# Transport Signatures of Fractional Quantum Hall Binding Transitions

Christian Spånslätt<EMAIL_ADDRESS>Department of
Microtechnology and Nanoscience (MC2),Chalmers University 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  Ady Stern  Department of Condensed Matter Physics,
Weizmann Institute of Science, Rehovot 7610001, Israel  Alexander D. Mirlin
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

(August 29, 2024)

###### Abstract

Certain fractional quantum Hall edges have been predicted to undergo quantum
phase transitions which reduce the number of edge channels and at the same
time bind electrons together. However, detailed studies of experimental
signatures of such a “binding transition” remain lacking. Here, we propose
quantum transport signatures with focus on the edge at filling $\nu=9/5$. We
demonstrate theoretically that in the regime of non-equilibrated edge
transport, the bound and unbound edge phases have distinct conductance and
noise characteristics. We also show that for a quantum point contact in the
strong back-scattering regime, the bound phase produces a minimum Fano-factor
$F_{SBS}=3$ corresponding to three-electron tunneling, whereas single electron
tunneling is strongly suppressed at low energies. Together with recent
experimental developments, our results will be useful for detecting binding
transitions in the fractional quantum Hall regime.

##  I Introduction

Edges of fractional quantum Hall (FQH) states [1, 2] are outstanding platforms
for strongly correlated electron physics. A FQH edge realizes the so-called
chiral Luttinger liquid [3, 4, 5, 6, 7, 8], which is a set of one-dimensional
conducting channels inheriting topological properties of the FQH bulk state.
The chiral Luttinger liquid has been successfully used to investigate a wide
variety of fundamental quantum phenomena, e.g. topological quantization,
charge fractionalization [9, 10, 11], anyonic statistics [12, 13], topological
quantum computation [14], or quantum phase transitions [15, 16, 17].

A particularly striking FQH edge quantum phase transition, called the binding
transition, was proposed by Kao et.al. [18], based on earlier work by Haldane
[19]. In the binding transition, pairs of oppositely propagating edge channels
localize due to an edge instability triggered by inter-channel particle
tunneling and strong interactions. The remaining edge channels may then carry
excitations with electrical charges different from those of the original edge;
charges that can be viewed as bound composites of electrons.

Binding transitions are possible only for so-called T-unstable FQH states,
defined as those states permitting charge-neutral and bosonic quasiparticle
excitations lacking topological content in their correlation function [19,
17]. On the edge, the creation and annihilation operators of such excitations
describe charge tunneling between edge channels, and appear in the edge
Hamiltonian without breaking any symmetries. Physically, it is only T-unstable
edges that permit pairs of oppositely propagating channels to localize.
Equivalently, T-unstable edges have low energy charge-neutral fixed points
with an equal number of neutral modes (we use the terms “mode” and “channel”
interchangeably in this work) propagating in each direction [17]. Importantly,
the binding transition does not alter the topological order of the FQH bulk
state, and is therefore a pure edge transition, amenable for detection in edge
experiments.

Figure 1: The FQH binding transition at filling $\nu=9/5$. The free edge phase
comprises two downstream propagating channels (right-pointing arrows), both
with filling factor discontinuities $\delta\nu=1$ and charge $q=e$ (here, the
charge $q$ is the minimum charge that can tunnel across vacuum into the
channel). There is also one upstream channel filling factor discontinuity
$\delta\nu=-1/5$ and charge $q=e$ (left-pointing arrow). This structure is
equivalent (after a $SL(3,\mathbf{Z})$ basis change) to a Laughlin edge state
with $\delta\nu=1/5$ composed of charge $q=3e$ composite particles, plus two
counter-propagating channels with $\delta\nu=\pm 1$ and $q=e$. The binding
transition is the phenomenon where inter-channel interactions and tunneling
cause the counter-propagating channels to localize (with characteristic length
scale $\xi_{0}$). In the bound phase, only the single composite channel
remains.

The simplest example of the binding transition was predicted for the edge at
filling factor $\nu=9/5$ (see Fig. 1). In the free, or unbound, phase, the
$9/5$ edge hosts three channels, which after the binding transition are
reduced to a single channel. Most remarkably, in this so-called bound phase,
single electron excitations become short ranged and do not participate in the
low energy edge transport. By contrast, excitations with charges $3e$ remain
long ranged and do contribute to the transport [18]. Despite such a striking
re-organization of the edge structure, the prospects of experimentally
observing a binding transition remain to large extent unexplored.

In this paper, we address this issue by proposing several experimental
signatures of the binding transition. Our work is motivated by novel
developments for probing FQH edges with quantum transport (a recent overview
is given in Ref. [20]). More specifically, a growing body of experiments have
demonstrated the existence of a wide range of different edge transport
regimes. These range from complete charge and heat equilibration of the edge
channels [21, 22, 23], to intermediate regimes with full charge but no heat
equilibration [24, 25, 26, 27, 28], to the extreme limit of non-equilibrated
charge transport [29, 30]. To detect a binding transition, access to non-
equilibrated transport regimes is of particular interest, since in these
regimes, the charge and heat conductances do not necessarily reflect the bulk
topological order. Instead, these quantities can reveal the total number of
edge channels and the charges they carry, quantities that both change across
the binding transition. In addition, recent experiments [31, 32], have
demonstrated robust, engineered FQH edges formed between regions with
different bulk fillings. For example, an “artificial” $9/5$ edge can be
synthesized by proximitizing regions with fillings $\nu=2$ and $\nu=1/5$. Such
structures, which might allow experimental control over Landau level spin
polarizations and thereby tunneling rates between edge channels, facilitate
the detection of binding transitions.

As a key result, we find that the two edge phases have different charge and
heat conductance characteristics (see Tab. 1). With decreasing level of
equilibration (i.e., with decreasing temperature $T$ and/or system size $L$),
the free phase conductances monotonously increase from the values for
equilibrated transport, which defines regime $\mathcal{III}$, to saturation at
the non-equilibrated values, defining regime $\mathcal{II}$. By contrast, the
bound phase is characterized by the existence of a localized regime, regime
$\mathcal{I}$, with similar characteristics as regime $\mathcal{III}$. A
transition between regimes $\mathcal{I}$ and $\mathcal{II}$ is possible for
strong interactions and give rise to the unusual situation of increasing
conductances with increasing temperature. Such an observation is a striking
hallmark of the existence of edge localization. Complementing the
conductances, we further argue that a current biased edge segment produces
shot noise, $S$, when the charge transport is equilibrated but the heat
transport is not. This feature occurs only for strong interactions, associated
with the bound phase, in regime $\mathcal{II}$.

We also demonstrate that in a quantum point contact (QPC) device in the strong
back-scattering (SBS) regime, the bound phase yields a minimum shot noise
Fano-factor $F_{SBS}=3$ corresponding to three-electron tunneling. No single
electron tunneling is possible at low energies. This result stands in stark
contrast to the free phase, where strong back-scattering favours single
electron tunneling, i.e., $F_{SBS}=1$. By the same token, in the weak back-
scattering (WBS) regime, the most relevant (in the renormalization group, RG,
sense) quasiparticle tunneling yields in the bound phase $F_{WBS}=3/5$, in
contrast to the free phase value $F_{WBS}=1/5$. Altogether, our set of derived
transport signatures present several possibilities for experimentally
detecting a FQH binding transition.

The remainder of this paper is organized as follows. In Sec. II, we review the
basics of the FQH edge theory and the binding transition. We also perform a
renormalization group (RG) treatment of the transition. In the main part of
this paper, Sec. III, we derive several transport signatures of the bound
phase and contrast them to those of the free phase. In Sec. IV, we discuss
possible experimental setups to detect the proposed signatures. We summarize
in Sec. V and also provide an outlook towards future studies.

Throughout this paper, we generally use units $e=\hbar=k_{\rm B}=1$, but we
restore important units for transport observables.

Regimes | Transport charact. | Free | Bound
---|---|---|---
$\mathcal{I}$ | $G/(e^{2}/h)$ | – | 9/5
$G^{Q}/(\kappa T)$ | – | 1
$S$ | – | $0$
$\mathcal{II}$ | $G/(e^{2}/h)$ | 11/5 | 11/5
$G^{Q}/(\kappa T)$ | 3 | 3
$S$ [$10^{-29}\rm{A}^{2}/(\rm{nA}\rm{Hz})$] | $\simeq 0$ | 0.25-0.7
$\mathcal{III}$ | $G/(e^{2}/h)$ | 9/5 | 9/5
$G^{Q}/(\kappa T)$ | 1 | 1
$S$ | $\simeq 0$ | 0
QPC Fano factors | $F_{SBS}$ | 1 | 3
$F_{WBS}$ | 1/5 | 3/5
Table 1:  Transport characteristics for the $\nu=9/5$ edge. For the free and
bound edge phases, values for the two-terminal electrical ($G$) and heat
($G^{Q}$) conductances, the shot noise $S$ of a current biased edge segment,
and QPC Fano actors for strong, $F_{SBS}$, and weak $F_{WBS}$ back scattering
regimes are given. There are three relevant transport regimes. Regime
$\mathcal{I}$: regime of localization, $\mathcal{II}$: no localization but
vanishing equilibration, and $\mathcal{III}$ full edge equilibration. The
emergence of regime $\mathcal{I}$ for the bound phase is a fundamental feature
of the binding transition. The quantized $G^{Q}$ values in regime
$\mathcal{II}$ are given under the assumption of vanishing interference in
edge-contact plasmon scattering: $L\gg L_{T}\sim T^{-1}$. The noise in regime
$\mathcal{II}$ is given under the condition (76) of efficient charge
equilibration and poor thermal equilibration. This condition holds for strong
interactions which is only the case in the bound phase.

##  II The Chiral Luttinger liquid and the binding transition

For completeness, we review here key aspects of the chiral Luttinger theory
and the FQH binding transition, closely following Refs. [16, 17, 18].

###  II.1 The chiral Luttinger liquid

At low energies, an Abelian FQH edge is well described by the chiral Luttinger
liquid ($\chi_{LL}$) model, specified by the pair ($K$, $\mathbf{t}$) [5].
Here, $K$ is an $n\times n$ integer valued symmetric matrix, and the charge
vector $\mathbf{t}$ is an $n$-dimensional vector of integers. The generic
$n$-channel action reads

$\displaystyle S_{\chi_{LL}}=-\frac{1}{4\pi}\int
dtdx\big{[}\partial_{t}\bm{\phi}^{T}K\partial_{x}\bm{\phi}+\partial_{x}\bm{\phi}^{T}V\partial_{x}\bm{\phi}\big{]},$
(1)

where $\bm{\phi}=(\phi_{1},\ldots,\phi_{n})$ is a set of bosonic fields
obeying the commutation relations

$\displaystyle[\phi_{i}(x),\phi_{j}(y)]=i\pi K^{-1}_{ij}\text{sgn}(x-y).$ (2)

The symmetric matrix $V$ in (1) parametrizes the channel velocities, $V_{ii}$,
and mutual, short-range Coulomb interactions, $V_{i\neq j}$. Generally, $V$
contains $n(n+1)/2$, independent, non-universal parameters, determined by
microscopic details in the edge electrostatic confinement.

The electrical charge enters the theory in the charge densities

$\displaystyle\rho_{i}\equiv t_{i}\frac{\partial_{x}\phi_{i}}{2\pi},$ (3)

which obey

$\displaystyle[\rho_{i}(x),\rho_{j}(y)]=\frac{i}{2\pi}t_{i}^{2}K^{-1}_{ij}\partial_{x}\delta(x-y).$
(4)

Several quantities determined by the bulk topological order appear in the edge
theory. The filling factor $\nu$ is given as

$\displaystyle\nu=\mathbf{t}^{T}K^{-1}\mathbf{t},$ (5)

and the “thermal quantum number”

$\displaystyle\nu_{Q}\equiv\text{Tr}(\eta)=n_{d}-n_{u}.$ (6)

Here, $\eta$ is the signature matrix corresponding to $K$, so that $\nu_{Q}$
equals the difference in the number of positive and negative eigenvalues of
$K$ (i.e., $\eta_{ij}=\pm 1\delta_{ij}$). In turn, this is equivalent to the
number of “downstream” $n_{d}$ and “upstream” $n_{u}$ propagating channels
(with respect to the chirality direction set by the magnetic field)
respectively [33, 34]. For the present theory of Abelian states,
$\nu_{Q}\in\mathbb{Z}$, while for non-Abelian states, other values of
$\nu_{Q}$ are possible. For example, Majorana edge channels allow half-integer
values $\nu_{Q}\in\mathbb{Z}/2$ [22]. As described below in Sec. III,
$\nu_{Q}$ is closely connected to the edge heat transport characteristics.

