text:
\section{}
\section{Introduction}\label{sec:introduction}
The Lieb Shultz Mattis (LSM) theorem \cite{Lieb:1961p407}, appropriately generalized to higher dimensions\cite{Oshikawa:2000p1535,Paramekanti2004luttinger, hastings2004lieb, Hastings:2005p824}, requires that a gapped spin system with fractional spin (eg. S=1/2) per unit cell possess excitations with fractional statistics (anyon) and fractional quantum numbers (topological order), if all symmetries (including lattice translations) are preserved. This has served as a powerful principle to diagnose exotic phases such as the fractional quantum Hall effect, and quantum spin liquids. Furthermore, in some cases the nature of the resulting topological order can be further constrained by the microscopic data \cite{Zaletel:2014p,cheng2015translational}.

In recent years there has been an explosion of activity on  symmetry protected topological (SPT) phases, which feature protected boundary modes although the bulk is short range entangled (SRE) and in contrast to the situation above, is free of anyon excitations. These include phases like topological insulators, which can be captured by free fermion models\cite{Qi:2011p1057,Hasan:2010p3045}, as well as intrinsically interacting phases \cite{Affleck:1987p799,Chen:2012p1604,chen2013symmetry} A natural question to ask is - are there setting where the microscopic data alone would enforce an SPT phase, in a fashion analogous to the LSM theorem? If so, for a particular set of microscopic data,  can we further characterize precisely which kinds of SPT orders are mandated?

These questions are answered in the present work. We show that SPT order {\em must} arise when the following conditions are met. The first ingredient is  magnetic translation symmetry, that is an enlargement of the unit cell due to the non-commutativity of the primitive translation operations. Second, we require that the primitive unit cell (ignoring the noncommutativity) does not admit a trivial insulating phase. This is arranged by requiring a projective representation at each lattice site. Finally, we need some compatibility conditions between these two ingredients that allow, among other conditions, that the enlarged unit cell to be effectively at integer filling, what admits a short range entangled ground state.  The latter is then shown to be an SPT. Furthermore  for 2+1D bosonic systems we explicitly calculate the allowed SPTs compatible with the microscopic specifications. In addition we construct exactly soluble lattice models of this phenomenon to demonstrate the validity of our conclusions. This general principle should aid in the search for SPTs in realistic settings and exposes anew aspect of the interplay between symmetry and topology.

To give some simple plausibility arguments as to how microscopic details can enforce SPT order, consider  free fermions in a magnetic field,  when the filling fraction (ratio of particle density to magnetic flux density) is an integer. Then, an integer number of  Landau levels will be filled, leading to a Chern insulator - which is a SRE topological phase with gapless edge states. Even in the presence of a lattice , one can establish a similar connection between the Hall conductance $\sigma_{xy}$, the flux $n_\phi$ and electron filling in the unit cell  $n_e$  \cite{dana1985quantised, LuRanOshikawa} which has been extended to the case of time reversal symmetric topological insulators \cite{wu2017symmetry}.

To state our result more precisely, we consider a two dimensional lattice where the unit translations obey: $T_xT_yT_x^{-1}T_y^{-1} = g$, where $g$ is an element of the symmetry group $G$. This generalizes the notion of a magnetic translation, particles acquire a phase factor depending on their $g$ charge. We assume $g$ is in the center of the symmetry group $G$ (i.e. commutes with all other elements), but otherwise consider a general $G$, which can either be discrete or continuous, Abelian or nonAbelian, and can include time reversal implemented by an antiunitary representatation. Furthermore, in each unit cell a projective representation of the symmetry group labeled by  `$\alpha$' is present. We derive a formula which provides a necessary and sufficient condition on these inputs to allow for a SRE phase, and determine constraints on the resulting SPT. Physically, this formula demands that a symmetry flux $g$ inserted into this system will precisely generate a projective representation that can screen `$\alpha$' \cite{wu2017symmetry}.

Let us give two physical pictures to view this filling and flux enforced SPTs. First we describe a vortex condensation based picture, for a system of lattice bosons with a conserved U(1) charge, with flux  $n_\phi$  and filling $n_b$  per lattice unit cell. Although our paper focuses on having projective representations per site (rather than fractional filling) this example will be useful to build intuition.  It is well known that a conventional insulator can be thought of as a condensate of vortices. However, for fractional filling $n_b$,  the vortices see a fractional flux per unit cell \cite{lee1989anyon}, and their condensate will break lattice symmetries. Similarly, the bosons themselves cannot condense without breaking lattice symmetries due to the fractional flux $n_\phi$. {\em However} the bound state of a vortex and $p$  bosons  may be able to propagate freely if: $n_b \pm p n_\phi \in {\mathcal Z}$ is an integer. The resulting object is a boson for $p$ even which can then condense giving rise to  a SRE and symmetric insulator. These are nothing but the Bosonic Integer quantum Hall insulators at $\nu=n_b/n_\phi = p$ \cite{chen2013symmetry,Lu:2013classification, senthil2013integer}. Note, here the condensing particle carries unit vorticity and hence the resulting insulator is free of topological order \cite{balents1999dual} and also preserves the U(1) symmetry since the condensing charge is attached to vorticity. A generalization of this result to include arbitrary symmetry groups is the main result of this paper. An interesting exception occurs for $p=1$, which is realized for example when one has bosons at half filling (or a projective representation of $U(1)\rtimes Z_2$), and a $\pi$ flux in each unit cell. The doubled unit cell is at integer filling. At first sight it appears we can obtain an insulator by condensing the vortex-charge composite which sees no net flux in the unit cell. However, this composite is a fermion and cannot be condensed. This is also seen by a flux threading argument \cite{LuRanOshikawa} that constrains such SRE phases to have $\sigma_{xy} = {\rm odd\ integer}$, which is impossible for a SRE topological phase of bosons \cite{Lu:2013classification,senthil2013integer}.  Interestingly, this result continues to hold if the $U(1)$ is broken to a discrete symmetry as shown below.

A second perspective is to begin in a topologically ordered phase with fractionalized excitations and consider confining all exotic excitations by an appropriate anyon condensate. For example, for bosons at half filling, one could obtain toric code (Z$_2$) topological order where the $e$ particle carries half charge \cite{Wenbook}.  The $m$ particle however sees the fractional charge density as background flux and cannot condense while preserving spatial symmetries. This is the situation in the absence of magnetic translations, where the LSM theorem enforces topological order for gapped symmetric states. However, once we allow for magnetic translations with $g$ charge, a way out to an SRE phase may become available. The $m$ particle, bound to a $g$ charge that sees the magnetic flux, forms  a composite object that may condense uniformly and confine the topological order. At the same time, this leads to an SPT phase since the condensing anyon carries nontrivial symmetry charge \cite{jiang2017anyon,duivenvoorden2017entanglement}. Indeed this picture will allows us to construct models of such LSM enforced SPT phases as we describe below.

Before discussing construction of models, it may be helpful to give a few examples. Consider a system of degenerate doublets (``S=1/2") on sites of a square lattice. This site degeneracy may arise from spin rotation invariance ($SO(3)$), or  even just as Kramers degeneracy protected by time reversal $Z_2^T$ symmetry. Now consider an additional $Z_2$ symmetry which is invoked in defining the magnetic translations, i.e. we have a fully frustrated Ising model on the same lattice. According to our results, in both these situations SRE ground states are possible but must be SPT phases. While the SPT phase is unique for the second case of Kramers doublets of $Z_T\times Z_2$, in the former case of $SO(3)\times Z_2$ there is more than one SPT phase possible. Interestingly, if we consider a minor modification of the $Z_T\times Z_2$ model, such that the doublets on each site are non Kramers pairs, protected by the combination of the two symmetries, then {\em no} SRE ground state exists (and hence no SPT exists) that respects all symmetries. These examples are discussed in detail in Section \ref{sec:examples} which also introduces models that realize them.

In constructing models, the first step is to begin in the deconfined phase of a discrete lattice gauge theory (or of a dimer model). Then, one way to obtain a confined phase is by decorating the electric field lines with domain walls of a global symmetry. This identification implies that we have condensed the composite of magnetic flux and symmetry charge. The resulting confined phase is potentially an SPT if the electric charges are associated with the appropriate symmetry fractionalization \cite{jiang2017anyon,duivenvoorden2017entanglement}. However, to obtain an LSM enforced SPTs the situation is different since they involve fractional spin on the sites. In a dimer model this corresponds to having an odd number of dimers associated with a unit cell, in which case we cannot decorate them with regular domain walls (which should be closed loops). {\em However} if the global symmetry is also associated with flux in the unit cell (for example a fully frustrated Ising model), the two kinds of frustration cancel each other out, and one can still achieve this decoration of electric field line. This is discussed explicitly  in Section \ref{sec:BFG}, for a specific model and the resulting state is shown to be the desired SPT. The model there is one of hardcore bosons on the Kagome lattice tuned to half filling by particle hole symmetry, previously introduced by Balents Fisher and Girvin \cite{balents2002fractionalization}. While their focus was on a Z$_2$ spin liquid phase, we decorate their model with an additional Z$_2$ symmetry realized by a fully-frustrated Ising model. The combination is shown to realize an LSM enforced SPT phase with gapless edge states, but a short range entangled bulk.

Finally in Section \ref{sec:MainResults} we discuss the problem for general symmetry groups, and derive the necessary and sufficient conditions for SRE phases to emerge and identify the class of SPTs that must be realized. Proofs can be found in the appendices.

\section{A simple model realizing SPT phase}\label{sec:BFG}
Our discussion starts from a concrete microscopic model realizing an SPT phase. The beauty of this model is its simplicity, which only includes two-spin and three spin interactions. It turns out that the crucial features of this model can be systematically generalized which form the main results of the current study.

The model constructed below (see Eq.(\ref{eq:decoratedBFG_full})) is based on the Balents-Fisher-Girvin(BFG) model\cite{balents2002fractionalization}. The original BFG model\cite{balents2002fractionalization} is a model with spin-1/2 residing on Kagome lattice. It is the low energy effective Hamiltonian if we take the $J_z\gg J_{\perp}$ limit of the following $XXZ$ Hamiltonian
\begin{equation}
H^{XXZ}=J_{\perp}\sum\limits_{\small\hexagon}[(\sum\limits_{i\in\hexagon}S^x_i)^2+(\sum\limits_{i\in\hexagon}S^y_i)^2-3]+J_z\sum\limits_{\hexagon}(\sum\limits_{i\in\hexagon}S^z_i)^2,
\end{equation}
which has a spin-liquid ground state for $J_z\gg J_{\perp}$ with deconfined spinons as confirmed by various numerical methods \cite{sheng2005numerical,Isakov2006spin}.

Let's then take a look at the low energy effective Hamiltonian. The limit $J_z\gg J_{\perp}$ ensures that $S^z_{\hexagon}=0$ for every hexagon and the resulting Hamiltonian in this low energy manifold takes the following ring-exchange form
\begin{equation}\label{eq:originalBFG}
H^{BFG}=-J_{\text{ring}}\sum\limits_{\bowtie}(\big|\raisebox{-4mm}\BFGB\big\rangle \big\langle\raisebox{-4mm}\BFGA\big|+h.c.),
\end{equation}
with $J_{ring}=J_{\perp}^2/J_z$.

\begin{figure}
\includegraphics[width=0.48\textwidth]{decoratedBFG.png};
\caption{(color online) Degrees of freedom in the decorated BFG model. The Ising d.o.f. $\sigma_I$ live on the honeycomb lattice and the spin d.o.f. $S_i$ lives on the Kagome lattice. The Ising coupling signs $s_{IJ}=+1$ on red bonds, and $s_{IJ}=-1$ on blue bonds. The thick red bonds represent the ``$y$-odd zigzag chains'' used in Eq.~\eqref{eq:BFG_magn_trans}}
\label{fig:decoratedBFG}
\end{figure}

Let's then decorate the $XXZ$ model by putting a layer of Ising spins $\sigma$ inside every triangle of the Kagome lattice, which comprises a honeycomb lattice. The Ising spins are in a transverse field, \textit{i.e.},
\begin{equation}
H^{Ising}=h\sum\limits_{I}\sigma_I^x,
\end{equation}
with Ising spin $\sigma_I$ living on the honeycomb lattices labeled by $I$.

