text:
\section{Introduction}

In a quantum many-body system,  distribution of quantum correlation in its  subparts often provides many insightful details of the underlying physical phenomena. In the past years,  there has been a plethora of works where bipartite quantum entanglement is considered to be an efficient probe to characterize quantum properties of  strongly correlated systems \cite{entanglement_many_body1,entanglement_many_body2,entanglement_many_body3,entanglement_many_body4,entanglement_many_body5,entanglement_many_body6,entanglement_many_body7}. In particular,  entanglement entropy quantifies the amount of entanglement present between a part of the system and its complement and  allows us to distinguish quantum phases that do not break any symmetry and hence can not be characterized using any local order parameter, e.g., topological states of matter \cite{topo_entropy1,topo_entropy2,topo_entropy3,topo_entropy4,topo_entropy5,entanglement_many_body6},  quantum spin liquids \cite{ent_entropy_spin_liquid1,ent_entropy_spin_liquid2,ent_entropy_spin_liquid3}, etc. Additionally, the area law for the entanglement entropy of low-energy states of local Hamiltonians has provided a new framework to describe quantum many-body systems in a more efficient way by exploiting tensor network theory \cite{entanglement_many_body4,entanglement_many_body5, ent_entropy_area_law,tensor_network_bunch1,tensor_network_bunch2,tensor_network_bunch3}.

The von Neumann entanglement entropy summarizes the information about the reduced density matrix for a part of the system into a single number. However, there are scenarios where the distribution of the eigenvalues of the reduced system, commonly known as {\it entanglement spectrum} (ES), can unveil more refined information than the entanglement entropy alone. This has attracted much interest in the community, as several studies have been  reported situations where the ES has been  very insightful. The list includes quantum criticality \cite{ent_spectrum_ref0,ent_spectrum_ref1,ent_spectrum_ref2,ent_spectrum_ref3,ent_spectrum_ref4}, symmetry protected phases of matter \cite{ent_spectrum_ref5,ent_spectrum_ref6,ent_spectrum_ref7,ent_spectrum_ref8,ent_spectrum_ref9},  many-body localization \cite{ent_spectrum_ref10,ent_spectrum_ref11,ent_spectrum_ref12}, etc.  As an example, it has been shown that the degeneracy in the reduced state of the spin-1 chain reflects the non-trivial {\em topological} nature of the Haldane phase of the model \cite{ent_spectrum_ref7}. The degeneracy is protected by the same set of symmetries which protects the Haldane phase. Very recently, a scheme to measure the entanglement spectrum using the IBM quantum computing interface has been demonstrated \cite{ent_spectrum_ref13}. Even further information can be provided by considering the reduced density matrix as a thermal state associated to a certain entanglement Hamiltonian, thus providing a direct route to relate the properties of the subsystem to the physical Hamiltonian of the system. For instance, physics at the edge of topologically ordered phases, e.g., fractional quantum Hall state \cite{ent_spectrum_ref5} and of symmetry-protected topological states of matter \cite{ent_ham_ref2,ent_ham_ref3,ent_ham_ref4,ent_ham_ref5,ent_ham_ref6,ent_ham_ref7} has been found to be imprinted in the entanglement Hamiltonian of a part of the system.  The properties of the entanglement Hamiltonian in conformally invariant systems have also been extensively explored in many recent works  \cite{ent_ham_conformal1,ent_ham_conformal2,ent_ham_conformal3,ent_ham_conformal4,ent_ham_conformal5,ent_ham_conformal6}.  For instance, in Ref.\cite{ent_ham_conformal2} it is shown that for two dimension (2D) CFT, the universal part of the spectrum of the entanglement Hamiltonian is always that of a boundary CFT with appropriate boundary conditions.

In most of the above cases, the entanglement spectrum and the entanglement Hamiltonian have been obtained for contiguous bipartitions of the state. Derivation of entanglement Hamiltonians for non-contiguous blocks of the system is a relatively hard albeit interesting case to explore. In this work, we derive the entanglement Hamiltonian for a topological state built on a spin-1 system for both contiguous and non-contiguous bipartitions of the state. We consider the periodic Affleck-Kennedy-Lieb-Tasaki (AKLT) state \cite{AKLT1} in 1D, which can be obtained as the unique ground state (GS) of the periodic spin-1 bilinear-biquadratic Heisenberg (BBH) Hamiltonian at the AKLT point within the Haldane phase \cite{AKLT2}. Then, we compute the entanglement spectrum for contiguous as well as non-contiguous bipartitions by employing the matrix product states formalism, which provides a more direct route than previous approaches \cite{korepin1,Hirano,korepin2}. We note that in both the cases, the low-energy component of the entanglement Hamiltonian can be approximated as an interacting Heisenberg Hamiltonian between the effective spin-1/2 particles at the boundaries of the blocks. Moreover, for the contiguous case, the low-energy properties of the entanglement Hamiltonian can be related to the edge properties of the associated physical Hamiltonian with open boundaries. This suggests the emergence of a bulk-edge correspondence in the model, similar to the one reported in earlier works \cite{ent_spectrum_ref5,ent_ham_ref2,ent_ham_ref3,ent_ham_ref4,ent_ham_ref5,ent_ham_ref6, ent_ham_ref7}.
In this article, we explore this relation in more depth and report that this bulk-edge correspondence persists even at other points in the Haldane phase, as it is manifested by the high overlap between the low-energy eigenstates of the entanglement and physical Hamiltonians, the identical scaling of their respective energy gaps and the same degeneracy structure of the energy eigenvalues corresponding to the low-energy part of the spectrum. Additionally, we show an intriguing relation between the scaling of both the energy gaps and that of the string order parameter (SOP)  \cite{SOP} in the vicinity of the AKLT point.  As it is well known, the SOP provides a hidden order parameter for the Haldane phase.  In particular, the scaling of SOP with system size remains akin to that of the energy gaps,  which eventually results in a  correlation length  identical to the characteristic length scale associated with the BBH model.

This article is organized as follows. In Sec.\ref{Sec1}, we discuss the matrix product state representation of the AKLT state and the form of the corresponding transfer matrix which will be useful to compute the relevant quantities that we wish to explore. In Sec. \ref{Sec2}, we present the analytical derivation of the ES of contiguous and non-contiguous bipartitions of the AKLT state. This is followed by the derivation and characterization of the corresponding entanglement Hamiltonian in both cases. On the other hand, Sec. \ref{Sec3} is devoted to the characterization of the bulk-edge correspondence that emerges between the low-energy spectra of the entanglement Hamiltonian and the corresponding physical Hamiltonian, which is shown to persist even at other points in the Haldane phase. Finally, in Sec. \ref{Sec4} we summarize our conclusions and discuss possible lines of future work.

\section{Preliminaries}
\label{Sec1}

We start with the matrix product states (MPS) representation of the periodic AKLT state, given by
\begin{eqnarray}
|\psi\rangle_{\text{AKLT}}&=&\sum_{i_1 i_2 \dots i_N} \text{Tr}(A_{i_1} A_{i_2}\dots A_{i_N}) |i_1 i_2\dots i_N\rangle,
\label{eqn:AKLT_MPS}
\end{eqnarray}
where the index `$k$' in $A_k$ ($k \in 0, 1, 2$) labels the standard spin-1 basis states in each physical site, with $A_0=\sqrt{\frac{2}{3}}\sigma^+$, $A_1=-\sqrt{\frac{1}{3}}\sigma _z$, $A_2=-\sqrt{\frac{2}{3}}\sigma^-$, $\sigma_z$ is the $z$-component of the Pauli spin vector and $\sigma^+(\sigma^-)$ is spin raising (lowering) operator.  Such a representation of the AKLT state guarantees its canonical form, $\sum_k A_k A_k^ {\dagger}=\mathbb{I}$ (right canonical), and $\sum_k A_k^\dagger A_k=\mathbb{I}$ (left canonical) \cite{MPS_Garcia}. For the sake of efficiency, it is customary to define a transfer matrix as
\begin{eqnarray}
E=\sum_k A_k \otimes A^*_k.
\label{eqn:Transfer_matrix}
\end{eqnarray}
 The transfer matrix $E$ is diagonalizable and can be expressed in terms of its left and right eigenvectors as follows $E=\sum_{i=0}^{D^2-1} \gamma_i |R_i\rangle\langle L_i|$. For the above choices of the $A_k$ matrices for AKLT state, the set of vectors $\{|L_i\rangle\}_{i=0}^{D^2-1}$ and $ \{|R_i\rangle\}_{i=0}^{D^2-1}$ present the following form in the computational basis,

\begin{eqnarray}
|R_0\rangle=|L_0\rangle&=&\frac{1}{\sqrt{2}} (|00\rangle +|11\rangle),\nonumber\\
|R_1\rangle=|L_1\rangle&=&\frac{1}{\sqrt{2}} (|00\rangle -|11\rangle),\nonumber\\
|R_2\rangle=|L_2\rangle&=&|01\rangle,\nonumber\\
|R_3\rangle=|L_3\rangle&=&|10\rangle,
\label{eqn:transfer_matrix_vec}
\end{eqnarray}
with $\gamma_0=1, \gamma_1=\gamma_2=\gamma_3=\gamma=-\frac{1}{3}$.

Along with this, here we introduce the quantum many-body Hamiltonian we will use in our work, known as spin-1 bilinear-biquadratic Heisenberg (BBH) Hamiltonian,
\begin{eqnarray}
H_{BBH}=\sum_i \cos(\theta)\, \vec{S}_i\cdot\vec{S}_{i+1} + \sin(\theta)\, (\vec{S}_i\cdot\vec{S}_{i+1})^2,
\label{eqn:BBH}
\end{eqnarray}
where $S^k$ are spin-1 operators, $k\in x,y,z$.  The AKLT state described above appears at $\theta_{AKLT}=\arctan(\frac{1}{3})$.

\section{Derivation of Entanglement Hamiltonian}
\label{Sec2}

Let us evaluate the entanglement Hamiltonian for both contiguous and non-contiguous subsystems of a periodic AKLT state. In general, the entanglement Hamiltonian $\mathcal{H}_E$ is defined in terms of the reduced density matrix $\rho$ of a quantum state,

\begin{eqnarray}
\mathcal{H}_E=-\ln(\rho).
\label{eqn:Ent_Ham}
\end{eqnarray}
The above representation often provides an efficient way to unveil the information in the reduced subsystem of the system \cite{ent_spectrum_ref5,ent_ham_ref2,ent_ham_ref3,ent_ham_ref4,ent_ham_ref5,ent_ham_ref6,
ent_ham_ref7,ent_ham_conformal1,ent_ham_conformal2,ent_ham_conformal3,ent_ham_conformal4,
ent_ham_conformal5,ent_ham_conformal6}. In particular, for experimental purposes where direct experimental measurement of the entanglement spectrum requires full state tomography, it may be more feasible to extract the entanglement properties by engineering the entanglement Hamiltonian \cite{ent_ham_exp1}. However, the derivation of the exact form of the entanglement Hamiltonian for a generic quantum many-body state is not straightforward. In our case, we first obtain the exact mathematical form of the reduced density matrices for different choices of the subsystems of the AKLT state and use Eq. (\ref{eqn:Ent_Ham}) to derive the corresponding entanglement Hamiltonian. The detailed methodology is discussed in the following paragraphs.

