text:
\section{Introduction}

Ultracold atoms in optical lattices are an ideal platform to simulate complex quantum many-body systems of condensed matter
physics~\cite{lewenstein_07,bloch_08,Lewenstein12BOOK,gross_17}. One paradigmatic example is the Bose-Hubbard model~\cite{fisher_89,jaksch_98}. Here,
the possibility to tune the model's coefficients by means of external fields permits one to experimentally characterize the quantum phase transition between superfluid (SF) {to} Mott insulator (MI)~\cite{greiner_02,bakr_10}.

The recent experimental realization of ultracold dipolar gases in optical lattices \cite{depaz_13,moses_15,covey_16,baier_16,reichsollner_17,moses_17,bohn_17} opens the perspective to study the quantum properties of matter in a lattice emerging from the interplay between long-range interactions, contact interactions, and quantum fluctuations. In the theoretical model, long-range dipole-dipole interactions give rise to additional terms in the Bose-Hubbard model. These are density-density repulsive interactions, frustrating the occupation of neighbouring sites, and correlated tunneling terms, including tunneling of pairs of bosons and hopping depending on the site density \cite{sowinski_12,luhmann_12,maik_13,jurgensen_14,cartarius_17}. Similar additional interaction terms also play a key role in determining the properties of superconductors~\cite{rainer_93,hirsch_94,amadon_96}.
Most theoretical studies of the phase diagram of dipolar bosons include solely the density-density repulsive term. These works showed that the competition of this term with tunneling and contact interactions can gives rise to density modulations ~\cite{goral_02,yi_07}. The corresponding phases are denoted by charge density wave (CDW), when the phase is incompressible, and by supersolid (SS), when the phase is superfluid (for a review see \cite{Lahaye09,Baranov12}). Moreover, in one dimension the same model for unit mean density hosts a topological Haldane insulator  phase \cite{Torre06,rossini_12}.

The phases emerging from correlated hopping are relatively unexplored.  They are known in condensed matter context, where for instance
density-dependent tunneling has being coined as bond-charge  interaction~\cite{hirsch_89}. In standard solid state materials, however, the
role of interaction-induced tunneling is experimentally difficult to observe. In contrast, ultracold dipolar gases in optical lattices provide the unique
possibility to fully characterize the role of correlated hopping in determining the quantum phases and dynamics~\cite{jurgensen_14,baier_16}.
Theoretical studies predict that correlated tunneling can strongly modify the phase diagram \cite{dutta_15}
and is responsible for the appearance of
novel SF phases~\cite{sowinski_12,johnstone_19}. To mention some,  it can give rise to superfluidity with complex order parameters~\cite{jurgensen_15,luhmann_16},
or suppress the SS phase and give rise to phase-separation~\cite{maik_13}. Investigations at fractional density 3/2
predict interesting anyonic excitations \cite{Wikberg12,Duric17}. The interplay between contact and dipolar interactions were analysed in one dimension in Refs. \cite{sowinski_12,cartarius_17,Biedron18,kraus_20}
taking also care of varying the parameters according to the physical constrains of the system. The studies of Refs. \cite{Biedron18,kraus_20}, in particular, determined the ground state properties by means of density matrix renormalization group (DMRG) approach \cite{Schollwoeck11} and reported so-called staggered superfluid phases, namely, superfluid phases whose order parameter is staggered. Moreover, in Ref. \cite{kraus_20} it was predicted that correlated and single-particle hopping can interfere destructively, resulting in incompressible phases even for relatively shallow lattice, where one would otherwise expect superfluidity.

In this work we theoretically analyse the phases of a two-dimensional lattice of ultracold dipolar bosons by means of a Bose-Hubbard model which consistently includes density-density interactions and correlated hopping. We are particularly interested in the fate of the interference between correlated and single-particle hopping in two dimensions and in a square lattice. We determine the quantum ground state using primarily the single-site standard Gutzwiller mean-field (SGMF) approach~\cite{rokshar_91,krauth_92,sheshadri_93} truncating the dipolar interactions first to nearest neighbours (NN) and then to next-nearest neighbours (NNN) terms. We then analyze the role of quantum fluctuations on the staggered phases using cluster Gutzwiller mean-field (CGMF) theory~\cite{buonsante_04,yamamoto_09,pisarski_11,mcintosh_12,yamamoto_12,luhmann_13}.

{This paper is structured as follows. In Sec.~\ref{model} we introduce the extended Bose-Hubbard model at the basis of our investigations and discuss the theoretical approaches
employed in the present work. The ground-state phase diagram at fixed average densities are presented in Sec.~\ref{pd_den}. In Sec.~\ref{pd_v}
we examine the phase diagram at fixed nearest neighbour interaction. Finally, we discuss the conclusions and outlooks in Sec.~\ref{conc}.}

\section{Model and methods}

{We consider a gas of ultracold bosons which are confined in two dimensions in a square optical lattice. The motion in the direction orthogonal to the plane is assumed to be frozen out. The bosons interact via the contact potential and the dipolar interactions. The dipoles are polarized perpendicularly with respect to the plane of the lattice, thus they interact repulsively and isotropically in the plane. In this section we introduce the extended Bose-Hubbard model at the basis of our investigation, we discuss the parameter regime on which we focus and describe the theoretical methods we employ in order to determine the quantum ground state.}

\label{model}
\subsection{Extended Bose-Hubbard model}

We assume a grand-canonical ensemble. The bosons are confined by a square lattice in the $x-y$ plane with $L=L_x\times L_y$ sites and periodic boundary conditions. We denote by $\hat b_{p,q} $ and $\hat b_{p,q} ^\dagger$ the operators annihilating and creating a particle at site $\{p,q\}$ with commutation relations $[\hat b_{p,q} ,\hat b_{p',q'}^\dagger]=\delta_{p,p'}\delta_{q,q'}$. {Here, $p$($q$) is the lattice site index along the $x$($y$) direction.} Let $\hat n_{p,q}=\hat b_{p,q} ^\dagger\hat b_{p,q}$ denote the corresponding number operator. The extended Bose-Hubbard model (eBHM) we consider is described by the Hamiltonian $ \hat{H}$ which we separate into the contribution of onsite, nearest-neighbour and next-nearest neighbour interactions:
\begin{align}
 \hat{H}=\hat{H}_{0}+\hat{H}^{(1)}+\hat{H}^{(2)}.
 \label{ham}
\end{align}
Here, $\hat{H}_0$ is the standard Bose Hubbard model:
 \begin{eqnarray}
 	\hat{H}_0 &= &-t\sum_{p,q}\left(
 	\hat{b}^{\dagger}_{p+1,q}\hat{b}_{p,q}
 	+ \hat{b}^{\dagger}_{p,q+1}\hat{b}_{p,q}
 	+ \rm{H.c.} \right) \nonumber\\
 	&& +\sum_{p,q} \hat{n}_{p,q}
 	\left[ - \mu
 	+ \frac{U}{2} (\hat{n}_{p,q} - 1) \right]\,,
 \label{BH}
 \end{eqnarray}
where $t$ is the hopping term due to single-particle effects and is isotropic, $U$ denotes the onsite interaction term, and $\mu$ is the chemical potential.
The nearest neighbor (NN) and next-to-nearest neighbor (NNN) contributions to the Hamiltonian  contain {the terms due to both the contact and} the dipolar interactions. In detail,
\begin{eqnarray}
  \hat{H}^{(1)} &= & V \sum_{p,q} \hat{n}_{p,q} \left(\hat{n}_{p+1,q} + \hat{n}_{p,q+1}\right) \nonumber \\
		      && - T \sum_{p,q}\bigg[\hat{b}^{\dagger}_{p+1,q} \big(\hat{n}_{p,q}
			 + \hat{n}_{p+1,q}\big)\hat{b}_{p,q} \nonumber \\
		      && + \hat{b}^{\dagger}_{p,q+1} \big(\hat{n}_{p,q}
		         + \hat{n}_{p,q+1} \big)\hat{b}_{p,q}
			 + \rm{H.c.} \bigg] \nonumber \\
		      && + \frac{P}{2} \sum_{p,q}
		       \left(\hat{b}^{\dagger 2}_{p+1,q} + \hat{b}^{\dagger 2}_{p,q+1} \right)\hat{b}^{2}_{p,q} + \rm{H.c.}\,,
\label{ebhm}
\end{eqnarray}
where $V$ is the density-density interaction between neighbouring sites, $T$ is the amplitude scaling density-dependent tunneling, and $P$ is the pair tunneling coefficient.
The Hamiltonian corresponding to the NNN coupling is specific to 2D square lattice and couples sites along the diagonals of the lattice.
Explicitly:
\begin{align}
\hat{H}^{(2)} =& V_{\rm \text{diag}}\sum_{p,q} \hat{n}_{p,q} \big(\hat{n}_{p+1,q+1} + \hat{n}_{p-1,q+1} \nonumber\\
&+ \hat{n}_{p+1,q-1} + \hat{n}_{p-1,q-1}\big) \nonumber \\
& - T_{\rm \text{diag}} \sum_{p,q}\bigg[\hat{b}^{\dagger}_{p+1,q+1} \big(\hat{n}_{p,q}
+ \hat{n}_{p+1,q+1}\big)\hat{b}_{p,q} \nonumber \\
& + \hat{b}^{\dagger}_{p-1,q+1} \big(\hat{n}_{p,q}
+ \hat{n}_{p-1,q+1} \big)\hat{b}_{p,q} + \rm{H.c.} \bigg] \,.
\label{ebhm_h2}
\end{align}
The coefficients scale the corresponding terms as in Eq. \eqref{ebhm}. The subscript ``diag" indicates that we accounted for the lattice geometry in evaluating the interactions between NNN, as we specify in the following.

\subsection{Parameters}
\label{Sec:Par}

The coefficients of the extended Bose-Hubbard model, Eq. \eqref{ham}, are here calculated by taking into account the algebraic scaling of the dipolar interactions, which is truncated at the NN or at the NNN, and the lattice geometry. Moreover, in determining the phase diagram we tune the strength of the contact interactions and of the dipolar interactions: all other coefficients are systematically scaled. The procedure extends the one implemented in Ref.~\cite{kraus_20} to a two-dimensional square lattice. We now provide some details of how we determine the coefficients, and refer the reader to Ref. \cite{kraus_20} for details.

The potential confining the bosons in the $x-y$ plane has the form $V_{\rm latt}(\textbf r) = V_0 [\sin^{2}(kx) + \sin^{2}(ky)]$ , with $k$ the wave number and $\textbf r=(x,y)$, while the motion in the $z$ direction is frozen out.  Moreover, the bosons interact via the contact interaction, $U_{g}(\textbf r)$, and the repulsive dipolar potential $U_{\rm dip}(\textbf r)$, which decays with the distance $r$ as $r^{-3}$. The bosons are at zero temperature and the lattice depth $V_0$ is  sufficiently deep so we may limit the analysis to the lowest band. We fix $V_0=8E_R$ with the recoil energy $E_R=\hbar^2k^2/2m$ and $m$ is the particle's mass.
The single-particle tunneling coefficient is given by the overlap integral \cite{jaksch_98}
\begin{equation}
	t_{ij} = - \int^{}_{} d\textbf{r}~w_{i}(\textbf r)
	         \left[ -\frac{\hbar^2}{2m} \nabla^{2} + V_{\rm latt}
		 (\textbf r) \right] w_{j}(\textbf r),
\end{equation}
where $w_i({\mathbf r})$ is a product of standard (real-valued) Wannier functions at  $i\equiv \{p,q\}$. Note that due to separability of $V_{\rm latt}$ the coefficients $t_{ij}$ are non-zero only along $x$ or along the $y$ directions. Therefore, ``diagonal'' tunnelings such as e.g. $i\equiv \{p,q\}$ and $j\equiv \{p+1,q\pm1\}$ are strictly vanishing. The situation is different for the interaction terms.

