arXiv:cond-mat/0510206v2 [cond-mat.str-el] 15 Mar 2006 Europhysics Letters PREPRINT A multi-channel fixed point for a Kondo spin coupled to a junction of Luttinger liquids V. Ravi Chandra 1 , Sumathi Rao 2 and Diptiman Sen 1 1 2 Centre for High Energy Physics, Indian Institute of Science, Bangalore 560012, India Harish-Chandra Research Institute, Chhatnag Road, Jhusi, Allahabad 211019, India PACS. 73.63.Nm – Quantum wires. PACS. 72.15.Qm – Scattering mechanisms and Kondo effect. PACS. 73.23.-b – Electronic transport in mesoscopic systems. Abstract. – We study a system of an impurity spin coupled to a junction of several TomonagaLuttinger liquids using a renormalization group scheme. For the decoupled S-matrix at the junction, there is a range of Kondo couplings which flow to a multi-channel fixed point for repulsive inter-electron interactions; this is associated with a characteristic temperature dependence of the spin-flip scatterings. If the junction is governed by the Griffiths S-matrix, the Kondo couplings flow to a strong coupling fixed point where all the wires are decoupled. Although the Kondo effect has been studied for many years and is one of the best understood paradigms of strongly correlated systems [1], it continues to yield new physics, such as in its recent manifestations in quantum dots [2, 3]. For quantum dots with an odd number of electrons, a Kondo resonance occurs at the Fermi level of the leads, which is seen as a peak in the conductance. If the leads are one-dimensional, inter-electron interactions turn them into TomonogaLuttinger liquids (TLLs); this changes the physics considerably. The Kondo effect has been studied for a system with two TLL leads [4–7] and for crossed TLL wires [8]. For weak potential scattering, the strong coupling fixed point (FP) consists of decoupled TLLs, the conductance vanishes at T = 0 via the usual TLL power law [9], and a spin singlet is formed if the impurity has spin 1/2 (Furusaki-Nagaosa point) [5]. Motivated by recent experiments which probe the Kondo density of states in a three terminal geometry [10], we will study what happens when an impurity spin is coupled to a junction of more than two quantum wires which are modeled as TLLs. The junction is characterized by an S-matrix which governs the connections between the wires, and the couplings of the Kondo spin are described by a J-matrix. (Although the S-matrix formalism for calculating the conductance is strictly valid only for non-interacting electrons, we will use it here because the interactions will be assumed to be weak and will be treated perturbatively [11, 12]). We obtain the renormalization group (RG) equations for the system, and find that the flow of the Kondo couplings depends on the form of the S-matrix. At the point where all the wires are decoupled from each other, we find that for a large range of initial values of the Kondo couplings, the system flows to a multi-channel FP lying at zero coupling. This FP is c EDP Sciences 2 EUROPHYSICS LETTERS associated with spin-flip scatterings of the electrons from the impurity spin whose temperature dependence will be discussed below. [Note that in the language of the standard N -channel Kondo problem, the scattering matrix S is the N × N identity matrix, and Jij = Ji δij ; anisotropy between the channels is introduced by having off-diagonal elements in the S and J matrices]. On the other hand, at the Griffiths S-matrix (defined below), there is no stable FP for finite values of the Kondo couplings, and the system flows towards strong coupling in two possible ways. In one case, the impurity is strongly and antiferromagnetically coupled to the electron spin at the junction. We then perform an expansion in the inverse of the Kondo couplings and find that the system is now near the decoupled S-matrix; hence it flows to the multi-channel FP. In the other case, the impurity is coupled strongly and ferromagnetically to the electron spin at the origin and antiferromagnetically to the neighboring electron spins; further analysis then depends on the values of N and