# Landscapes of an out-of-equilibrium anyonic sea

Gu Zhang<EMAIL_ADDRESS>Beijing Academy of Quantum Information Sciences,
Beijing 100193, China  Igor Gornyi<EMAIL_ADDRESS>Institute for Quantum
Materials and Technologies and Institut für Theorie der Kondensierten Materie,
Karlsruhe Institute of Technology, 76131 Karlsruhe, Germany  Yuval Gefen
<EMAIL_ADDRESS>Department of Condensed Matter Physics, Weizmann
Institute of Science, Rehovot 761001, Israel

###### Abstract

The low-energy dynamics of two-dimensional topological matter hinges on its
one-dimensional edge modes. Tunneling between fractional quantum Hall edge
modes facilitates the study of anyonic statistics: it induces time-domain
braiding that dominates signals from diluted anyon beams. We develop a
framework for characterizing out-of-equilibrium anyonic states and define
their effective potential and temperature, as well as the landscape of their
excitations. Unlike fermions, the effective anyon potential depends on the
type of the tunneling quasiparticles; non-equilibrium anyonic states are
underlain by power-law energy distributions. This allows “hot” anyons to
tunnel above the chemical potential of the source, which we capture by a
measurable universal witness function.

The foundational observations associated with non-conventional quantum
statistics (see e.g., Refs. Laughlin (1983); Arovas  _et al._ (1984); Kitaev
(2001); Mong  _et al._ (2014)), and the prospects for harnessing this for the
design of quantum information platforms Horodecki  _et al._ (2009); Nielsen
and Chuang (2010); Wilde (2013); Gühne and Tóth (2009); Streltsov  _et al._
(2017), render the physics of anyons a focal point of interest, with recent
experiments on fractional quantum Hall (QH) edge states Nakamura  _et al._
(2019, 2020); Bartolomei  _et al._ (2020); Nakamura  _et al._ (2022, 2023);
Kundu  _et al._ (2023); Glidic  _et al._ (2023a, b); Lee  _et al._ (2023);
Ruelle  _et al._ (2023) following a long line of theoretical study Thouless
and Gefen (1991); Safi  _et al._ (2001); Kane and Fisher (2003); Vishveshwara
(2003); Kim  _et al._ (2005); Law  _et al._ (2006); Feldman and Kitaev (2006);
Feldman  _et al._ (2007); Ponomarenko and Averin (2007); Rosenow and Halperin
(2007); Campagnano  _et al._ (2012, 2013); Rosenow  _et al._ (2016);
Campagnano  _et al._ (2016); Han  _et al._ (2016); Lee  _et al._ (2019);
Rosenow and Stern (2020); Rech  _et al._ (2020); Morel  _et al._ (2022);
Schiller  _et al._ (2022); Zhang  _et al._ (2022); Lee and Sim (2022);
Schiller  _et al._ (2023); Jonckheere  _et al._ (2023); Iyer  _et al._ (2023a,
b); Zhang  _et al._ (2023); Kivelson and Murthy (2024); Thamm and Rosenow
(2024). Remarkable breakthrough has been reported in identifying the anyonic
braiding phase employing Fabry-Perot interferometry Nakamura  _et al._ (2019,
2020, 2022, 2023), Mach-Zehnder Kundu  _et al._ (2023), as well as Hong-Ou-
Mandel (HOM) collider setups Bartolomei  _et al._ (2020); Glidic  _et al._
(2023a); Lee  _et al._ (2023); Ruelle  _et al._ (2023). With the latter, the
anyonic braiding phase was produced through “time-domain braiding” Schiller
_et al._ (2023), a generalization of “bubble-diagram braiding” Han  _et al._
(2016); Lee  _et al._ (2019), see Supplemental Information (SI) 111see
Supplemental Information for technical details Sec. S1 for illustration. This
process comprises an exchange between an incoming anyon (part of a source-
injected non-equilibrium beam) Han  _et al._ (2016); Lee  _et al._ (2019) and
a quasiparticle-quasihole pair generated at the collider; this is followed by
the annihilation of this pair at a later time, hence “time-domain braiding”.
Admittedly, the results reported in Refs. Bartolomei  _et al._ (2020); Glidic
_et al._ (2023a); Lee  _et al._ (2023); Ruelle  _et al._ (2023) do not provide
a direct observation of how time-domain braiding influences individual-anyon
tunneling events, in which the anyonic gas is maintained out of equilibrium.

The study reported here relies on a notion of Laughlin surface, the anyonic
counterpart of electrons’ Fermi surface. We investigate the non-equilibrium
landscape of excitations around the Laughlin surface, focusing on the impact
of time-domain braiding dynamics. Its inclusion radically sets our analysis
off from earlier studies (Refs. Haldane (1991); Wu (1994)), where equilibrium
distributions of anyons have been evaluated with standard statistical-
mechanics tools. The challenging question here is how the physics of a non-
equilibrium Laughlin surface, replacing Pauli blockade by anyonic statistics,
differs from that of a non-equilibrium Fermi landscape.

_Main results_ —Consider Fig. 1(a), representing three chiral anyonic channels
that may support quasiparticles of charge $\nu e$. Channel A is grounded at
its source; non-equilibrium current $I_{\text{A}}$ is injected from SA via a
tunneling bridge (“diluter”). The reference channel B is prepared at
equilibrium potential $V_{\text{B}}$ and temperature $T_{\text{B}}$. Our main
results are: (I). Time-domain-braiding introduces an effective shift,
$V_{\text{eff}}$, of the potential of channel A [Fig. 1(b)] and an effective
temperature $T_{\text{eff}}$ [Figs. 1(c) and 1(d)]. These are diagnosed by
channel B: tuning it such that there is no net charge or heat transport in the
collider connecting channels A and B implies $V_{\text{eff}}=V_{\text{B}}$
Note  and $T_{\text{eff}}=T_{\text{B}}$ (cf. Refs. Engquist and Anderson
(1981); Nosiglia  _et al._ (2018); Spånslätt  _et al._ (2019)). We further
show that $V_{\text{eff}}$ is generated by the non-equilibrium current
$I_{\text{A}}$; while $T_{\text{eff}}$ is influenced by both the current
noise, $S_{\text{A}}$ and the ambient temperature $T_{\text{A}}$ of A
(upstream of the diluter), see Fig. 2. (II). The non-equilibrium distribution
function of excited anyon modes decreases slowly (power law) at high energies
[Fig. 3(a)], in sharp contrast to the exponential decrease with energy of
equilibrium excitations. This implies finite non-equilibrium particle
excitations even above the “ceiling” of $\nu eV_{\text{SA}}$ (the energy above
which zero-temperature equilibrium particle distribution at the source
channel, SA, vanishes). Hereafter this is referred to as time-domain-braiding
generation of hot anyons. As a result, tunneling from channel A to channel B
is possible, even with $V_{\text{SA}}<V_{\text{B}}$. We construct a universal
witness function, $\mathcal{W}$ [Eq. (7) and Fig. 3(b)], characterizing this
phenomenon of hot-anyon generation. (III). At zero ambient temperature, charge
tunneling from the non-equilibrium channel to the equilibrium one is only
possible above the Laughlin surface (the surface of the anyonic sea) of the
latter; tunneling of anyons with energies below this Laughlin surface is
blocked, in analogy with Pauli blocking for fermions. (IV). The effective
potential depends on the type of quasiparticles that tunnel at the collider.
These remarkable features are unique to out-of-equilibrium anyonic landscapes,
which rely on the anyonic statistical braiding phase.

_The model and Hamiltonian_ —As shown in Fig. 1(a), the system we consider
consists of three chiral channels described by the free Hamiltonian
$H_{\text{edge}}=\frac{1}{4\pi}\sum_{\alpha}\int
dx(\partial_{x}\phi_{\alpha})^{2}$, where $\phi_{\alpha}$ is the dynamical
bosonic field operator, satisfying
$[\phi_{\alpha}(x),\phi_{\alpha^{\prime}}(x^{\prime})]=i\pi\delta_{\alpha,\alpha^{\prime}}\text{sgn}(x-x^{\prime})$,
with $\alpha=\text{SA},\,\text{A},\,\text{B}$. Channels SA (with temperature
$T_{\text{SA}}$) and B (with temperature $T_{\text{B}}$) are biased at
voltages $V_{\text{SA}}$ and $V_{\text{B}}$, respectively. The diluter
Hamiltonian ($x=0$) is
$H_{\text{A}}=t_{\text{A}}\psi^{\dagger}_{\text{SA}}(0)\psi_{\text{A}}(0)+\text{H.c.}$,
where $|t_{\text{A}}|^{2}=\mathcal{T}_{\text{A}}$,
$\psi^{\dagger}_{\text{SA}}(0)\psi_{\text{A}}(0)=F_{\text{SA}}^{\dagger}F_{\text{A}}\exp[-i\sqrt{\nu}(\phi_{\text{SA}}-\phi_{\text{A}})]/(2\pi\tau_{c})$,
with $\tau_{c}$ the ultraviolet cutoff, and
$F_{\text{SA}}^{\dagger}F_{\text{A}}$ the product of Klein factors. Downstream
of the diluter, channel A carries a non-equilibrium current $I_{\text{A}}$
with noise $S_{\text{A}}\equiv\int
dt[\langle\hat{I}_{\text{A}}(t)\hat{I}_{\text{A}}(0)\rangle-
I_{\text{A}}^{2}]$. The collider Hamiltonian (at $x=L$) is
$H_{\text{T}}=t_{\text{C}}\psi^{\dagger}_{\text{A}}(L)\psi_{\text{B}}(L)+\text{H.c.}$,
where $|t_{\text{C}}|^{2}=\mathcal{T}_{\text{C}}$. Hereafter, we set
$\hbar=k_{B}=1$.

Figure 1:  System model and effective parameters. (a) Channel A receives a
non-equilibrium current $I_{\text{A}}$ from source SA. Effective potential
$V_{\text{eff}}$ and temperature $T_{\text{eff}}$ of channel A are diagnosed
by an equilibrium channel B. Voltage biases and temperatures are marked out
for all three channels. (b) The tunneling-current spectrum
$\mathcal{I}_{\text{T}}(\epsilon)$, when
$V_{\text{B}}=V_{\text{eff}}=I_{\text{A}}\sin(2\pi\nu)/\nu^{2}e^{2}$, for
zero-temperature (the blue curve) and finite-temperature
($T_{\text{B}}=I_{\text{A}}/e$, the red curve). In both situations, the charge
tunneling current vanishes: $I_{\text{T}}\equiv\nu e\int
d\epsilon\mathcal{I}_{\text{T}}(\epsilon)=0$. (c) The heat current
$\mathcal{J}_{Q}(\epsilon)$ at energy $\epsilon$, when $V_{\text{B
}}=V_{\text{eff}}=I_{\text{A}}\sin(2\pi\nu)/\nu^{2}e^{2}$, for zero
temperature (the blue curve) and finite temperature
($T_{\text{B}}=9I_{\text{A}}/e$, the red curve). (d) The total heat current
$J_{Q}=\int d\epsilon\mathcal{J}_{Q}(\epsilon)$ as a function of
$eT_{\text{B}}/I_{\text{A}}$; $J_{Q}=0$ when $T_{\text{B}}=T_{\text{eff}}$.
(e) Particle and hole distributions in channel A, for $T_{\text{A}}=0$ (solid
curves) and $T_{\text{A}}=10eV_{\text{SA}}$ (dashed curves), when
$T_{\text{SA}}=0$.

_Correlation functions from time-domain braiding_ —When the dilution is
strong, i.e., $I_{A}/(\nu^{2}e^{2}V_{\text{SA}})\ll 1$, non-equilibrium anyons
emanating from the source SA rarely participate in tunneling at the collider.
Instead, they mainly influence tunneling indirectly, via the time-domain-
braiding processes Rosenow  _et al._ (2016); Han  _et al._ (2016); Lee and Sim
(2022); Lee  _et al._ (2023). These processes feature braiding between an
anyon-hole pair excitation generated at the collider and an injected (diluted)
anyon beam. The relevant building-block correlators are derived in Sec. S2 of
SI:

$\left.\begin{array}[]{c}\big{\langle}\\!\psi^{\dagger}_{A}(L,t)\psi_{A}(L,0)\\!\big{\rangle}_{\text{neq}}\\\
\big{\langle}\\!\psi_{A}(L,t)\psi^{\dagger}_{A}(L,0)\\!\big{\rangle}_{\text{neq}}\end{array}\\!\\!\\!\right\\}\\!\\!=\\!\\!\frac{e^{-\\!\frac{S_{\text{A}}}{\nu^{2}e^{2}}[1\\!-\\!\cos(2\pi\nu)]|t|\\!\pm\\!i\frac{I_{\text{A}}}{\nu
e}\sin(2\pi\nu)t}}{2(\\!\pi\tau_{c}\\!)^{1\\!-\\!\nu}\\!\sin^{\nu}[\pi
T_{\text{A}}(\tau_{c}\\!+\\!it)]/T_{\text{A}}^{\nu}},$ (1)

where “neq” indicates that correlations are evaluated in the non-equilibrium
state downstream of the diluter (for $\mathcal{T}_{\text{C}}=0$). Compared
with zero temperature Rosenow  _et al._ (2016); Han  _et al._ (2016); Lee and
Sim (2022); Lee  _et al._ (2023), finite temperature enters Eq. (1) through
$T_{\text{A}}$ and $T_{\text{SA}}$; both affect the zero-temperature noise-
current relation, $S_{\text{A}}=\nu eI_{\text{A}}$, rendering the ratio
$S_{\text{A}}/I_{\text{A}}$ temperature-dependent. Note also that Eq. (1) is
only valid when $T_{\text{A}}$ is smaller than $\nu eV_{\text{SA}}$: otherwise
thermal fluctuations undermine time-domain-braiding.

Channel B is characterized by the equilibrium correlation functions (labeled
by “0”; again evaluated at $\mathcal{T}_{\text{C}}=0$)

$\left.\begin{array}[]{c}\big{\langle}\psi^{\dagger}_{B}(L,t)\psi_{B}(L,0)\big{\rangle}_{0}\\\
\big{\langle}\psi_{B}(L,t)\psi_{B}^{\dagger}(L,0)\big{\rangle}_{0}\end{array}\\!\\!\right\\}\\!=\\!\frac{1}{2\pi\tau_{c}}\frac{(\tau_{c}\pi
T_{\text{B}})^{\nu}e^{\pm i\nu eV_{\text{B}}t}}{\sin^{\nu}[\pi
T_{\text{B}}(\tau_{c}+it)]}.$ (2)

With Eqs. (1) and (2), the tunneling current between channels A and B reads,
to leading order in $\mathcal{T}_{\text{C}}$, as

$\displaystyle I_{\text{T}}\\!\equiv$ $\displaystyle\nu
e\mathcal{T}_{\text{C}}\\!\\!\int\\!\\!dt\big{[}\langle\psi^{\dagger}_{A}(L,\\!t)\psi_{A}(L,\\!0)\rangle_{\text{neq}}\langle\psi_{B}(L,\\!t)\psi_{B}^{\dagger}(L,\\!0)\rangle_{0}$
(3)
$\displaystyle\\!-\\!\langle\psi_{A}(L,0)\psi^{\dagger}_{A}(L,0)\rangle_{\text{neq}}\langle\psi^{\dagger}_{B}(L,0)\psi_{B}(L,0)\rangle_{0}\big{]}.$

_Effective chemical potential—_ The effective potential $V_{\text{eff}}$ of A
is defined as the value of $V_{\text{B}}$, at which the net charge current at
the collider vanishes. Following this definition, we derive the effective
potential $V_{\text{eff}}=I_{\text{A}}\sin(2\pi\nu)/\nu^{2}e^{2}$, which is
manifestly temperature-independent. To gain insight into this temperature-
independence, we define the charge tunneling rate at energy $\epsilon$,
$\mathcal{I}_{\text{T}}(\epsilon)\equiv
n^{\text{A}}_{\text{p}}(\epsilon)n^{\text{B}}_{\text{h}}(\epsilon)-n^{\text{A}}_{\text{h}}(\epsilon)n^{\text{B}}_{\text{p}}(\epsilon)$,
with particle and hole distributions (extension of Ref. Zhang  _et al._ (2022)
to non-equilibrium cases)

$\left.\begin{array}[]{c}n^{\alpha}_{\text{p}}(\epsilon)\\\
n^{\alpha}_{\text{h}}(\epsilon)\end{array}\right\\}\equiv\int
dt\,e^{\\!-i\epsilon
t}\left\\{\begin{array}[]{c}\langle\psi_{\alpha}^{\dagger}(t)\psi_{\alpha}(0)\rangle_{a},\\\
\langle\psi_{\alpha}(t)\psi^{\dagger}_{\alpha}(0)\rangle_{a},\end{array}\right.$
(4)

where ${a=\text{neq},0,0}$ for ${\alpha=\text{A},\text{SA},\text{B}}$,
respectively. Although the distributions used in Eq. (3) are defined upstream
of the collider, their definition, Eq. (4), applies to any point between the
diluter and the collider. In the limit of small $\mathcal{T}_{\text{C}}$, this
also applies to positions downstream of the collider (SI Sec. S3).

