# Thermal conductance and noise of Majorana modes along interfaced $\nu=5/2$
fractional quantum Hall states

Michael Hein  Department of Physics, University of Konstanz, D-78457 Konstanz,
Germany  Department of Microtechnology and Nanoscience (MC2),Chalmers
University of Technology, S-412 96 Göteborg, Sweden  Christian Spånslätt
Department of Microtechnology and Nanoscience (MC2),Chalmers University of
Technology, S-412 96 Göteborg, Sweden

(August 28, 2024)

###### Abstract

Identifying the topological order of the fractional quantum Hall state at
filling $\nu=5/2$ is an important step towards realizing non-Abelian Majorana
modes in condensed matter physics. However, to unambiguously distinguish
between various proposals for this order is a formidable challenge. Here, we
present a detailed study of transport along interfaced edge segments of
fractional quantum Hall states hosting non-Abelian Majorana modes. With an
incoherent model approach, we compute, for edge segments based on Pfaffian,
anti-Pfaffian, and particle-hole-Pfaffian topological orders, thermal
conductances, voltage biased charge current noise, and delta-$T$ noise. We
determine how the thermal equilibration of edge modes impacts these
observables and identify the temperature scalings of transitions between
regimes of differently quantized thermal conductances. In combination with
recent experimental data, we use our results to estimate thermal and charge
equilibration lengths in real devices. We also propose an experimental setup
which permits measuring several transport observables for interfaced
fractional quantum Hall edges in a single device. It can, e.g., be used to
rule out edge reconstruction effects. In this context, we further point out
some subtleties in two-terminal thermal conductance measurements and how to
remedy them. Our findings are consistent with recent experimental results
pointing towards a particle-hole-Pfaffian topological order at filling
$\nu=5/2$ in GaAs/AlGaAs, and provide further means to pin-point the edge
structure at this filling and possibly also other exotic fractional quantum
Hall states.

##  I Introduction

Non-Abelian anyons are exotic excitations in two-dimensional condensed matter
systems with no counterpart in particle physics [1]. A system with non-Abelian
anyons has a ground state degeneracy, and interchanging – or “braiding – the
anyons can shift the system between different ground states. This shift
depends only on the order of exchanges and information about the braiding
process is stored globally in the ground state wavefunction. Besides
signifying a strongly correlated phase of matter, non-Abelian braiding forms
the basis for the appealing, but so-far speculative, idea of topologically
protected quantum computations [2, 3].

A most promising candidate system to host non-Abelian anyons is the fractional
quantum Hall (FQH) state [4, 5] at filling $\nu=5/2$ [6, 7, 8, 9]. The quantum
state that is realized at this filling is however still not fully understood
and poses a long standing question in condensed matter physics (see Ref. [10]
for a recent overview).

The purpose of this paper is to model FQH edge transport involving various
candidate states at this filling, in light of recent experiments [11, 12] that
have brought progress towards identifying the $5/2$ state. To date, the three
most prominent candidate states are the Pfaffian (Pf) [7], anti-Pfaffian (aPf)
[13, 14], and particle-hole-Pfaffian (phPf) [15, 16, 17, 18]. Whereas
numerical calculations have seemed to favor the aPf state [19, 20, 21, 22, 23,
24, 25], tunneling experiments are more consistent with the aPf, or the
Abelian SU(2)2, 331, or 113 states [26, 27, 28].

Figure 1: Edge structures for the Pfaffian (Pf), anti-Pfaffian (aPf), and
particle-hole-Pfaffian (phPf) candidate states at filling $\nu=5/2$. Solid,
double arrows (in red, blue, and green) describe integer modes labelled
$\phi_{1}$, $\phi_{2}$, and $\phi_{3}$. Black, solid lines describe fractional
$1/2$ bosonic modes $\phi_{4}$, and purple, dashed arrows are Majorana modes
$\psi$. The edge modes belong to the two lowest Landau levels (LLLs) or the
second (2LL) Landau level. Vertical arrows depict the expected Landau level
spin polarizations.

Although all three non-Abelian candidates are (by construction) compatible
with the observed Hall conductance $G_{H}/(e^{2}/h)=\nu=5/2$, they have
distinct bulk topological orders [29]. Via the bulk-boundary correspondence,
this order is in turn manifested by different edge structures, i.e., the
number, chirality, and type of channels (or “modes”) propagating around the
FQH edge. As depicted in Fig. 1, the proposed edge structures of the Pf, aPf,
and phPf states differ only in the second Landau level (2LL). This distinction
can be quantified by the topological quantum number

$\displaystyle\nu_{Q}\equiv c-\bar{c},$ (1)

where $c$ and $\bar{c}$ are the total central charges for the chiral and anti-
chiral sectors, respectively, in the underlying conformal field theory [30].
For Abelian FQH edges, $c$ ($\bar{c}$) equals the number of “downstream”
(“upstream”) channels (where the downstream direction is defined as that of
the equilibrated charge flow). For more exotic edge structures, $c$ and
$\bar{c}$ may take rational values, e.g., $c=1/2$ for a single non-Abelian
Majorana mode (MM). As can be seen in Fig. 1, $\nu_{Q}^{\rm Pf}=7/2$,
$\nu_{Q}^{\rm aPf}=3/2$, and $\nu_{Q}^{\rm phPf}=5/2$.

Quite remarkably, the abstract quantity (1) can be related to the
experimentally accessible edge thermal conductance $G^{Q}$ according to the
relation [31, 32]

$\displaystyle G^{Q}=\nu_{Q}\kappa_{0}\bar{T},$ (2)

in which $\kappa_{0}\bar{T}\equiv\pi^{2}k_{\rm B}^{2}/(3h)\bar{T}$ is the
quantum of heat (with $k_{\rm B}$ and $h$ the Boltzmann and Planck constants,
respectively, and $\bar{T}$ is the temperature).

Table 1: Summary of results for the two terminal thermal conductance
$G^{Q}_{\rm 2T}$, the thermal Hall conductance $G^{Q}_{H}$ as well as charge
current noise and delta-$T$ noises for FQH interfaces phPf-$n$, aPf-$n$, and
Pf-$n$, where $n\in\\{2,3\\}$. Expressions are given in the limits of
vanishing, $\alpha,\beta\to 0$, and fully developed, $\alpha,\beta\to\infty$,
thermal equilibration. We always assume full charge equilibration: $\delta\gg
1$ and no edge reconstruction. The units of noise are given in terms of
voltage bias $\Delta V$ or large thermal bias $\Delta T\gg\bar{T}$. The two-
terminal charge conductance $G_{\rm 2T}/(e^{2}/h)=1/2$ for all interfaces.
Interfaces with three modes: equilibration parameters $\alpha$ and $\beta$
---
Interface |  $G^{Q}_{\rm 2T}$ $[\kappa_{0}\bar{T}]$ |  $G^{Q}_{\rm H}$ $[\kappa_{0}\bar{T}]$
|  $\alpha,\beta\to 0$ |  $\alpha\to\infty$ $\beta\to 0$ |  $\alpha\to 0$ $\beta\to\infty$ |  $\alpha,\beta\to\infty$ | Universal value
phPf-$3$ |  5/2 |  1/2 |  3/2 |  1/2 | 1/2
aPf-$2$ and Pf-$3$ |  5/2 |  1/2 |  3/2 |  1/2 | 1/2
|  Voltage biased charge current noise $[\Delta Ve^{3}/h]$ |  Downstream excess delta-$T$ noise $[2G_{\rm 2T}k_{\rm B}\Delta T]$ |  Upstream excess delta-$T$ noise $[2G_{\rm 2T}k_{\rm B}\Delta T]$
|  $\alpha,\beta\to 0$ |  $\alpha,\beta\to\infty$ |  $\alpha,\beta\to 0$ |  $\alpha,\beta\to\infty$ |  $\alpha,\beta\to 0$ |  $\alpha,\beta\to\infty$
phPf-$3$ |  $\simeq 0.113$ |  $\simeq\exp(-\delta)$ |  $\simeq 0.865$ |  $\simeq 1$ |  $\simeq 0.365$ |  $\simeq\exp(-\delta)$
aPf-$2$ and Pf-$3$ |  $\simeq 0.152$ |  $\simeq 0.195$ |  $\simeq 1.361$ |  $\simeq 1$ |  $\simeq 0.438$ |  $\simeq 0.5$
Interfaces with two modes: equilibration parameter $\alpha$
|  $G^{Q}_{\rm 2T}$ $[\kappa_{0}\bar{T}]$ |  $G^{Q}_{\rm H}$ $[\kappa_{0}\bar{T}]$
| $\alpha\to 0$ | $\alpha\to\infty$ | Universal value
phPf-$2$ | 3/2 | 1/2 | 1/2
aPf-$3$ and Pf-$2$ | Universal: 3/2 | 3/2
|  Voltage biased charge current noise $[\Delta Ve^{3}/h]$ |  Downstream excess delta-$T$ noise $[2G_{\rm 2T}k_{\rm B}\Delta T]$ |  Upstream excess delta-$T$ noise $[2G_{\rm 2T}k_{\rm B}\Delta T]$
|  $\alpha\to 0$ |  $\alpha\to\infty$ |  $\alpha\to 0$ |  $\alpha\to\infty$ |  $\alpha\to 0$ |  $\alpha\to\infty$
phPf-$2$ |  $\simeq 0.086$ |  $\simeq\exp(-\delta_{23})$ |  $\simeq 0.899$ |  $\simeq 1$ |  $\simeq 0.338$ |  $\simeq\exp(-\delta_{23})$
aPf-$3$ and Pf-$2$ |  0 |  0 |  $1$ |  $1$ |  0 |  0
|  |  |  |  |  |

Two-terminal conductance measurements for QH edges, first demonstrated in Ref.
[33], have subsequently been performed for many FQH states in both GaAs/AlGaAs
[34, 35, 11, 36, 37] and graphene [38, 39, 40, 41]. In particular, Ref. [35]
reported $G^{Q}/(\kappa_{0}\bar{T})\approx 5/2$ at filling $\nu=5/2$, which
fits well with Eq. (1) for the phPf state (see Fig. 1). This value further
rules out Abelian candidate states, since those are incompatible with half-
integer $G^{Q}$. However, Eq. (2) is valid for the two-terminal conductance
only when the heat transport is fully equilibrated, i.e., when edge channels
exchange energy efficiently [42, 43, 44, 45, 46, 41]. This equilibration can
be quantified by a characteristic thermal equilibration length $\ell^{Q}_{\rm
eq}$, so that the condition for Eq. (2) to hold reads $\ell^{Q}_{\rm eq}\ll
L$, where $L$ is the edge length. Importantly, $\ell^{Q}_{\rm eq}$ is non-
universal and depends on microscopic details such as inter-channel
interactions, the edge disorder strength, and the temperature. It is worth
pointing out that an analogous condition, $\ell^{C}_{\rm eq}\ll L$, with
$\ell^{C}_{\rm eq}$ a characteristic charge equilibration length, is needed
also for robust charge conductance quantization for FQH edges with counter-
propagating modes ($c,\bar{c}\neq 0$). However, almost all FQH experiments to
date indicate that this condition is normally well fulfilled (see Refs. [47,
48] for exceptions).

The interpretation of the experiment in Ref. [35] as revealing a thermally
equilibrated phPf edge was therefore questioned, since
$G^{Q}/(\kappa_{0}\bar{T})\approx 5/2$ can be obtained also for a partially
equilibrated aPf edge [49, 50, 51, 52, 53]. The same value can, under certain
conditions, be obtained from models with random puddles of alternating non-
Abelian orders [54, 55, 56, 57, 58, 59], or from reentrant states due to
Landau level mixing [60].

Further progress has been made recently in GaAs/AlGaAs devices where the
$\nu=5/2$ state is interfaced with integer QH states [11, 12]. This results in
an effective $\nu=5/2-n$ edge, where $n=1,2,3$ (see Fig. 2). The basic idea is
that since all candidate states share two Abelian, integer edge channels
(coming from two filled LLs), successive elimination of them exposes the
remaining non-Abelian “$\nu=1/2$ structure” for which the states differ. This
elimination occurs either due to Anderson localization [61, 62, 63] or
efficient equilibration of the edge states. For $n=3$, a possible particle-
hole symmetry of the 2LL can be tested. In Ref. [11], the thermal conductance
for the $\nu=5/2-2$ and $\nu=5/2-3$ edges were both measured to
$G^{Q}/(\kappa_{0}\bar{T})\approx 1/2$. It was argued that this is only
possible if the underlying $\nu=5/2$ state is the phPf, thereby further
strengthening the case for this state. It has also been proposed that charge
conductance measurements of $5/2-n$ interfaces could distinguish between non-
Abelian candidate states [64].

Figure 2: Effective structures of interfaces between non-Abelian $\nu=5/2$
candidate states and integer states $n=1,2,3$. In this work, we focus on
$n=2,3$ (indicated by the frame), which are the integers that expose the
second Landau level. The interfaces with orange text below their name are
those for which the directions of equilibrated charge and heat transport are
opposite.

For the same type of interfaced structures but in another device, the authors
in Ref. [12] measured the excess noise for a current biased edge segment.
Previously, it was proposed [65, 66, 67, 53] that such noise discloses
important properties of the edge. More specifically, for full thermal and
charge equilibration of any edge structure, the noise $S$ scales with $L$ in
one of three possible classes: $S\simeq\exp(-L/\ell^{C}_{\rm eq})$ for
$\nu_{Q}>0$, $S\simeq\sqrt{\ell^{C}_{\rm eq}/L}$ for $\nu_{Q}=0$, or $S\simeq
const.$ for $\nu_{Q}<0$ [66]. This classification holds under conditions where
heat leakage into the QH bulk is negligible. If the heat leakage is efficient,
the noise is strongly suppressed in $L/\ell^{C}_{\rm eq}$ also for $\nu_{Q}=0$
and $\nu_{Q}<0$. By contrast, under conditions where the thermal equilibration
between downstream and upstream modes (if present) is negligible, the noise
scales as $S\simeq const.$, and absence of upstream modes implies identically
$S=0$. This noise classification is the result of the chiral nature of the
edge: For equilibrated charge transport, the voltage drop across a biased edge
segment occurs only close to one of the contacts (in the so-called hot spot)
whereas partitioning of charges, i.e., excess noise, is dominantly produced
close to the other contact (in the so-called noise spot), see Fig. 3. This
partitioning is enhanced with respect to the equilibrium noise only in the
presence of upstream modes that transport heat from the hot spot to the noise
spot. In turn, this heat transport depends strongly on $\nu_{Q}$ which leads
to the correspondence between $S$ and $\nu_{Q}$.

For an interfaced Pf edge, the classification above implies that only the
$5/2-3$ interface can generate excess noise, since only that interfacing
results in upstream modes. For the aPf, both $5/2-1$ and $5/2-2$ can result in
noise whereas $5/2-3$ cannot. Finally, for the phPf, all three interfaces host
counter-propagating modes and can therefore produce finite noise. However,
since $\nu_{Q}>0$ for the phPf interfaces [68], the noise is exponentially
suppressed in $L/\ell^{C}_{\rm eq}$ and finite noise should only emerge for
poor thermal equilibration, i.e., either for small $L$ and/or large
$\ell^{C}_{\rm eq}$.

Indeed, Ref. [12] reported finite excess noise for both $5/2-2$ and $5/2-3$
interfaces for short $L$, but the noise weakened significantly for larger $L$
(see Fig. 11 below). At the same time, the two-terminal charge conductance of
the interface, $G_{\rm 2T}$, was always measured to $G_{\rm
2T}/(e^{2}/h)=1/2$, indicating a well established charge equilibration between
downstream and upstream (if present) modes.

Taken together, the two experiments [11, 12] suggest that the $\nu=5/2$ state
in GaAs/AlGaAs is of phPf type. However, a detailed model of how the upstream
mode mediated noise is generated for interfaced $\nu=5/2$ edges remains
lacking. Moreover, the above interpretation of the noise measurements hinges
on the absence of edge reconstruction [69, 70, 71, 72]. This effect introduces
non-topological pairs of counter-propagating modes which complement the edge
structure from the bulk boundary correspondence. The addition of such modes
and conditions with poor thermal equilibration can result in noise generation
for any FQH edge, which complicates experimental interpretations.

In this work, we incorporate the qualitative noise and conductance analysis
above in a comprehensive theoretical model which further permits a
quantitative comparison with experimental data. To this end, we study
transport along interfaces between non-Abelian and integer $n$ edges with the
incoherent edge approach, recently developed in Refs. [44, 65, 66, 67, 53,
73]. We review the basics of this model in Sec. II. In Secs. III and Sec. IV,
we use the model to compute the thermal conductance and the noise,
respectively, for interfaced edge structures. We focus on $n=2,3$, since it is
those integers that expose the 2LL structure, see Fig. 2. A summary of the
calculations in these sections is given in Tab. 1. In Sec. V, we analyze the
temperature scalings of the thermal equilibration lengths. This scaling
permits us to analyze the temperature dependence of the noise and thermal
conductance. In Sec. VI, we compare our results to the experiments in Refs.
[11, 12] and provide estimates on thermal and charge equilibration lengths. We
then propose in Sec. VII a unified experimental setup allowing several
independent experiments for probing FQH edge structures in a single device. We
argue that this device is beneficial for ruling out edge reconstruction
effects as well as possible sample-to-sample differences between separate
devices probing noise and the thermal conductance. We summarize and conclude
our work in Sec. VIII. A number of technical calculations are delegated to
Appendices A-E.

