Conducting fixed points for inhomogeneous quantum wires: a conformally invariant boundary theory N. Sedlmayr,1, ∗ D. Morath,1 J. Sirker,1 S. Eggert,1 and I. Affleck2 1 arXiv:1311.0007v2 [cond-mat.str-el] 25 Jan 2014 Department of Physics and Research Center OPTIMAS, University of Kaiserslautern, D-67663 Kaiserslautern, Germany 2 Department of Physics and Astronomy, The University of British Columbia, Vancouver, BC V6T 1Z1, Canada Inhomogeneities and junctions in wires are natural sources of scattering, and hence resistance. A conducting fixed point usually requires an adiabatically smooth system. One notable exception is “healing”, which has been predicted in systems with special symmetries, where the system is driven to the homogeneous fixed point. Here we present theoretical results for a different type of conducting fixed point which occurs in inhomogeneous wires with an abrupt jump in hopping and interaction strength. We show that it is always possible to tune the system to an unstable conducting fixed point which does not correspond to translational invariance. We analyze the temperature scaling of correlation functions at and near this fixed point and show that two distinct boundary exponents appear, which correspond to different effective Luttinger liquid parameters. Even though the system consists of two separate interacting parts, the fixed point is described by a single conformally invariant boundary theory. We present details of the general effective bosonic field theory including the mode expansion and the finite size spectrum. The results are confirmed by numerical quantum Monte Carlo simulations on spinless fermions. We predict characteristic experimental signatures of the local density of states near junctions. PACS numbers: 73.63.Nm, 71.10.Pm, 73.40.-c I. INTRODUCTION Transport in quantum wires is a rich field bringing together conductivity experiments1–5 and Luttinger liquid theory which describes the crucial electron-electron interaction effects in one dimension.6–8 Scattering from a single impurity or other inhomogeneities, for example, becomes renormalized by the interaction and can lead to insulating behavior at low temperatures even for weak impurities.9–14 In order to determine the conductivity of a onedimensional wire it is necessary to couple it to some leads or reservoirs, normally a two dimensional electron gas (2DEG). Such a set up can be most readily described as an inhomogeneous wire, in which the 2DEGs are modeled as non-interacting wires. In this case the conductance is usually controlled by the parameters of the lead rather than of the wire,15–28 in contrast to what a naive calculation on an infinite interacting wire would suggest. The conductance for perfect adiabatic contacts and wires can be understood by the decomposition of an electron into fractional charges.16,29 Additional relaxation processes which take place within the interacting region of the wire do, however, lead to a resistance which is affected by the wire parameters. The resistance due to impurity scattering30 or phonon scattering28 within the interacting wire, for example, will in general depend both on the Luttinger liquid parameter of the leads and the wire. In this paper we consider the intrinsic scattering from the junctions between the wire and leads, which is generically present due to the abrupt change of parameters even for otherwise perfect ballistic connections. This scattering is renormalized by the interaction,30 leading to a vanishing dc conductance in the low temperature limit for repulsive interactions within the wire. However, perfect conductance is still possible by tuning the parameters on the two sides of the junctions as has been analyzed in detail for a particle-hole symmetric model.27 In this case a line of conducting fixed points in parameter space exists as only one relevant backscattering operator is permitted by symmetry which can always be tuned to zero. Here we generalize to the more experimentally relevant case where particle-hole symmetry is no longer present. Even in this more general case we still find a line of conducting fixed points provided the underlying microscopic theory has certain local symmetry properties. Even though the systems under consideration are inhomogeneous, it is possible to characterize the fixed points by a single conformally invariant boundary theory with a characteristic mode expansion and finite size spectrum. The results are confirmed by numerical quantum Monte Carlo (QMC) simulations on spinless fermions. Characteristic experimental signatures for the local density of states near junctions can be predicted. For conductivity experiments we must typically consider a system with two junctions, one at each end of an interacting wire where it is connected to the leads (e.g. 2DEGs). These junctions are intrinsic sources of inhomogeneity, but in most cases the junctions do not influence each other since the length of the wire is much larger than the coherence length uβ, where β is the inverse temperature and u the velocity of the collective excitations. For our purpose to make predictions for the backscattering and the local behavior near the leads, it is therefore sufficient to analyze one junction between a lead and a wire. As an introduction in Sec. II we consider an idealized 2 junction in a non-interacting lattice model and discuss the applicability of a narrow band approximation. In Sec. III we start from a microscopic interacting model and demonstrate how the backscattering terms arise, and then introduce the general effective bosonic field theory. Focusing on abrupt junctions connecting otherwise homogeneous wires, we examine the renormalization group flow of perturbing operators in the model. We discuss the locations of the unstable conducting fixed points in relation to the symmetry properties of the underlying microscopic model. Finally in Sec. IV we describe the conformally invariant boundary theory for the conducting fixed point and the scaling of the local correlation