arX

iv:1

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4588

v1 [

cond

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1

Density of states of relativistic and nonrelativistic two-dimensional electron gases in a

uniform magnetic and Aharonov-Bohm fields

A.O. Slobodeniuk,1, ∗ S.G. Sharapov,1, † and V.M. Loktev1, 2, ‡

1Bogolyubov Institute for Theoretical Physics, National Academy of Science of Ukraine,

14-b Metrologicheskaya Street, Kiev, 03680, Ukraine2National Technical University of Ukraine ”KPI”, 37 Peremogy Ave., Kiev 03056, Ukraine

(Dated: May 24, 2011)

We study the electronic properties of 2D electron gas (2DEG) with quadratic dispersion andwith relativistic dispersion as in graphene in the inhomogeneous magnetic field consisting of theAharonov-Bohm flux and a constant background field. The total and local density of states (LDOS)are obtained on the base of the analytic solutions of the Schrodinger and Dirac equations in theinhomogeneous magnetic field. It is shown that as it was in the situation with a pure Aharonov-Bohm flux, in the case of graphene there is an excess of LDOS near the vortex, while in 2DEG theLDOS is depleted. This results in excess of the induced by the vortex DOS in graphene and in itsdepletion in 2DEG.

PACS numbers: 03.65.-w, 73.20.At, 72.10.Fk

I. INTRODUCTION

The continuous linear energy dispersion E(k) =±~vF |k| of the Dirac quasiparticle excitations when thehomogeneous magnetic field B is applied perpendicularto its two-dimensional plane transforms into the discreteLandau levels (LLs)

En = ±ǫ0√2n, n = 0, 1, 2 . . . , (1.1)

observed in graphene. Here k is the momentum measuredfrom K± points, ǫ0 =

√

~v2F eB/c is the relativistic Lan-dau scale with vF being the Fermi velocity. The spectrum(1.1) is characteristic of Dirac fermions and the break-through in experimental studies of graphene is causednot only by its fabrication1, but also by the demonstra-tion of its unique electronic properties2,3 that follow fromthe unusual spectrum (1.1).The hallmark of this spectrum is the zero energy field

independent lowest LL which existence does not in factdepend on the homogeneity of the field.4 In general theinhomogeneous magnetic perturbation can be presentedas a sum of a constant (averaged over the system) fieldand field localized in some regions of the two-dimensionalsystem. A limiting case of the perturbation can be pre-sented by the Aharonov-Bohm field which is created byan infinitely long and infinitesimally thin solenoid.The purpose of the present paper is to study the elec-

tronic excitations in graphene in the field consisting of theAharonov-Bohm flux and a constant background mag-netic field. As in the first publication,5 where we stud-ied the Aharonov-Bohm flux only, our main goal is theinvestigation of the local density of states (LDOS). Wefind that demonstrated in Ref. 5 rather peculiar behav-ior of LDOS in Dirac theory with Aharonov-Bohm fieldpersists in the presence of the constant background field.We expect that this behavior can be observed in scanningtunneling spectroscopy measurements for graphene pen-etrated by vortices from a type-II superconductor on top

of it. We also compare the obtained expressions with thecorresponding results for two-dimensional electron gas(2DEG) with a quadratic dispersion, where the singularbehavior of the LDOS is absent.In practice such a magnetic field configuration may

be obtained when a type-II superconductor is placed ontop of graphene. In the previous publication we con-sidered idealized picture when the vortex is single andthere is no impact from other Abrikosov vortices. Nowthe constant background field is supposed to mimic theimpact of the other vortices penetrating graphene. It isworth to stress that devices like this with a supercon-ducting film grown on top of a semiconducting hetero-junction (such as GaAs/AlGaAs) hosting a 2DEG havein fact been fabricated twenty years ago,6,7 so it should bepossible to fabricate the graphene based devices. Whilenormally the 2DEG is buried deep in a semiconductingheterostructure which makes the LDOS measurementsproblematic,8 the graphene surface is open to the LDOSmeasurements. While initially the STS measurementswere done on graphene flakes on graphite9, recently thesemeasurements were carried out on exfoliated graphenesamples deposited on a chlorinated SiO2 thermal oxidetuning the density through the Si backgate.10 So far allthese measurements were done in a homogeneous mag-netic field and showed a single sequence of pronouncedLL peaks corresponding to massless Dirac fermions ex-pected of pristine graphene.In a wider context, the inhomogeneous vortex-like

field configurations arise due the topological defects ingraphene which result in the pseudomagnetic field vor-tices, see e.g. Refs. 11,12. Interestingly, even nonsingularpseudomagnetic filed configuration created by a curvedbump on flat graphene13 result in the profile of the LDOSsimilar to the LDOS induced by the Abrikosov’s vortex.5

Thus we hope that the combination of the vortex + con-stant background field considered in the present papershould be useful not only for the studies which involvereal magnetic field, but also for the problems which in-

2

volve the superposition of magnetic and pseudomagneticfields.The paper is organized as follows. In Sec. II we intro-

duce the model Hamiltonians and discuss the configura-tion of the magnetic field and the regularization of theAharonov-Bohm potential used in this work. Sec. III isdevoted to the nonrelativistic case, and the relativisticcase is discussed in detail in Sec. IV. The structure ofboth sections is the same: we consider the solutions of thecorresponding Schrodinger or Dirac equation which allowto write down a general representation for the LDOS inSecs. III A and IVA. Then a more simple analysis of theDOS is made in Secs. III B and III B, while the behaviorof the LDOS is studied in Secs. III C and IVC. In Sec. Vour final results are summarized. The method of the cal-culation of the LDOS is explained in Appendix A, whereas an example we firstly calculate the LDOS in a con-stant magnetic field for the nonrelativistic case. Since theproblem with Aharonov-Bohm vortex has to be treated inthe symmetric gauge, the calculation of the LDOS in Ap-pendix A involves the sum over the azimuthal quantumnumber which is calculated in Appendix B. The full DOSis calculated in Appendix C. The LDOS both in nonrela-tivistic and relativistic cases is expressed in terms of thefunction calculated in Appendix D. The Dirac equationin the magnetic field consisting of the Aharonov-Bohmflux and a constant background field is solved in Ap-pendix E.

II. MODELS AND MAIN NOTATIONS

As in the paper5 we consider both nonrelativisticand relativistic Hamiltonians. The 2D nonrelativistic(Schrodinger) Hamiltonian has the standard form

HS = − ~2

2M(D2

1 +D22), (2.1)

where Dj = ∇j + ie/~cAj, j = 1, 2 with the vector po-tential A, Planck’s constant ~ and the velocity of light cdescribes a spinless particle with a mass M and charge−e < 0.The Dirac quasiparticle in graphene is described by the

Hamiltonian

HD = −i~vFβ(γ1D1 + γ2D2) + ∆β, (2.2)

where the matrices β and βγj are defined in terms of thePauli matrices as

β = σ3, βγj = (σ1, ζσ2). (2.3)

Here ζ = ±1 labels two unitary inequivalent represen-tations of 2 × 2 gamma matrices in 2 + 1 dimension, sothat one considers a pair of Dirac equations correspond-ing to two inequivalent K± points of graphene’s Brillouinzone. The spin degree of freedom is not included neitherin Eq. (2.1) nor in Eq. (2.2). In Eq. (2.2) vF is the Fermi

velocity and ∆ is the Dirac mass (or gap). An overview ofits physical origin is given in5 (see also a review14). Herewe only point out that the presence of a finite ∆ allowsone to distinguish unambiguously positive and negativeenergy solutions.There are numerous studies of the Dirac fermions in

the field of a singular Aharonov-Bohm vortex (see e.g.Refs. 15–17) and, in particular, of this vortex and a uni-form magnetic field18,19 devoted to the mathematical as-pects of the problem such as self-adjoint extension of theDirac operator. As in the previous article to avoid themathematical difficulties related to a singular nature ofthe Aharonov-Bohm potential at the origin, we considera regularized potential20,21 which depends on the dimen-sional parameter R:

A(r) = Aϕ(r)eϕ, Aϕ(r) =Br

2+Φ0η

2πrθ(r−R), (2.4)

where r = (r, ϕ, z), Φ0η is the flux of the vortex expressedvia magnetic flux quantum of the electron Φ0 = hc/ewith η ∈ [0, 1[. The value η = 1/2 corresponds to theAbrikosov’s vortex flux. The corresponding magneticfield

B(r) = ∇×A =

(

B +ηΦ0

2πRδ(r −R)

)

ez. (2.5)

The radiusR of the flux tube determines the region r > Rwhere the regularized potential coincides with the poten-tial of the problem with Aharonov-Bohm potential, whilefor r < R it describes a particle moving in a constantmagnetic field. The solution of the problem is found bymatching the solutions obtained in these regions. Thelimit R → 0 can taken at the end and allows to avoid theformal complications. As was shown in Ref. 21, the finalanswer does not depend on the specific form of the reg-ularizing potential provided that the profile of the mag-netic field is nonsingular at the origin.We also mention recent works22,23 where the induced

by the Aharonov-Bohm field charge density and currentwere studied for the massless Dirac fermions. In the firstpaper22 an infinitesimally thin solenoid was considered.The regularization by a magnetic flux tube of a smallradius R as in the present work is considered in the sec-ond paper.23 It is shown that in the limit R → 0 theinduced current is a periodic function of the magneticflux irrespectively the magnetic field distribution insidethe flux tube and whether the region inside the flux tubeis forbidden or not for penetration of electron. Also thevalue of the self-adjoint extension parameter is fixed bythe regularization.

III. NONRELATIVISTIC CASE

In this section we consider the solutions of theSchrodinger equation

HSψ(r) = Eψ(r) (3.1)

3

in polar coordinates r = (r, ϕ) and using them obtain thefull and local DOS. These results are important not onlyfor comparison with relativistic case, but also because therelativistic result is constructed using the nonrelativisticone.

