Basic concepts related to Weyl semimetals be sharply delineated due to the presence of an energy gap, were reviewed by Turner and Vishwanath (2013) as well as by interesting connections to gapless states have begun to emerge. Witten (2015). Transport properties of Weyl semimetals were For one, the surfaces of topological insulators in 3D feature a reviewed by Hosur and Qi (2013) and Burkov (2015a), while gapless Dirac dispersion, analogous to the two-dimensional the extensive contributions of ab initio techniques to the semimetal graphene, but with important differences in the discovery of topological materials (including Weyl and number of nodal points. The transition between topological and Dirac states) were reviewed by Bansil, Lin, and Das (2016) trivial phases proceeds through a gapless state, for example, a and Weng, Dai, and Fang (2016). Kharzeev (2014) reviewed 3D topological to trivial insulator transition, in the presence of connections of Weyl fermions in solid-state physics to the chiral both time reversal and inversion symmetry proceeds through anomaly in quantum chromodynamics. A number of other the 3D Dirac dispersion (Murakami, 2007). If inversion shorter reviews on specific aspects of topological semimetals symmetry is lost, then the critical point expands into a gapless have appeared recently (Witczak-Krempa et al., 2014; Hasan, phase with Weyl nodes that migrate across the Brillouin zone Xu, and Bian, 2015; Burkov, 2016, 2017; Jia, Xu, and Hasan, (BZ) and annihilate with an opposite chirality partner, leading 2016; Hasan et al., 2017; Šmejkal, Jungwirth, and Sinova, to the change in topology. 2017; Yan and Felser, 2017; Syzranov and Radzihovsky, 2018) A direct manifestation of the topological aspects of Weyl and connections to related systems such as nodal supercon- fermions appeared with the realization that Weyl nodes lead to ductors have been reviewed by Vafek and Vishwanath (2014), exotic surface states in the form of Fermi arcs (Wan et al., Wehling, Black-Schaffer, and Balatsky (2014), and Schnyder 2011). The name “Weyl semimetal” (WSM) was introduced to and Brydon (2015). In this review, we attempt to summarize the describe a phase where the chemical potential is near the Weyl theoretical, material, and experimental situations of 3D Dirac nodes and a potential realization of such a state in a family of and Weyl semimetals with an emphasis on general features materials, the pyrochlore iridates, was proposed along with the independent of specific material systems. There have been prediction of a special all-in, all-out magnetic ordering pattern many interesting developments in this area in the last few years, (Wan et al., 2011). A subsequent proposed realization of Weyl in regards to the theoretical proposals, the development of new semimetals in magnetic systems includes the spinel HgCr2Se4 materials, and the study of experimental phenomena. Although (a double Weyl) (G. Xu et al., 2011), heterostructures of we have been attentive to matters of priority, the experimental ferromagnets and topological insulators (Burkov and Balents, data we include are not necessarily the first that were shown to demonstrate some phenomenon, but this is our estimation of 2011), and Hg1−x−yCdxMnyTe films (Bulmash, Liu, and Qi, 2014). Although a clear-cut demonstration of a magnetic that most illustrative effect. Unfortunately even in the relatively WSM remains outstanding, the search for inversion-breaking well-defined scope of the current topic, the literature is vast and Weyl systems as envisaged by Murakami (2007) and Halász we cannot hope to cover all work. Important omissions are and Balents (2012) reached fruition with the prediction and regrettable but inevitable. discovery of TaAs as a WSM (S.-M. Huang et al., 2015; Lv et al., 2015b, 2015c; Weng, Fang et al., 2015; Xu et al., II. PROPERTIES OF WEYL SEMIMETALS 2015b) [and other members in this material class (Xu et al., 2015a, 2015c; N. Xu et al., 2016)], and the observation of A. Background Fermi arc surface states attached to the bulk Weyl points. Here we first review two seemingly unrelated topics that For the case of 3D material systems described by the massless originated in the early days of quantum mechanics: the Dirac equation, the possibility of stable fourfold-degenerate problem of level repulsion and accidental degeneracies and Dirac points (DP) was raised by Abrikosov and Beneslavskii