Theory of Magnetotransport in the Dirac and Weyl Semimetals

THEORY OF MAGNETOTRANSPORT IN THE DIRAC AND WEYL SEMIMETALS

By MUHAMMAD IMRAN

A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2020 c 2020 Muhammad Imran I dedicate this thesis to Schrodinger’s Cats. ACKNOWLEDGMENTS Thanks to all the help I have received in writing about this thesis. I am thankful to my advisor Selman Hershﬁeld and the professors of my advisory committee. I also want to thank everyone who have contributed by any means to success of this work.

4 TABLE OF CONTENTS page ACKNOWLEDGMENTS...... 4 LIST OF FIGURES...... 7 ABSTRACT...... 9

CHAPTERS 1 INTRODUCTION TO DIRAC AND WEYL SEMIMETALS...... 11 1.1 Topological Insulators Dirac Fermions and Weyl Fermions...... 18 1.2 Detection of Weyl Fermions by Angle-Resolve Photo Emission Spectroscopy.. 21 1.3 The Mangnetotransport Measurements...... 23 1.4 Planar Hall Eﬀect in Dirac and Weyl Semimetals...... 25 1.5 Conclusions...... 27 2 THE SEMICLASSICAL EQUATIONS OF MOTION FOR THE DIRAC AND WEYL SEMIMETALS...... 28 2.1 Introduction...... 28 2.2 Quantum Mechanics in Phase Space...... 29 2.3 Multiplying Two Operators In the Quantum Phase Space...... 32 2.4 The Semiclassical Equations of Motion for the Weyl Semimetals...... 35 2.5 The Boltzmann Equation for the Weyl Gas...... 39 2.6 Incorporating the Berry Curvature in the Group Velocity and Force...... 41 2.7 The Semiclassical Theory of Magnetotransport in the Weyl Semimetals.... 43 2.8 Results and Discussions...... 49 2.9 Magnetoconductivity...... 50 2.10 Magnetic Field dependent transport time...... 54 2.11 Conclusion...... 55 3 THE QUANTUM THEORY OF MAGNETOTRANSPORT IN TILTED DIRAC AND WEYL SPECTRUM...... 57 3.1 Introduction...... 57 3.2 Theoretical formulation of the tilted Weyl spectrum...... 59 3.3 The Magnetortransport Properties...... 65 3.4 The Semiclassical theory of the planar Hall eﬀect...... 71 3.5 Results and Discussions...... 73 3.6 Conclusions...... 79 4 CONCLUSION OF THE THESIS...... 82 4.1 Conclusion...... 82

APPENDIX

5 A NO GROUND STATE IN PERPENDICULAR MAGNETIC FIELD...... 84

B TENSOR SIJ ...... 85 C EQUATION OF MOTION DERIVATION...... 90 REFERENCES...... 94 BIOGRAPHICAL SKETCH...... 97

6 LIST OF FIGURES Figure page 1-1 Dispersion for a massless Dirac fermion...... 11 1-2 An illustration of two electron beams...... 14 1-3 The degenerate Dirac cones...... 17 1-4 These Weyl cones are spin split by breaking time reversal symmetry...... 18 1-5 An Schematic illustration of topological insulator...... 19 1-6 The Weyl semimetal phase...... 20 1-7 The experimental data for observation of Weyl fermions...... 22 1-8 Negative magneto resistance...... 24 1-9 The Planar Hall effect...... 26 2-1 The average deviation in the distribution function...... 50 2-2 The magnetoconductivity of Weyl spectrum...... 51 2-3 The MC of Weyl spectrum...... 52 2-4 The MR of Weyl spectrum for the magnetic field independent transport time..... 53 2-5 The MR of Weyl spectrum for the magnetic field dependent transport time..... 55 3-1 Equation of circle with added anisotropy...... 58 3-2 Energy dispersion relations for type-I and type-II Weyl semimetals...... 60 3-3 The energy spectra of Weyl gases...... 64 3-4 The schematic illustration of the planar Hall effect...... 70 3-5 The energy density of states for the 3D Weyl gas...... 74 3-6 The energy density of states for the 3D Weyl gas in the presence of magnetic field.. 75

3-7 The transverse conductivity σxx...... 76

3-8 The Hall conductivity σxy...... 77

3-9 The longitudinal conductivity σzz...... 78

3-10 The transverse resistivity ρxx, and the Hall resistivity ρxy...... 79

3-11 The longitudinal resistivity ρzz...... 80

7 3-12 The planar Hall resistivity ρxz...... 81

8 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy THEORY OF MAGNETOTRANSPORT IN THE DIRAC AND WEYL SEMIMETALS By Muhammad Imran May 2020 Chair: Selman Hershfield Major: Physics In this thesis we study the magnetotransport of the Dirac and Weyl semimetals theoretically. These materials are exotic and exhibit unusual magnetic field dependence in their magnetotransport properties. The most prominent effect is the chiral anomaly effect. This is a Berry curvature effect. According to this effect the magneoconductivity increases in the direction of magnetic field. The chiral anomaly was first observed in Weyl semimetals TaAs. Now there are many materials that show chiral anomaly characteristics. We study these semimetals magnetotransport properties both from the semiclassical and quantum perspective. For the semicalssical theory, we develop an equation of motion theory for the Dirac and Weyl semimetals by using the Wigner’s method. Later we solve the distribution function to calculate formulas of magnetotransport. We limit ourself to the linear response of the electric field. The derived formulas are matched with the experimental results. This includes the chiral anomaly effect. For the quantum theory, we use the Kubo formalism. This is a linear response theory. We derive the magnetotransport properties in a strong magnetic field. The formulas of magnetotransport show quantization effect. This is consistent with the Shubnikov-de Haas oscillations in the magneto transport properties. Low energy Dirac and Weyl semimetals are allowed to have asymmetric energy dispersion. This asymmetry in energy dispersion allows a type of Hall effect, the Planar Hall effect. We derive the formula for the Planar Hall effect. The oscillations in the Planar Hall effect can become a fingerprint to allow one to detect

9 the anomalous transport in the Dirac and Weyl semimetals. The Planar Hall eﬀect has been observed in the Dirac and Weyl semimetals experimentally.

10 CHAPTER 1 INTRODUCTION TO DIRAC AND WEYL SEMIMETALS The quantum study of a relativistic fermion starts with the Dirac equation, which for a free fermion is

2 (c~α · ~p + βmc )|ψi = Es|ψi. (1-1)

Here α and β are Dirac matrices. c is speed of light. m, ~p, Es, and |ψi denote the mass, momentum, energy and wave function of the Dirac fermion respectively. The energy eigenvalues of the Dirac equation are

p 2 2 2 4 Es = s p c + m c , (1-2) where s = + refers to energy of a particle, and s = − refers to energy of an anti-particle. The energy dispersion of the Dirac fermion is gapped due to their mass. Herman Weyl extended the work of Dirac by studying relativistic fermions of zero mass. He realized that a Dirac fermion with zero mass had not only a gapless energy dispersion, but its energy eigenstates and chirality states became same. Chirality refers to the eigenvalue of pˆ · ~σ, where pˆ is the momentum direction and ~σ is the spin operator. It has value +1 if the particle momentum

Figure 1-1. Dispersion for a massless Dirac fermion. The blue cone refers to the energy dispersion of particles with right handed chirality, S=+1, and the red cone refers to the energy dispersion of a particles with left handed chirality, S=-1.

11 is parallel to the spin angular momentum, and it has value −1 if the particle momentum is anti-parallel to the spin angular momentum. Massless Dirac fermions are called the Weyl fermions. A schematic picture of the Weyl cone is shown in Fig. 1-1. Do Weyl Fermions exist? This is an interesting question. After the invention of the Dirac equation, it took just a few years to detect the electron anti-particle, the positron. This confirmed that Dirac fermions do exist. This does not apply to Weyl fermions. Decades have passed and relativistic massless fermions have still not been observed. Once neutrinos were considered as Weyl fermions, but they do have mass. However, the energy band structure of solids can mimic a type of Weyl spectrum that has a linear in wave vector dispersion and these bands cross at some point in wave vector space, but in condensed matter physics these are called the Dirac fermions. The notion of Weyl fermion is a little different in low energy physics than in high energy physics. As will be explained later, in low energy physics the Weyl fermions dispersion also needs to be non-degenerate. This means not every linearly dispersing and gapless energy dispersion refers to Weyl fermions in condensed matter physics. The Weyl fermions energy dispersion emerges from the Dirac fermions by breaking either or both of time reversal symmetry and inversion symmetry. The famous example is graphene that has a spin and valley degenerate Dirac dispersion. The study of Dirac and Weyl fermions (WF) is a very active area of research in condensed matter physics. The WF’s found in condensed matter systems are the first experimental realization of this kind of particle. Weyl fermions only exist in three dimensions, and these are topologically protected excitations. For example in graphene only two Pauli spinors are used

(σx, σy) to describe the pseudo spin of the two Dirac cones, whereas in WF’s all three of the

Pauli spinors are used (σx, σy, σz). Therefore a perturbation can not gap the WF spectrum. Rather a perturbation will shift the Dirac cone. Prototype model of three dimensional Weyl fermions starts with two bands in a solid described by Hamiltonian H(~k). These bands are a ~ function of wave vector k. At some wave vector magnitude |k0| these bands cross each other.

For simplicity we consider k0 = 0. Any 2 × 2 matrix can be expanded in Pauli matrices ~σ and

12 the identity matrix σ0. This is exploited to write the Hamiltonian of two bands as

~ ~ ~ ~ H(k) = f0(k)σ0 + f(k) · ~σ, (1-3) which has eigenvalues

q ~ 2 ~ 2 ~ 2 ~ E(s, |k|) = f0(k) + s fx (k) + fy (k) + fz (k). (1-4)

~ Here fi(k), i = x, y, z, are the bands parameters and s = ± is band index. For gapless and isotropic spectrum these bands parameters and eigenvalues are

~ ~ ~ ~ f0(k) = 0, f(k) = vF k (1-5) ~ E(k, s) = svf |k|,

here vF is Fermi velocity. It is interesting to note that the Weyl fermions emerge due to two or more bands crossing. These band crossings can happen in crystal structures that preserve time reversal and/or inversion symmetry. Also, the Pauli spinors used in the above Hamiltonian are used for describing the two bands. Those bands do not include the real spins of Weyl fermions. This means Weyl cones are spin degenerate. The degeneracy of spins along with the inversion symmetry degeneracy along the wavevector ~k directions combine to make a total of eight fold degeneracy in the energy dispersion. At this point we need to introduce the concept of Berry phase to better understand Weyl fermions in condensed matter physics. What is a Berry phase? Consider a beam of electron waves that follows two diﬀerent paths and interfere. This generates a pattern of bright and dark bands on screen. In Fig. 1-2 an schematic illustration is shown to realize this experimental setup. Now we make a twist in this set up. We make this beam cross a region where magnetic ﬁeld is zero, but the vector potential is not zero. This time the same beam of electron waves generates a pattern of bright and dark bands but the bands are shifted. This shift in bands on screen is called

13 Figure 1-2. An illustration of two electron beams passing through a region that only includes vector potential to form an interference pattern on the screen.

Aharonov-Bohm eﬀect. The underlying physics behind this phenomenon is created by the Berry phase. The Berry phase is a phase of a system that is evolving adiabatically in time or with other parameters. For instance, we take time as a parameter. Consider a Hamiltonian H for this system. If the Hamiltonian is time independent, then we can relate the eigenstates of this

1 R t system at any later time by a dynamic phase factor θn(t) = dtEn(t): ~ 0

|ψn(t)i = exp[−iθn(t)]|ψn(0)i. (1-6)

But if the Hamiltonian depends on time adiabatically, the energy eigenstates are multiplied by another phase factor, the Berry phase factor. For a time dependent Hamiltonian we express the energy eigenstates in orthonormal basis of time independent eigenstates.

X |ψ(t)i = cn(t)|ψn(t)i (1-7) n The time dependent Schroodinger equation for the above equation is,

14 Taking the inner product of the above equation with ψm, this gives

X ˙ (˙cm(t) exp(iθm(t) + cn(t)hψm|ψn(t)i) exp(iθn(t)) = 0 n (1-9) X ˙ c˙m(t) = cn(t)hψm|ψn(t)i) exp(−i(θn(t) − θm(t))). n ˙ We need to ﬁnd hψm|ψn(t)i. For this we take the time derivative of energy eigenvalue equation,

and take inner product with energy eigenstate ψm,

˙ ˙ ˙ ˙ hψm|H|ψn(t)i + hψm|H|ψn(t)i = Enδnm + Enhψm|ψn(t)i, (1-11)

which for n 6= m becomes ˙ ˙ hψm|H|ψn(t)i hψm|ψn(t)i = . (1-12) En − Em This gives

˙ ˙ X hψm|H|ψn(t)i c˙m(t) = −cmhψm|ψm(t)i − cn(t) exp(−i(θn(t) − θm(t))). (1-13) En − Em n6=m

In above equation we ignore the second term because it involves a change in the Hamiltonian and due to adiabatic approximation it is ignored. Thus, within the adiabatic approximation

˙ c˙m(t) = −cmhψm|ψm(t)i. (1-14)

The solution of this equation is

cm(t) = cm(0) exp(iγm(t)), (1-15)

where γm(t) is Berry phase Z t ˙ γm(t) = i dthψm|ψm(t)i. (1-16) 0

15 For a dynamic system that is evolving adiabatically in time, the time evolution of energy eigenstates also multiplies with the Berry phase factor.

|ψn(t)i = exp[iθn(t)] exp(iγm(t))|ψn(t)i. (1-17)

In the above example the parameter that is changing is the magnetic vector potential. The Berry phase associated with the magnetic vector potential is

Z R(t) ~ ~ ~ ~ ~ γ(R(t)) = i dR · hψ(R(t))|∇R~ ψ(R(t))i R(0) (1-18) e I γ(R~(t)) = −i d~x · A~(x). ~ The dynamics of the propagating electron beams can be studied by including the magnetic vector potential in Hamiltonian or in the wave function

H(~p − eA~(R~))|ψi = H(~p)(exp(iγ(R~))|ψi). (1-19)

Now we are ready to discuss the role of Berry phase in Weyl fermions physics. The prototype model Hamiltonian that we have discussed earlier for Dirac fermions has vector potential in wavevector space A~(~k). This vector potential creates a Berry phase exactly the same way as the magnetic vector potential does. Z γ(~k) = i d~k · A~(~k) (1-20) ~ ~ † ~ ~ ~ A(k) = iU (k)∇kU(k)

Here U(~k) is a unitary operator that diagonalizes the Dirac Hamiltonian. This vector potential is not physically observable. The physical observable quantity is the Berry curvature

kˆ Ω(~ ~k) = ∇~ × A~(~k) = − σ . (1-21) k |k|2 z

The Berry curvature take the form of a magnetic monopole ﬁeld in wavevector space. These monopoles always come in pairs. In present form they can not be observed because the net Berry curvature is zero. The positive magnetic monopole ﬁeld cancels the negative magnetic

16 Figure 1-3. The degenerate Dirac cones are split by breaking time reversal symmetry (Zeeman energy) ∆z and by breaking the inversion symmetry ∆I to form Weyl cones.

monopole field. These monopoles need to be split in order to be observable. That is why the creation of Weyl fermions requires breaking of either or both of time reversal and inversion symmetries. The degeneracy of Weyl cone is broken by considering an external magnetic field [6] and/or a crystal structure that lacks inversion symmetry [1]. The external magnetic field splits the Weyl cones in momentum space, and the inversion symmetry splits the Weyl cone in energy space. A schematic illustration is shown in Fig. 1-3. A linear in wave vector energy dispersion of a semimetal does not tell if it is topologically protected or not. A degenerate Dirac cone is not topologically protected excitation whereas a non-degenerate Weyl cone is topologically protected. So how can these be distinguished? The Weyl cones can be distinguished from their cousin the Dirac cones by the bulk boundary correspondence. This is done by mapping the boundary surface Brillouin zone of three dimensional semimetals. The surface states of the Weyl cone form a non-closed Fermi arc. A schematic illustration of the Weyl Fermi arc states for broken time reversal symmetry is shown in Fig. 1-4. The

Weyl nodes exist at Fermi energy EF = 0 and wave vector k = 0. The two Weyl nodes act as two magnetic monopoles with opposite charges. The ﬂux from the source is F = 1 and

17 sink is F = −1. The monopoles ﬁeld starts from the source and terminates at the sink. This makes the surface of the Weyl semimetals an open Fermi arc. The magnetic monopoles of the Dirac nodes lives at the same wave vector. Therefore the ﬂux due to them is always zero. The semimetals with the Dirac dispersions have close Fermi circles at the Brillouin zone of the boundary surface. 1.1 Topological Insulators Dirac Fermions and Weyl Fermions

In the preceding sections the splitting of Dirac nodes by applying an external magnetic field can be doubtful for a beginner of this field. The model Hamiltonian we discussed above do not say anything about spin angular momentum. This section should bridge this gap of lack of knowledge and should provide an insight about how real spin and momentum couple in Weyl semimetals systems. In topological insulators the real spin of the particles is coupled to their momentum. Here we derive the Dirac energy dispersion from a prototype model of topological insulator. Then we will consider the effect of a magnetic field to break the time reversal

Figure 1-4. These Weyl cones are spin split by breaking time reversal symmetry. Each Weyl cone can be considered as a magnetic monopole at zero Fermi energy. The ﬂux from magnetic monopoles are F = ±1. These Weyl nodes form an open Fermi arc on the surface of Weyl semimetal.