the impurity spin S. We begin with N semi-infinite wires which meet at a junction, where the incoming and outgoing fields are related by an N × N unitary S-matrix. For an electron which is incoming in wire i (= 1, 2, · · · , N ) with spin α (=↑, ↓) and wave number k (defined with respect to the Fermi wave number kF ), the wave function is given by ψiαk (x) = e−i(k+kF )x + Sii ei(k+kF )x on wire i , = Sji ei(k+kF )x on wire j = i . (1) Here k goes from −Λ to Λ. The second quantized annihilation operator corresponding to the above wave function is given by Ψiαk (x) = ciαk ψiαk (x). If an impurity spin is coupled to the electrons at the junction, the Hamiltonian is given by Λ H0 + Hspin = vF −Λ α i dk k c† ciαk iαk 2π Λ Λ −Λ −Λ + i,j α,β σαβ dk1 dk2 Jij S · c† 1 cjβk2 , iαk 2π 2π 2 (2) where the dispersion has been linearized (E = vF k), Jij is a Hermitian matrix, σ denotes the Pauli matrices, and we are assuming an isotropic spin coupling Jx = Jy = Jz for simplicity. Next, we consider density-density interactions between the electrons of the form Hint = 1 2 dx dy ρ(x) U (x − y) ρ(y) , (3) where the density operator ρ is given in terms of the second quantized electron field Ψα (x) = i dk/(2π)Ψiαk (x) as ρ = Ψ† Ψ↑ + Ψ† Ψ↓ . Writing the electron field in terms of outgoing ↑ ↓ and incoming fields as Ψα (x) = ΨOα (x) + ΨIα (x), the Hamiltonian in (3) takes the form Hint = [g1 Ψ† Ψ† ΨOβ ΨIα + g2 Ψ† Ψ† ΨIβ ΨOα Oα Iβ Oα Iβ dx α,β 1 g4 (Ψ† Ψ† ΨOβ ΨOα + Ψ† Ψ† ΨIβ ΨIα )]. Oα Oβ Iα Iβ 2 The parameters g1 , g2 and g4 satisfy some RG equations [11, 13], and are given by + (4) ˜ U (2kF ) 1 1 ˜ ˜ U (2kF ) + , g2 (L) = U (0) − ˜ 2 2 1 + U(2kF ) ln L πvF g1 (L) = ˜ U (2kF ) 1 + ˜ U(2kF ) πvF ln L ˜ , and g4 (L) = U (0), (5) V. Ravi Chandra , Sumathi Rao and Diptiman Sen : A multi-channel fixed point for a Kondo spin coupled to a juncti ˜ where L denotes the length scale, and U is the Fourier transform of U . (We have ignored umklapp scattering). The junction S-matrix satisfies an RG equation which was derived in Refs. [11, 12] in the absence of Kondo couplings. We find that the Kondo couplings Jij do not affect the RG flows of the S-matrix up to second order in the Jij . Since we are mainly interested in the flows of the Jij , we will assume for simplicity that we are at a FP of the RG equations for Sij . We will study what happens near two particular FPs of Sij . We use the technique of ‘poor man’s RG’ [14,15] to derive the RG equations for the Kondo couplings Jij . (The details will be presented elsewhere). To second order in the couplings Jij and the parameters ga (which are given in Eq. (5)), we find that 1 dJij = d ln L 2πvF [ Jik Jkj + k − 1 ∗ ∗ g2 (Sij Jik Sik + Sji Jkj Sjk ) 2 1 ∗ ∗ (g2 − 2g1 ) (Jik Skk Skj + Ski Skk Jkj ) ], 2 (6) We now consider two possibilities for the S-matrix. The first case is that of N disconnected wires for which the S-matrix is given by the N ×N identity matrix (up to phases). We consider a highly symmetric form of the Kondo coupling matrix (consistent with the symmetry of the S-matrix) in which all the diagonal entries are J1 and all the off-diagonal entries are J2 , with both J1 and J2 being real. Eq. (6) then gives dJ1 1 2 [J 2 + (N − 1)J2 + 2g1 J1 ] , = d ln L 2πvF 1 dJ2 1 2 [2J1 J2 + (N − 2)J2 − (g2 − 2g1 )J2 ] . = d ln L 2πvF (7) (For N = 2 and g1 = 0, Eq. (7) agrees with the results in Ref. [6]). Since g1 (L = ∞) = 0, Eq. (7) has a FP at (J1 , J2 ) = (0, 0). If ν ≡ g2 (L = ∞)/(2πvF ) > 0 (repulsive interactions), a linear stability analysis shows that this FP is stable to small perturbations in J2 . For small perturbations in J1 , this FP is marginal; a second order analysis shows that it is stable if J1 < 0 and unstable if J1 > 0, i.e., it is the usual ferromagnetic fixed point which is found for Fermi liquid leads. However, the approach to the fixed point is quite different when the leads are TLLs. At large length scales, the FP is approached as J1 ∼ −1/ ln L and J2 ∼ 1/Lν . From this, we can deduce the behavior at very low temperatures, namely, J1 ∼ − 1/(ln 1/T ) , and J2 ∼ T ν . (8) This is in