##  I Supplementary Information

Supplementary Figure 1 Integration contour for the terms of
$\eta_{1}=\eta_{2}=1$ and $\eta_{1}=-\eta_{2}=1$ in Supplementary Equation 3.
Magenta lines represent the real-axis integral line, while the black line a
proper contour for contour integral in the complex plane of $t$. Dashed blue
lines show branch-cut lines.

Supplementary Figure 2 Braiding phase of Type II process. (a) and (b), The two
interfering paths of the Type II process in Fig. 2. (c) and (d), Paths
equivalent, respectively, to those in (a) and (b) in the gauge chosen in
parallel to the extended edge scheme in Fig. 4 of the main text. In the
interference of the two paths (a) and (b), a thermal anyon (whose trajectory
is represented by dashed lines) excited at QPC2 or QPC3 effectively winds
around a voltage-biased anyon (solid line) injected from Edge 1 via QPC1. The
effective winding results in the braiding phase factor
$\zeta_{a}^{*}\zeta_{b}=e^{i2\pi\nu}$. The phase factors $\zeta_{a}$ and
$\zeta_{b}$ are derived from the commutation rules in Eqs. (11) and (12). For
the details, see the Supplementary Note 6.

### Supplementary Note 1 : Green’s function.

We provide the Green’s function of bosonic fields $\phi_{i}$. It is defined as

$G(t,x)=\langle\phi_{i}(t,x)\phi_{i}(0,0)\rangle-\langle\phi_{i}(0,0)\phi_{i}(0,0)\rangle,$
(1)

where $\langle\cdots\rangle$ is the ensemble average under the bare
Hamiltonian $H_{\textrm{edge},i}$. The expression of $G$ is known Delft  as
$G_{\beta}(t,x)=-\nu\ln\left[\frac{\sin\left(\pi(\tau_{0}+i(t-x))/\beta\right)}{\pi\tau_{0}/\beta}\right]$,
where $\hbar/\tau_{0}$ is the infrared energy cutoff and we use
$v_{\textrm{p}}\equiv 1$ and $\beta=(k_{B}T)^{-1}$. At zero temperature,
$G(x,t)=-\nu\ln(\tau_{0}+i(t-x)/\tau_{0})$.

To describe the non-equilibrium situation by the voltage $V$ at edge 1, we use
the Green’s function on the standard Keldysh contour with Keldysh ordering
$T_{K}$. The contour has two branches, the upper one denoted by Keldysh index
$\eta=1$ and the lower of $\eta=-1$. For two operators $A$ and $B$ on the
upper branch, $T_{K}(AB)=AB$ if $B$ is the earlier event than $A$, and
$T_{K}(AB)=BA$ otherwise; it is correct not to attach the phase factor for the
exchange statistics in the definition of Keldysh ordering Kim06 . For $A$ and
$B$ on the lower, $T_{K}(AB)=BA$ if $B$ is earlier than $A$, and
$T_{K}(AB)=AB$ otherwise. For $A$ on the upper and $B$ on the lower,
$T_{K}(AB)=BA$. For $A$ on the lower and $B$ on the upper, $T_{K}(AB)=AB$. The
Keldysh Green’s function $G_{\eta\eta_{0}}(t,x)\equiv\langle
T_{K}[\phi_{i}(t^{\eta},x)\phi_{i}(0^{\eta_{0}},0)]\rangle-\langle\phi_{i}(0,0)\phi_{i}(0,0)\rangle$
is obtained as Kim06

$\displaystyle
G_{\eta\eta_{0}}(t,x)=-\nu\ln\frac{\sin\left(\pi(\tau_{0}+i\chi_{\eta\eta_{0}}(t)(t-x)\right)/\beta)}{\pi\tau_{0}/\beta},$
(2)

where
$\chi_{\eta_{1}\eta_{2}}(t_{1}-t_{2})=[(\eta_{1}+\eta_{2})/2]\textrm{sgn}(t_{1}-t_{2})-(\eta_{1}-\eta_{2})/2$
and $\eta_{0}$ and $\eta$ are Keldysh indexes.

### Supplementary Note 2 : Perturbation theory.

We compute the charge current $I_{\textrm{D}_{3}}$ at drain D3 by the voltage
$V$ at edge 1 up to order $\gamma_{1}^{2}\gamma_{2}\gamma_{3}$, employing
perturbation theory and the Keldysh Green function in Supplementary Equation
2. Hereafter, we occasionally use $\omega_{0}\equiv e^{*}V$,
$\beta\equiv(k_{B}T)^{-1}$, $v_{\textrm{p}}\equiv 1$, and $\hbar\equiv 1$.

The current $I_{\textrm{D}_{3}}$ is expressed as
$I_{D_{3}}=ie^{*}[N_{3},H]=ie^{*}(\langle T_{2}^{\downarrow}\rangle-\langle
T_{2}^{\uparrow}\rangle+\langle T_{3}^{\downarrow}\rangle-\langle
T_{3}^{\uparrow}\rangle)$. $I_{\textrm{D}_{3}}$ is decomposed,
$I_{\textrm{D}_{3}}=I_{\textrm{D}_{3}}^{\textrm{dir}}+I_{\textrm{D}_{3}}^{\textrm{int}}$,
into direct current
$I_{\textrm{D}_{3}}^{\textrm{dir}}\propto\gamma_{1}^{2}\gamma_{2}^{2},\,\gamma_{1}^{2}\gamma_{3}^{2}$,
and interference current
$I_{\textrm{D}_{3}}^{\textrm{int}}\propto\gamma_{1}^{2}\gamma_{2}\gamma_{3}$
depending on $\Phi$ (the leading-order contribution).

We focus on the contribution of $\langle T_{2}^{\downarrow}\rangle$ onto
$I_{\textrm{D}_{3}}^{\textrm{int}}\propto\gamma_{1}^{2}\gamma_{2}\gamma_{3}$,
which is denoted by $\langle T_{2}^{\downarrow,\textrm{int}}\rangle$. It is

$\langle
T_{2}^{\downarrow,\textrm{int}}\rangle=\sum_{\eta,\eta_{0}\eta_{1}\eta_{2}=\pm
1}\eta\eta_{1}\eta_{2}\int_{-\infty}^{\infty}dt_{1}\int_{-\infty}^{\infty}dt_{2}\int_{-\infty}^{\infty}dt\langle
T_{K}[T_{2}^{\downarrow}(0^{\eta_{0}})T_{1}^{\rightarrow}(t_{1}^{\eta_{1}})T_{1}^{\leftarrow}(t_{2}^{\eta_{2}})T_{3}^{\uparrow}(t^{\eta})]\rangle.\\\
$

$\langle
T_{K}[T_{2}^{\downarrow}(0^{\eta_{0}})T_{1}^{\rightarrow}(t_{1}^{\eta_{1}})T_{1}^{\leftarrow}(t_{2}^{\eta_{2}})T_{3}^{\uparrow}(t^{\eta})]\rangle/\gamma_{1}^{2}\gamma_{2}\gamma_{3}$
is decomposed into two parts,

$\displaystyle\langle
T_{K}[\Psi^{\dagger}_{3}(0^{\eta_{0}},L)\Psi_{2}(0^{\eta_{0}},d)\Psi^{\dagger}_{2}(t_{1}^{\eta_{1}},0)\Psi_{1}(t_{1}^{\eta_{1}},0)\Psi_{1}^{\dagger}(t_{2}^{\eta_{2}},0)\Psi_{2}(t_{2}^{\eta_{2}},0)\Psi_{2}^{\dagger}(t^{\eta},d+L)\Psi_{3}(t^{\eta},0)]\rangle$
$\displaystyle=$ $\displaystyle\langle
T_{K}[e^{-i\phi_{3}(0^{\eta_{0}},L)}e^{i\phi_{2}(0^{\eta_{0}},d)}e^{-i\phi_{2}(t_{1}^{\eta_{1}},0)}e^{i\phi_{1}(t_{1}^{\eta_{1}},0)}e^{-i\phi_{1}(t_{2}^{\eta_{2}},0)}e^{i\phi_{2}(t_{2}^{\eta_{2}},0)}e^{-i\phi_{2}(t^{\eta},d+L)}e^{i\phi_{3}(t^{\eta},0)}]\rangle$
$\displaystyle\times\langle
T_{K}[F_{3}^{\dagger}(0^{\eta_{0}})F_{2}(0^{\eta_{0}})F_{2}^{\dagger}(t_{1}^{\eta_{1}})F_{1}(t_{1}^{\eta_{1}})F_{1}^{\dagger}(t_{2}^{\eta_{2}})F_{2}(t_{2}^{\eta_{2}})F_{2}^{\dagger}(t^{\eta})F_{3}(t^{\eta})]\rangle.$

