# Contacts, equilibration, and interactions in fractional quantum Hall edge
transport

C. Spånslätt  Department of Microtechnology and Nanoscience, Chalmers
Institute of Technology, S-412 96 Göteborg, Sweden Institute for Quantum
Materials and Technologies, Karlsruhe Institute of Technology, 76021
Karlsruhe, Germany Institut für Theorie der Kondensierten Materie, Karlsruhe
Institute of Technology, 76128 Karlsruhe, Germany Yuval Gefen  Department of
Condensed Matter Physics, Weizmann Institute of Science, Rehovot 76100, Israel
Institute for Quantum Materials and Technologies, Karlsruhe Institute of
Technology, 76021 Karlsruhe, Germany I. V. Gornyi  Institute for Quantum
Materials and Technologies, Karlsruhe Institute of Technology, 76021
Karlsruhe, Germany Institut für Theorie der Kondensierten Materie, Karlsruhe
Institute of Technology, 76128 Karlsruhe, Germany Ioffe Institute, 194021
St.Petersburg, Russia  D. G. Polyakov  Institute for Quantum Materials and
Technologies, Karlsruhe Institute of Technology, 76021 Karlsruhe, Germany

###### Abstract

We study electron transport through a multichannel fractional quantum Hall
edge in the presence of both interchannel interaction and random tunneling
between channels, with emphasis on the role of contacts. The prime example in
our discussion is the edge at filling factor 2/3 with two counterpropagating
channels. Having established a general framework to describe contacts to a
multichannel edge as thermal reservoirs, we particularly focus on the line-
junction model for the contacts and investigate incoherent charge transport
for an arbitrary strength of interchannel interaction beneath the contacts
and, possibly different, outside them. We show that the conductance does not
explicitly depend on the interaction strength either in or outside the contact
regions (implicitly, it only depends through renormalization of the tunneling
rates). Rather, a long line-junction contact is characterized by a single
parameter which defines the modes that are at thermal equilibrium with the
contact and is determined by the interplay of various types of scattering
beneath the contact. This parameter—playing the role of an effective
interaction strength within an idealized model of thermal reservoirs—is
generically nonzero and affects the conductance. We formulate a framework of
fractionalization-renormalized tunneling to describe the effect of disorder on
transport in the presence of interchannel interaction. Within this framework,
we give a detailed discussion of charge equilibration for arbitrarily strong
interaction in the bulk of the edge and arbitrary effective interaction
characterizing the line-junction contacts.

##  I Introduction

The interpretation of electrical conductance measurements in mesoscopic
conductors was intensively debated from the very onset of mesoscopic physics
up until the late 90s [1, 2]. The discussions mostly revolved around the role
of electrical contacts [3, *Landauer1989, 5, 6, 7, 8], with the focus
initially on noninteracting electrons. One conceptually essential point
recognized back then concerns the difference between the two- and four-
terminal measurements. Specifically, it was understood that, when electrons
are not reflected at the contacts (“perfect junctions”), the two-terminal
ballistic dc conductance $G$ is measured in units of $e^{2}/h$ per conducting
channel, with the resistance $1/G$ emerging entirely from relaxation processes
in the attached contacts (“reservoirs”).

As the discussions expanded to cover interacting electrons, the notion of
ballistic transport yielded a remarkable result for one-dimensional correlated
electrons in a ballistic (conserving both the total electron momentum and the
numbers of right- and left-moving electrons) Luttinger liquid (LL) [9].
Namely, it was established, from various theoretical perspectives [10, 11, 12,
13, *Egger1998, 15, *Alekseev1998, 17, 18, 19], that when a ballistic LL
quantum wire terminates in two Fermi liquid contacts, all signatures of
interaction inside the wire vanish from $G$. Under the assumption of
interactions inside the contacts being negligible, $G$ was understood to be
universally quantized at $e^{2}/h$ (per spin), largely in accordance with the
experimental observations [20, 21].

Our purpose here is to investigate transport of interacting electrons through
a fractional quantum Hall (FQH, fractional QH) edge, with emphasis on the role
of contacts and the universality of $G$ for the case when the edge hosts
several nonequivalent chiral conducting channels. An FQH edge is a strongly
correlated “chiral LL” [22, 23, *Wen1995, 25] that inherits its compositional
properties from the topological order of the bulk. For a given bulk filling
factor $\nu$, the topological constraint on the edge structure (the number of
edge channels and the channel filling factor discontinuities, hence the
channel chiralities) allows for multiple specific choices. Which of the
choices is realized is determined by the confinement-controlled
“electrostatics” of the edge.

An archetypical example of a multichannel edge, on which we focus in this
paper, is the “hole-conjugate” edge for $\nu=2/3$. The concrete model we
discuss corresponds to two counterpropagating channels with filling factor
discontinuties 1 and $-$1/3 [26, *Johnson1991], which is thought to be
appropriate for the case of a sufficiently sharp confinement (below, we refer
to these channels as channel 1 and channel 1/3, resp. modes 1 and 1/3). The
key ingredient in our story is interaction between charge densities in these
two channels. More complex edge structures with more channels emerge with
softening confinement as a result of “edge reconstruction” [28, 29, 30, 31,
*Wan2003, 33], with the emergence of fractional modes being characteristic of
both integer [30, 33] and fractional bulk phases, eventually approaching the
“coarse-grained” quasiclassical limit [34, 35]. Additional channels were
argued to play an essential role in certain experiments on the $\nu=2/3$ edge
[36]. The ideas that are central to the description of edge transport within
the model we focus on here are equally applicable to these, more involved
structures. Experimentally, there has been an immense effort, in the last
decade or so, to probe the structure of complex FQH edges, especially at
$\nu=2/3$, with evidence pointing towards the existence of counterpropagating
edge modes [37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52,
53, 54].

An FQH edge with counterpropagating channels represents an intermediate case
between a nonchiral quantum wire (in particular, a conventional LL quantum
wire or a symmetric QH line junction [55, 56]) and a single-channel Laughlin
edge (corresponding to $\nu=1/m$, with $m$ an odd integer), and is different
from both in an essential way. Specifically, on the one hand, its chiral
nature is manifest in the presence of a ballistic charge mode irrespective of
the presence of backscattering disorder inside the edge—in contrast to the LL
quantum wire. On the other hand, disorder-induced charge equilibration between
the channels generically affects the conductance—in contrast to the single-
channel edge. A more subtle difference from the LL quantum wire concerns the
nature of contacts. The nonchiral wire can terminate in the contacts, whereas
the chiral edge cannot. Therefore, contacts to the QH edge are necessarily
“side attached” (Fig. 1) [57].

Figure 1: Schematic illustration of a contact (orange) side-attached to a
chiral (not terminating in the contact) two-channel edge (blue). The contact
is connected to an external circuit and coupled to the edge modes through
tunneling bridges.

###  I.1 Contacts

For the single-channel case, both the two-terminal conductance $G$ and the
Hall conductance $G_{H}$ show “fractional quantization,” $G=G_{H}=1/m$
(hereafter, we set $e=\hbar=1$, except for the conductances, which are
measured, as in the formula above, in units of $e^{2}/h$), under the
assumption that the side-attached contacts are “ideal”; otherwise, the
quantization of $G$ at $1/m$ is lost [57, 58]. An ideal contact is defined as
fully absorbing the incident charge current (“black body”) and emitting
current that is independent of the incident one, with the contact playing the
role of a source at equilibrium. For example, a single tunneling link between
the edge and the reservoir does not meet these conditions [57, 58], nor does a
contact that emits several edge modes that are not at equilibrium with each
other [59]. The ideal contact is supposed to be characterized by a certain
electrochemical potential $\mu$ and temperature $T$ of electrons. A voltage
probe (in its invasive version that “thermalizes” electrons) is then
understood as an ideal contact with no net current flowing through it.

####  I.1.1 Nonideal contact

Calculating the edge conductance with ideal contacts in the clean case (where
“clean” means no charge scattering between channels if the number of channels
exceeds one) is thus tantamount to the identification of the edge electron
density excitations that are conjugate to $\mu$ [15, *Alekseev1998, 60]. For
the single-channel edge, the electron charge density mode is defined uniquely,
which yields the abovementioned $G=G_{H}=1/m$. From this perspective, the
quantization of $G=1$ for a clean interacting LL wire [10, 11, 12, 13,
*Egger1998, 15, *Alekseev1998, 17, 18, 19] (two chiral channels) stems from
$\mu$ being conjugate to the “bare” (noninteracting) right- or left moving
electron modes, and not the chiral, interaction-renormalized electron
eigenmodes. That is, the contacts that provide for $G=1$ in interacting LL
quantum wires are not ideal: the currents incident on the contacts from the
bulk of the wire, which are the eigenmode currents, are not fully absorbed by
the contacts, and the currents emitted by the contacts into the bulk are
delicate linear combinations of the eigenmodes.

The apparent dichotomy between the requirement of ideal contacts for the
quantization of $G$ in the Laughlin edge and the requirement of nonideal
contacts for the quantization in nonchiral quantum wires reflects the tacitely
assumed absence of interaction between electrons on different segments of the
QH edge separated by contacts. Bringing these segments in proximity to each
other, in a narrow Hall bar [61, 62] or on the sides of a narrow barrier in
the QH line-junction setup [55, 56], would also make the “nonideality” of the
contacts—in the same sense as in the LL case—a necessary condition for the
quantization of $G$ at $1/m$ for the single-channel edge. Conversely, assuming
the current-supplying contacts to be ideal, the interaction between the parts
of the Laughlin edge on the opposite sides of the Hall bar yields a
nonuniversal value of $G$ [61], with $G_{H}$ still quantized at $1/m$.

Focusing on the multichannel FQH edge in a Hall-bar geometry, we work with the
assumption that the interaction between electrons on the edge is sufficiently
short-range so as not to act between any points on the edge separated by a
contact. However, even with this assumption, the presence of local
interchannel interactions both outside the side-attached contact and beneath
it—which are not necessarily of the same strength—brings up a question about
the influence of these interactions on the conductance, which we will discuss
in detail in the main part of the paper.

It was shown in Ref. 63 that generically neither $G$ nor $G_{H}$ in the clean
case is quantized for the $\nu=2/3$ edge with counterpropagating channels, if
the contacts are considered as ideal. That is, both $G$ and $G_{H}$ depend
then on the strength of interchannel interaction. This is in similarity to a
LL quantum wire with ideal contacts. However, it is a widely held idea that
the contacts to a wire that terminates in reservoirs are, as already noted
above, generically not ideal—because of the mismatch in the interaction
strength inside the wire and in the contacts that are thought to be
noninteracting. The mismatch is understood here as a “sharp” one, with the
interaction strength changing nonadibatically fast with respect to the scale
of the characteristic wavelength of density excitations. The embodiment of
this idea is the model of the LL quantum wire with an inhomogeneous
(“vanishing at infinity”) strength of interaction [10, 11, 12].

A direct generalization of this model to the case of a QH edge is the model in
which the interacting edges, or the interacting edge channels for that matter,
split up from each other and are contacted beyond the region inside which they
run parallel (“elongated quantum point contact”) [64, 65, 66]. Within this
construction, the currents in the noninteracting “spacer” are incident on and
emitted by an ideal noninteracting contact. The nonideality of the “extended
contact,” which is the contact as such plus the spacer, is then associated
with Fresnel-like scattering of density excitations at the boundary between
the spacer and the interacting part of the edge. For the model of ideal
contacts attached to the noninteracting spacers, the two-terminal conductance
of the clean $\nu=2/3$ edge has a universal value independent of interchannel
interaction [66]: $G=4/3$, which should be contrasted with the result of Ref.
63.

This model, however, ignores the possibility of interaction between the edge
channels as they run past the contact: indeed, the contact to a QH edge is, as
already mentioned above, unavoidably side-attached (Fig.1) [57], hence
screening of the interchannel interaction by it is not necessarily perfect.
This might be particularly clear if one thinks about the contact being
attached “laterally,” as opposed to “on top.” If one imagines the limit of no
screening, the interaction strength is then homogeneous everywhere along the
closed path of the edge. This alone makes a difference between the side-
attached contact and the contact terminating a quantum wire, the consequences
of which we will explore in the main text. In particular, one of the questions
that arise in this connection is whether the two-terminal conductance for the
clean $\nu=2/3$ edge depends, instead of being quantized at 4/3, on the
strength of interchannel interaction beneath the contacts.

####  I.1.2 Line-junction contact

A very natural model for the side-attached contact that incorporates
interchannel interactions is that of a “line-junction contact” [57], which
represents a linear sequence of tunnel links connecting the edge and the
reservoir. On the phenomenological level, dynamics of electrons in the
reservoir is supposed to be fully incoherent and characterized by an
infinitely fast equilibration to a given thermal state (“ideal reservoir”).
One can supplement the picture by modeling the reservoir as a collection of
“incoherent” chiral noninteracting channels each of which supplies electrons
at a given common electrochemical potential [58].

In the limit of an infinite density of infinitesimally weak links with the
tunneling rate held fixed, the model is describable by a set of scattering
rates between different channels and the reservoir, and between the channels
themselves. In this limit, it was argued [57] that the distinguishing property
of a long line-junction contact to the edge with counterpropagating
channels—as opposed to that with copropagating channels—is that the current to
the reservoir is determined by the relative amplitudes of the nonuniversal
partial scattering rates. Moreover, assuming that the interaction strength
beneath the contact and outside it is the same, the argument was made [57]
that the conductance should depend only on the scattering rates, but not on,
separately, the interaction strength itself, as would be the case for the
model with ideal contacts analyzed in Ref. 63.

By taking this point further, we explicitly calculate $G$ and $G_{H}$ for the
model of line-junction contacts, for arbitrary interaction strengths beneath
the contacts and in the rest of the edge. One of the observation we make is
that—for arbitrary parameters of the line-junction contact—there exists a set
of edge modes that is at equilibrium with a given contact at its end points
(the interface between the part of the edge beneath the contact and the “bulk”
of the edge). This means that the contact can be viewed as an ideal one with
respect to these modes and characterized by a single parameter describing
charge equilibration in the contact region. Remarkably, this parameter plays
the same role as the strength of interchannel interaction in the ideal-contact
model adopted in Ref. 63. That is, the line-junction contact is characterized
by the strength of an effective interchannel interaction beneath it.

###  I.2 Disorder

Apart from categorizing the contacts, another essential ingredient for the
framework to describe charge transport through multichannel QH edges is
disorder-induced charge scattering between the channels. The role of this
scattering is twofold. First, it “equilibrates” the charge densities in
different channels on average. This establishes a ballistic propagation of the
total charge density, irrespective of the difference in the values and signs
of the channel velocities [67]. Second, it leads to mesoscopic fluctuations of
these densities. For edges with only copropagating channels, disorder-induced
charge equilibration plays, in the dc limit, the same role as equilibration by
a voltage probe and so does not affect the total edge current, with mesoscopic
fluctuations showing up only in the elements of the conductance matrix (in
channel indices) [68]. By contrast, for edges with counterpropagating
channels, mesoscopic fluctuations affect the conductance, and so does,
generically, charge equilibration, as we outline below.

####  I.2.1 Interchannel equilibration

Charge equilibration establishes, in the limit of full equilibration, the
universal quantization of disorder-averaged $G=\nu$ [67], independently of the
value of $G$ in the clean limit. The quantization of $G$ at the value of
$\nu$, irrespective of the interchannel interaction, in the charge-
equilibrated limit was also demonstrated [63] (and argued on more
phenomenological grounds [57]) specifically for $\nu=2/3$. Importantly, the
charge equilibration-induced quantization of $G$ for $\nu=2/3$ does not rely
on the decoupling of the charge and neutral modes at a specific value of the
interaction strength, with disorder affecting then only the neutral mode,
along the lines of the renormalization-group treatment [69, 70]. The
quantization results solely from a combination of the total charge
conservation and the local equilibration, in the spirit of hydrodynamics [67].

By providing for the conductance quantization at $\nu$, the edge segment
consisting of the contact proper and the charge-equilibrated parts of the
disordered edge on both sides of it is, as a whole, at equilibrium with the
bare (noninteracting) density modes. It was argued [57] that such a “compound”
contact may be viewed as a realization of Büttiker’s ideal contact [59] for
the edge with counterpropagating channels. It is worth noting, however, that
this is only true if there is no interaction between the channels (as was the
case in the Büttiker construction with copropagating channels). Indeed, as was
already mentioned above, the ideal contacts to a clean edge do not yield a
quantized conductance for interacting counterpropagating channels [63].

A crossover to the universally quantized $G=\nu$ [67] as the length of the
disordered edge increases was considered for two counterpropagating channels
both in the absence [71, 72] and in the presence [73, 74, 66] of interchannel
interaction. A model of local thermal equilibration [72], in which every pair
of adjacent tunneling links is separated by voltage and temperature probes in
each of the channels, was employed to explore, in the absence of interchannel
interaction, also thermal transport and shot noise [75, 76, *Spanslatt2020,
78]. Below, we will study charge equilibration particularly for the line-
junction contacts, for arbitrary interactions beneath and outside the
contacts.

####  I.2.2 Mesoscopic fluctuations

Turning to mesoscopic fluctuations for the case of counterpropagating
channels, it is important to distinguish two essentially different types of
the fluctuations. One is not related to the presence or absence of
interchannel interaction. Because of the chiral nature of the edge, mesoscopic
conductance fluctuations of this type self-average with increasing edge
length. For incoherent transport (see Sec. I.2.3), they vanish altogether in
the Gaussian limit of the Poisson distribution of interchannel tunneling links
in space between the contacts. The other relies on spatial inhomogeneity of
the interaction strength; in particular, at the contacts. If the interaction
strength changes at the contacts and is homogeneous otherwise, then Fresnel-
like scattering of the density modes at the contacts creates a Fabry-Pérot
resonator between the contacts, with charge transfer through its facets
depending on the particular realization of disorder.

