Weak-localization approach to a 2D electron gas with a spectral node
aa r X i v : . [ c ond - m a t . d i s - nn ] N ov Weak-localization approach to a 2D electron gas with a spectral node
K. Ziegler and A. Sinner
Institut f¨ur Physik, Universit¨at AugsburgD-86135 Augsburg, Germany (Dated: March 25, 2018)We study a weakly disordered 2D electron gas with two bands and a spectral node within theweak-localization approach and compare its results with those of Gaussian fluctuations around theself-consistent Born approximation. The appearance of diffusive modes depends on the type of disor-der. In particular, we find for a random gap a diffusive mode only from ladder contributions, whereasfor a random scalar potential the diffusive mode is created by ladder and by maximally crossed contri-butions. The ladder (maximally crossed) contributions correspond to fermionic (bosonic) Gaussianfluctuations. We calculate the conductivity corrections from the density–density Kubo formula andfind a good agreement with the experimentally observed V-shape conductivity of graphene.
PACS numbers: 05.60.Gg, 66.30.Fq, 05.40.-a
I. INTRODUCTION
The weak-localization approach (WLA) has been a very popular tool to estimate whether electronicstates in a weakly disordered system tend to localize or to delocalize on large scales. Moreover, it enablesus to calculate the magnetoresistance in the presence of a weak magnetic field and weak scattering. Acentral result of the WLA is that on large scales there might be diffusion due to one or more masslessmodes. This has been studied in great detail for conventional metals [1–3] and more recently for graphene[4, 5] and for the surface of topological insulators [6, 7], using a one-band projection for the two-bandsystem. T he existence of a diffusive mode, which is a necessary (but not a sufficient) condition formetallic behavior, has been debated for the one-band projected graphene model. It was found that eithera single diffusive channel exists [4, 6, 7] or no diffusion [5] in the presence of generic disorder. On the otherhand, the weak-scattering approach (WSA), where transport properties are studied within the expansionin powers of η/E b ( η is the scattering rate and E b is the bandwidth) [8], a non-Abelian chiral symmetrywas identified, describing diffusion in two-band systems due to spontaneous symmetry breaking [9]. Thisis also the origin of a massless fermion mode found for 2D Dirac fermions with a random gap in Ref.[10]. In the WSA disorder fluctuations of the two-band model are approximated by Gaussian fluctuationsaround a saddle-point of the original model, expressed in terms of a functional integral [8]. The saddlepoint is equivalent to the self-consistent Born approximation (SCBA) of the one-particle Green’s function,while the Gaussian fluctuations are equivalent to the WLA. The latter consists of one-particle and two-particle diagrams which are partially summed up in terms of geometric series (cf. Sect. III). Within theWSA it is also possible to analyze the fluctuations with respect to the non-Abelian chiral symmetry. Theprojection onto these fluctuations generates a nonlinear field which allows us to go beyond the Gaussianapproximation within the expansion in powers of η/E b . This idea is analogous to the nonlinear sigmamodel, derived originally for one-band Hamiltonians by Sch¨afer and Wegner [11]. The difference betweenthe one-band and the two-band Hamiltonians is that the former can be formulated either by a symmetricreplica