We compute the former part, using the identity (valid when the Hamiltonian of
$B$’s is quadratic) for bosonic operators $B$’s, $\langle
T_{K}[e^{B_{1}(t_{1})}e^{B_{2}(t_{2})}...e^{B_{n}(t_{n})}]\rangle=e^{\frac{1}{2}\sum_{j=1}^{n}\langle
T_{K}[B_{j}(t_{j})^{2}]\rangle+\sum_{i<j}\langle
T_{K}[B_{i}(t_{i})B_{j}(t_{j})]\rangle}$,

$\displaystyle\langle
T_{K}[e^{-i\phi_{3}(0^{\eta_{0}},L)}e^{i\phi_{2}(0^{\eta_{0}},d)}e^{-i\phi_{2}(t_{1}^{\eta_{1}},0)}e^{i\phi_{1}(t_{1}^{\eta_{1}},0)}e^{-i\phi_{1}(t_{2}^{\eta_{2}},0)}e^{i\phi_{2}(t_{2}^{\eta_{2}},0)}e^{-i\phi_{2}(t^{\eta},d+L)}e^{i\phi_{3}(t^{\eta},0)}]\rangle$
$\displaystyle=e^{2G_{\eta_{1}\eta_{2}}(t_{1}-t_{2},0)+G_{\eta_{1}\eta_{0}}(t_{1},-d)-G_{\eta_{1}\eta}(t_{1}-t,-d-L)+G_{\eta_{2}\eta}(t_{2}-t,-d-L)-G_{\eta_{2}\eta_{0}}(t_{2},-d)+G_{\eta\eta_{0}}(t,-L)+G_{\eta\eta_{0}}(t,L)},$

in which the Keldysh Green’s functions $G_{\eta\eta_{0}}$ in Supplementary
Equation 2 appear. The latter Klein-factor part is

$\displaystyle\langle
T_{K}[F_{3}^{\dagger}(0^{\eta_{0}})F_{2}(0^{\eta_{0}})F_{2}^{\dagger}(t_{1}^{\eta_{1}})F_{1}(t_{1}^{\eta_{1}})F_{1}^{\dagger}(t_{2}^{\eta_{2}})F_{2}(t_{2}^{\eta_{2}})F_{2}^{\dagger}(t^{\eta})F_{3}(t^{\eta})]\rangle$
$\displaystyle\quad\quad\quad\quad\quad=e^{-i\omega_{0}(t_{1}-t_{2})}\frac{\exp[{i\pi\nu\\{\chi_{\eta_{1}\eta_{0}}(t_{1})+\chi_{\eta_{2}\eta}(t_{2}-t)}\\}/2]}{\exp[{i\pi\nu\\{\chi_{\eta_{2}\eta_{0}}(t_{2})+\chi_{\eta_{1}\eta}(t_{1}-t)}\\}/2]}.$

It is obtained, by using $F_{1}(t)=F_{1}(0)e^{-ie^{*}Vt/\hbar}$,
$F_{2}(t)=F_{2}(0)$, $F_{3}(t)=F_{3}(0)$,
$F_{i}^{\dagger}F_{i}=F_{i}F_{i}^{\dagger}=1$, and the exchange statistics
rule in Eq. (12). The final expression is

$\displaystyle\langle T_{2}^{\downarrow,\textrm{int}}\rangle$
$\displaystyle=\gamma_{1}^{2}\gamma_{2}\gamma_{3}\sum_{\eta_{1},\eta_{2},\eta=\pm
1}\int_{-\infty}^{0}dt\int_{-\infty}^{\infty}dt_{2}\int_{-\infty}^{\infty}dt_{1}\frac{\eta_{1}\eta_{2}\eta
e^{-i2\pi\Phi/\Phi_{0}^{*}}e^{-i\omega_{0}(t_{1}-t_{2})}}{\left(\frac{\beta}{\pi}\sin(\frac{\pi}{\beta}(\tau_{0}+i\chi_{\eta_{1}\eta_{2}}(t_{1}-t_{2})(t_{1}-t_{2})))\right)^{2\nu}}$
$\displaystyle\times\frac{1}{\left(\frac{\beta}{\pi}\sin(\frac{\pi}{\beta}(\tau_{0}-i\eta(t+L)))\right)^{\nu}\left(\frac{\beta}{\pi}\sin(\frac{\pi}{\beta}(\tau_{0}-i\eta(t-L)))\right)^{\nu}}$
$\displaystyle\times\frac{\left(\sin(\frac{\pi}{\beta}(\tau_{0}-i\eta_{1}(t_{1}-t+d+L)))\right)^{\nu}\left(\sin(\frac{\pi}{\beta}(\tau_{0}-i\eta_{2}(t_{2}+d)))\right)^{\nu}}{\left(\sin(\frac{\pi}{\beta}(\tau_{0}-i\eta_{1}(t_{1}+d)))\right)^{\nu}\left(\sin(\frac{\pi}{\beta}(\tau_{0}-i\eta_{2}(t_{2}-t+d+L))\right)^{\nu}}.$

Computing the other terms in the same way, we obtain

$\displaystyle I_{D_{3}}^{\textrm{int}}$
$\displaystyle=\alpha\gamma_{1}^{2}\gamma_{2}\gamma_{3}\sum_{\epsilon\eta_{1},\eta_{2},\eta=\pm
1}\int_{-\infty}^{\infty}dt\int_{-\infty}^{\infty}dt_{2}\int_{-\infty}^{\infty}dt_{1}\frac{\epsilon\eta_{1}\eta_{2}\eta
e^{-i\epsilon
2\pi\Phi/\Phi_{0}^{*}}e^{-i\omega_{0}(t_{1}-t_{2})}}{\left(\frac{\beta}{\pi}\sin(\frac{\pi}{\beta}(\tau_{0}+i\chi_{\eta_{1}\eta_{2}}(t_{1}-t_{2})(t_{1}-t_{2})))\right)^{2\nu}}$
$\displaystyle\times\frac{1}{\left(\frac{\beta}{\pi}\sin(\frac{\pi}{\beta}(\tau_{0}-i\eta(t+L)))\right)^{\nu}\left(\frac{\beta}{\pi}\sin(\frac{\pi}{\beta}(\tau_{0}-i\eta(t-L)))\right)^{\nu}}$
$\displaystyle\times\left[\frac{\left(\sin(\frac{\pi}{\beta}(\tau_{0}-i\eta_{1}(t_{1}-t+d+L)))\right)^{\nu}\left(\sin(\frac{\pi}{\beta}(\tau_{0}-i\eta_{2}(t_{2}+d)))\right)^{\nu}}{\left(\sin(\frac{\pi}{\beta}(\tau_{0}-i\eta_{1}(t_{1}+d)))\right)^{\nu}\left(\sin(\frac{\pi}{\beta}(\tau_{0}-i\eta_{2}(t_{2}-t+d+L))\right)^{\nu}}\right]^{\epsilon},$
(3)

where $\alpha\equiv(e^{*}/\hbar)a^{4\nu}/[(\hbar v_{\textrm{p}})^{4\nu}(2\pi
a)^{4}]$. Supplementary Equation 3 is written as
$I_{D_{3}}^{\textrm{int}}\propto\cos(2\pi\frac{\Phi}{\Phi^{*}_{0}}+\theta)$ in
Eq. (5),

$\displaystyle\theta$ $\displaystyle=\arg[\sum_{\eta_{1},\eta_{2},\eta=\pm
1}\int_{-\infty}^{\infty}dt\int_{-\infty}^{\infty}dt_{2}\int_{-\infty}^{\infty}dt_{1}\frac{\eta_{1}\eta_{2}\eta
e^{-i\omega_{0}(t_{1}-t_{2})}}{\left(\frac{\beta}{\pi}\sin(\frac{\pi}{\beta}(\tau_{0}+i\chi_{\eta_{1}\eta_{2}}(t_{1}-t_{2})(t_{1}-t_{2})))\right)^{2\nu}}\frac{1}{\left(\frac{\beta}{\pi}\sin(\frac{\pi}{\beta}(\tau_{0}-i\eta(t+L)))\right)^{\nu}}$
$\displaystyle\times\frac{1}{\left(\frac{\beta}{\pi}\sin(\frac{\pi}{\beta}(\tau_{0}-i\eta(t-L)))\right)^{\nu}}\frac{\left(\sin(\frac{\pi}{\beta}(\tau_{0}-i\eta_{1}(t_{1}-t+d+L)))\right)^{\nu}\left(\sin(\frac{\pi}{\beta}(\tau_{0}-i\eta_{2}(t_{2}+d)))\right)^{\nu}}{\left(\sin(\frac{\pi}{\beta}(\tau_{0}-i\eta_{1}(t_{1}+d)))\right)^{\nu}\left(\sin(\frac{\pi}{\beta}(\tau_{0}-i\eta_{2}(t_{2}-t+d+L))\right)^{\nu}}].$
(4)

We omit the expression of the direct-current part
$I_{\textrm{D}_{3}}^{\textrm{dir}}$ of $I_{\textrm{D}_{3}}$ (of the order of
$\gamma_{1}^{2}\gamma_{2}^{2}$ and $\gamma_{1}^{2}\gamma_{3}^{2}$). The whole
expression of $I_{\textrm{D}_{3}}$ (up to the order of
$\gamma_{1}^{2}\gamma_{2}\gamma_{3}$) is zero at $V=0$, as expected.