functions at the boundary. In Sec. V we conclude. II. NON–INTERACTING MODELS Before considering the interacting model it is instructive to analyze the backscattering seen in inhomogeneous systems of free particles, where exact results are obtainable and can be compared directly with low energy approximations. We start with a lattice model of noninteracting spinless fermions described by the Hamiltonian † † [tj (ψj ψj+1 + H.c.) − Vj ψj ψj ] . (II.1) ˆ H0 − µN = − j † ψj creates a particle at site j, tj and Vj are the position dependent hopping elements and local potential energy respectively, and N the total particle number. We set = 1 and include in the following the chemical potential µ in the local potential energy Vj . Generically we consider situations in which we have two homogeneous regions on the left (j ≤ j ) and right (j ≥ jr ) side of the wire. In these asymptotic regions the plane-wave solutions have the same energy so the parameters are related by − 2t cos[k a] + V = −2tr cos[kr a] + Vr , (II.2) with k ,r the momenta, V ,r the potential, and t ,r the hopping on the left ( ) and right (r) side. We have also introduced the lattice spacing a. We consider a wavefunction incident from the left ψj = eik j + Re−ik j , j ≤ j . T eikr j , j ≥ jr It is natural to refer to R = 0 as “perfect transmission”, although this does not necessarily maximize |T |2 . A reasonable definition of perfect transmission would be maximizing the outgoing current on the right for a given value of the incoming current from the left, u ; that is, maximizing |T |2 ur /u . Noting that |T |2 ur /u = 1 − |R|2 we see that the condition for perfect transmission equivalently corresponds to minimizing |R|2 . This can also be seen by considering the Landauer transmission, see Appendix A. In general, accurate results cannot be obtained by ignoring states far from the Fermi energy. This can be seen from the fact that the off-diagonal components of the T matrix, Tk,k are non-negligible when |k | is not close to |k|. This implies a non-negligible mixing of low energy states with high energy ones due to scattering near the interface. However, in certain limits, a narrow band theory can be used, in which we keep only a narrow band of states, of width Λ kF , where kF is the Fermi momentum, and linearize the dispersion relation. This can be justified in one of two cases. a) If all potential energy terms Vi and all hopping terms ti are nearly equal, including the asymptotic ones t ≈ tr . This corresponds to the adiabatic limit where a local density approximation suffices. b) If there are one or more very weak hopping terms separating otherwise uniform chains. In this latter case the ratio t /tr can be arbitrary. These are the limits of weak backscattering or weak tunneling. Starting with the unperturbed basis of translationally invariant wave-functions, or wave-functions vanishing at the interface respectively, a small perturbation only mixes states with energy differences of order of magnitude of the perturbation. In these cases we may keep only a narrow band of states near zero energy and introduce left and right moving fields in the usual way, ψ √j ≈ eikF,x x ψ+ (x) + e−ikF,x x ψ− (x) , a with x = aj a continuous variable and kF x being the Fermi momentum in the left, kF,x≤aj = kF , or right, kF,x≥ajr = kF r , of the wire. Here we want to consider only the simplest model for a junction while various other types of junctions are discussed in Appendix B. In the simplest model two homogeneous regions are connected at one site such that (II.3) tj = (II.4) i = { , r}, and current conservation implies (1 − |R|2 )u = |T |2 ur . (II.5) t , j<0 tr , j ≥ 0 Vi = The region from j to jr is the region of inhomogeneity describing the junction. There are two velocities ui ≡ 2ati sin[ki a] , (II.6) V , j<0 Vr , j > 0 (II.7) and V0 is kept as a free parameter. The reflection amplitude is determined by the Schr¨dinger equation for the o central site and results in R=− a(V + Vr − 2V0 ) − i(u − ur ) a(V + Vr − 2V0 ) + i(u + ur ) (II.8) 3 The conditions for perfect transmission are therefore u = ur , and V0 = (V + Vr )/2 . (II.9) When these conditions are satisfied, R = 0 and |T |2 = 1. Curiously, the maximum possible value of |T |2 actually occurs when V0 = (V + Vr )/2 and ur = 0, in which case |R| = 1 and |T | = 2. But in this case the current is actually zero on both sides, so calling this perfect transmission would seem inappropriate. The existence of the two conditions (II.9) for perfect conductance is related to the breaking of particle–hole symmetry, see Sec. III A. Next, we consider the abrupt junction of Eq. (II.7) in a narrow band approximation setting t = t − δt and tr = t + δt, with |δt| t. When δt = 0 we obtain the usual free, translationally invariant Dirac fermion model, with uniform velocity u0 = 2at sin kF . Here we treat the δt term as a perturbation. Using the separation into right and left moving fields, Eq. (II.6), the backscattering at the junction is given by  −1 ˆ δ H ≈ −a  (2t eikF a − V )e2ikF ja + 2tr eikF r a j=−∞  ∞ † (2tr eikF r a − Vr )e2ikF r ja  ψ− ψ+ + H.c. (II.10) −V0 + j=1 Since ψ− (x) and ψ+ (x) are assumed to vary slowly on −1 the scale of kF /r the oscillating terms in the bulk cancel, leaving only the contributions at x = 0. We may then write the local backscattering at x = 0 as † ˆ δ H ≈ 2πiλψ− ψ+ (x = 0) + H.c. (II.11) with (see also Appendix E) t tr a − (II.12) 2π sin[kF a] sin[kF r a] a − (V cot[kF a] − Vr cot[kF r a]) 4π a = (t sin[kF a] − tr sin[kF r a]) , 2π a Im λ = (V + Vr − 2V0 ) , 4π Re λ = where we have used Eq. (II.2) to simplify the real part. We see that the scattering amplitude λ is real if the local potential energies are equal, V0 = V = Vr . This is surprising because for any non-zero local potential the problem is no longer particle-hole symmetric. In App. B we show that this is a special property of the junction (II.7) and does not hold in general. Finally, we can use the fact that we are treating the difference in hopping δt