Crucially, for anyonic systems, $n_{\text{h}}^{\alpha}\neq
1-n^{\alpha}_{\text{p}}$, unlike the fermionic case. Setting
$V_{\text{B}}=V_{\text{eff}}\equiv I_{\text{A}}\sin(2\pi\nu)/\nu^{2}e^{2}$,
the energy-dependent tunneling rate $\mathcal{I}_{\text{T}}$ displays the
symmetry ${\mathcal{I}_{\text{T}}(\epsilon-\nu
eV_{\text{eff}})=-\mathcal{I}_{\text{T}}(-\epsilon+\nu eV_{\text{eff}})}$ [see
Fig. 1(b) for $T_{\text{B}}=0$ and $T_{\text{B}}=I_{\text{A}}/e$], indicating
a vanishing tunneling current $I_{\text{T}}=\nu
e\int_{-\infty}^{\infty}d\epsilon\ \mathcal{I}_{\text{T}}(\epsilon)=0$.
Importantly, this observation remains valid even for finite $T_{\text{A}}$
and/or $T_{\text{SA}}$. Indeed, following Eqs. (1) and (2), as long as
$V_{\text{B}}=V_{\text{eff}}$, we have
$\langle\psi^{\dagger}_{A}(L,t)\psi_{A}(L,0)\rangle_{\text{neq}}\langle\psi_{B}(L,t)\psi_{B}^{\dagger}(L,0)\rangle_{0}=\langle\psi^{\dagger}_{B}(L,t)\psi_{B}(L,0)\rangle_{0}\langle\psi_{A}(L,t)\psi^{\dagger}_{A}(L,0)\rangle_{\text{neq}}$,
yielding zero for the integrand in Eq. (3). This implies that $V_{\text{eff}}$
depends on the state of the system only through the non-equilibrium current
$I_{\text{A}}$. The above features are a direct consequence of the particle-
hole symmetry $n_{\text{p}}^{\text{A}}[I_{\text{A}},\epsilon-
I_{\text{A}}\sin(2\pi\nu)/\nu
e]=n_{\text{h}}^{\text{A}}[I_{\text{A}},-\epsilon+I_{\text{A}}\sin(2\pi\nu)/\nu
e]$ [cf. Fig. 1(e), and more general cases in SI Sec. S6], valid for all
temperatures. Remarkably, this particle-hole symmetry is absent in the
fermionic case, where time-domain braiding is absent. Even for anyonic
systems, this symmetry is undermined when direct non-equilibrium anyon
tunneling processes are taken into account (SI Sec. S8). These processes are,
however, safely negligible in our limiting case of strongly dilute beams.

Figure 2:  Effective temperature of channel A. Here we fix $V_{\text{SA}}$,
and tune $T_{\text{SA}}$ or $T_{\text{A}}$. (a) When only $T_{\text{SA}}$ is
finite, the effective temperature $T_{\text{eff}}$ is proportional to the
noise $S_{\text{A}}$. (b) When $T_{\text{A}}$ is finite, the effective
temperature $T_{\text{eff}}$ approximately equals the sum of its ambient
temperature $T_{\text{A}}$, and the noise ($S_{\text{A}}$) induced effective
temperature.

_Effective temperature—_ To set the effective temperature $T_{\text{eff}}$ of
channel A, we fix $V_{\text{B}}=V_{\text{eff}}$ and calibrate $T_{\text{eff}}$
against $T_{\text{B}}$, at which no net heat current tunnels between A and B.
The heat current is given by $J_{Q}=\int_{-\infty}^{\infty}d\epsilon\
\mathcal{J}_{Q}(\epsilon)$, where heat refers to the energy counted from the
Laughlin surface, $\nu eV_{\text{B}}$, of channel B:

$\displaystyle\mathcal{J}_{Q}(\epsilon)\\!\equiv\\!\left[n^{\text{A}}_{\text{p}}(\epsilon)n^{\text{B}}_{\text{h}}(\epsilon)-n^{\text{A}}_{\text{h}}(\epsilon)n^{\text{B}}_{\text{p}}(\epsilon)\right]\left(\epsilon-\nu
eV_{\text{B}}\right).$ (5)

We start by setting $T_{\text{A}}=T_{\text{SA}}=0$. When $T_{\text{B}}=0$ [the
blue curve of Fig. 1(d)], $\mathcal{J}_{Q}(\epsilon)$ is positive for all
energies, indicating a net heat current from A to B. With $T_{\text{B}}$
finite, however, $\mathcal{J}_{Q}(\epsilon)$ can become negative at low
energies [$T_{\text{B}}=9I_{\text{A}}/e$, the red curve of Fig. 1(c)]. A sign
change of $J_{Q}$ is thus anticipated when $T_{\text{B}}$ becomes large enough
[Fig. 1(d); note that the heat flow vanishes when
$T_{\text{B}}=T_{\text{eff}}\approx 17.3I_{\text{A}}/e$ (SI Sec. S6E)].

To identify the sources of the effective temperature, one needs to consider
both vanishing and finite ambient temperatures. When
$T_{\text{A}}=T_{\text{SA}}=0$, one gets $S_{\text{A}}=\nu eI_{\text{A}}$ (SI
Sec. S2): the noise, which dictates the effective temperature, is solely
determined by the diluted current. We evaluate the heat current following
${J_{Q}=-2i\pi\mathcal{T}_{\text{C}}\int
dt\left\\{[\partial_{t}f_{1}(t)]f_{2}(t)-f_{1}(t)\partial_{t}f_{2}(t)\right\\}}$,
where

$\displaystyle\\!\\!\\!\\!f_{1}(t)$
$\displaystyle\\!\equiv\\!\big{\langle}\psi^{\dagger}_{\text{A}}(L,t)\psi_{\text{A}}(L,0)\big{\rangle}_{\text{neq}}\\!\exp\\!\left[-iI_{\text{A}}\\!\sin(2\pi\nu)t/\nu
e\right],$ (6) $\displaystyle\\!\\!\\!\\!f_{2}(t)$
$\displaystyle\\!\equiv\\!\big{\langle}\psi_{\text{B}}(L,t)\psi_{\text{B}}^{\dagger}(L,0)\big{\rangle}_{\text{0}}\,\exp\\!\left[iI_{\text{A}}\\!\sin(2\pi\nu)t/\nu
e\right].$

Using Eq. (6), the effective temperature, obtained by requiring $J_{Q}=0$, is
shown in Fig. 2, where $T_{\text{eff}}^{(0)}$ (chosen as the energy unit) is
the effective temperature when $T_{\text{A}}=T_{\text{SA}}=0$. For
$T_{\text{SA}}=0$ [Fig. 2(a)], the effective temperature $T_{\text{eff}}$
(doted points) is proportional to $S_{\text{A}}$ (the red curve), identifying
$S_{\text{A}}$ as the source of $T_{\text{eff}}$. In this case,
$T_{\text{eff}}=S_{\text{A}}/(s_{c}\nu^{2}e^{2})$, where $s_{c}$ is a
universal number that depends only on the filling fraction $\nu$ (SI Sec.
S6E). When $T_{\text{A}}\neq 0$, the effective temperature in A, downstream of
the diluter, has two contributions: the real temperature $T_{\text{A}}$ and
the temperature associated with the noise at the diluter ($S_{\text{A}}$)
through a Johnson-Nyquist relation (SI Sec. S10). The effective temperature
$T_{\text{eff}}$ is then approximately the sum of these two temperatures, as
illustrated in Fig. 2(b).

_Hot anyons from time-domain braiding_ —As shown above, non-equilibrium
current introduces an effective temperature in an anyonic channel. One would
thus naturally anticipate the presence of “thermal” quasiparticle excitations
in channel A around the shifted chemical potential. Indeed, as Fig. 3(a)
shows, the particle distribution is smeared by $T_{\text{eff}}$ and has a tail
for positive energies (counted from the effective chemical potential $\nu
eV_{\text{eff}}$). This tail, however, decays polynomially (SI Sec. S9), in
contrast to the exponential decay of equilibrium thermal distributions. Most
remarkably, the distribution remains finite even for energies larger than $\nu
eV_{\text{SA}}$, the Laughlin surface of source SA. We refer to these high-
energy anyons as hot anyons (similar to hot electrons in semiconductors and
superconductors Sze  _et al._ (1966); Asche  _et al._ (2006)) generated by
time-domain-braiding. In sharp contrast, for zero-ambient-temperature
fermionic systems, particle energies should _never_ exceed the Fermi surface
of the source.

Figure 3:  Hot anyons for $\nu=1/3$. (a) Particle distributions, with (red
curve) and without (black curve) a non-equilibrium current. Inset: with a non-
equilibrium current, particle distributions decay polynomially
$\sim(e\epsilon/I_{\text{A}})^{\nu-2}$ for positive energies. (b) The value of
$\mathcal{W}$ [Eq. (7)] as a function of the bias $V_{\text{B}}$. When
$e^{2}V_{\text{B}}\gg I_{\text{A}}$, $\mathcal{W}\to
4\tan(\pi\nu)(1-\nu)/\nu^{3}$, a universal value that depends only on the
filling factor.

We define a “witness function”, identifying the presence of hot anyons
generated by time-domain braiding:

$\mathcal{W}\equiv\left(e^{2}V_{\text{B}}/I_{\text{A}}\right)\,\left(\partial_{V_{\text{B}}}S_{\text{T}}\right)^{-1}\,\partial_{V_{\text{B}}}(eI_{\text{T}}+S_{\text{T}}/\nu).$
(7)

Here $S_{\text{T}}\equiv\int
dt[\langle\hat{I}_{\text{T}}(t)\hat{I}_{\text{T}}(0)\rangle-I_{\text{T}}^{2}]$
is the tunneling-current noise at the collider, with $\hat{I}_{\text{T}}$ the
tunneling-current operator. The function $\mathcal{W}$ is accessible by
measuring the current auto-correlations in channel B (SI Sec. S7). As shown by
Fig. 3(b), when the bias $V_{\text{B}}$ in channel B becomes much larger than
$I_{\text{A}}$, $\mathcal{W}$ approaches $4(1-\nu)\tan(\pi\nu)/\nu^{3}$, a
“universal value” that only depends on the Laughlin filling fraction, with the
factor $\tan(\pi\nu)$ stemming from the “braiding phase” and $(1-\nu)/\nu^{3}$
from the fractional charge.

As stated above, ${\mathcal{W}(\nu=1)=0}$, i.e., $\mathcal{W}$ vanishes for
fermions, indicating that any finite $\mathcal{W}$ is an anyon-specific
feature. In comparison to the “generalized Fano factor” of Refs. Rosenow  _et
al._ (2016); Lee and Sim (2022); Lee  _et al._ (2023), $\mathcal{W}$ has
several advantages. First, when $V_{\text{B}}>V_{\text{SA}}$, contributions to
$\mathcal{W}$ come entirely from quasiparticles whose energy is above the
“anyonic sea”, representing hot anyons. Second, to access the universal value
of $\mathcal{W}$, one requires $e^{2}V_{\text{B}}\gg I_{\text{A}}$, which is
more easily accessible experimentally than the limit of vanishingly small
transmission through the diluter. Focusing on $\mathcal{W}$ at large bias also
minimizes the effect of thermal contributions. Finally, $\mathcal{W}$ is
defined in terms of the differential current and noise instead of the total
current and noise. With this definition, one overcomes the difficulty of
dealing with energy-dependent transmission amplitudes experimentally (see
recent discussions in Refs. Veillon  _et al._ (2024); Schiller  _et al._
(2024b)).

_Chemical potential and tunneling quasiparticle charge_ —We have shown above
how the effective chemical potential and effective temperature originate from
the non-equilibrium current $I_{\text{A}}$. These results are obtained for the
setup, in which all three chiral channels are characterized by the same
filling factor $\nu$, with the same quasiparticles of charge $\nu e$ tunneling
at both the diluter and the collider. However, when different channels of the
system have different filling factors and different types of quasiparticles
may be involved in inter-channel tunneling, the effective chemical potential
also depends on the charge of tunneling quasiparticles. This is yet another
feature that distinguishes a non-equilibrium anyonic channel from a fermionic
one.

Figure 4:  An out-of-equilibrium anyonic channel A (with filling fraction
$\nu$ and non-equilibrium current $I_{\text{A}}$, after receiving charge $\nu
e$ quasiparticles from the not-shown source), whose effective chemical
potential is calibrated by an equilibrium channel B (with filling fraction
$m_{\text{B}}\nu$ and bias $V_{\text{B}}$). These two channels communicate via
quasiparticles with charge $m\nu e$.

To elaborate on this idea, we consider an out-of-equilibrium edge A, whose
effective chemical potential is calibrated by an equilibrium edge B (Fig. 4).
These two modes have different filling fractions, $\nu$ and $m_{\text{B}}\nu$,
respectively, with $m_{\text{B}}$ an integer. The non-equilibrium channel A
carries current $I_{\text{A}}$, provided by charge-$\nu e$ quasiparticles from
the source SA. We further assume that at the collider, these two channels
communicate via tunneling of charge-$m\nu e$ quasiparticles, with $m$ an
integer. In practice, $\nu$, $m_{\text{B}}$, and $m$ are determined by sample-
specific features, see SI Secs. S11 and S12 for the discussion of specific
cases. The tunneling from A to B (Fig. 4) is described by the operator
$\propto\exp[-i(m\sqrt{\nu}\phi_{\text{A}}-m\sqrt{\nu/m_{\text{B}}}\phi_{\text{B}})]$,
with $\phi_{\text{A}}$ and $\phi_{\text{B}}$ the corresponding bosonic fields.
The net tunneling current between A and B vanishes when
$n^{\text{A}}_{\text{p}}n^{\text{B}}_{\text{h}}=n^{\text{B}}_{\text{p}}n^{\text{A}}_{\text{h}}$,
which now happens for (SI Sec. S11)

$V_{\text{B}}=V_{\text{eff}}=I_{\text{A}}\sin\left(2m\pi\nu\right)/(m\nu^{2}e^{2}).$
(8)

Equation (S67) displays a notable feature: the generalized effective potential
induced by time-domain braiding, $V_{\text{eff}}$, depends on the charge $m\nu
e$ of quasiparticles that tunnel between channels A and B. This relies on the
fact that the channel-B voltage $V_{\text{B}}$ (used for calibrating
$V_{\text{eff}}$) appears in the correlation functions [generalizing Eq. (2)]
being multiplied with $m\nu e$. The type of quasiparticles that tunnel through
the collider is also important for the braiding phase that appears in the
exponential factor of the channel-A correlators [cf. Eq. (1)], where $2\pi\nu$
is replaced with $2\pi m\nu$. In particular, for $\nu=1/3$ and $m=m_{B}=3$,
the channels A and B communicate by tunneling of charge-$e$ fermions, which
results in the absence of time-domain braiding (anyons do not braid with
fermions) and, hence, the corresponding shift of the chemical potential
vanishes, $V_{\text{eff}}=0$.

At the same time, $m_{\text{B}}$ determines only the power in the denominator
of the equilibrium correlation function of channel B [cf. Eq. (2)] and is thus
irrelevant to determining the effective chemical potential. Remarkably, the
above insights indicate that, contrary to common wisdom for fermions, the
effective chemical potential of an anyonic channel only becomes well-defined
after specifying the type of quasiparticles with which the anyonic channel
communicates with other channels. Equivalently, the non-equilibrium
distribution functions (4) of anyons can only be accessed by probes that
involve tunneling of these same anyons. The above conclusions also apply to
setups where channel A has the filling fraction $m_{\text{A}}\nu$, with
$m_{\text{A}}$ an integer (SI Sec. S11).

This unique anyonic feature could be relevant to tunneling between out-of-
equilibrium anyonic edges with $\nu=1/3$ and $2/3$. In particular, at the
Kane-Fisher-Polchinski Kane  _et al._ (1994) fixed point, the $2/3$ channel
consists of a neutral node and a charge-$2e/3$ mode, where inter-channel
tunneling may rely on either charge-$e/3$ or $2e/3$ quasiparticles, with some
experimental tuning in place Goldstein and Gefen (2016), see details in SI
Sec. S12.

_Discussion and Perspective—_ We have generalized the concept of non-
equilibrium Fermi sea to anyonic sea, involving non-equilibrium landscape of
Laughlin surface. The definition and characterization of these new entities
establish the basis of new and rich physics: (i) Generalization to QH states
beyond the Laughlin fractions, e.g., for hole-like QH states, reconstructed
edges Meir (1994); Wan  _et al._ (2006, 2008); Wang  _et al._ (2013); Zhang
_et al._ (2014), and counter propagating modes. (ii) The dependence of the
above picture to the degree of decoherence and inter-mode equilibration (see,
e.g., Refs. Protopopov  _et al._ (2017); Nosiglia  _et al._ (2018); Spånslätt
_et al._ (2019); Park  _et al._ (2020); Asasi and Mulligan (2020); Srivastav
_et al._ (2021a, b); Spånslätt  _et al._ (2021); Dutta  _et al._ (2022); Kumar
_et al._ (2022); Melcer  _et al._ (2022); Khanna  _et al._ (2022); Das  _et
al._ (2022); Le Breton  _et al._ (2022); Yutushui and Mross (2023); Hein and
Spånslätt (2023); Park  _et al._ (2024); Kumar  _et al._ (2024); Manna  _et
al._ (2024)). (iii) An extension to the boundaries of non-Abelian QH phases.
(iv) Developing an effective linear response theory on top of the non-
equilibrium Laughlin surface, including effective fluctuation-dissipation
relations.

