Graphene under the influence of Aharonov–Bohm flux and constant magnetic field
Investigation of real two–dimensional systems with Dirac–like electronic behavior under the influence of magnetic field is challenging and leads to many interesting physical results. In this paper we study 2D graphene model with a particular form of magnetic field as a superposition of a homogeneous field and an Aharonov–Bohm vortex. For this configuration, electronic wave functions and energy spectrum were obtained and it was shown that the magnetic Aharonov–Bohm vortex plays the role of a charge impurity. As a demonstration of vacuum properties of the system, vacuum current, as well as an electric current, is calculated and their representation for particular limiting cases of magnetic field is obtained.
Recent investigations in condensed matter physics revealed a number of materials, that can be described by the effective two dimensional Dirac equation. The well known example of such systems is graphene, the planar monoatomic layer of carbon Novoselov et al. (2004, 2005); Katsnelson (2007); Geim (2009); Castro Neto et al. (2009), where the gapless nature is protected by sublattice symmetry. Band structure of the orbitals in graphene is formed by the two bands with the linear dispersion at the Fermi level at two inequivalent points of the Brillouin zone. The nearest–neighbour hoppings in graphene are much more stronger, than next to nearest–neighbor ones, thus electrons can be effectively modelled by using a continuum version of the tight-binding model, leading to the Dirac equation for massless fermions Wallace (1947); Semenoff (1984); Gusynin et al. (2007); Neto (2010).
Further researches of electronic behavior in 2D models by the Dirac equation with account for topological properties were made in Chamon et al. (2008a, b); Obispo and Hott (2014, 2015). In particular, the specified study in the application to graphene with different types of defects Lahiri et al. (2010); Huang et al. (2011); Ori et al. (2011) gave important results concerning nontrivial properties of transmission of propagating particles. Note also the recent theoretical studies of electronic transport through line defects in graphene Jiang et al. (2011); Ebert et al. (2014); Gunlycke and White (2014, 2015), which can be used to control the electronic transport in graphene. Furthermore, the electrons in the surface states of the 3D topological insulators Kane and Mele (2005a, b); Hasan and Kane (2010), where the crossing point is also protected by topology, and in 3D Weyl semimetals Wan et al. (2011); Burkov and Balents (2011); Wang et al. (2012) behave as two–dimensional Dirac particles as well. Therefore, investigations of the 2D materials with nontrivial topology with the use of the Dirac equation are of great importance nowadays.
On the other hand, applying of the electric field to Dirac systems is also challenging and leads to many important features. For example, recent investigations in graphene and graphene like models under the influence of AC Delplace et al. (2013); Yudin et al. (2015); Dutreix et al. (2016) and DC Novoselov et al. (2004); Castro et al. (2007) electric fields demonstrate a possibility to change the band structure of the system and the dynamics of quasiparticles. Magnetic fields of complicated configuration might also be the cause of various non-trivial effects. Exact solutions of relativistic wave equations for the 3D and 2D Dirac systems with the superpositions of Aharonov–Bohm, magnetic, and electric fields were obtained in Bagrov et al. (2001a, b). Further studies of the effect of the Aharonov–Bohm magnetic field, produced by a thin solenoid in plane structures, were made in Jackiw et al. (2009); Milstein and Terekhov (2011); Slobodeniuk et al. (2010, 2011) and resulted in calculations of the density of states, induced charge density and induced current. It is worth mentioning, that similar effect has been investigated in Sitenko (1999, 2000) in the context of quantum field theory.
In this paper we develop a continuum description of the Dirac model of graphene with the superposition of a constant and homogeneous magnetic field and an Aharonov–Bohm vortex of finite radius. We obtain the wave functions and energy spectrum of electrons and show that the Aharonow–Bohm vortex in graphene plays the role of a charged defect, which certainly affects the energy spectrum. We also derive, using the effective potential method, the vacuum current induced in the model, and consider the electric current in the strong magnetic field limit and show, that it is directed along the Aharonov–Bohm vortex.