space or a supersymmetric fermion-boson [12], whereas the latter can be expressed in terms ofa non-symmetric fermion-boson theory [10]. Therefore, in the derivation of a nonlinear sigma model itis crucial to take the two-band structure into account. A projection onto a single band would destroythe relevant symmetries of the system. In more physical terms, the two-band structure is essential forsupporting diffusion in a two-dimensional system, since it allows for Klein tunneling. The latter enables aparticle in a potential barrier to transmute to a hole, for which the potential barrier is not an obstacle. Ouraim is to establish a direct connection between the WLA and the Gaussian fluctuations around the saddlepoint for 2D Dirac fermions with a random gap, and to provide a general discussion about the existenceof diffusive modes due to ladder and maximally crossed contributions in two-band systems. Finally, theseresults will be employed to calculate the conductivity corrections, and the resulting conductivities willbe compared with experimental measurements in graphene. The results can also be applied to other 2Dtwo-band systems such as the surface of topological insulators [13].The paper is organized as follows. In Sect. II we introduce a general description for the two-bandHamiltonian and various types of random scattering. The main ideas of the WLA are discussed inSect. III, which includes the self-consistent Born approximation for the average one-particles Green’sfunction, the ladder and the maximally crossed contribution of the average two-particle Green’s function.In Sect. IV we study the long-range behavior of the average two-particle Green’s function for a one-bandHamiltonian (Sect. IV A) and for the two-band Hamiltonian (Sect. IV B). These results are used tocalculate the conductivity (Sect. V). And finally, in Sect. VI we discuss the connection of the WLAwith the WSA, the robustness of the diffusion pole structure with respect to a one-band projection ofthe two-band Hamiltonian and the symmetry properties of the inter-node scattering. II. MODEL: HAMILTONIANS, GREEN’S FUNCTIONS AND SYMMETRIES
Quasiparticles in a system with two bands are described by a spinor wavefunction. The correspondingHamiltonian can be expanded in terms of Pauli matrices σ , , , . Here we will consider either a gaplessHamiltonian H = h σ + h σ (1)or a gapped Hamiltonian H m = h σ + h σ + mσ . (2)The gapless Hamiltonian changes its sign under a chiral transformation σ H σ = − H , (3)which implies the continuous Abelian chiral symmetry e ασ H e ασ = H . The situation is more subtle for H m because its transformation properties depends on the properties of h , . We distinguish here two cases, namely h Tj = − h j (Dirac fermions, T is the transposition, acting onreal space), where σ H Tm σ = − H m (4)and h Tj = h j (e.g., bilayer graphene), where σ H Tm σ = − H m . (5)The transformation properties of the Hamiltonians imply a relation between the wavefunctions in theupper and in the lower band. In particular, Eq. (3) implies that Ψ − E = σ Ψ E , Eq. (4) implies thatΨ − E = σ Ψ ∗ E and Eq. (5) that Ψ − E = σ Ψ ∗ E .Disorder is included by an additional random term diag ( v , v ) in the Hamiltonians H and H m . In thefollowing we will consider the case of v = v (scalar potential), v = − v (random gap) and independentrandom diagonal elements v , v , assuming that the matrix elements v , have mean zero and variance g . III. DYSON AND BETHE-SALPETER EQUATIONA. Dyson equation
