# Anyon statistics through conductance measurements of time-domain
interferometry

Noam Schiller1 Yotam Shapira2 Ady Stern1 Yuval Oreg1 1Department of Condensed
Matter Physics
2Department of Physics of Complex Systems
Weizmann Institute of Science, Rehovot 7610001, Israel

###### Abstract

We propose a method to extract the mutual exchange statistics of the anyonic
excitations of a general Abelian fractional quantum Hall state, by comparing
the tunneling characteristics of a quantum point contact in two different
experimental conditions. In the first, the tunneling current between two edges
at different chemical potentials is measured. In the second, one of these
edges is strongly diluted by an earlier point contact. We describe the case of
the dilute beam in terms of a time-domain interferometer between the anyons
flowing along the edge and quasiparticle-quasihole excitations created at the
tunneling quantum point contact. In both cases, temperature is kept large,
such that the measured current is given to linear response. Remarkably, our
proposal does not require the measurement of current correlations, and allows
us to carefully separate effects of the fractional charge and statistics from
effects of intra- and inter-edge interactions.

Introduction.— It has been almost four decades since the initial proposal that
the elementary quasiparticles of fractional quantum Hall (FQH) systems obey
anyonic statistics [1]. Despite the apparent maturity of the field, the
pursuit to definitively observe the physical quantities and quantum numbers
characterizing anyons [2, 3] is constantly being reinvigorated [4, 5, 6, 7, 8,
9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20]. In particular, early 2020 saw
two major experimental steps forward: the observation of anyonic braiding in a
Fabry-Perot interferometer [21], and demonstration of a so-called “anyon
collider” [22, 23] using cross-correlation measurements.

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Figure 1: (a) Two counter-propagating edge modes ($u/d$) of a fractional
quantum Hall droplet at filling factor $\nu$ are connected by a quantum point
contact, through which quasiparticles of charge $e^{*}_{1}$ and scaling
dimension $\delta_{1}$ can tunnel. Current is measured at the lower edge’s
drain, denoted by $I$. A current of $I_{\mathrm{inj}}$ is injected into the
upper edge via a second, injection QPC, e.g. from a third auxiliary edge mode
($a$). The injection QPC is placed at a bias voltage of $V$, and allows
tunneling of quasiparticles of charge $e^{*}_{2}$ and scaling dimension
$\delta_{2}$. All other sources and drains are grounded. (b) The ratio between
$I/I_{\mathrm{inj}}$ in the dilute case and $I/I_{\mathrm{inj}}$ in the full
case, as a function of temperature, for $\nu=e^{*}_{1}/e=e^{*}_{2}/e=1/3$, and
for different scaling dimensions $\delta_{1}$. For the dilute case, we
$I_{\mathrm{inj}}=10\mathrm{pA}$, and assume $k_{B}T\ll eV$ for all relevant
temperatures, such that the contribution from $G_{\mathrm{direct}}$ to Eq. (1)
is negligible. In the full case, we use $V=10\upmu V$. Both cases use
$\xi=72\textrm{mK},\tau_{c}=10^{-13}\mathrm{s}$. When the dilute case
satisfies $\hbar I_{\mathrm{inj}}/e\ll k_{B}T\ll eV\ll\hbar/\tau_{c}$, and the
full case satisfies $\hbar I_{\mathrm{inj}}/e=\nu eV/2\pi\ll
k_{B}T\ll\hbar/\tau_{c}$, the ratio approaches an asymptote that does not
depend on scaling dimension, allowing extraction of the mutual statistics
$\theta_{12}$. Inset: $I/I_{\mathrm{inj}}$ for the dilute and full cases as a
function of temperature for $\delta_{1}=1/6$, the canonical value for a
Laughlin $1/3$ state.

Here we show that anyonic statistics can be inferred directly from conductance
measurements, without requiring current correlation measurements or explicitly
building an interferometer. The configuration we propose to obtain this result
consists of a quantum point contact (QPC) between two edges of a general
Abelian FQH state which are driven out of equilibrium. The edges may be driven
off-equilibrium by one of three methods: injecting a single quasiparticle into
one of the edges; injecting a Poissonian, dilute beam of quasiparticles into
one of the edges; and placing a finite bias voltage between the edges.

Our proposed setup, shown in Fig. 1(a), allows a smooth transition between the
dilute Poissonian beam and a full beam at finite bias voltage. This is
obtained by tuning a second, injection QPC from fully open (a differential
conductance, $G_{\mathrm{inj}}\equiv dI_{\rm inj}/dV$, satisfying
$G_{\mathrm{inj}}/\sigma_{xy}\rightarrow 0$) to fully closed
($G_{\mathrm{inj}}/\sigma_{xy}\rightarrow 1$). We henceforth refer to these as
the dilute and full limits, respectively.

We propose sweeping $G_{\mathrm{inj}}$ through this range, and measuring the
ratio $I/I_{\text{inj}}$, where $I$ is the measured current after the
tunneling QPC, and $I_{\text{inj}}$ is the injected incident current, as
defined in Fig. 1(a). Comparing the values at the dilute and full limits
cancels out non-universal constants, yielding the relation,

$\left[\frac{I(T)}{I_{\rm
inj}(T)}\right]_{\mathrm{dilute}}\\!\\!\\!\\!\\!\\!\\!=\frac{\nu e^{2}}{2\pi
e^{*}_{1}e^{*}_{2}}\sin{2\theta_{12}}\left[\frac{I(T)}{I_{\rm
inj}(T)}\right]_{\mathrm{full}}\\!\\!\\!\\!\\!+\frac{G_{\mathrm{direct}}}{G_{\mathrm{inj}}}.$
(1)

Here, $e^{*}_{1}$ is the tunneling quasiparticle charge, $e^{*}_{2}$ the
injected quasiparticle charge, $\delta_{1}$ is the tunneling quasiparticle
scaling dimension, $\theta_{12}$ is the mutual statistics phase between the
injected and tunneling quasiparticles, $T$ is temperature, and
$G_{\mathrm{direct}}$ is a residual conductance corresponding to direct
tunneling [24, 25, 26] through both QPCs. The full crossover between these two
limits is shown schematically in Fig. 1(b).

The mechanism leading to this result is a time-domain interferometer at the
tunneling QPC which is created by the dilute incident beam. The interference
is between two processes, in which a quasiparticle-quasihole excitation occurs
at the tunneling QPC either before or after the arrival of an injected
quasiparticle (see Fig. 2). A similar physical picture has been shown in Refs.
[27, 28, 25]. We further find that this interference is sensitive to the
mutual statistics phase between the injected and the tunneling quasiparticles,
$\theta_{12}$. We emphasize that these quasiparticles are not necessarily of
the same type, although they must be supported by the same FQH liquid.

Since our focus is the interference of two amplitudes which differ from one
another by the orderings of events, the key point of our analysis is the
identification of the phase differences between the two orderings. We find
phase differences that are determined by the quasiparticle charge $e^{*}$,
which is a fraction of the electron charge for non-integer values of $\nu$ [4,
5, 6]; the scaling dimension $\delta$, which defines the zero-temperature time
correlations of the quasiparticle via the relation
$\langle\psi^{\dagger}(\tau)\psi(0)\rangle\sim\tau^{-2\delta}$ [29, 30, 31,
32]; and the exchange statistics phase $\theta$, which for anyons take special
values beyond the fermionic $\pi$ and the bosonic $2\pi$ [1, 2, 3].

We are interested here in isolating the effect of $\theta$ from the other two
effects. In particular, we would like to separate it from the effect of
$\delta$. For non-interacting edges, in which all the modes propagate in the
same direction, $2\pi\delta=\theta$; however, in general $\delta$ is affected
by non-universal factors, such as intra-edge and inter-edge interactions,
$1/f$ noise or neutral modes [33, 34, 35, 36, 37, 38]. This in stark contrast
to the charge, exchange statistics phase, or filling factor, which are
universal.

We separate the effect of $\theta$ from that of $\delta$ by tuning the system
to a regime where $\delta$ only affects observables through a non-universal
prefactor, which then cancels out in the ratio of currents given in Eq. (1).
We arrive at this regime by employing a careful ordering of the various energy
scales in the system, such that $\hbar I_{\mathrm{inj}}/e\ll k_{B}T$
throughout the entire crossover of $G_{\mathrm{inj}}$. This ensures that the
current $I$ is given to linear response in $I_{\mathrm{inj}}$. We present an
analytic expression generalizing Eq. (1) outside of this regime in App A, Eq.
(20).

While in the full limit the edge that enters the tunneling QPC is in
equilibrium at chemical potential $V$, at the dilute limit we need the
injection QPC to reflect only a small fraction of the impinging electrons,
such that the resulting injection current is Poissonian and rare. Said
differently, the injected current in this limit must satisfy
$I_{\mathrm{inj}}\ll\sigma_{xy}V$. Furthermore, the beam must still be dilute
when arriving at the tunneling QPC. As such, the distance between the two QPCs
must be sufficiently small that no equilibration or dephasing occurs along the
way. Finally, we assume that tuning the injection QPC does not affect the
transparency of the tunneling QPC, to ensure that all non-universal constants
are cancelled when examining the ratio of the two limits. 111In practice this
may be difficult to implement experimentally, as the tuning of both QPCs is
most easily done through gating. This obstacle may be overcome by performing
the dilute beam measurements as shown in Fig. 1(a), while performing the full
beam measurements by biasing the lower edge through source $S_{d}$, and
measuring the tunneling current to the upper edge at drain $D_{u}$.

Easy extraction of $\theta_{12}$ requires $G_{\mathrm{direct}}$ to be sub-
dominant (see Eq. (1)). Quantitatively, this is the case if both $k_{B}T\ll
eV$ and $4\delta_{1}<2$ are satisfied. These constraints result from the
direct tunneling process being dominated by short time scales. Naive theories
describing quasiparticles may satisfy this condition even if the
aforementioned non-universal effects change the scaling dimension quite
significantly. For example, theory gives $\delta=1/2m$ for Laughlin
quasiparticles.