18 Figure 1-5. An Schematic illustration of topological insulator(TI) and ordinary insulator(OI) layers. The hopping energy between the top and bottom surface of the same topological insulator layer is ∆s whereas the hopping energy between the top surface of one topological insulator layer to the bottom surface of neighboring topological insulator layer is ∆D. symmetry and split the Dirac cones into two Weyl cones. In this section we will not start from an accidental band crossing between two energy bands but rather from the Hamiltonian of a topological insulator that relies on spin orbit coupling. Topological insulators are like ordinary insulators except that their boundaries are conducting and fault tolerant. The objective of discussing topological insulators is their unique connections with Weyl semimetals. The topological insulators and ordinary insulators provide the boundary where Weyl fermions exists. Imagine an experimental setup of several layers of topological insulators. A layer of ordinary insulator is used for separating two adjacent topological insulator layers. This is the setup considered in Ref. 1 to elaborate the connection between Dirac and Weyl semimetals derived from topological systems, see Fig. 1-5. The Hamiltonian for this setup is

X z ~ 0 z x 0 H = [vF τ (ˆz × ~σ) · k⊥δi,j + mτ σ δi,j + ∆sτ σ δi,j i,j (1-22) 1 + 0 − 0 † + ∆D(τ σ δj,i+1 + τ σ δj,i−1)]c ck ,j, 2 k⊥,i ⊥

z x + τ x+iτ y − τ x−iτ y where vF is Fermi velocity, τ , τ , τ = 2 , τ = 2 are Pauli matrices operating on pseudo spin basis, while ~σ, σ0 are also Pauli matrices but these are operating on real spin ~ basis. k⊥ is wave vector on surface of topological insulator layers, and m is magnetization. ∆s is the hopping energy of particles between top and bottom surfaces of any topological insulator

19 layers , and ∆D is hopping energy of particles between top surface of a topological insulator layer to the bottom surface of neighboring topological insulator layer. The summation index runs over the topological insulator layers. After taking the Fourier’s transformation of the above Hamiltonian, the Hamiltonian in k space is

z ~ 0 z x 0 H(k) = vF τ (ˆz × ~σ) · k⊥ + mτ σ + (∆s + ∆D cos(kzd))τ σ (1-23) y 0 −∆D sin(kzd)τ σ ,

where d is the spacing between two adjacent topological insulator layers. The energy eigenvalues of the above Hamiltonian are

2 2 2 2 ε (±, k) = vF k⊥ + (m ± ∆(kz)) , (1-24)

p 2 2 p 2 2 where s = ±, k⊥ = kx + ky, and ∆(kz) = ∆s + ∆D + 2∆s∆D cos(kzd). First we discuss a time reversal invariant energy dispersion (m = 0). In this energy dispersion the Dirac cones

Figure 1-6. The Weyl semimetal phase exists between boundary of ordinary insulator(OI) and topological insulator (TI). Here m = 0.5.

20 can be formed when

kzd = 0 and ∆s = −∆D or (1-25)

kzd = π and ∆s = ∆D.

Otherwise the energy dispersion is gapped. The energy dispersion is expanded near these points (kzd = 0, kzd = π) to form the Dirac cones

∆(kz) ∼ vF kz, (1-26)

√ where vF = d ∆s∆D. After locating the Dirac nodes in the energy dispersion, we split them by breaking time reversal symmetry m 6= 0:

2 2 2 2 ε (±, k) = vF k⊥ + (m ± vF kz) . (1-27)

The Hamiltonian for the above energy dispersion is

~ z H = χvF ~σ · k + mσ , (1-28) where χ = ± is the chirality of the particles. Thus the two energy bands model can be realized from the multiple topological insulator layers model. This is a concrete example where a degenerate Dirac cone split by breaking time reversal symmetry into two separate Weyl cones. However it was not the purpose of the authors to just derive this Hamiltonian [7]. The conclusion is more interesting. The Weyl states exist between boundaries of topological and ordinary insulators, provided time reversal symmetry is broken.

m ± ∆(kz) = 0 (1-29) 2 2 2 (∆s − ∆D) ≤ m ≤ (∆s + ∆D)

The above condition of Weyl phase is plotted in Fig. 1-6. 1.2 Detection of Weyl Fermions by Angle-Resolve Photo Emission Spectroscopy

Angle resolve photo emission spectroscopy (ARPES) is an experimental technique which determines the energy dispersion of a sample. The working principle of the ARPES

21 Figure 1-7. The experimental data for observation of Weyl fermions. Figs (a-d) are taken from Ref. [1], and Figs(e-g) are taken from Ref. [2]. Both of these experimental groups used ARPES technique to observe the Fermi arc states and the Weyl cone states in TaAs semimetal. technique is based on the photoelectric eﬀect. By using this technique both the surface Fermi arc states and the bulk Weyl cone states have been detected in several semimetals. In Fig. 1-7 the experimental data of TaAs semimetals is shown. These were among the ﬁrst experimental observations of Weyl fermions in semimetals. Both of these groups used the ARPES technique to detect the surface Fermi arc states and the bulk Weyl cone states. Fig. 1-7a is the experimental data for the surface Fermi arc states. This is matched with the schematic plot in Fig. 1-7b. These surface Fermi arc states connect with the bulk Weyl cone states. This is shown in Fig. 1-7c. Fig. 1-7d is an schematic illustration of the energy dispersion of Weyl cone in kz direction. This data is taken from the Ref. [1]. Figs. 1-7(e,f) are the experimental data of the surface Fermi arc states taken from Ref. [2]. These Fermi surface arc states also connect with the bulk Weyl cone states. This can be seen in Fig. 1-7g.

22 1.3 The Mangnetotransport Measurements

The electronic properties of semimetals can also be revealed by magnetotransport properties like the conductivity, resistivity, and magnetization. In a magnetotransport experiment the sample under study is exposed to external electric and magnetic ﬁelds. In the presence of electric and magnetic ﬁelds the free electrons and holes of energy bands are accelerated via F~ = eE~ + e~v × B.~ (1-30)

The electric field shifts the position of the Fermi surface in k-space and particles get an average displacement in or against the direction of applied electric field. These effects are experimentally observed in linear response theory,

~j = σE,~ (1-31) where ~j is current density and σ is conductivity tensor– the response of the electric field. A magnetic field tends to rotate the direction of the free particles. This causes the response of current density to an electric field non-diagonal. For isotropic energy bands (like the Dirac and Weyl energy dispersions) the different elements of the conductivity and resistivity matrices are

σ0 σxx = σyy = 2 (1-32) 1 + (ωcτtr) σ0ωcτtr σxy = −σyx = 2 (1-33) 1 + (ωcτtr)

σzz = σ0 (1-34) 1 ρxx = ρyy = ρzz = (1-35) σ0

ρxy = −ρyx = ωcτtr. (1-36)

Here σ0 is the conductivity of charge particles in the absence of the magnetic ﬁeld. ωc is cyclotron frequency and τtr is the transport time. It can be seen from the above equations that the longitudinal conductivity σzz and resistivity ρzz are independent of the magnetic ﬁeld. However in Dirac and Weyl semimetals this theory breaks down. In these semimetals the

23 Figure 1-8. Negative magneto resistance shown for different angle between electric and magnetic field (T=2K), and at different temperatures (θ = 0). [3]

longitudinal conductivity σzz(B) and resistivity ρzz(B) are functions of magnetic ﬁeld. This diﬀerence arises due to Berry curvature Ω~ . The Dirac and Weyl semimetals have anomalous magnetotransport due to the presence of Berry curvature. This anomalous magnetotransport has been experimentally observed in several Dirac and Weyl semimetals. In these semimetals the longitudinal conductivity increases quadratically

2 with magnetic ﬁeld σzz ∼ B , and therefore the longitudinal resistivity decreases quadratically

−2 ρzz ∼ B . Cd3As2 is a Dirac semimetal, and its magnetotransport properties are anomalous.

[3]. In Fig. 1-8 the experimental data for magnetotransport of Cd3As2 is shown. This data is obtained by changing angle between the electric and magnetic ﬁelds. The longitudinal

magnetoresistance ρzz is observed when both the electric and magnetic fields are parallel (θ = 0). For an isotropic Dirac energy dispersion the longitudinal resistivity should stay constant unless the Berry curvature effect is included. The magnetoresistance data also shows the angular dependence of the magnetoresistance. As the angle between electric and magnetic field is increased the magnetoresistance functional dependence on magnetic field is changed. For an orthogonal setup of electric and magnetic fields the magnetoreistance is shown to

24 depend linearly on magnetic ﬁeld, ρxx ∼ B. This is not consistent with the conventional theory with a Dirac energy dispersion, which requires the transverse magnetoresistance to

−1 be independent of magnetic ﬁeld ρxx = σ0 . These new experimental observations of magnetotranport in the Dirac semimetals make the study of magnetotransport interesting. The anomalous longitudinal magnetoresitance was measured at several temperatures. This is because the physical origin of negative magnetoresistance can be weak localization due to

quantum interference effect and magnetic scattering. But the studied sample (Cd3As2) is nonmagnetic and this negative magnetoresistance is shown to survive in room temperature (T ∼ 300k), where quantum interference effect giving rise to weak localization are practically absent. 1.4 Planar Hall Effect in Dirac and Weyl Semimetals

Another anomalous feature of the magnetotransport in Dirac and Weyl semimetals is the observation of Planar Hall effect. Unlike the usual Hall effect, the Planar Hall effect is the magnetotransport phenomena where the Hall voltage is measured in the plane of the electric and magnetic fields, Fig. 1-9(d). The Planar Hall effect is observed by tilting magnetic field towards the plane of the sample. One of the possible explanation of the Planar Hall effect in Dirac and Weyl semimetals is due to the chiral anomaly. For a setup of magnetotransport experiment where magnetic and electric fields are making some finite angle θ(x-z plane) than a Hall voltage is observed in this plane, provided the sample is Dirac or Weyl semimetal. By including the chiral anomaly effect in the magnetotransport theory the Planar Hall resistivity is [5]

2 ρxz ∼ B sin θ cos θ. (1-37)

The Planar Hall resistivity is observed in several Dirac and Weyl semimetals that support the anomalous magnetotransport. However, the Planar Hall resistivity is also observed in some semimetals that do not support the anomalous magnetotransport. In Fig. 1-9(a,b) the magnetotransport data is shown for the half Heusler GdPtBi Weyl semimetal. The angle φ is the angle between the applied electric and magnetic ﬁelds. Fig. 1-9a shows the angle

25 Figure 1-9. Figs. (a,b) are the data taken from the Ref. [4] and Figs. (c,d) are the data taken from the Ref. [5]. In Figs. (a,b) the Planar Hall effect are explained by chiral anomaly and the sample is Half Heusler GdPtBi Weyl semimetal. In Figs. (c,d) the Planar Hall effect can not be explained by the chiral anomaly and the sample is Type-II Dirac semimetal NiT e2 dependence of the Planar hall effect and Fig. 1-9b shows the negative magnetoresistance. This confirms the origin of the Planar Hall effect in GdPtBi is the chiral anomaly that arises due to Berry curvature. This is the not case with data of sample NiT e2. In Figs. 1-9 (c,d) magnetotransport data is shown for NiT e2. Here the angle θ is the angle between the applied electric and magnetic fields. The variation in the planar Hall effect with the angle is shown in Fig. 1-9c. However the longitudinal magnetoresistance data does not show negative magnetoresistance. This rules out chiral anomaly as a possibility of the Planar Hall effect in

NiT e2.

26 1.5 Conclusions

In this chapter we introduced Dirac and Weyl semimetals. The difference between the Dirac and Weyl semimetals arises due to breaking of the time reversal and inversion symmetries. Weyl semimetals require either or both of these symmetries to be absent. We have also discussed the two experimental techniques to detect these semimetals, angle-resolve photo emission spectroscopy and magnetotransport measurements . In ARPES the Weyl semimetals are distinguished from the Dirac semimetals by measuring the surface states of a semimetal sample. Since in Dirac semimetals the surface states form a close loop Fermi circle, whereas in Weyl semimetals the surface states form an open loop Fermi arc. Both of these surface states are emanating from the bulk energy dispersion. In the bulk they have linear energy dispersion and form either Dirac or Weyl cones. The magnetotransport properties of these semimetals is different than an ordinary three dimensional electron gas. These difference are experimentally observed in different components of the conductivity and resistivity matrices. The most prominent difference is seen in functional

−2 dependence of the longitudinal magnetoresistance in magnetic ﬁeld, ρzz ∼ B . This component of resistivity tensor is measured when the electric and magnetic ﬁelds are parallel to each other, but stay constant for an ordinary semimetal. This anomalous magnetoresistivity arises due to Berry curvature Ω~ . The functional dependence of the transverse component of

magnetoresistivity in magnetic ﬁeld also shows an unusual result, ρxx = ρyy ∼ B. In an

−1 ordinary semimetal this component of magnetoresistivity is constant, ρxx = ρyy = σ0 . The Planar Hall effect is experimentally observed in the Dirac and Weyl semimetals. The Planar Hall effect in these semimetals is explained by the chiral anomaly – a Berry curvature effect Ref. [4]. However there are examples that contrast with this conclusion Ref. [5].

27 CHAPTER 2 THE SEMICLASSICAL EQUATIONS OF MOTION FOR THE DIRAC AND WEYL SEMIMETALS 2.1 Introduction

In this chapter we derive the formulas of magnetotransport properties of the Weyl and Dirac semimetals. For calculating these formulas we utilize quantum theory in phase space. We start by discussing the underlying assumption of quantum mechanics in phase space. This phase space is formed by the conjugate variables position(time) and momentum(energy). We also derive an equation that defines the order operation between product of two operators in quantum phase space. We need this because the Boltzmann transport equation is derived by the product of two operators. First we use this equation in the quantum kinetic equation for any general operator. Then we specifically focus on the operator ‘the Keldysh Greens function’ since its quantum kinetic equation gives the Boltzmann equation. The equations of motion for the Dirac and Weyl semimetals are different than ordinary three dimensional electron gas. This difference enters from the Berry curvature. In the presence of external electric and magnetic field additional terms add up to the force and velocity equations of these gases. In this chapter we compare the relative effects of the Lorentz force and the force due to Berry curvature in magnetotransport derived from the Boltzmann equation. This is achieved by changing the relative angle between the electric field and the magnetic field. The diagonal elements of the transverse magnetoconductivity tensor decrease quadratically with the magnetic field due to the Lorentz force effect. Whereas the longitudinal MC increases with the magnetic field due to the Berry curvature force effect. Here transverse and longitudinal are used as a reference to the direction of the magnetic field. Therefore, by rotating the electric field away from the magnetic field axis, the diagonal elements of the magnetoconductivity tensor plot pass through a minimum point. At this minimum point the Lorentz force and the force of the Berry curvature effectively cancel each other. The situation is different for the magnetoresistance. The effect of the Lorentz force is absent in the diagonal

28 elements of the transverse magnetoresistance (TMR) tensor, and therefore a comparison between these forces is not possible. Interestingly the TMR data from Weyl semimetals does not stay constant with magnetic field. [8] The TMR is shown to increase linearly with increasing magnetic field [9], in contrast to a formal theory of Boltzmann equation in the relaxation time approximation. In this thesis we phenomenologically include the magnetic field dependence in the relaxation process of the distribution function in order to obtain a TMR which increases linearly with magnetic field. The magnetic field dependence of the TMR has been an active area of research in the Dirac and Weyl semimetals, both in the ultra quantum and a semiclassical limits of studies. [10, 11] 2.2 Quantum Mechanics in Phase Space

The mechanics of classical particles is described by the position(q) and momentum(p) of the particles in phase space. In quantum mechanics one can not specify the position and momentum of a particle simultaneously because the position and momentum operators do not commutes [ˆq , pˆ] = i~. Even though the position and momentum operators do not commute, one can define a type of distribution function in a position and momentum space, much like the classical phase space. However, there are some differences. The analog of the classical distribution function is not positive definite. In this section we show how quantum mechanics can be done rigorously in phase space. In the next section we apply this approach to the specific case of a gas of Weyl fermions. We shall derive the equations of motion of the Weyl Hamiltonian. The gapless dispersion relation of the Weyl fermions introduces changes in the velocity and force equations when compare to free electrons. Additional information on this approach can be found in Refs. [12] and [13]. In classical mechanics the average value of any property A (like energy, momentum etc.) is computed by a phase space integral:

Z hAiClassical = dq dp P (q, p) A(q, p). (2-1)

29 Here q, p, P (q, p), A(q, p) are the position, the momentum, the probability distribution, and the physical property, respectively. There is a similar relation in quantum mechanics. Speciﬁcally, there is a distribution function, W , which depends on position and momentum and the expectation value of a quantum operator can be written in a similar form to the classical average above: Z ˆ hAiQuantum = dq dp W (q, p) A(q, p). (2-2)

In the following we take the system to be in quantum state |ψi. First we construct the function W . Then we define A(q, p), and finally we show that the Eq. 2-2 is satisfied. The function, W , which is often called the Wigner distribution function, is defined in terms of the expectation value of the unitary operator Uˆ(λ, µ) = exp( i [λqˆ + µpˆ]). Here we denote ~ operators with a carat, e.g. pˆ, whereas the equivalent eigenvalue is just p. Using Uˆ the Wigner distribution function is defined as

1 Z i W (q, p) = dλdµ hψ|Uˆ(λ, µ)|ψi exp(− (λq + µp)). (2-3) (2π~)2 ~

The Wigner distribution function may be thought of as the Fourier transform in the diﬀerence coordinates. To see this we manipulate Eq. 2-3 using the identity

i i i iλµ exp( [λqˆ + µpˆ]) = exp[ λqˆ] exp[ µpˆ] exp(− ), (2-4) ~ ~ ~ 2~ which follows from the fact that both qˆ and pˆ commute with their commutator. Inserting a complete set of position eigenstates, |q0i, the Wigner distribution function is

