AC conductivity of a quantum Hall line junction

# AC conductivity of a quantum Hall line junction

Amit Agarwal and Diptiman Sen Center for High Energy Physics, Indian Institute of Science, Bangalore 560 012, India
July 26, 2019
###### Abstract

We present a microscopic model for calculating the AC conductivity of a finite length line junction made up of two counter or co-propagating single mode quantum Hall edges with possibly different filling fractions. The effect of density-density interactions and a local tunneling conductance () between the two edges is considered. Assuming that is independent of the frequency , we derive expressions for the AC conductivity as a function of , the length of the line junction and other parameters of the system. We reproduce the results of Phys. Rev. B78, 085430 (2008) in the DC limit (), and generalize those results for an interacting system. As a function of , the AC conductivity shows significant oscillations if is small; the oscillations become less prominent as increases. A renormalization group analysis shows that the system may be in a metallic or an insulating phase depending on the strength of the interactions. We discuss the experimental implications of this for the behavior of the AC conductivity at low temperatures.

###### pacs:
73.43.-f, 73.43.Jn, 71.10.Pm, 73.23.-b

## I Introduction

A line junction (LJ) renn (); oreg (); kane1 (); mitra (); kollar (); kim (); zulicke (); papa (); das () separating two edges of fractional quantum Hall (QH) states allows the realization of one-dimensional systems of interacting electrons for which the Luttinger parameter can be tuned gogolin (); giamarchi1 (); rao (). A LJ is formed by using a gate voltage to create a narrow barrier which divides a fractional QH state such that there are two chiral edges flowing in opposite directions (counter propagating) on the two sides of the barrier kang (); yang1 (); yang2 (); roddaro1 (); roddaro2 (). For a QH system corresponding to a filling fraction which is the inverse of an odd integer such as , the edge consists of a single mode which can be described by a chiral bosonic theory wen1 (). In a system with two QH states separated by a line junction, the edges on the two sides of the LJ generally interact with each other through a short-range density-density interaction (screened Coulomb repulsion); such an interaction can be treated exactly in the bosonic language. The physical separation between the two edges and, therefore, the strength of the interaction can be controlled by a gate voltage. In general, a LJ also allows tunneling between the two edges; if the LJ is disordered, the tunneling amplitude is taken to be a random variable. The tunneling amplitude is also dependent on the separation between the edges.

Recently, QH systems with a sharp bend of have been fabricated grayson1 (); grayson2 (). An application of an appropriately tilted magnetic field in such a system can produce QH states on the two faces which have different filling fractions and , since the filling fractions are governed by the components of the magnetic field perpendicular to the faces. The two perpendicular components can even have opposite signs if the magnetic field is sufficiently tilted. Depending on whether and have the same sign or opposite signs, the edge states on the two sides of the line separating the two QH states may propagate in opposite directions or in the same direction; we call these counter or co-propagating edges respectively. A QH system with a bend therefore provides a new kind of LJ in which the filling fractions can be different on the two sides of the LJ, and the two edges can be co-propagating.

In an earlier paper sen (), we developed a microscopic model for the direct current (DC) conductivity of a finite length LJ with either counter or co-propagating edges. The conductivity is expressed by a current splitting matrix which depends on the filling fractions and , the choice of current splitting matrices which provide boundary conditions for the bosonic fields at the two ends of the LJ, the tunneling conductance per unit length , and the length of the LJ. The Coulomb interaction between the edges was ignored in the calculation of the DC conductivity, but the effect of the interaction was then taken into account to study the renormalization group flow of the tunneling conductance and therefore the conductivity.

In this paper, we will generalize the results of Ref. sen () to find the alternating current (AC) conductivity along a LJ; the inter-edge interactions will be taken into account in this calculation. In the limits in which the AC frequency goes to zero, we will recover the results obtained in Ref. sen ().