perturbatively and approximate kF ≈ kF r ≈ kF in which case the real part of the scattering amplitude further simplifies, Re λ = − u − ur a δt sin[kF a] = , π 4π (II.13) where the difference in velocities on the two sides of the junction is given by ur − u = 4aδt sin[kF a]. We see that for V0 ≈ V ≈ Vr and u ≈ ur , required for the narrow band approximation to be valid, the result for the scattering amplitude λ is fully consistent with the exact result for the reflection amplitude (II.8) by using the general relation R = 4πλ/(ur +u ) between these two quantities in this limit. In Sec. III A we will discuss how the narrow band calculation for this type of junction can be extended to the interacting case using bosonization. III. INTERACTING MODEL As a microscopic interacting model we use the Hamilˆ ˆ ˆ ˆ tonian H = H0 + HI , where H0 is given by Eq. (II.1) and † † Uj : ψj ψj :: ψj+1 ψj+1 : ˆ HI = (III.1) j for interactions with a position dependent nearest neighbor interaction strength Uj . Normal ordered operators † † † are given by : ψj ψj := ψj ψj − 0|ψj ψj |0 , with |0 the ground state. It is assumed that the spatial variation of ˆ Uj , tj , and Vj in H0 , is consistent with the narrow band approximation explained in the preceding section. Later we will focus on the limiting case of an abrupt jump in the interaction and hopping parameters at the junction, as used elsewhere.16,17,27,29,30 In order to find the underlying low energy bosonic theory, we first need to linearize the spectrum. Analogously to the normal Luttinger liquid theory,6–8 one can linearize around the bulk band structure in the left and right regions of the wire.27 Linearization is performed around the Fermi momenta kF,x for left and right movers: ψ √j = ψ(x) = eiαkF,x x ψα (x) , a α=± (III.2) with the appropriate commutation relations † [ψα (x), ψβ (x )]+ = 0 and ψα (x), ψβ (x ) = + δαβ δ(x−x ). Here kF,x is defined by −2t cos kF,x +Vx = 0. Note that it is not necessary to assume that kF ≈ kF r . After linearization of the free Hamiltonian we find ˆ H0 = − − † atx eiακx ψα (x)∂x ψα (x) + H.c. dx α=± − + 2tx e−2iακx − Vx e−2iαkF,x dx α=± † ×ψα (x)ψ−α (x) , (III.3) where the Fermi momenta are determined by Vx = 2tx cos κ− , x (III.4) and we have defined κ− = kF,x+a (x + a) − kF,x x and x 2κ+ = kF,x+a (x + a) + kF,x x. Similarly, one can write x 4 the linearized interaction as ˆ HI = dxaUx † † : ψα ψα (x) :: ψβ ψβ (x + a) : α,β=± † † +e−β2ikF,x+a (x+a) : ψα ψα (x) :: ψβ ψ−β (x + a) : † † +e−α2ikF,x x : ψα ψ−α (x) :: ψβ ψβ (x + a) : +e−α2ikF,x x−β2ikF,x+a (x+a) (III.5) † † × : ψα ψ−α (x) :: ψβ ψ−β (x + a) : , keeping for the moment all of the terms. If the interaction acts homogeneously then many of the terms can be neglected as they are suppressed by the rapidly oscillating phases. Due to the inhomogeneity in Ux this is no longer true and all processes could in principle be important. In fact we find that umklapp scattering is generically irrelevant under renormalization group (RG) flow, see Appendix D, and to lowest order the backscattering only renormalizes the single particle backscattering already present in the non-interacting Hamiltonian. We bosonize using the local vertex operator31,32 ψα (x) = √ √ 1 eiα 4π[φα (x)] . 2πa (III.6) We use the following convention: φ(x) = φ+ (x) + φ− (x) ˜ and its adjoint φ(x) = φ+ (x) − φ− (x) with the conjugate ˜ momentum, Π(x) = ∂x φ(x). These fields obey i [φ+ (x), φ− (y)] = − , 4 iα [φα (x), φα (y)] = sgn(y − x), and 4 [φ(x), Π(y)] = iδ(x − y) . (III.7) dx ux 2 1 ˜ (∂x φ)2 + gx (∂x φ)2 gx . (III.8) To lowest order we can determine the renormalized velocity ux ≈ 2atx sin[κ− ] 1 + x Ux sin[κ− ] x πtx , (III.9) and the Luttinger parameter gx ≈ 1 − Ux sin[κ− ] . x πtx − ˆ H = 1 −i√4πφ(x)−2ikF,x x e−iκx ux e − Vx + H.c.. 2πi a sin[κ− ] x x=ja j∈Z (III.11) We keep the sum over x = ja here discrete in order to avoid ambiguity as to what the alternating terms are in the continuum limit. This also helps the precise calculation of these sums. A. (III.10) An abrupt junction Let us now focus on the simple junction considered already in the previous section for the non-interacting case where two semi-infinite wires are joined at x = 0 with tx<0 = t , tx≥0 = tr , and Ux defined equivalently. The local potential energy is taken to be uniform, Vj = V , except where explicitly said to the contrary. The Fermi momenta, kF,x , can also be written with a similar structure as kF,x<0 = kF and kF,x≥0 = kF r . In this system backscattering can be rewritten as √ ˆ H ≈ λe−i 4πφ(x=0) + H.c., − λ = −i x 1 −2ikF,x x e−iκx ux e − V a . (III.12) 2πa sin[κ− ] x With the help of appendix E, and noting that for an abrupt jump κ− = kF,x a, we have to lowest order in the x interaction λ ≈ Some further useful formulas for bosonization are given in App. C. ˆ ˆ ˆ The full Hamiltonian H = H0 + HI can be rewritten in the bosonic representation as a quadratic Hamiltonian, a local backscatterer, and umklapp scattering: ˆ ˆ ˆ ˆ H = Hb + H + HU , see App. C for details. As already mentioned, away from half-filling the umklapp scattering ˆ term HU becomes a local perturbation confined to the regions where Uj is varying, and is then irrelevant under RG flow. It is neglected in the following. We find the quadratic term to be ˆ Hb = The local backscattering from all processes in Eqs. (III.3) and (III.5) can be summarized in one term 1 t U tr Ur + − − 2π sin[kF a] π sin[kF r a] π V − [cot[kF a] − cot[kF r a]] , (III.13) 4π which generalizes the non-interacting √ result, Eq. (II.12). As λ is real we find that there is no sin[ 4πφ(0)] operator present at the boundary and the total backscattering is √ ˆ (III.14) H = 2λ cos[ 4πφ(0)] . √ The perhaps surprising absence of the sin[ 4πφ(0)] operator is connected to the local properties of the Hamiltonian in the vicinity of the boundary, see App. B. As such there remains only one condition to fulfill for the conducting fixed point: λ = 0 with λ real. For V = 0 when there is particle-hole symmetry present, corresponding to the mapping φ → −φ and √ ˜ ˜ φ → −φ, it is transparent that sin[ 4πφ(0)] is forbidden. For V = 0 we find that λ remains real for the specific junction considered—analytically to first order in the interaction U , see Eq. (III.13), and numerically for all interactions strengths, see below. We do not have a simple argument why this is the case and App. B shows that this is in fact not a generic feature of an abrupt junction. 