the relativistic wave equations for fermions. We will see that (1971a) and more recently by Wang et al. (2012) and Young these provide complementary perspectives on Weyl semimet- et al. (2012). Unlike the case of Weyl points, this degeneracy is als and are unified by the identification of their topological not topologically protected since its net Chern number is zero aspects. Toward this end we will review topological invariants and residual momentum-conserving terms in the Hamiltonian of insulators in the third part of this section. can potentially mix these terms and gap the electronic spectrum. However, in particular situations this mixing can be forbidden 1. Accidental degeneracies by space group symmetries in which case the nodes remain intact as symmetry-protected degeneracies. For instance this can As a starting point, consider the basic question of when occur at a phase transition between TI and non-TI phases in a accidental degeneracies arise in an energy spectrum (von crystal system that preserves inversion (Murakami, 2007; Neuman and Wigner, 1929). We focus on a pair of energy Murakami et al.,2007). However, as pointed out by Wang levels and ask if one can bring these levels into degeneracy by et al. (2012), Young et al. (2012),andSteinberg et al. (2014) a tuning Hamiltonian parameters. The energy levels (up to Dirac semimetal (DSM) can also appear as a robust electronic an overall constant) are determined by the most general 2 2 σ σ σ phase that is stable over a range of Hamiltonian control × Hamiltonian H ¼ f1 x þ f2 ypþffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffif3 z, with an energy 2 2 2 parameters. Such systems are called DSMs. A number of splitting between the levels ΔE ¼ 2 f1 þ f2 þ f3. In gen- material realizations of such symmetry-protected DSMs have eral in the absence of any symmetry, this cannot be
Rev. Mod. Phys., Vol. 90, No. 1, January–March 2018 015001-3 N. P. Armitage, E. J. Mele, and Ashvin Vishwanath: Weyl and Dirac semimetals in three- … accomplished by tuning just one parameter; degeneracy Let us now specialize to d ¼ 3. The gamma matrices are, as requires tuning all three terms to give zero simultaneously. Dirac found, now 4 × 4 matrices and can be represented as 0 i i 5 If we focus on real Hamiltonians with time reversal symmetry, γ ¼ I ⊗ τx, γ ¼ σ ⊗ iτy, and γ ¼ −I ⊗ τz. Again, if we we can exclude the imaginary Pauli matrix. Then, a pair of γ ψ ψ ψ identify chiral components 5 ¼ , where are levels can be brought into coincidence typically by tuning two effectively two component vectors, we have for the massless ϵ 0 parameters, since we can typically solve two equations x ¼ Dirac equation and ϵz ¼ 0 with two variables. But in the absence of any such symmetry, we need to tune three independent parameters to ∂ ψ ψ i t ¼ H ; achieve a degeneracy. As mentioned, these points of degen- ∓ ⃗ σ⃗ eracy in the extended two or three parameter space are termed H ¼ p · . ð3Þ diabolic points and have been discussed in the context of Berry’s phase (Berry, 1985) and not surprisingly will be Thus Weyl fermions propagate parallel (or antiparallel) to associated with a topological property as we discuss next. their spin, which defines their chirality. We will see that a single chirality of Weyl fermions cannot be realized in 3D, but 2. Weyl and Dirac fermions momentum separated pairs can arise. These are the Weyl semimetals. The Dirac equation in d spatial dimension and effective speed of light c ¼ 1 is 3. Topological invariants for band insulators μ ðiγ ∂μ − mÞψ ¼ 0; ð1Þ Band theory describes the electronic states within a crystal in terms of one particle Bloch wave functions junðkÞi that are where μ ¼ 0; 1; …;dlabel time and space dimensions, and the defined within the unit cell and are labeled by a crystal d þ 1 gamma matrices satisfy the anticommutation relation momentum k and band index n. The Berry phase of the fγμ; γνg¼0 for μ ≠ ν and ðγ0Þ2 ¼ −ðγiÞ2 ¼ I, where Bloch wave functions within a single band n is captured by i ¼ 1; …;d. I is the 2 × 2 unit matrix. The minimal sized the line integral of the Berry connection AnðkÞ¼ matrices that satisfy this property depend of course on the −ihunðkÞj∇kjunðkÞi, or equivalently the surface integral of 2kþ1 2kþ1 F ab k ∂ Ab − ∂ Aa dimension and are ( × )-dimensional matrices in both the Berry flux n ð Þ¼ ka n kb n. For a two-dimen- 2 1 2 2 xy spatial dimensions d ¼ k þ and k þ . sional