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

It has long been conjectured that for a wide range of quantum systems described by gapped local Hamiltonians, the entanglement entropy with respect to any bipartition of the system scales as the boundary area. Such entanglement entropy scaling is known as the \emph{entanglement area law}. An entanglement area law for the ground state of 1D quantum spin systems was first proved in the seminal paper by Hastings \cite{Hastings2007area}, and the scaling with respect to the spectral gap was improved by later work such as \cite{ALV12,AradKitaevLandauVazirani2013area}. Entanglement area laws have also been proved for degenerate ground state and low-lying eigenstates \cite{ALVV17}. Limited results are also available for higher-dimensional quantum systems, especially for the case where the Hamiltonian is frustration-free \cite{anshu2020entanglement,anshu2021area}. For 1D systems, whether a quantum state satisfies an entanglement area law is an important criterion for determining whether it can be approximated by a matrix-product state\cite{fannes1992finitely}, which is a key component in the density-matrix renormalization group (DMRG) algorithm \cite{white1993density} (see also \cite{Gharibian2015Hamiltonian}). Using theoretical tools constructed for proving the 1D area law, a polynomial time algorithm for computing the ground state of 1D gapped local Hamiltonians was proposed in \cite{landau2015polynomial}.

The aforementioned results are all proved in the setting of quantum spin systems, i.e., each lattice site is associated with a finite-dimensional local Hilbert space.
However, there are many quantum systems of practical interest that involve infinite-dimensional local Hilbert spaces. Examples of such include bosonic systems and gauge theories. When a quantum system involves bosons, each bosonic mode corresponds to an infinite-dimensional local Hilbert space, due to the fact that its occupation number can be arbitrarily large. A similar situation arises when we consider lattice gauge theories (LGTs), with the Hamiltonians constructed according to \cite{kogut1975hamiltonian}. Given a fixed lattice discretization of a gauge theory, each gauge link (an edge of the lattice) has a local Hilbert space that is spanned by all the elements of the symmetry group, which is infinite dimensional when the symmetry group contains infinitely many elements. The gauge theories that are of the greatest interests, i.e.\ the U(1), SU(2), and SU(3) theories, all fall into this category.

When the dimension of the local Hilbert space is infinite, the area law proof techniques developed for the finite dimensional case cannot be directly applied.
By the state-of-the-art 1D area law result in \cite{AradKitaevLandauVazirani2013area}, across a given cut in a 1D system described by a gapped local Hamiltonian, the entanglement entropy scales as $\mathcal{O}(\Delta^{-1}\log^3(d))$ where $d$ is the local Hilbert space dimension and $\Delta$ is the spectral gap.
This becomes infinity when the local Hilbert space is infinite-dimensional. Moreover, in quantum spin systems all local Hamiltonian terms can be rescaled to have operator norm at most $1$, whereas in the models we consider in this work the local Hamiltonian terms can be unbounded, making it impossible to normalize them.
For certain non-interacting bosonic systems, such as those involving infinite-dimensional local Hilbert spaces but are exactly solvable, the entanglement area law has been proven in Refs.\ \cite{CramerEisertEtAl2006entanglement,CramerEisert2006correlations,CramerEisertPlenio2007statistics} (see also \cite{Eisert2010arealaw}).
However, a general methodology remains elusive for establishing area laws for quantum systems with infinite local degrees of freedom.

\medskip
\noindent \textit{Main result.}
In this work we prove an entanglement area law for a class of 1D quantum systems that involve bosons or arise from gauge theories, using the approximate ground state projector (AGSP) framework developed in \cite{ALV12}. These quantum systems involve infinite-dimensional local Hilbert spaces. Examples of such quantum systems include the Hubbard-Holstein model, U(1) and SU(2) LGTs, all of which are defined on a 1D chain. For these models, we can introduce a notion of \textit{local quantum number}, which is the occupation number in the bosonic case, the electric field value in the U(1) LGT, and the total angular momentum in the SU(2) LGT.
Informally, our main result can be summarized as follows:
\begin{thm*}[Main result (informal)]
For the gapped ground state of the 1D Hubbard-Holstein model, or the 1D U(1) or SU(2) LGTs, the entanglement entropy across a cut scales like $\mathcal{O}(\poly(\Delta^{-1}))$, where $\Delta$ is the spectral gap, assuming that all coefficients in the Hamiltonian remain constant. In particular, this entanglement entropy scaling is independent of the system size.
\end{thm*}
A more precise statement can be found in Theorem \ref{thm:area_law}.
This result generalizes to other models as long as certain assumptions about the growth of the local Hilbert space dimension and quantum numbers are satisfied. We discuss these assumptions in detail in Section \ref{sec:the_abstract_model}.

\medskip
\noindent \textit{Proof strategy.}
Our result utilizes a local quantum number tail bound recently obtained in \cite{TongAlbertMcCleanPreskillSu2021provably}.
Essentially, this tail bound tells us that for the class of quantum systems we study, a spectrally isolated energy eigenstate can be well-approximated by a truncated state with low local quantum numbers.
This suggests that we may choose an effective local Hilbert space with finite dimensions and thereby apply the framework developed in \cite{ALV12,AradKitaevLandauVazirani2013area}.
To implement this proof strategy, we need to apply truncation to the local Hilbert space and by extension to the Hamiltonian terms, on each lattice site close to the cut. Through this procedure we will obtain a different Hamiltonian than the one we started out with; however, we show in Theorem~\ref{thm:robustness} that the truncation changes the ground state only by an exponentially small amount. This is similar to what is known as the robustness theorem \cite[Theorem 6.1]{AradKitaevLandauVazirani2013area}. Because of this we can show that the AGSP framework developed in \cite{AradKitaevLandauVazirani2013area} is robust to this truncation. By the ``off-the-rack'' lemma with approximate target space in \cite{OTR}, generalized from the case \cite{ALV12} of exact target state, we will prove the entanglement area law for our class of unbounded quantum systems.

As an intermediate step in our proof, we develop tools to show that the mean absolute value of the local quantum number can be bounded independently of the system size for many quantum systems without translation symmetry. We state the general version of our bound in Lemmas \ref{lem:local_quantum_number_bound_first_step} and \ref{lem:local_quantum_number_bound_general} and later specialize it to the Hubbard-Holstein model (Corollary \ref{cor:mean_abs_bound_Hubbard_Holstein}) and U(1) and SU(2) LGTs (Corollary \ref{cor:mean_abs_bound_LGT}) with open boundary conditions.

In proving area laws for unbounded quantum systems, it is tempting to plug the eigenstate tail bound from \cite{TongAlbertMcCleanPreskillSu2021provably} directly into the area law result for spin systems \cite{AradKitaevLandauVazirani2013area}. However, such a naive strategy does \emph{not} seem to work here.
The main reason is that the tail bound from \cite{TongAlbertMcCleanPreskillSu2021provably} only guarantees the proximity of the quantum states before and after truncation, but not the proximity of the corresponding entanglement entropies. In the finite-dimensional setting, one could invoke the Fannes' inequality to estimate the entanglement entropy difference \cite{watrous2018theory,wilde2013quantum}, but its explicit dependence on the dimension of the Hilbert space will ruin the area law scaling for unbounded quantum systems. In fact, the main contribution of our paper is to give a careful truncation of the local Hilbert space that does not blow up entanglement entropy of the ground state, overcoming the technical difficulties mentioned above.

Tensor network methods have been extensively applied to studying LGTs to obtain interesting numerical results \cite{PichlerDalmonteEtAl2016real,BanulsBlattEtAl2020simulating,BanulsCichyEtAl2017efficient,banuls2013mass,banuls2013matrix,silvi2017finite,papaefstathiou2021density,tagliacozzo2014tensor}. Entanglement area law is a prerequisite for the ground state to be efficiently approximable using a tensor network state, and hence a rigorous proof in the setting of LGTs is crucial to the theoretical foundation of these numerical results. Moreover, if more mathematical rigor is needed, we might want a provably accurate algorithm to compute the ground state of gauge theories in 1D, along the line of the rigorous renormalization group (RG) method proposed in Ref.\ \cite{ALVV17}. The current work can be seen as a first step towards that goal, since many of the proof techniques, such as the AGSP, are also useful components of the rigorous RG method.

\medskip
\noindent \textit{Organization of the paper.} The rest of the paper is organized as follows: in Section \ref{sec:gauge_theories_and_bosonic_systems} we will briefly introduce the class of models we consider, including the 1D gauge theories and the Hubbard-Holstein model. In Section \ref{sec:the_abstract_model} we will discuss the common structure of these models that enables a proof of area law. In Section \ref{sec:effective_dims} we will discuss properties of the ground state that can be used to truncate the infinite-dimensional local Hilbert space. In Section \ref{sec:mean_abs_local_quantum_number_bound} we prove that the local quantum numbers have a mean absolute value that can be bounded independently of the system size.
This fact removes an otherwise extra assumption in the proof of the area law.
In Section \ref{sec:ham_truncation} we will demonstrate that the ground state is robust to such truncations using properties derived in the previous section. In Section \ref{sec:area_law} we prove our main result of the entanglement area law scaling, using an ``off-the-rack'' lemma slightly tighter than the one from \cite{OTR}.
We conclude our paper in Section \ref{sec:conclusion} with a brief summary of the main result and a selection of questions for future work.

\section{1D gauge theories and bosonic systems}
\label{sec:gauge_theories_and_bosonic_systems}

In this section we first introduce some specific quantum systems that we will study, and then extract the common structure of these systems. These quantum systems are all discussed in \cite[Section~I]{TongAlbertMcCleanPreskillSu2021provably}.
We first consider the 1D Hubbard-Holstein model \cite{holstein1959studies},
a model describing electron-phonon interactions.

\vspace{1em}
\textit{The Hubbard-Holstein model.} This model is defined on a 1D chain of $N$ nodes.
Each node in the lattice, indexed by $x$, contains two fermionic modes (spin up and down) and a bosonic mode.
The Hamiltonian is
\begin{equation}
    \label{eq:ham_hubbard_holstein}
    H = H_f + H_{fb} + H_b,
\end{equation}
where $H_{f}$ is the Hamiltonian of the Fermi-Hubbard model \cite{Hubbard1963electron} acting on only the fermionic modes,
and
\begin{equation}

    H_{fb} =  g\sum_{x=1}^N (b_x^{\dagger}+b_x)(n_{x,\uparrow}+n_{x,\downarrow}-1), \qquad
    H_b = \omega_0 \sum_{x=1}^N b_{x}^{\dagger}b_x,
\end{equation}
are the boson-fermion coupling and purely bosonic parts of the Hamiltonian respectively. Here, $b_x$ is the bosonic annihilation operator on node $x$, and $n_{x,\sigma}$ is the fermionic number operator for node $x$ and spin $\sigma$.
\vspace{1em}

For gauge theories, we consider the Hamiltonian formulation of the U(1) and SU(2) LGTs \cite{kogut1975hamiltonian} in one dimension. The U(1) LGT is also known as the Schwinger model.

\vspace{1em}
\textit{The U(1) lattice gauge theory.}
The system consists of a chain of $N$ nodes with $N-1$ links between adjacent nodes. We denote each node by $x$, and the links by the node on its left end. The links are sometimes called \textit{gauge links}.

On each node $x$ we have a fermionic mode whose annihilation operator is denoted by $\phi_x$. Each link consists of a planar rotor, whose configuration can be described by an angle $\theta\in[0,2\pi]$.
The local Hilbert space is the vector space of square-integrable functions on $U(1)$.
An orthonormal basis of the local Hilbert space
can be chosen to be the Fourier basis.
More specifically we denote by $\ket{k}$ the Fourier mode $(2\pi)^{-1/2}e^{ik\theta}$, and $\{\ket{k}:k\in\mathbb{Z}\}$ form the basis we need.

We further define the operators $E_{x}$ and $U_{x}$, which act on the vector space of the links, through
\begin{equation}
\label{eq:operators_U(1)}
    E_{x}\ket{k} = k\ket{k},\qquad U_{x}\ket{k} = \ket{k-1}.
\end{equation}
The Hamiltonian for $U(1)$ LGT can then be described in terms of these operators via
\begin{equation}
\label{eq:LGT_ham}
\begin{aligned}
    H &= H_M +H_{GM} + H_E,\\
    H_M &= g_M\sum_{x}(-1)^x \phi_x^{\dagger}\phi_x, \\
    H_{GM} &= g_{GM}\sum_{x} (\phi_x^{\dagger}U_{x}\phi_{x+1}+\phi_{x+1}^{\dagger}U_{x}^{\dagger}\phi_{x}), \\
    H_E &= g_E\sum_{x} E_{x}^2.
\end{aligned}
\end{equation}
The three terms $H_M, H_{GM}, H_E$ describe the fermionic mass (using staggered fermions \cite{kogut1975hamiltonian}), the gauge-matter interaction, and the electric energy respectively.