One peculiar situation emerges when the interchannel interaction is strong and
the charge and neutral modes decouple [69, 70], with disorder in interchannel
tunneling not affecting the charge mode. For ideal contacts, mesoscopic
fluctuations of the two-terminal conductance $G$ are then strictly absent.
Otherwise, they are describable in terms of a disordered chiral sine-Gordon
($\chi$sG) model [79, 65, 66]. For the $\nu=2/3$ edge, composed of channels 1
and 1/3, with noninteracting spacers at the contacts (see above), fluctuations
with varying edge length are strong within this model. Specifically, $G$
fluctuates between 4/3 and 1/3 [66], or, equivalently, the diagonal element,
for channel 1, of the conductance matrix fluctuates between 1 and 1/2 [65]. To
the best of our knowledge, this (“coherent”) transport regime has not so far
been observed experimentally; on the contrary, the dependence of $G$ on the
edge length for $\nu=2/3$ was reported to show charge equilibration as the
length is increased, with $G$ approaching $2/3$ in an (arguably) smooth manner
[52].

####  I.2.3 Incoherent transport

In this paper, we restrict attention to the incoherent model and focus mostly
on the case of white-noise (weak and with a vanishing correlation length)
disorder in the amplitude of tunneling links. Within the incoherent model with
white-noise disorder, $G$ is a smooth function of the edge length with no
mesoscopic fluctuations by construction (which seems to be in agreement with
the experimental observations [52] mentioned above). As part of a brief
rationale for the model, let us first comment on the meaning of “incoherent”
in the context of the $\nu=2/3$ edge. A conceptually effective formalism to
solve the disordered $\chi$sG model is to map it onto the (pseudo)spin-1/2
dynamics of a chiral fermion in a spatially random Zeeman field [69]. The
coherent $\chi$sG and incoherent models differ then in that the former deals
with random rotations of spin over the Bloch sphere, whereas the latter with
spin flips between the up and down positions. As such, the incoherent model is
fully described by the occupation numbers for spin up and spin down [80].

Returning to the original problem, the spin flips correspond to flipping the
orientation of a charge dipole between channels 1 and 1/3. As a result, the
incoherent model is formalizable in terms of a linear equation of motion for
the “partial” charge densities. This picture is pertinent to the decoupled
charge and neutral modes. In the main text, we will discuss the dynamics of
charge within the incoherent model, for arbitrary interaction between the
channels, in more detail.

From this perspective, a question arises as to the effect of nonzero $T$ on
the neutral-mode dynamics within the disordered $\chi$sG model, in particular,
whether it may lead to a suppression of the mesoscopic fluctuations. It was
argued for the $\nu=2/3$ model [66]—and demonstrated for a related model [79],
with a proper adjustment for the $\nu=2/3$ case—that the property of transport
being coherent is insensitive to temperature as long as the neutral and charge
modes are decoupled (otherwise, an incoherent transport regime [80] emerges if
the edge length is sufficiently long). On the other hand, a different scenario
was proposed, in which a crossover to the incoherent regime, as $T$ is
increased, is governed by the ratio of $T$ and the characteristic energy
spacing for the density excitations on the scale of the edge length [65]. Yet
another condition for such a crossover was associated with the ratio of $T$
and the energy spacing on the scale of the “typical distance between
scatterers” [74].

While it is beyond the scope of this paper to go deeper into the story about
the coherent regime, let us briefly mention various mechanisms that may indeed
suppress mesoscopic fluctuations and justify the incoherent model,
irrespective of whether the charge and neutral modes are decoupled or not. In
particular, at nonzero $T$, one can think of the mechanisms that are based on
weakening the assumptions behind the conventional formulation of the $\chi$sG
problem, i.e., introducing perturbations to the coherent model.

For example, this may be nonzero curvature of the density-mode spectrum
(related to a finite range of interaction), which leads to a “self-averaging”
of the conductance given by a sum of contributions with different velocities
of propagation (to an extent, this is similar to the curvature-induced
suppression of interference in a Mach-Zender interferometer [81]). Nonzero
curvature of the electron spectrum (related to the shape of self-consistent
confinement) generically adds decay channels for the edge excitations. Note
that the nonlinearity of the spectrum may be enhanced close to the edge
reconstruction transition. Or it may be dephasing by the environment. One
particular source of it may be temporal fluctuations of the strength of
tunneling because of interactions with additional—due to edge
reconstruction—channels possibly running close to those taken into account
without exchanging charge with them.

Another mechanism of the suppression of mesoscopic flustuations, effective
also at $T=0$, may be due to the coherent random interchannel dynamics of
charge being extended from the “bulk” of the edge into the regions beneath the
line-junction contacts. Then, fluctuations tend to self-average because the
conductance is given by a sum over paths of different length for the density
excitations that are emitted and absorbed at different points along the
contacts.

Mesoscopic fluctuations may also be suppressed merely because of interchannel
interaction being so weak (or so strong, see below) that the edge finds
itself, even upon disorder-induced renormalization [69, 63], far away from the
point at which the charge and neutral modes decouple. It was argued in Ref. 66
that mesoscopic fluctuations are suppressed for weak interaction and
prominent, for not a too long edge, close to the decoupling point. However, as
discussed in Sec. II.1, there is a certain duality between the properties of
the edge for weak and strong interchannel interaction. A direct consequence of
it is that, for strong interchannel interaction—stronger than required for the
decoupling of the charge and neutral modes—disorder should be irrelevant in
the renormalization-group sense in the same manner as for weak interaction.
Moreover, dephasing of mesoscopic fluctuations at $T\neq 0$ should be
characteristic not only of the weak-interaction case, but the opposite case as
well.

###  I.3 Outline of the results

In Secs. I.1 and I.2, we discussed the multifaceted issues of contacts and
disorder in QH edge transport, by placing them in proper perspective with
regard to an FQH edge with counterpropagating channels. With this background
in mind, we investigate charge transport through such an edge in the presence
of both interchannel interaction and backscattering disorder. We focus on the
prime example for these purposes, namely the two-channel edge at $\nu=2/3$
with channels 1 and 1/3. Here, we do so within the incoherent model, defined
and rationalized in Sec. I.2.

One of our aims is to gain understanding of how edge transport is affected by
interchannel interaction when the interaction strength is different beneath
and outside the contacts. The particular model of the contact that we keep in
mind in the first place is a line-junction contact. Before proceeding to this
model, however, we first set up a general phenomenological framework in which
the contact is supposed to be at equilibrium with a particular set of edge
modes that are not the eigenmodes outside it. As such, this contact is
nonideal—in the precise sense discussed in Sec. I.1.

Our main results, some of which were already mentioned above, can be broadly
described as follows.

  * (1)

For the contacts represented—first at the model level—as thermal reservoirs
for the edge eigenmodes in the contact regions, the conductance of the clean
edge does not depend on the strength of interchannel interaction outside these
regions. Rather, the conductance depends on the strength of interaction
characterizing the modes that are at equilibrium with the contacts. The
dependence of transport on the latter is formalized by introducing
“generalized” boundary conditions on the facets of the contacts. The
independence of the conductance on interchannel interaction outside the
contacts holds irrespective of whether different contacts correspond to the
same or different equilibrium modes associated with them. The obtained
expressions for the conductance—for various arrangements of the measuring
terminals—extend the conventional picture of ideal contacts.

  * (2)

The long line-junction contact is fully characterizable by a single parameter,
which brings in the notion of “universality” in the classification of
different contacts attached to the edge. Remarkably, this parameter plays the
role of an effective strength of interaction inside the contact within the
above model of a thermal reservoir. In the microscopic picture, this effective
interaction strength is generically nonzero for the line-junction contact. The
parameter “labelling” a given contact is determined by the interplay of
backscattering between the edge channels and scattering between the reservoir
and the edge, but not the interaction strength either beneath or outside the
contact—apart from the interaction-induced renormalization of the scattering
rates.

  * (3)

Disorder-induced tunneling between edge channels is described in terms of
charge fractionalization. The framework of fractionalization-renormalized
tunneling, which emerges from this approach, yields the dependence of the
scattering rates on the interaction strength, formalized in terms of
electrostatic screening of charges created by tunneling. This dependence is
distinctly different from the conventional renormalization of the tunneling
strength by the interaction-induced orthogonality catastrophe. The picture of
fractionalization-renormalized tunneling reveals the physics behind the effect
of backscattering disorder on the edge eigenmodes, including the charge and
neutral modes when these decouple from each other. It also describes the
emergence of negative partial scattering rates for sufficiently strong
interaction. The thermodynamic constraint on tunneling is framed into the
fractionalization picture for an arbitrary strength of interaction.

  * (4)

Charge equilibration and the resulting dependence of the conductance on the
edge length are analyzed for arbitrary strength of interchannel interaction
both beneath the contacts and outside them. The strength of true interaction
is shown to cancel out from the conductance of a disordered edge apart from
the renormalization of the scattering rates. Instead, transport is affected by
the effective interaction that characterizes equilibration in the long line-
junction contacts, making the conductance of a short edge dependent on the
strength of this effective interaction. The existence and the properties of
the disorder-modified ballistic charge mode, responsible for the universal
quantization of the conductance in the limit of a long edge, are discussed for
arbitrary strength of interaction. The conventional notions of the contact and
bulk contributions to the two-terminal resistance are demonstrated to be
inapplicable to the QH edge with counterpropagating channels.

The remainder of the paper is organized in the following way. Section II is
devoted to transport through the clean edge, with emphasis on the formulation
of the boundary conditions for the contacts [items (1) and (2) in the above
outline of the results]. In Sec. II.1, we specify the model for the $\nu=2/3$
edge. In Sec. II.2, we discuss the generalized boundary condition [item (1)].
In Sec. II.3, we calculate the conductance for this type of the boundary
condition with various arrangements of the measuring terminals. In Sec. II.4,
we consider the model of the line-junction contact [item (2)]. In Sec. II.5,
we map the model of the line-junction contact onto the model of the
generalized boundary condition, and obtain the conductance for the line-
junction contacts. Section III deals with fractionalization-renormalized
tunneling [item (3)]. In Sec. III.1, we discuss fractionalization upon
tunneling into the edge. In Sec. III.2, we turn to fractionalization upon
tunneling between edge channels and formulate a general framework to describe
fractionalization-renormalized tunneling for the case of a single tunneling
link. In Sec. III.3, we analyze the strong-tunneling limit and the
thermodynamic constraint on tunneling for an arbitrary strength of
interchannel interaction. Section IV covers transport through a disordered
edge [item (4)]. In Sec. IV.1, we consider the emergence of negative
scattering rates for the case of strong interchannel correlations. In Sec.
IV.2, we discuss the disorder-modified eigenmodes of the edge. In Sec. IV.3,
we calculate the conductance of the disordered edge for arbitrary parameters
of the line-junction contacts and address the question of whether the contact
and bulk resistances in the two-terminal setup are generically meaningful
notions for a multichannel QH edge. Section V concludes with a succinct
summary. In Appendix, we discuss some aspects of the nonchiral LL model from
the perspective of the framework formulated for the chiral edge; in
particular, the line-junction contact and fractionalization-renormalized
tunneling.

##  II Clean edge

We begin by considering a clean edge, with no interchannel tunneling, and
specifically focus on the $\nu=2/3$ edge within its model discussed at the
beginning of Sec. I, namely the one composed of counterpropagating channels 1
and 1/3.

###  II.1 Model for the $\nu=2/3$ edge

The model is defined by the Hamiltonian density

$H_{0}=\pi\left(v_{1}n_{1}^{2}+3v_{2}n_{2}^{2}+2v_{12}n_{1}n_{2}\right)$ (1)

and the commutation relation at points $x$ and $x^{\prime}$ along the edge

$\left[\,n_{\textrm{i}}(x),n_{\textrm{j}}(x^{\prime})\,\right]=\frac{i}{2\pi}\delta_{\textrm{ij}}\,\delta\nu_{\textrm{i}}\,\partial_{x}\delta(x-x^{\prime})$
(2)

for the charge densities (in units of $-e$) $n_{1}$ and $n_{2}$ in channels 1
and 1/3, respectively. This is equivalent to the Lagrangian formulation in
Ref. 69. The constants $v_{1,2}$ are the speeds of propagation of the
densities $n_{1,2}$ in the noninteracting limit (here and below,
“noninteracting” means no interchannel interaction, whereas short-range
interactions within the channels are incorporated in the velocities
$v_{1,2}$). In Eq. (1), the densities $n_{1}$ and $n_{2}$ interact with each
other via a short-range potential with the zero-momentum Fourier component
$2\pi v_{12}$. The filling factor discontinuities $\delta\nu_{1}=1$ and
$\delta\nu_{2}=-1/3$ in Eq. (2) are of opposite sign, which encodes the
property of the modes $n_{1}$ and $n_{2}$ propagating in opposite directions
for $v_{12}=0$.

In accordance with Eq. (1), we assume everywhere below that the interchannel
interaction is point-like. For our purposes in this paper, this is an accurate
description of the edge on length scales larger than the screening radius of
Coulomb interaction, where screening is provided by a nearby metallic gate. It
is worth noting, however, that a nonzero radius of interaction leads to a
nonlinear dispersion relation for the density excitations, which may be
important on arbitrarily large length scales. For instance, one of the
consequences of the nonlinear dispersion, commented upon in Sec. I.2.3, is the
suppression of mesoscopic conductance fluctuations, which adds to the
justification of the incoherent description of the edge dynamics.

A diagonalization of the quadratic form in Eq. (1) represents $H_{0}$ in terms
of the eigenmode (“chiral”) charge densities $n_{\pm}$:

$H_{0}=\frac{\pi v_{+}}{g_{+}}n_{+}^{2}+\frac{\pi v_{-}}{g_{-}}n_{-}^{2}~{},$
(3)

with $n_{\pm}$ obeying

$\left[\,n_{\pm}(x),n_{\pm}(x^{\prime})\,\right]=\pm\frac{i}{2\pi}g_{\pm}\partial_{x}\delta(x-x^{\prime})~{}.$
(4)

Both $n_{1,2}$ and $n_{\pm}$ are defined as charge densities, so that

$n_{+}+n_{-}=n_{1}+n_{2}~{}.$ (5)

The eigenmodes propagate with the velocities $v_{\pm}$ (both defined
positively, as speeds) to the right (+) and to the left ($-$), and are
characterized by the dimensionless “conductances” $g_{\pm}\geq 0$ [63].
Parametrized in terms of the dimensionless interaction strength

$\alpha=\frac{2}{\sqrt{3}}\frac{v_{12}}{v_{1}+v_{2}}~{},$ (6)

$v_{\pm}$ are given by

$v_{\pm}=\frac{1}{2}\left[\,\pm(v_{1}-v_{2})+(v_{1}+v_{2})\sqrt{1-\alpha^{2}}\,\right]~{}.$
(7)

To represent $g_{\pm}$ in a compact form, it is convenient to introduce the
parameter [69, 63]

$\Delta=\frac{2-\sqrt{3}\alpha}{\sqrt{1-\alpha^{2}}}~{},$ (8)

in terms of which $g_{\pm}$ are written as

$g_{\pm}=\frac{\Delta\pm 1}{3}~{}.$ (9)

From Eqs. (3) and (7), the stability conditions are $\textrm{Im}\,v_{\pm}=0$,
which is $|\alpha|\leq 1$, and $v_{\pm}\geq 0$, which is [69]

$v_{12}^{2}\leq 3v_{1}v_{2}~{}.$ (10)

The latter is stronger than or identical to the former for arbitrary $v_{1}$
and $v_{2}$ (identical for $v_{1}=v_{2}$), so that the only condition is Eq.
(10). Note that $\Delta\geq 1$ [cf. Eq. (9)] for all $|\alpha|\leq 1$. The
point $\Delta=1$, at which the charge and neutral modes decouple [69],
corresponds to $\alpha=\sqrt{3}/2$. It is worthwhile to mention that if the
velocities $v_{1,2}$ differ strongly enough from each other, namely if $v_{1}$
is beyond the interval

$v_{2}/3<v_{1}<3v_{2}~{},$ (11)

an instability occurs with increasing $\alpha$ before the point $\Delta=1$ is
reached.

It is also worth noticing that the function $\alpha(\Delta)$ [which solves Eq.
(8)] is double valued for $|\alpha|\leq 1$, with two branches merging at
$\Delta=1$. That is, for the case of repulsive interaction ($\alpha>0$), the
range of $\Delta$ is not limited to the interval $1\leq\Delta\leq 2$, with the
noninteracting point at $\Delta=2$:

$0\leq\alpha\leq\sqrt{3}/2\quad\mapsto\quad 1\leq\Delta\leq 2~{}.$ (12)

On the other branch of $\alpha(\Delta)$, with

$\displaystyle\sqrt{3}$ $\displaystyle/2<\alpha\leq
2\sqrt{v_{1}v_{2}}/(v_{1}+v_{2})$ $\displaystyle\mapsto\quad 1<\Delta\leq
2(v_{1}+v_{2}-\sqrt{3v_{1}v_{2}})/|v_{1}-v_{2}|~{},$ (13)

the value of $\Delta$ grows with increasing $\alpha$. This means, in
particular, a nonmonotonic dependence of $g_{\pm}$ on $\alpha$. The
largest—within the model (1)—$\alpha$ in Eq. (13) corresponds to the upper
boundary (10), at which one of the speeds $v_{\pm}$ slows down to zero (or
both, if $v_{1}=v_{2}$).