We denote the interactions by the potential $U_{\rm int}(\textbf r) = U_{g}(\textbf r) + U_{\rm dip}(\textbf r)$. The corresponding
interaction coefficients in the site basis read
\begin{equation}
	V_{ijst} = \frac{1}{2} \int d{\textbf{r}}~d{\textbf{r}'}
			  w_i(\textbf{r})w_j(\textbf{r}')U_{\rm int}(\textbf{r} - \textbf{r}') w_s(\textbf{r}')w_t(\textbf{r}),
\end{equation}
and contribute to the interaction Hamiltonian term
\begin{equation} H_{\rm int}=\sum_{ijst} V_{ijst}\hat b^\dagger_i \hat b^\dagger_j \hat b_s \hat b_t \,.
\end{equation}
By truncating up to the NNN terms we then obtain terms such as pair tunneling and density-dependent tunneling terms.

In our calculations we keep the lattice depth constant. Due to our choice of $V_0$, standard approximations are justified, in particular a direct calculation shows that tunneling coefficients $t_{ij}$ between sites
separated by more than one lattice spacing are negligible (this is not so in more shallow lattices, see e.g. \cite{Trotzky12}). We note that  the strength of the coefficients of Hamiltonian~(\ref{ebhm}) and (\ref{ebhm_h2}) follow $|V| > |T| \gg |P|$ and that we can neglect terms where $V_{ijst}$ couple four different lattice sites. Moreover,
$|V_{\rm \text{diag}} |> |T_{\rm \text{diag}}|$. In our isotropic lattice geometry the overlap integral giving $P$ is orders of magnitude smaller than $V$ or $T$, moreover, it is much smaller than the coefficients appearing in $ \hat{H}^{(2)}$. We have checked explicitly that taking it into account  does not modify our results.
For that reason, from now on we omit the term proportional to $P$ in \eqref{ebhm} - this justifies also that  the corresponding term is also missing in \eqref{ebhm_h2} from the very beginning. For completeness, we remark that the pair tunneling coefficient can be enhanced by modifying the confinement in the $z$ direction \cite{sowinski_12}.

Let us repeat that
{in our calculations we vary the parameters by keeping the lattice depth constant and tuning both the contact and the dipolar interactions}. While the {contact interactions are tuned by means of} Feshbach resonances \cite{Chin10}, {dipolar forces can be varied by rotating the
orientation of dipoles as proposed by \cite{Giovanazzi02} and demonstrated in \cite{baier_16,Tang18}. A similar effect may be also} obtained - within the limitations mentioned above -  by changing the depth of the optical lattice and the contact potential keeping the strength of the dipolar interaction fixed.

In Ref.~\cite{kraus_20} we analysed the parameter regime where $T<0$. In particular, mean-field considerations predict that the density-dependent tunneling term scaling with $T$ can interfere destructively with the single-particle hopping scaling with $t$. Destructive interference is found when the condition
\begin{equation}
\label{T:t}t=|T|(2\rho-1)
\end{equation} is fulfilled at average densities $\rho>1/2$ \cite{kraus_20}. In Ref. \cite{kraus_20} we verified that this interference cuts the phase diagram into two topologically different superfluid phases. In particular, there where correlated hopping dominates the SF order parameter is staggered. This is the regime we analyse in this work.

\subsection{Gutzwiller mean-field theory}
\label{gucio}

We study the ground state properties of the model by means of the so called site-decoupled Gutzwiller mean-field or SGMF
approach~\cite{rokshar_91,krauth_92,sheshadri_93}. In its standard version the bosonic annihilation (creation)
operator is decomposed as $$\hat{b}_{p,q} = \langle \hat{b}_{p,q} \rangle + \delta \hat{b}_{p,q}\,,$$ where
$$\langle \hat{b}_{p,q} \rangle \equiv \phi_{p,q}$$ is the SF order parameter, which signals long-range phase coherence, and
$\delta \hat{b}_{p,q}$ is the fluctuation operator. In the presence of density-dependent tunneling and pair tunneling this approach has to be generalized
including additional order parameters connected with density{-dependent}  and pair hopping \cite{johnstone_19}, which we define below.

The many-body wave function is given by the Gutzwiller ansatz
\begin{equation}
 |\Psi_{\rm GW}\rangle = \prod_{p,q}|\psi_{p,q}\rangle =\prod_{p,q}
		       \sum_{n=0}^{n_{\text{max}}} c^{(p,q)}_n|n\rangle_{p,q},
 \label{gw}
\end{equation}
where ${|n\rangle_{p,q}}$ are the occupation number basis states,
$n_{\text{max}}$ is the maximum
number of bosons at each site, and $c^{(p,q)}_n$ are basis expansion coefficients
of  $|\psi_{p,q}\rangle$.  The state $|\Psi_{\rm GW}\rangle$ is normalized to unity by imposing $\sum |c^{(p,q)}_n|^2=1$.
Using Eq. \eqref{gw} the SF order parameter reads
\begin{equation}
  \phi_{p,q} = \langle\Psi_{\rm GW}|\hat{b}_{p,q}|\Psi_{\rm GW}\rangle=\sum_{n} \sqrt{n}~{c^{(p,q)}_{n-1}}^{*}c_n^{(p,q)}.
\end{equation}
It is finite
for the SF, SS and their respective staggered phases whereas it is zero for incompressible
phases.

The average density $\rho$ is given by $$\rho=\sum_{p,q} n_{p,q}/L$$ where $L=L_x\times L_y$ is the system size and
$n_{p,q}$ is the density at $(p,q)$th site:
\begin{equation}
  n_{p,q} = \langle\Psi_{\rm GW}| \hat{n}_{p,q}
            |\Psi_{\rm GW}\rangle
          = \sum_{n} n |c_n^{(p,q)}|^2.
\end{equation}
Finally, we introduce the density-assisted correlation order parameter, which reads
\begin{equation}
  \eta_{p,q} = \langle\Psi_{\rm GW}|\hat{n}_{p,q}\hat{b}_{p,q} |\Psi_{\rm GW}\rangle
	      = \sum_{n} \sqrt{n} (n-1)~{c^{(p,q)}_{n-1}}^{*}c_n^{(p,q)}\,.
 \label{eta_chi}
\end{equation}
Its behaviour allows us to identify staggered phases \cite{johnstone_19}.

Using these definitions the mean-field Hamiltonian of the system may be written as a sum
 of  single-site Hamiltonians $\hat{H}_{\rm MF} =\sum_{p,q} \hat{h}_{p,q}$. For NN interactions the individual summands $\hat{h}_{p,q}$  read

\begin{widetext}
  \begin{eqnarray}
    \hat{h}_{p,q}
            &=& -t\Big[\left(\phi^{*}_{p+1,q}\hat{b}_{p,q}
	      + \hat{b}^{\dagger}_{p,q}\phi_{p-1,q} - \phi^{*}_{p+1,q}\phi_{p,q}\right)
	      + \left(\phi^{*}_{p,q+1}\hat{b}_{p,q} + \hat{b}^{\dagger}_{p,q}\phi_{p,q-1}
	      - \phi^{*}_{p,q+1}\phi_{p,q}\right) + {\rm H.c.}\Big]
              \nonumber\\
	    &&+ \frac{U}{2}\hat{n}_{p,q}(\hat{n}_{p,q}-1)
	      - \mu\hat{n}_{p,q}
	      + V \Big[\hat{n}_{p,q}\Big(\langle\hat{n}_{p+1,q}\rangle
              + \langle\hat{n}_{p,q+1}\rangle + \langle\hat{n}_{p-1,q}\rangle + \langle\hat{n}_{p,q-1}\rangle \Big)
	      - \langle \hat{n}_{p,q} \rangle
             \Big(\langle\hat{n}_{p+1,q}\rangle
             + \langle\hat{n}_{p,q+1}\rangle \Big)\Big]
              \nonumber\\
	   &&- T \Big[\Big(\hat{b}^{\dagger}_{p,q} \eta_{p-1,q} + \phi^{*}_{p+1,q} \hat{n}_{p,q}\hat{b}_{p,q}
	     + \eta^{*}_{p+1,q} \hat{b}_{p,q} + \hat{b}^{\dagger}_{p,q} \hat{n}_{p,q} \phi_{p-1,q}
	     - \phi^{*}_{p+1,q} \eta_{p,q} - \eta^{*}_{p+1,q} \phi_{p,q} \Big)
              \nonumber\\
	   &&+ \Big(\hat{b}^{\dagger}_{p,q} \eta_{p,q-1} + \phi^{*}_{p,q+1} \hat{n}_{p,q}\hat{b}_{p,q}
	     + \eta^{*}_{p,q+1} \hat{b}_{p,q} + \hat{b}^{\dagger}_{p,q} \hat{n}_{p,q} \phi_{p,q-1}
	     - \phi^{*}_{p,q+1} \eta_{p,q} - \eta^{*}_{p,q+1}\phi_{p,q} \Big)
             + {\rm H.c.}
             \Big],

\label{ebhm_mf}
\end{eqnarray}
\end{widetext}
The mean field Hamiltonian corresponding to higher order (NNN) terms is found by applying the same procedure and has a similar form.
This form makes evident that density-assisted tunneling and single-particle hopping can interfere when both the SF order parameter and $ \eta_{p,q}$ are different from zero.

We solve the model by diagonalizing the single-site Hamiltonians coupled through the mean-field
self-consistently~\cite{jaksch_98,goral_02,zakrzewski_05,scarola_06,menotti_07,iskin_11,kuno_17,danshita_17,
bai_18,pal_19,bandyopadhyay_19,suthar_19,johnstone_19}.
In order to solve the on-site Hamiltonian~[Eq.(\ref{ebhm_mf})], we initialize the Gutzwiller
coefficients by $1/\sqrt{n_{\rm max}}$ on each-site and then {evaluate the corresponding initial order parameters}. {We then sequentially diagonalize the local Hamiltonians, the order parameters
are redefined using the ground states found from these diagonalizations}. With each diagonalization, the order parameters $\phi$,
and $\eta$'s are updated and this procedure is repeated until the convergence
criteria of the order parameter is satisfied. For the present work, we consider the
convergence criterion of $10^{-12}$ in $\phi$ for two consecutive iterations.
The ground-state of the
system is obtained by checking the initial order parameters with uniform {density} and {with} density wave orders,
and by then analyzing the energy of the system.