Starting point of the WLA is that the Green’s function G ( µ − iδ ) = ( H − µ + iδ ) − of the Hamiltonian H is perturbed by V and creates the Green’s function G = ( G − + V ) − . The latter relation is equivalentto the matrix identity G = G − G V G which connects the unperturbed Green’s function G with theperturbed Green’s function G . This equation can be iterated to give a geometric series G = G − G V G = G − G V G + G V G V G = ... = G X l ≥ ( − V G ) l . (6) ✗ ✗ ✗ ✗✗ ✗ ✗ ✗ ✗ ✗ ✗ ✗✗ ✗ ✗ ✗ FIG. 1: Diagrammatic representation of the fourth order terms for the ladder contribution of ( − t ) − andmaximally crossed contributions of ( − τ ) − , respectively. Now we assume that V is a random quantity with mean h V i = 0 such that the averaged Dyson equationbecomes h G i = G + G h V G V G i . (7)When we assume that the correlation between the Green’s function and V is weak, the factorization ofthe average h V G V G i ≈ h
V G V ih G i is possible, which creates a linear equation for h G i : h G i ≈ G + G h V G V ih G i , (8)whose solution reads h G i ≈ ( G − − h V G V i ) − . This result is known as the Born approximation withthe self-energy h V G V i . The self-consistent Born approximation (SCBA) is provided by the replacement h V i G ,ij V j i → h V i V j ih G ij i =: Σ ij (9)on the right-hand side of Eq. (8): h G i ≈ ¯ G = ( G − − Σ) − . (10) B. Bethe-Salpeter equation
The two-particle Green’s function G + G − , created from the two one-particle Green’s functions G ± (e.g., the adavanced and the retarded Green’s function), reads with the help of the corresponding Dysonequations (6) (using the summation convention in this Section) h G + ij G − kl i = h T − ik,mn i G +0 ,mj G − ,nl with T ik,mn = δ im δ kn − G +0 ,im ′ V m ′ m G − ,kn ′ V n ′ n . (11)On the left-hand side is the two-particle Green’s function, while the right-hand side depends only onproducts of G . This equation is known as the Bethe-Salpeter equation. Now we can perform the averagewith respect to the random scatterers V jj ′ , assuming that this is a Gaussian variable with zero mean.Here it is convenient to define γ ± im = G ± ,im ′ V m ′ m such that T ik,mn = δ im δ kn − γ + im γ − kn . (12)The expansion of T − leads to a geometric series in γ + m m γ − n n . Averaging this series with respect to aGaussian distribution can be achieved by what is known as Wick’s theorem: the average of the productof random variable h V m V n · · ·i is expressed as a sum over all possible products of pairs h V m V n i · · · . Thisseries includes a ladder contribution and a maximally crossed contribution [14] (cf. Fig. 1) as specialcases. We also include the contribution from the iterated Dyson equation of Sect. III A in terms of theSCBA and obtain eventually h G + ij G − kl i ≈ [ h T − ik,mn i L + h T − ik,mn i M − δ im δ kn − t ik,mn ] ¯ G + mj ¯ G − nl , (13)where the last two terms on the right-hand side are introduced to avoid overcounting in the geometricseries. Beginning with the ladder contribution, we obtain h T − ik,mn i L = ( − t ) − ik,mn with t m n ,m n = h ¯ G + m m ′ V m ′ m ¯ G − n n ′ V n ′ n i . (14)Then the maximally crossed contribution is created by re-arranging the order of factors in the geometricseries which after averaging results in h T − ik,mn i M = ( − τ ) − in,mk with τ m m ′ ,m m ′ = h ¯ G + m n V n m ¯ G − m ′ n ′ V n ′ m ′ i . (15)Thus, switching from the ladder contribution to the maximally crossed contribution is achieved simplythrough changing γ − by the transposition γ − m ′ m ′ → γ − m ′ m ′ .In the following we will discuss the two-particle Green’s functions of (13)–(15) and the SCBA for thetwo-band Hamiltonians of Sect. II with an additional random term diag ( v , v ), which is either a randomscalar potential, a random gap or independent random diagonal elements. For simplicity it is assumedthat the random terms are spatially uncorrelated. IV. SPECIAL CASES OF THE TWO-PARTICLE GREEN’S FUNCTION
