Transport through asymmetric two-lead junctions of Luttinger liquid wires
We calculate the conductance of a system of two spinless Luttinger liquid wires with different interaction strengths , connected through a short junction, within the scattering state formalism. Following earlier work we formulate the problem in current algebra language, and calculate the scale dependent contribution to the conductance in perturbation theory keeping the leading universal contributions to all orders in the interaction strength. From that we derive a renormalization group (RG) equation for the conductance. The analytical solution of the RG-equation is discussed in dependence on . The regions of stability of the two fixed points corresponding to conductance and respectively, are determined.
pacs:71.10.Pm, 72.10.-d, 85.35.Be
Introduction. Charge or spin transport in three-dimensional metallic systems may be described in terms of Landau quasi-particles. In strictly one-dimensional quantum wires quasi-particles are found to be unstable on account of the interaction between electrons. An important part of that physics is captured by the exactly solvable Tomonaga-Luttinger liquid (TLL) model. Giamarchi (2003); Gogolin et al. (1998) As is well known, transport through junctions of TLL wires is strongly affected by the interaction in the wires, in some cases leading to a complete blocking of transmission. The latter behavior can be traced back to the formation of Friedel oscillations of the charge density around the barrier, leading to an infinitely extended effective barrier potential in the limit of low energy. The suitable language to describe this situation is the renormalization group (RG) method, allowing to calculate the conductance as a function of (length or energy) scale. Quite generally, the transport behavior at low energy/temperature is dominated by only a few fixed points of the RG flow. In the neighborhood of these fixed points the conductance is found to obey power-law behavior as a function of temperature for the infinite system or as a function of system length at zero temperature. The problem of the two-wire junction has been studied first in the pioneering works Kane and Fisher (1992); Furusaki and Nagaosa (1993), using the method of bosonization. Later, their results have been confirmed by many other authors, see Safi and Schulz (1999) and references therein. For special values of the interaction exact results have been obtained Weiss et al. (1995); Fendley et al. (1995).
A purely fermionic formulation of the problem has been introduced in the limit of weak interaction by Yue, Glazman and Matveev Yue et al. (1994). We have extended that theory to the regime of strong coupling by summing up an infinite series of terms in perturbation theory, identified as the leading and universal contributions Aristov and Wölfle (2009). The results obtained with our method for the two-lead junction as well as the three lead junction with time-reversal symmetry Aristov et al. (2010); Aristov and Wölfle (2011) and with magnetic flux Aristov and Wölfle () are in agreement with exact theoretical results, where available. However, our results go beyond what has been obtained by other authors in various ways. The majority of the previous works considered the symmetric case of equal interaction strength in both half wires, with exception of the works Furusaki and Nagaosa (1996); de C. Chamon and Fradkin (1997) discussed below.
In this paper we generalize our previous treatment Aristov and Wölfle (2009) to the case of two wires with different interaction strength . This includes the case of a barrier at the end of a TLL wire, for which one of the interaction parameters is zero, e.g. . As before, we confine our considerations to spinless fermions.
The model. To illustrate the system we are interested in, we first consider a tight binding Hamiltonian of free spin-less fermions describing two quantum wires connected at a single junction by tunneling amplitudes:
Here creates a fermion in wire at site , and is the tunneling amplitude connecting the sites at the junction. The 22 -matrix characterizing the scattering at this junction has the structure (up to overall phase factors in the individual wires)
We choose this parametrization in terms of the transmission and reflection amplitudes , since it is readily generalizable to the case of multi-wire junctions ( wires, ). The above form of the -matrix is completely general (up to irrelevant phase factors) and, in fact, defines our model. Passing to the continuum limit, linearizing the spectrum at the Fermi energy and adding forward scattering interaction of strength in wire , we may write the TLL Hamiltonian in the representation of incoming and outgoing waves as
We put from now on. The range of the interaction lies within the interval , where serves as a ultraviolet cutoff and separates the domains of interaction and potential scattering on the junction; non-interacting leads correspond to large beyond . In terms of the doublet of incoming fermions the outgoing fermion operators may be expressed with the aid of the -matrix as . We express the interaction term of the Hamiltonian in terms of density operators , and , where , as
The matrices are given by and . A convenient representation of 22-matrices is in terms of Pauli matrices , , the generators of (see Aristov and Wölfle (2009)). Notice that the interaction operator only involves (besides the unit operator ). We note , , . Defining a two-component vector , we have , where the matrix has the properties , , and . The outgoing amplitudes will be expressed in terms of . With the aid of the the -matrix may be parametrized by three angular (Euler) variables (see Aristov and Wölfle (2009)), . For the case under consideration only two of these, , are relevant: . The corresponding elements of the -matrix have been given in Eq. (2).