### Supplementary Note 3 : Semiclassical Approximation.

We derive Eqs. (3), (4) and (6) in the main text, by performing the integrals
in Supplementary Equation 3 (or those in Supplementary Equation 4) in the
semiclassical regime of $e^{*}V\gg k_{B}T\gg\hbar v_{\textrm{p}}/L$.

In the semiclassical regime, the wave packet size of an injected anyon from S1
is $W\sim\hbar v_{\textrm{p}}/(e^{*}V)\ll L$, hence, the anyon is thought of
as particle-like (rather than wave-like). And the injection of hole-like
anyons (induced by thermal fluctuations) from S1 can be ignored. Thermally
excited virtual anyons also have size much smaller than $L$. Then, the
braiding of the anyons results in the well-defined exchange statistics phase
of $\pi\nu$.

In Supplementary Equation 3, there are 16 configurations of $\eta_{1}$,
$\eta_{2}$, $\eta$, and $\epsilon$. $\eta_{1}$ and $\eta_{2}$ $(=\pm 1)$ are
Keldysh indexes for tunneling at QPC1, while $\eta$ is for tunneling at QPC2
or QPC3. $\epsilon=\pm 1$ represents the winding direction of anyons around
Aharonov-Bohm flux $\Phi$. The terms of Supplementary Equation 3 are
classified, depending on whether $\eta_{1}=\eta_{2}$. In the regime of
$e^{*}V\gg k_{B}T\gg\hbar v_{\textrm{p}}/L$, some terms of
$\eta_{1}\neq\eta_{2}$ are represented by connected Feynman diagrams, and
contribute to Type I-1 and Type I-2. The other terms of $\eta_{1}\neq\eta_{2}$
and all the terms of $\eta_{1}=\eta_{2}$ are represented by vacuum bubbles.

Some of the vacuum bubbles of $\eta_{1}\neq\eta_{2}$ describe the virtual
anyonic excitations winding around a real anyon injected from S1 (similarly to
Fig. 1a in the main text). Each of such bubble diagrams has a partner diagram
among the terms of $\eta_{1}=\eta_{2}$, in which a vacuum bubble does not wind
around the real anyon (similarly to Fig. 1b in the main text); each of the
diagrams of $\eta_{1}=\eta_{2}$ is decomposed into two disconnected and
independent subdiagrams, one describing the real anyon, and the other for the
virtual anyonic excitations not winding around the real anyon. A bubble
diagram of $\eta_{1}\neq\eta_{2}$ winding around a real anyon, and its partner
diagram of $\eta_{1}=\eta_{2}$ cancel each other. The cancellation is only
partial, resulting in the interference current of Type II and Type III in Eqs.
(3) and (4), and disobeying the linked cluster theorem; the cancellation is
exact in the case of bosons or fermions. Note that there are the terms of
$\eta_{1}\neq\eta_{2}$ not involving any braiding effect between real anyons
and virtual anyons. These terms show divergence, but are fully canceled by
their partner diagrams of $\eta_{1}=\eta_{2}$, satisfying the linked cluster
theorem.

We now further compute Supplementary Equation 3 in the limit of $e^{*}V\gg
k_{B}T\gg\hbar v_{\textrm{p}}/L$. For example,

$\displaystyle\textrm{the}\textrm{ term of }(\eta_{1}=\eta_{2}=\pm
1,\epsilon=1)\textrm{ in Supplementary
Equation~{}\ref{Current_Total}}=2\alpha\gamma_{1}^{2}\gamma_{2}\gamma_{3}\sum_{\eta,\eta_{1}}\eta\int_{-\infty}^{\infty}dt\int_{-\infty}^{\infty}d\tilde{t}\int_{-\infty}^{\infty}d\tilde{T}$
$\displaystyle\times\frac{e^{-i2\pi\Phi/\Phi_{0}^{*}}e^{-i\omega_{0}2\tilde{t}}}{\left(\frac{\beta}{\pi}\sin(\frac{\pi}{\beta}(\tau_{0}+i\eta_{1}\textrm{sgn}(\tilde{t})2\tilde{t})))\right)^{2\nu}\left(\frac{\beta}{\pi}\sin(\frac{\pi}{\beta}(\tau_{0}-i\eta(t+L)))\right)^{\nu}\left(\frac{\beta}{\pi}\sin(\frac{\pi}{\beta}(\tau_{0}-i\eta(t-L)))\right)^{\nu}}$
$\displaystyle\times\left[\frac{\left(\sin(\frac{\pi}{\beta}(\tau_{0}-i\eta_{1}(\tilde{T}+\tilde{t}-t+d+L)))\right)^{\nu}\left(\sin(\frac{\pi}{\beta}(\tau_{0}-i\eta_{1}(\tilde{T}-\tilde{t}+d)))\right)^{\nu}}{\left(\sin(\frac{\pi}{\beta}(\tau_{0}-i\eta_{1}(\tilde{T}+\tilde{t}+d)))\right)^{\nu}\left(\sin(\frac{\pi}{\beta}(\tau_{0}-i\eta_{1}(\tilde{T}-\tilde{t}-t+d+L))\right)^{\nu}}\right]$
$\displaystyle\simeq
2\alpha\gamma_{1}^{2}\gamma_{2}\gamma_{3}\sum_{\eta,\eta_{1}}\eta\int_{-\infty}^{\infty}dt\int_{-\infty}^{\infty}d\tilde{T}\int_{C}d\tilde{t}$
$\displaystyle\times\frac{e^{-i2\pi\Phi/\Phi_{0}^{*}}e^{-i\omega_{0}2\tilde{t}}\left[1+\frac{2\pi\nu}{\beta}\tilde{t}\textrm{
sgn}(t-L)\left\\{\textrm{sgn}(\tilde{T}-t+d+L)\textrm{sgn}(\tilde{T}+d)-1\right\\}\right]}{\left(\frac{\beta}{\pi}\sin(\frac{\pi}{\beta}(\tau_{0}+i\eta_{1}\textrm{sgn}(\tilde{t})2\tilde{t}))\right)^{2\nu}\left(\frac{\beta}{\pi}\sin(\frac{\pi}{\beta}(\tau_{0}-i\eta(t+L)))\right)^{\nu}\left(\frac{\beta}{\pi}\sin(\frac{\pi}{\beta}(\tau_{0}-i\eta(t-L)))\right)^{\nu}}$

In the first equality, we put $\tilde{t}=(t_{1}-t_{2})/2$ and
$\tilde{T}=(t_{1}+t_{2})/2$ into Supplementary Equation 3. In the second, we
change the integral range of $\tilde{T}$ from $\tilde{T}\in(-\infty,\infty)$
to $\tilde{T}\in(-\infty+i\eta_{1}\beta/2,\infty+i\eta_{1}\beta/2)$ using the
contour shown in Supplementary Figure 1a, and that of $\tilde{t}$ from
$\tilde{t}\in(-\infty,\infty)$ to the vertical parts $C$ near
Re$(\tilde{t})=0$ of the contour in Supplementary Figure 1b. Note that the
integral of $\tilde{t}$ along the horizontal black line of Supplementary
Figure 1b is negligible in the limit of $e^{*}V\gg k_{B}T\gg\hbar
v_{\textrm{p}}/L$. Then

