A Shape of the broadening of separated Landau levels

Transversal magnetoresistance in Weyl semimetals


We explore theoretically the magnetoresistivity of three-dimensional Weyl and Dirac semimetals in transversal magnetic fields within two alternative models of disorder: (i) short-range impurities and (ii) charged (Coulomb) impurities. Impurity scattering is treated using the self-consistent Born approximation. We find that an unusual broadening of Landau levels leads to a variety of regimes of the resistivity scaling in the temperature–magnetic field plane. In particular, the magnetoresistance is non-monotonous for the white-noise disorder model. For the magnetoresistance for short-range impurities vanishes in a non-analytic way as . In the limits of strongest magnetic fields , the magnetoresistivity vanishes as for pointlike impurities, while it is linear and positive in the model with Coulomb impurities.

I Introduction

Topological materials and structures represent one of the central research directions in the modern condensed matter physics. One of the classes of such materials is topological insulators and superconductors which possess a bulk gap and topologically protected surface excitations with massless Dirac spectra. Another important class is gapless materials with topologically protected Fermi points or nodal lines. The most well-known example is graphene whose dispersion is characterized by two Fermi points where the valence and the conduction bands touch. Excitations in the vicinity of these points have a linear dispersion and can be viewed as two-dimensional Dirac fermions.

Three-dimensional counterparts of graphene are Dirac semimetals. In such a material the dispersion near the nodal point is characterized by Dirac Hamiltonian, i.e. the conductance and the valence bands have an additional twofold degeneracy. Experimental realizations of Dirac semimetals include CdAs M. Neupane et al. (2014) and NaBi Liu et al. (2014); further candidate materials have been recently discussed Gibson et al. (2015). The twofold degeneracy discussed above can be lifted if either spatial inversion or time-reversal symmetry is broken. The four-component Dirac solution then decouples into two independent two-component solutions representing two Weyl fermions of opposite chirality. Thus, each of the Dirac points then splits into two Weyl points without any additional degeneracies. Recent experiments provided evidence that TaAs Lv et al. (2015); S.-Y. Xu et al. (2015a) and NbAsS.-Y. Xu et al. (2015b) can be classified as Weyl semimetals S.-Y. Xu et al. (2015c); Yang et al. (2015). Further promising candidates for Weyl semimetals include pyrochlore iridates Wan et al. (2011), topological insulator heterostructures Burkov and Balents (2011), and CdAs with lowered symmetry Wang et al. (2013). In the rest of the paper we will use the term “Weyl semimetal” in a broader sense, including also the degenerate case of Dirac semimetals.

Transport properties of Weyl semimetals are highly peculiar (for various theoretical aspects of the problem, see, e.g., Refs. Burkov et al., 2011; Hosur et al., 2012; Vafek and Vishwanath, 2014; Burkov, 2015a; Ominato and Koshino, 2014; Sbierski et al., 2014; Son and Spivak, 2013; Biswas and Ryu, 2014; Syzranov et al., 2015a; Altland and Bagrets, 2015; Burkov, 2014, 2015b; Syzranov et al., 2015b; Sbierski et al., 2015; Ominato and Koshino, 2015; Rodionov and Syzranov, 2015; Wan et al., 2011 and references therein). In particular, weak disorder (with a strength below a certain critical value) has a negligible effect on the density of states. Specifically, the density of states vanishes quadratically in energy around the Weyl point despite the presence of disorder. Furthermore, the limits and (where is the frequency) are not interchangeable for the behavior of the conductivity (assuming that the chemical potential is at the Weyl point). While sending frequency to zero first results in a finite conductivity, the zero- ac conductivity vanishes Wan et al. (2011); Burkov et al. (2011); Hosur et al. (2012); Ashby and Carbotte (2014a) in the limit as . In the strong-disorder regime these singularities are eliminated.