\subsection{Cluster Gutzwiller mean-field theory}
In the SGMF theory, the inter-site coupling is
incorporated through the mean fields. For instance, the nearest neighbour
hopping is through the SF order parameter. This accounts for the poor
resolution of  inter-site correlations. This shortcoming can be partially overcome
by the application of the CGMF method. In CGMF one make partitions of the entire system, say
$L_{x} \times L_{y}$ lattice, into $W$ clusters of dimension $M \times N$.
Here,  $W = (L_{x} \times L_{y})/(M \times N)$ is an integer, and the system
Hamiltonian is written as a sum of the cluster Hamiltonians. A cluster
Hamiltonian has two types of terms. First, the intra-cluster terms involving
lattice sites from within the cluster only.
And, second, the inter-cluster terms which couple the lattice sites at the boundary
of the cluster with the sites of the neighbouring clusters.
Only the latter are treated in the mean field level as in SGMF. The
detailed implementation of the CGMF can be found in \cite{buonsante_04,
yamamoto_09,pisarski_11,mcintosh_12,yamamoto_12,luhmann_13,deng_15,sachdeva_17,bai_18,
suthar_19}. The Hamiltonian of a single cluster is
\allowdisplaybreaks
\begin{widetext}
\vspace{-1em}
\begin{eqnarray}
   \hat{H}_C = &-& t \sum_{p,q}^{'} \left(\hat{b}_{p+1,q}^{\dagger}\hat{b}_{p,q} +
                   \hat{b}_{p,q+1}^{\dagger}\hat{b}_{p,q} + {\rm H.c.} \right)
                 - t \sum_{p,q \in \delta C}\big(\phi_{p+1,q}^{*}\hat{b}_{p,q} +
                   \hat{b}_{p,q}^{\dagger}\phi_{p-1,q}
                 + \phi_{p,q+1}^{*}\hat{b}_{p,q} +
                   \hat{b}_{p,q}^{\dagger}\phi_{p,q-1} + {\rm H.c.}\big)\nonumber \\
                &+&\frac{U}{2}\sum_{p,q} \hat{n}_{p,q}(\hat{n}_{p,q}-1) -
                   \mu\sum_{p,q}\hat{n}_{p,q}
		 + V \sum_{p,q}^{'} \hat{n}_{p,q}\Big(\hat{n}_{p+1,q} +
                   \hat{n}_{p,q+1}\Big) \nonumber \\
	        &+&V \sum_{p,q \in \delta C}\hat{n}_{p,q}\Big(
                   \langle \hat{n}_{p+1,q} \rangle + \langle \hat{n}_{p,q+1}
		   \rangle + \langle \hat{n}_{p-1,q} \rangle + \langle \hat{n}_{p,q-1} \rangle \Big) \nonumber \\
	        &-&T \sum_{p,q}^{'} \Big(\hat{b}_{p+1,q}^{\dagger}
                   (\hat{n}_{p,q} + \hat{n}_{p+1,q}) \hat{b}_{p,q}
                 + \hat{b}_{p,q+1}^{\dagger}(\hat{n}_{p,q} + \hat{n}_{p,q+1})
                   \hat{b}_{p,q} + {\rm H.c.} \Big) \nonumber \\
	        &-&T \sum_{p,q \in \delta C}\Big[\Big(\hat{b}^{\dagger}_{p,q}\eta_{p-1,q} +
                   \phi^{*}_{p+1,q}\hat{n}_{p,q}\hat{b}_{p,q} + \eta^{*}_{p+1,q}\hat{b}_{p,q}
		 + \hat{b}_{p,q}^{\dagger}\hat{n}_{p,q}\phi_{p-1,q} \Big)\nonumber \\
	        &+&\Big(\hat{b}^{\dagger}_{p,q}\eta_{p,q-1} + \phi^{*}_{p,q+1} \hat{n}_{p,q}\hat{b}_{p,q}
	         + \eta^{*}_{p,q+1}\hat{b}_{p,q} + \hat{b}_{p,q}^{\dagger}\hat{n}_{p,q}\phi_{p,q-1} \Big)
		 + {\rm H.c.}\Big].

  \label{clus_hamil}
\end{eqnarray}
\end{widetext}
The prime in the summation of the intra-cluster terms is to
indicate that the lattice sites $(p+1,q)$ and $(p,q+1)$ are also within the
cluster. Here, $\delta C$ is the set of lattice sites at the boundary of the
cluster.

The ground state of the cluster Hamiltonian is
\begin{equation}
  \Ket{\Psi_c} = \sum_{l}C_l \Ket{\Phi_c}_\ell\,,
  \label{cluster_gs}
\end{equation}
where
\begin{equation}
   \ket{\Phi_c}_\ell = \prod_{q=0}^{N-1}\prod_{p=0}^{M-1} \ket{n}_{p,q}\,.
\end{equation}
where $\ket{n}_{p,q}$ is the occupation number basis at the $(p,q)$ lattice
site of the cluster, and $\ell \equiv \{n_{00}, n_{10}, \ldots, n_{M-1,0}, n_{01}, n_{11},\ldots
n_{M-1,1}, \ldots, n_{M-1,N-1}\}$ is the index quantum number to identify
the cluster state. The ground state of the entire system, as in the SGMF, is
the direct product of the cluster ground states
\begin{equation}
  \ket{\Psi^c_{\rm GW}} = \prod_k\ket{\Psi_c}_k,
\label{cgw_state}
\end{equation}
where $k$ is the cluster index and varies from 1 to $W$.
To solve the cluster Hamiltonian Eq.(\ref{clus_hamil}), we
initialize the order parameters in our model. We then diagonalize the cluster
Hamiltonians, and then update the values of the order parameters based on
the ground states obtained during the diagonalization. As in SGMF, this
procedure is repeated till {it converges}.

\subsection{Observables}
\label{Sec:Obs}

To examine the ground-state properties of the system, we compute the
single-particle correlation
\begin{eqnarray}
  M_{b} (\textbf{k}) &=& \frac{1}{L^2} \sum_{j,j'}
			  e^{i \textbf{k}\cdot(\textbf{r}_ j - \textbf{r}_{j'})}
		          \langle \hat{b}^{\dagger}_{j} \hat{b}_{j'} \rangle
\end{eqnarray}

as well as
the density-density correlation or the structure factor  defined as
\begin{equation}
  S (\textbf{k}) = \frac{1}{L^2} \sum_{j,j'}
	            e^{i \textbf{k}\cdot(\textbf{r}_j - \textbf{r}_{j'})}
		    \langle \hat{n}_{j} \hat{n}_{j'} \rangle.
\end{equation}
In the above definitions, $\textbf{k} = (k_x, k_y)$ is the dimensionless wave-vector and $\textbf{r}_ j\equiv (p,q)$, $\textbf{r}_{j'} \equiv (p', q')$.  Finite values of $M_b{({\bf k})}$ and $S(\pi,\pi)$ correspond to off-diagonal long range order and diagonal long range
order, respectively~\cite{otterlo_94,batrouni_00,hebert_01}.

Using the Gutzwiller wave function given by Eq.(\ref{gw}), $M_b(\textbf{k})$
 and $S(\textbf{k})$ can be rewritten as
\begin{eqnarray}
  M_{b} (\textbf{k}) &=& \frac{1}{L^2} \sum_{j,j'}
                         e^{i \textbf{k}\cdot(\textbf{r}_ j - \textbf{r}_{j'})}
                         \phi^{*}_{j} \phi_{j'} \nonumber \\
  S (\textbf{k})     &=& \frac{1}{L^2} \sum_{j,j'}
	                  e^{i \textbf{k}\cdot(\textbf{r}_j - \textbf{r}_{j'})}
		          n_{j} n_{j'}. \nonumber
\end{eqnarray}
 In the CGMF theory, the correlated wavefunction in
Eq.(\ref{cluster_gs}) and Eq.(\ref{cgw_state}) prevents the replacement of
the operators by their expectation values. Instead, the observables with
operators from within a single cluster are calculated using the
cluster wavefunction defined in Eq.(\ref{cluster_gs}) and the total
wave function (\ref{cgw_state}) is used to calculate the observables with
operators from different clusters.

The classification of various quantum phases can be obtained using the behavior of
correlation and structure factor at zero and finite momentum $\textbf{k}$. When truncating the Hamiltonian to NN, the effect of the interactions is to give rise to spatially periodic structures with $2\times 2$ cells. Periodic density modulations, in particular, can be characterized by their sublattice distributions $(n_a,n_b)$ with $a\equiv (p,q)$ and $b\equiv (p',q')$, where $(p',q')$ is the nearest neighbor lattice site of $(p,q)$. Similarly, correlated tunneling gives rise to periodic modulation of the SF order parameter of the phase~\cite{johnstone_19}. The onset of these structures is signalled by the finite values of $S$ and $M_b$, respectively, at the wave vector $\textbf{k}=(\pi,\pi)$.

The inclusion of NNN terms break this symmetry. In general, it is known that the NNN repulsion tends to stabilize striped order of supersolid phase. The coexistence of NN and NNN stabilizes various solid orders, for $V \geqslant 2V_{\rm diag} (V < 2V_{\rm diag})$, checkerboard (stripe) solid ordered state is formed~\cite{batrouni_95,scalettar_95,yamamoto_12}. Here, in order to distinguish between various orders of solid, supersolid and staggered phases,
we determine the Fourier transform of the single-particle correlations $M_b(\mathbf{k})$ and {the} structure factor $S(\mathbf{k})$ at $\textbf{k} = (\pi,\pi), (0,\pi)$ and $(\pi,0)$. A finite value of those observables at $(\pi,\pi)$ reflects checkerboard order whereas a finite value at $(0,\pi)$ or $(\pi,0)$ shows
striped order. {Correspondingly,} a finite value of $M_b(\mathbf{k})$ and $S(\mathbf{k})$ at $\textbf{k} = (\pi,\pi), (0,\pi)$ and $(\pi,0)$ is the characteristic property of a quarter-filled ordered superfluid phase~\cite{chen_08,ng_10}. Depending on the ratio between $M_b(0,0)$ and $M_b(\pi,\pi)$ the quarter-filled order superfluid phase is either a staggered or a normal superfluid. In this work we will find quarter-filled ordered staggered SS phases, which we will label by the acronym QF-SSS. Table~\ref{tab_op} summarizes the properties of different quantum phases in terms of the correlation and  order parameters.

\begin{table*}
  \caption{Classification of phases: Sublattice values of the occupation $\boldsymbol{n}=(n_a,n_b)$, single-particle correlations, and structure form factor. When single-particle correlations and structure form factor do not vanish, we report the value at which $\textbf{k}= \textbf{k}_{\rm max}$ at which their absolute value is maximum and which identify the phase. The details of the QF-SSS phase are discussed in Sec. \ref{Sec:Obs}. See also Ref. \cite{johnstone_19}.}
  \centering
  \begin{tabular}{l c c c c c c c c } \\
    \hline\hline \\
      Phase &  Acronym & $ {\boldsymbol{n}} $  &  $ M_b(\textbf{k})$ &  $ S(\pi,\pi) $\\ \\ [0.2ex]
	    \hline\hline \\
      Mott Insulator             & MI           & $ (n,n) $           & $ 0 $                & $ 0 $ \\
      Charge Density Wave        & CDW          & $ (n_a,n_b) $ &  $ 0 $        & $\neq 0$ \\
      Superfluid                 & SF      & $ (n,n) $     &  $ \textbf{k}_{\rm max}=(0,0)$  & $ 0 $\\
      Supersolid                 & SS  & $ (n_a,n_b) $  &  $ \textbf{k}_{\rm max}=(0,0)$ & $ \neq 0$ \\
      Staggered Superfluid       & SSF     & $ (n,n) $    & $ \textbf{k}_{\rm max}=(\pi,\pi)$        & $0$\\
      Staggered Supersolid       & SSS & $ (n_a,n_b) $ & $ \textbf{k}_{\rm max}=(\pi,\pi)$          & $\neq 0$\\
      Quarter-Filled Staggered Supersolid & QF-SSS & other  & other  & other \\[1.5ex]

      \hline
  \end{tabular}
\label{tab_op}
\end{table*}

Finally, for convenience in the following we also use the average SF order parameter
\begin{equation}
\label{phi:av}
\phi_{\rm avg} = \sum_{p,q} |\phi_{p,q}|/L\,,
\end{equation}
as well as the average  density-correlated order parameter
\begin{align}
		\eta_{\rm avg}=\sum_{p,q}|\eta_{p,q}|/L \, . \label{average_eta}
\end{align}
{ These two order parameters serve to define quantum phases of eBHM. Here, we consider the absolute values of
$\phi$'s and $\eta$'s as for staggered phases their distributions over the lattice sites alternate in sign and in particular
for staggered superfluidity these observables become zero otherwise.
}

\section{Phase transitions at fixed density}
\label{pd_den}

We numerically determine the ground states of the system in the grand-canonical ensemble and at fixed density.
For this purpose we first obtain the {values} of the chemical potential $\mu$ corresponding to a constant average density. The model parameters are
obtained using Wannier function formalism as described in Sec. \ref{Sec:Par}, where {we change $T$ and $V$ by varying} the
interaction strength, see Ref. \cite{kraus_20} for details. Below we choose to plot $V$ instead of the interaction strength. We remind the reader that finite values of $V$ {also imply} finite values of $T$.