##  II Model of edge transport

###  II.1 Charge and energy transport

Our starting point is the generic edge segment with two attached contacts,
depicted in Fig. 3. Charge transport along this segment is described by [44,
67]

$\displaystyle\partial_{x}\vec{V}(x)=\mathcal{M}_{V}\vec{V}(x).$ (3)

Here, $\vec{V}(x)\equiv\left[V_{1}(x),\ldots,V_{N}(x)\right]^{T}$ (with
$[...]^{T}$ denoting vector transpose) describes the local voltages of $N$
edge channels and the matrix

$\displaystyle\mathcal{M}_{V}=\begin{pmatrix}-\frac{\sum\limits_{n\neq
1}(l^{C}_{1,n})^{-1}}{\chi_{1}\nu_{1}}&\frac{(l^{C}_{1,2})^{-1}}{\chi_{1}\nu_{1}}&\dots&\frac{(l^{C}_{1,N})^{-1}}{\chi_{1}\nu_{1}}\\\\[3.0pt]
\frac{(l^{C}_{2,1})^{-1}}{\chi_{2}\nu_{2}}&-\frac{\sum\limits_{n\neq
2}(l^{C}_{2,n})^{-1}}{\chi_{2}\nu_{2}}&\dots&\frac{(l^{C}_{2,N})^{-1}}{\chi_{2}\nu_{2}}\\\\[3.0pt]
\vdots&\vdots&\ddots&\vdots\\\
\frac{(l^{C}_{N,1})^{-1}}{\chi_{N}\nu_{N}}&\frac{(l^{C}_{N,2})^{-1}}{\chi_{N}\nu_{N}}&\dots&-\frac{\sum\limits_{n\neq
N}(l^{C}_{N,n})^{-1}}{\chi_{N}\nu_{N}}\end{pmatrix}$ (4)

describes couplings between edge channels in terms of the channel chiralities
$\chi_{i}=\pm 1$ (with $+1$ and $-1$ corresponding to downstream and upstream
directions, respectively), filling factor discontinuities $\nu_{i}$, and
charge equilibration lengths $l^{C}_{i,j}=l^{C}_{j,i}$ between modes $i$ and
$j$. The microscopic content affecting these lengths can be obtained within a
chiral Luttinger liquid approach, see e.g., Refs. [46, 51, 39]. As follows, we
always label the modes of all edge structures in Fig. 2 with $i=1$ starting
from the top.

Figure 3: Schematics of the incoherent edge transport model for a single edge
segment with two attached contacts, L and R. A set of $N$ edge channels are
characterized by reservoirs with local voltages, $V_{n}(x)$, and temperatures,
$T_{n}(x)$, for $n=1,\dots,N$. Differences in these quantities drive inter-
channel charge and energy currents ($I_{n,m}^{\tau}$ and $J_{n,m}^{\tau}$
respectively) between the channels, which equilibrate the edge. For
equilibrated, chiral charge transport (defining the downstream, ds,
direction), Joule heating occurs only in the hot spot (red area) while noise
is dominantly produced in the noise spot (yellow area). Equations (3)-(6) are
obtained in the continuum limit $N\rightarrow\infty$, $\varepsilon\rightarrow
0$, with $N/\epsilon=const.$

The local electric currents
$\vec{I}(x)\equiv\left[I_{1}(x),\ldots,I_{N}(x)\right]^{T}$, corresponding to
$\vec{V}(x)$, obey a similar equation

$\partial_{x}\vec{I}(x)=\mathcal{M}_{I}\vec{I}(x),\qquad\mathcal{M}_{I}=\mathcal{D}_{I}\mathcal{M}_{V}\mathcal{D}^{-1}_{I},$
(5)

with the diagonal matrix
$\mathcal{D}_{I}=\text{diag}(\chi_{1}\nu_{1},\ldots,\chi_{n}\nu_{n})$. This
description of edge currents was presented also in Ref. [51] and can further
be mapped onto the capacitive circuit model in Ref. [74]. Coherence effects
between successive inter-channel scattering in all these descriptions are
neglected (e.g., due to thermal dephasing), which is a prerequisite for robust
conductance quantization [42, 43, 44, 67]. The present model therefore
describes incoherent transport along FQH edges, hence the name.

Similarly to the charge transport, edge energy transport is described by

$\partial_{x}\vec{T^{2}}(x)=\mathcal{M}_{T}\vec{T^{2}}(x)+\delta\vec{V}(x),$
(6)

where $\vec{T^{2}}(x)=\left[T_{1}^{2}(x),\ldots,T_{N}^{2}(x)\right]^{T}$ are
the local temperatures (squared), and the matrix

$\displaystyle\mathcal{M}_{T}=\begin{pmatrix}-\frac{\sum\limits_{n\neq
1}(l^{Q}_{1,n})^{-1}}{\chi_{1}n_{1}}&\frac{(l^{Q}_{1,2})^{-1}}{\chi_{1}n_{1}}&\dots&\frac{(l^{Q}_{1,N})^{-1}}{\chi_{1}n_{1}}\\\\[3.0pt]
\frac{(l^{Q}_{2,1})^{-1}}{\chi_{2}n_{2}}&-\frac{\sum\limits_{n\neq
2}(l^{Q}_{2,n})^{-1}}{\chi_{2}n_{2}}&\dots&\frac{(l^{Q}_{2,N})^{-1}}{\chi_{2}n_{2}}\\\\[3.0pt]
\vdots&\vdots&\ddots&\vdots\\\
\frac{(l^{Q}_{N,1})^{-1}}{\chi_{N}n_{N}}&\frac{(l^{Q}_{N,2})^{-1}}{\chi_{N}n_{N}}&\dots&-\frac{\sum\limits_{n\neq
N}(l^{Q}_{N,n})^{-1}}{\chi_{N}n_{N}}\end{pmatrix}$ (7)

in which $l^{Q}_{i,j}=l^{Q}_{j,i}$ are the thermal equilibration lengths (see
Appendix A for an example). The final term in Eq. (6) is the Joule heating
contribution

$\displaystyle\delta\vec{V}(x)=$ (8)
$\displaystyle\frac{e^{2}}{h\kappa_{0}}\sum_{n=1}^{N}\left(\frac{[V_{1}(x)-V_{n}(x)]^{2}}{l_{1,n}^{C}\chi_{1}},\dots,\frac{[V_{N}(x)-V_{n}(x)]^{2}}{l_{N,n}^{C}\chi_{N}}\right)^{T},$

and originates from the voltage drops between the edge channels. In the matrix
$\mathcal{M}_{T}$, the thermal conductance of the edge modes are described by
the numbers $n_{i}$, which equal $1$ for Abelian edge channels and $1/2$ for
Majorana edge channels. These numbers are related to Eq. (1) as

$\displaystyle c=\sum_{i:\chi_{i}=+1}n_{i},$ (9)
$\displaystyle\bar{c}=\sum_{i:\chi_{i}=-1}n_{i},$ (10)
$\displaystyle\nu_{Q}=\sum_{i}\chi_{i}n_{i}.$ (11)

The local temperatures are further related to local heat currents as

$\displaystyle J_{i}(x)=\frac{n_{i}\kappa_{0}}{2}T^{2}_{i}(x).$ (12)

The transport equations (3) and (6) must be supplemented by boundary
conditions that depend on the setup. Boundary conditions are discussed in more
detail in Sec. III.

###  II.2 Charge current noise

We now focus on the regime of efficient charge equilibration, which we denote
as $\delta\equiv L/\ell^{C}_{\rm eq}\gg 1$, where $L$ is the edge length and

$\displaystyle\ell^{C}_{\rm
eq}\equiv\text{max}(\ell^{C}_{i,j}),\quad\chi_{i}\times\chi_{j}=-1,$ (13)

is defined as the largest equilibration length in the set of pairs of counter-
propagating channels. As mentioned in Sec. I, this condition is normally
fulfilled on all FQH edges. Then, the excess charge current noise $S$, due to
a voltage bias, in any of the two contacts (equal due to charge conservation),
see Fig. 3, is to good approximation given by [65, 66, 75]

$\displaystyle S\simeq\frac{2e^{2}}{h\ell^{C}_{\rm
eq}}\frac{\nu_{-}}{\nu_{+}}(\nu_{+}-\nu_{-})\int_{0}^{L}dx\;e^{-\frac{2x}{\ell^{C}_{\rm
eq}}}\Lambda(x).$ (14)

A detailed derivation this formula can be found in Ch. 3 of Ref. [76]. In Eq.
(14), “$\simeq$” should here, and below, be understood as “equal in the limit
of very large $\delta$”.Furthermore,

$\displaystyle\nu_{+}=\sum_{i:\chi_{i}=+1}\nu_{i},$ (15)
$\displaystyle\nu_{-}=\sum_{i:\chi_{i}=-1}\nu_{i},$ (16)

are the total filling factor discontinuities of the downstream $(+)$ and
upstream $(-)$ edge modes respectively. They satisfy the relations
$\nu_{+}>\nu_{-}$ and $\nu=\nu_{+}-\nu_{-}$, where $\nu$ is the effective
filling factor of the edge structure. For a $5/2-n$ interface, we simply have
$\nu=5/2-n$ for $\delta\gg 1$. The exponential suppression in Eq. (14) follows
from the chiral nature of the edge [65]. It implies that the noise generation
is predominantly influenced by the region of size $\sim\ell^{C}_{\rm eq}$
close to the most upstream contact. We call this the noise spot.

The key quantity in Eq. (14) is the local noise kernel

$\displaystyle\Lambda(x)\equiv\frac{S_{\rm loc}[\delta
V(x),T_{+}(x),T_{-}(x)]}{2g_{\rm loc}[\delta V(x),T_{+}(x),T_{-}(x)]},$ (17)

where $S_{\rm loc}$ and $g_{\rm loc}$ is the local dc noise and the
(dimensionless) tunneling conductance, respectively. It is assumed that all
downstream and upstream edge channels charge-equilibrate separately very
efficiently, e.g., by emanating from the same (ideal) contact [77].
Importantly, both $S_{\rm loc}$ and $g_{\rm loc}$ depend on microscopic
details of the edge such as inter-channel interactions, the edge disorder
strength, the local voltage difference between the modes $\delta V(x)$, and
the thermal equilibration-induced effective temperatures $T_{\pm}$ of
downstream and upstream edge modes. Appendix B outlines how noise kernels are
obtained from a bosonization approach.

Our procedure to find the charge and heat flows along an edge segment is as
follows. We first solve Eqs. (3) and (6) with suitable boundary conditions.
These will depend on the type of setup. We then use these solutions to compute
charge and thermal conductances, or insert them first into Eqs. (17) and then
(14) to obtain the noise.

##  III Thermal conductance

###  III.1 Two-terminal thermal conductance

Figure 4: Schematic two-terminal setup for an interface between regions at
filling $5/2$ and integers $n$. For $n<5/2$, the round arrow depicts the
downstream direction (in which the charge current flows) while for $n>5/2$, it
depicts the upstream direction. Swapping the magnetic field direction in the
latter case restores the arrow direction to indicate the downstream direction.

(a)

(b)

Figure 5: (a) Two-terminal thermal conductance $G^{Q}_{\rm 2T}$ (in units of
the thermal conductance quantum $\kappa_{0}\bar{T}$) for the phPf-$2$
interface as a function of the degree of pairwise thermal equilibration
$\alpha$ [see Eq. (23)]. (b) $G^{Q}_{\rm 2T}$ as a function of the two
equilibration parameters $\alpha$ and $\beta$ [see Eqs. (23)-(24)] for
interfaces phPf-$3$, aPf-$2$ and Pf-$3$.

In this section, we compute the two-terminal thermal conductance for various
$5/2-n$ (with $n=2,3$) interfaces by applying Eq. (6) to the setup in Fig. 4.
To this end, we set the voltages in both contacts equal to zero:
$\bar{V}=\Delta V=0$. The solutions to Eq. (3) are then trivial:
$\vec{V}(x)=0$. For the thermal transport, the boundary conditions for the top
edge segment read

$\displaystyle T_{i}(0)=\bar{T}+\Delta T,\ $ for $\displaystyle\chi_{i}=+1,$
(18a) $\displaystyle T_{i}(L)=\bar{T},\ $ for $\displaystyle\chi_{i}=-1,$
(18b)

and for the bottom segment we have

$\displaystyle T_{i}(0)=\bar{T},\ $ for $\displaystyle\chi_{i}=+1,$ (19a)
$\displaystyle T_{i}(L)=\bar{T}+\Delta T,\ $ for $\displaystyle\chi_{i}=-1.$
(19b)

We obtain the heat currents on the top and bottom edge segments by solving Eq.
(6) for

$\displaystyle J^{\rm
top}_{Q}\equiv\sum_{i:\chi_{i}=+1}J_{i}(L)-\sum_{i:\chi_{i}=-1}J_{i}(L),$ (20)
$\displaystyle J^{\rm
bot}_{Q}\equiv\sum_{i:\chi_{i}=+1}J_{i}(0)-\sum_{i:\chi_{i}=-1}J_{i}(0),$ (21)

with $J_{i}(x)$ given in Eq. (12) [see also Appendix A.9 in Ref. [76] for
details in solving Eq. (6)]. Next, by differentiating the total edge current
with respect to the temperature difference $\Delta T$, we obtain the two-
terminal thermal conductance

$\displaystyle G^{Q}_{\rm 2T}\equiv\lim_{\Delta T\to 0}\left(\frac{d(J^{\rm
top}_{Q}+J^{\rm bot}_{Q})}{d\Delta T}\right).$ (22)

Let us start with the edge structures Pf-$2$, and aPf-$3$, which have only co-
propagating modes. For both these interfaces, we readily find $G^{Q}_{\rm
2T}/(\kappa_{0}\bar{T})=3/2$, independently of $L$ and $\ell^{Q}_{i,j}$, which
follows from the fact that the heat exchanged between co-propagating channels
never backscatters, and such processes can therefore not affect $G^{Q}_{\rm
2T}$.

For the other four interfaces aPf-$2$, Pf-$3$, phPf-$2$, and phPf-$3$, we
consider only pairwise thermal equilibration between the counter-propagating
channels, see Fig. 2. Thermal equilibration between co-propagating channels
can be ignored, as discussed in the previous paragraph. For example, for the
Pf-$3$ interface, we consider only the thermal equilibration between channel
pairs $1$-$2$ and $1$-$3$. As we will show, the thermal conductance then
depends on the degrees of thermal equilibration $L/\ell^{Q}_{i,j}$ between
these pairs. To parameterize the equilibration, we introduce two dimensionless
equilibration parameters $\alpha$ and $\beta$ as

$\displaystyle\alpha=\frac{L}{\ell^{Q}_{1,2}}\quad\text{and}\quad\beta=\frac{L}{\ell^{Q}_{1,3}}$
(23)

for the phPf-$2$, Pf-$3$, and aPf-$2$ interfaces and

$\displaystyle\alpha=\frac{L}{\ell^{Q}_{1,2}}\quad\text{and}\quad\beta=\frac{L}{\ell^{Q}_{2,3}}$
(24)

for the phPf-$3$ structure (note the slight difference in the definition of
$\beta$, due to their different structures). We plot the thermal conductances
as functions of $\alpha$ and $\beta$ in Fig. 5. For the phPf-$2$ interface,
Fig. 5a shows that $G^{Q}_{\rm 2T}$ has a step-like behavior and transitions
from $G^{Q}_{\rm 2T}/(\kappa_{0}\bar{T})=1/2\to 3/2$ with decreasing $\alpha$
(there is no $\beta$-parameter for this interface). For large $\alpha$ the
thermal transport is effectively mediated by a single, “collective mode” with
thermal quantum number $n_{\rm tot}=1-1/2=1/2$. For small $\alpha$, the two
edge channels are essentially decoupled and heat transport occurs in both
directions along the edge. The thermal quantum numbers of the channels then
add up, $n_{\rm tot}=1+1/2=3/2$, in their contribution to $G^{Q}_{\rm 2T}$.