Behavior of Weyl semimetals in the external magnetic field is also expected to be very nontrivial. This is related, first of all, to the unconventional Landau quantization of Dirac fermions. Furthermore, since a single Weyl node displays a chiral anomaly, a possibility to control the valley polarization as well as a large anomalous Hall effect are expected. Vafek and Vishwanath (2014); Burkov (2015a); Vazifeh and Franz (2013) Much attention has been recently put on the longitudinal magnetoresistance in Weyl semimetals that originates from the chiral anomaly. Son and Spivak (2013); Gorbar et al. (2014); Burkov (2014, 2015b); Goswami et al. (2015); Ghimire et al. (2015); Shekhar et al. (2015); C. Shekhar et al. (2015)

In this paper, we develop a theory of the transversal magnetoresistivity of a Weyl semimetal. We are particularly interested in the range of sufficiently strong magnetic fields, such that the Landau quantization is important. 1 One of the motivations for our work was a spectacular experimental observation of a large, approximately linear magnetoresistance in the Dirac semimetals CdAs and TlBiSSe in strong magnetic field. T. Liang et al. (2015); Feng et al. (2015); Ghimire et al. (2015); Novak et al. (2015) Quantum linear magnetoresistance has been obtained by Abrikosov in a seminal paper, Ref. Abrikosov, 1998, for Dirac semimetals in the extreme limit when only one Landau level is filled (i.e., the cyclotron frequency exceeds the temperature). The linear behavior was traced back to the magnetic-field dependent screening of charged (Coulomb) impurities.

We consider a general case of arbitrary relation between the magnetic field and temperature such that, depending on the regime, the magnetoresistance is dominated by contributions from the zeroth Landau level, other separated Landau levels, and overlapping Landau levels. We address two alternative models of disorder: (i) short-range impurities and (ii) Coulomb impurities. We show that for pointlike impurities the transition between the weak-disorder and the strong-disorder phases persists in the presence of magnetic field and explore singularities of the weak-disorder phase. Further, we find that an unusual broadening of Landau levels leads to a variety of regimes of the resistivity scaling in the temperature-magnetic field plane. The transversal magnetoresitance is found to be non-monotonous for the model of weak white-noise disorder. Remarkably, in the limit of , the magnetoresistance for short-range impurities shows a non-analytical behavior . For strong pointlike impurities and for charged impurities we find a positive quadratic magnetoresistance at . In the limit of strongest magnetic fields , the magnetoresistivity vanishes as for pointlike impurities, while it is linear and positive in the model with Coulomb impurities, in agreement with Abrikosov’s result Abrikosov (1998) and recent experimental findings.

The paper is organized as follows. In Sec. II, we outline the implementation of the Born approximation in the context of Weyl semimetals subjected to a magnetic field in the presence of pointlike impurities. We find that, in analogy with the zero-magnetic-field case, one has to distinguish between weak and strong disorder regimes separated by a phase transition. In Sec. III we develop the formalism of the self-consistent Born approximation (SCBA) and analyze the broadening of Landau levels due to disorder and magnetic field. Section IV presents the formalism for calculation of the conductivity in magnetic field for the model of white-noise disorder. In Sec. V, we extend our analysis to the case of charged impurities. In Sec. VI we use the obtained results to calculate and analyze the magnetoresistance for both models of disorder. Our findings are summarized in Sec. VII.

Ii Landau-levels spectrum in Weyl semimetals

ii.1 Clean case

In the presence of a constant homogeneous magnetic field in direction, electrons of a single Weyl node are described by the Weyl Hamiltonian


where denote the Pauli matrices and is the vector potential. (We have chosen the Landau gauge.) The eigenfunctions of this Hamiltonian have two components () in the space spanned by . Positions of the Landau levels (LLs) in a clean Weyl semimetal are given by


where is the magnetic length and is the distance between the zeroth and first LL. We set throughout the paper. In the following, we choose the energy bands such Abrikosov (1998) that the wave function of the clean zeroth LL has only component 1 in the pseudospin space:


for and for . Here are the normalized eigenfunctions of free electrons in magnetic field and denotes the Heaviside step function. The retarded bare Green function of the clean system is conveniently represented as a matrix in the pseudospin space of bands :


It is worth noting that the summation of eliminates the theta-functions in the term in Eq. (5), so that the integration over in what follows will always be performed from to .

Figure 1: Density of states of a clean Weyl semimetal. The blue solid line shows the DoS in a finite magnetic field, Eq. (7), while the red dashed line corresponds to the zero magnetic field, Eq. (6).

In the zero magnetic field the clean density of states (DoS) behaves quadratically in energy:


In the presence of magnetic field, the DoS acquires a sawtooth form with square root singularities originating from the one-dimensional () dispersion of each Landau band (cf. Ref. Ashby and Carbotte, 2014b)


The DoS is visualized in Fig. 1.

ii.2 Introducing disorder

We consider now the effect of disorder. The impurity scattering generates a self-energy in the (impurity-averaged) Green function,


which is a matrix in the pseudospin space (in which the Pauli matrices operate).

