Model of resistances in systems of Tomonaga-Luttinger liquid wires
In a recent paper, we combined the technique of bosonization with the concept of a Rayleigh dissipation function to develop a model for resistances in one-dimensional systems of interacting spinless electrons [arXiv:1011.5058]. We also studied the conductance of a system of three wires by using a current splitting matrix at the junction. In this paper we extend our earlier work in several ways. The power dissipated in a three-wire system is calculated as a function of and the voltages applied in the leads. By combining two junctions of three wires, we examine a system consisting of two parallel resistances. We study the conductance of this system as a function of the matrices and the two resistances; we find that the total resistance is generally quite different from what one expects for a classical system of parallel resistances. We will do a sum over paths to compute the conductance of this system when one of the two resistances is taken to be infinitely large. Finally we study the conductance of a three-wire system of interacting spin-1/2 electrons, and show that the charge and spin conductances can generally be different from each other.
pacs:73.23.-b, 73.63.Nm, 71.10.Pm
For non-interacting electrons, the conductance of a ballistic quantum wire is well-known to be quantized in units of at low temperatures buttiker (); datta (); wees (). This remains valid when interactions between the electrons in the wire are taken into account, provided that there are no sources of backscattering (such as impurities and junctions) and that the wire is connected to leads where there are no interactions maslov (); pono (); safi1 (); safi2 (); thomale (). Thus, if the wire is modeled as a Tomonaga-Luttinger liquid (TLL) and the interaction strength is given by the Luttinger parameter , the conductance of a clean wire does not depend on . This breaks down if there are impurities in a wire with interacting electrons. In that case, the impurity strengths vary with the length scale according to some renormalization group (RG) equations, and the conductance depends on and other parameters like the length of the wire, the distances between the impurities, and the temperature kane-furusaki (); expt (). A considerable amount of work has also been done on junctions of several quantum wires both theoretically sand (); nayak (); lal (); chen (); chamon (); meden (); das (); giuliano (); bella1 (); agar () and experimentally fuhrer (); terrones (). In these systems, the conductance matrix again becomes length scale dependent due to the interactions. A junction of three quantum wires with interacting spin-1/2 electrons has been studied in Ref. hou (), and it has been found that some of the fixed points of the RG equations exhibit different charge and spin conductances. Thus impurities in a single wire or a junction of three or more wires effectively give rise to a resistance which leads to power dissipation. There have been some studies of power dissipation on the edges of a quantum Hall system wen () and at a junction of quantum wires bella2 (). However, there has been relatively little discussion in the literature of the effects of an extended region of dissipation (a resistive patch) within the framework of TLL theory or bosonization which is the most efficient way to study the effects of interactions boson ().
To remedy this situation, we recently introduced a formalism which can combine the technique of bosonization with the classical notion of resistance; both a single wire and a system of three wires with a junction were studied using this formalism soori (). Our analysis was restricted to spinless electrons and zero temperature. In contrast to this, Ref. rech () considered the effect of an extended region of inhomogeneity in a quantum wire at low temperatures; it was shown that this leads to weak backscattering which gives rise to a resistance which is linear in the temperature.
It is useful to briefly recapitulate our earlier work soori (). We introduced the resistances phenomenologically using a Rayleigh dissipative function gold (). Our treatment was classical in the sense that the resistance was taken to be purely a source of power dissipation, and the microscopic quantum mechanical origins of the resistance were not specified. This is equivalent to treating scattering by the resistance as a phase incoherent process; a consequence of this is that the resistances of different patches add up in series with no effects of interference. We found expressions for the conductance of a single wire and of a three-wire system with a junction (this was described by a current splitting matrix which is orthogonal) in terms of the resistances in the wires. The conductance can be calculated even when the Luttinger parameter , the velocity of the quasiparticles and the resistivity all vary with the spatial coordinate in an arbitrary way in the different wires. Remarkably, we found that the conductance of the three-wire system is independent of the Luttinger parameter (which is determined by the strength of the interaction between the electrons near the junction) if the matrix is invariant under time reversal; this remains true no matter what the values of the resistances are in the three wires.