The thermal conductance of aPf-$2$, Pf-$3$ and phPf-$3$, is depicted in Fig.
5b. For these interfaces, we see that, depending on which pair of modes that
equilibrates most efficiently, the thermal conductance approaches different
limits. More specifically we have

$\displaystyle\begin{aligned}
\lim\limits_{\begin{subarray}{c}\alpha\to\infty\\\
\beta\to\infty\end{subarray}}&G^{Q}_{\rm
2T}=\frac{1}{2}\kappa_{0}\bar{T},\quad\lim\limits_{\begin{subarray}{c}\alpha\to\infty\\\
\beta\to 0\end{subarray}}&G^{Q}_{\rm 2T}=\frac{1}{2}\kappa_{0}\bar{T},\\\
\lim\limits_{\begin{subarray}{c}\alpha\to 0\\\
\beta\to\infty\end{subarray}}&G^{Q}_{\rm
2T}=\frac{3}{2}\kappa_{0}\bar{T},\quad\lim\limits_{\begin{subarray}{c}\alpha\to
0\\\ \beta\to 0\end{subarray}}&G^{Q}_{\rm
2T}=\frac{5}{2}\kappa_{0}\bar{T}.\end{aligned}$ (25)

Similar to the phPf-$2$ interface, the maximum value of the conductance is
obtained when there is no thermal equilibration and the contributions of all
edge channels along the interface add up: $n_{\rm tot}=1+1+1/2=5/2$, as
expected. The value $G^{Q}_{\rm 2T}/(\kappa_{0}\bar{T})=1/2$ is generated when
all edge channels being fully equilibrated ($\alpha,\beta\rightarrow\infty$)
and $n_{\rm tot}=1-1+1/2=1/2$, in accordance with Eq. (2), up to corrections
exponentially small in $L$. Alternatively, $G^{Q}_{\rm
2T}/(\kappa_{0}\bar{T})=1/2$ is produced for
$\alpha\rightarrow\infty,\beta\rightarrow 0$, where the edge structure becomes
a decoupled MM and a pair of two strongly equilibrated bosons: $n_{\rm
tot}=1/2+0$. The zero here corresponds to a diffusive correction which
vanishes as $\sim\alpha^{-1}$ (see, e.g., the Supplemental Material of Ref.
[66] for a detailed discussion). In contrast, $G^{Q}_{\rm
2T}/(\kappa_{0}\bar{T})=3/2$ is produced by two decoupled collective modes
generating $n_{\rm tot}=1/2+1=3/2$. The limits in Eq. (25) are clearly
idealized, and real devices have finite values of $\alpha,\beta$. However to
estimate the relative magnitude of $\alpha$ and $\beta$ requires detailed
microscopic information about inter-channel energy exchange mechanisms, which
generically depend on e.g., spin, orbital, or valley degrees of freedom of the
LLs. Incorporating these effects is a challenging problem which, however, lies
beyond the scope of the present work.

We end this subsection by pointing out that the experimental value of
$G^{Q}_{\rm 2T}/(\kappa_{0}\bar{T})=1/2$ for both $5/2-2$ and $5/2-3$ can, at
least in principle, be generated from a state other than the phPf. Consider an
edge structure similar to the aPf edge but with the MM direction reversed.
Then, for full thermal equilibration $G^{Q}_{\rm 2T}/(\kappa_{0}\bar{T})=1/2$
is indeed produced for both $n=2$ and $n=3$ ($n_{\rm tot}=1-1+1/2=1/2$ and
$n_{\rm tot}=1-1/2=1/2$, respectively). The non-equilibrated limits for those
interfaces are however $G^{Q}_{\rm 2T}/(\kappa_{0}\bar{T})=5/2$ and
$G^{Q}_{\rm 2T}/(\kappa_{0}\bar{T})=3/2$, respectively, which stand in
contrast to the phPf which has $G^{Q}_{\rm 2T}/(\kappa_{0}\bar{T})=3/2$ and
$G^{Q}_{\rm 2T}/(\kappa_{0}\bar{T})=5/2$ in this limit.

###  III.2 Thermal Hall conductance

Equation (22) describes the two-terminal thermal conductance which we have
shown to significantly depend on the degree of thermal equilibration. Only for
efficient thermal equilibration does $G^{Q}_{\rm 2T}$ take the universal value
as specified in Eq. (2). Note that in Eq. (22), the two edge current
contributions are added (cf., Fig. 4). By instead subtracting the two edge
currents, one can define a thermal Hall conductance as

$\displaystyle G^{Q}_{\rm H}\equiv\lim_{\Delta T\to 0}\left(\frac{d(J^{\rm
top}_{Q}-J^{\rm bot}_{Q})}{d\Delta T}\right).$ (26)

In the most general case, the top and bottom edge heat currents depend on all
mutual equilibration lengths between all pairs of counter-propagating modes.
However, within the model in Sec. II, we prove (see Appendix C for the proof)
that for any edge structure, universality emerges

$\displaystyle G^{Q}_{\rm H}/(\kappa_{0}\bar{T})=c-\bar{c}=\nu_{Q},$ (27)

provided the degrees of equilibration on the top and bottom edge segments are
equal, even if they are poor. Such a situation can, e.g., be achieved by
designing devices with equal top and bottom edge segment lengths. Indeed, such
a setup was presented in Ref. [37], in which the thermal Hall conductance was
measured at $\nu=2/3$ and was found to be in good agreement with the expected
value $G^{Q}_{\rm H}/(\kappa_{0}\bar{T})=0$. At the same time, it was found
that, $G^{Q}_{\rm 2T}/(\kappa_{0}\bar{T})\approx 0.82$, indicating an
incomplete thermal equlibration. In the present context, Figs. 1-2 suggests
that a Hall type of thermal conductance measurement for interfaces $5/2-n$
would give $G^{Q}_{\rm H}/(\kappa_{0}\bar{T})=7/2-n$, $3/2-n$, and $5/2-n$ for
the Pf, aPf, and phPf edges, respectively. A thermal Hall measurement
therefore unambiguously distinguishes between these three edge structures.
This conclusion holds even in the absence of interfacing, since already $n=0$
results in different $G^{Q}_{\rm H}$.

Let us end our treatment of the thermal conductance by emphasizing that for
edge states with co-propagating channels, there is no significant advantage
gained from a thermal Hall measurement. By contrast, for edges with counter-
propagating channels, $G^{Q}_{\rm H}$ provides information directly related to
the state’s topological order, i.e., $\nu_{Q}$, independent of the thermal
equilibration (which might be hard to accurately control experimentally).
Finally, we point out that it is simple to check, that within our model,
corresponding conclusions hold for a similar definition of the charge Hall
conductance as well, i.e.,

$\displaystyle G_{\rm H}\equiv\frac{I_{\rm top}-I_{\rm bot}}{\Delta
V}=\nu\frac{e^{2}}{h},$ (28)

when top and bottom charge equilibrations are identical. Here, $I_{\rm
top/bottom}$ are defined in perfect analogy with Eqs. (20)-(21).

##  IV Noise generation on interfaced edges

In this section, we compute the noise generated on a single interfaced edge
segment with two attached contacts. Such a setup has been realized in
experiments, see, e.g., Refs. [75, 36, 12]. We are here interested in three
cases: First, in Sec. IV.1, we take the two contacts to have equal
temperatures and impose a voltage bias. This is illustrated in Fig 6a. We then
move on to the case with no applied voltage bias, but let the two contacts
have different temperatures. Here, we first take the most upstream contact
(the left one in the figure) to be the hot one, see Fig 6b, and compute in
Sec. IV.2 the excess noise in the colder, downstream contact. We call this
downstream delta-$T$ noise. Finally, in Sec. IV.3 we consider the situation in
which the hot contact lies most downstream and the noise is measured in the
upstream, colder contact, see Fig 6c. We call this upstream delta-$T$ noise.

---

(a)

(b)

(c)

Figure 6: Schematic setups for measuring charge current noise (in the
amplifiers depicted in gray) of a single edge segment in response to (a) a
voltage bias, (b) a temperature bias applied in the “downstream” (ds) or (c)
“upstream” (us) direction.

###  IV.1 Voltage biased edge segment

For an applied voltage bias $\Delta V$, the injected downstream charge current
dissipates heat only close to one of the contacts, when the charge
equilibration is efficient (the hot spot location is independent of the
voltage bias direction [65]). We assume this in the following, which further
implies that in the noise spot region, all charged edge modes equilibrate to
the same electrochemical potential [65, 66]. We can then set
$\delta\vec{V}(x\lesssim\ell^{C}_{\rm eq})\approx\vec{0}$ [see Eq. (8)], with
negligible corrections $\sim\exp[-L/\ell^{C}_{\rm eq}]\ll 1$ [65]. We further
assume that $e\Delta V\gg k_{\rm B}\bar{T}$, so that in this subsection, we
may set the base temperature $\bar{T}$ to zero. Our computed noise then
amounts to the excess noise.

Efficient Thermal Equilibration |  |  No Thermal Equilibration
---|---|---
|  |
|  |

(a)

(b)

(c)

(d)

Figure 7:  (a) Excess (basis temperature $\bar{T}=0$) noise $S^{\rm exc}$ in
units of $e^{3}\Delta V/h$, where $\Delta V$ is the bias voltage. The noise is
computed vs the charge equilibration parameter $\delta=L/\ell^{C}_{\rm eq}$
for parameters causing efficient efficient thermal equilibration:
$(\alpha,\beta)=\\{(250,250),(100,100),(25,25)\\}$ [see Eqs. (23)-(24)], for
solid, dashed, and dotted lines, respectively. (b) same as (a) but in the
limit of absent thermal equilibration $\alpha,\beta\rightarrow 0$. (c) Excess
noise in the presence of a temperature gradient $\Delta T$ in downstream (ds)
or upstream (us) direction for very efficient thermal equilibration with
$(\alpha,\beta)=(100,100)$. (d) same as (c) but for absent thermal
equilibration. For the phPf-$2$ interface, we have used $\delta=\delta_{23}$,
see Fig. 8

Such excess noise is generated only if heat from the hot spot can propagate
upstream (see Fig. 6a). This possibility depends in turn on the edge structure
and on how well the edge channels thermally equilibrate. This feature is
captured within our model in Eq. (14). As follows, we consider the two limits
of either very efficient or very poor thermal equilibration.

####  IV.1.1 Efficient thermal equilibration

For efficient thermal equilibration, we start by solving Eq. (3) for the
boundary conditions $V_{i}(0)=\Delta V$ (we set $\bar{V}=0$) for $\chi_{i}=+1$
and $V_{i}(L)=0$ for $\chi_{i}=-1$. This is done for all the charged edge
channels. From the solution, we extract the voltage drops (8), which we then
insert into Eq. (6), which is finally solved with the boundary conditions
$T_{i}(0)=T_{i}(L)=0$. We give examples of voltage and temperature profiles in
Appendix D. The resulting temperature profiles are used in Eq. (14). This
integral is dominated by the temperatures at the noise spot, i.e.,
$T_{i}(x\lesssim\ell^{C}_{\rm eq})$. Moreover, due to the efficient thermal
equilibration, the channel temperatures are similar in this region. Performing
the integration in Eq. (14), these two features result in noise on Nyquist-
Johnson (NJ) form [78, 79]

$\displaystyle
S\simeq\frac{2e^{2}}{h}\frac{\nu_{-}}{\nu_{+}}(\nu_{+}-\nu_{-})k_{\rm B}T_{\rm
ns},$ (29)

in terms of an effective noise spot temperature $T_{\rm ns}$. This temperature
depends on the voltage bias $\Delta V$ and the way $T_{\rm ns}$ depends on $L$
stands is in one-to-one correspondence to the three possible cases of a
thermally equilbirated edge: $\nu_{Q}>0$, $\nu_{Q}<0$, and $\nu_{Q}=0$ [66].
This classification of the noise was described in detail in Sec. I.

We first analyze the noise generated on the Pf-$2$, and aPf-$3$ interfaces.
For these, we have that the noise vanishes identically, $S=0$, for any value
of $\delta$ and $\Delta V$. This happens simply because there are no upstream
modes present that can transport heat to the noise spot, and thus both
$\nu_{-}=0$ and $\Lambda(x)=0$ (the latter equality holds because we set
$\bar{T}=0$).

The other four interfaces have $\bar{c}\neq 0$, but the aPf-$2$ and Pf-$3$
have $\nu_{Q}<0$ whereas phPf-$2$ and phPf-$3$ have $\nu_{Q}>0$. For these
four interfaces, we plot the noise $S$ vs $\delta$ in Fig. 7a. We see that the
noise approaches a constant value $S\simeq const.$ for aPf-$2$ and Pf-$3$ but
decays exponentially $S\simeq\exp(-\delta)$ for phPf-$2$ and phPf-$3$. We also
see that with increasing thermal equilibration (increasing $\alpha,\beta$) the
noise is overall suppressed for $\nu_{Q}>0$, because the heat that reaches the
noise spot is suppressed even further. In contrast, when $\nu_{Q}<0$,
increasing thermal equilibration leads to enhanced noise since, in this case,
there is more heat propagating upstream to the noise spot.

####  IV.1.2 Absent thermal equilibration

For absent thermal equilibration, downstream and upstream edge modes are
generally at very different temperatures at the noise spot. To estimate these
temperatures, we follow the approach in Ref. [36] and model the hot spot as a
point-like heat source with power [67]

$\displaystyle P_{0}=\frac{e^{2}\Delta
V^{2}}{2h}\frac{\nu_{-}(\nu_{+}-\nu_{-})}{\nu_{+}}.$ (30)

We next assume that $P_{0}$ is equally divided among all edge channels. The
temperature of the modes propagating upstream from the hot spot is thus
approximated as

$\displaystyle
T_{-}\approx\sqrt{\frac{\bar{c}}{c+\bar{c}}}\sqrt{\frac{2P_{0}}{\kappa_{0}}}.$
(31)

Since we have set the temperature in the contacts to $\bar{T}=0$, the
downstream modes will have zero temperature at the noise spot

$\displaystyle T_{+}\approx 0.$ (32)

Within the approximations (31)-(32), we compute noise kernels $\Lambda(x)$ in
terms of finite and zero temperature bosonized Green’s functions. Details of
these calculations are given in Appendix B.

We present the thermally non-equilibrated noise in Fig. 7b. Similarly to the
case of efficient thermal equilibration, the Pf-$2$, and aPf-$3$ do not
generate any excess noise. For the other interfaces, the noises are
essentially constant as functions of $\delta$, since the heat reaching the
noise spot is now independent of this parameter. The relative magnitude
between the interfaces follow from different noise spot temperatures due to
differing pre-factors in Eqs. (14) and (31).

Figure 8: Edge structure of phPf-$2$ with an additional pair of counter-
propagating integer modes. These modes are responsible for Joule-heating in
the hot spot and charge partitioning in noise spot.

The case of the phPf-$2$ interface warrants here an extra discussion. The
noise of this interface was a crucial ingredient in the interpretation of the
experiment in Ref. [12]. Figure. 1 shows that this structure only has a single
downstream charge mode. Hence, no partitioning would be possible and no excess
noise could be generated. However, by recalling that on sufficiently small
length-scales, $x\ll\ell^{C}_{\rm eq}$, there are in fact counter-propagating
integer modes close to both source and drain contacts. These modes are
responsible both for Joule-heating and charge partitioning. To correctly
describe the noise, we include these modes, see Fig. 8. This procedure
introduces two charge equilibration lengths between each pair of counter-
propagating modes. We parametrize them as $\delta_{12}\equiv L/\ell^{C}_{12}$
and $\delta_{23}\equiv L/\ell^{C}_{23}$ in the channel basis
$\\{\phi_{1}$,$\phi_{2},\phi_{3},\psi\\}$ with $\chi=\\{+1,-1,+1,-1\\}$ and
$\nu=\\{1,1,1/2,0\\}$.

As our next step, we assume that the charge equilibration of the integer
channels is much faster than their equilibration with the bosonic 1/2 mode. In
terms of our equilibration parameters, this amounts to taking
$\delta_{12}\gg\delta_{23}$. The main contribution to the Joule-heating then
comes from the equilibration of integer channels. This limit is consistent
with a quantized expected charge conductance, since we have that

$\displaystyle\lim\limits_{\begin{subarray}{l}\delta_{12}\to\infty\\\
\delta_{23}\to 0\end{subarray}}G_{\rm 2T}=\frac{1}{2}\frac{e^{2}}{h},$ (33)

which was indeed observed experimentally for the $5/2-2$ interface [12]. With
this implementation, our model describes how the phPf-$2$ interface generates
noise in the thermally non-equilibrated limit, see Fig. 7d. We note that this
noise is similar in magnitude as the phPf-$3$, in accordance with experimental
observations [12], see also the discussion in Sec. VI below. We further note
that the outlined procedure bares a similarity with implementing edge-
reconstruction [69, 70, 71, 72]. Effects of edge-reconstruction are discussed
in more detail in Sec. VII.

###  IV.2 Downstream delta-T noise

Here, we consider the situation where the two contacts are at the same
potential, but at different temperatures. Excess charge current noise
generated only by a temperature gradient and in the absence of an average
charge current, is known as “thermally activated shot noise” or “delta-$T$
noise” [80]. This noise has attracted considerable attention lately (for
recent work on delta-$T$ noise for weak tunneling between FQH edges, see Refs.
[81, 82, 83, 84]). Moreover, delta-$T$ noise is, in fact, at play in two-
terminal thermal conductance measurements when the central contact is at zero
potential (see Sec. VII below).

We first take the hot contact to lie upstream from the cold contact, see Fig.
6b. The boundary conditions are then given in Eq. (18). We are interested in
the noise in the right, cold contact, which we call downstream delta-$T$ noise
(because the noise is measured downstream from the heat source). Note however
that the noise spot lies close to the hot contact (given that the equilibrated
charge flow is from left to right). The noise is given by a modification of
Eq. (14), namely

$\displaystyle S$ $\displaystyle\simeq\frac{2e^{2}}{h\ell^{C}_{\rm
eq}}\frac{\nu_{-}}{\nu_{+}}(\nu_{+}-\nu_{-})\int_{0}^{L}dx\;e^{-\frac{2x}{\ell^{C}_{\rm
eq}}}\Lambda(x)$
$\displaystyle+\frac{2e^{2}}{h}\frac{(\nu_{+}-\nu_{-})^{2}}{\nu_{+}}k_{\rm
B}(\bar{T}+\Delta T).$ (34)

Here, the second term takes into account thermal fluctuations coming from the
hot, left contact [36]. The thermal noise coming from the cold, right contact
is suppressed by a factor that is exponential in $\delta=L/\ell^{C}_{\rm
eq}\gg 1$ [65, 76]. This term can thus be safely neglected due to the
efficient charge equilibration. In contrast to the voltage biased edge segment
(where we assumed $e\Delta V\gg k_{\rm B}\bar{T}\approx 0$), we focus here on
excess noise obtained by subtracting from Eq. (IV.2) the equilibrium noise in
the cold contact, i.e.,

$\displaystyle S^{\rm excess}\equiv S-2G_{\rm 2T}k_{\rm B}\bar{T},$ (35)

where $G_{\rm 2T}=e^{2}(\nu_{+}-\nu_{-})/h$ is the two-terminal charge
conductance.

