Chiral Y-junction of Luttinger liquid wires at strong coupling: fermionic representation

Chiral Y-junction of Luttinger liquid wires at strong coupling: fermionic representation

D.N. Aristov “PNPI” NRC “Kurchatov Institute”, Gatchina 188300, Russia Institute for Nanotechnology, Karlsruhe Institute of Technology, 76021 Karlsruhe, Germany Department of Physics, St.Petersburg State University, Ulianovskaya 1, St.Petersburg 198504, Russia    P. Wölfle Institute for Nanotechnology, Karlsruhe Institute of Technology, 76021 Karlsruhe, Germany Institute for Condensed Matter Theory, Karlsruhe Institute of Technology, 76128 Karlsruhe, Germany
July 14, 2019
Abstract

We calculate the conductances of a three-terminal junction set-up of spinless Luttinger liquid wires threaded by a magnetic flux, allowing for different interaction strength in the third wire. We employ the fermionic representation in the scattering state picture, allowing for a direct calculation of the linear response conductances, without the need of introducing contact resistances at the connection points to the outer ideal leads. The matrix of conductances is parametrized by three variables. For these we derive coupled renormalization group (RG) equations, by summing up infinite classes of contributions in perturbation theory. The resulting general structure of the RG equations may be employed to describe junctions with an arbitrary number of wires and arbitrary interaction strength in each wire. The fixed point structure of these equations (for the chiral Y-junction) is analyzed in detail. For repulsive interaction () there is only one stable fixed point, corresponding to the complete separation of the wires. For attractive interaction ( and/or ) four fixed points are found, the stability of which depends on the interaction strength. We confirm our previous weak-coupling result of lines of fixed points for special values of the interaction parameters reaching into the strong coupling domain. We find new fixed points not discussed before, even at the symmetric line , at variance with the results of Oshikawa et al. The pair tunneling phenomenon conjectured by the latter authors is not found by us.

pacs:
71.10.Pm, 72.10.-d, 85.35.Be

I Introduction

Electric circuits of one-dimensional quantum wires are described by laws quite different from those of corresponding classical systems. These laws are non-local, often quantized, and are generally difficult to derive. An important first step is the exploration of the properties of junctions of wires. The linear response conductance of a (symmetric) two-terminal junction in the limit of zero temperature has been found to be either zero (repulsive) or unity (attractive interaction) in units of the conductance quantum Kane and Fisher (1992); Furusaki and Nagaosa (1993). Most of the existing theories employ the exactly solvable Tomonaga-Luttinger liquid (TLL) model, in which backward scattering is neglected Giamarchi (2003). Moreover, for simplicity the fermions are assumed to be spinless, in which case backward scattering only gives rise to a renormalization of the forward scattering amplitude. Furthermore, most of the early works on the problem of two TLL-wires connected by a junction employed the method of bosonization, or, for special values of the interaction parameter, used a mapping on to exactly solvable modelsWeiss et al. (1995); Fendley et al. (1995).

An alternative formulation using a purely fermionic representation in terms of scattering states and a renormalization group (RG) approach for the conductance has been pioneered by Yue et al. (1994). The latter formulation has several advantages over the bosonic approach. Perhaps the most important difference is that it allows to describe the physically relevant system of a quantum wire (with interaction between the fermions) adiabatically coupled to reservoirs (with negligible interaction), whereas the bosonic approach is formulated for the infinite system. The effect of a finite wire length has only been convincingly derived for a clean wire Maslov and Stone (1995); Ponomarenko (1996); Safi and Schulz (1995) ; in the case of wires connected by a junction a plausible, but unproven ad hoc procedure has been proposed Oshikawa et al. (2006). As will be discussed below we have reasons to suspect that the latter procedure is not always correct. The fermionic scattering state approach is generally applicable, allowing for the study of wires with two impurities Polyakov and Gornyi (2003), of multi-lead junctions Das et al. (2004); Lal et al. (2002) as well as the inclusion of spin, of backward scattering, Yue et al. (1994) of out of equilibrium situations, Devillard et al. (2008) point contact in a quantum spin Hall insulator Teo and Kane (2009). The original model calculation Yue et al. (1994) has been limited to weak coupling. It may be shown, however, that the method can be extended to work in the strong coupling regime as well, by summing up an infinite class of leading contributions in perturbation theory Aristov and Wölfle (2009). In all cases considered so far it has been found that the method provides results in agreement with any available exact results, which builds confidence in its validity. As shown in Aristov and Wölfle (2009), in the case of a two-lead junction, the contributions neglected in the above-mentioned partial summation are subleading and may be shown to vanish when the fixed points are approached.

