Join us for an evening of great food and drink, conversation and a presentation by Russell Dunn and Barbara Delaney, the featured speakers at this year's CMC dinner. Russ and Barb are the authors of a number of hiking guides for the Catskills, the Adirondacks, the Hudson River Valley, and the Taconics and Berkshires, all published since they visited Kaaterskill Falls for their first date. Notably for us, Russ wrote the popular Catskill Region Waterfall Guide and the recently published Rambles to Remarkable Rocks, which will be featured in the evening's presentation. They have presented talks throughout the Capital and Hudson River Valley regions and we are thrilled to have them join us to share their recent discoveries.


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text:
\section{Introduction}
Numerical simulations stand as a fundamental tool to analyze strongly
correlated quantum systems. Basically, two different approaches have
proven very powerful. The first one corresponds to Monte Carlo simulation,
which can be applied to most systems with the exception of those
that present a sign problem. A second approach
is based on introducing a structured ansatz for {\sl e.g.} the ground state of the
system that captures as much of its entanglement properties as possible.
The kind of structure which is best suited for a given system will depend
on its dimensionality, symmetry properties and geometry of interactions.

A widely used structured ansatz to handle one-dimensional systems corresponds to
Matrix Product States (MPS) \cite{Ostlund:1995,Perez-Garcia:2007}. The basic idea of this approach
consists of representing states via their correlations rather than their
coefficients in  position or momentum space. Quantum states that carry little
entanglement are well described by MPS \cite{VLRK,Vidal:MPS}. This is the case of one-dimensional
off-critical states. There, MPS produces a very precise approximation to all
the properties of the state. The MPS approach deteriorates at criticality since
conformal invariance is recovered and quantum correlations pervade the system.
Yet, MPS remains a useful tool in that regime. In higher dimensions, MPS must be
substituted with the so-called Projected Entangled Pairs (PEPs) \cite{Cirac:PEPS}.
A conceptually different structured ansatz corresponds to tensor networks
that do not encode the standard nearest neighbor topology but, rather,
a renormalization group tree. Tree tensors and, in particular, the Multiscale Entanglement Renormalization
Ansatz (MERA) \cite{Vidal:MERA} may well turn into a powerful method to deal with critical quantum systems
in any number of dimensions . In this paper, we shall concentrate on
finding exact results for the MPS approximation.

The issue of finding an optimal MPS representation for the ground state
of an infinite quantum system is not trivial.
A number of results can be found in the literature \cite{MPSexact} (and references therein)
concerning exact MPS in particular cases,  exploiting rotational
or translational invariance.
It is also possible, for instance,
to rewrite the Laughlin wave function coefficients as a product of matrices
\cite{ILO:2006}. But, in general, exact results are excepcional and
it is necessary to resort to
a specific numerical algorithm that delivers such a state. In the case
of infinite systems, two possibilities are
at hand: the Density Matrix Renormalization Group (DMRG) algorithm
\cite{White:1992} and the Euclidean evolution (iTEBD) algorithm \cite{Vidal:iTEBD}. Both methods have been proven to produce the same results on examples \cite{TOIL:2008}. Nevertheless, it remains
an open question whether both algorithms do find the optimal MPS approximation
to the ground state of a given Hamiltonian. It is also unclear how the MPS approximation
handles the symmetries present in a Hamiltonian.
We shall try to answer both of this questions in this paper.

Our approach will exploit the iTEBD approach in an analytical way and turn the
problem of finding optimal MPS to a minimization strategy.
Thus, the novelty of the method lies in the exactness of this minimization procedure
that allows for an explicit analysis of symmetries in the MPS context.
We shall present our method and results for the particular case
of the isotropic spin-$1/2$ anti-ferromagnetic Heisenberg chain.
This thoroughly studied model is suited for our purposes since the
MPS approximation will make a compromise  to handle its symmetries,
as we shall see later on.

We divide our paper as follows. First, in Sect. \ref{Sect:Heisenberg},
we recall some basic exact results for the Heisenberg spin chain that we
use as a testing ground for the MPS approach. Sect. \ref{Sect:iTEBD} is devoted
to present the elements of MPS states and the iTEBD algorithm.
Then, in Sect. \ref{Sect:exact}, we transform the iTEBD approach into
a minimization strategy and find exact solutions for MPS states.
The analysis on symmetry restoration is discussed
in Sect. \ref{Sect:symmetry}.

\section{Heisenberg spin chain}
\label{Sect:Heisenberg}

The isotropic infinite spin-$1/2$ Heisenberg quantum chain is described
by the Hamiltonian
\begin{eqnarray}
  H &=& \sum_m h_{[m,m+1]} \nonumber \\
          &=&\frac{1}{4}\sum_{m} \left( \sigma_m^X \sigma_{m+1}^X
            + \sigma_m^Y \sigma_{m+1}^Y + \sigma_m^Z \sigma_{m+1}^Z \right),
  \label{eq:ham}
\end{eqnarray}
where $\sigma_m^X$, $\sigma_m^Y$ and $\sigma_m^Z$ are
Pauli matrices at site $m$. This theory stands
as a reference model
to try new ideas to treat strongly correlated
systems. There are a several reasons that justify
its relevance. First, the ground state of the Heisenberg
spin chain displays critical
phenomena, that is, long distance correlations dominate the system and
entanglement entropy of a finite block scales logarithmically with its size
\cite{VLRK}.
Second, the conformal properties of the Heisenberg chain
are related to the rotational symmetry of the Hamiltonian.
This will be a key ingredient in our analysis.
Third, the Heisenberg model is exactly solvable using the Bethe ansatz
{\cite{Bethe:1931,Sachdev:1999}}.

Let us be more precise about some of the properties of the Heisenberg chain that
can be calculated exactly. The exact energy of the ground state of this model can be worked out via the Bethe ansatz,
$E = 1/4-\ln 2 \sim -0.443147181\dots$ .
The iTEBD algorithm that we shall present later on, will work as a variational method  based on energy minimization
strategy. Therefore, it should approach the above result from above.

The ground state of the Heisenberg Hamiltonian is translationally invariant and,
as a consequence of rotational invariance, can be chosen to be an eigenstate of the total $Z$-component of spin, with eigenvalue equal to 0. These properties constrain the form of the state and, thus, its
reduced density matrices. Let us analyze this point in more detail.

We start with  an arbitrary state $\vert \psi\rangle$.
Its 2-body reduced density matrix for two adjacent sites,
where all spins are integrated out but for the sites $m$ and $m+1$, reads
\begin{equation}
\rho_{\left[m,m+1\right]}=  \mbox{tr}_{\left[m,m+1\right]}\vert\psi\rangle\langle\psi\vert \ .
\end{equation}
This operator takes the general form
\begin{equation}
\rho_{\left[m,m+1\right]} = \left( \begin{array} {cccc}
a & d & d' & f \\
d & b & c & e \\
d' & c & b' & e' \\
f & e & e' & a' \\
\end{array} \right) \ . \label{eq:twobody}
\end{equation}
We can now compute the 1-body reduced density matrix as a function of the 2-body density matrix parameters. Tracing out spin $m+1$ we get
\begin{equation}
\rho_{\left[m\right]} = \left( \begin{array} {cc}
a+b & d'+e \\
d'+e & b'+a' \\
\end{array} \right) \ . \label{eq:onebody}
\end{equation}

It is clear that
the total amount of parameters that describe the 1-body and 2-body
ground state reduced density matrices should be reduced by the presence of symmetries, as it is the case in the Heisenberg chain.
Let us consider separately how translational invariance and
rotational invariance simplify the above general discussion for
a finite spin chain.

\subsubsection{Eigenstate of the total $Z$-component of spin}
An eigenstate of the total $Z$-component of spin with eigenvalue $\Lambda$ obeys
\begin{equation}
\sum_{m=1}^N{\sigma_m^Z} { | \psi \rangle } = \Lambda | \psi \rangle \ . \label{eq:eigenstate}
\end{equation}
It is possible to write such a state in the spin basis
\begin{equation}
\sum_{m=1}^N{\sigma_m^Z} | \psi \rangle = \sum_{i_1 \ldots i_N}^{\lbrace -1,1 \rbrace} { c_{i_1 \ldots i_N} \sum_{m=1}^N {\sigma_m^Z} { | i_1 \ldots i_N \rangle } } \ ,
\end{equation}
where the coefficients $c_{i_1 \ldots i_N}$ will be constrained. It is straightforward to see that, for a given $\Lambda$,
\begin{equation}
\left ( \sum_{m=1}^N {i_m} \right ) c_{i_1 \ldots i_N}  = \Lambda c_{i_1 \ldots i_N}
\end{equation}
implies
\begin{equation}
\sum_{m=1}^N {i_m} \neq \Lambda \longrightarrow c_{i_1 \ldots i_N} = 0 \ . \label{eq:vanish_c}
\end{equation}
Furthermore, we can write the 2-body density matrix as a function of the basis elements
\begin{equation}
\rho_{\left[m,m+1\right]} = \rho^{i_m i_{m+1}}_{i'_m i'_{m+1}} | i_m i_{m+1} \rangle \langle i'_m i'_{m+1} |
\end{equation}
as
\begin{equation}
\rho^{i_m i_{m+1}}_{i'_m i'_{m+1}} =c^\ast_{i_1 \ldots i'_m i'_{m+1} \ldots i_N} c_{i_1 \ldots i_m i_{m+1} \ldots i_N} \ ,
\end{equation}
where repeated indices  are implicitly summed over. Then, the constraint (\ref{eq:vanish_c}) imposes
\begin{equation}
i_m + i_{m+1} \neq i'_m + i'_{m+1} \longrightarrow \rho^{i_m i_{m+1}}_{i'_m i'_{m+1}} = 0 \ ,
\end{equation}
That is, we find
\begin{equation}
 \rho^{00}_{01} = \rho^{00}_{10} = \rho^{00}_{11} = \rho^{01}_{11} = \rho^{10}_{11} = 0 \ ,
 \label{eq:den_zeros}
\end{equation}
which implies
\begin{equation}
 d=d'=f=e=e'=0.
\end{equation}
Note, that there is no restriction on the rest of elements of $\rho_{\left[m,m+1\right]}$.