Note that it is the parameter $\Delta$ that is of prime importance by
determining the scaling dimensions of the correlation functions of the model
[69, 63]. In view of Eqs. (12) and (13), the model (1) possesses a duality
between the cases of small and large $v_{12}$ corresponding to the same value
of $\Delta$, with the only difference being the different velocities $v_{\pm}$
that are determined by $v_{12}$ itself. Perhaps only numerics can say if this
captures physics of a more realistic model [82]. Having made this cautionary
remark, we assume below, for definiteness, that $1\leq\Delta\leq 2$ with
$\alpha$ varying within the interval (12).

The Heisenberg equations of motion resulting from Eqs. (1) and (2), which have
the form of continuity equations $\partial_{t}n_{1,2}+\partial_{x}j_{1,2}=0$,
yield the partial charge currents

$j_{1}=v_{1}n_{1}+v_{12}n_{2}~{},\quad j_{2}=-(v_{2}n_{2}+v_{12}n_{1}/3)~{}.$
(14)

Similarly, from Eqs. (3) and (4), the chiral charge currents $j_{\pm}$ obey
$\partial_{t}n_{\pm}+\partial_{x}j_{\pm}=0$ with

$j_{\pm}=\pm v_{\pm}n_{\pm}~{}.$ (15)

The total edge current $j=j_{+}+j_{-}=j_{1}+j_{2}$.

###  II.2 Generalized boundary condition

The equations of motion introduced in Sec. II.1 need to be supplied with
boundary conditions at the contacts. As outlined in Sec. I.3, we first
consider—before turning in Sec. II.4 to the line-junction contact—an
instructive example in which the contact is at equilibrium with a set of
density modes that are, in general, not the eigenmodes outside the contact.
Specifically, let the contact be a thermal reservoir for the modes $n_{\pm c}$
[Eqs. (3) and (4), with $c$ for “contact”] corresponding to a given
interaction parameter $\Delta_{c}$ [Eq. (8)] not necessarily equal to the
parameter $\Delta$ for the edge outside the contact. If $\Delta_{c}=\Delta$,
the contact is, by definition in Sec. I.1, an ideal contact, for which the
two-terminal conductance was obtained as [69, 57, 63]

$G=g_{+}+g_{-}=\frac{2}{3}\Delta$ (16)

[this is a “cousin” of the equality $G=K$, where $K$ is the Luttinger
constant, for a LL wire with ideal contacts (see Appendix)]. As we will
demonstrate in Sec. II.5, this model of a thermal reservoir with a certain
$\Delta_{c}$ is directly related to the line-junction model discussed in Sec.
I.1.2. Let us, therefore, proceed to the case of $\Delta_{c}\neq\Delta$.

Figure 2: Schematic profile (solid line) of the parameter $\Delta(x)$ [Eq.
(8)] characterizing the strength of interchannel interaction beneath
($\Delta_{c}$, for $|x|<\delta$) and outside ($\Delta$, for $|x|>\delta$) the
side-attached contact. Because of screening of the interaction by the contact,
$\Delta_{c}$ and $\Delta$ are generically not the same. The difference between
them depends on the mutual position of the contact and the edge; in
particular, the ratio of two characteristic distances: the one between the
contact and the edge and the other between the channels. The dashed line for
$\Delta(x)$ illustrates the “point-splitting” procedure discussed below Eqs.
(29) and (30).

Imagine that the contact “covers” the part of the edge between $x=\delta$ and
$x=-\delta$ (Fig. 2), so that the boundary conditions for the densities
outside the contact are set at $|x|=\delta$. The chemical-potential shift
$\delta H$ of the Hamiltonian density of the edge, provided by the contact, is

$\delta H=-\mu n~{},$ (17)

where $n$ is the total charge density. According to the above, assume that
$\mu$ is conjugate to the densities $n_{\pm c}$. Specifically, writing

$n=n_{+c}+n_{-c}$ (18)

in $\delta H$ and minimizing $H_{0}+\delta H$ by varying it with respect to
$n_{+c}$ and $n_{-c}$, the two boundary conditions on the facets of the
contact are obtained as

$\displaystyle\mu=\left(\hat{\lambda}^{T}\hat{\Pi}^{-1}\right)_{++}n_{+}(\delta)+\left(\hat{\lambda}^{T}\hat{\Pi}^{-1}\right)_{+-}n_{-}(\delta)~{},$
(19)
$\displaystyle\mu=\left(\hat{\lambda}^{T}\hat{\Pi}^{-1}\right)_{-+}n_{+}(-\delta)+\left(\hat{\lambda}^{T}\hat{\Pi}^{-1}\right)_{--}n_{-}(-\delta)~{}.$
(20)

Here,

$\hat{\Pi}^{-1}=2\pi\left(\begin{array}[]{cc}v_{+}/g_{+}&0\\\
0&v_{-}/g_{-}\end{array}\right)$ (21)

is the inverse thermodynamic compressibility matrix in the bulk of the edge
(for $|x|>\delta$) in the $n_{\pm}$ basis [cf. Eq. (3)], and the matrix
$\hat{\lambda}$ relates $\textbf{n}_{c}=(n_{+c},n_{-c})^{T}$ and
$\textbf{n}=(n_{+},n_{-})^{T}$:

$\textbf{n}=\hat{\lambda}\,\textbf{n}_{c}$ (22)

($\hat{\lambda}^{T}$ denotes transpose of $\hat{\lambda}$). The matrix
$\hat{\lambda}$ is given by

$\displaystyle\hat{\lambda}$
$\displaystyle=\frac{1}{2}\left(\begin{array}[]{cc}1+\Delta&1+\Delta\\\
1-\Delta&1-\Delta\end{array}\right)$ (25)
$\displaystyle+\frac{1}{2}\sqrt{\frac{\Delta^{2}-1}{\Delta_{c}^{2}-1}}\left(\begin{array}[]{rr}1-\Delta_{c}&\,-1-\Delta_{c}\\\
-1+\Delta_{c}&\,1+\Delta_{c}\end{array}\right)~{}.$ (28)

By definition, $\hat{\lambda}^{-1}$ equals $\hat{\lambda}$ with the exchange
$\Delta\leftrightarrow\Delta_{c}$.

In terms of the currents (15), the boundary conditions (19) and (20) become

$\displaystyle\frac{\mu}{2\pi}=\frac{j_{+}(\delta)}{g_{+}}\,\lambda_{++}-\frac{j_{-}(\delta)}{g_{-}}\,\lambda_{-+}~{},$
(29)
$\displaystyle\frac{\mu}{2\pi}=\frac{j_{+}(-\delta)}{g_{+}}\,\lambda_{+-}-\frac{j_{-}(-\delta)}{g_{-}}\,\lambda_{--}~{}.$
(30)

Equations (29) and (30) impose two links on the four currents
$j_{\pm}(\delta)$ and $j_{\pm}(-\delta)$. Placing boundary conditions of this
type on each of the contacts fixes all currents between the contacts on the
closed loop along the edge. The boundary conditions (29) and (30) [or (19) and
(20) for that matter] generalize those for the ideal contact, for which
$\hat{\lambda}=\mathbb{1}$ is the identity matrix.

Although the generalized boundary conditions are significant by themselves,
the concern that one might have at this point is over the fact that they are
imposed on the facets of the contact where the interaction strength
experiences, by construction, a jump, corresponding to the change between
$\Delta$ and $\Delta_{c}$. It is therefore instructive to “point split” the
boundary conditions by placing them at $|x|=x_{0}+0$, where $x_{0}>\delta$
(Fig. 2), and assuming that the interaction parameter is given by $\Delta_{c}$
in the “spacer” between $|x|=\delta$ and $|x|=x_{0}$. The purpose is to
demonstrate that Eqs. (29) and (30) correspond to precisely this procedure
with $x_{0}\to\delta$.

The contact itself (covering, as before the point splitting, $|x|<\delta$) is
then ideal, with a simple boundary condition at $|x|=\delta$:

$\frac{\mu}{2\pi}=\pm\frac{j_{\pm c}(\pm\delta)}{g_{\pm c}}~{},$ (31)

where $j_{\pm c}(\pm\delta)$ and $g_{\pm c}$ are the outgoing eigenmode
currents (those emitted by the contact) and the eigenmode conductances,
respectively, corresponding to the interaction parameter $\Delta_{c}$. The
boundary condition at $|x|=x_{0}+0$ is thus a combination of the ideal-contact
condition (31) at $|x|=\delta$ and the matching condition at $|x|=x_{0}$. The
latter relates the densities and currents on the sides of the interface at
which the interaction parameter changes from $\Delta_{c}$ to $\Delta$ with
increasing $|x|$.

It is convenient to write the matching condition in the basis that does not
change across the interface. Let it be, as a transparent example, the
$n_{1,2}$ basis [Eqs. (1) and (2)]. Around the interface, the continuity
equations then read [cf. Eq. (14)]

$\displaystyle\partial_{t}n_{1}+\frac{1}{2\pi}\,\partial_{x}\\!\left[\left(\hat{\bar{\Pi}}^{-1}\right)_{11}\\!n_{1}+\left(\hat{\bar{\Pi}}^{-1}\right)_{12}\\!n_{2}\right]=0~{},$
(32)
$\displaystyle\partial_{t}n_{2}-\frac{1}{2\pi}\,\frac{1}{3}\,\partial_{x}\\!\left[\left(\hat{\bar{\Pi}}^{-1}\right)_{21}\\!n_{1}+\left(\hat{\bar{\Pi}}^{-1}\right)_{22}\\!n_{2}\right]=0~{},$
(33)

where $\hat{\bar{\Pi}}$ is the $x$ dependent thermodynamic compressibility
matrix in the $n_{1,2}$ basis [the bar is put to distinguish $\hat{\bar{\Pi}}$
from $\hat{\Pi}$ in Eq. (21)], with

$\hat{\bar{\Pi}}^{-1}=2\pi\left(\begin{array}[]{cc}v_{1}&\,v_{12}\\\
v_{12}&\,3v_{2}\end{array}\right)~{}.$ (34)

The matching condition is, therefore, the condition of continuity, upon
crossing the interface, of the partial “local chemical potentials”

$\displaystyle\mu_{\textrm{i}}=\sum_{\textrm{j}}\left(\hat{\bar{\Pi}}^{-1}\right)_{\textrm{ij}}n_{\textrm{j}}~{}:$
(35)
$\displaystyle\mu_{\textrm{i}}|_{|x|=x_{0}-0}=\mu_{\textrm{i}}|_{|x|=x_{0}+0}~{}.$
(36)

Equivalently, Eq. (36) is the condition of continuity of the currents
$j_{1}=\mu_{1}$ and $j_{2}=-\mu_{2}/3$ that are separately—beyond the total
current conservation—continuous across the interface. The continuity of
$j_{1,2}$ is achieved by the corresponding jumps in $n_{1,2}$.

By the same token, the continuity of the partial currents is maintained in any
basis that is not changed across the interface; in particular, in the basis of
$n_{\pm c}$. By representing: (i) $j_{\pm c}(x=\pm x_{0}\mp 0)$ in terms of
$\mu$ from Eq. (31) and (ii) $j_{\pm c}(x=\pm x_{0}\pm 0)$ in terms of
$j_{\pm}$ at the same $x$ by changing the basis according to Eq. (22), the
continuity condition for $j_{\pm c}$ becomes exactly Eqs. (29) and (30) after
$x_{0}\to\delta$, which was to be demonstrated.

###  II.3 Conductance for the generalized boundary condition

We now turn to the calculation of the two- and four-terminal conductances of a
clean $\nu=2/3$ edge for the boundary conditions (29) and (30), by placing
them on each of the contacts, in the dc limit. Nonequilibrium is then created
by “biasing” the current contacts with different chemical potentials (this is
a sufficient minimal model to calculate the measured conductance in a QH
system with a gapped bulk [83, 57]). Assume that all of the contacts are
characterized by the same $\Delta_{c}$ (the case of different contacts will be
considered in Sec. IV.3.2).

((a))

((b))

Figure 3: Measurement with two (a) and four (b) terminals, characterized by
the chemical potentials $\mu_{L}$ (left), $\mu_{R}$ (right), $\mu_{T}$ (top),
and $\mu_{B}$ (bottom), for the edge with two counterpropagating channels. The
total edge current in the two-terminal geometry is a sum of the currents in
the upper ($j_{u}$) and lower ($j_{l}$) parts of the edge. This is so for the
four-terminal setup as well, if the top and bottom terminals are the voltage
probes.

For the two-terminal setup [Fig. 3(a)], the currents $j_{u\pm}$ in the upper
part of the edge [running to the right (+) and to the left ($-$)] are given,
from Eqs. (29) and (30), by

$\left(\begin{array}[]{c}j_{u+}/g_{+}\\\
-j_{u-}/g_{-}\end{array}\right)=\frac{1}{2\pi}\left(\hat{\lambda}^{-1}\right)^{T}\left(\begin{array}[]{c}\mu_{L}\\\
\mu_{R}\end{array}\right)~{},$ (37)

where $\mu_{L}$ and $\mu_{R}$ are the chemical potentials of the left and
right contacts, respectively. For the total current at the upper edge
$j_{u}=j_{u+}+j_{u-}$, the combination of $g_{\pm}$ and $\hat{\lambda}^{-1}$
reduces to $g_{\pm c}$:

$j_{u}=\frac{1}{2\pi}\left(g_{+c}\mu_{L}-g_{-c}\mu_{R}\right)~{}.$ (38)

The total current at the lower edge $j_{l}$ is obtained from Eq. (38) by
exchanging $\mu_{L}\leftrightarrow\mu_{R}$, with the total current $j$ between
the contacts being $j=j_{u}-j_{l}$. The conductance $G=2\pi
j/(\mu_{L}-\mu_{R})$ is then found as

$G=g_{+c}+g_{-c}=\frac{2}{3}\Delta_{c}~{}.$ (39)

The strength of interchannel interaction for the edge outside the contacts is
thus seen to drop out from the two-terminal conductance (and from each of the
currents $j_{u,l}$ separately). That is, as a general rule, the conductance of
a clean edge depends only on the characteristics of the modes that are at
equilibrium with the contacts [cf. Eq. (16) for the ideal contact].

Extending the argument that led to Eq. (38) to compute the four-terminal
conductances for the setup in Fig. 3(b), the current flowing from the left
contact into the edge, $j_{L}$, is written as

$j_{L}=\frac{1}{2\pi}\left[\,g_{+c}\left(\mu_{L}-\mu_{B}\right)-g_{-c}\left(\mu_{T}-\mu_{L}\right)\,\right]~{},$
(40)

where $\mu_{L}$ is the chemical potential of contact $L$, etc. The currents
$j_{T,R,B}$ into contacts $T,R,B$ are obtainable from Eq. (40) by cyclic
permutation $L\to T\to R\to B\to L$. For the Hall measurement, the top and
bottom contacts are taken as voltage probes with $j_{T,B}=0$, so that the
source-drain current $j=j_{L}=-j_{R}$. The Hall conductance $G_{H}$ and the
source-drain conductance $G_{4}$ are then obtained as

$\displaystyle G_{H}=\frac{2\pi
j}{\mu_{T}-\mu_{B}}=\frac{g_{+c}^{2}+g_{-c}^{2}}{g_{+c}-g_{-c}}=\frac{1}{3}\left(\Delta_{c}^{2}+1\right)~{},$
(41) $\displaystyle G_{4}=\frac{2\pi
j}{\mu_{L}-\mu_{R}}=\frac{g_{+c}^{2}+g_{-c}^{2}}{g_{+c}+g_{-c}}=\frac{1}{3}\left(\Delta_{c}+1/\Delta_{c}\right)~{},$
(42)

with $G_{H}=\Delta_{c}G_{4}$. Similarly to $G$ in Eq. (39), $\Delta$ vanishes
from $G_{H}$ and $G_{4}$, which depend only on $\Delta_{c}$.

###  II.4 Line-junction contact

Having introduced the phenomenological model for the contacts in Secs. II.2
and II.3, we now turn to the line-junction contact model [57]. As will be seen
shortly, the two are directly mappable onto each other. As already discussed
in Sec. I.1.2, the line-junction contact model is formalized in terms of
scattering rates at which the edge channels exchange electrons with the
reservoir and, in general, also the rates at which the channels exchange
electrons among themselves. Specifically, for the $\nu=2/3$ edge, the
equations of motion for the currents $j_{1,2}$ beneath the contact (i.e., for
$|x|<\delta$) read

$\displaystyle\partial_{x}j_{1}+\frac{1}{2}\gamma_{c}(j_{1}+3j_{2})-\gamma_{1}\left(\frac{\mu}{2\pi}-j_{1}\right)=0~{},$
(43)
$\displaystyle\partial_{x}j_{2}-\frac{1}{2}\gamma_{c}(j_{1}+3j_{2})-\gamma_{2}\left(\frac{\mu}{2\pi}+3j_{2}\right)=0~{},$
(44)

where $\gamma_{1}^{-1}$ and $\gamma_{2}^{-1}$ are the scattering lengths for
the exchange of electrons between the reservoir and channels 1 and 1/3,
respectively, and $2\gamma_{c}^{-1}$ is the backscattering length for these
channels in the contact region (Fig. 4). The combination
$j_{1}+3j_{2}=\mu_{1}-\mu_{2}$ is given by the difference of the local partial
chemical potentials (35), a nonzero value of which produces the “transverse”
(between the channels) local current. The combination of $\mu$ and $j_{1,2}$
in the last terms in Eqs. (43) and (44) means that the currents $j_{1}$ and
$j_{2}$ tend to equilibrate, respectively, at $\mu/2\pi$ and
$-(1/3)\times\mu/2\pi$, where $\mu$ is the chemical potential of electrons in
the piece of metal that constitutes the side-attached contact, viewed as a
thermal reservoir for electrons.

