\section{Introduction}
\label{sec:intro}

There are very many situations in astrophysics where one object or a
group of objects is exposed to relatively strong radiation
produced by a nearby external source. Examples of such situations
include planets and moons irradiated by their host star,
a star irradiated by its companion in a binary system,
gas clouds irradiated by a nearby stellar cluster or by
an active galactic nucleus (AGN), and finally
an outer part of a flaring accretion disk irradiated by its inner part
or by the accretor.

Radiative heating caused by irradiation can change the irradiated object
in several ways, e.g., it could change its structure, shape, size
and the overall appearance. It  could also lead to a significant mass loss and
even acceleration of the cloud away from the source of radiation
(via the so-called rocket effect).
To some degree, similar changes could be caused by the radiation force
(i.e., without radiative heating).

Effects of irradiation are most profound (e.g., cause ablation and destruction)
in cases where the radiation energy is relatively high and the mass of
the irradiated object is small so that self-gravity is negligible.
Such cases are relevant in a variety of astrophysical
environments, e.g., in the interstellar medium (ISM) \citep[e.g.,][]
{OortSpitzer1955,Bertoldi1989,BertoldiMcKee1990,Bally1995},
in planetary nebulae \citep[e.g.,][]{Mellemaetal1998},
near the central region of AGN
\citep[e.g.,][and references therein]{Mathews1986}
and outside the AGN host galaxy, in the intergalactic
medium (IGM) \citep[e.g.,][]{DonahueShull1987}.

The key questions in this context include:
How does the radiation that is reflected, reprocessed, or transmitted
by a cloud compare to the original external radiation?
What aspects and to what degree can the observed structure and kinematics
be accounted for by irradiation?
What are the dominant processes responsible for
dispersing the gas that was initially collected in the cloud?
What are the dominant processes responsible for accelerating the cloud?
Can a cloud be significantly accelerated before it is dispersed?
How do the acceleration and dispersing rates compare with the cloud
formation rate?

Answering these questions is hard from both observational and theoretical
points of view because the evolution of real clouds is very complex
and too slow to be measured directly by observations. In addition,
only in a few cases are clouds well spatially resolved (i.e.,
those in the nearby ISM, e.g., \cite[][]{Bally1995}).
Proper modeling of the clouds is further complicated by the fact that many
time-dependent and multi-demensional processes and effects are
involved, e.g., radiative transfer (RT), gas photoionization and heating,
the subsequent development and propagation of ionization and thermal fronts
(IF and TF, respectively) and of shocks and discontinuities.
Generally, the cloud irradiation problem requires the simultaneous solving of
radiation-hydrodynamic (R-HD) equations
\citep[examples of such studies include][]{LeflochLazareff1994,
Mellemaetal1998,Gonzalezetal2007,Ragaetal2007}.
One important aspect of this problem is that
irradiation is anisotropic and optically thick clouds will cast shadows.
Therefore, the RT methods for solving the R-HD equation
have to treat the shadows accurately
\citep[e.g.,][]{Davisetal2012,Jiangetal2012}.
Effects of magnetic fields and dust increase the level of complexity
even more \citep[e.g.,][and references therein]{Krauseetal2012}.

\subsection{Simple Cases}
\label{sec:scases}

However, under some special and idealized conditions,
the solution to the problem of the  evolution of a spherical cold cloud
that is
suddenly exposed to external radiation could be quite trivial.
For example, in optically thin cases with pure absorption opacity,
the radiation would propagate very fast (faster than sound)
throughout a cloud. If the cloud was initially of constant density
and in pressure equilibrium with the uniform ambient medium, it
will be uniformily heated. Consequently, the cloud will
expand without changing its shape and without gaining net momentum (i.e.,
it will behave like an expanding balloon for weak irradiation or
an exploding sphere for strong irradiation).
We will refer to such cases as simple example~I.

Another class of trivial solutions exists for optically thin clouds with
pure scattering opacity. Then again the radiation would propagate
throughout the cloud very fast and afterward the cloud would
experience a uniform acceleration away from the radiation source.
The acceleration would be constant also with time (for
small clouds far from the radiation source) and the cloud would gain
momentum without changing its size and shape and without mass loss
(i.e., in some respects, it will behave like a bullet).
We will refer to such cases as simple example~II.

Another simple yet non-trivial example, is when a cloud is very optically
thick due to absorption opacity and it is exposed to weak radiation.
In such a case, the part of the cloud facing the radiation source
will be gently heated and the radiation will penetrate only a thin
layer of the cloud. The IF and TF would move very slowly across the cloud and
there would be no shock. In addition, the cloud would slowly lose mass from
its heated part and would be in a quasi-steady state. This simple case
and the early evolution of other special cases can be and have been
studied using analytic methods that have made various simplifying assumptions
in order to estimate for example, the mass loss rate, the final velocity
of the neutral cloud, and shape of the IF
\citep[e.g.,][]{OortSpitzer1955,Bertoldi1989}.

\subsection{Clouds in AGN}
\label{sec:agncl}

It is challenging to determine what type of clouds
are most relevant in a given environment. It is especially true
for the so-called Broad Line Regions (BLRs)
in AGN because they are spatially unresolved.
In most studies that aim at interpreting
or modeling the observed line emission and absorption produced
in the BLRs,
the cloud propriety -- such as the density, size, and shape --
have been assumed
\citep[e.g.,][and references therein]{Mathews1974,BlumenthalMathews1975,BlumenthalMathews1979,Capriottietal1981,Mathews1982,AravLi1994,Krolik1999}.

The presence of broad emission lines (BELs) and broad absorption lines
(BALs) in AGN spectra shows that AGN continuum radiation affects
the AGN's immediate environment.
BELs are one of the defining spectral features of AGN. They are observed
in optical and ultraviolet (UV) spectra and have line wings extending
to velocities up to $10^4~{\rm km~s^{-1}}$.
It is well established that the primary physical mechanism
for production of BELs is photoionization by the compact
continuum source of AGN \citep[e.g.][and references
therein]{KwanKrolik1981,FerlandElitzur1984,
Ferlandetal1998,HamannFerland1999,Krolik1999}. Detailed
photoionization calculations presented in these and other studies yield
relatively tight constraints on some physical conditions
of the emitting gas (e.g., the gas temperature,
$T_g\approx 10^4$~K, the number density, $n\approx10^9~{\rm cm^{-3}}$,
the column density $N\mathrel{\copy\simgreatbox} 10^{22}~{\rm cm^{-2}}$, and the ionizing
flux, $F_{ion}$ is so high that
the ratio between the radiation to gas pressure $\Xi\equiv F_{ion}/c n k T\approx 1$).

The width of BELs indicates that the emitting gas is highly {\it supersonic}.
The shape and position of the BEL profiles has been traditionally explained
by lines emitted in a cloudy region without a preferred velocity direction
and with a nearly spherical distribution
\citep[e.g.,][and references therein]{UrryPadovani1995,Krolik1999}.
We note that another possibility is that the BEL are produced
at the base of a wind from an accretion disk
\citep[e.g.,][]{Murrayetal1995,Bottorffetal1997}.

The key issues that any cloudy model for the BLRs faces are
stability and confinement of the clouds
\citep[see][for reviews]{OsterbrockMathews1986,Krolik1999,Krauseetal2012}.
In short, the clouds in the BLRs
are  hydrodynamically unstable, the nature of their confinement
is unclear, while the production of new clouds
appears to be inefficient and requires rather extreme conditions.
In addition, it has been argued that radiation would cause
significant shearing and destroy the clouds before they could
contribute to the line emission \citep[e.g.,][]{Krolik1988,MathewsDoane1990}.

As we mentioned earlier, modeling irradiated clouds is a very challenging
problem. Many previous studies of clouds in AGN used various simplifying
assumptions and the robustness and applicability of their results remain quite
uncertain. Perhaps the  most robust results of the previous
work is that BELs in AGN are produced by  a
photoionized and supersonic medium that is optically thick
(i.e., $N > 10^{22}~{\rm cm^{-2}}$)
\citep[e.g.,][]{KwanKrolik1981,FerlandElitzur1984,SneddenGaskell2007,
Reesetal1989}.
If this medium is indeed made of optically thick clouds
then one must accurately treat RT, in particular, shadows.
This requires solving self-consistently time-dependent,
multidimensional R-HD equations.

The direction of the flux is not known independently of the energy
density. Therefore,  methods based on the
diffusion approximation cannot represent shadows.
This has been demonstrated through the irradiation
of very optically thick structures with a beamed radiation field
\citep[e.g.,][]{HayesNorman2003,Gonzalezetal2007}.
However, there are other methods, like the variable Eddington tensor
(VET) method that can capture shadows accurately.

For example, \cite{Davisetal2012} have developed algorithms
to solve the RT equation using short characteristics to compute the VET.
The tests described in \cite{Davisetal2012}
show the accuracy of the RT solver for two radiation beams
(see Figure 6 of that paper).
This solver was implemented in the R-MHD code
Athena \citep{Stoneetal2008,Jiangetal2012}.
\cite{Jiangetal2012} presented some results from a few test runs
of the dynamical evolution of a cloud ablated by
an intense radiation field. These tests show that the HD
as well as MHD solvers implemented in Athena when coupled
with the VET method capture shadows correctly. In particular,
by using an input radiation field at two angles,  \cite{Davisetal2012}
and  \cite{Jiangetal2012}   explored tests where
both umbra and penumbra are formed.  This makes the tests
more difficult, because ad-hoc closures that capture only one direction for
the flux will not represent both the umbra and penumbra correctly.