We will assume that the disorder potential is diagonal in both spin and pseudospin indices and neglect scattering between different Weyl nodes. We will discuss this approximation and its limitations in the end of the paper, Sec. VII. Clearly, in the absence of internode scattering, the structure in the node space will be trivial for all quantities. We do not show it explicitly below; the calculated density of states and the conductivities are those per Weyl node.

We will first consider a model of pointlike impurities and later analyze generalization to the case of Coulomb impurities. The impurity potential has then the form


where is the unit matrix in the pseudospin space. In view of the matrix structure of the impurity potential , the impurity correlator becomes a rank-four tensor. The self-energy reads


For a diagonal impurity potential, the impurity correlator is diagonal as well, which is expressed as


where . The self-energy is diagonal in the energy-band space. However, in the presence of magnetic field, the self-energy is no longer proportional to the unit matrix:


This asymmetry originates from the asymmetry of states in the zeroth LL. In the clean case, the states of the zeroth LL are only present in one energy band. Later we will see that a strong impurity scattering eliminates this asymmetry.

We switch to LL representation so that and . The diagonal components of the matrix Green function (8) that determine the self-energy read:


In general, the self-energy depends on energy and on the LL index, . However, for a white-noise disorder, the dependences on and drop out.

ii.3 Born approximation

We start with the Born approximation, where we neglect the self-energies in Green’s functions (13) and (LABEL:G22) for the calculation of self-energies:


The summation over here should, in fact, be restricted by an upper cut-off , as will be discussed below. After shifting the summation over in , we see that the two self-energies differ only by the absence of the term in :


where, using , we have introduced


and is the bandwidth. The necessity to introduce the ultraviolet cut-off originates from the approximation of a true energy dispersion by the Dirac-fermion one, which is, in fact, a low-energy approximation. Our analysis is applicable for .

In view of Eq. (17), it is sufficient to calculate :


The imaginary part of the Born self-energy is produced by the Landau levels below , while the real part is due to the contribution of the Landau levels above . In Eqs. (19) and (20)


is the number of the Landau level below energy , and the symbol denotes the integer part of a number. For the evaluation of the sum in Eq. (20), we have introduced the upper cutoff which is determined by the ultraviolet energy cutoff in the following way: ( is the index of the highest Landau level within the bandwidth ).

The sum in is dominated by the upper limit : . Assuming no Landau quantization at the ultraviolet energies , we use the zero- result




The parameter quantifies the strength of disorder. For sufficiently strong disorder, the real part of Born self-energy becomes larger than , which clearly signifies the insufficiency of the simple Born approximation. As we will see in Sec. III.3 below, the self-consistent treatment of strong disorder yields a dramatic change of the behavior of the density of states for strong disorder.

In what follows, however, we mostly focus on the limit of weak disorder, . We absorb into the redefinition of the energy , and neglect the difference between and . For , we find


For higher energies, using the Euler-Maclaurin formula for the sum over , we express the imaginary part of the Born self-energy as


This result is illustrated in Fig. 2. The first term in the square brackets of Eq. (25) is responsible for the square-root divergency at the positions of LLs, whereas the last term yields the parabolic background similarly to the zero- case.

Figure 2: Disordered Weyl semimetal in Born approximation. in units of . Blue: Eq. (19); Dark blue, dashed: Eq. (25); Green: [minima in Eq. (25)]; Red: (result without magnetic field).

Iii Self-consistent Born approximation

We now turn to the self-consistent Born approximation (SCBA). The self-consistent treatment is motivated by the presence of square-root singularities in the Landau-level broadening (25) obtained within the Born approximation. The introduction of disorder-induced self-energies in Green’s functions should cut off such divergencies.

The SCBA equations (10), (13) and (LABEL:G22) with the disorder correlator (11) acquires the form (below ):


As above, we absorb the real parts of self-energies (determined by the ultraviolet cut-off ) into the shifts of energies . The density of states is given by


In the regime of well separated LLs (the corresponding conditions will be analyzed below), the sum over the Landau levels is evaluated as follows. Let us assume that the energy is close to the bottom of the -th Landau level . Then the main contribution of the sum over comes from the term . We thus single out the term of from the sum and evaluate it separately from the sum over the remaining Landau levels.