$\displaystyle\textrm{Sum of
the}\,\,(\eta_{1}=\eta_{2}=1,\epsilon=1)\,\,\textrm{term and
the}\,\,(\eta_{1}=\eta_{2}=-1,\epsilon=1)\,\,\textrm{term}$
$\displaystyle\simeq\alpha\gamma_{1}^{2}\gamma_{2}\gamma_{3}\int_{-\infty}^{\infty}dt\int_{-\infty}^{\infty}d\tilde{T}\left[1+\frac{2\pi
i\nu(1-2\nu)\textrm{
sgn}(t-L)}{\omega\beta}\left\\{\textrm{sgn}(\tilde{T}-t+d+L)\textrm{sgn}(\tilde{T}+d)-1\right\\}\right]$
$\displaystyle\quad\qquad\qquad\times\sum_{\eta}\eta\frac{A_{1}e^{-i2\pi\Phi/\Phi_{0}^{*}}}{\left(\frac{\beta}{\pi}\sin(\frac{\pi}{\beta}(\tau_{0}-i\eta(t+L)))\right)^{\nu}\left(\frac{\beta}{\pi}\sin(\frac{\pi}{\beta}(\tau_{0}-i\eta(t-L)))\right)^{\nu}},$
(5)

where
$A_{1}=\int_{-\infty}^{\infty}d\tilde{t}e^{-i\omega_{0}\tilde{t}}\left(\frac{\beta}{\pi}\sin(\frac{\pi}{\beta}(\tau_{0}-i\tilde{t})\right)^{-2\nu}\simeq\frac{2\pi}{\Gamma(2\nu)}\omega_{0}^{2\nu-1}$
and $\Gamma(2\nu)$ is the Gamma function. The divergence of Supplementary
Equation 5 is fully canceled by some terms of $\eta_{1}\neq\eta_{2}$. Next, we
consider

the $\displaystyle\textrm{ term of
}(\eta_{1}=1,\eta_{2}=-1,\epsilon=1)=2\alpha\gamma_{1}^{2}\gamma_{2}\gamma_{3}\sum_{\eta}(-\eta)\int_{-\infty}^{\infty}dt\int_{-\infty}^{\infty}d\tilde{T}\int_{-\infty}^{\infty}d\tilde{t}$
$\displaystyle\times\frac{e^{-i2\pi\Phi/\Phi_{0}^{*}}e^{-i\omega_{0}2\tilde{t}}}{\left(\frac{\beta}{\pi}\sin(\frac{\pi}{\beta}(\tau_{0}-i2\tilde{t}))\right)^{2\nu}\left(\frac{\beta}{\pi}\sin(\frac{\pi}{\beta}(\tau_{0}-i\eta(t+L)))\right)^{\nu}\left(\frac{\beta}{\pi}\sin(\frac{\pi}{\beta}(\tau_{0}-i\eta(t-L)))\right)^{\nu}}$
$\displaystyle\times\left[\frac{\left(\sin(\frac{\pi}{\beta}(\tau_{0}-i(\tilde{T}+\tilde{t}-t+d+L)))\right)^{\nu}\left(\sin(\frac{\pi}{\beta}(\tau_{0}+i(\tilde{T}-\tilde{t}+d)))\right)^{\nu}}{\left(\sin(\frac{\pi}{\beta}(\tau_{0}-i(\tilde{T}+\tilde{t}+d)))\right)^{\nu}\left(\sin(\frac{\pi}{\beta}(\tau_{0}+i(\tilde{T}-\tilde{t}-t+d+L))\right)^{\nu}}\right]$
$\displaystyle\simeq\alpha\gamma_{1}^{2}\gamma_{2}\gamma_{3}\int_{-\infty}^{\infty}dt\left(-A_{2}^{2}e^{i\pi\nu\textrm{sgn}(t-L)}-A_{2}^{2}e^{i\omega_{0}(t-L)}-A_{1}\int_{-\infty}^{\infty}d\tilde{T}e^{-i\pi\nu(\textrm{sgn}(\tilde{T}-t+d+L)-\textrm{sgn}(\tilde{T}+d))}\right.$
$\displaystyle\left.\times\left[1+\frac{2\pi i\nu(1-2\nu)\textrm{
sgn}(t-L)}{\omega\beta}\left\\{\textrm{sgn}(\tilde{T}-t+d+L)\textrm{sgn}(\tilde{T}+d)-1\right\\}\right]\right)$
$\displaystyle\times\sum_{\eta}\eta\frac{e^{-i2\pi\Phi/\Phi_{0}^{*}}}{\left(\frac{\beta}{\pi}\sin(\frac{\pi}{\beta}(\tau_{0}-i\eta(t+L)))\right)^{\nu}\left(\frac{\beta}{\pi}\sin(\frac{\pi}{\beta}(\tau_{0}-i\eta(t-L)))\right)^{\nu}}.$
(6)

Here
$A_{2}=\int_{-\infty}^{\infty}dte^{-i\omega_{0}t}\left(\frac{\beta}{\pi}\sin(\frac{\pi}{\beta}(\tau-
it))\right)^{-\nu}\simeq\frac{2\pi}{\Gamma(\nu)}\omega_{0}^{\nu-1}$,
$\tilde{t}=(t_{1}-t_{2})/2$, and $\tilde{T}=(t_{1}+t_{2})/2$. The integral
over $\tilde{t}$ is done in the regime of $e^{*}V\gg k_{B}T$, using the
contour in Supplementary Figure 1c.

The terms of $(\eta_{1}=1,\eta_{2}=-1,\epsilon=1)$ with coefficient $A_{2}$ in
Supplementary Equation 6 describe Types I-1 and I-2, which have no counterpart
in the terms of $\eta_{1}=\eta_{2}$. The divergence of some terms with
coefficient $A_{1}$ in Supplementary Equation 6 is fully canceled by the
corresponding terms of $\eta_{1}=\eta_{2}$. The remaining terms with $A_{1}$
are partially canceled by the corresponding terms of $\eta_{1}=\eta_{2}$,
describing Types II and III. Indeed, the difference of the terms with $A_{1}$
between Supplementary Equations 5 and 6 is proportional to

$\displaystyle-
A_{1}\int_{-\infty}^{\infty}d\tilde{T}\left(e^{-i\pi\nu(\textrm{sgn}(\tilde{T}-t+d+L)-\textrm{sgn}(\tilde{T}+d))}-1\right)$
$\displaystyle\quad\times\left[1+\frac{2\pi i\nu(1-2\nu)\textrm{
sgn}(t-L)}{\omega\beta}\left\\{\textrm{sgn}(\tilde{T}-t+d+L)\textrm{sgn}(\tilde{T}+d)-1\right\\}\right]$
$\displaystyle\quad=-2iA_{1}(t-L)e^{i\pi\nu\textrm{sgn}(t-L)}\left[1+\frac{2\pi
i\nu(1-2\nu)\textrm{ sgn}(t-L)}{\omega\beta}\right]\sin\pi\nu.$

The difference is proportional to $\sin\pi\nu$, and describes the topological
vacuum bubbles corresponding to the braiding discussed in Fig. 1 and Eqs. (3)
and (4). The term of $\eta_{1}=-1$, $\eta_{2}=1$, and $\epsilon=1$ describes
the injection of hole-like anyons from edge 1 to edge 2, and is negligible in
our regime of $e^{*}V\gg k_{B}T$. The terms with $\epsilon=-1$ are obtained in
the same way, as they are for the processes of winding $\Phi$ in the opposite
direction to those of $\epsilon=1$.

Now we collect all the terms of $I_{D_{3}}^{\textrm{int}}$,

$\displaystyle
I_{D_{3}}^{\textrm{int}}=\,\alpha\gamma_{1}^{2}\gamma_{2}\gamma_{3}\sum_{\eta}(-\eta)\int_{-\infty}^{\infty}dt\frac{1}{\left(\frac{\beta}{\pi}\sin(\frac{\beta}{\pi}(\tau_{0}-i\eta(t+L)))\right)^{\nu}\left(\frac{\beta}{\pi}\sin(\frac{\pi}{\beta}(\tau_{0}-i\eta(t-L)))\right)^{\nu}}$
$\displaystyle\times$
$\displaystyle[(-2i\sin(\pi\nu)A_{1}\\{1+f(t)\\}e^{i\pi\nu\textrm{sgn}(t-L)}(t-L)-A_{2}^{2}e^{i\pi\nu\textrm{sgn}(t-L)}-A_{2}^{2}e^{i\omega_{0}(t-L)})e^{-i2\pi\Phi/\Phi_{0}^{*}}$
$\displaystyle-(2i\sin(\pi\nu)A_{1}\\{1-f(t)\\}e^{-i\pi\nu\textrm{sgn}(t-L)}(t-L)-A_{2}^{2}e^{-i\pi\nu\textrm{sgn}(t-L)}-A_{2}^{2}e^{-i\omega_{0}(t-L)})e^{i2\pi\Phi/\Phi_{0}^{*}}],$
(7)

where $f(t)=\frac{2\pi i\nu(1-2\nu)\textrm{ sgn}(t-L)}{\omega\beta}$. The
integral of $t$ is further computed for $k_{B}T\gg\hbar v_{\textrm{p}}/L$; we
use
$\frac{\beta}{\pi}\sin(\frac{\pi}{\beta}(\tau_{0}-it))\simeq-i\textrm{sgn}(t)\frac{\beta}{2\pi}e^{\frac{\pi}{\beta}|t|}$
for $t\gg\beta$, $\int_{-\infty}^{0}dte^{\pi\nu
t/\beta}\left(\frac{\beta}{\pi}\sin(\frac{\beta}{\pi}(\tau_{0}+it))\right)^{-\nu}=(\frac{2\pi}{\beta})^{\nu}\frac{\beta}{2\pi}e^{i\pi\nu/2}\pi\csc(\pi\nu)$,
and $\int_{-\infty}^{0}dte^{\pi\nu
t/\beta}t\left(\frac{\beta}{\pi}\sin(\frac{\beta}{\pi}(\tau_{0}+it))\right)^{-\nu}=(\frac{2\pi}{\beta})^{\nu}(\frac{\beta}{2\pi})^{2}\pi
e^{i\pi\nu/2}(\psi(1)-\psi(\nu))\csc(\pi\nu)$, where $\psi(x)$ is the
polygamma function. The integration leads to Eq. (3),

