Internet & Web

Absence of interaction corrections in graphene conductivity

Description
Absence of interaction corrections in graphene conductivity
Categories
Published
of 4
All materials on our website are shared by users. If you have any questions about copyright issues, please report us to resolve them. We are always happy to assist you.
Related Documents
Share
Transcript
    a  r   X   i  v  :   1   0   1   0 .   4   4   6   1  v   1   [  c  o  n   d  -  m  a   t .  s   t  r  -  e   l   ]   2   1   O  c   t   2   0   1   0 Absence of interaction corrections in graphene conductivity Alessandro Giuliani, 1 Vieri Mastropietro, 2 and Marcello Porta 3 1 Universit`a di Roma Tre, L.go S. L. Murialdo 1, 00146 Roma, Italy  2 Universit`a di Roma Tor Vergata, Viale della Ricerca Scientifica 00133 Roma, Italy  3 Universit`a di Roma La Sapienza, P.le A. Moro 2, 00185 Roma, Italy  The exact vanishing of the interaction corrections to the zero temperature and zero frequencyconductivity of graphene in the presence of weak short range interactions is rigorously established. PACS numbers: 05.10.Cc, 05.30.Fk, 71.10.Fd, 72.80.Vp Graphene [1] has severalpeculiar propertiessrcinatingfrom its perfect two dimensionality and from the Dirac-like nature of its charge carriers at half-filling. In partic-ular, recent optical measurements [2] show that at half-filling and small temperatures, if the frequency is in arange between the temperature and the band-width, theconductivity is essentially constant and equal, up to afew percent, to  σ 0  =  e 2 hπ 2 ; such value only depends on thefundamental von Klitzing constant  h/e 2 and not on thematerial parameters, like the Fermi velocity. These ex-perimental results confirm the theoretical predictions [3]based on the description of graphene in terms of masslessnon-interacting Dirac particles [4, 5]; lattice effects have been taken into account in [6]. Since truly universal phe-nomena are quite rare in condensed matter (an exampleis provided by the Quantum Hall effect), it is importantto understand whether this apparently  universal value   is just an artifact of the idealized description in terms of non-interacting fermions or rather it is a robust propertystill valid in the presence of electron-electroninteractions,which are certainly present and expected to play a role inreal graphene. This question is entirely analogous to theone concerning universality in the quantum Hall effect[7], a notoriously difficult and still open problem.The effects of the electron-electron interactions onthe graphene conductivity have been investigated in theDirac approximation by perturbation theory both in thepresence of long- and of short-ranged interactions; how-ever, lowest order explicit computations have produced different   results [8–10], depending on the regularization scheme (momentum cut-off or dimensional regulariza-tion) chosen to cure the spurious ultraviolet divergencesintroduced by the Dirac approximation. In [9] it was pre-dicted that in the presence of electrostatic interactionsthe zero frequency conductivity tends to zero, while in[8, 10] it was argued that it