The phases are identified by analyzing the behaviour of correlations and order parameters, see Sec. \ref{Sec:Obs}. We first discuss the role
of NN interaction and density-dependent tunneling. We then investigate the role of NNN terms in determining the ground state properties.
Note that below we label the coordination number by the parameter $z$. In our lattice geometry $z=4$.

\subsection{Phase diagram for nearest-neighbor interactions}
\begin{figure}

  \includegraphics[width=\linewidth]{corr_den_1.png}
  \caption{Observables for the ground-state of the 2D eBHM at $\rho = 1$ for the mean-field Hamiltonian, Eq.\,\eqref{ebhm_mf}, truncated at NN. Here the ground state was calculated by means of the SGMF approach.
	   	Subplots (a) and (b) show the Fourier transform of the off-diagonal single-particle correlation $M_b$ at $(k_x,k_y) = (0,0)$
		and
	       $(k_x,k_y) = (\pi,\pi)$, respectively. The transition from MI-SF is identified by a finite value of $M_b(0,0)\geqslant M_b(\pi,\pi)$. In contrast the staggered superfluid phases correspond to the regions where $M_b(0,0)< M_b(\pi,\pi)$. Subplot (c) shows the average SF order parameter, Eq. \eqref{phi:av}. The structure factor $S(\pi,\pi)$ is shown in (d). Its finite value signals density modulated phases.
	  The system size is $L=12\times 12$, periodic boundary conditions are assumed and the maximum occupancy
	  per lattice site $n_{\rm max}$ is taken to be $8$.}
  \label{corr1}
\end{figure}

\begin{figure}

  \includegraphics[width=\linewidth]{corr_den_2.png}
  \caption{Observables for the ground-state of the 2D eBHM at $\rho = 2$ for the mean-field Hamiltonian, Eq.\,\eqref{ebhm_mf}, truncated at NN. Here the ground state was calculated by means of the SGMF approach.
		Subplots (a) and (b) show the Fourier transform of the off-diagonal single-particle correlation $M_b$ at $(k_x,k_y) = (0,0)$ and
		$(k_x,k_y) = (\pi,\pi)$, respectively. Subplot (c) displays the average SF order parameter, Eq. \eqref{phi:av}. The structure factor $S$ at
		$(k_x,k_y) = (\pi,\pi)$ is shown in (d). As for $\rho=1$, we consider $12\times12$ system with periodic boundary conditions
		 and $n_{\rm max}=8$.}

\label{corr2}
\end{figure}

\subsubsection{Average density $\rho = 1$}

We now discuss the phase diagram at density $\rho=1$. {We first recall that  the properties of the ground state are very
well studied in the absence of density-dependent tunneling (for $T=0$). In this limit the inclusion of NN density-density interaction to BHM leads to additional quantum phases which we shortly review {in the following}~\cite{kuhner_00,rossini_12,Lahaye09,Baranov12}. For large $V$ bosons occupy every alternate lattice site forming CDW phase, {an incompressible phase} which spontaneously breaks sublattice symmetry.
{When the NN interaction $V$ is} comparable to the on-site repulsion, the system breaks both $U(1)$ gauge symmetry and translational invariance to form a SS
phase. The SS phase of eBHM is found to be stable in wide range of interaction strengths~\cite{goral_02,kovrizhin_05,menotti_07,danshita_09}.}
{For unit filling or density the stability of SS of soft-core bosons has been demonstrated in Refs.~\cite{sengupta_05,ohgoe_12}.
When {$4V \ll U$}, the system exhibits MI-SF phase transition as $t$ increases.} At stronger NN interactions
{$4V\approx U$}, the MI phase becomes unstable and CDW phase replaces the MI
phase~\cite{iskin_11,kovrizhin_05,yi_07,ohgoe_12}. Numerical studies of one dimensional lattices reveal the existence of topological Haldane phase separating
MI and CDW phases~\cite{rossini_12,gremaud_17,sugimoto_19}. {The dependence of the hopping parameter $t$ on various phase transitions
for fixed densities has been discussed~\cite{kimura_11}. We report the corresponding phase diagram with $T=0$ in the Appendix.}

The effect of correlated hopping in determining the phases is now visible in Fig.~\ref{corr1}, which displays the correlation $M_b$ at $(0,0)$ and $(\pi,\pi)$, the average SF order parameter,
and the structure form factor $S(\pi,\pi)$. These quantities are reported in the  $t-V$ plane and have been obtained using the SGMF approach.
The properties of $M_b$ and $S(\pi,\pi)$ allow us to identify the phases. At {$V/U\leqslant 0.25$}, as $t$ increases, $M_b(0,0)$ remains nonzero
whereas it vanishes at $(\pi,\pi)$. In addition, $S(\pi,\pi)$ is zero in this region. The region where both $M_b(0,0)$ and $S(\pi,\pi)$ vanishes is MI, whereas the region with finite $M_b(0,0)$ is SF. At lower $V$ we recover the MI-SF transition of BHM~\cite{fisher_89,jaksch_98}.
 {The critical hopping strength of MI-SF transition $t_c$ increases for {$0\leqslant 4V \leqslant U$} with finite $T$.} This shift is due to the finite value of correlated hopping, which interferes destructively with single-particle tunneling.

At {$V/U\simeq 0.25$} we observe the onset of periodic density modulations, which are signalled by the finite value of  $S(\pi,\pi)$. The CDW phase corresponds to the large region with vanishing off-diagonal order, $M_b({\bf k})=0$. This region is centered about the line of perfect interference $|T|=t$, Eq. \eqref{T:t}. There is a direct MI-CDW transition for a finite range of values $t/U$ at {$V/U\simeq 0.25$}, which is characterized by a fast increase of $S(\pi,\pi)$ from zero to the maximum value. In the presence of SF this increase is gradual and characterised by the appearance of a finite value of $M_b(\pi,\pi)$. We identify this region with the SS phase since $|M_b(\pi,\pi)| < |M_b(0,0)|$, see Table~\ref{tab_op}. The boundary $t_c/U$ of the transition CDW-SS increases with $V$. We have checked that $t_c/U$ is shifted to larger values with respect to the phase boundary one obtains by setting $T=0$, see Appendix. The resulting domain of the CDW phase is larger. We attribute this effect to the destructive interference between single-particle and correlated hopping.

At larger values {$V \approx 2U$} and for vanishing $t/U$ we observe the appearance of finite off-diagonal long-range order with $|M_b(\pi,\pi)| > |M_b(0,0)|$, which we identify with a staggered supersolid (SSS). This phase is due to correlated hopping, which becomes dominant at large ratios $V/U$.

These behaviours can be understood on the basis of mean field considerations: correlated hopping favours staggered SF for $t/U\to 0$. The size of this region increases as $V$ (and thus the strength of the dipolar interactions) increases. At large ratios $t/U$, instead, the kinetic energy dominates and the phase is SF. The two contributions interfere destructively in an intermediate region, where the phase is incompressible. Diagonal long-range order is found for {$V/U\geqslant 0.25$}. We observe that the size of the SSS region at {$V/U\geqslant 0.25$} is now significantly smaller.

\subsubsection{Average density $\rho = 2$}

Figure~\ref{corr2} displays the phase diagrams of the relevant quantities in $t-V$ plane for the average density $\rho=2$.
Similarly to the unit filling case, for {$V/U<0.25$} the phase is either MI or SF. The phase boundary $t_c$ separating the two phases depends on the strength of the dipolar interactions due to the interference between correlated hopping and single-particle hopping. For {$V/U\gtrsim 0.25$} the structure form factor $S(\pi,\pi$ is different from zero and signals the onset of density modulations. As compared to quantum phases at unit filling [Fig.~\ref{corr1}], the size of the region of insulating CDW phase is reduced whereas the size of the SSS parameter region is significantly larger. A striking difference with respect to the phase diagram at unit filling is the appearance of SSF at {$V/U\sim 0.25$} and $t/U\to 0$.

The behaviour reported in Fig. \ref{corr2} qualitatively agrees with the phase diagram calculated for the same model but in one dimension and using DMRG~\cite{kraus_20}.
{ However, it is worth noting that the parameter regime of SSF is shifted to lower $V$ values as compared to 1D.
This is due to the larger coordination number of square lattices and for $d$-dimensional lattices the staggered superfluidity
exist at and around $2dV\approx U$, where $2d=z$ is the lattice coordination number.}
As in \cite{kraus_20}, the finite value of $T$ is responsible for the appearance of SSF at {$V/U\sim 0.25$} and $t/U\to 0$. The size of the SSF phase is now significantly smaller. In order to perform a systematic comparison with the one dimensional case, we now consider a small, fixed value $t/U$ and analyse the phases as a function of $V/U$. Figure \ref{corr_vs_t} displays the behaviour of $M_b(\pi,\pi)$ as a function of $V/U$ for (a) $t/U=0.002$, (b) $t/U=0.02$, (c) $t/U=0.23$, and (d) $t/U=0.35$. For  $t/U=0.002$ the phase is first MI. In the atomic limit, at {$V/U=0.25$} MI and several CDW phases are degenerate. Here, the phase becomes SSF due to the prevailing role of correlated hopping and is signalled by the peak of $M_b(\pi,\pi)$ at {$V/U\sim 0.25$}. This peak was also observed in one dimension and exhibits features of a continuous phase transition \cite{kraus_20}. At higher $V$ the value of $M_b$ increases again with $V$, together with the finite value of the structure form factor we identify this phase as SSS. At $t/U=0.02$, the phases SSF and SSS are separated by an incompressible CDW phase.
At higher $t$ in (c), single-particle hopping dominates over correlated tunneling. Here, the system is driven first from  SF to SS and then  to SSS with an intermediate insulating CDW phase. The width of CDW phase in between the SS and SSS phases increases and the region of SSS phase reduces with $t$. This is evident from the behaviour of $M_b$ at $t/U=0.35$ shown in Fig.~\ref{corr_vs_t}(d).

\begin{figure}
 \includegraphics[width=\linewidth]{fig3.pdf}
 \caption{Off-diagonal single-particle correlation $M_{b}(\pi,\pi)$ as a
	  function of $V/U$. The blue lines report the behaviour corresponding to Fig. \ref{corr2}(b) for hopping strengths $t/U=0.002,0.02,0.23,0.35$. The values of $t/U$ is reported in the subplots. The dashed red line is obtained using $2 \times 2$ cluster in CGMF theory. Here, we consider the maximum
occupancy per lattice site $n_{\text{max}} = 8$: this choice of
$n_{\text{max}}$ is sufficient to obtain converging results for densities considered.
}
 \label{corr_vs_t}
\end{figure}

In order to examine the effects of quantum fluctuations within our mean-field ansatz, we use the CGMF method and compute the $M_b(\pi,\pi)$ with a $2\times 2$
cluster. The resulting behaviour is reported by the dashed red line in Fig.\ref{corr_vs_t}. At $t/U = 0.002$ the domain of the SSF phase, signalled by
the peak in the $M_{b}(\pi,\pi)$, is the same as in the SGMF method, while the SSF-SSS transition becomes sharper. For $t/U = 0.02$, Fig.\ref{corr_vs_t} (b), the critical value of $V/U$
for the CDW-SSS transition shows a small increase as compared to SGMF transition. { When $t/U$ is increased to
$t/U = 0.23$, Fig.\ref{corr_vs_t}(c), we observe a shift in the peak of $M_b(\pi,\pi)$, signalling the SS phase, to higher values of $V/U$.
In detail, the peak occurs at $V/U = 0.3$ with SGMF, this shifts to $V/U = 0.36$ with the $2\times 2$ cluster. We note that the shift between
the boundary SF-SS predicted by CGMF and the one predicted by SGMF is consistent with the findings reported in \cite{ohgoe_12,suthar_19} for
the SF-SS transition. }