The paper is organized as follows. In Sec. II, we introduce a microscopic model for the LJ and discuss the general form of the current splitting matrix for both DC and AC. We obtain the condition for zero power dissipation. We also discuss the possible boundary conditions which can be imposed at the ends of the line junction; if we require that the commutation relations of the incoming and outgoing bosonic fields be preserved, we find that the boundary conditions must take one of two forms, which are described by matrices and . In Sec. III, we introduce short-range interactions (whose strength is given by a parameter ) and a local tunneling conductance (denoted by ) between the edges of the line junction. We then discuss the case of counter propagating edges and present the frequency dependent matrix for some simple choices of the filling fractions and velocities of the edge modes. In Sec. IV, we discuss the case of co-propagating edges and present the matrix , again for some simple cases. We present some plots of the elements of as functions of the AC frequency . In Sec. V, we discuss the implications of a renormalization group analysis for the low temperature behavior of , and how this may be checked experimentally. Section VI summarizes our results and discusses possible extensions of our work. In the Appendix we present the details of the calculations for the general case of arbitrary filling fractions and velocities of the edge modes.

## Ii Model for the line junction

We consider a LJ with two different QH liquids on the two sides. The edges of the QH liquids on the two sides of the LJ are assumed to be spatially close to each other; hence there are density-density interactions between the two edges, and electrons can also tunnel between the edges. For simplicity, we will assume that the interaction strength and the tunneling conductance have the same magnitude at all points along the LJ. We will also assume that the incoming and outgoing fields connect continuously to the corresponding fields at each end of the LJ.

Consider the counter propagating (co-propagating) LJ systems shown in Figs. 1 (a) and (b) respectively. The currents (voltages) in the two incoming edges are denoted as () and (), and in the two outgoing edges as () and (). Here edges and correspond to a QH system with filling fraction , while edges and correspond to a system with filling fraction . We also assume that the QH edge modes are locally equilibrated; discussions of equilibration at zero frequency have been presented in Refs. kane2 (); buttiker (). Namely, at each point , which may lie either on one of the outer edges 1-4 or inside the line junction (where goes from 0 to ), we assume, for small bias, that

 Ii(x,t) = e2h νi Vi(x,t). (1)

In the linear response regime (when the applied voltage bias is small), we expect the outgoing currents to be related to the incoming ones in a linear way. Let us denote the alternating current on edge by , where is a complex number in general. The numbers are related by a current splitting matrix as

 (α3α4) = Sac (α1α2), where    Sac = (r(ω)¯t(ω)t(ω)¯r(ω)). (2)

When all the edge states are in equilibrium, the power dissipated is given by the difference of the incoming and outgoing powers, namely,

 P = 12 [I1V1 + I2V2 − I3V3 − I4V4]. (3)

If there is no power dissipation in the system (we will see below that this is true if the tunneling conductance is zero all along the LJ), then the average over one oscillation cycle of the incoming energy must be equal to the outgoing energy. This imposes the following condition

 S†ac (1/ν1001/ν2) Sac = (1/ν1001/ν2), (4)

or, more explicitly,

 |r(ω)|2ν1 + |t(ω)|2ν2 = 1ν1, |¯r(ω)|2ν2 + |¯t(ω)|2ν1 = 1ν2, and    r∗(ω)¯t(ω)ν1 + t∗(ω)¯r(ω)ν2 = 0. (5)

When the incoming currents are DC in nature, the currents satisfy a linear relation and are related by a current splitting matrix . This is a real matrix which can be characterized by a single parameter (called the scattering coefficient); it has the general form wen2 (); halperin (); sen ()

 (I3I4) = ⎛⎜⎝1−2γν2ν1+ν22γν1ν1+ν22γν2ν1+ν21−2γν1ν1+ν2⎞⎟⎠(I1I2). (6)

The power dissipated is the difference of the incoming and outgoing energy flux (3), and it is given by

 P = e2h 2ν1ν2ν1+ν2 γ(1−γ) (V1−V2)2. (7)

The condition that requires that . No power is dissipated if or , and maximum power dissipation occurs when .