\subsection{Contiguous bipartition}
\label{Contiguous bipartition}
Let us consider the entanglement Hamiltonian for an $l$-site contiguous subsystem  of the periodic AKLT state.  We decompose the MPS representation of the AKLT state given in Eq. (\ref{eqn:AKLT_MPS}) in the   $A:B$ bipartition as follows
\begin{eqnarray}
|\psi\rangle_{\text{AKLT}}=&&\sum_{\alpha \beta} |\phi^l_{\alpha\beta}\rangle |\phi_{ \beta\alpha}^{N-l}\rangle,
\label{eqn:AKLT_contiguous}
\end{eqnarray}
such that sites $1,2\dots l \in A$ and $l+1,l+2\dots N \in B$ and $\alpha, \beta \in \{0, 1\}$. The intermediate steps to arrive at such decomposition from Eq.  (\ref{eqn:AKLT_MPS}) and the exact mathematical forms of the states $|\phi_{\alpha \beta}^l\rangle$ are given in Appendix \ref{AppendixA}. Though such  decomposition is akin to the Schmidt decomposition in the bipartition $l: N-l$, in general,  $|\phi_{\alpha\beta}^l\rangle$ (or $|\phi_{\alpha\beta}^{N-l}\rangle$) are not mutually orthogonal. For instance, for the AKLT state, the states  $|\phi_{00}^l \rangle$ and $|\phi^l_{11}\rangle$ have non-zero overlap,  which decays with block size $l$ as  $\langle\phi_{00}^l|\phi^l_{11}\rangle$=$\langle \phi_{11}^l |\phi^l_{00}\rangle$=$\gamma^l$. In that case, from the set of states $|\phi_{\alpha \beta}^l\rangle$ one needs to construct an orthogonal basis  $\{|\phi^l_{+}\rangle, |\phi^l_{-}\rangle, |\phi^l_{01}\rangle, |\phi^l_{10}\rangle\}$, where  $|\phi^l_{\pm}\rangle=\frac{1}{\sqrt{2}}(|\phi_{00}^l\rangle\pm |\phi_{11}^l\rangle)$.  We discuss the procedure in detail in Appendix \ref{AppendixA}.

Now if we trace out subsystem $B$, the exact form of the $l$-site reduced density matrix $\rho$  in its orthonormal eigenbasis is given by (see Appendix \ref{AppendixA} for a detailed derivation)
\begin{eqnarray}
\rho=\lambda_0|\phi^l_+\rangle \langle \phi^l_+| + \lambda_1|\phi^l_- \rangle\langle \phi^l_-|+\lambda_2|\phi^l_{01}\rangle\langle \phi^l_{01}|+\lambda_3|\phi^l_{10}\rangle \langle \phi^l_{10}|.\nonumber\\
\end{eqnarray}
The eigenvalues $\lambda_i$ for $N\gg1$ can be expressed as

\begin{eqnarray}
\lambda_0&=&\frac{{(1+3\gamma^l) (1+3\gamma^{N-l})}}{4},\nonumber\\
\lambda_1=\lambda_2=\lambda_3&=&\frac{{(1-\gamma^l) (1-\gamma^{N-l})}}{4}.\nonumber\\
\label{eqn:lambda_contiguous}
\end{eqnarray}
The above eigenvalues remain invariant under the exchange of subsystems, i.e., $l$ by $N-l$. Now if we keep total system size $N$  even, for $l$ even, we have $\lambda_0> \lambda_1= \lambda_2= \lambda_3$. Therefore, the spectrum of $\rho$ is comprised of two distinct eigenvalues, the lowest one is triply degenerate and the corresponding  eigenspace is spanned by the basis states  $\{|\phi^l_-\rangle, |\phi^l_{01}\rangle, |\phi^l_{10}\rangle\}$. On the other hand, the highest one is unique and the corresponding eigenstate is given by  $|\phi^l_+\rangle$. Similarly, for odd $l$, we have the opposite ordering of the eigenstates, i,e, in that case the lowest eigenvalue state is unique but the highest eigenvalue is triply degenerate. However, for moderately large values of $l$, this distinction vanishes and all the eigenvalues becomes degenerate and converge to $\frac{1}{4}$. This degeneracy is a signature of the symmetry protected topological (SPT) nature of the AKLT state \cite{ent_spectrum_ref8,SPT_AKLT}. Fig. \ref{fig:lambda_contiguous} depicts the behavior of $\lambda_0$ and $\lambda_1$ for even and odd values of $l$.

\begin{figure}[t]
\includegraphics[width=8.5cm]{Fig1}
\caption{Plot of the eigenvalues $\lambda_i$ of the reduced density matrix $\rho$ derived from the contiguous bipartition of the periodic AKLT state as expressed in Eq. (\ref{eqn:lambda_contiguous}). In both panels, we consider the total size of the system $N=50$. In panel (a) we plot the $\lambda_i$'s for even values of $l$ and we can see $\lambda_0\geq\lambda_1$ for all $l$. Whereas, in (b), we plot the $\lambda_i$'s for odd values of $l$ and we get $\lambda_1\geq\lambda_0$ for all $l$. For large $l$,  all the $\lambda_i$'s converge to $\frac{1}{4}$.}
\label{fig:lambda_contiguous}
\end{figure}

Once the exact form of the reduced density matrix $\rho$ has been derived, the expression of the entanglement Hamiltonian can be obtained using Eq. (\ref{eqn:Ent_Ham}). We propose

\begin{eqnarray}
\mathcal{H}_E =-\ln \rho= \varepsilon_0 + J^{(1)}_E\vec{\sigma_1} . \vec{\sigma_2},
\label{eqn:EH_contiguous}
\end{eqnarray}
where $\varepsilon_0$ is a constant, $J^{(1)}_E$ is a function of $l$ and $\vec{\sigma}$ is the spin-1/2 vector operator.
This suggests

\begin{eqnarray}
- \log \lambda_0 &=& \varepsilon_0 - 3  J^{(1)}_E,\nonumber\\
- \log \lambda_k &=& \varepsilon_0 +  J^{(1)}_E,
\label{eqn:eigen_ent_Ham}
\end{eqnarray}
with $k\in \{1, 2, 3\}$. At a moderately large value of $l$, $\varepsilon_0 \approx 2\log2+O(\gamma^{2l})$, and $J^{(1)}_E\approx \gamma^l$, which is consistent with the behavior of $\lambda_i$ in this limit. Hence, the  entanglement Hamiltonian remains non-trivial as long as the size of the subsystems remains moderately low such that $J^{(1)}_E$ is non-negligible. Notice that the even-odd dependence that we discussed earlier comes from the fact that $J^{(1)}_E\approx \gamma^l$ with $\gamma=-\frac{1}{3}$. Additionally, the constants $\varepsilon_0$ and $J^{(1)}_E$ can now be interpreted as the GS entanglement energy and entanglement coupling respectively. We note that for even $l$,  $|\phi^l_+\rangle$ becomes the unique GS of the entanglement Hamiltonian and the excited subspace is spanned by the triplet $\{|\phi^l_-\rangle, |\phi^l_{01}\rangle, |\phi^l_{10}\rangle\}$.
At this stage, in order to unveil characteristic features of the entanglement Hamiltonian, we perform a local transformation of the auxiliary basis, denoted by the index $\alpha$ and $\beta$ in $|\phi_{\alpha\beta}^l\rangle$ (see Appendix\ref{AppendixA}), such that
\begin{figure}[h]
\includegraphics[width=8cm]{Fig2}
\caption{Schematic of relation between parts of the AKLT state and the corresponding entanglement Hamiltonian. The entanglement Hamiltonian for a part of the state is similar to the Heisenberg Hamiltonian between the effective spin-1/2 particles at the boundaries of the blocks. The green line in $H_{int}$ represents couplings between inter-block effective spin-1/2 particles ($J_E^{(2)}$), while the red lines represent the coupling between intra-block spins ($J_E^{(1)}$).}
\label{fig:schematic_first}
\end{figure},
the eigenbasis of $\mathcal{H}_E$ under this transformation changes as
\begin{eqnarray}
|\phi^l_+\rangle=\frac{(|\phi_{00}^l\rangle+|\phi_{11}^l\rangle)}{\sqrt{2}}&\rightarrow&|\tilde{\phi}^l_+\rangle=\frac{(|\tilde{\phi}^l_{01}\rangle-|\tilde{\phi}^l_{10}\rangle)}{\sqrt{2}},\nonumber\\
|\phi^l_-\rangle=\frac{(|\phi_{00}^l\rangle-|\phi_{11}^l\rangle)}{\sqrt{2}}&\rightarrow&|\tilde{\phi}^l_-\rangle=\frac{(|\tilde{\phi}^l_{01}\rangle+|\tilde{\phi}^l_{10}\rangle)}{\sqrt{2}},\nonumber\\
|\phi^l_{01}\rangle&\rightarrow&-|\tilde{\phi}^l_{00}\rangle,\nonumber\\
|\phi^l_{10}\rangle&\rightarrow&|\tilde{\phi}^l_{11}\rangle.
\end{eqnarray}
Interestingly, a nice interpretation of the entanglement Hamiltonian emerges from the above derivation. In particular, $H_{int}=\vec{\sigma}_1\cdot \vec{\sigma}_2$ stands for the interacting Heisenberg Hamiltonian between the effective spin-1/2 particles at the end of the block (see Fig. \ref{fig:schematic_first}), and the corresponding GS
$|\psi\rangle_G=|\tilde{\phi}_{+}^l\rangle=\frac{|\tilde{\phi}^l_{01}\rangle-|\tilde{\phi}^l_{10}\rangle}{\sqrt{2}}$ can be regarded as a {\em singlet}-state formed in the auxiliary basis $\{|\tilde{\alpha}\rangle, |\tilde{\beta}\rangle\}$ (see Appendix \ref{AppendixA}). Similarly, the excited states of the interaction  Hamiltonian $H_{int}$  ($|\tilde{\phi}^l_-\rangle, |\tilde{\phi}^l_{00}\rangle, |\tilde{\phi}^l_{11}\rangle$) are an analogue to the {\em spin-triplet} state of the Heisenberg Hamiltonian. This interpretation is part of the edge-bulk correspondence that we shall discuss in more detail later on in Sec. \ref{Sec3}.

