Linear response theory

Transcription

1 Chap 3 Linear response theory Ming-Che Chang Department of Physis, National Taiwan Normal University, Taipei, Taiwan (Dated: April 6, 23) I. GENERAL FORMULATION A material would polarize, or arry a urrent under an external eletri field, P = χ e E () j = σe. If the eletri field is not too strong, then the eletri suseptibility the ondutivity are independent of the eletri field. They only depend on the material properties in the absene of the eletri field (in equilibrium). This type of response is alled the linear response. The average of an observable (suh as the eletri polarization) in equilibrium is A = {n} e βe{n} {n} A {n}. (2) Under an external field, the states {n} are perturbed to beome {n}, the average beomes A = e βe {n} {n} A {n} = A + δ A. (3) {n} Our job is to find out δ A. In the following, as long as there is no ambiguity, we will write the labels of a manybody state {n} simply as n. Before perturbation, H n = E n n, (4) E n n are eigne-energies eigenstates of the manybody Hamiltonian H. The external perturbation is assumed to be H = H + H (t)θ(t t ). (5) That is, the perturbation is turned on at time t. After the perturbation, H(t) n(t) = i n(t). (6) t In the interation piture, the perturbed states n(t) = e iht n I (t) (7) = e ih t U I (t, t ) n I (t ), t U I (t, t ) = i dt H I(t ) +, t (8) n I (t ) = n is the states before perturbation. Substitute Eqs. (7) (8) into Eq. (3), eep only the terms to linear order in H, we have A(t) (9) t = A i dt n [A I (t), H I(t )] n e βe n t n t = A i dt [A I (t), H I(t )]. t For example, if H (t) = then dv B(r) f(r, t), () operator C number δ A(r, t) () t = i dv dt [A I (r, t), B I (r, t )] f(r, t ) t = i dv dt θ(t t ) [A I (r, t), B I (r, t )] f(r, t ), This an be written as δ A(x) = dx χ ABα (x, x )f α (x ), (2) α x = (r, t), dx dv dt, χ ABα (x, x ) = iθ(t t ) [A I (x), B Iα (x )]. (3) Eq (2) is alled the Kubo formula, χ ABα is alled the response funtion. Notie that the operators are written in the interation piture. II. DENSITY RESPONSE AND DIELECTRIC FUNCTION A. Density response In this setion, we onsider the perturbation of eletron density aused by an external eletri potential. Before perturbation, H = T + V L + V ee, (4)

2 2 V L is a one-body interation, suh as the eletronion interation, V ee is the eletron-eletron interation. The perturbation an be written in the following form, H = dvρ e (r)φ ext (r, t), (5) ρ e = q s ψ s(r)ψ s (r) (q = e) is the eletron density, φ ext is an external potential. Beause of the external potential φ ext, eletron density ρ e ρ e = ρ e + δ ρ e. (6) Comparing with the Kubo formula, we find the following replaement neessary, A ρ e, (7) B ρ e, f φ ext. The Kubo formula gives δ ρ e (x) = dx χ ρe (x, x )φ(x ), (8) the response funtion is χ ρe (x, x ) = iθ(t t ) [ρ e (x), ρ e (x )]. (9) Remember that the operators are in the interation piture, but the subsript I is negleted from now on. If the unperturbed system H is uniform in both spae time, then χ ρe (x, x ) = χ ρe (x x ). (2) In this ase, the onvolution theorem in Fourier analysis tells us that δ ρ e (κ) = χ ρe (κ)φ ext (κ) (2) κ (q, ω), κx q r ωt, δ ρ e (x) = e