Generally, the transport behavior of one-dimensional systems at low energy/temperature is dominated by only a few fixed points (FPs) of the RG flow. The most common and intuitively plausible FPs are those with quantized conduction values, , in units of the conductance quantum (we consider systems of spin-less fermions). There may, however, appear additional FPs associated with noninteger conductance values, usually in the case of attractive interaction. A particularly well-studied case is that of a symmetric three-lead junction with broken chiral symmetry, as induced by a magnetic flux threading the junction. This latter problem has been studied by the method of bosonization in Chamon et al. (2003); Oshikawa et al. (2006), and by the functional RG method inBarnabé-Thériault et al. (2005). These authors have identified a number of new FPs, for attractive interaction. The discovery of these new FPs has sparked interest in the problem of mapping out the complete fixed point structure of the theory, even though most of the interesting new physics appears to be in the physically less accessible domain of strongly attractive interactions. Only the fully symmetric situation of identical wires (, see below) has been considered in Oshikawa et al. (2006); Barnabé-Thériault et al. (2005). As we have shown in Aristov and Wölfle (2012a) for weak coupling and will show below for any coupling strength, important new physics is missed by restricting the consideration to the symmetric case.

One result found by Oshikawa et al. (2006) deserves special attention: it is the claim that a new fixed point named (Dirichlet) exists for strongly attractive interaction (Luttinger parameter ), which is interpreted as describing multiparticle transport. As a consequence, the conductance values at this FP are outside the domain of values allowed by a single particle -matrix description. Put differently, this requires the excitation of particle-hole pairs at the junction, with say the particle moving into lead , while the hole is going into lead . In our view, at least at zero temperature, and in the linear response regime, such processes are not occurring, because the phase space for particle-hole excitations tends to zero in this limit. As we will show below, we do not find a fixed point with the characteristics of , but in contrast we find two new fixed points, not discussed in Refs. [Oshikawa et al., 2006; Hou et al., 2012].

In principle, the Y-junction set-up may be realized experimentally by a one-dimensional tunneling tip contacting a quantum wire. A magnetic flux threading the junction may be created by local magnetic moments at the junction, although this has not been realized experimentally so far.

In this paper we report results of an RG treatment of electron transport in the linear response regime through a junction of three spinless TLL wires threaded by a magnetic flux. We employ a fermionic representation as described in detail inAristov and Wölfle (2009). The method has formerly been used in Das et al. (2004); Lal et al. (2002) in the case of Y-junctions. More recently this approach has been used to derive the RG equations for the conductances of a Y-junction connecting three TLL wires in the absence of magnetic flux, for weak interaction Aristov et al. (2010), and in a recent work for any strength of interaction Aristov and Wölfle (2011). InAristov et al. (2010) it was found that even in weak coupling, but beyond lowest order, interesting new structures appear. Even a weak higher order contribution may change the RG-flow in a dramatic way. In the present paper we describe a similar effect: an asymmetry of the three wires of a chiral Y-junction to the effect that in the tip wire the interaction strength is different from that in the main wire, , allows to access certain regions in interaction parameter space where a whole line of fixed points rather than a single fixed point is stable. This happens for attractive interaction only ()  The line of stable fixed points is connecting two fixed points at two special manifolds of interaction values, (a) ; , or (b) whenever the condition , is met. Along these fixed point lines the FP values of the conductances are continuously varying. Results on this new aspect of transport through Y-junctions within weak coupling have been reported in Aristov and Wölfle (2012a). In the present work we extend our theory into the strong coupling regime. We derive a general expression for the scale-dependent contribution to the conductances, valid for a junction with any number of leads and any values of interaction in lead . This result allows to derive RG-equations for the set of independent conductances in the given case. We go on to evaluate these general expressions for the three-lead junction of the -type (interaction in the main wire and in the tip wire) in the presence of magnetic flux. Then we determine the six physically allowed FPs, and discuss their stability as a function of position in coupling constant space.