$\displaystyle I_{\textrm{D}_{3}}^{\textrm{I-1}}\,=$
$\displaystyle\,\frac{16\pi^{3}(2\pi)^{\nu}\alpha}{\Gamma(\nu)^{3}}\gamma_{1}^{2}\gamma_{2}\gamma_{3}(e^{*}V)^{3\nu-3}(k_{B}T)^{\nu}\cos(e^{*}VL/\hbar
v_{\textrm{p}}-\pi\nu/2)$ $\displaystyle\times
e^{-2L/L_{T}}\cos(2\pi\Phi/\Phi_{0}^{*}+e^{*}VL/\hbar v_{\textrm{p}}),$
$\displaystyle I_{\textrm{D}_{3}}^{\textrm{I-2}}\,=$
$\displaystyle\,-\frac{8\pi(2\pi)^{2\nu}\alpha}{\nu\Gamma(\nu)^{2}}\gamma_{1}^{2}\gamma_{2}\gamma_{3}(e^{*}V)^{2\nu-2}(k_{B}T)^{2\nu-1}e^{-2L/L_{T}}\sin^{2}\pi\nu\cos(2\pi\Phi/\Phi_{0}^{*}),$
$\displaystyle I_{\textrm{D}_{3}}^{\textrm{II}}\,=$
$\displaystyle\,\frac{4(2\pi)^{2\nu}\pi\alpha\csc(\pi\nu)}{\Gamma(2\nu)}\gamma_{1}^{2}\gamma_{2}\gamma_{3}(e^{*}V)^{2\nu-1}(k_{B}T)^{2\nu-1}\sqrt{1+h_{1}(T/V)}\frac{2L+CL_{T}}{\hbar
v_{\textrm{p}}}$ $\displaystyle\times
e^{-2L/L_{T}}\sin^{2}\pi\nu\cos(2\pi\Phi/\Phi_{0}^{*}+\pi\nu+h_{2}(T/V)),$
$\displaystyle I_{\textrm{D}_{3}}^{\textrm{III}}\,=$
$\displaystyle\,\frac{4(2\pi)^{2\nu}\pi\alpha\csc(\pi\nu)}{\Gamma(2\nu)}\gamma_{1}^{2}\gamma_{2}\gamma_{3}(e^{*}V)^{2\nu-1}(k_{B}T)^{2\nu-1}\sqrt{1+h_{1}(T/V)}\frac{CL_{T}}{\hbar
v_{\textrm{p}}}$ $\displaystyle\times
e^{-2L/L_{T}}\sin^{2}\pi\nu\cos(2\pi\Phi/\Phi_{0}^{*}-\pi\nu-h_{2}(T/V)),$

where $C\simeq\nu[\psi(\nu)-\psi(1)]/2$,
$\alpha\equiv(e^{*}/\hbar)a^{4\nu}/[(\hbar v_{\textrm{p}})^{4\nu}(2\pi
a)^{4}]$, and $a$ is the short-length cutoff; $C=0.43$ for $\nu=1/3$ and $C\to
0.5$ as $\nu\to 0$. Note that $f(t)\propto k_{B}T/\omega$ in Supplementary
Equation 7 leads to the higher-order corrections of
$h_{1}(T/V)\equiv\left(\frac{2\pi\nu k_{B}T}{e^{*}V}\right)^{2}$ and
$h_{2}(T/V)\equiv(1-2\nu)\arctan(\frac{2\pi\nu k_{B}T}{e^{*}V})$. These
corrections are ignored in Eqs. (3) and (4), considering the limit $e^{*}V\gg
k_{B}T$, while they are included in Eq. (6) for better comparison with the
numerical result in Fig. 3.

### Supplementary Note 4 : Reference signal.

In the main text, we suggest to measure a reference interference signal
$I^{\textrm{int}}_{\textrm{Ref, D}_{3}}$, by applying infinitesimal voltage
$V_{\textrm{ref}}/2$ at source S2 and $-V_{\textrm{ref}}/2$ at S3, but turning
off the voltage $V$ at S1. It is obtained Chamon  up to the lowest order in
$\gamma$’s,

$\displaystyle I^{\textrm{int}}_{\textrm{Ref, D}_{3}}$
$\displaystyle\simeq\frac{e^{*}}{\hbar}\frac{\gamma_{2}\gamma_{3}\,a^{2\nu}}{(\hbar
v_{\textrm{p}})^{2\nu}(2\pi
a)^{2}}e^{*}V_{\textrm{ref}}(k_{B}T)^{2\nu-2}e^{-2L/L_{T}}\cos\frac{2\pi\Phi}{\Phi^{*}_{0}}\quad\quad\textrm{for}\,\,k_{B}T\gg
e^{*}V_{\textrm{ref}}.$ (8)

Since the reference signal $I^{\textrm{int}}_{\textrm{Ref, D}_{3}}$ and the
main interference signal $I^{\textrm{int}}_{\textrm{D}_{3}}$ are obtained from
the same setup under the same external parameters (temperature, gate voltage,
magnetic field, etc), the possible side effects affecting the main signal,
including the external-parameter dependence of the size, shape, QPC tunneling,
and anyon excitations of the interferometry, appear in the reference in the
same manner. Therefore, by comparing the reference with the main signal, one
can exclude the side effects and unambiguously detect the interference phase
shift $\theta$ in Eq. (6), hence, the fractional statistics. This strategy
works well both in the pure Aharonov-Bohm regime and the Coulomb-dominated
regime.

### Supplementary Note 5 : Coulomb dominated regime.

We discuss about the Coulomb interaction between the edge and the bulk of the
interferometry. There are two regimes of Fabry-Perot interferometry, the pure
Aharonov-Bohm regime and the Coulomb dominated regime. In the former regime,
the interaction is negligible, while it is crucial in the latter Halperin .
The Fabry-Perot interferometers of recent experiments An ; Ofek ; McClure ;
Willett ; Camino  in the fractional quantum Hall regime are in the Coulomb
dominated regime. We below compute the interference current
$I^{\textrm{int}}_{\textrm{D}_{3}}$ in the presence of the Coulomb
interaction, and show that Equation (6) of the main text is applicable to both
of the pure Aharonov-Bohm regime and the Coulomb dominated regime. We then
generalize this to the quantum Hall regime of filling factor
$\nu^{\prime}=\nu+\nu_{0}$ where $\nu=1/(2n+1)$ and $\nu_{0}$ is a nonzero
integer.

We compute $I^{\textrm{int}}_{\textrm{D}_{3}}$, combining our chiral Luttinger
liquid theory with the capacitive interaction model Halperin  that
successfully describes the Coulomb dominated regime. The interferometer
Hamiltonian $H=\sum_{i}H_{\textrm{edge},i}+H_{\textrm{tun}}$ is modified by
the Coulomb interaction as

$\displaystyle H$ $\displaystyle\to
H+U_{\textrm{int}}Q_{\textrm{bulk}}(\int_{d}^{d+L}dx:\partial_{x}\phi_{2}(x):+\int_{0}^{L}dx:\partial_{x}\phi_{3}(x):)+U_{\textrm{bulk}}Q_{\textrm{bulk}}^{2}$
$\displaystyle=\sum_{i=1,2,3}\frac{\hbar
v_{\textrm{p}}}{4\pi\nu}\int_{-\infty}^{\infty}dx:(\partial_{x}\bar{\phi}_{i}(x))^{2}+H_{\textrm{bulk}}+H_{\textrm{tun}}.$
(9)

Here, $Q_{\textrm{bulk}}=\nu BA_{\textrm{area}}/\Phi_{0}+\nu N_{L}-\bar{q}$ is
the number of the net charges localized within the interferometer bulk (inside
the interference loop), $A_{\textrm{area}}$ is the area of the interferometer,
$N_{L}$ is the net number of quasiparticles minus quasiholes, and $\bar{q}$ is
the number of positive background charges induced by the gate voltage applied
to the interferometer. $U_{\textrm{int}}$ is the strength of Coulomb
interaction between the charges of the interferometer edge and the charges
localized in the interferometer bulk, and $U_{\textrm{bulk}}$ is the strength
of interaction between the bulk charges. In the second equality of
Supplementary Equation 9, we introduce a boson field $\bar{\phi}_{i}$ for each
edge $i$, $\bar{\phi}_{i}(x)=\phi_{i}(x)+\frac{2\pi\nu}{\hbar
v_{\textrm{p}}}U_{\textrm{int}}Q_{\textrm{bulk}}\int_{-\infty}^{x}K_{i}(x^{\prime})dx^{\prime}$,
where $K_{2}(x)=1$ for $d<x<d+L$, $K_{3}(x)=1$ for $0<x<L$, and $K_{i}(x)=0$
otherwise. The second term of $\bar{\phi}$ describes the charges
$-\frac{2\pi\nu}{\hbar v_{\textrm{p}}}U_{\textrm{int}}Q_{\textrm{bulk}}$
induced per unit length by the interaction. In Supplementary Equation 9, the
Hamiltonian is quadratic in $\bar{\phi}_{i}$ and has
$H_{\textrm{bulk}}=(U_{\textrm{bulk}}-\frac{2\pi\nu L}{\hbar
v_{\textrm{p}}}U_{\textrm{int}}^{2})Q_{\textrm{bulk}}^{2}$. Note that
$U_{\textrm{bulk}}-\frac{2\pi\nu L}{\hbar
v_{\textrm{p}}}U_{\textrm{int}}^{2}>0$. The Keldysh Green’s function of
$\bar{\phi}_{i}$ is identical to that of $\phi_{i}$ in Supplementary Equation
2.