5 0.54 For this we require the following integral ρ β 0.52 τ (x) ≡ 2 √ √ dτ cos[ 4πφ(x, 0)] cos[ 4πφ(0, τ )] 0 β 0.5 dτ e2π[G(x,0;τ )−G(0,0;0)] = (III.17) 0 0.48 = 1 T 4πT a ux g ¯ ux 2πT x sinh 2πaT ux −gx P−¯ (z) g which has been calculated using the Green’s function in Appendix D. We introduced 0.46 -40 0 Uℓ = 0 non-interacting x 40 z ≡ coth Ur = 1.8tr interacting 2πT x , ux (III.18) and Pl (z) is the Legendre function. This gives FIG. 1: (Color online) The full density including Friedel oscillations near the boundary, numerical results (filled circles) are fitted to the analytical result of Eq. (III.19) (lines) with tl = 1.308tr , U = 0, and Ur = 1.8tr . The local potential energy is V = 0.25tr and tr β = 10. Underneath a schematic of the system under consideration is shown. B. Local density and compressibility For the system with an abrupt jump in hopping and interaction strength it is possible to calculate a variety of properties perturbatively in the boundary operators using the exact Green’s function for the Hamiltonian (III.8), see Eq. (D.2) in the Appendix. In addition to the dc conductance one can also consider local properties such as the local density and compressibility of the wire. For abrupt changes in parameters the local density is known to show characteristic oscillations, the Friedel oscillations33 , which give information about the interacting correlation functions34–36 and the strength of the backscattering.27,37 The bosonized density operator for the fermions becomes 1 n(x) = n0 (x) − √ ∂x φ(x) π √ const. ∗ + sin[2kF,x x + 4πφx ] . π √ const. ∗ sin[2kF,x x + 4πφ(x)] π , In order to test the calculations we have developed a quantum Monte Carlo (QMC) code using a stochastic series expansion (SSE) with directed loops.38,39 In Figs. 1 and 2 we show a comparison of this analytical result with the outcome of QMC simulations on spinless Fermions. Even for a very large jump in parameters the fit remains very good. Note that what is seen in the local density and compressibility profiles, see below, is an interplay between the shape of τ (x) and the incommensurate oscilla∗ tions from sin[2kF,x x]. For the fitting procedure between the analytical and numerical results there are two parameters. The first is the amplitude of the effect due to the unknown constant in Eq. (III.16) and the cutoffs in the field theory. The second is a small offset in position, ρalt (x − a), due to an effective width of the scattering ¯ center, with a being of the order of the lattice spacing a. ¯ The Luttinger parameters g ,r can be found from Bethe ansatz.40–43 The local compressibility is defined as χx = − (III.15) As before we keep the local potential energy constant, Vx = V . The oscillating contribution to the density, i.e. the Friedel oscillations, which are given by ρalt (x) ≡ √ const. β ∗ ˆ dτ sin[2kF,x x + 4πφ(x)]H π 0 const. ∗ = −λ 2 τ (x) sin[2kF,x x] (III.19) π a ρalt (x) = − ∂ nx ˆ ∂δV , (III.20) δV =0 analogous to the local susceptibility in a spin chain.35 For the alternating contribution this yields ∗ χalt ∝ λxτ (x) cos[2kF,x x] . (III.21) Unlike the Friedel oscillations in the density this observable remains non-zero even for half-filling and is therefore in that particular case a more useful quantity to study. (III.16) C. ∗ will be calculated to first order in λ. kF,x is the renormalized Fermi momentum at finite temperatures which can ∗ be found from the bulk density: ρx ≡ n0,x = kF,x /π. Conducting fixed points In Sec. III A we have predicted that for the abrupt junction considered only one parameter needs to be tuned 6 0.5 0.01 V = - 0.4 tr 0.01 x 0.48 0.02 V = - 0.2 tr (-1) ρalt(x) ρ 0.46 0 0 (-1) ρalt(x) 0.44 0.4 0 x 10 20 30 40 0 10 20 V = - 0.6 tr 0.02 x 0.42 -40 0 30 V = - 0.8 tr 0.03 0.02 0.01 0.01 0 0 40 FIG. 2: (Color online) The full density including Friedel oscillations near the boundary, numerical results (filled circles) are fitted to the analytical result of Eq. (III.19) (lines) with tl = 1.31tr , U = 0, and Ur = 1.8tr . The local potential energy is V = 0.75tr and tr β = 10. in order to find a conducting fixed point. The low-order expansion for λ given by Eq. (III.13) is not sufficient however to find the location of the fixed points for the large interaction strengths we want to consider in general. Only in the limit Ux → 0, where we know the exact result, can we be confident of its predictions. An exception is the half-filled case where we have previously argued27 that the scattering amplitude λ vanishes for all interaction strengths if u = ur , with the velocities at half-filling known in closed form as a function of the interaction strength from Bethe ansatz.40,41 Instead, at generic fillings, we can find the locations of the solutions t∗ (V ) which solve λ(t = t∗ , V ) = 0, keeping Ux and tr fixed, by analyzing the local density or compressibility of the system by QMC simulations described in the preceding subsection. We find that, away from half-filling, these do not correspond to u = ur . For λ = 0 the density is determined entirely by the Hamiltonian Eq. (III.8), plus irrelevant perturbations. For λ = 0, on the other hand, the relevant backscattering term contributes. By plotting the density for different t in Fig. 3 we can find the places where the leading corrections vanish and λ changes sign,27 which typically can be observed in the range 5a x 10a. Since we can always identify a value of hopping where the leading contribution vanishes, there must be a line of conducting fixed points in parameter space. In turn the existence of a full line of fixed points demonstrates that there is only one condition for the conducting fixed point, λ = 0 with real λ. We want to stress though that even at such a point in parameter space there are still irrelevant backscattering processes present which only vanish in the zero temperature limit β → ∞. 