It is now necessary to impose that the ground state is not a generic eigenstate of the total $Z$-component of spin operator but, rather,
\begin{equation}
 \Lambda = 0 \ . \label{eq:eigen_zero}
\end{equation}
 However, no constraint on the 2-body reduced density matrix will follow from that unless 2-site translational invariance is assumed
\begin{equation}
 \rho_{\left[m,m+1\right]} = \rho_{\left[m+2,m+3\right]} \ \ \ \ \ \forall{m} \ .
 \label{eq:2_ti}
\end{equation}
Note that this condition is weaker than imposing complete translational symmetry, yet sufficient to constrain the elements of $\rho_{\left[m,m+1\right]}$
to obey
\begin{equation}
 \mbox{tr} \left( \left( \sigma_m^Z + \sigma_{m+1}^Z \right) \rho_{\left[m,m+1\right]} \right) = \Lambda = 0 \ \ \ \ \ \forall{m} \ .
\end{equation}
This further implies
\begin{equation}
\rho^{00}_{00} = \rho^{11}_{11} \equiv a \ . \label{eq:den_a}
\end{equation}

The whole set of constraints we have presented in
 Eq. (\ref{eq:den_zeros}) and Eq. (\ref{eq:den_a}) reduce the number of parameters of the 2-body reduced density matrix (\ref{eq:twobody}) that, now, reads
\begin{equation}
\rho_{\left[m,m+1\right]} = \left( \begin{array} {cccc}
a & 0 & 0  & 0 \\
0 & b & c  & 0 \\
0 & c & b' & 0 \\
0 & 0 & 0  & a \\
\end{array} \right) \ .
\label{eq:twobody_eig}
\end{equation}
Note that this form corresponds to the density matrix for even sites, $m=2p$, whereas for odd sites, $m=2p-1$, coefficients $b$ and $b'$ should be exchanged. Furthermore, we can compute the 1-body reduced density matrix by tracing out one spin from the 2-body one
\begin{equation}
 \rho_{\left[m\right]} = \left( \begin{array} {cc}
   a +b & 0 \\
   0 & b'+a \\
   \end{array} \right) \ ,
\label{eq:onebody_eig}
\end{equation}
which is diagonal by construction.

\subsubsection{Strict Translational Invariance}
The constraint of translational invariance
produces a different form of the ansatz.
For instance,
on the reduced density matrices, this condition implies
\begin{equation}
\rho_{\left[m,m+1\right]} = \rho_{\left[m+1,m+2\right]} \label{eq:1_ti} \longrightarrow
\rho^{i_mi_{m+1}}_{i_m' i_{m+1}'} = \rho^{i_{m+1}i_m}_{i_{m+1}' i_m'} \ ,
\end{equation}
that is,
\begin{equation}
\begin{split}
&\rho^{01}_{01} = \rho^{10}_{10} \equiv b \\
&\rho^{00}_{01} = \rho^{00}_{10} \equiv d \\
&\rho^{01}_{11} = \rho^{10}_{11} \equiv e \ . \\
\end{split}
\label{eq:den_ti}
\end{equation}
Consequently, the generic expression in Eq. (\ref{eq:twobody}) for $\rho_{\left[m,m+1\right]}$ will be casted into the form
\begin{equation}
\rho_{\left[m,m+1\right]} = \left( \begin{array} {cccc}
a & d & d & f \\
d & b & c & e \\
d & c & b & e \\
f & e & e & a' \\
\end{array} \right) \ \ \ \ \forall{m} \ . \label{eq:twobody_tr}
\end{equation}
Note that translational invariance reduces the set of parameters
in a quite different way as compared to rotational invariance.

The 1-body reduced density matrix can be obtained by integrating
out one of the spins in Eq. (\ref{eq:twobody_tr})
\begin{equation}
\rho_{\left[m\right]} = \left( \begin{array} {cc}
a+b & d+e \\
d+e & b+a' \\
\end{array} \right) \ . \label{eq:onebody_tr}
\end{equation}
This result remains the same whatever spin is chosen
to be integrated out, $\rho_{\left[m\right]} = \rho_{\left[m+1\right]}$.
Note that off-diagonal terms are now present, at variance with the
rotational invariant case. Therefore, these off-diagonal terms
can be seen as a signature of the
lack of rotational symmetry in the state.

\subsubsection{Quasi Translational Invariance}
The ground state of the Heisenberg spin chain turns out to verify
translational invariance up to a sign. That is, the ground state is invariant
under translating all spins by one position if, simultaneously, all of them
are flipped. This property is rooted at the combination of the anti-ferromagnetic
character of the Heisenberg interaction and the 2-body nearest neighbor
decomposition of the Hamiltonian. For the sake of simplicity,
we call this symmetry "quasi translational
invariance". It is, then, necessary to modify our discussion on translational symmetry in order to accommodate this effect. It is easy to see
that the generic restriction stated by Eq. (\ref{eq:den_ti}) gets modified
in the following way
\begin{equation}
\begin{split}
&\rho^{01}_{01} = \ \ \rho^{10}_{10} \equiv b \\
&\rho^{00}_{01} = -\rho^{00}_{10} \equiv d \\
&\rho^{01}_{11} = -\rho^{10}_{11} \equiv e \ . \\
\end{split}
\label{eq:den_ti_relax}
\end{equation}
To be precise, some off-diagonal terms include a minus sign. Let us note that the expectation value of the energy does not depend on this sign. Therefore, the new form of the 2-body reduced density matrix reads
\begin{equation}
 \rho_{\left[m,m+1\right]} = \left( \begin{array} {cccc}
 a & d & -d & f \\
 d & b & c & e \\
 -d & c & b & -e \\
 f & e & -e & a' \\
 \end{array} \right) \ , \label{eq:twobody_ti_relax}
\end{equation}
where $d \leftrightarrow -d$, $e \leftrightarrow -e$ should be exchanged for odd/even values of $m$.
The 1-body reduced density matrix is also different for odd/even values of $m$
\begin{equation}
\rho_{\left[m\right]} = \left( \begin{array} {cc}
a+b & -d+e \\
-d+e & b+a' \\
\end{array} \right) \ , \label{eq:onebody_tr_relax}
\end{equation}
with a sign exchange for $d$ and $e$. Note that, again,
off-diagonal terms are present, indicating that
no rotational symmetry is preserved by this ansatz in general.

\subsubsection{Exact ground state}
Finally, instead of considering symmetries separately, let us impose all rotational and quasi translational invariances simultaneously on a generic state. This is the relevant case of the exact ground state
of the Heisenberg model.
Then, the density matrix form is even simpler. From the combination of Eq. (\ref{eq:den_zeros}), Eq. (\ref{eq:den_a}) and Eq. (\ref{eq:den_ti_relax}) we  obtain the expression for the 2-body density matrix of the Heisenberg ground state {\cite{Wooters:2004}}
\begin{equation}
\rho_{\left[m,m+1\right]} = \left( \begin{array} {cccc}
a & 0 & 0 & 0 \\
0 & b & c & 0 \\
0 & c & b & 0 \\
0 & 0 & 0 & a \\
\end{array} \right) \ \ \ \ \forall{m} \ . \label{eq:twobody_hh}
\end{equation}
Only a few parameters $\lbrace a, b, c \rbrace$ remain undetermined by
symmetries.

The 1-body reduced density matrix turns out to be
\begin{equation}
\rho_{\left[m\right]} = \left( \begin{array} {cc}
1/2 & 0 \\
0 & 1/2 \\
\end{array} \right) \ \ \ \ \ \ \forall{m} \
\label{eq:onebody_hh}
\end{equation}
as a consequence of the unitarity of this reduced density matrix.
This reflects the fact that each individual spin in the system has equal probability to
take any value in any direction since $ \mbox{tr} \left( \sigma_Z \rho_m \right) =0$ ($Z\leftrightarrow X$,
$Z\leftrightarrow Y$), as consequence of rotational {\sl and} translational
invariance. Note that there is no distinction here between the constraints
coming from strict translational invariance or quasi translational invariance.

Later on, we shall use the 1-body and (adjacent) 2-body reduced density matrices as figures of merit to determine
how rotational symmetry or quasi translational invariance are maintained within the MPS approximation.

\section{Matrix Product States}
\label{Sect:iTEBD}

The Matrix Product State ansatz is a powerful tool to describe one-dimensional systems with a reduced number of parameters. The basic idea underlying this approximation is to
represent the set of $2^N$ coefficients that describe any state of  a $N$-spin chain as a product of  $N$ matrices that can be chosen as follows \cite{Vidal:MPS}
\begin{equation}
  | \psi \rangle = \sum_{i_1 \ldots i_N = 1}^2 \Gamma^{[1]i_1} \lambda^{[1]} \ldots \lambda^{[N-1]} \Gamma^{[N]i_N} |i_1
  \rangle \ldots |i_N \rangle \ .
\label{Eq:state}
\end{equation}
All $\Gamma^{[m]i_m}$ objects are $\chi \times \chi$ matrices and $\lambda^{[m]}$ are diagonal matrices of dimension $\chi$. The state can be pictured
as shown in Fig. \ref{fig:MPS}.

\begin{figure}[h]
  \resizebox{6cm}{!}{\includegraphics{MPS}}
  \caption{MPS diagrammatic representation. Each filled circle represents a $\Gamma$ and each line represents an index. The diamonds represent the Schmidt coefficients $\lambda$. }
\label{fig:MPS}
\end{figure}

The rational for this ansatz relies on the fact that the above construction
is the result of a systematic use of Schmidt decompositions. That is, at any
site $k$ we may divide the spin chain in two pieces. The state can then be
written as
\begin{equation}
  \label{Eq:schmidt}
  \vert \psi\rangle=\sum_{\alpha=1}^{\chi}\vert Left_\alpha\rangle \vert Right_\alpha\rangle \ ,
\end{equation}
where $\vert Left_\alpha\rangle$ and $\vert Right_\alpha\rangle$ stand for the states that form an orthogonal basis on the left
and right parts of the chain.

As a consequence of this Schmidt decomposition, several measures of entanglement can be constructed. The most elementary one corresponds to
the value of $\chi$ itself, called Schmidt number.
For instance, $\chi=1$ on a given link means that the state can be
divided there as a simple product of left and right states. In such a case, there is no
entanglement between the left and right parts of the system.
The opposite case of maximum   $\chi= 2^\frac{N}{2}$ corresponds to
maximally entangled states. From the Schmidt decomposition we can
also construct the Von Neumann entanglement entropy
\begin{equation}
S = -\sum_{\alpha=1}^{\chi}{\lambda_{\alpha}^2\log \lambda_{\alpha}^2 } \ .
\end{equation}
Zero entropy corresponds to uncorrelated partitions, whereas a linear scaling
of entropy with the number of sites in the smaller partition corresponds to
maximal entanglement.

Extensive efforts have been devoted to understand the amount of entanglement
which is present in the ground states of strongly correlated systems.
At quantum phase transitions, the entanglement entropy for a block of adjacent spins scales logarithmically with the number of sites for critical models
\cite{VLRK}.
The ground state of the Heisenberg spin chain displays this scaling property.

Note that the Schmidt decomposition produces naturally the following isometries
\begin{equation}
\sum_{i,\alpha}\lambda_{\alpha}^{[m-1]} \Gamma_{\alpha \beta}^{ \left[ m \right] i} \left( \lambda_{\alpha}^{[m-1]} \Gamma_{\alpha \beta'}^{\left[ m \right]i} \right)^* = \delta_{\beta \beta'} \label{eq:isom_left}
\end{equation}
and
\begin{equation}
\sum_{i,\beta} \Gamma_{\alpha \beta}^{ \left[ m \right] i} \lambda_{\beta}^{[m]}  \left( \Gamma_{\alpha' \beta}^{\left[ m \right]i} \lambda_{\beta}^{[m]} \right)^* = \delta_{\alpha \alpha'} \label{eq:isom_right} \ .
\end{equation}
This is an instrumental property to simplify the contractions
of states that have a MPS structure.

