Coherent Potential Approximation

Transcription

1 Coheret Potetal Approxato Noveber 29, 2009 Gree-fucto atrces the TB forals I the tght bdg TB pcture the atrx of a Haltoa H s the for H = { H j}, where H j = δ j ε + γ j. 2 Sgle ad double uderles deote atrces agular oetu space ad ste-agular oetu space, respectvely. The sze of each agular oetu block s detered by the deso of the bass cetered at each ste. I the case of 3d trasto etals e.g., the hybrdzed 3d-4s-4p valece bad spas a 9-desoal space 8 cludg sp. I ay cases the o-ste eergy blocks ε Eq. 2 are theselves dagoal, but ths s ot ecessary. The hoppg tegrals γ j are strctly ste-off-dagoal. The resolvet or statc Gree-fucto atrx of a gve syste descrbed by the Haltoa H ca be defed as Gz := z H 3 for ay z C at least where the verso ca be perfored. Supposg that the solutos of the egevalue equato, are kow, the the Haltoa atrx ca be wrtte as H = ε, 4 H = ε, 5 where rus over all egefuctos. Ths ples the spectral decoposto of the atrx Gz, Gz = z ε. 6 The fudaetal aalytc property of the resolvet, G z = Gz 7

2 s a corollary of ths decoposto. Aother fudaetal detty ca be derved fro defto: dgz dz = Gz 2. 8 Sce Gz s udefed at real eerges, ε, we have to approach the real arguets fro the agary drecto, G ± ε := l δ 0 G ε ± ıδ 9 =, 0 ε ε ± ı0 for ay ε R. Note that G + ε G ε f ε s the spectru of H. Equato 7 ples G + ε ] = G ε. The well-kow detty of geeralzed fuctos, ε ε ± ı0 = P ıπδ ε ε, 2 ε ε leads to the relatoshp, δ ε ε = G + ε G ε ] 2πı = G + ε G + ε ]. 3 2πı By coposg the trace of 3, the desty of states of the syste, ε, ca be expressed fro the Gree fucto as ε = δ ε ε = 2πı Tr G + ε G + ε ] = Tr G + ε Tr G + ε ] 2πı ε = π ITr G+ ε = π ITr G ε. 4 The expectato value of a observable A at zero teperature ca be calculated as A = = ε F ε b ε F ε b δ ε ε A dε 5 Tr δ ε ε A dε 6 A = εf π I so Gz ad the spectru of H cota the sae forato. ε b Tr A G + ε ] dε, 7 2

3 2 Perturbatos wth respect to a referece syste Suppose ow that H = H 0 + H, ad = syste. The z H 0 s the resolvet of the referece Gz = z H 8 z H 0 H Gz = I I z H Gz = z 9 Gz = O the other had, fro Eq. 9, Ths equato ca be solved teratvely: G 0 z = z I z H G0 z = z I H z. 20 G z = z + z H z Gz = z + z H Gz. 2 G 2 z = z + z H z + z H z H z. Gz = z + z H z + z H z H z Ths Dyso-equato ca be rearraged as ] Gz = z + z H + H z H +... z where T z s the so-called scatterg atrx, = z + z T z z, 23 T z = H + H Gz H = H + H z H + H z H z H +... = H + H T z. 24 Ths ca be rearraged to gve ] T z = I H z H = H I z H]. 25 It ca easly be show that the T atrx has slar aalytcal propertes as the resolvet, T z =T z, 26 dt z =T z dg z 0 T z, ad dz dz 27 T ± ε := l T ε ± ıδ δ