####  IV.2.1 Efficient thermal equilibration

For efficient thermal equilibration, we find that the first term in Eq. (IV.2)
for any edge structure, reduces to $2e^{2}\nu_{-}(\nu_{+}-\nu_{-})k_{\rm
B}(T+\Delta T)/(h\nu_{+})$. Adding this to the second term, we obtain the
downstream delta-$T$ noise as

$\displaystyle S_{\rm ds}$ $\displaystyle\simeq
2\frac{e^{2}}{h}(\nu_{+}-\nu_{-})k_{\rm B}(\bar{T}+\Delta T)$
$\displaystyle=2G_{\rm 2T}k_{\rm B}(\bar{T}+\Delta T).$ (36)

The corresponding downstream excess noise (35) reduces to

$\displaystyle S^{\rm excess}_{\rm ds}\simeq 2G_{\rm 2T}k_{\rm B}\Delta T.$
(37)

Hence, the excess downstream delta-$T$ noise for strong thermal equilibration
equals the excess NJ-like noise emanating from the hot contact. This result is
independent of $\nu_{Q}$. We also note that for $\Delta T=0$, the entire edge
segment and the contacts are in thermal equilibrium and $S_{\rm ds}=2G_{\rm
2T}k_{\rm B}\bar{T}\Rightarrow S^{\rm excess}_{\rm ds}=0$ as expected. In Fig.
7c, we plot, for phPf-$3$, phPf-$2$, Pf-$3,$ and aPf-$2$, $S^{\rm excess}_{\rm
ds}/(2G_{\rm 2T}k_{\rm B})$ vs $\delta$. We see that with increasing $\delta$
this ratio approaches unity as expected. For aPf-$3$ and Pf-$2$, where
$\bar{c}=0$, we have trivially $S^{\rm excess}_{\rm ds}=2G_{\rm 2T}k_{\rm
B}\Delta T$ for all $\delta$.

####  IV.2.2 Absent thermal equilibration

In case of absent thermal equilibration, edge modes can have very different
temperatures at the noise spot. When computing the noise in Eq. (IV.2) for
this case, we find that the excess noise still takes a form similar to the NJ
noise (37), namely

$\displaystyle S^{\rm excess}_{\rm ds}=2\lambda_{\rm ds}G_{\rm 2T}k_{\rm
B}\Delta T.$ (38)

Here, $\lambda_{\rm ds}\sim\mathcal{O}(1)$ is a correction factor [36], which
reflects the poor thermal equilibration. Its origin is the first term in Eq.
(IV.2): For poor thermal equilibration, this term does take the simple form
$\sim k_{\rm B}(T+\Delta T)$ that produced Eq. (IV.2.1). Instead, we compute
(see Appendix B) the noise kernels with distinct temperatures at the noise
spot and insert these kernels in Eqs. (IV.2)-(35). We then obtain Eq. (38)
with the correction factors $\lambda_{\rm ds}$.

In Tab. 2, we give the values of $\lambda_{\rm ds}$ for weak temperature bias
$\Delta T\ll\bar{T}$ and strong bias $\Delta T\gg\bar{T}$ (see Appendix B.2
for the derivation). Note that the values are obtained in the extreme limit of
no thermal equilibration, while full equilibration amounts to $\lambda_{\rm
ds}=1$ according to Eq. (37). Partial equilibration then corresponds to values
larger than those in Tab. 2, but still below $1$. We use the strong bias
values for the plots in Fig. 7d.

Table 2: Correction factors for thermally non-equilibrated downstream (ds) excess delta-$T$ noise [see Eq. (38)] for small and large applied bias $\Delta T$. $\lambda_{\rm ds}$ | phPf-$3$ | aPf-$2$ | phPf-$2$
---|---|---|---
$\Delta T\ll\bar{T}$ | $\displaystyle\frac{3}{4}$ | $\displaystyle\frac{5}{8}$ | $\displaystyle\frac{4}{5}$
$\Delta T\gg\bar{T}$ | $0.865$ | $0.774$ | $0.899$

###  IV.3 Upstream delta-T noise

We now analyze the situation where the hot contact lies downstream to the cold
contact, as depicted in Fig. 6c. The noise is measured in the cold, upstream
contact and we call this noise upstream noise (since noise is measured
upstream of the heat source). Note that also in this case, the noise spot is
close to the left contact. The boundary conditions are now given in Eq. (19)
and the noise reads

$\displaystyle S$ $\displaystyle\simeq\frac{2e^{2}}{h\ell^{C}_{\rm
eq}}\frac{\nu_{-}}{\nu_{+}}(\nu_{+}-\nu_{-})\int_{0}^{L}dx\;e^{-\frac{2x}{\ell^{C}_{\rm
eq}}}\Lambda(x)$
$\displaystyle+\frac{2e^{2}}{h}\frac{(\nu_{+}-\nu_{-})^{2}}{\nu_{+}}k_{\rm
B}\bar{T}.$ (39)

The difference between Eq. (IV.2) and Eq. (IV.3) lies only in the temperature
entering the last term: now the thermal charge fluctuations from the hot,
right contact are exponentially suppressed in $\delta=L/\ell^{C}_{\rm eq}$.
The definition of excess noise (35) holds also for the upstream delta-$T$
noise.

####  IV.3.1 Efficient thermal equilibration

Whether the noise spot acquires a temperature larger than $\bar{T}$ depends,
for efficient thermal equilibration, strongly on the thermal quantum number
$\nu_{Q}$ in Eq. (1). For $\nu_{Q}>0$, the upstream heat flow is exponentially
small in $\delta$ and the resulting noise is to excellent approximation given
as the NJ noise corresponding to the cold contact

$\displaystyle S_{\rm us}\simeq 2G_{\rm 2T}k_{\rm B}\bar{T}.$ (40)

This implies $S^{\rm excess}_{\rm us}=0$ for all edges with $\nu_{Q}>0$. For
$\nu_{Q}\leq 0$, the situation is similar to the voltage biased and thermally
equilibrated edge in Sec. IV.1.1. The only difference is that instead of heat
coming from dissipation in the hot spot, it comes from a heated contact.
Hence, the classification presented in Ref. [66] (this classification was
outlined in Sec. I) carries over to thermally equilibrated, upstream delta-$T$
noise.

We plot the upstream, equilibrated delta-$T$ noise in Fig. 7c. The interfaces
Pf-$2$, aPf-$3$, phPf-$2$, and phPf-$3$ all have $\nu_{Q}>0$ (relative to
their charge flows) and their upstream delta-$T$ noises therefore either
vanish identically $S=0$ (aPf-$2$ and Pf-$3$) or decay exponentially in
$\delta$ (aPf-$3$ and Pf-$2$). Due to the single Majorana mode on all
interfaces, none of them have $\nu_{Q}=0$. For aPf-$2$ and Pf-$3$, $\nu_{Q}<0$
and $S^{\rm excess}_{\rm us}$ reaches a constant value with increasing
$\delta$.

####  IV.3.2 Absent thermal equilibration

For very poor thermal equilibration, upstream excess delta-$T$ noise becomes
possible in the presence of upstream modes, $\bar{c}\neq 0$. For $\bar{c}=0$,
we have that $S_{\rm us}\simeq 2G_{\rm 2T}\bar{T}\Rightarrow S^{\rm
excess}_{\rm us}\simeq 0$. This is the case for Pf-$2$ and aPf-$3$ interfaces.

All other interfaces have $\bar{c}\neq 0$. We compute the noise kernels for
those in Appendix B. The resulting noise can be written as

$\displaystyle S^{\rm excess}_{\rm us}=2\lambda_{\rm us}G_{\rm 2T}k_{\rm
B}\Delta T,$ (41)

with $\lambda_{\rm us}$ given in Tab. 3, for both weak, $\Delta T\ll\bar{T}$,
and strong temperature bias $\Delta T\gg\bar{T}$. Since the thermally
equilibrated $S^{\rm excess}_{\rm us}$ depends on $\nu_{Q}$, we do not expect
a continuous limit for $\lambda_{\rm us}$, as was the case for $S^{\rm
excess}_{\rm ds}$. The strong bias values for $\lambda_{\rm us}$ are used for
the plots in Fig. 7d: The $\bar{c}\neq 0$ interfaces display constant noise vs
$\delta$.

Table 3: Correction factors for thermally non-equilibrated upstream (us) excess delta-$T$ noise [see Eq. (41)] for small and large applied bias $\Delta T$. $\lambda_{\rm us}$ | phPf-$3$ | aPf-$2$ | phPf-$2$
---|---|---|---
$\Delta T\ll\bar{T}$ | $\displaystyle\frac{3}{4}$ | $\displaystyle\frac{7}{8}$ | $\displaystyle\frac{7}{10}$
$\Delta T\gg\bar{T}$ | $0.365$ | $0.438$ | $0.338$

The results for the various types of noise computed in Sec. IV are summarized
in Tab. 1.

###  IV.4 Downstream-upstream symmetry of phPf-$3$ for absent thermal
equilibration

Figure 9: Noise kernel $\Lambda$ dependencies on the base temperature
$\bar{T}$ (here, we set $\bar{T}=0.1$ for concreteness) and applied bias
$\Delta T$, for poor thermal equilibration. The boxed areas correspond to
regions in which the approximation for weak or strong biases (wb and sb,
respectively) are valid, see Appendix B.4. The downstream and upstream noise
kernels are identical for all $\Delta T,\bar{T}$ for the phPf-$3$ interface.

Examination of $\lambda_{\rm ds}$ and $\lambda_{\rm us}$ exposes a unique
relation for the phPf-$3$ interface. For strong applied biases, we have for
this interface that $\lambda_{\rm ds}\equiv 0.5+\lambda_{\rm us}$. The
constant offset follows from Eqs. (IV.2) and (IV.3) together with the fact
that for the phPf-$3$ the noise kernels for excess delta-$T$ noise in the
down- and upstream bias configurations are identical, see Appendix B.4. By
computing the noise kernels for general $\Delta T/\bar{T}$, we find that the
above downstream-upstream symmetry holds for all $\Delta T$ (as long as the
thermal equilibration is poor). In Fig. 9, this result is visualized by
overlapping solid and dashed lines for all $\Delta T$ at constant $\bar{T}$
only for the phPf-$3$ (blue). We present an experimental method to test this
symmetry in Sec. VII.

##  V Temperature dependence of thermal equilibration lengths

Here, we discuss the impact of the base temperature $\bar{T}$ on the thermal
equilibration. The degree of this equilibration is determined by parameters on
the form $L/\ell^{Q}_{\rm eq}$. Thus, to investigate phenomena related to
thermal equilibration, one may either tune $L$, which requires advanced
devices (see, e.g., Ref. [36]), or tune $\ell^{Q}_{\rm eq}$ through its
temperature dependence. Hence, this dependence is an important edge
characteristic, as it determines

  1. i)

The value of the two-terminal thermal conductance $G^{Q}_{\rm 2T}$ for a given
temperature $\bar{T}$ and edge length $L$ [41].

  2. ii)

The sharpness of transitions between plateaus of quantized two-terminal
thermal conductances (see Sec. III.1).

  3. iii)

The magnitude of the noise generated on a single edge segment (see Sec.
III.2).

In the remainder of this section, we therefore discuss the temperature
dependence of various $\ell^{Q}_{i,j}$ for our considered interfaces.

In Ref. [46], the authors argued that most thermal equilibration lengths scale
as

$\displaystyle\ell^{Q}_{\rm eq}\sim\bar{T}^{-2},$ (42)

due to tunneling of particles between the edge channels. The exponent in (42)
comes from the fact that in most cases, the most relevant (in the
renormalization group sense) tunneling operators $\mathcal{O}$ take the form

$\displaystyle\mathcal{O}\sim e^{i\sum_{j}m_{j}\phi_{j}}\text{ or }\sim\psi
e^{i\sum_{j}m_{j}\phi_{j}}.$ (43)

Here, $\psi$ is a MM, $\phi_{j}$ are bosons, and $m_{j}$ are real-valued
numbers indicating the number and charges of particles that tunnel. When the
edge is sufficiently close to a strong disorder fixed point [85, 86, 87, 43],
the $m_{j}$ take on integer values which results in an integer scaling
dimension of $\mathcal{O}$, denoted $\Delta$ [30], and the exponent
$\alpha=2(\Delta-1)$ in (42) is thus also an integer. Away from such a point,
however, the exponents are expected to take on smaller, non-universal values
which depend on the inter-edge interaction strength [43].

An important exception from Eq. (42) is the equilibration length between a
counter-propagating boson and Majorana mode. The most relevant operator
coupling these modes is instead on the form [46]

$\displaystyle\mathcal{O}\sim\partial_{x}\phi\psi i\partial_{x}\psi.$ (44)

Essentially, (44) follows from the fact that it is impossible to construct
tunneling operators between a counter-propagating boson $\phi$ and a MM $\psi$
(the point is that only their combination can create electrons), and the
simplest operator exchanging energy between the modes is instead of density-
density type. This feature leads to the temperature scaling

$\displaystyle\ell^{Q}_{\rm eq}\sim\bar{T}^{-4}.$ (45)

We present a detailed derivation of this result in Appendix A. Equation (45)
suggests that, for edges with a counter-propagating boson and a MM, which is
the case for phPf-$2$ and phPf-$3$ interfaces (see Fig. 2), there can be
unusually sharp transitions between thermal conductance plateaus. They are
important signatures in thermal conductance measurements. For example, the
phPf-$3$ interface produces a thermal conductance transition from $G^{Q}_{\rm
2T}/(\kappa_{0}\bar{T})=1/2$ to $G^{Q}_{\rm 2T}/(\kappa_{0}\bar{T})=3/2$ with
decreasing temperature $\bar{T}$. The scaling of this transition is of the
sharper form (45), which might enhance the prospects of its experimental
observation at not too low $\bar{T}$.

Temperature scalings of the thermal equilibration lengths enter the incoherent
model (summarized in Sec. II), as [39]

$\displaystyle\ell^{Q}_{i,j}=\frac{a}{g_{i,j}\gamma_{i,j}(n_{i}-n_{j})}=\frac{a\kappa_{0}\bar{T}}{G_{i,j}^{Q}\gamma_{i,j}(n_{i}-n_{j})},\quad\
n_{i}\neq n_{j}$ (46a)
$\displaystyle\ell^{Q}_{i,j}=\frac{a}{g_{i,j}\gamma_{i,j}}=\frac{a\kappa_{0}\bar{T}}{G_{i,j}^{Q}\gamma_{i,j}},\quad
n_{i}=n_{j}$ (46b)

Here, $n_{i}$ and $n_{j}$ are the central charges of the two modes,
$\gamma_{i,j}$ is a parameter characterizing deviations from the Wiedemann-
Franz law [44, 51, 39], $a$ is the typical lengthscale for inter-channel heat
exchange, and $G^{Q}_{i,j}/(\kappa_{0}\bar{T})\equiv g_{i,j}$ are the
dimensionful and dimensionless thermal conductances of this exchange. The
latter quantities can be computed within the chiral Luttinger liquid model
(see e.g., Refs. [46, 51, 39]).

For the special case of the counter-propagating boson and MM, denoted
$\phi_{2}$ and $\psi$, respectively, we have

$\displaystyle\ell^{Q}_{\phi,\psi}=\frac{a}{g(n_{\phi}-n_{\psi})\gamma_{\phi,\psi}}=\frac{2a\kappa_{0}\bar{T}}{G_{\rm
int}^{Q}}$ (47)

with $\gamma_{\phi,\psi}=1$ and

$\displaystyle G_{\rm int}^{Q}=\frac{8b^{2}k_{\rm
B}^{4}\pi^{5}\Gamma_{0}^{2}}{35\hbar^{6}v_{\phi}^{2}v_{\psi}^{4}}\bar{T}^{4}\kappa_{0}\bar{T},$
(48)

as computed in Appendix A. Here, $\Gamma_{0}$ is the bare coupling strength
between the channels, which we assume to be weak, and $v_{\phi}$, $v_{\psi}$
are the propagation velocities of $\phi$ and $\psi$, respectively. The
parameter $b$ is an ultra-violet cutoff, with dimension of length.

Knowledge of the temperature scalings of $\ell^{Q}_{i,j}$, permits us to
analyze $G^{Q}_{\rm 2T}$ vs $\bar{T}$, rather than $\alpha,\beta$ (as we did
in Sec. III). As an instructive example, we show in Fig. 10 such a plot for
the phPf-$3$ interface. The characteristic feature is the presence of plateaus
with different quantized values of $G^{Q}_{\rm 2T}$. These plateaus are
associated to different regimes of equilibrated and non-equilibrated edge
modes. In this particular example, there are two inter-plateau transitions,
which we label $\Romannum{1}$ and $\Romannum{2}$. These transitions are
related to thermal equilibration by electron tunneling between $\phi_{1}$ and
$\psi\times\phi_{2}$, transition $\Romannum{1}$, and by a density-density
interaction between $\phi_{2}$ and the MM $\psi$: transition $\Romannum{2}$.
From the above analysis, we have different temperature scalings for these
transitions, namely $\ell^{\Romannum{1}}_{\rm eq}\sim\bar{T}^{-2}$ and
$\ell^{\Romannum{2}}_{\rm eq}\sim\bar{T}^{-4}$.