\subsection{Phase diagram for next nearest-neighbor interactions}
\begin{figure}
  \includegraphics[width=\linewidth]{corr_NNN_avgden_1.png}
  \caption{Observables for the ground-state of the 2D eBHM at $\rho = 1$ for the mean-field Hamiltonian, Eq.\,\eqref{ebhm_mf}, truncated at NNN, and calculated by means of the SGMF approach.
	   	Subplots (a), (b) and (c) show the Fourier transform of the off-diagonal single-particle correlation $M_b$ at $(k_x,k_y) = (0,0)$, $(0,\pi)$, and
	   	$(k_x,k_y) = (\pi,\pi)$, respectively. At $V/U \gtrsim 1.5$, the finite values of $M_b$ both at $(0,\pi)$ and $(\pi,\pi)$ signals the quarter-filled ordered phase. The phase is QF-SSS because
	   $|M_b(0,0)|<|M_b(0,\pi)|,|M_b(\pi,\pi)|$. Subplot (d) displays the average SF order parameter.}
  \label{corr1_nnn}
\end{figure}

\begin{figure}
  \includegraphics[width=\linewidth]{corr_NNN_avgden_2.png}
  \caption{The Fourier transform of the single-particle correlations $M_b$ for $\rho = 2$ with NNN at (a) $(k_x,k_y) = (0,0)$, (b) $(k_x,k_y) = (0,\pi)$ and
	   	(c) $(k_x,k_y) = (\pi,\pi)$. Unlike at unit filling, here for lower values of NN interaction a CDW-SSS transition occurs and at
	   	$V/U \gtrsim 1.5$, the system enters into QF-SSS phase.
		The QF-SSS phase region is characterized by a finite value of $M_b(0,\pi)$ (see (b)) and of  $M_b(\pi,\pi)>M_b(0,0)$ (see (c)). {Note the existence of the SSF phase at $V/U\sim 0.25$ in the presence of the NNN density-density interactions}. Subplot (d) shows the average SF order parameter in the vicinity of the SSF phase.}
  \label{corr2_nnn}
\end{figure}

\begin{figure}
  \includegraphics[width=\linewidth]{avgop.pdf}
  \caption{The average SF, Eq. (\ref{phi:av}), and density-correlated order parameter, Eq. (\ref{average_eta}),  as a function of $V/U$ for $\rho=2$ with NNN terms of the eBHM, Eq. \eqref{ebhm_h2}. Subplot (a) has been calculated for $t/U=0.02$ and
	   (b) for $t/U=0.3$. Here, the solid blue (dashed red) line is
	   $\eta_{\rm avg}$ ($\phi_{\rm avg}$).}
  \label{avgop}
\end{figure}

We now include the NNN terms in our mean-field treatment. We first recall that the quantum phases of eBHM due to competition between NN and NNN repulsion at various filling have been
studied before in the absence of the density-dependent tunnelings~\cite{ng_08,ng_10,kuno_14,heydarinasab_17}. We now include density-dependent tunneling and analyse the ground state of the model Hamiltonian in Eq.~\eqref{ebhm_mf} when the NNN terms of density-density repulsion and of correlated tunneling are included.

We first discuss the phase diagram at density $\rho=1$. Figure~\ref{corr1_nnn} displays  the single-particle correlations and the average SF order
parameter.  Some qualitative features are similar to the NN case, compare with Fig. \ref{corr1}. A striking difference from the NN case is the decrease of the size of
the insulating CDW phase. Moreover, correlated hopping here gives rise to a QF-SSS phase.

These features become more enhanced as the density is increased. Figure~\ref{corr2_nnn} shows the correlations and average SF order parameter for density $\rho=2$. With respect to the NN case (cif. Fig. \ref{corr2}) we observe that the SSS phase now disappears above a critical value $V/U\sim 1.5$ and becomes QF-SSS. The size of the QF-SSS phase is significantly larger than for unit filling. In particular, the phase boundary of CDW to QF-SSS transition weakly depends on $t/U$. These behaviours qualitatively agree with the one-dimensional phase diagram of Ref. \cite{kraus_20}. Important differences are that the SSF phase results to be smaller. Moreover, the nature of the phase, which is here QF-SSS, could not be uniquely identified in Ref. \cite{kraus_20}.

Figure~\ref{avgop} displays {the average SF order parameter, $\phi_{\rm avg}$, and the average  density-correlated order parameter $\eta_{\rm avg}$, Eq. \eqref{average_eta},
as a function of $V/U$ and fixed values of $t/U$. We first consider the value $t/U = 0.02$, subplot (a). Here, at $V=0$ the phase
is MI. As $V$ is increased, the transitions  MI-SSF-SSS take place at {$V/U\sim 0.25$} and is here signalled by a sharp increase and a local maximum of both parameters. Finally, at the transition SSS to QF-SSS $\eta_{\rm avg}$ increases whereas $\phi_{\rm avg}$ decreases. At $t/U=0.3$, the system shows SF-SS-CDW-SSS transition as $V/U$ is increased. The corresponding trends of $\eta_{\rm avg}$ and $\phi_{\rm avg}$ are visible in Fig.~\ref{avgop}(b).

\section{Phase transitions at fixed $V/U$}
\label{pd_v}
\begin{figure}
  \includegraphics[width=\linewidth]{corr_nn_vp25_v2.png}
  \caption{The off-diagonal single-particle correlations $M_{b}(0,0)$ and $M_b(\pi,\pi)$ as a function of $\mu/U$ and $t/U$ for
	   {$V/U=0.25$} (upper panels) and {$V/U=2$} (lower panels). The left panels show the contour plot of $M_{b}(0,0)$, the right panels of $M_{b}(\pi,\pi)$. The diagram has been calculated using the SGMF method and truncating the interactions to the NN terms.}
  \label{corr_nn}
\end{figure}

We now discuss  the ground state phase diagrams in $t-\mu$ plane and for fixed values of $V/U$. For these values the NN repulsion energy becomes comparable to the on-site interaction energy and the insulating phases are  CDW~\cite{iskin_11}. At {$4V=U$}, in particular, the MI phase with $n_0$ boson per site becomes degenerate
with the CDW with occupancies $(2n_0,0)$. Moreover, the CDW phases with $(n_0+1,n_0)$ and with  $(2n_0+1,0)$ are degenerate~\cite{iskin_11}.
We vary $\mu$ from $0$ up to $4U$ in order to include the density $\rho=2$. In the $t-\mu$ plane the line of destructive interference, Eq. \eqref{T:t}, moves to higher $t$ values as $\mu$ is increased, since the density $\rho=\rho(\mu)$ increases monotonically with $\mu$.

\subsection{Results using SGMF}

Figure \ref{corr_nn} displays the phase correlation functions $M_b(0,0)$ and $M_b(\pi,\pi)$ for {$V/U = 0.25$} (upper panels) and {$V/U = 2$} (lower panels). The phase diagram is separated into two regions by the sequence of insulating CDW lobes, which are localized along the values of $\rho$ and $t$ fulfilling Eq. \eqref{T:t}.  On the left side, where correlated hopping dominates, the phase is SSS. The size of this region increases with $V$, and thus with the value of $|T|$. The SSF phase is observed only for {$V/U=0.25$} and at sufficiently high values of $\mu$, here at $\mu\geqslant 3$, where $\rho\approx 2$. Here, a direct SSF-CDW transition is observed in a small parameter region at $\mu\approx3.2$. On the right side single-particle hopping is responsible for the emergence of the SS phases. At higher $t$, the system enters into the SF phase. At {$V/U=2$} we observe the SS-SF phase  boundary, which varies linearly as a function of $t$. This feature is consistent with the findings of quantum Monte Carlo studies of 2D eBHM~\cite{ohgoe_12}.
{It is important to note that the choice of boundary conditions strongly affects the phases and their transitions. As mentioned
previously, we consider periodic boundary conditions, while the use of open boundary conditions lead to nonuniform densities at the edges
which hinders the identification of quantum phases.}

\subsection{Comparison between SGMF and CGMF}

\begin{figure}
  \includegraphics[width=\linewidth]{pd_dit_t_mu.pdf}
	\caption{Ground state phase diagram of {Eq.~(\ref{ham})} with NN
	   interactions and density-dependent tunneling. The solid and dashed lines are phase boundaries
	   obtained using the SGMF and CGMF methods, respectively, for simulating {Eq.~(\ref{ham})}. The CGMF method
	   uses $2\times 2$ clusters. The CDW phases are indicated by their
	   sublattice occupancies $(n_a,n_b)$. The insets zoom into regions of the phase diagram where CGMF predicts an increase of the size of the SSF phase (a) and a decrease of the size of the SSS phase (b) with respect to the SGMF predictions.}
  \label{pd_nn}
\end{figure}

We explore  the
effects of quantum fluctuations and the intersite correlations on the
quantum phase transitions by means of the CGMF method using $2\times 2$ cluster. We restrict here to the NN case. Figure \ref{pd_nn} shows the ground state phase diagram of our model in the $t-\mu$ plane for {$V/U=0.25$} and {$V/U=2$}.
To illustrate the differences, the boundaries between various phase transitions
computed with the SGMF and CGMF methods are shown. For {$V/U=0.25$},
Fig.\ref{pd_nn}(a),
the CGMF method predicts smaller domains of the $(n_0,0)$ CDW lobes. For instance, the phase boundary  separating the CDW $(1,0)$ from the SS phase is at $t_c/U\approx 0.093$ for SGMF, and it is decreased to
$t_c/U \approx 0.089$ with the CGMF. A similar trend of the phase boundary $t_c/U$ for the CDW-SS transition was reported in
\cite{suthar_19} for {$V/U = 0.2$}.
It is to be noted  that the size of the SSF phase increases with the CGMF method.
This is visible in the inset of Fig.\ref{pd_nn}(a). At $\mu/U = 3.8$, the
SSF phase persists upto $t/U \approx 0.023$ using SGMF, and with CGMF this
is modified to $t/U \approx 0.03$. Our computations show that CGMF predicts an increase of the size of the SSF phase while the SSS-CDW phase boundary remains unaffected. This results in a reduction of the SSS phase domain.
Thus, the quantum fluctuations captured by the CGMF tend to correct the SGMF predictions by increasing the size of the uniform density phases like the SSF and the SF, while the domains of the structured density phases like the
CDW, SSS and SS are tendentially decreased. As with the {$V/U=0.25$} case, the CGMF results show a decrease in the domain of the SS phase in the
phase diagram with {$V/U=2$}, from the inset in Fig.\ref{pd_nn}(b). On the contrary, the domain of the CDW phase remains unaffected.

\section{Conclusions}
\label{conc}
 We have studied the zero temperature phase diagram of dipolar bosons in 2D optical lattice using a mean-field approach
and investigated the effects of the interplay between single-particle hopping and correlated tunnelings when these can destructively interfere.
The mean-field study, both by means of site-decoupled and  cluster Gutzwiller approach, confirms the findings of the 1D phase diagrams for density $\rho=2$ and obtained using DMRG \cite{kraus_20}. Moreover, it extends them to the grandcanonical ensemble: The interference cuts the phase diagram into two topologically different superfluid phases. In particular, there where correlated hopping dominates the SF order parameter is staggered. The comparison of the domains of SSS and SSF at fixed nearest neighbor interactions exhibits the suppression of SSS phase with the enhancement of SSF phase when the  inter-site correlations are accounted for. When next-nearest neighbour terms are included, a quarter-filled supersolidity appears at large dipolar strengths.
The present study has been performed for experimentally consistent parameters and can serve as a guide for a possible observation of staggered quantum phases of dipolar bosons in 2D optical lattices.
{The recent experimental realization of extended Bose-Hubbard model for ultracold gas with strong dipolar interaction is an ideal setup where the quantum states here studied could be observed~\cite{baier_16}.}

Finally, we have verified that the quantum phases and phase diagrams reported in the present work are robust to the system size, by repeating the calculations for $30x30$ lattice site system.