0 10 20 x/a 30 0 10 20 30 x/a FIG. 3: (Color online) Plotted are the Friedel oscillations for different local potential energies V calculated by QMC simulations, see main text for details, on the right hand side of the junction (x > 0). We only show the longer wavelength amplitude of the rapid oscillations. In each panel from top to bottom: t = 1.3tr for (black) circles, t = 1.4tr for (red) squares, t = 1.5tr for (green) diamonds, t = 1.6tr for (blue) up-triangles, and t = 1.7tr for (purple) down-triangles. We have used everywhere U = 0, Ur = 1.8tr and inverse temperature tr β = 10. IV. CONFORMALLY INVARIANT BOUNDARY THEORY In the preceding sections it has been demonstrated that it is possible to find an unstable conducting fixed point in two wires connected at a junction by appropriately tuning the bulk parameters of the wires. The existence of this fixed point immediately invites the question of the nature of the effective low energy theory. Obviously translational invariance is lost and it is also not possible to use mirror charges as would be the case for an open boundary condition. Therefore it is highly non-trivial to postulate a description in terms of a conformally invariant theory in this case. Nonetheless, as we will show in this section it is possible to characterize this fixed point in terms of mode expansions and two effective boundary Luttinger liquid parameters. Particular attention is paid to the case of half-filling where we can pinpoint the fixed point precisely. This allows convenient numerical checks of the results. A. Mode expansion and finite size spectrum In the absence of backscattering at a junction we have the bosonic Hamiltonian17,27,30 ˆ H= dx 1 2 1 ˜ (∂x φ)2 + gx (∂x φ)2 gx (IV.1) Compared to (III.8) the position, x, was rescaled on the two sides of the junction such that u , ur → 1. 7 The fields obey the canonical commutation relation: ˜ [φ(x), ∂y φ(y)] = iδ(x−y). Therefore we have the relation ∂t φ(x) = i[H, φ(x)] ˜ = gx ∂x φ(x) . (IV.2) 1 ∂x gx φ(x, t) = 0 (IV.3) φ(0− ) = φ(0+ ), φ(−L) = φ(L) − The corresponding Green’s function can be determined from Eq. (IV.1), see Eq. (D.2). Here we explore other properties of this boundary condition. We are interested in the solutions of the classical equation of motion, 2 ∂t − gx ∂x At the boundaries φ(x) and ∂x φ(x)/gx have to be continuous leading to the boundary conditions ∂x φ(−L) ∂x φ(L) = g gr ∂x φ(0 ) ∂x φ(0 ) = , g gr [φ(x), ∂t φ(y)] = igx δ(x − y) . g if − L < x < 0 gr if 0 < x < L . φ(x, t) = φ0 + g Πt Qxgx ¯ + + 2L 2¯ L γ g Πx ¯ 1 Qt ˜ ˜ + − φ(x, t) = φ0 + γ 2L 2gx L ¯ (IV.4) ∞ l=1 ∞ l=1 1 1 1 1 = + g ¯ 2 g gr , (IV.8) which describes the conductance.16,27,30 Interestingly, we find in addition a second boundary Luttinger parameter γ= ¯ 1 [g + gr ] , 2 (IV.9) which is important for other correlation functions as we will see below. Π is the field conjugate to √ 0 with φ [φ0 , Π] = i. As this field is periodic, φ0 → φ0 + π, it is clear that the eigenvalues of the conjugate field Π must √ be 2 πm, where m is √ integer. Q is the field conjugate an ˜ ˜ ˜ to φ0 and φ0 → φ0 + √4π so that the eigenvalues of the conjugate field Q are πn for integer n. The classical equation of motion (IV.3) has to follow from a classical least action principle from which the classical Hamiltonian 2L H= 0 dx 2 2 (∂t φ) + (∂x φ) . 2gx (IV.10) is determined. Substituting the mode expansion into the Hamiltonian, we may read off the finite size spectrum π 1 n2 E= − + + m2 g + ¯ L 12 4¯ γ ∞ l(me,l + mo,l ) . l=1 (IV.11) (IV.6) This leads to e−iπlt/L √ gx √ g cos(πlx/L)ae,l + √ i sin(πlx/L)ao,l + H.c. , ¯ γ ¯ 2πl √ e−iπlt/L g ¯ 1 √ i sin(πlx/L)ae,l + √ cos(πlx/L)ao,l + H.c. . gx γ ¯ 2πl As before we have the boundary Luttinger parameter (IV.5) The classical equation of motion (IV.3) has oscillatory solutions as well as solutions linear in x, see Appendix F for details. We may expand the field φ(x) in these solutions, while respecting the canonical commutation relation on a ring with circumference 2L where gx = + (IV.7) Here n and m are arbitrary integers while me/o,l are non-negative integers corresponding to the eigenvalues of a† ae/o,l . We have included the universal term in the e/o,l ground state energy −cπ/(12L) with c = 1 for a periodic system of length 2L. B. Scaling properties of the conducting fixed point As usual, since we have imposed the same boundary condition at both ends, we may read off the scaling dimensions of all single-valued boundary operators in the bosonized theory from the finite size spectrum. The scaling dimensions are ζm,n = n2 + m2 g + ¯ 4¯ γ ∞ l(me,l + mo,l ) . (IV.12) l=1 Each dimension corresponds to a different boundary op√ erator. m2 g corresponds to exp[im 4πφ(0)] with the ¯ m = ±1 operators being the leading relevant operators at the unstable fixed point. γ /4 is the dimension of the ¯ √ ˜ operators exp[±i π φ(0)], which effectively correspond to spin operators S ± (x = 0), see below. To analyze the scaling properties of the system, and compare the results with numerical calculations, it is convenient to introduce correlation functions for a spin system equivalent to our fermionic system. The mapping 8 g ¯ 0.3 gℓ gr γ ¯ g ¯ 0.1 γ ¯ Cz(τ) 0.1 C+-(τ) Cz(τ) gr gℓ 0.1 0.01 0.01 (a) β tr=0.5, 1, 2.5, 5, 10, 15, 20, 25 (b) 0.1 1 0.04 0.1 sin[πτ/β] 1 sin[πτ/β] FIG. 4: (Color online) The scaling of the local spin correlation functions Cz (τ ) and C± (τ ) at the fixed point: t = 1.518tr , U = 0, and Ur = 1.8tr . The magnetic field is zero (i.e. V = 0 for the corresponding fermion system) and the temperature is tr β = 25. (a) Numerical data, black circles, are compared to the predicted scaling f (τ )ν with f (τ ) ≡ | sin(πτ /β)|−2 and ν = g . (red curve). As a comparison we also plot, f (τ )ν with ¯ ν = g , gr , γ , see Eq. (IV.20). (b) Numerical