insulator, the Berry flux for each isolated band F n ¼ F n Weyl noticed that this equation can be further simplified in is effectively a single component object, and the net Berry flux is certain cases in odd spatial dimensions (Weyl, 1929). For quantized to integers values since 1 simplicity consider d ¼ . Then one needs only two anti- Z commuting matrices, e.g., the 2 × 2 Pauli matrices, e.g., γ0 ¼ d2k 1 F nðkÞ¼Nn. ð4Þ σz and γ ¼ iσy. Therefore the Dirac equation in 1 þ 1 2π dimension involves a two component spinor and can be 0 1 0 written as i∂tψ ¼ðγ γ p þ mγ Þψ, where p ¼ −i∂x. If one The quantized Hall conductance is obtained by summing over all were describing a massless particle m ¼ 0, this equation can occupied bands. In a three-dimensional crystal, the Berry flux abc ab be further simplified by simply picking eigenstates of the behaves like a dual magnetic field ϵ BcðkÞ¼F ðkÞ, γ γ0γ1 σx γ ψ ψ Hermitian matrix 5 ¼ ¼ ,if 5 ¼ . One then switching the roles of position and momentum (with suppressed has the 1D Weyl equation band index n). The semiclassical equations of motion for an electron now take the following symmetric form (Xiao, Chang, ∂ ψ ψ i t ¼ p . ð2Þ and Niu, 2010): r_ v − p_ B The resulting dispersion is simply E ðpÞ¼ p which ¼ × ; ð5Þ denotes a right (left) moving particle, which are termed chiral or Weyl fermions. Analogous dispersions arise at the one- p_ ¼ eE þ er_ × B; ð6Þ dimensional edge of an integer quantum Hall state, but are not allowed in an isolated one-dimensional system where chiral where v is an appropriately defined renormalized band velocity fermions must appear in opposite pairs. The fermion mass and E, B are externally applied electric and magnetic fields. term interconverts opposite chiralities. We will see that an Note the Berry flux restores the symmetry r ↔ p of these analogous situation prevails in 3 þ 1 dimensions and indeed equations of motion, which is otherwise broken by the Lorentz the analogy with 1D fermions will be a theme that we will force. However, there is an important difference between B and repeatedly return to. B. Unlike the physical magnetic field, the Berry field B is In any odd spatial dimension d ¼ 2k þ 1 one can form the allowed to have magnetic monopoles. We will see that these k 0 1 d Hermitian matrix γ5 ¼ i γ γ γ . This is guaranteed to precisely correspond to the Weyl points in the band structure. commute with the “velocity” matrices γ0γi, which can be simultaneously diagonalized along with the massless Dirac B. Topological aspects of Weyl semimetals equation. At the same time it differs from the identity matrix since it anticommutes with γ0. In even spatial dimensions, the In Weyl semimetals, the conduction and valence bands latter property no longer holds, since all the gamma matrices coincide in energy over some region of the Brillouin zone. are utilized, and their product is just the identity. Furthermore, this band touching is stable at least to small
Rev. Mod. Phys., Vol. 90, No. 1, January–March 2018 015001-4 N. P. Armitage, E. J. Mele, and Ashvin Vishwanath: Weyl and Dirac semimetals in three- … variations of parameters. A key input in determining conditions for such band touchings is the degeneracy of bands, which in turn is determined by symmetry. If spin rotation symmetry is assumed, e.g., by ignoring spin-orbit coupling, the bands are doubly degenerate. Alternatively, doubly degenerate bands also arise if both time reversal T and inversion symmetry P (r → −r) are simultaneously present or their combined PT ~ symmetry is. Then, under the operation T ¼ PT , crystal ~ 2 momenta are invariant and moreover T ¼ −1, which ensures double degeneracy. On the other hand, if only T is present, Excluded Volumes Vi bands are generally nondegenerate since crystal momentum is reversed under its action. Only at the time reversal invariant momenta (TRIM), where k ≡ −k, is a Kramers degeneracy present. Similarly, if time reversal is broken, and only inversion FIG. 1. Net chirality of Weyl nodes must be zero, which is a is present, the bands are typically nondegenerate. As discussed, consequence of the fact that the net Berry flux integrated over the the conditions for a pair of such nondegenerate bands to touch Brillouin zone (a closed volume) must vanish. can be captured by discussing just a pair of levels whose effective Hamiltonian can generically be expanded as Weyl nodes. While this makes a