Figure 4: Schematic portrait of a line-junction contact. Channels 1 and 1/3
exchange electrons, for $|x|<\delta$, through tunneling links (depicted as
dashed lines) with the contact viewed as a thermal reservoir of electrons at
the chemical potential $\mu$. The model is characterized by three inverse
scattering lengths: $\gamma_{1}$ and $\gamma_{2}$ for tunneling between the
contact and channels 1 and 1/3, respectively, and $\gamma_{c}/2$ for tunneling
between the channels [Eqs. (43) and (44)].

The origin of backscattering between the channels beneath the contact is,
generically, twofold. Firstly, it may be related to direct disorder-induced
tunneling between the channels. Importantly, the disorder strength may be
different from that outside the contact region. For example, manufacturing the
contact may introduce more disorder to the edge, as a result of which the
line-junction contact with disorder-induced $\gamma_{c}\neq 0$ may be
considered as connected to an otherwise clean edge. Secondly, virtual
tunneling into and from the contact results in backscattering if the initial
channel differs from the final one. However, in the limit of infinitesimally
weak individual tunneling links to the contact (Sec. I.1.2), these processes
do not contribute to $\gamma_{c}$. Specifically, for the Hamiltonian of a
single tunneling link

${\mathcal{H}}_{t}^{c}=t_{1}\psi_{c}^{\dagger}\psi_{1}+t_{2}\psi_{c}^{\dagger}\psi_{2}+\textrm{h.c.}~{},$
(45)

where $\psi_{c}$ and $\psi_{1,2}$ are the electron operators at the point of
tunneling in the contact and in channels 1 and 1/3, respectively, for the
tunneling amplitudes $t_{1,2}\to 0$ and the link concentration
$n_{c}\to\infty$ with $n_{c}|t_{1,2}|^{2}$ held fixed, the rates of tunneling
to and from the contact, related to $\gamma_{1,2}$, are finite, whereas the
contribution to $\gamma_{c}$ vanishes as $n_{c}|t_{1}t_{2}^{*}|^{2}\to 0$.

Note that Eq. (45) assumes that the tunneling link connects the reservoir to
both channels 1 and 1/3. In a more general modeling of the experimental setup,
the links to channels 1 and 1/3 need not be at the same points along the edge
and, moreover, their densities may differ. Nevertheless, in the Gaussian limit
of infinitesimally weak links, the virtual processes of tunneling into the
contact do not contribute to $\gamma_{c}$ in any case. That said, it may be
important to keep in mind that, beyond this limit, $\gamma_{c}\neq 0$ even if
disorder-induced backscattering in the contact region is viewed as negligibly
weak.

Equations (43) and (44) demonstrate a remarkable degree of universality with
regard to the strength of interaction. A sufficient, for the purpose of
determining the edge conductances, way to think of the source of
nonequilibrium is to view it as the difference between the chemical potentials
at different contacts represented as thermal reservoirs, as was already
mentioned in Sec. II.3. The static effect of possible interaction between
electrons inside the contact is then encoded in the difference between the
chemical and electrochemical potentials outside the edge and, as such, does
not show up explicitly in Eqs. (43) and (44). Interaction between the channels
beneath the contact manifests itself in the definition of the currents
$j_{1,2}$ in Eq. (14), but not explicitly in Eqs. (43) and (44). An immediate
consequence of this is that, if $\Delta_{c}=\Delta$, the strength of
interchannel interaction—beyond the effects of interaction-induced
renormalization of the constants $\gamma_{1,2}$ and $\gamma_{c}$—drops out
from the boundary condition at the contacts and the conductance itself [57].
Note also that interaction between electrons in the edge and electrons in the
contact does not affect the structure of Eqs. (43) and (44), including the
source terms. Below, we treat $\gamma_{1,2}$ and $\gamma_{c}$, which are
dependent on particular microscopic details of the contact arrangement, as
phenomenological parameters of the model [84].

By introducing $\bar{j}_{1}=j_{1}-\mu/2\pi$ and
$\bar{j}_{2}=j_{2}+(1/3)\times\mu/2\pi$, Eqs. (43) and (44) can be rewritten
as homogeneous ones:

$\left(\partial_{x}\mathbb{1}+\hat{\upgamma}_{c}\right)\left(\begin{array}[]{c}\bar{j}_{1}\\\
\bar{j}_{2}\end{array}\right)=0~{},$ (46)

where the matrix of inverse scattering lengths beneath the contact
$\hat{\upgamma}_{c}$ reads

$\hat{\upgamma}_{c}=\frac{1}{2}\left[\begin{array}[]{cc}\gamma_{c}+2\gamma_{1}&3\gamma_{c}\\\
-\gamma_{c}&-3(\gamma_{c}+2\gamma_{2})\end{array}\right]~{}.$ (47)

The eigenvalues $k_{1,2}$ of $\hat{\upgamma}_{c}$ are

$k_{1,2}=\frac{\gamma_{1}-3\gamma_{2}-\gamma_{c}}{2}\pm\frac{Q}{2}~{},$ (48)

where

$\displaystyle
Q=(\gamma_{1}+3\gamma_{2}-\gamma_{c})\left[\,1+\frac{6\gamma_{c}(\gamma_{1}+3\gamma_{2})}{(\gamma_{1}+3\gamma_{2}-\gamma_{c})^{2}}\,\right]^{1/2}~{},$
(49)

with $k_{1}k_{2}\leq 0$.

One of the useful ways to represent the solution of Eq. (46) is to view it as
a scattering problem (Fig. 5), by expressing the outgoing currents
$j_{1}(\delta)$ and $-j_{2}(-\delta)$ in terms of the incoming currents
$j_{1}(-\delta)$ and $-j_{2}(\delta)$:

$\left[\begin{array}[]{c}\bar{j}_{1}(\delta)\\\
-\bar{j}_{2}(-\delta)\end{array}\right]=\hat{\textrm{W}}\left[\begin{array}[]{c}\bar{j}_{1}(-\delta)\\\
-\bar{j}_{2}(\delta)\end{array}\right]~{},$ (50)

where the leaky current-scattering matrix $\hat{\textrm{W}}$ is given by

$\displaystyle\hat{\textrm{W}}=\,\,\frac{1}{Q\cosh(Q\delta)+\left(\gamma_{1}+3\gamma_{2}+2\gamma_{c}\right)\sinh(Q\delta)}$
$\displaystyle\times\left[\begin{array}[]{cc}Qe^{-(\gamma_{1}-3\gamma_{2}-\gamma_{c})\delta}&3\gamma_{c}\sinh(Q\delta)\\\
\gamma_{c}\sinh(Q\delta)&Qe^{(\gamma_{1}-3\gamma_{2}-\gamma_{c})\delta}\end{array}\right]$
(53)

(note that $Q$ may be of either sign depending on the sign of
$\gamma_{1}+3\gamma_{2}-\gamma_{c}$, with $\hat{\textrm{W}}$ being an even
function of $Q$). The term “leaky” in the definition of $\hat{\textrm{W}}$
means that there is a current through the contact $j_{c}=j_{c1}+j_{c2}$ (Fig.
5), with

$\left(\begin{array}[]{c}j_{c1}\\\
j_{c2}\end{array}\right)=\left(\mathbb{1}-\hat{\textrm{W}}\right)\left[\begin{array}[]{c}\bar{j}_{1}(-\delta)\\\
-\bar{j}_{2}(\delta)\end{array}\right]~{}.$ (54)

In the limit of $\hat{\textrm{W}}\to 0$,

$j_{c}=\bar{j}_{1}(-\delta)-\bar{j}_{2}(\delta)=j_{1}(-\delta)-j_{2}(\delta)-\frac{4}{3}\frac{\mu}{2\pi}~{},$
(55)

which means complete absorption of the currents $j_{1}(-\delta)$ and
$-j_{2}(\delta)$ that enter the contact.

Figure 5: Contact as a scatterer. Scattering is parametrized by the leaky
scattering matrix $\hat{\textrm{W}}$. The matrix relates the outgoing
[$j_{1}(\delta)$ and $j_{2}(-\delta)$] and incoming [$j_{1}(-\delta)$ and
$j_{2}(\delta)$] currents [Eq. (50)], and also expresses the “leaking” current
through the contact $j_{c}$ in terms of the incoming currents [Eq. (54)].

###  II.5 Conductance for the line-junction contacts

In the limit of $\hat{\textrm{W}}\to 0$ [Eq. (55)], combining two contacts at
the chemical potentials $\mu_{L}$ and $\mu_{R}$, respectively, as shown in
Fig. 3(a), the two-terminal conductance of a clean edge is obtained as

$G=4/3~{},$ (56)

independently of the interchannel interaction strength beneath the line-
junction contacts. Equation (56) relies on the currents $j_{1,2}$ being
continuous at the interface between the parts of the edge beneath the
contacts, where the currents are described by Eqs. (43) and (44), and the bulk
of the edge. Importantly, this condition is fulfilled irrespective of the
relation between $\Delta_{c}$ and $\Delta$, as demonstrated at the end of Sec.
II.2. That is, Eq. (56) is valid for the clean edge with the line-junction
contacts in the limit of $\hat{\textrm{W}}\to 0$ for arbitrary $\Delta_{c}$
and $\Delta$, being independent of both of them, also for
$\Delta_{c}\neq\Delta$. This is in contrast to $G$ in Eq. (39) for the
generalized boundary condition, where $G$ is independent of $\Delta$, but
depends on $\Delta_{c}$. The independence of $G$ in Eq. (56) on $\Delta$ is
thus in line with Eq. (39), but the line-junction contact with
$\hat{\textrm{W}}\to 0$ corresponds to the contact described in Sec. II.2 if
one puts an effective parameter

$\Delta_{c}^{\textrm{eff}}=2~{},$ (57)

regardless of the actual one, in the generalized boundary conditions (19) and
(20), or (29) and (30). Recall that, as was already mentioned in Sec. I.1.1,
Eq. (56) holds also for the model of noninteracting reservoirs [Eq. (39) with
$\Delta_{c}=2$] attached to noninteracting spacers [66].

The independence of $G$ for the line-junction contacts on the interaction
strength beneath them is a direct consequence of the fact that electron
tunneling between the contact and the edge in Eqs. (43) and (44) occurs
between the contact and channels 1 and 1/3 separately [85]. If tunneling
occurred between the contact and channels $\pm$, i.e., the charges that
constitute the eigenmode were tunneling to the contact from modes 1 and 1/3
simultaneously (and similarly for tunneling from the contact), the boundary
conditions on the contact facets in the limit $\hat{\textrm{W}}\to 0$ would be
those from Sec. II.2. The conductance $G$ would then be given by Eq. (39),
instead of Eq. (56). By contrast, local tunneling from either mode 1 or 1/3 to
the contact creates two density pulses running along the edge beneath the
contact in opposite directions, i.e., excites both eigenmodes (the
fractionalization dynamics will be considered in more detail in Sec. III).

The mixing of modes $\pm$ by tunneling between modes 1 or 1/3 and the contact
is the reason for the nonideality, as is manifest from Eq. (56), of the line-
junction contact. Indeed, the condition $\hat{\textrm{W}}=0$ means that the
contact is at equilibrium with modes 1 and 1/3, with
$\bar{j}_{1}(\delta)=\bar{j}_{2}(-\delta)=0$. The outgoing eigenmode currents
$j_{+}(\delta)$ and $j_{-}(-\delta)$ obey, then, Eqs. (29) and (30) with the
matrix $\hat{\lambda}$ corresponding to $\Delta_{c}=2$ [Eq. (57)], i.e., with
$\hat{\lambda}\neq\mathbb{1}$ unless $\Delta=2$ as well. Nonzero off-diagonal
elements $\lambda_{+-,-+}\neq 0$ produce partial reflection of the eigenmodes
incident on the contact by admixing them to the outgoing eigenmodes, while the
difference of the diagonal elements from unity, $\lambda_{++,--}\neq 1$, means
that the emitted eigenmodes are not at equilibrium with the contact. Both the
former and the latter signify nonideality of the contact.

The limit of $\hat{\textrm{W}}\to 0$ can be achieved by first putting
$\gamma_{c}\to 0$, which gives a diagonal matrix $\hat{\textrm{W}}$:

$\hat{\textrm{W}}\rightarrow\left(\begin{array}[]{cc}e^{-2\gamma_{1}\delta}&0\\\
0&e^{-6\gamma_{2}\delta}\end{array}\right)~{},$ (58)

and then taking the limit of $\delta\to\infty$. Equation (58) simply describes
an edge not equilibrated with the contact because of the finite size of the
latter. More interesting—and essential to our discussion—is the limit of a
long contact, $\delta\to\infty$, taken without making any assumption about
$\gamma_{c}$ or $\gamma_{1,2}$. In this limit, $\hat{\textrm{W}}$ is purely
nondiagonal:

$\hat{\textrm{W}}\to c\left(\begin{array}[]{cc}0&3\\\
1&0\end{array}\right)~{},$ (59)

characterized by a single parameter [86]

$c=\frac{1}{2+\zeta+\sqrt{(2+\zeta)^{2}-3}}~{},$ (60)

where

$\zeta=\frac{\gamma_{1}+3\gamma_{2}}{\gamma_{c}}~{}.$ (61)

The constant $c$ is constrained by $0\leq c\leq 1/3$, with the limiting values
of $c=0$ and $c=1/3$ obtained for $\zeta\to\infty$ and $\zeta\to 0$,
respectively.

With $\hat{\textrm{W}}$ from Eq. (59), the currents $j_{1,2}(\pm\delta)$ on
the facets of the left [Fig. 3(a)] contact are obtained, for the clean edge
outside the contact regions, as

$\displaystyle j_{1}(\delta)$
$\displaystyle=\frac{1}{2\pi}\,\frac{(1-c)\mu_{L}+c(1-3c)\mu_{R}}{1-3c^{2}}~{},$
(62) $\displaystyle j_{2}(\delta)$
$\displaystyle=-\frac{1}{2\pi}\,\frac{c(1-c)\mu_{L}+\frac{1}{3}(1-3c)\mu_{R}}{1-3c^{2}}~{},$
(63)

and

$j_{1,2}(-\delta)=j_{1,2}(\delta)|_{\mu_{L}\leftrightarrow\mu_{R}}~{}.$ (64)

The resulting conductance $G=2\pi j_{c}/(\mu_{L}-\mu_{R})$, with the current
through the contact $j_{c}=j_{1}(\delta)+j_{2}(\delta)-(\delta\to-\delta)$
[or, equivalently, Eq. (54)], is then given by

$G=\frac{2}{3}\,\Delta_{c}^{\textrm{eff}}~{},$ (65)

where

$\Delta_{c}^{\textrm{eff}}=1+\frac{(1-3c)^{2}}{1-3c^{2}}~{}.$ (66)

For $c=0$, Eq. (65) reduces to Eq. (56). As $\zeta$ [Eq. (61)] decreases with
increasing strength of backscattering beneath the contact, $G$ changes from
4/3 for $c=0$ to 2/3 for $c=1/3$.

As follows from Eqs. (65) and (66), there exists a surface in space of
$\gamma_{1}$, $\gamma_{2}$, and $\gamma$ on which

$\Delta_{c}^{\textrm{eff}}=\Delta~{}.$ (67)

On this surface, the contact is ideal, with the emitted currents being at
equilibrium with the contact, namely $j_{\pm}(\pm\delta)=\pm g_{\pm}\mu/2\pi$
[cf. Eq. (31)]. At the same time, the incident eigenmode currents
$j_{\pm}(\mp\delta)$ are completely absorbed. This construction is thus the
embodiment of the notion of an ideal contact—for an arbitrary strength of
interchannel interaction both beneath the contact and outside it.

Away from the surface on which Eq. (67) holds, the contact is at equilibrium
with the eigenmodes corresponding to the effective interaction parameter
$\Delta_{c}^{\textrm{eff}}$ from Eq. (66). As a consequence, the calculation
in Secs. II.2 and II.3 for the generalized boundary condition applies directly
to the line-junction model, with $\textbf{n}_{c}$ understood as the eigenmodes
for $\Delta_{c}$ substituted by $\Delta_{c}^{\textrm{eff}}$. The purpose of
making a “digression” to discuss the generalized boundary condition in Secs.
II.2 and II.3 has now become clear. As a starting point, it demonstrated in a
concise and precise manner that the strength of interaction outside the
contacts drops out from the conductance of a clean edge [Eqs. (39), (41), and
(42)]. Perhaps more importantly, when combined with the calculation for the
line-junction, it serves as a basis for the physical picture in which the
conductance does not depend on either $\Delta$ or $\Delta_{c}$, but is
determined by the effective strength of interaction beneath the side-attached
contact, parametrized by $\Delta_{c}^{\textrm{eff}}$ [Eq. (65)].

From this perspective, the conductance $G$ varies between 2/3 and 4/3—with the
value of $\Delta_{c}^{\textrm{eff}}$ varying between 1 and 2—being controlled
by the single parameter (60) that reflects the interplay of tunneling between
the edge channels beneath the contact and tunneling between the contact and
the edge. This establishes a significant degree of universality of the model
of long line-junction contacts. Instead of contrasting, as in Ref. 57, the
model of ideal contacts on the one hand and the model of line-junction
contacts on the other, this viewpoint emphasizes that the two models are not
at all different from each other in terms of universality. If a contact of
either type is viewed as a “black box” (with arbitrary microscopic “content”),
there is one single parameter that characterizes the contact as far as the
current flowing through it is concerned.