30 By using R dλ exp(− i (λ(q − q0 + µ )) = 2π δ(q − q0 + µ ) and performing the integral on dq0, ~ 2 ~ 2 we get: 1 Z µ µ i W (q, p) = dµ hψ|q + ihq − |ψi exp(− µp), (2-7) (2π~) 2 2 ~ which is indeed the Fourier transform in the diﬀerence of coordinates. The class of operators that we consider are combinations of the position and momentum operators. We write a general such operator, Aˆ, as

1 Z i Aˆ(ˆq, pˆ) = dλdµ a(λ, µ) exp[ (λqˆ + µpˆ)]. (2-8) (2π~)2 ~ a(λ, µ) is a function not an operator. For example the function a(λ, µ) for the position operator is,

Z i a(λ, µ) = dq dp q exp[− (λq + µp)], (2-9) ~ and similarly a(λ, µ) for the momentum operator is,

Z i a(λ, µ) = dq dp p exp[− (λq + µp)]. (2-10) ~

Once we have the function a(λ, µ), then we deﬁne the corresponding function in phase space, A(q, p), that will be used in Eq. 2-7,

Z i A(q, p) = dλ dµ a(λ, µ) exp[ (λq + µp)]. (2-11) ~ a(λ, µ) and A(q, p) are related by the Fourier transformation,

Z i a(λ, µ) = dq dp A(q, p) exp[− (λq + µp)]. (2-12) ~

Substituting Eq. 2-12 into Eq. 2-8 implies that

1 Z i Aˆ(ˆq, pˆ) = dλ dµ dq dp A(q, p) exp[− (λ(q − qˆ) + µ(p − pˆ))]. (2-13) (2π~)2 ~

31 The expectation value of a quantum operator is

ˆ ˆ hAiQuantum = hψ|A(ˆq, pˆ)|ψi (2-14)

1 Z i hAˆ(ˆq, pˆ)i = dλ dµ dq dp A(q, p) exp[− (λq + µp)]hψ|Uˆ(λ, µ)|ψi. (2-15) (2π~)2 ~ By substituting Eq. 2-3 into Eq. 2-15, we get the expectation value of an operator in terms of a phase space integral Z ˆ hAiQuantum = dq dp W (q, p) A(q, p). (2-16)

This is our desired result. 2.3 Multiplying Two Operators In the Quantum Phase Space

In this section I shall derive an equation that will Fourier transform the product of two operators (A,ˆ Bˆ) with their eigenvalues (A, B) in phase space. We use this technique to ﬁnd the equations of motion for the position, the momentum, and the distribution function of a Weyl gas. We begin by inverting Eq. 2-13 to ﬁnd an expression for A(q, p). This is done by multiplying Eq. 2-13 by U †(λ, µ) and then using the identity T r[Uˆ †(λ, µ)Uˆ(λ0, µ0)] =

2π~δ(λ − λ0)δ(µ − µ0).

1 Z i A(q, p) = dλdµ exp[ (λq + µp)] T r[Uˆ †(λ, µ)Aˆ(ˆq, pˆ)] (2-17) (2π~)2 ~

We next consider a case when the operator of interest is the product of two operators: Cˆ = AˆBˆ:

1 Z i C(q, p) = dλdµ exp[ (λq + µp)] T r[Uˆ †(λ, µ)Aˆ(ˆq, pˆ)Bˆ(ˆq, pˆ)]. (2-18) (2π~)2 ~

32 Substituting expressions for Aˆ and Bˆ in terms of A and B as in Eq. 2-13, we obtain the following rather lengthy expression for C.

1 Z C(q, p) = dλdµdλ0dµ0dq0dp0dλ00dµ00dq00dp00A(q0, p0)B(q00.p00) (2π~)5 i exp[ (λq − λ0q0 − λ00q00 + µp − µ0p0 − µ00p00)] (2-19) ~ T r[Uˆ †(λ, µ)Uˆ(λ0, µ0)Uˆ(λ00, µ00)]

To evaluate the trace in Eq. 2-19 we use the identity

i Uˆ †(λ, µ)Uˆ(λ0, µ0) = Uˆ †(λ − λ0, µ − µ0) exp[ (λµ0 − λ0µ)], (2-20) 2~

which allows us to take the trace i i exp[ (λµ0 − λ0µ)]T r[Uˆ †(λ − λ0, µ − µ0)Uˆ(λ00, µ00)] = exp[ (λ00µ0 − λ0µ00)] 2~ 2~ (2-21) 0 00 0 00 2π~δ(λ − λ − λ )δ(µ − µ − µ ).

The new expression for C is

1 Z C(q, p) = dλ0dµ0dq0dp0dλ00dµ00dq00dp00A(q0, p0)B(q00.p00) (2π~)4 i i exp[ (λ0(q − q0) + µ0(p − p0))] exp[ (λ00(q − q00) + µ00(p − p00))] (2-22) ~ ~ i exp[ (λ00µ0 − λ0µ00)]. 2~ Eq. 2-22 has the same form as Eq. 2-13 except for factor exp[ i (λ00µ0 − λ0µ00)]. We expand 2~ the exponential i i exp[ (λ00µ0 − λ0µ00)] = 1 + [ (λ00µ0 − λ0µ00)] ··· (2-23) 2~ 2~ The ﬁrst term in the expansion gives a simple relation of Fourier transformation between the product of operators and their eigenvalues,

A(q, p)B(q, p). (2-24)

33 For the second term we have 1 Z dλ0dµ0dq0dp0dλ00dµ00dq00dp00A(q0, p0)B(q00.p00) (2π~)4 i i exp[ (λ0(q − q0) + µ0(p − p0))] exp[ (λ00(q − q00) + µ00(p − p00))] (2-25) ~ ~ i [ (λ00µ0 − λ0µ00)]. 2~ λ00 can be obtained by a derivative ∂/∂q00, µ0 can be obtained by a derivative ∂/∂p0, and similarly for λ0 and µ00. The integrand in the Eq. 2-25 is thus

i i i exp[ (λ0(q − q0) + µ0(p − p0))] exp[ (λ00(q − q00) + µ00(p − p00))][ (λ00µ0 − λ0µ00)] ~ ~ 2~ ∂ i ∂ i = ~ [ exp[ (λ00(q − q00) + µ00(p − p00))] exp[ (λ0(q − q0) + µ0(p − p0))] i2 ∂q00 ~ ∂p0 ~ ∂ i ∂ i − exp[ (λ0(q − q0) + µ0(p − p0))] exp[ (λ00(q − q00) + µ00(p − p00))]]. ∂q0 ~ ∂p00 ~ Performing the integrals yields

∂ ∂ ∂ ∂ = ~ ( A(q, p) B(q, p) − A(q, p) B(q, p)). (2-26) 2i ∂p ∂q ∂q ∂p

for the second term in the expansion. Eqs. 2-24 and 2-26 give the ﬁrst two terms of the Fourier transformation between the product of operators and their eigenvalues. The higher order terms are found by using the same procedure. The general expression for the product of two operators is

i ←− −→ ←− −→ C(q, p) = A(q, p) exp[ [ ∂ p · ∂ q − ∂ q · ∂ p]]B(q, p). (2-27) 2~ ←− −→ Eq. 2-27 includes all orders in the expansion of the exponential of Eq. 2-23. Here ∂ ( ∂ ) means the gradient operating on the left(right) function. So far we have only considered the position and momentum operators, but the above formula is equally valid for time and energy operators. In this thesis we use two pairs of the conjugate variables; position, momentum, and time, energy {(x, p), (t, )}. To include the time and energy derivative in the Eq. 2-26, we

34 deﬁne the Moyal operator mˆ = exp[ iMˆ ], ~

ˆ ←− −→ ←− −→ ←− −→ ←− −→ M = ∂ x · ∂ p + ∂ · ∂ t − ∂ p · ∂ x − ∂ t · ∂ , (2-28) that connects the Fourier transformation between the product of operators and their eigenvalues. 2.4 The Semiclassical Equations of Motion for the Weyl Semimetals

In this section the equation of motion for an operator fˆ (later we will set this operator equals to the Keldysh Greens function fˆ = Gˆk) is derived. The quantum kinetic equation for fˆ is, [Gˆ−1, fˆ] = 0, (2-29) here Gˆ−1 = − Hˆ shows inverse Greens function. To get the transport equation we wish to work in a representation where the unperturbed Hamiltonian is diagonal. This is accomplished by unitary operator Uˆ. The transformed commutator of Gˆ−1 and fˆ is

Uˆ †Gˆ−1Uˆ Uˆ †fˆUˆ − Uˆ †fˆUˆ Uˆ †Gˆ−1Uˆ = 0. (2-30)

Now we write Eq. 2-30 in phase space by using the Wigner’s representation introduced in the previous section.

U †mG−1mUmU †mfmU − U †mfmUmU †mG−1mU = 0. (2-31)

We are now going to expand using the expansion in Eq. 2-28. This is quite complicated so we do it one step at a time. First consider the expansion of

i U †mG−1mU = G¯−1 + ~[U †MG−1U + U †G−1MU] 2 i = G¯−1 + ~[(∂~ U † · ∂~ G−1 − ∂~ U † · ∂~ Gˆ−1)U (2-32) 2 x p p x † ~ −1 ~ ~ −1 ~ +U (∂xG · ∂pU − ∂pG · ∂xU)].

The Moyal operator also has a time derivative. Since the Hamiltonian and the inverse of Greens function have no time dependence, the time derivatives of unitary operator and the

35 † −1 inverse Greens function are zero ∂tU = 0, ∂tU = 0, and ∂tG . The Hamiltonian of a Weyl gas is a matrix. We work in basis of the Hamiltonian eigenfunctions. A bar on top of any operator means following operation O¯ = U †OU. Equation 2-32 can be further simpliﬁed

~ † ~ using the Berry vector potential, A(p) ≡ iU ∂pU, and the electromagnetic vector potential, ~ † ~ A(x) ≡ iU ∂xU.

† −1 ¯−1 ~ ~ ~ ¯−1 ~ ~ ¯−1 U mG mU = G + [−A(x) · ∂pG + A(p) · ∂xG 2 (2-33) ~ ¯−1 ~ ~ ¯−1 ~ +∂xG · A(p) − ∂pG · A(x)]

To avoid the repetition of lengthy expressions we deﬁne the second term on the right hand side as G1

G ≡ ~[−A~(x) · ∂~ G¯−1 + A~(p) · ∂~ G¯−1 + ∂~ G¯−1 · A~(p) − ∂~ G¯−1 · A~(x)]. (2-34) 1 2 p x x p

The same procedure can be repeated for the operator fˆ. We end up with the following two equations,

† −1 ¯−1 U mG mU = G + G1 (2-35) † ¯ U mfmU = f + D1,

where

D ≡ ~[−A~(x) · ∂~ f¯+ A~(p) · ∂~ f¯+ ∂~ f¯· A~(p) − ∂~ f¯· A~(x)]. (2-36) 1 2 p x x p

With this notation the commutator in Eq. 2-31 reduces to 4 terms:

U †mG−1mUmU †mfmU − U †mfmUmU †mG−1mU = [G¯−1, f¯] (2-37) i + ~([G¯−1M¯ f¯− f¯M¯ G¯−1](2-38) 2 ¯−1 ¯ ¯ ¯−1 + [G MD1 − D1MG ](2-39) ¯ ¯ ¯ ¯ + [G1Mf − fMG1]). (2-40)

36 We now calculate the terms in Eq. 2-37-2-40 separately. Substituting in the operator M¯ from Eq. 2-28 , the terms in Eq. 2-38 become

i i ~[G¯−1M¯ f¯− f¯M¯ G¯−1] = ~([∂~ G¯−1 · ∂~ f¯− ∂~ G¯−1 · ∂~ f¯− ∂~ f¯· ∂~ G¯−1 + ∂~ f¯· ∂~ G¯−1] 2 2 x p p x x p p x ¯−1 ¯ ¯−1 ¯ ¯ ¯−1 ¯ ¯−1 +[∂G · ∂tf − ∂tG · ∂f + ∂tf · ∂G − ∂f · ∂tG ])

i = i ∂ f¯+ ~(−{∂i H∂¯ i f¯+ ∂i f∂¯ i H¯ } + {∂i H∂¯ i f¯+ ∂i f∂¯ i H¯ }). (2-41) ~ t 2 x p p x p x x p ~ ¯−1 The last line equality is obtained by taking derivatives of the inverse Greens function, ∂xG = ~ ¯ ~ ¯−1 ~ ¯ −∂xH and ∂pG = −∂pH. For the next two terms, we shall ignore the second and higher order derivatives operating on the G¯−1 and f¯, since those are higher order corrections. Making the same substitutions of M¯ in Eqs. 2-39, 2-40 we obtain the following

¯−1 ¯ ¯ ¯−1 i ¯−1 i i i ¯−1 i ¯−1 i i i ¯−1 [G MD1 − D1MG ] = {∂xG ∂pD1 + ∂pD1∂xG } − {∂pG ∂xD1 + ∂xD1∂pG }. (2-42)

¯ ¯ ¯ ¯ i i ¯ i ¯ i i i ¯ i ¯ i [G1Mf − fMG1] = {∂xG1∂pf + ∂pf∂xG1} − {∂pG1∂xf + ∂xf∂pG1} (2-43)

In expansion of the Moyal operator m, we only keep derivative up to linear order. Although both potentials, the electromagnetic vector potential and the Berry potential, are the first order derivative of the unitary operator yet we need to take another derivative of these potentials to get the physical fields, the electromagnetic fields and the Berry curvature. This does not violate the linear order theory in the external electromagnetic fields. Thus, in Eqs.

2-41-2-43 we replace the derivatives of G1 and D1 by

i ~ i j j ¯−1 j ¯−1 i j ~ i j j ¯ j ¯ i j ∂pG1 = {∂pA (p)∂xG + ∂xG ∂pA (p)} = − {∂pA (p)∂xH + ∂xH∂pA (p)} 2 2 (2-44) ∂i G = −~{∂i Aj(x)∂jG¯−1 + ∂jG¯−1∂i Aj(x)} = ~{∂i Aj(x)∂jH¯ + ∂jH∂¯ i Aj(x)} x 1 2 x p p x 2 x p p x

i ~ i j j ¯ j ¯ i j ∂pD1 = {∂pA (p)∂xf + ∂xf∂pA (p)} 2 (2-45) ∂i D = −~{∂i Aj(x)∂jf¯+ ∂jf∂¯ i Aj(x)}. x 1 2 x p p x

37 With Eqs. 2-44 and 2-45 the sum of Eqs. 2-42 and 2-43 becomes

¯−1 ¯ ¯ ¯−1 ¯ ¯ ¯ ¯ i ¯−1 i i i ¯−1 [G MD1 − D1MG ] + [G1Mf − fMG1] = [{∂xG ∂pD1 + ∂pD1∂xG }− (2-46) i ¯−1 i i i ¯−1 i i ¯ i ¯ i i i ¯ i ¯ i {∂pG ∂xD1 + ∂xD1∂pG }] + [{∂xG1∂pf + ∂pf∂xG1} − {∂pG1∂xf + ∂xf∂pG1}].

The sum of second and third terms on the right hand side of Eq. 2-46 is thus

{∂i G ∂i f¯+ ∂i f∂¯ i G } − {∂i G¯−1∂i D + ∂i D ∂i G¯−1} = {~{∂i Aj(x)∂jH¯ + ∂jH∂¯ i Aj(x)}∂i f¯ x 1 p p x 1 p x 1 x 1 p 2 x p p x p +∂i f¯~{∂i Aj(x)∂jH¯ + ∂jH∂¯ i Aj(x)}} − {∂i H¯ ~{∂i Aj(x)∂jf¯+ ∂jf∂¯ i Aj(x)}+ p 2 x p p x p 2 x p p x ~{∂i Aj(x)∂jf¯+ ∂jf∂¯ i Aj(x)}∂i H¯ }, 2 x p p x p (2-47)

which will turn out to be the Lorentz force. Deﬁning the magnetic ﬁeld as

B~ = −~(∂~ × A~(R)), (2-48) e R

the right hand of the Eq. 2-47 reduces to the Lorentz force.

~ ¯ ~ ~ ¯ ~ ¯ ~ ¯ ~ e[∂pH × B · ∂pf + ∂pf · ∂pH × B] (2-49)

In similar manner the sum of the ﬁrst and the fourth term on the right hand side of the Eq. 2-46 gives the analog of the Lorentz force in the reciprocal space.