This approach is closely related to the DMRG ansatz. Note that the DMRG algorithm produces
a MPS state, that is,
a state whose coefficients are represented as a product of matrices.
This can be translated into the form in Eq. (\ref{Eq:state})
using a local symmetry. Indeed, at any point between two sites
in the product of matrices,
we can insert $I=XX^{-1}$ and reabsorb $X$ and $X^{-1}$
on the adjacent left and right matrices respectively. This
(gauge) freedom can be used to convert
the form of the state in such a way that Eq.
(\ref{eq:isom_left}) and Eq. (\ref{eq:isom_right}) are fulfilled.

The challenge of finding the best set of matrices $\left \lbrace \Gamma^{[m]i_m} \right \rbrace$, $\left \lbrace \lambda^{[m]} \right \rbrace$ has already been faced using numerical algorithms. We shall here use the iTEBD algorithm proposed
in Ref. \cite{Vidal:iTEBD}. The basic idea consists in evolving some initial state in imaginary time.  Provided that its overlap with the ground state is different from zero, the imaginary time evolution converges towards
the ground state of the system. This strategy can be implemented
using the Trotter decomposition \cite{Trotter} that allows to decompose
the imaginary time evolution of the total Hamiltonian in small
steps, each one corresponding to the imaginary time
evolution of a piece of the Hamiltonian.
Moreover, whenever the Hamiltonian is made of two-body nearest-neighbor interactions, it can be written as a sum of two non-commuting pieces which are made of commuting terms. Within each non-commuting subspace, the action of the Hamiltonian on the evolution of the state reads as a product of exponentials acting on two adjacent sites. Consequently, the state can be described by only two different matrices \cite{Vidal:iTEBD}
\begin{equation}
\begin{split}
&\Gamma^{[2m-1]} \equiv \Gamma^{[A]} \ , \ \ \Gamma^{[2m]}  \equiv \Gamma^{[B]}\quad \\
&\lambda^{[2m-1]}  \equiv \lambda^{[A]} \ , \ \ \lambda^{[2m]}  \equiv \lambda^{[B]}\quad \\
\end{split}
\ \ \ \forall m \ .
\end{equation}
It follows that the full algorithm reduces to a two-step iteration over two sites.

Let us emphasize that the iTEBD algorithm distinguishes as
many sites as the number of terms involved in the local interaction.
For instance, a Hamiltonian made with interactions that involve
next-to-nearest-neighbor sites would be handled by iTEBD using
three type of matrices $A$, $B$ and $C$ and a three-step iteration euclidean evolution.
Note as well that, in our case, the iTEBD algorithm incorporates explicitly two-site translational invariance.

The choice for the parameter $\chi$ of the MPS, that is the
size of the matrices in the ansatz (often called $m$ in DMRG literature)
controls both the computational cost and accuracy of the approximation.
Some results for the energy and the entropy of the ground state
of the Heisenberg spin-1/2 infinite chain obtained with the iTEBD algorithm
as a function of $\chi$ are
shown in Table \ref{tab:t1}.

\begin{table}[h]
\centering
\begin{tabular}[b]{|c|c|c|}
\hline
$\chi$ & $E_{\chi}$ & $S_{\chi}$ \\
\hline
2 & -0.427908 & 0.48570 \\
3 & -0.435784 & 0.60866 \\
4 & -0.441058 & 0.76413 \\
5 & -0.442048 & 0.84778 \\
\hline
$\infty$ & -0.44314718 & $\infty$ \\
\hline
\end{tabular}
\caption[]{Comparison between the exact result (labeled as
$\chi=\infty$) and the $\chi$-dependent iTEBD/DMRG approximation for the energy $E_{\chi}$ and the Von Neumann entropy $S_{\chi}$ in the Heisenberg spin-1/2 infinite chain. All numbers in the table are precise up to the last figure.} \label{tab:t1}
\end{table}

\section{Exact Results for MPS}
\label{Sect:exact}
In this section we shall present an exact method for finding the
MPS approximation to the ground state of the Heisenberg Hamiltonian. Then, we shall study the lowest-$\chi$ cases in detail.

\subsection{Exact method}

Instead of resorting to a numerical algorithm in order to find the
MPS approximation to the ground state of the Heisenberg Hamiltonian, we shall construct the exact form of this approximation by exploiting the isometry constraints that the matrices $\left\lbrace \Gamma^{ \left [m \right ]i_m}, \lambda^{[m]} \right\rbrace$ obey. It is, thus, necessary to understand
the MPS approach as a sequence of Schmidt decompositions rather than
the result of a DMRG procedure. This is the key point to our
analytical strategy.

We first observe that the 2-body reduced density matrix for a MPS
can be written in a simple form as a function of the matrices $\left \lbrace \Gamma^{\left[ A, B \right]i}, \lambda^{[A, B]} \right \rbrace$. That is,
the combined exploitation of two-site translational symmetry
and isometries implies
\begin{equation}
\begin{gathered}
\rho^{ij}_{i'j'} =  \\
\textrm{tr} \left( \lambda^{[B]} \Gamma^{[A]i} \lambda^{[A]} \Gamma^{[B]j} \lambda^{[B]} \lambda^{[B]} \Gamma^{\dagger[B]j'} \lambda^{[A]} \Gamma^{\dagger[A]i'} \lambda^{[B]} \right) \ .
\label{eq:rho}
\end{gathered}
\end{equation}
Isometries are, indeed, necessary to seamlessly contract the
infinite chain except for the open physical indices of the reduced density matrix.
In other words, the above expression is only valid in the gauge
where the matrices forming the MPS are appropriately chosen to
fulfill the isometry constraints in Eqs. (\ref{eq:isom_left},\ref{eq:isom_right}).

Once we know the dependence of the 2-body reduced density matrix on the MPS matrices, we can easily compute any observable. In particular, let us write down the energy of the system which is the quantity we want to minimize.
First, we take a local term at sites $[m,m+1]$ for two spins in the Heisenberg Hamiltonian (\ref{eq:ham})
\begin{equation}
h_{\left[ m,m+1 \right]} = \frac{1}{4}\left ( \begin{array} {cccc}
1 & 0 & 0 & 0 \\
0 & -1 & 2 & 0 \\
0 & 2 & -1 & 0 \\
0 & 0 & 0 & 1 \\
\end{array} \right)
\end{equation}
and write its expected value using the form of the
2-body reduced density matrix in Eq. (\ref{eq:rho})
\begin{equation}
\left\langle h \right\rangle = {\rm tr}(\rho h)=\frac{1}{4} - \frac{1}{2} \textrm{tr} \left( \lambda^B M \left(\lambda^B M \right)^{\dagger} \right) \ ,
\end{equation}
where
\begin{equation}
M \equiv \Gamma^{[A]1} \lambda^{[A]} \Gamma^{[B]2} \lambda^{[B]} - \Gamma^{[A]2} \lambda^{[A]} \Gamma^{[B]1} \lambda^{[B]} \ .
\end{equation}
This expected value can also be written in a diagrammatic form, see Fig. \ref{fig:energy}.

\begin{figure}[h]
  \resizebox{5cm}{!}{\includegraphics{hamil.eps}}
  \caption{Diagrammatic representation for the expected value of a local term of the Hamiltonian. Closed lines represent summed indices.} \label{fig:energy}
\end{figure}

At this point the set of isometries $\left\lbrace \Gamma^{[A]i}, \Gamma^{[B]i} \right\rbrace$ are still general $\chi \times \chi$ matrices satisfying Eq. (\ref{eq:isom_left}) and
(\ref{eq:isom_right}). From now on we will impose some additional requirements on them so as to incorporate the symmetries of the problem.

For instance, it is possible to impose
\begin{equation}
\Gamma^{[A]i} = \pm \Gamma^{[B]i} \equiv \Gamma^i \ ,
\ \ \lambda^{[A]} = \lambda^{[B]} \equiv \lambda
\label{eq:t_inv}
\end{equation}
in order to implement (quasi) translationally invariance.
In such a case, it follows that
\begin{equation}
M_{\pm} = \Gamma^1 \lambda \Gamma^2 \lambda \mp \Gamma^2 \lambda \Gamma^1 \lambda \ .
\label{eq:M_ti}
\end{equation}
We, then, conclude that the option of quasi translational invariance produces a lower energy than the strict translational invariant case. This, as mentioned
before, is related to the anti-ferromagnetic character of the Heisenberg
interaction.

Alternatively, we can explicitly break translational invariance and set
\begin{equation}
\Gamma^{[A]i} = \left( \pm \right)^j \Gamma^{[B]j} \equiv \Gamma^i , \ \ \
\lambda^{[A]} = \lambda^{[B]} \equiv \lambda\ \, \label{eq:non_t_inv}
\end{equation}
where $i,j=\left\lbrace 1, 2 \right\rbrace$, $i \neq j$. Then,
\begin{equation}
M_{\pm} = \pm \left( \Gamma^1 \lambda \Gamma^1 \lambda \mp \Gamma^2 \lambda \Gamma^2 \lambda \right) \ .
\label{eq:M_nti}
\end{equation}
This case will later be seen on explicit $\chi$ examples to correspond to a necessary but not
sufficient condition to maintain rotational invariance. Again, in this case,
the minus sign is favored by minimization of the energy.

Both scenarios lead to the minimization of an expression
of the form
\begin{equation}
\left\langle h \right\rangle = \frac{1}{4} - \frac{1}{2} \textrm{tr} \left( \lambda M_{-} \left( \lambda M_{-} \right)^{\dagger} \right) \ ,
\label{eq:maxim}
\end{equation}
where $M_-$ is given by either Eq. (\ref{eq:M_ti}) or Eq. (\ref{eq:M_nti}).
In order to make further analytical progress, we now turn to
the particular cases of low values for $\chi$.

\subsection{\boldmath{} $\chi=2$ \unboldmath{}}
Let us now focus on the simplest case of MPS with $\chi=2$. We can exploit
in different ways the freedom in implementing the isometry conditions on matrices $\Gamma$ and $\lambda$ to simplify further the MPS ansatz. We shall
discuss in turn several possibilites that deliver the same
minimum energy, though not the same state.