Repeating the calculation for the case of
$U_{\textrm{int}}=U_{\textrm{bulk}}=0$ and $N_{L}=0$, we compute the main
interference signal $I^{\textrm{int}}_{\textrm{D}_{3}}$ and the reference
signal $I^{\textrm{int}}_{\textrm{Ref, D}_{3}}$. The reference signal is
written as

$\displaystyle\langle I^{\textrm{int}}_{\textrm{Ref,
D}_{3}}\rangle\propto\textrm{Re}\left[\frac{\sum_{N_{L}=-\infty}^{\infty}e^{-\beta
H_{\textrm{bulk}}}\exp(-i2\pi(\nu BA_{\textrm{area}}/\Phi_{0}+\nu
N_{L})+i2\pi\frac{\nu U_{\textrm{int}}2L}{\hbar
v_{\textrm{p}}}Q_{\textrm{bulk}})}{\sum_{N_{L}=-\infty}^{\infty}e^{-\beta
H_{\textrm{bulk}}}}\right].$ (10)

Here, $\exp(-i2\pi\nu N_{L})$ counts the phase $2\pi\nu N_{L}$ by the braiding
of an interfering anyon around the anyons of the interferometer bulk, and
$\sum_{N_{L}}\cdots$ is for the ensemble average over thermal fluctuations of
$N_{L}$. Substituting $Q_{\textrm{bulk}}=\nu BA_{\textrm{area}}/\Phi_{0}+\nu
N_{L}-\bar{q}$, we obtain

$\displaystyle\langle I^{\textrm{int}}_{\textrm{Ref, D}_{3}}\rangle\propto$
$\displaystyle e^{\frac{-\pi^{2}\left(1-\frac{2\nu L}{\hbar
v_{\textrm{p}}}U_{\textrm{int}}\right)^{2}}{\beta(U_{\textrm{bulk}}-\frac{2\pi\nu
L}{\hbar
v_{\textrm{p}}}U_{\textrm{int}}^{2})}}\frac{\textrm{Re}\left[e^{-i2\pi\bar{q}}\Theta_{3}(\frac{\pi
BA_{\textrm{area}}}{\Phi_{0}}-\frac{\pi\bar{q}}{\nu}+\frac{i\pi^{2}\left(1-\frac{2\nu
L}{\hbar
v_{\textrm{p}}}U_{\textrm{int}}\right)}{\beta\nu\left(U_{\textrm{bulk}}-\frac{2\pi\nu
L}{\hbar
v_{\textrm{p}}}U_{\textrm{int}}^{2}\right)},\exp(\frac{-\pi^{2}\beta^{-1}\nu^{-2}}{(U_{\textrm{bulk}}-\frac{2\pi\nu
L}{\hbar v_{\textrm{p}}}U_{\textrm{int}}^{2})}))\right]}{\Theta_{3}(\frac{\pi
BA_{\textrm{area}}}{\Phi_{0}}-\frac{\pi\bar{q}}{\nu},\exp(\frac{-\pi^{2}\beta^{-1}\nu^{-2}}{(U_{\textrm{bulk}}-\frac{2\pi\nu
L}{\hbar v_{\textrm{p}}}U_{\textrm{int}})}))}.$ (11)

$\Theta_{3}(z,q)=1+2\sum_{n=1}^{\infty}q^{n^{2}}\cos(2nz)$ is the Jacobi theta
function and obeys $\Theta_{3}(z,q)=\Theta_{3}(z+\pi,q)$. In the Coulomb
dominated regime of $\hbar v_{\textrm{p}}/(2\nu L)\simeq U_{\textrm{int}}$,
$I^{\textrm{int}}_{\textrm{Ref, D}_{3}}$ is independent of the magnetic field
$B$, as $\Theta_{3}$’s in the numerator and the denominator cancel each other.
In the pure Aharonov-Bohm regime, $I^{\textrm{int}}_{\textrm{Ref, D}_{3}}$ is
a periodic function of $B$ with periodicity $\Phi_{0}/A_{\textrm{area}}$.

On the other hand, $I^{\textrm{int}}_{\textrm{Ref, D}_{3}}$ shows periodic
oscillations as a function of the gate voltage in both the regimes, since
$\bar{q}$ and $A_{\textrm{area}}$ depend on the gate voltage. To see this, we
write $I^{\textrm{int}}_{\textrm{Ref, D}_{3}}$ as

$\displaystyle\langle I^{\textrm{int}}_{\textrm{Ref,
D}_{3}}\rangle\propto\frac{\textrm{Re}\left[\Theta_{3}(\left(\pi\bar{\alpha}-\frac{\pi\bar{\gamma}}{\nu}\right)V_{\textrm{G}}+\frac{i\pi^{2}\left(1-\frac{2\nu
L}{\hbar
v_{\textrm{p}}}U_{\textrm{int}}\right)}{\beta\nu\left(U_{\textrm{bulk}}-\frac{2\pi\nu
L}{\hbar
v_{\textrm{p}}}U_{\textrm{int}}^{2}\right)},\exp(\frac{-\pi^{2}\beta^{-1}\nu^{-2}}{(U_{\textrm{bulk}}-\frac{2\pi\nu
L}{\hbar
v_{\textrm{p}}}U_{\textrm{int}}^{2})}))e^{-i2\pi\bar{\gamma}V_{\textrm{G}}}\right]}{\Theta_{3}(\left(\pi\bar{\alpha}-\frac{\pi\bar{\gamma}}{\nu}\right)V_{\textrm{G}},\exp(-\frac{\pi^{2}\beta^{-1}\nu^{-2}}{(U_{\textrm{bulk}}-\frac{2\pi\nu
L}{\hbar v_{\textrm{p}}}U_{\textrm{int}})}))},$ (12)

where $\bar{\gamma}\equiv\frac{d\bar{q}}{dV_{\textrm{G}}}$ and
$\bar{\alpha}\equiv\frac{B}{\Phi_{0}}\frac{dA_{\textrm{area}}}{dV_{\textrm{G}}}$.
The oscillation period is $1/\bar{\gamma}$ ($=V_{\textrm{G},0}$) in the
Coulomb dominated regime of $U_{\textrm{int}}\simeq\hbar v_{\textrm{p}}/(2\nu
L)$ and also in the pure Aharonov-Bohm regime of $U_{\textrm{int}}\simeq 0$
and $\beta U_{\textrm{bulk}}\ll 1$. All these results are in good agreement
with the previous result in Halperin .

And we obtain the main interference signal
$I^{\textrm{int}}_{\textrm{D}_{3}}$, combining Supplementary Equation 3 and
the ensemble average over thermal fluctuations of $N_{L}$ which is considered
in Supplementary Equation 10,

$\displaystyle\langle I^{\textrm{int}}_{\textrm{D}_{3}}\rangle$
$\displaystyle\propto\frac{\textrm{Re}\left[e^{-i2\pi\bar{q}}e^{-i\theta}\Theta_{3}(\frac{\pi
BA_{\textrm{area}}}{\Phi_{0}}-\frac{\pi\bar{q}}{\nu}+\frac{i\pi^{2}\left(1-\frac{2\nu
L}{\hbar
v_{\textrm{p}}}U_{\textrm{int}}\right)}{\beta\nu\left(U_{\textrm{bulk}}-\frac{2\pi\nu
L}{\hbar
v_{\textrm{p}}}U_{\textrm{int}}^{2}\right)},\exp(\frac{-\pi^{2}\beta^{-1}\nu^{-2}}{(U_{\textrm{bulk}}-\frac{2\pi\nu
L}{\hbar
v_{\textrm{p}}}U_{\textrm{int}}^{2})}))\right]}{\Theta_{3}(\pi\frac{\pi
BA_{\textrm{area}}}{\Phi_{0}}-\frac{\pi\bar{q}}{\nu},\exp(-\frac{\pi^{2}\beta^{-1}\nu^{-2}}{(U_{\textrm{bulk}}-\frac{2\pi\nu
L}{\hbar v_{\textrm{p}}}U_{\textrm{int}})}))},$ (13)

Notice that Supplementary Equation 13 is identical to Supplementary Equation.
11 except $e^{-i\theta}$. We find that the phase shift $\theta$ in
Supplementary Equation 13 satisfies the expression in Supplementary Equation
4, hence, follows Equation (6) (see the main text) in the semiclassical regime
of $e^{*}V\gg k_{B}T\gtrsim\hbar v_{\textrm{p}}/L$. Supplementary Equations
11, 12, and 13 are used for drawing Fig. 3.

