A Non-Topological Approach to Understanding Weyl Semimetals

Antonio Levy1,∗ Albert F. Rigosi1, Francois Joint2, and Gregory S. Jenkins3 1National Institute of Standards and Technology, Gaithersburg, Maryland 20899, USA 2Department of , University of Maryland, College Park, MD 20742, USA and 3Laboratory for Physical Sciences, College Park, MD 20740, USA (Dated: January 31, 2021) In this work, chiral anomalies and Drude enhancement in Weyl semimetals are separately discussed from a semi-classical and quantum perspective, clarifying the physics behind Weyl semimetals while avoiding explicit use of topological concepts. The intent is to provide a bridge to these modern ideas for educators, students, and scientists not in the field using the familiar language of traditional solid- state physics at the graduate or advanced undergraduate physics level.

I. INTRODUCTION II. BACKGROUND ON WEYL

The concept of Weyl fermions originated from the fu- Weyl fermions (defined below) have historically been sion of two familiar topics. The first is the energy- of interest in answering fundamental questions about relation from relativity expressed as: E2 = the universe, particularly the observation of the matter- 2 (pc)2 + mc2 . The second is the time-dependent antimatter imbalance.1 The family of elementary parti- Schr¨odingerequation from expressed cles classified as fermions, or particles of half-integer spin, 2 2 } O ψ ∂ψ are important in the that unifies three as: − (2m) = i} ∂t . of the four known forces of nature. Within the model To incorporate relativity into a quantum mechani- are twenty-four families of fermions. Almost all of them cal formula, Dirac treated each quantity in the energy- are massive Dirac fermions. Within the family of Dirac momentum relation as an operator acting on a wavefunc- ∂ fermions lies a subset class known as Weyl fermions, the tion. Using p = −i}O and E = i} ∂t , the resulting equa- set of fermions that are massless. Those well-versed in tion is: the physics of the weak nuclear force will recall that those interactions are stronger for both left-chiral matter and  2  2 2 right-chiral antimatter than their counterparts of oppo- 1 ∂ 2 m c − + O ψ = ψ (1) site (chirality summary shown in Fig. 1). c2 ∂t2 }2 Weyl and 3D Dirac semimetals are the only systems in This is known as the Klein-Gordon equation. It does which signatures of Weyl fermions have been observed. not include spin and is therefore applicable to zero-spin Although they are not fundamental particles, the exci- bosons. tations in these material systems offer a unique play- Dirac transformed this equation by taking the square ground to study Weyl physics like the chiral root of the operators, which requires consideration of all .2-6 Novel properties predicted for Weyl semimet- three spatial dimensions and the time dependence. The als (WSMs) like significantly reduced scattering, Drude transformation involves 4 × 4 matrices, called the Dirac enhancement along applied magnetic fields, and long- , that are directly related to the Pauli lived spin-polarized currents in the presence of magnetic matrices −→σ that describe half-spin fermions. The Dirac fields could lead to myriad applications in fields like spin- equation can be written in matrix block form with a tronics and quantum computing. particle-hole (p-h) – or particle-antiparticle – : Most introductions to WSMs and Weyl Fermions are mathematically intense, making it difficult to develop an  ε  −→ −→    c − mc − p · σ ψp −→ −→ ε  = 0, (2) intuitive understanding of their physics. There are a − p · σ − + mc ψh few works that try to explain the concepts behind Weyl c fermions in an educational context.7-9 This paper aims where ε is the energy of the particle. On the other to provide a conceptual introduction accessible to educa- hand, rather than using a particle-hole basis, the Weyl tors, students, and scientists who have some understand- equation uses left- and right-chiral particles as the basis: ing of traditional .

