\section{Introduction}\label{sec:introduction}
\setcounter{equation}{0}

{\it Introduction.} The Standard Model (SM) explanation for Electroweak Symmetry breaking
(EWSB) -- that one condensing Higgs boson doublet carries all burdens of elementary
particle mass generation -- remains speculative.  Experimental results are consistent
with this simple explanation, albeit for an increasingly shrinking window of Higgs boson
mass. Precision electroweak data suggest that the Higgs boson must be less than $\sim
200\, {\rm GeV}$~\cite{LEPEWWG}, otherwise the virtual effects of the Higgs
boson cause theory predictions to be incompatible with the data.
Direct searches at LEP~II indicate that the Higgs
boson mass must be greater than $114\, {\rm GeV}$~\cite{:2001xx}.

Despite these passive successes of the SM Higgs boson, well-known theoretical concerns
involving naturalness and the hierarchy problem suggest that the simple SM explanation
is not complete. There have been an extraordinary number of interesting ideas that
have surfaced for a more palatable description of EWSB and elementary particle mass
generation, and each of them affects Higgs boson collider observables.  The effect could
be small, as in supersymmetry with very heavy superpartners and a SM-like
lightest Higgs boson, or large, as in strongly coupled theories of EWSB~\cite{Hill:2002ap}.

In this letter, we examine an enlargement of the Higgs sector. It resembles the Standard
Model in that it contains one light Higgs boson, but differs from the SM in the strengths
of this particle's interactions with other fields. There is a compelling reason for doing
this sort of analysis: soon data from the LHC, a \(pp\) collider with
\(\sqrt{s}=14\)~TeV, will teach us a great deal more about the theoretical nuances of
EWSB.  We begin our analysis by introducing a basic, model-independent parameterization
of the couplings of the light Higgs boson to SM particles.  We then examine how
modifications to these couplings can dramatically affect the collider observables most
relevant to Higgs searches.  In particular, we focus on the process \(gg\rightarrow h\rightarrow
\gamma\gamma\), which is one of the most promising channels for the discovery of a light
Higgs boson at the LHC, and demonstrate how such modifications to the Higgs couplings
affect the potential for discovery.

{\it Shutting off two-photon decays.} Higgs boson decays to two photons provide one of
the most important detection channels for a SM Higgs boson in the mass range
\(115\mathrm{~GeV}\lesssim m_h\lesssim 150\)~GeV.  Here, we wish to discuss the
possibility that the two-photon decay branching fraction of the Higgs boson is
significantly reduced compared to that of the SM.  This can occur for many reasons.  In
certain regions of parameter space within supersymmetry, for example, this is possible
when the partial width to the dominant mode, such as $h\rightarrow b\bar b$, is significantly
enhanced such that $B(h\rightarrow \gamma\gamma)= \Gamma(h\rightarrow\gamma\gamma)/\Gamma_{tot}\rightarrow 0$
because $\Gamma_{tot}$ is so large~\cite{Kane:1995ek}.  However, a rather extreme shift
in couplings is needed in general to suppress the $h\rightarrow \gamma\gamma$ branching fraction
to an insignificant level this way. A less extreme way that nature could reduce the
$h\rightarrow\gamma\gamma$ branching fraction is by simultaneously altering the various couplings
that enter the effective $h\gamma\gamma$ vertex such that a cancelation occurs.  We will
focus primarily on this second possibility.

The SM Higgs boson observables at the LHC involve tree-level interactions of the Higgs
boson with $WW$, $ZZ$, and $f\bar f$. A model-independent parameterization of these
interactions suggests that we multiply each of the vertices by an $\eta$-factor of \begin{eqnarray}
g^{\rm sm}_{\mathit{hWW}}\rightarrow \eta_W g^{\rm sm}_{\mathit{hWW}},~~ g^{\rm
sm}_{\mathit{hZZ}}\rightarrow\eta_Z g^{\rm sm}_{\mathit{hZZ}}, ~~ g^{\rm sm}_{\mathit{h\bar f
f}}\rightarrow \eta_f g^{\rm sm}_{\mathit{h\bar f f}} \nonumber \end{eqnarray}
The relevant
observables also rely crucially on the Higgs boson interacting at the loop level with
$gg$ and $\gamma\gamma$, and to a lesser degree of importance $\gamma Z$. These
interactions can be sensitive to new particles entering at one-loop order. Experimental
data presently have little bearing on the question of how large an effect the new
particles can have on the effective $hgg$ or $h\gamma\gamma$ vertices, and so we will
parameterize their effects through effective operators.  In general, deviations of the
$h\gamma\gamma$ and $hgg$ couplings arise two ways: deviations of
couplings of SM particles in the loops, and extra corrections due to exotic particles or
effects contributing to the effective interactions.  The former  can be
parameterized in terms of the $\eta$ coefficients described above; the latter can be
parameterized by introducing new variables $\delta_g$ and $\delta_\gamma$.  In the limit
that the top quark is much heavier than the other SM fermions, the resulting operators
are \begin{equation} \left( \delta_\gamma +\eta_WF_1(\tau_W)+\eta_t\frac{4}{3}F_{1/2}(\tau_t)\right)
\frac{h}{v}\frac{\alpha}{8\pi}F_{\mu\nu}F^{\mu\nu} \end{equation} and \begin{equation}
\left(\delta_g+\eta_tF_{1/2}(\tau_t)\right) \frac{h}{v}\frac{\alpha_s}{8\pi}
G^a_{\mu\nu}G^{a\mu\nu},\end{equation} where $v\simeq 246\, {\rm GeV}$ (SM Higgs VEV),
$\tau_i=4m_i^2/m_h^2$, and \begin{eqnarray}
F_{1/2}(\tau)&=& -2\tau[1+(1-\tau)f(\tau)] \\
F_1(\tau)& =& 2+3\tau+3\tau(2-\tau)f(\tau) \end{eqnarray} and \begin{equation}
f(\tau)=\left\{ \begin{array}{cc} {\rm arcsin}^2(1/\sqrt{\tau}) & \tau\geq 1 \\
-\frac{1}{4}\left[ \log (\eta_+/\eta_-) -i\pi\right]^2 & \tau<1
\end{array}\right.
\end{equation} with $\eta_\pm =(1\pm \sqrt{1-\tau})$~\cite{Gunion:1989we}.
One should note that $F_1(\tau_i)$ and
$F_{1/2}(\tau_i)$ can be complex if $m_h>2m_i$, as this corresponds to internal lines
going on shell.  Since any Higgs boson worth its name has the property $\eta_W\neq 0$,
one can express the condition under which the effective $h\gamma\gamma$ vertex vanishes
as
\begin{equation} \left( \frac{\eta_t}{\eta_W}\right) =
  -\frac{3}{4}\left(\frac{1}{F_{1/2}(\tau_t)}\left( \frac{\delta_\gamma}{\eta_W} \right) +
  \frac{F_1(\tau_W)}{F_{1/2}(\tau_t)}\right)
\end{equation} In fig.~\ref{eta
relations}, we plot the contours of $\Gamma(h\rightarrow \gamma\gamma)=0$ in the $\eta_t/\eta_W$
vs. $\delta_\gamma/\eta_W$ plane. The various lines of the plot correspond to various
values of $m_h$ within the range of Higgs boson masses that are considered applicable for
discovery through their $h\rightarrow\gamma\gamma$ decay channel.

\begin{figure}
\includegraphics[width=8.5cm]{Fig1.eps}
\caption{Lines in  $\eta_t/\eta_W$ vs. $\delta_\gamma/\eta_W$ represent where
$\Gamma(h\rightarrow \gamma\gamma)\rightarrow 0$ for various values of $m_h$ in the range
of Higgs boson masses where the two-photon decays are thought to be important
for discovery at the LHC.}
\label{eta relations}
\end{figure}

{\it Theory discussion.} From experience in supersymmetry, exotic physics contributions
to $\delta_g$ or $\delta_\gamma$ can decouple very rapidly and have small
effects~\cite{Kane:1995ek}. Possible exceptions to this statement include a radion in
warped geometry that has large $\delta_g$ and $\delta_\gamma$ contributions due to
quantum breaking of conformal invariance~\cite{Giudice:2000av}, or
non-renormalizable operators with low-scale cutoff~\cite{Hall:1999fe}. If mixed with
a condensing Higgs boson, the lightest mass eigenstate can have significant $\delta_g$
and $\delta_\gamma$ contributions. Nevertheless, if we approximate that $\delta_g$ and
$\delta_\gamma$ are zero, we are left with the reasonable conclusion that the top quark
coupling to the Higgs boson is greatly enhanced compared to its SM value. This is to be
expected in the case where the $h\rightarrow \gamma\gamma$ decay rate goes to zero, since the
$W$-induced amplitude contribution is about six times larger than the top-quark induced
amplitude contribution and of the opposite sign. Thus, one expects that a theory with
reduction of the $W$ coupling and simultaneous increase in the top-quark coupling has the
potential to reduce and even zero out the $h\rightarrow \gamma\gamma$ amplitude.

Reducing $W$ couplings is easy: arrange for several states or mechanisms to contribute to
EWSB. Any individual Higgs boson state will have reduced couplings to $W$ since couplings
are proportional to the contribution that the Higgs boson
makes to the mass of the $W$ boson. As
for the coupling to the top quark, increasing its value is not only possible, but can be
expected when EWSB is shared among many sectors.  If the top quark couples to only one
Higgs boson \(H\) that does not fully generate the $W$ masses, then the Yukawa coupling
of the top quark to that Higgs has to have a larger value to generate the requisite top
quark mass (i.e., $m_t/\langle H\rangle > m_t/\langle H_{sm}\rangle$).

The above considerations lead us naturally to think in terms of a type I two-Higgs
doublet scenario~\cite{Haber:1978jt,Gunion:1989we}.  In such a scenario, both (complex)
doublets, which we denote \(\Phi_{f}\) and \(\Phi_{\mathit{EW}}\), obtain VEVs and
contribute to EWSB and vector boson mass generation, but only one of the Higgs bosons,
\(\Phi_{f}\), gives mass to the fermions.
Among the eight degrees of freedom in $\Phi_{EW}$ and $\Phi_f$ three are eaten by
$W^\pm_L,Z^0_L$, two become charged Higgses $H^\pm$, one becomes pseudo-scalar $A^0$, and
two become scalars $h^0,H^0$, where $h^0$ is the lightest.  Such a theory framework can
be motivated by the dynamics of a strongly coupled sector~\cite{Hill:2002ap}
which contributes to EWSB and
to the vector boson masses.  It is well-known that giving mass to the heavy fermions
simultaneously is difficult, and so an additional scalar (or effective scalar) can be
added to the theory that gives mass to the fermions (somewhat reminiscent of other
approaches~\cite{Simmons:1988fu}). Of course,
this second sector will necessarily contribute to vector boson masses as well (one cannot
hide EWSB from the vector bosons), and a complicated mixing among the two Higgs doublets
ensues.