The paper is organized as follows. In Section II we introduce the Dirac equation for the system with the homogeneous magnetic field and the Aharonov–Bohm vector potential. Upon solving the equation of motion it was possible to obtain the eigenvalues and eigenstates of the problem and show that the vortex field affects the spectrum as the charged defect only in the case, when it is captured by the electronic trajectory. In Section III we introduce the effective potential and study the weak and strong field limit. Vacuum current is also obtained in this section for the both limits of magnetic field strength. Section IV is dedicated to calculation of the electric current in the strong magnetic field limit and finally, Section V contains a summary and conclusions.
Ii Dirac equation
We consider the magnetic field configuration to consist of two separate components: a homogeneous magnetic field and an Aharonov–Bohm field. In cylindrical coordinates a homogeneous magnetic field is directed along the axis (), which is perpendicular to the graphene surface. The vector potential of the Aharonov–Bohm field has only angular component and is regularized by the finite radius of the circle . Then, the vector potential of the total magnetic field configuration can be written as
where is the Aharonov–Bohm potential
with () as the total magnetic flux of the vortex, and
is the magnetic flux quantum of the electron (it is assumed that in what follows due to the appropriate choice of system of units). This vector potential produces the total magnetic field of the form
It should be mentioned here, that in our problem we consider the solution in the region only, due to the fact, that the radius of the Aharonov–Bohm vortex is usually small enough and comparable to the distance between the neighbouring atoms. Thus, we are interested in the effects that take place at a certain distance from this vortex .
We assume that motion of electrons is described by the planar Dirac equation
where the Dirac Hamiltonian operator
includes the gauge potentials and 111Vector potential , instead of having an electromagnetic nature, may be due to certain defects in the graphene structure. In this case, we consider the so called pseudopotential description of our model of a circular defect (for the description of a linear defect by pseudopotential note, in particular, that changes in the distance between the atoms and in the overlap between the different orbitals by strain or bending lead to changes in nearest–neighbor (NN) hopping or next–nearest–neighbor (NNN) hopping amplitude and this results in appearance of vector potentials (this coupling may take the form of a gauge field with the matrix structure of the Pauli matrices, and ) in the Dirac Hamiltonian Castro Neto et al. (2009); Neto (2010) (see also Ref. Ebert et al., 2014). with the charge , as Pauli-matrices in the pseudospin space 222The physical spin of electrons that is due to spatial rotation properties of the electron wave function has been neglected in our analysis, and the spinor nature of the wave function has its origin in the sublattice degrees of freedom called pseudospin., as the Pauli matrix in the valley subspace with the eigenvalues for the two Fermi points , corresponding to valleys at the corners in the first Brillouin zone, is the unit matrix in the same space 333Our choice of signs in front of momentum and vector potential components of the Hamiltonian essentially corresponds to the conventions of Novoselov et al. (2005); Neto (2010) (see also Ref. Ebert et al., 2014). It may differ from that of other papers due to different initial definitions adopted. However, the final results do not depend on it.. Fermi velocity in what follows is supposed to be equal to unity with the corresponding choice of the units. Here the 4–spinors in the 2D plane () consist of two 2–spinor components ()
describing electrons at the two sublattices () with eigenvalues of for two Fermi points . The expression Eq. (6) implies that the low-momentum expansion around the other Fermi point with gives rise to a time-reversed Hamiltonian. Note that the total effect of both valleys, as described in four–spinor notations (see Ref. Gusynin et al., 2007, and references therein), respects time–reversal invariance. We shall use the appropriate 4–spinor notations (7) in what follows to describe solutions of the problem more conveniently.