Quasiparticles (including the special case of the electron) created or
destroyed at position $(t,x)$ on the edge is described by vertex operators

$\displaystyle\mathcal{T}_{\mathbf{l}}(t,x)=\frac{e^{i\mathbf{l}\cdot\bm{\phi}(t,x)}}{(2\pi
a)^{m/2}},$ (7)

where $m$ is the number of involved bosons. A vertex operator is uniquely
determined by specifying an integer valued vector $\mathbf{l}$ which describes
how many of each quasiparticle species that are created or destroyed. The
associated exchange statistics angle $\Theta_{\mathbf{l}}$ and electric charge
$Q_{\mathbf{l}}$ of a vertex operator are given by

$\displaystyle\Theta_{\mathbf{l}}=\pi\mathbf{l}^{T}K^{-1}\mathbf{l}\quad\text{mod
}2\pi,$ (8) $\displaystyle Q_{\mathbf{l}}=\mathbf{t}^{T}K^{-1}\mathbf{l}.$ (9)

With the use of vertex operators, tunneling of particles between edge channels
is included in the theory by adding to (1) the term

$\displaystyle S_{\rm T}=\int
dtdx\;\xi(x)\mathcal{T}_{\mathbf{l}}(t,x)+\text{H.c},$ (10)

where $\xi(x)$ is the local tunneling strength at spatial location $x$. As
discussed below in Eqs. (28)-(30), the function $\xi(x)$ determines the type
of tunneling between the edge channels.

To compute various observables in the theory, correlation functions involving
$\mathcal{T}_{\mathbf{l}}(t,x)$ are needed. These are most easily obtained by
diagonalizing $S_{\chi_{LL}}$ which is done by first taking $K$ to its
signature matrix $\eta$ by a (non-unique) matrix $M_{1}$

$\displaystyle M_{1}^{T}KM_{1}=\eta.$ (11)

Second, a matrix $M_{2}$ (also not unique) is sought which diagonalizes $V$
into $\tilde{V}$ but at the same time preserves $\eta$:

$\displaystyle\eta=M_{2}^{T}M_{1}^{T}KM_{1}M_{2},$ (12)
$\displaystyle\tilde{V}=M_{2}^{T}M_{1}^{T}VM_{1}M_{2}.$ (13)

In the diagonal basis, the action (1) becomes

$\displaystyle S_{\chi_{LL}}=-\frac{1}{4\pi}\int
dtdx\big{[}\partial_{t}\bm{\tilde{\phi}}^{T}\eta\partial_{x}\bm{\tilde{\phi}}+\partial_{x}\bm{\tilde{\phi}}^{T}\tilde{V}\partial_{x}\bm{\tilde{\phi}}\big{]}$
(14)

and the theory is now expressible in transformed quantities as

$\displaystyle\bm{\tilde{\phi}}=M^{-1}\bm{\phi},$ (15a)
$\displaystyle\mathbf{\tilde{t}}=M^{T}\mathbf{t},$ (15b)
$\displaystyle\mathbf{\tilde{l}}=M^{T}\mathbf{l},$ (15c)

with $M\equiv M_{1}M_{2}$. The free, “diagonal”, bosons $\tilde{\phi}_{i}$ and
their densities

$\displaystyle\tilde{\rho}_{i}\equiv\tilde{t}_{i}\frac{\partial_{x}\tilde{\phi}_{i}}{2\pi},$
(16)

obey

$\displaystyle[\tilde{\phi}_{i}(x),\tilde{\phi}_{j}(y)]=i\pi\eta_{ij}\text{sgn}(x-y),$
(17)
$\displaystyle[\tilde{\rho}_{i}(x),\tilde{\rho}_{j}(y)]=\frac{i}{2\pi}\tilde{t}_{i}^{2}\eta_{ij}\partial_{x}\delta(x-y).$
(18)

Note that all topological properties (5), (6), (8), and (9) are independent of
the choice of basis. The total charge density is also preserved:

$\displaystyle\sum_{i}\rho_{i}=\sum_{i}\tilde{\rho}_{i}.$ (19)

In the diagonal basis, the action is quadratic and correlation functions of
vertex operators follow from the identity

$\displaystyle\langle
e^{i\tilde{\phi}_{i}(t,x)}e^{-i\tilde{\phi}_{i}(0,0)}\rangle=e^{\langle\tilde{\phi}_{i}(t,x)\tilde{\phi}_{i}(0,0)-\langle\tilde{\phi}^{2}_{i}(0,0)\rangle}$
$\displaystyle=\frac{a}{a+x-i\eta_{i}\tilde{v}_{i}t},$ (20)

upon use of the zero temperature correlation function

$\displaystyle\langle\tilde{\phi}_{i}(t,x)\tilde{\phi}_{i}(0,0)-\langle\tilde{\phi}^{2}_{i}(0,0)\rangle=-\log\left[\frac{a+x-i\eta_{i}\tilde{v}_{i}t}{a}\right].$
(21)

In Eqs. (II.1) and (21), $\eta_{i}\equiv\eta_{ii}$ and
$\tilde{v}_{i}\equiv\tilde{V}_{ii}$ is the chirality and the speed of mode
$\tilde{\phi}_{i}$, respectively. We also introduced a short distance
(ultraviolet, UV) cutoff $a$ on the order of the characteristic magnetic
length. At finite temperature $T$, the correlation function (21) changes to

$\displaystyle\langle\tilde{\phi}_{i}(t,x)\tilde{\phi}_{i}(0,0)-\langle\tilde{\phi}^{2}_{i}(0,0)\rangle$
$\displaystyle=-\log\left[\frac{\sin\left(\frac{\pi
T}{\tilde{v}_{i}}(a+x-i\eta_{i}\tilde{v}_{i}t)\right)}{\pi
aT/\tilde{v}_{i}}\right].$ (22)

By combining the single mode correlation (II.1) with the transformation rules
(15), the (zero $T$) correlation function of $\mathcal{T}_{\mathbf{l}}(t,x)$
is obtained as

$\displaystyle\langle\mathcal{T}_{\mathbf{l}}(t,x)\mathcal{T}_{\mathbf{l}}^{\dagger}(0,0)\rangle\sim\langle
e^{i\mathbf{l}\cdot\bm{\phi}(t,x)}e^{-i\mathbf{l}\cdot\bm{\phi}(0,0)}\rangle$
$\displaystyle=\langle
e^{i\mathbf{\tilde{l}}\cdot\bm{\tilde{\phi}}(t,x)}e^{-i\mathbf{\tilde{l}}\cdot\bm{\tilde{\phi}}(0,0)}\rangle=\prod_{i}\left(\frac{a}{a+x-i\eta_{i}\tilde{v}_{i}t}\right)^{((\mathbf{\tilde{l}})_{i})^{2}}.$
(23)

The long time behaviour

$\displaystyle\langle\mathcal{T}_{\mathbf{l}}(t,0)\mathcal{T}_{\mathbf{l}}^{\dagger}(0,0)\rangle\propto\frac{1}{t^{\kappa(\mathbf{l})}}\frac{1}{|t|^{2\Delta(\mathbf{l})-\kappa(\mathbf{l})}},$
(24)

where

$\displaystyle\Delta(\mathbf{l})\equiv\frac{1}{2}\mathbf{\tilde{l}}^{T}\mathbf{\tilde{l}}=\frac{1}{2}\mathbf{l}^{T}MM^{T}\mathbf{l},$
(25)
$\displaystyle\kappa(\mathbf{l})\equiv\mathbf{\tilde{l}}^{T}\eta^{-1}\mathbf{\tilde{l}}=\mathbf{l}^{T}K^{-1}\mathbf{l}$
(26)

defines the scaling dimension and the topological part of the correlation
function, respectively. The generic scaling dimension (25) is non-universal,
as it depends not only on the topological matrix $K$ but also on the
components of $V$. An exception occurs for so-called maximally chiral edges,
in which all channels propagate in the same direction, i.e., either $n_{d}$ or
$n_{u}$ equals zero. Then, the scaling dimensions of tunneling operators are
fully specified by $K$ alone.

Importantly, $\kappa(\mathbf{l})$ and $\Delta(\mathbf{l})$ obey the following
inequality [17]

$\displaystyle|\kappa(\mathbf{l})|\leq 2\Delta(\mathbf{l}),$ (27)

with equality for vanishing interactions (diagonal $V$). For maximally chiral
edges, Eq. (27) becomes an equality independently of interactions.

We now consider tunnel coupled edge channels, so that the total action
$S=S_{\chi_{LL}}+S_{\rm T}$. Depending on the nature of the tunneling events,
$S_{\rm T}$ becomes an RG relevant perturbation when

$\displaystyle
3-2\Delta(\mathbf{l})>0,\;\;\langle\xi(x)\xi^{*}(y)\rangle=D\delta(x-y),$ (28)
$\displaystyle 1-\Delta(\mathbf{l})>0,\quad\xi(x)=\Gamma_{0}\delta(x),$ (29)
$\displaystyle 2-\Delta(\mathbf{l})>0,\quad\xi(x)=\Gamma_{0},$ (30)

corresponding to Gaussian random (characterized by the strength $D$), single
point, and uniform tunneling, respectively. In this paper, we mainly focus on
the common situation of random tunneling due to edge disorder (point tunneling
as realized in a QPC is considered in Sec. III.4). Then, to guarantee the
relevancy of $\mathcal{T}_{\mathbf{l}}$, Eq. (28) implies that
$\Delta(\mathbf{l})<3/2$. By virtue of Eq. (27), this implies further

$\displaystyle|\kappa(\mathbf{l})|\leq 2\Delta(\mathbf{l})<3.$ (31)

In turn, any tunneling operator must be bosonic, which by use of Eq. (8)
implies that $\kappa(\mathbf{l})$ must be an even integer. This fact, combined
with Eq. (31), implies that

$\displaystyle\kappa(\mathbf{l})=-2,0,2,$ (32)

for random, relevant tunneling operators. As we shall see next, when a very
particular class of such tunneling operators exist and are relevant, a FQH
edge will undergo a binding transition.

###  II.2 Review of the binding transition

The binding transition is only possible for so-called T-unstable edges. These
are defined as those edges permitting a special kind of quasiparticles (which
we parametrize for convenience by $\mathbf{m}$ rather than $\mathbf{l}$),
satisfying the two constraints [18, 19]

$\displaystyle\mathbf{m}^{T}K^{-1}\mathbf{m}=0,$ (33)
$\displaystyle\mathbf{t}^{T}K^{-1}\mathbf{m}=0.$ (34)

A non-zero string $\mathbf{m}$ obeying these constraints is called a null-
vector, and Eqs. (33) and (34) are called the null conditions. They are
invariant under basis transformations.

The possibility to satisfy the null conditions can be traced to the existence
of counter-propagating neutral modes in the charge-neutral basis [17].
Physically, the null operators create charge-neutral and bosonic particles
without any topological part in their correlation function (24). Then, and
only then, is it possible for pairs of edge channels to undergo localization.
We may view this feature as the edge structure containing a non-topological
part which can be removed by the non-topological disorder and interactions.

We can readily check that no null-vectors exist for edges (those with non-zero
Hall conductance) with one or two channels, i.e., for $n=1,2$. For $n=1$, we
have $\mathbf{m}=m$, $\mathbf{t}=t$ and $K^{-1}$ is an odd integer. Eqs. (33)
and (34) then read

$\displaystyle K^{-1}m^{2}=0,$ (35) $\displaystyle tK^{-1}m=0,$ (36)

which is only trivially satisfied by $m=0$.

For $n=2$, we may choose without loss of generality

$\displaystyle K=\begin{pmatrix}1/\delta\nu_{1}&0\\\
0&1/\delta\nu_{2}.\end{pmatrix},$
$\displaystyle\mathbf{m}^{T}=(m_{1},m_{2}),\quad\mathbf{t}^{T}=(1,1).$ (37)

Here, $\delta\nu_{1,2}$ (the eigenvalues of $K^{-1}$) are known as “filling
factor discontinuities”, and specify jumps in the Hall fluid density close to
the edge. For example, the $\nu=2/3$ edge is specified by $\delta\nu_{1}=1$
and $\delta\nu_{2}=-1/3$ [15]. The null conditions (33)-(34) now read

$\displaystyle m^{2}_{1}\delta\nu_{1}+m_{2}^{2}\delta\nu_{2}=0,$ (38)
$\displaystyle m_{1}\delta\nu_{1}+m_{2}\delta\nu_{2}=0.$ (39)

These equations have only the trivial $m_{1}=m_{2}=0$ solution, under the
condition $\delta\nu_{1}\neq\delta\nu_{2}$, which must be satisfied for the
$n=2$ chiral Luttinger liquid [5]. With the present formalism, we can however
readily see that for a standard, spinless Luttinger liquid, i.e., for
$\delta\nu_{1}=-\delta\nu_{2}=1$, all charge conserving ($m_{1}=m_{2}$)
operators satisfy (38)-(39) and may cause instabilities localizing the edge
channels [35, 36] (see also Ref. [37] for a recent discussion).