contrast to the behavior of J2 for Fermi liquid leads, i.e., for g1 = g2 = 0. In that case, Eq. (7) again gives a FP at (J1 , J2 ) = (0, 0), but J2 approaches zero as 1/(ln 1/T )2 . Note that J2 (which measures the asymmetry between the channels) approaches zero faster than J1 for Fermi liquid leads; but for TLLs, it goes to zero much faster, i.e., as a power of T . ˜ ˜ Fig. 1 shows the RG flows for three wires for U (0) = U (2kF ) = 0.2(2πvF ). [This gives a value of ν which is comparable to what is found in several experimental systems (see [16] and references therein). In Figs. 1 and 2, the values of Jij are shown in units of 2πvF .] We see that the RG flows take a large range of initial conditions to the FP at (0, 0). For all other initial conditions, we see that there are two directions along which the Kondo couplings flow to infinity; these are given by J2 = J1 and J2 = −J1 /(N − 1) (with N = 3). This asymptotic behavior can be understood by analyzing Eq. (7) after ignoring the terms of order g1 and g2 . The second case that we study is called the Griffiths S-matrix. Here all the N wires are connected to each other and there is maximal transmission, subject to the constraint that 4 EUROPHYSICS LETTERS 0 4 3 2 J2 −−−−−−> 1 0 −1 −2 −3 −4 −4 −3 −2 −1 0 1 J1 −−−−−−> 2 3 4 ˜ ˜ Fig. 1 – RG flows of the Kondo couplings for three disconnected wires, with U (0) = U (2kF ) = 0.2(2πvF ). there is complete symmetry between the wires. The resultant S-matrix has, up to phases, all the diagonal entries equal to −1 + 2/N and all the off-diagonal entries equal to 2/N . We again consider a highly symmetric form of the Kondo coupling matrix, with real parameters J1 and J2 as the diagonal and off-diagonal entries respectively. Eq. (6) then gives dJ1 2 1 2 1 2 2 [J1 + (N − 1)J2 + 2g1 (1 − )2 J1 − 4g1 (1 − ) (1 − ) J2 ], = d ln L 2πvF N N N 4g1 1 2 2 2 dJ2 2 [2J1 J2 + (N − 2)J2 − = (1 − ) J1 + (g2 − 2g1 (1 − ) )) J2 ]. (9) d ln L 2πvF N N N (For N = 2 and g1 = 0, Eq. (9) agrees with the results in Ref. [5]). Eq. (9) has a FP at the origin (which is unstable for g2 (∞) > 0), and two strong coupling FPs as before. Fig. 2 shows a picture of the RG flows for three wires. The couplings are again seen to flow to infinity along one of the two directions J2 = J1 and J2 = −J1 /(N − 1). 4 3 J2 −−−−−−> 2 1 0 −1 −2 −3 −4 −4 −3 −2 −1 0 1 J1 −−−−−−> 2 3 4 ˜ Fig. 2 – RG flows of the Kondo couplings for the Griffiths S-matrix for three wires, with U (0) = ˜ (2kF ) = 0.2(2πvF ). U We will now see how the different S-matrices and RG flows discussed above can be interpreted in terms of lattice models as was done for the two-wire case in Ref. [5]. The case of N V. Ravi Chandra , Sumathi Rao and Diptiman Sen : A multi-channel fixed point for a Kondo spin coupled to a juncti 3 2 2 1 1 3 2 1 0 1 2 3 3 Fig. 3 – A lattice model for the S-matrices discussed in the text. disconnected wires can be realized by the lattice model shown in Fig. 3. The Hamiltonian is taken to be of the tight-binding form, with a hopping amplitude −t on all the bonds, except on the N bonds connecting to the junction site n = 0 where they are taken to be zero. (This is equivalent to removing the junction site from the system). The impurity spin is coupled to the sites n = 1 on the different wires by the Hamiltonian Ψ† (i, 1) α Hspin = F1 S · i α,β σαβ Ψβ (i, 1) + F2 S · 2 Ψ† (i, 1) α i=j α,β σαβ Ψβ (j, 1), (10) 2 where Ψα (i, 1) denotes the second quantized electron field at site 1 on wire i. (Eq. (15) below will provide a justification for this Hamiltonian). In Eq. (10), F1 and F2 denote amplitudes for spin-dependent scattering from the impurity within the same wire and between two different wires respectively. Now, we find that the Kondo coupling matrix Jij in Eq. (2) is as follows: all the diagonal entries are given by J1 and all the off-diagonal entries are given by J2 , where J1 = 4F1 sin2 kF , and J2 = 4F2 sin2 kF (11) for modes with redefined wave numbers lying close to zero. The RG flow of this is given in Eq. (7). In particular, the approach to the FP at (J1 , J2 ) = (0, 0) given