Scaling behavior of impurities in mesoscopic Luttinger liquids V. Meden1 , W. Metzner2 , U. Schollw¨ck3 , and K. Sch¨nhammer1 o o arXiv:cond-mat/0104336v1 [cond-mat.str-el] 18 Apr 2001 1 Institut f¨r Theoretische Physik, Universit¨t G¨ttingen, Bunsenstr. 9, D-37073 G¨ttingen, Germany u a o o 2 Institut f¨r Theoretische Physik C, RWTH Aachen, D-52056 Aachen, Germany, u 3 Sektion Physik, Universit¨t M¨nchen, Theresienstr. 37, D-80333 M¨nchen, Germany a u u (April 18, 2001) Using a functional renormalization group we compute the flow of the renormalized impurity potential for a single impurity in a Luttinger liquid over the entire energy range - from the microscopic scale of a lattice-fermion model down to the low-energy limit. The non-perturbative method provides a complete real-space picture of the effective impurity potential. We confirm the universality of the open chain fixed point, but it turns out that very large systems (104 -105 sites) are required to reach the fixed point for realistic choices of the impurity and interaction parameters. The low-energy physics of one-dimensional interacting electron systems with Luttinger liquid (LL) behavior is dramatically affected by the presence of a single impurity. [1–6] The problem is usually mapped onto an effective field theory using bosonization, where terms which are expected to be irrelevant in the low-energy limit are neglected. [1–5] Then forward and backward impurity scattering decouple, and the more important backscattering processes are modeled by a single amplitude VB . From a perturbative bosonic renormalization group (RG) calculation [4] and a boundary conformal field theory analysis [5] the following picture emerged: In a chain of spinless fermions [7] with repulsive interactions (LL parameter Kρ < 1) the backscattering amplitude VB is a relevant perturbation which grows as ΛKρ −1 when the flow parameter Λ is sent to zero, and the perturbative analysis breaks down. This behavior can be traced back to the power-law singularity of the 2kF density response function in a LL. [1,2] On the other hand a weak hopping tw between the open ends of two semi-infinite chains is ir−1 relevant and scales to zero as ΛKρ −1 . [4] Assuming that the open chain represents the only stable fixed point it was argued that at low energy scales and even for a weak impurity physical observables behave as if the system was split in two chains with open boundary conditions at the end points. [4] Here we focus mainly on the local spectral weight ρj (ω) for lattice sites j close to the impurity and energies ω close to the chemical potential µ. For ρj (ω) a power-law suppression ρj (ω) ∼ |ω|αB with the −1 boundary exponent αB = Kρ − 1 which only depends on the interaction strength and band filling, but not on the impurity parameters, was predicted. [4] Within the bosonic field theory the above conjecture was verified by refermionization [4], quantum Monte Carlo calculations, [8,9] and the thermodynamic Bethe ansatz. [10] To confirm the field theoretical scenario and the validity of the underlying assumptions for a microscopic fermionic system with LL behavior, numerical methods (exact diagonalization [ED], density-matrix renormalization group [DMRG]) were applied to the lattice model of spinless fermions with nearest neighbor interaction. [5,11–13] Comparing ED data for up to N = 23 sites with the field theoretical prediction for the finite size corrections of energies, the expected scaling was confirmed for both weak impurities and weak hopping. [5] However, due to the limited system size it was impossible to go beyond the perturbative (in either VB or tw ) regime. Later it was claimed that the full flow from a weak impurity to the open boundary fixed point (BFP) was successfully demonstrated, [11,12] although this strong statement is not really supported by the numerical data presented. The smallest temperature discussed in Ref. [12] corresponds to a system of around 300 