##  III Fractionalization-renormalized tunneling

As an interlude between the discussions of the clean (Sec. II) and disordered
(Sec. IV) edges, it is instructive to formulate the “electrostatic” approach
to disorder-induced tunneling for the $\nu=2/3$ edge. In essential terms, we
mean electrostatics of tunneling that combines two effects: (i) the creation
of screening charges in channels 1 and 1/3 by the tunneling electron and the
hole that it leaves behind and (ii) the ensuing fractionalization of the
created charges, resolved with respect to both the edge channels and
chirality. This is quite apart from the renormalization of the tunneling
strength that is due to the interaction-induced orthogonality catastrophe upon
tunneling (“zero-bias anomaly” in the tunneling density of states in the
presence of a scatterer) [63, 87]. The latter is formalizable in terms of the
infrared-singular correlator of quantum fluctuations of the electron density
around the tunneling link. By contrast, the fractionalization-induced
renormalization of the tunneling strength describes ultraviolet—short-range in
real space—correlations between the charges created by the tunneling event.
These correlations are crucial for determining the $\Delta$ dependent relation
between the tunneling rates for different channels. In particular, they are
behind the emergence of the neutral mode at $\Delta=1$ [69], irrespective of
the strength of disorder, from the point of view of electrostatics, as will be
seen below.

###  III.1 Fractionalization upon tunneling into the edge

Let us first consider fractionalization in the $\nu=2/3$ edge upon addition,
locally, of a unit charge to channel 1 or 1/3, without tunneling between them.
The matrix $\hat{\Lambda}$ that expresses the eigenmodes
$\textbf{n}=(n_{+},n_{-})^{T}$ in terms of
$\bar{\textbf{n}}=(n_{1},n_{2})^{T}$ for the interaction parameter $\Delta$,

$\textbf{n}=\hat{\Lambda}\,\bar{\textbf{n}}~{},$ (68)

is given by $\hat{\Lambda}=\hat{\lambda}|_{\Delta_{c}=2}$, with
$\hat{\lambda}$ from Eq. (28). Note that $\hat{\Lambda}$ preserves the total
charge [Eq. (5)]. The ratio $\eta_{+}$ of $n_{2}$ and $n_{1}$ in mode + and
the ratio $\eta_{-}$ of $n_{1}$ and $n_{2}$ in mode $-$ are, then, written as

$\displaystyle\eta_{+}$
$\displaystyle=\left(n_{2}/n_{1}\right)_{+}=\left.\left(\hat{\Lambda}^{-1}\right)_{21}\middle/\left(\hat{\Lambda}^{-1}\right)_{11}~{},\right.$
(69) $\displaystyle\eta_{-}$
$\displaystyle=\left(n_{1}/n_{2}\right)_{-}=\left.\left(\hat{\Lambda}^{-1}\right)_{12}\middle/\left(\hat{\Lambda}^{-1}\right)_{22}~{},\right.$
(70)

where

$\displaystyle\hat{\Lambda}^{-1}=$
$\displaystyle\,\,\frac{1}{2}\left(\begin{array}[]{rr}3&3\\\
-1&-1\end{array}\right)$ (73) $\displaystyle+$
$\displaystyle\,\,\frac{1}{2}\sqrt{\frac{3}{\Delta^{2}-1}}\left(\begin{array}[]{rr}1-\Delta&\,-1-\Delta\\\
-1+\Delta&\,1+\Delta\end{array}\right)~{}.$ (76)

The constants $\eta_{\pm}$ have the meaning of screening charges;
specifically, $\eta_{+}$ is the charge in channel 1/3 induced by a unit charge
in channel 1, when both run to the right (mode +), and $\eta_{-}$ is the
charge in channel 1 induced by a unit charge in channel 1/3, when both run to
the left (mode $-$). They are related to each other as

$\frac{\eta_{+}}{\eta_{-}}=\frac{1}{3}~{},$ (77)

with

$\eta_{+}=-\frac{\sqrt{(\Delta+1)/3}-\sqrt{\Delta-1}}{\sqrt{3(\Delta+1)}-\sqrt{\Delta-1}}~{}.$
(78)

While keeping in mind the relation (77), the fractionalization picture is most
clearly formulated in terms of both $\eta_{+}$ and $\eta_{-}$.

For $\Delta=2$, the screening charges vanish: $\eta_{\pm}=0$. For $\Delta=1$,
they are given by

$\eta_{-}=3\eta_{+}=-1~{}.$ (79)

Note that screening is perfect, with the screening charge $\eta_{-}=-1$ being
exactly the mirror charge, for $\Delta=1$ in mode $-$, which becomes then the
“neutral mode” identified in Ref. 69. The charge carried by mode $-$ vanishes
at $\Delta=1$ as

$n_{-}\to\sqrt{\frac{\Delta-1}{6}}\,(n_{1}+3n_{2})~{},$ (80)

where $n_{1}+3n_{2}\to 2n_{2}$ according to Eq. (79). In mode +, which becomes
for $\Delta=1$ the “charge mode” with $n_{+}=n_{1}+n_{2}$, the “screening
coefficient” $\eta_{+}=-1/3$, so that the unit charge propagating to the right
at $\Delta=1$ is split into the charge 3/2 in mode 1 and the charge $-1/2$ in
mode 1/3 [88].

Figure 6: Charge fractionalization upon addition of a compact charge to
channel 1 (a) or channel 1/3 (b). The relative amplitudes of the
fractionalized charge pulses are shown for $\Delta=26/23\simeq 1.13$ (which
corresponds to the screening coefficients $\eta_{+}=-2/9$ and
$\eta_{-}=-2/3$). The charge distribution is described by the
fractionalization matrix given by Eqs. (86) and (92) for cases (a) and (b),
respectively.

The fractionalization process in a clean edge is illustrated in Fig. 6. It is
convenient to introduce the “fractionalization matrix” $\hat{q}$, whose
elements are $q_{\pm,\textrm{i}}$ with $\textrm{i}=1,2$, where $q_{+,1}$ is
the charge running in channel 1 in mode +, etc., upon insertion of a unit
charge in channel 1. From Eqs. (69)-(76), $\hat{q}$ is obtained as

$\displaystyle\hat{q}$ $\displaystyle=$
$\displaystyle\frac{1}{1-\eta_{+}\eta_{-}}\left(\begin{array}[]{cc}1&\eta_{+}\\\
-\eta_{+}\eta_{-}&-\eta_{+}\end{array}\right)$ (83) $\displaystyle=$
$\displaystyle\frac{1}{4}\left(\begin{array}[]{rr}\left[\sqrt{3(\Delta+1)}-\sqrt{\Delta-1}\,\right]^{2}&\,\,\,-\left[\sqrt{3(\Delta+1)}-\sqrt{\Delta-1}\,\right]\left[\sqrt{(\Delta+1)/3}-\sqrt{\Delta-1}\,\right]\\\
-\left[\sqrt{\Delta+1}-\sqrt{3(\Delta-1)}\,\right]^{2}&\,\,\,\left[\sqrt{3(\Delta+1)}-\sqrt{\Delta-1}\,\right]\left[\sqrt{(\Delta+1)/3}-\sqrt{\Delta-1}\,\right]\end{array}\,\right)~{}.$
(86)

The unit local charge added to channel 1 thus splits, for $\Delta\neq 2$, into
two parts running in opposite directions and creates a growing dipole in
channel 1/3. The fractional charges obey charge conservation with
$q_{+,1}+q_{-,1}=1$ and $q_{+,2}+q_{-,2}=0$. Similarly, when a unit local
charge is inserted into channel 1/3, the fractionalization matrix
$\hat{\bar{q}}$ (where the bar is put to distinguish it from the
fractionalization matrix for insertion into channel 1) reads

$\displaystyle\hat{\bar{q}}$ $\displaystyle=$
$\displaystyle\frac{1}{1-\eta_{+}\eta_{-}}\left(\begin{array}[]{cc}-\eta_{-}&-\eta_{+}\eta_{-}\\\
\eta_{-}&1\end{array}\right)$ (89) $\displaystyle=$
$\displaystyle\frac{3}{4}\left(\begin{array}[]{rr}\left[\sqrt{\Delta+1}-\sqrt{3(\Delta-1)}\,\right]\left[\sqrt{\Delta+1}-\sqrt{(\Delta-1)/3}\,\right]&\,\,\,-\left[\sqrt{(\Delta+1)/3}-\sqrt{\Delta-1}\,\right]^{2}\\\
-\left[\sqrt{\Delta+1}-\sqrt{3(\Delta-1)}\,\right]\left[\sqrt{\Delta+1}-\sqrt{(\Delta-1)/3}\,\right]&\,\,\,\left[\sqrt{\Delta+1}-\sqrt{(\Delta-1)/3}\,\right]^{2}\end{array}\,\right)~{},$
(92)

with $\bar{q}_{+,1}+\bar{q}_{-,1}=0$ and $\bar{q}_{+,2}+\bar{q}_{-,2}=1$. Note
also the relation $q_{\pm,1}=\bar{q}_{\mp,2}$.

For $\Delta=1$, the matrix $\hat{q}$ takes a simple form, as discussed around
Eqs. (79) and (80):

$\hat{q}=\frac{1}{2}\left(\begin{array}[]{rr}3&-1\\\
-1&1\end{array}\right)~{},$ (93)

whereas $\hat{\bar{q}}$ for $\Delta=1$ is given by

$\hat{\bar{q}}=\frac{1}{2}\left(\begin{array}[]{rr}3&-1\\\
-3&3\end{array}\right)~{}.$ (94)

In Eqs. (93) and (94), perfect screening in the neutral mode is apparent in
that the sum of the matrix elements in the lower row is zero in both cases.
Note that the amplitude of the charge mode excited by adding a unit charge at
$\Delta=1$ to channel 1 is the same as by adding it to channel 1/3. By
contrast, the amplitude of the neutral mode is a factor of 3 larger in the
latter case.

The phenomenon of charge fractionalization has two facets, inherently related
to each other. Specifically, at the single-particle level, it refers to the
factorization of a single-particle propagator in the space-time representation
into a product of parts moving with different velocities (with proper care
taken in dealing with the ultraviolet cutoff [89, *Voit1995]), as is the case,
e.g., for electrons in a LL. At the level of two-particle correlations, this
results in splitting of a compact density pulse into parts characterized by
different velocities of propagation. The above picture of fractionalization in
the $\nu=2/3$ edge is a generalization—formalized in purely electrostatic
terms—of the fractionalization picture in a LL [91, 92, 93, 94] to the case of
two nonequivalent counterpropagating channels. Here, “nonequivalent
counterpropagating” signifies that two channels together constitute a chiral
system—the one characterized by $g_{+}\neq g_{-}$.

It is worth mentioning that, in the original sense, the notion of charge
fractionalization in a two-channel (spinless) LL refers to splitting of the
density pulse into two chiral parts [91, 92, 93] (in a spinful LL, the
“chiral” separation is complemented with spin-charge separation). In Eqs.
(86)-(94), the charge splits into four parts (spatially separated both in the
longitudinal and transverse directions), with fractionalization “resolved”
with respect to not only chirality but also channels. This is similar to the
charge-fractionalization picture in the single-channel edge for $\nu=1$, when
counterpropagating parts of the edge are brought in proximity to each other
and form together a nonchiral system akin to the LL [64, 94, *Horsdal2011, 96,
97], or in two copropagating channels in the edge for $\nu=2$ [64, 98, 99,
100, 101, 102, 54]. In the latter case, the fractionalized parts of the
density pulse run in the same direction, with the decoupling of the charge and
neutral modes being equivalent to spin-charge separation. For $\nu=2/3$,
charge fractionalization was probed experimentally [54] through time-resolved
scattering of current pulses off the interface between regions with zero and
nonzero strength of interchannel interaction (cf. the time-resolved
measurements of this type of scattering in an artificial LL [96, 97] and in
the two-channel edge for $\nu=2$ [54]).

###  III.2 Fractionalization upon intermode tunneling

Imagine now that a compact unit charge in channel 1 is incident on a tunneling
link in mode + (Fig. 7). As follows from Sec. III.1, this charge is
accompanied by the screening charge $\eta_{+}$ in channel 1/3. Either of the
two can tunnel upon hitting the tunneling link. Scattering of the right-moving
composite object consisting of the charges 1 and $\eta_{+}$ is thus a
combination of four processes: (i) tunneling of the charge 1 to channel 1/3
and its ensuing fractionalization are accompanied by (ii) fractionalization of
the charge $-1$ left in channel 1 and, similarly, (iii) tunneling of the
charge $\eta_{+}$ to channel 1 and its fractionalization are accompanied by
(iv) fractionalization of the charge $-\eta_{+}$ left in channel 1/3.

Figure 7: Fractionalization-renormalized tunneling. Scattering of a charge
pulse incident on a tunneling link (marked in red) from the left (a) and from
the right (b). Electron tunneling occurs between the right-moving channel 1
and left-moving channel 1/3. The relative amplitudes of the components of the
incoming and outgoing pulses in channels 1 and 1/3 are shown for $\Delta=3/2$
(which corresponds to the screening coefficients $\eta_{+}\simeq-0.1$ and
$\eta_{-}\simeq-0.3$). The outgoing pulses are shown for $R_{0}=3/2$ [Eq.
(113)] (which corresponds for $\Delta=3/2$ to the reflection coefficients
$R_{+}=1/4$ and $R_{-}=5/4$), with their scattering amplitudes given by Eqs.
(111) and (112).

Specifically, to describe fractionalization in the process of tunneling, it is
instructive to first consider the matrix $\hat{Q}$ with elements
$Q_{\pm,\textrm{i}}$, where $Q_{+,1}$ is the charge running in channel 1 in
mode +, etc. [cf. Eqs. (86) and (92)]. Scattering of the charge pulse in mode
+ is then formalized by

$\hat{Q}_{\textrm{out}}-\hat{Q}_{\textrm{in}}=\left(W_{1}^{+}-\eta_{+}W_{2}^{+}\right)\left(\hat{\bar{q}}-\hat{q}\right)~{},$
(95)

where

$\hat{Q}_{\textrm{in}}=\left(\begin{array}[]{cc}1&\eta_{+}\\\
0&0\end{array}\right)$ (96)

is the matrix of incoming right-moving charges, $\hat{Q}_{\textrm{out}}$ is
the matrix of outgoing charges, and $W_{1}^{+}$ and $W_{2}^{+}$ are the
scattering coefficients in mode + for tunneling from channels 1 and 1/3,
respectively. The matrix of scattered charges, proportional to the combination
$\hat{\bar{q}}-\hat{q}$, contains four terms, each of which describes one of
the four processes mentioned above. Namely, the part proportional to
$\hat{\bar{q}}$ stands for creation (the term with $W_{1}^{+}$) of the charge
1 and annihilation ($W_{2}^{+}$) of the charge $\eta_{+}$ in channel 1/3,
while the part proportional to $\hat{q}$ denotes annihilation ($W_{1}^{+}$) of
the charge 1 and creation ($W_{2}^{+}$) of the charge $\eta_{+}$ in channel 1.
For scattering of a charge pulse in mode $-$, with scattering coefficients
$W^{-}_{1}$ and $W^{-}_{2}$ for tunneling from channels 1 and 1/3 in mode $-$,
respectively, the analog of Eq. (95) is given by

$\hat{Q}_{\textrm{out}}-\hat{Q}_{\textrm{in}}=\left(W_{2}^{-}-\eta_{-}W_{1}^{-}\right)\left(\hat{q}-\hat{\bar{q}}\right)~{},$
(97)

where $\hat{Q}_{\textrm{in}}$ is chosen as the matrix of incoming left-moving
charges:

$\hat{Q}_{\textrm{in}}=\left(\begin{array}[]{lc}0&\\!0\\\
\eta_{-}&\\!1\end{array}\right)~{}.$ (98)

The explicit form of the scattering coefficients $W^{\pm}_{1,2}$ will be
discussed below [Eqs. (106) and (107)].