The end points of the LJ shown in Fig. 1 lie at and . At each of these ends, we have two incoming edges and two outgoing edges; two of these correspond to the outer edges marked , , , and , while the other two edges are internal to the LJ and are marked and . An important ingredient of the model for such a system is the boundary condition which should be imposed at the end points. In Ref. sen (), it was shown that there are two possible boundary conditions which can be imposed at each end; both of these allow us to smoothly connect the bosonic fields which may be used to calculate the currents in the system. The two possible boundary conditions correspond to using one of two matrices or to related the incoming and outgoing modes at each end, where

 S0 = (1001), and   S1 = 1ν1+ν2 (ν1−ν22ν12ν2ν2−ν1). (8)

In this paper, we will only consider the boundary condition corresponding to the matrix ; this is physically the more plausible boundary condition since it just connects the incoming edge to the outgoing one for each QH liquid separately.

## Iii Counter propagating case

We will now present a microscopic model of a LJ for the case of counter propagating edges. For simple filling fractions given by the inverse of an odd integer, each edge is associated with a single chiral boson mode. For the counter propagating case, shown in Fig. 1 (a), the mode on one edge propagates from to , while the mode on the other edge propagates in the opposite direction; let us call the corresponding bosonic fields as (right mover) and (left mover) respectively. In the absence of density-density interactions between these edges, the Lagrangian is given by

 L = 14πν1 ∫L0dx ∂xϕ1 (−∂t−v1∂x) ϕ1 (9) + 14πν2 ∫L0dx ∂xϕ2 (∂t−v2∂x) ϕ2,

where denotes the velocity of mode . The density and current fields are defined as

 ρ1 =∂xϕ1/(2π),J1 = −∂tϕ1/(2π),ρ2 =− ∂xϕ2/(2π),J2 =∂tϕ2/(2π). (10)

For a short-range density-density interaction between the two edges, the term in the Lagrangian is of the form

 Lint = λ4π√ν1ν2 ∫L0dx ∂xϕ1 ∂xϕ2, (11)

where is the interaction strength (positive for repulsive interactions) with the dimensions of velocity.

The equations of motion for the Lagrangian given in Eqs. (9) and (11), written in terms of the density and the current fields, are

 J1 − v1ρ1 − λν12√ν1ν2 ρ2 = 0, J2 + v2ρ2 + λν22√ν1ν2 ρ1 = 0. (12)

A model of tunneling at zero frequency between different edges or point contacts in a QH system has been developed in Ref. kane2 (). By adding a time derivative term in their expressions, we can model tunneling at finite frequencies between the two edges along the LJ using the following equations

 ∂tρ1 + ∂xJ1 = σhe2(J2ν2−J1ν1), ∂tρ2 + ∂xJ2 = σhe2(J2ν2−J1ν1), (13)

where is the conductance per unit length across the LJ. Physically, Eq. (13) is the continuity equations for the edge states with a source term kane2 (), the source term being the current tunneling into the system because of the voltage difference between the corresponding points on the line junction, . We will assume to be constant along the LJ. Unlike Eq. (12), the model of tunneling given in Eq. (13) cannot be derived from any Lagrangian since it is non-unitary, and a non-zero value of implies that there is dissipation in the system.

For the DC case, in a non-interacting system, the current splitting matrix is given by Eq. (6), and is given by sen ()

 γ=ν1+ν221−e−L/lcν2−ν1e−L/lc   and   l−1c=σhe2(1ν1−1ν2) (14)

when . For the special case kane1 () of , we obtain .

Now we solve the problem for the general case with interactions and for an arbitrary value of . We can combine Eqs. (12) and (13) to obtain

 + (λν12√ν1ν2∂x−αν2)J2=0, (∂t−v2∂x+βν2)J2 − (λν22√ν1ν2∂x+βν1)J1=0,

where

 α = σhe2 (v1+λν12√ν1ν2), β = σhe2 (v2+λν22√ν1ν2). (16)

Solving these equations with appropriate boundary conditions gives us the current splitting matrix . We will work with the boundary condition that connects the fields along the LJ continuously to the corresponding incoming and outgoing fields, i.e., the incoming field at and at . The most general case will have and . We solve Eq. (III) in its most general form in the Appendix and present the matrix . In this section we present results for some relatively simple cases.