In order to compare our results with the recent findings of Hou et al. (2012) for a fully asymmetric junction, we extend our analysis by allowing for full asymmetry of the S-matrix at the junction, while keeping equal interaction in the main wires. This minimal modification enables us to elucidate the differences between our conclusions and those in Hou et al. (2012). We show that the set of RG equations for the most general asymmetric chiral case can be easily derived in our formalism, and allows us to discuss the robustness of our results obtained for the partially asymmetric case () with respect to further asymmetry. A full analysis of the case is, however, beyond the scope of the present study.

Our findings for the 1-2-symmetric Y-junction are summarized in Fig. 1, where the regions of stability of the four FPs , , and are displayed in the space of interaction constants and . Fixed point corresponds to the complete separation of all three wires. The separation of the third wire from the perfectly conducting “main wire” 1-2 is described by FP . The chiral FPs , correspond to maximum chirality. They describe a kind of Hall effect situation (depending on the orientation of the magnetic flux), where the in-current from wire flows into wire () or (). Fixed point , finally, corresponds to the situation of maximum transparency. The stability of the -point in the region to the right, , (checkerboard pattern) is lost for infinitesimally small deviations from perfect 1-2 symmetry of the junction, in favor of a new -point ( or , meaning separation of wire 1 or 2 from the remaining “main wire”). The region of stability of the -point on the left, , remains (see the detailed discussion in Section VI and Fig. 6). It is worth noting that in certain regions two FPs (e.g., and , or and ) are stable and attract trajectories within their respective basins of attraction.

Figure 1: (Color online) The RG phase portrait, showing regions with different stable fixed points. Diagonal red stripes correspond to stable point, vertical green stripes - to point, horizontal blue stripes - to points, gray shaded and checkerboard regions indicate stable point.

Ii Perturbation theory for conductances

ii.1 The model

We consider a model of interacting spinless fermions describing three quantum wires connected at a single junction by tunneling amplitudes in the presence of a magnetic flux piercing the junction. In the continuum limit, linearizing the spectrum at the Fermi energy and including forward scattering interaction of strength in wire , we may write the TLL Hamiltonian in the representation of incoming and outgoing waves in lead (fermion operators , ) as

(1)

Here denotes a vector operator of incoming fermions and the corresponding vector of outgoing fermions is expressed through the -matrix as . In the chiral representation we are using positions on the negative (positive) semi-axis corresponding to incoming (outgoing) waves. We consider quantum wires of finite length , contacted by reservoirs. The transition from wire to reservoir is assumed to be adiabatic (i.e. produces no additional potential scattering). The junction is assumed to have microscopic extension of the order of the Fermi wave length. Inside the junction interaction effects are neglected. This is expressed by the window function , if , and zero otherwise. The regions are thus regarded as reservoirs or leads labeled . We denote the interaction constants from now on, and put the Fermi velocity . The various incoming and outgoing channels are illustrated in Fig. 1 elsewhere. Aristov and Wölfle (2011); Aristov and Wölfle (2012a) The interaction term of the Hamiltonian is expressed in terms of density operators , and , where and the density matrices are given by and . The -matrix characterizes the scattering at the junction and (up to irrelevant phase factors) has the structure (see Aristov and Wölfle (2012a))