\vspace{1em}
\textit{The SU(2) lattice gauge theory.}
We only consider the theory using the fundamental representation of SU(2), as done in \cite{BanulsCichyEtAl2017efficient}. The model is still defined on a 1D chain consisting of nodes and links, similar to the U(1) case. The Hamiltonian also takes the form \eqref{eq:LGT_ham}. One difference is that in the SU(2) theory, each node $x$ now contains two fermionic modes, whose annihilation operators are denoted by $\phi_{x}^l$, $l=1,2$. We write $\phi_x = (\phi_{x}^1,\phi_{x}^2)^{\top}$.
Each gauge link consists of a rigid rotator whose configuration is described by an element of the group SU(2) \cite{kogut1975hamiltonian}.  For the link Hilbert space, which is the vector space of square-integrable functions with respect to the Haar measure on SU(2), we can construct an orthonormal basis that consists of the quantum states $\ket{jmm'}$, where $j, m, m'$ are simultaneously either integers or half-integers with $-j\leq m,m'\leq j$. Here $j$ is the total angular momentum of the rotator, and $m$, $m'$ are the components of angular momentum along the $z$-axis in the body-fixed and space-fixed coordinate systems.

The Hamiltonian takes the form \eqref{eq:LGT_ham}, and is invariant under $\mathrm{SU}(2)$ transformations acting either from the left or from the right, which correspond to rotations of the rigid rotator with respect to space-fixed or body-fixed axes respectively.
The operators $E^2_{x}$ and $U_{x}$ are different from the U(1) case.
The operator $E^2_{x}$ is defined through
\begin{equation}
\label{eq:operators_SU(2)}
    E^2_{x}\ket{jmm'} = j(j+1)\ket{jmm'}.
\end{equation}
Because $\phi_x$ has two components, where each component is a fermionic mode,  $U_{x}$ is a $2\times 2$ matrix, where each of the 4 matrix entries is an operator acting on the link space
\begin{equation}
    U_{x} = \begin{pmatrix}
    U^{11}_{x} & U^{12}_{x}\\
    U^{21}_{x} & U^{22}_{x}
    \end{pmatrix}.
\end{equation}
An important property that we will use later is
\begin{equation}
    \label{eq:U_SU(2)_properties}
    \braket{j_1 m_1 m_1'|U^{ll'}_{x}|j_2 m_2 m_2'}=0,\ \text{if }|j_1-j_2|>1/2,\qquad \|U^{ll'}_{x}\|\leq 1,
\end{equation}
which follows from rules for the addition of angular momentum, given that $U_{x,n_i}$ transforms as the $j=1/2$ representation of $\mathrm{SU}(2)$. Here $\|O\|$ denotes the spectral norm of an operator $O$.

\section{The abstract model}
\label{sec:the_abstract_model}

We now consider a more general class of Hamiltonians that include all the previously discussed examples. We consider a 1D system consisting of $N$ sites indexed by $x=1,2,\ldots, N$. In the context of the Hubbard-Holstein model, each site $x$ contains one fermionic mode and one bosonic mode. In the context of the LGTs, each site $x$ contains a node $x$ and the gauge link $x$ to the right, except for the last site which only contains a node (unless periodic boundary conditions are assumed). As discussed in Section~\ref{sec:gauge_theories_and_bosonic_systems}, each node contains one or two fermionic modes for the U(1) and SU(2) cases respectively. The Hamiltonian is geometrically local:
\begin{equation}
\label{eq:ham_general}
    H = H_1 + H_2 + \cdots + H_N,
\end{equation}
where each $H_x$ acts on two adjacent sites $x$ and $x+1$.

The local vector spaces are allowed to be infinite-dimensional, but we need to keep track of which part of the space is relevant to our problem. To this end, we introduce the notion of the \textit{local quantum number}.
\begin{defn}
\label{defn:local}
For any $1\leq x\leq N$, we define an integer local quantum number $\lambda_x$ and corresponding Hermitian observable $O_x$ acting only on the local Hilbert space $x$, such that $O_x\ket{\lambda_x}=\lambda_x \ket{\lambda_x}$.  We further define the following:
\begin{enumerate}
    \item Let projector $\Pi_{S}^{(x)}$ be defined for any $S\subset \mathbb{R}$ such that $\Pi_S^{(x)} \ket{\lambda_x} = \ket{\lambda_x}$ if and only if $\lambda_x \in S$.  Further, we define $\overline{\Pi}_S^{(x)} = I- \Pi_S^{(x)}$ to be the projector onto the complement of $S$.
    \item We define for cutoff $\Lambda\in\mathbb{R}$ the \emph{local Hilbert space dimension} of a site $x$ to be $d(\Lambda)=\mathrm{rank}(\Pi_{[-\Lambda,\Lambda]}^{(x)})$.
    For simplicity we use the same cutoff $\Lambda$ for all sites throughout the paper where truncation is needed.
    \item We define for site $x$ the norm of the local Hamiltonian $H_x$ constrained to $[-\Lambda,\Lambda]$ to be $\mathcal{N}(\Lambda)=\|H_x \Pi^{(x)}_{[-\Lambda,\Lambda]}\| $ where $\|\cdot\|$ is the spectral norm.
\end{enumerate}
\end{defn}
Often the local observables of interest in a physical problem may have noninteger eigenvalues. If the eigenvalues are all integer multiples of a fixed number, up to some constant shift, then simply shifting and rescaling the observable can get the local quantum numbers we want. Otherwise we can apply a non-linear transformation to the local observable to map all eigenvalues to integers.
Also, henceforth we do not distinguish between $O_x$ and $\lambda_x$ and will sometimes use $\lambda_x$ as an operator.

Using the notation of Definition \ref{defn:local}, we can easily articulate a fundamental assumption for our work, namely, the norm of the local Hamiltonian as well as the local Hilbert space dimension grow modestly with the cutoff $\Lambda$.  This assumption is stated formally below.
\begin{assumption}[Local quantum numbers]
\label{assumption:local_quantum_numbers}  The scaling of the maximum local dimension $d(\Lambda)$ and the norm of the local Hamiltonian $H_x$ constrained to the local quantum number range $[-\Lambda,\Lambda]$ obey
\begin{equation}
    \label{eq:dims_and_norm_truncation}
    d(\Lambda),\ \mathcal{N}(\Lambda)=\mathcal{O}(\poly(\Lambda)).
\end{equation}
\end{assumption}

More conditions are needed for us to handle the infinite dimensional local Hilbert space. Following \cite{TongAlbertMcCleanPreskillSu2021provably}, we assume that:
\begin{assumption}[Growth of local quantum numbers]
\label{assumption:local_quantum_number_growth}
There exist non-negative real-valued constants $\chi$ and $r$ such that, for any $x$, the Hamiltonian can be decomposed as
\begin{equation}
    \label{eq:ham_walk}
    H = H^{(x)}_W + H^{(x)}_R,
\end{equation}
where $H^{(x)}_W$ and $H^{(x)}_R$ satisfy (for $\Pi_{\lambda}^{(x)}\equiv \Pi_{\{\lambda\}}^{(x)}$)
\begin{subequations}
\label{eq:m_conditions_ham_general}
\begin{align}
    &\Pi^{(x)}_{\lambda}H_W^{(x)} \Pi^{(x)}_{\lambda'}=0,\ \text{if }|\lambda-\lambda'|>1, \label{eq:m_conditions_ham_general_1}\\
    &\|H_W^{(x)}\Pi^{(x)}_{[-\Lambda,\Lambda]}\|\leq \chi(\Lambda+1)^{r}, \label{eq:m_conditions_ham_general_2}\\
    &[H_R^{(x)},\Pi^{(x)}_{\lambda}]=0~\text{for all~}\lambda, \label{eq:m_conditions_ham_general_3}
\end{align}
for all $x=1,2,\ldots,N$.
\end{subequations}
\end{assumption}
We note that the $H = H^{(x)}_W + H^{(x)}_R$ is a decomposition of the entire Hamiltonian for each site $x$, and neither $H^{(x)}_W$ nor $H^{(x)}_R$ is necessarily local.

These assumptions are satisfied by many physical models of interest. For example, for the Hubbard-Holstein model we can take our local quantum number to be the bosonic occupation number of each of the sites.
Furthermore, we can choose for any $x$~\cite{TongAlbertMcCleanPreskillSu2021provably}
\begin{equation}
    H_W^{(x)} = g(b_x^\dagger + b_x)(n_{x,\uparrow} + n_{x,\downarrow} -1),~\qquad H_R^{(x)} = \sum_{x'} \omega_0 b^\dagger_{x'} b_{x'} + \sum_{x'\ne x}g(b_x^\dagger + b_x)(n_{x,\uparrow} + n_{x,\downarrow} -1) + H_f.
\end{equation}
If we take our local quantum number, $\lambda_x$, to be the phonon number at a particular site with $O_x = b^\dagger_x b_x$ we then see that~\eqref{eq:m_conditions_ham_general_1} holds because the bosonic ladder operators only couple between adjacent occupancy states.  Using the commutation properties of $b_x$ and $b_x^\dagger$ it is easy to see that \eqref{eq:m_conditions_ham_general_2} holds with $r=1/2$ and $\chi \le 2g$.  Since bosonic ladder operators commute across different sites, we have that~\eqref{eq:m_conditions_ham_general_3} also holds for this model.  A similar decomposition can be shown to exist for U(1) and SU(2) lattice gauge theories in one spatial dimension.

While such decompositions exist for many cases of interest, they are obviously not necessarily unique.  In fact, the choice of decomposition may potentially impact the value of $r$ and $\chi$ that appears in~\eqref{eq:m_conditions_ham_general_2}. But for our purpose of proving area law, it suffices to take any such decomposition of the Hamiltonian.

\section{Truncating the local Hilbert space}
\label{sec:effective_dims}

Although we consider the setting where the local Hilbert spaces are infinite dimensional, we can approximate spectrally isolated eigenstates with states containing low local quantum numbers.
This fact is made rigorous in \cite[Theorem 12]{TongAlbertMcCleanPreskillSu2021provably} which we restate here in a slightly modified way:
\begin{thm}[Quantum number distribution tail bound]
\label{thm:tail_bound}
    Let $H$ be a Hamiltonian satisfying~Assumption \ref{assumption:local_quantum_number_growth}.
    Let $\ket{\Psi}$ be an eigenstate of $H$ corresponding to an eigenvalue $\varepsilon$ with multiplicity $1$, and $\varepsilon$ be separated from the rest of the spectrum of $H$ by a spectral gap $\Delta$. Moreover, for a fixed site $x$ we assume local quantum number $|\lambda_x|$ has a finite expectation value:
    \[
    \sum_{\lambda}|\lambda|\braket{\Psi|\Pi^{(x)}_{\lambda}|\Psi}=\bar{\lambda}_x<\infty,
    \]
    we then have that
    \[
    \|(I-{\Pi}^{(x)}_{[-\Lambda,\Lambda]})\ket{\Psi}\| \leq e^{-\Omega\left(\sqrt{\chi^{-1}\Delta(\Lambda^{1-r}-(2\bar{\lambda}_x)^{1-r})}\right)}.
    \]
\end{thm}

