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\begin{document}
\title{Supplementary material}
\maketitle
%\begin{widetext}
\section{$|\Psi\rangle_{\text{RVB}}$ has no $AA$ ($BB$) pairing}
In the main text, we have illustrated why the leading order contribution of NNN pairing vanishes in $|\Psi\rangle_{\text{RVB}}$. Here we would like to show that, indeed, all the $AA$ ($BB$) pairing vanish in $|\Psi\rangle_{\text{RVB}}$ exactly up to any order in $c$. As explained in the text, the pairing amplitude between two sites $i$ and $j$ is given by the sum of
all teleportation paths that connect $i$ and $j$. As seen in Fig. \ref{pairing}, the solid red line a is a typical teleportation path that contributes to the pairing $(ij)$. According to the local rule of the teleportation path (each path must turn at a vertex), it is not hard to see that any teleportation path contributing to the $AA$ ($BB$) pairing must go through an odd number of corners and an even number of links. On the other hand, for a given teleportation path a, we can always find a dual teleportation path b which goes though the same number of corners and links. For any pair of dual a and b paths, the sign contribution from all the links (corners) are the same (opposite). Therefore, the total contribution from the pair of teleportation paths a and b always vanishes. Thus, we have proved that $|\Psi\rangle_{\text{RVB}}$ has no $AA$ ($BB$) pairing.
\begin{figure}
\begin{center}
\includegraphics[width=5cm]{fig8}
\caption{(Color online) For any teleportation path contributing to the $AA$ ($BB$) pairing, there always exists a dual teleportation path with opposite sign. Therefore
the $AA$ ($BB$) pairing amplitudes are exactly zero in $|\Psi\rangle_{\text{RVB}}$.}
\label{pairing}
\end{center}
\vskip -0.5cm
\end{figure}
\section{String picture}
Since the RVB bonds only connect sites on different sublattices, we
can view such a RVB state as a liquid state of oriented
strings. Indeed, choosing a reference $\text{VB}_0$ configuration, any
$\text{VB}$ configuration can be viewed as a closed oriented string
configuration: the RVB bonds in the $\text{VB}$ configuration are
regarded as a piece of the closed string pointing from the A to the B
sublattice, while the RVB bonds in the reference $\text{VB}_0$
configuration are regarded as the complementary piece pointing from
the B to the A sublattice.
If such a superposition of closed orientable string states indeed
represent a \emph{liquid} state of closed strings, then the
entanglement entropy for a such a state in a region $A$ has the form
$S_A= a L_A - \frac 12 \text{ln} L_A+b$, where $L_A$ is the length of
the perimeter of region $A$. To understand such a result, we view a
string as a flux line and the close-string condition implies that the
flux is conserved. Therefore, the total flux going through the
perimeter of $A$ is zero. If we had ignored the flux conservation and
assumed that the flux could fluctuate freely and independently, then
the entanglement entropy would have an exact area-law form $S_A=
aL_A$ and the typical amount of flux through the perimeter would be
proportional to $\sqrt{L_A}$. So if we restrict the amount of flux through
the perimeter to be zero, the entanglement entropy will be $S_A= a L_A
- \text{ln} \sqrt{L_A}+b= a L_A - \frac 12 \text{ln} L_A+b$. The
ln$L_A$ dependence in the entanglement entropy implies that the liquid
of orientable closed strings must be gapless. This property is
confirmed by our numerical calculation.
\section{Finite size extrapolation of the optimum parameter $c$}
We provide here a more accurate determination of the optimum parameter $c$.
For this purpose, we use a quadratic function to fit the even sector ($|\Psi\rangle_e$)
energy of a finite width ($N_v$) strip as a function of parameter $c$ (around the minimum),
and we extract the optimum parameter $c_{\text{opt}}(N_v)$ and the
minimum energy $E_{\text{min}}(N_v)$ for $N_v=10,\cdots,24$. The
results are presented in Fig.~\ref{para}(a). Taking the optimum
parameter $c_{\text{opt}}(N_v)$ and extrapolating it to the thermodynamic
limit as a function of inverse width $1/N_v$, we find
$c_{\text{opt}}(\infty)=0.356(1)$ (see Fig.~\ref{para}(b)). Again,
taking the minimum finite width energy $E_{\text{min}}(N_v)$ and
extrapolating it to the thermodynamic limit as a function of $1/N_v$, we
obtain a thermodynamic energy $E_{\infty}=-0.48620(1)$ (see
Fig.~\ref{para}(c)).