(2)

A convenient representation of 33-matrices is in terms of Gell-Mann matrices , , the generators of (seeAristov and Wölfle (2011)). Notice that the interaction operator only involves and (besides the unit operator ). We note , , , and . We introduce a compact notation by defining a three-component vector , in terms of which the densities may be expressed as , where the matrix is defined as

(3)

and has the properties , . The outgoing amplitudes are expressed in the analogous form with replaced by . With the aid of the the -matrix may be parametrized by eight angular variables (see Aristov (2011) and Appendix A). For the case under consideration only three of these, , are relevant: . The corresponding elements of the -matrix are given by

(4)

As will be shown below, the S-matrix and therefore the angular variables , will be renormalized by the interaction.

ii.2 Linear response conductances

In the linear response regime, the conductances

relate the current in lead (flowing towards the junction) to the electrical potential in lead , . Here the combination means that the current is measured at , and the integration over for the incoming densities is over the length of the ideal leads, to which the electric potential is applied. Denoting the frequency of the applied external field by , we consider the d.c. limit , when the value of does not play a role. Aristov and Wölfle (2009) The quantum averaging involves taking the trace over the matrix products. The conductance matrix has only three independent elements, , , and , relating the currents , to the bias voltages , . In compact notation we may define a matrix of conductances

(5)

The connection between matrices and comes from the observation, that the nonzero elements of the matrix are essentially the reduced conductances: , , , where the numerical factors arise due to the physically motivated asymmetric definitions of the currents and voltages. It may be shown that the parametrization of the conductance tensor in terms of follows quite naturally, by observing that the initial conductances following from the Kubo formula are given by where the superscript denotes that the quantity is fully renormalized by interactions Aristov and Wölfle (2011). By expressing in terms of as , we find by comparison with the conductance matrix, , that has nonzero elements given by the conductance parameters introduced above:

(6)

From now on we will drop the superscript , with the understanding that all quantities are renormalized. Therefore, we may use to represent the conductances in the renormalization group analysis below.

By substituting the -matrix in the form (2), (4) into the definition of , we find the relation of the , , to the Euler angle variables: , , . We find therefore that are confined within the region , , . The physically allowed points in ---space, satisfying the conditions , and , lie inside a body labelled as shown in Fig. 2. We may trace the above restriction on the allowed values of the conductance back to our consideration of energy conserving scattering. In the linear response regime and at zero temperature only lead electrons at the Fermi level will be transported through the interacting wire, and particle-hole excitations are excluded. The restriction of the set of conductances to the domain of allowed values has important consequences for the stability of certain fixed points of the RG flow, as will be discussed below. We mention now and will show later that all fixed points are located on the surface of the body in Fig. 2.

For the isotropic case discussed below we have . One possible parametrization of the S-matrix for this case is given by Eq. (35) The corresponding subspace of allowed conductance values is the twodimensional domain enclosed by a deltoid curve, see Eq. (42) and Fig. 3.

Figure 2: (Color online) Allowed values of conductances, , lie inside the depicted body, . The location of the RG fixed points , , , is also shown. The line of fixed points connecting FPs and is indicated in yellow.
Figure 3: (Color online) Allowed values of conductance , for the isotropic case. The S-matrices (35) characterized by parameters , lead to two quantities and , as described by (43). We took 500 pairs , at random, these are blue points on the plot. As discussed in Appendix B, the domain is bounded by a deltoid curve.