converges to the free Dirac one, as a consequence of the  divergence   of the Fermi ve-locity; however, if screening or retardation effects aretaken into account, the Fermi velocity is known to satu-rate at low frequency [11–13], in which case it is unclear what to expect. The extreme sensitivity of the conduc-tivity computation to approximations or regularizations(see also [14]) calls for a rigorous analysis.In this paper we consider the Hubbard model on thehoneycomb lattice, as a model of monolayer graphenewith screened interactions. While in general the under-standing of the low temperature behavior of the Hubbardmodel is a formidable challenge for theoreticians, in thecase of the honeycomb lattice at half filling the methodsintroduced in [12] and based on constructive Renormal-ization Group have proved to be quite effective. Usingthese techniques, we rigorously establish  the exact (non-perturbative) vanishing of the interaction corrections tothe conducivity   in the zero frequency limit. All Feyn-man graphs contributing to the conductivity cancel outexactly in the limit, a statement analogous to the Adler-Bardeen theorem in quantum electrodynamics [7, 15]. We introduce creation and annihilation fermionic oper-ators  ψ ± x,σ  = ( a ± x,σ ,b ± x +  δ 1 ,σ ) =  L − 2   k ∈B Λ ψ ±  k,σ e ± i k x forelectrons with spin index  σ  = ↑↓  sitting at the sites of the two triangular sublattices Λ A  and Λ B  of a periodichoneycomb lattice of side  L ; we assume that Λ A  = Λ hasbasis vectors   l 1 , 2  =  12 (3 , ±√  3) and that Λ B  = Λ A  +   δ  j ,with   δ  1  = (1 , 0) and   δ  2 , 3  =  12 ( − 1 , ±√  3) the nearestneighbor vectors;  B  Λ  =  {  k  =  n 1  G 1 /L  +  n 2  G 2 /L  :0  ≤  n i  < L }  with   G 1 , 2  =  2 π 3  (1 , ±√  3) is the firstBrillouin zone (note that in the thermodynamic limit L − 2   k ∈B Λ → |B| − 1   B  d k , with  |B|  = 8 π 2 / (3 √  3)).The grand-canonical Hamiltonian at half-filling is  H  Λ  = H  0Λ  + UV  Λ , where  H  0  is the free Hamiltonian, describingnearest neighbor hopping ( t  is the hopping parameter): H  0Λ ( t ) =  − t   x ∈ Λ A j =1 , 2 , 3  σ = ↑↓ ( a + x,σ b − x +  δ j ,σ + b + x +  δ j ,σ a − x,σ ) (1)and  V  Λ  is the local Hubbard interaction: V  Λ  =   x ∈ Λ A  σ = ↑↓ ( a + x,σ a − x,σ −  12)+   x ∈ Λ B  σ = ↑↓ ( b + x,σ b − x,σ −  12)  . (2)In order to define the lattice current and the conductiv-ity, we modify the hopping parameter along the bond( x,x +  δ  j ) as  t  →  t x,j (  A ) =  t exp { ie   10  A ( x + s δ  j ) ·  δ  j  ds } ,where   A (  x )  ∈  R 2 is a periodic continuum field on S  Λ  =  { x  =  Lξ  1  l 1  +  Lξ  2  l 2  :  ξ  i  ∈  [0 , 1) } ; its Fouriertransform is defined as   A ( x ) =  |S  Λ | − 1    p ∈D Λ  A   p e − i  px ,where  |S  Λ |  =  3 √  32  L 2 and  D Λ  =  {   p  =  n 1  G 1 /L  +  2 n 2  G 2 /L  :  n i  ∈  Z }  (note that in the thermodynamiclimit  |S  Λ | − 1     p ∈D Λ →  (2 π ) − 2   R 2  d  p ). If we denote by H  ( A ) =  H  0Λ ( { t  x,j (  A ) } )+ UV  Λ  the modified Hamiltonianwith the new hopping parameters, the lattice current isdefined as     p  =  − ∂H  ( A ) ∂   A  p , which gives, at