Edge theory.— We now define the system’s Hamiltonian and derive the current.
As shown by Wen, the edge theory of a general Abelian FQH state can be
described by $n$-boson fields,
$\bm{\phi}(x,t)\equiv\left(\phi_{1},\phi_{2},\cdots\phi_{n}\right)^{T}$ [2].
These define the theory in conjunction with a charge vector, $\bm{q}$, which
determines the electric charge carried by each boson field, and the so called
$K$-matrix, which determines the commutation relations between the boson
fields,

$\left[\phi_{i}(x),\partial_{x^{\prime}}\phi_{j}(x^{\prime})\right]=i2\pi(K^{-1})_{ij}\delta(x-x^{\prime}).$
(2)

The filling factor is then given by $\nu=\bm{q}^{T}K^{-1}\bm{q}$, and the
charge density is given by
$\rho=-\frac{1}{2\pi}\bm{q}\cdot\partial_{x}\bm{\phi}$. In terms of these
fields, the Hamiltonian of a single FQH edge mode is given by

$\mathcal{H_{\rm edge}}=\frac{1}{4\pi}\sum_{i,j=1}^{n}\int
dx\partial_{x}\phi_{i}V_{ij}\partial_{x}\phi_{j},$ (3)

where $\hat{V}$ is a positive definite matrix describing the velocities of the
modes and intra-edge interactions. These edges support quasiparticles of the
form $\psi_{\bm{l}}\sim e^{i\bm{l}\cdot\bm{\phi}}$, where $\bm{l}$ is a vector
of integers. The charge of these quasiparticles is then given by
$e^{*}_{\bm{l}}=\bm{q}^{T}K^{-1}\bm{l}$.

The configuration of Fig. 1(a) involves two edges, $u$ and $d$, tunnel-coupled
by a QPC. This is described by two copies of the Hamiltonian $\mathcal{H_{\rm
edge}}$, time reversed with regard to one another, as well as a tunneling
term, $\mathcal{H}_{T}$, which we treat as a perturbation. Assuming only one
type of quasiparticle, denoted by the vector $\bm{l}_{1}$ and carrying charge
$e^{*}_{1}$, tunnels between the edges, this is given by

$\mathcal{H}_{T}=\xi\left[\hat{A}+\hat{A}^{\dagger}\right];\;\hat{A}(t)\equiv
e^{i\left(\bm{l}_{1}\cdot\bm{\phi}^{(u)}(0,t)-\bm{l}_{1}\cdot\bm{\phi}^{(d)}(0,t)\right)}.$
(4)

Here, $\xi$ is a small tunneling amplitude, which we assume to be real, and
$\bm{\phi}^{(u)}$ ($\bm{\phi}^{(d)}$) are the bosonic field operators on the
upper (lower) edge. We project the auxiliary edge $a$ out of the Hamiltonian,
as it is only used to “initialize” the state of the edge $u$.

The current that tunnels from the upper edge to the lower edge is then given
by the operator, $\hat{I}_{T}(t)=i\xi
e^{*}_{1}\left[\hat{A}^{\dagger}(t)-\hat{A}(t)\right]$. Since the lower edge
is grounded, we henceforth identify $I=\langle\hat{I}_{T}\rangle$. Expanding
to leading order in $\xi$, the current is given by

$I(t)=e^{*}_{1}\xi^{2}\int_{-\infty}^{t}dt^{\prime}\left<\left[\hat{A}^{\dagger}(t),\hat{A}(t^{\prime})\right]+\left[\hat{A}^{\dagger}(t^{\prime}),\hat{A}(t)\right]\right>.$
(5)

Here, $[\cdot,\cdot]$ denotes commutation, and expectation values are
calculated with respect to the Hamiltonian in the absence of tunneling.

Deviation from Equilibrium.— It is clear from Eq. (5) that one needs to derive
correlation functions such as
$\langle\hat{A}^{\dagger}(t)\hat{A}(t^{\prime})\rangle$. In equilibrium, at
temperature $T$, the system is particle-hole symmetric, and the correlation
functions are given by [2, 40]

$\displaystyle\langle\hat{A}^{\dagger}(t)\hat{A}(t^{\prime})\rangle_{0}$
$\displaystyle=\langle\hat{A}(t)\hat{A}^{\dagger}(t^{\prime})\rangle_{0}$ (6)
$\displaystyle=\left[\frac{\pi T\tau_{c}}{\sinh\left(\pi
T\left|t-t^{\prime}\right|\right)}\right]^{4\delta_{1}}e^{-i2\pi\delta_{1}\textrm{sgn}\left(t-t^{\prime}\right)},$

where $\delta_{1}$ is the scaling dimension of the quasiparticle $\bm{l}_{1}$,
and $\tau_{c}>0$ is a short time cutoff.

Two main features are carried over from Eq. (6) to the correlation functions
out of equilibrium - the exponential decay at time difference larger than
$\hbar/T$, and the phase $e^{2\pi i\delta_{1}}$ associated with an interchange
of the time arguments.

We now consider two non-equibrium cases. In the first we introduce a constant
bias voltage $V\equiv V_{u}-V_{d}$ between the edges. In the setup of Fig.
1(a), this corresponds to a fully closed injection QPC, i.e.
$I_{\mathrm{inj}}=\sigma_{xy}V$. The introduction of the voltages can be
formally absorbed into the boson fields by use of a simple gauge
transformation, which maps
$\bm{\phi}^{(u/d)}(x,t)\mapsto\bm{\phi}^{(u/d)}(x,t)+K^{-1}\bm{q}V_{u/d}\left(t\mp
x/v\right)/\hbar$. This accordingly modifies the correlation functions by a
phase factor

$\displaystyle\langle\hat{A}^{\dagger}(t)\hat{A}(t^{\prime})\rangle_{\mathrm{full}}$
$\displaystyle=\langle\hat{A}^{\dagger}(t)\hat{A}(t^{\prime})\rangle_{0}e^{i\frac{e^{*}_{1}V}{\hbar}(t-t^{\prime})},$
(7)
$\displaystyle\langle\hat{A}(t)\hat{A}^{\dagger}(t^{\prime})\rangle_{\mathrm{full}}$
$\displaystyle=\langle\hat{A}(t)\hat{A}^{\dagger}(t^{\prime})\rangle_{0}e^{-i\frac{e^{*}_{1}V}{\hbar}(t-t^{\prime})}.$

In the second non-equilibrium driving, we consider injecting a single
quasiparticle, denoted by the vector $\bm{l}_{2}$, into the upper edge at the
location $x_{\mathrm{inj}}<0$ and at time $t_{\mathrm{inj}}$. This is shown
schematically in Fig. 2(a). In view of the commutation relations (2), the
application of the quasiparticle creation operator
$e^{-i\bm{l}_{2}\cdot{\bm{\phi}^{(u)}(x_{\mathrm{inj}},t_{\mathrm{inj}})}}$ on
the edge creates a soliton in each of the boson fields,

$\bm{\phi}^{(u)}(x,t_{\mathrm{inj}})\mapsto\bm{\phi}^{(u)}(x,t_{\mathrm{inj}})-2\pi
K^{-1}\bm{l}_{2}\Theta\left(x-x_{\mathrm{inj}}\right).$ (8)

We assume here the injection happens instantaneously. This assumption will be
relaxed to find the subleading term of Eq. (1).

The fields at general times can then be obtained using the equations of motion
dictated by the Hamiltonian in Eq. (3). If all modes are chiral with the same
velocity $v$, this amounts to replacing $x-x_{\mathrm{inj}}\rightarrow
x-x_{\mathrm{inj}}-v\left(t-t_{\mathrm{inj}}\right)$. The soliton thus arrives
at the QPC, $x=0$, at time $t_{0}\equiv t_{\mathrm{inj}}-x_{\mathrm{inj}}/v$.

The $c$-number shift in the bosonic field of Eq. (8) leads to a phase shift in
the correlator Eq. (6). We see directly from the definition of the operator
$\hat{A}$ in Eq. (4) that

$\displaystyle\langle\hat{A}^{\dagger}(t)\hat{A}(t^{\prime})\rangle_{\mathrm{qp}}$
$\displaystyle=\langle\hat{A}^{\dagger}(t)\hat{A}(t^{\prime})\rangle_{0}e^{2\pi
i\bm{l}_{1}K^{-1}\bm{l}_{2}\left[\Theta\left(t-t_{0}\right)-\Theta\left(t^{\prime}-t_{0}\right)\right]},$
(9)
$\displaystyle\langle\hat{A}(t)\hat{A}^{\dagger}(t^{\prime})\rangle_{\mathrm{qp}}$
$\displaystyle=\langle\hat{A}(t)\hat{A}^{\dagger}(t^{\prime})\rangle_{0}e^{-2\pi
i\bm{l}_{1}K^{-1}\bm{l}_{2}\left[\Theta\left(t-t_{0}\right)-\Theta\left(t^{\prime}-t_{0}\right)\right]}.$

The phase we obtain is the standard definition of mutual braiding statistics
between two quasiparticles, $\theta_{12}\equiv\pi\bm{l}_{1}K^{-1}\bm{l}_{2}$
[2]. The expression in Eq. (9) shows that the product gains a phase of
$e^{2i\theta_{12}\mathrm{sgn}\left(t-t^{\prime}\right)}$ if the arrival time
$t_{0}$ is between the times $t^{\prime}$ and $t$, and a trivial phase of 1
otherwise. We emphasize how naturally this result came from the underlying
theory: the only assumptions necessary to obtain this are the commutation
relations, (2), and the existence of quasiparticles in the edge’s excitation
spectrum.

This result holds for different boson modes with different velocities if all
solitons arrive at the tunneling QPC more or less concurrently, avoiding
dephasing. This is the case if $|x_{\mathrm{inj}}|/\Delta v\ll\hbar/T$, where
$\Delta v$ is the velocity difference between the fastest and the slowest
modes.

Time-domain interferometry.— The appearance of the phase, $\theta_{12}$, can
be understood as time-domain interferometry of the two distinct $\pm
e^{*}_{1}$ quasiparticle-quasihole excitations, before and after the injected
$e^{*}_{2}$ quasiparticle arrives at the QPC. A similar physical picture has
been shown in Ref. [27, 28, 25].

To show this we consider the configuration of a single injected particle, as
described in Fig. 2(a). In this case the non-equilibrium correlation function
takes the form,

$\langle\hat{A}^{\dagger}(t)\hat{A}(t^{\prime})\rangle_{\mathrm{qp}}=\langle\psi_{\bm{l}_{2}}(t_{0})\hat{A}^{\dagger}(t)\hat{A}(t^{\prime})\psi^{\dagger}_{\bm{l}_{2}}(t_{0})\rangle_{0},$
(10)

i.e., the expectation value is calculated with respect to the state resulting
from exciting the ground state $\ket{0}$ with a single quasiparticle. Here we
omit the position variable from the quasiparticle injection operator
$\psi^{\dagger}_{\bm{l}_{2}}(t_{0})$, and assume it arrives at the tunneling
QPC $x=0$ at time $t_{0}$.