{∂i G¯−1∂i D + ∂i D ∂i G¯−1} − {∂i G ∂i f¯+ ∂i f∂¯ i G } = {−∂i H¯ ~{∂i Aj(p)∂j f¯+ ∂j f∂¯ i Aj(p)} x p 1 p 1 x p 1 x x p 1 x 2 p x x p −~{∂i Aj(p)∂j f¯+ ∂j f∂¯ i Aj(p)}∂i H¯ } − {−~{∂i Aj(p)∂j H¯ + ∂j H∂¯ i Aj(p)}∂i f¯ 2 p x x p x 2 p x x p x −∂i f¯~{∂i Aj(p)∂j H¯ + ∂j H∂¯ i Aj(p)}} x 2 p x x p (2-50)

In reciprocal space the Berry curvature plays the role of the magnetic ﬁeld

~ ~ ~ Ω = ∂p × A(p) (2-51)

38 using this deﬁnition the right hand side of Eq. 2-50 becomes

1 ~ ¯ ~ ~ ¯ ~ ¯ ~ ¯ ~ = [∂xH × Ω · ∂xf + ∂xf · ∂xH × Ω]. (2-52) ~

We can now sum all the terms in the commutator to obtain the Moyal expansion of the quantum kinetic equation

U †mG−1mUmU †mfmU − U †mfmUmU †mG−1mU = [G¯−1, f¯] i +i ∂ f¯+ ~({(−∂~ H¯ + e ∂~ H¯ × B~ ) · ∂~ f¯+ ∂~ f¯· (−∂~ H¯ + e ∂~ H¯ × B~ )} (2-53) ~ t 2 x p p p x p ~ ¯ 1 ~ ¯ ~ ~ ¯ ~ ¯ ~ ¯ 1 ~ ¯ ~ +{(∂pH + ∂xH × Ω) · ∂xf + ∂xf · (∂pH + ∂xH × Ω)}). ~ ~ We can further simplify this by deﬁning the force and velocity as

~ ~ ¯ ~ ¯ ~ F = −∂xH + e ∂pH × B (2-54) ~ ¯ 1 ~ ¯ ~ ~x˙ = ∂pH + ∂xH × Ω, ~ Our ﬁnal result for the commutator is i i [U †mG−1mUmU †mfmU − U †mfmUmU †mG−1mU] = [f,¯ G¯−1] ~ ~ (2-55) 1 +∂ f¯+ ({F~ · ∂~ f¯+ ∂~ f¯· F~ } + {~x˙ · ∂~ f¯+ ∂~ f¯· ~x˙}). t 2 p p x x

2.5 The Boltzmann Equation for the Weyl Gas

In the previous section we derived the quantum kinetic equation for a general operator f¯. In this section we apply these results to get a Boltzmann like equation. For this purpose we substitute the Keldysh Greens function in place of a general operator f¯ = G¯K in Eq. 2-55,

i ¯K ¯−1 ¯K 1 ~ ~ ¯K ~ ¯K ~ ~ ¯K ~ ¯K [G , G ] + ∂tG + ({F · ∂pG + ∂pG · F } + {~x˙ · ∂xG + ∂xG · ~x˙}) = 0, (2-56) ~ 2

where the Keldysh Greens function in position and time space is [14],

K † † G (x1, t1, x2, t2) = −ih[ψ(x1, t1)ψ (x2, t2) − ψ (x2, t2)ψ(x1, t1)]i. (2-57)

39 The quantum kinetic equation of the Keldysh Greens function in phase space depends on

x1+x2 t1+t2 average (x = 2 , t = 2 ) and diﬀerence (x1 − x2, t1 − t2) of coordinates. The diﬀerence of coordinates are Fourier transformed (p, ). Notice that GK depends on (x, t, p, ). The Boltzmann equation only depends on (X, T, p). To get the Boltzmann equation we integrate over , which is one common way to do this [14].

1 Z d f (X, p) = [1 + G¯K (X, p, )] (2-58) s 2 2πi

By substituting Eq. 2-58 into Eq. 2-56 we get the left hand side of the Boltzmann equation, ~ where fs is a form of distribution function. We next assume that F and fs are diagonal in spin space. Finally if there is no external time dependence, then ∂tfs = 0 and we are left with the following left hand side,

X ~ ~ ~ ~ [(F · ∂pfs(X, p) + x˙ · ∂xfs(X, p))]. (2-59) s=±

Next we derive the right hand side of the Boltzmann equation. We consider collisions between the Weyl gas particles and impurities. We assume short range uncorrelated impurities. The interaction between Weyl gas particles and impurities introduces a ﬁnite self energy Σ¯ in the inverse Green’s function. This correction is modeled by deﬁning a collision integral I[C] = i [Σ¯ , G¯]K . These collisions are assumed to be elastic and only exchange momentum. ~

¯ K ¯R ¯A ¯ R ¯ A ¯K I[C] = [Σ0 {G0 − G0 } − {Σ0 − Σ0 }G0 ]. (2-60)

Here the superscripts R,A, and K denote retarded, advanced and the Keldysh Green’s functions and subscript 0 denotes the noninteracting functions. The noninteracting Green’s functions and self energies are

¯R ¯A X G0 − G0 = −2πi δ( − (s, p)) (2-61) s=±

¯K X G0 = −2πi δ( − (sp))(1 − 2fs(X, p)) (2-62) s=±

40 X Z dp0 Σ¯ R − Σ¯ A = −2πin |u|2 δ( − (sp0)) (2-63) 0 0 i (2π )3 s=± ~ X Z dp0 Σ¯ K = −2πin |u|2 δ( − (sp0))(1 − 2f (X, p0)). (2-64) 0 i (2π )3 s s=± ~ By substituting Eqs. 2-61 ··· 2-64 into Eq. 2-60 we get the collision integral is

i X Z dp0 I[C] = (−2πi)2n |u|2 δ( − (rp0))δ( − (sp)) i (2π )3 ~ r,s=± ~ (2-65) 0 [1 − 2fr(X, p )) − 1 + 2fs(X, p))].

The interband transitions are forbidden due to the delta functions in the integral:

i X Z dp0 I[C] = − 8π2n |u|2 δ( − (sp0))δ( − (sp)) i (2π )3 ~ s=± ~ (2-66) 0 [fs(X, p) − fs(X, p )].

The full Boltzmann like equation is now

2 Z 0 X 2πni|u| X dp [(F~ · ∂~ f (X, p) + ~x˙ · ∂~ f (X, p))] = δ((sp) − (sp0)) p s x s (2π )3 s=± ~ s=± ~ (2-67) 0 [fs(X, p) − fs(X, p )].

We may further simplify the above equation. This is done by using the relaxation time approximation. In this study we consider forces and distribution functions are independent of the spatial dependence. This excludes the spatial dependence of the distribution function. Thus in the next section we shall solve the following transport equation,

X X δfs [F~ · ∂~ f (X, p)] = − , p s τ (2-68) s=± s=± s

where τs are the relaxation time for the Weyl gas particles. 2.6 Incorporating the Berry Curvature in the Group Velocity and Force

In the previous section we derived the force and velocity equations Eq. 2-54. These are ~ ¯ ~ ¯ coupled equations in ∂xH and ∂pH:

~ ¯ ~ ¯ ~ p~˙ = −∂xH + e ∂pH × B (2-69)

41 ~ ¯ 1 ~ ¯ ~ ~x˙ = ∂pH + ∂xH × Ω. (2-70) ~ ~ ~ Here ~x˙ is the prefactor of ∂xf in the transport equation, and similarly p~˙ is the prefactor of ∂pf. The key observation now is that the second terms in Eqs. 2-69 and 2-70 are smaller than the

ﬁrst term. Thus, to lowest order p˙ = −∂xH and x˙ = ∂pH. To next lowest order we substitute p˙ for −∂xH and x˙ for ∂pH in the right hand side of Eqs. 2-69 and 2-70, respectively. Keeping only the ﬁrst order corrections, Eqs. 2-69 and 2-70 become

~ ¯ ~ p~˙ = −∂xH + e ~x˙ × B (2-71)

~ ¯ 1 ~ ~x˙ = ∂pH + p~˙ × Ω. (2-72) ~ One can uncouple these equation by using vector cross product properties (A~ × B~ ) × C~ = (A~ · C~ )B~ − (B~ · C~ )A~:

1 e e ~x˙ = (~v + E~ × Ω~ + (Ω~ · ~v)B~ ). (2-73) 1 + eB~ ·Ω~ ~ ~ ~ 1 e2 ~p˙ = (eE~ + e~v × B~ + (E~ · B~ )Ω)~ (2-74) 1 + eB~ ·Ω~ ~ ~ ~ ¯ ~ ~ ¯ In the above Eqs. 2-73 and 2-74 we substituted ~v = ∂pH and eE = −∂xH. Although these equations of motion are unconventional, they are well known in the study of Dirac and Weyl semimetals. They were also derived by using the equations of motion for a

e2 ~ ~ ~ Bloch wave packet [15]. In these equations the Berry curvature force, FB = (E · B)Ω, is ~ an unconventional combination of electric and magnetic ﬁelds. This study focuses on the case when both the Lorentz force and Berry curvature force are important in the magnetotransport in Weyl fermions. The inclusion of Berry curvature in a semiclassical study modiﬁes its equations of motion by introducing a change in phase space, adding an anomalous velocity and a force that is directly proportional to the Berry curvature. We shall use the term Berry force for this force. [16, 17, 15,9] From this work we know that the negative magnetoresistance in Weyl fermions can be explained by Berry curvature. Along with this, the conductivity should

42 show a comparison between the Lorentz and Berry curvature forces within the plane of applied fields. Berry curvature is a physical property of a Bloch wave function defined for a crystal lattice structure that lacks either the inversion symmetry [6] or the time reversal symmetry. [18] We will be interested in the magnetotransport of a semiclassical WF gas described by the linearized energy spectrum of an energy band structure near the Dirac points. The Dirac points are degenerate due to the Kramer’s rule. By applying an external magnetic field, the Landau levels are formed, and Kramer’s degeneracy is lifted. 2.7 The Semiclassical Theory of Magnetotransport in the Weyl Semimetals

The well studied Hamiltonian of Weyl fermions is [19, 20, 21]

~ ~ Hk = χvF ~((k − χQ) · ~σ + σ0Q0). (2-75)

Here, the σ’s are Pauli spinors and denote real spins, and χ = ± is used for including chirality. ~ The parameter(s) Q(Q0) breaks time reversal symmetry(inversion symmetry) and splits ~ degenerate Dirac cones in momentum(energy) space. [19] We consider Q0 = 0, and Q = Qzzˆ. In a semiclassical theory the magnetic ﬁeld enters from the Boltzmann equation, since the Landau levels are not resolved. Thus, the Zeeman splitting is included exactly in the energy eigenstates, but the Lorentz force is included in the transport equation. The eigenvalues of this Hamiltonian are

Ek(±, χ) = ±~k(χ)vF q (2-76) 2 2 k(χ) = k⊥ + (Qz + χkz) ,

p 2 2 where χ = ±1, and k⊥ = kx + ky. The eigenstates of the Hamiltonain are   cos θ 1 ~ 2 ψ+(~x) = √ exp(ik · ~x)   2V  θ  sin 2 exp iφ   (2-77) sin θ 1 ~ 2 ψ−(~x) = √ exp(ik · ~x)   , 2V  θ  − cos 2 exp iφ

43 where the wave vector points in the direction of θ and φ. The unitary operator deﬁned earlier maps plane wave states with the spin up or down onto the |ψ+i and |ψ−i eigenstates. For a given k direction, it is equal to   1 ie Z cos θ sin θ U(θ, φ) = √ exp(i~k · ~x) exp[ A~(x) · d~x]  2 2  , (2-78) 2V ~  θ θ  sin 2 exp iφ − cos 2 exp iφ

where we have included the vector potential A~(x) semiclassically. Here V denotes the volume

~ † ~ of a sample. With this unitary operator the Berry potential A(k) = i(U ∂kU) and Berry ~ ~ ~ curvature Ω = ∂k × A(k) of this Hamiltonian are   1 −φˆtan θ φˆ + iθˆ A~(p) =  2  2k  ˆ ˆ ˆ θ  φ − iθ −φ cot 2   (2-79) kˆ 1 0 Ω~ = −   , 2   2k 0 −1

~ † ~ ~ and similarly for the vector potential A(x) = i(U ∂xU) and the magnetic ﬁeld are B = e ~ ~ − ∂x × A(x). ~

e e ~x˙ = A(~v + E~ × Ω~ + (Ω~ · ~v)B~ ) (2-80) ~ ~ e2 F~ = A(eE~ + e~v × B~ + (E~ · B~ )Ω)~ , (2-81) ~ where A = (1 + e B~ · Ω)~ −1, is a change in phase space for Berry curvature. ~ In Eq. 2-68 we have derived a type of Boltzmann equation for the Dirac and Weyl semimetals. To make a comparative analysis of the Berry curvature force and the Lorentz force, we write the semiclassical Boltzmann equation with uniform in space and time independent electric and magnetic ﬁelds perturbing a system of Weyl particles, which includes both electrons and holes. Electrons are accelerated opposite to the direction holes in presence

44 of electric ﬁeld. They both can have left or right handed chirality.

~ 1 ~ F · ∇kfχ = I[Coll] ~ (2-82) e F~ = −eA(E~ + (E~ · B~ )Ω~ + ~v × B~ ) ~

Here fχ is the distribution function, I[coll] is the collision integral of a Boltzmann equation, ~ ~ ˆ E and B are the electric and magnetic ﬁelds, and ~v = ±vF k is the group velocity of the Dirac spectrum. The force term has the usual electric and magnetic forces as well as a force

~ ~k arising due to Berry curvature (Ω = χ 2k3 ). This force also displaces the sphere of the Weyl gas out of equilibrium and therefore charge transport parameters such as the thermal and electrical conductivities depend on this force. We assume linear response in the electric ﬁeld for the nonequilibrium distribution function, and use the relaxation time approximation: I[coll] = − δfχ . τtr

∂f e2 ∂f −eE~ · ~v eq − (E~ · B~ )(Ω~ · ~v) eq ∂E ∂E k ~ k (2-83) ~ 1 ~ δfχ −e~v × B · ∇kδfχ = − . ~ Aτtr

−1 −1 Here feq = (1 + exp(β(Ek − µ))) is the Fermi Dirac distribution function, β = kBT denotes the thermal energy of the Weyl fermions, and the symbol µ is used for the chemical potential. In the above equation we have used the vector identity ~v × B~ · ~v ∂feq = 0 on the last ∂Ek term of the left hand side of a Boltzmann equation. As we are dealing with fermions, we make an ansatz that the deviation of the equilibrium distribution function from equilibrium is projected close to the Fermi energy, δfχ = gχ(−∂feq/∂Ek). The equation for gχ is

~ 2 ~ ~ k e vF EzBz k ~ 1 ~ gχ eE · vF + χ 2 = evF × B · ∇kgχ − . (2-84) |k| 2~|k| |k| ~ Aτtr

45 Here, the wave vector in spherical coordinates is kˆ = ~k/|k| = cos θzˆ + sin θ(sin φyˆ + cos φxˆ).

Expand gχ in Fourier harmonics:

X χ χ gχ = (αn cos(nφ) + βn sin(nφ)). (2-85) n

χ χ To find the coefficients αn and βn we use orthonormality of Fourier components. All coefficients with n > 1 are zero. Our final result for gχ is

2 2 2Ek cos θ + χeBz~vF gχ = − evF τtrEz 2 2 (2-86) 2Ek + χeBz~vF cos θ 2evF τtrEk sin θ − 2 2 2 2 2 (2eEkBzτtrvF ) + (2Ek + χeBz~vF cos θ) 2 2 (Ex(cos φ(Ek(2Ek + χeBz~vF cos θ))+

2 2 2 sin φ(2EkeBzτtrvF )) + (Ey(sin φ(Ek(2Ek+

2 2 2 χeBz~vF cos θ)) − cos φ(2EkeBzτtrvF ))).

Using this distribution function one can evaluate the density, currents, and conductivities. It should be noted that the particle number is conserved.

Z 3 X d k δfχ e X A−1 = ( )2E~ · B~ χ = 0 (2-87) (2π)3 τ h χ=± tr χ=±

∂ (N + N ) + ∇~ · (J~ + J~ ) = 0 (2-88) ∂t + − + − The Berry curvature force induces an imbalance between the number density of the diﬀerent chirality particles populations:

Z d3k |δF | = (δf − δf ), (2-89) (2π)3 + −

46 where δF is the average diﬀerence in the deviation of the distribution function due to the chirality. With the distribution function in Eq. (9) this chirality imbalance is

Z ∞ 4 evF τtrEzβ 4Ek |δF | = 2 3 3 dEk( 2 + (2-90) 2π ~ vF −∞ eBz~vF 2 2 4 2 2 2 (eBz~vF ) − 4Ek 2Ek + eBz~vF Ek 2 2 ln | 2 2 |) (eBz~vF ) 2Ek − eBz~vF 1 . 2 Ek−µ cosh (β 2 ) This average diﬀerence in the distribution function vanishes for B → 0, δF → 0 . The current ﬂowing through this system is

X Z d3k ~j = −e A−1~xδf˙ . (2π)3 χ (2-91) χ

Note that in the above formula for the current, we have included the change in phase space

2 2 2 factor due to the Berry curvature: A = 2Ek/(2Ek + χeB~vF cos θ). [16] The zz component of the conductivity is

2 Z ∞ Z 1 2 2 2 e τtrβ X (2E y + χeB v ) σ = dE dy k ~ F (2-92) zz 32π2 3v k 2E2 + χeB v2 y ~ F χ −∞ −1 k ~ F 1 × . 2 Ek−µ cosh (β 2 ) Holes are included into the conductivity formula from the Weyl particles gas spectrum by changing −e to e, and Ek to −Ek. In a similar manner the other components of the conductivity tensor are found to be:

2 Z ∞ Z 1 e τtrβ X σ = dE dy(1 − y2) (2-93) xx 16π2 3v k ~ F χ −∞ −1 4 2 2 Ek(2Ek + χeB~vF y) 1 (2E2 + χeB v2 y)2 + (2E eBτ v2 )2 2 Ek−µ k ~ F k tr F cosh (β 2 )

47 2 Z ∞ Z 1 e τtrβ X σ = − dE dy(1 − y2) (2-94) xy 16π2 3v k ~ F χ −∞ −1 2E5(eBτ v2 ) 1 k tr F . (2E2 + χeB v2 y)2 + (2E eBτ v2 )2 2 Ek−µ k ~ F k tr F cosh (β 2 )

From σzz, σxx, and σxy one can obtain all the elements of the conductivity matrix using

σyy = σxx, and σyx = −σxy. The diﬀerent elements of resistivity matrix are found by taking inverse of the conductivity

2 2 2 2 matrix: ρzz = 1/σzz, ρxx = σyy/(σxx + σxy), ρyy = ρxx, ρxy = σxy/(σxx + σxy), and

ρxy = −ρyx. In the limit of zero temperature, we give here the explicit formula for the conductivities, and compare them with previous published work. [22, 17, 23]