Having described the mechanism of fractionalization in the process of
tunneling in a detailed and pictorial way by means of Eqs. (95) and (97), we
now combine them in the form of scattering theory for an arbitrary two-vector
of the eigenmode charges $\textbf{Q}=(Q_{+},Q_{-})^{T}$, where
$Q_{+}=Q_{+,1}+Q_{+,2}=Q_{+,1}(1+\eta_{+})$ and
$Q_{-}=Q_{-,1}+Q_{-,2}=Q_{-,2}(1+\eta_{-})$. Specifically, the relation
between the vectors of incoming ($\textbf{Q}_{\textrm{in}}$) and outgoing
($\textbf{Q}_{\textrm{out}}$) charges is written as

$\textbf{Q}_{\textrm{out}}=\left(\mathbb{1}+\hat{\mathcal{T}}\right)\textbf{Q}_{\textrm{in}}~{},$
(99)

where the $T$ matrix for scattering at a single tunneling link reads

$\hat{\mathcal{T}}=\left(\begin{array}[]{rr}-R_{+}&R_{-}\\\
R_{+}&-R_{-}\end{array}\right)~{},$ (100)

with $R_{\pm}$ being the reflection coefficients for the charge pulses in
modes $\pm$. The matrix structure of Eq. (100) signifies charge conservation
in the tunneling process for arbitrary screening charges. From Eqs. (95) and
(97),

$\displaystyle
R_{+}=(q_{+,1}-\bar{q}_{+,1})\left(W_{1}^{+}-\eta_{+}W_{2}^{+}\right)~{},$
(101) $\displaystyle
R_{-}=(\bar{q}_{-,2}-q_{-,2})\left(W_{2}^{-}-\eta_{-}W_{1}^{-}\right)~{},$
(102)

where [Eqs. (86) and (92)]

$\displaystyle
q_{+,1}-\bar{q}_{+,1}=\frac{\sqrt{\Delta-1}}{2}\left[\sqrt{3(\Delta+1)}-\sqrt{\Delta-1}\,\right]~{},$
(103)
$\displaystyle\bar{q}_{-,2}-q_{-,2}=\frac{\sqrt{\Delta+1}}{2\sqrt{3}}\left[\sqrt{3(\Delta+1)}-\sqrt{\Delta-1}\,\right]~{}.$
(104)

Note that the matrix (100), which relates the compact charges, i.e., the time
integrals of the current pulses, is also understood, by extension, as the
current-splitting matrix that relates the incoming and outgoing eigenmode
currents $j_{\pm}$ on the sides of the tunneling link in a stationary state by

$\left(\begin{array}[]{r}j_{+}\\\
-j_{-}\end{array}\right)_{\textrm{out}}=\left(\mathbb{1}+\hat{\mathcal{T}}\right)\left(\begin{array}[]{r}j_{+}\\\
-j_{-}\end{array}\right)_{\textrm{in}}~{}.$ (105)

In the stationary case, the currents $j_{\pm}$ are spatially homogeneous on
the sides of the link and experience a jump at it, as given by Eq. (105), with
the sum $j_{+}+j_{-}$ being continuous across the link. This is how the
fractionalization physics, encoded in the interaction-renormalized matrix
$\hat{\mathcal{T}}$ through Eqs. (101) and (102), affects the time-averaged
current.

Now turn to the scattering coefficients $W^{\pm}_{1,2}$. Let, for
concreteness, the Hamiltonian for a single tunneling link be of the form
${\mathcal{H}}_{t}=t_{0}\psi_{1}^{\dagger}\psi_{2}+\textrm{h.c.}$ In the Born
approximation with respect to tunneling, $W^{\pm}_{1,2}$ read

$\displaystyle
W^{\pm}_{1}=2\pi|\tilde{t}_{0}|^{2}\frac{1}{v_{\pm}}\\!\left(\bar{\Pi}_{22}+\bar{\Pi}_{21}\right)~{},$
(106) $\displaystyle
W^{\pm}_{2}=2\pi|\tilde{t}_{0}|^{2}\frac{1}{v_{\pm}}\\!\left(\bar{\Pi}_{11}+\bar{\Pi}_{12}\right)~{},$
(107)

where $\hat{\bar{\Pi}}$ is the compressibility matrix already introduced by
defining the local chemical potentials in Eqs. (34) and (35), and the tilde in
$\tilde{t}_{0}$ denotes the renormalization of $t_{0}$ by the interaction-
mediated orthogonality catastrophe [63, 87]. In the limit of no interchannel
interaction, $W^{+}_{1}=W^{-}_{2}/3=|\tilde{t}_{0}|^{2}/3v_{1}v_{2}$ [these
two coefficients are then the only ones that enter Eqs. (101) and (102), with
$R_{+}=W^{+}_{1}$ and $R_{-}=W^{-}_{2}$].

The factor $1/v_{\pm}$ in Eqs. (106) and (107) comes from the amplitude of the
incident current, while the factors in the brackets are the total
thermodynamic compressibilities for placing a charge in channel 1/3
($W^{\pm}_{1}$) and 1 ($W^{\pm}_{2}$). Substituting Eq. (34) in Eqs. (106) and
(107), and the resulting expressions for $W^{\pm}_{1,2}$ in Eqs. (101) and
(102), taking account of Eqs. (103) and (104), the reflection coefficients
$R_{\pm}$ reduce to

$R_{\pm}=|\tilde{t}_{0}|^{2}\frac{1}{3v_{+}v_{-}}(\Delta\mp 1)~{}.$ (108)

Equation (108) is also representable in the form

$R_{\pm}=|\tilde{t}_{0}|^{2}\frac{g_{\mp}}{v_{+}v_{-}}=2\pi|\tilde{t}_{0}|^{2}\frac{1}{v_{\pm}}\Pi_{\mp\mp}~{},$
(109)

where $\Pi_{++}=g_{+}/2\pi v_{+}$ and $\Pi_{--}=g_{-}/2\pi v_{-}$ are the
elements of the eigenmode compressibility matrix (21). Note that $R_{+}$ and
$R_{-}$ are universally related to each other:
$R_{+}/R_{-}=(\Delta-1)/(\Delta+1)$, with this universality being entirely due
to fractionalization upon tunneling.

It is worth emphasizing that two types of interaction-induced renormalization
of $R_{\pm}$, one by fractionalization and the other by the orthogonality
catastrophe, are sharply distinguished in Eq. (109). Specifically, the former
is inherently linked to the behavior of the thermodynamic compressibility
$\Pi_{\mp\mp}$ with varying $\Delta$, whereas the latter is encoded in the
modification of $|\tilde{t}_{0}|^{2}$ by a factor that represents the
difference between the thermodynamic compressibility and the single-particle
tunneling density of states. While the simple form of Eq. (109) is suggestive,
it is the “unfolding” of Eq. (109) in terms of two factors in Eqs. (101) and
(102) that reveals the mechanism of fractionalization-renormalized tunneling
that is behind the emergence of the factor $\Delta\mp 1$ in Eq. (108).

The tunneling conductance for a weak tunneling link connecting channels 1 and
1/3 for arbitrary $\Delta$, which conforms with Eq. (108), was also obtained
by a direct perturbation theory for the tunneling current in Ref. 74. It is
worth noting that, for weak tunneling, the factor $|\tilde{t}_{0}|^{2}/3$ in
Eq. (108) reduces exactly to $|t_{0}|^{2}$ renormalized by the orthogonality
catastrophe upon tunneling in a nonchiral LL (assuming the same ultraviolet
cutoff) with a substitution of $\Delta$ for the Luttinger constant
(supplemented with a straightforward substitution of $v_{+}$ and $v_{-}$ for
the plasmon velocity in the right- and left-moving parts of the tunneling
operator, respectively). In particular, for a given infrared cutoff of the
renormalization, say, the temperature $T$, the factor $|\tilde{t}_{0}|^{2}$
for a weak tunneling link scales with $T$ as $T^{2(\Delta-1)}$. What is,
however, especially revealing in the structuring of $R_{\pm}$ into the product
of $|\tilde{t}_{0}|^{2}$ and the fractionalization-induced factor is that the
former does not show any peculiar behavior near $\Delta=1$ (apart from the
vanishing of the scaling exponent in the power-law renormalization, in exact
correspondence with the noninteracting case in a LL), in contrast to the
latter, as we discuss next.

One significant feature of the dependence of $R_{\pm}$ on $\Delta$ is the
vanishing of $R_{+}$ at $\Delta=1$. The insensitivity of the + mode to
disorder for $\Delta=1$ is a manifestation of the charge and neutral mode
decoupling [69]. Equations (101), (103), and (106) show precisely how the
decoupling occurs from the point of view of the underlying physics.
Specifically, the factor $\Delta-1$ in Eq. (108) is a product of two factors
$\sqrt{\Delta-1}$ of distinctly different origin. The factor $\sqrt{\Delta-1}$
in Eq. (103) reflects partial cancellation, to the right of the tunneling
link, of the screening charge ($\bar{q}_{+,1}$) for a fractionalized electron
that has tunneled to channel 1/3 and the charge of the fractionalized hole in
channel 1 ($-q_{+,1}$) that the electron left behind. At $\Delta=1$, the
cancellation is exact and, as a result, no mode $-$ is created upon mode +
hitting the tunneling link, so that mode + passes through the link without
distortion. The other factor $\sqrt{\Delta-1}$ stems from partial cancellation
of the tunneling current of the charge transferred from channel 1 to channel
1/3 ($W^{+}_{1}$) by the backflow tunneling current of the screening charge
from channel 1/3 to channel 1 ($-\eta_{+}W^{+}_{2}$). Again, at $\Delta=1$,
the cancellation is exact.

From Eqs. (97), (102), and (104), the $-$ mode for $\Delta=1$ is scattered off
a tunneling link according to

$\hat{Q}_{\textrm{out}}=(1-R_{-})\left(\begin{array}[]{rr}0&0\\\
-1&1\end{array}\right)~{},$ (110)

where the matrix on the right hand side stands for the incident $-$ mode [Eq.
(98)] and the term proportional to $R_{-}$ corresponds to flipping the
neutral-mode dipole, as was already discussed in Sec. I.2.3. Scattering in Eq.
(110) is purely in the forward direction. For $\Delta\neq 1$, the forward
scattering of the pulse in mode $-$, which is no longer neutral, is
accompanied by “firing” off a charge pulse in the opposite direction, the
amplitude of which scales for $\Delta\to 1$ as
$1+\eta_{-}\propto\sqrt{\Delta-1}$:

$\hat{Q}_{\textrm{out}}=(1-R_{-})\left(\begin{array}[]{cc}0&0\\\
\eta_{-}&1\end{array}\right)+R_{-}\frac{1+\eta_{-}}{1+\eta_{+}}\left(\begin{array}[]{cc}1&\eta_{+}\\\
0&0\end{array}\right)~{}.$ (111)

Conversely, for $\Delta\neq 1$, the charge pulse in mode + emits, upon hitting
the tunneling link, a charge pulse in the opposite direction:

$\hat{Q}_{\textrm{out}}=(1-R_{+})\left(\begin{array}[]{cc}1&\eta_{+}\\\
0&0\end{array}\right)+R_{+}\frac{1+\eta_{+}}{1+\eta_{-}}\left(\begin{array}[]{cc}0&0\\\
\eta_{-}&1\end{array}\right)~{}.$ (112)

The amplitude of the backscattered charge pulse in Eq. (112) scales for
$\Delta\to 1$ as $R_{+}\propto\Delta-1$.

A rather nontrivial point about tunneling between interacting channels is that
the scattering coefficients $W^{\pm}_{1,2}$ in Eqs. (106) and (107) are not
necessarily positive. In the absence of interchannel interaction, they are
positively defined. However, as an indication of the highly collective nature
of tunneling in the presence of interaction, one of them—but not both—may
become negative if interaction is strong enough, reflecting the emergence of a
negative value for one of the partial thermodynamic compressibilities [the
factors in the brackets in Eqs. (106) and (107)]. As we will see shortly, in
Sec. IV.1, this has ramifications for the scattering rates (per unit time)
which describe tunneling from channel 1 to channel 1/3 and vice versa, with
one of them becoming negative for sufficiently large $v_{12}$.

###  III.3 Strong-tunneling limit

The difference of the reflection coefficients for the charge incident from the
left and from the right ($R_{+}\neq R_{-}$) is a direct consequence of the
inequality $g_{+}\neq g_{-}$ [Eq. (109)], which precisely encodes the property
of the two-channel edge being a chiral system. The difference shows up also in
the maximum value the reflection coefficient can reach with increasing
strength of tunneling. Going beyond the Born approximation for tunneling [Eqs.
(106) and (107)] results in a substitution of $|\tilde{t}_{0}|^{2}/v_{+}v_{-}$
in Eqs. (108) and (109) by a certain number $R_{0}$,

$R_{\pm}=R_{0}\,\frac{\Delta\mp 1}{3}~{},$ (113)

where $R_{0}$ is bounded from above by conservation of energy upon tunneling.

In terms of the matrix (100), the thermodynamic constraint on the current
scattering matrix with no interchannel interaction in the incoming and
outgoing currents ($\Delta=2$ outside the scattering region) [68, 103, 71, 66]
corresponds to

$R_{+}\leq 1/2~{},\quad R_{-}\leq 3/2~{},$ (114)

with $R_{+}/R_{-}=1/3$. The condition (114), when applied to a tunneling link
between noninteracting channels 1 and 1/3 ($R_{0}\leq 3/2$), says that mode
1/3 can be fully reflected ($R_{-}=1$ is allowed), whereas mode 1 cannot
($R_{+}\leq 1/2$). The explicit expression for the tunneling conductance
$G_{t}$ between channels 1 and 1/3 for $\Delta=2$ from Ref. 58, when taken in
the strong-tunneling limit, gives $G_{t}=1/2$, which complies with the maximum
value of $R_{0}$ equal to $3/2$ for $\Delta=2$. This upper bound on $R_{0}$
follows directly from the duality between weak and strong tunneling [58, 104].

The latter inequality in Eq. (114) allows for $R_{-}$ to be larger than 1. In
the interval $1<R_{-}\leq 3/2$ (or, equivalently, when represented in terms of
the conductance matrix [66], for the negative value of the diagonal, for
channel 1/3, element of it), the system behaves in a nontrivial way. Namely,
in the two-terminal setting, it can perform as a step-up transformer [68, 103]
(in particular, in the form of an “adiabatic junction” [103]) if the source
for channel 1 has a higher electrochemical potential. Or, if the source for
channel 1/3 is biased in this way, the charge current in channel 1/3 can be
“sucked in” from—not absorbed by—the nominally drain reservoir for channel 1/3
(“nominally” means that the drain reservoir is at a lower electrochemical
potential than the source reservoir), so that both reservoirs simultaneously
supply current to channel 1/3. The suck-in effect in tunneling ($R_{-}>1$) is
phenomenologically similar to Andreev reflection [104].

The negative values of $W^{\pm}_{1,2}$, mentioned at the end of Sec. III.2,
and the values of $R_{-}$ in Eq. (114) larger than 1 are two startling
manifestations of the strongly correlated nature of tunneling, both of which
carry nontrivial connotations from the point of view of thermodynamics. It is
worth emphasizing, however, that they are distinctly different in nature and,
as such, not directly related to each other. The former occurs if interchannel
interaction is strong enough, irrespective of the strength of tunneling. By
contrast, the latter occurs if tunneling is strong enough, also (as is the
case discussed above) in the limit of no interchannel interaction.

In the scattering problem described by Eq. (99), in contrast to the assumption
made in Refs. 68, 58, 103, 104, 71, 66, the incoming and outgoing currents in
modes 1 and 1/3 are generically interacting with each other. The interchannel
interaction strength is supposed to be homogeneous in the vicinity of the
tunneling link. To generalize the thermodynamic constraint of the type (114)
to $\Delta\neq 2$, assume that the incoming currents are at thermal
equilibrium with $T=0$. Denote the incoming and outgoing energy currents
associated with charges in modes $\pm$ by $j^{\epsilon}_{\pm,\textrm{in}}$ and
$j^{\epsilon}_{\pm,\textrm{out}}$, respectively. These obey

$j^{\epsilon}_{+,\textrm{in}}+j^{\epsilon}_{-,\textrm{in}}\geq
j^{\epsilon}_{+,\textrm{out}}+j^{\epsilon}_{-,\textrm{out}}~{},$ (115)

where the inequality accounts for the possibility of tunneling-induced
inelastic scattering which produces chargeless excitations inside channel + or
$-$, which carry energy away from the tunneling link. The charge-related
energy currents $j^{\epsilon}_{\pm}$ and the charge currents $j_{\pm}$, both
incoming and outgoing, are related to each other by
$j^{\epsilon}_{\pm}=j_{\pm}\mu_{\pm}/2$, where $\mu_{\pm}=\pm 2\pi
j_{\pm}/g_{\pm}$ are the local chemical potentials of modes $\pm$ [cf. Eq.
(35) for the local chemical potentials in channels 1 and 1/3], i.e.,

$j^{\epsilon}_{\pm}=\pm\frac{\pi}{g_{\pm}}j_{\pm}^{2}~{}.$ (116)

The incoming and outgoing energy currents $j^{\epsilon}_{\pm,\textrm{in}}$ and
$j^{\epsilon}_{\pm,\textrm{out}}$ are related to each other by the same
current-scattering matrix as the charge currents in Eq. (105):

$\left(\begin{array}[]{r}j^{\epsilon}_{+}\\\
-j^{\epsilon}_{-}\end{array}\right)_{\textrm{out}}=\left(\mathbb{1}+\hat{\mathcal{T}}\right)\left(\begin{array}[]{r}j^{\epsilon}_{+}\\\
-j^{\epsilon}_{-}\end{array}\right)_{\textrm{in}}~{}.$ (117)

Substituting Eqs. (116) and (117) in Eq. (115), together with using Eqs. (105)
and (113), produces the constraint on $R_{\pm}$ for arbitrary $\Delta$:

$0\leq R_{\pm}\leq\frac{\Delta\mp 1}{\Delta}~{},$ (118)

with $R_{0}\geq 0$ generically attaining any value up to $3/\Delta$:

$0\leq R_{0}\leq 3/\Delta~{}.$ (119)

For $R_{0}=3/\Delta$, at which $R_{+}$ and $R_{-}$ simultaneously reach their
maximum allowed values, tunneling is “dissipationless” in the sense that the
sign between the two parts of Eq. (115) becomes “equals,” i.e., no chargeless
excitations are created upon tunneling. Note that the maximum values of
$R_{+}$ and $R_{-}$ behave differently as $\Delta$ varies from $\Delta=2$ to
$\Delta=1$: the former decreases (down to zero), while the latter grows. Note
also that $R_{+}$ is always smaller than 1, in contrast to $R_{-}$.