The current in Eq. (5) is then given by integration over multiple terms of the
form in Eq. (10). We define
$\Ket{t,t_{0}}_{-}\equiv\hat{A}(t)\psi^{\dagger}_{\bm{l}_{2}}(t_{0})\ket{0}$
and
$\Ket{t,t_{0}}_{+}\equiv\hat{A}^{\dagger}(t)\psi^{\dagger}_{\bm{l}_{2}}(t_{0})\ket{0}$.
Eq. (5) can now be re-written as

$I\propto-\int_{-\infty}^{t}dt^{\prime}\sum_{b=\pm}b\big{|}\ket{t,t_{0}}_{b}+\ket{t^{\prime},t_{0}}_{b}\big{|}^{2}.$
(11)

The expression above involves two interference terms. The term with $b=-$ is
an interference between creation of $-e^{*}_{1}$ quasiholes on the upper edge
at the QPC at times $t$ and $t^{\prime}$. The two interfering processes are
shown schematically in Fig. 2(b). As shown in the first row of Eq. (9), these
two processes are distinguished by a non-trivial phase of $e^{i2\theta_{12}}$
if the arrival time $t_{0}$ is in between the quasiholes’ creation times,
$t^{\prime}<t_{0}<t$. Combined with the equilibrium correlation function Eq.
(6), one finds that this interference gives a term proportional to
$\cos\left(2\theta_{12}-2\pi\delta\right)$. Using similar arguments, the term
with $b=+$ in Eq. (11), gives an interference term proportional to
$\cos\left(2\theta_{12}+2\pi\delta\right)$. The total contribution from the
two terms in Eq. (11) is thus proportional to
$\sin\left(2\theta_{12}\right)\sin\left(2\pi\delta\right)$ 222This would
appear to give zero for $\delta=1/2$; however, this cancels out with a
divergence resulting from integration over the coordinate $t^{\prime}$..

This interference happens entirely in the time domain, and along only one
edge. It is however crucial that this edge be part of a two-dimensional bulk.
This is important both because the second edge is required to absorb the
leftover quasiparticle or quasihole resulting from the pair creation at the
QPC, and because the injected quasiparticle must be created within a bulk FQH
droplet. Furthermore, the bulk is intimately related to the edge through bulk-
edge correspondence. This dictates that the statistical phase contributing to
time-domain interference along a single edge, which our setup measures, is the
same as the phase obtained from spatial exchange.

Figure 2: Time-domain interferometry. (a) I A quasiparticle is injected from
the sourced, left edge, through the injection QPC, and into the upper edge. II
The injected quasiparticle, by virtue of its chiral motion along the edge,
arrives at the tunneling QPC. III A quasiparticle-quasihole pair is created at
the tunneling QPC. (b) The two processes by which charge carriers may
ultimately arrive at the drain. The injected quasiparticle arrives at the
tunneling QPC either before (upper panel) or after (lower panel) the creation
quasiparticle-quasihole pair. These two processes interfere, with a relative
phase dictated by the mutual statistics phase, $e^{i2\theta_{12}}$.

It is easy to generalize this to injection of multiple quasiparticles: as long
as all injected quasiparticles are mutually independent, each injected
quasiparticle contributes a phase of $e^{2i\theta_{12}}$ if and only if the
arrival time at the point contact was between $t^{\prime}$ and $t$. If we
assume this is a Poissonian process, with a quasiparticle injection rate of
$I_{\mathrm{inj}}/e^{*}_{2}$, we obtain for $t>0$

$\displaystyle\frac{\langle\hat{A}^{\dagger}(t)\hat{A}(0)\rangle_{\mathrm{dilute}}}{\langle\hat{A}^{\dagger}(t)\hat{A}(0)\rangle_{0}}$
$\displaystyle=\sum_{n=0}^{\infty}\frac{(tI_{\mathrm{inj}}/e_{2}^{*})^{n}e^{-tI_{\mathrm{inj}}/e_{2}^{*}}}{n!}e^{2in\theta_{12}}$
(12)
$\displaystyle=e^{-tI_{\mathrm{inj}}/e_{2}^{*}\left(1-e^{2i\theta_{12}}\right)}.$

This is precisely the result given in Refs. [23, 25] for injection along a
single edge. Adding injected quasiparticles to the lower edge and generalizing
for $t<0$ are straightforward using the same arguments.

Currents.— The effect of driving the system out of equilibrium is completely
encapsulated in the correlation functions obtained above. These can then be
used to derive any observable of interest, such as charge or heat currents in
any of the system’s drains, or their respective auto- and cross-correlations.

For concreteness, we present the explicit results of such a calculation for
the charge current at the lower drain, denoted as $I$ in Fig. 1. We show that
a simple cohort of current measurements is sufficient to obtain the mutual
statistics $\theta_{12}$, without requiring correlation measurements.

We focus on the regime where the temperature is large compared to the injected
current $\hbar I_{\mathrm{inj}}/ek_{B}T$. For the full limit, this assumption
guarantees linear response in the voltage and in the injected current, which
in this limit is $I_{\mathrm{inj}}=\sigma_{xy}V$. For the dilute limit, the
exponential suppression of the equilibrium correlation function at times
larger than $\hbar/T$, guarantees that the exponent in Eq. (12) may be
expanded to first order in $I_{\mathrm{inj}}$. Consequently,

$\frac{\langle\hat{A}^{\dagger}(t)\hat{A}(t^{\prime})\rangle_{\mathrm{full/dilute}}}{\langle\hat{A}^{\dagger}(t)\hat{A}(t^{\prime})\rangle_{0}}\approx
1+i\omega_{\mathrm{f/d}}\left(t-t^{\prime}\right),$ (13)

where the frequencies $\omega_{\mathrm{f/d}}$ are given by

$\omega_{\mathrm{f}}=\frac{e^{*}_{1}V}{\hbar}=\frac{e^{*}_{1}}{\hbar}\frac{I_{\mathrm{inj}}}{\sigma_{xy}};\quad\omega_{\mathrm{d}}=i\frac{I_{\mathrm{inj}}}{e^{*}_{2}}\left(1-e^{2i\theta_{12}}\right).$
(14)

The zeroth order term corresponds to the equilibrium state and does not
contribute to the current. The ratio of the two first order contributions is
Eq. (1).

Explicit calculation of the resulting current in Eq. (5), given in App. A,
finds that

$I_{\mathrm{full/dilute}}=2\pi e^{*}_{1}(\xi\tau_{c})^{2}(2\pi
T\tau_{c})^{4\delta_{1}-2}\mathcal{B}\left(2\delta_{1},2\delta_{1}\right)\mathrm{Re}\left[\omega_{\mathrm{f/d}}\right],$
(15)

where $\mathcal{B}(x,y)$ is the Euler Beta function. It is thus immediately
apparent that by focusing on the ratio between the full and dilute beams, all
dependence on $\delta_{1},T$ and $\xi$ drops out. Examining the ratio
$I/I_{\mathrm{inj}}$, and noting that $\sigma_{xy}\hbar/e^{*}_{1}e^{*}_{2}=\nu
e^{2}/2\pi e^{*}_{1}e^{*}_{2}$ we thus obtain Eq. (1).

For general temperatures, the current can no longer be treated as a linear
response to the drive of the full or dilute beams. We hence obtain the typical
power laws characterizing tunneling in Luttinger liquids [42, 34, 43, 2].
Comparing measurements of the full and dilute limits at the low temperature
limit $T\ll e^{*}V,I_{\mathrm{inj}}$ can still give a quantity related to the
mutual statistics $\theta_{12}$, but will explicitly depend on the value of
$\delta_{1}$. We present general expressions for the current in this case in
App. A.

For a fermionic $\theta_{12}=\pi$, Eq. (15) gives no current at all for a
dilute electron beam. However, Landauer-Buttiker-Imry scattering theory [44]
tells us the current is given by the product of the transparencies of the two
QPCs along the electron’s path, regardless of whether they are close to full
transmission or full reflection. This requires accounting for the direct
tunneling term in Eq. (1), which now becomes the leading contribution.

We do this by accounting for the finite width of the soliton. This leads to
the required, Landauer-Buttiker-Imry consistent result of
$I_{\mathrm{dilute}}=4\pi^{2}\tau_{c}^{2}\xi^{2}I_{\mathrm{inj}}$. The
physical intuition behind the requirement of a finite soliton width is that
tunneling without time-domain interferometry, dubbed the direct tunneling
process in [25, 24], is dominated by short times. Performing these
calculations explicitly in App. B, we show that the ratio between the first
term in Eq. (1) and $G_{\mathrm{direct}}$ is
$\propto\left(T\tau_{s}\right)^{4\delta_{1}-2}$, where $\tau_{s}$ is the
soliton width. It has been shown [25, 24] that
$\tau_{s}^{-1}\propto\max\\{eV,k_{B}T\\}$; as such, to ensure
$G_{\mathrm{direct}}$ is sub-dominant, the dilute limit must be measured when
$k_{B}T\ll eV$ and $4\delta_{1}<2$.

Several contemporary experimental setups use the equivalent of non-interacting
fermionic formulae to reasonable success [45], corresponding to the limiting
value of $2\delta_{1}=1$. In this case, the second term of Eq. (1) is a
numerical coefficient of order one, which may depend solely on
$e^{*},\delta_{1}$ and $\theta_{12}$. For non-interacting fermions, this
coefficient is easily found by comparing to known Landauer-Buttiker-Imry
scattering theory [44], but it is straightforward to generalize. We discuss
this coefficient further in App. B.

Discussion.— We propose a simple method to extract anyonic exchange
statistics. Our system consists only of a single quantum Hall droplet with two
QPCs, which effectively create a time-domain interferometer, as can be
identified from current measurements. We thus avoid both current correlation
(or noise) measurements, and the need for a real space interferometer, making
the identification of the exchange statistics much more accessible than
existing experiments. All time-domain interferometry is between pairs of an
injected quasiparticle and a tunneling quasiparticle, and occurs at the same
edge, as previously proposed in Ref. [25].

Both the exchange statistics $\theta_{11}$ of the tunneling quasiparticle, and
$\theta_{22}$ of the injection quasiparticle, do not appear in our derivation.
Rather, it is the two particles’ mutual statistics, $\theta_{12}$ that affect
the modified correlation functions, and hence, the physical observables.
Likewise, the scaling dimension and electric charge which directly effect
observables are only those of the tunneling quasiparticle, $\delta_{1}$ and
$e^{*}_{1}$ (properties of the injected quasiparticles may implicitly enter
through the injection rate).