 ε −→ −→    mc c − p · σ ψL ε −→ −→ = 0, (3) c + p · σ mc ψR Massive particles in the Weyl equation prevent pure chiral eigenstates from emerging out of Equation (3). ∗ [email protected] Pure chiral particles must be massless. The eigenstates 2

the zero-momentum Weyl point. Chirality Due to quantum mechanics, the allowable energies at which electrons are permitted to exist are discrete (and Left-handed Right-handed by extension, are also discrete in k-space via Fourier transformation). These quantized energies make up the band structure of a crystal. For the remainder of this p p paper, it is assumed that the Fermi energy exists solely within the Weyl band and that other bands are suffi- ciently separated in energy so as not to perturb the Weyl state. With this last concept recalled, we are ready to describe the anomaly. Consider the divergence theorem as applied to a charge Q, charge density ρ and electrical current density j :

y dQ ∂ρ −→ −→ = + · j d3x (4) σ σ dt ∂t O

FIG. 1. A left-handed chiral particle has a spin σ that is Since charge is conserved, the integral extends over antiparallel to its momentum p. The spin and momentum are the entire volume of the system, leaving both sides of parallel for a right-handed chiral particle. Eq. (4) equal to zero. This expression indicates that the total amounts of charge entering and leaving the system are equal. Combining Eq. (4) with Eq. (3) for m = 0 of massless Weyl particles only involve the momentum under non-zero electromagnetic fields (which is done by −→ and a parallel or antiparallel spin as shown in Figure 1. replacing −→p with −→p − e A ), the difference between the In solid state physics, the dispersion relation is a math- c number of right- and left-chiral particles, nR − nL can be ematical description of the available energies for an elec- obtained after considerable effort: tron moving about in a material with a given momentum. It is frequently plotted as energy versus momentum (also y referred to as k-space). Further details on the basics of d −→ −→ (n − n ) ∝ E · B d3x (5) solid state physics can be found in standard textbooks.10 dt R L The relativistic energy-momentum dispersion relation for massless particles is E = pc . For particle-like excita- Equation (5) shows that if a population of Weyl tions (or ) moving through solid materials, fermions is subjected to non-zero applied electric and the velocity can be a constant other than the speed of magnetic fields, the chirality of the population will light while still obeying the Weyl equation. Casting the −→ −→ change with time. This is equivalent to saying that Weyl momentum in terms of wavenumber p = } k , the oper- fermions of a single chirality will be annihilated and re- ative Weyl dispersion relation becomes E = }kv, where placed with Weyl fermions of the opposite chirality. This the v is the Fermi velocity which plays a role similar to c is the chiral anomaly. in the relativistic equations. The zero-energy is defined The chiral anomaly in a crystal is best understood in- at zero momentum and is called the Weyl point, and its tuitively by considering the effects of applied magnetic presence in Weyl semimetals is responsible for the novel and electric fields on the Fermi surface with linear dis- physics they are predicted to exhibit. persions illustrated in Fig. 2. The momentum direction of any on a spherical Fermi surface is ra- dially outward. In the depicted generic WSM, there are III. CHIRAL ANOMALIES two locations in k-space around which the band struc- ture obeys the Weyl equations. Each location hosts one To appreciate how a chiral anomaly emerges, it will species of chiral particle where the spin is either radially help to recall a few concepts from statistical and solid- inward opposite the momentum (orange Fermi surface state physics. Within any material, electrons and other pocket on the left side of each subfigure) or purely aligned quasiparticles only have access to a limited number of radially outward with the momentum (green Fermi sur- momenta. Mapping out these allowable momenta cre- face pocket on the right side of each subfigure). As will ates what is commonly referred to as k-space (closely be discussed later, including only a single chiral Fermi related to the reciprocal lattice of a crystalline solid). pocket violates the conservation of energy. The highest-energy electrons, in some sense, define the So what happens to the Fermi surface if you apply a bounds of achievable momenta, and these bounds sketch magnetic field? It will distort, expanding or contracting out an object in k-space called a Fermi surface. The by an amount proportional to the dot product of the spin Fermi surface for a Weyl particle is a sphere centered on and the magnetic field (Zeeman effect). To intuitively un- 3 derstand these distortions, consider the left-handed chi- the application of fields. ral (orange) Fermi surface pocket in Figure 2(e-g) at the Consider for simplicity the case with both fields point- point where the spin of a quasiparticle is purely antipar- ing in the −x direction (as in Fig. 2). Then, for the right- allel with applied magnetic