connection to Weyl fermions HðkÞ¼f0ðkÞI þ f1ðkÞσ þ f2ðkÞσ þ f3ðkÞσ . To bring x y z with a fixed chirality [C ¼ signðv · v × v Þ] it remains the bands in coincidence we need to adjust all three coefficients x y z unclear why Weyl nodes should come in opposite chirality f1 ¼ f2 ¼ f3 ¼ 0 simultaneously, which by the arguments of the previous section requires that we have three independent pairs. To realize this we need a topological characterization of variables, i.e., that we are in three spatial dimensions. As we can Weyl nodes which is furnished by calculating the Berry flux then expect band touchings without any special fine-tuning, we on a surface surrounding the Weyl point. Furthermore, we can check that the Berry flux piercing any can readily argue that the existence of Weyl nodes is stable to surface enclosing the point k0 is exactly 2πC, where C is the small perturbations of Hamiltonian parameters. The location of chirality, e.g., Weyl points are monopoles of Berry flux. If we the Weyl nodes can be geometrically visualized as follows. We consider the sphere surrounding a Weyl point and consider its consider the real function f1ðkÞ and ask where it vanishes in 2D band structure, it has a nonvanishing Chern number momentum space; typically this will be a 2D surface that C 1. However, if we expand this surface so that it covers separates positive and negative values of the function. If we ¼ the entire Brillouin zone, then by periodicity it is actually demand a simultaneous zero of f2ðkÞ;f3ðkÞ, this specifies the equivalent to a point and must have net Chern number zero. intersection of three independent surfaces, which will typically Therefore, the net Chern number of all Weyl points in the occur at a point. Now, consider a perturbation that changes the Brillouin zone must vanish. This can be seen from Fig. 1 functions f by a small amount. This will also move the zero a where we isolate band touchings within the volumes V . The surfaces and their points of intersection by a small amount, but i integral of ∇k · B k 0 over this volume vanishes, but can the intersection will persist, just at a different crystal momen- ð Þ¼ be expressed as an integral over the surfaces of the excluded tum. The Weyl nodes cannot be removed by any small P H P volumes ∂ BðkÞ · dSk ¼ −2π Ci which must vanish. perturbation and may disappear only by annihilation with i Vi i another Weyl node. Next we describe a topological perspective In the continuum, one can define a single Weyl node, since the that makes this fact obvious. momentum space is no longer compact. However, in lattice Based on this reasoning it may appear that all we need to do model realizations of Weyl fermions the net chirality must to realize a WSM is to find a 3D crystal with nondegenerate vanish. This also shows that Weyl nodes can be eliminated bands by breaking appropriate symmetries. While indeed only by distortions to the Hamiltonian in a pairwise fashion, Weyl nodes are quite natural, typically one also imposes an e.g., by annihilation with another Weyl node of opposite additional requirement—that they be close to the Fermi chirality. Note also the Berry flux must be an integer multiple energy, so this also requires that we find candidates for which of 2π which allows, for example, band touching with C ¼ 2, “ ” f0ðkÞ is nearly zero. We can further discuss the generic which corresponds to double Weyl nodes, which do not dispersion near the band touching point, by expanding the have a linear dispersion in all directions. Hamiltonian about k ¼ δk þ k0. This gives To build intuition and make the possibilities more explicit in the space of 4 × 4 Hamiltonians (Burkov, Hook, and Balents, X 2011), we can consider a simple continuum system with two k ∼ k I v δkI v δkσa Hð Þ f0ð 0Þ þ 0 · þ a · ; ð7Þ orbitals plus spin, which describes the cases of WSMs, “line a¼x;y;z node” semimetals,1 as well as conventional gapped magnetic semiconductors. Expanding around the Γ point, we consider a v ∇ k μ 0 … 3 4 4 where μ ¼ kfμð Þjk¼k0 (with ¼ ; ; ) are effective × Hamiltonian matrix, velocities which are typically nonvanishing in the absence of additional symmetries. Note if we revert to the special limit 1 where v0 ¼ 0 and va ¼ v0aˆ (a ¼ 1; …; 3), we obtain the Weyl As pointed out elsewhere (Burkov, Hook, and Balents, 2011), this equation (3). We therefore refer to these band touchings as term is an oxymoron. Yet it persists.