For the case of LJ with interactions but no tunneling (), same filling fraction () and same velocity (), we find that

 t(ω) = ¯t(ω) = − λsin(kL)2[i~vcos(kL)+vsin(kL)], r(ω) = ¯r(ω) = i~vi~vcos(kL)+vsin(kL), (17)

where and . Note that there is no dissipation in this case, and the AC current splitting matrix satisfies Eq. (4). We also note that this solution is consistent with Eq. (11) of Ref. oreg () in the limit of , (our corresponds to their ). Similar expressions have also appeared in Refs. safi2 (); safi3 (). Also note that in the DC limit (), . This implies that conductance across the LJ, . This is expected and is consistent with Ref. maslov (); ponomarenko (); safi1 (); safi4 ().

For the case of same ’s and same velocities but with the tunneling switched on, we have

 t(ω)= 2(k2v2ν2+(α−iνω)2)kν(2vα+λ(α−iνω))cot(kL)+(−k2vλν2+2α(α−iνω)), r(ω)= 1cos(kL)+(−k2vλν2+2α(α−iνω))sin(kL)kν(2vα+λ(α−iνω)), (18)

where , and

 k = ω√v2−λ2/4 (1+2iσhνe2ω(v+λ/2))1/2. (19)

The expression for the most general case is given in the Appendix in Eq. (37) and (A). In the DC limit , Eq. (37) gives and , while Eq. (A) gives

 r(ω→0) = ν1−ν2ν1−ν2 eL/lc. (20)

Comparing with the expression in Eq. (6), we get the same value of as in Eq.(14).

In Fig. 2, we show the absolute values of the various reflection and transmission amplitudes as functions of the frequency (in units of which has been set equal to unity) for various choices of the filling fractions , velocities , length , interaction (in units of ), and tunneling conductance per unit length (in units of ). Figs. 2 (a) and (b) show the cases of (zero tunneling) for equal filling fractions and different filling fractions respectively; in Fig. 2 (a), and by symmetry. In both figures, we see prominent oscillations as a function of . This is clear from Eq. (17) where we see that is real and the different amplitudes oscillate with a wavelength . In the limit, for both cases, as is evident from Eq. (20). Both these cases are dissipationless and the amplitudes satisfy Eq. (4). In Fig. 2 (b), the curves for and coincide for all ; this can be shown to hold if , no matter what the filling fractions and velocities are. Figs. 2 (c) and (d) show the cases of for equal filling fractions and different filling fractions respectively; we have assumed for simplicity that itself does not depend on . In this case, is complex as shown in Eq. (19) and (37); the imaginary part of remains finite for large . Hence the different amplitudes show oscillations but they also decay as increases. In the limit, is given by Eq. (20).

Note that we have treated the fully interacting problem in this section, but in the DC limit , we recover the results for the non-interacting case given in Ref. sen (). This is because in that limit, the terms involving vanish in Eq. (13). The currents can then be found from Eq. (13) alone, and Eq. (12) becomes unnecessary. The values of the currents therefore do not depend on the interaction parameter and the velocities appearing in Eq. (12). However, for the AC case, Eq. (12) and (13) are both required to find the currents, and the results are different for the interacting and non-interacting cases in general. It is also interesting to note that if there is no tunneling (), the corrections to lowest order in the AC frequency for the reflection and transmission amplitudes in Eq. (17) are of order , but if there is tunneling (), the lowest order corrections are of order . This follows from Eq. (19) which shows that for small , if , but if .

## Iv Co-propagating case

In the absence of density-density interactions between the co-propagating modes, the Lagrangian is given by

 L = 14πν1 ∫L0dx ∂xϕ1 (−∂t−v1∂x) ϕ1 (21) + 14πν2 ∫L0dx ∂xϕ2 (−∂t−v2∂x) ϕ2.

Here both the edge modes are taken to be propagating from to . The corresponding density and current fields are defined as and . The short-range repulsive density-density interaction between the two edges takes the form

 Lint = − λ4π√ν1ν2 ∫L0dx ∂xϕ1 ∂xϕ2, (22)

where is the interaction strength (positive for repulsive interactions) with the dimensions of velocity.