field. The quasiparticle’s en- chiral location, we track the difference between right- and ergy will decrease with applied field. This decrease in en- left-chiral particles nR(θ) − nL(θ), which is proportional ergy, through the linear energy dispersion, causes the mo- to integrated subtraction of the density of states, ex- mentum to decrease. The decrease in momentum shifts pressed as gR(L)(εF ) = g(εF +(−)a1Bcos(θ)+a2Ecos(θ)) this point of the Fermi surface toward the center of the for the right and left pockets, where a1 and a2 are con- Fermi pocket sphere. Likewise, the opposite point, where stants. The difference in sign in front of the a1Bcos(θ) the field and spin are parallel, shifts to higher momen- is due to the opposite spin of the two pockets, high- tum away from the center. All the points where the spin lighting the antiparallel nature of the chirality. Ap- is perpendicular to the field do not change their energy proximating for small fields, gR(L)(εF ) can be rewritten: 0 or momentum. Visualizing the smooth interpolation be- gR(L)(εF ) = g(εF ) + (−)a1Bcos(θ)g (εF + a2Ecos(θ)) tween these points results in a Fermi surface distorted or, equivalently, we could say: gR(εF ) − gL(εF ) = 0 into an egg-like shape with the major axis aligned with 2a1Bcos(θ)g (εF +a2Ecos(θ). The difference in the pop- field as shown in Figure 2(e). ulation of right- and left- chiral Weyl fermions is given Similar effects on the Fermi sea occur from the charge by: of the quasiparticle subjected to an applied electric field. In this case, all negatively charged quasiparticles, regard- Z π less of their spin or location on the Fermi surface, de- (gR (εF , θ) − gL (εF , θ)) dθ ≈ crease their (vector) momentum along the direction of 0 π (6) the electric field causing a net momentum shift of the Z 0 2a1B cos (θ) g (εF + a2Ecos (θ)) dθ Fermi surface. 0 With either an applied electric or magnetic field, the changes in a singly-chiral Fermi pocket induces a net mo- This subtraction is reduced to a derivative of the den- mentum and therefore an electric current. This current sity of states for small fields. With additional also transports a net spin current. However, when both considerations, the bounds of the integral are halved as (oppositely) chiral spherical Fermi surface pockets are long as we take into account E → −E for the respective considered, the application of a magnetic field (with no halves of the Fermi pockets: electric field) causes two equal and oppositely-oriented distortions that negate the total current. In fact, this π Z 0 can be understood as the reason that all Weyl semimet- 2a1B cos (θ) g (εF + a2Ecos (θ)) dθ ≈ als must include pairs of oppositely-chiral Fermi pockets; 0 π a static magnetic field should not be able to generate Z 2  0 0  a perpetual current in a material system with a finite 2a1B g (εF + a2E cos (θ)) − g (εF − a2Ecos (θ)) 0 scattering rate like Weyl semimetals, as this would vio- cos (θ) dθ late the conservation of energy. Furthermore, in order π Z 2 not to violate conservation of energy, this current would 00 2 ∝ 2a1a2EB g (εF ) cos (θ) dθ have to be superconducting. At high fields, there would 0 be a strong current that would flow consistently in one (7) 0 direction, which would persist independent of the direc- where the approximation g (εF ± a2E cos (θ)) = 0 00 tion in which an external electric field was applied to g (εF )±a2E cos (θ) g (εF ) is used to go from the second the material. This would mean that carriers could flow line of Eq. (7) to the third. in the direction opposite to the bias applied to the mate- With these considerations, the origin of Eq. (5) rial, leading to negative power dissipation in the material becomes apparent. Since the first time derivative of as a result of an applied external electric field (provided (nR − nL) is a nonzero quantity under the experimen- that field was not large enough to fundamentally alter tal conditions, a net chiral current, the “chiral anomaly,” the band structure). emerges and begins pumping particles from the left-chiral To formulate an expression for the generated currents, Fermi pocket to the right-chiral Fermi pocket leading to a the quasiparticles removed and added at various mo- net accumulation of right-chiral particles. Furthermore, menta with applied fields must be properly counted. Re- the total number of right-chiral particles increases faster call the density of states, which is the derivative of the than the corresponding decrease seen in the left-chiral number of states with respect to energy in the system. particles. For static fields, this ever-growing chirality im- The density of electronic states (g, which we define as balance is eventually counteracted by quasiparticle scat- the number of states per unit energy at a specific point tering between and within the two chiral Fermi pockets. in k-space) is energy dependent and therefore becomes A chirality-imbalanced equilibrium value is established dependent on k-space location (θ, defined as the polar that depends on both pumping rate and (current relax- angle between the k-vector and the applied fields) with ation) scattering rate. 4