Since our problem has cylindrical symmetry, we use polar coordinates in the plane, then the Dirac equation splits into a system of two equations
One can see that there are relations between two components of the wave function with different valley indices , i.e. . According to these relations, the Dirac equation may be solved just for . For this reason, we shall omit “+”sign in the subscripts in the following formulas, so , etc. The role of other brunch of solutions with will be considered later with the use of Appendix A. We will search for solutions in the following form:
where is the orbital quantum number. Let us introduce a dimensionless variable , then the Dirac equations look like
where instead of the energy we introduced a dimensionless parameter (we assume here that ; for situation with antiparallel to , see Appendix A). One can substitute from the second equation to the first one and vice versa to obtain the following equations
The general solutions for these equations can be found (see also Ref. Slobodeniuk et al., 2011)
One can see that Eqs. (11)-(12) and their solutions (13)-(14) have the same form as for the pure homogeneous magnetic field Ternov (1995) (i.e. without the Aharonov–Bohm vortex), but for an additional summand , changing to . Since the flux is not apriory an integer number, we introduce its fractional part according to
If the confluent hypergeometric function behaves as , thus the hypergeometric series should be reduced to a Laguerre polynomial . To this end one should put the first arguments of the confluent hypergeometric functions in Eqs. (13)–(14) to be equal to the non-positive integer numbers. Unfortunately, it is impossible to reduce both confluent hypergeometric functions that enter the solutions (13)-(14) of the Dirac Eqs. (11)-(12) with coefficients and to a polynomial at the same time, since parameter is proportional to the energy squared and therefore has the same value for both equations, but , generally speaking, is not integer due to the presence of the non-integer term . Thus, one has to put either , or coefficient to be equal to zero. Here we take in order to keep the solution with a correct behaviour in the limit Slobodeniuk et al. (2011). Now, to reduce the hypergeometric series to a Laguerre polynomial we put the first argument of the confluent hypergeometric function in Eq.(14) to be equal to the non-positive integer number (), which can be written in the form of the energy spectrum
Then we go over to “modified” orbital quantum numbers
Consider two cases:
Let a) : Then, the energy spectrum (18) does not depend on the flux and is given by , where is the principal (Landau) quantum number. It is the well known result of appearance of quantized Landau levels in graphene in a magnetic field (see Ref. Katsnelson, 2012 chapter 2.2), when the energy depends on like 444This is drastically different from the case of free non–relativistic electrons in a magnetic field, when .. The wave function in this case is given by the following relation
For this trivial case, we obtained the zero–energy modes with , which are protected by the topology (see Ref. Katsnelson, 2012 chapter 2.3). This means, that the electron trajectory does not capture the Aharonov–Bohm vortex and is affected only by the constant magnetic field.
b) Let : Then, from Eq. (18) one has
where is the radial quantum number. In this way, the confluent hypergeometric function transforms to the Laguerre polynomial (see, e.g. Ref. Ternov, 1995). Now the energy spectrum is given by the relation in dimensionless terms, or 555Similar result for (3+1)-D massive fermions see Ref.Bagrov et al., 2001b.
where the principle number is where we recall that For this nontrivial case the wave function is given by
It is important that in this case there are no zero energy states, which means that the topology is changed, in other words now the electron trajectory captures the Aharonov–Bohm vortex inside, which affects the energy spectrum. The energy spectrum is similar to the case of charged impurity on the two dimensional graphene surface (Ref. Katsnelson, 2012 chapter 2.10), thus the Aharonov–Bohm vortex plays the role of the impurity or the defect.
Iii Effective potential and vacuum current
Here we focus our attention only on the nontrivial case, when electrons are affected both by a constant homogeneous magnetic field and the Aharonov–Bohm field. The effective potential of the model can be calculated using the Fock–Schwinger proper time method Fock (1937) (see also Refs. Bietenholz et al., 2003; Zhukovsky and Stepanov, 2012; Sokolov et al., 1988; Bogoliubov et al., 1959)
where is the (2+1)D volume, is the area of the 2D graphene sample ( is large but finite), and is the square of the energy (21), i.e. the eigenvalue of , for quantum states with numbers , and as the sign of the energy . We shall further consider separately two possible cases: and :
Let a) : Recall that according to , where ; ; we have for Landau quantum number. Thus summation over and can be replaced by summation over and . According to Ternov (1995) one can obtain the radius of the semiclassical circular electron trajectory , and consequently, the typical size of the localized states in Landau level, i.e. the so called magnetic length is equal to . At the same time, the deviation of the center of the trajectory from the origin in terms of quantum number is defined as .
