017 7 3 1 4 3 5 4 6 5 6 5.1............................... 7 5.................................... 7 5.3............................... 9 5.4................................. 10 5.5................................. 1 5.6................................. 13 5.7................................. 13 6 14 6.1............................. 14 6..................................... 15 6.3.................................. 16 1
6.4.................................... 17 6.5...................... 17 1 *1 B(t) B(t) = tr (ρ(t)b(t)) ρ(t) ρ(t) = e H/T Z (1) H T Z Z = tr ( e ) H/T H tot (t) = H + H ext (t) t ρ(t) 1. t = ρ() = e H/T /Z. ρ(t) ρ(t) = n e H/T Z n(t) n(t) () n(t) t i t n(t) = H tot(t) n(t) (3) i t n(t) = n(t) H tot(t) (4) *1
ρ(t) t ρ(t) = i [ρ(t), H tot (t)] (5) ρ H ext ρ(t) = ρ() + δρ(t) t δρ(t) = i[δρ(t), H] + i[ρ(), H ext ] (6) δρ(t) = e iht Ce iht C e iht [ t C(t)]e iht = i[ρ(), H ext (t)] t C(t) = ρ() + i dt e iht [ρ(), H ext (t )]e iht (7) t ρ(t) = ρ() + i dt e ih(t t ) [ρ(), H ext (t )] e ih(t t ) (8) t B(t) tot = tr (ρ(t)b(t)) t B(t) tot = B(t) + i dt [ H ext (t ), B H (t t ) ] (9) = tr [ρ () ] B H (t) = e iht B(t)e iht F (t) A H ext (t) = A F (t) (10) B(t) tot = B(t) + t dt ϕ BA (t t )F (t ) (11) ϕ BA (t) = i [ B H (t), A ] (1) 3
ϕ BA (t) B(ω) tot = χ BA (ω) = dte iωt B(t) tot = B(ω) + χ BA (ω)f (ω) (13) dt e iωt θ(t)ϕ BA (t) (14) χ BA (ω) ( ) B(ω) = dt eiωt B(t) G AB (τ) = T τ A H ( iτ)b (15) T τ G AB (τ) = θ(τ) A H ( iτ)b ηθ( τ) BA H ( iτ) (16) η = ±1 A B A,B η = 1 η = 1 τ < β G AB (τ + β) = ηg AB (τ) (17) τ > β τ < β τ > β β G AB (τ) = T n e iω nτ G AB (iω n ) (18) G AB (iω n ) = β 0 dτe iω nτ G AB (τ) (19) 4
ω n = { ν n = nπt, η = 1 ϵ n = (n + 1)πT, η = 1 (0) ω n 3 G α AB(t) = iαθ(ατ) A H (t)b ηba H (t) (1) α = 1 α = 1 G α AB(ω) = e iωt G α AB(t) = nm e E n/t ηe E m/t Z n A m m B n ω + E n E m + iα0 () n E n G ± AB (ω) = G (3) B A A = B G + = G AA AA G AB (iω l ) = β 0 dτe iω lτ G AB (τ) = nm e E n/t ηe E m/t Z n A m m B n iω l + E n E m (4) G α AB(ω) = G AB (iω l ω + iα0) (5) S AB (t) = 1 A(t)B + BA(t) (6) 5