first order in   A ,    p  =  ( P  )   p  +    d q  (2 π ) 2     ( D )   p, q  A  q  ,  (3)where    d q (2 π ) 2  is a shorthand for  |S  Λ | − 1   q ∈D Λ and, if    B d k |B|  is a shorthand for  L − 2   k ∈B Λ and  η j  p  =  1 − e − i p δj i  p δ j ,  ( P  )   p  =  − iet  σ,j   B d k |B|  a +  k +   p,σ b −  k,σ  δ  j η j  p e − i k (  δ j −  δ 1 ) − b +  k +   p,σ a −  k,σ  δ  j η j  p e + i (  k +   p )(  δ j −  δ 1 )   (4)is the  paramagnetic current   and     ( D )   p, q  lm  =  e 2 t  σ,j (  δ  j ) l (  δ  j ) m η j  p η j q   B d k |B|  a +  k +   p +  q,σ b −  k,σ · e − i k (  δ j −  δ 1 ) + b +  k +   p +  q,σ a −  k,σ e i (  k +   p +  q )(  δ j −  δ 1 )   (5)is the  diamagnetic tensor  . The conductivity, at Matsub-ara frequency  p 0  ∈  2 πβ  − 1 ( Z +  12 ) and in units such that    = 1, is defined via Kubo formula as [6] σ β, Λ lm  (  p 0 ) =  − K  β, Λ lm  (  p 0 ,  0)  p 0 |S  Λ |  (6)where, if Ξ = Tr { e − βH  Λ } ,  ·  = Ξ − 1 Tr { e − βH  Λ ·}  and O x 0  =  e H  Λ x 0 Oe − H  Λ x 0 for a generic operator  O , K  β, Λ lm  (  p 0 ,  p ) =    β 0 dx 0  e − ip 0 x 0   j ( P  ) x 0 ,  p,l  j ( P  )0 , −   p,m  +      ( D )   p, −   p  lm  (7)It is known that in general the interaction  modifies   thevalues of the physical quantities; for instance, the Fermivelocity  v F  , the wave function renormalization  Z   and thevertex functions are known to depend explicitly on theinteraction [12]; moreover, it was proven in [12] that  v F  , Z   and the vertex functions are  analytic   functions of   U   for | U  | small enough, uniformly as  β, | Λ | → ∞ . In this Letterwe prove a similar result for the conductivity. Moreover,we prove that in the thermodynamic, zero temperatureand zero frequency limit, the conductivity is  universal  ,i.e., it is exactly independent of   U  . Theorem.  There exists a constant   U  0  >  0  such that, for   | U  | ≤  U  0  and any fixed   p 0 ,  σ β, Λ lm  (  p 0 )  is analytic in   U  uniformly in   β, Λ  as   β, | Λ | → ∞ . Moreover, lim  p 0 → 0 + lim β →∞ lim | Λ |→∞ σ β, Λ lm  (  p 0 ) =  e 2 hπ 2 δ  lm  .  (8)Note that the limit  β   → ∞  is taken before the limit  p 0  →  0 + . In other words, the theorem says that theinteraction corrections to the conductivity are negligibleat frequencies  β  − 1 ≪  p 0  ≪  t . Proof.  The idea of the proof is based on the two mainingredients: (i)  exact lattice   Ward Identities (WI) re-lating the current-current, vertex and 2-point functions;(ii) the fact that the interaction-dependent correctionsto the Fourier transform of the current-current correla-tions are  differentiable   with continuous derivative (in con-trast, the free part is continuous and not differentiableat zero frequency). This last property follows from thenon-perturbative estimates found in [12], which we nowbriefly recall. The generating functional for correlationscan be written in terms of a Grassmann integral: e W  ( A,λ ) =    P  ( dψ ) e V  ( ψ )+( ψ,λ )+ B ( A,ψ ) (9)where, if   k  = ( k 0 , k ) with  k 0  the Matsubara frequency, P  ( dψ ) is the fermionic gaussian integration for  ψ ± k ,σ  =( a ± k ,σ ,b ± k ,σ ), with inverse propagator g − 1 ( k ) =  − Z  0   ik 0  v 0 Ω ∗ (  k ) v 0 Ω(  k )  ik 0   ,  (10)with  Z  0  = 1,  v 0  =  32 t  and Ω(  k ) =  23  j =1 , 2 , 3  e i k (  δ j −  δ 1 ) (note that  g ( k ) is singular only at the Fermi points  k  = k ± F   = (0 ,  2 π 3  , ±  2 π 3 √  3 )). Moreover,  B ( A,ψ ) ==  σ    β 0 dx 0  x ∈ Λ  − ieψ + x ,σ  A 0 ( x ) 00  A 0 ( x + δ  1 )  ψ − x ,σ +  j  ( t x,j (  A ) − t ) a +( x 0 ,x ) ,σ b − ( x 0 ,x +  δ j ) ,σ + c.c.   (11)and ( ψ,λ ) =   β 0  dx 0  x ∈ Λ [ ψ + x λ − x  + λ + x ψ − x  ]. The responsefunction  K  β, Λ ( p ) corresponds to the spatial componentsof the tensor   K  µν  ( p ) =  δ 2 δA µ ( p ) δA ν ( − p ) W  ( A, 0)  A =0 , with µ,ν   = 0 , 1 , 2. Performing the phase transformation ψ ± x  →  e ± ieα x ψ ± x  in Eq.(9), we find W  ( A + ∂α,λe ieα ) =  W  ( A,λ )  ,  (12)which implies the following lattice Ward Identity [16] 2  µ =0  p µ   K  µν  ( p ) = 0  ,  (13)for all  ν   ∈ { 0 , 1 , 2 } . On the other hand, the functionalintegral Eq.(9) can be evaluated in terms of an exactRenormalization Group (RG) analysis, described in fulldetail in [12]. We decompose the field  ψ  as a sum of fields  ψ ( k ) , living on momentum scales  | k  −  k ± F  | ≃  2 h ,with  h  ≤  0 a scale label; the iterative integration of thefields on scales  h < h ′  ≤  0 leads to an effective theorysimilar to Eq.(9) with a cut-off around the Fermi points of width 2 h and with a scale dependent propagator g ( ≤ h ) ( k )  3with the same singularity structure as Eq.(10), with  Z  0 and  v 0  replaced by  Z  h  and  v h , respectively (the effec-tive wave function renormalization and Fermi velocityon scale  h ). Moreover, setting for simplicity  λ  = 0, atscale  h  the interaction  V  ( ψ ) + B ( A,ψ ) is replaced by aneffective interaction  V  ( ≤ h ) ( ψ ( ≤ h ) )+ B ( h ) ( A,ψ ( ≤ h ) ), withthe  effective potential   V  ( ≤ h ) ( ψ ( ≤ h ) ) a sum of monomialsin  ψ ( ≤ h ) of arbitrary order, characterized at order  n  bykernels  W  ( h ) n, 0 ( x 1 ,..., x n ) that are analytic in  U   and de-cay super-polynomially in the relative distances  | x i − x j | on scale 2 − h ; moreover the  effective source   is given by B ( h ) ( A,ψ ) = 2  µ =0 Z  µ,h    d p (2 π ) 3 A µ ( p )  j µ ( p ) + ¯ B ( h ) (14)where  j 0 ( p ) =  − ie  σ    d k (2 π ) |B| ψ + k + p ,σ Γ 0 (  k,  p ) ψ − k ,σ ,  ( p ) =  − ie  σ    d k (2 π ) |B| ψ + k + p ,σ   Γ(  k,  p ) ψ − k ,σ , [Γ 0 (  k,  p )] ij  = δ  ij  exp {− ip 1 δ  i 2 }  and   Γ(  k,  p ) = 23  j  δ  j η j  p   0  − e − i k (  δ j −  δ 1 ) e + i (  k +   p )(  δ j −  δ 1 ) 0   . (15)Finally, ¯ B ( h ) is a sum of monomials in ( A,ψ ) of arbi-trary order, characterized at order  n  in  ψ  and  m  in  A  bykernels  W  ( h ) n,m ( x 1 ,..., x n ; y 1 ,..., y m ) that are analytic in U  , decay super-polynomially in the relative distances onscale 2 − h and are non-zero only if   m  ≥  1 ,n  ≥  0 and m  +  n  ≥  3; in particular, for all 0  < θ <  1, they satisfythe bounds (proved in [12]),    d x 2 ··· d x n d y 1  ...d y m | W  ( h ) n,m ( x 1 ,..., x n ; y 1 ,..., y m ) |≤  (const . ) | e | m 2 (3 − n − m ) h  1 − δ  m, 0  + | U  | 2 θh   .  (16)The bounds Eq.(16) are  non-perturbative   (i.e., they arebased on the  convergence   of the expansion for the kernels W  ( h ) ). They are obtained by exploiting the anticommu-tativity properties of the Grassmann variables, via a de-terminant expansion and the use of the  Gram-Hadamard inequality   for determinants, see [12]. The factor 2 θh inthe bound will play a crucial role in the following andreflects the fact that the  scaling dimension   3 − n − m  isalways negative for  n >  2. The  running coupling con-stants   Z  h ,v h ,Z  µ,h  satisfy recursive equations ( beta func-tion equations  ) that, due to the bound Eq.(16), lead tobounded and controlled flows, i.e.,  Z  ( U  ) = lim h →−∞  Z  h , Z  µ ( U  ) = lim h →−∞ Z  µ,h  and  v F  ( U  ) = lim h →−∞ v h  areanalytic functions of   U  , analytically close to their unper-turbed values  Z  0  =  Z  0 , 0  = 1 and  Z  1 , 0  =  Z  2 , 0  =  v 0  =  32 t ,see [12]. The analyticity of the kernels of the effectivepotential and of the  h  → −∞  limits of the running cou-pling constants implies the analyticity of the imaginary-time correlation functions (see [12]) and, similarly, theanalyticity of   σ β, Λ (  p 0 ) claimed in the main theorem.We are left with proving the universality result Eq.(8).To this aim, it is important to notice that  Z  h ,v h ,Z  µ,h  arerelated by Ward Identities. Indeed, proceeding as in [13],we consider a reference model defined in a way similarto Eq.(9), with the important difference that the Grass-mann integration  P  ( dψ ) is modified into  P  ≥ h ( dψ ), whosepropagator differs from the srcinal one by the presenceof a smooth infrared cutoff selecting the momenta  ≥  2 h ;performing the phase transformation  ψ ± x  →  e ± iα x ψ ± x  inthis functional integral, we find the analogue of Eq.(12),which implies Z  0 ,h Z  h = 1+ O ( U  2 θh )  , Z  1 ,h Z  h v h =  Z  2 ,h Z  h v h = 1+ O ( U  2 θh )(17)where the corrections  O ( U  2 θh ) come from the symme-try breaking terms due to the infrared cut-off function. Therefore, the effective parameters are related by ex-act identities  ; the vertex density renormalization  Z  (0) h  isequal, up to negligible terms, to the wave function renor-malization, and the current renormalization  Z  (1) h  is equalto the product of the effective velocity and the wave func-tion renormalization [17].We can write  K  µν   =  K  ( P  ) µν   + K  ( D ) µν   , where the two termsin the right hand side correspond to the paramagneticand diamagnetic contributions to  K  µν  , see Eq.(7). Notethat   K  ( D ) µν   (  p 0 ,  0) is independent of   p 0 ; using Eq.(16), wefind that  |   K  ( D ) µν   (  p 0 ,  0) | ≤  (const . ) | e | 2   0 h = −∞ 2 h , whichis finite. On the other hand  K  ( P  ) µν   ( x ) == 0  h = −∞  2 e 2 Z  µ,h Z  ν,h ( Z  h ) 2    d k d p (2 π ) 2 |B| 2 e i px F  h ( k , p ) ·  (18) · Tr { Γ µ (  k,  p ) C  h ( k )Γ ν  (  k  +    p, −   p ) C  h ( k + p ) } + H  ( h ) µν   ( x )   , where the first term corresponds to the zero-th order in  U  in renormalized perturbation theory ( F  h ( k , p ) is a suit-able smooth cutoff function constraining  | k  −  k ± F  |  and | k − k ± F   + p |  to be  ≃  2 h and such that  0 h = −∞ F  h ( k , p ) =1; moreover,  Z  h C  − 1 h  ( k ) is given by Eq.