The magnetic field strength in continuous regime can be taken to be of the order of and in superconductor devices , while in pulsed regime it can reach , the extremely strong field that could be obtained in the laboratory is in pulsed regime.
Consider comparatively weak fields , when the magnetic length . The radius of the Aharonov–Bohm vortex in graphene is comparable to the distance between the neighbouring atoms and can be approximated by . Then we may assume that . For instance, if and the ratio . At the same time the size of the graphene sheet may be .
The energy spectrum depends only on and does not include quantum number , which means, in classical terms, that the center of the circular trajectory of a particle can not be fixed, and hence summation over this number should be made with regard to this degeneracy. For the classical electron trajectory to ”embrace” the circular vortex of radius , we have a natural geometrical condition , which is equally true for , as well as for , or . In this case and in weak fields we have . For the realistic situation with magnetic fields as described above, we arrive at the condition for summation over and (recall that under the condition for the trajectory of the particle ”embrace”s the vortex, and the value is excluded from the spectrum)
where . Then summation over is made, assuming that the upper limit under the condition can be extended to infinity
and hence summation over results in
In the problem without a vortex for a uniform plane with a homogeneous magnetic field perpendicular to the plane this limitation for summation over should be omitted and the contribution of with consideration for only one spin state for should be included. Then, instead of (25) one can get
Here factor 2 accounts for two spin orientations in the state with , while in the ground state only one spin orientation is possible and this gives an additional second term in (27). Summation over gives the degeneracy factor
Thus we may arrive, after appropriate subtraction and renormalization, at just what we have for 4D space-time effective Lagrangian, i.e. the well known Heisenberg–Euler formula (see, e.g., Schwinger (1954); Sokolov et al. (1988)) – in a magnetic field
In this paper we consider the situation with an Aharonov–Bohm vortex, described by formula (26), and we use the fundamental restriction for the wave-function, which means that the semiclassical trajectory of the particle should ”embrace” the vortex. Moreover, the value is excluded from the spectrum to avoid particle penetration into the vortex.
One should note, that with these conditions taken into account, the results for the effective potential and for the induced current (which follows) can not go over to the corresponding limiting formulas with .
Under these restrictions, we obtain the effective potential that takes into account both the Aharonov–Bohm vortex and the homogeneous magnetic field.
This integral diverges at the lower limit , thus we introduced the cutoff parameter , such that . The main contribution to this integral is given by , then . We study the real material that has a periodical atomic structure and according to the Debye theory of solids we should introduce the physical restriction on the wavelength and in consequence on the value of the proper time lower limit that will regularize the result. This restriction should be based on the structure of the material, i.e. it should be related to the characteristic length of the model. Parameter has the dimension of and one may put it equal to the area of the Aharonov–Bohm vortex in graphene . Then and if , for we have and
Thus, the degeneracy factor, which is proportional to , is cancelled by in the denominator of Eq.(30), and the result is
In fact, we may apply the restriction and approximately use Eq. (25) for summation starting with
The integral in Eq. (III) can be computed exactly
where is the Error Function. Considering , and , one obtains
Subtracting the term that does not depend on the magnetic field , in the limit putting , we have
As an illustration of the vacuum structure of the model we calculate the induced current
where is the Aharonov–Bohm potential (2), defined for , so . Recall that we considered in this part of the article the contribution of the branch of the spectrum with . In order to further distinguish this contribution we now call the corresponding part of the current . We have from (III)
or considering , , with the assumption that , we have
In this case the solution for the induced current will look like
An interesting question arises: what happens when the Aharonov–Bohm flux vanishes. Since in our calculations, we assumed that only those solutions provide nonzero contribution to the current that “embrace” the solenoid, there is no simple way to go to the limit of vanishing starting just with Eq. (III). Indeed, for this purpose we have to take the sum
which gives (with the divergent at contributions (see Eq. (III)) cancelling each other). By the way, we see that the following symmetry takes place
which is in agreement with Jackiw–Milstein conclusion Jackiw et al. (2009); Milstein and Terekhov (2011) for the problem without homogeneous magnetic field. At the same time, the presence of the preferred direction in this problem (the magnetic field) diminishes parity property with and as a consequence there is no symmetry , and contrary to Jackiw–Milstein conclusion valid for only Aharonov–Bohm flux participating, there is no symmetry in this problem.