Here, using Athena we study the time evolution of clouds irradiated
by a radiation field as strong as the field believed to irradiate the
gas in AGN (i.e., radiation pressure to be comparable to gas
pressure). We consider clouds with a range of properties: from
optically thin to thick and with the opacity due to scattering
or absorption processes or both.
To explain BELs and BALs, clouds in AGN must
move with large velocities. Therefore, our primary focus
is on identifying physical conditions under which  a pre-existing
cloud could be significantly accelerated before it is dispersed.

Athena allows us to explore the above mentioned wide
range of conditions. Nevertheless, our treatment of clouds is quite
simplified. For example, our simulations are in two-dimensions
(2-D) and we assume the clouds to be highly ionized. Therefore,
we do not follow the development and evolution of the IF.
Moreover, we  do not include dust grains and magnetic fields.
These and other complications could be modeled with the code but are
beyond the scope of this preliminary work.

The outline of this paper is as follows. We describe our calculations
in
\S~\ref{sec:methods}.
In \S~\ref{sec:results}, we present our results.
We summarize our results and discuss them together with their
limitations in
\S~\ref{sec:summary}.

\section{Methods}
\label{sec:methods}
We solve the radiation hydrodynamic equations in the mixed frame with radiation source terms given
by \cite{Lowrieetal1999}. We assume local thermal equilibrium (LTE) and that the Planck
and energy mean absorption opacities are the same. Detailed discussion of the equations we solve can be found in
\cite{Jiangetal2012}. The equations are
\begin{eqnarray}
\frac{\partial\rho}{\partial t}+{\mbox{\boldmath $\nabla$}}\cdot(\rho \bv)&=&0, \nonumber \\
\frac{\partial( \rho\bv)}{\partial t}+{\mbox{\boldmath $\nabla$}}\cdot({\rho \bv\bv+{\sf P}}) &=&-{\bf \tilde{S}_r}(\bP),\  \nonumber \\
\frac{\partial{E}}{\partial t}+{\mbox{\boldmath $\nabla$}}\cdot\left[(E+P)\bv\right]&=&-c\tilde{S}_r(E),  \nonumber \\
\frac{\partial E_r}{\partial t}+{\mbox{\boldmath $\nabla$}}\cdot \bF_r&=&c\tilde{S}_r(E), \nonumber \\
\frac{1}{c^2}\frac{\partial \bF_r}{\partial t}+{\mbox{\boldmath $\nabla$}}\cdot{\sf P}_r&=&{\bf \tilde{S}_r}(\bP).
\label{dimequation}
\end{eqnarray}
Here, $\rho$ is density, ${\sf P}\equiv P\bI$ with $\bI$ the unit tensor
and $P$ gas pressure, and $c$ is the speed of light.
The total gas energy density is
\begin{eqnarray}
E=E_g+\frac{1}{2}\rho v^2,
\end{eqnarray}
where $E_g$ is the internal gas energy density.   We adopt an equation of state
for an ideal gas with adiabatic index $\gamma$, thus
$E_g=P/(\gamma-1)$ and $T=P/R_{\text{ideal}}\rho$, where
$R_{\text{ideal}}$ is the ideal gas constant.
The radiation pressure ${\sf P}_r$ is related to the radiation energy density $E_r$
by the closure relation
\begin{eqnarray}
{\sf P}_r=\bfr E_r.
\end{eqnarray}
where $\bfr$ is the VET, and $\bF_r$ is radiation flux. Finally,
${\bf\tilde{S}_r}(\bP)$ and $\tilde{S}_r(E)$ are the radiation momentum
and energy source terms, respectively.

Following \cite{Jiangetal2012}, we use a dimensionless set of equations and variables
in the remainder of this work.  We convert the above set of equations to the dimensionless
form by choosing fiducial units for temperature, pressure, and velocity
as $T_0$, $P_0$, and $a_0=\sqrt{P_0/T_0}$, respectively. Then
units for radiation energy density $E_r$ and flux $\bF_r$ are $a_rT_0^4$
and $ca_rT_0^4$. In other words, $a_r=1$ in our units. The dimensionless speed of light is $\Crat\equiv c/a_0$.
The original dimensional equations can then be written in the following dimensionless
form
\begin{eqnarray}
\frac{\partial\rho}{\partial t}+{\mbox{\boldmath $\nabla$}}\cdot(\rho \bv)&=&0, \nonumber \\
\frac{\partial( \rho\bv)}{\partial t}+{\mbox{\boldmath $\nabla$}}\cdot({\rho \bv\bv+{\sf P}}) &=&-\mathbb{P}{\bf S_r}(\bP),\  \nonumber \\
\frac{\partial{E}}{\partial t}+{\mbox{\boldmath $\nabla$}}\cdot\left[(E+P)\bv\right]&=&-\mathbb{PC}S_r(E),  \nonumber \\
\frac{\partial E_r}{\partial t}+\mathbb{C}{\mbox{\boldmath $\nabla$}}\cdot \bF_r&=&\mathbb{C}S_r(E), \nonumber \\
\frac{\partial \bF_r}{\partial t}+\mathbb{C}{\mbox{\boldmath $\nabla$}}\cdot{\sf P}_r&=&\mathbb{C}{\bf S_r}(\bP),
\label{equations}
\end{eqnarray}
where the dimensionless source terms are,
\begin{eqnarray}
{\bf S_r}(\bP)&=&-\sigma_t\left(\bF_r-\frac{\bv E_r+\bv\cdot{\sf P} _r}{\mathbb{C}}\right)+\sigma_a\frac{\bv}{\mathbb{C}}(T^4-E_r) ,\nonumber\\
S_r(E)&=&\sigma_a(T^4-E_r)+(\sigma_a-\sigma_s)\frac{\bv}{\mathbb{C}}\cdot\left(\bF _r-\frac{\bv E_r+\bv\cdot{\sf  P} _r}{\mathbb{C}}\right),
\label{sources}
\end{eqnarray}
while $\sigma_a$ and $\sigma_s$ are the absorption and
scattering opacities. Total opacity (attenuation coefficient)
is $\sigma_t=\sigma_s+\sigma_a$.
The dimensionless number $\Prat\equiv a_rT_0^4/P_0$ is a measure of
the ratio between radiation pressure and gas pressure in the fiducial units.
We prefer the dimensionless equations because the dimensionless numbers,
such as $\Crat$ and $\Prat$, can quantitatively indicate the importance
of the radiation field as discussed in \cite{Jiangetal2012}.

We solve these equations in a 2D $x-y$ plane with the recently developed
radiation transfer module in Athena \citep[][]{Jiangetal2012}. The continuity
equation, gas momentum equation and gas energy equation are solved with
a modified Godunov method, which couples the stiff radiation source terms to
the calculations of the Riemann fluxes. The radiation subsystem for $E_r$
and $\bF_r$ are solved with a first order implicit Backward Euler method.
Details on the numerical algorithm and tests of the code are described
in \cite{Jiangetal2012}. The variable Eddington tensor is computed from angular
quadratures of the specific intensity $I_r$, which is calculated from
the time-independent transfer equation
\begin{eqnarray}
\frac{\partial I_r}{\partial s}=\kappa_t (S-I_r).
\label{calEdd}
\end{eqnarray}
Details on how we calculate the VET, including tests, are given in \cite{Davisetal2012}.

Most of our simulations are performed using
our standard computational domain
$(x_{min}, x_{max}) \times (y_{min}, y_{max})$ which is
$(-0.5,0.5) \times (-0.5,0.5)$~ and standard resolution which is
$512\times512$ cells.
Initially the background medium has
density $\rho_0=1$ $\rm g/cm^3$ and temperature $T_0=10^6$~K.  An over-dense
clump is located in a circular region $r\equiv x^2/x_0^2+y^2/y_0^2\le
1$, with $x_0=y_0=0.05$~cm.  The density inside this
region is $\rho(x,y)=\rho_0+(\rho_1-\rho_0)/[1.0+\exp(10(r-1))]$, where
$\rho_1$ is a free parameter.  The clump is in pressure equilibrium
with its surroundings, so the interior is colder than the
ambient medium.  The initial radiation temperature is the same as
the gas temperature everywhere.  Here we consider opacities due to scattering, $\sigma_s=\sigma_{s,0}(\rho/\rho_0)$,
or absorption $\sigma_a=\sigma_{a,0}(T/T_0)^{-3.5}(\rho/\rho_0)^2$~$\rm cm^{-1}$,
or both $\sigma_t=\sigma_s+\sigma_a$.
The radiation flux $\bF_r$ is zero everywhere initially.  We use reflection
boundary conditions on both $y$ boundaries, outflow boundary conditions on
the right $x$ boundary. A constant radiation field with
temperature $T_r=2T_0$ is input through the left $x$ boundary.
At the left $x$ boundary, the gas temperature and
density are fixed to $T_0$ and $\rho_0$ respectively.
The dimensionless speed of
light $\Crat=3.3\times10^3$, and the parameter $\Prat=10^{-3}$.

\section{Results}
\label{sec:results}

We have performed over thirty different simulations exploring the
parameter space and numerical effects. Here we discuss in some detail
five simulations that illustrate the evolution of a  cloud in significantly
different physical regimes: a optically thin and thick cloud with
the pure scattering opacity (runs S10 and S200, respectively) and
mildly optically thick, optically thick, and very optically thick cloud with
absorption dominated opacity (runs A10, A40, and A80, respectively).
Our convection of naming the simulations is the following: the letter A
or S stands for the opacity type, i.e., dominated by absorption or scattering,
respectively. A number that follows the letter corresponds to
the initial cloud density, $\rho_1$.