_Acknowledgments—_ We are grateful to Gabriele Campagnano, Domenico Giuliano,
and Bernd Rosenow for discussions. G.Z. acknowledges the support from National
Natural Science Foundation of China, (Grant No. 12374158), and Innovation
Program for Quantum Science and Technology (Grant No. 2021ZD0302400). I.G. and
Y.G. acknowledge the support by the Deutsche Forschungsgemeinschaft (DFG)
through grant No. MI $658/10$-$2$. Y.G. acknowledges support from the DFG
Grant RO $2247/11$-$1$, the US-Israel Binational Science Foundation, and the
Minerva Foundation. Y.G. is the incumbent of the InfoSys chair.

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Supplementary Information for “Landscapes of an out-of-equilibrium anyonic
sea”
Gu Zhang, Igor Gornyi, and Yuval Gefen

(Dated: July 19, 2024)

In this Supplementary Information, we provide details on: (i) Illustration of
time-domain braiding; (ii) Finite-temperature correlation functions of an out-
of-equilibrium anyonic channel; (iii) Distribution functions, defined through
Fourier transformation of correlation functions, of equilibrium and non-
equilibrium anyonic channels; (iv) Interpreting tunneling features in terms of
zero-temperature distribution functions; (v) Criteria to define the effective
chemical potential of an out-of-equilibrium anyonic channel; (vi) Evaluation
of the heat current in various cases; (vii) Relation between the tunneling-
current noise and the current auto-correlation (the latter is more easily
accessible in experiments); (viii) Distribution functions after the inclusion
of direct tunneling of non-equilibrium anyons from diluted beams; (ix)
Comparison of distribution functions characterized by a real temperature and
an effective temperature; (x) Nyquist-Johnson noise in the considered non-
equilibrium anyonic system; (xi) General expression of the effective
potential, for Tunneling between channels with different filling fractions;
and (xii) Tunneling of different-charge quasiparticles between $\nu=1/3$ and
$2/3$ edges.

## S1. Illustrations of time-domain braiding

As explained in the main text, the so-called time-domain braiding is a unique
process that occurs when “braiding” anyonic tunneling operators at the
collider with those of the “passing-by” anyons from the non-equilibrium
(diluted) beam Han  _et al._ (2016); Lee  _et al._ (2019); Schiller  _et al._
(2023). In this section, we further illustrate time-domain braiding.

More accurately, time-domain braiding can be viewed as the “interference”
between two tunneling processes. Here, without loss of generality, we consider
the overlap between the ket ($|\Psi_{i}\rangle$) and the bra
($|\Psi_{f}\rangle$) states defined in Fig. S1. This overlap appears in the
expansion for the correlation function of the tunneling operators at the
collider to leading-order expansion in diluter’s transmission
$\mathcal{T}_{A}$:

$\langle\Psi_{f}|\Psi_{i}\rangle=\langle
T_{K}\psi^{\dagger}_{\text{A}}(L,t^{-})\psi_{\text{B}}(L,t^{-})\psi^{\dagger}_{\text{B}}(L,0^{+})\psi_{\text{A}}(L,0^{+})\psi^{\dagger}_{\text{A}}(0,s_{1}^{\eta_{1}})\psi_{\text{SA}}(0,s_{1}^{\eta_{1}})\psi^{\dagger}_{\text{SA}}(0,s_{2}^{\eta_{2}})\psi_{\text{A}}(0,s_{2}^{\eta_{2}})\rangle
e^{i\theta(\eta_{1},\eta_{2})}.$

Here, $s_{1}$ and $s_{2}$ refer to the time arguments of the anyon operators
at the position of the diluter, with $\eta_{1}$ and $\eta_{2}$ their
corresponding Keldysh indexes. The phase factor
$e^{i\theta(\eta_{1},\eta_{2})}$ with

$\theta(\eta_{1},\eta_{2})=\pi\nu[1-(\eta_{1}-\eta_{2})],$

being $\eta_{1}$ and $\eta_{2}$-dependent, refers to the difference in phase
[see Figs. S1(c) to S1(f)] between $\langle\Psi_{f}|\Psi_{i}\rangle$ and the
Keldysh-ordered correlation function. This phase difference is generated by
exchanging the anyonic operators [see Figs. S1(c) to S1(f)]:

$\psi^{\dagger}_{\alpha}(x)\psi_{\beta}(x^{\prime})=e^{-i\pi\nu}\psi_{\beta}(x^{\prime})\psi^{\dagger}_{\alpha}(x),\quad\psi^{\dagger}_{\alpha}(x)\psi^{\dagger}_{\beta}(x^{\prime})=e^{i\pi\nu}\psi^{\dagger}_{\beta}(x^{\prime})\psi^{\dagger}_{\alpha}(x),$

where the position $x^{\prime}$ is downstream of $x$.

Figure S1:  Illustration of time-domain braiding for the leading-order
expansion in the tunneling transmission of the diluter. Panels (a) and (b):
Positions of anyonic operators, for states $|\Psi_{i}\rangle$ and
$|\Psi_{f}\rangle$, at times 0 and $t>0$, respectively. Here, red dots refer
to operators of the anyons that are generated by the tunneling at the diluter
(filled dot: quasiparticle, empty dot: quasihole) and blue dots refer to the
quasiparticle (fillied dot) and quasihole (empty dot) operators involved in
the tunneling process at the central QPC. The state $|\Omega\rangle$ is the
vacuum state (no quasiparticles in all three channels). Panels (c) to (f):
Operators after Keldysh ordering, for different arrangements of the Keldysh
indexes $\eta_{1}$ and $\eta_{2}$. The exchange of the operators (as indicated
by arrows) is needed to reconstruct states $|\Psi_{i}\rangle$ and
$|\Psi_{f}\rangle$, thus leading to extra phases that are marked out next to
corresponding arrows. The exchange-induced total phase
$\theta(\eta_{1},\eta_{2})$ is shown for each case. In panels (c) and (e), the
exchange operations marked by solid arrows are taken before those indicated by
dashed ones.

In Fig. S1, anyons of the non-equilibrium beam created by tunneling from SA to
A at times $s_{1}$ [panel (a)] and $s_{2}$ [panel (b)], when moving with
constant velocity $v$ downstream of the diluter, would arrive at the collider
at times $s_{1}+L/v$ and $s_{2}+L/v$, satisfying $0<s_{1}+L/v,s_{2}+L/v<t$,
where $L$ is the distance between the diluter and the collider. As a
consequence, in the state $|\Psi_{i}\rangle$, in which the tunneling event at
the collider (generating a quasiparticle-quasihole pair shown in blue) occurs
at time zero, the non-equilibrium pair (shown in red) is still upstream of the
collider [Fig. S1(a)]. In the other state, $|\Psi_{f}\rangle$, when the
tunneling at the collider occurs at a later moment $t$, this pair instead
already moves past the collider [Fig. S1(b)].

To contract operators with time arguments $s_{1}$ and $s_{2}$ ($s_{1}\to
s_{2}$), these operators need to be moved next to each other, when evaluating
$\langle\Psi_{f}|\Psi_{i}\rangle$. As a consequence, one has to exchange the
“non-equilibrium” operators at time moment $s_{2}$ with the collider operators
at time $t$, leading to the exchanging phase $-i\pi\nu$, half in amplitude of
the “braiding” phase, $\pm i2\pi\nu$. The other phase contribution comes from
the phase difference $\theta(\eta_{1},\eta_{2})$ between
$\langle\Psi_{f}|\Psi_{i}\rangle$ and the Keldysh-ordered correlation
function. For clarity, in Figs. S1(c) to S1(f), we show operators with
different Keldysh indexes, and the resulting values of $\theta$. In these
figures, when the operator in the bra-state is upstream of that in the ket-
state, exchanging them leads to a phase factor $-i\pi\nu$, when both operators
are creation or annihilation operators. The phase is opposite if the operator
in the bra-state is downstream of that in the ket-state. Following these
figures, the total phase equals $-i\pi\nu(\eta_{1}-\eta_{2})$, as reported in,
e.g., Ref. Morel  _et al._ (2022).

## S2. Finite-temperature correlations in an out-of-equilibrium anyonic
channel

In Eq. (1) of the main text, we show correlation functions of operators in
channel A, for finite temperatures $T_{\text{A}}$ or $T_{\text{SA}}$ (see
Fig.1 of the main text for the schematics of the setup). In this section, we
explain how to obtain these expressions. The correction to the equilibrium
correlation function of the tunneling operators at position $x=L$ in channel
A, which arises at the first order in the diluter transmission
$\mathcal{T}_{\text{A}}$, involves averaging in the presence of tunneling
operators at the diluter ($x=0$) in channels A and SA:

$\displaystyle\big{\langle}\psi^{\dagger}_{\text{A}}(L,t^{-})\psi_{\text{A}}(L,0^{+})\big{\rangle}_{\text{neq}}^{(1)}$
$\displaystyle=\mathcal{T}_{\text{A}}\int_{-\infty}^{\infty}ds_{1}\int_{-\infty}^{\infty}ds_{2}\sum_{\eta_{1}\eta_{2}}\eta_{1}\eta_{2}\,e^{-i\nu
eV(s_{1}-s_{2})}$
$\displaystyle\times\big{\langle}\psi^{\dagger}_{\text{A}}(L,t^{-})\psi_{\text{A}}(L,0^{+})\psi^{\dagger}_{\text{A}}(0,s_{1}^{\eta_{1}})\psi_{\text{A}}(0,s_{2}^{\eta_{2}})\big{\rangle}_{0}\langle\psi_{\text{SA}}(0,s_{1}^{\eta_{1}})\psi^{\dagger}_{\text{SA}}(0,s_{2}^{\eta_{2}})\big{\rangle}_{0}.$
(S1)

Here, $\eta_{1}=\pm$ and $\eta_{2}=\pm$ are the two Keldysh indexes, the
subscript “neq” denotes the averaging over the non-equilibrium state, the
superscript “(1)” indicates the power of $\mathcal{T}_{\text{A}}$, and the
subscript “0” indicates that correlations are evaluated at
$\mathcal{T}_{A}=\mathcal{T}_{\text{C}}=0$. We first consider the correlation
functions when the temperature of the source reservoir $T_{\text{SA}}$ is
finite but the ambient temperature of channel A is zero, $T_{\text{A}}=0$, and
then generalize the results to the case when both $T_{\text{A}}$ and
$T_{\text{SA}}$ are finite.

Case I: $T_{\text{SA}}$ finite, and $T_{\text{A}}=0$

We begin by considering the case $T_{\text{A}}=0$, when $T_{\text{SA}}$ is the
only finite temperature in the setup. In this case, the correlation functions
involving operators $\psi^{\dagger}_{\text{A}}$ and $\psi_{\text{A}}$ are
power-law functions of time, whereas the correlations of
$\psi^{\dagger}_{\text{SA}}$ and $\psi_{\text{SA}}$ produce a power of the
hyperbolic function. Equation (S1) takes then the form:

$\displaystyle\text{Eq.~{}(S1)}=$
$\displaystyle\frac{\mathcal{T}_{A}}{(2\pi\tau_{c})^{3}}\,\frac{\tau_{c}^{\nu}}{(\tau_{c}+it)^{\nu}}\sum_{\eta_{1}\eta_{2}}\eta_{1}\eta_{2}\int_{-\infty}^{\infty}ds_{1}\int_{-\infty}^{\infty}ds_{2}\,e^{-i\nu
eV(s_{1}-s_{2})}$
$\displaystyle\times\frac{\tau_{c}^{\nu}}{\tau_{c}+i(s_{1}-s_{2})\chi_{\eta_{1}\eta_{2}}(s_{1}-s_{2})]^{\nu}}\,\frac{[\tau_{c}+i(t-s_{1}-L)\chi_{-\eta_{1}}(t-s_{1})]^{\nu}[\tau_{c}+(-s_{2})\chi_{+\eta_{2}}(-s_{2})]^{\nu}}{[\tau_{c}+i(t-s_{2}-L)\chi_{-\eta_{2}}(t-s_{2})]^{\nu}[\tau_{c}+(-s_{1})\chi_{+\eta_{1}}(-s_{1})]^{\nu}}$
$\displaystyle\times\left(\frac{\pi T_{\text{SA}}\tau_{c}}{\sin\left\\{\pi
T_{\text{SA}}[\tau_{c}+i(s_{1}-s_{2})\chi_{\eta_{1}\eta_{2}}(s_{1}-s_{2})]\right\\}}\right)^{\nu},$
(S2)

where $\chi_{\eta\eta^{\prime}}(s>0)=\eta^{\prime}$ and
$\chi_{\eta\eta^{\prime}}(s<0)=-\eta$, i.e.,

$\chi_{\eta,\eta^{\prime}}(s)=\eta^{\prime}\,\Theta(s)-\eta\,\Theta(-s),$

where $\Theta(s)$ is the Heaviside step function.

Without loss of generality, we first consider $t>0$. The contribution of time-
domain braiding to the correlation function is determined by the singularity
in the integrand of Eq. (S2) at (cf. Ref. Morel  _et al._ (2022))

$s_{1}\to s_{2}.$

For $0<s_{1}=s_{2}<t$, we obtain

$\frac{[\tau_{c}+i(t-s_{1}-L)\chi_{-\eta_{1}}(t-s_{1})]^{\nu}[\tau_{c}+(-s_{2})\chi_{+\eta_{2}}(-s_{2})]^{\nu}}{[\tau_{c}+i(t-s_{2}-L)\chi_{-\eta_{2}}(t-s_{2})]^{\nu}[\tau_{c}+(-s_{1})\chi_{+\eta_{1}}(-s_{1})]^{\nu}}\
\to\ \exp[i\pi\nu(\eta_{1}-\eta_{2})].$ (S3)

When $s_{1}=s_{2}$ is outside the segment $[0,t]$, the fraction in Eq. (S3)
becomes unity, i.e., it does not depend on $\eta_{1}$ and $\eta_{2}$.
Introducing $s=s_{1}-s_{2}$ in Eq. (S2), it is instructive to re-write the
time-domain-braiding contribution to the correlation function by performing
explicitly the summation over the Keldysh indexes:

$\displaystyle\text{Eq.~{}(S1)}_{\text{\,TDB}}=$
$\displaystyle\frac{\mathcal{T}_{A}\,t}{(2\pi)^{3}}\frac{\tau_{c}^{3\nu-3}}{(\tau_{c}+it)^{\nu}}\sum_{\eta_{1}\eta_{2}}\eta_{1}\eta_{2}\,\,e^{i\pi\nu(\eta_{1}-\eta_{2})}\int_{-\infty}^{\infty}\\!\\!ds\frac{e^{-i\nu
eVs}}{[\tau_{c}+is\chi_{\eta_{1}\eta_{2}}(s)]^{\nu}}\,\frac{(\pi
T_{\text{SA}})^{\nu}}{\sin^{\nu}\left\\{\pi
T_{\text{SA}}[\tau_{c}+is\chi_{\eta_{1}\eta_{2}}(s)]\right\\}^{\nu}}$
$\displaystyle=$
$\displaystyle\frac{\mathcal{T}_{A}\,t}{(2\pi)^{3}}\frac{\tau_{c}^{3\nu-3}}{(\tau_{c}+it)^{\nu}}\left\\{\\!\int_{0}^{\infty}\\!\\!ds\frac{e^{-i\nu
Vs}}{(\tau_{c}+is)^{\nu}}\frac{(\pi T_{\text{SA}})^{\nu}}{\sin^{\nu}\left[\pi
T_{\text{SA}}(\tau_{c}+is)\right]^{\nu}}+\\!\int_{-\infty}^{0}\\!\\!ds\frac{e^{-i\nu
eVs}}{(\tau_{c}-is)^{\nu}}\frac{(\pi T_{\text{SA}})^{\nu}}{\sin^{\nu}\left[\pi
T_{\text{SA}}(\tau_{c}-is)\right]^{\nu}}\right.$ $\displaystyle+$
$\displaystyle\int_{0}^{\infty}ds\frac{e^{-i\nu
eVs}}{(\tau_{c}-is)^{\nu}}\frac{(\pi T_{\text{SA}})^{\nu}}{\sin^{\nu}\left[\pi
T_{\text{SA}}(\tau_{c}-is)\right]^{\nu}}+\int_{-\infty}^{0}ds\frac{e^{-i\nu
eVs}}{(\tau_{c}+is)^{\nu}}\frac{(\pi T_{\text{SA}})^{\nu}}{\sin^{\nu}\left[\pi
T_{\text{SA}}(\tau_{c}+is)\right]^{\nu}}$ $\displaystyle-$
$\displaystyle\left.\int_{-\infty}^{\infty}ds\frac{e^{-i\nu
eVs}}{(\tau_{c}+is)^{\nu}}\frac{(\pi T_{\text{SA}})^{\nu}}{\sin^{\nu}\left[\pi
T_{\text{SA}}(\tau_{c}+is)\right]^{\nu}}e^{-2i\pi\nu}-\int_{-\infty}^{\infty}ds\frac{e^{-i\nu
eVs}}{(\tau_{c}-is)^{\nu}}\frac{(\pi T_{\text{SA}})^{\nu}}{\sin^{\nu}\left[\pi
T_{\text{SA}}(\tau_{c}-is)\right]^{\nu}}e^{2i\pi\nu}\right\\},$ (S4)