2 2 e µ τtr 2 1 σzz = 3 (1 + 4 ) 3π~ vF π 5 (lBkF ) e2µ2 τ 1 1 tr 2 (2-95) σxx = 3 (1 − (ωcτtr) + 4 ) 3π~ vF π 20 (lBkF ) 2 2 e µ τtr 2 3 1 σxy = − 3 (ωcτtr)(1 − (ωcτtr) + 4 ), 3π~ vF π 20 (lBkF )

where l2 = ~ denotes the magnetic length. For our perturbation theory to be applied here, B eBz 2 we consider ωcτtr < 1, and lBkF > 1, µ = 10 meV .[22] At zero magnetic ﬁeld our results exactly matches Refs. [24, 25]. For singled Dirac cone

2 2 P 1 τtr e µ we set = , and = δ(Ω), the conductivity is σxx = σyy = σzz = 3 δ(Ω). For finite χ 2 π 6π~ vF 2 magnetic field, our result for conductivity δσzz ∼ Bz qualitatively agrees with Refs. [23, 17], namely a quadratic increase in the conductivity with magnetic field due to the chiral anomaly. To do a comparative analysis of the forces entering in a Boltzmann equation due to the Berry curvature and the Lorentz force, we use a rotation matrix for rotating the electric field. [26, 27] These forces have different effects on the magnetoconductivity. The electric field is rotated from perpendicular to the magnetic field to parallel to the magnetic field. For some angles the magnetoconductivity will change its slope as a function of the magnetic field. For this particular value of the magnetic field, the Berry curvature and the Lorentz forces are

48 balanced. The conductivity matrix from the above equations is       jx σxx σxy 0 Ex             j  = σ σ 0  E  . (2-96)  y  yx yy   y       jz 0 0 σzz Ez

When rotating about the y-axis, the current, conductivity tensor, and electric ﬁeld are transformed: j~0 = σˆ0E~ 0, (2-97) where j~0 = Mˆ~j, σˆ0 = Mˆ σˆMˆ †, and the E~ 0 = Mˆ E~ . The rotation matrix is   cos θ 0 − sin θ     Mˆ =  0 1 0  . (2-98)     sin θ 0 cos θ

The comparison between these forces is discussed in the results and discussions section. 2.8 Results and Discussions

The magnetotransport problem in Dirac and Weyl semimetals is unconventional due to the effective magnetic monopoles. These effective magnetic monopoles create force in the presence of electric and magnetic fields. The resultant force increases the magnetoconductivity in the direction of magnetic field. This is called the chiral anomaly anomaly effect. This is not the case with Lorentz force. The Lorentz force suppresses the flow of current by deflecting it in the plane perpendicular to the electric and magnetic fields. This decreases magnetoconductivity by increasing magnetic field. The magnetoconductivity provides a platform to compare these two forces. However this comparison is not possible by the magnetoresistivity. This is because the diagonal elements of the magnetoresistivity matrix do not depend on the Lorentz force. The effective magnetic monopoles decrease the magnetoresistivity. This negative magnetoresistance is the signature of chiral anomaly. This is interesting to discuss here that the experimentally observed magnetoresistivity does not follow the magnetic field independent relaxation time approximation model. According to the magnetic field independent relaxation

49 Figure 2-1. The average deviation in the distribution function due to opposite chiralities and the MC due to chiral anomaly. The different parameters: βµ = 5, 2 2 2 3 3 2 β e~vF B = 90B/Tesla, and Λ = evF τtrEz/(2π ~ vF β ). time approximation the diagonal elements of the magnetoresistivity should stay constant in the presence of the Lorentz force. But the experimentally observed magnetoresistivity increases with the magnetic field. Below we discuss this in detail by providing numerical results and fitting experimental data. 2.9 Magnetoconductivity

In this section, we shall assume µ > ~ωc, kBT . The average deviation in the distribution function between opposite chiralities and the MC due to chiral anomaly is shown in Fig. 2-1. This deviation in the distribution function arises due to the chiral anomaly and makes the magnetotransport properties diﬀerent from the conventional semiclassical magnetotransport.

50 Figure 2-2. The magnetoconductivity of Weyl spectrum. The inset shows a magnified look of the MC vs the magnetic field characteristics. The different characteristics are plotted to show a comparative effect of the Lorentz force and the Berry force. The 2 different parameters: βµ = 5, βeτtrvF B = 20B/Tesla, and 2 2 β e~vF B = 50B/T esla.

This increases linearly with magnetic field and vanishes at B = 0T . The magnetoconductivity due to the chiral anomaly increases quadratically with magnetic field. In the plane perpendicular to the electric and magnetic fields the Lorentz force is balanced by the Coulomb force, and the hall conductivity is observed. The inclusion of Berry curvature in the magnetotransport introduces a new comparison between the Lorentz and Berry curvature forces. Therefore this study focuses in the plane of the electric and magnetic fields to see a comparison between the Berry curvature and Lorentz forces. The magnetoconductivity has different functional dependence for these forces. It decreases for the Lorentz force but increases for the other. By varying the angle between these fields, we show the change in slope of the MC. This is shown in Fig. 2-2. The electric field angle is varied from the magnetic

π field axis at θ = 0 to θ = 2 . For a magnetic field applied perpendicular to the electric field, the MC decreases with increasing magnetic field – the familiar Boltzman’s equation result for the MC. By rotating the magnetic field to align with the electric field axis the MC

51 Figure 2-3. The MC of Weyl spectrum. The different characteristics are plotted to show the relative strength of Lorentz force and the Berry force. The parameter: βµ = 5. slope starts increasing and when both of the fields are parallel to each other, the MC slope is maximum. This effect is explained by the chiral anomaly, which enters into Boltzmann’s equation from Berry curvature. It can be noticed that for some angles between the electric and magnetic fields the MC plot has a turning point. This turning point of the MC plot is an interesting point to see a comparison between the Lorentz force and the force arising due to Berry curvature. In Boltzmann’s equation, the MC in the direction of magnetic field has no magnetic field dependence, whereas in a plane perpendicular to the magnetic field the MC falls off with increasing magnetic field. This is the case for an electron gas with no chiral anomaly. But in a Weyl gas, the MC is dependent on magnetic field in every direction, as a result of chiral anomaly.

52 Figure 2-4. The MR of Weyl spectrum for the magnetic field independent transport time. The different characteristics show the MR are only dependent on the Berry force, not 2 on the Lorentz force. The different parameters: βµ = 5, βeτtrvF B = 20B/T, and 2 2 β e~vF B = 90B/T.

π The comparison of these two forces strength is shown in Fig. 2-3 at ﬁxed angle θ = 4 . q 2 evF B~ The ratio α = 2 2 shows the relative strength between these forces. The minima in (evF Bτtr) the MC is tuned by this ratio. In this ratio α, ~ enters due to the Berry curvature whereas

τtr enters due to the Lorentz force. To detect this tuning experimentally, the transport time

τtr can be changed by changing mobility of the sample. The mobility of Weyl semimetals is function of temperature and impurity concentration. [3] The MR in the semiclassical theory stays constant with changing magnetic field. However, it decreases due to chiral anomaly with increasing magnetic field. This is shown in Fig. 2-4. The Berry curvature modifies the phase space. [16] Its effect is shown in the

53 π magnetoconductivity plot for θ = 2 , here it should have constant value(zero slope) if the Berry curvature is zero. 2.10 Magnetic Field dependent transport time

The experimental results of Weyl semimetals for the MR are not consistent with a semiclassical theory for a magnetic field independent relaxation time. [28] Experimentally the MR is often fit with Kohler’s rule [29, 30] and shown to depend in detail on the shape of the Fermi surface. This explanation usually involves multiband models. In Dirac and Weyl semimetals the MR can not be explained with the conventional multiband theory. Here, we find that the experimentally observed MR results for Weyl semimetals is well reproduced by assuming a linear magnetic field dependence on the charge transport rate. We do this by phenomenologically including the magnetic field dependence in the transport time. Here we motivate this assumption. A full calculation of the spin dependent Boltzmann equation would be necessary to check this assumption. In support of this phenomenology of the magnetic field dependent relaxation time, we compare the spin precession time and momentum relaxation time of these materials. The momentum relaxation time depends on the mobility of the sample. High mobility results in a long relaxation time. Particularly in Weyl

5 cm2 −13 −14 semimetals of mobility µB ∼ 10 V −sec the relaxation time is τtr ∼ (10 − 10 )sec. The −12 magnetic ﬁeld of B ∼ 1T can precess a free electron within the time τs ∼ 5 × 10 sec. In Weyl semimetals of large Lande g factor g ∼ 102 the same magnetic ﬁeld should precess an

−14 −13 electron with this time τs ∼ (10 − 10 )sec. This implies the momentum relaxation and the spin precession times are of the same order τtr ∼ τs. This is proposed in the semiclassical range since ωcτtr ∼ 0.3 < 1 for the magnetic field B ∼ (3 − 5)T .[31] Another supporting argument in favor of the magnetic field dependent transport time in these materials within a semiclassical region is given in the Ref. [10]. The above motivates us to include the magnetic field dependence in the relaxation time of the Weyl particles gas in a plane perpendicular to the magnetic field 1 = 1 + ∆ , otherwise τ(B) τtr ~ it stays constant 1 = 1 . Here ∆ = gµ B is Zeeman parameter, g is the Lande’ g factor, τ(B) τtr B

54 Figure 2-5. The MR of Weyl spectrum for the magnetic ﬁeld dependent transport time. The diﬀerent parameters: βµ = 5, τtr = 0.9 ∗ B/T, τs 2 −1 −1 2 2 βeτtrvF B(1 + 0.9B/T ) = 20B(1 + 0.9B/T ) /T, and β e~vF B = 45B/T.

and µB is the Bohar magneton. By including this magnetic ﬁeld dependent relaxation time

for magnetoresistivity, the resistivity element ρxx increases with the magnetic ﬁeld, but the

resistivity element ρzz decreases due to the chiral anomaly. Therefore, the slope of resistivity changes by rotating electric ﬁeld from the magnetic ﬁeld axis. This is shown in Fig. 2-5.

π This maximum in the magnetoresistivity for angle θ = 6 between applied ﬁelds is due to the competition between the chiral anomaly and magnetic ﬁeld dependent transport time. The MR plot qualitatively agrees with the experimental data of Ref. [3]. 2.11 Conclusion

In this chapter, we have derived the semiclassical Boltzman equation for Weyl semimetals. These equations are diﬀerent because they include the Berry curvature term. Later we have

55 used it to calculate the MC with both the Lorentz force and the Berry curvature force. For a finite angle between the electric and magnetic fields, the MC starts decreasing with increasing the magnetic field due to the Lorentz force. It reaches the minimum point and then it starts increasing due to the chiral anomaly. When these fields are orthogonal to each other, the MC plots only shows a decrease with the magnetic field due to the Lorentz force. The MC increases rapidly with the magnetic field when both fields are in the same direction. This increase in the MC is an effect of the chiral anomaly (Fig. 2-2). This increase in the MC due to the chiral anomaly is quadratic (Fig.2-2). The relative strength between the Lorentz and Berry curvature forces depends on the transport time. The chiral anomaly effect is more prominent in a clean sample (Fig.2-3). The plot of the MR does not show the effect of the Lorentz force within a single band semiclassical theory. The diagonal elements of the MR only show an effect of the Berry

force. Therefore the transverse MR, ρxx, is almost constant (a slight change is caused by the modification of phase space with the magnetic field) (Fig. 2-4). In this study, the transverse MR function dependence on the magnetic field is made consistent with the experimental results by phenomenologically including the linear in magnetic field dependence in the transport time. [3, 32] This is a plausible approximation for the relaxation mechanism of a distribution function. These results are valid in high mobility samples with large Lande g factor. In these conditions, the momentum relaxation time and the spin precession time are of the same order within a semiclassical region. [31] We predict a linear increase in the transverse MR of high quality and large value of Lande g factor sample is due to the spin precession. We conclude by inviting experimentalist to detect the comparison between the Lorentz force and the Berry curvature force by avoiding the Weyl semimetals that have large Lande g factor. The MR element ρxx should be independent of magnetic field to detect the comparison between these forces within the semiclassical region.

56 CHAPTER 3 THE QUANTUM THEORY OF MAGNETOTRANSPORT IN TILTED DIRAC AND WEYL SPECTRUM 3.1 Introduction

In this chapter we discuss the quantum theory of magnetotransport of the three dimensional Weyl semimetals. The presence of a constant and large magnetic field quantizes the motion of Weyl fermions. These quantized bands disperse in the direction of magnetic field. We use Kubo linear response theory to calculate the quantum magnetotransport properties in a quantizing magnetic field. We are interested in the region where the impurity broadening and temperature smearing is smaller than the quantization gaps of different Landau bands. In the quantum limit where the energy scale of temperature (kBT ) and impurities broadening (Γ) are smaller than the Landau levels energy, the thermodynamic properties and the energy density of states show quantum oscillations as the magnetic field is changed or the particle number density is tuned. We are also interested in a more general Weyl energy dispersion relation. This interest triggers from the difference between the low energy and high energy physics. The speed of electrons and holes in the energy bands of solids is always lesser than the speed of light. Therefore this is not a surprise that these fermions violate the analog to the Lorentz symmetry in the condensed matter systems. [33] There are two types of fermions that violate the Lorentz symmetry in the low energy physics. [34] This classification is determined by the boundary where the conduction and valance bands meet. One of them is called type-I Weyl fermions with the property of their bands converging to a single point. The bands can also meet and form a non-closed isoenergic orbit, these are type-II Weyl fermions. Both types of Weyl fermions have topologically protected excitations. To elaborate the difference between these two types of Weyl fermions, a good example is the equation of circle with added anisotropy, R = ax + by + cpx2 + y2, see Fig. 3-1. The type-I Weyl fermions are excited if the energy dispersion tilt part is smaller than the energy dispersion isotropic part. In the circle equation if anisotropic part is smaller than the isotropic

57 Figure 3-1. Equation of circle with added anisotropy defines the difference between type-I and type-II Weyl fermions. Type-I Weyl fermions are the cases when anisotropic part of the circle equation is smaller than the isotropic part a, b < c. Type-II Weyl fermions are the cases when anisotropic part of the circle equation is greater than the isotropic part a, b > c. part, ax + bx < cpx2 + y2, the path is closed. This means type-I Weyl fermions form a close orbit for any fixed energy. On the contrary the type-II Weyl fermions are excited if the energy dispersion tilt part are greater than the energy dispersion isotropic part. In the circle equation, if the anisotropic part is greater than the isotropic part ax + bx > cpx2 + y2, it will enclose an open orbit. This means type-II Weyl fermions form an open orbit for any fixed energy. In this chapter we explore the various important roles of tilt velocity in the magnetotransport. In general, the tilt velocity modifies the energy dispersion. This is shown in the energy density of states and the magnetotransport properties. For type-II Weyl semimetals, the tilt velocity destroys the chiral ground state when it is perpendicular to the magnetic field. [34, 35] The

58 tilt velocity mixes different elements of the conductivity matrix. This also induces a Hall voltage in the plane formed by the perpendicular electric and magnetic fields, provided the tilt velocity makes finite angle in this plane. This is called the planar Hall effect. Even though the planar Hall voltage survives in the absence of the magnetic field, the magnetic field induces quantization signatures in the planar Hall effect of the 3D Dirac and Weyl semimetals. The semiclassical theory of the planar Hall effect in the 3D Weyl semimetals has already been studied, [36, 37] and experimentally observed. [4] We are studying the planar Hall effect in the quantum region of large magnetic field, where the charge particles motions are quantized. We are interested in a general system, where the tilt parameter has both the perpendicular and parallel component with respect to the magnetic field. 3.2 Theoretical formulation of the tilted Weyl spectrum

Tilt enters into the dispersion relation from the tilt velocity(~vT ). The tilt velocity can be

either smaller than vF , vT < vF , for type-I Weyl fermions, or larger than vF , vT > vF for type-II Weyl fermion. In either case the Hamiltonian for Weyl fermions is

0 i H = ~vT · ~pσ + vF piσ . (3-1)

i Here σ and pi denote the x, y, and z components of the Pauli matrix and the momentum, and the summation is assumed in the repeated indexes. The eigenvalues of this Hamiltonian are

(s, ~p) = ~vT · ~p + svF p, where s = ± denotes the band index. The energy dispersion for type-I

and type-II Weyl fermions is shown in Fig. 3-2. For no tilt, vT = 0, E vs pz would be two lines with slopes ±vF . For E > 0 the spin is in the direction of the momentum (positive chirality), while for E < 0 the spin is in the opposite direction of the momentum (negative chirality).

For small but non-zero vT the lines making up the dispersion curve no longer have the same magnitude of the slope, but they still have a deﬁnite chirality. Once vT > vF (type-II Weyl fermions) the dispersion curves for E > 0 no longer correspond to positive chirality because the energy is dominated by the ~vT · ~pσ0 term.

59 Figure 3-2. Energy dispersion relations for type-I and type-II Weyl semimetals. The velocity vz vz ratios for type-I and type-II Weyl fermions are T = 0.5 and T = 1.5 respectively. vF vF

The energy density of states in the presence of a tilt velocity is:

1 g X D(E) = − Im[Gr(E)] = δ(E − (s, ~p)), π V (3-2) p where

δ(θ − θ0) δ(E − (vT p cos θ + svF p)) = . (3-3) |vT p| sin θ

In the above equation θ is the angle between ~vT and ~p, and θ0 is the angle for which the argument of the delta function vanishes. For ﬁxed energy, E, we can determine the range of

60 the p integral in Eq. 3-3 using the range of cos θ0.