In our study we define a mixing angle \(\beta\) between the two Higgs VEVs,
\(\tan\beta=\langle\Phi_f\rangle/\langle \Phi_{EW}\rangle\).  A second angle, \(\alpha\),
which parameterizes the mixing between the gauge and mass eigenstates of the CP-even
Higgs, is defined by the relation \begin{equation} \left(
\begin{array}{c} H^0 \\ h^0 \end{array}\right) =
\left( \begin{array}{cc} \cos\alpha & \sin\alpha \\
-\sin\alpha & \cos\alpha\end{array}\right) \left( \begin{array}{c}
\sqrt{2} {\rm Re}(\Phi_{EW}^0) \\
\sqrt{2} {\rm Re}(\Phi_f^0)
\end{array}\right)
\end{equation} The deviations in the couplings to SM fermions and vectors bosons of $h^0$ compared
to the SM Higgs depend only on these angles:
\begin{equation}
\eta_f=\frac{\cos\alpha}{\sin\beta},~~~\eta_{W,Z}=\sin(\beta-\alpha). \label{eq:etas}
\end{equation}

The value of $\alpha$ is computed from the full potential of the theory is
model-dependent, as are the masses of $H^\pm$ and $A^0$.  The most stringent experimental
constraint on these parameters comes from \(b\rightarrow s\gamma\). The experimental
limit is \(BR(b\rightarrow s\gamma)=(3.3\pm0.4)\times10^{-4}\) and in type~I Higgs
doublet models~\cite{Barger:1989fj},
\begin{equation}
  \Gamma(b\rightarrow s\gamma)=\frac{\alpha G_{F}^2 m_b^2}{128\pi^4}
  \left|A_{W}+\cot^2\beta A_{H}(m_{H^{\pm}})\right|^2,
  \label{bsg eq}
\end{equation}
where \(A_{W}\) and \(A_{H}\) are the respective loop functions for the SM
and charged-Higgs-induced contributions.  \(A_{H}\) is a  function of
\(m_{H^{\pm}}\) that has opposite sign to that of $A_W$. When
\(H^\pm\) is light, the charged Higgs
contribution to \(b\rightarrow s\gamma\) can be large and \(\sin\beta\) is either constrained to be
close to one, or $\sin\beta$ is tuned to a smaller value such that
$A_{W}+\cot^2\beta A_{H}\simeq -A_W$, thereby leading to an acceptable prediction.
If $m_{H^\pm}\lower.7ex\hbox{$\;\stackrel{\textstyle>}{\sim}\;$}$ few TeV $A_H$ is effectively zero for any value of $\sin\beta\lower.7ex\hbox{$\;\stackrel{\textstyle>}{\sim}\;$} 1/3$.
In both of these cases (light or heavy $H^\pm$) a solution exists for our purposes,
and the $H^\pm$ does not substantively affect the two-photon decay rate of the Higgs boson.
We specify \(\alpha\) to be small (in order to be
precise in our discussion, we choose \(\alpha=0\)), which is also nicely consistent
with $m_A^2,m_{H^0}^2,m_{H^\pm}^2\gg m_{h^0}^2$ and
the model contains only one light Higgs boson.  An extremely small \(\alpha\) is by no means
required, however, and we note that the results that carry forward will exhibit the same
qualitative features when $|\alpha|\lesssim1/2$.

{\it Numerical results.} In fig.~\ref{BRratio} we compute the decay branching fractions
of a $140\, {\rm GeV}$ Higgs boson as a function of $\sin\beta$ relative to the SM decay
branching fractions.  In this, and subsequent calculations, SM quantities are
obtained using HDECAY~\cite{Djouadi:2000gu}, and we allow \(\sin\beta\) to vary from 1
down to \(\sim0.250\), below which point the perturbativity of the top quark Yukawa
(given by \(y_{t}=\sqrt{2}m_t/v\sin{\beta}\) in this model) becomes a concern.  For low
values of $\sin\beta$ the branching fraction to $b\bar b$ is enhanced significantly over
that of $WW^*$. As $\sin\beta\rightarrow 1$, which recovers the SM result, $WW^*$ wins out.

\begin{figure}
\includegraphics[width=8.5cm]{BR140alpha0new.eps}
\caption{Ratio with respect to the SM of decay branching fractions as a function of
$\sin\beta$ for the light Higgs of a type I two-Higgs doublet model. Here, we have taken
\(m_h=140\)~GeV and \(\alpha=0\).} \label{BRratio}
\end{figure}

More important than the branching fractions, however, is the total cross-section of
$pp\rightarrow h\rightarrow\gamma\gamma$, since that is what is measured at the collider. The largest
contribution to the production cross-section for this observable $\sigma_h(\gamma\gamma)$
is through gluon fusion, $gg\rightarrow h\rightarrow \gamma\gamma$. The amplitude for $gg\rightarrow h$ is the
same as for $h\rightarrow gg$ up to simple co-factors at leading order. Thus, we make a good
estimate of the relative size of the $pp\rightarrow h\rightarrow\gamma\gamma$ cross-section: \begin{equation}
\frac{\sigma_h(\gamma\gamma)}{\sigma_h(\gamma\gamma)_{sm}} =
\frac{\Gamma_h(gg)}{\Gamma_h(gg)_{sm}}
\frac{\Gamma_h(\gamma\gamma)}{\Gamma_h(\gamma\gamma)_{sm}} \left( \frac{\Gamma_h({\rm
tot})}{\Gamma_h({\rm tot})_{sm}}\right)^{-1} \end{equation}

In fig.~\ref{logobservables} we plot $\sigma_h(\gamma\gamma)/\sigma_h(\gamma\gamma)_{sm}$
as a function of $\sin\beta$ for $m_h=140\, {\rm GeV}$ (and we retain $\alpha=0$). For
$0.38\lesssim \sin\beta \lower.7ex\hbox{$\;\stackrel{\textstyle<}{\sim}\;$} 0.55$ we see that the total cross-section into two photons
is more than an order of magnitude lower than the SM cross-section, thereby severely
compromising the ability of the LHC to discover the light Higgs boson via that channel.
However, as we suggested above, the $t\bar t h\rightarrow t\bar t b\bar b$ cross-section is
greatly enhanced by nearly an order of magnitude above the SM cross-section.  Thus, it
may still be possible to discover such a Higgs boson.

\begin{figure}
\includegraphics[width=8.5cm]{crosssec140alpha0new.eps}
\caption{Ratio with respect to the SM of cross-section observables at the LHC
(\(\sqrt{s}=14\)~TeV) as a function of $\sin\beta$ for the light Higgs of a type I
two-Higgs doublet model, with \(m_h=140\)~GeV and \(\alpha=0\).} \label{logobservables}
\end{figure}

\begin{table}
  \begin{center}
\begin{tabular}{|c|c|c|}\hline \(m_h\) (GeV) & \(\Gamma_{\mathit{tot}}(h)\) (MeV) &
   \(\Gamma_{\mathit{tot}}^{\mathit{sm}}(h)\) (MeV) \\ \hline
     110 & 12 & 3.0 \\
     130 & 16 & 4.9\\
     150 & 23 & 17\\ \hline
\end{tabular}
\end{center}
\caption{In this table, we list the total width of the Higgs boson, evaluated when the
\(h\gamma\gamma\) effective coupling is adjusted to zero, for a variety of choices of the
Higgs mass.  The SM width is also shown for
comparison\label{tab:HiggsWidth}.}
\end{table}

To investigate this, we compute the significance of discovery, with
\(\mathrm{30~fb}^{-1}\) of integrated luminosity, of the $h^0$ Higgs boson in this model
by scaling the known significance values~\cite{Asai:2004ws} for discovering a SM Higgs
boson in various channels using the ATLAS detector.
In fig.~\ref{higgs significance} we assume that $\alpha=0$ and
as we vary the Higgs boson mass we select the value of $\sin\beta$ such that $\Gamma(h\rightarrow
\gamma\gamma)$, and hence $\sigma(gg\rightarrow h\rightarrow \gamma\gamma)$, is zero.  Any value of
$\sin\beta$ in the neighborhood of this point will yield similar significance of
discovery values for the channels displayed.  The other significance values are modified
from their SM values by the scaling of the Higgs couplings by \(\eta_{W,Z}\) and
\(\eta_{f}\) according to equation~(\ref{eq:etas}).  Our scaling of the significance
values is justified since the Higgs boson width remains narrow and
never exceeds the invariant
mass resolutions (e.g., $\Delta m_{\gamma\gamma}/m_{\gamma\gamma}\sim 1\%$~\cite{ATLAStdr})
of the relevant final states over the range of interest -- see table~\ref{tab:HiggsWidth}.

Higgs boson discovery channels involving one or more
couplings between the Higgs and EW gauge bosons (including all weak boson fusion
processes) are suppressed by factors of \(\eta_{W,Z}\) as well as by an increase in the
total width of the Higgs due to an increase in Higgs decays to \(\overline{b}b\), which
dominate the total width in the mass range of interest---see table~\ref{tab:HiggsWidth}.
In contrast, the \(\overline{t} th\) channel significance is dramatically increased.  As
a result, while the usual channels in which one would look to discover a Higgs boson in
the mass range \(115\mathrm{~GeV}\lesssim m_{h}\lesssim 150\)~GeV are suppressed below
the \(5\sigma\) level, there is still the opportunity to discover the Higgs boson in the
$t\bar th$ channel. The searches, of course, must extend themselves to Higgs boson mass
values well above what is normally thought to be the relevant range for this signature.

\begin{figure}
\includegraphics[width=8.5cm]{SigPlot.eps}
\caption{Significance of various light Higgs boson observables at the LHC with
$30\xfb^{-1}$ of integrated luminosity. This plot is made for $\alpha=0$ and
$\sin\beta\simeq 0.45$ where $\Gamma(h\rightarrow\gamma\gamma)\rightarrow 0$ in the type I two-Higgs
doublet model.} \label{higgs significance}
\end{figure}

{\it Conclusions.} Higgs decays to two photons provide one of the most important channels
through which one might hope to discover a light (\(115\mathrm{~GeV}\lesssim
m_{h}\lesssim 150\)~GeV) SM Higgs boson at the LHC. However, nature allows
a Higgs sector whose couplings differ from their SM values in such a way that
\(\Gamma(h\rightarrow\gamma\gamma)\) is suppressed into irrelevance.  This possibility
occurs readily in scenarios involving additional contributions to electroweak symmetry
breaking from sources that do not contribute to the generation of fermion masses. The
Higgs coupling to EW gauge bosons is decreased relative to the SM result as its
contribution to EWSB is decreased, and its coupling to the fermions is consequently
augmented, allowing for a cancelation in the coefficient of the effective
\(h\gamma\gamma\) vertex.  This is well illustrated
in type~I two Higgs doublet models and in a
variety of models where there exists a dynamical contribution to EWSB. If this is the
case, one cannot rely on the usual channels (weak boson fusion, \(h\rightarrow
WW^{\ast}\), and \(h\rightarrow\gamma\gamma\) itself) to discover a Higgs boson.
Nevertheless, such a Higgs could still be detected through processes like
\(\overline{t}th\), with \(h\) decaying to \(\overline{b}b\) or \(\overline{\tau}\tau\).


\section{Introduction}

Since the experimental discovery of the fractional quantum Hall effect, a tremendous effort has been devoted to the study of topologically ordered phases that can be realized in strongly correlated quantum many-body systems. In the past decade, substantial progress has been made in the characterization of topologically ordered phases in spin or boson systems, and a unified algebraic framework called `modular tensor categories' has been identified to describe these phases \cite{Kitaev}. From a physical point of view, a modular tensor category simply describes how the anyonic excitations fuse and braid with each other. Phrasing the properties of topologically ordered phases in the rigid algebraic language of modular tensor categories has allowed theorists to make significant progress in the study of bosonic topological phases.

Not only has our theoretical understanding of topologically ordered phases vastly improved, in recent years many important results have been obtained on how to numerically identify the type of topological order realized by a particular microscopic Hamiltonian. For example, it was realized that a non-trivial topological order leaves an imprint on the entanglement entropy of a spatial region in the ground state via the `topological entanglement entropy' term \cite{KitaevPreskill,LevinWen}. A later refinement of the topological entanglement entropy showed that the complete spectrum of the reduced density matrix of a spatial region contains information about the universal edge physics of the topological phase \cite{LiHaldane}. When the system of interest is put on a torus, it will necessarily have a ground state degeneracy if a a non-trivial topological order is realized \cite{WenNiu}. In Refs. \cite{ZhangGrover,Cincio}, a useful basis for the ground state subspace, the so-called Minimally-Entangled State (MES) basis, was identified and it was shown that this basis can be used to obtain the $S$-matrix, which contains information about the anyon braiding statistics, and the $T$-matrix, which gives access to the chiral central charge and the topological spins of the anyons. Later works showed that these MES also give access to the topological spins of the anyons via a quantity called the `momentum polarization' \cite{ZaletelMong,TuZhang}.