(10) with  Z  0 ,v 0 replaced by  Z  h ,v h ) and, for all  N   ≥  0 and suitable con-stants  C  N  , | H  ( h ) µν   ( x ) | ≤  C  N  | U  |  2 (4+ θ ) h 1 + (2 h | x | ) N   .  (19)As compared to the zero-th order contribution to  K  ( P  ) µν   ,the dimensional bound on  H  ( h ) µν   has an extra factor 2 θh ,following again from Eq.(16). From Eq.(19), | K  ( P  ) µν   ( x − y ) | ≤  (const . ) 11 + | x − y | 4  ,  (20)that is,  K  µν  ( x ) is absolutely integrable and, therefore, itsFourier transform in the thermodynamic and zero tem-perature limit is continuous at  p  =  0 . Combining thisremark with the WI Eq.(13), we find that   K  µν  ( 0 ) =0. In fact, setting, e.g.,  p 2  = 0,   K  11 (  p 0 ,p 1 , 0) =  4( −  p 0 /p 1 )   K  01 (  p 0 ,p 1 , 0); taking first the limit  p 0  →  0 andthen  p 1  →  0 in the right hand side, we get   K  11 ( 0 ) = 0;proceeding analogously, we find that   K  µν  ( 0 ) = 0 for all µ,ν   ∈ { 0 , 1 , 2 } .On the other hand Eq.(17) implies that  Z  1 ,h Z  h =  v F  ( U  )+ O ( U  2 θh ), so that  K  ( P  ) µν   ( x ) =  K  ( P  ;0) µν   ( x )+ K  ( P  ;1) µν   ( x ) where K  ( P  ;0) µν   ( x ) is the paramagnetic response function for themodel with Hamiltonian  H  0Λ ( 23 v F  ( U  )) and  | K  ( P  ;1) µν   ( x ) | ≤ (const . ) | U  | (1 +  | x | 4+ θ ) − 1 , with 0  < θ <  1. There-fore, the Fourier transform of   K  ( P  ;1) µν   is  differentiable  in  p  and its derivative is continuous at  p  =  0 . Asimilar decomposition can be performed in the diamag-netic term, so that, defining  K  (1) µν   =  K  ( P  ;1) µν   +  K  ( D ;1) µν  and using the WI Eq.(13),   2 µ =0  p µ   K  (1) µν   ( p ) = 0 fromwhich, setting, e.g.,  p 2  = 0, we obtain   K  (1)11  (  p 0 ,p 1 , 0) =(  p 0 /p 1 ) 2   K  (1)00  (  p 0 ,p 1 , 0); deriving with respect to  p 0  bothsides and taking first the limit  p 0  →  0 and next  p 1  →  0in the right hand side, we get  ∂   p 0   K  (1)11  ( 0 ) = 0; proceed-ing analogously, we find that  ∂   p ρ   K  (1) µν   ( 0 ) = 0 for all ρ,µ,ν   ∈ { 0 , 1 , 2 } .  Note the crucial role played by the continuity of the derivatives of   ˆ K  (1) µν   , which allowed us toexchange the zero frequency and zero momentum limits  ,as compared to the order in Eq.(8).In order to compute the conductivity, we are left withthe contribution associated to a free theory with Fermivelocity equal to  v F  ( U  ) that, for the 11 component, set-ting  v  =  v F  ( U  ), reads: σ 11  = lim  p 0 → 0 + lim β →∞ lim | Λ |→∞ σ β, Λ11  (  p 0 ) = 2 e 2 v 2 lim  p 0 → 0 +    dk 0 2 π  ··   B d k |B| Tr  S  ( k ) − S  ( k +  p 0 )  p 0 Γ 1 (  k,  0) S  ( k )Γ 1 (  k,  0)   . The latter integral is not uniformly convergent in  p 0 ; inparticular, it is well known that one cannot exchange thelimit with the integral [14]. The integral can be evaluatedexplicitly (using residues to compute the integral over  k 0 )and leads to Eq.