S AB (ω) = 1 dt e iωt A(t)B + BA(t) = π ( ) e En/T + e E m/t A nm B mn δ(ω + E n E m ) Z nm = π 1 + e ω/t Z e En/T A nm B mn δ(ω + E n E m ) (7) nm B = A η = 1 ( ω ) S AA (ω) = coth Im G T AA (ω + i0) (8) ω 0 S AA (ω) T ω Im G AA (+i0) (9) 4 ω n T n T n 1 1 iω n α iω n β = η f η(α) f η (β) α β G(iω n )G(iω n + ω + i0) = η dϵ πi f η(ϵ) { G (ϵ ω) [ G + (ϵ) G (ϵ) ] (30) + [ G + (ϵ) G (ϵ) ] G + (ϵ + ω) } (31) f η (x) = (e x/t η) 1 (η = 1) (η = 1) 5 H = d 3 xψ (x)h(x, i + ea)ψ(x), (3) A 6
5.1 E bias A bias E bias = t A bias H tot = d 3 xψ (x)h(x, i + ea tot )ψ(x), A tot = A + A bias, (33) ĵtot(x, t) ĵ tot (x, t) = δh = eψ (x, t)ˆv tot ψ(x, t), ˆv tot = H(x, i + ea tot), (34) δa tot ( i ) ĵ(x, t) ĵ1(x, t) (ĵtot = ĵ + ĵ1) ĵ(x, t) = eψ (x, t)ˆvψ(x, t), ĵ1(x, t) = eψ (x, t)ˆv 1 ψ(x, t), (35) ˆv = H(x, i + ea), ˆv 1 = RA bias + O(A ( i ) bias), (36) R ij = e H(x, i + ea), R ij = R ji, (37) ( i i ) ( i j ) [x i, i j ] = iδ ij ˆv = i[x, H(x, i + ea)], (38) 5. j 1 A bias = 0 j H = d 3 xψ (x)h(x, i + ea)ψ(x) d 3 xĵ(x, t) A bias + O(A ). (39) 7
A bias E E(ω) = dte iωt ( t A) = iωa(ω), (40) j d 3 x ĵ(ω) = (13) j(ω) j(ω) = K(ω)A(ω) K ij (ω) = j 1 [ d dtθ(t)e iωt 3 x i ĵi(x, t), dte iωt ĵ(x, t), (41) ] d 3 yĵ j (y, 0), (4) d 3 x j 1 (ω) = e ψ (x)rψ(x) A bias (ω), (43) ϕ α (x) ψ a (x) = α ϕ α (x)c αa, d 3 xϕ α(x)ϕ β (x) = δ αβ, (44) ψ a (x) = k e ik x c ka, (45) ĵ(t) := d 3 x ĵ(x, t) = e c α(t)v αβ c β (t), v αβ = d 3 xϕ α(x)ˆvϕ β (x), (46) αa 1 α K K ij (ω) = i [ ] dtθ(t)e iωt ĵ i (t), ĵ j (0), (47) 8
Q ij (τ) = 1 T τ ĵ i ( iτ)ĵ j (0), (48) K(ω) = Q(ω + i0) σ ij σ ij (ω) = e d 3 x ψ (x)r ij ψ(x) + K ij(ω) iω iω, (49) ω 0 5.4 (49) e iω c αa[r ij ] αβ ab c βb = e iω [R ij] αβ ab G βb;αa( 0), (50) [R ij ] αβ ab v αβ = d 3 xϕ α(x)[r ij ] ab ϕ β (x). (51) e iω tr[r ijg(τ = 0)] = e ω dϵ π f(ϵ)tr { R ij [ G + (ϵ) G (ϵ) ]}, (5) 5.3 * Q ij (τ) = e tr [v ig(τ)v j G( τ)], (53) Q ij (iν n ) = T 1 0 = e T m dτe iν nτ Q ij (τ) tr [v i G(iϵ m + iν n )v j G(iϵ m )], (54) * A. Bastin, C. Lewiner, O. Betbeder-matibet, and P. Nozieres, J. Phys. Chem. Sol. 3, 1811 (1971). 9
T 1 G αβ (τ) = T τ c α ( iτ)c β, G αβ (iϵ n ) = dτe iϵnτ G αβ (τ), (55) 0 G(z) = (z H) 1, (56) Ĝ; [z H(x, i + ea/c)]ĝ(x y, z) = δ3 (x y), (57) *3 ; [G(z)] αβ = d 3 xϕ α(x)ĝ(x, z)ϕ β(x). (58) iν n ω + i0 K ij (ω) = Q ij (ω + i0) = e dϵf(ϵ)tr [ v i G + (ϵ + ω) v j A(ϵ) + v i A(ϵ)v j G (ϵ ω) ], (59) A(ϵ) = i G+ (ϵ) G (ϵ), (60) π G + = G A(ϵ) A αα (ϵ) = 1 π ImG+ αα(ϵ). (61) A(ϵ) 5.4 ω = 0 K ij (0) = e dϵ πi f(ϵ)tr [ v i G + (ϵ)v j G + (ϵ) v i G (ϵ)v j G (ϵ) ]. (6) *3 10