We then couple these two layers through a binding term
\begin{equation}
H^{binding}=-\sum\limits_{\bindingterm}\lambda S_i^z\cdot(s_{IJ}\sigma^z_I\sigma^z_J),
\end{equation}
where the summation is over all the bonds $IJ$ on honeycomb lattice with $S_i$ at the bond center. The sign $s_{IJ}=\pm 1$ are frustrated in the sense that $\prod\limits_{I,J\in\hexagon}s_{IJ}=-1$. We have specifically chosen a choice of $s_{IJ}$ in Fig.~\ref{fig:decoratedBFG}. The binding term binds spin-up with Ising happy bond ($s_{IJ}\sigma_I^z\sigma_J^z=+1$) and spin-down with Ising un-happy bond ($s_{IJ}\sigma_I^z\sigma_J^z=-1$).

The full Hamiltonian we are considering is then given by (see Fig.~\ref{fig:decoratedBFG})
\begin{equation}\label{eq:decoratedBFG_full}
H=H^{XXZ}+H^{binding}+H^{Ising}.
\end{equation}

One can divide $H$ into two parts
\begin{equation}
\begin{split}
&H_0=J_z\sum\limits_{\hexagon}(S^z_{\small\hexagon})^2-\sum\limits_{\bindingterm}\lambda S_i^z(s_{IJ}\sigma^z_I\sigma^z_J).\\
&H_1=J_{\perp}\sum\limits_{\hexagon}[(S^x_{\hexagon})^2+(S^y_{\hexagon})^2-3]+\sum\limits_{I}h\sigma_I^x.
\end{split}
\end{equation}

Considering the the limit where $J_z,\lambda \gg J_{\perp},h$, we can first deal with $H_0$ and then treat $H_1$ as a perturbation. All the terms in $H_0$ commutes with each other and hence all the eigen-states and eigen-energies are known for $H_0$. In fact, there is a two-to-one mapping from the ground state sector to the low energy sector of the BFG model (i.e. $\{S_i^z\}$ configurations satisfying 3 $S^z=+1/2$  per hexagon). We will consider periodic boundary conditions, and the Hilbert space of the original BFG model has four topological sectors labeled by parities of the $\prod_{k\in \mathcal{C}}2S_k^z$ around the non-contractable loops $\mathcal{C}$ (which is just the non-contractable vison flux line\cite{fradkin2013field}). This mapping only map onto one specific topological sector since $\prod_{k\in \mathcal{C}}2S_k^z$ is identified with  $\prod_{IJ\in \mathcal{C}}s_{IJ}$ due to $H^{binding}$. The preimage of any low energy $\{S_i^z\}$ configuration inside this topological sector are two states $\ket{\{S_i^z,+\}}$ and $\ket{\{S_i^z,-\}}$ (related to each other by a global Ising flip).

It turns out that the effective Hamiltonian in the parameter regime where $J_z\gg \lambda\gg J_{\perp},h$ and $\frac{h^2}{\lambda^2}\gg \frac{J_{\perp}}{J_z}$ has the following form (see Appendix.~\ref{app:BFG} for detailed calculation)
\begin{equation}\label{eq:decoratedBFG}
H^{deco.BFG}=-\frac{10J_{\perp}^2 h^2}{9J_z\lambda^2}\sum\limits_{\bowtie}(\big|\raisebox{-4mm}\bowtieB\big\rangle \big\langle\raisebox{-4mm}\bowtieA\big|+h.c.).
\end{equation}

Note that the kinetic term in this effective Hamiltonian is the ring exchange term of four spins at the ends of each bowtie as in the original BFG model combined with the flipping term of the two Ising d.o.f. within this bowtie, such that the constraint $S_i^z(s_{IJ}\sigma^z_I\sigma^z_J)=1$ is still satisfied everywhere.

\begin{figure}
\includegraphics[width=0.48\textwidth]{vison_condensation.png}
\caption{(color online) (a) Schematic phase diagram of the decorated BFG model by tuning $\lambda$. We have already fixed $J_z\gg J_{\perp},h$. In the limit $\lambda\rightarrow 0$ the Ising layer is decoupled and the ground state is just that of the original BFG model with $Z_2$ topological order. This is an SET state with spinon carrying $S^z=1/2$. When $\lambda$ is tuned to be within the parameter regime where $J_z\gg\lambda\gg J_{\perp},h$ and $\frac{h^2}{\lambda^2}\gg \frac{J_{\perp}}{J_z}$, we have an SPT state with Ising defect carrying $S^z=1/2$ as discussed in the main text. There is a possible direct phase transition triggered by the condensation of Ising-odd visons at some intermediate $\lambda_c$. (b) A schematic view of vison condensation. The honeycomb lattice where Ising d.o.f. lives is shown and the spin d.o.f. lives in the bond center. Two visons are created at the ends $I,J$ of the string operator $\sigma_I^z\sigma^z_J\prod\limits^j_{k=i}\!\!\!\!\!\!\!\!\!\!\longrightarrow 2S^z_k.$ with $S_k^z$ runs over all the black dot shown in the graph. Alternatively we can view the string operator as the product of bond variable $S_i^z\sigma_I^z\sigma^z_K$ along the thick blue bonds. Due to the constraint $S_i^z(s_{IK}\sigma_I^z\sigma_K^z)=1$, the string operator will yield a factor (product of $s_{IK}$'s along the thick blue bonds) when acting on the ground state wave-function, which means the visons are condensed and the topological order is killed. Note that the condensed visons in the present case are dressed by local $\sigma^z$ operator and hence carry the quantum number of $Z_2$ Ising symmetry, which result in an SPT state.}
\label{fig:vison_condensation}
\end{figure}
We shall then prove that the ground state of the decorated BFG model is a symmetric short-range entangled SPT state. In fact, using the mapping $\mathcal{P}$ between the Hilbert space of the decorated BFG model and the original BFG model
\begin{equation}
    \mathcal{P}: (\ket{\{S_i^z,+\}}+\ket{\{S_i^z,-\}})/\sqrt{2}\rightarrow\ket{\{S_i^z\}}.
\end{equation}
Such a mapping is clearly an isometry. Next we notice that
\begin{equation}
    \mathcal{P}H^{deco.BFG}\mathcal{P}^{-1}=H^{BFG},
\end{equation}
with the identification $J_{ring}=\frac{10J_{\perp}^2h^2}{9J_z\lambda^2}$, which can be proven by directly comparing the matrix elements on the two sides.

Therefore the spectrum of $H^{deco.BFG}$ within the Ising-even sector is exactly the same as that of $H^{BFG}$ inside a specific topological sector, which is known to be gapped. And the ground state $\ket{\psi}$ of $H^{BFG}$, should also be mapped to the ground state $\ket{\psi^{deco.}}$ of $H^{deco.BFG}$. However there is still one possibility that there exists a state in the Ising-odd sector with exactly the same energy as $\ket{\psi^{deco.}}$, which features the Ising symmetry breaking. This possibility is ruled out because $\ket{\psi^{deco.}}$ has no long-range order in $\sigma^z$ as will be discussed in the context of vison condensation.

The ground state of the original BFG model has $Z_2$ topological order which supports vison and spinon excitations. In the original BFG model, the vison excitations live in the honeycomb lattice and are created at the ends of the string of $S_k^z$ operators\cite{balents2002fractionalization}, \textit{i.e.},
\begin{equation}
v_Iv_J=\prod^{j}_{k=i}\!\!\!\!\!\!\!\!\!\!\longrightarrow2S^z_k,
\end{equation}
where two visons are created at $I$ and $J$ (see Fig.~\ref{fig:vison_condensation}).

We can see that the visons are condensed in $\ket{\psi^{deco.}}$. The vison operator $v^{deco.}$ at site $i$ should now be dressed by the local Ising operators $\sigma^z_I$ with $I$ around $i$ to obtain
\begin{equation}
v^{deco.}_Iv^{deco.}_J=\sigma_I^z\sigma^z_J\prod^j_{k=i}\!\!\!\!\!\!\!\!\!\!\longrightarrow 2S^z_k.
\end{equation}

With the constraint that $S_i^z(s_{IJ}\sigma^z_I\sigma^z_J)\equiv 1$ and the fact that intermediate $\sigma^z$ squared to 1, we know that $v^{deco.}_Iv^{deco.}_J$ must yield a constant (depending only on the product of $s_{IJ}$'s along the vison string) when acting on $\ket{\psi^{deco.}}$, see Fig.~\ref{fig:vison_condensation} for an illustration.

Now it is clear that the correlator $\sigma_I^z\sigma_J^z$ is short-range because we have
\begin{equation}
\begin{split}
&|\bra{\psi^{deco.}} \sigma^z_I\sigma_J^z\ket{\psi^{deco.}}|=|\bra{\psi^{deco.}} \prod^j_{k=i}\!\!\!\!\!\!\!\!\!\!\longrightarrow 2S^z_k\ket{\psi^{deco.}}|,\\
&=|\bra{\psi}\prod^j_{k=i}\!\!\!\!\!\!\!\!\!\!\longrightarrow 2S^z_k\ket{\psi}|,
\end{split}
\end{equation}
where the last correlator exhibits exponential decay since visons are deconfined in the original BFG model (the last equality holds because $\mathcal{P}$ commutes with the string operator).

The above discussions feature the physical picture of the condensation of visons carrying Ising quantum number, which kills the $Z_2$ topological order. We will soon show that the resulting phase is an SPT phase, which is exactly a realization of the anyon condensation mechanism to obtain SPT phases proposed in Ref.~\onlinecite{jiang2017anyon}. In fact, if we start from the decoupling limit with $\lambda=0$ and gradually increase $\lambda$ with all other couplings fixed, we should be able to see two phases: an SET phase with spinon carrying $S^z=1/2$ when $\lambda$ small and an SPT phase resulting from the condensation of Ising-odd visons when $\lambda$ is in the parameter regime $J_z\gg\lambda\gg J_{\perp},h$ and $\frac{h^2}{\lambda^2}\gg \frac{J_{\perp}}{J_z}$. This two phases might be related by a continuous phase transition at some intermediate $\lambda_c$. See Fig.~\ref{fig:vison_condensation} for an illustration.

\begin{figure}
\includegraphics[width=0.48\textwidth]{Ising_defect_BFG.png};
\caption{(color online) An pair of Ising defects (only one is shown) is created at the end points of the branch cut (dashed black line) after modifying the original Hamiltonian $H$ in Eq.(\ref{eq:decoratedBFG_full}) into $H'$. The sign $s_{IJ}$ is flipped in $H'$ along the branch cut comparing with the original model. (red bond: $s_{IJ}=+1$, blue bond: $s_{IJ}=-1$) For any loop $\mathcal{C}$ enclosing the Ising defect as the gray loop shown here, the product $\prod s_{IJ}$ around the loop flips sign comparing with the original model.  In order that $H'^{binding}$ does not cost extra energy, the spin should be flipped wherever $s_{IJ}$ changes its sign. As a result, the total $S^z$ around the loop $\mathcal{C}$ is changed by an odd integer. The result is that Ising defect is topologically bound with a half-integer spin. See the discussion in the main text.}
\label{fig:Ising_defect_BFG}
\end{figure}

One way to see that the ground state of the decorated BFG model Eq.(\ref{eq:decoratedBFG}) is an SPT phase is to consider the Ising defects, which turn out to be topologically bound with $S^z=\pm 1/2$ --- a projective representation of the symmetry group (see discussion below). In order to introduce Ising defects we need to take a branch cut and modify the terms straddling the branch cut such that only one side is conjugated by the Ising symmetry $\sigma^x$. The net effect is that for the bonds $IJ$ crossed by the branch cut, the sign of $s_{IJ}$ is flipped. See Fig.~\ref{fig:Ising_defect_BFG} for an illustration. To compute the $S^z$ quantum number carried by the Ising defect it is convenient to introduce the equivalent hard-core boson description: $n_i\equiv S^z_i+1/2$. Now let's take a loop $\mathcal{C}$ enclosing one of the Ising defect and measure the total charge within the area $\mathcal{D}$ bounded by $\mathcal{C}$. This is done by the following U(1) transformation
\begin{equation}\label{eq:bosonnumber}
\prod_{\hexagon\in \mathcal{D}}(\prod_{i\in\hexagon}e^{i\frac{\theta}{2}n_i})=\prod_{i\in \mathcal{C}}e^{i\frac{\theta}{2}n_i}\cdot \prod_{i\in \mathcal{D}/\mathcal{C}}e^{i\theta n_i},
\end{equation}
from which we know that the fractional charge part is determined by boson numbers on the boundary only.