−1 ≤ cos θ0 ≤ 1 E v (3-4) −1 ≤ − s F ≤ 1 vT p vT E E ≤ p ≤ (3-5) svF + vT svF − vT The density of states integral reduces to

Z 1 2 δ(θ − θ0) D(E) = g 3 dφ dθ sin θ p dp , (3-6) (2π~) |vT p| sin θ

which only involves a p integral after integrating over the delta function. Our ﬁnal result is:

g D(E) = D0(E) . (3-7) (1 − ( vT )2)2 vF

E2 Here D0(E) = 2 3 3 is the energy density of states for the Weyl spectrum without any tilt 2π ~ vF velocity [34]. The symbol g refers to the number of type-I(II) Weyl nodes. The energy density of states has the parabolic energy dependence with the normalized multiplying constant. The

energy density of states diverges for vT → 1 (see Ref. [19] for discussions on this topic). vF From the very start we wish to emphasize that magnetic field can only quantize the motion of type-II Weyl fermions when the direction of tilt velocity is parallel to the magnetic field. [35] Type-I Weyl fermion motion can be quantized for any tilt direction. Keeping this in mind we consider a Hamiltonian with two components of tilt velocity, the type-II component of tilt velocity along the magnetic field and a possible component of the tilt velocity perpendicular to the magnetic field for type-I Weyl fermions. The constant magnetic field is applied along

~ z z-axis, B = Bzzˆ. This also ﬁxes the direction of the type-II tilt velocity along the z-axis vT . We take the perpendicular component for the case of type-I tilt Weyl fermions to be along the

x x-axis, vT . The resulting Hamiltonian is

61 x z H|ψi = (vT px + vT pz + vF ~σ · ~p)|ψi (3-8)   x z v px + v pz + vF pz vF (px + ipy) =  T T  |ψi. (3-9)  x z  vF (px − ipy) vT px + vT pz − vF pz

At this point the standard technique would be to use the harmonic oscillator raising and lowering operators to diagonalize a Hamiltonian in strong magnetic ﬁeld. However, the diagonalization of this Hamiltonian is a cumbersome task, since all elements of the above

matrix have an x-component of the momentum px, which includes both raising and lowering

θ operators. This complexity can be avoided by using hyperbolic operator exp[σx 2 ] on the Weyl wave equation, (E − H)|ψi = 0 → (E¯ − H¯ )|ψ¯i = 0. (3-10)

The operator (E − H) and wave function |ψi are transformed by the hyperbolic operator

¯ ¯ 2 θ θ ¯ 1 θ (E − H) = N exp[σx 2 ](E − H) exp[σx 2 ], and |ψi = N exp[−σx 2 ]|ψi). Here N is the normalization constant of the hyperbolic operator. The transformed Weyl wave equation is

z ¯ {γ(vT pz − E)σ0 + vF (¯pxσx + pyσy) + vF pzσz}|ψi = 0. (3-11)

After using the hyperbolic transformation operator the x-component of momentum is changed,

x px γη z 1 vT p¯x = + (v pz − E). Here γ = √ and η = − = tan θ. If vT > vF or γ vF T 1−η2 vF |η| > 1, this approach does not work. This is why we restrict ourselves to type-I Weyl fermions

x when vT 6= 0. We select the Landau gauge for solving this problem py → py − eAy and

Ay = Bzx. We are now ready to solve this Hamiltonain and ﬁnd eigenvalues and eigenstates. The transformed Weyl wave equation is       z vF pz v pz + vF (¯px + ipy) ψ1 ψ1  T γ    = E   . (3-12)  z vF pz      vF (¯px − ipy) vT pz − γ ψ2 ψ2

62 Here ψ1 and ψ2 are the state vectors that diagonalize the Hamiltonian. We use the following raising and lowering operators r γ lB aˆ = i (¯px + ipy) (3-13) 2 ~ r † γ lB aˆ = −i (¯px − ipy). (3-14) 2 ~

These operators obey the commutation algebra

[ˆa, aˆ†] = 1. (3-15)

The harmonic oscillator eigenstates ψ1 = anφn(x − x0), ψ2 = bnφn−1(x − x0) diagonalize

2 pylB Hamiltonian. The harmonic oscillator eigenstates are shifted by the displacement x0 = . ~

The symbol lB is used for the magnetic length, and it is related with the magnetic momentum √ 2~ pB by pB = . The Weyl wave equation with this substitution of raising and lowering lB operators is  q      z vF pz ~vF 2 vT pz + −i aˆ ψ1 ψ1  γ lB γ       q    = E   . (3-16) ~vF 2 † z vF pz i aˆ v pz − ψ2 ψ2 lB γ T γ This gives the energy eigenvalues s 2 z svF 2 2n~ E(pz, n, s) = vT pz + pz + 2 (3-17) γ γlB

with s = ±, and the corresponding energy eigenstates,   i exp( (pyy + pzz)) anφn(x − x0) ψ+ = ~ √   (3-18) 2   −ibnφn−1(x − x0)   i exp( (pyy + pzz)) bnφn(x − x0) ψ− = ~ √   . (3-19) 2   ianφn−1(x − x0)

63 Figure 3-3. The energy spectra of Weyl gases. We assume the Lorentz factor γ = 1, the type-I vz vz tilt velocity T = 0.9, and the type-II tilt velocity T = 1.1. vF vF

Here an and bn are normalization constants:

v vF pz u γ an = u1 + (3-20) t q 2 2 2 2 2n~ vF vF pz + 2 γlB

v vF pz u γ bn = u1 − . (3-21) t q 2 2 2 2 2n~ vF vF pz + 2 γlB

The ground state, n = 0, is the gapless state with eigenvalue E(pz, n = 0, χ) =

z (vT + sgn(χ)vF )pz, where χ = ±. For the case of type-I Weyl fermions χ corresponds to the chirality. For type-II Weyl fermions there is no well deﬁned chirality, and χ is just an index. The energy spectra of the Weyl gases are plotted in the Fig. 3-3. The ground state is chiral for type-I Weyl spectrum, whereas the type-II Weyl spectrum is independent of the chirality. [35]

64 The magnetotransport of the Weyl semimetals is unique because it supports a gapless ground state. We have already seen the gapless ground state exists even for the type-II Weyl semimetal. However it does not exist if the magnetic ﬁeld is perpendicular to the type-II tilt velocity, see the Appendix V.3. 3.3 The Magnetortransport Properties

We calculate the energy density of states in the presence of magnetic ﬁeld. The retarded Greens function for the Weyl gas is

† ~0 X ψn,s(~x)ψn,s(x ) Gr(E, x, x0, p , p ) = . (3-22) z y E − (s, n, p ) + iΓ n,s z

~ Here Γ = 2τ is used for the ﬁnite broadening of the Landau levels. The energy density of states is found using this Greens function. The Greens function in matrix form in the transformed basis as in Eq. 3-22 is   a2φ (x − x )φ (x0 − x ) 0 r 0 1 + n 0 n 0 G (E, x, x , pz, py) = (g(E, n (pz))   + 2  2 0  0 b φn−1(x − x0)φn−1(x − x0)   b2φ (x − x )φ (x0 − x ) 0 − n 0 n 0 g(E, n (pz))  ).  2 0  0 a φn−1(x − x0)φn−1(x − x0)

(3-23)

s −1 Here g(E, n(pz)) is equal to (E − (s, n, pz) + iΓ) . The energy density of states is

1 D(E) = − Im[Gr(E)] (3-24) π Z ∞ 1 g X dpz 1 X 1 = − Im[ ( + 2 )] π 2πl2 2π E − (s, n = 0, p ) + iΓ E − (s, n, p ) + iΓ B s −∞ ~ z n=1 z 1 g Z ∞ dp 2(E − vz p + iΓ) D(E) = − Im[ z { T z (3-25) 2 z 2 vF pz 2 π 2πlB −∞ 2π~ (E − vT pz + iΓ) − ( γ ) z X 2(E − vT pz + iΓ) + 2 2 2 }]. (3-26) z 2 1 2 2 npB vF n=1 (E − vT pz + iΓ) − γ2 (vF pz + γ )

In Eq. 3-26 the ground state, n = 0, has half the degeneracy of the other states. [38] We deﬁne ε ≡ vF pz. The integrand of the above equation has two poles in the complex plane of ε

65 (this includes n = 0),

γvz s sgn[1 − ( T )2] vz np2 v2 γvz 1 vF 2 T 2 2 B F T 2 εn = vz (−γ E + E γ − |1 − ( ) | + iΓ) (3-27) |1 − γ2( T )2)| vF γ vF vF γvz s sgn[1 − ( T )2] vz np2 v2 γvz 2 vF 2 T 2 2 B F T 2 εn = − vz (γ E + E γ − |1 − ( ) | + iΓ). (3-28) |1 − γ2( T )2| vF γ vF vF Here sgn[x] is the sign function with the property sgn[x] = 1 for x ≥ 0, and sgn[x] = −1 for x < 0. After the ε integration, we get the following expression for the energy density of states.

z z vT 1 vT 1 1 g −i2(E − v ε0) X −i2(E − v εn) D(E) = − Im[ { F + 2 F }] (3-29) π 4πl2 v ε1 − ε2 ε1 − ε2 B~ F 0 0 n=1 n n z nmax g γvT X Eγ D(E) = {(1 − (1 + nmax(nmax + 1))) + 2 } 4π2l2 v γ v q np2 v2 vz B F ~ F n=1 E2γ2 − B F |1 − γ2( T )2| γ vF

E Here nmax is the largest integer less than . As will be shown in the next section if one ~ωc plots D(E) one will get a series of inverse square root singularities. Changing either E or the magnetic field, which is in pB, causes the density of states to oscillate. This is the same physical origin as the de Hass van Alphen effect and the Shubnikov de Hass effect. The above formula for the energy density of state reproduces the result of Refs. [16,17] by turning off the

z tilt parameters(γ = 1, η = 0, vT = 0) and setting g = 1. Now we derive the formulas for the magnetoconductivity and the magnetoresistivity tensors. We use Kubo linear response theory for studying the quantum magnetotransport. The current-current correlation function is given below [39]

Z Z Z X dpz dpy Q (Ω) = 2e2T dx0 ij 2π 2π ω ~ ~ (3-30) 0 0 T r[¯viG(py, pz, Ω + ω, x, x )¯vjG(py, pz, ω, x , x)].

66 The current-current correlation function is invariant under the hyperbolic transformation because the expectation value of any operator Oˆ is independent of the hyperbolic transformation,

1 v¯ = v , v¯ = v , v¯ = γvz (σ + ησ ) + v σ (3-32) x γ F y F z T 0 x F z

At this point we ﬁnd it convenient to introduce a new tensor sij that only involves the Pauli matrices. Z Z Z X dpz dpy s (Ω) = 2e2T dx0 ij 2π 2π ω ~ ~ (3-33) 0 0 T r[σiG(py, pz, Ω + ω, x, x )σjG(py, pz, ω, x , x)]

To ﬁnd the static response of the external electric ﬁeld, we use the following relation sij =

sij (Ω) limΩ→0 iΩ . We present the details of the calculation to ﬁnd tensor sij in the Appendix V.4.

The resulting elements of the tensor sij are given below.

2 Z 2 e βτ X dpz s = [ ~ xx 16πl2 2π τ 2( (p ) + (p ))2 + 2 B n ~ n z n−1 z ~ β−(p ) β+ (p ) [a2 a2 (sech2( n z ) + sech2( n−1 z )) n n−1 2 2 + − 2 2 2 βn (pz) 2 βn−1(pz) +bnbn−1(sech ( ) + sech ( ))] 2 2 (3-34) ~2 + 2 2 2 τ (n(pz) − n−1(pz)) + ~ β−(p ) β− (p ) [a2 b2 ((sech2( n z ) + sech2( n−1 z ))) n n−1 2 2 β+(p ) β+ (p ) +a2 b2 (sech2( n z ) + sech2( n−1 z ))]] n−1 n 2 2

67 2 Z e βτ X dpz s = [a4 + b4 zz 16πl2 2π n n B n ~ 2 2 2 ~ (3-35) +2anbn 2 2 2 ] τ (2n(pz)) + ~ β−(p ) β+(p ) [sech2( n z ) + sech2( n z )] 2 2 2 Z e X dpz 1 s = ~ [ (a2 a2 xy 4πl2 2π ( + )2 n n−1 B n ~ n n−1 β−(p ) β+ (p ) [tanh[ n z ] − tanh[ n−1 z ]] 2 2 + − 2 2 βn (pz) βn−1(pz) +bnbn−1[tanh[ ] − tanh[ ]]) 2 2 (3-36) 1 + 2 (n − n−1) β−(p ) β− (p ) (a2 b2 [tanh[ n z ] − tanh[ n−1 z ]] n n−1 2 2 β+(p ) β+ (p ) +a2 b2 [tanh[ n z ] − tanh[ n−1 z ]])] n−1 n 2 2

We exploit symmetry of the Pauli matrices to ﬁnd the other elements of the tensor sij; syy = sxx, and syx = −sxy. The conductivity matrix σij is related with the tensor sij in the equation given below.   σxx σxy σxz     [σij] = σ σ σ  (3-37)  yx yy yz   σzx σzy σzz

v2 v2 σ = s F , σ = s F , σ = s v vz η (3-38) xx xx γ2 xy xy γ xz xx F T

2 z σyx = −σxy, σyy = syyvF , σyz = syxvF vT γη (3-39)

σzx = σxz, σzy = −σyz, (3-40) 2 z 2 2 z 2 2 2 σzz = szz(vF + (vT ) γ ) + sxx(vT ) γ η In the presence of a magnetic ﬁeld and a general direction of the tilt velocity, all the elements of the conductivity matrix are ﬁnite. The new components of the conductivity matrix enter due to the tilt velocity (σxz, σzx, σyz, σzy). These new components are directly proportional to

z the tilt velocity parameters η, vT . In the absence of any one component of the tilt velocity

68 x z (vT = 0, or vT = 0), the structure of the conductivity matrix does not show a signature of the tilted Weyl spectrum. However, the energy eigenvalues of the Weyl Hamiltonian, and the current-current correlation function still depend on the tilt velocity. The transverse conductivity

σxx, and the Hall conductivity σxy are renormalized by the Lorentz factor γ. The longitudinal

conductivity σzz is enhanced by the parallel component of the tilt velocity. The response of the transverse conductivity sxx is also added up with the longitudinal conductivity σzz. The equation of conductivity matrix in the absence of the perpendicular component of tilt velocity is   2 2 sxxv sxyv 0  ⊥ ⊥   2 2  [σij] = s v s v 0  . (3-41)  yx ⊥ yy ⊥   2 z 2  0 0 szz(vF + (vT ) )

The resistivity matrix is found by taking inverse of the conductivity matrix. The diﬀerent components of the resistivity matrix are given in the following equations.   ρxx ρxy ρxz     [ρij] = ρ ρ ρ  (3-42)  yx yy yz   ρzx ρzy ρzz

2 2 z 2 2 z 2 vF 2 (sxx + sxy)(vT ) η + sxxszz((vT ) + ( ) ) ρ = γ (3-43) xx 2 2 vF 2 z 2 vF 2 (sxx + sxy)szz( γ ) ((vT ) + ( γ ) )

1 sxx ρyy = 2 2 2 (3-44) vF sxx + sxy sxyγ ρxy = − 2 2 2 (3-45) (sxx + sxy)vF 1 vz η ρ = − T (3-46) xz z 2 vF 2 szz vF ((vT ) + ( γ ) ) 1 1 ρ = (3-47) zz 2 z 2 vF 2 szz γ ((vT ) + ( γ ) ) The remaining elements of the resistivity matrix are found by using the symmetry of the

system, ρyx = −ρxy, ρzx = ρxz , and ρyz = ρzy = 0.

69 Figure 3-4. The schematic illustration of the planar Hall eﬀect.

All elements of the resistivity matrix are either renormalized or combined with the tensor elements sij due to the tilt velocity. The transverse resistivity matrix ρxx contains all elements of the tensor sij. The Hall and longitudinal resistivities ρxy, ρzz are renormalized by the Lorentz factor γ and the tilt velocity. In the regular Hall effect the voltage is measured perpendicular to the current flow and the magnetic field. For the planar Hall effect the voltage is measured perpendicular to the current flow but parallel to the magnetic field. In the above formula the presence of both components of the tilt velocity makes the planar Hall effect ρxz finite. A schematic picture of the planar Hall effect in a slab of the 3D Weyl gas is shown in the Fig. 3-4. In general sample of a 3D Weyl semimetal with tilted spectrum has the planar Hall effect. The tilt velocity can be set in the plane formed by the perpendicular electric and magnetic fields. This accumulates the Hall

70 voltage in the plane of the electric and magnetic fields. The planar Hall voltage is present even in the absence of the magnetic field. 3.4 The Semiclassical theory of the planar Hall effect

In this section we derive the semiclassical formula of the Planar Hall eﬀect by using the Boltzmann equation. For the semiclassical study of magnetotransport in the Dirac and Weyl semimetals we solve the Boltzmann equation in the relaxation time approximation

∂fχ ~ ~ 1 ~ ~ δfχ + x˙ · 5xfχ + F · 5kfχ = − , (3-48) ∂t ~ τtr

where 1 e2 F~ = (eE~ + e~v × B~ + (E~ · B~ )Ω)~ 1 + e B~ · Ω~ ~ ~ (3-49) 1 e e ~x˙ = (~v + E~ × Ω~ + (Ω~ · ~v)B~ ). 1 + e B~ · Ω~ ~ ~ ~ ~ F is the force acting on Dirac and Weyl fermions gas, ~x˙ is the group velocity, τtr is the ~ kˆ transport time, and Ω = −χ 2k2 is the Berry curvature. The apparent difference in equations of velocity and force for the Dirac and Weyl gases enters due to the Berry curvature. For the study of planar Hall effect we ignore the effect of the Lorentz force [40] and assume a constant in time and space distribution function.

k δfχ(x, k, t) ≡ δfχ(k) (3-50)

We use the linear order in electric ﬁeld approximation for calculating the current.

k ~ e ~ ~ ~ ∂feq(k,χ) δfx = Aτtre(E + (E · B)Ω) · ~v(− ) (3-51) ~ ∂k,χ X Z d3k j = e A−1x˙ δf k (3-52) z (2π)3 z x χ

−1 −1 Here feq(k,χ) = (1 + exp(β(k,χ − µ))) is the Fermi Dirac distribution function, and β is the thermal energy. In the formula for the current we have included the phase space correction

71 factor A = 1 . The planar Hall conductivity is 1+ e B~ ·Ω~ ~ Z 3 2 X d k e ~ σzx = e τtr A(vz + Bz(Ω · ~v)) (2π)3 ~ χ (3-53) e ~ ∂feq(k,χ) (vx + Bx(Ω · ~v))(− ). ~ ∂k,χ The energy dispersion and the velocities of the tilted Dirac and Weyl semimetals are

x z (3-54) k,χ = χ(~kxvT + ~kzvT + s~|k|vF ),

x vx = χ(vT + svF cos φ sin θ)

vy = χsvF sin φ sin θ (3-55)

z vz = χ(vT + svF cos θ). The sum of the particles densities (N +, N −) and the currents (J +, J −) are conserved in the presence of the chiral anomaly and the tilt velocity.