In this work, we focus on fermion systems with Abelian topological order, meaning that the anyons form an Abelian group under fusion. In the algebraic languague, the most important difference between bosonic and fermionic topological orders is that the latter don't satisfy the same strict requirement of modularity as bosonic systems do. We explain this in more detail in the main text, where the algebraic frameworks for both bosonic and fermionic Abelian topological orders are reviewed. A fermionic system necessarily has a fermion parity symmmetry, and a useful tool in the study of fermionic topological orders is to gauge this symmetry \cite{Kitaev} (the approach of gauging a global symmetry has also proven to be very useful in the study of symmetry-protected phases \cite{LevinGu}). Importantly, since the microscopic fermion becomes a non-local gauge charge after gauging, the resulting theory is purely bosonic. Below, two main questions regarding Abelian fermionic topological orders (AfTO) are addressed: (1) How does one uniquely identify the most general AfTO in numerics?, and (2) Does the existence of a gapped edge for the ungauged fermionic topological order imply the existence of a gapped edge for the gauged topological order and vice versa?

Regarding the first question, we note that a lot of the above mentioned numerical techniques for identifying a topological order have been successfully applied to fractional quantum Hall systems and fractional Chern insulators \cite{Neupert,ShengGu,Regnault,ZaletelMong,Rezayi,Grushin}, which are of course fermionic in nature. These approaches relied crucially on the presence of a fermion number conservation symmetry, which allows for the definition of a Hall conductivity $\sigma_{xy}$. In fractional quantum Hall systems one knows the value of $\sigma_{xy}$ exactly from Galilean invariance (if there is no disorder), and in lattice systems it can be computed numerically as the many-body Chern number \cite{Niu,ShengHaldane}, or from the entanglement eigenvalues by an adiabatic flux insertion procedure \cite{Zaletel,Grushin} (see also Ref. \cite{Alexandradinata}). In this work, we outline an alternative numerical detection scheme which does not rely on computing the Hall conductance, and which can identify the most general Abelian fermionic topological order. In formulating this numerical scheme, we will make heavy use of the special structure of gauged AfTOs. Another important ingredient for our detection scheme is the momentum polarization technique \cite{ZaletelMong,TuZhang}, and we point out some properties of this numerical probe which --to the best of our knowledge-- have not been discussed previously in the literature and which are relevant for fermionic systems.

As for the second question regarding the edge physics of topological phases, we will show that the bosonic theory which is obtained by gauging fermion parity in an AfTO can have a (conventional, bosonic) gapped edge with the trivial vacuum if and only if the ungauged fermionic theory can have a gapped edge as well. To show that gauging fermion parity does not change whether a system admits a gapped edge or not, we use the bulk-boundary correspondence formulated in terms of Lagrangian subgroups as introduced in Ref. \cite{Levin}.

For completeness, we also mention some previous works which have studied fermionic topological orders and are relevant for the present work. Refs. \cite{GuWangWen,TianLan1,TianLan2} worked out an algebraic framework for general fermionic topological orders (both Abelian and non-Abelian), which is a generalization of the modular tensor categories for bosonic topological orders. Some ideas of these works will be used below. Refs. \cite{Belov,Cano} have studied Abelian fermionic topological orders with fermion number conservation symmetry using multi-component U$(1)$ Chern-Simons theories. The algebraic approach adopted in this work agrees with the U$(1)$ Chern-Simons approach of Refs. \cite{Belov,Cano} where the results overlap. And finally, Ref. \cite{Wang} has studied general gauged fermionic topological orders from a mathematical perspective, and conjectured that Kitaev's $16$-fold way \cite{Kitaev} has a natural generalization to the most general fermionic topological order. This being said, let us now turn to a short review of the algebraic framework behind bosonic and fermionic Abelian topological order.

\section{Abelian bosonic topological order}

In this section, we briefly review the properties of Abelian bosonic topological orders (AbTO) that are relevant for the main discussion below. For more details, the reader is referred to Ref. \cite{Kitaev}. In mathematical terms, an AbTO is equivalent to an Abelian modular tensor category. To specify such an Abelian modular tensor category, we need to provide a list $\mathcal{A}=\{a,b,c,\dots\}$ of $N$ anyon types, together with the corresponding Abelian fusion rules $a\times b = \sum_c N_{a,b}^c c$ such that $N_{a,b}^c\in \{0,1\}$, and topological spins $\theta_a$ \footnote{This is a slight abuse of terminology which is common in the literature. If we write $\theta_a=e^{2\pi i h_a}$, then it is more appropriate to call $h_a$ the topological spin. However, in this work it will be more convenient to simply refer to $\theta_a$ as the topological spin.}. Every AbTO has a unique trivial anyon, denoted as $1$, which has the properties $1\times a = a$ and $\theta_1=1$.  Often, we will write the fusion of two anyons $a$ and $b$ simply as $ab = a\times b$. In principle, we also need to provide the $F$-symbols, but they will not be important here so we omit them. We denote the braiding phase associated with moving an anyon $b$ counter-clockwise around anyon $a$ as $M_{a,b}$. By definition, the braiding phases are symmetric: $M_{a,b}=M_{b,a}$. The ribbon identity allows $M_{a,b}$ to be expressed in terms of the topological spins:

\begin{equation}
M_{a,b}=\frac{\theta_{ab}}{\theta_a\theta_b}
\end{equation}
The $S$ matrix is related to the braiding phases by $S_{a,b}=M_{a,b}\mathcal{D}^{-1}$, where the total quantum dimension $\mathcal{D}$ of an Abelian modular tensor category is given by $\mathcal{D}=\sqrt{N}$. Modularity requires that $S^\dagger S=\mathds{1}$.

An important connection between the bulk anyons and the boundary theory of an AbTO is given by the following relation \cite{Kitaev}:

\begin{equation}\label{kit}
\frac{1}{\sqrt{N}}\sum_a \theta_a = e^{2\pi i c_-/8}\, ,
\end{equation}
where $c_-$ is the chiral central charge of the boundary theory. This implies that the bulk anyons determine the boundary chiral central charge up to a multiple of $8$. This is the best one can do, as there exists an invertible bosonic topological phase, the so-called $E_8$ state \cite{KitaevE8}, which has no anyons but a chiral edge with $c_-=8$. One can thus always stack an $E_8$ state on top of the system of interest, which does not affect the bulk anyons but changes the boundary chiral central charge by $8$.

\subsection{Gapped boundaries}\label{GappedB}

As was shown by Levin, an AbTO admits a gapped edge iff (1) $c_-=0$ and (2) the bulk anyons have a bosonic Lagrangian subgroup \cite{Levin}. A bosonic Lagrangian subgroup $\mathcal{L}_b$ is a subgroup of anyons which have the following properties (see also Refs. \cite{JuvenWang1,JuvenWang2}):

\begin{itemize}
\item[1)] Every anyon in $\mathcal{L}_b$ has trivial topological spin,
\item[2)] All anyons in $\mathcal{L}_b$ braid trivially with each other,
\item[3)] Every anyon which is not in $\mathcal{L}_b$ braids non-trivially with at least one anyon in the Lagrangian subgroup.
\end{itemize}
Using Eq.~(\ref{kit}), we will show that the existence of a Lagrangian subgroup implies that the chiral central charge is a multiple of $8$. It thus follows that an AbTO allows for a gapped edge iff it has a Lagrangian subgroup, and the separate requirement of zero chiral central charge is redundant, provided that we are allowed to stack $E_8$ states on top of our system. Combined with the results of Ref. \cite{Levin2}, this implies that every AbTO with a Lagrangian subgroup has a string-net representation \cite{stringnet}.

To derive $c_-=0$ mod $8$ from the existence of a bosonic Lagrangian subgroup $\mathcal{L}_b$, we write the AbTO as $\mathcal{A}=\{\mathcal{L}_b,\mathcal{L}_b\times n_1, \mathcal{L}_b\times n_2,\dots\}$, where $n_i$ are a set of arbitrary anyons not in $\mathcal{L}_b$. Starting from Eq.~(\ref{kit}), we can now do the following manipulations:

\begin{eqnarray}
e^{2\pi ic_-/8} & = & \frac{1}{\sqrt{N}}\sum_{a\in\mathcal{A}} \theta_a \\
 & = &  \frac{1}{\sqrt{N}}\sum_{l \in \mathcal{L}_b}\theta_l +  \frac{1}{\sqrt{N}} \sum_{n_i}\sum_{l \in \mathcal{L}_b}\theta_{n_il}  \\
 & = &  \frac{1}{\sqrt{N}}\sum_{l \in \mathcal{L}_b}\theta_l +  \frac{1}{\sqrt{N}} \sum_{n_i}\theta_{n_i} \sum_{l \in \mathcal{L}_b}\theta_{l}M_{l,n_i} \\
 & = &  \frac{N_{L}}{\sqrt{N}}+  \frac{1}{\sqrt{N}} \sum_{n_i}\theta_{n_i} \sum_{l \in \mathcal{L}_b}M_{l,n_i} \label{sum}\\
 & = & \frac{N_{L}}{\sqrt{N}}\, ,\label{Lb}
\end{eqnarray}
where $N_{L}$ is the number of anyons in $\mathcal{L}_b$. In the third line we have used the ribbon identity, and in the fourth line we relied on the definition of a bosonic Lagrangian subgroup wich states that $\theta_l=1$ if $l\in \mathcal{L}_b$. To see why the second term in~(\ref{sum}) is zero, note that $M_{l_1,n_i}M_{l_2,n_i}=M_{l_1l_2,n_i}$, such that $M_{l,n_i}$ for fixed $n_i$ forms a representation of $\mathcal{L}_b$. Because there is at least one $l\in\mathcal{L}_b$ for which $M_{l,n_i}\neq 1$, this representation cannot be the trivial representation. Schur's orthogonality relations then imply that the sum of $M_{l,n_i}$ over all $l\in\mathcal{L}_b$  is zero. We have thus obtained the desired result that $c_-=0$ mod $8$ if there exists a Lagrangian subgroup. As a side-result, we also found that $N_{L}^2=N$, such that only AbTOs where the number of anyons is a square number can have a Lagrangian subgroup (this relation between $N_{L}$ and $N$ was also obtained previously in Ref. \cite{Levin}).

\section{Abelian fermionic topological order}

The main difference between a fermionic topological order (fTO) and a bTO, is that a fTO has a distinguished fermion excitation $f$ with the properties $f^2=f\times f=1$ and $\theta_f=-1$, which is `transparent', i.e. it braids trivially with all other particles. This is not allowed in a bTO because the existence of such a particle is a violation of modularity. In Refs. \cite{Cano,Cheng}, it was shown that every Abelian fermionic topological order (AfTO) $\mathcal{A}_f$ can be written as

\begin{equation}\label{prod}
\mathcal{A}_f = \mathcal{A}_b\times \{1,f\}\, ,
\end{equation}
where $\mathcal{A}_b$ is an AbTO. This result also follows from corollary A.19 of Ref. \cite{Drinfeld}. Note that the above factorization property does not hold for non-Abelian fTO, as is known from explicit counter examples \cite{TianLan1,TianLan2}.

It is important to keep in mind that in general, the factorization in Eq.~(\ref{prod}) is not unique. In particular, let us define the homomorphism $\beta: \mathcal{A}_b\rightarrow \mathbb{Z}_2$, where $\beta(a)$ takes values in $\{0,1\}$. The fact that $\beta$ is a homomorphism then implies that $\beta(a b) = \beta(a)+\beta(b)$ mod $2$. Using $\beta$, we can rewrite the factorization in Eq. \ref{prod} in a different way as $\mathcal{A}_f = \mathcal{A}_b^\beta\times\{1,f\}$, where $\mathcal{A}_b^\beta = \{a f^{\beta(a)}|a\in\mathcal{A}_b\}$. So the number of different factorizations is given by the number of homomorphisms from $\mathcal{A}_b$ to $\mathbb{Z}_2$.

A fermionic system necessarily has fermion parity symmetry. This implies that we can introduce a corresponding fermion parity flux or $\pi$-flux into the system. In the absence of U$(1)$ fermion number symmetry, there is no unique way of defining such a fermion parity flux. In particular, for a given parity flux $\phi_i$ we can always obtain a different parity flux by attaching an anyon in $\mathcal{A}_f$ to it. This of course assumes that the parity flux cannot `absorb' the anyons in $\mathcal{A}_f$, i.e. that $\phi_i \times a\neq \phi_i$ for $a \in \mathcal{A}_f$. While it is possible for $\phi_i$ to absorb the transparent fermion $f$, we will show in the next section that the parity fluxes cannot absorb the anyons in $\mathcal{A}_b$. The case where the fermion parity fluxes can absorb $f$ occurs in superconducting systems where a Majorana mode binds to the parity flux, like in the weak pairing phase of spinless $p-$wave superconductors  \cite{Volovik,ReadGreen}. When this happens, the parity fluxes are non-Abelian defects \cite{Ivanov}.