A. Solution of the Schrodinger equation, general

representation for the local density of states and its

limiting η = 0 case

Technically to obtain the solutions of Eq. (3.1) in theregularized potential (2.4) one should solve this equationin two regions r < R and r > R. Since in the first domainr < R the potential is nonsingular, only a regular in thelimit r → 0 solution of the radial differential equation isadmissible. In the second domain r > R the solution con-tains both regular and singular in the limit r → 0 terms.The values of the relative weights of them can be foundby matching radial components and their derivatives atr = R. Finally, it turns out that in the limit R → 0 onlythe regular solution survives and the wave-function takesthe form

ψn,m(r, ϕ) = An,meimϕy|m+η|/2e−y/2L|m+η|

n (y), (3.2)

which also follows from the Schrodinger equation witha singular vortex. Here the dimensionless variabley ≡ r2/(2l2) is expressed via the magnetic length l =(~c/eB)1/2, Lα

n(y) is the generalized Laguerre polyno-mial and the normalization constant An,m is given by

A2n,m =

n!

2πl2Γ(n+ |m+ η|+ 1). (3.3)

The corresponding to the wave function (3.2) eigenenergyis equal to

En,m =~ωc

2(2n+ 1 + |m+ η|+m+ η), (3.4)

where the cyclotron frequency ωc = eB/Mc, the radialquantum number n = 0, 1, . . ., and the azimuthal quan-tum number m = −∞, . . . ,−1, 0, 1, . . . ,∞. In what fol-lows it is convenient to express all energies of the nonrel-ativistic problem in terms of the energy E0 ≡ ~ωc/2.Having the wave function one can calculate the LDOS

using the representation

N(r, E,B) =∑

n,m

|ψn,m(r)|2δ(E − En,m). (3.5)

In contrast to the previous article5 the presence of a con-stant magnetic field makes all energy spectra discretethat demands some regularization of the δ-function inEq. (3.5). For this purpose we introduce widening of theLLs to a Lorentzian shape:

δ(E − En,m) → 1

πIm

1

En,m − E − iΓ, (3.6)

where Γ is the LL width. Such a simple broadening ofLLs with a constant Γ was found to be a rather goodapproximation valid in not very strong magnetic fields.24

To illustrate the method of calculation in Appendix Awe derive the LDOS for the simplest case (η = 0) of theconstant magnetic field without vortex

NS0 (E,B) = −N

S0

πImψ

(

1

2− E + iΓ

~ωc

)

. (3.7)

Here NS0 = M/(2π~2) is a free DOS of 2DEG per spin

and unit area and we omitted r-dependence of the LDOS,because it is absent in the homogeneous field. One canreadily obtain Eq. (3.7) in a much simplier way25,26 start-ing from the usual Landau spectrum,

En = ~ωc

(

n+1

2

)

(3.8)

which follows from the spectrum (3.4) for η = 0, whenone relabels n+ (|m| +m)/2 → n. Here the relabeled ncorresponds to the LL index rather than the radial quan-tum number. Nevertheless, in Appendix A we proceededfrom Eq. (3.4) to illustrate how deal with the spectrumwhich is also dependent on the azimuthal quantum num-ber m. As seen in Fig. 1 a on the dashed (red) curve,Eq. (3.7) describes the usual quantum magnetic oscilla-tions which result in the de Haas-van Alphen effect. Onecan extract them analytically using the reflection formula(A11).

In a similar fashion we obtain in Appendix A the ex-pression for the LDOS perturbation, ∆NS

η (r, E,B) =

NSη (r, E,B) −NS

0 (r, E) induced by the vortex

∆NSη (r, E,B) = − M

(π~)2sinπη

πIm

[∫ ∞

0

dβe−(δ+β)e−βz e−y coth(δ+β)

1− e−2(δ+β)

∫ ∞

−∞

dωe−y coshω/ sinh(δ+β) e−η(δ+β+ω)

1 + e−(δ+β+ω)

]

.

(3.9)

Here NSη (r, E,B) is the LDOS in the presence of the con- stant field and vortex andNS

0 (r, E,B) is the LDOS in the

4

0 1 2 3 4 5E@Ωc

1

2

3

4

5

6

NΗSHr,E,BLN0

Sa

Η=0

Η=12

0 1 2 3 4 5E@Ωc

1

2

3

4

5

6

N12SHr,E,BLN0

Sb

rl=5

rl=0.5

FIG. 1: (Color online) The normalized full LDOSNS

η (r, E,B)/NS0 as a function of E in the units of ~ωc. (a) For

η = 0 (no vortex and LDOS is r-independent) and η = 1/2for r = l. (b) Both lines for η = 1/2, r = 0.5l and r = 5l. Inall cases the width Γ = 0.05~ωc.

constant magnetic field without vortex (the argument ris present to distinguish the LDOS from the DOS). Thisexpression has to be calculated for z > 0 with the ana-lytic continuation z → −(E + iΓ)/E0 done at the end ofthe calculation. The representation (3.9) for the LDOSis our starting point for the analysis of the LDOS andDOS. In the next Sec. III B we begin with a simpler caseof the DOS and return to the LDOS in Sec. III C.

B. The density of states

While in the constant magnetic field because the LDOSis position independent and is related to the full DOS bythe 2D volume (area) of the system factor V2D, this isnot so in the presence of the vortex when the LDOS isposition dependent. Then the full DOS per spin projec-tion is obtained from the LDOS (3.5) by integrating overthe space coordinates

Nη(E,B) =

∫ 2π

0

dϕ

∫ ∞

0

rdrNη(r, E,B). (3.10)

The details of the derivation of the DOS difference,∆NS

η (E,B) = NSη (E,B) − NS

0 (E,B) with NS0 (E) being

the full DOS in the presence of the constant field withoutvortex, are given in Appendix C. We obtain

∆NSη (E,B) =

1

π~ωcIm

(

1

2+E + iΓ

~ωc− η

)

×[

ψ

(

1

2− E + iΓ

~ωc

)

− ψ

(

1

2− E + iΓ

~ωc+ η

)]

.

(3.11)

Since the digamma function ψ(z) has simple poles forz = 0,−1,−2, . . . it is easy to see in the clean limit Γ → 0the DOS difference (3.11) reduces to a set of δ-peakscorresponding to the LLs

∆NSη (E,B) =−

∞∑

n=0

(n+ 1− η)δ

(

E − ~ωc

(

n+1

2

))

+

∞∑

n=0

(n+ 1)δ

(

E − ~ωc

(

n+1

2+ η

))

(3.12)

The physical meaning of (3.12) is that27 on each LL En =~ωc(n+1/2), n+1−η states disappear and n+1 appearat the energy En = ~ωc(n+ 1/2 + η).The limit of zero field, B → 0 can be obtained from

Eq. (3.11) using the asymptotic expansion

ψ(z) = ln z − 1

2 z− 1

12 z2+O

(

1

z4

)

. (3.13)

Then in the limit Γ → 0 we reproduce the Aharonov-Bohm depletion of the DOS5,27,28 at the bottom of thespectrum

∆NSη (E,B = 0) = NS

η (E,B = 0)− V2DNS0 =

= −1

2η(1 − η)δ(E)

(3.14)

caused by an isolated vortex. Integrating Eqs. (3.14) and(3.12) (with an appropriate regularization) one can checkthat the total deficit of the states induced by the vortex

∆NSη ≡

∫ ∞

−∞

dE∆NSη (E,B) = −1

2η(1 − η) (3.15)

does not depend on the strength B of the nonsingularbackground field.

C. The local density of states

The regularization parameter δ in Eq. (3.9) is impor-tant for the calculation of the DOS made in Appendix B,the integrand of Eq. (3.9) remains regular even in thelimit δ → 0. Therefore we can take this limit and rewrite

5

Eq. (3.9) as follows

∆NSη (r, E,B) = − M

(π~)2sinπη

2π

× Im

[

I

(

y, z → −E + iΓ

E0, η

)]

,

(3.16)

where

I(y, z, η) =

∫ ∞

0

dβe−βz e−y cothβ

sinhβ

×∫ ∞

−∞

dωe−y coshω/ sinh β e−η(ω+β)

1 + e−(ω+β),

(3.17)

and the variable y describes the spatial dependence. Al-though the integrals in Eq. (3.17) can be evaluated nu-merically, this computation becomes troublesome whenthe analytic continuation from z > 0 to the complex val-ues z → −(E+ iΓ)/E0 is done before the numerical inte-gration. Thus our purpose is to derive such a represen-tation for I(y, z, η) that it can be easily computed afterthe analytic continuation is done. The function I(y, z, η)is found in the Appendix D and is given by

I(y, z, η) =

Γ

(

z + 1

2

)

Γ

(

z + 2η − 1

2

)

F(1−z−η)/2,(1−η)/2(y)

+Γ

(

z − 1

2

)

Γ

(

z + 2η − 1

2

)

F(2−z−η)/2,η/2(y),

(3.18)

where the function Fλ,µ(y) is given by Eq. (D18).The results of the numerical computation of the LDOS

on the base of Eqs. (3.16) and (3.18) are shown in Figs. 1and 2. We emphasize that in Fig. 1 we plot the full LDOSNS

η (r, E,B) as a function of energy E for fixed valuesof r and in Fig. 2 the same quantity is presented as afunction of the distance r from the vortex center for fixedvalues of E. Since Eq. (3.16) describes the perturbationof the LDOS ∆NS

η (r, E,B) by the vortex, to obtain the

value of the full LDOS NSη (r, E,B) we add to ∆NS

η itsη = 0 value which is given by Eq. (3.7). We note that incontrast to Ref. 5 plotting these figures we do not takeinto account the presence of the finite carried density in2DEG by shifting the energy origin. This makes morestraightforward a comparison with the Dirac case, wherelow carried densities are indeed accessible experimentally.Although the model we consider is suitable for all val-

ues of the distance from the center of the vortex r, thereare obvious physical limitations on the possible value ofr if the vortex penetrating graphene is coming from atype-II superconductor. First of all, r cannot smallerthan the vortex core which is at least on the order ofmagnitude larger than the distance scale r0 of the orderof the lattice constant. We remind that in the previouspaper5 the distance r was measured in the units of r0,because for B = 0 there is no such a natural scale as amagnetic length. Secondly, we replace the magnetic field

0 1 2 3 4rl

1

2

3

4

5

6

N12SHr,E,BLN0

S

E=2@Ωc

E=1.5@Ωc

E=@Ωc

E=0.5@Ωc

FIG. 2: (Color online) The normalized full LDOSNS

1/2(r, E,B)/NS0 as a function of the distance r measured

in the units of the magnetic length l for four values ofE/~ωc = 0.5, 1.5 (usual LLs) and E/~ωc = 1, 2 (vortex-likelevels). The width Γ = 0.05~ωc.