We summarize the properties of the five simulations in
Table~\ref{table:summary}.
Columns (2), (3) and (4) give the input physical parameters:
$\sigma_{a,0}$, $\sigma_{s,0}$, and $\rho_1$.
In columns (5), (6), (7), (8), and (9)
we list the following initial properties of the cloud:
the absorption optical depth, $\tau_{a}=2 x_o \sigma_a$,
the scattering optical depth, $\tau_{s}=2 x_o \sigma_s$,
the radiation diffusion time across the cloud,
$t_{dif}=4x_o^2\sigma_t/\Crat=\tau_t t_{fs}$, where $t_{fs}=2x_o/\Crat$ is
the free-streaming time\footnote
{$t_{fs}=6.06\times10^{-5}$ and it is the same for all the cases discussed.},
the thermal time scale inside the cloud, $t_{th}=P/(\Prat \Crat E_r \sigma_a)$,
and the sound crossing time
$t_{sc}=2 x_o/c_s$, where $c_s=\sqrt{\gamma P/\rho}$
is the adiabatic sound speed.
Finally, columns (10), (11), and (12) give
the numerical resolution, $n_x \times n_y$,
the Courant number, $C_0$,
and the final time at which we stopped each simulation, $t_f$.
For the five runs, we used our standard computational domain.

Figure~\ref{fig:gallery} gives an overview
of the cloud evolution in the five runs (columns from left
to right correspond to runs: S10, S200, A10, A40, and A80) .
Specifically, the figure shows sequences of density maps overlaid
with velocities at five different times (the
time increases from top to bottom; the actual
time is given in the top left corner in each panel).

Figure~\ref{fig:time} shows several cloud
properties as a function of time. The figure
illustrates more qualitatively the differences
between various runs
(the columns' correspondence to various runs is the same as
in Fig.~\ref{fig:gallery})
as well as dramatic changes in the clouds with time.

To illustrate how a given cloud would appear to an observer
measuring the radiation at the right side of the computational
domain, the panels in the top row present
the $x$ component of the normalized radiation field, $F_{r1}$,
as a function of the $y$ coordinate.
The second from  top row of panels show $F_{r1}$ at the right boundary
but only at $y=0$ (i.e, the $(x_{max},0)$ location) and its minimum value
along the right boundary (the dashed and solid lines, respectively).
The fluxes in these panels are normalized so that they are in units of
the maximum radiation flux along the right boundary at a given time.
The middle panels present the time evolution of the {\it cold} gas
in the computational domain:
the maximum density, total mass and
the mass loss rate of the cold gas
(the solid, dotted, and dashed lines, respectively).
We normalized the maximum density to the initial density of the cloud,
$\rho_1$. Our operational definition of ``cold'' gas
is the gas with the temperature less than 2 times
the initial temperature of the cloud.
Therefore, the middle row panels can be used to follow the cloud heating and
subsequent evaporation.

To follow the average cloud motion, the second bottom row of panels show
the $x$-position of the center of mass (CoM) of the cold gas, while
the bottom panels show
the $x$ component of the CoM velocity and the maximum velocity of the
cold gas (the solid and dash-dotted lines, respectively).

We start by discussing run~S10 that is related to one
of the simple cases mentioned in \S~\ref{sec:scases}: simple example~II,
with scattering opacity only
and relatively small optical depth (i.e., $\tau_s=0.2$).
The left columns of panels of Figs.~\ref{fig:gallery}
and ~\ref{fig:time} show that,
as expected, there is no compression (i.e., $\rho_{max}=1$)
no loss of the cold gas, and no shock formation. The
cloud is almost uniformly accelerated in the horizontal direction.
The implied mass loss of cold gas for $t \geq 27$ is simply the result of the
cold cloud being advected out of the domain.
In addition,  the cloud is almost co-moving with the ambient medium
We note that the acceleration due to the radiation force, $a_{rad}$
is almost position and time independent in this optically
thin case, i.e.,
$a_{rad}=\mathbb{P} \sigma_{s,0} F_{r1}\simeq 1.6\times10^{-3}$
(one can ignore the velocity dependent terms in the source term,
${\bf S_r}(\bP)$, in the momentum equation as these terms are very small, i.e.,
$v/\mathbb{C}<<1$).
This acceleration is relatively small. Specifically, the time for an optically
thin fluid element to travel a distance equal to the cloud diameter,
$t_{dyn}=(4 x_o/a_{rad})^{1/2}\approx 11$ is long
compared to $t_{sc}=0.24$.

Also as expected, the position of CoM is a quadratic function of time
and the maximum velocity of the cold gas is very similar to the velocity of
CoM (see the two lowest panels of the left hand side column in
Fig.~\ref{fig:time}).
The small but not zero optical depth means that the front side
of the cloud experiences a slightly stronger  push by radiation
than the back side (the acceleration at the back compared
to that at the front is smaller by a factor of $0.9$).
Consequently, the cloud is flattened by radiation.
This effect is much stronger in run~S200, where the cloud
is optically thick.

The second left column of panels in Figure~\ref{fig:gallery}
illustrate how the radiation initially 'squeezes' the cloud in S200:
the front side is pushed by radiation while the back side does not move.
The radiation pressure acts only from the left side and the gas pressure
inside the cloud is higher than the pressure of the ambient gas.
This pressure imbalance leads to lateral expansion that is noticeable
already by the time, $t=12$. The cloud evolution is further complicated
by the fact that the optical depth decreases (not always monotonically)
as a function of $y$
from the cloud center to its edge. For example,
as the cloud moves as a whole to the right
it also expands laterally and its shape
starts to resemble a crescent because the more transparent
edges are pushed more than the cloud center.

The top panel in the second left column in Figure~\ref{fig:time}
shows clearly that the shadow size increases with time. However,
the panel also shows that at the later times (i.e., $t\simgreat20$)
the cloud center is not the most opaque
part of the cloud. Instead the most opaque regions are near the edges where
the  horns of the crescent bend over toward the center
(the second top panel  shows this too: for $t\simgreat20$,
the minimum radiation field
along the $y$ direction at $x=x_{max}$ is not at $y=0$).

Overall the cloud in run~S200 moves slower than in run~S10
(compare the second to bottom and bottom panels in corresponding columns
in Fig.~\ref{fig:time}). In addition, for run~S200,
the CoM velocity is lower than the maximum velocity of the cold gas.
This difference in the velocities is an indication
of a non-uniform cloud evolution. The two just presented examples
show that even for a pure scattering opacity case, the cloud
evolution is fast and very different than
the movement of a bullet where there is no change in the shape and size
as the cloud moves. The evolution of a cloud with absorption dominated opacity
is even more dynamic and complex and also faster.

We start by discussing run~A10 that is related to another
simple case mentioned in \S~\ref{sec:scases}: simple example~I - the cloud
behaving as a balloon.
In this run, $\tau_a=2\times10^3$, $t_{dif}=1.2\times10^{-1}$,
while $t_{th}=2\times10^{-6}$. Thus, both $t_{dif}$ and $t_{th}$ are much
smaller than $t_{dyn}$. Therefore, the evolution of the absorption dominated
cases is much faster than the pure scattering cases. For example,
the middle column of panels of Fig.~\ref{fig:gallery}
show that the cloud expands within time less than 0.2.
During its evolution, the cloud does not gain much net momentum and
it remains almost spherical.

The total mass of the cold gas  drops to zero within $t=3\times10^{-2}$
(see the middle panel in the middle column in Fig.~\ref{fig:time}).
This time is of the order of $t_{dif}$.
The gas density in the cloud is initially increased by a factor of 2 but
for $t\simgreat0.05$ $\rho_{max}$
decreases below 0.01. We note that in this run,
the velocity of CoM does not correspond to the movement
of the whole cloud but rather to the change in the location
of the boundary between the cold and hot
(to the rapid propagation of TF).
Hence, the implied CoM velocity can exceed the maximal velocity in
these runs.
The bottom panel in the middle column of Figure~\ref{fig:time}
shows that for as long as the cold gas exists,
its velocity increases very fast with time.
After $t_{dif}$, the only gas left on the grid is
the transparent and hot gas moving almost radially away
from the center of the grid
where the initial cloud was located.

The cloud in run~A10 is initially optically thick. However,
once the fast TF passes the cloud, the optical depth drops to
about 0.6 due to an increase of the temperature alone (i.e.,
with small change in the cloud density). This quick, isochoric decrease
in the cloud optical depth is the main reason for
the evolution in run~A10 to resemble the evolution
of simple example~I which we referred to as an expanding balloon.
Runs with $\tau_a$ lower than in run~A10 show very similar evolution
but occurring on a longer time scale. However runs with higher optical depth
show substantial qualitative differences. A good indicator
of the expected difference in the evolution is the relation between
the diffusion and sound crossing times.

For run~A10, $t_{dif}< t_{sc}$ and the cloud behaves like a balloon
even though it is initially optically thick.
However, for run~A40 with $t_{dif}> t_{sc}$ (and $t_{th}$ shortened
by almost 4 orders of magnitude), we observe strong evaporation
from the irradiated side, the development and propagation of
a strong shock inside the cloud, and a few  other features unseen in run~A10
(compare the middle and second right columns of panels
in Figs.~\ref{fig:gallery}~and~\ref{fig:time}).