Z 3 k Z 3 X d k δfχ X d k ∂feq(k,χ) A−1 = e χ 2π3 τ 2π3 ∂ χ=± tr χ=± k,χ (3-56) x z e ~ ~ ~ [(vT Ex + vT Ez) + (E · B)(Ω · ~v)] = 0 ~ ∂ (N + + N −) + ∇~ · (J~+ + J~−) = 0 (3-57) ∂t However, the chiral anomaly and tilt velocity creates imbalance between population densities of diﬀerent chirality particles

Z d3k δN = N + − N − = A−1(δf k − δf k ). (3-58) 2π3 + −

In the absence of a magnetic ﬁeld this imbalance is created by the tilt velocity

+ − x z δN = N − N = −eτtr(vT Ex + vT Ez) Z 3 (3-59) d k ∂feq(k,+) ∂feq(k,−) 3 ( + ). 2π ∂k,+ ∂k,−

72 This imbalance in turn creates planar Hall eﬀect

Z 3 2 x z d k ∂feq(k) σzx = e τtrvT vT 3 (− ) (3-60) (2π) ∂k ∂f ( ) ∂f ( ) ∂f ( ) eq k = ( eq k,+ + eq k,− ). (3-61) ∂k ∂k,+ ∂k,−

The planar Hall effect arising due to chiral anomaly changes its functional dependence on magnetic field in the presence of tilt velocity, ρxz ∼ Bz,[41] since the planar Hall effect

x z without tilt velocity is proportional to BxBz. Because we have allowed for both vT and vT to be non zero, the planar Hall effect derived here exists even for zero magnetic field. For magnetic field along one axis like Bz the planar Hall effect due to chiral anomaly will vanish. Therefore the tilt velocity provides another fingerprint to detect chiral anomaly via the planar Hall effect. 3.5 Results and Discussions

In this work we have assumed the strong magnetic field region where the Landau levels are well resolved. The spacing between the two adjacent Landau levels is much greater √ ∆E = 2~vF ∼ 1eV than the thermal energy kB T ∼ 0.01 and the impurity broadening lB ∆E Γ ∆E ∼ 0.01. In our calculations for the current-current correlation function we have assumed the quasiparticle lifetime τq is of the same order as the transport life time τtr ∼ τq. In general the calculations of the transport time requires the self-consistent Born approximation, and this can differ by order of magnitudes with the quasiparticle lifetime. This introduces the significant new physics by updating the quasiparticle lifetime with the transport time. However in the domain of strong magnetic field where the Landau levels spacing is the largest magnetotransport parameter, the transport and the quasiparticle lifetimes are reasonably of the same order. [38] The case of overlapping Landau levels and intermediate temperature region, where the interaction corrections are important, is discussed in the Refs. [16, 17]. Experimentally the quasiparticle lifetime, Fermi velocity and the effective mass is measured by fitting the De Hass Van Alphen oscillations or the Shubnikov-de Hass oscillations with the Liftshitz-Kosevich formula. By using the experimental data of T aP [42] for the Fermi velocity

73 vF ∼ 0.125 (eV − nm), Fermi energy EF ∼ 0.039 eV , impurity broadening Γ ∼ 0.0023 eV ,

m2 −13 and the mobility µQ ∼ 0.36 V s , we found the quasiparticle lifetime τq ∼ 2.8 × 10 s, and the −13 transport time τtr ∼ 4 × 10 s are of the same order τq ∼ τtr. The energy density of states of a 3D Weyl gas has the parabolic energy dependence. A strong magnetic field quantizes it. This is shown in the Figs. 3-5 and 3-6. The different Landau levels are tuned by changing the chemical potential on top of the energy density of states for the 3D Weyl gas. [43,2] The energy density of states starts from the finite value at n = 0. In the presence of the tilt velocity the energy spectrum of the Weyl gas is updated. This also shifts the resonant peaks of the energy density of states. The thermodynamic properties of the Weyl gas depends on energy density of states. The magnetotransport properties show the signatures of the energy density

Figure 3-5. The energy density of states for the 3D Weyl gas.

74 Figure 3-6. The energy density of states for the 3D Weyl gas in the presence of magnetic ﬁeld. √ElB g Here X = , Γ = 0.01 eV , D0 = 2 2 , 2~vF 2π lB vF ~ z ∆D D γvT = + (1 + Y nmax(nmax + 1)), and Y = . D0 2D0 vF

of states. In this study we assume a finite lifetime of the Weyl gas particles that is smaller than √ the gap between the Landau levels 1 << 2~vF . Γ lB Now we shall discuss the different elements of the magnetoconductivity matrix. In a 3D Weyl gas a strong magnetic field quantizes the motion of charges in the plane, but the third direction is free for the charges to skip. This modifies the sharp resonant peaks of the conductivity seen in the two dimensional electron gas. This feature is shown as a background at the bottom of the resonant peaks shown in the Fig. 3-7. The transverse conductivity σxx

s resonates with the energy band µ = n, as the chemical potential crosses it. The ﬁnite width of the peak is due to the life time of the quasi particles (the scattering of quasi particles with

75 point like impurities should give this life time to them). In the presence of the tilt velocity the energy spectrum of the Weyl gas is changed. This shifts the peaks position in the transverse conductivity. The height of these peaks are enhanced by the Lorentz factor γ, see Eq. 3-38.

The Hall conductivity σxy also shows the quantization eﬀect. This is shown as kinks in the Hall conductivity plot. The steps are not as sharp as they are formed in the two dimensional electron gas. In the present situation the Hall edge states are integrated out in the third direction. This spoils the sharp quantized feature of the Hall eﬀect in the 3D Weyl gas. This can be seen in the Fig. 3-8. The presence of tilt velocity shifts the position of the kinks in the

Hall conductivity. The longitudinal conductivity σzz also shows the quantization eﬀect. This

Figure 3-7. The transverse conductivity σxx. The solid black curve is the plot for no tilt in the Weyl spectrum, whereas the blue dashed curve is the plot in the presence of the tilt z 2 vT e vF β −1 parameters, v = 0.2, and η = 0.2. Here σ0 = 32π2l2 , β = 100 eV , and √ F B 2~vF = 1 eV . lB

76 Figure 3-8. The Hall conductivity σxy. The solid black curve is the plot for no tilt in the Weyl spectrum, whereas the blue dashed curve is the plot in the presence of the tilt vz parameters, T = 0.2, and η = 0.2. The parameters used here are the same as vF used in the transverse conductivity σxx. has been experimentally observed in the several Dirac and Weyl semimetals. [28, 44, 45, 46]

σzz is shown in the Fig. 3-9. Here, the presence of the tilt velocity mixes the transverse conductivity with the longitudinal conductivity, and the peaks position are also shifted.

The in plane resistivity components ρxx and ρxy are shown in the Fig. 3-10. The oscillation in the resistivity elements are clearly visible. The transverse resistivity ρxx and the Hall resistivity ρxy show peaks and plateaus whenever the chemical potential crosses the energy band µ = n. Here, the tilt velocity mixes all the components of the tensor sij in the transverse resistivity ρxx, see Eq. 3-43. However the Hall resistivity is renormalized by the Lorentz factor γ, see Eq. 3-45.

77 The longitudinal and planar Hall resistivities, ρzz and ρxz, are plotted in the Figs. 3-11 and 3-12. The oscillations in the longitudinal resistivity is also shown in the Ref. [24]. The oscillations in the longitudinal resistivity is a unique feature of the 3D Dirac and Weyl semimetals. The planar Hall resistivity enters into the magnetoresistivity tensor due to both

x z components of the tilt velocity vT and vT . The planar Hall resistivity is directly proportional z x 2 vT vT γ to the longitudinal resistivity ρxz = − 2 ρzz. Therefore, the oscillations in the planar Hall vF resistivity provide another probe for the detection of the chiral anomaly. The planar Hall eﬀect can be detected in materials like T aP ,[42] and W T e2.[47] By allowing the tilt velocity to

Figure 3-9. The longitudinal conductivity σzz. The solid black curve is the plot for no tilt in the Weyl spectrum, whereas the sold blue curve is the plot in the presence of the vz tilt parameters, T = 0.2, and η = 0.2. The parameters used here are the same as vF used in the transverse conductivity σxx.

78 −1 Figure 3-10. The transverse resistivity ρxx, and the Hall resistivity ρxy. Here ρ0 = σ0 . The solid curves are the plots for no tilt in the Weyl spectrum, whereas the dashed vz curves are the plots in the presence of the tilt parameters, T = 0.2, and η = 0.2. vF The parameters used here are the same as used in the σxx. make a finite angle in the plane formed by the perpendicular electric and magnetic fields, the planar Hall effect signals can be detected. 3.6 Conclusions

We have studied the magnetotransport in the Weyl semimetals with a generalized energy spectrum that includes both components of the tilt velocity. The presence of the magnetic ﬁeld quantizes the motion of charge particles in the 3D Weyl gas. This is shown in the plot of the energy density of states in Fig. 3-5 and 3-6. The tilt velocity changes the spectrum of the Weyl gas. This shifts the position of resonant peaks in the energy density of states.

79 The transverse (σxx), longitudinal (σzz), and the Hall (σxy) magnetoconductivities are plotted in the Figs. 3-7, 3-8, and 3-9. The magnetoconductivities show resonant peaks whenever the chemical potential resonates with the energy band µ = n. The tilt velocity mixes the longitudinal conductivity with the transverse conductivity. This also renormalizes the transverse and longitudinal conductivities with the Lorentz factor, see Eqs. 3-38, 3-39, and 3-40. All the components of the magnetoresisitivity matrix also show the features of the energy density of states. The longitudinal ρzz, transverse ρxx, Hall ρxy, and planar Hall ρxz magnetoresistivities resonate whenever the chemical potential crosses the energy band µ = n.

Figure 3-11. The longitudinal resistivity ρzz. The solid black curve is the plot for no tilt in the Weyl spectrum, whereas the solid red curve is the plot in the presence of the tilt vz parameters, T = 0.2, and η = 0.2. The parameters used here are the same as vF used in the σxx.

80 Figure 3-12. The planar Hall resistivity ρxz. The parameters used here are the same as used in the σxx.

The tilt velocity shifts the position and renormalizes the weight of these peaks, see Figs. 3-10, 3-11, and 3-12. This also mixes the different elements of the conductivity matrix, see Eqs. 3-43, 3-45, 3-46, and 3-47. Especially the planar Hall effect is directly proportional to the tilt velocity. The planar Hall effect can become a fingerprint to detect nontrivial magnetotransport in the 3D Weyl gas with the tilted spectrum. The planar Hall effect is directly proportional to the longitudinal resistivity, which oscillates with magnetic field. [48] The oscillations in the planar Hall effect can identify a material that support the tilted Weyl spectrum with the unique anomalous transport features.

81 CHAPTER 4 CONCLUSION OF THE THESIS 4.1 Conclusion

In this thesis we have studied the transport properties of the Dirac and Weyl semimetals. The formulas are derived in the linear response regime. We have matched our derived formulas with the experimental results. Here we discuss a summary of work done in this thesis. In Chapter 1 we have introduced the three dimensional Dirac and Weyl semimetals. We have discussed two experimental techniques to detect these states of matter, angle resolve photo emission spectroscopy (ARPES) technique and magnetotransport transport measurements technique. The ARPES technique can distinguish between the Dirac and Weyl semimetals since the Weyl semimetals bulk states end on surface to form Fermi arcs, whereas the Dirac semimetals bulk states end to from a close Fermi circle. These differences emanate from the Berry curvature. The Berry curvature also modifies the magnetotransport measurements. This is observed as negative magnetoresistance in the direction of applied magnetic field and the planar Hall effect observed in the plane of applied electric and magnetic fields. In Chapter 2 we have derived the formulas of magnetotransport properties like conductivities and resistivities within the semiclassical region. To this end we have utilized the Moyal and Wigner technique to derive the semiclassical equation of motion for any operator fˆ (later we set this operator equals to the Keldysh Greens function) in the response of electromagnetic fields. These results are unconventional because they include the Berry vector potential. The inclusion of Berry potential modifies the velocity and force equations of charge particles. This has motivated us to study a comparison of the force that arises due to Berry curvature with the Lorentz force, since magnetoconductivity decreases with magnetic field for Lorentz force whereas it increases for the force arising due to Berry curvature. This comparison is done by considering a simple model of charge transport where the response of electromagnetic field is relaxed by impurities. However the experimental results does not fit in this model,

82 since the experimentally observed relaxation time is magnetic field dependent. We have phenomenologically included the magnetic field dependence in the transport time that reproduces the experimental results. Later we have suggested that a comparison of the Berry curvature force with the Lorentz force is possible in materials possessing magnetic field independent relaxation time. In Chapter 3 we have exploited the option of relaxing the Lorentz symmetry in low energy physics as compare to high energy physics. This is done by introducing a direction dependent tilt velocity. We have studied this aspect of relaxing Lorentz symmetry both in the quantum and semiclassical domains. The energy spectra of Dirac and Weyl semimetals become angle dependent in the presence of tilt velocity. This asymmetry induces a Hall voltage in the plane of applied electric field and the tilt velocity. The magnetic field induces the Shubnikov de Hass oscillation in the planar Hall effect. The derived formula of planar Hall effect is directly proportional to the longitudinal magnetoresistivity and therefore these oscillations are expected. This should be noted that the longitudinal magnetoresistivity of ordinary three dimensional electron gas does not show Shubnikov de Hass oscillation. In longitudinal resistivity these only arise in three dimensional Dirac and Weyl semimetals and have been observed experimentally.

83 APPENDIX A NO GROUND STATE IN PERPENDICULAR MAGNETIC FIELD Type-II Weyl semimetals with tilt velocity perpendicular to the magnetic ﬁeld does not have energy solutions. To show this we assume the magnetic ﬁeld is B~ = Bxˆ, and the tilt velocity perpendicular to it is ωz. We select the gauge pz → pz − eAz, here Az = By. The energy eigenvalue equation is

      (ω + v )(p − eA ) v (p + ip ) φ φ  z z z z ⊥ x y   1  1     = E   . (A-1) v⊥(px − ipy)(ωz − vz)(pz − eAz) φ2 φ2

Here φ1(2) are the energy eigenstates that diagonalize the Hamiltonian. We make the following

2 ± ± lB pz γ lB † † substitution in the above equation, γ (y − ) = √ (b + b ), v⊥py = iλ(b − b), ~ 2

+ ~(vz−ωz) − ~(vz+ωz) ~v⊥ γ = √ , γ = − √ , λ = √ , and v⊥px = Px. 2lB 2lB 2lB       − † † γ (b + b ) Px + λ(b − b) φ1 φ1     = E   . (A-2)  † − †      Px − λ(b − b) γ (b + b ) φ2 φ2

This gives the following equation for the ground state |φ1i = |0i .

4 4 2 2 2 2 2 2 2 2 2 4 2 E + Px + E (γ+ + γ− + 4γ+γ− − 2λ ) + 2Px (λ − E − γ+γ−) + 3γ+γ− + 3λ + 6λ γ+γ− = 0 (A-3)

The energy eigenvalue of the above equation becomes complex for ωz > vz.

ωz To check this we set Px = 0, γ+ = −λ(1 + c), γ− = λ(1 − c), here vz = v⊥, and c = . v⊥

E4 + 2λ2(3c2 − 2)E2 + 3λ4c4 = 0 (A-4)

E2 = (2 − 3c2) − p(2 − 3c2)2 − 3c4 (A-5)

q 2 The energy eigenvalue becomes complex for c = 3 . In our numerical results we are using c = 0.2 for the perpendicular component of the tilt velocity with respect to the magnetic ﬁeld .