For $n=3$, there exists several T-unstable FQH states. Specifically, we focus
as follows on the state at filling $\nu=9/5$. The corresponding edge theory is
defined by

$\displaystyle K=\begin{pmatrix}1&0&0\\\ 0&1&0\\\
0&0&-5\end{pmatrix},\quad\mathbf{t}^{T}=(1,1,1).$ (40)

Using the null conditions (33) and (34), it can readily be checked that

$\displaystyle\mathbf{m}^{T}_{1}=(-1,2,5),\quad\mathbf{m}^{T}_{2}=(2,-1,5),$
(41)

are the two possible null vectors (changing an overall sign does not count as
a new null-vector). We denote the corresponding null-operators by

$\displaystyle\mathcal{V}_{\mathbf{m}_{j}}(t,x)\sim
e^{i\mathbf{m}_{j}\cdot\bm{\phi}(t,x)},\quad j=1,2.$ (42)

With the scaling dimension formula (25), one can check that in the absence of
interactions, i.e., when $V$ is diagonal in basis (40), the scaling dimensions
of these operators are $\Delta(\mathbf{m}_{1})=\Delta(\mathbf{m}_{2})=5$,
whereas interactions reduce these values.

The binding transition becomes possible when at least one of the null-
operators (42) is RG relevant [18], i.e., $\Delta(\mathbf{m}_{j})<3/2$,
according to Eq. (28). This requires sufficiently strong edge interactions. As
follows, we now assume, without loss of generality [18], that
$\mathcal{V}_{\mathbf{m}_{1}}$ is a relevant operator and ignore all effects
from $\mathcal{V}_{\mathbf{m}_{2}}$.

To understand the binding transition, it is very useful to perform a basis
transformation with a matrix $W\in SL(3,\mathbb{Z})$. With the appropriate
choice of $W$ (the exact details of $W$ are not important, but its existence
follows from the existence of nullvectors), we can equally represent the $9/5$
edge as

$\displaystyle K^{\prime}=\begin{pmatrix}5&0&0\\\ 0&1&0\\\
0&0&-1\end{pmatrix},\quad\mathbf{t^{\prime}}^{T}=(3,1,1).$ (43)

In this basis, the nullvectors read

$\displaystyle\mathbf{m^{\prime}}^{T}_{1}=(0,1,1),\quad\mathbf{m^{\prime}}^{T}_{2}=(15,-2,7).$
(44)

We focus only on the null vector $\mathbf{m^{\prime}}_{1}$ and its associated
null operator $\mathcal{V}_{\mathbf{m^{\prime}}_{1}}$. Note that this operator
only couples two of the three modes. In the absence of interactions in the
basis (43), the scaling dimension of $\mathcal{V}_{\mathbf{m^{\prime}}_{1}}$
is $\Delta(\mathbf{m^{\prime}}_{1})=1$.

Eq. (43) suggests that the $9/5$ edge can be viewed as two counter-propagating
channels with filling factor discontinuities $\pm 1$ carrying unit charge, and
one downstream channel with filling factor discontinuity $1/5$ carrying
particles with charge $3e$ (see Fig. 1). These composite particles can be
thought of “Cooper-triplons”: bound states of three electrons. Unlike
conventional Cooper-pairs, however, the composite particles here are fermionic
and therefore cannot condense.

The binding transition is the manifestation of localization of the two equal
and counter-propagating channels due to interactions and disorder [35, 36].
After localization, only one channel remains. In some sense, this can be
thought of as an “inverted edge reconstruction”, where non-topological pairs
of channels vanish rather than appear. On length scales much larger than a
characteristic localization length (to be discussed in more detail below in
Sec. II.3), the edge structure is then effectively given as

$\displaystyle K=(5),\quad\mathbf{t}^{T}=(3),$ (45)

which is nothing but a $\nu=1/5$ Laughlin state made out of charge $3e$
composites. The peculiarity of the binding transition is that when two
counter-propagating but otherwise equal channels localize, the remaining
channel carries a changed charge as to preserve the topological quantum
numbers from the bulk. This picture clarifies also why $n=3$ is the minimum
number of edge channels required for a binding transition.

From Eqs. (43) and (44), we see that excitations (7) with vectors

$\displaystyle\mathbf{l^{\prime}}^{T}=(0,q,q),\quad q\in\mathbb{Z},$ (46)

will become localized on the edge, i.e., those $\mathbf{l^{\prime}}$
satisfying

$\displaystyle\mathbf{l^{\prime}}^{T}K^{\prime-1}\mathbf{m^{\prime}}_{1}=0.$
(47)

Due to the transformation rules (15), the condition (47) holds in any basis.
In contrast, an excitation on the form

$\displaystyle\mathbf{l^{\prime}}^{T}=(p,0,0),\quad p\in\mathbb{Z}$ (48)

will propagate freely along the edge. In more technical terms, in the original
three-dimensional vector-space of excitations, only excitations in the one-
dimensional subspace of vectors $\mathbf{l^{\prime}}$ satisfying

$\displaystyle\mathbf{l^{\prime}}^{T}K^{\prime-1}\mathbf{m^{\prime}}_{1}=0\text{
and }\mathbf{l^{\prime}}\text{ not proportional to }\mathbf{m^{\prime}}_{1}$
(49)

will propagate freely on the bound edge. From Eqs. (8) and (9), we see that
the statistics and the charge of the propagating excitations (48) are

$\displaystyle\Theta_{\mathbf{l^{\prime}}}=\pi\frac{p^{2}}{5},$ (50)
$\displaystyle Q_{\mathbf{l^{\prime}}}=\frac{3p}{5}.$ (51)

Since only $p=5N$, with $N$ an odd integer produces quantum numbers consistent
with the electron, it follows that only electron excitations in bunches of $3$
can propagate. In contrast, single electron excitations become localized on
the edge. The consequence of this effect is explored in more detail in Sec.
III.4. In passing, we note that for bosonic FQH states, $N$ must be even to
preserve Bose statistics. This implies that binding transitions for such
states always generate even multiples of bound composites.

The goal of this paper can now be formulated as investigating the transport
properties for the edge in the two different edge phases. To this end, we next
perform an RG analysis to find the phase diagram of the edge.

###  II.3 Scaling analysis of the binding transition

Our simple scaling analysis of the binding transition follows the approach in
Refs. [35, 36, 38]. Throughout this section, we assume that
$\mathcal{V}_{\mathbf{m^{\prime}}_{1}}$ [see Eq. (42)] is more relevant than
$\mathcal{V}_{\mathbf{m^{\prime}}_{2}}$ and analyze here only the influence of
$\mathcal{V}_{\mathbf{m^{\prime}}_{1}}$. The opposite situation can be treated
in a perfectly analogous manner.

Figure 2: Schematic log-log phase diagrams of the temperature ($T$) dependence
of equilibration lengths $\ell_{\rm eq}\sim T^{2-2\Delta_{0}}$. Here, $\ell$
is the length scale at which the system is observed, $\Delta_{0}$ is the bare
scaling dimension of a nulloperator, $\ell_{\varphi}\sim 1/T$ is the thermal
length in the presence of strong interactions, and $\xi_{0}$ is the zero
temperature localization length. The gray, dashed line is the characteristic
temperature scale for onset of localization:
$\ell_{\varphi}(\tilde{T})=\ell_{\rm eq}(\tilde{T})$. In $(a)$ and $(b)$,
localization occurs in regime $\mathcal{I}$ (yellow region), whereas regimes
$\mathcal{II}$ (blue region) and $\mathcal{III}$ (red region) correspond to
non equilibrated and equilibrated transport, respectively. In $(c)$,
localization is absent.

For convenience, we next redefine the disorder strength in Eq. (28) so that it
becomes dimensionless (see Appendix A). We thus set

$\displaystyle D\rightarrow D\times\frac{a^{3-m}}{v^{2}}.$ (52)

Here, $m$ is the number of bosonic fields involved in the null-operator
$\mathcal{V}_{\mathbf{m}_{1}}$, i.e., $m=3$, and $v$ is a characteristic
velocity (i.e., some combination of the $v_{i}$; the exact details are not
important here). The renormalization equation for $D$ then reads [35]

$\displaystyle\frac{\partial D}{\partial\ell}=(3-2\Delta_{0})D,$ (53)

where $\Delta_{0}$ is the initial scaling dimension of
$\mathcal{V}_{\mathbf{m^{\prime}}_{1}}$ and $\ell=\log(L/a)$, with $L$ the
system size and $a$ is the UV length cutoff (e.g., the magnetic length). As
follows, we ignore effects of renormalization of $\Delta_{0}$, as they are
assumed weak [36, 38]. In any case, the renormalization of $\Delta_{0}$ is
non-essential for the qualitative analysis presented here. Hence, as follows
we take

$\displaystyle\Delta\approx\Delta_{0}$ (54)

during the RG flow. We now study the consequences of Eq. (53) in the two
regions $\Delta_{0}>3/2$ and $\Delta_{0}<3/2$.

####  II.3.1 Weak interactions and disorder: $\Delta_{0}>3/2$

From Eq. (53), it follows that

$\displaystyle D(L)=D_{0}\left(\frac{L}{a}\right)^{3-2\Delta_{0}},$ (55)

where $D_{0}\equiv D(L=a)$ is the bare (non-renormalized) disorder strength,
assumed to be weak: $D_{0}\ll 1$. For $\Delta_{0}>3/2$, $D$ decreases upon
renormalization towards lower energies (i.e., with increasing $\ell$) and the
edge is in the free phase, described by Eq. (40) or (43). Tunneling between
the edge modes remains weak at all energies. This tunneling exchanges charge
and heat among the edge channels which causes equilibration. We denote the
characteristic length scales for charge and heat equilibration as $\ell_{\rm
C}$ respectively $\ell_{\rm Q}$. They are both non-universal as they depend on
microscopic details such as disorder, interactions, and the temperature.
Generically, a FQH edge is characterized by two such length scales for each
pair of edge modes [39, 40]. However, only tunneling between counter-
propagating channels can partition charge and energy flows and influence
transport coefficients. The quantities $\ell_{\rm C}$ and $\ell_{\rm Q}$
should then be understood as the dominating ones in the full set.

The temperature scalings of $\ell_{\rm C}$ and $\ell_{\rm Q}$ are expected to
be the same [38]. In principle, strong interactions can cause parametrically
different equilibration lengths [24], but that case is excluded for the case
$\Delta_{0}>3/2$. In the remainder of this subsection, as we are interested in
only the temperature scaling, we treat the length scales on the same footing
by setting

$\displaystyle\ell_{\rm C}\sim\ell_{\rm Q}\sim\ell_{\rm eq},$ (56)

and $\ell_{\rm eq}$ is to be understood as meaning both $\ell_{\rm C}$ and
$\ell_{\rm Q}$.

To find the temperature scaling of $\ell_{\rm eq}$, we first note that with
weakening disorder, the RG flow is cut at the thermal length

$\displaystyle L_{T}\sim T^{-1}.$ (57)

At this scale, the disorder has, according to Eq. (55), the strength

$\displaystyle D(L_{T})=D_{0}\left(\frac{L_{T}}{a}\right)^{3-2\Delta_{0}}\sim
T^{2\Delta_{0}-3}.$ (58)

At larger length scales $L>L_{T}$, $D(L)$ scales linearly (trivially) in $L$
as

$\displaystyle
D(L)\sim\frac{L}{L_{T}}D(L_{T})\sim\frac{L}{L_{T}}D_{0}\left(\frac{L_{T}}{a}\right)^{3-2\Delta_{0}},$
(59)

where used Eq. (58) in the final step. The equilibration length is defined as

$\displaystyle D(\ell_{\rm eq})\sim 1,$ (60)

which by use of Eqs. (58) and (59) becomes

$\displaystyle\ell_{\rm eq}\sim
D_{0}^{-1}a\left(\frac{L_{T}}{a}\right)^{2\Delta_{0}-2}\sim
T^{2-2\Delta_{0}}.$ (61)

Note that $2-2\Delta_{0}<0$ for weak interactions $\Delta_{0}>3/2$ so that
both $\ell_{\rm C}$ and $\ell_{\rm Q}$ increase with decreasing temperature
[16, 24].