Iii Renormalization group equations

The renormalization of the conductances by the interaction is determined by first calculating the correction terms in each order of perturbation theory. We are in particular interested in the scale-dependent contributions proportional to , where and are two lengths, characterizing the interaction region in the wires (see above). In lowest order in the interaction the scale dependent contribution to the conductances is given by Aristov and Wölfle (2012a)

(7)

where , the are a set of nine matrices (products of ’s are matrix products), is the matrix of interaction constants and the trace operation Tr is defined with respect to the matrix space of ’s. As argued in Aristov and Wölfle (2009) the leading terms in each order of perturbation theory are certain diagrams of the ladder type, which may be summed up analytically. The result is a renormalized interaction matrix replacing the bare interaction matrix  in Eq.(7). The components of are obtained from the following matrix equation (see appendix C)

(8)

We observe that the effective interaction is found to depend on the conductance components in a highly nonlinear way. The matrix characterizes the interaction strength and depends on the Luttinger parameters as

(9)

(Notice that for any interaction strength.) In order to remove the redundancy in the conductance matrix we now multiply with from the left and from the right to get the components of in the form

(10)

Differentiating these results with respect to (and then putting ) we find the RG equations up to infinite order in the interaction

(11)

Here the are a set of nine matrices and the trace operation Tr is defined with respect to that matrix space, whereas  and are scalars with respect to this space. The nine matrices are best evaluated with the aid of computer algebra, inserting the -matrix in terms of the quantities as given above (see also Appendix D). As a result one finds the following set of RG-equations

(12)

where , and

The RG-equations describe the flow of the conductances upon increasing until a stable fixed point is reached. The fixed points are found by putting the right hand sides (the -functions) simultaneously equal to zero. The approach towards any given fixed point is characterized by power laws. The fixed point pattern and the power law exponents depend on the interaction strength. We note that the -functions (the right hand sides of Eqs. (12) ) are highly nonlinear functions of and of , giving rise to a rich manifold of fixed points and RG flow patterns.

Iv Fixed points of RG flow

iv.1 Chiral isotropic case

In the isotropic case, meaning (i) equal interaction strength in all wires, , and (ii) isotropic tunneling amplitudes, we have and . The RG equations read

(13)

where . By putting the -functions equal to zero we find four fixed points, labeled , at positions

(14)

The location of these points in the -plane is shown in Fig. 4. We observe that all the FPs lie on the boundary curve of the domain of allowed conductances. We discard the unphysical solution , , because .

Figure 4: (Color online) RG flows for the fully symmetric chiral case, for different values of interaction. The four starting points are the same in all plots. The non-universal positions of the chiral points are shown by open circles.

Next we discuss the stability of these FPs. We start with the -point, located at a cusp of the boundary curve , see Fig. 2. The cusp is infinitely sharp, leaving only a single direction of approach from the inside of : , . Linearizing the RG equation for in the small quantity we find

It follows that the -point is stable for repulsive interaction, when .

We now turn to the -FPs, which are again located at a cusp of Curve , Fig. 3. Therefore, again there is only one allowed direction for the trajectory towards or away from : , where is found to obey

We thus see that the -FPs are stable in the interaction regime .

The -point is located at a flat section of curve and we therefore have a two-dimensional manifold of possible trajectories leading to it. Expanding the RG-equations in linear order around the -point we find

It follows that the -point is stable for . We note in passing that the -point cannot be discussed within the Abelian bosonization approach, as the criterion is not satisfied. Aristov (2011)

Whereas the -FPs have been described before, we find in addition a new pair of FPs here. The location of these FPs is not geometrical, but depends on the interaction. We first observe that the -points are physically allowed (i.e. are located within the open domain ) only for sufficiently strong attractive interaction, (they are always residing at the boundary curve ). At they merge with the -point. As increases the -points start to move away from the -point in opposite direction until they end at the -points when . Upon further increase of the -points move beyond the -points along the boundary of , until they end at , , in the limit . The stability analysis tells that the -points are never stable.