\subsection{Non-contiguous bipartitions}

Next, let us consider the entanglement Hamiltonian for non-contiguous subsystems of the AKLT model. In particular, we will decompose the AKLT state into the following bipartition,  $A_1: B_1: A_2: B_2$, such that

\begin{eqnarray}
&&[1, 2,\dots,  l_{A_1}] \in A_1,\nonumber\\
&&[l_{A_1}+1, l_{A_1}+2,\dots,   l_{A_1}+l_{B_1}] \in B_1, \nonumber\\
&&[l_{A_1}+l_{B_1}+1, l_{A_1}+l_{B_1}+2,\dots, l_{A_1}+l_{B_1}+l_{A_2}] \in  A_2, \nonumber\\
&&[l_{A_1}+l_{B_1}+l_{A_2}+1, \dots,   l_{A_1}+l_{B_1}+l_{A_2}+l_{B_2}]\in B_2,
\end{eqnarray}
where $l_{A_1}+l_{B_1}+l_{A_2}+l_{B_2}=N$, $A_1\cup A_2=A$, and  $B_1\cup B_2=B$.
To find the reduced density matrix in this case, we start with a similar decomposition of the AKLT state as given in Eq. (\ref{eqn:AKLT_contiguous})
\begin{eqnarray}
|\psi\rangle_{\text{AKLT}}&=&\sum_{\alpha \beta}|\phi_{\alpha \beta}^{l_{A_1}+l_{B_1}}\rangle  |\phi_{\beta \alpha}^{l_{A_2}+l_{B_2}}\rangle.
\end{eqnarray}
Further decomposition of the basis $|\phi_{\alpha \beta}^{l_{A_1}+l_{B_1}}\rangle$ and $|\phi_{\beta \alpha}^{l_{A_2}+l_{B_2}}\rangle$ yields
\begin{eqnarray}
|\psi\rangle_{\text{AKLT}}&=& \sum_{\alpha  \gamma  \beta \delta}  |\phi_{\alpha \gamma}^{l_{A_1}}\rangle   |\phi_{\gamma \beta}^{l_{B_1}}\rangle |\phi^{l_{A_2}}_{\beta \delta}\rangle|\phi^{l_{B_2}}_{\delta \alpha}\rangle,\nonumber\\
&=& \sum_{\alpha  \gamma  \beta \delta} \underbrace{ |\phi_{\alpha \gamma}^{l_{A_1}}\rangle   |\phi^{l_{A_2}}_{\beta \delta}}_{A}\rangle \underbrace{|\phi _{\gamma \beta}^{l_{B_1}}\rangle |\phi^{l_{B_2}}_{\delta \alpha}\rangle}_{B}.
\end{eqnarray}
The above representation of the AKLT state now becomes very similar to the contiguous case. Therefore, in order to obtain the reduced density matrix for the subsystem $A=A_1\cup A_2$, we can employ the same techniques as before. However, note that, since the indices $\alpha, \beta, \gamma, \delta$ can take two values, 0 and 1, the reduced state is spanned by sixteen basis states. Similarly to the contiguous case, we find that most of the basis states $|\phi_{\alpha \gamma}^{l_{A_1}}\rangle  |\phi^{l_{A_2}}_{\beta \delta}\rangle$ are not mutually orthogonal. The procedure of construction of mutually orthogonal basis states from these non-orthogonal states is much more complex than the previous case. We discuss the details in Appendix \ref{AppendixA}. Let us stress that the reduced density matrix obtained in this case depends on the parameters $l_{A_1}$, $l_{B_1}$, $l_{A_2}$ and $l_{B_2}$.

\begin{figure}[t]
\includegraphics[width=8.5cm]{Fig3}
\caption{Eigenvalues ($\lambda_1$, $\lambda_6$, $\lambda_9$, $\lambda_{10}$ , $\lambda_{13}$, $\lambda_{16}$) of the reduced density matrix $\rho$ for the non-contiguous bipartition of the periodic AKLT state as expressed in Eq. (\ref{eqn:lambda_noncontiguous1}). In panel (a) we consider the size of each $A$ blocks $\ell_A=2$. For moderately large $\ell_B$ these $\lambda_i$'s saturate to three non-unique values, given by Eq. (\ref{eqn:lambda_non_contiguous2}). In panel (b) we present the same plot but with a moderately large value of $\ell_A=20$. In this limit, for large $\ell_B$ all the $\lambda_i$'s converge to $\frac{1}{16}.$}
\label{fig:lambda_noncontiguous}
\end{figure}

We start with the case when the sub-blocks of each partition are of equal size, i.e., $ l_{A_1}=l_{A_2}=\ell_A, l_{B_1}=l_{B_2}=\ell_B$. The entanglement spectrum of the reduced density matrix obtained in this case is given by (see Appendix \ref{AppendixA} for detailed derivations)
\begin{eqnarray}
\lambda_i&=&\frac{1}{16}(1-\gamma^{\ell_A})^2 (1-\gamma^{\ell_B})^2, i \in 1, 2, 3, 4, 5,\nonumber\\
 \lambda_j&=&\frac{1}{16}(1-\gamma^{\ell_A})^2  (1+3\gamma^{\ell_B}) (1-\gamma^{\ell_B}), j \in 6, 7 , 8, \nonumber\\
 \lambda_9&=&\frac{1}{2}(r-\sqrt{r^2-4s}),\nonumber\\
  \lambda_k&=&\frac{1}{16}(1+3\gamma^{\ell_A})(1-\gamma^{\ell_A}) (1-\gamma^{\ell_B})^2, k\in 10, 11, 12, \nonumber\\
\lambda_l&=& \frac{1}{16}(1+3\gamma^{\ell_A})(1-\gamma^{\ell_A}) (1+3\gamma^{\ell_B}) (1-\gamma^{\ell_B}), \nonumber\\&&
l\in 13, 14 , 15, \nonumber\\
\lambda_{16}&=&\frac{1}{2}(r+\sqrt{r^2-4s}),
\label{eqn:lambda_noncontiguous1}
\end{eqnarray}
where
\begin{eqnarray}
s&=&\Big[\frac{1}{16}\Big((1+3\gamma^{\ell_A}) (1-\gamma^{\ell_A})(1+3\gamma^{\ell_B}) (1-\gamma^{\ell_B}) \Big)\Big]^2, \nonumber\\
r&=&\frac{1}{64} \Big((1+3\gamma^{\ell_A})^2(1+3\gamma^{\ell_B})^2+3(1+3\gamma^{\ell_A})^2(1-\gamma^{\ell_B})^2\nonumber\\
&+&3(1-\gamma^{\ell_A})^2(1+3\gamma^{\ell_B})^2+(1-\gamma^{\ell_A})^2(1-\gamma^{\ell_B})^2\Big).\nonumber\\
\end{eqnarray}
Hence, the entanglement spectrum in this case has six distinct eigenvalues. This is consistent with the expected degeneracy of the entanglement spectrum based on the SO(3) symmetry: $\frac{1}{2}\otimes \frac{1}{2} \otimes \frac{1}{2}\otimes \frac{1}{2}= 0 (2) \oplus 1 (3) \oplus 2(1)$. Among those $\lambda_{i}$'s, $\lambda_{16}$ is the highest one for all $\ell_B$. However, the lowest eigenstate depends on the parity of $\ell_B$. For even $\ell_B$, the degenerate set $\lambda_i$, with $i\in \{1,\cdots,5\}$ is the lowest one. Yet, for odd $\ell_B$, $\lambda_9$ becomes the lowest eigenvalue. Moreover, for a low $\ell_A$ and moderately large values of $\ell_B$ the number of such distinct eigenvalues decreases further and reduces to

\begin{eqnarray}
\lambda_i&\simeq& \frac{1}{16}(1-\gamma^{\ell_A})^2, i \in 1 ~\text{to}~9,\nonumber\\
\lambda_j&\simeq&\frac{1}{16}(1+3\gamma^{\ell_A})(1-\gamma^{\ell_A}), j\in 10 ~\text{to}~15, \nonumber\\
\lambda_{16}&\simeq&\frac{1}{16}(1+3\gamma^{\ell_A})^2.
\label{eqn:lambda_non_contiguous2}
\end{eqnarray}
We observe that these $\lambda_i$ can be identified as products of the eigenvalues derived for the contiguous case, given in Eq. (\ref{eqn:lambda_contiguous}). This suggests that  in this limit,  the mixed state describing  part $A$ becomes approximately factorized, $\rho \approx \rho_{A_1}\otimes \rho_{A_2}$.
Therefore, the reduced density matrix $\rho$  becomes diagonal in the $\{|\eta_k^{\ell_A}\rangle\otimes |\eta_p^{\ell_A}\rangle\}$ basis, with $|\eta_k^{\ell_A}\rangle, |\eta_p^{\ell_A}\rangle\in \{|\phi^{\ell_A}_+\rangle, |\phi^{\ell_A}_-\rangle, |\phi^{\ell_A}_{01}\rangle, |\phi^{\ell_A}_{10}\}\rangle$. The eigenstate with highest eigenvalue, i.e., $\lambda_{16}$ can now be written as  $|\lambda_{16}\rangle=|\phi^{\ell_A}_+\rangle \otimes |\phi^{\ell_A}_+\rangle$  Furthermore, if we now increase the size of subsystem $A$ to a moderately high value, all the $\lambda_i$ approach $\frac{1}{16} (\epsilon_0=4\log2)$. We present the behavior of the $\lambda_i$ with parameters $\ell_A$ and $\ell_B$ in Fig. \ref{fig:lambda_noncontiguous}.

We are now equipped with the necessary tools to construct the entanglement Hamiltonian in this case. Similarly to the contiguous case, we expand the entanglement Hamiltonian $\mathcal{H}_{E}$ in the eigenbasis of the reduced density matrix obtained above and propose

\begin{eqnarray}
\mathcal{H}_E&=&-\ln(\rho),\nonumber\\
&=&\epsilon_0 + \Big(J_E^{(1)}\vec{\sigma}_1.\vec{\sigma}_2+J_E^{(2)}\vec{\sigma}_2.\vec{\sigma}_3+J_E^{(1)}\vec{\sigma}_3. \vec{\sigma}_4 +J_E^{(2)}\vec{\sigma}_4.\vec{\sigma}_1 \Big),\nonumber\\
\label{eqn:EH_non_contiguous}
\end{eqnarray}
where in the above expansion we only consider terms up to   two-body interactions with nearest-neighbor  exchange and  all other interaction terms have been dropped considering those as leading order approximation. Now for large values of $\ell_A$ and $\ell_B$, we have $\epsilon_0\approx  4 \log 2+O(\gamma^{\ell_A})$, $J_E^{(1)}\approx \gamma^{\ell_A}$, and $J_E^{(2)}\approx \gamma^{\frac{\ell_A+\ell_B}{2}}$.
Hence, in this case, the second part of the entanglement Hamiltonian can be understood as an interacting Hamiltonian, $H_{int}=\Big(J_E^{(1)}\vec{\sigma}_1.\vec{\sigma}_2+J_E^{(2)}\vec{\sigma}_2.\vec{\sigma}_3+J_E^{(1)}\vec{\sigma}_3. \vec{\sigma}_4+J_E^{(2)}\vec{\sigma}_4.\vec{\sigma}_1 \Big)$, where the effective spins at the end of each block interact with coupling strength $J_E^{(1)}$ and the inter-block interactions is given by $J_E^{(2)}$.  The illustration in Fig. \ref{fig:schematic_first} depicts the relation between different bipartitions of the AKLT state and the configuration of the interacting Hamiltonian derived from those. In this regard, we consider two limiting cases for $\ell_B$ and analyze the properties of the corresponding entanglement Hamiltonians in detail as follows.