data, black ¯ circles, are compared to the predicted scaling f (τ )1/4ν with ν = γ (red curve). As a comparison we also plot, f (τ )1/4ν ¯ with ν = g , gr , g , see Eq. (IV.19). ¯ between spin operators and fermionic operators is given by the Jordan-Wigner transformation † + Sj = ψj eiπ l 0) where the scaling modulates from the non-interacting result at the boundary x = 0, to the bulk interacting value far inside the wire. In contrast to the density or compressibility of Sec. III B there is no proximity effect near the boundary in the non-interacting wire. 10 D. Fixed points and the g-theorem From the finite size spectrum, Eq. (IV.11), we may also read off the partition function in the scaling limit: γ g Z(β/L) = η −2 e−πβ/L θ3 e−πβ/[2¯ L] θ3 e−πβ2¯/L . (IV.26) Here we have introduced the Dedekind eta and Jacobi theta functions, ∞ η(q) ≡ q 1/24 (1 − q n ) and n=1 ∞ qn θ3 (q) ≡ 2 /2 . c i and hence g > 1. In this case gd < gd so this flow is also ¯ consistent with the g-theorem. It is also interesting to consider the flow starting from the insulating fixed point, but with a weakly connected resonant site in between the two wires: the resonant fixed point. Then for a range of Luttinger parameters an RG flow from the resonant to the conducting fixed point is expected. A necessary condition for the flow from resonant to conducting fixed points is that the tunneling operators from each chain to the resonant site are relevant, g ,r > 1/2. The ground state degeneracy of the resonant fixed point is bigger by a factor of 2 than that of the insulating fixed point due to the 2-fold degeneracy of the resonant site and (IV.27) r gd = 2(g gr )1/4 . n=−∞ In the thermodynamic limit, β/L → 0, this becomes γ /¯eπL/(3β) . ¯ g Z→ c gd = γ ¯ g ¯ 1 4 = (g + gr )1/2 . (4g gr )1/4 (IV.29) This may be compared to the ground state degeneracy for the insulating fixed point where the junction consists of the perfectly reflecting ends of two quantum wires with Luttinger parameters g and gr . This fixed point has11 i gd = (g gr )1/4 , Thus the ratio of ground state degeneracies of the resonant to conducting fixed points is (IV.28) Apart from the usual bulk free energy, F = −πL/(3β 2 ), there is also a “ground state degeneracy”, gd , associated with the two interfaces in the system. The factor for each interface is (IV.30) According to the “g-theorem”, boundary RG flows between fixed points can only occur when gd is reduced during the flow.51 Therefore, it is interesting to consider the ratio √ r c ¯ gd /gd = 2 g . CONCLUSIONS (IV.31) The g-theorem states that flow from the conducting to insulating fixed point is only possible when g < 1. This is ¯ consistent with the √ analysis here since g is the dimension ¯ of the operator cos[ 4πφ(0)] which drives the flow. The flow only takes place when the operator is relevant, cori responding to g < 1. For sufficiently large gd the renor¯ malization flow can occur from insulating to conducting fixed points. As shown in Ref. 11, the fermion operator, or equivalently spin raising operator, at the end of the open chain, has scaling dimension 1/(2g ) or 1/(2gr ) as appropriate. We might expect the flow from insulating to conducting when the tunneling between the two open chains is relevant which occurs when 1 1 1 + = <1 2g 2gr g ¯ (IV.34) √ r c We can see that gd /gd ≥ 2 whenever g , gr > 1/2 so the g-theorem is also obeyed by this RG flow. Even when λ is tuned to zero, corresponding to resonance, the next √ most relevant operators, exp[±2i 4πφ(0)] will still be present. This can drive the flow from the conducting to the resonant fixed points when √ becomes relevant, it c r ¯ i.e. for 4¯ < 1. Since gd /gd = 1/(2 g ) we see that this g flow is consistent with the g-theorem as it only occurs when g < 1/4. Therefore all expected RG flows are con¯ sistent with the g-theorem. As first observed by Kane and Fisher9 in the case g = gr , there is a range of Luttinger parameters where both conducting and resonant fixed points are stable. In this case they are separated by an intermediate unstable fixed point. V. 1 c i gd /gd = √ . g ¯ (IV.33) (IV.32) In conclusion, we have described a novel conducting fixed point in inhomogeneous quantum wires. This fixed point is reached by tuning to zero the amplitude of the leading backscattering operator at the junction between two homogeneous parts of the wire. We have, in particular, studied a lattice model of spinless fermions with nearest neighbor hopping and interaction in the critical regime. For the case of an abrupt junction we have derived the backscattering amplitude for all fillings in lowest order in the interaction. For the half-filled case it is even possible to give a condition for the vanishing of the backscattering amplitude valid for all interaction strengths. The prediction of a conducting fixed point were numerically confirmed by numerical QMC calculations of the Friedel oscillations in the local density and compressibility close to the boundary which vanish in leading order at the fixed point. 11 One of our main results is the derivation of the boundary conformal field for this novel unstable conducting fixed point. The conformally invariant theory for this case is highly unusual because the two parts of the wire are governed by different bulk Luttinger parameters g and gr . As a consequence, we find that the scaling dimensions of boundary operators are also governed by two different Luttinger parameters given by γ = (g + gr )/2 ¯ and g = 2g gr /(g + gr ). We showed, both analytically ¯ and numerically, that γ is controlling the transverse spin ¯ autocorrelation function while g controls the longitudi¯ nal one in the corresponding spin model. Experimentally, a test of the boundary exponents could possibly be obtained by scanning tunneling microscopy which would allow one to measure the local density of states which shows energy scaling with an exponent being determined by γ and g . ¯ ¯ It is convenient to change integration variables to the first integral and 2 in the second, giving: µL d 1 [1 − |R( 1 )|2 ] 2π I = −e 1 (0) µR 1 in (A.3) d 2 u ( 2) |T ( 2 )|2 2π ur ( 2 ) +e 1 (0) Since |T |2 u /ur = 1 − |R|2 this can be written as µL I = −e µR d [1 − |R( )|2 ] . 