In run~A40, the radiation heats the front much faster than it can diffuse
across
the cloud. Therefore, the cloud loses mass from the left side through
a hot outflow. In addition, a very fast and strong shock propagates to
the right of the cloud compressing it and changing its shape.
The changes in the shape are major: from the initial convex shape through
the concave one of the front side to the break up of the cloud
into two smaller elongated clumps.

The cloud evolution is far from being isochoric. For example,
the density on the left side of the cold part of the cloud is increased
by a factor of 8 (i.e., $\rho_{max}=8)$ at $t=0.9$.
This time corresponds to the maximum compression
of the cloud and the smallest shadow (see the fourth column of panels in
Figs.~\ref{fig:gallery} and ~\ref{fig:time}). After this time the cloud
re-expands and fragments (around $t=0.17$). Finally, at $t\approx 0.25$,
even the dense clumps become optically thin as they are heated and dispersed.

As expected, the cloud in run~A40 survives much longer than in run~A10.
Another consequence of the higher cloud density is that
the dense parts of the cloud can travel over some distance before they are
heated and dispersed. For example, the second bottom panel in the fourth
column of Fig.~\ref{fig:gallery} shows two dense and optically thick clumps
at $x=0.18$ for t=0.20 -- more than three radii of the initial cloud away
from the original location. This cloud movement to the right is caused by the
cloud hot outflow that moves to the left (i.e., the rocket effect).
The push by radiation pressure in this run as well as in runs A10 and A80 is
negligible because $t_{dyn}\approx 11$ which is orders of magnitude
longer than $t_{th}$.

To show the evolution of an even denser cloud, we carried out a simulation
for $\rho_1=80$ (run~A80). The right hand side panels of
Figs.~\ref{fig:gallery} and ~\ref{fig:time} show the results of this
simulation. Overall, the evolution of this denser cloud resembles that of
the cloud in run~A40. However, there are some quantitative differences.
In particular, in run~A80, the cloud stays optically thick for about
twice as long as in run~A40  (compare the second top panels in the second right
and right columns in Fig.~\ref{fig:time}). The two clumps that form after
the re-expansion phase are denser and more elongated.

The middle to bottom panels in the right column of the figure suggest that
the cloud disappears at $t\approx0.13$ and reappears around  0.24. However,
this is an artifact of our formal definition of the cold phase (the gas with
the temperature less than 2 times the initial temperature of the cloud).
The 'disappearance' of the cold phase corresponds the smallest cloud size
with the gas temperature increased due to compression rather than radiative
heating. However, as the dense cloud later re-expands, the gas temperature
drops below our temperature threshold. Later still,
the cloud moves to the right, fragments and continues to be heated
by radiation so that  at $t\approx 0.40$ it is totally dispersed.

\section{Summary and Discussion}
\label{sec:summary}

We have presented two-dimensional, radiation-hydrodynamic calculations
of time-dependent structure of  strongly irradiated clouds.
Our primary conclusions are the following:
1) even a relatively dense cloud that is radiatively heated
(i.e., with significant absorption opacity) does not move as a whole, and
instead undergoes a very dramatic evolution in its shape, size and physical
properties. In particular, the cloud and its remnants become optically thin
within less than one sound crossing time and
before they can travel over a significant distance  (a distance of a few
radii of the initial cloud) and
2) a cloud can be accelerated as a whole under quite extreme conditions, e.g.,
the opacity must be dominated by scattering. However, the radiation force
acceleration is relatively small and unless the cloud
is optically thin, the cloud quickly changes its size and shape.

A competition of several physical processes determines the cloud evolution.
In the cases that we have explored, the most important processes
are the radiation diffusion and heating, the radiation force acceleration,
and finally propagation of the sound speed.
Comparing the time scales corresponding
to each of these processes (see \S~\ref{sec:intro}),
one can predict the behaviour of a cloud with given initial
conditions. For example, as we noted in \S~\ref{sec:results},
for absoprtion dominated opacity and
$t_{dif} < t_{sc}$, a cloud will behave like a balloon (simple example~I)
even if the cloud is initially optically thick.
The radiative heating is usually the fastest process. Therefore,
simple example~II (a bullet like case) requires very special conditions:
scattering dominated opacity and the optical depth much less than one.

Our goal here was to carry out simulations in a simple and well-controlled way
so that we can assess the role of the type and magnitude of the opacity on the
cloud evolution. While achieving this goal, we found that Athena can handle well
quite a wide range of physical conditions. Not surprisingly, we also found that
the most difficult cases to compute are those with huge density and opacity
contrasts,  especially when the opacity is dominated by absorption. In these cases,
the thermal time scale can be extremely short so that the equation of energy is
stiff. To avoid spurious oscillations, we had to increase the numerical resolution
and reduce the Courant number (see columns 10 and 11 in Table~\ref{table:summary}).
For example, in run~A80, we reduced $C_0$ to 0.02 from its usual value of 0.8.

In the runs discussed here, we increased the temperature of the incoming radiation
by a relatively small factor of 2 ($T_r=2 T_0$). This corresponds to an increase
of the radiation pressure by a factor of 16. Results from our  various tests with
higher $T_r$ and also higher $\Prat$ confirm simple expectations that increasing
$T_r$ or $\Prat$ results in a faster cloud evolution because of the increases
in the heating rate and cloud acceleration.

As we mentioned in \S~\ref{sec:intro}, the clouds in the BLRs are optically thick
to absorption and optically thin to scattering. Therefore, the results from
runs A40 and A80 are most applicable to the BLRs of AGN. These results well
illustrate the fact that any cloudy model for the BLRs faces the issues
of stability and confinement.

These are very serious issues because in our simplified simulations, the cloud
velocity was not very different from the ambient velocity.
In particular, we did not assume that the cloud is on a Keplerian
orbit around the central black hole as it is assumed in some BLR models
\citep[e.g.,][and references therein]{Pancoastetal2011,Krauseetal2012}.
Therefore, here the cloud was not much affected by hydrodynamic instabilities
caused by velocity shear. Nevertheless, we found that the cloud can not
be significantly accelerated before it is dispersed (i.e., the rocket effect
is not efficient). The main process responsible for the dispersal is radiative
heating. One of the implications of these results is that clouds in the BLRs,
if they exist, likely take on many different and complex shapes and this
complication should be considered in detailed calculations of cloud line emission.

There are a number of limitations to our results. Probably the most important are
that we neglected gas photoionization and assumed frequency-independent (gray)
opacities. Additional limitations to our calculations are that they are 2-D
instead of fully 3-D and that we did not include thermal conduction,
magnetic fields, and the process or processes responsible for the cloud formation.
Moreover, in realistic simulations of the BLRs, one should also consider computing
the evolution of interacting multiple clouds with various properties, not just
one cloud as we did here. Our results indicate that such multi-cloud
simulations are feasible using Athena.

We plan to go beyond some of these limitations in the near future.
In particular, we plan to address the issue of cloud formation and subsequent
evolution. One possible approach would be to consider the radiative processes
and conditions for which the gas is thermally unstable so
that cold condensations can grow on relatively short time scales
despite the presence of a strong radiation field
\citep[e.g.,][and references therein]{Krolik1988, MoscibrodzkaProga2013}.
Such an approach, our intended next step,
will involve simulations quite different from those presented
here because among other things, here the gas is
thermally stable and initially it is not in a radiative equilibrium.

\section*{Acknowledgments}

This work was supported by NASA under ATP grant NNX11AI96G and NNX11AF49G.
D.P. thanks S. Lepp for discussions and also the Department of Astrophysical
Sciences, Princeton University for its hospitality during his sabbatical
when the work presented here was initiated. He also
wishes to acknowledge the National Supercomputing Center
        for Energy and the Environment, for providing computer resources
        and support.
Y.F.J. was provided by NASA through Einstein Postdoctoral Fellowship grant
number PF-140109 awarded by the Chandra X-ray Center, which is operated by
the Smithsonian Astrophysical Observatory for NASA under contract NAS8-03060.
\bibliographystyle{apj}


\section{Introduction}

Topological insulators\cite{Hasan2010,Moore2010,Qi2011} (TIs) support gapless boundary excitations in spite of a gapped bulk spectrum. The edge states are believed to be stable against any perturbation, as long as certain symmetries are preserved. When symmetries are broken, however, TIs can be continuously tuned into a trivial atomic insulator without phase transitions. Recently it's realized\cite{Chen2013a} that aside from weakly-interacting electrons, such phases generally exist in interacting bosons and they are dubbed ``symmetry protected topological'' (SPT) phases.

When symmetries are spontaneously broken, a gap can open up in the edge spectrum of SPT phases. There are always topological defects (kinks)\cite{Mermin1979} associated with spontaneous symmetry breaking, such as the domain wall excitation in an Ising ferromagnet. Usually by proliferating the defects one can restore symmetry, leading to a gapped symmetric state: {\emph{e.g.~}} the disordered phase of a transverse Ising model can be achieved by ``condensing'' the domain walls. Similarly can one achieve a gapped symmetric state on the edge of an SPT phase?

In this work we answer this question constructively, focusing on two spatial dimensions (2+1-D). We show that topological defects (kinks) on the boundary always carry fractional statistics\cite{Wilczek1990B} or symmetry quantum numbers, hence their proliferation is either forbidden or breaks symmetry. However, on a boundary between SPT phase and certain fractionalized phase (which hosts anyon excitations\cite{Wilczek1990B}), one can form a bosonic bound state of the kink on SPT side and anyon on fractionalized side. Proliferating this composite object will lead to a gapped symmetric boundary. This can be generalized to any spatial dimensions. Two examples are presented: 1) boundary between bosonic $Z_2$-SPT and double semion model, equipped with an exactly solvable model; 2) boundary between quantum spin Hall insulator (QSHI) and a fractionalized QSHI$^\ast$ phase. We show that a hybrid structure containing SPT and fractionalized phases support ground state degeneracy (GSD) on a torus (FIG. \ref{fig:torus}).