Only in the case where the injected and tunneling quasiparticles are
identical, $\bm{l}_{1}=\bm{l}_{2}$, do we obtain exchange statistics for a
single quasiparticle type. We remark that this is indeed the case in the
experiment of Ref. [22], where all quasiparticles are Laughlin $e^{*}=e/3$
anyons, and subsequent recreations for the $\nu=1/3$ and $\nu=2/5$ cases [26,
46, 47]. Interestingly, a recent experiment employing a similar setup, where
the injected quasiparticle was a $e/3$ anyon and the tunneling quasiparticle
was an electron, observed Andreev-like reflection [48]. This is consistent
with a mutual statistics phase of $\theta_{12}=\pi$, for which Eq. (1) gives
no time-domain interferometry signal.

Acknowledgements.— We thank Tomer Alkalay, Moty Heiblum, Changki Hong, June-
Young Lee and H.-S. Sim for insightful discussions and comments on the
manuscript. This work was partially supported by grants from the ERC under the
European Union’s Horizon 2020 research and innovation programme (grant
agreements LEGOTOP No. 788715 and HQMAT No. 817799), the DFG (CRC/Transregio
183, EI 519/7-1), the BSF and NSF (2018643), the ISF Quantum Science and
Technology (2074/19). N.S. was supported by the Clore Scholars Programme.

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##  Appendix A Finite temperature current from time-domain interferometry

Here derive explicit expressions for the tunneling current $I$ at finite
temperature $T$. This section neglects the contribution $G_{\mathrm{direct}}$
(see Eq. (1), which is discussed in App. B. We begin with the expression for
the current in Eq. (5). Writing this explicitly,

$I=e^{*}_{1}\xi^{2}\int_{-\infty}^{t}dt^{\prime}\bigg{\\{}\left<\hat{A}^{\dagger}(t)\hat{A}(t^{\prime})\right>-\left<\hat{A}(t^{\prime})\hat{A}^{\dagger}(t)\right>+\left<\hat{A}^{\dagger}(t^{\prime})\hat{A}(t)\right>-\left<\hat{A}(t)\hat{A}^{\dagger}(t^{\prime})\right>\bigg{\\}}.$
(16)

In the case where the edges are not driven out of equilibrium, we plug the
equilibrium correlation functions Eq. (6), and obtain $I=0$, as expected. A
similar expression can be written for the symmetrized current fluctuations,

$\left<\Big{\\{}\delta\hat{I}_{T}(t),\delta\hat{I}_{T}(t^{\prime})\Big{\\}}\right>=(e^{*}_{1})^{2}\xi^{2}\bigg{\\{}\left<\hat{A}^{\dagger}(t)\hat{A}(t^{\prime})\right>+\left<\hat{A}(t^{\prime})\hat{A}^{\dagger}(t)\right>+\left<\hat{A}^{\dagger}(t^{\prime})\hat{A}(t)\right>+\left<\hat{A}(t)\hat{A}^{\dagger}(t^{\prime})\right>\bigg{\\}},$
(17)

where we define
$\delta\hat{I}_{T}\equiv\delta\hat{I}_{T}-\langle\delta\hat{I}_{T}\rangle$. We
do not focus on current fluctuations in this work, but note that our methods
reproduce the known results of Refs. [23, 25].

We now want to obtain the current for each of the three methods of driving the
two edges out of equilibrium. Each of these leads to a corresponding
multiplicative factor to the correlation functions. A finite bias voltage $V$,
used for the “full” beam, gives the correlation functions of Eq. (7);
injection of a single quasiparticle gives the correlation functions of Eq.
(9); and a dilute, Poissonian beam of quasiparticles gives the correlation
functions of Eq. (12). Plugging in these appropriate correlation functions
gives after minor algebra and changes of variables

$\displaystyle I_{\mathrm{full}}$
$\displaystyle=2ie^{*}_{1}\xi^{2}\int_{0}^{\infty}d\tilde{t}\sin\left(\frac{e^{*}_{1}V}{\hbar}\tilde{t}\right)\biggr{\\{}\left[\frac{\pi
T\tau_{c}}{i\sinh\left(\pi
T\left(\tilde{t}-i\tau_{c}\right)\right)}\right]^{4\delta_{1}}-\left[\frac{\pi
T\tau_{c}}{i\sinh\left(\pi
T\left(-\tilde{t}-i\tau_{c}\right)\right)}\right]^{4\delta_{1}}\biggr{\\}},$
(18a) $\displaystyle I_{\mathrm{dilute}}$
$\displaystyle=2ie^{*}_{1}\xi^{2}\int_{0}^{\infty}d\tilde{t}\frac{\sin\left(\frac{I_{\mathrm{inj}}}{e^{*}_{2}}\tilde{t}\sin{2\theta_{12}}\right)}{\exp\left(\frac{I_{\mathrm{inj}}}{e^{*}_{2}}\tilde{t}\left(1-\cos{2\theta_{12}}\right)\right)}\biggr{\\{}\left[\frac{\pi
T\tau_{c}}{i\sinh\left(\pi
T\left(\tilde{t}-i\tau_{c}\right)\right)}\right]^{4\delta_{1}}-\left[\frac{\pi
T\tau_{c}}{i\sinh\left(\pi
T\left(-\tilde{t}-i\tau_{c}\right)\right)}\right]^{4\delta_{1}}\biggr{\\}},$
(18b) $\displaystyle I_{\mathrm{qp}}$
$\displaystyle=2ie^{*}_{1}\xi^{2}\int_{-\infty}^{t}dt^{\prime}\sin\left(2\theta_{12}\left[\Theta\left(t-t_{0}\right)-\Theta\left(t^{\prime}-t_{0}\right)\right]\right)\biggr{\\{}\left[\frac{\pi
T\tau_{c}}{i\sinh\left(\pi
T\left(t-t^{\prime}-i\tau_{c}\right)\right)}\right]^{4\delta_{1}}-\left[\frac{\pi
T\tau_{c}}{i\sinh\left(\pi
T\left(t^{\prime}-t-i\tau_{c}\right)\right)}\right]^{4\delta_{1}}\biggr{\\}}.$
(18c)

We proceed using the identity, correct at the limit $\tau_{c}\rightarrow 0$,

$i\biggr{\\{}\left[\frac{\pi T\tau_{c}}{i\sinh\left(\pi
T\left(\tilde{t}-i\tau_{c}\right)\right)}\right]^{4\delta_{1}}-\left[\frac{\pi
T\tau_{c}}{i\sinh\left(\pi
T\left(-\tilde{t}-i\tau_{c}\right)\right)}\right]^{4\delta_{1}}\biggr{\\}}=\begin{cases}-2\pi\tau_{c}^{2}\partial_{\tilde{t}}\delta(\tilde{t})&2\delta_{1}=1\\\
\left[\frac{\pi T\tau_{c}}{\sinh\left(\pi
T|\tilde{t}|\right)}\right]^{4\delta_{1}}2\sin{(2\pi\delta_{1})}\mathrm{sgn}(\tilde{t})&2\delta_{1}\neq
1\end{cases},$ (19)

where $\delta(t)$ is the Dirac delta function. This identity is necessary to
treat the case of $\delta_{1}=1$, which otherwise may lead to divergent
integrals.

Standard manipulations of these integrals then give results in terms of the
Euler Beta function and the incomplete Beta function,
$\mathcal{B}\left(x;a,b\right)\equiv\int_{0}^{x}t^{a-1}(1-t)^{b-1}dt$,
$\mathcal{B}\left(a,b\right)\equiv\mathcal{B}\left(1;a,b\right)$. We thus
obtain the general results

$\displaystyle I_{\mathrm{full}}$ $\displaystyle=2e^{*}_{1}\xi^{2}(2\pi
T)^{4\delta_{1}-1}\tau_{c}^{4\delta_{1}}\sinh\left(\frac{e^{*}_{1}V}{2T}\right)\mathcal{B}\left(2\delta_{1}+i\frac{e^{*}_{1}V}{2\pi
T},2\delta_{1}-i\frac{e^{*}_{1}V}{2\pi T}\right)$ (20a) $\displaystyle
I_{\mathrm{dilute}}$
$\displaystyle=-\frac{2\pi}{\cos\left(2\pi\delta_{1}\right)\Gamma\left(4\delta_{1}\right)}e^{*}_{1}\xi^{2}(2\pi
T)^{4\delta_{1}-1}\tau_{c}^{4\delta_{1}}\mathrm{Im}\left[\frac{\Gamma\left(\frac{I_{\mathrm{inj}}}{e^{*}_{2}}\frac{1-\cos\left(2\theta_{12}\right)+i\sin\left(2\theta_{12}\right)}{2\pi
T}+2\delta_{1}\right)}{\Gamma\left(\frac{I_{\mathrm{inj}}}{e^{*}_{2}}\frac{1-\cos\left(2\theta_{12}\right)+i\sin\left(2\theta_{12}\right)}{2\pi
T}+1-2\delta_{1}\right)}\right]$ (20b) $\displaystyle I_{\mathrm{qp}}$
$\displaystyle=4e^{*}_{1}\xi^{2}(2\pi
T)^{4\delta_{1}-1}\tau_{c}^{4\delta_{1}}\sin\left(2\theta_{12}\right)\sin\left(2\pi\delta\right)\mathcal{B}\left(e^{-2\pi
T\left(t-t_{0}\right)};1+2\delta_{1},1-4\delta_{1}\right).$ (20c)

Here, $\Gamma(a)$ is the Euler Gamma function, satisfying
$\mathcal{B}(a,b)=\frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)}$, and
$\mathrm{Im}[\dots]$ denotes the imaginary part.

The high temperature and zero temperature limits of the full beam and dilute
beam are then immediately reproducible. For $e^{*}V,\hbar
I_{\mathrm{inj}}/e^{*}_{2}\ll k_{B}T$, one expands to leading order in
$e^{*}V/T$ and $I_{\mathrm{inj}}/e^{*}_{2}T$, one obtains

$\displaystyle I_{\mathrm{full}}$ $\displaystyle\approx 2\pi
e^{*}_{1}(\xi\tau_{c})^{2}(2\pi
T\tau_{c})^{4\delta_{1}-2}\mathcal{B}\left(2\delta_{1},2\delta_{1}\right)\frac{e^{*}_{1}V}{\hbar},$
(21a) $\displaystyle I_{\mathrm{dilute}}$ $\displaystyle\approx 2\pi
e^{*}_{1}(\xi\tau_{c})^{2}(2\pi
T\tau_{c})^{4\delta_{1}-2}\mathcal{B}\left(2\delta_{1},2\delta_{1}\right)\frac{I_{\mathrm{inj}}}{e^{*}_{2}}\sin\left(2\theta_{12}\right).$
(21b)

We thus see that the mutual statistics are immediately extractable from the
dilute case. While this expression does depend on the non-universal $\xi$ and
$\delta$, as well as the temperature, these are all encoded in a prefactor
which appears in the full case as well. We can hence lose this unwanted
prefactor by examining the ratio between the two cases.