We note that the self-consistent treatment of the LL broadening is fully justified for weak disorder and . All the renormalization effects not captured by the SCBA affect the real part of the self-energy, see discussion in Ref. Ostrovsky et al., 2008 where the SCBA was employed for 2D Dirac fermions in graphene. In that case, disorder was marginally relevant and its effect could be incorporated through the renormalization of parameters (induced by the contributions of higher LLs) that enter the SCBA equations for a given Landau level. In the present 3D case, the renormalization of can be neglected for the case of weak disorder, . Even for the lowest LL, the SCBA density of states in two dimensions is parametrically correct. Ostrovsky et al. (2008) The exact shape can be calculated along the lines of Refs. Wegner, 1983; Brézin et al., 1984 In three dimensional systems for weak disorder, the extra integration over the momentum further reduces the difference between the exact and SCBA results in the limit .

iii.1 Energies close to the lowest Landau level

We first consider the case of lowest energies, (i.e., ) for weak disorder, . In this case, the asymmetry with respect to the zeroth LL should be taken into account and the imaginary parts of the two self-energies strongly differ from each other. When the lowest Landau level is well separated from the others (), the contribution of higher Landau levels to the sum over can be treated within the Born approximation, while the contribution of should be calculated self-consistently. Within this procedure, we immediately get and


This result coincides with the result of the non-self-consistent Born approximation, Sec. II.3. The zeroth LL is separated from the first as long as the condition is fulfilled. The density of states for is finite and, to the leading order, is energy-independent.

Using Eq. (29), we find the leading non-vanishing term in for :


Thus, in the limit of weak disorder, , we can neglect for . In fact, the condition is even softer: becomes of the order of only in the close vicinity of the first Landau level, .

iii.2 Energies at high Landau levels

We now consider high energies, . As we have seen in Sec. II.3, already within the Born approximation the average (as well as minimal) broadening of Landau levels increases with parabolically, as in the zero- case: , see Fig. 2. Therefore, the difference between and that comes from the contribution of can be neglected for energies away from the zeroth LL. In what follows we set for .



and setting , we arrive at the self-consistent equation for


where we have introduced the partial contribution of the th Landau level to the total broadening . Note that each term in the r.h.s. of Eq. (32) contains the total broadening rather than the partial . Further, the position of the th Landau level is shifted by disorder: appears only in combination


In the case of weak disorder, for all energies we have , so that Eq. (33) can be written as

For this yields and for we get . Thus, when crosses , the th Landau level gets an extra contribution to . The solution of the self-consistent equation (32) is shown for in Fig. 3.

Figure 3: Disordered Weyl semimetal in the self-consistent Born approximation for : in units of obtained by numerical solution of Eq. (32). Blue, red, and green curves correspond to , respectively. For all curves .

Let us now fix the Landau-level number and consider the range of energies around . Assuming well separated Landau levels below , we neglect in all terms with :


The contribution of Landau levels with is dominated by and can be neglected for weak disorder:


Finally, the contribution of the th Landau level (closest to the energy ) can be further simplified for :

In particular, exactly at we find


Using Eqs. (36) and (LABEL:GammaN), when is close to , the self-consistency equation takes the form:


where , so that the r.h.s. of Eq. (40) explicitly depends on only, as it should be.

Exactly at , the self-consistency equation reads:


We observe that for sufficiently small energies, the broadening is dominated by the self-consistent contribution of the same Landau level, whereas for large energies, the broadening is given by the zero- result stemming from lower Landau levels:




Below Landau levels are fully separated. Each peak in is non-symmetric with respect to , as inherited from the clean density of states. The shape of the LL broadening is analyzed in detail in Appendix A.

For high energies, the behavior in zero magnetic field should be recovered. Indeed, we can express the result for in terms of the energy as follows:


We see that the magnetic field has dropped out from this result, as expected. Thus the LL broadening is dominated by the result for In fact, taking into account the corrections to the broadening at , we will see in Sec. III.4 below that the Landau level quantization of the density of states remains intact in a finite range of energies above . This should be contrasted with the 2D case, where a single scale separates regimes of strong and weak Landau quantization. Finally, we note that magnetooscillations in Weyl semimetals were addressed in Ref. Ashby and Carbotte, 2013 with phenomenological energy-independent broadening. We find, however, that the energy dependence of is very rich.