by Eq. (8) at low temperatures implies that spin-flip scattering within the same wire or between two different wires will have quite different temperature dependences. The case of the Griffiths S-matrix can also be realized by the lattice shown in Fig. 3 and a tight-binding Hamiltonian. The hopping amplitude is now −t on all bonds, except for the N bonds connecting to the junction site where it is taken to be t1 = −t 2/N . We then find that the S-matrix is of the Griffiths form for all values of the wave number k. The impurity spin is then coupled to the junction site and the n = 1 sites by the Hamiltonian Ψ† (0) α Hspin = F3 S · α,β σαβ Ψβ (0) + F4 S · 2 Ψ† (i, 1) α i α,β σαβ Ψβ (i, 1) , 2 (12) where Ψα (0) denotes the electron field at the junction site with spin α. Then the Kondo coupling matrix Jij in Eq. (2) has all the diagonal entries equal to J1 and all the off-diagonal entries equal to J2 , where J1 = 4F3 2 4F4 4F3 + 2F4 [ 1 − (1 − ) cos 2kF ] , and J2 = + cos 2kF 2 2 N N N N (13) 6 EUROPHYSICS LETTERS for modes with wave numbers lying close to zero. Eq. (9) gives the RG flows of these parameters. Eq. (13) implies J1 − J2 = 2F4 (1 − cos 2kF ) , and J1 + (N − 1)J2 = 4F3 + 2F4 (1 + cos 2kF ). (14) N Since 0 < kF < π, 1 ± cos 2kF lie between 0 and 2. In the first quadrant of Fig. 2, we see that J1 + (N − 1)J2 goes to ∞ much faster than |J1 − J2 |; Eq. (14) then implies that F3 goes to ∞ and |F4 | ≪ F3 . In the fourth quadrant of Fig. 2, J1 − J2 goes to ∞ much faster than J1 + (N − 1)J2 ; hence F4 goes to ∞ and F3 goes to −∞. These flows to strong coupling have the following physical interpretations. In the first case, F3 flows to ∞ which means that the impurity spin is strongly and antiferromagnetically coupled to an electron spin at the junction site n = 0; hence those two spins will combine to form an effective spin of S − 1/2. In the second case, the impurity spin is coupled strongly and ferromagnetically to an electron spin at the site n = 0, and strongly and antiferromagnetically to electron spins at the sites n = 1 on each of the N wires to form an effective spin of S + 1/2 − N/2. We considered above two kinds of S-matrices and found that the Kondo couplings flow to infinity for many initial conditions. We will now show through an example that the vicinity of a strong coupling FP can be studied through an expansion in the inverse of the Kondo coupling [15]. Following the discussion after Eq. (14), let us assume that the RG flows for the case of the Griffiths S-matrix have taken us to a strong coupling FP along the line J2 = J1 ; thus the impurity spin is coupled to the electron spin at n = 0 with a large and positive (antiferromagnetic) value F3 , while its couplings to the sites n = 1 have the value F4 = 0. (The arguments given below do not change significantly if F4 = 0, provided that |F4 | ≪ F3 ). From the first term in Eq. (12), we see that the impurity spin couples to an electron at n = 0 to form an effective spin of S − 1/2; the energy of this spin state is −F3 (S + 1)/2. This lies far below the high energy states in which an electron at site n = 0 forms a total spin of S + 1/2 with the impurity spin (these states have energy F3 S/2), or the states in which the site n = 0 is empty or doubly occupied (these states have zero energy). We now perturb in 1/F3 . The unperturbed Hamiltonian corresponds to N disconnected wires along with the impurity spin coupled to the junction site n = 0. The perturbation Hpert consists of the hopping amplitude t1 on the N bonds connecting the sites n = 1 to the junction site. Using this perturbation, we can find an effective Hamiltonian [15]. If S > 1/2, we find that the effective Hamiltonian has no terms of order t1 , and is given by Ψ† (i, 1) α Heff = F1,eff Seff · i with F1,eff = F2,eff α,β σαβ Ψβ (i, 1) + F2,eff Seff · 2 8t2 1 =− , F3 (S + 1) (2S + 1) Ψ† (i, 1) α i=j α,β σαβ Ψβ (j, 1) 2 (15) and Seff denotes an object with spin S − 1/2. We thus find a weak interaction between Seff and all the sites labeled as n = 1 in Fig. 3. [If the impurity has S = 1/2, the electron