\acknowledgments
We thank K. Biedr\'on for discussions and acknowledge the support of PL-Grid Infrastructure, Poland and Vikram-100, the 100TFLOP HPC Cluster at
Physical Research Laboratory, Ahmedabad, India. K.S. and J.Z. acknowledge the support by  National Science
Centre (Poland) under project 2016/21/B/ST2/01086. R.K. and G.M. acknowledge the support by the German Research Foundation (the priority program
No. 1929 GiRyd) and by the German Ministry of Education and Research (BMBF) via the QuantERA project NAQUAS. Project NAQUAS has received funding
from the QuantERA ERA-NET Cofund in Quantum Technologies implemented within the European Union's Horizon 2020 program.

\section*{Appendix}
We discuss here the ground-state phase diagram of the model considered in the absence of the density dependent tunnelings
($T$ is put to zero by force).
In Fig.~\ref{corr_no_ddt}, the single-particle correlations $M_b$ at $(0,0)$ and $(\pi,\pi)$ are
shown for the average densities $\rho=1$ and $\rho=2$. We first discuss the quantum phase transitions
at $\rho=1$. The $M_b(0,0)$ which is a measure of off-diagonal long-range order, has a finite value for compressible
SF and SS phases whereas it remains zero for insulating MI and CDW phases. At low NN interaction
{$V/U \ll 0.25$}, there is MI-SF transition where the critical hopping $t_c$ is independent on $V$.
When NN interaction is comparable or overcomes to the onsite interaction {$V/U \geqslant 0.25$}, the transition between
density modulated quantum phases CDW-SS occurs. The $t_c$ of CDW-SS transition increases as a function
of $V$~\cite{kimura_11}. The finite value of $M_b(\pi,\pi)$ for SS phase clearly demarcates it from SF and
CDW phases, as evident from Fig.~\ref{corr_no_ddt}(b). At $t/U=0.05$, the MI-CDW transition of $\rho=1$ as a function
of $V$ is consistent to the previous quantum Monte Carlo study~\cite{sengupta_05}. At $\rho=2$, the qualitative features of various
phase transitions remain similar to $\rho=1$ case, however quantitatively the critical hopping varies [Fig.~\ref{corr_no_ddt}(c,d)].
\begin{figure}
  \includegraphics[width=\linewidth]{app_fig.png}
  \caption{The single-particle correlations with NN interaction of eBHM and $T=0$.
	   The correlations for $\rho=1$ are shown at (a) $(k_x,k_y) = (0,0)$ and (b) $(k_x,k_y) = (\pi,\pi)$,
	   and for $\rho=2$ these are shown at (c) $(k_x,k_y) = (0,0)$ and (d) $(k_x,k_y) = (\pi,\pi)$}
  \label{corr_no_ddt}
\end{figure}

\input{dip2d_v08.bbl}

\end{document}

redpajama_set_name: RedPajamaArXiv


text:
Week in review: Top photos of the week from AP
Here are some of the top photos of the week, as selected by the Associated Press.
APTOPIX Trump Impeachment Pelosi
Speaker of the House Nancy Pelosi, D-Calif., responds forcefully to a question from a reporter who asked if she hated President Donald Trump, after announcing earlier that the House is moving forward to draft articles of impeachment against Trump, at the Capitol in Washington, Thursday, Dec. 5, 2019. (AP Photo/J. Scott Applewhite)
J. Scott Applewhite
APTOPIX France Strikes
A man stands on a traffic light during a demonstration in Paris, Thursday, Dec. 5, 2019. The Eiffel Tower shut down, France's vaunted high-speed trains stood still and several thousand protesters marched through Paris as unions launched open-ended, nationwide strikes Thursday over the government's plan to overhaul the retirement system. (AP Photo/Thibault Camus)
Thibault Camus
APTOPIX Wintry Weather New Jersey
A car makes its way through a snowy landscape in Highland Falls, N.J., Tuesday, Dec. 3, 2019. The last of the snow is falling over parts of New Jersey after leaving behind power outages in the northwest part of the state. (AP Photo/Seth Wenig)
Seth Wenig
APTOPIX Brazil Gender Violence
Women perform the choreography, "Un violador en tu camino" or "A rapist in your path," in a demonstration against gender-based violence, at Cinelandia Square, in Rio de Janeiro, Brazil, Tuesday, Dec. 3, 2019. The blindfolded women chanted the words that include: "The fault wasn't mine, nor where I was, nor how I was dressed. You were the rapist, you are the rapist," originated by the Chilean feminist collective Las Tesis which has become a global mantra for women against gender violence. (AP Photo/Silvia Izquierdo)
Silvia Izquierdo
APTOPIX Germany Daily Life
An illuminated party ship cruises over the river Main after sunset in Frankfurt, Germany, Monday, Dec. 2, 2019. (AP Photo/Michael Probst)
Michael Probst
APTOPIX Panama Daily Life
Children play in a Christmas light installation at the Plaza de la Independencia in Panama City, Tuesday, Dec. 3, 2019. (AP Photo/Arnulfo Franco)
Arnulfo Franco
APTOPIX Wintry Weather Minnesota
Neighbors work together in an effort to clear out as much snow as possible from E. 8th Street, Sunday, Dec. 1, 2019, in Duluth, Minn. (Alex Kormann/Star Tribune via AP)
Alex Kormann
APTOPIX Britain NATO
NATO Secretary General Jens Stoltenberg, center front left, speaks with U.S. President Donald Trump, center front right, after a group photo at a NATO leaders meeting at The Grove hotel and resort in Watford, Hertfordshire, England, Wednesday, Dec. 4, 2019. NATO Secretary-General Jens Stoltenberg rejected Wednesday French criticism that the military alliance is suffering from brain death, and insisted that the organization is adapting to modern challenges. From left, French President Emmanuel Macron, Norway's Prime Minister Erna Solberg, German Chancellor Angela Merkel, Poland's President Andrzej Duda and Greek Prime Minister Kyriakos Mitsotakis. (AP Photo/Francisco Seco)
Francisco Seco
Timegoggles
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text:
⇑ back to "Devotionals"
4:00AM EDT 3/24/2022
Seek the Kingdom of God above all else, and live righteously, and he will give you everything you need. — Matthew 6:33
In a column, reporter Bob Greene once opined on "How to Become a Hero." He described a Texas gem dealer who went to a mineral show in Arizona. There he met a man who wandered creek beds, looking for interesting stones. This man was displaying some of his finds in plastic containers.
Among the smaller stones the dealer saw a large rock, and that's all the displayer thought it was--a big rock. But this gem dealer knew otherwise. He recognized it as the biggest star sapphire he had ever seen.
Knowing that the owner had no idea of the true worth of the stone, the gem dealer bargained with him and paid $10 for the rock. Later he reported that it was worth $2.5 million.
And, noted Greene, this gem dealer has become a sort of folk hero because he found something of value and purchased it at the lowest possible price, without hinting to the original owner what was going on. Greene went on to say, "This is just a particularly dramatic example of the way so many people are getting rich these days. They don't do anything of importance or value; they just manipulate and sidestep and feint and parry. They make people think they are going to do one thing, and then they do another. The lesson seems to be that only suckers believe in putting in a day's work for a day's pay. The smart boys are the tricksters."
Unfortunately, Bob Greene is right. People's lives today seem consumed with lotteries, long shots, and insider trading . . . and a continuing saga of greed, bribery, extortion, and embezzlement.
As Christians, however, we shouldn't be too surprised. Jesus said that his values are the opposite of the world's and that people who follow him will be misunderstood. True followers of Christ won't fit into society.
The Bible says that who we are on the inside is infinitely more important than our outward appearance, the way we achieve our goals is as important as the goals themselves, and that seeking Christ and his kingdom is more important than being the richest person on earth.
Greene concludes his column with these thoughts: "Let's imagine that the gem dealer sees the star sapphire and instead of purchasing it for $10, he tells the man of its real worth and urges him to get some advice about what a fair asking price is before selling it. I know, I know; that's not the way the world works. Maybe, though, he could have at least cut the poor rock-hunter in on his profits. But then he wouldn't have been a hero for these days, would he?"
"And what do you benefit if you gain the whole world but lose your own soul?" (Matt 16:26).
Don't worry about making it in today's world. Focus your attention on being a "hero" where it counts . . . in God's eyes.


What better setting for the acclaimed rock opera Jesus Christ Superstar than the nave of historic Lincoln Cathedral?
With the hours quickly decreasing until opening night, The Lincolnite had an exclusive seat at one of the show's final dress rehearsals.
The dramatic stage is set beneath towering ceilings and looming columns and the cast of hundreds is gearing up for the colourful performance.
After an extremely popular project launch, the community production sold out within six weeks, with an extra date being scheduled and 4,000 tickets in total going to lucky theatre goers.
The cathedral will present the show for the first time on July 23, and it will run until August 2.
With a cast of over 200 local people from various acting and singing backgrounds, hundreds of hours of preparation has gone into the show.
The performance will feature local people from the youngest actor (aged 6) to the oldest at age 86.
Producer Ben Poole (Jesus Christ), who is also a Lincoln businessman in the city, said: "I've done acting and singing for most of my life, both amateur and professional.
"This is the fruition of a dream I had as long as eight years ago and we have spend the last three years after gaining permission from the cathedral putting the show together.
"Hundreds of people have put in hours of their time in making this show happen. Our costume designer for example probably got a bit of a shock when it came to making 200 costumes.
"I knew it would be popular. I had to go with my gut feeling that Jesus Christ Superstar, in Lincoln Cathedral, rock music… I know I'm going.
"For the first time ever we have not had to advertise the show at all, and the extra date pretty much sold out over night.
Local carpenter Andrew Morrant will be appearing on stage for the first time.
"I was asked to make the rehearsal stage and agreed to come along and do that. I was then speaking with 'Jesus', and he said 'would you mind crucifying me every night?


text:
Q: Query shortest path from coordinates with pgrouting How do I calculate the path, pass coordinates and, if possible, in the return function, have the route coordinate list?
(I'm trying to plot the route with the ipyleaflet library in a jupyter notebook, and I need the return of the route's coordinate list. I'm new to this and I don't know how to do it)
I tried with an example query that I searched for in a book
but I got an empty answer
SELECT r.seq, r.node, w.name As street, w.x1, w.x2, r.cost::numeric(10,4), ( SUM(w.length_m) OVER(ORDER BY r.seq) )::numeric(10,2) As dist_m
FROM pgr_dijkstra('SELECT gid AS id, source, target, cost, reverse_cost FROM ways', (SELECT id FROM ways_vertices_pgr ORDER BY the_geom <-> ST_SetSRID( ST_Point(-3.7482, -38.5486), 4326) limit 1 ) , (SELECT id FROM ways_vertices_pgr
ORDER BY the_geom <-> ST_SetSRID( ST_Point(-3.7532, -38.5170), 4326) limit 1 ), true) AS r
LEFT JOIN ways AS w ON r.edge = w.gid;

A: Try this query if it works
     with source_tmp as (SELECT source
            FROM ways order by st_distance(the_geom, ST_SetSRID(ST_MakePoint(-38.5486,-3.7482), 4326)) limit 1),
    target_tmp as (SELECT target
            FROM ways order by st_distance(the_geom, ST_SetSRID(ST_MakePoint(-38.5170,-3.7532), 4326)) limit 1)

SELECT *
FROM pgr_dijkstra('SELECT gid AS id, source, target, cost, reverse_cost FROM ways', (SELECT source from source_tmp) ,
                  (SELECT target from  target_tmp), true) AS r
LEFT JOIN ways AS w ON r.edge = w.gid;

What I did here is selected source and target initially using lat long, you have to reverse the order in your st_makepoint function. Then I can the query