The general expressions in (14) and in (15) shall now be applied to specific cases. For a diagonalrandom matrix V mm ′ = V m δ m,m ′ we obtain t m n ,m n = ¯ G + m m ¯ G − n n h V m V n i , τ m n ,m n = ¯ G + m m ¯ G − n n h V m V n i . (16) A. Scalar Green’s function
Before we start to discuss the two-band Hamiltonians, the simpler one-band Hamiltonian with Fouriercomponents h ( k ) is used to explain briefly the main ideas of the WLA. For this case we consider thefollowing coordinates in real space: i = k = r , m = n = r ′ . Then the ladder and the maximally crossedcontributions read for uncorrelated disorder h V r V r ′ i = gδ r,r ′ t r r ,r r = g ¯ G + r r ¯ G − r r δ r ,r , τ r r ,r r = g ¯ G + r r ¯ G − r r δ r ,r , (17)and in the geometric series P l ≥ t lrr,r ′ r ′ ( P l ≥ τ lrr ′ ,r ′ r ) only t r r := t r r ,r r = g ¯ G + r r ¯ G − r r , τ r r := τ r r ,r r = g ¯ G + r r ¯ G − r r (18)contributes. Thus Eq. (13) reads h G + rr ′ G − rr ′ i ≈ X r ′′ [( − t ) − rr ′′ + ( − τ ) − rr ′′ − δ rr ′′ − t rr ′′ ] ¯ G + r ′′ r ′ ¯ G − r ′′ r ′ . (19)Moreover, with (18) the extra factor ¯ G + r ′′ r ′ ¯ G − r ′′ r ′ = 1 g t r ′′ r ′ = 1 g τ r ′′ r ′ , (20)can be replaced by t and τ , respectively, to provide h G + rr ′ G − rr ′ i ≈ g ( − t ) − rr ′ + 1 g ( − τ ) − rr ′ − g ( t rr ′ + t rr ′ + 2 δ rr ′ ) . (21)The Green’s function is symmetric in the absence of a magnetic field, such that the Fourier componentsof the Hamiltonian h ( k ) satisfy the relation h ( − k ) = h ( k ) . Then there is no difference between ladderand maximally crossed contributions. We consider q ∼ qt q ∼ g X r ¯ G + r ¯ G − r − q g X r r µ ¯ G + r ¯ G − r (22)and use the SCBA with the self-energy Σ ± : ¯ G + = G ( z ), ¯ G − = G ( z ) ∗ with z = µ − iδ + Σ + . This gives t ∼ − iδ/ ( z − z ∗ ), such that 1 / (1 − t q ) becomes the diffusion propagator11 − t q ∼ iδ/ ( z − z ∗ ) + q ( g/ P r r µ ¯ G + r ¯ G − r . (23)This result is remarkable because it implies that 1 / (1 − t q ) diverges like q − for q ∼ δ ∼
0, reflectingthe well-known massless two-particle mode for diffusion [1–3].
B. Spinor Green’s function
For the spinor Hamiltonian H m of Eq. (2) we introduce the coordinate r and the Pauli matrix index a = 1 , h V r,a V r ′ ,a ′ i = g aa ′ δ r,r ′ and with ¯ G + = G ( z ), ¯ G − = G ( z ) ∗ for z = µ − iδ − iη , where µ is the renormalized Fermi energy (i.e.,the bare Fermi energy which is shifted by the real part of the self-energy Σ + ) and η is the scattering rate(i.e., the imaginary part of the self-energy). Now we use the Fourier representation of H m and get with h = h + h for the Green’s function G ,k ( z ) = − z − m − h ( zσ + h σ + h σ + mσ ) , (24)and t q ; ab,cd = g cd Z k G k,ac G ∗ q + k,bd , τ q ; ab,cd = g cb Z k G k,ac G ∗ q + k,db . (25)For random scalar potential vσ (random gap vσ ) the prefactors read g = g = g = g ≡ g ( g = g = − g = − g ≡ g ), and for independent random diagonal elements g = g ≡ g , g = g = 0. Then we get from Eq. (25) for q = 0 the matrices t = α β sα sα β α , τ = α sα β β sα