created by the other vortices by a constant backgroundmagnetic field. This approximation may be appropriateif one considers a vicinity of the selected vortex whichimplies that r has to be less than the intervortex dis-tance lv. This distance is proportional to the magneticlength29, lv = c

√πl ≈ 1.77l, where c ≈ 1 is the geomet-

ric factor dependent on the Abrikosov’s lattice structure.Thus although one can investigate the regime r ≫ l the-oretically, in practice it is not accessible.In Fig. 1 a we compare the already discussed after

Eq. (3.7) case of the constant magnetic field with thecase when the Abrikosov vortex is also present (η = 1/2)for r = l. We observe that while for η = 0 [the dashed(red) curve is, obviously, r-independent] only the peaks athalf-integers E/~ωc are present, for η = 1/2 the weight ofthese peaks is reduced and a set of the new peaks at theintegers E/~ωc on the solid (blue) curve is developed.This behavior can be foreseen from the expression forthe full DOS difference (3.12) [or Eq. (3.11)] discussed inSec. III C. The case with the Abrikosov vortex is furtherexplored in Fig. 1 b, where we plot the energy depen-dence of the LDOS for r = 0.5l [the solid (blue) curve]and r = 5l [the dashed (red) curve]. Comparing the re-sults for r/l = 0.5, 1., 5. we find that as the distance rdecreases, the integer E/~ωc peaks are getting stronger,while for r = 5.l they practically disappear. This behav-ior allows to attribute the corresponding energy levels tothe vortex. On the other hand, the half-integer E/~ωc

peaks corresponding to the usual LLs (3.8) formed in aconstant magnetic field are getting weaker as the distancer decreases. We stress that even for an arbitrary vortexflux η the latter levels will not change the positions, whilethe levels related to the vortex will shift their energies.Analyzing Eq. (3.18) which was used to plot Fig. 1, we

observe that the positions of all peaks are controlled bythe gamma functions Γ(z) which contain simple polesfor z = 0,−1,−2, . . .. However, the intensity of the

6

peaks depends on the rather complicated modulatingfunction Fλ,µ(y). For example, we verified that despitethat the gamma function Γ[(z− 1)/2] in the second termof Eq. (3.18) contains the pole at the negative energyE = −~ωc/2, the final LDOS does not contain this pole.To gain more insight on the behavior of the LDOS wehave investigated its behavior in the limits r → 0 andr → ∞. Taking into account the y → 0 limit of ImI givenby Eq. (D23), we obtain that the value ∆NS

η (r = 0, E,B)is equal to the negative LDOS (3.7) in the constant mag-netic field. This implies that the full LDOS in the centerof the vortex is completely depleted,

NSη (r = 0, E,B) = 0. (3.19)

Formally this property reflects a simple fact that all so-lutions (3.2) of the Schrodinger equation are vanishingat the origin, ψn,m(r = 0, ϕ) = 0. This vortex induceddepletion of the LDOS in the nonrelativistic 2DEG wasalready seen in Ref. 5 and now we conclude that it shouldalso occur in the presence of the background magneticfield. This is exactly what we observe in Fig. 2, where allfour curves begin from zero. Two of these curves, viz. thesolid (blue) and the dash-dotted (black) are for the usualLLs with E/~ωc = 0.5, 1.5, and the other two [dashed(red) and dotted (violet)] are for the vortex levels withE/~ωc = 1, 2. For small r < l all curves increase lin-early as expected from the analytic results described inAppendix C if we take there η = 1/2. Since for the largey the function Fλµ decays exponentially [see Eq. (D26)],

the LDOS difference ∆NSη (r, E,B) ∼ e−r2/2l2 for r → ∞.

Accordingly, the large r behavior of the full LDOS de-pends on the contribution of the position independentLDOS (3.7). Thus the large r limit of all curves in Fig. 2is determined by the corresponding value of the LDOSin the dashed (red) curve in Fig. 1 a.

IV. RELATIVISTIC CASE

In Sec. IVA we consider the solutions of the Diracequation

HDΨ(r, ζ) = EΨ(r, ζ), (4.1)

where the wave function is now a spinor

Ψ(r, ζ) =

[

ψ1(r, ζ)ψ2(r, ζ)

]

, (4.2)

and the index ζ labels two inequivalent K± points. No-tice that in the Appendix E the definition (E1) for ψ2

explicitly includes the factor i. Using these solutions inSec. IVB we obtain the full DOS and the local DOS isconsidered in Sec. IVC.

A. Solutions of the Dirac equation, general

representation for the local density of states and its

limiting η = 0 case

The Dirac Eq. (4.1) with the regularized potential (2.4)is solved in Appendix E. A general strategy is the sameas described in Sec. III A, but the main difference is inthe matching conditions. While the radial componentsof the spinor Ψ(r) have to be continuous:

ψ1(R + 0, ζ) = ψ1(R − 0, ζ),

ψ2(R + 0, ζ) = ψ2(R − 0, ζ),(4.3)

their derivatives in contrast to the nonrelativistic casehave a discontinuity:

ψ′1(R + 0, ζ)− ψ′

1(R− 0, ζ) =ζη

Rψ1(R, ζ),

ψ′2(R + 0, ζ)− ψ′

2(R− 0, ζ) = −ζηRψ2(R, ζ).

(4.4)

The discontinuity of the conditions (4.4) follows fromEq. (E2) with the discontinuous potential (2.4). Anotherway to apprehend this discontinuity is to obtain a sin-gular ar r = R pseudo-Zeeman term squaring the Diracequation (see Ref. 5 for an overview).After the limit R→ 0 is taken we obtain the following

solutions:

Ψ(±)n,m(r, 1) =

1

2l√

πEn,m

[ √

En,m ±∆ ei(m−1)ϕJnm+η−1(y)

±i√

En,m ∓∆ eimϕJnm+η(y)

]

(4.5)

for m > 0,

Ψ(±)n,0 (r, 1) =

1

2l√

πEn,0

[ √

En,0 ±∆ e−iϕJn1−η(y)

∓i√

En,0 ∓∆ Jn+1−η (y)

]

(4.6)

for m = 0, and

Ψ(±)n,m(r, 1) =

1

2l√

πEn,m

[

√

En,m ±∆ ei(m−1)ϕJn|m+η−1|(y)

∓i√

En,m ∓∆ eimϕJn+1|m+η|(y)

]

(4.7)

for m < 0. Here the upper and lower signs ± correspondto the positive and negative energy solutions, E(±) =±En,m with the absolute value of the energy

En,m =√

∆2 + ǫ20λn,m,

λn,m = 2n+ |m+ η − 1|+m+ η + 1(4.8)

for n ≥ 0, the function Jnν (y) is defined by Eq. (E14)

with y as in the nonrelativistic case, and the relativisticLandau scale ǫ0 is defined after Eq. (1.1). The zero modesolution with E = −∆ is a hole-like

Ψ(−)0,m(r, 1) =

1√2πl

[

0eimϕ J0

|m|−η(y)

]

, m ≤ 0. (4.9)

7

Let us now compare the solutions of the Schrodinger andDirac equations with the zero azimuthal number, m = 0.One can see from Eq. (3.2) that ψn,0(r, 1) ∼ rη , because30

Lαn(0) =

(

n+ αn

)

=Γ(α + n+ 1)

Γ(α+ 1)n!. (4.10)

On the other hand, from Eqs. (4.6) and (4.9) for m = 0we observe that while the upper components are regularat r = 0, the lower components diverge as as ψ2n,0(r, 1) ∼r−η. Comparing these results with the behavior of thewave function in the Aharonov-Bohm field5 we observethat the presence of the background magnetic field doesnot change the asymptotics of the m = 0 solutions forr → 0. Also as expected5 the zero mode solution (4.9)for ζ = 1 and chosen direction of the field is holelike,E = −∆.

The solutions for the case ζ = −1 are the following:

Ψ(±)n,m(r,−1) =

1

2l√

πEn,m

[

∓√

En,m ±∆ eimϕJnm+η(y)

i√

En,m ∓∆ ei(m−1)ϕJnm+η−1(y)

]

(4.11)

for m > 0,

Ψ(±)n,0 (r,−1) =

1

2l√

πEn,0

[

±√

En,0 ±∆ Jn+1−η (y)

i√

En,0 ∓∆ e−iϕJn1−η(y)

]

(4.12)

for m = 0, and

Ψ(±)n,m(r,−1) =

1

2l√

πEn,m

[

±√

En,m ±∆ eimϕJn+1|m+η|(y)

i√

En,m ∓∆ ei(m−1)ϕJn|m+η−1|(y)

]

(4.13)

for m < 0. Again the signs ± correspond to the solutionsE(±) = ±En,m with n ≥ 0 and the energy En,m given byEq. (4.8). Now the zero mode solution

Ψ0,m(r,−1) =1√2πl

[

eimϕ J0|m|−η(y)

0

]

, m ≤ 0,

(4.14)is electron-like, E = ∆. Also the lower components of them = 0 solutions (4.12) and (4.14) are regular at r = 0,and the upper components diverge as ψ1n,0(r,−1) ∼ r−η.

Since the solutions of the Dirac equation are character-ized not only by the quantum numbers, but also by thesublattice label A and B, energy ±, and the valley indexζ = ±1, instead of directly writing an analog of Eq. (3.5)it is more convenient to construct the Green’s functionexpressing the LDOS via the combinations of its matrixelements. The eigenfunction expansion for the retarded

Green’s function reads

GDη (r, r

′, E + i0; ζ) =

∞∑

n=0

∞∑

m=−∞

(

Ψ(+)n,m(r, ζ)Ψ

(+)†n,m (r′, ζ)

E − En,m + i0

+Ψ

(−)n,m(r, ζ)Ψ

(−)†n,m (r′)

E + En,m + i0

)

.