For $\Delta=1$, the maximum value for $R_{-}$, according to Eq. (118), is 2.
From Eq. (110), it follows that the neutral mode incident on the tunneling
link with $R_{-}=2$ deterministically changes sign upon passing through it:

$\hat{Q}_{\textrm{out}}=\left(\begin{array}[]{rr}0&0\\\
1&-1\end{array}\right)=-\hat{Q}_{\textrm{in}}~{}.$ (120)

The strong-tunneling limit for $\Delta=1$ thus corresponds to flipping the
neutral-mode dipole with probability 1. The conductance $G$ in this limit only
depends on $N_{u}$ and $N_{l}$, where $N_{u,l}$ is the number of the tunneling
links in the upper ($u$) and lower ($l$) parts of the edge [Fig. 3(a)].
Specifically, $G$ can only take one of three values. If both $N_{u}$ and
$N_{l}$ are even, tunneling does not affect either the charge mode or the
neutral mode, so that $G$ is the same as for a clean edge [Eq. (65)]. If
$N_{u}$ is odd, the current $j_{u}$ in the upper part of the edge is obtained
from Eqs. (29) and (30) by imposing the condition that the ratio of $j_{-}$ at
the right terminal and $j_{-}$ at the left terminal is $-1$ (odd number of the
dipole flips), which yields [105]

$j_{u}=\frac{1}{2\pi\Delta_{c}^{\textrm{eff}}}(g_{+c}\mu_{L}+g_{-c}\mu_{R})~{},$
(121)

instead of Eq. (38). If $N_{l}$ is also odd, the current $j_{l}$ in the lower
part of the edge follows from Eq. (121) by exchanging
$\mu_{L}\leftrightarrow\mu_{R}$; otherwise, this exchange should be performed
on Eq. (38). The three values of the conductance, $G_{e-e}$, $G_{o-o}$, and
$G_{e-o}$, corresponding to the even-even, odd-odd, and even-odd
configurations, respectively, are then given by

$\displaystyle G_{e-e}=\frac{2\Delta_{c}^{\textrm{eff}}}{3}~{},\quad
G_{o-o}=\frac{2}{3\Delta_{c}^{\textrm{eff}}}~{},$ $\displaystyle
G_{e-o}=\frac{1}{3}\left(\Delta_{c}^{\textrm{eff}}+\frac{1}{\Delta_{c}^{\textrm{eff}}}\right)~{}.$
(122)

Note that, for $\Delta_{c}^{\textrm{eff}}=2$, the maximum and minimum values
of $G$ from Eq. (122) are 4/3 and 1/3—the same as mentioned above in
connection with the $\chi$sG model with weak disorder [79, 65, 66] (Secs.
I.2.2 and I.2.3). It is also worth mentioning that scattering for $R_{-}=1$
corresponds to flipping half of the dipole current, which, on average,
annihilates the neutral mode.

##  IV Disordered edge

Having considered the contacts and the clean edge in Sec. II, and scattering
at a single tunneling link in Sec. III, we are now prepared to discuss
transport through a disordered edge. As was already mentioned in the
introductory overview, we restrict our attention here to incoherent transport
(Sec. I.2.3). Between the contacts, the equation of motion for the disorder-
averaged densities $n_{\pm}$ in the presence of random tunneling between
channels 1 and 1/3 then reads

$(\partial_{t}\pm v_{\pm}\partial_{x})n_{\pm}\pm I_{t}=0~{},$ (123)

with the “collision” term

$I_{t}=\Gamma_{+}n_{+}-\Gamma_{-}n_{-}$ (124)

and the scattering rates

$\Gamma_{\pm}=n_{t}v_{\pm}R_{\pm}~{},$ (125)

where $n_{t}$ is the concentration of the tunneling links. The reflection
coefficients $R_{\pm}$ are given by Eq. (108) for weak tunneling through a
single tunneling link, or by Eq. (113) otherwise. Below, we focus on the
Gaussian limit of $n_{t}\to\infty$ and $t_{0}\to 0$ with $n_{t}|t_{0}|^{2}$
held constant, where $n_{\pm}$ in Eq. (123) are the exact density
distributions with vanishing mesoscopic fluctuations.

###  IV.1 Negative scattering rates

Before proceeding to the solution of Eq. (123), let us return to the
conceptually significant point, mentioned at the very end of Sec. III.2, about
the signs of the scattering rates for tunneling between channels 1 and 1/3.
Recall also that this peculiarity is substantially different from another
nontrivial, from the point of view of thermodynamics, point, namely the one
about $R_{-}>1$, as commented in Sec. III.3. The equation of motion for the
densities $n_{1,2}$ is obtained from Eq. (123) by using the transformation
(68):

$\displaystyle\partial_{t}n_{1}+\partial_{x}(v_{1}n_{1}+v_{12}n_{2})+\bar{I}_{t}=0~{},$
(126)
$\displaystyle\partial_{t}n_{2}-\partial_{x}\left(v_{2}n_{2}+\frac{1}{3}v_{12}n_{1}\right)-\bar{I}_{t}=0~{},$
(127)

where the collision term

$\bar{I}_{t}=\Gamma_{1}n_{1}-\Gamma_{2}n_{2}$ (128)

is proportional to that in Eq. (123). Specifically,

$\bar{I}_{t}=\sqrt{\frac{3}{\Delta^{2}-1}}\,I_{t}~{},$ (129)

and $\Gamma_{1,2}$ are written as

$\Gamma_{1}=\frac{1}{3}\gamma_{0}(v_{1}-v_{12})~{},\quad\\!\\!\Gamma_{2}=\frac{1}{3}\gamma_{0}(3v_{2}-v_{12})~{},$
(130)

where

$\gamma_{0}=\frac{n_{t}|\tilde{t}_{0}|^{2}}{v_{+}v_{-}}$ (131)

(with $v_{+}v_{-}=v_{1}v_{2}-v_{12}^{2}/3$). Note that $\bar{I}_{t}$ does not
describe the total-current relaxation (which distinguishes the chiral edge
from a LL):

$\bar{I}_{t}=\frac{1}{3}\gamma_{0}(j_{1}+3j_{2})$ (132)

[cf. Eqs. (43) and (44)].

Remarkably, the scattering rates $\Gamma_{1,2}$ need not be both positive
within the stability range of $v_{12}$ from Eq. (10), in contrast to
$\Gamma_{\pm}$. In particular, for $\Delta=1$, i.e., for

$v_{12}=\frac{3}{4}(v_{1}+v_{2})~{},$ (133)

the ratio $\Gamma_{1}/\Gamma_{2}$ is universal (independent of $v_{1,2}$) and
given by

$\frac{\Gamma_{1}}{\Gamma_{2}}=-\frac{1}{3}~{}.$ (134)

Since $\Gamma_{1}/\Gamma_{2}$ is positive at $\Delta=2$ and negative at
$\Delta=1$, it changes sign in between, with, as follows from Eq. (130),
either $\Gamma_{1}=0$ (for $v_{1}<3v_{2}$) or $\Gamma_{2}=0$ (for
$v_{1}>3v_{2}$) at that point. If the point $\Delta=1$ is reachable with
increasing $v_{12}$ [the condition for this to happen is Eq. (11)], it is
$\Gamma_{1}$ that changes sign at $v_{12}=v_{1}$. At the stability threshold
[Eq. (10)], $\Gamma_{1}/\Gamma_{2}=-\sqrt{v_{1}/3v_{2}}<0$.

The negative value of one of the scattering rates $\Gamma_{1,2}$ does not
spell instability, or contradict causality for that matter, because causality
only requires that

$\Gamma=\Gamma_{1}+\Gamma_{2}=\Gamma_{+}+\Gamma_{-}\geq 0~{}.$ (135)

This follows from the characteristic equation for the dynamics of $n_{1,2}$,
or $n_{\pm}$,

$(\omega-v_{+}q)(\omega+v_{-}q)+i\omega\Gamma-
iq(\Gamma_{-}v_{+}-\Gamma_{+}v_{-})=0~{},$ (136)

where the linear-in-$\omega$ term contains $\Gamma_{1,2}$ in the combination
$\Gamma=\Gamma_{1}+\Gamma_{2}>0$ [the stability condition (10) is stricter
than $v_{12}<(v_{1}+3v_{2})/2$, which guarantees the positive sign of $\Gamma$
within the range of $v_{12}$ from Eq. (10)].

It is worth noting that the ratio of $\Gamma_{1}/\Gamma_{2}$ (or
$\Gamma_{+}/\Gamma_{-}$ for that matter) is obtainable on rather general
grounds. In Sec. III, we presented a microscopic picture of fractionalization-
renormalized tunneling that produced both the explicit dependence of
$R_{+}/R_{-}$ on $\Delta$ [Eq. (108)] and the expression of $R_{\pm}$ in terms
of the thermodynamic compressibility matrix [Eq. (109)]. Alternatively, one
might have relied on gauge invariance in “constructing” the collision terms in
phenomenological equations of motion for the densities [67]. Specifically, the
collision terms should remain intact upon shifting the densities

$\bar{\textbf{n}}=\left(\begin{array}[]{l}n_{1}\\\
n_{2}\end{array}\right)\to\bar{\textbf{n}}+\hat{\bar{\Pi}}\left(\begin{array}[]{l}1\\\
1\end{array}\right)\delta\mu$ (137)

for arbitrary $\delta\mu$ [67], or similarly for the densities in any other
basis. Stating that $\hat{\bar{\Pi}}(1,1)^{T}$ is the zero mode of the
collision operator (whose structure in terms of $\Gamma_{1,2}$ follows from
conservation of the total charge),

$\left(\begin{array}[]{rr}\Gamma_{1}&-\Gamma_{2}\\\
-\Gamma_{1}&\Gamma_{2}\end{array}\right)\hat{\bar{\Pi}}\left(\begin{array}[]{l}1\\\
1\end{array}\right)=0~{},$ (138)

Eq. (138) fixes the ratio $\Gamma_{1}/\Gamma_{2}$ in terms of the elements of
the compressibility matrix [cf. Eqs. (34) and (130)].

Importantly, $\hat{\bar{\Pi}}$ in Eq. (138) is the thermodynamic
compressibility, not the single-particle tunneling density of states affected
by the orthogonality catastrophe. It is this distinction, as was already
mentioned in Sec. III.2, that makes the concept of fractionalization-
renormalized tunneling precise, by cleanly separating it from that of the
renormalization of tunneling by the orthogonality catastrophe.

###  IV.2 Tunneling-modified eigenmodes

As an essential part of the physical picture, let us discuss how tunneling
affects dynamics of the edge eigenmodes. It is the last (linear in $q$) term
in Eq. (136) that encodes the property of the edge being chiral. Specifically,
from Eq. (125), the combination of $\Gamma_{+}$ and $\Gamma_{-}$ within the
brackets can be represented as

$\Gamma_{-}v_{+}-\Gamma_{+}v_{-}=n_{t}v_{+}v_{-}(R_{-}-R_{+})~{},$ (139)

which is nonzero only if the reflection coefficients for the currents incident
on the tunneling link from the left and from the right are not the same [for
the $\nu=2/3$ edge, it is also representable as
$(2/3)n_{t}|\tilde{t}_{0}|^{2}$].

In the limit of small $q$ (specifically, for
$|q|\ll\Gamma/\max\\{v_{1},v_{2}\\}$ for any $\Delta$ between $\Delta=2$ and
$\Delta=1$), the solution of Eq. (136) gives two low-energy eigenmodes of the
edge in the presence of random tunneling. One is a weakly damped (if
$\Delta\neq 1$) ballistic mode with

$\omega=\omega_{\rho}\to v_{\rho}q-iD_{b}q^{2}~{},$ (140)

where the velocity of charge propagation (hence the subscript $\rho$)
$v_{\rho}$ is given by

$v_{\rho}=\frac{\Gamma_{-}v_{+}-\Gamma_{+}v_{-}}{\Gamma}=\frac{2}{3}\,\frac{3v_{1}v_{2}-v_{12}^{2}}{v_{1}+3v_{2}-2v_{12}}~{}.$
(141)

This mode is propagating ballistically only because of $R_{+}\neq R_{-}$ [Eq.
(139)]. The direction of propagation coincides with that for the + mode. Note
that, unless $\Delta=1$, the velocity of the right-moving eigenmode in the
small-$q$ limit is modified (ballistic propagation is slowed down) by
tunneling:

$v_{\rho}\leq v_{+}~{}.$ (142)

The designation “weakly damped” above Eq. (140) means a slow diffusive
broadening in real space on top of the ballistic propagation, with the
diffusion coefficient

$D_{b}=\frac{1}{\Gamma}(v_{+}-v_{\rho})(v_{-}+v_{\rho})~{}.$ (143)

Again, even when one of the scattering rates $\Gamma_{1}$ or $\Gamma_{2}$
becomes negative with increasing $v_{12}$, they enter $D_{b}\geq 0$ in the
combination $\Gamma=\Gamma_{1}+\Gamma_{2}\geq 0$. The other eigenmode in the
small-$q$ limit is gapped, with the gap determined, similarly to $D_{b}$, by
the total scattering rate:

$\omega=\omega_{n}\to-i\Gamma-v_{n}q~{},$ (144)

where $v_{n}=(\Gamma_{-}v_{-}-\Gamma_{+}v_{+})/\Gamma=v_{\rho}-v_{1}+v_{2}$
and the subscript $n$ stands for “neutral” (the meaning of which will become
clear in the next paragraph). At the stability threshold [Eq. (10)], both
$v_{\rho}$ and $D_{b}$ vanish to zero, with $\Gamma\to\infty$ [the numerator
of $D_{b}$ in Eq. (143) also vanishes].

The property of the tunneling-modified eigenmodes $\omega_{\rho}$ and
$\omega_{n}$ being propagating and localized, respectively, shows up in
splitting of a compact charge pulse into two parts: mobile (charged) and
immobile (neutral). Consider the evolution in time of
$\bar{\textbf{n}}=(n_{1},n_{2})^{T}$ with the initial condition at $t=0$
fixing a certain density profile $\bar{\textbf{n}}_{q}(t=0)$ in $q$ space.
Assume that compact charges $Q_{1,2}$ are created at the point $x=0$ at $t=0$,
i.e., $\bar{\textbf{n}}_{q}(t=0)=(Q_{1},Q_{2})^{T}$. Solution to the equation
of motion in the limit of small $|q|\ll\Gamma/\max\\{v_{1},v_{2}\\}$ gives,
for $\bar{\textbf{n}}$ in $(\omega,q)$ space:

$\displaystyle\bar{\textbf{n}}_{\omega q}\to$
$\displaystyle\,\,\frac{i}{\omega-
v_{\rho}q+iD_{b}q^{2}}\,\frac{Q_{1}+Q_{2}}{\Gamma}\left(\begin{array}[]{l}\Gamma_{2}\\\
\Gamma_{1}\end{array}\right)$ (147) $\displaystyle+$
$\displaystyle\,\,\frac{i}{\omega+i\Gamma}\,\frac{\Gamma_{1}Q_{1}-\Gamma_{2}Q_{2}}{\Gamma}\left(\begin{array}[]{r}1\\\
-1\end{array}\right)~{}.$ (150)

The physical picture that emerges from Eq. (150) is that the total charge
$Q_{1}+Q_{2}$ is all taken away by the ballistic mode. What is left behind is
a localized dipole (with opposite charges in channels 1 and 1/3) that is stuck
to the starting point $x=0$ and decays “on the spot.” More precisely, the
extinction of the dipole on distances exceeding the backscattering length from
the starting point is accompanied by a ballistic run at speed $v_{n}$ [Eq.
(144)]. Importantly, this “charge-neutral” separation occurs for arbitrary
$\Delta$. The existence of the nondecaying ballistic—in the limit of small
$q$—charge mode (140) is fundamental to the conductance quantization in the
limit of charge equilibration (Sec. I.2.1), a crossover to which from
ballistic transport is discussed in detail below, in Sec. IV.3.

Tunneling modifies the screening properties of the ballistic mode:
$n_{1}/n_{2}$ at the same spatial point in the running away diffusion-
broadened charge pulse in Eq. (150) is given by

$n_{1}/n_{2}=\Gamma_{2}/\Gamma_{1}~{},$ (151)

which nullifies $\bar{I}_{t}$, instead of Eq. (69) for the clean edge. This is
because the tunneling-renormalized ballistic mode is a combination of $n_{+}$
and $n_{-}$. Note that $n_{1}/n_{2}$ in Eq. (151) can be of either sign (Sec.
IV.1).