We can then identify two possible regimes in the free phase:

  * •

At lowest temperatures, such that $L\ll\ell_{\rm C},\ell_{\rm Q}\sim
T^{2-2\Delta_{0}}$, the three modes in the free phase are neither charge nor
heat equilibrated. The temperature scaling exponent depends non-universally on
inter-channel interactions and velocities. We call this regime $\mathcal{II}$:
the regime of absent localization and poor equilibration.

  * •

At higher temperatures: $\ell_{\rm C},\ell_{\rm Q}\ll L$ the three edge modes
become fully equilibrated, which we denote as regime $\mathcal{III}$. Here,
the edge is expected to be governed by hydrodynamic behaviour due to inter-
channel scattering [41].

We depict regimes $\mathcal{II}-\mathcal{III}$ for the free phase in Fig.
2(c).

####  II.3.2 Strong interactions and disorder: $\Delta_{0}<3/2$

For $\Delta_{0}<3/2$, Eq. (53) shows that $D$ grows during renormalization.
The characteristic length scale where the disorder becomes strong is defined
as [35, 36]

$\displaystyle D(\xi_{0})\sim 1.$ (62)

Using this condition in Eq. (53) gives the characteristic zero temperature
localization length

$\displaystyle\xi_{0}\propto aD_{0}^{-1/(3-2\Delta_{0})}.$ (63)

Approaching the critical value $\Delta_{0}=3/2$ from below, $\xi_{0}$
diverges, and no localization is possible for $\Delta_{0}>3/2$ as expected.

Remarkably, we find that the strongly interacting regime, $\Delta_{0}<3/2$, is
split into two parameter regions with very distinct behaviour: Eq. (61)
implies that $\ell_{\rm eq}$ increases with increasing $T$ if $\Delta_{0}<1$.
In contrast, when $\Delta_{0}>1$ $\ell_{\rm eq}$ decreases with increasing
$T$. These two distinct cases are depicted in Figs. 2(a),(b), respectively.
The existence of the regime $\Delta_{0}<1$ is a striking manifestation of the
fact that the $\nu=9/5$ edge is T-unstable and thus susceptible to the binding
transition. This is a key result of this paper.

Let us now analyze the onset of edge localization. At finite temperature, the
disorder is dressed by temperature dependent Friedel oscillations and
$\xi_{0}$ is replaced by $\ell_{\rm eq}$ in Eq. (61). The parameter $\ell_{\rm
eq}$ thus plays the role as a finite temperature localization length. The
temperature scale $\tilde{T}$ where the localization becomes strong is
determined by the condition

$\displaystyle\ell_{\rm eq}(\tilde{T})\sim\ell_{\varphi}(\tilde{T})$ (64)

where $\ell_{\varphi}$ is a characteristic phase coherence length. For this
length, we use the estimate [36]

$\displaystyle\ell_{\varphi}\sim L_{T}\sim T^{-1}.$ (65)

It is important to note that this estimate can only be true for strong
interactions (which is assumed here), as it is clear that non-interacting
electrons are subject to localization independently of the temperature:
$\ell_{\varphi}=\infty$. Hence, we expect also that [36]

$\displaystyle\tilde{T}\propto(\Delta_{0}-1)^{-2},$ (66)

since $\Delta_{0}=1$ at vanishing interaction. Here, the exponent $-2$ follows
from a more detailed analysis [36]. The scale $\tilde{T}$ marks the onset
where weak localization corrections to the conductivity start to dominate over
the classical Drude contribution and, as mentioned above, signals Anderson
(i.e., strong) localization.

We now analyze the edge under the assumption that $\Delta_{0}<3/2$. Three
possibles regimes appear:

  * •

At lowest $T\ll\tilde{T}$ and $L>\ell_{\rm eq},\xi_{0}$, the edge is in the
strongly localized or bound phase (45). This is depicted as regime
$\mathcal{I}$ in Fig. 2(a)-(b).

  * •

For $L<\ell_{\rm eq},\xi_{0}$ for all $T$, disorder remains weak and no
localization or equilibration occurs. We call this regime $\mathcal{II}$.

  * •

For $L>\ell_{\rm eq},\xi_{0}$ and $T>\tilde{T}$, dephasing suppresses the
localization and at the same time, the three edge modes are equilibrated. This
is regime $\mathcal{III}$.

The conclusion of the present analysis is that the bound phase is
characterized by three transport regimes: the localization regime
$\mathcal{I}$, and non-equilibrated and equilibrated regimes $\mathcal{II}$
and $\mathcal{III}$, respectively. This stands in stark contrast to the free
phase which exhibit only regimes $\mathcal{II}-\mathcal{III}$. The localized
regime $\mathcal{I}$ in the bound phase is depicted as the yellow regions in
Fig. 2. With this phase diagram in mind, we now move on to a transport
analysis for the regimes $\mathcal{I}-\mathcal{III}$.

##  III Transport signatures of the binding transition

Electrical and thermal transport on FQH edges are usually quantized,
reflecting a non-trivial bulk topological order [3]. Specifically, the
electrical Hall and two-terminal conductances, $G_{H}$ and $G$, are commonly
proportional to the bulk filling factor [see Eq. (5)]

$G_{H}=\nu\frac{e^{2}}{h},\qquad G=|G_{H}|.$ (67)

By contrast, thermal Hall and two-terminal conductances, $G^{Q}_{H}$ and
$G^{Q}$ are determined by $\nu_{Q}$ [see Eq. (6)] as

$G^{Q}_{H}=\nu_{Q}\kappa T,\qquad G^{Q}=|G^{Q}_{H}|,$ (68)

where the heat conductance quantum $\kappa T=\pi^{2}k_{\rm B}^{2}T/(3h)$, in
which $T$ is the temperature, and $k_{\rm B}$ and $h$ are the Boltzmann and
Planck constants, respectively.

For edges with counter-propagating channels, the quantizations (67) and (68)
hold only in the transport regime of full edge channel equilibration [38, 42].
To observe deviations from (67) and (68) requires poor equilibration, which
can be achieved by very low temperatures, strong inter-channel interactions
[24, 25], very small inter-contact distance, or a detailed control over the
inter-channel tunneling strength [29]. Since $\nu_{Q}$ takes negative values
on some edges (e.g., at filling $\nu=3/5$), it has further been predicted that
heat may flow in the opposite direction of the charge [33]. However, most
experiments measure $|\nu_{Q}|$. It was therefore proposed that the direction
of heat flow (i.e., the sign of $\nu_{Q}$) is in direct correspondence with
the scaling behaviour of the electrical shot noise with the edge length $L$
[43, 39, 44, 45, 46]. These insights have led to a deeper understanding of the
FQH edge structure. In particular, recent measurements of the heat conductance
[22, 31] and noise [32] now strongly point towards GaAs/AlGaAs hosting the
non-Abelian particle-hole-Pfaffian edge structure [47, 48, 49, 50] at filling
$\nu=5/2$ [51, 52].

As described in Sec. II, the binding transition preserves the topological
transport coefficients $\nu$ and $\nu_{Q}$. It is therefore clear that charge
and heat conductances in the fully equilibrated regime cannot distinguish
between the free (40) and bound (45) edge phases. Indeed, in both phases, the
equilibrated transport coefficients are given as

$\displaystyle G_{H}=G=\frac{9}{5}\frac{e^{2}}{h},$ (69) $\displaystyle
G^{Q}_{H}=G^{Q}=1\kappa T.$ (70)

However, by building upon the results in Sec. II.3 we will next argue that
signatures of the binding transition can be deduced from edge transport
experiments by accessing regimes with absent equilibration.

###  III.1 Two-terminal charge conductance

Figure 3: Two-terminal setup for depicting edge transport at filling $\nu=9/5$
in the free (a-b) and bound (c-d) edge phases. The device length is $L$. Two
contacts at different potentials $V_{L}\neq V_{R}$ but the same temperature
$T$ allows extraction of the two-terminal conductance $G$ (left figures). For
contacts at the same potential $V_{L}=V_{R}=V$ but different temperatures
$T_{L}\neq T_{R}$ (right figures) one can determine the two-terminal heat
conductance $G^{Q}$.

We first consider the two terminal charge conductance $G$ (see Figs. 3a and
3c). As follows, we assume for simplicity a sharp enough edge potential such
that no edge reconstruction occurs. We also assume that in contact regions,
screening causes the inter-channel interactions to vanish.

####  III.1.1 The free phase

When the edge is in the free phase, $\Delta_{0}>3/2$, we expect with
decreasing system size $L$ and/or decreasing temperature $T$, a crossover in
$G$. When $L\gg\ell_{\rm C}$, i.e., for equilibration (i.e., regime
$\mathcal{III}$, cf. Sec. II.3) we expect $G/(e^{2}/h)=\nu=9/5$ in accordance
with Eq. (67). Since $\Delta_{0}>3/2$, we have from Eq. (61) that $\ell_{\rm
C}$ increases with decreasing $T$. When $L\sim\ell_{\rm C}$, charge begins to
propagate upstream, which increases $G$. In the limit of $L\ll\ell_{\rm C}$,
i.e., no equilibration (regime $\mathcal{II}$), the upstream channels
transport charge upstream and also remain in equilibrium with the contact they
were emitted from. The individual channel conductance contributions then add
[38, 53], i.e., $G/(e^{2}/h)=1+1+1/5=11/5$. Hence, we expect the conductance
characteristics

$\displaystyle G/(e^{2}/h)=11/5\rightarrow 9/5,$ (71)

with increasing $L$ and/or $T$ in the free edge phase. This corresponds to
moving from the blue to the red region in Fig. 2(c).

####  III.1.2 The bound phase

In the localized regime $\mathcal{I}$, only a single channel transports charge
over distances $L>\ell_{\rm C},\xi_{0}$. Hence, Eq. (67) gives the charge
conductance $G/(e^{2}/h)=9/5$. The analysis of regimes $\mathcal{II}$ and
$\mathcal{III}$ proceed just as for the free phase and give charge
conductances $G/(e^{2}/h)=11/5$ and $G/(e^{2}/h)=9/5$, respectively.

We next analyze two important situations. For $1<\Delta_{0}<3/2$, Fig. 2(b)
indicates that, for fixed $L>\ell_{\rm C},\xi_{0}$ the conductance remains at
$G/(e^{2}/h)=9/5$ with decreasing $T$, i.e., transitioning from regime
$\mathcal{III}$ to regime $\mathcal{I}$. However, for $L<\ell_{\rm
C},\xi_{0}$, the conductance increases from $G/(e^{2}/h)=9/5$ to $11/5$ since
one channel begins to conduct more and more charge upstream. The other case is
$\Delta_{0}<1$, see Fig. 2(a). Just as in the previous case, a transition
between regimes $\mathcal{I}$ and $\mathcal{III}$ is not visible in the charge
conductance which remains at $G/(e^{2}/h)=9/5$. This is the blue, dashed line
in Fig. 2(a) However, we see that it is possible to crossover directly between
regimes $\mathcal{I}$ and $\mathcal{II}$. This is depicted as the red, dashed
line. Such a transition would give rise to a change in the conductance

$\displaystyle G/(e^{2}/h)=11/5\rightarrow 9/5,$ (72)

with decreasing temperature, which is a quite unusual situation, only possible
due to the existence of the localized regime. Observing this crossover would
be a strong hallmark of edge localization and the binding transition.

###  III.2 Two-terminal heat conductance

Here, we consider the setup in Figs. 3b and 3d and analyze the two-terminal
heat conductance $G^{Q}$. We do note that state-of-the-art measurements of
$G^{Q}$ use a different geometry (see, e.g., Ref. [54]). This does not however
change the validity of the results in this section.

In contrast to the topological quantization (68), in the regime of vanishing
heat equilibration, $G^{Q}$ becomes proportional to the total number of edge
channels

$\displaystyle G^{Q}=(n_{d}+n_{u})\kappa T.$ (73)

This holds under the condition $L<L_{T}$ [55], which we assume is fulfilled as
follows. Deviations from this assumption is commented upon below.

####  III.2.1 The free phase

Abelian FQH edge channels carry the same heat conductance regardless of their
filling factor discontinuity $\delta\nu_{i}$ and charge $t_{i}$. Simple
channel counting gives, for the free phase, the crossover

$\displaystyle G^{Q}/(\kappa T)=3\rightarrow 1.$ (74)

with decreasing $L$ and/or $T$. This corresponds to the two limiting values
(68) and (73) when moving from regime $\mathcal{III}$ to $\mathcal{II}$.
Analogously to the charge transport, the crossover is governed by a
characteristic, non-universal, heat equilibration length $\ell_{\rm Q}$
[$\sim\ell_{\rm eq}$ in Fig. 2(c)].