00 0 0 α , (26)where s = − s = 1 for random scalar potential, s = 0 for independent random diagonalelements. For Dirac fermions we have h j = k j and the matrix elements are the following expressions: α = gI ( z + m )( z ∗ + m ) , α = gI ( z + m )( z ∗ − m ) , α = gI ( z − m )( z ∗ + m ) , α = gI ( z − m )( z ∗ − m ) I = Z k | z − m − k | , β = g Z k k | z − m − k | . (27)Using the SCBA we obtain the integral Z k | z | + m + k | z − m − k | = 1 g − δgη , (28)which implies β ∼ − gI ( | z | + m ). Thus, besides two independent diagonal matrix elements, we gettwo eigenvalues for the non-diagonal 2 × − t and 1 − τ , namely λ ± L = 1 − h ( α + α ) ± p ( α − α ) + 4 β i , λ ± M = 1 − h s ( α + α ) ± p s ( α − α ) + 4 β i (29)with the parameters α + α = 2 gI ( | z | + m ), α − α = 2 gmI ( z ∗ + z ) and α + α = 2 gI ( | z | − m ), α − α = 2 gmI ( z ∗ − z ). For s = 0 the eigenvalues λ ± M are always massive: λ ± M = 1 ∓ β = (cid:26) gI ( | z | + m )2 − gI ( | z | + m ) (30)and for s = ± (I) At the Dirac node µ = 0: For the Hamiltonian H m of Eq. (2) we get the parameter β ∼ − g ( η + m ) I , and the eigenvalues of Eq. (29) now read λ ± L = 1 − gI ( η + m )] ∓ [1 − gI ( η + m )] , λ ± M = 1 − s ( η − m ) gI ± p [1 − gI ( η − m ) ][1 − gI ( η + m ) ] . (31)Thus, the ladder contribution λ + L is always massless, in contrast to λ − L and the maximally crossed con-tributions, which are all massive. independent random diagonal elements random scalar potential random gap s = 0 s = 1 s = − µ = 0 λ + L = 0 λ + L = 0 λ + L = 0 m = 0 λ + L = 0 λ + L = 0, λ − M = 0 λ + L = 0 m, µ = 0 none none noneTABLE I: Vanishing eigenvalues of − t and − τ (cf. Eqs. (26), (31), (32)). (II) Gapless spectrum m = 0: For the Hamiltonian H we have β ∼ − g | z | I , and the eigenvaluesread λ ± L = 1 − g | z | I ∓ (1 − g | z | I ) , λ ± M = 1 − sg | z | I ± (1 − g | z | I ) , (32)such that there is a massless mode λ + L = 0 for any s , like for the Dirac node, and an additional masslessmode from the maximally crossed contribution λ − M = 0 for s = 1. These results are summarized in TableI. C. Diffusion propagator
The findings of the previous Section can be used to evaluate the correlation function as h G rr ′ ,ab G ∗ rr ′ ,cd i ≈ g bd ( − t ) − rr ′ ; ac,bd + 1 g bd ( − τ ) − rr ′ ; ad,bc − g bd ( t rr ′ ; ac,bd + t rr ′ ; ac,bd + 2 δ rr ′ δ ac δ bd ) . (33)For the long-range behavior with | r − r ′ | ∼ ∞ it is sufficient to consider the two-particle propagators( − t q ) − and ( − τ q ) − for q ∼
0. Then we can focus on the massless (diffusion) modes as the mostimportant contributions to get ∼ ηδ + D t,τ q → η − iω + D t,τ q , (34)where the second expression is obtained by the analytic continuation δ → − iω . The diffusion coefficients D t,τ are obtained from the q expansion of the eigenvalues of (25) as D t = gη X r r µ [ G r , G ∗ r, + G r , G ∗ r, ] + o ( g )and additionally for s = 1 D τ = gη X r r µ [ G r , G ∗ r, + G r , G ∗ r, ] + o ( g ) . Both coefficients agree, at least up to terms of o ( g ), and give us D t = D τ = g πη (cid:18) ζ ζ arctan ζ (cid:19) + o ( g ) ( ζ = µ/η ) . (35) V. CORRECTIONS TO THE BOLTZMANN-DRUDE CONDUCTIVITY