Rev. Mod. Phys., Vol. 90, No. 1, January–March 2018 015001-5 N. P. Armitage, E. J. Mele, and Ashvin Vishwanath: Weyl and Dirac semimetals in three- …
0 FIG. 2. (Left to right) Energy spectra of εsμð0;py;pzÞ for the Dirac semimetal (m ¼ b ¼ b ¼ 0), magnetic semiconductor (m ¼ 1;b¼ 0.5;b0 ¼ 0), Weyl semimetal (m ¼ 0.5;b¼ 1;b0 ¼ 0), and line node semimetal (m ¼ 0;b¼ 0;b0 ¼ 1) for the Hamiltonian equation (8). From Koshino and Hizbullah, 2016.
τ σ k τ σ 0τ σ H ¼ v xð · Þþm z þ b z þ b z x where the energy bands are gapped in the range 0 E < m − b , and (center right) b > m that represents mI þ bσz þ b σx vσ · k j j j j j j j j j j ¼ ; ð8Þ the WSM where the middle bands touch at a pair of isolated vσ · k −mI þ bσ − b0σ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi z x point nodes k ¼ð0; 0; b2 − m2=vÞ. Further plots from where k ¼ðkx;ky;kzÞ is the momentum, and the τn’s are Pauli Tabert and Carbotte (2016) are shown in Fig. 3 that reflects matrices for the pseudospin orbital degrees of freedom. Here different regimes in the m and b parameter space. m is a mass parameter, and b and b0 are Zeeman fields that For the case of m ¼ b ¼ 0,butb0 finite, one obtains the physically can correspond to magnetic fields in the z and x eigenvalues directions, respectively. A number of interesting and relevant rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffih qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i cases can be obtained as a function of m, b, and b0.Forb0 2 ε k 2 2 2 2 μ 0 equal to zero, one obtains the eigenvalues sμð Þ¼s v kx þ v ky þ kz þ b ; ð10Þ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 2 2 2 2 2 2 2 whereqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi the zero energy contour becomes a circle at kx ¼ and εsμðkÞ¼s m þ b þ v k þ μ2b v kz þ m ; ð9Þ 2 2 0 ky þ kz ¼ b =v as shown in Fig. 2 (far right). The spectrum k 1 μ 1 where k ¼j j, and s ¼ and ¼ . The spectrum for is immediately gapped for kx away from 0. εsμð0;ky;kzÞ is plotted in Fig. 2 for cases (far left) m ¼ b ¼ 0, Although idealized here as T breaking fields in this which corresponds to a Dirac semimetal composed of a pair of continuum model, b and b0 are representative of the degenerate linear bands, which touch at k ¼ 0, (center symmetry breaking perturbations that may be encountered left) jmj > jbj describes a gapped magnetic semiconductor, in lattices, which may require different considerations; see,
FIG. 3. (Left) Band structure from Eq. (8) for values of m=b for the s ¼þand μ ¼ bands for increasing m=b in the WSM phase. For finite m the μ ¼þband is gapped, while μ ¼ − contains two Weyl nodes. (Center) The s ¼þand μ ¼ − bands near the phase transition at m=b ¼ 1. (Right) Phase diagram of Eq. (8).Atm=b < 1, the system is a WSM, while m=b > 1, a gapped semimetal exists. Along b ¼ 0, a degenerate massive DSM is observed. At m ¼ b ¼ 0, massless degenerate Dirac fermions exist. From Tabert and Carbotte, 2016.