Figure 10: Two-terminal thermal conductance $G^{Q}_{\rm 2T}$ (in units of the
thermal conductance quantum $\kappa_{0}\bar{T}$) for $5/2-3$ vs $\bar{T}$ for
FQH interfaces with three modes and two distinct thermal equilibration
lengths. For concreteness, we have chosen numerical values
$\ell^{\Romannum{1}}_{\rm eq,i}\sim
5\cdot\\{10^{-5},10^{-6},10^{-7},10^{-8}\\}\cdot\bar{T}^{-2}$ and
$\ell^{\Romannum{2}}_{\rm eq}\sim\\{10^{-16}\\}\cdot\bar{T}^{-4}$, such that
the transitions take place at temperatures below $10$ mK.

Figure 10 shows that for full thermal equilibration (high $\bar{T}$),
$G^{Q}_{\rm 2T}/(k_{\rm B}\bar{T})=1-1+1/2=1/2$ which transitions at lower
$\bar{T}$ to the value for absent thermal equilibration: $G^{Q}_{\rm
2T}/(k_{\rm B}\bar{T})=1+1+1/2=5/2$. However, an intermediate plateau emerges
in a temperature regime when there is a separation of the two equilibration
length scales $\ell^{\Romannum{1}}_{\rm eq}\ll\ell^{\Romannum{2}}_{\rm eq}$.
This plateau vanishes for $\ell^{\Romannum{1}}_{\rm
eq}\approx\ell^{\Romannum{2}}_{\rm eq}$ within the range of the transition
temperature. A similar analysis can be performed for any edge structure, with
its own specific set of plateaus and transitions. The set of these two
features can be viewed as a “pin-code” of the FQH edge structure [41].

In the above analysis, we have have neglected interference effects from
scattering of bosonic plasmon waves on contact-edge interfaces [88, 89, 43].
This amounts to assuming $L\gg L_{T}\sim\bar{T}^{-1}$, where $L_{T}$ is a
characteristic thermal length scale. Including the interference effects can
potentially generate additional plateaus [43, 36].

##  VI Comparison to experiments

In this section, we compare the results from our model to the recent
experimental findings in Refs. [11, 12]. We start with the thermal
conductance. In Ref. [11], the two-terminal thermal conductance at the
interface between the $5/2$ state and integer states $n$ was observed to obey
$G^{Q}_{\rm 2T}/(\kappa_{0}\bar{T})\approx\left|5/2-n\right|$. We see that
this result is incompatible with both the Pf and aPf edge structures, since
for Pf-$2$ and aPf-$3$, we have $G^{Q}_{\rm 2T}/(\kappa_{0}\bar{T})=3/2$,
independently of the thermal equilibration (see Sec. III.1). The measured
conductances can further be compared with Fig. 5, where we note that the
measured values are consistent with a phPf edge structure, if we assume an
efficient thermal equilibration when interfaced with integer modes. Indeed,
among the three non-Abelian candidates in Fig. 1, the phPf is the only one
compatible with the expected values of $G^{Q}_{\rm 2T}$ for all interfaced
structures.

We further note that with decreasing temperature, and thus decreasing degree
of equilibration (see Sec. V), the thermal conductance should increase and
saturate at $G^{Q}_{\rm 2T}/(\kappa_{0}\bar{T})=3/2$ and $5/2$ for the
phPf-$2$ and phPf-$3$ edges, respectively. As mentioned in the end of Sec.
III.1 the non-equilibrated limit is an important check for uniquely pin-
pointing the edge structure. This could be tested in future experiments
similar to that in Ref. [41].

We next move on to the noise. Ref. [12] reported measurements of excess noise
at interfaces of $5/2-n$ for an applied voltage bias. From our analysis in
Sec. III.2, we note that for both Pf-$2$, and aPf-$3$, which consist of only
co-propagating modes over long length scales, no excess noise is expected for
any degree of thermal equilibration. As finite, but with $L$ and $\bar{T}$
decreasing, noise was found for both $5/2-2$ and $5/2-3$, the most compatible
edge structure is, just as in Ref. [12], the phPf, this time assuming the
thermal equilibration to be neither full nor absent but rather in an
intermediate regime.

Our model allows a comparison on a quantitative level: We compare the measured
noise data with our Figs. 7a-7b. Due to the the well-established charge
conductance quantization in Ref. [12], we can safely assume efficient charge
equilibration $\delta\gg 1$. From our calculations, we find that the slope of
the noise vs bias voltage curve for the phPf-$3$ interface approximately
evaluates to

$\displaystyle 0\leq\partial_{\Delta V}S^{\rm exc}_{\rm phPf-3}\leq
0.113\frac{e^{3}}{h}\approx 0.70\times 10^{-30}\frac{A^{2}}{\mu V\ \rm Hz},$
(49)

between the two limits of absent and full thermal equilibration (see Appendix
E for conversion between units). For this comparison we have chosen
$\delta>10$. The only constraint on $\delta$ we have is to reproduce a
correctly quantized charge conductance of $G_{\rm 2T}/(e^{2}/h)\approx 1/2$ to
good accuracy. Since this is still the case $5<\delta<10$, our upper bound for
the noise is slightly enhanced for such choices. Access to more accurate
estimates of $\delta$ from experimental data would improve the accuracy of our
noise magnitudes. For the phPf-$2$ interface, our noise approximately
evaluates to

$\displaystyle 0\leq\partial_{\Delta V}S^{\rm exc}_{\rm phPf-2}\leq
0.086\frac{e^{3}}{h}\approx 0.54\times 10^{-30}\frac{A^{2}}{\mu V\ \rm Hz},$
(50)

i.e., it is of similar magnitude to the one for phPf-$3$. In comparison, the
experimentally observed noise characteristics for both $5/2-2$ and $5/2-3$
were [12]

$\displaystyle\partial_{\Delta V}S^{\rm exc}_{\rm measured}\approx 0.1\times
10^{-30}\frac{A^{2}}{\mu V\ \rm Hz},$ (51)

which lies within our estimate and is moreover compatible with interfaced phPf
edges that are not fully thermal equilibrated.

Figure 11: Length dependent noise $S(L)$ at the $5/2-3$ interface. The red
squares and dashed, black line are measured noise data and a fit,
respectively, from Ref. [12]. The data is obtained for interface lengths
$L=\\{28,38,48,58\\}\mu m$ and was normalized to $S(L=28\mu m)$. Blue and
green triangles is our computed noise for a phPf-$3$ interface. For the sets
$1$ and $2$, we used parameters $(\alpha,\beta,\delta)=(6,10,100)$ and
$(\alpha,\beta,\delta)=(4,8,25)$ at $L=28\mu m$ respectively, to obtain the
normalizing constants $S(L=28\mu m)=\\{0.023,0.026\\}\Delta V\cdot
10^{-30}A^{2}/(\mu VHz)$ and thus $S(L)/S(L=28\mu m)$.

We finally estimate the thermal equilibration lengths from noise data measured
at lengths $L=\\{28,38,48,58\\}\mu m$ in Ref. [12], see Fig. 11. For
concreteness, we use the phPf-$3$ edge structure and compute, for a given set
of $\alpha,\beta,\delta$, the temperature profiles for a given bias $\Delta
V\gg\bar{T}\approx 0$. These profiles are then used with Eq. (14) to obtain
the noise $S(\alpha,\beta,\delta)$. In Fig. 11, we present $S(L)$-profiles for
two sets of model parameters that reasonably reproduce the observed length
dependent noise. For set $1$ (blue triangles), we have taken
$(\alpha,\beta,\delta)=(6,10,100)$ and for set $2$ (green triangles), we took
$(\alpha,\beta,\delta)=(4,8,25)$. For the shortest length $L=28\mu m$, the
values $\delta=100$ and $\delta=25$ correspond to $\ell^{C}_{\rm eq}\approx
0.28\mu m$ and $\ell^{C}_{\rm eq}\approx 1.12\mu m$, respectively. Both these
charge equilibration lengths accurately produce $G_{\rm 2T}/(e^{2}/h)=1/2$, as
observed experimentally.

For both parameter sets, we find that thermal equilibration lengths

$\displaystyle 3\mu m\leq\\{\ell^{Q}_{12},\ell^{Q}_{23}\\}\leq 7\mu m$ (52)

fits the data well.

##  VII Combined conductance and noise measurements in a single device

In this section, we propose a device designed for measurements of both
$G^{Q}_{\rm 2T}$ and various types of noise in a single device, see Fig. 12.
This device is beneficial for two main reasons: First, it can be used to rule
out possible sample-to-sample differences between separate devices targeted
towards noise and conductance measurements. Second, it can be used to exclude
effects of edge reconstruction [69, 70, 71, 72]. To do so is particularly
important, since in its presence, and under conditions of poor thermal
equilibration, any edge structure can produce excess upstream noise. The
benefit with our device is that the two-terminal thermal conductance at poor
equilibration gives access to the total number of edge channels (with MM’s
counted as a “half-channel”). This number can can be used to ascertain that in
the noise measurements there is no upstream heat transport in non-topological,
spurious upstream modes from edge reconstruction.

Figure 12: Schematic setup to measure both the two-terminal thermal
conductance and noise for FQH interfaces. The source contacts $S_{1}$, $S_{2}$
and $S_{n}$ feed currents into the central contact $\Omega$, dissipating power
$P$ and establishing the electrical potential $V_{\Omega}$, see Eqs.
(53)-(54). The dissipated power results in an increased temperature
$T_{\Omega}$ which can be extracted from excess noise in the downstream
contact $A_{1}$. Heat is carried away from $\Omega$ by edge modes along
interfaces between gated regions (with fillings $\nu_{1}$, $\nu_{2}$, $n$, and
$\nu_{in}$) or with the vacuum (thick, black lines). Upstream noise for the
$\nu_{in}-n$ interface can be measured for either $P=0$, $V_{\Omega}\neq 0$ or
$P\neq 0$, $V_{\Omega}=0$, see the text in Sec. VII.

In describing our device below, we will also point out some subtleties in
$G^{Q}_{\rm 2T}$ measurement schemes for states with counter-propagating edge
channels, and how to remedy them.

In Fig. 12, four QH regions are connected to a central, floating contact,
$\Omega$, which can act as a hot reservoir when electrical currents impinge on
it. We take the three regions with fillings $\nu_{1}$, $\nu_{2}$, and $n$ as
the source regions: the source contacts $S_{1}$, $S_{2}$ and $S_{n}$ inject
currents $I_{1}=\nu_{1}V_{1}e^{2}/h$, $I_{2}=\nu_{2}V_{2}e^{2}/h$, and
$I_{n}=nV_{n}e^{2}/h$, respectively, and $V_{1}$, $V_{2}$, and $V_{n}$ are the
corresponding three bias voltages with respect to ground. The drain contacts
$D_{1}$, $D_{2}$, $D_{i}$, and $D_{in}$ are grounded, and the amplifiers
$A_{1}$ and $A_{i}$ are floating. Note that the currents that emanate from
$\Omega$ and enter the amplifiers are emitted to grounded contacts. Thus,
these currents do not propagate back into source contacts. The filling in the
region $\nu_{\rm in}$ is here tuned to $5/2$ (or to some other state of
interest), and this region is interfaced with a region with integer filling
$n$.

The contact potential $V_{\Omega}$ and the dissipated electrical power $P$ can
be tuned independently. They are given as

$\displaystyle
V_{\Omega}=\frac{h}{e^{2}}\frac{I_{1}+I_{2}+I_{3}}{\nu_{1}+\nu_{2}+\text{max}(\nu_{in},n)},$
(53) $\displaystyle
P=\frac{h}{2e^{2}}\Big{(}\frac{I_{1}^{2}}{\nu_{1}}+\frac{I_{2}^{2}}{\nu_{2}}+\frac{I_{n}^{2}}{n}-\frac{\left(I_{1}+I_{2}+I_{n}\right)^{2}}{\nu_{1}+\nu_{2}+\text{max}(\nu_{in},n)}\Big{)}.$
(54)

Strictly speaking, Eq. (54) holds when the sourced edges have $\nu_{Q}>0$. If
this is not the case, it is not evident that all dissipated power heats the
central contact. Instead, some Joule heat can leak out back on the source edge
via upstream modes, without heating the contact. This produces a correction to
Eq. (54) which will influence how the thermal conductance is extracted from
the experimental data. We refer to Ref. [36] for a detailed discussion on this
issue. However, this possible complication can be avoided by tuning the
fillings $\nu_{1}$ and $\nu_{2}$ to, e.g., integer values. Then, it is certain
that Eq. (54) is the dissipated power in the contact. In the following, we
assume that this is the case.

Two operational modes with the device are of particular interest for this
paper:

i) $P=0$ and $V_{\Omega}\neq 0$ and the noise in the amplifier $A_{i}$ is
measured vs $V_{\Omega}$. This configuration realizes the setup in Fig. 6a,
upon the identification $V_{\Omega}=\Delta V$ and assuming $n<5/2$. For
$n>5/2$, the charge flow direction along the interface is from $A_{i}$ to
$\Omega$ but this flow can be reversed by swapping the magnetic field
direction [12]. Since $A_{i}$ is floating, no net current is injected into
$\Omega$ for $n>5/2$. Note that the charge fluctuations emanating from the hot
contact in Fig. 12 is split between four edges. This requires a minor
modification of the downstream noise as compared to Secs. IV.1-IV.2.
Specifically, the device geometry is taken into account by substituting in
Eqs. (IV.2.1),(37), (40), and (41)

$\displaystyle G_{\rm 2T}\rightarrow G^{*}_{m},$
$\displaystyle\frac{1}{G^{*}_{m}}=\frac{1}{G_{m}}+\frac{1}{\sum_{k\neq
m}G_{k}}.$ (55)

Here, $G_{m}$ is the equilibrated charge conductance of edge $m$, where
$m=1,(\text{in}-n)$ labels the two outgoing edges with amplifiers and
$k=1,2,\text{in},(\text{in}-n)$ labels all outgoing edges.

ii) $P\neq 0$ and $V_{\Omega}=0$. This produces the situations considered in
Figs. 6b-6c. A measurement of the excess noise downstream of the central
contact (see Fig. 6b) allows to determine its excess temperature
$T_{\Omega}=\bar{T}+\Delta T$. In turn, access to this noise allows extraction
of the two-terminal thermal conductance [33]. Let us mention that
$V_{\Omega}=0$ is not a necessary condition when measuring the thermal
conductance. However, finite $V_{\Omega}$ leads to additional hot spots in the
device (close to all drain contacts downstream of $\Omega$). In the presence
of upstream heat flow, the generated heat at these points could potentially
effect the heating of the central contact. Such complications are absent for
$V_{\Omega}=0$. Ideally, the downstream noise for this purpose is probed on an
edge with $\bar{c}=0$ (e.g., at contact $A_{1}$ with $\nu_{1}$ tuned to an
integer). This choice avoids possible corrections to the downstream thermal
noise, compare Eq. (IV.2.1) and Eq. (38). In fact, it is very useful to
compare the downstream noises in $A_{1}$ and $A_{i}$ to investigate the
validity of the NJ relation [78, 79] for QH states with counter-propagating
channels and poor thermal equilibration. To illustrate this, we consider
$\nu_{1}$ having only downstream modes. The excess noise in $A_{1}$ is then
obtained from Eq. (IV.2.1) as

$\displaystyle\frac{S^{\rm excess,1}_{\rm ds}}{2G^{*}_{1}}=k_{\rm B}\Delta T.$
(56)

With the assumption of no upstream modes above, this expression holds
independently of any equilibration. In contrast, the excess noise in $A_{i}$
reads

$\displaystyle\frac{S^{\rm excess,i}_{\rm ds}}{2G^{*}_{\rm in-n}}=\lambda_{\rm
ds}k_{\rm B}\Delta T,$ (57)

where $\lambda_{\rm ds}$ does depends on the thermal equilibration of the
$\nu_{\rm in}-n$ interface. Only for full thermal equilibration is
$\lambda_{\rm ds}=1$.

The ratio between Eq. (56) and Eq. (57) allows extraction of $\lambda_{\rm
ds}$:

$\displaystyle\lambda_{\rm ds}=\Bigg{(}\frac{S^{\rm excess,i}_{\rm
ds}}{2G^{*}_{\rm in-n}}\Bigg{)}\times\Bigg{(}\frac{S^{\rm excess,1}_{\rm
ds}}{2G^{*}_{1}}\Bigg{)}^{-1}.$ (58)

We next move on to upstream noise measurements. By swapping $\nu_{in}$ and
$n$, the noise in $A_{i}$ corresponds to upstream delta-$T$ noise as presented
in Fig. 6c (again, the substitution (VII) is needed). Similarly to the
downstream delta-$T$ noise, the upstream delta-$T$ can be measured vs $\Delta
T$ where the latter is obtained from Eq. (56).