A G ± (ϵ) A = G± (ϵ) H A G± (ϵ) = eg ± (ϵ)vg ± (ϵ), (63) K ij (0) = e {( dϵ πi f(ϵ)tr [ G + (ϵ) G (ϵ) ] ) } v j, (64) A i A=0 R ij = v i A j = v j A i, (65) lim ω 0 ωσ ij(ω) = e = e dϵ π f(ϵ)tr {[ G + (ϵ) G (ϵ) ] } K ij (0) R ij + i { dϵ [G π f(ϵ)tr + (ϵ) G (ϵ) ] ( ) v j A i ( [ + G + (ϵ) G (ϵ) ] A i A=0 ) A=0 v j } = e dϵ A i π f(ϵ)tr {[ G + (ϵ) G (ϵ) ] } v A=0 j = i ĵ j A i, (66) A=0 d 3 x ĵ j = ψ (x)ev j ψ(x) = i dϵ π f tr [( G + G ) ev j ], (67) A ĵ j = 0, *4 p p + ea A *4 11
σ ij (ω) 1/ω σ ij (ω) = K ij(ω) K ij (0), (68) iω 5.5 σ i = σ ii (0) σ i = i K ii(ω) ω = πe ω=0 T = 0 σ i = πe Bastin [ ] A(ϵ) dϵf(ϵ)tr v i A(ϵ)v i, (69) ϵ [ dϵ f(ϵ) ] tr [v i A(ϵ)v i A(ϵ)] (70) ϵ σ i = πe tr [v ia(0)v i A(0)] (71) H = p /m τ < G(ω) = Bastin σ = πe k 1 ω ϵ k + iτ 1 (7) vx τ π δ(ϵ F ϵ k ) = ne τ m τ v x v F /3 ρ = 1 δ(ϵ F ϵ k ) = 3n mvf n k 1 (73) (74)
5.6 ω 0 σ ii (ω) Reσ ii (ω) = [ImK ii (ω) ImK ii (0)]/ω G ± = G v i = v i ImK ii (ω) = πe T = 0 [ ( dϵ f ϵ ω ) ( f ϵ + ω )] [ ( tr v i A ϵ ω ) ( v i A ϵ + ω )], (75) ω/ [ ( ImK ii (ω) T 0 πe dϵtr v i A ϵ ω ) ( v i A ϵ + ω )]. (76) ω/ vi v F /3 Reσ ii (ω) = ne m τ 1 ω, (77) + τ 5.7 ω = 0 σ ij (0) σ ij (0) = i K ij ω = i e ω=0 [ dϵf(ϵ)tr G + (ϵ) G (ϵ) v i v j A(ϵ) v i A(ϵ)v j ϵ ϵ : σ ij (0) = i e ]. (78) f(e α ) f(e β ) (E α E β + i0) (v i) αβ (v j ) βα. (79) 13
5.7.1 j B 5.7. 6 j µ, µ = 0, 1,, 3 (j 0, j 1, j, j 3 ) = ( ecρ, j x, j y, j z ), (80) ρ eρ, e > 0 c c = 1 (A 0, A 1, A, A 3 ) = (cϕ, A x, A y, A z ), (81) x µ µ (x 0, x 1, x, x 3 ) = (ct, x, y, z), ( 0, 1,, 3 ) = ( t /c, x, y, z ), (8) A µ B µ A µ B µ = A 0 B 0 + A 1 B 1 + A B + A 3 B 3, (83) 6.1 E = 4π(ρ ex + ρ in ) (84) B = 0 (85) E = 1 B c t (86) H = 1 E c t + 4π c (j ex + j in ) (87) ρ ex j ex ρ in j in D E D = E + 4πP (88) 14
D P D = 4πρ ex, j ex = 1 D 4π t, P = ρ in, j in = P t, ρ ex + j ex = 0 (89) t ρ in + j in = 0 (90) t B = 0 B = (A + A in ) A A in ( ) ( E + 1 c ) (A + A in ) = 0 (91) t E = 1 c (A + A in ) t (ϕ + ϕ in ) (9) ϕ ϕ in D = 1 A ϕ c t (93) 4πP = 1 A in ϕ in c t (94) D 6. j in,µ (r, t) j in,µ (r, t) = dr dt K µν (r, t; r, t )A ν (r, t ), (95) j in,µ = j µ χ A µ A µ + µ χ, (96) 15