####  III.2.2 The bound phase

Also in the bound phase, we have in regime $\mathcal{III}$ a heat conductance
$G^{Q}/(\kappa T)=1$ according to Eq. (68). In regime $\mathcal{II}$, Eq. (73)
gives $G^{Q}/(\kappa T)=3$. For $1<\Delta_{0}<3/2$, Fig. 2(b) indicates that,
for fixed $L>\ell_{\rm Q},\xi_{0}$ the heat conductance remains at
$G^{Q}/(\kappa T)=1$ with decreasing $T$. For $L<\ell_{\rm Q},\xi_{0}$, the
heat conductance instead increases from $G^{Q}/(\kappa T)=1$ to $G^{Q}/(\kappa
T)=3$. For strong interactions, $\Delta_{0}<1$, the transition between regimes
$\mathcal{I}$ and $\mathcal{III}$ is not visible as $G^{Q}/(\kappa T)=1$
across the transition; the blue, dashed line in Fig. 2(a). Crossing over from
$\mathcal{II}$ to $\mathcal{I}$ directly (the red, dashed line) yields the
crossover

$\displaystyle G^{Q}/(\kappa T)=3\rightarrow 1,$ (75)

with decreasing temperature, due localization. Similarly to the charge
conductance, such an unusual crossover is a hallmark of localization on the
edge.

Finally, we comment on the situation $L>L_{T}$. Then, plasmon scattering on
interfaces between edges and contacts lead to quantum interference effects
[55]. This interference reduces the conductance from the value in Eq. (73).
For example, the strongly interacting $\nu=2/3$ edge with $L_{T}<L<\ell_{\rm
eq}$ produces a heat conductance $G^{Q}/(\kappa T)=1$ [38, 25]. A prerequisite
for this interference effect is the presence of counter-propagating channels
which do not thermally equilibrate. The only regime where the interference
effect therefore could potentially have an impact is regime $\mathcal{II}$,
where it would reduce $G^{Q}/(\kappa T)$ from $3$ to slightly lower values.

###  III.3 Shot noise on a voltage biased edge segment

Figure 4:  Top panel: noise generation in regime of full charge equilibration
but no thermal equilibration [condition (76)] in regime $\mathcal{II}$ (cf.
Tab. 1). With voltage bias $V_{0}$, heat is generated by the voltage dropping
in a region of size $\sim\ell_{\rm C}$ to the right contact: the hot spot. The
dissipated power $P\sim V_{0}^{2}$ [see Eq. (92)]. Dc noise by partitioning of
the downstream (left to right direction) charge current is only generated in a
region of size $\sim\ell_{\rm C}$ close to the left contact. This happens if
heat is transported upstream (right to left) via the upstream channel, which
is here only possible in the absence of thermal equilibration. Bottom panel:
By dissipation in the hot spot and reflection of plasmons at contacts
(reflection coefficient $\mathcal{R}$), the downstream and upstream modes
acquire steady state temperatures $T_{+}$ respectively $T_{-}$ in the noise
spot.

Here, we analyze the shot (or excess dc) noise, $S$, generated on a single
edge segment, of length $L$, bridging two contacts (see Fig. 4). This setup
was studied theoretically in Refs. [43, 39, 44, 45] and has been realized
experimentally for both conventional [26] and interfaced [32] edge structures.

Under conditions of strongly equilibrated charge transport, $\ell_{\rm C}\ll
L$, noise in this setup is generated by an interplay between the charge and
heat transport characteristics: When a charge current is driven between the
two contacts, heat is generated only near the drain contact (in a region
called the hot spot; see right hand side in Fig. 4). This is a consequence of
downstream ballistic charge transport due to the efficient charge
equilibration. By contrast, excess noise can only be generated near the source
contact (in a region called the noise spot, see left hand side in Fig. 4) due
to thermal enhancement of current partitioning. Partitioning beyond this spot
leads, due to repeated charge scattering and the chiral nature of the edge, to
scattered particles ending up in the same contact and no noise is generated.
Non-zero shot noise in the contacts is therefore possible only if (i) the edge
hosts counter-propagating modes and (ii) there is an upstream heat flow from
the hot spot to the noise spot.

Let us apply this reasoning to the $\nu=9/5$ edge. In the localized regime
$\mathcal{I}$ there is no upstream heat flow and therefore no noise; $S=0$. In
regime $\mathcal{III}$, the edge is fully thermally equilibrated and the
upstream heat flow as well. The noise is then exponentially suppressed in $L$:
$S\propto\exp[-L/\ell_{\rm C}]\simeq 0$. These two regimes stand in stark
contrast to regime $\mathcal{II}$, under the additional condition

$\displaystyle\ell_{\rm C}\ll L\ll\ell_{\rm Q}.$ (76)

In this case, the edge has three charge equilibrated but not thermally
equilibrated channels, conditions which have been experimentally observed [25,
26]. Upstream heat transport is then possible. This heat reaches the noise
spot through ballistic upstream flow and noise is generated. The condition
(76) can arise due to strong interactions [24], so we expect that it holds
only for $\Delta_{0}\approx 0$, i.e., deep in the bound phase.

Our goal in the remainder of this subsection is to estimate the magnitude of
$S$ in regime $\mathcal{II}$, under the condition (76). In our analysis, we
consider a large voltage bias $eV_{0}\gg T$, which allows us to effectively
set $T\approx 0$ in the following calculations.

The shot noise in any of the two contacts [see Fig. 4] (equal due to current
conservation) can be written as [25]

$\displaystyle S$ $\displaystyle=\frac{2e^{2}}{h\ell_{\rm
C}}\frac{\nu_{-}}{\nu_{+}}(\nu_{+}-\nu_{-})\int_{0}^{L}dx\;\Lambda(x)e^{-\frac{2x}{\ell_{\rm
C}}}.$ (77)

Here, $\nu_{+}$ and $\nu_{-}$ are the combined filling factor discontinuities
of the downstream $(+)$ and upstream $(-)$ edge modes respectively. They
satisfy the relation $\nu=\nu_{+}-\nu_{-}$ where $\nu$ is the bulk filling
factor. For the $\nu=9/5$ edge in regime $\mathcal{II}$, we have $\nu_{+}=2$
and $\nu_{-}=1/5$. The exponential factor in the integral is a result of
chiral, equilibrated charge transport as described above. It indicates that
noise is dominantly generated in a region of size $\sim l^{C}_{\rm eq}$ close
to the upstream contact, i.e., the noise spot.

The key quantity to compute in Eq. (77) is the local noise kernel

$\displaystyle\Lambda(x)\equiv\frac{S_{\rm loc}[\delta
V(x),T_{+}(x),T_{-}(x),\Delta]}{2g_{\rm loc}[\delta
V(x),T_{+}(x),T_{-}(x),\Delta]}.$ (78)

It is composed of $S_{\rm loc}$ and $g_{\rm loc}$ which is the local electron
tunneling dc noise and the (dimensionless) tunneling conductance,
respectively. Importantly, the conductance $g_{\rm loc}\propto\ell^{-1}_{\rm
C}$, where the proportionality factor is the typical distance between
scattering points [43, 40]. Both $S_{\rm loc}$ and $g_{\rm loc}$ depend on
microscopic details of the edge: Inter-channel interactions, the edge disorder
strength, the local voltage difference between the modes $\delta V(x)$, and
the effective temperatures $T_{\pm}(x)$ of downstream and upstream edge modes.
Importantly, the interactions enter via the scaling dimension $\Delta$ of the
most relevant inter-channel charge tunneling operator (7).

Under the condition (76), both the voltage difference and the temperatures are
to excellent approximation constant across the noise spot:

$\displaystyle\delta V(x)\approx 0\quad\text{and}\quad T_{\pm}(x)\approx
T_{\pm}.$ (79)

The first approximation holds because the channels equilibrate to the same
voltage along the edge (except at the hot spot), whereas the second holds
because of assumed poor thermal equilibration. With the approximations (79) we
can write

$\displaystyle\Lambda(x)\approx\Lambda(T_{\pm},\Delta).$ (80)

By using Eq. (80) in Eq. (77), the integral can be trivially performed to give

$\displaystyle S=\frac{9e^{2}}{50h}\Lambda(T_{\pm},\Delta).$ (81)

We now want to find $\Lambda(T_{\pm},\Delta)$. The first step is to specify
the edge tunneling operator, which we take as the null operator (42) with
$\mathbf{m}_{1}$ from Eq. (41). We use the basis (40).

For this tunneling process, we next compute $\Lambda(T_{\pm},\Delta)$ with the
approach described in Ref. [26] (see also Sec. III.4 below). For weak
tunneling, a perturbative approach for the local noise and conductance gives

$\displaystyle S_{\rm loc}\propto 2\int dt\;\cos\left(\delta
Vt\right)\langle\mathcal{V}_{\mathbf{m}_{1}}(t,0)\mathcal{V}^{\dagger}_{\mathbf{m}_{1}}(0,0)\rangle$
(82) $\displaystyle g_{\rm loc}\propto i\partial_{\delta V}\left[\int
dt\;\sin\left(\delta
Vt\right)\langle\mathcal{V}_{\mathbf{m}_{1}}(t,0)\mathcal{V}^{\dagger}_{\mathbf{m}_{1}}(0,0)\rangle\right]_{\delta
V\rightarrow 0}.$ (83)

Here, the non-universal proportionality constant depends on a short distance
cutoff, but importantly, the constant is the same for $S_{\rm loc}$ and
$g_{\rm loc}$ and therefore cancels in $\Lambda$. Inserting the above
expressions into Eq. (78) and using $\delta V=0$ gives

$\displaystyle\Lambda(T_{\pm},\Delta)=\frac{\int
dt\;\langle\mathcal{V}_{\mathbf{m}_{1}}(t,0)\mathcal{V}^{\dagger}_{\mathbf{m}_{1}}(0,0)\rangle}{i\int
dt\;t\langle\mathcal{V}_{\mathbf{m}_{1}}(t,0)\mathcal{V}^{\dagger}_{\mathbf{m}_{1}}(0,0)\rangle}.$
(84)

In Eq. (84) the temperature dependence enters via the correlation functions

$\displaystyle\langle\mathcal{V}_{\mathbf{m}_{1}}(t,0)\mathcal{V}^{\dagger}_{\mathbf{m}_{1}}(0,0)\rangle\propto\prod_{j=1,2,3}\tilde{G}_{j}(t,0)^{\left((\mathbf{\tilde{m}}_{1})_{j}\right)^{2}},$
(85)

where the finite temperature Green’s functions

$\displaystyle\tilde{G}_{j}(t,0)=\frac{\pi
aT_{j}/\tilde{v}_{j}}{\sin\left[\frac{\pi
T_{j}}{\tilde{v}_{j}}(a-i\tilde{v}_{j}t)\right]},\quad j\in\\{1,2,3\\},$ (86)

Note that the Green’s functions are those of of the modes in the diagonal
basis (14). From Eq. (25), we have that the exponents in (85) satisfy

$\displaystyle\left((\mathbf{\tilde{m}}_{1})_{1}\right)^{2}+\left((\mathbf{\tilde{m}}_{1})_{2}\right)^{2}+\left((\mathbf{\tilde{m}}_{1})_{3}\right)^{2}=2\Delta.$
(87)

The formula for the noise (77) assumes that all downstream modes have the
temperature $T_{+}$, and the upstream modes are at $T_{-}$. We thus set

$\displaystyle T_{1}=T_{2}=T_{+},$ (88a) $\displaystyle T_{3}=T_{-}.$ (88b)

By plugging these temperatures into the Green’s functions (86), and expanding
to leading order in $a$, we can cast Eq. (84) on the form

$\displaystyle\Lambda(T_{\pm},\Delta)=$ $\displaystyle\frac{\int
dz\sin\left[\frac{T_{+}}{T_{-}}(\frac{\pi}{2}+iz)\right]^{-\delta_{+}}\cosh\left[z\right]^{-\delta_{-}}}{\int
dz\left(\frac{1}{2T_{-}}+\frac{iz}{\pi
T_{-}}\right)\sin\left[\frac{T_{+}}{T_{-}}(\frac{\pi}{2}+iz)\right]^{-\delta_{+}}\cosh\left[z\right]^{-\delta_{-}}},$
(89)

where the exponents

$\displaystyle\delta_{+}\equiv\left((\mathbf{\tilde{m}}_{1})_{1}\right)^{2}+\left((\mathbf{\tilde{m}}_{1})_{2}\right)^{2},$
(90a)
$\displaystyle\delta_{-}\equiv\left((\mathbf{\tilde{m}}_{1})_{3}\right)^{2}.$
(90b)