Finally, we discuss measurements of the phPf-$3$ downstream-upstream symmetry
of the delta-$T$ noise at poor thermal equilibration, as presented in Sec.
IV.4. To detect this symmetry, we propose to first measure the downstream
delta-$T$ noise (IV.2), but in contrast to the excess noise (35), one
subtracts in this case the hot contact thermal noise, i.e.,

$\displaystyle S^{\rm ds}_{\Delta T}\equiv S-2G^{*}_{\rm in-n}k_{\rm
B}(\bar{T}+\Delta T)=\frac{G^{*}_{\rm in-n}}{2}\Lambda^{\rm ds}.$ (59)

This quantity can then be compared with the upstream delta-$T$ noise (IV.3)
for which the cold contact thermal noise is subtracted

$\displaystyle S^{\rm us}_{\Delta T}\equiv S-2G^{*}_{\rm in-n}k_{\rm
B}\bar{T}=\frac{G^{*}_{\rm in-n}}{2}\Lambda^{\rm us}.$ (60)

In going from Eqs. (IV.2) and (IV.3) to Eqs. (59) and (60), we have again used
the conductance substitution (VII). The downstream-upstream symmetry is now
exposed in the ratio

$\displaystyle\frac{S^{\rm ds}_{\Delta T}}{S^{\rm us}_{\Delta
T}}=\frac{\Lambda^{\rm ds}}{\Lambda^{\rm us}}$ (61)

which uniquely equals unity for the phPf-$3$ interface. Comparison of
delta-$T$ noises for thermally non-equilibrated interfaces thus provides
another experimental signature to distinguish between non-Abelian candidate
states.

##  VIII Summary and conclusions

We have presented a comprehensive theoretical description of quantum transport
along interfaces formed between integer ($n=2$ and $n=3$) QH states and non-
Abelian Pf, aPf, and phPf candidates for the $\nu=5/2$ state. Such interfaces
isolate the “non-Abelian part” of the edge and was used in recent experiments
[11, 12], to distinguish between different candidate theories.

For such interfaces and experiments, we have here in detail determined the
impact of thermal equilibration on the edge thermal conductance as well as on
excess noise on voltage or temperature biased edge segments (so-called
delta-$T$ noise). In contrast to Abelian edges, non-Abelian $5/2-n$ interfaces
feature Majorana modes, which although charge neutral, may influence the
charge current noise generated on the edge. A major finding is that non-
Abelian interfaces with counter-propagating modes are highly sensitive to
thermal equilibration. This feature produces a significant length and
temperature dependence for the two-terminal thermal conductance. For the
noise, the degree of thermal equilibration influences the noise magnitude, the
length dependence of the noise, as well as the Nyquist-Johnson relation for
charge current noise emanating from a heated contact. Our more quantitative
results are summarized in Tab. 1.

In contrast to the two-terminal thermal conductance and the noise, we have
proved that for the thermal Hall conductance, as it was defined in Eq. (27),
the thermal equilibration dependency drops out for a generic edge structure,
provided the two involved edge segments have the same degree of thermal
equilibration. Such experimental conditions were put forward in Ref. [37]. Our
results suggest that a similar experiment performed at $\nu=5/2$ provides a
clear distinction between all candidate states, without any interfacing.

The authors of Refs. [11, 12], interpreted their noise and thermal conductance
measurements as pointing towards the phPf as the realized state in
GaAs/AlGaAs. On the qualitative level, our findings favor such an
interpretation as well. We further computed the voltage biased noise for
phPf-$2$ and phPf-$3$ interfaces, and our results are quantitatively
consistent with the experimentally obtained values. Importantly, our
calculations show that the measured noise magnitude is indeed consistent with
a FQH edge that is not fully thermally equilibrated, which was a crucial
feature for the interpretation in Ref. [12].

We would like to further emphasize that the interpretation in Ref. [12]
favoring the phPf state is based on the absence of edge reconstruction. If
this effect would be present, any edge can generate noise for poor thermal
equilibration. This severely complicates the interpretation of the
experimental data. Ideally, one would therefore like to confidently rule out
such effects. To address this problem, we proposed a device to do precisely
that, by allowing thermal conductance and noise measurements in the same
sample. Combining such measurements permits an unambiguous determination of
the edge structure. We also pointed out some potential issues (highlighted
previously in Ref. [36]) with the standard two-terminal thermal conductance
setup [33] for edges with counter-propagating modes. Here, we proposed
concrete improvements of the setup in order to mitigate these issues.

While our work targeted the famous $\nu=5/2$ state, we envision that it can be
generalized for the purpose of pin-pointing edge structures of other exotic
FQH states, such as the even-denominator states in graphene [90, 91, 92, 93,
94, 95, 96] or the state at $\nu=12/5$ [97, 98, 99].

###### Acknowledgements.

We thank Janine Splettstoesser for numerous enlightening discussions during
the course of this work, Therese Karmstrand and Oliver Hahn for their input on
Appendix C, and Matteo Acciai for his insights on the calculations in Appendix
A. We are also grateful to Jinhong Park for critical comments on the
manuscript. C.S. acknowledges support from the Excellence Initiative Nano 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).
Note: Some of the results in this paper were reported in the Master thesis
Ref. [76].

##  Appendix A Heat exchange between counter-propagating boson and Majorana
modes

The phPf-$2$ edge structure, depicted in Fig. 2, consists of two modes: One
bosonic mode $\phi$ and one Majorana mode $\psi$. This structure is valid for
lengthscales $L\gg\ell^{C}_{\rm eq}$, where the contribution of integer edge
modes can be neglected. A low energy effective description of the two counter-
propagating modes is then given by the free Lagrangian density

$\displaystyle\mathcal{L}_{0}=\frac{2}{4\pi}\partial_{x}\phi(\partial_{t}-v_{\phi}\partial_{x})\partial_{x}\phi+i\psi(-\partial_{t}-v_{\psi}\partial_{x})\psi,$
(62)

where $v_{\phi}$ and $v_{\psi}$ are the mode velocities. Our goal is to
compute the heat exchange between the two modes due to a coupling between
them. The edge electron operator reads $\psi e^{i2\phi}$, so we cannot
introduce electron or quasiparticle tunneling operators to couple these modes.
Instead, the simplest, most relevant (in the renormalization group sense)
operator coupling the channels is given by the density-density operator [46]

$\displaystyle\mathcal{O}=\partial_{x}\phi\psi i\partial_{x}\psi.$ (63)

We therefore add to $\mathcal{L}_{0}$ the point-like perturbation

$\displaystyle\mathcal{L}_{T}=\Gamma_{0}\delta(x)\partial_{x}\phi\psi
i\partial_{x}\psi,$ (64)

where $\Gamma_{0}$ is the coupling constant, assumed to be weak. We thus seek
the heat current and inter-mode thermal conductance induced by
$\mathcal{L}_{T}$, treated as a weak perturbation.

To do so, we consider the unperturbed bosonic energy current, which is given
in terms of the stress energy tensor of the upstream bosonic field
$\mathscr{T}_{\phi}\equiv\left(\partial_{x}\phi\right)^{2}$, as [32]

$\displaystyle
J_{Q,\phi}^{(0)}(d,t)=\hbar\frac{v_{\phi}^{2}}{2\pi}\mathscr{T}_{\phi}(\tilde{t}).$
(65)

Here, $\tilde{t}=t-\frac{d}{v_{\phi}}$ is a shifted time referring to the
transport of energy a small distance $d$ away from the point $x=0$. In the
interaction picture, the average heat current in the presence of
$\mathcal{L}_{T}$, can be written as

$\displaystyle\langle
J_{Q,\phi}(t)\rangle=\langle\mathcal{T}e^{\frac{i}{\hbar}\int_{t}H^{\prime}(t)}J_{Q,\phi}^{(0)}(\tilde{t})e^{-\frac{i}{\hbar}\int_{t}H^{\prime}(t)}\rangle,$
(66)

where $H^{\prime}$ is the Hamiltonian corresponding to $\mathcal{L}_{T}$, and
$\mathcal{T}$ denotes time ordering. We next expand the time-evolution
operators up to $\mathcal{O}(\Gamma_{0}^{2})$. Collecting terms corresponding
to the same order in $\Gamma_{0}$, the first and second order correction to
the heat current become

$\displaystyle
J_{Q,\phi}^{(1)}(t)=\frac{i}{\hbar}\int_{-\infty}^{t}\,dt^{\prime}[H^{\prime}(t^{\prime}),J_{Q,\phi}^{(0)}(\tilde{t})]$
(67)

and

$\displaystyle
J_{Q,\phi}^{(2)}(t)=\frac{i^{2}}{\hbar^{2}}\int_{-\infty}^{t}\,dt^{\prime}\int_{-\infty}^{t^{\prime}}dt^{\prime\prime}[H^{\prime}(t^{\prime\prime}),[H^{\prime}(t^{\prime}),J_{Q,\phi}^{(0)}(\tilde{t})]],$
(68)

respectively. The commutators in these expressions can be computed with
operator product expansions with the stress energy tensor [30]. After some
algebra, we find the corrections to the average energy current in terms of the
modes Green’s functions as

$\displaystyle\langle J_{Q,\phi}^{(1)}(t)\rangle=0$ (69) and
$\displaystyle\langle
J_{Q,\phi}^{(2)}(t)\rangle=\frac{\Gamma_{0}^{2}}{\hbar}\int_{-\infty}^{\infty}\big{(}\partial_{\tau}G_{\phi}(\tau)\big{)}\mathcal{G}_{\psi}(\tau)\,d\tau.$
(70)

In Eq. (70), the integrand is given in terms of the finite temperatures
Green’s functions, $G_{\phi}(\tau)$ and $\mathcal{G}_{\psi}(\tau)$ [30], which
read

$\displaystyle G_{\phi}(\tau)$
$\displaystyle\equiv\langle\partial_{x}\phi(\tau,0)\partial_{x}\phi(0,0)\rangle$
$\displaystyle=\left(\frac{\pi bk_{\rm B}T_{\phi}}{\hbar
v_{\phi}}\csc\left(\frac{\pi k_{\rm B}T_{\phi}}{\hbar
v_{\phi}}(b-iv_{\phi}\tau)\right)\right)^{2}.$ (71)

with $b$ the UV cutoff, and

$\displaystyle\mathcal{G}_{\psi}(\tau)$
$\displaystyle\equiv\langle\mathscr{T}_{\psi}(\tau,0)\mathscr{T}_{\psi}(0,0)\rangle$
$\displaystyle=\left(\frac{\pi bk_{\rm B}T_{\psi}}{\hbar
v_{\psi}}\csc\left(\frac{\pi k_{\rm B}T_{\psi}}{\hbar
v_{\psi}}(b-iv_{\psi}\tau)\right)\right)^{4}$
$\displaystyle+\frac{c^{2}}{36}\left(\frac{\pi k_{\rm B}T_{\psi}}{\hbar
v_{\psi}}\right)^{4},$ (72)

where $c=1/2$ and $\mathscr{T}_{\psi}=\psi i\partial_{x}\psi$ for the MM
$\psi$.

For completeness, we give also the correlation function for a tunneling
operator of the form $\mathcal{O}\sim\psi e^{i2\phi_{1}+\phi_{2}}$. From the
statistical independence of the involved fields we obtain

$\displaystyle\langle\mathcal{O}(\tau)\mathcal{O}^{\dagger}(0)\rangle\propto
G_{\phi_{1}}(\tau)G_{\phi_{2}}(\tau)G_{\psi}(\tau),$ (73)

with

$\displaystyle G_{\phi}(\tau)$ $\displaystyle\equiv\langle
e^{i\nu_{\phi}\phi(\tau,0)}e^{i\nu_{\phi}\phi(0,0)}\rangle$
$\displaystyle=\left(\frac{\pi bk_{\rm B}T_{\phi}}{\hbar
v_{\phi}}\csc\left(\frac{\pi k_{\rm B}T_{\phi}}{\hbar
v_{\phi}}(b-iv_{\phi}\tau)\right)\right)^{1/\nu_{\phi}}$ (74) in which
$\nu_{\phi}\in\\{1/2,1\\}$, and $\displaystyle G_{\psi}(\tau)$
$\displaystyle\equiv\langle\psi(\tau,0)\psi(0,0)\rangle$
$\displaystyle=\left(\frac{\pi bk_{\rm B}T_{\psi}}{\hbar
v_{\psi}}\csc\left(\frac{\pi k_{\rm B}T_{\psi}}{\hbar
v_{\psi}}(b-iv_{\psi}\tau)\right)\right).$ (75)

Next, we insert the expressions for $G_{\phi}(\tau)$ from Eq. (71) and
$\mathcal{G}_{\psi}(\tau)$ from Eq. (72) into Eq. (70) and shift variables
$\\{\tau\to\tau+i\frac{b}{v_{\phi}}-i\frac{\hbar\beta_{\phi}}{2}\\}$.
Following Ref. [100], the integral boundaries are switched back for a properly
introduced cut-off $b$ that satisfies $\hbar v_{\phi}\beta_{\phi}>b$. Setting
the temperatures $T_{\psi}=\bar{T}+\frac{\Delta T}{2}$ and
$T_{\phi}=\bar{T}-\frac{\Delta T}{2}$, gives to leading order in $\Delta T$
the interaction induced heat current

$\displaystyle\langle J_{Q,\phi}^{(2)}(t)\rangle=\frac{8b^{2}k_{\rm
B}^{6}\pi^{7}\Gamma_{0}^{2}}{105\hbar^{7}v_{\phi}^{2}v_{\psi}^{4}}\bar{T}^{5}\Delta
T\ .$ (76)

The corresponding interaction thermal conductance thus reads

$\displaystyle G_{\rm int}^{Q}=\lim_{\Delta T\to 0}\left(\frac{d}{d\Delta
T}\langle J_{Q,\phi}^{(2)}\rangle\right)=\frac{8b^{2}k_{\rm
B}^{4}\pi^{5}\Gamma_{0}^{2}}{35\hbar^{6}v_{\phi}^{2}v_{\psi}^{4}}\bar{T}^{4}\kappa_{0}\bar{T}.$
(77)

As our final step, we consider an array of points, distanced with the length
$a$, with couplings on the form (64) and take the continuum limit (see Refs.
[65, 67, 51, 39]). This procedure relates the conductance (77) to the thermal
equilibration length as

$\displaystyle\ell_{\rm
eq}^{Q}=\frac{a}{g\gamma_{\phi,\psi}(n_{\phi}-n_{\psi})}=\frac{2a}{g\gamma_{\phi,\psi}}=\frac{2a\kappa_{0}\bar{T}}{G_{\rm
int}^{Q}\gamma}\sim\bar{T}^{-4},$ (78)

which is Eq. (47). For the interaction (64), we have $\gamma_{\phi,\psi}=1$.

##  Appendix B Computation of noise kernels

Our approach to compute noise kernels $\Lambda(x)=S^{\rm loc}(x)/[2g^{\rm
loc}(x)]$ follows that in Ref. [75]. The local noise $S^{\rm loc}$ and local
tunneling conductance $g^{\rm loc}$ generically read

$\displaystyle S^{\rm loc}(x)\approx
4\int_{-\infty}^{\infty}\langle\mathcal{O}(\tau,0)\mathcal{O}^{\dagger}(0,0)\rangle\,d\tau$
(79) and $\displaystyle g^{\rm loc}(x)\approx
2i\int_{-\infty}^{\infty}\tau\langle\mathcal{O}(\tau,0)\mathcal{O}^{\dagger}(0,0)\rangle\,d\tau\
.$ (80)

Here, we have assumed zero voltage difference between the edge channels, since
at the noise spot, edge channels equilibrate to the same electrochemical
potential. Furthermore, $\mathcal{O}$ denotes the most relevant tunneling
operator coupling charged edge channels, and the corresponding correlation
function can be expressed as a product of Green’s functions [see Eqs.
(74)-(75)] as

$\displaystyle\langle\mathcal{O}(\tau,0)\mathcal{O}^{\dagger}(0,0)\rangle=\frac{\Gamma_{0}^{2}}{(2\pi
b)^{2}}\prod_{k}G_{k}(\tau,0).$ (81)

Here, $k\in\\{\psi,\phi_{i\in\\{1,2\\}}\\}$, $b$ is a short distance cut-off,
and $\Gamma_{0}$ is the bare coupling amplitude. As follows, we compute
$\Lambda(x)$ for various interfaces in the two limiting cases of efficient and
absent thermal equilibration. We emphasize that all $x$-dependence in
$\Lambda(x)$ enters in the mode temperature profiles $T_{k}(x)$.

###  B.1 Voltage biased charge current noise for absent thermal equilibration

####  B.1.1 phPf-$3$

For this interface, equilibrated charge transport is from right to left (see
Fig. 2). The hot spot and noise spot are therefore inter-changed in comparison
to Fig. 6a. Heat from the hot spot is transported upstream by the bosonic 1/2
mode. Absence of thermal equilibration leads, with the procedure outlined in
Sec. IV.1.2, to the temperatures $T_{\phi_{1}}=T_{\psi}=T_{+}=0$ and
$T_{\phi_{2}}=T_{-}=\sqrt{\frac{4P_{0}}{5\kappa_{0}}}$. Following the same
approach as in the previous section, using Eqs. (74) and (75), we arrive at
the noise kernel

$\displaystyle\Lambda^{\Delta V}_{{\rm
phPf-}3}=\frac{12\zeta(3)}{\pi^{2}}k_{\rm B}T_{-},$ (82)

where $\zeta(z)$ is the Riemann zeta-function.