Here, we set $s_{1}=s_{2}$ in the fraction (S3) and took into account the
configurations that produce a nontrivial phase factor
$e^{i\pi\nu(\eta_{1}-\eta_{2})}$ (otherwise the summation over $\eta_{1,2}$
yields zero). The integration over the center-of-mass time $(s_{1}+s_{2})/2$
produces then the total time $t$, while the integration over the time
difference $s$ can be safely extended to infinite limits, since the integral
over $s$ is dominated by $s\lesssim\text{min}[1/(\nu eV),1/T_{\text{SA}}]\ll
t$. Equation (S4) can be written in the form emphasizing the two tunneling
processes—tunneling from A to SA ($I_{\text{A}\to\text{SA}}$) and tunneling
from SA to A ($I_{\text{SA}\to\text{A}}$):

$\displaystyle\text{Eq.~{}(S1)}_{\text{\,TDB}}=$
$\displaystyle\frac{\mathcal{T}_{A}\,t}{(2\pi)^{3}}\frac{\tau_{c}^{3\nu-3}}{(\tau_{c}+it)^{\nu}}\,\int_{-\infty}^{\infty}ds\frac{e^{-i\nu
eVs}}{(\tau_{c}+is)^{\nu}}\frac{(\pi T_{\text{SA}})^{\nu}}{\sin^{\nu}\left[\pi
T_{\text{SA}}(\tau_{c}+is)\right]^{\nu}}\left(1-e^{-2i\pi\nu}\right)$
$\displaystyle+$
$\displaystyle\frac{\mathcal{T}_{A}\,t}{(2\pi)^{3}}\frac{\tau_{c}^{3\nu-3}}{(\tau_{c}+it)^{\nu}}\,\int_{-\infty}^{\infty}ds\frac{e^{-i\nu
eVs}}{(\tau_{c}-is)^{\nu}}\frac{(\pi T_{\text{SA}})^{\nu}}{\sin^{\nu}\left[\pi
T_{\text{SA}}(\tau_{c}-is)\right]^{\nu}}\left(1-e^{2i\pi\nu}\right).$ (S5)

Written in terms of the partial currents, Eq. (S5) becomes

$\displaystyle\text{Eq.~{}(S1)}_{\text{\,TDB}}$
$\displaystyle=\frac{1}{2\pi\tau_{c}}\frac{\tau_{c}^{\nu}}{(\tau_{c}+it)^{\nu}}\left[-\frac{I_{\text{SA}\to\text{A}}}{\nu
e}\left(1-e^{2i\pi\nu}\right)t-\frac{I_{\text{A}\to\text{SA}}}{\nu
e}\left(1-e^{-2i\pi\nu}\right)t\right]$ (S6)
$\displaystyle=\frac{1}{2\pi\tau_{c}}\frac{\tau_{c}^{\nu}}{(\tau_{c}+it)^{\nu}}\left[-\frac{S_{\text{A}}}{\nu^{2}e^{2}}[1-\cos(2\pi\nu)]\,t+i\frac{I_{\text{A}}}{\nu
e}\sin(2\pi\nu)\,t\right],$

where $S_{A}=\nu e(I_{\text{SA}\to\text{A}}+I_{\text{A}\to\text{SA}})$ is the
noise in the non-equilibrium channel A, after the diluter. It is defined as
$S_{\text{A}}\equiv\int
dt[\langle\hat{I}_{\text{A}}(t)\hat{I}_{\text{A}}(0)\rangle-
I_{\text{A}}^{2}]$, with $\hat{I}_{\text{A}}$ the current operator,
$\langle\hat{I}_{\text{A}}\rangle=I_{\text{A}}$. At zero temperature,
$T_{\text{SA}}=0$, Eq. (S6) becomes the leading-order correlation functions of
Refs. Lee and Sim (2022); Morel  _et al._ (2022); Schiller  _et al._ (2023),
after representing the tunneling noise as $S_{\text{A}}=\nu eI_{\text{A}}$.
For negative times, $t<0$, one replaces
$-S_{A}[1-\cos(2\pi\nu)]\,t/\nu^{2}e^{2}$ in Eq. (S6) with
$S_{A}[1-\cos(2\pi\nu)]\,t/\nu^{2}e^{2}$.

Now we go to higher-order expansions of $\mathcal{T}_{\text{A}}$. For the
zero-temperature situation, one can sum up contributions to all orders of
dilutions, if $\nu eV_{\text{SA}}>I_{\text{A}}/\nu e$. For a finite
temperature, the requirement of the resummation becomes modified, into $\nu
eV_{\text{SA}}>S_{\text{A}}/\nu^{2}e^{2}$. With this requirement satisfied, we
perform resummation over all orders of dilutions. More specifically, assuming
self-contraction (meaning independence of anyonic pairs) of $n$ pairs of non-
equilibrium anyons, the corresponding correlation becomes

$\displaystyle\frac{1}{2\pi\tau_{c}}\frac{\tau_{c}^{\nu}}{(\tau_{c}+it)^{\nu}}\frac{C_{2n}^{2}C_{2n-2}^{2}\cdot\cdot\cdot
C_{2}^{2}2^{n}}{2n!}\frac{1}{n!}\left[-\frac{S_{\text{A}}}{\nu^{2}e^{2}}[1-\cos(2\pi\nu)]t+i\frac{I_{\text{A}}}{\nu
e}\sin(2\pi\nu)t\right]^{n}$ (S7) $\displaystyle=$
$\displaystyle\frac{1}{2\pi\tau_{c}}\frac{\tau_{c}^{\nu}}{(\tau_{c}+it)^{\nu}}\frac{1}{n!}\left[-\frac{S_{\text{A}}}{\nu^{2}e^{2}}[1-\cos(2\pi\nu)]t+i\frac{I_{\text{A}}}{\nu
e}\sin(2\pi\nu)t\right]^{n},$

where $C_{m}^{n}=m!/[n!(m-n)!]$: the number of options to pick out $n$ items
out of in-total $m$ ones. In Eq. (S7), the factor
$C_{2n}^{2}C_{2n-2}^{2}\cdot\cdot\cdot C_{2}^{2}2^{n}$ refers to the number of
self-contraction options of non-equilibrium anyons, and the factor $1/2n!$ is
the prefactor of performing $2n$th order Keldysh expansion, and finally, the
factor $1/n!$ removes the number of options on the picking order of $n$
anyonic pairs. Resummation over $n$ then leads to

$\langle\psi_{A}^{\dagger}(t)\psi_{A}(0)\rangle_{\text{neq}}\Big{|}_{T_{\text{A}}=0,T_{\text{SA}}}=\frac{1}{2\pi\tau_{c}}\frac{\tau_{c}^{\nu}}{(\tau_{c}+it)^{\nu}}\exp\left\\{-\frac{S_{\text{A}}}{\nu^{2}e^{2}}[1-\cos(2\pi\nu)]|t|+i\frac{I_{\text{A}}}{\nu
e}\sin(2\pi\nu)t\right\\}.$ (S8)

Case II: $T_{\text{SA}}$ and $T_{\text{A}}$ both finite

In this case, Eq. (S2) is modified by using the hyperbolic functions of time
for all the two-point correlators involved:

$\displaystyle\text{Eq.~{}(S1)}=$
$\displaystyle\frac{\mathcal{T}_{A}}{(2\pi\tau_{c})^{3}}\frac{(\pi
T_{\text{SA}}\tau_{c}^{3})^{\nu}(\pi
T_{\text{A}}\tau_{c})^{2\nu}}{\sin^{\nu}[\pi T_{\text{A}}(\tau_{c}+it)]}\iint
ds_{1}ds_{2}\sum_{\eta_{1}\eta_{2}}\frac{\eta_{1}\eta_{2}e^{-i\nu
eV(s_{1}-s_{2})}}{\sin^{\nu}\left\\{\pi
T_{\text{sA}}[\tau_{c}+i(s_{1}-s_{2})\chi_{\eta_{1}\eta_{2}}(s_{1}-s_{2})]\right\\}}$
(S9) $\displaystyle\times\frac{1}{\sin^{\nu}\left\\{\pi
T_{\text{A}}[\tau_{c}+i(s_{1}-s_{2})\chi_{\eta_{1}\eta_{2}}(s_{1}-s_{2})]\right\\}}$
$\displaystyle\times\frac{\sin^{\nu}\left\\{\pi
T_{\text{A}}[\tau_{c}+i(t-s_{1}-L)\chi_{-\eta_{1}}(t-s_{1})]\right\\}\sin^{\nu}\left\\{\pi
T_{\text{A}}[\tau_{c}+(-s_{2})\chi_{+\eta_{2}}(-s_{2})]\right\\}}{\sin^{\nu}\left\\{\pi
T_{\text{A}}[\tau_{c}+i(t-s_{2}-L)\chi_{-\eta_{2}}(t-s_{2})]\right\\}\sin^{\nu}\left\\{\pi
T_{\text{A}}[\tau_{c}+(-s_{1})\chi_{+\eta_{1}}(-s_{1})]\right\\}}.$

The further derivation is analogous to that for $T_{A}=0$.When considering the
time-domain-braiding process, we again take $s_{1}\to s_{2}$, with which the
last line of Eq. (S9) produces the exchange phase, leading to (assuming $t>0$
first)

$\displaystyle\text{Eq.~{}(S1)}_{\text{ TDB}}$
$\displaystyle=\frac{\mathcal{T}_{A}}{(2\pi\tau_{c})^{3}}\frac{(\pi
T_{\text{SA}}\tau_{c}^{3})^{\nu}(\pi
T_{\text{A}}\tau_{c})^{2\nu}}{\sin^{\nu}[\pi T_{\text{A}}(\tau_{c}+it)]}t$
(S10) $\displaystyle\times\left\\{\int_{-\infty}^{\infty}ds\frac{e^{-i\nu
eVs}}{\sin^{\nu}\left[\pi
T_{\text{SA}}(\tau_{c}+is)\right]}\frac{1}{\sin^{\nu}\left[\pi
T_{\text{A}}(\tau_{c}+is)\right]}\left(1-e^{-2i\pi\nu}\right)\right.$
$\displaystyle\left.+\int_{-\infty}^{\infty}ds\frac{e^{i\nu
eVs}}{\sin^{\nu}\left[\pi
T_{\text{SA}}(\tau_{c}+is)\right]}\frac{1}{\sin^{\nu}\left[\pi
T_{\text{A}}(\tau_{c}+is)\right]}\left(1-e^{2i\pi\nu}\right)\right\\}$

Again, the expression above can be simplified, by noticing that to the leading
order in $\mathcal{T}_{A}$, the partial currents read as

$\displaystyle I_{\text{SA}\to\text{A}}$ $\displaystyle=\nu
e\frac{\mathcal{T}_{A}}{(2\pi\tau_{c})^{2}}(\pi^{2}T_{\text{A}}T_{\text{SA}}\tau_{c}^{2})^{\nu}\int_{-\infty}^{\infty}ds\frac{e^{i\nu
eVs}}{\sin^{\nu}\left[\pi
T_{\text{sA}}(\tau_{c}+is)\right]}\frac{1}{\sin^{\nu}\left[\pi
T_{\text{A}}(\tau_{c}+is)\right]},$ (S11) $\displaystyle
I_{\text{A}\to\text{SA}}$ $\displaystyle=\nu
e\frac{\mathcal{T}_{A}}{(2\pi\tau_{c})^{2}}(\pi^{2}T_{\text{A}}T_{\text{SA}}\tau_{c}^{2})^{\nu}\int_{-\infty}^{\infty}ds\frac{e^{-i\nu
eVs}}{\sin^{\nu}\left[\pi
T_{\text{sA}}(\tau_{c}+is)\right]}\frac{1}{\sin^{\nu}\left[\pi
T_{\text{A}}(\tau_{c}+is)\right]},$

for tunneling from SA to A ($I_{\text{SA}\to\text{A}}$), and for the opposite
direction of tunneling, A to SA ($I_{\text{A}\to\text{SA}}$). With Eq. (S11),
Eq. (S10) becomes

$\displaystyle\text{Eq.~{}(S1)}_{\text{\, TDB}}$
$\displaystyle=\frac{1}{2\pi\tau_{c}}\frac{(\pi
T_{\text{A}}\tau_{c})^{\nu}}{\sin^{\nu}[\pi
T_{\text{A}}(\tau_{c}+it)]}\left[-\frac{I_{\text{SA}\to\text{A}}}{\nu
e}\left(1-e^{2i\pi\nu}\right)t-\frac{I_{\text{A}\to\text{SA}}}{\nu
e}\left(1-e^{-2i\pi\nu}\right)t\right]$ (S12)
$\displaystyle=\frac{1}{2\pi\tau_{c}}\frac{(\pi
T_{\text{A}}\tau_{c})^{\nu}}{\sin^{\nu}[\pi
T_{\text{A}}(\tau_{c}+it)]}\left[-\frac{S_{\text{A}}}{\nu^{2}e^{2}}[1-\cos(2\pi\nu)]t+i\frac{I_{\text{A}}}{\nu
e}\sin(2\pi\nu)t\right],$

which, in comparison to Eq. (S6), contains a $T_{\text{A}}$ dependent
prefactor. In addition, the value of $S_{\text{A}}$ is also implicitly
modified by $T_{\text{A}}$. For negative times $t<0$, one again replaces
$-S_{A}[1-\cos(2\pi\nu)]t/\nu^{2}e^{2}$ by
$S_{A}[1-\cos(2\pi\nu)]t/\nu^{2}e^{2}$.

Now we go to higher orders in the diluter transmission. In comparison to that
of $T_{\text{A}}=0$, where only $\nu eV_{\text{SA}}>S_{\text{A}}/\nu^{2}e^{2}$
is required, here we further require $\nu eV_{\text{SA}}>T_{\text{A}}$.
Indeed, if $T_{\text{A}}$ becomes large enough, the time $t$ between two
tunneling events at the collider is bounded by the temperature inverse
$1/T_{\text{A}}$, thus presenting resummation over higher-order contributions.
With both requirements satisfied, we obtain (following the same steps as for
the case $T_{A}=0$):

$\displaystyle\langle\psi_{A}^{\dagger}(t)\psi_{A}(0)\rangle_{\text{neq}}\Big{|}_{T_{\text{A}},T_{\text{SA}}}=\frac{1}{2\pi\tau_{c}}\frac{(\pi
T_{\text{A}}\tau_{c})^{\nu}}{\sin^{\nu}[\pi
T_{\text{A}}(\tau_{c}+it)]}\exp\left\\{-\frac{S_{\text{A}}}{\nu^{2}e^{2}}[1-\cos(2\pi\nu)]|t|+i\frac{I_{\text{A}}}{\nu
e}\sin(2\pi\nu)t\right\\},$ (S13)
$\displaystyle\langle\psi_{A}(t)\psi^{\dagger}_{A}(0)\rangle_{\text{neq}}\Big{|}_{T_{\text{A}},T_{\text{SA}}}=\frac{1}{2\pi\tau_{c}}\frac{(\pi
T_{\text{A}}\tau_{c})^{\nu}}{\sin^{\nu}[\pi
T_{\text{A}}(\tau_{c}+it)]}\exp\left\\{-\frac{S_{\text{A}}}{\nu^{2}e^{2}}[1-\cos(2\pi\nu)]|t|-i\frac{I_{\text{A}}}{\nu
e}\sin(2\pi\nu)t\right\\},$

which is Eq. (1) of the main text.

An inspection of Eq. (S13) discloses an important message: the particle
correlation
$\langle\psi_{A}^{\dagger}(t)\psi_{A}(0)\rangle_{T_{\text{A}},T_{\text{SA}}}$,
and the hole correlation
$\langle\psi_{A}(t)\psi^{\dagger}_{A}(0)\rangle_{T_{\text{A}},T_{\text{SA}}}$
only differ by the imaginary phase in the exponential factor, even at finite
temperatures. This feature, remarkably, leads to the symmetry between the
particle and hole distributions, with respect to the energy
$I_{\text{A}}\sin(2\pi\nu)/\nu e$, see more details in Sec. S3 below.
Noteworthily, this symmetry between the particle and hole distributions is
present if taking into account only time-domain-braiding processes. Indeed,
this symmetry is sabotaged, if allowing non-equilibrium anyons to participate
in the tunneling correlation functions directly (i.e., when an anyon appears
at the position of the collider, at moment 0 or/and $t$). Nevertheless, non-
equilbirium anyons have little direct influence on correlation functions in
the strongly diluted limit—the limit addressed in the current work. We
conclude that the above symmetry between particle and hole distributions is an
anyonic-specific feature, because in fermionic systems time-domain braiding is
absent.

## S3. Distribution functions, for equilibrium and non-equilibrium channels

In the main text, we describe non-equilibrium anyonic channels by defining
particle and hole distribution functions

$\displaystyle n_{\text{p},\alpha}(\epsilon)\equiv\int dt\,e^{-i\epsilon
t}\langle\psi_{\alpha}^{\dagger}(t)\psi_{\alpha}(0)\rangle_{a},$ (S14)
$\displaystyle n_{\text{h},\alpha}(\epsilon)\equiv\int dt\,e^{i\epsilon
t}\langle\psi_{\alpha}(t)\psi^{\dagger}_{\alpha}(0)\rangle_{a},$

for three channels $\alpha=\,$SA, A and B with $a=0$, neq, and 0,
respectively. In this section, we present and compare distribution functions
for different situations. When considering tunneling between, e.g., channels A
and B, one encounters integral of the form

$\displaystyle\mathcal{T}_{\text{C}}\int
dt\langle\psi_{\text{A}}^{\dagger}(t)\psi_{\text{B}}(t)\psi_{\text{B}}^{\dagger}(0)\psi_{\text{A}}(0)\rangle_{\text{neq}}$
$\displaystyle=\frac{\mathcal{T}_{\text{C}}}{(2\pi)^{2}}\int dt\iint
d\epsilon_{1}d\epsilon_{2}\,e^{-i(\epsilon_{1}-\epsilon_{2})t}n_{\text{p,A}}(\epsilon_{1})n_{\text{h,B}}(\epsilon_{2})$
(S15) $\displaystyle=\frac{\mathcal{T}_{\text{C}}}{2\pi}\int
d\epsilon\,n_{\text{p,A}}(\epsilon)\,n_{\text{h,B}}(\epsilon).$

For a given value of energy $\epsilon$, Eq. (S15) can be interpreted as a
product of two distribution functions defined in Eq. (S14): the anyonic
particle distribution in A and the anyonic hole distribution in B.

