A Landauer formula for transmission

Conducting fixed points for inhomogeneous quantum wires: a conformally invariant boundary theory


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.

73.63.Nm, 71.10.Pm, 73.40.-c

I Introduction

Transport in quantum wires is a rich field bringing together conductivity experiments Liang et al. (2001); Javey et al. (2003); Yacoby et al. (1996); Steinberg et al. (2008); Tarucha et al. (1995) and Luttinger liquid theory which describes the crucial electron-electron interaction effects in one dimension.Tomonaga (1950); Luttinger (1963); Giamarchi (2004) 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.Kane and Fisher (1992a, b); Eggert and Affleck (1992); Furusaki and Nagaosa (1993); Pereira and Miranda (2004); Sedlmayr et al. (2011)

In order to determine the conductivity of a one-dimensional 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,Yue et al. (1994); Safi and Schulz (1995); Maslov and Stone (1995); Ogata and Fukuyama (1994); Wong and Affleck (1994); de Chamon and Fradkin (1997); Imura et al. (2002); (22); Rech and Matveev (2008a); Rech and Matveev (2008b); Gutman et al. (2010); Thomale and Seidel (2011); Sedlmayr et al. (2012); Sedlmayr et al. (2013) 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.Safi and Schulz (1995, 1999) 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 scattering Furusaki and Nagaosa (1996) or phonon scattering Sedlmayr et al. (2013) 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,Furusaki and Nagaosa (1996) 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.Sedlmayr et al. (2012) 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 , where is the inverse temperature and 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 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 non-interacting spinless fermions described by the Hamiltonian


creates a particle at site , and are the position dependent hopping elements and local potential energy respectively, and the total particle number. We set and include in the following the chemical potential in the local potential energy . Generically we consider situations in which we have two homogeneous regions on the left () and right () side of the wire. In these asymptotic regions the plane-wave solutions have the same energy so the parameters are related by


with the momenta, the potential, and the hopping on the left () and right () side. We have also introduced the lattice spacing . We consider a wave-function incident from the left


The region from to is the region of inhomogeneity describing the junction.

There are two velocities


, and current conservation implies


It is natural to refer to as “perfect transmission”, although this does not necessarily maximize . 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, ; that is, maximizing . Noting that we see that the condition for perfect transmission equivalently corresponds to minimizing . 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 -matrix, are non-negligible when is not close to . 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 , where is the Fermi momentum, and linearize the dispersion relation. This can be justified in one of two cases. a) If all potential energy terms and all hopping terms are nearly equal, including the asymptotic ones . 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 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,


with a continuous variable and being the Fermi momentum in the left, , or right, , 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


and is kept as a free parameter. The reflection amplitude is determined by the Schrödinger equation for the central site and results in


The conditions for perfect transmission are therefore


When these conditions are satisfied, and . Curiously, the maximum possible value of actually occurs when and , in which case and . 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.1.

Next, we consider the abrupt junction of Eq. (II.7) in a narrow band approximation setting and , with . When we obtain the usual free, translationally invariant Dirac fermion model, with uniform velocity . Here we treat the term as a perturbation. Using the separation into right and left moving fields, Eq. (II.6), the backscattering at the junction is given by


Since and are assumed to vary slowly on the scale of the oscillating terms in the bulk cancel, leaving only the contributions at . We may then write the local backscattering at as


with (see also Appendix E)


where we have used Eq. (II.2) to simplify the real part. We see that the scattering amplitude is real if the local potential energies are equal, . 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 perturbatively and approximate in which case the real part of the scattering amplitude further simplifies,


where the difference in velocities on the two sides of the junction is given by . We see that for and , required for the narrow band approximation to be valid, the result for the scattering amplitude is fully consistent with the exact result for the reflection amplitude (II.8) by using the general relation between these two quantities in this limit. In Sec. III.1 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 Hamiltonian , where is given by Eq. (II.1) and


for interactions with a position dependent nearest neighbor interaction strength . Normal ordered operators are given by , with the ground state. It is assumed that the spatial variation of , , and in , 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.Safi and Schulz (1995); Maslov and Stone (1995); Furusaki and Nagaosa (1996); Safi and Schulz (1999); Sedlmayr et al. (2012)

In order to find the underlying low energy bosonic theory, we first need to linearize the spectrum. Analogously to the normal Luttinger liquid theory,Tomonaga (1950); Luttinger (1963); Giamarchi (2004) one can linearize around the bulk band structure in the left and right regions of the wire.Sedlmayr et al. (2012) Linearization is performed around the Fermi momenta for left and right movers:


with the appropriate commutation relations and . Here is defined by . Note that it is not necessary to assume that .