iii.3 Strong Disorder

We now briefly discuss the regime of strong disorder, . As follows from the consideration of the weak-disorder case, see Eq. (30), for strong disorder the difference between the two self-energies, and , becomes inessential even at . After the evaluation of the sum over in Eqs. (26) and (LABEL:Sigma2SCBA), we find a qualitative change in the behavior of the imaginary part of self-energy (and thus of the density of states) at . This implies the existence of a critical disorder strength, separating the two regimes. In the absence of magnetic field, the emergence of such a critical disorder strength was reported in Refs. Ominato and Koshino, 2014, Sbierski et al., 2014, Kobayashi et al., 2014, Altland and Bagrets, 2015 and Syzranov et al., 2015b. Remarkably, the critical disorder strength which we find for the case of a strong magnetic field turns out to be equal to the zero-field value .

The solution of the SCBA equation for strong disorder for is given by


which is equal to the zero- result obtained in Ref. Ominato and Koshino, 2014. When is substantially larger than (i.e., ), the broadening becomes of the order of the ultraviolet cut-off,


which ensures that all Landau levels overlap. Further, at , the solution of the SCBA equations yields for the LL broadening


Thus, when the disorder is substantially stronger than the critical one, even the zeroth LL overlaps with the rest of the spectrum.

Within the SCBA, the real part of the self energy for is found to be


yielding .

In this paper, we do not discuss the critical regime near the transition from weak to strong disorder at . At zero magnetic field, the criticality was addressed using the -expansion within the renormalization group approach in Refs. Syzranov et al., 2015b, a. The effect of magnetic field near the transition remains a very interesting question for future work.

iii.4 Density of States

In Fig. 4, we plot the density of states obtained by a numerical solution of the SCBA equation in the case of weak disorder. The three figures illustrate the evolution of the density of states with the increasing value of the parameter (proportional to the disorder strength and to the square root of the magnetic field).

Figure 4: Density of states, , within self-consistent Born approximation in units of [Eq. (52)], as obtained from Eq. (32) for and Eq. (29) for . The curves corresponds to (a) , (b) , and (c) . The value was used.
Figure 5: Relevant energy scales and dominant contributions to the density of states of a disordered Weyl semimetal (for brevity, we have set ).

Even in the clean case, Landau levels are broadened due to the integration over , see Fig. 1, so that the divergent peaks at are located on top of the background density of states. However, these peaks are well resolved for all energies. Disorder leads to the suppression of the peaks and eventually Landau levels fully overlap at high energies. Let us discuss the characteristic values of Landau level index at which the behavior of the density of states changes qualitatively.

At , the broadening of Landau levels is dominated by the background (zero-) contribution. From Eq. (43), we see that the corresponding LL index decreases with increasing :


In order to find out whether LLs are resolved or not, we should check whether the condition is fulfilled. For energies , the width of high Landau levels is smaller than the distance between them. Therefore, at we have a situation when Landau levels are still resolved on top of the background, but the height of the small peaks , Eq. (41), is lower than the height of the background. The latter then dominates the broadening.

The neighboring peaks fully overlap in the regime , where the broadening is given by the zero- result:


The corresponding LL index


decreases with increasing magnetic field, similarly to . Let us emphasize that, contrary to conventional expectations, the number of separated LLs decreases with increasing magnetic field. We thus see that a large number of low-lying Landau levels are well resolved for weak magnetic field.

When the magnetic field increases for a fixed disorder strength, the Landau level index associated with the starting point of overlapping, becomes smaller. This behavior is very unusual, as it is opposite to that in the case of conventional semiconductors. The energy where the Landau levels start to overlap increases with as . With regard to the behavior with disorder strength, the obtained results qualitatively conform with intuitive expectations. Specifically, with increasing disorder, the number of separated Landau levels decreases and the corresponding energy range shrinks.

It is important to stress that, in the presence of magnetic field, the density of states at the Dirac point (zero energy) is finite even in the weak-disorder regime. Specifically, the value of the density of states at is linear in magnetic field:


It is worth noting that a finite value of the density of states at the degeneracy point will lead to a finite conductivity independently of the order of limits and . From this point of view, a finite magnetic field has the same effect as a strong disorder.

All the features of the density of states that we have found analytically (see Fig. 5) are perfectly observed in Figs. 4 a,b,c. First, one sees that for weak disorder and weak magnetic field many LLs are separated and that the number of separated Landau levels decreases with increasing magnetic field or with increasing disorder. Second, there is an intermediate range of energies where the density of states is dominated by the background value but Landau levels are well resolved. Third, one observes that the background density of states is equal to that in the absence of magnetic field (quadratic in energy). Finally, the magnetic field creates a finite density of states at the degeneracy point which depends on the magnetic field.