For $T\ll e^{*}V,I_{\mathrm{inj}}/e^{*}_{2}$, we use the identities

$\displaystyle\lim_{x\rightarrow\infty}\Gamma\left(x+a\right)$
$\displaystyle=\Gamma\left(x\right)x^{a},$
$\displaystyle\lim_{x\rightarrow\infty}\sinh(\pi
x)\mathcal{B}\left(a+ix,a-ix\right)$
$\displaystyle=\frac{\pi}{\Gamma(2a)}x^{2a-1},$

to obtain

$\displaystyle I_{\mathrm{full}}$ $\displaystyle\approx\frac{2\pi
e^{*}_{1}\xi^{2}\tau_{c}^{4\delta_{1}}}{\Gamma\left(4\delta_{1}\right)}\left(\frac{e^{*}_{1}V}{\hbar}\right)^{4\delta_{1}-1},$
(22a) $\displaystyle I_{\mathrm{dilute}}$ $\displaystyle\approx-\frac{2\pi
e^{*}_{1}\xi^{2}\tau_{c}^{4\delta_{1}}}{\cos{\left(2\pi\delta_{1}\right)}\Gamma\left(4\delta_{1}\right)}\left(\frac{I_{\mathrm{inj}}}{e^{*}_{2}}\right)^{4\delta_{1}-1}\mathrm{Im}\bigg{[}\bigg{(}1-\cos{\left(2\theta_{12}\right)}+i\sin{\left(2\theta_{12}\right)}\bigg{)}^{4\delta_{1}-1}\bigg{]}.$
(22b)

By tuning $2\delta_{1}\rightarrow 1$, we again obtain an expression from which
the mutual statistics are easily extractable, with an identical non-universal
prefactor appearing in both the full and dilute cases. However, once the
scaling dimension is tuned to this critical value, the contribution from time-
domain interferometry no longer dominates the direct tunneling process, as can
be seen from the calculation of $G_{\mathrm{direct}}$ in App. B.

We note that for temperatures larger than the source voltage, one has to
account for injection of both quasiparticles and quasiholes through the
injection QPC. This can be done by modifying the Poissonian correlation
function in Eq. (12) according to

$\displaystyle\frac{\langle\hat{A}^{\dagger}(t)\hat{A}(0)\rangle_{\mathrm{dilute}}}{\langle\hat{A}^{\dagger}(t)\hat{A}(0)\rangle_{0}}$
$\displaystyle=e^{-tI_{\mathrm{inj}}/e_{2}^{*}\left(1-e^{2i\theta_{12}}\right)}$
(23) $\displaystyle\rightarrow
e^{-tI_{\mathrm{inj}}^{qp}/e_{2}^{*}\left(1-e^{2i\theta_{12}}\right)}e^{-tI_{\mathrm{inj}}^{qh}/e_{2}^{*}\left(1-e^{-2i\theta_{12}}\right)},$

where $I_{\mathrm{inj}}^{qp}$ is the injection rate of quasiparticles, and
$I_{\mathrm{inj}}^{qh}$ is the injection rate of quasiholes. This is a similar
expression to the three QPC setup considered in [23] and [25]. Performing the
same algebra as in this section, and identifying $I_{\mathrm{inj}}\equiv
I_{\mathrm{inj}}^{qp}-I_{\mathrm{inj}}^{qh}$, one then reproduces Eq. (21) for
the high temperature limit.

Finally, it is instructive to consider the current due to the injection of a
single quasiparticle at time $t_{0}$, which was obtained in Eq. (20c). In this
case we must examine the explicit temperature dependence, as tunneling of a
single quasiparticle may be relevant, and we lack any other energy scale to
serve as a cutoff for the RG flow of the process. This current exhibits a
power-law decay for $t-t_{0}\ll 1/\pi T$, consistent with the orthogonality
catastrophe that characterizes injection into Luttinger liquid edges. For
$2\delta_{1}=1$, this results in
$\langle\hat{I}_{T}\rangle_{\mathrm{qp}}\propto\delta\left(t-t_{0}\right)$.
This gives some intuition as to what makes the $2\delta_{1}=1$ case so unique
- the QPC just scatters the incident particle with some probability, without
inducing any long-time correlations, resulting in the direct tunneling
process.

##  Appendix B Finite soliton width: restoring Landauer-Buttiker-Imry for
electrons and subleading corrections

The results of App. A are seemingly inconsistent with the known non-
interacting electron limits. Indeed, inserting $e^{*}_{1}=e^{*}_{2}=e$,
$2\delta_{1}=2\delta_{2}=1$ and $\theta_{12}=\pi$ into these results would
indicate that the dilute electron beam gives no current at all. This is in
direct contrast with the intuition of Landauer-Buttiker-Imry scattering
theory, which would indicate that the current should be given by the product
of the transparencies of the two QPCs along the electron’s path, regardless of
whether they are close to full transmission or full reflection.

The culprit of this result is a peculiarity of soliton physics. The boson
field $\phi$ is compact under $\phi\mapsto\phi+2\pi$. As such, a soliton of
height $2\pi K^{-1}\bm{l}_{2}$ would appear to leave the boson field
completely unperturbed if $K^{-1}\bm{l}_{2}$ is an integer. This corresponds
precisely to electron injection operators [2]. As such, our soliton
description is ill-equipped to treat electrons without modifications.

We solve this issue by introducing a finite width to the soliton, $\tau_{s}$.
To fully recreate the known non-interacting result, it is crucial to maintain
an order of limits such that the soliton width is larger than the short-time
cutoff, $\tau_{c}$. We note that we still take care to ensure that
$\tau_{s}<1/T,(I_{\mathrm{inj}}/e^{*}_{2})^{-1}$, i.e. the solitons are still
narrow compared to the larger time scales in the problem. Previous works [25,
24], performing a full Keldysh calculation, have shown the soliton width
(refered to in the cited papers as the temporal width) is given by the
voltage, $h/e^{*}V$, if $eV>k_{B}T$, and by the inverse temperature
$\hbar/k_{B}T$ if $eV\lesssim k_{B}T$; as such, the dilute limit must be
measured in the regime $I_{\mathrm{inj}}/e^{*}_{2}\ll k_{B}T\ll eV$.

Formally, this means that injecting a quasiparticle into the upper edge at the
location $x_{0}$ and time $t_{0}$ transforms the boson field according to

$\bm{\phi}^{(u)}(x,t_{0})\mapsto\bm{\phi}^{(u)}(x,t_{0})-2\pi
K^{-1}\bm{l}_{2}\left(\frac{1}{\pi}\tan^{-1}\left(\frac{x-x_{0}}{\tau_{s}}\right)-\frac{1}{2}\right).$

Accordingly, the correlation functions of Eq. (9) are now replaced with

$\displaystyle\langle\hat{A}^{\dagger}(t)\hat{A}(t^{\prime})\rangle_{\mathrm{qp}}$
$\displaystyle=\langle\hat{A}^{\dagger}(t)\hat{A}(t^{\prime})\rangle_{0}\exp{\left(2i\frac{\theta_{12}}{\pi}\left[\tan^{-1}\left(\frac{t-t_{0}}{\tau_{s}}\right)-\tan^{-1}\left(\frac{t^{\prime}-t_{0}}{\tau_{s}}\right)\right]\right)},$
(24)
$\displaystyle\langle\hat{A}(t)\hat{A}^{\dagger}(t^{\prime})\rangle_{\mathrm{qp}}$
$\displaystyle=\langle\hat{A}(t)\hat{A}^{\dagger}(t^{\prime})\rangle_{0}\exp{\left(-2i\frac{\theta_{12}}{\pi}\left[\tan^{-1}\left(\frac{t-t_{0}}{\tau_{s}}\right)-\tan^{-1}\left(\frac{t^{\prime}-t_{0}}{\tau_{s}}\right)\right]\right)}.$

One indeed sees that at the limit $\tau_{c}\rightarrow 0$, one reproduces the
immediate soliton results from the main text.

To find the correlation function in the presence of a dilute, Poissonian beam
of injected quasiparticles, we now must sum over the number of injected
quasiparticles, in a manner similar to Eq. (12). However, this is now
trickier, for two reasons. First, the accumulated phase explicitly depends on
the time of the injected quasiparticle. Second, injected quasiparticles
outside of the window $[0,t]$ can still affect the correlation function, due
to the long tails of the finite-width solitons.

So generalizing the methods that lead to Eq. (12), the correlation function
now changes to define

$\displaystyle\frac{\langle\hat{A}^{\dagger}(t)\hat{A}(0)\rangle_{\mathrm{fw}}}{\langle\hat{A}^{\dagger}(t)\hat{A}(0)\rangle_{0}}$
$\displaystyle=\sum_{n}\frac{\left((t+2c\tau_{c})\frac{I_{\mathrm{inj}}}{e_{2}^{*}}\right)^{n}e^{-(t+2c\tau_{c})\frac{I_{\mathrm{inj}}}{e_{2}^{*}}}}{n!}\left[\int_{-c\tau_{c}}^{t+c\tau_{c}}dt_{0}P\left(\textrm{Particle
injected at
}t_{0}\right)e^{2i\frac{\theta_{12}}{\pi}\left[\tan^{-1}\left(\frac{t-t_{0}}{\tau_{s}}\right)-\tan^{-1}\left(\frac{0-t_{0}}{\tau_{s}}\right)\right]}\right]^{n}.$
(25)

Here $c$ is some unitless cutoff, chosen such that injected quasiparticles
affect the correlation function only if they are injected in the window
$[-c\tau_{c},t+c\tau_{c}]$, which we will eventually take to be infinite. The
probability of injection at a particular time $t_{0}$ is given by

$P\left(\textrm{Particle injected at
}t_{0}\right)=\frac{I_{\mathrm{inj}}/e^{*}_{2}e^{-I_{\mathrm{inj}}t_{0}/e^{*}_{2}}}{\int_{-c\tau_{c}}^{t+c\tau_{c}}dt_{0}I_{\mathrm{inj}}/e^{*}_{2}e^{-I_{\mathrm{inj}}t_{0}/e^{*}_{2}}}.$
(26)