After linearization of the free Hamiltonian we find


where the Fermi momenta are determined by


and we have defined and . Similarly, one can write the linearized interaction as


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 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 operator Haldane (1981a, b)


We use the following convention: and its adjoint with the conjugate momentum, . These fields obey


Some further useful formulas for bosonization are given in App. C.

The full Hamiltonian can be rewritten in the bosonic representation as a quadratic Hamiltonian, a local backscatterer, and umklapp scattering: , see App. C for details. As already mentioned, away from half-filling the umklapp scattering term becomes a local perturbation confined to the regions where is varying, and is then irrelevant under RG flow. It is neglected in the following. We find the quadratic term to be


To lowest order we can determine the renormalized velocity


and the Luttinger parameter


The local backscattering from all processes in Eqs. (III.3) and (III.5) can be summarized in one term


We keep the sum over 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.

iii.1 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 with , , and defined equivalently. The local potential energy is taken to be uniform, , except where explicitly said to the contrary. The Fermi momenta, , can also be written with a similar structure as and . In this system backscattering can be rewritten as


With the help of appendix E, and noting that for an abrupt jump , we have to lowest order in the interaction


which generalizes the non-interacting result, Eq. (II.12). As is real we find that there is no operator present at the boundary and the total backscattering is


The perhaps surprising absence of the 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: with real.

For when there is particle-hole symmetry present, corresponding to the mapping and , it is transparent that is forbidden. For we find that remains real for the specific junction considered—analytically to first order in the interaction , 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.

iii.2 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 oscillations Friedel (1958), which give information about the interacting correlation functions Egger and Grabert (1995); Eggert and Affleck (1995); Söffing et al. (2009) and the strength of the backscattering.Rommer and Eggert (2000); Sedlmayr et al. (2012)

Figure 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 , , and . The local potential energy is and . Underneath a schematic of the system under consideration is shown.

The bosonized density operator for the fermions becomes


As before we keep the local potential energy constant, . The oscillating contribution to the density, i.e. the Friedel oscillations, which are given by


will be calculated to first order in . is the renormalized Fermi momentum at finite temperatures which can be found from the bulk density: .

For this we require the following integral


which has been calculated using the Green’s function in Appendix D. We introduced


and is the Legendre function. This gives


In order to test the calculations we have developed a quantum Monte Carlo (QMC) code using a stochastic series expansion (SSE) with directed loops.Syljuåsen and Sandvik (2002); Dorneich and Troyer (2001) 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 and the incommensurate oscillations from . 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, , due to an effective width of the scattering center, with being of the order of the lattice spacing . The Luttinger parameters can be found from Bethe ansatz.Takahashi (1999); Essler et al. (2005); Pereira et al. (2007); Sirker (2012)

Figure 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 , , and . The local potential energy is and .

The local compressibility is defined as


analogous to the local susceptibility in a spin chain.Eggert and Affleck (1995) For the alternating contribution this yields


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.3 Conducting fixed points

In Sec. III.1 we have predicted that for the abrupt junction considered only one parameter needs to be tuned in order to find a conducting fixed point. The low-order expansion for 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 , where we know the exact result, can we be confident of its predictions. An exception is the half-filled case where we have previously arguedSedlmayr et al. (2012) that the scattering amplitude vanishes for all interaction strengths if , with the velocities at half-filling known in closed form as a function of the interaction strength from Bethe ansatz.Takahashi (1999); Essler et al. (2005)

Instead, at generic fillings, we can find the locations of the solutions which solve , keeping and 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 . For the density is determined entirely by the Hamiltonian Eq. (III.8), plus irrelevant perturbations. For , on the other hand, the relevant backscattering term contributes. By plotting the density for different in Fig. 3 we can find the places where the leading corrections vanish and changes sign,Sedlmayr et al. (2012) which typically can be observed in the range . 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, with real . 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 .

Figure 3: (Color online) Plotted are the Friedel oscillations for different local potential energies calculated by QMC simulations, see main text for details, on the right hand side of the junction (). We only show the longer wavelength amplitude of the rapid oscillations. In each panel from top to bottom: for (black) circles, for (red) squares, for (green) diamonds, for (blue) up-triangles, and for (purple) down-triangles. We have used everywhere , and inverse temperature .