An important property of fermion parity fluxes is that they have a well-defined topological spin. This is different from $\mathbb{Z}_2$ fluxes in bosonic systems, which have a topological spin that is only defined up to a minus sign. The reason is that bosonic systems admit trivial particles with both even and odd $\mathbb{Z}_2$ charge. So if we attach such a trivial charge one object to a $\mathbb{Z}_2$ flux, we don't change the superselection sector, but we do change its topological spin by $-1$. In fermionic systems, however, all charge one particles have topological spin $\theta_f=-1$, so by attaching them to a parity flux we don't change the topological spin.

\subsection{Gapped boundaries}\label{GappedF}

Similarly to the bosonic case, Levin showed that an AfTO admits a gapped edge if and only if $c_-=0$, and there exists a fermionic Lagrangian subgroup $\mathcal{L}_f$, which is defined to have the following properties \cite{Levin}:

\begin{itemize}
\item[1)] All anyons in $\mathcal{L}_f$ braid trivially with each other,
\item[2)] Every anyon in $\mathcal{A}_b$ which is not in $\mathcal{L}_f$ braids non-trivially with at least one anyon in the Lagrangian subgroup.
\end{itemize}
The only difference between the definitions of $\mathcal{L}_b$ and $\mathcal{L}_f$ is that in the latter we do not require the anyons in $\mathcal{L}_f$ to have trivial topological spin. Note, however, that because $\theta_l^2=M_{l,l}=1$ for all $l\in\mathcal{L}_f$, it follows that the definition of a fermionic Lagrangian subgroup only leaves a sign ambiguity in the topological spins of the anyons in $\mathcal{L}_f$.

\section{Gauging fermion parity: modular extensions of an AfTO}\label{sec:gauging}

There exists a well-defined microscopic prescription to gauge the fermion parity symmetry in any fermionic lattice Hamiltonian \cite{Kogut}. After gauging, the system realizes a bosonic topological order in the bulk. This is because the gauging procedure promotes the parity fluxes to deconfined anyonic excitations, which braid non-trivially with $f$. This implies that the gauged theory is modular, and can be realized in a bosonic system. In mathematical terms, a GfTO is called a `modular extension' of the original fTO \cite{TianLan1,TianLan2}. In this section, we will show that for AfTO such modular extensions have a special structure. For this we consider the process where one creates an $a-\bar{a}$ anyon pair from the vacuum and braids one of them, say $a$, around a fermion parity flux. It is well-known from bosonic symmetry-enriched topological orders that braiding around a symmetry defect $g$ can permute the anyon types \cite{Bombin,Barkeshli,Tarantino,Chen}. This means that after braiding an anyon $a$ around $g$, it is possible for $a$ not to come back as itself, but as a different anyon $\pi_g(a)$. After gauging, the parity fluxes become deconfined anyonic excitations. Because braiding of anyons cannot change the anyon type, this implies that after gauging the anyons $a$ and $\pi_g(a)$ have to be identified as the same anyon \cite{Barkeshli,Tarantino,Chen}. However, we now argue that this cannot happen for a fermion parity flux, i.e. for fermion parity we always have $\pi_\phi(a)=a$. The reason is that fermionic Hilbert spaces have a superselection rule which states that every physical state needs to have well-defined fermion parity \cite{Wick}. So if fermion parity could permute anyons, then starting from an excited state with some localized anyons it would be possible to create an orthogonal state by acting with fermion parity, but this clearly violates the superselection rule.

As already anticipated above, we can now also argue that fermion parity fluxes cannot absorb anyons in $\mathcal{A}_b$. To see this, assume that a parity flux $\phi_i$ could absorb an anyon $a\in\mathcal{A}_b$. The only way this can happen consistently, is if all anyons $b$ with $M_{a,b}\neq 1$ get permuted when they braid with the parity flux. But as we just argued, this is impossible. Note that $f$ escapes this argument since it braids trivially with all anyons in $\mathcal{A}_f$.

When gauging a $\mathbb{Z}_2$ symmetry in bosonic systems, the trivial anyon $1$ before gauging splits into a trivial anyon and a non-trivial anyon after gauging. This is because the original symmetry-enriched topological order has trivial anyons with both even and odd $\mathbb{Z}_2$ charge. Under gauging, the trivial anyons with even charge remain trivial, but the trivial anyons with odd charge become the gauge charge anyons of the gauged theory. For fermionic systems, this does not happen because a fTO has no trivial anyons with odd fermion parity.

The above two arguments show that the anyons of a fTO do not get identified and do not split after gauging fermion parity, which implies that the original fTO is a subcategory of the GfTO. For AfTO, if $\mathcal{A}_f\times\{1,f\}$ is a subcategory of the GfTO, then $\mathcal{A}_b$ is obviously also a subcategory of the gauged theory. We can now use theorem 3.13 from Ref. \cite{Drinfeld}, which says the following:
\\ \\
\textbf{Theorem (Ref. \cite{Drinfeld}).} Consider a MTC $\mathcal{K}$, and assume that it is a fusion subcategory of $\mathcal{C}$, i.e. $\mathcal{K}\subset\mathcal{C}$. If $\mathcal{C}$ is a MTC, then $\mathcal{C}=\mathcal{K}\times\mathcal{K}'$, where $\mathcal{K}'$ is also a MTC.
\\ \\
Because $\mathcal{A}_b$ is modular, we can directly apply the above theorem to conclude that the GfTO takes the form

\begin{equation}\label{factor1}
GfTO = \mathcal{A}_b \times\{1,f,\phi,f\phi\}\, ,
\end{equation}
if the fermion parity fluxes are Abelian, and

\begin{equation}\label{factor2}
 GfTO = \mathcal{A}_{b}\times\{1,f,\phi\}\, ,
\end{equation}
if the fermion parity fluxes are non-Abelian. Both $\phi$ and $f\phi$ are anyons which correspond to deconfined fermion parity fluxes.

In the present context, it is not hard to prove the factorization of the GfTO, as given in Eqs.~(\ref{factor1}) and (\ref{factor2}), without invoking theorem 3.13 of Ref. \cite{Drinfeld}. To see this, note that since fermion parity fluxes $\phi_i$ cannot permute anyons, the process of making an $a-\bar{a}$ pair, braiding $a$ around $\phi_i$, and subsequently annihilating the anyon pair again is a well-defined adiabatic process for every parity flux $\phi_i$ and anyon $a\in\mathcal{A}_b$. Therefore, we can associate a Berry phase to it. Because there are no trivial particles with odd fermion parity, the Berry phases depend only on the anyon type and it makes sense to write them as $e^{i\gamma_i(a)}$ and $e^{i\gamma_i(af)}$ for every flux $\phi_i$ and $a\in\mathcal{A}_b$. The Berry phases satisfy the obvious properties $e^{i\gamma_i(a b)}=e^{\gamma_i(a)}e^{i\gamma(b)}$ and $e^{i\gamma_i(f)}=-1$. Because $\mathcal{A}_b$ is modular, we can use lemma 3.31 from Ref. \cite{Drinfeld}  (see also Ref. \cite{Barkeshli}, page 11), which states that for every function $e^{i\gamma_i(\cdot)}:\mathcal{A}_b\rightarrow U(1)$ that satisfies $e^{i\gamma_i(a b)}=e^{\gamma_i(a)}e^{i\gamma_i(b)}$, there exists a corresponding unique anyon $a_i\in \mathcal{A}_b$ such that

\begin{equation}\label{braidingthm}
e^{i\gamma_i(a)}=M_{a,a_i}\,,\;\;\forall a \in \mathcal{A}_b
\end{equation}
This implies that for every parity flux $\phi_i$, we can find an anyon $a_i\in\mathcal{A}_b$ such that $\phi_i \bar{a}_i$ braids trivially with all anyons in $\mathcal{A}_b$. Because the different $\phi_i$ are related by fusion with anyons in $\mathcal{A}_b$, and for every $\phi_i$ there is a unique anyon $a_i$ such that (\ref{braidingthm}) holds, we conclude that $\phi_i \bar{a}_i$ is independent of $i$. The parity flux in Eqs.~(\ref{factor1}) and (\ref{factor2}) is then simply defined as $\phi = \phi_i\bar{a}_i$ (a similar argument for the factorization of GAfTOs with fermion number symmetry was recently given in Ref. \cite{Lapa}).

As mentioned previously, the factorization of an AfTO $\mathcal{A}_f=\mathcal{A}_b\times\{1,f\}$ is not unique and we can obtain an equivalent factorization $\mathcal{A}_f=\mathcal{A}_b^\beta\times\{1,f\}$ using a homorphism $\beta:\mathcal{A}_b\rightarrow \mathbb{Z}_2$. So as consistency check, we show that for every $\mathcal{A}_b^\beta$, there exists a corresponding factorization of the GfTO as in Eqs.~(\ref{factor1}) and (\ref{factor2}). To see this, we can use the same lemma from Ref. \cite{Drinfeld} to conclude that for every homomorphism $\beta$, there must exist an anyon $\tilde{b}\in\mathcal{A}_b$ such that

\begin{equation}
(-1)^{\beta(a)}=M_{a,\tilde{b}}\,,\;\; \forall a \in\mathcal{A}_b
\end{equation}
So we see that now the GfTO factorizes as $\mathcal{A}_b^\beta\times \{1,\tilde{b}\phi ,f, \tilde{b}\phi f\}$ or $\mathcal{A}_b^\beta\times \{1,\tilde{b}\phi ,f\}$.

\subsubsection{Example: U$(1)_4\times$ $\overline{IQH}$}

Let us give an example to illustrate the factorization property of GAfTO. We consider the multi-component U$(1)$ Chern-Simons theory

\begin{equation}
\mathcal{L}=\frac{1}{4\pi}K_{IJ}\epsilon^{\mu\nu\lambda}a_\mu^I \partial_\nu a_\lambda^J + \frac{1}{2\pi}t_I \epsilon^{\mu\nu\lambda} A_\mu \partial_\nu a_{\lambda}^I\, ,
\end{equation}
with $K$-matrix
\begin{equation}
K = \left( \begin{array}{cc} 4 & \\ & -1 \end{array}\right)
\end{equation}
This describes a $\mathcal{A}_f=\mathcal{A}_b\times \{1,f\}=\mathbb{Z}_4\times \{1,f\}$ fermionic topological order, where the transparant fermion $f$ corresponds to the vector $l_f = \left(0,1\right)^T$, and $\mathbb{Z}_4=\{a,a^2,a^3,a^4=1\}$ is generated by anyon $a$ corresponding to vector $l_a=\left(1,0\right)^T$. The topological spins are given by $\theta_{a^p}=e^{ip^2\pi l_a^TK^{-1}l_a}=e^{ip^2\pi/4}$ and $\theta_f=e^{i\pi l_fK^{-1}l_f}=-1$.  $A_\mu$ is a probe gauge field for the global U$(1)$ particle number symmetry. If we require that the fermion has charge one, then this implies that $q_f=1=l_f^TK^{-1}t= -t_2$. So the only freedom left is the first component from the charge vector $t=(t_1, -1)^T$. This freedom determines the Hall conductance, which is given by $\sigma_{xy}=t^TK^{-1}t=t_1^2/4-1$, and the U$(1)$ charges of the anyons in $\mathbb{Z}_4$: $q_a = l_a^TK^{-1}t=t_1/4$. Let us take $t_1=1$, such that $\sigma_{xy}=-3/4$ and $q_a=1/4$. If a system has U$(1)$ fermion number symmetry, then there is a preferred way to create a parity flux by adiabatically inserting $\pi$ flux of the U$(1)$ particle number symmetry. Let us denote the fermion parity flux obtained via this adiabatic procedure as $\phi_A$. As is well-known, the topological spin of $\phi_A$ is fixed by the Hall conductance. In particular, it holds that $\theta_{\phi_A} = e^{i\pi\sigma_{xy}/4}$ \cite{Goldhaber,ChengZaletel}. If we apply this formula to our example with $t_1=1$, we learn that $\theta_{\phi_A}=e^{-i3\pi/16}$ and therefore $\theta_{\phi_A^2} = e^{-i3\pi/4}$. This implies that $\phi_A\times\phi _A= af$ or $\phi_A\times \phi_A = a^3f$, which is at odds with the proposed factorization property of the GAfTO because we cannot find a parity flux $\phi= \phi_A a^p$ such that $\phi\times\phi= f$. However, the choice $t_1=1$ is not allowed. This is because if $q_a=1/4$, then $q_{a^4}=q_1= 1$. This is not possible if $t$ is the charge vector of U$(1)$ fermion number symmetry, because the trivial anyon should always have even fermion parity. So $t_1=2t'$ has to be even. With this property correctly incorporated, we find $\theta_{\phi_A} = e^{i\pi (t'^2-1)/4}$ and $\theta_{\phi_A^2}=e^{i\pi(t'^2-1)}=(-1)^{t'+1}$. Under a shift $t'\rightarrow t'+4$, the anyon charges change as $q_a\rightarrow q_a + 2$, and $\theta_{\phi_A}$ remains invariant. So the only four remaining cases we have to consider are $t'=0,1,2,3$, corresponding to respectively $q_a = 0,1/2,1,3/2$. Let us work through the case where $t'=1$. With $t'=1$, it holds that $\phi_A\times\phi_A = a^2f$. We can now define $\phi = \phi_A a^3$, such that $\phi \times\phi = f$ and $M_{a,\phi}=M_{a,a^3}M_{a,\phi_A} = e^{i3\pi/2}e^{i\pi q_a} = 1$. Therefore, the GfTO factorizes as $\{a,a^2,a^3,1\}\times \{1,\phi_Aa^3,f,\phi_Aa^3f\}$. The factorization for other choices of $t'$ can be obtained in a similar way.

