Field-theory computation of charge- density oscillations in a Tomonaga-Luttinger model with impurities

Transcription

1 La Plata 99-6 arxiv:cond-mat/99290v2 [cond-mat.str-el] 6 Mar 2000 Field-theory computation of charge- density oscillations in a Tomonaga-Luttinger model impurities Victoria I. Fernández a,b and Carlos M. Naón a,b March 2000 Abstract We compute the charge-density oscillations in a Tomonaga-Luttinger model impurities for a short-range electron-electron potential. Pacs: Fk.0.Lm 7.0.Pm a Depto. de Física. Universidad Nacional de La Plata. CC 67, 900 La Plata, Argentina. b CONICET. victoria@venus.fisica.unlp.edu.ar naon@venus.fisica.unlp.edu.ar

2 In a recent paper [] we introduced an extended version of the Nonlocal Thirring model (NLT) [2] which includes a local interaction between the fermion current and a classical background field C µ. Thus, the NLT interaction Lagrangian density L int = 2 g2 d 2 yj µ (x)v (µ) (x, y)j µ (y), () is now modified by adding L imp = Ψ/CΨ M(x) ΨΨ (2) The name L imp refers to the fact that this density can be used to study the interaction between electrons and impurities [3]. Indeed, if one uses () in order to describe the forward scattering of one-dimensional spinless electrons [2] [4], then the first (second) term in the rhs of (2) models the forward (backward) scattering between electrons and impurities. In particular, if we choose C (x) = 0 together C 0 (x) = V δ(x 0 )δ(x d) = M(x) (3) where V is a constant, our model coincides the one recently considered in ref.[5] to study Friedel charge-density oscillations in a d Tomonaga-Luttinger liquid [6] an impurity located at x = d. We now start by considering the general case (C µ (x) and M(x) arbitrary). Using a convenient representation of the functional delta and introducing the vector field A µ (Please see ref. [7] for details), the partition function of the model under consideration can be written as Z = DA µ det(i/ + iγ 0 p F + g/a + /C M(x))e S[A], (4) S(A) = 2 d 2 xd 2 yv (µ) (x, y)a µ(x)a µ (y), (5) where V (µ) is such that d 2 z V (µ) (z, x)v (µ)(y, z) = δ 2 (x y). (6)

3 As shown in [] the partition function (4) is equivalent to a purely bosonic one, corresponding to a non-local Sine-Gordon model given by 2 L NLSG = 2 ( µϕ(x)) 2 + d 2 y µ ϕ(x)d (µ) (x, y) µ ϕ(y) + L imp, (7) L imp = F µ (x) µ ϕ(x) α 0(x) β 2 cosβϕ(x) (8) F µ (x) represents a couple of classical functions to be related to the C µ s and α 0 (x) is to be related, of course, to M(x). Indeed, it is straightforward to show that the corresponding partition functions coincide provided that the following identities hold: and g 2 π (p2 ˆV 0 () + p 2 ˆV (0) ) + p = β2 2 4π(p 2 + ˆd (0) p ˆd () p 2 ), (9) M(x) cos(2p F x ) = α 0(x) β 2 (0) 2iˆV (µ) ( p)ĉµ( p) p µc(p) 2ǫ µν p ν B(p) (p) 2g 2 p µ p µ B(p)ĥ( p) = (p) ˆF µ ( p)p µ β p 2 + ˆd (0) p ˆd () p 2 () where A(p) = g2 2π p2 + [ˆV (0) 2 (p)p2 + ˆV () (p)p2 0 ], (2) B(p) = [ˆV (0) 2 (p)p2 0 + ˆV () (p)p2 ], (3) 2

4 C(p) = [ˆV (0) (p) ˆV () (p)]p 0p, (4) and (p) = C 2 (p) 4A(p)B(p), (5) D(p, x i, y i ) = i (e ipx i e ipy i ). (6) As stated in the introductory paragraph, the path-integral identification depicted above allows to make contact recent descriptions of d electronic systems in the presence of fixed (not randomly distributed) impurities [5]. We shall now discuss this possibility. To be specific we shall consider the charge-density expectation value in a Tomonaga-Luttinger electronic liquid (ˆV () = 0), a non-magnetic impurity located al x = d, where ρ(x) = i π ϕ cos(2p F x ) cos(2 πϕ) LBos (7) L Bos = ( 2 µϕ(x)) 2 + d 2 y ϕ(x) V (0)(x,y) π ϕ(y) 2i d 2 yp F x (V (0) (x, y) ) y ϕ(y) + i π C 0 (x) ϕ C 0 (x) cos(2p F x ) cos(2 πϕ). It is convenient to have an even Lagrangian density. This is easily achieved through a translation in the field ϕ. The result is L Bos = ( 2 µϕ(x)) 2 + d 2 y ϕ(x) V (0)(x,y) π ϕ(y) C 0 (x) cos(2p F x ) cos(2 πϕ) where f is a classical function satisfying (8) and 0 f = 0 (9) 3