84 APPENDIX B TENSOR SIJ

Here we calculate the transverse component of the correlation function sxx. Z Z Z X dpz dpy s (Ω) = 2e2T dx0T r[σ G(p , p , Ω + ω, x, x0)σ G(p , p , ω, x0, x)] xx 2π 2π x y z x y z ω ~ ~ 2 Z Z Z e X dpz dpy = T dx0[((a2 g(ω, + (p )) + b2 g(ω, − (p )))(a2 g(ω + Ω, −(p ))+ 2 2π 2π m m z m m z n n z ω,m,n ~ ~

2 + 0 0 2 − bng(ω + Ω, n (pz))))φm(x − x0)φm(x − x0)φn−1(x − x0)φn−1(x − x0) + ((amg(ω, m(pz))

2 + 2 + 2 − 0 +bmg(ω, m(pz)))(ang(ω + Ω, n (pz)) + bng(ω + Ω, n (pz))))φm−1(x − x0)φm−1(x − x0)

0 φn(x − x0)φn(x − x0)] (B-1)

Perform the integrations R dx0 R dpy : 2π~ 2 Z e X X dpz s (Ω) = T [a2 a2 g(ω + Ω, −(p ))g(ω, + (p )) + a2 b2 g(ω + Ω, −(p )) xx 4πl2 2π n n−1 n z n−1 z n n−1 n z B ω n ~

− 2 2 + + 2 2 + − g(ω, n−1(pz)) + bnan−1g(ω + Ω, n (pz))g(ω, n−1(pz)) + bnbn−1g(ω + Ω, n (pz))g(ω, n−1(pz))

2 2 + − 2 2 + + 2 2 +anan−1g(ω + Ω, n−1(pz))g(ω, n (pz)) + bnan−1g(ω + Ω, n−1(pz))g(ω, n (pz)) + anbn−1

− − 2 2 − + g(ω + Ω, n−1(pz))g(ω, n (pz)) + bnbn−1g(ω + Ω, n−1(pz))g(ω, n (pz))] (B-2)

and do the Matsubara sum: Z X r s dω βω r s T g(ω, (pz))g(ω + Ω, (pz)) = tanh[ ]g(ω, (pz))g(ω + Ω, (pz)) (B-3) m n −2πi 2 m n ω

This is the standard technique of evaluating the Matsubara sums (see Ref. [31]). In this technique we ﬁrst convert the sum on the Matsubara frequency into the contour that includes

βω the pole of tanh 2 , while no poles of the propagators are included. Later on we shall exclude βω all the poles of the tanh 2 and include only the poles of the propagators in the contour region.

85 s 1 In the above correlation function we have two propagators: g(ω, n(pz)) = s i~ , ~ω−n(pz)+ 2τ sgn(α) s 1 s and g(ω, n(pz)) = s i~ , where n(pz) = E(pz, n, s) − µ. We make the partial ~ω+~Ω−n(pz)+ 2τ sgn(β) fraction of the propagators product.

r s 1 1 g(ω, (pz))g(ω + Ω, (pz)) = m n s r i~ 2 ~Ω − (n(pz) − m(pz)) + (sgn(β) − sgn(α)) 2τ (B-4) 1 1 [ − ] s i~ s i~ ~ω − n(pz) + 2τ sgn(α) ~ω + ~Ω − n(pz) + 2τ sgn(β) Finally we split the pole of the propagators between the upper and the lower half planes.

sgn(α) = −1 α < 0

sgn(β) = 1 β > 0.

X r s 1 1 T g(ω, (pz))g(ω + Ω, (pz)) = m n 2 Ω − (s (p ) − r (p )) + i~ ω ~ n z m z τ (B-5) βr (p ) β(s (p ) − Ω) [tanh[ m z ] + tanh[ n z ~ ]] 2 2

X r s s r T [g(ω + Ω, m(pz))g(ω, n(pz)) + g(ω + Ω, n(pz))g(ω, m(pz))] ω i~ r s ~Ω + β (pz) β (pz) ' τ [(tanh[ m ] + tanh[ n ])]− i~ 2 r s 2 (~Ω + ) − (m(pz) − n(pz))) 2 2 τ (B-6) r s i~ r β Ω ~Ω − (m(pz) − n(pz)) − β (pz) ~ [ τ sech2( m )+ 4 ~2 r s 2 2 τ 2 + (~Ω − (m(pz) − n(pz))) r s i~ s ~Ω + (m(pz) − n(pz)) − β (pz) τ sech2( n ))] ~2 r s 2 2 τ 2 + (~Ω + (m(pz) − n(pz))) Since we are interested in the static response of the correlation function, and we set Ω = 0 at

i the end. Now to ﬁnd sxx we multiply this function with Ω , set Ω → 0, and take the real value.

86 1 1 We shall avoid the inﬁnity Ωτ by setting τ = 0.

2 Z 2 − e βτ X dpz β (pz) s = [ ~ [a2 a2 (sech2( n )+ xx 16πl2 2π τ 2( (p ) + (p ))2 + 2 n n−1 2 B n ~ n z n−1 z ~ + + − 2 βn−1(pz) 2 2 2 βn (pz) 2 βn−1(pz) sech ( )) + bnbn−1(sech ( ) + sech ( ))] 2 2 2 (B-7) 2 − − ~ 2 2 2 βn (pz) 2 βn−1(pz) + 2 2 2 [anbn−1((sech ( ) + sech ( ))) τ (n(pz) − n−1(pz)) + ~ 2 2 β+(p ) β+ (p ) +a2 b2 (sech2( n z ) + sech2( n−1 z ))]]. n−1 n 2 2

Because of the symmetry syy = sxx, so we do not need to make a separate calculation for syy, but the situation is different for the z-axis. The magnetic field is along the z-axis. This makes the z-axis different from the other two axes x, y. Therefore we need to calculate szz separately: Z Z Z X dpz dpy s (Ω) = 2e2T dx0T r[σ G(p , p , Ω + ω, x, x0)σ G(p , p , ω, x0, x)] zz 2π 2π z y z z y z ω ~ ~ 2 Z Z Z e X dpz dpy = T dx0[((a2 g(ω, − (p )) + b2 g(ω, + (p )))(a2 g(ω + Ω, −(p ))+ 2 2π 2π m m z m m z n n z ω,m,n ~ ~

2 + 0 0 2 + bng(ω + Ω, n (pz))))φm−1(x − x0)φm−1(x − x0)φn−1(x − x0)φn−1(x − x0) + ((amg(ω, m(pz))

2 − 2 + 2 − 0 +bmg(ω, m(pz)))(ang(ω + Ω, n (pz)) + bng(ω + Ω, n (pz))))φm(x − x0)φm(x − x0)

0 φn(x − x0)φn(x − x0)]. (B-8)

We perform the integrations R dx0 R dpy : 2π~ 2 Z e X X dpz s (Ω) = T [(a4 + b4 )(g(ω + Ω, +(p ))g(ω, +(p )) zz 4πl2 2π n n n z n z B ω n ~

− − 2 2 + − (B-9) +g(ω + Ω, n (pz))g(ω, n (pz))) + 2anbn(g(ω + Ω, n (pz))g(ω, n (pz))

− + +g(ω + Ω, n (pz))g(ω, n (pz))].

87 i To ﬁnd szz we multiply the correlation function with Ω , set Ω → 0 and take the real value. We 1 avoid the inﬁnity by setting τ → 0. This gives:

2 Z 2 − + e βτ X dpz β (pz) β (pz) s = [a4 + b4 + 2a2 b2 ~ ][sech2( n ) + sech2( n )]. zz 16πl2 2π n n n n τ 2(2 (p ))2 + 2 2 2 B n ~ n z ~ (B-10)

Finally, we calculate the correlation function sxy: Z Z Z X dpz dpy s (Ω) = 2e2T dx0T r[σ G(p , p , Ω + ω, x, x0)σ G(p , p , ω, x0, x)] xy 2π 2π x y z y y z ω ~ ~ 2 Z Z Z e X dpz dpy = T dx0[((a2 g(ω, + (p )) + b2 g(ω, − (p )))(a2 g(ω + Ω, −(p ))+ 2 2π 2π m m z m m z n n z ω,m,n ~ ~

2 + 0 0 2 − bng(ω + Ω, n (pz))))φm(x − x0)φm(x − x0)φn−1(x − x0)φn−1(x − x0) − ((amg(ω, m(pz))

2 + 2 + 2 − 0 +bmg(ω, m(pz)))(ang(ω + Ω, n (pz)) + bng(ω + Ω, n (pz))))φm−1(x − x0)φm−1(x − x0)

0 φn(x − x0)φn(x − x0)]. (B-11)

We perform the integrations R dx0 R dpy : 2π~ 2 Z e X X dpz s (Ω) = T [(a2 a2 g(ω + Ω, −(p ))g(ω, + (p )) + a2 b2 g(ω + Ω, −(p )) xy 4πl2 2π n n−1 n z n−1 z n n−1 n z B ω n ~

− 2 2 + + 2 2 + − g(ω, n−1(pz)) + bnan−1g(ω + Ω, n (pz))g(ω, n−1(pz)) + bnbn−1g(ω + Ω, n (pz))g(ω, n−1(pz)))

2 2 + − 2 2 + + 2 2 −(anan−1g(ω + Ω, n−1(pz))g(ω, n (pz)) + bnan−1g(ω + Ω, n−1(pz))g(ω, n (pz)) + anbn−1

− − 2 2 − + g(ω + Ω, n−1(pz))g(ω, n (pz)) + bnbn−1g(ω + Ω, n−1(pz))g(ω, n (pz)))]. (B-12)

The correlation function sxy is directly proportional to the Hall conductivity. In the quantum region the Hall conductivity is nondissipative. We set both poles in the lower half plane.

sgn(α) = 1 α > 0

sgn(β) = 1 β > 0.

88 X r s s r T [g(ω + Ω, m(pz))g(ω, n(pz)) − g(ω + Ω, n(pz))g(ω, m(pz))] = ω s r r s (tanh[ βn(pz) ] − tanh[ β(m(pz)+~Ω) ]) (tanh[ βm(pz) ] − tanh[ β(n(pz)+~Ω) ]) 1 2 2 2 2 (B-13) [ r s − r s ] 2 ~Ω − (m(pz) − n(pz)) ~Ω + (m(pz) − n(pz)) s r ~Ω βn(pz) βm(pz) ' 2 2 r s 2 [tanh[ ] − tanh[ ]))]. ~ Ω − (m(pz) − n(pz)) 2 2 i To ﬁnd sxy we multiply the correlation function with Ω , set Ω → 0, and take the real value. This gives

2 Z − + e X dpz 1 β (pz) β (pz) s = ~ [ (a2 a2 [tanh[ n ] − tanh[ n−1 ]] xy 4πl2 2π ( + )2 n n−1 2 2 B n ~ n n−1 + − 2 2 βn (pz) βn−1(pz) +bnbn−1[tanh[ ] − tanh[ ]]) 2 2 (B-14) − − 1 2 2 βn (pz) βn−1(pz) + 2 (anbn−1[tanh[ ] − tanh[ ]] (n − n−1) 2 2 β+(p ) β+ (p ) +a2 b2 [tanh[ n z ] − tanh[ n−1 z ]])]. n−1 n 2 2

89 APPENDIX C EQUATION OF MOTION DERIVATION The formulas for the mean velocity and the Lorentz force of the Bloch wave packet is derived. The equation of motion for Bloch wave packet was derived by using variational time dependent principle. [8, 49, 50] Here we consider only the electromagnetic ﬁeld coupling with the Bloch wave packet. The Bloch wave packet is formed by integrating Bloch wave function

over the Bloch wave vector ~~q. Z |ψi = |a(q, t)| expi(~q.xˆ−γ(~q,t)) |u(~q)i, (C-1) ~q

where γ(~q, t) is the phase of Bloch wave packet, and |u(~q)i is the atomic part of the Bloch wave function. We are assuming a single band Bloch wave function so no index is needed

to deﬁne diﬀerent orbitals |un=1(~q)i = |u(~q)i. The Bloch wave packet is assumed to be normalized hψ|ψi = 1. The spatial integral over the Bloch wave packet consists of two parts the bigger part that involves the whole crystal (our sample under investigation) and the smaller R R R part that involves only the unit cell. ~x = cell × crystal. The formula for Lagrangian of the Bloch wave packet is written by following the time dependent variational principle.

d L = hψ|(i − H)|ψi (C-2) ~dt

The ﬁrst term gives

d Z ∂γ(~q, t) hψ|i |ψi = |a(~q, t)|2 ~dt ∂t ~q (C-3) ∂γ(~q , t) = c . ∂t

∂ R 2 In above equation we have used the fact that probability is not time dependent dt ~q |a(~q, t)| = 0, and the Bloch wave packet is narrowly peaked in the unit cell. The time derivative of the

∂ Bloch wave packet phase ∂t γ(~qc, t) is related with its mean position ~xc = hψ|xˆ|ψi and the ~q

90 ∂ dependence hu(~q)|i ∂~q u(~q)i,

Z i ∂ hψ|xˆ|ψi = δ(~q − ~q0)( |a(~q, t)|2 + |a(~q, t)|2 0 2 ∂~q ~q,~q (C-4) ∂ ∂ ( γ(~q, t) + hu(~q)|i u(~q)i)), ∂~q ∂~q

R ∂ 2 2 where the fact that probability is conserved also means ~q ∂~q |a(~q, t)| = δ|a(~q, t)| = 0.

Z ∂ ∂ hψ|xˆ|ψi = |a(~q, t)|2( γ(~q, t) + hu(~q)|i u(~q)i)) (C-5) ~q ∂~q ∂~q ∂ ∂ ~xc = γ(~qc, t) + hu(~qc)|i u(~qc)i (C-6) ∂~qc ∂~qc Adding a total time derivative function to a Lagrangian does not change the Euler’s Lagrange equation. d ˙ ∂ ∂ γ(~qc, t) = ~qc · γ(~qc, t) + γ(~qc, t) (C-7) dt ∂~qc ∂t ∂ d ˙ ˙ ∂ γ(~qc, t) = γ(~qc, t) − ~qc · ~xc + ~qc · hu(~qc)|i u(~qc)i (C-8) ∂t dt ∂~qc d This also implies dt (γ(~qc, t) − ~qc · ~xc) = 0. By using Eqs. (A3,A8):

d ˙ ˙ ∂ hψ|i~ |ψi = ~qc · ~xc + ~qc · hu(~qc)|i u(~qc)i. (C-9) dt ∂~qc

Now we shall include the coupling of electromagnetic ﬁelds [φ(~x,t),A(~x,t)] with the Bloch wave packet. This will modify the Bloch wave vector ~q into gauge invariant Bloch wave vector ~k = ~q + eA~(~x,t). Thus the expectation value of the Hamiltonian is

hψ|H|ψi = M~ − eφ(~xc, t), (C-10) where = − M~ · B~ and M~ is the orbital magnetization. The orbital magnetization gives M~ ~kc the perturbation correction in the energy. In our calculation for magnetotransport the orbital magnetization eﬀect is ignored.

91 By substituting Eq. A9 and Eq. A10 into Eq. A2, we derived the Lagrangian of Bloch wave packet.

˙ ~ ~˙ L(~xc, ~xc, kc, kc, t) = −M~ + eφ(~xc, t)+ (C-11) ˙ ∂ ~x˙ · (~k − eA~(~x , t)) + ~k · hu(~k )|i u(~k )i c c c c c ~ c ∂kc The Lorentz force acting on Bloch wave packet is derived by using Euler’s Lagrange equation ˙ of the canonical conjugate pair [~xc, ~xc].

d ∂L ∂L = (C-12) dt ∂~x˙ c ∂~xc

i ∂L ∂φ(~xc, t) i ∂A (~xc, t) j = e j − ex˙ c j (C-13) ∂xc ∂xc ∂xc i i d ∂L ij ˙ i ∂A (~xc, t) k A (~xc, t) j = δ (~kc − e − ex˙ c k ) (C-14) dt ∂x˙ c ∂t ∂xc ~˙ ~ ˙ ~ ~k = −e(E + ~xc × B) (C-15)

Also, the formula for the mean velocity of Bloch wave packet is derived by using Euler’s ~ ~˙ Lagrange equation for the canonical conjugate pair [~kc, ~kc].

1 d ∂L 1 ∂L = (C-16) dt ~˙ ~ ~ ∂kc ~ ∂kc

1 ∂L 1 ∂M~ j ij i = − i +x ˙ δ (C-17) ~ ∂kc ~ ∂k ~ 1 d ∂L d ~ ∂u(kc) ij = i hu(kc)| iδ (C-18) ˙ i j ~ dt ∂kc dt ∂kc ~ ~ ~ d ~ ∂u(kc) ˙ i ∂u(kc) ∂u(kc) i hu(kc)| i = ik (h | i− dt ∂kj c ∂ki ∂kj c c c (C-19) ~ ~ ∂u(kc) ∂u(kc) h j | i i) ∂kc ∂kc Deﬁne Berry curvature as

~ ~ ~ ~ k ∂u(kc) ∂u(kc) ∂u(kc) ∂u(kc) Ωn = i(h i | j i − h j | i i). (C-20) ∂kc ∂kc ∂kc ∂kc

92 The anomalous velocity enters into equation of motion of mean position of Bloch wave packet

~xc. ˙ ~˙ ~ ~xc = ~v − kc × Ω (C-21)

1 ~˙ ~ Here ~v = ~ k ~ . The anomalous velocity kc × Ω is Lorentz force in momentum space. ~ O M

~˙ ~ ˙ ~ ~kc = −e(E + ~xc × B) (C-22)

˙ ~˙ ~ ~xc = ~v − kc × Ω (C-23)

One can uncouple these equation by using vector cross product properties (A~ × B~ ) × C~ = (A~ · C~ )B~ − (B~ · C~ )A~ ˙ 1 e ~ ~ e ~ ~ ~xc = (~v + E × Ω + (Ω · ~v)B) (C-24) 1 + eB~ ·Ω~ ~ ~ ~ 2 ~˙ 1 ~ ~ e ~ ~ ~ ~kc = (eE + e~v × B + (E · B)Ω) (C-25) 1 + eB~ ·Ω~ ~ ~ ˙ ~˙ These are the formulas of the mean velocity ~xc and the Lorentz force ~kcof the Bloch wave packet.

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96 BIOGRAPHICAL SKETCH He was born in Islamabad, Pakistan. He received his Ph.D. in Theoretical Condensed Matter Physics from the University of Florida in 2020.