The equations of motion for the Lagrangian corresponding to Eq. (21) and (22), written in terms of the density and the current fields, are

 J1 − v1ρ1 − λν12√ν1ν2 ρ2 = 0, J2 − v2ρ2 − λν22√ν1ν2 ρ1 = 0. (23)

The tunneling along the LJ will be modeled using the following equations

 ∂tρ1 + ∂xJ1 = σhe2(J2ν2−J1ν1), ∂tρ2 + ∂xJ2 = −σhe2(J2ν2−J1ν1). (24)

We can combine Eq. (23) and (24) to give

 + (λν12√ν1ν2∂x−αν2)J2 = 0, (∂t+v2∂x+βν2)J2 + (λν22√ν1ν2∂x−βν1)J1 = 0,

where

 α = σhe2 (v1−λν12√ν1ν2), β = σhe2 (v2−λν22√ν1ν2). (26)

Solving these equations with the appropriate boundary conditions gives us the current splitting matrix.

For the DC case, in a non-interacting system, the current splitting matrix is given by Eq. (6), and is given by sen ()

 γ=1−e−L/lc2   and   l−1c=σhe2(1ν1+1ν2). (27)

We now turn to the AC case. For the simplest case of same filling fraction (), same velocity ()and and no tunneling (), we obtain

 t(ω)= 2(eiLk1−eiLk2)(ω−vk1)(ω−vk2)λω(k1−k2), r(ω)= (ω−vk2)k1e−iLk1−(ω−vk1)k2e−iLk2ω(k1−k2), (28)

where and . Note that this is the dissipationless case and the amplitudes satisfy Eq. (4), and in the DC limit, .

For the case of same filing fractions (), same velocities (), but , we get

 t(ω) = [2i(eiLk1−eiLk2)(α−iν(ω−vk1))(α−iν (ω−vk2))]/[(2vαν+λν(α−iνω))(k1−k2)] r(ω) = [(eiLk2−eiLk1)(2α(α−iνω)+λvν2k1k2) (29) +iλ(α−iνω)(k1eiLk1−k2eiLk2)+2ivαν (k1eiLk2−k2eiLk1)]/[iν(2αv+λ(α−iνω)) (k1−k2)]l,

where the values of are given by Eq. (41).

The expression for the most general case is given in the Appendix in Eq. (41) and (A). In the DC limit , Eq. (41) gives and , while Eq. (A) gives

 r(ω→0) = ν1+ν2 e−L/lcν1+ν2. (30)

Comparing with the expression in Eq. (6), we find the same value of as in Eq. (27).

In Fig. 3, we show the absolute values of the various reflection and transmission amplitudes as functions of the frequency (in units of ) for various choices of , , , (in units of ), and (in units of ). Figs. 3 (a) and (b) show the case of (zero tunneling) for equal filling fractions and different filling fractions respectively, while Figs. 2 (c) and (d) show the case of , again assuming that does not depend on . As in Fig. 2, we see prominent oscillations as a function of when , and both oscillations and decay when . Just as in Fig. 2 (b), we note that the curves for and coincide for all in Fig. 3 (b) also.

Once again we note that we have treated the fully interacting problem here, but in the DC limit we recover the non-interacting results given in Ref. sen (). The reasons for this are the same as those explained at the end of Sec. III. For the AC case, the expression for the current splitting matrix is different for the interacting and the non-interacting cases.

## V Renormalization group and experimental implications

Before discussing how measurements of the AC reflection and transmission amplitudes can provide information about the parameters of the system such as the tunneling conductance , the interaction strength and the edge mode velocities , we have to consider the renormalization group (RG) flow of . For the DC case, this has been discussed in Ref. sen (). Briefly, the tunneling amplitude is given by a term in the Hamiltonian density

 Htun = ξ(x) ψ†1(x)ψ2(x) + h.c., (31)

where denotes the electron annihilation operator at point on edge of the LJ. The tunneling conductance is proportional to when is small. The presence of impurities near the LJ makes a random complex variable; let us assume it to be a Gaussian variable with a variance . Then satisfies an RG equation; to lowest order (i.e., for small ), this is given by giamarchi2 (); kane1 ()

 dWdlnl = (3−2dt) W, (32)

where denotes the length scale, and is the scaling dimension of the tunneling operator appearing in Eq. (31). We will present expressions for below for both counter and co-propagating cases. There is also an RG equation for the interaction strength , but that can be ignored if is small.