lattice sites and the largest system considered in Ref. [11] was N = 52, while in Ref. [13] it was shown that N ≈ 102 lattice sites are clearly not enough to exclude an asymptotic behavior not governed by the BFP, even if one starts out with a fairly strong impurity. Recently functional RG methods, originally developed in a field theoretical context, have been introduced as a new powerful tool in the theory of interacting Fermi systems [14], with applications so far concentrating on translationally invariant two-dimensional systems. [15] In this letter we apply such a functional RG scheme to the spinless fermion model with site or hopping impurities. We compute the complete coupled flow of the renormalized onsite energies and the renormalized hopping amplitudes from the microscopic energy scale down to the infrared fixed point. The flow equations are non-perturbative in the impurity strength while perturbative in the electronelectron interaction. We treat the full functional form of the renormalized impurity potential as generated by the flow, instead of replacing it approximately by the scattering amplitudes at the Fermi level. Computing the local density of states near the impurity we convincingly confirm the universality of the BFP. However, it turns out that very large systems (104 − 105 sites) are required to reach the BFP for intermediate impurity and interaction parameters. Our RG scheme is checked against numerical exact DMRG data for systems with up to N = 768 sites. The one-dimensional lattice model of spinless fermions 1 only by a finite amount of order U 2 . [23] Hence, we can replace the renormalized two-particle vertex to leading order in U by the antisymmetrized bare interaction. In this way the exact hierarchy of flow equations gets truncated, and one obtains a simple one-loop flow equation for the self-energy Σ, where only the (full) propagator G and the bare electron-electron interaction U enter. Carrying out a Matsubara sum and choosing a real space representation of G and Σ, one obtains the flow equations (at temperature T = 0) with nearest neighbor hopping amplitude t = 1 and nearest neighbor interaction U is given by c† cj+1 + c† cj + U j j+1 H =− j nj nj+1 , (1) j in standard second-quantized notation. Here we focus on the half filled band case. This model is either complemented by a site impurity Hs = V nj0 or provided with a hopping impurity Hh = −tw (c†0 cj0 +1 + h.c.) on one of j its bonds. In a weakly interacting spinless LL with an open end the local density of states ρj (ω) near the boundary can to a surprisingly good approximation be obtained from a non-selfconsistent Hartree-Fock (HF) approximation. [16] It is instructive to consider also the impurity problem within HF, before turning to the RG treatment. The impurity leads to Friedel oscillations in the non-interacting density profile nj 0 which for large |j − j0 | behaves as R sin (2kF |j − j0 |)/|j − j0 |, where R is the reflection amplitude of the bare impurity. Similar oscillations are found in the matrix element c† cj+1 0 . Thus both the j Hartree potential U ( nj−1 0 + nj+1 0 ) and the Fock “hopping correction” are oscillating and of long range. One then has to solve a (non-trivial) one-particle problem within such a potential and with modulated hopping. Taking into account the Hartree term only, the resulting spectral weight for |ω| → 0 shows power-law behavior with an exponent which is proportional to the amplitude U R of the oscillations. [17] We have checked numerically (for systems of up to 106 lattice sites [16,18]) that this behavior is not changed when the Fock term is included. Thus due to the long range nature of the effective potential and the hopping modulation already HF yields a power-law for the spectral weight, but with an exponent which not only depends on U , but via R also on the bare impurity strength. It is tempting to extend the HF study using selfconsistent HF. [19] However, it turns out that