dr dt K µν (r, t; r, t ) νχ(r, t ) = 0, (97) χ νk µν (r, t; r, t ) = 0, (98) K µν r r t t ν K µν (r, t; r, t ) = 0, (99) µ j µ = 0 µ K µν (r, t; r, t ) = 0, (100) K µν (r, t; r, t ) = c d 4 q (π) 4 eiq µ(x µ xµ ) K µν (q, ω), (101) (q 0, q 1, q, q 3 ) = (ω/c, q x, q y, q z ) j µ (q, ω) = K µν (q, ω)a ν (q, ω). (10) q µ K µν (q, ω) = K µν (q, ω)q ν = 0, (103) 6.3 (93) D(q, ω) = i ω A(q, ω) iqϕ(q, ω) (104) c (103) j i (q, ω) = i c ω K ij(q, ω)d j (q, ω) (105) 16
0,1,,3, 1,,3 D E D i (q, ω) = ϵ ij (q, ω)e j (q, ω) ϵ ij j i (q, ω) = i c ω K ij(q, ω)ϵ jk (q, ω)e k (q, ω) (106) σ ij (q, ω) = i c ω K ik(q, ω)ϵ kj (q, ω) (107) 6.4 D = E + 4πP 4πP = (ϵ 1 1)D 4πj in = iω(ϵ 1 1)D j in = t P j in (105) j (j = j in ) ϵ 1 = 1 + 4πc ω K (108) K ij (90) ρ in = iq P = iq (D E)/4π = iq (1 ϵ 1 )D/4π (93) ρ in = iq (1 ϵ 1 )(iωa iqa 0 )/4πc cρ in = K 00 A 0 + K 0i A i ϵ 1 ij ϵ 1 ij = δ ij 4π q iq j q 4 K 00 (109) = δ ij 4πq j ωq K 0i (110) 6.5 17
6.5.1 *5 H = c mt mn c n, t mn = t nm (111) mn m, n t mn = t m n c m = 1 e ik m c(k), t m n = 1 e ik (m n) H(k), (11) N N k k H = k c (k)h(k)c(k) (113) H(k) H = [ ( ) m + n c m exp iea mn l ] (m n) t mn c n (114) j l = δh δa(l) = i e (m n)δ l,(m+n)/ c me iea(l) (m n) t mn c n (115) mn j p j d j pl = i e j dl = e (m n)δ l,(m+n)/ c mt mn c n (116) mn (m n)a(l) (m n)δ l,(m+n)/ c mt mn c n + O(A ) (117) mn *5 J. M. Luttinger and W. Kohn, Phys. Rev. 97, 869 (1955). 18
j p (q) = l = e e iq l j pl ( c k q ) H(k) (k k c + q ) k (118) j d (q) = l = e e iq l j dl kk c ( k q ) H(q ) q q q =(k+k )/ ( k k A ) ( c k + q ) (119) 6.5. k = i k π = i qa H = mn c mt mn c n = d 3 xψ (x) ( ) ( m n exp t mn δ x m + n ) ( n m exp ) ψ(x) (10) mn ψ(x); ψ(n) = c n H = H iqa [ ( )] m + n t mn t mn exp iq(m n) A (11) [ m ] exp iq dx A(x) n (1) 19
6.5.3 H = [ c mt mn exp iq(m n) A mn ( m + n )] c n (118), (119) M 6.5.4 g L atom S M atom M atom = µ B (L atom + gs) (13) µ B = eħ/(m e c) > 0 g g = 0