We emphasize that the distributions defined by Eq. (S15) apply to all points
between the diluter and the collider, for $\alpha=\,$A, B alike, not only to
the location of the collider, at which the tunneling current is evaluated. At
first glance, this seems counter-intuitive. Indeed, the non-equilibrium
distribution functions appearing in Eq. (S15) are strongly affected by the
phenomenon of time-domain braiding, which is generated by the presence of the
tunneling bridge (the collider) connecting channels A and B at a single point
in space. How can we get the same distribution function at the position
upstream of the collider when the time-domain braiding is induced by the
collider? To understand this for $\alpha=\,$A, we interpret Eq. (S15) as the
correlation between removing a particle at time 0 and position $x$ (that is,
between the diluter and the collider), and adding another one at time $t$ at
the same position, $x$. In order to probe the single-anyon distribution
function $n_{\text{p,A}}(\epsilon)$ at position $x$, one should introduce a
“measurement device” coupled to the single-anyon operator in channel $A$. This
probe can be introduced by adding an auxiliary channel, with the same filling
fraction as that in A, which is tunnel-coupled to channel A at position $x$.
With this auxiliary channel, time-domain braiding will take place between
nonequilibrium anyons in channel A and the spontaneously generated particle-
hole pair induced by the auxiliary channel at the point $x$. Any measurement
of the single-anyon distribution function inevitably introduces time-domain
braiding at the measurement position. As a result, the non-equilibrium
distribution function at the position of the probe will be affected by the
time-domain braiding in exactly the same way as it is affected at the collider
and, hence, will have the same form. This is why correlation functions of A
channel operators calculated at the collider, where time-domain braiding
occurs in the experimental setup, also apply to positions upstream of the
collider. Any probe of the distribution function with an auxiliary channel
could be considered analogous to the collider that couples channel A with
equilibrium channel B.

Importantly, the effect of time-domain braiding on the distribution function
does not depend, to the lowest order, on the transparency of the collider
probing the distribution function [cf. Eq. (S15), where
$\mathcal{T}_{\text{C}}\ll 1$ serves as a prefactor in the expression for the
tunneling current and does not enter the distribution functions]. Thus, in the
limit of weak tunneling, the probe will measure the same distribution function
at any position downstream of the diluter, also including the locations
downstream of the collider. The strength of the coupling to the probe will not
affect the measured distribution function in the weak-tunneling limit, whereas
time-domain braiding will be fully introduced by the probe. It is worth
reiterating that the probe should be coupled to single-anyon operators. If,
for example, we couple a $\nu=1/3$ channel A to a probe based on an integer
edge ($\nu=1$), such that the tunneling between the channel and the probe is
only possible for a triplet of anyons forming an electron, we would not able
to measure the single-anyon distribution function. In this case, time-domain
braiding is not effective (see discussion in Sec. S11 below) as electrons do
not braid with anyons. This implies that the coherent superposition of three
charge $e/3$ anyons cannot be described in terms of single-anyon distribution
functions: quantum correlations in such a triplet are crucially important.

Figure S2:  Distribution functions for an anyonic channel. An energy unit
$\epsilon_{0}>0$ is used as a reference energy to make all quantities
dimensionless, both for equilibrium and non-equilibrium distributions.
Functions $n_{\text{p,A}}^{\text{neq-braid}}$ and $n_{\text{h,A}}^{\text{neq-
braid}}$ refer, respectively, to the particle and hole distributions of non-
equilibrium channel A. Their superscript “neq-braid” indicates that only time-
domain-braiding processes are involved when evaluating the influence of non-
equilibrium anyons on correlation functions. Functions
$n_{\text{p,B}}^{\text{eq}}$ and $n_{\text{h,B}}^{\text{eq}}$ refer to
distributions of equilibrium channel B. In panels (a) - (e), all distributions
are obtained at zero temperatures,
$T_{\text{A}}=T_{\text{SA}}=T_{\text{B}}=0$. The thermal effects will be
discussed later, in Sec. S6. Panel (a): The particle and hole distributions
for an equilibrium anyonic channel are only finite for states with energies
below and above the Laughlin surface, respectively. Panel (b): With a finite
non-equilibrium current $I_{A}=e\epsilon_{0}$, both distributions are nonzero
for all energies. The two distributions are related to each other by mirror
symmetry with respect to the energy $\epsilon=\epsilon_{0}\sin(2\pi\nu)/\nu e$
(the gray dashed line). Panel (c): The non-equilibrium particle distributions
(red and blue lines) approach the equilibrium one (black line) when the non-
equilibrium current ($e\epsilon_{0}$ for the red curve and $e\epsilon_{0}/2$
for the blue curve) decreases. Panels (d) and (e): Particle distributions for
a non-equilibrium channel with $\nu=1/3$ and $\nu=1/5$, respectively. In both
cases, particle distributions decrease polynomially
$\sim(\epsilon/\epsilon_{0})^{\nu-2}$, for large energies
$\epsilon\gg\epsilon_{0}$. Panel (f): Particle distribution for an equilibrium
anyonic channel with temperature $T$. Inset: the log-log plot of the particle
distribution. The distribution decays exponentially $\sim\exp(-\epsilon/T)$,
at large energies.

One can use Eqs. (S14) and (S15) to describe features of tunneling between
different anyonic channels in the way conventionally employed for transport of
fermions Blanter and Büttiker (2000). At zero temperature $T=0$, the
equilibrium anyonic distribution functions of channel B or SA are given by

$\displaystyle n_{\text{p}}^{\text{eq}}(\epsilon,V;T=0)$
$\displaystyle=\frac{\tau_{c}^{\nu-1}(\nu
eV-\epsilon)^{\nu-1}}{\Gamma(\nu)}\,[1-\Theta(\epsilon-\nu eV)]$ (S16)
$\displaystyle n_{\text{h}}^{\text{eq}}(\epsilon,V;T=0)$
$\displaystyle=\frac{\tau_{c}^{\nu-1}(\epsilon-\nu
eV)^{\nu-1}}{\Gamma(\nu)}\,\Theta(\epsilon-\nu eV),$

where superscript “eq” emphasizes that channels B and SA are at equilibrium,
$V$ equals $V_{\text{B}}$ or $V_{\text{SA}}$, $\Gamma(\nu)$ is the gamma
function, and $\Theta$ is the step function. The voltage bias is defined with
respect to the chemical potential of channel A before the diluter.

The presence of Heaviside $\Theta$-functions in Eq. (S16) means that the
particle and hole distributions for a zero-$T$ equilibrium anyonic channel are
only finite for states with energies below and above the Laughlin-sea surface,
respectively. This feature is analogous to the property of fermionic
distribution functions. However, in contrast to the fermionic case, zero-$T$
equilibrium anyonic distributions are not flat when they are nonzero. A
related unique feature of anyonic distributions is that the sum of particle
and hole distributions is not a constant. In great contrast, for fermions, the
sum of particle and hole distributions equals one for all energies, a feature
arising from the fermionic anticommutation relation. The plots of Eq. (S16)
are shown in Fig. S2(a). Following Eq. (S16) and Fig. S2(a), the particle
(hole) distribution is finite when $\epsilon<\nu eV$, i.e., beneath (above)
the Laughlin surface at the energy $\nu eV$, and equals zero for energies
above the anyonic-sea surface. When approaching the Laughlin surface, both the
particle and hole distributions diverge polynomially.

Now we move to the non-equilibrium channel A, which has zero-temperature
($T_{\text{A}}=T_{\text{SA}}=0$) distribution functions

$\displaystyle n_{\text{p,A}}^{\text{neq-
braid}}(\epsilon,I_{\text{A}};T_{\text{A}}=0,T_{\text{SA}}=0)$
$\displaystyle=\frac{\nu^{1-\nu}\Gamma(1-\nu)}{\pi}\tau_{c}^{\nu-1}\text{Re}\left\\{e^{i\pi\nu/2}\left[\frac{I_{\text{A}}}{e}\left(1-e^{-2i\pi\nu}\right)-i\nu\epsilon\right]^{\nu-1}\right\\},$
(S17) $\displaystyle n_{\text{h,A}}^{\text{neq-
braid}}(\epsilon,I_{\text{A}};T_{\text{A}}=0,T_{\text{SA}}=0)$
$\displaystyle=\frac{\nu^{1-\nu}\Gamma(1-\nu)}{\pi}\tau_{c}^{\nu-1}\text{Re}\left\\{e^{-i\pi\nu/2}\left[\frac{I_{\text{A}}}{e}\left(1-e^{-2i\pi\nu}\right)-i\nu\epsilon\right]^{\nu-1}\right\\},$

where superscript “neq-braid” indicates that only time-domain braiding
(addressed by “braid”) processes are involved when evaluating the influence of
non-equilibrium (addressed by “neq”) anyons on correlation functions. In
contrast to Eq. (S16) where we used the voltage (chemical potential) to
parametrize the distribution functions, here we use the non-equilibrium
current $I_{\text{A}}$ as the second argument: now we fix the chemical
potential and tune the amplitude of $I_{\text{A}}$. Plots of these non-
equilibrium distribution functions are shown in Fig. S2(b). These non-
equilibrium distribution functions are further compared, in Fig. S2(c), with
equilibrium ones. Clearly, particle distribution gradually approaches the
equilibrium one (the black curve), when decreasing the non-equilibrium current
($I_{\text{A}}/e=\epsilon_{0}$ for the red curve and
$I_{\text{A}}/e=\epsilon_{0}/2$ for the blue curve).

With a non-equilibrium current in channel A, the zero-temperature
[$T_{\text{A}}=T_{\text{SA}}=0$, cf. Eq. (S17)] particle distribution in this
channel is finite even for states with positive energies, meaning that non-
equilibrium current excites particles into states with higher energies,
mimicking the feature of fermionic distributions induced by thermal
fluctuations. This is exactly the reason to define an effective temperature in
a non-equilibrium channel. In addition, as shown by Figs. S2(d) and S2(e),
particle distributions decrease in a polynomial manner with increasing energy
[as $\epsilon^{\nu-2}$, see Fig. S7]. This polynomial energy dependence,
remarkably, distinguishes the anyonic distribution characterized by an
effective temperature introduced by the non-equilibrium current, from an
equilibrium thermal distribution. Indeed, for a real temperature
$T_{\text{B}}$, equilibrium distribution functions of channel B read as

$\displaystyle n_{\text{p,B}}^{\text{eq}}(\epsilon,V_{\text{B}};T_{\text{B}})$
$\displaystyle=\frac{1}{2\pi\Gamma(\nu)}\left(\frac{1}{2\pi
T_{\text{B}}\tau_{c}}\right)^{1-\nu}\exp\left(-\frac{\epsilon-\nu
V_{\text{B}}}{2T_{\text{B}}}\right)\Bigg{|}\Gamma\left(\frac{\nu}{2}+i\frac{\epsilon-\nu
V_{\text{B}}}{2\pi T_{\text{B}}}\right)\Bigg{|}^{2},$ (S18) $\displaystyle
n_{\text{h,B}}^{\text{eq}}(\epsilon,V_{\text{B}};T_{\text{B}})$
$\displaystyle=\frac{1}{2\pi\Gamma(\nu)}\left(\frac{1}{2\pi
T_{\text{B}}\tau_{c}}\right)^{1-\nu}\exp\left(\frac{\epsilon-\nu
V_{\text{B}}}{2T_{\text{B}}}\right)\Bigg{|}\Gamma\left(\frac{\nu}{2}+i\frac{\epsilon-\nu
V_{\text{B}}}{2\pi T_{\text{B}}}\right)\Bigg{|}^{2},$

with their plots shown in Fig. S2(f) for $V_{\text{B}}=0$. The thermal
distribution functions decay exponentially away from the Laughlin surface,
$\sim\exp(-\epsilon/T)$ (cf. Fig. S7), much faster than Eq. (S17) that is
induced by a non-equilibrium current. A more detailed comparison is provided
in Sec. S9, to distinguish between non-equilibrium particle distributions with
effective parameters and equilibrium ones with real chemical potential and
temperature.

## S4. Interpreting zero-temperature tunneling current with distribution
functions

In this section, we analyze the zero-temperature tunneling current in the two
setups [shown in Fig. S3(a) and S3(d)] by considering the corresponding
particle and hole distributions (we will specify distributions by the label of
corresponding channels). We will simplify the notation for distribution
functions, by assigning them only one argument: the energy $\epsilon$. Before
addressing each of the two setups in detail, we notice, based on Eq. (S15),
that a finite tunneling current requires (i) finite particle distributions in
the channel where particles are from, and (ii) finite hole distributions in
the channel where particles tunnel into.

Figure S3:  Representation of zero-temperature inter-channel tunneling in
terms of anyonic distribution functions. The particle and hole distribution
functions are shown by red and blue solid curves, respectively. A finite
tunneling rate requires particle and hole distributions to be both finite for
each tunneling event. Panel (a): First setup – two equilibrium channels
communicate through a quantum point contact (red dashed line). Panel (b):
Tunneling from channel 1 to channel 2 (yielding a partial tunneling current
$I_{1\to 2}$) is determined by the product of particle ($n_{\text{p,1}}$ of
channel 1) and hole ($n_{\text{h,2}}$ of channel 2) distributions. $I_{1\to
2}=0$ since either particle or hole distribution equals zero, for all
energies. Panel (c): Tunneling from channel 2 to channel 1 ($I_{2\to 1}$) is
determined by the product of particle ($n_{\text{p,2}}$ of channel 2) and hole
($n_{\text{h,1}}$ of channel 1) distributions. $I_{2\to 1}$ receives
contributions for energies $0<\epsilon<\nu eV$, where both particle and hole
distributions are finite. Panel (d): Second setup – a non-equilibrium channel
(1’) and an equilibrium one (2’). The gray dashed line denotes a diluter,
through which a non-equilibrium current is provided to channel 1’. We choose,
without loss of generality, $I_{\text{neq}}=0.1e^{2}V_{\text{S}}$, thus fixing
the renormalized transmission of the diluter (which is, in fact, also a
function of $V_{\text{S}}$). Panel (e): Particle ($n_{\text{p,1'}}$ of channel
1’) and hole ($n_{\text{h,2'}}$ of channel 2’) distributions that are relevant
to the tunneling from 1’ to 2’ ($I_{1^{\prime}\to 2^{\prime}}$). The partial
tunneling current now receives contributions from energies $\epsilon>\nu eV$.
In great contrast to the equilibrium case (b), the non-equilibrium current
$I_{\text{neq}}$ in channel 1’ enables tunneling of anyons from 1’ to 2’.
Panel (f): Particle ($n_{\text{p,2'}}$ of channel 2’) and hole
($n_{\text{h,1'}}$ of channel 1’) distributions that determine tunneling from
2’ to 1’ ($I_{2^{\prime}\to 1^{\prime}}$). The partial tunneling current
receives contributions from energies $\epsilon<\nu eV$, where both
distributions are nonzero.

Now we move to two specific systems. The first setup is depicted in Fig.
S3(a), where two equilibrium channels are bridged by a tunneling quantum point
contact (the red dashed line). Channel 2 is biased by voltage $V$ with respect
to the grounded channel (channel 1). Tunneling between these two channels
involves two competing processes: tunneling of particles from channel 1 to
channel 2 [represented by the current $I_{1\to 2}$ in Fig. S3(b)] and
tunneling of particles from channel 2 to channel 1 [represented by the current
$I_{2\to 1}$ in Fig. S3(c)]. Based on Figs. S3(b) and S3(c), for equilibrium
cases, zero-temperature tunneling is only possible from the channel with a
larger bias to the one with a smaller one. Indeed, in this case, states with
energies $0<\epsilon<\nu eV$ have finite particle and hole distributions [Fig.
S3(c)] and, hence, the product $n_{p,2}(\epsilon)n_{h,1}(\epsilon)$ is finite
in this energy interval, enabling tunneling from 2 to 1. The tunneling current
in the opposite direction, from 1 to 2, is totally forbidden, as the product
of particle or hole distribution $n_{p,1}(\epsilon)n_{h,2}(\epsilon)$ vanishes
for all energies [Fig. S3(b)].