Rev. Mod. Phys., Vol. 90, No. 1, January–March 2018 015001-6 N. P. Armitage, E. J. Mele, and Ashvin Vishwanath: Weyl and Dirac semimetals in three- … for example, Carter et al. (2012). It is also important to note 1. Weyl semimetals with broken T symmetry that in general it is impossible to apply laboratory magnetic The simplest setting to discuss a WSM is to assume broken fields large enough to generate Zeeman splittings of the time reversal symmetry, but to preserve inversion. This allows Weyl nodes that are substantial fractions of the Brillouin for the minimal number of Weyl nodes, i.e., two with opposite 0 zone. Therefore, the fields b and b should be considered as chirality. Inversion symmetry guarantees they are at the same effective internal fields. energy and furthermore provides a simple criterion to diag- To gain further intuition into the physics, it is instructive to nose the existence of Weyl points based on the parity consider a real space model of a WSM. As a simplest realistic eigenvalues at the TRIM. T model for a broken WSM Burkov and Balents (2011) Let us discuss this in the context of the following toy model. considered a repeating structure of period d of normal We envision a magnetically ordered system so the bands have T insulators and TIs in which was broken through a no spin degeneracy, but with a pair of orbitals on each site of a Zeeman field. As shown in Fig. 4, one can model such simple cubic lattice. Further assume that the orbitals have system with two different tunneling matrix elements for opposite parity (e.g., s, p orbitals), so τ , which is diagonal in Δ z tunneling between surface states on the same layer ( S) the orbital basis, is required in the definition of inversion and between surface states on neighboring layers (ΔD). With z symmetry HðkÞ → τ Hð−kÞτz. The Hamiltonian is no magnetism this system shows a topological–normal band inversion transition as a function of the relative strength of ΔS HðkÞ¼tzð2 − cos kxa − cos kya þ γ − cos kzaÞτz and ΔD. The multilayer is a bulk 3D TI when ΔD > ΔS and a τ τ normal insulator for ΔD < ΔS. If the layers are magnetized þ txðsin kxaÞ x þ tyðsin kyaÞ y: ð12Þ giving a spin splitting b, one finds a WSM phase for values of ΔS near ΔD. The Weyl nodes are found at For −1 < γ < 1 we have a pair of Weyl nodes at location k0 ¼ð0; 0; k0Þ where cos k0 ¼ γ. The low-energy excita- k ≈ k q tions are obtained by approximating H ð Þ HPð 0 þ Þ π 1 Δ2 Δ2 − 2 q ≪ τ S þ D b where we assume small j j k0. Then H ¼ ava qa a kz ¼ arccos : ð11Þ d d 2ΔSΔD where v ¼ðtx;ty;tz sin k0Þ. Note at the eight TRIM momenta ðnx;ny;nzÞπ=a, where na ¼ 0, 1, only the first term in the Hamiltonian is active, and One can see that as a function of ΔS, ΔD, and b the Weyl if γ > 1 the parity eigenvalues of all the TRIM are the same nodes move through the BZ and can annihilate at the BZ edge and the bands are not inverted. However, at γ ¼ 0, the parity for a critical value of b giving a fully magnetized state and as eigenvalue of the Γ point shows the bands are inverted and it is discussed later for a quantized anomalous Hall conductivity. readily shown that this immediately implies Weyl nodes, i.e., A related construction is possible for P breaking WSMs an odd number of inverted parity eigenvalues is a diagnostic of (Halász and Balents, 2012). These layered models not only Weyl physics (Hughes, Prodan, and Bernevig, 2011; Turner suggest a possible route toward creating new WSM states but et al., 2012; Z. Wang, Vergniory et al., 2016). At the same also provide an alternate viewpoint on WSMs as a state of time let us compare the Chern numbers ΩðkzÞ of two planes in “ matter with a periodically inverted and uninverted local momentum space kz ¼ 0 and kz ¼ π=a. Then the Chern ” band gap. With this perspective, aspects such as the pre- number vanishes at Ωðkz ¼ π=aÞ¼0, but Ωðkz ¼ 0Þ¼1. sence of Fermi arcs and the anomalous Hall effect follow Starting at γ ¼ −1, Weyl nodes form at the BZ boundaries and naturally. Related schemes for building up 3D topological move toward each other before annihilating at the zone center semimetals by stacking one-dimensional primitives based on at γ ¼ 1.Asγ → 1, the entire Brillouin zone is filled with a the Aubry-Andre-Harper model have also been studied unit Chern number along the kz direction, and a three- (Ganeshan and Das Sarma, 2015). dimensional version of the integer quantum Hall state is realized (Halperin, 1987). Therefore the WSM appears