2π Now taking the limit µL → µR ≡ I→ e2 [1 − |R( 2π F, 2 F )| ](VL (A.4) we find: − VR ) . (A.5) 2 (A.6) Hence the linear conductance is Acknowledgments G= e2 dI = [1 − |R( dV 2π F )| ]. J.S., S.E., and N.S. acknowledge support by the Collaborative Research Center SFB/TR49 and the graduate school of excellence MAINZ. The research of I.A. was supported by NSERC and CIfAR. We are grateful for computation time at AHRP. This is another way of seeing that [1−|R|2 ] is the suitable measure of the transmission of the interface. Appendix A: Landauer formula for transmission In the main text, Sec. II we have considered the simplest possible junction, a jump between two homogeneous regions, in the non-interacting case. Here we want to present calculations for more general junctions to study the influence on the backscattering term. The significance of the measure of transmission R = 0 can be verified by considering the Landauer formula.52 Thus we imagine attaching the wire to reservoirs on the left and right side with different chemical potentials µL and µR . We consider particles emitted from the left reservoir with a thermal distribution with chemical potential µL = −eVL and from the right reservoir with a thermal distribution and chemical potential µR = −eVR . At zero temperature the Fermi wave-vectors on left and right sides, kF and kF r , are given by − 2t cos kF ,r −V ,r = µL,R . (A.1) Suppose that the bottom of the band on the left has higher energy than the bottom of the band on the right. Then the total current, at zero temperature, is kF I = −e 0 dk u (k)[1 − |R(k)|2 ] 2π −k2max +e −kF r Appendix B: Non-interacting calculations 1. Abrupt junction with additional local variation The calculation of Sec. II can be extended straightforwardly to a more general model where the hopping amplitude varies near the origin. Suppose, for example, that the hopping amplitude from site -1 to 0 is t−1 = t and from 0 to 1 is t0 = tr with the rest as given by Eq. (II.7). For simplicity we concentrate again on halffilling, Vj = 0. Then we may write the wave-function for an incoming wave from the left as in Eq. (II.3) with j = −1 and jr = 1. ψ0 is now a free parameter. Solving for the reflection amplitude as previously gives (A.2) dk u (k)|T (k)|2 . 2π The first term is the current emitted by the left reservoir and partially reflected at the interface. The second term is the current emitted by the right reservoir and partially transmitted. The maximum wave-vector for the second integral, k2max > 0, is given by 2 (−k2max ) = 1 (0) since lower energy incoming particles from the right have zero transmission probability. |R|2 = t2 u t2 tr2 t2 ur r 2 − t2 u t2 tr2 t2 ur r 2 + + a2 2 t2 t2 − tr2 t2 r 2 2− + a2 2 t2 t2 − tr2 t2 r 2 2− . (B.1) Solving for R = 0 one finds (t /t , tr /tr ) = √ 2(cos θ, sin θ) (B.2) with tan2 θ = u /ur . Thus maximal conductance can be achieved for any choice of energy , and thus any value of u /ur that can occur as is varied. 12 We see that the simple condition u = ur for perfect conductance is a special result, which only holds for the “abrupt junction” considered in the main text. In general, the condition u = ur can be regarded as removing the intrinsic scattering from a sharp jump between bulk values of the hopping. Additional variation on top of this will naturally result in scattering and an additional finetuning is required to reach the conducting fixed point. Note that the fact that two parameters, t and tr , need to be adjusted to achieve perfect conductance, in general is in accord with the renormalization group (RG) viewpoint. For non-zero energy , particle-hole symmetry is broken so the scattering amplitude λ can be complex. In the special case = 0 where particle-hole symmetry holds there is only one condition for perfect conductance t 2 /t2 = tr2 /t2 and only one parameter needs to be adr justed. Now let us consider the case with non-trivial ti within the narrow band approximation. We use the parametrization t = t + δt = t − δt + δt tr = tr + δtr = t + δt + δtr . To keep things as simple as possible we choose t2L , (i ≤ −2) . t2R , (i ≥ 0) (B.8) There is no particularly simple or natural choice for t2,−1 so it is kept as a free parameter. Let us assume that all the t1,i are close together and all the t2,i are close together so that the narrow band approach is applicable. Thus we write t1,i = t1L = t1 − δt1 , t1R = t1 + δt1 , t2,−1 = t2 + δt . t2,i = t2L = t2 − δt2 , t2R = t2 + δt2 , (B.9) A simple extension of the previous calculation gives πλ = −δt1 csc[kF a] − δt2 cot[kF a] + iδte2ikF a (B.10) for the back-scattering coupling constant. As a simpler special case, consider kF = π/2. Now πλ = −δt1 − δt2 − iδt , (B.3) In this case, there is another backscattering perturbation term t1L , (i ≤ −1) , t1R , (i ≥ 0) (B.11) and λ is complex in this case despite being at half-filling; this is natural since t2 breaks particle-hole symmetry at all fillings. † † ˆ δ H = −2δt ψ− ψ+ e−ikF a − 2δtr ψ− ψ+ eikF a + H.c. † † −(δt + δtr )2 cos[kF a](ψ− ψ− + ψ+ ψ+ ) (B.4) Appendix C: Bosonization details where ψ− and ψ+ are evaluated at x = 0 in all terms. Focusing on the backscattering term the perturbation becomes First we note the following useful relations: † ˆ ˆ δ H + δ H = 2πiλψ− ψ+ (x = 0) + H.c. (B.5) u − ur iδt e−ikF a + iδtr eikF a + . 