(4.15)

The LDOS for A and B sublattices is expressed in termsof the Green’s function (4.15) as follows

ND(A)η (r, E) = − 1

πIm [Gη11(r, r, E + iΓ; ζ = 1)

+ Gη11(r, r, E + iΓ; ζ = −1)] ,

ND(B)η (r, E) = − 1

πIm [Gη22(r, r, E + iΓ; ζ = 1)

+ Gη22(r, r, E + iΓ; ζ = −1)] ,

(4.16)

where similarly to the nonrelativistic case the LL widthΓ is introduced. Substituting the solutions of the Diracequation in the Green’s function (4.15), and using thedefinition (4.16) we obtain

ND(A,B)η (r, E,B) = −ND

0

1

πIm

[

E + iΓ±∆

ǫ0

× G

(

y, z → − (E + iΓ)2 −∆2

ǫ20, η

)]

,

(4.17)

where the upper (lower) sign corresponds to A (B) sub-lattice, the relativistic Landau scale ǫ0 is defined be-low Eq. (1.1), and the normalization constant ND

0 =ǫ0/(2π~

2v2F ) corresponds to the value of the free (η =B = Γ = 0) DOS for the Dirac quasiparticles per spinand one sublattice (or valley) taken at the energy E = ǫ0when the energy gap ∆ = 0. Writing Eq. (4.17) we in-troduced the function

G(y, z, η) =

3∑

i=1

gi(y, z, η) (4.18)

which consists of the three terms

g1(y, z, η) = −∞∑

n=0

∞∑

m=−∞

[Jn|m+η−1|(y)]

2

z + λn,m,

g2(y, z, η) = −∞∑

n=0

∞∑

m=−∞

[Jn|m+η|(y)]

2

z + λ′n,m,

g3(y, z, η) =

∞∑

n=0

(

[Jnη (y)]

2

z + 2(n+ η)−

[Jn−η(y)]

2

z + 2n

)

(4.19)

with λn,m defined in Eq. (4.8) and

λ′n,m = 2n+ |m+ η|+m+ η. (4.20)

Note that the function g3 contains the singular termswhich originate from m = 0 solutions of the Dirac equa-tions.

8

The g1,2 contributions are calculated in the same wayas was derived Eq. (3.9) in Appendix A, viz. exponen-tiating the denominators [see Eq. (A2)] and introducingthe regularizing factor δ > 0, and then using the sum(A4) we obtain

g1(y, z, η) = −∫ ∞

0

dβe−βz e−2(δ+β)

1− e−2(δ+β)e−y coth(δ+β)

×∞∑

m=−∞

e−(δ+β)(m+η)I|m+η|

(

y

sinh(δ + β)

)

,

(4.21)

where we also shifted the dummy index m → m + 1.As we saw in the nonrelativistic case, the presence of δis necessary for the calculation of the DOS, although itcan be omitted in the expressions for the LDOS. Theremaining sum over m can be found using Eq. (B7) fromAppendix B

∆g1(y, z, η) ≡ g1(y, z, η)− g1(y, z, 0)

=sinπη

π

∫ ∞

0

dβe−βz e−2(δ+β)

1− e−2(δ+β)e−y coth(δ+β)

∫ ∞

−∞

dωe−y coshω/ sinh(δ+β) e−η(ω+δ+β)

1 + e−(ω+δ+β),

(4.22)

where we introduced the function ∆g1 which describesthe perturbation by the the vortex. Similarly, for ∆g2we have

∆g2(y, z, η) ≡ g2(y, z, η)− g2(y, z, 0)

=sinπη

π

∫ ∞

0

dβe−βz 1

1− e−2(δ+β)e−y coth(δ+β)

×∫ ∞

−∞

dωe−y coshω/ sinh(δ+β) e−η(ω+δ+β)

1 + e−(ω+δ+β).

(4.23)

The case of g3 is even simpler, because there is no sum-mation over m. Using the sum (A4) we obtain an analogof Eq. (4.21). It contains the difference of two modifiedBessel functions, which can be expressed via the Mac-Donald function30

Kν(x) =π

2 sinπν[I−ν(x)− Iν(x)]. (4.24)

Finally we arrive at the result

g3(y, z, η) = −2 sinπη

π

×∫ ∞

0

dβe−βz e−(δ+β)η

1− e−2(δ+β)e−y coth(δ+β)Kη

(

y

sinh(δ + β)

)

.

(4.25)

Notice that since g3(y, z, η = 0) = 0, there is no needto introduce a function ∆g3. Having the functions ∆g1,2and g3 we can directly calculate the LDOS perturbation

by the vortex, ∆ND(A,B)η (r, E,B) = N

D(A,B)η (r, E,B) −

ND(A,B)0 (r, E,B).

The subsequent consideration is made in parallel to thenonrelativistic case. We consider first the LDOS in theconstant magnetic field (η = 0) when due to the transla-tional invariance it coincides with the DOS per unit area.The LDOS can be derived in a similar to Eq. (3.7) way,but a special care has to be taken, because in contrast tothe nonrelativistic case the cutoff parameter δ enters thefinal result,

ND(A,B)0 (E,B) = −N

D0

π

× Im

[

E + iΓ±∆

ε0

(

ln(2δ) + γ + ψ(z

2

)

+1

z

)]

,

(4.26)

where we kept only the divergent in the limit δ → 0terms It is more convenient to rewrite Eq. (4.26) in theform derived in Ref. 26, where instead of the cutoff δ thebandwidth W cutoff is used

ND(A,B)0 (E,B) =

ND0

π

Γ

ǫ0lnW 2

2ǫ20− Im

[

E + iΓ±∆

ǫ0

×(

ψ

(

∆2 − (E + iΓ)2

2ǫ20

)

+ǫ20

∆2 − (E + iΓ)2

)]

.

(4.27)

The advantage of the representation (4.27) is thatits B → 0 limit takes the usual form.26 The quan-tum magnetic oscillations of the LDOS, ND

0 (E,B) =

ND(A)0 (E,B) = N

D(B)0 (E,B) for ∆ = 0 are shown in

Fig. 3 a on a dashed (red) curve. Only the positive en-ergy region is shown, where the positions of the peaks,En/ǫ0 =

√2n are in accord with the Dirac spectrum (1.1).

The nonequidistant LLs along with the peak at E = 0 re-lated to the energy independent lowest LL are character-istic of the Dirac fermions. The reflection formula (A11)allows one to extract these oscillations analytically.26

The representation (4.17) for the LDOS, where thefunction (4.18) consists of the three terms (4.22), (4.23)and (4.25) is our starting point for the analysis of theLDOS and DOS in the relativistic case. In the nextSec. IVB we begin with the DOS and in Sec. IVC re-turn to the LDOS.

B. The density of states

The full DOS per spin projection is given by the spa-tial integral (3.10). Accordingly, the full DOS perturba-

tion by the vortex ∆ND(A,B)η (E,B) = N

D(A,B)η (E,B) −

ND(A,B)0 (E,B) for A and B sublattices takes the form

∆ND(A,B)η (E,B) = −ND

0 2l2Im

[

E + iΓ±∆

ε0×

∫ ∞

0

dy(∆g1(y, z, η) + ∆g2(y, z, η) + g3(y, z, η))

]

.

(4.28)

9

0.0 0.5 1.0 1.5 2.0 2.5 3.0EΕ0

1

2

3

4

5

6

7

NΗDHr,E,BLN0

Da

Η=0

Η=12

0.0 0.5 1.0 1.5 2.0 2.5 3.0EΕ0

2

4

6

8

10

N12DHr,E,BLN0

Db

rl=4

rl=0.5

FIG. 3: (Color online) The normalized full LDOSND

η (r, E,B)/ND0 (ǫ0) as a function of energy E in the units

of the relativistic Landau scale ǫ0. The LDOS is an evenfunction of E, so only the positive energy region is shown.(a) For η = 0 (no vortex and LDOS is r-independent) andη = 1/2 for r = l. (b) Both lines for η = 1/2, r = 0.5l andr = 4l. In all cases the width Γ = 0.05ǫ0, W/ǫ0 = 3.35 and∆ = 0.

The subsequent calculation on the base of Eq. (4.28)is similar to the nonrelativistic case considered in Ap-pendix C and gives [cp. Eqs. (C6) and (3.11)]

∆ND(A,B)η (E,B) = −Im

E + iΓ±∆

2πε20

[

2η

(

1 +1

z

)

+(z + 2η)

(

ψ(z

2

)

− ψ

(

z + 2η

2

))]

,

(4.29)

where z → −[(E+iΓ)2−∆2]/ǫ20. In the clean limit Γ → 0the DOS difference reduces to

∆ND(A,B)η (E,B) = ηδ(E ±∆)

+ (E ±∆)sgnE

[

∞∑

n=1

2nδ(E2 −∆2 − 2(n+ η)ǫ20)

−∞∑

n=1

(2n− 2η)δ(E2 −∆2 − 2nǫ20)

]

,

(4.30)

where except to the first proportional to η zero-modeterm each δ-functions corresponds to the both positiveand negative energy peaks. A comparison of this resultwith Eq. (3.12) for the nonrelativistic problem sheds thelight on the difference between these cases. We observedfrom Eq. (3.12) that all peaks associated with the usualLLs are depleted, while the peaks related to the vortexare developed. At first sight, Eq. (4.30) follows the same

pattern, viz. the LL peaks with E(±)n = ±

√

∆2 + 2ǫ20nwith n = 1, 2, . . . are depleted and the vortex-like levels

E(±)n = ±

√

∆2 + 2ǫ20(n+ η) with n = 1, 2, 3, . . . are de-veloped. However, the first term ηδ(E ± ∆) related tothe zero-mode solutions of the Dirac equation is presentfor any magnetic field configuration and the addition ofthe vortex only adds η to the weight of the correspondingpeak. This property is an illustration of the topologicalorigin of the lowest LL.32

The B → 0 limit can again be obtained using theasymptotic expansion (3.13) which for the expression inthe square brackets of Eq. (4.29) gives

2η

(

1 +1

z

)

+ (z + 2η)

(

ψ(z

2

)

− ψ

(

z + 2η

2

))

≈ −2η2

z+O

(

1

z2

)

.

(4.31)

Substituting Eq. (4.31) in Eq. (4.29) and making the an-alytic continuation z → −[(E+iΓ)2−∆2]/ǫ20 in the cleanlimit Γ → 0 we obtain

∆ND(A,B)η (E,B = 0) = ND(A,B)

η (E,B = 0)− V2DND(A,B)0

= η2δ(E ∓∆).

(4.32)

This result is in agreement with Refs. 5,33, where theDOS ρDη (E, ζ) for a separate K± point, but summed con-tributions for A and B sublattices was considered. Itsperturbation ∆ρDη (E, ζ) = ρDη (E, ζ)−V2DρD0 (E) with re-

spect to the free DOS per spin and one valley, ρD0 (E) =|E|θ(E2/∆2 − 1)/(2π~2v2F ), is equal to

∆ρDη (E, ζ) =− 1

2η(1 − η)[δ(E −∆) + δ(E +∆)]

+ ηδ(E + ζ∆), η > 0.