Next, let us assume that the phase de-coherence length (the length beyond which electrons lose phase coherence due to thermal smearing) is much smaller than both the length and the scattering mean free path of the LJ. Successive backscattering events are then incoherent, and quantum interference effects of disorder are absent. One can then show kane1 () that scales with the temperature as . It therefore seems that as if . However, it turns out that this is true only if , i.e., if is an irrelevant variable according to Eq. (32). If (called the metallic phase), one can simultaneously have (which justifies cutting off the RG flow at rather than at ), and , i.e., and , for some range of temperatures. Further, scales with temperature kane1 () as and . Thus throughout the metallic phase, as . But if (i.e., is a relevant variable), we have and ; hence (we call this the insulating phase).

The above analysis breaks down if one goes to very low temperatures where or . In that case, the RG flow of has to be cut off at the length scale or , rather than ; hence and therefore the scattering coefficient become independent of the temperature .

The scaling dimension can be computed using bosonization sen (). For the counter propagating case, we find that

 dt = 14K[(1+K2)(1ν1+1ν2)−2(1−K2)√ν1ν2], where  K = √v1+v2−λv1+v2+λ. (33)

Thus depends on the interaction strength and the velocities . (The stability of the system requires that ). For the co-propagating case, we have

 dt = 12ν1 + 12ν2, (34)

which is independent of and the .

We can now discuss the experimental implications of our results for the various reflection and transmission amplitudes as a function of the temperature. As mentioned earlier, a gate voltage can be used to control the distance between the two edges of the LJ. Making the gate voltage less repulsive for electrons is expected to reduce the distance between the edges; this should increase both the strength of the density-density interactions as well as the tunneling conductance kane1 (). In this way, one may be able to vary the scaling dimension across the value for the case of counter propagating edges in the LJ. We then see that quite different things should occur depending on whether the system is in the metallic phase () or in the insulating phase (). In the metallic phase, as ; based on Figs. 2 and 3, we expect that oscillations versus of the various amplitudes should become more prominent at low temperatures. In the insulating phases, as ; Figs. 2 and 3 then indicate that the oscillations versus of the different amplitudes should become less prominent at low temperatures. It would be interesting to check this qualitative prediction experimentally.

## Vi Discussion

In this paper, we have discussed the response of a LJ separating two QH states to an AC voltage in the linear response (or small bias) regime. Depending on the filling fractions on the two sides of the LJ, the edges of the LJ may be counter propagating or co-propagating. We have presented a microscopic model for the system which includes short-range density-density interactions and electron tunneling between the two edges. The AC response can be described by a current splitting matrix ; we have presented expressions for this matrix in terms of the AC frequency , the length of the LJ, the tunneling conductance, the strength of the interaction between the edges of the LJ, the filling fractions, and the velocities of the modes on the two sides of the LJ. In general, the elements of oscillate with the frequency ; the amplitude of oscillations depends on and the tunneling conductance across the LJ. We find the interesting result that the matrix does not depend on the interaction strength and the velocities in the DC limit , but does depend on those parameters for non-zero frequencies. [The fact that the DC conductivity is independent of the interaction strength has been observed earlier in the context of quantum wires modeled as non-chiral Tomonaga-Luttinger liquids safi2 (); safi3 (); maslov (); ponomarenko (); safi1 (); safi4 (); furusaki (); rech ().] The low temperature behaviors of the elements of can then be predicted based on a renormalization group analysis. In the case of counter propagating edges, we find that depending on the interaction strength, the system can be in either a metallic phase or an insulating phase. The two phases exhibit quite different behaviors of as we go to low temperatures.