In this paper, we will extend the ideas introduced in Ref. soori () in several ways. For the sake of completeness, we will briefly present our earlier results for a single wire in Sec. II and a three-wire system in Sec. III. We will then examine the issue of power dissipation in a three-wire system in Sec. IV. The dissipation occurs due to the resistances in the wires and the contact resistance in the leads. We will study the dependence of the power dissipation on the junction matrix and the relative magnitudes of the voltages applied in the leads of the system. In Sec. V, we will study a system of two resistances in parallel; the system consists of two junctions of three wires. We will make a detailed study of the total conductance in terms of the two junction matrices and the two resistances; the conductance will be calculated using a scattering approach. We find that the classical expression for the effective resistance of two parallel resistances is recovered only in one special case. In Sec. VI, we will carry out a sum over paths to calculate the conductance of the same system in the special case that one of the two resistances is infinitely large. In Sec. VII, we will generalize our results to the case of spin-1/2 electrons. We will argue there that the charge and spin conductances can generally be different from each other. We will summarize our results in Sec. VIII.
Ii Single wire
ii.1 Equation of motion
We start with the analysis of a single wire containing interacting spinless electrons. In the absence of any impurities, the bosonic Lagrangian for the system is given by
where and respectively denote the Luttinger parameter and velocity of the bosonic quasiparticles; we will allow and to vary with within a finite region which we will take to be . (For noninteracting electrons, , while for short-range repulsive interactions like a screened Coulomb interaction, we have ). The limit of noninteracting electrons, with and (the Fermi velocity), is used to model the two- or three-dimensional Fermi liquid leads situated in the regions . In the leads, the frequency and wave number of a plane wave are related by . The electron charge density and current can be expressed in terms of the bosonic field by the relations: and , where is the electron charge; these satisfy the equation of continuity . To describe resistances phenomenologically, we introduce a Rayleigh dissipation function
where the resistivity can vary with . We then obtain the following equation of motion as described in Ref. soori ()
Note that we have set , so that .
ii.2 Scattering approach
We will now derive an expression for the DC conductance for a general resistance profile , and will show that only the total resistance of all the resistive patches appears in the final expression for . Also, we will show that a -function resistance with the same integrated value of gives the same value of the conductance.
As mentioned earlier, we set and in the leads. However, and can have any profile in the wire region given by . Similarly, we will assume that the resistivity for , but can have any profile in the region such that [one such profile is illustrated in Fig. 1 (ii)]. As described in Ref. soori (), in the scattering approach, a plane wave with frequency is incident on the resistive patch, the reflection and transmission amplitudes are calculated as functions of , and finally the limit is taken to obtain the expression for . For a plane wave incident from left with , the spatial part of the solution outside the resistive patch is given by
In the second term in Eq. (6), we can replace by a constant (= up to zero-th order in ) since this term has a factor of already. Integrating this equation from to , we obtain which is the same as
One can easily redo the calculations from Eq. (4) to Eq. (8) and see that the result remains unchanged if we choose a -function resistivity profile described by , where . We will use this fact later in Sec. VI. We thus see that does not depend on the precise functional forms of , and , as long as and are equal to the constants 1 and in the leads, and is the total resistance of the wire. An implication of this is that the effective resistance of two or more resistive patches in series will be given by the sum of the individual resistances.
In this context, it is worthwhile to look at the phase coherent and phase incoherent transport in the literature (see for instance pp. 125-129 of Ref. datta ()). In general, the effective resistance of two resistances can have an extra term that depends on some phase factors at the two resistive patches in addition to the sum of the individual resistances. However, the effective resistance reduces to the sum of individual resistances in the incoherent limit. In this sense, our formalism assumes that the system is phase incoherent.
Iii Three wires with a junction
Let us consider a junction of three TLL wires as shown in Fig. 2. We will assume that each wire has three regions:
(i) — the wire region around the junction where and ; elsewhere and ,
(ii) — the dissipative region where ; elsewhere , and
(iii) — the semi-infinite leads.
Here labels the wires, and on wire , the coordinate runs from to , with denoting the junction point. The regions model the two- or three-dimensional leads which are assumed to be Fermi liquids with no interactions between the electrons; we therefore set and in those regions.
iii.1 Green’s function approach
We now follow Ref. maslov () and write
in the linear response regime, where is the Fourier component of the electric field on wire , and is the nonlocal ac conductance matrix. Using the Kubo formula, we then obtain
where , and
is the Fourier transform of the bosonic field in imaginary time, . The expressions for and in terms of and arise as follows. The actions in real (Minkowski) time and imaginary (Euclidean) time are given by
respectively. The requirement that the exponentials appearing in a path integral formulation be equal to each other, i.e., , implies that . Secondly, we want an outgoing plane wave in the lead of wire to be given by in terms of a real frequency and in terms of an imaginary frequency (this expression for an outgoing wave will be used in the boundary condition (iii) given below). The two planes will be identical if .