(4.33)

Integrating Eq. (4.32) one can find the total excess ofthe states induced by the vortex

∆NDη ≡

∫ ∞

−∞

dE(∆ND(A)η (E,B) + ∆ND(B)

η (E,B)) = 2η2.

(4.34)

As in the nonrelativistic case (3.15) one it turns out thatthe integral (4.34) does not depend on the strength B ofthe background field. This can be checked by integrating

10

the sum (4.30) and using an appropriate regularization.Completing our discussion of the DOS we note that thevalue ∆ND

η has to be distinguished from the induced by

the magnetic flux fractional fermion number34 which interms of the DOS (4.33) can be written as follows

Nη = −1

2

∫ ∞

−∞

dEsgnE∆ρDη (E, ζ) =ζη

2. (4.35)

C. The local density of states

The contributions ∆g1,2 to the relativistic LDOS givenby Eqs. (4.22) and (4.23) can be written in terms of thefunction I(y, z, η) defined by Eq. (3.17) which was usedin Sec. III C to express the nonrelativistic LDOS

∆g1(y, z, η) =sinπη

2πI(y, z + 1, η), (4.36)

and

∆g2(y, z, η) =sinπη

2πI(y, z − 1, η). (4.37)

Thus the only remaining term we have to find is g3 givenby Eq. (4.25). Changing the variable x = e−β we obtain

g3(y, z, η) = −2 sinπη

π

×∫ 1

0

dxxz+η−1

1− x2e−y(1+x2)/(1−x2)Kη

(

2xy

1− x2

)

,

(4.38)

Now using the integral (2.16.10.5) from31 (one can alsochange the variable to t via e−β = [t/(1 + t)]1/2 and usethe integral (D9))

∫ y

0

dxxα−1

x2 − y2exp

(

−by2 + x2

y2 − x2

)

Kν

(

2cx

y2 − x2

)

=− yα−1

4cΓ

(

α− ν

2

)

Γ

(

α+ ν

2

)

×W(1−α)/2,ν/2

(

b+√

b2 − (c/y)2)

×W(1−α)/2,ν/2

(

b−√

b2 − (c/y)2)

,

(4.39)

we can express g3 in terms of the Whittaker functionWλµ(z) as follows

g3(y, z, η) =sinπη

2πID(y, z, η) (4.40)

with the function

ID(y, z, η) = −1

yΓ(z

2

)

Γ

(

z + 2η

2

)

W 2(1−z−η)/2,η/2(y).

(4.41)

Thus the final expression for the relativistic LDOS per-turbation by the vortex takes the form

∆ND(A,B)η (r, E,B) = −ND

0

1

πIm

[

E + iΓ±∆

ǫ0

× ∆G

(

y, z → − (E + iΓ)2 −∆2

ǫ20, η

)]

,

(4.42)

where the function

∆G(y, z, η) =sinπη

2π

× [I(y, z + 1, η) + I(y, z − 1, η) + ID(y, z, η)](4.43)

is expressed via the defined above functions (3.18) and(4.41).

To complete the analytic treatment we consider thebehavior of the LDOS in the most interesting case of thesmall r, when we expect that the difference between therelativistic and nonrelativistic cases should be the mosttransparent. The observation (3.19) that in the nonrel-ativistic case the full LDOS in the center of the vortexvanishes turns out to be useful for better understandingof the relativistic case. Indeed, let’s consider the first twoterms of Eq. (4.18) with g1,2 which contribute to the fullLDOS (4.17). Since the numerators of g1,2 in Eq. (4.19)vanish at y = 0, the only term which governs the behav-ior of the full LDOS in the r → 0 limit is the functionID(y, z, η) which due to its origin from the m = 0 solu-tions is expected to be divergent.

The same result can be verified using the final expres-

sions (4.42), (4.43) for ∆ND(A,B)η (r = 0, E,B). For y = 0

the first two terms of Eq. (4.43) with the function I whichoriginate from ∆g1,2 [see Eqs. (4.36) and (4.37)] can becombined together

sinπη

2πIm[I(y = 0, z + 1, η) + I(y = 0, z − 1, η)]

= Im

[

ψ(z

2

)

+1

z

]

,(4.44)

where we used the value ImI(y = 0, z, η) established inEq. (D23) and then transformed the first digamma func-tion using Eq. (D22). Thus we find that in the limitΓ → 0 the contribution of these ∆g1,2 terms to the LDOS

difference ∆ND(A,B)η (r = 0, E,B) given by Eq. (4.42) is

equal to the negative LDOS (4.26) in the constant mag-netic field.

Let us now analyze the behavior of the functionID(y, z, η) in the r → 0 limit. Using the expansion ofthe Whittaker function (D19) in the limit y → 0, weobtain that

ID(y, z, η)

= −Γ(z/2)Γ2(η)

Γ(η + z/2)y−η +O(y0), y → 0.

(4.45)

11

Thus the full LDOS is divergent at the origin as

ND(A,B)η (r, E,B) ∼ r−2ηIm

[

E + iΓ±∆

ǫ0×

Γ

(

∆2 − (E + iΓ)2

2ǫ20

)

Γ−1

(

∆2 − (E + iΓ)2

2ǫ20+ η

)]

.

(4.46)

For η = 1/2 the divergence is ∼ r−1 as was in the ab-sence of the background field.5 As we discuss below, thepresence of this field makes the divergence of the LDOSstrongly energy-dependent.The results of the numerical computations of the full

LDOS on the base of Eqs. (4.42) and (4.43) are shown inFigs. 3, 4 and 5. Since Eq. (4.42) describes the perturba-tion of the LDOS ∆ND

η (r, E,B) by the vortex, to obtain

the value of the full LDOS NDη (r, E,B) we add to ∆NS

η

its η = 0 given by Eq. (4.27).In Fig. 3 a we compare the already discussed after

Eq. (4.27) case of the LDOS for a constant magnetic fieldwith the case when the vortex is also present (η = 1/2)for r = l. Since we consider the situation when ∆ = 0,there is no difference between sublattices, ND

0 (E,B) =

ND(A)0 (E,B) = N

D(B)0 (E,B) and the LDOS is an even

function of energy, so the positive energy region is plot-ted. We observe that compared to η = 0 [the dashed(red) curve] for η = 1/2 [solid (blue) curve] a set of thenew peaks at En/ǫ0 =

√2n+ 1 with n = 1, 2, . . . is de-

veloped and the lowest LL peak (n = 0) is enhanced.This behavior can be foreseen from the expression forthe full DOS difference Eq. (4.30) [or Eq. (4.29)] dis-cussed in Sec. IVB. The case with the Abrikosov vortexis further explored in Fig. 3 b, where we plot the energydependence of the LDOS for r = 0.5l [the solid (blue)curve] and r = 4l [the dashed (red) curve]. Comparingthe results for r/l = 0.5, 1., 4. we find that as the dis-tance r decreases, the peaks at En/ǫ0 =

√2n+ 1 with

n = 1, 2, 3 . . . related to the vortex are getting stronger.When r further decreases the peaks related to LLs growfaster than the vortex-like peaks. This behavior indeedallows to attribute the corresponding energy levels to thevortex. On the other hand, the peaks En/ǫ0 =

√2n with

n = 1, 2, . . . corresponding to the usual LLs (1.1) are get-ting weaker as the distance r decreases. We remind thateven for an arbitrary vortex flux η the latter levels willnot change the positions, while the levels related to thevortex will shift their energies. Fig. 3 also illustrates aspecial character of the lowest LL which is present evenin an inhomogeneous magnetic field (see also recent sim-ulations in Ref. 35), and, therefore, is getting stronger asr decreases.In Fig. 4 we consider the energy dependence of the

LDOS ND(A,B)1/2 (r, E,B) when there is a gap ∆ = ǫ0

in the spectrum. The distance form the vortex centeris r = l. The gap introduces asymmetry between theLDOS on A and B sublattices and also makes the LDOSasymmetric with respect to E = 0, so we have to plotboth negative and positive energy regions. Indeed we

-3 -2 -1 0 1 2 3EΕ0

2

4

6

8N12

HA,BLDHr,E,BLN0

D

N12HBLD

N12HALD

FIG. 4: (Color online) The normalized full LDOS

ND(A,B)1/2 (r, E,B)/ND

0 (ǫ0) as a function of energy E in the

units of the relativistic Landau scale ǫ0 for r = l. The gap∆ = ǫ0, the width Γ = 0.05ǫ0, W/ǫ0 = 3.35.

observe that the zero LL peak at E = ∆ is present only

in ND(A)1/2 (r, E,B), while the peak at E = −∆ shows up

in only in ND(B)1/2 (r, E,B). The vortex-like levels also be-

come asymmetric with respect to E = 0. All this illus-trates that the STS on graphene on a substrate which caninduce inequivalence of sublattices in graphene should re-veal these features.

From Eq. (4.46) we expect that the presence of thebackgroundmagnetic field makes r−1 divergence at r → 0of the LDOS strongly energy-dependent: it is empha-sized by the poles of the first Γ-function, when the en-

ergy E is close to the energies of the usual LLs, E(±)n =

±√

∆2 + 2ǫ20n with n = 0, 1, 2, . . . , and on the oppo-site, because the second Γ-function is in the denominator,when E is equal to the energies of the vortex-like levels,

E(±)n = ±

√

∆2 + 2ǫ20(n+ η) with n = 1, 2, . . . , the diver-gence is suppressed. This is exactly what we observe inFig. 5, where we show the dependence of the LDOS on thedistance r for fixed values of the energy (∆ = 0). Indeed,the solid (blue) and dash-dotted (black) curves which cor-

respond to energies E/ǫ0 =√2, 2 of the usual LLs have

divergent behavior at the origin. Obviously, this diver-gence is also present for E = 0 and the corresponding willabove the higher energy curves E/ǫ0 =

√2 and E/ǫ0 = 2.