\subsubsection{Large interblock spacing}

The first limit we consider is $\ell_B\gg \ell_A$, which results in $J_E^{(2)} \rightarrow 0$. Thus, the form of the GS of the $\mathcal{H}_E$ is given by
\begin{eqnarray}|\psi\rangle_G&=&|\phi^{\ell_A}_+\rangle \otimes |\phi^{\ell_A}_+\rangle, \nonumber\\&=&\left( \frac{|\phi^{\ell_A}_{00}\rangle+|\phi^{\ell_A}_{11}\rangle}{\sqrt{2}})\otimes (\frac{|\phi^{\ell_A}_{00}\rangle+|\phi^{\ell_A}_{11}\rangle}{\sqrt{2}}\right).
\end{eqnarray}
Now we carry out a similar basis transformation as the one in Eq. (\ref{eqn:Basis_transformation}) in Appendix\ref{AppendixA}, and write
\begin{eqnarray}|\psi\rangle_G&=&|\tilde{\phi}^{\ell_A}_+\rangle\otimes |\tilde{\phi}^{\ell_A}_+\rangle, \nonumber\\&=&\left( \frac{|\tilde{\phi}^{\ell_A}_{0 1}\rangle-|\tilde{\phi}^{\ell_A}_{10}\rangle}{\sqrt{2}})\otimes(\frac{|\tilde{\phi}^{\ell_A}_{01}\rangle-|\tilde{\phi}^{\ell_A}_{10}\rangle}{\sqrt{2}}\right).
\label{ground_state}
\end{eqnarray}
This provides a {\em spin-dimer}-like interpretation of the GS of the entanglement Hamiltonian $\mathcal{H}_E$. Therefore, the characteristic feature of the entanglement Hamiltonian remains consistent with the previous case. As we increase the number of boundaries of the subsystem of the AKLT state, the number of dimers in the GS of the entanglement Hamiltonian also increases proportionally.

The next excited state can be formed by breaking the dimer between any pair of sites (six such configurations, $\{|\tilde{\phi}^{\ell_A}_+\rangle \otimes |\eta_k^{\ell_A}\rangle, |\eta_k^{\ell_A}\rangle \otimes |\tilde{\phi}^{\ell_A}_+\rangle\}$, with $|\eta_k^{\ell_A}\rangle \in \{|\tilde{\phi}^{\ell_A}_-\rangle, |\tilde{\phi}^{\ell_A}_{01}\rangle, |\tilde{\phi}^{\ell_A}_{10}\}\rangle$) and finally the highest excited states are given by breaking both dimers (nine such configurations, $\{|\eta_k^{\ell_A}\rangle \otimes |\eta_p^{\ell_A}\rangle\}$, with $|\eta_k^{\ell_A}\rangle, |\eta_p^{\ell_A}\rangle \in\{|\tilde{\phi}^{{\ell_A}}_-\rangle, |\tilde{\phi}^{{\ell_A}}_{01}\rangle, |\tilde{\phi}^{\ell_A}_{10}\rangle\}$).

\subsubsection{Block size equals interblock spacing}

Let us now consider the case in which all the blocks have equal size, $\ell_A=\ell_B=l$, for even $l$, so the coefficients $J_E^{(1)}$ and $J_E^{(2)}$ become antiferromagnetic. In this case, the corresponding entanglement Hamiltonian  $\mathcal{H}_E$ becomes transnationally invariant and its GS matches exactly the GS of the physical Heisenberg Hamiltonian of four sites. Hence, we argue that the entanglement Hamiltonian derived from the equal-size non-contiguous bipartitions of the AKLT state eventually becomes an antiferromagnetic Heisenberg Hamiltonian in the auxiliary basis, which is a critical model. We elaborate on the analysis as follows.

\begin{figure}[h]
\includegraphics[width=7.cm]{Fig4}
\caption{Ratio between the coefficients $\chi_1$ and $\chi_2$ as expressed in Eq. (\ref{eqn:GS:Heisenberg}), as a function of the size of the $A$ and $B$ blocks $\ell_A=\ell_B=l$.}
\label{fig:coefficent_ratio}
\end{figure}

As a first step, we find that for $\ell_A=\ell_B=l$ the GS of the entanglement Hamiltonian given in Eq. (\ref{eqn:EH_non_contiguous}) can be expressed in the basis $\{|\phi^{l}_+\rangle, |\phi^{l}_-\rangle, |\phi^{l}_{01}\rangle, |\phi^{l}_{10}\rangle$ as
\begin{eqnarray}
|\psi\rangle_{G}&=& \chi_1 |\phi^{l}_+\rangle |\phi^{l}_{+}\rangle +\chi_2( |\phi^{l}_{-}\rangle |\phi^{l}_{-}\rangle+ |\phi^{l}_{01}\rangle |\phi^{l}_{10}\rangle\nonumber\\&+& |\phi^{l}_{10}\rangle |\phi^{l}_{01}\rangle),
\label{eqn:GS:Heisenberg}
\end{eqnarray}
where the ratio $\chi_1/\chi_2\rightarrow 3$ as we increase the size of the blocks, $l$. The behavior of the ratio $\chi_1/\chi_2$ with $l$ is depicted in Fig. \ref{fig:coefficent_ratio}. Now using $|\phi^{l}_{\pm}\rangle=\frac{1}{\sqrt{2}} (|\phi^{l}_{00}\rangle\pm|\phi^{l}_{11}\rangle)$ we can further decompose $|\psi\rangle_{G}$, obtaining
\begin{eqnarray}
|\psi\rangle_{G}&=&\frac{\chi_1 }{2}( |\phi^{l}_{00}\rangle+|\phi^{l}_{11}\rangle) ( |\phi^{l}_{00}\rangle+|\phi^{l}_{11}\rangle)\nonumber\\&+& \frac{\chi_2}{2}( |\phi^{l}_{00}\rangle-|\phi^{l}_{11}\rangle) ( |\phi^{l}_{00}\rangle-|\phi^{l}_{11}\rangle)\nonumber\\ &+&\chi_2(|\phi^{l}_{01}\rangle |\phi^{l}_{10}\rangle+ |\phi\rangle^{l}_{10} |\phi\rangle^{l}_{01}),\nonumber\\
&=& \frac{(\chi_1+\chi_2)}{2}(|\phi^{l}_{00}\rangle |\phi^{l}_{00}\rangle + |\phi^{l}_{11}\rangle |\phi^{l}_{11}\rangle)\nonumber\\&+&\frac{(\chi_1-\chi_2)}{2} (|\phi^{l}_{00}\rangle |\phi^{l}_{11}\rangle + |\phi^{l}_{11}\rangle |\phi^{l}_{00}\rangle)\nonumber\\&+&\chi_2(|\phi^{l}_{01}\rangle |\phi^{l}_{10}\rangle+ |\phi^{l}_{10}\rangle |\phi^{l}_{01}\rangle).
\end{eqnarray}
Using $\chi_1/\chi_2=3$, we get
\begin{eqnarray}
|\psi\rangle_{G}&=& 2\chi_2 (|\phi^{l}_{00}\rangle |\phi^{l}_{00}\rangle + |\phi^{l}_{11}\rangle |\phi^{l}_{11}\rangle)+\chi_2 (|\phi^{l}_{00}\rangle |\phi^{l}_{11}\rangle \nonumber\\&+&|\phi^{l}_{11}\rangle |\phi^{l}_{00}\rangle)+\chi_2 (|\phi^{l}_{01}\rangle |\phi^{l}_{10}\rangle+ |\phi^{l}_{10}\rangle |\phi^{l}_{01}\rangle).
\end{eqnarray}

Let us perform the basis transformation given in Eq. (\ref{eqn:Basis_transformation}), and rewrite the state $|\psi\rangle_{G}$ in the transformed basis as
\begin{eqnarray}
|\psi\rangle_{G}&=& 2\chi_2 \Big(|\tilde{\phi}^{l}_{01}\rangle |\tilde{\phi}^{l}_{01}\rangle + |\tilde{\phi}\rangle^{l}_{10}|\tilde{\phi}^{l}_{10}\rangle\Big)\nonumber\\&-&\chi_2\Big(|\tilde{\phi}^{l}_{01}\rangle |\tilde{\phi}^{l}_{10}\rangle +|\tilde{\phi}^{l}_{10}\rangle |\tilde{\phi}^{l}_{01}\rangle\Big)\nonumber\\&-&\chi_2\Big(|\tilde{\phi}^{l}_{00} \rangle|\tilde{\phi}^{l}_{11}\rangle+ |\tilde{\phi}^{l}_{11}\rangle  |\tilde{\phi}^{l}_{00}\rangle\Big).
\end{eqnarray}
Now from normalization, we get $\chi_2=\frac{1}{2\sqrt{3}}$, which finally yields

\begin{eqnarray}
|\psi\rangle_{G}&=& \frac{1}{\sqrt{3}} \Big(|\tilde{\phi}^{l}_{01}\rangle |\tilde{\phi}^{l}_{01}\rangle + |\tilde{\phi}\rangle^{l}_{10}|\tilde{\phi}^{l}_{10}\rangle\Big)\nonumber\\&-&\frac{1}{2\sqrt{3}} \Big(|\tilde{\phi}^{l}_{01}\rangle |\tilde{\phi}^{l}_{10}\rangle +|\tilde{\phi}^{l}_{10}\rangle |\tilde{\phi}^{l}_{01}\rangle\Big)\nonumber\\&-&\frac{1}{2\sqrt{3}} \Big(|\tilde{\phi}^{l}_{00} \rangle|\tilde{\phi}^{l}_{11}\rangle+ |\tilde{\phi}^{l}_{11}\rangle  |\tilde{\phi}^{l}_{00}\rangle\Big).
\end{eqnarray}
The above expression for $|\psi\rangle_G$ exactly matches the GS of a four site periodic spin-1/2 Heisenberg Hamiltonian, which in computational basis reads as

\begin{eqnarray}
|\psi\rangle_G^{Heisn}&=& \frac{1}{\sqrt{3}} \Big(|0101\rangle +|1010\rangle\Big)-\frac{1}{2\sqrt{3}}\Big(|0110\rangle+|1001\rangle\Big)\nonumber\\&-&\frac{1}{2\sqrt{3}} \Big(|0011\rangle+|1100\rangle\Big).
\end{eqnarray}
Therefore, from the above analysis, we conjecture that for any spin-1 quantum many-body state with an SPT order the entanglement Hamiltonian derived from its   non-contiguous partitions of equal size and with even number of sites is a  critical spin-1/2  antiferromagnetic Heisenberg Hamiltonian. If the model has $SU(2)$ symmetry  the entanglement Hamiltonian will belong to the
universality class described by the WZW model $SU(2)_1$. SPT phases  in 1 + 1 dimensions have been analyzed using boundary CFT (BCFT) in references \cite{ent_ham_conformal3} and  \cite{SPT_CFT}.  In this approach the entanglement spectrum corresponds to a bipartition of the system, while in our case the connection between SPT phases and CFT arises from non-contiguous ones.  One should expect  a relationship between these two approaches.