Performing this sum, and re-defining this integration with unitless variables,
we find that the new correlation function is given in integral form by

$\displaystyle\frac{\langle\hat{A}^{\dagger}(t)\hat{A}(0)\rangle_{\mathrm{fw}}}{\langle\hat{A}^{\dagger}(t)\hat{A}(0)\rangle_{0}}$
$\displaystyle=\exp{\left(-\left(t+2c\tau_{c}\right)\frac{I_{\mathrm{inj}}}{e^{*}_{2}}\left[1-\mathcal{I}_{\theta_{12}}\left(\frac{I_{\mathrm{inj}}}{e^{*}_{2}}\tau_{s},\frac{t}{2\tau_{s}}\right)\right]\right)},$
(27) $\displaystyle\mathcal{I}_{\theta_{12}}\left(a,b\right)$
$\displaystyle\equiv\frac{a}{2\sinh\left(a(b+c)\right)}\int_{-b-c}^{b+c}dxe^{-ax}e^{2i\frac{\theta_{12}}{\pi}\left[\tan^{-1}\left(x+b\right)-\tan^{-1}\left(x-b\right)\right]}.$

By plugging this new correlation function into the expression for the current
in Eq. (5), one now finds

$I_{\mathrm{dilute}}=2ie^{*}_{1}\xi^{2}\int_{0}^{\infty}d\tilde{t}\frac{\sin\left((\tilde{t}+2c\tau_{c})\frac{I_{\mathrm{inj}}}{e^{*}_{2}}\mathrm{Im}\left[\mathcal{I}_{\theta_{12}}\left(\frac{I_{\mathrm{inj}}}{e^{*}_{2}}\tau_{s},\frac{t}{2\tau_{s}}\right)\right]\right)}{\exp\left((\tilde{t}+2c\tau_{c})\frac{I_{\mathrm{inj}}}{e^{*}_{2}}\mathrm{Re}\left[1-\mathcal{I}_{\theta_{12}}\left(\frac{I_{\mathrm{inj}}}{e^{*}_{2}}\tau_{s},\frac{t}{2\tau_{s}}\right)\right]\right)}\\\
\times\biggr{\\{}\left[\frac{\pi T\tau_{c}}{i\sinh\left(\pi
T\left(\tilde{t}-i\tau_{c}\right)\right)}\right]^{4\delta_{1}}-\left[\frac{\pi
T\tau_{c}}{i\sinh\left(\pi
T\left(-\tilde{t}-i\tau_{c}\right)\right)}\right]^{4\delta_{1}}\biggr{\\}}.$
(28)

Careful re-application of the limit $\tau_{c}\rightarrow 0$ indeed replicates
our previous result in Eq. (12).

For general $\theta_{12}$, the integral
$\mathcal{I}_{\theta_{12}}\left(a,b\right)$ is difficult to solve
analytically. In the main text, this is circumvented by taking the limit
$\tau_{c}\rightarrow 0$, allowing use of Eq. (19), in conjunction with
replacing
$(\tilde{t}+2c\tau_{c})\frac{I_{\mathrm{inj}}}{e^{*}_{2}}\mathcal{I}_{\theta_{12}}\left(\frac{I_{\mathrm{inj}}}{e^{*}_{2}}\tau_{s},\frac{t}{2\tau_{s}}\right)\rightarrow-i\tilde{t}\omega_{d}$.
However, as noted previously, fermionic exchange statistics corresponding to
values of $\theta_{12}$ that are integer multiples of $\pi$ lead to
$\omega_{d}=0$, and hence give a vanishing current. As such, Eq. (28) must be
calculated in full while retaining a finite $\tau_{c}$.

To simplify these expressions, we assume that
$\left(\frac{I_{\mathrm{inj}}}{e^{*}_{2}}\right)$ is significantly larger than
any other time scale in the system. This makes sense from a physical
perspective as well, as it corresponds to the assumption that injection is
sufficiently rare such that solitons do not overlap. In this case, one can
assume the probability of injection which appears in Eqs. (25),(26) is
approximately uniform, i.e. $P\left(\textrm{Particle injected at
}t_{0}\right)\approx 1/(t+2c\tau_{c})$. One can now safely take the limit
$c\rightarrow\infty$ without artificial divergences, giving the simpler
result,

$\frac{\langle\hat{A}^{\dagger}(t)\hat{A}(0)\rangle_{\mathrm{fw}}}{\langle\hat{A}^{\dagger}(t)\hat{A}(0)\rangle_{0}}=\exp{\left(\frac{I_{\mathrm{inj}}}{e^{*}_{2}}\int_{-\infty}^{\infty}dt_{0}\left[e^{2i\frac{\theta_{12}}{\pi}\left[\tan^{-1}\left(\frac{t-t_{0}}{\tau_{s}}\right)-\tan^{-1}\left(\frac{0-t_{0}}{\tau_{s}}\right)\right]}-1\right]\right)}.$
(29)

Since we undertook this endeavor with the explicit goal of finding the correct
result for non-interacting electrons, we wish to find this integral for
$2\delta_{1}=1$, $\theta_{12}=\pi$, and $e^{*}_{1}=e^{*}_{2}=e$. This value of
$\theta_{12}$ allows one to significantly simplify Eq. (29) using trignometric
identities; plugging the resulting correlation function in Eq. (16), we obtain

$I_{\mathrm{dilute}}^{\theta_{12}=\pi}=2ie^{*}_{1}\xi^{2}\int_{0}^{\infty}d\tilde{t}\frac{\sin\left(\frac{I_{\mathrm{inj}}}{e^{*}_{2}}\frac{2\pi\tilde{t}(2\tau_{s})^{2}}{\tilde{t}^{2}+(2\tau_{s})^{2}}\right)}{\exp\left(\frac{I_{\mathrm{inj}}}{e^{*}_{2}}\frac{2\pi\tilde{t}^{2}(2\tau_{s})}{\tilde{t}^{2}+(2\tau_{s})^{2}}\right)}\biggr{\\{}\left[\frac{\pi
T\tau_{c}}{i\sinh\left(\pi
T\left(\tilde{t}-i\tau_{c}\right)\right)}\right]^{4\delta_{1}}-\left[\frac{\pi
T\tau_{c}}{i\sinh\left(\pi
T\left(-\tilde{t}-i\tau_{c}\right)\right)}\right]^{4\delta_{1}}\biggr{\\}}.$
(30)

As can be seen in Eq. (19), the expression in the curled brackets is
approximately zero for $\tilde{t}>\tau_{c}$. We can thus approximate the total
integral as the contribution from short times, $\tilde{t}\leq\tau_{c}\ll 1/\pi
T$. To leading order, this will be given by

$\displaystyle I_{\mathrm{dilute}}^{\theta_{12}=\pi,2\delta_{1}=1}$
$\displaystyle\approx
2ie^{*}_{1}\xi^{2}\tau_{c}^{2}\int_{0}^{\infty}d\tilde{t}\frac{I_{\mathrm{inj}}}{e^{*}_{2}}\frac{2\pi\tilde{t}(2\tau_{s})^{2}}{\tilde{t}^{2}+(2\tau_{s})^{2}}\biggr{\\{}\left(\frac{1}{i\tilde{t}+\tau_{c}}\right)^{2}-\left(\frac{1}{-i\tilde{t}+\tau_{c}}\right)^{2}\biggr{\\}}$
(31)
$\displaystyle=\frac{(2\tau_{s})^{2}}{(2\tau_{s}+\tau_{c})^{2}}4\pi^{2}\xi^{2}\tau_{c}^{2}I_{\mathrm{inj}}.$

Now taking the limit $\tau_{c}\ll\tau_{s}$, we compare to the electron case
in, say, Eq. (15) or Eq. (20). We find that the result we expect for non-
interacting electrons is indeed $4\pi^{2}\xi^{2}\tau_{c}^{2}I_{\mathrm{inj}}$.
This is consistent with - the current is linear in the injected current, and
in the transparency of the tunneling QPC (which is given by
$\xi^{2}\tau_{c}^{2}$).

For general values of $\theta_{12}$ and $\delta_{1}$ this integral is more
difficult to solve analytically. However, it is possible to re-write Eq. (29)
as

$\displaystyle\frac{\langle\hat{A}^{\dagger}(t)\hat{A}(0)\rangle_{\mathrm{fw}}}{\langle\hat{A}^{\dagger}(t)\hat{A}(0)\rangle_{0}}$
$\displaystyle=\exp{\frac{I_{\mathrm{inj}}}{e^{*}_{2}}\bigg{(}\sin\left(2\theta_{12}\right)t+f_{\theta_{12}}\left(t,\tau_{c})\right)\bigg{)}},$
(32) $\displaystyle f_{\theta_{12}}\left(t,\tau_{c})\right)$
$\displaystyle\propto\begin{cases}t&t\lesssim\tau_{s}\\\
\tau_{s}&t\gg\tau_{s},\theta_{12}\neq\pi\\\
(\tau_{s})^{2}/t&t\gg\tau_{s},\theta_{12}=\pi.\end{cases}$ (33)

Plugging this into the general expression for the current, and expanding to
linear response in $\frac{I_{\mathrm{inj}}}{e^{*}_{2}}$ we find

$I_{\mathrm{dilute}}=2ie^{*}_{1}\xi^{2}\frac{I_{\mathrm{inj}}}{e^{*}_{2}}\int_{0}^{\infty}d\tilde{t}\bigg{(}\sin\left(2\theta_{12}\right)t+f_{\theta_{12}}\left(t,\tau_{c})\right)\bigg{)}\biggr{\\{}\left[\frac{\pi
Thw\tau_{c}}{i\sinh\left(\pi
T\left(\tilde{t}-i\tau_{c}\right)\right)}\right]^{4\delta_{1}}-\left[\frac{\pi
T\tau_{c}}{i\sinh\left(\pi
T\left(-\tilde{t}-i\tau_{c}\right)\right)}\right]^{4\delta_{1}}\biggr{\\}}.$
(34)

The term proportional to $\sin\left(2\theta_{12}\right)$, as discussed at
length above, is the main interest of this paper. This is calculated in Eq.
(21). We see there that the time scales in the system contribute a leading
term of the form
$\propto(\xi\tau_{c})^{2}(T\tau_{c})^{4\delta_{1}-2}I_{\mathrm{inj}}$.