The Green’s function satisfies the equation
with the following three boundary conditions:
(i) is continuous at (where ) and ,
(ii) and are continuous at ,
The boundary condition in (iii) encodes the fact that the incoming and outgoing currents (and hence the bosonic fields) at the junction are related by the matrix . Various constraints at the junction such as current conservation and unitarity of the evolution of the system in real time (i.e., no power is dissipated exactly at the junction) imply that each row and column of must add up to unity and that must be orthogonal. It turns out that for a junction of three wires, the possible matrices must belong to one of two classes both of which are parameterized by a single parameter sand (); nayak (); lal (); chen (); chamon (); meden (); das (); giuliano (); bella1 (); agar (): (a) det and (b) det; these can be expressed as:
where , , and . We note that for any value of ; this relation will be used below.
Note that by introducing the orthogonal matrix , we have made the assumption that there is no dissipation exactly at the junction. The analysis is simpler if the junction, which governs how the incoming currents are distributed amongst the different wires, is separated from the dissipative regions which lie some distance away from the junction.
Solving Eq. (13) with the above boundary conditions and taking the limit , we obtain the following expression for the dc conductance matrix:
where is a diagonal matrix with ; we note that is simply the total resistance in wire . The conductance matrix relates the outgoing current to the potential applied in lead as . One can show in general that each row and column of must add up to zero; the columns adding up to zero is a consequence of current conservation ( must be zero), while the rows must add up to zero because each of the must vanish if the ’s have the same values in all the wires.
Eq. (15) can be understood as a combination of the conductance of a system with no resistances () and resistances on the three wires. Let us denote the conductance with no resistances by
If denote the potentials at the points (i.e., the points which lie after the interacting regions but before the dissipative regions), we have . Further, . Combining these equations with , we obtain
This relation will be used in Sec. VII below.
iii.2 Scattering approach
Eq. (15) can be derived in general using the equation of motion approach in the limit in the same way as described above for the single wire case. Let denote the bosonic field in real time which satisfies Eq. (3) in wire . Current conservation at the junction implies that which implies that . [Note that this is a different condition than the one used in the single wire case where was assumed to be continuous everywhere; for the case of more than two wires, it is more convenient to assume rather than the continuity of between different values of ]. We now assume that
where denote the incoming fields and denote the outgoing fields. Assuming that the Luttinger parameter and the velocity are given by and as and by 1 and as , we must have . In the limit , the field on wire is given by at and by as at zero-th order in , and . Since equal to a constant is a solution of Eq. (3) for , we must have . Next, the coefficients and are related by the current splitting matrix at the junction, . The coefficients and must be related by the conductance matrix in the leads, ; this follows from the statement that . Finally, we integrate Eq. (3) from to , ignoring the first term which is of order and setting the third term equal to , where . This gives the equation , where we have used the fact that and taken the limit . Using all these equations, we recover Eq. (15). We thus see that the precise profiles of , and in the different wires are not important; all that matters is that the values of and are given by and as and by 1 and as , and that .
iii.3 Conductance for the and classes
Within the class, the case is trivial because and . We now consider all other values of . We find that in general depends on , , and the resistances . [An exception arises for the case where we find that is independent of and depends only on the . As shown below, this occurs whenever which is true for and also for the class for any .] The dependence of on for the class is to be contrasted to the case of a single wire where the conductance is independent of maslov (); pono (); safi1 (); safi2 (); thomale ().
In the class, we find that although depends on and the , it is completely independent of . In Eq. (15), we write , where and . We can then use the relations that , and to show that does not depend on for any choice of and . The exact expression for turns out to be
where . We thus see that does not depend on , the Luttinger parameter in the wire regions.
Iv Power dissipation
In our model, there is no power dissipation exactly at the junction since the current splitting matrices and are orthogonal. Power dissipation occurs only at the resistive patches and in the leads due to the contact resistance. The power dissipation at the contact resistance occurs due to the energy relaxation of the electrons in the leads (reservoirs) which are maintained at some particular chemical potentials. Classically, if a current passes through a resistance , the power dissipated is given by . For a three-wire junction, we can define the power dissipated in two equivalent ways as follows
We have verified analytically that these two definitions give the same result. (A minus sign appears in Eq. (20a) because we have defined the to be outgoing currents).