####  B.1.2 aPf-$2$ and Pf-$3$

In treating the interfaces aPf-$2$ and Pf-$3$, we notice that they are
constructed by the same set of modes but in opposite directions. They are
therefore expected to generate the same charge current noise and thus to have
the same form of $\Lambda(x)$, also in absence of thermal equilibration. With
use of Eqs. (31)-(32), the modes’ temperatures are given by $T_{\phi_{1}}=0$,
$T_{\phi_{2}}\approx T_{\psi}=T_{-}=\sqrt{\frac{6P_{0}}{5\kappa_{0}}}$, we
arrive at

$\displaystyle\Lambda^{\Delta V}_{{\rm aPf-}2}=\Lambda^{\Delta V}_{{\rm
Pf-}3}\approx 1.604k_{\rm B}T_{-}.$ (83)

####  B.1.3 phPf-$2$

For the phPf-$2$ interface, we take into account an additional pair of integer
modes which are not thermally equilibrated (see the discussion in Sec. IV.1.2
and Fig. 8). The charge transport equation (3) then takes the form

$\displaystyle\partial_{x}\vec{V}(x)=\frac{\delta_{12}}{L}\begin{pmatrix}-\chi_{1}/\nu_{1}&\chi_{1}/\nu_{1}&0\\\
\chi_{2}/\nu_{2}&-\chi_{2}/\nu_{2}&0\\\
0&0&0\end{pmatrix}\begin{pmatrix}V_{1}(x)\\\ V_{2}(x)\\\
V_{3}(x)\end{pmatrix}$
$\displaystyle\phantom{\partial_{x}V(x)}+\frac{\delta_{23}}{L}\begin{pmatrix}0&0&0\\\
0&-\chi_{2}/\nu_{2}&\chi_{2}/\nu_{2}\\\
0&\chi_{3}/\nu_{3}&-\chi_{3}/\nu_{3}\end{pmatrix}\begin{pmatrix}V_{1}(x)\\\
V_{2}(x)\\\ V_{3}(x)\end{pmatrix}$ (84)

with $\nu_{1}=\nu_{2}=1$, $\nu_{3}=1/2$ and $\chi_{1}=-\chi_{2}=\chi_{3}=1$.
The boundary conditions are

$\displaystyle V_{1}(0)=\Delta V,\quad V_{2}(1)=0\ \ \text{, and}\ \
V_{3}(0)=\Delta V.$ (85)

To ensure a quantized charge conductance, we choose
$\delta_{12}\gg\delta_{23}$. In general it is not possible to give accurate
estimates on $\delta_{12}$ and $\delta_{23}$ without further experimental
data. For our purpose it is sufficient to adjust them such that $G_{\rm
2T}/(e^{2}/h)=1/2$ to good accuracy. Within this description, charge
partitioning occurs mainly within the region $0\lesssim x\lesssim\delta_{12}$.
Furthermore, the Joule heating contribution is non-zero and leads to a total
dissipated power of

$\displaystyle
P_{0}=\frac{3}{\pi^{2}}\frac{e^{2}V_{0}^{2}}{h\kappa_{0}}\int_{0}^{\delta_{12}}\big{[}\delta_{12}(V_{1}-V_{2})^{2}+\delta_{23}(V_{2}-V_{3})^{2}\big{]}\,dx,$
(86)

where we omitted the $x$-dependence of $V_{i}$ for notational ease. The
process of interest for the noise characteristics on the phPf-$2$ edge is the
charge tunneling between the bosonic channels of filling $\nu_{1}=1$ and the
electronic mode comprised of $\nu_{2}=1/2$ and the MM $\psi$. Then, the noise
kernel for absent thermal equilibration is the same as in the phPf-$3$ case
with the temperature $T_{-}=\sqrt{\frac{2P_{0}}{3\kappa_{0}}}$. Following the
same steps as before, the noise is computed vs the charge equilibration length
$\delta_{23}$, giving a noise of similar magnitude as for absent thermal
equilibration on the phPf-$3$ edge for a ratio
$3\lesssim\delta_{12}/\delta_{23}\lesssim 5$. The deviation from the expected
$G_{2T}$, is less than $0.5\%$ if $\delta_{23}\gtrsim 8$ which is what we
assume here.

###  B.2 Downstream delta-$T$ noise for absent thermal equilibration

For this type of noise, the appropriate boundary conditions are given in Eq.
(18). We consider the limiting cases of a weak bias (wb), $\Delta T\ll\bar{T}$
and strong bias (sb), $\Delta T\gg\bar{T}$ by expanding the expressions to
first order in $\Delta T/\bar{T}$ and setting $\bar{T}=0$, respectively. We
refer to the corresponding noise kernels as $\Lambda^{\rm(wb)}(x)$ and
$\Lambda^{\rm(sb)}(x)$, respectively. Following the same steps as in Appendix
A, using a shift of variables such that the mode with the largest temperature,
say $T_{m}$ and thus smallest $\beta_{m}\equiv[k_{\rm B}(\bar{T}+\Delta
T)]^{-1}$ fulfills $\hbar v_{m}\beta_{m}>b$ and performing the integrals, we
arrive at our desired expressions. In case of weak applied bias
$\bar{T}\gg\Delta T$, we find to first order in $\Delta T$

$\displaystyle\Lambda^{\rm(wb)}_{{\rm phPf-}3}(x)=2k_{\rm B}\bar{T}+k_{\rm
B}\Delta T,$ (87a) $\displaystyle\Lambda^{\rm(wb)}_{{\rm aPf-}2}(x)=2k_{\rm
B}\bar{T}+\frac{k_{\rm B}}{2}\Delta T,$ (87b)
$\displaystyle\Lambda^{\rm(wb)}_{{\rm phPf-}2}(x)=2k_{\rm
B}\bar{T}+\frac{6k_{\rm B}}{5}\Delta T,$ (87c)

which reduce to the equilibrium NJ form for vanishing applied bias $\Delta T$
as expected. For large bias $\Delta T\gg\bar{T}$ we obtain instead

$\displaystyle\Lambda^{\rm(sb)}_{{\rm
phPf-}3}(x)=\frac{12\zeta(3)}{\pi^{2}}k_{\rm B}\Delta T,$ (88a)
$\displaystyle\Lambda^{\rm(sb)}_{{\rm
aPf-}2}(x)=\frac{9\zeta(3)}{\pi^{2}}k_{\rm B}\Delta T,$ (88b)
$\displaystyle\Lambda^{\rm(sb)}_{{\rm
phPf-}2}(x)=\frac{17\pi^{4}}{60(\pi^{2}\log(4)+3\zeta(3))}k_{\rm B}\Delta T.$
(88c)

###  B.3 Upstream delta-$T$ noise for absent thermal equilibration

For the upstream delta-$T$ noise, we use the boundary conditions from Eq.
(19). The noise kernels are computed in a similar manner as for the downstream
delta-$T$ noise. To first order in $\Delta T$ we find for weak applied biases
$\Delta T\ll\bar{T}$

$\displaystyle\Lambda^{\rm(wb)}_{{\rm phPf-}3}(x)=2k_{\rm B}\bar{T}+k_{\rm
B}\Delta T,$ (89a) $\displaystyle\Lambda^{\rm(wb)}_{{\rm aPf-}2}(x)=2k_{\rm
B}\bar{T}+\frac{3k_{\rm B}}{2}\Delta T,$ (89b)
$\displaystyle\Lambda^{\rm(wb)}_{{\rm phPf-}2}(x)=2k_{\rm
B}\bar{T}+\frac{4k_{\rm B}}{5}\Delta T$ (89c)

which reduce to equilibrium noise for $\Delta T\to 0$. For strong bias $\Delta
T\gg\bar{T}$, we find

$\displaystyle\Lambda^{\rm(sb)}_{{\rm
phPf-}3}(x)=\frac{12\zeta(3)}{\pi^{2}}k_{\rm B}\Delta T,$ (90a)
$\displaystyle\Lambda^{\rm(sb)}_{{\rm
aPf-}2}(x)=\left(\log(4)+\frac{3\zeta(3)}{\pi^{2}}\right)k_{\rm B}\Delta T,$
(90b) $\displaystyle\Lambda^{\rm(sb)}_{{\rm
phPf-}2}(x)=\frac{\pi^{4}}{60\zeta(3)}k_{\rm B}\Delta T.$ (90c)

###  B.4 Downstream-upstream delta-$T$ noise symmetry of phPf-$3$ at poor
thermal equilibration

As depicted in Fig. 9, the most striking feature is that the downstream and
upstream noise kernels are uniquely equal for the phPf-$3$ interface (blue
curves in Fig. 9). This equality is clearly manifest also in the asymptotic
expressions: compare (87a) and (89a), respectively (88a) and (90a).

Mathematically, this “downstream-upstream symmetry” follows from the fact that
the exponent of the downstream and upstream sectors in the product of edge
mode Green’s functions [see Eq. (81)] are equal. More specifically, we have
for the phPf-$3$ interface that

$\displaystyle G_{\phi_{1}}(\tau,\bar{T}+\Delta
T)G_{\phi_{2}}(\tau,\bar{T})G_{\psi}(\tau,\bar{T}+\Delta T)$
$\displaystyle\propto
G_{\phi_{1}}(\tau,\bar{T})G_{\phi_{2}}(\tau,\bar{T}+\Delta
T)G_{\psi}(\tau,\bar{T}).$ (91)

Here, the first and second line correspond to downstream and upstream bias
conditions, respectively. Moreover, the proportionality factor in Eq. (B.4),
which includes powers of mode velocities and temperatures, crucially drops out
when dividing the noise with the conductance to obtain $\Lambda$, see Eq.
(17). Relations similar to (B.4) do not hold for any other edge structure in
Fig. 2. Our proposal to test this symmetry is presented in the end of Sec.
VII.

###  B.5 Some integrals and their computation

In computing the noise kernel for a voltage biased phPf-$3$ interface, we face
integrals

$\displaystyle S^{\rm
loc}\propto\int_{-\infty}^{\infty}\frac{\text{sech}^{2}(z)}{\left(\pi+2iz\right)^{2}}\,dz,$
(92) $\displaystyle g^{\rm
loc}\propto\int_{-\infty}^{\infty}\frac{\text{sech}^{2}(z)}{\pi+2iz}\,dz.$
(93)

Integrals of this kind can be solved by using Mittag-Lefflers’ theorem [101],
which amounts to expanding the hyperbolic functions as

$\displaystyle\text{sech}^{2}(z)=-\sum_{k=0}^{\infty}\left[\frac{1}{(z-A)^{2}}+\frac{1}{(z+A)^{2}}\right],$
(94)

where $A=i\frac{\pi}{2}(2k+1)$. Similar expansions can be found for other
hyperbolic functions, e.g., $\text{sech}(z)$. Inserting the series expansion
back into Eqs. (92)-(93), exchanging the order of integration and summation,
we arrive at

$\displaystyle S^{\rm loc}\propto\frac{\zeta(3)}{\pi^{2}}\quad{\rm and}\quad
g^{\rm loc}\propto\frac{\pi}{6}.$ (95)

For other integrals, complex contour integration and the residue theorem are
more useful. For example, in the case of delta-$T$ noise at phPf-$2$ we
encounter an integral

$\displaystyle S^{\rm
loc}\propto\int_{-\infty}^{\infty}\frac{1}{\cosh(z)^{3}(\pi+2iz)^{2}}\,dz.$
(96)

Substituting $z\to 2\pi t$ and manipulating the resulting expression leads to

$\displaystyle S^{\rm
loc}\propto\frac{1}{8\pi}\int_{-\infty}^{\infty}\frac{1}{\cosh(2\pi
t)^{3}(\frac{1}{4}-it)^{2}}\,dt.$ (97)

The right-hand-side can now be written as a derivative

$\displaystyle S^{\rm loc}$
$\displaystyle\propto-\frac{1}{8\pi}\int_{-\infty}^{\infty}\frac{\partial^{2}}{\partial
a^{2}}\left(\frac{\ln(a-it)}{\cosh(2\pi t)^{3}}\right)\Bigg{|}_{a=1/4}\,dt$
$\displaystyle\equiv-\frac{1}{8\pi}\frac{\partial^{2}}{\partial
a^{2}}J(a)\Big{|}_{a=1/4}.$ (98)

We next write the function $J(a)$ as an integral along the closed rectangular
contour $\gamma(z)$, defined by

$\displaystyle-R\to R\to R+i\to-R+i\to-R,\ R\in\mathbb{R}^{+},$ (99)

in the complex $z$-plane as

$\displaystyle J(a)$
$\displaystyle=-\oint_{\gamma(z)}\frac{\ln\left(\Gamma(a-iz)\right)}{\cosh(2\pi
z)^{3}}dz$ $\displaystyle=-2\pi
i\sum_{i}\text{Res}\left(\frac{\ln\left(\Gamma(a-iz)\right)}{\cosh(2\pi
z)^{3}},z_{i}\right),$ (100)

where we used the residue theorem in the final step with the two third order
poles $z_{1}=i/4$ and $z_{2}=3i/4$ enclosed by $\gamma(z)$. After some
additional algebraic manipulations, we arrive at

$\displaystyle J(a)$
$\displaystyle=\frac{1}{8\pi^{2}}\Big{[}\psi^{(1)}\big{(}a+3/4\big{)}-\psi^{(1)}\big{(}a+1/4\big{)}\Big{]}$
$\displaystyle+\frac{1}{2}\Big{[}\ln\big(\Gamma\left(a+3/4\right)\big{missing})-\ln\big(\Gamma\left(a+1/4\right)\big{missing})\Big{]},$
(101)

where $\psi^{(1)}(z)=\partial^{2}_{z}\ln\Gamma[z]$ is the trigamma function.
Combining Eqs. (B.5) and (B.5) gives

$\displaystyle\frac{1}{8\pi}\int_{-\infty}^{\infty}\frac{1}{\cosh(2\pi
t)^{3}(\frac{1}{4}-it)^{2}}\,dt=\frac{17\pi}{480}$ (102)

The same method be can be used to produce the following identity

$\displaystyle\int_{-\infty}^{\infty}\frac{1}{\cosh(2\pi
t)^{3}(a-it)^{n}}\,dt=\frac{(-1)^{n-1}}{(n-1)!}\frac{\,d^{n}J(a)}{\,da^{n}},\
n\in\mathbb{Z}^{+}.$ (103)

##  Appendix C Proof of the universality of $G^{Q}_{\rm H}$ for equal thermal
equilibration on top and bottom edges

Here, we prove Eq. (27), namely that the thermal Hall conductance (26) is
universal as long as the two edges have the same degree of equilibration (even
if it is poor).

We consider a generic edge structure with $k$ downstream (ds) and $N-k$
upstream (us) modes, and write the heat transport equation (6) (with
$\delta\vec{V}=0$, since we assume no voltage bias) as

$\displaystyle\partial_{x}\vec{\theta}(x)=\mathcal{M}_{T}\vec{\theta}(x).$
(104)

Here, we defined $\vec{\theta}(x)=\vec{T}^{2}(x)$, and the matrix
$\mathcal{M}_{T}$ (7) satisfies the heat current conservation law

$\displaystyle\sum_{j}(\mathcal{M}_{T})_{ij}=0,\quad\forall i.$ (105)

The general solution of Eq. (104) for an edge segment with length $L>0$ can be
written as

$\displaystyle\vec{\theta}(L)=e^{L\mathcal{M}_{T}}\vec{\theta}(0),$ (106)

which we express in ds and us blockform as

$\displaystyle\begin{pmatrix}\vec{\theta}_{\rm ds}(L)\\\ \vec{\theta}_{\rm
us}(L)\end{pmatrix}=\left(\begin{array}[]{c|c}\mathcal{A}&\mathcal{B}\\\\[3.0pt]
\hline\cr\\\\[-9.0pt]
\mathcal{C}&\mathcal{D}\end{array}\right)\begin{pmatrix}\vec{\theta}_{\rm
ds}(0)\\\ \vec{\theta}_{\rm us}(0)\end{pmatrix}.$ (110)

Here, the block matrices $\mathcal{A}\in\mathbb{R}^{k\times k}$,
$\mathcal{B}\in\mathbb{R}^{k\times(N-k)}$,
$\mathcal{C}\in\mathbb{R}^{(N-k)\times k}$, and
$\mathcal{D}\in\mathbb{R}^{(N-k)\times(N-k)}$. In terms of these matrices, the
heat conservation law (105) translates to

$\displaystyle\sum_{j}\mathcal{A}_{ij}+\sum_{j}\mathcal{B}_{ij}=1,\quad\forall
i,$ (111)
$\displaystyle\sum_{j}\mathcal{C}_{ij}+\sum_{j}\mathcal{D}_{ij}=1,\quad\forall
i.$ (112)

Next, we rearrange the terms in Eq. (110) as

$\displaystyle\begin{pmatrix}\vec{\theta}_{\rm ds}(L)\\\ \vec{\theta}_{\rm
us}(0)\end{pmatrix}=\left(\begin{array}[]{c|c}\mathcal{A}-\mathcal{B}\mathcal{D}^{-1}\mathcal{C}&\mathcal{B}\mathcal{D}^{-1}\\\\[3.0pt]
\hline\cr\\\\[-9.0pt]
-\mathcal{D}^{-1}\mathcal{C}&\mathcal{D}^{-1}\end{array}\right)\begin{pmatrix}\vec{\theta}_{\rm
ds}(0)\\\ \vec{\theta}_{\rm us}(L)\end{pmatrix}.$ (116)