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text:
Films and Filmmakers
Movies that you'd like to see remade? (I know...blasphemy!)
By traceyk65,
February 15, 2014 in Films and Filmmakers
traceyk65
This is the original trailer--makes it look pretty good. And it is, to a point. It's got a good mystery--Was Robert Forrest's death an accident? And if not, who killed him and why? Why won't the widow cooperate? What is she hiding? The trouble is that it gets bogged down in exposition and ant-fascist preachiness. And there's no love story between Tracy and Hepburn, which is really a waste of great chemistry.
I think this movie would actually benefit from a remake. Tighten up the plot, cut out the preaching (or at least tone it down) play up the suspense and the attraction between the two stars. Keep the comic relief and the cute kid and keep the claustrophobic dark atmosphere. Maybe have it star someone like Harrison Ford in the Tracy role and Cate Blanchett int he Hepburn part (I'm really terrible at casting modern actors) Set it in modern times and make the dead man a political candidate, rev up the paranoia and go!
Sooo--what other classic movies can you think of that are flawed that might be better if they were remade and why?
faceinthecrowd
LocationAlphaville
You're right -- it's blasphemy. No remake!
Dargo2
Not "blasphemy' at all, tracey! NOT at all, and the reason it isn't is EXACTLY for the reason(s) you mentioned in your proposal..."Keeper of the Flame" is a flawed film, and despite the fact it features two of the greatest films stars ever.
Yep, you're EXACTLY correct...in many ways it's "preacher" than what some people sit through on a Sunday Morning AND about as subtle as a sledgehammer in some other regards.
And now for MY proposed remake, and one which I've stated a few times around here previously and have also been met with cries of "blasphemy" by others: "The Petrified Forest".
And the reason I'd like someone to tackle this project is NOT as some naysayers erroneously conjectured because I want "more outlandish action", BUT because I believe Robert Sherwood's play about the collision of two distinctly different and intelligent men, one from the "right-side-of-the-tracks" but disillusioned with life and so anguished that life means nothing to him anymore, and the other from the "wrong-side-of-the tracks" who would do anything to keep on living, is SUCH a timeless theme that I would love to see it updated by use of more modern references.
(...and yeah, once again, even though the leads in the original of THIS one were ALSO some of the greatest of film stars ever)
oblige_me
I'd like a remake of Scaramouche simply because it's one of my favourite novels of all time. The 1923 version is good, but the 1952 version just didn't capture the book and took too many liberties. I always envisioned a film version of Scaramouche in the same vein as Richard Lester's Three and Four Musketeers.
A remake of Captain Blood and Of Human Bondage would be interesting, even though their predecessors are very fine films indeed.
On the subject of The Three Musketeers though, I NEVER want to see another adaptation of that in my life. It's one of my favourite books and all the modern (past 20 years or so) film versions have been so utterly awful. Richard Lester captured the books the best, and previous adaptations to those films are charming and fun.
Out of those adventure novels however, I feel Scaramouche has more to offer considering the complexity of the main character.
Terrence1
Many years ago, I read "The List of Adrian Messenger" for the first time and have re-read it many times since. It's a fascinating story and so well written. Because of this,I was eagerly anticipating the movie version. What a disapppointment! It was absolutely terrible. In the original ads, all they did was promote the gimmick of having celebrities on ther screen completely disguised. The basic plot was practially done away with. I'd love to see another attempt at filming this gripping suspense story.
Terrence.
Those are all good suggestions, especially Petrified Forest and Of Human Bondage. The plot and themes are still relevent today, maybe even more than when they were made. Forest especially could be tied into America's messed-up prison system and inequality issues and really could be interesting.
Not familiar with Scaramouche. The only reference I have is the Italian clown?
SansFin
I do not know if it could be said that it would truly be a remake - I would love to see a good version of: The Count of Monte Cristo written by A. Dumas.
There is one movie only which captures any of the best elements of the novel and that is a recent movie. It was forced to leave out many excellent aspects because of the constraint of squeezing such a sweeping story into a convenient length.
Do you know what Keeper of the Flame was really about and why it was so preachy? The film was actually about 4 years out of date when it was made.
johnm001
I've been waiting 50 years for a remake of MY FAIR LADY. I hate everything about that movie, and it's my favorite musical, I don't expect I'll see one in my lifetime, however.
>Do you know what Keeper of the Flame was really about and why it was so preachy? The film was actually about 4 years out of date when it was made
I assume it was warning about the dangers of European Fascism and the possibilty of it happening here. Was there something else going on then that Im missing?
I think it would be a good candidate for an updated remake--though it's almost inevitable that it might come out too Dan Brown-ish. Might be interesting to take the theories floating around out there right now (and there are a lot of them, ranging from, "OK, I can almost buy that" to "OMG This person is a grade A nut job!") about America moving towards a plutocracy and being secretly run by a handful of individuals. (I know... "You mean it isn't?" Yes, probably, but let me keep my illusions, ok?)
joefilmone
"Logan's Run" - the book is more complex than the movie and now you could cast it with younger actors.
ValentineXavier
LocationAnn Arbor, MI
I'd like to see a remake of *Attack of the Crab Monsters.* It has a great plot, some truly unique scifi elements, and a bit of an environmentalist message. With today's FX, it could be done even better, so long as they didn't go way overboard, like they did with the remake of *The Thing.*
*Attack of the Crab Monsters*
http://www.youtube.com/watch?v=tTVWQ8dxPW0
Man, that movie makes me hungry! I need a big vat of boiling water, some soft butter, a gallon jug of Tabasco sauce, some cocktail sauce, and about 50 boxes of Zatarain's Crab Boil.
And some good Cajun music
http://www.youtube.com/watch?v=b9V5Z5R97MA
Fred, you've got to remember that the crabs could not be touched. They were a bit out of phase with us, or something like that. But, they could eat us, absorb our brains, and speak telepathically in the voices of those they ate.
Along the same lines as "Crab Monsters" I would enjoy seeing a new special effects laden remke of some fifties movies - "Earth vs the Flying Saucers" "Incredible Shrinking Man" "This Island Earth" "Them!" or even "I Was a Teenage Werewolf".
aradia22
I have an answer that will probably be considered blasphemy (especially if there are any Brits in this forum) but I will say Brief Encounter. I saw it for the first time when it aired recently on TCM and it just didn't work for me. I didn't really love either of the characters or the way it was filmed but I could see the good movie hidden in there. From what I can tell, it wasn't very well adapted from the stage. The voiceover and the way it was staged made it lose something (even if I had liked the actors). So yes, I'm fully behind some director trying to redo Brief Encounter in a way that's faithful to the play and a period piece but that's filmed in a more interesting/compelling way. I take this all back if they're just going to hire Keira Knightley or Carey Mulligan for the lead female role.
Also, musicals. There are so many musicals I'd like to see remade or just put on screen the way they deserve to be preserved.
roverrocks
LocationMontrose, Colorado
There's a remake of GODZILLA coming out in 2014. I am not a big fan of remakes but the special effects in this trailer I saw today are pretty staggering and exciting. Who knows if the actual movie will be good or not. Here is the trailer:
>Along the same lines as "Crab Monsters" I would enjoy seeing a new >special effects laden remke of some fifties movies - "Earth vs the Flying >Saucers" "Incredible Shrinking Man" "This Island Earth" "Them!" or even >"I Was a Teenage Werewolf".
Didn;t they remake the teen werewolf in the 80's with Michael J Fox? If the point of the remake is better special effects, that might be a mistake. The reason amny of those old scifi movies are so enjoyable (for me anyway) is that the effects are so bad.
slaytonf
>traceyk65:
>The reason amny of those old scifi movies are so enjoyable (for me anyway) is that the effects are so bad.
The reason any of those old movies, such as Them!, and The Thing From Another World, are so enjoyable is because the story and characterization overcame the limitations of the effects. The mistake that the remakers make is assuming that the application of boffo special effects will make a successful movie. The general failure of the remakes of, say, King Kong, Godzilla, and The Day the Earth Stood Still, for example, highlight this misinterpretation.
I agree. Our modern FX are better, but tend to be overused, and too exaggerated. They should always be subordinate to the story.
>aradia22:
>I have an answer that will probably be considered blasphemy (especially if there are any Brits in this forum) but I will say Brief Encounter. I saw it for the first time when it aired recently on TCM and it just didn't work for me.
Now, I'm not one to have my thinking determined by others, and goodness knows I maintain enough positions at odds with popular thought. But I will say in this case, popular thought and I agree, that Brief Encounter is one of the great romantic movies, and is one of the early examples of David Lean's genius. It was remade in 1974, disastrously, with Sophia Loren and Richard Burton. The difference between the two can be summed up in that the first film was a story about an affair between ordinary people, elevated to a transcendent level by the actors and direction, and the later one told a story with glamorous movie stars slumming in the middle class, and made commonplace with pedestrian direction.
The Godzilla remake looks amazing- the director Gareth Edwards made an interesting indie sci-fi movie called "Monsters" (2010) and also created the films amazing FX. Modern special effects are not the only reason to remake a classic movie and I agree they must serve the story first. A remake should be done if the film makers have a different take on the material- not just a souped up re-hash.
Another question--are there any remakes you think worked?
I am a HUGE Katharine Hepburn fan. I can even sit through her less-than-good movies. And I LOVE The Lion in Winter. I expected to want to throw the remote at the 2003 remake, but I actually liked it, especially Glenn Close's Eleanor. Patrick stewart played Henry more or less as O'Toole had (it's not a terribly subtle character--I think you'd almost have to play it big and brash) but Close plays Eleanor more playfully than Hepburn, mostly, and may have actually done the dramatic scenes better. Here's the same scene played by both Close and Hepburn:
http://www.tcm.com/mediaroom/video/314330/Lion-In-Winter-The-Movie-Clip-Oh-My-Piglets.html
In addition, the remake's Richard and Phillip have actual chemistry (which Timothy Dalton and Anthony Hopkins did not).
One question related to remakes from a producers POV is if one should remake a movie that was well received (a first rate production), or one what wasn't.
Most people I meet assume it is better to remake a 'dud' (as long as the source material is strong). Often at this forum people will say 'why remake XYZ, that film was perfect!'. I can see a reason for either type of remake; each being a challenge to the producer in very different ways.
I haven?t seen the A Lion In Winter remake but I did see Glenn Close in the play Sunset Boulevard and she gave a great performance. I think people often get hung up when comparing performances as it relates to who was better. With Sunset Blvd it was a different interpretation and both were great.
Oh, and when I was a teen I saw Kate in a play and that was really special. Funny I have only see two major stars in plays and they are Kate and Close! This was the last stop on Kate's tour and she came out and talked to 'LA'. She said that some felt she didn't love Hollywood and she told us she loved the people in the area. She was warm and open and really made us feel special that night.
I think the Coen Brothers remake of TRUE GRIT was extraordinarily better than the John Wayne version decades ago but then I am not much of a Wayne fan. I think the Coen Brothers could remake anything and I would want to see it no matter what the original was. I have that much respect for this modern film-making duo. I did not see the Close/Stewart remake of THE LION IN WINTER and now really would like to see it.