(a) kz kz Left Hand Right Hand 2 Chiral (L) Chiral (R) ωD σD (ω) = 1  (8) 4π τ − iω k k kx y kx y

(b) (e) B (h) B As ω → 0 (when B = 0 ) the DC conductivity appears ω2 τ E as σ = D , where the term ω2 is referred to as the (R) (R) (R) DC 4π D (L) (L) (L) Drude weight. The Drude weight and conductivity of a material are properties that are generally well-described by the equations of motion for the charge carriers. For most metallic and semi-metallic systems in the E = 0 E = 0 E ≠ 0 semi-classical limit, application of a magnetic field par- (c) B = 0 (f) B ≠ 0 (i) B ≠ 0 allel to the driving electric field does not affect current ky ky ky B B since there is no Lorentz force. In the case of Weyl and E 3D Dirac semimetals, such an arrangement of applied fields is expected to increase the Drude weight and this was recently verified experimentally.11 Most explanations = A1 A2 k k k (L) (R) x (L) (R) x (L) (R) x of this very unusual effect rely heavily on the mathemat- ical consequences of the Berry’s phase curvature near the (d) (g) B (j) B ε ε ε E Weyl points. Such explanations, though rigorous in the semi-classical limit, do not provide an intuitive explana- tion of the origin of such an effect. (L) (R) (L) (R) (L) (R) Again, studying the Fermi surface of the Weyl state kx kx kx provides a non-topological approach to understanding this effect. Since the Weyl pockets are represented in FIG. 2. The chiral anomaly in Weyl semimetals is understood k-space as asymmetrically distorted (egg-shaped) Fermi by distortions and shifts of the Fermi surface caused by the surface pockets (Fig. 2 (h)), any applied magnetic field application of electric and magnetic fields. (a,b) With no ap- will have additional, distinct effects on both populations plied field, the Fermi surface is two spheres: one Weyl pocket of chiral particles, pushing more of the particles into the is left-handed chiral (orange) with spins perpendicular to the Fermi surface (pointing inward) and the other right-handed portion of the Fermi surface that contributes to the net chiral (green) with spins (red arrows) perpendicular to the current, as shown in Figs. 2(h) and 2(i). The Drude Fermi surface (pointing outward). (c) The kx−ky planar cut weight is proportional to the current generated by the through the center of the Fermi pocket spheres results in two applied external electric field. Integrating over the Fermi circles that demarcate allowable quasiparticle momenta. (d) surface of a pair of Weyl points in the semi-classical small- q The energy of the quasiparticles, with momenta taken as a 2}vxvy field limit, with l2  εF : kx− line cut through the centers of the circles, is given by the B linear dispersion E = }kxv (where velocity v is constant). (g) Application of an external magnetic field in the - x direction Z 2 2 −→ −→ k raises or lowers the energy of the quasiparticle depending on ωD ∝ dΩ[g+(εF , B ) + g−(εF , B )]vF (θ) the direction of the spin. Changes in energy are connected   (9) Z 1 00 2 to changes in momenta through the linear energy dispersion ∝ dΩ g (ε ) + g (ε )(a B cos (θ))2 vk (θ) resulting in (e,f) k -space shifts and distortions of the Fermi F 2 F 1 F surface. (j) The addition of an applied electric field inter- acts with the (negative) charge of the quasiparticle causing k where v (θ) = vF cos (θ) is the Fermi velocity compo- a vectoral increase of the momenta in the - x direction that F q }c either increases or decreases the quasiparticle kinetic energy nent parallel to the applied fields and lB = eB is the depending on the initial direction of the momentum. (h,i) magnetic length. The second term in (9) yields the B2 The combined effects of the applied fields distort and shift Drude weight enhancement obtained using the Berry’s the Fermi surface, generating currents that, in the depicted 12 case, are largest for the (green) right-handed chiral carriers. phase curvature.