at n = 0 forms a singlet with the impurity.] For S > 1/2, we see that Eq. (15) has the same form as in Eqs. (10-11), where the effective couplings J1,eff = 4F1,eff sin2 kF and J2,eff = 4F2,eff sin2 kF are equal, negative and small. With these initial conditions, we see from Eq. (7) and Fig. 1 that the Kondo couplings flow to the FP at (J1,eff , J2,eff ) = (0, 0). We thus obtain a picture of the RG flows at both short and large length scales. We start with the Griffiths S-matrix with certain values of the Kondo couplings, and finally end at the multi-channel FP of the disconnected S-matrix. But in this letter, we have restricted V. Ravi Chandra , Sumathi Rao and Diptiman Sen : A multi-channel fixed point for a Kondo spin coupled to a juncti ourselves to weak inter-electron interactions, since we use perturbative methods to analyze the effects of gi . Hence, the Luttinger parameter K is restricted to be less than but close to unity. An interesting question to address is whether this analysis is true for strong interelectron interactions when K is much less than unity. In the two-wire case, it was shown that at strong interactions, it is the two-channel antiferromagnetic Kondo fixed point which is stabilized for K < 1/2 [6]. An equivalent analysis for N wires is required [17]. To summarize, we have studied the Kondo effect at a junction of N quantum wires and find an interesting interplay of the Kondo logarithms and the TLL power laws. We find that the scaling of the Kondo couplings depends on the S-matrix at the junction. For the case of disconnected wires and repulsive interactions, there is a range of Kondo couplings which flow towards a multi-channel FP at (J1 , J2 ) = (0, 0). At low temperatures, we find spin-flip scattering processes with temperature dependences which are dictated by both the Kondo effect and the inter-electron interactions. It may be possible to observe such scatterings by placing a quantum dot with a spin at a junction of several wires with interacting electrons. At the fully connected or Griffiths S-matrix, we find that the Kondo couplings flow to a strong coupling FP, where their fate is decided by a 1/J analysis. There is a range of initial conditions which again lead to the FP at (J1,eff , J2,eff ) = (0, 0). ∗∗∗ SR thanks Y. Oreg and A. M. Finkel’stein for discussions. DS thanks the Department of Science and Technology, India for financial support under projects SR/FST/PSI-022/2000 and SP/S2/M-11/2000. REFERENCES [1] Hewson A. C., The Kondo Problem to Heavy Fermions (Cambridge University Press, Cambridge) 1993. [2] Oreg Y. and Goldhaber-Gordon D., Phys. Rev. Lett., 90 (2003) 136602. [3] Rosch A., Paaske J., Kroha J. and W¨lfle P., J. Phys. Soc. Jpn., 74 (2005) 118. o [4] Lee D.-H. and Toner J., Phys. Rev. Lett., 69 (1992) 3378. [5] Furusaki A. and Nagaosa N., Phys. Rev. Lett., 72 (1994) 892. [6] Fabrizio M. and Gogolin A. O., Phys. Rev. B, 51 (1995) 17827. [7] Fr¨jdh P. and Johannesson H., Phys. Rev. B, 53 (1996) 3211; Durganandini P., Phys. Rev. B, o 53 (1996) R8832; Egger R. and Komnik A., Phys. Rev. B, 57 (1998) 10620. [8] Le Hur K., Phys. Rev. B, 61 (2000) 1853. [9] Gogolin A. O., Nersesyan A. A. and Tsvelik A. M., Bosonization and Strongly Correlated Systems (Cambridge University Press, Cambridge) 1998; Giamarchi T., Quantum Physics in One Dimension (Oxford University Press, Oxford) 2004. [10] Leturcq R., Schmid L., Ensslin K., Meir Y., Driscoll D. C. and Gossard A. C., Phys. Rev. Lett., 95 (2005) 126603. [11] Yue D., Glazman L. I. and Matveev K. A., Phys. Rev. B, 49 (1994) 1966; Matveev K. A., Yue D., and Glazman L. I., Phys. Rev. Lett., 71 (1993) 3351. [12] Lal S., Rao S. and Sen D., Phys. Rev. B, 66 (2002) 165327; Das S., Rao S. and Sen D., Phys. Rev. B, 70 (2004) 085318. [13] Solyom J., Adv. Phys., 28 (1979) 201. [14] Anderson P. W., J. Phys. C, 3 (1970) 2436. [15] Nozieres P. and Blandin A., J. Phys. (Paris), 41 (1980) 193. [16] Lal S., Rao S. and Sen D., Phys. Rev. Lett., 87 (2001) 026801; Phys. Rev. B, 65 (2002) 195304. [17] Ravi Chandra V., Rao S. and Sen D. (work in progress).