As a result, we conclude, that for the only stable fixed point is the point, and not the unphysical point , suggested in Oshikawa et al. (2006). In Sec. 6.3 of that paper the authors say: “While the fixed points are stable against a small change in the flux, we do not know whether the fixed point has such a stability. The simplest assumption is that it does not. In this case the RG flows might go to the fixed points for any non-zero flux , starting from arbitrarily small . Alternatively, it is possible that the fixed point is stable against adding a small flux. In that case, there would have to be additional unstable fixed points defining the boundaries between the basins of attraction of the , and fixed points. So again, an ’economy’ principle suggests the simple picture with only three stable fixed points”, for . Our analysis shows, however, that this conjecture is not fully correct, as the new unstable fixed points appear for , allowing the point to become stable. Examples of flow trajectories for interaction strengths are shown in Fig. 4.

iv.2 Chiral anisotropic case

The above RG equations, Eqs. (12), describe the flow of towards the stable fixed points (FPs) inside the body of allowed conductance values (see Fig. 2), depending on the initial conditions. The fixed points are defined by the zeros of the -functions on the r.h.s. of Eqs. (12). Some of the fixed points are outside the allowed region in -space and will be discarded as unphysical. There are altogether six physical FPs (see table 1), which may be classified into two groups, universal (independent of the interaction) and non-universal FPs.

The three universal FPs are labeled . These FPs may be thought to be of geometrical nature, as the corresponding -matrix elements are or . The FP describes the total separation of the wires, meaning zero conductance in all components. By contrast, describes the separation of wire from the perfectly conducting wires 1–2. Finally, are the FPs generated by chirality (left- or right-handed).

The non-universal FPs are labeled . They only exist or are physical in a limited region of interaction constant space, mainly for attractive interaction. As shown in table 1 the conductances at these FPs depend in a complicated way on the interaction constants.

The locations of the and points have been given in Aristov and Wölfle (2011). We reproduce the results here in the present notation. Both points can simultaneously exist only for attractive interaction, . Notice that for equal interactions we have , (see below) and we return to (14).

The FPs and merge along a line in interaction constant space, defined by , where

(15)

The end point of this line is at , , and was named “tricritical” in Ref. [Aristov and Wölfle, 2011].

The location of the FPs is best represented in terms of the angular variables as

(16)

The points lie on the surface of the body of physically allowed conductances, since corresponds to the outmost points of our parametrization. The FPs are only in the physical domain if the interaction constants satisfy the requirement , implying , and leading to the bounds

The interval shrinks to zero when , and the above inequalities demand , which in turn gives . Remarkably, these values again correspond to and , dubbed the “tricritical” point above.

From the Table 1 one may also verify that the points merge with at

(17)

implying, in the symmetric situation, the condition .

FP
1 1 0
1 0
0
0
Table 1: The positions of the fixed points of the RG equations, (12). The quantity is defined in Eq. (15).

V Stability of fixed points

The above set of equations (12) allows to perform a rather straightforward analysis of the stability of FPs. Assuming that we have a fixed point , we may expand the RG equations in terms of the vector of small deviations , to linear order in :

(18)

The stability of the fixed point is defined by the direction of the RG flow, which is determined by the eigenvalues () of the matrix . All are positive for the fully unstable FP, all are negative for stable FPs and in the the saddle point case (some negative, some positive) the FP is again unstable.

Two remarks are in order here. First, the matrix is in general not symmetric, so that its eigenvectors are not orthogonal. Second, in the case of asymmetric interaction () the expressions for are not simple for the non-universal FPs, and should be analyzed numerically. Simplifications occur at , as was already demonstrated for the point in the non-chiral case in Aristov and Wölfle (2011).

We mention that not all directions of the vector are permitted. This is because all FPs lie on the surface of the body of physically allowed conductances. Certain displacements would take the point outside of the body. If the FP lies on a smooth part of the surface of the body, the requirement for to be inside the body is not very restricting, because it only selects half of all possible directions. But in our case the situation is more complicated by the fact, that some FPs lie on ridges of the body. This situation happens with the universal FPs and is discussed below.

In the following we present a detailed stability analysis for the different fixed points. The results are collected in Table 2 and in Fig. 5, showing the phase boundaries (i.e. the boundaries, separating different domains of stability) in the -plane.