Note that one can similarly consider another limit of the parameters $l_{A_1}, l_{A_2}, l_{B_1}, l_{B_2}$, such that $l_{A_1}=l_{A_2}=\ell_A$, $l_{B_1}=l_B$ and $l_{B_2} \rightarrow \infty$. In this limit, all the characteristic features remain as  above. The eigenspectrum for the $A_1A_2: B_1B_2$ bipartition is derived in Appendix \ref{AppendixA}.

\section{Bulk-edge correspondence}
\label{Sec3}

The characterization of the entanglement Hamiltonians in the previous section hints towards its underlying connection to the physical Hamiltonian of the system, which can be unveiled through the bulk-edge correspondence that we will discuss in detail in this section. In order to unveil the bulk-edge correspondence in our case, we first consider the entanglement Hamiltonian derived from a contiguous bipartition of the periodic AKLT state and compare its low-energy behavior to that of the AKLT model with open boundary conditions, which we call the {\em physical Hamiltonian}. For an open AKLT model, the GS is four-fold degenerate consisting of a spin-triplet and a singlet. This GS degeneracy is exact for all system sizes.
\begin{figure}[h]
\includegraphics[width=4cm]{Fig5}
\caption{Schematic of the edge degrees of freedom that appear at the end of the reduced block once we perform  the tracing operation in a periodic chain.}
\label{fig:schematic_second}
\end{figure}
The GS configuration, in this case, can be well-explained in terms of the edge-degrees of freedom appearing at the end of the open chain \cite{AKLT2}. Loosely speaking, for a finite length open AKLT chain, the spin-1/2 degrees of freedoms at the edges combine to form a singlet and a triplet configuration, yielding a unique GS in the thermodynamic limit.
Interestingly, from the derivations of the entanglement Hamiltonian as presented in Sec. \ref{Contiguous bipartition}, we observe that the low-energy part of the entanglement Hamiltonian coincides with that of the physical Hamiltonian. The low-energy subspace, in this case, is also spanned by a singlet and a triplet formed between the auxiliary spin-1/2 particles at the boundaries of the blocks (see Fig. \ref{fig:schematic_second} for a schematic representation). This justifies the emergence of the bulk-edge correspondence in this scenario. However, unlike the physical Hamiltonian, in this case, the degeneracy is not exact and the entanglement gap ($\Delta_{ent}$) closes only at moderately large value of   subsystem length $l$  (from Eq. (\ref{eqn:eigen_ent_Ham}) we can see that $\Delta_{ent}$  scales as $\Delta_{ent}\propto\gamma^l$).

We next move one step further and check whether the bulk-edge correspondence persists as we move away from the AKLT point. Towards this aim, we consider the bilinear-biquadratic Heisenberg (BBH) Hamiltonian, expressed in Eq. (\ref{eqn:BBH}). In our work, we mainly focus our study in the region $-\frac{\pi}{4}<\theta<\frac{\pi}{4}$, i.e., the Haldane phase of the model  \cite{SOP}.
The physical Hamiltonian, in this case, is the open BBH Hamiltonian for the region $-\frac{\pi}{4}<\theta<\theta_{AKLT}$, whose GS is always either a singlet for an even number of sites, and a triplet otherwise \cite{AKLT2}. In our case, without loss of generality, we always choose the size of the physical Hamiltonian ($N_O$)  to be even.

\begin{figure}[h]
\includegraphics[width=7cm]{Fig6}
\caption{Fidelity between the GS of the physical open BBH Hamiltonian and the GS of the entanglement Hamiltonian for the contiguous bipartition of the ground state of the periodic BBH Hamiltonian. Here,  the system size for the physical Hamiltonian is $N_O=4$ and entanglement Hamiltonian of size  $l=4$ is  obtained from a periodic chain with $N_P=8$.}
\label{fig:fidelity}
\end{figure}

To establish the bulk-edge correspondence at other points in the Haldane phase, we compute the reduced density matrix of the GS of the periodic BBH Hamiltonian in the region $-\frac{\pi}{4}<\theta<\theta_{AKLT}$.   We denote the size of the PBC Hamiltonian by $N_P$. Unfortunately, an analytical derivation of the reduced density matrix is not available away from the AKLT point. As a result, we first compute the ground state of the model by employing the density matrix renormalization group (DMRG) \cite{DMRG1,DMRG2}.  Here we use the finite-size algorithm with an adaptable number of kept states and a fixed tolerance on the discarded weights of the density matrix.  The entanglement Hamiltonian ($\mathcal{H}_E$) is then constructed from the reduced density matrix obtained from the contiguous partition of the ground state and its eigenenergies are derived from the eigenvalues of the reduced density matrix $\lambda_i$, $e^{ent}_i=-\log(\lambda_{m-i})$, where $i\in 0,1,2,\dots, m, m\approx 100$  and the $\lambda_i$ are arranged in ascending order. The low-energy part of the entanglement Hamiltonian is then analyzed and compared to that of the physical Hamiltonian of the same size ($N_O=l=N_P/2$).  We now discuss the results of such comparison in detail.

We find that the GS of the entanglement Hamiltonian for even $l$ is a singlet in
the considered Haldane region ($-\frac{\pi}{4}<\theta<\theta_{AKLT}$),
similarly to the physical Hamiltonian. The fidelity between the GS of the physical Hamiltonian $|\psi_0^{phy}\rangle$ and the entanglement Hamiltonian $|\psi_0^{ent}\rangle$, $f=|\langle \psi^{phy}_0|\psi_0^{ent}\rangle|^2$ in the region of interest is depicted in Fig. \ref{fig:fidelity}. From the plot we can see that the fidelity remains high throughout the considered region and decreases monotonously as $\theta$ moves away from the AKLT point. This provides a strong justification of the proposed bulk-edge correspondence: the high overlap of both GS points at the fact that the low-energy part of both Hamiltonians is structurally equivalent.\\

\begin{figure}[h]
\includegraphics[width=8.5cm]{Fig7}
\caption{(a)  Physical gap ($\Delta_{phy}$) and (b) entanglement gap ($\Delta_{ent}$) as a function of $\theta$ for various system sizes $N_O=l=N_P/2=10,12,14,16$. To compute $\Delta_{ent}$ we consider the length of the periodic BBH Hamiltonian ($N_P$) is double to that of the physical Hamiltonian ($N_O$).}
\label{fig:phy_gap_ent_gap}
\end{figure}

We next establish the bulk-edge correspondence through the identical behavior of the energy gaps obtained for both Hamiltonians. Away from the AKLT point the exact degeneracy of the GS of the physical Hamiltonian $H_{BBH}$ breaks, giving rise to an energy gap which decays exponentially with the system size, $\Delta_{phy} =a_0(\theta,N_O) \exp\(-N_O/\xi_{phy}(\theta,N_O)\)$ \cite{AKLT2}, where a correlation length is introduced, $\xi_{phy}$, with an explicit dependence on $\theta$ and $N_O$. In case of the entanglement Hamiltonian, the gap is defined as $\Delta_{ent}=e^{ent}_1-e^{ent}_0=\log(\lambda_{m})-\log(\lambda_{m-1})$.
We plot $\Delta_{ent}$ as a function of $\theta$ for different sizes $l$  and compare it with $\Delta_{phy}$ in Fig. \ref{fig:phy_gap_ent_gap}. One can note that both energy gaps remain very close to each other, hinting towards a similar decay factor. To confirm such behavior, we have performed a finite size scaling analysis of $\Delta_{ent}$, and observed that the entanglement gap scales as $\Delta_{ent}\propto \exp\(-l/\xi_{ent}(\theta, l)\)$.

\begin{figure}[t]
\includegraphics[width=8.5cm]{Fig8}
\caption{Physical gap ($\Delta_{phy}$) and entanglement gap ($\Delta_{ent}$) as a function of the system size ($N_O=l=N_P/2$)    for different regions of the Haldane phase (a) $\theta=0.18$, (b) $\theta=0.20$, (c) $\theta=0.24$, and (d) $\theta=0.26$. Notice that the vertical axis is scaled logarithmically.}
\label{fig:phy_gap_ent_gap_scaling}
\end{figure}

The scaling behavior of the energy gaps of both Hamiltonians with the system size  for a few points in the Haldane phase is depicted in Fig.  \ref{fig:phy_gap_ent_gap_scaling}, where panels (a)-(d) are obtained for four different values of $\theta$, eg., (a) $\theta=0.18$, (b) $\theta=0.20$, (c) $\theta=0.24$, (d) $\theta=0.26$, and the $y$-axis in all the plots is in log-scale. We observe that as $\theta$ approaches the AKLT point, the plots become more parallel to each other.
\begin{table}[h]
\centering
 \begin{tabular}{||c ||c|| c||}
 \hline
 \thead{$\theta$} & \thead{$\xi_{phy}(\theta, \infty)$} & $\xi_{ent}(\theta, \infty)$\\ [0.5ex]
 \hline\hline
 0.00 & 5.232&5.138 \\
   \hline \hline
0.06 & 4.549  &4.233 \\
  \hline \hline
0.10 & 4.070&  3.668\\
\hline \hline
 0.16&  3.323 & 2.908   \\
  \hline \hline
0.20 & 2.812  &2.445  \\
  \hline \hline
 0.24&2.293 & 2.006\\
  \hline \hline
   0.26&  2.028& 1.788 \\
   \hline \hline
 0.28&1.785 & 1.564 \\
 \hline \hline
 0.30& 1.511 & 1.320\\
 \hline \hline
 0.31& 1.369& 1.173\\
[1ex]
 \hline
 \end{tabular}
\caption{Correlation lengths $\xi_{phy}(\theta,\infty)$ and $\xi_{ent}(\theta,\infty)$ for increasing values of $\theta$, obtained through the fitting of  the corresponding  energy gaps.}
  \label{Table1}
\end{table}
This can be confirmed from the behavior of the quantities $\xi_{phy}(\theta,\infty)$ and $\xi_{ent}(\theta,\infty)$, which correspond to the correlation lenghts in the thermodynamic limit, obtained from the  fitting of the plots, as shown in Table \ref{Table1}. From the table, one can see that $\xi_{phy}(\theta, \infty)$ and $\xi_{ent}(\theta,\infty)$ converge to  almost the same value, $\frac{1}{\log(3)}=0.9102$ as we approach the AKLT point \cite{AKLT2}.  Hence, at the AKLT point though the system has an exact degeneracy in the physical spectrum, the correlation length remains non-zero, $\xi_{phy}(\theta_{AKLT}, \infty)=0.9102$. It is the coefficient $a_0$ in $\Delta_{phy}$  that vanishes at that point.