\subsubsection{Eigenstate of the total $Z$-component of spin}
As discussed previously, the constraint in Eq. (\ref{eq:non_t_inv}) allows for the choice of (up to a sign) identical matrices. The isometry property
leaves enough space to further restrict the form of the these matrices without loss of generality
to be
\begin{gather}
\Gamma^{\left[ A \right]1} = \left( \begin{array} {cc} a & 0 \\ 0 & 0 \\ \end{array} \right) \\ \Gamma^{\left[ A \right]2} = \left( \begin{array} {cc} 0 & b \\ b & 0 \\ \end{array} \right) \\ \label{eq:gamma2_nti}
\lambda^A = \lambda^B = \left( \begin{array} {cc} \lambda_1 & 0 \\ 0 & \lambda_2 \end{array} \right) \ .
\end{gather}
Then, the expression for the energy takes the form
\begin{equation}
\left\langle h \right\rangle = \frac{1}{4} - \frac{1}{2} \left( b^4 \lambda_1^2 \lambda_2^4 + \lambda_1^4 \left( a^2 \lambda_1 + b^2 \lambda_2 \right)^2 \right) ,
\end{equation}
and the isometry condition reduces to
\begin{equation}
b^2 \lambda_1^2 = 1 \ \ , \ \ a^2 \lambda_1^2 + b^2 \lambda_2^2 = 1 \ .
\end{equation}
These isometry conditions can be solved explicitly
\begin{equation}
b^2=\frac{1}{\lambda_1^2}\ , \ a^2=\frac{2 \lambda_1^2-1}{\lambda_1^4}\ ,
\end{equation}
where we have used the normalization condition $\lambda_1^2 + \lambda_2^2 = 1 $.
The energy of the system can, then, be written as a unique function of $\lambda_1$
\begin{equation}
\left\langle h \right\rangle = \frac{11}{4} - 2 \lambda_1^2 - \frac{1}{\lambda_1^2} - 2 \lambda_1 \sqrt{1-\lambda_1^2}+ \frac{\sqrt{1-\lambda_1^2}}{\lambda_1} \ . \label{eq:energy_l}
\end{equation}
After some algebraic manipulations are performed,
the minimum of this expression can be reduced  to
\begin{equation}
4 + 3\lambda_1^2-14 \lambda_1^4-8 \lambda_1^6+16\lambda_1^8=0 \ .
\end{equation}
One of the solutions of this quartic equation produces the lowest energy at
\begin{equation}
\lambda_{1,min} = 0.94586362...
\end{equation}
\begin{equation}
 E_{\chi=2} = -0.42790801...
\end{equation}
which is the same minimum obtained with numerical techniques (see
Table \ref{tab:t1}). This result confirms that DMRG and iTEBD
find numerically the correct minimum for the energy which is
consistent with a MPS ansatz.

We can also calculate the correlation length of the MPS from the transfer matrix $E = \sum_i \left( \Gamma^i \lambda \right) \otimes \left( \Gamma^{i\ast} \lambda^{\ast} \right)$. In this particular case, $E$ can be written as a function of $\lambda_1$:
\begin{equation}
E = \left( \begin{array} {cccc}
0 & 0 & 0 & 1 \\
0 & 0 & \frac{\sqrt{1-\lambda_1^2}}{\lambda_1} & 0 \\
0 & \frac{\sqrt{1-\lambda_1^2}}{\lambda_1} & 0 & 0 \\
\frac{1}{\lambda_1^2} - 1 & 0 & 0 & 2 - \frac{1}{\lambda_1^2} \\
\end{array} \right)
\end{equation}
whose eigenvalues define the correlation length $\xi = 1/\log \left( \frac{e_1}{e_2} \right)$. The explicit dependence of $\xi$ on $\lambda_1$ is
\begin{equation}
\xi^{-1}_{\chi=2} \left( \lambda_1 \right) = \log \frac{2\lambda_1^2 -1 + \sqrt{8\lambda_1^2 \left( \lambda_1^2-1 \right) +1}}{2\lambda_1 \sqrt{1-\lambda_1^2}} \ .
\end{equation}
Setting $\lambda_1 = \lambda_{1,min}$ we find
\begin{equation}
\xi_{\chi=2} = 0.93820167...
\end{equation}
The exact value of the half chain entropy for $\chi=2$ turns out to be
\begin{equation}
S_{\chi=2}=0.48570420...
\end{equation}

Finally, following the notation in Eq. (\ref{eq:twobody_eig}) we can also write down the 2-body density matrix of this MPS
\begin{equation}
\begin{split}
&a = 3-\frac{1}{\lambda_1^2}-2\lambda_1^2 \\
&b = -4+\frac{1}{\lambda_1^2}+4\lambda_1^2 \\
&b'= -1+\frac{1}{\lambda_1^2} \\
&c = \left( \frac{1}{\lambda_1} -2\lambda_1 \right) \sqrt{1-\lambda_1^2} \\
\end{split}
\end{equation}
and set $\lambda_1 = \lambda_{1,min}$
\begin{equation}
\rho_{\left[ m,m+1 \right]} = \left( \begin{array} {cccc}
0.0929 & 0 & 0 & 0 \\
0 & 0.6964 & -0.2708 & 0 \\
0 & -0.2708 & 0.1177 & 0 \\
0 & 0 & 0 & 0.0929 \\
\end{array} \right) \ .
\end{equation}
Tracing out one spin, the 1-body density matrix becomes
\begin{equation}
\rho_{\left[ m \right]} = \left( \begin{array} {cc}
-1+2\lambda_1^2 & 0 \\
0 & 2 - 2\lambda_1^2 \\
\end{array} \right) \ ,
\end{equation}
that yields
\begin{equation}
\rho_{\left[m\right]} = \left( \begin{array} {cc}
0.7893 & 0 \\
0 & 0.2107 \\
\end{array} \right) \ .
\end{equation}

Note that the chosen parametrization leads to an eigenstate of the total $Z$-component of spin (because of the block-diagonal form of the density matrices). However, the translational invariance is explicitly broken since $\rho_{\left[m-1,m\right]} \neq \rho_{\left[m,m+1\right]}$ and also $\rho_{\left[m\right]} \neq \rho_{\left[m+1\right]}$.

\subsubsection{Quasi Translational Invariance}
A second possibility we can follow corresponds to setting conditions (\ref{eq:t_inv}). In that case, we try to find out the minimum energy state with matrices $\Gamma$ parametrized as
\begin{equation}
\Gamma^{\left[ A \right]1} = \left( \begin{array} {cc} a & b \\ -b & 0 \\ \end{array} \right) \ \ \Gamma^{\left[ A \right]2} = \left( \begin{array} {cc} a & -b \\ b & 0 \\ \end{array} \right) \ ,
\end{equation}
and equal $\lambda$ matrices for all sites.
This new parametrization can be seen to produce the very same dependence of the energy on $\lambda_1$. Therefore,  the minimum energy we shall
obtain is the same than the first parametrization reached. Furthermore, the computation of the correlation length leads us to the same result as before. Despite these coincidences, the state is no more the same, which can be inferred from the 2-body density matrix of the system:
\begin{equation}
\begin{split}
&a = \frac{1}{4}+\frac{\sqrt{1-\lambda_1^2}}{2\lambda_1}-\lambda_1 \sqrt{1-\lambda_1^2} \\
&b = \frac{1}{4}-\frac{\sqrt{1-\lambda_1^2}}{2\lambda_1}+\lambda_1 \sqrt{1-\lambda_1^2} \\
&c = \frac{11}{4}-\frac{1}{\lambda_1^2}-2 \lambda_1^2+\frac{\sqrt{1-\lambda_1^2}}{2 \lambda_1}-\lambda_1 \sqrt{1-\lambda_1^2} \\
&d = e' = \frac{3}{4}-\lambda_1^2 \\
&d' = e = -d = -e' \\
&f = \frac{11}{4}-\frac{1}{\lambda_1^2}-2 \lambda_1^2-\frac{\sqrt{1-\lambda_1^2}}{2\lambda_1}+\lambda_1 \sqrt{1-\lambda_1^2} \ . \\
\end{split}
\end{equation}
Note that the equalities $d=d'$, $e=e'$ are maintained up to a sign, since the condition we require is precisely quasi translational invariance. Numerically, the 2-body reduced density matrix and also the 1-body reduced density matrix take the form:
\begin{equation}
\rho_{\left[m,m+1\right]} = \left( \begin{array} {cccc}
 0.1146 & -0.1447 &  0.1447 & -0.0216 \\
-0.1447 &  0.3854 & -0.2925 &  0.1447 \\
 0.1447 & -0.2925 &  0.3854 & -0.1447 \\
-0.0216 &  0.1447 & -0.1447 &  0.1146 \\
\end{array} \right)
\end{equation}
\begin{equation}
\rho_{\left[m\right]} = \left( \begin{array} {cc}
0.5 & 0.2893 \\
0.2893 & 0.5 \\
\end{array} \right) \ .
\end{equation}

The state represented by this MPS is not an eigenstate of the total $Z$-component of spin, but quasi translational invariance is recovered.
The two parametrizations we have analyzed deliver the same energy but
handle differently the original symmetries of the problem. The resulting
MPS will give different results on local observables attached to these symmetries. We may also consider a MPS which is not forced to take
the form imposed by any symmetry. Such new state would still deliver
the same energy and differ on other observables.

\subsection{\boldmath{} $\chi \geq 3$ \unboldmath{}}
We can also follow similar steps for higher values of $\chi$. In particular, let us apply the constraint discussed in Eq. (\ref{eq:non_t_inv}) for matrices of $\chi=3$. This requirement allows us to choose a rather simple form of the matrices
\begin{equation}
\Gamma^{\left[ A \right]1} = \left( \begin{array} {ccc} a & 0 & c \\ 0 & 0 & 0 \\ c & 0 & e \\ \end{array} \right) \ \ \Gamma^{\left[ A \right]2} = \left( \begin{array} {ccc} 0 & b & 0 \\ b & 0 & d \\ 0 & d & 0 \\ \end{array} \right) \
\end{equation}
which generalizes the form of matrices $\Gamma$ in Eq. (\ref{eq:gamma2_nti}) for $\chi=2$.

Note that the number of parameters that describes the state can be reduced by considering the isometry and normalization conditions
\begin{equation}
\begin{split}
&a^2\lambda_1^2 + b^2\lambda_2^2 + c^2\lambda_3^2 = 1 \\
&b^2\lambda_1^2 + d^2\lambda_3^2 = 1 \\
&^2\lambda_1^2 + d^2\lambda_2^2 + e^2\lambda_3^2 = 1 \\
&ac\lambda_1^2 + bd\lambda_2^2 + ce\lambda_3^2 = 0 \ . \\
\end{split}
\end{equation}
As a consequence of this reduction, the energy of the state can be expressed, after some algebra, as a function of only three parameters. Therefore, the problem of finding a good approximation to the ground state of the Heisenberg Hamiltonian turns into a three-variable function minimization. The solution is again the same as numerical algorithms reach with no constraint on the matrices
\begin{equation}
E_{\chi=3} = -0.435784... \label{eq:sol_chi3}
\end{equation}
and the reduced density matrices take the form
\begin{gather}
\rho_{\left[m,m+1\right]} = \left( \begin{array} {cccc}
0.0930 & 0 & 0 & 0 \\
0 & 0.6429 & -0.2788 & 0 \\
0 & -0.2788 & 0.1711 & 0 \\
0 & 0 & 0 & 0.0930 \\
\end{array} \right) \\
\rho_{\left[m\right]} = \left( \begin{array} {cc}
0.7359 & 0 \\
0 & 0.2641 \\
\end{array} \right) \ .
\end{gather}

We observe that again the $2(1)$-body reduced density matrix of the MPS clearly violates translational invariance. Nevertheless, in analogy to the $\chi=2$ case, we can require quasi translational invariance conditions on matrices $\Gamma$, such that Eq. (\ref{eq:t_inv}) is fulfilled. In this case, we have found the particular form
\begin{equation}
\Gamma^{\left[ A \right]1} = \left( \begin{array} {ccc} a & b & c \\ -b & 0 & d \\ c & -d & e \\ \end{array} \right) \ \ \Gamma^{\left[ A \right]2} = \left( \begin{array} {ccc} a & -b & c \\ b & 0 & -d \\ c & d & e \\ \end{array} \right)
\end{equation}
which produces the solution (\ref{eq:sol_chi3}) as well. However, despite obtaining the same minimum, the MPS is different from the previous one. This can be explicitly seen from the density matrices
\begin{gather}
\rho_{\left[m,m+1\right]} = \left( \begin{array} {cccc}
 0.1106 & -0.1180 &  0.1180 & -0.0176 \\
-0.1180 &  0.3894 & -0.2964 &  0.1180 \\
 0.1180 & -0.2964 &  0.3894 & -0.1180 \\
-0.0176 &  0.1180 & -0.1180 &  0.1106 \\
\end{array} \right) \\
\rho_{\left[m,m+1\right]} = \left( \begin{array} {cc}
0.5 & 0.2359 \\
0.2359 & 0.5 \\
\end{array} \right) \ .
\end{gather}
In this second case, quasi translational symmetry is preserved, but the rotational invariance is clearly broken.