as a transitional state between a trivial insulator and a TI. When the chemical potential is at EF ¼ 0, the Fermi surface consists solely of two points k0. On increasing EF, two nearly spherical Fermi surfaces appear around the Weyl points and a metal exists. The Fermi surfaces are closed two- dimensional manifolds within the Brillouin zone. One can therefore define the total Berry flux penetrating each, which by general arguments is required to be an integer, and in the present case is quantized to 1, which is a particular feature characteristic of a Weyl metal. When EF >E ¼ tzð1 − γÞ the FIG. 4. A heterostructure model of a Weyl semimetal of Fermi surfaces merge through a Lifshitz transition and the net topological and normal insulators. Doped magnetic impurities Chern number on a Fermi surface vanishes. At this point, one are shown by arrows. d is the real space periodicity of the lattice. would cease to call this phase a Weyl metal. This discussion ΔS and ΔD are tunneling between topological surface states on highlights the importance of the Weyl nodes being sufficiently the same topological insulator layer and between different layers, close to the chemical potential as compared to E . Ideally, we respectively. From Burkov, 2015a. want the chemical potential to be tuned to the location of the
Rev. Mod. Phys., Vol. 90, No. 1, January–March 2018 015001-7 N. P. Armitage, E. J. Mele, and Ashvin Vishwanath: Weyl and Dirac semimetals in three- …
Weyl nodes just from stoichiometry, as occurs for ideal graphene.
2. Weyl semimetals with broken P symmetry If T is preserved then inversion symmetry must be broken to realize a WSM. A key difference from the case of the T broken WSM is that the total number of Weyl points must now be a multiple of four. This occurs since under time reversal a Weyl node at k0 is converted into a Weyl node at −k0 with the same chirality. Since the net chirality must vanish, there must be another pair with the opposite chirality. Although poten- FIG. 5. (Left) Conventional type I Weyl point with pointlike tially more complicated with their greater number of nodes, Fermi surface. (Right) Type II Weyl point is the touching point such WSMs may be more experimentally compatible as between electron and hole pockets. Red and blue (highlighted) external fringe fields from a ferromagnet may be problematic isoenergy contours denote the Fermi surface coming from for angle-resolved photoemission spectroscopy’s (ARPES) electron and hole pockets with chemical potential tuned to the momentum resolution. Additionally, without the complica- touching point. tions of magnetism in principle some properties of the system should be simpler under strong magnetic field. “ ” A useful perspective on T symmetric WSMs is to view semimetallic phase has been termed a type II Weyl semimetal [or structured Weyl semimetal (Xu, Zhang, and them as the transition between a 3D topological insulator and “ ” a trivial insulator. When a 3D TI possesses both time reversal Zhang, 2015)], in contrast to a type I semimetal with closed symmetry and inversion there is a simple “parity” criterion to constant energy surfaces. Although type I and type II WSMs diagnose its band topology (Fu and Kane, 2007). To achieve a cannot be smoothly deformed into each other, they share transition between topological and trivial states, Kramers electronic behavior that derives from the presence of an doublets with opposite parities must cross each other at isolated band contact point in their bulk spectra. one of the TRIM. Since these pairs of states have opposite Interestingly the topological character of the Weyl point is parity eigenvalues, level repulsion is absent and they can be still fully controlled by the last term in Eq. (7) and persists made to cross by tuning just one parameter. At the transition even for type II Weyl semimetals. Thus type II Weyl point, a fourfold degeneracy occurs at the TRIM, leading semimetals support surface Fermi arcs that terminate on the generically to a Dirac dispersion. Hence the transition between surface projections of their band contact points which are the a trivial and a topological insulator, in the presence of signature of the topological nature of the semimetallic state. In inversion, proceeds via a Dirac point. However, on breaking addition to arcs, McCormick, Kimchi, and Trivedi (2017) inversion, the Kramers doublets at the TRIM points can no showed that type II WSMs support an additional class of “ ” longer cross each other while adjusting just a single tuning surface states they call track states. These are closed parameter. How then does the transition proceed? The key contours that are degenerate with the arcs but do not share observation is that the bands are now nondegenerate away their topological properties. They can be generated when the from the TRIM, and by our previous counting can be brought connectivity of Weyl nodes changes as one tunes the param- into coincidence by tuning the crystal momenta. Tuning an eters in a system with multiple sets of Weyl points. additional parameter to drive the transition involves moving Type II systems are also expected to support a variant of the the Weyl nodes toward each other and annihilating them as chiral anomaly when the magnetic field direction is well described by Murakami and Kuga (2008). For example, when aligned with the tilt direction, have a density of states different an inversion symmetry breaking staggered potential is applied than the usual form, possess novel quantum oscillations (QO) to the Fu-Kane-Mele model of the 3D TI, the transition due to momentum space Klein tunneling, and a modified ’ between weak and strong TI becomes a WSM phase. It was anomalous Hall conductivity (Soluyanov et al., 2015; O Brien, recently argued that the band closing transition of a semi- Diez, and Beenakker, 2016; Udagawa and Bergholtz, 2016; conductor lacking inversion symmetry always proceeds Zyuzin and Tiwari, 2016). It is proposed that tilting of the cones through a gapless phase, consisting of either Weyl points has a strong effect on the transport Fano factor F (the ratio of or nodal lines (Murakami et al., 2017). shot noise power and current) (Trescher et al., 2015). Type I and The velocity parameter v0 in Eq. (7) introduces an overall type II nodes of opposite chirality can be merged and tilt of the Weyl cone. Such a term is forbidden by Lorentz annihilate. And it is likely that some materials can undergo symmetry for the Weyl Hamiltonian in vacuum but it can type I to type II transitions with doping or under pressure. generically appear in a linearized long wavelength theory near Claims for a type II state have been made recently in MoTe2 an isolated twofold band crossing in a crystal (Wan et al., (Deng et al., 2016; L. Huang et al., 2016; Liang et al., 2016b; 2011; Soluyanov et al., 2015). Small v0 simply induces a Tamai et al., 2016; Jiang et al., 2017), WTe2 (C. Wang et al., crystal field anisotropy into the band dispersion near a Weyl 2016), their alloy MoxW1−xTe2 (Belopolski et al., 2016a, point. However, sufficiently large v0 produces a qualitatively 2016b), and TaIrTe4 (Belopolski et al., 2016d; Haubold new momentum space geometry wherein the constant energy et al., 2016; Koepernik et al., 2016), although the evidence surfaces are open rather than closed and the resulting electron in the case of WTe2 is controversial as discussed later (Bruno and hole pocket contact at a point as shown in Fig. 5. This new et al., 2016).
Rev. Mod. Phys., Vol. 90, No. 1, January–March 2018 015001-8 N. P. Armitage, E. J. Mele, and Ashvin Vishwanath: Weyl and Dirac semimetals in three- …
C. Physical consequences of topology surface of any three-dimensional insulator, where there is a finite energy gap throughout the entire Brillouin zone. We have seen that there are topological aspects of WSMs We can now discuss the nature of the surface states that which are most simply stated in terms of them being arise in WSMs, which at EF ¼ 0 are Fermi arcs that terminate monopoles of Berry curvature. Here we explore some of at k0. These are a direct consequence of the fact that Weyl the consequences of that topology. From experience with nodes are sources and sinks of Berry flux. Hence, we consider topological insulators and quantum Hall states, we are used to a pair of planes at kz ¼ 0 and kz ¼ π=a in the model given in two different manifestations of topology. The first is to look Eq. (12). Since they enclose a Weyl node, or Berry monopole, for nontrivial surface states, and the second is to study the there must be a difference in Berry flux piercing these two response to an applied electric and/or magnetic field. We planes that accounts for this source. Indeed, in the model of follow these general guidelines in this case. WSMs have Eq. (12) we see that kz ¼ 0 (kz ¼ π=a) has Chern number special surface states called Fermi arcs and an unusual C ¼ 1 (C ¼ 0). In fact, any plane −k0