iκx δ ρ e (κ), (22) κ δ ρ e (κ) = dxe iκx δ ρ e (x) ; φ ext (x) = e iκx φ ext (κ), κ φ ext (κ) = dxe iκx φ ext (x). The summation over should be understood as = dω 2π. (23) κ The Fourier expansion of the response funtion is q χ ρe (κ) = d(x x )e iκ(x x ) χ ρe (x x ) (25) = i d(t t )θ(t t )e iω(t t ) d(r r )e iq (r r ) [ρ e (r, t), ρ e (r, t )]. Sine the system is uniform in spae, one an perform an extra spae integral dr to the spae integral above, use dr d(r r ) = dr dr. (26) Then it is not diffiult to see that χ ρe (κ) = i dte iωt [ρ e (q, t), ρ e ( q, )]. (27) Notie that ρ e ( q, ) an also be written as ρ e(q, ). In the following, we may sometimes use the partile density ρ its response funtion χ ρ, whih are related to the eletron density its response funtion as ρ e = eρ, χ ρe = e 2 χ ρ. (28) Also, notie that these response funtions are related to, but not exatly the same as, the eletri suseptibility χ e introdued at the beginning of this hapter. B. Dieletri funtion The response funtion onnets δρ e with φ ext. However, the dieletri funtion onnets φ ext with the total potential φ, whih is the sum of φ ext the potential due to material response, The total partile density is ɛ(κ) = φ ext(κ) φ(κ). (29) ρ = ρ ext + δ ρ, (3) whih are related to the potentials via the Poisson equations (CGS), q 2 φ ext (κ) = 4πe ρ(κ) ext, (3) q 2 φ(κ) = 4πe ρ(κ). Notie that quantities suh as φ ext (κ) = φ ext (q, ω) is allowed to be frequeny-dependent. Also, if one prefers the MKS system, then just replaes 4π with ɛ. Combine the equations above, we get χ ρe (x x ) = κ e iκ(x x ) χ ρe (κ), (24) φ(κ) = φ ext + 4πe 2 χ ρ φ ext q 2. (32)

3 This leads to ɛ(κ) = + 4πe2 q }{{ 2 χ ρ, (33) } V (2) (q) The seond term in φ(r) is the indued potential, or the loal field orretion. That is, if one alulates the response to the total perturbing potential φ(r), then the unperturbed system is H, whih is non-interating. 3 in whih V (2) (q) is the Fourier transform of V (2) (r) = e 2 /r. Instead of using δ ρ = χ ρ φ ext, an alternative relation is It s not diffiult to see that δ ρ = χ ρφ, φ = φ ext + δφ. (34) χ ρ = χ ρ, (35) 4πe2 q χ 2 ρ ɛ(κ) = 4πe2 q 2 χ ρ. (36) The alulation of χ ρ is based on Eq. (27), in whih one averages over unperturbed manybody states (inluding eletron interations). A great advantage of using the alternative response funtion χ ρ is that, sine the loal field orretion has been inluded in φ, one may use non-interating manybody states in the alulation of the response funtion. This is justified as follows: The interation term is, apart from a one-body orretion (see Se. IV.B. of Chap ), V ee = dvdv V (2) (r r )ρ e (r)ρ e (r ). (37) 2 Using the mean field approximation, exp the harge density with respet to a mean value ρ(r) e, ρ e (r) = ρ e (r) + ρ e (r) ρ e (r). (38) δρ e (r) Negleting the (δρ e ) 2 term, we have V ee dvdv V (2) (r r )ρ e (r) ρ e (r ) (39) dvdv V (2) (r r ) ρ e (r) ρ e (r ). 