One remark is in order. In this problem, there is a preferred direction along the magnetic field, which we chose to be parallel to axis, i.e. with . At the same time we agreed to take the Aharonov–Bohm flux to be positive, when it is also directed along the axis and thus is parallel to . In the situation, when the magnetic field points in the direction opposite to , , while is still directed along , solutions of the Dirac equation change and for the energy we have (see Appendix A)
It is evident that in the inverse situation with the magnetic field still pointing in the direction, but the flux taken to be positive when it is antiparallel to and, hence, to , the solution (III) is also valid.
Iv Electric current in strong magnetic field
It is also possible to determine not only the vacuum current, but also the current of the real electrons in our model in the presence of a magnetic field. Description of the wave function of the real electrons as a superposition of the bare states (22) is a rather difficult procedure, because it is impossible to determine exactly in what state every electron is located. Nevertheless, there is one particular case, where it is possible to determine such wave package. Electrons under the strong magnetic field lay on the lowest energy level , which is infinitely degenerate in the quantum number . The energy spectrum in the limit of strong magnetic field corresponds to the trivial case in the absence of the vortex and is given by . Then, electronic state can be described by the wave function
because . Then, the electric current is given by the following equation
which means, that electric current of magnitude is directed along the Aharonov–Bohm vortex that plays the role of an impurity in this problem.
In this paper, the electronic wave functions and energy spectrum for the graphene model with the superposition of a constant magnetic field and an Aharonov–Bohm vortex were obtained. For the non-positive values of the “modified” orbital quantum number the energy spectrum does not depend on the magnetic flux of the Aharonov–Bohm field, therefore the trajectory of electrons does not capture the magnetic vortex. It is important that for this case the energy spectrum is degenerate, which allows us to calculate electronic current for the strong magnetic field limit. This current is directed strictly along the vortex, which was demonstrated in the last Section. For the positive values of , the energy spectrum explicitly depends on the Aharonov–Bohm flux and the magnetic vortex plays the role of a charged defect. Furthermore, using the effective potential method, the expressions for the vacuum current in the weak and strong field limits were obtained.
Appendix A Energy spectrum for the second valley and the opposite orientation of the magnetic field
a) Energy spectrum for the second valley:
Let us consider the case of . Then
where is the orbital quantum number. Recall . Then
where and . Let us suppose that points in the opposite direction (). This leads to change in the signs in front of “1” in the above equations and the following replacements and . Then instead of (10) we obtain equations
which is equivalent to replacement and and , leaving the energy spectrum (21) unchanged but for replacement
Acknowledgements.The authors would like to thank C. Dutreix, S. Brener and A.E. Lobanov for valuable discussions.
- Novoselov et al. (2004) K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, Science 306, 666 (2004).
- Novoselov et al. (2005) K. Novoselov, A. K. Geim, S. Morozov, D. Jiang, M. Katsnelson, I. Grigorieva, S. Dubonos, and A. Firsov, Nature 438, 197 (2005).
- Katsnelson (2007) M. I. Katsnelson, Materials Today 10, 20 (2007).
- Geim (2009) A. K. Geim, Science 324, 1530 (2009).
- Castro Neto et al. (2009) A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, Rev. Mod. Phys. 81, 109 (2009).