In order that the binding term after modification $H'^{binding}$ does not cost energy, the boson number $n_i$ on the bond $IJ$ across the branch cut should be changed by 1. Since $\mathcal{C}$ only crosses Ising defect line odd times, the total boson numbers around $\mathcal{C}$ should be changed by $1(mod\ 2)$. From Eq.~\eqref{eq:bosonnumber}, this amounts to the change of charge within $\mathcal{D}$ by $\frac{1}{2}(mod\ 1)$. This fact doesn't depend on the position of the branch cut or the position of $\mathcal{C}$ we have chosen, indicating that the extra $\frac{1}{2}$ charge is bounded with the Ising defect. In the original spin language it means Ising defect carries spin $S^z=\pm 1/2$.

\textbf{Summary:} Before proceeding let's pay close attention to the symmetry property of the Hamiltonian in Eq.~\eqref{eq:decoratedBFG_full}.
This model has the following $Z_{2g}\times(U(1)\rtimes Z_{2h}) $ onsite symmetry
\begin{enumerate}
\item Spin-rotation symmetry $U(1)$: $\prod\limits_ie^{i\theta (S_i^z+\frac{1}{2})},$ with $\theta\in [0,2\pi)$.
\item Spin-flip and Ising-flip symmetry $Z_{2h}=\{I,h\}$: $\prod\limits_i S_i^x\prod\limits_{I\in A}\sigma^x_I$, which flips all the spins on Kagome lattice and Ising d.o.f on $A$ sublattice of the honeycomb lattice. This symmetry operation leaves the Hamiltonian in Eq.~\eqref{eq:decoratedBFG_full} invariant.
\item Ising symmetry $Z_{2g}=\{I,g\}$: $\prod\limits_I\sigma^x_I$, which flips all the Ising d.o.f. on the honeycomb lattice.
\end{enumerate}

The onsite symmetry group has the direct product structure $G=Z_{2g}\times (U(1)\rtimes Z_{2h})$. In addition, every unit cell (three kagome sites and two honeycomb sites) carries a nontrivial projective representation of $U(1)\rtimes Z_{2h}$ (i.e. a half-integer spin). Naively one would suspect that the LSM theorem would rule out a symmetric short-range-entangled state in such a system. However, due to the frustrated nature of the gluing term, we actually have magnetic translation symmetry $T_x,T_y$ in the system instead of usual translation $T_{x}^{orig.},T_y^{orig.}$, which can be written as
\begin{equation}\label{eq:BFG_magn_trans}
\begin{split}
&T_x=(\prod\limits_{I\in\text{y-odd zigzag chains}}\sigma_I^x)T_x^{orig.},\\
& T_y=T_y^{orig.},
\end{split}
\end{equation}
which satisfy the magnetic translation algebra
\begin{equation}
T_xT_yT_x^{-1}T_y^{-1}=g.
\end{equation}
Therefore it is still possible for us to have an SPT state. And we also know that the $g$-defect carries $S^z=\pm 1/2$, which has the same projective representation as that carried by the unit-cell. Below we will find that this is not merely a coincidence and there is a deep connection between the patterns of short-range entanglement and the projective representation carried by a unit-cell related by the magnetic translation algebra.

\section{Main Results}
\label{sec:MainResults}
Our main results are captured in two theorems.  Theorem-I is easier to state but less general. Theorem-II is more general but is more mathematically involved to state.

Consider a two-dimensional \emph{bosonic} quantum system respecting an onsite symmetry group $G$ (which could contain time-reversal), and $g$ is a unitary symmetry element in the center of $G$ (i.e., $g$ commutes with any element in $G$). The system respects a ``magnetic'' translation symmetry group generated by $T_x,T_y$ satisfying the algebra:
\begin{align}
 T_xT_yT_x^{-1}T_y^{-1}=g,\label{eq:mag_transl}
\end{align}
where $T_x,T_y$ are assumed to be the usual translation operations combined with certain site-dependent onsite unitary transformations. We further assume that the physical degrees of freedom (d.o.f.) in each real space unit cell (\emph{not} the enlarged magnetic unit cell) form a nontrivial projective representation $\alpha$ of $G$, specified by a 2-cocycle: $\forall a,b\in G, \alpha(a,b)\in U(1)$ and $\alpha\in  H^2(G,U(1))$. Precisely, the unitary or antiunitary transformation $U_a,U_b$ of $a,b\in G$ satisfy:
\begin{align}
 U_a \leftidx{^a}U_b=\alpha(a,b) U_{ab},\label{eq:alpha}
\end{align}
where the left-superscript $a$ in $\leftidx{^a}U_b$ denotes the group action of $a$ on $U_b$: if $a$ is unitary (antiunitary), then $\leftidx{^a}U_b=U_b$ ($\leftidx{^a}U_b=U_b^{*}$, i.e. complex conjugation of $U_b$).

We ask the following question: is it possible for such a system to have a short-range entangled(SRE) gapped ground state without breaking symmetries?

Here we use the definition of SRE states following Ref.~\onlinecite{chen2013symmetry}; i.e., those are gapped quantum phases that can be deformed into the trivial product state via local unitary transformations. Note that if the system respects usual translational symmetries, this would be impossible: constrained by a generalized  Lieb Shultz Mattis theorem\cite{Lieb:1961p407,hastings2004lieb,Oshikawa:2000p1535}, the nontrivial projective representation per unit cell indicates that a gapped liquid ground state necessarily features topological order. Here because the system respects a magnetic translation symmetry, it is possible that a SRE liquid ground state exists. We give sufficient and necessary conditions for such a liquid phase to exist, and show that this SRE liquid phase must be an SPT phase.

In the presence of an onsite symmetry group $G$, focusing on bosonic systems specified by Eq.(\ref{eq:mag_transl},\ref{eq:alpha}), we have the following theorems:

\begin{framed}
\textbf{Theorem-I}: Here we further assume that $G=G_1\times Z_N$ where $Z_N$ is the finite abelian subgroup generated by $g$. The quantum system above can have a SRE liquid ground state if and only if the two conditions below are both satisfied. Such a liquid phase is necessarily a nontrivial SPT phase because the $g$-symmetry defect must carry the projective representation $\alpha$.
\begin{enumerate}
 \item $\alpha^N\simeq \mathbf{1} \in H^2(G,U(1))$, i.e. $N$ of these projective representations fuse into a regular representation.
 \item The group function (which maps elements of the symmetry group to phases, while preserving the group relations) $\gamma^{\alpha}_g(a)\equiv\frac{\alpha(g,a)}{\alpha(a,g)}$, $\forall a\in G$ is a trivial 1-cocycle (or equivalently, a trivial one-dimensional representation): \\
 $\gamma^{\alpha}_g\simeq \mathbf{1} \in H^1(G,U(1))$.
\end{enumerate}
\end{framed}
What is the physical meaning of these two conditions?  The first condition ensures that the enlarged magnetic unit cell does not have projective representations. Interestingly, this is not sufficient to ensure an SRE phase. Additionally, condition 2 must be satisfied, which essentially states that the symmetry involved in magnetic translations, $g$, can be chosen to commute with all other projective group actions in a proper gauge.

\begin{framed}
\textbf{Theorem-II}: Here we do not make extra assumptions on $G$. The quantum system above can have a SRE liquid ground state if and only if there exists a 3-cocyle $\omega_0$: $\forall a,b,c\in G,\omega_0(a,b,c)\in U(1)$ and $\omega_0\in H^3(G,U(1))$, such that the group function $\delta^{\omega_0}_g(a,b)\equiv \frac{\omega_0(g,a,b)\omega_0(a,b,g)}{\omega_0(a,g,b)}$ is 2-cycle equivalent to $\alpha^{-1}$: $\delta^{\omega_0}_g\simeq \alpha^{-1}\in H^2(G,U(1))$. Such a liquid phase is necessarily a nontrivial SPT phase because the $g$-symmetry defect must carry the projective representation $\alpha$. The possible nontrivial SPT phases form a coset from the classification point of view (see Remark below).
\end{framed}

\textbf{Remark}: it is straightforward to show that $\forall \alpha \in H^2(G,U(1))$, $\gamma^{\alpha}_g\in H^1(G,U(1))$. And similarly $\forall \omega \in H^3(G,U(1))$, $\delta^{\omega}_g\in H^2(G,U(1))$. The mappings $\gamma_g: H^2(G,U(1))\rightarrow H^1(G,U(1))$ and $\delta_g: H^3(G,U(1))\rightarrow H^2(G,U(1))$ reducing a $n$-cocycle to a $(n-1)$-cocycle are the so-called slant-products in mathematical context. $\gamma_g$ and $\delta_g$ preserve the multiplication relation in the cohomology group. In particular, there is a subgroup $\mathcal{A}_g\in H^{3}(G,U(1))$ such that $\forall\omega\in\mathcal{A}_g, \delta^{\omega}_g\simeq \mathbf{1}\in H^{2}(G,U(1))$, (i.e., $\mathcal{A}_g$ the kernal of the mapping $\delta_g$).

When the condition in Theorem-II is satisified, $\omega_0$ must be a nontrivial element in $H^3(G,U(1))$ because $\alpha$ is nontrivial by assumption. And the realizable SPT phases form a coset from the classification point of view. More precisely, the 3-cocyle characterizing the SRE liquid phase must be one of the element in the following coset: $\omega_0\cdot\mathcal{A}_g$.

\textbf{Outline of the proof}: The proof of these theorems is a combination of a pumping argument of entanglement spectra and derivations/constructions based on a recently developed symmetric tensor-network formulation\cite{jiang2017anyon}, which we outline here. Basically, if a SRE liquid phase exists, by the pumping argument of entanglement spectra one knows that \emph{the $g$-symmetry-defect in this phase must carry the projective representation $\alpha$, and consequently this phase must be an SPT phase}. This physical observation can be further justified by calculations based on symmetric tensor-networks, leading to the following mathematical result: if the 3-cocycle characterizing the SRE liquid phase as $\omega\in H^3(G,U(1))$, then magnetic translation symmetry dictates $\delta^{\omega}_g\simeq\alpha^{-1}$, which is exactly the same mathematical condition for the $g$-symmetry-defect carrying the projective representation $\alpha$. In addition, based on the symmetric tensor-network formulation, for any $\omega$ satisfying $\delta^{\omega}_g\simeq\alpha^{-1}$, a SRE liquid phase characterized by $\omega$ respecting the magnetic translation symmetry can be constructed. These prove that the conditions in Theorem-II are \emph{necessary and sufficient} for the SRE liquid phase to exist. In addition, when $\omega_0$ exists, because a 3-cocycle $\omega\in H^3(G,U(1))$ satisfies $\delta^{\omega}_g\simeq\alpha^{-1}$ if and only if $\omega\in \omega_0\cdot\mathcal{A}_g$, the coset structure in the Remark is also established.

Theorem-I is just a special case of Theorem-II. Namely when $G=G_1\times Z_N$, one can show that if and only if the two conditions in Theorem-I is satisified does the condition in Theorem-II is satisfied. The condition-(1) in Theorem-I is well anticipated. If condition-(1) is not satisfied, then physical degrees of freedom form a nontrivial projective representation of $G$ even in the enlarged magnetic unit cell ($N$ times larger than original unit cell), and the generalized  Lieb Shultz Mattis theorem\cite{Lieb:1961p407,hastings2004lieb,Oshikawa:2000p1535} already forbids a SRE liquid phase to exist. The condition-(2) is less obvious and more interesting, which puts additional constraints for the existence of a SRE liquid phase (see example-(4) below).

Before going into the details of the proof, let us consider a few simple examples to see the applications of the Theorems and the Remark.