Our next step is to find $T_{\pm}$ in terms of the bias voltage $V_{0}$. When
$n_{+}$ downstream and $n_{-}$ upstream edge modes are not thermally
equilibrated, the local downstream and upstream temperatures at the noise
spot, $T_{\pm}$, were computed in Ref. [26]. They are given as

$\displaystyle k_{\rm
B}T_{+}=\left(\frac{6hP}{\pi^{2}}\times\frac{\mathcal{R}(n_{-}+\mathcal{R})}{(n_{+}+n_{-})(n_{+}n_{-}-\mathcal{R}^{2})}\right)^{1/2},$
(91a) $\displaystyle k_{\rm
B}T_{-}=\left(\frac{6hP}{\pi^{2}}\times\frac{n_{+}(n_{-}+\mathcal{R})}{(n_{+}+n_{-})(n_{+}n_{-}-\mathcal{R}^{2})}\right)^{1/2}.$
(91b)

Here $\mathcal{R}\in[0,1]$ is the reflection coefficient between the contacts
and the edge. This reflection depends explicitly on the sharp change in
interaction strength between the contact region and the edge [38, 53, 26] (see
Fig. 4). We have assumed that both contacts have the same $\mathcal{R}$. Eq.
(91) also includes $P$, the power dissipated in the hot spot, which was
computed in Ref. [44] as

$\displaystyle
P=\frac{e^{2}V_{0}^{2}}{2h}\times\frac{(\nu_{+}-\nu_{-})\nu_{-}}{\nu_{+}},$
(92)

under the assumption (76). For the $\nu=9/5$ edge in regime $\mathcal{II}$, we
have $\nu_{+}=2$, $\nu_{-}=1/5$, $n_{+}=2$, and $n_{-}=1$. Plugging these
numbers into Eqs. (91) and (92), we find

$\displaystyle P=\frac{9e^{2}V_{0}^{2}}{100h},$ (93)

and

$\displaystyle k_{\rm
B}T_{+}=\frac{3eV_{0}}{5\pi}\sqrt{\frac{\mathcal{R}(1+\mathcal{R})}{4-2\mathcal{R}^{2}}},$
(94a) $\displaystyle k_{\rm
B}T_{-}=\frac{3eV_{0}}{5\pi}\sqrt{\frac{1+\mathcal{R}}{2-\mathcal{R}^{2}}}.$
(94b)

For $\mathcal{R}=0$, which corresponds to vanishing edge interactions, we have
$k_{\rm B}T_{+}=0$ and $k_{\rm B}T_{-}=3eV_{0}/(5\pi\sqrt{2})$. This is the
situation where the downstream channels remain in equilibrium with the left
contact (in our approximation, at $T=0$) and only the upstream channel is
heated by dissipation at the hot spot. Even in the absence of thermal
equilibration by edge impurities, a finite reflection probability at the two
contacts distributes the hot spot power to all edge channels.

Figure 5:  Noise $S$ vs the bias current $I_{0}$ for different values of the
scaling dimensions $\delta_{\pm}$ [see Eq. (90)] and the edge contact
reflection coefficient $\mathcal{R}$. The calculation is done for the
$\nu=9/5$ edge under the condition (76) for the edge equilibration length
scales.

For simpler comparison with the experimental convention of plotting $S$ versus
the source current $I_{0}$, we next convert the bias voltage $V_{0}$ to
$I_{0}$ via

$\displaystyle I_{0}=\frac{9}{5}\frac{e^{2}}{h}V_{0},$ (95)

which is valid under the condition (76).

We now have all ingredients to compute the noise in Eq. (81) for a given bias
current $I_{0}$. We evaluate the integrals in Eq. (84) numerically for various
values of $\mathcal{R}$ and $\delta_{\pm}$ and plot the result in Fig. 5.
Since we expect the condition (76) to be obtained for strong interactions
which also tend to increase $\mathcal{R}$, we limit the range of $\mathcal{R}$
to the range $\mathcal{R}\in[0.5,1]$, for consistency. For experimental
comparison, we have reinstated experimentally convenient units where noise is
measured in $10^{-29}\rm{A}^{2}/Hz$ and currents in $\rm{nA}$. Qualitatively,
we see that for fixed $\mathcal{R}$, the dependencies on $\delta_{\pm}$ are
quite weak, whereas the dependence on $\mathcal{R}$ is pronounced. This can be
understood on physical grounds since $\mathcal{R}$ strongly affects the
temperatures $T_{\pm}$ at the noise spot. By inspection, we give the rough
estimate

$\displaystyle S\approx 0.25-0.7\times
10^{-29}\frac{\rm{A}^{2}}{\rm{nA}\rm{Hz}}.$ (96)

This magnitude is around half of that detected for the $\nu=2/3$ edge in Ref.
[26]. The noise in regime $\mathcal{II}$ should therefore be detectable with
present technology.

###  III.4 Shot noise in a QPC device

As a complement to the transport characteristics in previous subsections, we
here compute the electrical shot noise generated by current partitioning in a
QPC device [56, 57, 58, 59]. We consider both the WBS regime (see Figs. 6a and
6c), allowing tunneling of fractionally charged quasiparticles through the FQH
bulk, and the SBS regime, which only allows electron tunneling across the non-
topological vacuum (Figs. 6b and 6d).

We begin by deriving formulas for the current and shot noise in point
tunneling between two counter-propagating edge channels. We then apply these
formulas to obtain Fano factors for the bound and free phases, in both SBS and
WBS regimes. We follow closely the perturbative Keldysh approach from Refs.
[53], [42] and [60].

(a)

(b)

(c)

(d)

Figure 6: Schematics of a quantum point contact (QPC) device at filling
$\nu=3/5$. (a)-(b): In the free phase, the weak back-scattering (WBS) regime
favours tunneling of quasiparticles with charge $e/5$. For strong back-
scattering (SBS), electrons with charge $e$ tunnel. (c) In the bound phase,
the WBS regime is dominated by tunneling of fractional charges $3e/5$. (d) In
the SBS regime in the bound phase, single electron tunneling is strongly
suppressed at low energies and three electron co-tunneling, i.e., charge
packets of $3e$ is the most dominant process. All four types of charges can be
detected in shot noise measurements.

####  III.4.1 Derivation of a general noise formula

The edge channels in the QPC device is described by the effective Hamiltonian

$\displaystyle H_{0}=\pi\int
dx\Big{[}v_{1}\frac{\rho_{1}^{2}}{t_{1}^{2}}+v_{2}\frac{\rho_{2}^{2}}{t_{2}^{2}}\Big{]},$
(97)

where $\rho_{i}=t_{i}\partial_{x}\phi_{i}/(2\pi)$ ($i=1,2$) are the
neigbouring two counter-propagating charge densities (with velocities $v_{i}$)
on each side of the constriction. For simplicity, we ignore the fully
transmitted or reflected channels in the free phase. Their influence in the
form of inter-channel interactions are discussed below Eqs. (106), (124), and
(126). We further ignore all interactions across the constriction. The bosons
$\phi_{i}$ and their densities obey the following set of commutation relations

$\displaystyle\left[\phi_{i}(x),\phi_{j}(x^{\prime})\right]=i\pi\delta_{ij}\delta\nu_{i}\;\text{sgn}(x-x^{\prime}),$
(98)
$\displaystyle\left[\rho_{i}(x),\rho_{j}(x^{\prime})\right]=\frac{i}{2\pi}\delta_{ij}t_{i}^{2}\delta\nu_{i}\partial_{x}\delta(x-x^{\prime}),$
(99)
$\displaystyle\left[\phi_{i}(x),\rho_{j}(x^{\prime})\right]=i\delta_{ij}t_{i}\delta\nu_{i}\delta(x-x^{\prime}).$
(100)

Here, the filling factor discontinuities $\delta\nu_{i}$ and the charge vector
entries $t_{i}$ are kept unspecified for full generality. To describe charge
tunneling in the presence of a voltage bias, we introduce chemical potentials
and point tunneling (at position $x=0$) with

$\displaystyle H_{V}=-\int dx\Big{[}\mu_{1}\rho_{1}+\mu_{2}\rho_{2}\Big{]},$
(101) $\displaystyle H_{\tau}=\int
dx\delta(x)\Big{[}\Gamma_{0}\mathcal{T}_{\mathbf{l}}(x)+h.c.\Big{]}.$ (102)

Here, $\Gamma_{0}$ is the tunneling amplitude and

$\displaystyle\mathcal{T}_{\mathbf{l}}(x)=\frac{e^{il_{1}\phi_{1}(x)+il_{2}\phi_{2}(x)}}{2\pi
a},$ (103)

is a tunneling operator parametrized by $\mathbf{l}=(l_{1},l_{2})$ [cf. Eq.
(7)]. The length $a$ is our UV distance cutoff, e.g., the magnetic length.
According to Eq. (9), the charge created by $e^{il_{1}\phi_{1}}$ is

$\displaystyle q_{1}=t_{1}\delta\nu_{1}l_{1}.$ (104)

Similarly, $e^{il_{2}\phi_{2}}$ creates charge

$\displaystyle q_{2}=t_{2}l_{2}\delta\nu_{2}.$ (105)

Conservation of charge in the tunneling process restricts $l_{1}$ and $l_{2}$
such that

$\displaystyle q_{1}=q_{2}\equiv q$ (106)

for a given set $\\{t_{1},t_{2},\delta\nu_{1},\delta\nu_{2}\\}$. Note that
since Eq. (9) is invariant under basis rotations, the charges $q_{1}$,
$q_{2}$, and $q$ are all independent of possible interactions with fully
transmitted and reflected channels in the free phase.

Next, the channel $i$ current operator is defined as

$\displaystyle I_{i}$ $\displaystyle=\frac{d}{dt}\left(\int
dx\rho_{i}\right)=\frac{1}{i}\left[\int
dx\rho_{i},H_{0}+H_{V}+H_{\tau}\right]$ $\displaystyle=\frac{1}{i}\left[\int
dx\rho_{i},H_{\tau}\right],$ (107)

where we used that $\rho_{i}$ commutes with the quadratic $H_{0}$ and $H_{V}$.
The commutator with $H_{\tau}$ yields

$\displaystyle I_{1}=-I_{2}\equiv
I=iq\Gamma_{0}\mathcal{T}_{\mathbf{l}}(0)+h.c.$ (108)

In the interaction picture with $H_{\tau}$ as the interaction Hamiltonian, the
operator $A$ time-evolve as

$A(t)=e^{i(H_{0}+H_{V})t}Ae^{-i(H_{0}+H_{V})t}.$ (109)

Applying this formula to $I$ and $H_{\tau}$, we find the time-evolutions

$\displaystyle
H_{\tau}(t)=e^{-i\omega_{0}t}\Gamma_{0}\mathcal{T}_{\mathbf{l}}(0)+h.c.,$
(110a) $\displaystyle
I(t)=iqe^{-i\omega_{0}t}\Gamma_{0}\mathcal{T}_{\mathbf{l}}(0)+h.c.,$ (110b)

where the characteristic “Josephson frequency” of the tunneling process

$\displaystyle\omega_{0}\equiv(t_{1}l_{1}\delta\nu_{1}\mu_{1}-t_{2}l_{2}\delta\nu_{2}\mu_{2})=q(\mu_{1}-\mu_{2}).$
(111)

In the second equality here, we used the charge conservation condition (106).
The relation between $\omega_{0}$ and the voltage across the constriction
reads $\hbar\omega_{0}=qe(V_{1}-V_{2})$.

Next, we compute the expectation value of the current $\langle I(t)\rangle$ on
the Keldysh contour

$\langle I(t)\rangle=\frac{1}{2}\sum_{\lambda=\pm 1}\langle
T_{C}\left[I(t^{\lambda})\exp\left(-i\int_{C}dt_{0}H_{\tau}(t_{0}^{\lambda^{\prime}})\right)\right]\rangle.$
(112)

Here, the Keldysh time-ordering operator, $T_{C}$ orders along the Keldysh
contour $C$: $-\infty\rightarrow+\infty\rightarrow-\infty$. We denote the
”upper” branch $-\infty\rightarrow+\infty$ by $\lambda=+1$ and lower branch
$+\infty\rightarrow-\infty$ with $\lambda=-1$. The ordering acts according to
$t^{-}>t_{0}^{+}$ for all $t$ and $t_{0}$; $t^{+}>t_{0}^{+}$ for $t>t_{0}$;
and $t^{-}>t_{0}^{-}$ for $t<t_{0}$. Since our $H_{\tau}$ only depends on a
single time argument, we have used a symmetric combination on both branches,
compensated for with the factor of $1/2$.