On the other hand, the dashed (red) and dotted (violet)

curves which correspond to the energies E/ǫ0 =√3,√5

of the vortex-like levels tend to go to a constant valueat r = 0. Strictly speaking r−1 divergence is suppressedonly when the function Γ−1(η + z/2) in Eq. (4.46) has azero, but for a small values of the level width Γ the diver-gence seems to be completely suppressed for the chosenvalues of the energy. For r ≫ l the behavior of the LDOSresembles the nonrelativistic case. Since in this limit theLDOS difference ∆NS

η (r, E,B) ∼ e−r2/2l2 , the large r be-havior of the full DOS is determined by the contributionof the position independent LDOS (4.27). Thus large r

12

0 1 2 3 4rl

2

4

6

8

10

12N12

DHr,E,BLN0

D

EΕ0= 5

EΕ0=2

EΕ0= 3

EΕ0= 2

FIG. 5: (Color online) The the normalized full LDOSND

1/2(r, E,B)/ND0 (ǫ0) as a function of distance r measured in

the units of the magnetic length l from the vortex for four val-ues of E/ǫ0 =

√

2, 2 (usual LLs) and E/ǫ0 =√

3,√

5 (vortex-like levels). The width Γ = 0.05ǫ0, W/ǫ0 = 3.35 and ∆ = 0.

limit of all curves in Fig. 5 is determined by the corre-sponding value of the LDOS in the dashed (red) curve inFig. 3 a.

V. CONCLUSIONS

The main motivation of this work was to addressa question whether one can distinguish graphene from2DEG measuring the LDOS near the Abrikosov vortexpenetrating them. In the first publication5 we investi-gated the simplest formulation of the problem with a sin-gle vortex. In the 2DEG the solutions of the Schrodingerequation in the presence of the Aharonov-Bohm field areregular and the LDOS near the vortex is depleted. Onthe other hand, the specific feature of the Dirac fermionsin the field of the Aharonov-Bohm flux such as the pres-ence of the divergent at the origin as r−η the m = 0solution of the Dirac equation results in the r−2η diver-gence of the LDOS in the vicinity of the vortex. There-fore, the LDOS enhancement near the vortex can reallydistinguish graphene from 2DEG.This positive answer obtained in the previous paper5

is now extended for the case of a more complicated mag-netic field configuration consisting of the the Aharonov-Bohm flux and a constant background field as one cansee just from a comparison of the Figs. 2 and 5. It turnsout that the character of the divergence in the Dirac caseremains the same, but it is now strongly modulated bythe energy dependent factor. The divergence is presentwhen the energy is equal to the energies of the usual LLs(1.1), including the lowest zero energy LL.Our main results can be summarized as follows.(1) We obtained analytic expression for the LDOS per-

turbation by the Aharonov-Bohm flux in the presence ofa constant background magnetic field in the nonrelativis-tic, Eqs. (3.16), (3.18) and relativistic Eqs. (4.42), (4.43)

cases. The nonrelativistic answer is written in terms ofthe function (3.18) which is expressed as a combinationof the Whittaker functions in Eq. (D18). The relativisticanswer (4.43) is expressed in terms of the same function(3.18) and a function (4.41) that describes the contribu-tion of the m = 0 solutions of the Dirac equation.(2) We show that in the vicinity of the vortex (r .

0.2r) the relativistic LDOS is governed by the function(4.41, so that in the limit r → 0 the LDOS is given byEq. (4.46).(3) We obtained compact analytic expressions for the

DOS perturbation by the Aharonov-Bohm flux. For thenonrelativistic this is Eq. (3.11), which in the clean limitreduces to the known27 result given by Eq. (3.12). Forthe relativistic case the corresponding expressions for theDOS are (4.29) and (4.30), respectively.We hope that the obtained results will be useful both

for the experimental STS studies of graphene and and forthe theoretical studies of the interaction effects in an in-homogeneous magnetic field similar to the recent work.35

Among possible extensions of the considered problem wemention the necessity to take into account a finite size ofthe vortex core, but this certainly demands more numer-ical work, while in the present paper the main goal wasto obtain some analytic results.

VI. ACKNOWLEDGMENTS

We thank V.P. Gusynin for stimulating discussions.This work was supported by the SCOPES grant No.IZ73Z0_128026 of Swiss NSF, by the grant SIMTECHNo. 246937 of the European FP7 program and by theProgram of Fundamental Research of the Physics andAstronomy Division of the National Academy of Sci-ences of Ukraine. A.O.S. and S.G.S. were also supportedby SFFR-RFBR grant ”Application of string theory andfield theory methods to nonlinear phenomena in low di-mensional systems”.

Appendix A: The calculation of the LDOS in

nonrelativistic case

Setting η = 0 in Eq. (3.2) one obtains the solution ofthe Schrodinger equation for B = const without vortex.Substituting this solution in the LDOS defintion (3.5)and taking into account the widening of the LLs (3.6) werepresent the LDOS as a double sum

NS0 (r, E,B) =

1

πIm

∞∑

n=0

∞∑

m=−∞

A2n,my

|m|e−y[L|m|n (y)]2

× 1

En,m + E0z,

(A1)

where in the second line we introduced the dimension-less variable z = −(E + iΓ)/E0 with the characteristic

13

energy E0 defined below Eq. (3.4). To calculate the sumin Eq. (A1) it is convenient to represent its last factor asan exponent

e−δ(2n+|m|+m+1)

En,m + E0z

=1

E0

∫ ∞

0

dβe−(β+δ)(2n+|m|+m+1)e−βz.

(A2)

Here we also introduced the regularizing exponential fac-tor with δ > 0 which makes the sum convergent andwhich will be set to 0 at the end. Then the LDOS ac-quires the form

NS0 (r, E,B) =

M

π2~2Im

[

∫ ∞

0

dβe−(δ+β)e−βz∞∑

m=−∞

y|m|e−ye−(β+δ)(|m|+m)∞∑

n=0

n!e−2(β+δ)n

Γ(n+ |m|+ 1)[L|m|

n (y)]2

]

. (A3)

We operate with the representation (A3) in the followingway. First we consider its analytic continuation for z > 0and perform the calculation. Then to obtain the LDOS

we return to the imaginary values z → −(E + iΓ)/E0

and evaluate the imaginary part. Using Eq. (10.12.20)from30

∞∑

n=0

n!

Γ(n+ α+ 1)Lαn(x)L

αn(y)z

n = (1− z)−1 exp(−z x+ y

1− z)(xyz)−

α

2 Iα

(

2

√xyz

1− z

)

, |z| < 1, (A4)

where Iα is modified Bessel function, we find the sumover n in Eq. (A3)

NS0 (r, E,B) =

M

π2~2Im

[

∫ ∞

0

dβe−(δ+β)e−βz e−y coth(δ+β)

1− e−2(δ+β)

∞∑

m=−∞

e−(δ+β)mI|m|

(

y

sinh(δ + β)

)

]

. (A5)

The remaining summation over m in Eq. (A5) can bedone using the property of the modified Bessel functionIm(x) = I−m(x), and that its generating function is30

∞∑

m=−∞

zmIm(x) = exp(x

2[z + 1/z]

)

. (A6)

We obtain

NS0 (E,B) =

M

(π~)2Im

[∫ ∞

0

dβe−(δ+β)e−βz

1− e−2(δ+β)

]

.(A7)

Notice that from the last expression one can explicitlyobserve that it does not depend on y, i.e. in a constantmagnetic field the LDOS is position independent. Intro-ducing a new variable x = 2(δ + β) we can rewrite the

last expression as follows

NS0 (E,B) = − M

2(π~)2Im

[

eδz∫ ∞

2δ

dxe−x − e−x(z+1)/2

1− e−x

−eδz∫ ∞

2δ

dxe−x

1− e−x

]

.

(A8)

In the limit δ → 0 the second term of Eq. (A8) remainsreal irrespectively the value of z, while the first term givesthe integral representation of the digamma function36

ψ(z) = −γ +∫ ∞

0

dte−t − e−tz

1− e−t, Rez > 0, (A9)

where γ is the Euler-Mascheroni constant. Thus we ob-tain

NS0 (E,B) = − M

2(π~)2Im

[

ψ

(

z + 1

2

)]

, (A10)

14

so that the final expression for the LDOS after the ana-lytic continuation z → −2(E + iΓ)/(~ωc) takes the formof Eq. (3.7). The oscillatory behavior of the LDOS canbe explicitly extracted from Eq. (A10) [or Eq. (3.7)] usingthe relationship

ψ(−z) = ψ(z) +1

z+ π cot(πz). (A11)

Now we generalize these results for the case when thevortex is present. Repeating the steps that led us fromEq. (A1) to Eq. (A5) we obtain

NS0 (r, E,B) =

M

π2~2Im

[

∫ ∞

0

dβe−(δ+β)e−βz e−y coth(δ+β)

1− e−2(δ+β)

∞∑

m=−∞

e−(δ+β)(m+η)I|m+η|

(

y

sinh(δ + β)

)

]

. (A12)

The sum over m in Eq. (A12) is calculated in Ap-pendix B. Using Eq. (B7) we obtain

∞∑

m=−∞

e−(δ+β)(m+η)I|m+η|

(

y

sinh(δ + β)

)

=

ey coth(δ+β) − sinπη

π

×∫ ∞

−∞

dωe−y coshω/ sinh(δ+β) e−η(δ+β+ω)

1 + e−(δ+β+ω).

(A13)

The first term on the RHS of the last equation corre-sponds to the LDOS without the vortex which was con-sidered above, so that we can concentrate on the secondterm. Substituting it in Eq. (A12) we arrive at Eq. (3.9)for ∆NS

η (r, E,B).

Appendix B: The calculation of the sum over the

azimuthal quantum number

The sum over the azimuthal quantum number

Σ(η) =

∞∑

m=−∞

e−β(m+η)I|m+η|(x). (B1)

can be found using the method described in Ref. 37.Using the integral representation of the modified Besselfunction38

Iν(z) =1

2πi

∫

C

ez coshω−νωdω (B2)

where C is a complex path beginning at −iπ + ∞ andending at iπ +∞, we obtain

Σ(η) =1

2πi

∫

C

dωex coshω

×[

∞∑

m=0

e−(β+ω)(m+η) +

∞∑

m=1

e−(ω−β)(m−η)

]

.

(B3)

Choosing the contour C to lie in such a way that thecondition Reω > β is satisfied, the series can be madeconvergent, so that

Σ(η) =1

2πi

∫

C

dωex coshω

[

e−(β+ω)η

1− e−(β+ω)+

e(ω−β)η

e(ω−β) − 1

]

.