4πa π (B.6) 1 † ψα (x)ψα (x) = ρα (x) ≡ − √ ∂x φα (x), π † ψα (x)∂x ψα (x) = αiπρ2 (x), and α iα −iα√4πφ(x) † ψα (x)ψ−α (x) = . e 2πa with λ= Although the variations in ti are small, they occur over only three sites, so this is not an adiabatic change. Note that while we were able to determine λ explicitly in this model, with all ti nearly equal, it may not be feasible to do so in all cases. In fact, a reduction to a narrow band model is not accurate in general, as discussed in the main text. 2. Next-nearest neighbor hopping (C.1) Due to the inhomogeneous nature of both the Fermi momentum and the interactions, the 2kF,x oscillating terms in the interaction can no longer be neglected, see Eq. (III.5). One finds a nonzero contribution to the backscattering around any region of inhomogeneity. These terms must be treated carefully, as an example we † † can take : ψα ψα (x) :: ψα ψ−α (x + a) :. Direct rearrange† † ment gives for ψα ψα (x)ψα ψ−α (x + a) either † † : ψα ψα (x) :: ψα ψ−α (x + a) : † † + 0|ψα ψα (x)|0 : ψα ψ−α (x + a) : (C.2) or Next-nearest neighbor hopping can also be added to the Hamiltonian, explicitly breaking particle-hole symmetry. We consider the Hamiltonian † † − : ψα (x)ψ−α (x + a) :: ψα (x + a)ψα (x) : (C.3) † † − : ψα (x)ψ−α (x + a) : 0|ψα (x + a)ψα (x)|0 , which are therefore equal. Then, using † † [−t1,j ψi ψi+1 − t2,i ψi ψi+2 + H.c.] . ˆ H0 = i (B.7) † 0|ψα (x + a)ψα (x)|0 ≈ iα/2πa , (C.4) 13 Solving this differential equation subject to the appropriate boundary conditions17,27 gives and expanding in the cut-off a this allows us to write † † : ψα ψα (x) :: ψα ψ−α (x + a) :≈ (C.5) † † − 0|ψα ψα (x)|0 : ψα ψ−α (x + a) : † − : ψα (x)ψ−α (x + a) : iα † + ρα (x) + aψα ∂x ψα (x) . × 2πa G(x, y; τ ) = φ(x, 0)φ(y, τ ) g ¯ = − ln sinh πT π + (D.2) |x| |y| + − iτ ux uy sinh πT L[x, y]gx ln π sinh πT Now keeping only the leading order terms we find √ : † ψα ψα (x) :: † ψα ψ−α (x e−iα 4πφ(x) . + a) :≈ 4π 2 a2 (C.6) Similar expressions hold for the other terms. The bosonized free and interacting Hamiltonians become − ˆ H0 = − αitx a2 eiακx (∂x φα )2 + h.c. (C.7) xα − xα iαtx −2iαkF,x x−iα√4πφ(x) −iακ− V x − e e π 2tx , and ˆ HI = a2 Ux − 1 − e−2iακx xα (∂x φ)2 2π √ +e 2iαkF,x x −e 4iακ+ x e 2iακ− x √ eiα2 4πφ(x) , (2πa)2 (C.8) + |y| uy |x−y| ux − iτ . − iτ We have introduced 2L[x, y] ≡ 1 + sgn[x] sgn[y] . The renormalization procedure is done in the standard manner by expanding the perturbation, exp(− dτ H ), to first order and integrating out the fields with fast Fourier components near the band-edge Λ < |k| < Λ. In order to recover the original form after re-exponentiating the action we rescale Λτnew = Λ τ , and define the new coupling constant λ, as λ(Λ ) = Λ ˜ λ(Λ)e−πG> (x=y=τ =0) . Λ (D.3) ˜ where G> is the Green’s function after integrating out the fast modes. Therefore for the RG equation what we need is the Green’s function summed over the fast modes. First let us change variables to r = x−y and R = (x+y)/2. Then, with u(r, R = 0) = 2ux uy /(ux + uy )|x=−y = 2u ur /(u + ur ) ≡ u, we have G(r, R = 0; τ ) = − 2eiα 4πφ(x) −1 (2πa)2 |x| ux g ¯ ln sinh πT π |r| − iτ u . (D.4) This is the same as the Green’s function for a homogeneous case, but with a new velocity and a new Luttinger parameter: 1 1 1 1 = + g ¯ 2 g gr respectively. Included in this is the irrelevant umklapp scattering . (D.5) Now we require ˆ HU = − x √ Ux a 2 cos 4κ+ + 2 4πφ(x) . x π a (C.9) For the abrupt jump of Sec. III A the Green’s function, at λ = 0, can be calculated exactly. We have (D.1) m 2 ωm ∂ ux ∂ − 2gx ux ∂x 2gx ∂x (D.6) dreikr G(r, R = τ = 0) . = Λ <|k|<Λ Thus integrating out the fast Fourier components results in a change of the Green’s function, g ¯ (D.7) G> ≈ d ln Λ , π which governs the renormalization in the usual manner: Appendix D: The Green’s function and renormalization group calculations eiωm τ Gm (x, y) and G(k, R = τ = 0) Λ <|k|<Λ Away from half-filling this is only a boundary contribution. G(x, y; τ ) = T G> (0, 0; 0) = Gm (x, y) = δ(x − y) . 1 dλ = 1−g. ¯ (D.8) λ d ln Λ We therefore expect that the effective backscattering renormalizes as a power law in the temperature R ∝ ¯ T g−1 , which in turn affects the conductance and other physical observables accordingly. This has been confirmed numerically.27 14 Appendix E: Useful sums for the boundary terms To find the coefficients of the backscattering terms several sums are needed. We want e−2ikF,x x F (x)O(x) I≡ (E.1) x=ja j∈Z Appendix F: The mode expansion and its correlation functions Let’s first consider the solutions of the classical equation of motion, Eq. (IV.3), subject to the boundary conditions Eq. (IV.5). There are two types of oscillating solutions (1) φk (x, t) ∼ eikt cos(kx) in the particular case where we can write F (x) = F Θ(−x − a) + Fr Θ(x) ≡F (x) (E.2) ≡Fr (x) e−2ikF i x Fi (x) I ≈ O(x = 0) (2) (E.3) x=ja i=1,2 j∈Z e−2ikF i x ≈ O(x = 0) x=ja i=1,2 j∈Z Zi [Fi (x) − Fi (x + a)] , 2 with Zi = 1 + i cot[kF i a]. Only a single term of each sum over x is non-zero and we find iF eikF a iFr eikF r a − . 2 sin[kF a] 2 sin[kF r a] I ≈ O(x = 0) ≡F (x) (E.5) ≡Fr (x) and we can independently change F0 on the central site. Then with I ≈ Ox=0 λ we find λ = F0 + iFr e−ikF r a iF eikF a − . 2 sin[kF a] 2 sin[kF r a] (F.2) with φ(x = 0) = 0. In addition there are solutions linear in x and t. The solutions linear in x have the form φx (x, t) ∼ gx x . (F.3) The bosonization formula (III.6) implies that the bosonic √ field φ is periodic with period φ + π. Furthermore we are considering solutions on a ring with circumference √ 2L thus φ(x) = φ(x + 2L) + πn. The oscillatory solutions of both types therefore must have k = πl/L for l = 1, 2, 3, . . .. For the solutions linear in x the same periodicity conditions imply √ (E.4) We may also be interested in the case where F (x) = F Θ(−x − a) + Fr Θ(x − a) +F0 δ(x) with ∂x φ(x = 0) = 0, and φk ∼ gx eikt sin(kx) with Θ(0) ≡ 1. Assuming Ox is slowly varying on a length scale of a, this allows us to write (F.1) (E.6) φx (x, t) = πn xgx . 2¯ L γ (F.4) ˆ Let’s now consider the mode expansion.Π is canonically conjugate to φ0 and the normalization of each term is fixed by requiring the canonical commutation relations to hold.For g = gr , we may expand in solutions of the classical equations of motion, while respecting the canonical commutation relations. This leads to the mode expansion given in Eq. (IV.7). Using the mode expansion we can first calculate the bosonic commutators in the ground state. 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