an iterative solution of the selfconsistent HF equation leads for all U to a charge density wave groundstate, [18,19] which is qualitatively incorrect since a single impurity cannot change bulk properties of the system. We now treat the problem using a fermionic functional RG approach. Cutting off the free propagator on a scale Λ and differentiating with respect to this flow parameter, an exact infinite hierarchy of coupled differential flow equations for the one-particle irreducible vertex functions can be derived. [20–22] For the impurity problem it is technically advantageous to use a frequency cut-off for the free propagator G0,Λ (iω) = Θ(|ω| − Λ)G0 (iω) U d Λ Σ =− dΛ j,j 2π d Λ U Σ = dΛ j,j±1 2π GΛ j+s,j+s (iω) (3) s=±1 ω=±Λ GΛ (iω) . j,j±1 (4) ω=±Λ The self-energy is frequency independent and tridiagonal, since the bare interaction is instantaneous and restricted to nearest neighbors. The full propagator GΛ on the right hand side of the flow equations is obtained by inverting −1 the matrix G0 − ΣΛ . The bare site and/or hopping impurity enter as initial conditions for ΣΛ at Λ = ∞. For a site impurity V at j0 , one sets ΣΛ=∞ = V , and j0 ,j0 for a hopping impurity between j0 and j0 + 1, one has ΣΛ=∞ = ΣΛ=∞ 0 = −tw , while the other matrix elej0 ,j0 +1 j0 +1,j ments are initially zero. The above flow equations are non-perturbative in the impurity parameters, in contrast to the perturbative bosonic RG. Written in momentum space the different scattering channels Σk,k′ are coupled. The self-energy at Λ = 0 can be given a simple physical meaning: ΣΛ=0 represents an effective one-particle j,j potential and ΣΛ=0 is an effective modulation of the j,j+1 hopping. To calculate ρj (ω) one determines the spectral weights of the remaining one-particle problem. For small impurity strength V , after transforming to momentum space and taking N → ∞, Eqs. (3) and (4) can be solved analytically, as long as ΣΛ stays small. For the backscattering this gives ΣΛF ,−kF ∼ Λ−η with k η = U [1 − cos (2kF )]/(πvF ) and the Fermi velocity vF . To leading order in U , the exponent η is just Kρ − 1 [16] which shows that the non-perturbative fermionic RG captures the power-law increase found in the perturbative bosonic RG. Numerically integrating the RG equations for finite systems we can go beyond the perturbative regime. In each step of the integration we have to invert an N × N matrix. If we assume open boundary conditions in H0 , −1 G0 − ΣΛ is tridiagonal in real space and the numerical effort is considerably reduced. [24] This allowed us to treat systems with up to 215 = 32768 lattice sites. For finite N the flow is effectively cut off on a scale of the order of 1/N . For smaller systems we also considered periodic boundary conditions. Fig. 1 shows typical results for ΣΛ=0 and ΣΛ=0 for a site impurity and lattice j,j j,j+1 sites close to j0 . Since Σ is symmetric around j0 mainly (2) where G0 is the free propagator without cut-off and ω the Matsubara frequency. Λ flows from ∞ to 0. For spinless fermions the electron-electron interaction is renormalized 2 −0.10 the region j < j0 is shown. Similar to HF the effective potential and hopping are oscillating and slowly decaying. The numerical data suggest that the oscillations again fall off as |j − j0 |−1 , but with an amplitude which compared to HF is slightly enhanced. The inset of Fig. 1 shows ΣΛ ,j0 as a function of Λ for different N . Obj0 viously the renormalized potential at the impurity site remains finite and the expected “cutting” of the chain does certainly not occur because a single on-site energy diverges, as one might guess if the bosonic RG is taken too literally. Singular behavior is only found in ΣΛ ′ for k,k momenta with k − k ′ ≈ ±2kF , which is associated with the long range oscillations in real space. 