\begin{figure}
\begin{center}
\includegraphics[width=6cm]{fig9}
%\caption{}
\caption{(Color online)(a) A quadratic fit of the even sector energy $E(N_v)$ of a strip as a function of $c$ around its minimum with $N_v=10,\cdots,24$. (b) Optimum parameter $c_{\text{opt}}(N_v)$ plotted as a function of $1/N_v$. A linear regression gives $c_{\text{opt}}(\infty)=0.356(1)$. (c) Minimum energy $E_{\text{min}}(N_v)$ plotted as a function of $1/N_v$. A linear regression gives $E(\infty)=-0.48620(1)$.}
\label{para}
\end{center}
\vskip -0.5cm
\end{figure}
%\end{widetext}
\section{Entanglement spectra and boundary Hamiltonian}
Entanglement spectra (ES) and boundary Hamiltonian reflect rich
information about the bulk of the system. Previous study has found
that a gaped bulk phase with local order corresponds to a boundary
Hamiltonian with local interactions, whereas critical behavior in the
bulk is reflected on a diverging interaction length of the boundary
Hamiltonian~\cite{ignacio}. Here we study both ES and boundary
Hamiltonian by partition an infinite cylinder into two semi-infinite
cylinders.
One of the virtue of PEPS description is that there is a duality
mapping between the bulk and its boundary. This exact mapping is
explained in detail in Ref.~\cite{ignacio,didier}, we only outline the
key relations in defining ES and boundary Hamiltonian.
The boundary Hamiltonian $H_b$ is defined from the reduced density
operator (acting on the edge degrees of freedom) as
$\sigma_b^2=\mbox{exp}(-H_b)$, and
$\sigma_b^2=\sigma_{bL}^2=\sigma_{bR}^2$, here $\sigma_{bL}^2$
($\sigma_{bR}^2$) is the reduced density operator of the left (right)
half cylinder taking the form of
$\sigma_{bL}^2=\sqrt{\sigma_{L}^{t}}\sigma_{R}\sqrt{\sigma_L^{t}}$
($\sigma_{bR}^2=\sqrt{\sigma_{R}^{t}}\sigma_{L}\sqrt{\sigma_{R}^{t}}$).
Here $\sigma_{L/R}$ are obtained by contracting the tensors of the
left (right) half cylinders column-wise from the cylinder boundary to
its edge. Note that to obtain $\sigma_{L/R}$ for an infinite cylinder,
one starts from an initial vector and continuously applies the
transfer matrix to the vector until it converges. The reduced density
matrix of the left (right) half cylinder is
$\rho_{L/R}=U\sigma_{bL/R}^2U^{\dagger}$, where $U$ is an isometry
that maps any operator defined on the bulk onto the virtual
spins~\cite{ignacio}. By definition $\rho_{L/R}$ and $\sigma_{bL/R}^2$
share the same spectra.
\begin{figure}
\begin{center}
\includegraphics[width=\columnwidth]{fig10}
%\caption{}
\caption{(Color online) Entanglement spectra of a $N_v=8$ cylinder as
a function of momentum $k$ winding around the cylinder and its total
spin $S$ for even and odd sectors without vison line in the
wavefunction. The normalization for both sector are
$\text{Tr}\{\sigma_{L,p}^t\sigma_{R,p}\}=1$, $p=e,o$. Plot is at the
best variational parameter $c=0.35$.}
\label{es}
\end{center}
\vskip -0.5cm
\end{figure}
Since the transfer matrix conserves the even or odd parity quantum
number, one can define a mixed reduced density operator by writing
$\sigma_{L/R}=\sigma_{L/R,e}+\sigma_{L/R,o}$ and
$\sigma_{bL}^2=\sqrt{\sigma_{L,e}^t}\sigma_{R,e}\sqrt{\sigma_{L,e}^t}+\sqrt{\sigma_{L,o}^t}\sigma_{R,o}\sqrt{\sigma_{L,o}^t}$,
(similar definition applies to $\sigma_{bR}^2$), here the equal
weight normalization condition is
$\mbox{Tr}\{\sigma_{L,e}^t\sigma_{R,e}\}=\mbox{Tr}\{\sigma_{L,o}^t\sigma_{R,o}\}=1$.
The boundary Hamiltonian is thus uniquely defined, which we denote as
$H_{\text{local}}$. The reduced density operator $\sigma^2_{bL/R,p}$
in each sector $p=e,o$ contains a finite fraction of zero-weight
eigenvalues.