V. QUANTUM MECHANICAL PICTURE OF IV. SEMI-CLASSICAL PICTURE OF DRUDE THE CHIRAL ANOMALY AND DRUDE ENHANCEMENT ENHANCEMENT

Another interesting phenomenon regarding WSMs is For a more comprehensive picture of Drude enhance- the existence of Drude enhancement. To get comfortable ment, we turn to the quantum limit of the Drude weight with this topic, one must recall the Drude model of AC of a Weyl pocket in 3D. The energy dispersion of the nth conductivity from solid state physics: excited Landau level (LL) for B k x is given by: 5

(see Fig. 3), the number of carriers in one of the two ε n = 0 LLs will decrease while the number in the other n = 0 LL will increase. The dispersion in the two pock- n = 3 ets’ n = 0 LLs are given by ε+ (kx) = +}vxkx and n = 2 ε− (kx) = −}vxkx for the left- and right-chiral cases, n = 1 respectively. Applying an electric field along the x- ε direction will deplete the population of one n = 0 LL F while increasing the population of the other n = 0 LL by the same amount. n = 0 B The transfer of charge from one pocket to the other oc- curs entirely through the chiral n = 0 LL of both pockets, E since charge will be redistributed within each excited LL under an electric field. The amount of charge transfer is proportional to the electric field component parallel to the magnetic field and the degeneracy of the n = 0 LL, k 1 x which is proportional to 2 due to Landau lB perpendicular to the magnetic field which, in our case, is the yz-plane. From this, the chiral anomaly will persist FIG. 3. The energy dispersion of the various Landau levels at both low and high field strengths in WSMs. along the magnetic field direction is illustrated. Left-handed The spectral weight in the chiral n = 0 LL is indepen- particles are pumped from one Weyl pocket to the right- dent of the Fermi energy. From the quantum mechani- handed pocket. The formation of Landau levels makes the dispersion quasi one-dimensional, due to the discretization of cal perspective, the increase in spectral weight with field kinetic energy perpendicular to the magnetic field (i.e., in the can be understood as transferring more and more carriers y and z directions). from slower bands (where they are closer to the vertex) to the faster, massless n = 0 LL. This quantum mechanical perspective on Drude enhancement gives a more nuanced understanding of the complexity of this effect without the need to rely on topological descriptions. s (2n + (1 + s)) }vyvz 2 2 2 εn (kx) = 2 + } vxkx (10) lB

where s = ±1 correspond to the two spin eigenstates. VI. CONCLUSIONS Note that this approach can convert the 3D system into a series of 1D systems. Every LL (except n = 0 ) has an It is rather uncommon to find approaches to the energy dispersion containing an effective . There- physics behind Weyl fermions in the literature without fore, only the n = 0 Landau level is chiral, making it involving a discussion on topology. Here, we explain the responsible for all of the nontrivial behavior associated concepts of chiral anomalies and Drude enhancement in with WSMs. Consider the chiral anomaly. For all n 6= 0 the context of Weyl semimetals intended for those not LLs, the application of an electric field simply shifts elec- in the field. Non-topological approaches offer a more trons within the LL. They do not change the population intuitive conceptual framework for understanding these of the pocket or even the LL itself. physical systems. This understanding is useful when de- Under an electric field parallel to the magnetic field signing, performing, or analyzing data from experiments.

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