In the setup with a non-equilibrium state in one of the channels [Fig. S3(d)],
the situation is different. Specifically, here, channel 1’ is grounded but it
carries a non-equilibrium (diluted) current $I_{\text{neq}}$ supplied (through
the diluter) by the channel 0’ biased by voltage $V_{\text{S}}$. With this
non-equilibrium current, channel 1’ has a finite particle distribution for all
energies. Even for positive energies, the particle and hole distributions in
channel 1’ are both finite, decreasing polynomially as $\epsilon^{\nu-2}$
[Figs. S2(d) and S3(e)]. Consequently, tunneling from 1’ to 2’, i.e.,
$I_{1^{\prime}\to 2^{\prime}}$, is non-zero as long as the hole distribution
$n_{h,2^{\prime}}(\epsilon)$ in channel 2’ is finite, i.e., for $\epsilon>\nu
eV$. Remarkably, $n_{p,1^{\prime}}(\epsilon)$ remains finite even for energies
larger than $\nu eV_{\text{S}}$ – the energy corresponding to the Laughlin
surface of the source channel 0’. This means that the partial tunneling
current $I_{1^{\prime}\to 2^{\prime}}$ is non-zero for $V_{\text{S}}<V$, i.e.,
when the source channel that produces the non-equilibrium state has a smaller
voltage than the equilibrium channel into which particles tunnel. Similarly,
tunneling from channel 2’ to channel 1’, which yields $I_{2^{\prime}\to
1^{\prime}}$, now involves states with a larger range of energy than in the
first setup with equilibrium channels: $\epsilon<\nu eV$ instead of
$0<\epsilon<\nu eV$.

## S5. Additional criteria to define the effective chemical potential

In the main text, the non-equilibrium channel A is calibrated by coupling it
to an equilibrium channel B. More specifically, the effective chemical
potential $V_{\text{eff}}$ of A is defined to be equal to the potential
$V_{\text{B}}$ applied to B, $V_{\text{eff}}=V_{\text{B}}$, such that no net
charge current tunnels between A and B for this value of $V_{\text{B}}$.
Following this definition,

$V_{\text{eff}}=I_{\text{A}}\sin(2\pi\nu)/\nu^{2}e^{2}$ (S19)

depends on the non-equilibrium current $I_{\text{A}}$ in channel A and the
filling fraction $\nu$. To support this conclusion, in this section, we
provide two more criteria to define the effective potential, which yields the
same value of $V_{\text{eff}}$ given by Eq. (S19).

### S5A. The neutrality requirement

Before introducing the neutrality requirement, we first define “modified”
particle and hole distributions

$\displaystyle\delta n_{\text{p,A}}^{\text{neq-
braid}}(\epsilon,I_{\text{A}};T_{\text{A}},T_{\text{SA}})\equiv
n_{\text{p,A}}^{\text{neq-
braid}}(\epsilon,I_{\text{A}};T_{\text{A}},T_{\text{SA}})-n_{\text{p,A}}^{\text{eq}}(\epsilon-\nu
eV_{\text{eff}},0;T_{\text{A}},T_{\text{SA}}),$ (S20) $\displaystyle\delta
n_{\text{h,A}}^{\text{neq-
braid}}(\epsilon,I_{\text{A}};T_{\text{A}},T_{\text{SA}})\equiv
n_{\text{h,A}}^{\text{neq-
braid}}(\epsilon,I_{\text{A}};T_{\text{A}},T_{\text{SA}})-n_{\text{h,A}}^{\text{eq}}(\epsilon-\nu
eV_{\text{eff}},0;T_{\text{A}},T_{\text{SA}}),$

as the difference between non-equilibrium distributions, and equilibrium ones,
after a shift in energy $\nu eV_{\text{eff}}$. Following Eq. (S20), we define
the neutrality requirement as

$\displaystyle\int_{\nu eV_{\text{eff}}}^{\infty}d\epsilon\ \delta
n_{\text{p,A}}^{\text{neq-
braid}}(\epsilon,I_{\text{A}};T_{\text{A}},T_{\text{SA}})=\int_{-\infty}^{\nu
eV_{\text{eff}}}d\epsilon\ \delta n_{\text{h,A}}^{\text{neq-
braid}}(\epsilon,I_{\text{A}};T_{\text{A}},T_{\text{SA}}),$ (S21)
$\displaystyle\int_{\nu eV_{\text{eff}}}^{\infty}d\epsilon\ \delta
n_{\text{h,A}}^{\text{neq-
braid}}(\epsilon,I_{\text{A}};T_{\text{A}},T_{\text{SA}})=\int_{-\infty}^{\nu
eV_{\text{eff}}}d\epsilon\ \delta n_{\text{p,A}}^{\text{neq-
braid}}(\epsilon,I_{\text{A}};T_{\text{A}},T_{\text{SA}}).$

With equation above, we are requiring two equalities: (i) between the extra
particle distribution above the energy $\nu eV_{\text{eff}}$, and the extra
hole distribution below the energy $\nu eV_{\text{eff}}$; and likewise, (ii)
between the extra particle distribution below the energy $\nu
eV_{\text{eff}}$, and the extra hole distribution above the energy $\nu
eV_{\text{eff}}$. Alternatively, Eq. (S21) requires the particle and hole
distributions to be symmetric to each other, with respect to the energy $\nu
V_{\text{eff}}$: the position of the shifted anyonic surface.

Both equalities are satisfied by choosing
$V_{\text{eff}}=I_{\text{A}}\sin(2\pi\nu)/\nu^{2}e^{2}$, which is exactly Eq.
(S19). To illustrate this, we show in Figs. S4(a)-S4(c) the values of $\delta
n^{\text{neq-
braid}}_{\text{p}}(\epsilon,I_{\text{A}};T_{\text{A}},T_{\text{SA}})$ and
$\delta n^{\text{neq-
braid}}_{\text{h}}(\epsilon,I_{\text{A}};T_{\text{A}},T_{\text{SA}})$, for
different temperatures. Clearly, $\delta n^{\text{neq-
braid}}_{\text{p}}(\epsilon-I_{\text{A}}\sin(2\pi\nu)/\nu
e,I_{\text{A}};T_{\text{A}},T_{\text{SA}})=\delta n^{\text{neq-
braid}}_{\text{h}}(-\epsilon+I_{\text{A}}\sin(2\pi\nu)/\nu
e,I_{\text{A}};T_{\text{A}},T_{\text{SA}})$, meaning $\delta n^{\text{neq-
braid}}_{\text{p}}$ and $\delta n^{\text{neq-braid}}_{\text{h}}$ are indeed
symmetric with respect to the energy
$\epsilon_{\mu}=I_{\text{A}}\sin(2\pi\nu)/\nu e$. Equalities of Eq. (S21) are
clearly satisfied, with the symmetry mentioned above.

Figure S4:  Two extra criteria to define the effective chemical potential.
Panel (a): Plots of the modified particle and hole distribution functions
defined in Eq. (S20) for $T_{\text{A}}=T_{\text{SA}}=0$. Panels (b) and (c):
Modifications of the particle and hole distributions for $T_{\text{A}}=0$,
$T_{\text{SA}}=10I_{\text{A}}/e$ and $T_{\text{A}}=10I_{\text{A}}/e$,
$T_{\text{SA}}=0$. For all three figures, the modified functions for particle
and hole states are symmetric to each other with respect to the energy
$I_{\text{A}}\sin(2\pi\nu)/\nu e$ (the gray dashed vertical line). Panel (d):
The log-log plot of the particle number within an energy strip (integrated
distribution), Eq. (S23). The integral decreases polynomially and thus
vanishes when the integration limits approach infinity, $E\to\infty$.

### S5B. Equality between “modified” and initial particle numbers

As another criterion, if the effective chemical potential $V_{\text{eff}}$ is
reasonably defined, we would like the particle numbers in the channel with a
non-equilibrium current $I_{\text{A}}$ to be equal to that of an equilibrium
channel, that has a relative shift $V_{\text{eff}}$ in the chemical potential,
i.e.,

$\displaystyle\int_{-\infty}^{\infty}d\epsilon\delta n^{\text{neq-
braid}}_{\text{p}}(\epsilon,I_{\text{A}};0,0)=\int_{-\infty}^{\infty}d\epsilon\delta
n^{\text{neq-braid}}_{\text{h}}(\epsilon,I_{\text{A}};0,0)=0,$ (S22)

for zero temperatures. To verify this requirement, we define another integral

$\delta\mathcal{N}(E,I_{\text{A}})\equiv e\int_{-E}^{E}d\epsilon\delta
n^{\text{neq}}_{\text{p}}(\epsilon,I_{\text{A}};0,0),$ (S23)

as the modification, introduced by the non-equilibrium current, of particle
number within the energy strip $(-E,\,E)$ compared to that of an equilibrium
channel. When $V_{\text{eff}}=I_{\text{A}}\sin(2\pi\nu)/\nu^{2}e^{2}$, we
check the requirement Eq. (S22) in Fig. S4(b). Following Fig. S4(b), the
integral in Eq. (S23) decays to zero polynomially, i.e., $\sim E^{-2/3}$. When
$E\to\infty$, $\delta\mathcal{N}\to 0$, indicating that Eq. (S22) is satisfied
by taking $V_{\text{eff}}=I_{\text{A}}\sin(2\pi\nu)/\nu^{2}e^{2}$.

We can further complement the above consideration by performing a Fourier
transformation:

$\displaystyle\lim_{E\to\infty}\delta\mathcal{N}(E,I_{\text{A}})$
$\displaystyle=e\int_{-\infty}^{\infty}d\epsilon\ [\delta
n^{\text{neq}}_{\text{p}}(\epsilon,I_{\text{A}};0,0)]$ (S24)
$\displaystyle=\lim_{t\to
0}[\langle\psi_{A}^{\dagger}(t)\psi_{A}(0)\rangle_{T_{\text{A}},T_{\text{SA}}}-\langle\psi_{B}^{\dagger}(t)\psi_{B}(0)\rangle_{T_{\text{B}}}]$
$\displaystyle=\lim_{t\to
0}\frac{\tau_{c}^{\nu-1}}{2\pi}\frac{\left(-\frac{S_{\text{A}}}{\nu^{2}e^{2}}\right)[1-\cos(2\pi\nu)]|t|}{(it)^{\nu}}\to
0,$

where we have set $V_{\text{B}}=V_{\text{eff}}\equiv
I_{\text{A}}\sin(2\pi\nu)/\nu^{2}e^{2}$ and used the fact that the difference
between the two correlation functions is not singular at $t\to 0$ (so that one
can take $\tau_{c}=0$ in the denominator).

### S5C. Role of direct tunneling of non-equilibrium anyons

Following previous section, when considering only the time-domain-braiding
contributions, the three criteria for defining the effective chemical
potential, namely (i) zero tunneling current, (ii) neutrality condition, and
(ii) equality between modified and initial total particle numbers, are
consistent. In this section, we exclude time-domain braiding, but include only
direct tunneling of non-equilibrium anyons. Here, we assume that channel A is
out of equilibrium and channel B is at equilibrium, as in the setup shown in
Fig. S2(d). More specifically, assuming channel A distributions are modified
by particles with energies between $0$ and $\nu eV_{\text{SA}}$, relevant
correlation functions now become

$\displaystyle\big{\langle}\psi^{\dagger}_{\text{A}}(t)\psi_{\text{A}}(0)\big{\rangle}_{\text{direct}}$
$\displaystyle\approx\frac{\tau_{c}^{\nu-1}}{2\pi(\tau_{c}+it)^{\nu}}\left[1+\mathcal{T}\left(e^{i\nu
eV_{\text{SA}}t}-1\right)\right],$ (S25)
$\displaystyle\big{\langle}\psi_{\text{B}}(t)\psi^{\dagger}_{\text{B}}(0)\big{\rangle}_{\text{direct}}$
$\displaystyle\approx\frac{\tau_{c}^{\nu-1}}{2\pi(\tau_{c}+it)^{\nu}}e^{-i\nu
eV_{\text{B}}t},$

with $\mathcal{T}$ the renormalized transmission probability through the
diluter. Let us now apply the above criteria to these correlation functions.

Criterion I: Zero tunneling current

As the first criterion, the effective chemical potential can be defined after
requiring a vanishing tunneling current, leading to

$\displaystyle\mathcal{T}(V_{\text{SA}}-V_{\text{B}})^{2\nu-1}=V_{\text{B}}^{2\nu-1}(1-\mathcal{T})$
(S26) $\displaystyle\Rightarrow$ $\displaystyle
V_{\text{eff}}=V_{\text{B}}=V_{\text{SA}}\left[1+\left(\frac{\mathcal{T}}{1-\mathcal{T}}\right)^{1/(1-2\nu)}\right]^{-1},$

at zero temperature. In the first line of Eq. (S26), the left- and right-hand
sides refer to tunneling from channel A to channel B and from B to A,
respectively.

Interestingly, in the strongly diluted limit, $\mathcal{T}\ll 1$,
$V_{\text{eff}}$ behaves differently depending on whether $\nu$ is larger or
smaller than $1/2$. Indeed, if $\nu>1/2$ (including the fermionic case,
$\nu=1$), $V_{\text{eff}}/V_{\text{SA}}\ll 1$ is a small number. In contrast,
if $\nu<1/2$ (including Laughlin states), $V_{\text{eff}}\approx
V_{\text{SA}}$. Actually, in the latter case, the tunneling from A to B is
greatly amplified by the large time cutoff
$\propto(V_{\text{SA}}-V_{\text{eff}})^{2\nu-1}$, which compensates for the
smallness of the transmission probability.

Criterion II: Equality of modified particle distributions
Now, we move to the second criterion. Following derivations of Eq. (S24), we
get

$\displaystyle\lim_{t\to
0}[\big{\langle}\psi^{\dagger}_{\text{A}}(t)\psi_{\text{A}}(0)\big{\rangle}_{\text{direct}}-\big{\langle}\psi^{\dagger}_{\text{B}}(t)\psi_{\text{B}}(0)\big{\rangle}_{\text{direct}}]\approx\lim_{t\to
0}\frac{i\tau_{c}^{\nu-1}}{2\pi}\frac{(\mathcal{T}\nu eV_{\text{SA}}-\nu
eV_{\text{B}})t}{(it)^{\nu}},$ (S27)

which yields zero, if $\nu<1$, for any finite value of $V_{\text{B}}$.
Similarly to the previous consideration for anyons with time-domain braiding,
the second criterion, i.e., zero modification of total particle number, is
always satisfied if $\nu<1$. When $\nu\geq 1$, the second criterion gives
instead $V_{\text{eff}}=\mathcal{T}V_{\text{SA}}$, which agrees with that from
the first criterion, Eq. (S26), only if $\nu=1$.

## S6. Heat current

This section provides details of calculating the tunneling heat current
between channels A and B. We define the partial heat current from channel A to
channel B as

$J_{\text{A}\to\text{B}}=\int
d\epsilon\mathcal{J}_{\text{A}\to\text{B}}(\epsilon),$

where

$\mathcal{J}_{\text{A}\to\text{B}}(\epsilon)\equiv\mathcal{T}_{\text{C}}\,n^{\text{neq-
braid}}_{\text{p,A}}(\epsilon,I_{\text{A}};T_{\text{A}},T_{\text{SA}})\,n^{\text{eq}}_{\text{h,B}}(\epsilon,V_{\text{B}};T_{\text{B}})\left(\epsilon-\nu
eV_{\text{B}}\right)$ (S28)

is the differential heat (defined as the energy with respect to the Laughlin
surface $\nu eV_{\text{B}}$, of B) carried by states with the energy
$\epsilon$. Here, $n^{\text{neq-braid}}_{\text{p,A}}$ and
$n^{\text{eq}}_{\text{h,B}}$ refer to the non-equilibrium particle
distribution in channel A and the equilibrium hole distribution in B,
respectively. Equation (S28) can be obtained after Fourier transformation of
the tunneling heat current derived in, e.g., Ref. Ebisu  _et al._ (2022). The
superscript of $n^{\text{neq-braid}}_{\text{p,A}}$ (“neq-braiding”) indicates
that the non-equilibrium distribution function involves only time-domain
braiding contribution. Direct tunneling of non-equilibrium particles is not
included. In addition, $n^{\text{neq-braid}}_{\text{p,A}}$ contains four
arguments: the particle energy $\epsilon$, the non-equilibrium current
$I_{\text{A}}$, and two temperatures $T_{\text{A}}$ and $T_{\text{SA}}$. The
other distribution function, $n^{\text{eq}}_{\text{h,B}}$, depends on three
variables: the particle energy $\epsilon$, the bias $V_{\text{B}}$ (defined
with respect to the chemical potential of channel A before the diluter), and
the temperature in channel B, $T_{\text{B}}$. Below, we focus on the situation
where $\nu eV_{\text{B}}=\nu eV_{\text{eff}}=I_{\text{A}}\sin(2\pi\nu)/\nu e$,
and study the heat current for different temperatures.