We present the ES $\{\xi_{\lambda}\}$ of the boundary Hamiltonian
$H_{\text{local}}$
($\sigma_b^2=V\text{exp}(-\text{diag}\{\xi_{\lambda}\})V^{\dagger}$)
as a function of momentum $k$ around the cylinder and its total spin
$S$ in Fig.~\ref{es}. The spin multiplet structure is inherited from
the spin $1/2\oplus 0$ representation of $\text{su}(2)$ symmetry of
the boundary Hamiltonian. One of the signatures of the ES is that the
lowest excitation energy $\Delta(S)=\xi(S)_{\text{exc}}-\xi_0$ does
not vanish at infinite cylinder when scales linearly as a function of
inverse cylinder parameter $N_v$, as depicted in Fig.~\ref{bh}(a),
which however is in sharp contrast to the the case of Kagome nearest
neighbor RVB state~\cite{didier} that has been proved to be a gaped
$\mathbb{Z}_2$ topological state~\cite{norbert1}.
To further explore the boundary theory, we expand the boundary
Hamiltonian into $3^{2N_v}$ orthogonal operators, each of them is a
product of any 9 local normalized orthogonal operators
$\{\hat{x}_0,\cdots,\hat{x}_8\}$ defined as in Ref.~\cite{didier}. The
weight of N-body terms for a $N_v=8$ cylinder is calculated and
plotted in Fig.~\ref{bh}(b). The boundary Hamiltonian is clearly
neither local nor short ranged, which is reflected from a flat fat
tail for N-body ($N>4$) terms in Fig.~\ref{bh}(b). This serves as a
side evidence that the bulk is not gaped.
\begin{figure}
\begin{center}
\includegraphics[width=\columnwidth]{fig11}
%\caption{}
\caption{(Color online) (a) The lowest energy gap
$\Delta(S)=\xi(S)-\xi_0$ as a function of inverse cylinder parameter
$N_v$. Fitted lines are of the form
$\Delta(S)=\Delta(S)_{\inf}+a_S/N_v$; the excitation gap at infinite
cylinder limit $\Delta(S)_{\inf}$ are finite for total spin
$S=0,1/2,1$. (b) N-body weight of the boundary Hamiltonian expanded
in local operators $\{\hat{x}_0,\cdots,\hat{x}_8\}$ for a $N_v=8$
cylinder. Both plot are at the best variational parameter $c=0.35$.}
\label{bh}
\end{center}
\vskip -0.5cm
\end{figure}
\begin{thebibliography}{99}
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}\fi \providecommand{\bibinfo}[2]{#2}
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\bibitem{ignacio}
\bibinfo{author}{\bibfnamefont{J.~I.~Cirac}},
\bibinfo{author}{\bibfnamefont{D.~Poilblanc}},
\bibinfo{author}{\bibfnamefont{N.~Schuch}} \bibfnamefont{and}
\bibinfo{author}{\bibfnamefont{F.~Verstraete}},
\bibinfo{title}{\bibfnamefont{Entanglement spectrum and boundary
theories with projected entangled-pair states}},
\bibinfo{journal}{Phys.~Rev.~B} \textbf{\bibinfo{volume}{83}},
\bibinfo{pages}{245134} (\bibinfo{year}{2011}).
\bibitem{didier}
\bibinfo{author}{\bibfnamefont{D.~Poilblanc}},
\bibinfo{author}{\bibfnamefont{N.~Schuch}},
\bibinfo{author}{\bibfnamefont{D.~Perez-Garcia}} \bibfnamefont{and}
\bibinfo{author}{\bibfnamefont{J.~I.~Cirac}},
\bibinfo{title}{\bibfnamefont{Topological and entanglement
properties of resonating valence bond wave functions}},
\bibinfo{journal}{Phys.~Rev.~B} \textbf{\bibinfo{volume}{86}},
\bibinfo{pages}{014404} (\bibinfo{year}{2012}).
\bibitem{norbert1}
\bibinfo{author}{\bibfnamefont{N.~Schuch}},
\bibinfo{author}{\bibfnamefont{D.~Poilblanc}},
\bibinfo{author}{\bibfnamefont{J.~I.~Cirac}} \bibfnamefont{and}
\bibinfo{author}{\bibfnamefont{D.~Perez-Garcia}},
\bibinfo{title}{\bibfnamefont{Resonating valence bond states in the PEPS formalism}},
\bibinfo{journal}{Phys.~Rev.~B} \textbf{\bibinfo{volume}{86}},
\bibinfo{pages}{115108} (\bibinfo{year}{2012}).
\end{thebibliography}
\end{document}