\subsection{Examples}\label{sec:examples} In these examples, the element $g$ in the magnetic translation algebra Eq.(\ref{eq:mag_transl}) generates a $Z_2^{Ising}\equiv \{I, g\}$ Ising symmetry group. For instance, a fully frustrated Ising model on the square lattice would satisfy this magnetic translation symmetry. The symmetry-enforced SPT phases in example-(1,2,3) will be demonstrated via a class of decorated quantum dimer models, which are exactly solvable at the Rokhsar-Kivelson points\cite{rokhsar1988superconductivity}.

\textbf{(1) $\mathbf{G=SO(3)\times Z_2^{Ising}}$, and a spin-1/2 per unit cell:} Namely, the projective representation $\alpha$ per unit cell is nontrivial because only the $SO(3)$ part is projectively represented, and the Ising and the spin-rotation still commute: $\alpha(g, a)=\alpha(a,g),\; \forall a\in G$. Clearly the two conditions in Theorem-I are both satisified. First, two spin-1/2's fuse into a regular $SO(3)$ representation, and $\gamma_g^{\alpha}(a)=1,\forall a\in G$.

\emph{According to Theorem-I, at least one SRE liquid phase can exist and must be an SPT phase in which the $g$-symmetry-defect carries a half-integer spin.} To understand how many SPT phases are possibly realized, one can follow the Remark. \footnote{SPT phases protected by $G=SO(3)\times Z_2^{Ising}$ form a group $H^3(SO(3)\times Z_2^{Ising},U(1))$. The Kunneth formula gives: $H^3(SO(3)\times Z_2^{Ising},U(1))=H^3(SO(3),U(1))\times H^3(Z_2,U(1))\times H^2(SO(3),Z_2)=Z\times Z_2\times Z_2$. Following the Remark, it is straightforward to show that only the $Z_2$ index in $H^2(SO(3),Z_2)$ is enforced to be nontrivial. (Namely $\omega_0$ in Theorem-(2) can be chosen to be the nontrivial element in $H^2(SO(3),Z_2)$,and the kernal $\mathcal{A}_g=H^3(SO(3),U(1))\times H^2(SO(3),Z_2)=Z\times Z_2$.)} The result is that among all possible SPT phases classified by $H^3(SO(3)\times Z_2^{Ising},U(1))=Z\times Z_2^2$, only one of the $Z_2$ indices is enforced to be nontrivial. And there are many distinct SPT phases that can be realized, which form a coset $\omega_0\cdot \mathcal{A}_g$, where $\mathcal{A}_g=Z\times Z_2$. \emph{In particular, after gauging the $Z_2^{Ising}$ symmetry, one may obtain either the toric-code or double-semion topological order, depending on which SPT phase is realized.}

\textbf{(2) $\mathbf{G=Z_2^T\times Z_2^{Ising}}$, and a Kramer doublet per unit cell:} Here $Z_2^T=\{I,\mathcal{T}\}$ is the time-reversal symmetrg group. Denoting the Ising and time-reversal transformations on the physical d.o.f. in one unit cell as $U_g$, and $U_{\mathcal{T}}$ (antiunitary), the projective representation $\alpha$ per unit cell satisfies:
\begin{align}
 U_g^2&=1,& U_{\mathcal{T}}U_{\mathcal{T}}^*&=-1, &U_{\mathcal{T}} U^*_g&=U_g U_{\mathcal{T}}.\label{eq:Kramer}
\end{align}
For instance, this algebra is satisfied if $U_g=\sigma_x$ and $U_{\mathcal{T}}=i \tau_y$ for a four-dimensional local Hilbert space (upon which $\sigma$ and $\tau$ Pauli matrices act). One can check that the two conditions in Theorem-(1) are both satisfied, and thus \emph{at least an SRE liquid phase can exist and must be an SPT phase in which the $g$-symmetry-defect carries the projective representation $\alpha$ (a Kramer-doublet) above.}

Naively this example is very similar to the example-(1). However there is an important difference. In this example, \emph{only one SPT phase can be realized} --- following the Remark, this is because the kernel subgroup $\mathcal{A}_g$ is the trivial $Z_1$ group. \footnote{Following the Kunneth formula: $H^3(Z_2^T\times Z_2^{Ising},U(1))=H^3(Z_2^T,U(1))\times H^3(Z_2,U(1))\times H^2(Z_2^T,Z_2)=Z_1\times Z_2\times Z_2$. In this example, the $Z_2$ index in $H^2(Z_2^T,Z_2)$ is enforced to be nontrivial, and the $Z_2$ index in $H^3(Z_2,U(1))$ is enforced to be trivial. This is because here $\omega_0$ in Theorem-2 is the nontrivial element in $H^2(Z_2^T,Z_2)$, and the kernel subgroup $\mathcal{A}_g=Z_1$.} \emph{After gauging the $Z_2^{Ising}$ symmetry, one must obtain a toric code topological order.} This realizable SPT phase is topologically identical to the one obtained by decorating Ising domain walls with the $Z_2^T$ Haldane chains\cite{chen2014symmetry}.

\textbf{(3) $\mathbf{G=Z_2^T\times Z_2^{Ising}}$, and a non-Kramer doublet per unit cell:} Here the projective representation $\alpha$ per unit cell satisfies:
\begin{align}
 U_g^2&=1,& U_{\mathcal{T}}U_{\mathcal{T}}^*&=1, &U_{\mathcal{T}} U^*_g&=-U_g U_{\mathcal{T}}.\label{eq:non_Kramer}
\end{align}
For instance, $U_g=\sigma_x$ and $U_{\mathcal{T}}=\sigma_z$ on a two-dimensional local Hilbert space would satisfy this algebra. One can check that the two conditions in Theorem-(1) are both satisfied, and thus \emph{at least an SRE liquid phase can exist and must be an SPT phase in which the $g$-symmetry-defect carries the projective representation $\alpha$ (a Kramer-doublet) above.}

Similar to example-(2), \emph{there is only one realizable SPT phase}. This SPT phase turns out to be the non-trivial Levin-Gu SPT phase protected by the $Z_2^{Ising}$ alone. \footnote{Now in this example, $\omega_0$ in Theorem-2 is the nontrivial element in $H^3(Z_2,U(1))$, and the kernel subgroup $\mathcal{A}_g=Z_1$.} Namely, here \emph{after gauging the $Z_2^{Ising}$ symmetry, one must obtain a double-semion topological order. }

\textbf{(4) $\mathbf{G=Z'_2\times Z_2^{Ising}}$, and a projective representation per unit cell:} In this example, $Z'_2=\{I,h\}$ is another unitary Ising symmetry group. The projective representation $\alpha$ satisfies:
\begin{align}
 U_g^2&=1, & U_h^2&=1,&U_gU_h&=-U_hU_g.
\end{align}
For instance, $U_g=\sigma_x$, $U_h=\sigma_z$ realize this algebra. Two of such projective representations fuse into a regular representation of $G$, so the condition-(1) in Theorem-(1) is satisified. But one can show that the condition-(2) is \emph{not} satisfied:
\begin{align}
 \gamma_g^{\alpha}(h)=-1,
\end{align}
i.e., $\gamma_g^{\alpha}$ is a nontrivial 1-cocycle. Therefore according to Theorem-(1), \emph{a SRE liquid phase is not possible.} Without breaking symmetry, this suggests that topological order is inevitable for gapped systems. This is a somewhat surprising result. If one views the system using the enlarged magnetic unit cell, there is no reason why a SRE liquid is not allowed.

\section{Decorated Quantum Dimer Models for SPT phases}
Closely related to the decorated-BFG model in Sec.\ref{sec:BFG}, in this section we describe a class of exactly solvable models realizing symmetry-enforced SPT phases. These models are constructed by decorating quantum dimer models(QDM) with relevant physical degrees of freedom, whose ground states can be exactly solved at the corresponding Rokhsar-Kivelson point\cite{rokhsar1988superconductivity}. Although this class of models can be generalized to other lattices, here we will focus on the decoration of the QDM on the triangular lattice\cite{moessner2001resonating,qi2015double}. In particular, we will construct models realizing the symmetry-enforced SPT phases in example-(1,2,3) in Sec.\ref{sec:examples}

\subsection{$G=SO(3)\times Z_2^{Ising}$, a spin-1/2 per unit cell}\label{sec:SO3_Ising_models}
Continuing with discussions in example-(1) in Sec.\ref{sec:examples}, in the presence of onsite global symmetry $G=SO(3)\times Z_2^{Ising}$, we consider quantum systems with one spin-$1/2$ per unit cell in two spatial dimensions respecting the Ising magnetic translation symmetry Eq.(\ref{eq:mag_transl}). Note that we will reserve symbols $T_x,T_y$ for the magnetic translations, and use $T_x^{orig.},T_y^{orig.}$ to represent the original translations.

We will construct two exactly solvable models (model-A and model-B) respecting the symmetry described above featuring SRE liquid ground states. Although the Ising defects in both models carry half-integer spins, the two models are in distinct SPT phases. The simplest way to understand their difference is that, after gauging the Ising symmetry, model-A has toric-code topological order while model-B has double-semion topological order.

\begin{figure}
\begin{tikzpicture}
\node {\includegraphics[width=0.48\textwidth]{honeycomb_triangular_lattices.pdf}};
\node at (-2.2,-2.2) [inner sep=0] {$T_x$};
\node at (-3.1,-1.45) [inner sep=0] {$T_y$};
\node at (-1.25,0.67) [inner sep=0] {$\sigma_I$};
\node at (-0.75,1.) [inner sep=0] {$\sigma_J$};
\node at (-0.6,0.2) [inner sep=0] {$\boldsymbol{\tau}_j$};
\node at (-1.3,1.45) [inner sep=0] {$\boldsymbol{\tau}_i$};
\end{tikzpicture}
\caption{(color online) The Ising d.o.f. $\sigma$ live on the honeycomb lattice and the spin d.o.f. $\boldsymbol{\tau}$ lives on the triangular lattice. The Ising coupling signs $s_{IJ}=+1$ on red bonds, and $s_{IJ}=-1$ on blue vertical bonds. The thick red bonds represent the ``$y$-odd zigzag chains'' used in Eq.(\ref{eq:Ising_mag_transl},\ref{eq:modified_Ising_mag_transl} ). The thick gray horizontal bonds on the triangular lattice represent the ``$y$-odd rows'' used in Eq.(\ref{eq:modified_Ising_mag_transl}). }
\label{fig:model-A}
\end{figure}

We start with constructing model-A. This model contains two sets of degrees of freedom (d.o.f.): the Ising-d.o.f. $\sigma_I$ which live on a honeycomb lattice and the spin-$1/2$-d.o.f. $\mathbf{S}_i=\boldsymbol{\tau}_i/2$ which live on the triangular lattice formed by centers of the hexagons, as shown in Fig.\ref{fig:model-A}. The Hamiltonian of model-A contains three terms:
\begin{align}
 H=H^{Ising}+H^{binding}+H^{A}.\label{eq:model-A}
\end{align}
where $H^{Ising}$ is simply a frustrated nearest-neighbor Ising model:
\begin{align}
 H^{Ising}=-K\sum_{\langle I J\rangle} s_{IJ}\sigma_I^{z}\sigma_J^{z}.\label{eq:H_Ising}
\end{align}
Here $\sigma_I^z,\sigma_J^z$ are the Ising spins living on the honeycomb sites labled by $I,J$, the coupling constant $K>0$, and $s_{IJ}=\pm 1$ defined as in Fig.\ref{fig:model-A}. $H^{binding}$ is an interaction between the Ising-d.o.f. and spin-$1/2$-d.o.f., which commutes with $H^{Ising}$:
\begin{align}
H^{binding}=-\lambda \sum_{\intersectinglinesA} \frac{1}{2}(1-s_{IJ}\sigma_I^z\sigma_J^z)\cdot \hat P_{\mathbf{S}_i+\mathbf{S}_j=0},\label{eq:H_binding}
\end{align}
where $\lambda>0$, $i,j$ labels the sites on the triangular lattice, and the summation of ``$\intersectinglines$'' is over all intersection points between the triangular lattice and the honeycomb lattice as shown in Fig.\ref{fig:model-A}. $\hat P_{\mathbf{S}_i+\mathbf{S}_j=0}\equiv \frac{1}{4}- \mathbf{S}_i\cdot \mathbf{S}_j$ is the operator projecting the two spin-1/2's on site-$i$ and site-$j$ into a spin singlet.