Parametrization of conductance matrix. We may define a matrix of conductances relating the current in lead (flowing towards the junction) to the electrical potential in lead : . It follows from the conservation of charge that and from invariance under a shift of the zero of energy that . Therefore, the conductance matrix has only one independent element , which relates the net current to the bias voltage , . We note the relation , while all other elements of are zero.
In the linear response regime the conductances are related to the -matrix by , where the label indicates that the quantity is fully renormalized by interactions. Defining and using the above relation of and , we see that the conductance components may be expressed in terms of as . Here and in the following bold faced quantities marked with overbar are matrices in the transformed space (indices ). It follows from the analysis of the transformed matrix given above that the matrix has block structure, the nonzero elements being given by the conductance parameter introduced above and by unity, , , . From the above relations we see that the parameter may be expressed by the angle in the above parametrization of the S-matrix as . We find therefore that is confined within the region .
Perturbation theory. We now calculate the conductance in perturbation theory in the interaction. In first order we have to evaluate the diagrams depicted in Fig. 1. Here solid lines denote Green’s functions in position-energy representation ( are fermionic Matsubara frequencies), . The double wavy lines denote the interaction operator, which in the lowest order is given by the matrix , with in the transformed or initial representations, respectively. At the ends of the interaction lines operators and are attached depending on whether or . As a result, one finds in lowest order in the interaction
Here the , and , are 22 matrices for each pair of (or ) and the trace operation Tr is defined with respect to that matrix space, whereas the , with are scalars. Notice that only one matrix element, , is nonzero.
We may extend this analysis into the strong coupling regime by summing up infinite classes of contributions in perturbation theory. It has been found in Aristov and Wölfle (2009) that the diagrams shown in Figs. 1 and 2 provide the leading terms in the neighborhood of the fixed points. Their contribution is universal in contrast to other higher order terms (see below). These may be interpreted as a renormalization of the bare interaction, . The contribution to conductance stemming from the first two diagrams (a) and (b) in Fig. 1 in this ladder approximation is given by
Here only the element of enters, corresponding to the fact that operators and are attached to the ends of the renormalized interaction line. The factor appears as . In the limit of zero temperature we may convert the summation over Matsubara frequencies to an integration along the imaginary axis. Another contribution to is obtained by reverting the arrows on the fermion lines in Fig. 1a, Fig. 1b, and doubles the above result. Performing the integrations on and on from to and taking the limit we find
It is useful to first calculate in the initial representation, where the interaction matrix is diagonal, but the matrix is nondiagonal. Then is found to satisfy the following linear integral equation for (Ref. [Aristov and Wölfle, 2009], note that the definitions of and for the strength of interaction there are different):
with the fermionic loop . Expressing via we have
with and . In order to solve this integral equation we first calculate a partial summation, defined by
. Then will be the solution of
and satisfies the integral equation
These integral equations are shown diagrammatically in Fig. 2.