\section{Numerical identification of Abelian fermionic topological order}

An important question is how one can numerically identify the type of topological order realized by a particular microscopic lattice Hamiltonian. In the past decade, it has become clear that the ground state wavefunctions contain a lot of (if not all) information about the anyonic excitations. The first example of a ground-state property that can be used to diagnose topological order is the topological entanglement entropy  \cite{KitaevPreskill,LevinWen}, which gives access to the total quantum dimension. By looking not only at the entanglement entropy, but at the entire spectrum of the reduced density matrix corresponding to some spatial region in the ground state wavefunction one also obtains universal information about possible gapless edge modes of the system \cite{LiHaldane}. This correspondence between the entanglement spectrum and edge spectrum has been worked out in full detail for free fermion systems in Ref. \cite{Alexandradinata,Fidkowski}. For systems on a torus, Ref. \cite{ZhangGrover} identified the ground states with a definite anyon flux through one of the holes of the torus as those which are `minimally entangled states' (MES) with respect to cuts wrapping the hole under consideration. In the MES basis, one can obtain both the $S$ and $T$ matrices by taking certain wave function overlaps \cite{ZhangGrover,ZaletelMong}. Finally, using the same MES on the cylinder, one can also find the $T$ matrix by calculating the `momentum polarization' \cite{ZaletelMong,TuZhang,Cincio,Zaletel,He,Wen}.

In section \ref{sec:mompol}, we first discuss the application of the momentum polarization technique to systems with non-trivial translational symmetry fractionalization, characterized by the presence of a non-trivial background anyon in each unit cell. Next to the entanglement contribution to momentum polarization as discussed in Refs. \cite{ZaletelMong,TuZhang,Cincio,Zaletel,He,Wen}, we identify an `eigenvalue contribution' which is determined by both the background anyon and the anyonic flux which labels the MES. The interplay of the entanglement and eigenvalue contributions to the momentum polarization is shown to lead to a physically intuitive picture for the behavior of the momentum polarization under a change of entanglement cut, from which one can numerically obtain the topological spin of the background anyon. Next, we discuss the application of the momentum polarization technique to fermion systems. It is shown that in fermionic systems not only non-trivial background anyons in $\mathcal{A}_b$, but also transparant `background fermions' give rise to a non-trivial eigenvalue contribution to the momentum polarization, which results entirely from the fermionic anti-commutation relations of the microscopic fermions. This eigenvalue contribution from background fermions also fits nicely with the physically intuitive picture for the dependence of the momentum polarization on the entanglement cut.

In section \ref{sec:numid} we use the results of section \ref{sec:gauging} to describe a minimal scheme to numerically identify a general AfTO in numerics, without relying on charge conservation symmetry. In particular, we make use of the special structure of GAfTO's to show that, compared to bosonic systems, the only additional piece of information that needs to be determined is $\theta_\phi$, i.e. the topological spin of the fermion parity flux. We provide a concrete (minimal) procedure to obtain $\theta_\phi$ by calculating the momentum polarization, for which one can rely on the results of section \ref{sec:mompol}.

\subsection{Momentum polarization in the presence of background anyons and its application to fermion systems}\label{sec:mompol}

\subsubsection{Bosonic systems}

Before turning to fermion systems, we first discuss the concept of momentum polarization in boson or spin systems,. Consider a MES on a cylinder with an anyon flux of the type $a$ through the hole of the cylinder. Let us denote this MES as $|\psi[a]\rangle$. We will call the direction along the axis of the cylinder the $x$-direction, and the direction wrapping the hole the $y$-direction. The size of the cylinder is given by $N_x\times N_y$ unit cells. We now choose a cut along the $y$-direction close to the middle of the cylinder, dividing the cylinder in two. The length of the left half is then $N_x^L$, while the length of the right half is $N_x^R$, such that $N_x^L+N_x^R=N_x$. With this cut, the translation operator in the $y$-direction can be written as a tensor product between the translation operator on the left half and the translation operator on the right half: $T_y = T_y^L\otimes T_y^R$. With these definitions in place, momentum polarization was defined in Refs. \cite{ZaletelMong,TuZhang,Cincio} as the expectation value $\langle\psi[a]|T_y^L|\psi[a]\rangle$. It was found that this expectation value scales with $N_y$ as

\begin{equation}\label{mompol}
\langle\psi[a]|T_y^L|\psi[a]\rangle = \exp\left(\frac{2\pi i}{N_y}\left( h_a -\frac{c_-}{24}\right) - \alpha N_y \right)\, ,
\end{equation}
where $\theta_a = e^{2\pi i h_a}$ is the topological spin of anyon $a$, and $c_-$ is the chiral central charge. The complex number $\alpha$ is non-universal.

One important assumption for the validity of Eq. (\ref{mompol}) is that there is no translational symmetry fractionalization. As we explain in more detail below, with non-trivial translational symmetry fractionalization we find that Eq. (\ref{mompol}) needs to be generalized to the following more general form:

\begin{equation}\label{mompolgeneral}
\langle\psi_b[C,a]|T_y^{C,L}|\psi_b[C,a]\rangle = \exp\left(\frac{2\pi i}{N_y}\left( h_a -\frac{c_-}{24}\right) + i\Theta_{a,b}N_x^{C,L} - \alpha_C N_y \right)\, ,
\end{equation}
where the notation for the MES $|\psi_b[C,a]\rangle$ now depends on two anyon labels $a$ and $b$, and on an integer $C$ which denotes the position of the entanglement cut. The anyon $a$ is again the anyon flux through the hole of the cylinder, measured along cut $C$, and $b$ is a `background anyon' which sits inside every unit cell \cite{Zaletel3,ChengZaletel}. The translation operator $T^{C,L}_y$ acts on the left of the cut labeled by $C$. On the right hands side, $N_x^{C,L}$ is an integer which corresponds to the length of the left half of the cylinder, and $\alpha_C$ is a non-universal complex number also depending on the cut. Although $\alpha_C$ is non-universal, we will argue that the difference $\Delta\alpha_C = \alpha_{C+1}-\alpha_C$ between two neighboring cuts is universal. The interpretation of $\Theta_{a,b}$ will be explained in the next paragraph.

To explain the general form of the momentum polarization formula Eq. (\ref{mompolgeneral}), it is useful to decompose the momentum polarization in an `eigenvalue contribution' and an `entanglement contribution'. Let us consider translationally invariant systems, such that $|\psi_b[C,a]\rangle$ is an eigenstate of $T_y$. The $y$-momentum of the MES is then determined by the anyon flux $a$ and the background anyon $b$ as follows:

\begin{equation}\label{Teig}
T_y|\psi_b[C,a]\rangle = e^{ i \Theta_{a,b} N_x}|\psi_a \rangle\, ,
\end{equation}
where $e^{i\Theta_{a,b}} = M_{a,b}$  \cite{Zaletel3,ChengZaletel}. This background anyon has to be non-trivial in systems where a Lieb-Schultz-Mattis-Oshikawa-Hastings (LSMOH) obstruction to a gapped trivial featureless phase is present \cite{LSM,Oshikawa1,Hastings,Oshikawa2,Zaletel3}. Equation (\ref{Teig}) holds irrespective of whether the system is bosonic or fermionic, although in general the types of topological orders which can satisfy the LSMOH obstruction are different in both cases \cite{FillingC}. From Eq. (\ref{Teig}) we can immediately identify the eigenvalue contribution to the momentum polarization as $e^{i\Theta_{a,b}N_x^{C,L}}$.

To identify the entanglement contribution to momentum polarization (as discussed in Refs. \cite{ZaletelMong,TuZhang,Cincio,Zaletel,He,Wen}), we write the MES as

\begin{eqnarray}
|\psi_b[C,a]\rangle & = & \sum_{\alpha,\beta} \Psi_{C,a,b}^{\alpha,\beta}|\alpha\rangle_L\otimes|\beta\rangle_R \\
 & = & \sum_{\mu} s_{C,a,b}^\mu\, |\mu\rangle_L\otimes |\mu\rangle_R
\end{eqnarray}
In the first line, we have used an arbitrary basis $|\alpha\rangle_L$ ($|\beta\rangle_R$) for the left (right) half of the cylinder. In the second line, the state is decomposed in the Schmidt basis. In the Schmidt basis, the action of translation in the $y$-direction can be written as

\begin{eqnarray}
T_y|\psi_b[C,a]\rangle & = & \sum_\mu s_{C,a,b}^\mu\,  T^{C,L}_y|\mu\rangle_L\otimes T_y^{C,R}|\mu\rangle_R \\
 & = &  e^{ i \Theta_{a,b} N_x}\sum_{\mu\lambda\sigma} s_{C,a,b}^\mu [U^{*C,L}_{a,b}]_{\mu\lambda} [U^{C,R}_{a,b}]_{\mu\sigma}|\lambda\rangle_L\otimes|\sigma\rangle_R\, ,
\end{eqnarray}
where in the second line, we have re-expanded $T^{C,L/R}_y|\mu\rangle_{L/R}$ in the Schmidt basis. Note that since $T_y^{C,L}$ and $T_y^{C,R}$ are unitary, so are $U^{C,L}_{a,b}$ and $U^{C,R}_{a,b}$. We have also separated out the eigenvalue factor $e^{i\Theta_{a,b}N_x}$. The MES $|\psi_b[C,a]\rangle$ is an eigenstate of $T_y$ if

\begin{equation}
\left(U_{a,b}^{C,L}\right)^\dagger S_{C,a,b}U_{a,b}^{C,R} = S_{C,a,b}\, ,
\end{equation}
where $S_{C,a,b} =$diag$(s_{C,a,b}^\mu)$. This implies that $U^{C,L}_{a,b}=U^{C,R}_{a,b}=U^C_{a,b}$, where $U^C_{a,b}$ commutes with $S_{C,a,b}$. The entanglement contribution to momentum polarization is then entirely given in terms of the Schmidt values and the unitary matrix $U^C_{a,b}$, and takes the form

\begin{equation}
\mathrm{tr}(S^2_{C,a,b}U^{C}_{a,b}) = \exp\left(\frac{2\pi i}{N_y}\left( h_a -\frac{c_-}{24}\right) - \alpha_C N_y \right) \, ,
\end{equation}
where we have again used the notation $\alpha_C$ to emphasize that $\alpha_C$ depends on the choice of cut. To understand how $\alpha_C$ depends on the cut, we note that because of the background anyon $b$ per unit cell, the anyon flux through the hole of the cylinder is not the same for every cut. The anyon fluxes for two neighboring cuts differ by the total background anyon charge enclosed by the two cuts, which is $b^{N_y}$. In other words, it holds that

\begin{equation}
|\psi_b[C,a]\rangle = |\psi_b[C+1,ab^{N_y}]\rangle
\end{equation}
The ribbon identity allows us to write $\theta_{ab^{Ny}} = \theta_a\theta_{b}^{N_y^2}M_{a,b}^{N_y}$. So given the momentum polarization for a particular cut, we can obtain the momentum polarization for the neighboring cut by shifting

\begin{equation}\label{intuitive}
h_a \rightarrow h_a + h_b N_y^2 + \frac{\Theta_{a,b}}{2\pi}N_y
\end{equation}
From this we conclude that $\alpha_C$ depends on the cut as $\alpha_{C+1}=\alpha_C +2\pi i h_b$. So, interestingly, even though $\alpha_C$ is non-universal, by calculating this coefficient for two neighboring cuts one can numerically obtain the topological spin of the background anyon $b$. Also, note that Eq. (\ref{intuitive}) implies that even though the anyon flux through the hole of the cylinder depends on the choice of entanglement cut if $b$ is non-trivial, the momentum polarization nevertheless allows one to obtain a topological spin $h_a$ which is independent of the choice of cut, because it relies on a scaling in the cylinder circumference $N_y$.