The phase $\theta$ results from the topological vacuum bubbles. $\theta$ is
detectable both in the Coulomb dominated regime and the pure Aharonov-Bohm
regime. In the Coulomb dominated regime of $U_{\textrm{int}}\simeq\hbar
v_{\textrm{p}}/(2\nu L)$,
$I^{\textrm{int}}_{\textrm{D}_{3}}\propto\textrm{Re}[e^{-i2\pi\bar{\gamma}V_{\textrm{G}}}e^{-i\theta}]$
and $I^{\textrm{int}}_{\textrm{Ref,
D}_{3}}\propto\textrm{Re}[e^{-i2\pi\bar{\gamma}V_{\textrm{G}}}]$ are
independent of the magnetic field. In this case, the interference pattern of
$I^{\textrm{int}}_{\textrm{D}_{3}}$ is shifted by $\theta$ from that of
$I^{\textrm{int}}_{\textrm{Ref, D}_{3}}$ as the gate voltage $V_{\textrm{G}}$
varies. On the other hand, in the pure Aharonov-Bohm regime of
$U_{\textrm{int}}\simeq 0$ and when $\beta U_{\textrm{bulk}}\ll 1$,
$I^{\textrm{int}}_{\textrm{D}_{3}}\propto\textrm{Re}[e^{-i\theta}e^{-i(2\pi
BA_{\textrm{area}}/\Phi_{0}-2\pi\bar{q}/\nu)}]$ and
$I^{\textrm{int}}_{\textrm{Ref, D}_{3}}\propto\textrm{Re}[e^{-i(2\pi
BA_{\textrm{area}}/\Phi_{0}-2\pi\bar{q}/\nu)}]$. Note that the period of the
magnetic-field dependence is determined by $\Phi_{0}\equiv h/e$, rather than
the anyon flux quantum $\Phi_{0}^{*}=h/e^{*}$, because the change of $N_{L}$
(the net number of anyonic quasiparticles minus anyonic quasiholes localized
inside the interferometry loop) is allowed in the model of Supplementary
Equation 9; see Ref. Chamon ; Kivelson . In this case, the interference
pattern of $I^{\textrm{int}}_{\textrm{D}_{3}}$ is shifted by $\theta$ from
that of $I^{\textrm{int}}_{\textrm{Ref, D}_{3}}$, as either the magnetic field
or the gate voltage varies.

The Coulomb interaction parameters chosen for the left panel of Fig. 3a are
$U_{\textrm{bulk}}=10\hbar v_{\textrm{p}}/(2\nu L)$ and
$U_{\textrm{int}}=0.1\hbar v_{\textrm{p}}/(2\nu L)$, while
$U_{\textrm{bulk}}=10\hbar v_{\textrm{p}}/(2\nu L)$ and
$U_{\textrm{int}}=0.9\hbar v_{\textrm{p}}/(2\nu L)$ for the right panel.

Next, we consider a more general quantum Hall regime of filling factor
$\nu^{\prime}=\nu+\nu_{0}$ where $\nu=1/(2n+1)$ and $\nu_{0}$ is a nonzero
integer, focusing on the situation that the edge channels from the integer
filling $\nu_{0}$ are fully transmitted through the QPCs while those from the
fractional filling $\nu$ participate in the Fabry-Perot interference as in Fig
2. As shown below, the presence of the edge channels from the integer filling
$\nu_{0}$ does not modify our main finding. Namely, in this case the
interference-pattern phase shift between $I^{\textrm{int}}_{\textrm{Ref,
D}_{3}}$ and $I^{\textrm{int}}_{\textrm{D}_{3}}$ is identical to the phase
$\theta$ of the $\nu_{0}=0$ case discussed in Equation (6) of the main text.

The Hamiltonian of this more general situation is the same as that in
Supplementary Equation 9 but with the replacement of $Q_{\textrm{bulk}}=\nu
BA_{\textrm{area}}/\Phi_{0}+\nu N_{L}-\bar{q}$ into
$Q_{\textrm{bulk}}=\nu^{\prime}BA_{\textrm{area}}/\Phi_{0}+\nu N_{L}-\bar{q}$.
The replacement $\nu\to\nu^{\prime}=\nu+\nu_{0}$ counts the fact that the
total amount of electron changes occupying the Landau levels of the
interferometer bulk region is now $\nu^{\prime}BA_{\textrm{area}}/\Phi_{0}$.
In the same way as before (see Supplementary Equation 10), we obtain the
reference signal

$\displaystyle\langle I^{\textrm{int}}_{\textrm{Ref, D}_{3}}\rangle\propto$
$\displaystyle e^{\frac{-\pi^{2}\left(1-\frac{2\nu L}{\hbar
v_{\textrm{p}}}U_{\textrm{int}}\right)^{2}}{\beta(U_{\textrm{bulk}}-\frac{2\pi\nu
L}{\hbar
v_{\textrm{p}}}U_{\textrm{int}}^{2})}}\frac{\textrm{Re}\left[e^{i2\pi(\frac{\nu_{0}BA_{\textrm{area}}}{\Phi_{0}}-\bar{q})}\Theta_{3}(\frac{\nu^{\prime}\pi
BA_{\textrm{area}}}{\nu\Phi_{0}}-\frac{\pi\bar{q}}{\nu}+\frac{i\pi^{2}\left(1-\frac{2\nu
L}{\hbar
v_{\textrm{p}}}U_{\textrm{int}}\right)}{\beta\nu\left(U_{\textrm{bulk}}-\frac{2\pi\nu
L}{\hbar
v_{\textrm{p}}}U_{\textrm{int}}^{2}\right)},\exp(\frac{-\pi^{2}\beta^{-1}\nu^{-2}}{(U_{\textrm{bulk}}-\frac{2\pi\nu
L}{\hbar
v_{\textrm{p}}}U_{\textrm{int}}^{2})}))\right]}{\Theta_{3}(\frac{\nu^{\prime}\pi
BA_{\textrm{area}}}{\nu\Phi_{0}}-\frac{\pi\bar{q}}{\nu},\exp(\frac{-\pi^{2}\beta^{-1}\nu^{-2}}{(U_{\textrm{bulk}}-\frac{2\pi\nu
L}{\hbar v_{\textrm{p}}}U_{\textrm{int}})}))}.$ (14)

We first discuss the magnetic-field dependence of the reference signal. In the
Coulomb dominated regime of $U_{\textrm{int}}\simeq\hbar v_{\textrm{p}}/(2\nu
L)$, the reference signal becomes $I^{\textrm{int}}_{\textrm{Ref,
D}_{3}}\propto\textrm{Re}[e^{2\pi
i(\nu_{0}BA_{\textrm{area}}/\Phi_{0}-\bar{q})}]$, showing magnetic-field
dependent oscillations with the period determined by $\nu_{0}$. This result
coincides with an expression obtained with a capacitive interaction model in
Ref. Halperin ; cf. Eqs. (24) and (25) of Ref. Halperin . In the pure
Aharonov-Bohm regime of $U_{\textrm{int}}\simeq 0$ and when $\beta
U_{\textrm{bulk}}\ll 1$, the reference signal becomes
$I^{\textrm{int}}_{\textrm{Ref, D}_{3}}\propto\textrm{Re}[e^{2\pi
i(\nu_{0}BA_{\textrm{area}}/\Phi_{0}-\bar{q})}(1+\frac{1}{2}\exp(\frac{\pi^{2}(2\nu-1)}{\beta\nu^{2}U_{\textrm{bulk}}})e^{-2\pi
i(\nu^{\prime}BA_{\textrm{area}}/(\nu\Phi_{0})-\bar{q}/\nu)})]$. This also
agrees with Ref. Halperin ; cf. the two dominant terms of Eq. (24) in Ref.
Halperin . For the integer quantum Hall regime of
$(\nu^{\prime},\nu_{0})=(2,1)$, this expression becomes
$I^{\textrm{int}}_{\textrm{Ref, D}_{3}}\propto\textrm{Re}[e^{-2\pi
iBA_{\textrm{area}}/\Phi_{0}}]$, which agrees with the experimental data in
Ref. Ofek  for Aharonov-Bohm oscillations in the Fabry-Perot interferometry in
the bulk filling factor $\nu^{\prime}=2$ and with one fully transmitting
channel ($\nu_{0}=1$). Next, we discuss the gate-voltage dependence of the
reference signal. The gate-voltage dependent part of the reference signal
becomes $I^{\textrm{int}}_{\textrm{Ref,
D}_{3}}\propto\textrm{Re}[e^{-i2\pi(\bar{\gamma}-\nu_{0}\bar{\alpha})V_{\textrm{G}}}]$
in the Coulomb dominated regime of $U_{\textrm{int}}\simeq\hbar
v_{\textrm{p}}/(2\nu L)$, and $I^{\textrm{int}}_{\textrm{Ref,
D}_{3}}\propto\textrm{Re}[e^{-2\pi
i(\bar{\gamma}-\nu_{0}\bar{\alpha})V_{\textrm{G}}}(1+\frac{1}{2}\exp(\frac{\pi^{2}(2\nu-1)}{\beta\nu^{2}U_{\textrm{bulk}}})e^{-2\pi
i(\nu^{\prime}\bar{\alpha}V_{\textrm{G}}/\nu-\bar{\gamma}V_{\textrm{G}}/\nu)})]$
in the pure Aharonov-Bohm regime of $U_{\textrm{int}}\simeq 0$ and $\beta
U_{\textrm{bulk}}\ll 1$. The gate-voltage dependences are again in good
agreement with Ref. Halperin .