Let us now consider the top edge, for which the boundary conditions read
$\vec{\theta}_{\rm ds}(0)=(\bar{T}+\Delta T)^{2}\times(1,\ldots,1)^{T}_{k}$
and $\vec{\theta}_{\rm us}(L)=\bar{T}^{2}\times(1,\ldots,1)^{T}_{N-k}$ (see
Fig. 4). Plugging these quantitites into Eq. (116), we write the top edge heat
current (20) as

$\displaystyle J_{Q}^{\rm top}$
$\displaystyle=\frac{\kappa_{0}}{2}\left(\sum_{i:\chi_{i}=+1}n_{i}\theta^{i}_{\rm
ds}(L)-\bar{T}^{2}\sum_{i:\chi_{i}=-1}n_{i}\right)$
$\displaystyle=\frac{\kappa_{0}}{2}\Big{(}(\bar{T}+\Delta
T)^{2}\sum_{i,j:\chi_{i}=+1}n_{i}(\mathcal{A}-\mathcal{B}\mathcal{D}^{-1}\mathcal{C})_{ij}$
$\displaystyle+\bar{T}^{2}\sum_{i,j:\chi_{i}=+1}n_{i}(\mathcal{B}\mathcal{D}^{-1})_{ij}-\bar{T}^{2}\sum_{i:\chi_{i}=-1}n_{i}\Big{)},$
(117)

where in the second line we used Eq. (116). For the bottom edge, we have
reversed boundary conditions $\vec{\theta}_{\rm
ds}(0)=\bar{T}^{2}\times(1,\ldots,1)^{T}_{k}$ and $\vec{\theta}_{\rm
us}(L)=(\bar{T}+\Delta T)^{2}\times(1,\ldots,1)^{T}_{N-k}$. The bottom edge
heat current (21) then reads

$\displaystyle J_{Q}^{\rm bot}$
$\displaystyle=-\frac{\kappa_{0}}{2}\left(\sum_{i:\chi_{i}=+1}n_{i}\theta^{i}_{\rm
ds}(L)-\bar{T}^{2}\sum_{i:\chi_{i}=-1}n_{i}\right)$
$\displaystyle=\frac{\kappa_{0}}{2}\Big{(}\bar{T}^{2}\sum_{i,j:\chi_{i}=+1}n_{i}(\mathcal{A}-\mathcal{B}\mathcal{D}^{-1}\mathcal{C})_{ij}$
$\displaystyle+(\bar{T}+\Delta
T)^{2}\sum_{i,j:\chi_{i}=+1}n_{i}(\mathcal{B}\mathcal{D}^{-1})_{ij}$
$\displaystyle-(\bar{T}+\Delta T)^{2}\sum_{i:\chi_{i}=-1}n_{i}\Big{)}.$ (118)

The crucial next step is the assumption that the degrees of thermal
equilibration on the top and bottom edges are identical. This translates to
identical block matrices $\mathcal{A}$, $\mathcal{B}$, $\mathcal{C}$, and
$\mathcal{D}$ for the two edges. Then, and only then, can we combine the two
currents as

$\displaystyle J_{Q}^{\rm top}-J_{Q}^{\rm
bot}=\frac{\kappa_{0}}{2}\left((\bar{T}+\Delta T)^{2}+\bar{T}^{2}\right)$
$\displaystyle\times\Big{(}\sum_{i,j:\chi_{i}=+1}n_{i}(\mathcal{A}-\mathcal{B}\mathcal{D}^{-1}\mathcal{C}+\mathcal{B}\mathcal{D}^{-1})_{ij}-\sum_{i:\chi_{i}=-1}n_{i}\Big{)}.$
(119)

Now, by using Eqs. (111)-(112), and identifying

$\displaystyle\sum_{i:\chi_{i}=+1}n_{i}=c,$ (120)
$\displaystyle\sum_{i:\chi_{i}=-1}n_{i}=\bar{c},$ (121)

the dependence on $\mathcal{A},\mathcal{B},\mathcal{C},$ and $\mathcal{D}$
cancels out, and we find that Eq. (C) reduces to

$\displaystyle J_{Q}^{\rm top}-J_{Q}^{\rm
bot}=\frac{\kappa_{0}}{2}\left((\bar{T}+\Delta
T)^{2}+\bar{T}^{2}\right)(c-\bar{c}).$ (122)

Finally, inserting this expression into the definition of $G^{Q}_{H}$, given
in Eq. (26), gives our desired result (27). Our proof generalizes the
theoretical analysis for $\nu=2/3$ in Ref. [37] to any edge structure.

##  Appendix D Computation of voltage and temperature profiles for the
phPf-$3$ interface

We compute the noise and thermal conductance of all considered edge structure
by using voltage and temperature profiles of the edge channels within the
incoherent tunneling model introduced in Sec. II. Below, we present in detail
their derivation for the phPf-$3$ interface using boundary conditions for
charge current as well as down- and upstream delta-$T$ noise. Other edge
structures are treated in a perfectly analogous manner.

We begin by labelling the edge channels as $1,2,3$ from top to bottom
according to Fig. 2. Transport along the edge is characterized by the channel
specific filling factors $\nu_{i}$, heat conductances $n_{i}$ and chiralities
$\chi_{i}$, given by

  1. 1:

$\nu_{1}=1\phantom{/2}$, $n_{1}=1\phantom{/2}$, $\chi_{1}=+$,

  2. 2:

$\nu_{2}=1/2$, $n_{2}=1\phantom{/2}$, $\chi_{2}=-$,

  3. 3:

$\nu_{3}=0\phantom{/2}$, $n_{3}=1/2$, $\chi_{3}=+$.

These values further specify the transport matrices $\mathcal{M}_{V}$ and
$\mathcal{M}_{T}$ in Eqs. (4) and (7) as

$\displaystyle\mathcal{M}_{T}$
$\displaystyle=\alpha\begin{pmatrix}-\chi_{1}n_{2}&\phantom{-}\chi_{1}n_{2}&0\\\
\phantom{-}\chi_{2}n_{1}&-\chi_{2}n_{1}&0\\\ 0&0&0\end{pmatrix}$
$\displaystyle+\beta\begin{pmatrix}0&0&0\\\
0&-\chi_{2}n_{3}&\phantom{-}\chi_{2}n_{3}\\\
0&\phantom{-}\chi_{3}n_{2}&-\chi_{3}n_{2}\end{pmatrix}$ (123) and
$\displaystyle\mathcal{M}_{V}$ $\displaystyle=\frac{1}{\ell^{C}_{\rm
eq}}\begin{pmatrix}-\frac{\chi_{1}}{\nu_{1}}&\phantom{-}\frac{\chi_{1}}{\nu_{1}}\\\
\phantom{-}\frac{\chi_{2}}{\nu_{2}}&-\frac{\chi_{2}}{\nu_{2}}\end{pmatrix}.$
(124)

###  D.1 Charge current noise, $\Delta V\neq 0$, $\Delta T=0$

Figure 13: Channel resolved voltage profiles $V_{1}(x)$ and $V_{2}(x)$ of the
charge carrying channels at phPf-$3$ using $\delta=20$.

For the charge current noise, the appropriate boundary conditions are

$\displaystyle T_{i}(0)$ $\displaystyle=0,\ $ $\displaystyle\text{for}\quad
i\in\\{1,2,3\\}$ (125a) $\displaystyle V_{i}(0)$ $\displaystyle=\Delta V,\ $
$\displaystyle\text{for}\quad\chi_{i}=+1$ (125b) $\displaystyle V_{i}(1)$
$\displaystyle=0,\ $ $\displaystyle\text{for}\quad\chi_{i}=-1.$ (125c)

Here, we assumed that $e\Delta V\gg k_{\rm B}\bar{T}$ so that we may set the
base temperature $\bar{T}\rightarrow 0$. The voltage profiles
$\vec{V}(\tilde{x})$ for the two charge carrying channels along the edge
segment are obtained by solving Eq. (3). They are visualized in Fig. 13.
Clearly the voltage drop occurs only on the right-hand-side of the edge
segment. This is the hot spot. Note also that the voltage profiles are
independent of the degree of thermal equilibration, since the processes
leading to charge and thermal equilibration are considered to take place at
different length scales. Knowledge of $\vec{V}(\tilde{x})$ allows us to
further compute the Joule-heating contribution in Eq. (8). For efficient
thermal equilibration, we next solve Eq. (6) and obtain temperature profiles
which depend on the pairwise degrees of thermal equilibration:
$\alpha=L/\ell^{Q}_{\rm eq,12},\beta=L/\ell^{Q}_{\rm eq,23}$ [see Eq. (24)],
as well as the charge equilibration parameter $\delta=L/\ell^{C}_{\rm eq}$.
Here, $L$ denotes the edge segment length and we define here $\tilde{x}=x/L$
as a rescaled coordinate along the segment. The channel resolved temperature
profiles are visualized in Fig. 14(a) for two different sets of thermal
equilibration lengths. We see that large $\alpha,\beta$ (red curves) are
needed to have all channels at similar temperatures at the noise spot (i.e.,
for $\tilde{x}\ll 1$) and thus to describe efficient thermal equilibration.
This is contrasted by the blue curves for which $\alpha$ is not large and the
channels are at different temperatures at the noise spot.

|  |
---|---|---
|  |
|  |

(a)  (b)

(c)  (d)

(e)  (f)

Figure 14: Channel resolved temperature profiles for channels $1$, $2$ and $3$
of the phPf-$3$ interface. The distance along the edge segment is parametrized
$\tilde{x}=x/L$. (a) Temperature profiles for voltage biased charge current
noise for two sets of parameters: Superscript (1) labels profiles for the
choice $(\alpha,\beta,\delta)=(100,100,20)$ (red lines ) and (2) are profiles
for $(\alpha,\beta,\delta)=(2,100,20)$ (blue lines). Both parameters sets
correspond to finite thermal equilibration (b) Same as (a) but for poor
thermal equilibration. (c)-(d) Same as (a)-(b) but for downstream delta-$T$
noise for strong temperature bias $\Delta T\gg\bar{T}$. (e)-(f) Same as
(a)-(b) but for upstream delta-$T$ noise for strong temperature bias $\Delta
T\gg\bar{T}$.

Absent thermal equilibration on the other hand corresponds to $\ell^{Q}_{\rm
eq,ij}\to\infty$ and Eq. (6) simplifies to

$\displaystyle\partial_{\tilde{x}}\vec{T^{2}}(\tilde{x})=\delta\vec{V}(\tilde{x}).$
(126)

The channel resolved temperature profiles for this solution are given in Fig.
14(b).

###  D.2 Downstream delta-$T$ noise, $\Delta V=0$, $\Delta T\neq 0$

This type of noise corresponds to the boundary conditions

$\displaystyle V_{i}(0)$ $\displaystyle=0=V_{i}(1),\ $
$\displaystyle\text{for}\quad i\in\\{1,2,3\\}$ (127a) $\displaystyle T_{i}(0)$
$\displaystyle=\bar{T}+\Delta T,\ $ $\displaystyle\text{for}\quad\chi_{i}=+$
(127b) $\displaystyle T_{i}(1)$ $\displaystyle=\bar{T},\ $
$\displaystyle\text{for}\quad\chi_{i}=-$ (127c)

which leads to trivial voltage profiles $V_{i}(\tilde{x})\equiv 0,\forall i$.
As a consequence, the energy transport equation Eq. (6) simplifies to

$\displaystyle\partial_{\tilde{x}}\vec{T^{2}}(\tilde{x})=\mathcal{M}_{T}\vec{T^{2}}(\tilde{x}).$
(128)

For efficient thermal equilibration, this equation is solved to obtain
temperature profiles in terms of $(\alpha,\beta)$. Here, we consider the
strong temperature bias limit, $\Delta T\gg\bar{T}$, and consider the
temperature profiles to leading order in $\Delta T/\bar{T}$. The resulting
temperature profiles are shown in Fig. 14(c). In case of absent thermal
equilibration, the energy transport equation reads

$\displaystyle\partial_{\tilde{x}}\vec{T^{2}}(\tilde{x})=0$ (129)

Hence, the temperature profiles of the channels will be constant and follow
immediately from the boundary conditions, see Fig. 14(d).

###  D.3 Upstream delta-$T$ noise, $\Delta V=0$, $\Delta T\neq 0$

For the upstream delta-$T$ noise, we use the boundary conditions

$\displaystyle V_{i}(0)$ $\displaystyle=0=V_{i}(1),\ $
$\displaystyle\text{for}\quad i\in\\{1,2,3\\}$ (130a) $\displaystyle T_{i}(0)$
$\displaystyle=\bar{T},\ $ $\displaystyle\text{for}\quad\chi_{i}=+$ (130b)
$\displaystyle T_{i}(1)$ $\displaystyle=\bar{T}+\Delta T,\ $
$\displaystyle\text{for}\quad\chi_{i}=-.$ (130c)

Hence, the voltage profiles are also trivial $V_{i}(\tilde{x})\equiv 0,\forall
i$ and we obtain the temperature profiles using Eq. (128) for efficient and
Eq. (129) for absent thermal equilibration. The resulting temperature profiles
are visualized in Fig. 14(e) and Fig. 14(f), respectively.

###  D.4 Connection of temperature profiles to noise

The phPf-$3$ interface has $\nu_{Q}>0$ so that for full thermal equilbiration,
the heat transport is dominantly in the downstream direction [see Fig. 2]. Any
upstream heat transport is exponentially suppressed in $L$ [66]. For the
situation of voltage biased charge current noise, in the regime of efficient
thermal equilibration, we find non-vanishing temperatures only in the region
$\tilde{x}\approx 1$. This is seen from the red curves in Fig. 14(a). This
characteristic region, called the hot spot, is a result of Joule-heating close
to the right (most downstream) contact and the dominantly downstream thermal
transport. We see that the effective temperature at the noise spot is very
small: $T_{\rm ns}\approx 0$, and we expect small noise as well $S\simeq 0$.
Indeed, by inserting the temperature profiles in the noise kernel
$\Lambda(x)=k_{\rm B}T(\tilde{x})$ in Eq. (14) and integrating, we obtain
exponentially suppressed noise $S\sim\exp(-\delta)$, plotted in Fig. 7(a).

For thermally equilibrated down- and upstream delta-$T$ noise, the boundary
conditions for an edge with $\nu_{Q}>0$ imply equal and constant temperatures
of all channels at the noise spot [see Fig. 14(c) and (e)]. This leads to a
hot spot temperature $T_{\rm ns}=\bar{T}+\Delta T$ and $T_{\rm ns}=\bar{T}$
for down- and upstream delta-$T$ noise respectively. The corresponding noise
profiles are shown in Fig. 14(c). Absent thermal equilibration on the other
hand leads to constant but different temperatures of the channels at the noise
spot for the three cases. We explore the out-of-equilibrium situation in the
noise spot by using the Green’s function method outlined in Sec. B.1 to
compute noise kernels $\Lambda(\tilde{x})$ with the full temperature profiles
in case of charge current noise. This leads to the noise plots in Fig. 7(b).
For thermally non-equilibrated down- and upstream delta-$T$ noise there is no
additional source of heating along the edge. Following the same approach as
for charge current noise in Sec. B.1, we find a modified NJ noise for down-
and upstream delta-$T$ noise, described by Eq. (38) and Eq. (41) respectively.
Adjusting $\mathcal{M}_{V}$, $\mathcal{M}_{T}$ as well as the boundary
conditions according to the transport properties of the involved channels, the
temperature and noise profiles of the other interfaces in Fig. 7 are obtained
following the same steps.

##  Appendix E Unit conversion without tears: from theorist to experimentalist
units

The voltage bias noise excess noise $S^{\rm exc}$ computed using our approach
is expressed in units of $e^{3}\Delta V/h$. Let us denote the noise in such
units as $S^{\rm exc}_{\Delta V}$. To connect these units to more
experimentally relevant units, we start by using the current-voltage relation
for chiral edge transport

$\displaystyle I_{0}=\nu\frac{e^{2}}{h}\Delta V.$ (131)

This relation holds for efficient charge equilibration, and is manifest
experimentally by robust charge conductance quantization. By using this
relation, we re-write the noise in terms of the source current $I_{0}$ with
the notation

$\displaystyle S^{\rm exc}_{I_{0}}=\frac{S^{\rm exc}_{\Delta V}}{\Delta
V}\frac{I_{0}}{\nu}\frac{h}{e^{2}}.$ (132)

For a voltage bias $\Delta V$ given in $\mu V$, the excess noise can then be
written as

$\displaystyle S^{\rm exc}\approx 6.20492\left(\frac{S^{\rm exc}_{\Delta
V}}{\Delta Ve^{3}/h}\right)\cdot 10^{-30}\frac{A^{2}}{\mu V\ \rm Hz}.$ (133)

Alternatively, for a bias current $I_{0}$ given in $nA$, we have

$\displaystyle S^{\rm exc}\approx 1.602\left(\frac{S^{\rm exc}_{\Delta
V}}{\Delta V}\cdot\frac{I_{0}}{\nu}\right)\cdot 10^{-28}\frac{A^{2}}{nA\ \rm
Hz}.$ (134)

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