(13). A similar computation shows that σ 22  =  σ 11  and that the off-diagonal terms vanish.The above analysis can be extended to the case of long range electromagnetic interactions; in such casethe wave function, density and current renormalizationshave a strong (anomalous) power law dependence on themomentum and the Fermi velocity increases up to thespeed of light [11, 13]. WIs similar to Eq.(17) are still valid and imply that, even if the effective parameters arestrongly momentum dependent,  the conductivity only de-pends weakly on the frequency   in the optical range.In conclusion, we rigorously proved the non existenceof corrections to the zero temperature and zero frequencylimit of the graphene conductivity due to weak shortrange interactions. The proof is based on a combinationof constructive Renormalization Group methods with ex-act lattice Ward identities. Remarkably, this is one of thevery few examples of universality in condensed matterthat can be established on firm mathematical grounds.A.G. and V.M. gratefully acknowledge financial sup-port from the ERC Starting Grant CoMBoS-239694. Wethank D. Haldane for valuable discussions on the role of exact lattice Ward Identities. [1] A. K. Geim and K. S. Novoselov,  Nature Materials   6 , 183(2007).[2] R. R. Nair et al.,  Science   320 , 1308 (2008); Z. Q. Li etal.,  Nature Phys.  4 , 532 (2008).[3] T. Ando et al.,  J. Phys. Soc. Jpn.  71 , 1318 (2002); V. P.Gusynin et al.,  Phys. Rev. Lett.  96 , 256802 (2006).[4] G. W. Semenoff,  Phys. Rev. Lett.  53 , 2449-2452 (1984)[5] A. Ludwig et al.,  Phys. Rev. B   50 , 7526 (1994).[6] T. Stauber, N. Peres and A. Geim,  Phys. Rev. B   78 ,085432 (2008).[7] K. Ishikawa and T. Matsuyama,  Nucl. Phys. B   280 , 523(1987).[8] I. Herbut, V. Juricic and O. Vafek,  Phys. Rev. Lett.  100 ,046403 (2008); arXiv:1009.3269v1; arXiv:0809.0725.[9] E. G. Mishchenko,  Phys. Rev. Lett.  98 , 216801 (2007); Europhys. Lett.  83 , 17005 (2008).[10] D. Sheehy and J. Schmalian,  Phys. Rev. Lett.  99 , 226803(2007);  Phys. Rev. B   80 , 193411 (2009).[11] J. Gonzalez, F. Guinea and M. A. H. Vozmediano,  Nucl.Phys. B   424 , 595-618 (1994).[12] A. Giuliani and V. Mastropietro, (a)  Comm. Math. Phys. 293 , 301 (2010); (b)  Phys. Rev. B   79 , 201403 (R) (2009);(c) Erratum to (b), arXiv:0901.4867.[13] A. Giuliani, V. Mastropietro and M. Porta,  Phys. Rev.B   82 , 121418(R) (2010).[14] K. Ziegler,  Phys. Rev. B   75 , 233407 (2007).[15] S. L. Adler and W. A. Bardeen,  Phys. Rev.  182 , 1517(1969).[16] F. D. M. Haldane, private communication (2009).[17] Eq.(17) should be compared with the result of Theorem2 in [12](b), where an unnatural definition of the currentswas adopted; this led to an asymmetry  Z  1   =  Z  2 , whichis an artifact of the unphysical definition of the currentsand that, correctly, does not show up here, see [12](c).
Search
Similar documents
View more...
Related Search
We Need Your Support
Thank you for visiting our website and your interest in our free products and services. We are nonprofit website to share and download documents. To the running of this website, we need your help to support us.

Thanks to everyone for your continued support.

No, Thanks