We will further need to apply this result to bound the error that emerges from applying the truncation bound to many sites.  The following corollary provides such a result in a convenient form.
\begin{cor}
\label{cor:multiple_truncation}
Let
$
\Pi' = \prod_{x=\ell+1}^{\ell+s}\Pi^{(x)}_{[-\Lambda,\Lambda]},
$
then under the same assumption as in Theorem~\ref{thm:tail_bound}, we have
\begin{equation*}
    \|(I-{\Pi}')\ket{\Psi}\| \leq \sqrt{s} e^{-\Omega\left(\sqrt{\chi^{-1}\Delta(\Lambda^{1-r}-(2\bar{\lambda})^{1-r})}\right)},
\end{equation*}
where $\bar{\lambda}=\max_{\ell< x\leq \ell+s} \bar{\lambda}_x$.
\end{cor}
\begin{proof}
Proof follows from~Theorem~\ref{thm:tail_bound} via a straightforward application of the triangle inequality and the sub-multiplicative property of the spectral norm:
\begin{align}
    \|(I-{\Pi}')\ket{\Psi}\| &= \sqrt{\bra{\Psi}(I-{\Pi}')^2\ket{\Psi}}=\sqrt{\bra{\Psi}(I-{\Pi}')\ket{\Psi}}\nonumber\\
    &=\sqrt{ \sum_{x=\ell+1}^{\ell +s} \bra{\Psi}\prod_{x'=\ell+1}^{x-1}\Pi_{[-\Lambda,\Lambda]}^{(x')}(I - \Pi^{(x)}_{[-\Lambda,\Lambda]})\ket{\Psi}}\nonumber\\
    &\le\sqrt{ \sum_{x=\ell+1}^{\ell +s} \|(I - \Pi^{(x)}_{[-\Lambda,\Lambda]})\ket{\Psi}\|}\le \sqrt{s} e^{-\Omega\left(\sqrt{\chi^{-1}\Delta(\Lambda^{1-r}-(2\bar{\lambda})^{1-r})}\right)}.
\end{align}
\end{proof}
A final important consequence of Theorem~\ref{thm:tail_bound} is a bound on the quantity $\|\Pi'H(I-{\Pi}')\ket{\Psi}\|$ where $\Pi'$ is defined as in Corollary~\ref{cor:multiple_truncation}.  We use the above corollary to derive such a bound below.
\begin{cor}
\label{cor:truncation_and_ham_error}
Let
$
\Pi' = \prod_{x=\ell+1}^{\ell+s}\Pi^{(x)}_{[-\Lambda,\Lambda]},
$
then under the same assumption as in Theorem~\ref{thm:tail_bound}, we have
\begin{equation*}
    \|\Pi' H(I-{\Pi}')\ket{\Psi}\| \leq s^{3/2} \mathcal{N}(\Lambda) e^{-\Omega\left(\sqrt{\chi^{-1}\Delta(\Lambda^{1-r}-(2\bar{\lambda})^{1-r})}\right)},
\end{equation*}
where $\bar{\lambda}=\max_{\ell< x\leq \ell+s} \bar{\lambda}_x$.
\end{cor}

\begin{proof}
We only need to use Corollary~\ref{cor:multiple_truncation} along with a bound for $\|\Pi' H(I-{\Pi}')\|$. We have
\begin{equation}
    \Pi' H(I-{\Pi}') = \sum_{x=1}^N \Pi' H_x (I-{\Pi}') = \sum_{x=\ell+1}^{\ell+s} \Pi' H_x (I-{\Pi}').
\end{equation}
The second equality is because $\Pi' H_x (I-{\Pi}')=\Pi' (I-{\Pi}') H_x =0$ if $x\notin\{\ell+1,\ldots,\ell+s\}$. Therefore
\begin{equation}
    \|\Pi' H(I-{\Pi}')\| \leq \sum_{x=\ell+1}^{\ell+s} \|\Pi' H_x (I-{\Pi}')\|\leq \sum_{x=\ell+1}^{\ell+s} \|\Pi^{(x)}_{[-\Lambda,\Lambda]} H_x\| \leq s\mathcal{N}(\Lambda).
\end{equation}
\end{proof}

We remark that the bound in the above corollary does not depend on the system size $N$. However, there is a dependence on $\bar{\lambda}$, an upper bound on the mean absolute value of the local quantum numbers on sites $\ell+1,\ell+2,\ldots,\ell+s$.
One might worry that $\bar{\lambda}$ will show up as an independent parameter in our expression of the entanglement entropy.
However, we will show in the next section that for the ground states of U(1) and SU(2) LGTs as well as the Hubbard-Holstein model, $\bar{\lambda}$ depends only on the coefficients in the Hamiltonian, and is thus independent of the system size.

\section{Bounding the mean absolute value of the local quantum number}
\label{sec:mean_abs_local_quantum_number_bound}

In this section we prove bounds on the mean absolute value $\bar{\lambda}_x=\braket{|\lambda_x|}$ of the local quantum numbers in the ground states of U(1) and SU(2) LGTs, as well as the Hubbard-Holstein model. $\lambda_x$ here is the local quantum number of site $x$ and $\braket{\cdot}$ denotes the ground state expectation value. We will drop the subscript $x$ in this section because we will focus on only a single bosonic mode or gauge link. These bounds are independent of the system size and only depend on the coefficients in the Hamiltonians.

In the following discussion, we view our lattice models as a bipartite system, where a subsystem $A$ is a gauge link or bosonic mode, and a subsystem $B$ is the rest of the system. We then bound $\braket{|\lambda|}$, where $\lambda$ is the local quantum number for $A$, using the variational principle.

\begin{lem}
\label{lem:local_quantum_number_bound_first_step}
Consider a bipartite system with Hilbert space $\mathcal{H}=\mathcal{H}_A\otimes\mathcal{H}_B$, where $\mathcal{H}_A$ and $\mathcal{H}_B$ are the Hilbert spaces for subsystems $A$ and $B$ respectively.
Let $H=H_A+H_{AB}+H_B$ be the Hamiltonian, where $H_A$ acts non-trivially only on $A$, and $H_B$ on $B$. Let $\ket{\Psi}$ be the ground state of $H$. Furthermore, we assume that there exists an operator $K_A$ acting non-trivially only on $A$ such that $|H_{AB}|\preceq K_A$. Then we have
\begin{equation}
\label{eq:local_expectation_value_bound}
    \braket{\Psi|\left(H_A-K_A\right)|\Psi}\leq\braket{\Psi_A|\left(H_A+K_A\right)|\Psi_A},
\end{equation}
for any $\ket{\Psi_A}\in\mathcal{H}_A$.
Here $\preceq$ denotes the partial order induced by the convex cone of positive semi-definite operators and $|O|=\sqrt{O^{\dagger}O}$ for operator $O$.
\end{lem}

\begin{proof}

Let $\ket{\Psi_P}=\ket{\Psi_A}\ket{\Psi_B}$ be a product state, where $\ket{\Psi_B}\in\mathcal{H}_B$ is the ground state of $H_B$, and $\ket{\Psi_A}\in\mathcal{H}_A$ is chosen arbitrarily. Then because $\ket{\Psi}$ is the ground state of $H$ we have
\begin{equation}
\label{eq:variational_principle}
    \braket{\Psi|H|\Psi}\leq \braket{\Psi_P|H|\Psi_P}.
\end{equation}

For $\braket{\Psi|H|\Psi}$ we have
\begin{equation}
\label{eq:variational_principle_LHS}
    \begin{aligned}
    \braket{\Psi|H|\Psi} &= \braket{\Psi|H_A|\Psi} + \braket{\Psi|H_{AB}|\Psi} + \braket{\Psi|H_B|\Psi} \\
    &\geq \braket{\Psi|H_A|\Psi} - \braket{\Psi|K_A|\Psi} + \braket{\Psi|H_B|\Psi} \\
    &\geq \braket{\Psi|\left(H_A-K_A\right)|\Psi} + \braket{\Psi_B|H_B|\Psi_B},
    \end{aligned}
\end{equation}
where in the first inequality we have used $|H_{AB}|\preceq K_A$ and in the second inequality we have used  $\braket{\Psi|H_B|\Psi}\geq \braket{\Psi_B|H_B|\Psi_B}$, which is true because $\ket{\Psi_B}$ is chosen to be the ground state of $H_B$.

For $\braket{\Psi_P|H|\Psi_P}$ we have
\begin{equation}
\label{eq:variational_principle_RHS}
    \begin{aligned}
    \braket{\Psi_P|H|\Psi_P} &= \braket{\Psi_P|H_A|\Psi_P} + \braket{\Psi_P|H_{AB}|\Psi_P} + \braket{\Psi_B|H_B|\Psi_B} \\
    &\leq \braket{\Psi_P|\left(H_A+K_A\right)|\Psi_P} + \braket{\Psi_B|H_B|\Psi_B}  \\
    &= \braket{\Psi_A|\left(H_A+K_A\right)|\Psi_A} + \braket{\Psi_B|H_B|\Psi_B}.
    \end{aligned}
\end{equation}

Combining \eqref{eq:variational_principle}, \eqref{eq:variational_principle_LHS}, \eqref{eq:variational_principle_RHS}, we have \eqref{eq:local_expectation_value_bound}.

\end{proof}

\begin{lem}
\label{lem:local_quantum_number_bound_general}
Under the same assumptions as in Lemma \ref{lem:local_quantum_number_bound_first_step}, we further let $\Xi$ be a Hermitian operator on $A$. Assume that $H_A-K_A\succeq L(|\Xi|)$ where $L(x)$ is a convex non-decreasing function for $x\in\mathbb{R}^+$ satisfying $L(x)\to+\infty$ when $x\to+\infty$. Then we have
\begin{equation}
\label{eq:mean_abs_bound_general}
    \braket{\Psi||\Xi||\Psi}
    \leq L^{-1}\left(\braket{\Psi|\left(H_A-K_A\right)|\Psi}\right)
    \leq L^{-1}\left(\min_{\ket{\Psi_A}\in\mathcal{H}_A}\braket{\Psi_A|\left(H_A+K_A\right)|\Psi_A}\right).
\end{equation}
\end{lem}

\begin{proof}
By the assumption that $H_A-K_A\succeq L(|\Xi|)$ and \eqref{eq:local_expectation_value_bound}, we have
\[
\braket{\Psi|L(|\Xi|)|\Psi} \leq \braket{\Psi_A|\left(H_A+K_A\right)|\Psi_A}.
\]
Because $L$ is convex, by Jensen's inequality $L(\braket{\Psi||\Xi||\Psi})\leq\braket{\Psi|L(|\Xi|)|\Psi}$. Because $L$ is non-decreasing and $\ket{\Psi_A}$ can be arbitrarily chosen, we have \eqref{eq:mean_abs_bound_general}.
\end{proof}

Note that the right-hand side of \eqref{eq:mean_abs_bound_general} is independent of the subsystem $B$. This allows us to bound the mean absolute value of the local quantum number in the lattice models in a way that is independent of the system size.
We will now apply this lemma to the case of gauge theories and the Hubbard-Holstein model.