\subsubsection{Fermionic systems}

For fermion systems on a cylinder, there are two types of boundary conditions: periodic and anti-periodic. For each type of boundary condition, one can find a set of MES. Let us start by considering MES in the anti-periodic sector.

\emph{Anti-periodic boundary conditions --}  With anti-periodic boundary conditions, the MES have anyon fluxes through the hole of the cylinder which are labeled by the anyons in $\mathcal{A}_f$, and \emph{not} by the different types of parity fluxes. The latter label MES in the \emph{periodic} sector, which we discuss below. So let us write a MES in the anti-periodic sector as

\begin{equation}
|\psi_{f^\sigma b}[C,f^\lambda a]\rangle\, , \;\;\;\sigma,\lambda\in \{0,1\}\,, \;\;a,b \in \mathcal{A}_b\, ,
\end{equation}
where $f^\sigma b$ again denotes the background anyon and $f^\lambda a$ the anyon flux through the hole of the cylinder measured at cut $C$. Note that both the background anyon and the flux through the hole of the cylinder are labeled with anyons in $\mathcal{A}_f=\mathcal{A}_b\times\{1,f\}$, even though the ground state degeneracy on the torus is only given by $|\mathcal{A}_b|$, i.e. the number of anyons in $\mathcal{A}_b$.

In the anti-periodic sector, the translation operator along the $y$-direction is defined as

\begin{eqnarray}
\tilde{T}_y: & c^\dagger_{(x,y)}\rightarrow c^\dagger_{(x,y+1)}\,,\qquad y\neq N_y-1\\
 & c^\dagger_{(x,N_y-1)}\rightarrow -c^\dagger_{(x,1)}
\end{eqnarray}
Using this twisted translation operator, the momentum polarization is defined as

\begin{equation}\label{mompolAP}
\langle\psi_{f^\sigma b}[C,f^\lambda a]|\tilde{T}^{C,L}_y|\psi_{f^\sigma b}[C,f^\lambda a] \rangle = \exp\left(\frac{2\pi i}{N_y}\left( h_{f^\lambda a} -\frac{c_-}{24}\right) + i\Theta_{a,b}N_x^{C,L} - \alpha_C N_y \right)
\end{equation}
As in bosonic systems, the MES satisfy the following property:

\begin{equation}
|\psi_{f^\sigma b}[C,f^\lambda a]\rangle = |\psi_{f^\sigma b}[C+1,f^{\lambda+\beta N_y} ab^{N_y}]\rangle\, ,
\end{equation}
which via the replacement $h_{f^\lambda a}\rightarrow h_{f^\lambda a} + N_y^2 (\sigma h_f +h_b) + \Theta_{a,b}N_y/2\pi$ implies that $\alpha_{C+1}-\alpha_C = 2\pi i (\sigma h_f +h_b) = i(\sigma \pi + 2\pi h_b)$. The dependence of $\Delta\alpha_C$ on $\sigma$ arises from the eigenvalue contribution to the entanglement polarization. To see this, consider the situation where $\mathcal{A}_b=1$. In this case, $\sigma$ corresponds to the fermion parity per unit cell, which means that  on the torus the fermion parity of $|\psi_{f^\sigma}[C,1]\rangle$ is given by $(-1)^{\sigma N_x N_y}$. The momentum in the $y$-direction on the torus (with periodic boundary conditions in the $x$-direction) is then given by

\begin{equation}\label{transeig1}
\tilde{T}_y|\psi_{f^\sigma}[C,1]\rangle = (-1)^{\sigma N_x N_y}|\psi_{f^\sigma}[C,1]\rangle
\end{equation}
This property can readily be checked for band insulators, where $\sigma$ is the number of filled bands modulo $2$, and also follows from fermionic tensor network descriptions of gapped ground states \cite{fMPS,fPEPS}. From Eq. (\ref{transeig1}) we can identify the eigenvalue contribution to the momentum polarization as $e^{i\sigma\pi N_x^{C,L}N_y}$, which indeed leads to the dependence of $\alpha_C$ on the cut as described above.

The last aspect of the MES in the anti-periodic sector that we need to comment on is the role of $f^\lambda$. The choice of $\lambda = 0,1$ is a non-universal property of the MES, as can be seen by noting that the value of $\lambda$ can be flipped by adding a single electron in a $k_y = \pi/N_y$ momentum state on the left of the cut, which changes the (eigenvalue contribution to the) momentum polarization accordingly by a factor of $e^{i\pi/N_y}$. This reflects the fact that the ground state degeneracy on the torus is given by $|\mathcal{A}_b|$, and not $|\mathcal{A}_f|$.

\emph{Periodic boundary conditions --} On a cylinder with periodic boundary conditions, the anyonic fluxes which label the MES are given by the different fermion parity fluxes. So, with anti-periodic boundary conditions, we can write the MES as

\begin{equation}
|\psi_{f^\sigma b}[C,\phi a f^\lambda]\rangle\, , \;\;\;\sigma,\lambda\in \{0,1\}\,, \;\;a,b \in \mathcal{A}_b\, ,
\end{equation}
where, as before, $\phi$ is the fermion parity flux which braids trivially with all anyons in $\mathcal{A}_b$. The momentum polarization in the periodic sector is then given by

\begin{equation}\label{mompolP}
\langle\psi_{f^\sigma b}[C,\phi a f^\lambda]| T_y^{C,L}|\psi_{f^\sigma b}[C,\phi a f^\lambda]\rangle = \exp\left(\frac{2\pi i}{N_y}\left( h_{\phi a} -\frac{c_-}{24}\right) + i\Theta_{\phi a,bf^\sigma}N_x^{C,L} - \alpha_C N_y \right)
\end{equation}
The by now familiar property of the MES:

\begin{equation}
|\psi_{f^\sigma b}[C,\phi a f^\lambda]\rangle = |\psi_{f^\sigma b}[C+1,\phi ab^{N_y} f^{\lambda+\sigma N_y}]\rangle
\end{equation}
implies that $\alpha_C$ depends on the cut as $\alpha_{C+1}-\alpha_C = i(\sigma \pi + 2\pi h_b)$. As a consistency check, let us again take $\mathcal{A}_b = 1$ such that $\sigma$ is the fermion number per site. This means that if we define the state on the torus with periodic boundary conditions along both cycles, then it holds that

\begin{equation}
(-1)^{\hat{F}}|\psi_{f^\sigma }[C,\phi ]\rangle = (-1)^{\eta+\sigma N_x N_y}|\psi_{f^\sigma }[C,\phi ]\rangle \, ,
\end{equation}
where $\hat{F}$ is the fermion number operator, and $\eta=1$ if $\phi$ is non-Abelian \cite{ReadGreen} and $\eta=0$ otherwise. With this definition, one finds that on a torus the momentum in the $y$-direction is given by

\begin{equation}\label{neighboringcuts}
T_y|\psi_{f^\sigma }[C,\phi ]\rangle = (-1)^{ \sigma N_x(N_y + 1)} |\psi_{f^\sigma }[C,\phi ]\rangle\, ,
\end{equation}
Again, this property can easily be checked for band insulators, and can also be seen in the fermionic tensor network formalism \cite{fMPS,fPEPS}. Eq. (\ref{neighboringcuts}) implies that the momentum polarizations for two neighboring cuts indeed differ by a factor $(-1)^{\sigma(N_y +1)} = e^{i\left(\Theta_{\phi,f^\sigma}+\sigma \pi N_y\right)}$, which arises entirely from the eigenvalue contribution to the momentum polarization.

Finally, we note that the momentum polarization with periodic boundary conditions is independent of $f^\lambda$. This agrees with the fact that the topological spins of the fermion parity fluxes are invariant under the addition of a transparent fermion.

In the appendix, we illustrate our general discussion of momentum polarization in fermionic systems by applying it to a Chern insulator and a topological $p+ip$ superconductor. In particular, we show that both in the anti-periodic and periodic sectors, the dependence of the momentum polarization on the choice of cut is indeed captured respectively by Eqs. (\ref{mompolAP}) and (\ref{mompolP}), with $\alpha_{C+1}-\alpha_C = i\pi$.

\subsection{Numerically determining the topological order}\label{sec:numid}

Having uncovered the special structure of GAfTO's in section \ref {sec:gauging}, and the details of momentum polarization in fermionic systems in section \ref{sec:mompol}, we now have all the necessary ingredients at our disposal to outline a minimal numerical detection scheme for the most general AfTO. First, by calculating the momentum polarizations for the different MES on the cylinder in the anti-periodic sector we obtain the topological spins of the anyons in $\mathcal{A}_b$, up to a minus sign ambiguity. Secondly, for the MES on the torus with anti-periodic boundary conditions along both cycles, we can use the formalism of Refs. \cite{ZaletelMong,ZhangGrover,Cincio} to obtain the unitary $S$-matrix of the MTC corresponding to $\mathcal{A}_b$, just as one does for bosonic systems. From the $S$-matrix, one obtains the fusion rules (i.e. the group structure) of $\mathcal{A}_b$ via the Verlinde formula \cite{Verlinde}

\begin{equation}
N_{a,b}^c = \sum_d \frac{S_{a,d}S_{b,d}S_{d,c}^*}{S_{0,d}}\, ,
\end{equation}
Once the group structure of $\mathcal{A}_b$ is obtained, this can be used to partially fix the sign ambiguity in the topological spins of the anyons in $\mathcal{A}_b$, by imposing that $S_{a,b} = N^{-1/2}M_{a,b} = N^{-1/2}\theta_{ab}/(\theta_a\theta_b)$, where $N$ is the number of MES (= the number of anyons in $\mathcal{A}_b$). This fixes the topological spins up to a homomorphism $\beta$ from $\mathcal{A}_b$ to $\mathbb{Z}_2$. This is the same homomorphism as discussed above, and different $\beta$ correspond to different ways of writing $\mathcal{A}_f = \mathcal{A}_b^\beta\times \{1,f\}$, where $\mathcal{A}_b^\beta = \{af^{\beta(a)}|a\in\mathcal{A}_b\}$. At this point, one has to choose a particular homomorphism to fix the topological spins.

To completely determine the topological quantum phase of the system of interest one does not only need to know $\mathcal{A}_f$, but also the complete algebraic data corresponding to the modular extension $\mathcal{A}_b\times \{1,f,\phi,f\phi\}$ (if $\phi$ is Abelian) or $\mathcal{A}_b\times\{1,f,\phi\}$ (if $\phi$ is non-Abelian). Let us first consider the case where the fermion parity fluxes are Abelian. Since $\phi$ is Abelian, it holds that either $\phi \times \phi = 1$ or $\phi\times\phi=f$. This in turn implies that $\theta_\phi^4=1$ or $\theta_\phi^4=-1$, which gives us eight different possible values for $\theta_\phi$. Once we know the topological spin of the Abelian parity flux, we have completely fixed which of the eight possible modular extensions is realized. When $\mathcal{A}_b=1$, this was shown by Kitaev as a part of his `$16$-fold way' \cite{Kitaev}.