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.

iv.1 Mode expansion and finite size spectrum

In the absence of backscattering at a junction we have the bosonic Hamiltonian Maslov and Stone (1995); Furusaki and Nagaosa (1996); Sedlmayr et al. (2012)


Compared to (III.8) the position, , was rescaled on the two sides of the junction such that . The fields obey the canonical commutation relation: . Therefore we have the relation


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,


on a ring with circumference where


At the boundaries and have to be continuous leading to the boundary conditions


The classical equation of motion (IV.3) has oscillatory solutions as well as solutions linear in , see Appendix F for details. We may expand the field in these solutions, while respecting the canonical commutation relation


This leads to


As before we have the boundary Luttinger parameter


which describes the conductance.Safi and Schulz (1995); Furusaki and Nagaosa (1996); Sedlmayr et al. (2012) Interestingly, we find in addition a second boundary Luttinger parameter


which is important for other correlation functions as we will see below. is the field conjugate to with . As this field is periodic, , it is clear that the eigenvalues of the conjugate field must be , where is an integer. is the field conjugate to and so that the eigenvalues of the conjugate field are for integer .

The classical equation of motion (IV.3) has to follow from a classical least action principle from which the classical Hamiltonian


is determined. Substituting the mode expansion into the Hamiltonian, we may read off the finite size spectrum


Here and are arbitrary integers while are non-negative integers corresponding to the eigenvalues of . We have included the universal term in the ground state energy with for a periodic system of length .

iv.2 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


Each dimension corresponds to a different boundary operator. corresponds to with the operators being the leading relevant operators at the unstable fixed point. is the dimension of the operators , which effectively correspond to spin operators , see below.

Figure 4: (Color online) The scaling of the local spin correlation functions and at the fixed point: , , and . The magnetic field is zero (i.e.  for the corresponding fermion system) and the temperature is . (a) Numerical data, black circles, are compared to the predicted scaling with and . (red curve). As a comparison we also plot, with , , , see Eq. (IV.20). (b) Numerical data, black circles, are compared to the predicted scaling with (red curve). As a comparison we also plot, with , , , see Eq. (IV.19).

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 between spin operators and fermionic operators is given by the Jordan-Wigner transformation


The leading correlation function at the boundary is, in bosonized form,


Using Eq. (IV.7) this results in


with the boundary exponent . For the operator we have after bosonization


The leading spin density waves are described by the autocorrelation function at the boundary


From this one finds


with the boundary exponent . Thus the boundary theory is described by two different boundary Luttinger parameters, and .

In the QMC simulations we consider finite temperatures in the limit of large system sizes and calculate the imaginary time correlation functions. In this case the results are most easily accessible by considering the Green’s function Eq. (D.2), and the equivalent correlation function for the adjoint field . Then we find




We compare the predicted scaling of these correlation functions with the results of QMC simulations. The predicted exponents are well verified, see Fig. 4. Not only can one clearly distinguish the two boundary exponents, but we have also checked that the bulk exponents do not fit the scaling. Note that the analytical formula are only valid in the asymptotic limit . The values of , and hence of and , can be found exactly from the results of Bethe ansatz.Takahashi (1999); Essler et al. (2005); Pereira et al. (2007); Sirker (2012)

Figure 5: (Color online) The scaling of the local spin correlation function at the conducting fixed point with , , and . The magnetic field is zero (i.e.  for the corresponding fermion system). Numerical data for different inverse temperatures (as indicated on the plot) are compared to the predicted scaling (lines). As temperature is lowered the field theory becomes more accurate.

Fig. 5 shows the temperature scaling of at the conducting fixed point , and Fig. 6 the scaling away from it. At the conducting fixed point the field theory, as expected, does not describe the data at high temperatures, such as or . As temperature is lowered the field theory becomes a better and better fit, showing good scaling already by . As we move away from the conducting fixed point the corrections to scaling are expected to grow while lowering the temperature but are only . This makes it impossible to see the approach to the insulating fixed point in . The Friedel oscillations of the density and compressibility considered in Sec. III.2 are, in principle, better to see the crossover to the insulating fixed point. However, the expected cross-over temperature is of order ,Sedlmayr et al. (2012) which is unfortunately well beyond the reach of the numerical QMC simulations.

Figure 6: (Color online) The scaling of the local spin correlation function away from the conducting fixed point with , , and . The magnetic field is zero (i.e.  for the corresponding fermion system). From top to bottom we plot different values of the inverse temperature .

iv.3 Local density of states

One possible experimental test on boundary exponents is the measurement of the local density of states with local spectroscopic tools, such as scanning tunneling spectroscopy.Blumenstein et al. (2011) Theoretically a characteristic depletion with the boundary exponent has been predicted,Kane and Fisher (1992a); Eggert et al. (1996); Eggert (2000); Kakashvili et al. (2006); Anfuso and Eggert (2003); Schneider and Eggert (2010) which may be corrected by irrelevant operators.Söffing et al. (2013) It is therefore interesting to calculate the characteristic signatures of the local density of states for this unusual fixed point.

The local density of states is defined as


Using the correlation functions calculated from the mode expansion, see Appendix F, and, neglecting the cut-off for the moment, this results in


The exponents are given by