First, for U(1) and SU(2) gauge theories with the Hamiltonian defined in \eqref{eq:LGT_ham}, we take $A$ to be a gauge link $x$. Then we let
\[
    H_A = g_E E_x^2,\qquad H_{AB} = g_{GM}(\phi_x^{\dagger}U_x\phi_{x+1}+\phi_{x+1}^{\dagger}U_x^{\dagger}\phi_{x}),\qquad H_{B}=H-H_A-H_B.
\]
A nice feature about this Hamiltonian is that $H_{AB}$ is bounded: $\|H_{AB}\|\leq 2|g_{GM}|$. Therefore we can simply choose $K_A=2|g_{GM}|$. The local quantum number $\lambda_x$ in this case is the electric field value in the U(1) case and the total angular momentum in the SU(2) case. We recall that $E_x^2 = \lambda_x^2$ for U(1) LGT and $E_x^2 = \lambda_x(\lambda_x+1)\succeq \lambda_x^2$ for SU(2) LGT (for the latter see \eqref{eq:operators_SU(2)}, also $\lambda_x\succeq 0$ for SU(2) LGT). Therefore we have
\[
H_A - K_A \succeq g_E \lambda_x^2 - 2|g_{GM}|
\]
for both cases. For the right-hand side of \eqref{eq:mean_abs_bound_general} we have
\[
\min_{\ket{\Psi_A}\in\mathcal{H}_A}\braket{\Psi_A|\left(H_A+K_A\right)|\Psi_A} = 2|g_{GM}|,
\]
where the minimum is attained by $\ket{\Psi_A}=\ket{0}$.
Combining the above facts, we apply Lemma~\ref{lem:local_quantum_number_bound_general} to get:
\begin{cor}
\label{cor:mean_abs_bound_LGT}
For the U(1) and SU(2) LGTs with the Hamiltonian defined in \eqref{eq:LGT_ham}, let $\lambda_x$ be the local quantum number on gauge link $x$, then $\braket{|\lambda_x|}\leq 2\sqrt{|g_{GM}|/g_E}$, where $\braket{\cdot}$ denotes the ground state expectation value.
\end{cor}
 Then let us consider the Hubbard-Holstein model with the Hamiltonian described in \eqref{eq:ham_hubbard_holstein}. We let $A$ be a bosonic mode $x$, and let $B$ be the rest of the system. We have
\[
H_A = \omega_0 b_x^{\dagger}b_x,\qquad H_{AB}=g(b_x^{\dagger}+b_x)(n_{x,\uparrow}+n_{x,\downarrow}-1),\qquad H_B = H-H_A-H_{AB}.
\]
Here $H_{AB}$ is no longer bounded, but we can still construct a $K_A$ such that $|H_{AB}|\preceq K_A$.
To simplify the discussion we introduce the position and momentum operators $X$ and $P$:
\[
X = \frac{1}{\sqrt{2}}(b_x^{\dagger}+b_x),\qquad P=\frac{i}{\sqrt{2}}(b_{x}^{\dagger}-b_x).
\]
Then $H_{AB}=\sqrt{2}g X(n_{x,\uparrow}+n_{x,\downarrow}-1)$. Because $\|n_{x,\uparrow}+n_{x,\downarrow}-1\|\leq 1$, we can define
\[
K_A = \sqrt{2}|g| |X|,
\]
which satisfies $|H_{AB}|\preceq K_A$. With this choice of $K_A$ we have
\[
\begin{aligned}
H_A - K_A &= \frac{\omega_0}{2}(X^2+P^2-1)-\sqrt{2}|g||X| \\
&\succeq \frac{\omega_0}{2}(X^2+P^2-1) - \frac{\omega_0}{4}X^2 - \frac{2g^2}{\omega_0} \\
&\succeq \frac{\omega_0}{4}(X^2+P^2-1) - \frac{\omega_0}{4}  - \frac{2g^2}{\omega_0} \\
&= \frac{\omega_0}{2}b_x^{\dagger}b_x - \frac{\omega_0}{4} - \frac{2g^2}{\omega_0}.
\end{aligned}
\]
The local quantum number here is the bosonic occupation number, which has to be non-negative. Therefore
\[
H_A - K_A \succeq \frac{\omega_0}{2}|\lambda_x| - \frac{\omega_0}{4} - \frac{2g^2}{\omega_0}.
\]
For the right-hand side of \eqref{eq:mean_abs_bound_general}, we have
\[
\min_{\ket{\Psi_A}\in\mathcal{H}_A}\braket{\Psi_A|\left(H_A+K_A\right)|\Psi_A}\leq \braket{0|\left(\omega_0b_x^{\dagger}b_x + 2|g||X|\right)|0} = \frac{2|g|}{\sqrt{\pi}},
\]
where we have used the analytic solution of the ground state of the harmonic oscillator in deriving the equality. Combining these results with Lemma~\ref{lem:local_quantum_number_bound_general} we have:
\begin{cor}
\label{cor:mean_abs_bound_Hubbard_Holstein}
For the Hubbard-Holstein model with the Hamiltonian defined in \eqref{eq:ham_hubbard_holstein}, let $\lambda_x$ be the local quantum number on site $x$, then
\[
\braket{|\lambda_x|}\leq \frac{1}{2}+\frac{4|g|}{\omega_0 \sqrt{\pi}} + \frac{4g^2}{\omega_0^2},
\]
where $\braket{\cdot}$ denotes the ground state expectation value.
\end{cor}

\section{Robustness of the ground state to truncation}
\label{sec:ham_truncation}

In this section, we show that the ground state, the ground state energy, and the spectral gap are all robust to the truncation of the Hamiltonian, in a way that we will specify later.
Following \cite{AradKitaevLandauVazirani2013area} we focus on the $s$ sites from $\ell+1$ to $\ell+s$. For convenience we relabel the sites so that the original site $x$ is now labelled $x-\ell$. The Hamiltonian can be rewritten as
\begin{equation}
\label{eq:local_ham_general}
    H = H_L + H_1 + \cdots + H_s + H_R,
\end{equation}
where $H_L = \sum_{x\leq 0} H_x$ and $H_R = \sum_{x\geq s+1} H_x$. We need to shift each Hamiltonian term ($H_L$, $H_1$, $H_2,\ldots,H_s$, $H_R$) by a constant to ensure that they are all positive semi-definite. As in \cite{AradKitaevLandauVazirani2013area} we look at the entanglement entropy across a cut between sites $s/2$ and $s/2+1$. If $\|H_x\|\leq 1$, the local Hilbert space dimension is $d$ and the spectral gap is $\Delta$, then it is known that the entanglement entropy scales as $\mathcal{O}(\log^3(d)/\Delta)$ \cite[Theorem 6.2]{AradKitaevLandauVazirani2013area}.

Now we want to consider the case where the local Hilbert space dimension is infinite. This compels us to truncate the local Hilbert space, and consequently the local Hamiltonian terms $H_i$ as well. We denote the truncation threshold, defined according to the local quantum number introduced in Section~\ref{sec:the_abstract_model}, by $\Lambda$, and correspondingly the truncated Hamiltonian term by $H_x'$. The truncated Hilbert space dimension is $d(\Lambda)$ and the truncated local term has a norm that is upper bounded by $\mathcal{N}(\Lambda)$.

\subsection{The two truncations}

We first clarify  in more detail what we mean by truncating the local Hilbert space. The original Hilbert space is $\mathcal{H}=\mathcal{H}_L\otimes \mathcal{H}_1 \otimes \cdots \mathcal{H}_s\otimes \mathcal{H}_R$. We consider a subspace $\mathcal{H}'=\mathcal{H}_L\otimes \mathcal{H}'_1 \otimes \cdots \mathcal{H}'_s\otimes \mathcal{H}_R \subseteq \mathcal{H}$, where each $\mathcal{H}_x$ has dimension $d(\Lambda)$.
We denote by $\Pi'_x$ the projection operator onto $\mathcal{H}'_x$, and define
\[
\Pi' = I_L \otimes \Pi'_1 \otimes \cdots \Pi'_s \otimes I_R,
\]
which is the projection operator onto $\mathcal{H}'$. $I_L$ and $I_R$ are the identity operators on $\mathcal{H}_L$ and $\mathcal{H}_R$ respectively.
The truncated Hamiltonian is a Hermitian operator $H'$ defined to be the \textit{restriction} of $\Pi'H\Pi'$ to the subspace $\mathcal{H}'$, by which we mean that $H'$ maps elements from $\mathcal{H}'$ to $\mathcal{H}'$. We do not directly define $H'$ to be $\Pi' H\Pi'$ because that would introduce an artificial eigenvalue $0$ corresponding to the part of the Hilbert space that is truncated out.
We can write $H'$ out as
\begin{equation}
    H'=H_L' + H_1' + \cdots + H_s' + H_R',
\end{equation}
where each $H'_x$ is the restriction of $\Pi' H_x\Pi'$ to the subspace $\mathcal{H}'$, and the same is true for $H'_L$ and $H'_R$.  Locality is preserved in this truncation as $H'_x$ still acts non-trivially on sites $x$ and $x+1$, $H'_L$ on sites to the left of and including site $1$, and $H'_R$ on sites to the right of site $s$.

The second truncation we consider comes from Ref.~\cite{AradKitaevLandauVazirani2013area}. We adopt the definition in \cite{AradKitaevLandauVazirani2013area}, for a Hermitian operator $A$,
\begin{equation}
A^{\leq t} = AP_t + \|AP_t\|(I-P_t),
\end{equation}
where $P_t$ is the projection operator onto the subspace spanned by eigenvectors of $A$ with eigenvalues at most $t$. Then the second truncation yields the Hamiltonian
\begin{equation}\label{eq:Hpp}
H'' = (H_L' + H_1')^{\leq t} + H_2' + \cdots + H_{s-1}' + (H_s' + H_R')^{\leq t}.
\end{equation}
 The goal here is to show that these two truncations (i) preserve the spectral gap up to a constant factor, and (ii) preserve the ground state up to an error exponentially small in $\Lambda$ and $t$.

\subsection{Truncation robustness of the ground state and energy}

\begin{defn}\label{epsdef}For a self-adjoint operator $A$ bounded from below, define the sequence $\epsilon_0(A)\le\epsilon_1(A)\le\ldots$ as follows. Let $\essspec(A)$ be its (closed) essential spectrum and let $K\in\mathbb{N}_0\cup\{\infty\}$ be the number of eigenvalues in $[-\infty,\min\essspec(A))$, including multiplicity. For $k<K$, let $\epsilon_k(A)$ be the $k$-th eigenvalue with multiplicity ($K$ eigenvalues as we start from $k=0$), and for $k\ge K$ let $\epsilon_k(A)=\min\essspec(A)$.\end{defn}
Each $\epsilon_k(A)$ is in the spectrum\footnote{I.e., the set of $\epsilon$ such that $A-\epsilon I$ has no bounded inverse.} of $A$ but is not necessarily an eigenvalue when $\epsilon_k(A)=\min\essspec(A)$. The min-max principle \cite[Theorem 4.10]{te2009mathematical} states that for $k\in\mathbb N_0$,
 \begin{equation}\label{minmaxthm}
 \epsilon_k(A)=\minmax{\in\mathcal{D}(A)}{k}{A},
 \end{equation}
 where $\mathcal{D}$ is the domain of $A$.

Writing $\epsilon_k=\epsilon_k(H)$, recall that we assume that the Hamiltonian $H$ has a non-degenerate lowest eigenvalue $\epsilon_0$ (the ground state energy), with a unique ground state $\ket{\psi_0}$. We also assume that $\epsilon_0$ is separated from the rest of the spectrum by a gap $\Delta=\epsilon_1-\epsilon_0$.

For truncated Hamiltonians $H'$ and $H''$, we will prove that there exist unique ground states $\ket{\psi_0'}$ and $\ket{\psi_0''}$, corresponding to non-degenerate lowest eigenvalues $\epsilon_0'$ and $\epsilon_0''$, for the two truncated Hamiltonians respectively. Write $\epsilon_k=\epsilon_k(H)$, $\epsilon_k'=\epsilon_k(H')$, and $\epsilon_k''=\epsilon_k(H'')$ and $\Delta'=\epsilon_1'-\epsilon_0'$, $\Delta''=\epsilon_1''-\epsilon_0''$.

\begin{thm}[Robustness to truncations]
\label{thm:robustness}
Let $\Pi'$ be the projection operator onto $\mathcal{H}'$.
Let $\delta_1 = \|(I-\Pi')\ket{\psi_0}\|$, $\delta_2=\|\Pi'H(I-\Pi')\ket{\psi_0}\|$, and
\begin{equation}
\frac{\delta_2}{1-\delta_1^2} \leq \frac{\Delta}{18},
\end{equation}
then for every cutoff of the local quantum number $\Lambda>0$ there exists
\begin{equation}
T=\mathcal{O}\left(\frac{\mathcal{N}(\Lambda)^2}{\Delta}\left(\frac{\epsilon_0+\delta_2/(1-\delta_1^2)}{\Delta}+1\right)\right),
\end{equation}
such that for all $t\geq T$,
\begin{itemize}
\item[(i)] For $H''$, there exists a non-degenerate ground state $\ket{\psi_0''}$ corresponding to the lowest eigenvalue $\epsilon_0''$.
\item[(ii)] $\Delta''=\Omega(\Delta)$;
\item[(iii)] The trace distance between $\ket{\psi_0}$ and $\ket{\psi''_0}$ can be bounded as follows:
\begin{equation}
D(\ket{\psi_0''},\ket{\psi_0})\le \sqrt{\frac{2\delta_2}{\Delta(1-\delta_1^2)}} + e^{-\Omega(t/\mathcal{N}(\Lambda))}.
\end{equation}
\end{itemize}
\end{thm}