\section{Edge field theory and ``fractional'' defects of SPT phases}

 SPT phases\cite{Chen2013a} in two spatial dimensions can be described\cite{Lu2012a} by multi-component Chern-Simons theory\cite{Read1990,Wen1992,Frohlich1991} with a symmetric unimodular matrix ${\bf K}$. In particular, 2+1-D SPT phases host gapless edge excitations, described by chiral bosons $\{\phi_i\}$ with the following effective field theory\cite{Wen1995}:
\begin{eqnarray}\label{edge action}
\mathcal{L}_{edge}=\sum_{I,J}\frac1{4\pi}{\bf K}_{I,J}\partial_t\phi_I\partial_x\phi_J-{\bf V}_{I,J}\partial_x\phi_I\partial_x\phi_J
\end{eqnarray}
where ${\bf V}$ is a positive-definite real symmetric matrix. Backscattering terms $\sim\cos(\sum_I l_I\phi_I)$ are generally not allowed by symmetry\cite{Lu2012a,Levin2012a} denoted by group $G_s$.

A simple example is topological paramagnet protected by $Z_n$ symmetry\cite{Levin2012,Lu2012a} where $G_s=Z_n\equiv\{{\boldsymbol{g}},{\boldsymbol{g}}^2,\cdots,{\boldsymbol{g}}^N={\boldsymbol{e}}\}$. Its edge structure is characterized by ${\bf K}=\bpm0&1\\1&0\end{pmatrix}$ in effective theory (\ref{edge action}), where under $Z_n$ symmetry operation the chiral bosons transform as\cite{Lu2012a}
\begin{eqnarray}\label{sym transf:Zn}
\begin{pmatrix}\phi_1\\ \phi_2\end{pmatrix}\overset{{\boldsymbol{g}}}\longrightarrow\begin{pmatrix}\phi_1\\ \phi_2\end{pmatrix}+\frac{2\pi}{n}\bpm1\\k\end{pmatrix},~~~1\leq k\leq n-1.
\end{eqnarray}
Here backscattering terms $\mathcal{H}_{bs}\sim\cos(\phi_{1,2}-\alpha_{1,2})$ are forbidden by the above $Z_n$ symmetry. Once symmetry is broken, edge states in (\ref{edge action}) can be gapped out by $\mathcal{H}_{bs}$.

Spontaneous symmetry breaking on the edge will pin chiral boson fields $\phi_{1,2}$ at certain classical values. Distinct classical values $\langle\phi_{1.2}\rangle$ correspond to different ways to break discrete $Z_n$ symmetry, and there are topological defects $-$ domain walls (or kinks) which spatially separates these different ``vacua''. Different kinks are classified\cite{Mermin1979} by homotopy group $\pi_0(Z_n)=Z_n$, {\emph{i.e.~}} there are $n$ distinct types of domain walls, including the trivial one $-$ no domain wall. All stable kinks can be generated by a fundamental domain wall, which in our case $G_s=Z_n$ can be written in terms of chiral bosons\cite{Chen2014}:
\begin{eqnarray}\label{domain wall:Zn}
\hat{D}_{Z_n|k}(x)=\exp\Big[\hspace{1pt}\mathrm{i}\hspace{1pt}\frac{k\phi_1(x)+\phi_2(x)}{n}\Big]
\end{eqnarray}
As implied by (\ref{edge action}) chiral bosons $\phi_{1,2}$ obey commutation relation
$[\phi_1(x),\phi_2(y)]=2\pi\hspace{1pt}\mathrm{i}\hspace{1pt}\cdot\theta(y-x)$.
It's straightforward to see the classical values $\langle\phi_{1,2}\rangle$ on two sides of domain wall (\ref{domain wall:Zn}) are related by the symmetry transformation (\ref{sym transf:Zn}), {\emph{e.g.~}} $\hat{D}_{Z_n|k}(x)\phi_1(y)\hat{D}_{Z_n|k}^{-1}(x)=\phi_1(y)+\frac{2\pi}{n}\theta(x-y)$.

A natural question is: can we restore symmetry ($G_s=Z_n$ here) simply by proliferating topological defects (kinks) and obtain a gapped symmetric edge of SPT phases? Remarkably the kink (\ref{domain wall:Zn}) is neither
a boson nor a fermion: generally it obeys fractional statistics\cite{Wilczek1990B,Wen1995}
\begin{eqnarray}
\hat{D}_{Z_n|k}(x)\hat{D}_{Z_n|k}(y)=\hat{D}_{Z_n|k}(y)\hat{D}_{Z_n|k}(x)e^{\hspace{1pt}\mathrm{i}\hspace{1pt}\frac{2\pi k}{n^2}\text{Sgn}(x-y)}\notag
\end{eqnarray}
with statistical angle $\theta_{Z_n|k}=2\pi\frac{k}{n^2}$. Moreover it has fractional mutual statistics with bosonic excitations $\{e^{\hspace{1pt}\mathrm{i}\hspace{1pt}\phi_{1,2}}\}$:
\begin{eqnarray}
&\notag\hat{D}_{Z_n|k}(x)e^{\hspace{1pt}\mathrm{i}\hspace{1pt}\phi_1(y)}=e^{\hspace{1pt}\mathrm{i}\hspace{1pt}\phi_1(y)}\hat{D}_{Z_n|k}(x)e^{\hspace{1pt}\mathrm{i}\hspace{1pt}\frac{2\pi}{n}\theta(x-y)},\\
&\notag\hat{D}_{Z_n|k}(x)e^{\hspace{1pt}\mathrm{i}\hspace{1pt}\phi_2(y)}=e^{\hspace{1pt}\mathrm{i}\hspace{1pt}\phi_2(y)}\hat{D}_{Z_n|k}(x)e^{-\hspace{1pt}\mathrm{i}\hspace{1pt}\frac{2\pi k}{n}\theta(y-x)}.
\end{eqnarray}
Its fractional statistics leads to destructive interference in the path integral when domain walls proliferate, therefore suppressing instanton events which create/annihilate domain walls. As a result in $Z_n$-SPT phases, it is impossible to disorder the symmetry-broken edges simply by proliferating solitons. More generally on the boundaries of $d+1$-D SPT phases, there are similar obstructions to proliferate symmetry-breaking topological defects\cite{Vishwanath2013}.

\section{Gapped symmetric boundary between a SPT phase and a fractionalized phase}

On the other hand, on the boundary between a SPT phase $\mathcal{S}$ and a fractionalized phase $\mathcal{F}$ ({\emph{i.e.~}} intrinsic topological order\cite{Wen2004B} which supports anyon excitations in the bulk), the bound state of a kink from SPT side and an anyon from fractionalized side can have bosonic statistics. This object in principle can proliferate and gap out all boundary excitations, restoring symmetry $G_s$. Notice that fractionalized phase $\mathcal{F}$ should also respect symmetry $G_s$, in other words it is a symmetry enriched topological (SET) phase\cite{Essin2013,Mesaros2013,Hung2013,Lu2013,Hung2013a}.

Given a 2+1-D SPT phase $\mathcal{S}$, only certain 2+1-D fractionalized phases can have a gapped symmetric boundary with $\mathcal{S}$. For example edge states of $\mathcal{F}$ and $\mathcal{S}$ must have the same number of chiral edge modes, {\emph{i.e.~}} same chiral central charge $c_-$. Meanwhile $\mathcal{F}$ must support anyon excitations with the same statistics and symmetry quantum numbers as topological defects (kinks) on the edge of $\mathcal{S}$.
We propose the following conjecture, which provides a way to look for fractionalized phase $\mathcal{F}$ sharing a gapped symmetric boundary with SPT phase $\mathcal{S}$:

\emph{An SPT phase $\mathcal{S}$ in any spatial dimensions always possesses a gapped symmetric boundary with a fractionalized (SET) phase $\mathcal{F}$, where $\mathcal{F}$ is obtained by gauging\cite{Levin2012} an Abelian discrete symmetry in $\mathcal{S}$.}
\begin{eqnarray}\label{conjecture}\end{eqnarray}

In the following we give a proof of this conjecture in 2+1-D based on Chern-Simons approach.