A similar reasoning can be followed in regard to higher $\chi$ MPS, looking for a generalized form of matrices $\Gamma$ and addressing the minimization of the energy by reducing the number of parameters that describes the MPS.
We shall quote some of the results obtained in the case of higher $\chi$ in the next section.

\section{Symmetry Restoration}
\label{Sect:symmetry}
The faithfulness of the MPS approximation to the ground state of a given Hamiltonian depends on the value of parameter $\chi$. We refer the reader to the extensive analysis of scaling relations
performed in Refs. \cite{TOIL:2008,andersson}. It is, there,
shown that
the finiteness of $\chi$  produces an effective
correlation length in the system
\begin{equation}
  \xi=\chi^\kappa \label{eq:scaling}
\end{equation}
with $\kappa=1.37(1)$ for the Heisenberg spin chain. In the present case, we shall
analyze a different effect, namely the breaking of the symmetries of the problem
as a function of $\chi$.

As discussed along the previous section, the MPS ansatz does not preserve all the symmetries of the exact state to be represented. Therefore, the symmetry breaking generated by this approximation should also be described as a function of $\chi$. In particular, for the limit $\chi \rightarrow \infty$ all the symmetries of the state described by the MPS should be completely restored.

In order to quantify the symmetry breaking of a MPS with parameter $\chi$, we need to construct a figure of merit from the representation of the state.
We shall present such a construction via an example. Let  us write down a
generic  MPS for $\chi=2$ obtained with the iTEBD algorithm with no restriction on the form of $\Gamma$ matrices
\begin{gather}
\rho_{\left[m,m+1\right]} = \left( \begin{array} {cccc}
 0.0972 & -0.0725 &  0.0554 & -0.0042 \\
-0.0725 &  0.6623 & -0.2751 &  0.0725 \\
 0.0554 & -0.2751 &  0.1433 & -0.0554 \\
-0.0042 &  0.0725 & -0.0554 &  0.0972 \\
\end{array} \right) \\
\rho_{\left[m\right]} = \left( \begin{array} {cc}
0.7595 & 0.1279 \\
0.1279 & 0.2405 \\
\end{array} \right) \ .
\end{gather}
Note that these reduced density matrices cast into the form stated by Eqs. (\ref{eq:twobody},\ref{eq:onebody}). In particular, let us focus on the 1-body reduced density matrix and write it as
\begin{equation}
\rho_{\left[m\right]} = \left( \begin{array} {cc}
\frac{1}{2}+x & y \\
y & \frac{1}{2}-x \\
\end{array} \right) \label{eq:1body_par}
\end{equation}
where the property $ \mbox{tr} \left( \rho_{\left[m\right]} \right) = 1$ has been used.

We see that the 1-body reduced density matrix is described by only two parameters $\left \lbrace x, y \right \rbrace$ that parametrize the deviation from the exact ground state. Indeed, with this notation, the 1-body reduced density matrix of the ground state of the Heisenberg chain corresponds to the setting $\left \lbrace x=0, y=0 \right \rbrace$ (see Eq. (\ref{eq:onebody_hh})). Therefore, these parameters account for the symmetries breaking of the MPS approximation.

This short description allows to compare again the two regimes studied above, namely the preservation of either rotational symmetry or quasi translational invariance. Let us summarize in Table \ref{tab:t2} the results of the previous section together with the generic 1-body reduced density matrix obtained with the iTEBD algorithm for $\chi=2$.

\begin{table}[h]
\centering
\begin{tabular}[c]{|c|c|c|c|}
\hline
\ & \textit{Gen} & \textit{Rot} & \textit{qTr} \\
\hline
$x$ & 0.2595 & 0.2893 & 0 \\
$y$ & 0.1279 & 0 & 0.2893 \\
$\Delta\mu\equiv 2\sqrt{x^2+y^2}$&0.2893&0.2893&0.2893\\
\hline
\end{tabular}
\caption[]{Comparison between the parameters of the 1-body reduced density matrix (see Eq. (\ref{eq:1body_par})) of a MPS with $\chi=2$ in the following cases: generic (\textit{Gen}), rotational invariance (\textit{Rot}) and quasi translational invariance (\textit{qTr}). Note that $\Delta\mu$ is identical
in all cases as it corresponds to the unique invariant coming from the
eigenvalues of the 1-body reduced density matrix.}
\label{tab:t2}
\end{table}

The results from Table \ref{tab:t2} show explicitly the differences between the states represented by MPS with different symmetries preserved. Nevertheless, all of them correspond to the very same values for the energy and the entropy. In such a case, we should be able to construct from the 1-body reduced density matrix an observable that shall not depend on the gauge.

Since the elements of the density matrix depend explicitly on the local basis, the unique parameters that are invariant under local unitary transformations shall be its eigenvalues, that we call $\left \lbrace \mu_1,\mu_2 \right \rbrace$. Furthermore, note that the trace of the 1-body reduced density matrix is also constrained to be equal to one. Therefore, the only gauge-invariant parameter that remains is the difference of the eigenvalues
\begin{equation}
\Delta \mu \equiv \mu_1 - \mu_2 = 2\sqrt{x^2+y^2}
\end{equation}
which can be read explicitly as a function of parameters $\left \lbrace x,y \right \rbrace$.

\begin{figure}[h]
  \resizebox{8cm}{!}{\includegraphics{delta_mu}}
  \caption{Restoration of the broken symmetries parametrized by
the difference of eigenvalues $\Delta\mu=\mu_1-\mu_2$ of the 1-body reduced density matrix gauge-invariant parameter.}
\label{fig:delta_mu}
\end{figure}

The quantity $\Delta \mu$ does not depend on the gauge and accounts for the faithfulness of the approximation. The freedom in implementing the isometry conditions on the matrices of the MPS allows to choose a particular form in which this approximation is performed. This choice is eventually a particular parametrization $\left \lbrace x,y \right \rbrace$ of the same amount of symmetry breaking $\Delta \mu$. Furthermore, this invariant observable shall depend on the parameter of the approximation, $\chi$. Some results are shown in Fig. \ref{fig:delta_mu}.
The best fit to our data corresponds to a scaling law of the form
\begin{gather}
\nonumber
\Delta\mu = a \chi ^{-b} \\
a = 0.88 \pm 0.03 \ \ \  b= 0.65 \pm 0.02 \
\end{gather}
with high accuracy as seen in Fig. \ref{fig:delta_mu}.

It is natural to expect the
exponent $b$ to be related to the general critical exponent $\kappa$ introduced in Eq. (\ref{eq:scaling}), since both of them parametrize the effects due to the finiteness of $\chi$.

\section{Discussion}

Beyond the well-known numerical strategies for computing a Matrix Product State approximation, exact results for the ground state of a low dimensional MPS can
 be found  using an analytical minimization strategy. The arbitrariness in the parametrization of matrices $\Gamma$ allows us to choose a simple structure for these matrices so that  the minimization problem can be solved analytically. This simplification is guided by the symmetries of the problem.

A number of results emerge from our anaytical approach. First, the exact minimum
found for low $\chi$ cases agrees perfectly with the numerical results obtained
with iTEBD or DMRG. This means that both methods avoid local minima and find
the optimal solution consistent with the MPS ansatz. Second,
a generic MPS does not preserve any symmetry of the problem beyond the 2-site translational invariance due to the fact that the algorithm looks for a minimum
of the Hamiltonian expectation value. Thus,
for the Heisenberg chain, the 1-site translational invariance of its ground state, and the property of being eigenstate of the total $Z$-component of spin are explicitly violated. However, it is possible to constrain the MPS to preserve one of them while remaining an absolute minimum for the energy. Yet, if both symmetries are imposed at the same time the MPS ansatz does not get down to the same minimum.

For low-$\chi$ MPS it is not difficult to find the suitable parametrization that allows us to preserve a chosen symmetry, as we have shown on examples. It is also possible to
quantify symmetry restoration using, as a figure of merit, the
difference of eigenvalues of the 1-body reduced density matrix. We observe
that symmetry gets restored when $\chi$ increases following a detailed scaling law.