Before we proceed with the proof we establish the following lemma, which follows from a similar reasoning as in \cite[Lemma 6.4]{AradKitaevLandauVazirani2013area} except we correct a minor mistake in their bound due to not taking the global phase into account.
\begin{lem}[Markov]
\label{lem:markov}
Let $H$ be a Hamiltonian with the lowest eigenvalue $\epsilon_0$ and all other eigenvalues at least $\epsilon_1>\epsilon_0$, and assume $\ket{\psi_0}$ is its unique ground state. Given a quantum state $\ket{\phi}$ with expectation value $\braket{\phi|H|\phi}=E$, we have
\begin{equation}
    |\braket{\phi|\psi_0}|^2 \geq \frac{\epsilon_1-E}{\epsilon_1-\epsilon_0}.
\end{equation}
\end{lem}

\begin{proof}
Since the eigenstate $\ket{\psi_0}$ is the unique ground state, the expectation value of $H$ in any other eigenstate must be at least $\epsilon_1$ by assumption.  Using the fact that $\epsilon_1-\epsilon_0>0$, we have from Markov's inequality
\begin{equation}
    E=\bra{\phi} H \ket{\phi} \ge \epsilon_0 |\braket{\phi|\psi_0}|^2 + \epsilon_1(1-|\braket{\phi|\psi_0}|^2).
\end{equation}
The result then follows by re-arranging the above expression.
\end{proof}

The proof of Theorem~\ref{thm:robustness} proceeds as follows: we first show that when we go from $H$ to $H'$, the spectral gap is preserved and the ground state is changed by a small amount, and then show the same is true when we go from $H'$ to $H''$. In the first step we obtain:

\begin{lem}
\label{lem:robustness_first_step}
Let $\Pi'$ be the projection operator onto $\mathcal{H}'$.
Let $\delta_1 = \|(I-\Pi')\ket{\psi_0}\|$, $\delta_2=\|\Pi'H(I-\Pi')\ket{\psi_0}\|$. Then if $\delta_1 < 1$ and
\begin{equation}
\label{eq:truncation_err_assumption_robust_lemma}
\frac{\delta_2}{1-\delta_1^2} \leq \frac{\Delta}{4},
\end{equation}
we have the following
\begin{itemize}
\item[(i)] For $H'$, there exists a non-degenerate ground state $\ket{\psi_0'}$ corresponding to the lowest eigenvalue $\epsilon_0'$.
\item[(ii)] $\Delta'=\Omega(\Delta)$.
\item[(iii)] The trace distance between $\ket{\psi_0}$ and $\ket{\psi'_0}$ can be bounded as
\begin{equation}
D(\ket{\psi_0'},\ket{\psi_0})\leq\sqrt{\frac{2\delta_2}{\Delta(1-\delta_1^2)}}.
\end{equation}
\item[(iv)] $\epsilon_0\leq \epsilon'_0\leq \epsilon_0 + \frac{2\delta_2}{1-\delta_1^2}$.
\end{itemize}
Here $D(\cdot,\cdot)$ denotes the trace distance.
\end{lem}
\begin{proof}
  By the min-max theorem (Equation \ref{minmaxthm}), we have for $k\in\mathbb N_0$
 \begin{equation}\begin{aligned}
 \epsilon_k=&\minmax{\in\mathcal{H}}{k}{H}\\\le&\minmax{\in\mathcal{H'}}{k}{H}=\epsilon_k'\le\min\essspec(H').
 \end{aligned}\label{minmax}\end{equation}
In particular we have $\epsilon_1\le\epsilon_1'\le\min\essspec(H')$. To establish a gap $\Delta'$ we need an upper bound on $\epsilon_0'$:
 \begin{equation}
 \begin{aligned}
      \epsilon_0 &= \braket{\psi_0|H|\psi_0} \\
      &= \braket{\psi_0|\Pi' H\Pi'|\psi_0} + \braket{\psi_0|(I-\Pi') H\Pi'|\psi_0} \\
      &\quad+ \braket{\psi_0|\Pi' H(I-\Pi')|\psi_0} + \braket{\psi_0|(I-\Pi') H(I-\Pi')|\psi_0} \\
      &\geq \epsilon'_0 \|\Pi'\ket{\psi_0}\|^2 - 2\|\Pi' H(I-\Pi')\ket{\psi_0}\| + \epsilon_0 \|(I-\Pi')\ket{\psi_0}\|^2 \\
      &= \epsilon'_0 (1-\delta_1^2) -2\delta_2 + \epsilon_0 \delta_1^2.
 \end{aligned}
 \end{equation}
 As a result,
 \begin{equation}
 \label{eq:new_ground_energy_bound}
     \epsilon'_0 \le\epsilon_0+ \frac{2\delta_2}{1-\delta_1^2}.
 \end{equation}
The bound $\epsilon_0\le\epsilon_0'$ is immediate from the variational principle (or \eqref{minmax}), so (iv) is established. The assumption \eqref{eq:truncation_err_assumption_robust_lemma} then yields \[\epsilon_0'\le\epsilon_0+\Delta/2\le\epsilon_1-\Delta/2\le\epsilon_1'-\Delta/2,\]
 which implies $\Delta'\ge\Delta/2$, hence (ii). In particular it implies (i), that $\epsilon_0'$ is a simple eigenvalue with eigenvector $\ket{\psi_0'}$.

To establish closeness between $\ket{\psi_0'}$ and $\ket{\psi_0}$ we apply Lemma~\ref{lem:markov},
 \[\braket{\psi_0'|\psi_0}^2\ge\frac{\epsilon_1-\braket{\psi_0'|H|\psi_0'}}{\epsilon_1-\epsilon_0}=\frac{\epsilon_1-\epsilon_0'}\Delta=1-\frac{\epsilon_0'-\epsilon_0}{\Delta}\ge1-\frac1\Delta\frac{2\delta_2}{1-\delta_1^2},\]
 where the last inequality follows from \eqref{eq:new_ground_energy_bound}. Claim (iii) follows since $D(\ket{\psi'_1},\ket{\psi_0}) \leq \sqrt{1 - |\braket{\psi'_1|\psi_0}|^2}$.
 \end{proof}

We are then ready to prove the main result.
\begin{proof}[Proof of Theorem~\ref{thm:robustness}]
The existence and uniqueness of the lowest eigenvalue $\epsilon_0''$ and the ground state $\ket{\psi_0''}$ in (i) can be proved similarly to the proof of Lemma \ref{lem:robustness_first_step} (i).
We rescale the Hamiltonian $H'$ by a factor $\mathcal{N}(\Lambda)$ and then apply \cite[Theorem 6.1]{AradKitaevLandauVazirani2013area}. That theorem tells us that $\Delta''=\Omega(\Delta')$. Combining this fact with Lemma~\ref{lem:robustness_first_step} (ii), we have (ii). \cite[Theorem 6.1]{AradKitaevLandauVazirani2013area} also tells us that there exists
\begin{equation}
    T = \mathcal{O}\left(\frac{\mathcal{N}(\Lambda)^2}{\Delta'}\left(\frac{\epsilon_0'}{\Delta'}+1\right)\right) = \mathcal{O}\left(\frac{\mathcal{N}(\Lambda)^2}{\Delta}\left(\frac{\epsilon_0+\delta_2/(1-\delta_1^2)}{\Delta}+1\right)\right),
\end{equation}
such that $D(\ket{\psi_0'},\ket{\psi_0''})=e^{-\Omega(t/\mathcal{N}(\Lambda))}$ for $t\geq T$, where $\mathcal{N}(\Lambda)$ comes from the rescaling. Here we have used Lemma~\ref{lem:robustness_first_step} (iv). Combining with Lemma~\ref{lem:robustness_first_step} (iii) and the triangle inequality, we have proved (iii).
\end{proof}

Now we can use Corollaries~\ref{cor:multiple_truncation} and \ref{cor:truncation_and_ham_error} to bound $\delta_1$ and $\delta_2$ in Theorem~\ref{thm:robustness}, which leads to the following robustness result:
\begin{cor}[Robustness to truncations]
\label{cor:robustness}
Assume that the Hamiltonian satisfies Assumptions~\ref{assumption:local_quantum_numbers} and \ref{assumption:local_quantum_number_growth}, and let
$
    \bar{\lambda} = \max_{1\leq x\leq s} \braket{|\lambda_x|}.
$
Then the truncated Hamiltonian $H''$ has a lowest eigenvalue $\epsilon_0''$ corresponding to a non-degenerate ground state $\ket{\psi_0''}$. And there exist constants $C_1$ and $C_2$ such that for any $\Lambda$ and $t$ satisfying
\begin{equation}\label{eq:minparams}
\Lambda^{1-r}\geq (2\bar{\lambda})^{1-r} + C_1 \Delta^{-1}\polylog(s,\Delta^{-1}),\qquad t\geq \frac{C_2 \mathcal{N}(\Lambda)^2}{\Delta^2},
\end{equation}
we have
\begin{itemize}
\item[(i)] $\Delta''=\Omega(\Delta)$;
\item[(ii)] The trace distance between $\ket{\psi_0}$ and $\ket{\psi''_0}$ can be bounded as follows:
\begin{equation}
D(\ket{\psi_0''},\ket{\psi_0})=\poly(s,\Delta^{-1},\Lambda)e^{-\Omega\left(\sqrt{\Delta(\Lambda^{1-r}-(2\bar{\lambda})^{1-r})}\right)} + e^{-\Omega(t/\mathcal{N}(\Lambda))}.
\end{equation}
\end{itemize}
\end{cor}

We recall that $r$ is the exponent involved in \eqref{eq:m_conditions_ham_general_2}, and that $r=0$ for LGTs, $r=1/2$ for the Hubbard-Holstein model. $\Delta$ here is the spectral gap, and $s$ is the size of the region around the cut that we want to pay special attention to in \eqref{eq:local_ham_general}.

If we assume $\bar\lambda=O(1)$, which we proved for the Hubbard-Holstien model and U(1) and SU(2) LGTs in Section \ref{sec:mean_abs_local_quantum_number_bound}, then to achieve $D(\ket{\psi_0''},\ket{\psi})^2\le\delta$, it suffices to require, for some constant $C$,
\[\sqrt{\Delta(\Lambda^{1-r}-C)}\ge C\log\frac{s\Lambda}{\delta\Delta}\quad\text{and}\quad t\ge C\mathcal{N}(\Lambda)\log(1/\delta)\vee\frac{C\mathcal{N}(\Lambda)^2}{\Delta^2},\]
where $\vee$ denotes the maximum, which can be satisfied by choosing
\begin{equation}
\label{eq:Ltchoices}
\Lambda=\poly(\Delta^{-1})\polylog(s/\delta)\quad\text{and}\quad t=\Theta( \mathcal{N}(\Lambda)\log(1/\delta)\vee{\mathcal{N}(\Lambda)^2}/{\Delta^2}).
\end{equation}

\section{Area law}
\label{sec:area_law}
In this section, we establish our main result of an entanglement area law for unbounded quantum systems.
We first recall the notion of AGSP from \cite[Definition 2.1]{AradKitaevLandauVazirani2013area}.
\begin{defn}
\label{defn:agsp}
$K$ is a $(\sigma,R)$-AGSP of a Hamiltonian $H$ on a bipartite system consisting of two parts $A$ and $B$ that has a non-degenerate ground state, if
\begin{enumerate}
    \item $K\ket{\Psi}=\ket{\Psi}$, where $\ket{\Psi}$ is the ground state of $H$.
    \item $\|K\ket{\Phi}\|\leq \sigma$ for any $\ket{\Phi}$ such that $\braket{\Phi|\Psi}=0$.
    \item There exist operators $K_j^A$ and $K_j^B$, acting on $A$ and $B$ respectively, $j=1,2,\ldots,R$, such that $K=\sum_{j=1}^{R}K_j^A\otimes K_j^B$.
\end{enumerate}
\end{defn}

An AGSP preserves the ground state, suppresses the excited states, and increases the entanglement by a finite amount.
The existence of an AGSP onto the exact ground state is known to imply a bound on the entanglement of ground state. More precisely, Corollary III.4 of \cite{ALV12} states that if $\shrink R\le1/2$ where $\shrink$ is the shrinking factor and $R$ is the entanglement rank of the AGSP, then the entanglement entropy of the ground state satisfies a bound of order $\log R$. For frustrated systems the target space of the AGSP becomes perturbed away from the exact ground state(s) as its construction involves spectral truncations. Analyses of such a situation are undertaken in \cite{AradKitaevLandauVazirani2013area} and \cite{ALVV17}. These tools were simplified in \cite{OTR}, where an ``off-the-rack'' lemma was stated which generalizes the one of \cite{ALV12} to perturbed and degenerate target spaces.