(B4)Now changing the variable ω → −ω in the second integralwe can write

Σ(η) =1

2πi

[∫

C

+

∫

C′

]

dωex coshω e−(β+ω)η

1− e−(β+ω), (B5)

where C′ is the contour symmetric to the contour C withrespect to the origin of coordinates. Joining the contoursC and C′ leads to an ω integral of the form

∫

C

dω +

∫

C′

dω =

∫ ∞+iπ

−∞+iπ

dω +

∫ −∞−iπ

∞−iπ

dω +

∮

C′′

dω,

(B6)where C′′ is a rectangle with the length larger than 2βand the width 2πi, centered in the origin, traversed anti-clockwise. Inside the contour C′′ the integrand has onlyone pole at ω0 = −β, so that this integral does not de-pend on η and corresponds to Σ(0). Therefore, we arriveat the final representation for the sum (B1)

Σ(η) = − sinπη

π

∫ ∞

−∞

dωe−x coshω e−(β+ω)η

1 + e−(β+ω)+Σ(0),

(B7)where

Σ(0) = ex cosh β. (B8)

Finally note that one can reproduce the value Σ(0) fromEq. (B8) using Eq. (A6) which for z = e−β reduces tothe sum (B1) with η = 0.

15

Appendix C: The calculation of the density of states

in nonrelativistic case

Substituting Eq. (3.9) in the definition (3.10) and in-tegrating over the spatial coordinates we obtain

∆NSη (E,B) = − sinπη

Ml2

2(π~)2Im

[∫ ∞

0

dβ

∫ ∞

−∞

dυ

e−βz

cosh(υ/2) cosh(β + δ − υ/2)

e−ηυ

1 + e−υ

]

,

(C1)

where we introduced the new variable υ = ω+β+δ. Thisdouble integral can be rewritten using the new variablest = e−2β, x = eυ as follows

∆NSη (E,B) = − sinπη

Ml2e−δ

(π~)2

× Im

[∫ 1

0

dtt(z−1)/2

∫ ∞

0

dxx1−η

(1 + x)2(1 + te−2δx)

]

,

(C2)

where the second integral can be calculated using theresidue theory

∫ ∞

0

dxx1−η

(1 + x)2(1 + te−2δx)

=π

sinπη

1− η + ηe−2δt− e−2ηδtη

(1− e−2δt)2.

(C3)

Then the remaining integral is expressed via the hyper-geometric function

∫ 1

0

dtt(z−1)/2 1− η + ηe−2δt− e−2ηδtη

(1 − e−2δt)2=

1− e−2δη

1− e−2δ

− (z + 2η − 1)

[

1

1 + z2F1

(

1,1 + z

2;3 + z

2; e−2δ

)

− e−2δη

1 + z + 2η2F1

(

1,1 + z

2+ η;

3 + z

2+ η; e−2δ

)]

.

(C4)

Now we use the series representation of hypergeometricfunctions in Eq. (C4)

e−2δη

z + 2η + 12F1

(

1,1 + z

2+ η,

3 + z

2+ η, e−2δ

)

=

=

∞∑

n=0

e−2δ(n+η)

z + 1 + 2η + 2n

= eδ(z+1)∞∑

n=0

∫ ∞

δ

dxe−x(2n+2η+z+1)

= eδ(z+1)

∫ ∞

δ

dxe−x(z+1+2η)

1− e−2x,

(C5)

where the first one in Eq. (C4) is recovered for η = 0. Weobserve that the presence of finite δ > 0 makes the hyper-geometric series well defined, but at the end of the calcu-lation the limit δ → 0 can already be taken. Then taking

into account the integral representation of the digammafunction (A9) [similarly to Eq. (A8)] one can express theDOS (C2) in the following simple form

∆NSη (E,B) =

Ml2

2π~2Im

(z + 2η − 1)

×[

ψ

(

z + 1

2+ η

)

− ψ

(

z + 1

2

)] (C6)

which after the analytic continuation z → −2(E +iΓ)/(~ωc) takes the final form (3.11).

Appendix D: Calculation of the function I(y, z, η)

As in Ref. 5 we observe that it is simpler to calculateintegrals with the derivative dI(y, z, η)/dy representingthe function I(y, z, η) in the form

I(y, z, η) = −∫ ∞

y

dI(Q, z, η)

dQ, (D1)

where we used that I(∞, z, η) = 0. The derivativedI(Q, z, η)/dQ contains two terms

dI(Q, z, η)

dQ=dI1(Q, z, η)

dQ+dI2(Q, z, η)

dQ, (D2)

where

dI1dQ

= −1

2

∫ ∞

0

dβe−β(z+η) e−Q coth β

sinh2 β

×∫ ∞

−∞

dωe−Q coshω/ sinh βe−(η−1)ω,

dI2dQ

= −1

2

∫ ∞

0

dβe−β(z+η−1) e−Q coth β

sinh2 β

×∫ ∞

−∞

dωe−Q coshω/ sinh βe−ηω.

(D3)

Using the integral representation of the MacDonald func-tion Kν(x) (Ref. 30)

Kν(x) =1

2

∫ ∞

−∞

e−x coshω−νωdω, (D4)

we obtain

dI1dQ

= −∫ ∞

0

dβe−β(z+η) e−Q cothβ

sinh2 βK1−η(Q/ sinhβ),

(D5)and

dI2dQ

= −∫ ∞

0

dβe−β(z+η−1) e−Q cothβ

sinh2 βKη(Q/ sinhβ).

(D6)Now introducing a new variable t via e−2β = t/(1+ t) weget

dI1dQ

= −2e−Q

∫ ∞

0

dtt(z+η)/2(1 + t)−(z+η)/2e−2Qt

×K1−η(2Q√

t(1 + t)),

(D7)

16

and

dI2dQ

= −2e−Q

∫ ∞

0

dtt(z+η−1)/2(1 + t)−(z+η−1)/2e−2Qt

×Kη(2Q√

t(1 + t)).

(D8)

To integrate over t in Eqs. (D7) and (D8), we use theintegral (2.16.10.2) from Ref. 31

∫ ∞

0

dxxρ−1

(x + z)ρe−pxKν(c

√

x2 + xz) =

1

czΓ(

ρ+ν

2

)

Γ(

ρ− ν

2

)

epz/2

×W1/2−ρ,ν/2 (z+/2)W1/2−ρ,ν/2 (z−/2) ,

z± = z(p±√

p2 − c2),

Re(p+ c) > 0, |arg z| < π, 2Reρ > |Reν|,

(D9)

where Wλ,µ(z) is the Whittaker function. To adaptEq. (D9) to the form of Eqs. (D7) and (D8), we haveto set z = 1, differentiate the result over p and then takethe limit p→ c. This gives

∫ ∞

0

dxxρ

(x+ 1)ρe−cxKν(c

√

x(x+ 1)) =

− 1

2Γ(

ρ+ν

2

)

Γ(

ρ− ν

2

)

ec/2G1/2−ρ,ν/2

( c

2

)

,

(D10)

where the function Gλ,µ(Q) is defined as follows

Gλ,µ(Q) =1

2QW 2

λ,µ(Q) +1

QWλ,µ(Q)W ′

λ,µ(Q)

+W ′′λ,µ(Q)Wλ,µ(Q)−W ′2

λ,µ(Q).

(D11)

Accordingly, we obtain that

dI1dQ

= Γ

(

z + 1

2

)

Γ

(

z + 2η − 1

2

)

G(1−z−η)/2,(1−η)/2(Q)

(D12)

and

dI2dQ

= Γ

(

z − 1

2

)

Γ

(

z + 2η − 1

2

)

G(2−z−η)/2,η/2(Q).

(D13)

Now using the differential equation

W ′′λ,µ(z) +

(

−1

4+λ

z+

1/4− µ2

z2

)

Wλ,µ(z) = 0 (D14)

and the recursion formula

zd

dzWλ,µ(z) =

(

λ− z

2

)

Wλ,µ(z)

−[

µ2 −(

λ− 1

2

)2]

Wλ−1,µ(z)(D15)

for the Whittaker function38 one can transform Gλ,µ(Q)to the form

Gλ,µ(Q) =µ2 + (λ− 1/2)2

Q2W 2

λ,µ(Q)

− [µ2 − (λ − 1/2)2]2

Q2W 2

λ−1,µ(Q)

− µ2 − (λ − 1/2)2

QWλ,µ(Q)Wλ−1,µ(Q)

−2λ− 1

2QW 2

λ,µ(Q)− (λ− 1/2)

(

W 2λ,µ(Q)

Q

)′

.

(D16)

To obtain the function Fλ,µ(y) = −∫∞

ydQGλ,µ(Q) we

employ the relationships

∫

dQ

QWλ,µ(Q)Wρ,µ(Q) =

1

ρ− λ[W ′

λ,µ(Q)Wρ,µ(Q)−W ′ρ,µ(Q)Wλ,µ(Q)],

∫

dQ

QWλ,µ(Q)Wλ,µ(Q) =

W ′λ,µ(Q)∂λWλ,µ(Q)− ∂λW

′λ,µ(Q)Wλ,µ(Q),

∫

dQ

Q2Wλ,ν(Q)Wλ,ν(Q) =

1

2ν(∂νW

′λ,ν(Q)Wλ,ν(Q)−W ′

λ,ν(Q)∂νWλ,ν(Q))

(D17)

which follow from the differential equation (D14) for theWhittaker function. Then using the the recursion for-mula (D15) we arrive at the following result

Fλ,µ(y) =µ2 + (λ− 1/2)2

2µy[Wλ+1,µ(y)∂µWλ,µ(y)−Wλ,µ(y)∂µWλ+1,µ(y)]

− [µ2 − (λ− 1/2)2]2

2µy[Wλ,µ(y)∂µWλ−1,µ(y)−Wλ−1,µ(y)∂µWλ,µ(y)]

+µ2 − (λ− 1/2)2

y[W 2

λ,µ(y)−Wλ−1,µ(y)Wλ,µ(y)−Wλ−1,µ(y)Wλ+1,µ(y)]

−2λ− 1

2y[2W 2

λ,µ(y)−Wλ+1,µ(y)∂λWλ,µ(y) +Wλ,µ(y)∂λWλ+1,µ(y)].