$H^{A}$ is more complicated and will be given in Eq.(\ref{eq:HA}). It is straightforward to checked that $H$ respects the Ising symmetry $U_g\equiv\prod_I \sigma_I^x$, the spin-rotation symmetry generated by $\sum_i \mathbf{S}_i$. $H$ also respects magnetic translation operations:
\begin{align}
T_x&=(\prod_{I\in \mbox{\scriptsize{y-odd zigzag chains}}}\sigma_I^x )\cdot T_x^{orig.},&T_y&=T_y^{orig.},\label{eq:Ising_mag_transl}
\end{align}
(see Fig.\ref{fig:model-A}), and the magnetic translation algebra Eq.(\ref{eq:mag_transl}) is satisfied. We will show that the ground state of $H$ is in a gapped liquid phase without topological order. According to our general results, this ground state must be in an SPT phase, which we will show momentarily.

The physical consequence of $H^{Ising}$ and $H^{binding}$ is to provide a highly degenerate low energy manifold, which will be lifted by $H^{A}$. To understand the low energy manifold, let us firstly consider $H^{Ising}$. Because every hexagonal plaquette frustrated, there will be at least one bond in each plaquette that is energetically unhappy. Namely the ground state manifold of $H^{Ising}$ is formed by all possible Ising configurations satisfying the ``one-unhappy-Ising-bond-per-plaquette'' condition.

$H^{binding}$ further constrains the spin-1/2 d.o.f. in the low energy manifold. It has effect only on the Ising unhappy bonds ($s_{IJ}\sigma_I^z\sigma_J^z=-1$), and energetically binds the two spin-1/2's near the unhappy Ising-bond into a spin singlet. The degenerate ground state manifold of $H^{Ising}+H^{binding}$ is now clear: it is formed by all such quantum states satisfying the ``one-unhappy-Ising-bond-per-plaquette'' condition and the two neighboring spin-1/2's normal to every unhappy Ising-bond form a spin singlet.

It is well-known that the Ising configurations satisfying the ``one-unhappy-Ising-bond-per-plaquette'' condition are intimately related to the Hilbert space of the QDM\cite{moessner2001ising,moessner2001short,moessner2000two}. Pictorically, any such an Ising configuration can be mapped to a dimer covering (with one dimer per site) on the triangular lattice by assigning a dimer crossing the unhappy Ising bond. The effect of $H^{binding}$ is simply to energetically binds the two spin-1/2's in each dimer into a spin-singlet. Namely if the Ising configuration is given, the state of spin-1/2 d.o.f. is also fixed.

Similar to the model in Sec.\ref{sec:BFG}, there is a two-to-one mapping from the ground state manifold of $H^{Ising}+H^{binding}$ to the QDM Hilbert space, since two low energy states differ by a global Ising transformation map into the same state in the QDM. Second, these states only map to one specific topological sector of the QDM: The parity of the number of dimers crossing a loop is simply given by the sign of the product $\prod_{\langle IJ\rangle\in loop} s_{IJ}$.

In fact, these relations between the Ising d.o.f. and the dimer d.o.f. can be viewed as the well-known duality mapping between quantum Ising models and $Z_2$ gauge theories\cite{kogut1979introduction}. The Ising paramagnet phase is dual to the deconfined $Z_2$ gauge phase in the QDM, while the Ising ordered phase is dual to the confined $Z_2$ gauge theory. More precisely, the ``unhappy-Ising-bond'' is dual to the electric flux line (i.e., the dimer) in the gauge theory, and the ``one-unhappy-Ising-bond-per-plaquette'' condition in the low energy manifold on the Ising side is dual to the ``one-$Z_2$-gauge-charge-per-site'' condition in the QDM. Here, the only new ingredient apart from this well-known duality is that there are also spin-1/2 d.o.f. on the Ising side. But due to $H^{binding}$, these spin-1/2's form a pattern of spin singlets fixed by the Ising d.o.f., and consequently the duality mapping is not modified after a convention of the spin-singlet signs is given (see below).

After introducing $H^{A}$, we will see that the degeneracy in the low energy manifold will be lifted and the unique ground state on a torus sample is formed. The usual QDM Hamiltonian on the triangular lattice is:
\begin{align}
 H^{TC}_{QDM}&=-t\sum_{\mbox{\scriptsize{plaquettes}}}\big(|\pA\rangle\langle\pB|+h.c.\big)\notag\\
 &+v\sum_{\mbox{\scriptsize{plaquettes}}}\big(|\pA\rangle\langle\pA|+|\pB\rangle\langle\pB|\big),\label{eq:TC_QDM}
\end{align}
where the summation is over all plaquettes (rhombi): ``$\pgA$'', ``$\pgB$'', ``$\pgC$''. The ground states of this model are exactly known at the RK-point given by $t=v>0$, and the superscript $TC$ is highlighting that the topological order is toric-code\cite{Kitaev:2003p2,moessner2001resonating} like in the deconfined phase (i.e., the usual $Z_2$ gauge theory). At this point $H^{TC}_{RK}$ can be rewritten as a summation of projectors\cite{moessner2001resonating}:
\begin{align}
 H^{TC}_{RK}&=t\sum_{\mbox{\scriptsize{plaquettes}}}\big(|\pA\rangle-|\pB\rangle\big)\big(\langle\pA|-\langle\pB|\big).
\end{align}
Clearly the equal weight superposition of all dimer coverings within any fixed topological sector $|\Phi^{TC}_{RK}\rangle=\sum_c |c\rangle$ ($c$ labels possible dimer coverings) is one ground state of $H^{TC}_{RK}$ since it is annihilated by all projectors.

Based on the duality mapping, $H^{TC}_{QDM}$ is mapped to $H^A$. Any dimer covering $|c\rangle$ will be mapped to two Ising configurations $|c_1\rangle$ and $|c_2\rangle$ distinct from each other by a global Ising transformation. One can further choose a translationally symmetric sign convention for the spin-singlets on the nearest neighbor bonds along the three orientations:
\begin{align}
 |\elA\rangle&\equiv \frac{1}{\sqrt{2}}\big(|\ellA\rangle -|\ellAm\rangle\big),\notag\\
 |\raisebox{-2mm}\elB\rangle&\equiv \frac{1}{\sqrt{2}}\big(\big|\raisebox{-2mm}\ellB\big\rangle -\big|\raisebox{-2mm}\ellBm\big\rangle\big),\notag\\
 |\raisebox{-2mm}\elC\rangle&\equiv \frac{1}{\sqrt{2}}\big(\big|\raisebox{-2mm}\ellC\big\rangle -\big|\raisebox{-2mm}\ellCm\big\rangle\big)\rangle.\label{eq:singlet_dimer}
\end{align}
With this sign convention, given a $c$, $|c_i\rangle$ ($i=1$ or $2$) fully determines a state in the ground state manifold of $H^{Ising}+H^{binding}$ by replacing the dimer configuration by the corresponding spin-singlet configuration. In addition, $\{|c_1\rangle,|c_2\rangle\}$ for all $c$ form a complete orthornormal basis in this manifold.

$H^A$ is defined as:
\begin{align}
 H^{A}=&-t\sum^{\sigma_I^z,\sigma_J^z=\pm1}_{\mbox{\scriptsize{plaquettes}}}\left(\big|\raisebox{-4mm}\pgelA\big\rangle \big\langle\raisebox{-4mm}\pgelB\big|+h.c.\right)\notag\\
 &+v\sum^{\sigma_I^z,\sigma_J^z=\pm1}_{\mbox{\scriptsize{plaquettes}}} \left(\big|\raisebox{-4mm}\pgelC\big\rangle \big\langle\raisebox{-4mm}\pgelC\big|\right.\notag\\
  &\;\;\;\;\;\;\;\;+\left.\big|\raisebox{-4mm}\pgelD\big\rangle \big\langle\raisebox{-4mm}\pgelD\big|\right)\label{eq:HA}
\end{align}
Note that the $t$-term also flips the two Ising spins inside the plaquette. At the RK point $t=v>0$, $H^A$ can again be written as a summation of projectors:
\begin{align}
 H^{A}_{RK}=t\sum^{\sigma_I^z,\sigma_J^z=\pm1}_{\mbox{\scriptsize{plaquettes}}}&\left(\big|\raisebox{-4mm}\pgelA\big\rangle -\big|\raisebox{-4mm}\pgelB\big\rangle \right)\notag\\
 &\cdot \left(\big\langle\raisebox{-4mm}\pgelA\big|-\big\langle\raisebox{-4mm}\pgelB\big|\right)\label{eq:A_RK}
\end{align}
To study the ground state of the total Hamiltonian $H$, it is suffice to focus on the degenerate ground state manifold of $H^{Ising}+H^{binding}$, and clearly $H^{A}$ acts within this manifold. In addition, it is straightforward to show that $|\Phi^A_{RK}\rangle=\sum_{c}(|c_1\rangle+|c_2\rangle)$, i.e, the equal weight superposition of all states in this manifold, is a ground state of $H^A_{RK}$ because it is annihilated by every projector in Eq.(\ref{eq:A_RK}). $|\Phi^A_{RK}\rangle$ is clearly a fully symmetric liquid wavefunction.

It is known that for the QDM Eq.(\ref{eq:TC_QDM}), the RK point of $H$ is exactly at a first-order phase transition boundary between a deconfined gapped liquid phase ($v<t$) and a staggered valence bond solid phase ($v>t$)\cite{moessner2001resonating}. Based on the duality mapping, the model-A is in a fully symmetric gapped liquid phase for $v_c<v<t$ with a unique ground state on torus. In the limit of $K,\lambda\gg v,t$, $v_c$ is given by the same critical value $v_c\approx 0.7 t$ as in the original QDM\cite{moessner2001resonating}. More precisely, in the global Ising-even sector of the Hilbert space of model-A, this mapping to the Hilbert space of the QDM is given by $1/\sqrt{2}(|c_1\rangle+|c_2\rangle)\rightarrow |c\rangle$, and clearly $H^A$ is mapped to $H^{TC}_{QDM}$. Namely, the full energy spectrum of model-A in the Ising-even sector has a one-to-one correspondence with the full energy spectrum of $H^{TC}_{QDM}$. In addition, in the Ising paramagnetic phase, ground state in the global Ising-odd sector of model-A has a finite excitation energy which is the same as the energy cost of a $Z_2$ gauge flux in the QDM.

Next we show that the liquid phase $v_c<v<t$ in model-A is an SPT phase because the Ising defects carry half-integer spins. Similar to the discussion in Sec.\ref{sec:BFG} (see Fig.\ref{fig:Ising_defect_BFG}), after a pair of Ising defects are spatially separated the original Hamiltonian $H$ is modified into $H'$. Comparing with $H$, the $s_{IJ}$ flips sign in $H'$ whenever the bond $I-J$ crosses the branch cut. Namely, for any loop on the honeycomb lattice enclosing a single Ising defect, the product $\prod s_{IJ}$ along the loop changes sign. In order not to cost $H'^{binding}$ energy, the parity of the number of dimers crossing this loop also flips. Consequently, an Ising defect is topologically bound with a monomer (an unpaired site on the triangular lattice). This monomer clearly carries a half-integer spin in model-A, which can be detected by the local spin susceptibility at low temperatures.