We now define , so that the inhomogeneity in the integral equation for is differentiable
The integral equation for may be converted into a second order differential equation
Since the matrix is diagonal, is diagonal and we have two uncoupled differential equations for the components , . Taking into account the boundedness of , the general solution is given by , where . It follows from the boundary conditions at that . The quantity is thus a diagonal matrix given by
We now return to the integral equation for . Since the kernel is separable it is easily solved. We define the auxiliary matrix functions , as well as the matrix . Multiplying the above integral equation (9) by and integrating over we find and we finally get
This result is now substituted into the expression for the conductance. Performing the integration over first we get . Next integrating over positive yields . Finally, the integration over produces the scale dependent logarithm . The conductance in the ladder approximation is found after transforming to the rotated basis (quantities with overbar) and taking the -element of the matrix,
Renormalization group equations. The renormalization of the conductances by the interaction is determined from the scale dependent contributions in perturbation theory. Differentiating these results with respect to (and then putting ) we find the RG equation for the quantity in the ladder approximation
Here and , such that and where , and is the usual Luttinger liquid parameter for wire . We define
(note that for any ), then the RG-equation takes the explicit form
RG flow and conductance. The fixed points of the above RG equation are labelled , at , , (complete separation of the wires) and at , (ideal conductance through the junction). In order to discuss the stability and to calculate the conductance we rewrite the RG-equation in terms of the conductance
Stability of a fixed point requires that the derivative of at the fixed point is negative. At fixed point this translates into , or more explicitly, , and in terms of the Luttinger parameters i.e. either both interactions should be repulsive, or one of them is attractive, but weak relative to the repulsive one. At fixed point the condition is , or and therefore , meaning that the interaction is predominantly attractive, but a weaker repulsive interaction in one of the wires is possible. The line separating the stability regions in the --plane is the hyperbola, , passing through the (no interaction) point . This hyperbola and corresponding stability regions are shown in Fig. 3
The RG-equation may be integrated to give
In the vicinity of fixed points and , respectively, we then find the power laws
Of special interest is the case , , when we have and the power laws are given by at and at . We may compare these findings with results obtained in Ref. [Furusaki and Nagaosa, 1996] for a non-symmetric position of an impurity in a TLL wire. When taking the impurity position sufficiently close to the interface between the TLL wire and a non-interacting lead, scaling exponents identical to the above ones were found in Furusaki and Nagaosa (1996), as can be read off from Eqs. (10) and (13) there.
Thus we recover the correct exponents in the scaling law for the conductance. We obtained them by summation of the ladder sequence for the renormalized interaction in the presence of the junction. As was shown in Aristov and Wölfle (2009), the first contributions to conductance beyond the ladder series appear in the third order of interaction and are of three-loop type (the two-loop RG contributions are absent). These latter terms do not contribute to above scaling exponents, but define the relation between the prefactors and . The Eq. (15) implies that both and depend on the initial conditions, encoded in the amplitude . It is easily seen, that the ladder approximation corresponds to the relation
It was shown Aristov and Wölfle (2009) that the three-loop corrections change this ratio to values, different from unity and depending on the strength of interaction and regularization scheme, i.e. non-universal.
Our expression (12) shows that the boundary exponents depend only on the sum . This is in precise agreement with Eq. (13) in Ref. de C. Chamon and Fradkin (1997), where the scaling exponent of the point contact between two chiral (Hall edge) states was derived. On the basis of this observation it was suggested there, that the combination of two chiral states, and , is equivalent to the well-known problem of impurity in TLL wire with , which can be fully solved. Kane and Fisher (1992) We note here that the coincidence of the scaling exponents at two fixed points and might not necessarily mean the coincidence of the full scaling form for the conductance. The non-universal three-loop terms (which are not discussed in the standard bosonization approach) may be different in cases , and .
Conclusion. In this paper we employed a fermionic description of a general two-wire junction of two TLL-wires to derive the renormalization group equation for the conductance, using the approach developed by us earlier Aristov and Wölfle (2009). We used an infinite summation of perturbation theory in the form of a ladder approximation, allowing for an analytical solution for arbitrary junction parameters and interactions in the wires. As demonstrated earlier the approximation is asymptotically exact in the vicinity of the fixed points. As in the well-studied case of a symmetric junction there exist two fixed points of the RG flow. Fixed point corresponds to a complete separation of the wires, i.e. the conductance vanishes. It is stable in a region of the -coupling constant plane which is predominantly repulsive, meaning that a weakly attractive component, say is permitted. At fixed point the conductance assumes its maximum value. It is stable in the complementary part of the coupling constant plane. The two stability regions in the -plane are separated by a hyperbolic boundary curve. The representation chosen is maximally general and may therefore be easily extended to junctions connecting more than two wires. The --symmetric three wire junction (“Y-junction”) has been considered by us in Refs. Aristov and Wölfle (); Aristov (2011); Aristov and Wölfle (2011); Aristov et al. (2010). Work on the four-wire junction is in progress.
We thank D.G. Polyakov for helpful comments on the manuscript and A.P. Dmitriev, I.V. Gornyi and V.Yu. Kachorovskii for useful discussions. The work of D.A. was supported by the German-Israeli-Foundation (GIF), the Dynasty foundation. The work of D.A. and P.W. was supported by the DFG-Center for Functional Nanostructures.
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