We have found that the bulk-edge correspondence  also manifests itself in the properties of the low-energy excited states. For instance, we note that the degeneracies of the five lowest eigenenergies for the physical Hamiltonian ($e_0^{phy}, \dots, e_4^{phy}$) and the entanglement Hamiltonian ($e_0^{ent}, \dots,  e_4^{ent}$) coincide: $(1)$, $(3)$, $(5)$, $(3)$, $(3)$. This, in other words, certifies that  not only the ground states but the lowest  15 $(1+3+5+3+3=15)$ eigenstates of both the Hamiltonians  are also indeed qualitatively similar. Let us recall that the first two degeneracy factors, $(1)$ and $(3)$, correspond to the GS   singlet and triplet configurations respectively, which we have discussed above in detail. We may interpret the remaining multiplets ($(5)$, $(3)$ and $(3)$) of the physical Hamiltonian as follows. The periodic physical Hamiltonian has a triplet excitation above the singlet. Thus, one could expect that for an open physical chain there this triplet will combine with the edge spins. These edge spins will be combined as a singlet and a triplet: 1/2 $\otimes$ 1/2=0 $\oplus$ 1. Further combination with the excited triplet state finally yields  (0 $\oplus$ 1) $\otimes$ 1 = 0 $\oplus$ 1 $\oplus$  1 $\oplus$ 2. Hence, we can relate the degeneracy factors with their total spin as $(d_f, s)\to (5,2), (3,1), (3,1), (1,0)$. Note that the singlet $(1, 0)$ arising in the above decomposition remains just above these multiplets ($(5)$, $(3)$ and $(3)$) of physical spectrum  and does not appear in the  entanglement spectrum.
\begin{figure}[h]
\includegraphics[width=8.5cm]{Fig9}
\caption{The five lowest energy eigenvalues of (a) the physical Hamiltonian ($e_0^{phy}, \dots, e_4^{phy}$) and (b) the entanglement Hamiltonian ($e_0^{ent}, \dots,  e_4^{ent}$) as a function of $\theta$. This certifies  a strong correspondence exists even up to 15 lowest eigenstates of both the Hamiltonians. Here, we consider $N_O=l=N_P/2=8$.}
\label{fig:eign_spectrum}
\end{figure}
Hence, starting from there, the degeneracy factors  of both  Hamiltonians differ.
It can be observed that not only the degeneracies coincide: the behavior of the set of eigenvalues as a function of $\theta$ remains qualitatively similar, as presented in Fig. \ref{fig:eign_spectrum}. Indeed, as we approach the AKLT point we observe how the eigenenergies of both Hamiltonians become degenerate following two different paths. Thus, the ground state ($e_0$) and first excited state ($e_1$) converge together whereas the next three excited energies ($e_2$, $e_3$ and $e_4$) follow a different route and finally converge as we approach the AKLT point. However, it is interesting to note that for the entanglement Hamiltonian, as we approach the AKLT point, the different paths are different from those of the entanglement Hamiltonian. This observation can be given a renormalization group (RG) interpretation, where the AKLT state represents the fixed point of an RG obtained by  keeping only the most relevant degrees of freedoms. As a result,  the angle $\theta$ in  Fig. \ref{fig:eign_spectrum} (b) can be identified with the RG direction since only the lowest eigenstates survives as $\theta \rightarrow \theta_{AKLT}$.

\begin{table}[h!]
\centering
\begin{adjustbox}{width=0.49\textwidth}
 \begin{tabular}{||c ||c|| c ||}
 \hline
 \thead{Low energy \\ properties} & \thead{$\mathcal{H}_E(l)$} & \thead{$H_{BBH}(N_O)$}\\ [0.5ex]
 \hline\hline
  \thead{Ground state configuration}  &\makecell{singlet (even $l$), \\ triplet (odd $l$)} &  \makecell{singlet (even $N_O$), \\ triplet (odd $N_O$)}\\
  \hline \hline
\thead{Scaling of energy gap}  & \makecell{$\Delta_{ent}\propto \exp\(-l/\xi_ {ent}\)$}& \makecell{ $\Delta_{phy}\propto \exp\(-N_O/\xi_{phy}\)$}  \\
  \hline \hline
   \thead{Correlation length} & $\xi_{ent}(\theta\rightarrow\theta_{AKLT},l)\rightarrow \frac{1}{\log(3)}$ &  $\xi_{phy}(\theta\rightarrow\theta_{AKLT},N_O)\rightarrow \frac{1}{\log(3)}$\\
\hline \hline
  \thead{Degeneracy factor (.)}  & (1), (3), (5), (3), (3) & (1), (3), (5), (3), (3)\\
  [1ex]
 \hline
 \end{tabular}
 \end{adjustbox}
 \caption{ In this table, we summarize the correspondence between  the low-energy  properties of the  entanglement Hamiltonian ($\mathcal{H}_E$) and corresponding physical Hamiltonian $H_{BBH}$ of same length $N_O=l=N_P/2$.}
  \label{Table:BE_summary}
\end{table}

Therefore, our analysis supports the conjecture that a bulk-edge correspondence emerges between the entanglement Hamiltonian derived from a part of our quantum many-body state and its corresponding physical Hamiltonian.
We summarize all the key features of the proposed bulk-edge correspondence discussed above in Table \ref{Table:BE_summary}.

\section{String order parameter}
\label{sec:SOP}
Let us compare the behavior of the energy gaps obtained above with other  relevant physical quantities computed for the model. In particular, we may consider the string order parameter (SOP) \cite{SOP} of the GS of the physical Hamiltonian, defined as
\begin{eqnarray}
\mathcal{O}_{\theta}(l, N_O)=\langle  S^z_1 e^{-i\pi \sum_{k=2}^{l+1}  S^z_{k}} S^z_{l+2} \rangle.
\label{eqn:SOP}
\end{eqnarray}
The Haldane phase of the BBH Hamiltonian presents a hidden topological order which can not be detected using any conventional local order parameter. Yet, the non-local SOP takes non-zero values over the whole Haldane phase and acts as an order parameter distinguishing it from the other phases. We plot the behavior of SOP ($\mathcal{O}$) for our region of interest, $-\frac{\pi}{4}< \theta < \theta_{AKLT}$ and for different sizes of the open chain, in Fig. \ref{figure:SOP}. From the figure we note that the SOP ($\mathcal{O}$) attains its minimum value at the AKLT point and increases as it moves away from it, in similarity to the energy gaps. The finite-size scaling of the SOP presents some subtle points. We first consider its behavior at the AKLT point, for which an exact analytical form of $\mathcal{O}(N_O)$ can be obtained as follows.
Indeed, we may apply Eq. (\ref{eqn:SOP}) to any matrix product state using the tranfer matrix formalism, obtaining \cite{SOP_AKLT}
\begin{eqnarray}
\mathcal{O}_{\theta}(l, N_O)= \text{Tr}(E^{N_O-l-2} \widehat{S^z} ({e^{-i \pi \widehat{S^z}} )^l\widehat{S^z}),}
\label{eqn:SOP_MPS}
\end{eqnarray}

where $E$ is the transfer matrix defined earlier in Eq. (\ref{eqn:Transfer_matrix}), $\widehat{S^z}=\sum_{kk'} (A_k \otimes A^*_{k'}) \langle k|S^z|k'\rangle$, and $\tilde{E}={e^{-i \pi \widehat{S^z}}}=\sum_{kk'} (A_k \otimes A^*_{k'}) \langle k|e^{-i \pi S^z}|k'\rangle$. Notice that, in similarity to the transfer matrix, $\tilde{E}$ can be decomposed in its right and left eignevectors as follows, $\tilde{E}=\sum_{i=0}^{D^2-1} \tilde{\gamma}_s |\tilde{R}_s\rangle\langle\tilde{L}_s|$, with

\begin{figure}[t]
\includegraphics[width=7.5cm]{Fig10}
\caption{String order parameter ($\mathcal{O}$) as defined in Eq. (\ref{eqn:SOP}) as a function of $\theta$ for different sizes of the open BBH chain,   $N_O =10$, $12$, $14$, $16$. In all the cases $\mathcal{O}$ have been computed for the spins situated at site '1' and the end site  '$N_O$'. Hence, here '$l=N_O-2$'.}
\label{figure:SOP}
\end{figure}