The method we have presented can be extended to any local Hamiltonian that displays
translational invariance. This is the key ingredient to reduce the computation to
a simplified minimization. Beyond nearest-neighbor interactions or higher
spins can also be
accommodated with straightforward extensions of the iTEBD algorithm.

\begin{acknowledgements}
We acknowledge fruitful discussions with G. Sierra, S. Iblisdir,
T. R. de Oliveira and J. M. Escart\'in.
We also ackowledge financial support from MICINN, UE QAP,
SRG.
\end{acknowledgements}

redpajama_set_name: RedPajamaArXiv


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Unfortunately it's been off the road for about 8 months and is overgrown with weeds etc now.. but this is how it once stood.
Meg 250 engine Clio, running 320/340.
This was my Megane powered Clio 197 that I built a few years back.
Miss this car as was good fun, may have to build another one soon.
There seems to be a bit of a myth that CS is solely for 172/182's.
Is this still true? I'm not on here much anymore but it seems the Mk2 numbers are dwindling these days.
Some really nice looking ones here, but the white cup racer looking one is bloody mega!!
Only a couple of bits left to do to it now, purple calipers, wheel retrim, and some point down the line a remap; mainly cause I have a non-res Scorpion and a predator manifold fitted and the flat spot can be pretty brutal and it runs rich, which is a pain as it's my daily.
Here we go then! FKU in all her glory!
My old 200, grey leather quilted recaros, black wheel centres and silver visions.
C'mon @Sunglasses_Ron how can you not like that spoiler?
That's a great pic @Gus. This thread shows what a bugger alien green is to capture properly in a photo.
There might not be that many 197/200s on here but the ones in thread all look stunning.
Here's mine, just been retired from daily so purely a weekend toy.
Picture just taken 10 minutes ago.
Felt like it needed another 100bhp but loved climbing into the Recaros and the handling was like nothing ive driven before or since.
Well might as well then.
You know what Ronald, at least you're consistent.
Needs some bits sorting but should look really nice once sorted.
Had mine a year today.
Get yourself along to the next North Weald meet mate.
And if you can make it Www.Cliosportfestival.Net is just about the best clio weekend of the year.


text:
More and more Italians are hesitant about taking the COVID-19 vaccine | Instant News
November 26, 2020 NewsDesk
Italy is preparing to launch a massive vaccination campaign in January, according to Health Minister Roberto Speranza.
The Italian government has appointed a task force to plan how and where the first dose will be given when international regulators approve a successful vaccine.
Scientists estimate that 60-90 percent of the population will need to be vaccinated – perhaps annually – to achieve herd immunity to the coronavirus and stop future outbreaks.
The Health Minister stated that the vaccine will be free in Italy and appears to be voluntary. And although the vaccine is not mandatory, doubts are growing among Italians about its safety.
Of the 800 people who took the recent survey by the research institute "SWG", nearly 40% said they would not take the risk.
Scientific research has broadened the choice of vaccine candidates, but even that has raised some doubts. The majority of people surveyed would rather have the Italian vaccine than the European or American vaccine.
According to the plan outlined by the government, health workers and medically vulnerable groups will be the first to receive it.
Italians may soon get a "vaccination passport" to keep track of their vaccination records, to be able to move domestically or travel abroad.
But experts are careful. Scientist Maria Rosaria Capobianchi, head of one of the first teams of virologists in Europe to isolate a viral DNA sequence, said it's not just the vaccine that will make the difference.
"This vaccine is not the only product available, we hope we will offer treatment accordingly. But the main tool is the behavior that each of us follows. This proved to be a very effective measure to contain the spread of the virus. "
It is unlikely that the vaccine will relieve people from wearing face masks or respect social distancing in the short term. Experts warn that it is too early to say how long it will take to fully return to normal.
Italy, among European countries now struggling with the second wave of the pandemic, has recorded 1.3 million cases and a death toll of nearly 50,000 since the coronavirus struck earlier this year.
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text:
Al Sharpton: 'Progressives' are shortchanging African Americans — again.
Democrats might care about issues that are important to us. But are they fueling African-American participation or interest? Not even close.
Al Sharpton: 'Progressives' are shortchanging African Americans — again. Democrats might care about issues that are important to us. But are they fueling African-American participation or interest? Not even close. Check out this story on USATODAY.com: https://usat.ly/2uoaiSL
Al Sharpton, Opinion contributor Published 5:00 a.m. ET Aug. 9, 2017 | Updated 1:24 p.m. ET Aug. 10, 2017
Hillary Clinton in 2016.(Photo: Carlos Osorio, AP)
When Jesse Jackson ran for president during the 1980s, as when I ran in 2004, there were progressives in America just like there are today. Those progressives were well-meaning individuals and politicians who shared our views and strongly believed in what we believed in. Despite this progressive political presence, our presidential campaigns were so important and necessary because the voices of black, brown and poorer white voters were not heard by the elites in American politics and government. Our agendas were not getting carried out. There was a great deal of talk back then, but no real action. That same dynamic holds true today.
The press speaks a great deal about the supposed fact that the "Democratic base" is riled up and activated by the state of play in America. This assessment ignores the most important segment of that base: the African-American voter. We are not motivated by anyone right now. While Sen. Bernie Sanders, I-Vt., did a remarkable job in the 2016 presidential primaries and went further than anyone thought possible, he did so without the African-American vote, losing among African-American voters by more than 50 percentage points.
More: Democratic 'Better Deal' robs from the future
More: Rep. David Cicilline: A Better Deal is a bold agenda
While that progressive coalition purported to speak for the African-American voter, it did not talk to African Americans. The so-called Hillary Clinton base of the party, while crushing Sanders, attracted substantially fewer black voters to turn out than in recent presidential primaries. And in the general election, with Clinton running against a novice, the black voter turnout rate declined for the first time in 20 years in a presidential election, falling 7 percentage points compared with 2012. Arguably, that disinterested black vote cost Clinton the presidency.
It would be unfair to claim that leaders like Clinton and Sanders do not care about issues that are important to people of color. They do. However, it is equally inaccurate to claim that the current progressive movement is fueling African-American participation or interest in our political process. It is not. Blacks largely sit on the sidelines while the game of politics is being played around us. In the post-Obama era, there is the sense Democrats feel that people of color — African Americans in particular — have had their chance, and that we should now take a back seat to new leadership and let them handle the politics of today. Such a sentiment is both foolhardy and wrong.
More: Jeff Sessions' Justice Department goes after affirmative action's institutional racism
The 21st century version of the rainbow coalition lacks vision and color. Remarkably, blacks still need to fight for a seat at the table and are too often simply stage props for allied elected leaders to make their points. Consider this: In 2016, when Sanders ran for president as a Democrat, there was one black chief of staff in the Senate working for a Republican, and none for Sanders or the Democrats.
Talk is not enough anymore to be on the righteous path for justice and black political participation. Nor is caring about, or sympathy over, unjust policies. An effective progressive movement is more than an intellectual exercise espousing policy goals: It requires action and results. And people of color need to be at the table in large enough numbers to help make that difference. We cannot depend upon action from well-meaning progressives or others who want to fight our fight for us. History proves that change comes too slowly when we rely on that model.
Perhaps it is time for another African-American presidential campaign to fuel black voter interest. Perhaps it's time to remind people that progressive politics cannot be advanced without results and a fully vibrant rainbow of colors working to make that difference. Two things are certain: African Americans will not be taken for granted again, and progressives invite failure yet again if they try.
The Rev. Al Sharpton is president of the National Action Network. Follow him on Twitter: @TheRevAl
Read or Share this story: https://usat.ly/2uoaiSL


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What's wrong with wearing our hair the same way most days?
"Hmmm. This is intesting. Why is it that Black women have a problem, managing their hair when it's past shoulder lenght, but not any other race? I have white. asian, hispanic frineds with hair down to their waist. All they do when it's in the way is tie it up in a bun, pony tail or pin it back.
Yes, we can do they same thing. I think it's because we as black women never learned how to take care of our hair from the begining, that we have all these struggles OR we're expecting TOO much from our natural hair. We have to have a different style every other day. Why?
I definitely don't hear people say that to my white, asian and hispanic friends and they just about wear their hair the same way 24/7.
Soooooo true!!! I mean, why is it we as black girls feel we have to change up our hairstyles every single freakin day? or every week…what's so wrong about having a stable go to style just like every other girl in the world has? Personally, I've gotten so used to having different styles throughout my childhood that there's no going back. I love changing it up, I love accessories I love extensions etc. etc. But now my hair is a bit longer, why can't I wear the same mohawk look as I did yesterday, and the day before, and the day before that? Maybe cos the whole natural thing is still new for me so I wanna try getting mileage out of my stretched out hair so maybe after a year or so, I'll start to lessen the number of different styles I do and only really go all out when there's a party or something special. But there I was, secretly hating on peeps who wear their hair the SAME way ALL the time but if that's how they want to wear it, why the hell not? I LOVED my look yesterday, it turned out very nicely so why not do it again today? Very interesting.


I read the post about the loose wire not allowing cold water into the machine. It is bull ****. The motor hums when i turn it on but then the code starts flashing. Tip: The max point reward for answering a question is 15. 0 characters Video Photos Link Replacement Parts Add Upload Upload × × Draw a box over the problem!!
Contact Us|Terms of Service|Privacy & Security|About Us|Our Network|Sitemap © 2003-2016 JustAnswer LLC JustAnswer UKJustAnswer GermanyJustAnswer SpanishJustAnswer Japan 51 6040809 Login|Contact Us Appliance Need Appliance Repair Help? The machine will not beep during this time. Correct problems.1 Beep/E10--Error: Dispenser drawer is open Cause: Washer will not operate if drawer is open.2 Beeps/E20--Error: Water dosen't drain. models) you will experience more starts and stops throughout the entire cycle.
Request unsuccessful. It is flashing and beeping three times every minute or so. Next look inside the valve where the hoses hook up and you will see the screens. All rights reserved. © Applianceblog.com 2004 - 2014 Help Remember Me?
Cancel replyLeave a CommentName *E-mail *Website Current [email protected] * Leave this field empty Notify me of follow-up comments by email. The washer door is still locked.
and other Appliance Specialists are ready to help you Ask your own question now Share this conversation Related Appliance Questions Question Date Submitted Samsung Refrigerator Model # = RSG257AARS screen is I have checked all of these things and these do not appear to be the problem. It shouldn't take you more than 5 minutes. ? · 2 years ago 0 Thumbs up 0 Thumbs down Comment Add a comment Submit · just now Report Abuse Add your Remember to remove power first.
Loading...Oops :Please try again. ⌂HomeMailSearchNewsSportsFinanceCelebrityWeatherAnswersFlickrMobileMore⋁PoliticsMoviesMusicTVGroupsStyleBeautyTechShopping Yahoo Answers  Sign in ✉ Mail ⚙ Help Account Info Help Suggestions Send Feedback Answers Home All Categories Arts & Humanities Beauty & Style Business Cause: Washer will not operate if door is open. Can't find a source for what these codes are. Please help!Reply 130 DIY Project Help Tips December 8, 2015 at 8:20 pmChristina Murray, Can you give us the model number of your Whirlpool washer?
prev next 3 Points Related Questions: 1 Answer Why does it beep through the entire. If the washer is not draining properly check the drain hose for kinks or clogs. Can you please advise how I might be able to fix this issue.Reply 13 Umar September 21, 2016 at 1:55 amI have a Toshiba washing machine and when i turn on October 10, 2016 Samsung Refrigerator OF OF Code On Display - How To Clear?
Frigidaire gallery professional series not working? Th... I changed to rapid wash with a 30 minute cycle and it worked okay. Or just drag it here!
What happens exactly when you get the LC error code? Can anyone help me to get the of location of temperature sensor in Imesa washing machine ASEP 2000.Reply 103 DIY Project Help Tips February 10, 2016 at 12:16 amNAREN, You may Rinse Aid. I can't find much information (or the manual) online, any chance you can help?
89 judy March 10, 2016 at 12:43 pmBosch front load washing machine ticking sound the last minute of cycle. You may have to register before you can post: click the register link above to proceed. To start viewing messages, select the forum that you want to visit from the selection below. You may NOT copy or distribute the content that appears on this site without written permission from Fixya Ltd. © 2005-2015, Fixya, Ltd.
Add your answer Sign in to answer a question 1 Answer from this member: The E10 error indicates that one of a group of error codes occurred. Cause: Washer will not operate if door is open. September 26, 2016 LG Washing Machine Error Code dE - How To Clear? Find the release ring below the locking mechanism.
Share on Facebook Share on Twitter Share on Google+ Share on Pinterest Share by Email × Question about Frigidaire Affinity ATF7000E Front Load Washer 8 Answers E10 error Shortly after starting That's why I said to make sure you're getting good water pressure to the washer. It keeps reading error message FA. The error code chart indicates that this could be a water flow problem, too low of a drain hose height or an oversudsing problem.
When you start the QC test be sure to listen for the lock engageing. If none of this works you may have an issue with the water inlet solenoids. June 24, 2016 Stove Top Temperature Control Knobs - What Do The Numbers Relate To? - Low, Medium, High June 21, 2016 Window Air Conditioner Leaking Water Into House - What The power continued but the machine would just hum and not do anything, I was turning it on and off until it started displaying the F sign.
Yes No Sorry, something has gone wrong. or the screens are clogged? Expert: Tyler Z. Check it to be sure it is not broken or has loose wires. It just sits there flashing the codes.
Make sure you're getting a good water supply to the washer on both hot and cold water line. Machine malfunctioned three days after the one year warranty expired. So we would use a small amount of concentrate and it was hit or miss on the error code popping up. Error code F 21 code shows up when washing halfway through.