1.6 Σ Λ j ,j +1 0 0 0 −0.11 −0.12 0 −0.6 0.0 0.6 ω U=1, t j =0.1 −0.13 10 0 −3 10 −2 10 −1 10 0 10 1 10 2 Λ FIG. 2. ΣΛ ,j0 +1 as a function of Λ for a hopping impurity j0 with tj0 = 0.1, U = 1, N = 1024, and j0 = 512. Inset: ρj0 −1 (ω) as a function of ω for a site impurity with the same parameters as in Fig. 1 (N = 1026). 1.60 Λ j0,j0 1.59 1.57 Λ 10 1.0 413 In Fig. 3 we show the negative of the logarithmic centered differences αI (N ) of W (N ) as a function of N for U = 0.5 and different V obtained from RG (N ≤ 32768) and DMRG (N ≤ 768). If W (N ) decays for N → ∞ as a power-law, αI (N ) converges to the respective exponent. For comparison we also calculated αB (N ) for the lattice site next to an open boundary (V = ∞). The DMRG and RG data are parallel to each other, which in addition to the analytical arguments is a strong indication that our fermionic RG captures the essential physics. For V = ∞ both methods produce the expected powerlaw behavior with boundary exponents αDMRG and αRG . B B αDMRG (N = 512) agrees up to 1% with the exact expoB nent αex ≈ 0.1609. αRG (N = 16384) which effectively B B is equal to αRG (N = ∞) deviates by roughly 6% from B αex since the RG is only correct to leading order in U . B The RG curves for finite V suggest that for N → ∞ the αRG (N ) converge to the universal (V independent) expoI nent αRG . This is in agreement with the field theoretical B prediction. It is remarkable that even for fairly strong impurities (V = 4) extremely large N = 104 -105 are needed to exclude non-universal (V dependent) fixed points with some certainty. Solely relying on DMRG data for a few hundred lattice sites would in this case give no definite result. [13] In Fig. 4 RG and DMRG data are presented for an intermediate impurity strength V = 1 and V = ∞ for different values of U . Due to higher order corrections in U , the difference between the RG and DMRG data increases with increasing U . For larger U , the αRG (N ) I approach αRG faster, but even for the largest U = 1.8 B considered here [26] (which corresponds to Kρ ≈ 0.58) very large N are needed. This demonstrates that for intermediate V and U , which are experimentally most relevant, very large systems are needed to observe the universal BFP physics. For chains which are not long enough a strong system size dependence of experimentally extracted exponents must be expected. Σ 1.58 1.2 ρj −1(ω) U=1, V=0.5 U=1, V=0.5 1.4 1.5 −2 10 −1 Σj,j −3Σj,j+1 463 j 513 FIG. 1. ΣΛ=0 for a site impurity with N = 1026 and j,j ′ j0 = 513. The inset shows ΣΛ ,j0 as a function of Λ for N = 66 j0 (circles), N = 130 (squares), N = 258 (diamonds), N = 514 (triangles), and N = 1026 (stars). In Fig. 2 we present results for the case of a weak hopping between two open chains. It shows ΣΛ ,j0 +1 as a j0 function of Λ for N = 1024. We have checked that the curve to a good approximation already presents the N → ∞ result. In contrast to a simplistic interpretation of the bosonization result, the renormalized hopping ΣΛ ,j0 +1 does not scale to zero. Similar to the case of a j0 site impurity, ΣΛ=0 shows long range oscillations in both j,j ′ the effective potential and the hopping. Again this and not the scaling of a single Vj or tj is the reason for the peculiar behavior of physical observables, as for example ρj (ω), discussed next. As an inset to Fig. 2 the density of states near the impurity, ρj0 −1 (ω), is presented for a site impurity. The data show a suppression of the weight for |ω| → 0 as expected. Each spike represents a δ-peak of the finite system. Instead of trying to fit a power-law to these data it is advantageous to analyze the finite size scaling of the spectral weight W (N ) at µ. [25] If ρj0 −1 (ω) follows a power-law as a function of frequency, we expect a power-law with the same exponent in the N dependence of W (N ). 