A non-Abelian parity flux satisfies $\phi\times\phi=1+f$. If $\mathcal{A}_b=1$, then Kitaev has shown that there exist exactly eight different modular extensions with non-Abelian parity fluxes \cite{Kitaev}. These eight different modular extensions correspond to the eight different Ising MTC's, and are uniquely identified by the topological spin of the fermion parity flux. In Ref. \cite{Wang}, it was conjectured that Kitaev's 16-fold way generalizes to all fTO's (Abelian and non-Abelian), i.e. it was conjectured that every fTO has exactly 16 different modular extensions, 8 of which have Abelian fermion parity flux and 8 which have non-Abelian fermion parity flux. When the fTO is Abelian, it was shown above that the GfTO factorizes as $\mathcal{A}_b\times\{1,f,\phi\}$, so there are indeed eight different non-Abelian modular extensions which again correspond the eight different Ising categories.

From the above discussion, we learn that to complete the numerical identification of an Abelian fermionic topological order we only need to know $\theta_\phi$. To access $\theta_\phi$, one first calculates the momentum polarizations of the MES on the cylinder with periodic boundary conditions. This provides a set of topological spins, which correspond to $\theta_{a\phi}$. Because $\phi$ braids trivially with all anyons in $\mathcal{A}_b$, these topological spins factorize as $\theta_{a\phi} = \theta_a \theta_\phi$. This implies that we can organize the topological spins in the periodic sector in an ordered vector, which is proportional to the ordered vector of topological spins in the anti-periodic sector. The proportionality constant obtained in this way is unique and corresponds to $\theta_\phi$, such that one can simply try all the possible permutations of the topological spins in the periodic sector until one finds one where the required proportionality is realized. To see why the proportionality constant is unique we need to show that we cannot permute the vector of topological spins in the periodic sector to obtain a vector that is proportional to (and different from) the original, unpermuted one. Since we are only interested in permutations that do not leave the vector invariant, there can be no element in the vector that is fixed under the permutation. This means that the permutation acts as $\theta_{a\phi} \rightarrow \theta_{ad\phi}$, where $d$ is an anyon in $\mathcal{A}_b$ which is not the trivial anyon. Because $\theta_{ad\phi} = \theta_{a\phi}\theta_d M_{a,d}$, the resulting vector is proportional to the unpermuted one if and only if $M_{a,d}$ is independent of $a$. But this is a violation of modularity, and therefore this cannot happen. This completes the procedure of how to uniquely characterize an AfTO in numerics.

Before concluding the discussion on numerical identification of fermionic topological orders, let us consider what happens if one would have made a different choice of homomorphism $\beta$. In that case, $\mathcal{A}_b$ becomes $\mathcal{A}_b^\beta = \{af^{\beta(a)}|a\in \mathcal{A}_b\}$ and one would interpret the set of topological spins obtained from momentum polarization in the periodic sector as $\theta_{af^{\beta(a)}\tilde{\phi}} = \theta_{af^{\beta(a)}\phi \tilde{b}}$, where $\tilde{b}$ is the unique anyon that satisfies $(-1)^{\beta(a)} = M_{a,\tilde{b}}$. Using this property, one can factorize the topological spins in the periodic sector as

\begin{equation}
\theta_{af^{\beta(a)}\tilde{\phi}}=\theta_{af^{\beta(a)}\phi \tilde{b}} = \theta_{af^{\beta(a)}}\theta_{\phi \tilde{b}} = \theta_{af^{\beta(a)}}\theta_{\tilde{\phi}} \, ,
\end{equation}
so we can again permute them in such a way that they become proportional --as a vector-- to the vector of topological spins $\theta_{af^{\beta(a)}}$ in the anti-periodic sector. Of course, the set of topological spins in the periodic sector obtained from momentum polarization is independent of our choice of $\beta$, as can easily be verified:

\begin{eqnarray}
\theta_{af^{\beta(a)}\tilde{\phi}} = \theta_{af^{\beta(a)}\phi \tilde{b}}=  \theta_{a\tilde{b}\phi}
\end{eqnarray}
So we find that the final identification of the AfTO is independent of our choice of $\beta$, as it should be of course. One only needs to keep in mind that when comparing two different AfTO with the same $\mathcal{A}_f$, one should always use the same of choice of $\beta$ to compare the topological spins of the fermion parity fluxes $\phi$.

\section{Gauging fermion parity with boundaries}

\subsection{Fermionic vs. bosonic gapped edges}

In this section, we address the question of what happens at the boundary of an AfTO after fermion parity is gauged. To gain some intuition about this question, let us consider a fTO which has a gapped boundary $\mathcal{B}_0$ separating it from the trivial phase. Now imagine gauging the fermion parity everywhere in the bulk, except in a narrow strip along the edge. After gauging, the GfTO in the bulk is separated by a boundary $\mathcal{B}_1$ from a narrow strip of the original ungauged fTO, which itself is separated from the trivial phase by the gapped boundary $\mathcal{B}_0$. See figure \ref{fig:B1}(a) for an illustration. If $\mathcal{B}_1$ is also gapped, then this construction gives a gapped boundary separating the GfTO from the trivial phase.

\begin{figure}
\begin{center}
\includegraphics[scale=0.5]{B1.pdf}
\caption{A GfTO obtained by gauging a fTO with a gapped boundary $\mathcal{B}_0$ to the trivial phase. The gauging is done such that a narrow strip along the boundary is unaffected and remains in the original fTO phase. The boundary separating the GfTO from the ungauged fTO is denoted as $\mathcal{B}_1$.}\label{fig:B1}
\end{center}
\end{figure}

To argue why we can always take $\mathcal{B}_1$ to be gapped, let us first consider the case of bTO. In bosonic systems, the `ungauging' procedure corresponds to condensing the gauge charges \cite{Barkeshli}, which are always bosonic and braid trivially with each other. Condensing the gauge charges results in confinement of the gauge fluxes, which means that after condensation the energy of a flux pair grows linearly with the spatial separation between the fluxes. Condensation of the gauge charges thus transforms the gauge fluxes into the symmetry defects of the ungauged phase \cite{Barkeshli}. Using the general relation between anyon condensation and gapped boundaries \cite{Levin,KitaevKong}, we can then always construct the gapped boundary $\mathcal{B}_1$ between the gauged and ungauged bTO as a domain wall where the gauge charges get condensed.

In GfTOs, the $\mathbb{Z}_2$ gauge charge $\tilde{f}$ is by definition a fermion. Because $\tilde{f}$ has non-trivial topological spin, it is impossible to construct a gapped boundary where $\tilde{f}$ gets condensed. However, at a domain wall between the GfTO and the original fTO, we can condense the bound state $\tilde{f}f$, where $f$ is the transparant fermion of the fTO. Because $\tilde{f}f$ is a boson, this will result in a gapped boundary. See Refs. \cite{Aasen,Wan} for more details on this construction.

At this point, we have obtained an argument that gauging fermion parity always preserves a gapped edge. Let us now connect this argument to the formalism of Lagrangian subgroups reviewed above in Secs. \ref{GappedB} and \ref{GappedF}, where it was stated that an Abelian bosonic (fermionic) TO admits a gapped edge iff it has a bosonic (fermionic) Lagrangian subgroup. Since the GfTO is modular it corresponds to a bTO, and so according to Sec. \ref{GappedB} we would expect it to have a bosonic Lagrangian subgroup if it admits a gapped boundary. However, this is far from clear from the argument presented above. In the `layered-boundary' construction the fermionic bulk gauge charge $\tilde{f}$ is bound to a transparant, microscopic fermion $f$  which only lives on the edge and the resulting bound state is subsequently condensed. We will refer to such a gapped edge which relies on the presence of microscopic boundary fermions as a `fermionic gapped edge'. The conventional notion of a gapped edge for a bTO as used in Sec. \ref{GappedB}, however, does not permit the use of microscopic fermionic degrees on the boundary, and requires all condensed particles to be bosons. We will refer to such a gapped edge which does not contain microscopic fermion degrees of freedom as a `bosonic gapped edge'. Using this terminology, Secs. \ref{GappedB} and \ref{GappedF} then simply state that an Abelian bosonic (fermionic) TO admits a bosonic (fermionic) gapped edge iff it has a bosonic (fermionic) Lagrangian subgroup. So we see that the `layered-boundary' argument only tells us that we can construct a fermionic gapped edge for the GfTO, but not necessarily a bosonic gapped edge.

In the next two sections, we will show the stronger statement that the GfTO always admits a bosonic gapped edge if the ungauged fTO admits a fermionic gapped edge. We will also show the inverse implication, i.e. that an AfTO has a fermionic gapped edge if the gauged theory has a bosonic gapped edge. One of the implications of this result is that any bosonic topological order which is obtained by gauging a fermionic topological order with a gapped edge, can be realized by a purely bosonic lattice Hamiltonian with a gapped edge.

\subsection{Abelian parity flux}

In this section, we will show that a GAfTO has a bosonic gapped edge iff the corresponding ungauged AfTO has a fermionic gapped edge, while assuming that the fermion parity flux is Abelian. The case with non-Abelian fermion parity flux will be discussed in the next section.

When the parity fluxes are Abelian it is not difficult to see that if we apply Eq.~(\ref{kit}) to the modular GAfTO, we get the following expression:

\begin{equation}\label{GM1}
e^{2\pi ic_-/8} = \frac{\theta_\phi}{\sqrt{N_b}}\sum_{a\in\mathcal{A}_b}\theta_a \,,
\end{equation}
where $N_b$ is the number of anyons in $\mathcal{A}_b$. Because gauging a discrete symmetry cannot change the chiral central charge, Eq. (\ref{GM1}) not only determines $c_-$ of the GAfTO (mod 8), but also of the original ungauged fTO \cite{TianLan1,TianLan2}.

If we apply Eq. (\ref{GM1}) to a different factorization $\mathcal{A}_b^\beta\times\{1,\tilde{b}\phi,f,\tilde{b}\phi f\}$ of the GAfTO, we get

\begin{eqnarray}
e^{2\pi ic_-/8} & = &  \frac{\theta_{\tilde{b}\phi}}{\sqrt{N_b}}\sum_{a\in\mathcal{A}_b^\beta}\theta_a \\
 & = & \frac{\theta_{\phi}\theta_{\tilde{b}}}{\sqrt{N_b}} \sum_{a\in\mathcal{A}_b}\theta_a (-1)^{\beta(a)} \label{GM2}
\end{eqnarray}
Because $M_{a,\tilde{b}}=(-1)^{\beta(a)}$ for all $a\in\mathcal{A}_b$, it follows that $\tilde{b}^2$ braids trivially with all anyons in $\mathcal{A}_b$, and must therefore be the trivial anyon. From $\tilde{b}^2=1$, we know that $\theta_{\tilde{b}}^4=1$. So by equating Eqs. (\ref{GM1}) and (\ref{GM2}), we find that the insertion of the minus signs $(-1)^{\beta(a)}$ in the sum of the topological spins of $\mathcal{A}_b$ changes the value of that sum by a multiplicative factor which is a fourth root of unity, and equals $\theta_{\tilde{b}}^*$. Now we are equipped to show the following result:
\begin{center}
\emph{A GAfTO has a bosonic Lagrangian subgroup $\mathcal{L}_b$ if and only if the corresponding ungauged AfTO has a fermionic Lagrangian subgroup $\mathcal{L}_f$, and an integer chiral central charge which is a multiple of eight.}
\end{center}
Using the bulk-boundary connection reviewed in the previous sections, this result then implies that a GfTO has a bosonic gapped edge if and only if the ungauged fTO has a fermionic gapped edge.