- Wallace (1947) P. R. Wallace, Phys. Rev. 71, 622 (1947).
- Semenoff (1984) G. W. Semenoff, Phys. Rev. Lett. 53, 2449 (1984).
- Gusynin et al. (2007) V. P. Gusynin, S. G. Sharapov, and J. P. Carbotte, International Journal of Modern Physics B 21, 4611 (2007).
- Neto (2010) A. H. C. Neto, arXiv preprint arXiv:1004.3682 (2010).
- Chamon et al. (2008a) C. Chamon, C.-Y. Hou, R. Jackiw, C. Mudry, S.-Y. Pi, and A. P. Schnyder, Phys. Rev. Lett. 100, 110405 (2008a).
- Chamon et al. (2008b) C. Chamon, C.-Y. Hou, R. Jackiw, C. Mudry, S.-Y. Pi, and G. Semenoff, Phys. Rev. B 77, 235431 (2008b).
- Obispo and Hott (2014) A. E. Obispo and M. Hott, Phys. Rev. B 89, 165405 (2014).
- Obispo and Hott (2015) A. E. Obispo and M. Hott, Phys. Rev. B 91, 035404 (2015).
- Lahiri et al. (2010) J. Lahiri, Y. Lin, P. Bozkurt, I. I. Oleynik, and M. Batzill, Nature nanotechnology 5, 326 (2010).
- Huang et al. (2011) P. Y. Huang, C. S. Ruiz-Vargas, A. M. van der Zande, W. S. Whitney, M. P. Levendorf, J. W. Kevek, S. Garg, J. S. Alden, C. J. Hustedt, Y. Zhu, et al., Nature 469, 389 (2011).
- Ori et al. (2011) O. Ori, F. Cataldo, and M. V. Putz, International journal of molecular sciences 12, 7934 (2011).
- Jiang et al. (2011) L. Jiang, X. Lv, and Y. Zheng, Physics Letters A 376, 136 (2011).
- Ebert et al. (2014) D. Ebert, V. C. Zhukovsky, and E. A. Stepanov, Journal of Physics: Condensed Matter 26, 125502 (2014).
- Gunlycke and White (2014) D. Gunlycke and C. T. White, Phys. Rev. B 90, 035452 (2014).
- Gunlycke and White (2015) D. Gunlycke and C. T. White, Phys. Rev. B 91, 075425 (2015).
- Kane and Mele (2005a) C. L. Kane and E. J. Mele, Phys. Rev. Lett. 95, 146802 (2005a).
- Kane and Mele (2005b) C. L. Kane and E. J. Mele, Phys. Rev. Lett. 95, 226801 (2005b).
- Hasan and Kane (2010) M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045 (2010).
- Wan et al. (2011) X. Wan, A. M. Turner, A. Vishwanath, and S. Y. Savrasov, Phys. Rev. B 83, 205101 (2011).
- Burkov and Balents (2011) A. A. Burkov and L. Balents, Phys. Rev. Lett. 107, 127205 (2011).
- Wang et al. (2012) Z. Wang, Y. Sun, X.-Q. Chen, C. Franchini, G. Xu, H. Weng, X. Dai, and Z. Fang, Phys. Rev. B 85, 195320 (2012).
- Delplace et al. (2013) P. Delplace, A. Gómez-León, and G. Platero, Phys. Rev. B 88, 245422 (2013).
- Yudin et al. (2015) D. Yudin, O. Eriksson, and M. I. Katsnelson, Phys. Rev. B 91, 075419 (2015).
- Dutreix et al. (2016) C. Dutreix, E. A. Stepanov, and M. I. Katsnelson, Phys. Rev. B 93, 241404 (2016).
- Castro et al. (2007) E. V. Castro, K. S. Novoselov, S. V. Morozov, N. M. R. Peres, J. M. B. Lopes dos Santos, J. Nilsson, F. Guinea, A. K. Geim, and A. H. C. Neto, Phys. Rev. Lett. 99, 216802 (2007).