Here we make a further improvement to \cite{OTR} to obtain the cleaner and slightly stronger statement of Lemma \ref{OTRlemma} below. For two subspaces $\mathcal Y,\mathcal Z\subset\mathcal H$, we say $\mathcal{Y}$ is $\delta$-viable for $\mathcal{Z}$ if $\braket{z|P_{\mathcal{Y}^\perp}|z}\leq\delta$ for all unit vectors $\ket{z}\in\mathcal{Z}$, where $P_{\mathcal Y^\perp}$ is the projection onto the orthogonal complement of subspace $\mathcal{Y}$. We write $\mathcal Y\approx_\delta\mathcal Z$ if $\bra z P_{\mathcal Y^\perp}\ket z\le\delta$ and $\bra y P_{\mathcal Z^\perp}\ket y\le\delta$ for unit vectors $\ket y\in\mathcal Y$ and $\ket z\in\mathcal Z$ respectively.

\begin{lem}\label{OTRlemma}
Let $\mathcal Z$ be a subspace of bipartite space $\mathcal H=\mathcal H_1\otimes\mathcal H_2$.
Let $\tilde{\mathcal Z}_n$ be a sequence of subspaces of $\mathcal H$ such that $\tilde{\mathcal Z}_n\approx_{\delta_n}\mathcal Z$ where $\delta_n$ is a sequence such that $\sum_{n=0}^\infty n\delta_n=O(1)$.

Suppose there exist $K_1,K_2,\ldots$ such that $K_n$ is an $(\shrink^n,R^n)$-AGSP with target space $\tilde{\mathcal Z}_n$ where $\shrink=\frac1{2R}$. Then the entanglement entropy of any state in $\mathcal Z$ is $O(\log R+\log\dim\mathcal Z)$.
\end{lem}
\begin{proof}
This follows from the proof of \cite[Lemma 4.7]{OTR} and the following strengthening of \cite[Lemma 4.6]{OTR}.
\end{proof}
\begin{lem}
Let $\mathcal Z$ be a subspace of $\mathcal H_1\otimes\mathcal H_2$ and suppose there exists a subspace $\mathcal V\subset\mathcal H_1$ with $\dim(\mathcal{V})=V$ which is $\delta$-viable for $\mathcal Z$. Pick any normalized state $\ket\psi\in\mathcal Z$ and write the Schmidt decomposition $\sum_{i}\sqrt{\lambda_i}\ket{x_i}\ket{y_i}$ with nonincreasing Schmidt coefficients. Then we have the tail bound $\sum_{i>V}\lambda_i\le\delta$.
\end{lem}
\begin{proof}
By the definition of $\delta$-viability, we have
\begin{equation}
	\langle z'|(P_{\mathcal{V}^\bot}\otimes I)|z'\rangle\leq\delta
\end{equation}
for all normalized state $|z'\rangle$ in $\mathcal{Z}$. In particular, this implies
\begin{equation}
	\langle z|(P_{\mathcal{V}^\bot}\otimes I)|z\rangle
	=\sum_{i}\lambda_i\langle x_i|P_{\mathcal{V}^\bot}|x_i\rangle
	=\mathrm{tr}\left(P_{\mathcal{V}^\bot}\sum_{i}\lambda_i|x_i\rangle\langle x_i|\right)
	\leq\delta.
\end{equation}
Here, $P_{\mathcal{V}^\bot}$ projects onto $\mathcal{V}^\bot$ with $\dim(\mathcal{V}^\bot)=\dim(\mathcal{H}_1)-V$. Now use Poincar\'{e}'s inequalities \cite[Corollary 4.3.39]{horn2012matrix} to conclude that
\begin{equation}
	\mathrm{tr}\left(P_{\mathcal{V}^\bot}\sum_{i}\lambda_i|x_i\rangle\langle x_i|\right)
	\geq\sum_{i=\dim(\mathcal{H}_1)-\dim(\mathcal{V}^\bot)+1}^{\dim(\mathcal{H}_1)}\lambda_i
	=\sum_{i=V+1}^{\dim(\mathcal{H}_1)}\lambda_i.
\end{equation}
This establishes the claimed bound.
\end{proof}
We apply Lemma \ref{OTRlemma} to the case of a simple ground state. Since $\operatorname{span}\ket{\psi_1}\approx_\delta\operatorname{span}\ket{\psi_2}$ where $\delta=D(\ket{\psi_1},\ket{\psi_2})^2$ we obtain
\begin{cor}\label{cor:OTR_simple}
Suppose there exist $K_1,K_2,\ldots$ such that $K_n$ is an $(\shrink^n,R^n)$-AGSP with target state $\ket{\psi_n}$ where $\shrink=\frac1{2R}$. If $\sum_{n=0}^\infty n D(\ket{\psi_n},\ket{\psi})^2=O(1)$ then the entanglement entropy of $\ket\psi$ is $O(\log R)$.
\end{cor}

The Hamiltonian $H''$ of \eqref{eq:Hpp} has spectral norm $O(s\mathcal{N}(\Lambda)+T(\Lambda))$. Under the conditions of Corollary \ref{cor:robustness}, $H''$ has a spectral gap $\Delta''=\Omega(\Delta)$, so we have
\[\Delta''/\|H''\|=\Omega\Big(\frac{\Delta}{s\mathcal{N}(\Lambda)+T(\Lambda)}\Big).\]
The Chebyshev polynomial of degree $\ell$ is bounded by $1$ on the unit interval but satisfies $T_\ell(1+\Delta)\ge\frac12(1+\sqrt{2\Delta})^\ell=\frac12\exp(\Omega(\ell\sqrt\Delta))$, so composing it with linear transformations yields a polynomial $f$ with $f_\ell(0)=1$ and $|f_\ell(\lambda)|\le2\exp(-\Omega(\ell\sqrt{\Delta/M}))$ for $\lambda\in[\Delta,M]$. Picking $M=\|H''\|$ we get that $K=f_\ell(H'')$ is an AGSP with shrinking factor
 \begin{equation}\label{eq:sf}\shrink\lesssim\exp\Big(-\Omega(\ell\sqrt{\Delta''/\| H''\|})\Big)=\exp\Big[-\Omega\Big(\frac{\ell\sqrt{\Delta}}{\sqrt{sN+T}}\Big)\Big].\end{equation}
 Assume $\bar\lambda=O(1)$. We may pick parameters as in Eq.\ \eqref{eq:Ltchoices} so that $D(\ket{\psi_0''},\ket{\psi_0})^2\le\delta$. Since $\mathcal{N}(\Lambda)=\poly(\Lambda)$, Eq.\ \eqref{eq:Ltchoices} implies
\begin{equation}\Lambda,\mathcal{N}(\Lambda),T(\Lambda)=\poly(\Delta^{-1})\polylog(s/\delta).\end{equation}
 Eq.\ \eqref{eq:sf} then becomes
 \[\shrink(\delta)\lesssim\exp\Big[-\Omega\Big(\ell\cdot\frac{\Delta^{O(1)}}{\sqrt{s}\cdot\polylog(s/\delta)}\Big)\Big].\]

$H''$ is a spin chain with a length-$s$ segment such that each qudit on the segment has local dimension $d(\Lambda)=\Lambda^{O(1)}$. The amortization bound from the proof of \cite[Lemma 4.2]{AradKitaevLandauVazirani2013area} yields that $K$ has entanglement rank:
\[R(\delta)=(\ell d)^{O(\ell/s+s)}\le\exp(C(\ell/s+s)\log(\ell\Lambda))\le\exp[C(\ell/s+s)\big(\log(\ell/\Delta)+\log\log(s/\delta)\big)].\]
  Let $\ell=s^2$:
 \begin{equation}R(\delta)=\exp\Big[Cs\big(\log s+\log(\Delta^{-1})+\log\log(1/\delta)\big)\Big],\qquad\sigma(\delta)=\exp\Big[-cs^{3/2}\cdot (\tfrac\Delta{\log(s/\delta)})^{O(1)}\Big].\label{Rs}\end{equation}

\begin{thm}[Area law]
\label{thm:area_law}
Under Assumptions~\ref{assumption:local_quantum_numbers} and \ref{assumption:local_quantum_number_growth} with $\bar\lambda=O(1)$, the ground state of $H$ satisfies an area law with entanglement entropy bounded by $\poly(\Delta^{-1})$.
\end{thm}
We remark that $\bar\lambda=O(1)$ is proved for the U(1) and SU(2) LGTs (Corollary \ref{cor:mean_abs_bound_LGT}), as well as for the Hubbard-Holstein model (Corollary \ref{cor:mean_abs_bound_Hubbard_Holstein}), in Section~\ref{sec:mean_abs_local_quantum_number_bound}.
\begin{proof}
For $k=1,2,\ldots$ consider $s=rk/\log(rk)$ and $\delta_k=k^{-3}$, chosen so that $\sum k\delta_k<\infty$. Then \eqref{Rs} yields entanglement bounds and shrinking factors
 \[R_k=\exp[C_0r k\log(\Delta^{-1})],\qquad\sigma_k=\exp[-c(rk)^{3/2}\cdot (\tfrac\Delta{\log(rk)})^{O(1)}].\]
In particular we have $R_k=R^k$ where $R=R_1$. To apply Corollary \ref{cor:OTR_simple} it remains to ensure that $\sigma_k\le(2R)^{-k}$ for all $k\in\mathbb N$
 \[(2R)^k\sigma_k\le\exp\Big[k+C_0rk\log(\Delta^{-1})-c(rk)^{3/2}\cdot (\tfrac\Delta{\log(rk)})^{O(1)}\Big].\]
 So we just need the expression in the square brackets to be negative. Dividing the exponent by $rk$ we see that it suffices to pick $r$ such that for all $k$,
  \[1/r+C_0\log(\Delta^{-1})\le c\sqrt{rk}\cdot \Big(\frac\Delta{\log(rk)}\Big)^{C_1}.\]
  Pick $r=\tilde\Theta(\Delta^{-2C_1})$ to achieve this for all $k\ge1$.
By Lemma \ref{OTRlemma} the ground state entanglement entropy is bounded by:
\[\log R=C_0r\log(\Delta^{-1})=\poly(\Delta^{^{-1}}).\]
This completes our proof.
\end{proof}

\section{Conclusions and discussion}
\label{sec:conclusion}

In this work we have rigorously established an entanglement area law for the gapped ground state of 1D quantum systems with infinite-dimensional local Hilbert spaces under natural assumptions \ref{assumption:local_quantum_numbers} and \ref{assumption:local_quantum_number_growth} on the local quantum numbers. In particular, our entanglement area law applies to U(1) and SU(2) LGTs and the Hubbard-Holstein model in 1D, and the result may be adapted to handle other bosonic and gauge theory models of interest. As an intermediate result, we develop a system-size independent bound on the ground state expectation value of local observables for Hamiltonians without translation symmetry, which may be useful in other contexts beyond our area law proof.

Proving the entanglement area law is an important step toward justifying the use of tensor network methods in classical simulation of quantum systems. The proof techniques also provide many useful tools for designing a rigorous RG algorithm for these quantum systems, which should be investigated by future work. It is also worth considering how the result in this work can be generalized to degenerate ground state or low-energy states as in Ref.\ \cite{ALVV17}.

When studying the entanglement area law scaling in LGTs, we considered the limit where the lattice size grows to infinity while the lattice spacing remains constant, which is needed to ensure that the coefficients in the Hamiltonian remain constant. This corresponds to the thermodynamic limit rather than the continuum limit, where the lattice spacing should also decrease to $0$. It remains an open question whether the entanglement entropy can be bounded in the continuum limit as well. It would also be interesting to investigate the scaling of other entropic quantities beyond the standard von Neumann entropy considered here.

Our result relies on the local quantum number tail bound proved in Ref.\ \cite{TongAlbertMcCleanPreskillSu2021provably}, which in turn follows from technical tools for analyzing the dynamical simulation of unbounded Hamiltonians on digital quantum computers. Along this line, previous work such as\ \cite{anshu2020communication,kuwahara2021improved,nachtergaele2006lieb,Tran2019powerlaw} have found other applications of quantum simulation techniques, to solving problems in quantum many-body physics beyond quantum computing. We consider finding further applications of this kind to be very interesting, as they demonstrate the immediate utility of studying quantum algorithms, while the building of scalable quantum computers remains a long-term goal.