We first show the `if' direction and assume that the fTO has a fermionic Lagrangian subgroup $\mathcal{L}_f$ and zero chiral central charge (mod 8). To start, we observe that because the anyons in $\mathcal{L}_f$ braid trivially with each other, it follows from the ribbon identity that their topological spins form a $\mathbb{Z}_2$ valued representation of $\mathcal{L}_f$. In other words, $\theta_{l_i}=(-1)^{\alpha(l_i)}$ such that $\alpha(\cdot):\mathcal{L}_f\rightarrow \mathbb{Z}_2=\{0,1\}$ is a homomorphism. In general this homomorphism cannot be extended to a homomorphism from $\mathcal{A}_b$ to $\mathbb{Z}_2$. Using this homomorphism, we define $\mathcal{L}'_f = \{l_if^{\alpha(l_i)}|l_i\in\mathcal{L}_f\}$ such that all anyons in $\mathcal{L}_f'$ have trivial topological spin. We now want to extend $\mathcal{L}'_f$ in such a way that it becomes a bosonic Lagrangian subgroup of the GAfTO. Because all anyons in $\mathcal{L}_f'$ braid trivially with $f$ and because $f$ itself cannot be in $\mathcal{L}_b$, the extended Lagrangian subgroup $\mathcal{L}_b$ will have to contain a fermion parity flux. At this point, it is clear that we can find a bosonic Lagrangian subgroup $\mathcal{L}_b=\mathcal{L}'_f\times\{1,\phi c\}$ of the GfTO, provided that there exists an anyon $c\in \mathcal{A}_b$ such that: (1) $M_{c,l}= (-1)^{\alpha(l)}$ for all $l\in\mathcal{L}_f$, and (2) $\theta_\phi = \theta_c^*$. Here, we have used the factorization property of GAfTO to define $\phi$ as the parity flux which braids trivially with all anyons in $\mathcal{A}_b$.

Let us first show that there exists an anyon $c$ satifying property (1), i.e. $M_{c,l}=(-1)^{\alpha(l)}$ for all $l\in\mathcal{L}_f$. If the homomorphism $\alpha: \mathcal{L}_f\rightarrow \mathbb{Z}_2$ is trivial, then property (1) is also trivial and $c$ is simply the identity anyon. Let us therefore focus on the case where $\alpha$ is non-trivial and write $\mathcal{A}_b = \{\mathcal{L}_f, \mathcal{L}_f\times c_1, \mathcal{L}_f\times c_2,\dots,\mathcal{L}_f\times d_1,\mathcal{L}_f\times d_2,\dots \}$, where $c_i, d_i$ are a set of anyons not in $\mathcal{L}_f$ of which the $c_i$ satisfy criterion (1), and the $d_i$ do not. Using Eq. (\ref{GM1}), we find

\begin{eqnarray}
\sqrt{N_b}\theta_\phi^* e^{2\pi ic_-/8} & = & \sum_{l\in\mathcal{L}_f}\theta_l + \sum_{i}\sum_{l\in\mathcal{L}_f}\theta_{lc_i}+\sum_{i}\sum_{l\in\mathcal{L}_f}\theta_{ld_i} \\
& = & \sum_{i}\theta_{c_i}\sum_{l\in\mathcal{L}_f}(-1)^{\alpha(l)}M_{l,c_i}+\sum_{i}\theta_{d_i}\sum_{l\in\mathcal{L}_f}(-1)^{\alpha(l)}M_{l,d_i} \\
& = & N_{L_f}\sum_{i}\theta_{c_i}\label{ci}\, ,
\end{eqnarray}
where $N_{L_f}$ is the number of anyons in $\mathcal{L}_f$. In the second and third line we have applied Schur's orthogonality relations to the 1D irreps of $\mathcal{L}_f$. From this result, we see that there must exist at least one anyon $c_i$.

To show that we can find an anyon $c$ satisfying both properties $(1)$ ($M_{c,l}=(-1)^{\alpha(l)}$), and $(2)$ ($\theta_c=\theta_\phi^*$), we proceed as follows. Because the $c_ic_j$ braid trivially with all anyons in $\mathcal{L}_f$, it follows from the definition of a fermionic Lagrangian subgroup that $c_ic_j \in \mathcal{L}_f$. This implies that

\begin{equation}
\theta_{c_ic_j} = (-1)^{\alpha(c_ic_j)} =  M_{c_i,c_ic_j}
\end{equation}
Applying the ribbon identity to the left-hand side of this equation, we find

\begin{equation}
\theta_{c_i}\theta_{c_j} M_{c_i,c_j} =  M_{c_i,c_i}M_{c_i,c_j} = \theta_{c_i}^2 M_{c_i,c_j} \Rightarrow \theta_{c_i} = \theta_{c_j}
\end{equation}
Because the topological spins of all the $c_i$ are the same, expression (\ref{ci}) for the chiral central charge becomes

\begin{equation}
e^{2\pi ic_-/8} = \frac{N_{L_f}}{\sqrt{N_b}}N_c \theta_\phi \theta_c \, ,
\end{equation}
where $N_c$ is the number of $c_i$. From the assumption that the chiral central charge $c_-$ is a multiple of eight, we find that the topological spin of the $c_i$ indeed satisfies $\theta_c = \theta_\phi^*$. Because $N_{L_f}=\sqrt{N_b}$, it also follows that $N_c=1$, i.e. the anyon $c$ satisfying $\theta_c=\theta_\phi^*$ and $M_{l,c}=\theta_l$ for all $l\in\mathcal{L}_f$ is unique.

The `only if' direction is almost trivial to show. If we assume that the GAfTO = $\mathcal{A}_b\times\{1,\phi,f,\phi f\}$ has a bosonic Lagrangian subgroup, we know that there exists a group of anyons $\mathcal{M}\subset\mathcal{A}_b$ that braid trivially with each other, and have the property that every $a\in\mathcal{A}_b$ which is not in $\mathcal{M}$ braids non-trivially with at least one anyon in $\mathcal{M}$. Because the anyons in $\mathcal{M}$ have trivial mutual braiding, it follows from the ribbon identity that their topological spins form a representation of $\mathcal{M}$: $\theta_{m_1}\theta_{m_2} = \theta_{m_1m_2}$ for all $m_1,m_2\in\mathcal{M}$. From the relation $\theta_{m}^2=M_{m,m}=1$ between topological spin and self-braiding, we also know that $\theta_{m}=\pm 1$ for all $m\in\mathcal{M}$. But this implies that $\mathcal{M}=\mathcal{L}_f$ is a fermionic Lagrangian subgroup of $\mathcal{A}_f=\mathcal{A}_b\times\{1,f\}$.

\subsubsection{Example: gauging U$(1)_4\times\overline{IQH}$}
To illustrate the general result we revisit the example discussed above, i.e. the $\mathcal{A}_f=\mathbb{Z}_4\times\{1,f\}=\{a,a^2,a^3,1\}\times\{1,f\}$ AfTO. Regardless of how we choose the charge vector $t=(2t',-1)$, this AfTO always has a fermionic Lagrangian subgroup given by $\mathcal{L}_f=\{1,a^2\}$. Recall that $\theta_{a^2}=-1$, so this would not be a valid Lagrangian subgroup if the system were bosonic. We first consider the case where the charge vector is given by $t=\left(2,-1\right)^T$. This implies that $\sigma_{xy}=0$, and $q_a=1/2$. This model is known to have a gapped edge, as it is equivalent to a fermionic $\mathbb{Z}_2$ gauge theory \cite{FillingC} (i.e. a fermionic toric code \cite{GuWang}). Let us again denote the fermion parity flux obtained by adiabatic flux insertion as $\phi_{A}$. Because $\sigma_{xy}=0$, $\phi_{A}$ is a boson. Because $a^2$ has charge one (recall that $q_{a^p} = pt'/2$), it also follows that $M_{\phi_{A},a^2}=-1$. From this we see that $\mathcal{L}_b=\{1,a^2f,\phi_{A},\phi_{A}a^2f\}=\{1,a^2f\}\times\{1,\phi_A\}=\mathbb{Z}_2\times\mathbb{Z}_2$ is a bosonic Lagrangian subgroup of the GAfTO.

Let us now repeat this analysis for the case where $t=\left(0,-1\right)$. This choice of charge vector implies that $\sigma_{xy}=-1$ and $q_a=0$. This fTO is just the stacking of a $\sigma_{xy}=-1$ IQH state, and a purely bosonic U$(1)_4$ topological order (because all anyons in U$(1)_4$ have trivial fermion parity charge). The parity flux obtained by adiabatic flux insertion now has topological spin $\theta_{\phi_{A}}=e^{i\pi\sigma_{xy}/4}=e^{-i\pi/4}$. The flux $\phi_A$ also braids trivially with all anyons in $\mathcal{A}_b=\{1,a,a^2,a^3\}$ because they have zero charge. It is now easy to check that $\mathcal{L}_b=\{1,a\phi_{A},a^2f,a^3\phi_{A}f\}=\mathbb{Z}_4$ is the Lagrangian subgroup of the GAfTO. So in this example, the anyon $a$ plays the role of the special anyon $c$ which occured in the general proof above.

\subsection{Non-Abelian parity flux}

If the fermion parity flux, and therefore also the GfTO, is non-Abelian, we have to use a generalization of Eq. (\ref{kit}) to determine the chiral central charge modulo eight from the bulk data. Writing the quantum dimensions of the anyons in the GfTO order as $d_a$, the general expression for $c_-$ becomes \cite{Kitaev}:

\begin{equation}\label{nonAb}
e^{2\pi i c_-/8} = \frac{1}{\mathcal{D}} \sum_a d_a^2 \theta_a\, ,
\end{equation}
where the total quantum dimension is given by $\mathcal{D}=\sqrt{\sum_a d_a^2}$. In a GAfTO $\mathcal{A}_b\times\{1,\phi,f\}$, the only non-Abelian anyons are $\mathcal{A}_b\times \phi$, and these are therefore the only anyons which have a quantum dimension different from one. From the fusion rule $\phi\times\phi = 1+f$, it follows that $d_\phi = \sqrt{2}$. Applying Eq. (\ref{nonAb}) to a non-Abelian GAfTO, we find

\begin{equation}\label{eqdouble}
e^{2\pi i c_-/8} = \frac{\theta_\phi}{\sqrt{N_b}}\sum_{a\in\mathcal{A}_b} \theta_a
\end{equation}
This is exactly the same expression as the one we obtained for Abelian fermion parity fluxes.

We can now easily argue that an AfTO with non-Abelian fermion parity fluxes can never have a gapped edge. First, we note that $N_b^{-1/2}\sum_{a\in\mathcal{A}_b}\theta_a$ is always an eighth root of unity. This is because Abelian topological orders have a multi-component U$(1)$ Chern-Simons description, such that the corresponding edge theories are chiral Luttinger liquids with integer chiral central charge \cite{XGWen}. On the other hand, if $\phi$ is non-Abelian, it has a topological spin $\theta_\phi = e^{2\pi i (2n+1)/16}$, where $n$ is an integer \cite{Kitaev}. So from Eq. (\ref{eqdouble}), it follows that $c_-$ is a half odd integer, which means that the edge is always chiral and cannot be gapped.

\section{Conclusions}
In this work we have explored the special structure of gauged Abelian fermionic topological orders, and we have exploited this structure to study both the numerical detection of such phases and the fate of the edge physics under the gauging process. We have outlined a minimal scheme to uniquely identify the AfTO realized by a microscopic lattice Hamiltonian, which does not make use of fermion number conservation symmetry. We have also shown that a gauged fTO can have a gapped bosonic edge to the vacuum if and only if the original, ungauged fTO admits a fermionic gapped edge.

An obvious question is of course how to generalize these results to systems with non-Abelian anyons. The mathematical framework required  to address non-Abelian fermionic topological orders is developed and discussed in Refs. \cite{GuWangWen,TianLan1,TianLan2,Wang,Aasen}, and it is substantially more involved than the simple arguments used in this work. However, the understanding of Abelian systems provides a clear, intuitive picture of the physics involved, and hopefully this will be helpful for a rigorous study of non-Abelian systems. We leave such a study for future work.

\subsubsection{Acknowledgements} I am grateful to Meng Cheng for pointing out theorem 3.13 from Ref. \cite{Drinfeld} to me, and for a collaboration on a previous project which was very useful for the present paper. I also want to thank Mike Zaletel for inspiring discussions which formed the motivation for the present work, and Johannes Motruk for discussions and for pointing me to some important references. During the completion of this work I was supported by the DOE, office of Basic Energy Sciences under contract no. DE-AC02-05-CH11231.