The term proportional to $f_{\theta_{12}}\left(t,\tau_{c})\right)$ contains
several contributions: at short times ($\tilde{t}\sim\tau_{c}$), we obtain a
contribution of order $(\xi\tau_{c})^{2}$; at long times ($\tilde{t}\sim 1/\pi
T$) we obtain a contribution of order
$(\xi\tau_{c})^{2}(\tau_{s}/\tau_{c})\left(T\tau_{c}\right)^{4\delta_{1}-1}$
for $\theta_{12}\neq\pi$ and
$(\xi\tau_{c})^{2}(\tau_{s}/\tau_{c})\left(T\tau_{c}\right)^{4\delta_{1}}$ for
$\theta_{12}=\pi$; and at intermediate times ($\tilde{t}\sim\tau_{s}$) we
obtain contributions of order
$(\xi\tau_{c})^{2}(\tau_{s}/\tau_{c})^{1-4\delta_{1}}$ and
$(\xi\tau_{c})^{2}(\tau_{s}/\tau_{c})^{2-4\delta_{1}}$.

We compare these contributions to the coefficients of Eq. (15) or Eq. (21),
which give the time-domain interferometry process, which is of order
$(\xi\tau_{c})^{2}\left(T\tau_{c}\right)^{4\delta_{1}-2}$. Utilizing
$\tau_{c}\ll\tau_{s}\ll 1/\pi T$, we see that the long time contribution is
always subdominant, but the short time dominates for $2\delta_{1}\geq 1$ \-
consistent with both Eq. (31) and the known electron result. This is
consistent with our physical intuition: direct tunneling dominates short
times, which give the main contribution for $2\delta_{1}\geq 1$, whereas time-
domain interferometry dominates long times, which give the main contribution
for $2\delta_{1}<1$.

Finally, if we indeed assume $2\delta_{1}<1$, the intermediate time
contribution dominates the entire direct process. In this case, the ratio
between the time-domain interferometry process and the direct process is given
by $\propto(T\tau_{s})^{4\delta_{1}-2}$. This again confirms that we must have
a soliton width smaller than the inverse temperature to ensure time-domain
interferometry

This method is also what we use to calculate the current for an almost full
beam, i.e. $\sigma_{xy}-G_{\mathrm{inj}}\ll 1$. Since in this case, the beam
can be treated as a conjoined full beam of fractional quasiparticles with a
dilute beam of $e^{*}=e$ holes, we have $2\theta_{12}=2\pi n$ regardless of
the tunneling quasiparticles. Defining the injection rate of holes as
$I_{\mathrm{inj}}^{\mathrm{holes}}\equiv\sigma_{xy}V-I_{\mathrm{inj}}$, we
combine the full beam correlation function of Eq. (7) and the regularized
Poissonian hole injection to obtain described in this section

$I_{\mathrm{|G_{\mathrm{inj}}-\sigma_{xy}|\ll
1}}=2ie^{*}_{1}\xi^{2}\int_{0}^{\infty}d\tilde{t}\frac{\sin\left(\frac{e^{*}_{1}V}{\hbar}\tilde{t}-\frac{I_{\mathrm{inj}}^{\mathrm{holes}}}{e}\frac{2\pi\tilde{t}(2\tau_{s})^{2}}{\tilde{t}^{2}+(2\tau_{s})^{2}}\right)}{\exp\left(\frac{I_{\mathrm{inj}}^{\mathrm{holes}}}{e}\frac{2\pi\tilde{t}^{2}(2\tau_{s})}{\tilde{t}^{2}+(2\tau_{s})^{2}}\right)}\\\
\times\biggr{\\{}\left[\frac{\pi T\tau_{c}}{i\sinh\left(\pi
T\left(\tilde{t}-i\tau_{c}\right)\right)}\right]^{4\delta_{1}}-\left[\frac{\pi
T\tau_{c}}{i\sinh\left(\pi
T\left(-\tilde{t}-i\tau_{c}\right)\right)}\right]^{4\delta_{1}}\biggr{\\}}.$
(35)

In the relevant limits, the same methods as previously mention allow us to
approximate the exponent in the denominator as $1$, and to expand the sine in
the numerator. We thus have the sum of two linear responses, one in in
$\frac{e^{*}_{1}V}{\hbar}$ and one in
$-\frac{I_{\mathrm{inj}}^{\mathrm{holes}}}{e}$. Taking, as in the Landauer-
Buttiker-Imry case, the limit $\tau_{s}\gg\tau_{c}$, i.e. a soliton width that
is larger than the short time cutoff, this can be re-written as

$I_{\mathrm{|G_{\mathrm{inj}}-\sigma_{xy}|\ll 1}}\approx
2ie^{*}_{1}\xi^{2}\int_{0}^{\infty}d\tilde{t}\left(\frac{e^{*}_{1}V}{\hbar}-2\pi\frac{I_{\mathrm{inj}}^{\mathrm{holes}}}{e}\right)\tilde{t}\biggr{\\{}\left[\frac{\pi
T\tau_{c}}{i\sinh\left(\pi
T\left(\tilde{t}-i\tau_{c}\right)\right)}\right]^{4\delta_{1}}-\left[\frac{\pi
T\tau_{c}}{i\sinh\left(\pi
T\left(-\tilde{t}-i\tau_{c}\right)\right)}\right]^{4\delta_{1}}\biggr{\\}}.$
(36)

Identifying
$\left(\frac{e^{*}_{1}V}{\hbar}-2\pi\frac{I_{\mathrm{inj}}^{\mathrm{holes}}}{e}\right)=\frac{2\pi}{e}\left(\sigma_{xy}V-I_{\mathrm{inj}}^{\mathrm{holes}}\right)\equiv\frac{2\pi}{e}I_{\mathrm{inj}}$,
we see that this is precisely the same integral that we had in Eq. (18a) for
the full beam case, with the replacement $\sigma_{xy}V\rightarrow
I_{\mathrm{inj}}$. We note that we used here $\sigma_{xy}=ee^{*}/h$, which is
correct only for Laughlin edge states, $\nu=1/m$; this is valid as Laughlin
edges are the outer level of heirarchal FQH fluids, and thus are the states of
interest for nearly full closed QPCs.


11institutetext:  Data Science Group, Paderborn University
Warburgerstrasse 100, 33098 Paderborn
11email<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
# topFiberM: Scalable and Efficient Boolean Matrix Factorization

Abdelmoneim Amer Desouki 11 Michael Röder 11 Axel-Cyrille Ngonga Ngomo 11

###### Abstract

Matrix Factorization has many applications such as clustering. When the matrix
is Boolean it is favorable to have Boolean factors too. This will save the
efforts of quantizing the reconstructed data back, which usually is done using
arbitrary thresholds. Here we introduce topFiberM a Boolean matrix
factorization algorithm. topFiberM chooses in a greedy way the fibers (rows or
columns) to represent the entire matrix. Fibers are extended to rectangles
according to a threshold on precision. The search for these "top fibers" can
continue beyond the required rank and according to an optional parameter that
defines the limit for this search. A factor with a better gain replaces the
factor with minimum gain in "top fibers". We compared topFiberM to the state-
of-the-art methods, it achieved better quality for the set of datasets usually
used in literature. We also applied our algorithm to linked-data to show its
scalability. topFiberM was in average 128 times faster than the well known
Asso method when applied to a set of matrices representing a real multigraph
although Asso is implemented in C and topFiberM is implemented in R which is
generally slower than C. topFiberM is publicly available from Github
(https://github.com/dice-group/BMF).

###### Keywords:

Boolean Matrix Factorization BMF Asso

##  1 Introduction

Boolean Matrix Fatorization (BMF) for a Boolean Matrix $I(n\times m)$ tries to
find two Boolean matrices $A(n\times k)$ and $B(m\times k)$ such that
$I\approx(A\circ B)$ where $(A\circ B)_{ij}=max^{k}_{l=1}min(A_{il},B_{lj})$
and k is the rank of the factorization. There are two types of problems in
Boolean Matrix factorization. The Discrete Base Problem (DBP)[11] and
Approximate Factorization Problem [5]. In the first problem it is important to
find the best coverage over small rank values while in the second problem it
is required to find the minimum rank to get error less than a defined
threshold[2]

###  1.1 Related Work

We introduce some of BMF methods:

_Asso_ [11]: is the most famous BMF method. It constructs a square matrix
(i.e. Association matrix) with the same number of columns as the data matrix
and employs a three parameters to form the desired factor matrices in a greedy
way. It is implemented in C by the authors.

GreConD [5] uses the Formal Concept Analysis (FCA)[8] to compute exact
decompositions. The algorithm can be used to provide approximate DBP solutions
by stopping it after computing the first k factors. Also it can provide
approximate solution to AFP by stopping it after the committed error E does
not exceed $\epsilon$ [2].

GreConD+ [5] is an extension to GreConD that can backwardly change factors
when new better ones are discovered. The results demonstrate that it
outperforms other methods in terms of quality and robustness to small changes
in data but this came on a price of being much slower.

NaiveCol [6] is a simple algorithm which searches for factors in a greedy way
in the columns of the input matrix I. The algorithm may be used for both AFP
and DBP[2].

For methods Hyper [12], GreEss [3] and PaNDa [9], we refer to a recent review
of BMF methods in [2].

Metrics used to evaluate BMF methods mainly include coverage which include the
two types of errors and is defined as follows:

$c=1-\lVert I-(A\circ B)\rVert/\lVert I\rVert$ (1)

We introduce topFiberM a BMF method which can be seen as an extension to
NaiveCol by including not only columns but also rows and by extending the
coverage of factors. topFiberM also includes parameters like _Asso_ that
parameters can be used to trade between recall and precision of results.
Moreover fibers (rows/columns) are rectangles with one side equal to unity.
topFiberM keeps track of uncovered ones in I and can eliminate factors when
better ones are discovered. topFiber rely on matrix operations and not single
bit-operations which makes it very fast when compared to other methods.