\section*{Acknowledgments}

This work is partially supported by the Department of Energy under the Quantum System Accelerator (QSA) program (Y.T.), by the National Science Foundation under the Quantum Leap Challenge Institute (QLCI) program through grant number OMA-2016245, NSERC Discovery Program (N.W.), Google Inc (N.W.) and by the Simons Foundation (N.A.). Y.S.\ thanks Andrew M.\ Childs, Abhinav Deshpande, Adam Ehrenberg, Alexey V.\ Gorshkov, Andrew Guo, and Minh Cong Tran for helpful discussions about the entanglement area law.  N.W.\ thanks Ning Bao and Yuta Kikuchi for useful preliminary discussions about area laws for unbounded operators.

\bibliographystyle{abbrvnat}

redpajama_set_name: RedPajamaArXiv


text:
Stimulus check: Here's how to make sure you get your child-tax-credit payments
By Emily Long
IRS adds tool to streamline the payment process
(Image credit: Getty Images)
Families eligible for the expanded child-tax credit, half of which is set to be paid out in advance monthly installments starting next week, can now update their direct deposit information (opens in new tab) online to ensure quick receipt of their stimulus benefits.
The Child Tax Credit Update Portal (opens in new tab) on IRS.gov now allows parents and caregivers with qualifying children to add or update their bank-account information. All that's needed is an account number and bank routing number — both of which can be found on a personal check or within an online bank portal.
Fourth stimulus check: Possible amount, status and latest news
There's a 'secret' stimulus payment you need to ask for
Plus: Scam alert: These fake child tax credit websites will steal your identity
(On paper checks, the routing number is nine digits long, usually printed on the bottom left in old-fashioned machine-readable type. It's set off by colons and will look like |: 123456789|:)
If the IRS already has direct-deposit information on file, child-tax credit benefits will be paid out to that account. Through the portal, recipients can check to make sure that the information is correct, can update the information with a different account, or can add a bank account if there's not one already linked. Any changes will apply starting with the August 13 payment.
Families not enrolled in direct deposit will receive paper checks. However, the IRS is urging eligible recipients to opt for direct deposit to speed up payments and reduce the likelihood of funds getting lost in the mail.
The portal also gives you the option to unenroll from advance child-tax-credit payments entirely. Families who do so will receive the balance of their credit as a lump sum when filing their 2021 taxes in early 2022. The deadline for unenrolling is August 2.
The following groups may consider opting out of advance payments:
Families who expect their 2021 income to be too high to qualify for the child-tax credit
Caregivers whose children are claimed on someone else's tax returns
Families whose primary residence was outside the U.S. for more than half of 2021
Who gets the child-tax credit?
The expanded child-tax credit, which was included in the Biden administration's American Rescue Plan, provides up to $3,600 per child ($300 per month) under age 6 and up to $3,000 per child ($250 per month) age 6 through 17.
Those benefits will drop by $50 for every $1,000 in income for single filers earning more than $75,000 per year and married filers making more than $150,000 per year. Families with higher incomes are still eligible for the regular pre-2021 credit of $2,000 per child under age 17, but they won't qualify for advance monthly payments.
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Emily Long
Emily Long is a Utah-based freelance writer who covers consumer technology, privacy and personal finance for Tom's Guide. She has been reporting and writing for nearly 10 years, and her work has appeared in Wirecutter, Lifehacker, NBC BETTER and CN Traveler, among others. When she's not working, you can find her trail running, teaching and practicing yoga, or studying for grad school — all fueled by coffee, obviously.
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Amazon just slashed $55 off the Samsung Galaxy Buds 2

redpajama_set_name: RedPajamaCommonCrawl


text:
USERTESTING APPOINTS TATYANA MAMUT, NEXTDOOR'S CHIEF PRODUCT OFFICER TO BOARD OF DIRECTORS
SAN FRANCISCO, Calif – June 23, 2020 – UserTesting®, a leading provider of on-demand human insights, today announced that Tatyana Mamut, Chief Product Officer of Nextdoor, has been named to the UserTesting Board of Directors.
Mamut brings more than two decades of experience as a results-oriented leader with an obsession for building customer-led products and services for leading companies including Amazon, IDEO, Nextdoor, and Salesforce. She is currently the Chief Product Officer of Nextdoor, where she is responsible for leading product management, design and insights.
"I am thrilled to join the Board of UserTesting, a company that has created an outsized impact on my product, research, and design teams at Salesforce and Nextdoor," said Mamut. "As organizations evolve their cultures to being customer-obsessed, all functions across the organization need the tools to get close to, listen, and respond quickly to evolving customer needs."
Prior to Nextdoor, Mamut was general manager and director of product at Amazon Web Services where she led the creation and development of a new-to-the-world software product. She is a start-up advisor and fellow at Sapphire Ventures and executive board member at Advancing Women in Product. She has won multiple design and advertising awards, and holds several patents for her inventions at Salesforce and Amazon. She authored the Human-Centered Design Toolkit to bring Design Thinking to the developing world, and has led product innovation on six continents.
"We are very fortunate to have the opportunity to add someone with Tatyana's expertise and passion for customer-driven innovation to UserTesting's Board of Directors," said Andy MacMillan, CEO of UserTesting. "I look forward to the insights Tatyana will bring to the UserTesting board, and the company overall, as we focus on global expansion and building better product experiences."
Mamut concluded by stating that, "UserTesting is leading the revolution toward putting the customer at the center of innovation and decision-making. In the future, every conference room—physical and virtual—will have a 'UserTesting button' so that decision-makers can get input from real customers on all important decisions, and I'm excited to serve in making this future a reality. Today, as the world rapidly changes around us, the ability for companies to listen closely to their current and aspirational customers is more important than ever."


text:
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We hope many folks listen and enjoy the 2018 ARRL Hurricane Webinar recording!
The webinar will go through an overview of the historic 2017 Atlantic Hurricane Season, lessons learned from the season, key Amateur Radio group's preparedness for the new season and how Amateur Radio Operators and SKYWARN Spotters can become involved. These Amateur Radio Groups include, WX4NHC, the Amateur Radio Station at the National Hurricane Center, which is now in its 37th year of service, the Hurricane Watch Net, the VoIP Hurricane Net, SATERN (the Salvation Army Team Emergency Response Network) and the American Radio Relay League (ARRL). Also, an Amateur Radio Operator who is the Warning Preparedness Meteorologist at the Canadian Hurricane Centre will be giving the meteorological overview of the 2018 season.
We hope many folks can join the 2018 ARRL Hurricane Webinar!
It never hurts to have preparations in place and review safety rules prior to any impending tropical system affecting the region. This info can help you stay informed and understand products issued by the National Hurricane Center and National Weather Service in Boston/Norton and prepare properly if Southern New England were to be affected by a tropical system this season.


It is a New Season and another round of shows are being held across the globe. For the exhibitor there is a hope that the economy is improving and dealers are anxious to sell their collections at shows. Often times the public takes these shows for granted and do not realize that the exhibitor on the floor is not there just to provide "entertainment" but is there to sell and/or engage in an enlightened conversation about their specialty. A dealer has spent an enormous time pulling together the concept of their space and executing the installation of the space. They have invested money with their inventory, advertising, booth costs, fees, transport, travel expenses and set up to be in a location for a limited amount of time. These people deserve respect for their efforts and I hope to point out here some examples of public behavior which is certainly annoying to all dealers.
Most often at shows there is a "Preview Party" or "Preview Benefit" for a known charity to bring in good sponsors and a group of big donors who have the means to shop at an art and antique fair. Often times these people do not actually "buy" on preview night in order to maintain privacy but they are going to "zoom" in on what they may come back to. During the days and evenings of a show there are often lectures, sponsor parties and other events which attract more visitors. Designers come with or without their clients and this is important for the dealer. I am presenting to you things which have happened to me as an exhibitor or complaints I have heard from dealers in other fields. It is amazing at each show there seems to always be another "story" that circulates of "Bad Manners" at a show.
1. Never stand with a group of people either in a booth or in the aisle having idle chit chat blocking the view of others to see the booth. Move to a location where the dealer has visibility from the aisle. I have had this done in or out of my booth so many times. It is amazing that these groups will be in one spot for long periods and no one can see in or have an inclination to enter an exhibitor's booth where this visitation is taking place.
2. Please do not engage the dealer just to impress your friends with your "knowledge". Engage the dealer with polite conversation if you wish and come back for a more private conversation. There is always a "know it all" who comes into my booth. Most really knowledgeable attendees are truly aware of their surroundings when engaging the dealer. I welcome these conversations!
3. Never embarrass the dealer by saying you have the "same thing" you purchased at auction or somewhere else and you paid a considerable lower price. If you choose to enter this discussion do it privately and at your own risk. Almost everything at these shows is a unique item and it is doubtful that you have the same thing or in the case of a painting by the same artist there are many other factors which enter the equation as to price.
4. Imagine this scene of a group of very attractive "committee" people each wearing clothes which cost thousands of dollars, the ladies sporting "ice cubes" on their fingers, wrists and ears carrying handbags which costs way upwards of $10,000.00 descending on a booth going from one thing to another asking one question after another and then suddenly one of them says "OH! You are soooo expensive I would never pay _____ for that!" Abruptly, they all leave whispering, laughing, posing as others join the circle out of curiosity with giggles and stares as they finally move on. Now, I ask the Party Chairman who is interested in filling up next year's soiree to raise money for his/her beloved charity if those are the good manners which would entice a dealer to be back if he/she did not do well at the show?
5. Please never pull the tags off the walls or remove descriptions.
6. Never touch a painting or the frame of a painting!
7. DO NOT pick up any "smalls" in a booth without asking the permission of the dealer.
8. I wish I had a video camera in my booth at shows to watch the number of people approach a locked piece of furniture and try to open it. French locks can be tricky and most people do not have a clue as to what they are doing which often results in a scene of breaking and entering! Ask the exhibitor to open the doors and drawers for you.
9. Never set food or drink on any surface in a dealer's booth. This activity can happen all during a show and it is amazing to watch this.
10. A dealer is not in his/her booth to collect your garbage. Never hand the dealer your garbage and ask them to throw it out. Your garbage is your responsibility.
11. Children in strollers need to be in the aisles and not weaving in and out of a booth of beautifully arranged furniture unless you ask the dealer if you can bring the stroller in.
12. Children not in strollers need to be controlled in a booth. Their hands need to stay to themselves. Please attend to your children. They are welcome but not at the risk of destroying a booth.
13. Never take out a pen, pencil and a piece of paper and proceed to write on the surface of any top of a piece of furniture. This happened to me in a New York Fair when a man proceeded to come into my booth to write his check for a purchase from the booth across the aisle. The restoration of this desk cost me $1500.00.
14. Please refrain from playing with the flowers in the booth and interrupting the dealer who is in conversation to ask "are these real" or "where did you get your flowers?"
15. Never assume that any chair or sofa in a dealer's booth is there for you to sit on. If you are tired there are usually places provided by the show for you to relax. If you are courteous, a dealer may even invite you to sit if you ask nicely.
16. When looking at jewelry please do not "occupy" the entire counter with an assembly of friends trying on for the sake of trying on if you KNOW that you are not in the market for another piece. Find another time to come back and "play" when the dealer may not be as busy. You are prohibiting the dealer from valuable viewing and selling time.
17. Please read the maps or signage at a show and do not interrupt a dealer who is talking to another person "where is the bathroom?" or "where is the café?" I never mind answering these questions if I am not busy and I expect a courteous thank you. Do not run off in silence as if the dealer was there to provide directions.
18. Often shows are a great opportunity for a dealer to add to their inventory. If you are trying to sell something at a show please approach the exhibitor while they are not busy. Pulling out your photographs immediately and engaging the dealer while there are many people in a booth who could be potential clients is not courteous. Just ask the dealer if you could meet at a specific time.
19. Shows are a great opportunity to meet restorers and conservationists. If you are in this occupation, introduce yourself with a card and ask when would be a good time to meet.
20. Antique shows with exhibitors bringing furniture are a great place for the "knock off" furniture makers or "cabinet makers" to attend. They are very sneaky, engaging in an endless conversation about the wood, construction, locks and patina. They may ask for a photograph and then eventually a dealer realizes that this person is gathering information. Sometimes these people actually bring the customer in the booth. As a seasoned dealer I can usually spot them and ask them to step out of the booth so others can view and engage with me.
21 Other exhibitors or a dealer from the outside should NEVER enter another exhibitor's booth and talk to anyone in the booth until that person has completely left the booth. Nor should another dealer call out to the person in any way to distract the visitor. Stalking the visitor down the aisle to "snake" them is obscene behavior.
22 Never put anything on hold unless you are quite sure that you are seriously interested in the piece. There is a limited amount of time for the dealer to sell the item and you must be considerate of this.
23 Never take something or have something delivered to you unless you are serious about needing the object. Once delivered, you MUST decide quickly because the dealer will need to take it back and try to sell it while He/she has an opportunity.
24 Never come to an agreement with a dealer about a purchase, take it home whether paid for or not (some dealers will give you terms) and then decide not to go through with the purchase for reasons which are not legitimate. If you have a good reason which is no fault of the dealer then you should be responsible to return the item at your expense. Do not "stick" the dealer with the transport.
25 Please do not collect catalogues and cards from dealers only to leave them as trash in the facility or on the sidewalk when leaving the show. These cost money to the exhibitor and it would be kinder if you do not take or accept what is offered in the booth if you are not interested.
These are just a few examples of behavior at Shows. I know there are other stories but I have mentioned the most frequent behavior. The purpose of these exhibitions is not only for a dealer to sell but to engage with an interested and courteous audience. We love to teach about paintings, furniture and the decorative arts. We welcome anyone who appreciates our knowledge and expertise and hope that we provide a feast for the eyes.