Before moving to specific cases, we begin by rewriting the heat current, Eq.
(S28), in another form that involves an integral over time. Let us consider a
general case where two channels, 1 and 2, with equal chemical potentials
$V_{0}$ (either effective or real potential), communicate through a quantum
point contact with transmission $\mathcal{T}_{\text{C}}$. In this case, the
partial heat current from channel 1 to channel 2, $J_{1\to 2}$ (not the net
tunneling heat current), becomes

$\displaystyle J_{1\to 2}$ $\displaystyle=\mathcal{T}_{\text{C}}\int
d\epsilon\ (\epsilon-\nu eV_{0})\
n_{\text{p,1}}(\epsilon)n_{\text{h,2}}(\epsilon)\ =\mathcal{T}_{\text{C}}\int
d\epsilon\ \epsilon\ n_{\text{p,1}}(\epsilon+\nu
eV_{0})n_{\text{h,2}}(\epsilon+\nu eV_{0})$ (S29)
$\displaystyle=\mathcal{T}_{\text{C}}\int d\epsilon\int dt_{1}\int dt_{2}\
\epsilon\
e^{-i\epsilon(t_{1}-t_{2})}f_{1}(t_{1})f_{2}(t_{2})=-i\pi\mathcal{T}_{\text{C}}\int
dt\left\\{\left[\frac{\partial}{\partial
t}f_{1}(t)\right]f_{2}(t)-f_{1}(t)\frac{\partial}{\partial
t}f_{2}(t)\right\\},$

where $n_{\text{p,1}}$ and $n_{\text{h,2}}$ are the particle distribution in
channel 1 and the hole distribution in channel 2, respectively. In Eq. (S29),
$\nu eV_{0}$ refers to the energy with respect to which the heat is defined.
For cases where the chemical potentials (effective or real) of the
communicating channels are equal, $V_{0}$ is chosen such that tunneling heat
current comes solely from thermal fluctuations (stemming from an effective or
real temperature). Functions $f_{1}$ and $f_{2}$ in Eq. (S29) are Fourier
transformed functions of these two distributions, after a shift $\nu eV_{0}$
in energy. Specifically,

$\displaystyle f_{1}(t)$ $\displaystyle\equiv\frac{1}{2\pi}\int d\epsilon
e^{i\epsilon t}n_{\text{p,1}}(\epsilon+\nu eV_{0})=\frac{e^{-i\nu
eV_{0}t}}{2\pi}\int d\epsilon e^{i\epsilon t}n_{\text{p,1}}(\epsilon),$ (S30)
$\displaystyle f_{2}(t)$ $\displaystyle\equiv\frac{1}{2\pi}\int d\epsilon
e^{-i\epsilon t}n_{\text{h,2}}(\epsilon+\nu eV_{0})=\frac{e^{i\nu
eV_{0}t}}{2\pi}\int d\epsilon e^{-i\epsilon t}n_{\text{h,2}}(\epsilon).$

During the derivation Eq. (S29), we have used the identity

$\int d\epsilon\ \epsilon\ e^{-i\epsilon t_{0}}=2\pi i\frac{\partial}{\partial
t_{0}}\delta(-t_{0}),$ (S31)

and integration by parts. One can similarly calculate the tunneling heat
current from 2 to 1, $J_{2\to 1}$, with which the net tunneling heat current
between channels 1 and 2 is given by

$J_{Q}=J_{1\to 2}-J_{2\to 1}.$

### S6A. Zero ambient temperatures

To begin with, we show details of the calculation of the heat current between
channels A and B, when
$V_{\text{eff}}=I_{\text{A}}\sin(2\pi\nu)/\nu^{2}e^{2}$, at zero temperatures,
$T_{\text{A}}=T_{\text{SA}}=T_{\text{B}}=0$. In this case, we simplify the
notation by writing $n^{\text{neq-
braid}}_{\text{p,A}}(\epsilon,I_{\text{A}};T_{\text{A}}=0,T_{\text{SA}}=0)\to
n_{\text{p}}^{\text{neq-braid}}(\epsilon,I_{\text{A}})$ and
$n^{\text{eq}}_{\text{h,B}}(\epsilon,V_{\text{B}};T_{\text{B}})\to
n^{\text{eq}}_{\text{h}}(\epsilon,V_{\text{B}})=n^{\text{eq}}_{\text{h}}(\epsilon-\nu
eV_{\text{B}},0)$. The heat current can be evaluated with Eq. (S29), where the
functions $f_{1}(t)$ and $f_{2}(t)$ are given by

$\displaystyle f_{1}(t)$ $\displaystyle\equiv\frac{1}{2\pi}\int
d\epsilon\,e^{i\epsilon t}\,n^{\text{neq-
braid}}_{\text{p}}\left(\epsilon+I_{\text{A}}\frac{\sin(2\pi\nu)}{\nu
e},I_{\text{A}}\right)=\langle\psi^{\dagger}_{\text{A}}(t)\psi_{\text{A}}(0)\rangle_{\text{neq}}\exp\left[-iI_{\text{A}}\frac{\sin(2\pi\nu)}{\nu
e}t\right]$ (S32)
$\displaystyle=\frac{1}{2\pi}\frac{\tau_{c}^{\nu-1}}{(\tau_{c}+it)^{\nu}}\exp\left[-\frac{I_{\text{A}}}{\nu
e}[1-\cos(2\pi\nu)]|t|\right],$ $\displaystyle f_{2}(t)$
$\displaystyle\equiv\frac{1}{2\pi}\int d\epsilon\,e^{-i\epsilon
t}\,n^{\text{eq}}_{\text{h}}(\epsilon,0)=\frac{1}{2\pi}\frac{\tau_{c}^{\nu-1}}{(\tau_{c}+it)^{\nu}}.$

The heat current, Eq. (S29), contains the time derivatives of both $f_{1}(t)$
and $f_{2}(t)$. These derivatives act, in particular, on the function
$(\tau_{c}+it)^{-\nu}$. It turns out that such contributions however cancel
out in Eq. (S29); the only term that survives in Eq. (S29) originates from the
time derivative of the exponential factor in $f_{1}(t)$. Note that this
exponential factor depends on the absolute value $|t|$. As a consequence, the
tunneling heat current from A to B takes the form

$\displaystyle J_{\text{A}\to\text{B}}$
$\displaystyle=i\pi\frac{I_{\text{A}}}{\nu
e}[1-\cos(2\pi\nu)]\mathcal{T}_{\text{C}}\,\frac{\tau_{c}^{2\nu-2}}{(2\pi)^{2}}\int
dt\frac{e^{-\frac{I_{\text{A}}}{\nu
e}[1-\cos(2\pi\nu)]|t|}}{(\tau_{c}+it)^{2\nu}}\text{sgn}(t)$ (S33)
$\displaystyle=\pi\frac{I_{\text{A}}}{\nu
e}[1-\cos(2\pi\nu)]\frac{\tan(\pi\nu)}{\nu}\frac{I_{\text{A}\to\text{B}}}{e}=\pi\tan^{2}(\pi\nu)\,V_{\text{eff}}\,I_{\text{A}\to\text{B}},$

where $I_{\text{A}\to\text{B}}$ stands for the tunneling charge current from
channel A (with non-equilibrium current $I_{\text{A}}$) to the equilibrium
channel B characterized by a finite potential $V_{\text{B}}$. We see now that
the factor

$\frac{I_{\text{A}}}{\nu e}[1-\cos(2\pi\nu)]\sim eV_{\text{eff}}$

plays the role of the effective temperature. Indeed, for two unbiased
channels, one with temperature $T$ and the other at zero temperature, the heat
current should be proportional to $TI_{\text{A}\to\text{B}}$.

Now, we consider the heat current produced by the tunneling processes from
channel B to channel A:

$\displaystyle J_{\text{B}\to\text{A}}$
$\displaystyle=\mathcal{T}_{\text{C}}\int d\epsilon\,n_{\text{h}}^{\text{neq-
braid}}(I_{\text{A}},\epsilon)\,n_{\text{p}}^{\text{eq}}\left(\epsilon-
I_{\text{A}}\frac{\sin 2\pi\nu}{\nu e},0\right)\left[\epsilon-
I_{\text{A}}\frac{\sin(2\pi\nu)}{\nu e}\right]$ (S34)
$\displaystyle=\mathcal{T}_{\text{C}}\int d\epsilon\ \epsilon\ n^{\text{neq-
braid}}_{\text{h}}\left(\epsilon+I_{\text{A}}\frac{\sin(2\pi\nu)}{\nu
e},I_{\text{A}}\right)n^{\text{eq}}_{\text{p}}\left(\epsilon,0\right).$

Noticing that [cf. Fig. S2(b)]

$n_{\text{h}}^{\text{neq-braid}}(\epsilon+I_{\text{A}}\sin(2\pi\nu)/\nu
e,I_{\text{A}})=n_{\text{p}}^{\text{neq-
braid}}(-\epsilon+I_{\text{A}}\sin(2\pi\nu)/\nu e,I_{\text{A}})$

and

$n^{\text{eq}}_{\text{p}}\left(\epsilon,0\right)=n^{\text{eq}}_{\text{h}}\left(-\epsilon,0\right),$

Eq. (S34) can be rewritten as

$\displaystyle J_{\text{B}\to\text{A}}$
$\displaystyle=\mathcal{T}_{\text{C}}\int d\epsilon\ \epsilon\ n^{\text{neq-
braid}}_{\text{h}}\left(\epsilon+I_{\text{A}}\frac{\sin(2\pi\nu)}{\nu
e},I_{\text{A}}\right)n^{\text{eq}}_{\text{p}}\left(\epsilon,0\right)$ (S35)
$\displaystyle=\mathcal{T}_{\text{C}}\int d\epsilon\ \epsilon\ n^{\text{neq-
braid}}_{\text{p}}\left(-\epsilon+I_{\text{A}}\frac{\sin(2\pi\nu)}{\nu
e},I_{\text{A}}\right)n^{\text{eq}}_{\text{h}}\left(-\epsilon,0\right)$
$\displaystyle=-\mathcal{T}_{\text{C}}\int d\epsilon\ \epsilon\ n^{\text{neq-
braid}}_{\text{p}}\left(\epsilon+I_{\text{A}}\frac{\sin(2\pi\nu)}{\nu
e},I_{\text{A}}\right)n^{\text{eq}}_{\text{h}}\left(\epsilon,0\right)=-J_{\text{A}\to\text{B}}.$

The total heat current $J_{Q}$ between channels A and B then reads

$J_{Q}=J_{\text{A}\to\text{B}}-J_{\text{B}\to\text{A}}=2\pi\tan^{2}(\pi\nu)\,V_{\text{eff}}\,I_{\text{A}\to\text{B}}.$
(S36)

We plot the heat current carried by the state with energy $\epsilon$ in Fig.
S5. Here, the first two plots, Fig. S5(a) and (b), show the results for
$T_{\text{A}}=T_{\text{SA}}=T_{\text{B}}=0$. The particle distributions are
symmetric to the corresponding hole distributions, with respect to the energy
of the effective chemical potential $\nu eV_{\text{eff}}$. As another
important feature, for the zero-temperature situation,
$\mathcal{J}_{\text{A}\to\text{B}}$ and $-\mathcal{J}_{\text{B}\to\text{A}}$
are both positive for all energies, indicating that the heat current is only
allowed to transport from A to B, even for nonzero charge transport from B to
A. This is because heat refers to energy defined with respect to the Laughlin
surface of the equilibrium channel B.

Figure S5:  Heat flow for different arrangements of temperatures in channels A
and B. Functions $n_{\text{p,A}}^{\text{neq-braid}}$ and
$n_{\text{h,A}}^{\text{neq-braid}}$ refer, respectively, to the particle and
hole distributions in the non-equilibrium channel A. Their superscript “neq-
braid” indicates that only time-domain braiding processes are included when
evaluating the influence of non-equilibrium anyons on correlation functions.
Functions $n_{\text{p,B}}^{\text{eq}}$ and $n_{\text{h,B}}^{\text{eq}}$ refer
to the corresponding distributions in the equilibrium channel B. For
convenience, temperatures are labeled on top of each panel and neglected in
legends. Panel (a): The particle distribution functions for
$T_{\text{SA}}=T_{\text{A}}=T_{\text{B}}=0$. The red curves represent the
particle distributions in channels A (solid red curve) and B (dashed red
curve). The blue curves represent the hole distributions in channels A (solid
blue curve) and B (dashed blue curve). Panel (b): Plots of the differential
heat flows $\mathcal{J}_{\text{A}\to\text{B}}$ (red curve) and
$-\mathcal{J}_{\text{B}\to\text{A}}$ (blue curve). Both curves are positive,
meaning that heat only flows from A to B, for
$T_{\text{SA}}=T_{\text{A}}=T_{\text{B}}=0$. Panels (c) and (d) show the
corresponding results for $T_{\text{SA}}=T_{\text{A}}=0$, but $T_{\text{B}}$
finite. Now, the differential heat flows can be either positive or negative.
Panels (e) - (h): Curves correspond to those in panel (a) - (d) but with a
finite temperature of channel SA, $T_{\text{SA}}=10I_{\text{A}}/e$. Panels (i)
- (l): Curves correspond to those in (a) - (d) but with a finite temperature
of channel A, $T_{\text{A}}=10I_{\text{A}}/e$. In panels (e) - (l), we take
$V_{\text{SA}}=30I_{\text{A}}/e^{2}$.

### S6B. Finite $T_{\text{B}}$ and $T_{\text{SA}}=T_{\text{A}}=0$

When the equilibrium channel B is at finite temperature $T_{B}\neq 0$, its
correlation functions are given by Eq. (S18). Again, we assume that channel B
is biased such that no charge current tunnels through the collider:
$V_{\text{B}}=V_{\text{eff}}=I_{\text{A}}\sin(2\pi\nu)/\nu^{2}e^{2}$. For
finite $T_{B}$, it is hard to obtain a compact analytical form of the heat
current. Therefore, in Fig. S5, we present results of the numerical
integration performed in Eq. (S29) with the known distribution functions given
by Eqs. (S17) and (S18). In Fig. S5(c), we show the particle (red curves) and
hole (blue curves) distribution functions for channels A (solid curves) and B
(dashed curves). Similar to the zero-temperature case [Fig. S5(a)], particle
and hole distributions at finite $T_{\text{B}}$, shown in Fig. S5(c), remain
symmetric to each other with respect to the Laughlin surface (at $\epsilon=\nu
eV_{\text{eff}}$) of channel B. Finite temperature $T_{\text{B}}$ “softens”
the equilibrium particle distributions compared to Fig. S5(a). As a
consequence, the corresponding differential heat current [Fig. S5(d)] can have
negative values at low energies, in great contrast to that in the zero-
temperature case [Fig. S5(b)].

Note that, in Fig. S5(d), the temperature in channel B,
$T_{\text{B}}=10I_{\text{A}}/e$, is chosen to be smaller than the temperature
[$\approx 17.3I_{\text{A}}$, see Fig. 1(d) of the main text] at which the
total heat current $J_{Q}=\int
d\epsilon\,[\mathcal{J}_{\text{A}\to\text{B}}(\epsilon)-\mathcal{J}_{\text{B}\to\text{A}}(\epsilon)]$
equals zero. Nevertheless, at this temperature,
$\mathcal{J}_{\text{A}\to\text{B}}(\epsilon)<\mathcal{J}_{\text{B}\to\text{A}}(\epsilon)$
for $\epsilon\leq 15I_{\text{A}}/e$, indicating heat flow from B to A at low
energies. This heat current from B to A, is balanced and exceeded by the
reverse heat current, from A to B, at large energies. Indeed, in Fig. S5(d)
[and likewise, Figs. S5(h) and S5(l)], curves with positive values decay much
slower than their negative counterparts, indicating that the total heat flow
is dominated by the partial heat flow from A to B at large energies.

### S6C. Finite $T_{\text{SA}}$ and $T_{\text{A}}=0$

Next, we consider the situation where $T_{\text{SA}}$ is finite and
$T_{\text{A}}=0$. In this case, Eq. (S13) leads to

$\displaystyle\left.\langle\psi_{A}^{\dagger}(t)\psi_{A}(0)\rangle_{\text{neq}}\right|_{T_{\text{A}}=0,\,T_{\text{SA}}}$
$\displaystyle=\frac{1}{2\pi\tau_{c}}\frac{\tau_{c}^{\nu}}{(\tau_{c}+it)^{\nu}}\exp\left\\{-\frac{S_{\text{A}}}{\nu^{2}e^{2}}[1-\cos(2\pi\nu)]|t|+i\frac{I_{\text{A}}}{\nu
e}\sin(2\pi\nu)t\right\\},$ (S37)
$\displaystyle\left.\langle\psi_{A}(t)\psi^{\dagger}_{A}(0)\rangle_{\text{neq}}\right|_{T_{\text{A}}=0,\,T_{\text{SA}}}$
$\displaystyle=\frac{1}{2\pi\tau_{c}}\frac{\tau_{c}^{\nu}}{(\tau_{c}+it)^{\nu}}\exp\left\\{-\frac{S_{\text{A}}}{\nu^{2}e^{2}}[1-\cos(2\pi\nu)]|t|-i\frac{I_{\text{A}}}{\nu
e}\sin(2\pi\nu)t\right\\},$

where $T_{\text{SA}}$ is implicitly contained in the noise of the diluted
channel

$S_{\text{A}}\equiv\int
dt[\langle\hat{I}_{\text{A}}(0,t)\hat{I}_{\text{A}}(0,0)\rangle-
I_{\text{A}}^{2}].$

Indeed, $S_{\text{A}}=\nu eI_{\text{A}}$ when $T_{\text{A}}=T_{\text{SA}}=0$,
and deviates from this value otherwise.

When $T_{\text{B}}=0$, the heat current from A to B can be similarly obtained
following Eqs. (S29) and (S32), after replacing $I_{\text{A}}$ in Eq. (S32)
with $S_{\text{A}}/\nu$, leading to

$J_{Q}=2\pi\frac{S_{\text{A}}}{\nu^{3}e^{3}}[1-\cos(2\pi\nu)]\,\tan(\pi\nu)I_{\text{A}\to\text{B}}.$
(S38)

Thus, when $T_{\text{SA}}$ is finite, the effective temperature becomes
proportional to the current noise $S_{\text{A}}$ in the diluted channel A,
which now combines the non-equilibrium and thermal parts of the tunneling
noise of the diluter.