2 C. Calulation of χ ρ We now drop the supersript subsript that refer to equilibrium states. Reall that χ ρ(κ) = i dte iωt [ρ(q, t), ρ( q, )]. (42) In the interation piture, ρ(q, t) = e ih t ρ(q)e ih t. The summation n I(q; t, ) (43) = n,m e βe n e βen n ρ(q, t)ρ( q, ) n ei(en Em)t n ρ(q, ) m m ρ( q, ) n, we have inserted a omplete set m m m, used e iht m = e iemt m. Sine both n m are manybody states of noninterating partiles, m a s a q,s n an be non-zero only if, when omparing to n, the m state has one more eletron at state (, s), but one less eletron at ( q, s). Therefore, E n E m = ε + ε q, a differene of two single-partile energies. This energy fator an now be moved outside of the m-summation, I(q; t, ) (44) = e βen e i(ε q ε )t n a q,s a sa s a q,s n n =,s = 2,s e i(ε q ε )t n e βen e i(ε q ε )t f(ε q )[ f(ε )], n a q,s a q,s( a s a s) n The mean-field Hamiltonian under perturbation is (dropping the seond term in Eq. (39)) H MF (4) = H + dvdv V (2) (r r )ρ e (r) ρ e (r ) + dvρ e (r)φ ext = H + dvρ e (r)φ(r), H = T + V L, φ(r) = φ ext (r) + dv V (2) (r r ) ρ e (r ). (4) f(ε ) = n e βen n a s a s n (45) is the Fermi distribution funtion (spin-independent here). It is left as an exerise to show that Similarly, one an show that I( q;, t) = 2 f(ε ) = + e βε. (46) e i(ε q ε )t f(ε )[ f(ε q )]. (47)

4 4 From Eq. (42), we have χ ρ(κ) = i = 2i dte iωt [I(q; t, ) I( q;, t)] (48) The integral over time is dte iωt e i(ε q ε )t [f(ε q ) f(ε )]. dte i(ω+iδ)t e i(ε q ε )t i = ω + iδ + (ε q ε ). (49) The positive infinitesimal δ is added to ensure the onvergene of the exponential at t =. Finally, χ ρ(q, ω) = 2 f(ε q ) f(ε ) ω + iδ + (ε q ε ), (5) ɛ(q, ω) = 4πe2 2 f(ε q ) f(ε ) q 2 ω + iδ + (ε q ε ). (5) This is alled the Lindhard dieletri funtion.. Low frequeny limit For frequeny as low as ω v F q, the ω in the denominator an be negleted, χ ρ(q, ) 2 f(ε q ) f(ε ) ε q ε. (52) For general wave length (in 3-dim), it an be shown that ( ) q χ ρ(q, ) D(ε F )F, (53) 2 F D(ε F ) is the density of states at the Fermi energy, F (x) = 2 + x2 4x ln + x x (54) is the Lindhard funtion (see Se. II.A). At long wavelength, χ ρ(q, ) 2 ( f ) = D(ε F ). (55) ε In this limit, the dieletri funtion is ɛ(q, ) = + 2 T F q 2, (56) T 2 F = 4πe2 D(ε F ) is the Thomas-Fermi wave vetor. 2. High frequeny limit The response funtion in Eq. (5) an be re-written as χ ρ(q, ω) = 2 2(ε q ε ) f(ε ) ω 2 (ε q ε ) 2. (57) For high frequeny long wave length (ω v F q), χ ρ(, ω) q2 2 mω 2 n is the partile density. Therefore, f(ε ) = q2 n mω 2, (58) ɛ(, ω) = ω2 p ω 2, (59) ω 2 p = 4πne 2 /m is the plasma frequeny. Notie that lim lim ɛ(q, ω) lim lim ɛ(q, ω). (6) q ω ω q That is, the dieletri funtion is not analyti at (q, ω) = (, ). III. CURRENT RESPONSE AND CONDUCTIVITY In this setion, we onsider the generation of eletron urrent aused by an external eletri field. Before perturbation, H = dvψ (r) p2 2m ψ(r) + V L + V ee, (6) V L is the one-body interation, V ee is the eletron interation. In general, the external eletri field depends on both salar vetor potentials, E(r, t) = φ A t. (62) For a stati field, it is ommon to use E = φ, as in Eq. (5). A stati uniform field then has φ(r) = E r. A disadvantage of this salar potential is that it is not bounded at infinity. To avoid suh a problem, one an hoose a gauge suh that E(r, t) = A t. (63) In this ase, a stati uniform field has A(t) = Et. After applying the eletri field, the Hamiltonian beomes ( p + e H = dvψ (r) A) 2 ψ(r) + V L + V ee (64) 2m = H + e dv ( ψ p Aψ + ψ A pψ ) 2m + e2 2m 2 dva 2 ψ ψ,