An Abelian discrete symmetry group always has the form of a direct product of cyclic groups:
\begin{eqnarray}\label{abelian discrete group}
G_s=\prod_{n\geq2}\big(Z_{n}\big)^{\alpha_n},~~~\alpha_n=0,1,2,\cdots.
\end{eqnarray}
Here we explicitly prove the conjecture for the case of a generic $Z_n$ symmetry (cyclic group of order $n$)

Consider an arbitrary 2+1-D SPT phase $\mathcal{S}$ with $Z_n$ symmetry. Since it's an Abelian 2+1-D phase its bulk effective theory is a multi-component Chern-Simons theory\cite{Wen2004B}
\begin{eqnarray}\label{bulk action}
\mathcal{L}_{bulk}=\frac{\epsilon^{\mu\nu\rho}}{4\pi}\sum_{I,J}a_\mu^I{\bf K}_{I,J}\partial_\nu a^J_\rho-\sum_I a_\mu^I j^\mu_I
\end{eqnarray}
where $j^\mu_I$ are the quasiparticle currents. In the long-wavelength limit its edge excitations are described by\cite{Wen1995,Lu2012a}:
\begin{eqnarray}\label{edge action}
\mathcal{L}_{edge}=\sum_{I,J}\frac1{4\pi}{\bf K}_{I,J}\partial_t\phi_I\partial_x\phi_J-{\bf V}_{I,J}\partial_x\phi_I\partial_x\phi_J
\end{eqnarray}
where ${\bf K}$ is a unimodular matrix and ${\bf V}$ is a real positive-definite matrix. Denoting the generator of $Z_n$ symmetry by ${\boldsymbol{g}}$ (${\boldsymbol{g}}^n={\boldsymbol{e}}$), the edge chiral bosons $\{\phi_i\}$ transforms as\cite{Lu2012a,Lu2013}
\begin{eqnarray}\label{sym transf}
\phi_I\overset{{\boldsymbol{g}}}\longrightarrow\phi_I+\delta\phi^{\boldsymbol{g}}_I;~~~~~n{\bf K}\vec{\delta\phi^{\boldsymbol{g}}}=0\mod2\pi.
\end{eqnarray}

First of all, what is the fractionalized (SET) phase $\mathcal{F}$ after gauging $Z_n$ symmetry? A well-defined way to gauge the symmetry in a lattice model\cite{Levin2012} is to couple the local degrees of freedom (which lives on lattice sites and transforms under $Z_n$ symmetry) to a dynamical $Z_n$ gauge field (which lives on links). In an effective field theory, the effect of gauging a symmetry is captured by deconfining the symmetry twist (or symmetry flux)\cite{Levin2012,Hung2012a,Lu2013} in the original SPT phase.
A symmetry twist has the following property: when a particle carrying symmetry quantum numbers goes around this symmetry twist once, it transforms under a symmetry operation. In the language of gauge field theory (\ref{bulk action}), particles are integer gauge charges of fields $\{a_\mu^I\}$: they are labeled by an integer vector ${\bf l}$ and represented by operators $e^{\hspace{1pt}\mathrm{i}\hspace{1pt}\sum_I{\bf l}_I\phi_I}$ on the edge. Notice that particle $e^{\hspace{1pt}\mathrm{i}\hspace{1pt}\phi_I}$ pick up a phase $e^{\hspace{1pt}\mathrm{i}\hspace{1pt}\delta\phi^{\boldsymbol{g}}_I}$ under $Z_n$ symmetry operation ${\boldsymbol{g}}$, therefore symmetry twists are nothing but gauge fluxes in (\ref{bulk action}). In a Chern-Simons theory, gauge fluxes are also gauge charges, which becomes transparent in the following equations of motion
\begin{eqnarray}
\notag\frac{\delta\mathcal{L}_{bulk}}{\delta a_\mu}=0\Longrightarrow j^\mu_I=\frac{\epsilon^{\mu\nu\rho}}{2\pi}\sum_{J}{\bf K}_{I,J}\partial_\nu a^J_\rho
\end{eqnarray}
As a result, in the fractionalized phase $\mathcal{F}$ obtained by gauging $Z_n$ symmetry, quasiparticles corresponding to (fractional gauge charge) vector
\begin{eqnarray}
{\bf l}^{\boldsymbol{g}}={\bf K}\vec{\delta\phi^{\boldsymbol{g}}}/2\pi,
\end{eqnarray}
are new excitations in SET phase $\mathcal{F}$. One can immediately see from (\ref{sym transf}) that $n{\bf l}^{\boldsymbol{g}}$ must be an integer vector. Moreover in a $Z_n$-SPT the symmetry transformations (\ref{sym transf}) form a faithful representation\cite{Lu2012a} of $Z_n$ group, meaning that at least one component of integer vector ${\bf l}^{\boldsymbol{g}}$ is $1/n$. Without loss of generality, we assume that ${\bf l}^{\boldsymbol{g}}_{p+1}=1/n$ where $\dim{\bf K}=p+1$. In other words we have
\begin{eqnarray}
{\bf l}^{\boldsymbol{g}}=({\bf v}^T,1)^T/n,~~~{\bf v}\in{\mathbb{Z}}^p.
\end{eqnarray}
As discussed in \Ref{Lu2013}, the fractionalized SET phase $\mathcal{F}$ obtained by gauging $Z_n$ symmetry is described by Chern-Simons theory (\ref{bulk action}) with matrix
\begin{eqnarray}
{\bf K}^{\boldsymbol{g}}={\bf M}^{-1}{\bf K}\big({\bf M}^{-1}\big)^T,~~~{\bf M}=\bpm1_{p\times p}&{\bf v}/n\\0_{1\times p}&1/n\end{pmatrix}\notag.
\end{eqnarray}
It's easy to see
\begin{eqnarray}
{\bf M}^{-1}=\bpm1_{p\times p}&-{\bf v}\\0_{1\times p}&n\end{pmatrix},~~~\det{\bf K}^{\boldsymbol{g}}=n^2\det{\bf K}.
\end{eqnarray}
We label chiral bosons on the edge of SET phase $\mathcal{F}$ by $\{\tilde\phi_I|1\leq I\leq p+1\}$. They have the following correspondence with the edge chiral bosons $\{\phi_i\}$ in SPT phase $\mathcal{S}$:
\begin{eqnarray}
\tilde\phi_I\leftrightarrow\sum_{J}{\bf M}_{J,I}\phi_I
\end{eqnarray}
Similar relations hold for their symmetry transformations $\{\tilde\delta\phi^{\boldsymbol{g}}_I\}$ and $\{\delta\phi^{\boldsymbol{g}}_I\}$.

In SET phase $\mathcal{F}$, mutual statistics between particle ${\bf l}$ and the new (fractional) particle ${\bf l}^{\boldsymbol{g}}$ is given by\cite{Wen2004B}:
\begin{eqnarray}
\theta_{{\bf l},{\bf l}^{\boldsymbol{g}}}=2\pi{\bf l}^T{\bf K}^{-1}{\bf l}^{\boldsymbol{g}}={\bf l}^T\vec{\delta\phi^{\boldsymbol{g}}}.
\end{eqnarray}
Generally $\delta\phi^{\boldsymbol{g}}_I/2\pi=p_I/q_I$ where $(p_I,q_I)$ are two mutually prime integers. It's not hard to check that the following set of backscattering terms between the SPT edge and SET edge gives rise to a gapped symmetric boundary between them:
\begin{eqnarray}
\notag\mathcal{L}_1=\sum_{I=1}^p\cos\big[q_I(\tilde\phi_I-\phi_I)\big]+\cos\big[n(\tilde\phi_{p+1}-\sum_I{\bf l}^{\boldsymbol{g}}_I\phi_I)\big].
\end{eqnarray}\\

In the case of gauging a continuous symmetry, the conjecture (\ref{conjecture}) is not valid anymore. An obvious example is $U(1)$-SPT in 2+1-D, {\emph{i.e.~}} bosonic integer quantum Hall effect\cite{Lu2012a}. The chiral central charge is always $c_-=0$ for such a $U(1)$-SPT phase $\mathcal{S}$. After gauging $U(1)$ symmetry, we obtain a fractionalized phase $\mathcal{F}$ described by $U(1)$ level-$\sigma_{xy}$ Chern-Simons term, which has chiral central charge $c_-=\text{Sgn}(\sigma_{xy})$. Apparently there cannot be a gapped edge between $\mathcal{S}$ and $\mathcal{F}$.

\section{Examples}

In this section we'll demonstrate the conjecture (\ref{conjecture}) in two examples.

\subsection{Topological Ising paramagnet $\Big(G_s=Z_2\Big)$}

The simplest example of 2+1-D bosonic SPT phases is the topological Ising paramagnet\cite{Levin2012} with $G_s=Z_2$ symmetry. In its edge effective theory (\ref{edge action}), ${\bf K}=\bpm0&1\\1&0\end{pmatrix}$ and Ising symmetry acts on chiral bosons $\phi_{1,2}$ as (\ref{sym transf:Zn}) with $n=2,k=1$. The domain wall operator $\hat{D}_{Z_2|1}$ in (\ref{domain wall:Zn}) is a semion\footnote{Strictly speaking the domain wall could also be an anti-simion, as a bound state of a semionic domain wall and a local spin flip.}, with fractional statistics $\theta=\pi/2$. In order to acquire a gapped symmetric edge, the corresponding fractionalized (SET) phase must support bulk semion excitations and non-chiral edge states. The simplest choice is the double semion state\cite{Levin2005} with ${\bf K}=\bpm2&0\\0&-2\end{pmatrix}$ in (\ref{edge action}). Note that gauging Ising symmetry in $Z_2$-SPT leads to\cite{Levin2012} nothing but double semion model. On its edge there are two branches of chiral bosons $\phi_{s,\bar{s}}$, where $e^{\hspace{1pt}\mathrm{i}\hspace{1pt}\phi_s}$ creates a semion and  $e^{\hspace{1pt}\mathrm{i}\hspace{1pt}\phi_{\bar{s}}}$ creates an anti-semion. Now the following tunneling terms between the SPT edge ($\phi_{1,2}$) and double semion edge ($\phi_{s,\bar{s}}$)
\begin{eqnarray}\label{boundary term:Z2}
\mathcal{H}_{t}=T_1\cos(\phi_1+\phi_2-2\phi_s)+T_2\cos(\phi_1-\phi_2-2\phi_{\bar s}).
\end{eqnarray}
can open up a gap on their boundary without breaking Ising symmetry. Here spins in double semion model doesn't transform under Ising symmetry. In contrast, on the ``pure" boundary between this $Z_2$-SPT phase and the vacuum, there is no way to get rid of gapless excitations without breaking symmetry\cite{Chen2013a,Levin2012}.
\begin{figure}
 \includegraphics[width=0.4\textwidth]{fig_model.pdf}
\caption{(color online) Exactly solvable model of a gapped symmetric edge between bosonic $Z_2$-SPT (diamonds denote spins $\{\vec\sigma_{\bf r}\}$ on triangular lattice) and double semion model (solid circles denote spins $\{\vec\tau_{{\bf r}^\prime}\}$ on edge centers of honeycomb lattice). Dashed green lines on triangular lattice represent domain wall configurations in ground state wavefunction of Ising paramagnet, using $\sigma^z=\pm1$ basis. Solid green lines denote (oriented) string-net configurations in the ground state of double semion model, where $\tau^z=1$ correspond to no string and $\tau^z=-1$ to occupied by a string. The boundary term (\ref{boundary ham:Z2}) guarantees that a $Z_2$ domain wall from upper half always forms a bound state with the end of a string on the lower half. Such a bosonic bound state can hop and condense, giving rise to a gapped symmetric edge.}\label{fig:model}
\end{figure}