Next, we demonstrate a different symmetry-enforced SPT phase using the model-B defined as follows:
\begin{align}
 H=H^{Ising}+H^{binding}+H^B.\label{eq:model-B}
\end{align}
Comparing with the model-A in Eq.(\ref{eq:model-A}), only the last term is modified:
\begin{widetext}
\begin{align}
 H^B=&v\sum^{\sigma_I^z,\sigma_J^z=\pm1}_{\mbox{\scriptsize{plaquettes}}} \left(\big|\raisebox{-4mm}\pgelC\big\rangle \big\langle\raisebox{-4mm}\pgelC\big|+\big|\raisebox{-4mm}\pgelD\big\rangle \big\langle\raisebox{-4mm}\pgelD\big|\right)+\sum^{\sigma_I^z,\sigma_J^z=\pm1}_{\hspace{-1mm}\pgA}\left(-it\big|\raisebox{-4mm}\pgelA\big\rangle \big\langle\raisebox{-4mm}\pgelB\big|+h.c.\right)\notag\\
 &+\sum^{\sigma_I^z,\sigma_J^z=\pm1}_{\hspace{-1mm}\pgB}\left(-it\big|\raisebox{-4mm}\pgelE\big\rangle \big\langle\raisebox{-4mm}\pgelF\big|+h.c.\right)
 +\sum^{\sigma_I^z,\sigma_J^z=\pm1}_{\hspace{-1mm}\pgC}\left(-it\big|\raisebox{-6mm}\pgelG\big\rangle \big\langle\raisebox{-6mm}\pgelH\big|+h.c.\right)
 \label{eq:HB}
\end{align}
\end{widetext}
One can straightforwardly check that the model-B defined in Eq.(\ref{eq:model-B},\ref{eq:HB}) also respects the Ising symmetry, the $SO(3)$ spin-rotation symmetry, and the magnetic translation symmetry Eq.(\ref{eq:mag_transl}). Below we show that in a finite parameter regime $v'_c<v<t$, the model-B is in a gapped liquid phase, and this phase is another SPT phase.

In the ground state manifold of $H^{Ising}+H^{binding}$, the duality transformation maps $H^B$ into the following QDM Hamiltonian:
\begin{align}
 &H^{DS}_{QDM}=v\sum_{\mbox{\scriptsize{plaquettes}}}\big(|\pA\rangle\langle\pA|+|\pB\rangle\langle\pB|\big)\notag\\
 &+\sum_{\pgA,\pgB,\pgC}-it\big(|\pA\rangle\langle\pB|+|\pC\rangle\langle\pD|+|\raisebox{-2mm}\pE\rangle\langle\raisebox{-2mm}\pF|\big)+h.c.\label{eq:DS_QDM}
\end{align}
This QDM was firstly introduced and studied in Ref.~\onlinecite{qi2015double}, where the exactly solvable RK point $t=v$ has been shown to be adjacent a gapped liquid phase for $v\lesssim t$. Interestingly, this phase was demonstrated to have a double-semion topological order (the superscript $DS$ here is to highlight this fact). By the duality mapping, we know that in a finite parameter regime $v'_c<v<t$, the model-B is in a gapped liquid phase.

Similar to previous disussion on the model-A, it is straightforward to show that the Ising defect in the gapped liquid phase of model-B also carries half-integer spin, so it is also an SPT phase. To see the difference from the SPT phase realized in model-A, let us consider the Ising symmetry only. It is known that Ising symmetry itself can protect two paramagetic phases: the trivial phase and the SPT phase. Levin and Gu pointed out\cite{levin2012braiding} that the duality mapping maps the usual Ising paramagnet to the toric-code topological order, while the nontrivial Ising SPT phase maps to the double semion topological order. Consequently, the SPT phases realized in model-A and model-B are different because the Ising symmetry alone already distinguishes them.

\subsection{$G=Z_2^T\times Z_2^{Ising}$}\label{sec:Z2T_Ising_models}
Here we demonstrate symmetry-enforced SPT phases outlined in example-(2) and (3) in Sec.\ref{sec:examples}. Unlike the $G=SO(3)\times Z_2^{Ising}$ case, here we show that symmetry conditions fully determine the SPT phase. The models below have the same Hilbert space as in the $G=SO(3)\times Z_2^{Ising}$ case (i.e., $\sigma_I$ on the honeycomb lattice and $\boldsymbol{\tau}_i$ on the triangular lattice), but with different symmetries defined.

\textbf{A Kramer doublet per unit cell}: A simple generalization is for example-(2) (i.e., a Kramer-doublet per unit cell) where we can recycle the model-A. Namely, defining the Ising symmetry $U_g=\prod_I\sigma_I^x$ as before and the antiunitary time-reversal symmetry as $U_{\mathcal{T}}=e^{i\pi\mathbf{S}^y}=i\boldsymbol{\tau}^y$, clearly model-A respect all the required symmetries. In addition, we have one Kramer doublet $\boldsymbol{\tau}$ per unit cell. According to our discussion in example-(2), this SRE liquid phase realized in $v_c<v<t$ must be an SPT phase in which the Ising defect carries a Kramer doublet, which is obviously realized in model-A. In addition, after gauging $Z_2^{Ising}$ symmetry, one necessarily obtains the toric-code topological order, which is confirmed in model-A.

On the other hand, model-B, gauging which gives double-semion topological order, explicitly breaks the time-reversal symmetry defined above.

\textbf{A non-Kramer doublet per unit cell}: Now let us move on to example-(3) in Sec.\ref{sec:examples}. In order to construct a model realizing the non-Kramer doublet projective representation defined in Eq.(\ref{eq:non_Kramer}), let us define the following symmetry operations:
\begin{align}
 U_g&=\boldsymbol{\tau}^z\prod_I\sigma_I^x ,&U_\mathcal{T}&=\boldsymbol{\tau}^x.\label{eq:U_non_Kramer}
\end{align}
Consequently we have one non-Kramer doublet $\boldsymbol{\tau}$ per unit cell. The model Hamiltonian in this example will be given by
\begin{align}
 H=H^{Ising}+\tilde H^{binding}+\tilde H^B,\label{eq:tilde_model_B}
\end{align}
where $H^{Ising}$ is given in Eq.(\ref{eq:H_Ising}), $\tilde H^{binding}$ is in Eq.(\ref{eq:tilde_H_binding}). $\tilde H^B$ has the same form as $H^B$ in Eq.(\ref{eq:HB}), but with a modified interpretations of the dimers: replacing the $|\elA\rangle$'s defined in Eq.(\ref{eq:singlet_dimer}) by $|\elAp\rangle$'s and $|\elApp\rangle$'s defined in Eq.(\ref{eq:dashed_singlet_dimer},\ref{eq:dotted_singlet_dimer}) depending the dimer positions (see discussions below and Fig.\ref{fig:tilde_symmetric_dimers}). Eventually we will show that in a finite regime $v \lesssim t$ this model features a SRE gapped liquid ground state which is the Levin-Gu Ising SPT phase, consistent with discussions in Sec.\ref{sec:examples}.

We will construct a model similar to $H=H^{Ising}+H^{binding}+...$, but respecting a magnetic translation operations different from Eq.(\ref{eq:Ising_mag_transl}) since $U_g$ is now different. In particular, we define magnetic translations:
\begin{align}
 T_x&=\big(\prod_{i\in \mbox{\scriptsize{y-odd row}}}\boldsymbol{\tau}_i^z\big)\big(\prod_{I\in \mbox{\scriptsize{y-odd zigzag chains}} }\sigma_I^x\big)T_x^{orig.},\notag\\
 T_y&=T_y^{orig.},\label{eq:modified_Ising_mag_transl}
\end{align}
(see Fig.\ref{fig:model-A}), which satisifies Eq.(\ref{eq:mag_transl}) with $g=\prod_i\boldsymbol{\tau}_i^z\prod_I\sigma_I^x$ following Eq.(\ref{eq:U_non_Kramer}). Although $H^{Ising}$ still respects all the symmetries, $H^{binding}$ does not respect $g$ and $T_x$. This is because the usual spin singlets $|\elA\rangle$ in the projector $\hat P_{\mathbf{S}_i+\mathbf{S}_j=0}=|\elA\rangle\langle \elA|$ does not respect $g$ and $T_x$. We therefore need to modify these dimer states and the projectors.

We define the following dimer states formed by the $\boldsymbol{\tau}$ spins:
\begin{align}
 |\elAp\rangle&\equiv \frac{1}{\sqrt{2}}\big(|\ellAp\rangle +|\ellApm\rangle\big),\notag\\
 |\raisebox{-2mm}\elBp\rangle&\equiv \frac{1}{\sqrt{2}}\big(\big|\raisebox{-2mm}\ellBp\big\rangle +\big|\raisebox{-2mm}\ellBpm\big\rangle\big),\notag\\
 |\raisebox{-2mm}\elCp\rangle&\equiv \frac{1}{\sqrt{2}}\big(\big|\raisebox{-2mm}\ellCp\big\rangle +\big|\raisebox{-2mm}\ellCpm\big\rangle\big)\rangle.
 \label{eq:dashed_singlet_dimer}
\end{align}
Clearly these dimer states are both Ising and time-reversal even according to Eq.(\ref{eq:U_non_Kramer}).

\begin{figure}
 \includegraphics[width=0.48\textwidth]{dashed_dotted_honeycomb_triangular_lattices.pdf}
 \caption{To construct model in Eq.(\ref{eq:tilde_model_B}), the dimer states living on the nearest neighbor bonds on the triangular lattice have a spatial dependent pattern: the dimer states living on the dashed bonds are defined in Eq.(\ref{eq:dashed_singlet_dimer}), and those living on the dotted bonds are defined in Eq.(\ref{eq:dotted_singlet_dimer}).}
 \label{fig:tilde_symmetric_dimers}
\end{figure}

Under the magnetic translation $T_x$ in Eq.(\ref{eq:modified_Ising_mag_transl}), dimer states with $|\raisebox{-2mm}\elBp\rangle$ and $|\raisebox{-2mm}\elCp\rangle$ connecting even and odd rows on the triangular lattice will transform into the following states:
\begin{align}
 |\raisebox{-2mm}\elBpp\rangle&\equiv \frac{1}{\sqrt{2}}\big(\big|\raisebox{-2mm}\ellBp\big\rangle -\big|\raisebox{-2mm}\ellBpm\big\rangle\big),\notag\\
 |\raisebox{-2mm}\elCpp\rangle&\equiv \frac{1}{\sqrt{2}}\big(\big|\raisebox{-2mm}\ellCp\big\rangle -\big|\raisebox{-2mm}\ellCpm\big\rangle\big)\rangle.
 \label{eq:dotted_singlet_dimer}
\end{align}
Note that these dotted dimer states are Ising even, but \emph{time-reversal odd} according to Eq.(\ref{eq:U_non_Kramer}). The magnetic translational symmetric assignment of the dimer states is given in Fig.(\ref{fig:tilde_symmetric_dimers}): the states $|\elAp\rangle$, $|\raisebox{-2mm}\elBp\rangle$, $|\raisebox{-2mm}\elCp\rangle$ are assigned on the dashed bonds, while the states $|\raisebox{-2mm}\elBpp\rangle$, $|\raisebox{-2mm}\elCpp\rangle$ are assigned on the dotted bonds.

By replacing the spin singlets in $\hat P_{\mathbf{S}_i+\mathbf{S}_j=0}=|\elA\rangle\langle \elA|$ by the corresponding $|\elAp\rangle\langle \elAp|$ and $|\elApp\rangle\langle \elApp|$ in the spatial dependent fashion in Fig.\ref{fig:tilde_symmetric_dimers}, we modify $H^{binding}$ naturally as
\begin{align}
 \tilde H^{binding}&=-\lambda \sum_{\intersectinglinesAp} \frac{1}{2}(1-s_{IJ}\sigma_I^z\sigma_J^z)\cdot \big(|\elAp\rangle\langle\elAp|\big)\notag\\
 &-\lambda \sum_{\intersectinglinesApp} \frac{1}{2}(1-s_{IJ}\sigma_I^z\sigma_J^z)\cdot \big(|\elApp\rangle\langle\elApp|\big),\label{eq:tilde_H_binding}
\end{align}

Now $H^{Ising}+\tilde H^{binding}$ respects all the required symmetries, but one still need terms like $H^A$ or $H^B$ to lift the ground state degeneracy to reach a SRE gapped liquid phase. In order to preserve the Ising and magnetic translation symmetry, it is natural to replace the dimer states $|\elA\rangle$ by the corresponding $|\elAp\rangle$ and $|\elApp\rangle$. Let us denote the resulting modified Hamiltonians as  $\tilde H^A$ or $\tilde H^B$. Both modified models $H^{Ising}+\tilde H^{binding}+\tilde H^A$ and $H^{Ising}+\tilde H^{binding}+\tilde H^B$ are solvable, by duality mapping to the $H^{TC}_{QDM}$ and $H^{DS}_{QDM}$ respectively. These two models both give SRE gapped ground states in the regime $v\lesssim t$ respecting the magnetic translation symmetry and the Ising symmetry, gauging which give toric-code and double semion topological order respectively.