\begin{align}
|\tilde{R}_0\rangle=|R_1\rangle, \;\; \langle \tilde{L}_0|=\langle L_1|, \;\;
|\tilde{R}_1\rangle=|R_0\rangle, \;\; \langle \tilde{L}_1|=\langle L_0|,\nonumber\\
|\tilde{R}_2\rangle=|R_2\rangle, \;\; \langle \tilde{L}_2|=\langle L_2|, \;\;
|\tilde{R}_3\rangle=|R_3\rangle, \;\; \langle \tilde{L}_3|=\langle L_3|,\nonumber\\
\end{align}
and $\tilde{\gamma}_k=\gamma_k$, for all $k$, where the set of eigenvectors  and eigenvalues $\{|R_k \rangle, | L_k\rangle, \gamma_k\}$ obtained for the AKLT state was defined in Eq. (\ref{eqn:transfer_matrix_vec}). Using this, we can further simplify Eq. (\ref{eqn:SOP_MPS}) as follows.
\begin{eqnarray}
\mathcal{O}_{\theta_{AKLT}}(l, N_O) &=&  \text{Tr} \(\[ \gamma_k^{N_O-l-2}|R_k\rangle \langle L_k|\]\widehat{S^z} \[\sum_{s=0}^{D^2-1}\tilde{\gamma}^l_s |\tilde{R}_s\rangle\langle \tilde{L}_s| \]\widehat{S^z}\), \nonumber\\
&=&\sum_{k=0}^{D^2-1}\( \gamma_k^{N_O-l-2} \langle L_k|\widehat{S^z} \[\sum_{s=0}^{D^2-1}\tilde{\gamma}^l_s |\tilde{R}_s\rangle\langle \tilde{L}_s|\]\widehat{S^z}|R_k\rangle\),\nonumber\\
  &=&\langle L_0|\widehat{S_z}|\tilde{R}_0\rangle\langle \tilde{L}_0|\widehat{S^z}|R_0\rangle\nonumber\\
  &+&\sum_{s=1}^{D^2-1} \tilde{\gamma}_s^{l}\langle L_0|\widehat{S^z}|\tilde{R}_s\rangle\langle \tilde{L}_s|\widehat{S^z}|R_0\rangle\nonumber \\
  &+&\sum_{k=1}^{D^2-1}  \gamma^{N_O-l-2}_k\langle L_k|\widehat{S^z}|\tilde{R}_0\rangle\langle \tilde{L}_0|\widehat{S^z}|R_k\rangle\nonumber\\
  &+&\sum_{k,s=1}^{D^2-1} \gamma_k^{N_O-l-2} \tilde{\gamma}^l_s \langle L_k|\widehat{S^z}|\tilde{R}_s\rangle\langle \tilde{L}_s|\widehat{S^z}|R_k\rangle.
\end{eqnarray}
Now using the action  of  $\widehat{S^z}$ on $|R_i\rangle, |L_i\rangle$ and $|\tilde{R}_i\rangle, |\tilde{L}_i\rangle$ (see Appendix \ref{AppendixB}) we can further show that
\begin{align}
\mathcal{O}_{\theta_{AKLT}}(l, N_O)=&-\(\frac{2}{3}\)^2 \( 1
  + \sum_{s=1}^{D^2-1} \tilde{\gamma}_s^{l} \delta_{s,0}
  + \sum_{k=1}^{D^2-1}\gamma_k^{N_O-l-2} \delta_{k,0} \right.\nonumber\\
  &  \left. + \gamma_1^{N_O-l-2} \tilde{\gamma}^l_1 \), \nonumber\\
=&-\(\frac{2}{3}\)^2-\(\frac{2}{3}\)^2  \(-\frac{1}{3}\)^{N_O-2}.\nonumber\\
   \end{align}
  Now as $\mathcal{O}_{\theta_{AKLT}}(l, N_O)$ does not depend on $l$, we can replace  $\mathcal{O}_{\theta_{AKLT}}(l, N_O)$ by  $\mathcal{O}_{\theta_{AKLT}}(N_O)$   and get
   \begin{align}
\mathcal{O}_{\theta_{AKLT}}(N_O)=&-\(\frac{2}{3}\)^2-\(\frac{2}{3}\)^2  \(-\frac{1}{3}\)^{N_O-2}, \nonumber\\
=&-\(\frac{2}{3}\)^2-\(\frac{2}{3}\)^2 \(-\frac{1}{3}\)^{-2}\exp\(-\frac{N_O}{\frac{1}{\log(3)}}\), \nonumber\\
=&\mathcal{O}_{\theta_{AKLT}}(\infty)+\mathcal{A}(\theta_{AKLT},\infty) \exp\(-\frac{N_O}{\xi_{\mathcal{O}}(\theta_{AKLT}, \infty)}\),
\label{eqn:SOP_fitting}
\end{align}
with $\mathcal{O}_{\theta_{AKLT}}(\infty)=-(\frac{2}{3})^2$, $\mathcal{A}(\theta_{AKLT},\infty)=-4$ and $\xi_{\mathcal{O}}(\theta_{AKLT}, \infty)=\frac{1}{\log(3)}=0.9102$. Hence, at the AKLT point, $\mathcal{O}_{\theta_{AKLT}}(N_O)$ approaches its thermodynamic limit $\mathcal{O}_{\theta_{AKLT}}(\infty)$ exponentially with the system size $N_O$. This behavior remains similar to the exponential decay of the energy gaps obtained above.  Interestingly, one can see that that the relation remains independent of the inter-spin distance $l$ and depends only on the total system size $N_O$.  We may conjecture is generic behavior of $\mathcal{O}_{\theta_{AKLT}}(l, N_O)$ still holds close to the $\theta_{AKLT}$, so Eq. (\ref{eqn:SOP_fitting}) can still be used to find $\xi_{\mathcal{O}}(\theta,\infty)$ for other points in the Haldane phase  lies in   the  neighborhood of  the AKLT point. Using this fact, we have obtained $\mathcal{O}_{\theta}(\infty)$, $\mathcal{A}(\theta,\infty)$ and $\xi_{\mathcal{O}}(\theta, \infty)$ for                    $\theta=0.28, 0.30, 0.31, 0.32$  in Table   \ref{Table3}. From the table, we can clearly see that as $\theta \rightarrow \theta_{AKLT}$, all the three-quantities  \big($\mathcal{O}_{\theta}(\infty)$, $\mathcal{A}(\theta,\infty)$, $\xi_{\mathcal{O}}(\theta, \infty)$\big) approach to their respective  values obtained at AKLT point.

\begin{table}[h]
\centering
 \begin{tabular}{||c ||c|| c||c||}
 \hline
 \thead{$\theta$} & \thead{$\mathcal{O}_{\theta}(\infty)$} & $\mathcal{A}(\theta,\infty)$& $\xi_{\mathcal{O}}(\theta, \infty)$\\ [0.5ex]
 \hline\hline
 0.28 & -0.438 &-1.27&1.106 \\
 \hline\hline
  0.30 & -0.443 &-1.39&1.062 \\
 \hline\hline
 0.31 & -0.444 &-1.57&1.027 \\
 \hline\hline
 0.32 & -0.444 &-2.19&0.944 \\
[1ex]
 \hline
 \end{tabular}
 \caption{Behavior of the coefficients $\mathcal{O}_{\theta}(\infty)$, $\mathcal{A}(\theta,\infty)$ and $\xi_{\mathcal{O}}(\theta, \infty)$, obtained using  Eq. (\ref{eqn:SOP_fitting}) for $\theta$ values close to the AKLT point.}
 \label{Table3}
 \end{table}

 The scaling of both energy gaps and the SOP allows us to argue that the characteristic scale of the BBH model remains close to the value  $\frac{1}{\log(3)}$ and  manifests in the scaling of relevant physical quantities associated to it.

\section{Conclusions and Further Work}
\label{Sec4}

In this work, we have investigated the properties of the entanglement Hamiltonian for contiguous and non-contiguous bipartitions of the AKLT state. In both cases, the entanglement Hamiltonian can be expressed as an interacting Hamiltonian acting between the auxiliary spins at the edges of the blocks. In particular, when the block sizes are equal  and comprises of even number of sites  the corresponding entanglement Hamiltonian becomes the antiferromagnetic  Heisenberg model, which is critical.  We conjecture that this behavior is generic for other SPT states.

We further observed that the low-energy properties of the entanglement Hamiltonian obtained for the contiguous bipartition of the AKLT state can be related to the edge properties of the physical Hamiltonian with open boundaries, hinting towards a bulk-edge correspondence that has been extensively studied in many earlier works  \cite{ent_spectrum_ref5,ent_ham_ref2,ent_ham_ref3,ent_ham_ref4,ent_ham_ref5,ent_ham_ref6}. To investigate this bulk-edge correspondence in our context in more detail, we also analyzed the properties of the entanglement Hamiltonian obtained for other states within the Haldane phase. By employing the DMRG technique, we obtained the GS of the periodic BBH Hamiltonian and derived the corresponding entanglement Hamiltonian from a contiguous subsystem. The low-energy properties are then analyzed and compared to that of the physical model of the same size.   As a first result, we showed that for the whole region considered, the GS of the physical and entanglement Hamiltonians remain structurally equivalent, which serves as even a stronger example of the existing bulk-edge correspondence. Additionally, similarly to the physical gap, we found that the entanglement gaps decays exponentially with the system size, with the associated correlation lengths obtained from a finite-size scaling analysis approaching the same value for the AKLT point. Along with this, we showed that the bulk-edge correspondence manifests itself the properties of a few excited states of both Hamiltonians, as we can see in the identical degeneracies for the first five low-energy eigenstates, and their behavior in the region of the Haldane phase we have considered in our work.

Finally, the behavior of the energy gaps is compared to that of the string order parameter. It is  shown that similar to the energy gaps, SOP decays exponentially towards its asymptotic as we approach the AKLT point. Moreover, the correlation length obtained at the AKLT point was found to be exactly $\frac{1}{\log(3)}$  and deviating slightly in its vicinity. This again matches the behavior obtained for the energy gaps and manifests the characteristic length scale associated with the physical model.  As a future work, we wish to extend our analysis for fermionic SPT phases and to deepen the relation
between the topological phases and CFT.

\section*{Acknowledgements}
This work has also been financed by the Spanish grants PGC2018-095862-B-C21, PGC2018-094763-B-I00, PID2019-105182GB-I00, QUITEMAD+ S2013/ICE-2801, SEV-2016-0597 of the ``Centro de Excelencia Severo Ochoa" Programme and the CSIC Research Platform on Quantum Technologies PTI-001.
\begin{widetext}

redpajama_set_name: RedPajamaArXiv


text:
Aleix Suñé
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CLEVELAND — The city has announced new locations for its Portable Camera Units.
The PCUs are moved periodically and their new positions are always made public.
There are 49 fixed camera sites and 15 portable sites in Cleveland.
Retired September 30,2013. What a gift that would be!!!!
dear Cleveland enough with the traffic trap bs. how about handing out jaywalking tickets on E55 instead or catching the muggers on euclid.
what a police state! i never remember getting a chance to vote on this!
I don't like tickets either…slow down and avoid them, like the rest of us do. It's not a police state…they are trying to save lives. I know someone who got hit and run on by a speeder and their life is absolute crud now. SLOW DOWN FOOLS.
Just started working in Cleveland, recd 2 tickets from the camera – thought speed limit was 35, not 25, not posted until you reach the camera, nice way to welcome people to your town, now you know why no one wants to come to Cleveland, 4 lane highways have 25MPH – operation safe Cleveland, — why would we want to go 25MPH to have anyone be able to walk up to our cars in unsafe areas – no cameras to catch the murderer of Aliza Sherman – I appreciate law enforcement, but camera's just to increase revenue should be illegal.
Get yourself a firearm, learn how to use it, and carry it with you! Do not depend on others for your personal safety.


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text:
Bert Lytell (24 de febrero de 1885 – 28 de septiembre de 1954) fue un actor teatral, cinematográfico y televisivo de nacionalidad estadounidense, cuya carrera inició en la época del cine mudo.

Biografía
Nacido en Nueva York, su nombre completo era Bertram Lytell. Como actor teatral, trabajó con Marie Dressler en 1914 en la obra  A Mix-up, representada en el circuito de Broadway. Actuó también en el género del vodevil en los años 1920 con un show en un acto, The Valiant.

Como actor cinematográfico, Lytell fue sobre todo activo en los años del cine mudo, participando en 49 producciones estrenadas entre 1917 y 1927. Solamente hizo cinco primeros papeles en el cine sonoro, el primero en la película de 1928 On Trial (de Archie Mayo, con Pauline Frederick), y el último en 1931 en The Single Sin (de William Nigh, con Kay Johnson).

Entre las actrices con las que trabajó figuran Betty Compson, Alice Lake, Barbara La Marr, May McAvoy, Irene Rich, Rosemary Theby, Virginia Valli, Lois Wilson y Claire Windsor, siendo esta última una de sus esposas, estando casado con ella desde 1924 a 1927.