##  2 topFiberM

The algorithm starts by initializing the factor search matrices As and Bs to
zeros, i to 1, input matrix $I$ is copied to X and SR parameter is set to not
exceed the minimum dimension. In each iteration the row sum and column sum is
calculated for the not yet covered ones in X. The fiber with the maximum
number of ones is considered. If it is a row ( value in mxr) then we try to
extend this by finding rows in X which are similar to mxr according to
parameter tP. The algorithm accepts first k fibers and stores information
about them in tf including index and gain. Note that when calculating the gain
the true positives are calculated depending on X (not yet covered) while the
false positives are calculated on the original matrix $I$. Afterwards the ones
that are covered in this iteration are set to zeros to exclude from further
processing. We used R function expand.grid to get the result of applying one
factor. This function finds the permutations of two vectors which are in this
case the indexes of ones in factors. The true positives will be the sum of
values in $I$ of the resulting set of indexes ( variable ix) and the false
positives will be $sum(!I[ix])$. When i is greater than k we compare the gain
to the minimum gain in tf (top fibers) if the gain is not more than the
minimum the fiber is marked as excluded. Otherwise we replace that fiber with
minimum gain in the tf list and recalculate the uncovered after considering
the new fiber. In each iteration the matrix X is checked if it is all zeros
then the algorithm terminates. Finally the A and B matrix are taken from the
As and Bs according to the indexes in tf. See Algorithm 1.

Data: I: $n\times m$ Boolean matrix, k: rank, tP: threshold on precision, SR:
search limit of factors default is k

Result: Boolean factors A: $n\times k$, B: $m\times k$ where $X\approx A\circ
B$

1

2SR = min(SR, m, n);

3 As = zeros(n, SR);

4 Bs = zeros(SR, m);

5 tf = array(struct(i, fiber type, fiber index, gain));

6 excluded_cols = rep(FALSE, m);

7 excluded_rows = rep(FALSE, n);

8 X = I;

9 i=1;

10 while  _( $i\leq SR$ )_ do

11 Find rowSums and columnSums of X;

12 Choose best Fiber (has maximum no of ones);

13 let best unexcluded fiber be a row with index mxr;

14 Bi = X[mxr, ] ;

15 rtp = rowSums(X[, Bi]);

16 //better to be calculated on original I rfp = rowSums(!I[, Bi]);

17 Ai = $ifelse(rtp=0,FALSE,((rtp/(rtp+rfp))\geq tP))$;

18 //revise B ctp = colSums(X[Ai,]);

19 cfp = colSums(!I[Ai,,drop=FALSE]);

20 Bi = ifelse(ctp=0,FALSE,((ctp/(ctp+cfp)) >=tP));

21 ix = as.matrix(expand.grid(which(Ai),which(Bi)),ncol=2);

22 gain = sum(X[ix]) - sum(!I[ix]);

23 Similarly when the best fiber is a column but with reverse role of Ai and
Bi.

24 As[,i] = Ai;

25 Bs[i,] = Bi;

26 if  _$i\leq k$_ then

27 X[ix] = FALSE;

28 add row to tf with i, fiber type ( 1 for a row and 2 for a column), mxr and
gain

29

30 else

31 if  _if gain $\leq$ min gain in tf_ then

32 mark fiber as excluded

33 else

34 replace row with min gain in tf with current fiber

35 X = uncovered ones in I

36

37 end if

38

39 end if

40 // test if all ones are covered

41 if  _sum(X) = 0_ then

42 break;

43

44 end if

45 i = $i+1$;

46

47 end while

48A = columns of As with index in tf;

49 B = rows of Bs with index in tf;

Algorithm 1 topFiberM: with backward correction

##  3 Evaluating topFiberM

###  3.1 Applying to literature datasets

We followed similar approach as in [2] to evaluate the performance of our
algorithm. The datasets used are Chess [1], DBLP [10], Firewall 1 [6],
Mushroom [1], and Paleo111NOW public release 030717, available from
http://www.helsinki.fi/science/now/[7]. The Boolean matrices was downloaded as
MATLAB mat files from http://datasets.inf.upol.cz/. The main properties of the
datasets are shown in Table 1.

Dataset |  Dimensions |  Number of 1s |  Density
---|---|---|---
Chess |  $3196\times 76$ |  118 252 |  0.487
DBLP |  $6980\times 19$ |  17 173 |  0.130
Firewall 1 |  $365\times 709$ |  31 951 |  0.124
Mushroom |  $8124\times 119$ |  186 852 |  0.193
Paleo |  $501\times 139$ |  3 537 |  0.051
Table 1: Datasets used

###  3.2 DBP view

Table 2 represents the results of topFiberM compared to three BMF methods that
performed best in the evaluation of BMF methods in [2] (GreConD+ outperforms
GreConD). Out of 20 values topFiberM achieved the best coverage in 16 and the
well-known BMF method _Asso_ achieved best in 5 times. The parameter used for
tP is 0.5, this means to minimize the overall error and equally cost of the
two types of error. The parameter SR value was set to 100. From these results
it appears that topFiberM outperforms other methods in DBP.

Dataset |  k |  Asso |  GreConD+ |  NaiveCol |  topFiberM
---|---|---|---|---|---
Chess |  1 |  0.497 |  0.447 |  0.119 |  0.610
|  2 |  0.574 |  0.506 |  0.177 |  0.625
|  5 |  0.628 |  0.621 |  0.323 |  0.667
|  10 |  0.703 |  0.710 |  0.461 |  0.724
DBLP |  1 |  0.111 |  0.131 |  0.111 |  0.187
|  2 |  0.217 |  0.238 |  0.217 |  0.293
|  5 |  0.475 |  0.468 |  0.475 |  0.495
|  10 |  0.738 |  0.692 |  0.738 |  0.738
Firewall 1 |  1 |  0.726 |  0.688 |  0.651 |  0.724
|  2 |  0.818 |  0.841 |  0.804 |  0.847
|  5 |  0.908 |  0.951 |  0.932 |  0.953
|  10 |  0.917 |  0.979 |  0.976 |  0.980
Mushroom |  1 |  0.226 |  0.131 |  0.129 |  0.253
|  2 |  0.323 |  0.235 |  0.234 |  0.305
|  5 |  0.461 |  0.504 |  0.398 |  0.425
|  10 |  0.555 |  0.613 |  0.512 |  0.522
Paleo |  1 |  0.027 |  0.027 |  0.027 |  0.027
|  2 |  0.047 |  0.047 |  0.047 |  0.049
|  5 |  0.105 |  0.106 |  0.105 |  0.106
|  10 |  0.182 |  0.181 |  0.182 |  0.182
Count best |  |  5 |  4 |  3 |  16
Table 2: Coverage of low ranks (DBP view), bold values are the best in row.
Results of Asso and NaiveCol were taken from [2]

###  3.3 AFP view

Table 3 represents the results of topFiberM compared to other three BMF
methods for AFP view as in the recent evaluation of BMF methods in [2]. Out of
20 values topFiberM achieved the best coverage in 17 then NaiveCol got 8,
_Asso_ was not able to run for high ranks of bigger datasets (scalability
problem) while GreConD+ achieved the best in 5 values. From these results we
can conclude that topFiberM can be used to solve AFP. The parameter used for
SR is k+10, for tP values are higher as shown in Table 3 column tP. As can be
seen from results the value of tP that gives best coverage increases with the
coverage (c). The parameter w in GreConD+ was set to 4 as suggested by [4].

Dataset |  _c_ |  Asso |  GreConD+ |  NaiveCol |  topFiberM |  tP
---|---|---|---|---|---|---
Chess |  0.8 |  21 |  19 |  21 |  19 |  0.7
|  0.9 |  NA |  34 |  34 |  33 |  0.8
|  0.95 |  NA |  48 |  46 |  45 |  0.8
|  1 |  NA |  130 |  72 |  71 |  1
DBLP |  0.8 |  12 |  13 |  12 |  12 |  0.7
|  0.9 |  15 |  16 |  15 |  15 |  0.7
|  0.95 |  17 |  18 |  17 |  17 |  0.7
|  1 |  19 |  21 |  19 |  19 |  0.7
Firewall 1 |  0.8 |  2 |  2 |  2 |  2 |  0.7
|  0.9 |  5 |  4 |  4 |  3 |  0.7
|  0.95 |  NA |  6 |  7 |  5 |  0.7
|  1 |  NA |  100 |  71 |  69 |  0.9
Mushroom |  0.8 |  47 |  29 |  32 |  34 |  0.8
|  0.9 |  NA |  46 |  47 |  50 |  0.9
|  0.95 |  NA |  62 |  62 |  65 |  0.9
|  1 |  NA |  120 |  110 |  109 |  0.9
Paleo |  0.8 |  84 |  86 |  83 |  83 |  0.7
|  0.9 |  109 |  110 |  107 |  106 |  0.7
|  0.95 |  125 |  127 |  122 |  121 |  0.7
|  1 |  NA |  151 |  139 |  139 |  0.8
Count best |  |  5 |  5 |  8 |  17 |
Table 3: Minimum ranks to get coverage (AFP view), bold values are the best in
row. Results of Asso and NaiveCol were taken from [2]

###  3.4 Applying to RDF data

Resource Description Framework (RDF)222https://www.w3.org/RDF/ is used to
represent data in form of triples i.e. subject, predicate , and object. We
applied topFiberM to Semantic Web Dog
Food333https://old.datahub.io/dataset/semantic-web-dog-food (SWDF) dataset and
also Asso. The dataset is about conferences and papers related to Semantic Web
during years 2001 to 2015. We presented the graph as a set of frontal slices
(i.e. each slice represents one predicate). For Asso it is required to remove
the columns which are all-zeros other wise it will not be able to finish. We
considered the predicates that are represented 1000 times or more and the rank
used was 100. The running times show that topFiberM is faster than Asso with
two order of magnitudes ( in average 128 times) and faster than GreConD on
average 838 while giving slight improvement in coverage (c=0.4535 for GreConD,
0.4460 for Asso, and 0.4538 for topFiberM). GreConD originally implemented
MATLAB we converted that code to R. We tried GreConD+ MATLAB implementation
with critical parts implemented in C but it turned out to be much slower than
GreConD (It took more than 20 hours in processing only the first two
matrices).

Figure 1: Run times on seconds for applying Asso, topFiberM, and GreConD to
SWDF dataset predicates.

The running times for the 31 predicates are shown in Figure 1. We used a
standard core i5 laptop of 16GB RAM running Windows 10 for the experiment.
From this figure it is clear that topFiberM is scalable in terms of number of
columns. The predicate the take the longest time by topFiberM is rdf:type
predicate which contains the maximum number of nonzeros.

##  4 Conclusion and Future Work

We proposed topFiberM a new BMF algorithm. topFiberM finds factors in rows or
columns in a greedy way then extends them to rectangles if possible. Our
algorithm showed better performance in AFP and DBP. topFiberM is much faster
than the famous _Asso_ algorithm. Future work include applications of our
algorithm in RDF graphs and extending it to Boolean Tensor Factorization.

##  5 Acknowledgements

We thank Martin Trnecka for providing the datasets and MATLAB code of GreConD
and GreConD+.

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