We also obtain the main interference signal
$I^{\textrm{int}}_{\textrm{D}_{3}}$ for the general case of
$\nu^{\prime}=\nu+\nu_{0}$,

$\displaystyle\langle I^{\textrm{int}}_{\textrm{D}_{3}}\rangle$
$\displaystyle\propto\frac{\textrm{Re}\left[e^{i2\pi(\frac{\nu_{0}BA_{\textrm{area}}}{\Phi_{0}}-\bar{q})}e^{-i\theta}\Theta_{3}(\frac{\nu^{\prime}\pi
BA_{\textrm{area}}}{\nu\Phi_{0}}-\frac{\pi\bar{q}}{\nu}+\frac{i\pi^{2}\left(1-\frac{2\nu
L}{\hbar
v_{\textrm{p}}}U_{\textrm{int}}\right)}{\beta\nu\left(U_{\textrm{bulk}}-\frac{2\pi\nu
L}{\hbar
v_{\textrm{p}}}U_{\textrm{int}}^{2}\right)},\exp(\frac{-\pi^{2}\beta^{-1}\nu^{-2}}{(U_{\textrm{bulk}}-\frac{2\pi\nu
L}{\hbar
v_{\textrm{p}}}U_{\textrm{int}}^{2})}))\right]}{\Theta_{3}(\frac{\nu^{\prime}\pi
BA_{\textrm{area}}}{\nu\Phi_{0}}-\frac{\pi\bar{q}}{\nu},\exp(\frac{-\pi^{2}\beta^{-1}\nu^{-2}}{(U_{\textrm{bulk}}-\frac{2\pi\nu
L}{\hbar v_{\textrm{p}}}U_{\textrm{int}})}))}.$ (15)

Again as in the previous case of $\nu_{0}=0$, Supplementary Equation 15 is
identical to Supplementary Equation 14 except $e^{-i\theta}$, and the phase
shift $\theta$ satisfies the expression in Eq. (6) (see the main text) in the
semiclassical regime of $e^{*}V\gg k_{B}T\gtrsim\hbar v_{\textrm{p}}/L$.
Therefore, $\theta$ is detectable also in the general case of
$\nu^{\prime}=\nu+\nu_{0}$ from the phase shift between the reference signal
and the main signal.

### Supplementary Note 6 : Discussion about Eqs. (3) and (4).

We discuss Equations (3) and (4) of the main text, focusing on the physical
interpretation of Type II which involves topological vacuum bubbles. We repeat
the expression for Type II in Eq. (3),

$\displaystyle I_{\textrm{D}_{3}}^{\textrm{II}}$ $\displaystyle\propto$
$\displaystyle\gamma_{1}^{2}\gamma_{2}\gamma_{3}(e^{*}V)^{2\nu-1}(k_{B}T)^{2\nu-1}\frac{2L+CL_{T}}{\hbar
v_{\textrm{p}}}e^{-2L/L_{T}}\sin^{2}\pi\nu\cos(2\pi\Phi/\Phi_{0}^{*}+\pi\nu),$

The factors $(e^{*}V)^{2\nu}$ and $(k_{B}T)^{2\nu}$ arise from the tunneling
of an anyon biased by $e^{*}V$ and that of a thermally excited anyon,
respectively. Other factors $(e^{*}V)^{-1}$, $(k_{B}T)^{-1}$, and
$e^{-2L/L_{T}}$ describe the reduction of wave packet overlap, hence the
reduction of interference amplitude. In particular the last one,
$e^{-2L/L_{T}}$, is the well-known dephasing factor. The factor $2L+CL_{T}$
comes from the phase space for anyon braiding of Type II. In the factor of
$\sin^{2}\pi\nu$, one power, $\sin\pi\nu$, originates from the braiding phase
$2\pi\nu$ and the vacuum bubble effect (see Supplementary Figure 2 and the
next paragraph), while the other $\sin\pi\nu$ originates from the exchange
between two thermally excited anyons. The other expressions of Types I-1, I-2,
and III in Eq. (4) can be understood in a similar way.

We below explain how the braiding phase $2\pi\nu$ effectively occurs in Type
II process.

  * •

The two interfering paths of a Type II process are shown in Supplementary
Figures 2a and b.

  * •

A gauge describing the exchange-statistics phase of anyons between different
edges (Edges 2 and 3) is chosen for tunneling operators at QPC2 and QPC3, as
shown in Equation (12) of the main text. We choose the gauge in parallel to
the extended edge scheme in Fig. 4 of the main text.

  * •

When the particle-like “virtual” anyon thermally excited on Edge 2 at QPC2
jumps to Edge 3 (see Supplementary Figure 2a), it gains the exchange-
statistics phase factor $\zeta_{a}=e^{-i\pi\nu}$ as its location is exchanged
with the voltage-biased “real” anyon injected from Edge 1; see Supplementary
Figure 2c. In the same gauge, the particle-like “virtual” anyon thermally
excited on Edge 2 at QPC3 in Supplementary Figure 2b gains no exchange-
statistics phase factor, when it jumps to Edge 3.

  * •

After the jump, the spatial ordering of anyons on Edge 2 is different between
Supplementary Figures 2a and b. For the paths 2a and b to interfere, the
anyons (the particle-like real anyon and the hole-like virtual anyon) for
example in Supplementary Figure 2b must be exchanged as in Supplementary
Figure 2d, for them to have the same spatial ordering as Supplementary Figure
2a. This results in another exchange-statistics phase factor of
$\zeta_{b}=e^{i\pi\nu}$ shown in Supplementary Figure 2d.

  * •

The two factors lead to the winding phase factor
$\zeta_{a}^{*}\zeta_{b}=e^{i2\pi\nu}$ in the interference signal.

  * •

The phase factors $\zeta_{a}$ and $\zeta_{b}$ are derived from the commutation
rules in Eqs. (11) and (12).

We further discuss Eqs. (3) and (4). The phase shift $e^{*}VL/\hbar
v_{\textrm{p}}$ in
$I_{\textrm{D}_{3}}^{\textrm{I-1}}\propto\cos(2\pi\Phi/\Phi_{0}^{*}+e^{*}VL/\hbar
v_{\textrm{p}})$ originates from the average of dynamical phase $2kL$ over the
energy window of $e^{*}V$, where $\hbar k$ is anyon momentum.
$I_{\textrm{D}_{3}}^{\textrm{I-2}}$ is proportional to
$\cos(2\pi\Phi/\Phi_{0}^{*})$ without the shift $\propto e^{*}V$. It is
because Type I-2 is the interference between the anyon of energy $e^{*}V$
injected from S1 and a thermally excited anyon so that the energy window for
the average of the dynamical phase is determined by
$\textrm{min}\\{e^{*}V,k_{B}T\\}=k_{B}T$.
$I_{\textrm{D}_{3}}^{\textrm{II}}\propto\cos(2\pi\Phi/\Phi_{0}^{*}+\pi\nu)$
and
$I_{\textrm{D}_{3}}^{\textrm{III}}\propto\cos(2\pi\Phi/\Phi_{0}^{*}-\pi\nu)$
do not have the phase shift $\propto e^{*}V$, because the interference occurs
by thermally excited anyons.

We point out that the phase shift $\theta$ is well described by Eq. (6) (see
the main text) in the regime of $e^{*}V\gg k_{B}T,\hbar v_{\textrm{p}}/L$
(beyond $e^{*}V\gg k_{B}T\gg\hbar v_{\textrm{p}}/L$). It is because Types II
and III share the common physics except only for the phase space ($2L+CL_{T}$
and $CL_{T}$, respectively) for anyon braiding.

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