Next, the results of the WLA will be used to evaluate the quantum corrections to the Boltzmann-Drudeconductivity. The latter is usually calculated in terms of the current-current correlation function for thereal part of the conductivity [4] σ µµ ∼ π ¯ h h Tr (cid:0) j µ Gj µ G † (cid:1) i . (36)The advantage of using this expression is that the current-current correlation function h Tr( h j µ Gj µ G † i ) i is closely related to the action of the corresponding nonlinear sigma model (1 / t ) R Tr( ∂ µ Q∂ µ Q ) [2], sincethe current operator j µ of a conventional one-band model is proportional to the momentum operator − i∂ µ .Therefore, the renormalization of the parameter t is similar to the renormalization of the current-currentcorrelation function. This relation, however, breaks down for Dirac fermions, where j µ is proportional tothe Pauli matrix σ µ . For this reason it is not obvious that the renormalization of the nonlinear sigmamodel is linked to the renormalization of the current-current correlation function. In this case it is betterto use an alternative Kubo formula, which is based on the density-density correlation function [15, 16]: σ µµ = − e h ω X r r µ T r h G r G † r i . (37)This expression is closely related to diffusion and the Einstein relation [9]. The classical approximationassumes a weak correlation between the two Green’s functions G , G † such that we can factorize theexpectation value as h G r G † r i ≈ h G r ih G † r i and obtain the Boltzmann-Drude conductivity as¯ σ = − e h ω X r r µ T r h G r ih G † r i . (38)Furthermore, the average one-particles Green’s functions are evaluated within the SCBA as h G r i ≈ ¯ G r .For the gapless case m = 0 and for the parameter χ = 2 µ/ω we can write X r r µ T r h G r ih G † r i ≈ X r r µ T r ¯ G r ¯ G † r ≈ − πω h χ (1 − χ ) log (cid:16) (1+ χ ) (1 − χ ) (cid:17)i for ω ≫ η πη h ζ (1 + ζ ) arctan ζ i for ω ≪ η . (39)In particular, for ω ≫ η the Boltzmann-Drude conductivity reads¯ σ = e πh (cid:20) χ (1 − χ ) log (cid:18) (1 + χ ) (1 − χ ) (cid:19)(cid:21) . (40)This expression is obviously not the Boltzmann-Drude conductivity of a conventional metal with one-band Hamiltonian. At the Dirac node µ = 0 it has a frequency independent conductivity ¯ σ = e /hπ anddecreases monotonically from e /hπ to zero as we move the Fermi energy µ away from the Dirac node.This indicates a cross-over from the optical conductivity of the two-band model at the Dirac node to theBoltzmann-Drude behavior of a conventional metal, where the optical conductivity is much lower thanin the two-band case. Here it should be noticed that there are additional corrections for ω ≫ η , whichincrease the optical conductivity to πe /
8. These are not taken into account here, since they disappearin the DC limit [16]. For ω ≪ η , on the other hand, the Boltzmann approximation is invalid, which isreflected by a negative Boltzmann-Drude conductivity that vanishes like ¯ σ ∝ ω . For this regime wemust consider the corrections δσ which can be evaluated from Eq. (33) aslim ω → ω X r ( r µ − r ′ µ ) h G rr ′ ,ab G ∗ rr ′ ,ab i ≈ g lim ω → ω X r ( r µ − r ′ µ ) [( − t ) − rr ′ ; aa,bb + ( − τ ) − rr ′ ; ab,ba ] . (41)With the help of the propagator in Eq. (34) and the diffusion coefficient (35) the conductivity correctionsthen read δσ = e h ω ∂ ∂q l η/g − iω + Dq (cid:12)(cid:12)(cid:12) q =0 = e hg ηD = e πh (cid:18) ζ ζ arctan ζ (cid:19) + o ( g ) ∼ e πh (1 + ζ / . (42)There is an additional factor 2 for s = 1 due to the extra massless mode from λ − M = 0 in that case. Thisresult, which is depicted in Fig. 2, agrees with previous calculations based on the WSA [9] as well aswith the V-shape conductivity with respect to µ in graphene [17, 18]. PSfrag replacements ζ c o ndu c t i v i t y [ e / π h ] FIG. 2: Conductivity as a function of ζ from the expression in Eq. (42). VI. DISCUSSION