Sin embargo, y como muchas otras estrellas del cine mudo, la carrera de Lytell se fue abajo tras la llegada del cine sonoro.  Tras retirarse del cine, él prosiguió como actor teatral. Actuó en Broadway hasta el año 1946, trabajando con actores de la talla de Kirk Douglas, Ruth Gordon, Otto Preminger y Margaret Sullavan. En 1941 formó parte de la comedia musical Lady in the Dark, pieza protagonizada por Gertrude Lawrence.

En sus últimos años Lytell fue también actor televisivo, medio para el cual trabajó entre los años 1948 y 1953, interviniendo en seis series televisivas, entre ellas Tales of Tomorrow.

Lytell fue presidente del famoso club de actores The Lambs entre 1947 y 1952, siendo considerado como un Immortal Lamb. Por su contribución a la industria del cine, se le concedió una estrella en el Paseo de la Fama de Hollywood, en el 6417 de Hollywood Boulevard.

Bert Lytell falleció en 1954 en la ciudad de Nueva York, a causa de complicaciones surgidas tras ser sometido a cirugía. Sus restos fueron incinerados. Su hermano menor era el actor Wilfred Lytell (1891–1954).

Teatro

Selección de su filmografía

Cine

Televisión

Galería fotográfica

Referencias

Enlaces externos

Bert Lytell en Virtual History

Actores de cine de Estados Unidos
Actores de cine mudo
Actores de teatro de Estados Unidos
Actores de televisión de Estados Unidos
Personas de Nueva York (estado)
Paseo de la Fama de Hollywood

redpajama_set_name: RedPajamaWikipedia


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redpajama_set_name: RedPajamaGithub


1272 Bassett Ave is a $457,500, 2,271 square foot, 3 bedroom, 3.0 bath home located in Louisville, KY.
This home is located on a quiet street in historic preservation area of Cherokee Triangle, near Cherokee Park entrance and within 1.5 blocks of Willow Park's playground and the popular gazebo. 1272 Bassett Avenue features 3 bedrooms, 2.5 baths, and a welcoming front porch. The front door offers lovely stained glass. The gracious foyer, the great room, and the dining room offer original wood flooring in great condition. Lots of fresh paint adds to the light and brightness of the house. A garden room opens to the remodeled kitchen and also opens to outside to the Trex Deck and beautiful landscaping. The remodeled kitchen has quartz counter tops, wood cabinets, and pull out cabinet drawers next to the stove and the lower pantry.
I am interested in 1272 Bassett Ave, Louisville, KY 40204.


On Monday, Apple officially announced that it would throw an event on 25th March. The company offered a teaser: It's show time. The invitation does not have much to look into the message, but it has the animated GIF. A graphical invitation starts with a counter like 4… 3… 2…, like a traditional short film. As per the Apple invitation, the event will take place in Cupertino, California, at the Steve jobs Theater. But it is not the first time the tech giant has used this tagline. The company used the same slogan in the event that took place in September 2006. At the time, the company announced that it would start showing movies via iTunes and launched iTV also.
During the event starting at 10:00 a.m. (PT), the company may launch its 'Netflix-style' subscription news and video services. In February, The Wall Street Journal reported that Apple is planning a revamped News product. It might include a cluster of subscriptions from significant news outlets that too under one roof. So Apple may launch a new streaming TV service. The company may offer free access to its users for watching original series. It may facilitate users to sign up for other streaming services like Starz and Showtime. Rumors reveal the streaming service will hit the market in April or May. There are fair chances that users could see the expected declarations of remodeled AirPods.
On the other hand, Apple may launch some new hardware. Before some months, the company updated its iPad Pro. But there are no updates in the classic, low-cost $329 iPad since last one year. Thus, Apple may announce a new iPad and replace the standard one. While there is a probability, Apple will update its iPads or Apple TV devices later. Ben Bajarin, an analyst with Creative Strategies, said the event would probably be Apple's first event with major media participation. Ben added hardware is not the primary focus of the company. Let's wait for the day, and see what Apple will cater to the event.


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Red square. Panoramic web camera online
The red Square covers an area of over 20,000 m2, and is the main squareMoscowandRussia. It is located near the Moscow Kremlin, adhering to its North-Eastern part. This is one of the favorite places for walks and excursions, numerous tourists, and the residents themselves. Concerts are held here and the Victory parades, as well as folk festivals and major city events. And in the cold season, the Red square turns into a fabulous ice rink in the center of which stands the main Christmas tree of the country. The history of the area begins in the late XV century, when on the waste land left after a huge fire outside the Kremlin walls, began to experience spontaneous markets. After the construction of the Pokrovsky Cathedral and the Spasskaya tower in the XVI century, market square received its current name. Today, Red square is a historical and cultural complex, has a huge amount of valuable attractions. One of the most striking religious monuments of red Square is St. Basil's Cathedral, or St. Basil's Cathedral. It is the center of the architectural ensemble of red square, the most valuable monument of history, recorded in the list of world heritage of UNESCO. The Cathedral was built on the site of Holy Trinity Church in honor of the capture of Kazan by the troops of Ivan the terrible. This event happened in celebration of the Cover. The Church St. Basil was added to the main structure later. Now the temple is a branch of the state Historical Museum of Moscow, but sometimes there are services. In front of the temple is a monument to K. Minin and D. Pozharsky was inaugurated in 1818. Another monument, dedicated to the expulsion of the Polish-Lithuanian invaders, is the Cathedral of the Kazan icon of the Mother of God, built in 1625, and in spite of numerous destruction and fires, which has kept its primary appearance up to our days. Speaking about the Historical Museum, it should be noted that it is one of the landmarks of the city. After all, a Museum exhibition located in a beautiful building, built in 1881 by V. O. Sherwood. In the years of Soviet power, the building's appearance has undergone many changes, but our century has undergone a series of reconstructions of the original appearance of this architectural monument. Another popular red square is the Moscow state Department store, which is open since 1893 and has kept its historic name until the present day, despite the fact that has long been private property. Another important object in the architectural look of red square is the Spasskaya tower with the huge clock – a symbol of the beginning of the New year. And, of course, the most popular and unique attraction is the Lenin mausoleum adjacent to the Kremlin walls. Earlier, at the entrance to the mausoleum was located the post # 1, replaced every hour. Which was later moved to the Alexander garden, to the memorial complex of the Tomb of the Unknown soldier. Among other important sights of red square, it should be noted the place of the skull once served to the public and important public event and the resurrection gate.
Web camera in real time shows a Sand alley in Sokolniki Park. Sand Avenue passes through the territory of the Park "Sokolniki" the Sokolnichesky of the passage Round to the West to Mitki travel.


Dans la futur, écrire un texte pour présenter mes idées vas être très important. Tu dois écrire des grands paragraphes dans l'université et si tu utilise tes propres idées, tu peux expliquer plus comment tu pense et impressionner les professeurs avec voir le sujet par un autre perspective. Je veux être un avocat alors écrire les papiers et les débattre va être un grand parti de mon vie professionnel. Tu peux aussi utilisé le débat dans les conversations avec les amis et ta famille pour montrer tes idées.
Si les robots prenaient contrôle de l'ecole, ils remplaceraient les professeurs. Ils changeraient le style de l'ecole avec leur technologie. Les étudiants auraient peur de les robots et feront tous leurs devoirs.
I used a mix of graphs and picture models to show what alcohol has done to people and how this substance can ruin peoples lives.
I used this way to show my data because its informative with visuals and conveys the message without much reading. Children and teens will notice it because there are pictures which show very clearly what happens to people who drink alcohol.
Drinking alcohol is like ingesting poison. It damages your liver and takes away your self control. People all over the world get addicted to this poison and some drink so much that it affects the way they think. This could result in car accidents, harming someone or yourself unintentionally and getting into fights. Lots of these outcomes could result in death of yourself or someone around you due to your unclear thinking. Alcohol damages your liver and causes people to get liver disease which could lead to death. This disease has affected many people and has shortened their lives due to this avoidable disease.
I believe that all my research has made me think about everyday life in a different perspective and wonder how we could use speed to find a way to travel in time or to speed up the way we do things on earth and maybe even travel to different universes. My inquiry question has made me think of other questions so I have lots of other inquiry questions and other things that my mind can wonder about.


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These are links and contact information of arganisations who have been, and are, fighting against honor killings and other forms of violence against women in Pakistan.
Shirkat Gah is committed to a just, progressive and democratic society where women are fully empowered, gender equality, human security and opportunities are ensured for all, peace prevails and resources are shared on a sustainable and equitable basis. It strives to ensure a responsible and sovereign state in a world order and equity between national and international interests.
www.shirkatgah.org
Women Action Forum (WAF) does active lobbying, advocacy on behalf of women in Pakistan. it holds demonstrations and public-awareness campaigns. It is committed to a just and peaceful society based on democracy. The issues picked up by WAF have included challenging discriminatory legislation against women, the invisibility of women in government plans and policies, the exclusion of women from media, sports and cultural activities, dress codes for women, violence against women and the seclusion of women. WAF's activism has led to the birth of many women's rights groups and resource centres thereby increasing its outreach. WAF considers all issues as "women's issues" and has taken positions on national and global developments. It allies itself with democratic and progressive forces in the country as well as linking its struggle with that of minorities and other oppressed peoples.
Human Rights Commission of Pakistan (HRCP) is an independent, voluntary, non-profit organisation which is not associated or affiliated with the government or any political party. It is committed to act with impartiality and objectivity in all matters. Among its main aims are spreading awareness of human rights among the people, mobilising public opinion, collecting information and disseminating knowledge about human rights abuses.
Human Rights Commission of Pakistan
Women Living Under Muslim Laws (WLUML) is an international solidarity network that provides information, support and a collective space for women whose lives are shaped, conditioned or governed by laws and customs said to derive from Islam.
The WLUML has in the past decades, has informed the women's movement in Pakistan to fight all forms of violence against women including the grave issue of 'honor' killings.
wluml.org


We work directly with the Funeral Home or Mortuary. Just let them know that you would like to have the funeral at: Our Lady of Perpetual Help in Daly City, and we'll take care of the rest.
Please contact the Parish Office, If you or a family member cannot attend Mass and would like to have a Homebound visit to your home. This service is only limited within the Parish Borders and/or Zone Boundaries.
A Memorial Mass should be arranged with a Priest.
To schedule a Memorial Mass, please call the Parish Office.
To save yourself a trip to the Parish Office, we have Mass Intention envelopes located at the vestibule of our church. You may place your envelopes in the Collection basket and they'll be processed on Monday. Be sure to include your number and phone number If we have any questions.
*Please note that we will not accept any telephone requests for Mass intentions.
To schedule an Anniversary Mass, please call the Parish Office.
If you would like to request a copy of your Baptismal or Confirmation Certificate. You may do so by calling our Parish Office or sending us an email.
*Please note: It may take some time to process as we have to research the records on file.