Finally, let us consider the time-reversal symmetry in Eq.(\ref{eq:non_Kramer}). Importantly, $|\elAp\rangle$'s are time-reversal even while $|\elApp\rangle$ are time-reversal odd. As shown in the pattern Fig.\ref{fig:tilde_symmetric_dimers}, any dimer resonant term like $|\pA\rangle\langle\pB|$ will involve an odd number of $|\elApp\rangle$ states. Consequently, \emph{only $\tilde H^B$ is time-reversal symmetric, while $\tilde H^A$ explicitly breaks the time-reversal.} In fact according to Theorem-I, it is impossible to have a SRE liquid respecting all the required symmetries and gauging the Ising symmetry gives a toric code topological order.

\section{Proof of Theorems}
Here we present a combination of physical argument and mathematical derivations based on symmetric tensor network formulation\cite{perez2010characterizing,Zhao:2010p174411,Singh:2010p50301,Singh:2011p115125, Singh:2012p195114,Bauer:2011p125106,Weichselbaum:2012p2972,jiang2015symmetric,jiang2017anyon}. We will focus on Theorem-II, and in Appendix\ref{app:theorem-i} we show that Theorem-I can be viewed as its special case.

We need to show the condition in Theorem-II is necessary and sufficient for a SRE liquid phase to exist, which must be an SPT phase. To show this condition is necessary, we consider such a SRE liquid phase and the pumping of the entanglement spectra during an adiabatic process in Sec.\ref{sec:entanglement_pumping}, leading to an observation that a $g$-symmetry defect must carry a projective representation $\alpha$. In a SRE liquid phase characterized by 3-cocycle $\omega$, we use the symmetric tensor-network formulation in Sec.\ref{sec:SPT_constraints} to establish that the projective representation carried by a $g$-symmetry defect is given by $(\delta_g^{\omega})^{-1}$. Since the projective representation carried by a symmetry defect is physical and independent of formulation, together with the pumping argument, the necessary condition in Theorem-II is established \emph{independent of formulation}.

As a complementary calculation, we also explicitly compute the projective representation carried by a $g$-symmetry defect in a SRE liquid phase representable within the symmetry tensor-network formulation in Appendix.\ref{app:mag_transl_proj_rep}, which turns out to be $\alpha$, consistent with the previous discussion\cite{zaletel2014detecting}.

To further show that the condition is also sufficient, we will show that for any 3-cocycle $\omega_0$ satisfying $(\delta_g^{\omega_0})^{-1}\simeq \alpha$, generic symmetric tensor network wavefunctions representing a SRE liquid phase characterized by $\omega_0$ can be constructed in Sec.\ref{sec:SPT_constructions}.

\subsection{Entanglement Pumping argument}\label{sec:entanglement_pumping}
Here let us assume a SRE liquid phase exist. It is straightforward to show that after a local unitary transformation (a site dependent $U_g$ action) without changing the physical action of $G$, one can always choose a gauge in which $T_y=T_y^{orig.}$ and $T_x=(\prod_{y-odd}U_g)\cdot T_x^{orig.}$. Then we consider putting this SRE liquid on a infinite cylinder $C$ along the $x$-direction, with $L_y$ number of unit cells across the $y$-direction loop. We choose $L_y=l_y\cdot N+d_y$ where $l_y, d_y$ are integers, $g^N=I$ and $0 \leq d_y<N$. Note that if $d_y\neq 0$ this choice of $L_y$ will explicitly break the $T_x$ symmetry.

This infinite long cylinder can be viewed as a one-dimensional system, and for a large enough $L_y$ the onsite symmetry $G$ will be respected. We will study the entanglement spectrum of this one dimensional system at a particular cut $x_0+1/2$. A SRE liquid respecting $G$ symmetry dictates that entanglement states at this cut carry a particular projective representation $\xi$ of $G$\cite{Pollmann:2010p64439,Chen:2011p35107}.

\begin{figure}
\includegraphics[width=0.48\textwidth]{Entanglement_Pumping.pdf}
\caption{Illustration of adiabatically separating a pair of g-defect/antidefect along the $x$-direction with $g^3=I$. For simplicity, one may imagine Hamiltonian to host nearest neighbor (NN) terms. Along the $x$-direction, due to the magnetic translation symmetry Eq.(\ref{eq:mag_transl}), the NN interactions on the vertical bonds have a three-unit-cell periodicity (solid,dashed and dotted bonds). While the $g$-defect crosses the entanglement cut at $x_0+1/2$, the Hamiltonian along the branch cut (dashed gray line) is effectively translated along $x$-direction by one unit cell. After separating such pairs of defects for every row, the final Hamiltonian is related to the original Hamiltonian by $T_x^{orig.}$.}
\label{fig:entanglement_pumping}
\end{figure}

Next, we adiabatically create a $g$-symmetry defect/anti-defect pair at a given $y$-coordinate and separate them to infinity along the $x$-direction. After repeating this procedure for every $y$-coordinate, we totally move $L_y$ number of $g$-symmetry defects acrosses the entanglement cut $x_0+1/2$. As shown in Fig.\ref{fig:entanglement_pumping}, the final Hamiltonian is related with the original Hamiltonian by the \emph{original} translation operation $T^{orig.}_x$. Therefore, the final entanglement states at the cut $x_0+1/2$ is equivalent to the initial entanglement states at a different cut: $x_0-1/2$.

Because the initial entanglement states at $x_0-1/2$ differs from the initial entanglement states at $x_0+1/2$ by a column of unit cells along $y$-direction, we conclude that the final entanglement eigenstates at $x_0+1/2$ must carry the $\alpha^{d_y}\cdot\xi$ projective representation. This pumping of the entanglement projective representation can only be explained by the projective representation $\tilde \alpha$ carried by a $g$-symmetry defect, and $(\tilde \alpha)^{L_y}\simeq \alpha^{d_y}$. But because $N$ $g$-defect fuse into a trivial object, we must have $(\tilde \alpha)^N\simeq 1$. Consequently $\tilde\alpha=\alpha$.

\subsection{Symmetry-enforced constraints on SPT cocycles}\label{sec:SPT_constraints}
Here we consider an SPT wavefunction represented using the symmetric tensor-network formulation\cite{jiang2017anyon}. The advantage of this formulation is that it allows us to introduce symmetry defects conveniently.

The local symmetry transformation of an onsite symmetry $a$ on a $g$-defect is given by the application of of $U_a$ inside a disk $D$ covering the $g$-defect, together with an application of unitary operations on the virtual degrees of freedom on the boundary of $D$. This boundary operation should be defined in such a way that after these two operations, no excitation is created near the boundary of $D$. In Appendix \ref{app:defect_proj_rep} we explictly constructed such boundary operations. With these boundary operations, we explicitly show that the projective representaion carried by the $g$-defect is given by $(\delta^{\omega}_g)^{-1}$ in an SPT phase characterized by the 3-cocycle $\omega$, which is given without proof in Ref.~\onlinecite{zaletel2014detecting}.

\subsection{Generic constructions of Symmetry-enforced SPT wavefunctions}\label{sec:SPT_constructions}
Our strategy here is to start from an SPT state characterized by a 3-cocycle $\omega$ with a regular representation of $G$ per unit cell and respecting the usual translation symmetry $T_x^{orig.} , T_y^{orig.}$. Such a state can be generically represented using the symmetric tensor-network formulation\cite{jiang2017anyon}. In particular, the symmetric tensor-network needs to satisfy a collection of algebraic equations (constraints). Then we show that after properly modifying these algebraic constraints, the new tensor-network will respect the magnetic translation symmetry $T_x,T_y$, at the same time the physical d.o.f. must carry a projective representation $(\delta^{\omega}_g)^{-1}$ per unit cell (otherwise the wavefunction vanishes). The tensor-network states satisfying these modified constraints are generic constructions of the SPT states in Theorem-II.

The details of the construction can be found in Appendix \ref{app:generic_construction}.

\section{Discussion}
Generalized Hastings-Oshikawa-Lieb-Schultz-Mattis theorems put strong constraints on possible symmetric quantum states of matter. In particular, in the presence of translation symmetry and a projective representation of the onsite symmetry group per unit cell, it is impossible to have a gapped short-range entangled (SRE) symmetric ground state. In this paper we discuss that in the presence of magnetic translation symmetry, gapped SRE symmetric ground states could exist, which are enforced to be symmetry protected topological (SPT) phases. Focusing on bosonic systems in two spatial dimensions, we provide the generic necessary and sufficient condition for such symmetry-enforced SPT phases to occur in Theorem-I and II. When the condition is satisified, we sharply characterize the coset structure of the realizable SPT phases in the Remark.

The condition-(2) in Theorem-I is particular non-obvious. It states that if symmetries protecting the fractional spin (projective representation) per unit cell and those generating the magnetic translations fail to commute with one another, then SRE liquid state is impossible even if the fractional spins fuse into an integer spin (regular representation) in the magnetic unit cell.

In addition, we design a class of decorated quantum dimer models realizing some of these symmetry-enforced SPT phases, which are exactly solvable at the corresponding Rokhsar-Kivelson points. A particularly simple model realizing a symmetry-enforced SPT phase is given in Sec.\ref{sec:BFG} by coupling a Balents-Fisher-Girvin spin liquid with a layer of pure-transverse-field Ising spins via three-spin interactions. This model also demonstrates the route to obtain SPT phases via condensing anyons in SET phases\cite{jiang2017anyon,duivenvoorden2017entanglement}.

It is interesting to consider the situation of fermions with magnetic translation symmetries, in which case (generalized) Hastings-Oshikawa-Lieb-Schultz-Mattis theorem apply for fractional filled systems with regular translation symmetries. In fact, earlier works\cite{dana1985quantised,LuRanOshikawa} establish the magnetic translation symmetry protected integer Hall conductivity, and a recent work by Wu et.al. studied the magnetic translation enforced quantum spin Hall insulators in fractionally filled fermionic systems\cite{wu2017symmetry}. Connecting with these works, the present work focuses on bosonic systems with projective representation per unit cell, but obtains systematic results.

YR would like to thank Yuan-Ming Lu and Masaki Oshikawa for inspiring related collaborations. XY, SJ and YR are supported by NSF under Grant No. DMR-1151440. AV is funded by a Simons Investigator award.


text:
https://www.nhregister.com/connecticut/article/Got-pink-Wear-it-Sunday-in-Seymour-to-support-11380399.php
Got pink? Wear it Sunday in Seymour to support breast cancer research
Seymour "Walk a Mile in Her Shoes" for breast cancer awareness. Melanie Stengel/Register
SEYMOUR >> The community's fight against breast cancer takes center stage next week with the 6th annual 'Seymour Pink Day.'
On Oct. 15, Seymour Pink Founder Mary Deming is encouraging as many residents and businesses to deck themselves and their storefronts in shades of pink. The goal: play up on the town's name and get people to literally "see more pink," while raising awareness, funds and support for breast cancer.
Deming founded the grassroots initiative in 2009, to pay homage to her own mother who lost her battle with the disease. Statistics show Connecticut is ranked 2nd in the nation for the most breast cases per year.
"The statistics are startling," Deming said. "Globally, every 2 ½ minutes a person hears those horrific words, 'you have breast cancer.' Every 13 minutes, someone's loved one loses their battle. Incredible progress has been made in both screening and treatment of this disease. But we cannot stop until we live in a world without breast cancer!"
Since 2009, Seymour Pink, Inc. has raised more than $500,000 for breast cancer through various fundraising events and activities.
Several events leading up to Seymour Pink Day include a cut-a-thon and mammogram screening from 8 a.m. to 3 p.m. Saturday at En V Hair Lounge & Wig Center, located at 225 West St. in the Oxmour Plaza. A shampoo and haircut will cost $20, with all proceeds to benefit Seymour Pink. Special raffles, prizes and other events are planned throughout the day.
Yale-New Haven Hospital's Digital Mobile Mammography Van will be in the salon's parking lot, offering mammograms from 8 a.m. to 2:30 p.m. For an appointment, call (203)688-1010.
On Sunday, downtown Seymour will come alive with men (and women and kids) parading around in high heels for the 6th annual Pink Promenade. The festivities will kick off at 12:30 p.m. at the Community Center, 20 Pine St., with the mile-long walk starting at 1 p.m. There is still time for walkers to join the fun by visiting the town's Facebook page at: http://www.seymourct.org/PINK-proMENade/ for more information
According to its website, Seymour Pink has donated nearly $40,000 to the Susan G. Komen Foundation, more than $11,000 to the Griffin Hospital's Hewitt Breast Wellness Center and $10,000 to Yale-New Haven's Smilow Breast Institute. Locally, the organization has helped more than 80 breast cancer victims and their families, with both financial and emotional support.
"On October 15th, we hope that the momentum will continue to grow with local businesses, town agencies and departments and families getting involved by wearing PINK or displaying PINK at their places of work and residence," Deming said. "Put on your PINK and show your support for our 'community's fight against breast cancer.'"
Have questions, feedback or ideas about our news coverage? Connect directly with the editors of the New Haven Register at AskTheRegister.com.