text:
Front Page » Government » Miami moves to regulate drones
Miami moves to regulate drones
Written by John Charles Robbins on December 2, 2015
Miami is moving to regulate drones as their popularity skyrockets.
City Commissioner Frank Carollo introduced a proposed ordinance, which passed on first reading Nov. 19.
Drones, also known as unmanned aircraft systems or UAS, need to be controlled, Mr. Carollo said.
He's worried the Miami sky will become filled with drones, endangering public safety.
Mr. Carollo pointed out that Miami, with warm and usually welcoming weather year around, has a host of popular outdoor events that attract thousands of participants. Those include the annual Calle Ocho Festival, Miami International Book Fair and the nationally-televised New Year's Eve show hosted by entertainer Pitbull.
"Mr. 305" is set to host the Dec. 31 show from Bayfront Park for the second year in a row, a concert that's televised live, according to Mr. Carollo.
Mr. Carollo says he believes drones will be one of the more popular gift items this Christmas, so the timing of his legislation is important. The administration was directed to have a finished version ready for a final vote Dec. 10.
Mr. Carollo said he wants to avoid a situation where several drones – fresh out of wrappings as Christmas presents – hover over the assembled masses at Bayfront Park on New Year's Eve.
He called drones "a motor vehicle of the air" and said police need ways to locate the owner-operator of a drone, particularly if it's violating the law or posing a threat to the public.
In addition, authorities need a tool to bring it down safely, he said.
"It's all about public safety," said Officer Marcos Perez with the Miami Police Department's Office of Emergency Management and Homeland Security.
At a terrorism awareness conference the office hosted in September, one topic was a briefing on drones from Federal Aviation Administration officials.
Mr. Carollo said he's working with Officer Perez and others to draft the legislation.
Officer Perez said Miami was one of the first US cities to use computers in police patrol cars, and the city can again be at the forefront by having one of the nation's first ordinances to control drones.
Referring to the New Year's Eve event in Bayfront Park, Officer Perez said "we can't afford to have a drone go down on Dec. 31."
Commission Chair Wifredo "Willy" Gort said now is the time to meet potential problems in advance.
"It's no longer a toy," he said of drones, "and could be very dangerous."
Mr. Gort asked police to be prepared to show commissioners how they plan to enforce the new rules. He said he'd like to require drone operators to register.
Mr. Carollo said the ordinance will get tweaked before the final vote, giving consideration to impounding drones temporarily versus confiscating them.
"The intent here is not the enforcement part, it's the public safety part," Mr. Carollo said.
The proposed ordinance notes that declining cost has made drones increasingly popular and available to citizens and hobbyists, and that they can be equipped with high-definition cameras, night-vision cameras and infrared scopes, "as well as with combustive or explosive composites and materials."
Flying a drone over or near heavily populated areas can create "an unnecessary risk of accident, opportunity for counter surveillance, the delivery of harmful substances, or a terror attack in a densely populated area," the proposed ordinance says.
The city would regulate drones within a half-mile of events in parks, stadiums, open spaces, plazas and streets that attract large groups. They couldn't be flown within or over sporting or large-venue special events, the ordinance says.
It states, "This ordinance is not intended to preempt FAA rules, but to operate in conjunction with those rules to promote public safety while recognizing the limitations in the FAA's enforcement capabilities."
County backs bill to expand police use of drones
Miami's Bayfront Park to be open to public many more days
Miami to rename Museum Park for Maurice Ferré
Miami parking price hike held hostage for marinas rate raise
Miami will try again soon to hike parking, marina rates
Dinner Key Marina may go to bid for private operator
Miami moves toward filling in FEC Slip
Juggernaut mall touted to replace Miami golf course
Commission probes police policy for active shooters
Miami wants bigger Bayside Ferris wheel receipt share


There are numerous accounts of the origin of the pizza. Some say it started with the Greeks, while others contend that the Egyptians made the first pizza. In Italy, it was once an inexpensive treat topped with oil, cheese, tomatoes, garlic, and anchovies. The first time the term "pizza" was used was in 997 in Gaeta.
Whatever the origin, the pizza is already on the list of favorite food of many Cebuanos. Due to this, we came up with another list where you can satisfy your pizza cravings while you are in Cebu.
Biaño's Homemade Pizza is another homegrown pizza place that has caught the fancy of many Cebuanos. The owner, Alberto Biaño, decided to learn how to make pizza after being dissatisfied with a pizza he ordered from a well-known pizza restaurant. His first customers were his neighbors, who gave good reviews for his pizza. The first branch of Biaño's Homemade Pizza opened its doors to the Cebu market in 2007. Biaño's Homemade Pizza already has numerous branches all over the archipelago. The prices of their pizza start at PHP49 for a six-inch Garlic and Cheese Pizza up to PHP220 for the 11-inch Full House pizza.
Similar to many homegrown pizza places, Calda Pizza started with one outlet to check if the market will respond well to their pizza. The first branch was located along Salinas Drive in Lahug. It was a combination of a pizza place and a convenience store. The Cebuano market responded well to the pizza offered by Calda Pizza and the company decided to open more branches around metro Cebu before expanding to other places around the country. The pizzas of Calda Pizza are priced between PHP165 for the Regular 11-inch Margherita and PHP2,050 for the Monster 48-inch Quattro Stagione.
La Bella Pizza Bistro offers authentic Italian dishes to residents and visitors to Cebu. Similar to restaurants offering authentic Italian dishes, the place uses only the best ingredients for their pizza. Formerly called La Bella Napoli, the pizza at the restaurant is handmade and customers have the chance to see how the pizza is created. La Bella Pizza Bistro has branches in Mabolo, Maribago, the Mactan-Cebu International Airport, and in Dumaguete. The prices of the pizza at the bistro start at PHP220 for the solo size of the Margherita up to PHP550 for the family size of the Prinipessa pizza.
Owned by Chef Giuseppe Genco, Giuseppe Pizzeria & Sicilian Roast is considered among the best Italian restaurants in the Queen City of the South. The pizzas offered at the restaurant use fresh ingredients and are cooked using wood-fired ovens. Giuseppe Pizzeria & Sicilian Roast is located in Paseo Saturnino, Maria Luisa Road, Banilad. The pizzas offered at Giuseppe Pizzeria & Sicilian Roast are priced between PHP400 for the Biancaneve All Aglio and PHP620 for the Gregorio.
Mr. A is more than just a place to hang out with friends over a cold bottle of your favorite brew since they also have a pizzeria. Mr. A Pizzeria offers a good selection of thin-crust pizzas for customers who want to enjoy this Italian dish while watching the sunset on the mountains of Busay. The prices of the pizza at Mr. A Pizzeria start at PHP300 for the Margherita up to PHP390 for the Meat Lovers pizza.