###  IV.3 Conductance

Now let us turn to the conductance of a disordered edge. In the dc limit, the
nonzero root $\partial_{x}\to\gamma\neq 0$ of the characteristic equation to
Eq. (123) (the other is trivially zero, because of conservation of
$j_{+}+j_{-}$) is

$\gamma=\frac{\Gamma_{-}}{v_{-}}-\frac{\Gamma_{+}}{v_{+}}=n_{t}(R_{-}-R_{+})~{},$
(152)

or, in terms of $\gamma_{0}$:

$\gamma=\frac{2}{3}\gamma_{0}~{}.$ (153)

The nonvanishing of $\gamma$ is only due to chirality of the edge ($R_{+}\neq
R_{-}$). Remarkably, the fractionalization-induced $\Delta$-dependent factors
in $R_{\pm}$ (108) cancel out in the combination $R_{-}-R_{+}$ in Eq. (152),
so that the scattering length $2\gamma^{-1}$ depends on $\Delta$ only through
the renormalization of $t_{0}$ (which, in particular, does not show any
singular behavior for $\Delta\to 1$, as was already discussed in Sec. III.2).
This can also be clearly seen from Eqs. (126) and (127), which—in view of Eq.
(132)—can be rewritten as [cf. Eq. (46)]

$\left(\partial_{x}\mathbb{1}+\hat{\upgamma}\right)\left(\begin{array}[]{c}j_{1}\\\
j_{2}\end{array}\right)=0~{},$ (154)

where the matrix $\hat{\upgamma}$ does not explicitly depend on $\Delta$:

$\hat{\upgamma}=\frac{1}{2}\,\gamma\left(\begin{array}[]{rr}1&3\\\
-1&-3\end{array}\right)~{}.$ (155)

####  IV.3.1 Identical contacts

For the line-junction contacts (Secs. II.4 and II.5), solution of Eq. (123) in
the dc limit, with the boundary conditions (50) and the current-scattering
matrix (59), gives for the currents $j_{u\pm}$ in the upper part of the edge
in Fig. 3(a)

$\displaystyle\left(\begin{array}[]{r}j_{u+}\\\ -j_{u-}\end{array}\right)$
$\displaystyle=\frac{1}{6\pi}\,\dfrac{1}{e^{\gamma
L}-\chi}\left[\,\left(\mu_{L}e^{\gamma
L}-\mu_{R}\,\chi\right)\\!\left(\begin{array}[]{r}\Delta+1\\\
\Delta-1\end{array}\right)\right.$ (160)
$\displaystyle\left.-\,(\mu_{L}-\mu_{R})e^{\gamma
x}\sqrt{(\Delta^{2}-1)\,\chi}\left(\begin{array}[]{r}1\\\
1\end{array}\right)\,\right]~{},$ (163)

where

$\chi=\dfrac{\Delta^{\textrm{eff}}_{c}-1}{\Delta^{\textrm{eff}}_{c}+1}$ (164)

and $L$ is the distance between the left and right contacts, which are,
respectively, at the chemical potentials $\mu_{L}$ and $\mu_{R}$. In Eq.
(163), both contacts are assumed to be identical and characterized by the same
effective-interaction parameter $\Delta_{c}^{\textrm{eff}}$ (66). For
$\Delta=2$ (noninteracting edge) and $\Delta^{\textrm{eff}}_{c}=2$ (ideal
contact for $\Delta=2$), Eq. (163) agrees with the expressions for the
currents from Refs. 71, 72, 76, *Spanslatt2020.

The first term in the square brackets in Eq. (163) does not depend on $x$,
whereas the second one is proportional to $e^{\gamma x}$. Note that the
currents in Eq. (163) are $x$ independent not only if $\Delta=1$, but also if
$\Delta_{c}^{\textrm{eff}}=1$. For $\gamma L\gg 1$, the first term is dominant
for almost the whole length of the edge for arbitrary $\Delta$ and
$\Delta_{c}^{\textrm{eff}}$, and describes the counterpropagating charge
currents equilibrated with each other at the chemical potential $\mu_{L}$.
Equilibration, described by the second term, occurs then in a narrow region
within the distance of the order of $\gamma^{-1}$ near the right contact
(emitting mode 1/3 at $\Delta_{c}^{\textrm{eff}}=2$). Specifically, for
$\gamma L\gg 1$,

$\displaystyle\left(\begin{array}[]{r}j_{u+}\\\
-j_{u-}\end{array}\right)\simeq\frac{1}{2\pi}\left[\,\mu_{L}\\!\left(\begin{array}[]{r}g_{+}\\\
g_{-}\end{array}\right)\right.$ (169)
$\displaystyle\quad\,\,\,\left.-\,e^{-\gamma(L-x)}\sqrt{g_{+}g_{-}\chi}\,(\mu_{L}-\mu_{R})\\!\left(\begin{array}[]{r}1\\\
1\end{array}\right)\right]~{}.$ (172)

Either of the currents $j_{u+}$ and $j_{u-}$, and either of two terms in Eq.
(163), depends on both $\Delta$ and $\Delta_{c}^{\textrm{eff}}$. Remarkably,
the total current $j_{u}=j_{u+}+j_{u-}$ depends only on
$\Delta_{c}^{\textrm{eff}}$ (apart from the dependence of $\gamma$ on
$\Delta$):

$j_{u}=\frac{1}{2\pi}\,\frac{2}{3}\,\dfrac{\,\mu_{L}-e^{-\gamma
L}\,\chi\,\mu_{R}\,}{1-e^{-\gamma L}\,\chi}~{}.$ (173)

The total current $j_{l}$ in the lower part of the edge follows from Eq. (173)
by exchanging $\mu_{L}\leftrightarrow\mu_{R}$ (assuming that the upper and
lower parts of the edge have the same length $L$ and are characterized by the
same strength of disorder). The two-terminal conductance of the edge $G=2\pi
j/(\mu_{L}-\mu_{R})$, with $j=j_{u}-j_{l}$, is then given by

$G=\frac{2}{3}\,\frac{1+e^{-\gamma L}\,\chi}{1-e^{-\gamma L}\,\chi}~{},$ (174)

as illustrated in Fig. 8. The dependence on $\Delta$ (besides that which is
encoded in $\gamma$) cancels out from $G$—as it does separately for $j_{u}$
and $j_{l}$. Recall that also the dependence on $\Delta_{c}$ cancels out for
the line-junction model of the contacts, assumed in Eq. (174), with
$\Delta_{c}^{\textrm{eff}}$ in Eq. (164) being the effective interaction
strength beneath the contacts (Secs. II.4 and II.5). For
$\Delta_{c}^{\textrm{eff}}=2$, Eq. (174) agrees with that from Ref. 74. In the
clean limit, Eq. (174) reduces to $G=2\Delta_{c}^{\textrm{eff}}/3$ [Eq. (65)].
In the limit of large $\gamma L$, the two-terminal conductance is quantized at
$G=2/3$, in accordance with the universal result $G=\nu$ for the limit of full
charge equilibration [67], as was already discussed in Sec. I.2.1. In
particular, no decoupling between the charge and neutral modes is needed for
the quantization. Equation (174) shows that the quantization holds
irrespective of not only the value of $\Delta$, but also the value of
$\Delta_{c}^{\textrm{eff}}$. The quantization is inherently related to the
existence of the ballistic mode (140), which carries over charge, as discussed
below Eq. (150).

Figure 8: Two-terminal conductance $G$ of the $\nu=2/3$ edge as a function of
$\gamma L$, where $L$ is the edge length and $2/\gamma$ is the backscattering
length, for different parameters $\Delta_{c}^{\textrm{eff}}$ [Eq. (66)]
characterizing the effective interaction strength beneath the contacts. Top to
bottom: $\Delta_{c}^{\textrm{eff}}=2$, 1.75, 1.5, 1.25, 1. The conductance
does not depend on either the interaction strength outside the contacts
($\Delta$) or the true interaction strength beneath the contacts
($\Delta_{c}$), apart from the interaction-induced renormalization of $\gamma$
and the scattering lengths defining $\Delta_{c}^{\textrm{eff}}$. In the
charge-equilibrated limit of $\gamma L\to\infty$, the conductance $G\to 2/3$
irrespective of the value of $\Delta_{c}^{\textrm{eff}}$.

It may be worth mentioning—particularly with regard to the experimental
results [52], which have demonstrated the quantization of $G\simeq 2/3$ for
large $L$ and a crossover to larger values of $G$ for smaller $L$—that varying
the control parameter of the experiment for given $L$ (primarily the magnetic
field in Ref. 52) may change not only $\gamma$ and $\Delta$, but also
$\Delta_{c}^{\textrm{eff}}$. Within the picture described by Eq. (174), the
variation of $\Delta$ may only show up indirectly, through the dependence of
$\gamma$ on $\Delta$, whereas the variation of $\Delta_{c}^{\textrm{eff}}$
manifests itself in the limiting value of $G$ for small $L$. Recall also that
the value of $\Delta_{c}^{\textrm{eff}}$ is determined by the interplay of
tunneling processes beneath the contact and may possibly be controlled in a
similar way as the tunneling length $1/\gamma$ outside the contact. In Ref.
52, $G\simeq 4/3$ was observed for small $L$, which corresponds to
$\Delta_{c}^{\textrm{eff}}\simeq 2$, for relatively large equilibration
lengths (stronger magnetic fields), while the data for smaller equilibration
lengths (weaker magnetic fields) may be compatible with the limiting value of
$G=2\Delta_{c}^{\textrm{eff}}/3$, as $L$ is decreased, substantially smaller
than 4/3.

Following the same route of cyclic permutation as for the derivation of Eqs.
(41) and (42), the four-terminal conductances $G_{H}$ (Hall) and $G_{4}$
(source-drain) [Fig. 3(b)] for the line-junction contacts (all four assumed to
be identical) in the presence of disorder are obtained as

$\displaystyle G_{H}=\frac{2}{3}\,\frac{1+e^{-\gamma
L}\chi^{2}}{\left(1-e^{-\gamma L/2}\chi\right)^{2}}~{},$ (175) $\displaystyle
G_{4}=\frac{2}{3}\,\frac{1+e^{-\gamma L}\chi^{2}}{1-e^{-\gamma
L}\chi^{2}}~{}.$ (176)

In Eqs. (175) and (176), for ease of presentation, each pair of adjacent
contacts is assumed to be separated by the same distance $L/2$. Similarly to
$G$ in Eq. (174), both $G_{H}$ and $G_{4}$ are equal to 2/3 in the limit of
large $\gamma L$, for arbitrary $\Delta$ and $\Delta_{c}^{\textrm{eff}}$,
which generalizes the result $G=\nu$ [67] to the case of four-terminal
measurements [106].

Apart from the spatial distribution of the eigenmode currents for a long edge
with $\gamma L\gg 1$ [Eq. (172)], it is worthwhile to also comment on that of
the densities in channels 1 and 1/3. For arbitrary strength of interchannel
interaction, $\bar{\textbf{n}}$ in the upper part of the edge is represented
in terms of $v_{12}$ (more compactly than in terms of $\Delta$) as

$\displaystyle\bar{\textbf{n}}$ $\displaystyle=\frac{1}{6\pi
v_{+}v_{-}(e^{\gamma L}-\chi)}\left[\,(\mu_{L}e^{\gamma
L}-\mu_{R}\chi)\left(\begin{array}[]{c}3v_{2}-v_{12}\\\
v_{1}-v_{12}\end{array}\right)\right.$ (179)
$\displaystyle\left.\qquad\,\,-\,(\mu_{L}-\mu_{R})e^{\gamma
x}\sqrt{\chi}\left(\begin{array}[]{c}v_{2}-v_{12}\\\
v_{1}-v_{12}/3\end{array}\right)\,\right]~{}.$ (182)

For $\gamma L\gg 1$, Eq. (175) reduces to

$\displaystyle\bar{\textbf{n}}$
$\displaystyle\simeq\hat{\bar{\Pi}}\left(\begin{array}[]{r}1\\\
1\end{array}\right)\mu_{L}-e^{-\gamma(L-x)}\sqrt{\chi}$ (185)
$\displaystyle\times\frac{1}{6\pi
v_{+}v_{-}}\left(\begin{array}[]{c}v_{2}-v_{12}\\\
v_{1}-v_{12}/3\end{array}\right)(\mu_{L}-\mu_{R})~{},$ (188)

where the first term represents the densities $n_{1}$ and $n_{2}$ equilibrated
with each other at the chemical potential $\mu_{L}$, and the second term
describes their equilibration in the vicinity of the right contact [cf. Eq.
(172)]. Note that for $v_{1,2}$ from the interval (11), the relative sign of
the contributions to $n_{1}$ and $n_{2}$ of the first term in Eq. (182)
changes—becomes negative—with increasing $v_{12}$ at $v_{12}=v_{1}$.
Similarly, the relative sign of the contributions of the second term changes
at $v_{12}=v_{2}$. The latter means that, for the case of strong interchannel
interaction ($v_{12}>v_{2}$), one of the densities $n_{1,2}$ goes sharply (for
$\gamma L\gg 1$) up when approaching the right contact, whereas the other goes
sharply down. The evolution of the spatial distribution of $n_{1,2}$ as the
strength of interaction increases is illustrated in Fig. 9.

Figure 9: Evolution of the distribution of the charge densities $n_{1}$ (dark
blue) and $n_{2}$ (dark red) for channels 1 and 1/3, respectively, and the
distribution of the total density $n=n_{1}+n_{2}$ (dashed violet, shown is
$n/2$) [Eq. (182)], as the strength of interchannel interaction increases. Top
to bottom: $\Delta=2$ (a), $1.5$ (b), $1$ (c). The charge densities are shown
in units of $\mu_{L}/2\pi v_{1}$ for the upper part of the edge in Fig. 3(a)
for $\mu_{R}=0$, $v_{2}=0.8v_{1}$, $\Delta^{\textrm{eff}}_{c}=1.75$, and
$\gamma L=6$.

As is seen from Eqs. (182) and (188), not only the partial densities $n_{1,2}$
but also the total density $n_{1}+n_{2}$ is inhomogeneous for a generic value
of $v_{12}$ [Figs. 9(a) and 9(b)]. The point to notice is that, for $\Delta=1$
[with $v_{12}$ from Eq. (133)], the total charge is exactly homogeneous [Fig.
9(c)], in accordance with the decoupling of the charge and neutral modes, with
the equilibration of the neutral mode showing up in a variation of the partial
densities on the scale of $\gamma^{-1}$ near the right contact. Specifically,
for $\Delta=1$, the ratio of the contribution to $n_{1}$ to the contribution
to $n_{2}$ is $-3$ in the first term of Eq. (182) and $-1$ in the second [cf.
Eq. (79)].

####  IV.3.2 Asymmetric setups

In Sec. IV.3.1, the contacts were assumed to be identical. Let us now discuss
the case of different contacts, i.e., characterized by different values of
$\Delta_{c}^{\textrm{eff}}$ (recall that, within the line-junction model, this
is the only parameter to characterize a contact). This is, arguably, useful to
make contact with experiment: in practice, the contacts may be much different
from each other, in particular, because of a different strength of disorder
beneath them, which may also be different from the strength of disorder
outside them (as was already mentioned in Sec. II.4). We will also discuss a
more conceptual issue of the contact and bulk resistances for the QH edge.

Assume, as a starting point, that the edge is clean and the left and right
contacts in the two-terminal setup [Fig. 3(a)] are characterized by
$\Delta_{cL}^{\textrm{eff}}$ and $\Delta_{cR}^{\textrm{eff}}$, respectively.
For $\Delta_{cR}^{\textrm{eff}}\neq\Delta_{cL}^{\textrm{eff}}$, Eqs. (62)–(64)
generalize to

$\displaystyle j_{1}(\delta)$
$\displaystyle=\frac{1}{2\pi}\,\frac{(1-c_{L})\mu_{L}+c_{L}(1-3c_{R})\mu_{R}}{1-3c_{L}c_{R}}~{},$
(189) $\displaystyle j_{2}(\delta)$
$\displaystyle=-\frac{1}{2\pi}\,\frac{c_{R}(1-c_{L})\mu_{L}+\frac{1}{3}(1-3c_{R})\mu_{R}}{1-3c_{L}c_{R}}~{},$
(190)

and

$j_{1,2}(-\delta)=j_{1,2}(\delta)\\!\left|{}_{\begin{subarray}{c}\mu_{L}\leftrightarrow\mu_{R}\\\
c_{L}\leftrightarrow c_{R}\end{subarray}}\right.~{},$ (191)

with $\Delta_{cL,R}^{\textrm{eff}}$ expressed in terms of $c_{L,R}$ by Eq.
(66) with a substitution of $c_{L,R}$ for $c$. The conductance $G$ then reads

$G=\frac{2}{3}\,\frac{1+\sqrt{\chi_{L}\chi_{R}}}{1-\sqrt{\chi_{L}\chi_{R}}}~{},$
(192)

where $\chi_{L,R}$ is given by Eq. (164) with $\Delta_{c}^{\textrm{eff}}$
substituted by $\Delta_{cL,R}^{\textrm{eff}}$, or, explicitly in terms of
$\Delta_{cL,R}^{\textrm{eff}}$:

$\\!\\!G=\frac{2}{3}\,\frac{\Delta_{cL}^{\textrm{eff}}+\Delta_{cR}^{\textrm{eff}}}{1+\Delta_{cL}^{\textrm{eff}}\Delta_{cR}^{\textrm{eff}}-\sqrt{\left[\left(\Delta_{cL}^{\textrm{eff}}\right)^{2}-1\right]\\!\left[\left(\Delta_{cR}^{\textrm{eff}}\right)^{2}-1\right]}}~{}.$
(193)

For $\Delta_{cR}^{\textrm{eff}}=\Delta_{cL}^{\textrm{eff}}$, Eq. (193) reduces
to Eq. (65). Note that $G=2/3$ if one of the parameters
$\Delta_{cR}^{\textrm{eff}}$ and $\Delta_{cL}^{\textrm{eff}}$ is equal to 1
(strong backscattering beneath the contact), irrespective of the value of the
other.

It is worth briefly discussing, in connection with the setup asymmetry, a
question of whether the notion of a “contact resistance” is generically
applicable to QH edges. Namely, the question is if the two-terminal resistance
in the clean case $R_{c}$—which is only nonzero because of dissipation in the
contacts—is representable as a sum of two terms, one of which, $R_{cL}$,
depends only on the properties of the left contact and the other, $R_{cR}$,
only on the properties of the right contact. For a single-channel QH edge, or
a clean LL quantum wire for that matter, $R_{c}=R_{cL}+R_{cR}$ is trivially
the case, with each of the terms being the contact resistance for one of the
contacts. As follows from Eq. (192), this is not so for the $\nu=2/3$ edge.