To second order in $\Gamma_{0}$, we find

$\displaystyle\langle
I(t)\rangle=\frac{q|\Gamma_{0}|^{2}}{2}\sum_{\lambda,\lambda^{\prime}=\pm
1}\lambda^{\prime}\int_{-\infty}^{\infty}dt_{0}\sin\big{[}\omega_{0}(t^{\lambda}-t_{0}^{\lambda^{\prime}})\big{]}\times$
$\displaystyle\langle
T_{C}\mathcal{T}_{\mathbf{l}}(0,t^{\lambda})\mathcal{T}_{\mathbf{l}}^{\dagger}(0,t_{0}^{\lambda^{\prime}})\rangle,$
(113)

where we used that the tunneling operators are normal ordered. Next, we use
the finite temperature Keldysh Green’s functions for the tunneling operators

$\displaystyle\langle
T_{C}\mathcal{T}(0,t^{\lambda})\mathcal{T}_{\mathbf{l}}^{\dagger}(0,t_{0}^{\lambda^{\prime}})\rangle=\frac{1}{(2\pi
a)^{2}}\times$ $\displaystyle\prod_{j=1,2}\left(\frac{\pi
a/(\beta_{j}v_{j})}{\sin\left[\frac{\pi\left(a+i\chi_{\lambda,\lambda^{\prime}}(t-t_{0})\left(v_{j}(t-t_{0})\right)\right)}{v_{j}\beta_{j}}\right]}\right)^{2d_{j}},$
(114)

where
$\chi_{\lambda,\lambda^{\prime}}(t-t_{0})=\text{sgn}(t-t_{0})(\lambda+\lambda^{\prime})/2-(\lambda-\lambda^{\prime})/2$,
$\beta_{j}=1/T_{j}$ is the inverse temperature of channel $j$, and
$d_{j}=l_{j}|\delta\nu_{j}|/2$ is the scaling dimension of
$\exp(il_{i}\phi_{i})$. The total scaling dimension is then
$\Delta(\mathbf{l})=\sum_{j}d_{j}$. Inserting the Green’s functions (III.4.1)
into Eq. (III.4.1), we obtain

$\displaystyle\langle I(t)\rangle=\frac{q|\Gamma_{0}|^{2}}{(2\pi
a)^{2}}\prod_{k}\left(\frac{\pi
a}{v_{k}\beta_{k}}\right)^{2d_{k}}\sum_{\lambda,\lambda^{\prime}=\pm
1}\lambda^{\prime}\times$
$\displaystyle\int_{-\infty}^{\infty}dt_{0}\frac{\sin\big{[}\omega_{0}(t^{\lambda}-t_{0}^{\lambda^{\prime}})\big{]}}{\prod_{j}\sin\left[\frac{\pi}{v_{j}\beta_{j}}\left(a+i\chi_{\lambda,\lambda^{\prime}}(t-t_{0})v_{j}(t-t_{0}\right)\right]^{2d_{j}}}$
$\displaystyle=\frac{q|\Gamma_{0}|^{2}}{(2\pi a)^{2}}\prod_{k}\left(\frac{\pi
a}{v_{k}\beta_{k}}\right)^{2d_{k}}\sum_{\lambda=\pm 1}\lambda\times$
$\displaystyle\int_{-\infty}^{\infty}dt_{0}\frac{\sin\big{[}\omega_{0}(t-t_{0})\big{]}}{\prod_{j}\sin\left[\frac{\pi}{v_{j}\beta_{j}}\left(a-i\lambda
v_{j}(t-t_{0}\right)\right]^{2d_{j}}},$ (115)

In the second equality, we used that when $\lambda=\lambda^{\prime}$, the
integrand becomes odd in $t-t_{0}$ and hence this contribution vanishes. We
next assume that the two channels are at the same temperature: $T_{j}=T$ and
also that $aT/v_{j}<1$ for $j=1,2$. We then change variables
$t-t_{0}\rightarrow\lambda(t_{0}+i/(2T))$. To lowest order in $b$, the
integral becomes

$\displaystyle\langle I(t)\rangle=\frac{q|\Gamma_{0}|^{2}}{(2\pi
a)^{2}}\prod_{k}\left(\frac{\pi
a}{v_{k}\beta_{k}}\right)^{2d_{k}}\sum_{\lambda=\pm 1}\lambda\times$
$\displaystyle\int_{-\infty}^{\infty}dt_{0}\frac{\sin\big{[}-\lambda\omega_{0}(t_{0}+i/(2T))\big{]}}{\prod_{j}\cosh(\pi
Tt_{0})^{2d_{j}}}$ $\displaystyle=\frac{2q|\Gamma_{0}|^{2}}{(2\pi
a)^{2}}\prod_{k}\left(\frac{\pi
aT}{v_{k}}\right)^{2d_{k}}\sinh\left(\frac{\omega_{0}}{2T}\right)\times$
$\displaystyle\int_{-\infty}^{\infty}dt_{0}\frac{\cos\big{[}\omega_{0}t_{0}\big{]}}{\prod_{j}\cosh(\pi
Tt_{0})^{2d_{j}}}$ $\displaystyle=\frac{2q|\Gamma_{0}|^{2}}{(2\pi
a)^{2}}\left(\pi
aT\right)^{2\Delta(\mathbf{l})}\prod_{k}\left(\frac{1}{v_{k}}\right)^{2d_{k}}\sinh\left(\frac{\omega_{0}}{2T}\right)\times$
$\displaystyle\int_{-\infty}^{\infty}dt_{0}\frac{\cos\big{[}\omega_{0}t_{0}\big{]}}{\cosh(\pi
Tt_{0})^{2\Delta(\mathbf{l})}}.$ (116)

The final integral can now be performed with the identity
$\int_{-\infty}^{\infty}dt\cosh(2yt)/\cosh^{2x}t=2^{2x-1}\Gamma(x+y)\Gamma(x-y)/\Gamma(2x)$,
for $\textrm{Re}\,x>|\textrm{Re}\,y|$ and $\textrm{Re}\,x>0$, with $\Gamma(z)$
the Gamma-function. We then find the time-independent tunneling current

$\displaystyle\langle I\rangle\equiv\langle
I(t)\rangle=2q|\Gamma_{0}|^{2}\left(2\pi
a\right)^{2\Delta(\mathbf{l})-2}T^{2\Delta(\mathbf{l})-1}\times$
$\displaystyle\prod_{k}\left(\frac{1}{v_{k}}\right)^{2d_{k}}\sinh\left(\frac{\omega_{0}}{2T}\right)\frac{|\Gamma(\Delta(\mathbf{l})+i\frac{\omega_{0}}{2\pi
T})|^{2}}{\Gamma(2\Delta(\mathbf{l}))}.$ (117)

In the high temperature regime $\omega_{0}\ll T$, we find the Ohmic behaviour

$\displaystyle\langle I\rangle=g_{\tau}\frac{e^{2}}{h}(V_{1}-V_{2}),$ (118)

with the tunneling conductance

$\displaystyle g_{\tau}\equiv q^{2}\frac{|\Gamma_{0}|^{2}\left(2\pi
aT\right)^{2\Delta(\mathbf{l})-2}}{\prod_{k}v_{k}^{2d_{k}}}\frac{|\Gamma(\Delta(\mathbf{l}))|^{2}}{\Gamma(2\Delta(\mathbf{l}))}.$
(119)

We next consider the symmetrized noise

$\displaystyle S(t,t^{\prime})=\langle I(t)I(t^{\prime})\rangle+\langle
I(t^{\prime})I(t)\rangle-2\langle I(t)\rangle\langle I(t^{\prime})\rangle.$
(120)

To leading order in $\Gamma_{0}$, $S(t,t^{\prime})$ is given on the Keldysh
contour as

$\displaystyle S(t,t^{\prime})\equiv S(t-t^{\prime})=\sum_{\lambda=\pm
1}\langle T_{C}\left[I(t^{\lambda})I(t^{\prime-\lambda})\right]\rangle$
$\displaystyle=2q^{2}|\Gamma_{0}|^{2}\sum_{\lambda=\pm
1}\cos\big{[}\omega_{0}(t^{\lambda}-t_{0}^{\lambda^{\prime}})\big{]}\times$
$\displaystyle\langle
T_{C}\mathcal{T}_{\mathbf{l}}(0,t^{\lambda})\mathcal{T}_{\mathbf{l}}^{\dagger}(0,t_{0}^{\lambda^{\prime}})\rangle.$
(121)

We are interested in the zero frequency (dc) noise $S(\omega=0)\equiv\int
d(t-t_{0})S(t-t_{0})$. The calculations proceed in perfect analogy to those
for $\langle I\rangle$, and using Eq. (III.4.1), we arrive at

$\displaystyle
S(\omega=0)=2q\cosh\left(\frac{\omega_{0}}{2T}\right)\frac{\langle
I\rangle}{\sinh\left(\frac{\omega_{0}}{2T}\right)}=2q\langle
I\rangle\coth\left(\frac{\omega_{0}}{2T}\right).$ (122)

In the limit $\omega_{0}\ll T$, the noise approaches the equilibrium Nyqvist-
Johnson noise

$\displaystyle S(\omega=0)=4g_{\tau}\frac{e^{2}}{h}k_{\rm B}T.$ (123)

In the shot-noise limit $\omega_{0}\gg T$, the Fano factor $F$ becomes

$\displaystyle F\equiv\frac{S(\omega=0)}{2\langle
I\rangle}=q\coth\left(\frac{\omega_{0}}{2T}\right)\xrightarrow[\omega_{0}\gg
T]{}q.$ (124)

Eq. (124) manifest the well-known result that weak tunneling reveals the
charges of the transferred particles. By use of Eqs. (104), (105), the formula
(124) manifests how the tunneling charge is affected by generic charge vector
entries $t_{1}$ and $t_{2}$. We emphasize again that $F$ is independent of
interactions in the free phase.

Since the tunneling operator $\mathcal{T}_{\mathbf{l}}$ so far has been
treated as general, we must now determine the most relevant tunneling
processes in the SBS and WBS regimes.

For point tunneling, the tree level RG equation for the tunneling amplitude
$\Gamma_{0}$ can be read of directly from $H_{0}+H_{\tau}$. The equation reads

$\displaystyle\frac{d\Gamma_{0}}{d\ell}=\left[1-\Delta(\mathbf{l})\right]\Gamma_{0},$
(125)

where $\Delta(\mathbf{l})$ is the scaling dimension of
$\mathcal{T}_{\mathbf{l}}$ and $\ell$ is the running length scale. Under the
assumption of no inter-channel interactions across the constriction, the
scaling dimension is obtained from Eq. (25) as

$\displaystyle\Delta(\mathbf{l})=\frac{1}{2}(d_{1}+d_{2})=\frac{1}{2}(|\delta\nu_{1}|l_{1}^{2}+|\delta\nu_{2}|l_{2}^{2}).$
(126)

This relation holds strictly only in the bound phase, where there is only a
single channel propagating. In the free phase, with three edge channels, the
scaling dimensions will be affected by inter-channel interactions. However,
since the free phase is characterized by weak interactions, we expect that the
difference between Eq. (126) and the true, “renormalized” scaling dimensions
is small and can thus be ignored.

For the case of identical edge channels
$|\delta\nu_{1}|=|\delta\nu_{2}|\equiv|\delta\nu|$, charge conservation
requires $l_{1}=l_{2}\equiv l$. In this case Eq. (125) becomes

$\displaystyle\frac{d\Gamma_{0}}{d\ell}=\left(1-l^{2}|\delta\nu|\right)\Gamma_{0}.$
(127)

Hence, the most relevant tunneling process is obtained for $l=1$. Physically,
this means that single particle tunneling events dominate over multi particle
events. In the next sections, we apply Eqs. (124), and (127) to determine the
most dominant Fano factors for the two edge phases Eq. (40) and Eq. (45).

####  III.4.2 Fano factors for the free phase

For a $\nu=9/5$ device in the free phase, we have for the innermost channels
$t_{1}=t_{2}=1$ and $\delta\nu_{1}=-\delta\nu_{2}=1/5\equiv\delta\nu$ [see Eq.
(40)]. In the WBS regime, fractional charges may tunnel across the
constriction through the Hall fluid, and, as stated above, the most relevant
tunneling operator $\mathcal{T}_{\mathbf{l}}$ is obtained for $l=1$. The
operator then describe the transfer of particles with charge

$\displaystyle q=t\delta\nu l=1\times\frac{1}{5}\times 1=\frac{1}{5}.$ (128)

Using Eqs. (126) in Eq. (119), we see that the tunneling conductance for this
process scales with temperature in the Ohmic limit as $g_{\tau}\sim T^{-8/5}$.
Note that this scaling law strictly holds in the absence of interactions.
However, interactions are expected to only lead to small deviations from this
value of the scaling exponent. The divergence at zero temperature signals the
instability of the system towards quasiparticle tunneling. Thus, this type of
tunneling is only visible at high temperatures/voltages.