For a more comprehensive understanding, we plot the distribution functions and
the differential heat flow, in Figs. S5(e) and S5(f), respectively, for
$T_{\text{SA}}=10I_{\text{A}}/e$ and $T_{\text{A}}=T_{\text{B}}=0$. Clearly,
when $T_{\text{B}}=0$, the spectral heat flows
$\mathcal{J}_{\text{A}\to\text{B}}$ and $-\mathcal{J}_{\text{B}\to\text{A}}$
are both positive for all energies, indicating positive heat flow from A to B.
This is in line with the fact that in channel B, particle distributions are
zero for energies larger than $\nu eV_{\text{B}}$.

When $T_{\text{B}}$ is finite, particle and hole distributions in channel B
are again softened by thermal fluctuation. This fact is shown in S5(g) for
$T_{\text{SA}}=T_{\text{B}}=10I_{\text{A}}/e$, where particle distribution in
channel B remains finite for states with heat ($\epsilon-\nu eV_{\text{B}}$)
within the window of the order of $T_{\text{B}}$. As a consequence, depending
on the state energy, the heat can either flow from channel A to channel B or
from B to A, as shown in Fig. S5(h). Note that, for energies above the
Laughlin surface of channel B, the particle distribution in B (the red dashed
curve) is influenced by temperature $T_{\text{B}}$ and decays much faster than
the particle distribution in channel A (the red solid curve), which is induced
by a non-equilibrium current. This is an important difference between a real
temperature and an effective one that is induced by a non-equilibrium current.

### S6D. Finite $T_{\text{A}}$ and $T_{\text{SA}}=0$

Finally, we consider the situation where $T_{\text{A}}$ is finite but
$T_{\text{SA}}=0$. In this case, the correlation functions in channel A read

$\displaystyle\left.\langle\psi_{A}^{\dagger}(t)\psi_{A}(0)\rangle_{\text{neq}}\right|_{T_{\text{SA}}=0,\,T_{\text{A}}}$
$\displaystyle=\frac{1}{2\pi\tau_{c}}\frac{(\pi
T_{\text{A}}\tau_{c})^{\nu}}{\sin^{\nu}[\pi
T_{\text{A}}(\tau_{c}+it)]}\exp\left\\{-\frac{S_{\text{A}}}{\nu^{2}e^{2}}[1-\cos(2\pi\nu)]|t|+i\frac{I_{\text{A}}}{\nu
e}\sin(2\pi\nu)t\right\\},$ (S39)
$\displaystyle\left.\langle\psi_{A}(t)\psi^{\dagger}_{A}(0)\rangle_{\text{neq}}\right|_{T_{\text{SA}}=0,\,T_{\text{A}}}$
$\displaystyle=\frac{1}{2\pi\tau_{c}}\frac{(\pi
T_{\text{A}}\tau_{c})^{\nu}}{\sin^{\nu}[\pi
T_{\text{A}}(\tau_{c}+it)]}\exp\left\\{-\frac{S_{\text{A}}}{\nu^{2}e^{2}}[1-\cos(2\pi\nu)]|t|-i\frac{I_{\text{A}}}{\nu
e}\sin(2\pi\nu)t\right\\},$

which contains two $T_{\text{A}}$-dependent factors: the equilibrium
correlation function in the prefactor, which explicitly depends on
$T_{\text{A}}$, and the noise $S_{\text{A}}$ that is implicitly influenced by
$T_{\text{A}}$.

When $T_{\text{B}}=T_{\text{SA}}=0$, the corresponding distribution functions
are provided in Fig. (S5)(i), for $T_{\text{A}}=10I_{\text{A}}/e$. In
comparison to those at finite $T_{\text{SA}}$ [Fig. S5(e)], distribution
functions of channel A in Fig. (S5)(i) are flatter, indicating a stronger
influence of thermal fluctuation resulting from finite $T_{\text{A}}$. This
result agrees with the observation above: $T_{\text{A}}$ influences the
correlation function in a two-fold way (explicitly and implicitly), while
$T_{\text{SA}}$ influences the correlation function only implicitly, through
the noise $S_{\text{A}}$. We also show in Fig. (S5)(j) the spectral content of
the heat current. Again, since $T_{\text{B}}=0$, the energy-resolved heat
current at all energies flow is from channel A to channel B.

When $T_{\text{B}}$ is finite, we plot distribution functions in Fig. S5(k).
Again, here distribution functions in A are flatter, in comparison to that in
Fig. S5(g), indicating a stronger thermal influence from $T_{\text{A}}$. The
corresponding differential heat flow is shown in Fig. S5(l).

### S6E. Effective temperature $T_{\text{eff}}$ of the non-equilibrium channel
A at $T_{A}=0$

Before ending this section, we derive the relation between the noise
$S_{\text{A}}$ of the non-equilibrium anyonic channel A, and its effective
temperature $T_{\text{eff}}$. Following Eq. (S29) and using correlation
functions Eq. (S37) for the anyon operators in channel A and

$\displaystyle\langle\psi^{\dagger}_{\text{B}}(t)\psi_{\text{B}}(0)\rangle_{0}$
$\displaystyle=\frac{1}{2\pi\tau_{c}}\frac{(\pi
T_{\text{B}}\tau_{c})^{\nu}}{\sin^{\nu}\left[\pi
T_{\text{B}}(\tau_{c}+it)\right]}e^{i\nu eV_{\text{B}}},$ (S40)
$\displaystyle\langle\psi_{\text{B}}(t)\psi_{\text{B}}^{\dagger}(0)\rangle_{0}$
$\displaystyle=\frac{1}{2\pi\tau_{c}}\frac{(\pi
T_{\text{B}}\tau_{c})^{\nu}}{\sin^{\nu}\left[\pi
T_{\text{B}}(\tau_{c}+it)\right]}e^{-i\nu eV_{\text{B}}},$

for the anyon operators in channel B, after setting
$V_{\text{B}}=V_{\text{eff}}=I\sin(2\pi\nu)/\nu^{2}e^{2}$, we write the
condition of zero heat current for $T_{\text{B}}=T_{\text{eff}}$ as

$\int\\!dt\\!\left\\{\frac{\partial}{\partial
t}\left[\frac{e^{-\frac{S_{\text{A}}}{\nu^{2}e^{2}}[1-\cos(2\pi\nu)]|t|}}{(\tau_{c}+it)^{\nu}}\right]\frac{1}{\sin^{\nu}[\pi
T_{\text{eff}}(\tau_{c}+it)]}-\frac{e^{-\frac{S_{\text{A}}}{\nu^{2}e^{2}}[1-\cos(2\pi\nu)]|t|}}{(\tau_{c}+it)^{\nu}}\frac{\partial}{\partial
t}\left[\frac{1}{\sin^{\nu}[\pi
T_{\text{eff}}(\tau_{c}+it)]}\right]\right\\}=0.$ (S41)

The equation above simplifies into

$\int
dt^{\prime}\frac{e^{-\frac{s_{c}}{\pi}[1-\cos(2\pi\nu)]|t^{\prime}|}}{(\tau_{c}^{\prime}+it^{\prime})^{\nu}\sin^{\nu}(\tau_{c}^{\prime}+it^{\prime})}\left\\{-\frac{i\nu}{\tau_{c}^{\prime}+it^{\prime}}+\frac{i\nu}{\tan(\tau_{c}^{\prime}+it^{\prime})}-\frac{s_{c}}{\pi}[1-\cos(2\pi\nu)]\text{sgn}(t^{\prime})\right\\}=0,$
(S42)

where $\tau_{c}^{\prime}\equiv\pi T_{\text{eff},\alpha}\tau_{c}$,
$t^{\prime}\equiv\pi T_{\text{eff}}t$, and

$s_{c}\equiv\frac{S_{\text{A}}}{\nu^{2}e^{2}T_{\text{eff}}}.$ (S43)

is a _dimensionless constant_ (given $\nu$ fixed), at which the integral in
Eq. (S42) vanishes. For the situation displayed in Fig. 1 of the main text,
$\nu=1/3$, we have $s_{c}\approx 0.174$. In that case, the factor
$s_{c}[1-\cos(2\pi\nu)]/\pi=0.083$ is a numerically small quantity, and we
expand the exponential in Eq. (S42) to leading order in
$s_{c}[1-\cos(2\pi\nu)]/\pi$, which yields an approximate expression for
$s_{c}$:

$s_{c}\approx\frac{\int
dt^{\prime}\dfrac{i\pi\nu}{(\tau_{c}^{\prime}+it^{\prime})^{\nu+1}\sin^{\nu}(\tau_{c}^{\prime}+it^{\prime})}\Big{[}(\tau_{c}^{\prime}+it^{\prime})\cot(\tau_{c}^{\prime}+it^{\prime})-1\Big{]}}{\int
dt^{\prime}\dfrac{1-\cos(2\pi\nu)}{(\tau_{c}^{\prime}+it^{\prime})^{\nu+1}\sin^{\nu}(\tau_{c}^{\prime}+it^{\prime})}\Big{[}\tau_{c}^{\prime}\
\text{sgn}(t^{\prime})+i(1-\nu)|t^{\prime}|+i\nu|t^{\prime}|(\tau_{c}^{\prime}+it^{\prime})\cot(\tau_{c}^{\prime}+it^{\prime})\Big{]}},$
(S44)

which equals $0.152$ for $\nu=1/3$, close to the result $0.174$, obtained by
numerically solving Eq. (S42). For smaller values of $\nu$, the factor
$s_{c}[1-\cos(2\pi\nu)]/\pi$ further decreases: following Eq. (S44), it
approximately equals 0.064 and 0.052 for $\nu=1/5$ and $1/7$, respectively. In
comparison, their exact values, obtained by numerically solving Eq. (S42), are
equal to 0.087 and 0.076, respectively. Clearly, following Eqs. (S42) and
(S43), the effective temperature

$T_{\text{eff}}=\frac{S_{\text{A}}}{\nu^{2}e^{2}s_{c}},$ (S45)

is proportional to the noise $S_{\text{A}}$ of the non-equilibrium channel A.

## S7. Tunneling-current noise and witness function $\mathcal{W}$

Experimentally, to measure the value of $\mathcal{W}$ [defined by Eq. (8) of
the main text], one needs to know the tunneling-current noise at the collider,

$S_{\text{T}}=\int dt\,\langle\delta I_{\text{T}}(t)\delta
I_{\text{T}}(0)\rangle,$

where $\delta I_{\text{T}}\equiv\hat{I}_{\text{T}}-I_{\text{T}}$ describes
fluctuations of the tunneling current. In general, measuring $S_{\text{T}}$ is
a demanding task. Here, we show, however, that in the considered setups [Fig.
1(a) of the main text], $S_{\text{T}}$ can be experimentally obtained by
measuring the auto-correlation noise in channel B. To demonstrate this, we
start by decomposing the tunneling current noise into several components
(assuming the collider located at the position $x=L$):

$\displaystyle\int dt\,\langle\delta I_{\text{T}}(t)\delta
I_{\text{T}}(0)\rangle=\int dt\,\langle[\delta I_{\text{B}}(x>L,t)-\delta
I_{\text{B}}(x<L,t)]\,[\delta I_{\text{B}}(x>L,0)-\delta
I_{\text{B}}(x<L,0)]\rangle$ (S46) $\displaystyle=\int dt\,\langle\delta
I_{\text{B}}(x>L,t)\,\delta I_{\text{B}}(x>L,0)\rangle+\int dt\,\langle\delta
I_{\text{B}}(x<L,t)\,\delta I_{\text{B}}(x<L,0)\rangle$ $\displaystyle-\int
dt\,\langle\delta I_{\text{B}}(x>L,t)\,\delta I_{\text{B}}(x<L,0)\rangle-\int
dt\,\langle\delta I_{\text{B}}(x<L,t)\,\delta I_{\text{B}}(x>L,0)\rangle,$

where $\delta I_{\text{B}}\equiv\hat{I}_{\text{B}}-I_{\text{B}}$ denotes
fluctuations of the current $I_{\text{B}}$ in channel B. In the second line of
Eq. (S46), the first term is the current auto-correlation in channel B
downstream of the collider and the second term is the noise upstream of the
collider. We neglect this term, as it depends only on the temperature
$T_{\text{B}}$.

Now we move to the last line of Eq. (S46), to show that it is zero at zero
temperature. We can simplify the first term of the last line of Eq. (S46) as
follows:

$\displaystyle\int dt\,\langle\delta I_{\text{B}}(x>L,t)\,\delta
I_{\text{B}}(x<L,0)\rangle$ $\displaystyle=\nu^{2}e^{2}\int
dt\langle\delta\hat{n}_{\text{p,B}}(x>L,t)\,\delta\hat{n}_{\text{p,B}}(x<L,0)\rangle,$
(S47)

where $\delta\hat{n}_{\text{p,B}}$ is the operator (hence, marked with a
“hat”) that refers to fluctuation (indicated by $\delta$) of the particle
distribution in channel B. This correlation function can be further cast in
the form of an integral over the energy:

$\int
d\epsilon\,\left[\langle\hat{n}_{\text{p,B}}(x>L,\epsilon)\,\hat{n}_{\text{p,B}}(x<L,\epsilon)\rangle-\langle\hat{n}_{\text{p,B}}(x>L,\epsilon)\rangle\,\langle\hat{n}_{\text{p,B}}(x<L,\epsilon)\rangle\right].$
(S48)

At zero temperature, Eq. (S48) is only nonzero if $\epsilon<\nu eV_{B}$, i.e.,
below the Laughlin surface of channel B. Indeed, otherwise (when $\epsilon>\nu
eV_{B}$) both the first and second terms of Eq. (S48) vanish, because there
are no particles with corresponding energies in channel B. However, when
$\epsilon<\nu eV_{B}$, tunneling from A to B vanishes, since the hole
distribution in B vanishes, i.e.,
$n_{\text{h,B}}(x<L,\epsilon)=\langle\hat{n}_{\text{h,B}}(x<L,\epsilon)\rangle=0$.
As a consequence, at the collider, only tunneling from B to A modifies the
particle distribution $n_{\text{p,B}}=\langle\hat{n}_{\text{p,B}}\rangle$,
leading to

$\langle\hat{n}_{\text{p,B}}(x>L,\epsilon)\,\hat{n}_{\text{p,B}}(x<L,\epsilon)\rangle=n_{\text{p,B}}(x<L,\epsilon)\,\big{[}n_{\text{p,B}}(x<L,\epsilon)-\tilde{\mathcal{T}}_{\text{C}}\,n_{\text{p,B}}(x<L,\epsilon)\,n_{\text{h,A}}(x<L,\epsilon)\big{]},$
(S49)

for $\epsilon<\nu eV_{B}$. Here, $\tilde{\mathcal{T}}_{\text{C}}$ refers to
the (renormalized) transmission observed in experiment, not
$\mathcal{T}_{\text{C}}$ in the tunneling Hamiltonian.

On the other hand, we have

$\displaystyle\langle\hat{n}_{\text{p,B}}(x<L,\epsilon)\rangle$
$\displaystyle=n_{\text{p,B}}(x<L,\epsilon),\quad\text{ if }\epsilon<\nu
eV_{B},$ (S50) $\displaystyle\langle\hat{n}_{\text{p,B}}(x<L,\epsilon)\rangle$
$\displaystyle=0\quad\text{otherwise};$
$\displaystyle\langle\hat{n}_{\text{p,B}}(x>L,\epsilon)\rangle$
$\displaystyle=n_{\text{p,B}}(x<L,\epsilon)-\tilde{\mathcal{T}}_{\text{C}}\,n_{\text{p,B}}(x<L,\epsilon)\,n_{\text{h,A}}(x<L,\epsilon),\quad\text{
if }\epsilon<\nu eV_{B},$
$\displaystyle\langle\hat{n}_{\text{p,B}}(x>L,\epsilon)\rangle$
$\displaystyle=\tilde{\mathcal{T}}_{\text{C}}\,n_{\text{h,B}}(x<L,\epsilon)\,n_{\text{p,A}}(x<L,\nu\epsilon)\quad\text{otherwise}.$

Following Eq. (S50), the product
$\langle\hat{n}_{\text{p,B}}(x>L,\epsilon)\rangle\,\langle\hat{n}_{\text{p,B}}(x<L,\epsilon)\rangle$
is only finite for $\epsilon<\nu eV_{B}$, where it is equal to the average of
the product of operators. We have thus arrived at the conclusion that, at zero
temperature,

$\int dt\,\langle\delta I_{\text{B}}(x>L,t)\,\delta
I_{\text{B}}(x<L,0)\rangle=0.$ (S51)

We emphasize that the vanishing of Eq. (S51) resorts to the fact that channel
B is equilibrium upstream of the collider; otherwise, Eqs. (S49) and (S51)
will not be exactly equal. Following similar steps, we can prove that the last
term of the last line of Eq. (S46) also vanishes:

$\int dt\,\langle\delta I_{\text{B}}(x<L,t)\,\delta
I_{\text{B}}(x>L,0)\rangle=0.$ (S52)

As a consequence, when $V_{\text{B}}$ is much larger than the experimental
temperature, the tunneling-current noise in the setup shown in Fig. 1(a) of
the main text can be obtained from measurements of the current auto-
correlations in channel B [second line of Eq. (S46)].

## S8. Distribution functions with non-equilibrium particle tunneling

In both the main text and the Supplementary Information (see Secs. S3 and S6;
in particular, Figs. S2 and S5), we have shown a remarkable property of
anyonic distributions. Specifically, if only time-domain braiding processes
are included in the correlation functions, the particle and hole distribution
functions are symmetric to each other with respect to the energy
$I_{\text{A}}\sin(2\pi\nu)/\nu e$ that corresponds to the effective chemical
potential $\nu eV_{\text{eff}}$.