Iv Conductivity at charge neutrality

In this Section, we calculate the conductivity of a disordered Weyl semimetal in the presence of a quantizing transversal magnetic field. Here we restrict ourselves to the case of weak disorder and to zero chemical potential, . We use the Kubo formula for the real part of the longitudinal conductivity,


Here is the bare current operator and is the current vertex dressed by disorder, see Appendix B. The effect of disorder manifests itself in the replacement of bare Green’s functions by impurity-averaged matrix Green’s functions (8) and in the appearance of the current vertex corrections . As discussed in Ref. Biswas and Ryu, 2014, the calculation of the conductivity in Weyl semimetals requires taking into account vertex corrections even for point-like disorder (similarly to graphene). In the absence of magnetic field, the inclusion of vertex corrections away from the Weyl point leads to the difference between the transport and quantum (coming from the single-particle self-energy) scattering times: Biswas and Ryu (2014) . In what follows, we first evaluate the conductivity without vertex corrections and then include the vertex corrections at the end of the calculation.

Starting from Eq. (IV), setting , evaluating the trace, and taking into account the orthogonality of wave functions of different LLs, we find that the Kubo formula for the conductivity without vertex corrections takes the following form in the LL representation:


Since the self-energies for the zeroth Landau level differ from those for higher Landau levels, we have to distinguish between the zeroth Landau level and the others. This is also true for the case of vertex corrections. For low temperature, , the conductivity is dominated by the contribution of the zeroth LL. For higher temperatures, excitations to higher LLs are possible and therefore, the conductivity is determined by the contributions of the zeroth LL, separated and overlapping LLs.

iv.1 Low temperatures : Zeroth Landau Level

We consider first the situation when the contribution of the zeroth LL is dominant. This is the case under the following two assumptions: (i) the zeroth LL is separated from the first one, which is the case if the condition is fulfilled; (ii) the temperature satisfies , so that excitations to higher LLs are suppressed exponentially. In this case, the integral over energy is dominated by the contribution of the zeroth Landau level. We note that for and weak disorder there is no room for the regime , since this would imply whereas . Furthermore, the current vertex correction for energies close to the Weyl node turn out to be small, , see Appendix B. Therefore, in the regime of the dominant zeroth LL contribution, we will ignore the difference between the quantum scattering time and transport scattering time.

Using and and disregarding the real part of self-energies, we get


Substituting Eqs. (55) and (LABEL:G22e0) into Eq. (54) and setting in Green’s functions, we arrive at ():


Neglecting (the condition of separation of the lowest Landau level) in the denominators for higher Landau levels, we see that the sum over converges and gives the contribution to the conductivity, whereas the term yields . Thus, for the total conductivity at and is dominated by the contribution of the zeroth Landau level and is given by:


The resulting conductivity Eq. (58) is proportional both to the disorder strength and to the magnetic field.

iv.2 High temperatures,

For higher temperatures, , energies are involved in the thermal averaging, so that we need to evaluate the contribution of high Landau levels to the conductivity. For we neglect the difference between the self-energies: . As before, we include the real part of self-energies into the shifted energy and drop the tilde everywhere. The imaginary part of the self-energy is written through the Landau-level broadening:

The Green functions for take the form:


yielding with


Substituting these in Eq. (54), we arrive at


The evaluation of the integral in Eq. (LABEL:Qne) then yields


Let us now include the vertex corrections. The total conductivity is then given by Eq. (63) with the replacement , where includes the dressing of the current operator by disorder (“transportization”). The vertex correction is calculated in Appendix B:


The inclusion of vertex corrections replaces with in the denominator of Eq. (65). In zero magnetic field this yields , in agreement with Ref. Biswas and Ryu, 2014. Below we will see that the effect of vertex corrections in magnetic field is captured by the replacement of with in the Drude-like formula for the magnetoconductivity.

A detailed evaluation of for is given in Appendix C. For , when the LL broadening is dominated by the background, , we have


As a result, the contribution of this energy region to the conductivity reads:


This expression can be cast in the form of a conventional Drude-like formula for the magnetoconductivity (for recent review see Ref. Dmitriev et al., 2012) with the -dependent transport scattering time and effective cyclotron frequency ,