1 p g Γ J L S g L Γ 6 1/ 0 1/ Γ 7 1/ 1 1/ /3 Γ 8 3/ 1 1/ 4/3 J atom = L + S M atom = g L µ B J (14) g L L + gs = g L J J L = L(L + 1), S = S(S + 1), J = J(J + 1), L S = [J(J + 1) L(L + 1) S(S + 1)]/ g = g L = 3 + S(S + 1) L(L + 1) J(J + 1) (15) g L g 1 M atom B H Z,atom = m c m( M atom B)c m = k c (k)( M atom B)c(k) (16) M atom m *6 k 0 *6 J. M. Luttinger and W. Kohn, Phys. Rev. 97, 869 (1955); M. Graf and P. ogl, Phys. Rev. B 51, 4940 (1995). 1
k = k 0 H(k) = H(k 0 ) + H(k 0) k 0 (k k 0 ) + 1 (k k 0) H(k 0 ) k 0 k 0 (k k 0 ) (17) n(k), H(k) n(k) = E n (k) n(k) m n kp H mn (k) = E m (k 0 )δ mn + v mn (k 0 )(k k 0 ) + 1 (k k 0)m 1 mn(k 0 )(k k 0 ) (18) m, n [m mn (k 0 )] ij m 1 mn(k 0 ) = [ vmm (k 0 )v m n(k 0 ) E m (k 0 ) E m (k 0 ) + v ] mm (k 0 )v m n(k 0 ) E n (k 0 ) E m (k 0 ) m m + H mn (k 0 ) k 0 k 0 (19) m m mm (k 0 ) (111) Roth (Peierls substitution *7 ) k k 0 π = i + ea *8 π B π i π j = 1 {π i, π j } + 1 [π i, π j ] = 1 {π i, π j } iϵ ijk eb k (130) k 0 m n H mn (π) = E m (k 0 )δ mn + v(k 0 )π + 1 [m 1 mn(k 0 )] ij π i π j = E m (k 0 )δ mn + v(k 0 )π + 1 + 1 [m 1 mn(k 0 )] ij [m 1 mn(k 0 )] ji [π i, π j ] [m 1 mn(k 0 )] ij + [m 1 mn(k 0 )] ji {π i, π j } (131) H mn (π) = E m (k 0 )δ mn + v mn (k 0 ) ( i ) + 1 ( i ) m 1 mn(k 0 ) ( i ) J mn A M mn B (13) *7 R. Peierls, Z. Phys., 80, 763 (1933). *8 k = k 0
J pmn (k 0 ) = e v mn(k 0 ) + e J dmn (k 0 ) = e m m m 1 mn(k 0 ) + m 1 [ 1 vmm (k 0 ) v m n(k 0 ) M mn (k 0 ) = im e µ B E m (k 0 ) E m (k 0 ) m 1 mn(k 0 ) + m 1 mn(k 0 ) T k (133) mn(k 0 ) T A (134) + v mm (k ] 0) v m n(k 0 ) E n (k 0 ) E m (k 0 ) (135) H(k 0 )/ k 0i k 0j ij Roth *9 k = k 0 H band (k 0 ) = diag (H 1 (k 0 ),, H M (k 0 )) (136) H i (k 0 ) = E i (k 0 )1 g i (k 0 ) g i (k 0 ) g i (k 0 ) E i (k 0 ) (13) (135) M(k 0 ) = M atom + M band (k 0 ) (M i (k 0 ) = P i (k 0 )M(k 0 )P i (k 0 ), P i (k 0 ) = g i (k 0 ) m=1 m; E i(k 0 ) m; E i (k 0 ) ) H i (k 0 ) H i (k 0 ) M i (k 0 ) B k 0 ( E1 (k H 1 (k 0 ) = 0 ) M 11 (k 0 ) B M 1 (k 0 ) B ( E (k H (k 0 ) = 0 ) M 33 (k 0 ) B M 43 (k 0 ) B. ) E 1 (k 0 ) M (k 0 ) B ) E (k 0 ) M 44 (k 0 ) B (137) (138) H(k 0 ) = U(k 0 )H band (k 0 )U (k 0 ) (139) H band (k 0 ) = U (k 0 )H(k 0 )U(k 0 ) M band (k 0 ) k 0 k 0 *9 L. M. Roth, B. Lax, and S. Zwerdling, Phys. Rev. 114, 90 (1959). 3