text:
Oradell man used knife to try to kill his father, police say
The man was arrested after cutting his father's throat in an attempt to kill him, police said.
Oradell man used knife to try to kill his father, police say The man was arrested after cutting his father's throat in an attempt to kill him, police said. Check out this story on northjersey.com: https://njersy.co/2x6mWVy
Stephanie Noda, North Jersey Record Published 3:27 p.m. ET Sept. 12, 2018 | Updated 4:32 p.m. ET Sept. 12, 2018
Dashcam video of NJ State Troopers arresting a municipal judge from Essex County on Route 80 in November 2016. The judge was later found not guilty. Michael V. Pettigano and Matt Kadosh, NorthJersey
Sergey Detinich.(Photo: Photo courtesy of the Oradell Police Department)
ORADELL — A 37-year-old borough man was arrested Tuesday after he allegedly tried to kill his father with a knife.
Police arrived at the Forest Avenue residence of Sergey Detinich around 3:40 p.m. in response to a 911 call involving an unknown medical issue, said Detective Sgt. Marc Fedorchak.
After officers arrived, they determined that Detinich had attempted to kill his father by cutting his throat with a knife, police said.
The father, who suffered non-life-threatening injuries, was taken to Hackensack University Medical Center at Pascack Valley in Westwood, treated and released, Fedorchak said.
Detinich was charged with attempted murder, aggravated assault and possession of a weapon for unlawful purposes. He was remanded to the Bergen County Jail and has a court hearing scheduled for Friday, Fedorchak said.
REPORT: Car dealership owners with Englewood Cliffs store sued by Nissan, Hyundai
DRUGS: Paterson man sold thousands of heroin pills disguised as Percocet
STABBING: Hasbrouck Heights man stabbed in Elmwood Park
Read or Share this story: https://njersy.co/2x6mWVy
Jason Vanderee rejects plea deal in fatal Wayne crash
Residents in Jefferson saw woman trying to flee shooting
Sears in Wayne to close in September; sales begin
What we know about reported ICE raids in NJ
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Approaching the C42, it looks deceptively small. It sits fairly close to the ground on a tri-gear undercarriage and it's only when you open up the clamshell doors, hinged at the top, and slide inside, that you realise the cockpit is more spacious than many Group A aircraft. It's much bigger than a cosy Cessna 152, for instance, where most adults are rubbing shoulders, and much more modern. The seats look as though they're from a Caterham sports car, reclined and shaped, although the padding is a bit thin.
The legroom is plentiful, even for tall people, though there is no adjustment for either the seats or the rudder bars-sturdy aluminium tubes. If you're short, presumably you have to bring a cushion to stuff behind your back. Elasticated netting side pockets in the doors are big enough for charts etc, and there's storage space under the seats, but there's not much space anywhere else in the cockpit. Between the seats is a high central structure, covering the main fuselage spar, on which the stick is mounted. It makes a handy armrest too.
The panel, while not stuffed full of instruments, looks more familiar than many microlights, with a recognisable vertical speed indicator, altimeter, turn ball and compass. UK distributor, Simon Du Boulay of Aerosport uk, says this is deliberate to make Group A pilots feel at home. The test aircraft didn't have an attitude indicator but the VLA version of the C42, with a 23 kg heavier weight limit, will. That extra 23 kg won't add much in terms of usable load but will allow a fuller panel. The Microair 760 radio and Mode C transponder are typically small (and light) microlight avionics but worked well enough.
Other instruments include three engine-monitoring gauges: oil pressure, oil temperature and cylinder head temperature, and a fuel computer. A choke knob for cold starts and cockpit heater knob complete the controls. There's no magneto switch because the sparks are by modern dual electronic ignition, and no carb heat because the twin carbs are heated by the engine's liquid cooling system.
Start-up was on the button, with the 100 hp four-cylinder liquid-cooled 1,352 cc Rotax 912S mounted in the conventional nose position, giving the typical Rotax shudder as it fired. It idled at 1,800 rpm, which was a bit fast, according to Paul. From the left seat, you can make a final visual fuel level check by opening a flap in the back 'wall' material to the translucent plastic fuel tank located behind the passenger.
Taxying across the grass was straightforward enough. Visibility from the two-seat, side-by-side cockpit is excellent, the nosewheel steers directly from the rudder bars, and the hydraulic disc brakes on the main wheels are simply operated by a single hand-lever on the central stick. No differential braking, but it isn't necessary when the aircraft can turn on a sixpence just by steering it. The throttle took a little getting used to; it's a slim aluminium tube between your legs, hinged to fold out of the way for getting in and out, and for positioning. There's one for both seats so the aircraft can be flown from either side.
The take-off was anything but conventional, at least by the standards of a C152. Paul instructed me to pull the stick back, apply full power... and after just a few seconds and about eighty metres, we were airborne. It's not surprising that the performance is so sprightly, with 100 hp pulling a max of 450 kg. The climbout was at seventy knots, with 900 fpm showing initially, picking up to 1,050 fpm. The electric trim is operated by buttons on the stop of the stick, and with the stick forces trimmed out we could continue hands-off. Paul was very keen on showing the hands-off capability.
Once we'd reached the circuit height of 500 feet, we lowered the nose until a cruise speed of 83 knots was indicated and brought the power back to 4,500 rpm, and trimmed out again to fly crosswind hands-off, touching the stick only when rolling into a 90° turn for the downwind leg.
The turn was indeed just 'rolling in', with hardly a touch of the very powerful rudder required, just a gentle back pressure on the stick halfway through the turn, to keep the nose up. The stick forces, which had been quite stiff on the ground, were very light in the air with just a gentle two-finger touch required. This was slightly at odds with the feel of the short, stubby stick, which suggested, rather like a German car's manual gear lever, that forceful movements would be the norm. But then the C42 was proving quite a surprisingly little aircraft in many ways.
Thermal activity had us bumping around a fair bit but, once again, Paul persuaded me to leave the controls alone and let the aircraft return to straight-and-level by itself, which it did. It seems that once trimmed, the C42 runs on rails.
And so it proved as the downwind leg brought us over the airfield and we turned onto base. With the power pulled back, the speed came back to the flap-limiting speed of 62 knots and we deployed the first stage of flap, 15º. The handle for operating the manual flaps is located in the roof and I found it too easy to go past stage one and straight to two. The trick, according to Paul, is to let go of the ratchet release lever as soon as the handle has cleared the gate-it will then slot into the first stage of flap and won't accidentally carry on. Once one stage of flap was deployed, we set power at 3,000 rpm and the speed settled at 58 knots, then, trim reset, it was time to go hands-off again.
Turning onto final, the second stage of flap, 40º, was deployed, speed reduced to 53 knots, retrim and... no, not hands-off this time because not even the C42 will land itself, but it was a bit of a non-event, even given the gusting crosswind from the right. Wing-down proved the simplest approach and with the pilot's seat being so low to the ground, judging the flare was easy. A little more difficult was co-ordinating limbs during the ensuing touch-and-go.
Landing with my right hand on the stick and left hand on the throttle, meant swapping hands on the stick to free my right hand to remove the flap using the central handle, then swapping hands again to add power and pull the stick back. Pressing the stick-mounted PTT button to make a radio call was a bit beyond me this first time. Off we went again for several more circuits, each flown a little more precisely as the aircraft's controls became more familiar.
The C42 has one big advantage as a training aircraft over some other fixed-wing microlights, namely the nosewheel mounting. Unlike some, it's not mounted to the engine frame or firewall, but directly to the main fuselage frame, a large diameter aluminium alloy tube running the length of the aircraft with triangulated alloy tube substructures bolted on-the bodywork hangs off these. This means that if the aircraft has a nose-heavy landing, not an uncommon occurrence for training and group aircraft, it's just the nosewheel assembly that needs checking, not the whole engine mount.
Back on the ground for a break before a second flying session away from the circuit, Paul took me on a walkround. The wings can be removed, taking about twenty minutes per wing, to occupy less hangar space or to put on a trailer. The biggest job is removing the top canopy fairing, while disconnecting the metal struts and spars is a simple job of removing cotter pins and locking clips. The struts stay with the wing, and there's no need for adjustment when reconnecting the flap and aileron push-rod operated controls. There's also a folding-wing version.
Looking closely reveals many nice touches. The fabric covering the wings, tailplane and rudder is a lattice cloth called GT-foil, claimed to be resistant to ultraviolet light, and it fits tightly over aluminium tube frames. Upswept composite wingtips are an option and are said to give better low-speed control and reduce drag. The same composite of glass fibre and Kevlar honeycomb is used for the fuselage bodywork and engine cowlings, all immaculately finished. The engine cowling is split into top and bottom halves with fifteen fasteners for the top, eight for the bottom. Fortunately, there's an inspection flap to check the oil level. If the aircraft has been standing overnight or longer, Paul recommends pulling the engine over by the prop (ignition off!) to pump oil back from the crankcase to the dry sump to make sure of a correct reading. It's possible to overfill otherwise.
Exposed tubing such as the struts and undercarriage frame has a streamlined section to reduce drag, and all the joints in the control push-rods are nicely machined from alloy. It's a shame though that they're simple hinge joints rather than rose joints. Still, there's remarkably little friction in movement of the controls. Streamlined struts are used for the main struts of the undercarriage too, with round tubes for the trailing-links going forward to a point on the main fuselage spar. Plastic dust caps cover all exposed bold ends. Gas-oil suspension units bolt onto a substantial, square-section aluminium cross-fuselage tube which is also the lower mount for the wing struts. The undercarriage alone, while looking a bit narrow, is far more substantial than on any other microlight I've seen and it certainly absorbed the bumps of Chiltern Parks grass runway and enhanced my less-than-perfect landings.
Behind the cockpit on the port side there's a removable panel giving access to the inside of the fuselage. It's big enough to stick your head and shoulders through to make a thorough inspection, and there's also a fabric luggage sack attached to the tubular frame that's just about big enough to take an overnight bag. The fifty-litre fuel tank can also be inspected.
The big side windows, extending down to leg height, give a great view, although the wing does obscure the view into a turn because you're sitting quite a long way behind the leading-edge. There's a optional clear panel in the roof which would help.
Paul explained that the 83-knot cruise is where the C42 feels most comfortable, as well being fairly economical-it's burning around 13 litres an hour. Importer Du Boulay says that with the more efficient Warp Drive propeller that comes down to 11 lph.
Upping the cruise to 90 kt, which the C42 handles easily, gives around 16 lph fuel burn, or you can go the opposite way and slow to 70 kt for 8 to 9 lph, or even 62 kt for 6 to 7 lph, though Paul says, "It starts to feel more like a microlight below 75 knots."
Stalls are another non-event. Both the clean stall and with full flap produced a gentle nodding of the nose with no tendency to drop a wing. Paul demonstrated flying through the stall with full flap and holding the stick right back, producing nothing more a loss of height and occasional dip of the nose. Recovery is immediate as you release the stick back pressure and add power. Maximum permitted angle of bank is 60º but, without an attitude indicator, that's a matter of judgement. Steep turns are straightforward, needing little rudder to roll into, just back pressure and power. The aircraft rolled from one turn to another very easily and crisply, again as if on rails.
Back to Chiltern Park, it was still quite thermally and we needed a long, wide turn to descend from an overhead join at 2,000 feet to the circuit height. The thermals were noticeable again as we turned final a bit high. Paul took control and demonstrated an aggressive sideslip to get us down to the threshold.
So, what to make of the Ikarus C42? Would a Group A pilot feel it was a 'real' aircraft? It's much more Group A than any other fixed-wing microlight, with the possible exception of the Dyn'Aéro BanBi, but that's a high-performing hot ship. The C42's closest microlight rival is the Rans S-6, an alloy tube and fabric aircraft with an excellent reputation for flying qualities and ruggedness, but with less performance and more of a kitplane appearance.
THE IKARUS C42 FB costs £55000.00 + VAT and delivery to anywhere on the UK mainland. That's ready to fly with a BMAA Permit to Fly. It will have been flight tested at the UK importers' HQ at Wolverhampton Airport.
average maintenance cost of 60p per hour.
An annual Permit to Fly, costs between £250 & £300 inc VAT, although if work is required to pass the inspection that'll be extra.
Fuel consumption is in the region of 11 to 13 litres per hour at a cruise speed of 83 kt, depending on the prop fitted. That's 2.9 to 3.4 usg per hour, or about £13 to £20 per hour using mogas (recommended for the Rotax engine; 100LL avgas can be used but prolonged use is to be avoided).
Insurance will come as a shock to microlight pilots, but not to Group A pilots. Broker Flemings has insured many of the C42s in the UK and quotes an average cost of £2,000 pa. That's for the full hull value, plus passenger cover and £1m liability. David Bacon of Flemings says the premium is high because many of the pilots are low-hours and have come from flex-wing microlights. The premium will be lower for more experienced pilots and they prefer pilots from a Group A background.


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