3 scales behave as if the chain is split in two parts with open boundary conditions at the end points. Our fermionic RG is non-perturbative in the impurity strength. Longrange oscillations in the effective impurity potential provide a simple real-space picture of the “splitting” mechanism. The accuracy of the finite site RG scheme was confirmed by a direct comparison to DMRG data. For realistic parameters very large systems are needed to reach the asymptotic open chain regime. Hence only special mesoscopic systems, such as very long carbon nanotubes, are suitable for experimentally observing the impurityinduced asymptotic open boundary physics. Our method can easily be generalized to the case of several impurities and e.g. resonance phenomena can be studied. [18] We would like to thank W. Apel, P. Durganandini, P. Kopietz, N. Shannon, C. Wetterich, and especially M. Salmhofer and H. Schoeller for valuable discussions. U.S. is grateful to the Deutsche Forschungsgemeinschaft for support from the Gerhard-Hess-Preis. αI(N) 0.16 0.12 0.08 10 2 10 N 3 10 4 FIG. 3. αI (N ) as a function of N for U = 0.5 and different V : V = 1 (circles), V = 2 (squares), V = 4 (diamonds), and V = ∞ (triangles). The filled symbols are DMRG data and the open ones obtained from the RG. The dashed-dotted line gives the exact boundary exponent αex . B 0.8 [1] A. Luther and I. Peschel, Phys. Rev. B 9, 2911 (1974). [2] D. C. Mattis, J. Math. Phys. 15, 609 (1974). [3] W. Apel and T. M. Rice, Phys. Rev. B 26, 7063 (1982); T. Giamarchi and H. J. Schulz, ibid. 37, 325 (1988). [4] C. L. Kane and M. P. A. Fisher, Phys. Rev. B 46, 15233 (1992). [5] S. Eggert and I. Affleck, Phys. Rev. B 46, 10866 (1992). [6] D. Yue et al., Phys. Rev. B 49, 1966 (1994). [7] Here we will only discuss spinless fermions. [8] K. Moon et al., Phys. Rev. Lett. 71, 4381 (1993). [9] R. Egger and H. Grabert, Phys. Rev. Lett. 75, 3505 (1995). [10] P. Fendley et al., Phys. Rev. Lett. 74, 3005 (1995). [11] S. Qin et al., Phys. Rev. B 54, R9643 (1996). [12] S. Rommer and S. Eggert, Phys. Rev. B 62, 4370 (2000). [13] V. Meden et al., Phys. Rev. B 57, 8878 (1998). [14] M. Salmhofer, Renormalization (Springer, Berlin, 1998). [15] D. Zanchi and H. J. Schulz, Phys. Rev. B 61, 13609 (2000); C. J. Halboth and W. Metzner, ibid. 61, 7364 (2000); C. Honerkamp et al., ibid. 63, 5109 (2001). [16] V. Meden et al., Eur. Phys. J. B 16, 631 (2000). [17] M. Klaus, J. Math. Phys. 32, 163 (1991). [18] V. Meden et al., to be published. [19] A. Cohen et al., Phys. Rev. B 57, 6223 (1998). [20] C. Wetterich, Phys. Lett. B 301, 90 (1993). [21] T. R. Morris, Int. J. Mod. Phys. A 9, 2411 (1994). [22] M. Salmhofer and C. Honerkamp, Prog. Theor. Phys. 105, 1 (2001). [23] J. S´lyom, Adv. Phys. 28, 201 (1979). o [24] The open ends are far away from the impurity site j0 and do not influence the behavior close to j0 . [25] For a precise definition of W (N ), see Ref. [16]. [26] The LL phase is stable only for U < 2. [27] We thank H. Schoeller for pointing this out to us. [28] Y. Oreg and A. M. Finkel’stein, Phys. Rev. Lett. 76, 4230 (1996). αI(N) 0.6 0.4 0.2 10 2 10 N 3 10 4 FIG. 4. αI (N ) as a function of N for V = 1 (dashed lines) and αB (N ) for V = ∞ (solid lines) for different U : U = 0.5 (circles), U = 1 (squares), and U = 1.8 (diamonds). Filled symbols are DMRG data, open ones RG results. The dashed-dotted lines give the exact U dependent boundary exponents αex . B We finally note that in the fermionic RG used in Ref. [6] flow equations were set up for a single parameter only: the transmission amplitude at the Fermi level. Our functional RG flow however indicates that in the nonperturbative regime different momentum channels are strongly coupled. Hence, we believe that it is important to take the whole renormalized impurity potential profile into account. The RG equations used in Ref. [6] can also be derived within our formalism, if one makes similar crude approximations. [27] Also we did not find signs of an enhanced spectral weight as predicted in Ref. [28]. In summary, by solving a functional flow equation in a fermionic representation we have shown that in a onedimensional lattice electron system with Luttinger liquid behavior an impurity makes observables at low energy 4