5 f + 2 π d 2 yv (0) (x, y) f y π C 0 + 2ip F d 2 yy xv 0(x, y) = 0 (20) + i Thus we get ρ(x) i π f = cos(2p F x ) cos( 4πf(x )) cos( 4πϕ) L Bos, (2) where cos(2p F x ) cos( 4πf(x )) is called the Friedel oscillation and A(x) = cos( 4πϕ) L Bos is the corresponding envelope. Since L Bos is not exactly solvable, we shall use the well-known self consistent harmonic approximation (SCHA) [2], which consists in replacing L Bos by π L = 2 ( µϕ(x)) 2 + m(x) 2 ϕ2 + dy x ϕ(x 0, x )U(x y ) y ϕ(x 0, y ) (22) where m(x) is an arbitrary function, related to the impurity, to be variationally determined. Also note that we have already considered the most interesting case of instantaneous potentials of the form V (0) (x, y) = δ(x 0 y 0 )U(x y ). Let us now compute the envelope A(x) in this approximation: A(x) = e d 2yJ(x,y)ϕ(y) L (23) J(x, y) = 2i πδ 2 (x y) (24) 4

6 We are interested in the case m(x) = mδ(x d). (25) Going to momentum space and doing a translation in the field ˆϕ(p) ˆφ(p) + a(p), a(p) a classical function, we find where a(p) satisfies A(x) = exp 2 a(p) = J(p) 2R( p) d 2 p (2π) 2J( p)a(p) = πa(x) ei (26) d 2 q (2π) 2m(p q) a(q) 2R( p) (27) R(p) = p2 2 + U(p ) p2 π m(k) = 2πmδ( k 0 )e ik d J(k) = 2i πe ik.x (28) As we can see, the calculation of A(x) is reduced to finding the classical function a(p), that is, to solving the integral equation (27). This is an integral equation degenerated kernel whose solution is [3] a(p) = J(p) 2R(p) + K(p 0)f( p) (29) f(p) = m 2 g(q ) = eiq d K( p 0 ) = e ip d R(p) 2π 2 dq g(q ) J( p 0,q ) R(p 0,q ) dq g(q )f(p 0, q ) (30) 5

7 It is straightforward to verify that a(p) can be written in a more compact way as where r = d x. a(p) = i π R(p) e ip.x [ meipr I(p 0, r) π + mi(p 0, 0) ] (3) I(p 0, r) = 0 dq cos(q r) p q 2 ( + 2U(q ) π ). (32) So we finally obtained A(r) as a functional of the electron-electron potential: A(r) = exp π ( ) dp 0 I(p 0, 0) mi2 (p 0, r). (33) π + mi(p 0, 0) This is the main formal result of our work. In order to check the validity of our formula we shall now consider a short-range interaction U(q ) = U = constant. It is convenient to express A(r) in the form where one has defined A(r) = exp (T + W)(r) (34) T = 2 π W(r) = m π 0 dp 0 I(p 0, 0) dp 0 I 2 (p 0, r) π + mi(p 0, 0) (35) These integrals are easily performed yielding T = 2 π 0 dp 0 0 dq p q 2 ( + 2U π ) = ln[ µ Λ ]λ (36) 6

8 and W = m I 2 (p 0, r) dp 0 π π + mi(p 0, 0) = λ e2µr λ Ei ( 2µr λ ) + λ e mr λ 2 E i ( mr ), (37) λ2 where E i (x) is the exponential integral function, µ 0 + and Λ are infrared and ultraviolet cutoffs respectively, and λ = + 2U. π At this point, in order to compare our results previous computations we consider the long distance regime defined by 2µr mr <<, >>. We get λ λ 2 A(r) = C(λ, µ, Λ) λ 2r λ e λ mr (38) which exactly coincides [] and [5] under the same regime. Acknowledgements The authors are supported by Universidad Nacional de La Plata (UNLP) and Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET), Argentina. They are grateful to the organizers of the XX Encontro Nacional de Fisica de Particulas e Campos (Brazil) where these results were presented. 7

9 References [] V. Fernández, K. Li, C. Naón, Physics Letters B 452, 98, (999). [2] C.Naón, M.C.von.Reichenbach, M.L. Trobo, Nucl. Phys. B 435 [fs], 567, (995). [3] C.L.Kane, M.P.A.Fisher,Phys.Rev.B46 5, 233, (992). [4] J.Voit, Rep. on Progr. in Phys. 58, 977, (995). [5] Q. Yuan, H.Chen, Y.Zhang, Y.Chen, Phys.Rev.B 58, 084, (998). [6] S.Tomonaga, Prog. Theor. Phys. 5, 544, (950). J.Luttinger, J. Math. Phys. 4, 54, (963). E.Lieb and D.Mattis, J. Math. Phys. 6, 304, (965). [7] Kang Li, Carlos Naón, J. Phys. A 3, 7929, (998). [8] H.J.Schulz, Phys.Rev.Lett.7, 864, (993). [9] K.Fujikawa, Phys. Rev. D 2, 2848, (980). [0] M.Manías, C.Naón, M.L.Trobo, Physics Letters B 46, 57, (998). [] R. Egger, H. Grabert, Phys.Rev.Lett.75, 3505, (995). [2] Y. Saito, Z. Phys. B 32, 75, (978). M. Stevenson, Phys. Rev. D 32, 389, (985). M.P.A. Fisher, W. Zwerger, Phys. Rev. B 32, 690, (985). A.O. Gogolin, Phys. Rev. Lett. 7, 2995, (993). N.V. Prokof ev, Phys. Rev. B 49, 2243, (994). B.W. Xu, Y.M. Zhang, J. Phys. A 29, 7349, (996). [3] See for instance: S.L. Sobolev, Partial Differential Equations of Mathematical Physics, Dover Pub. Inc.,