In the following we present an exactly solvable model for such a boundary between bosonic $Z_2$-SPT\cite{Levin2012} and double semion model\cite{Levin2005}. Its Hamiltonian consists of commuting local projectors:
\begin{eqnarray}\label{model}
 H=H_{{SPT}}\{\sigma_{\bf a}\}+H_{{SET}}\{\tau_{{\bf i}}\}+H_{BDY},
\end{eqnarray}
On $Z_2$-SPT side\cite{Levin2012}
\begin{eqnarray}
\notag H_{{SPT}}=\sum_{\bf a}\sigma_{\bf a}^x\big(\prod_{\langle{\bf a,b,c}\rangle}\hspace{1pt}\mathrm{i}\hspace{1pt}^{\frac{1-\sigma_{\bf b}^z\sigma^z_{\bf c}}{2}}\big)
\end{eqnarray}
where $\langle{\bf a,b,c}\rangle$ runs over all six (nearest neighbor) triangles containing ${\bf a}$. On the side of double semion model\cite{Levin2005}
\begin{eqnarray}
\notag H_{SET}=-\sum_{{\bf I}}\prod_{\text{legs of}~{\bf I}}\tau_{\bf i}^x+\sum_{\bf r}(\prod_{\text{edges of}~{\bf r}}\tau^z_{\bf j})(\prod_{\text{R-legs of}~{\bf r}}\hspace{1pt}\mathrm{i}\hspace{1pt}^{\frac{1-\tau^x_{\bf j}}{2}})
\end{eqnarray}
{\bf I} denotes vertices and ${\bf r}$ denotes hexagonal plaquette center. The boundary Hamiltonian connects the two sides symmetrically
\begin{align}\label{boundary ham:Z2}
& H_{BDY}=\sum_{\bf r\in \text{boundary}}-\tau^x_{{\bf r}+\hat{x}/2}\sigma^z_{\bf {r}}\sigma^z_{{\bf r}+\hat{x}}\\
&\notag+\sigma^x_{\bf r}(\prod_{\langle{\bf r,b,c}\rangle}\hspace{1pt}\mathrm{i}\hspace{1pt}^{\frac{1-\sigma_{\bf b}^z\sigma^z_{\bf c}}{2}})(\prod_{\text{edges of}~{\bf r}}\tau^z_{\bf j})(\prod_{\text{R-legs of}~{\bf r}}\hspace{1pt}\mathrm{i}\hspace{1pt}^{\frac{1-\tau^x_{\bf j}}{2}})
\end{align}
as shown in FIG. \ref{fig:model}. It's easy to verify that all terms commute with each other. Physically the 1st term in $H_{BDY}$ guarantees a domain wall from $Z_2$-SPT side is always bound to a string from double semion side, while the 2nd term provides kinetic energy which allows this bound state to hop on the boundary. Therefore they can ``condense" on the boundary and gap out the edge states without breaking Ising symmetry. The low-energy effective theory for boundary Hamiltonian $H_{BDY}$ is nothing but (\ref{boundary term:Z2}).

The Ising symmetry in this model is implemented by ${\boldsymbol{g}}=\prod\sigma^x$, thus $\vec\tau$ spins in double semion model is invariant under Ising spin flip. Now let's consider model (\ref{model}) on a torus (see FIG. \ref{fig:torus}), where a half of the torus hosts double semion model and the other half hosts $Z_2$-SPT phase. Both shared boundaries between $Z_2$-SPT and double semion model are gapped by Hamiltonian (\ref{boundary ham:Z2}). However there is a 2-fold ground state degeneracy (GSD) in such a hybrid structure on torus. This can be easily verified by comparing the number of independent stablizers\cite{Kitaev2003} (local commuting projectors in $H$) and the total number of spins, since there is a global constraint for the local projectors in model (\ref{model}) on torus:
\begin{eqnarray}
\notag\prod_{{\bf I}}(\prod_{\text{legs of}~{\bf I}}\tau_{\bf i}^x)\cdot\prod_{\bf r\in \text{boundary}}(\tau^x_{{\bf r}+\hat{x}/2}\sigma^z_{\bf {r}}\sigma^z_{{\bf r}+\hat{x}})=1
\end{eqnarray}
These two degenerate ground states can be labeled by eigenvalues of string ``order parameter''\cite{Levin2005}
\begin{eqnarray}
\notag\hat{O}_S=\prod_{\text{edges of}~S}\tau^z_{\bf j}\prod_{\text{R-legs}}\hspace{1pt}\mathrm{i}\hspace{1pt}^{\frac{1-\tau_{\bf j}^x}{2}}\prod_{\text{L-vertices}}(-1)^{\frac{(1-\tau_{\bf i}^x)(1+\tau_{\bf j}^x)}{4}}.
\end{eqnarray}
The closed string $S$ winds around a non-contractible loop of torus once, parallel to the boundary between $Z_2$-SPT and double semion model, as illustrated by the horizontal green loop in FIG. \ref{fig:model}. It's easy to verify $(\hat{O}_S)^2=1$, hence it has eigenvalues $\pm1$.

\begin{figure}
 \includegraphics[width=0.4\textwidth]{torus1.jpg}
\caption{(color online) Illustration of the hybrid structure on a torus, containing SPT phase $\mathcal{S}$ (blue) and fractionalized SET phase $\mathcal{F}$ (red). Both boundaries between the two regions are fully gapped without breaking any symmetry. The two oriented loops correspond to string order parameters $\hat{O}_S$ (green) and $\hat{O}_L$ (black). Unlike $\hat{O}_S$ which consists of operators only from the SET region, open string operator $\hat{O}_L$ also contains operators from SPT side: such as spin $\sigma^z$ on sites ${\bf r}_b$ and ${\bf r}_e$ in the example of topological Ising paramagnet. Both string operators preserve the symmetry of the system.}\label{fig:torus}
\end{figure}

A pure double-semion model on torus has 4-fold GSD labeled by two commuting string operators: $\hat{O}_S$ and $\hat{O}^\prime_S=\prod_{\text{R-legs}}\tau_{\bf j}^x$. However in the ground states of our model, the 1st term of boundary Hamiltonian (\ref{boundary ham:Z2}) fixes the eigenvalue of $\hat{O}^\prime_S$ to be 1. Therefore the original 4-fold GSD of double-semion model on torus reduces to 2-fold in the hybrid structure here. The two degenerate ground states can be alternatively labeled by another string operator
\begin{eqnarray}
\notag\hat{O}_L=\sigma^z_{{\bf r}_b}\big(\prod_{\text{R-legs}}\tau^x_{\bf i}\big)\sigma^z_{{\bf r}_e},~~~[\hat{O}_L,H]=0,~~~(\hat{O}_L)^2=1.
\end{eqnarray}
where the open oriented line $L$ starts on one boundary (${\bf r}_b$ is the closet $\vec\sigma$ spin on its r.h.s.) and ends on the other boundary (${\bf r}_e$ on its r.h.s.) in FIG. \ref{fig:torus}. Notice that open line $L$ crosses the closed string $S$ only once, hence $\{\hat{O}_T,\hat{O}_S\}=0$. Both string operators are \emph{even} under Ising symmetry. This means operator $\hat{O}_L$ tunnels between two $\hat{O}_S$ eigenstates in the ground states' manifold:
\begin{eqnarray}
\hat{O}_L|O_S=\pm1\rangle=|O_S=\mp1\rangle.\notag
\end{eqnarray}
As we shrink the SET region in FIG. \ref{fig:torus}, the length of line $L$ also decreases and ultimately $\hat{O}_L$ will become a local operator. In this limit the two $\hat{O}_S$ eigenstates start to mix by local interactions and the system picks up a unique ground state as a superposition of $|O_s=\pm1\rangle$ states. As a result we go back to the case of pure $Z_2$-SPT phase, which has no GSD on torus.