Our main result is that the ladder and the maximally crossed contributions are quite different forthe spinor Hamiltonians, in contrast to their agreement for the scalar Hamiltonian of Sect. IV A. Thissituation is reminiscent of the Bose-Fermi functional representation of the average two-particle Green’sfunction in the case of Dirac fermions with a random gap, where it was observed that the bosonic andthe fermionic propagators are distinct [8]. Similar to the expressions in Eq. (26), the inverse bosonictwo-particle propagator is also a 4 × a, ..., d = 1 , δ ac δ bd − g Z k G k,ac ( z ) G k − q,bd ( z ) , while its fermionic counterpart reads δ ac δ bd − g Z k G k,ac ( z ) G − k − q,db ( z ) . A straightforward calculation shows that only the fermionic propagator has a massless mode, very similarto − t q with s = − µ > G k,ab = U ∗ a ǫ ( k ) − z U b + U ∗ a − ǫ ( k ) − z U b → U ∗ a ǫ ( k ) − z U b with U = 1 √ (cid:18) κ ∗ − κ ∗ (cid:19) (43)and κ = ( k − ik ) /k ≡ e i Φ( k ) , which provides the expression G ≈ ǫ k − µ + iδ + iη
12 ( σ − σ k /k − σ k /k ) . (44)For the projected Green’s function the matrices (26) then read t = gI s s
01 0 0 1 , τ = gI s s
00 0 0 1 , I = Z k | k − z | , (45)where we get gI = 2 − δ/η from the SCBA. Thus the eigenvalues of − t are 0 , , − s/ , − s/ − τ are 1 − ( s + 1) / , − ( s − / , / , /
2. Like in the case of the two-bandGreen’s function of Sect. IV B there is one massless mode for − t for any value of s and one masslessmode for − τ if s = 1. This indicates that the result of the one-band projection preserves the structureof two-band result of Table I in terms of the number of massless modes. The agreement of the resultsfrom the one-band projected Green’s function and the two-band Green’s function reflects the fact thatthe type of diffusive modes is not sensitive to the scattering to the second band. A. Inter-node scattering
Finally, we briefly discuss the effect of inter-node scattering for the Hamiltonian H . For this purposewe introduce the extended Hamiltonian ¯ H = (cid:18) H vσ vσ H ∗ (cid:19) , (46)which describes inter-node scattering by the random scattering terms vσ .Now we could evaluate the inverse two-particle Green’s functions within the WLA and study thevanishing eigenvalues, which would be associated with diffusive behavior. Alternatively, we can also startfrom the symmetry argument and analyze the underlying chiral symmetry whose spontaneous breakingwould create a massless mode, analogously to the treatment of random scattering terms in Ref. [9].Following this concept, we first realize that the Hamiltonian ¯ H changes its sign under the transformationin Eq. (3) ¯ H → S ¯ H S = − ¯ H , S = (cid:18) σ − σ (cid:19) . (47)According to the general procedure of Ref. [9], this leads for the Hamiltonian ˆ H = diag ( ¯ H , ¯ H ) to thenon-Abelian chiral symmetry e ˆ S ˆ H e ˆ S = ˆ H , ˆ S = (cid:18) ϕSϕ ′ S (cid:19) (48)for independent continuous parameters ϕ , ϕ ′ , since ˆ H and ˆ S anticommute. This symmetry is sponta-neously broken due to the scattering rate η , causing the appearance of a massless mode. Here it should benoticed though that the appearance of a massless mode is only a necessary but not a sufficient conditionfor a diffusive behavior because the interaction of the nonlinear symmetry fields generated by ϕ , ϕ ′ , couldlead to Anderson localization [19]. VII. CONCLUSIONS