We emphasize that in the absence of any tunneling between the two edges, our calculation is valid in the linear response (small AC amplitude) regime and only for frequencies which lie within the linearization regime for each LL wire (i.e., ). In the presence of inter-edge tunneling also, we have assumed that the current is proportional to the potential (voltage) at every point with no phase difference between the two (Eq. (1)). This is only true if is less than the inverse of the relaxation time (for equilibration after each tunneling event). However, in the absence of a detailed theory of equilibration at finite frequencies, we do not know the precise form of . Another limitation of our calculation is that the intrinsic frequency dependence of is not known, although the RG analysis discussed in Sec. V gives an idea of the length scale dependence of arising due to the interactions. We note that our various expressions for and will remain valid even if we take to be frequency dependent. However, Figs. 2 (c-d) and 3 (c-d) have been made under the assumption that does not depend on .

Before ending, we would like to mention some other studies of QH systems with multiple filling fractions sandler (); lal (); rosenow1 (). It may be interesting to extend our studies of AC response to these systems. Finally, we note that the AC response of non-chiral Tomonaga-Luttinger liquids with disorder has been studied earlier in some papers maura (); fogler (); rosenow2 (); safibena ().

## Acknowledgments

A.A. thanks Yuval Oreg for suggesting this problem and CSIR, India for financial support. We thank Sumathi Rao for comments and DST, India for financial support under Project No. SR/S2/CMP-27/2006.

## Appendix A Details of calculations

We first find the current splitting matrix for the counter propagating case. We start with the following guess for the edge currents along the LJ,

 J1 = (a1eik1x+a2eik2x) e−iωt, J2 = (b1eik1x+b2eik2x) e−iωt. (35)

These guess solutions must satisfy Eq. (III) for all values of along the LJ. This gives us the following equations

 (−iω+iv1k1+αν1)a1+(ik1λν12√ν1ν2−αν2)b1=0, (−iω+iv1k2+αν1)a2+(ik2λν12√ν1ν2−αν2)b2=0, (36)

where and is given by

 k1/2 = 12~v2[ω(v2−v1)+i~v2l−1c±[(ω(v2−v1)+i~v2l−1c)2 (37) + 4~v2(ω2+iσhωe2(v1ν1+v2ν2+λ√ν1ν2))]1/2],

where , and . Now consider an incoming current incident on the LJ from edge 1, and no incoming current from edge 3. Then we have the following equations at the two end points the LJ corresponding to the matrix defined in Eq. (8),

 a1+a2 = 1,      a1eik1L+a2eik2l=r(ω), b1+b2 = t(ω),   b1eik1L+b2eik2l=0. (38)

Solving these six simultaneous equations gives us two elements of the AC current splitting matrix. For the most general case, we obtain

 t(ω) = [2ν2(−eiLk1+eiLk2)(−iω+ik1v1+α/ν1)(−iω + ik2v1+α/ν1)]/[eiLk2(−iω+ik2v1+α/ν1)(2α − iλk1√ν1ν2)−eiLk1(−iω+ik1v1+α/ν1)(2α − iλk2√ν1ν2)], r(ω) = [eiLk1(k1−k2)ν1(λν2(iα+ων1)+2iαv1√ν1ν2)] / [2(−1+eiL(k1−k2))α(α−iων1)√ν1ν2−k2ν1(λν2 × (iα+ων1)eiL(k1−k2)+2iαv1√ν1ν2)+k1ν1(λν2 × (iα+ων1+k2v1ν1(−1+eiL(k1−k2)))+2iαv1 × √ν1ν2eiL(k1−k2))].

Repeating this calculation for the case with an incoming unit current in wire 3 and no incoming current in wire 1 will give us the other two amplitudes, and , of the AC current splitting matrix.

For the co-propagating case, we again start from the guess solution given by Eq. (35). Substituting them in Eq. (23), we get the following equations

 (−iω+iv1k1+αν1)a1+(ik1λν12√ν1ν2−αν2)b1=0, (−iω+iv1k2+αν1)a2+(ik2λν12√ν1ν2−αν2)b2=0, (40)

where and is given by

 k1/2 = 12~v2[ω(v2+v1)+i~v2l−1c