redpajama_set_name: RedPajamaCommonCrawl


text:
Q: Excel to Word wrong range copied if sheet containing range not active I've found a weird quirk when copying a range from Excel to Word in Office 2016 (not tried in other versions) as an OLEObject. The VBA is in Excel. Here's the code, it locates a bookmark in the Word file, pastes in the table and resets the bookmark:
Set wordRange = wordDoc.bookmarks("Table").Range
wsService.Range("A1:B15").Copy
wordRange.PasteSpecial Link:=False, DataType:=0, Placement:=0
wordDoc.bookmarks.Add "Table", wordRange

This works if sheet wsService is active in Excel. If not Range("A1:B15") on the active sheet is copied instead. I've tried other data types (e.g. bitmap, metafile, shape) and they all work fine. Is this a bug?

A: Your description suggests all you're doing with wsService is pointing it to the ActiveSheet. For example:
set wsService = ActiveSheet

instead of saying which particular worksheet it should use:
set wsService = ThisWorkbook.Sheets("Sheet1")

This way, the worksheet need not be visible to copy anything from it.

redpajama_set_name: RedPajamaStackExchange


text:
USS LST-565 was a United States Navy  in commission from 1944 to 1946.

Construction and commissioning
LST-565 was laid down on 16 March 1944 at Evansville, Indiana, by the Missouri Valley Bridge and Iron Company. She was launched on 8 May 1944, sponsored by Mrs. Vergil P. Dyer, and commissioned on 25 May 1944.

Service history
During World War II, LST-565 was assigned to the Pacific Theater of Operations. She took part in the Philippines campaign, participating in the landings on Leyte in October 1944, the landings on Mindoro in December 1944, and the landings at Zambales-Subic Bay in January 1945. She then took part in the invasion and occupation of Okinawa Gunto in May 1945.

Following the war, LST-565 performed occupation duty in the Far East and saw service in China until mid-May 1946, when she departed for the United States.

Decommissioning and disposal
After returning to the United States, LST-565 was decommissioned on 13 June 1946 and stricken from the Navy List on 3 July 1946. She was sold for scrapping on 21 June 1948.

Honors and awards
LST-565 earned four battle stars for her World War II service.

References

NavSource Online: Amphibious Photo Archive LST-565

LST-542-class tank landing ships
World War II amphibious warfare vessels of the United States
Ships built in Evansville, Indiana
1944 ships

redpajama_set_name: RedPajamaWikipedia


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Cryptocurrencies Make Their Way to Campus, Bringing Flexibility and Risks
College students are using bitcoin and other digital currencies on campus, but are the downsides worth the rewards?
Erin Brereton
Erin Brereton has written about technology, business and other topics for more than 50 magazines, newspapers and online publications.
Attending college is an expensive endeavor. While balancing a demanding course load, extracurricular activities and blossoming social life, it can be hard for students to find time to battle the financial pressures of receiving a higher education.
In lieu of carving out time in an already packed schedule, a growing proportion — one in five college students, according to The Student Loan Report — are opting to invest in cryptocurrencies such as bitcoin, Ethereum and Ripple.
Recent research has shown younger Americans are more likely to invest in bitcoin and Ethereum, two popular cryptocurrencies, than older generations. According to LendEDU research, 36 percent of Americans age 18 to 24 say they plan to invest in bitcoin as an asset for the future; 32 percent intend to invest in Ethereum.
As this trend continues, college campuses are starting to adapt to the needs of their students.
Schools Let Students Swap Cash for Bitcoin
Rather than risk being left behind as the expansion of digitized money continues, some universities are tackling digital currency's rising presence on campus head on.
Some schools are also starting to let students use digital funds for daily purchases. The Georgia Institute of Technology, for example, began allowing students to add bitcoin funds to their campus debit cards to purchase food and other goods in 2014.
The Massachusetts Institute of Technology also began accepting bitcoin in 2014 — the same year a research group from MIT's Sloan School of Business distributed $100 in bitcoin currency to incoming freshman as part of a cryptocurrency spending project. The school uses payment service provider BitPay in its university bookstore, which also offered a bitcoin exchange.
King's College, a liberal arts university in New York City, has also allowed students to pay for their education via bitcoin since 2014.
Internationally, bitcoin and similar cryptocurrencies are being used by students not just to pay for campus essentials, but to be on campus at all.
The University of Nicosia in Cyprus, which began accepting bitcoin payments for tuition in 2013, gives students who use the cryptocurrency to pay their tuition a 5 percent net fee discount, according to the school's website.
The university told Forbes it expected the first students to use the bitcoin option would be those attending its online degree programs from countries where international payments can be slow and involve transmission fees that total 5 to 10 percent of the monthly tuition cost.
A number of other international schools, including the Lucerne University of Applied Sciences and Arts in Switzerland and the European School of Management and Technology Berlin in Germany, now accept cryptocurrency tuition payments.
Cryptocurrency Opens Up Security Risks as Well as Financial Options
While some students purchase and invest in cryptocurrency, others have decided to take an alternate approach and redirect their efforts towards mining instead.
If ransomware was the scourge of 2017, cryptocurrency mining could be the problem to watch this year — especially in higher education. In a recent Vectra analysis of the five industries showing cryptocurrency mining attacks, higher education had the majority of activity by far (85 percent).
In cryptomining, a student's computer and associated electrical power are used to verify cryptocurrency transactions in exchange for digital currency. This is a compute-intensive process that may be done on purpose (for financial reward) or without the student's knowledge if he or she visits a compromised website.
This activity creates problems for universities, including network noise that may hide security issues, a negative impact on IP address reputation, an open door to cybercriminals who may buy access to compromised computers, and drains on electricity and computing power.
Campuses are attractive targets because of their relatively open internet access and the presence of gamers and others who use CPU-intensive applications.
Vjom/Getty Images
Hackers Evolve Attacks Methods in Higher Education Breaches
Security, Privacy and Confidentiality: What's the Difference?


The import is using FileMaker's import function and automating it, and automating this process requires matching table names between the production and development databases. Any edits to table names must manually be made to both the production and development files so that both files have matching table names to avoid a failed import.
Be sure that for your solutions, the File Options -> Text -> Data Entry preferences are set to "Always use current system settings" as desired.


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2018-01-25 Asseco has adapted KSI ZUS to important changes in law
2017-12-05 Asseco Poland will create a digital data library for the Ministry of National Defense
2017-09-22 KSI ZUS among IT Leaders
2017-08-24 Asseco to support land and mortgage register system
Rankings and awards Public institutions
Asseco the "Leader of Cooperation" according to the Social Insurance Institution (ZUS)
On October 10, 2019, the Social Insurance Institution (ZUS) celebrated its 85th anniversary. The jubilee ceremonies were an opportunity to appreciate institutions and companies that cooperate with ZUS on a daily basis...
Asseco has launched documentation exchange system at ZUS
Since July 1, 2019, the Social Insurance Institution (ZUS) has been using a new information exchange system - Electronic Exchange of Social Security Information (EESSI) - which is to ensure cooperation with nearly 15,000...
Asseco will support SIA - the largest IT system at ARMA
The Agency for Restructuring and Modernization of Agriculture (ARMA) has concluded an agreement with Asseco Poland for the development and maintenance of the SIA system. It is the largest system of the Agency, which is...
Asseco has signed an agreement with ZUS for the development and maintenance of the Electronic Services Platform
Asseco Poland has signed another important contract with the Social Insurance Institution (ZUS). The Company will be responsible for the development and maintenance of the Client Portal and the Service Bus (ESB). They...
Uniformed services Public institutions
Asseco Opens a training center for operators of Unmanned Vehicles in Nigeria
Asseco has opened a certified training center for operators of unmanned aerial vehicles (UAV) along with a maintenance center in Nigeria. The Company has already conducted first training sessions for the operators. The...
Asseco supports the safety of road traffic
Asseco Poland has signed an agreement with the Chief Inspectorate of Road Transport (GITD) in the field of technical support and maintaining a software license for the Central Processing System operating in the Automatic...
Asseco has signed a framework agreement with ZUS
Asseco Poland has signed a framework agreement with the Social Insurance Institution (ZUS) for the modification and development of the Comprehensive IT System (KSI ZUS). The contract worth almost PLN 350 million gross...
Asseco continues to operate the central insurance system in the Agricultural Social Insurance Fund
The agreement for "Comprehensive operation of IT systems supporting the service of insurance and short-term benefits" was signed on 02.07.2018 and will be valid for the next 4 years. The gross value of the contract...
Asseco will develop a system to support farmers in KRUS
Asseco has signed an agreement with the Agricultural Social Insurance Fund (KRUS) for the maintenance and development of the Workflow system, which is a central component of the Integrated Information System (ZSI)...
Asseco signs an agreement for the maintenance of KSI ZUS during the transition period
Asseco Poland has signed an agreement to support the operation and maintenance of the Comprehensive IT System for the Social Insurance Institution (KSI ZUS) during the period in which a new contractor is to take over the...


What Kind of Gravel Do You Need?
No matter the size of your construction site, you're probably going to need an aggregate like gravel or sand. There are many different types of these aggregates and they're all useful in different ways. At Dawson Construction, we're happy to give you a bit of inside information.
Gravel, aggregates and sand come in all sizes and shapes, each with their own important features and uses. No matter what you're building, we can help you find the material you need. These are just some of the varieties of gravel and aggregates we offer clients in Kamloops and throughout Western Canada.
Pit run: naturally occurring gravel, with rocks of varying sizes. It's used as a base layer for roads and driveways and must be machine levelled.
Aggregates: different mixes of gravel or sand that are mixed with binders and then harden into materials like concrete or pavement. We offer different types depending on your needs.
Clear rock: grey rocks that are crushed into small uniform pieces. Typically used for drainage or for driveways or parking lots.
Select granular sub base or SGSB: either crushed or drawn directly from an aggregate source, SBSG forms the loadbearing layer of pavement.
Well-graded base course or WGBC: a finely crushed layer goes on top of the sub base and directly underneath the top layer of concrete or pavement.
Riprap: the largest type of gravel with pieces between 10 and 22 centimetres in diameter. The smaller size can be used as a base layer for roads, and the larger size is often used for erosion control along bodies of fast moving water.
Sand: finely crushed rocks. Can be used as an aggregate to create concrete, and when used without a binding agent, make a good base for play areas, patios and walkways.
Topsoil: the uppermost layer of soil. This nutrient rich earth is often used to start a garden or to improve lawns and gardens.
The type of gravel or aggregate you need depends on the project. Building a new road will use very different materials than what you'd use for landscaping. Your general contractor can tell you which type you need for your project.
If you need aggregates, gravel or sand, call Dawson Construction in Kamloops today. We're western Canada's expert road builders and we can help you with projects big and small.


After six weeks off to prepare for what was the most perfect wedding I could ever have dreamed of, and a much needed honeymoon in Cornwall with my lovely family, I'm back!
Thanks so much to my twenty five guest posters for holding the fort and sharing their wonderful wedding stories - and to you fabulous readers for not going anywhere!
Normal service is now set to resume and I'm raring to go!