The calculation of the ladder and maximally crossed contributions of the average two-particle Green’sfunction has revealed a characteristic diffusion pole structure. Depending on the type of randomness, theladder contributions always have one diffusion pole, provided the Fermi energy is at the Dirac node oraway from the Dirac node but in the absence of a gap. Moreover, for a random scalar potential thereis an additional diffusion pole from the maximally crossed contributions. All these results require theexistence of a nonzero scattering rate, obtained as a solution of the self-consistent Born relation (SCBA).No diffusion pole have been found away from the Dirac node in the presence of a one-particle gap.0The diffusion pole structure of the WLA is identical to that of the WSA, at least for the case of arandom gap. This enabled us to identify the origin of the diffusion poles with massless modes which arecreated by spontaneously broken symmetries. These are chiral symmetries, associated with the symmetryof the two bands. Such a symmetry exists also in the presence of inter-node scattering. Further work isnecessary though, to compare the relation between the WLA and the WSA for other types of disorderscattering.The diffusion pole structure of the two-particle Green’s function is preserved when we employ a one-band projection of the one-particle Green’s function by removing one pole of the latter. Although thisprojection changes the form of the diffusion coefficient (cf. [5, 6]), it may serve as a good approximationthat reduces the computational effort significantly.As already mentioned in the Introduction, the existence of a diffusion pole is a necessary condition butdoes not guarantee a metallic behavior. Higher order terms beyond the ladder and maximally crossedcontribution can destroy diffusion and eventually lead to Anderson localization. This was studied recentlyin terms of a strong scattering expansion [19], which revealed an exponential decay when the scatteringrate η is larger than the band width E b .AcknowledgementWe are grateful to E. Hankiewicz, D. Schmeltzer and G. Tkachov for inspiring discussions. [1] L.P. Gorkov, A.I. Larkin, and D.E. Khmelnitskii, Pisma v ZhETF , 248 (1979); JETP Lett. , 228 (1979).[2] S. Hikami, A.I. Larkin and Y. Nagaoka, Prog. Theor. Phys. (1980).[3] B.L. Altshuler et al., Phys. Rev. B , 5142 (1980); B.L. Altshuler and B.D. Simons, in Mesoscopic quantumphysics , eds. E. Akkermans et al. (North-Holland 1995).[4] T. Ando, Y. Zheng and H. Suzuura, J. Phys. Soc. Japan , 1318 (2002); H. Suzuura and T. Ando, Phys.Rev. Lett. , 266603 (2002); E. McCann et al., Phys. Rev. Lett. , 146805 (2006).[5] D.V. Khveshchenko, Phys. Rev. Lett. , 036802 (2006).[6] G. Tkachov and E.M. Hankiewicz, Phys. Rev. B , 035444 (2011).[7] D. Schmeltzer and A. Saxena, Phys. Rev. B , 035140 (2013).[8] K. Ziegler, Phys. Rev. B , 10661 (1997).[9] K. Ziegler, Eur. Phys. J. B , 391 (2013).[10] K. Ziegler, Phys. Rev. Lett. , 3113 (1998).[11] L. Sch¨afer and F. Wegner, Z. Physik B , 113 (1980).[12] K. Efetov, Supersymmetry in Disorder and Chaos (Cambridge University Press 1997).[13] S. Raghu, S.B. Chung, X.-L. Qi and S.-C. Zhang, Phys. Rev. Lett. , 116401 (2010); X.-L. Qi and S.-C.Zhang, Rev. Mod. Phys. , 1057 (2011).[14] J.S. Langer and T. Neal, Phys. Rev. Lett. , 984 (1966).[15] F.J. Wegner, Phys. Rev. B , 783 (1979).[16] K. Ziegler, Phys. Rev. B , 125401 (2008).[17] K.S. Novoselov, A.K. Geim, S.V. Morozov, D. Jiang, M.I. Katsnelson, I.V. Grigorieva, S.V. Dubonos, A.A.Firsov, Nature , 197 (2005).[18] Y. Zhang, Y.-W. Tan, H.L. Stormer, P. Kim, Nature , 201 (2005).[19] A. Hill and K. Ziegler, Physica E56