(D18)

17

The integral of each term in Eq. (D2) is expressed viavia Fλ,µ(y) with the prefactors given by Eqs. (D12) and(D13), so that we arrive at the final expression (3.18) forthe function I(y, z, η) which was defined in Eq. (D1).To complete our analysis we consider the asymptotic of

the ImI(y, z → −(E+ iΓ)/E0, η) in the limits y → 0 andy → ∞. To do this we use the following representationsof the Whittaker function:

Wλ,µ(y) ≈ y1/2−µ Γ(2µ)

Γ(1/2 + µ− λ)+O(y3/2−µ)

+ y1/2+µ Γ(−2µ)

Γ(1/2− µ− λ)+O(y3/2+µ), y → 0,

(D19)

and

Wλ,µ(y) ≈ e−y/2yλ [1 +O(1/y)] , y → ∞. (D20)

Substituting Eq. (D19) in Eq. (3.18) and omitting all realterms which will not contribute to ImI, we obtain

ImI(y → 0, z, η)

≈ − π

sinπηIm

[

ψ

(

z − 3

2

)

+4(z − 2)

(z − 1)(z − 3)

]

.(D21)

Now using the property of the digamma function

ψ(z) = ψ(z + 1)− 1

z(D22)

we arrive at the result

ImI(y = 0, z, η) = − π

sinπηImψ

(

1 + z

2

)

. (D23)

One can reproduce the same result directly fromEq. (3.17). Indeed, setting y = 0 in Eq. (3.17) we have

I(y = 0, z, η) =

∫ ∞

0

dβe−βz

sinhβ

∫ ∞

−∞

dωe−ηω

1 + e−ω. (D24)

Now the integral over ω is elementary and after replacing2β → β we obtain

I(y = 0, z, η) =π

sinπη

∫ ∞

0

dβe−β(z+1)/2

1− e−β. (D25)

Recognizing in this integral the imaginary part of thedigamma function (A9) we again arrive at Eq. (D23).The next order corrections to ImI(y → 0, z, η) can beobtained by expanding the function dI/dQ ar Q = 0, in-tegrating the result over Q and using the y = 0 result(D23). For y → 0 the expansion contains terms ∼ y1−η

and yη with the prefactors that make the resulting ap-proximate expression for the LDOS divergent at η = 0, 1.Substituting the asymptotic (D20) in Eq. (3.18) we

obtain that for y → ∞

Fλ,µ(y) ∼ e−yy2λ−3

(

1

4+ λ(λ− 1)− µ2

)

. (D26)

Appendix E: Solution of the Dirac equation

A positive energy solution of the time-dependent Diracequation has a form Ψ(t, r) = exp(−iEt/~)Ψ(r), wherethe components of a two-component spinor

Ψ(r, ζ) =

[

ψ1(r, ζ)iψ2(r, ζ)

]

(E1)

satisfy (r, ϕ) the following equations [cp. with Ap-pendix A of Ref. 5]

(E −∆)ψ1(r, ζ)

− ~vF e−iζϕ

(

∂

∂r− iζ

r

∂

∂ϕ+eζAϕ

~c

)

ψ2(r, ζ) = 0,

~vF eiζϕ

(

∂

∂r+iζ

r

∂

∂ϕ− eζAϕ

~c

)

ψ1(r, ζ)

+ (E +∆)ψ2(r, ζ) = 0.

(E2)

The vector potential Aϕ(r) in Eq. (E2) is given byEq. (2.4). From now on, we consider the specific caseζ = 1 (omitting the label ζ in the wave functions) andseek for a solution of Eq. (E2) in the following form

ψ1(r) = ei(m−1)ϕψ1(r), ψ2(r) = eimϕψ2(r). (E3)

Then the radial components of the spinor ψ1(r) and ψ2(r)satisfy the following system of equations

ψ1(r) =~vFE −∆

(

d

dr+m+ ηθ(r −R)

r+

r

2l2

)

ψ2(r),

ψ2(r) = − ~vFE +∆

(

d

dr− m+ ηθ(r −R)− 1

r− r

2l2

)

ψ1(r).

(E4)

Introducing the dimensionless variable y = r2/(2l2) anddenoting the ρ ≡ R2/(2l2) we rewrite the system (E4)for y ∈ [0, ρ]

ψ1(y) =~vF

√2

(E −∆)l

√y

(

d

dy+m

2y+

1

2

)

ψ2(y),

ψ2(y) = − ~vF√2

(E +∆)l

√y

(

d

dy− m− 1

y− 1

2

)

ψ1(y).

(E5)

Since there is no Aharonov-Bohm field for y < ρ the prob-lem in this domain is identical to that of the Appendix Din Ref. 39. For y ∈ [ρ,∞[ the system (E4) acquires theform

ψ1(y) =~vF

√2

(E −∆)l

√y

(

d

dy+m+ η

2y+

1

2

)

ψ2(y), (E6a)

ψ2(y) = − ~vF√2

(E +∆)l

√y

(

d

dy− m+ η − 1

2y− 1

2

)

ψ1(y).

(E6b)

18

The matching conditions (4.3) and (4.4) take the form

ψ1(ρ− 0) = ψ1(ρ+ 0),

ψ′1(ρ+ 0)− ψ′

1(ρ− 0) =η

2ρψ1(ρ),

(E7)

and

ψ2(ρ− 0) = ψ2(ρ+ 0),

ψ′2(ρ+ 0)− ψ′

2(ρ− 0) = − η

2ρψ2(ρ),

(E8)

where the derivative is taken over y. One can obtain fromthe system (E5) that for y ∈ [0, ρ] the spinor componentssatisfy the following second order differential equations:

d2

dy2+

1

y

d

dy− 1

4− (m− 1)2

4y2+λ−m

2y

ψ1(y) = 0,

(E9a)

d2

dy2+

1

y

d

dy− 1

4− m2

4y2+λ−m+ 1

2y

ψ2(y) = 0,

(E9b)

where we introduced λ = (E2 −∆2)l2/(~vF )2. The sec-

ond order differential equations for the domain y ∈ [ρ,∞[corresponding to the system (E6) can be obtained fromEq. (E9) by replacing m→ m+ η:

d2

dy2+

1

y

d

dy− 1

4− (m+ η − 1)2

4y2+λ−m− η

2y

ψ1(y) = 0,

(E10a)

d2

dy2+

1

y

d

dy− 1

4− (m+ η)2

4y2+λ−m− η + 1

2y

ψ2(y) = 0.

(E10b)

The equations can be reduced to the equations for thedegenerate hypergeometric function (see Eq. (6.3.1) ofRef. 36) and the solutions of Eqs. (E9a) and (E10a) aregiven by, respectively, by

ψ1(y) =Cmy|m−1|/2e−y/2Φ

( |m− 1|+m+ 1− λ

2, 1 + |m− 1|; y

)

, r < R (E11a)

ψ1(y) =Amy|m+η−1|/2e−y/2Φ

(

a+ − λ

2, 1 + |m+ η − 1|; y

)

+Bmy−|m+η−1|/2e−y/2Φ

(

a− − λ

2, 1− |m+ η − 1|; y

)

, r > R, (E11b)

where a± ≡ m + η + 1 ± |m + η − 1|, Am, Bm, and Cm

are constants, and Φ(a, c; z) is the confluent hypergeo-metric function. The solution (E11a) contains only oneterm due to the condition of square integrability and theabsence of the Aharonov-Bohm field for r < R. Writingthe solution (E11b) we used that for noninteger c the so-lution of Eq. (E10a) can be expressed via Φ(a, c; z) andz1−cΦ(a− c+ 1, 2− c; z). The coefficients Am, Bm, andCm to be found from the matching conditions (E7). Theconsideration of the limit R → 0 (ρ → 0) greatly simpli-fies the calculation, because one can expand the solutionsto the linear in ρ terms. Then one finds that

ψ1(y) = Amy|m+η−1|/2e−y/2

× Φ

( |m+ η − 1|+m+ η + 1− λ

2, 1 + |m+ η − 1|; y

)

.

(E12)

Since Φ(a, c; z) behaves as ey at large y unless a = −nwith n = 0, 1, 2, . . ., in order to have the square inte-grable solutions, the value λ should be equal to the eigen-value λm,n defined by Eq. (4.8). In this case Φ-function

is reduced to the generalized Laguerre polynomials (seeEq. (6.9.2.36) of Ref. 36)

Lαn(y) =

Γ(α+ n+ 1)

Γ(α + 1)n!Φ(−n, α+ 1, y). (E13)

Introducing the functions

Jnν (x) =

(

Γ(n+ 1)

Γ(n+ ν + 1)

)1/2

e−x/2xν/2Lνn(x), (E14)

one can rewrite the solution (E12) in a a more compactform

ψ1(y) = AmJn|m+η−1|(y). (E15)

The definition (E14) generalizes functions considered inRef. 40 for the case of the noninteger ν > −1. Thesefunctions satisfy the following orthogonality condition

∞∫

0

dxJnν (x)J

n′

ν (x) = δnn′ . (E16)

19

Then having ψ1(y) one can find ψ2(y) from Eq. (E6b)using the recursion formulae40

(x + ν)Jnν (x) =

[x(n+ ν)]1

2Jnν−1(x) + [x(n+ ν + 1)]

1

2Jnν+1(x),

2x(d/dx)Jnν (x) =

[x(n+ ν)]1

2Jnν−1(x)− [x(n+ ν + 1)]

1

2Jnν+1(x).

(E17)

Then demanding that the spinors obey the normalizationcondition

∫ 2π

0

dϕ

∫ ∞

0

rdrΨ†n′m′(r, ζ)Ψnm(r, ζ) = δn,n′δm,m′

(E18)we obtain the solutions (4.5), (4.6), and (4.7) for n > 0.The zero mode solutions have to be considered separately.

Analyzing the initial system (E6) one finds that the onlyallowed solution is the negative energy one E = −∆,m ≤ 0 with ψ1(r) = 0. The corresponding spinor is givenby Eq. (4.9). One can verify that for η = 0 these solu-tions transform up to the phase factors to the solutionsobtained in Ref. 39. To show this one should relabel thequantum numbers n+m → n for m ≥ 1 and n+ 1 → nfor m ≤ 0 and use the property Jn

ν (y) = (−1)νJn+ν−ν (y)

which is valid only when Jnν (y) is defined for the integer

values of ν as done in Ref. 40. After this relabeling ismade, the spectrum (4.8) acquires a conventional formdependent only on the LL index which for ∆ = 0 reducesto Eq. (1.1).

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