We know that when the voltages applied in all the three leads are equal to each other, there should be no current in any of the three wires, and hence the power dissipated should be zero. To incorporate this fact, we choose a new coordinate system for the ’s known as the Jacobi coordinates
In this coordinate system, the power dissipated does not depend on the voltage . Further, it turns out that and can be parameterized in such a way that depends only on one of two parameters; this parametrization is different for junctions described by and as we will see below. Since we know from Eq. (20b) that the power dissipated at the resistance is , for simplicity we will only consider the case of (for ) for further analysis.
iv.1 Power dissipated for the class
In this case, the -matrix is invariant under a cyclic permutation of the wires 1, 2 and 3, and the dissipated power turns out to be invariant under rotations in the plane. If we write and , the power dissipated is proportional to and does not depend on . However, depends on the parameters and since the conductance matrix depends on those two parameters. The dependence of on and is shown as a contour plot in Fig. 3. The choice decouples the three wires at the junction, making the currents in the three wires zero; hence, the power dissipated is zero for this choice. The power dissipated is maximum on the contour .
iv.2 Power dissipated for the class
Junctions described by -class are time reversal invariant (the -matrix is symmetric). In this case, the power dissipated is a constant over straight lines in the plane. If we define the variables , then the power dissipated is found to be independent of and is given by . Further, does not depend on in this case.
V Resistances in parallel
At the end of Sec. II we saw that the effective resistance of two or more resistances in series is the sum of the individual resistances in agreement with the classical result. By the word ‘classical’ we mean the result obtained for the effective resistance by using Kirchoff’s circuit laws. In this spirit, it is interesting to study whether the effective resistance of two resistances in parallel will agree with the classical result for a similar system.
To begin with, let us consider a general model illustrated in
Fig. 4. The three wires at each junction are labeled by the
index , and the two junctions are described by current splitting
matrices and . The coordinate runs from
(i) to on the wire on the left,
(ii) to on the wires ,
(iii) to on the wire on the right.
The resistive regions on the wires lie in the range , the total resistances being . In the leads, and as usual, while and in the regions shown in pink (solid lines). In the resistive regions, and can have any profiles and these do not affect the final result. The incoming and outgoing fields at the two junctions (at ) are related by the matrices and . A plane wave incident from the left has the solution given by:
A calculation similar to the one performed in Sec. II.2 relates (in the DC limit) the unknowns and (for and ) as
where . The amplitudes and for different ’s are related by the matrix as follows:
Once we know the final expression for the DC conductance ( in the limit ), we can obtain the effective resistance by subtracting out the contact resistance from the total resistance,
We have listed the results obtained for different choices of and in Table 1.
|Depends on , and|
|as shown in Figs. 5 and 6|
|Depends only on|
|as shown in Fig. 7|
|Depends only on|
|as shown in Fig. 7|
In an earlier paper, we observed that in the case of a three-wire junction, if time reversal invariance is broken at the junction (i.e., if is not a symmetric matrix), then the conductance matrix depends on soori (). In the model studied here, we find that the final DC conductance depends on only when time reversal invariance is broken at both the junctions (i.e., if both and belong to the class) and . The dependence of the conductance on () is shown in Fig. 5(6). The contour corresponds to the maximum value of . On this contour, does not depend on either or ; moreover we get an expression for the effective resistance which agrees with the classical result for .
The choice at a junction described by -matrix decouples all the three wires at the junction. Hence, the conductance of the system is zero (or equivalently ) if either of the two junctions have . We can see this in both Table 1 and Fig. 6. Another interesting case that results in infinite effective resistance arises when both and lie in the class and . The choices, (i) and lie in the class with and (ii) lies in the class and lies in the class, with and , yield the same -dependent expression for the conductance, as shown in Fig. 7 for one particular choice of the parameters. The case and is redundant since we have already analyzed the case and , and these two cases are equivalent. The equivalence of these two cases follows from the parity transformation , , since the conductance is parity invariant for this system.
Finally, as a special case, we look at a symmetric situation treating the wires 2 and 3 on the same footing, with both junctions being described by the same -matrix. The case