Physically how does string operator $\hat{O}_L$ tunnel between the 2 degenerate ground states? There is a clear picture based on effective theory (\ref{boundary term:Z2}). Since the bound states of kinks (from $Z_2$-SPT) and ends of strings (from double semion model) ``condense'' on the boundary, it's number doesn't conserve anymore. physical observables will be screened by the condensate.
The unscreened anyons must have trivial mutual statistics with the condensate, {\emph{i.e.~}} they must commute with the arguments of both cosine terms in (\ref{boundary term:Z2}). It turns out there is only one type of unscreened anyon here:
\begin{eqnarray}
\psi=e^{\hspace{1pt}\mathrm{i}\hspace{1pt}(\phi_s+\phi_{\bar s})}\cdot e^{-\hspace{1pt}\mathrm{i}\hspace{1pt}\phi_1}\simeq e^{\hspace{1pt}\mathrm{i}\hspace{1pt}(\phi_s-\phi_{\bar s})}\cdot e^{-\hspace{1pt}\mathrm{i}\hspace{1pt}\phi_2}\notag
\end{eqnarray}
When such a anyon is created from one boundary, another anyon must be created from the other boundary. The double-semion-model part of this operator ({\emph{i.e.~}} $e^{\hspace{1pt}\mathrm{i}\hspace{1pt}(\phi_s\pm\phi_{\bar s})}$ part) from both boundary can be brought into the bulk of double-semion-model region and annihilated\cite{Wang2012}, while the $Z_2$-SPT part of operator $\psi$ ({\emph{i.e.~}} $e^{-\hspace{1pt}\mathrm{i}\hspace{1pt}\phi_{1,2}}$ part) are left on both boundaries. Such a process of of creating (on the boundary) and annihilating (in the SET bulk) anyons is precisely realized by string operator $\hat{O}_L$.\\

\subsection{Quantum spin Hall insulators $\Big(G_s=U(1)\rtimes Z_2^{\boldsymbol{T}}\Big)$}

Now let's turn to a more familiar example of SPT phases of electrons: ${\mathbb{Z}}_2$ quantum spin Hall insulator\cite{Kane2005a,Bernevig2006,Konig2007} (QSHI). Its edge theory is (\ref{edge action}) with ${\bf K}=\bpm1&0\\0&-1\end{pmatrix}$. Let's denote the two branches of chiral bosons as $\phi_{R/L}$, and they transform under $U(1)\rtimes Z_2^{\boldsymbol{T}}$ symmetry as
\begin{eqnarray}
\notag&e^{\hspace{1pt}\mathrm{i}\hspace{1pt}\theta\hat{N}_f}\phi_{R/L}e^{-\hspace{1pt}\mathrm{i}\hspace{1pt}\theta\hat{N}_f}=\phi_{R/L}+\theta,~~~\begin{pmatrix}\phi_R\\ \phi_L\end{pmatrix}\overset{{\boldsymbol{T}}}\rightarrow\begin{pmatrix}-\phi_L\\-\phi_R+\pi\end{pmatrix}.
\end{eqnarray}
Electrons on the edge are $\psi_{R/L}\sim e^{\hspace{1pt}\mathrm{i}\hspace{1pt}\phi_{R/L}}$. A gap will open up on the edge specrum, by magnetic order $\hat{M}=M\cos(\phi_R-\phi_L+\alpha_M)$ which breaks time reversal ${\boldsymbol{T}}$, or by superconductivity $\hat\Delta=\Delta\cos(\phi_R+\phi_L+\alpha_\Delta)$ which breaks $U(1)$ charge conservation. Can we restore symmetry and obtain a gapped symmetric edge by proliferating defects of order parameters $\hat{M}$ and $\hat\Delta$? The answer is no. For example, domain wall of magnetic order $\hat{D}_{\boldsymbol{T}}=\exp\big[\hspace{1pt}\mathrm{i}\hspace{1pt}(\phi_R+\phi_L)/2\big]$ is a bosonic object, but it carries unit charge. Therefore proliferating this object will break $U(1)$ charge conservation!

Hence we need to play the same trick and consider the boundary between QSHI and a fractionalized SET phase. A convenient choice is the so-called QSHI$^\ast$ phase\cite{Ruegg2012}, obtained by coupling fermions in QSHI to a dynamical $Z_2$ gauge field\cite{Senthil2000}. As shown in Appendix \ref{app}, the effective edge theory of QSHI$^\ast$ is described by (\ref{edge action}) with ${\bf K}=\bpm0&2\\2&0\end{pmatrix}$. If we label its edge chiral bosons as $\phi_{s/c}$, they transform under $U(1)\rtimes Z_2^{\boldsymbol{T}}$ symmetry as
\begin{eqnarray}
\notag&e^{\hspace{1pt}\mathrm{i}\hspace{1pt}\theta\hat{N}_f}\begin{pmatrix}\phi_s\\ \phi_c\end{pmatrix} e^{-\hspace{1pt}\mathrm{i}\hspace{1pt}\theta\hat{N}_f}=\begin{pmatrix}\phi_s\\ \phi_c+\theta\end{pmatrix},\\
&\notag\begin{pmatrix}\phi_s\\ \phi_c\end{pmatrix}\overset{{\boldsymbol{T}}}\longrightarrow\begin{pmatrix}\phi_s\\-\phi_c\end{pmatrix}+\frac{\pi}{2}\bpm1\\1\end{pmatrix}.
\end{eqnarray}
Therefore $e^{\hspace{1pt}\mathrm{i}\hspace{1pt}\phi_c}$ is a charge-1 boson (``chargeon''), with mutual semion statistics with neutral boson $e^{\hspace{1pt}\mathrm{i}\hspace{1pt}\phi_s}$ (``spinon'' since ${\boldsymbol{T}}^2=-1$). Clearly $e^{-\hspace{1pt}\mathrm{i}\hspace{1pt}\phi_c}$ can form a neutral bosonic bound state with domain wall $\hat{D}_{\boldsymbol{T}}$, and its condensation will lead to a gapped symmetric edge between QSHI and QSHI$^\ast$. More precisely the boundary tunneling term is
\begin{eqnarray}
\notag&\mathcal{H}_t=T_c\cos(\phi_R+\phi_L-2\phi_c)+T_s\cos(\phi_R-\phi_L-2\phi_s).\\
\label{boundary term:QSHI}
\end{eqnarray}

When we consider the hybrid geometry in FIG. \ref{fig:torus}, there is a 2-fold GSD. Remarkably the two degenerate ground states are labeled by different electron number parity on each edge. Here the closed string operator becomes
\begin{eqnarray}
\hat{O}_S=e^{\hspace{1pt}\mathrm{i}\hspace{1pt}\oint_C\text{d}x~\partial_x\phi_c(x)}=e^{\hspace{1pt}\mathrm{i}\hspace{1pt}\oint_C\text{d}x~\partial_x\phi_s(x)}.
\end{eqnarray}
where the equality is enforced by boundary condition (\ref{boundary term:QSHI}). Meanwhile the physical meaning of open string operator $\hat{O}_L$ in FIG. \ref{fig:torus} is to create a pair of anyons
\begin{eqnarray}
\psi=e^{\hspace{1pt}\mathrm{i}\hspace{1pt}(\phi_c+\phi_s)}\cdot e^{\hspace{1pt}\mathrm{i}\hspace{1pt}\phi_R}\simeq e^{\hspace{1pt}\mathrm{i}\hspace{1pt}(\phi_c-\phi_s)}\cdot e^{\hspace{1pt}\mathrm{i}\hspace{1pt}\phi_L}
\end{eqnarray}
on each edge, move the $e^{\hspace{1pt}\mathrm{i}\hspace{1pt}(\phi_c\pm\phi_s)}$ part to the QSHI$^\ast$ bulk and annihilate them. Notice that in this process an extra electron $e^{\hspace{1pt}\mathrm{i}\hspace{1pt}\phi_{R/L}}$ is created on each boundary, causing the local change of fermion number parity. In supplemental materials we prove that gauging fermion number parity ($Z_2^f$) symmetry in QSHI leads to QSHI$^\ast$, again confirming our conjecture (\ref{conjecture}).

\section{Summary}

In this paper we investigate the question of how to proliferate topological defects (domain walls or kinks) on the symmetry-broken edge of an SPT phase, in order to achieve a gapped symmetric edge. We show that condensing these defects is either forbidden by quantum statistics or breaks symmetry. On the other hand, we can overcome these obstructions by considering a boundary between an SPT phase $\mathcal{S}$ and a fractionalized (SET) phase $\mathcal{F}$. We propose a conjecture (\ref{conjecture}) for how to find such fractionalized phases for a given SPT, which generalizes to all spatial dimensions. Two examples in two spatial dimensions are presented with effective field theory and exactly solvable models. We found that once this hybrid structure is put on a torus as in FIG. \ref{fig:torus}, there will be ground state degeneracy accompanying the two gapped symmetric edges between $\mathcal{S}$ and $\mathcal{F}$.

\emph{Note added}~~~Upon completion of this work. we became aware of \Ref{Wang2013}, where a different geometry containing both SPT and SET phases are considered. In their situation open string order parameter $\hat{O}_L$ only contains operators from SET side on one boundary, therefore it breaks symmetry in a nonlocal way.

\acknowledgements

This work is supported by DOE Office
of Basic Energy Sciences, Materials Sciences Division of the U.S. DOE under contract No. DE-AC02-05CH11231 (YML,DHL) and in part by the National Science Foundation under Grant No. PHYS-1066293(YML). We thank Taylor L. Hughes and Ashvin Vishwanath for helpful conversations. YML thanks the hospitality of the Aspen Center for Physics where he presented this work in 2013 summer program ``Disorder, Dynamics, Frustration and Topology in Quantum Condensed Matter", as well as Institute for Advance Study in Tsinghua University for hospitality where part of this work is finished.