5 5 H refers to the parts that do not depend on A. The partile urrent density operator J is related to the variation of the Hamiltonian as follows, δh = e dvj δa, (65) J [ ψ ψ ( ψ ) ψ ] + } 2mi {{} paramagneti urrent J p e m Aψ ψ. diamagneti urrent J A (66) We would lie to find out the onnetion between J A (to first order). After perturbation, a manybody state n n n + n, (67) n is of order A. Therefore, n J n = n J A n + n J p n + n J p n + O(A 2 ). (68) We have assumed, of ourse, that the equilibrium state arries no urrent, n J p n =. After taing the thermal average, the first term beomes J A = e A(r, t) ρ(r). (69) m The other two terms are evaluated using the Kubo formula in Eq. (2), with the following replaement, A J p α, (7) B J p, f e A. This gives us (reall that x = (r, t)) Jα(x) p = e dx χ p αβ (x, x )A β (x ), (7) χ p αβ (x, x ) = iθ(t t ) [Jα(x), p J p β (x )]. (72) After ombining with the diamagneti term in Eq. (69), the response funtion for the total urrent is χ αβ (x, x ) = δ αβ δ(x x ) ρ(x) m + χp αβ (x, x ). (73) Sine H is time independent, the response funtion χ αβ (x, x ) = χ αβ (r, r ; t t ). Applying the onvolution theorem to the time variable (see Eq. (2)), one has J α (r, ω) = e dv χ αβ (r, r ; ω)a β (r, ω). (74) The vetor potential is related to the eletri field as follows, E(ω) = i ω A(ω). (75) Therefore, for the eletri urrent density J e = ej, one has Jα(r, e ω) = dv σ p αβ (r, r ; ω)e β (r, ω). (76) The ondutivity tensor is σ αβ (r, r ; ω) = i e2 ω χ αβ(r, r ; ω). (77) Sine the ondutivity in general is a non-loal quantity, the urrent density at point r would not only depend on the eletri field at r, but also on neighboring eletri field. For a homogeneous material, σ αβ (r, r ; ω) = σ αβ (r r ; ω). (78) We an then apply the onvolution theorem to the spae variable get J e α(q, ω) = σ αβ (q, ω)e β (q, ω), (79) σ αβ (q, ω) = i e2 ω [ ] ρ(q, ω) δ αβ m + χp αβ (q, ω), (8) in whih (Cf. eq. (27)) χ p i αβ (q, ω) = dtθ(t t )e iω(t t ) [J α (q, t), J β ( q, t )]. (8) Notie that the diamagneti part diverges as ω. For usual ondutors insulators, this divergene would be anelled by part of the paramagneti term, so that the DC ondutivity remains finite. In a superondutor (whih is a perfet diamagnet), the paramagneti term vanishes in the DC limit, the ondutivity is purely imaginary, σαβ SC (q, ω) = i e2 ω δ ρ(q, ω) αβ m. (82) A purely imaginary ondutivity leads to indutive behavior, would not ause energy dissipation. A. Condutivity for non-interating eletrons We would lie to start from a formulation that does not presume spaial homogeneity: χ p αβ (r, r ; ω) = i [ ] dte iωt J V α(r, p t), J p β (r, ), (83) Jα(r, p t) = e iht Jα(r)e p iht. Therefore, n I αβ (r, t, r, ) (84) = n,m e βe n e βe n n J p α(r, t)j p β (r, ) n ei(e n E m )t n J p α(r, ) m m J p β (r, ) n,