- Bagrov et al. (2001a) V. G. Bagrov, D. M. Gitman, and V. B. Tlyachev, Journal of Mathematical Physics 42 (2001a).
- Bagrov et al. (2001b) V. G. Bagrov, D. M. Gitman, A. Levin, and V. B. Tlyachev, Nuclear Physics B 605, 425 (2001b).
- Jackiw et al. (2009) R. Jackiw, A. I. Milstein, S.-Y. Pi, and I. S. Terekhov, Phys. Rev. B 80, 033413 (2009).
- Milstein and Terekhov (2011) A. I. Milstein and I. S. Terekhov, Phys. Rev. B 83, 075420 (2011).
- Slobodeniuk et al. (2010) A. O. Slobodeniuk, S. G. Sharapov, and V. M. Loktev, Phys. Rev. B 82, 075316 (2010).
- Slobodeniuk et al. (2011) A. O. Slobodeniuk, S. G. Sharapov, and V. M. Loktev, Phys. Rev. B 84, 125306 (2011).
- Sitenko (1999) Y. A. Sitenko, Phys. Rev. D 60, 125017 (1999).
- Sitenko (2000) Y. A. Sitenko, Annals of Physics 282, 167 (2000).
- (39) Vector potential , instead of having an electromagnetic nature, may be due to certain defects in the graphene structure. In this case, we consider the so called pseudopotential description of our model of a circular defect (for the description of a linear defect by pseudopotential note, in particular, that changes in the distance between the atoms and in the overlap between the different orbitals by strain or bending lead to changes in nearest–neighbor (NN) hopping or next–nearest–neighbor (NNN) hopping amplitude and this results in appearance of vector potentials (this coupling may take the form of a gauge field with the matrix structure of the Pauli matrices, and ) in the Dirac Hamiltonian Castro Neto et al. (2009); Neto (2010) (see also Ref. \rev@citealpnumline).
- (40) The physical spin of electrons that is due to spatial rotation properties of the electron wave function has been neglected in our analysis, and the spinor nature of the wave function has its origin in the sublattice degrees of freedom called pseudospin.
- (41) Our choice of signs in front of momentum and vector potential components of the Hamiltonian essentially corresponds to the conventions of Novoselov et al. (2005); Neto (2010) (see also Ref. \rev@citealpnumline). It may differ from that of other papers due to different initial definitions adopted. However, the final results do not depend on it.
- Gradshtein and Ryzhik (1971) I. S. Gradshtein and I. M. Ryzhik, “Tables of integrals, sums, series, and products,” (1971).
- Ternov (1995) I. M. Ternov, Physics-Uspekhi 38, p.66 (1995).
- Katsnelson (2012) M. I. Katsnelson, Graphene: carbon in two dimensions (Cambridge University Press, 2012).
- (45) This is drastically different from the case of free non–relativistic electrons in a magnetic field, when .
- (46) Similar result for (3+1)-D massive fermions see Ref.\rev@citealpnumBGT1.
- Fock (1937) V. Fock, Sov. Phys 12, 404 (1937).
- Bietenholz et al. (2003) W. Bietenholz, A. Gfeller, and U.-J. Wiese, Journal of High Energy Physics 2003, 018 (2003).
- Zhukovsky and Stepanov (2012) V. C. Zhukovsky and E. A. Stepanov, Physics Letters B 718, 597 (2012).
- Sokolov et al. (1988) A. A. Sokolov, I. M. Ternov, V. C. Zhukovskii, and A. V. Borisov, Quantum Electrodynamics (Mir Publishing Moscow, 1988).
- Bogoliubov et al. (1959) N. N. Bogoliubov, D. V. Shirkov, and S. Chomet, Introduction to the theory of quantized fields, Vol. 59 (Interscience New York, 1959).
- Schwinger (1954) J. Schwinger, Proceedings of the National Academy of Sciences 40, 132 (1954).