text:
The IF Production Book
Posted inNews
ICAA launches My Cinema Premiere
by The IF Team May 5, 2015
Jeremy Sims' Last Cab to Darwin will be the first film to premiere under the new Independent Association of Australia initiative, My Cinema Premiere.
Both the initiative and the film's involvement were announced by ICAA CEO Adrianne Pecotic at the opening of the ICAA Conference, held from 5-7 May at Dendy Opera Quays in Sydney.
My Cinema Premiere is an extension of ICAA's hugely successful marketing platform, My Cinema, which was launched at the Conference last year.
Supported by Screen Australia's Enterprise Growth program, My Cinema Premiere has been created to allow Australian cinema audiences – particularly in regional areas – the opportunity to enjoy the excitement and glamour of an exclusive premiere event.
The initiative is a branded series of special events and preview screenings designed to build audiences for select films in ICAA Independent Cinemas. Hosted around the country by ICAA, the aim is to provide participating member cinemas with a targeted marketing and communications program.
"Independent Cinemas represent 30% of the Australian national cinema market and 80% of regional," said ICAA President, Scott Seddon. "In the last year, working together through the My Cinema marketing platform, we have had terrific success offering exciting national campaigns to independent cinema audiences. We have delivered greater box office success for films like Paper Planes, making independent cinema businesses more sustainable. The addition of My Cinema Premiere events will for the first time deliver a special event experience to patrons in many communities around Australia, especially in country areas."
My Cinema Premiere screenings of Last Cab to Darwin will run from 30 July to 5 August, before it is released by Icon on August 6.
Michael Caton stars as a taxi driver from Broken Hill who is diagnosed with terminal cancer, and sets off across the country with the goal of seeing out the rest of his life on his own terms.
The film also stars Jacki Weaver and features breakout performances from Mark Coles Smith, Ningali Lawford-Wolf and Emma Hamilton.
Greg Hughes, CEO Dendy/Icon Group said, "My Cinema Premiere is an exciting new initiative and Icon Film Distribution is delighted to be 'first cab off the rank' with Last Cab to Darwin. Filled with hope, humour and heart, this moving Australian film and the My Cinema Premiere events to launch it will be supported with a dedicated package of additional content, point of sale materials; a chance for patrons to win a trip for two to Darwin valued at $5,000 and runner up prize packs worth $100."
"We are thrilled to bring this new initiative to My Cinema sites all across Australia," added Adrianne Pecotic CEO of ICAA. "Participating indie cinemas will be able to roll out the red carpet and bring the glamour and excitement of a premiere film event to their local patrons and VIPs. New digital technology means that a My Cinema Premiere preview screening will include exclusive interview footage and Q&A with the stars of "Last Cab to Darwin" no matter how remote the cinema location."
Today's announcement of My Cinema Premiere follows the successful launch in 2014 of ICAA's My Cinema, a marketing platform that brings ICAA member exhibitors together with national promotions and strategic partnerships. Australian films feature in the My Cinema Premiere series with the support of the Screen Australia Enterprise Growth program.
Tagged: icaa, icaa-conference, last-cab-to-darwin, my-cinema, my-cinema-premiere
Luis Bayonas says:
I'm an independent filmmaker, work from Colac, Victoria. I premiered my two previous feature films, "Dancing Cows" and "Adios" in the Colac Cinemas. it went very well. My latest movie "Virtual Dogs and Loaded Guns" , also feature length, has been in production since january. Also to be premiered in the Region.
Just read bits and pieces about My Cinema Premiere – sounds good. About time something is being done to promote indie productions and indie cinemas!
So, what do we have to do to qualify? We are looking for any form of distribution. We have no grants, not funding, only the power of a whole community, our pocket money and COLLABORATION: a bunch of professionals helping each other. There is an up side to the daily hardships of being an independent: it allows us to make honest, free, no-strings-attached movies. 1
Luis Bayonas, writer/ director/producer
Jenny Sharp, producer


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One of the open questions about coaching, however, has to do with the nitty-gritty details of the programs: Is online coaching as effective as face-to-face? How does coaching an early grades. the.
but my teacher oscar hagerman was a particularly big influence on me in the early years. he opened my eyes to the social. this way opinions and questions of taste are not up for discussion – we're.
The staffing will build on the expertise of staff at the current Newbridge schools and will also give opportunities for new staff interested in SEND teaching, including Early Years Educators, teachers.
Young children are like sponges. Every day they learn skills that will help them become readers. Infants, toddlers, and preschoolers become aware of books and print and learn about sounds and letters. Preschool educators play a critical role in promoting literacy, preventing reading difficulties, and preparing young children for kindergarten.
The influential child psychologist Jean Piaget regarded early childhood education as. For most child care centers and preschools, practical questions about classroom procedures will be hugely.
The interview has been edited for length. supportive measures are taken, and high quality teachers are put in place. And then cutting across early childhood and the school years is doing something.
Dec 02, 2009 · BEd in Primary education- Early years interview Questions? I have an interview at a Uni for a BEd in Primary school education- Early years, next Wednesday. I have tonnes and tonnes of infromation on early years & loads about education in the news at the moment as it is a big topic.
Barack Obama, the 44th President of the United States, was born on August 4, 1961 in Honolulu, Hawaii to Barack Obama, Sr. (1936–1982) (born in Rachuonyo District, British Kenya) and Stanley Ann Dunham, known as Ann (1942–1995) (born in Wichita, Kansas, United States). Barack Obama spent most of his childhood years in Honolulu, where his mother attended the University of Hawaii at Manoa.
As part of our series on early. I spent 3 years as a teaching assistant for the introductory biology laboratories at my college, where I discovered how much I enjoyed helping other students learn.
In an interview with Boston Public Radio Friday. and center at a hearing Monday called by At-Large Councilor Annissa Essaibi-George." – "Key questions remain unanswered in Kraft's case," by Bob.
making it one of the main providers of teacher education in Wales. What does the award mean for the early years sector? I'm thrilled that this award recognises the importance and dedication of the.
Beverley Johnson was unlike any teacher I had met. He was the sort of teacher who is probably common nowadays, but was very unusual in the early Sixties. who came along like a breath of fresh air a.
New questions will begin in September. I'm kicking off this summer series with an interview. for over 18 years and I wrote the book in order to get dispel the urban myths and misconceptions about.
with a proper emphasis on the quality of teaching, rather than just published results. 3. In the light of the Rose and Cambridge reviews of primary education, what do you see as the priorities for the.
Early years teachers work with children from birth to five and are expected to meet the same entry requirements and pass the same skills tests as primary school teachers. The Teach First and School Direct programmes now cover early years education.
Mar 28, 2019 · Write down and practice at least three questions to ask the interviewer, showing how much you care about this kind of work. These should not be things that can be easily discovered through simple research (learn long before the hiring interview about the expected pay range and company benefits such as profit sharing plan, etc., as this interview is not the time for that).
10 Kindergarten Teacher Interview Questions and Answers. September 11, 2018 Staff. We spoke with an early-childhood educator about kindergarten teacher interview questions and answers you can prep for ahead of the big day. In five years I hope to have earned my Master's in School Leadership and plan on becoming an administrator within.
This article will list the most common interview questions for lecturer positions and suggest some possible answers to them. The examples of good answers are not the only options.
Rawley, a friend of Gronkowski's, said Monday in an interview with TMZ Sports. He coached linebackers early on in his career and served as defensive coordinator for the Badgers for two years before.
The unknown years of Jesus (also called his silent years, lost years, or missing years) generally refers to the period of Jesus's life between his childhood and the beginning of his ministry, a period not described in the New Testament. The "lost years of Jesus" concept is usually encountered in esoteric literature (where it at times also refers to his possible post-crucifixion activities.
• Talk me through how you have assessed the effectiveness of teaching and learning in your school. How have you used that assessment to inform best practice? • Tell me about an innovative teaching and learning strategy you have researched in the last year.
Over the years. he says in an interview with EdSurge. "They're basically becoming a cloud storage company because they're allowing us to save massive files in one link." He estimates that roughly.
Early years teacher: An early years teacher works with children aged 3-5. Tasks include planning lessons and activities and creating learning resources. You could be working in an infant or primary school, a children's centre, or a nursery. If you do want to work in a school, you will first need to gain qualified teacher status (QTS).
there are some early career teachers quietly packing up their desks and walking out the school doors with no plan to return next year. Some estimate the attrition rates in teaching to be as high as 30.
Janice Kiaski STEUBENVILLE — Never underestimate the impact a second-grade teacher. years. Since then, Hall has assumed.
The above interview questions also can be used for job title levels: entry level early years officer, junior early years officer, senior early years officer, early years officer assistant, early years officer associate, early years officer administrator, early years officer clerk, early years officer coordinator, early years officer consultant.
He would become one of the most familiar voices in prog, rivaled only by Yes' Jon Anderson and the Moody Blues' Justin Hayward, before embarking on a solo career in the early 1980s. Emerson Lake and.
Older children could even fill out the interview questions themselves. If you are looking for more fun New Years Eve activities for kids take a look at our New Years Eve party blowers, firework craft and our New Years Eve memory jar.


text:
Werner scores as RB Leipzig beat FC Koln in the Bundesliga
For Matchday 29 of the Bundesliga, RB Leipzig visited FC Koln searching for the three points. Despite starting behind in the scoring, the visitors managed to get a comeback in the first half and, in the end, won the game by 4-2
For Matchday 29 of the Bundesliga, RB Leipzig visited FC Koln searching for the three points. Despite starting behind in the scoring, the visitors managed to get a comeback in the first half and, in the end, won the game by 4-2.
Jhon Cordoba and Anthony Modeste scored for the hosts, while Patrick Schick, Christopher Nkunku, Timo Werner, and Dani Olmo scored for Leipzig.
The first half began with the visitors pressing and dominating. However, it was the home side who broke the deadlock. At the 6th minute, Cordoba took advantage of a rebound inside the box and put the ball into the back of the net. Goalkeeper Gulacsi even touched the ball but wasn't enough to avoid the goal.
The match didn't change after the first goal and the away side kept on putting pressure. At the 20th minute, it finally had an effect. Schick took advantage of Angelino's cross and headed to the goal, drawing the game.
With a better team on the pitch, it didn't take long before the visitors were able to get a comeback. At the 38th minute, Nkuku received a beautiful pass from Konrad Laimer and scored the second goal of Leipzig.
Fc Koln came back after the half time trying to draw the match again. However, in a fast play, the away team found another goal. At the 50th minute, goalkeeper Gulacsi found Werner free and the top scorer didn't disappoint, putting the ball into the back of the net.
Five minutes later, the hosts found their second goal. Modeste tried from outside the box and finished without chances for Leipzig's goalkeeper.
However, it didn't take long until Leipzig had the final say in the match. At the 57th minute, Olmo sent the ball past goalkeeper Timo Horn and scored the 4th goal.
The visitors still could have scored more goals, if Schick, Werner, and Sabitzer didn't waste their chances.
With this 4-2 victory, RB Leipzig reached 58 points and remain in third place. FC Koln, on the other hand, remains their bad streak and continues with the same 34 points.
#Anthony Modeste
#Christopher Nkunku
#Dani Olmo
#FC Koln
#Gulacsi
#Jhon Cordoba
#Patrick Schick
#Timo Werner
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