In the SBS regime, the left and right parts of the constriction are bridged by
a region fully depleted of Hall fluid. No FQH quasiparticles may exist in this
region, and hence only tunneling processes of electrons (integer charges)
couple the edges. The most dominant such process is obtained for $l=5$, which
gives a smallest possible integer charge of

$\displaystyle q=t\delta\nu l=1\times\frac{1}{5}\times 5=1,$ (129)

i.e., single electron tunneling. For this process, Eqs. (126) and (119) shows
that the tunneling conductance scaling with temperature becomes in the Ohmic
limit $g_{\tau}\sim T^{8}$. Also this scaling exponent may be modified by
interactions but we expect this effect to be small. The SBS regime is a stable
RG fixed point.

Direct application of Eq. (124) for these two types of processes give the Fano
factors for the free phase

$\displaystyle F_{WBS}=\frac{1}{5},$ (130) $\displaystyle F_{SBS}=1.$ (131)

Bases on these results, we predict that in the free phase, shot noise
measurements in the WBS and SBS regimes should reveal tunneling of fractional
charges $q=1/5$ and single electrons $q=1$ respectively.

####  III.4.3 Fano factors for the bound phase

In the bound phase, we have $t_{1}=t_{2}=3$ and
$\delta\nu_{1}=-\delta\nu_{2}=1/5$ [see Eq. (45)]. In the WBS regime, the most
relevant tunneling operator is again obtained for $l=1$, which amounts to
transfer of particles with charge

$\displaystyle q=t\delta\nu l=3\times\frac{1}{5}\times 1=\frac{3}{5}.$ (132)

In the SBS regime, we seek the most relevant operator transferring an integer
number of charges. From Eq. (104), we see that, once more, this operator is
found for $l=5$, which amounts to the tunneling of charge

$\displaystyle q=t\delta\nu l=3\times\frac{1}{5}\times 5=3.$ (133)

The conductance scaling laws in the WBS and SBS regimes are $g_{\tau}\sim
T^{-8/5}$ and $g_{\tau}\sim T^{8}$, respectively, i.e., the same as those
obtained for the bound phase. Application of Eq. (124) for the WBS and SBS
regimes gives

$\displaystyle F_{WBS}=\frac{3}{5},$ (134) $\displaystyle F_{SBS}=3.$ (135)

Eq. (135) is a key result of this section. It signals three-electron co-
tunneling as the most relevant process in the SBS regime. Since,
$t_{1}=t_{2}=t=3$, a process yielding $F_{SBS}=1$ is impossible at low
energies in the bound phase. Similarly, for WBS it not possible to observe
particles with charge $1/5$ in the bound phase. Measuring $F_{SBS}=3$ and
$F_{WBS}=3/5$ in the SBS respectively WBS regimes, is therefore a striking
manifestation of the $9/5$ edge in the bound phase.

####  III.4.4 Tunneling between the bound edge and a metal

As a consistency check of our equations, we consider finally also the
previously studied setup [18] of electron tunneling from an ordinary metal
(or, equivalently, an integer quantum Hall edge) into the bound $9/5$ edge. As
previously stated, only tunneling of electrons in bunches of three is
possible. We therefore have $t_{1}=1$, $\delta\nu_{1}=1$, and $l_{1}=3$,
respectively $t_{2}=3$, $\delta\nu_{2}=-1/5$, and $l_{2}=5$. Since
$q_{1}=t_{1}|\delta\nu_{1}|l_{1}=q_{2}=t_{2}|\delta\nu_{2}|l_{2}=3$, we
immediately find from Eq. (124) that $F=3$ for this tunneling. Furthermore, Eq
(126) gives the scaling dimension

$\displaystyle\Delta=\frac{1}{2}(1\times 3^{2}+\frac{1}{5}\times 5^{2})=7,$
(136)

for the most relevant tunneling operator. It follows that in the high-
temperature limit, Eq. (119) gives that the tunneling conductance scales as
$g_{\tau}\sim T^{2\Delta-2}=T^{12}$. In the low temperature limit we have from
dimensional analysis $\langle I\rangle\sim V^{2\Delta-1}=V^{13}$. These
scaling laws are precisely those derived in Ref. [18].

We end Sec. III by summarizing the key results in Tab. 1.

##  IV Discussion

Our analysis at $\nu=9/5$ can straightforwardly be adapted to other T-unstable
edge structures (several examples were given in Ref. [18]). Consider for
example an interface between bulk fillings $\nu=2/5$ and $\nu=1/7$. Here, we
assume that a FQH state and not a Wigner crystal forms at filling $\nu=1/7$ of
the electron gas.

The resulting edge structure is described by

$\displaystyle K=\begin{pmatrix}3&0&0\\\ 0&15&0\\\
0&0&-7\end{pmatrix},\quad\mathbf{t}^{T}=(1,1,1).$ (137)

According to Eq. (5), the “effective filling” of this structure is $\nu=9/35$.
The two null vectors are $\mathbf{m}_{1}=(1,10,7)^{T}$ and
$\mathbf{m}_{2}=(4,-5,7)^{T}$. Crossing over from the corresponding regimes
$\mathcal{III}$ to $\mathcal{II}$, we therefore expect conductance transitions
$G=9/35\rightarrow 19/35$ and $G^{Q}=1\rightarrow 3$ with decreasing
temperature.

A basis transformation of Eq. (137) on the form (15) leads to an alternative
representation

$\displaystyle K^{\prime}=\begin{pmatrix}35&0&0\\\ 0&1&0\\\
0&0&-1\end{pmatrix},\quad\mathbf{t^{\prime}}^{T}=(3,1,1).$ (138)

The binding transition amounts to the pair of counter-propagating integer
channels localizing. The remaining channel has a filling factor discontinuity
$\delta\nu=1/35$ and it is made out of $3e$ composites. Hence, the bound phase
is characterized by

$\displaystyle K^{\prime}=(35),\quad\mathbf{t^{\prime}}^{T}=(3).$ (139)

In the bound phase, we therefore expect conductances $G=9/35$ and $G^{Q}=1$ in
regime $\mathcal{I}$.

Tunneling in a QPC bridging two interfaced edges is geometrically complicated
and we do not analyse it here. However, electron tunneling into the interface
edge from a metal (e.g., an STM tip or a $\nu=1$ QH state) is conceptually
simpler and we use Eq. (126) to compute the scaling dimension of the tunneling
operator (7) transferring three electrons. The result is

$\displaystyle\Delta=\frac{1}{2}(1\times 3^{2}+\frac{1}{35}\times 35^{2})=22,$
(140)

where we used $t_{1}=1$, $\delta\nu_{1}=1$, and $l_{1}=3$, respectively
$t_{2}=3$, $\delta\nu_{2}=-1/35$, and $l_{2}=35$. Then from Eqs. (104), (105)
, and (106), we have
$q_{1}=t_{1}|\delta\nu_{1}|l_{1}=q_{2}=t_{2}|\delta\nu_{2}|l_{2}=3$. With
these values, Eq. (124) produces $F=3$ as expected for this tunneling. From
Eq. (119), we have that the tunneling conductance scales with temperature as
$g_{\tau}\sim T^{2\Delta-2}=T^{42}$. In the low temperature limit, the voltage
scaling of the current reads $\langle I\rangle\sim V^{2\Delta-1}=V^{41}$.

We now move on to discuss practical aspects of experimentally detecting the
binding transition. The transition at filling $\nu=9/5$ can be expected to
require tunneling between Landau levels with different spin-polarization.
Breaking spin-rotation symmetry, for example by spin-orbit coupling, is
therefore a necessary condition. A bi-layered device could be used to
facilitate the binding transition [18]. Alternatively, carefully designed
devices in the spirit of Ref. [29] where all channels have the same spin may
be another possibility.

In our view, the most standard probe of the binding transition should be a
measurement of the shot noise in a QPC device. From Sec. III.4, we anticipate
that, with decreasing temperature, a binding transition at $\nu=9/5$ amounts
to the crossover from $F_{SBS}=1$ to $F_{SBS}=3$ in the strong back-scattering
regime of the QPC. Similarly, for weak back-scattering, one expects a
crossover from $F_{WBS}=1/5$ to $F_{WBS}=3/5$. Changes in Fano factors with
decreasing temperature were recently measured in Ref. [61]. It would be
interesting to investigate whether these changes could arise due to a
combination of edge reconstruction and binding transitions.

##  V Summary and Outlook

We proposed quantum transport signatures for the FQH edge binding transition,
with focus on filling $\nu=9/5$. For this edge, we showed that interactions
and disorder conspire to generate a rich phase diagram (Fig. 2) with distinct
charge and heat transport regimes (see Tab. 1). The three regimes, labelled
$\mathcal{I}$, $\mathcal{II}$, and $\mathcal{III}$ displays localized, non-
equilibrated, and fully equilibrated characteristics. Probing the distinct
transport behaviour, in terms of charge and heat conductances, of these
regimes should be possible with present technology. As a complement to the
conductance, we also estimated the shot noise produced of a single current
biased edge segment. We demonstrated that such noise is only expected for the
thermally non-equilibrated edge (regime $\mathcal{II}$) under the strong
interaction condition (76) associated with the bound phase.

We also studied a QPC device in the strong and weak backscattering regimes and
derived shot noise Fano factors for tunneling processes across the
constriction. The bound phase does not allow single electron tunneling in the
strong back-scattering (SBS) regime. Instead of the typical Fano factor
$F_{SBS}=1$ corresponding to single electron tunneling, we therefore found
that the smallest Fano factor compatible with transferring an integer number
of charges is $F_{SBS}=3$. This corresponds to three-electron co-tunneling. In
addition, all higher order (but less relevant) tunneling processes of
electrons are necessarily integer multiples of 3. In the weak back-scattering
(WBS) regime, we found that the most relevant tunneling of quasiparticles
yields $F_{SBS}=3/5$. These SBS and WBS Fano factors are in stark contrast to
those for the free phase: $F_{SBS}=1$ respectively $F_{WBS}=1/5$. These
contrasting values serve as a clear signature for the binding transition.

We hope that our predictions and proposals will stimulate further theoretical
and experimental investigations of FQH binding transitions. While our present
analysis focused on Abelian FQH edges, binding transitions for non-Abelian
candidate edge theories for the state filling $\nu=5/2$ were studied in Ref.
[62]. An experimentally oriented analysis similar to the present work could be
useful for pin-pointing that state’s underlying topological order.

###### Acknowledgements.

C.S. thanks Jinhong Park, Gu Zhang, and Kyrylo Snizhko for discussions. C.S.
and A.D.M acknowledge support from the DFG grant No. MI 658/10-2 and the
German-Israeli Foundation grant I-1505-303.10/2019. C.S. further acknowledge
funding from the EI Nano Excellence Initiative at Chalmers University of
Technology. This project has received funding from the European Union’s
Horizon 2020 research and innovation programme under grant agreement No
101031655 (TEAPOT). This work was also supported by the European Union’s
Horizon 2020 research and innovation programme (Grant Agreement LEGOTOP No.
788715), the DFG (CRC/Transregio 183, EI 519/7-1), ISF Quantum Science and
Technology (2074/19).

##  Appendix A Dimensions of the disorder strength $D$

A generic tunneling operator $\mathcal{T}_{\mathbf{l}}$ has length ($L$)
dimensions

$\displaystyle L^{-m/2},$ (141)

where $m$ is the number of involved vertex operators ($m=3$ for the $\nu=9/5$
edge). The action (10) is dimensionless, so the random tunneling amplitude
$\xi(x)$ has dimensions

$\displaystyle L^{m/2-1}s^{-1}.$ (142)

where $s$ is the unit of time. From Eq. (28), we then have that the units of
$D$ are

$\displaystyle L^{m-1}s^{-2}.$ (143)

Hence, to obtain a dimensionless $D$ we let

$\displaystyle D\rightarrow D\times\frac{a^{3-m}}{v^{2}},$ (144)

with some characteristic velocity $v$ (a combination of all $v_{i}$ to various
powers such that the total power is $2$). This result is consistent with Ref.
[35] in which $m=2$.

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