text:
Renowned Cellist Maya Beiser and Dance Icon Wendy Whelan Perform in a Rare Collaboration Among Legends
Featuring an original score by Pulitzer Prize-winner David Lang and choreography by Lucinda Childs, The Day represents a landmark co-presentation between OZ Arts Nashville and TPAC in partnership with Nashville Ballet.
"America's greatest contemporary ballerina." – The New York Times on Wendy Whelan
"Like no other… Haunting and very, very beautiful" – Los Angeles Times
"A gut punch… nothing short of life itself: by turns hopeful, funny, surprising and tragic."
NASHVILLE, TENN. – Dec. 9, 2019 – The Day is a new music and dance work by renowned cellist Maya Beiser, New York City Ballet legend Wendy Whelan, seminal choreographer Lucinda Childs, and Pulitzer Prize-winning composer David Lang. A collaboration among masters from the worlds of ballet, contemporary dance, and music, The Day is a sensory exploration of two journeys—life and the eternal, post-mortal voyage of the soul. Inspired by reactions to 9/11, this bold, highly collaborative work explores universal themes through the shared language of music and dance.
Joining forces across artistic genres, four creative minds at the top of their disciplines bring their individual mastery to this soul-stirring performance. Louisville native Wendy Whelan, widely considered one of the world's leading dancers, spent 30 years as a principal dancer with New York City Ballet and originated numerous roles in new works by the world's most esteemed choreographers. Cellist Maya Beiser, who conceived the piece, has been described by the Boston Globe as "a force of nature" and by Rolling Stone as a "cello rock star," and is a veteran of the world's most revered stages. The two will be onstage all evening, embodying the iconic choreography of Lucinda Childs (a Commandeur in France's Ordre des Arts et des Lettres and 2018 inductee in Hall of Fame at the National Museum of Dance) set to original music by Pulitzer Prize-winner David Lang.
This rare, one night engagement on January 18 is made possible by an unprecedented collaboration between three of Nashville's leading arts institutions—co-presenters OZ Arts Nashville and Tennessee Performing Arts Center with partner Nashville Ballet.
OZ Arts Artistic Director Mark Murphy says, "It's thrilling that this groundbreaking collaboration by four renowned masters of their crafts has also become a chance to bring together three of Nashville's influential arts organizations. I was very impressed that TPAC and Nashville Ballet both instantly agreed to come on board when we approached them with this idea. Critical response for The Day has been astounding across the country and I couldn't be more excited to bring this beautiful work to Nashville right before its European debut in Paris."
"The Day is a look at the impact of September 11, the way we as a culture view life and death, and how a single moment in time can affect that view," says Christi Dortch, Executive Vice President for Programming and Sales at TPAC. "It just makes sense for our organizations to partner on something for which we both have a passion and felt deserved a chance to reach a larger audience."
Nashville Ballet Artistic Director Paul Vasterling says, "Wendy is one of the greatest American ballerinas of our time—she's the artistic director of New York City Ballet and she represents this incredible standard for exceptional choreography and artistry, so it's an honor to have her here in Nashville. Our city's arts community is growing and thriving because of partnerships like this—with organizations like Nashville Ballet collaborating with other groups like OZ and TPAC—and I hope it leads to even more inspiring dance engagements in Nashville."
The Day had its world premiere earlier this year at the esteemed Jacob's Pillow Dance Festival and has been shown at leading performance venues around the country, including UCLA's Center for the Art of Performance in Los Angeles and The Joyce Theater in New York. The one-night-only Nashville engagement allows local audiences an opportunity to experience the piece in person just before it's European premiere at Théâtre de la Ville in Paris.
The Day video trailer:
https://www.youtube.com/watch?time_continue=4&v=wBm21WjrB2Q&feature=emb_title
The Day, co-presented by OZ Arts Nashville and TPAC in partnership with Nashville Ballet, will be performed on January 18 at TPAC's Polk Theater. Tickets start at $37 and are on sale now at ozartsnashville.org, TPAC.org, and by phone at (615) 782-4040.
Maya Beiser (Cellist) is a 2015 United States Artists Distinguished Fellow in Music; a 2017 Mellon Distinguished Visiting Artist at MIT Center for Art, Science & Technology; and a Presenting Artist at the inaugural CultureSummit, held in 2017 in Abu Dhabi. Reinventing solo cello performance in the mainstream arena, Maya has been a featured performer on the world's most prestigious stages, including Lincoln Center, Carnegie Hall, Brooklyn Academy of Music, John F. Kennedy Center for the Performing Arts, Royal Albert Hall, Sydney Opera House, and the Beijing Festival, among many others. She has appeared with many top orchestras, including Detroit Symphony, BBC Concert Orchestra, Boston Pops, Sydney Symphony, China Philharmonic, and Swedish Chamber Orchestra. Maya's vast discography includes eleven solo albums and work as a soloist on many film soundtracks, including M. Night Shyamalan's The Happening and After Earth, The Great Debaters, Blood Diamond, and Snow White and the Huntsman.
Wendy Whelan (Dancer) was recently appointed Associate Artistic Director of New York City Ballet, following a three-decade career as a dancer with the company. In that time, she danced nearly all of the major Balanchine roles, worked closely with Jerome Robbins on many of his ballets, and originated roles in works by William Forsythe, Twyla Tharp, Alexei Ratmansky, Jorma Elo, Shen Wei, and Wayne McGregor. Her most notable choreographic collaboration was with Christopher Wheeldon, who created roles for Wendy in 13 of his ballets, including Polyphonia, Liturgy, and After the Rain. With Wheeldon's own company, Morphoses/The Wheeldon Company, Wendy also originated several roles, garnering Olivier Award and Critics Circle Award nominations for her work in 2007. She has been a guest artist with The Royal Ballet and the Kirov Ballet, and has performed all over the U.S., South America, Europe, and Asia. She received the Dance Magazine Award in 2007, an honorary Doctorate of Arts in 2009 from Bellarmine University, and both The Jerome Robbins Award and Bessie Award in 2011 for Sustained Achievement in Performance.
Lucinda Childs (Choreographer) began her career at the Judson Dance Theater in 1963. Since forming her company ten years later, she has created over fifty works, both solo and ensemble. In 1976, she was featured in the landmark avant-garde opera Einstein on the Beach by Philip Glass and Robert Wilson, for which she won an Obie Award. She subsequently appeared in a number of Wilson's productions, including I Was Sitting on My Patio This Guy Appeared I Thought I Was Hallucinating, Quartett, and White Raven, among others. In 1979, Childs choreographer one of her most enduring works, Dance, with music by Philip Glass, which continues to tour internationally and has been added to the repertory of the Lyon Opera Ballet. She has choreographed over thirty works for major ballet companies, including the Paris Opera Ballet and Les Ballets de Monte Carlo, as well as directing and choreographing a number of contemporary and 18-century operas. Childs holds the rank of Commandeur in France's Ordre des Arts et des Lettres, has received the Golden Lion award from the Venice Biennale and the Samuel H. Scripps American Dance Festival Award for Lifetime Achievement, and was recently inducted into the Hall of Fame of the National Museum of Dance.
David Lang (Composer) won the Pulitzer Prize for Music for his composition the little match girl passion, based on the Hans Christian Andersen story "The Little Match Girl." His extensive catalogue of opera, orchestra, chamber, and solo works make him one of America's most-performed composers. Lang's simple song #3, written as part of his score for Paolo Sorrentino's acclaimed film Youth, received numerous awards nominations in 2016, including the Academy Award and Golden Globe. His works have been performed around the globe by BBC Symphony, the International Contemporary Ensemble, New York Philharmonic, Boston Symphony, and Munich Chamber Orchestra, among many others. His music is regularly used for ballet and modern dance around the world, including Twyla Tharp, Paris Opera Ballet, New York City Ballet, Susan Marshall, and Nederlands Dans Theater. He is the recipient of numerous awards and honors, including Musical America's Composer of the Year, Carnegie Hall's Debs Composer's Chair, the Rome Prize, the BMW Music-Theater Prize, Bessie Award (1999, Susan Marshall's The Most Dangerous Room in the House), and Best Small Ensemble Performance Grammy Award in 2010 for the recording of the little match girl passion.
About OZ Arts Nashville's 2019-2020 Season:
The 2019-2020 season is the 7th season for OZ Arts and the first under its new Artistic Director Mark Murphy's direction. This dynamic season of its signature "Brave New Art" features a unique blend of influential contemporary artists and ensembles from around the world, as well as groundbreaking Nashville-based artists, with a special emphasis on multidisciplinary performances, and work that highlights unique new uses of visual and audio technology. The season is made possible with generous support from season sponsor Advance Financial.
Remaining Season Presentations After "The Day" Include:
Notes of a Native Song by Stew & Heidi with The Negro Problem (Jan. 31 – Feb. 1, 2020)
The Triangle by New Dialect (Feb. 20-22, 2020)
Plata Quemada (Burnt Money) by TeatroCinema – Chile (March 5-7, 2020)
Steal Away by Dave Ragland with Inversion Vocal Ensemble (April 17-18, 2020)
Grace and Mercy by Ronald K. Brown / EVIDENCE (May 8-10, 2020)
This Holding by Jana Harper (May 29-30, 2020)
Grand Magnolia by Jessika Malone and Collaborators (June 10-21, 2020)
One-minute highlight reel of 2019-2020 Season: OZ Arts 2019-2020 Season Highlight Reel
About OZ Arts Nashville
Since opening in 2014, OZ Arts Nashville, a 501(c)(3) contemporary arts center, has changed the cultural landscape of the city. Housed in the former C.A.O. cigar warehouse owned by Nashville's Ozgener family, OZ Arts, under the artistic leadership of Mark Murphy, brings world-class performances and art installations to the city, and gives ambitious local artists opportunities to work on a grand scale. The flexible 10,000 square-foot, column-free venue, nestled amidst five acres of artfully landscaped grounds, is continually reconfigured to serve artists' imaginations, and to challenge and inspire a diverse range of curious audiences.
OZ Arts regularly engages the community for participation with visiting artists and artworks – either directly, through school visits, workshops, master classes, school performances and/or curated programs led by local teaching artists. In addition, OZ Arts founded a program called OZ School Days, a daylong, multi-arts program presented in partnership with Centennial Performing Arts Studios that aims to engage students aged 5 – 15 years old on days when Metro Nashville Public Schools are out of session (ex: Columbus Day, Presidents Day).
For more information, please visit http://www.ozartsnashville.org/.
About Tennessee Performing Arts Center:
The nonprofit Tennessee Performing Arts Center, located in downtown Nashville, is dedicated to providing and supporting the presentation of the performing and cultural arts. Founded in 1980, TPAC serves several hundred thousand audience members each year. Performances include the HCA Healthcare / TriStar Health Broadway at TPAC series, a variety of special engagements, and the productions of three resident artistic companies – Nashville Ballet, Nashville Opera, and Nashville Repertory Theatre.
In addition, TPAC administers one of the largest and most comprehensive arts education programs in the United States, serving students from pre-school to high school, educators, and adults. Over almost four decades, TPAC has welcomed more than 12 million audience members and served more than 1.8 million students, educators, and adults with performances at TPAC, teacher resources, professional development opportunities, classroom residencies, and enrichment programs.
For more information, please visit https://www.tpac.org/.
Amy A. Atkinson
Amy Atkinson Communications on behalf of OZ Arts Nashville
Cell – (615) 305-8118
Email: amy@amyacommunications.com
Brittany Carlton
OZ Arts Nashville
Email: Brittany@ozartsnashville.org


As we conclude our sermon series this weekend, again considering how God has woven lessons about of life and faith even into the tiniest of creatures, we consider the spider. Of all our sermon insects: bees, ants, and locusts, spiders seem to raise the most emotional reaction even though we are all far more likely to be stung by a bee or to have an ant come crawling across our patio or porch than to have a spider cross our path. People seem to have a more visceral reaction to spiders. But I encourage you to think about the lesson of the spider or as we call it in our sermon title this weekend, the Spider's Secret.
SATURDAY NIGHT WORSHIP AT 5:00 in TRINITY HOUSE (air conditioned) – The Ten Simple Rules praise band will lead in music and Pastor Don Gebhard will share the sermon.
8:15 CHAPEL – Rev Paul Wells leads and preaches and his wife Jan will share a song.
9:15 SANCTUARY WORSHIP – The Meanz of Grace praise band leads in music, recently confirmed youth Richard Cort leads us in prayer and Linda Gonia will read the scripture lessons. I'll share the message. A reminder that in this service in the summer we have a Children's Time and Creative Activities (where Charlotte's Web will be shown) for kids from age 4 to grade 4.
10:45 SANCTUARY WORSHIP – Pastor Deb De Vos will lead worship and preach and I'll be the liturgist. We'll have solos from Jen Caldwell and Jan Wells.
In all services, you can sign up to serve in our Caravan of Hope, Media ministry or Thrift Shop.
I hope you'll make worshipping God for an hour a priority this weekend. Blessings and peace to you.


The authorities have imposed new changes with effect from 1 September 2018 on the rules governing traffic in the Turkish Straits.
Failure to comply with company procedures regarding entry into enclosed spaces can be fatal.
Skuld encourages all its members and assureds to conduct early planning regarding the Global Sulphur Cap 2020 limits.
Breaching INL from a charterers' perspective, both in terms of potential liability for damage to the vessel and in terms of potential cover issues.
Skuld today announces that Hilde Søbstad Løvskar has been appointed chief legal officer (CLO).
2018 saw a huge increase in bunker quality disputes, starting around March/April with bunkers stemmed in Houston and gradually spreading to different regions and countries.
Skuld today announces that it will underwrite all its hull and offshore energy business on the A-rated corporate paper of Skuld Assuranceforeningen, through Skuld UK in London and SMA in Oslo.
Skuld announces a positive outcome from renewals with a 4.3% net increase in mutual P&I gross tonnage year-on-year from 2018 to 2019, including committed tonnage for delivery throughout 2019.
Easy and secure access to your statistics, policies and certificates, claims, charter declaration and more.