Figure 5: (Color online) The RG phase portrait, showing regions with different set and character of the FPs. The region of attractive interactions is shown in more detail in the second panel. The numbers in the plot refer to its description in Table 2. The three dots correspond to the special points , , and , , discussed in the text.
I u s u - - -
II u s u u - -
III s u u u - -
IV s u u u - -
V s u u - - -
VI u u u s - -
1 u s u - - -
2 u s u - - u
3 u u s u - -
4 u u s s - u
5 u s u s u -
6 u s u s u u
3’ u u u s - -
Table 2: Summary of the RG phase portrait, depicted in Fig. 5. The existence and stability of the RG fixed points is shown below. FPs , , always exist, and the stability is indicated by an “s” symbol. The stable and unstable FPs , , are denoted by “s” and “u”, the “-” symbol is used if such a point does not exist.

A corresponding figure for the non-chiral case () has been presented as Fig. 4 in our previous publication Aristov and Wölfle (2011). By comparing the figures we observe that phases I–VI and 1,3’,5 are characterized by FPs with and were already discussed in the non-chiral case. The new phases are labeled 2–4, 6, involving either the stable chiral FPs or the new unstable FPs, .

It should be emphasized, that the scaling exponents obtained by our method for the “universal” FPs , , are in exact agreement with the results, found by boundary conformal field theory Hou et al. (2012). Our results for the non-chiral FPs , were reported earlier Aristov and Wölfle (2011). The particular expressions for the points below, Eq. (24), coincide with the exponents deduced from Table I of Ref. [Hou et al., 2012] for the scaling dimensions of the leading irrelevant boundary operators, . This becomes clear after the identification of our with their , and after putting their equal to our .

v.1 point

In this case the matrix is given by

(19)

Formally, there are three (left) eigenvectors , , , corresponding to eigenvalues , , , respectively.

However, as can be seen in Fig. 3, any displacement with is not allowed, as it would end outside the allowed domain. Therefore, when discussing the RG flows starting inside “the body”, we should discard any displacements along . We thus end with two exponents governing the RG flow towards

(20)

which may be interpreted as the weak link exponents in the main wire and between the main wire and the tip, respectively (eq. (46) in Aristov and Wölfle (2011)). These values are negative for , so that the point is stable in the case of repulsive interaction in all wires.

v.2 point

The matrix for the asymmetric point is given by

(21)

We have three (left) eigenvectors , , , corresponding to eigenvalues , , , respectively. The last eigenvalue is doubly degenerate, and therefore we again have only two exponents

(22)

which may be interpreted as i) the weak impurity exponent in the main wire and ii) the tunneling exponent in the main wire and the boundary exponent in the tip, respectively (eq. (47) in Aristov and Wölfle (2011)). At the “tricritical” point, , , the second exponent vanishes. This is a manifestation of the fact, that the tricritical point is the endpoint of the line of stable fixed points in the interaction parameter space, see Fig. 5.

v.3 point

We consider only one of these chiral points, , for which we obtain

(23)

The three left eigenvectors are , , , corresponding to the eigenvalues , , , respectively. The last eigenvector points out of “the body” and should be discarded, similarly to the situation at the point.

The other two physical exponents are

(24)

For we have both exponents equal :

When the and merge, which happens at , the second exponent in (24) vanishes.

For the form of the eigenvectors is not transparent, and it might be useful to restore the parametrization in terms of Euler angles. The coordinates of the points are then given by , , for arbitrary . For , we put and expand , , with . Then, linearizing the resulting RG equations, one finds a decoupled set of differential equations for

(25)

in correspondence with the above exponents for the conductance components.

v.4 , and points

As seen in Table 1, the position of these FPs is not universal, i.e. depends on the interaction. The linearization of the corresponding set of RG equations around these points leads to very cumbersome expressions for the matrix , which we do not show here. Instead we list the three scaling exponents for the conductances.

For fixed point we have (for ):

(26)

with the corresponding eigenvectors , , and defined in (15).

For point we obtain (again, for )