6 6 in whih we have inserted a omplete set m m m, used e ih t m = e ie mt m (see Se. II.C). The urrent density operator an be written as (see Chap ) J p α(r) = µ J () α (r) ν a µa ν, (85) J α () is a one-body operator to be speified later. From now on, assume the eletrons are non-interating. Substitute Jα(r) p into Eq. (84), we get terms with the form n a a 2 m m a 3 a 4 n, (86), 2 are simplified notations for single-partile state labels µ, ν. For this type of term to be non-zero, the single-partile states have to satisfy ( = 4, 2 = 3), or ( = 2, 3 = 4). They both lead to E n E m = ε ε 2 (the seond ase has ε = ε 2 ). The summation over m an now be removed, I αβ (r, t, r, ) =,2,3,4 e i(ε ε 2 )t J () α 2 3 J () β 4 (87) a a 2a 3 a 4 (δ 4 δ 23 + δ 2 δ 34 ). The thermal averages are (see Eq. (45)) a a 2a 2 a = f ( f 2 ), (88) a a a 2 a 2 = f f 2. f is the Fermi distribution funtion. As a result, one an show that = 2 Therefore, I αβ (r, t, r, ) I βα (r,, r, t) (89) e i(ε ε 2 )t J () α 2 2 J () β (f f 2 ). χ P αβ(r, r, ω) (9) = i 2 dte iωt [I αβ (r, t, r, ) I βα (r,, r, t)] = () J α (r) 2 2 J () β (f f 2 ) (r ), Ω + ε ε 2 Ω ω + iδ If the material is homogeneous, then χ P αβ(q, ω) = () J α (q) 2 2 J () β (f f 2 ) ( q), Ω + ε ε 2 2 The one-body operator is (see Chap ) J α () (q) = ( pα e iq r + e iq r ) p α 2m (9) (92). Uniform limit For the uniform ase (q = ), the ondutivity is σ αβ (, ω) (93) [ ] = ie2 ρ(ω) δ αβ ω m + m 2 (f µ f ν ) µ p α ν ν p β µ. Ω + ε µ ε ν (We have re-written, 2 as µ, ν.) The denominator an be deomposed as ( ɛ ± Ω = ε Ω ε ± Ω ), (94) ε ε µ ε ν. Substitute this to Eq. (93), then the first term of the deomposition would anel with the diamagneti term, beause of the following f-sum rule: (f µ f ν ) µ p α ν ν p β µ = mρδ αβ. (95) ε As a result, σ αβ (, ω) = e2 (f µ f ν ) µ v α ν ν v β µ, (96) i ε (Ω + ε ) v α = p α /m. This is sometimes alled the Kubo- Greenwood formula. 2. Uniform stati, Hall ondutivity Finally, we would lie to onsider the DC Hall ondutivity as an example. Aording to Eq. (96), it an be re-written as σ DC α β = e2 i µ v α ν ν v β µ µ v β ν ν v α µ f µ ε 2. (97) If the single-partile states are Bloh states (µ n), r n = e i r u n (r), u n (r) is the ell-periodi funtion, then one an show that σα β DC = e2 f n i n ( un u n un u ) n, α β β α Berry urvature Ω γ n (98) α, β, γ are yli. We will all this as the TKNdN formula (see Ref. 4). In 2-dim, for a filled b, C (n) 2π d 2 Ω z n (99) filled B must be an integer (see Se. II.B of Ref. 5). Therefore, the Hall ondutivity from filled bs is quantized, σ DC xy = e2 h filled n C (n), ()

7 7 we have put ba the expliitly. The quantized (topologial) nature of suh an integral is first pointed out by D.J. Thouless to explain the quantum Hall effet. Prob. Derive the f-sum rule, (f µ f ν ) µ p α ν ν p β µ = mρδ αβ. () ε Prob. 2 Start from Eq. (97), derive the TKNdN formula in Eq. (98). Referenes [] Chap 6 of H. Bruss K. Flensberg, Many-body quantum theory in ondensed matter physis, Oxford University Press, 24. [2] Se. I.5 of T. Giamarhi, A. Iui, C. Berthod, Introdution to Many body physis, on-line leture notes. [3] Chap 6: Eletron Transport, by P. Allen, in Coneptual Foundations of Materials: A stard model for ground exited-state properties, ed. by S.G. Louie K.L. Choen, Elsevier Siene 26. [4] D.J. Thouless, M. Kohmoto, P. Nightingale, M. de Nijs, Phys. Rev. Lett. 49, 45 (982). [5] D. Xiao, M.C. Chang, Q. Niu, Rev. Mod. Phys. 82, 959 (2).