4 at real eerges ε. By usg equatos 4 ad 23, we get the desty of states DOS of the perturbed syste wth respect to the referece syste, ε = 0 ε π ITr G + 0 ε T + ε G + 0 ε ]. 29 Usg propertes 8 ad 27, the tegratg wth respect to eergy, we arrve at the Lloydforula, whch gves the tegrated DOS of the perturbed syste, Nε := ε ε dε = N 0 ε + π ITr l T + ε O-ste purtes Case of a sgle o-ste purty: H = { H δ δ }, T = H + H H = { H + H G 0 H +... ] } δ δ = {t δ δ }, thus t = H + H G 0 t. 32 Now let H be a su of such o-ste dffereces: H = H. The T = H + H H +... = H + H + H H k ,j,j,k T = H δ + H G 0 H + k H G k 0 H k G k 0 H +... = H δ + k H G k 0 T k. 34 Both operator ad atrx sese, T = = = H + H ,j H + H +... j, where 36 4

5 := H + H H = H + H = H + H + H I H = H + H =t + t where t s forally a sgle purty T -atrx o the -th ste, Solvg equato 38 teratvely, 0 2 :=t =t + t =t +. =t +, 38 t = {t δ δ j }. 39 t + k t + k t t k 40 t t k Usg equato 37, we arrve at the ultple scatterg expaso of the T atrx, T = t + t + t t k k Sce all t have the structure of H, T =t δ + δ t G 0 + k t G k 0 t k G k k k =t δ + k t G k 0 δ k T k. 44 Defg the ste-off-dagoal part of the referece syste s resolvet, Ĝ 0 := { G k 0 δ k }, 45 T = t + t Ĝ0 T Ĝ0] T = t

6 O the other had, T =t + t Ĝ0t + t Ĝ0t Ĝ0t..., so 47 G = + T 48 s gve ters of ad t. 4 Checally dsordered systes 4. Bary alloys Let s cosder ow a two-state dsordered syste, e.g. a two-copoet bary rado alloy: H = ξ H A + ξ H B, 49 where ξ are depedet rado varables wth Beroull dstrbuto: { wth probablty P := c ξ =. 0 wth probablty P 0 = c By defto the expected values are thus the expected value of H s Eξ ξ = c, 50 H = ξ H A + ξ H B = c H A + c H B. 5 Idepedece eas that the jot probablty ass fucto of {ξ} decoposes to the product of the dvdual probablty ass fuctos: P {ξ} = N P ξ. 52 Of course P {ξ} s a probablty, sce trvally P {ξ} = P ξ =. 53 {ξ} The cofguratoal average of soe physcal quatty s the defed as = ξ =0 F {ξ} := {ξ} P {ξ} F {ξ} = ξ... ξ N P ξ... P N ξ N F ξ,..., ξ N. 54 Sce G = G {ξ, ξ 2,..., ξ N } G {ξ}, the ea of a physcal quatty A the TB pcture s A = π I fε Tr A G {ξ} ] dε 55 = π I fε Tr A G ] dε 56 where we suppressed the depedece of G o the eergy ε. 6

7 4.2 Coheret Potetal Approxato G = G0 + T G0 =: G c = z H c, 57 where we defed the effectve Haltoa H c ofte oted as Σ c, the self-eergy. Ths assupto s the coheret potetal approxato CPA. The 57 CPA codto ca oly be satsfed f H c = H c z s a fucto of the eergy, but t s by defto cofguratodepedet. Let us ow choose our referece syste to be H c, T =t + t Ĝct + +t Ĝct Ĝct H =H H c = { H H c, δ δ }, 59 where H c, are to be detered. A codto s gve by G =Gc = G c + G c T Gc 60 T = 0 6 t + t Ĝc t +... = Sgle-ste CPA: t := Cosderg ths, t Ĝ k k k c t Ĝ c t k Ĝ k c = k k = t Ĝ c =0 64 t Ĝ k c t k Ĝk c =0. 65 Thus eq. 63 satsfes the codto set by eq. 6 up to fourth order t. Sce t = ξ t A + ξ t B, 66 eq. 63 reads as t = c t A + c t B = Ths s fact a syste of equatos for H c, because ad G c ca be detered fro equato 57. t α = I H α G c H α, where 68 H α = H α H c,, α = A, B 69 7